Mass spectra in $\mathbf{{\cal N}=1\,\,SQCD}$ with additional colorless fields
and problems with Seiberg’s duality

Victor L. Chernyak

Budker Institute of Nuclear Physics and Novosibirsk State University,
Novosibirsk, Russia

Talk given at Session of Nucl. Phys. section of RAS, Novosibirsk, Russia,
10-12 March

Abstract

Considered is the direct ${\cal N}=1$ SQCD-like $\Phi$-theory with $SU(N_{c})$
colors and $3N_{c}/2<N_{F}<2N_{c}$ flavors of light quarks
${\overline{Q}},\,Q$. Besides, it includes $N^{2}_{F}$ additional colorless
but flavored fields $\Phi_{i}^{j}$ with the large mass parameter
$\mu_{\Phi}\gg\Lambda_{Q}$, interacting with quarks through the Yukawa
coupling. In parallel, is considered its Seiberg’s dual variant, i.e. the
$d\Phi$-theory with $(N_{F}-N_{c})$ dual colors, $N_{F}$ flavors of dual
quarks ${\overline{q}},\,{q}$ and $N_{F}^{2}$ elementary mion fields
$M^{i}_{j}\rightarrow({\overline{Q}}_{j}Q^{i})$.

In considered here vacua, the quarks of both theories are in the conformal
regimes at scales $\mu<\Lambda_{Q}$. The mass spectra are calculated in
sections 4 and 5. It is shown that they are different in the direct and dual
theories, in disagreement with the Seiberg hypothesis about equivalence of two
such theories.

Besides it is shown in the direct theory that a qualitatively new phenomenon
takes place: the seemingly heavy fields $\Phi$ ‘return back’ and there appear
two additional generations of light $\Phi$-particles with small masses
$\mu^{\rm pole}_{2,3}(\Phi)\ll\Lambda_{Q}$.

In Conclusions also presented comparison of mass spectra of these two theories
for such values of parameters when the direct theory is in the very strong
coupling regime, while the dual one is in the weak coupling IR-free
logarithmic regime. It is shown that mass spectra of these two theories are
parametrically different in this case.

###### Contents

  1. 1 Definitions and some generalities
  2. 2 Quark and gluino condensates and multiplicities of vacua at $\mathbf{3N_{c}/2<N_{F}<2N_{c}}$
  3. 3 Fions $\mathbf{\Phi^{i}_{j}}$ in the direct theory : one or three generations
  4. 4  Mass spectra in $\rm br2$ vacua. Direct theory
  5. 5  Mass spectra in $\rm br2$ vacua. Dual theory
  6. 6  Conclusions
  7. A ’t Hooft triangles, $0<{\rm\overline{b}}_{\rm o}/N_{F}=O(1)\,,\,0<({\rm\overline{b}}_{\rm o}-2n_{1})/N_{F}=O(1)$
  8. B Mass spectra in the standard ${\cal N}=1$ SQCD and its Seiberg’s dual at $N_{F}=N_{c}+1$.
  9. References

##  1 Definitions and some generalities

Direct $\mathbf{\Phi}$ \- theory

The field content of this direct ${\cal N}=1\,\,\,\Phi$-theory includes
$SU(N_{c})$ gluons and $3N_{c}/2<N_{F}<2N_{c}$ flavors of light quarks
${\overline{Q}}_{j},Q^{i}$. Besides, there are $N^{2}_{F}$ colorless but
flavored fields $\Phi^{j}_{i}$ (fions) with the large mass parameter
$\mu_{\Phi}\gg\Lambda_{Q}$.

The Lagrangian at the scale $\mu=\Lambda_{Q}$ in superfield notations looks as
(the exponents with gluons in the Kahler term are implied here and everywhere
below, ${\overline{N}}_{c}=N_{F}-N_{c}$):

$\displaystyle L=\int d^{4}x\int
d^{2}{\overline{\theta}}d^{2}\theta\,K(x,{\overline{\theta}},\theta)\,+\Bigl{(}{\mathcal{W}}+h.c.\Bigr{)},\quad{\mathcal{W}}=\int
d^{4}x\int d^{2}\theta\,{\mathcal{W}}_{tot}(x,\theta)\,,$ (1.1) $K={\rm
Tr}\,\Bigl{(}\Phi^{\dagger}\Phi\Bigr{)}+{\rm
Tr}\Bigl{(}\,Q^{\dagger}Q+(Q\rightarrow{\overline{Q}})\,\Bigr{)}\,,\quad{\mathcal{W}}_{\rm
tot}={\mathcal{W}}_{\rm gauge}+{\mathcal{W}}_{\rm
matter}\,,\quad{\mathcal{W}}_{\rm
gauge}=-\frac{2\pi}{\alpha(\mu,\Lambda_{Q})}S\,,$ ${\mathcal{W}}_{\rm
matter}={\mathcal{W}}_{Q}+{\mathcal{W}}_{\Phi}\,,\quad{\mathcal{W}}_{Q}={\rm
Tr}\,{\overline{Q}}m^{\rm tot}_{Q}Q={\rm
Tr}\,{\overline{Q}}(m_{Q}-\Phi)Q,\quad{\mathcal{W}}_{\Phi}=\frac{\mu_{\Phi}}{2}\Biggl{[}{\rm
Tr}\,(\Phi^{2})-\frac{1}{{\overline{N}}_{c}}\Bigl{(}{\rm
Tr}\,\Phi\Bigr{)}^{2}\Biggr{]}\,.$

Here : $\mu_{\Phi}$ and $m_{Q}$ are the mass parameters,
$S=W^{A}_{\beta}W^{A,\,\beta}/32\pi^{2}$, where $W^{A}_{\beta}$ is the field
strength of the gauge superfield, $A=1...N_{c}^{2}-1,\,\beta=1,2$,
$\alpha(\mu,\Lambda_{Q})=g^{2}(\mu,\Lambda_{Q})/4\pi$ is the gauge coupling
with its scale factor $\Lambda_{Q}$. Except for section 3, this normalization
of fields is used everywhere below in the text.

Therefore, finally, the $\Phi$-theory we deal with has the parameters :
$N_{c},\,N_{F},\,\mu_{\Phi},\,\Lambda_{Q},\,m_{Q}$, with the strong
hierarchies $\mu_{\Phi}\gg\Lambda_{Q}\gg m_{Q}$. Everywhere below in the text
the mass parameter $\mu_{\Phi}$ is in the range:
$\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,0}=\Lambda_{Q}(\Lambda_{Q}/m_{Q})^{(2N_{c}-N_{F})/N_{c}}$.
111 Here and below: $A\approx B$ means equality up to small corrections, $A\gg
B$ has to be understood as $|A|\gg|B|$, $A\sim B$ means the same power
dependence of A and B on small parameters $m_{Q}/\Lambda_{Q}\ll 1$,
$\Lambda_{Q}/\mu_{\Phi}\ll 1$ and $Z_{q}\ll 1$, up to a constant factor.

Dual $\mathbf{d}\Phi$-theory

In parallel with the direct $\Phi$-theory with $3N_{c}/2<N_{F}<2N_{c}$ , we
consider also the Seiberg dual variant [2] (the $d\Phi$-theory). The dual
Lagrangian at $\mu=\Lambda_{Q}$ looks as

${\overline{K}}={\rm Tr}\,\Bigl{(}\Phi^{\dagger}\Phi\Bigr{)}+{\rm
Tr}\Bigl{(}q^{\dagger}q+(q\rightarrow\overline{q})\,\Bigr{)}+{\rm
Tr}\,\frac{M^{\dagger}M}{f^{2}Z^{2}_{q}\Lambda_{Q}^{2}}\,,\quad{\overline{\mathcal{W}}}={\overline{\mathcal{W}}}_{\rm
gauge}+{\overline{\mathcal{W}}}_{\rm
matter},\quad{\overline{\mathcal{W}}}_{\rm
gauge}=\,-\,\frac{2\pi}{\overline{\alpha}(\mu=\Lambda_{Q})}\,{\overline{S}}\,,$
$\displaystyle{\overline{\mathcal{W}}}_{\rm
matter}={\mathcal{W}}_{\Phi}+{{\mathcal{W}}}_{M\Phi}+{\mathcal{W}}_{q}\,,\quad{{\mathcal{W}}}_{M\Phi}={\rm
Tr}\,M(m_{Q}-\Phi),\quad{{\mathcal{W}}}_{q}=-\frac{1}{Z_{q}\Lambda_{Q}}\,\rm{Tr}\Bigl{(}{\overline{q}}\,M\,q\Bigr{)}\,.$
(1.2)

Here : the number of dual colors is ${\overline{N}}_{c}=(N_{F}-N_{c})$ and
$M^{i}_{j}\rightarrow({\overline{Q}}_{j}Q^{i})$ are $N_{F}^{2}$ elementary
mion fields,
${\overline{a}}(\mu)={\overline{N}}_{c}{\overline{g}}^{2}(\mu)/8\pi$ is the
dual running gauge coupling (with its scale parameter
$|\Lambda_{q}|=\Lambda_{Q}$),
${\overline{S}}={\rm\overline{W}}^{B}_{\beta}{\rm\overline{W}}^{B,\,\beta}/32\pi^{2},\,\,B=1...({\overline{N}}_{c}^{2}-1)$,
${\rm\overline{W}}^{B}_{\beta}$ is the dual gluon field strength. The factors
$a_{f}={\overline{N}}_{c}f^{2}/2\pi$ and $Z_{q}$ in (1.2) are $O(1)$ at
${\rm\overline{b}}_{\rm o}/N_{F}=O(1)$ (and are omitted below in this case),
but are parametrically small at ${\rm\overline{b}}_{\rm o}/N_{F}\ll
1\,:\,a_{f}=O({\rm\overline{b}}_{\rm o}/N_{F})$, while $Z_{q}$ is
exponentially small. (And $Z_{q}$ is accounted for then, see Conclusions).

At $3/2<N_{F}/N_{c}<2$ this dual theory can be taken as UV free at
$\mu\gg\Lambda_{Q}$. We consider it below at $\mu\leq\Lambda_{Q}$ only where,
according to Seiberg’s hypothesis, it becomes equivalent to the direct
$\Phi$-theory.

Really, all $N_{F}^{2}$ fields $\Phi^{j}_{i}$ remain always too heavy and
dynamically irrelevant in this $d\Phi$-theory at $3N_{c}/2<N_{F}<2N_{c}$ and
$\mu<\Lambda_{Q}$, so that they can be integrated out once and forever and,
finally, we write the Lagrangian of the dual theory at $\mu=\Lambda_{Q}$ in
the form

$K={\rm Tr}\Bigl{(}q^{\dagger}q+(q\rightarrow\overline{q})\Bigr{)}+{\rm
Tr}\,\frac{M^{\dagger}M}{f^{2}Z^{2}_{q}\Lambda_{Q}^{2}}\,,\quad{\overline{\mathcal{W}}}_{\rm
matter}={\mathcal{W}}_{M}+{\mathcal{W}}_{q}\,,$
$\displaystyle{\mathcal{W}}_{M}=m_{Q}{\rm
Tr}\,M-\frac{1}{2\mu_{\Phi}}\Biggl{[}{\rm Tr}\,(M^{2})-\frac{1}{N_{c}}({\rm
Tr}\,M)^{2}\Biggr{]}\,,\quad{\mathcal{W}}_{q}=-\,\frac{1}{Z_{q}\Lambda_{Q}}\,\rm{Tr}\Bigl{(}{\overline{q}}\,M\,q\Bigr{)}\,.$
(1.3)

The gluino condensates of the direct and dual theories are matched in all
vacua, $\langle{-\,\overline{S}}\rangle=\langle
S\rangle=\langle\Lambda_{SYM}\rangle^{3}$, as well as $\langle
M_{j}^{i}(\mu=\Lambda_{Q})\rangle=\langle
M_{j}^{i}\rangle=\langle{\overline{Q}}_{j}Q^{i}(\mu=\Lambda_{Q})\rangle=\langle{\overline{Q}}_{j}Q^{i}\rangle,\,\,\langle{\overline{Q}}_{j}Q^{i}\rangle=\sum_{a=1}^{N_{c}}\langle{\overline{Q}}_{j}^{a}Q^{i}_{a}\rangle$.

Besides, the perturbative NSVZ $\beta$-function [3] for (effectively) massless
${\cal N}=1$ SQCD is used

$\displaystyle\frac{d}{d\ln\mu}\,\frac{1}{a(\mu)}=\beta(a)=\frac{1}{1-a(\mu)}\Bigl{[}\frac{{\rm
b_{o}}}{N_{c}}-\frac{N_{F}}{N_{c}}\gamma_{Q}(a)\Bigr{]}\,,\quad
a(\mu)=\frac{N_{c}g^{2}(\mu)}{8\pi^{2}}\,,\quad{\rm b_{o}}=3N_{c}-N_{F}\,,$
(1.4)

where $\gamma_{Q}$ is the quark anomalous dimension (and similarly in the dual
theory:
$N_{c}\rightarrow\,{\overline{N}}_{c},\,\,\gamma_{Q}\rightarrow\,\gamma_{q},\,\,\,a=N_{c}g^{2}/8\pi^{2}\rightarrow\,{\overline{a}}={\overline{N}}_{c}{\overline{g}}^{2}/8\pi^{2},\,a_{f}={\overline{N}}_{c}f^{2}/2\pi,\,{\rm
b_{o}}=(3N_{c}-N_{F})\rightarrow\,{\rm\overline{b}}_{\rm
o}=(3{\overline{N}}_{c}-N_{F})\,)$.

We take below (except for Conclusions): ${\rm b_{o}}/N_{F}$ and
${\rm\overline{b}}_{\rm o}/N_{F}$ as $O(1)$. Then $Z_{q}$ and $a_{f}$ are both
$O(1)$ and are omitted.

