DEEP INELASTIC LEPTON-HADRON INTERACTIONS AT HIGH ENERGIES
V.A. Matveev Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, USSR.
PREFACE
The purpose of these lectures - to state the main
ideas and elements of theoretical tools in studying the
deep inelastic processes of high energy lcpton-hadron
interactions.
This field of elementary particle physics has achieved
in the last years great successes both experimental and
theoretical. About ten years ago M. Markov put forward
an idea on point-like behaviour of total cross sections
of inelastic neutrino reactions. The first SLAC-MIT experi•
ments on deep inelastic scattering of electrons on proton
v/ere performed in 19C8 and the foregoing DESY experiments
supported this idea. A similar point-like picture of in•
elastic neutrino-nucleon interactions has been observed
in the CEBN experiments.
More over, the experiments on deep inelastic electro-
production reveal a remarkable regularity - the inelastic
form factors of this process are determined asymptotically
by functions of dimensionless ratios of kinematical varia•
bles and do not depend on any dimensional parameters.Such
a phenomena - called usually the Bjorken scaling - has
stimulated many theoretical investigations. There are now
a number of different approaches and models - more or less
sophysticated - which are trying to explain the main regu•
larities of various processes of high energy lepton-hadron internctions ( Feynman's pnrton model, the current algebra behaviour was also observed in pure hadronic inclusive
and li/rht-cone analysis, dual-resonance model and other). reactions, for the first time on the Serpuhov machine and
It is the principle of automodelity, proposed in 1969 then on the ISR and NAL. The theoretical studying of the
by Ii. Murodynn, A.Tavkhelidze and the author, which state inclusive processes was initiated by pioneer's work3 of
the scale invariant behaviour to be a universal model- A.A. Logunov and Collaborators.
independent property of all the deep inelastic processes An extension of the automodelity hypothesis to
determining by the physical similarity laws and the dimen• relativistic nucleus collisions has led A. Baldin to a
sional analysis. A similar approach which uses the dimen• prediction of the so-called commulative production effect,
sional analysis was recently proposed by T.D. Lee. confirmed in 1970 in Bubna.
As it was noticed by N.N. Dogolubov, the automodel or It will be extremely hard task to give a full history scale invariant behaviour of the inelastic form factors of of development of physics of deep inelastic interactions. deep inelastic processes is in an intimate analogy with This course of lectures is only an introductory the so-called automodel solutions of a problem of strong review and does not pretend to represent all of the vari• point explosion in hydrodynamics, well-known for a long ous aspects of the physics of deep inelastic lepton-hadron time. interactions at high energies. There are many excellent The successful applications of the automodelity reviews on this field to which we may refer the readers. principle to the analysis of the muon pair production
experiments performed in Brookhaven by the Columbia Uni• 1. INTRODUCTION. versity group and the recent experiments on the inclusive
hndron electroproduction show the fruitfulneas of this 1.1. Deep inelastic lepton-hadron processes as a tool for studying the elementary hypothesis in studying deep inelastic processes. particle structure. The recent fundamental investigations by N. Bogolubov,
A. Tavkhelidze and V. Vladimirov on automodel asymptotics In the recent, years the main interest in studying
in quantum field theory have provided a rigorous basis high energy interactions of particles have shifted from for the light-cone analysis, developed by R. Brandt, the field of elastic and quasielastic interactions to H. and others. Leutwyler that of essentially inelastic multiparticle processes. The
Notice, that the automodel or scale invariant reason is that in studying elastic processes one deals, in essense, with global properties of particles ( effec• structureless particles. tive radii, "transperance", etc.) whereas in studying the The"point-like" nature of lepton-hadron interactions inelastic multiparticle interactions one gets a possibility at high energies has led to an idee on automodel or scale- to investigate the local structure of particles. In essen• invariant behaviour of the deep inelastic differential tially inelastic processes when extremely large number of cross sections. channels are openning, we deal, in some sense, not with The automodelity principle consists in an assumption properties of individual particles, but with those of that the asymptotic behaviour of deep inelastic processes
"hsdron matter" ( an excitation of continium occurs). is independent of any dimensional quantities, such as
Processes of essentially inelastic lepton-hadron masses, "elementary lengths", etc.; therefore the struc• interactions at high energies and large momenta transfered ture functions or form factors of these processes arc from leptons to hadrons, accompanied by the secondaries functions of the invariant kinematical variables, the gene• production, are usually called the deep inelastic pro• ral form of which is defined by the dimensional analysis. cesses. Due to the local nature of weak and electromag• A number of approaches to the theoretical interpreta• netic interactions, deep inelastic lepton-hadron scatter• tion of automodelity or scale invariance for deep inelas• ing is an unique tool for studying the local properties tic scattering do exist. In what follows we will discuss of particlea, i.e. a behaviour of "hadron matter" at only some of them, which are based on the very attractive small distances. from the physical viewpoint idea on hadrons as composite
As experiment reveals one of the typical properties extended objects with internal degrees of freedom. Among of all the deep inelastic lepton-hadron processes is their such model heuristic approaches there is the parton' model
"point-lile" character. The essence of this property which is treating, in some sense, hadrons as a gas, of consists in the following. In the limit of high energies elementary constituents - partons. The parton model and large momentum transfers, the total deep inelastic allows one to obtain relations between cross sections and cross sections, having been summed over all open channels form factors of various deep inelastic processes, starting probability of each of them is considerably suppressed from concrete assumptions about a type of partons (valent, because of form factors of strongly interacting particles, quarks, quark-antiquart pairs, glyons., etc.) and on a are comparable by magnitude with those of point-like degree of compositeness of hadrons. The other more or less traditional approach of quantum field theory to studying To the second class one can assign the process of deep inelastic processes are developing such as vector electron-positron pairs annihilation into hadrons: dominance, model, current algebra and analysis of singula• 3. t~-*> battrons , rities of current commutators on the light cone ( "physics and the procesa of the muon-pair production in proton- on the light cone") and some other methods which will be nucleon collisions: mentioned in these lectures only in passing.
