# Challenges to Obtaining Results for Real QCD from SUSY QCD

Michael Dine  Yan Yu

###### Abstract

Recently there have been proposals to understand features of QCD such as
confinement and chiral symmetry breaking by considering supersymmetric
versions of the theory with various patterns of soft breaking. In this note we
recall that with small soft breakings, SUSY QCD is suggestive of observed
features of real QCD. But we outline some of the challenges to establishing
these features of the theory with large soft breakings. It appears difficult
to argue for confinement or chiral symmetry breaking; at best, one can say
that if the non-supersymmetric theory does not confine and/or break chiral
symmetry, phase transitions would appear inevitable as one increases the soft
breakings. We also discuss, at large N, where confinement would imply chiral
symmetry breaking, the challenges to establishing confinement.

##  1 Introduction and Overview

Much is understood about the dynamics of supersymmetric gauge theories. With
massless quarks, their moduli and pseudomoduli spaces have been elucidated and
patterns of symmetry breaking have been determined. With supersymmetric masses
for quarks, one can find discrete, supersymmetric ground states and determine
their properties. In certain theories which can be obtained by deformations of
$N=2$ theories, one can demonstrate confinement. In obtaining some of these
results, explicit, controlled computations are possible; in others, holomorphy
of the superpotential and gauge coupling functions are important tools.111The
literature on these topics is extensive. For reviews, with extensive
references, see, for example, [1, 2].

One might hope to break the supersymmetry and obtain results for real QCD (we
will use the term “Real QCD” for a gauge theory with three vector-like light
fermions in the fundamental representation). Adding small soft breakings is
generally straightforward, and typically yields results consistent with
expectations for the strong interactions[3, 4]. 222Reference [5] considers
such issues, particularly the problem of describing supersymmetric theories
including small soft breakings in the infrared. References [6, 7] considered
challenges to understanding breaking supersymmetry simultaneously in theories
related by Seiberg duality. But actually breaking supersymmetry sufficiently
to yield real QCD would seem challenging. Recently it has been argued that it
is possible to obtain information about QCD[8, 9] by considering a particular
type of soft breaking, anomaly mediation[10, 11], The ultraviolet
insensitivity of this mechanism suggests that such an analysis might be
particularly robust.

In this note we look critically at the question of establishing results in
real QCD starting with the supersymmetric theory and general patterns of soft
breakings. We recall that with small soft breakings:

  1. 1.

For $SU(N)$ QCD with $N_{f}$ flavors, one finds the symmetry breaking pattern
one expects in actual QCD[3, 4].

  2. 2.

For theories which can be obtained by deforming $N=2$ theories, either to
$N=1$ supersymmetry[12, 13], and then with small soft breakings to $N=0$
supersymmetry, one can demonstrate monopole condensation and confinement.

To turn these observations into something resembling a proof of chiral
symmetry breaking and/or confinement in real QCD, one would seem to need only
to establish that there is a trajectory in the soft breaking parameter space
along which there is no phase transition in these theories. One might imagine
a proof by contradiction: assume there is a phase transition to a phase with
unbroken chiral symmetries or without monopole condensation, and demonstrate
some inconsistency. But we will see that achieving such a result is a tall
order, Indeed, if one starts with the assumption that real QCD does not break
chiral symmetry and/or confine, one can advance strong arguments that there is
a phase transition. In other words, rather than being able to rule out a phase
transition to unbroken chiral symmetry or deconfinement, whether or not there
is a phase transition depends on whether real QCD does/does not break chiral
symmetry/confine, i.e. on what happens in a regime where one does not have
computational control over the theory. We will see that this is the situation
quite broadly, including the case of anomaly mediated supersymmetry breaking.
We will also consider the situation at large $N$. In large $N$, it has long
been known that it is enough to establish confinement to establish the pattern
of chiral symmetry breaking[14]333A recent paper elucidating aspects of the
large $N$ problem is [15].. We will describe a strategy for establishing
confinement at large $N$ through such deformations by considering a class of
Wilsonian effective actions, but the problem of phase transitions here turns
into a question about renormalization group flows in the far infrared.

This paper is organized as follows. In section 2, we review what it might mean
to find – or fail to find – evidence for a phase transition (between confined
and unconfined phases or chiral symmetry broken/chiral symmetry unbroken
phases) as one varies parameters. In section 3, we review some aspects of
supersymmetric QCD. We discuss theories with $N$ colors and $N_{f}$ flavors,
the non-perturbative superpotential and patterns of supersymmetry breaking. We
discuss confinement in $N=2$ theories broken down to $N=1$, and note that,
starting with these, one can demonstrate confinement in the $N=1$ pure gauge
theories with $SU(N)$ gauge group. In section 4, we consider the addition of
general small soft breakings, with the resulting patterns of chiral symmetry
breaking and the problem of confinement. We turn to the realistic problem of
large breakings in section 5. We consider the obstacles, for various choices
of soft breaking parameters, to maintaining any theoretical control as one
moves to regimes of badly broken supersymmetry. If one assumes that the theory
does not confine and/or does not break its chiral symmetries, we argue that it
is quite plausible that there is a phase transition. In section 6, we study
anomaly mediation specifically. We explain the microscopic origin of the soft
breaking potential in terms of the one loop gaugino mass, and explain why one
loses control of the computation as the mass becomes comparable to the QCD
scale. We consider the problem of confinement in the context of the large $N$
expansion in section 7. We outline two challenges. First, we note that it is
again quite plausible that there is a phase transition to a state of unbroken
chiral symmetry. In this case, one would also lose confinement. Alternatively,
one can focus on confinement directly. As we noted above, the problem can be
translated into one of renormalization group flows. It is quite plausible, and
one expects, that the flow is such that confinement occurs. Proving this is
likely challenging. We summarize in section 8.

Reference [16] considers some of these issues, focussing on anomaly mediation.
The analysis is limited to infinitesimal soft breakings, where one has the
greatest control. This paper offera speculations both pro and con on the
possibility of a phase transition. We will discuss this work in section 6,
explaining why one expects suppression of the potential and thus a phase
transition is plausible as the soft masses increase.

##  2 Possible Phase Structures of Softly Broken SUSY QCD

Whether one can learn about symmetry breaking and confinement by deforming
supersymmetric QCD to real QCD depends on whether one can rule out a phase
transition as one increases soft breaking parameters from zero to values large
compared to the QCD scale. To understand the possibilities, we consider,
first, the possible phase structure of a simple theory of a real scalar field
with an exact $Z_{2}$ symmetry, Suppose the potential is:

$V={1\over 2}\mu^{2}\phi^{2}+{\lambda\over 4}\phi^{4}.$ (2.1)

Then we have:

$\phi=\left(-{\mu^{2}\over\lambda}\right)^{1/2}$ (2.2)

This is singular at the origin of the parameter $\mu^{2}$; e.g. derivatives of
$\phi$ (and other quantities) diverge as $\mu^{2}\rightarrow 0$.

