REGNIRPS STCART NI NREDOM SCISYHP Ergebnisse der exakten Natur- wissenschaften 76 Volume Editor. .G H6hler Associate Editor: E.A. Niekisch Editorial Board. .S FIQgge .J Hamilton .F Hund .H Lehmann .G Leibfried .W Paul Springer-Verlag Berlin Heidelberg New York 3791 2 noitcudortnI 1. Introduction The crucial role of short distance behaviour in elementary particle theory si widely recognized. In the last years the experiments at SLAC on deep inelastic electron scattering 1, ,51 ,62 153, 130 have dramaticaly evidenced the fundamental role played by short-distance behaviour. In this case one si indeed exploring the behaviour at short distance of a product of two electromagnetic currents. Deep inelastic neutrino reac- tions 30, 125, ,93 134 are similarly related to the product of two weak currents. The forthcoming developments at NAL will provide addi- tional stimulus and a new bunch of data on a wider range. The short distance behaviour of an operator product of e.m. or weak currents ,si of course, not only important for deep inelastic scat- tering, but also plays an important role in the theoretical description of other phenomena (such as for instance non-leptonic decays, electron pair annihilation, etc.). Wilson 159 has given a deep and original discussion of the operator product expansion and of its applications. Wilson's expansion for the product of two currents has the form j~(x)jv(y ) = ~ C~v(X -- y) On(y ) . )1.1( n In Eq. (1.1) c"~v(x- y) are c-numbers and On(y) are local operators. The c-numbers are generally singular at short-distance. The expansion has been discussed in different theoretical frameworks, such as perturbation theory ,02- 163, 164, 159, 165 and in model field theories. There has been particular discussion of operator expansion in the Thirring model 154, 103, 159, 160, 117, 50. The conclusions strongly support the conjecture of the existence of an operator product expansion. In his important paper 159 Wilson expecially emphasized the idea of using the concept of operator product expansion together with the assumption of scale invariance at short distances. Scale invariance has been invoked since many years in elementary particle theory 106, ,801 ,811 119, 159. In the expansion in Eq. (1.1) scale invariance allows for the deter- mination of the degree of singularity of the coefficient c~v(x- .)y The determination si based simply on comparison of the "scale dimensions" of the left- and right-hand-sides in Eq. (1.1). It si then of preliminary importance to know whether the operators nO have definite "scale dimensions" or not. And, in case they have definite "scale dimensions", to know their actual values. Field theory investigations on this point have been very intricate, difficult, and not yet conclusive 31, ,34 ,061 148-151, ,44 ,19 161. On the other hand the set of operators nO is expected to contain at least some operator of well defined dimension, such as the currents Introduction 3 (weak and electromagnetic, or, more generally, the unitary currents), and the stress-energy tensor. The currents have dimensions 3 in mass units and the stress-energy tensor has dimension .4 The existence (necessary in any local covariant theory) of a stress tensor vu0 of definite dimension 4 has led to the suggestion that the scaling observed at SLAC is in fact explainable on the basis of the presence of 0,v among the operators O, in the right-hand-side of Eq. (1.1) 40. The most simple explanation in terms of a single tensor of dimension 4, namely 0,v, seems in conflict with the different scaling of neutron and proton data. A more detailed argument making use of the universality of the coupling of vu0 can however be developed and provides a more general justifica- tion 120 (a point which has particularly been emphasized by Wilson 1623). Once scaling is accepted, as it must anyway at least approximately because of experiment, it can be shown by different arguments that a subset of operators O, must exist, in Eq. (1.1), with dimensions d related to their spin by d = s + .2 Among them is the energy-momentum tensor with s = 2, and d = .4 A simple way to demonstrate such a statement is from the con- vergence of the Bjorken-Callan-Gross sum rules 31, ,31 33. Following the general treatment by Bjorken 13 one has the sum rules 2 f~ = y dco~"V,(co) o )/p ip (1.2.) O(xj,)O,x(xj_~lzpI , O n 2im 5- (- i)n j)O 1 d3 x 2/2q ,v for each n = ,1 ,3 ,5 ... In Eq. (1.2) oc is the scaling variable, oc = - in the usual notations, and Ft(co ) is the transverse scaling function; jx is the x-component of the e.m. current and p = (P0, 0, 0, )zP is the nucleon momentum. The positivity of F,(co) ensures that each momentum inte- gral f, in Eq. (1.2) is positive and that the f,'s form a monotonic de- creasing sequence. The relation d = s + 2 comes out in the following way. ~j,"tO/~j"?