Phase transitions between confinement and higgs phases in N=1\, SU(Nc) SQCD with 1≤ NF≤ Nc quark flavors

Considered is 4-dimensional N=1 supersymmetric SU(Nc) QCD (SQCD) with 1≤ NF≤ Nc quark flavors with masses mQ,i in the bi-fundamental representation. { The gauge invariant order parameter ρ is introduced distinguishing confinement (with ρ=0) and higgs (with ρ≠ 0) phases}. Using a number of independent arguments for different variants of transition between the confinement and higgs regimes in these theories, it is shown that { transitions between these regimes are not crossovers but the phase transitions}. In {FS} the very special SU(Nc) QCD theory with NF=Nc defective scalar "quarks" in the unitary gauge: Φ ⁱβ= δⁱβ(|v|= const) > 0,\,\, i,β=1,...,NF=Nc, was considered by E.Fradkin and S.H.Shenker. The conclusion of {FS} was that the transition between the confinement at 0 < |v|ΛQCD (according to {FS}) and higgs (i.e. with condensed quarks) at |v| ΛQCD regimes is the analytic crossover. And although the theory considered in {FS} was very specific, the experience shows that up to now there is a widely spread opinion that this conclusion has general applicability. This model {FS} is criticized as incompatible with and very different from the standard non-SUSY SU(Nc)\,\,NF=Nc QCD theory with standard scalar quarks φⁱβ with all 2 N²c their physical real degrees of freedom. It is emphasized that this model {FS} is really { the Stuckelberg SU(Nc) YM-theory with no dynamical electric quarks and massive all Nc²-1 electric gluons with fixed by hands nonzero masses $g|v| > 0$}. There is no genuine confinement in this theory, it stays permanently in the completely higgsed (i.e. condensed) by hands phase only. And this is a reason for a crossover in this theory. While in the theory with standard scalar quarks there is the phase transition between the confinement (at 0 < |v| ΛQCD) and higgs (at |v|ΛQCD) regimes. Besides, the arguments presented in {IS} by K.Intriligator and N.Seiberg for the standard direct SU(Nc), \,\,NF=Nc\, \,\, N=1 SQCD in support of the crossover from {FS} are criticized as erroneous.