Add to the potential a small $Z_{2}$ symmetry breaking term:

$V=\mu^{2}\phi^{2}+{\lambda\over 4}\phi^{4}+\epsilon\phi.$ (2.3)

Now there is a non-zero solution everywhere for $\phi$, and $\phi$ is a smooth
function of $\mu^{2}$. Derivatives of $\phi$ with respect to $\mu^{2}$ may be
large. For example, for $\mu^{3}\gg\epsilon^{2/3}>0$ (thinking of $\lambda$ as
a number of order $1$),

$\phi=-{\epsilon\over\mu^{2}}.$ (2.4)

Here, for large soft breakings (compared to the QCD scale), $\epsilon$ would
correspond to a term in the potential for the scalar fields arising from the
breaking of the chiral symmetry in real QCD. It might be proportional to the
quark condensate:

$\epsilon\propto\langle\bar{q}q\rangle$ (2.5)

If $\langle\bar{q}q\rangle\neq 0$ in real QCD, there would not be an actual
phase transition, though for large soft breakings one would find large
derivatives of physical quantities in regions of the parameter space.

We might well model $\epsilon$ itself as the expectation value of another
field, $\rho$, much lighter than $\phi$. $\rho$ might have an expectation
value in some regions of the parameter space, in others not. With small soft
breakings, in particular, the degrees of freedom of supersymmetric QCD are
quite different than those of ordinary QCD. As we increase the soft breakings,
they become identical. With large soft breakings, we can integrate out the
heavy fields, potentially leading to dynamics quite different than suggested
by a simple extrapolation of the soft breaking parameters from small to large
values. As a result, in some instances, we will be able to establish a
structure of this general type, but, without solving QCD, we won’t be able to
determine whether $\epsilon\neq 0$. Indeed, in the spirit of attempting a
proof by contradiction, we will see that the question of symmetry breaking in
QCD is precisely the question of whether $\epsilon\neq 0$; if $\epsilon=0$,
i.e. if real QCD breaks the chiral symmetry, there is no phase transition; if
$\epsilon\neq 0$, there is.

We consider a model with a continuous $U(1)$ symmetry and two complex fields
to illustrate some additional featues :

$V=\mu^{2}|\phi|^{2}+{\lambda\over
4}|\phi|^{4}-\epsilon|\rho|^{2}+\lambda^{\prime}|\rho|^{4}+(\lambda^{\prime\prime}\phi\rho^{3}+{\rm
c.c.}).$ (2.6)

with $\lambda^{\prime}\sim\lambda^{\prime\prime}\sim 1$. Then in the regime
where $\mu^{2}$ is large and positive, while $\epsilon$ is small and positive,
$\rho$ has an expectation value of order $\epsilon$, while $\phi$ has a small
expectation value, suppressed by $\epsilon\over\mu^{2}$. The Goldstone boson
is principally ${\rm Im}~{}\rho$ (for a particular choice of phases of fields
and couplings), with a small admixture (of order $\epsilon\over\mu^{2}$) of
${\rm Im}~{}\phi$. The symmetry breaking dynamics are controlled by the light
field, $\rho$. Again, we might expect that as we approach real QCD from
supersymmetric QCD by increasing the soft breaking masses for scalars and the
gaugino, the dynamics of the quarks dominate the question of chiral symmetry
breaking. All of this is consistent with what we might expect for large soft
breaking parameters in susy QCD, especially for the squark masses. Then we can
integrate out the squarks and gluinos. The real and imaginary parts of the
squarks, in particular, will have nearly degenerate masses, and any would-be
Goldstone fields must consist principally of the light fields.

Note, again, that there is no phase transition if $\epsilon$ is negative as we
vary $\mu^{2}$ from small negative to large positive values. If $\epsilon>0$,
there is a phase transition. In the following sections, we will consider how
the features of this model are realized in supersymmetric QCD with various
patterns of soft supersymmetry breaking, and ask similar questions about a
possible deconfinement phase transition.

##  3 A Brief Review of Supersymmetric Gauge Theories

In this section we consider supersymmetric gauge theories with $N=1$
supersymmetry from the perspectives of chiral symmetry breaking; we will
consider confinement when we discuss small soft breakings in the next section.

###  3.1 SUSY QCD with N colors and $N_{f}$ Flavors

Here we focus on $SU(N)$ theories with $N_{f}$ massless flavors. For
simplicity, we consider $1\leq N_{f}<N$. Classically the theory has symmetry
$U(N)_{L}\times U(N)_{R}\times U(1)_{R}$. One linear combination of symmetries
is anomalous; the actual symmetry is $U(1)_{B}\times SU(N)_{L}\times
SU(N)_{R}\times U(1)_{R}$. In this case, classically, there is a moduli space
of zero energy states. Up to flavor transformations,

$Q=\bar{Q}=\left(\begin{matrix}v_{1}&0&0&\dots\cr 0&v_{2}&0&\dots\cr
0&0&v_{3}&\dots\cr 0&0&0&v_{4}\end{matrix}\right)$ (3.1)

Non-perturbatively, a superspotential is generated:

$W_{np}=\Lambda_{hol}^{3N-N_{f}\over N-N_{f}}{\rm det}(\bar{Q}Q)^{-{1\over
N-N_{f}}}.$ (3.2)

The resulting potential tends to zero for large $|v_{i}|$. One can add quark
masses, in which case there are $N$ supersymmetric vacua.

In considering the theory with soft breakings, and especially with large soft
breakings, it will be helpful to have a more microscopic picture of the
dynamics. Consider, first, the case $N_{f}<N-1$. The the configuration of eqn.
3.1 breaks the gauge symmetry to $SU(N-N_{f})$. The low energy theory consists
of the gauge bosons and gauginos of $SU(N-N_{f})$ and a set of Goldstone
bosons and their superpartners. These can be described by an effective
lagrangian including, at low orders in $1/v$, the kinetic terms for these
fields, and a coupling

${1\over 16\pi^{2}}\int d^{2}\theta W_{\alpha}^{2}\log({\rm det}(\bar{Q}Q)).$
(3.3)

corresponding to the relation of the low energy and high energy gauge
couplings. As a result, the $\Lambda$ parameter of the low energy group is
related to that of the full gauge group by:

$\Lambda_{LE}^{3}=\Lambda^{3}\left({\Lambda\over v}\right)^{N_{f}/N}.$ (3.4)

In the low energy theory, gaugino condensation,

$\langle\lambda\lambda\rangle=\Lambda_{LE}^{3}$ (3.5)

generates, through the coupling of equation 3.3, the non-perturbative
superpotential. We will have occasion to review this further in section 6.

For the case $N_{f}=N-1$, the superpotential is generated by instantons[17].
The scale invariance is broken by the scalar expectation values, and the
characteristic instanton size scales with $v$. In slightly more detail, one
integrates over a set of approximate solutions. For very small instantons, one
neglects the scalar potential[18] and solves an equation

$D^{2}\phi=0,$ (3.6)

where $\phi$ stands for the $Q$ and $\bar{Q}$ fields and the covariant
derivative refers to the gauge field configuration in the absence of the
scalars, with the scale size $\rho$. The action of the resulting configuration
depends on $\rho$ as $\rho^{2}v^{2}$, which cuts off the scale size
integration, yielding contributions proportional to inverse powers of $v$ in
the effective action. The role of different scales will change as we add soft
breakings.