Because the range $3N_{c}/2<N_{F}<2N_{c}$ considered here is within the
conformal window $3N_{c}/2<N_{F}<3N_{c}$, both the direct and dual theories
(which are in the logarithmically weak UV free regime at $\mu\gg\Lambda_{Q}$)
enter smoothly the conformal regime at $\mu<\Lambda_{Q}$, with frozen
couplings:
$a(\mu<\Lambda_{Q})=a_{*}=O(1),\,\,{\overline{a}}(\mu<\Lambda_{Q})={\overline{a}}_{*}=O(1),\,a_{f}(\mu<\Lambda_{Q})=a^{*}_{f}=O(1)$
(until this conformal regime with effectively massless quarks, gluons and
mions $M^{i}_{j}$ is broken by particles masses at lower energies). Then, the
anomalous dimensions of all fields and so the corresponding renormalization
factors of all Kahler terms are known in the conformal regime:

$\displaystyle\beta^{(a)}_{\rm conf}(a_{*})=\beta^{({\overline{a}})}_{\rm
conf}({\overline{a}}_{*},a^{*}_{f})=\beta^{({a_{f}})}_{\rm
conf}({\overline{a}}_{*},a^{*}_{f})=0\,\,\rightarrow\,\,\gamma_{Q}(a_{*})=\frac{3N_{c}-N_{F}}{N_{F}},\quad\gamma_{\Phi}(a_{*})=-2\gamma_{Q}(a_{*})\,,$
(1.5)
$\gamma_{q}({\overline{a}}_{*},a^{*}_{f})=\frac{3{\overline{N}}_{c}-N_{F}}{N_{F}},\quad\gamma_{M}({\overline{a}}_{*},a^{*}_{f})=-2\gamma_{q}({\overline{a}}_{*},a^{*}_{f})\,,$

in the direct and dual theories respectively.

##  2 Quark and gluino condensates and multiplicities of vacua at
$\mathbf{3N_{c}/2<N_{F}<2N_{c}}$

To obtain the numerical values of the quark condensates (really, the mean
vacuum values)
$\langle{\overline{Q}}_{j}Q^{i}\rangle=\delta^{i}_{j}\langle({\overline{Q}}Q)_{i}\rangle$,
but only for this purpose, the simplest way is to use the known exact form of
the non-perturbative contribution ${\mathcal{W}}_{\rm non-pert}$ to the
effective superpotential in the standard SQCD (i.e. without the fion fields
$\Phi$). It seems clear that at sufficiently large values of
$\mu_{\Phi}\gg\Lambda_{Q}$ among the vacua of the $\Phi$-theory there will be
$N_{c}$ vacua of the standard SQCD in which, definitely, all fions $\Phi$ are
too heavy and dynamically irrelevant. Therefore, they all can be integrated
out and this only results in additional 4-quark term in the superpotential, so
that the exact effective superpotential will look as

$\displaystyle{{\mathcal{W}}}_{\rm eff}=\Biggl{[}{\mathcal{W}}_{\rm non-
pert}=\,-{\overline{N}}_{c}S=\,-{\overline{N}}_{c}\Bigl{(}\frac{\det{\overline{Q}}Q}{\Lambda_{Q}^{{\rm
b_{o}}}}\Bigr{)}^{1/{\overline{N}}_{c}}\Biggr{]}+m_{Q}{\rm
Tr}\,{\overline{Q}}Q-\frac{1}{2\mu_{\Phi}}\Biggl{[}{\rm
Tr}({{\overline{Q}}Q})^{2}-\frac{1}{N_{c}}({\rm
Tr}\,{\overline{Q}}Q)^{2}\Biggr{]},\,\,\,$ (2.1)

where the first non-perturbative term in (2.1) is well known in the standard
${\cal N}=1$ SQCD without fions.

Indeed, e.g. at $3N_{c}/2<N_{F}<2N_{c}$ and sufficiently large $\mu_{\Phi}$,
there are $N_{c}$ SQCD vacua in (2.1) with the unbroken $U(N_{F})$ global
flavor symmetry. In these, the last 4-quark term in (2.1) gives a small
correction only and can be neglected and one obtains the well known results

$\displaystyle\langle{\overline{Q}}_{j}Q^{i}\rangle_{SQCD}\approx\delta^{i}_{j}\frac{1}{m_{Q}}\Bigl{(}\Lambda_{SYM}^{(\rm
SQCD)}\Bigr{)}^{3}=\delta^{i}_{j}\frac{1}{m_{Q}}\Bigl{(}\Lambda_{Q}^{{\rm
b_{o}}}m_{Q}^{N_{F}}\Bigr{)}_{,}^{1/N_{c}}\,\,\,\langle
S\rangle_{SQCD}=\langle\frac{\lambda\lambda}{32\pi^{2}}\rangle_{SQCD}\approx\Bigl{(}\Lambda_{Q}^{{\rm
b_{o}}}m_{Q}^{N_{F}}\Bigr{)}^{1/N_{c}}.\,\,\,$ (2.2)

Now, using the holomorphic dependence of the superpotential (2.1) on the
chiral superfields $({\overline{Q}}_{j}Q^{i})$ and the chiral parameters
$m_{Q}$ and $\mu_{\Phi}$, the exact form (2.1) can be used to find the values
of the quark condensates $\langle{\overline{Q}}_{j}Q^{i}\rangle$ and
multiplicities of vacua in all other numerous vacua of the $\Phi$-theory and
at all other values of $\mu_{\Phi}\gg\Lambda_{Q}$. It is worth recalling only
that, in general, as in the standard SQCD without additional fields
$\Phi^{i}_{j}$, ${{\mathcal{W}}}_{\rm eff}$ in (2.1) is not the superpotential
of the genuine low energy Lagrangian describing lightest particles, it
determines only the values of the vacuum mean values
$\langle{\overline{Q}}_{j}Q^{i}\rangle$ and $\langle S\rangle$. (The genuine
low energy Lagrangians will be obtained below, both in the direct and dual
theories).

Useful relations from (2.1) for vacua with broken flavor symmetry look as

$\displaystyle\langle({\overline{Q}}Q)_{1}+({\overline{Q}}Q)_{2}-\frac{1}{N_{c}}{\rm
Tr}\,({\overline{Q}}Q)\rangle=m_{Q}\mu_{\Phi}.\quad\langle
S\rangle=\Biggl{(}\frac{\det\langle{\overline{Q}}Q\rangle}{\Lambda_{Q}^{\rm
b_{o}}}\Biggr{)}^{1/{\overline{N}}_{c}}=\frac{\langle({\overline{Q}}Q)_{1}\rangle\langle({\overline{Q}}Q)_{2}\rangle}{\mu_{\Phi}}.$
(2.3)

It follows from (2.1),(2.3) that there is a large number of various different
vacua in this theory. But as for a realization of the global flavor symmetry
$U(N_{F})$, there are only two types of vacua: those with unbroken $U(N_{F})$
and those with the spontaneous breaking $U(N_{F})\rightarrow U(n_{1})\times
U(n_{2}),\,n_{1}+n_{2}=N_{F},\,n_{1}\leq N_{F}/2$ .

As an example, we consider below only the br2-vacua (br=breaking) with
$\langle({\overline{Q}}Q)_{2}\rangle\gg\langle({\overline{Q}}Q)_{1}\rangle$
and $n_{2}>N_{c},\,n_{1}<{\overline{N}}_{c}$, and with the multiplicity
$N_{\rm
br2}=({\overline{N}}_{c}-n_{1})C^{n_{1}}_{N_{F}},\,\,C^{n_{1}}_{N_{F}}=N_{F}!/(n_{1}!\,n_{2}!)$.

##  3 Fions $\mathbf{\Phi^{i}_{j}}$ in the direct theory : one or three
generations

At all those scales $\mu<\Lambda_{Q}$ until the field $\Phi$ remains too heavy
and non-dynamical (while the light quarks and gluons are still effectively
massless and dynamical), i.e. until the perturbative running mass
$\mu_{\Phi}^{\rm pert}(\mu)>\mu$, the field $\Phi$ decouples and can be
integrated out, and the Lagrangian in the conformal regime takes the form at
the scale $\mu\ll\Lambda_{Q}$ (${\overline{Q}_{R}},Q_{R}$ are renormalized
fields)

$K=z_{Q}(\Lambda_{Q},\mu){\rm
Tr}\Bigl{(}Q^{\dagger}Q+Q\rightarrow{\overline{Q}}\Bigr{)}={\rm
Tr}\Biggl{(}Q_{R}^{\dagger}Q_{R}+(Q_{R}\rightarrow{\overline{Q}_{R}})\Biggr{)},\,\,z_{Q}(\Lambda_{Q},\mu\ll\Lambda_{Q})=\Bigl{(}\frac{\mu}{\Lambda_{Q}}\Bigr{)}^{\gamma_{Q}=\frac{3N_{c}-N_{F}}{N_{F}}\,>\,0}\ll
1\,,$ $\displaystyle{\mathcal{W}}_{Q}=\frac{m_{Q}}{z_{Q}(\Lambda_{Q},\mu)}{\rm
Tr}\Bigl{(}{\overline{Q}_{R}}Q_{R}\Bigr{)}-\frac{1}{2\mu_{\Phi}z^{2}_{Q}(\Lambda_{Q},\mu)}\Biggl{(}{\rm
Tr}\,({\overline{Q}}_{R}Q_{R})^{2}-\frac{1}{N_{c}}\Bigl{(}{\rm
Tr}\,{\overline{Q}}_{R}Q_{R}\Bigr{)}^{2}\Biggr{)}.\,\,$ (3.1)

Because the quark renormalization factor $z_{Q}(\Lambda_{Q},\mu)$ decreases at
smaller scale $\mu$, it is seen from (3.1) that the role of the 4-quark term
$\sim({\overline{Q}}_{R}Q_{R})^{2}$ increases with lowering energy. Hence,
while it is irrelevant at the scale $\mu\sim\Lambda_{Q}$ because
$\mu_{\Phi}\gg\Lambda_{Q}$, the question is whether it becomes dynamically
relevant at some lower scale $\mu=\mu_{o}$. For this, we estimate the scale
$\mu_{o}$ where this term becomes relevant in the conformal regime of the
(effectively) massless theory of quarks and gluons, with
$z_{Q}(\Lambda_{Q},\mu\ll\Lambda_{Q})\sim(\mu/\Lambda_{Q})^{\gamma_{Q}}$ :

$\displaystyle\frac{\mu_{o}}{\mu_{\Phi}}\frac{1}{z^{2}_{Q}(\Lambda_{Q},\mu_{o})}\sim\frac{\mu_{o}}{\mu_{\Phi}}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{o}}\Bigr{)}^{2\gamma_{Q}}\sim
1\quad\rightarrow\quad\mu_{o}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{1}{(2\gamma_{Q}-1)}}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{F}}{3(2N_{c}-N_{F})}\,>\,0}\ll\Lambda_{Q}\,.$
(3.2)

We recall that even at those scales $\mu$ when the running mass of fions
$\mu_{\Phi}(\mu)=\mu_{\Phi}/z_{\Phi}(\Lambda_{Q},\mu)\gg\mu$ and so they are
too heavy and dynamically irrelevant, the quarks and gluons remain effectively
massless and active. Therefore, due to the Yukawa interactions of fions with
quarks, the loops of still active light quarks (and gluons interacting with
quarks) still induce the power-like running renormalization factor
$z_{\Phi}(\Lambda_{Q},\mu\ll\Lambda_{Q})\sim(\mu/\Lambda_{Q})^{\gamma_{\Phi}<0}\gg
1$ of fions at all those scales until quarks are effectively massless, i.e.
$\mu>m^{\rm pole}_{Q}$ (see below).

It seems clear that the physical reason why the $4$-quark terms in the
superpotential (3.1) become relevant at scales $\mu<\mu_{o}$ is that the fion
field $\Phi$ which was too heavy and so dynamically irrelevant at
$\mu>\mu_{o},\,\mu_{\Phi}(\mu>\mu_{o})>\mu$ , becomes effectively massless at
$\mu<\mu_{o},\,\mu_{\Phi}(\mu<\mu_{o})<\mu$ , and begins to participate in the
renormgroup evolution, i.e. it becomes relevant. In other words, the 4-quark
term in (3.1) ‘remembers’ about fions and signals about the scale below which
the fions become effectively massless, $\mu_{o}=\mu^{\rm pole}_{2}(\Phi)$.
This allows us to find the value of $z_{\Phi}(\Lambda_{Q},\mu>\mu_{o})$:

$\displaystyle\frac{\mu_{\Phi}}{z_{\Phi}(\Lambda_{Q},\mu_{o})}\sim\mu_{o}\quad\rightarrow\quad
z_{\Phi}(\Lambda_{Q},\mu_{o}<\mu<\Lambda_{Q})\sim\Bigl{(}\frac{\Lambda_{Q}}{\mu}\Bigr{)}^{2\gamma_{Q}}\gg
1\quad\rightarrow\quad\gamma_{\Phi}=-2\gamma_{Q}<0\,.$ (3.3)

Because the propagator of the renormalized fion field looks as
$1/(p^{2}-\,\mu^{2}_{\Phi}(p^{2}))$ and
$|\mu^{2}_{\Phi}(p^{2})|\lessgtr|p^{2}|$ at $p^{2}\lessgtr\mu^{2}_{o}$, where
$\mu_{o}\ll\Lambda_{Q}$ (3.2), it is clear that there is a pole in the fion
propagator at $p^{2}=\mu_{2}^{\rm
pole}(\Phi)=(\mu^{2}_{o}-i\mu_{o}\Gamma_{\Phi})$, i.e. there is a second
generation of all $N_{F}^{2}$ fields $\Phi^{i}_{j}$ (the first one is at
$\mu^{\rm pole}_{1}(\Phi)\gg\Lambda_{Q}$).

It can be shown that the conformal regime remains the same even at scales
$m^{\rm pole}_{Q}<\mu<\mu_{o}$ where fion fields became relevant, and the
quark and fion anomalous dimensions $\gamma_{Q}$ and $\gamma_{\Phi}$ remain
the same. I.e., the perturbative running mass
$\mu_{\Phi}(\mu)=\mu_{\Phi}/z_{\Phi}(\Lambda_{Q},\mu\ll\Lambda_{Q})\ll\Lambda_{Q}$
of fions continues to decrease quickly with diminishing $\mu$ at all scales
$m^{\rm pole}_{Q}<\mu<\Lambda_{Q}$ until quarks remain effectively massless,
and becomes frozen only at scales below the quark physical mass $m^{\rm
pole}_{Q}$, when the heavy quarks decouple.

However, if $m^{\rm pole}_{Q}>\mu_{o}$ , there is no pole in the fion
propagator at scales $\mu<\Lambda_{Q}$. The reason is that quarks decouple as
heavy at $\mu<m^{\rm pole}_{Q}$. And because $m^{\rm pole}_{Q}>\mu_{o}$, all
fions $\Phi^{i}_{j}$ remain too heavy and irrelevant at this scale. Then, at
$\mu<m^{\rm pole}_{Q}$, the running fion mass remains frozen at the large
value $\mu_{\Phi}(\mu=m^{\rm pole}_{Q}>\mu_{o})>m^{\rm pole}_{Q}$. The fions
remain then dynamically irrelevant and unobservable as resonances in this case
at all scales $\mu<\Lambda_{Q}$.

