B)   At larger $\langle\Lambda^{SU(\nd)}_{{\cal N}=2\,\,SYM}\rangle\ll\mx\ll\, m$, the difference is that all $SU(\nd)$ adjoint scalars $X^{adj}_{SU(\nd)}$ are now too heavy, their physical $SU(\nd)$ phases induced interactions with light $SU(\nd)$ gluons fluctuate freely at all scales $\mu\gtrsim\langle\Lambda^{SU(\nd)}_{{\cal N}=1\,\,SYM}\rangle$, and they are not higgsed. Instead, they decouple as heavy much before, already at the scale $\mu<[\,\mx^{\,\rm pole}=|g^2\mx|\,]/{\rm several)}\ll m$, and there remains ${\cal N}=1 \,\, SU(\nd)$ SYM with the scale factor $\langle\Lambda^{SU(\nd)}_{{\cal
N}=1\,\,SYM}\rangle=[\,\wmu \langle\Lambda^{SU(\nd)}_{{\cal N}=2\,\,SYM}\rangle^2\,]^{1/3}$ of its gauge coupling, $\,\,\langle\Lambda^{SU(\nd)}_{{\cal N}=2\,\,SYM}\rangle\ll\langle\Lambda^{SU(\nd)}_{{\cal N}=1\,\,SYM}\rangle\ll\mx\ll\, m$. (As in section 2.3, the small nonzero value $\langle {\rm Tr\,} (X^{adj}_{SU(\nd)})^2\rangle=\nd\langle S\rangle_{\nd}/\wmu=\nd\langle\Lambda^{SU(\nd)}_{{\cal N}=2\,\, SYM}\rangle^2\ll\langle\Lambda^{SU(\nd)}_{{\cal N}=1\,\, SYM}\rangle^2$, see (<ref>), arises here only due to the Konishi anomaly, i.e. from one-loop Feynman diagrams with heavy scalars $X^{adj}_{SU(\nd)}$ and their fermionic superpartners with masses $\sim\mx$ inside, not because $X^{adj}_{SU(\nd)}$ are higgsed).

The multiplicity of vacua of this ${\cal N}=1 \,\, SU(\nd)$ SYM is also $\nd$ as it should be. There appears now a large number of strongly coupled gluonia with the mass scale $\sim\langle\Lambda^{SU(\nd)}_{{\cal N}=1\,\,SYM}\rangle$. All $SU(\nd)$ electrically  charged particles are still confined, the tension of the confining string is larger now, $\sigma^{1/2}_{SU(\nd)}\sim\langle\Lambda^{SU(\nd)}_{{\cal N}=1\,\,SYM}\rangle\ll\mx\ll\, m$.

The case with $m\ll\mx\ll\lm$ is described in section 8.2 of  [9].

Part II.  Large quark masses, $\Large\mathbf{m\gg\lm}$

Part II.  Large quark masses, $\Large\mathbf{m\gg\lm}$

§ BROKEN FLAVOR SYMMETRY

We present below in this section the mass spectra at $m\gg\lm$ of the direct (electric) $SU(N_c)$ theory (<ref>) in vacua  with spontaneously broken flavor symmetry, $U(N_F)\ra U(\no)\times U(\nt)$,  $N_c+1<N_F<2N_c-1$. There are the br1-vacua (br=breaking) with $1\leq\no< N_F/2\,,\,\, N_F=\no+\nt\,,$ in which $\langle\Qo\rangle_{N_c}\gg\langle\Qt\rangle_{N_c}$, see section 3 in [13] and (<ref>),(<ref>). The quark and gluino condensates look in these br1 vacua of $SU(N_c)$ as (the leading terms only, see the footnote <ref>)
⟨⟩_N_c≈m_3 ,  ⟨⟩_N_c≈m_3(/m_3)^2N_c-N_F/N_c-  ,  m_3=N_c/N_c-  m ,
⟨S⟩_N_c=⟨⟩_N_c⟨⟩_N_c/≈m_3^2(/m_3)^2N_c-N_F/N_c- ,  ⟨⟩_N_c/⟨⟩_N_c≈(/m_3)^2N_c-N_F/N_c-≪1 ,
while $\no\leftrightarrow\nt$ in (<ref>) in br2-vacua, but with $N_F/2<\nt<N_c$ in this case. The multiplicity of these br1 and br2 vacua is correspondingly $N_{\rm br1}=(N_c-\no)C_{N_F}^{\,\no}$ and $N_{\rm br2}=(N_c-\nt)C_{N_F}^{\,\nt},\,\, C_{N_F}^{\,\no}=C_{N_F}^{\,\nt}=[\,N_{F} !/\no!\,\nt!\,],\,\,\, 1\leq \no< N_F/2,\,\, \nt> N_F/2$. It is seen from (<ref>) that the discrete $Z_{\bb}$ symmetry is unbroken at $m\gg\lm$ in all these br1 and br2 vacua with $\no\neq\nd$. In contrast, $Z_{\bb}$ symmetry is (formally) broken spontaneously in special vacua with $\no=\nd,\,\,\nt=N_c$ and the multiplicity $N_{spec}=(\bb)  C_{N_F}^{\,\nd}$ (as it is really broken spontaneously in these special vacua at $m\ll\lm$).

Out of them, see section 3 in [13] and/or section 4 in  [14], $(\nd-\no)C_{N_F}^{\,\no}$ part of br1-vacua with $1\leq\no <\nd$ evolves at $m\ll\lm$ into the br2-vacua of section 2.1 with $\langle\Qt\rangle^{(\rm br2)}_{N_c}\sim\mx m\gg\langle\Qo\rangle^{(\rm br2)}_{N_c}$, while the other part of br1-vacua and br2-vacua evolve at $m\ll\lm$ into $Lt$ vacua with $\langle\Qo\rangle^{(\rm Lt)}_{N_c}\sim\langle\Qt\rangle^{(\rm Lt)}_{N_c}\sim\mx\lm,\, ,\, \langle S\rangle_{Lt}\sim\mx\lm^2$.

§.§ $SU(N_c),$  br1 vacua, smaller $\mx\,,\,\,\,\mx\ll\Lambda^{SU(N_{c}-\no)}_{{\cal N}=2\,\,SYM}$

In the case considered, $X^{\rm adj}_{SU(N_c)}$ breaks the $SU(N_c)$ group in the weak coupling regime at the largest scale $\mu\sim m\gg\lm$ as : $\,SU(N_c)\ra SU(\no)\times U(1)\times SU(N_c-\no)$, see (<ref>) below,
⟨X^adj_SU(N_c)⟩= ⟨  X^adj_SU()+ X_U(1)+ X^adj_SU(N_c-) ⟩ ,
√(2)  X_U(1)=a  diag( 1_ ;  c_N_c-  ) ,  c=-/N_c- ,  a≡⟨a⟩+â ,  ⟨a ⟩=m ,
⟨â⟩≡0 .
As a result, all quarks $Q^i_a,\,{\ov Q}_j^{\,a}$ with flavors $i, j=1...N_F$ and colors $a=(\no+1)...N_c$ have large masses $m_3=m-c\langle a\rangle=m N_c/(N_c-\no)$, the hybrid gluons and hybrid $X$ also have masses $m_3\gg\lm$, and they all decouple at scales $\mu\lesssim\, m_3$. The lower energy theory at $\mu<\, m_3$ consists of ${\cal N}=2\,\, SU(N_c-\no)$   SYM, then ${\cal N}=2\,\, SU(\no)$ SQCD with $N_F$ flavors of massless at $\mx\ra 0$ quarks $Q^{i}_{\rm b},\, {\ov Q}^{\,\rm b}_j$ with $\no$ colors, ${\rm b}=1\,...\,\no$, and finally one ${\cal N}=2\,\, U(1)$ photon multiplet with its scalar superpartner $X_{U(1)}$. The scale factor of the ${\cal N}=2\,\,\, SU(N_{c}-\no)$ SYM gauge coupling is $\langle\Lambda^{SU(N_{c}-{\rm n}_1)}_{{\cal N}=2\,\,SYM}\rangle\ll m$. The field $X^{\rm adj}_{SU(N_c-\no)}$ in (<ref>)) breaks this ${\cal N}=2\,\, SU(N_{c}-\no)$ SYM at the scale $\mu\sim\langle\Lambda^{SU(N_{c}-{\rm n}_1)}_{{\cal N}=2\,\,SYM}\rangle$ in a standard way, $SU(N_c-\no)\ra U(1)^{N_c-\no-1}$ [3], see also (<ref>),
⟨√(2)  X^adj_SU(N_c-)⟩∼⟨Λ^SU(N_c-)_N=2  SYM⟩ diag(  0_ ; k_1,k_2, ... , k_N_c-) ,  k_i=O(1) ,
⟨Λ^SU(N_c-)_N=2  SYM⟩^2≈(^(m_3)^N_F/(m_3)^2)^1/N_c-≈m^2_3(/m_3)^2N_c-N_F/N_c-≪m^2 ,
⟨S⟩_N_c-=⟨Λ^SU(N_c-)_N=2  SYM⟩^2=⟨S⟩_N_c ,
where $k_i$ are known numbers [3]. We note that $\Lambda^{SU(N_{c}-\no)}_{{\cal N}=2\,\,SYM}$ in (<ref>) and $\langle a\rangle=m$ in (<ref>) have the same charge 2 under the discrete $Z_{2N_c-N_F}$ transformations as $X$ itself, this is due to unbroken $Z_{\bb}$ symmetry. There are $N_c-\no$ physically equivalent vacua and $N_c-\no-1$ light pure magnetic monopoles ${\ov M}_n,\, M_n,\,\, n=1... N_c-\no-1$ with the $SU(N_{c}-\no)$ adjoint charges (massless at $\mx\ra 0$) in each of these ${\cal N}=2$ SYM vacua. The relevant part of the low energy superpotential of this SYM part looks as
W^ low_tot=W^(M)_SYM+W_+… ,
W^(M)_SYM= - ∑_n=1^N_c--1 ã_M, n[  M_n M_n+⟨Λ^SU(N_c-n_1)_N=2  SYM⟩(1+O(⟨Λ^SU(N_c-n_1)_N=2  SYM⟩/m)) z_n   ] ,  z_n=O(1),
where $z_{\rm n}$ are known numbers [3] and dots denote as always smaller power corrections.

All monopoles in (<ref>) are higgsed at $\mx\ne 0$ with $\langle M_{\rm n}\rangle=\langle{\ov M}_{\rm n} \rangle\sim [\,\mx\langle\Lambda^{SU(N_{c}-{\rm n}_1)}_{{\cal N}=2\,\,SYM}\rangle\,]^{1/2}$, so that all light particles of this ${\cal N}=2\,\, U(1)^{N_c-\no-1}$ Abelian magnetic part form $N_c-\no-1$ long ${\cal N}=2$ multiplets of massive dual photons with masses $\sim (\mx\langle\Lambda^{SU(N_{c}-\no}_{{\cal N}=2\,\,SYM})^{1/2}\rangle$ .

Besides, for this reason, all heavier original $SU(N_{c}-\no)$ electrically charged particles  with masses $\sim m$ or $\sim\langle\Lambda^{SU(N_{c}-{\rm n}_1)}_{{\cal N}=2\,\,SYM}\rangle$ are weakly confined (the confinement is weak in the sense that the tension of the confining string is much smaller than their masses, $\sigma^{1/2}_{SYM}\sim (\mx\langle\Lambda^{SU(N_{c}-{\rm n}_1)}_{{\cal N}=2\,\,SYM}\rangle)^{1/2}\ll\langle\Lambda^{SU(N_{c}-{\rm n}_1)}_{{\cal N}=2\,\,SYM}\rangle\ll m$).

And finally, as for the (independent) ${\cal N}=2\,\,SU(\no)\times U(1)$ part with $N_F$ flavors of original electric quarks ${\ov Q}^b_j,  Q^{\,i}_b,\,\, i,j=1...N_F,\,\, b=1...\no$. Its superpotential at the scale $\mu=m/(\rm several)$ looks as
W_=(  m-a ) Tr (Q Q)_ - Tr (Q√(2) X^adj_SU() Q) +Tr ( X^adj_SU())^2+
+  /2 N_c/N_c- a^2+(N_c-)[ (Λ^SU(N_c-n_1)_N=2  SYM)^2≈(^2N_c-N_F(m_3-c â)^N_F/[ m_3+(1-c)â ]^2)^1/N_c- ] ,  a≡⟨a⟩+â .

Because the type of color breaking $SU(N_c)\ra SU(\no)\times U(1)\times SU(N_c-\no)$ in (<ref>) is qualitatively different from $SU(N_c)\ra SU(N_F-N_c)\times U^{(1)}(1)\times U(1)^{\bb-1}$ in section 2.1 and
occurs here in the weak coupling regime $a(\mu\sim m)=N_c g^2(\mu\sim m)/8\pi^2\ll 1$, the non-perturbative instanton contributions do not operate in this high energy region. They originate and operate in these vacua only at much lower energies $\sim\langle\Lambda^{SU(N_{c}-{\rm n}_1)}_{{\cal N}=2\,\,SYM}\rangle$ from the SYM part, see (<ref>). For this reason, there are no in (<ref>) analogs of terms $\sim\delta_i$ in (<ref>).

From (<ref>) (neglecting here small power corrections in the quark condensate $\langle\Qo\rangle
_{\no}$ originating from the SYM part, see (<ref>) ), compare also with (<ref>),
⟨a⟩=m ,  ⟨X^adj_SU()⟩=0 ,  ⟨Tr (Q Q)_⟩=⟨⟩_+⟨⟩_≈m_3 ,
⟨⟩^SU(N_c)_=⟨Q^1_1⟩⟨Q^1_1⟩≈m_3≈⟨⟩^SU(N_c)_N_c ,  ⟨⟩_=∑_a=1^⟨Q^ a_2⟩⟨Q^2_a⟩=0 ,  ⟨S⟩_=0 ,
⟨Q^i_b⟩=⟨Q_i^ b⟩≈δ^i_b (m_3)^1/2,   b=1... ,  i=1...N_F ,   m_3=N_c/N_c- .

On the whole, all this results in the right multiplicity of these br1 vacua, $N_{\rm br1}=(N_c-\no) C_{N_F}^{\,\no}$, the factor $N_c-\no$ originates from ${\cal N}=2\,\,\, SU(N_{c}-\no)$ SYM, while $C_{N_F}^{\,\no}$ is due to the spontaneous breaking $U(N_F)\ra U(\no)\times U(\nt)$ by $\no$ higgsed quarks in the $SU(\no)\times U(1)$ sector.

It is worth emphasizing that the value $\langle a\rangle=m$ in (<ref>) is exact (i.e. there is no any small correction). This is seen from the following. At very small $\mx\ra 0$, the possible correction $\delta m\neq 0$ is independent of $\mx$, and so $\delta m\gg\mx$. Then quarks $Q^i_b, {\ov Q}_i^{\,b},\,\, i=1...N_F,\,\, b=1...\no$ will acquire masses $\delta m$, much larger than the scale of their potentially possible coherent condensate, $\delta m\gg (\mx m)^{1/2},\,\, \mx\ra 0$. In this case, these quarks will be not higgsed but will decouple as heavy ones at scales $\mu<\delta m$ and the flavor symmetry will remain unbroken. There will remain ${\cal N}=2\,\, SU(\no)\times U(1)$ SYM  at scales $\mu<\delta m$. This lowest energy ${\cal N}=2\,\, SU(\no)$ SYM will give then its own multiplicity factor $\no$, so that the overall multiplicity of vacua will be $\no (N_c-\no)$, while the right multiplicity is $(N_c-
\no)C^{\,\no}_{N_F}$, see (<ref>) and section 3 in [13]. To have this right multiplicity and spontaneous flavor symmetry breaking all $\no$ quarks have to be higgsed at arbitrary small $\mx$, and this is only possible when they are exactly massless in the limit $\mx\ra 0$, i.e. at $\delta m=0$.
[  These considerations clearly concern also all other similar cases.

As a result of higgsing of $\no$ out of $N_F$ quark flavors in $SU(\no)\times U(1)$ in (<ref>), the flavor symmetry is broken spontaneously as $U(N_F)\ra U(\no)\times U(\nt)$, the quarks $Q^{\,l}_b, {\ov Q}^{\,b}_l$ with flavors $l=(\no+1)...N_F$ and colors $b=1...\no$ will be the massless Nambu-Goldstone particles ($\,2\no\nt$ complex degrees of freedom), while all other particles in (<ref>) will acquire masses $\sim (\mx m)^{1/2}$ and will form $\no^2$ long ${\cal N}=2$ multiplets of massive gluons.

On the whole, the hierarchies of nonzero masses look in this case as
⟨M_n⟩∼(⟨Λ^SU(N_c-n_1)_N=2  SYM⟩)^1/2≪⟨Q^i_b=i⟩∼(m)^1/2 ,  ≪⟨Λ^SU(N_c-)_N=2  SYM⟩≪m ,
where ${\rm n}=1...N_{c}-\no-1\,,\,\,\, i=1...\no$ . There are no massless particles, except for the $\,2\no\nt$ Nambu-Goldstone multiplets.

The curve (<ref>) has $N_c-1$ double roots in these vacua. $N_c-\no-1$ unequal roots correspond to pure magnetic monopoles $M_{\rm n}$ in (<ref>) and $\no$ equal ones correspond to higgsed quarks from $SU(\no)$. The two single roots with $(e^{+}-e^{-})\sim\langle\Lambda^{SU(N_{c}-\no)}_{{\cal N}=2\,\,SYM}\rangle$ originate from SYM.

The calculation of leading power corrections to $\langle{\cal W}^{\,\rm low}_{\rm tot}\rangle$ is presented in Appendix A, and those to $\langle\Qo\rangle^{SU(N_c)}_{\no}$ are presented in Appendices A and B. These latter are also compared therein with the independent calculations of $\langle\Qo\rangle^{SU(N_c)}_{\no}$ using the roots of the curve (<ref>).

Now, let us recall some additional explanations about notations in (<ref>),(<ref>). As for higgsed $\langle a\rangle$, it looks as $\langle a\rangle=\langle a\rangle^{(\rm several) m}_{m/(\rm several)}=m$. In other words, its total mean value originates and saturates at $\mu\sim m$. If not,  all quarks will decouple as heavy at $\mu=m/(\rm several)$, the multiplicity of these br1 vacua will be wrong and $U(N_F)$ will remain unbroken. As for $\langle Q_a\rangle,\,\, a=1...\no$, of light quarks with $SU(\no)$ colors. - At scales $[\mu^{\rm low}_{\rm cut}=(\mx m_3)^{1/2}]<\mu<m/(\rm several)$, all $SU(\no)$ gluons are effectively massless and all light quarks with $SU(\no)$ colors move freely and independently. Therefore, the physical (i.e. path dependent) $SU(\no)$ phases of these quarks induced by their interactions with $SU(\no)$ gluons fluctuate freely in this higher energy region, so that
⟨Q^ b_j Q^i_a⟩_μ^low_cut=⟨Q^ b_j⟩_μ^low_cut⟨Q^i_a⟩_μ^low_cut=0 ,  a,b=1... ,  i,j=1...N_F   μ^low_cut=(m_3)^1/2 .
Therefore, in the whole region $0\leq \mu<m/(\rm several)$, the nonzero quark mean values in (<ref>), i.e. $[\langle\Qo\rangle_{\no}=\langle{\ov Q}^{1}_{1}\rangle\langle Q^{1}_{1}\rangle\,]
/\mx\approx [\langle a\rangle=m]N_c/(N_c-\no)]$ (the approximation here is controllable and consists in neglecting small power corrections originating from the SYM part, see (<ref>),(<ref>) and Appendix A) originate and saturate only in the region $g_Q (\mx m_3)^{1/2}/(\rm several)<\mu< (\rm several){\it g}_Q (\mx m_3)^{1/2}$, after $\no$ out of $N_F$ quarks are higgsed (i.e. form the coherent condensate), giving masses $g_Q (\mx m)^{1/2}$ to themselves and all $SU(\no)\times U(1)$ gluons. And there remain only $2\no\no$ massless Nambu-Goldstone multiplets in this sector at lower energies.

And there is no contradiction in the relation following from (<ref>)
⟨⟩_=(⟨Q^1_1⟩⟨Q^1_1⟩)^(several)g_Q (m_3)^1/2_g_Q (m_3)^1/2/(several)≈[N_c/N_c-(⟨a⟩^(several) m_m/(several)=m )=m_3 ],
in that the total mean values of $\langle a\rangle$ and $\langle Q^1_1\rangle$ originate and saturate in different energy regions, because this is not the equality between mean values of two equal to each other operators, but only the numerical equality between two total mean values of two different operators.

On the whole, the total decomposition of quark condensates $\langle (\qq)_{1,2}\rangle_{N_c}$ over their separate color parts look in these br1 vacua as follows.

I) The condensate $\langle\Qo\rangle_{N_c}$.

a) From (<ref>) and accounting for the leading power correction, see (<ref>), the (factorizable) condensate of higgsed quarks in the $SU(\no)$ part
⟨⟩_≈m_3[1-2N_c-N_F/N_c-(/m_3)^2N_c-N_F/N_c-] ,  m_3=N_c/N_c- m≫⟨Λ^SU(N_c-)_N=2  SYM⟩≫ .
b) The (non-factorizable) condensate $\langle\Qo\rangle_{N_c-\no}$ is determined by the one-loop Konishi anomaly for the heavy non-higgsed quarks with the mass $m_3$ in the $SU(N_c-\no)$ SYM sector. I.e., on the first "preliminary" stage, from the one-loop diagrams with heavy scalar quarks and their fermionic superpartners inside, transforming in the weak coupling regime at the scale $\mu\sim m$ the heavy quark operator $(\Qo)_{N_c-\no}$ into the operator $\sim (\lambda\lambda)_{N_c-\no}/m_3$ of lighter $SU(N_c-\no)$ gluino. And the mean value of this latter originates only at much lower energies $\mu\lesssim\langle\Lambda^{SU(N_{c}-\no)}_{{\cal N}=2\,\,SYM}\rangle$, see (<ref>),
_N=2  SYM⟩^2/m_3≈m_3(/m_3)^/N_c- .

