i'^e P~
Tr t\Q Text book on Elementary Particle Physics I t"r^ Manuel de Physique des Particules
WEAK MTERACTIOn/ MTERflCTIOfl/ FAIBLE/
This book has evolved from lectures presented at the International School of Elementary Particle Physics, Bdsko Polje (Yugoslavia). Ce manuel a pour origine une série de cours donnés à l'Ecole Internationale de Physique des Particules de Basko Polje (Yougoslavie).
Contributors Auteurs
Jean-Marc GAILLARD Mary K.GAILLARD Dieter HAIDT Louis JAUIMEAU Otto NACHTMANN Herbert PIETSCHMANN François VAIMNUCCI
Edited by Edité par IV?. K. Gaillard M. Nikolic
I IU2 P3 Institut National de Physique Nucléaire et de Physique des Particules 11, rue Pierre et Marie Curie 75231 Paris Cedex 05 CONTRIBUTORS : AUTEURS :
J.M. GAILLARD CERN, Geneva (Switzerland)
Laboratoire de l'Accélérateur Linéaire, Orsay (France)
M.K. GAILLARD CERN, Geneva {Switzerland}
Laboratoire de Physique Théorique et des Particules Elémentaires, Orsay (France)
a. HAIDT CERN, Geneva (Switzerland)
L. JAUNEAU Ecole Polytechnique, Palaiseau (Fran<
Laboratoire de l'Accélérateur Linéaire, Orsay (France)
0. NACHTMANN Institut for Theoretical Physics, University of Heidelberg (Federal Republic of Germany)
H. PIETSCHMANN Institut for Theoretical Physics, University of Wien (Austria)
F. VANNUCCI CERN, Geneva (Switzerland)
This book may be purchased from :
Service des Relations Extérieures I H? P3 iInstitut National de Physique Nucléaire et de Physique Jes Particules) 11, rue Pierre et Marie Curie 75231 Paris Celex 05 TABLE OF CONTENTS
AVANT-PROPOS .
PRÉFACÉ
PROLOGUE
HISTORICAL INTRODUCTION FROM H RADIOACTIVITY TO THE V-A HYPOTHESES
I. THE FIRST THEOSY OF ^-RADIOACTIVITY
1.1. Fermi Theory 1.2. Ferni and Ganov - Teller transitions 1.3. The general weak hami Etonian 1.4. The electron-neutrino angular correlation 1.5. Forbidden transitions 1.6. The Universal Fermi Interaction
I. THE DISCOVERY OF PARITY BREAK-DOWN
2.1. The r— puzzle 2.2. The CV experiment 2.3. The IT - |i • e decay chain 2.4. General form of the parity-viol a ting weak hamiltonian 2.5. Measurements of electron polarization in nuclear fl-decay . 2.6. Determination of the neutrino heliclty 2.7. Structure of the nucléon weak coupling
FURTHER DEVELOPMENTS
3.1. The two-component thtwry of neutrino 3.2. The V-A law 3.3. The current-current form of weak Interactions Neutrino interactions and lepton conservation The two neutrinos Classification of weak processes Isospin selection rules 3.8. The Cabhibo formulation of universality 3.9. Neutral kaons and the discovery of CP violation 3.10 The intermediate vector boson hypothesis 3.11 Higher crder processes and the renornalization problen .
REFERENCES .
PROBLEMS ... LEPTONIC INTERACTIONS
1. MUON DECAY
1.1. Introduction and kinematics , l.Z. General shape of the electron spectrum 1.3. Muon total decay rate •'.. 1.4. Decay of polarized muons , 1.5. Two-component neutrino assumed 1.6. Uncertainty on the muon decay matrix element ,
NEUTRINO-ELECTRON INTERACTIONS .
2.1. Introduction 2.2. Cross sections
2.3. General features of lepton scattering
3. TRI-LEPTON PRODUCTION BY NEUTRINOS
APPENDIX I : NOTATIONS AND CONVENTIONS
APPENDIX [I : THE FIERZ-MICHEL TRANSFORMATION
APPENDIX III : CALCULATION OF y„ - e AND v, - e CROSS SECTIONS .
REFERENCES .
CHAPTER III - SEMI-LEPTONIC INTERACTIONS
1. EXTENSION OF THE FERHI INTERACTION
1.1. Th'.' vector nature of the interaction ,
1.2. Strangeness-changing decays .
I. THE STRUCTURE OF THE STRANGENESS-CONSERVING CURRENT ,
2.1. The vector current : CVC hypothesis , 69
2.?. The isovector character of the axial current , /;, 2.3. Trie strength of the axial couplings : PCAC .
THI. CABIBBP THEORY OF CURRENTS
3.1. election rules for strangeness-changing decays , 3.2. Trie postulates of Cabibbo theory 3.3. Meson decays 3.4. Baryon decays 3.5. Some observable effects of the V-A struc.'.ve
3.6. What is the meaning of universality ?
FORM FACTORS IN BARVON DECAY
4.1. The vector matrix element .. •'
4.2. The anial vec'ar jtnx element
4.3. Rates, spectr- ..nd expérimenta' correlations ..... 4.4. Experiment' results and fit to the Cabibbo prédit 4.5. Further • operties of the axial current ,
>.tPRA OF CURREttrS
Universality Mon-renorTial !2a tion theorem, for vector cfccrqes ; norr-ùl izjtion of the
FOf'M FACTORS \>i KAOH DECAY
6.1. K, forr-, factors
6.3. The relation of PCAC to chiral syinmetry .
7. THE ABSENCE OF 'JFUTRAL CURRENTS
SEFED-'iCES
APPEtfm ; SU[3) WTIONS A'.D NOTATIONS
MOSLEMS
CHAPTM IV - NOfl-LEPTONIC INTERACTIONS
PV-EW£'IOL0".* f]f m-LfPTQ-JIC flECAYS . 1.1. The {'.S! < 2 rule 1.2. The li[| = 1/2 rule in K decays .
1.3. Hyperon decays .
GENERAL FEATURES OF THE CURRENT-CURRENT INTERACTION
2.1. ^election ruK-s and relative transition strengths ,
2.2. Charge conjugation ar-i U sMn .
2.3. '.eutral currents ar.c! the M = I/? rule .
3. SC.-T PION THEOREMS
3-1. K decay
3.2. FJaryon decay .
3.3. The origin of the "•.' = 1/? rule
4. RADIATIVE NYPERON DECAYS
4.1. The bremsstrahlunc amplitude
4.2. Direct emission amplitudes
4.3. Extraction of the amplitudes from experiment
5. PARITY VIOLATION IN NUCLEAR PHYSICS
REFERENCES
APPENDIX : U SPIN AND 5U(2) PHASE CONVENTIONS ,
PROBLEMS
THE NEITRAL WVi SYSTEM
I. THE Kj'-Kj SYSTEH
1.1. General formalism 1.2. K/^-K| mass difference
1.3. Sign of the rass difference
2- SEM;-LEPT0:IIC DECAYS OF KEUTRAL KAO'JS .
3. C? VIOLATION AND THE K£-K* SYSTEM 3.1. Evidence for CP violation 3.2 Interference experiments SEARCH FOR K| • 3n DECAYS
T VIOLATION I'l K- DECAYS
Cn VIOLATION UKDER CPT CONSERVATION
5.1. Pheno/nercological analysis 305 6.2. Charge asymnetry in K.fl leptonic decays . 206 6.3. Comparison with theory
7. CONCLUSION
REFERENCES
APPENDIX 1 : HEISSKOPF-WIGNER FORMALISM APPLIED TO THE K°-K° SYSTEM ,
APPENDIX I : NEUTRAL K DECAY RATES FOLLOWING A REGENERATOR
APPENDK 3 : MDDFLS OF CP VIOLATION ,
CHAPTER 71 - INTERACTION'S CF NEUTRINOS HITtf HADRONS
1. INTRODUCTION
2. THE INTERACTION LAGRANMAN AND A GENERAL THEOREM .
3. QUASI-ELA5TIC SCATTERING
3.1. Invariance principles 230
331
3.2. Anslyticity properties pf form factors . 234 3.3. Experimental results ?36
RESONA'.CE PR0DUCT!ON
4.1. General rsrarks 337 4.2. f. •; impie isobar node I 239
AOLER'S CVC AND PCAC TESTS
5.1. Theory with ne THE AOLER VJEI58ERGER AND ADLER SUM RULES ... 6.1. General remarks 6.?. The Adler Weisberger relation 6.3. Ailler ' s neutrino sum rule 'SUP HELAST1C NEL'TSIHÛ NUCLEON SCATTERI'.r, A'.D IhE PARTT, MOSEL 7.1. ieieral rerarics, kiner-atics 1.2. Scaling behaviour, the parton model 7.3. The qiiark parton ™del 7.4. Other parto.i models ... 7.5. Conclusion and outlook EXERCISES SOLUTION TO EXERCISES . CHAPTER V r T - HIGH ENERGY BEHAVIOUR OF WEAK INTERACTIONS AW RENDRMALIZABLE THEORIES INTRODUCTION PECOLLCCTIO'J OF ."IFUTRIND INDULED REALT \0"S Il IL '"JtrAfi.'TV D'-IIT fFFEC'S "F THE l'JTERICOIAFE VECTOR BOSO'i ^nnucnoN Arm DECAY or THE INTÎ^M-DIAÎE 3ISON -.1. Production of the intermediate boson '}.?. Tuo-particle decay of the '.V-boson 5.3. ~he total hadronic decay width of the H-boson THE WEAK 'JEITRU CURRENT 6.!. Phenomenology 6.2. Isospin properties of the neutral current 6.3. Deea inelastic neutrino sCJtterintj with neutral curren !N:FIED VODELS OF HEAK AND ELECTROMAGNETIC INFRACTIONS .... 7.1. The Higrjs nechanisn: 7.?. The Sala.Ti-li'einbern model 7.3. The proSletn of incl jding hadrons ?.l. lie schene <^f Glasgow. IHonoulos and Haiani 7.5. Ser.crnal izatton and the probien of triangle anomalies 7.6. Alternative models APPEfirif - IHTSODlJfTlON TO CURRENT ALGEBRA ""îi vii 1. REVIEW OF SU2 ANO S, PROPERTIED 337 1.1. General notions 337 1.2. Isospîn symmetry 339 1.3. SU, symmetry 342 ;.-•:. i hi'.'.pcrtiùn ••>:' th.- croup £'î/{ 342 ."./. " ' -fi'iztCs-. çf ? a-xi ï in t,:i-ma of SU., centrât 3 344 /./..' c: rsprc^r.fiti.wa. Pro-lat of rcpt^aentitiona 345 ;./.; Soltt/ni !r;^:-r. g^ncriura of tkc rcpular rcpivaanSaticn and atruotupe aonotroita 348 .'..•..• 'irjatu! .-.-wte-rr ".»* M, ivprceewatton 349 j /...-' •;.-: nif»^ 352 ;..:.-• ","J-»!-HI>-J-»I ^upH'ig 354 :..'.-• / *. .I'J.-.'1-J 356 2. OJRREMS 357 2.1. Introduction of isospin and SU currents 357 2.2. The Noether theorem 358 2.3. Quark currents 3Ç1 ?.4. Catibbc formulâtîin of universality 363 3. CURANT ALGEBRA 364 3.1. Why current algebra ? 364 3.2. Connutation relations of current tirae-components 365 3.3. Effectç of symmetry breaking 366 3.4. The problem of axial vector currents 367 3.5. S'j3 % Sll3 algebra 363 3.6. Camutators with current space-components. Schwinger terms 369 3.7. Universality and current algebra 3?0 3.S. A first example of application of current Jlgebra 372 REFERENCE:; 37i ! t AVANT - PROPOS L'idée IÎL' la redaction d'un manuel de physique des particules (TEPP) par une collaboration inti'i •n-> i'i^.ij(i'S dans I 'enseignement et dans li recherche est née à ("Ecole Interna tîan.'i?e do Physique ires. C.-tLc Ecole, fruit d'une ce 1 labors t ion iranco-yougoslave, s'est i!'aïiord ter.ue à llerceç-Nov: (You^slavie) de l'j'iî- fuis à Baskc-Polje-, près de Mdkarsk.n, après l';7l. i physique des particule. de nouveaux développements, liés ,i la tu se en servira c'iine nouvelle génération d'acr. qui existent sur ce sujet: se limitent le pl-js souvent à certains aspects. Oiiant aux f clés spe. i.il : =e s qu hjse qui fîïiil P-irfo '•• props e dan C d'f [poser les aspects fondamentaux Je ce domine nouvan;, ceux Qui sont le asins sour temps, et qui CnnSl tuent les outils de base do la recherche en physique des particules. Il .1 bénéfi' fait pour l.i p répara itînn des cours de l'Ecole de Herceg N'ovi-!Usk I.' Institut lal de Physique Nuclé; ire et de Physique des Particu!e5 (l \2 P3) .-> oris en char, emier vol un» faibles lies particules. Nous espérons que ce volume rendr; H et aux jei du travail que ses auteurs y one investi. r de l'I .,.'. Pj International School of Elementary Par • Physics organized by Che Institute of Physics, I'm the Centre de Keciirrehos Nucléaires, S •o-jrg, ws first held in 15*5 at Herses SCVL, Y.igosl During t':e peripc when i; was located at iiercef: tov îe school printed over a hundred leetur , in the fii energy physics which •-•ere widely û istributed arton<; p! lysicists and vhich. were subsequent 1 y pabl I ifd by Con: rli (Sew Y«rk) in thi» scries of books "Methods in Sub; AC the end of l9fiH the idci i>f jointly preparing a text boon on elementary [article physi.es origii iiins .inung physicists inuolvod in teaching and research in this field. Physicists from many Kuropeai lis<(iss tiiL- initial nta^js of the project called TKPP (Text Hook on Elementary Particle -hysics) uln tJ include books covering the basic aspects of the field : Kinematics, Symmetries and the S Matrix ., Stroi.g Interactions, Electromagnetic Interactions, and Weak Interactions. In 1971 the Internationa! School of tl'pentary Particle Physics moved to Basko Poljo and the 1971, vrre systematically devoted ti> teaching based r Th-ir.ks o the Institut National de Physique Nucléaire et de Physique des Particules at Paris, the to îiunit Professors J. Teillac and J. YOCCDZ of the I S2 Vi *"or their support of this publication ; General Mil PROLOGUE The intent sf ;his book is to provide the basic elements of weak interaction '.eory and phenomenology, and it ; addressed to advanced graduate students and young physicists, both in theory and experiment. While the fit-Id of weak graciions is at present rapidly evolving, some of its phenonenoiogical aspei •raccion, the Cabibbo theory of currents, and more recently and aore approx the parton aojcl for deep inelacl .tattering, are rather well established even though the underlying theory na 11 elude us. Wliiie ve have devoted ti attention to the more firmly established aspects of the field, we have also touch- on more speculative aspects or ur ain results so that the reader may become aware of the directions of possible pro- gress, and more importantly of where tlic problems lie. An attempt has been made to provide a sufficient sample of cal culation to serve as examples but not so many as to obscure the development of physical ideas. Problems are given at the end of most chapters to aid the student in familiarizing himself with the theoretical tools. Me have also att»>npted to bring out, again through specific examples, some of the methodology and problems involved in experimental measure- Chapter I i il description of the development of weak interactions pliysi theory of B decay, ; •ch highlights as the discoveries of parity violation, tne ; Chapter ri dt-als with purely leptnnic weak interactions in lowest order, u'hich is a ue\ J -understock and self- conwincd topic and lias pedagogical value as a clean laboratory for studying "eak interactions. In the discussion of Chapter III on s itri-loptonic decoys, we s ;e that the theory of hedronic currents leads to a considerable predictive power, in spite of the presence of poorly understood strong interaction effects. On the other hand for non-leptonic ilec-iys, the subject of Chapter IV, the ueak interaction structure is much more obscured by strong interactions, and we discuss on a more empirical level the patterns that have emerged and how they fit into the theoretical framework developed on the basis of leptonic .ind seroi-leptonic interactions. While :he intv action responsible for CP violation is still 3 aystery, the study of the neutral kaon systea in the presence of CP violation provides a beautiful example of the superposition principle in quantum mechanics. This is the subject of Chapter V. Chapter VI treats th-i s-jbject of neutrino scattering froo nucléons j the cost striking feature which has ercerjjcd frou high energy scattering at leptons is that at very high energies hadronic structure become;, simple, and in the case of neutrino scattering, this has allowed a "rediscovery" of the simple V-A coupling which emerged from early acuities of \i and fl decay. Finally, Chapter VU deals with the outstanding problem of veai*. interactions, namely the development of a rcnorcial izablo theory which can be tested beyond loueut order in the weak coupling. The problems arising ir. the simple piieiiameno logical theory arc discussed aid new theoretical and expérimental developments which point in tlv direction of ,i solution ro this probten jre outlined, although one must bear in mind that the ijr.tec is far from established at WIÎ are indebted to Jacques Prentki whose careful reading and construe Live criticism on both the scientific and pedagogical levels have been invalu3b!e. We havr also benefitted from discussion with il. Din and we thank A. Frenkel for bis contributio.i to aspects of CP violation. Finally, wc are fateful LO the typing services at CERK and t!ie laboratoire de l'Accélérateur Linéaire for their excellant uork. M.K. Gaillard M it or for HIM I CHAPTER I HISTORICAL INTRODUCTION FROM 0 RADIOACTIVITY TO THE V-A HYPOTHESIS L. JAUNEAT HISTOHTCAL INTRODUCTION IVTRODUCTION The first weak process was observed by Decquerel in 1896, at a time where nothing was known about atomic structu re : Rutherford proved the existence of atomic nucleus in 1911, and Heisenberg assumed in 1932 that the only two cons tituents of nuclei were the proton, and the neutron which had just been discovered. It was then necessary to under stand how electrons could be emitted by radioactive nuclei. Another problem arose from the observation of a continuous B-ray spectmm : one had either to give up the prir- cipje of energy conservation, or to admit that a fraction of the energy released in the 0-disintegration had escaped detection. [Pauli (1933)] suggested, that, together with the electron, a new particle was emitted which had no elec tric charge, a mass of the order or IL^S than the electron mass, and carried half a unit of angular raomentuŒ in order to balance angular momentum in nuclear 6-decay. [Ferai (1933)]'applied quanttci field theory to this problem, assuming that the election and the new particle, which he named neutrino, were created during the 6-decay process. The enission of the light particles was then descri bed in a way analogous to the emission of radiation from an excited atom. Forty years later, the Fenâ theory remains, with sone ninor changes, a fundamental element of our theoretical knowledge of weak processes. It has been applied with success to ehe weak Drocesses which were subsequently discovered fran u-decay to neutrino interactions. 1. TitE FIRST TltEOKT OF a-RADIOACTIVITY 1.1 Fermi theory The elementary process of 3-radioactivity is the neutron decay process : n + p * e" * v (In writing the weak reactions, we shall distinguish between the neutrino w and its anti-particle Z . We call neu trino the particle emitted with a positron, and antineutrino the or emitted with e". It will be shown later that v and v are actually different). Emission of the light particles e and v (leptons) in nuclear e-decay was described by Fepiii in a way analogous to the emission of a photon by a radioactive nucleus : A * A' + Y , i.e- the process n •* p + (e- v) is similar to P * P * Y (virtual). I;i quantum electrodynamics, the process p - p • y is described by an hamiltonian : û (P')T0 U (P) \ where u IJJ) is a Dirac spinor, u (P')Y U (p) is the proton current, a four-vector built from two spinors. —I I CHAPTER I \ is the electromagnetic four-potential (photon field). To describe the B-process n •+ p (e" «), hu is then replaced by a four-vector built from e and v fields, and the weak hamiltonian is written as : G [G(p') ;u u[p)] [Ge V uj where G is the weak interaction coupling constant (Fermi constant), which is very small. Another way to look at the Fermi interaction is to compare the so-called "etastic" neutrino reaction v + n •+ p + e" , which is deduced from n -» p + e" + v by crossing the antineutrino, and the electron-proton scatte ring process e" + p -* e" * p . In both cases, the initial and final states consist of a lepton and a nucléon. In the electromagnetic case, the nucléon does not change its charge. In the neutrino reaction, the nucléon charge is changed by one unit. The interaction of an electron with a proton is described, in a non relativistic lidit, as the interaction of the electron charge with the Coulomb potential produced by the proton. Assuming a point-like proton and no recoil, this yields the Rutherford formula. In the relativistic case, the charge and the Coulomb potential are respectively the tice-coEiponents of a 4-vector current and a 4-vector potential. A relativistic spin 1/Z particle is described by a Pirac spinor, and the electron current, a 4-vector built from two spinors, is ùg yu ufi {with û = u* Y,,). It can then he shown that the scattering matrix eler.ent is proportional to c2 Cû e THUe) ~ (ûp ïuUp) where q is the 4-momentum transfer : q2 = (P - p')2 This expression is the product of two currents, the electron current and the proton current, with a factor — . However, the proton is not point-like, so its current has a more complicated expression, involving two form fac tors. This leads to the Roscnbluth formula. The — factor stems from the - dependence of ihs Coulomb potential. It gives the -r- or •— dependence V r ^ sln"° 2 o£ the Rutherford formula. In relativistic quantim theory, it is associated with tne exchange of a ;ero mass particle, the photon : Kermi assumed that the weak interaction is a local interaction, that is, in the neutron decay for instance, the four fermions n, p, e~, Z interact at the same point (more precisely, the fields describing the four fermions inte ract at the same point of space-time). Therefore, no q2 dependent factor appears (such a d^enû.-nce has never been detected). Fermi wrote the local interaction of four fermions as the product of two currents. G (ûe YU uv) (Up yu uj HISTORICAL INTRODUCTION Let us consider a nuclear B-decay : Let i be the angular momentum between the e-v pair and the final nucleus A*. A transition with l = 0 is called an "allowed" transition. The probability of "forbidden" transitions ï / 0 decreases very fast with .ncreasing a . '..2 Fermi and Garaov - Teller transitions For an allowed transition t = 0 , the change of nuclear spin is entirely determined by the :;pin of the lepton pair which can be S = 0 (singlet state) or S = 1 (triplet state). If we define ûJ = |J- - Jf|, where Ji and Jf are respectively the initial and final nuclear spins, ùj = 0 for S = 0, and aJ = 0,1 for S = 1 . A Dirac spinor u describing a spin 1/2 particle reduces, when the velocity of this particle is snail, to a Z-com- poncnt PAULI spinov U. In nucUar e-decay, the final nucleus has a very small velocity and the nucLear currcr.t car be brought back to its non relativistic fora : reduces to U*U ieduces to 0 are the usual matrices (::)•••(:-:) So, the nuclear current reduces to a scalar and cannot change the nuclear spin : AJ 3 0 . Since aJ j* 0 transitions are observed, the Fermi description is not complete and the following possibilities must be considered : e-v pair nuclear spin Fermi transitions singlet state ÛJ = 0 Gamov-Teller transitions triplet state aj = 0,1 In fact. Ferai had followed too strictly the analogy with electromagnetic interaction. The onl» rule of the game is to build, with the Ilirac spinors of the four interacting fermions, an haailtonian which is a scalar Condition of Lorentz invariance) : any covariant built from two spinors and contracted with a covariant of the same kind gives a scalar. In a general matrix clement tOt, a Linear operator 0 is represented in Dirac spinor space by a 4 * 4 matrix. Such a matrix is a linear combination of 16 independent matrices which can be fonr.ed with the unit matrix 3nd the Pirac i matrices : 1 v - Y (j v » 7T (ïu>v yO (i.e. the antisymmetri:ed product y vv) i TS (i.e. the antisymmetriicd product y Y0Ï_) t (i.e. the ontisymctriied i ? t T ) Under a linear transformation in Lorentz space (rotation or Lorent: transformation), the bilinear TO.V transfont î a well defined way (Dir&c ccvariants) ?* SCALAR TT Y VECTOR Co yT TENSOR Î?T ïS¥ AXIAL VECTOR (i for hemiticity) ?T ? PSEUDOSCALAR 1.3 The ge.eral weak hamiltonian Thi? most gérerai hamiltonian can be written as : For the nuclear matrix element, one can use the no;: relativist!'; approximation : s "" ~~2> u*u V muu T .»„„>- , itra, A iûy Y5U —" P ùy.u * 0 Therefore S and V couplings corres"ind to Fermi transitions : AJ a 0 , T and A couplings correspond to Gamov-Teller transitions : ûJ = 0,1 . v2 The P coupling contributes only to nv.clcar 6-decay in the order —=• and can therefore be neglected. Three cases can be encountered in B-decay : a) a transition J. = 0 •* J, = D is a pure Fermi transition : S, or V, or S and V exanple : G™ - N^4 <• e* b) a transition with ûJ » 1 is a pure Gamov-Teller transition : T, or A, or T and A example : He* * Li6. + e" • v (NeZ3 ...) c) a transition AJ • D with J- t 0 is a mixture of Fermi and Gamov-Teller transitions example : n * p • e" * v or H? - He| * e" • Z ~Lct x' = a x. and T -* ?' "ST (traiiS format ion ot Dirac spinors). The Dirac equation sust be invariant under Lorentz transformation, so : The transformation properties of the TO^T can be easily derived from this relation. HISTORICAL IOTHCDL'CTIOS _ 7 Fur 8~-decay, the elenentiry process is : n - p * e~ + v We write the haniltonian as an integr.d over space of the pre uct ct" the four fermions fields tahen at the same point b * i /?p(x]0i Vx) Vx)0i TvW d3x the nucléon wave (unctions being zero cutsidi the nuclear volume. If we neglect the effect of Coulomb interaction on the electron, the outgoing leptons can r.e described by plane waves : ller.ue we have : ./nuclear volume where the integral is called the nuclear maLrix element, p and q are here the lepton momenta. Tht? energies involved in nuclear a-decay are such that p and q are of the order of m c , and electron Corpton wave length If T 0;t is non zero, one can therefore replace the exponential by 1 (allowed transition). If polarisations are not measured, summation over the e and u spin states and integration over the angle bet ween e and « yields the electron energy spectrum. For an allowed transition , the nuclear natrix element ^ is irJependent of p and q , and the energy spectrum is given by phase space and is called the "statistical spectrum" : *ere E is the available energy, i.e. the maximum electron energy if the recoil energy can be neglected, and E the observed electron energy. Actually, this spectrum is modified by Coulomb effects inside the nucleus. To check the statistical shape, one plots, after correction for Couloab effect, - I/T^ as a function of E (Kurie plot). Very often, one finds experi mentally simple straight lines, corresponding to allowed transitions. 1.4 The electron-neutrino angular correlation he have seen that, in nuclear B-decay, four different couplings S, V, T, A cai. contribute. Arc they all present 7 If S and V couplings both contribute to a pure Ferai transition, there will he an interference term codifying the statistical spectma. This is called a FIERZ interference. The statistical spectruo is then pultiplied by a factor of the form 1 * ~ (a being proportional to Cs Cy) which favors the emission of low energy electrons. An analogous modification occurs from T - A interference in pure Ganov-Teller transitions. As no such deviation of the spectrin of allowed transitions has ever been observed, one concludes that Ferai transitions are S or V coupling, and Gamov-Toller transitions are T or A coupling. Which kind of coupling has been chosen by nature ? It is possible, at least in principle, to answer this question by looking at the electron neutrino angular correlation. After sunning over the e and v spin states (no polarisa tion observed), one finds that the decay probability depends on the angle à between the electron and the neutrino through the function v 1 + A— COS? c The parameter i depends on the type of coupling and is equtil to : - 1, + 1 ,••?,- -r for S, V, T, A couplings (the T factor cores from the three possible orientations of the total lepton spin S = 1 , which tends to attenuate Che correlation) lixperi mentally, as the neud'ino carnet be observed, one had to measure the angular correlation between the elec tron and the recoiling nucleus. But, even in very thin layers of radioactive substance, the electron could undergo Cculccb SL^ttering, and the experuaents are very delicate. This is illustrated by the fact that physicists were con vinced that the iî-decay couplings were most probably S and T . In 1957, within a few ironths, this conclusion was invalidated,and the couplings were determined to be V and A . 1.5 f-'orhidden transitions the nuclear matrix element M as : M » H0 * H( * ^ + ... Each term is of the order of magnitude of the preceeding one multiplied by a factor "uÇ_pus M = eiectron Compton '.iravc length). It may happen that M = 0. Then, the 6-decay is dominated by the Mj terfi, and its rate will be less than the rate of art allowed transition by a factor a, t0~4 : this is a first order forbidden transition. And so on. In fact, one nakes a multipolar expansion of the plane wave e «Ht"*K in terms of angular mor.cntim eigenfunc- tions. A te*n (i, m} corresponds to the emission of the e-u pair in a state of orbital angular momentum l with res pect to the final nucleus, and z-componcnt n. Each tern has a p*q dependence, and, for forbidden transitions, the electron energy spectruni is different fraa the statistical shape. ^-transitions are compared by their "ft - values", where t is the period and f the integral of the electron energy spectmn. The inverse of the product of f am. t is manifestly proportional to the squared modulus of tJie nuclear ratrix cJr„iont. The radioactive nucleus \T , which has a very long life time (1.3 x 109 years} decays thrcuEh a 3 -order for bidden transition. The effect of phase space factor {proportional to At-r, 'where AM is the mass difference between the initial and final nuclei) and the existence of forbidden transitions explains the very wide range of life-times in nuclear u-decay. }.b The L'inversai Femi Interaction Another element of information on weak interactions was supplied by the mion. The mum was discovered in cosmic rays by ANDERSON in 1938, and its decay into an electron was observed by WILLIA*E and ROBERTS in 1940. In fact, this new cosmic particle was at first confused with the quantum of nuclear forces predicted by Yuknwn, and then called "u- neson". An experiment [Ccnversi, Panelni and Piccioni (1946)] showed that the u-particlo interacts weakly with natter, ami therefore cannot be the Yukawa meson : positive nions always decay when stopped in matter, but negative muons de cay in light olcrcnts anJ are preferentially absorbed in heavy elements. The decay ntodc of the cuon was established by observing that the election has not a well-defined energy, and that HISTORICAL INTRODUCTION the decay induces no materialisation electrons from T'S. SO it was assumed that the irmon decays into an electron and two neutrinos : „* - e* • » • ;; The u decay can be described by the weak coupling of four fermion fields. As in nuclear B-decay, the most gene- lal matrix element was assumed to be a superposition of live independent couplings : SS, W, TT, AA, PP. The global coupling constant C can be computed from the measured life-time T -V 2.10~& sec . Very good agreement was found between this value of G and the value deduced from the o'4 6-decay. The estimation agreed also with the observed weak capture rate of r -gativc anion, for instance : u" + p •* v + n Therefore the near equality of the effective coupling constants in nuclear B-decay, suon decay and nuon capture, led to rhc hypothesis of a Universal Fermi Interaction (UFI) [Klein, Puppi, Tiorano-Wheeler, Lce-Rosenblu'Ji-Yang (1'J48-1Wf>J^ between fo«:r fermion fields. One has only or.e global coupling constant, the Femi constant G : G = 1.4 10-49 erg x cm3 = 10"5 M*2 (with ft * c = 1) The Yukawa meson was discovered [Lattes, Occhialini,Powell (1947)] and called n-meson. Its decay into a muon and a neutrino, with a life-time of 2.6 10"^ sec, is another example of a weak process . Other weakly decaying particles were discovered at the same time : the V p.-.tides which later were named stran ge particles, K mesons and hyperons. The concept of a new quantum number, stranf ness, strictly conserved by strong and electromagnetic interactions but violated by weak interactions, was introduced [Gell-Mann] and [Nishijima], in order to explain the copious production of these particles in contrast to their lo-ig life-times, typical of weak interactions. These strange particles can decay either without emission of leptons, as K * Z-", 3T, A" - NI: ..., or with emis sion of a pair of leptons, as in B-Jecay : K* - "%%, .'.'*pe'I ... .'. DISaWEPY OF PARITY BREAK-DOVJJ Z.1 The i-O puizlr Before 1'JSh, it was assumed thac the physics of elementary particles was invariant under space reflection, i.e.- that parity was a good quantum number for all interactions. In the '•arly fifties, it was not yet clear whether all the decay modes of heavy mesons could be attributed to only one particle, [n particular, one had observed the decays 0* - n*»° and t* -• n*n*n~ . The growing similarity of measured masses and life-times forced physicists to admit thai. 0 and t were the same particle, a K meson of zero spin However, this hypothesis led to a difficulty, as the parities of 2a and J-n systems with total J = 0 are res pectively • 1 and - I . This was known as the "r-o puzile". [Lee, Yang (1956)] node a thorough examination of the proofs of parity conservation, and concluded that there was no experimental proof at all of parity conservation in weak interaction. They suggested a number of crucial expe riments in order to test this hypothesis. The principle was to look for pseudoscalar terms in the weak hamiltonian, whicn change sign under space ;eflection (or mirrur synnetry). For instance, in e-decay of polarized nuclei, parity violation leads to a correlation between the direction of the polarization und of the momentum of the emitted clcc- Thc best estimate of the u life-tinc is now : 6 ia • (2.1994 t 0.0006) HT 5ec tron, since momentum changes sign under space reflection, but polarisation does not. 2.2 The Co&0 experiment The experiment was performed [Miss Wu and collaborators (19S6)] on polarized Co nuclei. The specimen was a single crystal of cerito magnesium nitrate on the upper surface of which a thin crystalline layer containing Cobu h2d been grown. It was placed inside a cryostat, and cooled to a temperature of -r^ K, obtained by an adiahatic demagnetisation procedure. Then, a magnetic field, provided by a solenoid, induced the nuclear spin to line up, and decay electrons were detected by a thin anthracene crystal placed inside the cryostat, about 2 cm above the cobalt source (Fig. 1). The decay reaction is : jP. 5* I . Ni60 » 2y This allowed transition proceeds with ûJ = and no change of paTity, and is therefor*; a pure Gamcv-Teller tran- sition. The nuclear polarization is measured by observing the anisotropy oi y-emission, as the Y'S frco excited Ni are emitted preferentially in a direction perpendicular to the Ccfi® polarization. The specimen warms up, causing the nuclear polarization of Co60 and the Y-ray anisotropy to disappear in about 6 minutes. A large a asymmetry was observed by the experimenters (Fig. 2). This asynmetry changes sign upon reversal of the magnetic field, and disappears in the same way as Y-ray anisotropy. The negative sign of the correlation coefficient indicates that the emission of electrons is more favored in the direction opposite to that of nuclear spin. This re sult provides unequivocal proof of parity violation. 2.3 The n - u •» e decay chain Another place where parity violation effects were expected was r - p - e decay chain : Let us consider the n* rest system. Two cases ray occur since orbital angular momentum component along the u-i line of flight vanishes : s*- tinder space reflection : (a) * fb), so the parity eigenstates are taJ - IP/ , and, if parity is a good quantum number, the u cannot have a longitudinal polarization (pseudoscalar < o > .p ). Parity viol at ii can induce suJi a polarization of the muon. Then, in the subsequent v-decay, a correlation between this polarization and the electron di rection may be observed (pseudoscalar « j > -Pg). -10 a- TEPP - Weak Interactions Chapter 1 Historical Introduction -] 1 1 i— GAUMA-ANI50TRCPÏ H 1 1- GfiMMJ-ÛNISOTflCBT CALCULATED FRGM[0)a|t>) AHTHMiC£HE CftrSLM. Fie. I. Sthc:v..itic drawing of ihc lower i>nrt of ihe cryoslai. F,U:£^^:^:n;:i^;^s^' Fie. -I Variation of ?atcd J-J countin^ rale with rtMirni ,.lW:^™:^J^x;?^. current. The r--ili;! curve i« cnNputcd iron; an assume*! el. angular liMritmiinn 1-J corf, «iih counter anil çni ir resolution ['allied in. -10 b- f. vu WETAL SHIELD Ti'-.. 5 Experimental arrangement l'or analvzint; circular polar- I-'in. <•' Ki-nwiant-scnu.Teil y ravs -n I i2ation ol n-.nu:r. -caUi'ied vrais. Weisîit'uf Sin-Oj scaiu-rtr: laker. «i|h.in:ii>j:.TiK-iii >'.•..«n i.i l-iK 1 » 1850 jrrams LUHLT curve siiuws nuiin win.ini li.ickicruiin.l Irark^rtiiini'V c=i-:,,»i 7 Srij\s i |i;| - •• nil - - ii - ii HISTORICAL INTRODUCTION This test was quickly performed {Garwin, Ledennan, Heinrich (1956)] Mith a set up prepared for another experiment. The results were published in 1957 in the same issue of Physical Review as those of Wu et al. In this experiment (Fig. 3), muons produced by the decay in flight of n* in the vicinity of the Nevis cyclotron target were brought to rest in a carbon targit. The decay electrons from the muons were detected by a delayed counter telescope at •*• 90° from the TT* beam, within a tine interval of 0.75 to 2 microseconds after thj incident neson entered the carbon block. The carbon target was placed in a small magnetic field, perpendicular to the plane containing the besn direction and che counter telescope. Therefore, if parity is not conserved, muons are polarized and the polarization will precess with termor frequen cy. On the other hand, the direrrtion of emission of t1-*- lecay electron is correlated with the u polarization, and the electron angular distribution turns with the polarization. As the angle by which the polarization is rotated before the u decay is proportional to the magnetic field, one observes an oscillatory variation of the counting rate with respect to the applied field (Fig. 4). The maxinro e+ emission is observed in a direction opposite to the motion of the H*. Should parity be conserved, the electron angular distribution would be isotropic, and the counting rate flat. At about the same time, a similar asymretry in the »-u-e decay ch2in was detected in nuclear emulsion [Friedman, Telegdi (1957)]!1 As was pointed out [Lee, Oehce, Yang (195/)], these asymmetry experiments not only proved the violation of parity, but also the violation of charge conjugation invariance. The observc-d asymmetries come from interference between pari ty conserving and non conserving decay amplitudes. If the we*Jc hamiltonian is strictly invariant under charge conjuga tion and if the decay products can be considered as free particles (no final state interaction), it was shown, assu ming invariance under CFT, that the relative phase of the two decay amplitudes is 90°, and therefore there is no in terference. However, the experiments were compatible with CP invariance. 2.4 General form of the parity-violating weak hamiltonian Before 1956, the weak hamiltonian was written as a sum of Lorentz scalars obtained by contracting two covariants of the same parity. Jince parity is not conserved, one has to add pseudoscalar terms, obtained vy contracting two Dirac covariants of opposite parities. For n + p e" î : H Seal. H pseudoscal. s A u fVVs'Vi' c c l L%°i%> ("ew i ;<=P°iV 'VVsV The cost general matrix element will be : i • S, V, T, A, P . Let us assume that T invariance holds, so that C. and C! are real. Experimental results show that parity violation is maximum. therefore C! = ± C. , and one can write : = I! > =— ? Cu 0. u Û O.C S? i l p l n L e i h'ith the — factor, G is defined as before. What is the neaning of the expression (1 : » ) uti ? 1 A non relatii'istic spin T particle can be described by a 2 component Pauli spinor. The spin operator is J = Z . Thc longitudinal polarization, cr helicity, is -r_ *< "J.Î ;p; ^ = .- |,p >f where p is the unit vector along ? . Polariza- tion is right or left if the eigenvalue of a.p • 1. The projection operators on the + 1 and - 1 helici- ty states are : A relativi^tic spin 7 particle is described by 4 conponent Dirac spinor obeying the Dirac equation which can bs written : where a and a are the usual A x 4 matrices : Assuming a zero mass for the neutrino : E = |p), and (a.p)u = u .In order to define a spin projection operatcr, we r.eed the matrices U ; (which we note a') related to the a matrices by : Therefore, the projection operators for the + 1 and - 1 helicity states are : The (I : Y ) factor projects out the - 1 and + 1 helicity states from u . The same argument can be applied to an electron when E ?> m [~ Î 1). The Co60 experiment gave a first indication y ' concerning the lepton polarizations. The lepton pair has to t carry the AJ = 1 lest by the nucleus. As the electron is preferentially emitted backward with respect to the direc- ^ y. s= 1 tion of Co^ polarization, this means that, in this configu- + [[ ration, the longitudinal polarizations of e" and u are respectively - 4 and • y - Co60 N.C°" e- HISTORICAL INTRODUCTION 2.5 Measurements of electron polarization in nuclear B-decav Electron longitudinal polarizati. n can he measured in several ways. A first method consists in changing the longi tudinal into transverse polarization cither by an electrostatic deflection of the beam or by crossed electric and ma gnetic fields which twist the spin orientation without deflecting the beam direction. Then the transverse polarization is analysed by measuring the asymretry of electron-nucleus scattering. A second method uses the fact that, in electron electron scattering, the cross section is larger when the electron spins are antiparallel. The longitudinal polariza tion is then directly analysed by observing the electron scattering in magnetized iron. In the "Bremsstrahlung" method one detects the circular polarisation of 7 radiated by a decelerated longitudinally polarized electron. Many experiments were performed in 1957 in different countries. The conclusion was that electrons have negative helicity (left-handed) and positrons have positive helicity (right-handed). This fa;t being established, It was then possible to nake a connection hetween couplings and neutrino helicity. The matrix elesient u 0A1 * Y )u written above describes the annihilation of a neutrino and the creation of an electron (v - e") or equivalently the creation of a pair e+ u in a 6* decay process A -• A' + e* + v . We know from experiment that the emitted positron is right-handed or, equivalently, the absorbed electron is left-handed, so we can replace the electron spinor u by 5J u . û then becoces ù Si. and : u tY ]u u (1 CT Ju e °i° * 5 v * e —T~^ °i * 5 v where t = * l for a left-handed neutrino, e = - T for a right-handed neutrino. .<*••; > and 1 Y anticoDTCutc with y , one has, for V and A couplings : EY )u i = V, A 5<>i(H> ' u 0. (I » YJU if e = + 1 1 1 and Y Y commute with Y • Then for S and T couplings : "e "4^ °i t1 * **sK * "e °i (^P) f1 * "J " "e Oi (1 " Y5)"v if -" - " 1 Therefore V and A couplings iinply a left-handed neutrino, S and T couplings imply a right-handed neutrino. The crucial cxperinent was then to measure the neutrino helicity. Z.b Determination of the neutrino helicity The carli'-r data on electron-neutrino correlation, mainly froa He0 decay, seared to point to S and T couplings respectively _" - Ferai and Ganov-Teller transitions. In 1957 however, after the discovery of parity violation in the decay of polarized Co60, measurements on the decay reaction A35 - CI35 * c* * v [which is practically a pure Femi transition), showed that e+ and v are preferentially emitted in the same direction [Hermannsfeldt, Allen, Stahelin]'3: this urpljcs a correlation coefficient \ • + 1, that is V coupling. The relation between coupling and neutrino helicity can be sketched here in a very single man-.er. In a pure Fermi transition, the lepton pair is in a singlet state, S = 0, which means that e+ and v have opposite spins. On the other hand, the e* is right-handed. So, the most probable configurations are, for S and V couplings : s v * =-1 A = *i Therefore, l = +• 1 (V coupling) implies a left-handed neutrino. Réévaluation of the earlier He0 data (1958). and new measurements (1959) on Ne^3 (He and Ne25 are pure Ganov-Teller emitters) and Ne'9 (mixture of Fermi and Gamov-Teller) confirmed that Fermi and Gamov-Teller transitions were respectively V and A couplings, showing that the neutrino had a negative helicity. In 19S7, however, the neutrino helicity had been directly measured in a beautiful experiment [Goldhaher and col laborators]. The principle of that experiment is to replace the difficult detection of a recoil nucleus by the measu rement of the circular polarization of a Y emitted by the recoiling nucleus. M. Goldhaber used a radioactive isomer, EU,_ ra , which, by allowed orbital electronic capture, decays into an excited state of Sm]5 . The spin and parity of these nuclei being respectively 0~ and 1", the transition is of the Gamov-Teller type : »- i >" i The electron is alaost entirely captured in an i. = 0 state. Let 1 be the direction of the emitted neutrino (this neutrino is assumed to be of the same type as the one emitted in 6* decay). If the Z projection of the neutrino spin • , the spin projection m of the recoiling nucleus can only be 0 or - 1, and 0 or • 1 for a neutrino spin projection equal to - 4- • The excited state of Sm,7 decays to the ground state, of spin parity 0 , by y emission : Only the photons emitted forward with respect to the recoil (i.e. with the maximum pass Jle er :rgy) are selected : y recoil \> —^ c=£> <=£> <3=i As the Z projection of the total angular momentum of the t is * 1, it follows that the spin projections of the y and the recoil nucleus are equal. So, a left (right) circular polarization of the y means that the neutrino is left Handed (right-handed). The experimental set up (Fig. 5) is such that the detected photons are required to undergo a resonance scattering in a SzuO, scatterer (the resonance scattering consists of a resonance absorption and immediate re-emission) (Fig.6). The excited state So,. has a mean life of •>. 3.10"'4 sec., and the Y emission takes place before the HISTORICAL INTRODUCTION recoil nucleus has lost its energy. The scattered y must supply energy to the target nucleus in the scatterer, there fore only the forward emitted photons, which carry the energy of ecoil along with *he excitation energy, can undergo resonant scattering. The Y-rays coming from the radioactive source go through a block of magnetized iron before reaching the scatterer. In magnetized iror., some electrons have their spin oriented with respect to the magnetic field (in the opposite direction). On the other hand, a photon is more strongly absorbed when its spin is antiparaHe1 to the electron spin (then an electron can absorb the Y angular momentum by flipping its own spin). When the magnetic field has the same directicn as the y beam, the absorption is stronger for right-circularly polarized Y-rays. Thus, the y circular polari sation can be determined by observing the variation of t-ray intensity when reversing the nagnetic field. The experiment showed that the forward y-rays were left-circularly polarized, implying that the neutrino emitted in the deoy of EU.!, is left-handed. Therefore the coupling is A, and not T, for the Gamov-Teller transition, 2.7 Structure of the nucléon weak coupling Thus, within a year after the discovery of parity violation, a large aaount J£ experimental informations had been gathered, showing decisively that V and A couplings were responsible for nucl-ar B-decay. The matrix element of the general parity violating weak haniltonlan (2.4) reduces to : (with Oy = Y , 0, =• iy Y ) ' H ' - \ V„«V " Wn S.V * ^s'u « C,, and C. are the normalized vector and axial vector coupling constants : C, *• I, since the value of the Fermi cons- 14 14 tant G is deduced from the pure Femi transition 0„ •* N_ + e* + v, in good agreement with the value deduced from the u mean life. The ratio of the absolute values of C. and Cy determines the relative strengths of Fermi and Gasov-Tellev cou plings. It was determined by comparing the half-lives, or more exactly the ft-values, of 0 , which is a pure Ferai emitter, and n, which is a mixture of Femi and Gamov-Teller. One finds : | The sign of the ratio A is connected to the longitudinal polarization of the proton ir the neutron B-decay, the same way the sign in I i Y is connected to neutrino helicity. However the velocity of the pr con is too small I- x 0.04J to allow such a measurement. For urtpolarized nuclei, there is no interference between Fermi and Gamov-Tcller transi1 iins, because singlet and triplet states are orthogonal to each other. That is no longer true for polarized nuclei. In the case of pure Fermi radiation, the electron angular distribution is isotropic with respect to the nuclear spin direction : the e-v pair being in a singlet state takes no angular momentum from the nucleus and is not influen ced by its spin. On th' other hand, the anisotropy is maximum for a pure Gamov-Teller emitter such as Co . For a mix ture of Fermi and Gamov-Teller couplings, however, the Fermi part can contribute to the anisotropy of the electron an gular distribution through interference with the GT part. This provides - means of determining the sign of the inter- terence term. Experiments on the decay of polarized neutrons were performed in 1958 at Argonne (USA) and Chalk River (Canada). The observed electron angular distribution with respect to the neutron spin direction has a very small anisotropy A - - 0.11 i 0.02 in 1 + AP .-£ . There is a destructive interference between the Fermi and Gamov-Teller terms. C The best value of £-A i= now : 1.Z50 ± 0.009 RJRIUKR DEVELOPtgNTS Looking back at this period from 1956 to 1958, we have seen an amazingly rapid succession of decisive experiments and as rapid a change of our understanding of weak interactions. There was a convergence of new experimental possibili ties and new theoretical concepts started by the questioning of paiity conservation by Lee and Yang. On the experimen tal side, experiments were performed very quickly because new techniques happened to be available or experimental set up had been designed for other purposes. The Co^O experiment was made possible by very low temperature techniques, al lowing the "freezing" of nuclear spins. As soon as they heard of the Co60 preliminary results, Lederman et al were able to use their apparatus for studying the ïï-u-e chain. The Goldhaber experiment for detecting neutrino helicity needed a radijacti\'- process fulfilling very strict conditions. It just happened that Goldhaber, Grodzins and Sunyar had the radioisotope Lu from their previous investigations. And, at the Argonne National Laboratory, a highly polarized neutron beam had been successfully completed since 1957. 3.1 The two component theory of neutrino On the theoretical frcnt, the preliminary results of the Co60 experiment led [Salarc, Lee and Yang, Landau] to in troduce a two-component theory of the neutrino. We have seen (2.4) that, in order to take into account maximum parity violation, the lepton part of the matrix element was written as û 0,(1 * Y5)U , and —I_Ii are the - 1 and + 1 hclicity operators. This form could have been obtained from the beginning by assuming that the neutrino had only one spin state, or, if one prefers, only one helicity state, the spin being always parallel or antiparallel to p . There fore, the neutrino behaves like a screw. In this theory, parity violation in weak interaction is intimately connected to a particular neutrino property. A Dirac wave function has usually four components describing two spin states and the positive and negative energy states. If there is no mass term, only three anticommuting hemtitian matrices ars needed, and the neutrino can be represented by a 2-component spinor function. It is amusing to note that this kind of theory for a relativistic spin j particle was first suggested [Weyl (1929)] it was later examined by Pauli in 1933, but rejected because it implied an intrinsic violation of space inversion inva riance. By the end of 1957, it had become clear that the neutrino and its antiparticle were different : the neutrino was found to be left-handed and the antincutrino right-handed. Under space inversion or charge conjugation, a neutrino is therefore changed into an unphysical state : the two-component theory violates parity and charge conjugation invarian ce. But there is invariance under the combined CP operation. HISTORICAL INTRODUCTION 3.2 The V-A law The two-component theory of neutrino may be a consequence of a universal Fermi interaction between pairs of fer mions with equal amounts of vector and axial vector couplings. This V-A law was postulated in 1957 [Feynman and Gell-Mann, Marshak and Sudarshan, and Sakuraï] ', even before the experimental crt-jation had been settled. The fact that CA = - 1.2 Cy in nuclear a-decay was thought to be a "renormalization effect" due to strcng interactions. In the absence of strong interactions, one would have CA = - Cy , hence V - A - The almost perfect agreement between 14 the values of the Fermi constant deduced from u-decay and from O p-decay (i.e. Cv - 1] shows that there is little renormalization effect on the vector paTt of the current. This was interpreted by *-he conserved vector current hypo thesis (CVC) of Feynman and Cell-Mann (and also [Gerstein and Zel'dovich] , who wrote in 1955, when B-decay couplings were commonly taken as S and T : "It is of no practical significance, but only of theoretical interest that in the case of the vector interaction type V, we should expect the equality g„ = g' ..."). 3.3 The current-current form of weak interactions It is worth noting that, Z3 years after the original Fermi publication on B-decay, the main change in the theory was to replace y by > [T + T ). The only thing Fermi had not Foreseen was parity violation. The neutron decay is described by the product of two weak currents, the nucli»nic current which is approximately PY (1 * t,)n and the leptonic current ër (1 * T )V , the u-decay by the product of two leptonic currents Y v ant it (1 * «,) ' o 11 * *b)v. tach current is a sum of vector and axial vector currents. This description was generalised by [Feynman and Gell-Mann] as the current-current hyrx>thesis. They define a weak current J , which is the sum of two currents : 1 - a leptonic current : J - ër (1 • Y5)V * u>uO * YS)V (with the sane coefficient for the e and u teras, from the hypothesis of e-u universality), - a hadronic current : J => p> (1 * >5)n * teins taking account of strange particle decays. The current-current hypothesis states that all weak processes arise from the interaction of the current J with itself : The universality is a consequence of the self-interaction of the weak current J . This general form of the weak interaction can describe u-decay, nuclear B-decay and e" capture, u~ capture. It also describes the decay of strange particles, either to leptons, for instance : *J * pe~v by (pA°) (ëv) or to hadrons ; A* - p*~ by (pO(np) (np "»• •" by strong interaction). It also predicts, through diagonal terns, neutrino-clectrcn scattering, and a weak interaction between four nu cléons (which has been seen very recently in Nuclear Physics). Fcynman and Gcll-Mann deliberately ignored the possibility of neutral currents, for which there was no experimen tal evidence. 3,4 Neutrino interactions and lepton conservation The term (pn) (ëv) + h.c. of the weak hamiltonian describes as well : neutron fl-decay : n •+• p + e" + Z electron orbital capture : e~ • p - n + u and neutrino reartions : v * n -+ p • e" v * p •+ n + e* The last reaction was first observed by [Reines and Cowan {19S3)J[improved experiment in 1956). Antineutrino were produced in a nuclear reactor by 6" nuclear decays. The apparatus was made of five alternate layers of three scintil lation detectors and two target tanks consisting of polyethylene boxes containing a water solution of cadmium chloride. Lvents were selected by requiring a prompt pulse from c* annihilation into photons, and a delayed pulse from neutron capture in cadmium. The observed cross section was in very good agreement with the theoretical value. The concept of lepton conservation was first discussed by [Konopinski and Mahmoud (1953)J, and later by Lee and Yang in their paper on the two component theory of the neutrino. Defining the neutrino as the one emitted in B+ decay (left-handed), and the antineutrino as the one emitted in B~ decay (right-handed), it can be seen that the above pro cesses substitute a neutrino for an electron, or an antineutrino for a positron, or vice-versa. So, one is led to at tribute a leptonic charge (or lepton number) * 1 to the particles e~ and v, and - 1 to the antiparticles e* and v . The lepton conservation law forbids double B-decay processes (such as Te^.-, + X.^ + e~ + e") in which the neutrino emitted in the decay of a first neutron would be reabsorbed by another neutron, giving rise to a second electron. However, such a process is also forbidden by the fact that neutrino and antin°utrinc have opposite helicities. 3.5 The two neutrinos At first, it was thought that the muon had the same lepton number as the electron, and that it decayed by emit ting a neutrino - pntineutrino pair having the vacuum quantum numbers. In such a scheme, the muon was to be considered simply- as a heavy electron, and one would expect to see the decays : u~ * e" + e • e" or the capture process : u" * p + e~ + p The experimental upper limits of these processes are respectively found to be : 2.10"8 , 6.2 10~9 and 2.10"7 . This suggests that e and u must differ by some quantum minier, the conservatinn of which prohibits the above processes. Therefore, one has to introduce two different kinds of neutrinos, v and v (and Z and v ). This hypothesis, made independently in 1957 by Schwinger and Nishijima, was experimentally tested in IS62 when the first neutrino beam became available at the Brookhavcn 30 CeV accelerator [Danby et alj? The experiment was repeated thereafter at ŒRN [Bienlein et al (1964)]. In these experiments, the neutrino beam is produced by decay in flight of pions, the accompanying muons being ab sorbed in a very thick shielding wall. Neutrino interactions are observed in heavy spark chambers. They should produce equal nunbers of electrons and nuons if there were only one type of neutrino. The absence of electron showers in the spark cheers is a proof of the existence of two different neutrinos \>c and •.. . Here, pion-decays produce v which, by interacting, arc changed into nuons. One has to introduce an additional quantum number distinguishing the muon from the electron. We can therefore describe leptons in terns of two additive quantum numbers e and t having the follo wing values : HISTORICAL INTRODUCTION e" ue e* Ze u" v v u Z v i * 1 + 1 -1 -I 0 0 0 0 e t 0 0 0 0 • 1 + 1 - 1 - v Ail known decays are cocpatible with separate congélation of t and ï . However other forms of lepton conservation laws are not excluded- 3.6 Classification of weak processes There arc 3 main categories of weak processes : J.S.I. Leptonic piwceeiwa in which only leptons are involved. These processes are indui_?d by self-interac tion of the leptonic current : il-^'-iJ"*- The best known is the u-decay Other processes are possible, but have not yet been observed. First the so-called diagonal processe and e* + e~ - v • Z . which nay play a role in stellar evolution. e e The reaction v + e" •* u * e~ is forbidden in first order by lepton number conservation in the absence of neutral currents. The process u * e" * ' is possible, but could be observed only with very energetic neutri- nos, as the threshold energy is 11 GeV. S.û.2, Serri-lcptonic ppoocoana induced by the cross terns between leptonic and hadronic currents. They are further classified according to tj.c change of strangeness between the hadrons of the initial and final states : a) ^_°„n__trans'ilS"? . These transitions include : - a" and e* nuclear decays. The clcnentary pioccss is : n -* p • e~ + v - electron orbital capture : e" + p •* n * « - na«n weak capture : u~ • p * n + v - neutrino reactions, either "elastic" processes as v + n -• p * p~ or inelastic processes where more hadrons are produced. - n decay into uv and ev - also aS = 0 transitions between strange particles : Z + A0 e+ v t~ * A° e" v e b) ɧ_f_J transitions . This category contains : - all leptonic decays of hyperons as A° -» ne~v - neutrino reactions with ûS = 1 as Z * p * p + A° [£") - teptonic decays of K mesons. 3.6.1. Non leptonic (or purely hadronia) processes In this category one has the hadronic decays of A0, E~, H5 and (T hyperons, and the decays K - 2n, 3n. All are iS = : 1 processes. he nay surraarize by the following table : LEPTOSIC u ' decay neutrino - electron interactions as - 0 oS = 1 Baryons 6 nuclear decay u~ capture leptonic decays of hyperons SEMI - LEPTONIC E* •* A°e: neutrino neutrino nucléon interactions neutrino nucléon interactions Mesons " ~-* ev îeptonic decays of K mesons parity violation effects in hadronic decays of hypsrons IIADRONIC K - 2" nuclear physics K - 3" Table 1 : WEAK t.VTERACTIOSS 3.7 Isospin selection rules Isospin is not conserved in weak transitions. However, son» approximate selection rules have emerged from the ; lysis of experimental data. Tht^c rules are different for scmi-Ieptonic and purely hadronic transitions. J. ?. 1. Stmi-loptonia tranb'tior.a of the typo : A - A' • e (or u) + neutrino where A* can be a system of hadrons. From the Cell-Mann-.Nishijima relation Q = I* + it follows that the HISTORICAL INTRODUCTION variations of the quantum numbers Q, I, and S between the initial and final hadronic states A and A' are related by : AQ = ûl3 + ^ since ûB =• 0 (baryonic charge is strictly conserved). In the absence of neutral weak currents, ''Q| = 1. Two cases are to be considered : a) ^S_f_0 transitions Then |AI,[ =• 1 and a! can be 1, 2, ... Obviously, only ûl = 1 can contribute to the following decay proces ses : n •* p + e" + v : Al = 1, AI, = + 1 n* - u* + u : AI = 1, AI3 = - 1 The study of nuclear transitions indicates the absence of Al > 1. The proposed selection rule is therefore : 41 = 1, AI3 = ± 1 b) ].ûSj_=_l transitions AI is half-integer, for instance : A" + i: + e" + v Al = — ^ e Z K+ •* v* + y, Al = i More generally, with |AS| n 1 and |ûQj = 1, two cases are possible : - AS ° • AQ implies |ûl3| = j and AI » | , | , -| . Here, two selection rules have been proposed : - the AS • » AQ selection rule forbids I* •* n * e* • v . This decay has not been established, but E* • n * e" * « is observed ; n a - the semi-lcptonic A[ = -l selection rule, which is stronger than AS +AQ (since AS * AQ stems from ûl = 7). This rule implies relations iroong the decays of K , K~ and neutral K into niv , which are reasonably well satisfied experimentally. J. '.'. 2. Hon taptonia trtmoitionn Again AI is half-integer, for oarrple : A" - p + „- : AI - 2 » f K - 2" . ÛI • T » T I T For purely hadronie transitions : ûQ = 0, ûB = 0 and ûl, = - y- . The strangeness selection rule |ûS| = 1 implies |ûl,| = y and ûl can be -^ , ^ ••• The non leptonic ûl = j selection rule was introduced in order to account for the smailness of the decay rate of K* -* "*TT° compared to K° •* nn . FOr the ûl = » selection rule, K* + TT+TT° is a forbidden transition. It proceeds 3 5 through a small contribution of il = T and/or ûl = -j • The branching ratios of A° + Nn and K? •* tin, and the ratio of mean lives of =° and E", can be deduced from this selection rule, in good agreement with experiment. We may summarize the isospin selection rules in table 2. 3.8 The Cabbibo formulation of univeisality In the universal current-current theory of weak interaction, all weak processes are described through the inter action of the total weak current with itself, with only one coupling constant G. This theory was successful in inter preting all aS = 0 semi-leptonic transitions. However, it was in trouble with the leptonic decays of hyperans : the rates were found to be systematically lower, by a factor of about 20, than what could be predicted from the universal Fermi interaction, [t was not possible to explain such a large and systematic reduction factor in terms of strong in teraction renormalization effects, i.e. in terms of form factors. [Cabbibo (1963)]^ proposed a new definition of universality by assuming that the strangeness conserving and stran geness non conserving currents snter the total current through a normalized linear ambination : where 0 is known as the Cabbibo angle. This combination was interpreted within the framework of SU3. The comparison between the ûS = 1 and ûS = 0 processes allows the détermination of 0 since the branching ratios are proportional to tg2e. One obtains approximately sinz9 ^ 0.05. At the sane time, the Cabbibo foroolation was able to explain a small discrepancy which remained between the values G deduced from u-decay and Gfl deduced from 0 de cay, since G is identical to the Ferai constant G, and G. n G cosO. Let us irfntion that the extension of the universal current-current interaction to non ;„, lie transitions is not straightforward and aeets with sone difficulties. In particular, it is not easy to understand the origin of the ûl ° 7 non leptonic selection rule. Non leptonic AS = 1 transitions would be induced by the product of two currents j£ and J, which carry respectively isospin changes ûl * 1 and ûl = T . The combination of isospins I and i gives ~L as 1 3 " 2 well as j r and one is left with the problem of explaining why the ûl •> y part should be depressed relatively to ai • I . 3.9 Neutral kaons and the discovery of CP violation Conservation of strangeness by strong interaction implies that K" is different fron its antiparticle K° : X° an K" ire eigenstotes of strangeness with eigenvalues * 1 and - 1. [Gell-Mann and Pais (1954JJ' pointed out that, due to non conservation of strangeness by weak interaction, the physical particles, characterized iy a definite life time, were not K° and R° but charge conjugation cigenstates K" • , Kj - . After the discovery of P and C violation, K* and K" were defined as CP eigenstatcs. K° would decay into two pions, since a 2 pion state with i •» 0 has CP •> • 1, and K° into 3 pions (predominantly CP • - 1 state}. The prediction was that K° had a longer life-time than K° . The lonp lived KZ was discovered at Brookhavcn [Lande et al (1956)J in a cloud chamber experiment. Neutral K'J produced in a target inside the chamber decayed into two pions. When the target was 6 ireters away from the chamber, HISTORICAL INTRODUCTION only three body decay modes of neutral K mesons (called "anomalous" decay modes) were observed. The existence of common decay modes which are not CP eigenstates led to the interesting possibility of observing interference effects between K? and K° . In 1964, once more, the K meson shook the theory of weak inte action an it did with the T - o puzzle. The theory which had been developed was CP conserving, and, in such a theory, K, cannot decay into tvo pions. However, decays into v*v~ of long lived K mesons were observed at Princeton [Christenson et al (1964)J, showing that the long lived K" (thereafter called Kj" , in contrast with the short ljved K The weak theory was not violently shake» as in 1957, but suffers from a disease which has proved very difficult to aire. Several tentative explanations for saving CP invariance in weak interactions were successively ruled out by experiment. The hypothesis of a long range galactic field interacting with hypercharge implied that the K.° •+ n+n~ rate should depend on the K° energy. Experiments with K momentum varying from about 1 GeV to 10 GeV yielded compatible values of the branching ratio. Another possibility was to assume that K° decayed first into K" and a new particle of very snail mass, thus simulating a decay ¥.Z - ***', In such a model, in a superposition of K". and K, there could be no interference between the »*n" decay amplitudes. This interference, leading to oscillation in the decay rate, was observed, and afforded the best way to measure the K° mass difference. A podel also was considered in which the CP violation was only apparent, due to a violation of Bose statistics by the piois. If that were true, one should not observe the decay K° •* rDjt°, since CF is always + 1 for a two r° state. Several experiments performed in 1967 and 1968 showed that such a decay does exist- Dsspite veiy active research, the CP violation has been observed only in neutral K meson decays : 1°) decay of long lived K.° into n*v~ and *°»0 ; 2") charge asyranetry in the leptonic decay nodes K°, and K°, : r(K£ - .-*+v) / r(K£ •* *+£-3 The asynnetry -— is the order of ID-3 . r* * r" Where docs the CP violation effect coae fron ? It can be in the cocposition of the long lived KL , which is V, plus a very small admixture of K^ , or in the 2* decay oode, allowing the decay K| * 2t, This question will be oiscussed quite extensively in the chapter on CP violation. All that can be said here is that it is very difficult to incorporate CP violation into the framework of the universal current-current theory. 3.10 The intermediate vector-boson hypothesis In writing the weak hacu 1 Ionian, we have assumed a local, or zero range, interaction. As i consequence, the neu trino total cross section increases linearly with energy, leading to a difficulty with uni rarity. The weak interaction a.iy have a non zero range. For example, it might be mediated by an intermediate boson. This idea was first due to [Yukawa (1935)]v Shortly after the introduction of Fermi's theory, Yukawa made the suggestion that nuclear fl-decay and the strong nuclear force could be explained by the same mechanism of meson exchange, this mesen decaying into an electron and a neutrino. However, the vector transformation properties of the weak current are not compatible with pion exchange. The intermediate boson of weak interaction must he of a vector character (spin one), charged (with possible a neutral component) and massive (the mass being inversely proportional to the range). The total four Eermion Fermi-interaction is then replaced by a semi-weak Yukawa type interaction : H - gGIj W* • h.c.) •V exchange introduces a propagator -v —* . For increasing q , this factor will prevent the neutrino cross-section from increasing linearly with neutrino energy, as it will do in a structureless local theory. For small q2 [s-decay processes), one has : >•¥ The experimental lower limit for Mj is now ^ 10 GeV. 3.11 Higher order processes and the renormalization problem The reaction v„ * e" 1 e is forbidden in the absence of neutral weak currents, but it can proceed through second order interaction for Ferni interaction for intermediate boson interaction The evaluation of such terms leads to infinities, unless one applies a raonentun cut off when integrating over the loop. Using the so called unitary cut off, the second order terra is found to be of the sane aognitude as the first or der term, which is unacceptable, since this would violate the known selection rules. The intermediate boson does not help to solve this difficulty. One has to adnit that the weak interaction is described by an "effective lagrangian" which is valid only in the first order approximation [where it gives very good results). Very recently attemps have been made to build a renormalliable theory of weak interactions, which at the same ti ne, unify the description of weak and electromagnetic interactions. The possibility of renonraliiation seems to exist but these new theories imply neutral currents, or new heavy leptons, or both, and all have difficulties in taking into account strongly interacting particles. It is gratifying that neutral currents have been recently observed. One cannot soy now what will be the final result of this very a;tiv* research, but it shows at least that the history of weak in teraction processes is not yet closed. HISTORICAL INTRODUCTION REFERENCES 1. Pauli, W., Septière Conseil Solvay, 1933 (Gauthier-Villars, Paris 1934). 1. II Nuovo Cicento Jl, 1 (1934). La Ricerca Scientifica 4_ (Z), 491 (1933). Zeitschrift fur Phyzik 88, 161 (1934). 3. Conversi, M., E. Pancini and 0. Piccioni, Phys. Rev. 68_, 232 (1945), Phys. Rev. 71_, 209 (1947). 4. Klein, 0., Nature 161, 897 (1948). Lee, T.D., M. Rosenbluth and C.N. Yang, Phys. Rev. ^5, 90S [1949). Puppi, G., Nuovo Ciroento 5_, 587 (1948). Tiotrara, J. and J.A. Wheeler, Rev. of Hod. Phys. £1» 144 (1949). 5. Lattes, C.M.G., G, Occhialini and C.F. Powell, Nature 160, 453 and 486 (1947). 6. Gell-Mann, M. , unpublished note : On the classification of particle-. (1953) - Suppl. Nuovo Cim. £, 848 (1956). 7. Nishijima, K., Progr. in Theor. Phys. (Japan) J£, 107 (1954), J3, 285 (1955). 8. Lee T.D. and C.N. Yang, Phys. Rev. JD4, 254 (1956). 9. Ambler, E., R.W. Jlayward, O.D. Hoppes, R.P. Hudson and C.S. Ku, Phys. Rev. J05_, 1413 (1957). IP. Garvin, R.L., L.M. Lederman and M. Weinrich, Phys. Rev. 105, 1415 (1957). 11. Friedman, J.1. and V.L. Telegdi, Phys. Rev. JOS, 1681 (1957). 12. Let, T.D., R. Oehme and C.N. Yang, Phys. Rev. JQ6, 340 (1957). 13. Allen, J.S., K.B. Hennannsfeldr and P. Stahelin, Phys. Rev. J07, 641 (1957). 11. Goldhaber, M., L. Grodzins and A. Sunyar, Phys. Rev. J09, lt),s 0958). 15. Landau, L.D., JETP 32, 407 (1957). Lee, T.D. and C.N. Yang, Phys. Rev. JOS, 1671 (1957). Salan, A., Nuovo Cimento 5_, 29 (1957). 16. Weyl, H., Zeitschrift fur Phyzik S6, 330 (1929). 17. Feyonan, K.P. and M. Gell-tann, Phys. Rev- J09, 193 (19S8). Marshak, R.E. and E.C.G. Sudarshan, Phys. Rev. J09, 1860 (1958). Sakuraî, J... Nuovo Circonto ]_• °49 (1958). 18. Gerstein, S.S. and J.B. Zel'dovich, JETP 2, 576 (1957). 19. Feynaan, R.r. and M. GelJ-Mann, Phys. Rev- J09, 193 (19S8). 2D. Cowan, C.L., H.K. Cnise, F.B. Harrison, A.D. Mc Guire and F. Reines, Science 124, n° 3Z12.103 (1956). Reines, F. and CL. Cowan, Phys. Rev. 92, 830 (19S3). 21. Konopinski, E.J. and H.M. Mahmoud, Phys. Rev. 92, 1045 (19S3). - 26 - CHAPTER I 22. Panby, G-, J.M. Gaillard, K. Goulianos, L.M. Lederman, M. Mistry, M. Schwartz and J. Steinberger, Phys. Rev. 9, 36 (1962). 23. Bienlein, J.K., A. Bohm, G, von Dardel, H. Faissner, F. Ferrern, J.H. Gaillard, H.J. Gerber, B. Hahn, V. Kaftanov, F. Krienen, M. Reinharz, R.A. Salmeron, P.G. Seller, A. Staude, J. Stein and H.J. Steiner, Phys. Letters J3, 80 (1964). 24. Cabbibo, N.,Phys. Rev. Letters J0_, 531 (1963). 25. Gell-Mann, M. and A. Pais, Phys. Rev. 97, 1387 (1955). 26. Booth, H.T., W. Chinowski, J. Impeduglia, K. Lande and L.M. Lederraan, Phys. Rev. _105, 1901 (19S6). 27. Christenson, J.H., J.W. Cronin, V.L. Fitch and R. Turlay, Phys. Rev. Let. J3_, 138 (1964). 28. Yukawa, H-, Proc. Math. Soc. Japan J7_, 48 (1953). j I | GENERAL KEFERENŒS Allen, J.S. : The neutrino (Princeton University Press 1958). « Kcnopinski, E.J. : The experimental Clarification of the laws of S-radioactivity, Annual Rev. of Nue. Se. ^, (19S9). Kabir, P.K. : The development of weak interaction theory (Gordon and Breach 1963). Killlen, G. : Elementary particle physics (Addison-Wesley 1964). Lee, T.D. and C.S. Wu : Weak interactions, Annual Rev. of Nue. So 1_5_ and J6_ . Wu, C.S. and S.A. Moszkowski : e-decay (Interscience Publishers, New-York 1966). Harshak, R.E., Riazuddin and C.P. Ryan : Theory of weak interactions in particle physics (Wiley-Interscience 1969). HISTORICAL INTRODUCTION - 27 " PROBLEM 1. Consider, in the non relativistic approximation, the scattering of a point-like charged particle by a potential. Show that the qz dependence (q = four momentum transfer) of the matrix element is i_ for a Coulomb potential, and __J for s. Yukawa potential. What is the inteipretation of these factors 7 qa - m2 Z. Calculate explicitly the non relativistic approximation of the five Dirac «variants [1-3J. 3. Derive the expression of the statistical spectrum in nuclear a-decay. Show that the decay probability is proportio nal to (ûM)5, where iM is the mass difference between the initial and final nuclei. 4. FIBBZ interference. Show that, if S and V couplings both contribute to a pure Feroi transition, the statistical spectrum is mltiplied by a factor Cj + -=• , with e => jp . e e 5. Electron-neutrino angular correlation. Show that, in the case of a pure scalar or vector interaction, the decay probability, sunned over e and v spin states, depends on the angle e through a factor 1+ -£ cos 9 . 6. What is the difference between the nuclear capture of a negative pion and that of a negative mon ? 7. What are the parities of 2* and 3w systems with total J = 0 ? Give the diarge conjugation quantum nunber for neu tral Zc or 3i systems. n 8. Defining a = JT (T Ï - Y Y ) and denoting a' the space components of o , show that a' - Y5« " (2.4). u 9. Lepton conservation laws. In the so-called multiplicative law, the sura t + iu and the nuonic parity (- 1)* are separately conserved. What processes would allow for a distinction between additive and multiplicative laws ? 8 10 ' i. Explain in a qualitative way why the mean lives of K" , K. and Kz are respectively •* 10~ sec., ^ I0" sec. and ^ 5.10-e sec. CHAPTER H LEP~ONIC INTERACTIONS L. JAUNEAU LEETONIC INTERACTIONS 1. MON DECAY 1.1 Introduction and kinematics Leptonic weak interactions are those in which only leptons parcipate, without the interplay of strong interac tions. Muon decay was for a long time the only purely leptonic weak process which had been observed. It is interesting to know whether the main features of nuclear 6-d.ecay : strergth of the interaction, parity vio lation, electron helicity, V-A structure, are present in muon decay. It will be seen'that, although muon decay is com patible with V-A, lending support to the hypothesis of a universal weak interaction, it is not possible, in this case, to détermine completely the form of the interaction. In muon decay, one can measure : - the u life-time, proportional to the Fermi constant squared ; - the energy spectrum of the electron for an unpolarized muon beam ; - the decay electron polarization ; - the angular correlation function between the electron momentum and the muon polariïation. However, one element of infon-ation is missing : the neutrinos are not observed and their energy spectra and pola rizations ^annot be determined. Therefore, we will have to use assumptions which are supported by the study of B-radio- activity. We note p, p', k, k' the energy-momentum four-vectors : p = p' + k + k' The metric is such that p2 n m , p'2 = mz (m = electron mass). We assume that the u decays at Test : p = (m ,Q) The electron energy is : n2 + m2 - (k+ k'V E iz ajxisazn for (k * k') minimum, i.e. (k + k')2 •» (n *• n ,) . Assuming mass less neutrinos, we have : ™2 . «* EMax = The observed value is compatible with "vanishingly small oass" for both neutrinos, but experimental error ti-jes not allow to get good limits on neutrino mass. On the other hand, the shape of the end of the electron spectrun is sensiti ve Co neutrino mass, however the determination would be difficult, due to large radiative corrections. The best present limit c^ < 60 eV comes from a very accurate determination of the end of tritium 6 spectrum. The vu mass limit is obtained from a measurement of u momentum in n * uu decay. The pion mass has therefore to be known ; with a very good accuracy, and is deduced from energy measurement of n-mesic X-rays. The present l mit is mw < 1.2 MeV. 1.2 General shape of the electron spectrum The man decay probability is calculated to first order in the Fermi interaction (the possible effect of an inter mediate boson will be discussed in ch. VII). For a given configuration : i Ei (2")> (Z.)S (2n)3 It) is the invariant Feynman anplitude. In the muon rest system : Eu = rau , hence : (2.0 s (m2 will disappear with a factor — in the sum of |TÏÎ| ov?r spin states). Let us write explicitly : but the distinction between v and v is not necessary for the calculation. We assume a non derivative local Fermi interaction. The Feynman amplitude W is a Lorentz invariant built with the following Dirac spinors : for the incoming particle u for the mtgoing particle e~ for the outgoing particle v n v (£') u- (- k') for the outgoing antipartide Ze (u. is the negative energy solution of the Dirac equation). As was done in nuclear B-decay.TO is written as a sum of five terms obtained by contracting tw> Dirac covariants of the same kind. However, for u decay with a local interaction, there is some arbitrariness in ordering the spinors. We have two possibilities : - the so-called charge retention order : . G £ T*l i «e °i "* V °i (Ci + Ci ^ V which simplifies calculations ; - the charge exchange order (analogous to the form used in a-radiaactivity) in which lepton number conservation is explicit, i.e. u" •* vu , e" - ve : Of course, these tvm forms of TTt are not independent. One can be deduced from the other by means of the Fieri-Michel reordering theorem. This theorem expresses the C. (C[) coupling constants in terms of the Cj (Cp : *Thc pi factors are normalization factors for the four-fermion fields. Another normalization convention can be used, : which the neutrino moss never appears (see Appendix I). LEPTONIC INTERACTIONS i, j = S, V, T, A, P [see Appendix II). An important property is that the V-A form of & interaction is invariant by Fierz-Michel transformation. Til is the most general parity non conserving matrix element. No assumption has been made concerning helicity and mass of the neutrinos. Therefore, 7)1 depends upon 19 real parameters (10 complex C- and C! with an arbitrary global phase). Note that C. and C! are real if there is time reversal invariance, l i Calculation of the muon decay probability is very lengthy in the general case. We will indicate only the main steps, and give the important formulae. Detailed calculations can be found in the books of [Marshak et al (1969)] and [Kallen (1%4)]'. a) Neutrinos are not observed. Therefore, one has to sum |U1| over the neutrino spin states, using standard trace techniques (an example of which will be given later), and to integrate over the momenta k and k' ; b) if the decaying muon is not polarized, one has to average over the muon spin states. If the electron polariza tion is rot observed, one has to sum over the electron spin states. This is done once more by using standard trace techniques. After these integrations and summations have been performed, we are left with the variable p', and, after angular integration, we get the electron energy spectrum. The shape of this spectrum depends upon two parameters o a.-*! n which are function», of coupling constants C= and cj : E 3tW E J p 4E W 3 nfW E } -***- " -^ Tv %e e P e e f e ~ e * e ~ f e ' " — V —e " e } dfl dEe 12." \ 3 ^ Bj Ee { m(W - EJ o is the MICJIEL parameter : til. n is called the low-energy parameter, since the n term is negligi ble except at very low electron energy. e The expressions of p and r as functions of C- and C- are given in Table 1. Experimental values of o and n are (see Roscnfeld tables) : 0 « 0.7517 t 0.0026 o - - 0.12 ! 0.21 Neglecting the electron mass and defining a dimensionless variable x • , one gets the simpler form : _*!_ . S. X2 f3(, - X) + I P(4X . 3)1 dfl dx T92i»u L 3 J TABLE 1 MJON DECAY PARAMETERS »i • iCii * iqi î S, V, T, A, P A » aj * 4^ » 6a,. » 4aA * ap a' • Z Re (Cj Ç* • cj C,*) b' " 2 Re 1CV CÀ* * S/ CA*> Ferai constant 3a • 6 ^ • 3a v A A "s" 2\ • îaA " ^ A - 3a ' - tb' * 14c A 3b' - 6c' 3a' * 4b' - 14c' a' * 4b* * 6c" LEFÎQMC INTERACTIONS 1.3 MLion total decay rate Integrating over dn and dx gives the total decay rate (note that P disappears in the total decay rate) This expression has to be multiplied by a numerical factor taking into account radiative corrections. T is the muon life-time : t = [2.1994 i 0.0006)iO-6 s. In order to compute the numerical value of G from the muon life-time, we must remember that we use particular units such that"h" ° c = 1 . First, we note that the Fermi constant has the dimensions of the product of an energy and a volume : the weak hamiltonian for muon decay (which has the dimension of an energy) is written as the integral over space of the product of four fermion fields, each one having the dimension . Therefore, one has : Wd3xfeH Let us choose the proton rest mass M:2 as a"unit of energy, the proton Compton wave length as a unit of length, and as a unit of time : Mc2 Mc2 = 938 MeV , -=2.lx 10"' He It follows that G= ( K—] —J- - isi .a dimensionless constant (that is GM! in our unit system with 1Î = c = 1). Then we nay write : i 192 "!\N/ ' ' • 192 »>\M/ [_ V*/ * J * dimcnsionlesa constant and G %* HT""(it5 (2- f •* * Taking into account radiative corrections : G • (1.023 : 0.001)I0"5 M~2 in units K » c » 1 Froa (1) : Cî 1.4 lO*'*9 erjs x OÊ2 Vhc30 corrections have been calculated for the local four feraion interaction. If there is an intermediate boson, the result is significantly changed and is DodcL-dcpcndcnt. 1,4 Decay of polarized muons Let P be the tnuon polarization, •. the angle between electron momentum and muon polarization. Neglecting the electron mass, tlia angular correlation function can be written as : _J*_ = -£L m p En 3(K -EJtîp (4Ep - Sri) - P -Ê cos 9 tfcW - E J • ^ 4 (4Ep - 3W)1 I e e e da d£e 12." ^«e^ 3 Ee L 3 -1J or : jv = — x2 J 3(1 - x) • - p (4x - 3) - P cos e t [(1 - x) • ^ i(4x - 3)1 L da dx 192 i* ) 3 L 3 JJ This function contains two new parameters Ç and 6 (see table 1). The calculation for u* decay gives the same result with a change of sign in front of t • Anisotropy is definitely observed, implying violation of parity in muon decay. The present experimental values (including radiative corrections) are : f. = • 0.972 ± D.013 S = + 0.7551 ± 0.0085 The polarization h of the electrons from dtcaying u-particles has been measured by several authors. That can be done by aeasuring the polarization of their Brcmnstrahlung or by using the spin dependence of electron-electron scattering on a target containing polarized electrons.(For e* craning from u* decay, it is possible to treasure the polarization of the photons eraitted in the annihilation process, or to study the positron-electron scattering on polarized electrons). ïiithin the experimental errors, the present results are compatible with - 1 for electrons and + 1 for positrons, in good agreement with what was observed in nuclear 8-decay. In conclusion, nuon atcay depends upon 19 real parameters if time reversal invariance is not assumed (complex C- and Cp and upon 10 real parameters (Cj and CJ real) assisting time reversal invariance. Only six functions of these parameters con be cicasured experimentally : the Ferai constant G, the spectrum parameters c, S, -., 4, and the electron polarization h. One has learnt that parity and charge conjugation are violated, electron helicity is negative, suggesting V-A, and the values of the parameters arc compatible with V-A (as will be seen below). However, it is Jt possible to determine com pletely the matrix element of ntuon decay without injecting information that has been gained in the study of nuclear 6~decay. Note : We have written the matrix clement in the charge retention order : i "e°i % "v °i (Ci * Ci T5,vv JVIOLI.T expression can be found in the literature : \ ÛC t8i • «; ,s) 0; uv Û„U o, vUo It is cosy to check thai the correspondence is : %-s «p-s «s "S ^"^ 8v-cv *A"CA H-'ck >kmCi : EPTONIC INTERACTIONS 1,5 Two component neutrino assumed In nuclear 6-decay, the neutrino helicity has been experimentally determined and is found to be negative. We assu me now that the same property holds for vu , i.e. \> and « have negative helicity, v and v have positive helicity. Therefore, we can write : 1 * *s v for v (annihilation of «e) The matrix e'ement becomes : ÇG o „ î _-î . (c. •cj, )—- _0,(, T5,T c i ]i Vu 0 s !i.»i».(-VK^ * «* ( tt II follw.s that tiie only possible couplings are V and A (since 0S , Q_ , 0p connute with Ts,at*l (1 - t^) - ï5J 0 Kith the .wo co.Tponent neutrino hypothesis, the matrix element for nuon decay reduces to : CC C Y )u y T "e *P V " A s i= \ v C * slvVfi with C^Cy and C^-CA. This implies (frcn table I) : c? • (ic/ > :c/J "V* - 'CA'' I -&— • C' nS «J f(3 - ilj - o>s 1 1 P(2x - 1)1 The asyro-ctry tokos the very sirple form : I * rs ». Is chosen in such a vay that projects out negative helicity states (left-handed statcj). 