CERN 914)1 10 January 1991
ORGANISATION EUROPÉENNE POUR LA RECHERCHE NUCLÉAIRE
CERN EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
1988 CERN SCHOOL OF PHYSICS
Lefkada, Greece
8 September - 1 October 1988
PROCEEDINGS
GENEVA 1991 \ Propriété littéraire et scientiOt]iie réservés pour Literary and scientific copyrights reserved in ail tous les pays du monde. Ce document ne peut countries of the world, This report, or any part être reproduit ou Iradin! en (oui (ju en partie of it. may- not be reprinted or translated without iaiu l'autorisation écrite du directeur gênerai du written permission or the copyright holder, the CERN, titulaire du droit d'auteur. Dans les cas Director-tîcneral or CERN, However, permis appropriés, cl s'il s'ap,it d'utiliser ttî document à sion will be freely granted for appropriate des fins non commerciales, ecllc autorisation non-Lommereial use, sera volontiers accordée. II' any patentable invention or registrable design Le CERN ne revendique pas la propriété des is described in the report, CERN makes no claim inventions brevetabJes et dentins on mode let to properly rights in il but offers it for the free susceptibles de dépôt qui pourraient être décrits use of research instilullons, manufacture]' and dans le présent document; ceux-ci peuvent être others. CERN, however, may oppose any librement utilisés par tes instituts de recherche, attempi by a user 10 daim any proprietary or les industriels et autres intéressés. Cependant, le patent rights m such inventions or designs as CERN se réserve le droit de s'opposer à toute may be described in (he present document. revendication qu'un usager pourrait faire de la propriété scientifique ou industrielle de toute invention et tout dessin ou modèle décrits dans le présent document,
ISSN 0531-4283 ISBN 92-9083-032-8 CE RN 91-01 10 January 1991
ORGANISATION EUROPÉENNE POUR LA RECHERCHE NUCLÉAIRE
CERN EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
1988 CERN SCHOOL OF PHYSICS
Lefkada, Greece
8 September - 1 October 1988
PROCEEDINGS
GENEVA 1991 CERN-Service d'information scientifique-RD/815-- 700-mars 1991 1000-novembre 1991 ABSTRACT PREFACE
The CERN School of Physics is intended to give young The 1988 CERN School of Physics was held from 18 September to 1 October on the experimental physicists an introduction to the theoretical aspects beautiful and unspoilt island of Lefkada off the west coast of Greece, It was attended by or recent advances in elementary particle physics. These 106 students; all but four came from laboratories or institutes in the CERN Member Proceedings contain reports of lecture series on the following States. topics: introduction to field theory and to weak interactions, heavy icn collisions, perturbative QCD, the standard model, proton- Our sincere thanks are due to the lecturers and discussion leaders for their active antiproton collider results and detectors, cosmology. participation m the School and for making the scientific programme so stimulating. The high attendance at the lectures in spite of the warm and sunny weather testifies to the excellence of their work.
The School was organized by CERN in conjunction with the Nuclear Research Center 'Demokritos', Athens, and we are indebted to the Center and particularly to the Scientific Director, Professor N. Antoniou, for financial support. Our warmest thanks are extended to Dr. E.N. Argyres ('Demokritos') who, as Director of the School, ensured the smooth running of all the essential practical details of the day-to-day organization. Our particuiar thanks go also to his wife, Antigone, for her untiring efforts, especially for the liaison with the photocopy shops, and for her ever-cheerful disposition. Dr. Argyres was ably assisted by his Greek colleagues on the Organizing Committee, Dr. G.K. Leontaris of the University of Ioánnina and Dr. N. Tracas of the Technical University of Athens who competently handled ail the financial matters. Lastly, the CERN Organizing Secretary, Miss S.M. Tracy, efficiently co-ordinated all the preparations for the School.
The School was head in the Xenia Hotel, Lefkada, and our thanks go to the Manager, Mr. E. Vagenas and his staff for making us welcome and comfortable. The evening meal was taken in the Restaurant 'Adriatica' in Lefkada, thus giving the participants the chance to walk through the charming old town before dinner, when the social life was most active. The owner, Mr. A. Argyris, spared no efforts to meet the healthy appetites and tastes of the multi-national student body, who greatly enjoyed his hospitality.
A varied social programme was organized by Dr. Argyres, with the aid of his friends and colleagues of his home town, especially the Mayor, Mr. E. Margelis. The School ended with a farewell banquet, attended by Professor Antoniou, Director of NCSR 'Demokritos', Professor H. Schopper, the Director-General of CERN, Mr. P. Valakis, the County authority, the Mayor of Lefkada, and the Right Reverend Nikiforos, Lord Bishop of Lefkada, amongst others; it was enlivened by a spirited folklore performance by the 'Orpheus' dancers of Lefkada.
W.O. Lock on behalf of the CERN Organizing Committee Ji !
".» . CONTENTS
Page
Preface, WO. Lock iv
Elementary field theory, G. Tiktopoulos 1
Phenomenology of electroweak interactions, G. Altarelii 51
Heavy ion collisions, W. Willis 101
QCD and collider physics, W.J. Stirling (Lecture notes by R.K. Ellis
and W.J. Stirling/ 135
Beyond the Standard Model, D.V. Nanopouhs 237
Collider experiments, P. Jenni 297
Particle physics and cosmology, G. Lazarides 353
Organizing Committee 379
List of Participants 380 ELEMENTARY FIELD THEORY or, equivalently, the time development of state vectors in the Schrôdinger picture according to
G. Tiktopoulos National Technical University Athens, Greece 1.3 The path integral The main dynamical problem for the quantum system is: given that at time t = ti, q = qi (i.e. the system is in an eigenstate of q with eigenvalue qi), what is the amplitude that at a later time t = t2 one will measure q to be equal to q ? We can write this amplitude as LECTURE ONE: FIELDS AND PARTICLES e In this lecture we introduce the method of path integrals and use it to calculate the Green's <*.Mvt.> „-
1.1 Action principle in classical mechanics Consider a simple system with one degree of freedom described by the Lagrangian (q = dq/dt) Ordinarily, it is obtained by solving the Schrôdinger equation as a differential equation. Here, we will express it as a. path integral, a remarkably fruitful way of writing amplitudes. From the completeness of the eigenstates |q,t) at any time t we deduce the "reproducing property" A classical trajectory q(t) makes the action A = J«Jt L(^4; stationary, i.e. Dividing the time interval (t,t') into a large number N of equal intervals (t,ti), (ti.tz),
(tN- i,t') we write
This is the differential equation of motion. In our case
For small time intervals we may use the approximation (see Appendix) 1.2 Canonical quantization To quantize the simple system of Section 1.1 (i.e. to construct a quantum system which in the classical limit ti—• 0 goes over to the system of Section 1.1) we introduce the canonical momentum p conjugate to the coordinate q
which allows us to write
and we express the quantity qp - L, the energy, as a function of q and p A -, i(A* •AH/H.,+ -+Avyt
We go over to the quantum system by postulating that q and p are operators obeying the (equal where time) commutation relation In-cl-^ k.K-i - L(^pV(^) et
and that the Hamiltonian H = p /2m + V(q) represents the energy operator. Recall that H determines the time development of operators in the Heisenberg picture according Since the sum £k Ak.k- I in the exponent would approximate the classical action if q, qi to q„- l, q' lay on a smooth trajectory we write this, symbolically, i'A(l>/k
1 2 3 4
For this problem an important quantity is
1 The path integral form for (q',t'|q,t) is valid even when the potential V(q) depends on time. Therefore, if ^o(q) is the ground-state wave function of the oscillator we have O^ N :ii»i€
One way to calculate is to expand the path integral in powers of J: as an integral over all "paths" i.e. classical trajectories from (q,t) to (q',t'). This is the famous path integral introduced by Feynman (1948).
1.4 The driven harmonic oscillator For the simple harmonic oscillator of frequency w and mass = 1 (from now on we set h = 1) we have:
L = - -^ l tl r1 > P-1 •je '** ^je " The Hamiltonian is "diagonalized" by introducing the ladder operators a and af according to The first term in this expansion equals one, since in the absence of J we have il* = il ~, whereas the second term vanishes because the integrand is odd in q(ti). Thus we have 3" *.- J T«o where we have introduced the "time-ordered product" of two time-dependent operators A(t) and B(t) to mean in terms of which V t») , ÏVJ-*- The first, second, etc. excited eigenstates of H are obtained by acting with a* on the ground state |îî) From the known Heisenberg-picture time; dependencdependence oof q(t)q(t):: once, twice, etc. Each time af acts, it raises the energy by a>. Let us now assume that during the time interval - Ti < t < Ti an external, time-dependent force - J(t) acts on the oscillator, the new Lagrangian being •vIZuT L = L6 + T(*H = i; -<\- i«V t Tit) i • we find The solution of this equation gives <£|T<]«o<|(toU> = 2.W The time-ordered product is the probability amplitude that a force 5(t - t2) (when ti > t2) will -.-tolt-t'/ raise the oscillator from its ground state to its first excited state and that later a force 6(t - ti) will bring it back down to the ground state. Armed with patience one may proceed to calculate the general term in the expansion of (0+ |fl ~ ) j in powers of J in much the same way. An alternative and more convenient way to calculate (0+ |0~)j is to "discretize" the path so iD(t - t') is identical to the time-ordered product (0]Tq(t)q(t' )|0). So finally we find a closed integral and do it as a multiple Gaussian integral: expression for (SÎ+ |Q ~ )j: + J Important point: As it stands the multiple integral is ill-defmed because the exponential is purely defines GN as the N-point causal Green's function for the system. oscillatory. We must make the replacement For the special case of the harmonic oscillator we have that G2n+ i vanishes and that G2„ is a sum of products of n D's. CO o-t'i Un'-fk t •= rta-6 ou*.J- hof/f*/* Diagrammatically: which makes the integral absolutely convergent, compute the integral and then let e go to zero. This is t, the origin of the it's in quantum mechanics! G'aa U,,U^ij > •• The integral can be done by a change of variables which diagonalizes the matrix Bk ,• The general formula is -H--* A, CAti»v> Accordingly, we get T T t, <^ lv >T - <^^r^0**ti\^nomrLmj t _ (i*+w -.-c')'pCt-t'j * «set-to 7 where AV is the lattice cell volume. The canonical commutators are 1.5 The free real scalar field We shall use the space-time coordinate notation: * ; V ~ (*«j *«•;*;>-.*•»-'*•) [ Now we take the limit 4V -> 0. Denoting by T(X) the limit of î/AV we obtain the continuum X *J= IA.J, expressions is specified by a 4 x 4 matrix A such that AAT = 1. A scalar field #(x) transforms under Lorentz transformations according to / / i-' Consider now a real-valued classical scalar field where OOj, ^ JUSm? does not involve products of fields at different space-time points. The equal-time commutation relations for The variables T(XJ,X0) canonically conjugate to the 0(XJ,XO)'S are xi*.-,*.)- ii—- - AV?(V*0 The Hamiltonian expressed in terms of the a^'s and a£'s reads "àd> CXJJ*.:* This clearly shows that the quantum field is equivalent to an infinite set of uncoupled harmonic The particles described by this theory are: 2 2 i) Spinless. This is obvious since one-particle states are completely specified by their momentum, oscillators—one for each wave vector k, its frequency being uk = Jk + m . However, we cannot leave H as it stands because the sum of the ground-state energies oik/2 of the ii) Bosons. From [at, at'] = Oit follows that |k',k) = |k,k')etc. oscillators diverges. So we drop it and redefine H as iii) Non-interacting. This is because the number of particles and their individual momenta are time-independent: if |i/-), = o = |ki, £2 k„) then |i£), = exp - i (Eu„)t • |k,, k2, ..., k„). It is an important fact that in relativistic local field theories the relation between spin and H = Jctk to, a. a_ statistics is a theorem. The fact that our spinless particles turned out to be bosons is just a special case k U It of the general theorem. Another important physical quantity is the momentum of the field P which is also simply expressed in terms of the a's and a + 's: 1.6 Interaction with an external source Let us now add to the quadratic Lagrangian density £0 a linear term \A k atat That this is indeed the momentum operator can be easily established by verifying the relations where J(x) is an external, given c-number "source" —actually, a force density acting on the oscillators of the field operator a^ destroys (or annihilates) a particle of momentum k. '.T 2 2 More explicitly, denoting by D(x, - x2) the kernel of (d - m + ie)~ ' we have A t fD(xrxv; = Table 1 C2r04L kW-Lt States of the real scalar field Oscillator Particle State symbol Energy Momentum _ e . _ e description description |vac) 0 5 All oscillators in Vacuum state their ground state (no particles) |k) = ai|vac) Wk k Oscillator "k" in State of one ("NOTATION : kx»L-x-k.x. 1st excited state. All particle of momentum others in ground state. k and mass m jCk^^Lai.a^lvac) Oscillators "ki" and State of two Uk, + Wk2 ki + k2 "kV in 1st excited particles of momenta state. All others in ki and £2 and ground state. mass m Note that the necessary ie is introduced according to the prescription: "Replace m2 by m2 - ic" 10 4 11 12 iD is the 2-point causal Green's function or the Feynman propagator for the field - L (ix, ci^TC*.)!) (x, -XOT (x-0 <•{** f ••>•< x just as we did for the harmonic oscillator. We have The N-point causal Green's functions of our theory are defined as the "coefficient" functions GN(XI, X2,..., XN) in the expansion of (ft + |ft ~ )j in powers of J: ÇiOS/O = ^(V*0 = This means that, for x2o > xio say, the Feynman propagator is the probability amplitude for the creation of a particle at xi.xio by the localized "source" 5M)(x - Xi), and its propagation to and (, destruction at x2,x20 by the localized "source" 6 "(x - x2). It is instructive to look at the asymptotic phase of iD(xi - x2) for large time-like separations + xi - x2, i.e. for From our closed expression for (ft |Q ) j we conclude that G2n+ 1 = 0 and that G2n is a sum over products of n D's: lx„-*,„| > I V*J »*~ • Ga0cox,; = iT> U.-X^ From the integral expression f , ., , c[lt.(vx.J-VM.Vt \X>.-Xz.)l The Green's functions are represented by diagrams in which the Feynman propagator iD(xi - x ) is a 2 we find that the asymptotic phase comes from the value of k that makes the exponent stationary. line joining point xi to point x2: at L 2 2 givesfc" = p where p/Jm + p = (xi - x2)/|xio - x20| is the velocity of a free classical particle moving from xi,Xio to x2,x2o. Therefore *3 is the correct de Broglie phase. For large space-like separations X! - x2 there is no value of k making the phase stationary, i.e. there is no classical trajectory joining x to x . As a result D(xi - x ) becomes exponentially small, of *, - l 2 2 2 order exp [ - m„'(xi - x2) ]. 1.8 Appendix LECTURE TWO: SELF-INTERACTING SCALAR FIELD PROOF that To describe interacting particles we must introduce in the Lagrangian density terms that are higher than quadratic in the field variables namely, cubic, quartic, and so on. In general, it is not possible to get a closed expression for the Green's functions of an interacting field theory. Instead one looks for approximate results. A systematic approach is the perturbative •stwe. one, i.e. to regard the interaction (= non-quadratic) part of the Lagrangian as a small perturbation and to try to expand the Green's functions in powers of this small part. where The perturbative approach has succeeded spectacularly for Quantum Electrodynamics where it yields an expansion of experimentally measurable quantities in powers of the fine structure constant H = -^^+V( , a = 1/137. V It must be stressed that consistent perturbation expansions are possible only for a limited class of field theories which are called renormalizable (see below). In this lecture we discuss the dynamics of the self-interacting real scalar field 4>(x) based on the Lagrangian density The equality really holds when both sides are integrated over a smooth function ^(q). The left side then gives 0^ = < +• o£TH1. L = \*ik' P\ V *- 'Jc: A *Cr - % and the right side gives For this theory we shall obtain a set of rules called Feynman Rules which enable us to write the terms of the perturbation expansion of the Green's functions as multiple integrals. We shall indicate the relation between Green's functions and T-matrix elements (i.e. scattering amplitudes and decay amplitudes). 2.1 Perturbative expansion of (1)+ j() )j We begin with the path integral expression: [<&[»*?++:!>] e «-> (,|-iiv(q))(-i+<:Mi_,)N'(±) is the ground-state wave function as a function of the field configurations . j (a*0 (4401>»1 £(*•)«- ^.Ou 13 15 ib [Here 6/êJ(x) is a functional derivative. Recall that the functional derivative of a fucntional î(J(x)) i 3" T r t N ,i i f defined in terms of its change 5Ï when J(x) is varied infinitesimally by 5J(x): „ I = c^,»t. \ jcJx^cf^JxjdJx, • X c oT . (A *L we» "1 T T x J J > <5T(*> .T(i'1)TCx1.)3CK3JJC'<,/)D(A1-xjD(.xt-x;o(. 3- ';C>(xy-x) K> X —£V -* X.-T-A o, -I1"! , We can now expand the first exponential to get (Q + |fl~)j as power series of the "small" parameter X. If we also expand the second exponential in powers of J and carry out the functional derivatives we get, to each order in X, a finite number of terms. The terms are products of propagators D(x - x') and currents J(x") integrated over all space-time points x, x', x", ..., which The integrals associated with the Feynman diagrams are called Feynman integrals. occur in their arguments. It is mnemonically convenient to represent the integrand of each such term by a Feynman diagram. 2.2 Disconnected diagrams The diagram for a term of order Xv with n factors of J consists of Feynman diagrams corresponding to disconnected diagrams, such as that of example (d) above, v internal vertices Xi, X2, ..., x» are essentially equal to the product of the Feynman integrals of their connected parts. In fact, it is not n external vertices x i, x» , •••, x + „ v + + 2 v difficult to show that and a number of lines joining 2 vertices so that exactly 4 lines join at each internal vertex and 1 line ends at each external vertex. The integrand corresponding to a given diagram is a product of factors i) iD(xj - xt) for each line joining Xj to Xk ii) J(x() for each external vertex x,. There is also an overall multiplicative constant proportional to X" which we will not specify here. where Thus CJTKKtcM {<& arm*. S J 2.3 Vacuum diagrams Examples Connected diagrams with no external vertices (no factors of J) such as ta> Q _^ T^= cjr. \ÇA T>(o;T>L») correspond to contributions proportional to J d"x, which is, of course, infinite. To see what this means let us replace the infinite space by a large 3-dimensional box^of volume V Also let us assume that the interaction acts only for a large (but finite) time T. Then j d x = TV. \J_ _^ I = <*.it X \c£x ix.dtx, T(x,; X(*o • Consider now the vacuum-to-vacuum amplitude for J = 0 (i.e. "in the absence of the external 0»; T source"). We have Mil) cTVE where we have set W(0) = TVe since it is the sum of connected "vacuum" diagrams. Clearly, V-e is the energy shift of the vacuum state due to the interaction so that G4 (\%>-j*tj*l) '• ~ dx t>(*rx ) D Ivo D (?*--* .)£> (*,-*__) Xv Xj This is then why in the limit TV -» x 2.4 Causal Green's functions ~ x IdxdV TXxrx; D (**-*;> D (_x, -x; . The causal Green's functions GN(xi, x2 xN) for the interacting theory are defined just as for < the non-interacting theory in terms of the expansion of the vacuum-to-vacuum amplitude x*CK *l , The expansion of W(J) in powers of J: .•D\X-X ;D(X,-*')'D(*I--0 The physical meaning of the amplitudes associated with these Feynman diagrams is suggested by their topology if we recall the meaning and properties of the propagator D. defines GN(X,, ..., xN), the (N-point) connected causal Green's function. It is given by the sum over all connected diagrams with N external vertices (see section 2.2), from which we have removed the N Consider, for example, the following diagram factors of J together with the corresponding integrations. Examples X, A, 0 X * x which represents a contribution to the 6-point function G6(xi,x ,x ,x4,X5,X6). Let us assume that the _Q_Q_ 2 3 points xi,X2 are in the remote past (ti,t2 -» -«) and the points X3,X4,x5,X6 in the remote future (U,U,U,U, -» + »). Then the amplitude represents the creation of two particles at Xi and x2 and their propagation as free particles to yi and y2, respectively, the emergence of four particles at y3,y4,y6 and their propagation as free particles to x3,X4,x5,x« where they are annihilated. Thus the external "legs" correspond to propagation from t= - °o to t = +°° while the rest of the diagram gives the collision amplitude. ^ 1 Although we shall not use Heisenberg field operators in what follows, we briefly mention here - x ^\d*' T> (>,-*; t>V*o D c*'-*o their close relation to GN(XI, ..., XN) just to make the connection with the standard field operator approach. Just as we did for the simple harmonic oscillator and the non-interacting field theory, if we expand the integrand of the path integral for (0 + |Q "" ), in powers of J we obtain the formula 17 18 19 20 -ii-rn. . H -P C i x xl»^< 4.,.., *•,.-»+~ where T<£(X[) ... 0(XN) is the time-ordered product of the Heisenberg field operators 4>(xi), ..., <£(xN) defined by ( kx = k-x -{VnN-S -X. ~) Note the use of complex conjugates of plane waves for final particles. If we now introduce, for the external-leg propagators, their F.T. according to We derived the formula for G2(xi,X2) in Section 1.7 for the non-interacting field theory for which, of course, |0 + ) = |Q"). f } i iH*-x'Jr _; we find that our amplitude equals 2.5 Collision amplitudes The causal Green's functions GN(XI,X2, ..., xN) contain all the information needed to calculate amplitudes for collision experiments. To extract it let us first make explicit the N external-leg propagators in GN by writing where GN is the F.T. of G™ according to 0H (*, A,-^ ) = S&;...4< '1)(v<)-'D(vO 0?U,*l,~< ) • ~TR ( f v TR i 21 p, x • This defines the truncated causal N-point function G™(xi,X2, ..., XN). It is obviously given by the same diagrams as GN but without including the external-leg propagators. To be concrete, suppose we want to calculate the amplitude for a collision process in which two Note that k1.k2.k3, ..., kN are the physical 4-momenta of the initial and final particles, i.e. particles with momenta ki and E2 collide to produce N-2 particles with momenta ic3,k4 icN in 2 2 the final state. kj0 = + Jm + k , j = 1, 2, ..., N. Clearly, we must evaluate GN(XI,X2 XN) for values of Xi,X2 XN appropriate to the experiment which begins with the "initial" particles 1 and 2 being far apart in the remote past: translations, i.e. T6 Tt X4.jX,.-»-«> , |x,-5ct |-> «> and ends with the final particles 3, 4, ..., N being far apart in the remote future As a result of this, G™ has an energy-momentum conservation delta function as a factor. It is convenient to display this by setting G (A^A^-A,)- C^KV^AV-A^OVAITIW Since initially and finally our particles are far apart from each other, they are out of each other's range of interaction, so they move as free particles. This free motion is described by the propagators associated by the external legs of the Feynman diagrams. This defines T as the (Lorentz invariant, reduced) transition matrix or T-matrix. As it stands GN(XI,X2, ..., XN) is an amplitude appropriate for initial particles created at xi,X2 and final particles destroyed at X3,x4, ..., XN. To get the collision amplitude for particles of definite momenta k*i, k~2, ..., EN we must extract the appropriate Fourier component: 2.6 Cross-sections where By squaring the modulus of the collision amplitude and multiplying by the phase space differential d3k ... dJk we get the differential collision probability 3 N 1Soj= —— ;s 4u* F.T. A nx*; -t-Mo-t-t \ I = the number of internal lines of g. 0'< \^f.2ui. F, = iX • 4!(2T)4S (sum of 4-momenta flowing out of vertex a) is a factor for vertex a. S = "symmetry" factor = number of permutations of internal lines which leave the diagram topological^ unchanged while keeping the external lines fixed. Note that after doing all the integrations that can be done with the help of the vertex delta functions in Fi,F one is left with one energy-momentum conservation delta function as a factor Had we used, instead of the infinite space-time, a large space-time box with periodic boundary 2 conditions as an intermediate device, we would have gotten, instead of (2T)45(0), the 4-dimensional for each connected part of the diagram. Thus, in general, we may factor out an overall 4-momentum volume of the box. Therefore dividing by this we obtain the differential collision probability per unit conservation delta function by writing time and unit volume: .1 where l^ducc,i has no delta function factors for connected graphs g. » Obviously, the sum Dividing this by the "relative velocity" |vi - v | of the initial particles we get the differential 2 3 cross-section da. Since |v, - v | = J(pip )2 - m4/u - u , we have 2 2 kl k2 i over all diagrams g, evaluated for "physical" 4-momenta pi,p2, ..., PN (i.e. for Pjo = ± Jm + pj, j = 1, 2, ..., N) gives T-matrix elements for various collision processes involving N (initial plus final) particles. Examples _ R Consider first some low-order diagrams and their contributions to G2 (Pi,P2 = -pi) a quantity whose importance will be seen in the next lecture. 2.7 Feynman Rules We have seen that the perturbation expansion of various Green's functions of the theory, i.e. their expansion in powers of X, may be expressed as a sum over Feynman integrals. To be concrete, consider the expansion for the truncated functions 1 T. = 4.'X (2irA k'W-ilt To calculate the v-th order term XTS1' one draws all Feynman diagrams with N external lines and v vertices such that exactly 4 lines are joined at each vertex. <*- ,.,. i [' «A, aV To each topological^ distinct diagram g there corresponds a Feynman integral lg according to rules which follow from our path-integral discussion in Section 2.1 and are given below. P \ ^ t* JC2tr)Mc (Zr)V Rules for GJ5R - Assign to the N external lines the 4-momenta pi,P2, ..., PN as outgoing. - Assign to each internal line j a 4-momentum kj as "flowing" in some arbitrarily chosen direction (i.e. if the line joins vertex vi to vertex v2 one chooses to regard k, as flowing from v, to v2 or vice versa). \r^-ii t\+vi-u (p-t,-io+->vi-«-'t Then R Next consider GÎ (pi,p2,P3,P4 = -pi-p2-P3>. As we have seen it is related to the process of two-particle scattering, e.g. with - pi, - p as incoming momenta and pa, P4 as outgoing momenta. o S }(i«)\ <2*)\ (infi 2 ll 22 24 23 rU LECTURE THREE: RENORMALIZATION «<*. X, 4'.-> 3.1 Divergences in the perturbation expansion Note that "three" diagrams, i.e. diagrams with no loops have no divergences because V = T-t- 1 - L. r«*. rcJ. ho. r^ vu>.»-£ K*. ~% -€*-*J.t I. = 4!^ v«rri CA, tTiTT -6 *v«. 1 f'+wNl'fc Thus for L = 0, the number of integrations is equal to the available delta functions. Examples: vtJl. dk, ^v z \K (Zrr; X /^ s\ /K 911 t^W-it fri+h-^,)-*- -^ Next we look at one-loop diagrams: There are obvious problems with some of these examples: i) The integrals are divergent. The divergence stems from the fact that the integrands do not vanish fast enough for large values of the integration momenta, ii) In our fourth example Ifd becomes infinite for pi = -m2, i.e. when P2 is a physical 4-momentum. But it is for physical external 4-momenta pi,P2,P3,p< that we need to evaluate GjR in order to calculate T-matrix elements and cross-sections. In the next lecture we shall see how one deals with these problems by the renormalization procedure. The result of renormalization will be a perturbation expansion of the Green's functions which is finite order by order in X. Theories, such as the \ Q. I df *=- _ __^ c^utdrtj:. dJv«v<^-ctJ- X2x ~ f^ H - V"'•Him' . <&Mvt*ryeht f, v ><. ,Pl ~ idt-lc. J— _1_ J_ * orvu-cv^-t^f J v- ^ k*- 2 i r ^ L. i i M ^2. —^ C*viv/«v~J-«.t<_f". But, for example, 1 XzC^?oV = ^^ Z- - c«nw It is easy to see that /or a/y L, 1 PI diagrams with 2 e*f. fines are overall quadratically divergent and diagrams with 4 ext. lines are overall logarithmically divergent. because Thus: 1PI diagrams with 2 and 4 ext. lines are the "primitively divergent" diagrams of X04 theory. The fact that there are only a finite number of primitively divergent types of diagrams is l connected with the fact that the coupling parameter X is dimensionless. That is also why the overall .t-,,..«1 degree of divergence of a diagram does not change when more vertices are added (keeping the number of external lines fixed). Suppose we added a cubic term in the Lagrangian density such as Recapitulation of types of Green's functions: G (fi>— PO •" «//diagrams a «" h We would then also have vertices joining only 3 lines c (S fPu— t>0 » connected diagrams \ TR ~,-) <»•-• —*• It is straightforward to verify that such a vertex would not increase the divergences of the theory. G„ CRJ-JP-) ~ ^ (fi)"- ^ ^V^H OVjPO *• connected amputated diagrams That is connected with the fact that g has the dimensions of [mass]1. If, however, we had a vertex such as gi^6, say, where We take one more step toward simplicity. We define the one-particle irreducible Green's function: -Z. <&»«%= f>***l connected, the situation would be vastly worse. amputated, one-particle irreducible There would be primitively divergent diagrams with any number of ext. legs! It turns out that in 4 this case the divergences could not be controlled and the theory is not meaningful. with the (2T) «(EP) factored out. Going back to the \ «.3. <^)— is Q. •= \ oo = I'É ; n TE, (JO « o ) IÛL * " AC- 26 27 28 3.2 Mass and wave function renormalization o- Q ,'> i-t itr We saw that the interaction changes the "free" propagator iût* (^ ^JUII J,inofej«^«*4- What are the TN'S good for? Let us look at G2 : The full propagatoor no longer has a polepo/e at p2 = - m2 bbuu t at some other value of p2: J2_ + 0 0 + Q 0 0. Thus 1 ^•Cv-*t:> -^«IÇ'VM l*.-x»„i 1 rt rx AiVMfT. + —©— 1- -^I>^@- +- so the physical mass of our particles is not m but M! Also called self-energy part * 0 • * So: where •H . jj_i'(w;f' * /.I- <- ' P-* so that ^* AVMPr J4W\ tS-Al'1- i.e. we only need to calculate 1PI graphs, to get E(p) and then G2. wrong normalization! We must define renormalized Green's functions: \ (A.? C. (A3 ^ A <*• s A -•> «> A->oo So up to terms of order X we have l{ t*^- *3 A ^<"X( 1 ( ii » How do we compute E(p)? What is m? J-J. A. t- ^3- n _ ^ I(h; = _Q_ + -^ -1 but ^ ?-[I-^.(AJI] ^ ^=yU\rn(A;+Co(A) 1 3.3 Coupling constant renormalization l*ji k' How should we choose X? Suppose that we know that the world is described by a \4-type field theory but we do not know Introduce a momentum cut-off parameter A, e.g. EN X. We perform scattering experiments and we measure the value of rf (pi,p2,P3,P4) for specific values of pj = py. kh-vua ^Vu1- fc*<-Av 1- — — — *\ •4 kiUl»)!-1!] Then: We have -Q = T>^ ^( ,)= x + )o(* 1 f P Z ^ 2. -j 2- 2 4 3 Ç\ = Tt(Aji0 = c0K>+- C,(A^^\^J+- P(F>J) where = X-H T(s) +Ht; +X0») 29 30 31 32 where To see roughly what is going on we look at a non-primitively-divergent amplitude. Consider, for example, the contributions to F"" coming from the class of diagrams of the type - drawn below. s = CjT.-t-f^P*" J "t -= (fx+fi/ j *- = (P.^PO* Write V2> We have c Gi = ZDF («.** ^F #** ' ^ This specifies the "vertex renormalization constant" Zx to order X. -1 „v*d Finally, so that Thus up to order \? we set 1 1 \z~".X z = v ^ a « % * o (^j and KM. (because Z = 1 + 0(X)). r = fWt« So we obtain & finite expression for IT" (expanded in powers of Xr). In general: 1 i v ! \* ^X^A = f'«;t« finite LECTURE FOUR: INTERACTING FIELDS FOR SPIN 0, 1/2 AND 1 PARTICLES Thus we get fwo Ar;'n Action: but we also can consider complex J's: *• .a Obviously, J,J* would excite ("create") particle states such as The action is invariant under /*,->-£(>>-'"^,9 Classical eqs.: We may call the - states "antiparticles" and the + states "particles". A state of N + particles and N _ antiparticles carries a "charge" of G? = N+-^_"| The quantum theory describes particles of mass m and spin 0. c^ Dirac field \ji Propagator Action: v " " 6; Complex scalar field can be described in terms of two real fields 4>i,02: i A = /<*: c-1 (39,/_ ^.yj <- /& (- * ^ - sto 33 34 35 36 Under Lorentz transformations x, -» A,„J<„ i/<(x) transforms like a "spinor" Important: To use the path integral method to discuss the quantum Dirac field \ji we must integrate over \p and ii * not as ordinary numbers but as anticommuting numbers!! -/ Thus Note M(A) = e ' '""'" for infinitesimaill €,,t,.,, anand A,A^, =— &„6^r + + 6e^, and we get different results for Gaussian integrals: Describes particles of spin 1/2. Propagator ÎWW -e. ;â (*-*)= îliàV -\2TOV iX+tn-i'e .(dU-'Sj-e. Particle states: We get two kinds of particles: satisfies "Particles" |k,s, + ) etc. "Antiparticles" |k,s,-) etc. Plane-wave solutions of (7-5 + m)^ = 0: with Fermi-Dirac statistics, e.g.: r ,p* ,- l,w;lv*.*>--|W3^o+> • Note: where , \ Kjj-j^.^;». _ <# Vector field V„ Subsidiary condition d„V» = 0 => 3 spin degrees of freedom. We have T~ /]• > . ,-•, r Action and T ,. ,... .T,_. —>+«, Classical eqs. Under Lorentz transformations A more interesting symmetry if Xi = X2 = X, i.e. x —» A^u xu The real vector field describes particles of maxs m and spin I. Then we have invariance under Propagator <+> —1& Co>d <^ —•} SiyCo4> +• Cojfl «^ It satisfies: Another way to display the symmetry is to introduce the complex field *. .«• 4.2 Interacting Field Theories We obtain Interacting Field Theories when we add to the quadratic actions we have been describing a number of non-quadratic terms. There is a large variety of such non-quadratic terms that can be considered. They are limited however by certain requirements in order to be interesting as models, i) Invariance under Lorentz transformations (and space-time translations), which is invariant under ii) Incorporation of prescribed internal symmetries. iii) Renormalizability. [The ultimate purpose is, of course, to guess and study the field theory that describes accurately 4> —* «- $ all observable phenomena!) Examples q> —£ -e, Form drfpix) = 0 it follows that the "charge" 1 3 Obviously relativistically invariant. is time-independent (a conservation law). All interaction terms are of dimension (mass)4 -> renormalizable. For the free theory Q is simply proportional to the number of particles minus the number of Internal symmetries? antiparticles. 37 38 39 40 It is easy to see that Q defined this way is still conserved in any process of the interacting theory because the basic interaction vertex is (for truncated Green's functions) Draw all (topological^) distinct diagrams with vertices corresponding to the interaction \ (.<*>*<*> f (non-quadratic) terms in the action. X Internal line factors. where * • » y corresponds to either ^particle going from x to y or an antiparticle going from y to x. « k spin 0 -Is 3) Next let us consider non-quadratic terms involving Dirac fields. kW- Lorentz covariance requires us to use the bilinear combinations: k I spin 1/2 \ • -.'€ '«(» ? ' spin 1 h Vertex factors: * We may try to use only Dirac fields: 3- ' 6 &«r these have dimension (mass) and are non-renormalizable ("four-fermi interactions"). s J We can combine a bilinear with a scalar or vector field, e.g. 'wn Writing full Lagrangians: For each diagram: Assign the external momenta pi,..., p„ to the n external lines and one integration momentum kj to each one of the internal lines. Multiply all internal line and vertex factors and integrate over all kj with measure Y "YHUW* cU Multiply by 1/S where S ("symmetry" factor) is the number of permutations of internal lines and vertices leaving the diagram unchanged when the extenal lines are held fixed. Multiply by a factor of ( - 1) for each fermion loop. Include an extra sign equal to the signature of the permutation of external lines. Convergence of integrals at large momenta (P, > m) Spin 0 propagator: 1 71 LECTURE FIVE: GAUGE FIELD THEORIES Spin 1/2 propagator: . Î 5.1 Vector particles and gauge fields Consider a model theory in which a massive vector field V, couples to a spinor ^: Spin 1 propagator: {». fo/lM* oc°=-i.F F - M\ V Note that the spin 1 propagator is peculiar because at large momenta the mass is still there! So the dimensional argument for convergence fails. -ip (y-3 ^J^ + 'j^y^^u Questions for vector particles: How can one control the bad large momentum behaviour of the vector propagator? Consider the two diagrams below for the 4-fermion Green's function What happens when m = 0? k —# 's The integrands are common except for the 3 fermion propagators on the right and the 2 vertices between them. Denoting their contribution by G,„ we have •if- + -L— 'ï. ( -k cy.f>+m liv. (jj+k'j+vu o'y. (p-rfc+k'j+k. ,yv cV^ft^ iv.(j»ft.)-t-in l'y. (^tt-k-bO-t-K* 4- l"Y- -T-l1 + u« : y. (p+ k'; +K. t'y. (^p+X^tO-t-^ 41 42 43 44 Owing to this cancellation It is worth mentioning the analogous result for a model in which V„ is coupled to a complex scalar field 4>: OW; k.^ (fW) =. o i z. p+iu =o Thus the kek,/^2 term of the vector propagator carrying the momentum k does not contribute to physical amplitudes and may be dropped. This result is completely general: the k,k,/V2 piece can be dropped from all vector propagators. nruj I ! I Î <* -"H<^ " PS» ~ (V'^> • (V^* ~ " &» =» The model is renormalizable. MINIMAL COUPLING: The above identity is simply an expression of the fact that in this theory the massive vector field V„ is coupled to a locally conserved current J* 7 J* â f r What about n -» 0? Since k„k»/^2 does not contribute to the physics we may just drop it. Then take n = 0 in the [Thus it would not work if we had, for instance, two fields tpui'i of different masses mi,m2 and, for example, propagators: The conservation of this current is related (by Noether's theorem) to the invariance of £ under U(l) transformation — -'"•7T Ap -J •«- T Alternatively we try to make sense of the Lagrangian with ji = 0, So: Renormalizability imposes symmetry! (The converse is not true.) by introducing it in the path integral! It is a multiple integral in which the integrand does not depend on a subset of the variables of integration, e.g. But where Vol. is the multidimensional volume of the y's integration domain. In the case of the functional integral over A„ the y's are all the longitudinal A's and of course and the inverse of Vol. = oo. A way to fix this is to introduce a multidimensional 5(y) or more generally take as the value of the integral - To understand what happened note that /'n the absence of the vector mass term ti2V„V„ we have A surface y = f(x) can be written more generally as F(x,y) = 0 so that invariance under local U(l) or gauge transformations which is independent of the choice of F. (See Coleman's Erice Lecture "Secret Symmetry".) The above is the idea behing the Fadeev-Popov Ansatz: .? Note that then •> [&*A3 fy-Ww^^fa-MW Take, for instance, F(A;>= \A/*~$ F -^> F Of course, gauge invariance alone does not determine the form of £. For example, we could add terms such as Then V^^> f>„ 4F<^-'<*&-*&->* but they would spoil renormalizability. Now we can understand the problem with the functional integral. We insist on integrating over JUi" ("à J = c^ui-l-cvJ- all V„(x)'s, whereas for V„'s related by a gauge transformation the integrand exp i A(V„) is the same! 45 46 48 Example SU(2j ;[5-£&etfJ ^r - S**l -e f " ' w *•'*'>- This amounts to .^ -.«* Now the differential operator of the quadratic piece does have an inverse Afore- 1) Gauge invariance does not allow a mass term /i2A,A, for the vectors. 2) In contrast with the Abelian case the vector fields couple directly with each other and the theory is not a free (non-interacting) theory even in the absence of matter fields. 3) Because of the coupling of the vectors to conserved currents the vector propagators (after gauge -* Propagator: fixing*') are well-behaved at large momenta. 4) Quantum Chromodynamics is an example of an (unbroken) locally symmetric theory based on the groupe SU(3) (of colour). 5) The only way to construct a renormalizable theory of massive vector mesons in which they couple directly with each other is by a spontaneous breakdown of gauge symmetry of a non-Abelian k*-;6 gauge theory. The prime example is the standard SU(2) x U(l) model of the electroweak interactions. For p = 1 we have the Feynman gauge, for p = 0 we have the Landau gauge, etc. Note that for any value of p the large momentum behaviour of the propagator is O.K. renormalizable model! 5.2 Non-Abelian gauge symmetry Now, do the same as above (try to get rid of the k,k, pieces) for a more general system involving scalars, spinors, and several massive vector fields. Of course, one obvious case is when each one of the vector fields is coupled to its own Abelian U(l) current. But one finds that, in general, even non-Abelian types of symmetry work. This type of theory (the massless version) was discovered by Yang and Mills in 1954 on the basis •) For a derivation of the Feynman Rules from the Path Integral Formalism see, for example, of the analogy with Quantum Electrodynamics. Coleman's notes. An example of how the need to control large momentum behaviour (or equivalently the \i -» 0 limit) leads to gauge symmetry in the non-Abelian case. b a + S d c *- I ^akf ^cdi + 3*< n"h 34 wu u»-eXWc Unless gjbc all zero, Ihey must be the structure constants of a non-Abelian group. 'V> 51 52 PHENOMENOLOGY OF ELECTROWEAK INTERACTIONS corresponding to m« = 81 GeV. This very large value for the W (or the Z) mass makes a drastic- difference, compared with the massless photon and the infinite range of the QED force. The G. Altarelli experimental limits on the photon mass [3] are listed in the following. From a laboratory experiment, CERN, Geneva, Switzerland H one obtains mY < 10" eV by a method based on the vanishing of the electric field inside a cavity with conducting walls, predicted by Gauss' law. In fact, the exact r"2 behaviour of the electric field corresponds to my = 0. From the observed distribution of planetary magnetic fields (the field should 1. INTRODUCTION be damped by an extra factor e "^ if mY ^ 0) the Pioneer probe to Jupiter obtained mT < 6 x These lectures on electroweak interactions start with a summary of the Glashow-Weinberg- 10" 16 eV. Finally, indirect evidence from galactic magnetic fields indicates that m,<3x 10"27 eV. Salam theory [1] and then cover more advanced subjects of present interest in phenomenology: the Thus, on the one hand, there is very good evidence that the photon is massless. On the other hand, Higgs sector and the open problem of the experimental investigation on the origin of the Fermi scale the weak bosons are very heavy. A unified theory of electroweak interactions has to face this striking ,/2 of mass GF ; the experimental tests of the theory; LEP physics; and finally, the physics of flavour, difference. flavour mixing, and CP violation. Another apparent obstacle in the way of electroweak unification is the chiral structure of weak The modern electroweak theory inherits the phenomenological processes of the (V-A) ® interactions: in the massless limit for fermions, only left-handed quarks and leptons (and right- (V-A) four-fermion low-energy description of weak interactions [2], and provides a well-defined handed antiquarks and antileptons) are coupled to W's. This clearly implies parity and charge- and consistent theoretical framework including weak interactions and quantum electrodynamics in a conjugation violation in weak interactions. unified picture. The universality of weak interactions and the algebraic properties of the electromagnetic and As an introduction, in the following we recall some salient physical features of the weak weak currents [the conservation of vector currents (CVC), the partial conservation of axial currents interactions. The weak interactions derive their name from their intensity. At low energy the strength (PCAC), the algebra of currents, etc.] have been crucial in pointing to a symmetric role of of the effective four-fermion interaction of charged currents is determined by the Fermi coupling electromagnetism and weak interactions at a more fundamental level. The old Cabibbo universality constant GF. For example, the effective interaction for muon decay is given by for the weak charged current [4]: a eak £«ff = (GF/\/2) [?»7„(1 -yi)n][ey (\ -75)"el , (1.1) J" = ivYa(l -ys)li + ÏV7a( 1 ~7s)e + cos0c Û7o(l - 7s)d + sinOc Û7<,(1 ~75)s + ... , (1.6) with [3] suitably extended, is naturally implied by the standard electroweak theory. In this theory the weak gauge bosons couple to all particles with couplings that are proportional to their weak charges, in the 5 2 GF = 1.16637(2) X 10" GeV" . (1.2) same way as the photon couples to all particles in proportion to their electric charges (d' = cos 0c d + sin Be s is the weak-isospin partner of u in a doublet). -2 In natural units ft = c = 1, GF has dimensions of (mass) . As a result, the intensity of weak Another crucial feature is that the charged weak interactions are the only known interactions 2 interactions at low energy is characterized by GFE , where E is the energy scale for a given process that can change flavour: charged leptons into neutrinos or up-type quarks into down-type quarks. On (E = m,, for muon decay). Since the contrary, there are no flavour-changing neutral currents at tree level. This is a remarkable property of the weak neutral current, which is explained by the introduction of the GIM mechanism 2 2 2 5 2 GFE = GFm (E/mp) = 10" (E/mp) , (1.3) [5] and has led to the successful prediction of charm. The natural suppression of flavour-changing neutral currents, the separate conservation of e, n, where mp is the proton mass, the weak interactions are indeed weak at low energies (of order mp). and T leptonic flavours, the mechanism of CP violation [6] through the phase in the quark-mixing The quadratic increase with energy cannot continue for ever, because it would lead to a violation of matrix, are all crucial features of the Standard Model. Many examples of new physics tend to break unitarity. In fact, at large energies the propagator effects can no longer be neglected, and the the selection rules of the standard theory. Thus the experimental study of rare flavour-changing current-current interaction is resolved into current-W gauge boson vertices connected by a W transitions is an important window on possible new physics. propagator. The strength of the weak interactions at high energies is then measured by gw, the In the following sections we shall see how these properties of weak interactions fit into the W-,z-><, coupling, or, even better, by av = gw/4x analogous to the fine-structure constant a of standard electroweak theory. QED. In the standard electroweak theory, we have 2 aw = V2Grml//ir = a/sin 0W = 1/30. (1.4) 2. GAUGE THEORIES In this section we summarize the definition and the structure of a gauge Yang-Mills theory [7, That is, at high energies the weak interactions are no longer so weak. 8]. We will list here the general rules for constructing such a theory. Then in the next section these The range rw of weak interactions is very short: it is only with the experimental discovery of the results will be applied to the electroweak theory. W and Z gauge bosons that it could be demonstrated that r. is non-vanishing. Now we know that Consider a Lagrangian density £[ A A A B For 9* infinitesimal, U^) = 1 + ig EA S*!*, where T are the generators of the group r of F „ = dX - d,V - gC*BcV V? . (2.11) transformations (2.1) in the (in general reducible) representation of the fields B AB tr(t\ ) = V26 . (2.2) The complete Yang-Mills Lagrangian, which is invariant under gauge transformations, can be written in the form The generators satisfy the commutation relations £VM = - \ SF^F"*" + £[0, D. A general no longer invariant under the gauge transformations U[9 (x,)], because of the derivative In this case, the ¥f, tensor is linear in the gauge field V,, so that in the absence of matter fields the terms. Gauge invariance is recovered if the ordinary derivative is replaced by the covariant derivative: theory is free. On the other hand, in the non-Abelian case the FA„ tensor contains both linear and quadratic terms in VA, so that the theory is non-trivial even in the absence of matter fields. D, = dr + igV„ , (2.5) where VA are a set of D gauge fields (in one-to-one correspondence with the group generators) with 3. THE STANDARD MODEL OF THE ELECTROWEAK INTERACTIONS the transformation law In this section, we summarize the structure of the standard electroweak Lagrangian and specify the couplings of W* and Z, the intermediate vector bosons (IVBs). v; = uv„in ' - (i/ig)o,u)U- '. (2.6) For this discussion we split the Lagrangian into two parts by separating the Higgs boson couplings: For constant 6*, V reduces to a tensor of the adjoint (or regular) representation of the group: (3.1) v; = uv„u- ' = v, + ig[ We start by specifying £symm, which involves only gauge bosons and fermions: which implies that 3 A A £symm = - V4 S F ,F " - V4B„,B"" + &.iVD,tfL + ^RiVD^R . (3.2) V;c = Vf - gCABC^V8 , (2.8) A= 1 This is the Yang-Mills Lagrangian for the gauge group SU(2) (g) U(l) with fermion matter fields. where repeated indices are summed up. Here As a consequence of Eqs. (2.5) and (2.6), Df A A A B B„, = dfK - 3,B„ and F „ = 3„W - 3»W - g CABC W W^ (3.3) (D„0)' = U(D„*). (2.9) are the gauge antisymmetric tensors constructed out of the gauge field B, associated with U(l), and Thus £[4>, D,0J is indeed invariant under gauge transformations. In order to construct a gauge- A W corresponding to the three SU(2) generators; 6ABC are the group structure constants (see A invariant kinetic energy term for the gauge fields V , we consider Eqs. (3.6)] which, for SU(2), coincide with the totally antisymmetric Levi-Civita tensor (recall the familiar angular momentum commutators). The normalization of the SU(2) gauge coupling g is [D„ D„] 54 55 56 2 1 2 The fermion fields are described through their left-hand and right-hand components: where t* = t' ± it and W* = (W ± iW )/V2, we obtain the vertex Vw» = 0y, [(t VV2) (1 - )/2 + (t VV2) (1 + 7s)/2]0W,- + h.c . (3.11) 0L,R = [(1 =F 7s)/2]0 , 0L,R = m ± Ts)/2] , (3.4) g L 73 R with 75 and other Dirac matrices defined as in the book by Bjorken-Drell [9]. In particular, 75 = 1, In the neutral current (NC) sector, the photon A, and the mediator Z, of the weak NC are 75 = 75- Note that, as given in Eq. (3.4), orthogonal and normalized linear combinations of B, and Wj: A, = cosfl B, + sinfl„ Wj and Z, = -sin0 B, + cos0»Wj. (3.12) h = ^70 = ^[(1 -7sV2]70 = food -7s)/2]70 = 0[(1 + 75>/2] . w w Equations (3.12) define the weak mixing angle 0.. The photon is characterized by equal couplings to The matrices P* = (1 ± 75)/2are projectors. They satisfy the relations P*P* = P*, P*P» = 0, left and right fermions with a strength equal to the electric charge. Recalling Eq. (3.9) for the charge P+ + P- = 1. The sixteen linearly independent Dirac matrices can be divided into 75-even and 75-odd matrix Q, we immediately obtain according to whether they commute or anticommute with 75. For the 75-even, we have gsinflw = g' cos0„ = e , (3.13) £r ^ = WWR + y^r ^L (r ^ 1, i , <,,,), (3.5) E E E 7s or equivalently, whilst for the 75-odd, tg0„ = g'/g, (3.14) e = gg'/Vg2 + g'2. 0To0 = ^ITO^L + ^TOIAR (TO = 7M. 7,7S) • (3.6) In the Standard Model the left and right fermions have different transformation properties under the Once 0, has been fixed by the photon couplings, it is a simple matter of algebra to derive the Z gauge groups. Thus, mass terms for fermions (of the form 0L0R + h.c.) are forbidden in the couplings, with the result symmetric limit. In particular, all 0R are singlets in the minimal Standard Model. But for the Twz = (g/2cos0.)