2. The kinematics of deep inelastic processes. 4. p>+ N-*> jtt*+ fjT+ battrons •
2.1. The clasflification of deep inelastic lepton- To the last process there is closely related the
hadron processes. process of the charged lepton pairs production
via, as one usually believe, the generation of the In respect with that whether the four-dimensional intermediate charged fir -mesons: momentum ^, trensfered from leptons to hadrons is space- jfc . like ( g
be separated into two classes:
1. Í <-° :"excitation" of hadrons by leptons; In the last years there arises a special logic in
2. £ >0 ; annihilation and production lepton pairs studying pure hadronic inelastic reactions, namely, inclu
with participation of hadrons. sive experiments, i.e. the multiparticle production
To the first class such processes as the inelastic electro- processes in high energy hadron collisions when one or production or neutrino/antineutrino production on nucléons more particles in the final state are identified while belong the rest being undetected. The classic deep inelsstic 1. ¿~+N-t eT+ herons, processes of the types 1, and 2 are in fact, the inclu• sive type processes when a secondary lepton is identified 2. + M -*) ^ + hadrona, in final states ( so called " one arm" experiment).
y# +. { + hadrons, / = e~ f*~ . An extension of inclusive description on hadron The studying of these processes put basis of the system in the final states of deep inelastic lepton theory of deep inelastic reactions. hadron collisions leads to a special type of deep 2.2. Inelastic electroproduction on nucleón inelastic processes, for example, an inclusive electro- or neutrino/antineutrino/ - production on nucléons: The matrix element of deep inelastic electroproduc•
tion on nucleón £ .• • « corresponding to the
6. •£ +tV-> €~+ Á¿+ ÁActroHS : diagramm, pictured on Fig. i ,
or a process of inclusive annihilation of electron- ELECTRON t FINAL HADRON positron pairs into hadrons: STATE
PROTON FIG. 4. ro where  j - is some particular hadron.
is determined by the expression For 811 the processes listed above the lowest approximations in electromagnetic and weak interactions are usually assumed to be hold. Besides, some deep inelastic processes of lepton-hadron interactions are where discussed which require to appeal to more higher orders in the electromagnetic or the weak coupling constants. We are interested in the total differential cross section of this process summed over all the possible final hadron
states, symbolically:
die where cross line denotes an integration over a phase space contains all the dynamical information on the excitation
of real secondary particles. of an unpolarized nucleón by a vertual phuton with the
Averaging over the polarization of the initial nucleón momentum £ .
and electron we get, in the approximation fHe = 0 , for Using the evident equality
the differential cross section of deep inelastic electro-
production
and the condition of the translation invariance
SfitlJpCK)/p>= e • iTNl Jfjt (*)(/>>> where S — Cp+*~ )*~ ond
one can easily represent the tensor M/HV in the
"local" form:
/ r T " - J Cty 4V +*i,k¿ -Jfcr-**') - Ot. ') Sf>th is the result of averaging over the initial electron The current product in this expression can be inter• polarization and summing over the final electron polariza• changed at £0 > 0 by the current commutator. tion; Due to the hermitian property of the hadron electro- -r + f
magnetic current j = and the continuety equation
Spin, — Jufx)-0. the quantity WuV should satisfy the N following conditions: iy. tyt> fat) = 0 , tyv (ht) « Cht)
Starting from these conditions one can expand the tenaor
Wuy on the gauge-invariant structures: the inelasticity parameter m ^. In accordance
with a choice of invariant variables, an element of the
phase volume of the final electron takes the form: The form factors W\ , ^5 which are usually called the structure function of deep inelastic electro- production, depend on the invariant variables' ^ and i. -Jjl. jtzji ;
v EK> s-M* Y=H>
Notice, that the quantity -f (p,k,k') which deter• mines a spectrum of the final electron momenta, is a ¿ 2. di'm e'je'm - E m*tn • function of three independent inveriant variables, e.g.:
, pk' , kk'. The region of the deep inelastic scattering is determined
The invariant variables can be chosen in a number of by ways, for example:
2 L t i. s-fr+k) -, t' =(k-k') , v> = pi = p(*-kO;
The character of this asymptotic limit in terms of 2. E,E, 0 - energies of an initial various invariant variables is discussed in Appendix I. and a final electrons,and Nov; we find the form of the function ^ in the
scattering angle in the limit of high energies.
laboratory system (P " 0 ), Using ( J. I ) and ( J?. J ) we obtain
respectively.