This analysis represents a very safe application of ideas of effective field
theory, as only the light degrees of freedom are described by this effective
action. If we turn on small soft breakings, we we do in the next section, we
will be able to treat their effects as small perturbations of this effective
lagrangian. This will no longer be the case as the soft breakings become
large. Some degrees of freedom included in the would-be low energy theory may
become heavy, while additional degrees of freedom may become light, so not
only the action but even the degrees of freedom may not be accounted for
properly by the effective theory for large values of the soft breakings.

###  3.2 Confinement in Theories with $N=1$ Supersymmetry Obtained by
Deforming $N=2$ Theories

It is remarkable that one can essentially prove confinement in gauge theories
obtained as perturbations of $N=2$ supersymmetric theories[12, 13]444For a
clear and concise introduction to Seiberg-Witten theory, see [1].. The
analysis starts with consideration of the moduli space of $N=2$ theories (for
simplicity, without hypermultiplets). In the language of $N=1$ supersymmetry,
in addition to the gauge bosons and gauginos, there is a chiral multiplet,
$\phi$, in the adjoint representation of the group. Studying the singularities
of the gauge coupling function, one can demonstrate that there are points in
the moduli space of vacua at which monopole-antimonopole pairs become
massless. For simplicity, consider the case of $SU(2)$, where

$\phi=\left(\begin{matrix}v&0\cr 0&-v\end{matrix}\right).$ (3.7)

Calling the members of the monopole-antimonopole pair $M$ and $\bar{M}$, and
$u={\rm Tr\phi^{2}}$, there is a superpotential:

$W=b(u-u_{0})M\bar{M}.$ (3.8)

If we take $M$ and $\bar{M}$ to have dimension $1$, $b$ has dimension $-1$.
Now we can perturb this system, first breaking the supersymmetry to $N=1$, by
introducing a mass for $\phi$, ${m_{\phi}\over 2}\phi^{2}$ in the
superpotential. (One can see that his breaks the symmetry down to $N=1$ as the
two adjoint fermions now have different mass). One can solve the equations

${\partial W\over\partial u}={\partial W\over\partial M}={\partial
W\over\partial\bar{M}}=0,$ (3.9)

subject to the condition that the auxiliary $D$ fields vanish for the dual
group. One has

$u=u_{0};~{}~{}M=\bar{M}=\left({m_{\phi}\over b}\right)^{1/2}.$ (3.10)

This is the phenomenon of monopole condensation. One remarkable feature of
these expressions is that we can take $m_{\phi}$ as large as we like,
demonstrating confinement in $N=1$ supersymmetric QCD, with no matter fields.
In making this statement, it is crucial that we know that the $N=1$
supersymmetry is unbroken. Then we can describe the effective low energy
theory for large $\phi$ in terms of a supersymmetric effective action. The
theory possesses a a non-anomalous $R$ symmetry under which $\phi$ has charge
$0$ and $m$ acts as a spurion of charge $2$. So there is only the tree level
superpotential for $\phi$. It should also be noted that the superpotential
gives mass to one linear combination of $M$ and $\bar{M}$, while, as a result
of the gauge interactions, the vev’s for $M$ and $\bar{M}$ yield masses for
the other linear combination and the gauge fields. So as expected in a
confining theory, the spectrum is gapped.

At first it may seem surprising that the scale of the monopole condensate
grows with $m_{\phi}$. But, as we elaborate further in the next subsection,
the scale, $\Lambda_{QCD}$, of the low energy gauge theory grows with
$m_{\phi}$ as $m_{\phi}^{1/3}$. Indeed, as we explain in the next subsection,
the expectation value of the superpotential should scale with
$\Lambda_{QCD}^{3}$ and it does. In fact the expectation value of the gaugino
condensate matches the expectation value of $\langle W\rangle$ for the
superpotential of equation 3.8.

We can break the supersymmetry down to $N=0$ in various ways. If we add a
small gaugino mass term, $m_{\lambda}$, for example, we expect that the
monopole potential will be modified by terms proportional to $m_{\lambda}$.
For small enough $m_{\lambda}$, the monopoles will still condense. Quite
generally, this follows from the existence of a mass gap. For large
$m_{\lambda}$, the potential is hard to compute, but we might expect it to be
a complicated function of $m_{\lambda}$. Absent other arguments, there is the
logical possibility that at some value of $m_{\lambda}$,
$m_{\lambda}=m_{\lambda}^{0}$, the condensate disappears, as does the gap in
the spectrum. We will discuss the various possibilities further shortly.

Refs. [8, 9] consider a different perturbation, which breaks the $N=2$
supersymmetric theory directly to $N=0$, motivated by “anomaly mediation”.
These authors modify only the scalar potential, adding a term:

$\delta V_{am}=m\left({\partial W\over\partial\phi}-3W\right).$ (3.11)

This generates a minimum for the $\phi$ potential at $u=u_{0}$. and, again, a
monopole condensate.

###  3.3 Matching Gaugino Condensation and Monopole Condensation

In the pure gauge theory, the expectation value of $W$ should be given by the
gluino condensate, i.e.

$\langle W\rangle=\langle\lambda\lambda\rangle={\rm c}\Lambda_{LE}^{3}.$
(3.12)

Here $\Lambda_{LE}$ is the holomorphic Lambda parameter of the low energy
gauge theory, a theory with, for definiteness, gauge group $SU(2)$ and no
matter fields. ${\rm c}$ is an order one constant.

We can actually be more precise, determining the constant ${\rm c}$, following
reference [19]. For $N_{f}=N-1$, these authors showed that the non-
perturbative superpotential for the massless theory, generated by instantons,
is

$W={\Lambda^{2N+1}\over{\rm det}\bar{Q}Q}+m\bar{Q}Q.$ (3.13)

Here we have added a mass term for the quarks. Exploiting the holomorphy of
the superpotential, we can take $m$ arbitrarily large. Then we can determine
that

$\langle W\rangle=2\Lambda_{LE}^{3}.$ (3.14)

Here we have used:

$\Lambda_{LE}=me^{-{8\pi^{2}\over b_{LE}g^{2}(m)}}=me^{-{8\pi^{2}\over
g^{2}(M)}-{b_{HE}\over b_{LE}}\log(M/m)}=m^{1/6}\Lambda^{5/6}$ (3.15)

where $b_{LE}$ is the lowest order $\beta$ function coefficient of the low
energy theory, in this case $6$, while $b_{HE}$ is the corresponding
coefficient of the high energy $\beta$-function, here $b_{HE}=4$. $M$ is some
high energy scale, well above the expectation value of $\phi$.