~ In Eq. (1.2) the matrix element of carries n + 1 momenta. The leading term in the commutator is then contributed by some opera- tor, which we call Oul.. u, + ,, which has spin s = n + 1 and dimensions, in mass units, d = n + .3 Thus d = s + ,2 that is, one has a particular relation to satisfy between spin and dimension. The relation would indeed easily be satisfied in a quark model through the (uncritical) assertion that each local operator keeps its "canonical" dimensions, as in free field theory, where such operators are constructed as Wick products of free fields. But the situation is not expected to be so simple in renormalized field theories 159. The problem is indeed a complicated dynamical one 31, 43, 160, 148-151, 44, 91, 161. 4 Introduction From work of Callan, Coleman and Symanzik (see Refs. 31, 43, 148-151, 44) it appears that, under some assumptions, one obtains renormalized dimensions (i.e. not necessarily canonical dimensions, but at least definite dimensions) in the so-called deep euclidean region and possibly also in the situation when the four-vector x,-y, vanishes (the tip of the light-cone). The extension of such arguments to the whole light-cone is not yet known. For the operators relevant to an expansion on the whole light-cone 21, ,22 ,92 95, 23, ,17 ,18 ,28 100, 114, 115, 25, ,27 167, ,37 147, ,78 45 one would have to prove much more, than just definite dimensions, namely, the relation d = s + 2 (canonical dimensions). The argument showing that the scaling behaviour results from the configuration space behaviour on the light cone is well known (see for instance Ref. 95). In the Fourier transform S eiqx P< ju(x),j~(0) p) d4x )3.1( of the current correlation function one can write the exponent asymp- totically as qx = vx o - x3(v + talon) by choosing the rest frame for the nucleon (p = ,m( ))0 and the z direction along the vector q. For v ~ eo the integrand at oX ~ x3 gives the domi- nant contribution, and because of the causal structure of the com- mutator this means that one is indeed exploring the light-cone. (For discussion on the relations between the p ~ oe method with equal time commutators and the light cone, see for instance Refs. 113, ,2 102, ,211 45.) In fact one is exploring the whole light cone and not only its tip. Note however that the positivity conditions play a fundamental role in that they allow to formulate restrictions on the operator product expansion, only by making use of its behaviour near the tip of the light- cone. This shows the power of the equal-time-commutator approach 33, 46, ,31 ,43 47 and suggests the predominant role of the stress- energy tensor in the problem of scaling 40, 120. When the naive quark commutators are assumed 73, 45, 87 both approaches, the one using equal time commutators or small distance behaviour of operator products, and the one postulating a quark-model light-cone expansion (possibly including gluon coupling 11, 24, ,37 87, 45) reproduce a full set of testable relations 116, 121, 73, 45, ,78 49, which show the almost complete identity of such models to the more explicit parton model by Feynman 68, 69 and Bjorken 12, 14 (see Refs. 52-60, ,38 136, 137, 171). In the parton model the virtual photon interacts very rapidly with the constituent as compared to the typical interaction time among constituents, which can therefore be assimilated to free particles. The parton model predictions are however more de- tailed in that they are based on particular parton distributions. Introduction 5 The approach using the algebraic concept of scale invariance was advocated by Wilson in connection to operator product expansion 159 and soon after adopted in the theory of deep electroproduction 40, ,021 ,97 ,08 152. Previous work on the algebraic notion of dilatation in- variance had been