2 Recall that the srcio arguocnt was used to dCTJcnstrate V-A coupling in R-dccay (chapter 1, 2.5). where the asyranetry parameter a is a well defined function of Eg : 3 - 2x Integrating over energy gives : I (cos e) = - ft - - ç P cos el The expression of the electron polarization h can be easily derived in the case of V and A couplings. Let us write : • «V -A 's'"e '» • 5 S " CA ^ • «V " CA ' -y^ * «V ' CA ) y- Therefore, C, - C, is the amplitude of the left-handed stace, Cy * C, the amplitude of the right-handed state. The longitudinal electron polarization in therefore : ; C h , 'Cy* • O' - IV - O' , •*• V A "v* ' CA*I! * iV - CA*I2 I If furthermore one assumes V-A interaction, i.e. C. = - Cy : h = - 1 : the decay electron is cccipletely polarized e~ has negative helicity in u - e + v + « e has positive helicity in u -* e •>"„*% This result is independent of cuon polarization. t » • 1 : the angular distribution is proportional to : [3 - 2x - cos 6 P [2* - 1)] For conplctely polarized nuons, at the maximum energy (x •> 1) no electrons are cxiitted in the direction of j polarization, and there is a caximun probability for the electrons to nove away in exactly the opposite direction. Two component neutrino \fcA ^ V, A i.e. CA - - C, 2 2 . M • ISJ r » 0 2 |Cv]2 . |Cfl|> 2 Re Cy* C. r. - - h • f*—2-T f. - * 1 (h - electron helicity) h - - 1 LEPTOKIC INTERACTIONS 1.6 Uncertainty on the muon decay iuatrix eJement he have seen that experimental results are compatible with predictionsof the V-A postulate. However, it cannot be claimed that V-A has been experimentally proved for cuon decay. It has been shown [Jarlskog (1966)] that the strongest restriction which can be placet! on the fom of the muon decay interaction comes from the determination of electron helicity. Taking the values suggested by experiment : h = - 1, p = « -| , $ = +1 it can be shown that the matrix element must be of the form : =e V * *s>uu \ \,«V - CA »s> \ Therefore, the V'-A structure is established only for the charged lepton part, but, if the neutrinos are undetected, the ratio of the V and A constants in the neutrino part remaî.iç unknown. A conclusive test of V-A would require an electron neutrino coincidence experiment, which is well beyond the foreseeable experimental possibilities. If we assume now that « and \>u have negative helicity, their coupling is also pure V-A ; only in this case is the overall coupling V-A when written in the usual charge-exchange current-current form (since V-A is invariant by FIERZ- MiaiEL transformation). If the two neutral particles are assumed to be identical two coiçonent particles obeying Femi statistics, it may be shown easily that o a 0. The probability density for maximum energy electron is proportional to p (see e spectrum with x = 1). This corresponds to the configuration wh**re the two neutrinos are emitted in the same direction opposite to electron aornentum and have zero relative momentum : Furthermore, their wave function must be antisymmetric. The assumption of two component identical particles implies that they are in a triplet spin state S * 1, which is symmetric. Therefore, the decay probability must go to zero at x = 1, so that o = 0. It follows from table I that only S and P couplings can then occur. Obviously, the value c = 0 is in con tradiction with experiment. Furthermore, the decay u * e + v + v would lead to v - e + y unless the two neutrinos are different (see chapter I , 5.5. ). The existence of ve and v has been established in neutrino-nucleon interactions. .'. SEUTRINq-ËLECTRON INTERACTIONS i. 1 Introduction The current-current hypothesis predicts, through self-interaction of the leptonic current, the possibility of neu trino interactions with electron targets, and of tri-lcpton production (trident) by neutrinos in the elecuvjnagnetic field of a nucleus. The allowed neutrino-electron processes are the inverse processes of muon decay : and the so-called diagonal processt those which arise from the self-coupling of a single term i° *^<* current, such as elastic scattering : The corresponding graphs, in which factorization is made explicit, are drawn so as to illustrate lepton number conserva tion in each current coupling where only charged currents are assumed (the dotted line may represent propagation of a virtual intermediate boson [W] ; then lepton numb?r is conserved at each vertex). A common feature of these reactions is a very low value of the available energy in their center of mass system, due to the low mass of the target. Let s be the Mandelstam variable : (center of r. • energy)' E„ (GeVT • pwr ?-„K- This incites very low crass sections and a high threshold for reactions (1} and (2) tthreshold,i5/î1|(,eV The processes : % • e" * 3„ * c" (6) are forbidden to first order in the Fermi interaction if all weak currents are charged currents (more precisely : char ge exchanging currents). They could proceed via second order weak interaction, for instance : Neutral currents allow reactions (5) and (b) and add contri bution to (3) and (4). However, due to lepton charge conservation at the current - "W" vertex, reactions (1) and (Z) can recsivo no contribution from neutral currents. If the W exists, the processes (3) and (4) can occur in order aG : ue «--I—' wc ye %-^"^ w y - sT e c e e . d Diagonal processes nay play a role in astrophysics, through photoproduction i * c~ * i c*c~ ' v Z , and "plasmon" decay into a pair v • v . UPTONIC INTERACTIONS 2.2 Cross sections We want to calculate the cross sections of the processes : and • U = e, u) The expression of the cross section for a reaction 1 + 2 * (n - 2) fermions is : & P, ! TH| s" Cfi„ " P,„H,) F " 'J n-Z {2.)3E, F is the invariant flux factor : r» |/(p, • p2r - "\ < For a neutrino-electron reactioi F = k.p „,_] L „„ „* fm'£iL &Lt> (k.p-k'-p'j ,2,)' k.p e J k' p' S.C.I. Neutrino-electron reaction Let us consider first the neutrino-electron reaction : ^00 * e'Cp) - ^(k') * i"tP'J Assuming a V-A interaction, the catrix element Wl can be written (in the charge-exchange order) : Û (£') (1 • Y )U (p)ù (£*) ï (1 * Y )U tf) •m v Yji 5 e £ x 5 v t ^ Mnce polarization is not observed,the transition probability has to be averaged over initial states and summed over fi nal states. Calculation of L |7Tl| over spin states of the initial and final leptons is performed by using standard trace technique (see Appendix III).The result is : 2 c HWl'.i „ '- 16 (k.pl (k'.p1) Wc compute (k.p) and (k'.p') by writing : s • (k • p)' • (k- • p')" with : (using the metric such that p: = Eï - p = m ; note that; for convenience, the opposite metric is used in Appendix 111) Thus : S-m? k.p-- k'.p- k.p and k'.p' , being independent of the final state variables, can be taken out fron the integral : , 1 Si _L_ 16 LLÈ fel Éel .- (k.,p - k' - p') - 2 s"2 ^ J K "o The integral is just the tho-body invariant phase space factor, equal to 2i - s and (2*)4 A factor - cones from the average over initial states. Conversely, it can he said that, because of the 7-A structure of the we-ik interaction, only the negative helicity state of the target electron contributes to ',Tfi] . An unpolarizetl target electron has a prohr-hility - to be left. do _ C' (s - "fr da (2ii)2 s C' ' <»„ •«"*»„• »"> ' <2>)2 ci (s - cjy In the center of nass system, the charged leptun angular distribution is isotropic. -.2.2. Ar.iLneutrirji-clectron reaction The antincutrino-elcctron reaction : k + p " k' • p' results fron crossing v and v in the neutrino-electron reaction : (- k') • p - (- k) . p' LEPIONIC INTERACTIONS Therefore, we may use the formulae obtained for v + e~ •+ v + X by changing k into - k* and k' into - k . We get : ni2 !i_ zW m— (k'.p)(k.P') ra„ m m2 It is shown in Appendix III, that : 16(k'.pMk.p'J - -~ (s - n£)[s - nj) [(s - n£)i "ej [(s ' ml'>a's 6 * s * "tj where e is the angle, in the center of mass system, of the final neutrino momentum with respect to the incoming neutrino. He can write (rememberinI g that F = k.p = s - mi I\ : f(s, cose) — i^ a* (k • p - k' - p') K Po d3k' =• k'? d |k'| dfl , and we do not integrate over dn . f(s, cosO) is independent of the other variables of the final stdte and can be taken out from the integral. We are left with the phase space factor 1 - . So we get ; do G* (S _ mO 1 ., ,. — = — f{5, cosd) dfl (2*)2 s3 2 As above, we have another factor y if the target electron is not polarized : n (s 1138 . O . _£L i!_J*_ f(s. „>)cosB , ! L e e] [ " "I)" * s * mij (2.)= 4s to be compared with : C2 (s-mj)' = („,•.--...«-)• <2.)J The angular distribution is not isotropic in the case of antineutrino-electron reaction : dn c (2')' s dn 5 G' < - "V '"oV' . foe "tV Therefore, the reaction vL • c" -* vc + e" is pure S-wave, and the charged lepton is emitted isotropically. The reac tion Zc * e" * v( + i" is predominantly P-wave, with a small admixture of S-wavc in the ratio ^2—Ï , and the charged lepton is predominantly emitted backward in the center of mass system. For s very large compared to m* , the differential cross sections take very simple forms : i ) - —— i (1 * cose)2 dn <2")2 4 do ,,--.(—J a 2 The fact that -t- (v • e - v * t ) = 0 for 8 i (s >=• m ) can be readily understood from helicity conser vation. There is no restriction from helicity conservation in the case of neutrino-electron reaction, as can be seen be low (the electron has purely negative helicity in t" >f zero mass, i.e. when ra? « s) : -i>= initial state (no contribution from angular momentum) In the case of antincutrino-electron reaction, one has : initial state This diffcrei.' ial cross section may be expressed as function of q2 , where q *• k - p' : q-' » (k • P I2 " "i| - -tup" 5 = n2 _ - "* f(S - njjeose * s • ra*l )t s large conrpared to ra; : q* = - — (1 * cosG) and i- - i dq- is strictly independent of q* for a local interaction between two point-like particl'.-s (in electromagnetic inter action, Coulomb potential, or, conversely, the exchange of a photon, would give ^.," — ). This shows that, for dq- q' veTy high energy and suificicntly high monentuzi transfer, the weak interaction may compete with electrcnagnetism. LEPTONIC INTERACTIONS Therefore the amplitude has to be proportional to 1 + cose in order to conserve helicity. 2. 2. 2. Integrated avons sections By integrating over the angles, one gets ; mf) + (s + mp{s + n~) n S 45 [3 For s very large compared to m? : Ir. our calculations, we have assumed a local (îero range) weak interaction between tvo point-like (structureless) parti cles. This has a remarkable consequence : o and o increase linearly with s , i.e. with 0 since s ï 2m Ey . It will be shuwn later (chapter VII) that this property leads to difficulty with unitarity. Due to the very small mass of the target electron, the effective value of s attained in the laboratory is never large, and the elastic cross sections a and -~ are very small. Let us write •' where M is the proton mass. We have seen (1.3) that G % NT5 M*2 , so : 2 G MP M2 M a squared length, so n is a length (in the c " ÎÏ • I systra), that is rrr . The cor^letc expression is : G2M, - \H:J ffc* G-"M , It results that t.hen Kv is expressed in GcV, hence : o _ -- lO"1*' err for [• -- 1 GeV 2.5 General features of lepton scattering (n section J.J.I., we calculated the cross section for : , we obtained the cross section for : using crossing sycmctry. The matrix element of the first reaction is : (3J «e P 5 e I p s e In Appendix III, we show how the matrix element for (2) may b.ï written using the substitution u + v when parti cle * antiparticle. However, instead of negative energy spinors, fce could use positive energy fiirac spinors for the an- tiparticles (which are related to the fanner by charge conjugation). Since the antiparticles have positive helicity, their coupling is of the V + A fora : uf ,u(l - Y5)U= The h'ier: reordering thcorec allows us tD write reaction (2) as the coupling of a V + A (v - v.) current with a V - A (e - O current, that is : u£ TU(1 - TJuJ x û, Y (1 * vju W vt u s ve l " 5 e If the differential cross section for (1) is ; p p J •ktPtfi ' e ' z ' Pvm and nasses c-'.i be neglected, the cross section for (2) is obtained from : *(p- . pc. p. . P£> by changing the sign of the parity-violating V, A interference term (i.e. the i term.see Appendix III). In the limit of point-like fermons, local interaction and high rmenta, any particle-particle or antiparticle- particle scattering cross section can be obtained fron (3) and (4) by substituting the appropriate momenta. This is par ticularly useful when crossing cannot be applied, such as in the comparison : v, • p - l . n The salient features which follow from the assumptions of point-like particles and a local V-A interaction are : I) - lint-ir increase of cross sections with energy ; " an isotropic distribution for particle-particle scattering and a (I * cosO) behavior for particle-antiparticle scattering; J) a ratio ! for "("tiparticlc-particlcf 3 o(particle-p3rtii:le) These features will prove to be relevant in the application of parton node!s to deep inelastic neutrino scattering Uhapter VIII). I LEPTONIC INTERACTIONS - 47 3. TRI-LETTON PRODUCTION BY NEUTRINOS This process consists of a virtual transition : followed by the interaction of a charged lepton with the electromagnetic field of a nucleus. The following reactions can be expected : for instance : but v *Z«Z*u » e *e could be due only to neutral currents. Crc-is sections are proportional to n2G2 (a » 1/137), and increase approximately as E . For high energy neutri nos, the dominant process is a coherent interaction with the nucleus of charge 2, and the cross section goes as Z2 [Cry: et al {1964)]"*. An order of magnitude is I0-**1 cmz for Fe and Ë ^ 10 CeV. It should be possible to observe these processes with the new accelerators of several hundred GoV. NOTATIONS AND CONVENTIONS i. rouR-Vhcrois AND ftnucs X, , X, . X3 , X4 - it (c = \) lour-vectors : A = (A„ , Â") A - (A, A, = i At A.B = A D = g AB A.fl =6 A B = A = A . B - A„ t-V * Suv Av) Encrgy-nancntun tour-vector : P - (E, ft p - (P. ili) making kincnatical calculations easier. IHHAC HjUVTIOM A\D KVTRIŒS In a non covariant fashion, the Dirac equation is : [E - o.p -d a\i, - D (a for =) '•C3 ••(::) LEPIONIC INTERACTIONS In a covariant way : in X-space : (iv X-— - Jml * lit CXi = 0 (ï.p - m)u(p) =• 0 ti t.p • m)u(pj = 0 p.otdtion : jf * T.p) - The Hirac matrices obey the anticommutation relations : V-j + 1Ju = Zg^ I YuYv • YyYu ° 2Spv he have chosen the following explicit representations for the f-matrices (other representations may be used}: (with w ric ©, one has to distinguish Y and TU) T'S are heinitian matrices hemitian antihemitian 1 •lefitted in such a way that the projector onto négative h-licity states is always : ': 01';' \l)oint spinor -Vljoir.t Hir.i- cqiiiCtœi : ùf> - n] - 0 u(i -.p • n) • 0 3. PROJECTION OPERATORS - NORWLI2ATIOM OF STATES - The Dirac equation has positive and negative energy solutions : - A U+ (P) - « I _ , \E+m vhere a and b are two-conponent Pauli spinors. Particle states : u(p) = u+[p) Antiparticle states : v(p] = u_(p) (E > 0) From Feynman rules for spin j particles : - an incoming particle is described by u(p) ; - an outgoing particle is described by u(p) ; - an incoming antiparticle is described by v(p) = û-(-p) ; - an outgoing antiparticle is described by v(p) a u_(-p). - The projection operators onto the particle and antiparticle states are respectively : A (pi • . r UÛ *Ky' spin states *-tt»> " " spin states ™ Normal uat ion of states : {>-f\F) Another noma li;at ion can be used : uu = 2-0 (K = '|Ef • ci) - Lxpllcit forms of projection operators are given for both retries and both normalization conventions Normalization of states Projection operators Feynman factor Space phase © © «.M • °- iy.l ». (p) 'jp'* a {• ::: ] LEPTONIC INTERACTIONS - 51 - APPENDIX II THE FIERZ -MICHEL TRANS TO RMYT ION There are two manners of writing thi most general matrix element for muon decay : - the charge exchange order : (v u) (ë v ) 12 3 4 - the charge retention order : [ë y) (v ^ ) 5 2 î 4 One can write : ]G.t Oj h)l} OjCCj * Cj Y5)*, = [G3 0t *2j», O-ttj * C! Y^ i, j • S, V, T, A, P hhere the C- and C- coupling constants are of course not independent. The reordering will not affect the parity of the natrix element. Therefore : c i itfrj 0, (,^(5, Oi *J - j Cjrt, Oj *2)Cvj Oj ij or, core explicitely : 16 16 _ c ( r rH r *3 r *2)(», rr »„) - r.JT ^,(5, rr, *a) (*3 rr, *,] where the r are the sixteen basic 4x4 matrices 1, Y , a , i^ y and y, . Since the components it- are independent, we obtain an equality betveen the coefficients of each product of compo nents [*i)a {i,2)B (J,)T {*„)fl Multiplying both sides by (r^J» (which is a number), one gets : Smriing over indices a and Y and using the gcnorsl property of the sixteen Dirac matrices : .5, (V,« cr "Va* • J I- K- "V- 'r rr'>„« <» [no riicxutlcn over r) relation *hich c;in he condensed by returning to the indices i, J : C- 0. = - z- C.(0. 0. 00 ii 4 J 3 J i y (no summation over i). Let us remark that we can gc one step further if we multiply [1) by (r ) and sum over a and 6 (no simulation over rj Th's shows that the coupling coj. *ants C and C are related by linear transformation : The A., can be computed by using equation (Z). The result is : A double Fitrz transformation is an identity, so A* - 1, which can be checked on the ejplicit form of ihe A-aatrix. [f •'. is the transformation matrix of the coupling constants C. , the matrix elements transform under A . Let TTÎ be a general ratrix elcraent : ™- \h A • j cjmj ' 5 5 *ji ai ™j so, identifving the factors of C. , one gets : TO. A.. TÏÎ. j "j (m) . AT (TO) , and, ! M- »*(*} Defining <,!, - t *u , one sees iizsediately that th coupling constants C. and C! transform in the same way. The relation between C and C is true for wave functions. However, if the (.'s are field operators, there is a global mi nus sign coning from the permutation of Femi fields. It is easy to check that, if the aatrix clcoent CALCULATION OF v.-e AVD Z.-e CJOSS SECTIONS For convenience, we adopt here a metric different from the one used in the text. A four vector i so p2 = p2 - E3 = - m2 (p2 = m2 in the text, making kinematical calculations easier). This metric, which has the inconvenience of introducing an imaginary fourth component, removes the distinction between upper and lower indices. The anticomnutation relation obeyed by Dirac matrices is then : y are hermitian matrices, v = 1 , y ° Y y y, y , y = 1 . The adjoint spinor ïs defined as ù = u* y . In order to describe : », * e" - »e • .- k + p « k' • p' -•e have to înt'. iduce t!ie following Dirac spinor s : u (k) fci the i.looming neutrino « i- (pj for the incoming electron e~ \ (P') for the outgoing particle l~ u (k'J for the outgoing neutrino v Forgetting about the factor — , the matrix element 1ft is written (in the charge-exchange order for instance) as : /T ™ - ûv(k')ï,n • v5)ue(P) ^(P')y,(i * T5JUVW e- * ve «, - t" Let BA • uj(p')yv(l * Y îu^fk) . Then : *W- uv(k')Y.B{t * y5)ue(p) and "W - ûe(p)(1 - Y5)Y„ Y-B* yk uv(k') Let us wrire : fl - (g, B4) , B* » (I*, B4*) Y.B* =• y.fi" • Y4 B4* B n y,. >- " Ï, - Y.B** y,, B4* - - Y.B' with 0' - (5*, - B4*) With this new no'jtion : 1Y1* » - û (p) (1 - yJy.B'u (k') 2 , |7rt| = The way to handle this expression is based on the two following properties : 1°) Let there be a complete orthanormal set of states |i> : |i> 2 m Therefore : spin states "" = all states A*&,)uG = h+® of positive energy For a neutrino, rt..(p) reduces to —i^E „ one can sun over the neutrino spin states, since a (1 • >,) factor 2m s u 1 + Ï5 insures that the neutrino is left-handed (we recall that is a projection operator onto the negative helicity state]. 2 Thus we get : „ states^' * " "\W« " ^^^f") '•B° * V«W Za) Let A be a linear operator, represented in the basis of states fi> by a matrix A.. ^ * ifA|j > : I < i|A|i > - I Aa - Tr A We can write [ û A u " Tr A if we sum over all spin and energy states of the electron. In order to do so, we intro duce the projection operator —"v*"1^ onto the positive energy states. Thus, we have shifted our problem to one of confuting the trace of a product of Dirac matrices : • -^- Tr «f(I , - YJY.B' Y.k* Y.B (m- iy.pH 2m m^ *. If instead wu use the normalization : E£0 uû = m - iy.p the wave function normalisation is the same as for bosons : 1 "i — instead of — 2EA Et and die « mass never appears in the calculation (sec Appendix I). LEPTONIC INTERACTIONS B since Y anticomnutes with Y. (>• 1, 2, 3, 4) and (1 - Y5) = 2(1 - Y5). Let us recall now the elementary rules of trace computing : a] The trace of a product of matrices is invariant by circular permutation of these matrices. In particular : Vr AB = Tr BA . This implies Tr AB = 0 , if A and B anticommute. Another consequence is : Tr A BA = Tr B . bj The Y are defined as traceless matrices, and Try, 3 0 from its definition : but (fron a) : Tr Y i Y Y =• Tr Y Y. Y„ Y, c) Tr Y Y = 44 . Therefore . Tr T.a v.b = 4a.b, where a,b = 4-vectors d) The trace of the product of an odd nittiber of Y matrices is zero. e) Y Y Y Y = : Y, if ail ii.lices are different. The trace can be non zero only if indices are equal two by Tr Y Y Y Y B 4 (a 6 -6 & +5 6 \ Tr(Y.aHY.b)(Y.c){Y.d) - 4 [(a.b)(c.d) - (a.c)(b.d) + (a.d)(b.c)] i Tr Y^YUYU • 0 Tr YSY ÏVTO = 0 g] Let c v be the corpletely antisyimetric tensor : el234 • * ' * Lx vo " f" ^ where n is the number of index pemutations, We can write •' TrY5(Y.a)(Y.b)(Y.c)(Y.d) » 4 c^ a^ bo cy dn (cj a^ b cu d is the determinant, or "mixed product", of the 4-vcctors a, b, c, d, generalization of 'ijk"ibjck' (îx6j.îj . Applying these rules, one gets : I FT --2- ((B'.k')(B.p) - (B'.B)Ck'.p) • (B'.p)(k'.B) - t B; k' B p } B has been defined as : B, - ûjtp'jYjCl • r5KW Î* • ii>)^l * v5)J Y„ u,(P') - - i^dOO - Y;)Y" U;(p') 3; - (5*. - B*),. - û fk)C1 - Y5)Y( ut(p') Let us consider the first term in J | : (S'.k')fB-p) • - Ui,(k)(l - ,5),.k' ^(p'jCjtp'jY.pd • Yju^fk) m - iY.p' he apply once nore the sane procedure : introducing the projector 5- in front of u,(p') and of u (k), and simrc.n£ over alt states of I and v yieljs the trace : — Tr[- (1 - Y,)Y.k'(m - lY.p'Jr.pO • Y,)r.k] 4m't, <\ s s Taking into account the other terms give r. •m|' . -^-ij Tr |(1 - Y5) [- r.k'fm, - iY.p')Y.p * -.JCOJ " iY.p'^CK'.p) • Y-P(n( - iY.p'JY.k' • tluv|> Yx(m; - iY.p')Yv \ pp](l • »s)>.kj After some calculations, one gets facto- Si ). lie said ['-J-JJ that r|fn| can be deduced from the above result hy changing k into - k' and k' into - k . We get : r.|m|' , (k'.pKk.p-) ninij a. This result can be found by a diiict calculation quite similar to the one performed above. Lei us mention onh some LEPÏONIC INTERACTIONS di f îerences. In order to describe the reaction : • e - v, • £ k + p = k' * p' we have to introduce : vfkj = û_(-k} for the incoming antineutrino v (u_ denotes a negative energy state), u.Cp) for the incoming electron, v(k'J = u_(-k'} for the outgoing antineutrino v , u, (p'J for the outgoing particle l and "W can he written as : ™ = v(k)Y,(1 * r^CpJÙ^p'jY^I * >5)v(fc') H is now defined as : Bx = ùjtp'h^I * T5)v{k') jîTÎI " is found to he : Sew uc have : *„- iy. t-M , u_[-k)u_(-k) - iT.k spin states 2% Im^ (the minus sign comes from normalization of the states : ù(p)u(pJ IPj Introducing ^^ in front of u (pj and calculating the trace gives : l \m\' - -J- /(fl'.k)(B.p) - (B'.BXk.p) • (B\p)(k.B) - e. B.' k B p } ai - '*-P' In the nr*t step, we replace ut(p')ut(p*) by • but the projection operator applied to u_(-k") «u • i».f-k'J iv.fc" Then the t.iloil.icion goes unchanged. \ow, for kmwnatical cal rions, we ccce hack to the more convenient metric such that p; = V.: - $• = * m: Multiplying p' - Ii * p - • liy k , we set (k: • 0} ; k.p' ' k.p - k.k' In the same way : k'.p = k'.p' - k.k' From s = (k + p] = (k1 + p')~, we have : c * p = k' » p' Remembering that, in the center of mass of a two-body reaction 1 + 2 -* 3 • ( 2 / Ï)2 and the sazie for (1,2) - (3,4], with E* * s, one has : , , , , , s • mz [s - ra2l (s - m») k.p' - k.p - k.k' - -^ Si (l - OJ56) 2 4s ' I (s - m£)ros6 • 5 • mj , , , s - oj (s - a2) (s - »t) k'.p - k'.p' - k.k' -t- - - '— ii .1 - cose) Z 4s s " nj r T - - l(s - mz)coso * s * m2 4s l ' 2 (k.p'J(k'.p) '— (s - n'){s - np |(5 - rfjcose . s . m ]f(s - ajjcjse • s • m\\ . EFTONIC INTERACTIONS 1. Harshak, R.E., Riazuddin and C.P. Ryan, Theory of Weak Interactions in Particle Physics, Wiley-Interscience (1969). 2. Kallen, G-, Elementary Particle Physics, Addison-Wesley (1964). 3. Jarlskog, C., Nucl. Phys. 7S, 6S9 (1966). 4. Czyz, K., C.C. Sheppey and J.D. WalecKa, Nuovo Cimcnto 34, 404 (1964). REFERENCES TOR APPENDIX II Fieri, M., Zeitschrift fur Physik _104, SS3 (1937). Kdllen, G., Elementary Particle Physics, 377. Marshak, R.E., Riazuddin and C.P. Ryan, Theory of Weak Interactif ; in Particle Physics, 82. ). Calculate the maximum energy of the rauon decay electron (1.1). 2. Calculate the statistical energy spectrum of the mion decay electron. Investigate the influence of a neutrino mass on the shape of this spectrum. 3. Show, from helicity conservation arguments, that for completely polarised muons the electrons of maximum enorg are emitted in a direction opposite to that of muon polarization (1.5). 4. Calculate the threshold of the reaction v + e~ - v • u" (2.1). 5. Let v be the probability of transition per unit tice from an initial state to a given final state : 2 u w = r»* fm; , TT -i , î i. 6 (pc. - p. ) K ' ' ' fermons P bosons 7P \ fin in/ fci ikj where Yfl is the Feyrean asiplitude and the factors — and — cone frcn normalization of the states (App. I). Show that the cress section a for reaction 1 + 2 - 3 * 4 * • may be written : n - 2 particles , . SislL IT b f ffm;' "IT, JÎ2L_ j./p . p. -, (b » - l'or basons, f - m. for fcmionsl where the so-called flux factor is F' ' fPl'P?)' " mi "h • n've tno expressions for F in the laboratory sys tem and in the center of mass system. 6. Show that, for the reaction u * c~ - v • l~ , trie two-body invariant phase space factor is erial to S - Dis In L (J.2.1). 7. Show that the first three terns of the trace in l I'D]* (App. III.l) reduce to : Tr|(l - ^[^(k'.r.) * ZHp'.yh-V * 2i(k'.p')Y.P] (1 * \h-'k) 8. Show Uiat calculation of thia trace yields : lb 1 [(fc'-p')(k.p) * (k'.k)(p' p)] 9. Calculate the contribution of the (, tcra in Z W , 10. Using the algebra of Dirac catrlccs, derive eJtpticitly the catrix A of the FIERZ-MlCHfl transformation (App.II). CHAPTER HI SEMI-LEPTONIC INTERACTIONS J. M. GAILLARD, M.K. GAILLARD and F. VAKNUCCI SEMI-LEPTONIC INTERACTIONS 1. EXTEHS10H OF THE FtSHl INTERACTION In the last chapter we extensively studiea the four-point current-current coupling of leptons . j<2) • ïir^l • i^Vj , 1 - u,e .hich in particular is res|*cnsiiilc for men decay. In Oiapter I we saw how a long series of experiments established that the effective interaction .^sponsible for B decay can be written in the very sij^ilar fom v, . C ,u , suggesting a principle ot universality [Puppi (1918), Klein (1946), Tiens» (1949), Le« (1949)]. However, wc know that strong couplings allow a pion, for example, to disassociate into 3 virtual nucléon- antinucleon pair, so the coupling (1.21 nnst also induce the process -" - c'v (Fig- 1): e\ Fig. ) Earapie of ccchaoisa inducing n* •* c*v„ is net I i3 ->: - •>" * C'M, arul processes involving strange particles: _ r - rxrv t =" - =°c 0e . e f * t e"vtf . It* - KVv . Ccr.-cncty, .i direct coupling of pions or of strange particles to the leptan current would induce 6 decay through itrcny c**plln;»i to ch** -iclccrs. In contrast to Icptonic transitions, where corrections to the Born approximation iri: electromagnetic ' -t order in the weak interaction, the prinary coupling of the haJronit current is expected to bf .ira.-seica!Ly mt ay strong interaction effects; for this reason we have referred to the interaction of !*|. li-il i'» "uffvctlKe". (gncrirs;, for the present, the possible radiation of weak interactions by vector bosons, we write the icmtltpecrUc current -current interaction as «SL-^jJ"^-^. . (1.3) whtTc J , ïhe hajrente c ur r en !. is a Iteiscnbcrg field operator with respect to the strong interactions. We know thjt Li mu.it contain .» vector part (Feml transitions) and an axial part (Ganow-Tcl 1er transitions): r>.e Interaction. (1.3) lnh:CC3 all the? processes lintnl aîwvc as well as u capture in nuclei and - » uv . In partlcuLjr, frtxi the dtscimion in Quptcr I, wc bm that the matrix clcienï of the hadronic current between nnçlcon. socc» ti of the fam • ) Theau£hoiit this chapter, wo ecrploy metric 1 of Ouptcr II, ilpfwnltx I. Cur spinors are normalized so that Z utï(E "0) • p • n and the phAïc-spJte factor is ii'Ti/ZHZ")' for femions and bosons. CHAPTER III (p|J 'jn) = ù_Yu(l + 1.2 Y.lu • velocity-dependent terms . (1.5) I.1 The vector nature of the interaction To what extent has it been established that processes other thar. 6 decay are of a vector and axial vector nature? There ;ire not yet sufficient data on the Z-A transition ro provide a conclusive study as in 6 decay; other processes involving strange particles have not yet been observed. Let us see what has been learned fro» pion decay. 1. !. I r„_ decay Th.' pion is a pseudoscalar meson and thus necessarily decays into a J =• 0 final state, ijnplying that the two lep- tojis nust have the SXK helicity- he know that for the V-A coupling occurring in g decay, the leptons are ettitted with opposite hclicities, so the ti, decay is forbidden in V-A theory. In fact, helicity is a conserved quantum nunber only in the liiit of sasslcss particles. The wave function (1 * Yj)u£(p) will always contain a component with positive helicity, with relative strength proportional to the lepton mass. There fore, if V-A theory is valid, and if the pion is coupled with equal strength to the nuon and the electron currents, the transition matrix elements will be in the ratio Mr n» Ihis strong suppression of the electron matrix element is partially compensated by the available phase space. For the purpose of illustration we shall outline the calculation of the decay rates . From Lorentz cov3riance, Che matrix element of the hadronic current between a single pion and the vacuum must be of the form <|Jh(x)h-[q)) =• iq^Mx.q) + x^(>.,q} where f_(x.q) and fL,(x,q) are Lorcnt; scalars. Under the translation operation we have the property c'E^M c-'f"" - jj(x • a) and for any state n of nonentin p e'f-a ntp» • c'l» ,n(p)) Then f_(».i|) and R^x.q) mst satisfy f„(*.q] - eiqa f,(x • a,q) , g„(x,q) = 0 . for any a; takinti a • -x: f,(x,q) - e""* f„(Ofq) . As the only tarent: scalar which can be formed frcn q is qJ • a^, is a constant which characterizes The strength of the pion coupling: to the axial current: i(,x (U^xJ'-r-tq)» - ifT c" qu . 11.M Then the transition r.acrU element for • - Sv is of the fom ,, i, C-.)*4-(q - pE - Pv)M-N • ^Jd x(£Oe|jJ(x)J (x)|!-) • J For fomilac ;ind derivations in inorc duplicated cases [-i.fi. three-bydy decays) the rtaJer is referred for cxsmlc to Hirshak (l'Jb9). SENI-LEPTONIC INTERACTIONS Using the Dirac equation (p J liEue(p£) - V* l and the assauption of massless neutrinos, we obtain n 1 MrT£i iVW + YS)VU - Sjiraning over spin states gives the result spins . 4G'fJnJ(p;-pv) 2 P,'PV • (P, * Pv)' - p't - Pi • »J " "J In the pion rest frane E ?i • "Pv = P . P • %' ", ~ ( E E E Pt"Pv " f v ' P* " P' t * P) * P", P " K * nlJ/2nn The phase-space integral in the pion rest frarae is 1 f.i.,.iv, „ _ , d'M'Pv _ p _ (1 - ng/mj) ME^Ey 8-ni^ 16rnv Coritinini! (1.8) and (1.91 we obtain for the ratio of decay rates: r(nl>' ' Bn, . r(7 « (1 - nj/rrj,' c;1 D 1.232 * 10-' After inclusion of radiative corrections [Kinoshila (1959)], the ratio (I.10) is noJii'ieJ to R - 1.284 « 10-* . rfcc TOSÏ recent (and nost precise) r»asurerxnt gives the value fDi Capua (1964)]: R - (1.247 . 0.023) • 10'* . Oiis is J atrikinj; confirmation of the conhlned assicpticis of a vector (V,A) interaction a-I of u-e universality. If we h.kl inste.id assur-ed J universal scalar (S,P) coupling, the natrix clencnts would be sirply » • -^ f'u.U • &>,lv . £ - -,c .ni :he ratio of the rjtrl* clcrxnts would be unity. Then the electron dcc.iy node would be the r»re copious as it is fjveured by phone spice. A tensor coupling („• for the leptons) is r.ot possible here, as only cue four-vector (n. ) is .ivjil.iMe for censtmetir.,: the nlcn wane function, jrnl *e CJIUSJI construct .in ar.tisymctric tensor. - 66 - CHAPTER III The arguments leading to the ratio (1.10) are in fact independent of the neutrino helicity. If the lepton matrix elements were of the form b tf l = v e <*v!jy !) = ûj(a + V va > > 2 2 where vu is a four-component spinor, the only change in th.. calculation would be a multiplicative factor (a + b )/2 in Fq. (1.3). The ratio (1 '0) is unchanged as lor.g as the u aial e couplings are identical. To test the hypothesis that the leptoii J current involved here is the saw as in $ decay we rust determine the neutrino helicily or, equivalently, the charged lepton helicity, since conservation of angular momentum requires then; to be the same. If the neutrino is a two-tomponent massless fermion, V-A theory requires a helicity -1 for the v (n+ decay) and *1 for the v (n- decay). The helicity of the charged lepton is correspondingly ±1 for I' (which is op posite to the electron helicities in fl decay). Existing measureraeMs of the muon polarization from n decay confirm this prediction within experimental accuracy [Alikanov (1960), Backe.istoss (1961), Bardon (1961)]. 1.1.2 TT - *• * a\ivg) The branching ratio for the n decay is "ery small T7(Mn •(1-02 s °-l'7) "10"" and detailed studies of the decay mechanism have not yet been feasible. However, if we assume that the decay is given by the universal current-current interaction of Eq. (1.3), we nay test some consequences of this assumption. Consider first the hadrcnic matrix element. Since there are two independent four-vectors, p_, p , Lorcntz in variance allows TWO independent tenus in the matrix elenent: 0 1<ÏJt <« (pa)U5wt"-(p_)) - e" ov»(p.qî * quf-CP,q)i , a.ID where H Pu (P- * Po)u % ~ (P- " P»V ' The "form factors" f. are Lorenti invariant and therefore depend only on the Lorentz scalars qJ , qP , V1 . However, only one of these variables is inuependeat: q'P • nj- - n^j . P? * u* -Jri^ . 1 he take q as the Independent variable, rtoccntua conservation in the decay process requires q = pp » pv, so wher. we contract (l-ll) with the lepton r .rix clement, the second tenn in 'i.ll) gives The secend tern RIVCS a contribution proportional to the electron cass which can he neglected relative to tl>e total energy release nn_ - o.a - 4.6 HcV . The qJ dependence of f, ij dotemincd by strong interacMons and is expected to be characterized by a typical hadron rrasa (say the p meson). That Is we expect Mq1) ' f,(0)fl • ^ • ... 1 For i decay, q' Is in the range SEMI-LEPTONIC INTERACTIONS - 67 - so this dependence is also negligible. Therefore, aside from an over-all factor, the decay distributions are completely determined by the V-A hypothesis. In particular, the electron spectrum can be calculated and the experimental data [Dunaitsev (1965)] are compatible with the predicted spectrum. A uiich more striking confirmation of the theory comes from the prediction of the total decay rate which makes use of the additional hypothesis of a conserved vector current, to be discussed in Section 2. Since the picn is a pseudoscalar, only the axial vector part of the hadronic current can contribute to TT^ decay: s>- and only the vector part can contribute to n : So both modes mist be studied for a complete test of the theory. The experimental data on n decay supports the evidenc. from neutron e decay in favour of a universal weak current-current coupling. Î.2 Strangeness-changing decays In the introduction to this section we listed reactions which are expected to occur if 6 decay occurs, due to the interplay of strong interactions. However, there is another class of observed decays which cannot be induced from the coupling of Eq. (1-2), namely those which change strangeness by one unit (AS =• il): E,A - Nlv E - Mv K * Iv + nn , n = 0, 1, 2 . he may ask whether these decays conforo. to the universality principle which evolved from the study of seni-leptonic de cays with US =• 0. I.Z.I Thé V-A otntcture of the interaaticn There is not yet sufficient data on hyperon decays to detemine their space-time structure. However, the study of K decay supports the V-A hypothesis; the arguments are identical to those used in discussing n decay. For K. decay, V-A theory predicts r(K R a »> B îâ (I - ce/mfo* Ituvcver, since n| « ni , E = e, u there is no phase-space enhancement for the electronic node and the branching ratio is corrt»pondingly much more diffi cult ro erasure. The predicted branching ratio (up to radiative corrections) R •» 2.S9 « 10-* is confimed within experimental accuracy; the most precise measurement to date gives fciark (1972)J R - (2.4Z • 0.42) - HP1 , whcr>; radiative c rrecelons have been accounted for. The amplitude for K- ure, as in r decay, determined by Lorcntz covarinncc and translation invariance in terms of two form factors; for a vector coupling we have a(Kt)) - C(q)!jJ(0)|K(k))G]l>ij(l • Ys)vy («(q)jjJ(«)|K(k)> - fjt)(k . q)v * f.ft)(k - q)u , t - (k - q)' . (1,12) CHAPTER III Numbtr of Evenlf /IOM«V»IOM«V P, (Me*) Te (M«VJ Fig. 2 Tests of the lector nature of che Kj, catr: element: a) measured electron specttun ^Eshtruch (1968)] and predict it ns for V, S and T couplings; b> electron spectra for fixed pion energies — prediction jpling and experimental data £Ch: {1971)3- The ana'ysis for the manic mode is canpl icated ; since the rauon mass is not negligible, both form factors confibutc. Only one form factor (ft) contributes significantly in the electron rode. However, the t dependence can no longer be neglected in contrast to the c i5e for r decay. The range of variation 2 2 n£ S t i (MK - m,,) = (360 MeV) (1.13) allows t values which are not negligible on a hadronic nass scale. However, we can still test the vector coupling hypo thesis by studying the matrix elenent at fixed t; in the kaon rest frasie t»tk-qj*-*^*mî- ZM^ . (1.1-1) The vector ccupHnjj hypothesis completely determines the electron spectrum for fixed pion energy in the K rest fra-oe. Experimental studies of this spectrum give a striking confirmation {see Fig. 2) of the V-A hypothesis [Chien (19""1>, Eiscle (1971), Eshtruth (1%8)]. l,C.2 7nc ocuplimj strength While the space-time structure of K decays confir s the picture of a universal V-A interaction, the strength of strangeness-changing semi-leptonic couplings does not. Amplitudes for hyperon decay and K decay to charged lepton pairs arc consistently lower -- by a factor of about 1/5 ~ than strangeness-^ iservinjj decays. Then the primary weak inter action responsible for all the decays discussed so far appears to be of the form *W = * J , J (J J )3 h.c. | U (u) y *3 u W ••• -y spoiling the sircple picture of a universal coupling strength. We shall sec in our discussion of the Cabibbo theory that with a suitable definition of the hadronic current, the universality principle remains intact and many processes may be accosted for in a unified way. First, we shall study in more detail the structure of the strangeness-conserving current. SEHI-LEPTONIC INTERACTIONS 2. THE STRUCTURE OF THE STftANGENESS-OKiSEfiVING CURRENT 2.1 The vector current: CVC hypothesis The postulate of universality was based on th^ similarity in structure and in strength of the amplitudes for 6 de cay and u decay. While u decay can be calculated in the Born approximation, using the Hamiltonian density (1-1), the Hamiltonian density (1.2] is only a phenomenological representation of our knowledge of the 6 decay matrix element. Whatever we assume cor the primary weak coupling, the matrix element (1-5) corresponds, in perturbation theory, to the sum of all graphs involving weak interactions where the wavy line represents the momentum cairied by the current. Then, what is the content of the assumption of a universal coupling constant? The situation is very similar for the electromagnetic interaction. The matrix elenent for elastic electron-positron scattering can be calculated [up to Coulocb corrections) in the Bom approximation, where as e-p scattering involves the interplay of strong interactions. Yet the raatrix elements are equal in the limit of vanishing r In analogy with the conserved vector current which is the source of the electromagnetic field, the proposal was made [Gerstein (19So), Fcynnan (19S8j"l that the hadronic current of weak interactions is also a conserved vector current ;CVQ 3^00 = 0 (2.1) implying that there is a conserved quantum nucber ;.ssociated with the "charge" operator F(x0) = /d'x V0(x) , (2.2) which f=ust be tune independent - F(xa) - Jd'x i^- - Jd'x PVuM - 0 , (2.3) since the integral over ?*V(x) vanishes identically. The charges (2.12) are therefore invariant operators with respect to strong interactions. As.dc from charge, baryon number and strangeness, the only known operator, 'hich leave the strong interactions tnvit'iant jre the generators of isotopic spin rotations The CVC hypothesis is the |>o5tuJate that the charged vector currents coupled to Icptons are the conserved isospin cur- runts; thus wc make the identification V - V1 • iV: = V* t • U, U (^J VT - V1 - iV3 : V . The associated "ch;ir>;rs" are Just the isospin raising and lowering operators I. - 1. t il (2.5 Ij • j d»x .0(x) , i - 1, 2. 3 . CHAPTER III The identification (2.4] by itself implies that semi-leptonic processes with AS = 0 must satisfy the selection rules AQ = AI3 = ll , |il| = 1 . (2.6) Mareover: ' a) the charge raising and lowering currents are members of the same isotopic spin triplet ein|* V*. e"iirIa - -V" ; (2.7) b) the neutral member of this triplet, V', is just the isovector part of the conserved electromagnetic current; if V" is the isoscalar part, we nay write (2-8) The postulate of universality is now the statement that the currents (2.4) are coupled to leptons with the same strength as the coupling for u decay. 0.1.1 Pion 8 decay The hadronic matrix eleaent for r was determined in Section 1 from Lorentz covariance and translation invariance. hhat further restrictions are imposed by the CVC hypothesis? a) Current conservation implies M B r 3 W»|\r(x)K> = -iqM Fran the definition of the form factors, Eq. (1.11), we obtain q*Pf+(q') * q'Mq*) - o - However, q*P = »*• - n>!0 » 0 , ignoring the electromagnetic mass difference (electromagnetic effects violate current conservation, since charge cocrutes only with Ti). So we must have Mq1) - 0 . (2.9) This prediction is of course academic, since as discussed before f_ does not contribute significantly to the pica decay. b) The important prediction is in the norcaliiation of f+(q-'J. The matrix elecwnt of the charge associated with V„(x) is now s iqx I d'x (n |Vft(x)|n-> - /d'x c' P0f«.(q>) - (2n)>6(i5)Pof+(0) , (2.10) since for q - 0, qo » t. - Ett • 0. The isospin raising operator has the property* nB n , a 1 ( (pfl)|IJ "tp_)) - ^ ^2"2Eii{2n) 6(pD - pj , fz.ii) The CVC hypothesis SJVS thac we siould identify Eqs. (2.10) and (2.11) (where P » E j • E . • IE ) , giving MO) - n . (2.12) The pion decay rnacr-x clement is now completely dctcmincd up to q* effects which are negligible. Uv obtain a prediction for tht .»tc [including radiative corrections, Terent'ev (IS63), Qiang (1963)3: r„ - (0.393 : 0.002) sec-1 , which has a striking experimental confirmation. The nost precise measurement gives [Depommier (1968)]: r„ - [0.38 *'*"! sec-1 . •) We use the covariant nomaliiation of states: 2.1.2 The matrix element for neutron S decay Once again we use «rentz invariance and translation invariance to write the general form of the matrix element - .-*•» ojy,[,•) - i^Û owvqv » t,lq>K),„ (2.13) with q = p - p . Covariants involving p + p reduce to these three upon application of the Dirac equation. Vector current conservation implies -iq^plV'Wln) - 0 . '2.14) The first term in the expansion of Eq. (2.14) in terms of the form factors f:(qJ) is proportional to V1"" " Qp(°n " V» " ° ' neglecting the electromagnetic mass difference. The second tern vanishes identically because of the antisymm«-:ry of a . So we are left with the condition q'f.Cq1) ' or fjtf) • 0 . The matrix element at zero momentum transfer v f is again determined by the matrix element of the charge (pljd'x V((x)|n> = {p|I+|n> - 2£p(2TT)'6(q) , so that fi(D) " 1 • U - The observed similarity of vector coupling constants in u decay and in 6 decay becomes very significant the context of the CVC hypothesis. How similar in fact are they? We may compare the values of the Fermi constant G as determined from u decay and from Fermi transitions in 6 decay [Wilkinson (1972)3 G = (1.4335 t 0.00031 erg cm1 G_ - (1.4132 * 0.0016) erg cm1 , where Coulomb corrections have been included. There is some uncertainty in other radiâti/e corrections, i iot enougii to account for the small discrepancy between these two numbers. This slight failure of universality as i< Li ted here 15 just sufficient to allow the incorporation of strangeness-changing decays in the theory, ,i.. will be de- i-J in Section 3. Thus far we have determined the matrix element at vanishing momentum transfer; the CI-IKTJI matrix i is de- tcrmincd by the knowledge of the nucléon electromagnetic foiti factors. The isoscolar part oi the electromagr^t ~ cur rent is the same for the proton and neutron; then the isovector pan is given by Using the properties f-.B), we obtain 1 (Piv' -: • (Pl[i..jf]ln) • (rU™IP> - c "• n> . So th-: fom factors of Eq. (2.13) are related to the electromagnetic forn factot .'"p bv fj(q') - fjV) " *J(qS) • [2.17} In particular we obtain for the "Weak carnet Isa" contribution CHAPTER III f,tO) =Up-%, (2.18) where v is the anor-ajus magnetic moment. The predictions (2.17) and '" IB) can be tested in neutrino experiments where the matrix element can be probeo at high q2. Existing data are compat^ole with the theory, but there is not yet suf ficient evidence for a exclusive test, 2,1.3 The triad "fl-"c-iaff It was pointed out by Gel1-Mann (19S8) that the decay of the 1* isospin triplet (1!B,l!C,l2N) to the 0* isoscalar ground state of l:C provides a good place to test the weak magnetism correction. Only the isovector part of the electro magnetic current can contribute to the (1!C) •+ iaC transition. This transition is therefore related to the vector part of the 6 decays of 12B and 12C by isotopic spin (up to electromagnetic corrections) Since we are oonsideruig a iJ = 1 transition, the leading ("allowed") vector contribution (Fermi transition: ûv u •* il» does not contribute, so we are testing the magnetic coupling of the vector current. The fading transition is Gamow- Teller (Gy i u - (5)), so the lowest-order (first forbidden) contribution of weak magnetism will appear through the V,A interference terra. If we do not study spin correlations, we cannot observe a parity-odd correlation. Therefore, the significant term is the double interference term ^Wlccns " <*'•*> leptons . (2-20) In the decay of !,B an e~$ pair is emitted with matrix decent V„« * \>% • For l2S decay, an e'o pair is emitted with matrix rlencnt since felicity charges sign when we substitute aniiparticlc for particle [u - uc). Thus, the V,A interference term channel sign fcr the leptons, giving a sign change to the double interference *-rm (2.20). The effect of weak magnetism is that the allaw-xl spectrum is corrected by a factor of the fom C(P.) • 1 ; ? aE for 0* emission where |{^ lb the i^unow-k'llcr coupling strength and u is the transition trament for (,!C) decay in units of ihc proton Bohr magneton, Mien various corrections are taken into account [see hu (196-1) and references therein], one obtains tin- predictions £C(E) - Ojj. - :(0-^S : 0.09)(t/McV] • 10"' to he compared with the exj.trmental results [Lee (l'Jbj)] Triri il *f».S5 : D.10)|t/MeVl - 10"' for à' LLU.) - ij ,(JéJ, . o.P6>(E/HcV) - 10" for 6* . n.'.' decay of I \\ li dJrurtatcd by the n-p transitiez (as oppose 1 to pion exchange effects I; empirically the tranjutien -nur-eni of t»\. (;..'!) satisfies int-ni rrovitîcs a rou^h ccr.firrjticn of tq. (2.M). ibwevcr, as thi • -pertics l.'-l')) art- iikicpeiiJent .ktmir; the rc?4ilt is a norc r,cne..il confirmation of the assist ion oi I^s. (2.s;. SEMI-LEPTONIC INTERACTIONS 2.1.4 £" - Ae'v From the point of view of isotopic spin this decay is similar to the one just discussed. The space-time structure of the matrix element is identical to that for neutron decay, 5q. (2.IS). Since the masses are not the same, current conservation requires, instead of Eq. (2.15), 2 [Hj- - MA)f({q ) * q^tq') = 0 . (2.22) Then we have f.(0) = 0 , in conformity with the fact that -he charge operator cannot connect neefcers of different isospin mltiplets < A ÏI * | E ! > = U - As the form factor fs will contribute a tera proportional to the electron cass, and as the tern in ft is "second for bidden" f,lq-') = 0 * 0(q*) , the leading vector contribution to the decay will be the weak magnetism term, wnjnh is related to the magnetic tran sit icn in analogy with Eq. (2.18) fi(0î D vl^ri • 2.2 The isovector character of the axial current As noted above, observed semi-leptonic interactions satisfy the selection rules ÛQ = AI, - :1 . The OX assumption for the vector current implies the stronger restrictions, tqs. (2.6) and (2.7). In the framework of a universal V-A coupling it is natural to attribute the sase properties to the axial current. Then we may write the total haUronic current as jj : J* - V* • ** .. (2.25) J*f H J" - V" * A" U H U li with the iioEupu spin property c'"'i j£ e-i". . -j; . C2.-4) Sin^e m!i'f (he suce tr-uu"forr.uion i'1-"1' ipi - In» . c1""*' In) - -p) . (2.2S) [hi-, -u-uanp: i-i: ' x-i urport.int l^ilitatl^ni for the é decay of rjrror nuclei; their decay uiatrix elcnents are equal up to ,i p.la;-.c (jnd up to cue t rcrugnef le effects). Thus, for c.xarple, for the A • 12 nuclei discussed in Section 2.1 ("C..J* !'H> • -{,;CJ""IJN> tmlsprmlsr.tlv at the nutlet; JCC.TV r-.cch.ini5". Similarly for the Z-\ tr.-rr .it ion* (/, J*=I*> • 1.\:J'JZ'> (2.2b) irplytn,; tNr «iiulitv ci both vector ù^d -'l.il fer-i •"-> tcrs. • I OIT convent-ion ii «such thjs (-I*rE*,r") are the corrponsnt* of an l lospin triplet; then c '|r* >. In KLTcnl. o''[' I.I,) - (-)<*I'![,-!,), CHAPTER III Let us consider in detail the matrix element of ihe axial current in neutron B decay. The general form is 2 The Lsotopic spin properties (2.24) and (2.25) imply the relation (plp.sJlA^Wp'.s')) = where p, p' and s, s' are momenta and spins. Hermitian conjugation of the right-hand side gives No-, consider the first tern in the matrix element (2.27); as the neutron and proff1^ spinors are indistinguishable in the limit of equal masses we nay vr'"e u s = + [0(p'.s')-Vl)Tf» tP» '3" " (P-S)YIYJYJU(P'PS') . Using the properties YB " y0 ' y\ " l ' Vu " \ • YS ° YS ' 'VV ° ° ' Me obtain [ùtp'.sOïyï^Cp.s)]" - û(p,shuYsu(p',s'} . Stmlarly [ûtp'.s'jo^Y^ulp.s)]* » -Q(p,s)ouvv^(p'fs') [u(p'.5')Tf^'.p,s>]* = -û(p.s)ysu(p',s') . Bearing in nind that q •* -q for p *— p', and equating tern by tern the natrix elements of Eq. (2.28), we obtain the conditions 8,(1*) " g"(q*) . *t,(q') • -B*(qJ) . g,ta*) n g^fq*) - Since tunc reversal requires the fori) factors to be real we have the constraint gjCq'l - 0 . (2.29) The «c argument could be used to show that f^q1) • 0 in the vector matrix element (2-13), independently of current conservation. 2.3 The strength of trio axial coupling: PCflC The iTtial current is not conserved. a3 is lllustra.cd by the fact that the pion decays into a lepton pair. Taking the divergence of the matrix element (1.6), we cbt.iin u 1 l <|jJ(aM*"(q)> - m£fr c""* t 0 . C-30 rhc matrix element ID nonrvantahlng; however. It us prcportlonji to the plt-n .tass which is viry snail on a hadrontc irem scale. Therefore one la led to the notion that the axial current night he 'yxirtially conserved" (PO. '. - p.'irttally conserved axial current). If wo set [Gell-Hlcui (1940)J ,*V • fjf.s.. , c.îi) whore 5 t<î the fleticrijcrg field operator which iccilhllatcn a pton What would axial current conservation ijnply? Since the axial "charges" yh*0) s/d'x Aj(x) (2.32) connect states of opposite paritv, corservation of the charges (2.32) would seem to ijnply "parity doubling", i.e. the existence of degenerate states of opposite parity, which is not even appruxijnately realized in nature. However, in the lirait of vanishing picn coss this is not necessary; operation with tha axial r".arge can now change the parity of a state without changing its energy by the addition of a zero energy pion. However, in the broken syncetry case the operator equation (2.31) is without physical content, since the Heisenberg operator is coupled to 3*, NN, etc., as well as to the single pion state- Thus, we must supplement Eq. (2.31) with the dynamical assumption that 4r is dominated by the pion pole (at least near the pole). Once this assumption is made we need not even introduce a pion field. The usual assumption of PCAC is the assumpt'cn that Eatrix elements of the axial current are doninated by the pion pole tor low monentun transfer: U (A!3 A |B> s where u = a_. Equation (2.33) is illustrated in Fig. S. oi(q)\ Fig. 3 DlaararEJtii: representation of the pica pole dqoinanee aosuaption (PCAC) Z.J.I Pic GoldburGor—Tzvimcn rctatitm the PCAC assumption [h%, (2.33)] gives a scans of predicting the strength of the axial vector coupling to nucléons. Tallinn * •IIU* ^ to be proton and neutron, we have at q* * 0: if\y*\\n) ' -iûd^ug^O) • iO^ • .\)u>(uK,IO) • (2.U) Tt-.c onîri* e Irisent en the righc-ttsul 3 We cf Eq. (2.J3) I* Just (he pten-nutlccn coupling constant ï*>. far qJ • (i wo 'OtJtn thr CoMscrgcî'TirlAin relaïicn [Coltdcffier (13S8)] s'"" :'SA'V^' '"'•*" i::ur.i (!u* «pcrt.T.-nt.tL f.i::,:r of :!*; pton coupling I'lVl ^, •'?[»-* :::::)• we obtain for the right-hand jide of (Z.36) |gAl = 1.33 t 0.03 (2.37) to be cocpared with a direct measurement of the axial coupling ui 6 decay (see Section 4.4): gA - -1.250 i 0.009 . (2.38) The PCAC hypothesis predicts the strength of couplings to the strangeness-conserving axial current in tents oi the pion coupling. Foi example, we have a similar relation for LA decay 6>)-^. (2.39) However, this hypothesis, in contrast to CVC, does not determine the scale of The couplings; in other words, it does not determine f^. The scale can only be determined by non-linear constraints, as are imposed by the algebra of currents 2.3.2 PCAC vcraua PDDAC In deriving the Goldberger-Trciman relation we made use only of tne assumption of pole dominance of the divergence of the axial current (PDDAC). This assunption in fact is not manifestly related to an approximate symmetry of strong interactions as implied by the torn PCAC. As the pion is the lowest lying state which can contribute to a dispersion rci3tion in q! for the matrix element (2.34), it is not surprising that the pion pole should dominate. The cut con 1 2 tribution has q i 9Li, so on dynamical growxJs alone we night expect pole dominance at q = 0 to be good to 10Ï, as is borne out in the comparison of (2.37) and (2.38)- What is the relevance of current conservation? If the pion mass were exactly zero, and the current were exactly conserved, Eq. (2.3t) would be exact. In this limit we have [Eq. (2.27)] U I î 0 - fpia^Ajn) - -iq (p!Ajn> - iu[(!^ * Mn)g|(q') - q gj(q )>1U . (2.40) Although the pion decouples fran the divergence when its mass vanishes, the matrix element of the axial current has a pole at qJ • u1 -0 (fig. -If, giving a .cntribution to g,; S.to1) ' "^T "* * fi--Ute tems . (2.41) q 1 JrucTtir& (.ML) in (2.JO) and taking the Jbiit q - 0, we obtain the relation (2.36). Thus (.jpFroxinsit'.') oxiat current ccrvicrvaticri assures the validity of the Goldberger-Treinan relation. However, i= it a need-wiry ^ofidltton? It has been argued that it is [Dasher» (19i>9)], since in the limit q; * 0, the mass de- pendt-nee disappears from the PCAC relation, Eq. (2.33). If high pseudoscalar states are coupled to the axial currt.it J.-1 itr ft';. * Picn p.:U <3ft*»ituetcn to tl-.c 4jaU SEHI-IEPTONIC INTERACTIONS 2.3.5 The paeudoBaalar form factor: U capture We saw that there is a pion pole contribution (Fig. 4) to the pseudoscalar form factor (gj). In general, this form factor is related to g (q2) by the PDDAC (or PCAC) hypothesis, Eq. (2.33), over the Tange of q* for which pole dominance is assumed valid If we define the non-pole contribution, gj(q2), by > q* - u* we obtain 2 The variation of the axial coupling g at low q is expected to be determined by the hadronic mass scale (say the At nass) g,(q*) = S,IO)(I-^-- ...). So for low qJ Clearly, the pseudoscalar Tern factor will be determined by the pole tern at low q*. This fom factor cannot be ncasured in fl decay, as it contributes a tern proportional to both the lepton mass C*tU \ • (V2I L*H! the ncsentuia transfer to the Icr-teas ii K-iorin;; noi.-polc contrihition.1, we evaluate the pcuascalar fcro factor; f-K-r-, -9.09 ; Q.i! 8,(0.93.*) - - ^2— - . (J u' • O.Sfc^ r^ iinnc tlie h.-.il» .ntrnicstan n punctual, the capture rate li proportional to the prcoabtllty that the mon iinJ protoi lv.ilu.it mu the matrtt o kiarnt far the ,lppf07Triatc- values of the cxrenta, the resultant rate will depend 1 initial spin iMCc of the „p .iton ['ti.-sfvik (Wâ'»)] * • J3 ice-1 (1 • r where a and 6 depend on the nucléon form factors. FOT the singlet state one finds ^^L- . „ - 36 = [f, * 3g, • i^ (3f, * g, * Zf, * V,)]' . The transition rate is dominated by the "allowed" vector and axial couplings (f, = gv, g, = • ;«). However, for capture from the triplet state there is a cancellation of the leading terms A-p %G»G,3 28 sec"1 c, - fi - Si - Ji'. f. 2H, G, " 2i * Since tie axial and vector constants always appear in the combination f, - g, = -0.2 , the sensitivity to the magnetic and pseudoscalar form factors is enhanced. In practice, extraction of the form factors from experiment is complicated by the formation of mesic molecules (pup a.id dud) and by nuclear effects if the capture takes place in complex nuclei. Nevertheless, oy studying v capture in various nuclei one can, in principle, test the full structure of the matrix element, as sensitivity to different form factors varies from one process to another. The bulk of the data is compatible with ue universality and vith pv = -g., which prcd J ILf « Ag . Using the values of gv and g, detemined from B uecay, the best value extracted for the pseudoscalar form factor is [Zavattini (1971)]: m^tO.yn*) - (10 t 1.6)gA - -12.2 * 2 , to be compared with the prediction of Çq. (2.43). 3. THE CJ9IBBQ THEORY OF CURRENTS «Is briefly discussed n ' tic L, all observed scni-lepionic decays may be classified into rvo groups according to the quantu.il numbers ci. tt-j t, *..ic hadronic current a) £Q - Ai, • n-pcvc i * * ir evc US • 0 r * Jicve . • . i». I" * nlv£ E- . (A.I )«v, Far the siran(tcnes--conservin;î drçaya, |tl,| • 1 Inpllin |il| ; 1. Only a errent with lAl I • I can contribute to the L.notopic ipln triinsltioa-v '/t • '/. (i » p) and I * 0 (I • \, ? • vacuo). A ja|| • 2 current could contribute to plan J da..iy (I * Hi however, the iucceai of the CVC prediction (2.N) provides strong evidence that this transi tion Ka *Lio only \&l; * I. Eurthcrnorc, there is evidence frtm. nuclear decay that '.Q • il currents are nenhers oi .i ^urmjn iwipi» triplet with tF.c self-conjugation property (£.£•). This property alio requires the ^^jallty of J-T>H- tudri In ' » .1 Jccjy £E*|. (J.Jfr)], iinplylrnt thai the ratio of the dtriy rates is dctemlr-oJ by the piiasc-spicc ratios If.61 r(i- - :.) SEMI-LEPTONIC INTERACTIONS The experunental value [Chounet (1972)3 R = 0.67 * 0.15 . although imprecise, supports this prediction. The-efore we may sumnarize uie selection rules for strangeness-conserving decays by US = 0 , AQ « Ai, = ±1 , jAr|=1. 13.1) ..et us examine the second class (b) of observed decays. 3.I Selection rules for strangeness-changing decays First wr „re the absence of transitions with |AS| > 1. ir-e decays - * Mfv (AS = I) have not been observed. The experimental limits o.i be displayed as [Particle Data Croup (1974)] ris° * F>-"\ < i.o rf=" - Ae-v) Recalling th.it for a \-A coupling there is a phase-space enhancenent proportional to the fifth power of tnt -nergy release L(M= - «A)J =' this implies limits in the amplitude ratios cf 0.20 and 0.76, respectively. Secondly, all observed decays satisfy the rule AS - AQ . The suppression cf decays with AS • -AQ is illustrated by the lioits {3icunet (1972}] w iff* - *r> < o.0i9 W * rjy> < 0.095 r(E- * vu c) and [Buurc[uin (1971)] :^21< 0.013 . rhc AS • zQ rule alio implies that in tho K. axle of neutral lam drcay, the IT"C*V final state arises only frcn K'* deer/, while T*C"5 arises tfron K9 decay. As will be shoo in Chapter V, this prediction can be tested by staying the time distribution of the decays «inco the ttsv dupcnderncc of th* K* ,%* admixture of a neutral K Icra is known if its IntE.at it.ite .•» known. The csl.*ElE-n data [Oiounet (J9?J)] provide a lirut for the amplitude ratio which caSTi;»ioj»l9 to j lrnte af (?.Sl Ivt the rate ratio. Th? cfurge -»)rawtr/ tn flciït-lejucnlc t.. decay- tilers a caM- •iurnwnc of the re.il p.irt vt thl» ratio. Th? ram prccl-w remit tca$* [Gcwtnificr f!9~8)] Re x « o.cc:i • o.nia . du Known Ji.aJroivU it.itci sjClafy t.*ic relation between ejuntun meftern Q • t, • ) . where if B I*Î h.iryon number, U * 3 • 0 la hypercfiJfRr. Tftcrs, alrsc !i in cc?i;crvcsi !fjr *rj! tnlcuetier.i. ne hjvc *.S * .',Y jrd we oht.itn CHAPTER III 41,-aq-f - £-*-"* AQ = AY = ±1 . In other words there is no component with AI, = ±3A- It is then natural to ask if one might have the more restrictive selection rule |ÛII • % This rule has been tested only in K. decay. In terms of isotopic spin the KTI transition is Vz •* 1 which could occur via AI = 'A or Vj. If only AI = % is present, the matrix elements foT different charge states are related by Clebsch- Gordai. coefficients Since CP invariance requires the equality of rates for K° •* n~e*y and KD •* TI*C"\J, we obtain for the decay of the long- lived state rot. r2v) = ^{r(K3 - iTZv) + r(Kfl -» T*iy)l = 2"(K* - Viv) up to phase-space (1.351} and radiative (IS) correction;; [Rubbia (1969)]. IVe obtain the prediction that the form fac tors in K, and K" decay must be the sace (up to the over-all nomalization factor) and show equal variations with the momenoxn transfer t between hadrons. Although the K. data are not yet completely understood, there is no serious dis agreement with these predictions, and the comparison of the slopes of the form factors in K* and K. supports this idea (Fig. 5)• For K decay the analysis is more complicated as the final 2n system can be in more than one isospin state: I = 0, 1, 2. However, the AI = 'A rule predicts the absence of I = 2 final states and relates the form factors of different charge modes for transitions to I = 0, 1 final states. These can be separated by their angular rumen tirn; Bose statistics for the pions requires 1=0 for £ even and 1=1 for £ odd. At present only the K* -* r*r"ev mode has been extensively studied. 1 —— I —i mean value + i mean value l ,, ,,i 11 i l i 111 l i ' i ' ' ' 11 i i ' I -0.10 -0.05 0 0.05 0.10 0.15 X. -0.10 -0.05 0 0.05 0.10 0.15 \. Kej Ke] Fig. 5 Measured values of the slope of the farm factor In K decay SEMI-LEPTONIC INTERACTIONS To conclude there is serious evidence in favour of the selection rules ÛY = ÛQ= ±1 , [AI | = 'A . (3.3) 3.2 The postulates cf Cabibbo theory We are now in a position to draw a unified picture of semi-leptonic decays through the assumptions of Cabibbo theory [Gcll-Manii (1960), Cabibbo (1963)]. a) The components of the hadron current belong to a single self-conjugate representation of SU(3). This postulate is based on the observation that the selection rules (3.1) and (3.3) are the requisite properties for operators which are the charged components of an SU(3) octet, as illustrated in Fig. 6, where the well-known octet of pseudoscalar states is also displayed. We define the strangeness-changing currents by u " Jy *P \i (3. i) ,h(4S=l)+ : 3v ' where v and a^ denote the separation into vector and axial currents. The self-conjugation property of postulate (a) follows from hermitkity of the ..cak Hrailtoniar: and the requirement that currents with AQ = il belong to a common octet. b) Universality: the leptonic current f_Eq. (1-1)] is coupled to a single hadronic current of unit length: J = cos a J • sin 8 j (3.S) with the universal coupling strength G: The effective Fermi constant for 6 decay is now Ce - Gu cos e . Comparing the measured values ot" GR and G [Section 2.1.1), we obtain cos 0 = 0.9858 i 0.0011 sin ». = 0.1677 ± 0.006(> . 1 K-J* u TfJ- T), n re'.j* -1 -1/2 0 1/2 1 .-1 K° Fig. 6 FL'O) weight diagraa for pseudost charged currents CHAPTER III However, we should ^rtually write = G cos 6 1 4° where Aa represents model-dependent radiative corrections; it depends both on the non-local structure of the weak inter action (it is divergent for a local interaction) and the elementary structure of the hadron current. If the current is constructed from the usual fractionslly-charged quarks (see Section 3.5), and is coupled to leptons through a charged vector boson in the mass range ! GeV : 500 GeV , \ : one obtains [Wilkinson (1972)] Aa = H-S î 1) * 10""* cos I = 0.978S ± 0.D0S0 sin 6 = 0.2064 ± 0.0236 . Although the effect of the uiicertain-y on the B-decay coupling strength is slight, there is an appreciable effect on the determination of sin 0. Since the model used to determine &a is not established na should probably increase the error en sin 9; we take sin Ô = 0.21 ± 0.03 . (3.7; In any case we expect a suppression of strangeness-changing processes by a factor of about five; this is just the empirical suppression discussed in Section 1.2. c) Generalized CVC; in the linit of SU(3) symmetry, the charges associated with the vector currents are the time- independent isospin and «•: • Jigeness-changing operators. We have seen in Section 2 that the AS = 0 vector currents Vf, along with the 'sovector component VJ of the elec tromagnetic current, form an isotopic spin triplet of conserved vector currents. The strangeness-changing vector cur rents are not conserved; their associated charges: ' / d3x V±(x) (3.8) do not commute with the strong Hœniltonian which is only approximately imvriant under SU(3). However, the assumption that the "--rrents v" and V" are members of a comcon octet is well defined only in the limit of ^'(3) syranctry. In this linit, the charge operators of Eq. (3.8) are also time independent and generate infinitesinal SU(5) transformations. They arc strar.neness raising and lowering operators; in the notation of Gell-Mann (1961): S1 = Fi t iF., (3.9) and their matrix elements between octet states are determined by the structure constants of SIi(3) : 3 where f... is a totally antisyouctrie tensor. The :cro momentum transfer linit of matrix eler^nts of v" are detennined by F.q. (3.10) up to symmetry-breaking effects, lie shall sec in Section S that the corrections are in fact second o."der in the symmetry-breaking strength. 3.3 Meson decays For pseudoscalar to pseudoscalar transitions SEHI-LEPTONIC INTERACTIONS only the vector current contributes. The general form of the matrix elements has been given in Eqs, (1.11) and (1.12), In addition to the il = % relation 'Z.2), the charge-conjugation property of the vector currents: a-c-1 = -vj , cv^c-1 = -vj requires the equalities {r'lV^lO - -(-°|V;!r*> + (3 U) U*\V*\m =- = (*'|v~|K°> (where in the last relation we have used the usual convention CP|KD) = |K°), SO ClK°> = -|KC)). Consider now the matrix element for the divergence of the strangeness-changing current 2 Hia'VjK) = (M - u*)f+(t) + tf_(t) = (M* - u')f(t) , (3.12) where M and u are the kaon and pion nasses, respectively, and t = (p„ - p_)*. Current conservation requires in the SU{3) limit (M = M) f_(t) = 0 • 0(e) , (5.15) where & is a parameter which characterises the strength of the 5U(3)-breaking interaction. However, the coefficient of f in lij. (5.12) is also first .inter in the symnetry breaking and it is numerically very larf.e f-^£ • « ' 0(O " 1 • (3.14) Given such 3 large symmetry breaking in the masses, the significance of the prediction (3.13J is not clear. However, the prediction for the "charge" matrix elements is expected to be valid to second order in e. In the SU(3) limit we have, for example, + e s :'fT tp)i = (V(p)) . so ir'lS + X) = (2n)'2En6(p!T - fy •= (2")^^ - PK)[f+Ct)(ET. * ^ • f.(t)(E_, - E^] , 2 where tor p •-' pK, t =• (En - Ej,) + Û in the limit of equal masses, anu we obtain f^(0) = 1 + 0(e2) for K7 * if+£v - (3.15) The normalization for the other Kaj decay modes is then determined by the conditions (3.2) and (3.11). Symmetry- breaking corrt-ctions to (3.15) will be discussed in Section 5. This prediction enables us to deteniit'e the Cabibbo angle from the comparison of K. and n decay rates. One finds [Chounet (1972)] sin 3 = 0.?12 ± 0.005 , (3.1o) to be compared with (3.7). Another determination of the Cabibbo angle is fron the comparison of n, and K. rates : C|A-*C0)J-i(p)) = ifrpu (3.17) (|a"(0)[K-(p)> = ifKpu . Under the octet assumption of Cabibbo theory, an SU(3) transformation such that - 84 - CHAPTER III will also transform the currents Then SU(3J invariance requires -A= 1 *0(e) . (Î.18) If symmetry-breaking effects are neglected, we obtain froci the experimental rates [Chounet (1972)] sin 9 = 0.2691 ± 0.005 . (3.19) Another way to nake the comparison is to assune that the theory is correct and to consider the amplitude ratio a(KAz) fK a(Ttla)a(Kej) " cos e fflf+(0) ' Since cos 6 (as opposed to sin 6) is accurately determined fron nuclear 6 decay, we may eliminate it. Then we find 1.27 i D.D5 = 1 + 0(e) . (5.20) Considering the Jarge symmetry-breaking effects which may be expected for processes involving pseudoscalar mesons, this result is in good agreement with the assumptions of Cabibbo. 3.4 Baryon decays These decays provide a imch more powerful tool for testing Cabibbo theory than the meson decays for several reasons: a) Since SU(3) is broken in the masses to only about 101 one may reasonably hope that the vertex corrections will not be too large. b) For zero momentum transfer, all decay amplitudes are determined in terms of only two parameters in addition to the Cabibbo angle. c) There are four non-vanishing form factors and they are constrained by the theory to satisfy certain relations. We defer discussion of the form factors to Section 4. Here we consider only predictions for the amplitudes in the zero momentum transfer limit. In this lijnit the general matrix element for the decay E[ •* B; + iv is of the form The vector matrix eleiwnt (Bi|VjB) = gvPi_*0(qu) (_ = P-J (Pi * P^g ' % " CP! " Pl)y is determined by the assumption that the "charges" are generators of 5U(3). In analogy with the deteminaticn of the vector form factor in S decay [Eq. (2.16)], the vector couplings for the strangeness-changing decays are determined up to (second-order) SU(3)-breaking effects by the properties (3.9) and (3.10) of the associated charges. If g*J is the vector coupling for B. * B-, then SEMI-LEPTONIC INTERACTIONS - 85 - (B.iS^B.) = gJj2E(2n)'5[q) 1 J ^ (3.23) 1 + J gj" = iCf^j ± fsij) 0(e ] (iS = ±1) . For the axial vector matrix elements we have only the constraint that the current transforms like an octet. This implies that its matrix elements between octet states are completely detemined by the Wo parameters which charac terize the strength of the antisymmetric (F) and symmetric (Pi couplings of two octets to form a third octet where k = l,Jl=2 for AS = 0 k = 4 , t = 5 for ûS = il . The explicit values (see Appendix) of the tensor elements f... and d... give the following predictions for the obscr .•'le matrix elements of the weak hadronic current (3-5) at zero momentum transfer (Aîjjlr) = (A|jJ:r*) = /| D cos e "V,i (3 25b) 25c) (3 25(1) l (S|jj!="> « sin 6 STJ3 * (3F - D)YJl/.^ (3 25g) (3 25h) These predictions are of course subject to symmetry-breaking corrections. There are several predictions, however, which provide crucial tests of the theory, because they are not modified by symmetry-breaking effects, a) The predictions concerning the EA transitions, discussed in Section 2 [Eqs. (2.22) and (2.26)],should be valid to order a = e2/4n. b) The relation (3.2Sh) between amplitudes for 2 •* E£v is a consequence of the Al = Vi property of the strangeness- changing current and is not modified by strong interactions. c) Tlte AS = -AQ rule rust be valid up to second order in the weak interaction if the postulates of Cabibbo are cor rect. 3.5 Some observable effects of the V-A structure The ir.atrix elements at zero momentum transfer given in the previous section exhibit a composite vector — axial vector structure. In practice, as we shall see in Section -1.2, this very simple structure is somewhat obscured for finite momentum transfer and additional terms must be introduced. It is, however, rather instructive to utilize this simplified form to give a very concrete illustration of the observable effects of the V-A structure, keeping in mind that the simple formule given in this section will have to be completed with recoil corrections and the intro duction of other form factors. CHAPTER III Consider the decay of a completely polarized baryon \ •* Bi + e + v. The helicities of the leptons are fixed and the decay proceeds Ss indicated in Fig. 7. There, x is the ratio between the axial and the vector couplings for the particular decay; for instance in the caye of E •+• nev, x = (F - D), Vn"-7 x) \ | (Ux) f I HI-1 J) 'V7 u Fig. 7 The process B * flj • e + û. f indicates the spin direction, || indicates the momentum direction; x » G^/Cv. The factor •'ï for the spin-flip ampli tude in the axial term is just a Clebsch-Gordan coefficient. It is a simple matter of counting states to determine from Fig. 5 several observable quantities. The decay rate is proportional to C£fl + 3xJ) . The lepton-neutrino correlation coefïiciont a. , defined by W(cos 6Ev) = \ (1 + aey cos e&v} , (3.26) where 0» is the angle between the lepton and the neutrino directions, is given by (3.27) lv 1 + 3x2 1 + 3x* The correlations a. between the directions of the decay particles and the initial baryon polarization lead to intensity distributions of the form Kn) = | CI'* "iV^) . (5.^8) where q is the unit momentum of the relevant particle and Pg, is the polarization of the initial baryon. Because of the strict helicity properties of the neutrino and the lepton in the decay, one gets immediately from Fig. S: l J a - 'Ad * *>' - Ml - *)' - 2x - -Zx(x - 1) (329, £ 1 * 3x2 1 • 3xz = 'Ad * *)z * Zx' - 'Ml - x)' _ 2x(x * 1) ' 1 + 3x2 I * 3x2 SEMI-LEPTONIC INTERACTIONS Finally, the decay baryon polarization is given by 1 + 3%2 We emphasize again that those formulae are only valid for zero momentum transfer. They are given to indicate how, without complete and complex calculations one can get a concrete idea of how the observable effects arise. Complete expressions will be given in Section 4.3. 3.6 What is the aeaning of universality? The original universality postulate was inspired partly by the nea' V-A structure (gy * -g^) of the hadronic matrix element in neutron s decay. However, it is apparent from Eqs. (5.Z5) i.hat this feature does not persist for hyperon decays. This simply reflects the fact that the baryons are no more fundamental than the mesons in defining the current, in fact the neutron decay ratio gA/gi( = -1 appears here as something of an accident. In particular, the total normali zation of the atial current is not defined. The universal V-A structure can be defined most simply in a model in which only fermions couple to the weak current- This is the case for the quark model. If the strong interaction Lagrangian contains no derivatives the hadronic current is jj » MpYyd * YS)<^ , (3-32) where a/ = cos 6 qn + sin e q (3.33) is a neutral quark state related to the neutron quark by a U spin rotation. Thus the current is formed by the V-A coupling of a single neutral quark to a single charged quark, in complete analogy with the muon and electron currents. All of the postulates of Cabibbo follow froa Eq. (5.32). The postulate that Eq. (3.32) is the fundamental coupling defines the strength of thu axial current. However, in order to relate this normalization of the axial current to constraints on matrix elements between physical states, it is necessary to introduce current conmutators. Once the commutation relations are established, they arc sufficient in themselves to define all the relevant properties of the currents. In other words, we may simply postulate the com mutation relations; the quarks are a mathematical device but may have no dynamical meaning. The current commutators will be discussed in Section 5. A. FORM FACTORS IN BARTOW DECAV The general matrix elenent for the weak current between spin-1/; haryon states is given by for AS - (J) and qu = (p, - p2)u . Time reversal invariance implies the reality of f and g. At zero momentum transfer, this expression reduces to the amplitudes discussed in Section 3.4 [Eq. (5.21)3 f,(0).gv, g,(0)--gA. (4.2) Under the assumption that the current J transforms accorjing to the octet representation of SU(3), the form factors for different decays are related in the symmetry limit. However, for the broken symmetry case it is difficult to sepa rate kinematical effects arising from the mass splittings from dynamical corrections to invariant amplitudes. For exemple : CHAPTER III -1 a) The coefficients of a in Eq. (4.1) have the dimension mass . Should one apply SU(3) constraints to f2 or to f2/(M, + Mj)? Or to some other parameter? b) In the syiiiiietry limit the form factors must have the same t dependence. However, the singularities in the un- physical region which deteimine the t dependence through dispersion relations occur at different values when the symmetry is broken. Thus p dominance for the matrix elements of V" predicts a slope for the form factor fi'(O) 1 — « — - 1.7 GeV~2 , (US = 0) (4.3) f,(0) m* whereas K dominance for the matrix elements of v~ gives fÏCO) 1 -±— = — = 1.3 CeV1 , (US = ±1) . 1.4.4) fl(0) TTfc Sirple pole dominance does not give a good description of the electromagnetic form factors; however, Oqs. (4.3) ar.o (4.-1) serve to illustrate the problen. c) i\s will be discussed in Section 4.4, the uncorrected Cabibbo predictions of Eqs. (3.25} give an excellent fit to all existing data relative to the determination of the coupling constants (4.1). The predictions for g . are valid to second order in the symmetry breaking, but gA is expected to have first-order corrections. Roos '.1971) has made che ad hoc: assumption that the lowest_order correction is just SÂ'' " '"A'''. Sif ' (4-5) ,l where (gï )0 is the uncorrected Cabibbo value for the axial coupling in the decay Bl •* Bz + Iv. Although in the worst case (£" - nllv) the correction factor is 11%, the fit obtained using the predictions (4.5) is as good as the fit using uncorrected predictions. In the following subsections we shall discuss the constraints of the theory and the expected validity of different types of predictions. 4.1 The vector matrix element In this case the assumption that the currents are related to the electromagnetic current by SU(3) trans formerions proviJcs strong predictive power. The matrix element of the electromagnetic current between identical spin- % baryon states is of the form CBCpaJ|j™C0)|B(Pll) = 5,{v; - ~^~ i\ * Vf}», H.6) % = (pi " pA* Current conservation requires fcf. Eqs. (2.15)-(2.15)] fj(q') - Û . (4.7) In the Cabibbo tlieoiy the weak vector currents are part of an SU(3) octet together with the electromagnetic current Sea-LEPTONIC INTERACTIONS Then the weak form factors can be completely determined from the neutron proton electromagne ic form factors up to symmetry-breaking corrections. For the matrix element of any vector current V*(i = 1. •••, 8) we may write k k ra = 1, 2, 3 , i, j, k = 1, .... 8 . J Eat. . set o£ form factors f* is determined in terms of two independent form factors Fm, Dn, which characterize the antisymmetric and symmetric couplings, respectively. Since there are only two independent amplitudes in each case, they arc determined by the knowledge of the nucléon electromagnetic form factors; explicit evaluation of the tensors f^jk and di.jc (see Appendix) gives P = F + D = D f m T m • C " T * C4.ll) m m 3 m m 3 m ^ ' (These form factors of course also determine the electromagnetic form factor of the hypcrons in the synmetry limit.) 4.1.1 The veator form factors These are related tJ the charges as discussed in Section 3. The charge matrix elements [Eq. (3.9)] are totally antisymmetric. The electric charge is just % = f J(0) . Then comparison with Eq. (4.11) gives 0, • 0 ; D,(0) - 0 • 0(e'j (4.12) Op - 1 ; F,(0) = ! • 0(c!) . 4.1.2 Weak rrggr.etiem The electromagnetic form factor fz is related to the anomalous ragnetic moment: «B " ' obtain from Fq. (4.11) 0,(0) - - j>n • 2.870 . 0(e) (4.14) F.(0) •>,•,»,"p 2" "n •" 0.83b • 0(E) . Using these values, explicit predictions for f: in weak decays may be read off from Eqs. (3.25), since I- and 1) parameters figure in exactly the same way as for the determination of the axial coupling constants g.. •1.1.3 The acalai' form factor From the vanishing of f,(qJ) in Eq. (4.6) st(q;) » f^(q') = 0 . it follows that this form factor vanishes for all baryon decay amplitudes F (|:) » 0 • O(c) (4.15) U (q!) • 0 . 0(c) CHAPTER III In Eqs. (4.12)-(4.1S) we have indicated the order of symmetry breaking to which the predictions are valid. The moracntum dependence of D_(q3) and F fq23 is also completely de*-^rmined by the momentum dependence of the electromagnetic fom factors in the synmetry liait. However» as mentioned above, the interpretation of this prediction is ambiguous when the synmetry is broken. On the other hand, the form factors for neutron decay are related to the electromagnetic form factors by isotopic spin alone. Therefore the OJC predictions of Section 2.1.2 are expected to be valid up to electromagnetic corrections. In Section 2.4.1 we derived from CVC a relation between the vector and seal;: orm factors in EA decay : 1 (Mr - MA)f(Q ï • q^fo ) = D * 0(a) . Although we have cne further requirement that ffAC03 = 0 + 0(a) . we expect f (q1) / D far q t 0 [just as for the neutron charge forn factor f"(q*) J 0; fn(0) = 0]. Thus, we obtain the prediction a + ff(0) = -(Mj. - MA)^f?(q )| 2 0(a) • (4.16) The deviation from the SU(3) prediction [Eq. [4.15)] f^A(0) = 0 is determined precisely by the Z, A mass splitting Mj. - Mft = 0 * 0(e) . This result illustrates that kinematic effects can play a role in determining symmetry-breaking corrections to in variant vertex functions. 4.2 The axial vector matrix element In this case there is no observed axial current on which predictions cay be based. However, we may use the iso topic spin property [Eq. (2.24)] of the AS = 0 currents A" to constrain the matrix elements. Using SU(3) invariance and the octet assumption for the currents, we may again express all of the form factors in terms of tut) "reduced" form factors. We write «AV-Mvi*-^^ ««", Cabibbo theory requires 1 tfV) - -ifijkF;(q-) * Oijktfo ) * OM At vanishing momentum transfer, the axial form factors g of exprcssio i (4.18) reduce to the axial couplings if Eqs. (3.25) Fj(0) = F , D[(0) = D . (4.! In Section 2.2 we saw that the assumed isospin properties of the strangeness-conserving axial current requires the vanishing of g; in neutron decay sÇV) = F>!) * D=(qî) = ° • The sane reasoning holds for the I = Vi •* ' - V:, iS = 0 transition E~- E°ey. Thus we obtain gf°S"Cq*) - D'(q') - F'(qa) - 0 , SEMT-LEPTONIC INTERACTIONS implying that the "pseudotensor" form factor gz mist vanish in oil decays up to symmetry-breaking effects 4.3 Rates, spectra and experimental correlations The complete natrix element for the decay B1 - Ba + £ + v has been given in Eq. (4.1). Before discussing which measurements can be performed 1?t us.recall the basic properties arid assumptions about the form factors. T invariance implies that the f, and g. are real and in the limit of exacr SU(3) the absence of "second-class current" implies = = ^i gj °- ^tice that the fact that f3 vanishes is also a consequence of the CVC hypothesis. According to that hypothesis the weak vector current and the electromagnetic current are members of the saœe SU(3) octet, thus f2(0) is determined by the anomalous magnetic noraents of the proton and the neutron and is called weak magnetism. The contribu tion of the g, term is proportional to the lepton mass; it is negligible for electron decay modes. For the nuonic modes its contribution can be estimated from the PLAC hypothesis (see Section 4.5.1). Kith these assumptions the analysis of the present data is made in terms of three naraneters 8, F and D; the q2 dependence of the form factors is generally neglected. The matrix elements are those of Eqs. (4.1). The weak magnetism term is written as f2/(Mj + M2J because CVC relates directly fa defined in that way to the fz of the electromagnetic form factor. In the experimental analysis the more cumbersome definition fJ/Mt is generally used. In the Cabibbo theory for the observed leptonic decays one gets the following relations between f (0), g.(0), and fa CO.» : -* pev 8,/f, f,/f, - <"p - "il) •* A£v f,/g, f/E; = -Pn/(%0) ~ptv g/f, f,/f, '"v - nlv 8,/f, f/f, • f"p * 2un> f - Aev e/ , VJD f,/f, -• zcev» 8,/f, D Vf, The experimental results will be given in terms of fj = (fîMI)/(M, + M;). From the matrix element (4.1), ore derives the decay rates and the distributions experimentally measured. Most of those derivations, too lengthy to be reproduced hers, can be found in the article of Willis and Thompson (1968)- We will discuss the final equations relevant to the analysis of the data on hyperon decays. The experimental efforts have been concentrated so far in determining the decay rates and the g^f, (or t'i/gi) ratio for the various decays. The weak mag netism term introduces in general only a small correction; we will keep terms up to first order in 6 = (Mj - Mj)/(M, + Mz) unless otherwise indioted. The mass of the lepton is neglected throughout. With these approximations the decay rate for B, -*• B2 + 9. * w is given by [Harrington (I960)] R = G2 -gll (ff * 3gJ)(l -35) , (4.22) oUn where Am = Mi - M2 and G is the universal Fermi coupling constant. The magnitude of g,/f. (or f,/g,) is determined primarily by measurements of the lepton1 neutrino correlation a. with CHAPTER III L-2« By comparing with (3.27), notice the importance of the recoil correction; for instance for the decay L -* nev, a£v = O.S and 26 = Q.Z. The sign of gi/£l (or ^/gj) can be detemined using three independent sources of information: - polarization of the decay baryon for unpolarized hyperons - decay asynmetry of decay products with polarized hyperons - shape of the lepton spectrum for unpolarized hyperons. The polarization of the decay baryon in its own rest system and for unpolarized hyperons is given by [Linke (1969)3 m £ 2 + -1 \ U i\ Sig,!*) ]! Re [f*gl)a + | iBiTB + -J Im (f^jy] (4.25) kj x kv "" *,•«„: ' " i^r^i ' ' : ih " Ki ' k , k, are the unit vectors along the lepton and the neutrino momenta; (a, 3, y) is an orthonormal basis. The first component is sensitive to the sign of g,/f. (or f,/g,)- 1m (f^,) ;* 0 would indicate a violation of time-reversal invariance. The polarization information has been used for £ •+ Aev, where the subsequent A •+ p + r.