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Register a new userMass spectra in ${cal N}=1$ SQCD with additional colorless fields and problems with Seibergs duality
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Authors
Victor L. Chernyak
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Considered is the direct ${cal N}=1$ SQCD-like $Phi$-theory with $SU(N_c)$ colors and $3N_c/2< N_F<2N_c$ flavors of light quarks ${overline Q},,Q$. Besides, it includes $N^2_F$ additional colorless but flavored fields $Phi_{i}^{j}$ with the large mass parameter $mu_{Phi}ggLambda_Q$, interacting with quarks through the Yukawa coupling. In parallel, is considered its Seibergs dual variant, i.e. the $dPhi$-theory with $(N_F-N_c)$ dual colors, $N_F$ flavors of dual quarks ${overline q},,{q}$ and $N_F^2$ elementary mion fields $M^i_jrightarrow ({overline Q}_j Q^i)$. In considered here vacua, the quarks of both theories are in the conformal regimes at scales $mu<Lambda_Q$. The mass spectra are calculated in sections 4 and 5. It is shown that they are different in the direct and dual theories, in disagreement with the Seiberg hypothesis about equivalence of two such theories. Besides it is shown in the direct theory that a qualitatively new phenomenon takes place: the seemingly heavy fields $Phi$ `return back and there appear two additional generations of light $Phi$-particles with small masses $mu^{rm pole}_{2,3}(Phi)llLambda_Q$. In Conclusions also presented comparison of mass spectra of these two theories for such values of parameters when the direct theory is in the very strong coupling regime, while the dual one is in the weak coupling IR-free logarithmic regime. It is shown that mass spectra of these two theories are parametrically different in this case.
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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 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.
This paper continues our previous study of similar theories in cite{ch5}. We also consider here the ${cal N}=1$ SQCD-like theories with $SU(N_c)$ colors (and their Seibergs dual with $SU(N_F-N_c)$ dual colors) and $N_F$ flavors of light quarks, and with $N_F^2$ additional colorless flavored fields $Phi^j_i$, but now with $N_F$ in the range $2N_c<N_F<3N_c$. The multiplicities of various vacua and quark and gluino condensates in these vacua are found. The mass spectra of the direct and Seibergs dual theories in various vacua are calculated within the dynamical scenario which assumes that quarks in such ${cal N}=1$ SQCD-like theories can be in two {it standard} phases only. These are either the HQ (heavy quark) phase where they are confined or the Higgs phase. The word {it standard} implies here also that, in such ${cal N}=1$ theories without elementary colored adjoint scalars, no {it additional} parametrically light solitons (e.g. magnetic monopoles or dyons) are formed at those scales where quarks decouple as heavy or are higgsed. 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. The mass spectra of these direct and Seibergs dual theories calculated within this framework were found to be different, in general. These parametrical differences of mass spectra of direct and dual theories show, in particular, that all those tests, which were used as checks of the Seiberg hypothesis about the equivalence of the direct and dual theories, although necessary, may well be insufficient. Besides, the mass spectrum of the dual theory with $SU(N_F-N_c)$ colors and $N_c+1<N_F<3N_c/2$ dual quark flavors was calculated. And finally, considered is the direct ${cal N}=2$ SQCD with $SU(N_c)$ colors and $N_c+1<N_F<3N_c/2$ flavors of light quarks...
N=1 SQCD with SU(N_c) colors and N_F flavors of light quarks is considered within the dynamical scenario which assumes that quarks can be in two different phases only. These are: a) either the HQ (heavy quark) phase where they are confined, b) or they are higgsed, at the appropriate values of parameters of the Lagrangian. The mass spectra of this (direct) theory and its Seibergs dual are obtained and compared, for quarks of equal or unequal masses. It is shown that in all cases when there is the additional small parameter at hand (it is 0<(3N_c-N_F)/N_F << 1 for the direct theory, or its analog 0<(2N_F-3N_c)/N_F << 1 for the dual one), the mass spectra of the direct and dual theories are parametrically different. A number of other regimes are also considered.
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