Similarly, we can determine $\langle W\rangle$ from monopole condensation.
Here, following reference [20], for example, we have

$u_{0}=2\Lambda^{2}$ (3.16)

Then

$\langle W\rangle=m_{\phi}u_{0}=2\Lambda_{LE}^{2}$ (3.17)

Here we have used that $\Lambda_{LE}^{3}=m_{\phi}\Lambda^{2}$. So we have
complete agreement; the expectation value of the superpotential can be
understood either in terms of gaugino condensation or monopole condensation.
This is quite satisfying.

Once supersymmetry is broken, the holomorphic behavior of the superpotential
is lost and this connection will be modified. Still, we have to entertain (or
rule out) the possibility that if, for example, a large gaugino mass
suppresses or eliminates the gluino condensate, it might also suppress or
eliminate the monopole condensate.

##  4 Perturbing with small soft breaking parameters

To the theory with $N_{f}$ flavors, one can add small soft breakings, such as
small masses for squarks and gauginos. In this case, one can minimize the
resulting potential for the squarks, and find stable minima at relatively
large values of the fields, where one has good control over the theory (in the
limit that the squark masses tend to zero, these minima must move off to
infinity). The region of small fields is selected by large mass. This is the
interesting region for studying real QCD. We will consider the challenges to
such a study shortly.

We can think of small squark masses in terms of a supersymmetric spurion
field, $\Phi$, with $\Phi=\theta^{2}F$, and we take $\Phi$ to have dimension
$1$ (so $F$, the supersymmetry breaking parameter, has dimension $2$). Then
the mass terms can be written as

$m_{\tilde{Q}_{fg}}^{2}\tilde{Q}^{*}_{f}\tilde{Q}_{g}={\gamma_{fg}\over
M^{2}}\int d^{4}\theta\Phi^{\dagger}\Phi Q_{f}^{\dagger}Q_{g}.$ (4.1)

Note

$\tilde{m}_{Q}^{2}\sim{|F_{\Phi}|^{2}\over M^{2}}.$ (4.2)

Other supersymmetry breaking operators can be written similarly. Consider, in
particular,

${\beta\over M}s\int d^{2}\theta\Phi W.$ (4.3)

Note that $\beta$ is a holomorphic quantity. This gives the sort of potential
correction considered by Murayama[8]. At low orders in the soft breaking
parameters, there are restrictions on the types of couplings due to
holomorphy. In particular, we can assign an $R$ charge to $\Phi$,
$R_{\Phi}=-2$. Higher powers of $\Phi$ which might appear in the effective
action will be constrained by these symmetries. The squark mass terms of
equation 4.2, however, are allowed.

These and similar operators may, in particular, limit the ability to constrain
the theory as the soft breking masses ($F_{\Phi}$) become large. But for small
values of these soft breakings, we can obtain a good ground state and a
definite pattern of symmetry breaking. In particular, if we take all of the
squark masses identical, we preserve the $SU(N_{f})\times SU(N_{f})\times
U(1)_{R}$ of the supersymmetric theory, and obtain a good ground state.

In more detail, in equation 3.1, we can take $v_{i}=v$, a common expectation
value. The potential is now:

$V=|{\partial W\over\partial Q_{i}}|^{2}+|{\partial
W\over\partial\bar{Q}_{i}}|^{2}+m_{\tilde{Q}}^{2}(|Q_{i}|^{2}+|\bar{Q}_{i}|^{2}).$
(4.4) $~{}~{}~{}~{}\sim\Lambda^{6N-2N_{f}}v^{-{4N_{f}\over
N-2N_{f}}-2}+m_{\tilde{Q}}^{2}v^{2}.$

So

$v^{2}\sim\Lambda^{2}\left({\Lambda^{2}\over
m_{\tilde{Q}}^{2}}\right)^{N-N_{f}\over 2N}.$ (4.5)

For small $m_{\tilde{Q}}^{2}$, the system sits at large values of the fields
($v$), where the coupling is weak and the approximations are reliable.
Particularly relevant is that there is a clear separation of mass scales.
While there are particles at the scale $v$ and particles at the scale
$\Lambda$, the effective lagrangian describes particles with masses at the
much lower scale, $m_{\tilde{Q}}^{2}$. As $m_{\tilde{Q}}^{2}$ becomes
comparable to or larger than $\Lambda$, this will no longer be the case. The
symmetry is broken to $SU(N)$. There are a set of Goldstone bosons; their
would-be superpartners have mass scaled by the supersymmetry-breaking
parameters.

Anticipating our subsequent discussion, note that if we were to continue to
use this effective theory for $m_{\tilde{Q}}^{2}\gg\Lambda^{2}$, the
expectation values of the squarks would be small, but the symmetries would
still be broken. This is the paradox alluded to above. In this limit, we can
integrate out the squarks. Both the real and imaginary parts of the scalars
will have large mass; any Goldstone bosons arise from the fermionic quarks,
not the squarks.

##  5 What happens when the soft breakings become large?

We’d like to consider combinations of soft breakings for which, as they become
large, the system reduces to ordinary QCD. We might ask: if we assume chiral
symmetry is unbroken in real QCD, does this lead to a contradiction, once we
consider the known features of supersymmetric QCD. Similarly, if assume real
QCD does not confine, does this lead to a contradiction with features of
supersymmetric QCD? The situation, as we will see, is similar to our toy model
of equation 2.3, with $\epsilon=0$. Phase transitions are quite plausible.

###  5.1 Chiral Symmetry Breaking

We first consider the question of chiral symmetry breaking. Start by including
masses for the squarks and the gauginos. As these become larger than the QCD
scale, the theory indeed reduces to ordinary QCD. These fields can be
integrated out. Indeed, examining the microscopic theory, it is the pattern of
symmetry breaking at low energies which will control the expectation values of
the squarks. How might one model the approach to this limit?

One simple approach leads to a paradox opening the possibility of a phase
transition. Suppose that one simply takes the potential arising from the non-
perturbative superpotential, equation 3.2, and adds squark masses, allowing
these to be large compared to $\Lambda$. Then for all values of the squark
masses, the chiral symmetry is spontaneously broken, albeit with smaller and
smaller meson decay constants. As a result, while we have massless Goldstone
fields, we also have massive real scalar fields. But if the gaugino mass is
large (not necessarily as large as the squark mass, but large compared to the
QCD scale) this can’t be. For sufficiently large squark mass, we can integrate
out the complex scalars, leaving ordinary QCD. Whether or not there are
Goldstone bosons is a question of the dynamics of this theory. But if there
are, they are not composed of the massive scalars, but instead are the result
of the (fermionic) quark dynamics. In other words, the expectation values of
the squarks may vanish, or may be small, induced by the quark condensate, as
in our discussion of section 2.

To understand the issues more microscopically, consider the class of theories
with $N>N_{f}+1$. In the supersymmetric limit, these theories have flat
directions, in which the low energy theory is a pure gauge theory with gauge
group $SU(N-N_{f}).$ Gaugino condensation in this low energy theory generates
the non-perturbative superpotential.

As we turn on soft breakings, we might expect suppression of the non-
perturbative superpotential for two reasons:

  1. 1.