~ decay gives the following distributions for the proton: B(p.S) • £ [l • 4 «, j-SEJfp p-d] (4.26) "»•» • 7 D ' ! «A J717FH <4-27' «»•?)" Ï [l * î «A 3-T]^î P-î] . C«-Z8) where p is the unit vector along the proton direction, a. is the A asymmetry parameter and y = fj/g;- For polarized hyperons the correlation between the directions of the decay particles and the hyperon polarization R, is described by the distribution 1(3) -\ (1 +aiPY*q) , (4.29) where q and 5. are the unit momentum and the asymnetry parameter for the relevant decay particle. The up-down asymmetry parameters with respect to the production plane are approximately given by [Frarpton (1971)] «f,h - # •i*(f, *e,) * * 2f,(f, • 8,)] f; * 3ss 2(f,g, * eï) -ï*[2Cf, "S,) ' * 21,(f, " S,)] - I f|g| * I '['Cf. ' f,)l,] _ whe. e <5 => (M - M )/(M, + M2) and only the terms up to first order in & have been kept. SEHI-LEPTONIC INTERACTIO* Fig. B Shape of the electron spectrum for t" + the term (f, * £2)g1l»y - tiH)/M£ allows ofg./f,. For the lepton spectnen, a complete expression has been given by Harrington (I960) and we find, neglecting terns in {n,/Cg}2 and keeping only icro- and first-order terras in 6 in that expression J W(2] •«. BZ^I - :) {C1 + «zjff • (3 - 46 + 206z)gf - 4«'l - 2z)(f, + f2)g.} , (4.53) where 6 is the lepton velocity and : = ^./E»^ ^e deterrainati°n °f ^e sign °f fi/gi DV this method requires a good knowledge of the shape of the lepton spectrum including the effects of all instrumental biases. It will only be used with large statistics. Figure 8 shows the expected effect in the t •* r.eû case for various values of gi/f,- 4.4 Experimental results and fit to the Cabibbo predictions*' Although the initial systematic cieasurements of hyperon leptoric decays date from 1964, the amount of information is still very liraited. For only three decays are the statistics above 1000 events: 1 For a recent review of the experimental results see Chounet (1972) and Kleinknecht (197-5). CHAPTER III n -*• pe"v A° + pe~v . The value of jgj/f,| (often called also GA/Gy) for the neutron has been deduced from measurements of three independent quantities: - neutron lifetime Ig^fJ = 1.243 t 0.011 11.26 ± 0.02 - neutron spin-electron correlatijn lÊj/fJ 11.27 ± 0.02S - electron-neutrino angular correlation lg,/f,l = I-242 i °-041 • giving an average value of g,/f,| = 1.2S0 ; 0.009 . By convention, the sign of gi/fj is generally taken to be positive for the neutron, which fûtes all the other signs, Hie g,/fj value for the neutron gives a first relation between F and D F + D = 1.250 i Û.ÛÛ9 . In the .'ic dcr.ay case both the sign and the magnitude of gi/f 1 ha»'" boen obtaino.. The sign is determined by measuring the asyraietry parameters ae, a^ and A_, and Fig. 9 shows the variation of :. ese quantities as a function of g,/f,, together with the experimental values. The sensitivity of the measurement depends directly on the magni tude of the polarization and fortunately A0 can be produced with a higii degree of polarization in the reaction :i + N - A0 + K around 1 GeV/c . il 1 II 1 1 1^ -" 1 +CERN-HEIDELBERG |CHICAGO-ANL —7^ 0.8 — 7"«. _ / g 0.4 = — h- "~~^ \ / 2 \ / 9, ", = 071 ce< \ * ; a î 0 \ 1 —-—;_ * tr "*• ¥/ UJ -•- S -04 - \ A -0.8 — / .s 1 . .—"\" 1 1 1 j 1 -0.6 -0.2 0.2 0.6 1.0 9,", Fig. 9 Variation of the electron, neutrino and proton a3yinm:îtri< of gi/£•. Measure- ment9 from two experiments are shown. SEMI-LEPTONIC INTERACTIONS - 95 - For the f - ne"v decay, the recent development of high-energy negative hyperon beams has increased the number o'.' fully analysed events by a factor 100 compared to previous bubble-chamber results. These beams are produced by the irter- action of an extracted proton beam on a target and they are based on two simple ideas: a) As a function of the incident proton energy E^, the shielding length against hadrons, and therefore the beam length, increases only as the logarithm of Ep, while the hyperon decay length is proportional to Ep. b) At high momentum and in the forward direction the leading baryon should dominate the prodnction. The beam? obtained by this technique should improve by about two orders of magnitude at 300 GeV and provide =", if, etc. Unfortunately, the hyperons produced in the forward direction are not polarized- In all analyses it has been assumed that second-class currents are negligible. In general, the measured quantities are very insensitive to the value of the weak magnetism term, which is therefire set to the value predicted by CVC. The <".y exception is the decay A0 •* pe"v, where all correlations being measured a value fj/fj = 1-0 * 0.5 is obtained in agreement with the CVC prediction of 1.0. The data on muonic decays is extremely limited and will not be taken into account in what follows. The experimental values of the relevant parameters are given in Table 1 together with the results of a three-parameter fit due to H. Roos (1974). Table 1 One-angle Cabibbo fit SUi 6 = 0.250 ± 0.003 D/CD + F) = 0.6E8 ± 0.007 X2 d.f. = 8.4/8 Quantity Experiment Fit T, • r(A - pfTv) [8.13 ± 0.29) * lO""1 8.13 x 10— T„ • rw * pu-v) (1.57 ± 0.35) * 10"1* 1.34 x 10-fc Tj- • r(E" - ne-v) (1.082 ± 0.038) x 10"3 1.07 x 10"3 -11 4.95 x 10-" TE- • r(r" - nii-v) (4.47 ± 0.43) * lO 3 T-_ • rCE- ~ Ae"y) (i-« : I:") • *"" 0.46 x 10" 3 0.S5 * 10~* TH. • r(=" - J,e-u) (0.68 t 0.22) x io- - s -r _ • r(E~ * Aev) s 6.98 * 10" E (6.04 ± 0.60) x 10" -i T » • rp* - Ao*u) s 2.2B x 10 £ (2.02 ± 0.47) x I0' g,/f,(ii ~ pe~v) 1.250 i 0.009 input = F + D B,/f,(A- ps-JI +0.6S8 ± 0.054 0.702 g,/F,tr - ne-v) i0.435 ± 0.035 -0.394 2.62 x io_li sec 1.48 x 10-'° sec V input V 0.80 x ID"10 sec i 65 « 10-10 sec CHAPTER III I y/ 10 - //t- -**ne"v - - A —• pe-v"—^^^ '"^fcw' I ^ I ^"^ Fig. 10 Constraints on D/F ratio Fr< ;]/f[ alone independent of ; Fig. 11 Constraints on ratio of F and D co-jpling from ratio T(B. •+ B f v) SEMI-LEPTONIC INTERACTIONS The relations between F and D deduced frcm the data for each reaction are shown on Figs. 10 and 11. The width of each strip in those graphs corresponds to ±1 standard deviation around the measured values. The sign of g(/f, when it is not yet measured has been assumed to agree with the Cabibbo predictions. Figure 10 corresponds to the data which are independent of the valje of 6: angular correlations between momenta, and between spin and momenta, spectra, etc. Figure 11 corresponds to the rate measurements with the relevant sin2 0 or cos1 6 divided off. The fitted value sin e = 0.230 : 0.003 (-1.35) has been used to draw that graph. The fit is excellent between the present data and the Cabibbo model- The agreement ma/ even be too good, as the theoretical predictions are subject to symmetry-breaking corrections expected to be "J 10-20Î. The value obtained for sin 0 agrees reasonably well with the values obtained from meson decay Eqs. (3,16) and (3.19). We shall see in Section 4.S.2 that the fitted value p-5-p = 0.6SS ± 0.007 (4.36) is close to the SU(b) prediction. 4.5 Further properties of the axial current •?. 6.1 Generalised PCAC In Section 2.3 we showed that the assumption that the axial current A* is doninatcd by the pion pole allowed us to relate the axial coupling g^ in 3 decay to the ~\. decay constant fn. We may also consider the hypothesis of pole dominance for the divarger.ce of the strangeness-changing current (A|3uu*|B> = nherc g...- is a strong kaon-baryon coupling constant, and M^ is a mass which characterises the variation of the axial forr. factor at low q1: j a j sf (q ) = eî f°)[i • qVM£A + ...] . Mow ruch validity can we attribute to the assumption of pole dominance for kaons? J a) From the point of view of dynamics, the dispersion relation in q' for the matrix elements of 3 ^U has a cut start ing at qJ = (H + 2u)!. Then at q! c 0, the contribution of the cut relative to the pole should be less than compared with the relative cut contribution in the pion case m'/m • ••» • Thus where we might expect a 10Î validity for pion pole dominance, we would expect only •}<)% accuracy for the Kaon. hi Fro-n the point of view of symmetry, wc saw that pion PCAC becomes exact in the lir.it of axial current conservation, which requires a vanishing pion mass CHAPTER III 3MA^ = 0 , U' = 0 . (4.40) Since the pion niass is indeed very small on a hadronic scale, the idea that the current may in fact be nearly con served does not seem unreasonable. Similarly, Vnon PCAC would become exact for aV] = 0 , M* = 0 . (4.41) The kaon mass is not small. However, hadron physics shows an approximate invariance under SU(3) which requires 3V = 0 , M2 = u3 . (4.42) The mass condition in Eq. (4.42) is not much better satisfied than that of Eq. (4.41). So it could be that con servation of v and of a are on an equal footing. If the interpretation of symmetry breaking as measured by meson masses is correct, we expect conservation of A" to be a better concept than either (4.41) or (4.42). Now let us examine the PCAC predictions frr the axial coupling constants [Eqs. (2.36) and (4.38JJ. The meson nasses disappear from these relations B,B, ^PSPM, ,, ,^ Since, empirically, the ratios 1 + Ofe) a.-e all equal to within about 20%, gA as determined by Eq. (4.43) will satisfy SU(3) constraints to the extent that the strong couplings da. A particular consequence of Eq. (4.43) is that in the symmetry limit the F/U ratio for gA must be the same as the F/D ratio for the strong couplings. The experiment?! determination of th« K baryon coupling constants is subject to uncertainties; however, there is some indication [Ebel (1970)] that gj_j. is too low compared with the value predicted using kaon PCAC and experi mental input for fK and g™. As the baryon decay rates agree well with SU(3) values for the axial couplings [that is, a fit with only three free parameters (F, D and 6) fits the data well], this discrepancy would imply a failure of both kaon PCAC and of SU(3) invariance for strong couplings. However» pion PCAC alone is sufficient to detennine both F and 0 by cccfcining the Goldberger-Treiman relation which determines -gJP - F • D - 1.250 : 0.009 with the analogous relation for £A decay The expérimental fit to baryon decays gives the values (sec Table 1) F " 0.427 : 0.012 , D = 0.823 * 0.012 . (4.4S) Thus we predict from (4.44) 12.0 t 0.2 *KM Est;.natcs of the coupling constant extracted from strong interactions lie in the range [Ebel (1970)] 11.S < g,,,, < 16.2 . Finally, we remark that while meson pole dominance is compatibic with SUCS) invariant couplings [Eq. (4.43)J in spite of the large rK mass splitting, the predictions for the q1 dependence of the scalar fom factors at low q; «•HM««MWU*«| SEMI-LEPTONIC INTERACTIONS gj(q*) ^ 1 + flj. , AS = 0 a % 1 ss(q ) + rf . AS- i . are quite divergent. A straightforward application of SU(3) would require them to be the same. 4.5.2 The determj&tion of F/D: SU(6) •n the previous subsec'ion we related the F/D ratio of the axial currents to the F/D ratio of strong couplings. Can theory say anything about this ratio in icself? Cabibbo theory alone cannot, but we might consider the quark model which provides a mathematical basis for Cabibbo theory (and also for current algebra). In the non-relativistic quark model the baryons are formed by three two-component quarks in a state which is com pletely symmetric under exchange of spin and internal quantun numbers. It follows that the spin-'/i baryons along with the spin-Vi resonance decuplct form the representation 56 of SU(6). In the non-relativistic limit the vector current in baryon decay reduces to the Fermi operator [i = 1 8): v\ = qve 4 n = i+-r q «•«) and the axial current reduces to the Gasww-Teller operator (i = 0, 1, ..., 8)*' 1 A = qrYs -»r 1 = The 8 operators t-l.loj and the 27 operators (4.47) are the 3S generators of infinitesimal SU(6) transformations; their matrix elements may be read off using SU(6) Clebsch-Cordan coefficients. One finds D = 0 for the vector matrix elements since they are determined by the SU(3) subgroup of SU(6) and [Cursey (1964), Feynman (1964)] Jj.f or a = pfj • 0.6 (4.48) for the axial matrix elements. This prediction is in fair agreement with the value a = 0.658 ± 0.007 .. (4.49) extracted from the fit to hyperon decays, Eq. (4.36). In this picture one also obtains for the neutron 0 decay parameters gv = 1 , gA -- | - (4.SO) The value for ga/gv is rather high compared to the measured value. This can be understood in terms of a «normalization of the quark axial coupling since the operators (4.50) are not (and cannot be in any relativistic theory? tune indepen dent, and therefore do not correspond to a conserved quantity. In fact the success of the prediction (4.46) is much harder to understand theoretically than the success of the SU(3) predictions or .hose following froc the charge algebra. 5. THE ALGEBRA ÛT CURRENTS In Section 3.S we noted that the postulates of Cabibbo theory follow immediately if we express the hadronic cur rent as a V-A coupling of quarks dj - ty/l • Y.)qn - In this model the full octet of vector currents are defined by i - *i Vj(x) » q(x)Yu -y q(x) (5.1) and the octet of axial currents by i SU(3) singlet; *g 5 /Î/J is proportional ro the unit matrix. CHAPTER III AJ(X) MWY^jqW i (5.2) where the matrices *i (i = 1, ..., 8) form a complete set of traceless Hennitian 3" J matrices (see Appendix) which satisfy the confutation relations Oj.lj] - 2ifiJK»k • (5.3) lhe quark fields q(x) are 12-component objects; they have four Dirac components and three SU(3) components (q.^.q,) - (yv^ • fS.4) Among the 16 currents dsTined in Eqs. (S.r and (5.2) those that have been observed in nature are the electrcoagnetic current *F ' the weak current .,,S „ ,ii , ..5^ and its Menait tan conjugate j[j . Using the canonical commutators of the quark fields (^w.vw)i(x» - *>= V%w'«tx. - y») = ° (5.7b) a,6 = 1, 2, 3, 4, a,b = 1, 2, 3 and the commutation relations (5.3) of the ' matrices, the equal time comnutators of the currents are determined to be [Getl-Mann (1962), (1965)] E>î(x),vj(y)]5(x0 - y„) = [AJCX),AJ(Y)]5(X0 - ya) = ifijkvj(x)6-(x - y) . (5.8a) £vj(x),AjJ(y)]S(x( - y0) - [AJ(jO,VjJ(y)]6{xfl - y,) = ifijkAj(x)^(x - y] . (S.8b) Independently of the quark model for hadrons, we may simply postulate the commutators of Eqs. (S.8). They can be derived in many other field theory models. To what extent does this postulate go beyond the original postulates of Cabibbo theory? a) The assumption that the charges associated with the vector currents generate infinitesimal SU(5) transformations iisplies that they satisfy the SU(3) charge algebra (5.9) [VFj]cVt' where F. = Jd'x v}(x) (5.10) is a time-independent operator in the 5U(3) limit. In many models of symmetry breaking, fy. (5.9) still holds if F. (x0J and F.(xa) are taken at equal tines. This assumption requires that the vector-vectoi commutator of tq. (5.8a) be true up to terns which vanish under space integration [Schwingcr terns"*']- b) The assu=!ption that the axial currents transform according to the octet representation of 5U(3) implies [FpAJM] - iCijk!$W (5.11) •) The neutral current observed in neutrino experiments with nS = 0 is probably different from (5.S), but its properties with respect to parity and SU(3) have not yet been determined. • ) Such terms are knam to be present in the cccaitators [Jj.Jj], i = 1, 2, 3 [Schwingcr (1059)J; they will not be relevant to our applications of current algebra. SEMI-LEPTONIC INTERACTIONS in the SU(3) limit. The assumption that this relation persists at equal times for a broken symmetry gives a pre cise meaning to the octet assumption. Then the [V0,Ap] coraiutators in Eq. (5.8b) must be valid up to Schwinger A terms, and the [ 0»VU] commutators are determined by Lorentz covariance. c) The assumption which really goes beyond Cabibbo theory is that for the axial-axial commutators, which serves to define the normalization of the axial current. Under this assumption the current algebra closes to form the alge bra of chiral SU(3) & SU(3). In the following subsections we shall discuss some icroediate implications of the current algebra. 5.1 Universality We may now define the concept of universality independently of a specific Lagrangian model. Let us define the conmitator of the weak current with its Heimitian conjugate [JD0O.jJ(y)]5(x0 - y0) - -l"J«M.Ju(y>>(x0 - y0) ; Zjjs"(x - y) . (S.12) Using the commutators (5.8) and the definition of the weak current (5.6) one verifies the relations [J>)»yy)]6(x0 - y0) = JU(X)6*(X - y) (5.13a) [j;Cx),J+(y)]5(xe - y6) = -Jj(x)5'(x - y) . (5.13b) The three currents J„, jj and J' satisfy a closed algebra which is an SU(2) algebra. This is not surprising since the weak current J„ may be obtained from the isospin current J' + id* by a U spin rotation in SU(3) space under which qn •» q'. The commutators arc invariant under SU(3) transformations. The leptonic currents satisfy the same algebra, cs can easily be seen if we note that the lepton pairs (ve,e) and (v ,ii) may be considered as spinors in some SU(2) space. Then the loptonic currents may be written in the form nf I spin currents :W + ' LYU(1 + Y,)T L , L = (^) . Thus universality may be formulated as the requirement that the hadronic and leptonic currents satisfy the same algebra. This defines the normalization of the hadronic currents, since under a change of scale J * AJ , the comnu- 3 tator in (S.12) changes: J' •* A JU, and Eqs. (5.13) are no longer satisfied. 5.2 Non-renormalizatlon theorems for vector charges The charges associated with the strangeness-changing currents [Eq. (3.8)] satisfy The relation + _ [s (x0),S (xo)] = F, + /ÏFB = Q • Y , (S.IS) where the right-hand side is tirce independent, since electric charge (Q) and hypercharge (Y) are conserved. Sandwiching Eq. (S.ISI between single-particle states [o) of nauenta p and p' gives the sum rule Z |(o|S*|fl)[3 - L |(a|S-|6')|3 = 2fc (în)1^ • YJ5(p - p') . (5.16) 0 0' In the limit of SU(31 symmetry the charge operators have non-vanishing matrix elements only between nembers of the sane SU(3) rultiplct. If, for cxairple, we take a to be the neutron there is only one non-vanishing term in the sura (5.15) and we obtain 3 E 3 j'^T0^yT(n(?)|S*|E-(£)}{r(k)|S-!n(j5')> = 2E„(2") Cg^ ") *Mp - p*) - ^(Q,, * Yn)(2i) 6(p - p') (S-17) The corrections to this result are of two kinds. One is kinematical: since the masses are different when the sym metry is broken, the mm^ntum transfer in Bq. (5.17) is generally non-zero. We may eliminate this difficulty by taking the neutron momentum to infinity. For p = E the moaentim transfer is (p - W|i - (Ea - E,, . 0, 0, 0) -{^T^ . 0, 0. o) ^ 0 . (5.19) The second correction is dynamical; "off-diagonal" or "leakage" matrix elements of S* which connect the neutron to non- octet states contribute. However, since only the squares of these matrix eleaents enter, this correction is second order in the symmetry breaking [Rjbini (1965)]: {gf}2 -1*0U*) (5.20) so that the correction 6g s <5(g!)/2g to '" vector coupling at zero momentum transfer is of second order in the symmetry breaking [Memollo (1964)]. The same reasoning may be applied for K„ decay. If |a> is the n+ state, only the K° intermediate state contri butes in the symmetry limit and we obtain <71+( S+ F( >( )|S |Tr+ P = 2E S( 2 / 2E(C(*0* ^! l ^ ^ " ^' i. 5 " qOtfiOW * Yn+) + 0(e ) (5.21) and in the limit q = k •* » we obtain for the matrix element at zero momentum transfer f+(0) = 1 * O(e') . (S.Z2) Let us consider in more detail the sum rule for K. decay. The normalization condition, Eq. (5.22), was obtained by evaluating the .sum rule (5.16) for infinite momentum of thx external state and separating out the state which is allowed in the SU(3) limit. For arbitrary momentum transfer the charge matrix element is given by + <*-(q)|S |K*"(k)> o (2:r}i«(q - k*)[ft(t) (q, * kfl) + f.(t) (kfl - q0)] , where q = k implies (k, - %)' , K-«l > q. »' (i-(q)|S*|K'(k)> = (k„ • q0,HZi)'«(q - k)f(t) , where f(t) is the form factor for the divergence matrix elements, Eq. (3.12). The sum rule (S.16) may now be written [ ïq.œ'CqKZnJ'Sfi - q') • E_ l<"*|S*|B*>|' - Z l<"*|S-|B>|! - 0(e!) . B'^K» B For 0 t A - ™ the momentum transfer varies over the range (M - u)1 - t I 0, which covers the entire decay region. Thus by studying the behaviour of f(t) we may determine the contribution of the SU(3) forbidden matrix elements as a function of 8;. The kinematic factor in Eq. (5.2Ï) is also a seeot.J-oroer correction since in the decay region: SEHI-LEPTONIC INTERACTIONS - -f 0 as M - -j However, this correction becomes as large as ZQ\ for t -* (M - u)2. One may then wonder whether 5GJ should be expected to be small. {The analogous kinematical correction for £ decay is [(M_ - NLl/CM- + *K^¥ * l-^t.) In principle, the sun rule (S.16) allows a test of the current algebra hypothesis which is assumed to be exact in the presence of arbitrarily large symmetry breaking. One may also derive more general sum .-ules using the commutators of current densities, Eqs. (S.8). In practice, these tests are very difficult to realize. They could be done in neu trino experiments, provided it were possible to identify strangeness-changing events [which are suppressed by a factor (tan S)1 * 1/25]. The same ideas nay be applied to the evaluation of radiative corrections tc the vector coupling constant in 6 de cay. Although vector current conservation is violated by electromagnetic interactions, the equal time commutator of the charges r = F, i iFj remains time independent [I*.I ] - 21, - (5.24) Sandwiching the cannutator between proton states and separating out the neutron intermediate state, one obtains the sum rule 2EC2^*6(q)|gPn|a - ZE<2n)»6(5) * £ l(tt|l+|p>|a - £ l(0|I"|p)|! • The matrix elements on the right may be written s S| «.|[i ,n]!p) where H is the total llarjiltonian. Neglecting higher-order weak corrections, the only part of il which docs not centime with the charges T" is the electromagnetic interaction iPfx.) • ejd'x j™(x)au(«) , where a is the electromagnetic field operator. Since tlie total Hamiltonian is time independent, we ^->y write ! em ! [•'(x.l.llfr.jj - [l (x0),H (x„)] - e/d x V*(x)a"(x) . llie last equality follows from the cctmitation relations (2.8) and the fact that ^ commutes with hadronic operators. To lowest order in e, the operator (5.25) connects the proton to states containing one photon: a = a' + a So we have the equality EU(,i'|V*(0)|p> (aMrlp) • e/dW* ^^ = ,2^ J The matrix clement on the right-hand side of (5.26) is the hadronic part of the amplitude for inelastic vp scattering (with aS = 0). Therefore, the lowest-order corrections (one-photon exchange) to the C'C prediction can be directly related to neutrino-proton total cross-sections. 5.3 The Adlei'-Weisberger sum rules; normalization of the axiaT current The equal time commutator of the US = 0 axial charges 3 l±s) = Jd x (h\ ± ÏAÏ) is identical to the commutator (5.24) of vector charges: Sandwiching Eq. (5.28) between proton states gives the sura rule 1 r ElMi^jlU! - El One contribution is from the neutron intermediate state 2 r^TTTtPtP'JU^jlnMXnt^ll^jIptp')) - UiO^tp - F')4upY0YsW + M)Y€YSU • 2LXZn)>6(p- - p-'lsjv , where v = p/E is the proton velocity. This is directly measurable in B decay. The remaining terms are related to vp cross-sections as discussed by Adler (1966); this will be presented in more detail in Chapter VI. Since Eq. (5.29) is inhomogeneous in the squares of charge matrix elements this relation determines their over-all normalization. However, with the limited neutrino data now available, a direct test of the current normalization is not possible. In order to give predictive power to the sum rule (S.29), the additional assumption of PCAC (or PDDAC), introduced in Section 2.5, is used. Then the matrix elements of the axial charge can be related to rp scattering amplitudes: i(E l 5^ '"I'i.ji'Wo " P - "V^'M > "I*'* (pi^a.om, - wtG - Pi)fy ./.""(p'.'p)- • A description of the way in which the sum rule is implemented will he given in the chapter on neuli ino interactions. Here we only state the result 1 ^/^KV- where o (qo) is the total pn cross-section at laboratory energy q . Since fn and p. are linearly related by the Goldbergor-rreiman relation, we may eliminate one to obtain • prediction for the other. Using cross-section data, Adlcr (196t. i and Neisberger (1966) found in independent calculions |gAl = 1.24 and 1.15 , (! respectively, in striking agreement with the direct determination, Eq. (2.38). Several authors have also evaluated sum rules of the Type (5.29) otitained from the commutators of strangeness- changing axial charges tS(„.S( ,]-Q*Y : i /dJx (A* t iA*) s 10 Assuming fcaon pole dominance and using cross-sect:„n and resonance data and sane SU(3) approximations, the following results we i" obtained: |gjj°! - |F + D| • 1.23 i 0.10 [(Veisbcrger (I960)] { 0.7S t 0.L0 [l\'cisberger (19fa6)] 0.75 [Arcati (196?)] 0.63 [Levir.scn (196S)] . S.-MI-LEPTONIC INTERACTIONS In spite of the questionable validity of Kaon pole dominance, these results ?re in surprisingly good agreement witt the values obtained from decoy rates, Eqs. (Z.38) and (4.49). It would of course be desirable to have a direct test of the connutators fron neutrino experiments, rather than reiy on th" pole dcainance assumptions. 6. FORH FACTORS IN KAON DECAY The algebra of currents provides an elegant framework in which the postulates of Cabibl» theory and the principle of a universal V-A coupling take on a well-defined meaning. However, a decisive test of the current algebra requires a complete knowledge of nuetrino-nucleon total cross-sections. Nevertheless, the normalisation of the axial coupling was successfully determined through the use of the suppleaientary hypothesis of pion pole deninance. In this Section we shall discuss further predictions from the combined postulates of current algebra and PCAC which will be seen to constrain the behaviour of form factors in semi-leptonic kaon decay. Consider the general matrix element: M(q) = i fd3X elqX (A|TOPAj; where T indicates the time-ordered product: T(A(x),B(y)) = Q(x0 - y0)A(x)B(y) • e(y0 - x0)B(y)A(x) (6.2) anjy'(O) is any local Heisenberg field operator. M(q) is the full amplitude for the transition A where momenta q^ and p„ = (pa - p^ - q) are carried by the fields a^A* and '-£, respectively, as illustrate in Fig. 12, where the shaded region represents a sum over graphs contributing to all orders in the strong interact!1":'. Fig. 12 Diagrammatic representation of Fig. 13 Pion pole contribution to matrix matrix element o£ a time-ordered element of Fig. i2 product The amplitude M(q) has a pole at q2 = uz = nri, illustrated in Fig. 15 with residue 2 Rn = M(M) = . , + M'(q) , (b.4) l.2 - qJ whore M' is finite at q1 - |iJ- The PCAC assumption is the assertion that the non-pole part is negligible for low q! R. 1 M(q) = ; ; _ ; , 0 S q' < u . (b.5) The predictive power of PCAC lies in the fact that the algebra of currents allows us to cvalulate M(q) at q, = 0 when /(O) is a current operator. We may rewrite ?l(q) by performing a partial integration: CHAPTER III u 1 1 e^ H*t)>\M - ^V"* H*,»\Ml - iq e(xc) e ^ A^ - [JL 8(„()] e ^ ^ . As £ 1M - - A 8(-t) = 4 lqx M u l Then we may write (the last term vanishes upon integr'*:-f-n) : M(q) = -iq^fq) - iC(q) , (6.6) where 3qX Mp(q) = - i/d*x e C(q) = /d3xe-i^^(A|[Ai(-t,0),^(D)]|B). Now consider the linit a •* Û. The first tern on the right in Eq. (6.6) will vanish unless there is a pole in M(q) at Q = 0. This can happen only when the oirrent operator A* acts on an external line (Fig. 14). The processes which we shall consider in this section involve only pseudoscalar mesons as external particles. Then parity forbids contri butions of this type, so we nay set lira q^yq) - 0 . (6.7) Evaluating the last term in (6.6) at q * 0 gives lim CQ) = f d'x 1 M(0) -4" fn If y*(0) is a weak current JJ ' VJ • AJ , P U U commutation with I, * gives an iscspin partner of that current since a V-A current has the sane properties with respect to vector and axial charges tijo-ii-cviJ- (6.10) Fig. 14 Possible singularity arising in the soft pion limit SEMI-X.EPTONIC INTERACTIONS Using this property we are able to relate a process involving pions K * tat + £v to a process with one less pion: K - (n - l)u + Ev . 6.1 K£ form factors Since the form factors for different decay modes are related by isotopic spin and charge conjugation, we need only consider one decay process; we take K7 + n*î"v. Furthermore, if the hypothesis of u-e universality is correct one does not expect differences in the form factors involving the two kinds of leptons I. Then the soft pion theorem, Eq. (6.9) gives the relation + B 3 : (T (q)|v*jK (k)) = ku[f+(M ) + f.(M )] * "r where <•[';,]•«;] <6-12> is the isotopic spin partner of the iS = 1 axial current. Then, we have for the matrix clcr.cnt z + f f„(M ) f_(M*) = fK/ n • ^. 14) The prediction for the form factors is at an unphysical V3lue of the Cionentura transfer: t = (k - q)2 = k1 = M3 for o=0. Then, how can it be tested? Ke expect the form factors to be analytic functions which satisfy dispersion re lations. Sir.ce the current is not conserved, both spin 1 and spin 0 states can contribute to the integral over the absorptive part. Therefore, it is convenient to consider combinations of the form factors which arc anplitudes with well-defined angular monentum. Consider the transition amplitude in the lepton centre-of-mass system: U + 5 î V U + YS)VV(TI !V*|K "> = -Û£Y(1 + vï)vvU*|v*'|K > + Ûzy0{l * y^v^^'K*) . (6.15) The lepton matiix elements on the right-hand side correspond to total angular momentum one and zero, respectively. Therefore, the amplitude for spin 1 exchange is proportional to f+: 5 (T/IV^IK ") - (k + ajf+(t) , (p = 0) (6.16) and the amplitude for spin 0 exchange is given by the fom factor of the divergence matrix element, Eq. (3.1Z): % - E, h - \ where for p = 0, t = (Ej; - Ej,)*. It is clear in fact that matrix elements of iS'Vy, which is a Lorentz scalar, can have contributions only frca spin 0 exchange. Since the total tacnentira of the exchanged state is p = k - o , the pure spin I amplitude must satisfy PA'"' • « "£"'' - WvvlK> - Mï ,„|»V|K) • f.(t)[(k • „)u - «L-Uii pJ . «,.1S, The effective mass of the exchanged states which contribute to the absorptive part is p2 = t * (M * u)1. Since the amplitudes ft(t) and f(t) are governed by different angular momentum contributions, they satisfy un coupled dispersion relations and are expected to be smooth functinns in the decay region CHAPTER III m* < t s (M - u)2 = V . The cut is over the region t > (H + u)2 B Zlu2 . and the soft Dion prediction (6.14) is for the point t = M2 » 13|i2 . The variation of the form factors below the cut is expected to be roughly determiried by the masses of the lowest lying exchanged particles f+(t) * f+(0)[l + t/M^ + ...] (6.19a) f(t) = f(0)[l * t/M* + ...] . (6.19b) Note that f(0) = ft(0). The experimental determination of f (t) and f(t) will be discussed below. In any case, one does not expect a rapid variation of the form factors in the decay region. NON let us consider the divergence form factor at the point t = M3 f a) ore expects an error of order u2 (^ ICI) in the predictions of PCAC; bj the form factor f (t) is determined independently in K. decay and is compatible with the form 2 f+(t) = f+[0)[l + ^t/u ] , X+ = 0.03 the quadiatic term in the expansion being quite snail. Then extrapolating to t = M", and using the experimental ratio (5.20), Eq. (6.14) predicts M!!l«i. Therefore, if PCAC is correct, the difference between (6.14) and (b.lû) should be considerably less than lOi, the expected accuracy of PCAC predictions. 5incc f+(D) = f(0), we now obtain the prediction iflgl- fK Q l.W » 0.03 + cf-U . (6.21) f(0) f^fjo) UnJ l ; If f(t) is a nor.otonic function below the Kn scattering threshold [t = (M •* u)*], as expected, it must have a positive slope. In a linear approximation one obtains f(t) = f(D)[l • *.t/u}] . *„ " U-D? - (.b.22) The experimental test of this prediction is discussed below. Its importance will he tliscusscd more fully in Section 6.3. C.I.I Définition of the experimental pavameterB i) There is nc compelling evidence, given in the experimental data, for a t dependence more complicated than linear. So it is reasonable to expand f+ and f_ according to f+(t) = f+(0)(l * A+i) f.(t) = f_(0)(i • ^-jr}- We vill not consider the possibility of quadratic terms which is discussed elsewhere [Chounet (L972}]. SEMI-LEFTONIC INTERACTIONS One usually introduces a new parameter f (t) A simple variation is also assumed for Ç(t): Ç(t) = Ç{0) + A ^ . This last hypothesis is compatible with a linear variation of ft and f_ only if \+ proves to be small, which is the case. Then the following relations •* Id: f.CO) at» = A = £(»)(*_ - A+) - ii) The fora factor f{t) is ex)>anded in the same way f ft) = f(0)(l * \ ^-) - But here a linear dependence for both f and ft necessitates A, = 0. It is easy to check that A_ induces a quadratic term in the variation of f. From the definition f(t) = f+(t) * Hi \ 2 f_(t) , one gets 2 ;u; - M* - u ftt) - f»(t) 1 MO The snoothncss of Ç[t} at the origin requires f(0} = Mnh which is compatible with experiment. Expanding f(t) and f*[t) one then gets v2 1 • *+t/u2 A = -CC0J** • In sumnary, limiting oneself to linear variations of the form factors, one has four quantities to determine: M°) . f_(0) , A+ and A_ . An alternate choice is given by the following set: n MO > U ) > A and A+ . 6.1.2 Experimental techniques Sow the problem arises as to translating the theoretical expressions into measurable distributions. The quantities available to the experimentalist are Tho decay rates, the energy spectra, and the lepton polarization, i) p^ça^_ratç_rççasurçments Integrating over all possible configurations the squared matrix elenient comnartding the decay, one obtains the total decay » ite. For instance the rate for IT •* n°e"w is given by J r(K± j = G sin' 6 [f (0)1^(0.573 + 0.13BAJ . One sees that ff[0) appears in the expression through the combination f+(0)sin9. The knowledge of the Cabibbo angle is still uncertain and so the absolute scale remains unknown. Measurements of K, fom factors will give only the relative nagnitude of nroccsses together with t dependences, and we are t?fi with only three parameters for further investigation C(0) . A+ and A . CHAPTER III ii) ?Mitzjjlgt_analyjis The Dalitz plot is the distribution of K. events in a two-t unensional histo^ JII (E ,E.), the energies being measured in the K centre-of-mass system. The predicted density of events is easily computed, and we find in terms of f+(t) and f,(t): T^-- |f Ct) sin e|a[A + B«t) • CÇ2(t)] , dhj-db; + where A = -2»K(EJ * E„Ee) • Z(n£ - „•)£, • («> • | »;}B, - [mj - ^) E^ - r,^ L E* ES^l B - -n,[E£ * -j- - RK * -j-J C-^(E, -•£•*). E^3* is the caxixtian energy available for the picn in the K c.n. systm. In the case of K decays, the mass of the lepton being negligible, the expression simplifies to A = -2mK(E* * E^) * m^(2Ee * EF - jf") B • C = 0 . The Dalitz plot density docs not depend on Ç and apart from f+(0), only one parameter remains, A+. For K decays, the mass of the lepton can no longer be neglected. The expression remains complete and the study of the density allows, in principle, the measurement of £(0), A and *+. iii) Lçgton_p£larization_measurement A A muon polarization transverse to the plane pb Pn would be evidence for time-reversal violation in this process. No such polarization has been experimentally observed. One finds that the muon is completely polarized along a direction depending on En, E. and ç.'and assuming T in variance, one gets the expression E m ^ ° " Ail»n, - l_mf~ j B * ni~g |pJT^TTJ ( t„ " *V> + (%h ~ ETt )J J1 l,Uh A =• mJmjJ - mKE,j(l - Ç) * ^ (1 - O*] c B m£mK[(l - OfE^ - ig - 2EJ . Measuring the direction of J' allows one to determine £(Q) and A. in) rifv,)/! tKel) ratio The integration of the density of events over the Dalitz plot gives the total rate for K„3 processes, as was seen in the case of K by neglecting the terms induced by the mass of the lepton. The unknown scale f+(0)sùi8 drops out in a ratio of rates. Furthermore, u-e universality is assumed so that the same parameters appear in K and K , and we get the following numerical expressions: TOC,) 0.646 + 3.801A+ • 6.812** + 0.127Ç(Û) • 0.476t(0)*+ + 0.019£(0)* rfK*j) ° 1.0 + 3.700*+ + S.478A* r(K^) _ 0.645 * 3.5J6A+ + 5.932AJ • 0.I2SÇfO) • 0.4375(0)*+ * Q.019Ç(013 . B = "(K () 1.0 • 3.457).,. * 4.779*J SEMI-LEPTONIC INTERACTIONS The measurement of the ratios provides a quadratic relation between £(0) and A+, allowing the determination of ç(C) when one assumes \+ to be known. he summarize the different neans of analysis in Table 1. For core complete discussions the reader is referred to Qiounet (1970, 1972). Table 2 Methods Parameters Dalitz plot in X X ftj * Dalitz plot in K *+. ^t°l. * Lepton polarization ao), A rtV/r(1W relation [^(0), »J An c ample of K°3 Dalitz-plot analysis i) Wc SLW that the study of the K Dalits plot provides a means of measuring the slope parameter A+ of f+(t). Let us examine in some detail the measurenent of A+ in the CERN-Heidelberg experiment [Eisele (1973]]. The apparatus allows the selection of the decays K° * nev by signing the electrons with a Cerenkov counter and rejecting the muons with a special hodoscope. A spectrometer using proportional chambers measures the momenta of the charged particles TT and e, allowing one to enter each event in the Dalitz distribution (E^.Eg) or equivalently (TF)Te) where T is the kinetic energy. ii) Actually the experiment uses an incident K, beam of unknown momentum. As the neutrino escapes detection, the reconstruction leads to a two-fold ambiguity in the K|_ momentum, corresponding to forward and backward emission of the missing neutrino. Two solutions may be adopted to solve this problen: - One event can be entered twice in the Dalitz plot with weights for both solutions given by Monte Carlo technique. This method needs an accurate knowledge of the It momentum spectrum. In this experiment the spectrin is found by studying the K° - T*V~ decays. - Another possibility is to keep only events for which the two solutions in Tïï differ by less than a reasonable bin size. The resolution in T„ is 3.2 MeV and the bin size is chosen equal to 10 MeV. About 195,OOD events verify this criterion, out of a total sample of about 450,000 events in the Dalitz plot. iii) tfe saw that the Dalit2 distribution is of the fom jjgj.A.taelr.CJI». where A depends only on kinematical factors. To find the experimental distribution one must include the detection ef ficiency of the apparatus e 2 N(T„,Te) « A sin B |f+(t)i e(Tn,Tc) . Any mistake in the Monte Carlo program would directly affect the expected distribution and would induce a false t dependence for f+. Instead of fitting the total Dalitz plot, one studies the projection over the Tn and Te axes: - the electron spectrum, for constant T„, allows Che measurement of the radiative corrections which affect :hc process; CHAPTER III Hlr" Weighted solutions ^ • .•»* ^ .•fi l*'\ \y HfA > t,f irf f 0.04 006 0.06 t (GeV)J Fig. 16 Detennii i o£ f+(t) by the CEKN-Heidclbecg experiment [EÎE (L973)] - then knowing these corrections for each T^ bin the pion spectrum is used to measure the slope of the form factor. Indeed rewriting t, one sees that the momentum transfer depends linearly on T^ t - 2 = m ra ! " (PK Pvt C K " flJ " ^V* and the projection of the Daliti plot onto the Tn axis gives directly the t dependence of f+. The experimental distribution is compared with the expectation obtained in assuming a constant" f+ U+ =* D) in 25 bins or T_. Also here one must rely on the Monte Carlo prediction. The t dependence is then extracted by (At) . /555 V N(Tfl) Figure IS shows this variation for the two methods referred to. A linear dependence of f+ gives a good fit to the distribution and the result reads 1+ = 0.031 Î 0.002S . 6.2.3 -.atua of the expérimental recuite he will sketch the experimental situation by listing the different results obtained from the different ncthods. [Data are taken from Chounet (1972), Winter (1972) and Pascual (19721.] i) Mçasurçn;ent_of_*t s Figure S shows the time evolution of the A+ measurements both in the K and in the K decays. The compatibility between the early results is rather poor; however their average: \+ = 0.032 t 0.0054 . is in .-igrccecnt with the high statistics result already quoted: At = 0.P31 ± 0.0025 . The discrepancy between K results is less pronounced and the world average reads; SEMI-LEPTONIC INTERACTIONS •• 0.029 ± Û.D05 . Also K decays allow one to measure X+, but here A+ and f,[0) appear to be strongly correlated and one has to be very cautious in discussing the results. A new high statistics K° experiment has been published lecently [Donaldson (1974)]; its result is X+ = 0.030 ± 0.003 . ii) t^asûrement of_££0j The only method giving reasonable data is the analysis of the Dalit: distribution for K.j5 decays; £{0) depends very decisively on A+ and different results for A+ give striking discrepancies in Ç(0). As results are incompatible one can only say at present that the tendency seems to give -1 < £(0) < 0. A high statistics K° experiment [Donaldson (1974)] gives A„ = 0.019 ± 0.0O4 , which is equivalent to mf - m2 K C(0) = , - (*D - K) = -0.11 • m_ £..'-•/ 'c.-.parinon uith theoretical predictions Several predictions discussed earlier are subject to experimental tests. O Simnle_SU{31_Erediçtions Currents responsible for the different Kp3 decays belong to a common octet of SU(3). As discussed in Section 3.3 this predicts the relation between form factors: _ . - Mt)[K$,] f±(t)[Kt*,]- - y . We found an excellent argeement between A+ measured in K° decays and in K* . Now assuming A+[K* ] = At[K°] the com parison of the decay rates yields IMoiugJl =0.957i0,051 l^(Q)I^Ql = 0.984 ± 0.078 . The near equality of slopes and the compatibility with 1 of the above ratios strongly supports the seie-iion rule UlJ = % for leptonic decays. Furthermore, the comparison of slopes for K° and K° does not contradict the idea of L.-C universality. If SU(3) were a good sy=nctry we should have f,(0)[K«J - 1 , f.(t) <= 0 , £(0) * 0 , f(0) = fJO) . The magnitude of Ç can be considered as a measure of the breaking of SU(3) symmetry, but here the experimental situa tion is still too unclear to reach any fim conclusion. îi) ÇiSKE5J9D,r£l3ÎÎ9D5 Dispersion theory is useful Tor describing certain properties of form factors. Intermediate states appear through pole terns and their contribution can be explicitly evaluated. The pole for the lowest particle that can occur cay dor.mate the behaviour of the form factor, if its position is near the physical region and the continuum sufficiently far away. The general form of an unsubtracted dispersion relation is f(t) = I f J» f(f) dt' . » J t' • t - ic The absorptive parts of f+ and f are determined, respectively, by spin 1 and spin 0 intermediate IT-K states: f+ and f satisfy uncoupled dispersion relations. p Experimentally there exists a resonance X with spin parity J = 1" whose mass is mt = 890 MeV. Assuning pure K (890) dominance the corresponding pole saturates the dispersion relation for f , while f has no singularities and remains constant. One obtains f+(t) - f+(0) ~^- * * P; - t nï m, - t giving ni ^ - X. * ~ tn; c(oj = - mK ~ "*" . If one assumes the existence of a K state of mass Di+ with j" = 0* one can also assume pole dominance for f and obtain f(t) = f+(0) -^- , ml ~ ml m* - ml f.(t) = -MO) * _ —. - * m; n« m* 5(0) = K-OK -O _ In this case C(t) will always be citht positive or negative in the physical region, depending on the mass mt compared with a.. Putting the cass n, - 890 MoV, the prediction of x" dominance alone reads: \+ = 0.023 U0) " -0.284 These predictions are not far from the experimental results, which actually can be analysed in terms of thr.- mass of an effective K*. For instance, the ŒRN*Heidelberg [Eisele (1973)] result would give a. - (840 * 3S) McV . If one also assures the existence of 3 * pole with n+ = 1.6 GcV, one gets \„ - 0.036 C(0) - -0.116 . SEMI-LEPTONIC INTERACTIONS The experimental results are too etude to indicate anything concerning this assunption. iii) The soft pion relation has been derived above f+(oj[) • f_(m£) = |£ • Not only is this relation valid in the limit of massless pions, but it gives a prescription at an unphysical point in Romentun transfer t = ml and the extrapolation to the physical region may be uncertain. The relation is even simpler in terms of the scalar form factor f f(n# = f>£) ! f-K) It is the spirit of PCAC to neglect ml/ml. The" f(m*) = f+(nj>) < f,(m*) RemeciDering that f[0) n f+(0) the Callan-Trciman relation gives f at a second point in t, allowing one to determine the slope of f(t] if a linear interpolation between t = 0 and t * m£ is justifiable. Experimentally one obtains the result ^= f+(0)(1.27 ± 0.03) , rf(ra£) l mi ni a Fixing Xt 0.031 we also get a value for £(0) £(0) » -0.117 . The experimental situation concerning the soft pion relation has been confused. The high statistics K° experi ment of Donaldson (1974) tests this prediction and the authors present an extrapolation of these data to the soft pion point. This is showi in Fig. 16 and one can see that the agreement is astonishingly good. w Collon-Treiman point §1.2 2 '--W+V****^*" i.o| 0.8- I I I I I I 0 12 3 4 5 6 -^s t/mi Fig. 16 Experimental doterairu of the forra factor HO [Donaldson U974>3 b.Z Keit decay Both the axial and vector currents contribute to the decays K + imev. We write the general form of the matrix decent for K* •* r+n~ev as: + <"*Cq*)"-Cq-)l^(OJ|K M> = ^{(q+ + qJpF • (q+ - qJuC * (k - + v D t ( where by convention i The form factors F, C, r and h ore functions of the invariant variables: 2 2 T = (4* * qJ . S = (qt + q_ - k) , t± - (k - qj* , three of which are independent: (6.25) The form factor r cannot be 'etermined in K decay as its contribution is proportional to the lepton mass. The experimental analysis of the remaining form factors is simplified if one assumes that only the lowest partial waves, s and p, contribute. To separate out different partial waves we examine the momentum dependence in the dipion system; 4+ = -q_ : q( E+ = E. = E. In this frame we define Qv as the angle of the relative pion momentum with respect to the kaon line of flight. Nc take as independent variables: A = t, - t_ = -4q*jc = -qk cos en S^ = 4£* = 4(p* * q2) (6.26) s£ = M* - 4k0E + s„ . Then we may expand the form factors in powers of cos G„ as F(5q,5£,A} =• fs(sn,Sj) + a cos 9 fp[sn,sE) + ... C[s„,sK,fl) = gpt^.Sî) + ... (6.27) h(s.,se,A) = hptSn.Sj) + ... , where î a • {(I - 4nVsff)C(M» - S„ - sl) - 4s„sjl /Z (6.28) la a potential barrier factor which ensures that the p-wavc amplitude becomes proportional to |q"| as 5 * 0. Note that while the leading term in the expansion of F is an s-wavc amplitude, the leading terms in G and h are p waves, as a linear q dependence is contained in the kinematic factors: (% - U euvcok (q» * 9 f K, - f - p •fip , h = hn where the partial wave amplitudes are those for the T*TT" mode, Eqs. (6.27). Just as soft pion theorems relate K-, form factors to K, at the unphysical point t » M*, K, form factors can be related to K. at unphysical points for the variables (6.24), These relations will be discussed in the following paragraph. The soft pion limit, h-wevcr, sava nothing about the vector form factor, since the matrix clement (6.23b) vanishes when cither pion momentum vanishes SEHI-LEPTOHIC INTERACTIONS 6.2.1 Soft pion theorema Applying the results of Eqs. (6.1)-{6.10), we obtain the following soft pion theorens for the axial vector Matrix elements : + (n ,r|a-m ^t0 ^ The matrix cliuent on the right-hand side of Eq. (6.Z9a) is related to the K. amplitude by isospin: + + (* ]v°|K ) = _[f+(k * q+)u + f.[k - q4)p] . [6.30) Using the definitions of the K form factors, Eq. (6.25a), and of the invariant variables, Eq. (6.24), we obtain the following constraints: s£ = tt { t_ - M* S U r/M - -[f+(s<) * Ms.fl/f, ! "~ (A - s4 - ! 3 ? F - G = r =• 0 for SE = t_ , t+ = M , s„ = V . i = M - s4 . (6.31b) The extrapolation to the physical decay region is complicated here since there are three variables involved. If we simply assume that F and G are constant we obtain [Callan ;i966)] f.M f = G = - -f- • (6.3Z) 2 [Eq. (6.28)] and a£ < sz £ (M - Zu)* = 3.2 u . Empirically, |fp/fsl S 1 and \+ = 0.03. So the neglect of fp and of \. introduces errors of the order of 10%. One finds for the ratio [Basile (1971), Beicr (1973)] ; while the absolute value of fg is higher thrn the prediction (6.32) by about 40Î. This disagreem;.... ; not very dis turbing in view of the approximations made. On the other hand, it is evident from Eqs. (6.31) that r is not a slcwly varying function, unless f+(s.) + * f_(Sj) " 0. which contradicts the soft pion theorem for K,3 decay. The variation in r may be understood from the fact that there is J pole in r at Sj • M*. In the following paragraphs we shall see that a careful analysis of the pole contribution leads to self-consistent results when soft pion theorems arc systematically applied to K , K and Kr scattering ar.pUtudes. tn particular, Eqs. (6.31) will be seen to inply the Callan-Treiman relation for K, . 6.2.2 Analyoia of tha pole contribution* The pole contribution is illustrated in Fig. 17. Since the intermediate K couples to the current through its nementun (k' = k - q^. - q_), the pole contributes only to the fom factor r. The residue at the pole is given by the t"K* scattering amplitude, MirK*(t+it_), where t+ is the squared ccntre-of-mass energy, sn the momentum transfer and t. is equivalent to the Mandelstam variahle u. he separate out the pole contribution to r as follows: A(t_,t ) A'(t+,t_) r ° (t* " tJ 7i^— + MTTT- + Rt'-t-.»lJ - ^6.33) *) This section is especially intended for the theoretically inclined reader. \ / —x~v<— M„-ic Fig. 17 Pole contribution to K * VTiv where lia (s, - M')R(t,,t.,s.) = D (6.34) and A and A' are defined to be symmetric functions of t+ and t_. The first two terras on the right-hand side of Eq. (6.35) are then proportion^, respectively, to the antisymmetric and the symmetric parts of the i:K scattering amplitude: A'(t+,t_)/M = i ^KOVK* + VK-t] (6.35a) + 6 35b (t+ - t.)A(t„t_)= I fK[>VK " VK*J • C - ) since the symmetry in the pion momenta must be the same as their isospin symmetry: M K+(t+,t_) = M+K+(t_,t*). Com parison of Eq. (6.33) with the soft pion constraints gives 2 i-^Ll + Rstse,M*,se) = -A(s£lM*) + RA(SrH ,s£) = - \ [f+(s£) + f^jjM/f, , (6-36) where R$ and R. are, respectively, the symmetric and antisymmetric parts of R. Since the KtJ form factor cannot have 2 a pole at st = M (such a pole could only come from a < meson degenerate with the fcaon), Eq. (6.36) requires A'fMSM*) = 0. (6.37) 1 The point t+,t_,s£ -* M is just the double soft pion point: qT.q_ •* 0. The scattering amplitudes in this limit have been analysed by Weinberg (1966) who considers amplitudes of the type (6.1) where ^(0) =• 3UA1(0) is also an 1 = 1 axial current divergence. Applying PCAC to both divergences gives, for example: 1 V) -/d-x 6-y e "-* e*™ (K*.-|K*»-> • M,-^ . Thr '-.ft pion theorens are obtained by performing partial integrations with respect to both x and y. The symmetric part of the amplitude turns out to be proportional to the "a commutator": I 1 + ^A' The vanishing of this term is implied by chiral SU(2) symmetry. d^Ar = 0. The antisymmetric part is determined to be g (t. - tJACM'.M') = - ^ (,. - qJ"(K*|[IJs),A;]|K*) . (6.40) SEMI-LEPTONIC INTERACTIONS Frtm the algebra of currents we obtain 4 7 since V' is the conserved isospin current and t+ - t_ = 2k(q_ - q+). Inserting (6.41) into (6.40) we obtain for tht antisymmetric amplitude: miQ. . iS. . (6.421 v M 2fJ • • This result is equivalent to the Adler-Keisberger type relation, Eq. [5.31). (llie convergence of the integral over cross-sections may be inferred from the Peneranchuk theorem) An unsubtracted dispersion relation for M/E^, whe-e f^ is the incident pion momentum in the K rest frame, takes the form: H.K . 1 fdE Im H.K _ 1 f dE . ., f 1 . 1 1 ,, ... •ET-;_JTE^" J T""M^E; rr^J- <6-43- The imaginary part of the forward elastic scattering amplitude is determined by the optical theorem to be Im ^ - THENCE,,-} , (6.44) ! where cnj;(Eïï) s the total TTK cross-section in the K rest frame. For elastic scattering in that frame we have t+ - t_ = 2k-(q. - qt) = 4ME,, , since the incoming pion momentum is -q+. Then taking H in Eq. (6.43) to be the antisymmetric amplitude of Eq. (6.55b) 7 W-K* " >W) = 4MErft(%>U/M . we obtain in the limit of zero pion energy, using (6.42) and (6.44) which is just the analogue of Eq. (5.31) without the tenr in g?, since the axial cui rent cannot connect K* to a single pseudoscalar state. Comparing the result (6.24) with the soft pion constraint (6.36) we recover the K£J soft pion relation J f+(M ) * f_(M*) = •p , (6.46) since the antisymmetric amplitude Rft vanishes for t+ = t_. Finally, if A, A' and R in Eq. (6.33) are assumed constant (RA = 0), one obtains the explicit expression [Weinberg (1966a), [liopoulos (1967)] r - fK J *+ - t._ .1 H 2f* ] VP - st 7 ' This approximation can be valid only to the extent that f+fSj) + f.(Sn) is also constant. 6-H.3 Pion scattering phase ehifte The study of KQli decay allows a determination of the s-wave phase shift for nu scattering. As will be seen ex plicitly in Section 7, Eqs. (7.1)-[7.3), the matrix clement for K,,» decay satisfies the unitarity relation 2 Im (w.Wjc'ulT.K*) -£0Vie*v|T[n) I IE, Fig. 18 Contributii i the imaginary part of the amplitude for Keb decay since the loptons do not interact in this order. We obtain upon integration over nomenta in the intemc.iiate state where Next we note that, upon contraction of the hadronic matrix clement with the leptonic matrix element, the coefficients of the different form factors defined in Eq. (6.23) project out different momentum and spin dependence for the leptons. Since Eq. (b.48) holds for all values of the lepton variables, it also holds separately for each form factor. Thus 2 Im F-.'s-, s , cos ! "> "/4ÎPÏÏW <'.',m»W)F'[slF s„. COS 0') and similarly for G and h- The scattering matrix clement depends only an the scattering angle and may be expanded in partial waves: ,:T|r;.;) . £ 2 sin 5, ci6' /^f^ P,[co* (8 - 0')] • Upon a simitar expansion of the form factors [Eqs. (6.27)], and integration over the intermediate picn momenta, he obtain [Watson's theorem): ]m f. = sin à, e and similarly for g. and h.. In the approximation where only s and p waves are retained, the relative phase shift may be extracted from experiment by an analysis of lepton angular correlations, as discussed ,*»y Pais and Trciman [Pais (1969)]. SEMI-LEPTONIC INTERACTIONS For s^ i M5, we expect 6 « 6S, so that in effect one is measuring the s-wave phase shift. The value of the s-wave phase shift near threshold, s,, = 4uz, also provides a test of PCAC which determines the -n- scattering ampli tudes at vanishing pion four-momentum. The derivation is similar to that for Kir scattering amplitudes, tqs. (6.38)- (6.42). Preliminary results [Basile (1971), Beier (1973)] indicate values which may be high with respect to the soft pion prediction [Weinberg (1966)]: ao = O.Zfcn , where *., is the Compton wavelength of the pion. 6.3 The relation of PCAC to chiral symmetry In Section 2.3.Z we showed that the Goldb^rger-Trein^n relation becomes exact in the limit u? + 0 if the axial current becomes conserved in that limit. The same statement is true for the soft pion theorems in K decay. In the limit of a conserved axial current M(q) = 0 In fcq. (6.6) and the amplitude M has a pole at q =0, illustrated in Figure 19. The residue of the pole is just: -iqf which is identical to Eq. (6.9). Fig. 19 Pion pole diagram in the chiral syraetric limit he have thus shown that current conservation is a sufficient condition for the validity of the soft pion theorem. Is it also a necessary condition? To answer this question we consider the case where the operator vin Eqs. (6.I)-(6.9) is the divergence of the strangeness-changing weak current operator. Then PCAC, Eq. (6.9), gives the prediction L u i The lçt't-hand sides of Eqs. (6.50) satisfy the relation l t, (Air |ia Ju(x)|B> - p^A^lyxJlB) , (6.SI) where p^ = (pR - p, - q) = (pB - pA) for q = 0. For the right-hand sides of (6.50) we have the relation CHAPTER III M = i3 fA|[lj5)Cx0).ju(x)3|B) - i(A|[ijs)(xo),jo(x)]|6> - = Pu Therefore, Eqs. (6.50) are in general compatible with each other only if ïjs) = ! 3\(x) d'x = 0 , (6.S3) that is, if the current is conserved. Since we are dealing with the matrix element of a AS / 0 current divergence (6.50a), the last term in Eq. (6.52) is in fact negligible with respect to the first term only if conservation of A^ is good to a much better approximation than conservation of jp. If all the vector and axial currents defined in Section S were conserved we would have a full SU(3) » SU(3) sym metry (called "chiral SU(3)" symmetry). The generalized "charges" Q,1 » Fj ± F[S) , i = 1 8 satisfy the algebra of two carenuting SU(3) algebras: [ he know that the full synrcetry is broken. However, if the p« ^udoscalar meson masses give the correct clue to symmetry breaking, it nay be that the full chiral sytaetry breaking is only as bad as ordinary SU(3) breaking, and to a better approximation we have a chiral SU(2) symmetry. The charges QÎ = I1 s ij.-,. i = 1, 2, 3 satisfy the algebra of isotopic spin [<£.<£ = ieiJkQ* • !f this is not the case, soft pion theorems are expected to fail. Since the test of soft pion theorems is an iaportant tool for understanding chiral symmetry breaking, one nay worry about the extrapolation of these theorens to physical values of kinematic variables. In the case of K.3 decay, it can be shown under quite general assumptions that it «Mild be difficult to reconcile a negative slope for the font factor f(t) with approximate chiral SU(2) sym metry [Gaillard (1971) and references therein]. 7. THE ABSENCE OF NEUTRAL CURRENTS In Section 3 we saw how the observed currents fit into an octet pattern. However, one may ask whether the neutral m<37.bers of the octet play a role in weak interactions. If they should enter only through the self coupling of hadrons, which will be discussed in the chapter on non-leptonic interactions, it will be very difficult to observe their matrix elements, which can be separated fron strong interaction effects only through the production and decay of neutral inter mediate bosons. If US = 0 neutral currents couple to leptons, they are also difficult to observe in decay processes because they compete with electromagnetic decays to photonic or leptonic final states. Currents coupling to two neutrinos can be observed through vp scattering; recent tests for their presence have given positive results. The data indicate, however, that they must be suppressed by a factor of two or three in scattering cross-sections, relative to charged couplings. SEMI-LEPTONIC INTERACTIONS However, there are very strong limits on the coupling of strangeness-changing neutral currents to hadrons: B r(KL * uV)/r(K* •* yv) < D.7S * i + s [Clark (1971)] T(KL - e*e")/r(K •* ev) < 3 * 10" + r(Ks * uV)/r(K * uv) < 0-7 x lo-" [Gjesdal [1973)] ! r(K* * T.e*e-)/r(K* - nev) < 0.8 * 1Û" [Cline (1971)] + + P(K - TiuV)/r(K - nuv) < D.7 x 10"' [Bisi (1967)] r(K* - Trw)/r(K* * irev) < 3 * 10"s [Kleins (197:)] T(E+- pe+e-}/r(Z" - ne"v) < 0.D1 [Ang (1969)] . The most striking limit is for the decay K^ •* vu- In fact, this decay has been observed [Carithcrs (1975)]; although there is some controversy over the value of the branching ratio it is below the bound quoted above. Equivalent!)', ro^ - uu)/rL < 2 - io-" . In the absence of neutral currents which couple directly to a lepton current, neutral couplings arc expected to arise through a) higher order weak interactions (for example, as in Fig. 20), and b) electromagnetic effects (Fig. 21). W", n (W7P KL \ w* Fig. 20 Seeood-ordoi ributiori to K, •* Up Fig. 21 First-order veak, fourth-order electi nagnetic contribution to K, -* LU There is a contribution fron Fig. 21 in which the two photons are on their nass shell. The sum of such contributions determines the absorptive part of the decay amplitude which can be calculated. Unitarity of the S imtrix: gives the relation SS* - (1 * iT)(l - iT+) • 1 -i(T - T+) • TT+ . (7.1) Now ici us consider the decay K, •* uu. Sandwiching Eq. (7.1) between the final and initial states we obtain + -l[(uw|T|KL> - (UPIT+JKL)] =£ and we may writo Eq. (7.Z) as 2 Im The sun over intermediate states includes all states allowed by ene^py and momentum conservation and Cf invariance: YYi 3", *VI\J, IT. + y, etc. However, the contributions of all these state.-- have been carefully considered [Dolgov (1972), Stern (1973) and references therein] and it is found that only the two-photon intermediate state gives a significant con tribution. Thus the icjginary part of the amplitude is determined by the amplitude for K. •* YY which has been experi mentally determined, and by the amplitude for YY •* vu which can be calculated from quantum electrodynamics (Fig. 7.2). As the iraginary and real part^ of the amplitude cannot Fig. 22 Diagram for YY * UU interfere, the imaginary part provides a lower bound. One finds [Quigg (1968), Seghal (1966), Martin (1970)]: IlM.^*^ 1^,(6,0.4)* 10"'. The observed level of decay is not very far above th's bound. Estima..s of the contribution of the dispersive (off- shell) contribution of graphs of the type in Fig. 21 lead to an estimate -±- = ? x in-B which is already too high. One also expects significant contributions from higher order weak effects (Fig. 20) at this level; they can of course interfere destructively with those of Fig. 21, It will be seen in a later chapter that neutral couplings play an important role :n the possibility of building a theory of weak interactions which would allow a valid description to all orders in perturbation theory. The low limits on strangeness-changing neutral couplings set stringent constraints on the types of models which may be con- s Î tic red. SEMI-LEPTONIC INTERACTIONS REFERENCES Ademollo, M-, and R. Gatto, Phys. Rev. Letters tt, 264 (1964). Adler, S.L., Phys. Rev. Ul, U« (1966). Alikanov, A.I., Yu.V. Galaktionov, Yu.V. Gorodkov, G.P. Eltscev and V.A. Lyubûrov, Soviet Phys. JETP U.. 1580 (1960). Amati, D., C. Bouchiat and J. Nuyts, Phys. Letters ^£. 59 (1965). Ang, C, H. Ebcnhdh, F. Eisele, R. Englemann and H. Filthuth, 2. Phys. 22B_, LSI (1969). Sackcnstoss, G-, B.D. Hyams, C. Kirop, P.C. Marin and U. 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Martin, 9.R., E. de Rafael and J. Smith, Phys. Rev. D 2, 179 (1970). Pais, A., and S.B. Trciman, Phys. Rev. 168, 1858 (1968). Particle Data Croup, Rev. Mod. Phys. 45_, No. 2, Part II (1973). Particle Data Group, Phys. Letters 50B, No. 1 (1974). Pascual, P., GIFT 12/1972 (1972). Qjigg, C., and J.O. Jackson, Lawrence Radiation Laboratory preprint UCRL 18487 (1968J, Roos, M-, Phys. Letters 36B, 13U (1971). Roos, M., private caiminication in Kleinknccht (1974). Rubbia, C., Proc. Topical Conf. on Weak Interactions, CERN 69-7 (1969), p. 227. Schwinger, J., Phys. Rev. Letters 3_ 296 (i959). Seghal, L.M., Nuovo Cincnto £5A, 785 (1966). Stern, H., and M.K. Gaillard, Ann- Phys. (NYj 76, 580 (1973). Suzuki, M., Phys. Rev. Utters lb, 212 (1966). Tercnt'cv, M.V., 2h. Ekspcr. Teor. Fu. 44, 1320 (1963); Transi.: Soviet Phys. JETP |7, 890 (19b3). Tiarrco, J., ana J. Hhceler, Rev. Mod. Phys. 21, 144 (1949). "1 SEMI-LEPTONIC INTERACTIONS Weinberg, S., Phys. Rev. Letters 2?. 616 (1966). Weinberg, S., Phys. Rev. Letters V7_, 336 (1966a). Keisberger, W.I., Phys. Rev. 143, 1302 (1966). Wilkinson, D.H., Ccrnents tticlear Particle Phys. 5, 13S (1972). Willis, W,, and J. Thompson, Advances in Particle Physics (eds. R.L. Cool and R.E. Marshal) (Interscience, New York, 1968), Vol. 2» p. 295. Winter, K., Proc. Int. Conf. on Elementary Particles, Amsterdam, 1971 (North Holland, Amsterdani, 1972), p. 333. Zavattini, E., Lecture given in the Academic Training Progranme of CERN, 1971 (unpublished). CHAPTER III SU(3) NOTIONS AND NOTATIONS An 5U(3) octet is built from the fundamental representation 3 and its adjoint representation 3: 5 ©3 = 1 + 8 . (A.l) We nay characterize these representations by three-dir.ensional vectors 3 : € = «a) , 5 : I = (£a) , a = 1, 2, 3 . Then the 3*5 matrix "« • vB (*•*> contains the representations a 1 : M0 » Tr M = £a£ (repeated indices are understood to be siCTied) and 8: MS = Mfi - 3 4S M„ , Tr M = 0 . (A.3) Since M is traceless it may be written as a linear combination of a complete set of traceless Heimitian matrices \, i = 1, ••-. 8: Mj - £ M-fipJ The eight independent components of the octet representation ma/ thus be taken as the eight elements of the matrix M or as the components of an eight-dimensional vector. The X^ are normalized by Tr (AjXj) o 2&ij . (A.5) So the vector components are related to the matrix components by M. = j Tr (AjM) . (A.6) The tensor elements f-.. and d^ are defined by the conrnutators of the Aj: The totally antisymmetric elements f are the structure constants of SU[3). They determine the commutation relations of the generators of infinitesimal SU(3) transfornations in all representations. However, the anticomnutation relations (A.7b) are unique to the fundamental representation (3). Under an infinitesimal transfomaticn U of SU{3) (U ï c ' '' in the representation 3], vc have *i r - uç =• ç • joj -f i , - - + - - *: Ç - QF » Ç - iojÇ j- , (aL « 1) . [A.8) Tlien the octet transformation property is M * M • iOjUr iM 1 • (A-9) he obtain the transfonration of the vector closent Mi by taking the trace of Eq. (A.91 with >-j/2. Hj - Hj . i., Tr (^ C»i.»k>'kl • >'j " °ifjik"k • ,A-10) SEMI-LEPTONIC INTERACTIONS If Fj is the generator in the representation 8 and H is a vector representing an octet state, we have under the same transformation (U = em* i in the octet representation) ! M -* M + ia;F.M M M.~n.} * iaL(FiM)j -Hj * iaitFi)^. (A.1Z) Comparing (A.H) and (A. 10) we determine the matrix decent of F : a Y we may form an octet by contracting either the inner indices or the outer indices T,l - MeM*J - i SJMJM'6 , (A. IS) The syenetric and antisvmctric representations are defined hy Sa 'la '!« (A.16) Tto • T,l " T,a • tvriting M and M' in the form {A.4), we obtain from the coronation relations (A-7): T S • I T. Mi";i»i.*j) - J E Ï. MiMJTrHj.»^ • dul(MLMj>.k ±> r , ' ' (A.17) TA • 11 "i";[»i.>j] • "ijkWj • Then the vector ccnponcnts of the symétrie and antisyrnetric representations arc given by TA" 7Tr ^Vi» B ^ijk^V'k • (A"18) It follows that the matrix clement for the k component of an octet operator 0 between octet states n and m must be of the form <»j!<>kl°i> "V"" ifiJkF ' (A-19) *hcrc H and F are r"duced matrix clcccnvs which can be detcmined only by dynamics. In ordt-r to evaluate the d—^ «ml f^ we cust specify the X matrices. The condition Tr i- • 0 restricts the mnber of independent diagonal clcr-cnts to two; we ray choose two of the *• diagonal, and a convenient choice is C "J '-AM CHAPTER III The remaining X matrices shall be chosen as follows: *.-(;::)-G !D ^ - G ; -3 - tquations (A.20) and (A.7) determine the d--^ and fj^ for which we list the non-iero ciments [Gell-Mann (1961)} ijk fijk ijk ijk "ijk "ijk LZ3 i 118 1/3 366 -1/2 K7 1/2 146 1/2 377 -1/2 156 -1/2 157 1/2 448 -1/(2A) M 6 1/2 22B 1/A 558 -1/(2/5) 257 1/2 247 -1/2 668 -1/(2/3) 345 1/2 256 1/2 778 -1/(2/5) 367 -1/2 338 1/A 858 -1/A 458 A/2 344 1/2 67B A/2 The haryon and meson states are identified in terms of octet vector components by: in - I/A;B, • iB,) !-•) • l//2|p, • iP, ) in = 1/AIB, • iB,> !""> • 1/ZîlPj - IP,) |I«> • |B,> !-•> - iPi> lp> • 1//2IB. + iBs> |K*> • 1//2|P, + iPs) !n) - 1/A!B. • iB,) iK*> • l//2|Pt « iP,> |H'> • ]//5~|B, - iB,) |K*> * 1/A|P, - ip,> ;r> • J/.7;B. - iBj) |K") • I//TIP, " 'V W> • !B,> ii> • IP.) Note that, for example, the boson field P+ * iP: creates the state |P; + iPjHn*) and annihilates the state |P, - iP.) (f"]. Similarly the baryon field i( - i 1. a] Perform the phase-space integral in Eq. (1.9). b) Show that the three-body phase integral , = C d'p,d»P,j'p, (2„,.6.; , , P) ', J (2n)'8E,E!E, * ' '" ^ r, reduces to a uniform integral over the Dalits plot >'-ih I •*•••**• ï c) Taking m, = m2 and r.3 = 0, express the integral tj in terns of the invariant s = (p> + pi) and the angle between p, and p, ir the diparticle rest frame: p*, + p2 = 0. Evaluate the integral to obtain -rf- where M* = P2. .Show that covariants of the form Pi«Vn • P»V*Un . pvVu^n ' iP»V^Vn ' where P - p + pn, may be expressed as linear combinations of the terras appearing in Eqs. (Z.13) and {2.11). j) The spinor uc(jS), representing a positive energy antiparticlc state (c+,v) of momentum p, is related to the spinor v(p), representing a ncgaLive energy particle (e~,v] of momentum -p, by uc(p) - C(v((î))T , where C is a unitary antisymmetric catrix with the properties: C-'YUC = -ij , C-'Y,C = YI . Show that the amplitude for omission of an e*v pair: ûp(P)Yu(l - T,)^CP') may be expressed in the form: %(P')>U(1 • ^)vc(p) . hhat is the physical interpretation of this equality? b) Show that the non-relativistic lirait of the i»cak magnetism term is the same as the non-relativistic limit of the Gamov-Tellcr operator: It follows that, in this approximation, nuclear effects arc the same for both contributes. Show that to lowest order in ,q! • |p * p\, , the amplitude fir l-'H(l7N) decay may be written: a - is.; Neglecting terms proportional to n* show th.it the correction to the electron spectrum is ni von by I.q. C^ll. CHAPTER III Using Lq. (2.24) and time-reversal invariance {f^fj.f, real), show that f-j(q*) = 0 in the expression (2-13). Neglecting the electron mass and the weak magnetism contribution, calculate the branching ratio TV* * *e+") = Q.fl . r(Z" •* Ae-v) Derive Eqs. (3.25). Derive Eqs. (6.31). a) Show that, to all oiders in strong interactions, lowest order in weak interactions, and neglecting electro magnetic corrections, the only contribution to I • unitarity sum [6.47) is that illustrated in Fig. 9. b) Enumerate the contributions to the unitarity sum (7.3) which are of order Gpa!. Are there any contributifs of order Gp? c) Retaining only s anil p waves for the dipion system in Ke, decay, show that the unitarity relations for fs, fp, g^ and hp decouple [see Eqs. (6.27)] and may be written in the form (6.49). CHAPTER IV NON-LEPTONIC INTERACTIONS J. M. GAILLARD ana M. K. GAILLARD ! • NON-LEPTONIC INTERACTIONS 1. PHENOMENOLOGY Or HON-LEPTOHIC DECAYS A sir.ple extension of the universal Fcnni coupling described in Chapters I to III is the postulate that all weak transitions arise from the self-coupling of a single current. The conventional weak Hamiltonian can be written as Hw • 7! " h K-Jif• • "-11 where Jw is a current composed of an electronic, a muonic and a hadronic part: •>;--j{e) «ji10 •jjw • (1-2) (The reason for writing the current-current coupling in the form of a commutator will become apparent in Chapter VII.) The hadronic part is decomposed, according to the Cabibbo hypothesis, as J^W = cos 9 JCh)(AS = 0) * sin 0 J,th,(|âSJ - 1} . (1.3) By construction, the Iiamiltonian (1.1) forbids |ûS| > 2 decays to first order in the weak interaction coupling con stant. This property is known as the |aS| < 2 rule. Now consider transitions with |AS] = I, From the general relation for hadrons Q-I.-Ï and the fact that hadronic charge is conserved in non-leptonic transitions, we obtain «, .-f = .f = ±|. So there will be a net change of isospin: |ùl| = '/2, %, % ... . However a coupling which is bilinear in the Cabibbo currents admits transitions only with |AS| = % and %; higher isospin transitions can be induced by electromagnetic x corrections, as will be discussed in Section 2. Experimentally, |ûl| - /2 transitions are found to be dominant, lead ing to the institution of an empirical |ul| = V; rule, whici. is not completely understood theoretically. In the re mainder of this section we shall review the phenomei igy of non-leptonic kaun and hyperon decay, with emphasis on l experimental tests of the ]AS| < 2 and |AI] = /s rulu- Their theoretical interpretation will be discussed in sub- se The search for the decays =" •* nr" and 5° - nr" has given upper limits: 1A 10 3 ?-"- 1 Tn- * * " [Dauber (1969)] (1.4) ffVc ^ ^f^r < 3-6 " 10"5 (901 confidence level) [Geweniger (1975)] (l.S) These limits are still very much above the expected second-order weak contribution. Another test of the |AS] < 2 rule is given by the value of the K£-Kg mass difference in = (0.497 i 0.0u2)rs [Kleinknecht (1974)] . If ,ASi = 2 transitions between K° and K1 occurred to first order in the weak interaction, the mass difference between 5 Kj_ = U//THK' - K°l and K| = (.1/^2")(K* • K ) would also be a first-order term. The width differer.ee (rg - VL) -v. '.$ is a second-order tcm in veak interaction. Since the mass difference is about D.STg ami not a factor 10^ li'~ ,er, tnis is a very strong argument against J AS) = 2 transitions. The study or" Q~ decays should provide additional limits in a not too distant future. 1.2 The |&I| = '/; rule in K decay 1.2.1 % •* 2T. decays For these decays the |ûl| = '/2 rule predicts relations between the decay rates. Since kaons have spin zero, the two pions in the final state must be in a state of zero-angular momentum. Since the total wave function of the two pions has to be symmetrical, the isospin part of the wave function must alto be symmetrical and the two pions must be in the states I = 0 or 2. The Ij = 0 states corresponding to the neutral kaon are: |1 - 0, I, = 0) » -^ f*V + 7,°;,° + TT-7T*) (1.6) |l = 2, I, = 0> -jg [n+ ïT - ZV\* * TT>+) . (1.7! The 2n final state in the decay K: •+ i-^0 has jlj| =1 and it can be only in the |I 2 2, 1. = ±1) state. Since the isotopic spin of the kaon is Vi, the existence of the 2" decays of charged kaons inplies a violation of the |ûl| = '/, rule; the possible transitions correspond to |il| = % and ,'ûl| = %. The content of the states |l,IsJ in terms of these transition amplitudes can be found by the "spurion" technique. The spurion is a fictitious object invented for the purpose of carrying the isospin change. In Section 1.3.1, we show with the example of hyperon decay that in the limit of tine-reversal invariance the phase of a matrix element is determined by the final-state scattering phase shift. For the purpose of the present discussion, we wish to define isospin amplitudes with phase shift removed. Thus we define the amplitudes A2 and A0 AI = tU'lilHjK") o"ifiI , (1.8) where <5j is the phase shift introduced by the final-state interactions. The amplitude Aj may lurther he decomposed in terras of a reduced matrix element for each change of isospin amplitude |AI| = v= amplitude |ol| • % amplitude |ûl| % and a Clebsch-Gordan coefficient l1/,,!,; at, l'Ail,IJ> for combining the isospins of the kaon, I'/j.Ii) and the spurion. |sl,;'A), to form the isospin of the final-state pion system, |l,IJ) A, • <^A; Vi,-'/.!I • 0, I, • 0>a, K9 spurion A, • ('/!,'/,; Vj.-'/ilI "2,1, = 0)a, * «Vî.'/îi '/..-'All - 2, I, - 0)as A. -^a, (1.9) Ai • jj (a, * a,) . (1.10) Similarly for K* - * IT0 decays, one gets NON-LEPTONIC INTERACTIONS Using I:qs. (1.6), (1.7) and (1.9) to (1.11), and remembering that K| = (1/^2) (Kù • KD), one gets iS U A(K» - 'V) > ,f[J] a, e < . y'f (a, • i%) C '] (1.12) 1 5 A(KJ --•-•) = ^"(v'|a, e" - /| (», * »,) e' ') d-") A(K' --*-») = £[», -f 0,]e"= . (!.") (•"ram (1.1-) ;md (1.14) one gets: S T 1 B = * ' ' = til * 6/TRe -A cos (62 - d.)l C, the phase-space factor, is equal to 0.986i c3 and 60 the rtr S-wave phase shifts have been measured to give [Kleinknecht (1974)]: &2 ~ «S0 = "(55 ï 6) degrees . (1.16) The average of the jnca sûrement s corrected «.here necessary for the new value of the K lifetime, gives [Particle Data Croup (1374)3: R = Z.2Q : 0.03 (1.1") and by using [1.15) one pets Re (jrj) = (4.57 - 0.44) * UT2 . (1.18) flic measurements of the CP violating parameters (see Chapter V) set a very stringent limit on Im (A2/i\0). Using (1.18) and that I unit one gets: "" (sr) S °-01 Bc (&] • Time invariance requires a,, a, and as to be relatively real. As no T violation has been observed outside the K° system and as, even in that case, as shown by Eq. (1.19) the combination (a3 + a4) is nearly real with respect to a,, we will assume all the a's to have the same phase. The common pha.ce is taken to bc ÎITO by convention. From the K* and Kg decay rates one gets "•CKj;* I 2a, I ' ' and the measured average values comibed with (1.18) leads to £i = (4.54 * 0.16) * 10-= (1.2:) — = (0.03 î 0.28) « 10-' . (1.23) In (1.20) we have chosen the sign which give? it » as. The opposite sign would lead to a, a. — - -0.88 * 10"* , -*- = 5.45 • 10"2 . (l.M) There are s priori no good reasons to choose one solution rather than the other; however, as we shall see in the next paragraph, the K •* 3T data need ]AIj = Vi terras, but do not show any compelling evidence for |AIj = %. One night x also argue that the as/a, term is somehow a second-order violation of the |AI ! = h rule (see Section 2.1.5). A priori three pions can be in the isospin states I =0, J, 2 or 3. Obviously, for charged kaons, I = 0 is for bidden, ("or neutral kaons, symmetry arguments permit us to reduce suhstantially the number of possible states. A three- pion state has a G parity equal to -1. The G-parity operation is the product of a 180° rotation arouiid the axis 1 1 ly (c " Y) and the charge conjugation operator C. For a neutral system, since the isospin vector lies in the x-y plane (I, " Ij), a rotation around Iv can be replaced by a rotation around any vector in that plane, in particul.-.- I. From G = -], one therefore gets for the neutral 3- system: ,1+1 1 C = (-1) ci.:5;) For the ("*"~-°) state, if l is the relative orbital noncntum between r* and ^" since Cv' , the charge-conjugation opcration is equivalent to a parity transformation on the charged dipion system and c = (-ir . (1.26) A state with I > 0 is suppressed by the centrifugal barrier. For I ; , or 3, E. = 0 is permitted, but for 1 = 0 or 2, 0 I must be odd, thus at least 1. Furthermore the (T/T^TT ) r n-odd states ha.'e CP = +1 aMd at the level where CP violation can be neglected they contribute only to Kj - 3n. For the totally symmetric 3n° state, C = +1 and 1 is odd. Therefore the |AI| = V- rule requires that the 3^ final state in Kj anJ K± decays correspond to 1 = 1. As in the 2" case the rule gives relations between decay rates, but in addition it relates the Dalit: plot densities, he will derive those relations and conpare then with the experimental results. The 1 = 1 basic ingredient is the isospin vector and f(P]tPj.PjJ is a function only of the pion momenta. To choose the correct matrix element for each transition, the rules arc: Pick the correct combination. Take Bosc statistics into account. Remember that, similarly to the K% in the 2* case, = (K KL jr " " **> may be replaced by ^2 K° in the amplitudes for 1-even final states. spur ion K° The amplitudes for the various decays arc f A(KL ^ n%-n«j - ^ fP,.P_»P,J * f(P..P..P0JJ 0 A(KL - v -'""*} =• -=. (f(l,2,3) • permutations [6 terms)] NON-LEPTONIC INTERACTIONS A(K- * ,Mff*) = -^{f(l,2,±) * f[2,l,f)J (1.31) A(K: - 4*fr') = ± |f(a,i,b) • f(b,;,a) • f(;.a,b) + f(=,b,a)} (1-52) Convenient variables are y. = (3Ti - Q)/Q (1.33) m = E where the Tj are the pion kinetic energies, Q = Pfj - -C n) (T;) and X, * y3 M'a = 0 . (1.34) The combination F(l,2,3) = f(l,2.3) * f(2,l,3) , (1.35) which is symmetric in (1,2), Kin depend only upon y}. Although precise experiments indicate the presence of quadra tic terns which are at most 7% of the linear term, we will assisse in what follows a dependencj: F{1,2,3) = A * 3y, . (1.36) Tabic I i;ivoH the matrix elements and the corresponding rates and slope parameters normalized to K? - T I nQ values. To compare the \Al\ = Va rule predictions with the experimental results, one must take into account the relative phase- space factors which result from K'-K' and ir-n0 mass differences. For the same reason, the square of the matrix element is usually written as : M = 1 * g(s, - s0)/m^ * ... , (1.37) 3 + With Sj = (pK - pj) and s„ = (s, • s? + s3)/3. For the case of K •* n n n"p where all pion masses are equal, one 'us the relation m 5] - s0 = - y fflj - (1.38) The values of the rate:;, the phase-space factors and the slopes (g) are given in Table 2 [Particle Data Group (1974)]. The rate i-alues are obtained from an over-all fit to the experimental data- The |AI| = Vi rule predicts that the following five quantities should be equal to zero [Must (1969)]: -i * o.osi * n.M6 3 r+.„ i r,. ; -1 = -0.07S -• 0.024 4 -^ -1 * 0.2S1 i 0.038 (I-Jl) TS(*°0) * g(j;.-) - 0.04? : 0.011 (1.4:) K(-0) • Ri*:.) - 4 R(*00) = 0.135 * 0.024 . (1.43) The :'"s should be understood as corrected by the phase-space factors. IJ(u;iticnS (1.3?) and (1.40) test the existence of isospin I = 3 in the final state. Equation (1.41) is a measure of the f.J - % admixture in the I = 1 state. Uiu.it km (!.4;i ti'sts the presence of 1 = 2 in the fir^il state and ]!q. (1.43) gives information on the ni = 1 ;imp]irudc. Table 1 of = % predictions for K •+ 3TT decays Decay Matrix clement Rate Slope (normalized) (normalized) A * By„ 1 1 L (*-0) Kf - ."itV /3/ZA 0 (000) % 1/^2(A * By,) 1 (100) '/, (•-) /7(A - By,/') 2 -'A Table 2 Experimental results on K -* 3n Decay Rate Phase space Slope (106 sec-1} (-0) 2.31 ± 0.07 1.279 +0.610 ± 0.021 (000) J.11 ± 0.13 1.444 (+00) 1.40 • 0.04 1.147 +0.SZ2 * 0.020 (ÎÎÏ) 4.516 t 0.023 1.000 -0.214 ; 0.004 Some caution must be taken in interpreting these results. Because the slopes are big and the mass corrections important the rate tests could differ from zero by as much as 0.1 f_ Bouchiat (1969)]. Therefore, Eqs. (1.39) and (1.40) do not constitute evidence for an 1 n 3 final state. Equation (1.41) ùuplies a |AII = % admixture in the I - 1 amplitude. As in the 2n case, one can define «3»),„,|HJK'> (1.44) (1.45) A."'* ' «3-)MIKJK*> A,""° " jf If Eq, (1-41) is taken literally, one gets ^ = (1.7 ± 1.2}% (1.48) to be compared with Eq. (1-22) in the 2r case. For the slope tests, the results depend sonewhat upon the paracetriiaticn. Equation (1.43) yields a |AI| = % ad^Luure similar to that given by Eq. (1.48), and Eq. (1.42) indicates the need for a snail 1=2 amplitude. To sunr-arize, the |Al| = V2 admixture is similar to that derived in the 2~ case and there is no evider.ee far a |ûl| = % contribution. 1.3 Hyperon decays The |ûl| a */i rule again gives relations between the rates for the decays: Y * B • * . (1.49) Because parity is not conserved, the final state is a mixture of s and p waves and the relations are valid for each of the amplitudes separately. Mien the hyperons are polarized, the interference between the s and p waves gives asymmetries in the angular distribution of the decay products. These asymmetries permit us to determine completely the s and p amplitudes, as their relative phase can be inferred from T invariance. 1.3.1 ConeequenceB of T invariance Up to corrections due to CP violation, which are too small to be detected in hyperon decays, the weak Hamiltonian is invariant under time reversal T: T~V =Il • U-S0) T acting on the states gives 1 ^(Bilr - |Br).n (1.51) and T|Y> = where <5( is the phase shift due to final-state interactions. The amplitude a. is given by °«-out,l",l,VIY>- (1'5J) Applying complex conjugation to (1.5J), one gets a (B | ;-ouf "i *.i">' (1-55) and using, successively, F.qs. ( 1 .SO) to (1.53) ,l6t o[ - e" aB • (l.Sfa) thus T invariance tcplies that if there is no final-state interaction the s Lnd p amplitudes in hypcron decays ar rvat. Kith final-state interactions the phase of the anplitude is equal to the (":B) phase shift determined from •.ow-encrgy scattering. Consider the decay of a hyperon Y of polarization P . As the inital hyporon has a spin Vi, the final -B state can be written as |Bn>m-»11*1 IB-). CHAPTER IV I Bit) = S|J = %, % = 0) + P|J = %, 1 = 1) (1.57) where s and p are respectively the s- and p-wave amplitudes in the decay. There exists an operator U such that: |J = Va, * = J * Vz> = U|J » 'A, I = J - %); °*58) U is unitary. Since the states are spinors, it must be a 1 * 2 matrix, i.e. a linear combination of o matrices and the unit matrix. As it pnust be invariant by rotation and the two states have opposite parities, U is a pscudoscalar. The only vector in the system is the unit vector q of the decay taken,, for instance, along the haryon direction: (1.59) \A\ where q is the baryon momentum. The only operator which fulfils all the conditions is U = o-q , (1.60) The phase, which is arhitrary. is chosen according to the cocmonly used sign convention [Particle Data Group (19?-!)] liquation (1.57) can be rewritten as |BO - (S • PÔ*q)|J = % , I = 0) (i.6i) For £ = 0 in the final state, J is determined by the spin of the decay baryon, which by conservation of angular mo mentum must be the same as the initial spin. Therefore the state |.I = '/;. £ = 0) may be replaced by the inital hy peron spin state | i). By definition, the polarisation of the hyperon is given by - . (i ,o-| i ) (1.62) The distributions of the decay products can only depend upon their orientation relative to the hyperon nolari- tion. The baryon angular distribution is given by: <&i •D-> « 2v US) where C is the angle between the baryon momentum and the hyperon polarization. Replacing |B-) by the expression (1.61) and |J = '/;, i = D> by |i), one gets ! tiU'Sl' * [PI') * 2 Re (SV)Q.q|i) |S|' • |l>|' 1(C) = 7 (1 • aPy cos 0) , where •) is the angle between the baryon direction and the hyperon polariiatii . 2 Re S*P !S!' • 11' i..'..' r.^.w-i^ ;_ of e*V decay baruon the final baryon polarization is decomposed along the three orthogonal vectors: Î - q, J - ?v - q. k • n - (PY - <1) . The -t-tond U*TI;I change:, sign under a T transformation. lixcept for the final-state interaction discussed in StfCtifii !.>.! 'he poIari;.ir ion along that axis nust be :cro. NON-LEPTONIC INTERACTIONS The final baryon polarization is given by (BT>!3|B"> As examples, we write two decompositions of a, q expressions: o(5*q) = q - ÏO * q (c'q)o(3*q) = 2(5*q)q - 3 . By deriving the other decompositions, one gets * _ [[a * Pr-qjq- * 3?y M Mq • (Py » q)j B (1 * aPy-q) whore a is defined by I:q. [I.6S) and Z Im rs'pi |SP w ISP - w IPI1 a-" • 8* • >J » 1 . (1.72) Lquation (1.70) .shows that P = 0 if s and P are real, i.e. in the abse-cc of final-state interactions. For the purpose of testing T invariance, the following definitions are convenient: & = (1 - a') i sin e Y 3 (1 - a') cos * , f where T invariance ii^plics * - 5p - 6S . (1.74) •*_ and A. arc the p:, and si, ("B) phase shifts. Iqu;itikjns (l.&JJ and (1,69) can also be derived by using the density r^itrlx o which describes the h)7>cron initial 0 » \ (1 * a-Py) . (1.75) the po!ari:a'ion of the final baryon is measured from the asymmetries in scattering on carbon, for example, or, in the case of ; decay, the subsequent A' decay provides a direct measurement of its polarization as shown by Eq. (1.6-1). l./.'l Tuata of the |ûi*| ° '/; rulc_ Tlie previous paragraphs have shown how to relate the s an.! p amplitudes to the measured quantities. The relations predated for both amplitudes b> the Jul! = V; rule can be derived with the spur ion technique. Alternatively, one can im Ihc property [TMtiM|nI/i] ,1,1,) > o . (i.-ài hyperon where I' i* (he isospin lowering operator and 11' '" * is the 'AI,' • '/, part of the HamiIranian, which in the case of tivjierfu deca' corresponds also to il, • -ZS/2 » -Vi. Let .is demonstrate Eq. (1.76) in the z* case, for example, he define H^ûîl"'^ [1,1,1 = h'l.l,; '/r,-1/;) h commutes with T", since all the isospin action of rV. is given by the (VÎ,-'/Î) addition. Then [riHJuJ|='/z]jli:,> = Th|l,l; !&,-»£> - HT|1.1> = /2h|l,0; %,-%> - /?H|1,0> = D . i) £_dççays There arc three niembcrs of the isospin multiplet z*,te,T~, three weak decays: I* -» n • n* u:i r - P » i' (IÎ) Z~ - n * rt" (Q and one electromagnetic decay Es * A9 + r . The 1° weak decays correspond to decay rates about 1IT1S slower. They have not yet been neasured. Usine the property (1.76), one can write three independent relations between the decay amplitudes* : (nnBi[T-,H3|ï+) = 0 {p^-![T-.H]U*j * /ïipr'WT.