Therefore, on the whole
⟨⟩_N_c=⟨⟩_+⟨⟩_N_c-≈m_3[ 1+-N_c/N_c-(/m_3)^/-N_c] ,
as it should be, see (<ref>).

II) The condensate $\langle\Qt\rangle_{N_c}$.

a) From (<ref>), the (factorizable) condensate of non-higgsed massless Nambu-Goldstone particles in the $SU(\no)$ part
⟨⟩_=∑_a=1^⟨Q^ a_2⟩⟨Q^2_a⟩=0 .
b) The (non-factorizable) condensate $\langle\Qt\rangle_{N_c-\no}$ is determined by the same one-loop Konishi anomaly for the heavy non-higgsed quarks with the mass $m_3$ in the $SU(\nd-\no)$ SYM sector,
⟨⟩_N_c-=⟨S⟩_N_c-/m_3≈m_3(/m_3)^/N_c- .

Therefore, on the whole
⟨⟩_N_c=⟨⟩_+⟨⟩_N_c-≈m_3(/m_3)^/N_c- ,
as it should be, see (<ref>).

§.§ $U(N_c)$, br1 vacua, smaller $\mx\,,\,\, \mx\ll\Lambda^{SU(N_{c}-\no)}_{{\cal N}=2\,\,SYM}$

When the additional $U^{(0)}(1)$ is introduced, the Konishi anomalies look as in (<ref>) but $\langle\Qo\rangle_{N_c}$ is dominant in br1 vacua, i.e. (see the footnote <ref>, the leading terms only)
⟨+⟩_N_c=m ,  ⟨⟩_N_c=m-⟨⟩_N_c≈m ,
⟨⟩_N_c≈m(/m)^2N_c-N_F/N_c-≈m ⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m^2 ,
⟨a_0⟩/m=1/m  N_c⟨Tr ⟩_N_c≈[ /N_c +-/N_c(/m)^2N_c-N_F/N_c- ]
≈/N_c .
(Put attention that $\langle\Lambda^{SU(N_{c}-\no)}_{{\cal N}=2\,\,SYM}\rangle$ in this section is different from those in section 4.1, see (<ref>)).

Instead of (<ref>) the superpotential of the $SU(\no)\times U^{(0)}(1)\times U(1)$ part looks now as
W^ low_matter=_+_a_0,a+… ,
W_=( m-a_0-a)Tr(Q Q)_-Tr(Q√(2) X^adj_SU() Q)+Tr( X^adj_SU())^2,
_a_0,a=/2N_c a^2_0+/2N_c/N_c- a^2 ,
where dots denote smaller power corrections.

From (<ref>) (for the leading terms)
⟨a_0⟩≈/N_c m ,   ⟨a⟩=⟨m-a_0⟩≈N_c-
/N_c m ,    ⟨Tr (Q Q)⟩_≈N_c⟨a_0⟩≈N_c/N_c-⟨a⟩≈m ,
⟨X^adj_SU()⟩=0,  ⟨⟩^U(N_c)_=∑_a=1^⟨Q^ a_1 Q^1_a⟩=⟨Q^ a_1⟩⟨Q^1_a⟩≈m≈⟨⟩^U(N_c)_N_c,   ⟨⟩_=0.

The qualitative difference with section 4.1 is that one extra ${\cal N}=1$ photon multiplet remains massless now in this $U(N_c)$ theory while corresponding scalar multiplet has the smallest mass $\sim\mx$ because ${\cal N}=2$ is broken down to ${\cal N}=1$ at the level $O(\mx)$ .

The calculation of leading power corrections to $\langle{\cal W}^{\,\rm low}_{\rm tot}\rangle$ in these $U(N_c)$ br1 vacua is presented in Appendix A, and those to $\langle\Qo\rangle^{U(N_c)}_{\no}$ are presented in Appendices A and B. These latter are also compared therein with the independent calculations of $\langle\Qo\rangle^{U(N_c)}_{\no}$ using the roots of the curve (<ref>).

§.§ $SU(N_c)$, br1 vacua, larger $\mx\,,\, \Lambda^{SU(N_{c}-\no)}_{{\cal N}=2\,\,SYM}\ll\mx\ll m$

Nothing changes significantly in this case with the $SU(\no)\times U(1)$ part in (<ref>). The mean value $\langle a\rangle=m\gg\mx$ of the field $\langle\sqrt{2}\,X_{U(1)}\rangle$ in (<ref>) stays intact as far as $\mx\ll m$, and so the contributions $\sim (\mx m)^{1/2}$ to the masses of $\no^2-1$ long ${\cal N}=2\,\,\,SU(\no)$ multiplets of massive gluons and to the mass of the long ${\cal N}=2\,\,\,U(1)$ multiplet of the massive photon due to higgsing of $\no$ quarks remain dominant. At this level the ${\cal N}=2$ SUSY remains unbroken in this sector. It breaks down here to ${\cal N}=1$ only at the level $\sim\mx$ by the additional smaller contributions $\sim\mx\ll (\mx m)^{1/2}$ to the masses of corresponding scalar multiplets.

The situation with $SU(N_c-\no)$ SYM in (<ref>),(<ref>) is different. At $\langle\Lambda^
{SU(N_c-\no)}_{{\cal N}=2\,\,SYM}\rangle\ll\mx\ll m$, the adjoint scalars $X^{adj}_{SU(N_c-\no)}$ of the $SU(N_c-\no)$ subgroup in (<ref>) become too heavy and too short ranged. Their light $SU(N_c-\no)$ physical (i.e. path dependent) phases induced interactions with effectively massless at all scales $\mu\gtrsim\langle\Lambda^{SU(N_c-\no)}_{{\cal N}=1\,\,SYM}\rangle\,\, SU(N_c-\no)$ gluons fluctuate freely at the scale $\mu>\mx^{\rm pole}/(\rm several)={\it g}^2\mx/(\rm several)$, and they are not higgsed, i.e. $\langle X^{adj}_{SU(N_c-\no)}\rangle_{\mx^{\rm pole}/(\rm several)}=0$. Instead, they decouple as heavy already at the scale $\mu<\mx^{\rm pole}/(\rm several)\ll m$ in the weak coupling region, can all be integrated out and do not affect by themselves the lower energy dynamics in this $SU(N_c-\no)$ sector. There remains ${\cal N}=1 \,\, SU(N_c-\no)$ SYM with the scale factor $\langle\Lambda^{SU(N_c-\no)}_{{\cal N}=1\,\,SYM}\rangle=[\,\mx \langle\Lambda^{SU(N_c-\no)}_{{\cal N}=2\,\,SYM}\rangle^2\,]^{1/3}$ of its gauge coupling, $\,\,\langle\Lambda^{SU(N_c-\no)}_{{\cal N}=2\,\,SYM}\rangle\ll\langle\Lambda^{SU(N_c-\no)}_{{\cal N}=1\,\,SYM}\rangle\ll\mx\ll\, m$,
⟨S⟩_N_c-=⟨Λ^SU(N_c-)_N= 1  SYM⟩^3=m_3^2(/m_3)^2N_c-N_F/N_c-,  ⟨Λ^SU(N_c-)_N=1  SYM⟩/⟨Λ^SU(N_c-n_1)_N=2  SYM⟩∼(/⟨Λ^SU(N_c-n_1)_N=2  SYM⟩)^1/3≫1 .
The small (non-factorizable) value $\langle{\rm Tr\,} (X^{adj}_{SU(N_c-\no)})^2\rangle=
(N_c-\no)\langle S\rangle_{N_c-\no}/\mx=(N_c-\no)\langle\Lambda^{SU(N_c-\no)}_{{\cal N}=2\,\, SYM}\rangle^2 \\ \ll \langle\Lambda^{SU(N_c-\no)}_{{\cal N}=1\,\, SYM}\rangle^2$ arises here not because $X^{adj}_{SU(N_c-\no)}$ are higgsed, but only due to the Konishi anomaly. I.e. from one-loop diagrams with heavy scalars $X^{adj}_{SU(N_c-\no)}$ and their fermionic superpartners with masses $\sim\mx$ inside. On the first "preliminary" stage these loop effects transform, in the weak coupling regime at the scale $\sim \mx$, the heavy quark operator ${\rm Tr\,} (X^{adj}_{SU(N_c-\no)})^2$ into the operator $\sim {\rm Tr\,}(\lambda\lambda)_{SU(N_c-\no)}$ of lighter ${\cal N}=1\,\, SU(N_c-\no)$ SYM gluinos. And the nonzero mean value of this latter originates and saturates only in the strong coupling and non-perturbative regime at much lower energies $\sim\langle\Lambda^{SU(N_c-\no)}_{{\cal N}=1\,\, SYM}\rangle\ll\mx$ .

There is now a large number of strongly coupled gluonia with the mass scale $\sim\langle\Lambda^
{SU(N_c-\no)}_{{\cal N}=1\,\,SYM}\rangle$ in this ${\cal N}=1$ SYM. The multiplicity of vacua in this sector remains equal $N_c-\no$ as it should be. All heavier original electric particles with masses $\sim m$ or $\sim\mx$ charged with respect to $SU(N_c-\no)$ remain weakly confined, but the string tension is larger now, $\sigma^{1/2}_{{\cal N}=2}\ll\sigma^{1/2}_{{\cal N}=1}\sim\langle\Lambda^{SU(N_c-\no)}_{{\cal N}=1\,\,SYM}\rangle\ll\mx\ll m$.

§.§ $SU(N_c)$ ,   special vacua

The values of quark condensates in these special vacua with $\no=\nd,\, \nt=N_c$, see [13, 9] and (<ref>),
⟨+-1/N_cTr ()⟩_N_c=(1-=/N_c)⟨⟩_N_c=m ,
⟨⟩_N_c=N_c/m=m_3 ,  ⟨⟩_N_c= ,  ⟨S⟩_N_c=⟨⟩_N_c⟨⟩_N_c/=
=(⟨⟩_N_c/^2N_c-N_F^N_c )^1/=⟨S⟩_2N_c-N_F=⟨Λ^SU()_N=2  SYM⟩^2=m_3 ,
are exact and valid at any values $m\gtrless\lm$. Therefore, the discrete $Z_{\bb}$ symmetry is formally broken spontaneously therein at large $m\gg\lm$ also, as it is really broken at $m\ll\lm$. But practically they behave at $m\gg\lm$ as the br1 vacua described above in section (4.1) with $\no=\nd\,$. I.e., the factor $\bb$ in their multiplicity $N_{\rm spec}=(\bb) C^{\,\nd}_{N_F}$ originates at $m\gg\lm$ from the multiplicity $\bb$ of the $SU(\bb)\,\,{\cal N}=2$ SYM at $\mx\ll\langle\Lambda^{SU(\bb)}_{{\cal N}=2\,\,SYM}\rangle$ or ${\cal N}=1$ SYM at $\langle\Lambda^{SU(\bb)}_{{\cal N}=1\,\,SYM}\rangle\ll\mx\ll m$, after the breaking $SU(N_c)\ra SU(\nd)\times U(1)\times SU(\bb)$ by $\langle X_{U(1)}\rangle$ at the scale $\mu= m_3\gg\lm$, see (<ref>). The factor $C^{\,\nd}_{N_F}$ originates due to higgsing of $\no=\nd$ original electric quarks in the $SU(\nd)$ sector, $\langle\Qo\rangle_{\nd}=\langle{\ov Q}^{\,1}_1
\rangle\langle Q^1_1\rangle=\mx m_3$, see (<ref>), and spontaneous breaking $U(N_F)\ra U(\nd)\times U(N_c)$. The two single roots with $( e^{+}-e^{-})\sim\langle\Lambda^{SU(\bb)}_{{\cal N}=2\,\,SYM}\rangle$ of the curve (<ref>) at small $\mx$ originate from the ${\cal N}=2\,\,SU(\bb)$ SYM. There are no massless particles at $\mx\neq 0$, except for $2\no\nt$ Nambu-Goldstone multiplets.

§ UNBROKEN FLAVOR SYMMETRY,   SYM VACUA,   $SU(N_C)$

These SYM vacua of the $SU(N_c)$ theory have the multiplicity $N_c\,$. $\langle X^{\rm adj}\rangle_{N_c}\sim
\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle\ll m$ in these vacua, so that all quarks have here large masses $m\gg\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle$ and decouple in the weak coupling regime
at $\mu<m/(\rm several)$. There remains at this scale the ${\cal N}=2\,\, SU(N_c)$ SYM with the scale factor of its gauge coupling: $\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle^2=(\lm^{\bb} m^
{N_F})^{1/N_c}=m^2\, (\lm/m)^{(\bb)/N_c},\,\,\,\lm\ll\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle\ll m$.

If $\mx\ll\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle$, then the field $X^{\rm adj}_{SU(N_c)}$ is higgsed at the scale $\mu\sim\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle$ in a standard way for the ${\cal N}=2$ SYM, $\langle\sqrt{2} X^{\rm adj}_{SU(N_c)}\rangle\sim\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle\,{\rm diag}\,(k_1,k_2,\,...\,k_{N_c}),\,\, k_i=O(1)$ [3], $SU(N_c)\ra U^{N_c-1}(1)$. Note that the value $\sim\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle$ of $\langle X^{\rm adj}_{SU(N_c)}\rangle$ is consistent with the unbroken $Z_{\bb}$ discrete symmetry.

All original electrically charged gluons and scalars acquire masses $\sim\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle$ and $N_c-1$ lighter Abelian pure magnetic monopoles $M_k$ are formed. These are higgsed at $\mx\neq 0$ and $N_c-1$ long ${\cal N}=2$ multiplets of massive dual photons are formed, with masses $\sim (\mx\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle)^{1/2}\ll\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle$ (there are non-leading contributions $\sim\mx\ll (\mx\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle)^{1/2}$ to the masses of corresponding scalars, these break slightly ${\cal N}=2$ down to ${\cal N}=1$ ). As a result, all original electrically charged quarks, gluons and scalars $X$ are weakly confined (i.e. the tension of the confining string, $\sigma^{1/2}_2\sim (\mx\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle)^{1/2}$, is much smaller than their masses $\sim m$ or $\sim\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle\,$).

All $N_c-1$ double roots of the curve (<ref>) correspond in this case to $N_c-1$ pure magnetic monopoles (massless at $\mx\ra 0$), while two single roots with $(e^+ - e^-)\sim\langle
\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle$ also originate from this ${\cal N}=2 \,\,SU(N_c)$ SYM.

The mass spectrum is different if $\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\ll\mx\ll m$. All scalar fields $X^{\rm adj}_{SU(N_c)}$ are then too heavy and not higgsed. Instead, they decouple as heavy at scales $\mu<\mx^{\rm pole}/(\rm several)$, still in the weak coupling regime, and there remains at lower energies ${\cal N}=1\,\, SU(N_c)$ SYM with its $N_c$ vacua and with the scale factor $\langle\Lambda^{SU(N_c)}_{{\cal N}=1\, SYM}\rangle$ of its gauge coupling, $\langle\Lambda^{SU(N_c)}_{{\cal N}=1\, SYM}\rangle=[\,\mx(\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle)^{\,2}\,]^{1/3},\,\, \langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle\ll\langle\Lambda^{SU(N_c)}_{{\cal N}=1\, SYM}\rangle\ll\mx\ll m$. A large number of strongly coupled gluonia with the mass scale $\sim\langle\Lambda^{SU(N_c)}_{{\cal N}=1\, SYM}\rangle$ is formed in this ${\cal N}=1$ SYM theory, while all heavier original charged particles are weakly confined (the tension of the confining string, $\sigma^{1/2}_1\sim\langle\Lambda^{SU(N_c)}_{{\cal N}=1\, SYM}\rangle$, is much smaller than the quark masses $\sim m$ or the scalar masses $\sim\mx$).

§ VERY SPECIAL VACUA WITH $\LANGLE S\RANGLE_{N_C}=0$ IN $U(N_C)$ GAUGE THEORY

We consider in this section the vs (very special) vacua with $\no=\nd,\,\nt=N_c,\, N_c < N_F<2 N_c-1$ , $\langle S\rangle_{N_c}=0$ ( as always, $\langle S\rangle_{N_c}$ is the bilinear gluino condensate summed over all its colors) and the multiplicity $N_{vs}=C^{\nd}_{N_F}=C^{N_c}_{N_F}$ in the $U(N_c)=SU(N_c)\times U^{(0)}(1)$ theory, when the additional Abelian $U^{(0)}(1)$ with $\mu_0=\mx$ is added to the $SU(N_c)$ theory (see e.g. [23] and references therein, these are called $r=N_c$ vacua in [23]). Note that vacua with $\langle S\rangle_{N_c}=0$ are absent in the $SU(N_c)$ theory with $m\neq 0,\,\, \mx\neq 0$, see Appendix in [13].

The superpotential of this $U(N_c)$ theory looks at high energies as
W_matter=[Tr (X_0)^2=1/2N_c a_0^2]+Tr (X^adj_SU(N_c))^2+Tr [ (m-a_0) Q Q-Q√(2) X^adj_SU(N_c) Q ]_N_c ,
√(2) X_0=a_0  diag(  1_N_c ) ,  X^adj_SU(N_c)=X^A T^A ,   Tr (T^A T^B)=1/2 δ^AB ,   A, B=1, ..., N_c^2-1 .

From (<ref>)
⟨a_0⟩=⟨Tr(Q  Q)⟩_N_c=⟨⟩_N_c
+N_c⟨⟩_N_c/N_c ,
⟨Tr (√(2) X^adj_SU(N_c))^2⟩=(2N_c-N_F)⟨S⟩_N_c+⟨m-a_0⟩⟨Tr ()⟩_N_c .

Note that, as a consequence of a unique property $\langle S\rangle_{N_c}=0$ of these $U(N_c)$ vs -vacua, the curve (<ref>) at $\mx\ra 0$ for theory (<ref>) has not $N_c-1$ but $N_c$ double roots: $\,\nd$ equal double roots $e_k= - m$ corresponding to low energy non-Abelian $SU(\nd)$, and $2N_c-N_F$ unequal double roots $e_j= -m+\omega^{j-1}\lm,\, j=1...\bb$, see e.g. [32]. There are no single roots of the curve (<ref>) in these vacua.

As previously in section 1, taking $\mx$ sufficiently large
The scale factor $\Lambda_0$ of the Abelian coupling $g_0(\Lambda_0/\mu)$ is taken sufficiently large, the theory (6.1) is considered only at scales $\mu\ll\Lambda_0$ where $g_0$ is small, the large $\mx$ means here  $\lm\ll m\ll\mx\ll\Lambda_0$ if $m\gg\lm$ (and only here, while $\mx\ll\lm$ in all other cases).
and integrating out all $N^2_c$ scalars $X$ as heavy, the superpotential of the $U(N_c)\,\,\, {\cal N}=1$ SQCD looks then as
W_matter=m Tr  ( Q Q)_N_c-1/2 ∑_i,j=1^N_F (Q_j Q^i)_N_c(Q_i Q^j)_N_c ,  (Q_j Q^i)_N_c=∑_a=1^N_c(Q_j^ a Q^i_a) ,
while, instead of (<ref>), $\w^{\,\rm eff}_{\rm tot}$ looks now as
^ eff_tot=W_matter-S_N_c ,  S_N_c=(()_N_c/^2N_c-N_F^N_c)^1/ ,  ⟨S⟩_N_c=(⟨⟩^⟨⟩^N_c/^2N_c-N_F^N_c)^1/_N_c.
From (<ref>), in these specific vs-vacua with $\langle S\rangle_{N_c}=0$, the equations for determining quark condensates look as
⟨Q_k∂^eff_tot/∂Q_k⟩= m⟨()_k⟩_
N_c-⟨()_k⟩^2_N_c/-⟨S⟩_N_c=0 ,  k=1, 2 ,
⟨+⟩_N_c=m ,  ⟨S⟩_N_c=⟨⟩_N_c⟨⟩_N_c/ .
The only self-consistent solution for these vs-vacua with $\no=\nd,\, \nt=N_c,\,\, \langle S\rangle_{N_c}=0$ looks as, see (<ref>)-(<ref>),
⟨⟩_N_c=m,   ⟨⟩_N_c=0,
⟨Q√(2) X^adj_SU(N_c) Q⟩_N_c=0,
⟨a_0⟩=⟨⟩_N_c/ = m ,  ⟨Tr (√(2) X^adj_SU(N_c))^2⟩=0 ,   ⟨^ eff_tot⟩=
⟨W_matter⟩=/2N_c ⟨a_0⟩^2=/2N_c m^2 .
(The variant with $\langle\Qo\rangle_{N_c}=\mx m,\,\,\langle\Qt\rangle_{N_c}=0$ and $\langle a_0\rangle=\nd m/N_c$ from (<ref>), will result at $m\gg\lm$ in all heavy non-higgsed quarks with masses ${\hat m}=\langle m-a_0\rangle=(\bb) m/N_c\gg\lm$, the lower energy $SU(N_c)$   ${\cal N}=2$ SYM, the multiplicity of vacua equal $N_c$, $\,\langle S\rangle_{N_c}=\mx (\lm^{\bb}{\hat m}^{N_F})^{1/N_c}\neq 0$, the unbroken flavor symmetry, and the massless photon. There will be not $N_c$ but $N_c-1$ unequal ${\cal N}=2$ SYM double roots at $\mx\ra 0$. This is incompatible with $\langle S\rangle_{N_c}=\langle\Qo\rangle_{N_c}\langle\Qt
\rangle_{N_c}/\mx\ra 0$ from (<ref>), and the multiplicity is wrong. While at $m\ll\lm$ this variant with $\langle a_0\rangle=\nd m/N_c$ from (<ref>) will be at least incompatible with (<ref>),(<ref>)).