Gaugino and squark masses will suppress the coupling of the would-be modulus
to the low energy gauge group.

  2. 2.

The gaugino mass will tend to suppress the size of the condensate.

Both effects are important. The superpotential and the non-perturbative
correction to the potential, for small soft breakings, are proportional to
$\langle\lambda\lambda\rangle\propto\Lambda^{3}$. $\Lambda$ behaves as an
inverse power of $\langle Q\rangle$ for $\langle Q\rangle>m_{\tilde{Q}}$. For
still lower values of the squark vev, the dependence of the condensate on
$\langle Q\rangle$ is weak. As a reult, there is no longer enhancement of the
potential for small fields. For large soft breakings, one might expect the
squark soft breaking terms to be far larger than the non-perturbative terms in
the squark potential, and the vev’s to vanish at some critical value of the
squark mass, modulo the couplings of the squark fields to a quark condensate.
For large gluino mass, as we will consider shortly, one might expect further
suppression of the squark potential. In other words, as one increases the
squark and gaugino masses, there is the likelihood of a phase transition as a
function of $m_{\lambda}$ to an unbroken phase if real QCD does not break the
chiral symmetry ($\epsilon=0$ in equation 2.3).

As a model for the behavior of the condensate with $m_{\lambda}$, we can start
with the Veneziano-Yankielowicz action[21] for the field $S=W_{\alpha}^{2}$,
with a modification of the kinetic term and with a soft breaking term for the
scalar component of $S$:

${\cal L}_{S}=\int d^{4}\theta\Lambda^{-4}S^{\dagger}S+\int d^{2}\theta
S\left(\log(S^{N}/\Lambda^{3N}\right)-N)+m_{\lambda}^{2}\Lambda^{-4}|S^{2}|.$
(5.1)

With $m_{\lambda}^{2}=0$, the equation ${\partial W\over\partial S}=0$ yields
$S=\Lambda^{3}e^{2\pi ik\over N}$, $k=1,\dots N$, corresponding to the
expected gaugino condensate, which breaks the discrete $R$ symmetry. Adding
the soft breaking term, we consider the potential,

$V(S,S^{\dagger})=\Lambda^{4}|N\log(S/\Lambda^{3})|^{2}+m_{\lambda}^{2}\Lambda^{-4}|S^{2}|.$
(5.2)

The equation for the minimum of the potential is:

${\partial V\over\partial S^{\dagger}}=0={N\Lambda^{4}\log(S/\Lambda^{3})\over
S^{\dagger}}+m_{\lambda}^{2}\Lambda^{-4}S$ (5.3)

or

${m_{\lambda}^{2}\over\Lambda^{2}}|{S\over\Lambda^{3}}|^{2}=-N\log(S/\Lambda^{3}).$
(5.4)

For small $m_{\lambda}$, $S$ decreases with $m_{\lambda}$. This is consistent
with the expectation that the non-perturbative effects responsible for the
scalar potential turn off with increasing $m_{\lambda}$, supporting the
possibility of a phase transition.

While not reliable, we can explore the behavior of this equation for large
$m_{\lambda}$ in order to get some sense of possible behaviors of the system.
Defining $\hat{s}=S/\Lambda^{3}$, and $a=Nm_{\lambda}^{2}/\Lambda^{2}$, this
becomes:

$|\hat{s}|^{2}=-{1\over a}\log\hat{s}.$ (5.5)

For large $a$, given that the log does not grow rapidly with $s$, we have
that, roughly,

$\hat{s}={{\rm number~{}of~{}order~{}1}\over a^{1/2}}.$ (5.6)

So for large $a$, the size of the condensate decreases as a power of $a$. This
translates into an effective potential decreasing as a power of $m_{\lambda}$.

Returning to the full theory, we now have a potential for the moduli (squark
fields) behaving very differently than in the supersymmetric limit. Instead of
blowing up for small $\langle\bar{Q}Q\rangle$ (due to the fact that the
effective coupling of the theory blows up), it now goes to zero. Of course, we
can’t trust the precise form of this result – there is no systematic treatment
which justifies this analysis –but suppression of the non-perturbative effects
which generated the potential for zero and small gaugino mass at large
$m_{\lambda}$ seems almost inevitable. Again, we note that for large squark
and gaugino masses, we can integrate out these fields, leaving only a weak
dependence of the system on the expectation values of the squarks and the
gaugino condensate. These observations mean, in any case, that we can’t rule
out the possibility that there is a phase transition to a phase of unbroken
chiral symmetry as we now increase the soft breaking masses of the squarks,
without a full understanding of symmetry breakdown in real QCD.

###  5.2 $N_{f}=N-1$

In the case $N_{f}=N-1$, the non-perturbative superpotential for zero soft
mass terms is generated by instantons, which are approximate solutions of the
equations of motion. One integrates over a range of instanton scales from zero
to roughly the gauge boson Compton wavelength, proportional to $\langle
Q\rangle$. The largest instantons in this range dominate. Now as we turn on
soft breakings the typical instanton size will be less than
$m_{\tilde{Q}}^{-1}$, This follows by noting that for instantons of scale size
$\rho<m_{\tilde{Q}}^{-1}$, the instanton action, $e^{-{8\pi^{2}\over
g^{2}(\rho)}}$ no longer grows with $\rho$ as an inverse power of $\rho$,
while the scale size integration involves a large positive power of $\rho$. So
just as in the case of $N_{f}<N-1$, we would see a significant suppression of
the non-perturbative potential for large soft breakings. Again, we would
expect a phase transition unless the non-supersymmetric theory breaks the
chiral symmetry by itself.

###  5.3 Breakdown of the Effective Action

Critical to the derivation of the non-perturbative superpotential is the
ability to work with an effective action written strictly in terms of the
light fields. As we make the gaugino and squark masses large compared to the
QCD scale, there is no justification for keeping these fields in the effective
action below some scale slightly higher than the QCD scale. These degrees of
freedom are more massive than states of non-supersymmetric QCD which we are
not including. In section 6, we will confront this directly.

###  5.4 Confinement

We have seen that, starting with $N=2$ theories and perturbing to $N=1$, and
with small perturbations to $N=0$ (or simply perturbing to $N=0$ directly),
one can demonstrate monopole condensation and confinement. To demonstrate
confinement in theories like real QCD (here the pure gluonic theory). Starting
with the (confining) $N=1$ pure gauge theory, we need gluino masses large
enough that the degrees of freedom are just the gauge bosons. We would expect
the monopole potential to be sensitive to $m_{\lambda}$, in ways which are
difficult to estimate. One could well imagine that this potential, once
$m_{\lambda}\sim\Lambda_{QCD}$ develops a positive curvature near the origin
for some critical value of the gluino mass, $m_{\lambda}^{0}$, with the
monopole condensate (and with it the mass gap associated with the dual gauge
bosons) disappearing at that point. Of course, confinement might persist and
this is indeed what we expect, but it is not clear that we have the
theoretical tools to assess this question. We will consider the problem from
the point of view of the large N approximation shortly.