£v, [tftl-Ts) + t (l+7 ) - 2Qsin20»]^Z', (3.15) moment, by 4>L,R we mean a column vector, including all fermions in the theory that span a generic R 5 reducible representation of SU(2) ® U(l). The standard electroweak theory is a chiral theory, in the where T^z is a notation for the vertex. In the minimal Standard Model, tR = 0 and t^ = ± '/i. sense that 0L and i/-R behave differently under the gauge group. In the absence of mass terms, there In order to derive the effective four-fermion interactions that are equivalent, at low energies, to are only vector and axial vector interactions in the Lagrangian that have the property of not mixing the CC and NC couplings given in Eqs. (3.11) and (3.15), we anticipate that large masses, as 0L and ^R. Fermion masses will be introduced, together with W* and Z masses, by the mechanism of experimentally observed, are provided for W* and Z by £HI . For left-left CC couplings, when the symmetry breaking. KS momentum transfer squared can be neglected with respect to m£ in the propagator of Born diagrams The covariant derivatives D„0L,R are explicitly given by with single W exchange, from Eq. (3.11) we can write D,0L,R = [d, + igAS t*RW* + ig"/2YL.RBj>k,R , (3.7) 2 + £3f = (g /8n4) [07,(1 -75)tL 0][f/(l - 75)tL" 4>] • (3.16) where t* „ and 'AYL.R are the SU(2) and U(l) generators, respectively, in the reducible By specializing further in the case of doublet fields such as v-e~ or v-iiT, we obtain the tree-level representations VT.,R- The commutation relations of the SU(2) generators are given by t f relation of g with the Fermi coupling constant GF measured from \i decay [see Eq. (1.2)]: [t£, tf] = i €ABC if and [tR, tR] = i eABc tf . (3.8) 2 GF/V2 = g /8n4 . (3.17) We use the normalization Eq. (2.2) [in the fundamental representation of SU(2)]. The electric charge By recalling that g sin0 = e, we can also cast this relation in the form generator Q (in units of e, the positron charge) is given by w mw = /iBom/sin0w , (3.18) Q = t[ + '/2YL = tR + V2YR . (3.9) Note that the normalization of the U(l) gauge coupling g' in Eq. (3.7) is now specified. with All fermion couplings to the gauge bosons can be derived directly from Eqs. (3.2) and (3.7). The Mo™ = (xa/V2G )1/2 = 37.281 GeV , (3.19) charged-current (CC) couplings are the simplest. From F 2 2 2 2 + where a is the fine-structure constant of QED (a = e /4x = 1/137.036). g(t'wi + t W ) = g([(t' + it )/V2] [(Wj - iWj)A/2] + h.c.) = g([(t W„-)/\/2] + h.c.) , (3.10) In the same way, for neutral currents we obtain in Born approximation from Eq. (3.15) the Here as is the QCD coupling. We can define as according to the MS prescription. In this case the effective four-fermion interaction given by second-order correction is also known [11]. From experimental determinations of AQCD [12], we obtain a value of as(raz) in the range £% = y/2GTQoh> [•••] WV [•••] i , (3.20) as(mz) = 0.11 ±0.01 . (3.28) where In the case of W decays, formula (3.25) refers to the sum of all down-quarks associated with a given 3 2 2 [...) = t (l - 75) + 4(1 +T5) - 2Qsin 0» (3.21) up-quark. For a particular down-quark, q', a factor |Vqq.| would appear, where V,,- is the relevant term of the Cabibbo-Kobayashi-Maskawa matrix (see Section 6) (£,< IV,,,, |2 = 1 by unitarity). Note and that by writing TBO™ » GFM3, instead of Tiion. « g^M » aM/sin2fl., we make the Born approximation more precise (i.e. S*,z smaller). In fact the radiative corrections are large on both a 2 2 eo = m^/m|cos 9w. (3.22) and sin 0». On the other hand, there are no leading logs in the scale dependence of the Fermi z coupling GF [13]; 5* are very small (Sf « O.lVo) if mt « mz and there are no additional 2 All couplings given in this section are obtained at tree level and are modified in higher orders of heavy-fermion generations [provided the physical values of mw.z and sin 0w are inserted in perturbation theory. In particular, the relations between mw and sinflw [Eqs. (3.18) and (3.19)] and Eqs. (3.25) and (3.26)]. The electroweak radiative corrections become considerably larger if m, is the observed values of e (e = Co at tree level) in different NC processes, are altered by computable large or if weak isospin multiplets with large mass splittings exist [10]. With the exception of this electroweak radiative corrections, as discussed in Section 5. case, for example, we obtain The gauge-boson self-interactions can be derived from the F,, term in £symm, by using Eq. (3.12) 1 2 3 and W* = (W ± iW )/V2. Defining the three-gauge-boson vertex as in Fig. 1, we obtain (V = y,Z) r(W-ee) = 233(mw/81GeV) MeV, (3.29) + 3 rw-w+ v = igw-w+ v [g„,(q - p)x + g^(p - r), + g„x(r - q)„] , (3.23) r(Z^e e")= 86(mz/92GeV) MeV , (3.30) with 3 T(Z - i;«»e) = 170 (mz/92 GeV) MeV . (3.31) gw-w+7 = gsin0w = e and gw-w+z = gcos0». (3.24) 2 2 Note that the dependence of T(Z -• ee) on sin 0w is very small for sin 0w = 0.23. If a careful 2 The partial widths for the decay of W and Z into a massless fermion-antifermion pair, including determination of the partial widths is required, one should implement the dependence of sin 0W on first-order strong and electroweak radiative corrections, are given by mz given by Eq. (5.8) and take 5" and dt into account, which are large if m, > mz. For W -» tb and Z -» tt (and perhaps Z -» bb) the quark-mass corrections cannot be neglected. T(W - ft) = N(GFm^,/6W2)(l + «*) (3.25) The widths in Born approximation are then modified as follows [14]: 2 2 2 2 2 2 ! ! T(Z - ff) = N(GFeomJ/24W2)[l + (1 - 4|Qf| sin 0.) ](l + 6f), (3.26) rBon,(W - tb) = (GFm^/2xV2)(([(m v- m - m / - 4ra m ]/<]" 2 2 2 where eo = 1 for doublet Higgses; S* and Sf are known electroweak radiative corrections, given X (1 - (mf + ml)/2ml- V2[(m + m )/mw] ]) explicitly, for example, in Ref. [10]; and 2 = (GFm^/2W2) (1 - c) (1 + e/2), (3.32) N = 1 for leptons , 3 2 2 2 (3.27) Tsom (Z - QQ) = (GFeom^m/ï) W + (1 - 4|QQ| sin 0.) V2 0Q - (3 )] , (3.33) N = 3(1 + ajit + ...) for quarks. 1/2 withe = mf/mwand/3 = [1 - (4mQ/mz)] . The QCD corrections of order as are also modified with respect to the massless case and are given in Ref. [15]. 2 The leptonic branching ratios are independent of mw and mz (at fixed sin 0w). We obtain (p*q->- r «o) BR(W -> ei>) - 0.089 , m, = 40 GeV , (3.34) BR(W-er) = 0.109, m, > mw - mb, rAVT qvW" BR(Z - e+e") = 0.034 , (3.35) Fig. 1 The three-gauge boson vertex S7 58 59 60 BR(Z-> £ i-iïi) = 0.20, (3.36) This transformation does not alter the general structure of the fermion couplings For i = e,*i,T quarks, the Cabibbo-Kobayashi-Maskawa unitary transformation relates the mass eigenstates d, s, 2 if m, > mz/2 and sin 0w = 0.23. and b to the CC eigenstates d', s', b', i.e. the states coupled by W emission to u, c, and t, respectively. The total widths Tw and Tz are given to a good approximation by summing up the rates for The NC is then automatically diagonal in flavour at tree level (GIM mechanism [5]). In the case of V -• ff discussed above, because the rare decays of W/Z in the Standard Model are really rare and leptons, if the neutrinos are massless then clearly there is no mixing. can be neglected. For example, for mt > mw - mb, we obtain If only one Higgs doublet is present, the change of basis that makes M diagonal will at the same time diagonalize also the fermion-Higgs Yukawa couplings. Thus, in this case, no flavour-changing 3 Tw = T(W - ei/)/BR(W - ev) = 2.14(mw/81 GeV) GeV . (3.37) neutral Higgs exchanges are present. This is not true, in general, when there are several Higgs doublets. But one Higgs doublet for each electric charge sector—i.e. one doublet coupled only to We now turn to the Higgs sector [16] of the electroweak Lagrangian. Here we simply review the u-type quarks, one doublet to d-type quarks, one doublet to charged leptons—would also be all right formalism of the Higgs mechanism applied to the electroweak theory. In the next section we shall [17], because the mass matrices of fermions with different charges are diagonalized separately. For discuss in more detail the physics of the electroweak symmetry breaking. The Higgs Lagrangian is several Higgs doublets it is also possible to generate CP violation by complex phases in the Higgs specified by the gauge principle and the requirement of renormalizability to be couplings [18]. In the presence of six quark flavours, this CP violation mechanism is not necessary. In fact, at the moment, the simplest model with only one Higgs doublet seems adequate for £H«, = (D„*)t(iy under SU(2) ® U(l), contains, at most, quartic terms in 3 M = ^LSTO^R + IÂR3EVL , (3.40) Q|v) = (t + 72Y)|v) = 0. (3.45) with The charged W mass is given by the quadratic terms in the W field arising from £HLMS, when 0(x) is replaced by v. We obtain 3K = T-v. (3.41) + 2 + 2 mwW, W- ' = g |(t v/i/2)| W; W- ' , (3.46) In the minimal Standard Model, where all left fermions i/x are doublets and all right fermions \tR are singlets, only Higgs doublets can contribute to fermion masses. There are enough free couplings in T, whilst for the Z mass we get [recalling Eq. (3.12)] so that one single complex Higgs doublet is indeed sufficient to generate the most general fermion 3 2 mass matrix. It is important to observe that by a suitable change of basis we can always make the 'Am^Z* = |[g cos 8W t - g' sin 0W (Y/2)]v| Z,Z", (3.47) matrix 9TC Hermitian, 75-free, and diagonal. In fact, we can make separate unitary transformations on \{IL and ^R according to where the factor of V2 on the left-hand side is the correct normalization for the definition of the mass of a neutral field. By using Eq. (3.45), relating the action of t3 and V2Y on the vacuum v, and 2 2 3 2 2 2 3 2 and consequently y2m = (g cos 0W + g' sin 0„) |t v| = (g /cos 0„) |t v| . (3.48) 3tt-> 9ÏT = L^SIItV. (3.43) For Higgs doublets 4. THE HIGGS AND BEYOND: THE PROBLEM OF THE FERMI SCALE .*V V = (v)- (3'49) The gauge symmetry of the Standard Model was difficult to discover because it is well hidden in nature. The only observed gauge boson that is massless is the photon. The graviton is still unobserved we have even at the classical level of gravitational waves; the gluons are presumed massless but unobservable because of confinement, and the W and Z weak bosons carry a heavy mass. Actually the main + 2 2 3 2 2 |t v| = v , |t v| = 'Av , (3.50) difficulty in unifying weak and electromagnetic interactions was the fact that e.m. interactions have infinite range (m = 0), whilst the weak forces have a very short range, owing to rnw.z ^ 0. so that y The solution of this problem is in the concept of spontaneous symmetry breaking, which was borrowed from statistical mechanics. m^'/jgV, m2, = V gV7cos20. . (3.51) 2 Consider a ferromagnet at zero magnetic field in the Landau-Ginzburg approximation. The free energy in terms of the temperature T and the magnetization M can be written as Note that by using Eq. (3.17) we obtain 2 2 2 F(M, T) = F„(T) + 'V(T)M + V4X(T)(M ) + ... . (4.1) v = 2~,M GF 1/2 = 174.1 GeV . (3.52) This is an expansion which is valid at small magnetization, and which is the analogue in this context It is also evident that for Higgs doublets of the renormalizability criterion; X(T) > 0 is assumed for stability; F is invariant under rotations, i.e. all directions of M in space are equivalent. The minimum condition for F reads go = m£/m2 cos20„ = 1 . (3.53) 3F/3M = 0, l/(T) + X(T)M2]M = 0. (4.2) This relation is typical of one or more Higgs doublets and would be spoiled by the existence of Higgs triplets etc. In general, There are two cases. If /*2 > 0, then the only solution is M = 0, there is no magnetization, and the rotation symmetry is respected. If n1 < 0, then another solution appears, which is 2 2 2 2 2 eo = S [(ti) - (tf) + t,]v /Z 2(tf) v (3.54) 2 |M0| = V/X. (4.3) for several Higgses with VEVs Vj, weak isospin ti, and z-component tf. These results are valid at the tree level and are modified by calculable electroweak radiative corrections, as discussed in The direction chosen by the vector M° is a breaking of the rotation symmetry. The critical 2 Section 5. temperature Tcrit is where /x (T) changes sign: If only one Higgs doublet is present, then the fermion-Higgs couplings are in proportion to the fermion masses. In fact, from the Yukawa couplings g*Ft (ïi.4>fR + h.c), the mass mr is obtained by /(Ten.) = 0 . (4.4) replacing JEHiKs, by the replacement symmetric under the infinitesimal transformations V V*° (x)/ Vv + (HA/2)/ 2 [so that (D,*)* (jy 2 2 2 2 2 (,dV/d We have thus completed our summary of the standard electroweak theory and of the W*, Z The symmetry of V implies that couplings. 5V = (3V/30i) 5 61 62 63 64 By taking a seconde derivative at the minimum 2 3 V/30ka*i (0i = *?Hij*? = 0 , (4.10) the fermions, as well as to ensure the correct high-energy behaviour required by renormalizability. However a more profound physical reality could be hidden behind or accompany the Higgs 2 The second derivatives Mki = (d V/d^)[d0iK0i = *?) define the squared mass matrix. Thus the above formalism. In fact, the Higgs mechanism is at present without experimental support. Actually the equation in matrix notation can be read as clarification of the physical origin of the electroweak symmetry breaking is one of the most important problems for experimental particle physics in the next decade. There are arguments MV = 0, (4.11) indicating that the minimal Standard Model with fundamental Higgs fields cannot be the whole story and that some kind of new physics must necessarily appear near the Fermi scale. The most famous which shows that if the vector (t0°) is non-vanishing, i.e. there is some generator that shifts the argument of this type is based on the so-called 'hierarchy problem', which we now describe. ground state into some other state with the same energy, then t 2 2 £ = - V4Fj, + |(3, - ieA„)4>| + VyjV* - (X/4)(0*0) . (4.12) remains whether unification without the inclusion of quantum gravity is really plausible. It is clear that the inclusion of gravity must induce major changes in the physics of the Standard Model at Note the 'wrong' sign in front of the mass term for the scalar field 4>(x) - (1/V2)[e(x) + v] exp [if(x)/v] , A„(x) - A, - (l/ev)(3„f(x). (4.14) vector-boson masses are all specified in terms of the VEV of the Higgs field v, according to Eqs. (3.41) and (3.51). The value of v is determined by the curvature scale of the Higgs potential V: Then Q — 0 is the position of the minimum, and the Lagrangian becomes V(0) = - 'V 0V + (X/4X0V)2, (4.17) 2 2 2 2 2 2 £ = - V4F K + V2e v A + VzeVA . + e evA„ + £(e). (4.15) according to The field f (x), which corresponds to the would-be Goldstone boson, disappears, whilst the mass term VieMA2. for A„ is now present; Q is the massive Higgs particle. v = ^/VX. (4.18) The Higgs mechanism is realized in well-known physical situations. For a superconductor in the Landau-Ginzburg approximation the free energy can be written as The observed values of the masses require for v (and, therefore, roughly for /i as well) that v = 102GeV [see Eq. (3.52)]. 2 2 2 4 15 19 F = Fo + V2B + |(V - 2ieA)0| /4m - a|<*>| + 0|<*>| . (4.16) If A « (10 -10 ) GeV, then we face the problem of justifying the presence of two so largely different mass scales in a single theory (the so-called hierarchy problem). In general, if A is very large Here B is the magnetic field, | Poincaré generators, which specify the spinorial transformations of Qa under Lorentz generally, the existence of new strong forces. For example, the electroweak symmetry could be 1/2 transformations and its invariance under translations, the essentially new relation is the broken by condensates of new fermions attracted by a new force with Ancw = mF = GF (An<:w anticommutator being the analogous quantity of AQCD), as in technicolour theories [23]. Or the Higgs scalar can be a composite of new fermions bound by a new force [24]. In the last two cases there are unsolved (Qa.Qo) = -2(7,)aeP', (4.21) problems related to the fermion masses. Or the SU(2) ® U(l) gauge symmetry can be a low-energy fake [25]. At high energies S» mT, the W and Z° would be resolved into their constituents. where P' is the energy-momentum four-vector, which generates space-time translations. Another possibility is that the Higgs mass becomes very large [26] [0(1 TeV)]. Then, as we shall If all fundamental symmetries are gauge symmetries, then also SUSY is presumably a local see, the weak interactions become strong and a spectroscopy of tightly bound weak interaction symmetry. This immediately leads to the realm of gravity. In fact, the product of two local SUSY resonances appears (e.g. WW, WZ, or ZZ states). transformations is a translation with space-time-dependent parameters, as follows from Eq. (4.21). But a translation with space-time-dependent parameters is a general coordinate transformation. As 65 66 67 68 However, it is fair to say that the compositeness alternative is not at all so neat and clearly formulated as the supersymmetric option. On the contrary, in many respects the compositeness way is not well defined at all and leads to many unsolved problems. mH (GeV Of course, the two avenues are not necessarily mutually exclusive, and theoretical frameworks where both appear have been considered. Searching for the standard Higgs particle appears to be a good way to organize the experimental solution of the symmetry-breaking problem. Note that the information we have regarding the mass of the standard Higgs is not very abundant. The experimental lower bound [27] on the Higgs mass from atomic and nuclear physics (absence of long-range forces), mH Ï* 15MeV, (4.23) is rather solid, unless perhaps for mH < 2m*. The limit mH » 325 MeV, which was claimed in the 10' 100 + past to derive from K -» T* + H, was later criticized [28]. In a recent reanalysis [29] of the /nt,GeV problem, the excluded domain was reported to be 50 < mH < 140 MeV. However, some model Fig. 2 The dashed region is dependence is clearly unavoidable in this case (from hadronic matrix elements). The most important the allowed region for the experimental input is the lower limit on mH that can be obtained from the search for the decay T -• H minimal Higgs mass from + 7 [30]. If we assume the validity of the theoretical prediction from the Wilczek [31] formula plus vacuum stability [33] first-order QCD corrections [32], then CUSB data lead to mH > 5 GeV (or mH < 2m,,). However, the QCD corrections are large so that the higher orders can also be of importance. Also, corrections to the non-relativistic approximation could be sizeable. It is therefore difficult to fix a precise value for the lower bound. However, I think that the data are by now good enough to make a minimal standard Higgs of a few GeV unlikely. Of course, it is sufficient to introduce two Higgs doublets in order to evade this limit (by taking suitable values for the ratio of VEVs). m, IGeV] Vacuum stability can be considered as a criterion for deriving theoretical lower limits on the Fig. 3 The allowed parameter space of mH and mt for various Higgs mass [33]. At tree level the scalar potential is given by embedding scales A (see Ref. [38]). The area around the origin bounded by the various curves is allowed. The curve labelled V( 1 2 4 4 2 computations of Fig. 3). These are certainly very interesting results. The problem is that the limit V( requiring perturbation theory to be valid up to A = mP leads to [36] 2 4 7 = (3ml + 6mt- + mH - 12m?) / (64TT V ) . (4.27) mH « 200 GeV . (4.28) (The extra factor of three in front of m? is, of course, due to colour). From the requirements that |$| Similarly, from the requirement that problems due to the possible triviality of the X<£ theory be = v be a minimum and that V(tf>) -• + oo for | mH « 125 GeV (4.29) was obtained [37]. The upper curves of Fig. 3, taken from Ref. [38], are obtained merely by imposing At LEP 2 [42], with Vs = 160-200 GeV, the important process is that the theory remains non-trivial up to a given scale A. The same picture has been essentially confirmed by computer simulations of the electroweak theory on the lattice [39]. However, if ITIH is e+e" -> ZH (4.33) made to increase, no physical contradiction is actually met. All that happens is that at ma * I >bb 1 TeV, the Born amplitudes for longitudinal gauge-boson scattering violate unitarity [26], manifesting the breakdown of perturbation theory. The helicity-zero state of gauge bosons is The discovery range is easily extended up to mH — 70 GeV. With more difficulty, one could possibly obviously connected to symmetry breaking, because it does not exist for massless vector bosons. For get up to mH " rnz- In this range of mass the additional problem is the confusion of the Higgs signal mu — 1 TeV the weak interactions become strong. The Higgs boson becomes very broad, with the W and Z peaks in the relevant mass distributions. Only supercolliders can continue the Higgs search beyond mH = mz: hadronic supercolliders l 3 TH « h (mH) (TH, mH in TeV) . (4.30) such as the Superconducting Super Collider (SSC) (VF = 40 TeV), or the Large Hadron Collider (LHC) (Vs = 16-17 TeV), or e+e~ future linear colliders such as the CERN Linear Collider (CLIC) We have already remarked that the search for the standard Higgs particle appears to be a good (VF = 2 TeV). The problem of producing and detecting the Higgs particle at supercolliders has been way to organize the experimental programme for a solution to the problem of the origin of the Fermi much studied recently in connection with the SSC, the LHC, and CLIC. In particular, the problem of scale. On the basis of the previous arguments, it is believed [40] that if a set of experiments are observing the Higgs at the LHC and CLIC was analysed in great detail at the La Thuile-CERN sensitive enough to be able to detect the Higgs with mass up to 0(1 TeV), then a great discovery will Workshop on Physics at Future Accelerators [43], and a comparison with the SSC was made. The in any case be made. Either one finds the Higgs, or new physics, or both. At the very least, one would luminosity for the SSC, the LHC, and CLIC was in most cases assumed to be L = 10" cm-2 s~l. observe the onset of a new regime with strong weak-interactions. The results obtained at the La Thuile Workshop on the Higgs problem can be summarized as The best opportunity to find the Higgs in the near future is offered by LEP experiments. At follows. The relevant range of Higgs masses must be separated into two intervals: the intermediate- LEP1 or at the SLAC Linear Collider (SLC) we will look for the Higgs in Z decays. The most mass Higgs: mw/z < mH « 200 GeV, and the heavy Higgs: mH 5= 200 GeV. The problem of important modes are observing the Higgs is completely different in the two cases. In fact, for an intermediate-mass Higgs, the main decay mode is into the heaviest pair of quarks allowed by phase space, whilst a heavy Higgs Z-HM (4.31) decays mainly into WW or ZZ pairs. The observation of an intermediate-mass Higgs is certainly possible at e+e" linear colliders with and (to a lesser extent) L = 1033 cm~2 s_1 and VF = 1-2 TeV. On the contrary, at hadron-hadron colliders the observation of an intermediate-mass Higgs presents extremely difficult problems because of the QCD Z - H7 . (4.32) background. This is true for both the single-Higgs production (pp -• HX) and the associated Higgs-W production (pp -» HWX). Since the intermediate-Higgs production cross-section (mainly The corresponding branching ratios are shown in Fig. 4. With an integrated luminosity of around due to gluon fusion) is quite large (at VF = 40 TeV, a = 102 pb for mH « 150 GeV and is 3-4 times 500 pb" ', the Higgs can be found if m « (40-50) GeV [41]. H smaller at VF = 16 TeV), one can hope to make use of the rare decay modes of the Higgs, e.g. H -» T 1 F 1 1 i — i 77 and T*T~ , or UU. The possibility of seeing the rare decay modes is for mH < 2m,. Then the main \ S decay mode of the Higgs is H -> bb, its total width is not too large, and the rare branching ratios are \ \ riz'-KVv-) enhanced. Even in this favourable case the detection of rare modes is very difficult, the 77 mode \^.T(ZE-M*M-I - being the most promising. Concerning the heavy Higgs, the most reliable strategy appears to be the search for the channel H -» ZZ -* I* t~ vv (with t = e, n). This method has the advantage of avoiding \ \ \ all problems connected with QCD backgrounds, which are always present with quarks in the final \ ••• states of WW or ZZ decays. Also, no additional problems are induced if m, > mw and the main \ - decay mode of the t-quark is t -» Wb. In this case, for each tt pair produced, there would be a WW \ —— ' •—-^.^ \ ^ pair and the background to H -» WW would substantially increase. The disadvantage is that there is ^^"-^^ \ a reduced range of observable Higgs masses because of the small branching ratio BR « 8 X 10"3 for A \*^ \ "a- ! ^\\ H -> ZZ -» t* (~vv. The conclusion at La Thuile was that the Higgs can be discovered through this J ™ - 2 1 \\ channel at the LHC for only mH « 0.6 TeV (with L = 10" cm" s" ). \\ \ \ The discovery range guaranteed by the leptonic modes would be extended if the hadronic modes \ \ \\ \\ could be disentangled from the background. Whilst H -> 4 jets is hopeless, it was pointed out at La Thuile that H -» WW -» Iv + 2 jets could be extracted from the background if an efficient system of j i i quark tagging could be implemented. In fact, the dominant production mechanism for heavy Higgses 10 20 30 40 50 60 70 m„ (GeV) is through WW fusion (Fig. 5). The two outgoing quarks that radiated the incoming W pair have a Fig. 4 The branching ratios r(Z° — HVp")/rwand r(Z° - H°7)/rw as functions of mH, for 2 transverse momentum of order mw and large longitudinal momentum. If both these quarks could be sin 0w = 0.23. 69 70 71 72 2 ii) The quantity sin 0w derived from the gauge-boson mass matrix by the relation 2 2 eo = n4/m cos 0w = 1 . (5.2) iii) The parameter sin20„ which appears in the NC effective interaction. Schematically, 2 2 £eH = V2 GF6o (h - 2 sin ». I,.) . (5.3) Fig. 5 The Feynman graph for WW or ZZ fusion into an on-shell Higgs boson The equality at tree level of these three 'definitions' of sin2 9» is the signature of the minimal Standard Model. Nearly all conceivable departures from the minimal Standard Model remove the degeneracy 2 of the above 'definitions' of sin 0w- Thus precise measurements comparing NC couplings with detected at very small angles (typically 0 » 5° at the LHC), then the background processes could be measurements of weak gauge-boson masses are crucial tests of the Standard Model and could well sufficiently suppressed. Clearly, this poses formidable problems for the calorimetry, because of the lead to the discovery of new physics. enormous level of radiation near the beam pipe. However, a quantitative analysis of the background The relations (5.1) to (5.3) are also modified, in a calculable way, by radiative corrections [47, rejection factor that can be obtained by tagging the quarks was done by Kleiss and Stirling [44], Their 48]. In particular, at tree level conclusion was that the tagging method is a plausible possibility at the LHC. The discovery range for 2 m could then be extended at the LHC to m = 0.8 TeV with L = 10" cm" s" ' (at least for m, < 2 H H go = m£/nizCOS 0w = Q,H = g« = eeq = ••• . (5.4) raw). With higher luminosity [45], from H -• ZZ -• till, one could reach at the LHC: i.e. e is the same for all NC processes and is related in a simple way to W and Z masses. When 2 2 radiative corrections are included in Eq. (5.3), go ~» eo + 8g and sin 0w -• k sin 0«, with different 6Q M ! _l mH »S 0.7 TeV (L = 5 X 10 cm" s , I = y), and k for each channel, so that in particular 34 2 2 2 mH « 0.8 TeV (L = 5 X 10 cm" s~', I = e,/i), m£/m cos 0w * e„N * Q* * e=q ••• • (5-5) It is useful to fix the definition of sin20» by imposing [49] that, at all orders, 33 2 1 2 2 mH « 0.6TeV (L = 5 x 10 cm" s" ,1 = e,/i). C = m^/m cos 0w = go , (5.6) A recent reappraisal [46] of the discovery potential of the heavy Higgs at the SSC, with with go = 1 for doublet Higgs bosons. Thus by including radiative corrections, we can write L = 1033 cm"2 s"1 and no quark tagging, reached the conclusions that a) the Higgs particle can be observed up to mH « 0.8 TeV (at most) from H -• ZZ -» tilt, t = e, /j; b) the possibility of reaching e»N = go(l + Sg„N), g« = go(l + SQ„), etc. (5.7) larger Higgs masses by using H -» ZZ -> llri depends on detailed studies of the calorimeter hermeticity and response (owing to the large Higgs width and the small signal-to-background ratio 2 2 2 With the same definition of sin 0w, also the relation between sin 0„ and m ^ is modified. The for mH large). corrected expression of Eq. (5.1) can be conveniently written in the form It is thus clear that the operation of the LHC at high luminosity, L « 1-5 x 1034 cm"2 s"1, and/or the possibility of realizing the quark tagging technique, would push the LHC discovery 2 sin 0w = D4,„/(1 - Ar)] (I An J) , (5.8) potential for the heavy Higgs substantially closer to that of the SSC in the basic configuration. Finally, we recall that at CL1C the heavy Higgs can be detected in the hadronic mode H -» 4 jets or equivalently 33 2 up to a mass mH «S 0.6-0.8 TeV with L = 10 cm~ s~'. 2 2 2 sin 0„cos 0» = [/4,ra/(l -Ar)] (l/m ), (5.9) 5. PRECISION TESTS OF THE ELECTROWEAK THEORY where MBO™ is given in Eq. (5.1). In the minimal standard electroweak theory at tree level, three—in principle, different — In order to measure go accurately from the NC data, or to extract sin20* precisely from a 2 quantities, all identified with sin 0w, coincide: measurement of m^, we must compute the radiative corrections ôg„N, 6g«, ..., and Ar that appear in i) The quantity sin20. = e2/g2, which specifies the ratio between the positron electric charge and Eqs. (5.7), (5.8), and (5.9). However, even assuming that we have three fermion families and one the SU(2) gauge coupling. This is obtained experimentally by measuring mw: Higgs doublet, the radiative corrections still depend sensitively on m, and mH. Usually, in most existing computations, one makes a guess: typically, mt = 45 GeV and mH = mz. With these values a 2 m sin 9. = OraA/2 GF) ( 1 An£) = >4on/ w • (5.1) result for Q0 is obtained from the data, which according to two very recent analyses [47, 48) is as from the CDHS and CHARM experiments at CERN, and from the FMMF and CCFRR experiments follows: at Fermilab, we obtain (including electroweak radiative corrections), _ 0.998 ± 0.0086 (Amaldi et al.), (sin2»»)^ = 0.233 ± 0.003 ± 0.005 . (5.13) (e°'"p- 1.001 ±0.007 (Costaetal.). (510) 2 The agreement between these two determinations of sin 0w is the most precise test of the electroweak Clearly, even if the minimal electroweak theory is correct, (eoW could still deviate from one, theory available at present. Given the behaviour of Ar with mt, the agreement would be spoiled for 2 simply because the input values of mt and IHH are not correct. In particular, eo - 1 is quadratically too large values of m, (with the definition of sin 0, adopted here, the radiative corrections to KN increasing with mt and logarithmically varying with IDH. Since (eo)«*P—obtained from the data when deep-inelastic scattering are less sensitive to m, [52]). This argument leads to an independent the radiative corrections are computed with 'light' mt and mH — is remarkably close to one, it can be confirmation that m, « 200 GeV. deduced that the Higgses are (at least dominantly) doublets and that an upper bound on m, can be The sensitivity of the electroweak radiative corrections to m, and to other possible new heavy given (which is [47, 48] around m, « 200 GeV for any value of mH « 1 TeV). Also, for 'light' m, and particles, is a very important feature that makes precision tests very relevant in the search for signals mH, the value of Ar in Eqs. (5.8) and (5.9) is essentially coincident with [a(mw)/a(m,)] - 1, i.e. it is of new physics. In QED or QCD, heavy particles decouple. For example, the running of a or as is not determined by the running of the QED coupling from the defining scale of order m, (the electron affected by the presence of particles of mass m much larger than Q, the scale where a or a, is mass) up to a scale of order mw. A precise calculation for m, = 35 GeV, mH = 100 GeV, and mz = measured. In spontaneously broken gauge theories, heavy particles do not decouple because there are 93 GeV leads to [50] couplings that increase with masses. For example, this is the case for the couplings of longitudinal vector bosons to fermions. Recall that the longitudinal modes would not be present in the symmetric Ar = 0.0710 ± 0.0013. (5.11) limit and are in fact generated by the Higgs mechanism. Another source of dependence on heavy masses is the presence of anomalies [19]. For the cancellation of the axial anomaly, a complete For increasing mt the value of Ar decreases dramatically and goes through zero for generation of fermions is required. If m, -• oo, the removal of the t-quark has the effect of m, = 220 GeV or so (Fig. 6) [50]. If we use the value of Ar given in Eq. (5.11), together with the UA1 resurrecting the anomalous behaviour. and UA2 measurements of mw (see, for instance, Refs. [47, 48]), we obtain from Eq. (5.8), A concise summary of the present status of precision tests of the standard electroweak theory 2 expressed in terms of sin 0W is presented in Table 1. 2 (sin 0.)mw = 0.228 ± 0.002 ± 0.007 . (5.12) Soon, mz will be measured at LEP (and at the SLC) with a precision of Smz = ± 50 MeV or so [53], without transverse polarization. Actually, a transverse polarization P, with P => (10-20)%, This value is in very good agreement with the measurement of sin20» from NC data. The most precise which is relatively easy to implement, can considerably improve the measurement of mz [19], which is NC experiments are those on cN deep-inelastic scattering [51]. Combining the corresponding data the reference quantity, i.e. the starting point, for precision tests of the electroweak theory. This is Table 1 0.080 2 2 Values of sin 0w = 1 - (m^/m .) obtained from C.070 different experiments. This is a simplified version of a similar table given in Ref. [47]. Statistical and 0.060 systematic errors have been added in quadrature. Radiative corrections have been applied as computed 0.050 from m, = 45 GeV and mH = 100 GeV. 0.010 0.030 '?' N deep-inelastic scattering 0.233 ± 0.006 W/Z masses 0.228 ± 0.007 0.020 '?* p (elastic) 0.210 ± 0.033 C 010 'i^e (elastic) 0.223 ± 0.018 Atomic parity violation 0.209 ± 0.023 SLAC eD 0.221 ± 0.020 10 60 80 100 120 U0 160 180 BCDMS nC 0.250 ± 0.080 m, (GeV) All data 0.230 ± 0.005 Fig. 6 Ar as a function of the top mass for various mH (for mz = 91 GeV) 73 74 75 76 because with transverse polarization and polarimeters the uncertainty arising from the energy scale | M; : n GeVJ determination can be significantly reduced by electron spin resonance calibration. It was concluded m„ = 100 GeV [54] that at LEP the accuracy on mz can be decreased from 5mz = ± 50 MeV without polarization, to 6mz = ±20 MeV with polarization. If the radiative corrections (i.e. Ar) were known, one would m, = 180 GeV 2 2 obtain sin 0w from Eq. (5.9) with a precision of & sin 9w = ±0.0004 corresponding to 6mz = ±50 MeV, far superior to that of any other existing experiment. However, Ar is not known well enough, because it depends on the so far unknown parameters m, and niH (even assuming the validity of the minimal Standard Model with three generations). The electroweak radiative corrections to all processes also depend on m, and DIH. As we have seen, the dependence on mt is particularly large. It 6^, it 0.003 (Th: 2 0.0033! is eagerly hoped that mt will soon be known (presumably from hadron collider experiments). The 6My = 100MeV (Th:î25MeVI measurement of m,, apart from its intrinsic interest, would remove the largest masking effect in the connection between interesting physics and precision electroweak tests. Anyway, the best that can be done at the moment is to measure other quantities with adequate precision (typically, when mH is m,= 90GeV varied between 10 GeV and 1 TeV, the corresponding relative charge of sin20„ is S sin20„/sin20,» = ± 1%). In principle, we need three quantities in addition to mz: two are needed to eliminate m, and m„=1TeV mn, and the third would finally give a precise test of the theory. Of course this is a bit academic, because, for example, given realistic precisions, mH cannot be 'eliminated'. In other words, a certain redundancy in the number of precisely measured quantities would be useful to compensate for the 2 2 limited precision attainable in practice (with respect to the reference goal 8 sin 0»/sin 0w « 1%). On the other hand, very few precision experiments can be expected in the near future outside the realm of + e e~ annihilation (in fixed-target and hadron collider experiments). The CHARM 2 Collaboration is Fig. 8 Same as Fig. 7, for mw Fig. 9 Same as Fig. 7, for ALR at present taking data on (?ie elastic scattering with the aim of measuring sin2 9» with a precision of 2 5 sin 0w = ±0.005, free from hadronic uncertainties that limit the precision that can be attained in vN deep-inelastic scattering. Here and in the following we often quote, for indicative purposes, the uncertainties, the most important being the energy calibration error. For example, at the CERN pp 2 Collider, with 20 pb" ' of integrated luminosity (which should be obtained before the end of 1991), by precision on sin Sw that would be achieved if mt and mH were precisely known (i.e. obtained by combining UA1 and UA2 results it is planned to obtain 6(m /m ) = ±0.002, which from sin20 = neglecting all uncertainties related to radiative corrections). A more physically relevant criterion is w z w 1 - mw/mî directly implies S sin20 = ±0.004 (with no ambiguities from m, and m ). From the presented, for example, in Figs. 7-9. w H 2 At present pp colliders the most precise measurement of interest for testing the electroweak separate measurements of mw and mz, one can at most obtain h sin 9, = ± 0.005. In the same period theory is the determination of the ratio mw/mz that is free from several sources of systematic of time, comparable accuracies can be achieved at the Fermilab Tevatron (where an integrated luminosity of about 10 pb~ ' will presumably be collected). [H; * 92 GeV | In the following we shall see that at LEP 1, with no polarization, two very precise measurements can be made, i.e. the forward-backward asymmetry AFB in e+e~ -» \i?'/T (or e+e~ -» ff and the / N. m„=100GeV + polarization asymmetry Apoi in e e~ -» T*T~ (where the T helicity is reconstructed by T -» in>). With 6 2 m,=<.5Gev/ \ m,=180GeV 10 Z events, one expects [55] precisions of the order of & sin 0w = ±0.002 from AFB and Ap<,i (for given mt, mH). With longitudinal polarization one could, in addition, measure the difference over the sum of the total cross-sections (i.e. collecting statistics from all final states) from left- and right-handed electrons, i.e. the left-right asymmetry ALR. With reasonable values of the 0 021 0.023 0 025 0.027 2 A» polarization, the fantastic level of 6 sin 0» = ±0.0004 could be reached! In addition, very important 1 I • 1 information is also obtained by the polarized forward-backward asymmetry API and by ALR for 6AfB = ï0 002 [Th: =0.0016) different final states, e.g. bb or cc. Fig. 7 The values of AFB that correspond, in the Finally, at LEP 2 [42], one will be able to measure mw with a precision of around 5mw = 2 minimal Standard Model, to a fixed value of mz ± 100 MeV. Given mz with 6mz = ±50 MeV, this corresponds (from sin 0w = 1 - mw/ml) to 2 y \ m, = 60GeV (92 GeV) are given in the upper part for 45 GeV ^ 6 sin 9w = ±0.003 without uncertainties from radiative corrections. + m, < 180 GeV and mH = 100 GeV, and in the As already mentioned, in e e~ annihilation at the Z resonance the optimal tests are provided by m„ = 1IeV / \ m^lOGeV lower part for 10 GeV < mH < 1 TeV for m, = asymmetries. The most important ones are listed below. 60 GeV [56]. The experimental uncertainty 6AFB i) The forward-backward asymmetry AFB. It can be measured in any channel e+e~ -• ff, where f is expected at LEP is shown for comparison, either a quark or a lepton. For precision tests the most convenient channels are f = b, c, fi~ 0.021 0.023 0025 0 027 together with the associated theoretical error. *F8 (asymmetries for quarks are also discussed in Ref. [55]). If by 'forward' we denote the initial e 2 direction and by ap(f) the cross-section for finding the fermion f in the forward hemisphere For charged leptons, Vf is small because sin 0w is close to '/». For unpolarized beams, i.e. PC = 0, we (within a given fiducial region), then obtain 2 AFB = [oriO " orfOl/MD + "KOI • (5.14) |AFB| = 3, , (5.22) Often AFB is also indicated as the charge asymmetry, |Apoil = 27,,. (5.23) ii) (Final) polarization asymmetry Apoi. In e+e~ -* ff, this is given by This difference in the dependence on the small parameter t)f has a strong effect on the respective 2 2 2 Apoi = [ 2 where polarization from the pion spectrum in the decay T -* TV. 2 ISApoil = 86(sin 0w). (5.25) iii) The left-right asymmetry ALR. A longitudinal polarization for the e" beam is needed in this case. 2 2 The asymmetry ALR is given by For 0.22 « sin 0w * 0.24, AFB is very small; and the smaller it is the less it is sensitive to sin 0„. For sin20» = 0.23, ALR = ( 2 5(sin 0w) = '/2|ÔAFB|. (5.26) where CTL.R are the cross-sections for el.R + e+ -» X (X can be any channel). In particular, the experiment can be done in a totally inclusive way. 2 On the other hand, for all values of sin 0w, With no polarization, Ara and Apoi can be used for precision tests. We shall see that Apoi is more 2 2 sensitive than AFB to sin 0„ but can be measured with less accuracy. With longitudinal polarization 6(sin 0w) = V8 |ôAp„i| . (5.27) there are substantial advantages. Firstly, ALR is obviously easier to measure than Apoi and carries 2 equivalent information. Secondly, the sensitivity of AFB to sin 0w is much improved by a longitudinal Everything becomes even better if a sizeable longitudinal polarization Pc is present. In fact, the polarization. These statements are easily understood from the tree-level expressions of the dependence of AFB on ijf becomes nearly linear, as is seen from Eq. (5.17), so that its size and asymmetries. At the top of the Z resonance the diagram with Z exchange is completely dominant the 2 sensitivity to sin 0„ are much increased. At the same time ALR, which is nearly as good as Apoi as far 2 2 relative corrections to the asymmetries from the photon terms being of the order of (r,„,/mz) . as sensitivity to sin 0w is concerned, can also be measured. From Eq. (5.18) we obtain Neglecting such terms, the expressions of the asymmetries at the resonance (after integration over angles) are given in the Born approximation by 2 Ô(sin 0„) = (1/8P) |6ALR| . (5.28) IAFBI = 3i,f [(i,. + % P,)/(l + 2Pe^)] , (5.17) Clearly, ALR has considerable advantages over Apoi. It can be measured for any final state, with a large gain in statistics compared with Apoi, which in practice can only be measured for T*T~ final |ALR| = 2Pe>)c, (5.18) states and only via the T -» *v decay mode (the already small cross-section being further reduced by the efficiency of measuring T'S and their helicity). |Apoi| = 2vf, (5.19) In addition, the structure of radiative corrections to ALR is simpler than for AFB [56]. This is important because the radiative corrections have to be controlled to a degree of accuracy that is not where Pe is the e" longitudinal polarization and ijf is given by at all trivial to achieve. The radiative corrections are particularly simple when they can be approximately factored into e+e~ -• Z (with real Z) times Z -» f. Then most of the radiative 2 2 ni = VfAf/(V + A ), (5.20) corrections drop away from the asymmetry ratio. This is the case for ALR but not for AFB because the latter quantity is based on a correlation between the initial and the final state. Vf and Af being proportional to the vector and axial-vector couplings of the Z to the fermion f. In the Another advantage of ALR is its smooth energy dependence near the Z peak, approximately standard electroweak theory, linear with a small slope. This is not the case for AFB and is important because it leads to small corrections when moving around the Z (because of bremsstrahlung, avoiding spin resonances, etc.). 2 Vf = 1 - 4|Q,| sin 0w , A summary of the sensitivity to m, and IHH of precision tests of the electroweak theory based on (5.21) AFB, mw, and ALR, and the corresponding precision that can be [55] realistically obtained are Af = 1 . presented in Table 2 and in Figs. 7, 8, and 9, respectively. In each figure we assume that a given value for mz has been determined (with negligible error). For example, the value mz = 92 GeV was chosen. 77 78 79 80 Table 2 The reason for the primes is that the states d', s', b', ..., with definite weak interaction properties, do Predicted accuracies (statistical plus systematic) not in general coincide with the eigenstates of the mass matrix (to which the unprimed quark labels that are expected at LEP (from Ref. [55]) refer). In general, a change of basis, i.e. a unitary transformation, connects the primed states, with definite weak couplings, and the unprimed states, which diagonalize the mass matrix, 2 SA ê(sin 0„) d' À s' S Without polarization b' = U b (L = 200 pb-'): ^' j J, AU ±0.0035 ±0.0017 Ah 0.007 0.0012 Since U is unitary and commutes with T2, T3 (the weak isospin generators) and Q (the neutral-current couplings) are diagonal in both the primed and the unprimed basis. This is the GIM mechanism [5] AC 0.007 0.0015 FB that automatically ensures flavour conservation in the NC couplings. Conversely, it is simple to b A FB 0.005 0.0009 realize that the general criterion for obtaining natural flavour conservation in the NC is that all states with the same Q also have the same transformation properties under the weak group, i.e. the same T2 Apol 0.011 0.0014 and T3. With polarization For N generations of quarks (i.e. 2N flavours), U is an N x N unitary matrix that can, in general, be parametrized in terms of N2 real numbers. However, the 2N - 1 relative phases of quarks (L = 40pb-';(PL) = 0.5): 2 0.003 0.0004 are not observable and can be fixed arbitrarily (the overall phase has no effect on U). Therefore N - ALR 2N + 1 = (N- l)2 real numbers are left. On the other hand, the most general N X N orthogonal Ate 0.0006 matrix involves N(N- l)/2 real numbers. Thus the (N- l)2 parameters can be split into N(N- l)/2 matrix mixing angles and (N - 1)(N - 2)/2 phases. For N = 2, as in the old four-quark model, there was one mixing angle (the Cabibbo angle) and no phases. For N = 3, as in the present orthodoxy, there are three mixing angles and one phase. This phase is important and welcome, because it allows for an elegant accounting of CP violation in the theory [57], For N = 4, if one more generation would be found, there would be six mixings and three phases, and so on. Note that the known weak For each pair of fixed values of the two unknowns m, and ITIH, the prediction of the minimal properties of light quarks and leptons, plus the condition of natural flavour conservation of the NC Standard Model (with three generations and one Higgs doublet) can be derived for AFB or mw or by itself, imposes the standard classification of all I^L in doublets and all I^R in singlets. In turn, the ALR. In the upper part of each figure we show the range of variation of, for example, AFB for fixed model with three quark doublets and three lepton doublets is automatically free of the 75 anomaly mH = 100 GeV when m, is varied in the range 45 GeV < mt < 180 GeV. Similarly, in the lower part through the miraculous cancellation of the quark and lepton contributions to that anomaly. 1 say of the same figure, mt is fixed at 60 GeV and mH is varied in the range 10 GeV « mH * 1 TeV. Clearly, the ranges obtained are indicative of the minimum precision that is required in order to miraculous because a deeper understanding of this cancellation is still missing. The U matrix is the CKM matrix. For N = 3, we can define the three mixing angles and the obtain significant information, for example on mH. The sensitivity to mH can be considered as a monitor for the precision needed to probe the electroweak symmetry-breaking sector of the theory. phase in many different ways. Given the experimental values of the quark mixing angles, the most Actually, in many cases the sensitivity to new physics is greater than to mH. In each figure, the convenient parametrization of the CKM matrix is the one proposed by Maiani [58]. In comparison, planned experimental precision is also indicated (together with an estimate of the theoretical error on the original proposal by Kobayashi and Maskawa [57] leads to cumbersome expressions for the the prediction of the relevant quantity for given raz, mi, and mH, arising from ambiguities in physically relevant transition amplitudes. radiative corrections, the values of other parameters, etc.). We denote the three down partners of u, c, t, by d', s', b', respectively, in the weak CC left doublets. Following Maiani, we write |d') =c |d ) + s«ei*|b), (6.1) 6. FLAVOUR PHYSICS: s c THE CABIBBO-KOBAYASHI-MASKAWA (CKM) MATRIX where eg s cos/3, sj = sin/3 (analogous abbreviated notation will be used in the following); dc is the In the standard electroweak theory, all left-handed quarks qL are doublets and all right-handed Cabibbo-rotated down-quark quarks qR are singlets: r ~~\ r ~\ |dc) = c»|d> + s»|s> . (6.2) U c t qR: all singlets. d' s' b' Note that in a four-quark model the Cabibbo angle fixes both the ratio of the u-to-d coupling with We now consider the available experimental information on A, j, and Cabibbo-rotated strange quark 2 BRSL1 BS rB|Vbc| = °" =(2.9±0.6)x 10" s , (6.11) Zc(l + [r(b - u)/T(b - c)]) |sc> = - s#|d) + c»|s> , (6.3) where we used BRSL = 0.117 ± 0.006 [60] and Zc = 4.0 ± 0.6, the latter value being taken from and as Ref. [61]. For B°-B° mixing, only the combination TBA2 is important, and it can be obtained directly from Eq. (6.11) [T(b -» u/r(b -• c) is small, see Eq. (6.15)]. Otherwise, one can obtain A by using |v)= -see-^ldcj + celb). (6.4) [60] TB = (1.11 ± 0.16) X 10~'2s: The angle y is defined by the physical combinations of sc and v coupled to c and t quarks: A = 1.05 ± 0.17 . (6.12) 2 2 |s'> = c,|sc> + sT(-sse-'*|dc> + ce|b» , Note that, since A « 1, |VCb| = |Vts| = AX indeed turn out to be of order X . (6.5) The quantity g is fixed by the ratio R = T(b -» u)/T(b -> c) [where T(b -• u) means T(b -• uec), |b'> = -sT|sc> + M-s9e-'*|dc} + cs|b», i.e. the semileptonic width into charmless final states]: 2 2 From experiment (0.47 ± 0.02) R = |Vub/Vcb| = (X6) , (6.13) s» = X = 0.221 ± 0.002 . (6.6) where the numerical factor is obtained from the parton model plus phase space and QCD corrections, and R is determined by the electron spectrum near the end-point. Clearly 2 3 Also, sy = X and s Neglecting terms of order X", a considerable simplification is obtained: dr/dE, = T(b -» u)[[l/T(b - u)][dr(b - u)/dE«]) + T(b -* c)([l/r(b -* c)][dT(b - c)/dEc]) . (6.14) |u) |c> It) i4 (6.7) A priori there is some model dependence in the calculation of the normalized spectra. Various models |dc) + sge |b) |sc> + sy|b> |b>-s»-sse-'*|d) J have been tried. A posteriori the differences are found to be smaller than one could expect. The most stringent results are obtained by CLEO and are confirmed by ARGUS, Crystal Ball, and CUSB. The result is that quite reliably [62] v v v c» ud us ub s» sae" V = Vcs -Se vcd vcb = Ce Sy (6.8) |Vb»|/|Vbc| «0.14 or e<0.6. (6.15) v„ V.b^ srss —sge lv,d 1 However, in the following—in order to be really careful—we also present results for |Vbu|/lVbc| < 2 2 Finally, following Wolfenstein [59] (but not completely: our e is his Je + q ), it is convenient to s 0.20 or e < 0.9. A lower limit on Q can be obtained, if we assume that there are three families, from the observed 2 3 s^ = AX and sfl = AX e • (6.9) CP violation (i.e. the experimental values of e and t'/e). We find [62] Then V takes the simple form |Vbu|/|Vbc| > 0.02 or e>0.1. (6.16) 2 3 1 1 - (X /2) X AX e e * Finally, the existing limitations on 4> are not very stringent. We will discuss this issue later in these 2 -X I - (X /2) AX2 + 0(X4), (6.10) lectures. 3(i - ee"1*) -AX2 1 which will be used in the following. 81 82 83 84 7. FLAVOUR MIXING AND CP VIOLATION We define the ratio r of total (i.e. integrated over time) probabilities: For a stable free particle at rest, the quantum mechanics time evolution is given by ^ = e~|M1. 2 2 For an unstable particle at rest, this is modified into B) = UÏ|A(B - B)| dt]/[JJ|A(B -> B)| dt] , (7.10) with M and T real, positive numbers. For several coupled states, M and T become Hermitian matrices with positive eigenvalues (i.e. the analogue of real, positive numbers). In particular, for where T is a conveniently large time. We obtain directly from Eqs. (7.9) and (7.10), B°-B° (or any other similar system), we have r = |(1 - 0/(1 + OP M - ir/2 MU - ir,2/2 r r r, r r r 'H' (7.1) x (Jïdt[e- '" + e" 2' - 2e- cos(Amt)])/(fJdt[e- >' + e~ * + 2e" ' cos (Am t)J] (7.11) MÎ2 - ir,2/2 M - ir/2 where T = '/2(rt + T2) is the average width of Bi and B2, and Am = Mi - M% is their mass 0 Note that i) 'H' is not Hermitian (since probability is not conserved within the B -B° system, because difference (we define 1 and 2 so that Am ^ 0). By performing the integrals for T -* 00, we finally of the decays); ii) Hn = H22 by CPT; iii) H12 >* 0, H21 ^ 0 because of the weak interactions that obtain violate the conservation of quark flavours; iv) Im M12 ^ 0, Im Ti2 <* 0 because of CP violation. The eigenvalues of 'H' can be written down in the form r = |(1 - 0/(1 + 0|2 [(x2 + y2)/(2 + x2 - y2)] , (7.12) 2 (7.2) B,,2 = [(1 + 0 B° ± (1 - «)B°]/J2(1 + |t| ). where x = Am/r and y = AIV2r with AT = Ti - T2. Similarly, we could have obtained Note that Bi and B2 are not orthogonal because 'H' is not Hermitian. If t = 0, CP is conserved in the r = |(1 + 0/(1 " 0|2 [(x2 + yV(2 + x2 - y2)] , (7.13) wave functions. In general, e depends on the phase convention chosen. Thus, for example, e pure imaginary does not lead to any CP violation because it can be removed by a redefinition of the where relative B°-B° phase. A simple calculation immediately leads to the following results for t and the eigenvalues of M and T: r = [P(B-» B)/P(B - B)] . (7.14) i) s (1 - f)/(l + é) = „'(Mi2 - ir,2/2)/(Mn - ir,2/2) , (7.3) Clearly, when CP-violation effects are neglected, M 1,2 = M ± ReQ, (7.4) r = r = [(x2 + y2)/(2 + x2 - y2)] , (7.15) ri,2 = r =F 2ImQ, (7.5) so that the asymmetry where M, r, M12, and Tn are defined in Eq. (7.1) and a = (r - r)/(r + r) (7.16) (7.6) Q = J(Ml2 - ir12/2)(Mu - iT12/2)'. is a well-known measure of CP violation. Note that 2 The B -B° oscillations are caused by the different evolution in time of the eigenvectors Bi and B2. 