§.§ $m\gg\lm\,,\,\, \mx\gg\lm^2/m \,,\,\, N_c < N_F < 2 N_c-1$

Consider first this simplest case with $0<\mx\ll\lm,\,\,m\gg\lm$ and $\mx m\gg\lm^2$. Note that $\langle a_0\rangle=m$ "eats" all quark masses $m$ in (<ref>) in the range of scales $m/(\rm several)<\mu<(\rm several) m$, see (<ref>) (as otherwise all quarks will decouple as heavy at $\mu<m/(\rm several)$ resulting in the wrong lower energy theory). In this region of parameters, $\nt=N_c$ quarks are higgsed at $\mu\sim (\mx m)^{1/2}\gg\lm$ in the weak coupling regime, e.g. (compare with (<ref>),(<ref>)),
⟨Q_i=b+^  b⟩=⟨Q^i=b+_b⟩= (m)^1/2≫ ,  b=1...N_c ,  i=+1...N_F ,
⟨X^adj,A_SU(N_c)⟩=√(2) Tr [⟨Q⟩T^A⟨Q⟩]/=0 ,  A=1...N_c^2-1 ,  ⟨a_0⟩=m ,  ⟨S⟩_N_c=0 .

Higgsed quarks with $N_c$ colors and $\nt=N_c$ flavors give in this case large masses $\sim (\mx m)^{1/2}\gg\lm$ to themselves and to all gluons and scalars, and simultaneously prevent the adjoint scalars (which are light by themselves, in the sense $\mx\ll\lm$) from higgsing.

From (<ref>),(<ref>), the multiplicity of these vs-vacua is $N_{vs}=C^{N_c}_{N_F}$, the factor $C^{N_c}_{N_F}$ originates in this phase from the spontaneous flavor symmetry breaking, $U(N_F)\ra U(N_c)\times U(\nd)$, due to higgsing of $\nt=N_c$ out of $N_F$ quarks. This multiplicity shows that the non-trivial at $2N_c-N_F\geq 2$ discrete $Z_{\bb}$ symmetry is unbroken.

As far as $\mx$ remains sufficiently larger than $\lm^2/m$, the mass spectrum includes in this phase: $N_c^2$ long ${\cal N}=2$ multiplets of massive gluons and their ${\cal N}=2$ superpartners with masses $\mu_{\rm gl}\sim (\mx m)^{1/2}\gg\lm$ and $2\no\nt=2\nd N_c$ massless Nambu-Goldstone multiplets (these are remained original electric quarks with $\nd$ flavors and $N_c$ colors). There are no heavier particles with masses $~\sim m$.

§.§ $m\gg\lm\,,\,\, \mx\ll\lm^2/m\,,\,\, N_c < N_F < 2 N_c-1$

Consider now the case of smaller $\mx$, such that $(\mx m)^{1/2}\ll\lm$, while $m$ remains large, $m\gg\lm$. The quark condensates still look as in (<ref>), and $\,\langle a_0\rangle=m$ stays intact and still "eats"  all quark masses $"m"$ in (<ref>). Therefore, because the $SU(N_c)$ theory is UV free, its gauge coupling grows logarithmically with diminished energy and, if nothing prevents, it will become $g^2(\mu)<0$ at $\mu<\lm$. Clearly, even if quarks were higgsed as above in section 6.1, i.e. $\langle Q^2\rangle=\langle{\ov Q}_2\rangle\sim (\mx m)^{1/2}$ (while $\langle X^{adj}_{SU(N_c)}\rangle
\ll\lm$), see (<ref>), this would result only in appearance of small particle masses $\sim (\mx m)^{1/2}\ll\lm$, so that all particles would remain effectively massless at the scale $\mu\sim\lm\gg (\mx m)^{1/2}$ (this is especially clear in the unbroken ${\cal N}=2$ theory at $\mx\ra 0$). Therefore, this will not help and the problem with $g^2(\mu<\lm)<0$ cannot be solved in this way (these quarks ${\ov Q}_2, Q^2$ are really not higgsed now at all, i.e. $\langle Q^i_b\rangle=\langle{\ov Q}_i^{\,b}\rangle=0,\, i=\nd+1...N_F,\, b=1...N_c$, see below).

As explained in Introduction, to really avoid the unphysical $g^2(\mu<\lm)<0$, the field $X^{adj}_{SU(N_c)}$ is higgsed necessarily in this case at $\mu\sim\lm$. Because the non-trivial $Z_{\bb\geq 2}$ discrete symmetry is unbroken in these vs-vacua, this results, as in section 2.1, in $SU(N_c)\ra SU(\nd)\times U^{(1)}(1)\times U^{\bb-1}(1)$, and main contributions $\sim\lm$ to particle masses originate now from this higgsing of $X^{adj}_{SU(N_c)}$,
⟨  X^adj_SU(N_c)⟩=⟨X^adj_SU()+X^(1)_U(1)+ X^adj_SU()⟩,
⟨√(2) X^adj_SU()⟩∼ diag ( 0_ ;
ω^0, ω^1, ... ω^-1_  ) ,  ω=exp{2πi/} ,
√(2)  X^(1)_U(1)=a_1 diag(  1_ ;  c_1
_),  c_1=- /,   ⟨Q^i_k⟩=⟨Q_i^ k⟩=0,  k=+1... N_c,  i=1... N_F.

Now, vice versa, higgsed $\langle X^{A}_{SU(\bb)}\rangle\sim\lm$ prevent quarks $Q^i_k,\, {\ov Q}^{\,k}_i,\, k=\nd+1...\,N_c,\, i=1...\,N_F$ from higgsing, i.e. $\langle Q^i_k\rangle=\langle{\ov Q}_i^{\,k}\rangle=0$. The reason is that these quarks acquire now masses $\sim\lm$, much larger than a potentially possible scale $\sim (\mx m)^{1/2}$ of their coherent condensate, see (<ref>). These heavy quarks are too short ranged now and it looks unrealistic that they can form the coherent condensate with  $\langle Q^i_k\rangle\neq 0$ as before in section 6.1. Besides, for independently moving heavy ${\ov Q}$ and $Q$, their physical (i.e. path dependent) phases induced interactions with the light $U^{2N_c-N_F}(1)$ photons fluctuate independently and freely at least at all scales $\mu > \mu^{\rm low}_{\rm cut}=(\rm several){\rm max}\{(\mx m)^{1/2},\, (\mx\lm)^{1/2} \}$. I.e., in any case, $\langle Q^i_k\rangle_{\mu^{\rm low}_{\rm cut}}=\langle{\ov Q}_i^{\,k}\rangle_{\mu^{\rm low}_{\rm cut}}=0$. Clearly, this concerns also all other heavy charged particles with masses $\sim\lm$. Therefore, after integrating out the theory (<ref>) over the interval from the high energy down to the scale $\mu^{\rm low}_{\rm cut}$, all particles with masses $\sim\lm$ decouple as heavy already at the scale $\mu<\lm/(\rm several)$, i.e. much above $\mu^{\rm low}_{\rm cut}$, and definitely the heavy quarks by themselves do not give any contributions to masses of remaining lighter degrees of freedom in the Lagrangian (<ref>) because $\langle Q^i_k\rangle_{\mu^{\rm low}_{\rm cut}}=\langle{\ov Q}_i^{\,k}\rangle_{\mu^{\rm low}_{\rm cut}}=0$.
And after the heavy particles decoupled at $\mu<\lm/(\rm several)$, they themselves do not affect the lower energy theory. This is exactly the same situation as with heavy quarks with masses $\sim\lm$ discussed in detail in section 2.1 (and qualitatively the same as with $X^{adj}_{SU(\nd)}$ with masses $\sim\mx\gg (\mx m)^{1/2}$ in section 2.4), and only the parameter $m\gg\lm$ in this section while $m\ll\lm$ in sections 2.1 and 2.4, and always $\mx\ll\lm$.

The lower energy original electric ${\cal N}=2\,\,SU(\nd)$ theory at the scale $\mu=\mu^{\rm low}_{\rm cut}\ll\lm$, with $2\nd<N_F<2N_c-1$ flavors of remained light quarks $Q^i_a, {\ov Q}_j^{\,a},\,\, a=1...\nd,\,\, i,j=1...N_F$, is IR free and weakly coupled. The whole matter superpotential has the same form as (<ref>), but with  omitted $a_2$ and $SU(\nd-\no)$ SYM part, $\no=\nd$, $m\ra (m-a_0)$, and with the addition of $\sim\mx a_0^2\,$,
_N_c=_ +_D+_a_0, a_1 ,
_  =(m-a_0-a_1)Tr (Q Q)_-Tr (Q√(2)X_SU()^adj Q)_+(1+δ_2)Tr (X^adj_SU())^2 ,
_D=( m-a_0-c_1 a_1 )∑_j=1^D_j D_j -∑_j=1^ a_D,jD_j D_j
- ∑_j=1^ω^j-1 a_D,j+L ( ∑_j=1^ a_D,j) ,
_a_0, a_1=/2N_c a^2_0+/2N_c/ (1+δ_1) a^2_1+
N_cδ_3  a_1(m-a_0-c_1 a_1)+N_cδ_4  (m-a_0-c_1 a_1)^2 ,
where $\delta_1$ and $\delta_2$ are the same as in (<ref>) (and $\delta_3=0$, see (<ref>)).

Proceeding now similarly to section 2.1, we obtain from (<ref>)
⟨a_0⟩=m ,  ⟨a_1⟩=0 ,  ⟨a_D,j⟩=0 ,
⟨X_SU()^adj⟩=0 ,
⟨D_j D_j⟩=⟨D_j⟩⟨D_j ⟩= - ω^j-1+⟨L⟩,   ⟨L⟩=m,   ⟨Σ_D⟩=∑_j=1^⟨D_j⟩⟨D_j ⟩=()m,
⟨⟩_=⟨Q^1_1⟩⟨Q^1_1⟩=m=⟨⟩_N_c,   ⟨⟩_=∑_a=1^⟨Q^ a_2⟩⟨Q^2_a⟩=0=⟨⟩_N_c ,
⟨Tr (Q Q)⟩_=⟨⟩_+N_c⟨⟩_=m ,   ⟨S⟩_=⟨⟩_⟨⟩_=0.

From (<ref>), the multiplicity of these vs -vacua in this phase is $N_{vs}=C^{\nd}_{N_F}=C^{N_c}_{N_F}$ as it should be, the factor $C^{\nd}_{N_F}$ originates now from the spontaneous flavor symmetry breaking, $U(N_F)\ra U(\nd)\times U(N_c)$, due to higgsing of $\nd$ quarks in the $SU(\nd)$ color sector.

On the whole, the mass spectrum looks now as follows.

1) Due to higgsing of $X^{adj}_{SU(\bb)}$ at the scale $\sim\lm$, $\,\,SU(N_c)\ra SU(\nd)\times U^{(1)}(1)\times U^{\bb-1}(1)$, there is a large number of original pure electrically $SU(\bb)$ charged particles with the largest masses $\sim\lm$. They all are weakly confined due to higgsing at the scale $\mu\sim (\mx m)^{1/2}$ of mutually non-local with them $\bb$ BPS dyons $D_j, \ov D_j$ with the nonzero $SU(\bb)$ adjoint magnetic charges, the string tension is $\sigma^{1/2}_D\sim \langle{\ov D}_j D_j\rangle^{1/2}\sim (\mx m)^{1/2}\ll\lm$. These original electrically charged particles form hadrons with the mass scale $\sim\lm\,$. Clearly, higgsed $\langle X^{adj}_{SU(\bb)}\rangle\sim\lm$ does not break by itself the global flavor symmetry. All heavy quarks with $SU(\bb)$ colors have equal masses and are in the (anti)fundamental representation of the non-Abelian $SU(N_F)$, they form e.g. a number of $SU(N_F)$ adjoint hadrons with masses $\sim\lm$. All original heavy gluons and scalars are flavorless, and e.g. the $SU(\bb)$ adjoints have different masses. There is no color-flavor locking in this heaviest sector : the global $U(N_F)$ is unbroken while $SU(\bb)\ra U^{\bb-1}(1)$.

2) Due to higgsing of these dyons $D_j, {\ov D}_j$, there are $\bb$ long ${\cal N}=2\,\, U(1)$ multiplets of massive photons, all with masses $\sim \langle{\ov D}_j D_j\rangle^{1/2}\sim (\mx m)^{1/2}$.

3) There are $\nd^{\,2}$ long ${\cal N}=2$ multiplets of massive gluons with masses $\sim \langle\Qo\rangle^{1/2}_{\nd}\sim (\mx m)^{1/2}\ll\lm$ due to higgsing of $\nd$ flavors of original electric quarks with $SU(\nd)$ colors.

4) $2 \no\nt=2\nd N_c$ Nambu-Goldstone multiplets are massless (in essence, these are remained original electric quarks with $\nt=N_c=N_F-\nd$ flavors and $SU(\nd)$ colors). There are no other massless particles at $\mx\neq 0$.

5) There are no particles with masses $\sim m\gg\lm$.

The $U(N_c)$ curve (<ref>) has in this phase at $\mx\ra 0$   $\nd$ equal double roots $e_k= - m$ corresponding to $\nd$ out of $N_F$ higgsed original electric quarks with $SU(\nd)$ colors, and $\bb$ unequal double roots $e_j= -m+\omega^{j-1}\lm$ corresponding to dyons $D_j$.

It is worth recalling once more that higgsed $\langle\, X^{adj}_{SU(\bb)}\rangle\sim\lm$ in (<ref>) does not break by itself the global flavor symmetry, it is really broken only in the $SU(\nd)$ color sector by higgsed $\nd$ quarks with $N_F$ flavors and $SU(\nd)$ colors.

In essence, the situation here with the heavy non-higgsed quarks $Q^i_k, {\ov Q}^{\,k}_i,\, i=1...N_F,\, k=\nd+1...N_c$ with masses $\sim\lm$ and with their chiral bilinear condensates (really, it is more adequate to speak about mean values, using the word "condensate" only for "genuine", i.e. coherent, condensates of higgsed fields) is the same as in br2 vacua of section 2.1, see (<ref>)-(<ref>) and accompanying text. In the lower energy phase of this section at $\mx\ll\lm^2/m$, the nonzero mean values of non-factorizable local bilinears $\langle(\qq)_{1.2}\rangle_{\bb}$ of these heavy quarks with masses $\sim\lm$ originate from: a) the quantum loop effects in the strong coupling (and non-perturbative) regime at the scale $\mu\sim\lm$, transforming bilinears operators of heavy quark fields into bilinears operators of light $SU(\nd)$ quark fields and the operator $a_1$; b) and finally, from formed at much lower scales $\mu\sim (\mx m)^{1/2}\ll\lm$ the "genuine" (i.e. coherent) condensates of light higgsed quarks, $\langle{\ov Q}^1_1\rangle=\langle Q^1_1\rangle\neq 0$. I.e., the equations (<ref>),(<ref>) and (<ref>),(<ref>) (with $A=-2$) remain the same and only the values of condensates are different in br2 and vs vacua, see (<ref>).

In this section, the corresponding values of condensates of heavy quarks with $SU(\bb)$ colors, $N_F$ flavors, and masses $\sim\lm$ in these vs-vacua with $\nt=N_c,\, \no=\nd$, and $\langle S\rangle_{N_c}=0$ differ from those in the br2 vacua with $\nt>N_c,\, \no<\nd$ and $\langle S\rangle_{N_c}\neq 0$ in section 2.1 only by: 1) $m_1\ra m$ due to $SU(N_c)\ra U(N_c)$,   2) the absence of non-leading power corrections originating in br2 vacua from the SYM part, see (<ref>),(<ref>), compare with (<ref>), (<ref>),(<ref>),
⟨⟩_N_c=[⟨⟩_=m ]+[⟨⟩_=∑_a=1^⟨Q^ a_2⟩⟨Q^2_a⟩=0 ]=m ,
⟨⟩_N_c=[⟨⟩_=⟨Q^ 1_1⟩⟨Q^1_1⟩=m ]+[⟨⟩_= - m ]=0 ,
⟨Tr ⟩_=N_c⟨⟩_+⟨⟩_=⟨Σ_D⟩=(2N_c-N_F)m ,
while $\langle \sqrt{2} X^{adj}_{SU(\bb)}\rangle$ in (<ref>) is the same as (<ref>) and differs from (<ref>).

There is one important qualitative difference between the vs vacua and br2 vacua. Because (<ref>),
(<ref>),(<ref>),(<ref>) remain the same at $m\lessgtr\lm$, and $\langle a_0\rangle=m$ still "eats" all quark masses $"m"$ in (<ref>) at all $m\lessgtr\lm$, the whole physics in vs vacua is qualitatively independent separately of the ratio $m/\lm$ and depends only on a competition between $(\mx m)^{1/2}$ and $\lm$. For this reason, the lower energy phase with $m\gg\lm,\, (\mx m)^{1/2}\ll\lm$ in vs vacua of this section continues smoothly into the region $\mx\ll m\ll\lm$. This is in contrast with the br2 vacua with $\mx\ll m\ll\lm$ in section 2.1. In the latter, all properties depend essentially on the ratio $m/\lm$ and they evolve into the qualitatively different br1 vacua at $m\gg\lm$.

Comparing the mass spectra at $(\mx m)^{1/2}\gg\lm$ in section 6.1 with those in this section at $(\mx m)^{1/2}\ll\lm$ one can see the qualitative difference between these two cases. The lower energy phase with $\mx\ll\lm^2/m$ in this section is, in essence, the phase of the unbroken ${\cal N}=2$ theory because it continues down to $\mx\ra 0$. While the higher energy phase of section 6.1 with $\mx\gg\lm^2/m$, because it cannot be continued as it is down to $\mx\ra 0$, is, in essence, the phase of the strongly broken ${\cal N}=2$ theory (i.e. already the genuine ${\cal N}=1$ theory in this sense). Remind that the applicability of the whole machinery with the use of roots of the curve (<ref>) is guaranteed only at $\mx\ra 0$. I.e., it is applicable only to the lower energy phase of this section, and clearly not applicable to the higher energy phase of section 6.1, in spite of that $\mx\ll\lm$ therein also.

And these two phases at $\mx\gtrless\lm^2/m$ are qualitatively different, compare e.g. (<ref>) and (<ref>). (Otherwise, if e.g. $\langle X^{adj}_{SU(\bb)}\rangle$ were higgsed at $\mx\gg \lm^2/m$ in section 6.1 as in (<ref>), it would give then in particular the nonzero masses $\sim\lm\ll (\mx m)^{1/2}$ to $2\nd (\bb)$ out of previously massless $2\nd N_c$ Nambu-Goldstone multiplets in section 6.1. And this will be clearly wrong because the number of massless Nambu-Goldstone multiplets has to be exactly $2\nd N_c$).

In the higher energy phase in section 6.1 all charged gluons from $SU(\bb)$ have exactly equal masses $| g (\mx m)^{1/2}|$ in the whole wide region from $(\mx m)^{1/2}\gg\lm$ down to $(\mx m)^{1/2} >(\rm several)\lm$, see  (<ref>).

On the other hand, in the lower energy phase of this section, the masses of these $SU(\bb)$ gluons differ greatly at $\mx\ra 0:\, \mu_{\rm gl}^{ij}\sim |(\omega^{i}-\omega^{j})\lm |\,,\,\, i\neq j,\,\,i,j=0...\bb-1$. This shows that the dependence of masses of these gluons on $\mx m$ is non-analytic. Besides, the mean values of $\langle X^{adj}_{SU(N_c)}\rangle$ and $\langle Q^i_k\rangle=\langle{\ov Q}_i^{\,k}\rangle,\, k=\nd+1...\,N_c,\, i=1...\,N_F$ behave non-analytically, compare (<ref>) and (<ref>). In this sense, we can say that there is a phase transition somewhere at $(\mx m)^{1/2}\sim\lm$. This is in contradiction with a widespread opinion that there are no phase transitions in the supersymmetric theories because the gauge invariant condensates of chiral fields ( e.g. $\langle (\qq)_{1,2}\rangle_{N_c},\,\langle S\rangle_{N_c},\, \,\langle{\rm Tr\,}(X^{adj}_{SU(N_c)})^2\rangle$ in (<ref>),(<ref>)), depend holomorphically on the superpotential parameters.

This phenomenon with the phase transition of the type described above occurring at very small $\mx\ll\lm$ but $\mx m\gg\lm^2$ to the strongly broken ${\cal N}=2$, i.e. ${\cal N}=2\ra {\cal N}=1$, where the use of roots of the curve (<ref>) becomes not legitimate, appears clearly only at $m\gg\lm$. Let us start e.g. with the nearly unbroken ${\cal N}=2$ theory at sufficiently small $\mx$ where, in particular, the whole machinery with the use of roots of the curve (<ref>) is still legitimate. In this section, when $\mx$ begins to increase, the phase transition occurs at $\mx\sim (\lm^2/m)\ll\lm$. But in other cases, see sections 7 and 8 below, it occurs even at parametrically smaller values of $\mx$ than in this section.
In br2 vacua of section 7 it occurs at $\mx\sim \langle\Lambda_{SU(\nt)}\rangle^2/m,\, \langle\Lambda_
{SU(\nt)}\rangle\ll\lm$, see (<ref>), and in S- vacua of section 8 it occurs at $\mx\sim \langle\Lambda_{SU(N_F)}\rangle^2/m,\, \langle\Lambda_{SU(N_F)}\rangle\ll\lm$, see (<ref>).