In the laboratory system it is convenient some times to use instead of the energy of a final electron E f m where it follow,') that that gives for the differential cross section the result:
JB'JSI Ml* I * * * J
As it follows from this formula for the differential Thu3, tho croas section of inela3tic eieetroproduction
2 cross section, the studying of the angle distribution of in the limalimit of highi"h énergies ( S ^ M ) ia determined the secondary electron momenta allows to separate the by the expression : invariant form factors -If/ and W¿ , or, what is the same, to separate the contributions of the transverse and the longitudinal polarizations of the virtual photon. which by using the variables ¿¡ and ^ = / — — is transformed to the form: Photoabsorption croas sections:
A relation between the form factors Iff and ^¿ can be expressend in terms of the total cross sections of the photoabsorption on a nucleón:
The expressions ( 3. 3 ) and (4 ) have the evident relativistic invariant forma and can be' applied in an corresponding to the various directions of the virtual arbitrary reference aystem. In analysis of the inelastic photon polarization: electroproduction in the laboratory system ( P ~ 0 ) the variables E , £ and 9 are the most convenient. In terms of these variables we have *•* 'ft.x Choose, following the tradition, the i tux factor, Notice, that due to the positivity of the cross sections which coincides with -ftux of "equivalent" real
Then define the normalized polarization vectora of Let us introduce the parameter R, which deter• virtual photon i which satisfy the gau^e condi- mines the ratio of the photoabsorption cross sections
with the longitudinal and the transverse polarizations : tion f/,^-* :
f yT£ t 6'j 1') * i ) 0) - transverse polarization;
éj~ ~ .—— ff „ 0, 0. to) - longitudinal polorizct i en. ' /-Tí or where = ( 19/ fi, 0, f¿) - is the moment of the virtual
photon in the laboratory system: M^0 * )? , //f^ - •
Thus we obtain the result:
For the case of point-like particles in connection
S = 0 : CT j = O ; s oo j
S> I :
0~7JZ the form factors )V/ and are 4vM yty» JB'M * expressed as follows:
where r* - *Vl £' I Jn'M t* B 1-6
Under conditions
We cite here the expressions for the form factors and where G~yM -total cross section of the real cross sections for electron scattering off point-like photon absorption on unpolarized nucleón, it follows particles with spins: S • O and 1/2 . that
Point-like spinless particle. S=û ;
Using the expression for the electromagnetic current of point-like spinless particle with an electric charge Obviously, that under the same conditions R. -* O , as Q : Sp'tJj, (•> lp > * Q • (t> '+!>)ft The contributions of the various virtual photon we find polarizations to the differential cross section of
inelastic electroproduction are described by the Hand's formula: 4v ^J?p0 we find
= &f(?+U>) [fy ft fa'{»- wrfp ' from where it follows
Thus, at Y ^* /V ¿ the form factors are determined by
The double differential croas section has formally the scale-invariant ( = automodel) form:
For the differential cross sections we get the result
Point-like particle with spin S =1/2.
Using the expression for the electromagnetic current In conclusion we write down the contribution of en elastic of point-like Dirac particle with charge 6) : electron scattering on nucleón to the total cross section
of electroprodbction.
S/>'t7M (o) /fi> = 0 lifr) £ U(f>) , Single- nucleón term Under condition, that at-£l^> Ml » 7"¿^ ^> C-p
as for tne oint we find approximately ¿ W( «s V ^ P The electromagnetic current of nucleón is determined like spin 1/2 particle. by