0 < x < +00 Starting at t = 0 from a pure B° state, (7.17) 2 2 0 ^ y = [ By using Eq. (7.2) we can eliminate |Bi) and |B2) and write |i/<(0) as a superposition of B° and B°. The coefficients are the transition amplitudes: A(B -» B) and A(B -» B). We immediately obtain r = x/(l-x) or x = r/(l+r). (7.19) iM 1/2r W1T A(B^B) = V2[e- »'e- '' + e'^'e" V], From Eq. (7.18), it follows that 0 s; x $ V2. (7.9) - iMl 1/2r iM2t A(B - B) = (1 - f)/(l + c) 72 [e ' e" '" + e Typically, r is measured through the ratio v,, w v.* t v„ *_ -* • t> —>— = [N(BB) + N(BB)]/[N(BB) + N(BB)] , (7.20) J 1, w e , 1 1«—• d i —.— V,", W v,* V t V,, where N(BB) is the number of BB final states observed in a sample of events from a process where a ( M BB pair is produced; N(BB) is identified by some convenient final state (e.g. two negatively charged leptons: ("£"); N(BB) and N(BB) are experimentally the same but are kept separate to remind us of Fig. 10 Box diagrams for Ba-ËS mixing with t-quark exchange the possibility of a double flip. A double change of flavour, Ab = 2, as in B°-B° mixing, clearly requires the action of two charged currents. The two emitted W's must be reabsorbed, so that a total of four weak vertices are matrix that fixes the relative importance of the different virtual quark exchanges. For example, for involved. Quite in general, including all effects of strong interactions, we can write Bd-Bd mixing, Mn is dominated by the t-quark exchange, because |VtbVjd| is of the same order as |VcbV*d|. The same is also true for B?-B? mixing. On the other hand, Ml2 would vanish if all , 1 M12 = S dx dy dz D^x - y)Dc^z) space-time points where the W's are emitted or reabsorbed. If it can be proved that the four points at having to cope with QCD at scales of order mexi. Thus if the external meson is light (e.g. the kaon), some level of approximation can be confused with a single point, then the effective interaction the problem is not easy. One tentative solution is to evaluate the matrix element by vacuum insertion becomes local. The QCD effects can then, at least in principle, be taken into account perturbatively. (or saturation) [63]. This is a kind of valence approximation. The vacuum saturation approximation A simple situation is when mc*t •* mç •« mw, where nuxl is the external meson mass and ITIQ is the is expected to work better and better as the mass of the external meson increases. This is because of exchanged virtual quark mass. The W propagator D^u) is significantly different from zero only at the Zweig rule: gluon radiation from the quark legs is suppressed when the relevant transverse distances u * 1/mw. momentum, of order of the meson mass, is large. On the other hand, at short distances x-y * 1/mw, an operator expansion is valid, and In the Standard Model, all complex phases enter through the CKM matrix (see Fig. 10). The J^xyfr) approaches a local four-fermion operator. Thus the four-point function in Eq. (7.21) can phases determine Im M« versus Re M12 and Im Tu versus Re Ti2- be reduced to a two-point function when terms of order U1„/BIW and mo/mw are neglected. For The width r[2 is given by the absorptive part of the diagrams of Fig. II. A cut indicates the heavy enough virtual quarks, the operator expansion technique can, in an analogous way, be applied on-shell particles of the final state after decay (with the integrations over the available phase space). to the resulting product of two local four-fermion operators. The heavy virtual quark lines can also The cut in Fig. 11a corresponds to spectator decays, whilst the cut in Fig. lib has to do with be shrunk to a point. The effective Hamiltonian is then finally given in terms of a single local W-exchange decay modes. Clearly, only final states that are common to both B and B decay four-fermion operator: contribute to F12. Examples of such final states are given in Fig. 12. Obviously, only light quarks can Htff = GF dL 7» bL dL / bLc(mb, mq, mw, ...), (7.22) where B> B' QL = [d - 7s)/2] q , (7.23) and c is a coefficient function. Since the QCD coupling as is small at short distances, the coefficient function c can be computed in the approximation of neglecting the strong interactions. This corresponds to the evaluation of the box diagrams (Fig. 10). The QCD corrections are then, in principle, computable and are determined by the anomalous dimensions of the different operators that enter in the short distance expansion. When niQ * mw, the dominance of short distances remains true. What is to be rediscussed is the appropriate limiting procedure (e.g. mw, mQ -> °°, with Spectator Annihilation mo/mw fixed). In order that the above strategy can apply, it is necessary to have an argument for the dominance b) of heavy virtual quarks in the case of interest. As we shall see, it is the structure of the quark mixing Fig. 11 Contribution to Ti2 for the B°-B° system 85 87 AmB = 2|M,2| and ArB = 2|r,2|. (7.29) For kaons the situation is different, as we shall now summarize. For kaons [60], 12 Ann = mKL - mKs = (3.521 ± 0.014) X 10" MeV , TS = (0.8923 ± 0.0022) X 10" ,0 s , (7.30) 7L = (0.5183 ± 0.0040) X 10"7s. As a consequence of Ts > IY, we have rK = V2(rs + rL) = V2rs, with ArK = rL - rs = -rs, (7.3i) 3 andyK = Ar/2r = - 1 (actually yi = [(1 - 4IY/rs) = 1-7 X 10~ ].Also Fig. 12 Examples of common final states in B° and B° decays xK = AmK/rK = 0.954 . (7.32) appear in the final state. In the limit when all quarks are massless except for the b and t quarks, r 1: Finally, for B mesons turns out to be proportional to mb. Actually, in this limit 2 2 2 rK = (XK + ji)/(2 + x K - y K) =1-7x10"'. (7.33) 2 2 r,i« [Vubv;d + VcVjd] ml = (Vlbvr„) mb , (7.24) Moreover, for the kaon system the CP-violation parameter |e| is where the last equality is due to the unitarity of the 3 x 3 CKM matrix. Two important consequences follow for B mesons: i) Tu has almost the same phase as M12 (both 3 |e| = (2.28 ± 0.02) X 10~ . (7.34) being determined by VtbV*d); ii) |r 12I < \M,2\ or Ar •* Am [see Eq. (7.6)] in the same proportion as m <« mf. Thus for B mesons, a quite good approximation is b The small absolute value of e implies that for kaons, r = xV(2 + x2). (7.25) ImMi2<«ReMi2 and Imr,2 •* Re r)2, (7.35) Also, in the limit that M12 and Ti2 have the same phase, e is purely imaginary, as is seen from because e = 0 if M]2 and ri2 are both real [Eq. (7.3)]. From Eqs. (7.4) and (7.5), we see that for Eq. (7.3). IfM = iMule^andTn = Ir^e*, then 12 kaons, (1 - «)/(! + e) = e" (7.26) Amie = 2 Re Ml2 and ArK = 2 Re r,2. (7.36) In this case Re t = 0 and the effect of t can be rotated away by a phase redefinition. Thus there are Note how Eqs. (7.36) differ from Eqs. (7.29), which are valid for the B system. two ways of having small effects of CP violation [64]: either Im M12 and Im Vn are small, then |é| is Starting from the general expression for e given by Eq. (7.3), we can expand in the small small; or M12 and Ti2 can have nearly the same phase. In the CKM phase convention the first option imaginary parts Im M12 and Im T^: is realized for the kaon system, the second for the B meson system. Going back to Eqs. (7.3) to (7.6), we see that in general (1 - e)/(\ + e) = 1 - 2c = /Re Ml2 ~ 'Im Ml2 ~ (i/2) Re r'2 ~ '/2^m F'2 \ReM,2 + iImMi2 - (i/2)Reri2 + V2Imri2 Am = 2 Re Q and Ar = - 4 Im Q . (7.27) _ j _ JImMi2 + V Imri (7.37) For B mesons, M12 and Ti2 have nearly the same phase. Then 2 2 ReM12 - (i/2)Rer, Thus for kaons, Q = |M,2| -(i/2)|r,2| (7.28) e = [i Im M + V ri ]/(Am - i Af/2). (7.38) so that 12 2 2 We have already seen that AmK = - AFK/2. Also for kaons, Im rK = 0. This follows because only The very recent experimental result by the NA31 Collaboration at CERN [65] provides us with u quarks can contribute to kaon decay (as opposed to c and t quarks). But Vud and Vus can be chosen the value to be real, so that Im TK = 0 for kaons. Hence we obtain e'/e = (3.3 ± 1.1) X 10-3. (7.46) /4 e = e" [(Im M12)/(V2 Am)] . (7.39) An explicit calculation of the first of the two box diagrams in Fig. 10 leads to the following effective Actually the experimental definition of e is different from that given by Eq. (7.2) because of a four-fermion operator [66] : different phase convention. The measured value of « [Eq. (7.34)] corresponds to the definition 2 H«ft = (GI/4T ) m? [A(t|)/i,] X? d yf [(1 - yi)/2] b d 7* [(1 - y5)/2] b , (7.47) + + ij+- = A(KL-> T T")/A(KS-T 0 = t + t' , (7.40) with t) = m?/mw. 7,00 = A(KL - x°7r°)/A(Ks - *\°) = € - It . Once the effective Hamiltonian has been written in operator form, there is no need to compute the second diagram. Note that, in fact, in the limit of vanishing external momenta the loop integrals The two definitions of e coincide for the phase choice Im Ao = 0, where Ai = |AjJe'*I = A[K° -» are identical in the two diagrams. The contribution of the second diagram corresponds to a different Will I = 0, 2 being the isospin. Instead, the phase definition chosen here was specified by the contraction of the fields in the Hamiltonian with the external quarks. We will take both contributions requirement that |K°) = CP|K°). By defining into account when computing the matrix element (Bo|Hcrf|B°). We now compute the above matrix element in the vacuum saturation (or valence) i = ImAo/ReAo, (7.41) approximation. When the matrix element of a V-A current is taken between the vacuum and a pseudoscalar meson, only the axial current contributes (one cannot make a pseudovector out of only the correction to Eq. (7.37) is given by [64] one momentum pA). For ir -> \iv, we define iT/4 t = (e /N/2)[(Im M12/Am) + £] . (7.42) (0|Û7,y5d|T> = (i p, fj/jâ^). (7.48) Similarly With this normalization, experimentally f, = 130 MeV (similarly, fa = 160 MeV). Thus, by restricting the sum over a complete set of intermediate states to the vacuum only, we obtain in the B° (f'/e) = (l/V2)(l/e) exp (i[ô2 - ô0 + (x/2)])[(Im A2/Re Ao) - (Re A2/Re A0) £] . (7.43) rest frame, i! s + (x/2)1 /4 0 Note that experimentally 62 - 60 + (r/2) » (48 ± 8)°, so that e ^" ° = e" . The experi (B |jJ'lB°) = 2 V4 (1 + 73) (l/2m„) ml f|. (7.49) mental validity of the AI = V2 rule implies that We now explain, one by one, the factors that appear in the last expression. The factor of 2 is there oj = Re A2/Re Ao = 0.045 . (7.44) because we can choose the current on the B° side in two ways. The factor of '/» is there because each current contains the projector (1 - 75>/2. The factor V3 arises because the vacuum state can be According to Eq. (7.43), e' is determined by the imaginary parts of the AS = 1 K -» TT-K inserted in two ways. This corresponds to the two original diagrams (a) and (b) in Fig. 10, which amplitudes. These are zero in the Standard Model at tree level. At one loop, the imaginary parts are differ by the exchange of the two b-fields. A (V-A) ® (V-A) product of currents is invariant under generated by the famous penguin diagrams (Fig. 13). These diagrams contribute only to the AI = V2 Fierz rearrangement for colourless quarks. For coloured quarks, the colour indices have also to be amplitudes (hence to Ao and not to A2), so that Im A2 = 0. Thus rearranged, and we obtain [see Eq. (7.23)] A A |e'/e| = (1/V2e)a.|£| = 14|£| (7.45) diL7»b,L d2Lyb2L = V3 diL7»b2L d2L7*biL + 2 S diL7»t b2L d2L7*t biL , (7.50) A B AB (e'/e is nearly real and is positive for £ negative, as is most probably the case in the Standard Model). where tï} are the 3x3 colour generator matrices with normalization Tr t\ = V2 S [to derive Eq. (7.50) is a nice exercise for you to do]. The octet-octet term cannot contribute when each octet is sandwiched between the vacuum and the colour-singlet B state. Thus the two contractions, differing by bi <-* b2, would contribute (1 + 1) = 2 for colourless quarks and (1 + V3) = 4/3 for triplet coloured quarks. Thus for coloured quarks, we set u.c.t \ /u.c.t 2 (B°|[d7, (1 -75/2) b] |B°) = V, BB f| mB , (7.51) 9 q —« '—»— q Fig. 13 Penguin diagram with gluon exchange 89 90 91 92 2 3 order-of-magnitude estimates lead to (Am/r)D = 10 -10 . Experimentally, at 90% CL, TD * 1.0 1.4% (ARGUS) and rD « 0.6% (TPS). 0.1 Summarizing, for B mesons the t-quark contribution is by far the only one that matters. For kaons, the charm contribution, similar to Eq. (7.52), is sufficient for Re M12, whilst for Im M12 i 0 6 (which determines e), we have also to compute the term with c and t quarks on the two sides of the box. The calculation is entirely analogous to the one performed here, and the result can be found in o.t the literature [67]. 0.2 8. B°-Ë° MIXING IN THE STANDARD MODEL By combining Eqs. (7.29) and (7.52), we finally obtain the t-quark exchange contribution to 0 50 100 150 200 250 m,., IOeV) Am/r for a Bq meson (q = d, s): 2 2 xq = (Am/DBq = (GÊ/6* ) m? TB, BBq f|q mBq |VtbV;q| [A(T,)/T,] Î)QCD , (8.1) 2 Fig. 14 The function A(ij)/i) (ij = m /mw) defined in Eqs. (7.47) and (7.52) where A(r;)/i] is given by (see Fig. 14) [A(,)/,] = RI/4) + (9/T,)(1/1-„)- (3/2) [1/(1 -,)2] -(3/2)[(,,2ln„)/(l-»,)3]]; (8.2) where BB is a factor that is inserted to take into account all possible deviations from the vacuum saturation approximation. Finally, we have and T)QCD is a QCD corrective factor: 2 M,2 = (G|m?/127r ) BB fB mB X? [A(,)/T,] , (7.52) 6/23 4/7 DQCD = [as(mb)/as(m,)r [(3/2) [as(m,)/as(mw)r (8.3) where A(TÏ)/IJ is a slowly decreasing monotonie function of rj, plotted in Fig. 14, which is 1 at IJ = 0, 2 7 8/, 3 - MmO/asOn*)] ' + (1/2) [as(m,)/as(mw)] î • /4 at i) = 1, and 'A at TJ -> ». The values obtained for the mixing angles and the calculation of box diagrams allow some This formula has been derived for mb < mt <« mw. Strictly speaking, perturbative QCD with a given important semiqualitative statements. number of massless flavours can only be applied far away from quark thresholds. In the regions 1) For kaons, the dominant charm contribution to AMbox is proportional to (Vc V )2m2, d cs across the thresholds, we can only guess at reasonable extrapolations. whilst the t-quark contribution is proportional to (VÛV, )2m?. Since |v; V | = X, whilst |V«V | = s d cs U When m, increases up to mw and beyond [not too much beyond, because we know that m, « AX5|l - Qe~'*\ the t-quark term is negligible for all practical values of m, (i.e. for m, * (2-3)/mw], the interplay between the logarithmic terms for the anomalous dimensions and the mass rr^/X" = 500-600 GeV). terms [the function A(IJ)/IJ] becomes more complicated. The logs resummed by TJQCD are still there. The expression for IJQCD will change a bit but not much (for example, the running will occur all the 2) Going back to Eqs. (6.5) [or by unitarity, plus the fact that |VÛdVas| is real], we see that way with f = 5). However, the function A(T))/); could be deformed by additional QCD corrections 2 5 because I?QCD is the right multiplying factor in the limit D -+ 0. However, since A(t|)/ij is a slowly Im(|Vcdvy) = -Im(|Vldv;s|) = A X e sin* . varying function and TJQCD is almost a constant (TIQCD *> 0.80-0.85) for m, * 40-200 GeV, it is As presumably correct to use the product VQCD[MV)/V] for the whole physically interesting range of mt. Recall that rq = x|/(2 + xq) [Eq. (7.25)]. The first thing to observe is that the ratio xd/xs can be Im z2 = 2 Re z 1m z , predicted with little ambiguity. From Eq. (8.1), we in fact obtain then 2 x,i/xs = |V,d/Vts| [1 + SU(3)n breaking] , (8.4) [Im(|V vy)2/Im(|V V* |)2] = - [Re(|V v; |)/Re(|V v; |)] = -A2X4(1 - cos 0). (7.53) td cd s I(1 s cd s e where the d •" s symmetry-breaking correction arises from all quantities which in Eq. (8.1) carry a q label, apart from the mixing angles that are explicitly factored out. Although the corrective term may As a consequence, the t-quark contribution to t (i.e. to 1m M12) is important as soon asm,* m /X2 c well be sizeable (and is probably negative), it is safe to state that the main factor in Eq. (8.2) is the = 25-30 GeV. Note that the short distance approximation is reasonably justified for e and less so for 2 ratio |Vtd/V,s| « 1/5. Namely, xd is Cabibbo-suppressed with respect to xs (in fact, the present Amic. bound roughly corresponds to |Vld| « 29c |VK|). This is why rd was expected to be small. Then from 3) For charmed mesons (D° = cu), the D°-D° mixing is predicted to be very small and cannot be the experimental value of rd, we immediately deduce that rs must be near one. Precisely, rd > 0.13 2 computed reliably by the box diagram. In fact, the b term is proportional to (Vcbv;b) mb, hence it is implies rs > 0.75-0.80, depending on the assumed input maximum value of |V,d/V,sj. The resulting 2 2 so small that the (non-perturbative) s-quark exchange of order X ms can still win. Typical Table 3 (90V.C.L Limits) •Slandord Model BK: 0.33 ± 0.2 SU(3) + chiral inv. + experiment [72] Fig. 15 The experimentally allowed region in the x [x is defined in [Eq. (7.19)] and the CKM model constraint. The fl <;+0.37 U.D_o.l7 Lattice QCD [70] shaded area is the region allowed by the present experiments and the CKM model. 0.70 ± 0.07 1/N expansion [74] 0.1 0.2 v 03 Ot 0.5 *s BD: 1 +0.25 1-0.15 Lattice QCD [70] allowed region in the rd-rs plane is plotted in Fig. 15 together with the available experimental information. We see that the MARK II limit [68] is potentially dangerous for the Standard Model 0.98 ± 0.25 Lattice QCD [71] with three families. However, the interpretation of the MARK II results in terms of rd and rs requires an assumption, on the probabilities Pd and Ps, that the produced b quark picks up a d or s BBd: companion. For example, the allowed region for the Standard Model with three families is not 0.98 ± 0.25 Lattice QCD [71] greatly restricted by the MARK II result if Pd = 0.35 and P, = 0.10 (which, however, leaves a rather generous 20% for be and b-carrying baryons). estimated mainly by QCD sum rules and lattice calculations. The available results are collected in The value of x is more uncertain. We can write the approximate expressions that are valid for d Table 4. The tentative conclusion that was derived in Ref. [75] is given by [76-82] mt ^ mw: 2 BBd fBd = (140 ± 40) MeV . (8.9) 2 16 2 2 x„ = 0.15 [(rB|Vld| )/(3.3 X lO" s)][(BBd fid)/(0.14 GeV) ][m,/40 GeV] (8.5) The implications on m are illustrated in Fig. 16. It is obtained that almost certainly m > 45 GeV. or t t The most likely range for m, is given by 90 GeV < mt < 150 GeV. 2 2 2 xj = (0.26 ± 0.05) (1/4) (1 + e " 2ecos4>)[(BBdfid)/(0.14GeV) ][mI/40GeV] . (8.6) Table 4 f, = 130 MeV, fK = 160 MeV Recall that experimentally ARGUS and CLEO [69] find rd = 0.21 ± 0.06 and xd = 0.73 ± 0.10. (8.7) fD: - 220 (MeV) QCD sum rules [761 1/2 The main unknowns are TB]V,d|\ B fB, and m,. The errors on rB and |Vtd| are related, so the two - 165 QCD sum rules [77] factors should be kept together; |V,d| is maximum for an extreme value of the Kobayashi-Maskawa - 170 QCD sum rules [78] phase. However, at extreme values of the phase, all CP-violation effects vanish. The experimental 170 ± 20 QCD sum rules [79] values of t and e'/t therefore impose some constraints on how large |V | can be. Only mild td 220 ± 25 QCD sum rules [80] restrictions are found (which become weaker with increasing m ) if we take B « 1 and |V /V | « t K bu bc 180 ± 25 Lattice QCD [71] 0.2. With these assumptions we find for T |V |2 the range [see Eqs. (6.8) and (6.11)] B ld 128 ± 25 Lattice QCD [72] <290 Exp. MARK III [81] 2 6 7-B|V,d| = (0.02-6.2) X 10-' s . (8.8) fB: The factor BBf| contains all the uncertainties arising from the hadronic matrix element. The ~ 140 QCD sum rules [76] parameter B should approach one when the mass of the meson increases. In fact, the vacuum - 95 QCD sum rules [77] saturation approximation should be better when gluon emission from the quark legs is inhibited by - 130 QCD sum rules [78] large transferred momenta of order of the meson mass (Zweig rule). This trend appears to be 190 ± 30 QCD sum rules [82] supported by some recent lattice evaluations of B and B [70, 71]. 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It 1.1 States of ths Vacuum may even be that we have already found such an example in the discoveries presented in Chapter 5. There is powerful evidence for the existence of quarks and gluons from scattering The experiments relevant to the physical vacuum fall into two distinct categories: those experiments involving nucléons, pions and other hadrons. It is widely accepted that the failure to relating to deconfinement at high energy densities and those seeking evidence of the specific observe free quarks is due to the very large energy which would be necessary to place such a response of the physical vacuum to colour excitations, and these are dealt with separately in the particle in the physical vacuum, the ordinary vacuum found in the laboratory [1]. An attempt to following sections. draw a free quark into the vacuum then creates particle-antiparticle pairs so that the net colour quantum number remains zero, and only the ordinary hadrons remain. This picture is supported 1.2 High Energy Density Phenomena by attempts to evaluate the states of quantum chromodynamics (QCD) numerically, but reliable results are so far difficult to obtain. The origin of this effect is the highly complicated but ordered 1.2.1 Successive Stales of the Vacuum states of virtual excitations of the physical vacuum in QCD, analogous to vacuum polarization in Here we will describe in a little more detail the successives states through quantum electrodynamics, but so strong in this theory as to produce a qualitative change in the which our universe is claimed to have passed, and which we hope we can produce in a physical vacuum from the simple empty state of a world with very weak interactions. laboratory. We must deal with volumes which are sufficiently extended and long-lived to be described as fluids with well defined properties. We then expect to find at least three Furthermore, the theory indicates that this ordered state of the physical vacuum can be different fluids in successive stages of expansion of a high energy density state. This is destroyed by the presence of a sufficient density of quarks or gluons, reverting to a state better shown in Fig. 1 as snap shots of a small region at different times. At the highest density described as a simple vacuum, usually referred to as the 'perturbative vacuum', with quarks and we see the gas of deconfined quarks and gluons known to be predicted by QCD in very high gluons moving freely, except for their mutual interactions. This high density of particles, or energy density conditions. Since these particles carry colour charge, we may describe it equivalent^ a high energy density, must be present for a sufficiently long time for this change in as a plasma, the quark-gluon plasma (QGP). Although there are none of the usual hadrons the vacuum to develop. present, there could still be excitations corresponding to clusters of quarks and gluons carrying the quantum numbers of hadrons. At the other extreme, at low energy density, Since the energy density in our universe was high at short times after its formation, the we see a gas of hadrons, of mesons and baryons. This gas is a system whose properties can universe should have retained the perturbative vacuum up to times of about 10~6 seconds, then presumably be predicted reliably at low density, since we know the low energy scattering completing a transition to the physical vacuum containing a gas of hadrons at around 10"5 seconds. amplitudes experimentally. The hadrons condensed to nuclei at a much later time. This phase transition can have important consequences for the limit on the allowed baryon density in the Universe, and thereby on the In terms of QCD, this gas of hadrons consists of quarks grouped in colourless question of "dark matter" [2]: it can be that the density fluctuation associated with a first order combinations, maintained by the confinement mechanism of the physical vacuum QCD phase transition changes the nucleosynthesis in the universe in such a way as to require no described earlier. A phenomenological picture of this confinement which allows one to dark matter! draw an easily understandable picture, is the bag model. In this model the quarks are confined to small cavities in the physical vacuum which contain the perturbative vacuum, The Physics described above has little direct experimental support. The direct tests of QCD the bags. In this picture the hadron gas consists of the physical vacuum with a small are at high energies, where the confinement effects play no direct role, being handled fraction of its volume occupied by the bags containing quarks. When this phenomenology phenomenologically as "hadronization". It is almost universally accepted that the structure of QCD is used to explain the observed distributions of hadron masses, one concludes that there is is so well defined that there is little doubt that the above picture is correct, and that experimental a large energy associated with the creation of a bag. In the states of thermal-equilibrium 101 102 103 104 shown in Fig. 1, this bag energy is equivalent to a large latent heat needed to go from the An object of initial experiments is to show how efficient high energy projectiles are in state with a physical vacuum to that with the perturbative vacuum. A mixed phase will creating the desired states. The measurements related to energy deposition are relatively necessarily arise in the cooling of the quark gluon plasma in this case, which will contain straightforward and have allowed results to be obtained in a short space of time. both deconfined quarks and hadrons. To draw a picture of this state requires much more knowledge than we possess at the present time, but the ingredients it must contain are 1.2.3 Measurements Sensitive to the Evolution of High Enemy Density States shown in Fig. 2. Here we see different regions of physical vacuum imbedded in the The observables which are sensitive to the nature of the state produced are original perturbative vacuum. Inside these regions we see bags, corresponding to difficult to measure and the results of the recent experiments are not yet complete in this hadrons. In the regions of perturbative vacuum which are simultaneously present, we see respect. In these lectures I shall emphasize the simpler observables such as the quarks and gluons which are not confined but still have strong mutual interactions. If the transverse momentum and angular distribution of different particle types and the phase transition has indeed a large heat, it is a first order transition and the phases must information on the space time structure which can be obtained by studying the intensity be spatially separated as shown. We can only speculate about the characteristic size of interferometry by correlations of two identical particles. In principle these these regions and the nature of their boundaries. We note that the temperature of this measurements are sensitive to the collective expansion of the system, and can measure the state is held constant by the latent heat of the confinement transition while it expands. spatial distribution of the matter at the time the pions are formed, and the length of time over which they are emitted. This can, in turn, give us information on earlier stages in 1-2.2 Collision of the High Energy State in the Laboratory the evolution of the matter. I shall not discuss the important questions of the thermal There may be signs of the transition between the two vacuum states in the radiation of photons, or virtual photons appearing as pairs of leptons, or of the effect of early period of universe, still existing today [2], but we are interested in investigations the environment on the production of bound states of quarks, such as the heavy vector in the laboratory. The only way to achieve the energy densities required, that is resonances. comparable to the energy density of a hadron, of order 1 GeV/fermi3, is in the collisions of high energy particles or nuclei. Our strategy is to find suitable high energy collisions 1.3 Response of the Vacuum to Soft Excitations which provide a high energy density over a sufficiently large volume in order to obtain the needed thermal equilibrium and long lifetime. For a given accelerator, it seems clear The previous section describes attempts to destroy the order of the physical vacuum and cause the that we will gain in this respect by colliding nuclei rather than hadrons. These transition to the perturbative vacuum by increasing the temperature or quark density of the considerations have led to programmes of physics with nuclear beams at the CERN SPS and environment. On the other hand many of the most characteristic and interesting condensed matter the Brookhaven AGS [3]. phenomena occur at low temperature. In this section, we turn our attention to the possibilities for investigating the response of the physical vacuum in its normal, cold, environment. The techniques of equilibrium thermodynamics demand systems large compared to the scattering length of the constituents. Because of our inability to calculate We might think of doing that by the precision spectrometry of quark-antiquark states, such as in non-perturbative QCD, we do not know the interaction length of the soft constituents the JA|r. Unfortunately, it has been found that the energy levels of the observable spectrum are not very and consequently we cannot say how large a system is required. This must be determined sensitive to these questions [4]; in essence, as the energy of two quarks is increased, the continuum from experiment, using arguments described in the next section, where we will find that arrives before the states are sensitive to the details of the vacuum outside the bag. Use of hadron high energy collisions of hadrons certainly show some signs of thermal equilibrium, but scattering for this purpose seems even less likely to succeed. We know that the hadrons and hadron equally certainly are not fully in thermal equilibrium. This can lead us to hope that a resonances can be described to a good approximation as elementary, colourless objects, so that their relatively small increase in the size of the projectiles, for example moving from one to excitations of colour modes in the vacuum must be very weak. tens of nucléons, might be sufficient to attain our goal. These considerations show us that we must search for experiments where free colour is exposed An element of pessimism was present in the early consideration of whether in the interaction, and then look for probes which are somehow distinguishable from the complexities of the initial collisions between two nuclei would adequately transfer energy into a system ordinary hadron interactions. The prototype of a reaction where their colour is certainly present is the not too far from thermal equilibrium. This concern reflected a knowledge that as hadron production of quark-antiquark pairs by the annihilation of e+e-, Fig. 3(a), the quark and antiquark energy increased the absorption of the energy in the nuclear target became less complete. carry opposite signs of colour. The quarks evolve into a cascade of quarks and gluons in the process of hadronization, for which there is no real theory, and the colour is gradually neutralized as hadrons are energy to be retained by a single baryon in the final state. This indicates a tendency for the incident formed. For some time, until this neutralization has taken place, the physical vacuum is exposed to nucléon to retain its identity and a good fraction of its energy, as the "leading baryon". The rest of the colour excitation multipoles around the creation point of the quark-antiquark pair. We could imagine energy is carried by a relatively large number of particles, usually pions. The situation is then selecting events where there were no hadrons along a specific segment of the quark path thereby qualitatively the same as that for an electron losing energy by bremsstrahlung, which forms a useful selecting events with a certain degree of coherent colour excitation. Somewhat less clean, but simple picture. Consider an electron which enters an absorber with a unique energy. As it traverses the equivalent, colour excitation must be present in any high energy interaction which generates the quark absorber, its average energy falls and it develops a spread in energy due to the fluctuations in energy gluon cascade and hadronization, in distinction from diffractive reactions that merely excite hadron loss. After about 0.7 radiation length, its energy distribution is approximately uniform from the initial resonances. energy down to zero. Deeper in the absorber, the energy distribution peaks at zero with an asymmetric tail toward higher energies. After a proton-proton collision, the energy spectrum of the leading baryon Given the availability of colour excitation, where can we find distinctive probes? The first thing resembles that of an electron which has traversed about half a radiation length. This analogy remains to notice is that the effects we seek must necessarily be those with wavelengths larger than the size of useful when the target is a nucleus rather than a proton, corresponding to a thicker absorber. The the bag, of order one fermi, because there is no physical vacuum inside that volume. This argument can baryon distributions then peak well below half of the initial value, for a heavy target nucleus. be reversed, there are no conventional hadronic effects with wavelengths larger than the size of hadrons, except for the decay of long-lived resonances, a process which can be taken into account. It seems clear 2.3 Kinematics then that we must seek effects which have wavelengths which would be considered anomalously long for ordinary hadronic interactions. The most obvious and probably the cleanest probe is the radiation of If the incident energy is very high, the average energy of the leading baryon can still be quite direct photons with wavelengths greater than one fermi or so. In this case we have the advantage that the relativistic even if it has lost a major fraction of the incident energy. For this reason it is more background from hadronic interactions indicated in Figs. 3(b) and 3(c), can be calculated with some convenient to use a different variable to describe its energy distribution. The variable most suited for precision. this situation is the rapidity variable defined by E+ p y = —1 ,o g / z-l Investigations on these lines will be reported in a later chapter and indeed some interesting y 2 (E- Pz) effects have appeared. It is seen that y is a logarithmic energy variable, and has the property that the shape of 2. ENERGY DEPOSITION distributions in this variable are invariant under Lorentz transformations along the beam axis, z. 2.1 Introduction To illustrate its use we show in Fig. 4(b) the y distribution of an electron after 0.5, 1. and 1.5 radiation lengths compared with the energy distributions in Fig. 4(a). It is seen that the shape of the y The experimental study of energy deposition is the first task in the study of high energy density. distributions is not changing as drastically as in the energy variable, as the absorber thickness is It is also a subject of interest in that it allows us to learn something about the space time evolution of changed. The energy loss of the electron can be described by a shift Ay between the rapidity of the high energy particle systems which have interacted one or more times. This can be studied already with incident electron and the peak of the degraded electron distribution. hadron-nucleus collisions, whereas in the nucleus-nucleus collisions we can study the space-time evolution of more complex configurations which have already been disturbed by the previous collisions. The energy which is lost by the incident baryon reappears mainly in the form of many "soft" Since there are phenomenological models for these subjects but no real theory, the information we mesons (i.e. with energies small compared to the beam), mostly pions. It is convenient to describe the obtain is of basic interest. many particles produced by a global variable which is directly measurable and closely related to the energy density produced, the "transverse energy" defined by 2.2 Kinematics and Geometry ET = ]TEiSin8; Already from the experiments on proton-proton interactions, we see most of the physical where i is summed over all particles. This quantity can be measured directly by a "calorimeter" or total features which will be important in the more complex reactions. The first of these is that despite a very absorption counter, which absorbs all the energy of each particle and provides a signal proportion to its wide fluctuation from event to event, there is a strong tendency for a major fraction of the incidenl "available energy". The calorimeter must be segmented into many small elements, each characterized by 10? 106 107 108 polar angle 6j (and usually the azimuthal angle <>). The elements may then be summed with weights calculations which include phase changes between different states, accurate description of sin9ii and if its output is linear, the sum over particles is achieved automatically. This technique works the meson gas, and so on. They are useful for a qualitative interpretation of data described no matter how large the number of particles in the interaction may be. The measurement yields the by global variables. "energy flow" in two dimensions. More convenient than 6 and $ are T\ and "pseudorapidity", a logarithmic angle variable. 2.5.2. The Fermi Model Here some fraction of the initial centre-of-mass energy is transferred to a r\ = 1 log 1 + œse = - logtane/2 system which can be described as a fluid which dissipates isotropically, in some 2 1 - cose combination of radial expansion and thermal motion [6]. In the pseudorapidity Note that for pj » m, t\ = y, explaining why it is a most convenient variable, though since the peak of the approximation, the distribution of transverse energy flow is then PT distribution of the produced particles is near m, care must be exercised in its use. ——î= (hyperbolic COS(TI)) 2.4 Formation Zone dn It was long ago discovered that this model does not describe the particle It is natural to ask why the emphasis is put on the energy loss of the leading particle: should not production in hadron-hadron collisions at any energy. the produced particles also lose energy as they traverse a large nucleus? After all, we are familiar with the development of a cascade shower when an electron traverses an absorber, but is precisely in the case 2.5.3. The "Shurvak-Riorken Model" of the electromagnetic cascades that it was pointed out that the energy loss of the produced particles is A free expansion at velocity c along the beam direction, appropriate for very suppressed at sufficiently high energies: the formation zone effect of Landau, Migdal and Pomeranchuk. high energy where the produced particles are distributed over many units of rapidity, From uncertainty principle arguments it is clear that for sufficiently large Lorentz time dilations, the while the average transverse momentum, pj, is small and independent of rapidity [7]. time At between the successive collisions between target particles is less than the time AE/*t required Both dN/dy and dEy/cm. are constant in this model, while dN/cfn has a dip of kinematic for the produced particles, characterized by some appropriate energy AE in the rest frame of the origin. The distribution of particles produced in pp collisions at centre-of-mass energies incident particle, to manifest themselves as distinct entities. of 540 GeV fits these distributions, as is predicted by the multiperipheral model of hadron interactions at very high energies. This leads to a suppression of cascading at large incident energies. In the case of hadronic collisions, it is major effect throughout the energy region of interest, since AE must be of the order of 2.54. The "Landau Model" m , and m^/hc is comparable to the spacing between target nuclei. It becomes important for n The deposited energy is left in a disc, with dimension along the beam small electromagnetic cascades above present accelerator energies, however. compared to the transverse dimensions [8]. This disc then expands longitudinally, at a rate large compared to the transverse, expansion, in such a way that the PT of the particles The effect is illustrated in Fig. 5 [5], which compares the distribution of charged particle observed in different rapidity "regions is the same. The dN/dy or dEy/dii is a Gaussian. It multiplicity for hydrogen and lead targets. For the high energy half of the rapidity range, the collisions describes well the nucleon-nucleon data at centre of mass energies up to 63 GeV. The with lead show no higher multiplicity than those with protons, while a cascading of secondaries would previous two models represent approximate limiting cases of this one, at low and high have increased the multiplicity. energies. 2.5 Fate of Deposited Energy 2.6 Energy Deposition bv protons 2.5.1. Several very simple pictures have long been used to provide a basis for We examine first data on "stopping", the rapidity shift between the incident proton and the understanding the gross feature of the produced particles, especially their energy and leading proton exiting from the collision, AY. The comparison between proton-proton and proton-lead number flow. Although this is properly a dynamical question, these simple models collisions is shown in Fig. 6 in terms of AY [9]. There is indeed much more energy lost in traversing the emphasize a kinematic picture, appropriate for different cases. They can be described as lead nucleus. In interpreting this figure, it must be remembered that geometry plays an important role kinematic hydrodynamic models, to distinguish them from real hydrodynamical in the hadron collisions with a heavy target, as we learn immediately by the fact that the total cross- 2 3 sections vary as (ATAR) ' If we treat the target as a sphere containing A nucléons, and ask the mean 1 djr number of collisions a proton will make with the nucléons in the target, v, assuming it travels always along the incident direction, a geometrical calculation leads to v = Actpp/o-pA, and for lead v = 2.7. It is clear that many of the p-A interactions correspond to collisions with the outer part of the nucleus, with T0 an unknown constant, usually taken on dimensional grounds to be one fermi/c. If we follow this where v is closer to one, so that the central collisions, where v > 3, must have larger energy losses than practise, we may insert the largest values of dEjldr\ obtained in the data corresponding to sulphur- in the average represented in Fig. 6, leading to the two bands shown. The conclusion is that a central uranium in Figs. 10 and 11, dEj/dTi = 140 GeV and obtain e = 3 GeV/fermi3. collision with a lead nucleus takes away on the average considerably more than half the energy of a proton. This energy appears in the form of produced particles, as shown in Fig. 7 [10], showing Ej Aside from the lack of knowledge of the value of x0, this estimate suffers from the obvious integrated over rapidity, for proton-lead collisions. A feature of this curve is that it extends far beyond disagreement of the model with the shape of dEj/dri, a narrow Gaussian instead of a wide plateau. the kinematic limit for ET produced in p-p collisions, showing the importance of collisions with many Anticipating the result of 3.1, that the pj of the produced particles is independent of y, at least for 200 target nucléons. Superposition of independent nucleon-nucleon collisions can reproduce many features GeV/nucleon beams, we see that it is appropriate to use at least the kinematics of the Landau model (the of the data, but not all, for example the dEy/dii distributions as shown in Fig. 8 [10]. Important dynamical aspect is irrelevant here). In this way, it has been found that the observed highest values of cascading of the produced particles in the target must be present, showing that it is not much suppressed dEj/dTi correspond to essentially all the energy of the incident beam being deposited in a Landau by the "formation zone" effect. "fireball". Since the energy is then known, one can try to compute t by an estimate of the fireball volume on geometrical grounds. Its transverse size is just the area of the smaller nucleus. The 2.7 Nucleus-Nucleus collisions longitudinal extent requires a dynamical assumption. If indeed full stopping has been achieved in the most extreme collisions, it seems correct to assume it has been Lorenz contracted as in the Landau model Here the gross features of the collisions are even more fixed by the geometry. Consider the proper. The e obtained could be as high as10 GeV/(fermi)3 [13]. Either of these estimates is large interactions of nuclei with matter density given by the charge density measured in electron scattering. compared with the energy density of a hadron, justifying the hopes that the high energy nuclear Now compute the distribution of the number of nucleon-nucleon collisions by convoluting the two collisions would reach a new regime of energy densities. To investigate whether the equilibrium implied nuclear density distributions for random geometries, and using the free nucleon-nucleon cross-section, by such a large energy has indeed been achieved, we must proceed to study more detailed observables. to yield Fig. 9 [11]. The collisions with few interactions correspond to a grazing geometry of the two nuceli collisions, and the abrupt drop for high numbers of collisions corresponds to the case with the 3. SPECTPA OF CHARGED MESONS central axes of the nuclei aligned so that the maximum number of nucleon-nucleon collisions geometrically possible has been achieved. The unusual flat portion of the curve is just a property of the 3.1 Transverse Momentum geometry. In the previous chapter, we considered the most global description of high energy The data on Ej, Fig. 10 [12], show an obvious resemblance to the geometrical convolution collisions, the energy flow, and showed that interesting physics can be gleaned from it. The next step is indicating that a geometrical superposition of nucleon-nucleon collisions will explain the general to study the ensemble of particles which carry away the energy deposited. At high energies, these are features of the data. The dEj/dii distribution seen in Fig. 11 [12] shows that at 200 GeV/nucleon, we mostly mesons, pions and kaons. We often characterize them by their "Lorentz invariant inclusive are far from the ultra-high energies which would give a flat plateau. In fact, the data fit the Gaussian ot cross-sections" the Landau model, with a width only about twice as wide as that of the extreme limit of the Fermi model. J3 J3 d o _ da This will have important consequences in the next section. d 2 3P dyd pT In high energy proton-proton collisions, as well as in low energy nucleus-nucleus collisions, the 2.8 Energy Density data show a greater regularity when plotted against mj= (Pj + m2)1'2 rather than pj. We show this in Figs. 12 [14] and 13 [15]. We see that the curves are good fits to exponentials when using mj, and The purpose of the nucleus-nucleus collisions was to reach high energy densities, e. even with the same slope in the case of the p-p collisions, whereas there is a distortion for py < m, Measurement of the true energy densities achieved will require subtle measurements, but rough essentially due to a phase space factor, where pj is the variable. This good fit to an exponential has a estimates may be attempted from the global variable dE/di\. Bjorken [7] pointed out that in the natural explanation, in that a thermal distribution has just this form in these variables. We see at conditions of the model of 2.5.3, one can compute a scaling of the energy density 109 no Ill 112 higher mj, the cross-section for high energy p-p collisions become flatter, approaching a power law to display better the small differences in the spectra. We see that the changes from p-p to p-w are behaviour expected for the hard scattering of individual quarks, a regime we shall not discuss. enhanced for heavier projectiles, but the main conclusion is a substantial constancy in the mid-pi range for a great variation of beam and target. It is not at all certain to which degree the ensemble of particles is really thermal, and further, in most of the cases discussed below, there is not a simple exponential even in the low mj region. We shall In Fig. 16, the dependence of the average py in the mid-region on the Ey measured in the not speak of temperatures, but of the slope of the distribution, between defined limits. This would be associated event is shown [16]. Again the main impression is of a remarkable stability of the mid-pj proportional to 1/T in the truly thermal case. slope. The small increase noted is compatible with the change in the influence of the Cronin effect as the geometry of the collisions is changed by varying ET: the more peripheral events should resemble p-p 3.2 Collective flow collisions. It will be noticed that the slopes for pions and protons are different in Fig. 13 but the same in Fig. A fit has been made to the pion py spectrum from oxygen collisions in the streamer chamber, 12. The difference in the case of nuclear collisions is known to be due to a collective outward flow of the with a model incorporating a more complex pattern of radial flow than a simple boost, superimposed on a hot matter produced in the collision. This is inevitable in the expansion of a hot system into vacuum. thermal distribution [18]. The radial velocity is not a constant but depends on the square of the distance The change in the distribution due to giving a thermal distribution a fixed radial Lorentz boost, with a from the central point in the fireball, reaching vc at the outer edge. The result is no longer an velocity of vc, has been calculated by Siemens and Rasmussen [15]. For a massless particle, the effect is exponential but a curve which can fit the data, if vc attains the rather large value of 0.78 as shown in just to increase the inverse slope, Teft, by a factor yc (1-vc) cos h (yC). For a massive particle, for an Fig. 17. The streamer chamber has measured neutral strange particles as well, and in Fig. 18, their py initial Tj « m, the effect is to produce a peak at the energy corresponding to the radial velocity vc. A fit spectrum is shown as well, with a fit to the radial expansion with the same velocity and temperature. to pions and protons to Fig. 13 gives vc = 0.3c, and the Teff for pions from the inverse slope, 100 MeV, The agreement is satisfactory. corresponds to a thermal T; = 30 MeV. This is an instructive cautionary lesson on interpreting inverse slopes as temperature, but also is most illuminating in regard to the dynamics of the collision. The alert reader will have noticed that the curvature in the spectrum which is here fitted by the form of radial expansion is very similar to the Cronin effect in p-nucleus spectra which is usually The fact that no difference in slope is seen between pions and protons produced in high energy p-p interpreted as a hard scattering effect. There seem to be three possibilities. Either the Cronin effect is collisions is susceptible to two explanations. Often it is assumed that there is just not a system of also really a radial expansion effect, or the close resemblance between the p-nucleus and oxygen- sufficient size to produce the degree of equilibration necessary to give radial flow. Another possibility is nucleus spectra is accidental, or the curvature is not really a radial flow effect. that in this system the energy is carried by quarks and gluons which are not free to expand, due to the confining effect of the physical vacuum surrounding the collision. If this is the case, making the system 3.4 Thermal and Flavour Equilibrium larger will not lead to the appearance of radial motion. Since we do see radial motion at low energies, there would have to be some energy region where the radial motion disappeared, more or less sharply. In considering what conclusions to draw from these measurements, we draw attention to the arguments of E. Shuryak [19], who proceeds from examining the hypothesis that the particles 3.3 Results with nuclear targets and beams produced in hadron-hadron collisions are far from thermal equilibrium. One would then expect as the size of the collision volume and the number of particles in it are increased, that there will be more Measurements of the negative particle (mostly pion) spectra from proton and nuclear beams on a collisions between particles in the final state, inevitably moving the particles toward thermal 16 W target are shown in Fig. 14 [16]. The proton-proton results are shown as curves with arbitrary equilibrium. Since estimates show that the number of final state collisions in 0-W collisions should normalization. The difference between the nuclear target data and the p-p spectrum is similar for the be considerably larger than in p-p collisions, on the hypothesis, we should then see a considerable different beams. We may say that the slope is the same as in p-p for the region around pj = 300 MeV/c, change in the pr mid-region unless the non-equilibrium hadron distribution and the thermal where the majority of the pions are found, with a region of steeper slope for pj = 100 MeV/c, and a distribution were the same by coincidence. The precision of the constancy of the pj slope with beam, flatter slope at high p-r. The latter, "Cronin effect" [17] is usually assumed to be a hard parton target and ET is such that such a coincidence seems unlikely, and thus we are tempted to conclude that the scattering phenomenon related to multiple quark interactions and the modified nuclear structure distribution in nucleon-nucleon collisions must be close to thermal. function inside the nucleus. In Fig. 15, the ratio of the spectra for the different beam particles is shown This conclusion can be strengthened by independent evidence for the multiple interactions speaking, the behaviour of the correlation of the spatial component of the difference of two meson four- characteristic of a large system. Such evidence is available in the evolution of flavour with interaction momenta gives the spatial dimensions of the source, while the energy difference gives the lifetime. volume [20]. The rate for production of strangeness and charm are relatively slow, so it is much more Further consideration is required of coherent motions and the effect of long-lived resonances [27], likely that thermal equilibrium will be achieved in nucleon-nucleon collisions than that the strangeness or charm will reach equilibrium values, since each collision alters momentum, but only a small fraction The difficulty with applying this method is that only a small fraction <1%, of all pion pairs fall change strangeness. We may consequently expect to see strangeness enhancements even if no change in close together in phase space, in the region where the correlation is important. Few experiments have PT slopes is seen. This is indeed the case: at AGS energies, where the K+/n+ increases from = 5% proton had enough data to give accurate results, or to allow an investigation of Ihe dependence of the correlation collisions to 20% in 28Si-Au collisions [21]. In the streamer chamber experiment on S-S collisions, a on other event parameters. In Fig. 21 we show the result with the largest statistics, a p-p experiment strong increase in the fraction of strange particles with accompanying multiplicity is seen, as shown in [28], in terms of the two pion correlation function Fig. 19 [17]. Whether such an increase can be foreseen in a hadron gas, or requires the higher rates 2 dCT predicted in a deconfined state, involves models which need to be further constrained by other 12 measurements, but the strong effect of rescattering, and the equilibration is unequivocal [20,22]. C(Q). *ldP2 da, ca2 dpi dp2 Similar remarks apply to the striking suppression of J/ 4. PION INTERFEROMETRY In the case of hadron interactions, no information has been extracted from multi-variable analyses which was not already implied in the C(Q) distribution. Geometry is more important for nuclear collisions, and one may hope to measure three variables [29], One choice of variables is that The use of second-order interferometry using intensities rather than amplitudes was suggested shown in Fig. 22. The picture is drawn in a particular frame where, one pion is perpendicular to the by Hanbury-Brown and Twiss [23]. The idea can be explained for two sources of monochromatic light, beam axis. The second pion may then be along the beam, "longitudinal", along the direction of the first Fig. 20. The Bose-symmetry of the amplitude for two spin zero mesons of momentum khc requires the pion, "out", or the other direction in the transverse plane, "side". Hydrodynamic calculation shows that amplitude of the form the ratio of the two transverse dimensions is sensitive to the passage of the system through a first order Ayy= exp [ik (R-n + R22)] + exp [ik (R12 + R2i)l phase transition with a large latent heat. In that case, the system remains for a relatively long time in which, for small 9, gives the mixed phase while the deconfined quarks assemble into hadrons. The R0ut, which measures this time, I Ayyl 2 = 1+ COS (Lkq). can then be much larger than RS|DE, which measures the transverse size of the system at "freeze-out" the last interaction of the pions. Evidently, the symmetry of the amplitude leads to an interference term in the intensity even though the phase is lost. Observation of the oscillatory term in the two photon correlation function allows one to measure 9. This is of course a mainstay of radio astronomy and has also been applied in the optical The streamer chamber experiment has succeeded in the difficult task of measuring these region. correlations in the high multiplicity environment of 160-Au interactions, as shown in Fig. 23 [30], The results are given in Table 4.1, for two different rapidity intervals, 2 < y < 3, the region of the The idea can be applied to measurements with identical hadrons, but the details become much fireball, and 1 < y < 2, which may be considered a control region. The quantity A measures the strength more complicated [24]. The fact that the source is strongly time-dependent must be taken into account of the correlation: ,\ = 1 corresponds to a Gaussian with C(0) = 2. [25]. Coulomb and strong interactions corrections must be evaluated [26]. It turns out that, roughly 113 114 115 116 Table 4.1 2. The region between the projectile and the fireball is observed to contain a pion flux consistent with leading quark hadronization or with a Landau expansion. Future Pion source parameters extracted from the streamer chamber analysis of the 200 GeV/n 0 + Au -> rut particle composition studies may determine which description is more appropriate. sid e 0 correlation data with separation of RT into R T and R ^'. Rapidity intervals at midrapidity (2 < y < 3) and backward of midrapidity (1 < z < 2) were analysed. 