Let us compare now the corresponding parts of unbroken non-Abelian global symmetries of two phases, those in section 6.1 and here. It is $SU(N_c)_{C+F}\times SU(\nd)_F$ in section 6.1, where "C+F" denotes the color-flavor locking. The higgsed massive electric quarks $Q^{i=b+\nd}_b, {\ov Q}^{\,b}_{i=b+\nd},\,\, b=1...N_c,\, i=\nd+1...N_F$, with masses $m_Q\sim (\mx m)^{1/2}\gg\lm$, together with massive electric $SU(N_c)$ gluons and $X^{adj}_{SU(N_c)}$ scalars form the long ${\cal N}=2$ multiplet in the adjoint representation of  $SU(N_c)_{C+F}$. The $2\nd N_c$ Nambu-Goldstone multiplets are bifundamental.

The corresponding non-Abelian global symmetry in this section is $SU(\nd)_{C+F}\times SU(N_c)_F$ in the lighter sector, and higgsed massive electric quarks $Q^{i=a}_a, {\ov Q}^{\,a}_{i=a},\,\, a,i=1...\nd$, with masses $m_Q\sim (\mx m)^{1/2}\ll\lm$, together with $SU(\nd)$ gluons and $X^{adj}_{SU(\nd)}$ scalars form the long ${\cal N}=2$ multiplet in the adjoint representation of $SU(\nd)_{C+F}$, see the point "3" just above. The $2\nd N_c$ Nambu-Goldstone multiplets are also bifundamental. But, in addition, all $N_F$ flavors of original heavy quarks with $SU(\bb)$ colors have equal masses, so that the global $SU(N_F)$ is unbroken in this heavy sector. In the limit $\mx\ra 0$ these heavy quarks are not confined and form the (anti)fundamental representation of $SU(N_F)$. All original heavy unconfined at $\mx\ra 0$ $\,SU(\bb)$-adjoint and hybrid gluons and scalars with masses $\sim\lm$ are clearly flavor singlets. At small $\mx\neq 0$ all these heavy charged particles are weakly confined and form hadrons with masses $\sim\lm$, but this confinement does not break by itself $SU(N_F)$. In particular, the heavy quarks form a number of $SU(N_F)$ adjoint hadrons.

The difference of adjoint representations $SU(N_c)_{C+F}\leftrightarrow SU(\nd)_{C+F}$ at $\mx m\gtrless
\lm^2$ was considered in [24],[23] as an evidence that all $SU(\nd)$ particles cannot be the same original pure electric particles and, by analogy with [18], it was claimed that they are all dyons with nonzero magnetic charges. As it is seen from the above, this line of reasonings is not right. The reason for different adjoint representations is that the phase and mass spectrum are determined by $N_c$ out of $N_F$ higgsed quarks of $SU(N_c)$ at $\mx m\gg\lm^2$ (and $X^{adj}_{SU(N_c)}$ is not higgsed), while they are determined by higgsed both $X^{\rm adj}_{SU(2N_c-N_F)}$ and $\nd$ out of $N_F$ quarks of $SU(\nd)$ at $\mx m\ll\lm^2$ (both phases in section 6.1 and here at $m\gg\lm$, and the case $\mx\ll m\ll\lm$ clearly corresponds to the regime $\mx m <\lm^2$, see section 6.3 below).

Besides, it is worth noting the following.  It is erroneous to imagine that, because $\langle\Qt\rangle_{N_c}=\mx m$ still in the $\mx m\ll\lm^2$ phase of this section, then all $n_2 = N_c$ quarks are still higgsed as previously in section 6.1 at $\mx m\gg\lm^2$, i.e. $\langle{\ov Q}^{\,b}_{i=b+\nd}\rangle=\langle Q^{i=b+\nd}_b\rangle=(\mx m)^{1/2},\,\,b=1...N_c,\,\,i=\nd+1...N_F$, and so give corresponding contributions to particle masses, but because $(\mx m)^{1/2}\ll\lm$ now, this will be of little importance. It is sufficient to notice that, if it were the case, then in particular such higgsed quarks would give not only small corrections  $\sim (\mx m)^{1/2}\ll\lm$ to masses $\sim\lm$ of heavy particles, but all $2\nd Nc$ massless Nambu-Goldstone multiplets in (<ref>) would also acquire additional masses $\sim (\mx m)^{1/2}$ in this case, and this is clearly unacceptable.

And finally, we would like to especially emphasize (and this concerns the whole content of this paper, not only those of this section) that, for higgsed fields giving nonzero contributions to particle masses, the mean values of fields themselves should be nonzero, i.e. $\langle Q\rangle=\langle{\ov Q}\rangle\neq 0$ or $\langle X\rangle\neq 0$,
But even this can be sometimes insufficient as explained e.g. at the end of section 2.4.
not only something like $\langle Q^{\dagger} Q\rangle^{1/2}\neq 0$ or $\langle X^{\dagger} X\rangle^{1/2}\neq 0$ for the D-terms, or $\langle{\ov Q} Q\rangle^{1/2}\neq 0,\, \langle X^2\rangle^{1/2}\neq 0$ for the F-terms (which all are, as a rule, nonzero even for those fields which are really not higgsed).

This is especially clearly seen  e.g. from the mass terms of fermionic superpartners. For the D-terms of higgsed quarks or $X$ these look as:
Tr [ ( ⟨Q^ †⟩  χ+⟨Q^ †⟩ χ )λ ]  or  Tr [ ⟨X^†⟩ ψλ ] ,
and for the F-terms they look as
Tr [ ( ⟨Q⟩ χ+⟨Q⟩ χ )ψ ]  or  Tr [⟨X⟩ χ χ ] .

§.§ $m\ll\lm$

Comparing with the case $\mx\ll\lm,\,\, m\gg\lm$ in sections 6.1-6.2, the case $\mx\ll m\ll\lm$ corresponds clearly the regime $\mx\ll\lm^2/m$. The quark condensates remain the same as in (<ref>), and $\langle a_0\rangle=m$ remains the same as in (<ref>) and still "eats" all quark masses $"m"$. The phase remains the same as in section 6.2, and at $m\ll\lm$ the difference with the case $m\gg\lm,\,\, \mx\ll\lm^2/m$ in section 6.2 is only quantitative, not qualitative, see (<ref>). The main difference is that $\langle D_j\rangle\sim (\mx m)^{1/2}\gg (\mx\lm)^{1/2}$ at $m\gg\lm$, while $\langle D_j\rangle\sim (\mx\lm)^{1/2}\gg (\mx m)^{1/2}$ at $m\ll\lm$. Still, there are no particles with masses $\sim m$, the only particle masses independent of $\mx$ are $\sim\lm$.

§.§ $N_F=N_c$

Now, in short about the vs vacuum with $N=N_F=N_c=\nt,\, \nd=0=\no$ at $m\lessgtr\lm$ and $\mx\ll\lm$. There is only one such vacuum. And really, there is nothing extraordinary with this case.

§.§.§ Higher energy phase at $(\mx m)^{1/2}\gg\lm$

As always in all vs vacua, see (<ref>),(<ref>), $\langle\qq\rangle_{N_c}=\mx m,\, \langle S\rangle_{N_c}=0$ in this vacuum, and $\langle a_0\rangle=m$ "eats" all quark masses. The whole color group $U(N_c)$ is broken in this higher energy phase as before in vs-vacua of section 6.1 by higgsed quarks at the scale $\mu\sim g(\mx m)^{1/2}\gg\lm$ in the weak coupling regime, while $\langle X^{adj}_{SU(N_c)}\rangle=0$ is not higgsed, see (<ref>). All $SU(N_c)$ gluons have equal masses $\sim (\mx m)^{1/2}$ in this phase in the whole wide region from $(\mx m)^{1/2}\gg\lm$ down to $(\mx m)^{1/2} >(\rm several)\lm$. There is the residual global non-Abelian $SU(N)_{C+F}$ symmetry (i.e. color-flavor locking). $N^2-1$ long ${\cal N}=2$ multiplets of massive gluons are formed in the adjoint representation of this remained unbroken global symmetry $SU(N)_{C+F}$. There are no particles with masses $\sim m$.

The global $SU(N)_{F}$ is substituted by global $SU(N)_{C+F}$, so that there are no massless Nambu-Goldstone particles in this vacuum. And there are no massless particles at all in this phase.

§.§.§ Lower energy phase at $(\mx m)^{1/2}\ll\lm$

As in all vs-vacua, the total mean value of higgsed $a_0=\langle a_0\rangle^{(\rm several) m}_{m/(\rm several)}+[\,{\hat a}_0^{\rm soft}\,]^{m/(\rm several)}$ originates and saturates at $\mu\sim m$ and "eats" masses "m" of all quarks
⟨a_0⟩=⟨ Tr ⟩_N_c/N_c=m ,  ⟨a_0⟩=[ ⟨a_0⟩^(several) m
_m/(several)=m ]+[ ⟨â_0^soft⟩^m/(several)_μ_cut^lowest=0=0  ] .
In this phase, at all $m\gtrless\lm$, to avoid entering the unphysical regime $g^2(\mu<\lm)<0$ in this UV free theory, the non-Abelian color $SU(N_c)$ is also broken, but now in the strong coupling and non-perturbative regime at the scale $\mu\sim \lm$ by higgsed $\langle X^{adj}_{SU(N_c)}\rangle\sim\lm,\, SU(N_c)\ra U^{N_c-1}(1)$,
X^adj_SU(N_c)=⟨X^adj_SU(N_c)⟩^(several)_/(several)+[  X̂^adj,soft_SU(N_c)  ]^/(several) ,
⟨√(2) X^adj_SU(N_c)⟩^(several)_/(several)∼⟨⟩ diag (
ρ^0, ρ^1, ..., ρ^N_c-1 ),  ρ=exp{ 2πi/N_c } ,  ⟨X̂^adj,soft_SU(N_c)⟩^/(several)_μ_cut^lowest=0=0 .

The unbroken non-trivial discrete symmetry is $Z_{2N_c-N_F}=Z_{N\geq 2}$ in this vacuum, and it determines the standard pattern of this color symmetry breaking by higgsed $\langle X^{adj}_{SU(N_c)}\rangle$ giving the largest contributions $\sim\lm$ to particle masses. All original charged particles acquire masses $\sim\lm$, while $N_c=N$ light flavorless BPS dyons $D_j$ (massless at $\mx\ra 0$) are formed at the scale $\mu\sim\lm$.

Clearly, higgsed flavorless $X^{adj}_{SU(N_c)}$ does not break by itself the flavor symmetry $SU(N_F)$, i.e. there is no flavor-color locking. The global non-Abelian flavor symmetry $SU(N_F)=SU(N)_F$ remains unbroken, while the non-Abelian color $SU(N_c)=SU(N)_C$ is broken down to Abelian one.

As before in section 6.2 for heavy quarks with $SU(\bb)$ colors, all quarks have equal masses, but they are too heavy and too short ranged now and cannot form the coherent condensate. I.e., they are not higgsed in this phase, $\langle Q^i_a\rangle=\langle{\ov Q}^{\,b}_k\rangle=0,\, i,k,a,b=1...N$.  All equal mass quarks are clearly in the (anti)fundamental representation of unbroken global $SU(N)_F$, while the non-Abelian color $SU(N)_C$ is broken at the scale $\mu\sim\lm$ down to Abelian $U^{N-1}(1)$, and masses of all charged flavorless gluons (and scalars) differ greatly in this lower energy phase at $\mx\ra 0:\, \mu_{\rm gl}^{ij}\sim |(\rho^{i}-\rho^{j})\lm |\,,\,\, i\neq j,\,\,i,j=1...N$, see (<ref>). There is the phase transition at $(\mx m)^{1/2}\sim\lm$.

After all heavy particles with masses $\sim\lm$ decoupled at $\mu=\lm/(\rm several)$, the lower energy superpotential at the scale $\mu=\lm/(\rm several)$ looks as
_N_F=_D+_a ,
_D= (m-a_0)∑_j=1^ND_j D_j -∑_j=1^N a_D, jD_j D_j -∑_j=1^Nρ^ j-1 a_D, j+L (∑_j=1^N a_D, j)
_a=/2 N  a_0^2+N δ̂_4  (m-a_0)^2 .

From (<ref>)
⟨a_0⟩=m ,  ⟨a_D, j⟩=0 ,  ⟨D_j⟩⟨D_j ⟩= -ρ^ j-1+m ,
⟨Σ_D⟩=∑_j=1^N⟨D_j⟩⟨D_j⟩=Nm=⟨Tr ⟩_N_c .
All $N$ dyons are higgsed at small $\mx$ at the scale $\mu_D\sim g_D\langle{\ov D}_j D_j\rangle^{1/2},\, \mu_D\sim g_D\max [ (\mx m)^{1/2},\,(\mx\lm)^{1/2}]$ (at $m\gg\lm$ below in this section for definiteness), in the weak coupling regime $g_D\ll 1$, and $N$ long ${\cal N}=2$ multiplets of massive photons are formed with masses $\sim \mu_D\ll\lm$. All original heavy charged particles with masses $\sim\lm$ are weakly confined, the string tension is $\sigma^{1/2}\sim\langle{\ov D}_j D_j\rangle^{1/2}\ll\lm$. In particular, the heavy flavored quarks form a number of $SU(N_F)$ adjoint hadrons with masses $\sim\lm$. There are no massless Nambu-Goldstone particles because the global flavor symmetry $U(N_F)$ is unbroken.

As for non-factorizable condensates of heavy unhiggsed in this lower energy phase quarks with masses $\sim\lm$, (<ref>) looks similarly here, with $(\bb)\ra N_c$, as well as (<ref>). Instead of (<ref>) we have here
[ Tr ()_N_c ]^m/(several)_/(several)=[ Tr ()_N_c ]^(several)_/(several)=[ d̂_1
â^soft_0+Σ_D ]^/(several) ,  d̂_1=O(1) .
This operator expansion leads to the numerical equality
⟨Tr ()_N_c⟩^(several)_μ_cut^lowest=0=⟨Σ_D⟩=⟨Σ_D⟩^(several)g_D (m)^1/2_g_D (m)^1/2/(several) ,  ⟨â^soft_0⟩^/(several)_μ_cut^lowest=0=0 ,  m≫
as it should be, see (<ref>),(<ref>). As before in section 6.2 for $\langle{\rm Tr\,}\qq\rangle_{\bb}$ at $\nd\geq 1$, the nonzero total mean value of the heavy quark non-factorizable bilinear operator originates and saturates not at the scale $\mu\sim\lm$, but only at much lower energies $\mu_D\sim g_D (\mx m)^{1/2}$.

Moreover, at $\mx\ra 0$, because $\langle a_0\rangle=m$ "eats" all quark masses, all charged solitons will have masses either $\sim\lm$ or zero. But the $U(N_c)$ curve (<ref>) has only $N_c$ unequal double roots $e^{(D)}_j= - m +\rho^{j-1}\lm,\, j=1...N_c$ in this vacuum corresponding to $N_c=N$ massless BPS dyons $D_j$, and this shows that there are no additional charged ${\cal N}=2$ BPS solitons massless at $\mx\ra 0$. In particular, there are no massless pure magnetic monopoles, they all have masses $\sim\lm$. And there are no massless particles at all at small $\mx\neq 0$. Besides, there are no particles with masses $\sim m\gg\lm$, the largest masses are $\sim\lm$.

In any case, there is no confinement or charge screening due to higgsed quarks or dyons in the limit $\mx\ra 0$ of unbroken ${\cal N}=2$ SQCD. Therefore, because pure magnetic monopoles with $SU(N_c)$ adjoint charges are flavorless, the free flavored quarks cannot decay into these flavorless monopoles. And this remains true in this phase at $\mx\ll\lm^2/m,\,\, m\gtrless\lm$, when flavorless dyons are higgsed and these heavy non-higgsed quarks in the (anti)fundamental representation of unbroken $SU(N)_F$ become weakly confined. Remind that higgsed $\langle\, X^{adj}_{SU(\bb)}\rangle\sim\lm$ in (<ref>) also does not break by itself the global flavor symmetry $SU(N_F)$ in the $SU(\bb)$ color sector, it is broken spontaneously only in the $SU(\no)$ color sector by light higgsed $\no=\nd\geq 1$ quarks with $N_F$ flavors. And similarly in section 6.3 and in br2 vacua of section 2.1 with $1\leq\no<\nd$. But $\no=\nd=0$ in this section. And on the whole, it is clear that the only real difference between the vs-vacua in section 6.2 and vacua of this section is that $\nd=N_F-N_c\geq 1$ in section 6.2, so that there remains the light $SU(\nd)$ sector, while it is absent in this section with $\nd=0$. But all properties of the heavy sectors, i.e. $SU(\bb<N_c)$ in section 6.2 and $SU(\bb=N_c)$ here, are the same.

The above described picture in section 6.2 and in this section, with the phase transition at $\mx m\gtrless\lm^2$ (to avoid entering the unphysical regime with $g^2(\mu<\lm)<0$ without higgsed $\langle X^{adj}_{SU(\bb)}\rangle\sim\lm$), is in a clear contradiction with the proposed by M.Shifman and A.Yung  a smooth (i.e. analytical) crossover in the region $\mx m\sim\lm^2$ to the "instead-of-confinement" phase with still unbroken non-Abelian $SU(N)_{C+F}$ global symmetry at $\mx m\ll\lm^2$, as it was in section 6.4.1 above. (And not only for $N_F=N_c$ vs-vacua of this section but for all vs-vacua). See e.g. the most recent paper [33] and refs therein to their previous papers with, in particular, this "instead-of-confinement" phase in vs-vacua of section 6.2 and in br2 vacua of section 2.1.

According to [33], in this "instead-of-confinement" phase in the region $0 < (\mx m)^{1/2}\ll\lm$ of this section. -

a) The light charged particles (massless at $\mx\ra 0$) are solitonic BPS dyons ${\ov d}_j, d_j$ which have the same charges as quarks ${\ov Q}^{\,j}_j, Q^j_j,\, j=1...(\bb)$  ($\bb=N$ in this section), but they are not quarks but solitons. These dyons are all higgsed at $\mx\neq 0$. For comparison, let us put attention that with higgsed $\langle X^{adj}_{SU(N_c)}\rangle\sim\lm$ (<ref>), all BPS solitons with even one out of $U^{N-1}(1)$ pure electric charge have large masses $\sim\lm$.

b) In the vs-vacua with $N_F=N_c=N$ considered in this section, all original quarks ${\ov Q}^{\,a}_i, Q^i_a$ and all original adjoints became heavy, with mysteriously appeared large equal masses $\sim\lm$. And the unbroken $SU(N)_{C+F}$ global symmetry implies that, as in section 6.4.1 above, they still form the long ${\cal N}=2$ adjoint multiplet of the global $SU(N)_{C+F}$. It is not even attempted in [33] to recognize the real dynamical mechanism responsible for the origin of these large equal masses $\sim\lm$. And it is not attempted to answer the question: are the properties of this dynamical mechanism compatible with the unbroken global $SU(N)_{C+F}$ symmetry ? Recall that the described above in this section dynamical mechanism responsible for the appearance of large masses $\sim\lm$ of all original charged particles, i.e. higgsed $\langle X^{adj}_{SU(N)}\rangle\sim\lm$ (<ref>), is incompatible with the unbroken global $SU(N)_{C+F}$ requiring equal masses of all $N^2-1$  $\,SU(N)_{C+F}$ adjoint particles. E.g., all charged $SU(N)$ adjoint gluons have different masses.

c) In order not to spoil global $SU(N)_{C+F}$, the unrecognized dynamical mechanism giving largest equal masses $\sim\lm\gg (\mx m)^{1/2}$ to all original particles has to be $SU(N)_{C+F}$ singlet. Moreover, the smooth crossover implies the analytical behavior of all quantities at $\mx m\gtrless\lm^2$. In particular, all quarks will still be higgsed as in section 6.4.1, i.e. $\langle{\ov Q}^{\,a}_i\rangle=\langle Q^i_a\rangle=\delta ^i_a\,\mx m,\,\,i,a=1...N$, giving equal smaller contributions $\sim (\mx m)^{1/2}\ll\lm$ to masses, both themselves and to all $SU(N)$ original adjoints. As was pointed out in section 6.2 with $\nd\neq 0$ and with the same regime in the $SU(\bb)$ sector, these still higgsed quarks will give then nonzero masses to Nambu-Goldstone particles, and this is clearly unacceptable. $\nd\neq 0$ in section 6.2 allows to see this problem with the smooth crossover instead of the non-smooth phase transition.

d) Besides, unlike the remaining light ${\cal N}=2\,\,\, U^{N_c-1}(1)$ photon multiplets when $X^{adj}_{SU(N_c)}$ is higgsed as in (<ref>), the natural appearance of such additional massless at $\mx\ra 0$ ${\cal N}=2\,\, U^{N_c-1}(1)$ photon multiplets in this "instead-of-confinement" phase also remains unexplained. (The unbroken global $SU(N)_{C+F}$ requires equal masses $\sim\lm$ to all $N^2-1$ members of the long ${\cal N}=2\,\, SU(N)_{C+F}$ adjoint representation).

e) It is stated that quark decays into a monopole-antimonopole pair in this "instead-of-confinement" phase. Let us recall that, in any case, the considered theory is not at the arbitrary point on the formal moduli space existing at the mathematical point $\mx\equiv 0$ only, but stays in definite isolated vacua - those at $\mx\neq 0$, even in the formal limit $\mx\ra 0$. And e.g. massive BPS quarks are at least marginally stable at $\mx\ra 0$ in vacua considered in this paper, and can only formally dissolve "emitting" massless quanta. And all BPS quarks and magnetic monopoles have masses $\sim\lm$ in this section at
$\mx\ra 0$.

