Renormalization of local polynomials. Short-distance expansion (SDE)

This chapter discusses two related topics: the renormalization of local polynomials (or local operators) of the field, and the short-distance expansion (SDE) of the product of local operators, in space dimension 4 for simplicity. Both problems are related, since one can consider the insertion of a product of operators at different point as a regularization by point splitting of the product at the same points. Therefore, in the limit of coinciding points, one expects that the product is dominated by a linear combination of the local operators which appear in the renormalization of the product, with singular coefficients, functions of the separation, replacing the usual cut-off dependent renormalization constants. We first discuss the renormalization of local polynomials is first discussed from the viewpoint of power counting. Callan–Symanzik (CS) equations are derived for the insertion of operators of dimension 4 in the φ4 quantum field theory (QFT). Field equations are shown to imply linear relations between operators. The existence of a SDE for the product of two basic fields is established. A CS equation is derived for the Fourier transform of the coefficient of the expansion at leading order. The generalization of this analysis to the SDE beyond leading order, to the SDE of arbitrary operators and to the light-cone expansion (LCE), which appears in the study of the large momentum behaviour of real-time correlation function, are briefly discussed.

Short-distance HLbL contributions to the muon anomalous magnetic moment beyond perturbation theory

The hadronic light-by-light contribution to the muon anomalous magnetic moment depends on an integration over three off-shell momenta squared ($$ {Q}_i^2 $$ Q i 2 ) of the correlator of four electromagnetic currents and the fourth leg at zero momentum. We derive the short-distance expansion of this correlator in the limit where all three $$ {Q}_i^2 $$ Q i 2 are large and in the Euclidean domain in QCD. This is done via a systematic operator product expansion (OPE) in a background field which we construct. The leading order term in the expansion is the massless quark loop. We also compute the non-perturbative part of the next-to-leading contribution, which is suppressed by quark masses, and the chiral limit part of the next-to-next-to leading contributions to the OPE. We build a renormalisation program for the OPE. The numerical role of the higher-order contributions is estimated and found to be small.

QCD CORRECTIONS IN Γsl(B)

Short-distance expansion of the total semileptonic B widths is reviewed for the OPE-conformable scheme employing low-scale running quark masses. The third- and fourth-order BLM corrections are given and the complete resummation of the BLM series presented. The effect of higher perturbative orders with running quark masses is found to be very small. Numerical consequences for |Vcb| are addressed.