But when $m^{\rm pole}_{Q}\ll\mu_{o}$, there will be not only the second
generation of fions at $\mu=\mu^{\rm pole}_{2}(\Phi)$, but also a third
generation at $\mu=\mu^{\rm pole}_{3}(\Phi)\ll\mu^{\rm pole}_{2}(\Phi)$.
Indeed, after the heavy quarks decouple at the scale $m^{\rm
pole}_{Q}\ll\mu_{o}$ and the renormalization factor
$z_{\Phi}(\Lambda_{Q},m^{\rm pole}_{Q})$ of fions becomes frozen in the region
of scales where the fions already became relevant, the frozen value
$\mu_{\Phi}(\mu<m^{\rm pole}_{Q})=\mu_{\Phi}/z_{\Phi}(\Lambda_{Q},\mu=m^{\rm
pole}_{Q})$ of the fion mass is now: $\mu_{\Phi}(\mu=m^{\rm pole}_{Q})\ll
m^{\rm pole}_{Q}$. Therefore, there is one more pole in the fion propagator at
$\mu=\mu^{\rm pole}_{3}(\Phi)=\mu_{\Phi}(\mu=m^{\rm pole}_{Q})\ll m^{\rm
pole}_{Q}$.

On the whole, in a few words for the direct theory.
a) The fions remain dynamically irrelevant and there are no poles in the fion
propagator at scales $\mu<\Lambda_{Q}$ if $m^{\rm pole}_{Q}>\mu_{o}$.
b) If $m^{\rm
pole}_{Q}<\mu_{o}\sim\Lambda_{Q}\Bigl{(}\Lambda_{Q}/\mu_{\Phi}\Bigr{)}^{\frac{N_{F}}{3(2N_{c}-N_{F})}}\ll\Lambda_{Q}$,
there are two poles in the fion propagator at scales $\mu\ll\Lambda_{Q}$ :
$\mu^{\rm pole}_{2}(\Phi)\sim\mu_{o}$ and $\mu^{\rm
pole}_{3}(\Phi)=\mu_{\Phi}/z_{\Phi}(\Lambda_{Q},m^{\rm pole}_{Q})\ll\mu^{\rm
pole}_{2}(\Phi)$. In other words, the fions appear in three generations in
this case (we recall that there is always the largest pole mass of fions
$\mu^{\rm pole}_{1}(\Phi)\sim\mu_{\Phi}\gg\Lambda_{Q}$). Hence, the fions are
effectively massless and dynamically relevant in the range of scales $\mu^{\rm
pole}_{3}(\Phi)<\mu<\mu^{\rm pole}_{2}(\Phi)$.

Moreover, once the fions become relevant with respect to internal
interactions, they begin to contribute simultaneously to the external
anomalies ( the ’t Hooft triangles in the external background fields).

##  4  Mass spectra in $\rm br2$ vacua. Direct theory

${\rm\overline{b}}_{\rm o}/N_{F}=O(1),\,\,0<({\rm\overline{b}}_{\rm
o}-2n_{1})/N_{F}=O(1),\,\,\,\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,0}=\Lambda_{Q}(\Lambda_{Q}/m_{Q})^{(2N_{c}-N_{F})/N_{c}}$

The general scheme for calculations of mass spectra both in the direct and
dual theories looks as follows.

1) From the exact ${{\mathcal{W}}}_{\rm eff}$ in (2.1) the values of the quark
and gluino condensates at $\mu=\Lambda_{Q}$,
$\langle({\overline{Q}}Q)_{i}\rangle$ and $\langle S\rangle$, as well as
multiplicities of vacua, can be found in each vacuum.

2) From this and from the knowledge of all anomalous dimensions in the
conformal regime, all renormalization factors
$z_{i}(\Lambda_{Q},\mu\ll\Lambda_{Q})$ for all fields in the Kahler terms are
also known. Then the potentially possible values of pole masses of quarks,
$m^{\rm pole}_{Q}=\langle m_{Q}^{\rm tot}\rangle/z_{Q}(\Lambda_{Q},m^{\rm
pole}_{Q})$, or possible gluon pole masses $(\mu^{\rm pole}_{gl})^{2}\sim
z_{Q}(\Lambda_{Q},\mu_{gl})\langle{\overline{Q}}\rangle\langle Q\rangle$ for
higgsed quarks can be found (and, using the Konishi anomalies [4] and matching
$\langle M^{i}_{j}\rangle=\langle{\overline{Q}}_{j}Q^{i}\rangle,\,\langle
S\rangle=-\langle{\overline{S}}\rangle$ similarly in the dual theory).

3) The hierarchies between them determine then the realized phase states and
real mass spectra in each vacuum at given values of Lagrangian parameters.
E.g., if (see below) for dual quarks with $U(n_{1})$ flavors
${\overline{\mu}}^{\rm pole}_{\rm gl,1}\gg\mu^{\rm pole}_{q,1}$, then these
quarks are higgsed, i.e.
$\langle({\overline{q}}q)_{1}\rangle=\sum_{a=1}^{{\overline{N}}_{c}}\langle{\overline{q}}^{1}_{a}q^{a}_{1}\rangle=\langle{\overline{q}}^{1}_{1}\rangle\langle
q^{1}_{1}\rangle\sim m_{Q}\Lambda_{Q}$, and the dual color symmetry is broken:
$SU({\overline{N}}_{c})\rightarrow SU({\overline{N}}_{c}-n_{1})$. While if for
quarks ${\overline{Q}}^{a}_{1},Q^{1}_{a}$ in the direct theory $m^{\rm
pole}_{Q,1}\gg\mu^{\rm pole}_{gl,1}$, then these quarks decouple as heavy at
$\mu<m^{\rm pole}_{Q,i}$ and are not higgsed but confined. The confinement
originates from the unbroken color $SU(N_{c})\,\,{\cal N}=1$ supersymmetric YM
(SYM) with its only dimensional parameter $\langle\Lambda_{SYM}\rangle=\langle
S\rangle^{1/3}$, so that the string tension is
$\sigma^{1/2}\sim\langle\Lambda_{SYM}\rangle$.

From ${{\mathcal{W}}}_{\rm eff}$ in (2.1),(2.3) the condensates of quarks in
the direct theory look as ($m_{1}=m_{Q}N_{c}/(N_{c}-n_{2})$ ):

$\displaystyle\langle({\overline{Q}}Q)_{2}\rangle=m_{1}\mu_{\Phi}-\frac{N_{c}-n_{1}}{N_{c}-n_{2}}\langle({\overline{Q}}Q)_{1}\rangle,\,\,\langle({\overline{Q}}Q)_{1}\rangle\approx\Lambda_{Q}^{2}\Bigl{(}\frac{\mu_{\Phi}}{\Lambda_{Q}}\Bigr{)}^{\frac{n_{2}}{n_{2}-N_{c}}}\Bigl{(}\frac{m_{1}}{\Lambda_{Q}}\Bigr{)}^{\frac{N_{c}-n_{1}}{n_{2}-N_{c}}},\,\,\frac{\langle({\overline{Q}}Q)_{1}\rangle}{\langle({\overline{Q}}Q)_{2}\rangle}\sim\biggl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,0}}\biggr{)}^{\frac{N_{c}}{n_{2}-N_{c}}}\ll
1,\,\,\,\,$ (4.1)

in br2 - vacua with the spontaneous breaking $U(N_{F})\rightarrow
U(n_{1})\times U(n_{2})$, $n_{2}>N_{c}\,,1\leq n_{1}<{\overline{N}}_{c}$ . The
largest among the masses smaller than $\Lambda_{Q}$ are masses of $N_{F}^{2}$
second generation fions, see (3.2),

$\displaystyle\mu^{\rm
pole}_{2}(\Phi_{i}^{j})=\mu_{o}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{F}}{3(2N_{c}-N_{F})}}\ll\Lambda_{Q}\,,\quad
i,j=1...N_{F}\,,$ (4.2)

and all $N_{F}^{2}$ fions become dynamically relevant at scales $\mu<\mu_{o}$
(the cases when there are additional non-perturbative contributions to the
masses of fions have to be considered separately, see below).

Some other possible characteristic masses look in this vacuum as 222 Here and
below, $m^{\rm pole}_{Q,1},\,m^{\rm pole}_{Q,2}$ in the direct theory and
${\overline{\mu}}^{\rm pole}_{\rm gl,1},\,\mu^{\rm pole}_{\textsf{q},2}$ in
the dual one are the pure perturbative pole masses of quarks or gluons, i.e.
ignoring confinement with the string tension
$\sigma^{1/2}\sim\langle\Lambda_{SYM}\rangle$.

$\displaystyle\langle m^{\rm
tot}_{Q,1}\rangle=\frac{\langle({\overline{Q}}Q)_{2}\rangle}{\mu_{\Phi}}\approx
m_{1}\,,\quad m^{\rm pole}_{Q,2}\ll m^{\rm pole}_{Q,1}=\frac{\langle m^{\rm
tot}_{Q,1}\rangle}{z_{Q}(\Lambda_{Q},m^{\rm
pole}_{Q,1})}\sim\Lambda_{Q}\Bigl{(}\frac{m_{1}}{\Lambda_{Q}}\Bigr{)}^{N_{F}/3N_{c}}\ll\mu^{\rm
pole}_{2}(\Phi_{i}^{j})\,,$ (4.3) $\displaystyle\mu_{\rm gl,2}^{2}\sim
z_{Q}(\Lambda_{Q},\mu_{\rm
gl,2})\langle({\overline{Q}}Q)_{2}\rangle\gg\mu_{\rm
gl,1}^{2},\,\,z_{Q}(\Lambda_{Q},\mu_{\rm gl,2})\sim\Bigl{(}\frac{\mu_{\rm
gl,2}}{\Lambda_{Q}}\Bigr{)}^{\frac{3N_{c}-N_{F}}{N_{F}}}\ll
1\,\rightarrow\,\mu_{\rm gl,2}\sim\langle\Lambda_{SYM}\rangle\ll m^{\rm
pole}_{Q,1},\,$ (4.4)

where $m^{\rm pole}_{Q,1}$ and $m^{\rm pole}_{Q,2}$ are the pole masses of
quarks ${\overline{Q}}_{1},Q^{1}$ and ${\overline{Q}}_{2},Q^{2}$ and $\mu_{\rm
gl,1},\,\mu_{\rm gl,2}$ are the gluon masses due to possible higgsing of these
quarks. Hence, the largest mass is $m^{\rm pole}_{Q,1}$. The overall phase is:
all heavy quarks, i.e. not higgsed but confined,
$\langle{\overline{Q}}_{1}\rangle=\langle
Q^{1}\rangle=\langle{\overline{Q}}_{2}\rangle=\langle Q^{2}\rangle=0$.

After the heaviest quarks $Q^{1},\,{\overline{Q}}_{1}$ decoupled at
$\mu<m^{\rm pole}_{Q,1}$, the lower energy theory has $N_{c}$ colors and
$N_{F}^{\prime}=n_{2}>N_{c}$ flavors of still active lighter quarks
${\overline{Q}}_{2},Q^{2}$. In the range of scales $m^{\rm
pole}_{Q,2}<\mu<m^{\rm pole}_{Q,1}$ it will remain in the conformal regime at
$2n_{1}<\,{\rm\overline{b}}_{\rm o},\,\,{\rm\overline{b}}_{\rm
o}=(3{\overline{N}}_{c}-N_{F})>0$, while it will be not in the conformal but
in the strong coupling regime at $2n_{1}>{\rm\overline{b}}_{\rm o}$, with the
gauge coupling $a(\mu\ll m^{\rm pole}_{Q,1})=(m^{\rm
pole}_{Q,1}/\mu)^{\nu\,>\,0}\gg 1$. We do not consider the strong coupling
regime here and for this reason we consider $2n_{1}<{\rm\overline{b}}_{\rm o}$
only.

It follows from the exact ${{\mathcal{W}}}_{\rm eff}$ in (2.1) that the flavor
symmetry is broken spontaneously in these br2 vacua as $U(N_{F})\rightarrow
U(n_{1})\times U(n_{2})$. It follows then from this that quarks
${\overline{Q}}_{2},Q^{2}$ are not higgsed. If they were higgsed, then
$U(n_{2})$ would be further broken spontaneously due to the rank restriction
because $n_{2}>N_{c}$, this would contradict the exact (2.1). Therefore the
quarks ${\overline{Q}}_{2},Q^{2}$ in this case are not higgsed but confined.

In the lower energy theory at $\mu<m^{\rm pole}_{Q,1}$ the pole mass of quarks
${\overline{Q}}_{2},Q^{2}$ looks as

$\displaystyle m^{\rm pole}_{Q,2}=\frac{m^{\rm
pole}_{Q,1}}{z^{\,\prime}_{Q}(m^{\rm pole}_{Q,1},m^{\rm
pole}_{Q,2})}\Biggl{(}\,\frac{\langle({\overline{Q}}Q)_{1}\rangle}{\langle({\overline{Q}}Q)_{2}\rangle}\Biggr{)}\sim(\rm
several)\Lambda_{SYM},\quad z^{\,\prime}_{Q}(m^{\rm pole}_{Q,1},m^{\rm
pole}_{Q,2})\sim\Bigl{(}\frac{m^{\rm pole}_{Q,2}}{m^{\rm
pole}_{Q,1}}\Bigr{)}^{\frac{3N_{c}-n_{2}}{n_{2}}}\ll 1\,.$ (4.5)

Hence, after integrating out as heavy the quarks ${\overline{Q}}_{1},Q^{1}$ at
$\mu<m^{\rm pole}_{Q,1}$ and then quarks ${\overline{Q}}_{2},Q^{2}$ and
$SU(N_{c})$ gluons at $\mu<\langle\Lambda_{SYM}\rangle$ (these last through
the Veneziano - Yankielowicz procedure [5]), the Lagrangian of fions looks as,
see (4.5),