3. The black region shows the initial volume occupied by the deposited energy in the Landau model, determined by geometry in the full stopping case. This leads to energy density of order 10 GeV/fermi3 if the equilibrium is indeed established, as we Rapidity Gaussian believe is probably indicated by the large increase in strange particle production. Interval RT(fm) RL(fm) A This will be ascertained by observation of thermal radiation of lepton pairs when experiments in the appropriate kinematic region are done. o +0.09 1 R°ut = 4.4 + 1 ,o fm - " 5. The great stability of the pion mid-range PT slope could indicate a latent heat is involved in the evolution of the deposited energy. Measurements of low py 2 The results are very suggestive. The dimensions in the control region are comparable to the 7. The target fragments are observed in the 'plastic-ball' detector [33], which 16 geometrical size of 0, R = 3 fermi, but in the fireball region, large values are seen. The Rout is finds the target to be completely disrupted, indicating some mechanism of particularly large, which might optimistically be taken as a sign of the long-lived state predicted with a transferring the necessary energy, which is still a small fraction of the total E-r. phase transition. Before reaching a firm conclusion, more information is needed particularly since it has been shown that somewhat similar results could be simulated by long-lived resonances. It will 8. The time scale of pion emission is especially important: the first measurement probably be essential to examine the correlations as a function of the PT of the pions, as well as reported in Table 4.1, indicates that the lifetime of the emission is order 12. providing more information on the level of resonance production, particularly of the p and G>°, which fermi/c, long compared to the ordinary spatial scale of a few fermi/c in the direction could partially simulate this effect [31]. of the effect predicted for a strong phase transition. It is important to strengthen this conclusion. 5. St JMMARY OF CONCLUSIONS ON NUCLEI IS-Nt ICLEtJS COLLISIONS 6. PROBING THE SPACE AROUND AN INTERACTION Much of the information gained up to now is displayed pictorially in Fig. 24, which shows a The ideal version of such an experiment was described in the introduction: colour fields are central oxygen-lead collision just after the last interaction of the produced pions, in the centre of mass generated in a hadronic interaction, and electromagnetic radiation is observed from the area outside the of the "fireball". The labels in Fig. 24 refer to the following points: usual interaction volume, Fig. 3. Several experiments of this type have been performed, studying either real photons or virtual photons internally converting into a pair of leptons, starting some years ago with 1. The dotted outline shows that the empty volume which would be occupied by the the observation of 'anomalous lepton pairs' occurring at low mass, at a surprising large rate [33]. Only projectile: emulsion experiments which can measure down to very small angles find more recently has it become clear that the spectrum of mass and px extended to such low values as to no projectile fragments when high Ej events are selected [32]. exclude a phenomenon which originates inside the one or two fermi radius of a hadronic interaction [34]. One tries hard to find more mundane sources for the radiation. It is true that decaying This aspect has been much strengthened by observation of the copious production of the corresponding resonances, p, w° and so on, fill a large volume, similar to that required, but radiation from their real photons with transverse momenta as low as 10 MeV or less [35]. charged decay particles has already been included in the calculation of inner bremsstrahiung, ignoring interference between the different sources of radiation. This approximation should be checked. The cross-sections for real and virtual photons may be compared by using the theory of soft internal conversion, used to display a compilation of photon and lepton pair data in Fig. 25, as a function Clearly further experimental and theoretical investigations of this radiation are eagerly awaited. of the transverse mass variable. We see a spectrum which rises steadily as mj approaches zero, whereas it was generally supposed that such a phenomenon, of hadronic origin, should level off at the hadronic scale of about 300 MeV. REFERENCES There is an uninteresting source of radiation, inner bremsstrahiung, due to the radiation of [ 1 ] E.V. Shuryak, the QCD vacuum, Hadrons and the Superdense Matter; (World Scientific, charge by the outgoing particles created in the interaction. This radiation extends to very long Singapore, 1Ï wavelengths since the path of the outgoing particles in effectively infinite, leading to a spectrum proportional to (Ey)~1, somewhat similar to the data, but, as shown in Fig. 25 the bremsstrahiung [2] C. Alcock, Ann. Rev. Nucl. Particle Phys. 38 91988) and Cosmological Consequences of the Quark-Hadron Phase Transition, plenary talk at "Quark Matter '88", Lenox, Mass. 1988 (to be explains only a quarter or less of the data for Ey above 10 MeV [37]. published). [3] See "Quark Matter '87", Z. Phys. C38 (1988). The excess becomes much greater when one selects only events with a high number of charged tracks. Note that the net charge of the particles created in the collision is zero, so that the radiation is [4] F. Karsch, in Ref. 3, p. 147. due to fluctuations in the number of charged particles in a given solid angle. The radiation varies as the [5] J.E. Elias et al., Phys. Rev. D22, 139 (1980); square of the net charge in the relevant solid angle; the mean net charge varies as the square root of the K. Braune et al. Z. Phys. C17 (1983) 105-112. number of charged particles; and thus it is found that the number of photons varies linearly with the [6] E. Fermi, Prog. Theor. Phys. 5, 570 (1950); Phys. Rev. 81, 683 (1951). number of charged particles. The ratio of the number of photons in a given energy range to the number [7] E.V. Shuryak, Phys. Lett. B78, 150 (1978); J.D.Bjorken, Phys. Rev. D27, 140 (1983). of charged particles in the same solid angle is a number which can be calculated rather precisely, small effects due to measured particle correlations can be included, and it is found that the result is [ 8 ] L.D. Landau, in Collected papers of L.D. Landau, éd. D. Ter Haar (Gordon and Breach, New York, 1965); P. Carruthers and Minn Duong/van, Phys. Rev. D8, 859 (1973). approximately the same for different reactions and incident (high) energies. [9] W. Busza and A.S. Goldhaber, Phys. Lett. B139, 235 (1983). The data, presented in Fig. 26 for real photon to pion ratios as a function of multiplicity of [10] T. Âkesson, Z. Phys. C38, 397 (1988). charged particles [38], present a very different picture, with an approximately linear variation which [11] T. Âkesson et al., Z. Phys. C38, 383 (1988) corresponds to a quadratic dependence of photon radiation on particle number. Such a dependence (seen also in the virtual photons) is characteristic of radiation from charge confined in a volume, and this is [12] T. Âkesson et al., as quoted in [16] and submitted to Phys. lett. B. believed to be such a case, but then the long wavelength of the observed radiation implies that the volume [13] P. Braun-Munzinger and J. Stachel, Phys. Lett. 216, 1 (1989). of the box is very large, on a hadronic scale. [14] B. Alpher et al., Nucl. Phys B100, 237 (1975); K. Guttler et al., Phys. Lett. 64B, 111 (1976). 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Heinz, TPR-88-33. 117 118 119 120 [19] E.V. Shuryak, Phys. Lett. B207, 345 (1988). QG PLASMA [20] B.L. Fridman, GSI-88-65, to appear in "Quark Matter '88". [21 ] P. Vincent, E-802, Results in "Quark Matter '88". MI*ED [2 2] M. Jacob, CERN PRE 88-071, Summary Talk to appear in "Hadron '88: Hadronic Matter in PHASE Collision, Tucson, Arizona 1988. HADRON GAS [23] R. Hanbury/Brown and R.Q. Twin, Phil. Mag. 45, 663 (1954). o o [24] G. Goldhaber, S. Goldhaber, W. Lee and A. Pais, Phys. Rev. 120, 300 (1960). o [25] G. Gyulassy, S.K. Kaufman and L.W. Wilson, Phys. Rev. C20, 2267 (1979). CO [2 6] G.I. Kopylov and M.I. Podgoretsky, Sov. J. of Nuclear Phys. 18, 336 (1974). V o [27] T. Muller, Proc. XIV Int. Symp. on Multiparticle dynamics, World Scientific, Singapore 1984, p.528. Fig. 1 Schematic drawings of the three states passed through successively, starting with very high energy density. [28] T. Âkesson et al., Z. Phys. C36, 517 (1987). [29] S. Pratt, Phys. Rev. D33, 1314 (1986); A. Makhlin and Y. Sinykov, Kiev preprint ITP-87-645 (1987); G. Bertsch, M. Gong, M. Tohyama, Phys. Rev. C37, 1986 (1988). [30] A. Bamberger et al., Phys. Lett. B203, 320 (1988), and Ref. [17]. [31 ] M. Gyulassy and S.S. Padula, LBL-26077 (1988). To appear in Phys. Rev. Letters. [3 2] G. Romano, Summary of emulsion results, in "Quark Matter '88". [3 3] K.J. Anderson et al., Phys. Rev. lett. 37, 799 (1976). [34] T. Âkesson et al., Phys. Lett. 192B, 463 (1987); V. Hedberg, Thesis, Univ. Lund, LUNFD6 (NFFL-8037) (1987). Fig. 2 Arbitrary assumptions have been used to create a speculative drawing of the mixed state at a [35] P.V. Chliapnikov et al., Phys. Lett. B141, 276 [1981]. point midway in its expansion. The shaded region denotes physical vacuum in which coloured particles are confined. The remainder is the perturbative vacuum. [36] W.J. Willis, PANIC Conference Kyoto, Japan (1987); U. Goerlach, Lepton-Photon Conference, Munich (1988); A. Pfeiffer, Thesis, Univ. Heidelberg, (1988). [37] R. Ruckl, Phys. Lett. 64B, 39 (1976). (a) [3 8] See Ref. [36] and HELIOS Collaboration "Direct Soft Photons in 450 GeV/c p-Be and p-AI collisions" to be published. Y [3 9] L. Van Hove, CERN-TH. 5236/88, to appear in Ann. Physics. Fig. 3 (a) Schematic illustration of a hypothetical soft colour excitation of the physical vacuum. (b) The trivial background to (a) coming from inner bremsstrahlung of the charged particles produced. (c) The still more trivial background of photons (and soft lepton pairs) from meson decay. 4 (a) The energy spectrum of an electron after traversing 0.5, 1.0 and 1.5 radiation lengths of matter, (b) The same spectra but in terms of rapidity shift from the incident electron. p—Emulsion 50 GeV TT" 200 GeV 50 GeV ref. [10] xtrapolated from ref. [9] --'___ 200 GeV ref. [12] Ng=3...5 -1 Fig. 6 Probability distributions to lose rapidity -Ay in collisions with protons or lead nuclei, and approximate estimates of the distributions of collisions with lead in particular impact parameter ranges from Busza and Goldhaber [9]. The ratio of the pseudorapidity distributions of charged particle production lor "lead" and hydrogen targets, showing that the multiplicity does not change for h > 3, the forward hemisphere in the centre-of-mass syslem. The data are from Elias et al. and several emulsion experiments as analysed by Braune et al. [5]. 121 122 123 124 "i r m "i r w V77777 wzzA 0.4 / 3.8GeV±ET<7.8GeV 7.8GeV=ET< 11.8 GeV i 1 1 r 1 1 1- V7P77?. y///// ^fezzzf mf\^ •zzzzzzzzz. / 0.4 — / 19.0GeV2ET<23.0GeV 27.8GeVSET<31.8GeV WM V777jm w////// / / 0.4 -/ / 31.8GeV±ET<35.8GeV 35.8GeV±ET I I I L I I I L 1.0 1.4 1.8 2.2 1.0 1.4 1.8 2.2 Fig. 8 The distribution of transverse energy in pseudorapidity for various Ej values, for 200 GeV proton lead collisions [10]. Fig. 7 The distribution in transverse energy for 200 GeV proton-lead collisions [16]. J 10 : I 1 1 II 1 ; HELIOS w Transverse energy differential cross-section \\ Ag Al dff/dE, in -0.1 < 77 < 5.5 for 10' 200 GeV/nuc. "S-nucleus collisions B ^| * Ag v W • Pt E 10 T o Pb ] D U N —- 7 T3 "~T\ - \] 10-'—- i 11 1 ! 10" 1 ...1 iiii 20 40 60 80 100 120 140 fî Fig. 9 Geometrical cross-section as a function of the value of the number of collisions N = appQ, where Q. is the overlap integral Jpi x p2dS, for oxygen collisions with AI, Ag, and W nuclei [11]. 100 200 300 400 500 Fig. 10 Cross-seclion as a function of transverse energy in the interval -0.1 < r| < 5 for 200 GeV/ nucléon sulphur collisions with different targets [12]. 125 126 128 -i 1 1 1 1 8O0MeV/A Ar'KCI Blast-wave model 0 2 3 4 5 6 70 114 7< r_. <1J7 6 60 50 40 " 30 20 1 10 '/ 1 ! ! 200 400 600 800 E*(MeV) Fig. 11 The distribution of Ey in pseudorapidity for two values of Ey, for 200 GeV/nucleon sulphur-W collisions [12]. The smaller value of Ej is in the "plateau" region of Ihe Ey distribution while Fig. 13 Differential cross-sections as a function of kinetic energy in the centre-of-mass (equal to m the larger value is in the tail of the distribution for central collisions. T within an additive factor and consequently suitable for comparison with the previous Figure [15]). The reaction is 800 MeV/nucleon (laboratory energy) 20Ne on NaF. The curves are a fi! to an expanding fireball. da' Do1 ' 1 " 1 "V T" *T • • - • | • • • ' < . a) . ' ' b) c) •» r p -* W 200 CeV/n "0 -» W 200 C V/n _ . n«q porl«l i0.8< ij Fig. 14 Cross-section dN/dp| of negative particles in the region 0.8 < z < 2.0 for a) p-W, Fig. 12 Differential cross-sections as a function of mj for proton-proton collisions at the ISR [14]. In (b) 160-W, (c) 32S-W reactions at 200 GeV/n [16]. The dashed lines are arbitrarily the centre-of-mass, the charged particles are measured at 90° to the beams, and the energy is normalized parametrizations of pp data. 45 GeV/c. ~- 1 1 u_ S -— W p — W ? i ....Q. .- T5 100 1 _ C O 0 -*• W p —W r ZOO GeV/n 0 + Au -> negatives i -se c i 10 1 10J cros s s — w o 0 — W c o 10 1 ... J... "I i î - m , , pT (GeV/c) Fig. 15 Ratio of the cross-sections dN/dp| as a function of pT for negative particles in the region 0.8 < y < 2.0 [16]. Upper part: S-W divided by p-W. Middle part: O-W divided by p-W. Lower part: S.W. divided by O-W. The relative normalization is arbitrary. 10U < p, < ?000 McV 400 < p, < 2UO0 McV nisuincu (to/dp, - p, a) assumed «Jd/cJp, = p, i>) r - w ?oo Gtv/H "S - W Î0Q GtV/U c 1 10 rnT Trr TTT TTT Tr rrn 'T ^ "l""l""l" ' ' \ T T T T : Z00 GtV/n O + Au -> ANT1-A \ 200 CcV/n 0 + Au -> A '• PRELOOKAJtr PREUJDHARÏ 0 5 10 15 20 ?5 ]0 J5 to t5 50 îo wo 150 :oo no \ 10' • A D»U 10' ~ « AnU-A d*l« E, IGeVI E, IGeV! — \ ~ : ^ -— - A lUdi«l now. fll to n 3, - AnU-A R.dl*J now. fit to i Fig. 16 Ej dependence of the average transverse momentum of negative particles in (a) p-W and (b) ~ ^4 ;--.A rrttlof .-•Anll-A Frlllol 32 - S-W reactions, extracted from the slope of an exponential distribution fitted to the data in the 0 _ region 400 < p < 2000 MeV/c [16]. r~ 10' T \H 1 - ASL - ;••"!••• •!•• " " 1" • -Ï - \ *rv - 200 C«V/n S + S > I») - (-) • -1 _ 10" 10 j ~1 - B Radl.. n \ it -2 _ to -z r_~ 10 C "1 _- \ - , i 1 i i i i i i i i M i i 10 -3 J^ .," 10" h 0.5 1 1.5 2 2.5 : b i Pi (GeV/c) ...il..., 1 1.5 Streamer chamber results for transverse momentum distributions for central collisions ol 200 Pi. (G=V/c) Pi (GcV/c) GeV/n O+Au for the particles indicated on the figures [18]. Also shown are predictions ol a radial expansion model (solid curve) fitted to the distribution of negatives in a) and the Fig. 17 Results from the Streamer Chamber for transverse momentum distributions for central Lund/Fritiof model (dot-dash curve), intended to represent the shape of the data in nucléon collisions of 200 GeV/n S + S for a) negatively-charged particles (2 < z < 3) and b) the nucléon collisions. difference of positively- and negatively-charged particles assuming they are protons, together with a fit to both spectra with a model assuming a radial motion of a thermal syslem [18]. 129 130 131 132 3.0 3.0 r"rT^T",riir 200 CeV/n 5 •• S -> A + X 2.5 - PRELIMINARY 20 A <: o Fig. 22 1.5 — A suitable coordinate system for measuring V two-pion correlations produced by relativistic particles, which are along the LONGITUDINAL vertical direction in the picture. The rest- frame of the figure is such that the first pion is at 90° to the beam, and the second pion is analysed along the three axes indicated. 300 Fig. 19 For 200 GeV/n S+S interactions, streamer chamber observations of the mean multiplicities ot a) A'S and b) K°'s as a function of the mean charged particle multiplicity in each event sample [18]. the solid lines are predictions of the Lund/Fritiof model and the dashed lines those of an independent nucleon-nucleon model. The double line is the prediction of a hadron gas model for central collisions and the triple line that for a parton gas. > (b) 100 200 300 400 500 I a.,,0.,1 < 100 MeV/c) Fig. 20 Diagram defining the geometry for two-photon interference and showing the two states (a) and (b), which must appear in a symmetrical wave fun ction. Fig. 23 An example of the correlation measured for two negative particles in oxygen-gold collisions at 200 GeV/nucleon, showing the enhancement low values of the Q-value of the two charged particles [30]. 20 ffm —(?) 0.4 0.6 Q(GeV/cl Fig. 24 A pictorial summary of the results of the 200 GeV per nucléon experiments to date, as Fig. 21 The two-pion correlation function for pions in the central region of proton-proton collisions at interpreted in terms of a space-time snapshot at the time of pion freeze-out. A key in the text 63 GeV in the centre of mass [28]. explains and justifies the numbered features. ~i i i i 111 q i i r i 111 ii 1—i—i i i > HI i i r 10" • p-ViTN 225GeV/c, /s=20 GeV Anderson el al. • e*e~pN 13 GeV/c,/s = 5GeV Mikamo el al. • e'e" jTp 16GeV/c, /s= 6GeV Blockus el al. 10" « e*e~ n'p 17GeV/c, x AF£ \^T, n e'e'p^Be V9 GeV/c Vs = 3 3GeV >- "O D Roche el al. (DLS) H CL c ID 10- 2 « * c m '^ 10 -3 > 10- dn dy y = 3.5 Fig. 26 The ratio of photons production to pion production as a function of the number of charged particles is shown for data for different particles and beams. Inner bremsstrahlung would give a constant of this plot [36]. . ï K'p 70 GeV/c. /s = 11.5GeV 10" Lhliapnikov el al. * Y p*Be 450 GeV/c, Ys=29GeV o i p.p 200 GeV/c, /ifw^rKGeV v ï AFS-R808 10- _l i—_J_J_1_I-L1_ 1 10 100 1000 my I'ig. 25 A compilation of data on soft photon and virtual photon production at different energies and by different particles, displayed as a function of the mx of the photon or pair. The formula for internal conversion has been used to display real and virtual photons together. Also shown is the production for the background process of inner bremsstrahlung which has not been f.btracted [36]. 134 135 136 QCD AND COLLIDER PHYSICS'» and the indices A,B,C run over the eight colour degrees of freedom of the gluon field. It is the third 'non-Abelian' term on the right-hand-side of Eq.(1.2) which distinguishes R.K. Ellis, QCD from QED, giving rise to triplet and quartic gluon self-interactions and ultimately Fermi National Accelerator Laboratory, Batavia, 111., USA to the property of asymptotic freedom. N.J. Stirling The sum over the flavours runs over the nf different flavours of quarks, g is the coupling Dept. of Phys. and Math. Science, Univ. Durham, England constant which determines the strength of the interaction between coloured quanta, and JABC {A,B,C = 1,...,8) are the structure constants of the SU(3) colour group. The 1. Fundamentals of Perturbative QCD quark fields qa are in the triplet representation of the colour group, (a = 1,2,3) and D is the covariant derivative. Acting on triplet and octet fields the covariant derivative takes In this set of lectures we shall describe the use of perturbative methods to investigate the form the behaviour of strong interactions at short distances. Perturbative methods are appli c c cable because of the property of asymptotic freedom which will be described in the first (A.L = à*6* + ig (t°-<)„„> {Da)AB = dJAB + ig(T A a)AB, (1.3) lecture. Subsequent lectures describe how short distance cross sections are calculated where t and T are matrices in the fundamental and adjoint representations of St7(3) and how the results of these calculations compare with experiment. respective])': The treatment of perturbative QCD is developed in analogy with perturbative QED A B ABC c A B ABC c A ABC and an understanding of perturbative QED is therefore a prerequisite for this course. [t ,t ] = if t , [T ,T ]=if T , (T )BC = -if . (1.4) D in Eq.(l.l) is a symbolic notation for j^D" and the spinor indices of -yu and qa have 1.1 Lagrangian of QCD been suppressed. Otherwise we follow the notation of Bjorken and Drell [1] with metric given by gaS = diag( 1,-1 ,-1,-1) and set h = c = 1. By convention the normalisation of the We begin with a brief description of the QCD Lagrangian and the Feynman rules which SU(N) matrices is chosen to be, can be derived from it. This is a practical guide which does little more than introduce notation and certainly does not do justice to the elegant structure of quantum field theory. A B AB TTt t = TRS , r„ = i. (i.5) For more details, the reader is referred to the standard texts [1,2]. Introductions to perturbative QCD can be found in refs.[3,4,5,6,7]. With this choice the SU(N) colour matrices obey the following relations, Just as in Quantum Electrodynamics, the perturbative calculation of any process Ç££ = C S C = ^^ = \, (* = 3) (1-6) requires the use of Feynman rules describing the interactions of quarks and gluons. The F m r Feynman rules required for a perturbative analysis of QCD can be derived from an effective C D Tr T T = y-fABCfABD = CA gCDt CA = N = 3. (1.7) Lagrangian density which is given by We cannot perform perturbation theory with the Lagrangian of Eq.(l.l) without the C = -\FaBFf+ Y, 5"(iè " m)°1'9'' + £«*«^-««i». + A*».- (1-1) flflVOIUl gauge fixing term. It is impossible to define the propagator for the gluon field without making a choice of gauge. The choice, This Lagrangian density describes the interaction of spin-1/2 quarks of mass m and mass- less spin-1 gluons. FA is the field strength tensor derived from the gluon field AA, Cw^-bin. = ~(9' ABC F& = [^ - d,Ai - gf AlA<^\ (1.2) fixes the class of covariant gauges and A is the gauge parameter. In a non-Abelian the ory such as QCD this covariant gauge-fixing term must be supplemented by a ghost Lagrangian, which is given by *) This written version is based on the lectures given at the CERN and CERN-JINR Schools of Physics. It appeared in preprint form as FERMILAB-Conf.-90/164-T in •£<*„.. = daVA f {DABI") • (1-9) A.HMist 1990. Here r\A is a complex scalar field which obeys Fermi statistics. The derivation of the A, a p B,/3 0 -sT + (i - 2 form of the ghost Lagrangian is best provided by the path integral formalism [8] and the p +ie P2 + is procedures due to Fadeev and Popov [9]. For an explanation of the physical role played \S13SISISL3SLSSISU by ghost fields, the reader is referred to ref. [10]. t A p B p2+t£ 1.2 Feynman rules bj (p-m + ie)^ Eqs.(l.l), (1.8) and (1.9) are sufficient to derive the Feynman rules which should be used in weak coupling perturbation theory in a covariant gauge. The Feynman rules are defined from the action operator # = t JC d*x rather than from the Lagrangian density. We can separate the effective lagrangian into a free piece C , which normally contains all the ABC a0 7 a a -9f [g (P - $ = $o = i f^xCoix), 2 XAD XBC The practical recipe to determine the Feynman rules is that the inverse propagator is -i9 f f (gaeg,6-ga,gffi) derived from — $0, whereas the Feynman rules for the interacting parts of the theory -ig*fX^fXC°(ga,g0s-galgs,) which are treated as perturbations are derived from <£/. This recipe (including the extra minus sign) can be understood [11] by considering the A, a following two different approaches to the quantisation of a theory. For simplicity, consider a theory which contains only a complex scalar field In the second approach K is regarded as the free Lagrangian, $0 = c,3 *-T + (T)«-(Ï)*(7)'(7)'(T)+--ÏTF <'•«> Table 1: Feynman rules for QCD in a covariant gauge Note that with the choice of signs described above the full propagator of the Using the free piece C0 o{ the QCD Lagrangian given in Eq.(l.l) one can readily obtain of the propagator is added to preserve causality, in exactly the same way as in QED[lj. the quark and gluon propagators. Thus, for example, the inverse fermion propagator in Similarly the inverse propagator of the gluon field is found to be momentum space can be obtained by making the identification d" = — ip" for an incoming field. The two point function of the quark field becomes ru£.«*>(?) = iS*B \p2g<,0 - (i - T)P«P0 (1.14) 2 , It is straightforward to check that without the gauge fixing term this function would have ri k (p) = -i^6(p-m), (1.13) 138 139 140 no inverse. The result for the gluon propagator A is as given in Table 1: This first order partial differential equation is solved by implicitly defining a new function - the running coupling as(Q) - as follows: rM«,««(p) tfacihyb) = S*B3-, (1.15) —-, a /*=««• (1.20) „ , M \\Pf>Pt s *Xéc.fiyib)\W = 6BC- (1.16) /. P By differentiating Eq.(1.20) we can show that Replacing derivatives with the appropriate momenta, Eqs.(l.l), (1.8) and (1.9) can be used to derive all the rules in Table 1. 1.3 The running coupling constant and hence that R(l,as(Q)) is a solution of Eq.(1.19). The above analysis shows that all of the scale dependence in R enters through the running of the coupling constant as(Q). In order to introduce the concept of the running coupling, consider a dimensionless phys It follows that knowledge of the quantity R(l,as), calculated in fixed order perturbation ical observable R which depends on a single energy scale Q. By assumption the scale Q theory, allows us to predict the variation of R with Q if we can solve Eq.(1.20). In the is much bigger than all other dimensionful parameters such as masses. We shall therefore next section, we shall show that QCD is an asymptotically free theory. This means that set the masses to zero. (This step requires the additional assumption that R has a sensible as(Q) becomes smaller as the scale Q increases. For sufficiently large Q, therefore, we zero mass limit.) Naive scaling would suggest that because there is a single large scale, can always solve Eq.(1.20) using perturbation theory. R should have a constant value independent of Q. This result is not however true in a renormalisable quantum field theory. When we calculate R as a perturbation series in the 2 1.4 The beta function and the A parameter in QCD coupling as = jr /4ff, (defined in analogy with the fine structure constant of QED), the perturbation series requires renormalisation to remove ultra-violet divergences. Because The running of the coupling constant as is determined by the renormalisation group this renormalisation procedure introduces a second mass scale /i - the point at which the equation. In QCD, the f3 function has the perturbative expansion subtractions which remove the ultra-violet divergences are performed - R depends in gen eral on the ratio Q/fi and is not therefore constant. It follows also that the renormalised 0(as) = -4a|(l + b'as + 0(a|)) coupling aj depends on the choice made for the subtraction point fi. (33 - 2n,) (153-19n7) 6= b (L22) However y. is an arbitrary parameter. The Lagrangian of QCD makes no mention of 12* ' - 2*(33-2»,)' the scale fi, even though a choice of \i is required to define the theory at the quantum where n/ is the number of active light flavours. An alternative notation which is sometimes level. Therefore, if we hold the bare parameters fixed, physical quantities such as R cannot used is depend on the choice made for \L. Since R is dimensionless, it can only depend on the ratio Q2/ii2 and the renormalised coupling as- Mathematically, the /i dependence of R 0£\n may be quantified by «-J-SM;4f f ) 9 OQ 00 = 4TT6 = 11 - -n/, Pi = 16TI-266' = 102 - —n/, ... (1.23) 9 9a d 2 2 s R=0. (1.17) dfi2 dp3 das ^S'^ The /3 function coefficients can be extracted from the higher order (loop) corrections to To rewrite this equation in a more compact form we introduce the notations the bare vertices of the theory, as in QED. Here we see for the first time the effect of the non-Abelian interactions in QCD. In QED (with one fermion flavour) the f) function is t = H%), «-) = ^, (1.18) PQEDH = ^-aJ + ... (1.24) 07T and rewrite Eq.(1.17) as and thus the b coefficients in QED and QCD have the opposite sign. -*+«->£ R=0. (1.19) From Eq.(1.21) we may write, as 'the' fundamental parameter of the theory the value of the coupling constant at a convenient reference scale which is large enough to be in the perturbative domain, Mz 2 ^|P = -ba s(Q) [l + b'as(Q) + O(aKQ))]. (1.25) for example. An alternative approach - which was adopted historically and is now the de facto standard for specifying the strength of the strong interaction - is to introduce a di- If both a (fi) and a (Q) are in the perturbative region it makes sense to truncate the s s mensionful parameter directly into the definition of ats(Q). By convention this parameter series on the right-hand-side and solve the resulting differential equation for a (Q). For s is called A and is a constant of integration defined by example, neglecting the 6' and higher coefficients in Eq.(1.25) gives the solution Q2 i°° dx f°° dx ~v = " jLw) W) = jLw Mi + *-* + ...)' (O0) This gives the relation between cts{Q) and as(/i), if both are in the perturbative region In effect, A represents the scale at which the coupling cts(Q) becomes strong. The ar bitrariness of the integration constant is reflected in the fact that replacing A by A x Evidently as t becomes very large, the running coupling as(Q) decreases to zero. This constant in Eq.(1.30) still gives a solution to the differential equation for as(Q). is the property of asymptotic freedom. The approach to zero is rather slow since as only decreases like an inverse power of logQ2. Notice that the sign of 6 is crucial. With the The introduction of A allows us to write the correct asymptotic solution for as. In opposite sign of 6 the coupling would increase at large Q2, as it does in QED. leading order (LO), i.e. retaining only the b coefficient in the /? function, we can perform It is relatively straightforward to show that including the next-to-leading order coef the integral in Eq.(1.30) to obtain ficient V yields the solution as(Q) = . (1.31) SKW> 61n(QVA3) ' +b Note that this is now an implicit equation for e«s(Q) as a function of t and as(n). In A is extended to next-to-leading order (NLO) by including also the 6' coefficient in the practice, given values for these parameters, as(Q) can easily be obtained numerically to integral: any desired accuracy. -J—+ yinf b'asiQ] ) = bln(^\. (1.32) Returning to the physical quantity R, we can now demonstrate the type of terms Again, this allows a numerical determination of as(Q) for a given value of A. Alterna which the renormalisation group resums. Assume that in perturbation theory R has the tively, we can obtain an approximate solution of Eq.(1.32) in terms of inverse powers of expansion log(Qs/A2): R = a + ... (1.28) ! 2 s " 6'lnln(Q /A ) <*s(Q) = 2 1- -r- 1.33) Mn(Q7A ) 6 ln(QVA') where ... represents terms of order a| and higher. The solution R(l,as(Q)) - for the Note, however, that this expression corresponds to a slightly different definition of A to special choice of R given by Eq.(1.28) - can be re-expressed in terms of as(n) using J 2 Eq.(1.26): Eq.(1.32). The true expansion of as{Q) in inverse powers of log(Q /A ) would contain a term of order constant/log2. However the freedom to multiply A by a constant can be 2 R(l, ccs(Q)) = as(ti) V(-l)>(a5(/i)6t)' = as(ii) [l- as(n)bt + as(li)(bt) + ... ] (1.29) used to remove this term. Specifically, if we call Eqs.(1.32) and (1.33) definitions 1 and 2 respectively, then for the same value of as{Q) the two A's are related by Thus order by order in perturbation theory there are logarithms of Q2/V2 which are auto (1.34) matically resummed by using the running coupling. Higher order terms in R - represented Ai= (-V'A2iEl.l48A2, (n/ = 5) by the dots in Eq.(1.28) - when expanded give terms with fewer logarithms per power of as- An explicit example of how this works in practice will be discussed in the next It will be clear from the above discussion that the use of the parameter A as the chapter. fundamental parameter of QCD presents a number of traps which can ensnare the unwary. Perturbative QCD tells us how the coupling constant varies with the scale, not the First, A can be defined to leading or next-to-leading order and in each case multiplying A by a constant gives an equally acceptable definition. The differences induced in a {Q) are absolute value itself. The latter has to be obtained from experiment. Thus we can choose s 141 142 143 144 4 flavours 5 flavours 500 LO 0.234 0.255 Comparison of Lambda for 4 and 5 flavours NLO 0.184 0.206 matching at mb=5 GeV. 400 Table 2: as{Q) for Q = 5 GeV and A = 200 MeV one order higher in perturbation theory. Nowadays, all precision QCD phenomenology is performed at next-to-leading order. Either Eq.(1.32) or Eq.(1.33) can be used to define > 300 A in this case, and both definitions are used in the literature. Since in practice it is S usually as which is measured experimentally, it is important when comparing A values to check that the same equation has been used to determine A from the coupling constant. 200 Differences between the results obtained using different conventions - although small - can be comparable to present-day measurement errors. A second difficulty with the above definitions is that A depends on the number of 100 — active flavours. Values of A for different numbers of flavours are defined by imposing the continuity of as at the scale /i = m, where m is the mass of the heavy quark. This is illustrated in Table 2 where the LO and NLO couplings are calculated using Eqs.(1.31) I I I I I I i i i i i i i i i 100 200 300 400 500 600 and (1.33) respectively. The correct matching prescription is determined by the conditions A(4) [MeV] that for all values of the momenta the coupling constant must be both a solution of the renonnalisation group equation and also a continuous function. From Eq.(1.33) for fi > mt Figure 1: Comparison of A for 4 and 5 light quark flavours, with matching at mi, = 5 we have, GeV. <*s(/*,5) = 1 - ... (1.35) 6(5)ln(^/A(5)s)l The third troubling property of A is that it depends on the renormalisation scheme. For m < /i < m(,, the coupling evolves with four active flavours, and the correct form to c Consider two calculations of the renormalised coupling constant which start from the use is 1 6(4)lnU7A(5)2) same bare parameters. + constant (1.36) ots(M,4) [l-. as = Z as where the square bracket is the same as in Eq.(1.35). The constant is fixed by the = ZBa% (1.39) continuity condition, as(mb,i) = as(mb,5). (1.37) The two schemes start from the same bare coupling a"s. The infinite parts of the renor A B Using the next-to-leading order form for as(Q) one can show then that malisation constants Z and Z must be the same in all orders of perturbation theory. Therefore the two renormalised coupling constants must be related by a finite renormali sation: *w-*.)fcS0>Q0p (1.38) af=a2(l + ClQ$+ ...)• (1.40) Fig.(l) illustrates the relation between A(4) and A(5) graphically. In summary, it is Note that the first two coefficients of the /3 function, b and b', are unchanged by such important when comparing different A values to establish the number of light quark a transformation. They are therefore independent of the renormalisation scheme. From flavours assumed and also whether the LO or NLO expressions have been used. This is Eq.(1.30) we see that the two values of A are related by, illustrated in Table 2. ( determined by the one loop calculation which fixes cx: AB = A^exp| (1.42) Nowadays, most calculations in fixed order QCD perturbation theory are performed in the modified minimal subtraction renormalisation scheme. In this approach, ultra-violet loop divergences are regulated by reducing the number of space-time dimensions to n < 4: where e = 2 — y. Note that the renormalisation scale u preserves the dimensions of the couplings and the fields. Loop integrals of the form d"k/[k2 + m'Y then lead to poles at e = 0. The minimal subtraction renormalisation prescription is to subtract off these poles and to replace the bare coupling by the renormalised coupling cts(fi). In practice 5 the poles always appear in the combination •° r~ I i i i i i 11 i I i i i i M I i 3 10 30 100 M [GeV] - + ln(47r) - -IE, (1.44) ("IE is Euler's constant) and in the modified minimal subtraction scheme these additional Figure 2: Measurements of as compared with predictions for various values of A(5). constants are subtracted off as well. These two schemes are therefore examples of schemes A and B introduced above, and it is straightforward to show using Eqs.(1.40) and (1.42) that 2. QCD in e+e" —• Hadrons Ak = AMseW4"-7a)- (1.45) Many of the basic ideas and properties of perturbative QCD can be illustrated by Lastly, the expression of the experimentally measured coupling as in terms of A leads considering the process e+e~ —* hadrons. We begin by discussing the total cross section. to an error which is both exponentially magnified and asymmetric. This is mathematically We show how the order as corrections are calculated, and how renormalisation scheme correct but depressing for an experimenter since most experiments actually measure as- dependence enters at order as. The total hadronic cross section also provides one of the A partial compilation of measurements is shown in Fig.(2). The errors in Fig.(2) are too most precise measurements of the strong coupling, and we quote the latest experimental + large to conclude that as has a logarithmic fall-off with /i, but analysis of jet data in e e~ results. annihilation demonstrates that as does decrease with scale (see later). Perturbative QCD also predicts a rich 'jet' structure for the final state hadrons. We Guided by Fig.(2), for the phenomenological predictions made in the following lectures show how jet cross sections can be defined, and how the predictions compare with exper we shall assume iment. The property of colour coherence is also discussed. 100 MeV < Aj^ô) < 250 MeV. (1.46) This corresponds to about a 20% uncertainty at the mass of the Z: 0.10 < a {M ) < 0.12. + s z 2.1 The total cross section for e e~~ —> hadrons Lack of knowledge of as directly translates into an uncertainty in the prediction of the size of QCD cross sections. Thus we should expect errors in the prediction of cross sections One of the theoretically cleanest predictions of perturbative QCD is R' ' , the ratio of which begin in order as of about 20%. the total e+e" hadronic cross section to the muon pair production cross section. We begin by considering the high energy 2 —> 2 process e+e~ —» // with / a light charged fermion, / ^ e. In lowest order the process is mediated by either a virtual photon or a Z° in the 145 146 147 148 j—channel. Denoting the centre-of-mass scattering angle of the final state pair by 9, the tions which change quarks and gluons into hadrons modify the outgoing state, but they differential cross section is: occur too late to modify the probability for an event to happen. da In leading order perturbation theory, therefore, the total hadronic cross section is ira* 2 [I + cos 9)(Q) - 2Q,VmVfXl(,) + (A\ + V?){A) + V/)X,(')) dcosS ~2J~ obtained by simply summing over all kinematically accessible flavours and colours of quarks: (2.1) + cos6(-4Q{A (2.2) The 0(as) corrections to the total hadronic cross section are calculated from the real 4ffa I and virtual gluon diagrams shown in Fig.(3). For the real gluon emission diagrams shown and (V/, A/) are the vector and axial couplings of the fermions to the Z given explicitly in Fig.(3b) it is convenient to write the three-body phase space integration as in Eq.(6.11). The Xi term comes from the square of the Z-exchange amplitude and the 3 1 <2 Pl cPpi cPk Xi term from the photon-.Z interference. Now at centre-of-mass scattering energies y/s d$ à [a — Vt •P2~k) 3 2x)5 2ET. 2E 2E vv F far below the Z peak, the ratio sjM\ is small and so 1 » Xi ^* X2- This means that the 7 k weak effects - manifest in the terms involving the vector and axial couplings - are small -d$\d cos Ô\d where a,/3,7 are Euler angles, and X\ — 2E\j^/l and x2 = 2Ei/y/s are the energy da fractions of the final state quark and antiquark. Integrating out the Euler angles gives a -^(1+ «.'«). (2.3) Is matrix element which depends only on xt and x2 and the contribution to the total cross Integrating over 8 and setting Qj = —1 gives the total cross section for eTe~ —> /i+M~ section is 2as x\ + x\ where -y/s is the total centre-of-mass energy. When an electron and a positron annihilate they can also produce hadrons in the final state. Although the formation of the observed final state hadrons is not governed by perturbation theory the total cross section for the production of hadrons can be cal a) culated using perturbative methods. Why would one expect perturbation theory to give an accurate description of the total hadronic production cross section? The answer can be understood by visualising the event in space-time. The electron and positron form a photon of virtuality Q = •Js~ which fluctuates into a quark and an antiquark. By the uncertainty principle this fluctuation occurs in a space time volume 1/Q, and if Q is large the production rate should be predicted by perturbation theory. Subsequently the quarks b) and gluons form themselves into hadrons. This happens at a later time characterised by the scale 1/A, where A is the typical mass scale of the strong interactions. The interac- Figure 3: Feynman diagrams for the 0(as) corrections to the total hadronic cross section in e+e~ annihilation where the integration region is: 0 < Xi,x2 < 1, x, + x2 > 1. Unfortunately, we see that renormalisation, in the MS scheme for example, the 0(a|) coefficient depends on the the integrals are divergent at x; = 1. These singularities come from regions of phase space renormalisation scale y: where the gluon is collinear with either quark, fl,s —• 0, or where the gluon is soft, Eg —» 0. r33 365 Evidently we require some sort of régularisation procedure - to render the integrals finite R' 1 + -2"'ln + 7T 12 s "2T - before the calculation can be completed. A variety of methods are suitable. One can i give the gluon a small mass, or take the final state quark and antiquark off-mass-shell 2 f*s(n)\- •-n 'tf-^ n<] 1 4- 1 (2.11) by a small amount. In each case the singularities are then manifest as logarithms of the regulating mass. and ((3) = 1.2021. Note that the /i-dependence of the second order coefficient is exactly as 2 A more elegant procedure is to use dimensional régularisation, with the number of specified by the renormalisation group equation, i.e. the coefficient of ln(/i /a) is exactly space-time dimensions now n > 4. With the three-body phase space integrals now cast b-K, where 6 is the /3 function coefficient defined in Eq.(1.22). Specialising to the case of in n dimensions, the soft and collinear singularities appear as poles at n — 4. Details of p. = -Js and n/ = 5, Eq.(2.11) becomes how the calculation proceeds can be found for example in ref.[4]. The result is that the + 2 cross section of Eq.(2.7) becomes jr-- = 3^{i+^^ + i.4ii(2£(^) +...}. (2.12) l „* = .o 3 £3 ^ H(e) [i - 1 + ± + 0(e)], (2.8) What can one say of the higher order terms in this perturbation series? Before per forming an explicit calculation all we can say is that they will be of 0(a|). A calculation where H{t) = 1 + 0(e). of the third order coefficient in this perturbation series has been performed [14], but the The virtual gluon contributions shown in Fig.(3a) can be calculated in a similar fash results are now known to be in error [15]. ion, with dimensional régularisation again used to render finite the infra-red divergences In general the coefficients of any QCD perturbative expansion depend on the choice in the loops. The result is made for the renormalisation scale p.. As p is varied, the change in the coefficients exactly compensates the change in the coupling ûs(/i) in such a way that the physical predictions The cancellation of the soft and collinear singularities between the real and virtual R(1)(HFAC) = -R(2Wc)- (2.13) gluon diagrams is not accidental. Indeed there are theorems - the Bloch, Nordsieck [12] On the other hand, the principle of minimal sensitivity [17] suggests a scale choice p. = and Kinoshita, Lee, Nauenberg [13] theorems - which state that suitably defined inclusive quantities will be free of singularities in the massless limit. The total hadronic cross section y-PMS, where is an example of such a quantity, whereas the cross section for the exclusive qq final state, y~R^{p)\ = 0. (2.14) i.e. 2 + These two special scales can be identified in Fig.4. It is important to remember that there The 0(a s) corrections to R' '~ are also known. At this order we encounter the ultra-violet divergences associated with the renormalisation of the strong coupling. After are no theorems that prove that any of these schemes are correct. All one can say is that 149 150 151 152 +1 i ! | i i i |—i—i—i—|—i—i—i—|—i—i—i— the theoretical error on a quantity calculated to 0(as) is 0(a3 ). Varying the scale is Deviation from QPM result in QCD simply one way of quantifying this uncertainty. + : for e e total cross-section, S=1000 GeV* Finally, Fig.5 shows a recent fit [18] to data on R' '' over a broad energy range. The A<5) (two loop) = 230 MeV. weak and QCD contributions are displayed. The fitted value of as, in the MS scheme and using the second order QCD prediction, is as(34 GeV) = 0.158 ± 0.020 (2.15) il which corresponds to A& = 440^220 MeV" (2-16> 2.2 Jet cross sections The expression given for the total hadronic cross section in the previous section is very concise, but it tells us nothing about the kinematic distribution of hadrons in the final -1 1 L J I L I _]_ J L 20 40 60 80 100 state. If the hadronic fragments of a fast moving quark have limited transverse momentum M [GeV] relative to the quark momentum, then the lowest order contribution, (e+e~ —» qq), can Figure 4: The quantity R = [.RU>/.RQPM - l] as a function of the scale /i, where #« naively be interpreted as the production of two back-to-back jets. In this section we denotes the QCD prediction for R*+'~ truncated at 0(<4) investigate how higher order perturbative corrections modify this picture. Consider first the next-to-leading process e+e~ —• qqg. From Eq.(2.7) in the previous section, we have x 7.0 i_ff_ = ?^£ l + *l (2 17) Recall that the cross section becomes infinitely large when either (a) the gluon is collinear with one of the outgoing quarks, or (b) the gluon momentum goes to zero. This corre sponds to (a) only one and (b) both of the i; approaching 1 respectively. In other words the gluon prefers to be soft and/or collinear with the quarks. If the gluon is required to be well-separated in phase space from the quarks - a configuration corresponding to a 'three jet event' - then the cross section is suppressed relative to lowest order by one power of as- It would appear, therefore, that the two jet nature of the final state is maintained to next-to-leading order, since both the preferred configurations give a final state indistinguishable (after parton fragmentation to hadrons) from that at lowest order. This qualitative result holds in fact to all orders of perturbation theory. Multigluon emis 10.0 20.0 30.0 MO.O SO.O 60.0 sion leads to a final state which is predominantly 'two-jet-like', with a smaller probability Vs (GeV) (determined by as) for three or more distinguishable jets. A more complete discussion can be found in reference [19]. Figure 5: Combined QCD-electroweak fit to R* '~, from reference[18] To quantify this statement we need to introduce the concept of a jet measure, i.e. a procedure for classifying a final state of hadrons (experimentally) or quarks and gluons (theoretically) according to the number of jets. To be useful, a jet measure should be free Now in general we have of soft and collinear singularities when calculated in perturbative QCD, and should also /<*s{y/i)} /<*s{V*)> be relatively insensitive to the non-perturbative fragmentation of quarks and gluons into Ri+2(vs,y) = f—-—J > ciAy)(—-—J> » > o, hadrons. oo One of the most widely used jet measures is the 'minimum invariant mass' algorithm. V-Rn = I- (2.21) Consider a qqg final state. A three jet event is defined as one in which the invariant masses of the parton pairs are all larger than some fixed fraction y of the overall centre-of-mass Note that since the jet fraction criterion y is dimensionless all the energy dependence energy: of the jet fractions is contained in the coupling cts(i/s). One can therefore exhibit, at least in principle, the running of the strong coupling by measuring a decrease in R3 (p; + Vif > ya, i,j = q,q,g. (2.18) as i/s increases. The effect is clearly visible in Fig.