On the Application of Conformal Symmetry to Quantum Field Theory*

To leading order in CX~ ( Q2), conformal symmetry specifies the eigensolutions of the evolution equation for meson distribution amplitudes, the wavefunctions which control large-momentum-transfer exclusive mesonic processes in &CD. We find that at next to leading order, the eigensolutions in various field theories depend on the regularization scheme, even for zero p-function. This is contrary to the expectations of conformal symmetry. Conformal symmetry expresses the extended invariances of the Lagrangian of a renormalizable Lorentz-invariant theory which has no intrinsic length scale [ 11. Classical relativistic field theories which are scale invariant and have a renormalizable renormalizable Lagrangian are also conformal symmetric; i.e., invariant under the conformal group, which consists of the translations, boosts and rotations of the Poincare’ group, together with dilatations (zp -+ Xsp) and conformal transformations; i.e., inversion(xp + -xp/x2) X translation X inversion. Scale invariance, and therefore conformal symmetry, is destroyed in quantum field theory by masses, and by the renormalization procedure, which inevitably introduces a renormalization scale. In a field theory with zero P-function (e.g. QCD to one loop-order with Nf = y N,), the coupling constant cannot introduce any scale dependence, so one expects conformal symmetry (with possibly anomalous dimensions for the fields) to be valid at short distances where mass effects are negligible. A general, all orders, proof that conformal symmetry is satisfied asymptotically in renormalizable field theories with zero P-function has in fact been given by Parisi [2]. Th is result, however, may only be true for specific ultraviolet regulators(see below). In this letter we consider the -application of conformal symmetry to the operator product expansion of two fields at short distances, +(;) $(-?l) 2 en(z2 iczg) 2 ri)Z,, Zam O(n)al"'ama(0) (1) n m=n