$\displaystyle K=z_{\Phi}(\Lambda_{Q},m^{\rm pole}_{Q,1})\,{\rm
Tr}\,\Bigl{[}\,(\Phi_{1}^{1})^{\dagger}\Phi_{1}^{1}+(\Phi_{1}^{2})^{\dagger}\Phi_{1}^{2}+(\Phi_{2}^{1})^{\dagger}\Phi_{2}^{1}+z^{\,\prime}_{\Phi}(m^{\rm
pole}_{Q,1},m^{\rm
pole}_{Q,2})(\Phi_{2}^{2})^{\dagger}\Phi_{2}^{2}\,\Bigr{]}\,,$ (4.6)
$\displaystyle z_{\Phi}(\Lambda_{Q},m^{\rm
pole}_{Q,1})\sim\Bigl{(}\frac{\Lambda_{Q}}{m^{\rm
pole}_{Q,1}}\Bigr{)}^{\frac{2(3N_{c}-N_{F})}{N_{F}}}\gg
1,\quad{\mathcal{W}}=N_{c}S+{\mathcal{W}}_{\Phi},\quad{\mathcal{W}}_{\Phi}=\frac{\mu_{\Phi}}{2}\Bigl{(}{\rm
Tr}\,(\Phi^{2})-\frac{1}{{\overline{N}}_{c}}({\rm Tr}\,\Phi)^{2}\,\Bigr{)}\,,$
(4.7) $\langle m^{\rm
tot}_{Q,1}\rangle=\frac{\langle({\overline{Q}}Q)_{2}\rangle}{\mu_{\Phi}}\,,\quad\langle
m^{\rm
tot}_{Q,2}\rangle=\frac{\langle({\overline{Q}}Q)_{1}\rangle}{\mu_{\Phi}}\,,\quad
m^{\rm tot}_{Q}=(m_{Q}-\Phi)\,,\quad S=\Bigl{(}\Lambda_{Q}^{{\rm b_{o}}}\det
m^{\rm tot}_{Q}\Bigr{)}^{1/N_{c}}\,,$ $\langle\Lambda_{SYM}\rangle^{3}=\langle
S\rangle=\Bigl{(}\Lambda_{Q}^{{\rm b_{o}}}\det\langle m^{\rm
tot}_{Q}\rangle\Bigr{)}^{1/N_{c}}\approx\Lambda_{Q}^{3}\Bigl{(}\frac{\mu_{\Phi}}{\Lambda_{Q}}\Bigr{)}^{\frac{n_{2}}{n_{2}-N_{c}}}\Bigl{(}\frac{m_{1}}{\Lambda_{Q}}\Bigr{)}^{\frac{n_{2}-n_{1}}{n_{2}-N_{c}}},\quad
m_{1}=\frac{N_{c}m_{Q}}{N_{c}-n_{2}}.$

From (4.6),(4.7), the main contribution to the mass of $n_{1}^{2}$ third
generation fions $\Phi_{1}^{1}$ gives the term
$\sim\mu_{\Phi}(\Phi_{1}^{1})^{2}$,

$\displaystyle\mu^{\rm
pole}_{3}(\Phi_{1}^{1})=\frac{\mu_{\Phi}}{z_{\Phi}(\Lambda_{Q},m^{\rm
pole}_{Q,1})}\sim\mu_{\Phi}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{2(3N_{c}-N_{F})}{3N_{c}}}\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,0}}\Bigr{)}^{\frac{({\rm\overline{b}}_{\rm
o}-2n_{1})}{3(n_{2}-N_{c})}\,>\,0}\langle\Lambda_{SYM}\rangle\ll\langle\Lambda_{SYM}\rangle\,.$
(4.8)

As for $n_{2}^{2}$ third generation fions $\Phi_{2}^{2}$, the main
contribution to their masses comes from the non-perturbative term $\sim S$ in
the superpotential (4.7)

$\displaystyle\mu^{\rm pole}_{3}(\Phi_{2}^{2})=\frac{\langle S\rangle}{\langle
m^{\rm tot}_{Q,2}\rangle^{2}}\frac{1}{z_{\Phi}(\Lambda_{Q},m^{\rm
pole}_{Q,1})z^{\,\prime}_{\Phi}(m^{\rm pole}_{Q,1},m^{\rm pole}_{Q,2})}\sim
m^{\rm pole}_{Q,2}\sim\langle\Lambda_{SYM}\rangle.$ (4.9)

$2n_{1}n_{2}$ third generation hybrid fions $\Phi_{1}^{2},\Phi_{2}^{1}$ are
massless: $\mu^{\rm pole}_{3}(\Phi_{1}^{2})=\mu^{\rm
pole}_{3}(\Phi_{2}^{1})=0$, they are Nambu-Goldstone particles of the
spontaneously broken global flavor symmetry: $U(N_{F})\rightarrow
U(n_{1})\times U(n_{2})$.

##  5  Mass spectra in $\rm br2$ vacua. Dual theory

${\rm\overline{b}}_{\rm o}/N_{F}=O(1),\,\,0<({\rm\overline{b}}_{\rm
o}-2n_{1})/N_{F}=O(1),\,\,\,\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,0}=\Lambda_{Q}(\Lambda_{Q}/m_{Q})^{(2N_{c}-N_{F})/N_{c}}$

In these vacua with $n_{2}>N_{c}\,,1\leq n_{1}<{\overline{N}}_{c}$, using the
Konishi anomalies [4] and matching $\langle
M^{i}_{j}\rangle=\langle{\overline{Q}}_{j}Q^{i}\rangle,\,\langle
S\rangle=-\langle{\overline{S}}\rangle$, see also (4.7), the condensates of
mions and dual quarks look at $\mu=\Lambda_{Q}$ as:

$\displaystyle\langle M_{2}\rangle=\langle({\overline{Q}}Q)_{2}\rangle\approx
m_{1}\mu_{\Phi},\,\,\langle
M_{1}\rangle=\langle({\overline{Q}}Q)_{1}\rangle\approx\Lambda_{Q}^{2}\Bigl{(}\frac{\mu_{\Phi}}{\Lambda_{Q}}\Bigr{)}^{\frac{n_{2}}{n_{2}-N_{c}}}\Bigl{(}\frac{m_{1}}{\Lambda_{Q}}\Bigr{)}^{\frac{N_{c}-n_{1}}{n_{2}-N_{c}}},\,\,\frac{\langle
M_{1}\rangle}{\langle
M_{2}\rangle}\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,0}}\Bigr{)}^{\frac{N_{c}}{n_{2}-N_{c}}}\ll
1,\,\,$ (5.1)
$\langle\Lambda_{SYM}\rangle^{3}=-\langle{\overline{S}}\rangle=\frac{\langle({\overline{Q}}Q)_{1}\rangle\langle({\overline{Q}}Q)_{2}\rangle}{\mu_{\Phi}}=\frac{\langle
M_{1}\rangle\langle M_{2}\rangle}{\mu_{\Phi}},\,\,\langle
N_{1}\rangle=\langle({\overline{q}}q)_{1}\rangle=\frac{\Lambda_{Q}\langle
S\rangle}{\langle M_{1}\rangle}=\frac{\Lambda_{Q}\langle
M_{2}\rangle}{\mu_{\Phi}}\approx
m_{1}\Lambda_{Q}\gg\langle{(\overline{q}}q)_{2}\rangle.$

From these and (1.5), the heaviest are $N_{F}^{2}$ mions $M^{i}_{j}$ with the
pole masses

$\displaystyle\mu^{\rm
pole}(M^{i}_{j})=\frac{\Lambda_{Q}^{2}/\mu_{\Phi}}{z_{M}(\Lambda_{Q},\mu^{\rm
pole}(M))}\sim\Lambda_{Q}\Bigl{(}\frac{\Lambda_{Q}}{\mu_{\Phi}}\Bigr{)}^{\frac{N_{F}}{3(2N_{c}-N_{F})}}\sim\mu^{\rm
pole}_{2}(\Phi_{i}^{j})\gg{\overline{\mu}}^{\rm pole}_{\rm gl,1}\,,$ (5.2)
$z_{M}(\Lambda_{Q},\mu^{\rm pole}(M))\sim\Bigl{(}\frac{\mu^{\rm
pole}(M)}{\Lambda_{Q}}\Bigr{)}^{\gamma_{M}=-2\gamma_{q}=-2\frac{3{\overline{N}}_{c}-N_{F}}{N_{F}}}\gg
1\,,$

while some other possible characteristic masses look as

$\displaystyle\Bigl{(}{\overline{\mu}}^{\rm pole}_{\rm gl,1}\Bigr{)}^{2}\sim
z_{q}(\Lambda_{Q},{\overline{\mu}}^{\rm pole}_{\rm
gl,1})\langle{\overline{q}}^{1}_{1}\rangle\langle
q^{1}_{1}\rangle,\quad{\overline{\mu}}^{\rm pole}_{\rm
gl,1}\sim\Lambda_{Q}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{N_{F}}{3N_{c}}}\sim
m^{\rm pole}_{Q,1}\gg{\overline{\mu}}^{\rm pole}_{\rm
gl,2},\quad{\overline{\mu}}^{\rm pole}_{\rm gl,1}\gg\mu^{\rm
pole}_{q,2}\gg\mu^{\rm pole}_{q,1},\,\,$ (5.3)

where ${\overline{\mu}}^{\rm pole}_{\rm gl,1,2}$ are the gluon masses due to
possible higgsing of these quarks. Hence, the largest mass is
${\overline{\mu}}_{\rm gl,1}$ and the overall phase is ${\bf
Higgs_{1}-Hq_{2}}$ (i.e. higgsed quarks $q_{1}$ and confined quarks
$\textsf{q}_{2}$ with non-higgsed colors). The quarks
${\overline{\textsf{q}}}^{2},\,\textsf{q}_{2}$ with the $U(n_{2}>N_{c})$
flavor symmetry are not higgsed due to the same rank restriction as the quarks
${\overline{Q}}_{2},\,Q^{2}$ of the direct theory.

After integrating out all massive gluons and their scalar superpartners, the
dual Lagrangian at $\mu={\overline{\mu}}_{{\rm gl},\,1}$ looks as

$\displaystyle K=z_{M}(\Lambda_{Q},{\overline{\mu}}_{{\rm gl},\,1}){\rm
Tr}\,\frac{M^{\dagger}M}{\Lambda_{Q}^{2}}+z_{q}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1}){\rm Tr}\,\Bigl{[}\,2\sqrt{(N_{1}^{1})^{\dagger}N_{1}^{1}}+K_{\rm
hybr}+\Bigl{(}(\textsf{q}_{2})^{\dagger}\textsf{q}_{2}+({\textsf{q}}_{2}\rightarrow{\overline{\textsf{q}}}^{\,2})\Bigr{)}\,\Bigr{]}\,,\,\,$
(5.4) $K_{\rm
hybr}=\Biggl{(}(N_{1}^{2})^{\dagger}\frac{1}{\sqrt{N_{1}^{1}(N_{1}^{1})^{\dagger}}}N_{1}^{2}+(N_{1}^{2}\rightarrow
N_{2}^{1})\Biggr{)},\,\,z_{q}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})\sim\Bigl{(}\frac{{\overline{\mu}}_{{\rm
gl},\,1}}{\Lambda_{Q}}\Bigr{)}_{,}^{\frac{{\rm\overline{b}}_{\rm
o}}{N_{F}}}\ll 1,\,\,z_{M}=z^{-2}_{q},\,\,{\rm\overline{b}}_{\rm
o}=3{\overline{N}}_{c}-N_{F},$ $\displaystyle{\overline{\cal
W}}=\Bigl{[}-\frac{2\pi}{{\overline{\alpha}}(\mu)}{\overline{\textsf{S}}}\Bigr{]}-{\rm
Tr}\,\Bigl{(}{\overline{\textsf{q}}}^{2}\frac{M_{2}^{2}}{\Lambda_{Q}}{\textsf{q}}_{2}\Bigr{)}+{\mathcal{W}}_{MN}+{\mathcal{W}}_{M},\,\,{\mathcal{W}}_{M}=m_{Q}{\rm
Tr}\,M-\frac{1}{2\mu_{\Phi}}\Biggl{[}{\rm Tr}\,(M^{2})-\frac{1}{N_{c}}({\rm
Tr}\,M)^{2}\Biggr{]},\,\,$ (5.5)
${\mathcal{W}}_{MN}=\frac{-1}{\Lambda_{Q}}\,{\rm
Tr}\,\Bigl{(}M_{1}^{1}N_{1}^{1}+M_{2}^{1}N_{1}^{2}+M_{1}^{2}N_{2}^{1}+M_{2}^{2}N_{2}^{1}\frac{1}{N_{1}^{1}}N_{1}^{2}\Bigr{)},\,\,N^{1}_{2}=(\langle{\overline{q}}^{1}\rangle
q_{2}),\,\,N^{2}_{1}=({\overline{q}}^{2}\langle q_{1}\rangle),$

where $n_{1}^{2}$ nions (dual pions) $N_{1}^{1}$ originate from higgsing of
${\overline{q}}^{1},q_{1}$ dual quarks, while the hybrid nions $N_{1}^{2}$ and
$N_{2}^{1}$ are, in essence, the dual quarks ${\overline{q}}^{2}$ and $q_{2}$
with higgsed colors. ${\overline{\textsf{q}}}^{2},\textsf{q}_{2}$ are still
active quarks ${\overline{q}}^{2},\,q_{2}$ with non-higgsed colors.
$\overline{\textsf{S}}$ is the field strength squared of remained light dual
$SU({\overline{N}}_{c}-n_{1})$ gluons.

