The reactions and in PT with an isosinglet scalar resonance

Arbin Thapaliya; Daniel R. Phillips

doi:10.1140/epja/i2017-12401-8

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Hadrons and Nuclei

Eur. Phys. J. A (2017) 53: 206
https://doi.org/10.1140/epja/i2017-12401-8

Regular Article - Theoretical Physics

The reactions $\pi\pi \rightarrow \pi\pi$ and $\gamma\gamma \rightarrow \pi\pi$ in $\chi$ PT with an isosinglet scalar resonance

Arbin Thapaliya¹^,2 and Daniel R. Phillips²^*

¹ Department of Chemistry and Physics, Franklin College, 46131, Franklin, IN, USA
² Institute of Nuclear and Particle Physics and Department of Physics and Astronomy, Ohio University, 45701, Athens, OH, USA

^* e-mail: phillid1@ohio.edu

Received: 16 June 2017
Accepted: 28 September 2017
Published online: 24 October 2017

Abstract

The lowest-lying resonance in the QCD spectrum is the $0^{++}$ isoscalar $\sigma$ -meson, also known as the $f_{0}(500)$ . We augment SU(2) chiral perturbation theory ( $\chi$ PT) by including the $\sigma$ -meson as an additional explicit degree of freedom, as proposed by Soto, Talavera, and Tarrús and others. In this effective field theory, denoted $\chi$ PT_S, the $\sigma$ -meson’s well-established mass and decay width are not sufficient to properly renormalize its self energy. At $O(p^{4})$ another low-energy constant appears in the dressed $\sigma$ -meson propagator; we adjust it so that the isoscalar pion-pion scattering length is also reproduced. We compare the resulting amplitudes for the $\pi\pi\rightarrow\pi\pi$ and $\gamma\gamma\rightarrow\pi\pi$ reactions to data from threshold through the energies at which the $\sigma$ -meson resonance affects observables. The leading-order (LO) $\pi \pi$ amplitude reproduces the $\sigma$ -meson pole position, the isoscalar $\pi \pi$ scattering lengths and $\pi \pi$ scattering and $\gamma\gamma\rightarrow\pi\pi$ data up to $\sqrt{s} \approx 0.5$ GeV. It also yields a $\gamma\gamma\rightarrow\pi\pi$ amplitude that obeys the Ward identity. The value obtained for the $\pi^{0}$ polarizability is, however, only slightly larger than that obtained in standard $\chi$ PT.