developed earlier on the basis of independent theo- retical speculations 106, 108, 118, 119, 159. The extension from dilatations to the entire conformal algebra (see Refs. ,15 ,98 126, 127, 158, ,47 ,57 104, ,09 105, ,73 107, 109, ,83 122, ,23 ,81 ,88 152, ,53 1411) may be justified on the basis of a number of reasons (none of them being however compelling): )i( Lagrangian field theories which are formally invariant under dilatations are often invariant also under special conformal transforma- tions. For instance, a sufficient condition (but not necessary) si the absence of derivative couplings. (ii) Conformal transformations leave the light-cone invariant. (iii) The conformal algebra provides for a natural homogeneization of the inhomogeneous Poincar6 algebra. The algebraic implications of conformal invariance on the light cone were studied, in Ref. 41, within the formalism using equal-time com- mutators. The requirement of covariance under the infinitesimal gen- erators of SU(2,2) (the covering group of the conformal group) can directly be imposed on an operator product expansion on the light-cone 63, ,46 66. To such purpose one has first to analyze the transformation properties of the infinite set of local operators which provide a basis for the operator expansion 641. It will be interesting to preliminarily illustrate two aspects of light- cone expansions which finally turn out to be (rather misteriously at first sight) connected to conformal invariance. They are: )a( causality, and, )b( translation invariance on a hermitean basis. Let us consider the expansion A(x) )O(B = ~ c,(x) O,(0) )4.1( n where c,(x) are c-number functions and O,(x) form a complete set, extending the concepts and definitions of Wilson's work 159. When we commute with some arbitrary local operators C(y) we obtain A(x), C(y) B(0) + A(x) B(0), C(y) = ~ c,(x) O.(0), C(y) )5.1( n One notes that for uY spacelike, that is for y2< ,0 each commutator O,(0), C(y) vanishes. So, taking 2y < ,0 each term on the right hand side vanishes, whereas on the left hand side B(0), C(y) vanishes, but not necessarily does so the first term, A(x), C(y) B(0). The latter term vanishes if in addition (x - 2)y < ,0 which, on the light cone amounts to requiring y2< 2xy. There si no paradox, because of the infinite sum- 6 Introduction mation on the right-hand-side. However it would be better to have an improved operator product expansion which formally exhibits the causality properties in each of its terms. This is the problem we have indicated under (a). Let us now show what the problem specified under (b) is. Again let us consider a light cone expansion of the form A(x) B(O) = (- x 2 + ixo)-' ~ l~x ... x "~ 1~O ..... (0). (1.6) tl The tensors ~O ...... are symmetric traceless tensors. They can always be chosen to be hermitean 159, 25. The commutator A(x), B(0) has then the correct support required by causality. On Eq. (1.6) one makes a number of algebraic steps. One first translates by -x, then changes x into -x, expands ~O ...... (x) in a power series around x =0, and finally takes the hermitean conjugate. The expression one finds can be com- pared with the expression one had started from, Eq. (1.6), and one dis- covers that to obtain consistency the following infinite set of relations has to be satisfied ,~O .~.(x)= ~_-- (- m)-l r ,~?~ ~O .... (x). (1.7) "' ...=0 (n-m)! '"" "" It is interesting that, for n = odd, Eq. (1.7) tells that O, ...... is a sum of derivative operators (which therefore have vanishing forward matrix elements). It turns out that the imposition, to the general operator product expansion on the light-cone, of the requirement of covariance under infinitesimal special conformal transformations results in a very stringent set of limitations 63, 64, 66. They essentially amount to fixing the relative coefficients in the expansion of each derivative term c~+, ... a~~ ..... (x) with respect to the non-derivative term ~O ...... (x). The expansion in- cluding such restrictions is found to exhibit a very compact form in terms of a confluent hypergeometric function. The interesting circum- stance becomes then apparent, that the two problems we have mentioned under the headings (a) and (b) above, are in fact automatically solved with the new form of the operator product expansion, essentially