So it would appear, on the one hand, that it is difficult to formulate a proof
of confinement perturbing the $N=2$ theory to $N=0$, while on the other hand,
if one assumes that pure QCD does not confine, then a phase transition would
seem inevitable.

##  6 Anomaly Mediation

Reference [8] (and subsequently [16]) studies a particular soft breaking
pattern, referred to as “anomaly mediation”555We put anomaly mediation in
quotes because there are really various components to this structure,
including a particular, so called “sequestered” form of the Kahler
potential[10] and certain quantum corrections. . Anomaly mediation is
ultraviolet insensitive. In the present context, however, our concerns are
principally with effects in the infrared, as we approach non-supersymmetric
QCD in a particular manner. We imagine specifying these parameters ini a
Wilsonian effective action at scales where QCD is weakly coupled.

###  6.1 Microscopic Understanding of Anomaly Mediation in SUSY QCD with $N$
Colors and $N_{f}$ Flavors

Consider the leading corrections to the potential

$\delta V=m\left(\phi_{i}{\partial W\over\partial\phi_{i}}-3W\right)$ (6.1)

In the case of SUSY QCD, we should be able to understand this more
microscopically. If $N_{f}<N-1$, without the soft breakings, the low energy
theory is a pure supersymmetric gauge theory with a set of Goldstone
supermultiplets, The superpotential arises from gaugino condensation in this
theory. For simplicity in the writing, consider $N_{f}=1$. The theory has flat
directions with

$Q=\bar{Q}=\left(\begin{matrix}v&0&\dots&0\end{matrix}\right),$ (6.2)

up to a phase. The gauge coupling of the low energy theory depends on $v$,
corresponding to an effective action:

${\cal L}={1\over 32\pi^{2}}\left({8\pi^{2}\over g(M)^{2}}+{1\over
2}\log(\bar{Q}Q/M^{2})\right)W_{\alpha}^{2}.$ (6.3)

Correspondingly, in terms of the component fields, there is a term:

$\langle\lambda\lambda\rangle{1\over 64\pi^{2}}\left({F_{Q}\over
Q}+{F_{\bar{Q}}\over\bar{Q}}\right).$ (6.4)

So we see that (given that $W$ is described by a single power of $\bar{Q}Q$:

$\langle\lambda\lambda\rangle={64\pi^{2}}{\partial W\over\partial
Q}Q={64\pi^{2}\over N-N_{f}}W.$ (6.5)

Now using the anomaly-mediated formula for the gaugino mass[10, 11],

$m_{\lambda}=(3N-N_{f}){g^{2}\over 16\pi^{2}}m$ (6.6)

we can write the leading shift in the potential:

$\delta V={m_{\lambda}\over 2}\langle\lambda\lambda\rangle={3N-N_{f}\over
64\pi^{2}(N-N_{f})}{64\pi^{2}}mW.$ (6.7)

This is to be compared with the result from equation 6.1:

$\delta V={3N-N_{f}\over N-N_{f}}W$ (6.8)

i.e. we have complete agreement an a microscopic understanding of equation 6.1
in this framework.

###  6.2 Implications of the Microscopic Understanding

Note that the gaugino mass is important even though nominally a loop effect.
But note, also, that to obtain “real QCD”, we require that the gaugino mass
and the squark masses be large compared to $\Lambda_{QCD}$.666For the squark
masses, the prediction for the squark masses depends on other assumptions. For
a so-called “sequestered” Kahler potential, the squark massess are of the same
order in coupling as the gaugino masses, but they can well be larger[10, 11].
As we have seen, in this case, we expect a suppression of the gaugino
condensate, and correspondingly of the superpotential. It is then again hard
to make any firm statements about the pattern of symmetry breaking, and in
particular, to rule out the possibility of a phase transition. The presence of
a phase transition or its absence would appear to be completely contingent on
the behavior of the non-supersymmetric theory.

We can ask similar questions about confinement. One possible test is provided
by large N. For a theory with $N_{f}\ll N$, it is known that confinement
implies chiral symmetry breaking[14]. The inability to establish chiral
symmetry breaking than implies a similar obstacle to establishing confinement.
For theories with a single adjoint field, we have seen that at low energies
(if the adjoint is very massive) we have a confining pure supersymmetric gauge
theory. Within anomaly mediation, the effect of supersymmetry breaking is to
give mass to the gauginos. One cannot then readily determine what happens to
the monopole condensate as the gaugino mass is increased.

##  7 Large N

The large N limit potentially provides another approach to these questions.
The basic assumption of the large N limit is that the limit is smooth and that
the physics of this limit is in some sense represented by a formal sum of the
Feynman diagram expansion. One might hope, contingent on these assumptions, to
prove chiral symmetry breaking and or confinement by deforming supersymmetric
systems. In particular, confinement is a phenomenon of the far infrared. One
might hope that a Wilsonian action, computed at some scale well below the QCD
scale and the gaugino mass scale, would be characterized by features common to
both the small and large gaugino mass cases. Proving this is, in fact, the
obstacle. If one could demonstrate the existence of a mass gap in the large
gaugino mass limit, one could likely prove this universality. But this is as
difficult as (and presumably equivalent to) proving confinement in any case.

In this section, we consider two aspects of the problem. We discuss chiral
symmetry breaking with small soft masses at large N. Here we see the expected
pattern of soft breakings, but, as for fixed, finite N, constructing an
argument that there is no phase transition is challenging. We consider also
the problem of confinement at finite $N$ in the deformed supersymmetric CPN
models. Here there is a mass gap in both limits, and one can consider a
Wilsonian action in the far infrared in each case. One finds that these
actions are universal, differing only in a numerical coefficient and leading
to confinement in both cases. We finish the section with some comments on the
challenges to proving confinement in four dimensions.

###  7.1 Chiral Symmetry Breaking at Large $N$

We first consider the problem of ruling out a phase transition to unbroken
chiral symmetry. Consider, first, $N=1$ supersymmetric QCD with $N_{f}\ll N$
and $N\rightarrow\infty$. With small soft breakings, elaborating slight on the
work of [22], one can show:

  1. 1.

The $SU(N_{f})\times SU(N_{f})$ chiral symmetry is broken to $SU(N_{f})$.

  2. 2.

$\theta$ and $\eta^{\prime}$ dependence (in the presence of very small quark
masses) is as predicted for large $N$; physical quantities are functions of
${\theta-\eta^{\prime}/f_{\pi}\over N}$.

The question is: what happens as we turn on large soft breaking masses for
squarks and gauginos. Here the situation is similar to that we have
encountered at finite $N$. Ruling out a phase transition as we increase the
soft breakings is a challenge.

In large N we have the additional handle that, if one can prove confinement,
one can prove chiral symmetry breaking in the non-supersymmetric theory[14].
So we turn to the question of confinement.

