n}_{1}\mu_{\rm x}m_{1}\,,$ $\langle({\overline{Q}}Q)_{1}\rangle_{{\rm
n}_{1}}=\langle{\overline{Q}}^{\,1}_{1}\rangle\langle
Q^{1}_{1}\rangle\approx\frac{N_{c}}{N_{c}-{\rm n}_{2}}\,\mu_{\rm
x}m\approx\mu_{\rm
x}m_{1}\approx\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}\,,\quad\langle({\overline{Q}}Q)_{2}\rangle_{{\rm
n}_{1}}=\sum_{a=1}^{{\rm n}_{1}}\langle{\overline{Q}}^{\,a}_{2}\rangle\langle
Q^{2}_{a}\rangle=0\,.$

The multiplicity of these br2 vacua in the case considered is $N_{\rm
br2}=(N_{c}-{\rm n}_{2})C^{\,{\rm n}_{1}}_{N_{F}}=(N_{c}-{\rm n}_{2})C^{\,{\rm
n}_{2}}_{N_{F}}$, as it should be. The factor $(N_{c}-{\rm n}_{2})$ originates
from the $SU(N_{c}-{\rm n}_{2})$ SYM. The factor $C^{\,{\rm n}_{1}}_{N_{F}}$
is finally a consequence of the color breaking $SU({\rm n}_{2})\rightarrow
SU({\rm n}_{1})\times U(1)^{{\rm n}_{2}-{\rm n}_{1}}$ with the unbroken
$Z_{2{\rm n}_{2}-N_{F}}=Z_{{\rm n}_{2}-{\rm n}_{1}}$ discrete symmetry, and
spontaneous breaking of flavor symmetry due to higgsing of ${\rm n}_{1}$
quarks flavors (massless at $\mu_{\rm x}\rightarrow 0$) in the $SU({\rm
n}_{1})$ color subgroup of $SU({\rm n}_{2})$, $\langle
Q^{i}_{a}\rangle=\langle{\overline{Q}}^{\,a}_{i}\rangle\approx(\mu_{\rm
x}m_{1})^{1/2}$, $a=1...{\rm n}_{1},\,\,i=1...{\rm n}_{1}$.

Note also that if the non-trivial discrete symmetry $Z_{{\rm n}_{2}-{\rm
n}_{1}\geq 2}$ were broken spontaneously in the $SU({\rm n}_{2})$ subgroup at
the scale $\mu\sim\langle\Lambda_{SU({\rm n}_{2})}\rangle\gg(\mu_{\rm
x}m)^{1/2}$, this will lead then to the factor ${\rm n}_{2}-{\rm n}_{1}$ in
the multiplicity of these vacua, and this is a wrong factor.

Due to higgsing of ${\rm n}_{2}-{\rm n}_{1}$ dyons with the non-zero $SU({\rm
n}_{2}-{\rm n}_{1})$ magnetic charges,
$\langle{\overline{D}_{j}}\rangle\langle D_{j}\rangle\sim\mu_{\rm
x}m\gg\mu_{\rm x}\Lambda_{2}\gg\mu_{\rm x}\langle\Lambda_{SU({\rm
n}_{2})}\rangle$, all original pure electrically charged particles with
$SU({\rm n}_{2}-{\rm n}_{1})$ charges and masses either $\sim m$ or
$\sim\langle\Lambda_{SU({\rm n}_{2})}\rangle$ are weakly confined, the string
tension is $\sigma^{1/2}\sim(\mu_{\rm x}m)^{1/2}\ll\langle\Lambda_{SU({\rm
n}_{2})}\rangle\ll\Lambda_{2}\ll m$. Besides, ${\rm n}_{2}-{\rm n}_{1}$ long
${\cal N}=2$ multiplets of massive photons with masses $\sim(\mu_{\rm
x}m)^{1/2}$ are formed. Remind that, due to higgsing of $N_{c}-{\rm n}_{2}-1$
magnetic monopoles $M_{\rm n}$ from $SU(N_{c}-{\rm n}_{2})$ SYM at $\mu_{\rm
x}<\langle\Lambda^{SU(N_{c}-{\rm n}_{2})}_{{\cal N}=2\,SYM}\rangle$, all
original pure electrically charged particles with $SU(N_{c}-{\rm n}_{2})$
charges and masses either $\sim m$ or $\langle\Lambda^{SU(N_{c}-{\rm
n}_{2})}_{{\cal N}=2\,SYM}\rangle$ are also weakly confined, but this string
tension $\sigma^{1/2}_{2}\sim(\mu_{\rm x}\langle\Lambda^{SU(N_{c}-{\rm
n}_{2})}_{{\cal N}=2\,SYM}\rangle)^{1/2}\ll(\mu_{\rm x}m)^{1/2}$ is much
smaller.

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

Remind that the curve (1.2) has $N_{c}-1$ double roots in these br2 vacua at
$\mu_{\rm x}\rightarrow 0$. Of them: ${\rm n}_{2}-{\rm n}_{1}$ unequal roots
correspond to ${\rm n}_{2}-{\rm n}_{1}$ dyons $D_{j},\,\,{\rm n}_{1}$ equal
roots correspond to ${\rm n}_{1}$ higgsed quarks $Q^{i}$ of $SU({\rm n}_{1})$,
the remaining unequal $N_{c}-{\rm n}_{2}-1$ double roots correspond to
$N_{c}-{\rm n}_{2}-1$ pure magnetic monopoles $M_{\rm n}$ from $SU(N_{c}-{\rm
n}_{2})\,\,{\cal N}=2$ SYM. The two single roots with
$(e^{+}-e^{-})\sim\langle\Lambda^{SU(N_{c}-{\rm n}_{2})}_{{\cal
N}=2\,SYM}\rangle$ 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 $\mu_{\rm x}\sim\langle\Lambda_{SU({\rm
n}_{2})}\rangle^{2}/m$. E.g., all $SU({\rm n}_{2}-{\rm n}_{1})$ charged gluons
have equal masses in the whole range of the higher energy phase of section 7.1
at $\mu_{\rm x}>(\rm a\,few)\langle\Lambda_{SU({\rm n}_{2})}\rangle^{2}/m$,
while masses of these gluons differ greatly at $\mu_{\rm x}\rightarrow 0$ in
the lower energy phase of this section,
$\mu^{ij}_{gl}\sim|(\tau^{i}-\tau^{j})\langle\Lambda_{SU({\rm
n}_{2})}\rangle|,\,i,j=0...{\rm n}_{2}-{\rm n}_{1}-1,\,i\neq j$. And $\langle
X^{adj}_{SU({\rm n}_{2}-{\rm n}_{1})}\rangle$,
$\,\langle({\overline{Q}}Q)_{2}\rangle_{{\rm n}_{2}}$ also behave non-
analytically at $\mu_{\rm x}\gtrless\langle\Lambda_{SU({\rm
n}_{2})}\rangle^{2}/m$. But, unlike the phase transition at $\mu_{\rm
x}\sim\Lambda_{2}^{2}/m$ in vs vacua of section 6, the phase transition in
these br2 vacua is at much smaller $\mu_{\rm x}\sim\langle\Lambda_{SU({\rm
n}_{2})}\rangle^{2}/m\,,\,\,\langle\Lambda_{SU({\rm
n}_{2})}\rangle\ll\Lambda_{2}$, see (7.1.6).

And finally, remind that the mass spectra in these br2 vacua with ${\rm
n}_{2}<N_{c}$ and $m\gg\Lambda_{2}$ depend essentially on the value of
$m/\Lambda_{2}$ and all these vacua evolve at $m\ll\Lambda_{2}$ to Lt vacua
with the spontaneously broken discrete $Z_{2N_{c}-N_{F}}$ symmetry, see
section 3 in [13] or section 4 in [14].

##  8 Large quark masses $\mathbf{m\gg\Lambda_{2}}$,
$SU(N_{c}),\,\,\,N_{F}<N_{c}$

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
$U(N_{F})\rightarrow U({\rm n}_{1})\times U({\rm n}_{2})$ 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
$N_{F}<N_{c},\,\,m\gg\Lambda_{2}$ 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 $m\gg\Lambda_{2},\,\mu_{\rm x}\ll\Lambda_{2}\,$.

The quark and gluino condensates look in these $SU(N_{c})$ S-vacua as, see
(1.3),(1.4) and the footnote 4, the leading terms only,

$\displaystyle\langle{\overline{Q}}Q\rangle_{N_{c}}\approx\frac{N_{c}}{N_{c}-N_{F}}\,\mu_{\rm
x}m\approx\mu_{\rm x}m_{1}\,,\quad m_{1}=\frac{N_{c}}{N_{c}-({\rm
n}_{2}=N_{F})}\,m\,,$ (8.1) $\langle
S\rangle_{N_{c}}=\Bigl{(}\frac{\Lambda_{2}^{2N_{c}-N_{F}}\mu_{\rm
x}^{N_{c}}}{\det\langle{\overline{Q}}Q\rangle}\Bigr{)}^{\frac{1}{N_{c}-N_{F}}}\approx\mu_{\rm
x}m_{1}^{2}\Bigl{(}\frac{\Lambda_{2}}{m_{1}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-N_{F}}}\ll\mu_{\rm
x}m_{1}^{2}\,.$

It is seen from (8.1) 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 $\sim
m\gg\Lambda_{2}$ in the weak coupling regime, $SU(N_{c})\rightarrow
SU(N_{F})\times U(1)\times SU(N_{c}-N_{f})$, see (8.7) below,

$\displaystyle\langle X^{adj}_{SU(N_{c})}\rangle=\langle
X^{adj}_{SU(N_{F})}+X_{U(1)}+X^{adj}_{SU(N_{c}-N_{F})}\rangle,$ (8.2)
$\sqrt{2}\,X_{U(1)}=a\,{\rm
diag}\,(\,\underbrace{\,1}_{N_{F}}\,;\,\underbrace{\,\widehat{c}}_{N_{c}-N_{F}}\,),\quad\widehat{c}=-\,\frac{N_{F}}{N_{c}-N_{F}}\,,\quad\langle
a\rangle=m\,,$

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 $m_{Q}=\langle
m-{\widehat{c}}\,a\rangle=N_{c}m/(N_{c}-N_{F})=m_{1}$. After they are
integrated out, the scale factor of the $SU(N_{c}-N_{F})$ SYM gauge coupling
is, see also (8.1),