*) - ,'3(pn-:ii!v: ) = o ^.78) Lliminating the terms corresponding to the E° decays one gets ti.e relation; JlT.% * Z* = Zl . (1,80) The !il, = VÏ rule implies a triangular relation between the decay amplitudes as it is valid For both s and p waves. In this case also, one must eliminate the effrct due to the small mass differences between members of the same isospin multiplets. This is done by defining invariant dimcisionless amplitudes A and B related to S and P by: | -(J) [2M(E s m)]Va , (1.81) where M is the hyperon mass, U and m are, respectively, the total energy and the mass of the final baryon. Because only the relative sign of the s and p amplitudes is determined by the measurements, one must also choose a sign convention. Tor the decay A* * p*", A is taken as positive. The signs of the other s amplitudes are chosen to be compatible with Lq. (1.80) and the Lcc-Sugawara relation which will he discussed below: -lAfZ* - p«°) * A(A° - pu") - 2A[E" * A0"") = 0 . The measurement and the values derived f.--r A and B summarised in Table 3 [Particle Data Group (1974)]. ft>r the Z cas>., Fig. 1 shows the aapl itudes in the AB plane and from (1.80) the \fil\ • '^ rule imp1 ics that the triangle should close. Assuning that all the violation of the |il| <• V; rule is due to |A!| •= %, one gets STL* * It • T.Z " -3>/| a, , il.S2) where a, is the |A!| - % amplitude. Taking the values of A and B frora Table 3 gives -~ ° -0.060 * a.Mb (1.85) -f - -0-074 : 0.030 - A, and B, arc the |a1| • % s and p amplitudes. ») In this section wc use the phase convention of the Particle Uata Group: E^ transforms like |I = 1, Ij NON-LEPTONIC INTERACTIONS Table 3 Hypcron data and decay amplitudes Decay Lifetine Branching ratio a A B (NT10 sec) [48-4 * 0.7)* 0.066 ± 0.016 0.06 ? 0.02 19.05 ± 0.12 ^ 0.800 ; 0.006 (51.6 Î 0.7)1 -0.979 ± 0.016 1.48 : 0.05 -12.04 1 0.59 c; - n-- 1.482 : 0.017 -0.D69 ï 0.008 1.93 : 0.01 -0.65 î 0.08 (6-1.2 * 0.5)1 0.647 : 0.013 1.48 t 0.01 10.17 i 0.24 > 2.624 t 0.014al uo/a_ = ,, (55.S : 0.5)1 1.00O i 0.006DJ -1.08 t 0.02 -7.28 i 0.58 =; - AV 2.96 t 0.12 -0.441 ± 0.078 1.53 • 0.03 -5.90 ± 1.11 ,:-,,.- 1.652 i 0.023 -0.393 t 0.023 2.04 t 0.02 -6.73 • 0.41 a) Updated value, see Kleinknccht (1974). hi u0/a_ is obtained directly in the experiment of Olsen (1970) by comparing the proton and neutron distr buttons in AB decay. 20 1 \ 1 \ i i i i \£ t'o 10 ji: I X L . . .. z: 1 1 1 2 M fig. 1 1-dccay amplitudes. The point L corresponds to the right- hand side of Eq. (1.80) and the point N to the left-hand side LH represents the violation of the |ûl[ - '£ rule. ii) A°_decay Using {1.9b) one gets «J»-* •* C-= ° • <1-*» The jilj = Vj rule predicts r0/r_ => O.JO and a0 = a_. Using the experimental results (in this case |Alj = % cannot contribute), one gets: A3 3- = 0.027 î D.O0B CI.86) p- = 0.DS0 i 0.037 . (1.87) Although it is only loosely connected with the |al| = V rule, it is worth mentioning here that the Aa •* pu" decay has given one of the most precise tests of T conservation. Measurements of the angle £ defined in (1.73) and (1.74) give [Review of Particle Properties (1974)3: $ = (-6.S * 3.5) degrees , (1.88) where T conservation combined with tie |AI| = % rule predicts [Barnes (I960), Roper (196S)j * =• -5* - ôg"'^ •> (-6.5 i 1.5) degrees . (1.89) iii) =_decay in this case the decays are =° - rt° + v9 and (1.76) gives for the |ilj => '/- prediction: H" - /23° » 0 . (1.90) Again there is no contribution fnra |Al| = % and one finds ^ = -0.035 J 0.017 (l.yi) and ~ - -0.1Z ; 0.15 . (1.92) w) (T .decay The dominant decay is or * I;'K" . The |ûl| - xh rule predicts a relation between the tr~=° and TT°E" channels, which together account for about 501 of the decay rate f±9ll- * /I (i--no • 0 . (1.91) The number of observed events is not yet sufficient to allow a test of the rule. With higher energy machines much larger statistics should soon becomr available. Since the IT decays represent transitions between the SU(3) baryon dccuplct and the SJ(3) baryon octet there nay be a surprise there. After all nobody knows the real origin of the very well satisfied |Al| » '/t rule. from the present data on hyperon decay the 'ill - 'A amplitudes arc limited to about 51 of the corrc'pondinp Al| • % amplitude, a nur.bcr very similar to what is obtained in the case of K decays. 1.3.S The Lea-Surxnxwa i-olation The relation [Lee (196-1) and Sugawara (1964)} between the parity-violating s amplitudes (see Section 2.2.1): .flAj. • A^ - ZA=_ (1.94) is obtained by prorating the spur ion frcn the status of a fictitious object I I • Vj, I, • ;%>. which transforms like the K° under isospin transformations, to an object which transforms like the K° under all SU(3) transformations. This nssuflipticn leads to Eq. (1.94), which is well satisfied by the data: NON-LEPTONIC INTERACTIONS AA = -0.04 1 0.12 (1.95) and it is also surprisingly adequate for the p waves ÛE = *2.8 ± 2.1 . (1.96) 2. GENERAL PROPERTIES OF THE CURRENT-CURRENT INTERACTION In the first section of this Chapter we studied the phenomenology of non-leptonic decays. In the remaining sec tions we shall try to see how the empirical features of these decays fit into the picture of a universal Ferai ccupîing as enfcodied in Eq. v.1.1). For purely leptonic interactions, the matrix eleocnts of the ttuniltonian (1.1) may be calculated in the Born ap proximation, and the experimental analysis of y decay supports this fom of the interaction. While the Bora approxima tion is not valid for the hadronic part of the matrix element in semi-leptonic processes, it is still possible to test aj the vector-axial vector nature of me coupling by separating cut the leptonic matrix element and b) the universal coupling strength through the additional hypotheses of CVC and current algebra (which gives the Adler-to'eisberger noimalization condition for the axial coupling]. However, for purely hadronic processes it is not possible to sépara e out the space-time structure of the weak interaction from the effects of strong interactions. If we use the exenange of an intermediate boson to illustrate the wean interaction, a decay such as A * rN can proceed via many graphs, as illustrated in Fig. 2. The best we can do then, is to study the general features of the non-leptonic interaction and try to deduce testable consequences. w re ^ |vp XL v V . \ M /> . _A.—r4—^ w A V"\t M A W/\K K, Fig. 2 Diagrams contributing to .1 - Nn In the notatJjn of Chapter 111, wv write th" .ladronic part of the current as IT • cos a J* • sin 0 j* . (2.1) The non-lept.nic part of (1-L) may then be decomposed into its strangeness-changing and strangeness-conserving part^: "Si " jj k [cosî bK*jl,"I * sin* 9c(j*.Jy"l * "" ° «s e{ i*-Ju~ * Jy-U")] • U-Z) which we shall discuss in turn. 2.1 Selection rules ami relative transition strengths 2.1.1 The absence of \àS\ - 2 The most clear-cut feature of the Hamiltonian (1.4) is that it predicts the selection rule |AS; S i. Transitions with AS = 2 can occur only in second order in the Fermi constant through the tera 1 1 y Çcos fi sin e(j;j " + j^-)* . One way in which to observe transitions with |AS| = 2 is in the Kg-K, mass difference. Since K~ and K, are coupled to different states, one generally expects the weak contribution to their self masses (e.g. as in Fig. 3) to be different. Ignoring the small CP violation (which gives a relative contribution of 10""3 to the self masses) we write: .- 0 K° + F v K° - F The only contribution to the mass difference arises from an effective Hamiltonian with AS = ±2, that is one which allows K0 •*— K^. If the K^-Kg raass difference arises solely from second-order effects of the type illustrated in Fig. 3, the mass difference and the lifetime are the real and imaginary parts of a single amplitude: , iH = An - iar/2 = / d*x (. CL|7-(JCi_(x)),JCw(0))|KL) - (L - S) and they are expected to be of comparable magnitude. As discussed in Section 1.1, this is confirmed by experiment. 6m. = 1 K, /-;o-\ K, K mass Fig. 3 riograms contributing to the IV~ S difference 2.2.2 The A5 s g amplitudeù The strangeness-conserving amplitudes arise from two different contributions. The tern is the synrjtric coupling of two cocponents of an isospin vector with I5 = ïl and therefore gives contributions with |All ° 0 and 2. The tera \ tan1 6 - 0.04 relative to |Al| - 0 and 2 [Dashen (1964)]. Therefore, the experimental study of isospin selection rules in strangeness- conserving interactions could provide a test of the interaction (2.2), as will be discussed in Section 5. NON-LEPTONIC INTERACTIONS _ \i,q - 2.1.3 |AT| - 1 implitudea In this case we have the coupling of an isovector current with an isospinor: CJ*,jW"l + (j*,JP"} , (2.3) giving rise to transitions with [fill = Vz and % in roughly equal strength. However, the data show a clea- enhancement oE [AI| = % over |AlJ = Vz. For exanple in A decay, the |AI| = *& rule is satisfied to a few per cent (see Section 1.3.4). Such a small violation suggests that corrections to the |AI| = Vi rule might in fact be electromagnetic in origin. If this were the case the interaction (2,2) would have to be modified to eliminate the ,AI| = Vi contribution Since the electromagnetic current has |AI| = 0, 1 and electromagnetic corrections to hadronic processes are of second order in the electromagnetic coupling, they have [Al| £ 2. So, if the weak Hamiltonian has only [il] = Vj we expect electromagnetic corrections to give rise to |AI| = Vz and % transitions in roughly equal strength. In A decay the final Sn system has I = Vi or Vij therefore, only transitions with |AI| = '£ or ]AI| = % can occur from the initial (I = 0) state. Similarly, the decay H •+ ATT (Vi * 1) can occur only with |AI| = %, %. the decay I •+ Kit (1 - Vz or %) can occur with |AI| = Vz, % and %. However, there are two independent amplitudes with |M| = Vi and with AI = Vz (corresponding to the two final states). So there arc five independent amplitudes and only three observable decays (E" •* ir-n, Z* - p^*). nearly, one cannot sort out the different transition amplitudes; one may only observe that |AI| = Vi amplitudes (giving one relation between the three observable decays) appear to dominate to within about Si. K decay provides a better mechanism for studying the presence of |Al| = % amplitudes. For K + 2n (Vi •* 0, 2) we have three possible amplitudes which have been determined as discussed in Section l.Z.l. While there is a significant violation of |A£| = Vi. of the order of S%, the data are compatible with no |AI| = % amplitude. For K - 5n, transition amplitudes up to |AI| n % may contribute and they may all be determined. Here the problem of different final-state configurations complicates the analysis. The totally symmetric final state can have 1=1 (so |ûl| = Vi, Vz) or I ° 3 (|ùl| o %, %), The data discussed in Section 1.2.2 indicate that the contribution of I = 3 final states is less than 3% in the amplitudes (and compatible with zero), while in the 1=1 final state there is a significant contribution of AI = %, again of the order of 5%. Although the analysis of non-syitmetric final states is somewhat ambiguous because of electromagnetic mass differences, one finds that |AI| = Vi contributes between 41 and 101 (t3t) to the non-synmetric I = I final state, and there is a comparable contribution from I = 2 final states ('AI| = Vz, Vi). If we assume that the violation of the |Al| = Vj rule is purely electromagnetic in origin we are faced with the task of explaining why it is as large*'' as 5 or 61 in K decay, and why it is predominantly |AI| = Vs- On the other hand, if the |AI| - Vi transitions arise from the weak interaction [as expected from Eq. (2.2)], we must explain why they are so strongly suppressed relative to the |ûl| = '/* part, as is especially illustrated in A decay. Finally, wc comment on the over-all strength of the strangeness-changing transitions. According to Eq- (2.2), they are suppressed by a factor sin 6, and one would expect non-leptonic decay rates of strange particles to be roughly of the 52T.c orcer as their leptonic decays. However, the amplitudes for these decays are in fact cor.parabte to the amplitudes for AS = 0 6 decay. That is, their effective strength i. of order G rather than G sin 6. This has led to the speculation that the |AI| • % part proceeds at the expected strength, whereas the |£I| = Vz part is enhanced by soce dynamical nechanisn. However, if the enhancement is of order (sin G)"1 ; 5, one expects only a 201 validity for the JAI| - % rule, which is not sufficient to explain the data. Explicit calculation of A - "N in the Born approxi mation (the first tern in the expansion of Fig. 2) leads to a M • V» contribution considerably larger than the observed amplitude [Kobsarcv (1965)3- Thus a suppression of |il| = Vz as well as an onhanccreent of |ûl| =• '/z is needed lu ex plain the data. • ) It is perhaps relevant to note that the n-—1° nass difference -- presumed to be of electromagnetic origin ~- is 3.SÎ of the pion nass. 2.2 Charge conjugation and U spin So far we have discussed isotopic spin and strangeness-selection rules. Ke may ask if there is any way in which we can test the space-time structure of the weak Bjmiltonian responsible for non-leptonic transitions. In other words, can we test the current-current hypothesis? One feature can be deduced from the U-spin properties (see Appendix) of the current operators. The currents (-j*,J. ) and (JJ-J form doublets 'tnder U spin; then they satisfy the properties ' ei!,U* J1 e"i7,Uî = -J- Jv u Therefore, the strar.^cness-changing part of the non-leptonic Hamiltonian (1.4): KèS'i = sin e cos 9 i [{VjW~> * ,VJ"+}] (2àA) satisfies the property: : DMJ~"22 If --"U-iirUîj ,=_ V It transforms like the first coiEponent of a U-spin vector: XâS=I =JC* *3C~ ~ 2JC.' (2"6J where JC = JC. - iJC2 are operators which carry U = 1, Uj = ±1. Using the relation (2.S), as well as the properties [analogous to Eq. (1.7&)]: + [U",JCÛS=]3 = [U",JC +JC"] = [U'.JC*] = -2JC, (2.7) where.Ki is the third component of the U-spin triplet (-JC ,JCa,X ). h'e may derive U-spin • -lations for weak amplitudes. in complete analogy to the relations following from the AI = % rule, discussed in Section 1.3.4. However, as U-spin interchanges hyperons and nucléons, a physical process, such as Y * Nir , is related to an unphysical process, N * YTT . To obtain useful relations, we must also make use of crossing symmetry and the charge-conjugation properties of the weak interaction. To this end we divide the operator (Z.4) into its parity-conserving (pc) and parity-violating (pv) parts: each of which separately 'ntisfies (2.5). Since PC is conserved byJC.-, wc have the charge-conjugation property: CJE4S.l C" * Kpc " -V • (2-81 Let us examine the consequences of (2.S), (2.7) and (2.B) Z.B.I b, •- B - » In ti.is section wc shall derive the Lce-Sugawara relation discussed in Section 1.3.5. Although this relation is usually derived under the assur.ption that the non-leptonic weak interaction transforms like a component of an 9)(3) octet, we shall not make the assumption from the start, in order to see more precisely under what conditions the re lation is valid. •) In th,' remainder of this chapter we use the 9J(2) conventions defined in the /Ippendi*. NON-LEPTONIC INTERACTIONS First we consider the general matrix element: (BjPlK^lBi) = G2(an +bîlYs)u, , (2.9) where if p and s denote momentum and spin, u^ = Uj(p-,s-). P represents a pseudoscalar meson; therefore a is the pv amplitude (s wave) and b is the pc amplitude (p wave). For a three-particle vertex the amplitudes are functions only of the external masses and therefore are constant for particles on the mass shell. Under charge conjugation, we have from the property (2.8): ) C2 101 (B.-PlJClB,) = (B2P|Cjec-'iB,) - Sow let us take F to be an eigenstate of charge conjugation (P = n°, n; C|P) = |P<)- From crossing syanetry the ampli' tudes for Bt -+ B2P are given in terns of the amplitudes for B2 •* i^P by V (2.11) Vj = -YftQi?* . (2.12) From Eqs. (2.9) and (2.10) we obtain ûj[a3, + b^YsJu, = vl(all - bl3ys)v3 . (2.15) Using Eq. {2.12} and the properties C"V = _YÏ • c'l^c = yl • -T--c aou = (ÛOU) ' = uWIu' , where 0 is any Dirac matrix, w may write (2.13) in the form u ui(a2, + b2lY5)u, = ûj(-a12 + b12y,) ? • Since there is no difference between u,(p.,s.) and u,(p,,5.), we obtain the relations between amplitudes for Bi * B2P and B, •* BjP: Now we may consider properties under U spin. For baryon states, U-spin doublets are formed hy (-p,I ) and (E~,=~), a triplet is formed by [-n, B,, 1°) with B, = (-E0 * ^A)/2 , (2.15a) and B0 •= {/It* • A)/2 , (2.1Sb) is a U-spin singlet. Similarly, P, - (-IT» • ^Tn)/2 (2.16) P0 = Gfti" + n)/2 arc, respectively, the third component of a U-spin triplet and a U-spin singlet. Using the property: iT,Ul U_U, e |U,U3) = H |U,-UD> (2.17) and the analogous property (1.5) for the Haniltonian, we derive the following relations: U u (ppuixir.*) • t-) P[j = P(U,Uj - 0] . U " 0, 1 . Frein the relations (2-14) and (2.IS) we then obtain + a(=- - rp„] = a(i: * PP0) = 0 (2.18) + b(E" * E~P3) = b(E - PP3) = 0 . Similarly, for the neutral decays where +(-) refers to pcfpv). Using further the property (2.7) we obtain The first term on the right represents a transition (U = 1, U3 = 0) ^t (U = 1, U3 = 0) , which has a vanishing Clcbsch-C>ordan coefficient. So we obtain 0 (BjP0|.ie' = > = - Combining this result with (2.19) for the pv amplitudes we obtain a(=° * BJPJ) = :a(B, - nPft) = 0 . (2.21) Wo may therefore eliminate the n part and obtain the prediction a(=° * BjiT") = -a(B3 * nn°) . (2.22) This is itill not a testable relation, however, since it involves the energetically forbidden decay ~° •* E°FD. To go further, we must invoke the |Al| = V2 rule. If this is a universal rule, applicable to B, •* B: + n, we have the re lations (pfl.'JClr*) =• -/î(,nn\3C \Z°) (2.23) (Z"n|jC| = -> ' -/S(Z°fi|JC|s°> - Solving Eqs. (2.19) to eliminate the amplitudes for =• * An and A * n-;, we obtain: ((Ee * ^5\)-i\3Cia") • j This relation is usually cast in the forn 2a(E" - .V) * a(.l - pO - ^a(I* * P'q) . (2.25) which is equivalent under the |AI| • Vj rule. Relation (2.2S) is •• > -;fied within experimental accuracy [about 10%; one expects deviations at the level of SU(3) breaking effects], might be taken as evidence for the current-current hypothesis. However, it is a curious onpirical fact that the p-wa-c amplitudes satisfy the sane relation to within slightly more than a standard deviation [Particle Data Croup (1973)]. In order to derive the saxe relation for p waves we would need a llamiltonian transforming, for example, 1tk^ the second (rather than the first) component of a U-spin vector: Ul inlJl 2Xa • -KJE* - JC") , e'^ JC, e" . X, . NON-LEPTONIC INTERACTIONS - I 53 - 2.2.2 K •* 2TT Next we consider the constraints on K decay. If P, and P2 are n° or n, we have from "Jie charge-conjugation pro perty (2.8) 5 (P|P2|JE]K*> = (PtPjIXIX ) , (2.26) since only the pv part contributes and by convention"' CjK°) = -|K°>. Under U spin (-K°,Pj,-Kp) form a triplet; using the properties (2.S) and (2.17), we obtain the constraints U U b (PuVlJCIP) .-(-) * '(Pl|Pu/|y-|K ) . (2.27) Comparison with (2.2b) gives (P0P0|je!K) = where the last relation is obtained by crossing symmetry and the fact that the vertex function depends only on the ex ternal ^uissrs which are .il! equal in the SU(3) licit (U spin is a valid concept v.ily in this linit). Using the faci that (-K"*,-*) form a U-spin doublet, we obtain (nK*|JC|**) = (•""ntJC |-'J - 0 and together with (2.28) we predict the vanishing of all decays to neutral final states; in particular (n°n°|JC|K> = 0 . (2.30) Fran the |il| = V3 rule it alro follows that Then for the charged decay, wc have from experiment: a(K* - rV) * ± a(Ks * vu) - O(0.00S) » 0(a) . However, the approximation of copiai masses in the vertex function is not very close to reality (i.e. it is rnuch worïe than 101), and ii. fact the Kg decays as iapidly as do tie hyperons. (It is difficult to compare effective coupling strengths, however, as the amplitudes are of different dmension and mass paracoters must be introduced). 2.3 Neutral currents and the al • % rule In order to derive £qs. 12.25), (2.30) and (2.31) we assumed, on the one hand, properties derived frcn the Ittmiltanian (2. l) and on the other hand, validity of the ol • 'k rule for unobscrvable amplitudes involving the n. The latter assumption does not follow from (2.1), but was node on the basis of the empirical rule observed in other decays. If this rule derives fron sane dynamical r%chanisn due to strong interactions, and if this mechanise; also operates for A - B + n, our derivation is valid. If, on the other hand, the |M| = % rule is a property of the weak Ikniltonian, then, as stated above, the interaction (2.J) raist be nodificd. Then KC night no longer have the property (2.S). •) The convention is chosen so that Kg • Ki, where Ki • (K° • K^)//2 has positive CP. With this convention we musc define |M) • -|P* - 1P?>//T; the other pscudoscalar mesons are taken as defined in the Appendix i f Chapter III. CHAPTER IV If we wish to construct a current-current interaction incorporating the [ùl| = '/i rule we must add to (Z.4) a l piece containing the isospin partners of j" and J" in such a way that the full interaction transforms like |ûl| =• /îm Ke have an isotopic vector (-J ,J',J~) which couples to two isospin doublets (j ,j°) and (-j°,j~), where in SU(3) nota tion: K • Jî*ij; " (i'X • Tnen the construction U 1 transforms like a component of an isospin doublet. What arc the U-spin properties of the added terms? If J"j is the eighth component of the octet of currents Hi = K * Au*' then i-'K'K'ty £om a u"sPin triPlet with and 3; ; (AJ^ • J;>/2 is a U-spin singlet. Then we may write Jj - C^J* - 3*)/z and the neutral current part of U-3Z) is The first term in (2.33) clearly transforms like the first component of a U-spin vector since j° + j°+ does, and "'u a ^u+ *s a s^nB^ct• However, the second term is the symmetric combination of two components of a U-spin vector; it must have U = 0 or 2, and since Uj •> il, U = 2. Therefore, we will not in general have the property (Z.5). However, since J" is an isotopic singlet the A! = Vi rule is maintained if we add any amount of the coupling U" ,F*I * h.c. = \ (t^ïj» * j£), J»"0) * h.c. (2.34) Ke could add this coupling in such a way as to eliminate the second tern in (2.33). What would he the physical moti vation for this extra codification? In the framework of SU(3) it may be core reasonable to suppose th>t the weak Lagrangian transforms like a simple representation of SU(3) than of SU(Z). The lowest representation which can be formed by hadrcnic operators is 8; this assicr.ption that the weak non-leptonic Hamiltonian transforms like an octet assures both iaS| < 2 and |Al| D Vi (as well as 'AI| < 2 for the strangeness-conserving part). Then we rust also have AU| < 2, so if the interaction is constructed fron the octet of currents J . the strangeness-changing part is neces sarily of the fom: Conversely, since there is only one combination of the products (Z.33) and (Z.34) which eliminate the U » 2 conponent, this combination must belong to the octet representation. Lee and Yang (1960) were the first to construct an interaction incorporating the \ùl\ » '/a rule. They used inter mediate bosons which they referred to as "schizons" since they couple sometimes as isospin doublets (W ,K ) and (-h'°,W"), and sometimes as a triplet _f *" , (WB * Vle)/J2, W] in such a way as to ensure |ûl] » Vj • Later an elegant rajdcl was proposed by d'Espagnat (1963) in the framework of SU(3). lie assumed an SU(3) triplet of intermediate bosons (W .K'.W11 ] and its tlemitian conjugate. Their coup] jig to hadrons was constructed in such a way as to form a semi-weak inter action trans fom ing like the representation 3 of SU(3): MON-I£PTONIC INTERACTIONS Then the effective non-leptonic interaction ^.NL "*" XuK ^ 3 ® 3 = 1 © 8 necessarily transforms like an octet (plus a singlet which does not contribute for |ûlj = 1). A feature of this evdel is that for ùS = 0 it predicts |5l| = Û, |ûl| » 1 amplitudes of roughly equal strength, in contrast to the interaction (2-2) (se^ Section 2.1.2). However, the above models must be reconsidered in the light of recent developments concerning gauge theories, to be discussed in a later chapter. Renormalizability requires that neutral currents enter in such a way as to ensure a symnutry of the total Hamiltonian. Further, the absence of semi-leptonic transitions with AS " tl, ûQ = 0 (Section 7, Chapter III), requires that j° is not coupled. These restrictions are such that the neutral currents required for re- nornalizability do not easily reproduce a non-leptonic interaction which satisfies the |ûl| = % rule. On the other hand, if the |£l| • % rule arises fron a dynamical mechanism, and to the extent that SU(3) is a good symetry of the strong interactions, the U-spin property of the weak interaction cust be preserved. Therefore, c l the enhancement of its octet part will be valid to the sane extent as the enhancement of its |M| /2 part, since we have seen that the U 3 1, 1 = V: tern of the current-current product is necessarily in the octet representation. 3. son Pion THEOREMS in Section b of Chapter III we derived a soft pion theorem for the matrix element*' of any local operator [cf. tqs. (to.l)-(6.9)]. (B^Wr/TOHA)-* 4^ 1 Chapter III). Ifs) i'. the cJurge operator associated with the axial currents A : In this six tic v shall consider the case where y((J) ; JC^., (0) is the effective non-le|.ionic weak liarailtonian. If the weak interaction is constructed fron the V-A currents J1 with the property when: T* is "in isotapic spin operator, then the Itailtonian satisfies the sane relation [li,)'JCM.]'[''-XNL]' <5-5' or, cijuivjlcntly (as tr*- a*lal charge changes parity): ni,i-Vn''V- n*.e first tern on the right-hand side of Eq. (^.1) is then T~ (Bl[i;,,.-KSi3M) - - ^ KBIjlJC^IA)- (BIJC^I^A)) . * (3.4) A non-lcptiinic transition Involving (at least) one pion is thus related to a transition with one less pion. >) The nprooliiatlon i-> such that (|JA'|t')' u!t-'. then since A," • A Î lA , we rwst have, for j 3.) X decay As the axial current cannot act on an external pseudoscalar iseson line, Mu(q) in Eq. (Ï.1) has no pole at the origin for processes involving only these mesons. Therefore, the soft pion theorem for K + nn takes the form: (ir * (n - l)*|je(0)|K) -7*. ^1 <(n - l)n|[l :,JC(0)]lK) (3.5) We neglect CP violation and write: *S /I • KL ft • The K system foras rua isospin doublets (K*,K°) and (K'.lT), Then r|Ko> = |r>//3 , r|KL) = ±|r>//z (3.6) Let us first consider the amplitude for K. * n n "°; _ o aL[P*»P..PQ) '=• (i*" '' iX(0)lKL> . (3.7) If we make use of the fact that the amplitudes for K. * 2n vanish by CP invariance of JC, we obtain the following soft pion theorems [Elias (1966), Hara (1966)]: fl aL(0.p_.pD) - •~i- E aL(P-D.ps) 7^- + aL(p+.p_.01 - j^~ Relation (3.8b) follows from (3.8a) by CP inv;. oiicc which requires, on the one hand, the symmetry property of (3.71: a a(p..P-,Po) a(p_,p,,p0) , and, on the other hard, the equality of the right-hand sides of (3.8a) and (5.8b). So we have two independent con straints on the amplitude (3.7). In order to make a prediction for the physical decay region, we must assume something about the momentim dependence. toi it: plot analyses arc generally compatible with a linear dependence of the K -* 3n matrix elements on pion energy, altliough there is sane evidence for a quadratic tem in the amplitude (3.7). However, we shall assume a linear depen dence and write for the matrix elenent: a(P,.Pi.P») " 41 * ?p] » (3.9) 5i " (h( " Pi)J and u ts the pion nass. For particles on the mass shell, u. is directly related to the pion energy in the K rest frame: SON-LEFTONIC INTERACTIONS and is defined so as to vanish in the centre of the Dalitz plot where all pions have equal energy: E. = m.,/3. Taking 1 p3 to be the n° momentun, Eq. (3.9) is the only Form possible [up to terms in p? - v , which are assumed negligible in the soft pion approximation) under the linearity assumption since synnetry of TT* and v~ requires their energy depen dence to be of the fom ID* + or = -w". For, say, Pi + 0 we have Sj •* n£; for the non-vanishing pion raonenta energy and momentum conservation requires p; + Pj = piç, so that 2 ! sa = p* = v , s, = p* = u and we obtain for p •+ 0: a, = -2uj = -Ztijj = ~ (m£ - us) . Ther, using the pararictrization (3.9), together with the relations (3-8). we obtain the constraints: 1 + ( i P- "K ~ "*)] - -ias(i.*ir-)/^f, (3ata) 3 A[l - ^% {n* - u ]l = ia+(nV)//Jfn , [3.10b) where a^i^,:^) is the amplitude for K. •* JIJ.TTJ. These relations allow us to determine both tne slope parameter o and the decay rate. Empirically, the right-hand side of [3.10b) is only about 5% of (3.10a), reflecting the approximate validity of the ]ûl| ° % rule. If we set a+[TT*n°) = 0 we obtain o = 6uV[m* - u2] " 0.6 in good agreement with the observed slope. The total decay rate as determined by (3.10a) also agrees well with the experimental value to within the expected accuracy of soft pion theorems MOI]. Once the |il, = l/i rule is assumed, the rates and the slope parameters for the other decay modes are completely determined by the amplitude for K, •* TI+TI-TI0. We note that if for some reason the |ûl| = Vi rule dominates for K -* 2ir, the soft pion theorems imply that it must also dominate for K + 3n, since their amplitudes are related by the commutation of the weak Kami)tonian with an isospin operator £8q. (3.3)] which cannot change the isospin representation. A soft pion analysis of K * 2T shows that at the point where both pion momenta vanish, only [Al| = Vï can contri bute to the a.-splitude. This is because in the expression a(p1)Pl] _* ^^I ^iW^XH*) —~ =à rnoracntua at the weak vertex, we cannot do so here: p, + p2 j* p% if pjv « njL and p,, p, * 0. The soft pioi theorems require only that the ûl • % part of the amplitude vanish for vanishing pion oomenta; an off-shell amplitude of the form a^tPi.pj) a (Pi * PI)"PKA IS allowed. Extrapolating to the physical point we have (p, * PJ)'PK " mK» which is not small. Fig. A p B,(p,) B3(p,-q)Viy B.(p,) B,(p1)V3'B' Fig, 5 Pole contribution to Bj •* B; + A^ 3.2 Baryon decay In the analysis of soft pion theorems for baryon decay, B, •* Bj + n, we must take into account the pole terms in the amplitude H,(q) of Eq, (3.1) which correspond to the diagrams of Fig. S. Since the axial current connects baryons in the sar (3.12) where A and B are defined by and the g. are the axial couplings defined in Section 3.4 of Chapter III, In the limit a •* 0, for M3 = Mj ;, the pole terms are proportional to q^/p'q» which is indeterminate. However, in extrapolating the amplitude for B! - 52 * v c we from rhc soft pion point a = 0 to the physical point, qu (p, - pa)u, nust also take into account the variation due to the poles in this amplitude, illustrated in Fig. 6. Let us separate out the pole term frora the amplitude for B, * B2 * -i: a(q) - P(q) * R(q) • The pole contribution is given by where g^j is a strong pion-baryon coupling constant. Then if we assume that the remaining part of the amplitude does not vary significantly between the soft pion point and the physical point: R{0) » R(p, - p,) • We may write a(q) a lin a(q) • P(q) - lim P(q) . q-o o+o V / -*~&-—»—Cù—— • -•—Cù—•—^L*— ,tp,) B3(p,-q) VyB2(P!) B,Cp,) V^B^Pj.q) B,(p2) fig. 6 Pole contributions to B, * 1, ' = NON-LEPTONIC INTERACTIONS The first term on the right may be evaluated using the soft pion theorem (3.1), giving the prescription f_ Brown (1966)J: a[Q) = 4^ <*2JD5(°),Je(0)J)B,> + P(q) - lin UqVfq) * P(q)> . (3.15) * q*o Since (cf. Sections 2.3.1 and 4.5.1 of Chapter III) the coupling constants gff.. are related to the axial couplings J 8A by PCAC, which is the basic assumption of soft pion theorems, one may show that the last term in Eq. (3.15) is well defined in the linit a •* 0. The first term in (3.15) is determined by the property (3.4) in terns of amplitudes of the type (3.13) Khich are just the amplitudes figuring in tho pole terns. Let us new exaaine the amplitudes (3.13). The pc and pv amplitudes correspond to A and B, respectively. Using the property (2.8), ve may show in complete analogy to Eq. (2.14) that A» " A,, , B11 - -B,, , (3.16) the U spin properties (2.5), (2.IS) and (2.17) give the relations t (B3|Je|H'> - - {Bj|Je|H0) » -(n|Je|Bj> . (3.18) These constraints together with the property (3.16) forJC imply that B must vanish for all the above amplitudes. If we invoke the ]M| n Vi rule, the vanishing of t* * p and =" •* Z' transitions also implies the vanishing of ï° + n and B =" » Z°, which then implies that the p amplitudes for = -* B0 and B, •* n also vanish. However, we may 3lso note that in terms of SU(3) representations the current-current interaction is of the form 8®8-ie8f@8dQ10SÏÔ©27 . rtiwcvcr, as the interaction is a symétrie cccbination of octet currents, only the syanetric representations 1, 8, and n actually are present. The representation 1 does not contribute here, so the matrix elements (3.13) are determined in terms of only three reduced amplitudes ~ one for 27 and two (f,d) for 8. Therefore, the constraints <3.1o)-(3.18) ir.pl y (3.19) ij for all ij, independently of the \h\\ * % rule. 3.2.1 a-uavc decaya Because of the result (3.19), the pole terms will not «..-ntributc to s-wave (pv} decays in the SU(3) limit. There fore, we have from Eqs. {3.IS) and (3.3) [Sugawara (196S), Suzuki (1965)] i <^BJ|Kpv|Bl) -u.aj.u, - -•^ Consider first the decay A •* N«: i {rtN|jepvU>=^(N|[l ,JCpc]|A> . The right-hand side (I - 0 » ] » '/i) is a pure |A1| • '£ anplitudc; therefore, the left-hand side will also satisfy the |ûl| •* '/) rule. The sane argunent applies'for the decay S * An. For t * NIT, we obtain from (3,Z0j "iii--Tr"ni- " T7A »\ which differs fica the prediction of the [AI| = % rule "anT * anl- + /2afl<^pE " = 0 , only by the sign of 3^+. Since empirically a*-.* * 0, these relations coincide. Thus we might conclude that we have derived the |il| = Vi rule from soft pion theorems without imposing it a priori. However, the soft pion theorem for t* -* n«* is of the form (w'lJCjji'i-fidiKi-.jepjii*) a Since the transition I* - n has Als %, the right-hand side must be purely [ûl| = % (or higher). The fact that it vanishes means that this aœplitude is not present (or is somehow suppressed). Equation (3.17) gives the relations: The additional U-spin constraint (B0!JC|=°) = together with Eq. (3.16), gives Combining (3.22) and (3.24), we have «AS'" Then using the results, derived from (;. V = T7 AnA ' wv obtain the further sua rule which again Jiffcrs Erora the sum rule (2.25) only by the presence of the amplitude for Z* -* un* which is experimentally nearly vanishing. Ibwevcr, since there is only one reduced matrix element for the 27 part of the amplitude (3.13), the vanishing of a*,.* implies that 27 does not contribute, which is just what was required to obtain the sum rule (2.25) (sec Sections 2.2.1 and 2.3). Finally, if we assune octet dooinancc, the entire set of s-wave anplitudes is determined in terras of only two independent parameters, corresponding to F and D couplings. Since the effective non-lcptonic interaction transforms like the sixth component of an octet (i.e. like the first component of a U-spin vector), we have Au - 'W * d.ijd • <5-26> where f and d arc free parameters. Equation (3.26) gives ai. excellent fit to the s-wave data with [Gaillard (197J)]: f • -0.19 « 10' GeV/(rn_ sec) ^ 2 d - 0.06 » 10' GeV/(Ein sec) . The ratio d/f - -0.34 (3-28) is ranarkably close to the d/f ratio which characteriIOS the SU(3) nass splitting of the baryon octet; (d/Om« • -0.31 . (3.25) tiON-LEPTONIC INTERACTIONS For p-wave decay, the comaitator term in Eq- (3.15) vanishes, since [>;„, v [•'• V • by virtue of Eq. (3.19). Therefore, in the soft pion approach the p waves are detemined entirely by the pole terms fBadier (1966), Kara (1966a)]. Using the experimentally determined values for the F and D parameters of the axial cur rent couplings [see Section 4.4, Chapter III}, the pole terms (3.12) and (3.14) may be evaluated in terms of the ampli l tudes (3.1?). In order to obta-' the |AI| =• /t rule t*j are forced to assume it a pvioi-i. As discussed above, this is equivalent to octet dominance for the amplitudes (3.13); they are then of the form (3.26). If we now attempt to cal culate the p-wave amplitudes using the values for f and d obtained from the fit to J waves, Eqs. (3.Z7), we obtain a very poor fit to the p-wavc data. In particular, we predict which implies that the total rate for Z* - nr* must be nearly vanishing, in contradiction with experinent. This result is due to the fact that the d/f ratio, Eq. (3.28), is very close to the d/f ratio for the masses, Eq. (3.29) . This can he understood as follows. Suppose that the parity-conserving |AS| = 1 weak interaction K = (G//2)X* and the "medium strong" interaction •K ~- %~J&* transform like the sixth and eighth components, respectively, of an octet of operators X1. The total parity-conserving non-leptonic Kamiltonian is then of the form: X -X, . ^x* • j., X> • Xm • JCS" , (3.30) where JC0 is an SU(3) invariant. Then there will be no parity-conserving transitions with iS ^ D, because we may re define JC by means of a U-spin transformation £a = 0(G/g^) « l]: U = eiaUa = 1 • iaU; * 0(G2) . (3.31) Both JCg and .K are invariant LUider U spin; then we have 1 6 0) J UXir » X0 * E^U* * i~.[Nïf JE»]] * £ K * JCm • JC^ * 0(G ) . (3.32) Using the property lUj.tf'J B [F?.JC'3 - i y JC* and taking a • /SCZ/Sf- , we may eliminate 3C* from the Hamiltonian by a simple redefinition of the eighth a/is in 91(3) space. Now, to lowest order in the sytnetry breaking we must have "MKJ'V •%Iif'i/'"Vi (3.J3) If we attempt to calculate the pc amplitude for Bj + B.n to any order in perturbation theory using the internet ion (3.30) the result will vanish. This is because a unitary transformation of the Hxailtonian cannot change S-natrix elements, and these amplitudes vanish identically for the transformed tlaniltonian (3.32). In particular, this is true if we calculate In the pole model, including pscudosc i^r rwsfn pcTes (Fig. 7). Therefore the ncson poles nust exactly B, B, B' V/ B, fig. 7 Heson and bafyon poles for 1 cancel the baryon poles. Since there is no strong coupling Z* •* n + octet ir.eson, the baryon pole terms themselves inust vanish for this amplitude. As the calculation depends only on the d/f ratios, this result will hold whenever (d/f)pc - Wf)mss . If we ignore the s-wave data, a reasonably good fit to the p-wave datii can be obtained with f o -0.42 * 105 GeV/Cm- sec) !î V C3-34) d •» 0.33 x 10s GeV/Cm,, sec) '* . However, it is not possible to fit both s and p waves in terns of the two parameters f and d. Various attonpts have been made to icprove this situation by including symmetry-breaking effects, for example by taking R-. j1 0. On the other hand, there is reason to believe that the soft pion analysis for p waves may be a poor approxi mation. As in the case for K * 2n it is not possible to go to the soft pion limit while maintaining four-rocnentum con servation at the weak vertex. This necessitates the introduction of an (inobservable amplitude. The general off-shell amplitude is of the form: aCB» * B2 + ir) - u2[a + by5 • W - K)(a' + b'ï^Ji, , (3.35) whe/e k = pi - pi - q. At the soft pion limit we have q = 0, k = pi - p2, a = G;[f - a'(M, - M2) * {b + b'(M, + Mj)hs]u, , (3-36) while at the physical point, k = 0, q = p, - p2J and a = û2[a + a'plj - H:) * {b - b'{Mt * M^hjj, C3-37) For the pv amplitude, the difference between (3.36) and (3.37) is proportional to the baryon mass difference; it is of the same order as other symmetry-breaking effects which have been neglected. However, for the pc amplitude there may bft an appreciable variation, aside from that given by the pole terms. Another way to make the same argument is to note that for p waves the physical amplitude is proportional to the pion momentum in the decay rest frame, and vanishes in the soft pion limit; thus no information is obtained for the physical amplitude in this limit. 3.3 The origin of the M °.„7;, rule In the light of the above soft pion analysis, we might ask whether any insight has been gained as to the origin of the ]âl| = % rule. For p-wave baryon decay, the [ût| ° l/j rule nust be ass'jeed a priori. For s waves we derived |ûl| a Vi frcci the interaction (2.2) in A and = decay, but only to the extent that PCAC and SU(3) are good approximations. This docs not account for the 31 validity observed in A decay. Furthermore, the same argument applied to 11* * n-r* shows that the |ot] = % part of the interaction (2.2) does not contribute to JC... , at least for matrix elements between s:ngle baryon states; this fact still remains to be explained. It will be of interest to determine whether the non-leptonic decay modes o£ the fi" satisfy the |ûl| = '/a rule. This decay proceeds via p and d waves which vanish in the soft pion limit,and there is no simple dynamical argument which would predict the predominance of |M| n Vj amplitudes for this decay For K •* in, the lol| •= % rule is implied by soft pion theorems to the extent that it is valid for K *• 2n, fhe arguments used to "derive" this rule for the latter case are not convincing. However, the fact that the violation of ]Al| = V* is significant in K decay may provide a means of testing its origin, [f the Hxniltonion (2.4) is the correct one, its |AIj = xh and |ûl| = % components separately satisfy the re lations On the other hand, if the |ûl| • % amplitudes oL;erved in K decay arise from elcctranagnetii. corrections, (?-38) will not be satisfied for the effective Interaction which induces these amplitudes, since it involves the electromagnetic current which is a pure vector rather than V-A. If we assume the interaction (2.4), then using the techniques described in Section 3.1, we can relate the |M| ° % anplitudcs in K + 3n to those in K + 2-n. One obtains the following results [Eouchiat U967), Elias (1967), 'jolgov (1P68)]: NON-LEPTONIC INTERACTIONS 2A+(000) . A+Ç++-] - A+Ç00+) 2,^(0+) AL(*-O) AL(bOD) Tgm while the lûl| : i rule requires each of the above expressions to vanish. Experimentally, they are all significantly different from 2 j [V ^ (20 ± 4)%}, and roughly equal . Furthermore, the predictions for the slopes: -2c+H 1- give a marked iaproveocnt in comparison with the data over the |al| = 7? prediction (V = 0)- These results, if they stand up with future data, may be evidence in favour of the current-current interaction (2.4). Recently, sone light has been thrown on a possible dynamical origin of the il = % rule [Gaillard (1974), Altarelli (1974)3, in the context of gauge theories of strong and weak interactions, which will be introduced in Chapter VII. 4. RADIATIVE hTPERO» DECAYS Another possible probe of the nature of the non-leptonic weak interaction is in the study of radiative hyperon iïç make several preliminary observations. a) If the |il, = '/} rule is of dynamical origin, one could anticipate that the responsible mechanism might not play a role in radiative decays, thus providing a test of the origin of this rule. However, it turns out that the |AI| = Vî rule provides no constraints on the amplitudes [4.1). Furthermore, octet dominance imposes no restric tions other than those which follow from the U-spin properties of the current-current interaction (2.4). There- s Tore, these decays cannot throw any light on the origin of the |AI| = /3 rule. 1) On the other hand, since the eltctromagnetic current is a U-spin scalar, the U spin and charge conjugation pro perties of the current-current interaction lead to strong restrictions on both parity-conserving and parity- violating amplitudes. To the extent that the approximation of SU(3) symmetry is valid, we have a direct test of these properties since no other assumptions are needed. c) In Section 2 we found that the soft pion approximation for s waves gives a good description of the data, with the interesting result that the d/f ratio for matrix element» of JC between baryuns ("weak mass") is very close to the d/f ratio characteristic of the semi-strong baryon mass splittings. However, this conclusion depends on the soft pion theorem; the general validity of such theorems depends in turn on the validity of chirai symmetry as an approximate symmetry of the strong interactions (see Section 6.3, Chapter III). On the other hand, the analogous soft photon theorems [Low (1958)] are valid as a consequence of charge conservation. For this reason the decays (l.l) can possibly serve as a better means of determining the haryon "weak mass". It turns out that only neutral baryon decay is useful jn this respect. Vc shall consider separately the contributions to the decay amplitudes corresponding to the pole terns of ("ig. 8 (internal hrensstrahlungj and the retaining contribution ("direct emission") illustrated in Fig. 9. If an arbitrary amplitude far photen emission A - B * y is expanded in powers of the photon energy aA-B. " a-i'k * as * a^ * '"• * f4'-' V Y » ' »— —»—• » B; B, Fig. B BrcB! Fig. 9 Direct piic •) For Ks - 2' therr is a singificant phase which is due to the strong T interaction in the I > is correctlw accounted for in the analysis. the first two terms are determined completely by the non-radiative amplitude A •* B by the soft photon theorem. The first term in [4.2} arises only from the graphs of Fig. B and is gauge invariant by itself. The second term contains a contribution from Fig. 8 as well as one from Fig. 9 which must be added to ensure gauge invariance. Terms of order k and higher cannot be determined a priori from gauge invariance (that is, from charge conservation], because one can always add to the amplitude a gauge-invariant expression v {k Ey - e kv)M" = btk + b^ * ... , where t; is the photon polarization and the b, are arbitrary coefficients. What we shall refer to as "internal bremsstrahlung" is actually the part of the amplitude which is determined by the non-radiatLve process (a_,/k + a0); we shall refer to the indeterminate part (a,k + ak! + ...) as "direct emission". In the problem of interest here the photon energy is not an independent variable; in the Jccay rest franc For process (4.1J, k = (Mj - Mj)/2M, s M, - M2. So what we are really doinR is expanding the ar.plitude in powers of the 31(3) breaking parameter M, - M2. ht,- then hope that terms of lowest order in this parameter will be dominant. 4. I The br&ii55trahlunp amplitude First let us consider the general (off-mass shell) amplitude for the weak baryor.-baryon vertex B] - B2. It may be written in the form of a Dirac matrix: hiiCpi, P2) = \ * B0Y5 * it2 - Ha)(At * B1Ï5) * (A, * BjY,)^ - M,) * [^ - Ha) (A, * B,ys) (ji, - M,) , (4. 3) where A. and B. are invariant functions: 111212 (4.4) 2 Bt = B^P?. Pi» (P, -P;) ] - Tor comparison with non-leptonic decays we ire interested in the on-shell amplitude: h,,(0) î Aa(Mf,M^,0) + B,(M;^I|,0)YS = A2I + B2]Y5 . (4.SI The first graph of Fig. s gives the contribution ( } . sG;[e,r- - 2&p] \ . "' I*,; "' [Aa[M?,M;,0) • B0(MJ.M:l(»YJ (4.6) + [A,OIÎ,Mïf0J * B^IÎ.HÎ.OJÏBJIU, , where e is the electric charge (in units of the positron charge) and u is the nanalous magnetic moment. Fixpamling the amplitude (-l.b) we obtain : A-K.fPj * k)*,0] - A0(Mf,M^0) * 2pa-k ^ A.tMÏ.p'.OÏpï,^ * 0(V ) , etc,, and A similar expression U cbtain-d for the second graph in Fig. 8. A general result given by Ixih (1958) s that the terms which have no singularity (> l/p*k) as k - 0 must be exactly cancelled by a contribution from Fig. ''• in order to ensure gauge invariance (i.e. invariance under the replacement e •* e • *kj.j- fhc denonstrat ion rests on the fact that singularities at the origin car. only occur when the photon is emitted from an external lino as in Fig. 8. The re sult is that no unphysica] amplitude is introduced (to order k) by the fact that one bar}"011 'n the heak transition of l:ig. S is off shell. T?r decays involving neutral bai/ons, the graphs of Fig. 8 involve ï-A radiative transitions which have .singularities of the form (Ml - M* ; 2p*k)-1. If we treat T-f mass difference as an effect of order k, the above argument proceeds in the same way. NON-LEPTQNIC INTERACTIONS First consider the parity-violating amplitudes for the pole terms. In this case the residue of the pole vanishes to second order in SU(3) breaking. In Section 3.2 we showed that B = 0 in the symmetry limit; furthermore, for baryon = = e states of equal masses, Û2Y5UJ ° for Pi Pj- Th vanishing of the residue at the pole is reflected in the expression for the corresponding amplitude; Eq. (4.6) may be written: V "pf^'pf 21% = e" &^ " ^ ^ Uî + ' " " The parity-violating term is finite for M, = M2. Therefore the pv pole contribution vanishes to second order in SU(5) breaking [the term in 1/k vanishes identically by charge conservation; the first non-vanishing term has coefficient c kuoW/2p.fc = o(l)]. Next consider the parity-conserving amplitudes for charged baryon decay. Explicit calculation [Graham [1965]] gives for the sum of pole terms: However (£~,E") and (-p,E+) form U-spin doublets. As the electromagnetic current is a 'J-spin invariant, we must have u-* = UD> M=~ ° Vv~ *" t'ie SUf3) 1'iiit (M, = M-t. The vanishing of these amplitudes in the symnetry limit can be under stood on physical grounds. The i.nly model dependence is in the multiplicative constant A,,. ftowevor, we know of one model in which all AS = *l parity-violating transitions vanish in the SU[3) limit, namely that discussed in Section 5.2 [Eqs. (3.3(1J-(3.33)3. In that model A.. / 0; therefore, the model-independent coefficient must vanish identically. The pole amplitude for charged decay is thus of order P(k) = o(£ * fiMJ = 0(k) and therefore not enhanced relative to the direct emission term of Fig. 9. for neutral decays, however, the pole terras arc -odel dependent. Tl>e contributions to z6 ~ Ay are illustrated in Tig. 10. Since two different matrix elements of the weak Kami 1tonian are involved (A-a„ A-o-o), the mplitudc docs not vanish except in the special case where these matrix elements are related to the baryon mass splitting as in the model o Section 3.2 [liqs. (3.33)3- ^r us assune that X transfonns as an octet at least for matrix olencnts between baryon states. Then, if the f/d ratio of these matrix elements is the same as for the SU(3) mass splittings, one does not expect pole terns to be enhanced relative to direct emission anrplitudes. If, on the other hand, these f/d ratios are very different, one can expect an cnhanceincnt of neutral decays relative tc charged decays through the pole terms which contribute only to the fomer- If .e write the general amplitude for B, •* B^ * y as uV «BjYlJCjaJB,» - icciikvûo [Pï * a2 + B,ys]u , (4.9) explicit calculation of the pole amplitudes for neutral decays gives [Gaillard (1971)]: 1 Pro "• /3P, = -(3.2Sf * 10.6d) CeV" 1 h (J.10) (M= - M.) (Ms - Mr) where the second equality is obtained by neglecting deviations fron the foil-Mann - Okubo mass fon^ia: 3U * ZV,. • 31, * M. . A ="\5> A lq. to Brcoostralung diagra: CHAPTER IV Since the amplitudes contain the f and d parameters in a fixed combination [they must vanish for d/f = (**/^mass x "0,31' it is not possible to measure f and d separately. If the pole terms are indeed dominant (P; » a-), Eqs. (4.10) will be satisfied for the neutral p-wave amplitudes. If they are only nildly dominant [as is the case for the f and d values obtained from a pole fit to p-wave hyperon decay, Eq. (3.34)], we must separate them from the direct emission amplitudes a-. This will be discussed in the next subsection in conjunction with tests of U-spin properties of the weak inter action. 4.2 Direct emission amplitudes The general amplitude for the direct radiative transition B. -* Bj + Y is of the form Since the amplitudes a, 3 contain no singularities in 1/k, the amplitude (4.11) is a priori or order k = M. - M., that is, of first order in the symnetry breaking. In conformity with our previous expansion in k, we treat the invariant amplitudes a and 8 in the symmetry limit. As stated above, the invariance of the electromagnetic current under U spin allows us to constrain the amplitudes using only the U-spin property of the interaction (2.4). For this purpose, we must first examine the behaviour of the amplitudes under charge conjugation. Because the photon is odd under charge conjugation, we have from the pro[iei-ty (2.8): (BjYi-K^jBj) = -(BJYICX^C-'IBJ) - -lee^û-oUv^ - B^Y^lu. . Then in the same manner as we obtained (2.14) and (3.16) we find As the photon is a U-spin invariant the matrix elements (4.11) satisfy the same constraints as the tutrix elements (3.13) of.lC,„ between baryon states, that is, Kqs. (3.17), (3.18) and (3.23). If we write a.. = a., B.. for the observable decays we obtain [ibra (1964), Lo (1965), Gourdin (1967)]: p E Bn " -Bft = -\t/-fi (4,13) V " <2Qn * V* ' The vanishing of fl for charged decays is analogous to the vanishing of the "induced pseudoxensor" form f;u gj m B decay (Section 1.2, Chapter III), he could write the amplitude for radiative decay as tin1 murix element i effec tive current: n S (yj = [d*x T(j= (x)..lCrt,(y]]] ^ 3* •:," The properties of the effective currents 3"(AS • fl, AH, • ;1) under charge-conjugation, parity and II spin are identn to the properties of the strangeness-conserving currents J" under charge conjugation, runty and I spin. In contrast with the natrix elements (j.M), we cannot conclude here tha: ' : i ill aispl ' cudc;. The effect i- current 3. contains many SU(3) representations. Even in the case where JCv. tr,i; ITTIS JÏ. an cict, .i:ii isyrrjctru re presentations can contribute since j111 and JC arc not nenhers of a cordon octet. In.«ever, the representation 8- is fonr^xl by the ccrrutator of the constituent octets in the matrix representation (sec 'i -:>enJiji, Chapter 111). Since XNL -- v, and f™ -v /l> f * *^, they have ccrruting Tutrices, so i are left with the .-presentations B,, 10, TTi and J?. The Kiryon states cay combine in two ways to forn an octet, giving in all five independent amplitudes. NON-LEPTONIC INTERACTIONS The U-spin property of the weak interaction is closely related to its properties under charge conjugation. For any SU(3) representation whose ûl3 = AY = 0 (= ûU,) cenbers are eigenstates of charge conjugation, the first component of a U-spin vector will be an eigenstate with the same eigenvalue and the and component will have the opposite eigen value. Therefore, CP invariance of the weak interaction requires that its |fiU| = 1, iU, = ±1 components transform ac cording to e ' Jt c * -nJC , where n = il is the CP eigenvalue of the AY =» ilj = 0 members of the representation of K . Any interaction constructed from the Cabibbo currents will have .| = +1. (ftowever, if there is a |ûU] = 2 component -- see Section 2.3 — it will not have the same transformation property.) An example of a different model is one in which the non-leptonic interaction is formed from quark scalar and pseudo- scalar densities*'' XSL ' V^«q * Spv^TÏsq " CP invariance requires that the pv interaction transforms as the seventh, rather than the six.n, component cf an octet, i.e. as the second component of a U-spin vector. Thus tests of U-spin properties provide indirect tests of the space-time structure of the interaction. Radiative hyperon decays are particularly constrained by these properties and therefore are a good place to lest then. Jis- cussed in Section 4.1, they also allow the determination of a particular combination of the "weak mass" parameters f and d, which in turn can provide insight into the dynamics of the decays B •* B' * u. Let us denote the total pc ampli tude by a' = P * a; if experimental amplitudes satisfy the relation [cf. Eqs. (4.13)]: one may conclude that the pole amplitudes arc suppressed, as expected if d/f = -0.3. If on the other hand we find that [cf. Lqs. (4.100= a[ = algj/Z = -a'/O.So , we nay conclude that the pole terra are doninant, requiring a very different d/f ratio. In the intcrcediatc case, where the pole and the direct amplitudes are comparable, if ve assume a the validity of Eqs. (4.13) for parity-violating anptieurfes, and h) octet dominance for the pc anplitudes (3.1ÏJ, there are three independent pc amplitudes (say o , a. and P.), which determine the three ohservahlc neutral decays and, :n principle, they may all be extracted fron the data. 4.3 fxtrdcUqn of the amplitudes from experiment Let us define the general amplitude foi B, - Bj * Y by M u k aUU a !i " ll v"j < * hTiJui • (4.1i) In rh-.* decay rc^t frame, Eq. (4. 14) reduces to the fom M,, - x![10<* ' ^">*^ " Wîf*•«]>[, 1M, "- x*»*, (4.IS) • ) If chiral SI){3) » SU(3) symmetry ts broken only by quark cunsos, this interaction c;>r he eliminated by a i t ran» foin.it ton which redefines the quark rasscs in the sarx way as we used an 3J(1) tmnsfomat ion to el u pc interaction tn the nodcl of Section 3.2, fqa. (3.30)-;3.33). ••) [Tie onlv measured decay ?o far Is E* * m where one finds [Gershwin (]%9lJ fl/o' - 1.03 *l\H. m disagree with Fi)s. M.l.'i by two standard Jeviatlons. where k is the photon momentum and e its polarization vector (c-k = 0) ; xj is a two-component spinor and M is a 2 « 2 matrix acting in baryon spin space. If the initial-state baryon has polarization P([P\ < 1), the final-state distri bution is given by which may be written in the form dI = d C 5 a " 4°- ^Xi^V , (4.17) where T is a density ratrix siting in the space of both bar>-on spin and photon polarization. In general, a spin 1 object nay be described in terms of a three-component colunn matrix •0 If v>e define a basis of linrarJy polarized states c;(i = 1, 2, 3} by • J J then the most general 3-3 matrix acting in spin space is a linear combination )C the following operators: a scalar : £.cî = 1 (the on t matrix) a vector S- = -ie; .. E:£I. a symmetric, traccless tensor of rank 2: T^: = Cjct * c-c- - VJ^HI - However, as the photon has only two degrees of freedom, these operators are projected into a two-dimensional subspace. If ne chose the z-axis along the photon momentum, this subspacc is defined by ej n l>, and the non-zero operators art- Si , T, ^ T,, - TI2 , T2 s Tlz * T,, , (4.18) and the 2-2 unit matrix. Jf we choose as basis vectors stites of circular polarization: t, =• -(E. * iEj)//2 , the operators (5.18) redin-c to *» " t» » TI,Î D -Ti.i , (4.10} where the i arc the usual P3uli natrices (to be distinguished here from the o- acting in baryon spin space). The pro jection operator for a state of positive (negative) helicity is: P. = '/;(! -' S,) (.!,20) and the projection operator for linear polarization with polarization vector e P(0) - %(1 * cos 20 T, • sin 20 Tj) . (4.21) In terns of these operators, the density ratrix defined in Fjq. (4.17) takes the form T - 'Ada!» • |b|:)U • S,(P-k) - S,(3-^) - (3-k)(P>k) * a[P-E - î'k • S, - S,(5*k)(P-k)] - ïE(o,P) - 3E(o\k • h) t H'--' ••here J , . ' if t"""'_ B . -' '° ""-'I v . ! I' - '"!' M -,, laM • lb!' " |a|' • |b|' • |a|' - lb|' ' NON-LEPTOSIC INTERACTIONS and E(A,BJ - t^B, - AjBjjT, + (A,Ba + A BJTj represents a linear photon polarization of magnitude |AB| in the direction (A * B)/;A * B|. As the photon necessarily has helicity S. = ±1, when spins are measured along the axis defined by the photon no- nei.tun, conservation of the z-coctponent of angular raomentun requires as illustrated in Fig. 11 (S. is the z-cotrponent of spin ot B.). Therefore, the measurement of the photon helicity, the final-state baryon helicity and the decay angular distribution, from an initially polarized baryon, all determine t^e snr.L- parameter: (4.24) -k ^^ k S2 5V FÎQ. 11 Schematic rvpre Senear, ion of radiative bar yon decay. The total decay rate detemines the combination of unplitudes la|! * \b\* through the relation: r • ^ [ d cos 6 Tr T = ^ (lal1 • lb!1) . (4.25) If only the parameters |a!* * !b|J and a are detemined, there remains an ambiguity under the interchange o: a-Tpl itudes j --b" or a *- b . The only way to remve this anbiguity is through a triple polarization correlation measurement involving linear photon polar nation, transverse polarization of B7 and P. Note that a non-vanishing value of B would not imply violation of timc-reversal invariance. An iraginary component of the anplitudc niay be induced by the presence of a real inter mediate state (cf. Chapter III, Sections 6.2.3 and 7), for example B, - B, • " - B3 • y . Contributions of this type have been est ira ted by r'arnr (1971). S. PARITY VIOLATION W NUCLEAR PHYSICS Parity-violation in nuclear physics provides a curans of nc.isuring the narity-violiting amplitude for N - \r. The determination of this amplitude cay ihrcr- further light on the nature of the non-lcptonic veaV coupling. Consider tire current-current interaction (2.2). As discussed in Section 2.1.2, there Is a icrr: with hi' - 0, ;: JCj - cos' 0 4 (J*.J'U) (5.1, and a tern with ,il| * D, 1: CHAPTER IV If we define the operators K] 5 sin* 8 \ {jjo*"1) JC^ ï cos! 6 i Jjj"1 , (S.3) he nay separate the strangeness-conserving part af the non-leptonic interaction into its isospin components: •^SS-o " X, • JC, • JC, , with *, " I (JeJ * JCJ> * 7 (xj * 3cj> JCi * 1 tJcj -*3' (!-'1) JC, = ^ (JCj - 2JCjt . Under rotations about the second axis in isotopic spin space, the operators (5.4) satisfy: c'"1- JC, e-in,> - C-j'jCj . (S.S) Now consider the parity-violating amplitude for N, * NjP, where P • i or n. Charge- conjugation and crossing sym metry yive the relations [see Section 2.2.1, Eqs. (2.1i)-(2.14)]: The couplings to neutral nesons (Nt = N,) vanish identically [Barton (1961)]: For the charged pion coupling, the property (5.5), corhûied with the general properties for isospin states: cIîTl |I,I,) » l-)1"1'!!.-!^ (5.8) and with Eq. (S.o), gives the constraint: the consequence of tq. t5.9) is that only JC, can contribute to the pii coupling. Since the Jfl[| = 1 interaction is suppressed by a factor sin1 9, one ejtpects the amplitude to be very snail, [n contrat, since vector mesons have negative charge conjugation, transitions with |Al| • 0, 2 contribute to the îCto coupling, and the D0, u, f couplings arc not forbidden. Ajsucilng i>c are .••Me to dctemine the .*Xt coupling froo c- Cu:.0*.j"">] • -Cu!.fJj.J"u»] • Hj^.J-u» , (5.10) hlcTc U are rfio U-5p|n raiMnj JTJJ lowering eperjinr*. The operator on the right in Lq. (5.10) j*. prnpnrt icnaJ To the :.S « fl pirt of she 5tra.i£cne^5-chongtrTS Itollïonian (J.-3): it* " -iin 0 co* 0 i I}1rJ'u) . ? only >l ccr-:ribuic^ to the pJrity-vtoljtlr-,; Î^J coupling we ray write: NON-LEPTONIC INTERACTIONS where we define the operator JCj = J [JCj - tan2 6 JCj) Then the relations (5,10) imply [t^.JCj = ± tan 6 JO" . (S.15 As JC j transforms like the third component of a U-spin vector, we have the further p-operty: ei7:U2 X, e"inU2 = -JC, . (S.14 Then using the U-spin properties of baryons and mesons, we obtain from (5-13): ran e (rm*|JC*!r.*> = -/z"(B,ir*|JC, |ï*> • tan e <=«Jt*\K~\Z-/ » -(3°K*-| JC31£*> * ^tB,n*|Jfc,|£*> ar-J fron (5.1-1): tnr*[X,|p) = -(E°K*|JCj|l*) . Cabining the e relations, together with the constraint frca charge conjugation and crossing syrcetry: (rVIJC^IÎ*) • -(I-n-IJC^IS'i hi' obtain a relation between parity-violating ainplitudes a(p - n-*) - -a(n - p*") =• tan e [a(Z* - m*) - a(=9 - I'*-)] . If we new assume the -£li - Vi rule for strangeness-changing transitions: a'S" * IN") = a(5° * I0'5) - ~ a(E" - r*a) aC - n"*) • a(ZB - nn0) * ~ a(E* - pnfl) , wv nay list- bis. (2,2t) to obtain [Tadic (1966)] a(p - n**) - -.(n - p^~) » tan 0 ([4a(S° * rtn°) - a(A - nn°)]/^ + • 2a(I* - n**) * ~ a(Z* - pi0)} = • 2a(A * pr')] * 2a(E+ * n*)l . (S.1S) ' | vl [»(=-• n the last cKprcision is obtained by using the U-spir. constraint (2.25) and the [al| « '/t rule for 5 and A decay. In model1-, involving neutral currents the AS • 0 vertex can be similarly related to hypcron decay amplitudes, but the coefficient will in general be different fron tan 0. For example. In the nodel of d'Espagnat (see Section 2.3), the relation is Che sas* as Eq. (S.IS) but with tan 0 replaced by cot 0; thus, in this inodel the |AI| = 1 parity- violJtlnu transitions are enhanced by a factor of about 25 relative to the nodcl of Eq. (2.2). Once the vertex function 19 known, an effective parity-violating Yukawa potential V can be calculated as il lustrated In Fig. 12. In the presence of such a potential, the wave function of a state with parity P and angular inmncntun J will be altered by: v(J,P) - ;(J.PJ *l-.crc r -i. (V ) Jepcndu «n nuclear structure as nçll as on ct't'-ctiv.; potential V . Th? parity Irpurlty In nuclear nt.itei lc;nJ.i tc parity-*lolat 1rs effect* In n^slcar truwl- tioi-T. -nth .i:> thnr cnis^lcn of circularly potanird photcna. rhe f.ut th.it the r.uelcorji fcourd In the nuclei arc slightly off evil if-cll invalidates the CNJVU arulysli to iccc ca- tt-r.t. rrar.ittlens wjtîi ]cl\ * 1 c contribute to pica ^uupttniii Mtf- off-sr.oll riKlccns. Strcc the IMl t I tr-r-^i- ttom irr Jtvjln.ir,! fax the curtcr.t-currait lr.tcr;;;ttcn [S.J|. P*9. 12 Plcn-cKChnrg- «IdlaiUs nu. they may modify the result (5.15) by as much as 20Î [Fischbach (1973]], in addition to uncertainties expected from SU(3) breaking effects. However, these corrections should not cask the order of oagnitude difference between the pre diction (S.151 and that of the d'Espagnat model, for example. In order to calculate observable effects, one nust take into account potentials arising from couplings other than one-pion exchange [direct NNy couplings, 2n exchange, etc.). The most important of these is vector meson exchange. In this case there are no observed transitions to which the potential can be related. The weak NNp amplitudes have been calculated using current p'gebra techniques and o dominance of matrix elements of the isospin vector current [see, for example, Fischbach (1973)]. Once various effects have been accounted for, the theoretical predictions obtained using the interaction (2.2) turn out to be systematically lower than the measured effects fFischbach (1972)], However, with the present state of theore tical understanding, this is not conclusive evidence against the charged current coupling. A recent determination [Yegorov (1973)] of the circular polarization of photons from the reaction n + p •+ d * y shows an effect larger than theoretical expectations by one or more orders of magnitude. The |M| = 1 potential cannot contribute to this effect, and the |AI| = 0 amplitude does not vary greatly from one rxjdel to another. The gauge theories of weak interactions, to be discussed in Chapter VII, modify to some extent the form of the strangeness-changing weak interaction [flailin (1973)], and the suppression of the I = 1 amplitudes is expected to be less severe in those theories. However, the effective nNN coupling in that case can no longer be directly related to hypcron decay amplitudes, since the currents involved in the extra contribution are not pure V-A. On the other hand, gauge theories of str-ng interactions, mentioned in Section 2.3, in connection with the û! = 7: rule, may also pro vide the needed enhancement of both the |Mj = 1 and ]4l| = 0 transitions [Altarelli (1975)J. NON-LEPTOKIC INTERACTIONS - 173 - REFERENCES Altarelli, G., and L. Maiani, Phys. Letters S2B, 351 (1964). Altarelli, G., R.K. Ellis, L. Maiani and R. Pëtronzio, Nuclear Phys. B88, 215 (1975). Badier, S., and C. Bouchiat, Phys. Letters ^0, 529 (1966J. Bailin, A., A. Love, D.V. Nanopoulas and G.G. Ross, fiicleai- Phys. B59. 177 (1973). Barnes, S.K., B. Rose, G. Giacomelli, J- Ring, K. Mikaye and K. Kinsey, Phys. Rev. 1J£, 226 (I960). Barton, C., Nuovo Cimento lj), 512 (1961). Bouchiat, C, and Ph. Meyer, Phys. Letters 22, 198 (1966). Bouchiat, C, and Ph. Meyer, Phys. 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Rosenfeld, T.G. Trippe, F. Uchiyama, N. Barash-Schmidt, P. Soding and M. Roos, Phys. Letters SOB, 1 (197-1). This review of particle properties gives detailed references to the experimental data. Roper, L.D., R„M. Wright and B.T. Fejd, Phys. Rev. U8, B190 (1965). Sugarawo, H., Prcgr. Theor. Phys. 51, 215 (Î56-»). Sugarawa, H., Phys. Rev. Letters JLS, b70 (1965). Suzuki, M., Phys. Rev. Letters 15, 986 (1965). Suzuki, M., Phys. Rev. W, 1154 (1966), Tadic, D., Fhys. Rev. 17£, 1694 (1968). h^jcicki, S., Proc. 16th Internat. Conf. on High-Energy Phys., Chicago-Batavia, 1972 (National Accelerator Laboratory, Batavia, 1972), Vol. 2, p. 209. Yegorov, A.I., D.M. Kaainkcr, V.A. Knjarkov, V.M. Lobashov, N.A. Lo:ovo>-, V.A. Naiarenko, L.F. Sayenko, L.M. Snotritski and G.I. kharkevitch, to be published in Nuclear Phys. NON-LEPTONIC INTERACTION APPENDIX U SPIN AND SUI2) PHASE CONTENTIONS The isotopic spin properties of the members of an SU(3) octet are easily determined by considering the properties of SU(3) triplets (quarks), which are shown diagrammatically in Fig. 15a (representation 3) and Fig. 13b (representa tion 3) \/3/2Y V3/2Y X (X) • P (a) (b) Fig. 13 SU(3) weight diagrams for quarks (a) and antiquaries (b) and I-spi.n content In parentheses we have indicated the transformation properties under isotopic spin. The states (a,, a. , ..., a_j) transform as |an) = |I,I, = n). Their phase is determined by the conventional property: ljl.lj) = /ifl + 1) - Ud» ± DlI.I, ± 1) • The phase convention for the particle states in the triplet representation is chosen to meet the requirement that pp + nn be an isospin invariant. If q= I n) and q = (p ": I) are vectors in SU(3) space, we may determine the transformation properties of, say, the pseudoscalar mesons by making the identification*-* (see Appendix to Chapter III for the definition of the physical states TT, K, n in terms of the P.): • pi^7h- The pseudoscalar mesons are shown diagrammatically in FI'T, 14, where their composition in terms of q, q is given, as well as their isospin transformation properties. In particular, n° must transform like the third component of an isospin vector CA.l) and the composition of the n is determined by the requirements that it be isoscalar and traceless with respect to quark indices: (A- 2) Just as isotopic spin is the group SU(2) of unitary transformations p +-* n, U spin is the SU(2) group of trans formations n •*— A. The properties of particle states under U spin can be obtained immediately from their I-spin properties by a 120° rotation of the diagrams in Figs. 13 and Id. The resultant diagrams are shown in Figs. 15 and 16 where now (a^.a^j, ..., a.y) indicates the transformation property of |an) = |U,U3 - n) under U spin. In analogy with (A.l) and (A.Z) we must have for the Uj = 0 vector and scalar states: •• |P6 - iP,)//2", opposite to that used in the *ext. See footnote, \f3/2Y K°(nX) K'(pX) ( K\K°) (-ir,Tt (-K°,K") K'(Xp) Fig. 14 S0(3) weight diagrsn f. and I-spin content -\f3/2Q -V3/2Q P X n (n.X) Cp) • "U3 _U3 » (P) n X (-X.fi) P Fig. 15 SU(3) diagram for quarks witf U3 and Q axes, and U-spin content -\/3/2Q K"(Xp) rc'(np) (n",K-) K°(Xn) Pi P„ K It'(pn) K- (pX) (-K'.if) Fig. 16 Weight diagram for mesons with U-spin content NON-LEPTONIC INTERACTIONS P, - (fin - A.\)//7- C-"° * /3n)/Z 0 P3 = (an * U - ppl,'^ • -(v^.. + n)/2 . (The over-all phase of tne invariant Pe is irrelevant.) The transfonr.atio:i properties of other octet states or operators can now be read off frcn those for the pseudoscalar states according 1.0 thejr position on the I3, Y (or U3, Q) plor: p \ K* I+ "V Tl* "V J* , U ' Finally, just as die generators of isospin in the fundamental representation (3) of SU(3) are the À matrices whit act on the p, n subspaee. I = \.(2 , < 1, the generators of U spin are the >. matrices which act on the n, \ subspace (Appendix, Chapter III): Ihpcrchargc is an I-spin invariant and charge is a U spin invariant. PROBLEMS 1) Establish the relation (1.10). 2) Demonstrate that the G-parity of a system of n pions is (-1) , 3) Lstablish the tsospin content of the left-hand side of Eqs. (1.40) to (1.43). 4) Derive Eq. (1.63) by using the spurion technique. 5) If A is a unit vector show that: 30-A) = A" - ia * A" (S'3t)3(o>A) = 2(c-A")A" - o , which are Eqs. (1,68). Derive also expressions for: (o-A)o , S x (a x A) . Show that o = (o-Â)K + S x (5 x X) . 6) In the case of the decay 5" ->• AD + it" followed by AD •+ p + TT~, show that in the A0 system the proton angular distributions with respect te i, j and k defined in Eq. (1.66) arc given by: 1(0^ » |[l + a_aA cos 0S] ICOj) ^[l+jB^Mse/] «°kl " Ï t1 * Ï YH°APr- ^ °J • 7) Derive Eq. [1.80) and Eq. [1.82) using the spurion teclinique. 8) d} Assuming a weak coupling of the form (a, b constants): JC = : aïi'î'ïi|'IP * bj^^^jP : + h.c. where 'i> and P are baryon and mean fields, respectively, show that in the Born approximation the amplitudes for Bi - B2P and B2 * BjP satisfy the crossing relation (2.11). b) Show that, in general, the amplitudes a and b defined in Eq. (2.9) depend only on external masses and are therefore invariant under thr; substitution p, *- p2 in the limit of SU(3) invariance [e\i = m2). 9) Assuming a weak interaction of the form [2.4), show that the decay Kj - TY is forbidden by U-spin in the SU {3} limit. The decay K, * fin is not forbidden (without additional assumptions) and gives a non-vanishing contribution to the imaginary amplitude for K, •* YY: K, + it+n~ •+ yy (see Chapter III, Sections 6.2.3 and 7). Why is this not in contradiction with the U-spin constraint? 10) a) Derive the relations (2.23). b) Show that the AI a Vi rule implies that there are only two independent amplitudes for = - tf and derive the three constraints relating the five amplitudes (E~ - I"""0, j}^'\ ~° * L'..~, etc). c) Derive from Eqs. (2.18)-[2.24) the following relations which together with (2.22) give the Lee-Sugauara re lation for s waves; 0 0 a(=° - EV) + ^aCE" •* An ) = -a(E° * wt°) - -4 n{Z* - p* ) s(5° - EV) + -^ a(E~ - E-TT°) = -a(Z° - mr°) - /3a(A - mr') b(H° * Ï°TT°) * /3b(S° * An0) = b(Z" * n*0) - JL b(E+ - pn°) b(5° - IS0) - i b(E" * r-n0) = b[E° * nn°) + ^3b(A •• mi0) . NON-LEPTONIC INTERACTIONS 11] Show directly that, if the Hamiltonian (2.4) is the correct one, dynamical enhancement of Ai % with respect to AI = % implies [in the SU(3) limit] enhancement of 8 with respect to 27. 12) Give an explicit derivation of Eq. (3.19). 13) Assume that the weak Hamiltonian is given by Kq. (2.4). Then amplitudes for K decay contain AI = % and Al = V2 parts: a(K -+ nn) = ai, + a . n }h/ 2 Imperically, |av : •"< | a,, |. Neglecting terms of order (a3,/3y) , derive the soft pion relations {3.3 14) a) Derive the expression (4.8) for the parity-conserving pole contribution to I+ •* py. b) Using the Gell-Mann-Okubo mass formula, derive the relations (4.10) relating P_D, P. and P . 15) a) Show that gauge invariance requires that the amplitude for B •+ B' + y be of the form (4.9), with P2 •* u3 = a and Bj = 6 as arbitrary constants, b) Derive the expression (4.22) for the density matrix defined in (4.17). What measurements are required to remove the ambiguities a *—• b^and a —' b. 16) a) Derive expressions (4.12) and (4.13) using the methods of Section 2.2. b) In Section 2.2 of Chapter II, we showed that gz = D in e decay by invoking time-reversal invariance which re quires that gj be real. However, the amplitudes a and B are complex; imaginary amplitudes correspond to on- sholl intermediate states, e.g. A * Nn •+ ny- This is related to the fact that the current 3 defined in Section '.2 is not Hennitian because of the time-ordered product in its definition. Thus: where '•> and.C are Hermitian current operators. Time-reversal invariance requires Tab = Tb'a' ' where a' and h' are the states obtained from a and h under time reversal (reversal of spins and momenta). Assuming that b = Bi and a = (B2y) can be taken as eigenstates of time reversal, show that time-reversal in variance imposes separate reality conditions on the matrix elements of :n and A . Show that this is sufficient + to conclude directly from the vanishing of ga in fi decay that B = 0 [[in the SU(3) limit] for the decay Z •* py. CHAPTER V THE NEUTRAL KAON SYSTEM J. M. GAILLARD NEUTRAL KAON SYSTEM 1. THE Kj-Kj SYSTEM In this section we will study some general properties of the neutral kaon system which are not directly related to CP violation: ma^s difference, regeneration, sign of the mass difference. However, as CP violation effects will intervene at different levels, we will anticipate the results discussed in Section 5. For this section, it will be sufficient to know that the long-lived K" decays into two pions, and that the amplitude ratio: A(K| ~ . „ ) -3 with |n+_| = 2 * 10 and *+_ = (45.0 ± 1.3) degrees. 1.1 General formalism^ The state of definite strangeness K0(KD) can be produced in strong interactions through various reactions; for instance: TT~ + p - A0 + K° . K° is the antipan.icle oi KD. However, when the K°(K°) meson decays, through weak interactions, it appears as the superposition of two states of definite lifetimes: a short-lived component Kj, and a long-lived one K°: B TKO = 6 * 10~ sec . Furthermore, the iiun-leptonic decay modes are two n mesons for the Kj and three TT mesons for the K§. The fact that the phase space available for the three-pion decay is smaller than that for the two-pion mode explains why K° decays much faster than .>2. Historically, the existence of two independent states K° and K£ was first postulated by Ge11-Mann and Pais [Cell-Mann (1955)] as a consequence of C invariance. Païs and Piccioni [Pa'is (1955)] pointed out that the difference in lifetimes should correspond to a mass difference between K° and x£ of the order of the Kj mass width. Evidence for the existence of the K° was first obtained by Lande et al. [Lande (1956)]. Kaons played a major role in the dis covery of parity violation [Lee (1956)] with the T-9 puzzle; an elegant solution was found, according to which the T and 0 were eigenstates of the operator CP. Lee, Oehme and Yang [Lee (1957)] have analysed the K°-K£ system under all possible assumptions of P, C, T violations, assuming CPT to be conserved. The experimental results on neutral K mesons led to the Kj-Kj formalism in which the Kj, a CP = +1 state, decays into two pions, and the Kj, a CP = -1 state, decays into three pions. A neutral two-pion state of total angular momentum J = 0 is an oigenstate of CP, with eigenvalue +1, Under CP conservation, choosing the phase of |K°) and |Ka> such that CP|KD) = |P) , (1.1) the eigenstate with CP = +1 is given by |K?> =^[|K°) • IK3)] (1.2) and the state |Kj> orthogonal to |K°) has CP = -1: |K°) »^[|K'> - |F">] - (1.3) To include the small CP violation, we will adopt a description in terms of K° and i which are eigenstates v ith a definite mean life: A iKji - a • |E|»J-* CIK»> • E|K;>] (J .4) |K"> - (1 • |E|')"'*[|K;> • e|K«>] , (1.5) •) See Appendix 1 for a systematic study of the neuTal kaon system formalism. where |K°> is defined by Eq. (1.2) as the only state which under CP conservation could decay into two pions. thaï CP violation effects will be important, we shall romment on them. Consider now the evolution with time of a pure beam of K°'s created at t = 0: K°> •^WW], -t/2xs !K°Ct)> -^[i*S(t)>< |K£(t)> e" CI-7) wher !K°>, |K£(t)), |K£(t)> are unit vectors, TS and T, are the lifetimes of K° and K.° [T- • : T.), respectively. tor t : the K° beam becomes an almost pure beam of K?. Typically, the Kg lifetime corresponds to a distance of a few centimetres, and nearly all experiments working with neutral beams originating in the machine are dealing with pure K? beams (y 1000 Ki mean lives away from the target). In a few other experiments a KD beam is produced directly in the detector (bubble chamber) by a beam of K~'s, and one can directly observe the evolution of the initially pure K beam. Table 1 gives a summary of the important observed decays of K% and K,°. K| and K? decay modes Mean life Decays (sec) Partial mode Fraction IT Tl (68.77 * 0.26)1, TI€TI0 (31.23 ± 0.26)1 6 ID < 0.3 x 10- 0.895 ± 0.003 * 1G- V u "S e e < 35 « 10"s u it y 2.0 ± 0.4 x lO"3 YY < 0.4 x 10-3 (21.3 ± 0-6 )% irVn° (11.9 ± 0.4 )% TOJV {27.5 ± 0.5 }\ nev (39.0 ± 0.6 ]l itevy ( J.3 ± 0.8 )% nV (0.195 ± 0.005)r> K 5.179 t 0.040 x 10"B Tf°Tf° (0.089 i 0.019)1*) £ 3 V 71 Y < 0.4 x 10" 7iDYY < 2.4 x 10-" YY (4.9 ± 0.4) x 10- ey < 1.6 x 10-' 9 y P i, 9 x 10- e e < 1.6 x 10'* 1 mass = 497.7i i 0.13 MeV/cz. i Updated values from Kleinknecht (1974) NEUTRAL KAON SYSTEM 1.2 K[-K£ mass difference Ite have seen that the neutral kaonr. are created as K5 (or K°'s) When they interact strongly they behave as K0ls (or K°'s). In their weak decays they appear as Ko'- and K°"s. This behaviour gives one of the best illustrations of the superposition principle in quantum mechanics and provides a cieans of measuring the S?~Kj cass difference. Experi ments have been performed to measure that mass difference by two nethods. a) Evolution of a K0(KC) bean. Strangeness oscillation nethod*' At a distance x frazi its point of creation, a K3 becanes Lksx -x/2A, ''S » KJ ^L* „-X/2AL] _ JS " 8CT,S «, - 6tYTL kc,k. are the wave numbers of K£,K? We consider times t « T,; if we rewrite Eq. (1.8). as a functii £ = x/As s • (\ - ks> "s • ik X e S f . •4/2 H *M mKp - mf;o is expressed in units of II/T-; 1/A, has been neglected compared to 1/Ag. Rewriting Vi(x) as a function of K° and K°, one obtains -1/2 i«SH . „-t/2 _ „-ifi4 ^1 LkgXl + 0 YCx) > Various experiments have been performed to measure the W intensity as a function of t in an initially pure K° beam. This intensity is an oscillating function of I: IjriS.*-) - j(l • e-t - .e"a/2 cos 6£J (l.li; Its measurement for different Vs leads to a value of 5. Various experiments, mostly in bubble chambers, have neasured <5 using this method, with a precision not better than 20Î. b) Regeneration of Kg mesons"**-1 The KS-K? mass difference can also be determined by measuring the intensity of K_'s regenerated in a pire K? beam which traverses a certain artount of material. K°'s interact more strongly with natter than J^'s; therefore after tra versal of a certain thickness of material, the initial bean of K?'s will contain fewer K^'s'yan K°'s, and will be 2 nixture of Kf's and Kg's. These KS's will be coherent with the K?'s, i.e. thgy will travel ^n the direction of the initial beam, with a slightly different momentum. In addition to the coherent régénérâtion^n K|'s there is a re generation by diffraction. The effect of matter on a K? beam can be represented simply, as in Fig. 1. This effect was first suggested by Pais and Piccioni [Pai's [J955)], and analysed in detail by Cose (1956) anu '.- •) Fitch (1961), Canierini(1962), Meisner (1966), Chang (1967), Aubert (1964) and (1965)» Baldo-Ceolin (1965). .•) Recall that p = hk; k = 1013 cm"1 for p = 2ÛD MeV/c. ...) Good (1961), Christeiison (1965), Fujii (1964), Bott-Bodenhausen (1966), Alff-Steinberger (19f>G), Mischke (1967), Aronson (1970), Cullen (1970), Carnegie (1971), Geweniger (1974). -- K; DIFFRACTED > COHERENT DIFFRACTED F1g. 1 Schematic of K° Let us consider a thin slab of material (thickness L) in which multiple scatterings of K0,s are negligible. We use the notation number of nuclei per cm3 in material f(0) scattering amplitude per nucleus for K°'s ?(£.) scattering amplitude per nucleus for K*'s. (1.12) f,, = S,t =|[!' Q • (1.13) At a point x inside the regeneration slab, the Ki amplitude in the forward direction is given by •"f„l - 2'iMsf,,C) [l " e"i(-i6*V')] -No x/2 , stxJ ; — r; t T K CO) & -is » % where £ = x/Ac. Kg's arc regenerated a- various depths in a coherent fashion but with a slight difference in phase awing to the mass difference between K| and K£. A K| regenerated at x will have travelled as a K? with a wave number k. up to that point, and after that as a K| with a wave number ks- The nass difference expressed in units of the K?. mass width h/Tc is given by (1.15) jhe intensity of Kji's emerging at the end of the regenerator is obtained by putting x = L and squaring Ki(L", : 1|^(L) = " .; ^ y> x [1 * e - 2c cos «J [K£(0)J e 1 , where Ac = 2ir/kc. As the nuclear attenuation of K? and K| is the same throughout the regenerator: For K~ diffracted at an angle C* from the bean direction, the waves emitted from different nuclei in the regenerator cannot stay in phase; the various amplitudes add incoherently. At the end of the regenerator and in the forward direction: „£ (dr>/dfi)e=D 1 P5 - NAglf^CO): (1 " e *) = IK,(L; , 0.1») where dD/dfl is the diffracted intensity per unit solid angle. The ratio is proportional to N and independent of the regeneration amplitude f2I. In the various experiments which have been perforaed to measure the K|-K£ mass difference using the regeneration of K2, typical values are »t "1D"' ^"«"W = S * 1(,"S • I the K| direc Measuring p£ for various thicknesses of the regenerator, one can compute $. This method has been used in the first measurement of the mass difference of Good (1961). They obtained Jn general, to apply this method one needs tin accurate knowledge of the intensity of K?'s, the absorption in the re generator, and the value of ~'Si in order to eliminate sytematit errors. The imaginary part of f}[ can be determined from the K* and K" cross-sections on nucléons. From charge symmetry: o(K° + p) - o(K* * n) • o(K° + n) » o(K' + p) oCK" • p) » 3{K" * n) a(K° * n) » o(K" + p) . For the real parts, one usually assumes that the K° amplitude is purely imaginary; the K° real part is deduced from measurements of the real part of she K* amplitude in the forward direction Re f. f~~f- * 0.4 at 1 GeV/c . Ira f+ To calculate f,, for nuclei from the nucléon data, a detailed computation making using of the optical model is ncccssar-, A measurement of the mass differei.ee has been made by Christenson (1965) using two regenerators of constant thickness and varying the gap betseen ;hem. This method, proposed by Fitch, gives a Ks intensity varying as a function of the gap. The absorption is constant; in th? other experiments, changing the regenerator thickness modifies the INTERFEROMETER OECAV REGION J REGENERATOR ni;""'"! REGENERATOR , 2 Configuration of Che regenerators u£ Che experimei of Cullen et al. [Cullen (1970)]. The interfer. between Che two blocks 1 and 2 is observed in th.._e _jntral part ct the decay volume. The other two pares are used to determine separately the :z of the regeneration intensities from each one of the cvo blocks. absorption, and an appropriate correction is needed in evaluating the data. Thus this experiment can be considered as more reliable and yields 6 = 0.5f> '•: P 10. All the re»"ncration experiments have to deal with the delicate problem of multiple scatterings in a thick regenerator. CP violation effects lead to non-negligible corrections, and they have been taken into account in the analysis or the reanrlysis of the d ni(t) ' {|pi(t)l2 • |nCt)|2 + 2 Re [n*(t)pi(ti]} (1.20) Mt) = c * llpjft)!2 + |n(t)|2 + 2 Re [n*(t)Pa(t)]} 2 3 niaft) - c x {|p,(t)l + \pt(.t)\ + !n(t)M + 2 Re [p*(t)p2(t)] -< + 2 Re [n*(t)p,Ct]] + 2 Re [n*(t)Pl(t)]} (1.22) no(t) » c x |n(t)|2 , (1.23) *) Bott-Bodenhause.; (1966), Alff-SteinbergeT (1066), Mischke (1967). .) Aronson (1970), Cullen (1970), Carnegie (1971), Geweni^er (1974). NEUTRAL KAON SYSTEM vhere p,ft), p,(t), and n(.