(6). Note that experimentally the algorithm is applied to final state hadrons rather than parions. However studies using It is immediately clear that this region of phase space avoids the soft and collinear sin parton shower/fragmentation Monte Carlos have shown that - at least at high energy - the gularities of the matrix element. In fact in terms of the energy fractions, Eq.(2.18) is fragmentation corrections are small and therefore the QCD parton-level predictions can equivalent to be reliably compared with the experimental data (20). An example of such a comparison is shown in Fig.(7). 0 < Xi,x3 < 1 -y, xi + x3 > 1 + y. (2.19) If we define iî3 and R3 to be the two and three jet fractions then to O(as) we obtain +2ta, + * - £[<'-*>4i^) (r^) !-*-i*' 30 M I M I I M I II I I | II II | M I ) M I I II 1 II I j I I I I I I | | | Energy dependence of three jet production - JADE Ri = l-R3 (2.20) o Mark II Note that the soft and collinear singularities reappear as large logarithms in the limit 25 - TASSO y —* 0. Clearly the result only makes sense for y values large enough such that R S> R , t 3 x TRISTAN so that the 0{as) correction to iij is perturbatively small. * OPAL The generalisation to multi-jet fractions is straightforward. Starting from an n-parton i' final state, identify the pair with the lowest invariant mass squared. If this is greater than a i ys then the number of jets is n. If not, combine the lowest pair into a single 'duster'. 20 Then repeat for the (n — l)-parton/cluster final state, and so on until all parton/clusters have a relative invariant mass squared greater than ys. The number of clusters remaining is then by definition the number of jets in the final state. Note that an n-parton final state can give any number of jets between n (all partons well-separated) and 2 (for example, ot,=const two hard quarks accompanied by soft and collinear gluons). I I I I 1 I I I I I I I I I I I I I I I M I I I I M I h M I I I I I I I I I I I 15 10 20 30 40 50 60 70 80 90 100 Since a soft or collinear gluon emitted from a quark line does not change the multi Ecm [Gev] plicity of jets, the cancellation of soft and collinear singularities that was evident in the total cross section calculation can still take place, and the jet fractions defined this way Figure 6: The energy dependence of three jet production[21] are free of such singularities to all orders in perturbation theory. 153 154 155 156 100 OPAL b M g - Pi.kk.Pi ~ \k\> Ui W \GC- G bl) \ ' Ecm=91GeV m m^- "^" 2-\<* _ 80 where \k\ represents the energy of the soft gluon. The eikonal factor in Eq.(2.22) is the same as the factor obtained in the soft photon approximation in QED[l]. The expression : \ S • •xA2-,3-,4-, 5-jetdata in braces contains the collinear pole at Ci = 0 but not that at Cj = 0. Furthermore, when 13 «o averaged over the azimuthal angle fa around the direction of hard parton i, it vanishes • •• »/ QCDO(a^): outside the cone Ci = Cij- In fact i2*!23]) ^ = .0017 Ecm,AM?= 110 MeV I 40 ^'= E* ,Aj£ = 230MeV ; ••A = m "S Hence, averaging each term with respect to azimuth around its direction of singularity, *(J \ *""--..^30et we may wnte, Q Ci) + 0(Cij Ci) (2 24) V ''X^»--xi.-Jet * —'•- M = iêr ^ ~ i^7 " - - 0.0 0.05 0.10 0.15 Eq.(2.24) has the same form as the incoherent radiation emission result but with a dy y eut namically imposed angular constraint on the phase space. Figure 7: Jet fractions from the OPAL collaboration at LEP [21]. Perturbative QCD fits A heuristic explanation of the reason for angular ordering can be obtained [25] using a with different choices for the renormalisatiori scale fi are shown simple uncertainty principle argument. Consider an incoming virtual photon which decays into an electron-positron pair. An additional soft photon of momentum k is subsequently radiated from the electron-positron pair. The virtual state consisting of an electron and 2.3 Colour coherence a positron differs in energy from the final state containing an electron, a positron and a + soft photon by an energy A£, For the case of three jet events in e e~ annihilation the coherence of the radiation from the hard partons leads to the string effect [22,23]. In the language of perturbative QCD, AE = (JE; + Ej + Ek) -(£,-+/, + Ej) the string effect is a result of constructive and destructive interference. Of course, it is entirely unremarkable that such interference effects should be observed in quantum = y1pi|2 + m2 + |fc|-J(pi + fc) +m». (2.25) field theory. However, it is interesting to note that the experimental evidence indicates that such interference effects survive the hadronisation process, a phenomenon which the In the limit of very large p; and small 6ik this becomes, authors of ref.[23] call local parton-hadron duality. AS-!*>*• (2-26) At sufficiently high energy, the colour structure of the hard final state partons will determine the pattern of associated radiation. Because the distribution of this radiation By the uncertainty principle the virtual electron state lives for a time At which is approx is not significantly altered by hadronisation the observed pattern of the hadrons which imately given by lie between the jets will depend on the colour of the partons participating in the hard At~^~^, (2.27) scatter. where X ~ l/«r ~ l/(«*ifc) is tne transverse wavelength of the emitted soft photon. In We illustrate the derivation of the angle ordered approximation in the process e+e- —» T this interval of time At the electron and positron separate a transverse distance given by qqg. Soft gluons are emitted only inside certain angular regions around the directions of the hard partons q, q and g. We introduce the angular variables Ci = 1 — cos S{, where #,- Ad = Atfl0 = ^ii. (2.28) is the angle between the soft gluon and the hard parton i, and Cij = 1 — cos 8{j where 6{j is the angle between hard partons t and j. In terms of these variables the eikonal factor which describes the emission of soft radiation may be written, If 6ik > #,3, the separation of the electron and positron is less than the transverse wave 3.1 Deep inelastic scattering and the parton model length of the emitted soft photon. The emitted soft photon perceives the electron-positron pair as an unresolved charge neutral object and no radiation occurs. If, on the other hand, Consider the scattering of a high energy charged lepton off a hadxon target. If we label the the emitted photon lies within the cone described by the electron positron pair, Oik < 0^, incoming and outgoing lepton four-momenta by k" and V respectively, the momentum of the radiation is uninhibited. the target hadron (assumed hereafter to be a proton) by p* and the momentum transfer by q" = Jf — W, then the standard deep inelastic variables are denned by: This example indicates the reason for angular ordering in QED. The generalisation of 2 2 this argument to QCD is complicated by the fact that the gluons themselves carry colour Q2 = -«*, p = M charge, but the angular ordering result persists. Q1 Q1 It is an interesting property of the theory that the emission of gluons in the final state 2p • q 2M{E - E') can, to a good approximation, be represented by a semi-classical parton 'branching' or y = 1 - E'/E , (3.1) 'cascade' picture, i.e. the quarks emit gluons which in turn emit more gluons etc. This k-p property is evident for example in Eq.(2.24) where it is shown that the eikonal factor where the energy variables refer to the target rest frame. If the lepton is an electron or obtained from the interference of Feynman diagrams can be approximately represented muon, then the scattering is mediated by the exchange of a virtual photon, Fig.(8). as a sum of probabilities. The quarks produced at the photon vertex after an e+e~ The structure functions Fi(x,Q2) - which parametrise the structure of the target as annihilation have 'virtuality' (i.e. are off mass shell) of the order of the total centre-of- 'seen' by the virtual photon - are then defined in terms of the lepton scattering cross mass energy. Parton branching then takes place, reducing the virtualities, until all the sections. For charged lepton scattering, Ip —> IX, final state partons have virtualities of the order of the hadronic mass scale (0(1 GeV)). 2 This part of the fragmentation can be described in terms of QCD perturbation theory. d?c ST:a ME 1 + (1- ^L\2xFT Finally, the partons 'hadronise' to give final states made up of pions, kaons and other dxdy hadrons. The hadronisation of the partons cannot be described perturbatively, but instead +(l-y)(Fr-2xFr (M/2E)xyF? (3-2) can be modelled, the parameters being determined by fitting to the data. In this way jet fragmentation Monte Carlos are constructed. Different ways of performing the non- and for neutrino (antineutrino) scattering, vp —• IX, perturbative hadronisation lead to different models [26] which can be compared with G\ME •"H experimental data. (i-y-^y)F: dxdy 3. Deep Inelastic Scattering and Parton Distributions W*F?P) + (-)y(i-y/2)*F?P) • (3.3) The original, and still the most powerful, test of perturbative QCD is the breaking of Bjorken scaling in deep inelastic lepton-hadron scattering. Nowadays, deep inelastic structure function analyses not only provide some of the most precise tests of the theory but also determine the momentum distributions of partons in hadrons for use as input in predicting cross sections in high energy hadron collisions. In this lecture we begin by discussing deep inelastic scattering and the 'naive' parton model. We then show how QCD modifies the simple Bjorken scaling property of the parton model, and discuss how these 'scaling violations' can be calculated in perturbation theory. We compare the theoretical predictions with experimental data, and calculate the asymptotic behaviour of the parton distributions at small x. Finally, we describe the generalisation of the parton picture for general hard scattering processes involving quarks and gluons. Figure 8: Deep inelastic charged lepton-proton scattering 157 158 159 160 The Bjorken limit is defined as Q2,p • q —» oo with x fixed. In this limit the structure (3.6) functions obey an approximate scaling law, i.e. they depend only on the dimensionless £|Jff = 2e,V^ variable x: 2 Using Eq.(3.1) we can substitute for the deep inelastic variables: t = —Q , u = s(y — 1) *;(*, Bjorken scaling implies that the virtual photon scatters off pointlike constituents, since This result suggests that the structure function F2(x) 'probes' a quark constituent with otherwise the dimensionless structure functions would depend on the ratio Q/Qo, with momentum fraction x. Now clearly the measured structure function is a distribution in l/Qo some length scale characterizing the size of the constituents. The 'parton model' x rather than a delta function, suggesting that the quark constituents carry a range of picture of deep inelastic scattering is most easily formulated in a frame in which the momentum fractions. proton is moving very fast — the infinite momentum frame. In this frame, we consider a The above ideas are incorporated in what is now known as the 'naive parton model' simple model where the photon scatters off a pointlike quark which carries a fraction { of 1291: the proton's momentum. Setting M2 — 0, we can rewrite Eq.(3.2) as • l(0^( represents the probability that a quark q carries momentum fraction between J 4ira 2 { i and ( + d( [1 + (1 - y) ^ + -^-T^(F2 - 2xi\) (3.5) dxdQ* Now the spin-averaged matrix element squared for massless eq —• eq scattering is obtained • the virtual photon scatters incoherently off the quark constituents simply by crossing the corresponding matrix element for e+e~ —> qq considered in the previous lecture. In terms of the usual Mandelstam variables s,t,u we have Thus I ' I ' I ' a 2 •4 1- Q [GeV ] _] (3.9) $ « * 1.5 • 3.0 a » 4.0 u * 5.0 and so for the scattering of a charged lepton off a proton target, .3 • T * 8.0 F2"(x) = 2x d(x) + s(x) + ù(x) + c(x) + (3.11) I • I • I • I A complete list of the most commonly encountered structure functions is given below. .1 .2 .4 .5 .6 n. 8 . .9 Figure 9: The F2 structure function from the SLAC-MIT and BCDMS collaborations F," = 2x[d+s + û+c} 10 I ' ' ' I ' ' ' 2 xF% = 2x[d + a - û - c] M =10 GeV* 3 —. Ff = 2x[u + c + d+s] xF% = 2x[u +c-d-a) 1 — --& *7" = x[l(u + u + c + c) + i( 2xFl = F2. (3.12) This last result evident in Eq.(3.8) follows from the spin-i property of the quarks. With sufficient number of measured structure functions, the above relations can be inverted to give the quark distribution functions themselves. From such an analysis, the following picture emerges. The proton consists of three valence quarks (uud) which carry the electric charge and baryon quantum numbers of the proton, and an infinite sea of light qq pairs. When probed at scale Q, the sea contains all quark flavours with m, < Q. Thus at a scale of 0(1 GeV) we have Figure 10: Quark and gluon distribution functions at Q2 = 10 GeV2 u(x) = uv(x) + S(x) d(x) = dy(x) + S{x) 3.2 Scaling violations and the Altarelli-Parisi equations û(x) = d{x) = S{x). (3.13) In the 'naive' parton model the structure functions scale, i.e. F(x,Q2) —* F(x) in the asymptotic (Bjorken) limit: Q2 —* oo, x fixed. In QCD, this scaling is broken by log with the sum rules arithms of Q. To see how this Q2 dependence arises, consider the O(cts) corrections to the eq —> eq scattering process considered in the previous section. An explicit calculation f dx uv(x) = 2, / dx dv{x) = 1 gives dx x(q(x) + q(x)) ~ 0.5. (3.14) 1 2 S{z ?/ h(x,Q ) •*) The last of these is an experimental result. It indicates that the quarks only carry about o (3.15) 50% of the proton's momentum. The rest is attributed to gluon constituents. Although P( )in^ + C(-) + l{ the gluons are not directly measured in deep inelastic lepton hadron scattering, their presence is evident in other hard scattering processes such as large transverse momentum where P, C are calculable functions and K is a regulator (for example, the quark virtuality 2 2 jet and prompt photon production (see later). Fig.(10) shows a typical set of quark and K = —p ) which is introduced to control the collinear divergence which arises when the gluon distributions extracted from fits to deep inelastic data, at y.2 — 10 GeV3. gluon is emitted parallel to the incoming quark. This divergence is not subject to the theorems for cancellation of singularities discussed in the second lecture, because the Closer examination of Fig.(9) reveals a systematic deviation from exact Bjorken scal virtual photon can resolve a quark and a collinear quark-gluon pair carrying the same ing: the structure function decreases with increasing Q2 at large x and has the opposite overall momentum. behaviour at small x. In the following section, we discuss how these scaling violations are understood in perturbative QCD. If we again integrate the above result with the quark distribution function ?({) and choose to define Q2-dependent quark distributions by 2 2 Fi(x,Q ) = ^e .xq(x,Q), (3.16) 161 162 163 164 then we find to 0(as), Retaining only the first term in this expansion gives precisely the result in Eq.(3.18), with P = P<°>. ,(x ) = ,(») + g ^ ^,(0 j J^)ln g + C(|)| + .... (3.17) lM In fact the above derivations are strictly only correct for differences between quark distributions, q — ç; — qj. In general, the Altarelli-Parisi (AP) equation is a matrix How can we interpret the limit K2 —» 0? Exactly as for the renormalisation of the cou equation, pling constant, we can regard q(x) as an unmeasureable, bare distribution. The collinear d_ S q(x,t) \ = os(0 /* d{ / P„ ff,a5(0) Pu (f.«s(0) \ / «(«,*) singularities are absorbed into this bare distribution at a 'factorisation scale' /i0, which (3.24) plays a similar role to the renormalisation scale. There is therefore no absolute prediction dt[g(x,t) 2* J. ( \P„(f,as(0) -P„ff,as(0) / U(M) for the 'renormalised' distribution q(x,fi). What the theory does tell us, however, is how The AP kernels P[°\x) have an attractive physical interpretation as the probability of the distribution varies with y}. Thus if we define t = ln(^3//xj) and take the ^-derivative finding parton i in a partem of type j with a fraction x of the longitudinal momentum of of Eq.(3.17) we obtain the parent parton and a transverse momentum much less than fi. The interpretation as probabilities implies that the AP kernels are positive definite for x < 1. They satisfy the S^.0=^/f««.0P(f). (3.18) following relations: This equation - known as the Altarelli-Parisi equation - is the analogue of the 0 function dxP$\x) = 0 equation describing the variation of as(t) with t. / The above derivation is rather heuristic, but a more complete treatment confirms and dxx[p£K*) + p£K*)] = o extends the result. The full prediction of the theory is most easily cast in terms of the / moments (Mellin transforms) of the distributions: rfxX [271^(1) + PW(X)]=0. (3.25) / 1 1 These equations correspond to quark number conservation and momentum conservation q{j,t) = I dx x q{x,t). (3.19) in the splittings of quarks and gluons. In terms of these moments, the t dependence of the quark distribution function is given The kernels of the AP equations are calculable as a power series in the strong coupling by as. Both the lowest order terms [30] and the first correction [31] to the evolution kernels ^=7„(j,«sW)?(j,'). (3.20) have been calculated. The lowest order approximations to the evolution kernels are: We next define Pqq as the inverse Mellin transform of "Yggi P£KX) = cF a 5 1 f 1 — Pqq{x,as) = —: dj x- tqq{j,as), (3.21) Z7T ZTTl f 2 2 x +(l-x) , Tfl = P£K*) = TR where the integration contour in the complex j plane is parallel to the imaginary axis and i + (i-x)»i to the right of all singularities of the integrand. Taking the inverse Mellin transform of pj&X*) = CF Eq.(3.20), we obtain in x space, 1 — x JllN-4n,T ) P£>(x) = 2N + + x(l -x) + 5(1- R a ^^ " ^J^à(^dz6(x-iz)P„{z,as{tM(,t) (l-*)+ dt 2TI (3.26) d( x °s(0 £t -j p«Q,*s{tj)4i,o f (3.22) 2TT The 'plus prescription' on the singular parts of the kernels is defined as Pqq has a perturbative expansion in the running coupling, t dx /(x)[s(x)]+ = tdx (f(x)-f(l))g(x). (3.27) as P,q{z, as) = P<°>(;) + £P£K*) + • • • (3-23) In terms of moments these four evolution kernels take the form 1 1 ' 1. 1 1 1 ! 1 I 1 010 -rÏÏ'Ci) = cF + 2 • x = 003 0 30 • • • • • 2 JûTT)~ ^k EMC ; 2 (2 + j+i2 0 1.0 _ F," (x.U l 030 " • • • • x = 0 05 i(j + l)(j + 2) = o to 0 30 - - "(2 + j + r x - ooe . 7 singlet (in flavour space) such as q; — (3.34) (3.32) £ = £(* + *)• From Eq.(3.24), which holds for all flavours of quarks, we derive the equation for the Inserting the lowest order form for the running coupling, we find the solution flavour singlet combination of parton distributions, •MJ) V(j,t) = V(j,0) as(0) 7ff(i) (3.33) ^ [i*°> 2 + 2»,*>£> ® g] + O («J (r)) «5(0 - 165 166 167 168 Note, however, that the approach to the asymptotic limit is controlled by t ~ In /x2 and is I / 2(2) as(t) I -C,| /E(2) (3.36) therefore quite slow. For a tabulation of the eigenvectors and eigenvalues of the moments it I g(2) 2x CF\ =F 1^(2) of Eq.(3.35) we refer the reader to reference [5). Figs.(12) - (16) show the scale dependence The eigenvectors and corresponding eigenvalues of this system of equations are of the quark and gluon distributions. + 0 (2) = E(2)+5(2) Eigenvalue^ 0"(2) = 2(2) - £-g(2) Eigenvalue: (3.37) ,-'•? 1 i—i i IIIIII|—i i IIIIII|—i 11nni|—i i iiinij—i i iiuiij—i i Mini]—i 11inn Note that the combination 0+, which corresponds to the total momentum carried by the Up valance distribution .5 = x-10 ' quarks and gluons, is independent of t. The eigenvector 0~ vanishes at asymptotic t: T«m ' o- .,â"^o,,( ) = -^^) (3.38) (2) 2 2iri So that asymptotically we have 2(2) = nj__ Nn, 2 - (3.39) g(2) iCF 2(7V -I) The momentum fractions carried by the quarks and gluons in the p. —» co limit are therefore i ' mini i i i mill i ' '""'I ' i mml I i mini I I I mill n 4C 10' 10' 10" 10* _ 1,0" 10° 10 10 2(2) f \ j,i! ( V (3.40) [Gevj 5(2) = iCF + njj' |t=00 [ACF + nf Figure 13: The scale dependence of the valence up distribution t i i MIII»—i i IIIIII|—i i niiiij—r HI] i1 I1 1llllll mill—j i 111 uni IMIITIJ I i iinin i i iiniij niiiiij 1 1 llllllj i i iinii Gluon distribution ; Down valence distribution .3 =_ on"3 — x==i(f 2 ' .1 — x= JO"' .- x" •a .03 k - X .01 x==0. 6 .003 - - 1 I I I I mill I I I mill I I """1 ' I mini I ' "I I I I , I mini 1 II ml i 1 linn! 1 i ! mil I mini i i • 1 i i nun 3 6 ' 10' 10* 10 10* 10 10* 10' 10' 10' 102 103 10•0*, i° 10 10 10 S [GeV4] Figure 12: The scale dependence of the gluon distribution Figure 14: The scale dependence oi the valence down distribution 3.3 QCD fits to deep inelastic data In the the previous section we saw that perturbative QCD predicts the Q2 evolution of the structure functions, rather than the size and shape of the functions themselves. Quantitatively, the variation with Q2 is controlled by as(Q) and hence by the QCD scale parameter A. Deep inelastic scattering data of the type shown in Fig.(11), therefore, provide one of the 'precision' tests of QCD and, arguably, the most accurate determination of AMS' Although the theoretical predictions appear simplest when expressed in terms of struc ture function moments, it is very difficult to extract such moments from the data. This is because the measurements do not extend to very large and very small x, and some form of ad hoc extrapolation is required to construct the moment integrals. A more practical and accurate method is to choose a reference value Qo and parametrise the parton distri a b butions at that value, e.g. q(x,Q0) = Ax (l — x) . These distributions are then evolved numerically, using the Altarelli-Paxisi equations, to obtain values for the Fi(x,Q2) in the kinematic regions where they are measured. Note that in this approach the rate of change with Q2 of the structure function at a given x depends only on the structure function eval uated at £ > x, cf. Eq.(3.24). Finally, a global numerical fit is performed to determine the 'best' values for the parameters, including A. The extent to which the measured value of A depends on the other parameters can also be quantified and used to derive a systematic error. The above procedure is not, however, without problems. The most serious of these are: • In QCD, the structure functions have 'higher twist' power corrections, which are much more difficult to estimate quantitatively: F(x,Q2) = FW(x,Q2) + Fi,){*;Q2) + ..., (3.41) where the superscripts on the right-hand-side refer to the 'twist' = (dimension — spin) of the contributing operators. To avoid these complications, the analysis must be performed at large Q2 where the power suppressed terms are negligible. • The structure function F2 can be decomposed into singlet and non-singlet ('sea quark' and 'valence quark') parts, which dominate at small and large x respectively. Hence, except at large x, the Q2 dependence of F] is sensitive to the a priori unknown gluon distribution and there is potentially a strong A-gluon correlation. • Non-singlet structure functions do not suffer from the gluon correlation problem (see Eq.(3.31)), but these are only measurable experimentally by constructing dif ferences between cross sections, e.g. a** — o<*n. This inevitably introduces additional systematic and statistical uncertainties. 170 171 172 The most recent generation of deep inelastic experiments partially solve these problems l I by collecting high statistics data at large x and Q2. In fact the precision of contemporary b) BCDMS H2 -BCDMS H2 data demands that the next-to-leading order QCD predictions are used in the fits. Beyond ^ Q!>20GeV2 o— BCDMS C 2 leading order a specific renormalisation scheme must be chosen, and in practice this is Q » 20 GeV' usually the MS scheme. For this reason the results quoted in the literature almost always -0 1 - refer to Apjg. \ Some of the most precise recent data comes from the BCDMS collaboration [28,33]. x\ \ ASs = 100 MeV \f As an example, Fig.(17) shows the structure function Ft measured in deep inelastic muon- Afis= 205 MeV-^ 2 hydiogen scattering. The measurements extend up to x values of 0.75 and Q values of TD -02 A(r5 = (.00MeV^ \ several hundred GeV2. Fig.(18) shows the corresponding logarithmic Q2 derivative of logFj as a function of x. Note that the derivatives in this region are negative, consistent with a structure function which decreases with increasing Q2. Also shown are the predic tions of next-to-leading order QCD for three different values of A^. A detailed fit gives -0 3 j i_ [33] 02 0U 06 080 02 04 06 08 A<^ = 220 ± 15 ± 50 MeV (3.42) 2 This result for A^ÂS is compared with determinations from other processes in Fig.(2). Figure 18: Logarithmic Q derivative of the F? structure function in the previous figure with QCD fits, from BCDMS Deep inelastic experiments measure quark densities over a broad range in x up to about fi — 15 GeV. Knowing A^j, these can then be evolved to higher /i and used for hadron collider phenomenology. Instead of laboriously integrating the Altarelli-Parisi equations each time a parton distribution is required, it is useful to have an analytic approximation, valid to a sufficient accuracy over a prescribed (x,fi) range. Several such parametrisations are available. The widely used Duke and Owens parametrisations [34], for example, are of the form Ax"{l + cx){l - x) 1 A = Aa + Aïs + A2S etc. _ An(Q2/A2)\ > 0, (3.43) with the parameters A<>, A\, ... fitted to an exact leading order evolution to give an accu racy of a few per cent. Because deep inelastic scattering does not significantly constrain the gluon distribution, it was usual - in the past - to include in the parametrisations a choice of gluon distributions, typically a 'hard gluon' and a 'soft gluon', each with its own A value. Nowadays, high precision fixed-target prompt photon experiments are able to constrain the gluon, particularly in the medium x range, and 'hard gluon' parametrisa tions are ruled out [35]. The most recent generation of parton distributions - for example the HMRS sets [36] - are obtained from next-to-leading order QCD fits to a wide variety of deep inelastic data, as well as data from prompt photon and lepton pair production. The distributions cover a wide range in x and /x, and are ideal for making quantitative Figure 17: Data on the structure function F2 in muon-hydrogen scattering, from BCDMS predictions for present and future hadron-hadron and lepton-hadron colliders. 3.4 Small x behaviour of the parton distributions Notice that the dependence on the starting distribution enters via the j0th moment of g. Therefore at fixed (jy the initial information enters only as an overall factor. From Fig.(12), we see that the gluon distribution grows rapidly at small x. In the asymp A topic which is presently under active investigation [37] is the mechanism which totic limit where x —» 0 and n —> oo it is possible to determine the behaviour of the limits the growth of the gluon distribution. In the infinite momentum frame the gluon distributions directly from the Altarelli-Parisi equations. momentum distribution G(x,t) gives the number of gluons per unit of rapidity with a The x —> 0 limit of the parton distributions is controlled by the behaviour of the transverse size greater than l/fi. If the number of gluons grows so large that the partons anomalous dimensions -y(j) near j = 1. Considering the gluon only we have start to overlap inside the nucléon new effects will come into play. A crude estimate of when this begins to happen is provided by, jt9(j,t) = ^£\j)g(j,t) (3.44) Area of hadron ,, Î „.,-,•,• i , where from Eq.(3.28), G(x,t) = ,iV~M 25GeV-2, 3.54) Area of parton ^%)^~- (3-45) where r ~ 1/m» is the radius of the hadron. At presently attainable values of x the value In this limit the solution for the moments of the gluon distribution is, of G(x,t) does not exceed 3 or 4, so, if the above estimate is correct, the saturation limit is beyond the range of the present colliders. 9{j, *) = 9{i, to) exp ( . ^ A, (3.46) 4. The QCD Parton Model in Hadron-Hadron Collisions and ( is defined by, t = b f dt'as(t'). (3.47) In this lecture we shall consider the application of the parton model to processes involving two hadrons in the initial state. To return to x space we perform the inverse Mellin transform as given by Eq.(3.21) G(x,t) = xg(x,t) = J-fdjx-U-VgUS) (3.48) 4.1 The QCD improved parton model The high energy interactions of hadrons are described by the QCD improved parton model. = 2^/#s(i.to)«p[/(i)] fti) = J^y + o(j -j0y, j0 = i + !LK y= HÇin(i/»). (3.51) V xo y y Tzb The parton model for hard scattering events is depicted in Fig.(19). The momenta of We therefore find for the asymptotic solution the partons which participate in the hard interaction are px = XjPt and pa = X3P3. The characteristic scale of the hard scattering is denoted by Q. This could be, for example, the G(x,t) = g{jo,t0)exp y5^, (3.52) mass of a weak boson or heavy quark, or the transverse momentum of a jet. The functions which expressed in the original variables yields fi{x,ii) are the usual QCD quark or gluon distributions, defined at factorisation scale fi. The short distance cross section for the scattering of partons of type i and j is denoted AN. ln 2/A2, 1 » , , 33-2n, g(x) ~ - exp , M r by <7jj. Since the coupling is small at high energy, the short distance cross section can be A -rlni 2JA,ln~> iV = 3' b= T^ -• 3'53 173 174 175 176 Finally, it should be emphasised that Eq.(4.1) is not a description of the bulk of the events which occur at a hadron-hadron collider, but as we shall see, it can be used to describe the most interesting classes of events which involve a hard interaction. 4.2 Factorisation of the cross section The property of factorisation allows us to use the QCD parton model to describe inelastic processes. In this section we shall present a simple classical model that illustrates why the factorisation property holds and when it should fail. As an example of a hard process we consider the production of a massive vector boson V - in practice a massive photon, W or Z - in the collision of two hadrons, i?i(A) + H2{P2) -^V + X. (4.3) This is in many respects the simplest hard process involving two hadrons, since the ob Figure 19: Schematic of the parton model description of a hard scattering process served vector boson in the final state carries no colour and its leptonic decay products are observed directly. It is therefore the easiest to analyse theoretically and consequently has received the most theoretical attention. calculated as a perturbation series in the running coupling as. Therefore the nth order A very important theoretical issue in this process is whether the partons in hadron Hi, approximation to the short distance cross section is given by through the influence of their colour fields, change the distribution of partons in hadron (, Hi before the hard scattering occurs, thus spoiling the simple parton picture. Soft gluons * = C<°>a*(l+y\: Vs (4.2) which are created long before the collision are potentially troublesome in this respect. where the c"' are functions of the kinematic variables. We shall argue that soft gluons do not in fact spoil the parton picture, using a simple In the leading approximation (n = 0) the short distance cross section is identical to model [39] from classical electrodynamics. The vector potential due to a current density the normal parton scattering cross section calculated in exactly the same way as the cross J is given by [40] section for a QED process. In higher orders, the short distance cross section is derived ,J , A"(t,x)= [dt'dx iy*;h(t' + \x-x'\-t), c=l, (4.4) from the parton scattering cross section by removing long distance pieces and factoring / \x — x \ them into the parton distribution functions. The remaining cross section involves only where the delta function provides the retarded behaviour required by causality. Consider high momentum transfers and is insensitive to the physics of low momentum scales. In a particle with charge e travelling in the positive z direction with constant velocity /3. particular, the short distance cross section does not depend on the details of the hadron The non-zero components of the current density are wave function or the type of the incoming hadron. It is a purely short-distance construct and is calculable in perturbation theory because of asymptotic freedom. This factorisation J*(t,x) = c8(z-r(t)) property of the cross section can be proved to all orders in perturbation theory. For more J'{t,x) = e/3S(x-r(t)), r[t) = /3tz, (4.5) details, see for example reference [38]. A heuristic argument for the validity of factorisation is given in the next section. It is a fundamental property of the theory which turns QCD where z is a unit vector in the z direction. The charge passes through the origin at time into a reliable calculational tool with controllable approximations, distinguishing it from t = 0. At an observation point (the position of hadron Hi) described by coordinates the 'naive' parton model of Feynman [29]. x,y and z, the vector potential at time t due to the passage of the fast moving charge is The scale /i in Eq.(4.1) is an arbitrary parameter. It should be chosen to be of the obtained by performing the integrations in Eq.(4.4) using the current density of Eq.(4.5). order of the hard scale Q which characterises the parton-parton interaction. The more The result is terms are included in the perturbative expansion, the weaker the dependence on fi. e~j We may define the parton luminosity as follows: A (t X) ' ' J[x2+y2+12(pt-zy] Jr.. -t j A'(t,x) = 0 4.9 T~~dr' = 1 + 6-- [ dxi e-iB If à depends only on the product xxxi the parton cross section can be written as, A 'M = V[xl+y^^t_zyy (4-6) 1 dLjj (4.10) T 3 dr where 7' = 1/(1 — j32). The observation point can be taken to be the target hadron fcfr° Hj which is at rest near the origin, so that 7 =s s/m1. Note that for large 7 and fixed where à = xxxi» and the sum now runs over all pairs of partons {ij}. The first object non-zero (/3i — z) some components of the potential tend to a constant independent of 7, in square brackets has the dimensions of a cross section. The second object in square suggesting that there will be non-zero fields which are not in coincidence with the arrival brackets is dimensionless and is approximately determined by couplings. Hence knowing of the particle, even at high energy. However at large 7 the potential is a pure gauge the luminosities we can roughly estimate cross sections. As an example we can estimate piece and hence does not lead to £ or B fields. The implication of this result is that the cross section for the production of two gluon jets with pT > 1 TeV at v/i = 40 TeV. a covariant formulation which uses the vector potential A will not be the most efficient We assume that %/I = 2 TeV and from Fig.(20) we find method to handle this problem, since we will have large fields which ultimately have no physical effect. 1 dLjj 103pb. (4.11) To show that these large terms in the vector potential have no effect we compute the s dr field strengths from Eq.(4.6). The leading terms in 7 cancel and the field strengths are of The gluon jet cross section can be calculated to be approximately 10 pb after including order I/72 and hence of order m* /s*. For example, the electric field along the z direction two powers of as ~ 0.1. £*(t,*) = F'-^ + ^ = fZfci) r. (4 7) y at dz [X2 + y2 + 12(0t_zyyï 10 Thus the force experienced by a charge in the hadron H2, at any fixed time before the 2 10* arrival of the quark, decreases as m*/s . There are residual interactions which distort the 10' distribution of quarks in hadron Hi, but their effects vanish at high energies. A breakdown 2 10' of factorisation at order I/a is therefore to be expected in perturbation theory and has 10' been demonstrated explicitly in ref. [41]. Note that these effects are due to the long range 10' nature of the vector field. In the realistic case of an incoming colour neutral hadron there 10' are no long -range colour fields. It is therefore possible that the factorisation property is 10' even better in the full theory than in perturbation theory. In the next lecture we will -pp. Vs=40 TeV\ 10 consider vector boson production, dropping all terms suppressed by powers of s. The -pp, Vs=17 TeV * 10' r pn Vs=6 TeV \ QCD improved parton model will provide a valid description of this process. 10 -pp , Vs=l.B TeV ^ pp , Vs«0.63 TeV ! 10 4.3 Parton luminosities 10 100 «300 1000 3000 Vs [GeV] Since partons only carry a fraction of their parent hadron's momentum the available centre of mass energy of a parton-parton collision is less than the overall hadron-hadron collision Figure 20: Luminosity plot energy. A convenient way to quantify this is to define parton luminosities. Consider a 2 generic hard process initiated by two hadrons of momenta PI and Pj and s = (P1 + P,) , dx dx2 °"^) = Y. I ' [ fi(xuV)fAx2,n) àij(xiPi,x7P2,as{n)). (4.8) 177 178 180 ». I !*• | i i i i 111 i | i i i i M | I | i i i i i | 10" ' 1 [IIIIII 1 | 1 1 1 1 Mj i | i Mill 10" ^ v*v- ^C *-. g £(q+q) luminosity -t 10' r uu luminosity "i 10' \ r \ ""- X~X- ~t 10' Ns.*,, 1 \ •- X N X\ - 10 r \ ""-. v "'• Xv "1 10' 1 10* r \ '•-. X •••-X. 1 10' 10* r \ \ X"-C\ 10* LII I Lm i 10s 10' L/d T [pb ] r \ \ \ •- X \% I 10' -1 10' X \ \ \ « \ \ \., \\ - 10' I — --pp. Vs=17 TeV \ \ \ \ ^\ T 10' r -—pp, Vs=17 TeV \ '. N .X, i 10° T — --pp , Vs=6 TeV \ \ \ \f 10° pp , Vs=8 TeV \ \ \ - -pp , Vs=1.8 TeV \ \ \ r —-pp , Vs=1.8 TeV ', \ \ \ 10"' r \ ~t io- \i -pp , Vs=0.63 TeV \ \ \ M pp , Vs=0.63 TeV < \ \ - 10 ' r io- \ \ \ \~l 1 '• \ \~i IIIIIII! I 1 i i i i i il". i h i i k 1 1 1 ' ! 1 i 1 1 II 1 ' 1 11 10"' - I n ïr io- F i i fir 100 A300 1000 3000 100 .300 1000 Vs [GeV] Vs [GeV] Figure 21: Luminosity plot Figure 23: Luminosity plot 10 ï i | i i i i 11] i ] i i i TIT] 1 1—r- in, 1 i i i i 111 i j i m 1 10 -- uu luminosity 1 10' dd luminosity 10 "1 io7 10' 1 10* K-- 10' 1 10* ltf 1 10* : io 1 10' 10' 10' P pp. Vs=40 TèV \ \ 10 % j pp. Vs-17 TeV \ "\ \ '\\ i io' pp. Vs=17 TeV\ ~-v \ 10' \. pp , Vs-6 TeV \ \ \ 10° , pp . Vs=6 TeV \ - pp . Vs=1.8 TeV \ \ \ 10' Nj io- pp , Vs=1.8 TeV \ E pp , Vs=0.63 TeV \ \ \ -pp , Vs=0.63 TeV 10' 10- 10' " ' 1 1 ' 1 1 1 • * • i \> i i ' i i 111 n i i i ir 3000 100 .300 1000 3000 10000 100 A300 1000 10000 Vs [GeV] Vs [GeV] Figure 22: Luminosity plot Figure 24: Luminosity plot 10' ~| 1 I I I I M 1 5. Large pj- Jet Production in Hadron-Hadron Collisions 10' dd luminosity + + + 7 The scattering processes e e~ —» e e~, e e~ —» 77, ... provide fundamental tests io of QED. The analog processes for QCD, qq —» qq, qq —» gg, ... can be studied in the 10' production of large transverse momentum jets in hadron-hadron collisions. After denning ~ 10' some kinematics, we show how the jet inclusive cross section is calculated in the QCD ± io* improved parton model. We study the pr and angular distributions, and compare the £ 10' theoretical predictions with the experimental data. We extend the discussion to include Î io' -pp, Vs=40 TeV \ multijet cross sections, and finally describe the related process of direct photon production. \ 101 pp, Vs=17 TeV 10° pp , Vs«6 TeV pp . Vs=1.8 TeV 10"' 5.1 Kinematics and jet definition pp , Vs=0.63 TeV 10"' As described in the previous lecture, the scattering of two hadrons provides two broad 10"3 i I I 100 .300 1000 3000 10000 band beams of incoming partons. These incoming beams have a spectrum of longitudinal Vs [GeV] momenta determined by the parton distribution functions. The centre of mass of the parton-parton scattering is normally boosted with respect to the centre of mass of the Figure 25: Luminosity plot two incoming hadrons. It is therefore useful to classify the final state in terms of variables which transform simply under longitudinal boosts. For this purpose we introduce the rapidity y, the transverse momentum px and the azimuthal angle The rapidity y is therefore defined by io'i- — 10' * 10' I. io3 and is additive under the restrictive class of Lorentz transformations corresponding to a boost along the z direction. Rapidity differences are boost invariant. 10' -pp, Vs=40 TeV \ 10: —pp, Vs=17 TeV In practice the rapidity is normally replaced by the pseudorapidity t), 10' ^_ pp t Vs=6 TeV pp , Vs=1.8 TeV 10" 77 = -lntan(^), (5.3) pp , Vs=0.63 TeV 10' •t - r which coincides with the rapidity in the m —» 0 limit. It is a more convenient variable 10 experimentally, since the angle S from the beam direction is measured directly in the 100 A300 1000 Vs [GeV] detector. It is also standard to use the transverse energy rather than the transverse momentum for similar reasons. Many methods can be used to define what is meant Figure 26: Luminosity plot by a jet. There is no best definition, but one must be sure that both theoretical and experimental analyses use the same definition. A commonly used definition of a jet is a cluster of transverse energy ET in a cone of size AR, where Ai? = Vi(Aj,)2 + (A^)2i. (5.4) 181 182 183 184 In the two-dimensional y, 5.2 Two-jet cross sections / In QCD, two-jet events result when an incoming parton from one hadron scatters off an incoming parton from the other hadron to produce two high transverse momentum (b) partons which are observed as jets. From momentum conservation the two final state —*—fJLSUU partons are produced with equal and opposite momenta in the subprocess centre-of-mass ¥ frame. If only two partons are produced, and the relatively small intrinsic transverse —*-Ujl&v momentum of the incoming partons is neglected, then the two jets will be back-to-back (c) in azimuth and balanced in transverse momentum in the laboratory frame. For a 2 —> 2 parton scattering process (d) parton;(pi) + parton^pj) -» partonk(p3) + parton,(p4), (5.5) Figure 27: Diagrams for jet production described by a matrix element M, the parton cross section is 4 E%EiiP& 1 1 ^=^,.,.2 „*, Process £W/s B' = TT/2 ï^îpT = 2JÏI^i:W * (Pl + » - » - *). (5.6) 4 p+i2 q q' -> g q' 2.22 All parton processes which contribute in lowest order can be derived from the diagrams 9 i2 shown in Fig.(27) by including other diagrams which are related by crossing. Expressions 4 J2+û2 s2 + t\ 8 s2 3 9 9 — 9 9 3.26 for the leading order matrix elements squared £|M| , averaged and summed over initial 9 v i2 ' ù2 ' n i.i 2 and final state spins and colours are given in Table 3 in the notation i = (pi + P2) , 2 2 4 î + û 2 2 q q -. q' q' 0.22 * = (Pi - Pz) and û = (p, - p ) . 2 3 9 S The two-jet cross section may be written as a sum of terms each representing the 2 2 2 2 2 4 s + û t + û 8 û 2.59 qq-*qq 2 2 contribution to the cross section due to a particular combination of incoming (i,j) and 9 ^ t ' ê ' 27 it outgoing (k,l) partons. Using Eq.(5.6) the result for the two jet inclusive cross section is, 2 2 2 2 32 i + û 8t + û 1.04 9Q-* 9 3 27 tù 3 s2 2 2 2 2 1 i + û 3 t + ii 0.15 9 9 -> 9 9 6 tû 8 s2 where the /;(Z,/J) represent the number distributions for partons of type i (i = u,û,d,d,g,... etc.), 2 2 2 2 4 s +û û + s 6.11 9 9 -»S 9 2 evaluated at momentum scale /i, and ^3 and y4 represent the laboratory rapidities of the 9 iû i outgoing partons. For massless partons the rapidities and pseudorapidities may be used 9 tû su ât 30.4 interchangeably. The Kronecker delta function introduces the statistical factor necessary 9 9 ~> 9 9 2 ^ ~ ~P~ JÏ ~ û2' for identical final state partons. If we assume that the detector and jet algorithm are 100% efficient, the rapidities and p of the outgoing jets may be identified with those of T Table 3: The invariant matrix elements squared YNAfl for two-to-two parton subpro- the outgoing partons. cesses with massless partons. The colour and spin indices are averaged (summed) over initial (final) states. We now consider the kinematics of the two produced jets in detail. The laboratory Note that this result corresponds to massless quarks and gluons and that no distinction rapidity (j(bo<*<) of the two-parton system and the equal and opposite rapidities (±y*) of is made between quark and gluon jets. the two jets in the parton-parton centre-of-mass system are given in terms of the observed rapidities by: 5.3 Comparison with experiment JfbooK = (y3 + y*}/2, y' = {y3 - y<)/2. (5.8) For a massless parton the centre of mass scattering angle 8' is given by, Although large px jet production has been studied at different machines over a period of many years, the definitive data are from the high energy pp colliders, i.e. from the UA1 „*. Pi sinhjy') y3-yt and UA2 collaborations at the CERN pp collider (Ji = 546 GeV and 630 GeV) and from cos» = —- = ——- = tannl—-—) , y5.9) E' cosh(i/*) 2 ' the CDF collaboration at the FNAL Tevatron collider {^/i = 1.8 TeV). It appears that only at these very high collision energies does the identification and measurement of large where y' = y3 — î/boo.t- The measurement of the rapidity difference of the two jets in the px jets become relatively unambiguous. At lower energies it is difficult to separate the laboratory frame determines the subprocess centre of mass scattering angle 8'. jets from the other 'underlying' hadrons in the event. The longitudinal momentum fractions of the incoming partons Xi and x3 in Eq.(5.7) Two quantities are particularly useful for comparing theory with experiment. The are given in terms of pr,ys and y4 by momentum conservation: first is the jet pr distribution, obtained from the inclusive cross section by n n X! = z e »" cosh(y'), x, = x e~ '"' cosh(j/*), j/ . = - In —, (5.10) 3 3 T T boo t Ed a- _ d a 1 d?tr 2 x2 2 d?p ~ d pTdy ' 27r£r dETdr} ' ' where XT = 2yjjs/a. Lastly, the invariant mass of the jet-jet system can be written as, where the third term follows if we assume that the jets are approximately massless. 2 M)j = s = ApT cosh (y'). (5.11) Fig.(28) shows the jet ET distribution in pp collisions at y/i = 1.8 TeV, from the CDF collaboration. The curve is the QCD prediction, calculated in next-to-leading order (i.e. Given a knowledge of the parton distributions from deep inelastic scattering experi 0(a|)) by S. D. Ellis et al. [42] and using the HMRSB parton distributions from reference ments, Eq.(5.7) may be used to make leading order QCD predictions for jet production in [361. The next-to-leading order contributions considerably reduce the dependence on the hadron-hadron collisions. For example, the inclusive jet cross section at the parton level scale parameter /i, and allow a more precise treatment of effects due to the finite width may be obtained by integrating Eq.(5.6) over the momentum of one of the jets. of the jet. The agreement is excellent, especially considering that there are essentially no free parameters in the theoretical prediction. Note that at this energy about half the Ed3â d3â 1 1 ^= ,, ,.,. . -^ = ^^ = 2l8^EW2^ + i + ^ (5.12) cross section comes from quark-gluon scattering, the other half coming from gluon-gluon scattering at the lower ET end, and quark-(anti)quark scattering at the high ET end. where t and û are fixed by i and the centre of mass scattering angle, The second quantity of interest is the jet angular distribution. In the parton-parton centre of mass, the angular distribution is sensitive to the form of the 2 —• 2 matrix i = -- (1 - cos 9*) elements. The differential cross section for a jet pair of mass MJJ produced at an angle 8' to the beam direction in the jet-jet centre of mass can readily be obtained from Eq.(5.7) û = -- (1 + cosfl'). (5.13) using the transformation Again assuming that the detector and jet algorithm are 100% efficient, so that p^., = dpTdy3dyt = -dx\dx2dcos8' (5.16) ne Ppirtom ' single jet inclusive cross section is obtained from Eq.(5.12) by folding in the to give parton distribution functions: 3 1 ^ / dxxdx2 fi(x1,ii)fj{xi,n) 6(xixt3 - M] Ejd à 1 „ f dxx dx2 d* 16TT23 fe _/) n x pj u q s 2 TJ^ dLij(rj,ii) da" (5.17) s drj d cos i y|AT(iy-.W)|2—ij-^i + t + û). (5.14) i—l 1 + Ou 185 186 187 188 i—I—i—I—I—I—i—i—I—1—I—i—i—r i | i i i | i i i ! , , i j i i ! Inclusive jet cross section (AR=0.7) M„ > 200 GeV - CDF data, statistical errors only 10 — I Normalisation uncertainty 8 - •a 6 — as 4- 1- T - , 4 \ -*- 1: - - - 2 n i 1 i i i 1 i i i 1 i i i 1 i i i 100 200 300 400 500 4 6 8 10 ET [GeV] X Figure 28: Jet ET distribution from the CDF collaboration, compared with Figure 29: x distribution from the CDF collaboration compared with the leading order next-to-leading order QCD prediction from [42] QCD prediction with TJ = MJJ/S and d(T{' 1 ^ ^oiF ^Ç^jjD^'^i'rr^ (5.18) In the small angle limit (x —• oo) the cross section differential in x is then do- (5.22) Note that for each subprocess the dcr/dcos9' is symmetrised in t and û (unless k = I). — ~ constant. <*X Thus, for example, Data on the angular distribution from the CDF collaboration are shown in Fig.(29), with d&" ™ 4 4 + (l + cosfl,):i 4 + (l-cosg,);> the leading order QCD prediction. Again, there is excellent agreement. Note that these s (5.19) d cos 6' 2Mh 9 (1-COS0')2 (1+COS0*)' data automatically rule out certain other quark scattering mechanisms. For example, a model in which quarks scatter by exchanging a scalar gluon would give a less singular Numerically the most important subprocesses are gg —* gg, gq —» gq and qq —» qq. For behaviour (sin-î(#*/2)) at small angle. each of these, the 8' distributions have the familiar Rutherford scattering behaviour at It is also interesting to note that the angular dependences of the dominant subpro small angle, characteristic of the exchange of a vector boson in the t-channel: cesses are very similar. Fig.(30) shows the cos 6' dependence of the qg —> qg and qq —» qq db subprocesses normalised to gg —» gg. These ratios are evidently rather constant at the nu (5.20) dcos0' sin4(Ç)' merical values 4/9 and (4/9)2 respectively. This can be understood in terms of the colour structure of the Feynman diagrams. Thus to a good approximation the gg —t gg subpro It is convenient to plot the data in terms of the variable x> which removes the Rutherford cess can be used as the 'universal' subprocess in the result given in Eq.(5.17), i.e. the singularity [46], i + cos e* angular dependence effectively factors out leaving a convolution of parton distributions. (5.21) This is called the single effective subp-rocess approximation [46]. 1 — cos i (A) q(pi) + q'(pi) - q(ps) + q'(p*) + g(k) 1 1 ' 1 1 1 i (-B) 9(?i) + ?M - ?(P3) + 9(P4) + s(*0 (C) q(p„) + q{pb) -* g(pi) + s(?a) + ff(ps) _ (4/9) (D) g(pi) + g(p,) -* gfa) + g(p*) + gfa)- (5.24) — qg - qg The momentum assignments for the partons are given in brackets. All other matrix gg - gg elements for two-to-three parton amplitudes may be obtained by crossing from the above four processes. - The matrix elements squared for the processes (A — D), averaged (summed) over the (4/9)8 initial (final) colours and spins are given below. We have set the masses of the quarks - qq - qq - equal to zero. With the momentum assignments of Eq.(5.24) the matrix element [45] for — gg •* gg — process (A) is, £M" - «£("+"£•'+''•) («KM* w) • >.»] (5.25) 1 I , 1 1 i .2 .4 . .6 The kinematic variables are defined as follows, |cos e I 2 2 2 Figure 30: Quark-antiquark and quark-gluon angular distributions, normalised to that * = (Pi+P2) , t = {p1-p3) , u = (p1-pi) , for gg -» gg 2 2 *' = (P3+P4) , t' = (p2-P4) , u={p2-V3)\ (5.26) For compactness of notation we have introduced the eikonal factor [ij] which is defined as, 5.4 Multyet production (5 27) As long as the jets are required to be well separated in phase space, multijet cross sections ^£m- - can be calculated from scattering processes involving many quarks and gluons in the final We have also defined the following sum of eikonal terms, state. In this way one defines an n-jet cross section a" for producing n jets which satisfy, say, pj, > p?in, 1^1 < 7)""" and ARij > Aiïw. for i,j = l,...,n. In leading order QCD, [12;34] = 2[12] + 2[34] - [13] - [14] - [23] - [24]. (5.28) these cross sections are calculated at the parton level from 'tree-level' Feynman diagrams, Note that this combination is free from collinear singularities. In Eq.(5.25) the dependence i.e. diagrams without any internal loops. The general expression is again obtained from on the SU(N) colour group is shown explicitly, (C = N = 3,C = 4/3). Eq.(4.8): A F n i k In the same notation the result for process (B) with four identical quarks [45] may be S, + S + + U each n-jet cross section is proportional to aj, the cross sections fall roughly geometrically £|jtf->|' = ^c,^ + ^ + f + ^^([13l + [M])+i[12 ;34. Events with three jets at large transverse energy are described in QCD by amplitudes JV I 2uu' I l ru J L " N with two incoming partons and three outgoing partons. Very elegant results for the two- 2g*C f(S_ ,»){„•-tV-u<)\ f h to-three parton scattering processes have been given by Berends et al. [44]. For a complete F + 2CFm + [3i]) + l2;U JV2 \ itt'uu' j\ u " Nl description it is sufficient to consider the following four processes. (5.29) 189 190 191 192 To write the results for the remaining two processes we introduce a compact notation I ! I I i i : : ; i i i ; 111 700 for the dot product of two momenta, 600 — m Vi-Vi- (5.30) Using the momentum assignments of Eq.(5.24) the result for process (C) may be written 1000 500 M as [44], g\N*-l)(* {»»}{K}({«}»+ {«}') \ 400 — 4JV< I 2 <{al}{a2}{a3}{&l}{42}{63} I EM T > I % {al}{b2} + {a2}{bl} 300 — •1 T — {ab}+N'l{ab}- •? {12} 500 — 200 N* {Q3}{t3}({gl}{t2} + {a2}{il})\ — ] \ V- (5.31) "{ai} {23}{31} ? " The sums run over the three cyclic permutations P of the momentum labels of the final 100 state gluons. 11/11 11111111111 MM1 Using the momentum labels of Eq.(5.24) the result for process (D) is [44', .7 .8 .9 3 g°N {ii (5 32 EM = 240(iV2 - 1) JJny y£l2}{23}{34H45}{51} II > ' ' ) Figure 31: Distribution in the variable x3 and z< in a sample of three jet events, as The sums run over the 120 permutations of the momentum labels. measured by the CDF collaboration. The solid, dashed lines are the predictions from QCD, phase space respectively These matrix elements display the typical bremsstrahlung structure with the emission of soft and collinear gluons predominating. This is particularly clear from the form of the In Eq.