Operator product expansion and analyticity.pdf

a r X i v : h e p - p h / 9 9 0 8 4 8 4 v 1 2 7 A u g 1 9 9 9 PRA-HEP 99/04 Operator product expansion and analyticity Jan Fischer1 Institute of Physics, Academy of Sciences of the Czech Republic, CZ-182 21 Prague 8, Czech Republic and Ivo Vrkoc?2 Mathematical Institute, Academy of Sciences of the Czech Republic, Z?itna? 25, CZ-115 67 Prague 1, Czech Republic Abstract We discuss the current use of the operator-product expansion in QCD calcula- tions. Treating the OPE as an expansion in inverse powers of an energy-squared variable (with possible exponential terms added), approximating the vacuum ex- pectation value of the operator product by several terms and assuming a bound on the remainder along the euclidean region, we observe how the bound varies with increasing deflection from the euclidean ray down to the cut (Minkowski region). We argue that the assumption that the remainder is constant for all angles in the cut complex plane down to the Minkowski region is not justified. Making specific assumptions on the properties of the expanded function, we ob- tain bounds on the remainder in explicit form and show that they are very sensitive both to the deflection angle and to the class of functions considered. The results obtained are discussed in connection with calculations of the coupling constant αs from the τ decay. PACS numbers: 11.15.Tk, 12.38.Lg, 13.35.Dx July 1999 1e-mails: fischer@fzu.cz, Jan.Fischer@cern.ch 2e-mail: vrkoc@matsrv.math.cas.cz 1 Introduction The operator-product expansion (OPE) [1, 2], i ∫ dxeiqxA(x)B(0) ≈∑ k Ck(q)Ok, (1) represents the product AB of two local operators as a combination, with c-number co- efficients, of the local operators Ok. Here, q is the total four-momentum of the system considered and q2 = s = ?Q2. The singularities of the product are contained in the coefficient functions Ck(q), which are ordered according to the increasing exponent k in s?k. In local quantum field theory, the product A1(x 1)...Aa(x a) of two or sev