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.

THE LOCAL -POLYNOMIALS OF CLUSTER SUBDIVISIONS HAVE ONLY REAL ZEROS

Athanasiadis [‘A survey of subdivisions and local $h$-vectors’, in The Mathematical Legacy of Richard P. Stanley (American Mathematical Society, Providence, RI, 2017), 39–51] asked whether the local $h$-polynomials of type $A$ cluster subdivisions have only real zeros. We confirm this conjecture and prove that the local $h$-polynomials for all the Cartan–Killing types have only real roots. Our proofs use multiplier sequences and Chebyshev polynomials of the second kind.