According to [33], due to higgsed light pure electrically charged dyons ${\ov d}_j, d_j$ (and, by default, due to the smooth crossover, still higgsed as in section 6.4.1 quarks, i.e. $\langle{\ov Q}^{\,a}_i\rangle=\langle Q^i_a\rangle=\delta ^i_a\,\mx m,\,\,i,a=1...N$ ), these heavy magnetic monopoles with masses $\sim\lm$ are confined in this "instead-of-confinement" phase at small $0 <\mx\ll\lm^2/m$, and the pair monopole-antimonopole to which quark decays forms a "stringy meson".

Comparing properties of this "instead-of-confinement" phase from [33] with those described above in this section (and in section 6.2), it is seen that these properties are, so to say, "orthogonal" to each other.

§.§ $m\gg\lm\,,\,\, 2N_c-N_F=1$

At $\mx m\gg\lm^2$ the behavior of all vs-vacua is the same and is described in section 6.1. Therefore, we consider below only the case $\mx m\ll\lm^2$ when the vs-vacua with $\bb=1$ and $\bb\geq 2$ behave differently.

The pattern of flavor symmetry breaking remains fixed, $\no=\nd=N_c-1\,,\,\, \nt=N_c\,,\,\, U(N_F)\ra U(\no)\times U(\nt)$, as well as the multiplicity $N_{vs}=C^{\,\nt=N_c}_{N_F}=C^{\,\no=\nd}_{N_F}$. And $\langle a_0\rangle=m$ also remains the same in all vs -vacua (really, at all $m\gtrless\lm$) and still "eats" all quark masses $"m"$. But the discrete symmetry $Z_{2N_c-N_F}$ becomes trivial at $\bb=1$ and gives no restrictions on the form of $\langle X^{\rm adj}_{SU(N_c)}\rangle\sim\lm$ (remind that $\langle X^{\rm adj}_{SU(N_c)}\rangle\sim\lm$ is higgsed necessarily at $\mx m\ll\lm^2$ to avoid $g^2(\mu<\lm)<0$ ). As above in section 6.2, the right flavor symmetry breaking implies the unbroken $SU(\nd)$ lower energy group and a presence of $N_F$ flavors of quark-like particles massless at $\mx\ra 0$, then $\no=\nd$ of them will be higgsed at small $\mx\neq 0$ and there will remain $2\nd N_c$ Nambu-Goldstone multiplets.

However, it is not difficult to see that a literal picture with light BPS original electric particles
of remained unbroken at the scale $\mu\sim\lm$ electric subgroup $SU(\nd)$ cannot be right at $\bb=1$. Indeed, if this were the case, then $\langle X^{\rm adj}_{SU(N_c)}\rangle\sim\lm\, {\rm diag}(\,1... 1\,;\, -\nd\,),\,\,\nd=N_c-1$. But all quarks $Q^i,\,{\ov Q}_i$ with $SU(N_c)$ colors would acquire then masses $\sim\lm$ and decouple at $\mu\lesssim\lm$. The remained ${\cal N}=2\,\, SU(\nd)$ SYM with the scale factor $\sim\lm$ of its gauge coupling is not IR free and, to avoid unphysical $g^2(\mu\lesssim\lm)<0$, it is higgsed necessarily as $\langle X^{\rm adj,A}_{SU(\nd)}\rangle\sim\lm,\,\, SU(\nd)\ra U^{\nd-1}(1)$, in contradiction with the assumption that it remains unbroken at $\mu\lesssim\lm$.
Clearly, at $m\ll\lm,\, \mx\ra 0$ and $\bb=1$, this argument concerns also $SU(N_c)$ br2 vacua with $0<\no<\nd$ and the multiplicity $(\nd-\no)C^{\no}_{N_F}$, and S vacua with the multiplicity $\nd$, both with $\langle S\rangle_{N_c}\neq 0$ and with the $SU(\nd=N_c-1)$ gauge group at lower energy $\mu<\lm$.

Therefore, the vs-vacua with $2N_c-N_F=1$ are not typical vacua, they are really exceptional. The absence of the non-trivial unbroken $Z_{2N_c-N_F}$ symmetry is crucial.
Considered in [18] example of $U(N_c)$ vs-vacua ($r=N_c$ vacua in the language of [18]) with $N_c=3,\, N_F=5,\, \no=\nd=N_c-1=2$ and $\langle S\rangle_{N_c}=0$ belongs just to this exceptional type of vs-vacua with $2N_c-N_F=1$.
The possible exceptional properties of vacua with $\bb=1$ at $\mx\ra 0$ are indicated also by the curve (<ref>), its form is changed at $\bb=1$ : $m\ra m+(\lm/N_c)$ [8].

For this reason we do not deal with such vacua in this paper.

§ $\MATHBF{SU(N_C),\,\,M\GG\LM,\,\, U(N_F)\RA U(\NO)\TIMES U(\NT)},$    BR2 VACUA

§.§ Larger $\mx$

The quark condensates in these br2 vacua with $\no\geq 1,\, \nt<N_c$ are obtained from (<ref>),(<ref>) by the replacement $\no\leftrightarrow\nt$, i.e. : $\langle\Qt\rangle_{N_c}\approx\mx m_1\gg\langle\Qo\rangle_{N_c},\, m_1=m N_c/(N_c-\nt)$, see sections 3 and 11 in [13]. The discrete $Z_{\bb\geq 2}$ symmetry is also unbroken in these vacua and the multiplicity is $N_{\rm br2}=(N_c-\nt) C^{\,\nt}_{N_F}$. They evolve at $m\ll\lm$ to Lt-vacua with $\langle\Qo\rangle_{N_c}\sim\langle\Qt\rangle_{N_c}\sim\mx\lm$, see section 3 in [13].

Similarly to br1 vacua, the scalars $X^{adj}$ are also higgsed at the largest scale $\mu\sim m\gg\lm$ in the weak coupling region, now as $SU(N_c)\ra SU(\nt)\times U(1)\times SU(N_c-\nt),\,\, N_F/2<\nt<N_c$, so that (the leading terms only)
⟨⟩_N_c≈m_1 ,  ⟨⟩_N_c≈m_1(/m_1)^2N_c-N_F/N_c-  ,  m_1=N_c/N_c-  m ,
⟨S⟩_N_c=⟨⟩_N_c⟨⟩_N_c/≈m_1^2(/m_1)^2N_c-N_F/N_c-≪m^2 ,
⟨⟩_N_c/⟨⟩_N_c≈(/m_1)^2N_c-N_F/N_c-≪1 ,
⟨X^adj_SU(N_c)⟩≡⟨  X^adj_SU()+ X_U(1)+ X^adj_SU(N_c-) ⟩ ,
√(2)  X_U(1)=a  diag( 1_ ;  c_N_c-  ) ,  c=-/N_c- ,  ⟨a ⟩= m ,
⟨Tr (√(2) X^adj_SU(N_c))^2⟩=(2N_c-N_F)⟨S⟩_N_c+ m ⟨Tr ()⟩_N_c≈m (m_1) .
As a result, all quarks charged under $SU(N_c-\nt)$ and hybrids $SU(N_c)/[\,SU(\nt)\times SU(N_c-\nt)\times U(1)\,]$ acquire large masses $m_1=m-{\wh c}\,\langle a\rangle=m N_c/(N_c-\nt)$ and decouple at scales $\mu<\,m_1$, there remains ${\cal N}=2\,\, SU(N_c-\nt)$  SYM with the scale factor $\Lambda^{SU(N_c-\nt)}_
{{\cal N}=2\, SYM}$ of its gauge coupling, see also (<ref>),
⟨Λ^SU(N_c-)_N=2  SYM⟩^2=(^(m_1)^N_F/(m_1)^2
)^1/(N_c-)=m^2_1(/m_1)^/(N_c-)≪^2 ,  m_1=N_c/N_c- m ,
⟨S⟩_N_c-=⟨Λ^SU(N_c-)_N=2  SYM⟩^2≈⟨S⟩_N_c .

At $\mx\ll\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=2\, SYM}\rangle$ it is higgsed in a standard way [3], $SU(N_c-\nt)\ra U^{N_c-\nt-1}(1)$, due to higgsing of $X^{adj}_{SU(N_c-\nt)}$
⟨√(2) X^adj_SU(N_c-)⟩∼⟨Λ^SU(N_c-)_N=2  SYM⟩  diag (0_ ; k_1, ... k_N_c-_N_c-) ,  k_i=O(1) .
Note that the value $\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle$ of $\langle X^{adj}_{SU(N_c-\nt)}\rangle$ in (<ref>),(<ref>) is consistent with the unbroken $Z_{\bb}$ discrete symmetry.

As a result, $\,N_c-\nt-1$ pure magnetic monopoles $M_{\rm n}$ (massless at $\mx\ra 0$) with the $SU(N_c-\nt)$ adjoint charges are formed at the scale $\mu\sim\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=2\, SYM}\rangle$ in this SYM sector. These correspond to $N_c-\nt-1$ unequal double roots of the curve (<ref>) connected with this SYM sector. Note also that two single roots with $(e^{+} -e^{-})
\sim\langle\Lambda^{SU(N_c)}_{{\cal N}=2\, SYM}\rangle$ of the curve (<ref>) also originate from this SYM sector. Other $\nt$ double roots of the curve at $\mx\ra 0$ originate in these vacua from the $SU(\nt)$ sector, see below.

These $N_c-\nt-1$ monopoles are all higgsed with $\langle M_{\rm n}\rangle=\langle {\ov M}_{\rm n}\rangle\sim (\mx\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=2\, SYM}\rangle)^{1/2}$, so that $N_c-\nt-1$ long ${\cal N}=2$ multiplets of massive photons appear, with masses $\sim (\mx\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=2\, SYM}\rangle)^{1/2}$. Besides, this leads to a weak confinement of all heavier original $SU(N_c-\nt)$ electrically charged particles, i.e. quarks and all hybrids with masses $\sim m$, and all ${\cal N}=2\,\, SU(N_c-\nt)$ SYM adjoint charged particles with masses $\sim\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=2\, SYM}\rangle$, the tension of the confining string is $\sigma^{1/2}_2\sim (\mx\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=2\, SYM}\rangle)^{1/2}\ll\langle\Lambda^
{SU(N_c-\nt)}_{{\cal N}=2\, SYM}\rangle\ll\lm\ll m$. The factor $N_c-\nt$ in the multiplicity of these br2 vacua arises just from this ${\cal N}=2\,\, SU(N_c-\nt)$  SYM part.

At larger $\mx$ in the range $\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=2\, SYM}\rangle\ll\mx\ll m$, the scalars $X^{adj}_{SU(N_c-\nt)}$ become too heavy, their light  $SU(N_c-\nt)$ physical phases fluctuate then freely at the scale $\sim\mx^{\rm pole}=g^{2}(\mu=\mx^{\rm pole})\mx$ and they are not higgsed. Instead, they decouple as heavy in the weak coupling region at scales $\mu<\mx^{\rm pole}/(\rm several)$. There remains then ${\cal N}=1\,\, SU(N_c-\nt)$ SYM with the scale factor $\Lambda^{SU(N_c-\nt)}_{{\cal N}=1\, SYM}$ of its gauge coupling
⟨S⟩_N=1  SYM=⟨Λ^SU(N_c-)_N=1  SYM⟩^3
=⟨Λ^SU(N_c-)_N=2  SYM⟩^2,
⟨Λ^SU(N_c-)_N=2  SYM⟩≪⟨Λ^SU(N_c-)_N=1  SYM⟩≪≪m .
Therefore, there will be in this case a large number of strongly coupled $SU(N_c-\nt)$ gluonia with the mass scale $\sim\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=1\, SYM}\rangle$, while all original heavier charged particles with $SU(N_c-\nt)$ electric charges and masses either $\sim m\gg\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=1\, SYM}\rangle$ or $\sim g^{2}\mx\gg\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=1\, SYM}\rangle$ still will be weakly confined, the tension of the confining string is larger in this case, $\sigma^{1/2}_1\sim\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=1\, SYM}\rangle$. This ${\cal N}=1\,\,SU(N_c-\nt)\,$ SYM gives the same factor $N_c-\nt$ in the multiplicity of these vacua.

Now, about the electric $SU(\nt)$ part decoupled from $SU(N_c-\nt)$ SYM. The scale factor of its gauge coupling is
⟨Λ_SU()⟩=( ^/(m_1)^2(N_c-) )^1/2-N_F=m_1(/m_1 )^/2-N_F≪ ,  m_1=N_c/N_c- m .
Note that the value of $\langle\Lambda_{SU(\nt)}\rangle$ in (<ref>) is also consistent with the unbroken $Z_{\bb}$ discrete symmetry.

What qualitatively differs this $SU(\nt)$ part in br2 vacua at scales $\mu<\, m$ from its analog $SU(\no)$ in br1 vacua of section 4.1 is that $SU(\no)$ with $N_F>2\no$ is IR free, while $SU(\nt)$ with $N_F<2\nt$ is UV free, and its small ${\cal N}=2$ gauge coupling at the scale $\mu\sim m\gg\lm$ begins to grow logarithmically with diminished energy at $\mu<m$. Therefore, if nothing prevents, $\,\,X^{adj}_{SU(\nt)}$ will be higgsed necessarily at the scale $\sim\langle\Lambda_{SU(\nt)}\rangle$, with $\langle X^{adj}_{SU(\nt)}\rangle\sim\langle\Lambda_{SU(\nt)}\rangle$, to avoid unphysical $g^2(\mu<\langle\Lambda_{SU(\nt)}\rangle)<0$ (see section 7.2 below).

But if $(\mx m)^{1/2}\gg\langle\Lambda_{SU(\nt)}\rangle$, the phase will be different. The leading effect in this case will be the breaking of the whole $SU(\nt)$ group due to higgsing of $\nt<N_c$ out of $N_F$ quarks $Q^i,\,{\ov Q}_i$ in the weak coupling region, with e.g. $\langle Q^{i=b+\no}_b\rangle\sim \delta^{i}_b (\mx m)^{1/2},\,\, b=1...\nt,\,\, i=\no+1... N_F$. This will give the additional factor $C^{\,\nt}_{N_F}$ in the multiplicity of these br2 vacua due to spontaneous flavor symmetry breaking $U(N_F)\ra U(\no)\times U(\nt)$, so that the overall multiplicity will be $(N_c-\nt)C^{\,\nt}_{N_F}$, as it should be. The corresponding parts of the superpotential will be as in (<ref>),(<ref>), with a replacement $\no\ra\nt$,
W_=(  m-a ) Tr (Q Q)_ - Tr (Q√(2) X^adj_SU() Q)_ +Tr ( X^adj_SU())^2+/2 N_c/N_c- a^2 .

From (<ref>) in the case considered (no summation over $i$ or $j$ in (<ref>) )
⟨a⟩=m,   ⟨X^adj_SU()⟩=0,  ⟨⟩_=
∑_b=1^⟨Q_2^ b Q^2_b⟩=⟨Q_2^ 2⟩⟨Q^2_2⟩≈N_c/N_c-m=m_1 ,
⟨⟩_=∑_b=1^⟨Q_j^ b⟩⟨Q^j_b⟩=0 ,   j=1...,  ⟨ S⟩_=1/⟨⟩_⟨⟩_=0 .

$2\no\nt$ massless Nambu-Goldstone multiplets are formed as a result of the spontaneous flavor symmetry breaking, $\Pi_j^i= ({\ov Q}_{j} Q^i)_{\nt}\,\ra\,({\ov Q}_{j}\langle Q^i\rangle)_{\nt},\,\,\Pi_i^j= ({\ov Q}_{i} Q^j)_{\nt}\,\ra\,(\langle{\ov Q}_{i}\rangle Q^j)_{\nt},\,\, i=\no+1...N_F,\,\, j=1...\no,\,\,
\langle\Pi^i_j\rangle=\langle \Pi^j_i\rangle=0$ (in essence, these are non-higgsed quarks with $\no$ flavors and $SU(\nt)$ colors). Besides, there are $\nt^2$ long ${\cal N}=2$ multiplets of massive gluons with masses $\sim (\mx m)^{1/2}$.

We emphasize that higgsing of $\nt$ quarks at the higher scale $(\mx m)^{1/2}\gg\langle\Lambda_{SU(\nt)}
\rangle$, see (<ref>), (<ref>), prevents $X^{adj}_{SU(\nt)}$ from higgsing, i.e. $\langle  X^{adj}_{SU(\nt)}\rangle_{\nt}$ is not simply smaller, but exactly zero (see also section 6.1 for a similar regime).

On the whole for these br2 vacua in the case considered, i.e. at $(\mx m)^{1/2}\gg\langle\Lambda_{SU(\nt)}
\rangle$, all qualitative properties are similar to those of br1 vacua in section 4.1.

§.§ Smaller $\mx$

Consider now the behavior of the $SU(\nt)\times U(1)$ part in the opposite case of smaller $\mx,\, \mx\ll\langle\Lambda_{SU(\nt)}\rangle^2/m,\, \langle\Lambda_{SU(\nt)}\rangle\ll\lm\ll m$. As will be seen below, in this case even all qualitative properties of mass spectra in this part will be quite different (the behavior of the $SU(N_c-\nt)$ SYM part decoupled from $SU(\nt)\times U(1)$ was described above in section 7.1).

Now, at smaller $\mx$, instead of quarks, the scalar field $X^{adj}_{SU(\nt)}$ is higgsed first at the scale $\sim\langle\Lambda_{SU(\nt)}\rangle$ to avoid unphysical $g^2(\mu<\langle\Lambda_{SU(\nt)}\rangle)<0$ of UV free ${\cal N}=2\,\, SU(\nt)$. Although $\langle\Lambda_{SU(\nt)}\rangle$ respects the original $Z_{\bb}$ discrete symmetry of the UV free $SU(N_c)$ theory, see (<ref>), because the $SU(N_c-\nt)$ SYM part is now  decoupled, there is the analog of $Z_{\bb}$ in the $SU(\nt)$ group with $N_F$ flavors of quarks, this is the unbroken non-trivial $Z_{2\nt-N_F}=Z_{\nt-\no}$ discrete symmetry with $\nt-\no\geq 2$. Therefore, the field $X^{adj}_{SU(\nt)}$ is higgsed at $\nt-\no\geq 2$ at the scale $\mu\sim\langle\Lambda_{SU(\nt)}\rangle$ (in the strong coupling and non-perturbative regime), qualitatively similarly to (<ref>),
(<ref>) in section 2.1 : $SU(\nt)\ra SU(\no)\times U^{(1)}(1)\times U^{\nt-\no-1}(1)$. There will be similar dyons $D_j$ (massless at $\mx\ra 0$) etc., and a whole qualitative picture will be similar to those in section 2.1, with evident replacements of parameters. The only qualitative difference is that the additional SYM part is absent here because $\no$ quark flavors are higgsed now in $SU(\no)$,
⟨X^adj_SU()⟩=⟨X^adj_SU()+X^(1)_U(1)+ X^adj_SU(-)⟩,
⟨√(2) X^adj_SU(-)⟩∼⟨Λ_SU()⟩ diag ( 0_ ; τ^0, τ^1, ..., τ^--1_- ;
0_N_c- ) ,  τ=exp{2πi/-} ,
√(2)  X^(1)_U(1)=a_1 diag (  1_ ;   c_1
_- ;0_N_c- ),  c_1=- /- .  Therefore, similarly to (<ref>),(<ref>), in this case the low energy superpotential of this $SU(\nt)\times U(1)$ sector can be written as
In other words, in this case the $SU(N^\prime_c=\nt)\times U(1)$ part of $SU(N_c)$ with $m\gg\lm$ and with decoupled $SU(N_c-\nt)$ SYM part, is in its own vs-vacuum of section 6.2, with $\langle\,{S}\,\rangle_
{N^\prime_c}=0$ and with $a$ of $U(1)$ in (<ref>) playing a role of $a_0$ in (<ref>), etc.
_=_+_D+_a, a_1+… ,
_ =(m-a-a_1)Tr (Q Q)_-Tr (Q√(2)X_SU()^adj Q)_+(1+δ_2)Tr (X^adj_SU())^2 ,
_D=( m-a-c_1 a_1 )∑_j=1^-D_j D_j -∑_j=1^- a_D,jD_j D_j -Λ_SU()∑_j=1^-τ^j-1 a_D,j+L (∑_j=1^- a_D,j),
_a, a_1=/2N_c/N_c- a^2+/2
/-(1+δ_1) a^2_1+δ_3 a_1(m-a-c_1 a_1)+δ_4  (m-a-c_1 a_1)^2 ,
where $\wh{L}$ is the Lagrange multiplier field, $\langle\wh{L}\rangle=O(m)$ and dots denote small power corrections. The additional terms with $\wh{\delta}_{1,2,3,4}$, as previously in br2 vacua in section 2.1 and in vs -vacua in section 6.2, originate from integrating out all heavier fields with masses $\sim\Lambda_{SU(\nt)}$ in the soft background. Here these soft background fields are $(m-a-\wh{c}_1 a_1)$ and $[(1-\wh{c}_1) a_1+\sqrt{2}X^{\rm adj}_{SU(\no)}]$ ( while $a_1$ plays here a role of $a_1$ in (<ref>) ). These constants $\wh\delta_i$ differ from (<ref>) only because the number of colors is different, $N_c\ra N^\prime_c=\nt$, i.e. $\wh\delta_1=[\,- 2N^\prime_c/(2N^\prime_c-N_F)]
=[\,-2\nt/(\nt-\no)\,],\,\, \wh\delta_2=-2$ (while $\wh\delta_3=0$ as in br2 vacua, see (<ref>)).