###  7.2 The Problem of Confinement at Large $N$

We can ask the same questions about confinement. We can consider, for example,
a theory which in $N=1$ language contains a single adjoint chiral field. As
for finite $N$, adding a supersymmetric mass term for the chiral field, one
can demonstrate confinement[12, 13]. This mass can be arbitrarily large,
demonstrating confinement in the $N=1$ supersymmetric gauge theory without
matter. We will, indeed, provide further evidence for this in section 3.3.
Then if we add small masses for the gauginos this theory confines, as it does
for finite $N$. The question becomes: what happens as $m_{\lambda}$ becomes
large? Can we rule out a phase transition to a deconfined phase?

Heuristically, we might be tempted to argue that in computing, say, the
expectation value of a Wilson loop for very large quark-antiquark separation,
$R$, we are interested in the behavior of the theory in the far infrared, and
the Feynman diagrams are quite similar. But if we try to be more precise, we
run into obstacles. While formally, order by order, the diagrams have the same
structure, the diagrams have a different coupling. As the coupling is strong,
it is not clear what conclusions we can draw from this formal equivalence. The
issue is sharper if we phrase in terms of a Wilsonian effective action defined
at scales below $m_{\lambda}$. Framed in this way, it is not even clear what
are the appropriate degrees of freedom in the low energy theory, and what
should be considered the leading operator. Most strikingly, if real QCD did
not have a mass gap, then given that SUSY QCD does, we could have quite
dramatic differences in the degrees of freedom in the low energy theory.

As we now explain, the workings of confinement in the $CP^{N}$ models, with
and without supersymmetry, provide a possible scenario for QCD itself.

###  7.3 Supersymmetric CPN As a Model

The supersymmetric $CP^{N}$ theories, at large $N$, provide a model for these
phenomena[23, 24]. The theories are confining at large $N$. It is possible to
break the symmetry explicitly. For small breaking, the confining scale is only
slightly altered. For large breaking, the purely bosonic theory confines, as
is well known. In each case, the linear potential behaves as

$V(R)=c_{i}\Lambda_{i}^{2}R~{}~{}~{}i=1,2$ (7.1)

where $i=1$ is the supersymmetric case and $i=2$ is the non-supersymmetric
case. $\Lambda_{i}$ is the lambda parameter appropriate to the two limits, and
is, in fact, identical. In this case, there is no phase transition between the
supersymmetric and non-supersymmetric phases. The computation of the linear
potential can be described in the Wilsonian language we have used above. The
action at low energies is

${\cal L}_{LE}=-{1\over 4g^{2}}F_{\mu\nu}^{2},,$ (7.2)

with $g^{2}$ now a dimensionful quantity, determined by a one loop
computation. This leads to a linear potential between heavy charged objects.
The quartic and higher order terms are suppressed at small momentum transfer.

The model is discussed in more detail in Appendix A. But there are a few
features that are instructive. First, in the supersymmetric and non-
supersymmetric theories, and points in between, the theory is gapped. In a
Wilsonian action at scales below the gap, the theory is described in terms of
gauge degrees of freedom, $A_{\mu}$. In two dimensions, these are non-
dynamical, determined by the sources. The action can be written as a sum of
terms,

${\cal
L}=\sum_{n=1}^{\infty}{c_{n}\over\Lambda^{4n-2}}\left(F_{\mu\nu}^{2}\right)^{n}$
(7.3)

In the far infrared, relevant to the heavy charged particle/antipartcle
potential, all but the leading term in the sum may be ignored, and the first
term gives rise to a linear potential between these heavy objects

We might expect that real QCD is similar. First consider pure $N=1$
supersymmetric gauge theory. This theory we know confines and has a mass gap.
In the far infrared, in a gauge like Coulomb gauge, we would see only non-
propagating modes of the gauge field, and these would be responsible for the
linear force between quarks. These modes would be described by a simple
action. If the non-supersymmetric theory is gapped, as we expect it is, then
the situation would be similar.

##  8 Conclusions

As we have reviewed, supersymmetric QCD with small soft breakings leads to the
pattern of symmetry breaking we see in nature in the strong interactions. For
theories which can be obtained as deformations of $N=2$ supersymmetric
theories, one can exhibit monopole condensation and confinement. Breaking to
$N=1$, one can demonstrate this with large soft breakings, proving that, for
example, a theory of gauge bosons and gauginos is confining. Breaking to $N=0$
confinement persists with small soft breakings. To establish that these are
properties of real QCD requires proving that there is no phase transition as
one gives large masses to squarks and gauginos. It would be enough to
establish that there is no such phase transition along any trajectory in the
soft supersymmetry breaking parameter space which leads to real QCD.

We have asked whether assuming that QCD does not break chiral symmetries
and/or does not confine, leads to any contradictions. The answer to this
question appears to be no; instead, we have seen that with either of these two
assumptions, a phase transition would seem almost inevitable. For the question
of chiral symmetry breaking, we have argued that as one increases the gaugino
mass, for example, the non-perturbative effects responsible for the
interesting, theoretically controlled dynamics of susy gauge theories are
suppressed. Correspondingly, then, increasing the scalar masses, should lead
to vanishing vev’s for the scalar fields at some point, and correspondingly a
phase transition to unbroken chiral symmetry. For the question of confinement,
if one assumes that there is no confinement in real QCD, it is quite plausible
that there is a deconfining transition as one increases soft breakings. We
have noted that this is the case for general patterns of soft breaking,
including for anomaly mediated supersymmetry breaking.

We strongly believe that real QCD breaks chiral symmetry and confines. This
because these are features of nature, which otherwise seems to be well
described by QCD, and also from results of lattice simulations and large N
considerations. As a result, as we have outlined, there is not likely to be a
phase transition. It has long been appealing to take softly broken
supersymmetric gauge theories as a model for these phenomena. Using
deformations of these theories to prove these properties of real QCD remains a
daunting challenge.

Appendices

##  Appendix A The Supersymmetric CPN Model

The CPN model, without supersymmetry, provides an interesting model for
QCD[23]. The theory is confining at large $N$, and exhibits non-trivial
$\theta$ dependence. The model has a supersymmetric version. This theory also
is confining. We can ask about passage from the supersymmetric to the non-
supersymmetric theory, and establish that there is no phase transition. We’ll
use the notation of [23].

###  A.1 Features of the Supersymmetric and Non-Supersymmetric Models

We’ll formulate the theory as in [23]. There are a set of bosons, $n_{i}$,
with two dimensional Dirac fermionic partners, $\psi^{i}$. At tree level there
is a gauge field, $A_{\mu}$, with no kinetic term, and a similar fermion,
$\chi$. The lagrangian density is:

${\cal L}={N\over
g^{2}}\left[|(\partial_{\mu}-iA_{\mu})n_{i}|^{2}+\bar{\psi}_{i}(i\partial_{\mu}-A_{\mu})\gamma^{\mu}\psi^{i}+{1\over
2}(\sigma^{2}+\pi^{2})\\\
-{1\over\sqrt{2}}\bar{\psi}_{i}(\sigma+i\pi\gamma_{5})\psi^{i}+\lambda(|n_{i}|^{2}-1)+\bar{\chi}n_{i}\psi^{i}+\bar{\psi}_{i}n^{i*}\chi.\right]$
(A.1)

The lagrange multiplier fields $\lambda$ and $\chi$ enforce the constraints

$|n_{i}|^{2}=1;~{}~{}~{}\psi^{i}n_{i}=0.$ (A.2)

The fields $A_{\mu}$ and $\sigma+i\pi$ are auxiliary fields, which can be
eliminated from the lagrangian by solving their (algebraic) equations of
motion.