2.3. Deep inelastic neutrino reactions.
The weak interactions of leptons with hadrons are
a where where .— — . ro where 7/*« — £ /^/l/1- an<^ and -are the weak charged currents of hadrons: 6ML = 10~5x/.¿>¿ . The matrix elements of the processes of deep inelas• ¿» W » ñ M +F2 fa) tic neutrino and antineutrino scattering on nucleón are the total electric and magnetic Sacks form factors of nucleón, normalized at by V, + N -> S + -.. £•£ (o) = charge; 74 + N -» -f-h ... G-tf (o) - magnetic moment ( in units of — ) in accordance with the diagram ( Fig. 2): are determined by the expressions: to ON where -rp - the corresponding charged lepton currents: The quantity ^juv^/ftí) rePre3ents tne result of summing over all the final hadronic states The total cross sections of interactions for unpolarized nucléons summed over all the possible final states of hadrons are described by and contains all the information on the dynamics of neutri- no/antineutrino interactions with unpolarized nucleón. Starting from the hermitisn conjugation property of the weak currents of hadrons and Jy , the Here invariance under the time revision ( T -invariance ), and locality ( CPT -invariance) one can find the fol- lowing symmetry properties of the tensor For the differential cross sections describing the momentum spectra of the final leptons one gets i.e. M^' -are all real; ? - •*/•••/ & 2- tyi%%> = ¿[tyVcrtin]* where signs + for eigher ju - V= O or - for eigher ¡u = 0¡ ^ ¿7 or z. Using the variables ¿f = and h = (t r>) ro ON we find and hence W¿ ' s ö ; 3- \v Cht) - - Wy. (f>,-l)<- In terms of the variables ¿-' and ^ ,defined from where it follows the cross symmetry relations bet• f in the laboratory system ( p * 0 ) , the expressions for ween the form factors: the differential cross sections take the form: JEM ÏÏ ct^/y j9\M i j J • Under neglecting the lepton masses the form factors 7 4-JLfiJi• nrt V* m(£+e>)aft. n¡ j . do not contribute to the cross sections, so as Introduce by an analogy with the electroproduction ^&,¿,S ^ 0 follow the inequalities on the form =ß Co, I, it O) - right factors: ef'jL (0,1,1,0) - *• scalar or longi• We v/rite down here as an example the form factors tudinal polariza• and cross sections for the neutrino and antineutrino tion interactions with a point like Dirne particle, say, an electron: w where f = (¿,,0,*,^) . roint-like Dirac particle: ( The absorption cross sections which correspond to these polarisations of the currents are expressed through the invariant form factors by the following way: z where for the pure conditional case of an absorption of H a masaless particle with the quantum numbers of the Jr (l + í ) - £>r \>( e - it-esHer-tn^ -mesons, coupled to the weak hadron currents a coupling constant ^ , is equal: Integrating over all transfered momenta in the physical w e find the region C £ /g S » total cross sections of Yf { -and V¿ t - interactions where 3 n For the total cross section of annihilation process, summed over all the final hadronic states, we get 2.5. Annihilation of lepton pairs into hadrons: The annihilation of electron-positron pairs into hadrons is a simplest example of a deep inelastic process with the time like transfered momenta. In the one-photon approximation ( Fig. 3) the matrix elements of this process is given by: where éué? - - Spfy,C*-+»e)K (£+-»<) * represents the result of averaging over the polarizations of electron and positron, and Pfivti) - ¿fa) fifo-Up (Of a/ > Ca//J„ fi)/o> N J e >tX and the cro33 section * S * SoUp C*) Jv(o)(0 > = is a gauge-invariant quatity , which contains all dynamical information on transfering of a virtual photon into hadrons. 2. Muon pairs production in hadron-hadron Thus, in the approximation /Hg B0 , we have collisions. Î . ro Conpider the process of an inelastic collision of two ON Notice, that at ^ > 0 the quantity^^K coinsides hadrons and t> , with production of a lfipton I with the Fourier trasform of the vacuum expectation pair, say > and some system of hadrons A/ value of the commutator of the electromagnetic currents with the total momentum P^ : of hadrons, i.e. P^Om '^/[Jp (x), J** <• J„(.)]fr> I In the lowest order in the electromagnetic interaction the amplitude of the process, which corresponds to the Write down here, as an example, the form of the spectral diagram function Pff?-) for the «ase of the muon pair production in the final state: An element of the differential cross section for collision of unpolarized hadrons, summed over all the final hadron states and polarizations of the lepton pair: is determined by ro where, as early, the cross line denotes an integration oo / i where A, = U(k)fy iT(k') is the electro- over a phase space of real hadrons ; Tin and in - magnetic current of the lepton pair masses of the colliding hadrons A. and ¿ , respec• tively, Here we use the notations *) 'In this expression for the amplitude we have used the "in" and "out" - representations of hsdron states because, in contrary to one-particle states, for many-particle states the "in" and "out"- representations are uneqivalent. where the averaging over hadron polarizations has to be done; We discuss now the properties of the tensor , which contains all the dynamical information on the lepton pair production in collisions of unpolarized n ç*tî?