(5.33) the variable ifi is the angle between the plane containing jet-2 and jet-3 and result given in Eqs.(5.25,5.29) where the dominant contributions come from the region in which the eikonal factors are large. From the tree graph results one can also show that the plane containing jet-1 and the axis defined by the incoming partons. the same effective structure function which is relevant for two-jet production is also to a There is again excellent agreement between the above theoretical predictions and the very good approximation valid for three-jet production [47]. experimental data. As an example, Fig.(31) shows the distribution in the variable x3 For three final-state (massless) partons the final-state parton configuration, at fixed measured by the CDF collaboration. The solid line is the prediction from QCD, based on centre of mass energy, is specified by five independent variables. Two variables are required the 2 —> 3 parton scattering amplitudes, and the dashed line is the prediction from phase to specify how the available energy is shared between the three final-state partons, and space alone. The data clearly favour the former. two variables serve to fix the orientation of the three-jet system with respect to the axis defined by the colliding partons. The last variable is an overall azimuthal angle. If z , 3 5.5 Direct photon production £«, and xs are the energies of the outgoing partons scaled such that i3 + xt -r xs = 2 and ordered such that x3 > x4 > x5 and 6{ is the angle between parton i and the beam High transverse momentum direct photon production and high transverse momentum direction, then the subprocess differential cross section can be written using the three jet production are two closely related phenomena. From an experimental point of view, particle massless phase space of Eq.(2.6): the study of direct photon production has several advantages with respect to the study of jets: the energy resolution of the electromagnetic calorimeter is generally better for d*â 1 2 (5.33) photons than it is for hadrons, and systematic uncertainties on the photon energy scale are dx dxtdcos ôidij; (1024TT4) 3 ^E^ smaller. Furthermore, since photons do not fragment, the direction and energy of photons is straightforwardly measured in the calorimeter without the need for a jet algorithm which is required to reconstruct a jet. Only the relatively low rate for the production of direct 2 10 photons and the non-negligible background from jet production processes have limited :''''' the usefulness of the direct photons for making quantitative QCD tests. - The leading order subprocesses are (a) the annihilation process qq —» ~fg and (b) the r Compton process qg —• 79 shown in Fig.(32) The invariant matrix elements squared are 10 given in Table 4. Depending on the nature of the colliding hadrons and on the values of ; X v/ï and J>T{= PT)I either of these two subprocesses can dominate. For example, in proton- proton or proton-nucleus collisions at medium pr the Compton process dominates while .a : XJ - a. in proton-antiproton collisions at high pr the annihilation process is more important. All direct photon data show good agreement with QCD over a large energy range. The most precise data is from the WA70 collaboration [48]. Fig.(33) shows WA70 data on pp —> -fX at y/s = 23 GeV. The curves are the fully-corrected QCD cross sections, X WA70 pp —1SX based on the next-to-leading order calculation of Aurenche et al. [49], using the latest 10 HMRS(E,B) parton distributions [36]. In fact the gluon distributions in these two sets HMRS(B) ] are chosen to fit the WA70 data. HMRS(E) - " JWW\/ 10 p (GeV/c) v> T UUUULSUL- ^ Figure 33: Direct photon pr distribution measured by the WA70 collaboration. The a) curves are next-to-leading order QCD calculations, as described in the text >/WW\/ 6. The Production of Vector Bosons in Hadronic Collisions b) In this lecture we review the physics of vector boson production in hadron-hadron Figure 32: Diagrams for direct photon or vector boson production at large pr- collisions. We begin by discussing the production of lepton pairs by quark-antiquark annihilation into a virtual photon - the Drell-Yan process. After a brief review of the standard electroweak model, we next discuss the phenomenology of W, Z production in Process swvw pp collisions, with special emphasis on perturbative QCD effects. (TV2 - 1) t2 + v? + 2s(s + t + u) 1 9 -» 7" g N2 tu 6.1 The Drell-Yan mechanism 1 s2 +u2 + 2t(s + t + u) 9 9 -• 7' 1 The cross section for quark-antiquark annihilation to a lepton pair via an intermediate N su massive photon is easily obtained from the e+e~ —> qq cross section presented in the second lecture: Table 4: Lowest order processes for virtual photon production. The colour and spin *(ç --e+e-) = ^X. (6.1) indices are averaged (summed) over initial (final) states. For a real photon (j + ( + ti) = 0. 5 3i N 193 194 195 196 Note that the time-reversed process, qq —> e+e~ is smaller by a colour factor of 1/jV2 = dXidX2dz6{xiX2Z T) because of the averaging over the colours of the initial quarks. The differential cross i^ ikl ~ section for the production of a lepton pair of mass M is therefore given by [£«(»(*i,*»)fc(*»,/0 + [1 - 2])] [5(1 - z-^ +' as{ll) /,« 2x + [y)QÏ(*(*i,*0(*(*»,/0 + &(*».!»)) + ll« 2])] [^-f. (6.8) The overall colour factor of 1/N is due to the fact that only when the colour of the quark matches with the colour of the antiquark can the annihilation into a colour singlet where the correction terms are [50,51] final state take place. In the centre-of-mass frame of the two hadrons the components of momenta of the incoming partons may be written as 4TT' •ln(l-z)> 6{1„z){l + ^y6^z+{^)+ + 2{1 + z,)(^)+ Pi = ^-(x„ O.O.n) (z> + (l-zy)ln(l-z)+3--Sz+9-z1 (6.9) P2 = ^(x ,0,0,-x ). (6.3) 2 3 and the plus distributions are denned as in Eq.(3.27). The square of the parton centre-of-mass energy i is related to the corresponding hadronic The size of the O(as) correction depends on the lepton pair mass and on the overall quantity by J = xix7s. Using Eq.(4.1), the parton model cross section for this process center-of-m&ss energy. At fixed-target energies and masses the correction is large and can be written as positive, of order 50% or more. In this regime of relatively large r, the (negative) con tribution from the quark-gluon scattering term is quite small. However at pp collider = dxidxié x s ^ L ^ ^ - ^{y.^li^i^^^,^ + {l «-2])]. (6.4) energies, where r is much smaller, the fg term is more important and the overall correc tion is smaller. For W and Z production the O(as) correction increases the lowest order Apart from the mild logarithmic behaviour in the distribution functions, the lepton pair cross section by about 25% - 30%. 2 cross section exhibits scaling in the variable r = M /s: Several important pieces of information can be obtained from Drell-Yan data. Low 3 mass lepton pair production in high energy hadron collisions is sensitive to the small x M dcr SWr /* r_ , IAF = ~lN~ } dx^dx^ix^ ~ r)[5] From Eq.(6.3) the rapidity of the produced lepton pair is found to be y = l/21n(xi/i2), tributions in the latest MRS fits [36]. Fig.(35) shows data from the E605 collaboration and hence [52], compared with the next-to-leading order QCD calculation using the HMRS(E,B) ij = ./re", x-i = y/re~v. (6.6) distributions. Equally important is the fact that the distributions of quarks in pions can be extracted from Drell-Yan data in irp and -KN collisions. A comprehensive review of The double differential cross section is therefore MHy = S [Ç QIM^M*»») +11 ~ 21)] • (6.7) J^X/v^v + + ^ + jr^ with x\ and x2 given by Eq.(6.6). By measuring the distribution in rapidity and mass of the produced lepton pair one can in principle measure the quark distribution functions of the incoming hadrons. In QCD there exists a systematic procedure for calculating the perturbative corrections -TnnH—*-— to all orders. The next-to-leading order corrections are obtained from the graphs shown in Fig.(34): Figure 34: The leading and next-to-leading order diagrams for the Drell-Yan process Drell-Yan phenomenology can be found in reference [53]. Fig.(36) shows the predictions 10 for lepton pair production at collider energies, including the effects of the Z pole. Fig.(36) also illustrates the influence of higher order corrections. 10' 6.2 W and Z production The discovery in 1983 of the W and Z weak bosons provided dramatic confirmation of the Glashow-Salam-Weinberg electroweak model. In the remainder of this lecture we 10 r discuss the physics of W and Z production in pp collisions, starting with an elementary introduction to the electroweak model. The Lagrangian for the Glashow-Weinberg-Salam model of the electroweak interac -a E605 pN — nVX 3 1 tions is HMRS(B) HMRSIE) \l cows = -^r y^t"(i - s)(T+w:+T~w;)^ -e $>*7"*4. 0-1 gw-j-yvn^-An^iZ., (6.10) 0-1 0-2 03 0-4 05 2 cos ew i^ + where 6W is the Weinberg angle and gw = e/ ûa8w. T and T are the isospin raising and lowering operators and the vector and axial couplings of the Z are given by Figure 35: Drell-Yan data from the E605 collaboration with next-to-leading order theo retical predictions 2 V, = t3L(i) - 2Ç, sin (6w), -4, = t3L{i), (6.11) where t3L(i) is the weak isospin of the fermion (+~ for «,- and i/<, -i for di and e;), and 1000 En i—i—i—j—i—i—i—i—I—i—i—i—i—i—i—i—i—r=3 Qi is the charge of the fermion in units of the positron charge. At tree-graph level the e e pairs from DY and Z Z VS = 1.8 TeV with 0(a ) _^ Fermi constant can be written in terms of the coupling gw- 100 4- s VS = 1.8 TeV without 0(as) | 9w _ GF VS = 0.63 TeV with 0(a ) ^ (612) F\ s 8MJ, vT 10 >OJ o ~ The electromagnetic and Fermi coupling constants are measured to high precision .a - U. 1 using the Josephson effect and the muon lifetime respectively: '—5 ' T3 a"1 = 137.03604(11) h -o .1 =- 5 2 GF = 1.16637(2) x 10" GeV" . (6.13) .01 — Using the value for the Weinberg angle derived from charged and neutral deep inelastic 2 neutrino-nucleon total cross sections, sin 8W = 0.23 [54], we obtain the leading order predictions for the masses: 50 100 150 200 250 300 M [GeV] = /|^U_i_*78GeV, Mw LGfv^JsinÔH, Figure 36: The predicted e+e pair production cross section in pp collisions Mz = J^~89GeV. (6.14) 197 198 199 200 The most recent measured values [55,56,57] for the masses are I I I I I I [ I I I I I I Mw = 79.91 rb 0.35(stat) ± 0.24(sys) ± 0.19(scale) : CDF(ei/) I UA1 (1989) I UA1 (1989) 2.5 -i UA2 (1990) .25 •t UA2 (1990) Mw = 79.90 ± 0.53(stat) ± 0.32(sys) ± 0.08(scale) : CDF(fiiy) Ï CDF (1990) I CDF (1990) 7/ Mw = 80.79 ± 0.31(stat) ± 0.21(sys) ± 0.81(scale) : UA2(ef) 2 2 + C •S Mz = 91.49 ± 0.35(stat) ± 0.12(sys) ± 0.92(scale) : VA2(e e~) M = 91.150 ±0.032 :LEP + SLC z v u 1.5 v .15 T î The differences between the predictions in Eq.(6.14) and the experimental measurements N are due to higher order electroweak perturbative corrections. When these are taken into account [58], the agreement between theory and experiment is excellent. In analogy with the Drell-Yan cross section in the previous section, the subprocess cross sections for W and Z production are readily calculated to be .05 2 ,*-* = ^V2GFMw\Vqq,\ S(s-Mw) ' 1 ' ' ' I ' ' ' 1 I ' ' 0 1 1 1 1 1 1 1 1 ' ' ' ' 1 ' ' z 500 1000 1500 2000 â"^ = lV2G MHv; + A\)6(à-Ml), (6.16) 500 1000 1500 2000 F Vs (GeV) Vs (GeV) where Vni is the appropriate Kobayashi-Maskawa matrix element. Figure 37: Comparison of W.and Z cross section measurements with theoretical predic The O(ccs) perturbative QCD correction to the W and Z cross sections is the same as tions the Drell-Yan correction (for a photon of the same mass) discussed in the previous section - the gluon is 'flavour blind' and couples in the same way to the annihilating quark and antiquark. Fig.(37) shows the theoretical predictions for the W and Z cross sections (times leptonic branching ratios - see next section) compared with measurements from 6.3 W and Z decay properties the -pp collider experiments [59,60,61]. Note that the systematic and statistical errors on the measurements have been combined in quadrature. The values of the masses of the At leading order in electroweak perturbation theory the partial widths of the W and Z vector bosons have been chosen to be Mw = 80 GeV, Mz = 91.16 GeV. The parton bosons are given (in the standard model) by distributions are the HMRS(B) set, with the scale choice ft = Mw,z- We have included a ±10% error band on the theoretical prediction to allow for the uncertainties due to T(W+ -> //') = N- 6V2* the parton distributions, to higher order electroweak corrections, and - most significantly G M . - to the only partially known 0(0:5) QCD corrections [62]. Evidently the agreement is T(Z° - //) N F Zn 2iV + A}), (6.17) very good. Note that this constitutes a non-trivial check on the evolution of the parton 6V2TT distributions, since in this calculation they are being evaluated at much higher /x values where TV is a normalisation factor which is 1 for leptons and 3 for quarks. For the latter, than the deep inelastic scattering data. the W+ decay rate refers to the sum of the decays to a given quark of charge § and all antiquarks of charge |, e.g. W+ -* ud + us + ul. For any individual mode there is an additional factor from the Kobayashi-Maskawa mixing matrix. Using these relations we can calculate the branching ratios for the observed decay modes. By counting decay modes we obtain for the W (if the top quark is heavy: m, > mw — mb), 1 BR(W+ ->e+û,n+i>,T+i>) 11.1% Because the total widths (and hence the branching ratios) of the W and Z depend 3 + 3 + 3 on m.( and (for the Z) on the number of light neutrino species, measurements of the pro BR(W+ -» ud + uâ + ub) 33.3% duction and leptonic decay rates can provide information on these quantities. Nowadays BR(W+ -.cd + cs + cb) 33.3%. (6.18) we know from precision measurements of the Z width at LEP that Nu = 3, and from direct searches by the CDF collaboration at the Tevatron pp collider that mt > 89 GeV For the Z we obtain [64]. It is important nevertheless to check that the collider W and Z measurements are consistent with these results. It is not impossible, for example, that a light top quark + e e~ id vtve tin with non-standard decays could evade direct discovery while still contributing to the total 2 2 (6.19) [l + (l-4sin 0^) J 2 [2] 2 2 [l + (l-|sin ««r) ] W decay width. [l + (l-fsm + + theoretical input [65]. The idea is to express the ratio R of the number of observed W BR{Z° -» e+e-,M ^-,r r-) 3.4% and Z decays in terms of the ratio of production cross sections and branching fractions: BR(Z° — £V-*) 20.4% Number of decays W —» ev BR(W BR(Z° -» uû.cc) 11.8% R = = Rv • -RBR Number of decays Z —» ee (TZ BR(Z — ee) BR(2° ->dd,s3,bb) 15.2%. (6.20) B(W -+ lu) T(W -* lu) T(Z -> all) RBR = (6.24) B(Z ->l+l~) T(W -> all) T(Z — 1+1-) Note the large branching fraction of the Z boson into neutrinos and bottom quarks. The ratio R& is calculable theoretically, with a certain error due to ignorance of the input Although the hadronic decay modes are enhanced relative to the leptonic modes, at parton distributions. In Fig.(38) theory is compared with experiment. The theoretical hadronic colliders there is a very serious background from normal QCD two-jet production. A statistically significant signal has been reported by the UA2 collaboration [63]. The 1 1 1 ' 1 ' i i 1 ' I ' I ' 1 1 W decay mode into tb is of great interest since it offers the possibility of observing the - VS=0.63 TeV VS=1.8 TeV top quark. Taking the mass of the top quark into account, (but setting the mass of the - N„=5 bottom quark equal to zero), the partial width of the W into top and bottom quarks is 12 1? — reduced from the expression given for g;g • above. The correct result is A=5 --- - // N„=4 - T[W+ -> tb) *W 31 V«, rw)(l (1 + nv)), (6.21) ///N„=4 T(W+ -> e+v.) 11 — 11 ' — IIII N„=3 -- where TW = m^/M^. Counting up all modes we see that the branching ratio into a given - /////N„=3 + leptonic channel, such as e i/e, is 1 1 10 — 10 l T{Z° R " 1 ,1,1,1 1 R " 1 ,1.1,1 , 2 2 ^- = Vl _ 4rjj fl + (1 - \ sin 6wf + 2rz((l - *- sin Bwf - 2)1 (6.23) 40 60 80 100 120 T{Z° 40 60 80 100 120 mt [GeV] mt [GeV] where TZ = m\lm z Figure 38: Thcoreticed values of the R ratio compared with data. 201 202 203 204 predictions are shown as functions of m,, for N„ = 3,4,5. The band on each prediction is indicative of the theoretical uncertainty from parton distributions [36]. The most recent experimental measurements for R are [60,66}.- R = 9.38+j[;£(stat) ± 0.25(sys) : UA2 R = 10.2 ± 0.8(stat) ± 0.4(sys) : CDF. (6.25) The results are evidently consistent with the Nv = 3, m( > 90 GeV hypothesis. S 40 6.4 Lepton angular distribution in W and Z decay Another important test of the theory concerns the V — A structure of the weak charged current, Eq.(6.10). For the process - d(Pd) + ù(pu) -+ e (p«) + v(pv), (6.26) where the momentum labels are shown in brackets, we obtain (using the couplings derived from the electroweak Lagrangian), -.8 -.4 0 Figure 39: Angular distribution of leptons from W-boson decay. Likewise, for the charge conjugate process, we have .12 "i—i—i—i—|—i—r i—i—i—i—r Acceptance corrected cos 6 distribution CDF Preliminary where now p„ is the momentum of the incoming u quark etc. If we define 6' to be the e+(e~) angle of emission in the W rest frame, measured with respect to the direction of the incident p(p), and if we assume that all incoming quarks (antiquarks) are constituents of the proton (antiproton), then for both of the above matrix elements we have (Pu-P=)2 ~(1 + cos 6'f. (6.29) Thus the cross section is maximal when the outgoing electron (positron) moves in the direction of the incoming proton (antiproton). There is a simple angular momentum argument for this. In the standard model, the W couples to negative helicity fermions and positive helicity antifermions. Angular momentum conservation therefore requires the outgoing fermion (electron) to follow the direction of the incoming fermion (quark), which is usually the direction of the incoming proton. I I l l I I I I The lepton asymmetry is clearly visible in the data. Fig.(39) shows 6' distributions 0 I ' ' L. -1 -.5 0 . from the UA1 and UA2 collaborations [59,67]. The data are consistent with the V - A cosé? hypothesis. Note, however, that since there are two VT-fermion-fermion vertices in the Figure 40: Angular distribution of leptons from Z-boson decay, from CDF scattering amplitude, the arguments are unchanged if the (l —p5) coupling is replaced by 1000 (1+7B). W + W production at large pT The situation is more complicated for Z decay. Because the coupling of the Z to VS = 1.8 TeV, CDF preliminary fermions is a combination of left- and right-handed pieces, the lepton angular distribution is an admixture of (1 ± cosfl*)3 terms, the relative amounts being determined by sintfw. 100 — Fig.(40) shows the angular distribution from the CDF collaboration [68]. The curve shows the standard model prediction with sin2 6 = 0.231. > W u \ XI £ 10 6.5 W and Z transverse momentum distributions a. \-a Most W and Z bosons are produced with relatively little transverse momentum. However, b •a part of the total cross section corresponds to the production of large transverse momentum bosons. The relevant mechanisms are the 2 —» 2 processes qq —• Vg and qg —» Vq. The diagrams are identical to those for large pT direct photon production, Fig.(32), and the annihilation and Compton matrix elements are (Table 4) J L J I L ^\ 1 w 9 tu 50 100 150 Pi [GeV] 2 + 2 + 2t V|M—f = ,asV-2GFM^ i * * ^, (6.30) z—'' i 3 —su Figure 41: W transverse momentum distribution from the CDF collaboration, with with similar results for the Z obtained by changing the overall couplings. The W trans next-to-leading order QCD predictions verse momentum distribution is then obtained by convoluting these matrix elements with parton distributions in the usual way. Data on the px distribution of the W from the CDF collaboration [70] are shown in where the A; are coefficients of O(l). The higher order terms are evidently important Fig-(41). The curve is a next-to-leading order QCD prediction from Arnold and Reno when [69], using HMRS(B) parton distributions. The agreement is very reasonable over the M2 a {p\) log2 -ï- ~ 1. (6.33) complete px range, although it is clearly not possible yet to use such data for a precision s VT ^MS measurement. The UA2 collaboration have, however, derived a value for 05 to leading order by comparing the relative rates of W + 1 jet and W + 0 jet events [71]: In practice, this corresponds to pr values less than about 10 — 15 GeV/c. Fortunately, the large logarithms in Eq.(6.32) can be resummed to all orders in perturbation theory. For as(MS,n = Mw) = 0.13 ± 0.03(stat)± 0.03 (exp.sys) ± 0.02 (th.sys). (6.31) more details see reference [38]. The result is a 'Sudakov' form factor which regulates the cross section at small pT. The small px QCD cross section is most naturally expressed as From Fig.(2) we see that the result is consistent with measurements from other processes. a Fourier transform. Introducing the two-dimensional 'impact parameter' vector b, which At small transverse momentum, the theoretical cross section in Fig.(41) diverges. This is the Fourier conjugate of pr, the cross section is given, schematically, by is a reflection of the infra-red singularity in the matrix element (i.e. the poles at t = 0 1 da- and u = 0 in the expressions given in Eq.(6.30)). As the transverse momentum becomes dbbJo(bpT) exp(-5(6,M)) a dp\ 0 f smaller, the emission of multiple soft gluons becomes important. The generic expression M2 x 1 for the cross section in this limit is: dx! dx2S(x1x2 ) q(xitb ) q(x2lb ). 6.34 / 2 1 <^ . as(pT) AP a|(yj.) 3 M , A lo A To the extent that the exponent S in Eq.(6.34) depends on as and hence on A^jj, the ~j-r - i—-2— g-r + ?—5—log — -1-..., (6.32 adpT p\ p\ fT fr small px distribution can in principle be used as a test of QCD. In practice, however, 205 206 207 208 As long as the jets are required to be well-separated from each other and from the beam, the cross sections can be calculated from the matrix elements for the tree-level W+W at Vs=l.B TeV parton processes: ij -• V + fc, .. .k„, where V = W,Z and i,j,K = q,9- Details of how CDF preliminary data the matrix element calculations are performed, together with references to earlier work, can be found in reference [43]. As an illustration, Fig.(43) shows the standard model Resummed predictions for the jet fractions /„ defined by > the 0,..., 4 jet fractions are well-parametrised by /„ = /0(0.19)". Given the complexity of the multijet calculations, it is surprising that the final predictions are related in this simple way. PT [GeV] Figure 42: W transverse momentum distribution at small pJT, from the CDF collabora tion, with resummed QCD predictions from ref. [72] there are some difficulties - for example, some non-perturbative cut-off or smearing must be included to make the b integral in Eq.(6.34) converge at large 6. This introduces some theoretical uncertainty. It is also difficult to make an accurate experimental measurement when the transverse momentum is of the same order as the missing transverse energy resolution. Fig.(42) shows an example of a comparison [72] of theory with data from the CDF collaboration [70]. The solid line is the resummed QCD prediction and the dashed line is the 0{a\) fixed order prediction. Note that the latter is singular at pr = 0, in accordance with Eq.(6.32). Experiment and theory evidently agree quite well. 6.6 Multijet production with W and Z One of the most important standard model processes in high energy hadron-hadron col lisions is the production of a W or Z with accompanying hadron (quark or gluon) jets. Essentially any new physics process (heavy quarks, SUSY,...) can be mimicked by the production of vector bosons in association with jets. It is therefore important to be able to estimate these backgrounds accurately. In addition quantitative tests of QCD are pos sible - we have already mentioned in an earlier section the measurement by the UA2 collaboration of the strong coupling as from the relative rate of W + 1 jet and W + 0 jet production. Figure 43: Predictions for the jet fractions in W production 7. The Production of Heavy Quarks i—i—i—]—I—i—I—!—|—i—i—i—i—i—i—r 10' =- Total top quark width vs. mass The production of heavy quarks is an important issue. One of the motivations for collider experiments is to discover new heavy objects, such as the top quark. It is there 10" fore important to test our understanding of such production processes by predicting the production rates for the known heavy objects, such as the bottom quark. Because the cross sections are large, hadronic interactions offer the potential to produce the large 10"' = number of bottom quarks necessary to study their decays in detail. For example, with 4) sufficient 6's it may be possible to observe CP violation in the 6 system. The disadvantage o 10 — of hadronically produced i's is that they have to be distinguished from a large background of other hadrons. It is therefore necessary to find an efficient way to isolate the bottom events from the background. This is done by using the special properties of their decay 10 products. 10 7.1 The decays of heavy quarks 10 The existence of hadrons containing heavy quarks is deduced by observation of their 100 150 200 250 m [GeV] decays. Therefore any experiment which measures the cross section for the production of t hadxons containing heavy quarks makes assumptions concerning the branching fraction to the observed mode. Figure 44: The total width of the top quark We shall start by considering the decays of a free top quark in the standard model. In both cases the top branching ratio to leptons is given in the simplest approximation We shall consider the case m > mw as well as the experimentally less favoured case t by counting modes for the W decay. Assuming the decay channel to ih is forbidden because mt < mw- Consider the decay of a very massive top quark which decays into an on-shell m, > mw — mi,, the branching ratio is given by counting the decay modes eP«, £iP„, TVT W boson and a b quark. This process has a semi-weak decay rate. In the limit in which and three colours of ud and cs. "it 3> mw the total t width is given by, + + 1 BR(W -* e i>) = SB 11% (7.3) 2 3 + 3 + 3 T(t - bW) = f=^ |Vt6| « 170 Mevf-^V. (7.1) 8irV2 I ran- / ' All direct searches for the top quark make assumptions about the branching ratio into When the top quark is so heavy that the width becomes bigger than a typical hadronic leptons. It is important to investigate unconventional decays of the top quark, especially if scale the top quark decays before it hadronises. Hadrons containing the top quark are they alter the branching ratio into the leptonic decay mode. As an example, we consider a never formed. simple extension of the standard model which involves the introduction of a second Higgs This should be compared with the top quark decay for m, < raw — m-b which is a doublet. Top quark decay in this model has been investigated in ref. [75]. In order to avoid scaled-up version of /i decay. In this case the partial width into tv is given by, strangeness changing neutral currents [76] one must couple all quarks of a given charge to only one Higgs doublet. After spontaneous symmetry breaking we are left with one 2 m charged physical Higgs ij and three neutral Higgs particles. If mt > m, + mk the dominant T(t - bei>) = ^n£ \Vtb\ « 2.3 kevf ' V. (7 2) decay mode of the top quark is not to a leptonic mode, but rather to the charged Higgs, Fig.(44) shows the width of the top quark for general values of the top mass. The dashed 2m,m)\{m2 ,ml,m2 (7.4) lines show the asymptotes derived from Eq.(7.1) and Eq.(7.2). 47rtim, i t T) 209 210 In this equation X(a,b,c) = ^((a2 - b2 - c2f - 46V) and v is the normal vacuum Process EW/s4 expectation value. As an extreme example, for m, = 30 GeV, m„ = 25 GeV and mb = 4.7 GeV it is found [75] that T > OAMeV. It is clear that for a large range of parameters, qg^QQ ,+ the semi-weak width Eq.(7.4) greatly exceeds the weak width as determined from Eq.(7.2). f(^ §) The decay modes of the 77+ determine the signature of the light top quark in this model. + The r; decays predominantly to cJ and f i/T. If the vacuum expectation values of the two 9 9->QQ Higgs fields are taken to be equal the branching fraction into cs is 64% and into fv is r {6riT2-i)(^^^-/Tj 31% for m„ = 25 GeV [75]. We may also treat the decays of the B-meson in analogy with the decay of a free muon. Table 5: Lowest order processes for heavy quark production. £]|M|2 is the invariant This is called the spectator model, since the quarks which accompany the 6 quark in the matrix element squared. The colour and spin indices are averaged (summed) over initial B-meson play no role in the decay. However in this case the finite masses of the u and c (final) states. quarks, to which the 4 decays, must be taken into account. In addition strong interaction corrections can be appreciable because a {m ) is large. For further details and references s b in a compact form, we have introduced the following notation for the ratios of scalar to the original papers see ref. [77]. products, 2 2pi.p3 2p22-P.p33 4m . j T z i = —:—1 P-- -T-, s = (pi +p2) . (7.6) 7.2 The theory of heavy quark production s In leading order the short distance cross section is obtained from the invariant matrix The leading order processes for the production of a heavy quark Q of mass m are, element in the normal fashion [1]: 1 4 +p (7.5) *« = û (2$k (2§k^ ^ > - * - *> £i"«i (7.7) The first factor is the flux factor for massless incoming particles. The other terms come where the four momenta of the partons are given in brackets. The Feynman diagrams from the phase space for two-to-two scattering. which contribute to the matrix elements squared in 0(g4) are shown in Fig. (45). The invariant matrix elements squared [78,79] which result from the diagrams in Fig. (45) are We shall now illustrate why it is plausible that heavy quark production is described given in Table 5. The matrix elements squared have been averaged (summed) over initial by perturbation theory [80]. Consider first the differential cross section. Let us denote the momenta of the incoming hadrons, which are moving in the z direction, by P^ and (final) colours and spins, (as indicated by £ ). In order to express the matrix elements 2 Pj and the square of the total centre of mass energy by a where 3 = (P1 + Pi) . The short distance cross section in Eq.(4.1) is to be evaluated for parton momenta px = xi-Pi, Pi = I-ÎPÎ and hence the square of the total parton centre of mass energy is i = Xix23, if we ignore the masses of the incoming particles. The rapidity variable for the two final state partons is defined in terms of their energies and longitudinal momenta as, a) y — — In (7.8) " 2 E-p. Using Eqs.(4.1) and (7.7) the result for the invariant cross section may be written as, da (7.9) 2 -j-TJ^Zl/t(zi,M)*2/i(:E2,/i)^|.Mi32 : i •~l~HJUUUU « * OUUULSJL/ dy3dytd pT 167r j b) The energy momentum delta function in Eq.(7.7) fixes the values of Xi and Xi if we know Figure 45: Lowest order Feynman diagrams for heavy quark production the value of the pr and rapidity of the outgoing heavy quarks. In the centre of mass system of the incoming hadrons we may write the components of the parton four momenta as When a light quark is produced by these diagrams the lower cut-off on the virtuality of the propagators is provided by the light quark mass, which is less than the QCD scale A. Since propagators with small virtualities give the dominant contribution, the production PI = G/2{x ,0,0,x ) y 1 1 of a light quark will not be calculable in perturbative QCD. In the production of a heavy p2 = v/I/2(i„0,0,-x2) quark, the lower cut-off is provided by the mass m. It is therefore plausible that heavy quark production is controlled by 0:5 evaluated at the heavy quark scale. p3 = (mTcoshy3,pr,0,mi- sinhy3) Note also that the contribution to the cross section from values of pr which are much p = (mrcoshvi,— pr,0,mrsinhy ). (7.10) 4 4 greater than the quark mass is also suppressed. The differential cross section falls like Applying energy and momentum conservation we obtain, 1/nij. and as raj increases, the parton flux decreases because of the increase of x1 and 12 according to Eq.(7.11). Since all dependence on the transverse momentum appears in the w , x-, = ^|(e +e»«), I, = I^(e-" +e-» )1 i = 2m'T(l + cosh Ay). (7.11) transverse mass combination, the dominant contribution to the cross section comes from transverse momenta of the order of the mass of the heavy quark. :! The transverse mass of the heavy quarks is denoted by mr = A/(m +pJ.) and Ay = y3 — yt Thus for a sufficiently heavy quark we expect the methods of perturbation theory to is the rapidity difference between the two heavy quarks. be applicable. It is the mass of the heavy quark which provides the large scale in heavy Using Eqs.(7.9) and (7.11), we may write the cross section for the production of two quark production. The heavy quarks have transverse momenta of the order of the heavy massive quarks calculated in lowest order perturbation theory as, quark mass and are produced close in rapidity. The production is predominantly central, because of the rapidly falling parton fluxes. Final state interactions which transform the dyidy'epr = 64^n4(l + co.h(Ay))' Ç "^•") "M*»*) Y)Mi^ " <7"12) heavy quarks into the observed hadrons will not change the size of the cross section. A possible mechanism which might spoil this simple picture would be the interaction Expressed in terms of m, mr and Ay the matrix elements for the two processes in Table 5 of the produced heavy quark with the debris of the incoming hadrons. However these are, interactions with spectator partons are suppressed by powers of the heavy quark mass [81]. For a sufficiently heavy quark they can be ignored. The theoretical arguments summarized above do not address the issue of whether the charmed quark is sufficiently heavy that the hadroproduction of charmed hadrons Note that, because of the specific form of the matrix elements squared, the cross section, in all regions of phase space is well described by only processes (a) and (b) and their Eq.(7.12), is strongly damped as the rapidity separation Ay between the two heavy quarks perturbative corrections. becomes large. It is therefore to be expected that the dominant contribution to the total cross section comes from the region Ay < 1. Heavy quarks produced by qq annihilation 7.3 Higher order corrections to heavy quark production are more closely correlated those produced by gluon-gluon fusion. We now consider the propagators in the diagrams shown in Fig.(45). In terms of the The lowest order terms presented above are the beginning of a systematic expansion in above variables they can be written as, the running coupling, ^,m')=^%W,,4) (7.16) 2 (P1+P2) = 2pi.p2 = 2m£(l + cosh Ay) Eq.(7.16) completely describes the short distance cross-section for the production of a (Pi-Pif-m2 = -2pj.p = -TT4(1 + e"Av) 3 heavy quark of mass m in terms of the functions Ti$, where the indices t and j specify 2 A the types of the annihilating partons. The dimensionless functions Tij have the following {Pi ~ ViY ~ m = -2p2.p3 = -774(l + e "). (7.15) perturbative expansion, Note that the denominators are all off-shell by a quantity of least of order m1. It is this fact which distinguishes the production of a light quark from the production of a heavy quark. ) +4lra 1) ) M"'â =^ (") ^)[4 (^) + ^ (/')ln(^r)] +0{g*) (7.17) 213 214 215 216 y2—a (y) = -6a|(l + b'ccs + . • j£ sJUJUL, » s 33 - 2n, 153 - 19n, (7.20) 6 = 12ir , o = 2ir(33 - 2n/) OLSJtS-W^_ and the Altarelli-Parisi equation, Real emission diagrams ''£«*.*> = ^Ç/T«'W4<* (7.21) This illustrates an important point which is a general feature of renormalisation group V-AJUUH—*i improved perturbation series in QCD. The coefficient of the perturbative correction de pends on the choice made for y, but the y dependence changes the result in such a way that the physical result is independent of the choice made for y. Thus the y dependence OULSJL OUUjU is formally small because it is of higher order in as. This does not assure us that the y dependence is actually numerically small for all series. A pronounced dependence on y is Virtual emission diagrams a signal of an untrustworthy perturbation series. We shall illustrate this point by showing the y dependence found in two cases of current Figure 46: Examples of higher order corrections to heavy quark production interest. First, in Fig.(47), we show the y dependence found for the hadroproduction of a 120 GeV top quark in leading and next-to-leading order. The inclusion of the higher order terms leads to a stabilisation of the theoretical prediction with respect to changes in where p is defined in Eq.(7.6). The functions ^/' are completely known [82,83]. Ex y. The situation for the bottom quark is quite different. In Fig.(48) the scale dependence amples of the types of diagrams which contribute to ptV are shown in Fig.(46). The 50 l—i—i—i—I—i—i—i—!—|—i—i—i—i—|—i—r full calculation involves both real and virtual corrections. For full details we refer the reader to ref. [82]. In order to calculate the Tij in perturbation theory we must perform Top cross—section vs. scale /u, both renormalisation and factorisation of mass singularities. The subtractions required VS =1800 GeV, mt = 120 GeV for renormalisation and factorisation are done at mass scale y. The dependence on y of DFLM, Ae= 0.170 GeV the non-leading order term is displayed explicitly in Eq.(7.17). L+NL 40 As discussed in previous lectures y is an unphysical parameter. The physical predic L tions should be invariant under changes of y at the appropriate order in perturbation J3 theory. If we have performed a calculation to 0(a|), variations of the scale y will lead to a. corrections of O(a^), =0(a45) (7.18) "V - 30 In this equation cr is the hadronic cross section as determined by Eq. (4.1). Using Eq.(7.18) we find that the term !p which controls the y dependence of the higher order pertur- bative contributions is fixed in terms of the lower order result ^°): (7.19) 20 j i I i u_i_ -k. JP Zi J? z ^W = i? 2 100 150 200 250 M [GeV] In obtaining this result we have used the renormalisation group equation for the running coupling, Eq.(1.22) Figure 47: Scale dependence of the top quark cross section in second and third order 80 1 1 1 : ; i : 1 i i i i j ! i i ; i , i i Bottom cross-section vs. scale /J., 70 VS =1800 GeV, mb = 5 GeV 60 — \ DFLM, As= 0.170 GeV — a) Example of flavour excitation graph 50 \. L+NL ^\^^ L 3, 40 — — b 30 - - 20 - b) Graphs containing spin-one exchange in the t-channel 10 Figure 49: Graphs relevant for discussion of flavour excitation 0 1 1 1 i 1 i i i i 1 i i i i 1 i i i i 1 i i i i 0 5 10 15 20 25 M [GeV] shown in Fig.(49b). Does the flavour excitation approximation accurately represent the Figure 48: Scale dependence of the bottom quark cross section in second and third order results of these diagrams? In particular is the 1/q* pole, which is the signature of the presence of the flavour excitation diagrams, present in these diagrams? We shall now indicate why the 1/g4 behaviour is not present in the sum of all three of the predicted bottom quark cross section is shown. The cross section is approximately diagrams displayed in Fig.(49b). Let us denote the 'plus' and 'minus' components of any doubled by the inclusion of the higher order corrections, which do nothing to improve vector q as follows: the stability of the prediction under changes of p.. It is apparent that the predictions for bottom production at collider energies are subject to considerable uncertainty. q+ = q° + q\ + mass shell. If we denote the momentum transfer between the two incoming partons as (p,-9)' = 0, ? = -4, (7.23) 2p q, the parton cross section will contain a factor 1/q* coming from the propagator of the 2 since in the centre of mass system pt ~ P2 ~ •/*• In the low ?2 region the 'minus' exchanged gluon. Therefore these graphs appear to be sensitive to momentum scales component of q is determined from the condition that production is close to threshold, all the way down to the hadronic size scale. This casts doubt on the applicability of perturbative QCD to these processes. (p1 + qy^4m\ q-^^r. (7.24) In the following we shall sketch an analysis 180] which leads to an important conclusion. Pi When considering the total cross section, flavour excitation contributions should not be q~ is therefore also small in the fragmentation region in which p\ ~ •/*• ^e therefore included. The net contribution of these diagrams is already included as a higher order find that in the fragmentation region of the upper incoming hadron, correction to the gluon-gluon fusion process. This analysis begins from the observation 2 + 7 that the flavour excitation graph is already present as a subgraph of the first two diagrams q = q q~ - qT • qT « -qT • qT ( -25) 217 218 219 220 The current J to which the exchanged gluon of momentum q couples is determined cross sections is therefore not possible. In preparing the curve for charm production We by the upper part of the three diagrams. In the fragmentation region only the 'plus' have taken the lower limit on /i variations to be 1 GeV. component is large. The dependence on the value chosen for the heavy quark mass is particularly acute + + + for the case of charm. In fact, variations due to plausible changes in the quark mass, q"Jli = q J- + q~J - qT • JT = 0, J s ÎL-ll. (7.26) q- Eq.(7.28), are bigger than the uncertainties due to variations in the other parameters. where the Ward identity is a property of the sum of all three diagrams. The explicit term We shall therefore take the aim of studies of the hadroproduction and photoproduction proportional to qr in the amplitude shows that one power of the l/q2 is cancelled in the of charm to be the search for an answer to the following question. Is there a reasonable amplitude squared. value for the charm quark mass which can accommodate the majority of the data on hadroproduction? In Fig.(50) we show the theoretical prediction for charm production. This cancellation only occurs when the soft approximation to J+ is valid. This requires Note the large spread in the prediction. Also shown plotted is a compilation of data taken the terms quadratic in q to be small compared to the terms linear in q in the denominators from ref. [85] which suggests that a value of m = 1.5 GeV gives a fair description of the in the upper parts of the diagrams in Fig.(49b). The momentum q~ must not be too small, c data on the hadroproduction of D's. After inclusion of the O(a^) corrections, the data q2 < 2p+q~ ^m2. (7.27) can be explained without recourse to very small values of the charmed quark mass [841. This conclusion is further reinforced by consideration of the data on photoproduction We therefore expect the soft approximation to be valid and some cancellation to occur 2 of charm. The higher order 0(aa s) corrections to photoproduction have been considered 2 2 when q < m . For further details we refer the reader to ref. [80]. The calculation of in ref. [86]. After inclusion of these higher order terms we obtain predictions for the ref. [82] provides an explicit verification of this cancellation in the total cross section. total cross section as a function of the energy of the tagged photon beam. The principal uncertainty derives from the value of the heavy quark mass, so we have plotted the 7.4 Results on the production of charm and bottom quarks minimum cross section which is obtained by varying A and the scale fi within the range The value of the heavy quark mass is the principal parameter controlling the size of the 1000 cross section. This dependence is much more marked than the 1/m2 dependence in the short distance cross section expected from Eq.(7.16). As the mass decreases, the value of x at which the parton distributions are evaluated becomes smaller (cf. Eq.(7.11)) and the 300 cross section rises because of the growth of the parton flux. The approach which we shall take to the estimate of theoretical errors in heavy quark 100 cross sections is as follows [84]. We shall take A to run in the range given by Eq.(1.46) with corresponding variations of the gluon distribution function. We shall arbitrarily choose Xi to vary the parameter /t in the range m/2 < fi < 2m to test the sensitivity to p. Lastly, Ji 30 we shall consider quark masses in the ranges, b 10 1.2 < mc < 1.8 GeV 4.5 < mi, < 5.0 GeV. (7.28) 3 We shall consider the extremum of all these variations to give an estimate of the theoretical error. 1 We immediately encounter a difficulty with this procedure in the case of charm. Vari 0 10 20 30 40 50 60 70 VS [GeV] ations of fi down to m/2 will carry us into the region y. < 1 GeV in which we certainly do not trust perturbation theory. A estimate of the theoretical error on charm production Figure 50: Data on hadroproduction of D/D compared with theory 1 GeV < fi < 2m for three values of the charm quark mass. The comparison with the they yield a constant cross section, independent of energy [82]. Naturally these high î data on the photoproduction of charm [87,88], shown in Fig.(51), indicates that charm contributions are damped by the small number of energetic gluons in the parton flux, but quark masses smaller than 1.5 GeV do not give an acceptable explanation of the data. at collider energies the region %/! 3> m makes a sizeable contribution to the bottom cross section. The fact that this constant behaviour is present in both JF*1' and F indicates In conclusion, within the large uncertainties present in the theoretical estimates, the the sensitivity of the size of this term to the value chosen for fi. There is therefore an DjD production data presented here can be explained with a charm quark mass of the interplay between the size of this term and the small x behaviour of the gluon distribution order of 1.5 GeV. This is not true of all data on the hadroproduction of charm, especially function. the older experiments. For a review of the experimental situation we refer the reader to ref. [89]. At fixed target energies the cross section for the production of bottom quarks is theo retically more reliable. The fi dependence plot has a characteristic form similar to Fig.(47) As emphasised above, the theoretical prediction for bottom quark production at col and it is possible to make estimates of the theoretical errors. A compilation of theoretical lider energies is very uncertain. The cause of this large uncertainty is principally the very results [90] and estimates of the associated theoretical error is shown in Table 6. The small value of x at which the parton distributions are probed. In fact, at present collider experimental study of the production of bottom quarks in hadronic reactions is still in energies the bottom cross section is sensitive to the gluon distribution function at values its infancy, but Table 6 also includes the limited number of experimental results on total of x < 10~J. Needless to say the gluon distribution function has not been measured at bottom production cross sections. such small values of x. An associated problem is the form of the short distance cross section in the large i region. The lowest order short distance cross sections, J^-°\ tend The calculations of ref. [82] also allow us to examine the pr and rapidity distributions to zero in the large i region [82]. This is a consequence of the fact that they involve at of the one heavy quark inclusive cross sections. Although the prediction of the total most spin | exchange in the ^-channel as shown in Fig.(45). The higher order corrections bottom cross section at collider energy is uncertain, it is plausible that the shape of the to gg and gq processes have a different behaviour because they involve spin 1 exchange transverse momentum and rapidity distributions is well described by the form found in in the t-channel. The relevant diagrams are shown in Fig.(49b). In the high energy limit. lowest order pertubation theory. The supporting evidence [95] for this conjecture is shown in Fig.(52), which demonstrates that the inclusion of the first non-leading correction does not significantly modify the shape of the transverse momentum and rapidity distributions. T I I I I l I I l I I i I I ] I I I I I i i—i—r At a fixed value of ft, the two curves lie on top of one another if the lowest order is Photoproduction of charm multiplied by a constant factor. Similar results hold also for the shape of the top quark distribution [95]. The UA1 collaboration have investigated the transverse distribution of m6 [GeV] tr (theory) Theoretical error Experimental data v/i = 41 GeV, pp 4.5 23 nb +21 -15 5.0 9 nb +8.4 -5.9 ^s = 62 GeV, pp 4.5 142 nb +98 -80 BCF[91j, 150 < cr < 500 nb 5.0 66 nb +47 -38 yf» = 630 GeV, pp 4.5 19 fib +10 -8 UA1[92], 10.2±3.3 fib 5.0 12 fih +7-4 v» = 24.5 GeV, -KN 4.5 7.6 nb +4.7 -3.8 WA78[93], y/s= 24.5 GeV, 4.8 ±0.6±1.5 nb E, (GeV) 5.0 3.1 nb + 1.5 -1.5 NA10(94], V«= 23 GeV, 14+7-6 nb Figure 51: Data on photoproduction of charm compared with theoretical lower limits Table 6: Cross section for bottom production at various energies. 221 222 224 ~i—i—i—1—i—i—r pp, VS = 1.8 TeV m„ = 5 GeV y = 0, 3, 4 LO + NLO u LO times 2.5 \ .a ©I t- •o •o \ .001 -bo .0001 0001 ' ' I ' I I I I I ' 1 ' 1 ' 1 "'I'- 10 15 0 10 20 3p 40 50 60 70 80 kT [GeV] pT (min) [GeV] Figure 52: The shape of the cross-section for bottom quark production Figure 54: The cross-section for bottom quark production at FNAL energy the produced bottom quarks. In Fig.(53) comparison of the full a3 prediction with UA1 t I ' 1 ' I ' 1 ' 1 ' 3 s T_ data is made. The data are plotted as a function of the lower cutoff pj-(min) on the pp collisions, VS = .63 TeV, |y|<1.5, PT>PT (min) 10' l/l l 1 transverse momentum of the b quark. The agreement is satisfactory. \^r rnB=4.75 GeV, A*=260 MeV, E 2 2 The corresponding prediction for the shape of the bottom production cross section at 10" -_ |>N DFLM. n0 = V(mB +Pi ) _ the Tevatron is shown in Fig.(54). : %, 4.5 : evidence. The theoretical motivation is that complete families are required for the can lO. - UA1 data ^^<> cellation of anomalies in the currents which couple to gauge fields. Hence the partner of 10 — * J/Y -» /J. [i~ sample ^^I>- ~= the 6, T and i/T must exist to complete the third family. I OHigh mass dimuons ^^r-<.-- - lin n An anomaly occurs in a theory because symmetries present at the classical level are 10 - • Low mass dimuons ~~-^T^-H^| destroyed by quantum effects. They typically involve contributions to the divergence of a • Muon-jet samples (1990) T ~" current which is conserved at the classical level. If the gauge currents are anomalous, the 10"° I- t 1,1,1,1,1,= Ward identities, which are vital for the proof that the gauge theory is renormalisable, are 10 20 30 40 50 60 destroyed. Pr (min) [GeV] Anomalies occur in the simple triangle diagram with two vector currents and one axial Figure S3: The cross-section for bottom quark production at CERN energy vector current. Elimination of the anomalies for a particular current in the lowest order triangle diagram is sufficient to ensure that the current remains anomaly free, even after the inclusion of more complicated diagrams. If the currents which interact at the three corners of the triangle couple to the matrices L°,Lb and Lc for the left-handed fields, and to the matrices Ra,Rb and Rc for the right-handed fields, the vector-vector-axial vector triangle anomaly is proportional to, A = Tr [R'iR'',^}] - Tr [£•{£», Z«}]. (7.29) For the specific case of the SU(2)L x U(l) theory of Glashow, Weinberg and Salam (GSW) we have the following weak isospin and hypercharge assignments for the third family (Ç = T3 + Y), tL,T3 = \,YL = \, tR, T3 = 0, YR = \, >>L,T3 = -\,YL = \, bR, T3 = O, r* = -i, »L, T3=l,YL = -\ TL, T = -i,y = -|, T ,T = 0,Y = -1. (7.30) 3 L R 3 R 100 150 mt<>p [GeV] Substituting these couplings into Eq.(7.29), with all combinations of the SU(2) matrices T° or the U(l) matrices Y we obtain the form of the anomaly for the gauge currents of Figure 55: The cross section for top quark production at CERN and FNAL the GSW theory. Two of the resulting traces of the couplings vanish for each fermion separately, b c The simplest hypothesis is that the bottom quark is in an SU(2) doublet with the top TiT°{T ,T } = 0, TTT'{YL,YL} = 0. (7.31) quark, although more complicated schemes are certainly possible. The other two traces vanish only for a complete family [96] Thus assured that the top quark exists, it only remains to find it. The expected cross 3 i> Tr(^-Fi ) = 0, TrrL{T°,T } = 0. (7.32) section for the process p + p^t + t + X (7.34) It should be noted that there are still anomalies in global (non-gauged) currents in the GSW model. For example the normal isospin current corresponding to a global symmetry is shown in Fig.(55). The cross section is calculated using the full O(o|) calculation (in the absence of quark masses) is anomalous. It is this anomaly which is responsible for of [82] and the method of theoretical error estimate described in the previous sections, 7rQ decay. (cf. [84]). In addition, production of top quarks through the decay chain W —» tb is also shown. Note the differing proportions of the two modes at CERN and FNAL energies. At The experimental reason to believe in the existence of the top quark is the measurement ^/s = 1.8(0.63) TeV the tt production is due predominantly to gluon-gluon annihilation of the weak isospin of the bottom quark. The forward backward asymmetry of 6-jets in for Tji( < 100(40) GeV. On the other hand the W production comes mainly from qq e+e~ annihilation [97] is controlled by A„A , the product of the axial vector coupling to b annihilation at both energies. This explains the more rapid growth with energy of the tt the electron and the b quark. The produced b and 6 quarks are identified by the sign of production shown in Fig.