$\displaystyle\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=2\,SYM}\rangle^{2}=\Bigl{(}\frac{\Lambda_{2}^{2N_{c}-N_{F}}{(m_{1})}^{N_{F}}}{(m_{1})^{2N_{F}}}\Bigr{)}^{\frac{1}{N_{c}-N_{F}}}={m^{2}_{1}}\Bigl{(}\frac{\Lambda_{2}}{{m_{1}}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-N_{F}}}\ll\Lambda_{2}^{2}\,,\quad
m_{1}=\frac{N_{c}}{N_{c}-N_{F}}m\,,$ (8.3) $\langle
S\rangle_{N_{c}-N_{F}}=\mu_{\rm x}\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=2\,SYM}\rangle^{2}\approx\langle S\rangle_{N_{c}}\,,$

while those of $SU(N_{F})$ SQCD is

$\displaystyle\langle\Lambda_{SU(N_{F})}\rangle=\Bigl{(}\,\frac{\Lambda_{2}^{2N_{c}-N_{F}}}{(m_{1})^{2(N_{c}-N_{F})}}\,\Bigr{)}^{\frac{1}{N_{F}}}=m_{1}\Bigl{(}\frac{\Lambda_{2}}{m_{1}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{F}}}\ll\Lambda_{2}\,.$
(8.4)

If $\mu_{\rm x}\ll\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal N}=2\,SYM}\rangle$,
the scalars $X^{adj}_{SU(N_{c}-N_{F})}$ of ${\cal N}=2$ SYM are higgsed at
$\mu\sim\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal N}=2\,SYM}\rangle$ (8.3) in a
known way, $SU(N_{c}-N_{F})\rightarrow U^{N_{c}-N_{F}-1}(1)$ [3], giving
$N_{c}-N_{F}$ vacua and $N_{c}-N_{F}-1$ massless at $\mu_{\rm x}\rightarrow 0$
magnetic monopoles ${\overline{M}}_{\rm n},M_{\rm n}$. These all are higgsed
at $\mu_{\rm x}\neq 0$ giving $N_{c}-N_{F}-1$ $\,{\cal N}=2$ long multiplets
of dual massive photons with masses $\mu_{a_{M,{\rm n}}}\sim\langle M_{\rm
n}\rangle\sim(\mu_{\rm x}\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=2\,SYM}\rangle)^{1/2}$. 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 $\sigma^{1/2}_{2}\sim(\mu_{\rm
x}\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=2\,SYM}\rangle)^{1/2}\ll\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=2\,SYM}\rangle$.

On the other hand, if e.g. $\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=2\,SYM}\rangle\ll\mu_{\rm x}\ll m$, see (8.3), all scalars
$X^{adj}_{SU(N_{c}-N_{F})}$ are too heavy and not higgsed, they all decouple
and can be integrated out at $\mu<g\mu_{\rm x}$, there remains then ${\cal
N}=1\,\,SU(N_{c}-N_{F})$ SYM with $\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=1\,SYM}\rangle=[\,\mu_{\rm x}\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=2\,SYM}\rangle^{2}\,]^{1/3}\gg\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=2\,SYM}\rangle$. 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 $g\mu_{\rm
x}$ are still weakly confined, but the string tension in this sector is larger
now, $\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=2\,SYM}\rangle\ll\sigma^{1/2}_{1}\sim\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=1\,SYM}\rangle\ll\mu_{\rm x}$.

Now, as for the $SU(N_{F})$ sector. At scales
$\langle\Lambda_{SU(N_{F})}\rangle\ll\mu\ll m$ this is the UV free ${\cal
N}=2$ SQCD with $N_{F}$ flavors of massless quarks
${\overline{Q}}^{\,a}_{j},Q^{i}_{a},\,\,i,a=1...N_{F},\,\,\langle
m-a\rangle=0$. When $\langle\Lambda_{SU(N_{F})}\rangle^{2}/m\ll\mu_{\rm x}\ll
m$, see (8.4), then the low energy superpotential in this sector will be as in
(7.1.7) with ${\rm n}_{2}=N_{F}$. I.e., all quarks in this $SU(N_{F})$ sector
will be higgsed (while $\langle X^{adj}_{SU(N_{F})}\rangle=0$) and $N_{F}^{2}$
long ${\cal N}=2$ multiplets of massive gluons (including $U(1)$ with its
scalar ”a”) with masses $\mu_{gl}\sim
g\langle{\overline{Q}}Q\rangle^{1/2}_{N_{F}}\sim g(\mu_{\rm x}m)^{1/2}$ will
be formed, see (7.1.8). The global symmetry $SU(N_{F})_{C+F}$ is unbroken. 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 $(\mu_{\rm x}m)^{1/2}\ll\langle\Lambda_{SU(N_{F})}\rangle$ the 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 $\langle
X^{adj}_{SU(N_{F})}\rangle\sim\langle\Lambda_{SU(N_{F})}\rangle$. The
situation in this phase is qualitatively similar to those in section 7.2, the
difference is that now the whole $SU(N_{F})$ color group will be broken at the
scale $\sim\langle\Lambda_{SU(N_{F})}\rangle$, 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

$\displaystyle\langle\sqrt{2}X^{adj}_{SU(N_{F})}\rangle\sim\langle\Lambda_{SU(N_{F})}\rangle\,{\rm
diag}\,\Bigl{(}\,\underbrace{\rho^{0},\,\rho^{1},\,...,\,\rho^{N_{F}-1}}_{N_{F}}\,;\underbrace{0}_{N_{c}-N_{F}}\,\Bigr{)},\quad\rho=\exp\\{\,\frac{2\pi
i}{N_{F}}\,\\}\,,$ (8.5)

and the whole $SU(N_{F})$ color group is broken, $SU(N_{F})\rightarrow
U^{N_{F}-1}(1)$. All quarks with $SU(N_{F})$ 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,
$m_{Q}\gg(\mu_{\rm x}m)^{1/2}$, so that they are not higgsed but decouple as
heavy at $\mu<\langle\Lambda_{SU(N_{F})}\rangle$, as well as all charged
$SU(N_{F})$ adjoints. Instead, $N_{F}$ massless at $\mu_{\rm x}\rightarrow 0$
dyons ${\overline{D}}_{j},D_{j}$ are formed at the scale
$\sim\langle\Lambda_{SU(N_{F})}\rangle$. The relevant part of the low energy
superpotential of this sector looks in this case as

$\displaystyle{\cal W}_{N_{F}}={\cal W}_{D}+{\cal W}_{a}+{\cal W}_{SYM}\,,$
(8.6) ${\cal
W}_{D}=(m-a)\sum_{j=1}^{N_{F}}{\overline{D}}_{j}D_{j}-\sum_{j=1}^{N_{F}}a_{D,\,j}{\overline{D}}_{j}D_{j}-\mu_{\rm
x}\langle\Lambda_{SU(N_{F})}\rangle\sum_{j=1}^{N_{F}}\rho^{\,{j-1}}a_{D,\,j}+\mu_{\rm
x}{L^{\,\prime}}\,\Bigl{(}\sum_{j=1}^{N_{F}}a_{D,\,j}\Bigr{)}\,,$ ${\cal
W}_{a}=\frac{\mu_{\rm x}}{2}\frac{N_{c}N_{F}}{N_{c}-N_{F}}\,a^{2}+\mu_{\rm
x}N_{F}\,{\delta}^{\,\prime}_{4}\,(m-a)^{2}\,,$ ${\cal
W}_{SYM}=(N_{c}-N_{F})\mu_{\rm x}\Bigl{(}\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=2\,\,SYM}\Bigr{)}^{2}=(N_{c}-N_{F})\mu_{\rm
x}\Biggl{(}\frac{\Lambda_{2}^{2N_{c}-N_{F}}(m_{1}-{\hat{c}}\,{\hat{a}})^{N_{F}}}{[\,m_{1}+(1-{\hat{c}}){\hat{a}}\,]^{2N_{F}}}\Biggr{)}^{\frac{1}{N_{c}-N_{F}}}\approx$
$\approx\,\mu_{\rm x}\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=2\,\,SYM}\rangle^{2}\Biggl{[}(N_{c}-N_{F})-N_{F}\frac{(2N_{c}-N_{F})}{N_{c}-N_{F}}\,\frac{\hat{a}}{m_{1}}\,\Biggr{]}\,,\quad
m_{1}=\frac{N_{c}}{N_{c}-N_{F}}m\,,\quad a=\langle a\rangle+{\hat{a}}\,.$

From (8.6)

$\langle a\rangle=m\,,\quad\langle
a_{D,\,j}\rangle=0\,,\quad\langle{\overline{D}}_{j}\rangle\langle
D_{j}\rangle=-\mu_{\rm
x}\langle\Lambda_{SU(N_{F})}\rangle\rho^{\,{j-1}}+\mu_{\rm
x}\langle{L^{\,\prime}}\rangle\,,\quad\langle\Sigma_{D}\rangle=\sum_{j=1}^{N_{F}}\langle{\overline{D}}_{j}\rangle\langle
D_{j}\rangle=N_{F}\mu_{\rm x}\langle{L^{\,\prime}}\rangle,$
$\displaystyle\langle\Sigma_{D}\rangle\approx N_{F}\mu_{\rm
x}m_{1}\Bigl{[}\,1-\frac{2N_{c}-N_{F}}{N_{c}-N_{F}}\frac{\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}_{1}}\,\Bigr{]}\approx N_{F}\mu_{\rm
x}m_{1}\Bigl{[}\,1-\frac{2N_{c}-N_{F}}{N_{c}-N_{F}}\Bigl{(}\frac{\Lambda_{2}}{{m_{1}}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-N_{F}}}\,\Bigr{]}\,,$
(8.7) $\langle S\rangle_{N_{F}}=0,\quad\langle{\overline{D}}_{j}\rangle\langle
D_{j}\rangle\approx\mu_{\rm
x}m_{1}\Bigl{[}\,1-\frac{2N_{c}-N_{F}}{N_{c}-N_{F}}\frac{\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}_{1}}\,\Bigr{]}-\mu_{\rm
x}\langle\Lambda_{SU(N_{F})}\rangle\rho^{\,{j-1}}\,,\quad j=1...N_{F}\,.$

The curve (1.2) 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
$\mu_{\rm x}\rightarrow 0$ 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 $\mu_{\rm x}\rightarrow 0$ dyons $D_{j}$. The
two single roots with
$(e^{+}-e^{-})\sim\langle\Lambda^{SU(N_{c}-N_{F})}_{{\cal N}=2\,SYM}\rangle$
(8.3) originate at $\mu_{\rm x}\rightarrow 0$ from the
$SU(N_{c}-N_{F})\,\,{\cal N}=2$ SYM sector. The global flavor symmetry is
unbroken and there are no massless Nambu-Goldstone particles. And there are no
massless particles at all at $\mu_{\rm x}\neq 0$. All original charged
electric particles are confined in the lower energy phase at $\mu_{\rm
x}<\langle\Lambda_{SU(N_{F})}\rangle^{2}/m_{1}$, see (8.4). But the phase
changes in the $SU(N_{F})$ sector already at very small $\mu_{\rm
x}:\,\langle\Lambda_{SU(N_{F})}\rangle^{2}/m_{1}<\mu_{\rm
x}\ll\langle\Lambda_{SU(N_{F})}\rangle\ll\Lambda_{2}\ll m$. Instead of higgsed
$\langle
X^{adj}_{SU(N_{F})}\rangle\sim\langle\Lambda_{SU(N_{F})}\rangle,\,\,SU(N_{F})\rightarrow
U^{N_{F}-1}(1)$ and subsequent condensation of $N_{F}$ dyons $D_{j}$ (8.7),
all $N_{F}$ flavors of original electric quarks
${\overline{Q}}^{\,a}_{i},\,Q^{i}_{a},\,\,a,i=1...N_{F}$ are then higgsed as
$\langle{\overline{Q}}^{\,a}_{i}\rangle=\langle
Q_{a}^{i}\rangle\approx\delta^{i}_{a}(\mu_{\rm
x}m_{1})^{1/2}>\langle\Lambda_{SU(N_{F})}\rangle$ (and clearly not confined),
forming $N_{F}^{2}$ long ${\cal N}=2$ multiplets of massive gluons.