*tk?' i ,f . , * hadrons. Due to the electromagnetic current conservation this tensor should satisfy the gauge invariance condition fy/Opy = ívfyv = 0 nnd from the hermiticity p* - r\ it follows that the real part of this tensor must be symmetrical and its imaginary part antisymmetrical where JJl - is an element of solid angle, related to under replacing the indices ft y> . the directions of a momentum of one of the muons in In accordance with the current conservation the the sentre of mass system of the muon pair ( £ » Jc — o tensorp CPifi'i.') csn ^e decomposed into the five indepen- masa tt « JC* )» "JA - °f muon. dent gauge-invariant structures : Performing the integration over the angles in the ( sentre of mass system of muon pair, where +ß, -fa. ft'+ft ?/) + >/>g. • (<$ ft'- ft r¿) > where we find All the scalar form factors f>j are real functions which depend on four independent invariant variables constructed from the vectors p f p ' and £ . For example, one can choose the square of the virtual photon The c.m.s. of the muon pair. The axis ¿- is directed momentum ( £^ -muon invariant mass squared), along the momentum p , and and the ordinary Mandelstam's variables J , £ and the momentum p ' lies in U : the production plane Xê- . The normal to the production ^- plane is directed along the y -axis. where the invariant V^ff. in tne laboratory frame system ( p • 0 ) is proportional to the energy of the virtual photon, ^ =. ~ (PPO ~ *s an energy Then the corresponding four-dimentional polarization 1 of projectile hadron in the laboratory frame system. ro Z } vectors can be represented in a covariant form: Introducing one more invariant variables ty.. = rr Ar i the invariant final hadron mass squared, we get a linear relation between five invariant variables: It is convenient ot decompose the tensor fjuV into the structures corresponding to definite virtual photon polarization vectors: •€ ^T'\ ê ^Tl)sxA € in the rest frame system of a virtual photon, £ = 0 , i.e. in the centre of mass system of the muon pair ( see It is easy to see that the polarization vectors are Fig. 4 ) orthogonal to the virtual photon momentum ^. and one to another, their norm is , i.e. Pr, Pxx 0 P> tA O ; ryy 0 = Pr. Ci) C*) . |V 0 A h < - TI}T2} ¿ ; Up to a normalization this is the density matrix of the virtual photon given in a linear basis. and the completeness condition Now, the differential cross section of the process can be represented in an invariant form using the for• mula: ro 7Î, 7i Z —3 / t = holds. where The triple differential cross section of the process, when in the final state only one muon pair with definite tL , A3- and ( P.- = P . - fi. - O ) and of the form factors with the definite polarizations. m PCs,t\ At) x (»Tmfis-C*-»') where is the total cross section of production of a virtual #* photon with the mass ^^^in the process: The distribution over the squared effective mass of * + 6 -» Y*'+ /lirons the muon pair is obtained by integrating over A AZ and d the muon mass we get the following formula for di - muon mass spectrum: 1 In the limit of high energies, when cr ^ ftl ¡ ft , we get For the purpose of applying the vector dominance hypo• thesis it is convenient to represent the mass spectrum where £ = £ -— , and formula in the form transfered energy in the laboratory system. Notice, that the total cross section of the muon production process determines only the sum of the inva• riant form factors with definite polarizations and Pyf +Ptí + P¿ » does not depend on Pr¿ Measuring of all the form factors separately ( except P ^¿ ) can be one in studying the angular where distributions in directions of a momentum of one of the muons in the centre of mass system, where ^= O : We notice in conclusion, that the form factors 'T,>' >TT¿ have kinemntical singularities. Its relations to the form factors P¡tí 5- introduced above, which are where Û is given by fi_ +P , ß * ^Jïbàsfr free ( under some conditions) of kinematical singulari• is the muon velocity in the c.m.s. ties are given by : The form factor Pj£~^ csn be found studying a polarization of one of the muons along the normal to the production plane ( along y -axis on Fig. 5)• 0T Integrating over J4 oltoS& we find the distribution in only $ or 0 , respectively: »her, + /»'i fff) 'J h ¿ ; Wf,) '4 fa X> Saw) +r\6-r*'«yj- tions of the geometrical similarity, ratio and propor• tion already known by Greeks. For the first time this apparently has been done by Galilëg in determining the beam stability of a given material as depending on its 3.1. The concept of similarity and the dimensional linear size. Many great scientists starting from Mariotte , analysis Newton, Fourrier, etc. have contributed to developing of methods of the dimensional theory and to their basic concept of physical similarity. As experimental data indicate, one of the most The dimensional theory is based on the fundamental general tendencies observed in superhigh energy behaviour property of invariance of the physical laws and their of particle interactions is exposing of definite proper• mathematical formulations with respect to the concrete ties of similarity which connect a behaviour of ampli• choice of units for physical quantities. tudes and cross sections at different values of energy Transformations of similarity related to changes and momentum transfer. of scales, i.e. units of the main m dimensional physical Formulation and interpretation of these similarity quantities, are called the dimension transformations or properties is one of the most important problems of the scale transformations. The concept of dimensionality of contemporary theory of elementary particles. physical quantities is tightly related to the dimension Various approaches to solving this problem do exist. or scale transformations. In