*) are the time-dependent 11*11" decay amplitudes from regenerators 1 and 2, and without re generator; n,(tl, n2 (t), n12(t) are the ir*ir" decay rates after regenerators 1, 2, and 1 and 2; na(t) is the rate without regenerator. As the regenerators 1 and Z are of equal thickness and are made of the sanie material, the re generation phase is the sane for the two blocks. Therefore the argument of p*(t)pj(t) is equal to the phase difference between K. and K's propagating frcm regenerator 1 to regenerator Z: arg [p*Ct)p;[t)J = ûïi-T , where T is the proper tir.e which has elapsed between generators 1 and 2. Using Eqs. (1.200-[1.23], one gets ûn[t) = nuCt) - n,[t) - n2[t) * n0(t) = 2c - Re [pî(t;0z(t]] - CI.24) = 2c x ,p,(t)j * |pj(t).' * cos (ÔT) . For all vaLues of t, in(t) is zeru for T = -i/2in, and the r*r~ decay rate can be integrated over any interval of tûr.e. The integrated rate will also vanish for T = -/lus.. It is convenient to normalize the rate dividing Ein(t) by E[n,(t)n,(t)] ". For n(t) = 0, this procedure wculd give exactly cos (6T). The normalized rate, after integration over t, is derated as iN/2^M1N.. Tne experiment was performed in a 7° neutral beam derived from an external target at the CFRN PS. The decay products '--ere detected in a wire chamber spectroneter. The success of the experiment rests or. the ability to select the 2r decays of forward-going K: free of background from other SOUR S. Leptonic decays are eliminated, the electrons being identified by Cerenkov counters and the muons from the penett. Lon through a thick iron absorber. A tight selection on the reconstructed transverse momentum of the kaon reduces the background to a very low level, and the remaining contamination is subtracter by extrapolation of the transverse momentum distribution under the forward peak. The distributions of &ti/2/ti}Ht are plotted in Fig. 3 f\,r two samples of events obtained for two distances between the regenerator blocks. The zero crossing point gives the mass difference with high precision. The average of all experiments gives ûm = (0.536 ± 0.002) * 101D sec*1 . 1.3 Sign of the mass difference Four experiments have deteT*nined the sign of the mass difference between Kg and K? *•*. Their results, which agree, prove that K? is heavier than K*£. lVe shall only describe briefly the experiment of Mehlhop (196B), which gives the most significant results. Figure 4 shows the experimental set-up. A K* beam produces by charge exchange a pure K° beam. Some distance away from the target, the neutral beam enters a regenerator. KS's arc regenerated from K?'s, and can inter fere with the Kc's which were still left in the initial beam. The target is a sandwich of spark chambers and three layers of copper. The spark chambers provide a knowledge of the K° production point. Charged two-pion decays are observed in a large spark chamber placed inside a magnet. Data arc taken for various positions of the taTget along the beam; the regenerator remains in a fixed position. If D is the distance between the K° origin and the regenerator, the K| intensity af.-T the regenerator of thickness L is given The regenerator thickness is chosen in such a way inat for the various values of D, e ' s = |p|2- The intensity of Kg (correctly normalized) will exhibit a pronounced minimum for arg (p) - (6D/AS) = ±180°. In Appendix 2 we have derived the expression of 0- r • i.\') A l„m i-~—r, • (1-26' .) Mehlhop (1968), Meisncr (1966), Hill (1971), Jcvanovitch (1966). \ GEOMETRY 1 \ GEOMETRY 2 DISTANCE BETWEEN BLOCKS l=529'im \ OlSTANCE BETWEEN BLOCKS l = 77B.Bm T\ !\, \T NJ ftm = (0 536-00093)110''° see'' um=lOW6IO.D062)»10' sec h- A «0 .lO^sec Vn " PROPER TIME ,,f t\ K \- -03!- • 3 Détermination of the K^-Kg mass difference in the experi :nt of Culler [Culldn (1970)]. The zero crossing point occurs at a ti ! T •= n/2ûm. Fi". A Experinental set-up of Kehlhop et al. [Mehlhop (1968) J to aeosure the sign of BOSS .fference » NEUTRAL KAON SYSTEM From the interference between Coulomb and nuclear scattering one obtains the real part of f21(0). Total cross- sections of K* and \Ç give the imaginary part of f21, and therefore the phase of ftl is known. Other phases are known provided the magnitude and thi; sign of 6 = mj(? - ui^o are assumed. Under the conditions of the experiment the minimum should be reached at D/Ag = 3.06 for 6 = +0.5 D/«s = 7.4 for 6 = -0.5 . Figure 5 shows part of the re;ults which have been obtained; it clearly demonstrates that K? is heavier than Ki. 2. SEMl-LEPTONIC DECAYS OF NEUTPAL KAONS In this section we will study the time dependence of the semi-leptonic decays. Ne will not be concerned with the detailed dynamical structure of the decays and the problem of form factors which is discussed in Chapter III, Section 6.1. Assuming invariance with respect to CPT, the semi-leptonic decays can be described by two complex amplitudes f and g: f = (uV\i|T|KB> g = (TtVvlTlK5) - U*13 6 cm DATA JF2H =1061 F AT P = 775MeV/c f = 155° K« LIFETIMES Fig. 5 Best fil i the interference region i no interference effect CPT invariance implies that for the weak Hamiltonian % ' Hlj • where a is the initial state and j the final state; a and j are the antiparticle states of a and j. Ii CPT invariance gives f* = (irVv|T!F> g* = (TTVV|T|K°) . The CP operation transforms K° into K° and T* into n~. Therefore CP conservation in the decay implies which means f and g must be real. The empirical rules AS = AQ and |Al| = % also have consequences for f and g: K° —* ITVV S = +1 S = 0 I - \6. 1 = 1, l3 - -Vi l3 - -i Q = 0 Q » -1 . This decay satisfies the AS = AQ rule; it corresponds to JAIJ = '/;, but |Al| can be '/2 or %: S = -1 S = 0 r = V* , i = i, i3 » % i, = -l Q - 0 Q » -1 . 3 This decay does not satisfy the US = AQ rule; it corresponds to |AI,| = /2 and t»""-^fore to I AI | = Vs. In general the notation x = g/f is used. A non-zero value of x corresponds to a violation of the AS = ùQ rule. Im x different from zero indicates CP violation. Let us consider a pure K°, produced for instance by K* charge ex change. To write down the decay rates, we have to express |K°) in terms of |K|) and |K?), the decay eigenstates. Using equations (1.2) to (1.5), |K£> and |K£} are written as: 1 5 1^) = (1 • M )"'* [|K°> • rlK "}] (24] |K») - (1 + |r|I)"V2 [|K°> - rjF>] , where r ° 1 - It. and e is the CP violation parameter, and we get .) . (1 * 1*1')'* [lug, . [H.)] , !K With the definitions r - Rate (K° + H + v • i ) , r = Rate (K° - I * \ f+ = Rate fK° * l* + v * O > f" = Rate (K5" - l~ + \ 1 e « |c| e ^, x = |x| e^. V*S NEUTRAL KAON SYSTEM ! ! rst r* = j|l • x| - 4|c| [|x| cos Mc » y * |x| cos «jj e" + z 2 V * l|l - x| • 4|e| [|x| cos ttE * 4X) - |x| cos ij\ e~ ^ * 2 z * 2j[l - |x| » 4|c| |x| cos »£] cos rtat) - (rs rL)t/2 - [2|x| sin ^ - *|e| |x| sin »£ * y] sin (4mt) | e" * rst r" » l|l * x| = - 4|c| [cos «E * |x| cos t»E * *x)]U" * ! r t * l|l - x| - 4|c| [cos *e - |x| cos »c * •,)]U" I' - 2 - zkl - |x| - 4Je| COS *eJ cos (ûmt) + (r r t 2 * [z|x| sin ix - 4|e| |X| sin ftc » ^j] sin (AM)! e" S* L> , where only terms of first order in e have been retained. If one neglects the e terms, then r* = |1 « x|! o"rst * |1 - x|! e"rLt * + 2[(1 - |x|2) COS Apt - 2 Im x sin imt] e s L r , |1 *x|! e 'S' » |1 - x| =.- v • 2[(1 - !x|!) cos Amt + 2 Im x sin unit] e s L r* is obtained by replacing |c| by -|E| and $ by -« in T"; and r~ is obtained by the same changes in l"+. A list of experiments on AS/AQ rule is given in the references. The initial experiments were done in bubble chambers before the discovery of CP violation; e and Im x were neglected in the analysis. The number of events was in general small, and in many cases the analysis was made without looking at the sign of the electron, adding r+ and r~ which leads to the simple expression: r-.r-.jK^.-W-.-V. The ratio between the values of (r* + r~) for large t and for t = 0 gives R c Jl + x|V|l - x|2, A value of R corre-,ionds to two possibilities x and 1/x, but the fact that the AS = -AQ amplitude is sniall removes the ambiguity. The value of x 'ias been controversial for a long time. Precise determinations of x require n >t only good stati stics, but correct selection and identification of the events and unbiased measurements of the decay-time and of the efficiency as a function of the eigentimc. These difficulties have been overcome to a very large extent by the most recent experiments. The average values are Re x • 0.021 ± 0.022 (2.12) Im x " 0.008 ± 0.016 . (2.13) In addition to these experiments, the electron charge asymmetry has been measured after a regenerator by Bennett et al. (1972). This experiment gives jl"J*|I = °'96 * °"05 * (2,14} and as Im x is small, Re x = -0.020 ± 0.025 . (2-15) The iS = AQ rule is very well verified by the neutral kaon decays. 3. CP VIOLATION AND THE K£-K| SYSTEM 3.1 Evidence for CP violation The state K£, which has .J = 0 and CP =• -l, cannot decay into two plons if CP in conserved. Early experiments [jBardon (1958), Neagu (1961)] had set an upper limit o' 3 « HT3 for the fraction of K.° which decays into two charged pions. The experiment of Christenson, Cronin, Fitch and Turlay [Christenson (1964)] found that the K? mesons do decay into two charged pions, and that R " K{ - au"cto^eJ nodes " (2'° * 0'4' * l0" • Figure 6 shows the plan view of the detector arrangement used in that experiment. It was performed at the Brookhavcn National Laboratory AGS, in a neutral beam emitted from an internal target at an angle of 30°. The mea sured momentum distribution of the K.° beam including the highly selective efficiency of the apparatus was found to he pj. = 1100 ± 100 MeV/c. Figures 7 and 8, which are taken from Christenson (1964), illustrate well the difficulties of the experiment. Only an accurate definition of the angle between the vector sum of two measured momenta and the K? beam direction combined with a good measurement of the invariant mass m* permits the separation of the two-pion events from a huge background due mostly to K and K decays of K.°. Fig. 6 Detector for the experiment of Christenson et al. Qchristenson (1964)] -NEUTRAL KAOIJ SYSTEM (0) • DATA: 5211 EVENTS - MONTE-CARLO CALCULATION (b) VECTOR ~ -0.5 COO DATA too 120 VECTOR 7" -0.5 110 400 100 90 300 SO 1 70 50 rrirT' 40 30 ZO 10 _^^r~^>^ 0 300 330 400 450 900 350 600 M*V 0.999 cost? Fig. 7 a) Experimental distribution in m* compared with Honte Carlo calculation; ra is the invariant computed under the assumption that the two ctiarged particles which are observed are two pioi CP violation is indicated by the discrepancy between the Mante Carlo calculation and the dai b) Angular distribution of the observed events in the range 490 MeV < m* < 510 MeV. 484 < m* < 494 ,,-J^I •idv 494 504 Fig. 6 Angular distribution in three mass ranges for events with coe 6 > 0.9995. The strong forward peak for the central mass interval corresponds The K? direction was geometrically defined by a collimator with a spread of ±2 mrad. A calibration of the ap paratus with regenerated K?'s gives a global standard deviation of 3.4 ± 0.3 mrad for the direction measurement. In that calibration the mass distribution leads to an average mass (498.1 ± 0.4) MeV/c2 and a standard deviation of (3.6 ± D.2) MeV/c2 Alternative explanations of the experimental result in terms of Kg regeneration, K or K decay, K° •+ ÏÏ+ÏÏ~Y contribution were either impossible or would have required an extremely singular behaviour of the neutral particle spectrum in the decays. Shortly after the results of Christenson et al., further evidence for the decay K? •*• TT* + IT" were obtained by Aba^hian et al. (1964). One way to maintain CP conservation would be to assume that an external long-range field interacts with KQ and K° in opposite ways. This possibility has been analysed independently by Bell and Perring [Bell (1964)] and by Benistein, Cabibbo and Lee £Bernstein (1964)]. Such a field of galactic origin would probably make the ratio R pro portional to y1 = (E^B/m^o)1; this seems to be the only simple possibility corresponding to a potential which changes sign from K° to K°. Two experiments [Galbraith (1965), Do Bouard (1965)J were performed at higher K? momenta than the original one. The results of these experiments are given in Table 3. These experiments rule out the possibility of a yz de pendence of R, and at the same time give a very good confirmation of the result of Christenson (1964). The violation of CP is expressed by the amplitude ratios <"'"'IT|K£) _ i*,. n |T|K|) --(,-,-|T|lfi>" '"*-'' (r'j'.'TiK/) , . i«„ n"°(='-!Tl4i*;n"'e ' "-2> ant arc ^ I"+_,' i I'Sfljl* ° • *cj the four quantities which are accessible to experinents. A review of the experimental situation fjGaillard [1971 jj contains all pertinent references except for uie most recent results. The quantity |n+_.' had hcen reneasured in many experiments before 1967 and the average value was 3 |n+.| = (1.91 ± 0.06) * 10" . (3.3) Recent experiments [Geweniger (1974a), Messner (1973)3, more precise and using more careful normalizations, have radically modified the value of this supposed well-deternined paramter. The average value given above corresponds to the pre-1967 experœents. The new group of post-1973 experiments gives an average value: |n+_| = (2.279 ± 0.025) * 1Q-3 . (3.4) This value and the most recent experimental results on CP violation are presented and discussed by Kleinknecht (1974). + The phase c+_ has been measured by studying the interference between K, •+ :I TT" and Ks - n*-t~ decays. 3.2 Interference experiments K.' - n* » a" and K* - TI* * TI" occurring together le^d to interference effects provided the K? and K' waves arc coherent. In general, if A and B are the decay amplitudes corresponding to these two decays in an experiment, the in tensity for K° - T* « n" will take the form I (-**") = I'M* • !B|J * 2 Re (A*E) = |A|a + |B|3 + 2|A||B| cos (to - $•) . (5.5) The Kl can he regenerated in a K? beam, and the interference is then observed after the regenerator. Another possi bility is to study the 2n decay rate near the production target in a region where the remaining K£ amplitude is still large enough to interfere strongly with the K? •* 2v amplitude. 3.2.1 Interference betueon Kj-'s and regenerated K^'s We have seen in Section 1 that when a beam of K£'s traverses a slab of material, a beam of KÎ's, coherent with the original K? beam, is regenerated. After traversing a thickness x of regenerator, the K£/K£ amplitude ratio can be written as NEUTRAL KAON SYSTEM ili>sy;.i(»)r,(0) r. -xC-i6»'/j)»s" •h l where the various parameters have been defined in Section 1. Many experiments have been performed on the TT IT" decay mode to determine the phase of n+_ using this method. The experiments give directly $ = (*+_ - $ ), where * is the regeneration phase, and the results on <)>+_ have been unstable for a long time because of a poor knowledge of the phase *f of f21. This phase has been evaluated using the measure ment of total cross-sections for K1 interacting with copper to deduce the imaginary part of the regeneration amplitude. It has alSD been measured directly by studying the charge asymmetry for K decays behind a regeneration. The average between these various determinations is «f = -(48.2 ± 3.5) durées for K° momenta between 2 and 3 GeV/g and for copper. In most of the experiments a copper regenerator has been used. For other cases the phase was either measured directly or deduced from the copper pr>ase by optical model calculations. The avirage phase obtained from these experiments is \_ = £44.1 ± 2.2) degrees . (3.7) 3.2,2 Interference in a direct beam of neutral kgona A K° beam, with a small contamination of K°, is produced by protons ûnpinging on a target. The interference between K? and K£ decays is maximum around 12 lifetimes away from the target, when the K£ amplitude has been reduced by a factor of -* nt_. Because of the long distance from the target to the observation region, the dependence of the result on the value of ts\, the mass difference between K| and K£, is important. Typically, a change of 0-02 * 101C sec-1 in the value of the mass difference leads to a change of 10° for o+_. This is du** *•-'". ^ct that the interference term is of the form |A||B| cos (am - *+_) , (3.6) where T X 10"5 sec is the proper time counted from the target. The interference term has the same structure for K° and K", but its sign is changed; a contamination of K5" will just reduce the effect. With the very precise measurements of the mass difference, the influence of the mass difference accuracy is not too severe. The three experiments which have used that method give an average value *+_ = (45.6 t 1.6) degrees . (3.9) Combining the values obtained by both methods one gets o+_ = (45.0 i 1.3) degrees . (3.10) 3.3 Kf - 2r° decay The parameters of the 2TI° decay code are substantially more difficult to measure than those of the charged pion mode. Experimentally one observes the showers initiated by the conversion of some or all gama rays produced in the chain reactions /Y The main problems are those commonly associated with shower measurements: identification, lack of accuracy in energy and direction of gaina rays. Furthermore, as there are three possible ways to combine four ganmas into 2~3, one needs to discriminate between them. The background is essentially due to the decay K.° •* Sit0; for a small frac tion of those decays, only four gfntnas are detected out of the six gammas produced. Although the fraction is small, the branching ratio being cuch larger than for Kf •* 2r°, the nimber of contaminating events is very large. The eli mination relics upon the Vinematical fit for these events. In addition, there is a problem of normalization: either one uses a different mode K: •+ S-." and the relative efficiency has to be evaluated, or the normalization is given by an auxiliary run with a regenerator, and a careful monitoring of the two independent runs is needed. Thi; KD mass re solutions range from 25 to SO MeV/c2 (FWtW). In all experiments the four gammas are constrained to fit two n^'s. The error on the Ka transverse momentum is typically 30 NfeV/c. Because of the difficulties involved, the value of |n0.| has been controversial for some time. A list of the ex periments which have measured |n00| is given in the references. We will describe in some detail one of the last two experiments which have settled the question of the value of |nool« The method used by Holder [1972).is an attempt to overcome the difficulties encountered by other experiments. The apparatus is shown in Fig. 9; it consists of wire chambers and thin converters (D.9 radiation length in total) followed by a 61-cell lead-glass counter. With this set up the angular precision in projection is A9 = ±12 mrad and the energy definition AE/E = ±4.5$ for electrons of 1 GeV/c. The K? beam has an average momentum of about 4 GeV/c. The K? * 2fl° decays are observed in a decay region, and for the normalization the idea is to fill the decay space uniformly with K~ by moving a regenerator across it. If I and IR are the intensities of K°, and N and K the numbers of K° •+• 2ir observed in the free decay case (F) and in the re generator case (R), one gets ln |z _ l N^ L_ ao R C3.ll) IPP " IC N* AC ' where L is the length of the decay region. That formula har to be corrected for edge effects and for diffractive re generation (llî). The determination of IR and I~ is based on the observation of by events, where at least three gammas are converted in the lead converters, the others being observed in the lead-glass counter. For each regenerator position, the 3TT decays come from the region between the regenerator and the chambers^ in the free decay runs the whole decay region contributes all the time. Knowing the decay point of each K°, it is easy, as the regenerator moves uniformly, to weigh each free decay 6y event in such a way that T M^n Ip Nn h Hi' where Np is the number of weighted free decay events. The 6y events selected have a reconstructed KD angle less than 17 mrad with respect to the beam direction; in order to account for diffracted KÎ, a substantial correction (^ 301) has to be applied to NJT\ Figure 10 shows the observed 6y invariant mass distribution for free decay and for regenera tor. In Fig. 11 the 3TT decay vertex distribution is compared to the same distribution for regenerator events. The good agreement is a check of the weighting procedure. Figure 12 shows the mass distributions obtained for icgenerator and free decay, selecting the best of the three combinations; the FKIM is ^ 25 MeV/cz. For the free decay the signal is taken between 480 MeV/c1 and 520 MeV/c with a small background subtraction. The result based upon 167 fTee decay events is expressed as and when this value is averaged with the result of a similar experiment, one gets Iloo/n+_| =• 1-013 ± 0.046 . (3.13) has The phase $00 of n0o been measured by three experiments [Chollet (1970), Wolff (1971), Barbiellini (1973)]]. The method in all cases has been to Study the interference between K? and regenerated KS. The results are in good agreement and give an average: • (46.5 ± 13) degrees . (3.14) To summarize, all parameters of K, * 2ir decays have been measured. Recent experiments have substantially changed the value of what seemed to be the best-established parameter |nt_|. The currently accepted values of the parameters are [Kleinknecht (1974)]. *+. = (45.0 ± 1.3) degrees 400 - (46.5 ± 13) degrees (3.15) -3 ln+_| - (2.279 ± 0.0Z5) « 10 h0B/n-l - 1.013 ± 0.046 . NEUTRAL KAON SYSTEM SIDE VIEW OF T|OT- DETECTOR 61 Lecdgloss Blocks Counter Hodoscnpe AnticartEr S h Fig. 9 Experimental set-up of Holder fct al. ^Holder (1972)3 Eor the detennin; of KJ • Moving Regenerator Kt—*~3TI0 events Decay Vv1« Oisiribulion - No Regenerator Moving Regenerator 8>30 6y INVARIANT MASS (MeV/c'l DISTANCE FROM ANTI- COUNTER Mass distribution for 6y events in the Vertex distributions of .he 3n° decays for free experiment of holder ct al. [Holder (1972)] decay and regenerator events in the experiment of Holder et al. [Holder (1972)]. The tree decay events have been weighted according to the pro cedure described in the text. Free decay A> events |200r ~ 18o(- | m~ "tM300 40 0 500 600 3 DO 400 500 600 Fia. 12 Scatter diagram of the mass of the 4y ( 'ersus the maximum y transferse momentum for re generator and free decay events in the experiment of Holder et al. [.-»lder (1972)] of the 4y s, the Monte Carlo prediction For the K£ •+ 3n° background is indicated. 0.08 006 Q0A £ 0.02 X 0 I • I -T- 10 20 ^ -0.02 K1 DECAV TIMEx'(10"'°sec) -0.04 -006 0.08 V Fig. 13 Charged asymmetry as a function of the apparent decay time T' for the K decays. The i i ca*-> is the best fit to the data. _î NEUTRAL KAON SYSTEM 4. SEARCH FDR K| - 3r DECAYS Since the kaon has a spin zero, the total angular ranentum of the 3JT system is zero. In the Kc rest system let i be the orbital socentun between two pions [n* and n~ in the case of TI*TT~TT° decays} and L the orbital nonentirn of the third pion with respect to the system of the ether two; L + Z is equal to zero. For the 3tr° system, because it is completely symmetrical, L = I = 0. The parity of the 3rc system is P - C-l)1 • t-DE+L= -1 • (4.1} The C operation exchanges n* and TI", therefore it is equivalent to a parity operation an the charged dipion system: C(n+uV) = C-li* (4.2) COrVir0) = -1 . (4.3} CP(IF it ÏÏ») = (-I)* (4-4} CTOrVir0) = -1 . (4.5} D Neglecting terms of order z r,/rs, the decay K's * 3TT j? strictly forbidden under CP conservation. The decay JÛ •+ TT+TT-7r° violates CP for even values of E. It is ailowed for odd values of £, but since in that case the pions have to he at least in a p wave (£2 1), the decay will be strongly suppressed by the resulting centrifugal barrier. Consider now the isospin space with axis x, y, z corresponding to ITJ, T The observed states are The G-parity operation is defined as G - e1I7la C , (4.7) where e 2 is a rotation of 1B0° around the y axis. Applying the G-parif operation on the pion states, one gets G is a multiplicative quantum number. Therefore GfnVn*) u -1 . (4.9) Since tht- charge of the 3ir system is zero, the isospin vector is in the x, y plane; a rotation around any vector in that plane will give the same result as a rotation around I,; in particular one can take the total isospin vector I: G(»Vr°) * cl"h C(TTVV) - e"1 C[**nV) . (4.10) Since I D 0, 1, Z, 3 and G = -1, the equality (4.10) gives I+1 CCUVT.") - (-i) . r4.ll) Comparing Eq. (4.11} to Eq. (A.A), we get for t.ie ir+T~ir* system of total angular momentum zero the following quantum numbers : CP - +1 , I - 1, 3 ... I = 0, 2 , CP = -1 , 1 = 0, 2 ... I - 1, 3 . he will neglect all terms with I > 0 because of the centrifugal barrier effects. With that approximation, Kg * TT*TTTT° can occur only via CP violation. The CP = -1, I = 3 state can be reai_hed only through a AI * %, % transition. Ke define A(Kg - TIV-D) AfXg * n'li'»6) n+ 9 n = e 0 4 12 -° " A(KL * v*fv ) ' »" A(KL + r°n TT ) * ( - > There have been various measurements of n+_0i listed in the references, which give average values: 3 Re n+_0 0.10 + 0.10 [4.15) Im n^,, - 0.14 ± 0.16 . The real and imaginary parts of n+_„ are compatible with zero. For noao, one experiment [Barmin (1972)] gives an upper limit: Insool < 1-1 (901 confidence level). (4.14) As AI = Vi or 7/j transitions are extreœely unlikely, one can also ignore the I = 3 final state for the three pions. In th;s case there is only the 1=1 amplitude which can contribute, and from the Clebsch-Gordan coefficient one gets n,** = v A n^, - (4.is) 7 Mien the absence of AI =• %, /i transitions is assuoed, n.j«g can be derived from the n,_a measurements. Jn conclusion there is no evidence of CP violation in K|i -*- 3n decays. 5. T VIOLATION IN K° DECAYS The violation of CP clearly implies that T or (and) CPT are violated. It has been shown by several authors*' that the experimental results can be used to test T and CPT invariance. In the present analysis we shall follow the general treatment of Schubert [1970) which does not assume either CPT or T conservation. The time evolution of the neutral kaon is described by the equation ^[!;HC;] With the K°, W b~. is, the 2 * Z matrix M can be written as: 5A - | - eA + EA y • SA where w - (M,, • ^>/2, 4 - ^ - Mj, and M^ - «L_s - irL>s/2. The eigenvalue equation M|I(J> a m|ifi> has two solutions: Kg •=-£[(! * e + «) K' + (1 " c - 5) K3] , K? =^r[(l • e - C) K° - (1 - e + 6) F] . E and & are complex paramete. s, and using the Wigner-hfeisskopf formalism it has been shown by Caneschi and Van Hove fJCaneschi (1967)] tlut CPT invariance implies 6 = 0, while time reversal requires e = 0. wc .-ill need to use the relation derived from unitarity by Bell and Steinberger [Bell (1966)]. Let *(t) be an arbitrary mixture of K| und K£, and T the transition matrix for the decay of the neutral kaon system into final states f. The conservation of probabilities at any given time t leads to the equation •) Charpak (1967), Casellc (1969), Kabir (1968). NEUTRAL KAON SYSTEM £ <*wiMt)> * £ ! when. £f indicates the stftnation oyer all final states. Since this equation must be verified for any mixture of K£ ard K£, it yields four relations: 2 i(l^ - Mj) - £ | ! iff^-^) - £| itf.^ - M^)(KJ|K|) - £ ifl^ - M^J(K||K^) • J] (f |T|K|>* (f|T[Kj> . (5.8) f The two relations (5.5) and (5.6) are trivial. Relation (5.8) has been derived initially by Bell and Steinberger; relation (S.7) is the complex conjugate of (5.8). Using the Schwarz inequality |£(f|T[K|>* (f|T|Kj)|! < 2^ l<£|T|K£>t= * £ |({ TK|>|! , (5.9) f f f one gets from lelations (5.7) and (5.8) [(ISLlii)' . (^ . V»] |(K||K'>|' < Vl , (5.10) and using the experimental values for r-, r. and (m. - m_) : |(K||K£>| < 6 * W~! . (5.11) We define <2ir, I - 0|T[K|! " * (2., I - 0|T|KJ) (5-12) and E î Introducing the very good approximation rg = |(2ir,0|T|K^)| , and replacing i [»!,_ - Mj] (2 Re e - 21 Im S)= (eo * C)rs . (5.14) Using Eqs. (5.3) and (5.12), one finds a second relatior l, . e - 6 • a» . (5.15) where no " (Ao - A,)/(A, * 50) l6 lâ Ane ° = <2n,0iT|K») , A> ° = (2u,0|T|K°) ; 50 is the I = 0 mr phase shift; A0, AB are complex amplitudes. Talcing a phase convention such that A„ and Â0 have the same phase, for instance the Wu and Yang convention [Wu (1964)], one gets Im a0 = 0. Then under CPT invariance oa = 0. We can separate e0 into two parts: i) e, T-violating and CPT-;onserving; ii) S = 6 - a0, T-conserving and CPT-violating. Using the experimental data, the two complex equations (5.14) and ( 5.15) allow a determination of the two complex parameters e and 6 = 6 - afl. To a good approximation [Bell (1966)]: co - Vj n+_ + Vs HDO • (5.16) With the data of the previous section, one gets 3 Re e0 = (1.60 ± 0.11) * 10" (5.17) Im e, = (1.63 ± 0.11) x 10-3 , where all errors except the one on *nn give negligible contributions; the errors on Re eB and Im E0 are therefore completely correlated. We then proceed to the evaluation of the various possible contributions to c. From Gobbi (1969) one finds for the two pions in the isotopic state 1=2: - Re c2u ? ° (0.0 ± 0.3) x ID " , Ira C;ir ; = (-0.5 ± 0.2) x 10"" . For the 3n states, only the totally symmetric three-pion state 1=1,3 give non-negligible contribution [Gaillard (ly67)]. As indicated in Section 4, the assumption that AI =• % is absent in CP-conserving and in CP-violating tran sitions permits one to derive ç from n+_ alone: Re ç » (+0.5 ± 0.5) * 10"* , Im C]Tr = (-0.7 ± 0.8) x 10~" . For Çi__tt the limit is harder to get because none of the K. experiments have been analysed without assuming CPT con servation. Schubert et al. (1970) have reanalysed the results of two experiments [Weber (1968), Bennett (1972)] to get 1 (-2.4 * 2.1) x 10 Im çlept = (0.9 ± 3.4) x lO" " , The error on the latter limit will be greatly improved if the new AS/AQ experiments are also analysed without assuming , CPT. The contributions of other decay modes to ç are negligible; combining all the results given above, one t,ets Re ç = (-1.9 ± 2.2) * 10~* , Im ç = (-0.1 ± 3.5) x IO"" , (5.18) Re e o ( 1.54 ± 0.23) « 10-3 (5.19) In c - ( 1.49 *. 0.24) x 10"3 (S.20) Re Ô = (-0.06 * 0.25) x 10"S (5.21) In I = (-0.14 ± 0.25) x lO"1 (5.22) It is cle-- rom these values that e, the CPT-conserving T-violating amplitude, is different from zero and that 6, the CPT-violating T-conserving amplitude, is compatible with zero. HEUTRAL KAON SÏSÎSH A clear illustration of the numerical results can be given using the Vu and Yang Argand diagram (see Fig. 14). For 5=0, Eq. (5.15) gives es = e, and neglecting the ç contribution, Eq. (5.14) shows that arg E argfiO^ - M£)] = 43.8° . Furthemore, Eq. (S.14) shows that 6 is orthogonal î*> e. The quantity ce = % n+_ + Vj n00 is the difference between two vectors E and ô. The results given on E and on fi show that the error on 6 is about % of the value of E. Therefore in the Wu and Yang diagram, eD lies within a cone of half-angle ^ ±10° around e. 6. CP VIOLATION UNDER CPT CONSERVATION 6.1 Phenomenoloqical analysis Under CPT invariance the two states of the K°-K° system observed in weak decays can be written as (6-1) 8 K° =-^[(1 + O K° - (1 - E) K ] . There are two levels at which the CP violation can be introduced in K decays : a) The mass matrix can contain off-diagonal terms which induce the transition K° *-* KD, i.e. e f 0. b) By convention, the I = 0, K* + 2n decay amplitude is taken to be real; this is the dominant amplitude in IT de cays. If the 1=2 amplitude A, has a non-zero imaginary part, its interference with the 1=0 amplitude leads to CP violation. More formally, the K° decay into two pions is described by four anplitudes corresponding to the isotopic states I a 0 and I = 2 of the two pions. he define three amplitude ratios: _ <0|T|KL> _ f J_ <2|T|KL) J_ (Z|TjKs> C = <0|T|KS> ' " ft <0|T|KSÏ ' " ft <0[T|Ks> ' The measured quantities are: - the amplitude ratio • (E * E')/(1 + ») ; • the amplitude ratio (e - 2E')/U - 2u) ; (6.4) l z R= r(KS* 2TT°)/rfKs* v*V) = /i|(l - 2W)/(1 > W)| ; (6.S) - the charge asymmetry in K? leptonic decays, which is obtained form Eqs. (6.1) and froa the definitions of Section Z: •v -N»- 1 - I*'1 6 - -S - 2 Re E (6.0) a NJ+ • N£. |1 - x| The quantity e' is given in function of A0 and A2 by «/.JLJlil .!<«»-«.) where &t imd 5 are the un s-wave phase shifts. With CPT conservation, the unitarity relation (S.14) becomes ICHL - M^) * 2 Re e - (e + &TS . (6.8) For t. = 0, the phase of e is the "natural" phase *o = argCitML " M^)] = (43.8 = 0.2) degrees . (6.9) How cuch can the phase of e differ from "•e "••-*£ • -}fc • <6-10' The linits on ç are substantially reduced for the leptonic part when CPT is assined, and we get û*c < S° . (6.11) The measured roiue [Gobbi (1969)] of R •= 0.453 ± 0.007 gives u - (3.1 • 0.9) * ID"2 e'C-39"8'" , (6.12) J where the fac that E' C 10' has been used to neglect Ira A2 compared to Re A2. 6.2 Charge asy netry In Kf leptonic decays From Eqs. (6.1) and from the definitions of Section 2, one derives the charge asymmetry in KL leptonic decays: N„+ - N,- 1 - jx'2 2 N£+ + N£- Jl - x\ where NJ,+(^_) are the number of positive (negative) leptons observed. Bennett et al. (1972) have determined the factor (1 - |ÎCfa>>*11 - xjJ by measuring the lepton charge asymmetry after a regenerator: - 0.96 ± O.0S . (6.14) U- r The measured values of Re x and Im x given by Eqs. (2.12) and (2.13) can also be used to compute (1 - |x|2)/|l - x|J •= 1.042 i 0.05. Several experiments (see references) have measured the charge asymmetry. Very large statistics have been collected, but as the effect is extTemely small, these experiments require a careful investigation of the parasitic asymnetry effects. The corrections to be applied are a sizeable fraction of the measured asynmetry. Figure 13 shows the tine distribution of the charge asymoetry of the decay K° •+ ir~e~v observed in the recent experiment of T-eweniger (1974b). The results of the measurements give an average value 6 = (3.30 ± 0.12) x 10~s , (6.15} which leads to Re e « (1.65 ± 0.08) x 10"3 , (6.16) to be compared with the value obtained from nt_ and nfl0: ] Re e - e cos *e - |Vi n,_ • Vj n00| cos $£ = (1.67 ± 0.04) " 10" . (6.17) The very good agreement between these two determinations of Re e is also shown in the tto and Yang diagram of Fig. 14 which summarizes the experimental situation. S.3 Comparison with theory x s CP violation effects are very small (|n+_l ^ 2 10~ )( and despite massive expérimental efforts no sign of CP violation has been uncovered elsewhere than in the neutral kaon system. It has been proposed to introduce CP viola tion as a perturbative effect in either of the known classes of interactions: strong, electromagnetic, weak. The general scheme consists in adding a CP-violating term H_ to the usual CP-conserving Hamiltonian: NEUTRAL KAON SYSTEM 2-10' 1-10J - Fig. 14 Wu and Yang Argond-diasrno of i meters in K° decays. The value taken foi is the "natural1 phase *0 - 43.8°. H -H. • 1 • where Hd represents the strong and electromagnetic interaction Haniltonian, and H is the weak interaction Hamiltonian. The coupling constant of H_ is adjusted to yield the expericental value of e. Although it is not impossible in general to invent ad hoa theoretical circumstances to explain why no other effect of CP violation would have been seen, these theories often becorae less attractive because of their increased complexity. Wolfenstein [1964) has proposed that a new type of interaction could be responsible for CP violation. The superweak theory assures that the CP-violating Hamil tonian H_ is a |iS| = 2 term H . This Hamiltonian produces, to first order, unequal non-diagonal terms in the neutral kaon rass matrix because it can induce K° ; K9 transitions. In this theory the CP violation comes only from the mass aatrix, thcrcfoie E' ° 0. To produce the tern c£/n in the mass oatrix [see Eq. (4.2)], the superweak coupling constant G rcust Le related to the weak coupling constant G by 5SM » IcJ ^ (6.18) nth (^ « ID'5 and |e| = 1 ' the relation gives Because of the sroallness of G , the only detectable effects of CP violation will occur for neutral kaon decays into two pions. The supen.eak theory gives definite predictions for the (KL * 2rc/Kg -+ 2n) amplitude ratios: s n+_ n00 = E (&.20) $ = arctg [2&n/(rs - I"L)] = C43-S ± 0.2) degrees . (6.21] The relation (6.21) is obtained by setting t = 6 = 0 and e - e in the unitarity relation (5.14). The predictions of the superweak theory are well verified by the experimental results. As shown in Eq. (6.7) the phase of E' is equal to TT/2 + (62 - 6B), where S2 and ô0 are the s-wave nir phase shifts for the 1=2,0 isotopic states. Recent measure ments reported by Kleinknecht (1974) give arg (e') = 37 ± 6° . This value can be used in connection with the values (3.15) of the CP parameters to set a one standard deviation limit: |E'|/|E| < 0.02 . But the predictions of the supcrweak theory are essentially valid for most theories where the CP violation is intro duced as a milliwcak effect, H_ being a |iS| n 1, CP = -1 Hamiltonian. 7. CONCLUS TON The neutral kaon system has provided beautiful illustrations of the wave aspecr of particles through regeneration of K§ from K£ and measurements of the KL~K§ interference via CP violation. The discovery of the 2TT decay of K? has generated an intense experimental activity on the neutral kaon s>stem. CP violation and T violation have been proved, but although superfine experiments may eventually disprove the superweak theory, in the present situation the pro spects of finding more about the origin of CP violation seem rather gloomy. 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Solmitz, F.S. Crawford Jr. and M. Alston-Gam jost, Phys. Rev. Letters 21, 498 and 715 (1968); see also B.R. Webber, Ph.D. Thesis, report UCRL-19226 (1969). Webber, B.R., F.T. Solmitz, F.S. Crawford and M. Alston-Garnjost, Phys. Rev. D J., 1967 (1970). Webber, B.R., F.T. Solmitz, F.S. Crawford Jr. and M. Alston-Gamjost, Phys. Rev. D 3, 64 (1971). Wnlfenstein, L., Phys. Letters 13, 562 (1964). Wolff, B., J.C. Chollet, J.-P. Rëpellin, J.-M. Gaillard, M.R. Jane and K.R. Schubert, Phys. Letters 36B, 517 (1971). Wu, T.T., and C.N. Yang, Phys. Rev. Letters 13, 380 (1964). f WErSSKOPF-WIGNER TORMAUSM APPLIED TO THE K'-K" SYSTEM Strangeness is not conserved in weak decays. As a consequence K" and W, the states of strangeness *1 and -1, can decay into the same channels and 3re not eigenstates for the decay processes. As there are two particles Ka and K1*, let us assume the existence of two independent linear combinations of these states having an exponential deer/ law. The 'Vcisslopf-Wigner formalism permits the determination of the relevant linear combination am. their decay constants. Let a(t)|X°) • bOOlK3") (Al.l) be the general fom of these two linear congélations ; then The superposition principle leads to the wave function |*(t)> =a(t)|K°) +b(t)|F)+J FjCtJlFjî , (Al .3) j where the F-'s represent the decay products. The zero of energy being taken as the K° (K°) rest energy, the tune-dependent Schrodinger equations for a, b, and Hi's are lg - Z »ajM FJK..J .-*- ,,U.4, i^^-^DljaM^MjbMb]. (A,-<" are where H, :(<•!)• %]M the decay matrix elements and H. = H*-, Hjb = t^.:. The integration of the set of equations (A1.6) with respect to time will determine the F-'s: ) t ilA Xt/Z : J dF.(u,t) - Dlja(«)a(Uj + Hjb(u)b(0)] f e ' dr . tïith the initial conditions V-.(u,0) =• 0, one gets „iut-At/Z , IFjC.t) - CHjaM»(0) • HjbMb(.)] ;„ . >/2- ' . These values of the F.'s are then introduced into Eqs. (Al.'I) and (A1.5) giving (A1.3) j." ^iut-At/2 . 1 - Tp • E C%M V"""" * V»> "jbWX '] " "'"'"""-iw - \/2~ - =-»"-«/" NEUTRAL KAON SYSTEM Sunning over all energy states, one gets: - Tt*1 - £ K [V> "ja(°)a(t) " %(») I'jbtOjbd.)] • J , ,N CA1.U) • r j /a. ,j : C"»'M ""Ma(t) * ""M HftMbw]}]. where p^ is the density of states per unit da, and the two formulae: T e"iut lim / F(u) j^m du B -2ni FCO) and lijn / v + te- = Pè-i" 5(x) c+0 J x IE x have been used. P stands for "principal part", and the approximation X •* 0 has been made on the right-hand side (A is second order in the weak coupling). Similarly - TF • Y, n i'i,m v°)ii(t} * vo) Hjb(°)b(t'] * tMa2 . P { fj» p. i ["bi'"' »iaM»W * 'l, M HibMb(t)]n ^ > The coupled differential equations (Al.ll) and (AÎ.12) lead to the eigenvalue equations: a ziM b »» • »aa * »W' * Cab ' ab' Ma5, *b - (rb3 • 2i^a)a * (rbb . 2^b)b with r 7 0) 0) „B " * I »j V V j j The tho natriees r and M are Hermitian as a consequence of the hermiticity of H. M is the mass matrix. Ke will now derive the solutions of the eigenvalue equations (A1.13) in the special but general case where the validity of the CPT theorem is assumed. It follows froa this assumption that H . = H-r; therefore '*>"* (A1.1S) Maa • «a, • ftjr convenience, one usually takes the auxiliary variables iM'a. b z rba , ... rab ..,* 1 * "a" * ^a " T * '"ab ' The solutions of the eigenvalue equations {A1.13) are then given by *• " raa * 2il,aa * 2pq! ». " *IK°> * «lliWCIpl2 • |q|')'A A- " raa * 2fflaa " 2pq "' »- * (PlK'» " iFWCIpl' * lql!)'/! . >.+ and k_ are both complex parameters, and there is one arbitrary phase in the system, •'he choice is often made such that Xt is real and A_ is complex. The two states v+ and ty_ are usually called K|, the short-lived component, and K5, the long-lived coitponent. With the abov^-mentioned phase convention: r x K ' s . _ ' rL + 2iAn , (Alil8) where in = mri - mj;' is the mass difference between K£ and K|, Tg and rL are the K? and K£ decay widths, respectively. If the validity of the CPT theorem had not been assured, the solution of (A1.13) would have introduced four para meters p, q, r, s, and one would get |K£> =p|K°) + qjF) |K<>- r|K») + s|K*>. NEUTRAL KAON SYSTEM APPENDIX 2 NEUTRAL K DECAY RATES FOLLOWING A REGENERATOR The cross-section foT K°'s is larger than for K0,s because more channels are open. As a consequence, after tra versing some material, the composition of a neutral K beam in terms of K| and K? is modified by the strong i In the vacuum the propagation of a neutral K" beam is described by the two equations ik x K°(x) . e( L -«/2»L) . .'CO) (A2.1) ik x 2fl KJOO = e( s -^ s' . qm , corresponding to the two equations of motion -ïr-K-î'iJ'îM In the case of propagation through ciatter, Eqs. (A2.2) are transformed into ^ . A K-W • B K-M dKs(x) dx Let us consider a thin slab of material (thickness L) in which multiple diffusions of K0ls are negligible. We use the notation N = number of nuclei per cm' in material» f(D) = scattering amplitude per nucleus for K°'s; F(9J =• scattering amplitude per nucleus for K°'s. Let us first consi-ier the amplitudes in the forward direction. A contains» in addition to the terms on the right- hand side of Eq. [A2.21, the scattering amplitude for K£ -* K? Ion the nuclei A- ikL - j AL » 2ii»)fj>(0)/kL , (A2.< f^(0) -7[f«» * λ)] • B = 2tiNf,î(0)/kL C - ikj - \ Aj * 2iriNf,, (OJ/kj D = 2niNfji(0)/ks , fu(0) =i[f(0) • f(0)]- f,s(n) fi.(0) - -fnfO) -i[f - J(0)] , The coupled differential equations (42.3) can be fornallv integrated. To get a better physical insight into the phenomenon, we want to determine the anplirude of outgoing Kg in a différer.: way. Let us compute the differential snplitude of K§ created between z and z + dz ami its propagation to the regsrerator exit. We will neglect double re generation processes, for example: «L * «S * N. • because the regeneration amplitude is always small. Then as Kg(0) = 0, to a very good approximation we can neglect the contribution of K<-(z) to the change in KL;z), giving where o™. is the total cross-section for K.° and we have neglected the K£ decay length AL » x. The second equation i'A2.3) can then be rewritten as ik z M.cVl]tDe-^ KL(0) e L . The first term on the right-hand side of Eq. [A2.9J describes what is happening in the interval (z, z + dz) to the K^'s which have been created before z. The amplitude of K^'s created between z and î + dz is given by the second terra: d.-Wz) - D C-W2 KO(0J ei^ dE , (A2.10) At the end of the regenerator that amplitude will be modified by decay, absorption, and by a phase change, the factor being e-N{x-2)°T/2 ^(x-z) e-tx-z)/2As (A2.ll) (oT(.K|) = oT(K£) - \ C»T(K") * OjCÎ?)]] . In the forward dir _tion, the amplitudes coming from all dz intervals are coherent; they have to be added, giving for the K? amplitude at the end of the regenerator: -N(x-z)0 /2 ik Cx-z) -(x-z)/2A, K|W • T s S d^M 1zCk -k ) -(x-z)/ZA -Nxcr/2 ikg; J D Kj(0) e L s s ^(--DK^xl/c-'^^^-^e-'^'^dz The nass difference i, expressed in units of the K| mass width n/ig, is given by i , (%. - HK.]/(VTS) - fc. - kL)BYcTs . The integration of Eq. (A2.12) gives , 2-W.Sf„[0) [l - e-L(-i&*V,)-\ kj -is • % V*' • where £ = x/A~. NEUTRAL KAON SYSTEM In the forward direction Kg are regenerated coherently along the regenerator length with a slight difference in phase due to the nass difference between Kg and K?. A Kg regenerated at a point z will have travelled as a K£ with a wave nizrber k^ up to that point, and after as a Kg with a wave nuober kg- Let us new consider the Kl regeneration at an angle Ô with respect to the beam direction. Everywhere in the equa cos e e Tlie tions the difference k, - kg should be replaced by kL - kg = ^L ~ ^S * ^5 ' *^- coherence could be main tained in the forward direction because the K^-Kg1 mass difference is small. For example, in the case of a regenerator of length flg, 51 = 0,5 rad. Only for angles such that e* s zp^. k^/ks^Kr1" will the coherence condition be fulfilled for scattered kaons. The limiting angle is extremely small; therefore the scattering centres act incoherently for the diffractive regeneration. Consider a thin regenerator, such that multiple scattering of kaons can be neglected. The diffracted Kg amplitude produced in the interval (z, + dz) at an angle 9 is given by 2 ik Z d'J^fz) - f,,(e) «.-'""T'' Kj(0) e L dz . (A2.1S) The scattering angles are small: , _ 300 MeV/c p A'A ' where p is the kaon mesentum and A is the atomic number of the regenerator nuclei. Optical model calculations show that the single scattering angular distributions of K. and Kg are well represented for small-angle scattering by z Gaussians of almost equal width. |f2|(0)| is given as a function of the forward scattering amplitude by 2 2/2b ;fzi(S)|* = |f3lC0)| e~° , (A2.16) where b is the characteristic diffraction angle of K3's on the material nuclei. It is inversely proportional tu the momentum and can be evaluated by optical model computations. As the contributions from different scattering centres are incoherent, the intensity of diffracted Kg at the end of the regenerator in the forward direction and per unit solid angle is given by 1 e IDCx) = NAglfziCO)! (1 - e" ) IKj>(x) . (A2.17) Taking the square of Eq. (A2.14), one gets the intensity of the forward coherent K2's: , , CNX^j- |f„(0)|' , e-1 . 2 e-m cos „, , (x, , „, w . In most of the experiments with regenerator, a thick piece of material is used; we will therefore study the question of multiple diffusion. This problem has been considered in detail by Good [1961}. The regeneration of KT from K<- can always be neglected. In the small-angle approximation, the n scattering problem reduces to the combination of n Gaussians of equal width t>- The net effect is to widen the distribution by a factor Sn, leading to a distribution: where the factor l/lwrib* has been introduced to normalize the integral over the angles to unity. Because of CP violation, we will include thL amplitude for K£ decay into two pions to get a general expression for the 2n decay intensity after a regenerator. Let Ag and AL be the 2TT decay amplitudes for Kg and K£; then for a beam multiple scattered n times, the ZTT decay rate after a regenerator relative to the transmitted K, intensity and for a scattering angle 9 is given by XX X n ïjj£i - Clto„) J dz, J dz, ... J dzn MO C„(B) , (AZ.20) Vi 1 where cn = Znb ;f „ (0)|* is the K£ diftractive cross-section: ^ -K-i-az ^x'l° •"CMiV"s] *» ^r • i-l where s- is the point where the Ks is regenerated, ûMoMKc -MK» - i(rL- rg)/2, T is the proper time counted from the regenerator exit face, and ••'W.w1'^1' (A"2) Looking at Eqs. (A2.20) and (A2.21), one sees that for n = 1 the integration will lead to the r-suit (A2.17), pro vided the term A^ is emitted and the term p is neglected. The latter term corresponds to regeneration along the particle path followed or preceded by a scattering; it Ls really a second-order term which could be neglected to establish Eq. [A2.17). In wTiting the full expression of the intensity of 2s decays after a regenerator, ve shall usu the following re lations and appropriations: Oj. = ZX lm [fîz(0]] Sir.ce fjjCO) is nearly pure inaginary, we will raake the approximation that |Im fmO]|a - lfia(0)]2 , then 1 2 Zirb • oD • Wi will use the following notation: •« =arg [f„(0)/i] * * arg (p) a *cp arg (ALAg) 1 £ J H - Zi* » WlgU - e" ) |£zi(D)| . perfoiming thr integral (A2-20) and integrating over angles gives: e rS /2 S * 2K*sl " ' [HT " «' * «0 - »CT> " -iSTTTT Ip <°s <«* * »p - *a> " •») j where the notation a"'"'/[n - m)! means that the term docs not exist for n < m. 1 NEUTRAL KAON SYSTEM In general, for the experinents each scattering order has a different efficiency, because the angular distributions have not the same widths. Those efficiencies have to be multiplied in before suming over the different orders. To get the total IT decay intensity I(T), with the efficiencies all set to unity, we sum the In(T). starting with n = 0 which corresponds to the transmitted Kg: r T l(i) = ^jlAJ' • IAj.1» |p|> (1 • B - tafa . fcj/o*) - 2 For orientation purposes: $zz is very small (a few degrees) In addition to the diffractive and coherent regeneration, there is a contribution from inelastic kaon interactions. This effect can be estimated, and within the angular range accepted in regeneration experiments it amounts to a few per cent of the diffractive regeneration contribution. It can be neglected in all the experiments which have been per formed. 1 r - 220 - CHAPTER V APPENDIX 3*} KODELS OF CP VIOLATION Various nodcls have been proposed to explain CP violation. Host of then are now eliminated by the fact that no CP violation has been observed outside the K° system. It is however instructive to recall briefly how the models were constructed and some of their consequences. As mentioned in Section 6.3 the Hamiltonian is generally assumed to taie the form: H = H0 + 1^ + H. where Ho corresponds to the usual strong and electromagnetic interactions, U^ is the weak interaction Hamiltonian. Both Il0 and \\f are CP conserving. The extra term IL violates CP, and the models can be classified according to the additional properties of H_ into four general categories: millistrong, electromagnetic, milliweak, and superweak. In each case the strength of the coupling constant of H_ must be adjusted to yield the correct magnitude of the CP viola tion in KD decays. In Table 1 we have given the ratio F/G between the coupling constant of H_, the Hamiltonian which induces CP violation and the weak coupling constant C = 10""s nC2. Table 1 CP-violating Neutron Type Properties of H~ amplitude in K° •* 2-i decays electron dipole moment US = 0, ÛI = 0 conserves P 10"'' e x cm Millistrong F/C = 10 ' violates C AS = 0 Electromagnetic conserves P OU* • KL HT1' e » an violates C F/G = 10' A5 = I 13 Mil îweak 1Ù' e « cm violates CP F/G s 10"3 AS = 2 H_ !9 9 10" e x cm Superweak violates CP F/G = 10- For a detailed description of the consequences of these models we refer the reader to the rapporteur talk of J. Prentki at the Oxford International Conference on Elementary Particles [196SJ. Experimentally expected interference effects between the C-violating Hamiltonian and other terms 'live been carefully investigated without any success. For example In n" * n*n"n° the C-violating Hamiltonian H_ could b^1 expected to produce asymmetries in the charged pion dis tributions by interfering with the electromagnetic interaction responsible for the decay. No significant effect has been observed at the one per cent level. [f the oillistrong and electromagnetic C-violation should be eliminated on the basis of the experimental evidence, the situation is not as clear for the milliweak theories which do not differ much in their measurable predictions from the superweak theory except for the value of the electric dipole moment of the neutron (jee last column of the table). A non-zero electric dipole Jioraent of the neutron can arise only if T and P are both violated. Therefore the dipole monent cL, could correspond to a term with the CP-violating in* iction (CPT is assuned), a weak P-violating interaction and an electromagnetic interaction. In the electrotsagneti :1s the first and the third interactions can correspond to the samp vertex. From a simple dimensional estimate of iiie electric dipole moment •) This appendix is based in part on material prepared by Prof. A. Frenkel. NEUÏRAL KAOH SYSTEM "„ = Il «p « e where ?. is the product of the necessary interactions one gets the values of Table 1. The present experinental licit is d„< 5" lu'*1 on « e and seems to rule out all models except the superweak and the milliweak ones- CHAPTER VI INTERACTIONS OF NEUTRINOS O. NACHTMANN INTERACTIONS OF NEUTRIMOS 1. INTRODUCTION In this chapter we want to discuss the phenomena which occur when neutrinos scatter off nuclcons. As projectiles experimentalists use muon- and electron-neutrinos and antineutrinos. The target can consist of free protons, e.g. in a hydrogen bubble chamber, but in most experiments done so far the target consisted of heavy nuclear material such as freon (CFjBr) or iron (Fe). The use of heavy nuclei increa'ds the statistics at the expense of introducing nuclear effects, which have to be taken into account if one wants to extract cross-sections for protons and neutrons from the experimental data. We will treat only the "classical" neutrino reactions involving charged currents. The reactions for muon-neutrinos are written down in formulae (1.1). The reactic'is for electron-neutrinos are obtained by replacing v and u by v and e: v..00 + N(pl * iTCk') * X(p') U ^ (1.1) Vytk) + N(p) * u*(k') + X(p') . The four nonenta of the particles are indicated in brackets. N stands for the nucléon target, X for the final hadronic state. A diagrammatic representation of the reactions (1.1) is shown in Fig. 1, where 6 is the scattering angle in the laboratory system. The neutrino scatters and turns into a muon. One unit of charge and a four-momentum q = k-k' li(k') X(p') Fig. 1 Heutrino-nucleon scattering arc transferred to the nucléon, which can either recoil — a quasi-elastic event, or break up into several hadrons — an inelastic event. Froo the way the nuclcon recoils or breaks up, we hope to leam something about the structure of the nucléon. For neutrinos of laboratory energy E in the GeV range, cross-sections are typically of the order 10">8 c-'. This follows from a simple dimensional consideration. The cross-section for the reactions (1.1) must be proportional to G1, where G * 10"! GcV-1 is Fermi's constant. Multiplying with factors of the order of one in units of GeV to get the dimensions of a cross-section, wc find indeed - 10-' GeV1 • 10": It is clear that the measurement of such small cross-sections presents a formidable challenge to experimentalists. The reaction (1.1) presents quite different aspects depending on how much energy and momentum is transferred from the leptons to the nucléon. Convenient kinematic variables are the energy transfer in the laboratory system v = pq/H, where M is the nucléon cass, the four-momentum transfer squared Q2 = -q2, and the invariant nass squared of the final hadronic system N* = p'1 *'. These three variables are not independent of each other but are related in the following vay. -H' 2Mv - The v-Qz plane is shown in Fig. Z. , - (1) (2) - /^T ^^ (5) //- 1 Mmi^smm^^mmi^mim^^m^k^smsémmmsm^. 1 2 3 15 6 v (GeV) Fig. 2 The v-Q! plane. (1) quasi-elastic line; (2) inelastic threshold; (3) resonance region; W region of Adler's CVC and PCAC teata; (5} deep inelastic region If the nuclcon recoils without breaking up, we have a quasi-elastic event. Since the finai hadronic state con sists of a single nucléon, we must have K* • M1 and 2Mv •= Q* [line (1) of Fig. 2]. The physics is described by fom factors which arc completely analogous to the electromagnetic form factors of the nucléon. tn the next sioplest process the nuclcon gees over into on.' of its excited states — a resonance is produceu which then, of course, decays again, giving most of the time a nuclcon and one or more pions. Such processes are the dominant feature in region (3) of Fig. 2. Clearly the threshold for such inelastic processes is • M* 2Mu - QJ = (M + m^)2 , (1.2) where nu, is the pion mass [line (2) of Fig. l"). For very snail values of Q1 some subtle effects occur in inelastic collisions, which allow us to test the pr; ciples of CVC and PCAC already introduced in Chapter III [region (4) of Fig. 2]. .) We will use metric 1 of Chapter II, Appendix I, throughout this chapter. INTERACTIONS OF NEUTRINOS If finally both v and Q2 become large [(5) of Fig. 2j, we have a so-called deeply inelastic collision. In such collisions the nucléon is forced to absorb a lot of energy and momentum, and one might hope thereby to learn something about the fundamental structure of the nucléon. If we hit it hard enough the nucléon should, so to speak, reveal what its building blocks are. These deeply inelastic collisions are quite analogous to the large-angle collisions of a particles on atoms which allowed Lord Rutherford to gain fundamental insights into the structure of the atom. Hopefully some equally fundamental insight into the structure of the nucléon will be obtained from the present-day deep inelastic scattering experiments with electrons, muons, and neutrinos. In the following sections we will study the various regions of the v-Q* plane fFig. 2) in turn, but before doing so we have to write down the interaction Lagrangian which will be the basis of our discu ••ion, and acquaint ourselves with some general theorems. We also want to cite Marshak (1969] and Llewellyn Snith (1972) for further reading. Z. THE INTERACTION LAGRANGIAN AND A GENERAL THEOREM In Chapter III we studied extensively semi-leptcnic weak decays. An example is neutron beta-decay: n* p *e~ • 0e . (2.1) If we move the antineutrino in formula (2.1) from the right-hand side of the equation to the left-hand side and change it into a neutrino, we obtain a neutrino reaction (inverse beta-decay) ve + n -* p * e~ . (Z.Z) A very general principle of local relativistic quantum field theory, the substitution law £Jauch (1959)j, tells us that processes which are related by such a "crossing" transforaation arc described by the sane interaction Lagran gian. The basis of our discussion of neutrino reactions will therefore be the effectivt -eak Lagrangian for seni- leptonic decays which was found to be of the current * current type in Chapter HI: where G is the Fermi constant, j| the lepton current, and j| ' the hadron current in the form given by the Cabibbo theory: ,(h) 6zh cos e J sijl fl ' c J * c h where e is the Cabibbo angle; Jr are currents which conserve strangeness, j" change strangeness by one unit. Ir. terras of the octet of vector and axial vector currents introduced in Chapter III, we have h (v;. ivp • CA; * iA») (2.6) where 1, 2, 4, 5 are SU(3) indices. In the calculation of amplitudes for neutrino reactions [formula (1.1)] the interaction Lagrangian [Eq, (2.3)] wilL be used in first order only (that is why it is called an effective Lagrangian) and we will in general neglect radia tive corrections. This cannot be correct fur arbitrarily high energies of the incident neutrino since unitarity eventual ly forces us to consider higher-order weak effects. It turns out that these effects must be significant for neutrino energies in the laboratory system E J 5 • 10" GeV. At present, energies up to E - 200 GeV are available so we are still far away from this critical energy of weak interactions. We shall see in the following that so *ar all experimental results arc well described by the effective Lagrangian [Eq. (2.3)3- Higher-order weak effects will he further discussed in Chapter VII. CHAPTER VI Equations (2.3)-(2.6) contain already many assumptions about the structure of weak interactions which can be tested in neutrino reactions. In Eq. (2.4) we have assumed muon-electron universality. This has been tested for both neutrino and antineutrino scattering on complex nuclei [Eichten (1973)]. The ratios of the total cross-sections for incident neutrino energy less than 10 GeV are found to be —- = 1.26 ± 0.23 —- = 1.32 ± 0.32 . "V Within the large experimental uncertainty, electron-muon universality is confirmed. In Eq. (2.6) the strangeness-conserving (AS n 0) currents J" are assumed to be Ai = 1 operators, i.e. members of an isotriplet. Tests for this assumption will be discussed in Section 3. The strangeness-changing [AS = 1) cur rents j" are assumed to the AI = Vi operators, i.e. members of (two different) isodoublets. Owing to the smallness of the Cabibbo angle (B = 13°), neutrino reactions with AS = 1 are hard to observe and the properties of the strangeness- changing interaction are poorly tested at high energies. For most of the following we will restrict our attention to the strmgoness-conserving (AS = 0) neutrino reactions. Next we want to discuss a test of tlie current x current forn of the Lagrangian [Eq. (2.3)] fPais (1962), Lee (1962)]. Consider, for exa-iple, a neutrino-induced reaction v/k) + K(p) -n'Ck') * X(p') . (2.8) According to the rules of the arc [see, for example, Bjorken (1S65)] the S-natrix eleaicnt tor this reaction is obtained froci the interaction Lagrangian [Eq. (2.3)] in the following way*': Sfi = i(2i0*6tV *p' - k - p)(u~(k')X(p')!Leff (0)!vi|(k)N(j») ) . (2.9) If we substitute the current * current Lagrangian Eq. (2.3) in Eq. (2.9) the natrix element factorizes into a leptcnic and a hadronic part. The leptonic part is easily evaluated using tne explicit forn for the lepton current [Eq. (2.4)]; the hadronic part contains all the dynamics of strong interactions and will be the main object of our study later on. After a little algebra w find h)t| 5fi = -i -| (2r)"6(k' + p' - k - pju/k'W'Vl • Ys)Uv(k)(X(p')]j( K(p)> , (2.10) where u (k) and u (k') are the spinors of the incoming neutrino and outgoing muon plane waves. As a concrete example we imagine a neutrino expericent with an unpolarizcd target where we observe the energy and direction but not the polarization of the outgoing iruon. he will furthermore assume that wc have sonc device v-hich selects only those events where the hadronic final state falls into a given class r of hadronic states, (h'e could, for exanplc, select all events with one pion, etc.). The- differential cross-section with respect to the energy E' and solid angle Cl' of the outgoing nuon in the labora tory system is now easily derived fp*n Eq. (2.10). Ke can write it as product of a lepton tensor t " and a hadron tensor W ,: >.r) 3E'3n' Sir2 E P* ' J 3 ») Ke will normalize our states according to = 2p0(2n)'6 (p' - p) and the spinors according to u Lpins fP)"(p) - I* • m , I in5 v(p)v(p) *f - m . INTERACTIONS OF NEUTRINOS where E is the energy of the neutrino and k' the momentum of the muon, both in the laboratory system. Tor antineutrino reactions we obtain a similar formula with u replaced by v. The lepton tensor can be calculated in an elementary way using the formulae of Chapter II, Appendix I, to perform the sum over the polarizations of the muon. We find: where E .„ is the totally antisymmetric tensor with e01. The hadron tensor is defined by KS'n = I ^6(P' " P " q) from first principles. We have to resort to building models and we will discuss seme of them later on. For the mo- nient we want to see what we can learn about W from basic principles such as Lorentz invariance alone. Suppose that our device which selects hadronic final states if they fall into the class r is indifferent to the elementa and polarizations of the hadrons, i.e. the sum in Eq. (2.13) includes 3 sum over polarizations and integration over all momenta of the final-state hadrons considered: V could, for example, be the set of all hadronic states or the set of all hadronic states with a given nurcber of pions, etc. Then the Lorentz tensor W , [Eq. (2.13)] must be constructed fnxi p and q, the only four-vectors remaining. Its general form must therefore be K(v,n ., , Ppfr „ * W (VA * VA>W> * é "y* - VJ)K- • (2-14) where the scalars K. (i = '., .... 6) depend of course on V and on the two invariants we can form cut of the two vectors p and q, i.e. v = p*q/M and Q* = -q1: Wi -H^'^tv.Q*) (i - 1 6) . (2.15) For a sum over all hadronic final states, W& is zero if time reversal invariance holds (see Exercise 1). Now we observe the following: W. [Eq. (2.14)] is only of second order in p. Similarly, if we express I [tq. (2.12)] in term» of k + k' and q - k - k', it is only of second order in (k + k'). Therefore apart from an over all factor the cross-section [Eq. (2.11)] can be at most of second order in (p, k + k') with coefficients being func tions of v and Q*. It is convenient to use instead of (p, k + kf) the variable s-u defined as follows: s = (p + k)2 u = (p - k')2 s-u n 2(p, k + k'] - m* = 2M(E* E'] - n* . (2.K.) For this class of final states r we obtain then the following form for the cross-section: ..ELJHHJAf».->f. JC»,Q'): A, B, and C are easily expressed in terms of W. (i = 1, , 6} [Eq. (2.14)3 (Exercise 2). The important result is that the cross-section is a polynomial in (s-u) if we keep v and QJ fixed. This is a direct consequence of the current x current form of the Lagrangian [Eq. (2.3)], which allowed us to factorize the S-matrix element and the cross-section into lepton and hadron parts. The fact that the highest power occurring is (s-u)2 re flects directly the vector nature of the currents. For scalar currents only A1 ' ' would survive. Equation (2.17) is known as a test of the local action of the lepton current. This is, however, somewhat mis leading since all what matters is that an object of spin £ 1 is exchanged between leptons and hadrons (in the t-channel) [Gourdin (1962)]. In practice, Eq. (2.17) is slightly modified by radiative corrections. For experiments on nuclei one might fear thjt large Coulomb effects might destroy Eq. (2.17) completely. This was investigated and the correction: were found to be only of the order of a few per cent [Nachtmann (1970)]. Despite its importance as a direct test of the vector nature of weak interactions, no good experimental Uieck of Eq. (2.17) exists at present. The limited number of events available in neutrino experiments forces us to make some model for the v and Qz behaviour of A, E, and C before we can compare the data with Eq. (2.17). It will follow from the discussion in Section 7 that everything is perfectly compatible with Eq. (2.17) and therefore the vector nature of weak interactions. There exist further tests of the spin of the object exchanged in the t-channel which involve the observation of momenta of final-state hadrons. The interested reader is referred to the original work [Pais (1970)]. 3. QUA31-ELASTIC SCATTER!KG 3.1 Invariance principles 3.2.2 Lorenta invariance Vie now turn to a discussion of the simplest neutrino reactions where the final hadronic state consists of a single nucléon. There are two reactions to be considered: vy + n * u- + p v + p •+ u+ + n . (3.1) From Eq. (2.10) we see that these reactions are governed by the matrix elements of the AS = 0 weak currents J^ [Eq. (2.S)] between nucléon states. Explicitly we have (p(p')|J*in(P)) -ûCp'ir'te'.pMp) _ _ (3.2) (n(p')lJA[p(p)> »ù(p')rx(p',p)u(p) , where r7(p',p) are matrices which have to be constructed out of the available four-vectors. From Lorentz invariance alone we can expand I\(p',p) in the following way: •CifjiB.ta'ï * w°xpq^( All other terms which we might write down, e.g. a term proportional to (p' • p),, reduce to the terms of Eq. (3.3) when sandwiched between the spinors of Eq. (3.71- Heraiticity of the effective Lagrangian [Eq. (2-J)] requires J " (JI'f and relates rï t0 rx: INTERACTIONS 0? NEUTRINOS r~(p',p) = f*(p,p') , (3.4) where the bar means the Dirac adjoint. The functions f.(q') ar.d g-(qJ) (i = 1, 2, 3) are called the weak form factors of the nucléon. They could in general be cocplex functions. From the transformation properties of the Dirac spinors under parity transformations it is easy to see that f- are the form factors of the vector pan V., g- the form factors of the axial vector part A, of the weak current J" = V~ • A:. We have already encountered these form factors in the discussion of beta-decay of the neutron in Chapter III. There q was a negative tùne-like four-vector [note the change of sign in the definition of q, Eq. (3.3), as compared to Chapter III]. Neutrino reactions allow us to extend the study of form factors to space-like four-mc/wntum trans fers. As a consequence of the already mentioned substitution rule [Jauch (1959)3, tIie ^OTri factors h^sured in these two types of processes are related to each other by analytic continuation. We will have more to say on the analyticity properties of form factors in Section 3-2. Ideally we would now go and ask experimentalists to determine the form factors without further theoretical pre judice. This is, however, not possible since in present-day neutrino experiments polarizations cannot be observed, and we can therefore only measure three real combinations of form factors. This is a consequence of the general theoren, Eq. (2.17). To extract meaningful information from experiments, ve have to inject sone more physics, some more ideas on the structure of the weak current. Since we describe weak interactions tc 1ouest order and neglect electromagnetic effects, the matrix elements of Eq. (3.2) involve only strong interactions which show empirically invariance under many symétries such as isospin, etc. If we knew the transforcation property of the current under these syranetries, we could derive restrictions on the fom factors from Eq. (3.2) since the transformation properties of the nucléon states are known from strong inter actions. At this stage ve will resort to a model of strong interactions which has been extremely successful, the quark model [ûe11-Mann (1964), Zweig (1964)]. All hadrons are thought to be built of three subparticles, quarks, which we denote by u, d, s for up, down and strange (another frequent notation is p, n, A). To have something definite in mind we could, for example, imagine the quarks interacting with neutral particles, "gluons", in a way which is invariant under all syinnetries of strong interactions such as isospin, time reversal invariance, charge conjugation, etc. The AS •= 0 weak current is constructed out of quark fields in the following way (cf. Chapters II and III): •>; - 5ïXa • Ts)d ^ •>l • (JÎ'T • 3.1.2 Charge Btirmetru, time reuereal invariance, aecond-alage currents Consider now a synmetry of strong interactions such as charge symmetry which asserts invariance under a rotation by n around the second axis in isospin space: P, » e'"1' . (3.6) The transformation properties of quark fields and nucléon states are easily derived frcci their isodoublet Pjlp> • -In) P,|n) - |p> PjUPJ1 > -d P,dP;' - u . This implies for the currents [Eq. (3.5)3 wr • < • At this stage we might just postulate Eq. (3.9) to hold for the true current and forget the model. In any case it is now easy to derive the implications of Eqs. (3.7) and (3.9) for the form factors [Eq. (3.3)]. Hfe find the fol lowing conditions (see Excercise 4): ^(q1) - f-(q') (i = l,2) f,Cqa) --fJCl8) Bi(q!) =8^(q2) U * i, 3) e^q1) = -g,V) (3.10) nhere q1 is always negative. In a completely analogous manner we can derive that tine reversal invariance requires all form factors to be real (Exercise 4); fi(q2) = fj(*ia1 gj(q2) =g-(q2] i = 1, 2, 3 . (3.11) Another of the invariance properties of strong interactions is G parity. The corresponding operator is the pro duct of the charge symmetry and charge conjugation operators, G = P.C. It is easily found that the vector and axial vector parts of the m-xiel veak current [Eq. (3.5)] transform as follows under G: OCT' *1 CATC- Currents transforming according to Eq. (3-1Z) are called first-class currents [Weinberg (1958)]. Second-class currents V, A, have by definition the signs reversed: GVJC- - -vj GÀ*G"' - ÀJ . (3.13) If we take our model current [Eq. (3.5)] seriously we have to postulate absence of second-class currents. This requires f,(q*) -gjCq*) - 0 . (J.145 Experimentally there is no evidence for second-class currents from decays [Alburger (1971)]. Scattering experi ments are not yet precise enough to allov a definite conclusion to be drawn. looking back we see that the ccefcined requirements of charge synretry and tine reversal invariance leave us with 2 ! t four real form factors: f,(q ), f,(q ), gliH), g,(q*). Together they already icply absence of second-class currents [ET. (3.14)3- This is not an accident (see Exercise 5). 3.1. S CVC and PCAC he will now discuss two theoretical ideas, the CVC and PCAC priiu.tp.les which we already encountered in Chapter III. We recall them here both for completeness nnd because we will use the relevant formulae extensively in later sections. The conserved vector current (CVC) hypothesis [Gerstein (19S6), Feynman (1958)] assumes aVj(x) - 0 , (3.IS) 2 which implies vanishing of the form factor f}(q ) £Eq. (3.3)]: fjCq1) - 0 . (3.16) In a further step one makes the hypothesis that VT and the isovector part of the electromagnetic current v}1' are the isotriplct of currents which generate isospin rotations (isotriplet hypothesis). This allows us to relate the weak vector form factors to the electromagnetic form factors of the nucléon- It is easily seen that our model current Eq. (3.S) satisfies the fsorriplet hypothesis. INTERACTIONS OF NEUTRISOS Let us recall sone elenentary facts concerning the electromagnetic properties of the nucléon. The electromagnetic current is thought to consist of an isovector and isoscalar piece JT-vi-'-eJif. C3.n, where jj ' is the generating current of hypercharge transformations. The electromagnetic form factors of the proton arid the neutron are defined as follows: r D r where r = p, n stands for proton or neutron!utron. After some simple isospin algebralgebraa we find that the isotriplet hypothesis relates the weak vector form factors t:oo the electromagnetic ones in the followinfollowij g way: f^q2) = ff(q') " fjfq2) (i - 1, 2) , For qJ = 0 we obtain f,(0) = 1 , f,(0) = uj - un = 3.70 , (3.20) where y' are the anomalous nagnetic moaents of the proton and the neutron. The experimental evidence for Eq. (3.20) was already discussed in Chapter III. In principle, neutrino reactions c^-ild test the isotriplet hypothesis [Eq. (3,19)] over a wide range of four-aceentun transfer q- At present, however, Eq. (3.19) is taken as input in the analysis of the data (see Section 3.3). Another idea which has been very successful in hadron physics is trie hypothesis of partial conservation of the axial vector current (PCAC) £Cell-Hann (I960)]. Again this has already been discussed in Chapter III so we only re call the basic principles. In a field theoretic version, PCAC asserts that the divergence of the axial vector current is proportional to the pion field «a (this is the so-called "strong" version of PCAC): . a f ID* » Aj =-Jr-*J (a = 1, 2r 3) , (3.21) where mfl is the pion mass and f^ the pion decay constant, f^ = 0.932 fflTI+ [Ebel (1970)]. Note that Eq. (3.21) has non-trivial content only if we know already what the pion field is. This is the case in certain models such as the a-model [Cell-Mann (I960)]. In general, Eq. (3.21) must be considered as a definition of the pion field. Let us recall how we derive the Goldberger-Treiman relation from Eq. (3.21). We define the pion current j by ID • m*)03(x) = ja(x) (a = 1, 2, 3) , (3.22) and the pion-nucleon vertex function g(q*) through the following matTix element: a a Ê2 By cocliining Eqs. (3.3), (3.21), (3.22), and (3.23) we obtain the following relation: ^f^gfq2) SMB.Cq1) - q'gjCV) n — , • (3.2s) mJ - q2 If vc set q1 <• 0 in Eq, (3.25) and make the assumption that gfq1) varies little between q* * 0 and qa = m', g(0) = g(m') • * , (3.26) we obtain the Goldberger-Treiman relation ^Goldberger (1958)] which was already discussed extensively in Chapter III: £„=«• (3-27, Ihe crucial dynamical assumption is Eq- (3.26). It is the virtue of the definition of the pion field, as in Eq. (3.21), to suggest such dynamical assumptions. This would turn into a defect if experiment contradicted these assumptions. The extrapolation made in Eq, (3.26) looks however very reasonable since m2 = 0.02 GeV2 is indeed almost zero on a typical hadronic mass scale, taking as unit e.g. Mz = 1 GeVz. 3.2 flnalyticity properties of form factors A good starting point for discussing the analyticity properties of the weak and electromagnetic form factors of the nucléon is to consider a model with elementary pion and nucléon fields $ and N which are coupled through a Yukawa interaction: L' = ig0N$TYsN . (3.28) In zeroth order the photon has only a direct coupling to the proton corresponding to the diagram of Fig. 3a, FJg. 3 Two contributions to the electromagnetic structure of the proton in Che Yukawa theory This gives a form factor f^tq1) a 1. In higher orders there are correction terms due to diagrams like Fig. 3b, where the photon interacts with the "pion cloud" of the nucléon. A simple calculation shows that the form factor corresponding to this diagram is an analytic function in q2 with a cut on the positive real axis starting at q1 = 4m3. (see Exercise 6). It turns out that this result generalizes to all orders of perturbation theory [see, for example, Bjorken (1965)3- In this framework it can be proved *,iat the- form factors fjfq2), gjfq*) of Eq. (3.3) are analytic functions of q' except for poles and cuts on the positive real axis (Fig. 4): Imq2 -à}?»»»»>»»»»»»»m. Req2 Fig. 4 Singularity struct i the complex q plane INTERACT IOSS OF HEirTHIXOS The position of the first singularity- s is determined by the lightest real intermediate state which can couple to the current in question. For the vector current this is the two-pion state (Fig. Sa); therefore.the threshold for fi(q2) [Eq- (3.3)] is s = 4m*. For the axial current the lightest intermediate state is one pion, leading to a pale at q* = p*; however its contribution must be proportional to q. (Fig. 5b). Therefore only gj(q2) bas a pion pole. J 1 The three-pion intermediate state (Fig. 5c) leads to a *•• in g1(q ) and g (q ) starting at s = 9m^: j\n^r*risvv\ris4ffl AA/\A/\AAA/\^_ — _ _ _ A Fig, S Intermediate i i coupling co currents A simple model for the isovector electromagnetic form factors could be constructed as shown in Fig. 6. The isovector part of the photon is assumed to couple directly only to o mesons [Sakurai (I960), Cell-Mann (1961)]. If the photon-p meson and the o meson-nucleon coupling con stants are assumed to be independent of qz, then all the q1 variation comes from the p-meson propagator leading, for example, to Fig, 6 Vector dominance model for farm f ff(q!) - f>'> " jHh txperirentally this does not work well. Instead one finds to within 101 dipole behaviour for the so-called electric and magnetic Sachs fora factors: C^(q') • ff(q') • ffCq'i G?(q') = 0 (3.31) CHAPTER VI kith a dipole mass c^ = 0.71 GeV' [Wilson, R. (1972), Bartoli C1972)], No convincing theoretical explanation of this experûnental finding exists. If CVC holds then the weak vector forrj factors must have the sane behaviour: (^"V) » f,(q') * $r Mi'l - <%&*) - <£"%') - f,(q2) • f,(qa) • <$(q!) - cgfa') weak , ^M (q J ( n*Y2 1 "p - v„ l V A behaviour similar to Eq. (3.32) is usually assumed for the *eak axial vector form factor g,(qa) of Eq. (3.3): B, (0) s,(q') ; F¥' with 8,(0) = -gA = 1.2S0 i 0.009 [Chaloupka (1974)]. hhat about g (qz) [Eq. (3.3)], the induced pseudoscalar form factor? Let us rewrite the PCAC relation Eq. (3.25) as follows: ZMg, (q*) z 6,Cq') • i—j-[/îf^Cq») - 2%(q )] . (3.34) • mj; q2(q2 - m*) From our discussion of the analyticity properties we know that g (q*) is regular at q2 = Û. This is ensured by the "conspiracy" relation ^gfO) - 2Mg,(0) - 0 (3.35) which leads to the Golùuerger-Treiman relation Eq. (3.27). Equation (3.34) also shows that in the chiral symmetry 1 ! ] limit (m£ = 0), gj(q*) falls off faster with q than g,(q ). In any case, g3(q ) is hard to measure since its contri bution to the cross-section is proportional to the lepton mass squared [Jsee Eq. (3.37) below]. 3.3 Experimental results We are finally in the position to write down the cross-section for the reactions (3.1) in a fom which cui be usefully compared with experiments. Assuming charge syimetry and time reversal invariance for the fom factors tqs. (3.ID) and (3-11), we find (Exercise 6) do^^aSH^'W^^]'> GgM' } Of course the cross- "ction is of the general fora of Eq. (2.17) with *«• • f.f..(£ ] B(Q!) • (f, ' f,)S, $ . G = G cos 8 - (1.4132 t 0.0016)10-,,s erg cm1 INTERACTIONS OF NEUTRINOS The quasi-elastic reactions [Eq. (3.1)] were studied in a mraber of experiments. In one of then [Mann (1973)] the interactions of neutrinos of mean energy ^ 0.7 GeV were observed in a big bubble chamber filled with deuterium. Che hundred and sixty-six events of the reaction v + n * u~ • p satisfied all selection criteria and could be used for the analysis. The cross-section as a function of the neutrino energy is shown in Fig. 7 where corrections for deuterium effects have been taken into account. E„ |GeV) Fig. 7 Cross-section for V+ n * U~ * p as function of the neutrino energy E [wann (1973)]. The solid line corresponds to the best fit for the axial aass nft •> 0.95, the dashed lines in dicate one standard deviation. Tiiis was compared to the theoretical cross-section [Eq. (3.36)] integrated over the kineratically allowed range of Qz. The vector form factors f (q3) were parametrized according to the CVC relations [Eq. (3.32)]. The induced pseudoscalar g (qz) was dropped, since a reasonable estimate indicated that its contribution was very small (< 11). After parametrizing the axial form factor g (qz) as in Eq. (3.33), a single free parameter, the axial mass n., re gained. The best fit obtained was QA = 0.95 * 0.1Z GeV , (3.58) which is quite cLose to the dipole mass m- n 0.84 GeV [Eq. (3.31)] of the vector form factors. The fact that a good fit was obtained with a reasonable axial maps gives support to all the assumptions such as CVC which went into the anal ysis, but of course further experimental studies aTe needed in order to prove them. Closely related to the quasi-elastic reactions [Eq. (3.1)] is single hyperon production e.g. v i * p - u* * A . (3.39) Events of this type are very rare owing to the suppression by the sine of the Cabibbo angle. Nevertheless they have been observed [Eichten (1972)]. Through the study of these reactions one can check the Cabibbo theory which worked so well for hyperon decays, as discussed in Chapter III, over a much wider range of momentum transfer. 4. RESONANCE PRODUCTION 4.1 General remarks The total cross-sections for pion-nucleon scattering have a remarkable structure, he can see from Fig. 8, where wo show the total cross-sections for the scattering of positive and negative pions on protons, that there are big bumps for laboratory nomentin Pi,.»- s 2 GeV corresponding to a centre-of-mass energy W s 2 GeV. For higher energies on the contrary the total cross-sections show a very snooth behaviour. Th"ic innps are attributed to the excitation of resonances, unstable relatives of the nucléon. Similar features arc observed in photoproduction and electreproduction [see, for example, Chaloupka (1974), Stein (1975)]. In neutrino 7T-p A / (GeV/c) (GeV/c) Fig. 8 Experimental data for the total crods-sections for tr+p and TT p scattering [chaloupka (1974)3 reactions, resonance production is again the dominant feature for centre-of-mass energies W £ 2 GeV [region (3) of Fig. 2], and most pronounced is the excitation of the A(1232) Tesonance which decays into a nucléon and a pion: vg + N * u + AQ232) U N' + TT • (4.1) We will concentrate on the production of this resonance in the following. Different strategies have been pursued in the theoretical analysis. In the isobar model the A resonance is treated as a quasi-stable partie.^ The reaction (4.1) is then quite simi lar to the quasi-elastic reactions (3.1) and can be discussed in terms of transition form factors which are the analogs of the form factors of Eq. (3.3). The A has, however, spin %; its relativistic description involves Rarita-Schwinger spinors [sec, for example, Veltman (1966)] and one finds that the reaction (4.1) is described by eight form factors in stead of six for quasi-elastic reactions. Clearly it is a formidable theoretical task to make a model for all these form factors. Oie possibility is to use sooe version of the quark nodel, where the nucléons and the nucléon resonances are treated as bound states of quarks, in much the sxx way as nuclei arc treated as bound states of nucléons. Once the scattering of the neutrino on a quark is given, the quasi-elastic or transition fora factors can be calculated since the nodel provides us with explicit wave functions for nucléons and resonances, he shall not pursue this approach much further here, but in Exercise 10 this program is sketched for the non-relativistic quark model. For a treatment of re sonance excitation by neutrinos in a relativistic quark model we refer to the original works [e.g. Ravndal (1972)]. In the following we will outline a simple version of the isooar model due to Bell [Bell (1971]]. It turns out that for snail recoil mcoontun of the A in the reaction (4.1) the nurcber of form factors reduces to two. We will be able to relate these two fora factors to the coupling constants of photon- and pion-induced û production by invoking the CVC and PCAC principles. More sophisticated treatments of the reaction (4.1) exploit the analyticity properties of the amplitudes for one- pion production. It is beyond the scope of this article to discuss this so-called dispersion theory. The interested reader is refer.ed to the original works [e.g. Adler (1668), Zucker (1971)]. We note, however, that even these theories are not without free parameters. Before turning to a discussion of Bell's theory we want to point out that singlc-pion production allows a test of th? isospin assignment of the strangeness-conserving charged weak currents J7 [Eq. (2.5)]. A simple application of the Clcbsch-Cordan algebra shows that the following inequalities for cross-sections must hold if the currents J7 have iso- vector (AI - 1) character (Exercise 7): INTERACTIONS OF NEUTRINOS o(vp •» U'p" ) . o(vn + u"P"3) * o(vn •+• u"nn*) o(vn •» u ra") 3 . u o(vp - i.*nr0) • a(ûp - \i pr~) A violation of these bounds would indicate the presence of an isotensor (ûl = 2~) component in J7. In a recent experiment [Garfinkel (1975)], single-pion production by neutrinos was studied in a bubble chamber filled with deuterium. The number of events found in three reactions and in identical kinematic domains is shown in Table 1, where corrections fûT nuclear effects have been made. Table 1 Results of a neutrino experiment [Garfinkel (1975)] Reaction Number of events vp •+• u"pTTf 120 ± 13 vn -* u"p^° 38 ± 9 vn - uwnv* 37 ± 7 This leads to 1.6 t 0.3 , (4.3) onsistent with the absence of an isotensor current. Of course this experiment does .jt exclude such a current. hhercas we have rather definite ideas about the Lorentz and internal syrnnetiy properties of the charged weak current, the corresponding properties of the neutral weak current arc unknown and are the subject of current expericental studies {see Chapter VII). Performing tests of the type discussed above far resonance production by neutral currents will help jn answering these questions [see, for example, Lee (1972), Adler (1974), Sakurai (1975)]. he will now turn to a discussion of Bell's theory of b. production. 