The charges of fields and parameters entering (<ref>) under $Z_{\nt-\no}=\exp\{i\pi/(\nt-\no)\}$ transformation are : $q_{\lambda}=q_{\rm\theta}=1,\,\, q_{X_{SU(\no)}^{\rm adj}}=q_{a}=q_{a_1}=q_{a_{D,j}}=q_{\rm m}=q_{\wh{L}}=2,\,\,q_{Q}=q_{\ov Q}=q_{D_j}=q_{{\ov D}_j}=q_{\Lambda_{SU(\nt)}}=0,\,\, q_{\mx}=-2$. The non-trivial $Z_{\nt-\no\geq 2}$ transformations change only numerations of dual scalars $a_{D,j}$ and dyons in (<ref>), so that $\int d^2\theta\,\w_{\nt}$ is $Z_{\nt-\no}$-invariant.

From (<ref>):
⟨a⟩=m ,  ⟨a_1⟩=0 ,  ⟨a_D,j⟩=0 ,
⟨X_SU()^adj⟩=0 ,  ⟨S⟩_=0 ,
⟨D_j D_j⟩=⟨D_j⟩⟨D_j ⟩≈-⟨Λ_SU()⟩τ^j-1+⟨L⟩ ,  ⟨Σ_D⟩=∑_j=1^-⟨D_j⟩⟨D_j ⟩=(-)⟨L⟩,
⟨L⟩≈N_c/N_c-  m=m_1 ,   ⟨Tr (Q Q)⟩_=⟨⟩_+⟨⟩_≈m_1 ,
⟨⟩_=⟨Q^ 1_1⟩⟨Q^1_1⟩≈N_c/N_c- m≈m_1≈⟨⟩_N_c ,  ⟨⟩_=∑_a=1^⟨Q^ a_2⟩⟨Q^2_a⟩=0 .

The multiplicity of these br2 vacua in the case considered is $N_{\rm br2}=(N_c-\nt)C^{\,\no}_{N_F}=(N_c-\nt)C^{\,\nt}_{N_F}$, as it should be. The factor $(N_c-\nt)$ originates from the $SU(N_c-\nt)$ SYM. The factor $C^{\,\no}_{N_F}$ is finally a consequence of the color breaking $SU(\nt)\ra SU(\no)\times U(1)^{\nt-\no}$ with the unbroken $Z_{2\nt-N_F}=Z_{\nt-\no}$ discrete symmetry, and spontaneous breaking of flavor symmetry due to higgsing of $\no$ quarks flavors (massless at $\mx\ra 0$) in the $SU(\no)$ color subgroup of $SU(\nt)$, $\langle Q^i_a\rangle=\langle{\ov Q}^{\,a}_i\rangle\approx \delta_a^i (\mx m_1)^{1/2}$,  $a=1...\no,\,\,i=1...N_F$.

Note also that if the non-trivial discrete symmetry $Z_{\nt-\no\geq 2}$ were broken spontaneously in the $SU(\nt)$ subgroup at the scale $\mu\sim\langle\Lambda_{SU(\nt)}\rangle\gg(\mx m)^{1/2}$, this would lead then
to the factor $\nt-\no$ in the multiplicity of these vacua, and this is a wrong factor.

Due to higgsing of $\nt-\no$ dyons with the nonzero $SU(\nt-\no)$ adjoint magnetic charges, $\langle{\ov D_j}\rangle\langle D_j\rangle\sim \mx m\gg\mx\lm\gg\mx\langle\Lambda_{SU(\nt)}\rangle$, all original pure electrically charged particles with $SU(\nt-\no)$ colors and masses either $\sim m$ or $\sim\langle\Lambda_{SU(\nt)}\rangle$ are weakly confined, the string tension is $\sigma^{1/2}\sim (\mx m)^{1/2}\ll\langle\Lambda_{SU(\nt)}\rangle\ll\lm\ll m$. Besides, $\nt-\no$ long ${\cal N}=2$ multiplets of massive photons with masses $\sim (\mx m)^{1/2}$ are formed. Remind that, due to higgsing of $N_c-\nt-1$ magnetic monopoles $M_{\rm n}$ from $SU(N_c-\nt)$ SYM at $\mx < \langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=2\, SYM}\rangle/(\rm several)$, all original pure electrically charged particles with $SU(N_c-\nt)$ charges and masses either $\sim m$ or $\sim\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=2\, SYM}\rangle$ are also weakly confined, but this string tension $\sigma^{1/2}_2\sim (\mx\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=2\, SYM}\rangle)^{1/2}\ll (\mx m)^{1/2}$ is much smaller.

Due to higgsing of $\no$ flavors of original electric quarks in the IR free color $SU(\no)$ with $N_F>2\no$ flavors, there are $\no^2$ long ${\cal N}=2$ multiplets of massive gluons with masses $\sim (\mx m)^{1/2}\ll\langle\Lambda_{SU(\nt)}\rangle$, while the original non-higgsed quarks with $\nt$ flavors and $SU(\no)$ colors form $2\no\nt$ massless Nambu-Goldstone multiplets. There are no other massless particles at $\mx\neq 0$. And there are no particles with masses $\sim m\gg\lm$ in this $SU(\nt)\times U(1)$ sector.

Remind that the curve (<ref>) has $N_c-1$ double roots in these br2 vacua at $\mx\ra 0$. Of them: $\nt-\no$ unequal roots correspond to $\nt-\no$ dyons $D_j,\,\, \no$ equal roots correspond to $\no$ higgsed quarks $Q^i$ of $SU(\no)$, the remaining unequal $N_c-\nt-1$ double roots correspond to $N_c-\nt-1$ pure magnetic monopoles $M_{\rm n}$ from $SU(N_c-\nt)\,\, {\cal N}=2$ SYM. Two single roots with $(e^{+}-e^
{-})\sim\langle\Lambda^{SU(N_c-\nt)}_{{\cal N}=2\, SYM}\rangle$ originate from this ${\cal N}=2$ SYM.

Besides, similarly to vs-vacua in section 6, there is a phase transition in these br2 vacua in the region $\mx\sim\langle\Lambda_{SU(\nt)}\rangle^2/m$. E.g., all $SU(\nt-\no)$ charged gluons have equal masses in the whole wide range $(\rm several)\langle\Lambda_{SU(\nt)}\rangle < (\mx m)^{1/2}\ll m$ of the higher energy phase of section 7.1, while masses of these gluons differ greatly at $\mx\ra 0$ in the lower energy phase of this section, $\mu^{ij}_{gl}\sim |(\tau^i-\tau^j)\langle\Lambda_{SU(\nt)}\rangle|,\,i\neq j,\, i,j=0...\nt-\no-1$. And $\langle X^{adj}_{SU(\nt-\no)}\rangle$, $\,\langle\Qt\rangle_{\nt}$ also behave non-analytically at $\mx\gtrless\langle\Lambda_{SU(\nt)}\rangle^2/m$. But, unlike the phase transition at $\mx\sim\lm^2/m$ in
vs-vacua of section 6, the phase transition in these br2 vacua is at much smaller $\mx\sim\langle\Lambda_
{SU(\nt)}\rangle^2/m\,,\,\,\langle\Lambda_{SU(\nt)}\rangle\ll\lm$, see (<ref>).

And finally, remind that the mass spectra in these br2 vacua with $\nt<N_c$ and $m\gg\lm$ depend essentially on the value of $m/\lm$, and all these vacua  evolve at $m\ll\lm$ to Lt-vacua with the spontaneously broken discrete $Z_{\bb}$ symmetry, see section 3 in [13] or section 4 in [14].

§ LARGE QUARK MASSES $M\GG\LM,\,\,\, SU(N_C),\,\,\, N_F<N_C$

The vacua with the unbroken $U(N_F)$ flavor symmetry (here and in what follows at not too small $N_c$ and $N_F$) are in this case the SYM-vacua with the multiplicity $N_c$ and S-vacua with the multiplicity $N_c-N_F$, see section 2 in [13] and section 3 in [14]. The vacua with the broken $U(N_F)\ra U(\no)\times U(\nt)$ symmetry are the br1 and br2 vacua with multiplicities $(N_c-{\rm n}_i)C^{\,{\rm n}_i}_{N_F},\, i=1,\,2$.

Except for S-vacua, all formulae for this case $N_F<N_c,\,\, m\gg\lm$ are the same as described above for br1 vacua in section 4, for SYM vacua in section 5 and for br2 vacua in section 7 (and only $N_F<N_c$ now). Therefore, we describe below only the mass spectra in S-vacua with $m\gg\lm,\,\mx\ll\lm\,$.

^ eff_tot(Π)=m Tr (Q Q)_N_c-1/2[  ∑_i,j=1^N_F (Q_j Q^i)_N_c(Q_ i Q^j)_N_c-1/N_c(Tr (Q Q)_N_c)^2]+(N_c-N_F) S_N_c  ,
S_N_c=(^2N_c-N_F^N_c/()_N_c)^1/N_c-N_F ,
the quark and gluino condensates look in these $SU(N_c)$ S-vacua with $\no=0,\, \nt=N_F$ as,
⟨⟩_N_c≈m_1[1-N_c/N_c-N_F(/m_1)^2N_c-N_F/N_c-N_F  ],    m_1=N_c m/N_c-N_F m,
It is seen from (<ref>) that the non-trivial discrete symmetry $Z_{2N_c-N_F\geq 2}$ is not broken.

The scalars $ X^{adj}_{SU(N_c)}$ are higgsed at the highest scale $\sim m\gg\lm$ in the weak coupling regime, $SU(N_c)\ra SU(N_F)\times U(1)\times SU(N_c-N_f)$, see (<ref>) below,
⟨X^adj_SU(N_c)⟩=⟨X^adj_SU(N_F)+X_U(1)+ X^adj_SU(N_c-N_F)⟩,
√(2)  X_U(1)=a diag (  1_N_F ;   c
_N_c-N_F ),  c=- N_F/N_c-N_F ,  ⟨a⟩= m ,

The quarks in the $SU(N_c-N_F)$ sector and $SU(N_c)/[SU(N_F)\times SU(N_c-N_F)\times U(1)]$ hybrids have large masses $m_Q=\langle m-{\wh c}\, a\rangle=N_c m/(N_c-N_F)=m_1$. After they are integrated out, the scale factor of the $SU(N_c-N_F)$ SYM gauge coupling is, see also (<ref>),
⟨Λ^SU(N_c-N_F)_N=2  SYM⟩^2=(^(m_1)^N_F/(m_1)^2 N_F)^1/N_c-N_F=m^2_1(/m_1)^/N_c-N_F≪^2 ,  m_1=N_c/N_c-N_F m ,
⟨S⟩_N_c-N_F=⟨Λ^SU(N_c-N_F)_N=2  SYM⟩^2=⟨S⟩_N_c ,
while those of $SU(N_F)$ SQCD is
⟨Λ_SU(N_F)⟩=( ^/(m_1)^2(N_c-N_F) )^1/N_F=m_1(/m_1 )^/N_F≪ .

If $\mx\ll\langle\Lambda^{SU(N_c-N_F)}_{{\cal N}=2\, SYM}\rangle$, the scalars $X^{adj}_{SU(N_c-N_F)}$ of ${\cal N}=2$ SYM are higgsed at $\mu\sim\langle\Lambda^{SU(N_c-N_F)}_{{\cal N}=2\, SYM}\rangle$ (<ref>) in a known way, $SU(N_c-N_F)\ra U^{N_c-N_F-1}(1)$ [3], giving $N_c-N_F$ vacua and $N_c-N_F-1$ massless at $\mx\ra 0$ magnetic monopoles ${\ov M}_{\rm n}, M_{\rm n}$. These all are higgsed at $\mx\neq 0$ giving $N_c-N_F-1$ $\,{\cal N}=2$ long multiplets of dual massive photons with masses $\sim (\mx\langle\Lambda^{SU(N_c-N_F)}_{{\cal N}=2\, SYM}\rangle)^{1/2}$. All heavy original particles with masses $\sim m$ and all charged $SU(N_c-N_F)$ adjoints with masses $\sim\langle\Lambda^{SU(N_c-N_F)}_{{\cal N}=2\, SYM}\rangle$ are weakly confined, the string tension in this sector is $\sigma^{1/2}_2\sim (\mx\langle\Lambda^{SU(N_c-N_F)}_
{{\cal N}=2\, SYM}\rangle)^{1/2}\ll\langle\Lambda^{SU(N_c-N_F)}_{{\cal N}=2\, SYM}\rangle$.

On the other hand, if e.g. $\langle\Lambda^{SU(N_c-N_F)}_{{\cal N}=2\, SYM}\rangle\ll\mx\ll m$, see (<ref>), all scalars  $X^{adj}_{SU(N_c-N_F)}$ are too heavy and not higgsed, they all decouple in the weak coupling regime and can be integrated out at $\mu<g^2\mx/(\rm several)$, there remains then ${\cal N}=1\,\, SU(N_c-N_F)$ SYM with $\langle\Lambda^{SU(N_c-N_F)}_{{\cal N}=1\, SYM}\rangle=[\,\mx\langle
\Lambda^{SU(N_c-N_F)}_{{\cal N}=2\, SYM}\rangle^2\,]^{1/3}\gg\langle\Lambda^{SU(N_c-N_F)}_{{\cal N}=2\, SYM}\rangle$. There will be a large number of strongly coupled $SU(N_c-N_F)$ gluonia with the mass scale $\sim\langle\Lambda^{SU(N_c-N_F)}_{{\cal N}=1\, SYM}\rangle$. All original heavier $SU(N_c-N_F)$ charged particles with masses $\sim m$ or $g^2\mx$ are still weakly confined, but the string tension in this sector is larger now, $\langle\Lambda^{SU(N_c-N_F)}_{{\cal N}=2\, SYM}\rangle\ll\sigma^{1/2}_1
\sim\langle\Lambda^{SU(N_c-N_F)}_{{\cal N}=1\, SYM}\rangle\ll\mx$.

Now, as for the $SU(N_F)\times U(1)$ sector. At the scale $\mu= m/(\rm several)$ this is the UV free ${\cal N}=2$ SQCD with $N_F$ flavors of massless quarks ${\ov Q}^{\,a}_j, Q^i_a,\,\,i,a=1...N_F,\,\,
\langle m-a\rangle=0$, its gauge coupling grows logarithmically with diminished energy. The dynamics of this part is the same as in vs-vacua of section 6.4. When $\langle\Lambda_{SU(N_F)}\rangle^2/m\ll\mx\ll m$, see (<ref>), then the low energy superpotential in this sector will be as e.g. in (<ref>) with $\nt=N_F$. I.e., all quarks in this $SU(N_F)$ sector will be higgsed in the weak coupling regime in this higher energy phase (while $\langle X^{adj}_{SU(N_F)}\rangle=0$), and $[(N_F^2-1)+1]$ long ${\cal N}=2$ multiplets of massive gluons (including $U(1)$ with its scalar "a") with masses $\sim \langle\qq\rangle^{1/2}_{N_F}\sim (\mx m)^{1/2}$ will be formed, see (<ref>). The global symmetry $SU(N_F)_{C+F}$ is unbroken in this higher energy phase, and $(N_F^2-1)\,\, {\cal N}=2$ long multiplets form the $SU(N_F)_{C+F}$ adjoint representation.
There are no particles with mass $\sim m$ and there are no massless particles. The multiplicity of vacua in this phase is determined by the multiplicity $N_c-N_F$ of $SU(N_c-N_F)$ SYM.

But at $(\mx m)^{1/2}\ll\langle\Lambda_{SU(N_F)}\rangle$ the lower energy phase is different. To avoid $g^2(\mu<\langle\Lambda_{SU(N_F)}\rangle)<0$, the main contributions to masses originate now in this UV free $SU(N_F)$ sector from higgsed $\langle X^{adj}_{SU(N_F)}\rangle\sim\langle\Lambda_{SU(N_F)}\rangle$, in the strong coupling and non-perturbative regime. The situation in this phase is also qualitatively similar to those in section 6.4 (or in 7.2 with $\no\geq 1$, the difference is that $\no=0,\, \nt=N_F$ now as in section 6.4, so that the whole $SU(N_F)$ color group will be broken at the scale $\sim\langle\Lambda_{SU(\nt=N_F)}\rangle$, see below). Because $SU(N_c-N_F)$ sector is decoupled, there is the residual unbroken discrete symmetry $Z_{2N_F-N_F\geq\, 2}=Z_{N_F\geq\, 2}$. Therefore, $X^{adj}_{SU(N_F)}$ are higgsed as
⟨√(2) X^adj_SU(N_F)⟩∼⟨Λ_SU(N_F)⟩ diag (
ρ̃^ 0, ρ̃^  1, ..., ρ̃^  N_F-1_N_F ;0
_N_c-N_F ),  ρ̃=exp{ 2πi/N_F } ,
and the whole $SU(N_F)_C$ color group is broken, $SU(N_F)_C\ra U^{N_F-1}(1)$. All quarks with $SU(N_F)_C$ colors acquire now masses $m_Q\sim\langle\Lambda_{SU(N_F)}\rangle$ which are large in comparison with the potentially possible scale of their coherent condensate, $m_Q\gg (\mx m)^{1/2}$, so that they are not higgsed but decouple as heavy at $\mu<\langle\Lambda_{SU(N_F)}\rangle/(\rm several)$, as well as all charged $SU(N_F)_C$ adjoints. Instead, $N_F$ massless at $\mx\ra 0$ dyons ${\ov D}_j, D_j$ are formed at the scale $\sim\langle\Lambda_{SU(N_F)}\rangle$.

The relevant part of the low energy superpotential of this sector is in this case qualitatively the same as in vs-vacua of section 6.4, see (<ref>), and only $\w_{SYM}$ is added
^ (N_F)_low=_D+_a+_SYM ,
_D= (m-a)∑_j=1^N_FD_j D_j -∑_j=1^N_F a_D, jD_j D_j -⟨Λ_SU(N_F)⟩∑_j=1^N_Fρ̃^ j-1 a_D, j+ L̃
(∑_j=1^N_F a_D, j) ,
_a=/2N_c N_F/N_c-N_F a^2+N_F δ̃_4  (m-a)^2 ,
_SYM= (N_c-N_F)(Λ^SU(N_c-N_F)_N= 2  SYM)^2= (N_c-N_F)(^2N_c-N_F(m-ĉ a)^N_F/[ (1-ĉ) a ]^2 N_F)^1/N_c-N_F≈≈ ⟨Λ^SU(N_c-N_F)_N=2  SYM⟩^2[(N_c-N_F) -N_F (2N_c-N_F)/N_c-N_F â/m_1 ] ,  m_1=N_c/N_c-N_F m ,  a=⟨a⟩+â .

From (<ref>)
⟨a⟩=m ,  ⟨a_D, j⟩=0 ,  ⟨D_j⟩⟨D_j ⟩= -⟨Λ_SU(N_F)⟩ρ̃^ j-1+⟨L̃⟩ ,  ⟨Σ_D⟩=∑_j=1^N_F⟨D_j⟩⟨D_j⟩=N_F⟨L̃⟩,
⟨Σ_D⟩≈N_Fm_1[ 1-2N_c-N_F/N_c-N_F⟨Λ^SU(N_c-N_F)_N=2  SYM⟩^2/m^2_1 ]≈N_Fm_1[ 1-2N_c-N_F/N_c-N_F(/m_1)^/N_c-N_F] ,
⟨S⟩_N_F=0,  ⟨D_j⟩⟨D_j⟩≈m_1[ 1-2N_c-N_F/N_c-N_F⟨Λ^SU(N_c-N_F)_N=2  SYM⟩^2/m^2_1 ]-⟨Λ_SU(N_F)⟩ρ̃^ j-1 ,  j=1...N_F .

As an example, we present also the analog of (<ref>) in these S-vacua. From (<ref>),(<ref>),(<ref>)
⟨ ∂/∂m_matter⟩=⟨ ∂/∂m
^ (N_F)_low⟩⟨Tr ⟩_N_c=⟨Σ_D⟩+
⟨ ∂/∂m_SYM⟩≈
≈N_Fm_1[ 1-N_c/N_c-N_F(/m_1)^/N_c-N_F] ,
this agrees with (<ref>).

The curve (<ref>) has $N_c-1$ double roots in these $N_c-N_F$ S-vacua. From these, $N_c-N_F-1$ unequal double roots correspond to massless at $\mx\ra 0$ magnetic monopoles $M_{\rm n}$ from $SU(N_c-N_F)\,\, {\cal N}=2$ SYM, and remaining $N_F$ unequal double roots correspond to massless at $\mx\ra 0$ dyons $D_j$. Two single roots with $(e^{+}-e^{-})\sim \langle\Lambda^{SU(N_c-N_F)}_{{\cal N}=2\, SYM}\rangle$ (<ref>) originate at $\mx\ra 0$ from the $SU(N_c-N_F)\,\,{\cal N}=2$ SYM sector. The global flavor symmetry $U(N_F)$ is unbroken and there are no massless Nambu-Goldstone particles. And there are no massless particles at all at $\mx\neq 0$. At the same time, as in section 6.4.2, the color is broken in this sector, $SU(N_F)_C\ra U^{N_F-1}(1)$, and masses of charged $SU(N_F)_C$ adjoints are different, see (<ref>). There is no color-flavor locking in this lower energy phase.