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

##  9 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 number of
their various vacua with the unbroken non-trivial $Z_{2N_{c}-N_{F}\geq 2}$
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
$m,\,\mu_{\rm x},\,\Lambda_{2}$. A crucial role in obtaining these results was
played by the use : a) the unbroken $Z_{2N_{c}-N_{F}\geq 2}$ symmetry, b) the
pattern of spontaneous flavor symmetry breaking, $U(N_{F})\rightarrow U({\rm
n}_{1})\times U({\rm n}_{2}),\,0\leq{\rm n}_{1}<N_{F}/2,\,$ c) the knowledge
of multiplicities of various vacua. Besides, the knowledge of the quark and
gluino condensates
$\langle({\overline{Q}}Q)_{1}\rangle_{N_{c}},\,\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}$,
and $\langle S\rangle_{N_{c}}$ was also of importance. In addition, the 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 $\mu_{\rm x}\neq 0,\,m\neq 0$. All
this appeared to be sufficient to calculate mass spectra in vacua considered
at small but non-zero values of $\mu_{\rm x}$, i.e. $0<\mu_{\rm x}<M_{\rm
min}$, where $M_{\rm min}$ is the independent of $\mu_{\rm x}$ mass scale
appropriate for each given vacuum, such that the phase and mass spectrum
continue smoothly down to $\mu_{\rm x}\rightarrow 0$. In other words, the
theory stays at $\mu_{\rm x}<M_{\rm min}$ in the same regime and all
hierarchies in the mass spectrum are the same as in the ${\cal N}=2$ theory at
$\mu_{\rm x}\rightarrow 0$.

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 $m\ll\Lambda_{2}$
and $0<\mu_{\rm x}\ll M^{(\rm br2)}_{\rm
min}=\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle\ll m\ll\Lambda_{2}$. 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 $\mu\sim\Lambda_{2}$, $\,\langle
X^{adj}_{SU(2N_{c}-N_{F})}\rangle\sim\Lambda_{2}$, $\,SU(N_{c})\rightarrow
SU(N_{F}-N_{c})\times U^{(1)}(1)\times U^{2N_{c}-N_{F}-1}(1)$, this pattern is
required by the unbroken discrete $Z_{2N_{c}-N_{F}\geq 2}$ symmetry and is
necessary to avoid $g^{2}(\mu<\Lambda_{2})$ in this effectively massless at
the scale $\mu\sim\Lambda_{2}$ ${\cal N}=2$ UV free theory. The second stage
is in the $SU(N_{F}-N_{c})$ sector at the scale $\mu\sim m$, $\,\langle
X^{adj}_{SU(N_{F}-N_{c})}\rangle\sim m\ll\Lambda_{2}$,
$\,SU(N_{F}-N_{c})\rightarrow SU({\rm n}_{1})\times U^{(2)}(1)\times
SU(N_{F}-N_{c}-{\rm n}_{1})$ . And the third stage is in the
$SU(N_{F}-N_{c}-{\rm n}_{1})$ sector at the scale
$\mu\sim\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle\ll m$, $\,SU(N_{F}-N_{c}-{\rm n}_{1})\rightarrow
U^{N_{F}-N_{c}-{\rm n}_{1}-1}(1)$.

The global flavor symmetry $U(N_{F})$ is broken spontaneously at $\mu_{\rm
x}\neq 0$ at the scale $\mu\sim(\mu_{\rm x}m)^{1/2}$ in these vacua in the
$SU({\rm n}_{1})$ sector by higgsed original pure electric quarks,
$\,\langle{\overline{Q}}^{\,a}_{i}\rangle=\langle
Q^{i}_{a}\rangle\sim\delta^{a}_{i}\,(\mu_{\rm x}m)^{1/2}\,,\\\ a=1...{\rm
n}_{1}\,,\,\,i=1...N_{F}$, as : $U(N_{F})\rightarrow U({\rm n}_{1})\times
U({\rm n}_{2}),\,1\leq{\rm n}_{1}<N_{F}-N_{c}$.

It was shown that the lightest charged BPS particles (massless at $\mu_{\rm
x}\rightarrow 0$) are the following. -
1) $2N_{c}-N_{F}$ flavorless dyons $D_{j},{\overline{D}}_{j}$ (this number
$2N_{c}-N_{F}$ is required by the unbroken $Z_{2N_{c}-N_{F}\geq 2}$ discrete
symmetry which operates interchanging them with each other), with the two non-
zero pure electric charges, the $SU(N_{c})$ baryon charge and $U^{(1)}(1)$ one
and, on the whole, with non-zero all $2N_{c}-N_{F}-1$ independent
$SU(2N_{c}-N_{F})$ adjoint charges (with non-zero magnetic parts). These dyons
are formed at the scale $\mu\sim\Lambda_{2}$, at the first stage of the color
breaking, and are mutually local between themselves and with respect to all
particles of the $SU(N_{F}-N_{c})$ sector. 2) All original electric particles
of remaining unbroken at $\mu_{\rm x}\rightarrow 0$ $\,SU({\rm n}_{1})$
subgroup of original $SU(N_{c})$. 3) ${\overline{N}}_{c}-{\rm n}_{1}-1$ pure
magnetic monopoles with the $SU({\overline{N}}_{c}-{\rm n}_{1})$ adjoint
magnetic charges. These last are formed at the low scale
$\mu\sim\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle\ll m\ll\Lambda_{2}$ in the ${\cal
N}=2\,\,SU({\overline{N}}_{c}-{\rm n}_{1})$ SYM sector, with the scale factor
$\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle$ of its gauge coupling. In addition, there are $N_{c}-1$
Abelian ${\cal N}=2$ photon multiplets massless at $\mu_{\rm x}\rightarrow 0$.

Massless at $\mu_{\rm x}=0$ $\,2N_{c}-N_{F}$ dyons $D_{j},\,{\rm n}_{1}$
original pure electric quarks $Q^{i}_{a}$ from $SU({\rm n}_{1})$, and
${\overline{N}}_{c}-{\rm n}_{1}-1$ pure magnetic monopoles are all higgsed at
small $\mu_{\rm x}\neq 0$, and no massless particles remains (except for
$2{\rm n}_{1}{\rm n}_{2}$ Nambu-Goldstone multiplets due to spontaneous
breaking of global flavor symmetry, $U(N_{F})\rightarrow U({\rm n}_{1})\times
U({\rm n}_{2})$). The mass spectrum was described in these br2 vacua at
$0<\mu_{\rm x}\ll\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle\ll m\ll\Lambda_{2}$.

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 $\mu_{\rm x}$ was increased then in two different stages :
1) $\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle\ll\mu_{\rm x}\ll m$ in section 2.3 ; 2) $m\ll\mu_{\rm
x}\ll\Lambda_{2}$ 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 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 corresponding colored adjoint
scalars $X^{\rm adj}$ at those even lower scales where this ${\cal N}=1$
conformal regime is broken by non-zero particle masses (see also the footnote
3). 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 $m,\,\mu_{\rm x},\,\Lambda_{2}$
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
are presented in two important Appendices. These last calculations 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 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 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 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
$\langle({\overline{Q}}Q)_{1}\rangle_{{\rm n}_{1}}$ 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. These calculations confirm independently
in a non-trivial way a self-consistency of the whole approach.

##  Appendix A Power corrections to quark condensates

1) br1 vacua of $SU(N_{c}),\,\,m\gg\Lambda_{2}$ in section 4.1

From (1.3),(1.4), in these vacua (up to even smaller power corrections)

$\displaystyle\langle({\overline{Q}}Q)_{1}\rangle_{N_{c}}=\mu_{\rm
x}m_{3}-\frac{N_{c}-{\rm n}_{2}}{N_{c}-{\rm
n}_{1}}\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}\,,\quad\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}\approx\mu_{\rm
x}m_{3}\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\,,$ (A.1) $\langle
S\rangle_{N_{c}}=\frac{\langle({\overline{Q}}Q)_{1}\rangle_{N_{c}}\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}}{\mu_{\rm
x}}\approx m_{3}\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}\,,\quad
m_{3}=\frac{N_{c}}{N_{c}-{\rm n}_{1}}m\,.$

From (2.1.9),(A.1), the leading power correction to $\langle{\cal W}^{\,\rm
eff}_{\rm tot}\rangle$ is

$\displaystyle\langle\delta{\cal W}^{\,\rm eff}_{\rm
tot}\rangle\approx(N_{c}-{\rm n}_{1})\mu_{\rm
x}m^{2}_{3}\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\approx(N_{c}-{\rm n}_{1})\langle S\rangle_{N_{c}}\,.$ (A.2)

On the other hand, this leading power correction to $\langle{\cal W}^{\,\rm
low}_{\rm tot}\rangle$ originates from the $SU(N_{c}-{\rm n}_{1})$ SYM part,
see (4.1.3),(4.1.2),

$\langle\delta{\cal W}^{\,\rm low}_{\rm tot}\rangle\approx\mu_{\rm
x}\langle\,{\rm Tr\,}\Bigl{(}X^{adj}_{SU(N_{c}-{\rm
n}_{1})}\Bigr{)}^{2}\,\rangle\approx(N_{c}-{\rm n}_{1})\langle
S\rangle_{N_{c}-{\rm n}_{1}}\approx(N_{c}-{\rm n}_{1})\mu_{\rm
x}\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}\approx$ $\displaystyle\approx(N_{c}-{\rm
n}_{1})\mu_{\rm
x}m_{3}^{2}\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{(N_{c}-{\rm
n}_{1})}}\,,\quad\rightarrow\quad\langle\delta{\cal W}^{\,\rm low}_{\rm
tot}\rangle=\langle\delta{\cal W}^{\,\rm eff}_{\rm tot}\rangle\,,$ (A.3)

as it should be.

