QCD-like Theories on R_3\times S_1: a Smooth Journey from Small to Large r(S_1) with Double-Trace Deformations

Abstract

We consider QCD-like theories with one massless fermion in various representations of the gauge group SU$(N)$. The theories are formulated on $R_3\times S_1$. In the decompactification limit of large $r(S_1)$ all these theories are characterized by confinement, mass gap and spontaneous breaking of a (discrete) chiral symmetry ($\chi$SB). At small $r(S_1)$, in order to stabilize the vacua of these theories at a center-symmetric point, we suggest to perform a double trace deformation. With these deformation, the theories at hand are at weak coupling at small $r(S_1)$ and yet exhibit basic features of the large-$r(S_1)$ limit: confinement and $\chi$SB. We calculate the string tension, mass gap, bifermion condensates and $\theta$ dependence. The double-trace deformation becomes dynamically irrelevant at large $r(S_1)$. Despite the fact that at small $r(S_1)$ confinement is Abelian, while it is expected to be non-Abelian at large $r(S_1)$, we argue that small and large-$r(S_1)$ physics are continuously connected. If so, one can use small-$r(S_1)$ laboratory to extract lessons about QCD and QCD-like theories on $R_4$. Comment: 49 pages, 2 figures

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Work supported in part by US Department of Energy contract DE-AC02-76SF00515
UMN-TH-2632/08,FTPI-MINN-08/03, SLAC-PUB-13120
QCD-like Theories on R3× S1: a Smooth
Journey from Small to Large r(S1)
with Double-Trace Deformations
M. Shifmanaand Mithat¨Unsalb,c
aWilliam I. Fine Theoretical Physics Institute,
University of Minnesota, Minneapolis, MN 55455, USA
bSLAC, Stanford University, Menlo Park, CA 94025, USA
cPhysics Department, Stanford University, Stanford, CA,94305, USA
Abstract
We consider QCD-like theories with one massless fermion in vari-
ous representations of the gauge group SU(N). The theories are for-
mulated on R3×S1. In the decompactification limit of large r(S1) all
these theories are characterized by confinement, mass gap and sponta-
neous breaking of a (discrete) chiral symmetry (χSB). At small r(S1),
in order to stabilize the vacua of these theories at a center-symmetric
point, we suggest to perform a double trace deformation. With these
deformation, the theories at hand are at weak coupling at small r(S1)
and yet exhibit basic features of the large-r(S1) limit: confinement
and χSB. We calculate the string tension, mass gap, bifermion con-
densates and θ dependence. The double-trace deformation becomes
dynamically irrelevant at large r(S1). Despite the fact that at small
r(S1) confinement is Abelian, while it is expected to be non-Abelian
at large r(S1), we argue that small and large-r(S1) physics are contin-
uously connected. If so, one can use small-r(S1) laboratory to extract
lessons about QCD and QCD-like theories on R4.
February 2008
Submitted to Physical Review D

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Contents
1 Introduction2
2 QCD and QCD-like theories on R4and
R3× S1: general aspects
QCD with one fundamental fermion
3.1 Nonperturbative effects and the low-energy
Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2Bifermion condensate . . . . . . . . . . . . . . . . . . . . . . . 21
8
3 15
4 QCD with one bifundamental fermion
4.1 Deformed orbifold QCD . . . . . . . . . . . . . . . . . . . . . 23
4.2Nonperturbative low-energy effective Lagrangian . . . . . . . . 24
4.3Vacuum structure and chiral symmetry realization . . . . . . . 29
4.4 Mass gap and confinement . . . . . . . . . . . . . . . . . . . . 30
22
5 QCD with one AS fermion
5.1 Deformed orientifold QCD . . . . . . . . . . . . . . . . . . . . 32
5.2 Nonperturbative effects . . . . . . . . . . . . . . . . . . . . . . 33
31
6θ dependence38
7Remarks on planar equivalence40
8 Conclusions and prospects:
Abelian vs. non-Abelian confinement42
Appendix: Center stabilization 44
References 46
1

