Pseudoscalar vertex, Goldstone boson and quark masses on the lattice

Abstract

We analyse the Structure Function collaboration data on the quark pseudoscalar vertex and extract the Goldstone boson pole contribution, in 1/p 2 . The strength of the pole is found to be quite large at presently accessible scales. We draw the important consequences of this finding for the various definitions of quark masses (short distance and Georgi-Politzer), and point out problems with the operator product expansion and with the non-perturbative renormalisation method. Continuum model for the quark pseudoscalar vertexIt is well known that the quark pseudoscalar (PS) vertex contains a non-perturbative contribution from the Goldstone boson, in the continuum. On the lattice, the use of a non perturbative renormalisation scheme [2] makes this contribution manifest, although it should go to zero for large momentum transfers. The purpose of this letter is to extract it from lattice simulation data, and to show that it is not negligible for presently accessible scales. In particular, it must be subtracted when evaluating the short distance quark masses from the lattice through the non perturbative method of ref. [1]. This method uses the off-shell axial Ward identity (AWI) and renormalises the mass in the momentum subtraction (MOM) scheme of ref.[2], which can afterwards be related to the MS scheme. The MOM renormalisation involves the pseudoscalar vertex, whence the necessity of the subtraction of the Goldstone contribution to extract the short distance quantity 2 . For physical u, d quarks, the Goldstone contribution becomes very large, larger than the perturbative part; this corresponds to a very large dynamical u, d mass, larger than the usual current mass at the scales accessible to standard lattice calculations.The expected behaviour of the pseudoscalar vertex in the continuum has been described as follows in the 70's in the works of Lane and Pagels [4,5] and others. Near the chiral limit, the one-particle-irreducible PS quark vertex Λ 5 can be described through a perturbative contribution plus a non-perturbative Goldstone boson contribution.The perturbative contribution is of course 1 × γ 5 , with QCD radiative corrections, which according to the renormalisation group lead to a logarithmic behaviour [α s (p 2 )] 4/11 for N F = 0.As to the non-perturbative contribution, firstly, according to PCAC, the Goldstone boson must dominate other pole contributions in the PS vertex near the chiral limit: the coupling of the pion to the PS vertex indeed gives a pole ∼ 1/(q 2 + m 2 π ), where q is the momentum transferred at the vertex; other poles (radial excitations) contributions are suppressed in the chiral limit. At q = 0, in terms of the quark mass m, this gives a 1/m pole contribution. We emphasize that this pole contribution explodes at q = 0 in the chiral limit, i.e. it is singular 1 Laboratoire associe au CNRS-URA D00063. 2 The problem would be quite different if other methods are used to extract quark masses, see e.g. ref.[3].

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