4.2 A simple isobar model 4.2.1 Deacription of gpi'n '/; and epin Vi particles A spin xli particle at rest is described by the familip' ..o-componcnt Pauli spinor Xr (4.1) whose components are the amplitudes for spin orientation in the positive or negative z direction. The relativistic description of a spin % particle involves a four-component Dirac spinor u(p). However we can always choose some parti cular Lorcnts frarcc and decompose the Dirac spinor into its "big" and "snail" components [see, for example, Bjorkcn I196S)]: u(p) . Under rotations x transforms like a Pauli spinor. If we insert Eq. (4.5) for example into the expression for thr matrix clement of the electromagnetic current between nucléon states [Eq. (3.18)], we can rewrite everything in tenrs of Pauli spinors (Exercise 8), Of course the expressions reraain fully rolativistic; they arc only written in a special lurent: frame. CHAPTER VI The 4(1232) resonance is an object of spin % and positive parity. Sich a particle at rest can be described by a vector-spinor R*, i.e. each component R^ (j = 1, 2, 3) is a Pauli spinor of the type of Eq. (4.4). Under rotations S transforms according to the product of a vector and spinor representation which contains spin % and spin Vz [see, for exanple, Edmonds (I960)]: D(l) - D(^) = D(%) +D['/i) . (4.6) he want a wave function corresponding to pure spin %, therefore we must eliminate the spin '/2 part of R. This can be done by imposing a subsidiary condition: SR = 0 (4.7) Se will normalize R to one: fà. = 1 - (4.6) 4.2. Z Coupling consUmtB Consider now the following three reactions: n*fo) * P(p) * û++(p') (4.9u) Y(q) * P(p) * A+(p') (4.9b) + + wutk) P(p) * P'dt'l A**(p') . (4.9c) In the spirit of the isobar model we will treat the A as a quasi-stable particle, i.e. make a zero width approximation. All three reactions (4.9) are then governed by matrix elements of currents which can be expanded in terms of form factors. The S-ratrix element for the pion-induced reaction (4.9a) can be written in terms of the matrix element of the pion current already introduced in Eq. (3.22): + Sfi = i Equation (4.10) can be derived using the reduction technique [see, for example, Bjorken (1965)]. The reader not fami liar with this method is invited to verify Eq. (4.10) in first-order perturbation theory for the Yukawa coupling [Eq. (3.28)]. The pion has spin and parity 0", the nucléon Va*, and the A %*. This information together with Lorcnt: and parity invariance of strong interactions fixes the nat/ix element in Eq. (4.10) up to a constant- rndeed, let P be the Pauli spinor of the proton, ft* the vector-spinor of the A , and q n p' - p the fouri^orentum of the pion; then we must have in the rest fraise of the A: t q - p' (4.11) where g^„ is a coupling constant. It should be clear fnwi the discussion in Section 4.2.1 that the use of Pauli spinors in Eq. (4.11) does not icply a non-rclativistic approximation. The rate for the decay A** » p-1* is now easily calcu lated: r(A** where M is the mass of the A and q\ the morentura of the pion in the rest frame of the A. Taking M - 1.232 GeV and T(A** - pn*) - 0.116 GeV ("chaloupka (1974)], we obtain n l2-Rl 16.8 . (4.1; ^^tew INTERACTIONS OF NEUTRINOS Another easy exercise is to calculate the cross-section for ù production by pions [Eq, (4.9a)]. We find 3 o,CW'}=^!grR!'i^60V -Mp h'z - (P * q)* , [4 where the 6-function occurs because we treated the A in the zero width approximation. Let us next consider photoproduction of the A [Eq. (4.9b)]. This reaction is governed by the following matrix clercent of th • electromagnetic current: <û*(p')|Jl,]p(p)i 2 where q = p' - p, q = 0, and f R is again a coupling constant. We can obtain a numerical value for f „ from the radia tive decay width of the A*. An easy calculation gives J !qYl r(A* - PY) l m~ YRi where ti is the momentum of the photon in the A rest frame. The radiative decay width as deduced from photoproduction experiments is [Dalitz (1966), Berends (1971)]: T(A* * PY) = 0.6S • 0.02 McV ; this lCJds to 1.14 . (J.18) fcH' Weak A production [Eq. (4.9c)] is determined by the matrix elements of the weak vector and axial vector currents. Making, as for photoproduction, the static approximation, i.e. keeping only the terras with the lowest power of the tenentua transfer q, we find (A**(p')!v;!P(P)) - 0 + > we work again in the A rest frame. The isoinplct and the PCAC hypotheses discussed in Section J.1.3 allow us to relate the vector cou,ling constant to the elcctrcTviagj'.ctic one, and the axial vector coupling constant to the pionic one. Siriple Clcbsch-Gordan algebra and application of Eqs. (3.21) and (3.22) give l • AC. VR YR or with use of the Goldberger-Treiman Telation [Eq. (3.27)]: fAR *^gi(0) -22. = 5l' , (4.22) *i-R 8r 4.2.3 Comparison with experiment We can now calculate the cross-section fur weak û production [Eq. (4.9c)] in our model theory. We have to insert the expressions for the matrix elements [Eqs. (4.19) and (4.20)3 ^nV3 tne formula for the hadronic tensor W . [Eq. (2.13)] and sum over all spin orientations of the A. Using finally Eq. (2.11) we obtain G§lte„[W'] I ! ! aqW 16ï E M |qc| ! + Q1! . „ilfAR|Y fW! - K2 - 0! "Un » ZQ'M' u„ fVRfAR fs - ul FjV*M* *m; - 2(WW *V'r^ *MV)] Ka = (P * q)* , QJ = -q1 , (4.23) where we have used the expression for the pion cross-section Eq. (4.14) and neglected the nuon nass. Up to now we have worked in the narrow resonance approxination. We can repair this defect by using the experi mental pion-nucleon cross-section in Eq. (4.23). A convenient parametrization of the ^*p total cross-section in the region of the A(1232) resonance is provided by the static model [Chew (1956)]. According to this theory and to the fit by Hamilton [sec Fig. 14 of Hamilton (1967)] ve have Oi. 'q-ri' cot & a1 - M* 3 o_(K*) = -£l- sin 6 . -^- — " 0.085 M - 0.11S M - (4.24) A line corresponding to Eq. (4,24) would go right through the data points for the *• p total cross-section in Fig, 8 up to a beam Domcntun of •>. D.B rcV. Equation (4.23) is identical to the static model cress-section given by Bell and Llewellyn Sr.ith [Bell (197(1)3 and almost the saw as the result of the first calculation [Bell [1962)]. The general structure of the cross-section fonrula (4.23) is again of the type of Eq. (2.17), as it should he, Since we kept only thr lowest-order terms in q in Eqs. (4.19) and (4.20), we can only trust oui- result for small values of Q1. For higher values of QJ wc might, as a first approximation, replace the constants f.— and f^ by di- pole expressions (1 + qVnJ)1 fAR 0 * QVm*)2 1 INTERACTIONS OF NEUTRINOS This amounts to having a relativisr.ic theory where only two out of the eight possible form factors have been kept — hopefully the most important ones. If time reversal invariance holds, fVR and f,R must be relatively real (Exercise 9). This leaves the relative sign open which can, for example, be abstracted from the non-relativistic quark model (Exercise 10). It turns out positive. The cleanest experimental study of wck A production [Eq. (4.9c)] has been done with neutrinos of average energy ^ 1 CeV in a bubble chamber filled with hydrogen and deuterium [Campbell (1973)]. The analysis could be based on 153 events. In Fig. 9 we shew the expericental data for the total cross-section as a function of the neutrino energy. 1.0 - on _ 0.6 : \^" A* 0A 1 0.2 J 1 1 1 1 1 1 . 3 EffieV) Fig. S Total cross-section for the reaction up •* u A as function of the neutrino energy E, [Cacpofll (1973)]. Full line: theoretical curve corresponding to Bell's model as explained in the text, with mA - 0.95 GeV. Dotted line: Adler's nodel with aiA * 0.V6 GeV as quoted by Schretner and von HippeL [Schreiner (1973)]. To compare this experimental finding with our theory we have to integrate the formula for the cross-section [Eq. (4,23)] over the region of the û(1232) resonance in W* and the allowed kinematic domain in Q*. The result of this integration is represented by the solid line in Fig. 9. Let us recapitulate the assumptions and numerical values of the parameters which went into the theory: i) We made o small recoil approximation, this reduced the number of form factors to two. ii) The values of these two form factors at Q1 = 0 were determined by CVC and PCAC. Numerically we find from Eqs. (4.13), (4.18), and (4.21): 1^ L 0.24 (4.26) The Q1 dependence of the form factors was parametrized as in Eq. (4.25) with r = 0.84 GeV and n = 0.95 GeV as sug- gested by quasi-clastic scattering (Section 3). CHAPTER VI iiij The pion cross-section was parametrized according to the Chew-Low theory [Eq. (4.24)]. The success of Bell's simple isobar model suggests tn&t one can understand weak û production by invoking essen tially the CVC and PCAC principles. More detailed comparisons of various theories with the experimental data have been made [Schrciner (1973)]. The general conclusion is that Adler's model, which has the CVC and PCAC constraints built in and is quite close in spirit and practice to the isobar model outlined in this section, fares very well. But it is also found that a model due to Zucker [Zuckcr (1971)] which violates the PCAC constraint can describe the data. More experimental data is needed in order to decide if the PCAC hypothesis is valid or not. Finally, we want to remark that simple isospin considerations allow us to describe weak & production by neutrinos on neutrons or by antineutrinos on protons and neutrons by the same parameters as the reaction [4.9c) (Exercise 7). 5. APLER'.S CVC AND PCAC TESTS 5.i Theory with neglec* of the rauon mass AUlcr has shown that inelastic neutrino reactions with the rauon emerging in the forward direction (ù = 0 in Fig. 11 provide tests of the CVC and PCAC hypotheses which we encountered already in Section 3.1.5 [Adlcr [1964)]. It is clearl> of great importance that one thus has further possibilities of checking these two basic principles experimentally. Let us consider the neutrino-induced reaction vy(lc) + N(p) - iT(k') + X(p') (5.1) with the r.uon emerging in the forward direction. If we neglect the muon mass, then k, k', and the four-momentum trans fer q = k - k' are parallel four-vectors. For an inelastic reaction q ^ 0, and both k and k' are proportional to q, which is the only independent four-vector remaining on the lepton side. But then the matrix element of the lepton cur rent in the current * current Lagrangian [Eq. (2.3)] must also he proportional to q and this projects out the diver gence of the hadromc weak curiont. This kinematic property of the lepton current therefore allows us to check if the divergence of the n+(q) * N[p) * X(p') , [S.2) and similarly the antineutrino and ""-induced reactions must be elated. This is tlie PCAC test. To see now this works in detail we go back to the expression for the S-matrix clement [l^q. (2.K))] corresponding to the reaction (5.1) and neglect the muon mass. In this approximation the factor (1 + y5) in The lepton current implies that the muon is always produced with negative helicity. For such a muon emerging in t\\r forward direction we can easily calculate the matrix element of the lepton current in Fq. (2.10) using the explicit forms for the spinors given in 1^. [4.5). he find A Û [k')Y [l + V5)U fk}| = 4q* - p. . (5.3) Note that A0kJ/q0 is a ixarentz-invariant expression for zero muon mass. Next we insert liq. (5.3) in Eq. (7.10) and convert q» to a derivative acting on the hadron current. Restricting ourselves ro strangeness-conserving reactions, we find A (^-(k')X(p')IS|VlJ(k)N(F;)| - -(2T)"S(p' - p - q) -| 4 -£i. where G- = G cos 6 and q = k - k'. Now we invoke the CVC hypothesis [Eq. (3.15)] and set the divergence of the vector current in Eq. (5.4) to zero. This implies absence of all parity-violating vector-axial vector interference terms on the hadron side. To see what chis moans for an experimentalist, we imagine a neutrino experiment on a nucLecn target N(p) where we trigger on a firward-going muon and measure the rate to observe some final hadronic state X(p'). Consider now a reflection R on an arbitrary plane containing the origin and the direction of the neutrino beam, "learly we have Equations (5.4) with 3 V* = 0 together with Eq. i5-S) iaply that we will observe the state X(p') with the target N(p) at the same rate as the mirror image state RX(p') with the airror image target RMp). *n other words: there are no parity-violating correlations among the hadrons. This is the precise formulation of the CVC test. Next we invoke the PCAC formula |_Eq. (3.21)3 to rewrite Eq. (5.4] in terras of the pion current defined in Eq. (3.22) Ke find GR 4A0k; i (/(k'),X(p'):S|v(k),N(p))| - »*6{p' - p - q) -| — f, uhere we have used that q3 = 0 for 0 = 0 in the lijiit of zero nuon mass. Equation (5.6) looks very nuch like the S- matrix element for the pion-induced reaction (5.2). Indeed with the help of the reduction technique already mentioned in Section t.Z.Z, we find fX(p').S|r*(q)N(p)> = i(2-)"5(p' - p - q) where, however, we now have q1 = n3 for a pion on the mass shell. The crucial dynamical assumption typical of all applications of the PCAC hypothesis is that the nati be element of the pion current as defined by Eqs. (3.21) and (3.22) varies little between q* = 0 and q3 = c?. Makiîg :his assurâtion in our case we find proportionality between the matrix elements for tin; neutrino- and pion-induccd reactions (5.1) and (5.2) which implies proportionality for the cross-sections A simple calculation gives where h' is the cncigy of the muon and v = q0 the energy transfer of the neutrino reaction which equals the energy of the pion. For zero muon mass the combination E'3/v is Lorentz invariant; iu real life where the muo.i has a mass we will take the laboratory energies for definiteness. Lquation (5.B) should hold for arbitrary final state X. This means two things, i) 'Ihe cross-section for a neu trino reaction (5.!) with the muon emerging in the forward direction with less energy than the incident neutrino li.c. 1. - i:' • v f 0) and summed over all hadronic final states is related by a known factor to the tot^l cross-section of a positive pion of laboratory energy u on the same target, ii) All correlations among the hadrons arc the stum; in the two reactions. The experimental check of points (i) and (ii) above constitutes the PCAC test. 5.2 The CVC test he will now give two example illustrate how the CVC test works. First consider two-pion production; v (M * "(p) - P"(k') * p(pt) + n*(p2) + TT{p3) (5.9) and let us work in the laboratory systen where the target neutron is in an cigenstate of parity. With the muon in the ïomacA direction there should then be no parity-violating correlation among the hadrons in the final state. There can, for example, be no correlation of the type where q = it - k*' is the momentum transfer. In practical terms this means that the T~ distribution must be reflection symmetric with respect to the plane spanned by q- and the momentum of the T*. As a second example consider associated production: vp(k) + n(p) * u-(k') • A(Pl) * K*(p2) . (5.11) Absence of parity-violating effects for forward-going muons means that there can be no correlations between the spin of the A and any of tl:e polar vectors available, e.g. q or pt. There am, however., he a correlation with a pseudovector, e.g. q * p,. To see the consequence* of this, consider the reaction (5.1!) J" i-iw rest frame ot the ft (p, = 0). The polarization state of the A is described by a density matrix PA = 7 a • PAS) . \\\ * i . (s.iz) where V, is the polarisât ion vector. It f'.llws from the above discussion that P, can have no components proportional to ^ or Pj, but that it must be proportional to q * p%, which is the only independent axial vector available on the hadron side (remember that we are always concerned with forward-going muons and unpolarized target): PA - q * p\ • (5.13) Now the A has the nice property of analysing its polarization state by its decay. If a A of polarization state as given in Eq. (5.12) decays into a proton and a pion, rt •+ pjr", the angular distribution of the decay proton is given by 1 * oPAP , (5.14) where p is a unit vector in the direction of flight of the proton, and a the asymmetry parameter (see Chapter III) nhich has the experimental value [Chaloupka (1974)] a ' 0.647 i 0.013 . (5.IS) tquations (5.13) and ÇS.M1 imply that the distribution of the decay proton is axially symmetric with respect to the nornal of the reaction plane. We recall that these statements refer to the rest frame of the A hyperon. 5.3 The PCAC test We will now make some remarks concerning corrections for finite muon mass and finite scattering angle 0. Let us consider the axial vector contribution to the (AS •= 0) neutrino reaction (5.1), i.e. the matrix element (X(p')!A*|N(p)> , (5.16) and split it into two parts, one where the axial current couples to an off-shell pion which is then absorbed by the nucléon, and a second part including ail the rest, the non-pionic contribution (Fig. 10). This corresponds to defining a "tion-pionic" piece of the axial current in the following way: •f'^'lv1 CO-1.2. 3). (5.17, where *a is the pion field defined in Eq. (3.21) from which we find d*Aj[a --£ jj (a - 1, 2, 3) . (5.181 It is left as an exercise to the reader to convince himself that a matrix element of A' has indeed no pion pole. The pion-pole term (Fig. 10a) is proportional to q, and therefore drops out if the muon mass is neglecte--'. PCAC relates the non-pionic part of the matrix element [Eq. (5.16)] at q2 a 0 to the corresponding pion matrix element- INTERACTIONS OF ÎJEUTRIKOS The sane situation was found in Section 3 for quasi-elastic scattering. The pion pole contributes only to the foro z ! 1 iu:tor Ej(q ), but PCAC relates g((q l at q = 0 to the pion-nucleon coupling constant via the Goldberger-Treir-an relation Eq. (3.27). From the corrections for finite mucn mass, the pion-pole term will be the most inportant one. Indeed, the pion propagator in Fig. 10a becomes very big for small q! since we approach the pole at q1 - mjj. But this term is easily calculated. Using Eq. (5.17) we find i*ifl* + UCp'îlAA|N(p)> (X(p')l jf 0; * ij^|N(p)> + If we insert Eq. (5.19) in Eq. (2-10) and neglect the muon mass in all terms except the pion-pole contribution, we find for forward-going muons, instead of Eq. (S.6) (see Exercise 12), i Gs 4^ c r. V _"h*-'l! -L (u-(k'M(P')!s!u h!fc!.N(FÏJ -W5CP' " This leads to a cross-section (5.20} »E" r m^o •* - 1 —-U — C(TI*N - X) - J aE'an' v L 22(ro*GK' ' + mfvi,VHJ Equation (5.21) replaces Eq. (5.8) in the approximation of keeping the ouon mass only in the pion-pole term. This result and the analogous one for antineutrinos, where r is replaced by *~, are the original formulas given by Adler [Adlcr (1964)]. It is clear that all the discussion following Eq. (5.8) applies equally to Eq. (5.21). Experimentalists wanting to check Eq. (5.21) are still faced with a problem. If they restrict their sample to iero scattering angle they will have no events; therefore they must allow finite scattering angles for which /idler's theorem is not strictly valid. To see what sort of correction terms we might expect, we turn to the discussion of resonance production given in Section 4, Since we used both O/C and PCAC as input for cur model of û production, we should be able to verify Adler's formula explicitly and compute also the correction terms. This is left as an exercise It turns out that the relevant parameter to describe Thr rar,gc or validity of Adler's formula is not the angle 9 but rather the four-momentum transfer squared Q*. For values of the neutrino energy E of 1 to 2 GeV, we find for A pro duction, from Exercise 13, that the correction terms to Adler's formula reach ">- 100Î for Q: = 0.1 to 0.2 GeV*. The correction terms for other inelastic neutrino reactions have also been estimated [Piketty (1970)]. One finds a similar range in Q! for which one can hope to see something of Adler's PCAC formula. [experiments which have been done to check the CVC and PCAC relations discussed in this section are inconclusive so far. In Section 4.2.3 we mentioned that the data on ù production was consistent with PCAC but could not rule out a model violating the PCAC constraint. Another experiment fBorer (1969)], where one looked foi some subtle effects which should occur in nuclei If PCAC holds [Bell (1964)], was similarly inconclusive. The data of this experiment has been compared with theory by Bell and Llewellyn 3nith [Bell (1970)]. Fig. 10 Contributi > to the matrix clement Eq. (S.16); (a) pion-pole CHAPTER VI 6. THE ACUR HEISBEREER AKD ADLER SUM RULES 6.1 General remarks This section is devoted to a discussion of two sum rules which allow tests of the basic algebraic properties of the weak hadronic currents, the famous current algebra relations of Cell-Mann [Gell-Mann (19o4a)J. We start by considering our favourite source of inspiration, the quark model (see Section 3). In this model we can construct isotriplets of vector and axial vector currents in the following way VjM = q(x) Ç YAq(-0 , Aj(x) = q(x) ^ Yj^qOO a = 1, 2, 3 (6.11 where q is the spinor for the non-strange quarks fuï q = ldj and T" are the usual Pauli matrices of isotopic spin- The vector currents arc the local currents associated with iso- spm transformations according to Noether's theorea and therefore conserved as long as the model is isospin symmetric. The .ixial currents are in general not conserved. It is easy to derive the equal time coroitators of the currents Eq, (6.1]. All that we need to know are the canonical anticonnutation rule? of quark fields at equal tnncs which should be true not only for free quarks but also for interacting quarks as long as the interaction Lagrangian contains no derivatives. In this manner we find e.g. for the zero components of the currents: [<(x.t),v5cy,t)] = i eabc V^(x,t)6'(x - y) D£(x,t),A5(?,t)3 - i Eabc A={S,tî«JCE - y) (6.2) [A?(x,t),Ab(y,t)J = i eabc vC Ia <= / d'x \*(x,t) . la(t) = j' dJx A°(x,t) (6.3) [la,Ib] = i eabc Ie (6.4a) [la.Ib(tl] - i zabc lÇ(t) (6.4b) a abC C [I (t).l5(0]= i£ I • (6.4c) Lquations (6.2) and (6.4) are the local and integrated versions of Gell-Mann's current algebra. At this point we couk' abstract just the relations Eqs. (6.2) and (6.4) from the model and forget the derivation which involved manipulations of such elusive quantities as quark field operators. Adopting this philosophy, our basic assumption is that the measurable ûS = 0 charged weak currents and the isovector part of the electromagnetic current arc constructed according to Eqs. (2.6) and (3.171 out of vector and axial vector currents V?, A? (a = 1, 2, 3) satis fying the relations Eqs. (6.2) and (6.4). Note that the third axial current A' is not measurable in conventional weak processes but probably plays a role in neutral current reactions (sec Chapter VII). Before turning to experimental tests of the current algebra relations we have to make several comments. Equation (6.4a) so/s that the vector charges, which arc time independent since the vector currents are conserved, are the generators of isospin transformations. The implications of this relation for weak processes are already con tained in the isotriplet hypothesis. Equation (6.4b) says that the axial charges which are tine dependent since the axial currents are not conserved, form an isotriplet. For weak processes this does not imply anything lew with respect to the ûl = 1 rule for the AS = 0 weak axial current. The important new hypothesis is Eq. (6.4c) which assisses that the algebra of charges closes. The local relations Eq. (6.2) imply of course the relations for the charges Eq. (b.4). The reverse is not true. There could he terms proportional to gradients of 6-functions in Eq. (6.2) which would drop out on integration. In INTERACTIONS OF NEUTRINOS certain cases such terms must be present in the commutators of two local currents [Schwinger (1959)]. In our appli cations we will ignore such terms. This can be justified on general grounds. For a discussion of this point and many other topics of current algebra we refer to the book by Adler and Dashen [Adler (1968a)]. 6.2 The Adler Heisberger relation he will now derive the famous sum rule for g, = -g1(0) the axial vector coupling constant of neutron beta de;ay IVuiler (19b5), Wcisberger (1965), (1966)]. The starting point is the conrutation relation for the axial charges i; = \\ * -i; = f d3x A:d,t = 0) (6.5) which follows irmediately from Iq. (6.4c): [I*,i;] = 2I1 . (6.6) Sow we sandwich this operator equation between proton states and insert a complete set of internediate states in the jonisitator. Averaging over ttie proton spin directions this leads to x x i = Z]i where gA • -g,(0]. Tor states X) different from the one nucléon states we will convert the natrix elements of the axial charges !, to rcatri-i elements of the pion field. Using translation invariance we find easily fren Iii|. (0.S) and fron the de finition of the divergence (X(p")i3AAj(0);p(p)> = L(p" - p)X(X(pwl!AjfO)!p(plî (6.10) Now we observe the following. The nucléons are the lightest states with baryon number one. l\ changes neither baryon number nor the three-momentum. Therefore all states |X(p")> (different from the one nucléon states) which have a non-vanishing matrix element Eq. (6.9) must have higher energy than the proton state, p" > p0. YOT such states we have therefore w I U(p")|A;(i»!p(p)>| - ~~ - |p"-p »(P0 " PD) |p"=p ltp^'-pj ' /2 |p»=p (0.11) where we have used the definition of the pion field Eq. (3.21). Inserting Hqs. (6.8) and (6.11) in là]. (6.7) w find after some simple algebra: S(p )6 ( (b.12) where we have introduced an integration over q0 and a 6 function 6(p£ - p0 - q0) which cancel each other exactly. .Now he define the following quantities: Sr(v.q'l • » 2J tjj£ W - P - q)! Since our states are normalized covariantly and the sum in Eq. (6.13) runs over a complète set of states, the quantities S" are Lorentz invariant and can only depend on the two invariants v and qJ as indicated in Eq. (6,13). The integrand in Eq. (6.12) is given essentially by S'fv.q3) [Eq. (6.13)] evaluated in a reference frame where the proton four-momentum p and the four-vector q are given by p • (Po.ft . q = (q0,o) • The invariant variables are then - •&* q2 » ^a = (—1 y2 • ÎÛ.14) "•pflj Nov. we insert liq. (6.15) in Eq. (6.12) and change the integration variable frao qc to v according to Eq. (6.14). The result is: T JT4[4 where u, is determined from the energy of the lightest intermediate state consisting of a nucléon plus one pion vhiih contributes to the sum in Eq. (6.12) ! 2 (p * q) = pi * m^) 2Mv. = UpSoT ^ ' Remember that the proton energy p0 is still arbitrary in Eq. (6.IS). Khat we got is a whole fanily of sum rules relating g^ to integrals over the quantities S" [Eq. (6.13)]. For fixed p0 the integration path is a parabola in the --q3 plane (Fig. 11] q7 (Gev') OS oo Pc = » M 005 / y ^^.pD=20M -s^"^ - ^ -3 -2 -1 Î 3 v(GeV> Fig. 11 Integration path for Eq. (b.15) in the v-q1 plane. r-or finite p, all the parabolas go off to large positive values of q1 and we have no idea what the behaviour of the quantities S:(v,qJj [Eq. (6-13)] might be in this region, h'c see, however, that for p„ - - the parabola1; approach 1 the line q • 0. The trick of Fubini and Furlan [Fubini (1965)] is now to take pa to infinity and nsî-imo tliat this limit and thr integration over v in Eq. (6.15) can be interchanged. In this way wo obtain a fixed ^ sum rule: >k'l'C4J ^^CS*(v.,'-0)-S-(v,,» -0,] . It remains to express S"(v,qJ n 0) in terms of the pion proton total cross-sections. From the definition of the i current [iq. (3.22)] we find, using translation invariance K-q'KXfp")! ^ (il t i0J)lp(?)> D U(p")| j; (Ji '- ij;)|p(?)> re a • n" - p. Substituting in Eq. (6.13) wc obtain for a1 = 0 INTERACTIONS OF NEUTRINOS £ (Zn)'6(p" - p -q):(Xfp")| j= (j^ ± j'Kptp))]* = ~ ^Kq* » 0) where we have defined the total cross-section for pions of ïero mass scattering on a proton by applying the usual rules also for such off-shell particles. Inserting in Eq- (6.15a) we obtain finally the celebrated Adler-Weisberger relation .•l4JW "t " - 2M (6.1') After using the (exact) Goldcrger-Treiran relation [Eq. (3.25) with q: = n] we can rewrite i-q. (6.L?) as follows vt Equation (f>.I8) expresses me renormalization of the axial vector coupling constant entirely in terms of strong inter action parameters, the pion nucléon coupling constant and the pion nucléon cross-section, both however. Cor zero mass pions. In strong interactions we can measure the coupling constant and the cross-section for real pions, i.e. for q' « m'. Making as in sections 3 and 5 the assumption of slow variation between q2 = 0 and q7 = n^ B(0) = gr (6.19) we can use the expérimental data for pion proton scattering shown in Fig. 8 to compute |g.|. A careful numerical study of Eq. (6.18) has been made by Adler [Adler (1965)]. For the integral over the on-shell cross-section he finds 2M1 1 f dv J — 'v" - "H Ky^P " VpCv'<^ = °'254 (6'20> where the A(1232) gives a large positive contribution of 0.43 whereas the higher energy region gives a negative con tribution (compare Fig. 8). The region u 2 20 GeV finally contributes very little, only -0.01L. Note that the con vergence of the integrals tqs, (6,18) and (6.20) is assured by the so called Poneranchuk theorcn ^liich says that particle and antiparticlc total cross-sections on the sane target become equal for v + •». To estimate the corrections for the extrapolation fron q1 = 0 to q1 = m* one has to adopt a model. ''or finds that these corrections core mainly from the A(1232) region and amount to *0.09<1 for 1 - g'J. Putting everything to gether he finds a theoretical value for g. with an estimated error ^theory " U" ' °-°! (f'•2,, in excellent agreement with experiment [Chaloupka (1974)]: lg,| = 1.250 : 0.009 . (6.2:) " exp The sura rule Eq. (6.17) can also be tested directly. Ko have seen in Section 5 that inelastic neutrino rcactons with the rruon emerging in the forward direction allow one to measure the cross-sections for zero mass pions, i.e. exactly the quantities entering in Eq, (6.17). Fron Bq. (S.S) wc find indeed: D VD(v,q' ' 0) .*" " ~TT KP - M" • alljl (6.23) J " P G|fjE' 3E'3ft' " |0D» and a similar equation relating the -"p cross-section to antineutrino scattering. The pjrist will remark that we neglected the nuon mass in deriving Eq. [5.8). A careful analysis shows however that the correction terns to Eq. (6.23) vanish in The limit of fixed v, £' •+ = and are in any case negligible it we use electron neutrinos instead of muon neutrinos- Experiments are not yet good enough to test Eq. (b.17) in this way. Finally we want to emphasize that our derivation of the sum rules Cqs. (6.17), (6.18) contains a weak point; the assunpticn that the limit p; •* « can be taken inside the integral in Eq. (6.IS). Unfortunately rtore sophisticated treatments differ from similar weak points . The numerical success of the Adler-Weisherger relation gives us sane confidence both in the conmutation relation [Eq. (6.6)] and in the validity of the p0 * * method for this case. The implications of the sim rule for the question of the universality of weak interactions have already been discuss:^ i.i Chapter III. 6.3 fldler's neutrino sura rule We have seen in the previous section that the algebra of charges Eq. (6.4) receives experimental support from the gv sum rule. Khat about the algebra of charge densities Eu. (6.2)? Since it is more specific than the algebra of charges as discussed in Section 6.1, we should be able to obtain more predictions from En. (6.2) than from Eq. (6.4). This is indeed the case as shown by Adlcr, who used liq. (6.2'* to derive a very interesting sum rule for neutrino scat tering [Adler (1966)]. We will now sketch a derivation of this sum rule which runs along the same line; as the deriva tion of the sum rule for g. in the previous section. We will therefore be rather brief. The starting point is the commutator of the charged weak currents J" fEq. (2.6)] which follows from Eq. (6.2). [J*(x,0),j;(y,0l] = 4[V*(x,0) * A"(x.u)]63(S - y) . (b.24) Now we take the matrix element of Eq. (6.24) for protons and Fourier-transform. Setting y = 0 we find J lqX /d x< e~t (p(p)|(X(x"1U),j;(0)]!p(p)) - 4 If ue average over the spin directions of the proton, -V drops out and the right-hand side of liq. (6.2?) is given by the matrix element of the third component of the isospin current at zero momentum transfer, i.e. by the third component of the isospiii quantum number of the proton [see E.qs. (3.17), (3.18)]. On the left-hand side of liq. (0.25) we intro duce spurious integrations over variables q0 and t to rewrite Lq. (6.25) in the following way /£/• (6.2b) To convince ourselves that Eq. (6.2b) is a consequence of Eq. (6.25) wc note that the integration over qa produces a i-function 4(t). The next step is to Ktroduce a complete set of intermediate states in the commutator ami to use translation in variance to perform the integrations over t and 3. This leads to ~ I ^IJ^— i6 -6{p' - p * q) hhere q - (qa,ii). We observe that Eq. (6.27) holds for arbitrary momentum p* of the proton and arbitrary q\ If ue look hack to .Section 2 w0 find that the integrand of hiq. (6.27) is exactly the component A5: of ïl;e haJrcruc tcn>or 1% [lq. (2.13)] with the sum running over a'1 haJronic states and the weak lurrcnt replaced by it* .'*S = n part. i he expanded in the manner shown in Eq. (2.14) i*ith h ' o* ' 6 •) lor a general discussion of the p0 -* - and the so called dispersion method to derive sum rules wc rcf'i AJIcr (l'.'bHa). INTERACTIONS OF NEUTRINOF w(vp,0p)(piq) „ 2J -Î^ii 6(p' - p - q) where v = pq/M and Qz = -q3 = 32 - q*. Now we choose p and q* such that p-q = 0 " M - N i (6.29) substitute Eq. (6.28) in Eq. (6.27) and convert the integration over q0 to an integration over v. Observing that the first term in the brackets of Eq. (6.27) contributes only for q0 > 0 the second term only for qD < 0 we find after some simple algebra the following family of sum rules: : -[.(il]V;^(v.,Q') . W'^W) • (il)'^)>P)Cv,Q-) , 2(^] jï W^W)]} • 2 (6.30) 3 3 where Q depends on v as shown in Eq. (6.29) and pD and !q| are fixed and still arbitrary. The integration path is again a parabola in the v-q* plane as shown in Fig. 12. q (GeV1) y>,= 2" 1 1 / ' i -4 \J -2 1 / 1 2 y t. v[GeV) /-I -\ p0=4M ' -2 -3 (p0==>) q' = 2Mv q* = -2Mv Fig. 12 [ntcgrntion path in tlie v-q! plane for Eq. (6.30) The trick is now exactly the same as in Section 6.2. he note that for finite p0 the parabolas go off to large positive q* where wc have no experimental information. If, however, we take the limit p0 * ™ and assume that we can do this inside the integral in Eq. (6.30) we find a sum rule for fixed Q3 = |q*|J: (6.31) This is Adlcr's neutrino sum rule which should be valid for all Q' Note that the integral in liq. (6.31) receives contributions only from v * Q3/2M (see Exercise 14). With certain assucptions we can derive an interesting relation for the neutrino proton cross-section itself. If we neglect the muon nass and if the structure functions W'-tv.Q1) (i = 1, 2, 3) of Eq. (6.28) have a reasonable behaviour for v - » at fixed Q1 we find from Eq. (6.31) [see Exercise IS) lim 2Η- (E,Q') (6.32) &«. I dQ< ^W)"niQT f liS=14S-0 " fixed where do/dQ* is the cross-section for a sun over-all hadronic channels with strangeness S = 0. Adler's sum rule has not yet been tested by experiment. A confirmation would give us confidence in the ccctmta- tion relation Eq. (6.2d). A failure could, however, always be blamed on our doubtful procedure of interchanging an integration with the linit p, •* =>. In more sophisticated terns Adler's sura rule rests on the assumption that a certain amplitude satisfies an unsubtracted dispersion relation for which unfortunately there is little theoretical justification p ! l up î The sum rule Eq. (6.51) shows a remarkable property. The structure function WV '(v,Q ) - W ; ^(u,Q ) depends in genera] on both variables v and Q1 in a non-trivial way, therefore we would expect the integral over v to be a non- trivial function of Q2. Equation (6.31) tells us thnt in fact this integral is independent oî Q* if the local algebra of charge densities Eq. (6.2) is true. The value of the integral would be exactly the same for bare nucléons, i.e. for simple point particles without structure- Equation (6.32) is even more surprising, since we have seen in Section 5 that the quasi-elastic form factors and therefore also ti.^ cross-section for quasi-elastic neutrino scattering do/dQ3 (quasi-elastic) fall off rapidly with Q2. Equations (t.31) and (6.32) can only be true if the cross-section For 1 2 inelastic neutrino scattering a'o/auSQ (inelastic) which is governed by the functions Wi(v,Q ) Eq. (6.28) falls off much less rapidly with Q2 (see Exercise IS). Observations of this kind led Bjorken to some very interesting speculations about the substructure of the nucléon and initiated an extensive theoretical development to which we will turn in the next section. Finally we observe that the current algebra relations Eqs. [6.2) and (6.4) can be extended to the octets of vector and axial vector currents by replacing the structure constants ea c of SU(2) by those of SU(3), i.e. by f^ . The techniques discussed in this section can be used to derive sum rules for strangeness changing neutrino reactions [Adler (1966)] (see Exercise 16). 7. DEEP INELASTIC NEUTRIHO-NIJCLEON SCATTER1HG AfJD THE PAfiTON MODEL ?.l General rcmrks. kinematics In this section we will discuss neutrino reactions on nucléons [Eq. (1.1)] with large monentin transfer Q* and large energy transfer v [region (5) of Fig. Z]. Fron a study of these deep inelastic collisions we nope to get insight into the fundamental structure of the nuclcon. The questions to be asked are e.g. the following: does inelastic scat tering die out as quickly as quasi-elastic scattering with increasing QJ or is the behaviour different? In the first case we would consider the nucléon as a "soft" particle. A large cross-section for inelastic scattering on the other hand could mean that there are some "hard" structures inside the nucléon which absorb a lot of momentum and energy if the neutrino scatters on them"J. In the second case which s>,ems to be realûed in nature we have immediately further questions, hhat are the hard structures? Is the nucléon built out of subparticles, as the nucleus is built out of nucléons? If this is true, what are these subparticles, i.e. what are their quantum numbers? Before we can try to answer these questions we have to make a thorough study of the relevant kinematics. It will be important to consider together with ncutrino-nuclcon also clcctron-nuclcon collisions. 7.1.1 Kincnatiag of olectron-nuclvon oeattcritfj We will now study the following reaction: e(k) * Nfp) - e(k') • X(p') . (7.1) •) Imagine two bags full of some soft ituff like cotton where one contains some little steel spheres. You would he asked to find out the bag with the steel spheres by shooting with a gun at the bags and observing the scattered bullets. Only the bag with the steel spheres would be able to deflect bullets by a large angle. INTERACTIONS OF NEUTRINOS e(k') The corresponding diagram is easily obtained from Fig. 1 by replacing v and u by electrons. The wavy line corresponds then to the exchange of a virtual photon of four-Eomentiin q (Fig. 13). Since we are interested in the region of large Q1 and u we can safely neglect the muon and electron masses in treating the reactions Eqs. (1.1) and (7.1). The kine matics are then identica' for the two cases. In Table 2 we have listed the important kinematic variables sone of which have already been introduced in Sections 1 and 2. The S-eatrix decent for the reaction Eq. (7.1) is X(p-) easily obtained froei Fig. 15. Similarly to neutrino scat tering [Eq. (2.10)] we find that the lepton vertex can be calculated explicitly since it involves just the interaction of electrons and photons. The hadronic vertex is described Kinematic variables for deep inelastic lepton-nucleon scattering. If not stated otherwise, non-covariant quantities refer to the rest frame of the nucléon. The lepton masses have been neglected. Variable Meaning E Energy of the initial lepton •* F.' •£ Energy of the final lepton 0 Scattering angle of the leptons G. . k - k' Four-momentum transfer 1 J - -q Q Four-momentum transfer squared = 4EE' 5inJ | V . Ef . E - E' Energy transfer ] w - (P • q)J Invariant Class squared of the final hadronic state K Energy of a real photon needed to produce a hadronic final state of invariant mass h' Relative amount of longitudinal £ to transversal polarization of -M> •£)«•*!]" the virtual photon r •$H(1-«- Flux of virtual photons Momentum of the nucléon in the p - Mv/Q Br*it frame :U-'.SL Bjorken's scaling variable « 2M\; }•* - ,„.,-. - Q' Bloon-Gilman scaling variable 2l»i • «1* Y •È Fractional energy transfer CHAPTER VI by a matrix element of the hadronic part of the electromagnetic current and cannot be calculated from first principles. A simple consideration gives: where q = k - k' and a is the fine structure constant. As in Eq. (2.10) the matrix element factorizes into a leptonic and a hadronic part*-*. In most of the investigations done so far experimentalists have used an unp-*larized electron beam, an unpolarized target and observed only the angle and energy of thp outgoing electron. The cross-section for this situation is easily obtained from Eq. (7.2). We find factorization into a lepton tensor and a hadron tensor [compare Eqs. (2.11), (2.12) and (2.13)] 3*o (eN,eX) -isiJ^Jifc^PwtjN) (7.3) 3E'3S3' WSN) = E ^" *(P' " P " qKN(p)|Jpx(p')>(X(p')M:;m!Kû>)> (7-5) X where the sum in Eq. (7.5) runs over a complete set of hadronic states and an average over the spin directions of the initial nucléon is understood. The electron mass has been neglected in Eq. (7.4). The hadronic tensor [fcq. (7.5)} depends only on the two four-vectors p and q and can be expanded in a way similar to Eq, (2.14). he know however that electromagnetic interactions respect parity invariance and That the electromagnetic current is conserved. Using these facts and Lorentz invariance it is not difficult to show that W*? ' [Eq. (7.5)] can be expanded in the following way (Excercise 17) »r where lvCl ^(v.Q1) are structure functions depending only on the invariants v and Q' as indicated [Drcll (1964)]. In serting Lq. (7.6) in Eq. (7.3) wo find easily for the cross-section cN : >(v,Q>) s.n'f .»"" W)<:os'f}. (7-7) Another interpretation of the reaction Eq. (7.1) is often used. In the scattering process (Fig. 13) the electron emits a virtual photon whose four-momenttn q and state of po'arization, di^.rihed by the density matrix î| J [Eq. (7.4)] we can determine in an actual expcriinent from the energy of the initial electron and the energy and scattering angle'of the Final electron. The process we arc really interested in is then »he absorption of the virtual photon by the nucleoli. An experiment which does not look at the final hadronic state is in this interpretation an experiment to measure the total absorption cross-section of the virtual photon on a nucléon. To make these considerations more quantitative we introduce pclariiation vectors for the virtual photon of four- mr>mentum q. We rark in the rest frame of the nuclcon ind choose the --axis parallel to q, U'e have then p* - fM,0,0,0) qA - 1^,0,0,151) - £ (!q1,0,0,v) *) Equation (7.2) is only true to lowest order in a, in the so called one photon exchange approxuna•;on. Radiative corrections arc very hxportant for electron INTERACTIONS Of NEUTRINOS where et,e_,E( are the polarization vector* for a right-handed, left-handed and longitudinally polarized virtual photon. The normalization is -(e:,£+) = -(zl,c-) = (C;,E0) = 1 . (7.9) Note that for a virtual photon the longitudinal polarization is a physical degree of freedom in contrast to the cose of real photons. We will now define cro^s-sections for transversely and longitudinally polarized virtual photons in complete ana logy to the definition of the off-shell pion-proton cross-section in Eq. (6.1Ô). We will however use for all values of Q* the flux factor appropriate for the absorption of real photons on nucléons leading to a hadronic final state of the same invariant mass K. This is purely conventional [Hand (19&3)J. We have then: *• - -" - [7.ID) where K is defined in Table 2 and the equali.v of the absorption cross-sections for right- and left-handed plwtons on an unpolarized nucléon follows from parity invariance. A simple calculation gives the expressions of the structure functions WIj2 Eq. (7.6) in terms of o_ and o,: ">(U'Q!' = jfc^^ t°T] • <7-«> By its definition W , [fcq. (7.6)J is a positive semi-definite tensor. The positivity conditions following from this fact are (Exercise 17) 1 3 Ojfv.Q ) > 0 , oL(v,Q ) > D . (7.U) ihe cress-sections Eq. (/.10) are also very useful for discussing the limit Q2 - 0 i.e. the relation of electro- and photo-production. From Eqs. (7.S) and (7.10) it is easily seen that the transverse cross-section gots over into the absorption cross-section for real photons <,<<*> 7 15} Q^n OLCV.Q1) « Q* (7.11) for w > 0 and Q: - 0 (ILxorcisc 18). It is now a trivial exercise tu express the cross-section Eq. (7.7) in terms of o_ and o.. he find 2 j^ - r[>T(»,Q') • EcL(v.Q )] (7.15) where T and E (sec Table 2) can be interpreted ÛS the flux of virtual photons and the relative amount of longitudinally to transvcrsalJy polarized photons. 7.1.2 Kirwatieo of ncutrino-nuoleon ocattci'iig " Turning :jw to neutrino nuclcon scattering [Eq. (1.1) and Fig, l] we not? that we have already discussed most of the necessary kinematics in Sections 1 and 2. In this section we will restrict ourselves to strangeness conserving reactions*1. *) The extension of all the following considerations to ûS t 0 reactions is straigh.forward and left as an exercise to the reader. We imagine a neutrino experiment with an unpolarized nucléon as target where we observe only the energy and direc tion of the outgoing muon and sum over all hadronic final states. The cross-section for this case is then easily ob tained from Eq. (2.11): -* u~xl — • aKvJAfytvN.vN) = &[%: E PA -^E'^wf^W) sin1!* •K^'^Wl cos* | , W^W) *±*- sin> || (7.10) where we have neglected the muon mass. The tensor K\^",v' ' is given in Eq. (6-28) for a proton target and is defined in an analogous way for a neutron target. Several comments have to be made. The weak current J* ••* V~ + A" contains a vector and an axial vector piece. The product of the two matrix elements of the currents in Eq. (6.28) contains therefore W, AA, and VA terms. The structure functions IV. (i = 1,2,4,5) in Eq. (6.28) receive contributions only from the W and M terms since they are coefficients of polar tensors. The VA terms lead to a pseudotensor and contribute therefore only to Wj. The weak currents J7 are not conserved therefore W,, and Ws are independent structure functions in Eq. (6.28) and not related to W, and W„ as for electron scattering [Eq. (7.6)]. The contribution of Vh and K$ to the cross-section vanishes, howver, if we neglect the muon mass (compare Exercise 2). As in Section 3 we will assume that the AS = 0 weak current has the transformation property given in Eq. (3.9) under a charge symmetry operation. This relates the structure functions for protons and neutrons in the following way: wW^KQ1) = Wp^v.Q1) H^tv.Q1) • W^v.Q») (i = 1, .... S) . (7.17) [t will be important to check these relations experimentally, since their breakdown could indicate the appearance of new quantum numbers [de Rujula (1973)]. We will now focus our attention in Fig. 1 on the hadronic vertex where the weak current acts on the nucléon as the electromagnetic current does in Fig. 13. Thiâ suggests a definition of cross-sections in analogy to Eq. (7.10) where, however, we rep1ace 4ra by one: If a K-boson mediating weak interactions exists, the analogy between Fi^s, I and 13 would be corcpletc and the cross- sections Eq. (7.18) could be interpreted as absorption cross-sections fnr virtual h'-bosons of right, left and longitudii polarization . Note that the absorption cross-sections c, for right- and left-handed W-bosons are not equal since the weak current is a vector-axial vector mixture. A simple calculation (Exercise 19) gives the expressions of W, ; 5 in terms of o+ . : „(vN,iN) = K _ K(vS.W) , This concludes our discussion of kinematics. •) A virtual W-boson can also have scalar polarization, e « q in contrast to a virtual photon. For :ero ciuon mass however these enntributions vanish in the cross-section Eq. (7.16). INTERACTIONS OF NEUTRINOS 7.2 Scaling behaviour: the parton model 7.2.1 Experimental results on pealing behaviour The subject of deep inelastic iepton-nucleon scattering was started by Bjorken [Bjorken (1966), (1969)] who pre dicted on theoretical grounds that the cross-sections in the deep inelastic region should be large, of the order of cross-sections for point-like particles. Specifically he predicted that in the limit Q2 -* », v •+ ~, keeping the ratio x = QV2Mv fixed, the following structure functions of rqs. (6.28) and (7.6) should approach non-trivial limits: ZMW^v.Q1) -* F,(x) (7.20) vW,(v,q*) * Fa(x) ^«,Cv,qa) *F,(x) . Extensive experir.ental studies of deep inelastic eiectron-nucleon scattering were undertaken. Surprisingly enough the scaling behaviour Eq. (7-20) which was predicted to hold for very large Q! and \> seems to work well already for moderate values of Q2 and v, in a region given approximately by Q2 > 2 CeV2 W! = M1 + 2Mu - Q* > 4 CeV2 . (7.21) This phenomenon was called precocious scaling. In Figs, lfl and IS we show experimental results from the electron-proton and electron-neutron scattering. as lit '0| as V» , Q7 ' A, 4'" „•>* 0.1 \ 1 1 1 0.3 f 02 0.1 aio i it* "" i * 111 ,M'}i c O/JB à * 0.06 i/V v, * Q04 0.02 •« q 6 8 .0 0.1 0.2 Q3 Q1 Q5 06 0.7 QG 09 1.0 Fig. 14 Experimental results far the structure Fiç. 15 a) on/op versus x. functions 2HW and \>U2 [Eq. (7. \)] of the proton [Miller U972)j. Data arc b) v The scaling behaviour of the structure functions Eq. £7-20) is reflected by the fact that the experimental data points for different Q1 and v seen to fall roughly on the same line when plotted versus w = 2Mv/<3! or x = 1/iu. From Fig. 16 we get a feeling over which Tegion in the v - Qz plane Bjorken scaling behaviour has been verified expérimentaII)'. A fit was made to the experimental data for vWj'-^'f.VjQ2) and the contour lines of the function obtained in this way are shown. If Bjorken scaling [Eq. (7.20)] holds, the contour lines must be straight lines radiating from the origin. This is approximately true. 7.2.S The aesuwptions oj" the partem medal The question is now, how these remarkable experimental results can be understood. A simple and intuitive physical picture was developed by Feynman, Bjorken and Paschos and subsequently by many others [Feyniiian (1969), (1972); Bjorken (1969a), (1970); Landshoff (1971), Kuti (1971)]. The key idea is to assume that all observed hadrons are made out of point particles, partons in Feyroan's terminology, which have a simple coupling to the electromagnetic and weak currents"' Ke will now develop the most naive parton model in some ex tension- Following Feynman we look at the electron nucléon col lision [l3q. (7.1)] in the Ereit fra'nc, where the four-raomenta of the nucléon and the virtual photon >*, exchanged in Fig. 13, are given by A (•?= * M:,D,0.P) q* = (0,0,0,-zpx) P = Mu/Q , x = Q!/2Mv . Note that P •* < in the Bjorken limit, i.e. for Q2 - a 'J -* ™, -X fixed. The basic assumptions uf the parton model which will allow us to make calculations, to be confronted with experi 6 8 JO *Z It ment, are the following: Enirgy Tianshtr V (Qev) i) A fast moving nucléon looks like a box of free partons F>,. iur lines of the prot structure fm 1 vH A deep inelastic clcctron-nucleon collision in the parton picture is illustrated in Fig. I". Several comments have to be made. According to assumption (i) wc can consider the nucléon as a superposition of free parton states. There is no reason to think that only states with one fixed number of partons (e.g. three) would contribute. If the nucjeon is in a state with n partons of masses m, (i = I, ..., n), the four-momenta of these partons are according to d) given by • ) flic sophisticated reader can think of partons as hare field quanta. INTERACTION OF NEUTRINOS hadrons Nucléon partons fig. 17 Deep inelastic electron nucléon callisir in the parton picture Nucléon Collision of a nucléon ui irtual photon in the Breit fraae; (a) state before; (b) after the collision A • ['w * PIT * -i • hi' hp] S, 2 0 (i - 1 n) .-Lt. Pi T ' where F, is the fraction of loi.^itudinal noaenticn of the nucléon carried by the i'th partcn and the :ran:-vi'r-;e momenta arc assur-ed to be snail, [f we neglect russes and transverse momenta in the Bjorken lioit (P - <•>), ve :'nxl that the i'th parton carries approximately a fraction £, of the total four-norcentixa of the nucléon A pj * £kP - (7.24( This docs not imply that the nass of the i'th parton is a fraction f,. of the mass of the rjiclcon. In (li) we have assuncd the so called iirtpulsc appro* ir.at ion which should be valid as long as the p;n >n i. .^ was hit by the lepton is transferred to a state of motion very much different from the state of motion of all the other partons inside the nucléon which were not hit. This is indeed the case for a deep inelastic collision as wc shall sec In rhe Breit fra/sc in whicl< ve arc working, the oH'.ion of the electron with the nuch i.e. the absorption of the virtual photon y* by the nucléon takes a very sinplc fom. According to Eq. (7.22) tin i lual photon transfers no energy, and according to assumption (ii) the parton which absorbs the virtual photon is to be co idered as free. This implies that a virtual photon y* of Bjorken ratio x • Q'/lMu can only be absorbed by a parton carrying a fraction x of the longitundinal -anentun of the nucléon. Aftcv the collision the parton lus the same energy '>it the momentum reversed if *c neglect transverse nonenta (Exercise 20). For x > 0 it is then imL'-tl in a state of :-Mtion différent from that of all other partons as illustrated in Fig. IS. Having stated our assumptions, it is easy to make calculations. We are interested in electrjn nucléon scattering [£q. (7.1)]. According to our rules we have first to calculate the absorption rate of the visual photon on a point particle. Let us assure that we are dealing with a point particle of spin Vi, mass m. and charge e.. The absorption rate is then given by a tensor K , as in Bq. (7.5) but with the nucléon replaced by a parton and the sum over inter mediate states replaced by a sum over one parton states. An easy calculation gi.os (Exercise 21) *s>^> - •*«*., * ^. ^rs * (pip - &$*) k - up) ±\ where p. is the four-momentum of the parton and the 6-function occurs because the parton scatters elastically. According to Eq. (7.24) all partons trove essentially in the same direction as the nucléon, 'the distribution of partons inside the nucléon is then described by functions N.(£) where 0 - £ ^ 1 and the index i runs over all parton species, such that N-(Od£ is the probability to find a parton of the i'th kind with a momentum fraction between K and i * dç. The next step is to average the absorption rates on the individual partons over the parton distribution of the nucléon. Ke note however that the tensors W. Eqs. (7.5) and (7.25) give the absorption rates of a virtual photon by a nucléon or parton in their respective rest frames. If we consider both in the Breit frame we have to multiply vit!» Lorent: factors which take into account relativistic tiae dilatation. Keeping this in mind we find easily i 0 1 'Pi=CP where the sun runs over all parton species. If we assune that all charged partons have spin 0 or spin '/i we can use tq. (7.25) and the results of Exercise 21 to obtain after some simple algebra: eN eN, 2Mkï W) =Ff 00 - £ e?Nitx) spin'/i ^eN)(v,Q*) = FjcN)(x) =X £ e? N^x) * x £ e[ ^(x) (7.27) spi.i% spin» where the sums run over all parton species of spin 0 or '^ as indicated. We note that the pa n mass has dropped out. To our great satisfaction we find Bjorken scaling behaviour [Eq. (7.20)] for the structure functions. The scaling functions have a simple interpretation: they tell us the chance to find a parton with momentum fraction x multiplied with the charge squared. If we could get sone knowledge on the distribution functions from other sources, the scaling functions [Eq. (7.27)] would tell us what the charges of the partons are. ?.2.3 The apin of tha charged par-tone Equation (7.27) tells us already sonethinj! vcrv interesting on the spin of the charged partons. If all charged ^TiC.ns hav" spin 0 wc find F<<*>M - o F,(c,fl(x> f 0 , if all charged partons have spin % ve find instead t«N),, P °(x) t 0 -<>*),.•'<«>, -. «F;"™'(»,Ffc»>r, ) • n . !'.») We can understand these results in a sterile vay free Fig. 13. he have seen that the parton which absorbs the photon v* has tts nomentun reversed (.c neglect transverse oor-cnta). But then angular momentum conservation tells us immediately that a spin 0 parton can only absorb a longitundinal photon. The case of a spin '/: parton is 3 little rare complicated. We observe that the electromagnetic coupling is ts invariant. This implies that ir> leading order in the ratio mass/energy a fast parton of spin '/, does not change its helicltv when it absorbs the virtual photon. Angular momentum conservation requires then the virtual photon to be transvcrsally polarized (Fig. 19). INTERACTIONS OF NEUTRINOS he recall now the definition of the cross-sections Eq. (7.10) which we express in terras of scaling functions Spin 0 V*(eL ) by using the scaling lipit of Eq. (7.11). Our considera tions lead us then immediately to the following conclusions. If charged partons have spin 0 exclusively we must have - 0 Spin 1/2 Y (£* ) if charged partons have spin Vi exclusively we must have instead Fig. 19 Absorption of a virtual photon y' 0 by a spin 0 and spin '£ parton TârW This is in complete agreenent with the results of the formal calculation Eqs. (7.ZS) and (7.29). The connection between the spin of the partons and the behaviour of the observable cross-sections, [Ei;s, (7.50), (7.31)3 was first derived by Callan and Cross [Callan (1969)3 on the basis of somewhat more abstract considerations. If we believe in partons at ill, such a sirple means to determine the spin of the charged partons is clearly of the greatest importance. Experimentalists find in electron proton scattering for the average of the ratio R = o./o-r over the region where oT and o have been determined separately [Bloom (1974)3 R = 1,-168 ± 0.014 • (7.32) The snallness of R suggests that at present energies spin Vi partons are responsible for most of the deep inelastic electron nucléon scattering. Theoretically it would be very attractive to have charged partons of spin /"; exclusively. Ke could then interpret the finite value of R which we observe today as a manifestation of the fact that the present energies are not yet high enough for our asymptotic theory to apply strictly and make the prediction that R should vanish for v - « at fixed x • Q*/2Mv. A behaviour R = -J r(x) r[x) » (0.09 ± 0.01) —• (7.33) is indeed consistent with experiment but other possibilities are not excluded [Bloom (1974)]. 7.2.4 Juotifiaation of tha parton nodcl Before we go on with a discussion of the parton erodeI we have to make some remarks on the *vo basic assumptions (t) ajid (ii) which define the nodcl. Up to now nobody has succeeded in deriving these assumptions from more basic principles, e.g. from a local rclatlvlstic quantum field theory. It is even very doubtful that this is at all possible, fchat can be done 13 to give scsc plausible arguments [toyman (1972)3 and to make calculations in model theories with other tti hoa assu^i't ion* [Prell (1971) and references cited therein, Landshoff (1971)]. Oic plausible argument goes as follows, he take first a look at a nucléon at rest, he will see partons moving inside the nucléon scattering on each other, being created and annihilated in pairs etc-, all governed by a finite true scale if the interaction energy is finite. Now we look at a nuclccn in notion. Owing to the Lorentz time dilatation wc will sec the internal eoticn of the partons slowed down, the nore, the faster the nucléon. For an infinitely f**t nucU-cn the internal ration will be completely frotcn and wc see a jet of non-interacting partons. Another plausible argurrnt assures that it rakes sense to split the strong interaction [(amiltoman II . ,.ar* II, describing the notion of free partons and an interaction torn - 264 - CH*Tïi« VI We can then expand the nucléon state |N(p)) in terms of free parton states |n) in the following way: |N(pJ) • £ 'yg» In) (7.35) where E^ is the energy of the nucléon and E the energy of the parton state HsC|N(p)> = EjjlNCp)) H0|n) = En|n) . (7.36) Since V is an integral over all space of the interaction energy density, we see immediately that the total three momentum of the state |n) must equal the three momentum of the nucléon. How will the parton states |n) look like which contribute significantly to the wave function of the nucléon? One of the striking features of all high energy hadron-hadron collisions is that the transverse mc. <:nta of the produced particles are limited, with a mean value E E ipl M % K p)1 + + m (7 37) N " n - * '' " £ L" i PiT i^ * ' i=i In the Bjorken limit, i.e. for P * m,fL - E becomes small only if all partons move in the same direction as the nu cléon. Indeed, if S, 2 0 (i = 1, .... n), E^ - E •* i/P. If some partons move vigorously opposite to the nucléon on the other hand we find E^ - E = P. Therefore the chance to find partons moving opposite to the nuclcon is vanishingly small tor P * - . To reproduce the parton model results we have to make one fi.ial hypothesis. IVe will assume th3t for P •* ™ the wave function depends only on the fractional longitudinal moment;. Ç. of the partons and on the transverse momenta p*-, in absolute units. Such an assumption can again be justified by results from hadron-hadron collisions [Feynman (1909), (1972)]. For partons of small momenta, of the order of a CeV or su, the above hypothesis is not thought to apply. This so called "\icc" region would merit special Attention. For the nt ;n features of deep inelastic scattering, however "wee" partons play no role, therefore we will not treat them further here but refer to Feynman's book [Feynman (1972)]. Last but not least we have to raise the question of what happens with the partons after the collision, flow are the hadrons formed out of scattered partons (Fi^. 17)? This is a very difficult problem and no good answer exists at present. If we are only interested in the cross-section summed over alt hadronic final states we can however hope to sweep this problem under the carpet. The hope is that we can replace in Eq. (7.5) the sum over a complex set of states with physical particles (r-nesons, nucléons etc.) by the sum over a complete set of states with free partons 7.3 The quark parton codeI In order to extend the parton ideas to neutrino nucléon scattering [fljorken (1970), Gross [19b9), Llewellyn Smth. (1970)] Wc navc W atopï soo: model which specifies how the weak current acts on the partons. Our first choice will again be the quark nodel already discussed in Section 3. In this node! there arc three fundamental fermons u, J, s with quantum nuctbers as shown in Table 3. There ray or nay not be sone p-trtons in addition *ith I • Q « S • H • n which do not participate in weak and electromagnetic interactions, he will see wh.it experinent says on the presence of such "gluons". INTERACTIONS OF NEUTRINOS The electromagnetic and US = 0 weak currents are < pressed in terms of quark fields in the following way Quantum numbers of quarks. I = Lsospin, . 2 , Q = charge, S = strangeness, = d 1 5 B = baryon number „ t V -1 V - 3 V J* =ÛYA(1 * Ys)d (7.38] I I] q s • J~ - aÏAu + YS)U . u 'A 'A 7, 0 '/, J % -V. -V, 0 '/, Now it is a simple exercise to apply the rules of the par- s 0 0 -v> -1 ton model and to derive expressions for the weak structure '/, functions of the nucléon similar to Eq. (7.27) (Exercise Zl). Everything can be expressed in terms of the six distribution functions for quarks and antiquarks inside the proton since the distribution functions for the neutron are obtained by a charge symmetry operation. The distribution of "up" quarks in the proton equals the distribution of "down" quarks in the neutron, etc. Table 4 Expressions for structure functions in t^rrcs of quark and antiquark distribution functions of the proton Sd(x) N (x) N-(x) Njfx) 'V*' s NgM FfP'lx) V, '/, % V, V. "A Ff'M '/. V, V. '/, % % f,(v|"(x> 0 2 0 ; 0 0 F!*"M : 0 0 0 2 0 F;-"(«I 0 •2 0 ; 0 0 F V.-x structure functions of the neutron not listed in Table -1 are obtained fron the charge symetry relations tq. (".171 . he note that quarks and antiquarks contribute with opposite sign in the vector-axial vector interference tem F,. The jtructurc functions. F,(x) are given by the Cal Ian-Cross relation in our ooJcl sir.ee wv have only spin Vj partoius participating in electromagnetic and wait interactions. In analogy to Ec,. ;7.2*)| we have (*Si.a*), . . .c'vN.v.N) F "(x (x) (7.39) \ • proton, neutron . If wi- look .it Table -Î wc ice thj t not all the six str^turc functions H(xl listed are independent quark Tiod.-l predicts on? rilatlcu a^ong thm [Llewellyn Ssith (19"0)3 i^'u» 4F!C!"(: ''(X)]. "•I ir.-r.ui-i. Mi.it o!si- tun fce suy unout the structure functions F(») In our rxxlel? Giar^e Is a conserved qiuntin nu-inr, therefore Che tot.il cLirRe CJttlcJ hy the pjrtons In the proton rust cqu.ii the proton chnrjie. Die charp.e carried by purts-ns with r.onontun fruition between x and X » Jx Is L *„ \,C*> J. i li'o recall tint w? afe ccniiJ^rlnp only .\S • 0 ncuïrtr.c reactions. CHAPTER VI where q = u, d, s, û, d, s and e are the quark charges obtained fron: Table 3. To get the total charge we have to add the contributions of partons of all raonenta. This leads to a sura rule: ;*z Similar sum rules can be derived by considering the total strangeness and total baryon number of the proton, of the structure functions F.(x) this leads to two sum rules / dxOj^M -F[vp)(x)] = 2 (7.42) flp) P(vp) : f dx x S (x) = 1 - (xN > ?/ r o : (j.vri i i where txN.) is the fraction of longitudinal noccntuzi carried by gluons. Equation (7.44) gives us the possibility to check experimentally iT there arc gluons in the nuci on. Finally the distribution functions must be positive. Actually we can do a little better and derive a set of in equalities using isospin invariance and positivity [Nachtr.ann (1971a), (1972)}. They ore sinunari:cd in Table 5, their derivatioi left j ; ar> exercise (Exercise 25) Table E*T«rùre> liaqualitlcs for the distribution The relations of Fqs. (7.39), (7.40), (7.42), (7.43), functions of the proton in the (7.44) and Tahle 5 can be considered as the essence of the quark parton nodei quark parton nodcl. They do not depend on any assumn'ions for the behaviour of the distribution functions. Should any 1 i\(» ) - V» J 0 of these relations be ccntraJicted by experiment, the quart, Xjl.l : o parton node I would fall, lu- will <:ow turn lo a detailed comparison with cxpcrinent. -V" V" vi.l : o ï.'? b^j'in by considering the ratio of the structure func tions for electron neutron and electron proton scattering. Fron Tables 4 and 5 wç can easily derive the following t>ound> [\achtr--ann (1971a). (1972); M.ijir-J.ir fl9~l)]. r INTERACTIONS OF NEUTRINOS (Exercise Z4), The experimental ratio for the cross-iections*'1 shown in Fig. 15 is within these limits. Tor x -* 1. however, the lower bound seems to be approached. A violation of this bound would be a death blow for our simple quark- parton mo-'?l. Next we turn to neutrino experiments and ask if their findings can be understood in the framework of the quark parton model. It is convenient to rewrite the differential cross-section for ncutrino-nucleon scattering [Eq. (7.lu]] in terms of scaling variables x and y [Table 2]. Since most neutrino experiments in the deep inelastic region have been done on complex nuclei up to now we write the cross-section averaged over protons and neutrons, assuming scaling behaviour for the structure functions [Eq. (7.20)]. A sicple calculation gives for the neutriiio and amineutrino cross- sections in the limit E » M: a ~— --^-|xN(x) •.xN(x)(l - y) + 2xNLfx)(l - y» (7.46a) il2_L = ££!! jxN(x)(1 „ y]a + j^{x) + 2xNL(x)(l - y)\ [7.46b) Ktx) -^F^x) • F^fx) - F[VP)W - F^n>(x)] NU) - Jpf'M * r^hx) * Fj^W * F^(x]] \M * i Ik ^M * £ rlm)W - F[VP](X) - F{™»(x)] . (7.47) For I; >> M the kinematic domain is 0 < x < 1 , C'yïl . (7.48) he note that X, N, N. arc essentially the cross-sections [Eq. (7.18)] for absorption of a left-handed, right-handed and longitudinally polarized ueak current. To see this we have only to t; : the scaling limit of Lq. (7.19). The poaitivity conditions are therefore (sec also Exercise 3): S(x) = ^ o. > 0 N(x) - '-£ «. > 0 KL(x) « ^ oL > 0 . (7.J-J) The cross-sect ions [tq. [7.4b)} are quadratic foms in y = v/E. This is nothing other than our general theorem [tq. (J.17)] in a new form. It is no* t-asy to cimputc the total cross-sections. Integrating over x and y we find Tram 1^. (7.4b); °vf£) • -4-|<»s> * T(XS> * <3CVr <*N> « I dx xN(x) (xS> • J* dx xS(x) *) Ltperlmcnt-ill}- the ratio R - °iVo-j *s sea il for both prêtons and neutrons [sec £q. (7.3J) and Moon (1Q7J)]. riiercforc *c can corrparc the crosi-scctions directly with tq. (7.45). CHAPTER VI Equation (7.50) is a remarkable result. We have found that the scaling hypothesis Eq. (7.20) implies a linear rise of the total neutrino and antineutrino cross-sections with the energy of the incident nputrino. This is something which experimentalists can check rather easily. In Fig. 20 we show results from a neutrino experiment where the target was freon, CFjflr, in the big bubble chamber Gargamelle [Eichten (1973a)]. The data based on 2510 neutrino and 947 anti- neutrino events is indeed compatible with a linear rise of the total cross-sections. Making corrections for the neutron to proton ratio in freon and for iS = 1 contributions w; find from Fig. 20: Fig. 20 Total neutrino ai function of the i • E [Eichten (197ij)j CcMt C (E) - Ar- (0.J7 ! 0.02) û Gbil V 0 (E) . _£_ (0.18 * 0.01) 2J£i - 0.38 • O.02 So far he have only exploited the scaling hypothesis [Eq. (7.20)] Sow w p; on to test the quark-parton node]. Frcn ra.'tc -J t>c finJ cjsily that in the qturk-pjrtan csxlcl the functions N(x), S(x), N. (x) arc given by INTERACTIONS OF NEUTRINOS NW = Nu(x) + Nd(x) N(x) = NQ(x) * Nj(x) NL(x) = 0 . (7.5: [f the nucléon contains only quarks and no antiquarks which implies that the nucléon is always in a state with three v v quarks, the ratio o lE)/c (E) should equal % as we see imnedia'iely from Eq. (7.SO). Since N(.-) and S'L(x) are non- negative, they ca: only increase the ratio as we can also see from Eq. (7.50). Experimentally the ratio nu(£)/oVf is close to '/j [Eq. (7.52)]. Therefore the contribution of R[st) and N,(x) to the total cross-section cannot be JO big. This is analysed quanti tat i'.ely in Fig. 21. . . 010 i Fig. 21 Antiquark and longitudinal contributions to the total cr< N )> [see Eq. (7.5PJ C - <*5)/U(N * 3 + t;L)). n r quite consistent with the snallness of the longitudinal contribution in electron proton scattering [li). (7.52)]. This support;- afiatn the ar-surjition that all partons *hich participate in elcctrorjgnet it and weak interactions have spin V;. In the following he hill rxgicct the longitudi.ial contribution altogether. This should be kept in ninJ as a warning not to take the results which w will derive too literally. Ke have to wait for better data to nakc a norc complete analysis. fce will no*, discuss the dependence of the diffcrruial cross-sect ions [lq. (7.J(,)J on y at fixed x. Tor y • 0 the ^ro^r.-ictt ic-ns for neutrino* and ontincutrinos [ms. (7.4oa,h)] are equal. This is a consequence of the charge if^-x-try relations fi). ("M7I. .Vn c«pcrl=enta. check of this point constitutes therefore a check of" the charge syn- netry properties of the *cak cuircnt. There are IrJications that at high energies the equality of neutrino i>id .uili- neutrmo cross-sections for y • 0 rjy he violated [iiuhcri (1974)]. This ray indicate the appearance of new ..uantun mmhers in h;ulrrn physics but the data is too poor to draw definite conclusions. If we neglect the longitudinal and antiquark contributions in Eq. (7.46) altogether we find a distribution in dependent of y for the neutrino cross-section and proportional to (1 - y)z for the antineutrino cross-section. The smallness of the integrals To answer this question we rewrite Eq. (7.44) in the following way using Eqs. (7.51) and (7.53) (xN) + As experimental information we have the data for the total neutrino cross-sections [Eq. (7.52)] which determines and (xN) if we neglect (xNL> in Eq. (7.50). Furthermore we have the data on electron nucléon scattering [Miller (1972), Bodek (1973)] from which we can compute the following two moments i Y = f dx xF[ep5(x) = 0.167 * 0.008 0 (7.56) -f dxxFÎ* J(x) = 0.126 i 0.006 . where we have put an estimated error of 51. Expressing these moments in terms of quark distribution functions using Table 4 we find the following experimental values for the fraction of longitudinal momentum of the proton carried hy different parton species: (x(N- • Ng)) = 0.03 t 0.01 It is not unexpected that a large fraction of momentum is carried by "up" and "down" quarks. After all, in the naive shell model of hadrons the proton is a pure state of three quarks, p ^ uud. What is surprising is the laTge fraction of momentum carried by gluons. f) TJiÇ_9yê!ÎL^Ï5?li!ïy£,î23_fyDÇ.£i2B§ From the data on the differential cross-sections for neutrino and antineutrino scattering we can in principle ex tract the distribution functions N(x), N(x), N,(x) of Eq. (7.46) for all x. With the available data experimentalists had to make some simplifying assumptions. The longitudinal contribution was neglected. Since a large fraction of the data is in the resonance region a scaling variable x' (see Table 2) was used. The scaling functions in this variable arc supposed to describe also the resonance region on the average [Bloom (1970)] , The resulting distribution functions are shown in Fig. 22. The antiquarks are seeseenn to concentrate at snail ; With the help of these distribution fuXtions we can also check the Gross-Llewellyn Smith sun rule [Eq. (7.43)] which we can rewrite as / dx'[N(^') - N(.<'j] = 3 .) In the true asymptotic region, i.e. for ^ -* « and v •* » the choice of x or x' as scaling variable would be im material. At the presently available moderate values of Q! and v however the difference between x and x' is not negligible. The data on electron proton scattering seems to prefer scaling in x'. INTERACTIONS OF NEUTRINOS QUARK WO AWnOUAHK MOMENTUM DISTRIBUTIONS N(x7 m number Of quarts with momentum »' McElh*nty&Tu*n * AltiftUi tj *| undinoff a Pôlbnghom* Fig. 22 The distribution functions x'N(x') and x'N(x') defined in Eq. (7.53) [Deden (1975)J. The lines correspond to various redictions based on the electron-nucleon scattering data ELantishoft' (1.971a), McElhaney (1973), Altjirelli (1974)]. using x' as scaling variable. Experimentalists find for the integral in Eq. (7.58) a vali'c of 3.0 ± 0.6 [Deden (1975)] in excellent agreement with the quark parton model. g) LÇïJ2S5X..9y§ï!î§-§rS_ï!3ÇÏ?_iQ_5!2ejHJÇÏÇ0D? To answer this question we consider the following moment [compare Eq. (7.29) and Table 4]: f . F epJ( J ^ — ïï (q = u,d,s,Û,3,I) (7.60) where (Nu) is the total number of "up" quarks in the proton and similarly for the other quarks. If the number of quarks in the nucléon is finite, the integral on the left-hand side of Eq. (7.S9) must be finite which implies F'G|)'(X) * 0 tep) ep) for x - 0. The experimental results for Fa •= v»J [Fig. W] indicate however that F^Cx) approaches a non-iero value for u = x~l •* =. Should this behaviour be substantiated by future experiments, we would have to infer that the proton contains an infinite number of quarks and antiquarks. This concludes our discussion of tire quark parton model. All experimental findings of electron scattering for energies E S 20 CeV which vc have mentioned seem to be consistent with the quark parton model where the quarks carry fractional charges as shown in TM,1? I. More detailed comparisons with experiment support this view and give stringent ;redictions for as yet urateas'jrcd quantities [e.g. Kachtmann (1973), de Rujula (1974), Ktihnelt (1975)]. A simple a'.sat: foi the wave function of the proton is often used. The proton is magined as a system of three "valence" quarks plus an isospin symmetric infinite "sea" of quark-antiquark pairs ip> -v îuud)|SU(2) symmetric sea) . (7.bl) Such a ^odel is suggested by the shell model of hadrons. It is consistent with the data fron deep inelastic lepton- nucleon scattering (Exercise 25) but it cannot be deduced frcm this data alone. This is understandable since deep inelastic scattering is only sensitive to the one quark distribution functions. The sua rules of deep inelastic scat tering discussed in Section 7.3.1 imply only that the total iumber of u-minus û-quarks in the proton pust be two, d-minus 3-quarks one and s-ninus s-quarks zero as in the naive picture, where p ^ uud (N • N.) « 2 To get more detailed information on the wave function of the proton one has to consider other processes. Mast relevant in this context are the attempts to connect the shell model of hadrons and the pan.on picture [Gel1-Mann (1972), Melosh (IS™)» Hey (1974) and references cited therein]. 7.4 Other parton models What about other parton models? The quark-parton nodel is simple and attractive but quarks carry fractional charges which opens a whole Pandora's box of problems. Let us therefore investigate if the data allows an alternative. We will now forrajlate the parton model in more general terns than is usually done [Nachtmann (1972a)]. Suppose that partons which couple in electromagnetic and weak interactions have only spin xh as indicated by experiment [Eqs. (7.32) and (7.54)]. Let ty be the parton spinor and let us make the following ansatz for the vector, axiaL vector .ind hypercharge currents: K - ^* (a = 1, 2, 3) A » - «vsi> A - ^Y* where Ta, T3 and V are scce natrices acting in the space of internal quantum numbers c£ partons. will assume that the electronagnetic current is given by the Gell-Mann - Nishijica relation J™ = h^ Q = TJ *\ Y . (7.64) The current algebra relations [Eq. (6.2)] require the matrices 1/2(T^ ± T3) and Y to satisfy the commutation re lations of the group SU(2) " SU(2) * Y, i.e. we must have [\ (T3 s T?) , \ (T* * TÎ)] = ieabc 1 (Tc • TÇ) [\ ff t TÎ) , | (Tb T TÎJJ - 0 [| 0* î Tf) , y] = 0 . (7.65) Therefore we can classify all parton models which satisfy the current algebra relations [Eq. (6-2)] according to the irreducible representations of SU(2) x SU(2) * Y occuring in the decomposition of the representation given by the matrices T3, T?, and Y. How can we decide if a given parton model is compatible with experiment? It can be shown that to a given parton model, i.e. to a given representation of SU(2) x SU(Z] * Y, there corresponds a certain allowed domain for the obser vable structure functions F(x) [Eq. (7.20)]. These domains can be calculated explicitely using techniques similar to those which led to the inequalities of Table S [Nachtmann (1972a)]. If the exper'jnental values for thr structure func tions lie outside the allowed domain of a particular parton model, this model can be excluded. n As an example we consider all parton models wr re partons have isospin T < '/2 and integral charges Q with |Q| < 2. In Fig. 23 the allowed domain for the total neutrino and antineutrino cross-sections [Eq. (7.50)] in thess models is shown, where we have taken the experimental values for the moments Y [Eq. (7.56)] as input. It is seen that all these models can be excluded. INTERACTIONS OF NEUTRINOS YP -V, p *Tn Fig. 23 The allowed domain for 2 •= in all novels where partons carry ieospin T S V2 and integral charges |Q| £ 2. The experimental values for Y n [Eq. (7.56)] are input. The experinental values for Z, Z [Eq.' (7.52)3 arc indicated by the cross. Intuitively we can understand this result as follows. In electron-nucleon scattering one measures the charge z ! squared Q of the partons, in neutrino scattering the catrix element squared of the isospin raising operator |T+; . 1 The rather large value of the neutrino cross-section as compared to the electron scattering data implies Q < !T+|*. If we insist on parton isospin %, then {T+l = 1 and we must have fractional charges. If we want integral charges, Q* = 1, we must allow for partons with isospin > %. Partons of isospin 1 give, e.g. |T+| = A detailed comparison of parton models where partons carry integral charges and isospin < 1, with experimental data has been made [Kiihnelt (1975]]. It was found that such models are compatible with the data from deep inelastic electron ard neutrimrnucleon scattering. All such models are nevertheless in great trouble. Consider a parton theory where the octets of vector and axial vector currents are constructed in the manner of Eq. C7.03) and satisfy SU(3) * * SU(3) current algebra relations. Then the following can be shown [Kuhnelt (197?)]. If the theory contains only integrally charged partons and describes correctly the data on deep inelastic scattering and the amplitude for the decay n"1 -* Zy, we predict for the ratio of the cross-sections for electron pos-ûron annihilation into hadrons versus muon pairs: , a(e*c~ -* hadrons) . 11 , (7.66) o(e*e" - uV) This is rtuch too high compared to recent experiments, which give R *\, 5 [^Augustin (1975)]. Therefore it seens that we have to live with fractionally charged partons if we believe in partons at all . We have then a great puizle. The constituents of hadrons seen to be quasi free in deep inelastic Scattering but neverthe less rcust be permanently kept inside the hadrons or else we would obseive physical particles with fractional charges. The study of possible "quark confinement" mechanisms is a topic of current rerearch. •) These fractional charges could be only "effective" charges. A part of the charge could be inoperativi at present energies and only show up above some threshold [Han (1965), Lipkin (1972)]. "Il _ 274 - CHAPTER VI 7.5 tond us ion and outlook Up to now we have discussed the simplest ideas which have been put forward in order to understand deep inelastic phenomena. The picture turned out to be surprisingly simple. In a reference frame where the nucléon moves very fast we see essentially a jet of fast quarks accompanied by a cloLd of slow but probably infinitely many quark-antiquark pairs. Can the quark-parton model bo derived from some more fundamental theory? To answer this question one has to uie nore sophisticated techniques than we have developed in this section. It can be shown that the behaviour of the struc ture functions K.(v,Q!) [Eqs. (6.28), (7.6)] in the Bjorken limit reflects the nature of the singularity of the com mutator [J (x), J,(0)] of the electromagnetic or weak currents for x2 •+ 0, i.e. at almost light-like distances [Frishman (lî>70), Leutuyler (1970), Brandt (1971)]. It turns out that the quark-parton model corresponds to free field singu larities for x1 •* D. All the results of the quar.k-parton model can in fact be rederived by postulating free field be haviour of the current commutators near the light cone [Gell-Mann (1971), Cornwall (1971)j. For an excellent review of all these matters we refer the reader to K. Wilson (1971). Cur question can now be rephrased in the following way. Are there theories where the product of currents has free field behaviour near the light cone? it seems that in the franework of conventional local relativistic renormali2a)>le field theory no acceptable interacting theory has this property [parisi (1973), Call-Ji (1973)]. The theories shich cone closest to free field behaviour and Bjorken scaling [Eq. (7.20)] are the so called asymptotically free theories [Gross [1975), Folitzer (1973)], where bjorken scaling is only violated by logarithms. The experimental study of deep inelastic and related phenomena like electron-positron annihilation into hadrons is therefore of great interest. 5hould exact Bjorken scaling behaviour turn out to hold in nature then conventional field theory would be in great trouble. Should violations of Bjorken scaling he observed then we would have to study the precise nature of these violations which would indicate what kind of field theory, if any, describes hadron physics. At present some exploratory experiment.5: on deep inelastic scattering of muons and neutrinos off nucléons with energies up to E '-- 200 GeV have been made. It is remarkable th.it to a first approximation [t 20*.} Bjorken scaling and the smple quark-parton model seem to work at these higher energies [Benvenuti (1974), h'atanabe (1975), Barish (1975)]. But there arc also indications that the simple picture which we developed in this section is not the whole story. New hadronic degrees of freedom may show up [sec, e.g. Benvenuti (1975)] and/or Bjorken-scaling may lie violated [Chang (1975], It is too early to draw a definite conclusion except that the subject of deep inelastic lepton-hadron scattering will remain in the forefront of the interest of high energy physicists for some time to come. Acknowledgements It is a pleasure to thank J.S. Bell, H.K. Gaillard, D. Haidt and J. Prentki for valuable discussions and sug gestions and the CEHN Typing Service for its excellent work. INTERACTIONS OF NEUTRINOS - 275 - REFERENCES Ailler, S.L., Phys. Rev. 135, 8963 (1964). Adler, S.L., Phys. Rev. Letters 14, 1051 (1965); Phys. Rev. 140, B736 (1965). 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Symposium on Electron and Photon Interactions at High Energies (ed. N.B. Mistry) (Cornell University Press, Ithaca, \Y, 1972), p. 97. Zuckcr, P., Phys. Rev, D4, 3350 (1971). Zweig, C, reports CERN TH-4D1, 412, 1964 (unpublished). INTERACTIONS OF NEUTRINOS that Vt mist be :ero in this case if time reversal invariance holds. ,:) Express A, B, C of Eq. (2.17) in terms of W. (i = 1, ..., 6) of Eq. (2.14). Why does H6 drop out? What correla tion would one have to observe in order to measure W&? 5) The hadron tensor K , is by its definition in Eq. (2.13) a positive secidefinite quadratic form. Derive the positivity conditions for W. (i = 1, .... 