To calculate the leading power correction
$\delta\langle({\overline{Q}}Q)_{1}\rangle_{{\rm n}_{1}}\sim\mu_{\rm
x}\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}/m$ to
the low energy quark condensate $\langle({\overline{Q}}Q)_{1}\rangle_{{\rm
n}_{1}}$, 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 $SU(N_{c}-{\rm
n}_{1})$ SYM contribution to $\delta{\cal W}^{\,\rm low}_{\rm tot}$,
see(4.1.3),(4.1.2),

$\delta{\cal W}^{\,\rm low}_{\rm tot}\approx(N_{c}-{\rm n}_{1})\mu_{\rm
x}\,\delta\,\Bigl{(}\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\Bigr{)}^{2}\approx(N_{c}-{\rm n}_{1})\mu_{\rm
x}\,\delta\,\Bigl{(}\frac{\Lambda_{2}^{2N_{c}-N_{F}}(m_{3}-c\,{\hat{a}})^{N_{F}}}{[\,m_{3}+(1-c){\hat{a}}\,]^{2{\rm
n}_{1}}}\Bigr{)}^{\frac{1}{N_{c}-{\rm n}_{1}}}\approx$
$\displaystyle\approx\,\mu_{\rm x}\langle\Lambda^{SU(N_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}\,\Bigl{[}\,-{\rm
n}_{1}\frac{(2N_{c}-N_{F})}{N_{c}-{\rm
n}_{1}}\,\Bigr{]}\,\frac{{\hat{a}}}{m_{3}}\,,\quad
m_{3}=\frac{N_{c}}{N_{c}-{\rm n}_{1}}m\,.$ (A.4)

As a result, from (4.1.3),(4.1.4),(A.4),(A.1),

$\delta\langle({\overline{Q}}Q)_{1}\rangle_{{\rm n}_{1}}\approx\frac{1}{{\rm
n}_{1}}\langle\frac{\partial}{\partial{\hat{a}}}\delta{\cal W}^{\,\rm
low}_{\rm tot}\rangle\approx-\frac{(2N_{c}-N_{F})}{N_{c}-{\rm
n}_{1}}\,\mu_{\rm x}\frac{\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m_{3}}\,,$
$\displaystyle\langle({\overline{Q}}Q)_{1}\rangle^{SU(N_{c})}_{{\rm
n}_{1}}=\langle{\overline{Q}}^{1}_{1}\rangle\langle
Q^{1}_{1}\rangle\approx\mu_{\rm
x}m_{3}\Biggl{[}1-\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}\,\frac{\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}_{3}}\Biggr{]}\approx$ (A.5) $\approx\mu_{\rm
x}m_{3}\Biggl{[}1-\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\Biggr{]}\,,\quad\langle({\overline{Q}}Q)_{1}\rangle^{SU(N_{c})}_{N_{c}}\approx\mu_{\rm
x}m_{3}\Biggl{[}1-\frac{N_{c}-{\rm n}_{2}}{N_{c}-{\rm
n}_{1}}\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\Biggr{]}\,.$

It is seen from (A.5) that, although the leading terms are the same in
$\langle({\overline{Q}}Q)_{1}\rangle_{{\rm n}_{1}}$ and
$\langle({\overline{Q}}Q)_{1}\rangle_{N_{c}}$, the leading power corrections
are different.

2) br1 vacua of $U(N_{c}),\,\,m\gg\Lambda_{2}$ in section 4.2

From (1.4),(4.2.1),(4.2.2),(4.2.3) (keeping now leading power corrections)

$\displaystyle\langle({\overline{Q}}Q)_{1}\rangle_{N_{c}}=\mu_{\rm
x}m-\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}\,,$ (A.6)
$\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}=\mu_{\rm
x}\Lambda_{2}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\Bigl{(}\frac{\langle({\overline{Q}}Q)_{1}\rangle_{N_{c}}}{\mu_{\rm
x}}\Bigr{)}^{\frac{{\overline{N}}_{c}-{\rm n}_{1}}{N_{c}-{\rm
n}_{1}}}\approx\mu_{\rm
x}m\Bigl{(}\frac{\Lambda_{2}}{m}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\Biggl{(}1+O\Bigl{(}\frac{\langle\Lambda^{SU(N_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}}{m^{2}}\Bigr{)}\Biggr{)}\,,$ $\langle
S\rangle_{N_{c}}=\frac{\langle({\overline{Q}}Q)_{1}\rangle_{N_{c}}\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}}{\mu_{\rm
x}}\approx\mu_{\rm
x}m^{2}\Bigl{(}\frac{\Lambda_{2}}{m}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\Biggl{(}1+O\Bigl{(}\frac{\langle\Lambda^{SU(N_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}}{m^{2}}\Bigr{)}\Biggr{)}\,,$
$\displaystyle\frac{\langle a_{0}\rangle}{m}=\frac{1}{N_{c}\mu_{\rm
x}m}\langle{\rm Tr}\,{\overline{Q}}Q\rangle_{N_{c}}\approx\frac{{\rm
n}_{1}}{N_{c}}+\frac{{\rm n}_{2}-{\rm
n}_{1}}{N_{c}}\Bigl{(}\frac{\Lambda_{2}}{m}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\,,\quad\langle a\rangle=\langle m-a_{0}\rangle\,,$ (A.7)
$\displaystyle\langle{\cal W}^{\,\rm low}_{\rm
tot}\rangle\approx\Bigl{[}\langle{\cal W}_{{\rm
n}_{1}}\rangle=0\,\Bigr{]}+\Biggl{[}\langle{\cal
W}_{a_{0},a}\rangle=\frac{\mu_{\rm x}}{2}\Bigl{(}\,N_{c}\langle
a_{0}\rangle^{2}+\frac{{\rm n}_{1}N_{c}}{N_{c}-{\rm n}_{1}}\langle
a\rangle^{2}\Bigr{)}\Biggr{]}+\langle{\cal W}_{SU(N_{c}-{\rm
n}_{1})}^{\,SYM}\rangle\,,$ (A.8) $\langle{\cal W}_{SU(N_{c}-{\rm
n}_{1})}^{\,SYM}\rangle=\mu_{\rm x}\langle{\,\rm
Tr\,}\Bigl{(}X^{adj}_{SU(N_{c}-{\rm n}_{1})}\Bigr{)}^{2}\rangle=(N_{c}-{\rm
n}_{1})\langle S\rangle_{SU(N_{c}-{\rm n}_{1})}=(N_{c}-{\rm n}_{1})\mu_{\rm
x}\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}=$
$=(N_{c}-{\rm n}_{1})\mu_{\rm
x}\langle\Biggl{(}\frac{\Lambda_{2}^{2N_{c}-N_{F}}(m-a_{0}-c\,a)^{N_{F}}}{[\,(1-c)a\,]^{2{\rm
n}_{1}}}\Biggr{)}^{\frac{1}{N_{c}-{\rm n}_{1}}}\rangle\approx(N_{c}-{\rm
n}_{1})\mu_{\rm
x}m^{2}\Bigl{(}\frac{\Lambda_{2}}{m}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\,.$

From (A.6)-(A.8), the leading power correction $\langle\,\delta\,{\cal
W}^{\,\rm low}_{\rm tot}\rangle\sim\mu_{\rm x}\langle\Lambda^{SU(N_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}$ to $\langle{\cal W}^{\,\rm low}_{\rm
tot}\rangle$ looks as

$\langle\,\delta\,{\cal W}^{\,\rm low}_{\rm
tot}\rangle=\Bigl{(}\langle\,\delta\,{\cal
W}_{a_{0},a}\rangle=0\Bigr{)}+\langle\,\delta\,{\cal W}_{SU(N_{c}-{\rm
n}_{1})}^{\,SYM}\rangle\approx$ $\displaystyle\approx(N_{c}-{\rm
n}_{1})\mu_{\rm x}\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}\approx(N_{c}-{\rm
n}_{1})\,m\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}\approx(N_{c}-{\rm
n}_{1})\mu_{\rm
x}m^{2}\Bigl{(}\frac{\Lambda_{2}}{m}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\,.$ (A.9)

Instead of (2.1.9),(2.1.10) we have now, see (2.2.7),

$\displaystyle{\cal W}^{\,\rm eff}_{\rm tot}(\Pi)=m\,{\rm
Tr}\,({\overline{Q}}Q)_{N_{c}}-\frac{1}{2\mu_{\rm
x}}\Biggl{[}\,\sum_{i,j=1}^{N_{F}}({\overline{Q}}_{j}Q^{i})_{N_{c}}({\overline{Q}}_{\,i}Q^{j})_{N_{c}}\Biggr{]}-{\overline{N}}_{c}\Biggl{[}\,S_{N_{c}}=\frac{\langle({\overline{Q}}Q)_{1}\rangle_{N_{c}}\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}}{\mu_{\rm
x}}\,\Biggr{]},\,\,\,\,\,$ (A.10)

and using (A.6) (with the same accuracy)

$\langle\delta{\cal W}^{\,\rm eff}_{\rm tot}(\Pi)\rangle\approx(N_{c}-{\rm
n}_{1})m\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}\approx(N_{c}-{\rm
n}_{1})\mu_{\rm
x}m^{2}\Bigl{(}\frac{\Lambda_{2}}{m}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\approx\langle\delta{\cal W}^{\,\rm low}_{\rm tot}\rangle\,,$

as it should be.