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1 Introduction
Analyzing QCD and QCD-like theories on R3× S1provides new insights in
gauge dynamics at strong coupling and offers a new framework for discussing
various ideas on confinement. The radius of the compact dimension r(S1)
plays a role of an adjustable parameter, an obvious bonus and a welcome
addition to a rather scarce theoretical toolkit available in strongly coupled
gauge theories. As the circumference L of the circle S1varies, so does the
dynamical pattern of the theory. For instance, at L ? Λ−1in some instances
the theory becomes weakly coupled. On the other hand, in the decompactifi-
cation limit, L ? Λ−1, we recover conventional four-dimensional QCD, with
its most salient feature, non-Abelian confinement.
A qualitative picture of confinement in terms of the Polyakov line was
suggested by Polyakov and Susskind long ago [1, 2]. Assume that the com-
pactified dimension is z. The Polyakov line (sometimes called the Polyakov
loop) is defined as a path-ordered holonomy of the Wilson line in the com-
pactified dimension,
?
where L is the size of the compact dimension while V is a matrix diagonalizing
U,
U = diag{v1,v2,...,vN}.
According to Polyakov, non-Abelian confinement implies that the eigenvalues
viare randomized: the phases of viwildly fluctuate over the entire interval
[0,2π] so that
?TrU? = 0.
The exact vanishing of ?TrU? in pure Yang–Mills is the consequence of the
unbroken ZN center symmetry in the non-Abelian confinement regime. In-
troduction of dynamical fermions (quarks) generally speaking breaks the ZN
center symmetry at the Lagrangian level.1
fluctuations of the phases of vi’s remains intact. Therefore, it is generally ex-
pected that ?1
1It is still an emergent dynamical symmetry in the multicolor limit [3, 4]; however, we
limit ourselves to small N. In this paper parametrically N is of order one.
U = P expi
?L
0
azdz
?
≡ V UV†
(1)
(2)
(3)
However, the picture of wild
NTrU? is strongly suppressed even with the dynamical fermion
2

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QCD*
LL
C
L
QCD
Figure 1: Quantum chromodynamics as a function of compactified direction cir-
cumference before and after surgery (QCD and QCD∗, respectively). Lc is the
point of a phase transition.
fields that respect no center symmetry, ?1
ported by lattice simulations at finite temperatures [5] demonstrating that
?TrU? is very close to zero at large L (low temperatures).
On the other hand, in QCD and QCD-like theories2at small L (high
temperatures) the center-symmetric field configuration is dynamically disfa-
vored. In many instances the vacuum is attained at ?1
case, the effective low-energy theory is at strong coupling, and it is as hard
to deal with it as with QCD on R4. Typically, the small and large-L domains
are separated by a phase transition (or phase transitions). For instance, for
S/AS with even N this is a Z2 phase transition. Numerical studies show
that for N ≥ 3 there is a thermal phase transition between confinement and
deconfinement phases. Similar numerical studies detect a temperature Tχ
at which the broken chiral symmetry of T = 0 QCD gives place to restored
chiral symmetry of high-T QCD. The phase transition at Tχis that of the
chiral symmetry restoration (the lower plot in Fig. (1)).
In this case small-L physics says little, if at all, about large-L physics,
our desired goal. We would like to create a different situation. We would
like to design a theory which (i) in the decompactification large-L limit tends
to conventional QCD and its QCD-like sisters; (ii) at small L is analytically
tractable and has both confinement and chiral symmetry breaking; and (iii)
NTrU? ∼ 0. This expectation is sup-
NTrU? = 1. In this
2By QCD-like theories we mean non-Abelian gauge theories without elementary scalars,
e.g., Yang–Mills with fermions in the two-index symmetric or antisymmetric representa-
tion, to be referred to as S/AS, see below.
3

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has as smooth transition between the small and large-L domains as possible
(the upper plot in Fig. (1)). If this endeavor — rendering small and large-L
physics continuously connected — is successful, we could try to use small-L
laboratory to extract lessons about QCD and QCD-like theories on R4.
We will argue below that the goal can be achieved by performing a so-
called double-trace deformation of QCD and QCD-like theories.3To this end
we add a non-local operator
P[U(x)] =
2
π2L4
[N
?
2]
n=1
dn|TrUn(x)|2
for SU(N), (4)
to the QCD action,
∆S =
?
R3
d3xLP[U(x)], (5)
were dn are numerical parameters to be judiciously chosen. The theories
obtained in this way will be labeled by asterisk. In minimizing S + ∆S the
effect due to deformation (4) is two-fold. First, it tends to minimize |TrU(x)|.
Second it tends to maximize the distance between the eigenvalues of U. It
is necessary to have a polynomial of order [N/2] to force the eigenvalues
of the Polyakov line to be maximally apart from one another, i.e. to push
the theory towards the center-symmetric point depicted in Fig. 2. Here [x]
stands for the integer part of x. To stabilize the vacuum sufficiently close to
the center-symmetric configuration the coefficients dnmust be large enough,
presumably, dn∼ 1. Some technical details are discussed in Appendix.
At large L, the deformation switches off and has no impact on the theory,
i.e. QCD∗≈ QCD. However, at small L the impact is drastic. Given an
appropriate choice of dn’s the deformation (5) forces the theory to pick up
the following set4of the vacuum expectation values (VEVs):
vk= e
2πik
N ,k = 1,...,N, (6)
(or permutations), see Fig. 2. If we define
eiaL≡ U, (7)
3The double trace deformations were previously discussed in the context of gauge/string
theory dualities in [6, 7, 8, 9], as well as in field theory [10, 11, 12].
4More exactly, the set of VEVs will be very close to (6).
4