this section we will present the main ideas of the method called the automodelity principle, which provides Dimensionalities: an unique description of all the processes of deep in• Let among the given variable and constant quantities elastic lepton-hadron interactions. characterizing some class of phenomena we can pick out a The automodelity principle is based on the concept minimum number of basic quantities E/ , E¿ ... E% , of similarity and on the dimensional analysis. for which independent units or scales denoted as ^¡~[Et] Notice, that the dimensional theory has emerged as I = /,2, ; are determined. a result of application to physical phenomena of concep• V/e will say that a physical quantity /> has a dimensionality^^ f^V/*--- f¿* , where = (*,,tf„ ..., L , • t¿ ••• tk WHICH STAY INVARIANT und ET- THE SEULE? T R ANS FORI»«-4- I ONA OF THE DIMENSIONAL QUANTITIEC. The main laws and functional relations characterizing THEN ANY FUND I ©V-3 RELATION OF THE THEORY, OF the given range of physical phenomena do not depend on a R GONER*3 FORM ^("£,,..., Fk ; ¿V,.-, ^-*) ''INDEPENDENT OF h CHOICE choice of basic units and consequently do not change OF BASIC UNITS, CAN BE REDUCED TO A FUNCTIONAL RELUT ION their form under the scale transformations of dimensional BETWEEN SMALLER NUMBER OF THE DIN.ENRI c-NLESS VARIABLES: quantities. This property of mathematical forms of physi• f fa h,-j I**-* ) = c cal lwas results in one of the main theorems of the THE ION» IS DIFFERENCE BETWEEN NUMBERS OF «11 dimensional theory - the so-called 7f - theorem. DIMENSIONAL QUANTITIES AND BASIC OUANT I I. IEE WITH INDEPEN• - theorem. DENT DIMENSIONALITIES, THE LARGER SIMPLICITY IS ACQUIRED BY THE FUNCTIONAL RELATIONS OF THEORY BP n RESULT OF APPLI• Let the given class of phenomena be characterized CATION OF THE 7[ -THEOREM. IN PARTICULAR CF-SE, WHEN by a set of Ï) dimensional quantities among which K tl-k ~ I , THE FUNCTIONAL RELATION BECOMES OF THE EXTREMELY quantities have independent dimensionalities and can be SIMPLE FORM AND CAN BE REDUCED TO THE FORM chosen as " primary" fundamental quantities E2" , defining the basic units: WITH ONE ARBITRARY DIRAENSIONLESS FACTOR. WE WOULD REMIND THAT, FOR EXAMPLE, IN MECHANICS Then a set of others, " secondary" quantities P^'Bftjt ,H-K THE CGS SYSTEM WITH THREE INDEPENDENT DIMENSIONALITIES will have dimensionalities determined via basic ones in IS USUALLY USED the following way: (i) K=3 : ttng+J, ¿ fa* ) ; maís M fy) ; iine T (sec) . o/k /= /,2,,.., h-k. In elementary particle physics, due to the presence Not lat via formulate the concept of physical similari• of fundamental physical constants "k and C determin• ty. ing the nature of micro-world^there does exist only one Twn physical phenomena or processes are similar, if independent dimensionality ( JC = /) ." they cnn be connected through the scale transformation of length Z or mass M . bft-iio <\ in-Hns íonil quantities characterizing these phenomena There is however an interesting possibility of or processes. increasing of a number of basic dimensionalities when The nece33nry nnd sufficient condition for similarity using the so-called vector units of length A - (¿Xj/y,/}), of tAro phenomena is the universality of numerical values This possibility corresponds introducing of different of e«aential dimensionless parameters, defining a character scales for measurement of lengths along various space of the given processes. directions. Such a possibility, considered apparently Notice, that, for example, in hydrodynamics 3uch firstly by Williams in 18... , proved to be extremely dimensionlcss parameters, giving a criterion of similarity fruitful in applying the methods of the dimensional theory of processes of equilibrium flow of liquid ire the Reynolds to an analysis of multiparticle productions in hadron col• number ( £¿ = í£-£i° ), Mach number ( M - ) , etc¿ lisions, within the framework of the automodelity princi- .Vhen the given .set of phenomena iJoes not depend on pie.*5 dome dimensional quantities or parameters, one can say about self-similarity of the phenomena with respect to the scale transformations which do not touch these quantities or An application of the generalized dimensional analysis parameters. with vector units of length for description of the multi- In problems of ga3 dynamics and hydrodynamics in the particle process behaviour in hadron collisions is dis• corresponding cases, when processes are independent of any cussed in the lectures by Prof. R.Muradyan. .1 i mensiorinl parameters, the corresponding solutions are said to be autouiodel ( 3elf-3imilar ). A classical example of such a situation is the problem of intensive "point" explosion, the solution of which in the limit of infinitely small 3ize of a charge ha3 the automodel character. According to the Jf -theorem of the dimensional theory, an extra decrease of the number of essential dimen- Automodelity = the self-similarity, connected with sionless variables or parameters describing the process in degeneracy with respect to some set of dimensional variables or paramet• question as compared with their maximal total number( t)- < ) ers. is peculiar to the automodel solutions. Consequently, the automodel solutions can be considered as partial solutions 3.2. The automodelity principle for deep of the task which depend on smaller numer of variables and inelastic lepton-hadron interactions. parameters as compared with their total number ( degenera• tion with respect to the parameters.). In studying the processes of deep inelastic