As for the leading power correction
$\delta\langle({\overline{Q}}Q)_{1}\rangle_{{\rm n}_{1}}\sim\mu_{\rm
x}m\Bigl{(}\Lambda_{2}/m\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm n}_{1}}}$ to
the quark condensate, we have now instead of (A.4)

$\delta{\cal W}^{\,\rm low}_{\rm tot}=\delta{\cal W}_{a_{0},a}+\delta{\cal
W}_{SU(N_{c}-{\rm n}_{1})}^{\,SYM}\,,\quad a_{0}=\langle
a_{0}\rangle+{\hat{a}}_{0}\,,\quad a=\langle a\rangle+{\hat{a}}\,,$
$\delta{\cal W}_{SU(N_{c}-{\rm n}_{1})}^{\,SYM}=(N_{c}-{\rm n}_{1})\mu_{\rm
x}\,\delta\,\Bigl{(}\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\Bigr{)}^{2}=(N_{c}-{\rm n}_{1})\mu_{\rm
x}\,\delta\,\Bigl{(}\frac{\Lambda_{2}^{2N_{c}-N_{F}}(m-{\hat{a}}_{0}-c\,{\hat{a}})^{N_{F}}}{[\,m+(1-c){\hat{a}}\,]^{2{\rm
n}_{1}}}\Bigr{)}^{\frac{1}{N_{c}-{\rm n}_{1}}}\approx$
$\displaystyle\approx\Bigl{[}\,-N_{F}{\hat{a}_{0}}-\frac{{\rm
n}_{1}(2N_{c}-N_{F})}{N_{c}-{\rm n}_{1}}\,{\hat{a}}\,\Bigr{)}\mu_{\rm
x}\frac{\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m}\,,$ (A.11)

while from (4.2.3),(A.7)

$\displaystyle\delta{\cal W}_{a_{0},a}\approx({\rm n}_{2}-{\rm
n}_{1})\Bigl{[}\,{\hat{a}}_{0}-\frac{{\rm n}_{1}}{N_{c}-{\rm
n}_{1}}\,{\hat{a}}\,\Bigr{]}\mu_{\rm x}\frac{\langle\Lambda^{SU(N_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}}{m}\,.$ (A.12)

Instead of (A.5) (with the same accuracy), on the one hand, see
(4.2.3),(4.2.4),(A.6),

$\langle{\,\rm Tr\,}{\overline{Q}}Q\rangle_{{\rm n}_{1}}={\rm n}_{1}\mu_{\rm
x}m+\delta\langle{\,\rm Tr\,}{\overline{Q}}Q\rangle_{{\rm n}_{1}}={\rm
n}_{1}\Bigl{(}\mu_{\rm x}m+\delta\,\langle({\overline{Q}}Q)_{1}\rangle_{{\rm
n}_{1}}\Bigr{)}\,,$ $\displaystyle\delta\langle{\,\rm
Tr\,}({\overline{Q}}Q)_{1}\rangle_{{\rm n}_{1}}=\frac{1}{{\rm
n}_{1}}\langle\frac{\partial}{\partial{\hat{a}}_{0}}\Bigl{(}\delta{\cal
W}_{SU(N_{c}-{\rm n}_{1})}^{\,SYM}+\delta{\cal
W}_{a_{0},a}\Bigr{)}\rangle\approx-2\mu_{\rm
x}\frac{\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m}\,,$ (A.13)

while on the other hand,

$\delta\langle{\,\rm Tr\,}({\overline{Q}}Q)_{1}\rangle_{{\rm
n}_{1}}=\frac{1}{{\rm
n}_{1}}\langle\,\frac{\partial}{\partial{\hat{a}}}\Bigl{(}\delta{\cal
W}_{SU(N_{c}-{\rm n}_{1})}^{\,SYM}+\delta{\cal
W}_{a_{0},a}\Bigr{)}\approx-2\mu_{\rm x}\frac{\langle\Lambda^{SU(N_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}}{m}\,,$

as it should be.

Therefore, on the whole for these $U(N_{c})$ br1 vacua, see (A.6) for
$\langle({\overline{Q}}Q)_{1}\rangle^{U(N_{c})}_{N_{c}}$,

$\displaystyle\langle({\overline{Q}}Q)_{1}\rangle^{U(N_{c})}_{{\rm
n}_{1}}=\langle{\overline{Q}}^{1}_{1}\rangle\langle
Q^{1}_{1}\rangle\approx\mu_{\rm
x}m\Bigl{[}\,1-2\frac{\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,SYM}\rangle^{2}}{m^{2}}\,\Bigr{]}\approx\mu_{\rm
x}m\Bigl{[}\,1-2\Bigl{(}\frac{\Lambda_{2}}{m}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\,\Bigr{]}\,,$ (A.14)
$\langle({\overline{Q}}Q)_{1}\rangle^{U(N_{c})}_{N_{c}}\approx\mu_{\rm
x}m\Bigl{[}\,1-\frac{\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}}\,\Bigr{]}.$

We can compare also the result for
$\langle({\overline{Q}}Q)_{1}\rangle^{U(N_{c})}_{{\rm n}_{1}}$ in (A.14) with
those from (2.2.10). 141414 In these simplest br1 vacua of $U(N_{c})$ the
charges and multiplicities of particles massless at $\mu_{\rm x}\rightarrow 0$
are evident, these are original electric particles from $SU({\rm n}_{1})\times
U^{(0)}(1)$ and $N_{c}-{\rm n}_{1}-1$ magnetic monopoles from $SU(N_{c}-{\rm
n}_{1})$ SYM.  To obtain definite predictions from (2.2.10) one has to find
first the values of roots entering (2.2.10). In the case considered, to obtain
definite values of all roots of the curve (1.2) 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 (1.2)

$\displaystyle e^{\pm}=\pm 2\sqrt{\langle S\rangle_{N_{c}}/\mu_{\rm
x}}\,,\quad e^{\pm}_{c}=\frac{1}{2}(e^{+}+e^{-})=0\,.$ (A.15)

One can obtain then from the $U(N_{c})$ curve (1.2) the values of ${\rm
n}_{1}$ equal double roots $e^{(Q)}_{i}$ of original electric quarks from
$SU({\rm n}_{1})$ and $N_{c}-{\rm n}_{1}-1$ unequal double roots $e^{(M)}_{k}$
of magnetic monopoles from $SU(N_{c}-{\rm n}_{1})$ SYM. We find from
(1.2),(A.15) with our accuracy, see (A.6),(A.8),

$e^{+}=-e^{-}\approx
2m\Bigl{(}\frac{\Lambda_{2}}{m}\Bigr{)}^{\frac{2N_{c}-N_{F}}{2(N_{c}-{\rm
n}_{1})}}\Biggl{(}1+O\Bigl{(}\frac{\langle\Lambda^{SU(N_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}}{m^{2}}\Bigr{)}\Biggr{)}\approx
2\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal N}=2\,\,SYM}\rangle,\quad
e^{(Q)}_{i}=-m,\,\,i=1...{\rm n}_{1}\,,$ $\displaystyle e^{(M)}_{k}\approx
2\cos(\frac{\pi k}{N_{c}-{\rm n}_{1}})\langle\Lambda^{SU(N_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle-\frac{{\rm n}_{2}-{\rm n}_{1}}{N_{c}-{\rm
n}_{1}-1}\frac{\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m}\,,\quad k=1...(N_{c}-{\rm n}_{1}-1)\,.$ (A.16)

Let us note that these roots satisfy the sum rule, see (A.7),

$\displaystyle\sum_{n=1}^{N_{c}}\phi_{n}=\frac{1}{2}\sum_{n=1}^{2N_{c}}(-e_{n})=N_{c}\langle
a_{0}\rangle\,,$ (A.17)

as it should be.

Therefore, from (2.2.10),(A.16)

$\langle({\overline{Q}}Q)_{1}\rangle^{U(N_{c})}_{{\rm n}_{1}}=-\mu_{\rm
x}\sqrt{(e^{(Q)}_{i}-e^{+})(e^{(Q)}_{i}-e^{-})}\approx$
$\displaystyle\approx\mu_{\rm
x}m\Bigl{(}1-2\frac{\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}}\Bigr{)}\approx\mu_{\rm
x}m\Bigl{[}\,1-2\Bigl{(}\frac{\Lambda_{2}}{m}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\,\Bigr{]}\,,\quad i=1...{\rm n}_{1}\,,\,$ (A.18)

this agrees with (A.14).

Besides, we can use the knowledge of roots of the curve (1.2) for the
$U(N_{c})$ theory and of $\langle a_{0}\rangle$, see (A.7),(A.16), to obtain
the values of $\langle({\overline{Q}}Q)_{1}\rangle^{SU(N_{c})}_{{\rm n}_{1}}$
condensates in the $SU(N_{c})$ theory. For this, the curve (1.2) for the
$U(N_{c})$ theory

$\displaystyle
y^{2}=\prod_{i=1}^{N_{c}}(z+\phi_{i})^{2}-4\Lambda_{2}^{2N_{c}-N_{F}}(z+m)^{N_{F}}\,,\quad\sum_{i=1}^{N_{c}}\phi_{i}=\frac{1}{2}\sum_{n=1}^{2N_{c}}(-e_{n})=N_{c}\langle
a_{0}\rangle{\rm\,\,\,in\,\,\,U(N_{c})}\,,$ (A.19)

we rewrite in the form of the $SU(N_{c})$ theory

$\displaystyle
y^{2}=\prod_{i=1}^{N_{c}}({\hat{z}}+{\hat{\phi}}_{i})^{2}-4\Lambda_{2}^{2N_{c}-N_{F}}({\hat{z}}+{\hat{m}})^{N_{F}}\,,\,\,\sum_{i=1}^{N_{c}}{\hat{\phi}}_{i}=\frac{1}{2}\sum_{n=1}^{2N_{c}}(-{\hat{e}}_{n})=0\,,\,\,{\hat{m}}=m\,(1-m^{-1}\langle
a_{0}\rangle)\,.\,$ (A.20)

But clearly, it remained the same $U(N_{c})$ theory with the same condensates
(A.18). Therefore, for the $SU(N_{c})$ curve (1.2) the quark condensates will
be as in (A.18) 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 (A.7),

$\frac{\langle a_{0}\rangle}{m}\,\,\rightarrow\,\,\approx\Biggl{(}\,\frac{{\rm
n}_{1}}{N_{c}}+\frac{{\rm n}_{2}-{\rm
n}_{1}}{N_{c}}\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\,\Biggr{)}\,,$ $\displaystyle m\rightarrow{m^{\prime}}\approx
m_{3}\Biggl{(}1+\frac{{\rm n}_{2}-{\rm n}_{1}}{N_{c}-{\rm
n}_{1}}\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\Biggr{)}\,,\quad m_{3}=\frac{N_{c}}{N_{c}-{\rm n}_{1}}\,m\,.$ (A.21)

Then, from (A.18),(A.21),

$\displaystyle\langle({\overline{Q}}Q)_{1}\rangle^{SU(N_{c})}_{{\rm
n}_{1}}\approx\mu_{\rm x}m_{3}\Biggl{(}1+\frac{{\rm n}_{2}-{\rm
n}_{1}}{N_{c}-{\rm
n}_{1}}\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\Biggr{)}\Bigl{[}\,1-2\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\,\Bigr{]}\approx$ (A.22) $\approx\mu_{\rm
x}m_{3}\Bigl{[}\,1-\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\,\Bigr{]}\,,$

this agrees with (A.5).