(55). the observed muons to which they decay. The measurement is therefore subject to a small correction due to B° - B° mixing. Assuming that the axial coupling to the electron has From Fig.(55) the range of top quark masses which can be investigated in current its standard value the measured weak isospin of the left-handed b quark is [97], experiments can be derived. In a sample of 5 inverse picobarns about 2500 tt pairs will be produced if the top quark has a mass of 70 GeV. One can observe the decays of the top T3 = -0.5 ± 0.1. (7.33) quark to the efi channel or to the e+ jets channel. With a perfect detector the numbers of events expected is, 225 226 227 228 Number of e^ events = 2 x .11 x .11 x 2500 s= 60 7.6 Heavy quark in jets Number of e + jet events = 2 x .11 x .66 x 2500 « 360. (7.35) Another question of experimental interest is the frequency with which heavy quarks are found amongst the decay products of a light quark or gluon jet. Since hadtons containing The e plus jets channel gives a more copious signal and does not require muon detection, heavy quarks have appreciable semi-leptonic branching ratios such events will often lead but the background is larger due to the process pp —> W + jets. This background may to final states with leptons in jets. If we wish to use lepton plus jet events as a signature become less severe with increasing top mass as the jets present in top decay become more for new physics we must understand the background due to heavy quark production and energetic. decay. The current lower limit on the mass of the top quark is 89 GeV [64]. If the efficiency This issue is logically unrelated to the total heavy quark cross section. As discussed of extracting the signal from the data does not change with the mass of the top quark, above, the total cross section is dominated by events with a small transverse energy of the we can expect to improve the limit by an additional 40 GeV above the present limit, by order of the quark mass. Jet events inhabit a different region of phase space since they increasing the luminosity accumulated at the Tevatron by a factor of 10. Note however contain a cluster of transverse energy ET 3> mc,mi,. This latter kinematic region gives a that the efficiency of the e+ multi-jets channels will increase for a heavier top quark. As small contribution to the total heavy quark cross section. A gluon decaying into a heavy the mass of the top quark increases the 6 quark jets occurring in its decay will be recognised quark pair must have a virtuality k2 > 4m2 so perturbative methods should be applicable in the detector as fully-fledged jets. This occurs with no extra price in coupling constants. for a sufficiently heavy quark. The number of QQ pairs per gluon jet is calculable [99] The background due to normal W+jets production, discussed in the previous lecture, is using diagrams such as the one shown in Fig.(57). The calculation has two parts. First suppressed by a power of as for each extra jet. It will become less important in the 2 2 2 one has to calculate 7ia(j£ , fc ), the number of gluons of off-shellness k inside the original channel with an electron and/or three and four jets. The results of a detailed study of the gluon with off-shellness E2. Secondly, one needs the transition probability of a gluon with prospects for top quark discovery are shown in Fig.(56), taken from ref. [98]. The limits off-shellness A;2 to decay to a pair of heavy quarks. are based on the expected performance of the DO and upgraded CDF detectors. The number of gluons of mass squared A:2 inside a jet of virtuality E2 is given by [99], 2 2 2 \n(E /A 'exPv/[(2AV7ri)ln(.EVA )] n,(E',k2) = (7.36) ln(fc2/A2 exvy/{(2N/*b)ln(k*/\2)]' where 2 1 (N - 1) (7.37) — i+ —H 2iV2 41 3TT6V and 6 is the first order coefficient in the expansion of the j3 function, Eq.(7.20). The correct calculation of the growth of the gluon multiplicity Eq.(7.36) requires the imposition of jL I L_J i I . I i l i I i I i I 80 100 120 140 160 180 200 220 240 mt [GeV] Figure 56: Required luminosity to discover top at 1.8 TeV in various decay modes Figure 57: Heavy quark production in jets the angular ordering constraint which takes into account the coherence of the emitted soft for the number of bottom quarks per jet with A(4) = 260 MeV. The data point shows gluons [100] as discussed in the second lecture. the number of D' per jet as measured by the UA1 collaboration [101] and by the CDF collaboration [102]. Note that these results depend on the values used for the branching Define RQQ to be the number of QQ pairs per gluon jet. Ignoring for the moment ratios (V —> Dtr) and {D —• Kir). CDF uses the values of the Mark III collaboration gluon branching calculated above, we obtain [103] whereas UA1 uses the values quoted by the Particle Data Group. In order compare 3 2 2 2m these numbers with the cc pair rates, a model of the relative rates of D and D' production dz z + (l-z) + (7.38) <--£££•"•>£ is also needed. For example, if all spin states are produced equally one would expect the charged D' rate to be 75% of the total D production rate. The points in Fig. 58 need to where the integration limits are given by z = (1 ± /3)/2 with /3 = ^/(l - Am'/k2). The ± be corrected for unobserved modes before they can be compared with the curves for the term (z -f (1 — z)3)/2 is recognisable as P° , the Altarelli-Parisi branching probability for total cc pair rate. massless quarks. 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The road to D = 4 low-energy physics is paved with an effective no-scale supergravity theory, and I explain a potentially interesting way to determine This is a very brief set of notes to facilitate the reading of my transparencies. dynamically all coupling constants and masses in such theories. Finally, I discuss the output of a very interesting string model: flipped SU(5) X U(l). Hierarchical fermion masses, small mixing angles, m, = 100 GeV, nib " 3m , (quasi-)stable protons, supersymmetric masses = (100-300 GeV) Lecture 1 T are some of the interesting phenomenological consequences of such a model. At the end, an A general introduction to the idea of Unification is given. A panoramic view of all attempts at overview of the whole picture, from the string down to the Standard Model, is provided. unification, from electroweak to superstrings, is shown. Then the essentials of the Standard Model are given and experimental data supporting it are presented. In the next stage, the basic problems of the Standard Model are discussed, focusing in particular on the gauge hierarchy problem; this leads unavoidably to a new structure at the TeV range—compositeness or supersymmetry—and to explaining the reasons why I prefer supersymmetry. Then an elementary introduction to Grand Unified Theories (GUTs) is presented, which continues in the next lecture. SHORT BIBLIOGRAPHY Lecture 2 C. Kounnas, A. Masiero, D.V. Nanopoulos and K.A. Olive, Grand Unification with and without Here a more substantial discussion of GUTs is given, and certain predictions concerning sin2 8», supersymmetry, and cosmological implications (World Scientific, Singapore, 1984). nib/mr, and the proton lifetime, are analysed in detail. Certain GUT models, including SU(5), seem A.B. Lahanas and D.V. Nanopoulos, The road to no-scale supergravity, Phys. Rep. 145 (1987) 1. 2 to be already excluded by the data. In particular, the value of sin 0w indicates more structure than the simplest possible one, limiting naturally to supersymmetry, which is discussed in the following M.B. Green, I.H. Schwartz and E. Witten, Superstring theory (The University Press, Cambridge, lecture. 1987). Lecture 3 Supersymmetry is analysed in detail. The quantum mechanical origin of two kinds of particles, fermions and bosons, and their natural incorporation into supermultiplets, is discussed. The supersymmetry algebra is given, and the simplest example of a supersymmetric field theory is described. Then, the physics applications of supersymmetry are given. The low-energy world should be described by N = l supersymmetry and the particles should be doubled: photinos, gluinos, Winos, Zinos, squarks, sleptons, and Higgsinos should exist with masses smaller, in principle, than 0(1 TeV). We already have stringent lower bounds on these masses from 50-100 GeV, and the origins of these limits are discussed in detail. Lecture 4 Here I concentrate mainly on supergravity: the local version of supersymmetry, which automatically and naturally includes Einstein's gravity. First the reasons for moving from global to local supersymmetry (supergravity) are given; then some elementary discussion of supergravity is undertaken. A specific form of supergravity, no-scale supergravity, is discussed in detail, since we may determine dynamically, by quantum effects, all mass scales, including the electroweak- breaking scale and the supersymmetry-breaking one. Furthermore, no-scale supergravity is the low-energy limit of superstrings. Lecture 5 Despite all efforts, point-like field theories — supersymmetric or not — cannot provide a consistent, finite quantum gravity theory. We have to move to extra-dimensions and/or extended objects. Again I explain that point-like field theories are not enough, and thus our last resort is superstrings: one-dimensional extended objects, which is the focus of my last lecture. 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ÏRMW J E-H«-.«(?i( O^ecfr t -x> ( T->*>) r~ -- -v N-.i. ^tacMnry J j^io ; ^Ç'-ti^joaV + ex(ta kv,, u/L i~£ï!L I aM.-.,fer 7 ù.^f*afr <° <*'-*; £Vto«£ j [<*>?, & FLAUooft \ î^f|ie(Ç<»L APPROACH ?e Sn*mt> si-bxsttytXoaj M„ft COLLIDER EXPERIMENTS collector ring (AC) to the existing antiproton accumulator (AA), and the "six-bunch" operation of the SPS in pp storage mode. The first Collider run using AC took place as scheduled in P. Jenni November and December 1987. The peak luminosity is expected to increase by a factor 10, or CERN, Geneva, Switzerland even more, over the past performance, reaching about 5 x 1030 cnr2s '. The experiments expect data samples corresponding to L > 10 pb_1 from the future Collider runs at Vs = 630 Prologue GeV during the next few years. The performance reached during the first run at the end of 1987 The objective of these lectures was to discuss some aspects of experimental physics at (peak luminosity ~ 3 x 1029 cnrV and L ~ 50 nb"1 only) was however far below design existing and future colliders. Particular emphasis was given to the physics results already values due to various technical problems both at the antiproton collector and accumulator obtained and the prospects of the CERN and FNAL pp colliders. Only a few selected topics of complex (AAC) and the SPS itself. These difficulties have been overcome in a very convincing the LEP physics were touched, restricting the discussion essentially to e+e" physics at the way for the most recent running period which started in Autumn 1988. A peak luminosity of > highest LEP energies. Some highlights of the HERA ep physics were mentioned. Finally, the 2 x 1030 cnrV1 has been obtained and a data sample of about 2.7 pb1 was accumulated by the challenge of collider experiments to reach the TeV mass range was briefly described. Obviously upgraded UA2 experiment by the end of the year. such an extensive programme could only be covered in a very superficial way, and this is even The highest-energy hadron collider for the coming years is the Fermilab Tevatron pp more true for this written version of the lectures which is confined to the pp part only. Collider which produced its first collisions in 1985 at Vs = 1.6 TeV. The first physics run at Fortunately many excellent detailed summary articles, lecture notes and workshop proceedings this machine took place in 1987 with an integrated luminosity of about 30 nb1 for the CDF exist for all these topics. Reference to such articles will be made heavily in the following instead experiment at Vs = 1.8 TeV. A very successful data taking period has started at the FNAL pp of trying to write yet another account of the same physics. Collider in Summer 1988. The machine is regularly reaching peak luminosities in excess of 1030 cmV1 and the CDF experiment has accumulated data for an integrated luminosity of more 1. Experiments and Physics at the Present pp Colliders. than one pb ' at the end of the year. It is expected that the machine, with improved cooling, will The CERN pp Collider has played a very important role in particle physics during the reach its goal of Vs = 2 TeV with peak luminosities of > 5 x 1030 cirrV, making it possible last years, in particular with regard to the discovery of the Intermediate Vector Bosons (IVBs) for the experiments to accumulate data samples of L = 10 pb"1 in about 1990. Besides and the study of hard parton scattering in jet phenomena. After the excellent and so far producing collisions for a few specialized experiments, the Tevatron Collider hosts two large unmatched high-luminosity performance of the CERN Intersecting Storage Rings (ISR), which experiments: the Collider Detector at Fermilab (CDF), and the DO. pioneered pp collisions at Vs up to 63 GeV, it is really the CERN pp Collider at Vs = 546 GeV and 630 GeV which has proved the ability and high potential of hadron colliders to probe 1.1 The pp Collider Experiments physics at the highest mass and momentum transfer scales. This will be illustrated by the The CERN pp Collider operated in its first phase (1981-1985) for the two large general- physics results mentioned in these lectures. The CERN pp Collider operated during the years purpose detectors UA1 [1] and UA2 [2], as well as for the UA4 experiment [3] specifically 1981 to 1983 at Vs = 546 GeV, during which time the UA1 and UA2 experiments collected data designed to measure the total cross-section and the elastic scattering, and for the streamer for an integrated luminosity L of about 150 nb-1. The collision energy was increased to Vs = chamber experiment UA5 [4]. 630 GeV for the more recent running periods. The excellent performance of the CERN pp The improved CERN pp Collider is being exploited by four experiments. One of them, Collider then allowed the experiments to accumulate L - 750 nb1 during the years 1984 and UA6 [5], is strictly speaking not a collider experiment as it uses either the circulating proton or 1985. For completeness it is worth noting that in 1985 the machine was operated in a so-called antiproton beams with an internal hydrogen gas jet target to study inclusive electromagnetic ramping mode for exploratory experimentation, reaching Vs up to 900 GeV but yielding only L final states at large transverse momentum and A production in pp and pp interactions at Vs = ~7nb-'. 24.3 GeV. Another experiment, UA8 [6], is studying jet structure in high mass diffractive Since then the machine has been improved considerably. The major items of the processes with sets of mini-drift wire chambers installed in so-called Roman-pots on both sides improvement of the CERN pp Collider complex are the addition of a new separate antiproton of the interaction region housing UA2. These forward detectors measure the recoil protons 297 298 299 300 (antiprotons) while the central energy flow is examined using the information of the UA2 1.1.2 The Upgraded UA2 Experiment detector. In the following, we concentrate on the upgrade programmes of the two large-solid- The upgrading of the UA2 experiment, described in detail elsewhere [9], aimed at angle general-purpose detectors UA1 and UA2. mainly two aspects: i) full calorimeter coverage, and ii) better electron identification at low transverse momenta (pj). Full e.m. and hadron calorimeter coverage is achieved with the 1.1.1 The Upgraded UA1 Experiment addition of new end-cap modules covering the angular regions 6° < 9 < 40° with respect to the The main emphasis of the improvement programme for the UA1 experiment [7] is on the beam directions. These end-cap calorimeters consist of lead/scintillator samplings for the e.m. replacement of the old central and forward electromagnetic (e.m.) calorimeters (gondolas and part and iron/scintillator samplings for the hadronic part, both read out by the wavelength shifter bouchons) by a uranium/tetramethyl/pentane (TMP) calorimeter (Fig. 1). The use of uranium (WLS) technique. They are assembled in wedge-shaped modules subtending 30° in azimuth allows about 2.6 interaction lengths (K) to be accommodated in the space occupied by the old (Fig. 2a). The construction, performance, and granularity (A<|> x Arj = 15° x 0.2; r| is the e.m. calorimeter alone (= IX). The U/TMP calorimeter will therefore not only act as the e.m. pseudorapidity) are chosen to match the old central calorimeter [10]. This matching is calorimeter but, on the average, also absorb a large fraction of the hadronic energy. The particularly important for the new multilevel triggering scheme, which provides selective calorimeter consists of 4250 cells, each subtending an angle A6 x A muon detection is then provided in the angular range from 15° to 165° with respect are complemented in the forward regions by proportional tube chambers, including a 2 X0 lead to the beam axis. radiator for preshower detection, mounted on the calorimeter end-caps. All these detector b. A new data acquisition system with a multilevel trigger structure using information components have given very satisfactory test-beam results. A background rejection of at least a both from the calorimetry and the muon detection system allows UA1 to cope with factor of 20 is expected for the inclusive electron measurements as compared with the previous the increased collision rate expected for the future runs. UA2 performance. This will be very important at low p-j- (typically 10-15 GeV), where most of the signal electrons from a possible t-quark coming from W -> t B decays are expected. Finally, The upgraded UA1 detector is expected to be fully operational in 1990, whereas the time-of-flight counter arrays located on both sides of the interaction region provide a fast vertex present UA1 pp running is already undertaken with the improvements (a) and (b) above. localization and discriminate against beam-gas backgrounds. The upgraded UA2 detector has been assembled in time to meet the initial running period 1.2 Large Cross-section Physics of the improved CERN pp Collider complex at the end of 1987. All the detector components as The physics of soft processes at pp Collider energies has been reviewed extensively by well as the new data acquisition and triggering system have been brought into operation Refs. [ 14,15]. The first key feature is the rise of the total cross-section as shown in Fig. 5. The successfully, demonstrating the readiness of the experiment for high luminosity running. most precise contribution to these measurements comes from the UA4 experiment [3,16] which measured at ^s = 546 GeV simultaneously the forward elastic scattering at low four-momentum 1.1.3 The FNAL pp Collider Experiments transfers t and the total inelastic rate. By using the optical theorem one can write the differential The two major detectors at the FNAL pp Collider are the CDF (Collider Detector at elastic rate at low t as Fermilab) and the DO experiments. 2 2 The CDF experiment [12] has been taking data since the start-up of the collider. It is a dNei/dt = L[a tot(l + p )/16jc] exp (bt), large multi-purpose detector (Fig. 3). This very complete and powerful instrument has been designed to study the general features of pp collisions at Vs = 2 TeV, and it is well suited for where p is the ratio of the real to the imaginary part of the forward elastic scattering amplitude, continuing the study of all the physics processes investigated so far at the CERN pp Collider. b is the slope, and L is the machine luminosity. Moreover, In the central region the CDF consists of vertex time projection chambers surrounded by a large-volume drift chamber inside a superconducting solenoid, followed by lead/scintillator and Nel + Nin = L °tot. iron/scintillator sampling calorimeters and muon chambers. Particular care has been taken with the instrumentation of the forward regions. This includes end-plug calorimetry (10° < 0 < 30°) where Ne) and N;n are the elastic and inelastic rates, respectively. Combining the two previous with Pb and Fe samples, read out by proportional-pad chambers. Separate but similar forward expressions give calorimeters (2° < 0 < 10°) are followed by a forward muon detection system covering 2° < 0 < 17° with magnetized iron toroids. The CDF Collaboration already has a significant upgrade 2 otot = [1671/(1 + p )] x (Nel + Nin)"i [dNei/dt]t = o. programme for the next few years. They will complete the facility by the addition of a silicon microvertex detector and by an extension of the muon detector coverage. The total cross-section can thus be obtained without the need of an independent determination of For the design of the DO experiment [13], experience gained with the CERN pp the machine luminosity. The ratio p = 0.24 ± 0.04 has been measured [17] by the same Collider has been assimilated into the final detector layout (Fig. 4). Major emphasis has been experiment extending the elastic scattering detection to very small angles, called the Coulomb placed on good hermetic calorimetry and lepton identification. For the calorimeters (which interference region, where the hadronic amplitude interferes with the Coulomb scattering. consist of a central part, end-caps, and end-plug modules), the uranium/liquid-argon technique The total pp cross-section at Vs = 546 GeV measured by UA4 is atot = 60 ± 2 nb [17]. has been chosen, resulting in a compact design with excellent transverse and longitudinal The UA5 Collaboration has determined 0tot at Vs = 900 GeV from the total inelastic cross- granularity and good energy resolution. Furthermore, it should be possible to obtain very good section alone by measuring the ratio of the inelastic rates at Vs = 200 and 900 GeV when the stability of the calibration, helped by the fact that the liquid-argon read out will not suffer CERN pp Collider operated in the pulsed mode [18]. The idea is that in this case at least the radiation damage. In addition to identification by the highly segmented e.m. calorimeter, the ratio of the unprecisely known luminosities is tightly constrained by the machine properties. electron identification will be strongly enhanced in the central and forward regions by layers of Assuming that ctot is known at Vs = 200 GeV from fitting existing data UA5 has evaluated transition radiation detectors sandwiched between drift chambers with dE/dx measuring atot = 65.3 ± 0.7 ± 1.5 mb (statistical and systematic errors) at Vs = 900 GeV. One is eagerly capabilities. The muon detection system, with magnetized iron for the muon sign measurements awaiting the first atot measurements from the Fermilab experiments at Vs = 1.8 TeV in view of up to about 300 GeV momenta, covers all angles down to 11° with respect to the beams. The a possible small inconsistency [19] of the p and ctot measurements related by dispersion start-up of the fully installed DO experiment is unfortunately expected to take place only around relations. 1990. Another topic of soft physics at the pp Collider is the study of the differential cross- 301 302 303 304 section for elastic scattering at large values of 111 with the diffractive dip structure measured by from the ambiguous assignment of low momentum final state particles to a given jet which may the UA4 Collaboration [20]. Concerning standard (minimum bias) inelastic events, much partially overlap with other jets and with spectator hadrons not directly related to the sub- detailed information on general particle production properties has been obtained by the streamer process of interest. Further systematic uncertainties are due to instrumental effects, such as in chamber experiment UA5. Further contributions came from the UA1 and UA2 experiments. As the detailed knowledge of the calorimeter response to hadronic jets. an example, Fig. 6 shows the UA5 results on charged particle pseudorapidity (T| = -In tan 6/2) The largely increased statistical precision from the jet data samples expected from the distributions for inelastic non-diffractive pp collisions. As discussed in Refs. [14,15,21,22] improved CERN pp Collider will therefore not directly reflect into much higher precision QCD several properties such as the mean charged particle multiplicity and transverse momentum are tests in most cases where the systematic uncertainties are already important. Nevertheless, observed to increase approximately proportionally to In Vs going from Vs = 53 GeV (CERN significant improvements can be expected over the present data. A few typical jet and QCD ISR) to pp Collider energies at CERN and FNAL. physics topics are considered in the following to illustrate the present status and the expected improvements from the forthcoming Collider data at CERN and FNAL. 1.3 Jet Physics and OCD Studies 1.3.1 Introduction 1.3.2 Inclusive Jet Cross-section The CERN pp Collider has provided a very rich harvest of results on hadronic jet The inclusive jet cross-sections have been published by UA1 [29] and UA2 [30] physics, confirming that available lowest-order QCD calculations are able to describe the gross already, without including the data from the 1985 running period. The data from both features of the strong parton interactions with large momentum transfers. experiments agree with each other, and Fig. 8a shows as an example the data from UA2. The An introduction to jet physics at the hadron colliders can be found for example in Refs. global systematic uncertainties are evaluated to be ± 70% and ± 45% for UA1 and UA2 [23-26]. Experimental jet results at the pp Colliders are based essentially on calorimetric respectively. This uncertainty precludes a very quantitative comparison with QCD calculations. measurements of the transverse energy flow and its clustering within limited regions of azimuth It is remarkable though that the shape of the inclusive jet cross-section as a function of the jet <)> and pseudorapidity T|. Important considerations for such measurements are therefore the transverse momentum pj is so well described over several orders of magnitudes by lowest- granularity and the resolution of the calorimeters as well as the algorithms for jet energy order QCD calculations. For the future runs, UA2 has estimated that with L = 10 pb_1 they will determinations [27]. As an example, Fig. 7a from UA2 shows that the event topology at the extend the pj range to about 200 GeV/c and that the systematic errors could be reduced to CERN pp Collider for events with a large total transverse energy IEJ emitted within the central +35% by an improved control of calibration effects. However much higher pj's are reached by region -1 < r\ < 1 is predominantly two-jet like. Most of the IEJ is shared equally between the the CDF experiment at the TEVATRON pp Collider due to the larger collision energy. This is two largest transverse energy clusters Ej1 and Ej2 defined as adjacent calorimeter cells with illustrated in Fig. 8b which compares the calculated inclusive jet cross-sections at Vs of 0.63 energy Ecen > 400 MeV. As expected from transverse energy balance these clusters are back- and 2 TeV. to-back in the transverse projection, a feature visible in the event display of Fig. 7b which It is interesting to search for deviations at large jet pj from the QCD behaviour because shows the transverse energy distribution in the UA2 calorimeter cell map A 1 2 recognize as the example shown with Ey - = 100 GeV. transfers well below the characteristic energy scale Ac describing the strength of this The relevance, but also the limitations, of jet physics to QCD have been discussed at this conjectured interaction. Finite values of Ac would produce an excess of events compared to School by Ref. [28]. Very detailed quantitative tests of QCD with Collider data suffer from ordinary QCD predictions (corresponding to A^. = °°) at large pj. From the inclusive jet cross- many systematic limitations. From the theoretical side there is a lack in understanding of higher sections the present 95% CL lower limit is Ac > 400 GeV [29,30]. This limit is expected to order effects (in the strong coupling constant o^) and of all "soft" effects due to low momentum improve to about 650 GeV with the future runs at the CERN pp Collider, but it cannot compete transfers which cannot be treated perturbafively. The momentum transfer scale Q2 is not with the limit A^. > 1.6 TeV, which can be obtained with even a modest data sample (L=l pb"1) unambiguously defined, and the fragmentation of scattered partons into hadronic final states is at Vs = 2 TeV. The preliminary inclusive jet cross-section from the first CDF run is shown in not rigorously described. From the experimental point of view uncertainties arise at some level Fig. 8c, this data is based on L - 25 nb"1 only. systematic uncertainties in the cross-section measurements are expected to cancel, should yield a 1.3.3 Two-jet Cross-section and Angular Distribution determination of as. The calculated cross-section ratio can be adjusted to agree with the A study of the two-jet mass and angular distributions allows one to investigate in more experimental value by varying as. The calculations are however affected by higher-order as detail the parton hard scattering mechanism. Eight basic 2 -> 2 processes can contribute in corrections (expressed as K2 - and K3 - factors for the two - and three-jet cross-sections 2 lowest order (as ) to the two-jet cross-section, their respective importance depending on the respectively) implying that the three - to - two jet ratio is proportional to as (K3/K2). Another structure functions describing the initial quark (q) and gluon (g) fluxes [32]. The expected theoretical uncertainty arises from the ambiguous choice of the Q2 scale for the two - and three- fractions of gg, qq and (qg + qg) final state two-jet events in the pseudo-rapidity range -2 < ri jet events. Initial 2 -» 2 and 2 -> 3 parton scatterings are not the only contribution to < 2 are shown in Fig. 9 as a function of the subprocess centre of mass energy Vs (two-jet experimentally observed two-jet and three-jet cross-sections. Due to incomplete detector invariant mass). Only the future high luminosity running will enable the experiments to acceptance and overlapping jets, additional contributions come from multijet events. This means accumulate significant data samples with Vs £ 300 GeV where qq states clearly dominate. With that further systematic uncertainties in the data analysis arise from model dependent corrections such a data sample one could attempt to measure changes in the angular distribution as a like fragmentation effects, four-(or more) jet production, underlying (spectator) event function of Vs due to the changing qq, gg and (qg + qg) fractions. fluctuations, structure functions and so on. 2 The two-jet angular distribution is also sensitive to hypothetical contact interactions [31 ] The present results on ocs (K3/K2) using the same Q scale for two - and three-jet events already mentioned in the previous sub-sections, which would cause an enhancement at large are 0.23 + 0.02 ± 0.04 from UA1 [35] and 0.23 + 0.01 ± 0.04 from UA2 [36], where the first scattering angles 9 over the pure QCD distribution. Figure 10 shows the UA1 data [33] in the error is statistical and the second one describes the systematic uncertainties. The Q2 used is the variable X = (1 +cosG)/(l - cosO) and illustrates the effect of Ag = 300 GeV. Their present 95% maximum jet transverse momentum squared, giving average values of 4000 GeV2 and 2 CL lower limit on Ac from the % distribution is 415 GeV, slightly better than the one from the 1700 GeV for the two experiments respectively. inclusive jet cross-section alone. Similarly, somewhat improved limits over the ones quoted in The value of ct^KyKj) as a function of Vs has been studied by UA1 and UA2. As an the previous section can be expected from the angular distributions from the future large data example Fig. 11 shows the most recent data of UA2. The expected variation of as is shown as 2 2 samples. well (obtained by computing the variation of as as a function of Q at the value of the mean Q for each Vs bin and assuming that Kj/K2 remains constant). The present accuracy of the data is 1.3.4 Three-jet Events and ctj obviously not sufficient to demonstrate the "running" of as. The expected improvement of the 3 statistical error at high values of Vs in the future data sample is illustrated in Fig. 11 for the The study of three-jet events allows one to extend the QCD tests to order as processes, dominated by gluon bremsstrahlung [34]. Various angular and energy distributions of three-jet upgraded UA2 experiment for two 10 GeV wide Vs bins. Provided that the data span a 2 events have been successfully compared to QCD expectations by UA1 [35] and UA2 [36]. sufficiendy large Vs range, the Q dependence should become observable whereas the absolute Further detailed studies will be possible with the increased statistics and the larger acceptances value will likely remain dominated by the systematic uncertainty. of the upgraded detectors (in particular of UA2) and of CDF. As an example, it will become possible to pursue a study of the cos!; = (p2*-p3*) / Pi* distribution [36], which describes the 1.3.5 Multi-jet Production asymmetry of the energies found in the softer two jets (pj*, p2*, P3* are the three jet momenta Four-and more parton final state events are of particular interest because there are many in their common centre of mass system in decreasing order of energy). The distribution of cosÇ "new physics" processes for which such events are an important background. Multi-jet 4 is characteristic of gluon bremsstrahlung off a qq pair and provides a test of the spin of the production occurs not only due to higher order (ocs ) corrections to the two-jet processes but gluon. Though in the pp case many different subprocesses contribute to the three-jet there is also an interesting new class of interactions, namely the multi-parton scattering, where production, the cosÇ distribution remains a sensitive probe of the QCD dynamics. two pairs of incoming partons from the same nucléons interact [37]. The most topical interest of the three-jet studies lies in the possibility to extract as. The No multi-parton signal has been observed yet at the pp Colliders. The present four-jet basic idea follows from the fact that the two - and three-jet cross-sections are approximatively data [38] are well described by a 2 -> 4 scattering QCD model alone. The new multi-level 2 3 proportional to as and as respectively, and that therefore the ratio, for which many triggering system of the upgraded UA2 experiment aims at reaching lower jet transverse 305 306 307 308 momenta where the multi-parton signal is expected to be enhanced with respect to normal QCD. somewhat similar electron identification criteria, at least for the W and Z physics, where the It is thereby hoped that a signal can be revealed in the future high statistics data sample. main goal is to identify efficiently high p-r- isolated electrons. The cuts used by UA1 and UA2 for the W -> ev analysis are summarized in Table 1. The efficiency of these electron selection 1.3.6 Direct Photon Production criteria is estimated to be about 75% in both experiments with a rejection against QCD jets of The direct photon production pp -> y + X is of particular interest for QCD tests because about 50000. This efficiency does not apply to electrons inside or near to jets, such as those of two main advantages. First, it is experimentally possible to measure direcdy the photon originating from semileptonic decays of heavy flavours, which would be largely rejected by the transverse momentum in the electromagnetic calorimeters with much better accuracy than for implicit and explicit isolation criteria applied in the analysis. Very significant improvements are jets, and second fragmentation effects are absent. A complete QCD calculation in next-to- expected for the upgraded UA1 and UA2 experiments. 2 leading order (as ) is available [39]. However, the rates expected are about three orders of Muon identification. Muon identification is possible in the UA1 and CDF experiments magnitude smaller than for inclusive jet production. The present data [40] on direct photon and is foreseen for DO. The electromagnetic and hadronic calorimeters and magnetized iron are production are indeed limited by their statistical precision and are confined to transverse used as a hadron filter. The signature for a muon is a charged track in the central detector momenta well below 100 GeV. The systematic uncertainties on the cross-section are about matching a track in the muon chambers outside the shielding. The backgrounds from K and JC ±20%. The data are expected to improve significantly with the increased integrated luminosity decays are suppressed by rejecting tracks with kinks. Muons from heavy-flavour decays are in the near future. suppressed by explicit isolation requirements. Although clean IVB signals are observed in W -> \iv and Z -> mi, the resolution is very much worse than for electrons. In UA1, a track of 1.4 Intermediate Vector Boson Physics 40 GeV momentum (typical for a W -» p.v decay) and perpendicular to the magnetic field, has a The transformation of the CERN Super Proton Synchrotron (SPS) into a pp Collider in measurement error of Ap/p = 20%, which is an order of magnitude worse than for electrons of 1981 was motivated by the search for the Intermediate Vector Bosons (IVBs) of the electroweak the same momentum. interactions. Only two years later the first observations of the W and Z particles were published Tern identification. The decay chain W -> xvx, x -> vx + hadrons is expected to result in by the UA1 and UA2 Collaborations. Clear evidence for observation of IVBs has been obtained purely hadronic events with large missing transverse momentum p-p due to the undetected also at the FNAL Tevatron pp Collider at Vs = 1.8 TeV by the CDF Collaboration. neutrinos (see below). Since the x energy is in this case generally much larger than the x mass In order to optimize the experimental signal over background ratio all the experiments the hadrons from the x decay appear as a highly collimated jet with a very low charged panicle have chosen initially to identify the IVBs by their leptonic decay modes in spite of the smaller multiplicity. UA1 has presented a W -> xvx signal [41] based on a series of cuts which select branching ratios compared to the decays into qq pairs. The latter would appear as two-jet events events with only one narrow, isolated hadronic jet and well measured pj. in addition to the large continuum QCD background from hard parton scattering (see Sect. Neutrino identification. The presence of non-interacting high-pT neutrinos in the final 1.4.4). state is characteristic of all leptonic W decays. Since a large fraction of the total energy in the interaction is at small 0 and therefore remains inside the vacuum pipe, only the transverse 1.4.1 Lepton Detection momentum of the neutrino can be accurately measured. The signal for a high-p-j. neutrino is the Some basic considerations about lepton detection at the pp colliders are sketched below. apparent p-j- imbalance in the event. The pj of the neutrino is measured by the p-j- imbalance in The details depend on the particular detector and can be found elsewhere [41-44], The UA2 the event : experiment has no instrumentation to detect muons. V I V- T-1 ~* Electron identification. The aim of the electron identification cuts is to select isolated = = n PT PT " • T i electrons and reject backgrounds arising from QCD jets. In fact, the main background arises from pathological jet fragmentations, (i.e. from jets containing a large transverse momentum n° where Eji is the energy in the i* calorimeter cell, and n j is the unit vector that points from the v or 7t°'s, where Jt~jr° spacially overlap), or photon conversions which occur in the beam pipe or vertex to the cell i. A good measurement of pT at the present pp energies requires hadronic the detector, resulting in fake electron candidates. As a consequence, the experiments use calorimetry down to polar angles of (5 - 10)°. For the IVB results from the 1982-1985 CERN pp Collider runs the detectors performed higher top mass would improve the agreement between theory and experiment for d£ w. On the as follows. In UA1, the hadronic calorimetry extended down to very small polar angles but other hand, the experimental errors on G£ w are at present too large to provide a firm constraint there were cracks of A0 = ± 4° with respect to the vertical. If one excluded events where the on mt. angle between pjv and the vertical is less than 15° and asked that there be no high-py tracks The main features of IVB production at hadron colliders are described by [47] and within 5° of the vertical, then the missing pj resolution obtained can be parametrized by summarized in Fig. 13. In spite of the clear rise in cross-section for W, Z production (ow, oz) when Vs increases, we expect that precision measurements in the W, Z sector will become 2 e increasingly difficult at higher Vs. Comparing, for example, FNAL (Vs = 1.8 TeV) with CERN d^j-2 A * • (Vs = 630 GeV), we expect a factor of 3 increase in aw and oz, but also a factor of 10 increase et where A = 0.7 Vz Ej, with Ej in GeV. In UA2 there was no calorimetry coverage below 20°, in QCD two-jet production, o"jetj with p-pl = 40 GeV and h"|jetl < 2. This results in a worse and between 20° < 9 < 40° there was no hadronic calorimetry. This caused non-Gaussian tails signal to background ratio, which might affect precision measurements using W -> ev or in the measurement of the missing pjj however, the observation of large missing p-r- still gave a W -> |iv decays. In contrast the background from fake electrons or muons under the Z -» ee or significant rejection against background in the search for W -» ev. The probability of losing H|i peak is negligible). In addition, as shown by Figs. 13b and 13c, the gain in rate at one jet in a two-jet event was 10% (2%) for a jet with Ex = 15 (40) GeV. higher Vs is ruined by a broader pyw distribution and by less central production of the W's. The performance of the CDF detector with respect to pjv measurements has been reported [45] to be comparable to UA1. The upgrades of UA1 and UA2 have aimed to improve 1.4.3 Lepton Universality strongly the hermeticity of the detectors. Recent results from UA2 [46] show that this goal has All three expected leptonic decays of the W and the Z° -> e+e" and n+p." decay modes been reached. have been measured by UA1 [41]. The cross-sections times branching ratio measurements shown in Fig. 12 provide an experimental test of the lepton universality expected in the 2 2 1.4.2 IVB Production Cross-sections Standard Model for the weak charged and neutral current couplings at Q = m lVB. Defining the The UA1 and UA2 Collaborations have measured the product of the inclusive IVB weak charged coupling constants by (g/gj)2 = T(W -> £ ;V)/T(W -> £ jV) = a£ wj/0£ Wj and production cross-sections times the leptonic branching ratio G£IVB from the rate of observed similarly for the weak neutral couplings lq (i,j charged-lepton types), UA1 have obtained the leptonic decays, the measurement of the integrated luminosity, and the knowledge of the following results combining the Vs = 546 GeV and 630 GeV data samples detector acceptance and efficiency. The results from UA1 and UA2 are summarized in [41,43] and displayed in Fig. 12. There is good agreement between the two experiments on the cross- gtl/ge= 1.00 ±0.07 ±0.07 section measurements for the electron decay modes which are measured with the highest gT/ge= 1.01 ±0.10 ±0.06 accuracy. The UA1 data also show good consistency among all the observed leptonic decay yke= 1.02 ±0.15 ±0.04, modes. The preliminary result from the CDF experiment [45] is also shown in Fig. 12. The systematic errors for the experimental results are dominated by the systematic uncertainties in where the first error is statistical and the second one is due to systematic uncertainties. These the measurements of the integrated luminosity. measurements give support to the validity of lepton universality in the IVB decays to a level of The data are in satisfactory agreement with theoretical predictions. Shown in Fig. 12 as a about 15%. function of Vs are the calculations of Ref. [47], which were obtained under the assumptions of Much larger data samples are expected from the present and the forthcoming runs at the a top-quark mass mt = 40 GeV and IVB masses of mw = 83 GeV and mz = 94 GeV. The errors CERN and FNAL pp colliders. UA1 and CDF will improve significantly the accuracy of these of these theoretical predictions reflect uncertainties in the structure functions and ambiguities in measurements, without, however, being able to ever reach the precision of experiments at lower 2 the choice of the Q scale. A detailed study of the dependence of the theoretical predictions on Q2. 2 2 the structure functions, higher-order (as ) strong corrections, sin 6w, IVB and top-quark masses, and the number of light neutrinos has been discussed by Ref. [48]. One can note that a 309 310 311 312 1.4.4 Hadronic Decays 2 2 Therefore, both a good two-jet mass resolution and a large integrated luminosity are required in since the vector coupling of the Z° to leptons is proportional to 1 - 4 sin 0w and sin 8w is searching for IVB -> jet + jet events. The UA2 Collaboration [49] have recently reported the close to 0.25. The UA1 Z° decay angular distribution for the combined Z° -> e+e" and result of such a search, which is shown in Fig. 14. A signal of about 3 standard deviations Z° -> ji+ji" channels is shown in Fig. 15c together with the expectation for two values of 2 2 significance is observed above a large QCD background. The peak contains 632 ± 190 events sin 8w. The data provide a measurement of sin 8w = 0.24 _0'M consistent with the more compared with an expectation of 340 ± 80 events from W and Z° decays (excluding modes with precise determinations from the IVB masses discussed in Section 1.4.8. In the future, with a larger sample of Z decays, the statistical error may be greatly reduced but a significant a top quark). The fitted W mass is mw = 82.312'4 GeV. The UA2 result is consistent with the expectation from the Standard Model. However, stronger evidence for a signal and a significant systematic error from the uncertainty in the ratio of the u-quark and d-quark structure functions quantitative measurement of the IVB branching fractions into quark-antiquark pairs will require will remain. the collection of a much larger data sample. It will be very interesting to see whether UA1, UA2, CDF and DO, with their different 1.4.6 Transverse Momentum Distributions calorimeters, will be able to clearly establish the presence of the W, Z -> qq decays above the The production of the IVBs by quark-antiquark annihilation (Drell-Yan process) is in QCD background, with the much larger integrated luminosities expected in the near future. Jet- general accompanied by the radiation of gluons from the incoming partons giving rise to a non jet spectroscopy will be one of the crucial experimental features at future hadron colliders, and zero IVB transverse momentum p-i-IVB. The radiation can include hard (large-transverse- IVB the present pp colliders are an ideal testing ground for the feasibility of high precision and high momentum) gluons observable as hadronic jets causing a significant tail in the pT granularity calorimeters, within the high luminosity environment of future hadronic machines. distributions. z Experimentally only pT can be measured directly from the transverse momenta of the 1.4.5 Angular Distributions two final-state leptons whereas for pjw one must also measure the transverse momentum of the At the CERN pp Collider energy the IVB production arises mainly from the annihilation recoiling hadrons. The best precision is reached for the Z° -» e+e" channel for which UA2 [43] z of valence quarks and antiquaries. The W's are almost fully polarized along the p direction have evaluated an accuracy for pT of ± 2 GeV, dominated by the uncertainty of the energy z according to the Standard Model expectation. Assuming a universal V - A coupling of the W to measurement for the electrons. An even better accuracy of ~ 0.2 GeV is obtained for the pT fermions, one expects from helicity conservation a distinct charge asymmetry in the W -> £ v component pn2 perpendicular to the bisector of die e+ and e" transverse-momentum vectors, decay, with a lepton angular distribution of the form dn/d(cos6*) « (1 + q cos8*)2, where 8* which is mostly effected by the small angular measurement errors. The UA2 data correspond to is the angle of the emitted charged lepton with respect to the W polarization in the W rest frame an average value w is not measured) and that the definition ofcos9* is affected by the intrinsic pT of the W. shown for py > 15 GeV, the region where UA2 evaluates that this quantity can be measured The UA1 [41] and UA2 [43] W -> ev decay angular distributions for well-measured reliably, given the estimated experimental uncertainty for p-pw of about 5 GeV due to the events are given in Figs. 15a and 15b together with the Standard Model prediction. The measurement errors on the recoil hadrons. The curves in Fig. 16 show the behaviour expected expected forward peaking is clearly seen, and the small deviations from the simple (1 + cosO*)2 from QCD calculations as reported by UA1. The solid line below 25 GeV corresponds to a full law arise mainly from the presence of W bosons produced from sea quarks. From the UA1 data QCD calculation from Ref. [47] using a soft gluon resummation technique. The shaded band between 25 and 60 GeV results from a perturbative calculation (order as) normalized by the measured directly from the W -> £ v decays because of the unknown longitudinal momentum w lowest-order W cross-section, whereas the expectation shown for the region pT > 60 GeV component of the neutrino. Only its transverse component can be inferred from the missing corresponds to a Monte Carlo extrapolation (ISAJET) which has been adjusted in the region transverse momentum JJJ in the events. The W mass has been evaluated from fits to the 25 < p^-w < 60 GeV. transverse-mass distributions defined as The two measurements agree with the QCD expectation. Special attention has been given x w i li to the events with very large p-p . The three events (two from UA1 and one from UA2) with m-j- = [2pj fa(l - cosA<)>)] , Pjw > 60 GeV are statistically compatible with the QCD expectation of about one event. However, the two events from UA1 both contain a W recoiling against a two-jet system jj. In where pi is the electron, muon, or tau-jet transverse momentum, jij the missing transverse both cases the Wjj mass is in excess of 250 GeV, and the jj mass is compatible with an IVB momentum in the event due to the neutrino(s), and A<)> the angle between pj and p-p. The 2 decaying into a quark-antiquark pair. This topology cannot be easily explained by order as distributions of m-p depend only weakly on the W production mechanism. Nevertheless, Monte QCD calculations, which predict only 0.05 + 0.03 events, or by Standard Model boson-pair Carlo techniques are needed when extracting mw from mT in order to simulate the expected W production contributing even fewer events. The future pp runs at CERN and FNAL are longitudinal-and transverse-momentum distributions and to take into account the detector expected to clarify this question soon. characteristics. The W mass can alternatively be determined from the p-p* distribution. This The study of IVB production in association with hard jets offers in turn an interesting method is less sensitive to the JS-T- evaluation, but depends directly on a detailed knowledge of w new possibility to determine the strong coupling constant as. The basic idea is to compare the the shape of the pj distribution. experimental ratio of IVB + n jets to IVB + (n-1) jets (n = 1,2,...) with QCD expectations for a The transverse-mass distributions for the W -» £ v channels and the lepton-pair mass + given as. Using a sample of 251 W -> ev decays, of which 29 are produced in association with distributions for the Z° -> 2 £ ~ decays are shown in Figs. 17 and 18 for the UA1 data [41] one observed hard jet (pyjet > 10 GeV), UA2 [51] recently obtained as = 0.13 ± 0.03 (stat.) and in Fig. 19 for the UA2 data [43]. Estimated background contributions as well as expected ±0.03 (syst.,) ± 0.02 (syst.2). The first systematic error is experimental and is strongly IVB signal distributions are indicated. In the case of the muonic W decay, Fig. 17b, UA1 affected by the rather low jet threshold used in the analysis where jet energy scale uncertainties choose to present the inverse transverse-mass distribution (Mmj), which reflects more are large. The second systematic error is theoretical reflecting that this measurement uses adequately the errors in the muon-track momentum determination. Preliminary CDF mass approximate expressions of the K-factors [51] necessary to extract as from the data. One may distributions [45] are shown in Fig. 20. assume for the future that these approximate expressions will become exact, thus reducing this The IVB mass determinations from both CERN experiments agree with each other and source of uncertainty in the measurements. With an integrated luminosity of L = 10 pb"1 are summarized in Table 2 and displayed graphically in Fig. 21a. The most precise values are sufficiently large data samples will become available that the jet energy threshold can be obtained from the electron channels in spite of the fact that here the errors are already dominated increased in order to minimize the experimental systematic errors. One can hope to extract as by the systematic energy-scale uncertainties of the UA1 and UA2 electromagnetic calorimeters, using this method with similar experimental and theoretical (structure functions, higher orders) which are estimated to be about ± 3% and ± 1.5% respectively. The UA2 Collaboration have uncertainties of approximately 10%. checked their energy-calibration procedure by remeasuring a sample of their central calorimeters cells (40 out of 240) in a test beam after the last data taking. The result is given in Fig. 21b, 1.4.7 Mass Measurements where the ratio of the reconstructed energy to the beam energy is shown for 40 GeV electrons. The W and Z masses (mw, mz) are the two parameters of the Standard Model which can Further systematic uncertainties can arise in the case of mw from possible systematic biases in be measured directly by the collider experiments. For the Z, the mass can be evaluated in a the evaluation of {W- as quoted separately in Table 2 for the UA2 results. straightforward way from the lepton-pair mass distribution in the e+e" and \i+\i~ final states. The mass distributions of the electronic decay channels have also been used to extract Monte Carlo methods are used to extract mz in order to take into account the effects of the 90% CL upper limits on the total widths r1VB- The results are Tw < 5.4 GeV and Tz < 5.2 GeV experimental mass resolution on the relativistic Breit-Wigner shape of the Z peak, its finite from UA1 and Tw < 7.0 GeV and Tz < 5.6 GeV from UA2. width and the asymmetric production due to the structure functions. The W mass cannot be Future precision mass measurements at hadron colliders will clearly concentrate more 313 314 315 316 + and more on mw, because experiments at the e e~ colliders operating at the Z pole (SLC and where LEP) will determine mz to an accuracy of better than 50 MeV [52], which will never be /2 approached by the pp experiments at CERN and FNAL. This arises from the fact that present A = (wxW2 GF)' = 37.2810 + 0.0003 GeV. mass measurement errors are already dominated by the systematic uncertainties on the energy scale, which are directly linked to the absolute calibration of the electromagnetic calorimeters. The quantity Ar accounts for the effects of the one-loop radiative corrections on the IVB masses The goal of the existing calorimeters (UA2, CDF) and of the forthcoming more ambitious ones and has been calculated to be [53,54] (UA1, DO) is to reach the level of an uncertainty of ±1% on the absolute energy scale, which would result in errors of the order of ± 1 GeV on the W, Z mass measurements. In the Ar = 0.0711 ±0.0013, following, we shall assume that the Z mass and width will have been precisely measured at SLC and LEP by 1990. assuming that mt= 35 GeV and that the Higgs boson mass mH = 100 GeV. From a At that time, the pp experiments will have typically accumulated 2000 to 4000 well- measurement of the ratio mw/mz, for which in first order uncertainties of the calorimeter measured reconstructed W -> ev decays and 300 to 600 well-measured Z -» e+e" decays, calibration cancel, a direct measurement of sin^w is provided by the relation depending on calorimeter fiducial cuts and taking into account representative electron selection + 2 efficiencies (70% for W -> ev and 90% for Z -» e e"). The estimated statistical errors on mw sin Gw = 1 - (mw/mz)2. (3) and mz are similar, about 200 to 250 MeV, owing to the fact that the W mass cannot be 2 completely reconstructed. The determination of sin 0w from Eq. (3) is independent of other experiments and of This leads to a discussion of the systematic uncertainties inherent in the method used to theoretical uncertainties. On the other hand, a best fit to Eqs. (1) and (2) using accurate extract the W mass. Known theoretical and experimental uncertainties concern the knowledge of measurements of A and the calculation [53,54] for Ar gives a more precise measurement of 2 w the W width, of the W transverse momentum pj , as well as the measurement of the neutrino sin 0w. The results from UA1 [41] and UA2 [43] are summarized in Table 3 for both methods. 