Advances in High Energy Physics

Volume 2015, Article ID 803232, 4 pages

http://dx.doi.org/10.1155/2015/803232

## Scalar Form Factor of the Pion in the Kroll-Lee-Zumino Field Theory

^{1}Centre for Theoretical & Mathematical Physics and Department of Physics, University of Cape Town, Rondebosch 7700, South Africa^{2}Instituto de Física, Pontificia Universidad Católica de Chile, Casilla 306, 22 Santiago, Chile

Received 28 August 2014; Accepted 4 January 2015

Academic Editor: Alexey A. Petrov

Copyright © 2015 C. A. Dominguez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The renormalizable Kroll-Lee-Zumino field theory of pions and a neutral rho-meson is used to determine the scalar form factor of the pion in the space-like region at next-to-leading order. Perturbative calculations in this framework are parameter-free, as the masses and the rho-pion-pion coupling are known from experiment. Results compare favorably with lattice QCD calculations.

#### 1. Introduction

The scalar form factor of the pion [1, 2], and particularly its quadratic radius, plays an important role in chiral perturbation theory (CHPT) [3, 4]. This form factor is defined as the pion matrix element of the QCD scalar current ; that is, where . The associated quadratic scalar radius is given by where is the pion sigma term

The scalar radius fixes , one of the low energy constants of CHPT, through the relation where MeV is the physical pion decay constant [5]. The low energy constant , in turn, determines the leading contribution in the chiral expansion of the pion decay constant; that is, where is the pion decay constant in the chiral limit.

This scalar form factor is not accessible experimentally, but it has been determined from lattice QCD (LQCD) [6–8], or hadronic models [9, 10].

Theoretically, the ideal tool to study this form factor, independently from LQCD, is the Kroll-Lee-Zumino Abelian renormalizable field theory of pions and a neutral -meson [11, 12]. This provides the appropriate field theory platform for the phenomenological vector meson dominance (VMD) model [13, 14], allowing for a systematic calculation of higher order quantum corrections ([15], this paper, has a misprint in equation (15) (the sign of the first term in curly brackets should be negative), with the remaining equations being correct. The electromagnetic square radius of the pion quoted in the paper is incorrect; the correct value is fm^{2}, in much better agreement with data than naive (single ) VMD.) [16]. Due to the renormalizability of the theory, predictions are parameter-free, as the strong coupling, , is known from experiment. In spite of this coupling being a strong interaction quantity, perturbative calculations in the scheme make sense because the effective expansion parameter turns out to be .

The KLZ theory has been used to compute the next-to-leading order (NLO) correction to the tree level (VMD) electromagnetic form factor of the pion in the space-like region with very good results [15]. In fact, it agrees with data up to GeV^{2} with a chi-squared per degree of freedom , as opposed to VMD which gives . In addition, the mean-squared radius at NLO is fm^{2}, compared with the experimental result [5] fm^{2}, and the VMD value fm^{2}.

In this note we compute in this framework the scalar form factor of the pion at NLO in the space-like region and compare with current results from LQCD.

The KLZ Lagrangian is given by where is a vector field describing the meson (), is a complex pseudoscalar field describing the mesons, is the usual field strength tensor: , and is the current: . In spite of the explicit presence of the mass term in the Lagrangian, the theory is renormalizable because the neutral vector meson is coupled to a conserved current [11, 12]. Figures 1 and 2 show, respectively, the LO and the NLO diagrams, where the cross indicates the coupling of the current to the two pions. Notice that while the Lagrangian (6) contains a quartic coupling, this term only contributes in this application at NNLO and beyond.