3) br2 vacua of $SU(N_{c}),\,\,m\ll\Lambda_{2}$ in section 2.1

The calculations of leading power corrections to $\langle\,\delta{\cal W}_{\rm
tot}^{\,\rm low}\rangle$ and $\langle\,\delta{\cal W}^{\,\rm eff}_{\rm
tot}\rangle$ have been presented in (2.1.15). Therefore, we present here the
calculation of the leading power correction
$\delta\,\langle({\overline{Q}}Q)_{1}\rangle_{{\rm n}_{1}}\sim\mu_{\rm
x}\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}/m$ to
the quark condensate. For this, see (2.1.4),(2.1.15), we write
($a_{1,2}=\langle a_{1,2}\rangle+{\hat{a}}_{1,2}$)

$\delta{\cal W}^{(SYM)}_{SU({\overline{N}}_{c}-{\rm
n}_{1})}\approx({\overline{N}}_{c}-{\rm n}_{1})\widetilde{\mu}_{\rm
x}\,\delta\,\Bigl{(}\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\Bigr{)}^{2}\approx({\overline{N}}_{c}-{\rm
n}_{1})\widetilde{\mu}_{\rm
x}\,\delta\,\Biggl{(}\frac{\Lambda_{SU({\overline{N}}_{c})}^{2{\overline{N}}_{c}-N_{F}}(m_{2}-{\hat{a}}_{1}-c_{2}\,{\hat{a}}_{2})^{N_{F}}}{[\,m_{2}+(1-c_{2})\,{\hat{a}}_{2}\,]^{\,2{\rm
n}_{1}}}\Biggr{)}^{\frac{1}{{\rm n}_{2}-N_{c}}}\approx$
$\displaystyle\approx\widetilde{\mu}_{\rm
x}\frac{\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m_{1}}\,\Bigl{[}\,N_{F}{\hat{a}}_{1}+\frac{{\rm
n}_{1}(2N_{c}-N_{F})}{N_{c}-{\rm
n}_{2}}\,{\hat{a}}_{2}\Bigr{]},\,\,m_{1}=\frac{N_{c}}{N_{c}-{\rm
n}_{2}}m=-\,m_{2},\,\,\widetilde{\mu}_{\rm x}=-\,\mu_{\rm x},$ (A.23)
$\langle{\,\rm Tr\,}{\overline{Q}}Q\rangle_{{\rm n}_{1}}={\rm n}_{1}\mu_{\rm
x}m_{1}+\delta\langle{\,\rm Tr\,}{\overline{Q}}Q\rangle_{{\rm n}_{1}}={\rm
n}_{1}\Bigl{(}\mu_{\rm
x}m_{1}+\delta\,\langle({\overline{Q}}Q)_{1}\rangle_{{\rm
n}_{1}}\Bigr{)}\,,\quad\Lambda_{SU({\overline{N}}_{c})}=-\Lambda_{2}\,,$

From this, see (2.1.5), and (2.1) for
$\langle({\overline{Q}}Q)_{2}\rangle^{SU(N_{c})}_{N_{c}}$,

$\delta\,\langle({\overline{Q}}Q)_{1}\rangle_{{\rm n}_{1}}=\frac{1}{{\rm
n}_{1}}\langle\frac{\partial}{\partial{\hat{a}}_{2}}{\cal
W}^{(SYM)}_{SU({\overline{N}}_{c}-{\rm
n}_{1})}\rangle\approx\widetilde{\mu}_{\rm
x}m_{1}\Bigl{[}\,\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{2}}\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}_{1}}\,\Bigr{]},\,\,\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm
n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}_{1}}\approx\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}},$
$\displaystyle\langle({\overline{Q}}Q)_{1}\rangle^{SU(N_{c})}_{{\rm
n}_{1}}=\langle{\overline{Q}}^{1}_{1}\rangle\langle
Q^{1}_{1}\rangle\approx\mu_{\rm x}m_{1}\Bigl{[}\,1+\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}\,\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}}{m^{2}_{1}}\,\Bigr{]}\approx$ (A.24)
$\approx\mu_{\rm x}m_{1}\Bigl{[}\,1+\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}\,\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Bigr{]}\,,\quad m_{1}=\frac{N_{c}}{N_{c}-{\rm n}_{2}}m\,,$
$\langle({\overline{Q}}Q)_{2}\rangle^{SU(N_{c})}_{N_{c}}=\mu_{\rm
x}m_{1}-\frac{N_{c}-{\rm n}_{1}}{N_{c}-{\rm
n}_{2}}\,\langle({\overline{Q}}Q)_{1}\rangle_{N_{c}}\approx\mu_{\rm
x}m_{1}\Bigl{[}\,1+\frac{N_{c}-{\rm n}_{1}}{{\rm
n}_{2}-N_{c}}\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}_{1}}\,\Bigr{]}\,.$

It is seen that, as in (A.5), the leading corrections to
$\langle({\overline{Q}}Q)_{1}\rangle_{{\rm n}_{1}}$ and
$\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}$ are different.

4) br2 vacua of $U(N_{c}),\,\,m\ll\Lambda_{2}$ in section 2.2

Keeping the leading order corrections
$\sim\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}/m$, from (2.2.2),(2.2.4),(2.1.15) (put attention that
the quark and gluino condensates, and $\Lambda^{SU({\overline{N}}_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}$ are different in br2 vacua of $SU(N_{c})$ or
$U(N_{c})$ theories, see (A.24),(A.25),

$\displaystyle\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}=\mu_{\rm
x}m-\langle({\overline{Q}}Q)_{1}\rangle_{N_{c}}\,,\quad\langle({\overline{Q}}Q)_{1}\rangle_{N_{c}}\approx\mu_{\rm
x}m\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\approx\mu_{\rm
x}\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m}\,,$ (A.25) $\frac{\langle
a_{0}\rangle}{m}=\frac{1}{N_{c}\mu_{\rm x}m}\langle{\rm
Tr\,}{\overline{Q}}Q\rangle_{N_{c}}\approx\frac{{\rm n}_{2}}{N_{c}}+\frac{{\rm
n}_{1}-{\rm
n}_{2}}{N_{c}}\,\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\,.$ $\displaystyle\delta{\cal W}_{SU({\overline{N}}_{c}-{\rm
n}_{1})}^{\,SYM}=({\overline{N}}_{c}-{\rm n}_{1})\widetilde{\mu}_{\rm
x}\,\delta\,\Biggl{[}\Bigl{(}\Lambda^{SU({\overline{N}}_{c}-{\rm
n}_{1})}_{{\cal
N}=2\,\,SYM}\Bigr{)}^{2}=\Biggl{(}\frac{\Lambda_{SU({\overline{N}}_{c})}^{2{\overline{N}}_{c}-N_{F}}(-m-{\hat{a}}_{0}-{\hat{a}}_{1}-c_{2}\,{\hat{a}}_{2})^{N_{F}}}{[\,-m+(1-c_{2})\,{\hat{a}}_{2}\,]^{\,2{\rm
n}_{1}}}\Biggr{)}^{\frac{1}{{\rm n}_{2}-N_{c}}}\Biggr{]}\,\,\,$ (A.26)
$\approx\widetilde{\mu}_{\rm
x}\,\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m}\,\Bigl{[}\,N_{F}({\hat{a}}_{0}+{\hat{a}}_{1})-{\rm
n}_{1}\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}\,{\hat{a}}_{2}\,\Bigr{]},\,\,\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm
n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}}=\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}},\,\,\Lambda_{SU({\overline{N}}_{c})}=-\,\Lambda_{2}\,,$
$\delta{\cal W}_{a}\approx\mu_{\rm
x}\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m}\,\Bigl{[}({\rm n}_{1}-{\rm
n}_{2}){\hat{a}}_{0}-\frac{N_{F}}{2N_{c}-N_{F}}{\hat{a}}_{1}-{\rm
n}_{1}\frac{{\rm n}_{2}-{\rm n}_{1}}{{\rm
n}_{2}-N_{c}}\,{\hat{a}}_{2}\,\Bigr{]}\,,\quad a_{i}=\langle
a_{i}\rangle+{\hat{a}}_{i},\,i=0,\,1,\,2\,,\,$
$\delta\langle({\overline{Q}}Q)_{1}\rangle^{(U(N_{c})}_{{\rm
n}_{1}}=\frac{1}{{\rm
n}_{1}}\langle\frac{\partial}{\partial{\hat{a}}_{2}}\Bigl{(}\delta{\cal
W}_{SU({\overline{N}}_{c}-{\rm n}_{1})}^{\,SYM}+\delta{\cal
W}_{a}\Bigr{)}\rangle\approx-2\mu_{\rm
x}m\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}}\approx-2\mu_{\rm
x}m\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\,.$

Therefore, on the whole for these $U(N_{c})$ br2 vacua after accounting for
the leading power corrections, see (A.25),(A.26),

$\displaystyle\langle({\overline{Q}}Q)_{1}\rangle^{U(N_{c})}_{{\rm
n}_{1}}=\langle{\overline{Q}}^{1}_{1}\rangle\langle
Q^{1}_{1}\rangle\approx\mu_{\rm
x}m\Bigl{[}1-2\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Bigr{]},\,\,\langle({\overline{Q}}Q)_{2}\rangle^{U(N_{c})}_{N_{c}}\approx\mu_{\rm
x}m\Bigl{[}1-\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Bigr{]}.\,\,\,\,$ (A.27)

It is seen that, as above in $SU(N_{c})$ br2 vacua (A.24), the non-leading
terms in $\langle({\overline{Q}}Q)_{1}\rangle_{{\rm n}_{1}}$ and
$\langle({\overline{Q}}Q)_{2}\rangle_{N_{c}}$ are different.

We can compare now the value of
$\langle({\overline{Q}}Q)_{1}\rangle^{U(N_{c})}_{{\rm n}_{1}}$ from (A.27)
with those from (2.2.10) (see however the footnote 7, in these br2 vacua the
charges of massless at $\mu_{\rm x}\rightarrow 0$ particles are not trivial
and not obvious beforehand, as well as their multiplicities). But we know the
charges and multiplicities of massless at $\mu_{\rm x}\rightarrow 0$ particles
from sections 2.1 and 2.2. To obtain definite predictions from (2.2.10) for
the quark condensates $\langle({\overline{Q}}Q)_{1}\rangle_{{\rm n}_{1}}$ we
need the values of ${\rm n}_{1}$ double roots $e^{(Q)}_{k}$ corresponding to
original electric quarks from $SU({\rm n}_{1})$ and the values of two single
roots $e^{\pm}$. Similarly to (A.15),(A.16), these look here as

$\displaystyle e^{(Q)}_{k}=-m\,,\quad k=1...{\rm n}_{1}\,,\quad
e^{+}=-e^{-}\approx 2\langle\Lambda^{SU({\overline{N}}_{c}-{\rm
n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle\,,\quad\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm
n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}}\approx\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\ll 1\,.\,\,\,$ (A.28)

From this

$\displaystyle\langle({\overline{Q}}Q)_{1}\rangle^{U(N_{c})}_{{\rm
n}_{1}}=-\mu_{\rm
x}\sqrt{(e^{(Q)}_{k}-e^{+})(e^{(Q)}_{k}-e^{-})}\approx\mu_{\rm
x}m\Bigl{[}\,1-2\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\,\Bigr{]}\,,$ (A.29)

this agrees with (A.27).