2 transverse momentum. UA2 estimates, after a careful study, that the systematic error on the W All the measurements are in excellent agreement with the average value sin 9w = 0.232 ± 0.004 mass, due to the method used to extract its value, will not be smaller than ± 200 MeV. ± 0.003 obtained in neutrino experiments [55], where die first error is experimental and the 1 second one is due to theoretical uncertainties. This weighted mean is evaluated on the basis of With 10 pb" UA2 therefore expect to obtain 5mw = (± 0.22 ± 0.20 ± 0.81) GeV (errors due to statistics, method, and calibration, respectively), assuming an uncertainty of the experimental errors, assuming a charm-quark mass n\. = 1.5 GeV, and ignoring the ± 1% on the absolute energy scale. This uncertainty cancels in the measurement of R = uncertainties on the theoretical error due to rriç. mw/mz, which will be measured with an error of 5R = ± 0.003(stat.) ± 0.002(syst.). This Deviations from the minimal Standard Model could be detected in particular in the leads to 8mw = ± 0.35 GeV, for a known value of mz. This accuracy on mw is unlikely to be quantity [56] improved significantly until LEP operates above the threshold for W pair production. 2 2 2 p = mw / mz cos 8w, (4) 1.4.8 Standard Model Parameters 2 The IVB masses are two important parameters of the Standard Model, which relates which was assumed to be p = 1 in the above formalism to determine sin 6w. Using the 2 them in its minimal expression to the fine structure constant a, the Fermi constant GF, and the measurements of mw, mz, and the value of sin 8w deduced from Eq. (1), the values given in 2 weak mixing angle sin 6w by the following relations [53]: Table 3 have been derived. They are consistent with the minimal Standard Model hypothesis p = 1. Finally the relations (1) and (2) can also be used to measure the radiative correction 2 2 mw = A2/[(l - Ar) sin ew] ( 1 ) parameter 2 2 2 2 mz = A /[(l - Ar) sin 6w cos 9w], (2) 2 2 2 2 1 - Art = (A /mw )/[l - (mw /mz )]. (5) through gluon-gluon fusion via a heavy quark loop, with subsequent decay of the Higgs boson 2 A more precise value (Ar2) may be obtained using the sin 9w value from the neutrino into the heaviest fermion pair, i.e. into bb, if the Higgs mass is larger than 10 GeV. The experiments as additional input. Both estimates (Table 3) are consistent with the expected production rates at CERN are observable up to Higgs masses of about 50 GeV, and up to radiative corrections, even though the experimental errors are still too large for a definite around 100 GeV at the FNAL pp collider. However, the enormous background from hadron conclusion. The experimental results are also summarized in Fig. 22a where correlations jets (a factor of 105 above any possible H -> bb signal) precludes the observation of H between the uncertainties of the mw and mz measurements are shown in the (mz, mz - mw) production through gluon-gluon fusion, even with an experiment optimized for b-quark plane. tagging. A more promising production process is that in association with a W (or Z) boson, the Assuming the expected precision for future mass measurements mentioned in the so-called Higgs bremsstrahlung mechanism [59], which is the dominant source of H production 2 2 previous Section, the direct measurement of the weak mixing angle, sin 6w = 1 - (mw/mz) = at SLC and LEP. This process is smaller in cross-section than the gluon-gluon fusion 1 - R2, will have a statistical error of ± 0.006 and a systematic error of ± 0.004. The effect of mechanism, and yields observable rates only up to Higgs masses of 30 GeV (50 GeV) at the radiative corrections to the IVB masses will be determined as illustrated in Fig. 22b. The CERN (FNAL). However, in order to get rid of the background from hadron jets, one would improvement with respect to the present situation will be substantial, allowing such a be led to require associated production of a Higgs boson and a real Z, with the latter decaying measurement to provide a stringent test of the Standard Model, in particular with respect to the into a lepton pair. This reduces the observable rate by an order of magnitude. Therefore, it mass of the top quark, if it has not been measured by then. seems very unlikely that the Higgs boson could be discovered at existing hadron colliders. 1.4.9 IVB Pairs and Higgs 1.5 Heavy Flavour Physics The physics potential of studies of electroweak gauge-boson pair production was 1.5.1 B-quark Physics recognized a long time ago. In the Standard Model, there are important cancellations in the The UA1 results from the analysis of single and dimuon data [60] have shown that, amplitudes for WW and WZ production, which depend upon the gauge structure of the WWZ contrary to electrons, muons are identifiable inside or near to hadron jets. This opens up an trilinear coupling. The rate for Wy production is sensitive to the W magnetic moment. exciting field of heavy-flavour physics, leading to the first measurement of Bs-Bs mixing. This Unfortunately, the rates for boson pair production, at the present pp colliders are too low to field should clearly be pursued, in particular with the upgraded UA1 muon detector, which will allow detailed studies of these potentially very interesting events. be by far the most performant muon detector at pp colliders in the future. Recent summaries of Multijet backgrounds from QCD will swamp any purely hadronic signal for boson pair the UA1 b-quark physics results are available in [61]. production. Therefore only events where at least one gauge boson decays into ev or ee can be considered. The expected signal rates for 10 pb"1 are less than one event at CERN and less than 1.5.2 Search for the Top Quark 10 events at FNAL for the WW, WZ and ZZ channels. Even in this case, it has been pointed One of the great experimental challenges at the CERN and FNAL pp Colliders is the out [57] that the background from Standard Model W, Z + 2 jets production, where the two-jet search for the top quark. For small top masses (mt < mw) a clear signature is expected from the invariant mass is required to be within ± 10 GeV of the W, Z mass, will be almost two orders W -> tb decay followed by a semileptonic top-quark decay yielding events with an isolated of magnitude above the boson pair signal. Since the requirement of a double leptonic decay (pp lepton accompanied by two (or more) jets and a neutrino. A second source of top events is -> WW -> evev, pp -> WZ -> evee or pp -> ZZ -> eeee) will greatly reduce the already small provided by direct QCD tT production via gluon fusion and quark-antiquark annihilation. A rates, one has to conclude that the observation of heavy gauge-boson pair production is very comprehensive report on a search for the top quark has been given recently by UA1 [62] excluded at the present pp colliders. using both their electron and muon data samples. Whereas the expected contribution from The search for the neutral Higgs (H) boson is one of the main topics of physics studies W -> tb can be evaluated reliably from the measured W production cross-sections, the estimate at future accelerators [58]. The various H production mechanisms and experimental signatures of the tT channel is affected by theoretical uncertainties. Using the most recent calculation [63] are the subject of intensive studies in view of detector design for multi-TeV machines. the UA1 mass limit on the top quark is mt > 41 GeV (95% CL). At the present colliders, the dominant production mechanism for H production is Figure 23, reproduced from [64], shows the inclusive t (or T ) production cross-section 317 318 •MS 320 as a function of mt at the CERN (Vs = 0.63 TeV) and at the FNAL (Vs = 1.8 TeV) pp colliders. large missing Ej multijet sample before applying a cut A(|> < 140° (23 events) where A<(> is the Below mt - 80 GeV the contribution from W -» tb is clearly visible. From Fig. 23 it is clear azimuthal angle between the two highest transverse energy jets in the event. This distribution is that hadronic decays of the top quark would be abundant at CERN and FNAL for the integrated strongly peaked towards A 4. Colliders to Reach the TeV Mass Range 16) UA8 Collaboration, "Experiments at CERN in 1987", Geneva (1987) p.101. 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Peccei, Physics at LEP, Proc. 1986 CERN School of Physics, CERN 87-02 (1987) 209. 325 326 327 328 Table I Table 2 Electron identification criteria IVB mass values m (GeV) Variable UAl selection UA2 selection Source Channel w Ej cluster > 15 GeV > 11 GeV UAl W->e 82.7 ± 1.0 ±2.7 W->nv 81.8 + 6.0 +5-3 ± 2.6 Shower profile Consistent with electron Similar (longitudinal and lateral) W->TV 89 ±3 ±6 Charged track Required Required UA2 W -• ev 80.2 ± 0.6 ± 0.5 ± 1.3 (except IA<(>I < 15°) 1/p - 1/E <3a <4G UA1+UA2 W->ev(a) 81.0 ±2.0 (forward only) (b) 80.8 ± 1.3 Preshower Large charge Track-shower match Source Channel mz (GeV) Isolation < 3.2 GeV Ey in < 3 GeV in cone - 40° cone ~ 15° UAl Z->e+e" 93.1 ± 1.0 ±3.0 Efficiency -75% -75% Z -> n+n" 90.7 +|;g ±3.2 UA2 Z -» e+e- 91.5 ± 1.2 ± 1.7 UA1 + UA2 Z -> e+e- (a) 92.0 ± 2.4 (b) 92.0 ± 1.8 Source Channel mz - mw (GeV) UAl e-channel 10.4+ 1.4 + 0.8 +7 4 (i-channel 8.9 .77 ±1.9 UA2 e-channel 11.3 + 1.3 ±0.5 ±0.8 Note : The first errors on the UAl and UA2 data are statistical, followed by the systematic uncertainties which are given in the case of mw from UA2 separately for the uncertainties in the transverse-mass determination and for the energy scale. In the combined values, statistical and systematic errors are added linearly (a) or in quadrature (b) for each experiment. a) Table 3 Standard Model parameters Source Channel Direct measurements 2 sin 6W [Eq. (3)] UAI e-channel 0.211 ±0.025 |i-channel 0.187 ±0.148 ±0.033 UA2 e-channel 0.232 ± 0.025 ± 0.010 Source Channel Indirect measurements sin2 6W [Eqs. (1), (2)] p [Eq. (4)] UAI e-channel 0.219 ± 0.005 ± 0.014 1.010 ± 0.028 ± 0.020 b) ^.-channel 0.223 ^^±0.014 1.05 ± 0.16 ± 0.05 UA2 e-channel 0.232 ± 0.003 ± 0.008 1.001 ± 0.028 ± 0.006 Source AT! Ar2 UAI 0.037 ±0.100 ±0.067 0.128 ±0.023 ±0.060 UA2 0.068 ± 0.087 ± 0.030 0.068 ± 0.022 ± 0.032 Note : The first errors are statistical and the second ones describe the systematic uncertainty. Fig. 1 a. The UAI detector in its original form showing the central and forward (end-cap) em calorimeters (shower counters) that are to be replaced by U/TMP calorimeters, b. General layout of the new central (supergondolas) and forward (superbouchons) U/TMP calorimeters. 329 330 331 332 CtN'HU- DtrCCTOS WjMPWfSISTOfl ^ .-. LOWB£I*0U*0S -/«_*»*^ A.* Pun—ct«>- ™*'- :LEC™OM*G«TIC t SSH HADRON CALOWMETE" Fig. 3 Isometric view of the CDF detector. Fig. 2 a. General layout of the UA2 calorimeters. b. Longitudinal cross section of the UA2 detector. The apparatus is symmetric with respect to the centre of the collision region. Fig. 4 Isometric view of the DO detector. pp pp 65 * A. Serpukhov 1 [ 1 1 __, 1 , . FNAL ° ISR (1979) UA2 1983 , , ~\~ o • ISR (R210) o . ISR IR211I .-++++H- " -*- 4 - 60 M UAi, - ^^_ "*" h, = IE', . E',)/IET - — — 55- . + -+- ^ -^ -»- hi = Ej/EE, 50- -«- —- ~ -•- -^ \ PP _•_ -•- _ ""*" \ u - W- I 1 1 1 1 1 100 150 -i E E, (OeVI b) 35 L FULL 5 10 50 100 500 1000 nEI (!oO-246 7 (CC) = L'08 I .(FD). 29 6 ,(BD)= 9 0 CALORIMETER qhts on CC em=l 18. h1 = l 00, h?=l 06 EC em=l 00. bd=l 00 4 900 GeV * 54.6 GeV UAS * 200 GeV GENCL Fig. 7 a. The fraction hi and h2 of the total transverse energy contained in the largest and in the two largest clusters respectively within the central region -1 < r\ < 1. b. Transverse energy distribution in the UA2 calorimeter cell map A(f> x AT] for a typical large Ex two-jet event Fig. 6 Charged particle pseudorapidity distributions from UA5 [18] for various Vs. The solid line corresponds to the UA5 cluster model [15], 333 334 335 336 FRACTION OF GG:QG.OO 2-JET EVENTS VS SUBPROCESS CMS ENERGY 103 /s= 630 GeV -2 2 -T . . -, r r 10 a) 101 «i »,_,„.* j • A = 6J0 CtV ./i, S16 G.V 19831 •r io° 2C0 — /*-- iij c*v ; -- /l. S44 G*V • o f 10- •ê" io- v -g- ^v •- x t^'x ^ 10- '• X- -• 150 200 io- /"s (GeV) "-•H L!>C -4 -'• io- 10" Fie 9 The fractions of qq, gg and gq + qg final state two-jet events in the range 0 100 200 300 too SOC -2 < Tl < 2 as a function of the subprocess centre of mass energy Ms. p, IQeVI • :DF ;pr, Fig. 8 Inclusive jet cross-sections at T) = 0. As an example the UA2 data [30] at Vs = 546 GeV and 630 GeV. The effect of a \ \ hypothetical new contact interactions is also shown (see text). t* A comparison of the expected inclusive cross-sections at Vs = 630 GeV and 2TeV. Preliminary inclusive jet cross-sections from the first CDF run at Vs = 1.8 TeV. 10'1, " 1 100 200 E, (GeV) Fig. 10 The two-jet angular distribution for high mass events from UA1 [33]. UA2 1965 Upgraded UA2 n<2 (2 points with 10 ptf1) o is 1 b) 063 TeV - - 0. 10 - U TtV / "\\ V - 0.05 Fig. 11 The values of s(K3/K2> as a function of Vs from UA2 [36]. The expected _ improvement on the statistical precision with future runs (L = 10 pb"1) m Ttv - is indicated for two Vs bins of 10 GeV width. - •'•'. 20 TtV 1 ' i ' ' ' ' I ' ' ' channel IVB production cross sections 2 3 1. 5 6 8 10 20 30 to 50 electron ' qT (GeV) - A muon UA1 . a tau ! • electron UA2 * electron COF (pre iminary) Fig. 12 . 1st at errors o nlyl Intermediate Vector Boson production cross-sections times ^^2ë leptonic branching ratios in pp s*~ collisions. The data [41,43] Fig. 13 - y y^^ ^-"**^ s y correspond to Vs = 546 and 630 W production properties as predicted by ^ DD-^ W*-X - ll-lv s y GeV for the electron and muon Ref. [47]. L-I'.» ; /y channels, and to a luminosity a. Total cross-section for W production " ir>'«/ Us=0 43TeV y 1 - weighted average for the two Vs for (W+ and W", m = 83 GeV) and for . A// w // the tau channel. The preliminary hypothetical W's of heavier mass as a " / ,—— — result from the CDF experiment function of Vs. [45] is also shown. Only statistical b. Ratio R = [da/dq dy]/(dc/dy) at y = 0 " -^"~~~"^-~~~~~~~~^\ T ,--"" ^^^~^-~-~~~~~I errors are shown. The theoretical as function of the W transverse predictions [47] are indicated with momentum qT for various Vs. \ ^^^ ^~~ their estimated uncertainties over the y c. Normalized rapidity distributions - \ y y* s-' - pp — z\x CERN and FNAL pp collider /S = 10TtV [(da/dy)/a] for various Vs and L-,.,- : energy ranges. - ' mw = 83 GeV. À y - ' SPS Tevatr n 10 /s ITeVI 337 338 339 340 i i i l i i 250 - UA2 W,Z — qq" -1,200 - ai S e (a) \\(b) ï 150 - 3.8 -0.1 •n. -1.0 -0.6 -0.2 0.2 0.6 1.0 o q cos 0* cos 0* I100 - - 1 r UA1 W Z 1.5 i +i - 50 - Acceptance corrected 33 Events I i ,11, , , pz < 15 GeV/c 40 50 60 80 100 120 150 200 *- 1.0 and 0. well measured m(GeV) Fig. 14 Two-jet invariant mass distribution from UA2 [49]. The smooth curves are the results of the best fits to the QCD background alone (curve a) or including two Gaussians describing W and Z decays (curve b). -0.9 -0.6 -0.3 0 0 3 0 6 0 9 cos 0* Fig. 15 Intermediate Vector Boson decay angular distributions. The decay electron (positron) dn/d(q cos 6*) distributions are shown in (a) and (b) for the UA1 [41] and UA2 [43] data respectively for W -» ev decays with unambiguous measurements of the charge q and the angle 6* in the W rest frame with respect to the incident proton direction. Superimposed is the expectation from W decays including higher-order QCD effects. The Z decay angular distribution from UA1 [41] is displayed in (c) for the combined electron and muon data. The curves show the expectation for two values of sin 6W. 40 —I 1 1 1 T 200 120 80 50 30 (GeV) b) W transverse momentum a) UA1 W- —i—i r~ • UA1 Ef> 30GeV bl UA1 U6 W- o UA2 (p?>15 GeV only) 30 Ef> 30GeV H9 Events .20 10 8 16 24 32 40 3 50 70 90 110 1/n>r (10- /GeV) mT (GeV) c) UA1 W— TV 32 Events (LT»0) ^\ 10" ^0 80 120 PT (GeV) t,imh„\„„.„„*,i,lllin* Fig. 16 W transverse-momentum distributions from UA1 [50] (full points) and UA2 [43] 60 80 100 120 K0 (open points). Only data with pjw > 15 GeV are shown for UA2. The curves are nij (GeV) QCD predictions as reported in Ref. [50] based on calculations of Ref. [47] and extrapolated to high pjw. Fig. 17 Transverse-mass mj distributions from UA1 [41] for the leptonic W decays: a) W -> ev, b) W -> nv, and c) W -> TV. The histograms show the data and the curve shows the best fits. Background contributions are shown in the case of the W -> |iv 341 342 343 344 90 Mass (GeV) 20 40 60 80 100 120 1 1 IDT (GeV) - b) UA1 ~i r n 1 1 r 20 b) UA2 153 events background J' % \ 39 events \ (1.3 events background] i"~—^ LA. 40 60 80 100 120 140 160 N h MASS (OeV/c2) I ^•ftfl-^n Ri 0 20 30 U0 50 60 70 80 90 100 Fig. 18 Dilepton-mass distributions from UA1 [41] which were used to evaluate the Z mass m„ (GeVI + mz from (a) the Z -> e e* decay and from (b) the Z -» ji+jr decay. Fig. 19 Intermediate Vector Boson distributions form UA2 [43]. a) Transverse-mass distribution for the W -> ev channel showing all the data (histogram), the fitted W -> ev signal (dash-dotted curve), and the overall fit (solid curve) including all background contributions. b) Electron-pair-mass distribution with the expected background curve reproducing well the low-mass events. The hatched region corresponds to the sample of 25 events used in the evaluation of m,. 1 ' 1 ' 8 — n stat. error IVB masses - - Hr-^r^-^-l syst. error ~r ~r - ^^-1 * i^w-< W—— ev OJ UA1 H^-i-u W—- |iv \ . ^sssssjsn^r- » K^t^M*^! W—-TV V) If - c UA2 W—ev >OJ LiJ 2 Z —e*e' 'SSSSSI » f •-'S* \ UA1 , Z-— ti'n" i i Z —*-e*e" 1*0 60 80 100 80 90 Transverse mass (GeV/c2) m (GeV) UA2 b) . mean 0.993 rms 0.021 recalibration after 5 years 1 UF 0 4/- 20 1*0 60 80 100 0.9 1.0 e*e~(f) Invariant mass (GeV) ^expected Fig. 20 Preliminary CDF results [45]. Fig. 21 a) A comparison of the IVB-mass measurements showing in an explicit way the a) Transverse mass distribution for ev pairs, compared with a Monte Carlo statistical and systematic errors (added linearly), prediction. b) Results of a recalibration in a beam of 40 cells of the UA2 central electromagnetic b) Invariant mass distribution for e+e" or e+e"y candidates. The absence of events at calorimeter (out of 240 cells) after about 5 years of operation [42]. masses under 40 GeV is the effect of a very small acceptance due to the cuts used in the analysis. 345 346 347 348 1 ci. i. 1 1 Star, errors Stat. •• syst. errors a) -• UA1 + UA2 ^—^ •\ / / ) \^ ^\ / '' ^~— I sin20 = ^-^ / l^i -—J w 12 n 0.231*0.006 10 I ! JJK-J < Ar=0.0713 \ * ' .,— Ar=0 8 - 68% CL contours i i i 88 92 96 100 mz [GeV] Stat, errors Stat. • syst. errors h) U 12 0 231 +0.006 U0 50 60 70 60 90 100 110 120 m, IGeV/c2] 10 Fig. 23 Inclusive t (or t) production cross-section vs. mt at Vs = 0.63 and 1.8 TeV from [64]. The broken curvesand the full curves for mt > 80 GeV are QCD predictions [63] on tt production. The full _ curves for mt > 80 GeV include the contribution from W —» tb decay. 68% CL countours 92 96 100 mz (GeV) Fig. 22 Confidence contours (68% level) in the (mz, mz-mw) plane [54]. The dashed ellipses show the statistical errors and the solid ellipses show the statistical and systematic errors combined in quadrature. The hatched region corresponds to 2 sin 6W = 0.231 ± 0.006 as allowed from the average of recent neutrino scattering measurements. The solid curve is the Standard Model prediction for p = 1 with radiative corrections Ar, whereas the dash-dotted curve is the expectation ignoring all radiative corrections. a. The present UA1 [41 ] and UA2 [43] results. b. Expected precision from the future runs at the CERN pp Collider. 500 lOpb lOOpbHOp/ \/ b \/Ip b \ I i 400 v/s"= 1800 GeV —i 1 1 1 1 r Ipb vT= 630 GeV Data (23 Events) 20 Expected Contributions 300 . SUSY 16 200 12 100 ^ 100 200 300 400 500 600 cut .-r-R-n-ra- Fig. 25 Cross-sections for production of q and g at CERN (Vs = 630 GeV) <.0 BO 120 160 and FNAL (Vs = 1800 GeV), summed over q q, q g and g g AO (Jet 1 - Jet2) production, as a function of the q and g masses. Fig. 24 Distribution of A<(>, the azimuthal angle between the two highest transverse energy jets in the event, for the large missing transverse energy event sample (histogram). The dashed curve shows the expectation for conventional (standard model) processes plus background, normalised to the data. The dot-dashed curve shows the A(j> distribution (multiplied by a factor 10) for squark and gluino production for a squark mass of 60 GeV and a gluino mass of 70 GeV. From UA1 [67]. 349 350 351 0 100 200 300 1.00 500 600 700 800 900 M.., IOeV/c!) Fig. 26 Production of additional charged (W') and neutral (Z') vector bosons at CERN and FNAL as a function of their mass, assuming standard couplings to leptons and quarks. the universe, as it cools after the big bang, undergoes a series of phase trans formations. During these transitions the observed baryon asymmetry of the universe can be produced '. We can also have the production of topologically PARTICLE PHYSICS AND COSMOLOGY stable extended objects ' such as magnetic monopoles ', strings ' and domain walls . Magnetic monopoles, which exist in all Grand Unified Theories, can lead into cosmological problems ;. However, there are inflationary ; Grand Unified Models which avoid these cosmological difficulties. These models G. Lazarides solve the horizon and flatness problems of the universe. Strings produce the necessary primordial density fluctuations which lead to galaxy formation Physics Division, School of Technology whereas domain walls are ' absolutely catastrophic and should be avoided. In University of Thessaloniki some models one can have more complicated objects like walls bounded by GR-540 06 Thessaloniki strings and strings bounded by monopoles '. Under some circumstances GREECE 141 cosmic strings may behave like superconducting wires ' and provide us with a mechanism ' which can create the voids found ' in recent deep surveys of the northern sky. Galaxies are then formed by fragmentation of the shells ABSTRACT around the voids. The early stages of the universe evolution are discus Superstring inspired models lead to phase transitions in the early uni sed in the context of Grand Unified Theories and super- verse which differinmany ways from those predicted by GUTs . In particular, string inspired models. Phase transitions in the these transitions take place at cosmic temperatures on the order of the early universe and the creation of baryon asymmetry are electroweak scale rather than of a superheavy scale. As a consequence, new summarized. We review the topologically stable ex mechanisms ' for the production of the baryon asymmetry of the universe tended objects which may have been produced in the are needed. Also, the new character of the phase transitions in the early early universe. In particular, we analyze the primordial universe allows us to relax the stringent upper bound on the axion decay magnetic monopole problem and summarize the inflationary 191 20) attempts to solve it. Strings, domain walls as well as constant . Finally, in a three generation superstring model, we can have more complicated extended objects are also described. magnetic monopoles with three times the Dirac magnetic charge. These monopoles The properties and cosmological significance of super may exist in our galaxy at the level of the Parker bound. conducting cosmic strings and new possibilities for 2. STANDARD BIG BANG COSMOLOGY axions are discussed. 21 ) The standard cosmological model ' is based on the assumption that space is homogeneous and isotropic. The universe is then described by the 1. INTRODUCTION Robertson-Walker metric In recent years, the development of Grand Unified Theories (GUTs)1' of 2 strong, weak and electromagnetic interactions and of the superstring inspired 2 2 2 2 2 2 2 ds = -dt +R (t) [_^lT+r (de +sin edcp )] , (2.1) models has provided us with the necessary particle physics framework for 1-kr discussing the early stages of the universe evolution. where r,e and 353 354 355 356 neous (radial) distance between the origin and a point with coordinates r Z 2 e and ip is given by T " ÎS • I -?) Here v=(8nc/3)1/2 and M EG~1/2 - 1.2xl019GeV is the Planck mass. 'PP1I We R(t) | d% .,„ . (2.2) thus see that R «T"»t'^, and the universe expands starting at t=0 (l-kr2)1/2 (big bang)from a state with zero instantaneous distances between all its points. (Of course, this should not be taken literally since the classical All distances between objects fixed with respect to the comoving coordi theory of gravity is not expected to hold for t^tpi;Mp| - 10" sec.) nate system are proportional to the scale factor of the universe and vary Event horizon of a point is a sphere around it with radius equal with time. to the distance i„ travelled by light since the begining of time. From Einstein's field equations written in the Robertson-Walker metric eq.(2.1), we get that give (2 8) H2, (!)2 8n -^ , . ) 'H-wfffn • - = Gp (2 3 o For the radiation dominated universe, £u=2t and the event horizon grows where the dot denotes time derivative, G is the Newton's gravitational faster than the scale factor of the universe. This means that, as the constant, H is the Hubble "constant" and p the energy density of the universe expands, more and more galaxies come in causal contact with universe. For the radiation dominated era and at cosmic temperature T, each other. 2 3. PHASE TRANSITIONS IN THE EARLY UNIVERSE p =3S (V FV1" E CT4> (2-4> Grand Unified Theories ' of strong, weak and electromagnetic inter where N. (Nf) is the number of bosonic (fermionic) degrees of freedom at actions provided us with the necessary framework for discussing the T. The entropy density is given by early stages of the universe evolution. The simplest GUT model (which is, however, ruled out experimentally) is based on the gauge group - - ^rc /i, . 7,, ,x3 ,„ c. * " 7T"UV FV • t2-5' SU(5) . The symmetry breaking goes as follows, Assuming adiabatic evolution of the universe, the entropy in any comoving volume (fixed at the comoving coordinate system) must remain constant. 2 SU(5) 4 > [SU(3)xSU(2),] ®U(1)V ^ >SU(3)»U(1)_ 3 I5 c L L This means that S=R s=const. and, consequently, M %10 GeV ' M^KTGeV x W (3.1) RT = const. (2.6) Here, the succesive breakings at the GUT and the electroweak mass scale In the early universe, where R(t) is very small, the second term in the are achieved by the vacuum expectation values (VEVs) of a Higgs 24-plet right-hand side of equation (2.3) is subdominant and the curvature of the c and a Higgs 5-plet respectively. The left-handed fermions belong to universe is unimportant. The early universe can be considered essentially 5's and 10's of SU(5). flat and we can choose k=0. From eqs.(2.3) and (2.4), one then obtains GUTs together with the standard cosmological model, which is based the relation between cosmic time and temperature for the radiation domi on classical gravity, can, in principle, describe the early universe 1 -44 after the Planck time t^, 10 sec since after this time quantum nated universe, PI gauge bosons is decomposed under SU(3) xSU(2). as follows: gravitational fluctuations become unimportant. The prediction is that the universe, as it expands and cools after the big bang, undergoes a 22) series of phase transformations ' during which the original gauge 24 = (8,1)+(1,3)+(1,1) +(3,2)+(3,2) . (3.3) symmetry of grand unification breaks down in stages to the present day The first three terms correspond to the gluons, W~,Z and the photon and gauge symmetry of confining QCD and electromagnetism. For instance, in remain massless after the GUT transition. The rest (called X and Y bo the simplest SU(5) model, at cosmic time tM0~ sec.(at a critical temoe- 15 sons) acquire masses of order M and decay out of thermal equilibrium rature T MO GeV) the Higgs t starts developing its VEV and the full to a quark and a lepton or two antiquarks with branching ratio r and gauge symmetry breaks down to [SU(3) xSU(2) ]®U(l) . Later at tM0"1Csec c L y 1-r. These two channels have different baryon numbers B,=1/3 and (TMO GeV) we have the electroweak phase transition during which the Higgs B?=-2/3 respectively. The antiparticles of X and Y bosons decay to an 5-plet develops its VEV and the gauge symmetry further breaks to antiquark and an antilepton or two quarks with branching ratio r and SU(3) ®U(1) . In more complicated OUT models the universe can have a 1-r. Assuming CP violation we can have r^r and a net baryon number much richer history with many intermediate phase transitions. At proportional to (r-r)(B.-B2) can be produced in the universe. Higgs tMO" sec (TMOOMeV) the universe undergoes the confining transition ' boson decays can also be used for the same purpose instead of gauge of QCD during which the quarks get confined into hadrons. At even later boson decays. The so produced baryon asymmetry in the universe is not stages of the universe evolution, we have more transitions which, sufficiently large for the simplest SU(5) model. In more complicated however, do not belong to the realm of particle physics. Such transitions models, however, one can easily achieve the observed ratio of baryon and are nucleosynthesis at tMOO sec (TMMeV) and recombination of the photon number densities, n„/n MO -10 12 hydrogen atoms at tMO sec. During the phase transitions in the early universe topologically We will now describe the phase transition where the unifying gauge stable extended objects may be produced . These may be points,1ines or symmetry SU(5) breaks down to [SU(3)xSU(2)]®U(l) . This transition takes surfaces where the higher temperature phase is preserved by a topologi- 15 Y S) fi) place at a critical temperature T MM /g)M0 GeV and, for the time cal conservation law. They are called magnetic monopoles , strings being, we will assume that it is a second-order transition (g is the and domain walls respectively. Consider a symmetry breaking SU(5) gauge coupling). The finite temperature effective potential V(*) of the Higgs 24-plet $ has its absolute minimum at 4=0 at any T^T G —^<*~> H , (3.4) and SU{5) is unbroken. For T the homotopy group nMG/H). Thus, 35fo8 r monopoles to exist, n-(G/H) must 359 360 be non-trivial. A string (domain wall) is a deviation from the vacuum is not greater than the temperature T.i.e. localized around a closed line (surface) with thickness M , enerqy per 2 3 3 unit length (area) of order M /a(M /a) and •=(> at its central line(surfa- (Ç) E ÛV ce). They are classified by n^G/H) (nQ(G/H)) as is easily seen by a 24;) reasoning analogous to the monopole case. The T at which equality holds is called the Ginzburg temperature T_ . For T n2(G/U(l)emx...) = n^UCD^x...) = Z x .... (4.1) the Higgs correlation length at TG, where Z is the set of all integers. G h2T c 5. MOMOPOLE PRODUCTION IN THE EARLY UNIVERSE Within each such region the Higgs field is essentially aligned but its Monopoles are produced at the phase transition where the unifying values in different regions are uncorrelated. (At the boundaries of gauge symmetry SU(5) breaks down to [SU(3)xSU(2)]®U(l)v. This transition these regions, $ varies smoothly from one value to the other.) At the 15 takes place at a critical temperature T -\,(M /gHlO GeV and, for the corners where several regions meet we will occasionally have monopo- 4) time being, we will assume that it is a second-order transition. les '. Thus , the number density of monopoles produced at Tr is At temperatures of the universe below T , the difference in free energy density between the SU(5) symmetric maximum at $=0 and the asymme 6 n^pÇ^Ph ^ . (5.4) tric minimum at AVMi2^4 . (5.1) Here p is a geometric factor of order 1/10. Instead of n„, we customa rily use the "relative monopole density" For temperatures just below T , AV is very small and thermal fluctuati ons of the Higgs field * back and forth across the local maximum at r = -^ . (5.5) $=0 are very common. The condition for this to happen is the following. r The free energy needed for if to fluctuate from the minimum of VU) back to (f = 0 in a sphere of radius equal to the Higgs correlation length Ç=m[] The above estimate, then, gives for the initial relative monopole densi T > T -v 1012 GeV (6.2) ty, f 6 For T tainties in the values of h and TG. It is thus important to find a lower bound on r- which is more general and does not depend on so many 9 9 If r.n > 10" then rf1n ,. 10' , details. Whatever the details of the transition are, the Higgs field cannot be correlated over distances greater than the particle horizon 9 Ifrin<10- then rfip -, r-n . (6.3) 2t (t is the cosmic time) at Tr. So, the linear dimension of the This result together with the causality bound on the initial monopole regions in which $ is aligned cannot be bigger than 2t at Tr. We, thus, derive the causality bound ' on the initial monopole density density implies 10 rfin>10- . (6.4) (5.7) ^ ^(2t)3 or, equivalently, 7. OBSERVATIONAL BOUNDS ON r 10 rin>10' . (5.8) We can obtain a bound on the present value of r from the require ment that the mass density due to monopoles does not exceed the limit on the mass density of the universe imposed by the observed values of 81 6. SUBSEQUENT MONOPOLE ANNIHILATION the Hubble constant and the deceleration parameter. This bound reads ; After monopole production at T , the monopole number density 24 r r < 10~ . (7.1) follows the equation ' now ^ So, there is a big discrepancy between this constraint and our previous theoretical estimates. "dt = " DnM " 3R nM • (6-n However, one may argue that, since we do not know the details of The second term in the RHS of this equation describes the dilution of monopole history in the late universe, the above observational bound on monopoles due to the cosmoiogical expansion. The first term describes r cannot be compared with our theoretical estimates. (Monopoles may monopole-antimonopole annihilation. The annihilation process is cnara- have been accumulated in special places such as cores of galaxies or cterized by the coefficient D and has been studied by Preskill . The massive stars where they have beer, annihilated.) monopoles diffuse towards antimonopoles through the plasma of light There is another limit on r which does not depend on the details charged particles, capture each other in Bohr orbits end finally anni of the late universe. The standard scenario of nucleosynthesis requires hilate. This process goes on as long as the mean free path of the that the monopoles do not dominate the energy density of the universe 81 monopole is shorter than the capture distance, i.e., at T-lMeV. This yields the cosmoiogical bound ' 361 362 363 364 vacuum never coalesce and the transition is not completed . Even if rfT^lMeV^lO"19 , (7.2) we achieve completion of the transition, we end up with a very inhomo- geneous universe. which is still in great disagreement with our theoretical predictions. 271 281 GUIs are in trouble. They predict too many magnetic monopoles in Linde as well as Albrecht and Steinhardt ' have suggested a the universe, at least, within the standard hot big bang cosmology. very clever variant of the above scenario (new inflation). They consider SU(5) with the CW potential and they observe that is also suppressed. Unfortunately, this scenario has many problems. As Hawking temperature TH=H/2ir\.10 -10 GeV the H2 H (i)2 = *L Gp . (8. 1) ry models. Reheating is a little problem for this model since after the c 12 rolling down of <, this field oscillates with frequency m -vlO GeV. This is solved by Efficient reheating can be achieved through production of colour triplets in H. This requires that the mass of these triplets is 1/2 R(t) = exp(xt) , x = (^Go0) . (8.2; 11 12 ^10 GeV-10 GeV. This can be arranged but it requires an extra but and leads to exponential expansion (inflation) of the parts of the relatively mild fine tuning. universe that are in the false vacuum. Then the bubbles of the true Apart from the primordial magnetic monopole problem, inflation also solves two long standing problems of the standard cosmological model, Spin(10) ^ >Ho;Spin(6)®Spin(4) = SL'(4)c®[SU(2)LxSU(2)R] namely the horizon and the flatness problem. The horizon problem can be 15 K -vl0 GeV summarized as follows. Observations on the background cosmic radiation x of 2.7 K have shown that this radiation is isotropic to a very good 126 accuracy. Indeed, the experimental bound on temperature fluctuations in [SU(3)cxSU(2)L]fi>U(l)Y. (8.5) the cosmic radiation from various directions in the sky is 5T/T< 10"^-10"^ M -\,1012GeV But the radiation we observe now on earth has been emitted at recombina tion (t^lO yr.) from regions in the universe that they were not yet in Inflation is achieved again by a gauge singlet Higgs field $ with a VEV causal contact. The problem is, then, how could these regions be in of order 10 GeV. Durinq the first stage of symmetry breaking we have 34) thermal equilibrium with each other. Inflation, of course, solves na the production of Z~ magnetic monopoles '. This is due to the fact that the (-1,-1,-1) element of the Pati-Salam subgroup H = SU(4) «[su(2),x turally this problem since a period of exponential expansion can Q SU(2) ] of Spin(10) leaves invariant the 16=(4,2,1)+(4,1,2) and, hence, thermalize the universe in great distances (the horizon length t also D grows exponentially during inflation as is easily seen from eq.(2.8)). all the representations of Spin(10). It is, therefore, indistinguishable The flatness problem is related to the observation that the present day fror^ the identity transformation and should be identified with it. The density p of the universe is very close to the critical density secor.d horotopy group of Spin(10)/H is, then, given by 3H2 , „, n2(Spin(10)/Ho)=n1(Ho) = Z2 . (8.6) P c= 8nG > (0.3) which corresponds to spatially flat universe (H is the present day The non trivial element of this Z„ corresponds to loops which interpolate Hubble constant). From eq.(2.3), one obtains between (1,1,1) and (-1,-1,-1) in H . These Z~ monopoles have one unit of Dirac charge and are inflated away. In the second step of symmetry brea 2 2 king in ea.(8.5), we have production of "lighter" monopoles which carry ¥-'-^ *« «T- , (8.4) " SnGpR" two units of Dirac charge. These Schwinger monopoles are diluted but not completely inflated away since they are produced late in the inflationa- 35) where Q=p/p . Extrapolating from the present bounds on Q, one finds that ry phase. They can occur at or below the Parker bound 'from the galactic in the early universe C must have been very close to one. For example,at magnetic fields and their flux in our galaxy is possibly measurable.Note T-vlO GeV, 0-1+0(10" ). The problem is why the universe started being that these Schwinger monopoles do not catalyze baryon decay and, spatially flat to such an accuracy. Inflation provides a natural solu therefore, do not have to satisfy the more stringent neutron star tion since it flattens up the universe by exponentially expanding it. bound37'. One can show that in the model of ref.(31) one gets enough inflation to The most interesting solution to the primordial monopole problem is solve the horizon and flatness problems. certainly new inflation. There are models that can implement this solu 33) tion but none of them is totally satisfactory. Moreover ordinary new Finally, Lazarides and Shafi ~J> have proposed a successful inflatio inflation leads to the unfortunate conclusion that there are no monopo nary model which solves the monopole problem but leads to a possibly les in the universe. It is, therefore, interesting to know that there measurable monopole density in contrast to the other new inflationary models which lead to a complete absence of monopoles. The model is based are also new inflationary models that lead to a may be measurable mono- on Spin(10) (the universal covering of S0(10)) which breaks to the pole density. gauge group Of the standard model in two steps: 365 366 3b7 368 39) A very simple and elegant example of a string producing theory 9. COSMIC STRINGS AND DOMAIN WALLS is based on the gauge group Spin(10). The symmetry breaking goes as The existence of strings in Grand Unified Theories is not unavoid fol lows: able as the existence of magnetic monopoles but, as we will see, one can M M readily construct GUT models with strings. As suggested by Zel 'dovich10', SpindOj-jfg-» SU(5)xZ2-3|-> [SU(3)cxSU(2)]flU(D they may be of great cosmological importance. They have the right prope rties needed to produce the primordial density fluctuations that lead to galaxy formation. Consider a gauge group G which breaks down to H by x Z2 -jjlj-» SU(3)c 8 U(l)em x Z2. (9.1) the vacuum expectation value of a scalar field ction, we must look at the topology of the vacuum manifold V=G/H.Strings discrete factor Z?=(l,-1) contained in the centre of Spin(10) which is a are produced iff the fundamental group n.(G/H) is non-trivial, i.e.,iff I. subqrouo generated by iT . Here F" =i r P ...T is the "chirality there exist loops in G/H that cannot be continuously deformed down to a operator" and f1's are the generalized Dirac Matrices in 10 dimensions. point. Sucn loops are called homotopically non-trivial. A strinq is a One can show that n.(Spin (10)/SU(5)x Z„) = Z?. Thus a superheavy string -l -?g tube of thickness d^M '-10 cm. Outside this tube <;>E G/H. AS one network is produced during this phase transition. The strings remain un describes a loop around the string, «j > describes a non-trivial loop in affected down to T=0 since the Z~ never breaks. G/H. This guarantees the topological stability of the strinq. On the Domain Walls.7 )' can appear if n (G/H) is non-trivial. In this case the string axis, <^>=0 and the symmetry G is unbroken. The string carries vacuum manifold G/H is not connected. This happens when a discrete symmetry ? ?n is spontaneously broken. It is very easy to construct models with this an energy per unit length o-vM /cMO gm/cm. -35 At a cosmic time t~40 sec or at a critical temperature property. 15 Tc'JVg--10 GeV, G breaks down to H and a network of superheavy strings Domain Walls in the universe lead to gravitational collapse and, thus, is produced. Initially, the scale of this network (i.e., the mean dis they are absolutely catastrophic . They provide an important constraint on tance between two neighbouring strings) is -vM . It, then, grows very model building: One should either make sure that there are no spontaneously rapidly and soon becomes of the order of the particle horizon t and broken discrete symmetries in the theory or arrange the model so that the remains so thereafter '. we, thus, essentially have one string piece walls are inflated away. 3 per horizon volume ^t . This produces a density fluctuation Bp/p-v.p /p , Topologically unstable closed or open strings with magnetic monopoles where p is the radiation energy density and p the energy density due at their ends and/or topologically unstable domain walls bounded by strings -3 to strings. This fluctuation has the right magnitude ^10 for galaxy can also be produced in some theories ' 12 10. SUPERCONDUCTING STRINGS formation. 5p/p remains constant until decoupling at td^10 sec; only the scale of the fluctuation grows as t. After matter domination and Few vears aqo, Witten ' suggested that cosmic strings may, under certain 15) decoupling 5p/p grows as t , becomes -\.l at t^lO sec, and galaxies circumstances, become superconducting. Ostriker, Thompson and Witten had, are formed. subsequently, argued that such superconducting strings could explain the The space-time about gauge strings has been found to be flat, with a creation of the voids found ' in deep surveys of the northern sky. Galaxies conical defect at the core and a deficit angle 6(p^Go^lO" around the string can then be formed by fragmentation of the shells around the voids. The idea So strings are expected to behave like gravitational lenses producing double of this mechanism can be summarized as follows. We must first assume that there images of distant astronomical objects. are primordial magnetic fields in the universe. Then any closed superconducting cosmic string, which has been produced in the early universe during some phase transition, encloses a certain amount of primordial magnetic flux. This flux faints away by cosmological expansion and a supercurrent j is induced on The Dirac equation becomes the string. Subsequently, as the string shrinks down, its current increases till it reaches a maximal value j , and the energy losses of the oscillating (ft-f^Wz.tH) , (10.4) max loop through electromagnetic radiation soon overtake the energy losses through which gives gravitational radiation. The frequency of this electromagnetic radiation a(z,t) = f(z+t) . turns out to be much smaller than the plasma frequency at that time. Thus,the Thus the fermions are trapped in transverse zero modes that travel in the -z surrounding plasma gets heated and expands creating a cosmic void. In order direction with the velocity of light and are called left-movers (L-movers). to obtain the observed radius, filling factor and other characteristics of the For an one dimensional observer living on the string, these are massless left- voids, we must have superconducting strings with mass scale M ^ lO^-^-lO^ 7 handed chiral fermions (v°Y a(z,t)=-a). By the same token, one can show that GeV which can carry a maximal current jmax ^ M with lifetime T ^ lO* years. left-handed fermions which obtain masses by coupling to ip* instead of V get It has been shown^' that superstring inspired models with superconducting trapped in transverse zero modes travelling in the +z direction (R-movers). cosmic strings satisfying these requirements can be constructed. The effective field theory describing charged fermions trapped in transverse Let us now discuss the physics of string superconductivity for a minimal 14) zero modes on a string, in the simplest anomaly free case with one L-mover and string where a scalar field ip changes phase by 2n as we go around the string. one R-mover with the same electric charge q, is given by the action The string can become superconducting if, for example, there are charged fermi ons trapped in lognitudinal zero modes along the string '. Jackiw and Rossi ' 4 MV 10 5 I=- -J fd xFMvF + (dzdtiKz.thVD^z.t) • < - > have shown that such modes can arise in models of interest. Consider a toy model ' based on the group U(l)Qx U(1)R, where U(l)., is the Here ip(z,t) is a two dimensional Dirac field and i=0,3. By using the technique usual gauge group of electromagnetism and the extra U(1)R can be either local of bosonization, or global. U(1) can be broken to {1) by the VEV of a scalar field ip with R $vV- E1j3,*(z,t), (10.b) charges Q=0 and R=1. This model contains topologically stable strings. Now /n J introduce left-handed fermions ik and Xi w^h charges CL=q, R.=r and Q =-q, eq. (10.5) becomes R =-r-l respectively. These fermions obtain their masses through coupling to nates and the absolute minimum of the potential VT( where M. is the four dimensional Minkowski space-time and K is a six-dimensio dVT(4>) * + 3Hi=- —^— , (11.2) nal CY manifold (i.e.,a three complex dimensional analytic manifold with a Kâhler metric and SU(3) holonomy). By introducing a background SU(3) ( with 2 -7 where the dots denote time derivatives and H=u /M ,^10 GeV< -? -1 9 9 9 r q3 , T "»f M , with the time 5t needed for $ to gro w from 4M vto *'M*. We ' g g ' *• 3 cg produced at or before the phase transition. The only noticable effect of find this refinement is for axions which are produced during the decay of the <*> field and experience only a small part of the total dilution. The result is M*=T (l f"2M /T ) . (11.6) c + s c that f,^10 -10 GeV and, if this inequality is saturated, axions can be 19) an appreciable component of the dark matter in the universe ;. For f=1/5, M/T =1/3, the above equation gives M*=10T and the g's decay out Finally, I would like to argued that superstring models can predict the 20) of thermal equilibrium producing a baryon asymmetry. In specific models ', existence of magnetic monopoles with a perhaps measurable flux . As an 54) one finds that (nE/n Jinitial ^10" . This baryon asymmetry is, subsequently, example, let us consider the three generation superstring model based on 55) diluted by a factor û'tlO because of the entropy production following the the CY manifold constructed by Tian and Yau '. The four dimensional gauge decay of the ; field. Thus, we finally obtain nR/n MO group is G=SU(3)cxSU(3)LxSU(3)R and breaks to H=SU(3)cxSU(2)Lxll(l)Y at an 9 19 For M.-MO GeV, the dilution factor is big and no baryon asymmetry that intermediate mass scale MTM0l5GeV. It has been shown ' that, during the 1 Mi was produced before the decay of $ survives. In this case we must produce the phase transition at T MTeV where G —>-H, there is production of magnetic baryon asymmetry after the completion of the transition. Possible mechanisms monopoles with magnetic charge three times bigger than the minimal Dirac for this are discussed in ref. (18). magnetic charge. These monopoles have mass of order 10 GeV and are not 48) We now turn to the discussion of the strong CP problem ' and the expected to catalyze36' proton decay since, among other things, they are not 56) axions ' in superstring inspired models. The best known way to solve the even spherically symmetric . 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