The lower energy theory at $\mu<{\overline{\mu}}_{{\rm gl},\,1}$ has
$({\overline{N}}_{c}-n_{1})$ colors and $n_{2}>N_{c}$ flavors,
$0<{\rm\overline{b}}_{\rm o}^{\,\prime}=({\rm\overline{b}}_{\rm
o}-2n_{1})\,<\,{\rm\overline{b}}_{\rm o}$. We consider here only the case
${\rm\overline{b}}_{\rm o}^{\,\prime}>0$ when it remains in the conformal
window. The fields $N_{1}^{1},N_{1}^{2},N_{2}^{1}$ and
$M_{1}^{1},M_{1}^{2},M_{2}^{1}$ are frozen and do not evolve at
$\mu<{\overline{\mu}}_{{\rm gl},\,1}$, while the value of the pole mass
$\mu^{\rm pole}_{q,2}$ in this lower energy theory is

$\displaystyle\mu^{\rm pole}_{\textsf{q},2}=\frac{\langle
M_{2}\rangle}{\Lambda_{Q}}\frac{1}{z_{q}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})z^{\,\prime}_{q}({\overline{\mu}}_{{\rm gl},\,1},\mu^{\rm
pole}_{\textsf{q},2})}\sim(\rm several)\langle\Lambda_{SYM}\rangle\,,\quad
z^{\,\prime}_{q}({\overline{\mu}}_{{\rm gl},\,1},\mu^{\rm
pole}_{\textsf{q},2})\sim\Bigl{(}\frac{\mu^{\rm
pole}_{\textsf{q},2}}{{\overline{\mu}}_{{\rm
gl},\,1}}\Bigr{)}^{{\rm\overline{b}}_{\rm o}^{\,\prime}/n_{2}}\ll 1\,.$ (5.6)

Finally, after integrating out remained non-higgsed (but confined) quarks
${\overline{\textsf{q}}}^{2},\textsf{q}_{2}$ (confinement originates in this
case from the $SU({\overline{N}}_{c}-n_{1})\,\,{\cal N}=1$ SYM sector with its
scale factor $\langle\Lambda_{SYM}\rangle$) as heavy ones and then ${\cal
N}=1$ $SU({\overline{N}}_{c}-n_{1})$ SYM gluons at
$\mu<\langle\Lambda_{SYM}\rangle$ (these last through the Veneziano -
Yankielowicz procedure [5]), the lowest energy Lagrangian of mions and nions
looks as, see (5.4),

$\displaystyle K=z_{M}(\Lambda_{Q},{\overline{\mu}}_{{\rm gl},\,1}){\rm
Tr}\,K_{M}+z_{q}(\Lambda_{Q},{\overline{\mu}}_{{\rm gl},\,1}){\rm
Tr}\,\Bigl{[}\,2\sqrt{(N_{1}^{1})^{\dagger}N_{1}^{1}}+K_{\rm
hybr}\,\Bigr{]},\,\,\,\,$ (5.7)
$K_{M}=\frac{1}{\Lambda_{Q}^{2}}\Bigl{(}(M_{1}^{1})^{\dagger}M_{1}^{1}+(M_{1}^{2})^{\dagger}M_{1}^{2}+(M_{2}^{1})^{\dagger}M_{2}^{1}+z^{\,\prime}_{M}({\overline{\mu}}_{{\rm
gl},\,1},\mu^{\rm
pole}_{\textsf{q},2})(M_{2}^{2})^{\dagger}M_{2}^{2}\Bigr{)},\,\,z^{\,\prime}_{M}({\overline{\mu}}_{{\rm
gl},\,1},\mu^{\rm
pole}_{\textsf{q},2})\sim\Bigl{(}\frac{{\overline{\mu}}_{{\rm
gl},\,1}}{\mu^{\rm
pole}_{\textsf{q},2}}\Bigr{)}^{\frac{2\,{\rm\overline{b}}_{\rm
o}^{\,\prime}}{n_{2}}}\gg 1,$
${\mathcal{W}}=-{\overline{N}}_{c}^{\,\prime}\,{\cal
S}+{\mathcal{W}}_{MN}+{\mathcal{W}}_{M},\,\,{\cal
S}=\langle\Lambda_{SYM}\rangle^{3}\Biggl{(}\det\frac{\langle
N_{1}\rangle}{N_{1}^{1}}\det\frac{M_{2}^{2}}{\langle
M_{2}\rangle}\Biggr{)}^{1/{\overline{N}}_{c}^{\,\prime}},\,\,{\overline{N}}_{c}^{\,\prime}=({\overline{N}}_{c}-n_{1}),\,\,\langle\Lambda_{SYM}\rangle^{3}\approx
m_{1}\langle M_{1}\rangle.$

From (5.7), the ”masses” of mions at the low scale look as

$\displaystyle\mu_{low}(M_{1}^{1})\sim\mu_{low}(M_{1}^{2})\sim\mu_{low}(M_{2}^{1})\sim\frac{\Lambda_{Q}^{2}}{z_{M}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})\mu_{\Phi}}\sim\Bigl{(}\frac{\mu_{\Phi,0}}{\mu_{\Phi}}\Bigr{)}{\overline{\mu}}_{{\rm
gl},\,1}\gg{\overline{\mu}}_{{\rm
gl},\,1}\,,\quad\frac{\mu_{low}(M_{1}^{1})}{\mu^{\rm pole}(M)}\ll 1\,,$ (5.8)
$\displaystyle\mu_{low}(M_{2}^{2})\sim\frac{\Lambda_{Q}^{2}}{z_{M}(\Lambda_{Q},{\overline{\mu}}_{{\rm
gl},\,1})z^{\prime}_{M}({\overline{\mu}}_{{\rm gl},\,1},\mu^{\rm
pole}_{\textsf{q},2})\mu_{\Phi}}\sim\Bigl{(}\frac{\mu_{\Phi,0}}{\mu_{\Phi}}\Bigr{)}^{\frac{3N_{c}-n_{2}}{3(n_{2}-N_{c})}}\,{\overline{\mu}}_{{\rm
gl},\,1}\gg{\overline{\mu}}_{{\rm
gl},\,1}\,,\quad\frac{\mu_{low}(M_{2}^{2})}{\mu_{low}(M_{1}^{1})}\ll 1\,,$
(5.9)

while the pole masses of nions $N_{1}^{1}$ are

$\displaystyle\mu^{\rm
pole}(N_{1}^{1})\sim\mu_{\Phi}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{2(3N_{c}-N_{F})}{3N_{c}}}\sim\mu_{3}^{\rm
pole}(\Phi^{1}_{1})\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,0}}\Bigr{)}^{\frac{({\rm\overline{b}}_{\rm
o}-2n_{1})}{3(n_{2}-N_{c})}\,>\,0}\langle\Lambda_{SYM}\rangle\ll\langle\Lambda_{SYM}\rangle\,.$
(5.10)

$2n_{1}n_{2}$ hybrid nions $N_{1}^{2},N_{2}^{1}$ are massless: $\mu^{\rm
pole}(N_{1}^{2})=\mu^{\rm pole}(N_{2}^{1})=0$, they are Nambu-Goldstone
particles of the spontaneously broken global flavor symmetry:
$U(N_{F})\rightarrow U(n_{1})\times U(n_{2})$.

The large mion ”masses” (5.8),(5.9) are not their pole masses but simply the
frozen low energy values of their running masses. The reason is that all
$N_{F}^{2}$ mion fields $M^{i}_{j}$ are light and dynamically relevant only at
scales $\mu^{\rm pole}(M)<\mu<\Lambda_{Q}$, see (5.2). They become too heavy,
dynamically irrelevant and decouple at scales $\mu<\mu^{\rm pole}(M)$.
Nevertheless, their renormalization factors continue to grow with diminished
energy due to couplings with lighter dual quarks. They become frozen for
$M_{1}^{1},M_{1}^{2},M_{2}^{1}$ only at $\mu<{\overline{\mu}}_{{\rm gl},\,1}$
after the quarks ${\overline{q}}^{1},q_{1}$ are higgsed, and at $\mu<\mu^{\rm
pole}_{\textsf{q},2}$ for $M_{2}^{2}$ after the quarks
$\overline{\textsf{q}}^{2},\textsf{q}_{2}$ decouple as heavy. The only pole
masses of all $N_{F}^{2}$ mions $M^{i}_{j}$ are $\mu^{\rm
pole}(M)\sim\Lambda_{Q}\Bigl{(}\Lambda_{Q}/\mu_{\Phi}\Bigr{)}^{N_{F}/3(2N_{c}-N_{F})}$
in (5.2).

##  6  Conclusions

A). The qualitatively new phenomenon was found in the direct theory due to the
strong power-like renormgroup evolution in the conformal regime. - The
seemingly heavy and dynamically irrelevant $N_{F}^{2}$ fion fields
$\Phi_{i}^{j}$ ‘return back’ and there appear two additional generations of
light $\Phi$-particles with small masses $\mu_{3}^{\rm
pole}(\Phi)\ll\mu_{2}^{\rm pole}(\Phi)\ll\Lambda_{Q}$. Moreover, the third
generation fields $\Phi^{1}_{2}$ and $\Phi^{2}_{1}$ are massless, they are
Nambu-Goldstone particles of the spontaneously broken global flavor symmetry
$U(N_{F})\rightarrow U(n_{1})\times U(n_{2})$.

B). Let us compare now the mass spectra (for particle masses
$M_{k}<\Lambda_{Q}$) in the direct theory and in Seiberg’s dual one at
$3N_{c}/2<N_{F}<2N_{c}$ and
$\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,0}=\Lambda_{Q}(\Lambda_{Q}/m_{Q})^{(2N_{c}-N_{F})/N_{c}}$.

Part I : Mass spectra at $0<{\rm\overline{b}}_{\rm
o}/N_{F}=O(1)\,,\,0<({\rm\overline{b}}_{\rm o}-2n_{1})/N_{F}=O(1)$

1) The largest masses $\mu_{2}^{\rm
pole}(\Phi^{i}_{j})\sim\mu_{o}\sim\Lambda_{Q}(\Lambda_{Q}/\mu_{\Phi})^{N_{F}/3(2N_{c}-N_{F})}$
in the direct theory have $N_{F}^{2}$ second generation scalar fion
superfields $\Phi^{i}_{j}$ (4.2), and $N_{F}^{2}$ scalar mion superfields
$M^{i}_{j}$ (5.2) with parametrically the same pole masses in the dual one
(here and below: up to possible constant factors independent of $m_{Q}$ and
$\mu_{\Phi}$ which are hard to control). Therefore, these two sets look
undistinguishable (with our accuracy).

It is also worth noting that when all $N_{F}^{2}$ fion fields $\Phi^{i}_{j}$
become relevant at $\mu<\mu_{o}$ in the direct theory, then all $N_{F}^{2}$
mion fields $M^{i}_{j}$ become irrelevant in the dual one (and vice versa at
$\mu>\mu_{o}$).

2) The next scale is $m_{Q,1}^{\rm pole}\sim{\overline{\mu}}_{{\rm
gl},\,1}^{\rm
pole}\sim\Lambda_{Q}(m_{Q}/\Lambda_{Q})^{N_{F}/3N_{c}}\ll\mu_{2}^{\rm
pole}(\Phi^{i}_{j})$, (4.3),(5.3). Because all quarks with $n_{1}$ and $n_{2}$
flavors are confined in the direct theory and $m_{Q,1}^{\rm pole}\gg
m_{Q,2}^{\rm pole}$, there are e.g. : a) many flavored quarkonia with
different spins, with this scale of masses, made either from quarks
${\overline{Q}}_{1},\,Q^{1}$ with $n_{1}$ flavors or e.g. from
${\overline{Q}}_{1}$ and $Q^{2}$ quarks. On the other hand, in the dual theory
with higgsed (i.e. not confined but screened) ${\overline{q}}^{1}$ and $q_{1}$
dual quarks with $SU(n_{1})$ dual colors and with such scale of masses, there
are e.g. only fixed numbers of equal mass bosons with fixed quantum numbers:
$n_{1}(2{\overline{N}}_{c}-n_{1})$ massive dual gluons and the same number of
their scalar superpartners.

Therefore, the mass spectra at this scale are clearly distinguishable in the
direct and dual theories.

3) The next scale is $m_{Q,2}^{\rm pole}\sim\mu^{\rm
pole}_{\textsf{q},2}\sim\mu^{\rm
pole}_{3}(\Phi_{2}^{2})\sim\langle\Lambda_{SYM}\rangle\ll m_{Q,1}^{\rm pole}$,
(4.5),(4.9),(5.6). There are many gluonia in both direct and dual theories
with such scale of masses and it seems these can be undistinguishable.
Besides, there are e.g. many flavored quarkonia with different spins, with
masses of this scale, made from confined quarks ${\overline{Q}}_{2},Q^{2}$
quarks in the direct theory and from confined quarks
$\overline{\textsf{q}}^{2},\textsf{q}_{2}$ in the dual one. These two sets of
quarkonia can also be undistinguishable. But there are additionally
$(n_{2}^{2}-1)$ elementary $SU(n_{2})$ adjoint scalar superfields
$\Phi^{2}_{2}$ with this scale of masses in the direct theory. And supposing
that the number of scalar quarkonia $({\overline{Q}}_{2}Q^{2})$ and
$(\overline{\textsf{q}}^{2}\textsf{q}_{2})$ is the same in the direct and dual
theories, these extra $(n_{2}^{2}-1)$ elementary scalars $\Phi_{2}^{2}$ will
distinguish these two theories.

4) And finally for particles with nonzero masses, there are $n_{1}^{2}$ (i.e.
$(n_{1}^{2}-1)\,\,\,SU(n_{1})$ flavor adjoints plus one singlet) third
generation lightest elementary scalar fields $(\Phi^{\rm
pole}_{3})_{i}^{j},\,i,j=1...n_{1}$ with $\mu^{\rm
pole}_{3}(\Phi_{1}^{1})\ll\langle\Lambda_{SYM}\rangle$ in the direct theory
and the same number and the same (up to possible factors $O(1)$ ) mass dual
pions (nions) $N^{i}_{j},\,i,j=1...n_{1}$ in the dual one, (4.8),(5.10). These
two sets look undistinguishable (with our accuracy).

5) $2n_{1}n_{2}$ fion fields $\Phi^{1}_{2}$ and $\Phi^{2}_{1}$ of the third
generation in the direct theory, and the same number of nions (dual pions)
$N^{1}_{2}$ and $N^{2}_{1}$ in the dual theory have the same quantum numbers
and are all massless, they are Nambu-Goldstone particles of the spontaneously
broken global flavor symmetry $U(N_{F})\rightarrow U(n_{1})\times U(n_{2})$.
These two sets are clearly undistinguishable.

On the whole, the mass spectra of the direct and dual theories in this region
of the Lagrangian parameters are different (this is especially clearly seen in
the point ‘2’), in disagreement with the Seiberg hypothesis about equivalence
of such two theories.

Part II : Mass spectra at $0<{\rm\overline{b}}_{\rm o}/N_{F}\ll
1\,,\,0<(2n_{1}-{\rm\overline{b}}_{\rm o})/N_{F}\approx 2n_{1}/N_{F}=O(1)$

There is now the additional small parameter $0<{\rm\overline{b}}_{\rm
o}/N_{F}\ll 1,\,{\rm\overline{b}}_{\rm
o}=(3{\overline{N}}_{c}-N_{F})=(2N_{F}-3N_{c})$, and this allows to see
parametric differences between mass spectra of the direct and dual theories.