Besides, proceeding similarly to br1 vacua in (A.19)-(A.22), we obtain for
$\langle({\overline{Q}}Q)_{1}\rangle^{SU(N_{c})}_{{\rm n}_{1}}$ in br2 vacua
of $SU(N_{c})$, see (A.25),

$\frac{\langle a_{0}\rangle}{m}\,\,\rightarrow\,\,\approx\Biggl{(}\,\frac{{\rm
n}_{2}}{N_{c}}+\frac{{\rm n}_{1}-{\rm
n}_{2}}{N_{c}}\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\,\Biggr{)}\,,$ $\displaystyle m\rightarrow{m^{\prime}}\approx
m_{1}\Biggl{(}1+\frac{{\rm n}_{1}-{\rm n}_{2}}{N_{c}-{\rm
n}_{2}}\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Biggr{)}\,,\quad m_{1}=\frac{N_{c}}{N_{c}-{\rm n}_{2}}\,m\,,$
(A.30)

and from (A.29),(A.30)

$\displaystyle\langle({\overline{Q}}Q)_{1}\rangle^{SU(N_{c})}_{{\rm
n}_{1}}\approx\mu_{\rm x}m_{1}\Biggl{(}1+\frac{{\rm n}_{1}-{\rm
n}_{2}}{N_{c}-{\rm
n}_{2}}\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Biggr{)}\Bigl{[}\,1-2\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\,\Bigr{]}\approx$ (A.31) $\approx\mu_{\rm
x}m_{1}\Bigl{[}\,1+\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\,\Bigr{]}\approx\mu_{\rm
x}m_{1}\Biggl{[}\,1+\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}_{1}}\,\Biggr{]}\,,$

this agrees with (A.24).

##  Appendix B Power corrections to dyon condensates

Consider first the br2 vacua of $\mathbf{U(N_{c})}$ with $m\ll\Lambda_{2}$ 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 (1.2), one
needs to use not only (A.15), but also the sum rule (A.17). The sum rule
(A.17) looks in these br2 vacua as, see (A.25),

$\displaystyle N_{c}\langle a_{0}\rangle\approx\Bigl{(}{\rm n}_{2}m+({\rm
n}_{1}-{\rm n}_{2})\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}}{m}\Bigr{)}\approx\Biggl{(}{\rm
n}_{2}m+({\rm n}_{1}-{\rm
n}_{2})m\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Biggr{)}\approx$ (B.1) $\approx\Bigl{(}{\rm
n}_{1}m+({\overline{N}}_{c}-{\rm
n}_{1}-1)A^{(M)}_{c}\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm
n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m}-\frac{1}{2}(e^{+}+e^{-}=0)+(2N_{c}-N_{F})C^{(D)}_{c}\Bigr{)}\,,$

where ${\rm n}_{1}(-e^{(Q)}_{i})={\rm n}_{1}m$ is the contribution of ${\rm
n}_{1}$ equal double roots of original electric quarks from $SU({\rm n}_{1})$,
the term with $A^{(M)}_{c}$ is the contribution from the centre of
${\overline{N}}_{c}-{\rm n}_{1}-1$ 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
(1.2) together with (B.1) (with our accuracy)

$\displaystyle A^{(M)}_{c}=\frac{{\rm n}_{2}-{\rm
n}_{1}}{{\overline{N}}_{c}-{\rm n}_{1}-1}\,,\quad C^{(D)}_{c}=\frac{{\rm
n}_{2}-{\rm
n}_{1}}{2N_{c}-N_{F}}m\Bigl{(}1-2\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}}{m^{2}}\Bigr{)}\,,\,\,\,$ (B.2)
$e^{(M)}_{k}\approx 2\cos(\frac{\pi k}{{\overline{N}}_{c}-{\rm
n}_{1}})\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle-\frac{{\rm n}_{2}-{\rm n}_{1}}{{\overline{N}}_{c}-{\rm
n}_{1}-1}\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m}\,,\quad k=1...({\overline{N}}_{c}-{\rm
n}_{1}-1)\,,$
$e^{(D)}_{j}\approx\omega^{j-1}\Lambda_{2}\Biggl{(}1+O\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{2N_{c}-N_{F}}\Biggr{)}-\frac{{\rm
n}_{2}-{\rm
n}_{1}}{2N_{c}-N_{F}}\,m\Biggl{(}1-2\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Biggr{)}\,,\,\,j=1...(2N_{c}-N_{F})\,,$

while $e^{\pm}$ are given in (A.28).

From (2.2.10),(B.2) we obtain now for the dyon condensates in these br2 vacua
of $U(N_{c})$ at $m\ll\Lambda_{2}$ (with the same accuracy)

$\displaystyle\langle{\overline{D}}_{j}D_{j}\rangle_{U(N_{c})}=\langle{\overline{D}}_{j}\rangle\langle
D_{j}\rangle\approx\mu_{\rm x}\Biggl{[}\,-\,\omega^{j-1}\Lambda_{2}+\frac{{\rm
n}_{2}-{\rm
n}_{1}}{2N_{c}-N_{F}}\,m\Biggl{(}1-2\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Biggr{)}\Biggr{]}\,,$ (B.3)
$\langle\Sigma^{U(N_{c})}_{D}\rangle\equiv\sum_{j=1}^{2N_{c}-N_{F}}\langle{\overline{D}}_{j}D_{j}\rangle_{U(N_{c})}=\sum_{j=1}^{2N_{c}-N_{F}}\langle{\overline{D}}_{j}\rangle\langle
D_{j}\rangle\approx\mu_{\rm x}m\,({\rm n}_{2}-{\rm
n}_{1})\Biggl{(}1-2\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Biggr{)},\,\,\,j=1...2N_{c}-N_{F}\,.$

Now, using only the leading terms $\sim\mu_{\rm x}m$ of (B.3),(2.2.3),(2.2.5),
(2.2.8), we can find the value of $\delta_{3}=O(1)$ in
(2.1.2),(2.1.5),(2.2.4),(6.2.3). From (2.2.4)

$\displaystyle\langle\frac{\partial}{\partial a_{0}}{\cal W}^{\,\rm low}_{\rm
tot}\rangle=0=-\langle{\rm Tr\,}{\overline{Q}}Q\rangle_{{\rm
n}_{1}}-\langle\Sigma^{U(N_{c})}_{D}\rangle+\mu_{\rm x}N_{c}\langle
a_{0}\rangle-\mu_{\rm x}N_{c}\delta_{3}\langle a_{1}\rangle=-\mu_{\rm
x}N_{c}\delta_{3}\langle
a_{1}\rangle\rightarrow{\boldsymbol{\delta}}{\mathbf{{}_{3}=0}}.\,\,\quad$
(B.4)

(The same result follows from $\langle\partial{\cal W}^{\,\rm low}_{\rm
tot}/{\partial a_{1}}\rangle=0$ in (2.2.4) ).

As for the non-leading terms
$\delta\langle\Sigma^{U(N_{c})}_{D}\rangle\sim\mu_{\rm
x}m(m/\Lambda_{2})^{\frac{2N_{c}-N_{F}}{{\rm n}_{2}-N_{c}}}$ of
$\langle\Sigma^{U(N_{c})}_{D}\rangle$ in (B.3), with $\delta_{3}=0$ from
(B.4), these can now be also calculated independently from
(2.2.4),(A.25),(A.26), i.e.

$\langle\frac{\partial}{\partial{\hat{a}}_{0}}\delta{\cal W}^{\,\rm low}_{\rm
tot}\rangle=0\rightarrow\delta\langle\Sigma^{U(N_{c})}_{D}\rangle\approx\Bigl{[}-\delta\langle{\rm
Tr}{\overline{Q}}Q\rangle_{{\rm n}_{1}}=2{\rm n}_{1}\mu_{\rm
x}m\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Bigr{]}_{Q}+\Bigl{[}({\rm n}_{1}-{\rm n}_{2})\mu_{\rm
x}m\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Bigr{]}_{\delta{\cal W}_{a_{0}}}\,\,\,$
$\displaystyle+\Bigl{[}\langle\frac{\partial}{\partial{\hat{a}}_{0}}\delta{\cal
W}_{SU({\overline{N}}_{c}-{\rm n}_{1})}^{\,SYM}\rangle=-N_{F}\mu_{\rm
x}m\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Bigr{]}_{\delta{\cal W}_{SYM}}\approx-2\mu_{\rm x}m\,({\rm
n}_{2}-{\rm
n}_{1})\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\,,$ (B.5)

this agrees with (B.3).

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 (B.4), it can be found now
directly from (2.1.5),(2.1.12),(A.23),(A.24)

$\displaystyle\langle\frac{\partial}{\partial{\hat{a}}_{1}}{\cal W}^{\,\rm
low}_{\rm
tot}\rangle=0\rightarrow\langle\Sigma^{SU(N_{c})}_{D}\rangle\approx\mu_{\rm
x}m_{1}({\rm n}_{2}-{\rm n}_{1})\Biggl{[}1+\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Biggr{]}\,,\,\,m_{1}=\frac{N_{c}}{N_{c}-{\rm
n}_{2}}\,m\,.\,\,\,$ (B.6)

(Really, the power suppressed term in (B.6) can be found independently of the
value of $\delta_{3}$ because it is parametrically different and, unlike (B.5)
in $U(N_{c})$ theory, there are no power corrections to $\langle a_{1}\rangle$
in $SU(N_{c})$ theory, see (2.1.6)).

It can also be obtained from (B.3) in br2 vacua of $U(N_{c})$ proceeding
similarly to (A.30),(A.31) for the quark condensates. We obtain from (A.30)
and (B.3)

$\langle\Sigma^{SU(N_{c})}_{D}\rangle=\sum_{j=1}^{2N_{c}-N_{F}}\langle{\overline{D}}_{j}\rangle\langle
D_{j}\rangle\approx\mu_{\rm x}m_{1}\Biggl{(}1+\frac{{\rm n}_{1}-{\rm
n}_{2}}{N_{c}-{\rm
n}_{2}}\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Biggr{)}({\rm n}_{2}-{\rm
n}_{1})\Bigl{[}\,1-2\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\,\Bigr{]}\approx$ $\displaystyle\approx\mu_{\rm x}m_{1}({\rm
n}_{2}-{\rm n}_{1})\Biggl{[}1+\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Biggr{]}\,,$ (B.7)

this agrees with (B.6).