6) of Eq. (2.14) which follow from this fact. 4) Use the transformation properties of quark fields and nucléon states under a charge symmetry, time reversal in variance and charge conjugation operation to verify Eqs. (3.10), (3.11), ard (2.14). The transformation properties of Dirac fields and Dirac spinors are discussed, for example-, in [Bjorken (1965)]. For the discussion of charge conjugation invariance we also have to know that the substitution rule relates the weak form factors of anti- nucleons to those of nucléons in the following way: W)|jjlp(i>)> = -v(p)r:;c-p,-p'Mp') , where I\(-p,-p') is the analytic continuation of I\(p',p) [Eq. (3.3)] to negative time-like vectors. 5) TCP invariance is another of the fundamental principles of local relativistic quantum field theory [Lûders (1957); see also Bjorken (1965), Streater (1964)]. The assertion is that there exists an antiunitary symnetry operator 3 = TCP which transforms the currents and the nucléon field (i(x'J in the following way: (OJ^(x)Q-,l+ = --^(-x) 1 t (frMx}Q- ) = vs*(-x) • Does this imply any restrictions on the form factors QEq. (3.3)]? The existence of TCP symmetry also explains why charge symmetry and time -eversal invariance combined imply absence of second-class currents, since we can express the G-parity operator as follows: G = PjT'OP"1 . 6) Calculate the quasi-elastic cross-section (do/dQ2)v,u and verify Eqs. (3.36) and (3.37). Show that the form factor fP(q*) corresponding to the diagram of Fig. 3b is an analytic function in q2 with a cut starting at q1 = 4m£. 7) Prove the inequalities Eq. (4.2). Discuss isospin relations among the following reactions: v^p •* u~a+* - u"p-+ v n * U~û+ "* U~pnfl , ti"nn* vbP - uV •+ uV~ . u*nv° V ~ v*à~ * »*nr~ • (tow do thv ratios (u"pn*) : lu"pTB : (u~n«*) for pure a production compare with the experimental values cited in Table 1. 6) Rewrite the matrix element of the electromagnetic current between nucléon states [Eq. (3.19)] in the non- relativistic limit using Eq. (4.5) and keeping only the lowest-order terms in the momenta p and p'. 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B59, 641 (1973). Piketty, C.A., and L. Stodolsky, Nuclear Phys. B15, 571' (1970). Pilkuhn, H., W. Schmidt, A.D. Martin, C. Michael, F. Steiner, B.R. Martin, M.M. Nagels and J.J. .ie Swart, Nuclear Phys. B65, 460 (1973). Politzer, H.D., Phys. Rev. Letters 30, 1346 (1973). Ravndal, F., Nuovo Cimento Lettere 3_, 631 (1972). Sakurai, J.J., Ann. Phys. (NY) 11_, 1 (I960), Sakurai, J.J., Phys. Rev. Letters 35, 1037 (1975). Schreiner, P.A., and F. von Hippel, Phys. Rev. Letters 3J), 339 (1973); Nuclear Phys. BS8, 333 (1973). 9) Show that time reversal invariance implies relatively rerl form factors fym and f»p in Eqs. (4.19) and (4.20) ,flint: use the transformation properties of the weak currents as in Exercise 4 and derive the transformation property of the A by considering how the angular momentim is changed under a tine reversal operation. 10) This exercise is only suggested for readers having some knowledge of the non-relativistic, SU(6) sycnietric quark model. Use the wave functions for nucléons and A resonances [see, for example, Thirring (1965), van Royen (1967)] to derive values for u^/(l + <:'), gA = -g.fO), grR/gr, f^, f^, [Eqs. (J.20), (3.3), (4.11), (4.19), (4.20)]. Cocpare this to the experimental values quoted in the text. 11) Show that natrix elements of Afa defined in Eq. (5.17) have no pion pole. 12) Verify Eqs. (Ù.10) and (5.21) by perfortring the substitutions indicated in the text. 13) Expand the cross-section formula for A ..reduction [Eq. (4.23)] for small Q2 and verify the general PCAC result [Eq. (5.8)]. Take the numerical values of the form factors [Eq. (4.26)] and estinmie the value of Q2 where the correction terms become as important as the PCAC term. 14) Separate the nucléon Born term i.e. the quasi-elastic contribution in Eq. (6.31). Use Eq. (2.17) and the results of Exercise 2 and Section 5 to show that Eq. (6.31) reduces to Eq. (6.17) for Q2 = 0. 15) Derive the cross-section 32o/3v3Q2 for the reactions Eq. (1.1) with a sum over all hadronic states of strangeness S = 0 using Eqs. (2.17), (6.28) and the results of Exercise 2. Express the cross-section in teiros of the vari ables v, Q2 and E and neglect the muon mass. Integrate 3*o/3v3Q2 for fixed E and Q* over the allowed kinematic interval of v to obtain do/dQ2 (E,Q2). Assume that the structure functions K-(v,Q2) (i = J, 2, 3) of Eq. (6.2B) have the folic JJ:I~ behaviour for v •+ », Q2 fixed: il* D^W) •h*'P>cv,o')] = o litn-CufP'fv.Q1) - WW („,<)•)] = 0 . V+o) Show that with these assumptions which are consistent with the expectations of Regge theory, we can derive Eq. (6.32) from Eq. (6.31). 16) Use the commutation relations for the strangeness changing weak currents jZ [Eq. (2.6)] which follow from the extension of Eq. (6.2) Co the octets of vector and axial vector currents to derive the analogs of Eqs. (6.31) and (6.32) for AS > 0 neutrino nucléon scattering. 17) Use Lorenta, parity invariance and the conservation of the electromagnetic current to show that the tensor w' ' pA E14. (7.5) can be expanded as shown in Eq. (7.6). N\, J is by definition [Eq. (7.5)] a positive semi-definite tensor. Show that this implies the positivity conditions Eq. (7.12) and vice versa. 18) Assume that W\ ' [Eq. (7.5)] is finite for QJ +0 (v > 0) and use ihis together with current conservation to show z 2 2 that oT(v,Q ) stays finite and o,(v,Q ) •* Q in this limit. 19) Use the definition of wJjN,vN) and of cj^*^ [Eqs. [6.28) and (7.18)] to verify Eq. (7.19). 20) Use energy and momentum conservation to discuss the absorption of a virtual photon y* by a parton in the Breit frame [Eq. (7.22)]. Show that only a parton carrying a fraction x of the momentum of the nucléon can couple to a photon with Bjorken ratio x = Q!/ZMv and that the parton gets its nratentun reversed, rv fleeting transverse momenta. Zl) Calculate the hadronic tensor K , Eq. (7.5) for point particles of spin 0 and spin '% , fass n- and charge e-. Verify Eq. (7.27). 22) Apply the parton model rules and use Eq. (7.38) to verify the results given in Table 4. INTERACTIONS OF NEUTRINOS 23) Derive the inequalities for the quark distribution functions given in Table S. Hint: consider a Fbck space of free quarks and 3ntiquarks. In this space ve can define a Schrtidinger operator u(x) (u (x)) which annihilates (creates) an "up" quark carrying mcner.rum fraction x and similarly for the other quarks and antiquarks. The distribution functions can then be expressed in the following way: \,G0 = (pli^bOuMlp) , etc. Deccmpose the state u(x)jp) into eigenstates of isospin and use positivity. 24) Use the results given in Tables 4 and S to verify Eq. (7*45). 25) Assume that the wave function of the proton is given by Eq. (7.61). What does this special quark parton model predict for the integral / * Is the result specific for the wave function Eq. (7.61)? SOLUTIONS OF EXERCISES 1) If tine reversal invariance holds, there exists an antiunitary operator T such that the S-matrix element for the reaction (1.1) satisfies (X(p'),'j(k')!S|v(k),N(p)) = where the time reversed states Tv, Tu etc. have ncaenta and spins reversed and "in" and "out" states interchanged, teglecting radiative corrections we find ftori the current * current lagrangian Eq. (2.3) together with Eq. (S.l) + h t(£)A h From the explicit expression for the lepton current [Eq. (2.4)] we find 1 t + X (TjP(5Tft)r ) = j W (x>t) (S-3) + U) CZ) tfhlA h) U(p')|J |NCp)> = {TN(p)]J.[ )TX(P')) (S.4) Note that the one nucléon state transforms under T as foll-rvs !TN(p,s)) = (-l),/,'s|N(-f,-s)) (S.5) where s = t'/s stands for a nucléon with spin in the positive or negative z-direction. Inserting Eq. (S.4) in Eq. (2.13) we find that the tensor W\^(p,q) with a sunration over all hsdronic final states and an average over the polarizations of the initial nucléon must satisfy the following relation: W^Cp.q) - Wtv'A"(p.q) P~ u u Cs*« Comparing with Eq. (2.14) we find immediately that Eq. (S.6) implies W6 = 0. Similarly we can conclude that W„(».!•(£v * n)' must be zero whenever we sum over a set of hadronic final states which is invariant under T, e.p. all quasi elastic final states. 2) The expressions for A, B, C of Eq. (2.17) in terms of W- [Eq, (2.14)] are: A - 2(0" • oJ)«, - [ - MOW, * C = Ç W, (S.7) where the upper Closer) sign applies for neutrinos (antineutrinos). Since Wfi corresponds to a T-odd term it can only show up if we look for a T-odd correlation, i.e. a correlation which changes sign under a reversal of nmenta and spins. Such a term is, e.g. the polarization of the i-'-nal atuon normal to reaction plane, corresponding to the correlation o(\ x k') . (S.8) Let 9zo;v* V3E'3îî' be cross-sections for observing muons with polarization parallel (antiparallel) to Ï * it' where we sum over a set of final hadronic states r such that W^* ' has an expansion as shown in Eq. (2.14). A simple cal culation shows indeed that ; (v r] 3 q- ' _ G* njh'l' sinew(v,r) _ 3E'3fl' 3E'3ÎÎ' 2TT* M The polarization of the muon can be analysed by observing its decay, 3) The positivity conditions for the functions W. [Eq. (2.14)] are: phr1».-«.][«,•£». *£«- -S*.]1*?,^15*.-2<»> • «.)!• • 4) a) Çharge_sj™etry The transformation properties of current and states under a charge symmetry operation [tqs. (3.7), (3-9)] lead immediately to r*(p\p) = r~(p',p) . (! Together with Eq. (3.4) this implies Eq. (3.10). b) ïiH£_ï!ÇÏÇE5?i_iî!Y9ri9i!££ iitplies that there is an antiunitary operator T which transforms the current and the states as follows: (TjJ(x,t)T-')+ = J**(x,-t) S (: TlNfp.sll = (-11 ^~ lN(-p;-<;H l Cs = * /2; . From the properties of an antiunitary operator we deduce The Dirat -,iors have the property [see, e.g. Bjorken (1965)3 u(-p.-s) = (-1) Vï+%ùT(P,s) (S INTERACTIONS OF NEUTRINOS where "G is a matrix satisfying S-VB- YJ l trY5?_: = -YJ TV- -S= -t," • li* - -Î.T (S.1S) 2 with our explicit representation of Y matrices (metric 1 of Appendix I, Chapter I[) 6 = ivsY . Inserting in Hq. fs. 13) and using Eq. (3.3) we find rjfp'.p) -B"I(rl*(p',p))TB (S.16) c) G-parity. The transformation properties cf the nucléon states under a G-parity transfornation are: Gip) - -|n> (S.l?) G|n) - |p> where momenta and spins are unchanged. If the currents transform as in Eq. (3.12) we find easily the following restrictions on the vector and axial vector parts of r. (p',p) [Eq. (3.3)]: TT(P',P) ' -e-'(ry*(-p,-p'))Te A * (S.18) rj'cp'.p) •e-'(if*<-p,-p'))Te where C is a matrix satisfying: uT e-'yWe - -Y -1 e Ysc = YJ er^cp.si = v(p",s) [s.i9) In our representation C = ÎY2Y°- Equations (S.18) and (S.19) lead immediately to Eq. (3.14). 5) The TCP operator 0 transforms the states as follows 0|N(p,s)> = (-l)'A"S|N(pVs)) . (S.20) Usir.g the properties rf the Dirac spinors 1,4 S v(p,-s) = Ys(-l) " u(P,s) CS.21) we find easil> that TCP symmetry implies rj(p',p) = -Ysr*(-p',-p)Ys • (S.22) This is automatically satisfied by virtue of the substitution rule. 6) The contribution to the form factor fJ'Cq1) of the proton from the diagram of Fig. 3b is f^(q') = -Jig* [ [ [ da, da., da3 5(1 - a, - i2 - as) • (S.23) ? t- tr-? *| r - z i J "TvHIV * 2MlaiJ [£ï + ^a\at- mJ(°i * °ïï " "^J This is a divergent integral which r-ust be subtracted at q1 = 0 (vertex renormalization). If wc are only interested in the ar-alyticity properties wc can as well study the derivative df^q^/dq2 for which we find easily CHAFJ'ER VI -,— = —^r J / I da, da, da. 6[1 - a. - a, - a,) dq* iWJJj ' 2 3 * aia ZHza|3,aj 1 { 2 + : tt a [Va^ï * m^Cot, + o2) M'ÙJ] [-q a, 7 * "^«i * i> * "^D'I i analytic function of q1 fxcept for a cut which is given by the zeros of the denominate 3 _ ni(a, + a2) * M*Oj : 0 ; : 1 . (S.Z51 This implies indeed a cut starting at q3 = 4nr. The proof of the inequalities [Eq. (4.2)] is straightforward. Turning to A production we observe that the coupling CJnstants for all the reactions listed are islated by Clebsch-Gordan coefficients. In the cross-section, however, the V-A interference tern enters with opposite sign in neutrino and antineutrino reactions. Therefore we find for the cross-sections only the relations: aivyn * y*ù~) *.L-J^: (S.26) Taking into account the decay of the a we find for pure A productif (li-pn*) : (u>.°) : (u"m*l -9:2:1. (S.27) The experimental ratios (Table 1) show therefore that there is a substantial I = VÎ contribution. The expressions for the matrix elements of the electromagnetic current [Eq. (3.18)3 in tne non-relativistic limit (.I'aS'JlJflNOS)) - Mt'* —!fi N (N'(p')lJ |K(P)> • 2MN' •tJi. i^^-è^H^B^M] where N, N' are the non-relativistic i-auli spinors-isospinors for the nucléons. The term = p' + p" is called the convection current, the term * q * o the magnetic moment current. For the matrix elements of the magnetic moment operator for nucléons and A at rest we find: The electromagnetic coupling of the £ [Eq. (4.15)] which we have in our theory corresponds therefore to a magnetic dipole transition. T.is is consistent with experiment [Dalitï (1966), Berends (1971)]. 9) The transforation property of the angular montentun operator £ under a tine reversal transformation is CTJT-1)1" = -3 (S.50) Let |A('/:,m)) (m = -%, •-•. Vj) be the canonical basis for the states of the A at rest with the usual phase con ventions [sec, e.g. Edmonds (I960)]. By a suitable choice of over-all phase we have therefore T|A(%,mj) = (-U^lûtV^-m)) . (S.31) &à*t& INTERACTIONS OF NEUTRINOS for an arbitrary Û state at Test we write % m-'/2 (s % > »/i î = E E E cm<1't i '*»l%.»>*,lls m=-% i=-i s=-'A where we have used the usual Clebsch-GoTdan coefficients and canonical base vectors for spinor and vector re presentations. Ke find now from Eq. (S.31) and from the transformation property of the weak current under T [Eq. (S.12)]: T;0(S)> - !o(S')) S' = e-it" U"!J*!p(p)> • -(TS"|Jh|Tp(p)>* . (S.34) Intersert inb . s. (4.19), £4.201 and using the transformation properties of the states [Eqs. (S.12), (S.33)] leads indeed to rVR • fV R ' 10) From the explicit wave functions for nucléons and the Experiment A resonance we find the tabulated values, applying the 31(b) rules of the SU(6) symmetric non-relativistic quark -1.25 SA -% model. Since the model neglects the N-A mass dif 1 -0.685 ference, the predictions for the transition form fac "n" * »p> -'/, tors are somewhat ambiguous since we could multiply £AR 2 2,35 than by factors (M/M.) to some power. For a discus f 1 2 79 W * "p " ' 4.05 sion of corrections due to SU{6) symmetry breaking we 8„l/6 3/275 • 0.85 1.24 refer to Ualitz (19b6). r 11) Suppose a r... trix element of A.J between states a, g had a pion pole. Then we would have q a > ..a . q •••_ (s.; q = p' - p where the dots indicate terns not singular for q* = jnjj. Fran Eq. (5.13) we conclude then that also the rtatrix elerent of j., has a pion pole. But a iratrix element of the pion source has by definition no pion pole and we found a contradiction. 12) The solution of this exercise involves straightforward algebra. The result is given in the text. 13) The expansion of the cross-section formula for A production [Eq. (4.23)] for snail Q* is E"«C b SEW o m>)h •«Mâf (*-M!>' (S.37) . ifyRl' ("' - M']' * ^(E . E')' ^ p_ fyjf^ Ç4W(E . E') • 0(QlU ISCRI where we have used Eq. (4.21). For Q2 = 0 we find indeed E (1 * ...> of Eq. (S.3?) a result consistent with Eq. (5.B). Inserting numerical (CeV) CQ' in GeV2) values [Lq. (4.26)] we find for typical values of E and 1 1 • QVO.IZ W = 1.232 GeV for the expression in curly brackets in 2 1 * QV0.24 1*1. (S.37) the results shown in the table. 3 1 * OV0.33 14) Separating the nucléon pole term in Eq. (6.31) we find f,«') * fî«") i * sW * / [«Î"1"(". where fj ; and gt arc the form factors defined in Eq. (3.3) and Q* + 2*ta,. » m£ V<^ = ÏT—1' ts-39) Fron Eqs. (2.17), (S.7) and (5.8) we find for scattering angle e = 0 in the limit E •* •» keeping v fixed (v f 0): ! T 71 E_„ E* 3E 3(1 I, - + p , u* + x[s = 0j J 2r' 2^ " v P- P Setting Q2 = 0 in Eq. (S.38) arid inserting the corresponding values of the form factors f,(0) =1. 8,(0) - -gA and of W: [Eq. (S.40)] we arrive immediately at Eq. (6.17). 15) From Eqs. (2.17) and (6.28) we find easily, neglecting the muon mass: S} • The kinematic domain in v for fixed Q1 and E is: S - » - E " 4E • = K J 'h'Lf^W) -K,W(v.q!>] . Q'/2M Inserting Eq. (6.31) we verify Eq. (6.32). 16) The comnutation Tclations of the currents j, [Eq. (2.6)] are Dttx'.U.jJy.t)] - 4J£ (Vj(î,t) . Aj(î,t)) • ^ (VJrjS.t) • A;(î,t))|o'(S - 5) . Define structure functions K(vN,vM)| 1 I iStn INTERACTIONS OF NEUTRINOS for the strangeness changing case in a way completely analogous to Eq.(6.28) but with J" replaced by j" (i = 1, ..., 5; N = proton or neutron). Using exactly the sa»e techniques as in Section 6.3 we derive easily the fol lowing sum rule N) tVN) /civ [Wp (v,q') - Wa (^Q')]4Sj,„ " 2 Sunning the AS - 0 and US j1 0 contributions we obtain 2 2 , , j dv fwj^tv.Q ) - ^(w.Q'Otota! = tff* - sin 6C)(N, I |.V> * f sin' ec (N|Y|.V>] . (5.-16} Making assumptions about the behaviour of IV.{v.Q2} . (i = 1, 2, 3) for v -+ », Q2 fixed analogous to the ones of Exercise 15) we derive from Eq. (S.46): t )(N|l3|K) *4sina e (N-Y!N)] . (S.47) £[e- c c Q: fixed £W£W shown in Eq. (2.14) but with Wj = 0. The electromagnetic current is conserved which implies This leads irenediately to Eq. (7.6). To prove the positivity conditons [Eq. (7.12)] it is best to express Kvf ' in the helicity basis [l-.q. (".8)], where the tensor is diagonal, he find easily: (cN) (eN) - eN) K -r£ c* . c c* 1 -Ï- 0 » c e -*-c< . (S £ C c e li "pj L .0 .> «-o^-jJ ,„Ja °T op oJ 4i.'o L 18) IVe find from Eqs. (7.8) and (7.10) using current conservation [Eq. (S.48)] „«*•) , 4af« „(o.N) „(eN) _ «n't, <£ „(eN) CS.50) hhcre the coordinate system is the same as in Eq. (7.8). 19) Equation (7.19) is verified by straightforward algebra. 2D) Neglecting transverse momenta, the four-momenta p,, p. of the parton before and after absorption of the virtual photon are: ! pj - (A • P;,0,0,P,) (S.S1) P* • (A* • P",O,O,PJ) where m is the parton mass. Energy and momentum conservation requires P, * 1 • P: M' • Pj - A' • Pj P, - 2Px • Pj (S.52) from which we conclude -P, • (S.S3) 21) The tensor VrJ' [Eq. (7.S)] for a point particle of four-roomcntiTti p., spin 0, mass m-, and charge e^ is •«V„ . e=«(2piq• q.,(PlB - ^)(Pu - *f±) ±. Equation (7.27) is easily verified using Eqs. (7.25), (S.54) anc (7.26). 22) Let us e.g. calculate the structure functions for vp scattering. First we calculate the tensor W^ [Eq. (6.28)] for scattering of neutrinos on point quarks. There are two reactions to be considered v + d •+ u • u Let p be the four-momentum of the d- or û-quark, m their mass, then we find easily for unpolarized d- or û-quark in the limit m/p0 + 0 w(vd,vû),Si 2 (5 S6) "(P.q) = i «2M * q )| SpA V * 2PpP, ± «^V * Pp% * V»*} ' - In the Breit system [Eq. (7.22)] in which we are working we will have distribution functions for d- and û-quarks with positive and negative helicity (t or +) »dt(E) , «d4K> , N.,tO , Nfit(0 . For an unpolarized proton we have (S.57) "dN,,K» > - V,,tt' ' 2 »AB"d N s t ,c totaI where N.(0 = d*(0 * *VU^' * ' distribution function for d quarks and similarly for u quarks. The factor (1 • Ys) in tne expression for the weak current [Eq. (7.381] tells us that for n/p0 * 0 only d-quarks with negative helicity and û-quarks with positive helicity con couple to neutrinos. Therefore we find K(v,d.) , Zh.(vc) pX p). KCM» « 0 pA K(v.M . 2„(v.Qi . (s.sa) Inserting now in Eq. (7.26) we hav; i di a,, s 5 £ N^'CP.,] • / «|- [Kdim«£- hi.«) • Ns,(5)w^' e-*]P.£P • < - " Using Eqs. (S.56), (S.S7) and (S.SB) we find indeed F["P'^) " 2N,;x) • ZNQ(x) VP, FS (X) -»[2Nd(x) • 2N-(x)] (up, F, (x) - -2Nd(x) • 2N0(X) . 23) Kc can write the quark distribution functions in the following way: t NUM " + NdW • Decooposing into cigenstates of isospin we have: INTERACTIONS OF NEUTRINOS u(x)|p> = -v/| |1,0> * J\ |0,0) d(x)lp> = |l,l> . Therefore the positivity constraints are N,(x) = (1,1|1,1) = {1,0|1,0> = Nd(x) > 0 N8(x) = (0,0)0,0) = 2Nu(x) - Nj(x) I 0 . Sinilarly ue prove the inequalities for the antiquark distribution functions: N,(x) - NQ(x) > 0 RB(x) = 2Nd(x) - Stj(x) > 0 . exi N( 0(x) introduced in Exercise 23. c(ep) 2„J.1KI.1K. . 1 J . 1 ; . 1 ,, Ffen) = IT NO + \ N> * è Ns+1 s°+1 *>+ i Ns • (s-661 Since all the ,\"s are non-negative, at the extrena of the ratio FJCT'/F} only one N is different from zero. From Hq. (S.66) ve see now iinedlately what the extrera are: only N0 f 0 F[«n)/F(ep) , 1 1 only N9 * 0 F^/pf* = 4 . If therefore for a certain value of x experimentalists find f\enïx)/Fj-ep^(x) = 'A (compare Fig. 15] ne can deduce the following: i) At this value of x the distribution functions of the proton vanish for all quarks and antiquarks CÏ 'pt the "up" quark. ii) the partons moving to the right after the absolution of the virtual photon by the proton (Fig. 18b) are in a state of isospin zero. 25) If the wave function of the proton is as shown in Eq. (7.61) we can split the distribution functions of the proton into a "valence" and a "sea" quark contribution. V) 5) Nu(x) - NJ (X) + ^ (x) N.(x) = N.(v'(x) +NJ5)(X) Ns(x) - U^'M N (x) - Q ltf°M Nj(x) • N,(S>(*> q , N-(x) • 4S)M where we require isospin and particlc-antiparticle sycc«try for the "sea" distribution functions. N CS) S .NS (X).N| '(X) . For the valence qiarks we have the sun rules J dx NJ/'tx) • 2 / dx NJV)(X) • 1 . ^•^Stew Fron £qs. (S.67), (5.68) and (S.69) we derive easily the following sun rule /dx[F[«P'w-Ff«"(x)]-i. This result was first derived in the framework of the non-relativ^stic quark model [Gottfried {1967}]. Experi mentally, Eq. (S.70) is hard to test since it requires accurate data at small values of x but a somewhat smaller value then % seems to be preferred by the data [Stein (197S)]. The value V, for the integral of Eq. (S.70) is not specific for the wave function of Eq. (7.61). It is easy to construct wjve functions of the proton with three quarks and a quark-antiquark pair in a state of isospin one which give also the sun rule Eq. (S.7Û). CHAPTER VII HIGH ENERGY BEHAVIOUR OF WEAK INTERACTIONS •AND RENORMA LIZA BLE THEORIES H. PIETSCHMANN and D. HAIDT HIGH ENERGY BEHAVIOUR 1. INTRODUCTION Weak interaction physics provides a.\ almost ideal testing ground for the cross-fertilization of theory and experi ment in science. Fran 1958 to 1973, a "pa'cdigm" has been developed which allows for a vast number of predictions: the "classical" V-A theory of weak interactions. Whenever those predictions were confronted with experiment, they met success. On the other hand, the non-renormalizability of this standard theory prevents the calculation of processes which arc forbidden to first order in the coupling constant C as well as higher order corrections to lowest order pro cesses. As a consequence, the "standard" V-A theory as it has been developed and tested is --ssentially a "low-energy model"; "iow-energy" being defined as the energy domain below 10* GeV, say. The phenomenon of neutral weak currents discovered in 1973 was not included in this theory, and although it could easily be incorporated ad hoc, we shall see that it emerges naturally in a renorraalizable theory which goes beyond lowest order phenomenology. The path of fate has led to a climax of excitement due to a threefold coincidence. Neutral weak currents were discovered, new accelerators were opening up an energy domain beyond the 100 GeV region and s new theoretical insight allowed at least for a provisional understanding or even prediction of the new facts, lie refer, of course, to tJie "unified" or "gauge" models of weak and electromagnetic interactions. The exciting endeavour of exploring these new areas is far from being closed- In this part of the book, we want to provide the reader with the necessary theoretical tools to participate in this endeavour. We shall avoid those mathcratical complications which are not essential to the general understanding. However, we shall try to be explicit enough in all computations so that it should be easy to actually re-do each calculation without any unclear steps. In other words, we understand the theory of weak interaction as a phenomenalogical tool rather than a mathematically rigorous and complete description. 2. RECOLLECTION OF NEUTRINO INDUCED REACTIONS In order to study the high-energy behaviour of weak cross-sections, let us briefly recall their derivation within the standard V-A theory. The weak S-operator in its current-current form is J S = I - i / d"x -KK(x) + 0(G J (2.1} with JCw(x) = -| jJcx)J*(x) (2.2) JA(x) = tA(x) * cos 0 j*(x) • sin 0 jffx) (2.3) where 9 is the Cabibbo angle. The leptonic current is *A0O - L »£(X)YJ[1 + Y,)*v (xl (2.4) using the convertion of metTic 1, Chapter II, Appendix I, for the sign of Y5. We shall distinguish between "neutrino" v and "neutretto" v , using the term "neutrino" also when both are referred to. The stated assumptions of the "classical theory" already lead to some selection rules for neutrino lepton scat tering due to the absence of neutral currents: The processes v e / v e and v, e f v e are forbidden in lowest order. (The same holds true, of course, for up 4- veu and vu + vu but the latter cannot be searched for experimental 1>- becausc of ttie lack of a muon target.) Similar selection rules hold for neutrino baryon scattering. Ke define the T-matrix element for a process i •• f by (corresponding to the second normalization of states in Chapter II, Appendix I; the first normalization was used in Chapters III-VT) , f,, The cross-section for scattering of a + b into n final particles is 1 1 f dJp. d^ 0 = ! ffe .. I'Pn *W(P, - p.l Y" IT,-!' (7.7) .Vta (2')'"-' J 2E„ 21= (Pf P>' 4i. ' f1' ' ' where v-n is the flux of the incoming particles. The prime on the sum over spins denotes the fact that we sum over final spins but average over initial spins. However, we have to keep in mind that neutrinos have only one allowed spin orientation so that no averaging is necessary. Using the above presented formalism, we can write down the T-matrix element for the process ve(p2) + e-(p,) - e"(p[) + Oe(pJ) (2.8) 1 2 9 •ve /i —e-v-vi-wix * The neutrino mass m^ is îcept for normalization purposes only and shall drop out of all final results. Standard trace techniques allow for a calculation of the spin sun/average of the absolute square of T- : 2 £' I^P « 64G'(p;PjUpJpi) = 16G>| - u) (2.10) spins where " = tPi " P0Z • t2-11) Let us compare Eq, (2.10) with the analogous equation for the process veCp,) • e-(Pl) - e-(p[) + va(pj) (2.12) which is 2 ! 2 £ ' IT^I « 64G (Plp2)(p[pJ) •= 16G Cs - ap' (2.13) spins where Z 5 « (P, + P2) • (2.14) It is seen, that the differential cross-section for neutrino-electron scattering shows no angular dependence whereas the cross-section for antineutrino-electron scattering does. let us continue with the process (2.8) now. In the laboratory frame, the flux VJJ, is unity. Adopting Eq. (2.7) appropriât ;ly and inserting Eq. (2.10) gives, after son» el.-ientary manipulations where T€ is the kinetic recoil energy of the electron, i.e. Te = E^ -me . (2.16) We have given the differential cross-section with respect t* the recoil energy. In the laboratory frame, s '• me'iiie • 2Eç) = 2a^ (2.17) and the total cross-section becomes G* (s - ml) , „ 2G*m„ rm£\ Likewise, the total cross-section for process (2.12) is computed to be HIGH ENERGY BEHAVIOUR The interesting observation is that both cross-sections rise linearly with the laboratory energy. This has to be a lot» energy phenomenon, otherwise the crojs-sections would become arbitrarily large. In other words, higher order correc tions, which we cannot compute in our present theory, have to compensate part of the first order cross-sections at higher energies so that the sum of all contributions stays below the limit given by unitarity. For numerical evaluation we use the value for G from Roos (1970) and rewrite it in the convenient units G = 4.S4 x HT31 cm* « , (2.20) = 1.094 x 10~5 jÇ . E^. (2.20) allows for an immediate estimate of the magnitude of weak cross-sections. They are always proportional tr C1. One of the G's gives 10"" cm2, the second gives 10"s if energies are measured in GeV. Therefore, a typical value for neutrino-nucleon cross-sections is 10~3S cm*. For neutrino-electron cross-sections we have electron mass inste: ! of the nucléon mass which contributes another 10~3 so that cross-sections in the GeV range are typically of order lO"*1 cm2. More precisely, 1 2 ove = 1-68 " HT* Ey(GeV) on . (2.21) The cross-section for anti-r.eutrino scattering on electrons is just one third of Eq. (2.21) as can be inferred from Eqs. (2.18) and (2.19). This ratio of a third plays an important role in deep inelastic neutrino scattering (see the contribution of 0. Nachtnann). Let us therefore try to understand it a little better! This ratio is only true in the liir.it of vanishing electron mass. In this limit, only left-handed particles and right-handed anti-particles interact. Thus in neutrino electron scattering, two particles of apposite helicity, i.e. opposite angular momentum, hit each other (we consider the process in its centre of momentum frame). It is essential that the interaction s assumed to be point-like, i.e. no boson mediates the currents. This is because in such a case there is no orbital angular momenum. The two angular momentum vectors compensate to yield no angular dependences. In the case of anti-neutrino electron scattering we have a right-handed antiparticle and a left-handed particle. Thus the two angular momentum vectors are parallel and add up to a total angular momentum 1. In a classical cneory, only forward scattering would be possible and the angular distribution would be a 6-function in the forward direction. Quantum mecl. ir.ically, there is a probability amplitude of '/2(1 + cos 9) to find the angular momentum vector actually in a position of angle 8 with respect to its classical value. This leads to a forward peak in the angular distribution. l Quantitatively, we have to square the probability amplitude and integrate over d cos 6 to get exactly the factor /s. A description of neutrino scattering on nucléons can be found in Chapter VT. 3. THE UNITAPITY LIMIT Equation (2.13) shows that the cross-section for the process '2.12) is a sir^ple S-wave cross-section. Thus from the fact that the absolute square of a partial wave amplitude cannot exceed unity, we obtain the lir-it S-wave s mgEy ' (3.1) If this is compared with Eq. (2.19) we obtain an upper limit fPomeranchuk (1970)} of applicability of the low energy theory (2.1): G ' **v ' n^G ' (3. The bound is improved by a factor of two by considering tin,- inelastic channel v e" -* u~ve (see Problem 1), giving the so-called "unitarity liciit" for thn cross-section. Naturally, we expect deviations from the low energy approximation much before the stringent limit (3.2) because the cross-section is not expected to rise linearly up to the unitarity limit and abruptly turn down in its vicinity. Numerically, the unitarity limit is reached for a centre-of-mass energy of about 360 GeV. Clearly, this is far beyond the region even of the new super high energy accelerators. Let us therefore try to get a rough idea at what energies we might expect deviations from simple first order calculations. CHAPTER VII To this end, let us begin by computing the first correction to the process (2.8). It is shown in Fig. 1. Accord ing to standard Fcyranan rules, the T-rcatrix element for the graph in Fig. 1 is Fig, 1 Second order contribution to v e scaccerin) a TjJ - -^ Am^JHt-p^ll * Ys)vCpi)tXov(p,)T (l + Ys)u(p,) 1 Ao ~ Jl /(^v^^v ^^}- The integral of Eq. (3.4] diverges quadratically, which is precisely the reason that the theory given by Eq. (2.1) cannot be applied to high energies. Hence we have to introduce a cut-off parameter A in an ad1 hoc fashion. It is clear that this spoils the consistency of the theory and is here done only to obtain qualitative ideas about the be haviour of higher orders If we neglect the lepton mass, the tensor t. in Eq. (3.4) has to be proportional to the metric tensor g, because + the integral is symmetric in Pj and p2 and terms of the form (pt ' Pz)i(Pi P^a can be converted to lepton masses by means of energy-momentum conservation and the Dirac equation. Hence the addition of the second order correction to T^g gives a correction factor of the form Tw " TLC1 _ cGA*) (3.S) where c is a numerical constant. A computation of the integral (3.4) yields 12>^n2 We see that we do not gain very much by adding the next order contribution. If we sum up all the "chains of bubbles" we get a geometrical series whose sum gives the correction JTI ..bubbles _ ve f1 * " 1 * cCA* " l" This is still no great improvement because it only changes the slope of the linearly rising cross-section. We can force the cross-sectijn to stay below the unitarity bound by multiplying it with a damping factor HIGH ENERGY BEHAVIOUR where c' is a numerical constant which we assume to be of the order of unity. This gives us an estimate of the energy at which deviations from the straight first order approximation exceed 1(H. This rough estimate gives an energy Ss of the order of 100 GeV so thn^ there is limited hope that the new generation of accelerators may reach a domain where de viations from the predictions of the present theory become observable. An alternative not mentioned so far is to give up the "current-current" fom of the Hamiltonian, Eq. {2.2), and insert an intermediate vector boson (rVB)- This does not cure but it improves the situation. Let us therefore turn to the rVB or K-boson. 4. EFFECTS OF THE INTERMEDIATE vECIDR BOSON The present standard theory of weak interactions can be formulated in (at least) two different ways which cannot be distinguished in the low energy domain. Instead of the Hamiltonian (2.2) we may also use : • gJ*(x)w\x) *rre Cx) * h.c. (4.1) where Vi (x) is the field operator 5or the IVB or K-meson. If this meson had a low mass (up to the order of 1 GeV, say), its effects would be observable at low energies. Besides, the meson could be produced by "weak bremsstrahlung" with rather _low energy beans via x' • (4.2) where N and S' may be complex nuclei. Ke shall see presently that a high mass value for the K-meson masks its existence at low energies. Hence the limit for the mass of a possible W-meson is again some boundary up to which the present theory- has been tested and is thus applicable. If, for the moment, we forget about processes with W-mesons actually present in the physical states, we can regard the field W^(x) in Hq. (4.1) as representing the action of another current (in anology to the Mfiller potential which re presents the potential stemming from the scattered electron in electron-proton processes). To do this quantitatively, we have to invert the equation of motion for the W 3 V> * "wVx) " eV*> (4.3) where cuv(x) =• avwu(x) - auwv(x) In this way, we express the field W by its source, the current J . In momentum spare, Eq. (4.3) reads »V[k,A - ft' - MjlSuv] • gJ„ • The tensor on the left-hand side of Eq. (4.5) can be inverted by means of the identity tCk»kJ)/W'] - g"1 . CV. So that wc obtain Inserting Eq. [4.7) into the HamiLtonian (4.1) gives the effective Hamiltonian for lowest order weak processes which do not contain real intermediate basons. -^M^-^l^HkVl v More properly, we should write the configuration space Kamiltonian '00 = -±~r d"k—-5 gnV - ±-£-\ (4.10) K") is, of course, the propagator cf the intermediate buson and we have inserted the proper ie to define the pole in the causal fashion. Notice, that the form of the propagator is uniquely fixed by the identity (4.6). It leads to the bad high-energy behaviour of a massive vector particle since lim K v(k) •* const . Let us now compare the "IVB - llamiltonian" (4.8) with the current-current form, Kq. (2.Z). In the limit of in finite h'-mass, liq. (4.10) becomes lin KXu(x) = -WV^x). (4.11) v "* Inserting th'.s in Lq. (4.9) and comparing with Eq. (2.1) givss Two observations should be made: Firstly, Uq. (4.12) will be a good approximation for energies snail ccTpared to the h'-mass. Secondly, we do not know the coupling constant g which may be quite large if the W-mass is appropriately big. Having set down the defining scheme of the IVB version of the standard theory of weak interactions, we can now start to compute characteristic differences from the former version given by Eq. (2.2). Although we have stressed that deviations should occur primarily in the high energy regions let us first tum to a typical low energy process, namely muon decay. If we compute the spectrum N(x,cos 0) for electrons with energy % i{xo\, CI. 13) from the decay of muons with degree of polarization P, the result is N(x,cose)- -H- x*1'3 - Zx) +f^l x(Z - x) * P cos 6 [l - 2x -f^l xOl + eft] (4.14) 9on!^ I *• % J L •> %/ J J \.N^ i where 0 is the angle between mu .i polarization and direction of emission of the electron. In Chapter Z, the Michel spectrum for u-decay for a local four fermion coupling was discussed. It is of the general form •Siicteil**°5 8) • x'{c'(l x) * c>(l » -1) * p »- o[c>» -x) * c'(i * - A]] • (4-15> where the constants c1 depend on the coupling type (SVTAP), and their experimental determination gives an excellent fit to the values • -acted fron the "classical V-A theory". Vic see fron Eq. (4.14] that the corrections contain terms not included in (4.1SJ due to the W-propagator. Never theless, the paraxctri:ation (4.15), can bo used to obtain information on the intermcdi;Ue boson mass. One shows that the effective value of the Michel parameter p extracted from a least squares fit of the two different spectra [Lee (19S7)] is modified by: HIGH ENERGY BEHAVIOUR The best experimental value for o is [Derenzo (1968)] O = 0.7518 i 0.0026 . (4.17) exp Fron this value, we can derive a iywer limit for the W-nass but it is not even 1 CeV within the quoted error. In principle, however, we have learned from this investigation that high precision experiments of low energy processes may also yield information on high en^.-gy aspects of the theory. (Another good example for this fact is the famous "g-2 experiment" in quantum electrodynamics.) The electron mass has been neglected in Eq. (4.14). Spectra with non-vanishing electron mass can be found in the literature [Pietschmann (197D)]. The discussion so far has completely neglected radiative corrections, which can of course modifiy the muon spec trum and which will he much more important than \1- propagator effects. This does not affect the test (4.16) as long as radiative corrections are calculated. In the "classical" theory without intermediate bosons radiative corrections to u-decay are finite and are found to change the measured value of the p-paraneter by 5.61 [Kinoshita (1959)]. When the interr-ediate boson is introduced radiative corrections are no longer finite. However, in Section 7, we shall learn abcut gauge models, which are renormalizable. Let us anticipate their existence here so that we can write down finite radiative corrections also in the case of intermediate bosons. In the Salan-Neinberg gauge model, the muon lifetime including radiative corrections is [Sirlin (1974)] J 1 (4.18) 192TT J 5l%J 2TI U }J Radiative corrections amount to a few percent of (4.18). Because the intermediate bosons*-' are extremely heavy in this model the corrections to momentum spectra are essentially unmodif'ed with respect to the four-fermion coupling case. For details of the calculation the reader is referred to the extensive literature cited in Sirlin (1974). Let us now turn to high energy processes and investigate the effects of the IVB on neutrino-Icpton scattering. The transition natrix element for this process reads T 1 tVt " S'l^Ai'^MnC * TsMPl>=tP!)Y*(I * f,)u(p,) " " S BfeJKl • Y,)U(P,)Û(P;H1 * Y,]u(Pl)l —- ! — . (4.101 "ft J (PÎ - P,)! - The attribution of four-momentaa too the initial and final particles is as inn Eq.. (2.12)(2.12). With the standard tiace tech niques, the sum over spins of thee absolute square of the transitionn matrix elementt bebecomeo s (4.20) If we now neglect all lepton masses as compared to the h-nass, we compute the differential centre-of-nomentum (CI) cross- section to be an 2"' [| (1 • cos e) • M£| where 6 is the angle between incident and scattered lepton. The integration over the angle 6 can easily be carried out, giving the total cross-section •) It is an inherent feature of these models that radiât:ve corrections cannot be considered separately froa higher order wont. «'Tocts. We immediately observe the following characteristic difference with the cross-section (2.19]. Whereas both cross- sections rise linearly with s at small energies, the cross-section (4.22) flattens ; and approaches a finite value for infinitely large energies. The limiting value for the cross-section is G**tf ,.(-)*^ (4.23) The cross-section (4.21) also violates the unitarity, but at a much higher energy than for the four fermi coupling (see Problem 2). It has to be kept in mind that aïl calculations are done in lowest order. Thus we should not trust its pre dictions at very high energies. The qualitative difference between the cross-sections (2,15) and (4.22} is shot,n in Fig. 2 [Cundy (1974)]. I 1 1 | GGM | GGM V V ® HPW ® HPW • C l.T • C.l.T. Àr : " Propagator term / A = 16 GeV j$ 1 i* I/T ^opagator term '\ s \y a3 = 0 26 'fy A = 15 GeV ^a = 078 yft v y y i ' Fig. 2 Total v-nucleon (a) and v-nucleon (b) cross-sections as a function ot the neutrino energy. In the parton model the energy dependence is the same as in v-lepton (v-lepton) scattering Cdue to the "point-lil:enesE" of the partons). The effect of a propagator with a vector boson Bass of 15 GeV is indicated fjtaken from Cundy (1974)]. 5. PRODUCTIQH AHD DECAY OF THE INTERMEDIATE BOSOM 5.1 Production of the intermediate boson Photons are frequently produced by means of bremsstraitlung (Think of X-ray tubes, to give a pratical example). Because of the close similarity between weak and electromagnetic interactions, intermediate bosons - if wiey exist should also he produced by "weak bremsstrahlung". A typical process of this kind would De v. + N-K-*tf+*-X fft.la HIGH ENERGY 3EHAVI0UR where N is 3 target nucléon (or nucleus) and X can be any system with appropriate quantum numbers. In the low energy region, it would be just the recoiling target, but at higher energy, the target can be excited and break up into as many pieces as is allowed by the available energy. Naturally, weak brensstrahlung can also be induced by antineutrinos in the process 5£ + N*l***T + X. (S.lb) Indeed, processes (S.l) are the favoured ones to investigate the existence of intermediate bosons and to give lower limits to their mass. Mass limits for the intermediate bosons obtained from various analyse., of neutrino experiments are shown in Table 1, taken from Cline (1974). The search for N-production process (5.1) is most sensitive to the leptonic decay mode, but a meaningful limit requires the knowledge of the branching ratio (see Section S.3 and Problem 3). Indirect evidence through propagator modification of the Q*-dependence of the neutrino cross-sect ioi> is independent of the branch ing ratio and can set higher mass limits since the W need not be produced. However its reli-jility depends on the validity of the Bjorken scaling hypothesis. Table 1 Mass limits on new particles [Cline (1974)] Part icle Decay mode Mass limit Remarks or signature W+ - u\ > 8.0 GeV Assume B = O.ZS W* * hadrons Î 5 CeV B + 1 Propagator modification Ï 10 GeV Assumes Bjorken scaling Very low The cross-section for vy + 2 -* W°v„Z is very small The process could, of course, also be induced by the charged leptons, leaving the neutrino or antineutrino in the final state, but this reaction is much harder to detect because of the overwhelming background due to electro magnetic processes. U N in bqs. (5.1) ore complex nuclei, the reaction can take place eîLÎiei incoherently on one particular nucléon of the nucleus, or coherently on the nucleus as a whole object. Feynroann diagrams giving rise to process (S.la) are shown in Fig. 3. ~^w ./•" Fin. 3 Feynmon diagrams for the process vN * £N*W CHAPTER VII It is seen that the target nucléon (nucleus] acts as an external field to account for the energy momentum balance just as in ordinary bremsstrahlung. Therefore, the cross-section on a single nucléon is of the order of gJa2 or GaJ. If the reaction takes place on a nucleus with Z protons, the incoherent cross-section is of the order of GZaJ because it is just the sua over all protons. In the coherent cases the sura has to be taken in the amplitude rather than the cross-section because the contributions of all protons can now interfere. Thus the coherent cross-section is of the order of CI2a2. For large nuclei, the factor Z2 brings about a substantial enhancement of the cross-section. But the higher the energy, the rare incoherent processes will contribute. Let us now ask the question whether production processes can be conceived which are not suppressed by additional factor» of a. Tl« nost direct production of the intermediate boson would be via the reaction a + b ~ IT CS.2) where a and b arc any pair of particle antiparticle which has the correct quantum numbers of the W in the final state. + Lxamples art; (ve,e ), (ve,e~), tPii) and so on. This direct production would be observable as a resonance in the cross- section for production of the W decay products at the exact mass of the W. The difficulty of such an experiment lies in the relatively narrow width of such a resonance, which in most cases will be smaller than the experimental energy resolution. We shall see in the second part of this section that the total decay width of a W-boson witn mass in the 10 GeV range is of the order of 10 MeV; it Tises with the third power of the W-boson mass. Although this width is large, energy resolutions are wide at high energy also. Suppose the (normalized) pro bability distribution that a beam with nominal energy E has an actual energy between E' and C' + d£' is The measured crrss-section is then • J dE'pfE - %in uticre o(E) is the resonance cross-section. (For actual experùnents, it has to be radiatively corrected but for our discussion we shall omit this additional complication here.) The area under the peak is unchanged by the finite energy resolution AE, but its naxiram is reduced. It is im portant to estimate this reduction in order to determine whether the effect can be seen at all above background. The reduction factor will, of course, depend an the ratio of the width r to the energy resolution AE. In the following, we shall always assume tnat both r and ÛE are negligibly small compared to the mass of the intermediate boson. Suppose that o(E) can he represented by a Breit-Wjgner form 1 o(E) (5.6) 2n (E - M)1 + fa/4 K = J o(E)dE . (5.6) %in Suppose further, that the distribution (5.3) is Gaussian: P(E) •is'"'*™'1' <5-') The integral (5.4) can be done, but it leads to a lengthy result. Let us just state the important ratio of the maxi mum cross-section with and without energy spread, i.e. the desired reduction factor , B. fle' /tW][l.tttfJLH •i^J^ HIGH ENERGY BEHAVIOUR where erf(x) is the error function defined by x erf(x) = ~ j e_t* dt . (5.9) o In order to check that Eq. CS.8) really goes to one when AE vanishes we can use the asymptotic representation of the error function erf(x) 'l-~e"x for x— . (5.10) ]f iE and ? are equal, the maxinun of the resonance peak is reduced to jbout -14Î [Abranowitz (1965)]. Just like any other charged particle the intermediate boson can be pair-produced in purely electromagnetic pro cesses. The most direct example is e* + e" - K* * W (5.11) but photoproduction is as good an example: y * N - K' + W+ • r . (5.12) Cross-sect ions for process (5.11) are of order a! and for (5.12) of the order of aJ. There are, of course, additional kincmatical suppression factors due to the large mass of the W. Finally, "weak bremsstrahlung" can be induced by hadrons also. In this case, the leptons of processes (5.1) have to be replaced by the appropriate hadrons. Obviously, this increases the background — due to strong interactions -- considerably. '.Vc have not attempted actual computations of production cross-sections. In most cases, these computations arc cumbersome and involve nuclear physics considerations ir; cases where a complex nucleus is used as a target. he refer the iv.idcr to the original literature [Fearing f1972]] for details. Much more important is the way in which the short-lived h'-boson is detected, if at all. This is only possible indirectly, through the decay products, because of the short lifeline of the heavy particle. Let us therefore turn with sone care to the decay of the h-boson. 5.2 Two-particle decay of the W-boson The most general two-particle decay of the K-boson is shown graphically in Fig. 4. Following the general procedure, we write the decay-rate Fig. i The two-particle decay of the W-boson no..* .-L-i-fïIîLiaLs'-'ik-,, .,, Y. |T(i,,,B)l> where the prirce on the sin denotes the averaging over initial spins. CHAPTER VII The T-matrix clement is given by T(H,aS) - ge^M* hhcrc- E is the polarization vector of the Vf-bosons and" M* = (2TT)'2/E^,u,b|jX(01|0> . Computing the sum over spins gives spins " spins There are only 2 independent vectors, k and (q, - q.). Because of the condition V - » KÇ can make the following decomiwsition spins The dots stand for terms which do not contribul- to Eq. (5.16). Inserting Eq. (S.18) in Eq. (5.16) gives 1 2 £' lT(W,a6)|^|ji[^.m|,n1è)] A-• :3 B i I TO- " ° " ~ spins where w(a,b,c) is the notorious totally symmetric function occurring in kinematical analysis [Pietschmann (1974)] w(a,b,c) = (.a2 - c\- 2(ab • ac * be)]1*4 = [(a * b - c)z - 4ab]'/* = [a - Cb" + .-^^[a - (Vf - A)1^2 . (5.20) The right-hand side of Bq. (S.I9) is now independent of the integration variables of Er,. (S.13) and we can perform the phase space integral with the help of fPietschmann (1974)J it (t , 1 / t^ " - '-'' -ï^*« r0 aGl 3B N '' " ïiïk48^ {è.|MJ t"^'°i'°b'^ - "< w'»^"b>} [t remains to find the expressions A and B for the various particular final states by means of Eq. (S.18). They arc collected in Table 2 together with the decay rates. The form factors in the Nfl decay i-ve to be taken at M^. For the HY decay, a is the coefficient of the following decomposition of the matrix element (5.15) Y HJ = a^eqjUq, - qa)^ - 2(qtq2)eJ * Meq^k* (5.23) where e^ is the polarization vector of the photon. In perturbation theory, a is given by Vr«-f|- <5-M> In Table 2, we have also collected purely hadronic final states but they are strongly suppressed by form factor effects. The cost interesting two-particle decay is, of course, the Icptonic mode. Neglecting the lepton ciass com pared to the W-boson nass, we can write 1 r(W * £*4)(MeV) - j*- M^ - 0.44 * It)" ^(GeV) . (5-2S) so that the partial width for a leptonic decay of a W-boson cf SO GeV is about S5 HeV. HIGH ENERGY BEHAVIOUR Table 2 Decay rates for W + a + b aB A B r/n, 2 1 6 -4 -4("J-nJ) et -© *© ] cos 8 1 cos!6|f (Mj)|! D s' ' if (M>,i>ri it" ")']''' w 48„ lf^"'ip 4(^JJ Kn sin!e|£.((^)|! 0 1 -cos o[2F| • 2Gj - 3^F|] -COSZ8[2P;MJ* 2G;(K^- «%) * * 8(F, • Hf,)F,iyg -e:-"î- 3 1 1 2 1 ! n - a M£ cos a -a tM^ - n*) cos 6 5.3 The total hadronic decay width of the W->-oson The T-mairix element for the decay of a W-bosnn into a hadronic final state of four-momentum k is given by TfW.hadr) => g(2Tr) ^^e^hadrCk) | jA(0) |0> (5.26) where e, is the polarization vector of the W-boson and j is the total weak hadronic current. From Eq. (5.2b), the total hadronic decay width can be computed following standard procedures. «.•tolrl.ig'l,,,.!*),»» (5.2 where we have defined the "spectral function" pXv(k) --(2-)1 £$(,,)(k " k') n and the sum goes over the complete set of hadronic intermediate states. The invariant decomposition of the spectral function con be written in the following way In Eq. (5.27), energy momentua conservation requires k1 « M£ (5.3 so that insertion of Eq. (5.29) yields T(W * hadr) - £ç o$$) - -| n^|^) . (5.3 It remains to connect pfM^) to experimentally accessible quantities. The electromagnetic analog of Eq. (5.Z8), where the tîcak current is replaced hy the electromagnetic current, defines an electromagnetic spectral function p0". It is connected to the famous ratio R of electron positron annihilation into hadrons versus muon pairs by . ,(eV -hadron). u£ en M = p + C5.32) o(e+e * u*y~) 5 l s ' where s is — as usual — the square of the total energy in the centre-of-mass system. The total hadronic current as given by Eq. (2-3) consists of strangeness-conserving and strangeness-changing parts. Lach part, in turn, receives contributions from vector and axial vector so that we may write p(M2) = cos* 6 (pV + oA) • sin2 6 (oV + oA) (5.33) where 3 is the Cabibbo angle. l-ollowing Li and Paschos [Li (1971)], we neglect the strangeness-changing parts o and o and assume that vector and axi3l vector parts of the strangeness-conserving contribution are about equal. Let us further assume that the electromagnetic spectral function entering in Eq. (5.32) consists mainly of isovector contributions, i.e. that the isoscalar part can be neglected at high energies . The conserved vector current hypothesis tells us then (5.34) so that Lq. (5.31) becomes . Oft | (5.55) A comparison with liq. (5.25) allows us to express the important branching ratio of hadronic versus leptonic decays by [*[W * hadrons] . R^) (5.36) r(w* uu) + r(w-e« j ' At present maximal energies, R reaches a value of about 5. Thus the total hadronic decay width of a h'-boson of about 50 CeV can reach several hundred MeV, with a leptonic branching ratio B = B = VJ7. Similar considerations can also be performed for the decay of a neutral intermediate boson. The reader is re ferred to the original literature [Bertlmann (1976)}. 6. TH£ WEAK NEUTRAL CURfiENT 6,1 Phenomenology So far, we have only discussed th™ "classical V-A" theory, h'e have said right after Eq. (2.4) that there are no neutral lepton currents in this theory. In other words, leptons always appear in one of the combinations fv ,i0 CP.vJ (c.vc) u Cvo) [6.11 .lie way this rule was tested was to look at semi-leptonic decay processes. (Sec Qiapter III.) However, only strangeness-changing decay processes are accessible in this way, because all other processes are overshadowed by com peting channels ol electromagnetic nature. Thus the rule should more precisely read: "There are no neutral lepton currents in strangeness-changing processes". 'Vhat about strangeness-conserving processes? AS was just pointed out they are not accessible through weak decay cxper ments. Therefore one has to look at weak scattering processes. Again, care has to be taken that strange inter action do not induce the effect. Thus, the best way to perform these experiments is by means of neutrino scattering. Another possibility is to look for parity violating effects which are by their very nature always due to weak inter action. (!ood places to look for those effects are nuolcar transitions fJocnim (19?5)}, atomic transition*; £pouchiat (1974). rcinhcrg (11174), Bcmabcu (ly74j] and electron-positron (or electron-electron) scattering [Budny (1975), Dass (1973)]. In the case of nuclear transitions, it is, of course, difficult to separate possible neutral current *) Thi> should at least give the correct order of magnitude [see Problem 3). HIGH ENEKGï BEHAVIOUR - 307 - effects fron those of charged currents. In the latter two cases, charged currents dû not contribute. It is the inter ference between neutral weak current effects and electromagnetic effects which gives the nain contribution to parity violation in these cases. After Eq. (Z.ZO), we have estira ted neutTino-nucleon cross-sections in the GeV energy ronge to be of the order of 10~!S cm1. Neutrino-electron cross-sections in the same energy range are even smaller i.e. lCT1-1 cm2. Clearly, these experiments require a refined technique in shielding, v-intensity, background esti. -lion and so on. Advantage can be takrn from the fact that total cross-sections rise linearly with energy so that an order of magnitude in cross-section is gained with each order of magnitude increase in beam energy. Weak neutral currents have indeed been found both in the case of purely leptonic scattering on electron targets as well as in the case of semi-lcptonic scattering on nucléons [and nuclei). To be more specific, the process v_ + e" •* \>u » e" (6.2a) has been seen Qiascrt (1973)]]- It cannot be constructed as a pair-wise combination from the set of Eq. [6.1). In principle, reaction (6.2) could be explained by a charged lepton pair combination which violates separate conservation of muonic and electronic lepton number, namely the pair (O^e). he shall not pursue this possibility further but as sume separate conservation of both lepton numbers. In this case, the observation of reaction [6.2) suggests the exist ence of the neutral lepton pair combinations ' (5,el C6-3a) (Vv) • and by means of niuon-electron universality, which we also assume, we can immediately infer the existence of the two corresponding combinations (5e,e) , (C,u) . [o.5b) This finding is independently checked by locking at — strangeness conserving! — semi-leptonic weak reactions. In deed, the hypothesis is supported by the discovery of processes of the type [F.J. Hasert (1973), A. Benvenuti (19"4), U.C. Barish (1974), S.J. Barish (197S)] vu + N ' uu * x (6-4a) and vp • N •* \ • X . (6.4b) These processes have been observed first in the inclusive channels, where X is an abitrary, unspecified state and later in the exclusive channel where X represents the KN system. Of course, "observed" has to be taken with a grain of salt. It is certainly not possible to identify the neutrino in the final state. Thus, speculations that it is actually something else something hitherto unthought of, cannot be excluded. Ke shall, however, always assume that the most natural explanation through Eqs. (6.4) is indeed the truth. Since neutral currents aie not included in the "classical" theory of weak interactions, we have no guide line to assume their properties and test those assumptions- Experimentally we know that they exist, their strength and some distributions. Let us therefore try to formulate questions about the neutral current in order of importance. The first question is the same one that plagued weak interaction physicists in the "prc-classical" era before 1958: What is the Lorent2-structure of the weak neutral current? In other words, ir. it a scalar, vector or tensor abject? If we reserve the term "current" for vector objects, the question has to be posed in the form: "Is the weak neutral effect caused by a neutral current?". Like any question in physics, also this question can ultimately only be answered by experiments, and there is already some prclininary data in this direction, based on an extension of the successful parton model to v * N * y + * anytliing. Under the parton model assumption, the most general four fermion coupling gives a y distribution of the form (sec Problem 5): .) Reaction (4.2) coulu also occur in second order in the conventional theory (see Problem 4). *£*» CHAPTER VII where the g's represent the strengths of the S, V, T, A, P couplings (see Chapters I and II) averaged over the partons. The Caltech group has measured the y distribution in the dichromatic neutrino and antineutrino bears at the Fermi laboratory. The experimental distributions have been compared with the theoretical distributions in thTee ex treme cases: î) dominant V-A; ii) dominant V*A; iii) dominant S or P. Cases (ii) and (iii) give very bad fits. However, V-A with an admixture of V+A gives a reasonable fit [B.C. Barish (I97Sa), L. Kblfenstein (1965)]. There is also a theoretical prejudice in favour of V,A type couplings; the reasoning that led to the V-A theory of weak interactioi , was the argument of the two-component neutrino or the argument of definite neutrino helicity. Ne know, that in charged current reactions, only left-handed neutrinos and right-handed antineutrincs interact. This is expressed hy means of the chirality projection operators through 2 V (6.6) *v = \ *VU " YS) • It follows that only vector (and/or axial vector) currents occur because *v*w "J %(* - Ys]tl * yJK = ° and similarly for pseudoscalar and tensor objects. Thus, if the helicity of neutrinos is given independently of their interactions, that is to say that neutrinos never flip their spin, the neutral current has to be of vector (and/or axial vtctor) nature also. We shall henceforth assume that this is the case. Next, we must ask whether the weal; neutral current violates parity! In view of the argument just given this seems to be an obsolete question because the two-component neutrino automatically violates parity in a maximal way. However, it is perfectly possible that the hadronic part still conserves parity ^Bfig (1973), Sakurai (1974)3. Indeed, the weal; neutral current could be of the form j?" Ev»d *y^i *3" • (6.7) where J, is cither parity conserving as a whole or it splits into a hadronic part which conserves the parity and a part stcsning from the charged leptons which need not conserve parity. This is of extreme ir.portance because weak neutral currents shall contribute both to nuclear Corces [Ragutt (1974), Donelly (1974)] and to the atomic potential [Bouchiat (1974), Bcrnabcu (1974)]. In both cases these contributions can only be detected through parity violation because of the much stronger electromagnetic contribution to the parity conserving potential. (In the case of the nuclear forces one is also up against strong interaction contributions!). Let us suppose, the matrix element of the weak current between proton (or neutron) states (N|jJ(0]|N) = VA(VN > ANYi)uN (6.8) where N is a r"j;leon, i.e. either a protc-n or a neutron. \\, and AJJ are fuur constants which we leave open here. They should speci", the particular kind of weak neutral current existing in nature, but they are, of course, fixed in any- particular model for the weak neutral current. HIGH ENERGY BEHAVIOUR Let us ask the Question: What is the parity violating contiibution of the weak neutral current to the atonic potential? To find this out, let us also define the matrix element of the weak neutral current between electrons. + Notice that only one more constant is necessary because the second can be absorbed in an overall constant contained in Kq. (6.8). In general, the coupling of the two matrix elements £6.8) and (6.9) can be mediated by a vector particle of mass 2 n2, say. Since we are working in the very low energy domain, we can approximate the propagator by a constant, K^ , which leads to a 6-function potential in configuration space. Thus we are interested in the parity violating low energy part of Z l ' For the atomic potential, we can neglect the nuclear spin dependent terms. The static approximation of the Dirac spinors leads then to (r and r' denote the spins of the nucleus) iy^iv + 5x0<5rr' (6.11a) W*ur - hk^rr1 (6-nb» which is well-known from nucle3r beta-decay problems. We see that the entire axial vector of the nucléon part can be dropped because it gives rise to terms depending on the nuclear spin. Hence, for the parity violating potential, only the electronic part needs to be retained. Moreover, only the time component survives as is seen from Eq. (6.11a). Dirac algebra tells us that Û(P2)Y0YSU(PI) » ^ {(3?,) * tfp,)l (6.12) Putting everything together, we arrive at the parity violating contribution to the atomic potential where the writing of the curly bracket indicates where initial and final electron variables have to be inserted. Z and N are proton and neutron number in the nucleus under consideration. In deriving Eq. (6.13), we hove of course assumed that the weak neutral current does indeed violate parity. Other wise, the normalization of Eq. (6.9) is not defined (i.e. there could be no axial vector!). Effects of a parity violating potential of this kind can possibly be found in muonic atoms as well as in ordinary atoms provided the most advantageous transitions are investigated [Bouchiat (1974), Bemabeu (1974)]. 6.2 Isospin properties of the neutral current The next question about the neutral current is its isospin properties. We know that the charged current which conserves strangeness obeys the selection rule AI = 1. What about the neutral cuirent? From the Ce 11-Mann Nishijicna relation and the presumption*' that the neutral current conserves strangeness, it follows that it changes isospin by an integer number (including zero). For simplicity, let us discard changes i" isospin greater or equal to Z. In fact, we shall *) The data of Dcdcn (1975) imply a limit for the neutral current strangeness changing interaction o(v + N-* v + |S| - I). «• 5i o(v + N-» v * |S| = 0) ~ at a 9(U confidence level [Wolfcnstein (1974)]. CHAPTER VII later construct weak currents front quarks and find out that this automatically restricts isospin properties to those considered here. Under this assumption, we can decompose the weak neutral current into an isoscalar and an isovector part (6.IS) We have settled the question of Lorentz-structure of the weak neutral current in the sense that we shall assume it to be of vector (axial vector) nature. Therefore, we can in turn split each part of Eq. (6.IS) into its vector and axial vector contribution. • n,v -n.v.V ^ .n.v.A h 'h * V • Let us now put all these parts of the (hadronic) weak neutral current together again. In doing so, we take c plicitely certain weight paranreters a^ -n _ -n.i h - am This is 3 general ansatï for the hadronic weak neutral current. The parameters a; will assume particular values in every particular theoretical model. In this way, the isospin properties of the (hadronic) weak neutral current wii be defined. Certain crude restrictions can be tested in a first generation of experiments. For example, if the weak neutral current is either pure isovector or pure isoscalar, this will show up soon. An obvious experiment is for instance the pioduction of a pion nucléon state from a nucléon and the investigation of the isospin contributions to the final state. If the A-resonance is not produced at all, this is a clear indication for the absence of isovector parts in the weak neutral current. There are experimental results from h\c ANL 12' Bubble Chamber Group [Hyman (1975)], the old CERN propane experinent [Rousset (1974)], CERN Garganelle Freon experiment [Hasert (1975), Bertrand-Coreraans (1976)], Colunbia-Illinois-Rockefeller Collaboration at ENL [Lee, Paris (1975)]. Whereas the first experiment suffers from low statistics, the other experiaents deal with interactions occurring on complex nuclei and are consequently diffi cult to interpret. Only one experimental result (Bertrand-CoTeroans) disagrees with the hypothesis of pure isoscalar ai. rent. The success of the conserved vector current theory suggests that, in the same spirit, one might attempt to identify the terms in Eq. (6.17) with known currents. Note that the two vector pieces of Eq, (6.17) have the same isospin transformation properties as the two isospin parts of the electromagnetic current jj"->;•£• (6-18> Thus we might try to identify those pieces of the two currents. The current j. * *' transforms just as the neutral nember of an isotriplet whose other components are just the two "classical" hadronic chargud axial currents entering in Eq. [2.2). Let us denote it by ^•'•A - tf . (6.20) The remaining piece of Eq. (6.17), j"'S' , has no analog among "classical" currents. It is therefore a very interest ing experimental question, whether this "new" type of current does exist among weak neutral phenomena. We might also try the hypothesis that the isovector axial currents appear in the same V-A combination as in charged currents, and that the isoscalar vector current occurs only through j0"; then the four parameters of Eq. (6.17) reduce to three. HIGH ENERGY BEHAVIOUR + p j [6.21) J) P.JiJA. ^^h v * »>J:* .\ote that the V-A isovectur part of (6.21) is obtained by a rotation of the charged current in isospin space. Although (b.21) is by no means the most general fom for the neutral weak current — not even in the context of gauge models dis cussed in Section 7 — it will be seen that in the particularly sijnply Keinberg-Salam model, the neutral current is of the form (6.21) with Pj = 0. In order to obtain the whole neutral current, we have to add che leptonic part to Eq. (6.21). Taking into account the answer to our questions 1 and 2 of this section, the neutrino part of the weak neutral current should be of definite chirality, whereas the charged leptons may have different weights in their vector and axial vector parts. We shall, however, assume strict electron-muon universality so that the same factors entar for muons and electrons. We can thus write the weak neutral current in the following way: 1 * Ys h 6R ^r1) *«}+ vr^ * *Mw + jJw]+ Before we go on to discuss the underlying theory of these phenomena, let us turn to the information we may obtain from deep inelastic neutrino scattering by moans of neutral currents. The analysis of the processes like vu * N "" uu * x (6.23a) vu * N - vu • X (6.23b) is entirely similar to the corresponding processes with charged currents which are discussed in detail in the contri bution of 0. Nachtmann. The mass of the muon has been neglected there and in our case this is even strictly true. Let us assume that scaling in the sense of Bjorken also holds for neutral current reactions. The double dif ferential cToss-section for processes (6.23) can then be written in the usual nomenclature as = ££ Bljxy'Fjtx) + (1 - y)F,(jO * [y ' j Y*)xVlA (6-24) dxdy t where M is always the nucléon mass. Recall that the scaling structure functions Fi and Fj arise from terms in the inter action which are squared vectors and squared axial vectors whereas F3 arises fron the interference between vector and axial vector. This allows for an immediate experimental consequence: If the neutral current does not violate parity, F3 vanishes. If the final and initial state neutrinos are identical, herniticity requires [see for example Sakurai (1975)]: r«T, :VT,. F^W - F^CxI 1, 2, 3 (6-25) for any target of 1. Hence in the case of a parity conserving neutral current, cross-sections for neutrino scattering and antineutrino scattering must be equal- It is customary to give the ratios of neutral to charged current reaction cross-sections rather than the neutral current cross-sections themselves. Accordingly one defines otot(5 N - U*X) where N represents an isoscalar target. For charged current reactions, we use the ratio •-t.ich is close to V, and rather const Thus for parity conserving neutral currents, we should find or R roughly 3 times R. The present experimentally measured valves for R and R are shown in Table 3, and they appear incompatible with the hypothesis of a parity conserving neutral current. Experimental results on the ratio R of neutral currents over charged currents Group R R Reference CERN-Gargamelle 0.28 ± D.04 0.38 ± 0.05 Brisson (1976) 1 < 0.31 ± O.06 HPWF 0.39 * 0,10 Benvenuti (1976) 1 > 0.24 ± 0.04 CIT 0.2S ± 0.04 0.36 * 0.11 Bodek (1976) If the scattering of neutrinos is effected only by spin '/i constituents of the target, the Callan-Gross [Callan (1969)3 relation holds: 2xF,(iO - F,M • (6.29) In this case, several othei consequences can easily be obtained.- Let us define C do(v N - ., do(vJu < - yt) , 1 <»NC " }y dy ^V " "'" and similarly N for antineutrinos. Note that the quantities N and N art independent of the inconing neutruio spec- trun because cross-sections always appear in the numerator and in the denominator. The five quantities R, R, r, N and S have To obey the following Z consistency conditions [Ecker (1976), Sehga! (1973) and (1974)] HIGH ENERGY BEHAVIOUR They are completely independent of the form of the neutral current, except that it is of vector (axial vector) nature. A test of Eqs. (6.31) thus tests relation (6.29) and the assumption of Bjorken scaling for the neutral current. Under the same assumptions one can find another nice test. Define Mr) : and similarly for \ From the above assumption, it follows that : 1 - J~ = 0.42 (Û.34) Thus by measuring the y-distributions of inclusive neutrino and/or antineutrino scattering at one energy, one can test the vector nature of the weak neutral current, Eq. (6.29) and Bjorken scaling for t*»-» w»ak neutral current. In all our dwellings throughout this chapter we have simply ignored the neutral intermediate boson. The reason we are allowed to do so is that it is roost probably very heavy. Indeed, processes (6.23) may eventually also serve to put lower limits on the nass of the intermediate neutral boson just as the corresponding charged current reactions put lower limits on the mass of the W-bosons [Gross? (1974)]. 7. wintù MODELS OF WEAK AW ELECTROMAGNETIC INTERACTIONS Let us now return to the intermediate boson, and assume its existence throughout the rest of the discussion- After Eq. (4.1Z), we have remarked that neither the mass nor the coupling constant of the intermediate boson is known. It is therefore possible to speculate that there is no separate "semi-weak" coupling constant, but that the electric charge (or the fine structure constant) is the only dimensionless quantity in the realm of lepton physics. In Eq. (4.12), we would then hove to insert e instead of g and compute the mass of the (charged) intermediate bosons, fraa the weak coupling constant G, to be 102 CeV, Together with the (classless) photon, the two heavy charged inter mediate bosons would then form a kind of triplet. An intermediate boson which is heavier than a silver atco is in itself a suspicious object. Besides, we have not won anything significant by this scheme, hfe have superficially "unified" weak and electromagentic interactions, but we have not been able to achieve the imnense advantage of electromagnetism, namely the renormalizability. The non- vanishing ma" of the intermediate boson stands in our way, because the propagator (4.10) behaves too badly at large momenta as long as the second term in the bracket of Eq. (4.10) is present. A unification of weak and electromagnetic interactions thus has to start on a deeper level if we wish to gain re normalizability for all the interactions, h'e are forced to start from a Lsgrangian with all vector bosons massless or we will immediately run into the trouble of badly behaving propagators and thus harmful (i.e. noii-renoraali:able) di vergences in higher orders. Obviously, the cass of the weak intermediate bosons has to be generated somehow, for we know them not to be massless. Itow can this be done? Let us first learn the basic mechanism before we go into the detailed calculation. 7.1 The Hlgq.5 mechanism he start from a Lagrangian which describes the interaction of a scalar particle * with a massless vector particle A . This Lagrangian is perfectly renormal liable. The interaction term in this Lagrangian is Vj - gAlVy . (7.1) CHAPTER VII In the usual theory, the vacuum expectation value of the fields vanishes. Suppose now, that the scalar field has t non-vanishing vacuum expectation value A, i.e. (0|*|0) = 0 . (7.3) We can then insert Eq. (7.2) in Eq. (7.1) to obtain /j = g\A\ * gA^* . (7.4) The first tern in this Lagrangian is nothing but a r.ass term and we have in this way generated a mass for the vector particle oi w,1 -- 2gX . (7.S) It remains to be explained, of course, how we obtain the non-vanishing vacuum expectation value for the scalar field. We shall presently go through a more complete calculation. Following Higgs and Kibble [Higgs (1964), Higgs (1966), Kibble (1967)], we write down the complete Lagrangian of a scalar field * with quartic self-interaction and with an interaction with a massless vector fieli Al(. where In the normal case, u1 is positive and is the mass of the scalar particle. The Lagrangian (7.6) is rcnorcalizablc and is invariant under the gauge transforations •\ - \ • •'•/< « * « eieA (7.8) «+ - «t e"icA . Let us new invoke the non-vanishing vacuum expectation value of the "lliggs-Kibble" field Formally, we can always write <0|*il0> = 0 , i = 1, 2 . (7. Insertion of l.q. (7.10) into the Lagrangian (7.6) gives UV a + #« const - *,A(y* + hA*) + A FUUF + \ Oytj)* + \ C3p*a) w 1 2 • ^- t^/f - eXA (3u*j) - \ c'fjj * 3hA*) - \ $J(u* • h\ ) - y - hA*,(*J • **) * eA (3ll*il<., - eA^s^)^ » e'A^*, - so far, this is a purely formal rewriting. It becomes meaning through the requirement (7.11) which tells us that \ is indeed the vacuum expectation value of *. However, this can only be true if the vacuum expectation value of tt HIGH ENERGY BEHAVIOUR and * are indeed zero if computed from the Lagrangian (6.12). In lowest order perturbation theory this requires the linear term in 0, to vanish. Since we want to keep \ non-vanishing, the only alternative is ua + hAa - 0 . (7.13) We can always choose X to be real because that just fixes a phase. On the other hand, we want h to be positive other wise we would have an unstable "potential" in Eq. (7.6). Hence uJ has to be negative in order to fulfil Eq. (7.13). It chus has no longer the irnediate meaning of the mass square of the * field but is just another parameter in the theory It is by allowing ix2 to take on negative values that we achieve the non-vanishing vacuum expectation value of *- This is seen if we draw the "potential" V = u1»1 >h{*. (7.14) Figure 5 shows tiiis potential in the case of negative u1. Clearly, the minimum is now no longer at vanishing *min ~ ~/f ~ V *âT which coincides with Eq. (7-13). Fig. 5 The potential with i shing vacuum expectai Let us now turn to the problem of masses. As anticipated from our short sketch of the mechanism, we have generated a mass of the vector particle MA = eX . (7.16) Using Eq. (7.13), we infer that the scalar particle < (the genuine "Higgs-Kibble" particle) has acquired a mass m? - ZhX1 . (7.17) The other scalar field *2 is massless because of Eq, (7.13). In fact, this was thought to be an unavoidable conse quence of any theory with non-vanishing vacuum expectation value [Goldstone (1961), Coldstonc (1963)] until Higgs and Kibble [Higgs (1964), Higgs (1966), Kibble (1957)3 actually showed that these massless particles are unphysical ghosts in a theory with gauge invariance. Presently we shall show that they can be completely transformed away by a gauge transfonr-ation. The situation is reminiscent of quantua electrodynamics, where the covariant formalism (in the Lorentz gauge, say) exhibits longitudinal and scalar "ghost" photons. They do not appear in the transversal (radiation) gauge which is, however, often more ciciberscnr: in higher order computations. The fomalism which we have used so far is often called the "R-formalisra" because renonnali:aticn is more easily proved and handled in this way. Let us now turn to the "U-fonnalisn", in which there are no ghosts and unitarity of the S-natrix is thus explicit- For lowest order computations, this formalism is usually preferable. CHAFTEft VII Instead of splitting the scalar field into real and imaginary parts, we now split it into absolute value ar.l phase, so that we have, instead of Eq. [7.10) tt * Xl c (i/*)6 [7.18) n Notice that A is a real nunber, whereas x a^ 9 are hennitian fields. The advantage of Eq. (7.18) is that we can now eliminate the field 6 by choosing the gauge A«-ie. (7.19) eX This is immediately clear from Eqs. [7.8). In the gauge (7.19), we simply have to replace • in Eq. (7.6) by [* * x)/^ to obtain the Lagrangian in the Li fo rmal ism: Uw 2 z J •Su = const * J FiivF • | i*vx) * \ eWtyP * \ c^WXx * x ) - hlV - h*X - £ hx" (7.20) hhere we have used Eq. (7.13) to eliminate u*. We can read off the same masses for .he physical particles as given in Bqs. (7.16) and (7.17) but the unphysical ghost particle docs not show up at all in the Lagrangian [7.20). 7.2 The Salam-Meinberq model The model discussed so faT is unrealistic in the sense that it only contains a single vector meson. In nature, how ever, we should have at least 3, the photon and two charged heavy intermediate bosons. When we try to build a realistic mode), we face the additional difficulty of non-Abelian gauge symmetry [Veltmann (1970), t'ltooft (1971)]. This, however, presents no immediate problem in lowest order calculations. Let us therefore follow Salam and Weinberg £Salam (196<1), Salam (1968), Weinberg (1966) and [1971)] who presented the first and best studied realistic gauge model of unified weak and electromagnetic interactions. We want to take advantage of the Higgs mechanism to generate the masses of the intermediate bosons in a theory which is renormalizable at the outset. In the U-formalisn, we shall end up with a Lagrangian describing the interaction of massive vector fields with fermions [as well as Higgs-bosons). Consider the graph depicted in Fig. 6. If we conpute the corresponding cross-section, we find that it is proportional to s (square of centre-of-nonentim energy); thus it exceeds the unitarity linit at high energy (cf. Section 3). he can also elucidate this trouble by observing that a box graph composed of the graph of Fig. 6 and its mirror (see Fig. 7) is so divergent that the theory is not renomalizablc . The connection between the two statements is clear when we recark that the forward amplitude for the process of Fig. 7 can be replaced as a dispersion integral over the total cross-section for the process of Fig. d. Since the latter is proportional to s, wc need two subtractions in the dispersion integral. The subtraction constants are undetermined parameters which tells us that the theory is not renormalizable. -*-w" i" — W" . 6 A typical graph leading to a crosa-aectioi •ergont hat graph which exceeds the unitarity Limit In a renormaliiablc theory, this difficulty cust be renovable order by order. Ke conclude that it is not pos sible to construct a rcnormalizablc gauge nodel out of photons, charged W-bosons and the known leptons alone because there is no possibility to cancel the bad high energy behaviour for the graph of Fig. 6. Wc can achieve this by HIGH ENERGY BEHAVIOUR adding a neutral intermediate boson and/or heavy leptons. The resulting graphs capable of cancelling the bad high energy behaviour of Fig. t> are shown in Fig. 8, The required cancellation can, of course, only be achieved if the various coupling constants obey certain relations. It turns out, that these relations are exactly the structure re lations of a gauge group. For details we refer the reader to the original literature [Llewellyn Sïaith (I973), Bell (1973), Cornwall [1973)]. Fig. 8 Cancellation of the hard high-ei ïrgy behaviour of Fig. 6 by (a) neutral intermediate boson and (b) heavy lepton. For the construction of a specific model, a choice for the underlying gauge group must be made. Since neutral currents have been found experimentally, the incoporation of a neutral intermediate boson 2° is natural. "Hie question whether additional heavy leptons should also be added can untimately be answered only experimentally. Presently, the 'Minimal" choice is to add a :°-boson only. This is the choice of the Salam-Weinberg model. The underlying gauge group is SU(2) ® U(l) . Since the gauge vector bosons belong to the regular representation of the gauge group, we have a triplet plus a singlet of bosons in agreement with the assumption of four vector bosons, the photon, the :D-boson, and the two charged W-bosons. (he shall see later that the photon and the 2°-boson are in fact mixtures of the neutral triplet and singlet states.) To begin iiith, let us restrict our attention to non-hadronic systems, leaving the additional problem of hadrons to an extra subsection. Also, the doubling of leptons due to electronic and nnionic type of particles is an unnecessary explication in writing the equations which we shall leave aside in the beginning. The extension to the two kinds of leptons is a trivial generalization of the final result. In the Salam-lv'einberg model, the neutrino and its charged Icpton counterpart form a doublet in "weak isospin space" in which the gauge group acts. Since only positive chirality projections occur in weak interactions, we define the "left- handed" lepton doublet spinor by Notice in passing that "positive chirality" and "left-handedness" coincide for we start with a Lagrangian in which all particles are nassless. The weak currents of Eq. (2.4) are easily constructed with Eq. (7.21) by ^(x) « ë(x)fX(l * Y5)ve(x) - 2L{X)YAT_L(X) (7.22) where ij are the usual Pauli-matrices, acting in the "weak isospin space" spanned by the spinors (7.21). The electron also enters with its right-handed components in the electromagnetic current, because the latter con serves parity. Hence we also define a "right-handed" electron singlet spinor (7.23) • now easy to construct the usual electroragnetic current out of the doublet (7-21) and the singlet (7.231 _ 318 - CHAPTER VII This current coiuists of a weak isotopic singlet, the piece £ (LYWL) * RYUR (7 and the 3rd ccnponent of a weak isotopic triplet. The two remaining components of this triplet are just the weak current (7.22) and its hemitian conjugate. Thus we can write down an interaction Uigrangian which is invariant in weak isospin space. '•f-x = f LYJJTIJ^ • g'fi LYljL + RY KV (7.2b) Here we have aade use of the freedom to give the singlet interaction and the triplet interaction a different coupling constant. Tt shall turn out that the ratio of these coupling constants is the only additional free parameter we have in the theory and it is quite necessary for adjusting to empirical findings. In Eq. (4.26) we had to introduce 4 vector bosons. Nhile it is quite clear that the usual charged intermediate bosons can be identified with ^-components in the following way Wj(x) =± {A»(x) ± iAJ[(x)} , neither of the other 2 bosons is the physical photon. In order to work out the coupling of the electromagnetic current (7.24), we have to form linear combinations of the 2 neutral bosons by A' = cos *• Z-, - sin* A u ^ V [7.2S) B^ •= siniJiZy + cos $ A^ where î is called "Salam-Weinberg nixing angle". Insertion of Eqs. (7.28] into the Lagrangian (7.26) gives u ¥\ » charged part + Z jcosc| LyyT,L • sinçg'fi LY^L + RY^R] |* • A°lcos$g'(! tYuL + RYIJR) - sinôf LV,L| • (7.29) Since Ap shall be identified with the physical photon, we can uanediatoly read off the connection of the electric charge c with the Salam-K'einhcrg angle: e = g sin + = g' cos $ (7.31) so that the Salam-Weinbcrg angle now appears as the arctangent of the ratio of the singlet and triplet coupling con stants. Let us now turn to the even more interesting question as to what is the coupling of the new boson 2 . It couples of course, to the predicted weak neutral current whose form can be obtained from Eq. (7.29) by simple algebra Cy - 1 - 4 sin' 4 . (7.: Notice in passing, that neutrinos couple in the standard V-A fashion thus keeping their definite helicity. Charged leptons couple with an unequal mixture of vector and axial vector, depending on the value of 4. In fact, for sins n O.S, a rather syrxetric situation emerges. Neutrinos are coupled to the neutral weak current with equal V HIGH ENERCY BEHAVIOUR and A, to the electromagnetic current not at all of course. Charged leptons would couple to the electrcaagnetic current as pure vector and to the weak neutral current with pure axial vector. However, the neutral current is different for the case of quarks as we shall see subsequently. Thus this "symmetry" would be restricted to the realm of leptons and is therefore not so attractive. Let us now invoke the Higgs-Kibble mechanism. The only difference with the model of Section 7.1 is that we now have a larger gauge group. We can take advantage of this fact and define a d ublet of Higgs-Kibble fields by -(?)