At these values of parameters, the qualitative difference is that regimes at
$\mu<m_{Q,1}^{\rm pole}$ are not conformal now. The direct theory is in the
very strong coupling regime with $a(\mu\ll m_{Q,1}^{\rm pole})\gg 1$, while
the dual theory at $\mu^{\rm pole}_{\textsf{q},2}<\mu<{\overline{\mu}}^{\rm
pole}_{\rm gl,1}$ is in the weakly coupled infrared free logarithmic regime.
Not going into details, we note below only few qualitatively important points
and give some results.

i) In the direct theory. According to Seiberg’s view of the standard direct
(i.e. without fields $\Phi^{i}_{j}$) ${\cal N}=1$ SQCD at
$N_{c}+1<N_{F}<3N_{c}/2$, with the scale factor $\Lambda_{Q}$ of $SU(N_{c})$
gauge coupling 333 and the same at $\mu=m_{Q,1}^{\rm pole}$ for the direct
$\Phi$-theory considered here with $N_{F}\rightarrow
N^{\prime}_{F}=N_{F}-n_{1}=n_{2}$ and
$\Lambda_{Q}\rightarrow\Lambda^{\prime}=m_{Q,1}^{\rm pole}$ and direct quarks
with $m_{Q}=0$ (or with $m_{Q}\ll\Lambda_{Q}$), the regime of the direct
theory at $\mu<\Lambda_{Q}$ is in this case: ‘confinement without chiral
symmetry breaking’ (as far as small $m_{Q}\neq 0$ can be neglected). And the
dual theory is considered as the lower energy form of the direct theory. This
means that all direct quarks remained massless (or light), but hadrons made
from these massless (or light) quarks and direct gluons acquired large masses
$\sim\Lambda_{Q}$ due to mysterious confinement with the string tension
$\sigma^{1/2}\sim\Lambda_{Q}$, and decoupled at $\mu<\Lambda_{Q}$. Instead of
them, there mysteriously appeared massless (or light) composite solitons.
These last are particles of the dual theory.

This picture was questioned in [6] (see section 7 therein). It was argued
that, with the unbroken chiral flavor symmetry $SU(N_{F})_{L}\times
SU(N_{F})_{R}$ and unbroken R-charge, it is impossible to write at
$\mu\sim\Lambda_{Q}$ the nonsingular superpotential of the effective
Lagrangian of massive flavored hadrons with masses $\sim\Lambda_{Q}$ made from
direct massless (or light) quarks. 444 This is similar to our ordinary QCD
with confinement, with massless (or light) quarks but without chiral symmetry
breaking. It is impossible then e.g. to have in the effective hadron
Lagrangian at $\mu\sim\Lambda_{QCD}$ the massive nucleons with the mass
$\sim\Lambda_{QCD}$, as the term $\sim\Lambda_{QCD}{\overline{N}}N$ in the
potential is incompatible with the unbroken chiral symmetry. And the situation
in ${\cal N}=1$ SQCD is even more restrictive because the superpotential is
holomorphic and due to additional R-charge conservation.

We also recall here the following. There is no confinement in Yukawa-like
theories without gauge interactions. The confinement originates only from the
unbroken YM, or ${\cal N}=1$ SYM in ${\cal N}=1$ SQCD-like theories. And
because ${\cal N}=1$ SYM has only one dimensional parameter
$\langle\Lambda_{SYM}\rangle=\langle S\rangle^{1/3}$, the string tension is
$\sigma^{1/2}\sim\langle\Lambda_{SYM}\rangle$. But in the standard ${\cal
N}=1$ SQCD the value of $\Lambda_{SYM}$ is well known:
$\Lambda_{SYM}=(\Lambda_{Q}^{\rm b_{o}}\det m_{Q})^{1/3N_{c}}\ll\Lambda_{Q}$.
Therefore, such SYM cannot produce confinement with the string tension
$\sim\Lambda_{Q}$ (and there is no confinement at all at $m_{Q}\rightarrow
0$). 555 And the same for the direct SQCD-like $\Phi$-theory considered here:
$\langle\Lambda_{SYM}\rangle=(\Lambda_{Q}^{{\rm b_{o}}}\det\langle m^{\rm
tot}_{Q}\rangle)^{1/3N_{c}}\ll\Lambda^{\prime}=m_{Q,1}^{\rm pole}$. Therefore,
such SYM cannot produce confinement with $\sigma^{1/2}\sim m_{Q,1}^{\rm
pole}$, only with $\sigma^{1/2}\sim\langle\Lambda_{SYM}\rangle\ll m_{Q,1}^{\rm
pole}$.

For these reasons, we used below the picture described in section 7 of [6].
I.e., in our case here with ${\rm\overline{b}}_{\rm o}/N_{F}\ll 1$, after the
direct quarks ${\overline{Q}}_{1},Q^{1}$ decoupled as heavy at
$\mu<m_{Q,1}^{\rm pole}$, the remained direct theory with light $SU(N_{c})$
gluons and $n_{2}>N_{c}$ light quark flavors, $1<n_{2}/N_{c}<3/2$, enters
smoothly at lower energy into the perturbative (very) strong coupling regime
with $a(\mu\ll m_{Q,1}^{\rm pole})\sim(m_{Q,1}^{\rm
pole}/\mu)^{\nu=\frac{3N_{c}-2n_{2}}{n_{2}-N_{c}}}\gg 1$, and with all its
colored particles effectively massless at $m_{Q,2}^{\rm pole}<\mu<m_{Q,1}^{\rm
pole}$. (And NSVZ $\beta$-function [3] allows this). The anomalous dimension
of quarks ${\overline{Q}}_{2},Q^{2}$ in the range $m_{Q,2}^{\rm
pole}<\mu<m_{Q,1}^{\rm pole}$ is in this regime:
$\,\gamma^{\prime}_{Q,2}=(2N_{c}-n_{2})/(n_{2}-N_{c})>1$ [6, 8], while those
of $\Phi_{2}^{2}$ is $\gamma^{\prime}_{\Phi_{2}^{2}}=-2\gamma^{\prime}_{Q,2}$.
At $\mu<m_{Q,2}^{\rm pole}\sim(\mu_{\Phi}/\mu_{\Phi,1})m_{Q,1}^{\rm
pole}\gg\langle\Lambda_{SYM}\rangle$ the quarks ${\overline{Q}}_{2},Q^{2}$
decouple as heavy in the (very) strong coupling regime, and there remains
${\cal N}=1\,\,SU(N_{c})$ SYM with its scale factor
$\langle\Lambda_{SYM}\rangle$ determined from matching of couplings at
$\mu=m_{Q,2}^{\rm pole}$ :

$\Bigl{[}\,a_{+}(\mu=m_{Q,2}^{\rm pole})=\Bigl{(}\frac{m_{Q,1}^{\rm
pole}}{m_{Q,2}^{\rm
pole}}\Bigr{)}^{\nu=\frac{3N_{c}-2n_{2}}{n_{2}-N_{c}}}\,\Bigr{]}=\Bigl{[}\,a_{SYM}(\mu=m_{Q,2}^{\rm
pole})=\Bigl{(}\frac{m_{Q,2}^{\rm
pole}}{\langle\Lambda_{SYM}\rangle}\Bigr{)}^{3}\,\Bigr{]}\,\rightarrow$
$\rightarrow\,\langle\Lambda_{SYM}\rangle^{3}=\Lambda_{Q}^{3}\Bigl{(}\frac{\mu_{\Phi}}{\Lambda_{Q}}\Bigr{)}^{\frac{n_{2}}{n_{2}-N_{c}}}\Bigl{(}\frac{m_{1}}{\Lambda_{Q}}\Bigr{)}^{\frac{n_{2}-n_{1}}{n_{2}-N_{c}}}\,,$

as it should be, see (4.7).

ii) In the dual theory. This enters into the IR-free weakly coupled
logarithmic regime at $\mu^{\rm pole}_{\textsf{q},2}<\mu<{\overline{\mu}}^{\rm
pole}_{\rm gl,1}$, and the dual quarks
$\overline{\textsf{q}}^{2},\textsf{q}_{2}$ with $({\overline{N}}_{c}-n_{1})$
non-higgsed colors and $n_{2}>N_{c}$ flavors decouple as heavy at
$\mu<\mu^{\rm pole}_{\textsf{q},2}$. There remains ${\cal
N}=1\,\,SU({\overline{N}}_{c}-n_{1})$ SYM with the same scale factor
$\langle\Lambda_{SYM}\rangle\ll\mu^{\rm pole}_{\textsf{q},2}$.

The parameter $Z_{q}$ of the dual theory is exponentially small at
${\rm\overline{b}}_{\rm o}/N_{F}\ll 1$. Its value is determined from matching
at $\mu=\mu^{\rm pole}_{\textsf{q},2}$ of couplings ${\overline{a}}_{+}$ of
higher energy ${\cal N}=1$ SQCD with $SU({\overline{N}}_{c}-n_{1})$ colors and
with quarks $\overline{\textsf{q}}^{2},\textsf{q}_{2}$ with $n_{2}$ flavors,
and ${\overline{a}}_{-}$ of lower energy $SU({\overline{N}}_{c}-n_{1})$ ${\cal
N}=1$ SYM (see (6.3),(6.4) ) :

$\Biggl{[}\frac{1}{{\overline{a}}_{+}}\approx\frac{1}{{\overline{a}}_{*}}+\frac{2n_{1}-{\rm\overline{b}}_{\rm
o}}{{\overline{N}}_{c}-n_{1}}\log\Bigl{(}\frac{{\overline{\mu}}^{\rm
pole}_{\rm gl,1}}{\mu^{\rm
pole}_{\textsf{q},2}}\Bigr{)}\Biggr{]}=\Biggl{[}\frac{1}{{\overline{a}}_{-}}\approx
3\log\Bigl{(}\frac{\mu^{\rm
pole}_{\textsf{q},2}}{\langle\Lambda_{SYM}\rangle}\Bigr{)}\Biggr{]},\,\,\,\frac{1}{{\overline{a}}_{*}}=\frac{N_{F}}{{\rm\overline{b}}_{\rm
o}}\,\,\rightarrow\,\,Z_{q}\sim\exp\\{-\frac{{\overline{N}}_{c}-n_{1}}{{\rm\overline{b}}_{\rm
o}}\\}\ll 1.$

A) Strongly coupled direct theory

a) All $N_{F}^{2}$ masses of second generation fions $\mu^{\rm
pole}_{2}(\Phi_{i}^{j})=\mu_{o}$ remain the same as before (4.2).

b) The masses of $m_{Q,1}^{\rm pole}$ and $\mu^{\rm pole}_{3}(\Phi_{1}^{1})$
are frozen at $\mu<m_{Q,1}^{\rm pole}$ and so remain the same as before
(4.3),(4.8) (but now, at $(2n_{1}-{\rm\overline{b}}_{\rm o})>0,\,\,\mu^{\rm
pole}_{3}(\Phi_{1}^{1})\gg\langle\Lambda_{SYM}\rangle$ in (4.8)).

c) The mass of $m_{Q,2}^{\rm pole}$ looks now as:
$\langle\Lambda_{SYM}\rangle\ll m_{Q,2}^{\rm
pole}=(\mu_{\Phi}/\mu_{\Phi,1})m_{Q,1}^{\rm pole}\ll m_{Q,1}^{\rm pole},\\\
\mu_{\Phi,1}=\Lambda_{Q}(\Lambda_{Q}/m_{1})^{(2N_{c}-N_{F})/N_{c}}$, compare
with (4.5).

d) The masses of $n_{2}^{2}$ fions $\mu^{\rm pole}_{3}(\Phi_{2}^{2})$ are
parametrically smaller now than before, they become the smallest masses among
all other nonzero masses, compare with (4.9)

$\displaystyle\mu^{\rm
pole}_{3}(\Phi_{2}^{2})\sim\Bigl{(}\frac{\mu_{\Phi}}{\mu_{\Phi,0}}\Bigr{)}^{\frac{2(2n_{1}-{\rm\overline{b}}_{\rm
o})}{3(n_{2}-N_{c})}\,>\,0}\langle\Lambda_{SYM}\rangle\ll\langle\Lambda_{SYM}\rangle\,.$
(6.1)

e) $2n_{1}n_{2}$ fion fields $\Phi^{1}_{2}$ and $\Phi^{2}_{1}$ of the third
generation are massless as in Part I above.

B) Weakly coupled dual theory,
$({\overline{N}}_{c}-n_{1})/{\rm\overline{b}}_{\rm o}\gg 1$

For simplicity, we ignore logarithmic factors of the dual quark RG-evolution
at $\mu^{\rm pole}_{\textsf{q},2}<\mu<{\overline{\mu}}^{\rm pole}_{\rm gl,1}$.

a) All $N_{F}^{2}$ equal mass $\mu^{\rm pole}(M^{i}_{j})$ mions of the dual
theory and $N_{F}^{2}$ equal mass $\mu^{\rm pole}_{2}(\Phi_{i}^{j})$ of second
generation fions in the direct theory have now parametrically different
masses, compare with (4.2),(5.2), 666 Here and below we trace only factors
which are the exponentially small (or large) in their dependence on the small
parameter ${\rm\overline{b}}_{\rm o}/N_{F}\ll 1$, i.e.powers of
$Z_{q}\sim\exp\\{-({\overline{N}}_{c}-n_{1})/{\rm\overline{b}}_{\rm o}\\}\ll
1$, and ignore preexponential power-like in ${\rm\overline{b}}_{\rm o}/N_{F}$
factors. Besides, $Z_{q}$ does not compete in any way in its smallness with
e.g. $m_{Q}/\Lambda_{Q}\ll 1$ or $\mu_{\Phi}/\mu_{\Phi,0}\ll 1$.