On the whole, the dyon condensates in br2 vacua of $SU(N_{c})$ look at
$m\ll\Lambda_{2}$ as, $j=1,...,2N_{c}-N_{F}$,

$\displaystyle\langle{\overline{D}}_{j}D_{j}\rangle^{SU(N_{c})}\approx\mu_{\rm
x}\Biggl{[}-\omega^{j-1}\Lambda_{2}\Biggl{(}1+O\Bigl{(}\frac{m}{\Lambda_{2}}\Bigr{)}^{2N_{c}-N_{F}}\Biggr{)}+\frac{{\rm
n}_{2}-{\rm n}_{1}}{2N_{c}-N_{F}}\,m_{1}\Biggl{(}1+\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Biggr{)}\Biggr{]}.\,\,\,$ (B.8)

Finally, not going into details, we present below the values of roots of the
$\mathbf{SU(N_{c})}$ spectral curve (1.2) in br1 vacua of section 4.1 and br2
vacua of section 2.1.

1) br1 vacua. There are at $\mu_{\rm x}\rightarrow 0$ $\,{\rm n}_{1}$ equal
double roots $e^{(Q)}_{i}$ of massless original electric quarks $Q_{i}$, then
$N_{c}-{\rm n}_{1}-1$ unequal double roots $e^{(M)}_{k}$ of massless pure
magnetic monopoles $M_{k}$ and two single roots $e^{\pm}$ from ${\cal
N}=2\,\,SU(N_{c}-{\rm n}_{1})$ SYM. They look (with our accuracy) as, see
(A.3),

$e^{(Q)}_{i}=-m\,,\,\,i=1...{\rm n}_{1}\,,\quad e^{\pm}\approx e^{\pm}_{c}\pm
2\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle,\quad\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle\approx
m_{3}\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{2(N_{c}-{\rm
n}_{1})}},$ $\displaystyle e^{\pm}_{c}=\frac{1}{2}(e^{+}+e^{-})\approx
m_{3}\Biggl{[}\,\frac{{\rm n}_{1}}{N_{c}}+\frac{{\rm n}_{2}-{\rm
n}_{1}}{N_{c}-{\rm n}_{1}}\frac{\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}_{3}}\,\Biggr{]},\quad
m_{3}=\frac{N_{c}}{N_{c}-{\rm n}_{1}}m\,,$ (B.9)
$e^{(M)}_{k}\approx\Biggl{[}\,e^{\pm}_{c}-\frac{{\rm n}_{2}-{\rm
n}_{1}}{N_{c}-{\rm n}_{1}-1}\frac{\langle\Lambda^{SU(N_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}}{m_{3}}\,\Biggr{]}+2\cos(\frac{\pi
k}{N_{c}-{\rm n}_{1}})\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle,\quad\frac{1}{2}\sum_{n=1}^{2N_{c}}(-e_{n})=0\,.$

It should be emphasized that instead of $e^{\pm}_{c}=0$ in $\mathbf{U(N_{c})}$
theory, see (A.15), it looks in $\mathbf{SU(N_{c})}$ theory as, see
(A.21),(B.9),

$e^{\pm}_{c}\approx\Bigl{[}\frac{\langle
a_{0}\rangle}{m}\Bigr{]}\Bigl{[}\,m\,\Bigr{]}\,\rightarrow\,\Biggl{[}\,\frac{\langle
a_{0}\rangle}{m}\,\,\rightarrow\,\,\,\approx\Biggl{(}\,\frac{{\rm
n}_{1}}{N_{c}}+\frac{{\rm n}_{2}-{\rm
n}_{1}}{N_{c}}\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\,\Biggr{)}\,\Biggr{]}\Biggl{[}m\,\rightarrow\,{m^{\prime}}\approx
m_{3}\Biggl{(}\,1+\frac{{\rm n}_{2}-{\rm n}_{1}}{N_{c}-{\rm n}_{1}}$
$\displaystyle\cdot\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\Biggr{)}\,\Biggr{]}\approx m_{3}\Biggl{[}\frac{{\rm
n}_{1}}{N_{c}}+\frac{{\rm n}_{2}-{\rm n}_{1}}{N_{c}-{\rm
n}_{1}}\Bigl{(}\frac{\Lambda_{2}}{m_{3}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}}\Biggr{]}\approx m_{3}\Biggl{[}\,\frac{{\rm n}_{1}}{N_{c}}+\frac{{\rm
n}_{2}-{\rm n}_{1}}{N_{c}-{\rm n}_{1}}\frac{\langle\Lambda^{SU(N_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}}{m^{2}_{3}}\,\Biggr{]}\,.$ (B.10)

2) br2 vacua. There are at $\mu_{\rm x}\rightarrow 0$ $\,{\rm n}_{1}$ equal
double roots $e^{(Q)}_{i}$ of massless original electric quarks $Q_{i}$, then
${\overline{N}}_{c}-{\rm n}_{1}-1$ unequal double roots $e^{(M)}_{k}$ of
massless pure magnetic monopoles $M_{k}$ and two single roots $e^{\pm}$ from
${\cal N}=2\,\,SU({\overline{N}}_{c}-{\rm n}_{1})$ SYM, and $2N_{c}-N_{F}$
unequal double roots of massless dyons $D_{j}$. They all look (with our
accuracy) as, see (A.24),(A.25),(A.30),

$e^{(Q)}_{i}=-m\,,\,\,i=1...{\rm n}_{1}\,,\quad e^{\pm}\approx e^{\pm}_{c}\pm
2\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle,\quad\langle\Lambda^{SU({\overline{N}}_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle^{2}\approx
m^{2}_{1}\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}},$
$e^{\pm}_{c}=\frac{1}{2}(e^{+}+e^{-})\approx\Biggl{[}\,\frac{\langle
a_{0}\rangle}{m}\,\rightarrow\,\,\approx\Biggl{(}\,\frac{{\rm
n}_{2}}{N_{c}}+\frac{{\rm n}_{1}-{\rm
n}_{2}}{N_{c}}\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\,\Biggr{)}\,\Biggr{]}\Biggl{[}\,m\,\rightarrow\,\,{m^{\prime}}\approx
m_{1}\Biggl{(}1+$ $\displaystyle+\frac{{\rm n}_{1}-{\rm n}_{2}}{N_{c}-{\rm
n}_{2}}\Bigl{(}\frac{m_{1}}{\Lambda_{2}}\Bigr{)}^{\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}}\Biggr{)}\,\Biggr{]}\approx m_{1}\Biggl{[}\,\frac{{\rm
n}_{2}}{N_{c}}+\frac{{\rm n}_{2}-{\rm n}_{1}}{{\rm
n}_{2}-N_{c}}\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}_{1}}\,\Biggr{]},\quad
m_{1}=\frac{N_{c}}{N_{c}-{\rm n}_{2}}m\,,$ (B.11)
$e^{(M)}_{k}\approx\Biggl{[}\,e^{\pm}_{c}-\frac{{\rm n}_{2}-{\rm
n}_{1}}{{\overline{N}}_{c}-{\rm
n}_{1}-1}\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m_{1}}\,\Biggr{]}+2\cos(\frac{\pi
k}{{\overline{N}}_{c}-{\rm n}_{1}})\langle\Lambda^{SU({\overline{N}}_{c}-{\rm
n}_{1})}_{{\cal N}=2\,\,SYM}\rangle,\quad k=1...({\overline{N}}_{c}-{\rm
n}_{1}-1)\,,$ $\displaystyle
e^{(D)}_{j}\approx\omega^{j-1}\Lambda_{2}+\frac{N_{F}}{2N_{c}-N_{F}}\,m\,,\quad
j=1...2N_{c}-N_{F}\,,\quad\frac{1}{2}\sum_{n=1}^{2N_{c}}(-e_{n})=0\,.$ (B.12)

Now, with (B.9)-(B.12), we can perform checks of possible applicability of
(2.2.10) 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 $\mu_{\rm x}$ (clearly, this applicability is far not
evident beforehand, see the footnote 7).

a) From (B.9) for the quark condensates in br1 vacua of section 4.1

$\displaystyle\langle({\overline{Q}}Q)_{1}\rangle^{SU(N_{c})}_{{\rm
n}_{1}}=-\mu_{\rm
x}\sqrt{(e^{(Q)}_{k}-e^{+})(e^{(Q)}_{k}-e^{-})}\approx\mu_{\rm
x}m_{3}\Biggl{[}\,1-\frac{2N_{c}-N_{F}}{N_{c}-{\rm
n}_{1}}\frac{\langle\Lambda^{SU(N_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}_{3}}\,\Biggr{]}\,,$ (B.13)

this agrees with the independent calculation in (A.5).

b) From (B.11) for the quark condensates in br2 vacua of section 2.1

$\displaystyle\langle({\overline{Q}}Q)_{1}\rangle^{SU(N_{c})}_{{\rm
n}_{1}}=-\mu_{\rm
x}\sqrt{(e^{(Q)}_{k}-e^{+})(e^{(Q)}_{k}-e^{-})}\approx\mu_{\rm
x}m_{1}\Biggl{[}\,1+\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}_{1}}\,\Biggr{]}\,,$ (B.14)

this agrees with the independent calculation in (A.24).

c) From (B.11),(B.12) for the dyon condensates in br2 vacua of section 2.1

$\langle{\overline{D}}_{j}D_{j}\rangle^{SU(N_{c})}=\langle{\overline{D}}_{j}\rangle^{SU(N_{c})}\langle
D_{j}\rangle^{SU(N_{c})}=-\mu_{\rm
x}\sqrt{(e^{(D)}_{j}-e^{+})(e^{(D)}_{j}-e^{-})}\approx$
$\displaystyle\approx\mu_{\rm x}\Biggl{[}-\omega^{j-1}\Lambda_{2}+\frac{{\rm
n}_{2}-{\rm n}_{1}}{2N_{c}-N_{F}}\,m_{1}\Biggl{(}1+\frac{2N_{c}-N_{F}}{{\rm
n}_{2}-N_{c}}\frac{\langle\Lambda^{SU({\overline{N}}_{c}-{\rm n}_{1})}_{{\cal
N}=2\,\,SYM}\rangle^{2}}{m^{2}_{1}}\Biggl{)}\,\Biggr{]}\,,\quad
j=1...2N_{c}-N_{F}\,,$ (B.15)

this agrees with the independent calculation in (B.8).

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