Field Redefinitions and Wave Function Renormalization to O ( p 4 ) in Heavy Baryon Chiral Perturbation Theory

Abstract

Massand wave function renormalization is calculated to order p4 in heavy baryon chiral perturbation theory. Two different schemes used in the literature are considered. Several technical issues like field redefinitions, non-transformation of sources as well as subtleties related to the definition of the baryon propagator are discussed. The nucleon axial-vector coupling constant gA is calculated to order p 4 as an illustrative example. * Work supported in part by Schweizerischer Nationalfonds, by the EEC-TMR Program, Contract No. CT98-0169, and by VEGA grant No.1/4301/97. 1. Heavy baryon chiral perturbation theory (HBChPT) [1, 2] allows for a systematic low energy expansion of one-nucleon Green functions. However, the matrix elements calculated in HBChPT are frame dependent. In order to obtain Lorentz invariant Smatrix elements, the fully relativistic nucleon propagator has to be worked out too. [3] So-called heavy nucleon sources cannot be neglected but yield non-trivial contributions to the nucleon wave function renormalization ZN already at order p . In this article we extend the work of ref. [3] to order p. Mass and wave function renormalization – a prerequisite for any p calculation [4],[5] – is thus determined to this order. Two HBChPT lagrangians widely used in the literature are considered. These are the Lagrangian given in [2, 6] (called BKKM hereafter) and the form appearing in [7] (called EM). The difference consists in the absence of equation of motion (EOM) terms in EM, which have been eliminated by nucleon field redefinitions. While the corresponding differences in ZN have been discussed to order p 3 in [3], several additional issues enter when going beyond this chiral order. 1 In particular, a new EOM-transformation at the level of the relativistic Lagrangian is introduced which allows for a direct and elegant evaluation of ZN. We also comment on the non-transformation of nucleon sources. Our results are tested for consistency by calculating the nucleon axial-vector coupling constant gA to order p 4 in the two schemes considered. 2. The starting point for the derivation of HBChPT is the generating functional of relativistic Green functions e = N ∫ [dudΨdΨ̄] exp{i(S̃M + SMB + ∫ dx(η̄Ψ+ Ψ̄η))} . (1) j, η, η̄ denote the sources of mesonic and baryonic fields, respectively. S̃M is the mesonic action – the tilde reminds us of the nucleon degrees of freedom having not been integrated out – and SMB is the action corresponding to the pion nucleon Lagrangian [8] LπN = Ψ̄(i 6∇ −m+ ġA 2 6uγ5)Ψ + . . . , (2) where m and ġA denote the nucleon mass and axial-vector decay constant in the chiral limit, respectively. The ellipsis in (2) stand for higher order terms. A systematic low energy expansion is obtained by the frame dependent decomposition of the nucleon field Ψ(x) = e(Nv +Hv)(x) (3) with v being a unit time-like four-vector and P v Nv = Nv, P − v Hv = Hv, P ± v = 1 2 (1± 6v) . (4) Wave function renormalization to order p in the BKKM case was treated recently in [9]. The emphasis in this article is on different aspects than in the present work.

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Cite this paper

@inproceedings{Mojs1998FieldRA, title={Field Redefinitions and Wave Function Renormalization to O ( p 4 ) in Heavy Baryon Chiral Perturbation Theory}, author={Martin Moj{\vz}ǐs}, year={1998} }