$\displaystyle\mu^{\rm pole}(M^{i}_{j})\sim Z^{2}_{q}\,\mu^{\rm
pole}_{2}(\Phi_{i}^{j})\ll\mu^{\rm pole}_{2}(\Phi_{i}^{j})\,,\quad
Z_{q}\sim\exp\\{-\frac{{\overline{N}}_{c}-n_{1}}{{\rm\overline{b}}_{\rm
o}}\\}\ll 1\,.$ (6.2)

b) ${\overline{\mu}}^{\rm pole}_{\rm gl,1}$ is parametrically smaller now than
before, compare with (4.3),(5.3),

$\displaystyle{\overline{\mu}}^{\rm pole}_{\rm gl,1}\sim
Z^{1/2}_{q}\,m_{Q,1}^{\rm pole}\ll m_{Q,1}^{\rm pole}\,.$ (6.3)

c) $\mu^{\rm pole}_{\textsf{q},2}$ looks now as, compare with (4.5),(5.6),

$\displaystyle\mu^{\rm
pole}_{\textsf{q},2}\sim\frac{1}{Z_{q}}\Bigl{(}\frac{\mu_{\Phi,0}}{\mu_{\Phi}}\Bigr{)}^{\frac{2n_{1}-{\rm\overline{b}}_{\rm
o}}{3(n_{2}-N_{c})}\,>\,0}\langle\Lambda_{SYM}\rangle\gg\langle\Lambda_{SYM}\rangle\,,\quad\mu^{\rm
pole}_{\textsf{q},2}\sim\frac{1}{Z_{q}}\,m_{Q,2}^{\rm pole}\gg m_{Q,2}^{\rm
pole}\gg\langle\Lambda_{SYM}\rangle\,.$ (6.4)

$\mu^{\rm
pole}_{\textsf{q},2}\sim\Bigl{(}\frac{\mu_{\Phi}}{Z^{3/2}_{q}\mu_{\Phi,0}}\Bigr{)}\,{\overline{\mu}}^{\rm
pole}_{\rm gl,1}\ll{\overline{\mu}}^{\rm pole}_{\rm gl,1}\,,\quad\mu_{\Phi}\ll
Z^{3/2}_{q}\mu_{\Phi,0}\,.$

Both direct quarks ${\overline{Q}}_{2},\,Q^{2}$ and dual ones
$\overline{\textsf{q}}^{2},\,\textsf{q}_{2}$ are weakly confined (i.e. the
string tension originating from corresponding SYMs is parametrically smaller
than quark masses, $\sigma^{1/2}\sim\langle\Lambda_{SYM}\rangle\ll
m_{Q,2}^{\rm pole}\ll\mu^{\rm pole}_{\textsf{q},2}$) and form a large number
of various quarkonia. But quarks $\overline{\textsf{q}}^{2},\,\textsf{q}_{2}$
are non-relativistic and weakly coupled inside low lying quarkonia in the dual
theory, so that the mass splittings between adjacent levels of dual quarkonia
are parametrically small, $\delta m/m\sim O({\rm\overline{b}}_{\rm
o}^{\,2}/N_{F}^{2})\ll 1$, while there is nothing similar in the strongly
coupled direct theory.

d) $n_{1}^{2}$ fields $N_{1}^{1}$ of the dual theory and $n_{1}^{2}$ fields
$\Phi_{1}^{1}$ of the third generation fions of the direct theory, both sets
with the same quantum numbers, also have now parametrically different masses,
compare with (4.8),(5.10), but now at $(2n_{1}-{\rm\overline{b}}_{\rm o})>0$,

$\displaystyle\mu^{\rm pole}(N_{1}^{1})\sim\frac{1}{Z_{q}}\mu_{3}^{\rm
pole}(\Phi^{1}_{1})\gg\mu_{3}^{\rm
pole}(\Phi^{1}_{1})\gg\langle\Lambda_{SYM}\rangle\,.$ (6.5)

e) The low energy frozen ”masses” of mions are also changed. $z_{M}^{\prime}$
factor in (5.9) is only logarithmic now (and is ignored). Therefore, now
instead of (5.8):

$\displaystyle\mu_{low}(M_{i}^{j})\sim\Bigl{(}\frac{Z^{3/2}_{q}\mu_{\Phi,0}}{\mu_{\Phi}}\Bigr{)}{\overline{\mu}}_{{\rm
gl},\,1}\gg{\overline{\mu}}_{{\rm gl},\,1}\,.$ (6.6)

f) $2n_{1}n_{2}$ nion fields $N^{1}_{2}$ and $N^{2}_{1}$ (dual pions) of the
dual theory are massless as in Part I above and are undistinguishable from the
$2n_{1}n_{2}$ third generation massless fion fields $\Phi^{1}_{2}$ and
$\Phi^{2}_{1}$ of the direct theory. All these particles are Nambu-Goldstone
particles of the spontaneously broken global symmetry $U(N_{F})\rightarrow
U(n_{1})\times U(n_{2})$.

It is seen that at the left end of the conformal window, i.e. at
${\rm\overline{b}}_{\rm o}/N_{F}\ll 1$ in this Part II, in addition to clear
qualitative differences in point ‘2’ of Part I above at
${\rm\overline{b}}_{\rm o}/N_{F}=O(1)$, all corresponding nonzero mass scales
of the direct and dual theories are now parametrically different in this
region of the Lagrangian parameters: they differ at least by powers of the
parametric factor
$Z_{q}\sim\exp\\{-({\overline{N}}_{c}-n_{1})/{\rm\overline{b}}_{\rm o}\\}\ll
1\,.$ 777 And there are no particles now in the dual theory with the scale of
masses similar to $\mu^{\rm pole}_{3}(\Phi_{2}^{2})$ in (6.1).  And
logarithmic factors of the RG-evolution of
$\overline{\textsf{q}}^{2},\,\textsf{q}_{2}$ quarks present in the dual theory
(and absent in the direct one) result in additional parametric differences of
corresponding masses.

Therefore, there are no reasons for these corresponding masses to become
exactly equal at ${\rm\overline{b}}_{\rm o}/N_{F}=O(1)$ in Part I above.

On the whole, we conclude that, although clearly surprisingly similar in a
number of respects, the direct and Seiberg’s dual ${\cal N}=1$ SQCD-like
theories have different mass spectra and are not equivalent. As was shown
above, this is especially clearly seen at the left end of the conformal window
at $0<{\rm\overline{b}}_{\rm o}/N_{F}\ll 1$ considered here, where the
corresponding mass scales are parametrically different.

Recall that methods of mass spectra calculations used e.g. in [7, 8] and in
all cases considered above satisfy all those tests which were used as checks
of the Seiberg hypothesis about equivalence of the direct and dual theories.
This shows that all those tests, although necessary, are not sufficient. (And
similarly at both ends of the conformal window, at the left end
$(3{\overline{N}}_{c}-N_{F})/N_{F}\ll 1$, or at the right end
$(3N_{c}-N_{F})/N_{F}\ll 1$ in the standard ${\cal N}=1$ SQCD and its
Seiberg’s dual, i.e. both without fields $\Phi$, see [8]).

In addition, we see no any reasons to interpret the dual theory as ”the low
energy solitonic magnetic” form at $\mu<\Lambda_{Q}$ of the direct fundamental
electric theory. This is evident e.g. in the standard ${\cal N}=1$ SQCD within
the conformal window, where the UV free direct theory enters smoothly at
$\mu<\Lambda_{Q}$ into the perturbative conformal regime with all its quarks
and gluons remaining effectively massless. And as was argued e.g. in [6, 8]
and above in this section, at $N_{c}<N_{F}<3N_{c}/2$ also. The only difference
is that the regime at $\mu<\Lambda_{Q}$ will be not conformal but very strong
coupling one at $N_{c}<N_{F}<3N_{c}/2$. 888 In the case considered here this
happens also at $\mu<m_{Q,1}^{\rm pole}$ and $0<{\rm\overline{b}}_{\rm
o}/N_{F}\ll 1\,,\,0<(2n_{1}-{\rm\overline{b}}_{\rm o})/N_{F}\approx
2n_{1}/N_{F}=O(1)$ .  The ”dual” theory has to be considered simply as a
definite independent theory. And both theories can be compared at
$\mu<\Lambda_{Q}$ to see whether they are equivalent or not.

On the other hand, it seems clear that, indeed, there is some hidden symmetry
(broken by $m_{Q}\neq 0$ and, in our case here, by
$\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,0}=\Lambda_{Q}(\Lambda_{Q}/m_{Q})^{(2N_{c}-N_{F})/N_{c}}$)
which makes direct and Seiberg’s dual ${\cal N}=1$ SQCD-like theories,
although not equivalent, but very similar, see Appendix A. And described above
methods of mass spectra calculation for such theories at (very) strong
couplings demonstrate this. And, from our viewpoint, just this is most
important. This shows that we understand the dynamics of such theories
sufficiently well.

Much more examples can be found in [7]. See also [8] and Appendix B about mass
spectra in the standard SQCD and similar problems with its Seiberg’s dual
variant. 999 Recall that at $N_{c}<N_{F}<3N_{c}/2$ the standard ${\cal N}=1$
SQCD with light quarks, $m_{Q}\ll\Lambda_{Q}$, and its Seiberg’s dual (i.e.
both without fields $\Phi$) also have qualitatively different mass spectra
[8]. In the strongly coupled at $\mu<\Lambda_{Q}$ direct theory the quark
masses are $m^{\rm
pole}_{Q}\sim\Lambda_{Q}(m_{Q}/\Lambda_{Q})^{{\overline{N}}_{c}/N_{c}},\,\,{\overline{N}}_{c}=(N_{F}-N_{c})$.
These strongly coupled but weakly confined quarks decouple at $\mu<m^{\rm
pole}_{Q}$ and there remains ${\cal N}=1$ $SU(N_{c})$ SYM with
$\Lambda_{SYM}=(\Lambda_{Q}^{\rm b_{o}}m^{N_{F}}_{Q})^{1/3N_{c}}$. Up to
additional logarithmic factors, in the IR-free at $\mu<\Lambda_{Q}$ dual
theory the weakly coupled and weakly confined dual quarks have masses
$\mu^{\rm pole}_{q}\sim m^{\rm pole}_{Q}$. After they decouple, there remains
dual ${\cal N}=1$ $SU({\overline{N}}_{c})$ SYM with the same $\Lambda_{SYM}$
and $N_{F}^{2}$ lighter mions $M^{i}_{j}$ with masses $\mu^{\rm
pole}(M^{i}_{j})\sim\Lambda_{Q}(m_{Q}/\Lambda_{Q})^{(N_{c}-{\overline{N}}_{c})/N_{c}}\ll\Lambda_{SYM}\ll\mu^{\rm
pole}_{q}$. There is no analog of these light mions in the direct theory. In
direct theory, the case $N_{F}=N_{c}$ is qualitatively different from
$N_{c}<N_{F}<3N_{c}/2$ because quarks are higgsed at $\mu\sim\Lambda_{Q}$.

##  Appendix A ’t Hooft triangles, $0<{\rm\overline{b}}_{\rm
o}/N_{F}=O(1)\,,\,0<({\rm\overline{b}}_{\rm o}-2n_{1})/N_{F}=O(1)$

The quantum numbers of various fields with respect to the global
$SU(N_{F})_{L}\times SU(N_{F})_{R}$ chiral symmetry are the following. -

a) The direct quarks are $Q^{L}$ and ${\overline{Q}}_{\overline{R}}$, i.e. $Q$
realizes the fundamental representation $N_{F}$ of $SU(N_{F})_{L}$, while
${\overline{Q}}$ \- the antifundamental representation ${\overline{N}}_{F}$ of
$SU(N_{F})_{R}$.

b) The fions $\Phi^{i}_{j}$ are $\Phi_{\overline{L}}^{R}\,$.

c) The dual quarks are $q_{\overline{L}}$ and ${\overline{q}}^{R}$, while
mions $M^{i}_{j}$ are $M^{L}_{\overline{R}}$ and the nions (dual pions) are
$N^{R}_{\overline{L}}$.

It is worth noting also that ’t Hooft triangles have different values at
different ranges of scales because the chiral flavor symmetries are broken not
spontaneously but explicitly by $m_{Q}\neq 0$ and
$\Lambda_{Q}\ll\mu_{\Phi}\ll\mu_{\Phi,0}$.

1) The range $\mu_{2}^{\rm pole}(\Phi^{i}_{j})\sim\mu_{o}\sim\mu^{\rm
pole}(M^{i}_{j})\ll\mu\ll\Lambda_{Q}$.

All particles of the direct and dual theories, except for $N_{F}^{2}$ fions
$\Phi^{i}_{j}$, are relevant in this range. The triangle $SU^{3}(N_{F})_{L}$
is $N_{c}$ in the direct theory, while in the dual one it is
$(-{\overline{N}}_{c})$ from dual quarks and $N_{F}$ from $N_{F}^{2}$ mions
$M^{i}_{j}$, i.e. also $N_{c}$ on the whole. This case was checked by Seiberg
in [2].

2) The range $m_{Q,1}^{\rm pole}\sim{\overline{\mu}}_{{\rm gl},\,1}^{\rm
pole}\ll\mu\ll\mu_{2}^{\rm pole}(\Phi^{i}_{j})$

All particles of the direct and dual theories, except for $N_{F}^{2}$ mions
$M^{i}_{j}$, are relevant in this range. The triangle $SU^{3}(N_{F})_{L}$ is
$(-{\overline{N}}_{c})$ in the dual theory, while in the direct one it is
$N_{c}$ from direct quarks and $(-N_{F})$ from $N_{F}^{2}$ fions
$\Phi^{i}_{j}$, i.e. also $(-{\overline{N}}_{c})$ on the whole.

3) $m_{Q,2}^{\rm pole}\sim\mu^{\rm pole}_{\textsf{q},2}\sim\mu^{\rm
pole}_{3}(\Phi_{2}^{2})\sim\langle\Lambda_{SYM}\rangle\ll\mu\ll m_{Q,1}^{\rm
pole}$

a) The triangle $SU^{3}(n_{1})_{L}$. The nions $N^{1}_{1}$ give $(-n_{1})$ and
nions $N_{1}^{2}$ give $(-n_{2})$ in the dual theory, i.e. $(-N_{F})$ on the
whole. In the direct theory: fions $\Phi^{1}_{1}$ give $(-n_{1})$ and fions
$\Phi^{2}_{1}$ give $(-n_{2})$, i.e. also $(-N_{F})$ on the whole.

b) The triangle $SU^{3}(n_{2})_{L}$. In the dual theory: quarks
$\textsf{q}_{2}$ give $(-{\overline{N}}_{c}+n_{1})$ and nions $N^{1}_{2}$ give
$(-n_{1})$, i.e. $(-{\overline{N}}_{c})$ on the whole. In the direct theory:
quarks $Q^{2}$ give $N_{c}$, fions $\Phi^{1}_{2}$ give $(-n_{1})$ and fions
$\Phi^{2}_{2}$ give $(-n_{2})$, i.e. also $(-{\overline{N}}_{c})$ on the
whole.

4) The range $\mu_{3}^{\rm pole}(\Phi^{1}_{1})\sim\mu^{\rm
pole}(N_{1}^{1})\ll\mu\ll\langle\Lambda_{SYM}\rangle$

a) The triangle $SU^{3}(n_{1})_{L}$. In the dual theory: nions $N^{1}_{1}$
give $(-n_{1})$ and $N^{2}_{1}$ give $(-n_{2})$, i.e. $(-N_{F})$ on the whole.
In the direct theory: fions $\Phi^{1}_{1}$ give $(-n_{1})$ and $\Phi^{2}_{1}$
give $(-n_{2})$, i.e. also $(-N_{F})$ on the whole.

b) The triangle $SU^{3}(n_{2})_{L}$. In the dual theory: nions $N^{1}_{2}$
give $(-n_{1})$, while in the direct theory fions $\Phi^{1}_{2}$ also give
$(-n_{1})$.

5) The range $0\leq\mu\ll\mu_{3}^{\rm pole}(\Phi^{1}_{1})$.

a) The triangle $SU^{3}(n_{1})_{L}$. In the direct theory: the fions
$\Phi_{1}^{2}$ give $(-n_{2})$. In the dual theory: nions $N^{2}_{1}$ also
give $(-n_{2})$.

b) The triangle $SU^{3}(n_{2})_{L}$. In the direct theory: the fions
$\Phi^{1}_{2}$ give $(-n_{1})$. In the dual theory: nions $N^{1}_{2}$ also
give $(-n_{1})$.