All original charged electric particles are confined in the lower energy phase at $\mx<\langle\Lambda_
{SU(N_F)}\rangle^2/m_1$, see (<ref>). But the phase changes in the $SU(N_F)$ sector already at very small $\mx :\,\langle\Lambda_{SU(N_F)}\rangle^2/m_1<\mx\ll\langle\Lambda_{SU(N_F)}\rangle\ll\lm\ll m$. Instead of higgsed $\langle X^{adj}_{SU(N_F)}\rangle\sim\langle\Lambda_{SU(N_F)}\rangle,\,\, SU(N_F)\ra U^
{N_F-1}(1)$ and subsequent condensation of $N_F$ dyons $D_j$ (<ref>), all $N_F$ flavors of original electric quarks ${\ov Q}^{\,a}_i,\,Q^i_a,\,\,a,i=1...N_F$ are then higgsed as $\langle{\ov Q}^{\,a}_i
\rangle=\langle Q_a^i\rangle\approx\delta^i_a (\mx m_1)^{1/2}>\langle\Lambda_{SU(N_F)}\rangle$
(and clearly not confined), forming $N_F^2$ long ${\cal N}=2$ multiplets of massive gluons.

Remind that, at $N_F<N_c$ and $m\gg\lm$, the mass spectra in these S vacua with the multiplicity $N_c-N_F$, as well as in the SYM vacua with the multiplicity $N_c$ in section 5, depend essentially on the value of $m/\lm$, and all these vacua with the unbroken global flavor symmetry $U(N_F)$ evolve at $\mx\ll m\ll\lm$ to L-vacua with the multiplicity $2N_c-N_F$ and with the spontaneously broken discrete $Z_{\bb}$ symmetry, see sections 3.1 and 12 in [14].

§ CONCLUSIONS

We presented above in this paper the calculations of mass spectra of softly broken ${\cal N}=2\,\, SU(N_c)$ or $U(N_c)$ gauge theories in a large number of their various vacua with the unbroken non-trivial $Z_{\bb\geq 2}$ discrete symmetry. The charges of light particles in each vacuum were determined, as well as forms of corresponding low energy Lagrangians and mass spectra at different hierarchies between the Lagrangian parameters $m,\,\mx,\,\lm$. A crucial role in obtaining these results was played by the use : a) the unbroken $Z_{\bb\geq 2}$ symmetry,  b) the pattern of spontaneous flavor symmetry breaking, $U(N_F)\ra U(\no)\times U(\nt),\, 0\leq \no<N_F/2,\,$ c) the knowledge of multiplicities of various vacua. Besides, the knowledge of the quark and gluino condensates $\langle\Qo\rangle_{N_c},\,\langle\Qt\rangle_
{N_c}$, and $\langle S\rangle_{N_c}$ was also of importance. In addition, two dynamical assumptions "A" and "B " of general character formulated in Introduction were used. They concern the BPS properties of original particles and the absence of extra fields massless at $\mx\neq 0,\, m\neq 0$. All this appeared to be sufficient to calculate mass spectra in vacua considered at small but nonzero values of $\mx$, in particular at $0<\mx<M_{\rm min}$, where $M_{\rm min}$ is the independent of $\mx$ mass scale appropriate for each given vacuum, such that the phase and mass spectrum continue smoothly down to $\mx\ra 0$. In other words, the theory stays at $\mx<M_{\rm min}$ in the same regime and all hierarchies in the mass spectrum are the same as in the ${\cal N}=2$ theory at $\mx\ra 0$.

Within this framework we considered first in detail in section 2.1 the br2 vacua of $SU(N_c)$ theory with $N_c+1<N_F<2N_c-1$ (these are vacua of the baryonic branch in [8] or zero vacua in [22, 23]), at $m\ll\lm$ and $0<\mx\ll M^{(\rm br2)}_{\rm min}=\langle\Lambda^{SU(\nd-{\rm n}_1)}_{{\cal N}=2\,\,SYM}
\rangle^2/\lm\ll\langle\Lambda^{SU(\nd-{\rm n}_1)}_{{\cal N}=2\,\,SYM}\rangle\ll m\ll\lm$. The original color symmetry is broken spontaneously in these vacua by higgsed $\langle X^{adj}_{SU(N_c)}\rangle\neq 0$ in three stages. The first stage is at the highest scale $\mu\sim\lm$, $\,\langle X^{adj}_{SU(2N_c-N_F)}\rangle\sim
\lm$, $\,\,SU(N_c)\ra SU(N_F-N_c)\times U^{(1)}(1)\times U^{2N_c-N_F-1}(1)$, this pattern is required by the unbroken discrete $Z_{2N_c-N_F\geq 2}$ symmetry and is necessary to avoid $g^2(\mu<\lm)$ in this effectively massless at the scale $\mu\sim\lm$ ${\cal N}=2$ UV free theory. The second stage is in the $SU(N_F-N_c)$ color sector at the scale $\mu\sim m$, $\,\langle X^{adj}_{SU(N_F-N_c)}\rangle\sim m\ll\lm$, $\,SU(N_F-N_c)\ra SU(\no)\times U^{(2)}(1)\times SU(N_F-N_c-\no)$ . And the third stage is in the $SU(N_F-N_c-\no)$ color sector at the scale $\mu\sim\langle\Lambda^
{SU(\nd-{\rm n}_1)}_{{\cal N}=2\,\,SYM}\rangle\ll m$, $\,SU(N_F-N_c-\no)\ra U^{N_F-N_c-\no-1}(1)$.

The global flavor symmetry $U(N_F)$ is broken spontaneously at $\mx\neq 0$ in these br2-vacua at the scale $\mu\sim (\mx m)^{1/2}$ in the $SU(\no)$ color sector by higgsed original pure electric quarks, $\,\langle{\ov Q}^{\,a}_i\rangle=\langle Q^i_a\rangle\sim \delta_a^i\,(\mx m)^{1/2}\,,\,\, a=1...\no,\,\, i=1...N_F$, as : $U(N_F)\ra U(\no)\times U(\nt),\, 1\leq\no< N_F-N_c$.

It was shown that the lightest charged BPS particles (massless at $\mx\ra 0$) are the following. -

1) $\bb$ flavorless mutually local dyons $D_j, {\ov D}_j$ (this number $\bb$ is required by the unbroken $Z_{\bb\geq 2}$ discrete symmetry which operates interchanging them among one another), with the two nonzero pure electric charges, the $SU(N_c)$ baryon charge and $U^{(1)}(1)$ one. The nonzero magnetic parts of charges of all these dyons are $SU(\bb)$ adjoints. The whole $\bb$ set of these dyons is coupled with $\bb$ Abelian ${\cal N}=2$ photon multiplets, $U^{\bb-1}(1)$ dyonic and $U^{(1)}(1)$ pure electric. These dyons are formed at the scale $\mu\sim\lm$, at the first stage of the color breaking, and are mutually local with respect to all particles of the $SU(N_F-N_c)$ sector and between themselves.

2) All original electric particles of remaining unbroken at $\mx\ra 0$ $\,SU(\no)$ subgroup of original $SU(N_c)$.

3) $\nd-\no-1$ pure magnetic monopoles with the $SU(\nd-\no)$ adjoint charges. These last are formed at the low scale $\mu\sim\langle\Lambda^{SU(\nd-{\rm n}_1)}_{{\cal N}=2\,\,SYM}\rangle\ll m\ll\lm$ in the ${\cal N}=2\,\,SU(\nd-\no)$ SYM sector, with the scale factor $\langle\Lambda^{SU(\nd-{\rm n}_1)}_{{\cal N}=2\,\,SYM}\rangle$ of its gauge coupling. These magnetic monopoles are coupled with $U^{\nd-\no-1}(1)$
${\cal N}=2$ Abelian magnetic photon multiplets.

4) In addition to original pure electric ${\cal N}=2\,\, SU(\no)$ adjoints, there are $U^{\nd-\no-1}(1)$  pure magnetic, $U^{\bb-1}(1)$ dyonic, and one $U^{(1)}(1)$ pure electric ${\cal N}=2$ Abelian photon multiplets, all massless at $\mx\ra 0$.

Massless at $\mx\ra 0$ $\,2N_c-N_F$ dyons $D_j,\,\no$ original pure electric quarks $Q^i_a, {\ov Q}^a_i,\, a=1...\no,\, i=1...N_F$ from $SU(\no)$, and $\nd-\no-1$ pure magnetic monopoles are all higgsed at small $\mx\neq 0$, and no massless particles remains (except for $2\no\nt$ Nambu-Goldstone multiplets due to spontaneous breaking of global flavor symmetry, $U(N_F)\ra U(\no)\times U(\nt)$), in the $SU(\no)$ color sector due to higgsed original pure electric quarks with $N_F$ flavors). The mass spectrum was described in these br2-vacua at smallest $0<\mx\ll\langle\Lambda^{SU(\nd-{\rm n}_1)}_{{\cal N}=2\,\,SYM}\rangle^2/\lm$.

The material of this section 2.1 served then as a basis for similar regimes in sections 3, 6, 7, and 8.

The mass parameter $\mx$ was increased then in two different stages :  1) $\langle\Lambda^{SU(\nd-{\rm n}_1)}_{{\cal N}=2\,\,SYM}\rangle\ll\mx\ll m$ in section 2.3 ;  2) $m\ll\mx\ll\lm$ in section 2.4, and changes in the mass spectra were described. In all those cases when the corresponding ${\cal N}=1$ SQCD lower energy theories were weakly coupled, we needed no additional dynamical assumptions.  And only in those few cases in section 2.4, when the corresponding ${\cal N}=1$ SQCD theories were in the strongly coupled conformal regime at $3N_c/2<N_F<2N_c$, we used additionally the assumption of the dynamical scenario introduced in [25]. In essence, this assumption from [25] looks here as follows : no additional parametrically light composite solitons are formed in this ${\cal N}=1$ SQCD without decoupled  colored adjoint scalars $X^{\rm adj}$ at those even lower scales where this ${\cal N}=1$ conformal regime is broken by nonzero particle masses (see also the footnote <ref>). As was shown in [25, 14, 9], this dynamical scenario does not contradict to any proven properties of ${\cal N}=1$ SQCD, satisfies all those checks of Seiberg's duality hypothesis for ${\cal N}=1$ SQCD which were used in [15], and allows to calculate the corresponding mass spectra.

Many other vacua with the broken or unbroken flavor symmetry and at different hierarchies between the Lagrangian parameters $m,\, \mx,\, \lm$ were considered in sections 3-8 and corresponding mass spectra were calculated within the above described framework. Besides, calculations of power corrections to the leading terms of the low energy  quark and dyon condensates in a number of vacua 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. These agreements serve as the independent numerous checks of a self-consistency of the whole approach.

We consider that there is no need to repeat in detail in these conclusions a whole context of the paper (see the table of contents). But at least two points have to be emphasized. 1) As shown e.g. in section 6.2 (and similarly in sections 6.4, 7, and 8), the widespread opinion that the holomorphic dependence of gauge invariant chiral condensates (e.g. $\langle (\qq)_{1,2}\rangle_{N_c},\, \langle S\rangle_{N_c},\, \langle {\rm Tr\,}(X^{adj}_{SU(N_c)})^2\rangle\,)$ on parameters of the superpotential implies the absence of phase transitions in supersymmetric theories is not right. 2) In contradiction with the appearance of
"instead-of-confinement" phase proposed in a number of papers by M.Shifman and A.Yung, see e.g. the latest paper [33] and references therein to their previous papers with this phase, we have found no such phase in all cases considered (see e.g. section 6.4 with the simplest and most clear case).

Clearly, this article does not pretend on strict proofs as the two dynamical assumptions of general character presented in Introduction were used (i.e., the assumption "A" about the BPS nature of original electric particles, and "B" about the absence of extra massless particles). But, to the best of our knowledge, the results obtained look self-consistent, satisfy a large number of independent checks, and do not contradict to any proven results.

In comparison with corresponding results from recent related papers [22, 23, 33] of M.Shifman and
A.Yung (and from a number of their previous numerous papers on this subject), our results are essentially different. In addition to critical remarks given in Introduction and in section 6, an extended criticism of a number of results from [22, 23] is given in section 8 of [9] .

Power corrections to the quark and dyon condensates

As examples, we present below in two Appendices the calculation of leading power corrections to the low energy quark condensates $\langle\Qo\rangle_{\no}$ in br1 vacua of $SU(N_c)$ and $U(N_c)$ in sections 4.1 and 4.2, and in br2 vacua of $SU(N_c)$ and $U(N_c)$ in sections 2.1 and 2.2, and to the dyon condensates in sections 2.1 and 2.2. The results agree with also presented in these Appendices independent calculations of these condensates using roots of the Seiberg-Witten spectral curve. These agreements serve as the independent numerous checks of a self-consistency of the whole approach.

§ CALCULATIONS OF QUARK CONDENSATES

1) br1 vacua of $SU(N_c),\,\, m\gg\lm$ in section 4.1

From (<ref>),(<ref>), in these vacua (up to even smaller power corrections)
⟨⟩_N_c=m_3-N_c-/N_c-⟨⟩_N_c ,  ⟨⟩_N_c≈m_3(/m_3)^2N_c-N_F/N_c- ,
⟨S⟩_N_c=⟨⟩_N_c⟨⟩_N_c/≈m_3⟨⟩_N_c ,  m_3=N_c/N_c- m .
From (<ref>),(<ref>), the leading power correction to $\langle\w^{\,\rm eff}_{\rm tot}\rangle$ is
⟨δ^ eff_tot⟩≈(N_c-)m^2_3(/m_3)^2N_c-N_F/N_c-≈(N_c-)⟨S⟩_N_c .
On the other hand, this leading power correction to $\langle\w^{\,\rm low}_{\rm tot}\rangle$ originates from the $SU(N_c-\no)$ SYM part, see (<ref>),(<ref>),
⟨δ^ low_tot⟩≈⟨ Tr (X^adj_SU(N_c-))^2 ⟩≈(N_c-)⟨S⟩_N_c-
≈(N_c-)⟨Λ^SU(N_c-)_N=2  SYM⟩^2≈≈(N_c-)m_3^2(/m_3)^2N_c-N_F/(N_c-) ,    ⟨δ^ low_tot⟩=⟨δ^ eff_tot⟩ ,
as it should be.

To calculate the leading power correction $\delta\langle\Qo\rangle_{\no}\sim\mx\langle\Lambda^
{SU(N_{c}-\no)}_{{\cal N}=2\,\,SYM}\rangle^2/m$ to the low energy quark condensate $\langle\Qo\rangle_{\no}$, we have to account for the terms with the first power of quantum fluctuation ${\hat a}$ of the field $a=\langle a\rangle+{\hat a},\,\,\langle{\hat a}\rangle=0$, in the $SU(N_c-\no)$ SYM contribution to $\delta \w^{\,\rm low}_{\rm tot}$, see (<ref>)-(<ref>),
δ^ low_tot≈(N_c-)  δ  (Λ^SU(N_c-)_N=2  SYM)^2≈(N_c-) δ (^2N_c-N_F(m_3-c â)^N_F/[ m_3+(1-c)â ]^2)^1/N_c-≈≈ ⟨Λ^SU(N_c-)_N=2  SYM⟩^2 [ -(2N_c-N_F)/N_c- ] â/m_3 ,  m_3=N_c/N_c- m .
As a result, from (<ref>),(<ref>),(<ref>),
δ⟨⟩_≈1/⟨∂/∂âδ^ low_tot⟩≈- (2N_c-N_F)/N_c- ⟨Λ^SU(N_c-)_N=2  SYM ⟩^2/m_3 ,
⟨⟩^SU(N_c)_=⟨Q^1_1⟩⟨Q^1_1⟩≈m_3[1-2N_c-N_F/N_c-  ⟨Λ^SU(N_c-)_N=2  SYM ⟩^2/m^2_3]≈
≈m_3[1-2N_c-N_F/N_c-(/m_3)^2N_c-N_F/N_c- ] ,  ⟨⟩^SU(N_c)
_N_c≈m_3[1-N_c-/N_c-(/m_3)^2N_c-N_F/N_c- ] .
It is seen from (<ref>) that, although the leading terms are the same in $\langle\Qo\rangle_{\no}$ and $\langle\Qo\rangle_{N_c}$, the leading power corrections are different.

2) br1 vacua of $U(N_c),\,\, m\gg\lm$ in section 4.2

From (<ref>),(<ref>),(<ref>),(<ref>) (keeping now leading power corrections)
⟨⟩_N_c=m-⟨⟩_N_c ,
(1+O(⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m^2 )) ,
⟨S⟩_N_c=⟨⟩_N_c⟨⟩_N_c/≈m^2(/m)^2N_c-N_F/N_c-(1+O(⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m^2 )) ,
⟨a_0⟩/m=1/N_cm⟨Tr ⟩_N_c≈/N_c  +-/N_c(/m)^2N_c-N_F/N_c- ,  ⟨a⟩=⟨m-a_0⟩ ,
⟨^ low_tot⟩≈[⟨_⟩=0 ]+[
⟨_a_0,a⟩=/2( N_c⟨a_0⟩^2+N_c/N_c-⟨a⟩^2)]+⟨_SU(N_c-)^ SYM⟩ ,
⟨_SU(N_c-)^ SYM⟩=⟨ Tr ( X^adj_SU(N_c-))^2⟩=
(N_c-)⟨S⟩_SU(N_c-)=(N_c-)⟨Λ^SU(N_c-)_N=2  SYM⟩^2≈≈(N_c-)⟨(^2N_c-N_F(m-a_0-c  a)^N_F/[ (1-c) a  ]^2 )^1/N_c-⟩≈(N_c-)m^2(/m)^2N_c-N_F/N_c- .
From (<ref>)-(<ref>), the leading power correction $\langle\,\delta\,\w^{\,\rm low}_{\rm tot}\rangle\sim\mx\langle\Lambda^{SU(N_{c}-\no)}_{{\cal N}=2\,\,SYM}\rangle^2$ to $\langle\w^{\,\rm low}
_{\rm tot}\rangle$ looks as
⟨ δ  ^ low_tot⟩=(⟨ δ _a_0,a⟩=0)+
⟨ δ _SU(N_c-)^ SYM⟩≈≈(N_c-)⟨Λ^SU(N_c-)_N=2  SYM⟩^2≈(N_c-) m⟨⟩_N_c≈(N_c-)m^2(/m)^2N_c-N_F/N_c- .

Instead of (<ref>),(<ref>) we have now, see (<ref>),
^ eff_tot(Π)=m Tr (Q Q)_N_c-1/2[  ∑_i,j=1^N_F (Q_j Q^i)_N_c(Q_ i Q^j)_N_c]-[ S_N_c=⟨⟩_N_c⟨⟩_N_c/ ],
and using (<ref>) (with the same accuracy)
⟨δ^ eff_tot(Π)⟩≈(N_c-)m⟨⟩_N_c≈(N_c-)m^2(/m)^2N_c-N_F/N_c-≈⟨δ^ low
_tot⟩ ,
as it should be.

As for the leading power correction $\delta\langle\Qo\rangle_{\no}\sim \mx m\Bigl (\lm/m\Bigr )^
{\frac{2N_c-N_F}{N_c-\no}}$ to the quark condensate, we have now instead of (<ref>)
δ^ low_tot=δ_a_0,a+δ_SU(N_c-)^ SYM ,  a_0=⟨a_0⟩+â_0 ,  a=⟨a⟩+â ,
δ_SU(N_c-)^ SYM=(N_c-)  δ  (Λ^SU(N_c-)_N=2  SYM)^2≈(N_c-) δ (^2N_c-N_F(m-â_0-c â)^N_F/[ m+(1-c)â ]^2)^1/N_c-≈≈[ -N_Fâ_0-(2N_c-N_F)/N_c- â  ) ⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m ,

while from (<ref>),(<ref>)
δ_a_0,a≈(-)[ â_0-/N_c- â ]⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m .

Instead of (<ref>) (with the same accuracy), on the one hand, see (<ref>),(<ref>),(<ref>),
⟨ Tr ⟩_=m+δ⟨ Tr ⟩_=(m+δ ⟨⟩_) ,
δ⟨ Tr ⟩_=1/⟨∂/∂â_0 (δ_SU(N_c-)^ SYM+δ_a_0,a)⟩≈-2 ⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m ,
while on the other hand,
δ⟨ Tr ⟩_=1/⟨ ∂/∂â(δ_SU(N_c-)^ SYM+δ_a_0,a)≈-2 ⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m  ,
as it should be.

Therefore, on the whole for these $U(N_c)$ br1 vacua, see (<ref>) for $\langle\Qo\rangle^{U(N_c)}_{N_c}$,
⟨⟩^U(N_c)_=⟨Q^1_1⟩⟨Q^1_1⟩≈m[ 1-2⟨Λ^SU(N_c-)_N=2 SYM⟩^2/m^2 ]≈m [  1-2(/m)^2N_c-N_F/N_c- ] ,
⟨⟩^U(N_c)_N_c≈m[ 1-⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m^2 ].