lepton- According to that said above, we will use the concepts hadron interactions at high energies and large momentum of automodelity and automodel behaviour as an expression transfers, there emerges the so-called "point-like" of self-similarity which is a result of dropping of defi• picture of behaviour which is characterized by disappear• nite dimensional quantities or parameters out of the physi• ance of the dimensional parameters, defining the particle cal picture of a phenomenon. effective radii, in the differential cross sections of Thus, the theory of similarity and the dimensional these processes. analysis employ the following notions and definitions: Physically it means, that in the processes of weak Dimensionality = a degree of homogeneity of a and electromagnetic interactions leptons play the role of physical quantity with respect test particles with zero sizes, i.e. they behave as point• to the given set of basic units; like particles. Scale The effective sizes of hadron targets disappear, in transformations = transformations connected with summing over all the open channels, what results in the a change of scales of basic units; above mentioned "point-like" behnviour of deep inelastic Similarity = an invariance under scale trans• lepton-hadron interactions. The asymptotic behaviour of formations of the general form; form factors of these processes are determined by func• tions depending on the dimensionless ratios of invariant Self- variables only. Such an asymptotic behaviour of form aimilarity = the invariance under scale trans• formations not touching a definite factors of inelastic weak and electromagnetic processes set of dimensional quantities; has a character similar to the so-called automodel solu- tions of a number of classical problems of hydrodynamics, from leptons to hadrons and with momenta of interacting for example, as was mentioned above, the task of strong hadrons ¿>¿ , having a dimensionality >1 , i.e. point explosion . One can expect that at high energies and large momentum transfers, when the particle masses can be neglected, the situation is very simplified and, in under the scale transformations some sense, becomes "hydrodynamical" one. Starting from this analogy, we formulate here the automodelity principle for the processes of weak and electromagnetic interactions behaves as a homogeneous function at high energies and large momentum transfers, which provides a unique approach to the description of all deep inelastic lepton-hadron interactions by using the methods of similarity and dimensional analysis. An applicability range of the automodelity principle and following from this properties of scale invariance The automodelity principle. is given by the conditions: Let us suppose that the asymptotic behaviour of deep /fV, >V»£W ; -/VA. >AT; ± > •i- - ¿i*ect. inelastic lepton-hadron interactions does not depend on Below we shall demonstrate the applications of the any dimensional parameters fixing a scale of lengths or automodelity principle to description of various processes momenta, such as masses, dimensional coupling constants, of deep inelastic lepton-hadron interactions. The following "elementary lengths", etc.; and, consequently, the struc• agreement concerning dimensionalities of basic quantities ture functions of these processes would depend only on the will be employed: kinematical invariant variables. Then under the scale Aie take the system of units where ^s C =1 transformations, connected with changing of the measure• and , consequently, the velocity and action are dimension- ment scale of momenta, all physical quantities such as the less. As a basic dimensional quantity we take some mass cross sections, or form factors transform as homogeneous yn . It is evident that an element of cross section functions of the corresponding physical dimensionalities. will have dimensionality £d where C is some dimensionless constant. When defining dimensionalities for various quantities In the lowest approximation in the electromagnetic of the theory, such as current matrix elements, structure interaction, where functions and form factors, we should take into account that the dimensionality of currents equals: ^V->Aéu/rm.s Wm -JZPU*)' the dimensionless constant 0 is determined by the [fy] m 1*}*] m[fy*r+] m > limit: and the dimensionality of the one particle state vectors /which are normalized in the relativistically inva• riant way: Applying the dimensional analysis directly to the quantity i) defined by the expression can be taken as follows : P C\ l) & Î "fa +1fÍ>) =jjXt )ÍX<0 lju(X)Jyf)IO> we find, that the quentity p x) is dimensionless and consequently under the scale transformations £ —5 % £ transforms as a homogeneous function of zero dimension, Annihilation of lepton pairs: i.e. In accordance with the automodelity principle the total cross section of electron-positron pair annihilation what is in accordance with the formula (3.1. ). into hadrons: hadrons at fZpAfZ, where Af - For the pure conditional example, when the final is some mass parameter of the order of the nucleón mass, state of annihilation process coincides with a pair of + is a function of the variable only and does not point-like Dirac particles, say f* p~ _Pair» we nave depend on any dimensional fixed parameters. Then, from the dimensional analysis it follows Notice here an interesting circumstance, that the automodel where we use the definition: asymptotica of the total cross section of the -pair annihilation into hadrons corresponds, with respect to the character