• (7.34) In fact, we shall need this freedon to keep the photon nassless while the other three vector particles acquire susses. As outlined in Section 7.1, we add to the total Lagrangian the term #t n V|*|* " hU1" (7-35) where [•I2 = 1**1* + '*"[" • (7.3b) The gauge invariant coupling of the Higgs fields to tiie vector mesons is, of course, slightly more involved now be cause of the larger gauge group. It reads The signs are such that by means of Eqs. (7.28) and (7.31) the neutral part of Eq. (7.37) becomes -l £ T A + l ^_ R $-—a- M ,„ (7.38) z I » V 2 V) 2 COS * I u* * which tells us that the photon does jiot couple to the neutral Higgs field. In order to cbtain a non-vanishing vacuum expectation value, we allow again u* to take on negative values. We use the U-fornalism and define he have replaced the four real fields of 4 by the four real fields 1% and x. X is again the non-vanishing vacuum ex pectation value. It is a good exercise to show that Eq. (7.39(7.39)) iiss indeed a non-singular transformation! Just as in our simple model, the phase fields Can bhe eliminateeliminate*d by fixing the gauge. We then have to replace HA. J" (7.40) the electron has not yet obtained any mass. In order to remedy this, we add yet another piece to the Lagrangian A . X (R*+L - L9R) . (7.41) Like all the other parts, this piece is invariant in weak isospin space and it is formed in such a way that the neutrino stays mass less because of Liq. (7.40). Finally, we have to add, of course, the free Lagrangian for the vector bosons and Leptons r. ' Ï VV * 1 Bu/" * l** * 'OL (7.42) where •\v • \K - aA * «Pu" vl ('•«) B™ • w - Vu • <'•<•<) _ 320 - CHAPTER VII The full Lagrangian of the Salam-Weinberg model is then simply the sum of all pieces 1=0 In the U-fomalisms, we replace $ according to Eq. (7.40] to obtain the Lagrangian of the Salan-Keinberg model. It is trivial to add the piece for rauonic leptons and we shall do so. The result is 5 "sw - *. * Z {[û V * n>V* * i..c] * eî,u«A». £ e As usual, the sum goes over electronic and muonic types of leptons. Cy is defined in Eq. (7.33). We can now read off Eq. (7.46) several important relations 3 «rç (7.47a) \-¥ (7.47b) (7.47c) ^ cos 0 m * \m . (7.47(1) X Combining them with Eq. (7.3D) gives G 1 . e1 n 2X2 8f& sin1* he see now how the original idea of combining g and e is modified in the context of a gauge model. In addition to a factor 8 (which is just a coefficient due to the group) we have the sin1 MN>(47lG)/ï-57-3CeV (7.49) 1^ > 74.6 GeV . Eq. (7.47c) allows for a new interpretation of the nixing angle. Its cosine is just the ratio of charged to neutral intermediate boson trusses. The predicted values of M. and M- for varying values of sin* are plotted in Fig. 9. In Eq. (6.22) vc have given a general phenoncnotogical ansatz for the weak neutral current. We can now infer that Che Salam-Keinbcrg model predicts the following values for the parameters in the leptonic part of this current: gL • 2 sin' » - 1 (7.SO) a, . 2 sta'» 3&£^ HIGH ENERGY BEHAVIOUR Mw (GeV) Fig. 9 Allowed values of Htf and H^r in the Salacr-Veinberg ctodel. Pataoeter3 on the curve are sin *. Before Eq. (6.2) we have said that a crucial experiment to establish the existence of a neutral weak current was neutrino-electron scattering. We can now examine processes of the kind of Eq. (6.2) more closely in the context of the Salam-Weinberg model. Let us first turn to scattering of electronic neutrinos as discussed in Section 2. The two possible reactions are given by Eqs. (2.B) and (2.151 and the differential cross-section with respect to the recoil energy1 of process (2.8) is given — within the "classical" V-A theory — in Eq. (2.15). In the Salaro-Weinberg lodels, there is an additional contribution due to the possible exchange of the neutral intermediate boson. The contributing graphs are shown in Fig. 10. Graph fa) of Fig. 10 is the one which also occurs in the "classical" theory. Its matrix element can be written in the following way if we make use of a Fier2-transformation (see the contribution of L. Jauneau) Xfa) X 'IVJ»* Ys}veêY (l + Ys)e (7.51) where we used Eq. (4.12) in the approximation of a very heavy K-boson. Fig. 10 ue scattering in the Salam-Wcinberg model CHAPTER VII The new contribution, due to the neutral vector boson Z^ (Fig. 10b) can be read off the Lagrangian (7.46) ' (7.5Z) where Cy is defined by Eq. (7.33). Using Eqs. (7.47a) and (7.47c) we can recast this contribution into the form -iJfVaU'Y^Ve^CV'iele (7.53! so that the sum of both contributions becomes „«,&*>» , JL W1 . Ti,VvA(ceve • c^,,)e (7.s4, with (7.56) Ceve = I LA 2 ' The cross-section for antineutrino-electron scattering for the "classical" V-A theory is given in Eq. (2.15). In the case of gauge models, where V and A may contribute with different strengths, Eq. (J.IS) has to be gcneraiized to read do 0e^) . S (7.56) ^[ccv-cA)-(cv>cA4i-|) (cv - cA) ^1 where Te is defined by Eq. (2.16). Finally, let us turn to muonic neutrino-electron scattering. Tn the "classical" V-A theory, this process is al together forbidden. Indeed a graph like that of Fig. 10a cannot be constructs - -jr.J the process must go via a neutral current. But this renders it easy to read off the corresponding values of Cv and CA from Eq. (7.53) adapted to this • 2 sin3 $ As has been said before Eq. (6.2), this is the cleanest way to establish the existence of neutral current couplings. The CHRN-Cargamelle Group has observed three events interpreted as v^e" •* ÏLe" (Morfin 197S) giving a cross-section o = (0.15 ± 0.08iLy 10""1 craVe" . This value is model dependent. The Weinberg-Salan model has been applied. The result is shown in Fig. 11. In the neutrino exposure one candidate for v e" - v e has been observed [Brisson, Flaine (1976)3- However, the background calculation has not yet been completed and hence this candidate has not been included in Fig. 11. a Weinberg-Salam model Fig. 11 The experimental constraints and theoretical predictions for purely Leptonic reaction! [taken Erom Sakurai (1975)]. ||g[ Allowed region ^ HIGH ENERGY BEHAVIOUR We have already explained that the Salam-Weinberg model is the minimal model which contains one neutral inter mediate boson, one Higgs scalar but no heavy leptons. It js quite clear that there are no limits to the imagination of model building once one goes beyond the oinioal choice. In addition (or alternatively) to heavy leptons, a larger nunber of intermediate bosons and/or Higgs bosons can be introduced. In this way, the model gradually looses its pre dictive power. For example, a larger maker of Higgs bosons means that the masses of neutral and charged intermediate bosons are no longer correlated as in Fig. 9. he do not wish to dwell upon these more complex possibilities here. Reviews in which many references to alter native models can be found, are for instance Abers (19751 or Taylor (1976). 7.3 The problem of including hadrons When we try to extend the Salam-Weinberg model to include hadronr, we face serious difficulties. After £q. (6.1) we have fornulated the empirical rile: "There are no neutral lcpton currents in strangeness-changing processes!". In a semilcptonic process, a neutral lcpton current always goes along with a neutral hadron current because of over-all charge conservation. Thus we infer that the hadronic neutral current obeys the selection rule AS =0. In a straight forward attempt to extend the Salaffi-Y.'einbei'g model to include hadrons, we cannot incorporate this selection rule. To see this, let us conventionally describe the structure of the hadronic cirrent in terms of quarks. The "classical" triplet is -0-0 (7.58) where we have given the two most popular notations. All hadronic weak processes are thought of originating in a weak interaction of a pair of quarks. Thus we can consider the quark process given in Fig. 12 in analogy with the leptonic process of Fig. 6. Again, the corresponding cross-section for the fictitious quark scattering process is proportional to tlw square of the ccntre-of-moroentun energy and thus eventually exceeds the unitarity lirait. Kithin a gauge model, cancellations must occur. If no new quarks are postulated, the cancellation must stem from a quark sicilar to that of Fig. 8a, i.e. the coupling of the neutral intenaediate boson to the quark pair (Xn) which leads to strangeness changing neutral current interactions. Let us go through this argunent more quantitatively now. We define the "Cabibbu rotated" quark state n,- by n^. = n cos 6 • X sin e (7.59) where B is the Cabibbo angle. Suppressing y-matrices, the ordinary Cabibbo current is then simply (pnc). Mien we want to incorporate this in i gauge model, we define a left-handed doublet 1 • LU) (7.60) and right-handed singlets. (7.61a) (7.61b) (7.01C) Unlike the case of leptons, we now also need the right-handed components of n and A quarks because they carry charge and show i,p m the electromagnetic currant. In con^lcte analogy to the leptonic case, Eq. (7.22), we can now write the charged hadronic current (7.59) in the following way: Fig. 12 Diagram which can induce PYo-U * Ys)"c - 2VroT+Ln • (7.62) US J< 0, ÛQ » 0 transition! - 324 - CHAPTER VII The t/fird component of this triplet-current contains a piece of the following form (suppressing y-matrices again) "c"c s "n cos2 9 + ** si"'3 9 + ^ (Xn + fU) sinz 0 . (7.63) The last term in £q, (7.63) is unacceptable because it is a strangeness violating neutral current. It cannot be can celled by a piece of the electromagnetic currc.it either, because the latter obviously cannot contain strangeness- changing terms. This is the difficulty we have described in the beginning of this subsection. In writing down the electromagnetic current, one has to undo the Cabibbo rotation so that a term with Ac = -n sine + A cos 9 (7.04) is required among left-handed fields. There is no counterpart to p for this field in the triplet quark scheme. Thus the formulation of the electromagnetic current and the interaction Lagrangian would be aesthetically unsatisfactory. There are various Hays to cure these problems. The "minimal" nodel is that of Glashow, Iliopoulos and ftaiani [Glashow (1970)]. It needs only one additional quark, carrying j new quantum number, called "charm". With this new quark, it is possible to cancel the bad behaviour of the graph given in Fig. 12 by a similar graph with the p-quark replaced by the charmed quark. 7.4 The scheme cf Glashow. Iliopoulos and j-.'aiani Let us now turn to the model of Glashow, Iliopoulos and Maiani in a more quantitative way. It is the 'minùnal" model which excludes strangeness-changing neutral currents in lowest order in the sense that it predicts only one new quark (cf. Section 7.5). It is distinguished by the new quantum number "charm", which wc shall denote by C. Table 4 gives the quantum numbers of all four quarVs. The charm quantum number C is defined in such a way that the Gell-Mann -Nishijina relation reads Q-I,+J(B + S + Q (7.65) Table 4 Quantum numbers of the quarks The fourth quark now serves as the missing counterpart to X , Eq. (7.64) and we can define another left-handed doublet by Before we enter into the details of the interaction, let us nake sure that we can now avoid strangeness-changing neutral currents. To this end, let us extend the quark triplet (7.58) to the quadruplet ••m- Suppressing y-matriccs, we can write the charged hadronic current by means of the matrix HIGH ENERGY BEHAVIOUR 0 0 -5U-sil1l 6 COS 9 cos e sin P ( 0 0 0 cos in the following way 0 0 0 0 0 0 qC+q = p'C-sin Dn + cos 8X) * p(cos 6n + sin 8 A) {7.69) where 6 is again the Cabibbo angle. In a gauge model, the neutral current is obtained by cccraiting C+ with its trans pose. We find ( 10 0 O") 0 10 0 There are indeed no off-diagonal elements and thus no strangeness-changin0 0-1 0 g contributions as in Eq. (7.63). 0 0 0 -I J Let us now also fill in the y algebra, fhe current (7.69) can be rewritten by means of the doublets (7.601 and (7.66) + + + PV„{1 Y5)nc + p'Yu(l * YS)*C = 2ÔnYuT Ln VUT\) • (7-71 In order to formulate the electromagnetic current, we need the three right-handed singlets (7.61) as well as The electromagnetic current then becomes .em 2 -, „, , 2 - „ 1- 1 7 , ->v 3 p V * ï *V ~ 3 nV ' 5 AV = *§ VW -5W • <7-73> The remainder is very similar to the case of the leptons, Section 7.2. tie couple the triplet conponents to the triplet bosons Â" and the singlet to B^ to obtain the Lagrangian * 41 ^ * 3 WP - j VA * To bring out explicitely the coupling of the physical neutral vector bosons \ and 2^, we have to utilise Eqs. (7.281 again. After some algebra, Eq. (7-74) becomes u 2"CDl| • charged particle * eA jjj™ » * i cos» Z1J{PTl'(C^ * v')p * P'VCV * T>)p' " - nvu(C{ » Y,) n - Syu(Cj • Y>)»l (7.75) where « - 1 - 4 sin" « 3 (7.76) Cf • 1 - | sin= « . he have thuj also specified the neutral hadronic current in the Glashow-Iliopoulos-Maiani scheme. CHAPTER VII U,sing Eqs. (7.47), the effective four fermion coupling [for (momentum transfer)2 « M£] generated by the coupling (7.75) takes the form, analogous to Eq. (7.53): •^VoYAd*?.)" i £ 5 ^ (Gf • Y*)q £ 3 ^ (Oj • Ys)q| • Ke see that the axial couplings are completely determined and the vector coupling depends on a single parameter, sin! + 2LAYIJT LX = P'YU[1 • YS)A Cos6 - p\H * 75)n sine (7.77) where 6 is the Cabibbo angle. The selection rules for these new currents of the order of cos* G and sin2 9 are 0(C05* 6) , ûS = AQ = AC = 1 AI = 0 (7-76) 0(sin* 9) . AS = 0 AQ = AC AI = Vi • Thus the dec--./ of a charmed particle (containing a p' - quark) is predominantly into strange channels because non- strange final states are suppressed by tan* 9 . For details of the predicted new particles, their production and decays, the reader is referred to the original literature [Gaillard (1975)]. So far, we have only considered lowest order processes. A strangeness-changing neutral current process may, how ever, effectively be induced by a second order charged process. Tn a renormalizable gauge model, the coupling of weak bosons and photons is of similar strength, first order weak processes being suppressed by the large factor M& or Mi in the denominator. Thus the danp,er is inherent that higher order processes are only down by a factor a with respect to firsï order processes, contradicting the very low experimental limits. An example for the famous process K» •* v* + (I- (7.79) is shown in Fig. 13. It is one of the virtues of the Glashow-Iliopoulos-Maiani scheme that these induced neutral cur rents with AS ? 0 are indeed suppressed by the factor ^i_ (7.80) in agreement with experiment. Details can be found in Gaillard (1974). A new hadronic degree of freedom has been invented in order to explain the experimentally observed suppression of AS j! 0 neutral currents, like K •* u*u~- Due to the sznallness of the mass difference between Kg and h'£ t'te lQ,,,esC had ronic state with the new quantum number (charm) is expected to be around 2 GeV. About 10 years after this theoretical conjecture two discoveries have been made. In neutrino interactions final states with two ruons have been observed [Benvenuti (1965'']. The second muon is interpreted as being due to the seni- leptonic decay of a charmed hadron. The other phenomenon was the production of a narrow resonance J/Vi [Aubert (1974a), Augustin (1974)], interpreted in the quark model as a bound state of a channed quark pair (cc). The dilepton pheno menon is now observed in several experiments at low and high neutrino energies (HPWF [Rubbia (1975)], CIT [B.C. Barish (1976)], Gargamelle [Blietschau (1976)], 15' H2/Ne [von Krogh (1976), Cundy (IP/6)]). The bubble chamber experiments have revealed that in dilepton events excessive strange particle production ocoirs. The hadronic decay modes of charmed hadrons aro expected to be much more abundant than the seroileptonic ones. One candidate, vp •+ u~rtTt+TT*TT*r" [Palmer (1975)], is interpreted as a hadronic decay of a quasi-elastically produce chained baryon. No experimental support from other experiments exists at present [Cundy (1976)]. HIGH ENERGY DEHAVIOUR 7.5 Renormali2ation and the problem of triangle anomalies So far, we have concentrated mainly on the phenomenolagical aspect of unified gauge models. But \m have said in the beginning of this chapter that the ultimate goal is that of a renormalizabie theory. Is this goal achieved? The original gauge Lagrangian is certainly renormalizabie. The question remains, however, whether the Higgs mechanism does not introduce features which render the final version of the theory unrenoraalizable. In fact, renormalizability of the Salam-Weinberg model even after the Higgs symmetry breaking has been shown but the proof is too technical to be repeated here. The reader is once again referred to the origiral literature [t'Hooft (1972), Lee (1973), Lee (1974)]. Ifeie, we only vint to point out those features which lead to physical consequences. We have seen in Section 7.1 that the R-formalism contains unphysical scalar particles. In all physical a.Tplitudes, their effects must, of course, cancel. In most cases this is indeed so except for the notorious triangle graphs. Con sider a process as depicted in Fig. 14, It is a contribution to photon-photon scattering. The broken line which con nects the two fermion triangles can either be a B-boson or the *z scalar of the R-formalism, Since the *2-scalar is unphysical, its pole must be cancelled by a corresponding pole in the B propagator. It has been shown by Boucliiat, Iliopoulos and Meyer [Bouchiat (1972)] that this is only true under certain conditions. Suppose we write Che interaction Lagrangian in a general form xi " lyii a" * ^ C-B")L * iVs'™"'" (7'81) where L is a left-handed spinor comprising aU fermions and R is the corresponding right-handed object. In the Salar.- Wcinbcrg case we MDuld have e -ï C„ = I (7.82) D *\ (1 - T,) . The charge matrix is sinply Q - -D - £ (C, - C„) - (7.B5) The condition for absence of triangle anomalies is, according to Bcuchiat, Iliopoulos and Meyer, TJrC'Q - 0 . (7.84) In most physical models, C3 will be the unit matrix so that the condition simply tells us that the sum of all fermion charges must vanish. In the Salam-Kcinborg model amended by the Glashow-Iliopoulos-Maiani scheme the sum of lepton charges is -2 whereas the sum of quark charges is Z/3. Hence the total charge does not vanish. In order tD cure this disease one need not devise a new tool for the number of quarks has been tripled for other reasons already. These are the questions of quark statistics and the problen of n°-decay. Both problems suggest that there are in fact three sets of quarks with the quantum numbers as given in Table 2. They are distinguished by yet anctl.er new quantun number: colour. The three sets of quarks carry the quantum nunbers "red", white" and "blue" and const;Lute a new internal symmetry, colour-SU(5). The normal particles are singlets with respect to this colour-group [for a review see GBll-Mnnn (1972)]. 14 Graph with feroion triangles wl.-ieh may causi problems for renormalizjEior. (th*1 broken lii represents a B-boson or a Higgs scalar). CHAPTER VII CJjce this scheme is accepted, it automatically cures our problem because the total charge of the 12 quarks is then *Z thus exactly cancelling the charge of the leptons. It is quite clear that the possible observation of a new pair of heavy leptons (with the usual couplings) would automatically require the prediction of more quarks unless WE withdraw the requirement of renormalinability. The whole philosophy of renormalizability of weak and electromagnetic interactions rests on the fact chat the Lagrangian is invariant under the gauge group. Symmetry breaking can only occur spontaneously through the Higgs mechanism. It follows that also strong interactions must respect the symmetry of the weak gauge group. This opens a completely new view on the interconnection of interactions. For references, the reader is once more referred to [Abers (1973)] and [Taylor (1976)]. 7.6 Alternative models We have gone through the Salam-Weinberg model and the Clashow-Iliopoulos-Maiani scheme in all details because they represent the minimal (most economical) models. There are, however, a multitude of competing models suggested in the literature. Once the reader has familiarized himself with the typical tricks and techniques used in gauge ntodels on the examples given here, he should not find it difficult to go through the similar steps i-. other models. It is thus neither necessary nor possible to exhaust all examples given in the literature. In constructing new nrdels, it is important to keep in mind the main obstacles we have met in our exanples. Not only must there be no strangeness-changing neutral current to lowest order, but also higher orders must be suppressed, otherwise we hould face strangeness violation in neutral currents to order a, in contradiction to experiments. Rather arbitrarily, we shall pick the model of Beg and Zee [Bég (1973)] to demonstrate a few of the alternative possibilities for gauge models offered in the literature. In Section 7.5 We have introduced the notion of colour. The question arises, can this new quantun nimber be used instead of charm to explain the absence of strangeness-changing neutral currents? This is precisely what the model of Bég and Zee attempts to achieve. It turns out to be necessary to treat the various coloured quarks unsymnetrically* . In this way, universality of the weak coupling to quarks and leptons gets lost. In order to restore it, a "heavy neu trino" has to be introduced so that the coupling can be shared by the massless and the heavy neutrino in such a way that the usual universal strength is regained. Thus Beg and Zee introduce 2 left-handed and one right-handed doublets plus 2 left-handed singlets (not counting muonic type of leptons). 1 2l where E° is the heavy neutrino. Indeed, they achieve a neutral current which is of the form *) This model ha? the difficulty that parity violation in hadronic transitions might occur with the same strength as electromagnetic transitions whereas in models where the weak interactions are "colour blind" such transitions are shown to be suppressed to a level consistent with experimental observation [Weinberg (1973) and [1974), Nanopoulos (1974)} HIGH ENERGY BEHAVIOUR - 329 - where the neutral part of the interaction is 2 S?K = charged particle • off}™ * gZ^jJJ - (7.88] î is again the mixing angle and the sum gees over electronic and muonic type of neutrinos. The neutral current (7.86) is relevant for our discussion in Section 6.1 for it is precisely of the fom (6.7). Its non-neutrina part is indeed parity conserving. The reader who is interested in rcore models should consult the literature. Sone references are [Gupta (1974)], [Ross H974)], [Pati (1974)], [Eliezer (197-1)J, [Fayet (1974)], [FrenV.el (1974], [Fayyazuddin (1974)], [Meshkov (1975)], [Herbert (1975)] and [Georgi (1974)]. Earlier references are quoted in these papers. To sumsrize the rceihodology involved, we reproduce in Table S a recipe far model building given by Leo (1972). [t remains to wish the reader good luck on his further way through unified gauge theories. Table 5 Recipe for gauge models £Lee (1972)] 1. Choose a gauge group G. 2. Choose Higgs scalar fields 3V(«)/3$[$=V - 0 . The little group of the vâcuurc ° IWU- 4. Choose chiral spinor fields i>L, i^. 5. Form Yukawa couplings I^WR * n-c- 6. Couple gauge bosons to Higgs *, v^, and i . 7. Quantize 1 2 A- 7 Ow* * I^W +J (gV) V* . B. Y+L*C»R * h.C. -* yvW. C. Some of Higgs scalars •* longitudinal components of irassive vectors. Acknowledgements Part of this work was done whi'e one of the authors (MP) was NORDITA-professor at the Institute for Theoretical Physics in Gtttebcrg. He would like *o thank Jan Nilsson for the warm hospitality in Go'teberg and NORDITA for financial support. We both wisli to thank H.K. Gaillard for help in revising and completing the manuscript. CHAPTER VII REFERENCES Abers, U.C., and B.N. Lee, Phys. Reports 9C, 1 (1975). Abranowit:, M., and I. Stegun, Handbook of ftethematical Functions (Dover Publ, lnc.: New York, 1964). Aubert, B., et al., Phys. Rev. Letters 32, 1457 (1974). Aubert, J.J., et al., Phys. Rev, Letters 33, 14D4 (1974a) Augustin, J.-E., et al., Phys. Rev. Letters 33, 1406 (1974). Barish, S.J.r et al. Phys. Rev. Letters 33, 448 (1974). Barish, B.C., et al. Phys. Rev. Letters 34, 558 (197S). Barish, B.C., CUT 68-535 (1975a). Barish, B.C., et al., Phys. Rev. Letters 36, 939 (1976). Beg, N.A., and A. Zee, Phys. Rev. Letters 30, 675 (1973). Boll, J.S., Nuclear Phys. 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Wolfenste in, Proc. Internat. Symposium on Lepton and Photon Interactions at High Energies, p. 613 (Stanford, 1975). HIGH ENERGY BEHAVIOUR PROBLEM5 1) Derive the unitarity bound (3.1) from the unitarity condition on the S-matrix element for vee" -* vee". Show that the inelastic process v e" * u~ve is bounded by cinel < T by considering the unitary condition on the S-matrix element for v e" •+ v^e". {Recall that the u~ve intermediate state contributes to the imaginary part of the elastic v^C scattering amplitude.) 2) In contrast to the four ferai coupling cross-section of Eq. (2.19), which is pure s-wave, all partial waves con tribute to the vl scattering if a W-bOson is exchanged. By first projecting out the s-wave contribution to (4.21), via the partial wave decomposition —^— = Izf.p.Ccose)!1 d cos e * * || (1 * cose) * M*| show that 1 "wJ and consequently that the unitarity limit for v e" + u~ve is satisfied for Note that these expressions reduce to the point-like case for }\. * ». 3) In analogy with the parton model for deep inelastic scattering {Chiptcr 7), it is conjectured that at sufficiently high energy the total cross-section for e*e" •* hadrons is given by the incoherent sun over cross-sections for c*e" - quark * antiquark, where the quarks are treated as point-like, and all quark types are included in the sun. Then the ratio R defined in Eq. (5.32) is given by R • E R for Cabibb0 thc01 r r(W + uw^J • r(W-* eve) lo.9R in the CIM model (Section 7.4) . If quarks are fractionally charged, what is the numerical value if each quark comes in three colours (Section 7.5)? 4) Uraw the lowest order Feynmann diagram which can give rise to the reaction ve" •* v e" if only charged intermediate boson are exchanged. Using the intermediate boson propagator derived in Section 4: k2 - M2 l"w M> I show by examining the large k behaviour of the integrand that the Fcynmann integral is quauraticall/ divergent giving the behaviour for tlie amplitude hherc A is the upper lirait on tlie loop momentum integration. Tn rcnormalisablc theories of the type discussed in Section 7, the integral is finite, with the momentum effectively being cut off at the mass of the intermediate boson: A1 - Mi. Show that the contribution of the Z-h' exchange relative to single Z-exchange is therefore of the order of electromagnetic corrections. CHAP1ER VII 5. Using the valence parton model [no antipartons, see Chapter VI), show that the inclusive cross-section for v * N •+• v + anything Takes the form of Eq. (6.5) where if u and d denote the valence partons, with couplings 4'- <*;>"/rfc> a, • (g^'/Fjwdx . Show further that the contribution of antipartons gives an expression of the form (6-5), but with g^ •+ -gj for i = A, S, P. 6. Show that in the h'einberg-Salam model (Section 7.2) the proton contribution to the parity violating nuclear poten tial [Eq. (6.13)] vanishes for sin* * = J, where * is the angle defined in Eq. (7.2B). 7. Assuming that the neutral current is hermctian, derive the relation (6.25). 8. In the simplest version of the Keinberg-Salam model (with one Higgs doublet), what is the coupling of the physical Hipgs meson to fermions? To vector bosons? 9. By examining the Feynmann diagram for the process * * n - u+u", show without calculations that the estimate (7.80) is plausible. Invent a gauge model for leptons and quacks using the recipe of Table 4. Investigate some of its phenomenalogical consequences. Are induced strangeness-changing currents sufficiently suppressed (Section 7.4)? Do anomalies (triangle diagrams, Section 7.5) cancel? APPENDIX INTRODUCTION TO CURRENT ALGEBRA L. JAUNEAU ^B^ INTRODUCTION TO CURRENT ALGEBRA - 337 - 1. REVIEW OF SU, AND SUj PROPERTIES 1.1 General notions Symmetries play a very important role in particle physics. The most familiar example is invariance under rotations of all systems of particles whatever their interactions. Another example is invariance under space reflection, which is broken by weak interactions. Experience has shown that strong interactions have more symmetries than electromagnetic and weak interactions. The stronger the interaction, the core symétrie it is. Invariance of a system of particles under a group of operations %• (said to be a sysmetry group) ceans that the hamiltcmian conmtes with all operations of ^§. , The consequences are the following : a) There exists a set of conserved quantum numbers, such as angular momentum j and component m in the case of the rotation group. b) Energy levels of the system are degenerate : Let ty be an eigenvector of H corresponding to eigenvalue E. Under an operation S of 5 , • transforms into *' Q S*. Since H and S commute : H (SO * SH* = E(S*) All vectors S^ 3re eigenfunctions of H with the same eigenvalue E. For example, the atomic levels are 2j * 1 degenerate. c) The eigenvectors of a degenerate multiplet form a basis of an irreducible representation of group ^- : Let i, , *. j be the n orthonorraal eigenfunctions corresponding to the sane value E. S*. corres ponds also to E and is therefore a linear combination of the n orthonormal functions : If s and t are two operations of 'ù, , it can be easily shown that : Therefore, the setof S matrices form a group, whi 1 is a linear representation of M~ . If s is unitary, S is a unita ry matrix. For example, the irreducible representations of the rotation group are t-'e groups of transformations of spinor [J = -L), vector (J a 1) and more generally tensor fields. In particle physics, the words "multiplet" and "representation" are very often interchanged. The nost interesting groups in particle physics are the continuous groups (infinite continuum of operations) which depend upon a finite number of parameters. They are known as HE GROUPS. The number of independent parameters is cal led order of the group. For a given representation, the generators are the infinitesimal operators of the group. The funJaj^ental representation is the one of the lowest dimension (representation of dimension 1, i.e. group invariant, excepted). The regular representation is the one of dimension equal to the ordeu of the group. A representation G of a group 6. is a group of linear transformations (or, équivalent!/, of matrices) such that to each operation of group ^ corresponds one linear transformation of G. Two representations G and G* are said to be equivalent if one can be deduced from the other by a change of basis, i.e. G' = AGA"'. It may happen that, following a change of basis, a representation of dimension n splits into separate subgroups of linear transformations, that is the matrices take the form of block matrices. Such a representation reduces to several representations of lower dimension. If this cannot be achieved by any change of basis, the representation is said to he irreducible. The rotation group is a group of order 3. The fundamental representation is the group of transformations of spi- nors (i.e. the fundamental multiplet is a doublet). The regular representation is the group of transformations of vectors. The transformation of a field * of dimension n under an infinitesimal rotation around the z-axis can be written as : !{.' = fl + id9J2J* or, for a rotation dB around an axis defined by unit vector n : •j' = (1 + ideï.n)^ J , Jy , J, , components of the angular momentum operator, are the generators of the rotation group for the represen n tation of dimension n . For a doublet j = 1 (spinor) : J = - . For a triplet j 1 (vector), Jx , Jy and Jz are the 3 hermitian and traceLess matrices : 0 0 o\ /U 0 i\ /0 -i 0' 0 0 -i J = 0 0 0 Jz = I i 0 0 0/ \-i 0 0/ \0 0 0, More generally, Jx , Jy and J are n x n matrices, but the number of generators is always equal to the order of the group. For 11 to commute with all operations of group tf , it is sufficient that H commutes with the generators of this group, since every operation of We inflow that the operators J , Jy , J do not commute. They obey a set of comnutation relations : I J 1 J [• i. J- which are independent of the dimension of the representation but characterize the group itself (this is an application of the general LIE theoreo). A consequence is the ûnpossibility to find simultaneous eigenfunctions of Jx , J„ , Jz . However, J- commutes wich \ , Jy . Jz . It follows that J and only one generator, J2 for instance, can be simultaneously diagonal!zed.The number of generators of group "Ç which can be simultaneously diagonalized is called the rank of the Lie group- The rota tion group has rank one. Invariançe_^_çgnseryatign_law If ^ is a symmetry group of H : [S, H] = 0 for each SÉ^ For a Lie group, it is sufficient that H commîtes with all generators F. : [h •»]•<• Let F be a diagonalizeci generator, F and H have the same eigenvectors : F)n > - ajm > Let us consider a transition from an initial state |i > to some final state |f > : < f]FH - HF|i > = (af - ap < f|H|i > = 0 = 0 INTRODUCTION TO CURRENT ALGEBRA - 339 - < f|H|i > = 0 if jJO or af = ai The quantum number a is conserved in all transitions. The saine argument applies to operators of the J2-type (Casimir operators) which commute with H and with ail generators. Invariant under rotations inplies conservation of two quantum numbers : j (eigenvalue of J2 = j(j + 1)) specifies a multiplet or a representation (note that j is not additive) ; m , eigenvalue of J, , is an additive quantum number which specifies an element within a given multiplet. Therefore, we are able to classify all the levels of a particle system by means of the invariance group quantum numbers. Let us remark that, instead of speaking of rotations, we could have started from the transforantions in the 2- diccnsional repiesentatior, space ("i.e. spinor space). These transformtions form a group of order 3 and rank 1 called SU, (special unitary group, i.e. group of Z-dimensional unitary and unijnoùular transformations). 0- and SU^ have the sare representations and the sace generators. 1.2 Isospin symmetry The concept of isospin was introduced by Heisenberg in 1932, at a time where the only known hadrons were the pro ton and the neutron. When the number of hadrons increased, isospin appeared as a very powerful tool for classifying them and describing their interactions. Looking at the different particles as levels of some fundamental object, masses being energy levels, one sees that particles cluster in quasi-degenerate multiplets, the elements of which have vcy -"lose masses and differ by their electric charge Q. The situation is analogous to the one of atomic multiplets specified by total angular momen tum j, the elements of which are characterized by angular momentum projection m. The difference is that atomic multi plets are naturally degenerate (but degeneracy is removed by a perturbation which breaks rotational invariance, as a magnetic fieldj, whereas there is a natural splitting of mass levels. One can therefore think of an underlying symme try, a property of strong interactions, broken by electromagnetic interaction which is sensitive to the charge of the pa-ticles. Furthermore, the well-known property of charge independence of nuclear forces shows that interactions between protons and neutrons may be considered as invariant under unitary transformations in a 2-dimensional space whose basis vectors are P and N. Such a unitary transformation can always be written as : where H is a 2x2 hermitian matrix. H may be expressed in tt.mi of four independent matrices for which we take the unit matrix and the 3 Pauli matrices T. : Œ H u0 1 + «I T^ (u0 , uj «= real para^îters) Aside from rhe gauge transformation (>' - elu°i> , one is left with unitary and uniraodular transformations (de''"' = 1), which are written in analogy with ordinary spinor transformations : A "* * *• •* e2 * Therefore, the nucléon doublet is considered as the fundainental doublet, and the syacetry group is the SU, iso spin group. To this SU2 group correspond? a rotation group 0, in a fictitious 3-dimensional space called "isospin space" or "charge space". For a multiplet of dimension n , infinitesimal transformations are : *' « (1 * i "i lj)* (i = 1, 2, 3) where [. , I, , I, are the generators, or isospin operators. In the space of the n-dimensional representation, they are described by n x n matrices. For the fundamental representation : Isospin operators obey the sase confutation relations as the angular momentum operators : L e L-i • 'j] - ijk 'k These relations ore characteristic of the SU, group, whatever its physical interpretation. I, (analogous to J) is connected to electric charge (which is an additive conserved quantum number) through the empirical QiLL-MANN and NISHIJIMA relation : « = h * ! where hypercharge Y is defined as: Y => B * S (B = haryonic number, S = strangeness). Furthermore, isospin invariance implies conservation of total isotopic spin I (analogous to J). The three pion states form an isovector n analogous to the sp^ce vector x [regular representation of SU-,). The eigenstates of 1, are the charged states -*, *°, »" . The coqponents corresponding to x, y, z are r. -, , «_ : „• . "l " "Z „- . "1 * "2 /Z ' fi » " B "3 (n+, n", JI° are analogous to the spherical components OJ." r. Note that the above convention is different from the one used for spherical harmonics). Produçt_of_represçntatigns Let if be a vector of a representation R of a group %- . If Ft is called a covariant ^presentation, the contrava riant representation B is defiled in such a way that ïc- is an invariant. In the particular case of v'tar/ transforma tions, R is the conjugate repi mentation since : .w - Ï u* v * =• n a Wo denote a "vector" of representation 2 by ta "Jid a "vector" of conjugate representation 2 by t, (a = 1,2) (this notation will be refill for SU.), and we associate the antinucleon doublet to the conjugate representation 2. However, for SU. , R and R are equivalent representations. It can be easily seen that the conjugate of the funda mental représentât ion i - + i ~ •** i -, - j u n.T -ju n.i =• ûi n'.T XT » e^ is if = e l - if where n" is a unit vector deduced from it by 180° rotation around the second axis in isotopic space. Another way to see this is to introduce the antisymmetric tensor c „ . This tensor is a group invariant since : INTRODUCTION TO CURRENT ALGEBRA Therefore, if we define na = e . e , na transform.- as £ , We now perfora products of representations by means of "tensor products", for instance, the product 2x2 is a tensor HQB = ç0 £, . This representation of dimension 4 is reducible and can be split into two irreducible representations by taking symmetric and antisymmetric combinations of conponents (under an SU, transfonction, symmetric and antisynsetric conponents transform separately arong thenselves). One writes : antisymmetric synmetric 4 ooaponents -•• 1 component H * H) * ^ pn - np I pn * np Q n I n [ nn - 1 .Vow, the produ.'t I x 2 is the tensor MQ => ta K • Cft t° <• TrM is an invariant (representation 1). One is left with a traceless ntisor : H'a * ^ çB " 7 6a Tl*' (representation 3) In terms of nucléon and antinucleon states, the isospin normalised eigenstates are : 1 ° Q ; W ' "" > pii I, = + 1 1 *T I np - 1 he remark that an irreducible tensor of the regular representation can be written, either as an isovector *,*_,*" or as a tracelcss matrix : pn 2 p - PF - n5 2 This can be applied to the pion triplet, which can be written : as an isovector : i traceless matrû "l " ±n2 " "3 P i on_ nuc 1 eo n_ co up1 ing Isospin conservation implies that the vertex function must be an isoscalar, i.e. we have to saturate the indices of pion matrix s with nucléon isospins : -N t n Let us note that, with isospinors N and N, we have formed : an isoscalar (I =• 0) : NN = pp + fin an isovector {1 = 1) : S î N The isospin part of the pion-nucleon vertex function is the scalar product of isovectors N T N and n. In the case of *-t coupling, one has to form an isoscalar with three isovectors. The only possible coupling is : n . (I X E] 1.3 SU3 symmetry 1.3.1 Properties of the group SU, In fact, two additive quantum numbers are necessary to specify a particle state : electric charge Q, and strange ness S (or hypercharge Y) which is strictly conserved by strong and electromagnetic interactions. Both appear in the GELL-WNN and NISHIJ1M\ relation (conservation of baryonic charge B can be associated with the gauge transformation *' = c ° cO- Therefore, a group of rank two would be more convenient. On the other hand, it is tempting to enlarge isospin symmetry in order to cluster particles in supermultiplets, like the baryon octet V, a, c, S. After some un- succesfuJ attenpts, this was achieved by extending the fundamental doublet symmetry SU. to a fundamental triplet sym metry SU, . This triplet consists of an isospin doublet of zero strangeness and of an isospin singlet of strangeness -1. It could be the Sakata triplet (P, N, A0) which is a natural extension of the nucléon doublet (but this Sakata model is in violent conflict with experience), or the quark triplet (p, n, \) of ŒLL-M1NN and 2WIÎIG : nr : D [t is then assumed that there are two parts in strong interaction : very strong interaction for which SU, is a good symmetry, and semi-strong interaction which breaks SU, invariance, and splits SU, supennultiplets into isospin multiplets. Let U • e be a unitary transformation in the fundamental triplet space. The 3x3 hermitian matrix It can be ex pressed as a linear combination of the unit matrix 1 and 8 independent traceless and hermitian matrices *;: (i = 1, 2, , , 8) Again, is a simple gauge transformation (associated with the conservation of B), and one is left with the group SU- of unitary and uiiunodular transformations IT = e J * (Tr*. implies dcttf = 1). The orthogonality and normalization conditions for the ij matrices are the same as for Pauli matrices : (It implies that the 6 *• c-...iccs are independent). In the explicit fora given by Gell-Mann, the i- are generalizations of Pauli matrices : INTRODUCTION TO CURRENT ALGEBRA -i 0 1 0 0 0 0 0 -1 0 0 0, la o i 0 0 0 0 0 000 \>= (° o 0 0 0 0 0 0 0 -i V ° (0 i 0. >,= ^1. . 8y analogy with isospin transformations, one writes : SU, is a group of order 8. It has 8 generators F, . An infinitesimal transformation of the n-dimensional represen tation is : *' - (l + ibij F.U The F. are n x n matrices. For the fundamenta1 representation J : F..ii 1 2 The regular representation of SU, has dimension 8. Caanu t at i on_ ru les ifi- , Aj1 is a 3 x 3 traceless and hermitian matrix, and can therefore be expressed as a linear combination, with real coefficients, of the 8 independ.nt \- 's. More generally : ft • Fj] • ' fuk fk i»j»k = 1, Fron the Lit theorem, it follows that these confutation rules hold for all representations of SU, . The f- b are the structure constants of SU. , real and completely antisymmetric with respect to the three indices. From the form of A, ,i, , .*, , one sees that : cijk £ijk for (i.j.k) » (1,2,3) , so SU, is a group of rank 2 : tuo generators can be simultaneously diagonalized. In the explicit form chosen by ŒLL-MANN, *, and i„ are both diagonal. Therefore, there are two additive quantum numbers, which are eigenvalues of F, and Ffi , or, otherwise, electric chaigc Q t..J hypercharge Y which have to be connected in some way with F, and Ffi . Ant i commut at ion_ nil es They di'pend on the representation. We write them for the fundamental representation, *l is an homitian but not traceless matrix if i = j, and can be written : [\ ' *jj - " *ij ' * Z dljk » Taking the trace gives a - -? , hence : 2 (\ • 'j] • Ï «y ' • ijk k <£$&* For SU , the corresponding anticommutation rules are : l'i • 'il= 2 6iJ The d. -. constants are real and completely sycrnetric in their 3 indices. They are given in Table 1. TABLE 1 Non-zero elements of f... and d. -, iii fijk 123 1 147 1/2 156 - 1/2 246 1/2 257 1/2 345 1/2 367 - 1/2 458 J%n 678 A/2 m. fui iii ilk 118 1//3 355 VI 146 1/2 366 - 1/2 157 1/2 377 - 1/2 228 1/5 448 - 1/2.T 247 - 1/2 558 - 1/2.7 256 1/2 668 - 1/2 1.3.2 Definitions of q and Y in terms of SU3 generators It is convenient first to introduce SUj subgroups. It is obvious that generators x , \ , A_ define the isospin sub^oup and commute with »g (which is a unit matrix for the non-:cro part of »| , A, , *,]. The isospin subgroup operations mix P and N, for instance, in the Sakata triplet, and conserve strangeness, so Ag is connected to strangeness. Another subgroup, called the tr-spin suhnrcuu, mixes S and A and leaves P unchanged. Therefore, it conserves elec tric charge. Its generators, chosen in analogy to isospin generators, are : 0 0 o' 0 0 1 (0 Ï 0t The eperator u which ctmitcs with u. is found to be : ted to electric charge. Cccp-iring this rcl.-.cion : INTRODUCTION TO CURREOT ALGEBRA u« *T *o /T 2 2^Z rith the GELL-MANN and NISHIJIM* relation : Q = I3 • : 3 and remembering that I, =• — ( one gets : Q o _r + _2_ + 2 2^ In the Sakata model, the fundamental triplet has integral charges, and Q and Ï are defined as follows : 1 A 3 .. 2FR 2B where B is the baryon number. In the eightfoldway and in the quark model, Q and ï are simply defined : Y =- •T In the eiRhtfoldway, some representations (for instance 3) are ahsent. Baryons N, A, z, s form an octet of the regular representation 8. In the quark model, one assures "quarks" as fundamental constituents. I S y Quarks '3 Q B •> Y-S P 1/2 »l/3 2/3 0 1/3 1/3 n Ml -1/2 -1/3 0 1/3 1/3 X 0 0 -1/3 -1 -2/3 1/3 Antiquarks have opposite values of 1, , Q, S, Y and B. Therefore, the lowest baryon states ure issumed to be qqq (B - 1) and the lowest /states qq (these states have integral values of Q and Y). meson 1.3.3 SU, representations. Product of representations To each representation R (covariant) of SU, we associate a conjugate representation R (contravariant). In gene ral, R and H are not equivalent representations : R t R . Representations for which R •• R arc said to be self-conjuga te. . H J - c being a transformation of the fundamental representation 3, the corresponding transformatr of the i w.x. conjugate representation 3 is VJ* = e " .As in the case of SU, , we denote by ft ) (a = 1,2,3) a "vector" of representation 3 and by (eaJ a "vector" of representation 5. It can easily be seen that 3 > 3 : it is inpossible, for instance, to find a matrix A such that - *„ = tog A (since X* = x„). Another way to see this is to introduce the invariant tensor c „ (a,8,Y = 1,2,3). For SU, , c 8 o 3SY £ GP (a,B = 1,2) is the index raising or lowering operator which switches z.a into ç_ • For SU, , e does not allow us to lover one index only : e . K* " n ~ is antisymaetric with respect to indices a and 6. In the same way : F ç =• na antisymnetric in a and S All representations of SU, can be built up by taking tensor products of representation 3 and 3. a) Let us start with ; 3 0 3 that is we perform the tensor product : 9 components i (." = TrM is an invariant (representation 1). We are left with a traceless tensor : M'8 a *. K& - ^TrM (representation 8) 3 S 3 - 1 « i An irreducible tensor is therefore H' for representation R, and M'a for representation 8, so 9=8 : 8 is a seIf-conjugate representation. The relation 3ffl3n left implies that, in a quark model, mesons, which are qq states, must belong to representa tions I a."d 8. b) For 3 ffl 3, we make the tensor product : »w • t. H 9 components We now reduce this representation of dimension 9 by taking synwetric and antisyimetxic combinations of components (as was done for Z x 2 in the isospin case) : M„6 = V) * "(««) nunber of components : 9 3 6 Mr lis a representation 3 as wc have seen above U° v n.», = tY) , thus : | 3ft 3 • 3 »o| c) Let us go un with 6*3 (18 cleraents). Ne fora the tensor product : We again symmetrize and antisymnetrlze : M. , is an irreducible representation of dimension 10. Therefore, Mf(_3ivi is a representation 8. Indeed we have : '"'"[....rr^ 6 and ff - TlM • c " Mr, , i • 0 o [(naif] INTRODUCTION TO CURRENT ALGEBRA (since M is symmetric anil £ antisymmetric with respect to indices a and b) 6 © 3 = 8 « 10 d) For 6 © 3 , we form : All we can do here is to take the trace : rfj g, '• M : representation 3 We are left with a traceless tenser MJ . which belongs to the irreducible representation 15 ; e) In the same way, one finds ; 3 0 3 0 3= 1 « From the tensor M , we extract indeed Mr (representation 1) and M. 1 representation 10. The other two tensors of nixed symmetry belong to representations 3. This relation is important in a quark model : if baryon states are made of three quarks, they must belong to repre sentations 1, S and 10. f) Another unportant relation is : ( = 1 G 8» 8' © 10 « 10 9 27 (Note that the regular representation 3 appears twice in the product 8 0 8, whereas, in the case of SU, , 3 appears only once in the product 303= 1 • î • SJ. In the eightfoldway model, where mesons are BB compound states, nesonic states can be found in representations as high as 27. Representation 27 is M T . . 5U- representations up to 27 are sumrari-ed in Table 2. Thus, by using the completely antisymmetric tensor e or its conjugate, it is possible to cast all irreducible SU, tensors into the form : symmetric indices symmetric indices The SU, irreducible representation acting on this tensor is denoted : % (P.<0 It nay be shown by simple combinatory analysis that the dimension of *V(p,q) is : N • j Note that all irreiijciblc representations of SU. can be written as : v (P) " w g i P indices, each one taking values 1 or 2 The dimension of fl(p) is p * 1 - 21 • 1 , p - 21. 'Jrii^ftj». called Casirair operators, their eigenvalues are functions of p and q). On the other hand, an element of a multiplet is specified by three numbers : eigenvalues of the diagonal genera tors F3 =• Ij and F^ ( or alternatively Q and Y), and isospin I. Therefore, altogether five nuifcers are necessary to specify a state in the SU, symmetry. 1.3.4 Relation between generators of the regular representation and structure constants a) A relati >n between structure constants of a LIE group can be derived by applying the following identity bet ween operators : [[A,B],C] + [[B,C],A] + [[C,A],B] ; 0 to generators ¥• : [MKHlV^'i] * [[Vd-'j] ' ° (i,j,k » 1,2 8 for SU3), From ccaimutation relations : one gets (since generators are linearly independent operators) : fïji fekm + £jkt flim * fkii fejn a ° Remembering that the f's are antisymmetric with respect to the three indices, this can be written : lf i£ t-'fij.X-'W " (-«kjiH-iW " lk* (" yJ Let us define now 8x8 matrices, the elements of which are : Therefore, the above relation can be written as : [fi- %} • "ik. % These Tutrices obey the connutation relations of the group anO are therefore the generators of the regular representa tion. In the case of SU. , one has : C*i)jk • - ' CW This gives the three operators of infinitesimal rotations : /D 0 0\ /b 0 A /0 -i o' *J, - I 0 0 -i J lf2 - I 0 0 0 ^T3 • I i 0 0 Vo i 0/ V-i ° oj Vp o o. h) Applying now the following identity : [(A.BJ.C] • [fB,Cl,A] + [{C,A},B] * 0 to the generators of the fundamental representation of SU, : [ft-'jM'[M*]*[IV,l|'Fj],C and using the anticommutation relations : INTRODUCTION TO CURRENT ALGEBRA .1 . one gets relations between the d and f constants : dij£ f*km + djke feim + dki£ faj Ones defines new 8x8 matrices : "Vjk " dijk Thus, the above relation can be written as : 1.3.5 Physical content of SUj representations We want to know more precisely the content of an SU, multiplet, that is the quantum numbers of its elements. This can be done conveniently by means of a graphical method, using Z-dimensionol diagrams of SU3 representations. Such diagrams are obtained by plotting eigenvalues of the two diagonal generators F, = I, and F„ = -=—Y . The diagram for representation 3 is : 1 F8 = 4Yo\ (For representation 2 of SU2 one has a linear diagram : n I \ P^^" :\\ f\ /•i We note that the representative points Very often, one plats hypercharge Y instead of eigenvalue of Fg ; O V' AH, © It can be seen that the diagram of conjugate representation R is deduced from the R diagram by a reflection with res pect to the origin of coonlinates. Therefore, self-conjugate representations have symmetric diagrams. The diagram of a product of two representations con be easily obtained bf noting that I, and Y are additive quantum numbers. To each element of one multiplet of the product is associated each clement of the other multiplet, i.e. quantum numbers of the two elements are added. This can be done graphically. For instance, from 3 © 3 • 1 • 8, one obtains the diagram of the representation 8 : i •'•••••»»; v}.A\...L / ! \ i/i' J, \ ; o / \ '4 Representation 8 •'£^ w\i r- ."--'• '•• An octet contains an isospin singlet, two isodoublets and one isotriplet. Proceeding in the sar • way, one gets dia- grams of representations 6, 6, 10, Î5, 1S, Ï5, 27 ... (see Table 2). Multiplets of representations 3, 6, 15 and their canjugatc have non integer values of Q and Y and therefore are not used in the eightfoldway model. Physical particles, having integral values of Q and Y, can be in the multiplets I, 8, 10, ID, 27 ... . In the quark model, baryons are qqq states and have to be in 1, 8 and 10, but not in 10 and 27. The "stable" baryons of spin parity I (N, A, I,*) fill an octet. A decuplet containing N*(1236), Y*(1385) and =*(1530) had to be completed with a new particle for which a mass % 1675 HeV and strangeness -I were predicted. The discovery of n" was a big success for the SU, syuraetry. On the other hand, mesons (or at le ait the lowest mesonic states) are qq states, and can only be SU, singlets or octets. For otarie, one knows one octet and one singlet of pseudoscalar mesons. Hv ^ Y \ \\<{ h un 0 K-\ *K° "u-; y=o,i l^sgin_ multiplets £_ Particles of a given IT-spln multiplet have the same electric charge- The fundamental SU, triplet contains a "If-spin doublet (n,i) and a V-spin singlet (p). An SUj octet (for instance the pseudoscalar meson octet) contains a V-spin singlet, tw>T>spin doublets (K*"+ and K~t~) and a IT-spin triplet containing K° (V, - *l) and K" (U, • -1). However, the singlet M° and the central elcnent Mj of the triplet are not »• or n" but linear combinations of these 3tatcs. ^^Tiftjr INTRODUCTION TO CURRENT ALGEBRA REPRESENTATIONS OF SU- Representation 9(P.q) ^(3,0) *(2.1J (traceless) ati) The elesrsnt 1. = 0 of the isotriplet is "° = iV- ~ '"• . By analogy, the elements V, = 0 of thelT-spin triplet will be : Vi K1 tf It will be shown below (section 1.3.6) that the isoscalar is : a n = — (pp • nn - U) Sb ' ™ 1=0 1=0 Identifying the coefficients yields : rfT „ M- = -JL+^ The TT-spin scalar is deduced trom the expression of the isoscalar n° : M0 = -J- (nn + U - Zrô) i.e. (with a change of sign) : H„ = — n" * 2- The same formulae apply to elements |U"= 1, U, = 0 > and |TJ = 0 > of the baryon octet. 1.3.6 Octet matrix >ie want now to derive a condensed expression for the octet matrix similar to the SU, triplet matrix written as J-r. r . Let us cone back to the product 3 x 3 = 1 • 8 . In tenns of q and q triplets, the irreducible traceless tensor of representation 8 can be explicitely written as : f2pp - nii - A pn P» (O " («« <" " i ™) » I "P ^ - » - PP ni V »P . »s 2» - PP - "° > Let us assume that the qq states are pseudoscalar mesons : pn - . pi - K np . n" n\ « K if • K" (I - 1) (*-i) H) Only two diagonal ter.i-j tre independent. For instance 1 a 2nn - XA - pp Ke know from SU2 that : pp - nn A - B 9 a a (I - 1, I, - 0) 1Ï-. other independent linear combination is therefore A + B, and the normalized state is : , . pp*m-2AÎ . j£ (A + B) (I n 0) ^fc* INTRODUCTION TO CURRENT ALGEBRA and the pseudoscalar meson matrix can be written : fi /G The SU, singlet state is : : CPP + nn + *A) Therefore, the pseudoscalar octet can be described either by an 8-dimensional vector (r , « , a°, „", K , Kc, K , K°J or by a matrix. As was done for the pion triplet, we define "cartesian" components P- by : • P1 * i p, . Pi. - i p7 — Pft * * P7 .'»P5 -!_ ». P. JS JÎ [summation over ij Let us summarize : [SU-,) pion matrix [SUT) Pseudoscalar meson matrix Jl In the sine way, the — baryo.i octet can be described either by an 6-dinensional vector or by a craceless matrix — ^ Bi , where the B^ components are related to the physical fields N, A, t, » in the sane way the P. "s are rela ted to i, i, K, K. 41 - C - i- A", <"*** 1.3.7 Meson - baryon coupling This is a generalization of pion-nucleon coupling. In the absence of SU, symœetry, there are 12 independent cou pling constants between 7 baryons and O-roesons- SU_ reduce? this nunber to 2. SU, invariance implies that the vertex function must be St/, invariant. Two independent invariants can be formed with three octets as : 8*8" 1»8«8'« 10 e Ï5 • 27 and only the resulting 8 and 8' (or symmetric and ant iS) Timet rix combinations 8„ and 8 ) can be contracted with the third octet to give a singlet. Therefore, there are two couplings, called antisymmetric and symmetric, or F and D, couplings. It is convenient to write explicit expressions of these couplings by using the method of tensor products, he wite the octets in the matrix forn [Ba) and (Ps) . We have to saturate indices of matrices H, B and P , and we can build two independent invariants : («*s the trace of a product of matrices is invariant under cyclic permutation of these matrices, there are only two in dependent traces of the product of B, B and P). Or, taking the sum and the difference of these quantities : Tr (B, B] P symmetric or D-coupling Tr [B, B] t> antisymmetric or F-coupling (see note next page). These couplings can be written in a somewhat different way, which will prove to be useful later on. Introducing octet components BJ through : matrix B =• — *. B. ft * l and using the confutation rel- tions between *'s, one gees : Tr[B,B]p = -i- Trfx. Bi B. P 2i/T . L * L 2i f.. Tr* A. vector 'ements, not matrices XT The conjugate rotrix is in fact B •> B Y_, , the B-fields being Dirac spinor fields, but here we shall forget Dirac natriccs. B is the ncnaitii1 conjugate of matrix B, so we take 5 as : The SU, scalar (BB)0 is : INTRODUCTION TO CURRENT ALGEBRA - 355 . / Introducing the generators of the regular representation : s £ - ijk • Wy one has : Tr[B,B]P= - fi (B Fy a)pfc In the same way : Tr with (Jj.^1 • 4 SJJ • 2 dij£ », . Thus : THB,B)P » ^dijk5iBjPk he have defined 8x8 matiices (section 1.3.J) : so : Tr{B,B)P = /l(û Dk fl)Pk This shows why the couplings are called F and D couplings. Since (BF;B)P. and (BDJB^PJ are SU, invariants and the P- 's transform as components of an octet, the quanti ties BF-B and BDjB transform also as components of an octet. Note that BF,B corresponds to N 7 N in SU,, NOTE It is necessary to be more careful if one wants to write explicitly some coupling, for instance NN'n°. K Be " T Al TrEB (i,e' trac«less tensor (EB)O) and B^ B* - ~ These expressions can be written explicitly in tens of the physical particles. If summations arc performed ov^r all indices : Y j P° » P° - Tr P - 0 and we find tne simplified expressions TrBBP and TrBBP. Another expression may bo given : Tr[5,B]p « TrfM]^ Pi where Pj are vector elements. Therefore, the auantities Tr[B,B]A- and Tr(B,B|*, can be used in place of BF-B and BDjB . ;*t&S*q* whereas BD^B has no equivalent in isospin . Mor? generally, one can define ÏF.+ , where * is a "vector" of a representation R of generators F, . Let us she that the quantities : •A - *Fi* Ci = 1, 1, -..8) behave as components of an octet. Under an infinitesimal operation of the group, J> transforms as : * - (1 • i^Fj)* and ï as : t> " ï(1 - ia.F.) (F. being hermitian) Therefore : •j - *J » Ï0 - iu^Fj) F-0 • iujFj)* = ÏF^ * iw-Ï^.FjJ* where second order terms in u are neglected. We know that tac commutation relations are independent of the representation : F f PV J] •* «k h Thus : hriting : f f ' ijk " - » ;ik " (^ IVe get : Therefore, the set of quantities 4^ behave as a multiplet in the space of the regular representation of the group. This theorem is general for any LIE group. 1.3.8 LIE algebra In fact, an isospin operation which mixes proton and neutron states is unphysical since electric charge o • a supersclcctiou rule (whereas a rotation can mix spin states). However, all physical properties citr. be deduced : the commutation relations of the generators, without considering the group itself. The set of generators F- of a LIE group allows us to define a vector space of quantities :.i F- (a- ben.,' (3-dimensional for SU2 , B-dimensional for SUj). A product in this space is defined by the commutator [Fj.F/l. *nere- fore, to a LIE group ^. is associated a real vector space of quantities fa.F^ , closed under a multiplication def;- ned by [Fj.^j] " i fiifc F. : this is said to be the LIE ALGEBRA of the group. In fact, the F- thccsclves (for any representation) transform as coirponents of an octet, the sar.e way J , J ., for ar.y value of J, transform as components of a vector (regular representation \ or SU,}. It foliotas from the ccrccutation relations : D F £ [ i- j] • ' ljk \ that the f>- transform also as components of an octet. INTRODUCTION TO CURRENT ALGEBRA. 2. CURRENTS 2.1 Introduction of isospin and SU. c\.Tents Particle currents were introduced in order to describe el^ctromagneti. _nd weak transitions. For instance, elec tron-proton scattering is described as the interaction of the electron current with the proton current. The proton current is py p , where p stands tor the Dirac spinor u ; tie neutron current is rty n. These electromagnetic currents are purely vector currents. [n weak interactions, the nucleonic current is a sua of vector and axial vector terms : pY n and py y n for n + p (for instance n + p e~v ) + iwj? and ST ÏSP for p - n (B radioactivity] In the transitions described by these currents, there is a change of quantum numbers, in particular isospin, bet ween initial and final states. The cuirent is said to carry the quantum numbers which characterize particle states. For electromagnetic transitions : P -*P | superposition of il = 0 and si ' 1, al. n •* n J For weak transitions : n * p M - 1, M3 » + 1 p + n ûl = 1, M3 = - 1 Therefore, wc have a set of 3 currents carrying a! = 1, with ûl, = + 1, 0, - 1, analogous, as far as isospin is concerned, to particles i , «", n". In order to bring into evidence the isospin properties, let us introduce the iso- spinor N 3 [^J , the components of vhich are Dirac spinors (sore precisely, we are dealing with direct product of Lorentz space and "lospin space}. Taking linear combinations of clectronagnetic currents, we can write : 7 (PY..P * m n) » 7 N» N isoscalar current j (Pvup - ntun) » $vu ^ N M " 1, alj = o On the other hand, we write the vector part of weak currents : p\ n » S, ,*.«j . sr — — N &[ - 1, &I- - • 1 u 2 J 'i - ii2 ni p - ! Ï t"N - &y — N M - t, ûl, » - 1 u u " 2 * Therefore, the last three currents can be considered as components of an isospin current : 5> - S which is ii J-ircior in Lorcnt: space and an iscvector in isospin space. This i:.. triplet hypothesis has very far-rea ching conséquences. Since elcctrcaagnetic current is conserved [i.e. electric charge is conserved), it italics censor- .atIon of the vector weak current (CVC hypothesis). This explains why the vector coupling constant is not affected by Currents considered here arc ratrix olracnts of current operators, as : * pMj^Pl >' û(pJ»uu(p') in the ltnit q- - 0 (q • p - p')- strong interaction effects (near equality of G and G ). Furthermore, the nucleonic currents (and more generally all hadronic currents] have a more complicated structure than uy u (nucléons are not point-like particles and have an anomalous magnetic moment). It follows from rhe isotriplet hypothesis that the vector weak currents have the same struc ture as the isovector part of the electromagnetic current. The electromagnetic current is a combination of the ûl. = 0 component of the isovector current (ai = 1] and of the isoscalar current (M - 0). The space integral of the time-conponent of the electrocagnetic current is the electric charge Q, which is expressed by the Gell-Mann and Nishijina relation : isovector isoscalar Thus, it can be said that generator I, is the space integral of the time-component of current J (third component of isospin current J*). In the some way, Y* is connected to the integral of the time-component of isoscalar current. We consider now s'rangeness changing weak transitions (such as .'." •* p e"v 1. Let us assume that |ûS| = ] semi lep- to. ic weak transitions obey the selection rule : i.e. [ûSf = 1 hadronic currents (or their matrix elements) carry isospin ~ as K mesons. In other words, these stran geness changing currents transform like isospinors. The strangeness conserving isoscalar and isovector currents, and the strangeness changing isospinor currents have the quantum numbers of an octet of currents, extension to !aj of the isotriplet of vector currents. The generalization of : Â'Y y N N = nucléon doublet — = generators of representation 3 &iu Ft B (i - 1, 2, ... 8) where 8 denotes the baryjp octet and F, arc the the generators of the regular representation of SU_. Z.Z The Noethor thco'ea This theorem states that, in a Lagrangian theory, invariance under a LJE group implies the existence of conserved currents- . ., •. Let 4 I*,, T-M be the Lagrangian density, function of fields i-;(x) and of their first derivatives. The field eq. it ions [equations of notion) are : s». ('-'*) Let us assume that fields 4j forn an n-dinensional multiplet, the transformations of which define an n-dimensional representation of the group : «#» tn such a trans tarnation : For the sake of simplicity, we as*unc that the parameters u. do not depend en space-tiEc coorditurn In the general dcrvn^tr.ition of Soother thMrce, the u^ 's are assumed to he functions of x : the transformation is thai 3atd to be local. Such local transforations arc considered in the gauge theories unifying electromagnetic and weak interactions. INTRODUCTION TO CURRENT ALGEBRA ''I * H (Fk)y »j s i i F "»*i" » *ii "kC k)iJ Vj = 0 (equ.. :ons of motions) T577 (Fk)ij *j (the Ffc 's do not depend O.T the fields or their derivatives). The currents of the group (for the representation of generators F^) are defined to be : ! - nfc (Ft)jj « j I The number of currents is equal to the order of the group. Invariance of the Lagrangian : ii.„ H the currents arc divergenceless, i.e. are conserved currents. If the symmetry is broken, the divergence 0- current j is given by it Hxanrplcs . a) Let us consider first the nuclcon doublet. TIw Lagrangian density : is manifestly SU, invariant, 3inco -\N Is an SU, scalar. (Note that the field equation is here the Dlrac equation}. An tsoapin tranafornaiicn acta on Lh. nuclcon doublet as : lf,N'. (l'i^N We have : _l£. s FT v F, = IK From the Noether theorem, the isospin current is : This current is conserved in the hypothesis of strict SU, invariance. ^) E±2n_îï£El£ï The Lagrangian density ; is SU, invariant (ii.n is an SU2 scalar). The field equation is here the Klein-Goidon equation. - T— hence : aVi 3xu F. are the generators of the regular representation. We remember that : so that : >l • - L %Ç x ' c) Let us consider now the SU, fundozicntaj quark triplet. The Lagrangian density : £ • q (qq is an SU} Invariant). x The (icncratora of the fundiciental representation of SU, are — . The quark currents ore : i - *i J/qt^q (i - t. 2..-.B) d) If the fields jre ihe baryen fields (regular representation of Sty, one finds : These oificnta arc torccrvcJ in the case of urtct SU. invariance. Eut one knows that SU, is badly broken by s strong ineoFJtftlorj. so tftjt tfto |iS| » 1 curcrnis ore no: conserved- INTRODUCTION TO CURRENT ALGEBRA Charges [generalization of electric charge) The tiEe-coEyonent of a current is the charge density. Therefore, to each current in Lorentz space corresponds a ch;rgs defined as the integral over 3-dimensional space of the current tirce-conponent : Qk(« <= /jo &t)d*x In the case of strict invariance under the operationsof a group, it follows froa current conservation that these char ges are constants of notion : —-- 0 Jt Therefore : Ml • ° but these charges do not necessarily commute among themselves. 2.3 Quark currents The SU, currents in the fundamental representation (quark triplet) are : -6) [p and n denote now proton quark and neutron quark). Fron the explicit form of GelJ-Mann matrices, one gets : >l ' T >l • " I & \ " - " \ P' * S \ "I N jj • 7 (">„>* ; '„ ") iijl jj • —s (P >„ P • " t. n - Ji ,u i) Current quantum numbers * \ Behaves ûY ûl al, like K • >l • o; J„ • J„ - 'iu • <• »„ p J° • >i ' 1 , n> \ • >t • >)l • p ». > T »"u • JJ - oj • >„ P -: • >". • »: • •-•„• *: • ^ - o; - ^ .„ » !) I Therefore, quark currents form an octet similar to the meson octet Components 1, 2, 4,5 are coupled through weak interactions, and a particular combination of components 3 and 8 is coupled through electroma gnetic interactions. K <""> Components 6, 7 are not coupled (no neutral jûS| n 1 weak currents) It can be seen that, if weak currents are S!L currents, semi-leptonic weak transitions obey the selection rules : ûl =1 for aS = 0 semi-leptonic ùl = y for |flS| •> 1 ûS = + AQ In the same SU, framework, the electromagnetic current is written as : corresponding to the definition of the electric charge in t ms of SIL generators « • F3 * -h F„ •. ~/Ï *, , so i is called "hypercharge current". The quark electromagnetic current is therefore : «i ^ \ % We can also consider lf-spin rultiplets of currents. There arc : *Q two doublets of charged currents as-o USl-i a triplet of ncuti.il currents : u and which are not coupled to leptonic currents INTRODUCTION TO CURRENT ALGEBRA (r.o neutral |ûS| » 1 seni-leptonic transitions, as manifested by the absence of Kf -* v p"), and u" CûS - n) ; a scalar of U-spin, which is the electromagnetic current j3" (we recall that, in the octet of pseudoscalar mesons, the TT-spin scalar is : As another example, let us write the baryon electromagnetic current : F, and FQ are here generators of the Tegular representation. Thvir matrix elements are given by : and the octet components B- have been defined in section 1 i.6. One gets : 1 5 YU F3 B = 4 a \ P - à TH N) + \ (P *p =-• - F ,u z") * (F ^u v* - r vu r) [gene ralizf.t ion to SU3 of N -^ N » j (PP - Ss)) S ^ F8 B = 4 ^ Yu P + S Yw N) ~ "T (? \ " * ^ Yu ^ Therefore : which is the straightforward definition of the electromagnetic current. Axial vector currents Since parity is not conserved in weak interactions, weak currents have, in Lorentz space, a vector part V and an axial part A . These axial vector currents will be seen to belong to an oc.et of currents. 2.4 Cabibbo forrailation of universalis We have seen that the octet o curn s contains two positively charged currents j* and v (which carry the same quantum nuirbcrs as » a; .',, *• two negatively charged currents j" and v~ . Semi-lcptonic transitions >VS B 0 and |&S[ • 1 arise through the coupling of these currents with leptomc currents. However, the coupling constants for aS • P and |aS| • 1 transitions appear to bo different, since, as it has been said earlier, the rates of leptanic de cays of hypcrons arc systematically lower than what could be predicted from a universal coupling with a unique cou pling constant C . Cabibbo gave a solution to this [robtea by considering the oast general charged current as a linear combination of the SU, charged currents : I* - a y • bv! ASM) [flSI-1 The new universality assumption inplles that the sun of the squares of the ûS - 0 and |aS[ - I (AS - * it}) unrcnor- 7-alUcd coupling constants Is cquJl to the squared coupling constant for p-div:ay, i.e. : a: * b2 - 1 Introducing a new pjrocoter a,- called Cabibbo angle, one h33 : }t " 3» C'SSV* vu sinflv| (It can Dt said that J-~ is deduced from the charged parts of isospin current by an 5U, operation). The hamiltonian for semi-leptonic interactions is then taken to be : i] t't h.c. fit» M with G = G (z = leptonic current). The consequences are : a) Cy- '-"= deduced from 0 B-decay) * G cosOy b) |iS| = 1 and i& » 0 decay rates (for instance K* and n* decays) are in the ratio tg26y. . Experience has yielded : sina^ ">- 0.2 . The sanie formulation can be applied to axial vector charged currents, with an angle e» . Experimental values of e^. and e, are found to be cooyatible- 3. CUKREOT ALGEBRA 3.1 Why current algebra ? By cocparing the definitions of electric charge and elcctronagnetic current : J J u u /% vo see that "generalized charges" (i.e. integrals over space of current time-components) may be defined and may be considered as "realizations" of SU- generators. Therefore, "charges" must obey the connut ition rules of SU. generators [Q,.QJ - i fyk Qk These coaptation rules define a charge algebra. In the cose of an exact syaeetry, the currents defined fnn ,"Cocther*s thcorea are conserved, and charges arc r~"tstants of nation. If tho syecetry is broken, at least sone charges will be functions of tine. The hope is that the syncctry breaking is such that comitation relations arc still valid. Then, one can say that charge algebra is a Denary of the initial symctry. This assumption con be ncaninfiful only if it is a tine independent s tat ment. It can easily be shown that, although charges arc no nore constants of potion, co=cutaticn relations arc still Independent of tine. Lot us ass\~4> that these relations hold at a tine t : |Q,W^(t)) • 1 ll)k Q„(t) One can VTito : J [Qltt.de).Q>(tMt)] - [Q1(l) * QjttJdt.QjU) • Qj(fdt)] - i fiJk Qk(t) * (pj.Qj] * [Q,.Qjj) t ; «<> «rst order in ' ' JI •3r[«i-<'j]-"ijk«i, • ' fuk ik("dt) INTRODUCTION TO CURRENT ALGEBRA They still holtl at time t + tit . 3.2 Commutation relations of current time-components Let the vector currents be : V^ = ï Y, F- where J. is a multiplet of field operators. We want tu calculate the equal time-commutator : [vj(x,t),^(x\t)] that we cai write : [*+0Ï., i+0'i] + t* = * fM , 0 = Fi , 0' = Fj). Expliciting spinor indices gives : ft °« v *v °T« *J= °«B °* h v sjj x,t x'.t [the matrix elements 0 and O' are numbers, y and $* are operators). Using the following operator identities : [AB,C] » A[B,C] + [A,C]B [A,BC] = [A,B}C - B{A,C] one has : This decomposition allows to nafce use of the canonical commutation rules of spin 4- field operators Therefore : [** 0», •* 0'*] " 0oa 0^ (»* (.j «6T - ** t8 1^ S(x - J') =• Y* [0, û']c aCx - x') Thus : [** Fj »> ** F. tl _ - «.*p=j, F,l* i(H - x') " S fijk ** Fk * s<* - J'> We obtain therefore equal tine comcaitation relations between time-components of vector currents : Il [vj(xtt), Vj(x',t)] = i fijkVj&t) 6(x - ï') By integrating over x' [vj(5,t),Qj(t)] =ifijkvj(î,t) By integrating once more, we find the commutation relations between vector charges : |[Qi(t).QJW] • i lm qtW | 3.3 Effects of symmetry breaking An SU, invariant lagrangian density is : Isospin invariance is broken by an electromagnetic interaction term, for instance : e 3 YU —J—i N Ay However, as this term [minimal electromagnetic interaction! does not contain field derivatives, the current expression is unchanged, and the commutation relations are still valid. W ¥£cS9I_^irSDts_in_a_guark_mociel An invariant lagrangian is : <£ = q , -JL. + mqq • $\j invariant interaction terms An SU_ transformation is : q - (l + io>i -ijq The Noether theorem allows us to define vector currents : f = qy — q u ^ v 2 and vector charges Q^ which are constants of motion. However, we know that SU, is broken by semi-strong interactions. This SU, breaking term conserves isospin and strangeness, and therefore behaves as Y = 0, I = 0 components of SU, representations. The simplest assumption is that it behaves like the eighth/of an octet. In a quark model, such a term is ita q '.B q (it induces a mass difference bet ween strange * and non strange quarks p and n). Again, this term does not affect the expression of the currents and therefore their commutation relations, but it induces a non zero divergence for some currents. Under an SU^ translocation : A + w x q *a q * q(i - \ •* *k) B(i \ k k)q + w x to £irst order in = q *B q j k q[*e» k]q "k so that, from Noether's theorem : SV* 6<^ i IT = TT = 1 m <\> *kh INTRODUCTION TO CURRENT ALGEBRA From Tabic 1, it can be seen that *g commutes with *. for k = 1,2,3,8 and does not commute for k = 4,5,6,7 (strange currents}. Therefore, when SU, symmetry is broken by semi-strong interactions : isospin currents k = 1,2,3 are still conserved, and Q., Q2, Q3, Qs are constants of motion hypercharge current k = 8 strange currents k = 4,5,6,7 are not conserved, and Q., Qg, CL, Q7 are functions of time. However, the commutation relations are still valid for all currents. 3.4 The problem of axial vector currents One can derive axiai currents from Noether' : theorem by introducing chiral transformations which differ from the ordinary SUT transformations by a factor y . For the quark triplet, such transformations are : Noether's theorem yields : \ = q Yu Y5 ~J q Axial currents are not generally conserved even in the case of a strict SU- invariance. Let us consider the free quark lagrangian density : 'Jnder a chiral transformation : sqq •* mq+ (l - i^ -| yjy, (l + i^ -j yjq •- mqq + Zioi. raq-y5q so that the axial current divergence is : v~ = 2imq — Y5 q j' 0 unless the quark mass vanishes. The divergence of an axial current is in general a pseudoscalar density. One can define axial charges Qv which are functions of time. Let us consider the commutation relations between vector and axial charges. In "he above demonstra tion (section 3.2), rp-,F,| is then replaced by : [V -S xj] " 2i fijk Ts \ In the case of commutators between two axial charges, one has : 2i f [\ H- 'S »j] • ijk \ Therefore, we have two more sets of relations : [Qj(t). qj(t)] -ifijk^t) [ot(t).Qj(t)].ifijkQk(t) and the corresponding relations between charge and current time-component, and between current time-components. 3.5 SU3 « SU3 algebra Relation (I) expresses sizply the fact that the 8 vector currents (and the 8 vector charges Q|) transform like an SU, octet. I! is the coûirutation law of SU- generators. Relation (II) stems from the face that the 8 axial currents (and Che charges Qv) form an SU, octet. Therefore, the commutator of a member of an octet with Qj, which is an SU, generator, belongs to the same octet. Relation ([II) is not a mere consequence of SU-. The commutator Qv, Qvj transforms a priori as the direct product of two regular representations of SU, , and could have parts in multiplets 10, 10 and 27. It is a very strong constraint to assume that this commutator is still an octet of 5U, . We have derived this last relation in a very specific model, the quark model- The basic assumption of current alge bra, which gives its predictive power, is that these confutation relations are more general. To sunnariie, one can say that the ccsnutation relations between the 16 charges Q- and Qy define a closed algebra which is called SU, ©SU, algebra. One can introduce the following linear combinations : It follows that : K-^] but, due to the orthogonality of 1+Y5 and 1-y. factors : [0I.Q?] - « SU, O SU algebra is then the direct product of two SU, algebras (a left group with 1*YS> a right group with I-YJ. Let as write a transformation of the SU, O SU, group for the fundamental representation : q * (l • i»j -T('*YS) • 1=4 '-j (1-Y5))q One can also define chiral currents : q Y„ (1 s YS) -\ q For the SU- current in a free quark model, all the current divergences are determined by the quark masses- For example (with ,jp • m^ : i • 1,2,3 k - *i a A = tm + m q Y k =• 4,5,6,7 a u p -J 5 -J q >„ VJ ' (ni, - m,) q -4 q k - 4,5,6,7 If SU, is a good syxjnctry and the quark masses are large, we will have : Imp - mj « 2y the vector currents are all nearly conserved and the axial currents are not. Chiral symmetry is badly broken down to SU, which is wo;.ic than SU., : SU, S SU, - SU, * SU, INTRODUCTION TO CURRENT ALGEBRA On the other hand, the success of soft pion theorems can only be understood if the axial current (i=t,2,3) is near ly conserved, requiring nu very small. An analysis in terms of pseudoscalar meson masses suggests : nsTi$ • " V = Then we have a symmetry breaking pattern : su3 © su3 + su2 © su2 + SII2 The choice between the two possible patterns is still open- 3.6 Comnutators with current space components. Schwinger terms We have written commutators between : - charges, - charge and charge density {time-component of current), - charge densities. We consider now the following commutator : [jj(5,t)f jjtx'.tj] a - 1,2,3 The commutator [o, 0'] used in demonstration of section 3.2 is now : h. 1, A. y^ Y 1 for vector currents fAi ^i' *i yi* Y Yrl ^or ax^a^ currents As Y, Y commutes with 1, and Y4 Y Y, with Y, > one has : [jj&t), jj(x\t)] = i fijk j*(x,t) 5(X - X') {IV} (same relation for [V,V] - y, [v„t] * A, [A,A] -+V). By integrating over x' , then over x , one finds the previous commutation relations. However, this result can al so be obtained when some quantity is added to the right hand side of (IV), the integral of which is zero. Such terms are known as Schwin^er terms. It is possible to show by a simple argument that such terms must be present in the above commutation relations. Let us write such a relation for i => j . Since f..^ = 0 : [jit*'1)» iî&'V] =0 [« = 1.2,3) Taking the spatial divergence of j with respect to x' , we get : [j„Cx,t), div5(ï',t)] =0 The condition of conservation of current i : \K ' ° yields : divîcî-.t) --J^cî'.t) --!^(3È-,t) 3j + We write — » [H: jj and we l't now x' be equal to x . Calculating the vacuum expectation value of fjQ, [H, j ]] : 1 H j H H + H 0 * OlDo'P ' y]f° " ' « °'Jo J0 " o Jo " Jo *o Jo Jo' " ° ° we ^et : « 0) Jo II j0|o > = 0 (since M|0 ». 0) We insert new complete 5ets of interaediate states : nîm * 0|i°|n " n|"|a ".' ml-Jal° ' " ° EnSnm hence : Jl<0|j>>|^0 Therefore < 01 j |n > = 0 for any state n. The consequence is j = 0 . This shows that commutation relations (IV) cannot be true. There nrust be additional terms which are pr.-iportional to derivatives of del ta-function and disap pear by integration. These Schwinger terms are CMxJel-dc-pendent, Therefore, conumitation relations between charges are less restrictive than those between tine-cenponents and space-conponents of currents. Commutation relations between current spatial components are even more sensitive to models. Let us consider, in a quark model, a commutator such as : [j*(*.t), jjtx.t)] = [•* yh ya A. *, / ï^ T6 Xj *] (a,B = 1,2,3) From relation : [1(1* 0 J-, 1/ 0' i|>] = i[i+[o, O'jifi 6(X - X*) we can write : or x • »*(h- j\ |wv.!'(v >j j ['. v \ ie})> One finds : [v j'J "i fijk 5aB 'J * dijk ** °<.B *k * This commutator generates a new "current" (the aag term) which does not belong to the SU, x SU, algebra. Here, we have to take into account the anticoninutation relations between SU» generators ; they depend upon the representation and are, therefore, a fortiori, very strongly model-dependent (for instance, the result is different for the quark model 01 the eightfoldway). 3.7 Universality and current algebra Current algubra gives a set of non linear relations between charges, that is space integrals of currents. Therefo re, it fixes the normalization cf these currents and allows a definition of universality, The weak hamiltonian for semi-leptonic transitions has been written : with the total curre. t J defined as : j, - J1 • Jh 1 where J is the leptonic current : j = â Y (1 + -r l* INTRODUCTION TO CURRENT ALGEBRA a) For the sake of simplicity, let us forget for the moment ûS = 1 transitions. In a quark model, for instance, the hadronic current J is A priori proportional to quark current with some constant a : The definition of universality is the statement that a = 1. Let us define the charges Q- and Qv as space integrals of \ and A time components. The first set of commutation relations [Q-, Q-] = i?.. ., Q. can be satisfied only for a = 1. The relation IQ-, Q- being normalized, the third relation Q., Qvl = ie-jk Q^ fixes the normalization of QV and thus gives a mea ning to the V-A hypothesis : it says that the fundamental hadronic on rent is "V ± A . Nature has chosen the V + A combination. b) Let us consider now the full hadronic current with a strangeness changing part. A priori : v A A 3 i A •>!! • ' 'P v • <=)"„ * » >P v * *="« • °( ù • ' rl • l ' ' ')* W **^ * K * l) with arbitrary constants a and a. In Cabibbo theory, this J is assumed to be deduced from J (aS = 0} by an SIL. operation which, being unitary, does not affect the normalization. Therefore : a2 + e2 c 1. We define noh the charges corresponding to the total hadronic current J : JjJ {x)d^ - F^t) • i F2(t) and we postulate that they obey the usual isospin commutation rules : [FjCt), Fj(t)] = icijk Ffc(t) (i,j,k - 1.2,3) (1) One has obviously : F, • iF2 - .(Q, • >Q2 . q* • iq*) • 8(0, • iQ5 • qj • i<£) or, using the definition Q. *> Q. + Qv : F^t) = a Q*(t) + 6qJ(t) F^Ct) - « Qj(0 * BQjtt) Let us define F- by the commutation relation : F iF [ [Q\W, qttO] - i ^ijfcQjW CiJ,k - 1,2....8) gives : 2 F3[t) « (»* + 50 Q*Ct) - a fl Q*(t) + £ e With this explicit expression, one calculates ; [F,, F,] . i(„2 • S2)F2 [F2> Fj = i(.* * «F, Therefore, relations (1) imply : where 6 is a new parameter (Cabibbo angle). Normalization has thus been fixed by the commutation li-litions. Note that the Cabibbo quark current can be written as : jj - * YuO + Y5)(n cosec + A sina^ Furthermore, current algebra yields a new formulation of V-A hypothesis. The weak V-A current is defined with 1 + Y5 I and its associated charges belong to the Q^ charge algebra. It there fore commutes with QT : [jh(x,t), Q"(o] - 0 QI " Qi " ^i ' hence : In the current-current hypothesis, the weak hamiltonian for non leptonic transitions is proportional to J J and therefore Ilnon lePtonic conmutes also with Q? , i.e. : fjjnon leptonic QAI _ ["..non leptonic 0 1 This relation has important applications for non leptonic transitions, since the conserved charge Q. is an isospin operator. 3.8 A first application of currant algebra It consists in a demonstration, given by Fubini and Furlan, of the Ademollo and Gatto theorem. This theorem states that, in the presence of a symmetry-breaking interaction, renonnalization effects occur only to second oTder in this interaction. Let us consider the case of isospin invariance. The nucléon isospin current is : î„ = Sv„rN (t.|) the three coirponents of which are the currents : r p + p N T„ I N 3 \ n •+• n N Y I N : il3 = + I n-*p N Y I" N : il, = - 1 p •+• n I and I are the charge-raising and charge-lowering operators : Tl ± "a Is = I ± INTRODUCTION TO CURRENT ALGEBRA - 373 - The If electromagnetic interaction is tumed off, there is strict isospin invariance and the all three currents are conserved. If electromagnetic interaction is tumed on, the j~ currents are not conserved, I and I" are no more constants of motion, and the coupling constant is modified by the electromagnetic interaction. Current algebra may give some in formation on this renormalization effect. The matrix element of I, jetweer. two proton states is : (x) -N^'tx) =N<5'(X) .^5'(X)