In the supersymmetric case, one can compute the effective action at one loop,
integrating over $\psi^{i}$ and $n_{i}$. One can look for stationary points of
$\lambda$ and $\sigma+i\pi$. Reference [23] finds, with ultraviolet cutoff
$M$:

$\lambda=\sigma^{2}={1\over 2}M^{2}e^{-{2\pi\over
g^{2}}}\equiv\Lambda^{2};~{}~{}~{}\pi=0.$ (A.3)

As explained there, the $n_{i}$ and $\psi_{i}$ have identical, supersymmetric
masses. The gauge field and the field $\chi$ acquire non-trivial kinetic
terms:

${\cal L}_{kin}={N\over 16\pi\Lambda^{2}}F_{\mu\nu}^{2}$ (A.4)

As a result, there is a linear force between two static charges proportional
to $\Lambda^{2}$.

This is similar to what happens in the non-supersymmetric case, where one has
only the lagrange multiplier $\lambda$ and the fields $n_{i},~{}A_{\mu}$. In
fact, because the one loop determinant from the integration over the $n_{i}$’s
is the same, the scale, $\Lambda$, which emerges is also the same. However,
the effective gauge coupling, computed at one loop, now has contributions only
from bosons, not from the fermions. This leads to a different coefficient of
the linear potential.

###  A.2 Breaking Supersymmetry and Passing to the Non-Supersymmetric Theory

In two dimensions, we can break the supersymmetry by adding mass terms for the
fermions, $\psi^{i}$,

$\delta{\cal L}=\sum_{i=1}^{N}m_{i}\psi^{i}\bar{\psi}_{i}.$ (A.5)

We assume that the $m_{i}$’s are comparable, differing by ${\cal O}(1)$
factors. With this choice, at large $N$, the $\sigma$ vev will not
significantly alter the typical masses of the fermions. For $m_{i}\gg\Lambda$,
the fermion contributions to the gauge boson kinetic terms become negligible;
only the bosonic contribution is important. Correspondingly, the coefficient
of the linear potential, in terms of $\Lambda$, changes.

## Acknowledgments

We thank P. Draper and N. Seiberg for conversations. This work was supported
in part by U.S. Department of Energy grant No. DE-FG02-04ER41286.

## References

  * [1] M. E. Peskin, Duality in supersymmetric Yang-Mills theory, in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 96): Fields, Strings, and Duality, pp. 729–809, 2, 1997.  hep-th/9702094.
  * [2] M. Dine, Supersymmetry and String Theory: Beyond the Standard Model.  Cambridge University Press, 1, 2016.
  * [3] O. Aharony, J. Sonnenschein, M. E. Peskin, and S. Yankielowicz, Exotic nonsupersymmetric gauge dynamics from supersymmetric QCD, Phys. Rev. D 52 (1995) 6157–6174, [hep-th/9507013].
  * [4] N. J. Evans, S. D. H. Hsu, and M. Schwetz, Controlled soft breaking of N=1 SQCD, Phys. Lett. B 404 (1997) 77–82, [hep-th/9703197].
  * [5] S. Abel, M. Buican, and Z. Komargodski, Mapping Anomalous Currents in Supersymmetric Dualities, Phys. Rev. D 84 (2011) 045005, [arXiv:1105.2885].
  * [6] H.-C. Cheng and Y. Shadmi, Duality in the presence of supersymmetry breaking, Nucl. Phys. B 531 (1998) 125–150, [hep-th/9801146].
  * [7] N. Arkani-Hamed and R. Rattazzi, Exact results for nonholomorphic masses in softly broken supersymmetric gauge theories, Phys. Lett. B 454 (1999) 290–296, [hep-th/9804068].
  * [8] H. Murayama, Some Exact Results in QCD-like Theories, Phys. Rev. Lett. 126 (2021), no. 25 251601, [arXiv:2104.01179].
  * [9] C. Csáki, A. Gomes, H. Murayama, and O. Telem, Demonstration of Confinement and Chiral Symmetry Breaking in SO(Nc) Gauge Theories, Phys. Rev. Lett. 127 (2021), no. 25 251602, [arXiv:2106.10288].
  * [10] L. Randall and R. Sundrum, Out of this world supersymmetry breaking, Nucl. Phys. B 557 (1999) 79–118, [hep-th/9810155].
  * [11] G. F. Giudice, M. A. Luty, H. Murayama, and R. Rattazzi, Gaugino mass without singlets, JHEP 12 (1998) 027, [hep-ph/9810442].
  * [12] N. Seiberg and E. Witten, Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19–52, [hep-th/9407087]. [Erratum: Nucl.Phys.B 430, 485–486 (1994)].
  * [13] N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484–550, [hep-th/9408099].
  * [14] S. R. Coleman and E. Witten, Chiral Symmetry Breakdown in Large N Chromodynamics, Phys. Rev. Lett. 45 (1980) 100.
  * [15] R. Sato, Chiral Symmetry Breaking and Quark Bilinear Condensate in Large N QCD, arXiv:2202.07664.
  * [16] A. Luzio and L.-X. Xu, On the Derivation of Chiral Symmetry Breaking in QCD-like Theories and S-confining Theories, arXiv:2202.01239.
  * [17] I. Affleck, M. Dine, and N. Seiberg, Supersymmetry Breaking by Instantons, Phys. Rev. Lett. 51 (1983) 1026.
  * [18] G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D 14 (1976) 3432–3450. [Erratum: Phys.Rev.D 18, 2199 (1978)].
  * [19] D. Finnell and P. Pouliot, Instanton calculations versus exact results in four-dimensional SUSY gauge theories, Nucl. Phys. B 453 (1995) 225–239, [hep-th/9503115].
  * [20] Y. Tachikawa, N=2 supersymmetric dynamics for pedestrians.  12, 2013.
  * [21] G. Veneziano and S. Yankielowicz, An Effective Lagrangian for the Pure N=1 Supersymmetric Yang-Mills Theory, Phys. Lett. B 113 (1982) 231\.
  * [22] M. Dine, P. Draper, L. Stephenson-Haskins, and D. Xu, $\theta$ and the $\eta^{\prime}$ in Large $N$ Supersymmetric QCD, JHEP 05 (2017) 122, [arXiv:1612.05770].
  * [23] E. Witten, Instantons, the Quark Model, and the 1/n Expansion, Nucl. Phys. B 149 (1979) 285–320.
  * [24] A. D’Adda, A. C. Davis, P. Di Vecchia, and P. Salomonson, An Effective Action for the Supersymmetric CP(N-1) Model, Nucl. Phys. B 222 (1983) 45–70.