Softly broken ${\cal N}=2$ SQCD: mass spectra in vacua with unbroken $Z_{2N_c-N_F}$ symmetry

TL;DR

This paper analyzes the mass spectra and low energy dynamics of softly broken ${ m f N}=2$ SQCD with unbroken $Z_{2N_c-N_F}$ symmetry, providing detailed calculations and confirming results through spectral curve analysis.

Contribution

It presents a comprehensive study of vacua with unbroken discrete symmetry in softly broken ${ m f N}=2$ SQCD, including new calculations of condensates and mass spectra, and clarifies differences from prior results.

Findings

Mass spectra characterized for various vacua with unbroken $Z_{2N_c-N_F}$ symmetry.

Power corrections to condensates calculated and confirmed via spectral curve roots.

Results differ from previous literature, with explanations provided.

Abstract

Considered are ${\cal N}=2\, SU(N_c)$ or $U(N_c)$ SQCD with $N_F<2N_c-1$ equal mass quark flavors. ${\cal N}=2$ supersymmetry is softly broken down to ${\cal N}=1$ by the mass term $\mu_{\rm x}{\rm Tr}\,(X^2)$ of colored adjoint scalar partners of gluons, $\mu_{\rm x}\ll\Lambda_2$ ($\Lambda_2$ is the scale factor of the $SU(N_c)$ gauge coupling). There is a large number of different types of vacua in these theories with both unbroken and spontaneously broken global flavor symmetry, $U(N_F)\rightarrow U({\rm n}_1)\times U({\rm n}_2)$. We consider in this paper the large subset of these vacua with the unbroken non-trivial $Z_{2N_c-N_F\geq 2}$ discrete symmetry, at different hierarchies between the Lagrangian parameters $m\gtrless\Lambda_2,\,\, \mu_{\rm x}\gtrless m$. The forms of low energy Lagrangians, charges of light particles and mass spectra are described in the main text for all these vacua. The calculations of power corrections to the leading terms of the low energy quark and dyon condensates are presented in two important Appendices. The results agree with also presented in these Appendices independent calculations of these condensates using roots of the Seiberg-Witten spectral curve. This agreement confirms in a non-trivial way a self-consistency of the whole approach. Our results differ essentially from corresponding results in e.g. recent related papers arXiv:1304.0822, arXiv:1403.6086 and arXiv:1704.06201 of M.Shifman and A.Yung (and in a number of their previous numerous papers on this subject), and we explain in the text the reasons for these differences. (See also the extended critique of a number of results of these authors in section 8 of arXiv:1308.5863).