reducing to some known properties of the confluent hypergeometric function. That is, the imposition of conformal invariance directly eliminates the two problems of causality support and translation invariance on a her- mitean basis (the reverse however is not true). We have already discussed why the imposition of conformal sym- metry on the light cone seems to be a reasonable requirement. At this stage one is working within well-defined limitations: )i( one only deals with infinitesimal conformal transformations, and (ii) the symmetry is supposed to hold only on the light cone, but not necessarily for the noitcudortnI 7 complete theory. If conformal invariance is predominantly spontane- ously broken, then the requirement of covariance of the operator product expansion under the conformal group may correspondingly enjoy of a larger domain of validity. A most elegant derivation of a manifestly conformal covariant oper- ator product expansion can be given by exploiting the isomorphism between the conformal algebra and the 0(4,2) orthogonal algebra. The derivation is uniquely and in a straightforward way extensible off the light-cone onto the entire space-time. Such expansions, manifestly con- formal-covariant over the entire space-time, should apply to the skeleton theory, in Wilson's sense t59, provided it enjoys of the property of conformal invariance, beyond the postulated scale invariance. As we have said, simple Lagrangian theories which are invariant under dilata- tions turn out, under some general assumption to be also invariant under special conformal transformations; this may justify the hypothesis of a skeleton theory which is fully conformally invariant. An important result appears to be a general theorem 63, 67, which we call the theorem of spin-dimension correlation, which exhibits the dynamical content of the relation d = s + ,2 equivalent to the requirement of scaling in the Bjorken limit, in terms of a set of generalized partial conservation equations. By this we mean that the divergences of those tensors ~O ....... which contribute to the structure functions in the Bjorken limit, are annihilated by the generators of special conformal transformations .~K Of course, the special situation provided by the quark model commutators is a particular case of the above results, which can be regarded as a statement of the necessary and sufficient conditions for scaling (through the mechanism of canonical dimension for the relevant set of operators). The manifestly covariant formulation in a six-dimensional coordinate space, based on the isomorphism with 0(4,2), is particularly useful in allowing for a direct and simple construction of vacuum expectation values of field products 67. Among these, in particular, the three-point- functions, which are completely fixed (except from a constant) 128, 138, 145, are directly related to the covariant form of the operator product expansion. Precisely, the two problems, of constructing a conformally covariant operator product expansion, and of constructing the general covariant three-point-function are essentially equivalent formulations of the same problem 128, ,56 66. This holds both on the light cone 128, 65 and off the light cone 66. More generally, conformal co- variance restricts the form of the n-point function, although only for n = 3 is the restriction capable of a unique prediction (apart from a con- stant). The simplest case, n = ,2 i.e. the correlation function for two local operators, appears to be subject to very stringent limitations, which take 8 Introduction to the Conformal Group in Space-Time the form of selection rules. That is, the two-point function vanishes unless the spins and dimensions are correlated in a precise way, if one assumes only conformal invariance on the light-cone; and it vanishes quite generally unless the operators have same spin and dimension, under the stronger assumption of full conformal covariance. The implications of these results seem rather powerful, as they point out to strongly limited possibilities for a conformally invariant skeleton theory. In addition to these restrictions one has to recall that the cau- sality limitations for the commutator of two local observables imposes rather strict constraints to a theory invariant under finite conformal transformations. For this problem however we refer to a comprehensive investigation by Kastrup et al. 110. Furthermore, whenever gauge invariance constraints apply to con- formaly covariant vacuum expectation values of products of local observables one generally obtains a stronger set of selection rules 1283. It therefore appears that much work has still to be carried out in order to develop a comprehensive understanding of the structure of a (broken) conformally invariant theory. 