We can compare also the result for $\langle\Qo\rangle^{U(N_c)}_{\no}$ in (<ref>) with those from (<ref>).
In these simplest br1 vacua of $U(N_c)$ the charges and multiplicities of particles massless at $\mx\ra 0$ are evident, these are original electric particles from $SU(\no)\times U^{(0)}(1)$ and $N_c-\no-1$ magnetic monopoles from $SU(N_c-\no)$ SYM.
To obtain definite predictions from (<ref>) one has to find first the values of roots entering (<ref>). In the case  considered, to obtain definite values of all roots of the curve (<ref>) in the br1 vacua of $U(N_c)$ theory one needs only one additional relation [30, 31] for the two single roots $e^{\pm}$ of the curve (<ref>)
e^±=±2√(⟨S⟩_N_c/) ,  e^±_c=1/2 (e^++e^-)=0 .
One can obtain then from the $U(N_c)$ curve (<ref>) the values of $\no$ equal double roots $e^{(Q)}_i$ of original electric quarks from $SU(\no)$ and $N_c-\no-1$ unequal double roots $e^{(M)}_k$ of magnetic monopoles from $SU(N_c-\no)$ SYM. We find from (<ref>),(<ref>) with our accuracy, see (<ref>),(<ref>),
e^+=-e^-≈2 m(/m)^2N_c-N_F/2(N_c-)(1+O(⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m^2 ) )
≈2⟨Λ^SU(N_c-)_N=2  SYM⟩,  e^(Q)_i= - m,   i=1... ,
_N=2  SYM⟩--/N_c--1⟨Λ^SU(N_c-)
_N=2  SYM⟩^2/m ,  k=1...(N_c--1) .
Let us note that these roots satisfy the sum rule, see (<ref>),
∑_n=1^N_cϕ_n=1/2∑_n=1^2 N_c(-e_n)=N_c⟨a_0⟩ ,
as it should be.

Therefore, from (<ref>),(<ref>)
⟨⟩^U(N_c)_= - √((e^(Q)_i-e^+)(e^(Q)_i-e^-))≈≈m (1-2⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m^2 )≈m [  1-2(/m)^2N_c-N_F/N_c- ] ,  i=1... ,
this agrees with (<ref>).

Besides, we can use the knowledge of roots of the curve (<ref>) for the $U(N_c)$ theory and of $\langle a_0\rangle$, see (<ref>),(<ref>), to obtain the values of $\langle\Qo\rangle^{SU(N_c)}_{\no}$ condensates in the $SU(N_c)$ theory. For this, the curve (<ref>) for the $U(N_c)$ theory
y^2=∏_i=1^N_c(z+ϕ_i)^2-4^(z+m)^N_F ,  ∑_i=1^N_cϕ_i=1/2∑_n=1^2 N_c(-e_n)=N_c⟨a_0⟩    in    U(N_c) ,
we rewrite in the form of the $SU(N_c)$ theory
y^2=∏_i=1^N_c(ẑ+ϕ̂_i)^2-4^(ẑ+m̂)^N_F ,   ∑_i=1^N_cϕ̂_i=1/2∑_n=1^2 N_c(-ê_n)=0 ,
m̂=m  (1-m^-1⟨a_0⟩) .

But clearly, it remained the same $U(N_c)$ theory with the same condensates (<ref>). Therefore, for the $SU(N_c)$ curve (<ref>) the quark condensates will be as in (<ref>) with $m$ replaced by
${m^\prime}=m/(1-m^{-1}\langle a_0\rangle)$, both in the leading terms and in all power corrections, i.e., see (<ref>),
⟨a_0⟩/m    ≈( /N_c+-/N_c(/m_3)^2N_c-N_F/N_c- ) ,
mm^'≈m_3(1+-/N_c-(/m_3)^/N_c-) ,  m_3=N_c/N_c- m .
Then, from (<ref>),(<ref>),
⟨⟩^SU(N_c)_≈m_3 (1+-/N_c-(/m_3)^/N_c- )[  1-2(/m_3)^2N_c-N_F/N_c- ]≈
≈m_3 [  1-/N_c-(/m_3)^2N_c-N_F/N_c- ] ,
this agrees with (<ref>).

3) br2 vacua of $SU(N_c),\,\, m\ll\lm$ in section 2.1

The calculations of leading power corrections to $\langle\,\delta \w_{\rm tot}^{\,\rm low}\rangle$ and $\langle\,\delta \w^{\,\rm eff}_{\rm tot}\rangle$ have been presented in (<ref>). Therefore, we present here the calculation of the leading power correction $\delta\,\langle\Qo\rangle_{\no}\sim \mx \langle\Lambda^{SU(N_{c}-\no)}_{{\cal N}=2\,\,SYM}\rangle^2/m$ to the quark condensate. For this, see (<ref>),(<ref>), we write ($a_{1,2}=\langle a_{1,2}\rangle+{\hat a}_{1,2}$)
δ^(SYM)_SU(-)≈(-)  δ (Λ^SU(-)_N=2  SYM)^2≈(-) δ (Λ_SU()^2-N_F
(m_2-â_1-c_2 â_2)^N_F/[ m_2+(1-c_2) â_2 ]^ 2)^1/-N_c≈≈⟨Λ^SU(N_c-)_N=2  SYM
⟩^2/m_1 [ N_Fâ_1+(2N_c-N_F)/N_c- â_2 ],
m_1= N_c/N_c-m= -  m_2,   =- ,
⟨ Tr ⟩_=m_1+δ⟨ Tr ⟩_=(m_1+δ ⟨⟩_) ,  Λ_SU()=- ,
From this, see (<ref>), and (<ref>) for $\langle\Qt\rangle^{SU(N_c)}_{N_c}$,
δ ⟨⟩_=1/⟨∂/∂â_2^(SYM)_SU(-)⟩≈m_1[ 2N_c-N_F/N_c-⟨Λ^
SU(-)_N=2  SYM⟩^2/m^2_1 ],  ⟨Λ^SU(-)_N=2  SYM⟩^2/m^2_1≈(m_1/)^2N_c-N_F/-N_c,
⟨⟩^SU(N_c)_=⟨Q^1_1⟩⟨Q^1_1⟩≈m_1[ 1+2N_c-N_F/-N_c
⟨Λ^SU(-)_N=2  SYM⟩^2/m^2_1 ]≈
≈m_1[ 1+2N_c-N_F/-N_c (m_1/)^2N_c-N_F/-N_c] ,  m_1= N_c/N_c-m ,
⟨⟩^SU(N_c)_N_c=m_1-N_c-/N_c- ⟨⟩_N_c
≈m_1[  1+N_c-/-N_c⟨Λ^SU(-)_N=2  SYM⟩^2/m^2_1 ] .
It is seen that, as in (<ref>), the leading corrections to $\langle\Qo\rangle_{\no}$ and $\langle\Qt\rangle_{N_c}$ are different.

4) br2 vacua of $U(N_c),\,\, m\ll\lm$ in section 2.2

Keeping the leading order corrections $\sim\langle\Lambda^{SU(\nd-\no)}_{{\cal N}=2\,\, SYM}\rangle^2/m$, from (<ref>),(<ref>),(<ref>) (put attention that the quark and gluino condensates, and $\Lambda^{SU(\nd-\no)}_{{\cal N}=2\,\, SYM}$ are different in br2 vacua of $SU(N_c)$ or $U(N_c)$ theories, see (<ref>),(<ref>),
⟨⟩_N_c=m-⟨⟩_N_c ,  ⟨⟩_N_c≈m( m/)^2N_c-N_F/-N_c≈⟨Λ^SU(-)_N=2   SYM⟩^2/m  ,
⟨a_0⟩/m=1/N_cm⟨Tr ⟩_N_c≈/N_c+-/N_c
( m/)^2N_c-N_F/-N_c .
δ_SU(-)^ SYM=(-) δ [(Λ^SU(-)_N=2   SYM)^2≈(Λ_SU()^2-N_F(-m-â_0-â_1-c_2 â_2)^N_F/[ -m+(1-c_2) â_2 ]^ 2)^1/-N_c]
≈ ⟨Λ^SU(-)_N=2   SYM⟩^2/m  [ N_F(â_0+â_1)- 2N_c-N_F/-N_c â_2 ],   ⟨Λ^SU(-)_N=2  SYM⟩^2/m^2=
(m/)^2N_c-N_F/-N_c,   Λ_SU()=-  ,
δ_a≈⟨Λ^SU(-)_N=2   SYM⟩^2/m [(-) â_0-N_F/2N_c-N_F â_1--/-N_c â_2 ] ,  a_i=⟨a_i⟩+â_i,  i=0, 1, 2 ,
δ⟨⟩^(U(N_c)_=1/⟨∂/∂â_2(δ_SU(-)^ SYM+δ_a)⟩≈- 2m ⟨Λ^SU(-)_N=2   SYM⟩^2/m^2≈- 2m(m/)^2N_c-N_F/-N_c .
Therefore, on the whole for these $U(N_c)$ br2 vacua after accounting for the leading power corrections, see (<ref>),(<ref>),
],   ⟨⟩^U(N_c)_N_c≈m[1-(m/)^2N_c-N_F/-N_c].

It is seen that, as above in $SU(N_c)$ br2 vacua (<ref>), the non-leading terms in $\langle\Qo\rangle_{\no}$ and $\langle\Qt\rangle_{N_c}$ are different.

We can compare now the value of $\langle\Qo\rangle^{U(N_c)}_{\no}$ from (<ref>) with those from
(<ref>) (see however the footnote <ref>, in these br2 vacua the charges of massless at $\mx\ra 0$ particles are non-trivial and not obvious beforehand, as well as their multiplicities). But we know the charges and multiplicities of massless at $\mx\ra 0$ particles from sections 2.1 and 2.2. To obtain definite predictions from (<ref>) for the quark condensates $\langle\Qo\rangle_{\no}$ we need the values of $\no$ double roots $e^{(Q)}_k$ corresponding to original electric quarks from $SU(\no)$ and the values of two single roots $e^{\pm}$. Similarly to (<ref>),(<ref>), these look here as
e^(Q)_k= - m ,  k=1... ,  e^+=-e^-≈2⟨Λ^SU(-)_N=2   SYM⟩ ,  ⟨Λ^SU(-)_N=2   SYM⟩^2/m^2≈(m/)^2N_c-N_F/-N_c≪1 .
From this
⟨⟩^U(N_c)_= - √((e^(Q)_k-e^+)(e^(Q)_k-e^-))≈m [ 1-2(m/)^2N_c-N_F/-N_c ] ,
this agrees with (<ref>).

Besides, proceeding similarly to br1 vacua in (<ref>)-(<ref>), we obtain for $\langle\Qo\rangle^{SU(N_c)}_{\no}$ in br2 vacua of $SU(N_c)$, see (<ref>),
⟨a_0⟩/m    ≈( /N_c+-/N_c(m_1/)^2N_c-N_F/-N_c ) ,
mm^'≈m_1(1+-/N_c-(m_1/)^/-N_c ) ,  m_1=N_c/N_c- m ,
and from (<ref>),(<ref>)
⟨⟩^SU(N_c)_≈m_1 (1+-/N_c-(m_1/)^/-N_c)[  1-2(m_1/)^2N_c-N_F/-N_c ]≈
≈m_1 [  1+/-N_c(m_1/)^2N_c-N_F/-N_c ]≈m_1[ 1+2N_c-N_F/-N_c⟨Λ^SU(-)_N=2  SYM⟩^2/m^2_1 ] ,
this agrees with (<ref>).

§ CALCULATIONS OF DYON CONDENSATES

Consider first the br2 vacua of $\mathbf{U(N_c)}$ with $m\ll\lm$ in section 2.2. Because, unlike the br1 vacua of $U(N_c)$ theory in section 4.2, there are now in addition massless dyons in br2 vacua, to obtain with our accuracy the definite values of roots from the $U(N_c)$ curve (<ref>), one needs to use not only (<ref>), but also the sum rule (<ref>). The sum rule (<ref>) looks in these br2 vacua as, see (<ref>),
N_c⟨a_0⟩≈(m+(-)⟨Λ^SU(-)_N=2   SYM⟩^2/m )≈(m+(-) m(m/)^/-N_c)≈
≈(m+(--1)A^(M)_c⟨Λ^SU(-)_N=2   SYM⟩^2/m-1/2(e^++e^-=0)+(2N_c-N_F)C^(D)_c ) ,
where $\no(-e^{(Q)}_i)=\no m$ is the contribution of $\no$ equal double roots of original electric quarks from $SU(\no)$, the term with $A^{(M)}_c$ is the contribution from the centre of $\nd-\no-1$ unequal double roots of magnetic monopoles, and the term with $C^{(D)}_c$ is the contribution from the centre of $2N_c-N_F$ unequal double roots of dyons. We obtain then from the curve (<ref>) together with (<ref>) (with our accuracy)
A^(M)_c=-/--1 ,  C^(D)_c=-/2N_c-N_F m ( 1-2 ⟨Λ^SU(-)_N=2   SYM⟩^2/m^2 ) ,
_N=2  SYM⟩--/--1⟨Λ^SU(-)
_N=2  SYM⟩^2/m ,  k=1...(--1) ,
e^(D)_j≈ω^j-1(1+O(m/)^)--/2N_c-N_F m(1-2(m/)^/-N_c) ,   j=1...(2N_c-N_F) ,
while $e^{\pm}$ are given in (<ref>).

From (<ref>),(<ref>) we obtain now for the dyon condensates in these br2 vacua of $U(N_c)$ at $m\ll\lm$ (with the same accuracy)
⟨D_j D_j⟩_U(N_c)=⟨D_j⟩⟨D_j⟩≈[ -  ω^j-1+-/2N_c-N_F m
(1-2(m/)^/-N_c) ] ,
⟨Σ^U(N_c)_D⟩≡∑_j=1^2N_c-N_F⟨D_j D_j⟩_U(N_c)=∑_j=1^2N_c-N_F⟨D_j⟩⟨D_j⟩≈m (-)(1-2(m/)^/-N_c),     j=1... .

Now, using only the leading terms $\sim \mx m$ of (<ref>),(<ref>),(<ref>),
(<ref>), we can find the value of $\delta_3=O(1)$ in (<ref>),(<ref>),(<ref>),(<ref>). From (<ref>)
⟨∂/∂a_0^ low_tot⟩=0= - ⟨Tr ⟩_-⟨Σ^U(N_c)_D⟩+N_c⟨a_0⟩-N_cδ_3⟨a_1⟩= - N_cδ_3⟨a_1⟩δ_3=0.
(The same result follows from $\langle\partial \w^{\,\rm low}_{\rm tot}/{\partial a_1}\rangle=0$ in (<ref>) ).

As for the non-leading terms $\delta\langle\Sigma^{U(N_c)}_D\rangle\sim \mx m (m/\lm)^{\frac{\bb}{\nt-N_c}}$ of $\langle\Sigma^{U(N_c)}_D\rangle$ in (<ref>), with $\delta_3=0$ from (<ref>), these can now be also calculated independently from (<ref>),(<ref>),(<ref>), i.e.
⟨∂/∂â_0δ^ low_tot⟩=0δ⟨Σ^U(N_c)_D⟩≈[-δ⟨Tr⟩_=2m(m/)^/-N_c]_Q+[(-)m(m/)^/-N_c]_δ_a_0
+[⟨∂/∂â_0δ_SU(-)^ SYM⟩= - N_Fm(m/)^/-N_c]_δ_SYM≈-2m (-)
(m/)^/-N_c ,
this agrees with (<ref>).

As for the value of $\langle\Sigma^{SU(N_c)}_D\rangle$ in br2 vacua of $\mathbf{SU(N_c)}$, with $\delta_3=0$ from (<ref>), it can be found now directly from (<ref>),(<ref>),(<ref>),(<ref>)
⟨∂/∂â_1^ low_tot⟩=0⟨Σ^SU(N_c)_D⟩≈m_1(-)[1+/-N_c(m_1/)^/-N_c ] ,   m_1=N_c/N_c- m .
(Really, the power suppressed term in (<ref>) can be found independently of the value of $\delta_3$ because it is parametrically different and, unlike (<ref>) in $U(N_c)$ theory, there are no power corrections to $\langle a_1\rangle$ in $SU(N_c)$ theory, see (<ref>)).

It can also be obtained from (<ref>) in br2 vacua of $U(N_c)$ proceeding similarly to (<ref>),(<ref>) for the quark condensates. We obtain from (<ref>) and (<ref>)
⟨Σ^SU(N_c)_D⟩=∑_j=1^2N_c-N_F⟨D_j⟩⟨D_j⟩≈m_1(1+-/N_c-(m_1/)^/-N_c)(-)[  1-2(m_1/)^2N_c-N_F/-N_c ]≈≈m_1(-)[1+/-N_c(m_1/)^/-N_c ] ,
this agrees with (<ref>).

On the whole, the dyon condensates in br2 vacua of $SU(N_c)$ look at $m\ll\lm$ as, $j=1,...,\bb$,
⟨D_j D_j⟩^SU(N_c)≈[-ω^j-1(1+O(m/)^)+-/2N_c-N_F m_1(1+/-N_c(m_1/)^/-N_c) ].

Finally, not going into details, we present below the values of roots of the $\mathbf{SU(N_c)}$ spectral curve (<ref>) in br1 vacua of section 4.1 and br2 vacua of section 2.1.

1) br1 vacua. There are at $\mx\ra 0$ $\,\no$ equal double roots $e^{(Q)}_i$ of massless original electric quarks $Q_i$, then $N_c-\no-1$ unequal double roots $e^{(M)}_k$ of massless pure magnetic monopoles $M_k$ and two single roots $e^{\pm}$ from ${\cal N}=2\,\, SU(N_c-\no)$ SYM. They look (with our accuracy) as, see (<ref>),
e^(Q)_i= - m ,  i=1... ,  e^±≈e^±_c±2⟨Λ^SU(N_c-)_N=2  SYM⟩,  ⟨Λ^SU(N_c-)_N=2  SYM⟩≈m_3(/m_3)^2N_c-N_F/2(N_c-),
e^±_c=1/2(e^++e^-)≈m_3[ /N_c +-/N_c-⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m^2_3  ],   m_3=N_c/N_c- m ,
e^(M)_k≈[ e^±_c--/N_c--1⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m_3 ]+ 2cos(πk/N_c-)⟨Λ^SU(N_c-)
_N=2  SYM⟩,  1/2∑_n=1^2 N_c(-e_n)=0 .
It should be emphasized that instead of $e^{\pm}_{c}=0$ in $\mathbf{U(N_c)}$ theory, see (<ref>), it looks in $\mathbf{SU(N_c)}$ theory as, see (<ref>),(<ref>),
e^±_c≈[⟨a_0⟩/m][ m ]  [ ⟨a_0⟩/m      ≈( /N_c+-/N_c(/m_3)^2N_c-N_F/N_c- ) ][m   m^'
≈m_3( 1+-/N_c-
·(/m_3)^/N_c-) ]≈m_3[/N_c+-/N_c-(/m_3)^2N_c-N_F/N_c- ]≈m_3[ /N_c+ -/N_c-⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m^2_3  ] .

2) br2 vacua. There are at $\mx\ra 0$ $\,\no$ equal double roots $e^{(Q)}_i$ of massless original electric quarks $Q_i$, then $\nd-\no-1$ unequal double roots $e^{(M)}_k$ of massless pure magnetic monopoles $M_k$ and two single roots $e^{\pm}$ from ${\cal N}=2\,\, SU(\nd-\no)$ SYM, and $2N_c-N_F$ unequal double roots of massless dyons $D_j$. They all look (with our accuracy) as, see (<ref>),(<ref>),(<ref>),
e^(Q)_i= - m ,  i=1... ,  e^±≈e^±_c±2⟨Λ^SU(-)_N=2  SYM⟩,  ⟨Λ^SU(-)_N=2  SYM⟩^2≈m^2_1(m_1/)^2N_c-N_F/-N_c,
e^±_c=1/2(e^++e^-)≈[ ⟨a_0⟩/m    ≈( /N_c+-/N_c(m_1/)^2N_c-N_F/-N_c ) ][ m   m^'≈m_1(1+
+-/N_c-(m_1/)^/-N_c) ]≈m_1[ /N_c+-/-N_c
⟨Λ^SU(-)_N=2  SYM⟩^2/m^2_1  ],   m_1=N_c/N_c- m ,
e^(M)_k≈[ e^±_c--/--1⟨Λ^SU(-)_N=2  SYM⟩^2/m_1 ]+ 2cos(πk/-)⟨Λ^SU(-)_N=2  SYM⟩,  k=1...(--1) ,
e^(D)_j≈ω^j-1+N_F/2N_c-N_F m ,  j=1...2 N_c-N_F ,
1/2∑_n=1^2 N_c(-e_n)=0 .

Now, with (<ref>)-(<ref>), we can perform checks of possible applicability of (<ref>) not only to softly broken $\mathbf{U(N_c)}$ $\,{\cal N}=2$ SQCD, but also directly to softly broken $\mathbf{SU(N_c)}$ $\,{\cal N}=2$ SQCD at appropriately small $\mx$ (clearly, this applicability is far not evident beforehand, see the footnote <ref>).

a) From (<ref>) for the quark condensates in br1 vacua of section 4.1
⟨⟩^SU(N_c)_= - √((e^(Q)_k-e^+)(e^(Q)_k-e^-))≈m_3[ 1-2N_c-N_F/N_c-
⟨Λ^SU(N_c-)_N=2  SYM⟩^2/m^2_3  ] ,
this agrees with the independent calculation in (<ref>).

b) From (<ref>) for the quark condensates in br2 vacua of section 2.1
⟨⟩^SU(N_c)_= - √((e^(Q)_k-e^+)(e^(Q)_k-e^-))≈m_1[ 1+2N_c-N_F/-N_c⟨Λ^SU(-)_N=2  SYM⟩^2/m^2_1 ] ,
this agrees with the independent calculation in (<ref>).

c) From (<ref>),(<ref>) for the dyon condensates in br2 vacua of section 2.1
⟨D_j D_j⟩^SU(N_c)=⟨D_j⟩^SU(N_c)⟨D_j⟩^SU(N_c)= - √((e^(D)_j-e^+)(e^(D)_j-e^-))≈≈[-ω^j-1+-/2N_c-N_F m_1(1+/-N_c⟨Λ^SU(-)_N=2  SYM⟩^2/m^2_1) ] ,  j=1...2 N_c-N_F ,
this agrees with the independent calculation in (<ref>).

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