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Register a new userMass spectra in softly broken ${cal N}=2$ SQCD
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Authors
Victor L. Chernyak
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Considered are ${cal N}=2, SU(N_c)$ or $U(N_c)$ SQCD with $N_F<2N_c-1$ quark flavors with the quark mass term $m{rm Tr},({bar Q} Q)$ in the superpotential. ${cal N}=2$ 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, $mu_{rm x}llLambda_2$ ($Lambda_2$ 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 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 nontrivial $Z_{2N_c-N_Fgeq 2}$ discrete symmetry, at different hierarchies between the Lagrangian parameters $mgtrlessLambda_2,,, mu_{rm x}gtrless m$. The forms of low energy Lagrangians, quantum numbers 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. These calculations confirm additionally in a non-trivial way a self-consistency of the whole approach. Our results differ essentially from corresponding results in 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 numerous previous 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).
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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, $mu_{rm x}llLambda_2$ ($Lambda_2$ 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 $Z_{2N_c-N_Fgeq 2}$ discrete symmetry, at different hierarchies between the Lagrangian parameters $mgtrlessLambda_2,,, mu_{rm x}gtrless m$. 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).
Considered is the ${cal N}=1$ SQCD-like theory with $SU(N_c)$ colors and $0< N_F<2N_c$ flavors of equal mass $0< m_QllLambda_Q$ quarks. Besides, it includes $N^2_F$ additional colorless but flavored fields $Phi_{i}^{j}$, with the large mass parameter $mu_{Phi}ggLambda_Q$. The mass spectra of this $Phi$-theory are first directly calculated at $0<N_F<N_c$ where the quarks are weakly coupled, in all different vacua with the unbroken or spontaneously broken flavor symmetry $U(N_F)rightarrow U(n_1)times U(n_2)$. Further, the mass spectra of this direct $Phi$-theory and its Seibergs dual variant, the $dPhi$-theory, are calculated at $3N_c/2<N_F<2N_c$ and various values of $mu_{Phi}/Lambda_Qgg 1$ (in strong coupling regimes), using the dynamical scenario introduced by the author in his previous article cite{ch3}. This scenario assumes that quarks can be in two different standard phases only: either this is the HQ (heavy quark) phase where they are confined, or they are higgsed. Within the used dynamical scenario, it is shown that mass spectra of the direct $Phi$ and dual $dPhi$ - theories are parametrically different. Besides it is shown in the direct $Phi$-theory that a qualitatively new phenomenon takes place: under appropriate conditions, the seemingly heavy and dynamically irrelevant fields $Phi$ return back and there appear two additional generations of light $Phi$-particles with small masses $mu^{rm pole}(Phi)llLambda_Q$. Also considered is the $X$-theory which is the ${cal N}=2$ SQCD with $SU(N_c)$ colors and $0< N_F<2N_c$ flavors of light quarks, broken down to ${cal N}=1$ by the large mass parameter of the adjoint scalar superfield $X, , mu_XggLambda_2$. The tight interrelations between these $X$ and $Phi$ theories are described, in particular, the conditions under which they are equivalent.
We analyze the moduli space dynamics of domain walls in $SU(N)$ QCD at $bartheta=pi$, by softly breaking ${cal N}! =!1$ SQCD with sfermion mixing. In the supersymmetric limit, BPS domain walls between neighbouring vacua are known to possess non-translational flavour moduli that form a $mathcal{C} P^{N-1}$ sigma model. For the simplest case with gauge group $SU(2)$ and $N_f=2$, we show that this sigma model also exhibits a Hopf term descending from the bulk Wess-Zumino term with a quantized coefficient. On soft-breaking of supersymmetry via sfermion mixing that preserves the flavour symmetry, these walls and their moduli-space dynamics survives when $bartheta=pi$ so that there are two degenerate vacua.
We consider the ${cal N}=1$ $SU(N_c)$ SQCD-like (direct) theory (and its Seibergs dual with $SU(N_F-N_c)$ dual colors), and with $N_F$ flavors of light quarks ${overline Q}_j, Q^i$ with the mass term in the superpotential $m_Q{rm Tr}({overline Q} Q),,, m_QllLambda_Q$. Besides, there are $N_F^2$ additional colorless but flavored fields $Phi^j_i$ with the large mass parameter $mu_{Phi}ggLambda_Q$. But now considered is the region $N_c+1<N_F<3N_c/2$ where the UV free direct $SU(N_c)$ theory is strongly coupled at scales $mu<Lambda_Q$. The mass spectra of this direct theory in various vacua and at different values of $mu_{Phi}ggLambda_Q$ are calculated within the dynamical scenario introduced by the author in [10]. This scenario assumes that quarks in such ${cal N}=1$ SQCD-like theories can be in two standard phases only. These are either the HQ (heavy quark) phase where they are confined or the Higgs phase. It is shown that due to the strong power-like RG evolution, the seemingly heavy and dynamically irrelevant at scales $mu<Lambda_Q$ fields $Phi^j_i$ can become light and relevant at lower energies, and there appear then two additional generation of light $Phi$-particles with masses $mu_{2,3}^{rm pole}(Phi)llLambda_Q$. The calculated mass spectra of this strongly coupled at $mu<Lambda_Q, SU(N_c)$ theory are compared to those of its weakly coupled at $mu<Lambda_Q$ Seibergs dual $SU(N_F-N_c)$ variant and appeared to be parametrically different. It is worth to recall that the dynamical scenario from [10] used in this article satisfies all those tests which were used as checks of the Seiberg hypothesis about the equivalence of the direct and dual theories. This parametrical difference shows, in particular, that all these tests, although necessary, may well be insufficient.
This paper continues our studies in arXiV:1608.06452 [hep-th] of ${cal N}=1$ gauge theories in the strongly coupled regimes. We also consider here the ${cal N}=1$ SQCD-like theories with $SU(N_c)$ colors (and their Seibergs dual), with $N_F$ flavors of light quarks and $N_F^2$ additional colorless flavored scalars $Phi^j_i$, but now with $N_F$ in the range $N_F>3N_c$. The mass spectra of these direct and dual theories in various vacua are calculated within the dynamical scenario introduced by the author in [8]. It assumes that quarks in such ${cal N}=1$ SQCD-like theories without elementary colored adjoint scalars can be in two {it standard} phases only. These are either the HQ (heavy quark) phase where they are confined or the Higgs phase. Recall that this scenario satisfies all those tests which were used as checks of the Seiberg hypothesis about the equivalence of the direct and dual theories. Calculated mass spectra of the direct $SU(N_c)$ theory are compared to those of its Seibergs dual $SU(N_F-N_c)$ variant and appeared to be parametrically different.
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