of asymptotic behaviour, the upper bound on the Taking into account that the constant of weak interac• total cross section of annihilation in the lowest €. tion (r is dimensional quantity approximation, which follows from the unitarity condition ( see Appendix ) : we find that the quantity p weaJc is dimensionless €. e -» Akrons £ ¿ and in accordance with the automodelity principle must Thus, the automodel asymptotics for the total cross behave at large transfered momenta £ as constant,i.e. , section of annihilation of -pairs into hadrons does ro ¿ o not contradict, in the framework of £ -approximation, As an example, we notice that for pure leptonic * the unitarity condition only when the following inequality processes of the type holds: It is not difficult to generalize these results for the the cross sections at large momenta are case of annihilation of charged lepton pairs into hedrons; Vf / ktclrthS ; As early, the unitarity condition and the optical theorem the cross sections of which in the lowest approximation lead, in the framework of the lowest approximation in weak in weak interaction coupling constant & are determined coupling constant, to an upper bound on the total cross by the expression section of charged lepton pair annihilation into hadrons: Due to this bound there arises a limitation on the Thus, in the region allowed region of energies for which the consideration has a sense, namely the function F{S,¿,\ ¿) depends on the dimensionleas kinematical variables c and ti = (I- —J=¡//+- — J S ç£ only P{S,T>,¿)= F Deep inelastic scattering of electron ft,}) ' and neutrino In the one-photon approximation o structure of the quanti• ty F becomes much more simpler: The differential cross section of deep inelastic J electroproduction ^*^X/t , where 4~ — is dimensional FCÉT,Y)^ TV^[CI-^Fl(ti)+ ï¿F2 (6)]'> variable, can be represented in a general form as where F/ and F¿ are related to the contributions of the transvers and longitudinal polarizations of the ¿¿DA T* virtual photon. / To illustrate this point, apply the automodelity where the function F(£/^''/^) - dimensionless quantity . principle immediately to the tensor \z(f>, %) which Using the automodelity principle, we find that under the corresponds the Fourier transform of the electromagnetic scale transformations of the virtual photon and nucleón current commutator: momenta Using the agreement on the dimensionalities of currents the quantity F transforms as a homogeneous function and state vectors, we find with dimensionality.equal to zero, i;e. tions of a virtual photon, are the functions Under scale transformations the quantities ^I » ^2. Fy — fi Fj — transverse polarization; undergo the transformations F¿ = —F. - JlFf — longitudinal polarization, 1 v, ¿y, *) -=> ^ (*y, a *) - ») '• in terms of which the differential cross section of deep inelastic scattering reads: Wit?,*)w^^v^V- a"**^^- For this reason, at sufficiently large energies and A generalization to the case of deep inelastic neutrino transfered momenta the form factors W¡' behave as scattering requires only to consider in addition the follows form factor Wo , which determines in the matrix element 1 ro of the commutator of weak charged hadron currents an ro 1 *l£i\v)-¿F2(A)í interference of the vector and the axial parts of interaction, i.e. where p> (¿j - dimensiónless functions of the variable For the scattering off the point-like particle we As the form factor has a dimensionality have £fY¡J - , the automodelity principle leads to the behaviour of the type: t0T s in s= 0 Fi (A) = 0 - P K rv, y) « j F3 (t). Introducing the combinations of the automodel functions -1 ¿(/-¿I) - for spin 5» ^ * corresponding to the "right", "left" and the scalar In general case the scale-invariant ( automodel) functions polarizations of the current of lepton pair: which correspond the transverse and longitudinal polariza• S ~ -g- « (Soo &tv) reaches values forbiden by the unitarity in the framework = scalar or longitu of the lowest approximation in weak interactions. F$m ~£F¡ '+ gF± dinal one represents the differential cross section of deep In conclusion we would like to stress that inelastic neutrino scattering in the form: 1. the automodelity principle provides a general method applicable to all the processes of deep inelastic lepton-hadron collisions. 2. the automodelity principle is not based on any The distribution in the invariant transfered momentum specific dynamic mechanism of deep inelastic processes, squared is determined by it supposes only a degeneration of physical quantities with respect to dimension characteristics of particles, such as masses, effective sizes, eta.The predictions of the automodelity principle are based on the general con• where the dimensionless function is a polinom cepts of symilarity and dimensionality and, consequently, of second order degree in respect to the variable £/¡ are model-independent. For example, for pure leptonic processes one gets: 3. deviations from predictions of the automodelity principle if exist would have a fundamental character and must point apparently to the existence of an " elemen• tary" length or " superhigh" mass, which will require Integrating the distribution Jtvfj^- over all the the cardinal reconsideration of our ideas on elemen• r momentum transfers in the region Oil £ / g S i tary particles. we find the total cross sections of neutrino processes: Received by Publishing Department on June 13, 1973. Thus, the cross section of a neutrino reaction must grow and at sufficiently large energies