2. Introduction to the Conformal Group in Space-Time 2.1. The Conformal Group The conformal group provides for an extension of the Poincar~ group into a higher dimension homogeneous orthogonal group (see Refs. 51, 89, 126, 127, 133, 158, 92, 74, 75, 104-109, 90, ,61 37, 70, 38, 122, 123, 32, 88, 152, 35). The conformal generators JA~(A,B=O, 1 ... 6) satisfy the commutation relations JAg, Jc D = i(gA~J.c + gBCJAD -- gACJ.D -- )CAJ)ZBg (2.1) where gAB is diagonal with gAa = ( + , +). One has JAB = -- ABJ giving a total of 51 independent generators. In terms of the 10(3, 1)- generators M.., Pu, plus the new generators D and K u one has the correspondence 3 5 6 0 M#v (2.2) BAJ = 3 ~ I,,I K-~,P s The Conformal Group 9 The algebra satisfied by the generators, Eq. (2.1), is the 0(4,2) algebra. It is isomorphic to the spinor algebra SU(2,2). Written in terms of Muv, P,, D, and K s the commutation relations in Eq. (2.1) become: Poincar6 subalgebra (2.3) M,~,P~=i(gvoP,-gu~P 0 P,, P,, = 0. Lorentz behaviour of D (dilation generator) M,~, D = 0 (i.e., D is Lorentz scalar), (2.4) P,, D = iP, (P, acts as a step-up operator (2.5) with respect to D). Lorentz behaviour of K, (special conformal generator) Mu~, Ko = - i(gQu K~ - ~Qg K~) (K s is a four-vector), (2.6) P., K~ = i2(gu~D - MuO . (2.7) K u - D commutators K., K3 = o, (2.8) D, K. = i K u (K u acts as a step-down operator (2.9) with respect to D). Linear realizations of the group of conformal transformations can be obtained in six-dimensions. The transformations are those which leave invariant a bilinear form with metric (+ , +). In four dimensions (Minkowski space) the realization is non-linear. One has: x'. = a. + A~ ~x (generators of infinitesimal transformations: (2.10) Pu, M.O, x'. = e;~xu 2 real (dilatations: generator D), (2.1 l) xtt + ctt 2X ~'x - 1 + 2c~x ~ + cZx 2 ' c. real (special conformal trans- (2.12) formations: generators K.). The independent parameters are: 4(a~) + 6(A~) + 1 (2) + 4(c.) = 15. Special conformal transformations can be thought as products of (inversion) x (translation) x (inversion): X u/X 2 -- #C Xlz -t" CuX 2 xu~ x./x 2--* xu/x 2 + c.~ (xu/x2 + c.)2 = 1 + 2cx + c2x 2 The group has two abelian subgroups: one generated by P. the other by K s. It has two Poincar6 subalgebras: (Mu~, P.) and (M.~, K.) (2.13) 01 noitcudortnI to eht lamrofnoC Group ni emiT-ecapS 2.2. Mass Spectrum From P., D = iP. one has p2, D = pu Pu, O + P", D ~P = 2iP 2 . (2.14) Suppose there is a discrete state of mass m whose normalized ket is P), where p2= m .2 From Eq. (2.14) (P I p2, D p) = 2i<pl p2 p) = 2ira 2 , <pl PZ, Dlp) = (pm2 D - Dm2lp) =O. Thus: m 2= .0 Discrete massive states are impossible, unless the sym- metry is broken. Continuum massive states can however exist. The argument here is similar to that of ordinary quantum mechanics showing that the spectrum of p is continuus from q, p = .i Again <p'l q,PIP') = 0 = i(p'lp') ; however (p'lqp-pqlp") = (p"-p')(p'tqlp") -- i(p'lp") = i6(p'- p") showing that q = i Ofi3p. Indeed (p" - p') i(~/Op") (P'IP") = i(p" - p') (,3/Op") 6(p' - p") = i6(p' - p") . 2.3. Representations One uses the method of induced representations 124, 93, 1013 (Mackey). Let us call G anyone of the generators of the stability subgroup at x, = 0 : G -- (M,,., D, K,). The remaining generator is ,~P which acts like Pu, pc (x) l = - i ?( u oq (x) . )51.2( To evaluate ~o(x), G we use Eq. (2.15) ,)x(o~ G = e iex 9(0) e-ie~, G -- e iex 9(0), G e-'ex = e-iVXGe iPx . )61.2( Let us calculate G. One has = Z (- i)" xu" p.~, pu= P~'~ G, . n leiden-" X'u~ . . . . . . . . . If G=- Mu~, M~ = M. -ix., P"', M.~ - ~' x.~ xu; U", P"=, M~,v, + .'. )71.2( = - x., P. - g.1 Pv) = (xv P. - x. Pv). If G=_D L) = D - ix., PU', D - (cid:1)89 x., x~,;P"' P"=,D,+ .... D + xz P ~ . (2.18) If G~K, P~ = K. - ix.~ P", K,, -~ X//l 2~C)" P"', P"=, K,,, + ...