It is seen that triangles are the same in this case, while there are
differences in the mass spectra (see Part I Conclusions above). This shows
once more that the equality of triangles, although necessary, is not
sufficient to speak about the equivalence of two theories.

At ${\rm\overline{b}}_{\rm o}/N_{F}\ll 1$ in Part II Conclusions above, there
appear parametric intervals of scales within which the corresponding masses of
the direct and dual theories are different. And within these intervals the
triangles of the direct and dual theories are also different.

##  Appendix B Mass spectra in the standard ${\cal N}=1$ SQCD and its
Seiberg’s dual at $N_{F}=N_{c}+1$.

According to [8] (see also the arguments in Part II of Conclusions and the
footnote 9), the mass spectrum of the direct theory is a smooth continuation
to $N_{F}=N_{c}+1$ of those at $N_{c}+1<N_{F}<3N_{c}/2$. I.e., the quark
masses are $m_{Q}^{\rm
pole}=C_{Q}\Lambda_{Q}(m_{Q}/\Lambda_{Q})^{1/N_{c}}\ll\Lambda_{Q},\,\,C_{Q}={\cal
O}(1)$. After they decouple at $\mu<m_{Q}^{\rm pole}$, there remains ${\cal
N}=1\,\,SU(N_{c})$ SYM with its scale factor
$\Lambda_{SYM}=\Lambda_{Q}(m_{Q}/\Lambda_{Q})^{(N_{c}+1)/3N_{c}}\ll m_{Q}^{\rm
pole}$. This is all. The $SU(N_{c})$ gluonia with the mass scale
$\sim\Lambda_{SYM}$ are the lightest particles.

Now, about Seiberg’s dual theory. It was proposed by Seiberg in [9] that all
$SU(N_{c})$ gluons and all light direct quarks with masses
$m_{Q}=m_{Q}(\mu=\Lambda_{Q})\ll\Lambda_{Q}$ and $N_{F}=N_{c}+1$ flavors
acquire dynamical masses $\sim\Lambda_{Q}$ and decouple $\mu<\Lambda_{Q}$.
Instead of them, there appear light colorless solitons (dual particles):
mesons $M^{i}_{j}$ and baryons $B_{i},\,{\overline{B}}^{j},\,\,i,j=1...N_{F}$.
Their proposed Lagrangian at $\mu=\Lambda_{Q}$ has the form

$\displaystyle K_{\rm dual}={\rm
Tr}_{N_{c}+1}\,\frac{M^{\dagger}M}{\Lambda_{Q}^{2}}+{\rm
Tr}_{N_{c}+1}\,(B^{\dagger}B+{\overline{B}}^{\dagger}{\overline{B}})\,,$ (B.1)

${\cal W}_{\rm dual}=m_{Q}{\rm Tr}_{N_{c}+1}(M)+{\rm
Tr}_{N_{c}+1}\,({\overline{B}}\frac{M}{\Lambda_{Q}}B)-\frac{\det_{N_{c}+1}M}{\Lambda_{Q}^{2N_{c}-1}}\,.$

It was pointed out in the subsequent paper [2] that the Lagrangian (B.1) of
the dual theory with $N_{F}=N_{c}+1$ can be obtained e.g. as follows. One can
start with the direct $SU(N_{c})$ theory with $N_{F}^{\prime}=N_{c}+2$ flavors
of direct quarks and with the Lagrangian at $\mu=\Lambda_{Q}$

$\displaystyle K_{direct}={\rm
Tr}_{N_{c}+2}\,(Q^{\dagger}Q+Q\rightarrow{\overline{Q}})\,,\,\quad{\cal
W}_{direct}={\cal W}_{gauge}(SU(N_{c}))+m_{Q}{\rm
Tr}_{N_{c}+1}\,({\overline{Q}}Q)+\Lambda_{Q}({\overline{Q}}_{0}Q^{0})\,.$
(B.2)

Integrating out at $\mu<\Lambda_{Q}$ the last heavy quarks with the mass
$\Lambda_{Q}$, one obtains the desired direct $SU(N_{c})$ theory with
$N_{F}=N_{c}+1$ light flavors of direct quarks $Q^{i},\,i=1...N_{c}+1$, with
masses $m_{Q}\ll\Lambda_{Q}$.

On the other hand, the proposed in [2] Lagrangian of the dual
$SU({\overline{N}}_{c}=2)$ theory with $N_{F}^{\prime}=N_{c}+2$ dual quark
flavors looks at $\mu=\Lambda_{Q}$ as

$\displaystyle K_{\rm dual}={\rm
Tr}_{N_{c}+2}\frac{M^{\dagger}M}{\Lambda_{Q}^{2}}+{\rm
Tr}_{N_{c}+2}\Bigl{(}\,q^{\dagger}q+(q\rightarrow{\overline{q}}\,\Bigr{)}\,,$
(B.3)

${\cal W}_{dual}={\cal W}_{gauge}(SU(2))+m_{Q}{\rm
Tr}_{N_{c}+1}(M)+\Lambda_{Q}M^{0}_{0}-{\rm
Tr}_{N_{c}+2}\Bigl{(}\,{\overline{q}}\frac{M}{\Lambda_{Q}}q\,\Bigr{)}\,.$

From (B.2) and Konishi anomalies [4], the mean vacuum values of mions
$M^{i}_{j}\rightarrow({\overline{Q}}_{j}Q^{i})$ and dual quarks look at
$\mu=\Lambda_{Q}$ as

$\displaystyle\langle M^{i}_{j}\rangle=\delta^{i}_{j}\frac{\langle
S\rangle}{m_{Q}}=\delta^{i}_{j}\Lambda_{Q}^{2}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{1}{N_{c}}}\,,\quad\langle
M^{0}_{0}\rangle=\frac{\langle
S\rangle}{\Lambda_{Q}}=\Lambda_{Q}^{2}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{N_{c}+1}{N_{c}}}\,,\quad\langle
S\rangle=\Lambda_{Q}^{3}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{N_{c}+1}{N_{c}}}\,,$
(B.4)

$\langle{\overline{q}}^{0}q_{0}\rangle=\frac{\langle
S\rangle\Lambda_{Q}}{\langle
M^{0}_{0}\rangle}=\Lambda_{Q}^{2}\,,\quad\langle{\overline{q}}^{j}q_{i}\rangle=\frac{\langle
S\rangle\Lambda_{Q}}{\langle
M^{i}_{j}\rangle}=\delta_{i}^{j}m_{Q}\Lambda_{Q}\,,\quad i,j=1...N_{c}+1\,.$

It is seen from (B.4) that the condensate
$\langle{\overline{q}}^{0}q_{0}\rangle^{1/2}$ of last dual quarks is much
larger than their mass $\langle M^{0}_{0}\rangle/\Lambda_{Q}$. Therefore, they
are higgsed and broke the whole dual $SU({\overline{N}}_{c}=2)$ group at the
scale $\sim\Lambda_{Q}$. After integrating all heavy particles of the dual
theory with masses $\sim\Lambda_{Q}$, there remains the IR-free dual theory
with $N_{F}^{2}$ light mions $M^{i}_{j}$ and $N_{F}=N_{c}+1$ light quarks
$q_{i},\,{\overline{q}}^{j}$ with one screened color. Let us reassign them as
$B_{i}=i\epsilon_{\alpha\beta}\langle q_{0}^{\alpha}\rangle
q_{i}^{\beta}/\Lambda_{Q},\,\,\,{\overline{B}}^{j}=i\epsilon^{\alpha\beta}\langle{\overline{q}}^{0}_{\alpha}\rangle{\overline{q}}^{j}_{\beta}/\Lambda_{Q},\,\,\,\alpha,\beta=1,2$.
In the range of scales $\mu^{\rm pole}(B)<\mu<\Lambda_{Q}$ all these particles
are effectively massless and the Lagrangian looks as 101010 We account in this
range of scales of the IR-free theory (B.5) for the parametric logarithmic
renormalization factors $z_{M}$ and $z_{B}$ of $M$ and $B$.  (compare with
(B.1))

$K_{\rm dual}=z_{M}(\Lambda_{Q},\mu){\rm
Tr}_{N_{c}+1}\frac{M^{\dagger}M}{\Lambda_{Q}^{2}}+z_{B}(\Lambda_{Q},\mu){\rm
Tr}_{N_{c}+1}\Bigl{(}\,B^{\dagger}B+(B\rightarrow{\overline{B}}\,\Bigr{)}\,,$

$\displaystyle{\cal W}_{\rm dual}=m_{Q}{\rm Tr}_{N_{c}+1}M+{\rm
Tr}_{N_{c}+1}\Bigl{(}\,{\overline{B}}\frac{M}{\Lambda_{Q}}B\Bigr{)}-\int
d{\tilde{\mu}}\,\frac{\det_{N_{c}+1}({\overline{B}}B)}{\Lambda_{Q}^{2N_{c}-1}}\,,$
(B.5)

where $d{\tilde{\mu}}$ is the corresponding measure. The term
$\sim\det_{N_{c}+1}({\overline{B}}B)$ in (B.5) is a shorthand for the
instanton contribution in the form of the multiquark ’t Hooft operator. At
$\mu<\mu^{\rm pole}(B)$ the baryons decouple as heavy and the Lagrangian looks
as (compare with (B.1))

$K_{\rm dual}=z_{M}(\Lambda_{Q},\mu^{\rm pole}(B)){\rm
Tr}_{N_{c}+1}\frac{M^{\dagger}M}{\Lambda_{Q}^{2}}+z_{B}(\Lambda_{Q},\mu^{\rm
pole}(B)){\rm
Tr}_{N_{c}+1}\Bigl{(}\,B^{\dagger}B+(B\rightarrow{\overline{B}}\,\Bigr{)}\,,$

$\displaystyle{\cal W}_{\rm dual}=m_{Q}{\rm Tr}_{N_{c}+1}(M)+{\rm
Tr}_{N_{c}+1}\Bigl{(}\,{\overline{B}}\frac{M}{\Lambda_{Q}}B\Bigr{)}-\frac{\det_{N_{c}+1}(M)}{\Lambda_{Q}^{2N_{c}-1}}\,.$
(B.6)

The pole masses of baryons $B_{i},\,{\overline{B}}^{j}$ and mions $M^{i}_{j}$
from (B.6) are (compare with the footnote 9 at
${\overline{N}}_{c}=N_{F}-N_{c}=1$)

$\displaystyle\mu^{\rm
pole}(B,{\overline{B}})=\frac{\Lambda_{Q}}{z_{B}(\Lambda_{Q},\mu^{\rm
pole}(B))}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{1}{N_{c}}}\ll\Lambda_{Q},\,\,\mu^{\rm
pole}(M)=\frac{\Lambda_{Q}}{z_{M}(\Lambda_{Q},\mu^{\rm
pole}(B))}\Bigl{(}\frac{m_{Q}}{\Lambda_{Q}}\Bigr{)}^{\frac{N_{c}-1}{N_{c}}}\ll\Lambda_{SYM},\,\,$
(B.7)

It is seen from (B.7) that the mass spectrum at $\mu<\Lambda_{Q}$ of the dual
theory with $N_{F}=N_{c}+1$ flavors is also its smooth continuation from
$N_{c}+1<N_{F}<3N_{c}/2$ to ${\overline{N}}_{c}=N_{F}-N_{c}=1$, see the
footnote 9, and only heavy dual gluons are absent, while the baryons
$B_{i},{\overline{B}}^{j}$ are really the remained light dual quarks.

On the whole, the mass spectra of the direct and dual theories with
$N_{F}=N_{c}+1$flavors look as follows (see [8], Part II of Conclusions and
the footnote 9 for the mass spectra of the direct theory).
1) In the direct theory. - There is a number of flavored hadrons with the mass
scale $\sim m_{Q}^{\rm pole}\sim\Lambda_{Q}(m_{Q}/\Lambda_{Q})^{1/N_{c}}\\\
\ll\Lambda_{Q}$, with different spins and other quantum numbers. In addition,
there is only a number of gluonia with the mass scale
$\sim\Lambda_{SYM}=\Lambda_{Q}(m_{Q}/\Lambda_{Q})^{(N_{c}+1)/3N_{c}}\ll
m_{Q}^{\rm pole}$.
2) In the dual theory. - There are $2N_{F}$ baryons $B_{i}$ and
${\overline{B}}^{j}$ (= dual quarks). Their masses are $\mu^{\rm
pole}(B)=\Bigl{(}\Lambda_{Q}/z_{B}(\Lambda_{Q},\mu^{\rm
pole}(B))\Bigr{)}\Bigl{(}m_{Q}/\Lambda_{Q}\Bigr{)}^{1/N_{c}}$. And there are
$N_{F}^{2}$ lightest mions $M^{i}_{j}$ with masses $\mu^{\rm pole}(M)=\\\
\Bigl{(}\Lambda_{Q}/z_{M}(\Lambda_{Q},\mu^{\rm
pole}(B))\Bigr{)}\Bigl{(}m_{Q}/\Lambda_{Q}\Bigr{)}^{(N_{c}-1)/N_{c}}\ll\Lambda_{SYM}\ll\mu^{\rm
pole}(B)$. It is seen that the mass spectra of the direct and dual are
qualitatively different.

As for the ’t Hooft triangles. -
1) In the range of scales (ignoring logarithmic factor $z_{B}$): $m_{Q}^{\rm
pole}\sim\mu^{\rm pole}(B)\ll\mu\ll\Lambda_{Q}$. All triangles of the direct
and dual theories are the same [10],[11],[9].
2) In the range $\Lambda_{SYM}\ll\mu\ll m_{Q}^{\rm pole}$. - To the triangles
$R^{3}$ and $R$ contribute $SU(N_{c})$ gluinos of the direct theory and
fermionic partners $\psi^{i}_{j}$ of mions $M^{i}_{j}$ in the dual one. The
triangles in two theories are different. There are no contributions to other
triangles in the direct theory, while there are contributions from
$\psi^{i}_{j}$ to the $SU(N_{F})^{3}_{L}$ and $SU(N_{F})^{2}_{L}\times
U(1)_{R}$ triangles.
3) In the range of scales $\mu^{\rm pole}(M)\ll\mu\ll\Lambda_{SYM}$. - There
are no contributions to all triangles in the direct theory, while still there
are contributions from $\psi^{i}_{j}$ to the
$SU(N_{F})^{3}_{L},\,\,SU(N_{F})^{2}_{L}\times U(1)_{R}$, $R^{3}$ and $R$
triangles.

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