Page 1

arXiv:hep-th/0701229v2 27 Feb 2007

SWAT/07/505

FTPI-MINN-06/38

CERN-PH-TH/2007-013

UMN-TH-2528/06,

A note on C-Parity Conservation and the

Validity of Orientifold Planar Equivalence

Adi Armoni,aMikhail Shifman,band Gabriele Venezianoc,d

aDepartment of Physics, Swansea University,

Singleton Park, Swansea, SA2 8PP, UK

bWilliam I. Fine Theoretical Physics Institute,

University of Minnesota, Minneapolis, MN 55455, USA

cTheory Division, CERN

CH-1211 Geneva 23, Switzerland

dColl` ege de France, 11 place M. Berthelot, 75005 Paris, France

Abstract

We analyze the possibility of a spontaneous breaking of C-invari-

ance in gauge theories with fermions in vector-like — but otherwise

generic — representations of the gauge group. QCD, supersymmet-

ric Yang–Mills theory, and orientifold field theories, all belong to this

class. We argue that charge conjugation is not spontaneously bro-

ken as long as Lorentz invariance is maintained. Uniqueness of the

vacuum state in pure Yang–Mills theory (without fermions) and con-

vergence of the expansion in fermion loops are key ingredients. The

fact that C-invariance is conserved has an interesting application to

our proof of planar equivalence between supersymmetric Yang–Mills

theory and orientifold field theory on R4, since it allows the use of

charge conjugation to connect the large-N limit of Wilson loops in

different representations.

Page 2

1Introduction

There are very few tools that enable us to explore QCD in the nonpertur-

bative regime. Recently, building on an earlier idea due to Strassler [1], we

suggested a new tool for analyzing nonperturbative QCD [2, 3]. We argued

that one-flavor QCD can be approximated, within a 1/N error, by N = 1

super-Yang–Mills theory [3] (for a review see [4]). The relation between QCD

and super-Yang–Mills was established by observing that SU(N) gauge theory

with the Dirac two-index antisymmetric fermion (to be referred to as the ori-

entifold field theory) is nonperturbatively equivalent to super-Yang–Mills in a

well defined bosonic subsector at N → ∞. Planar equivalence led to several

strong predictions concerning QCD. Among them are the value of the quark

condensate [5] and the degeneracy of the σ and the η′mesons [3] in one-flavor

QCD. These predictions were supported by recent lattice simulations [6, 7].

In Ref. [8] we gave a formal proof of planar equivalence (see Ref. [9] for a

lattice strong-coupling version of the proof). Our proof assumes, implicitly,

charge conjugation invariance. The proof does not hold on compact spaces,

such as R3× S1, as was first demonstrated in [10].

Recently it was pointed out [11] (for earlier works see [12]) that a nec-

essary and sufficient condition for orientifold planar equivalence to hold is

the absence of spontaneous breaking of charge conjugation symmetry. To

further explore this observation the authors of [11] considered the orientifold

field theory on R3×S1(with a small radius of S1, so that the one-loop analy-

sis can be trusted) and demonstrated that C-parity is spontaneously broken

in this case, the order parameter being the Polyakov line in the compacti-

fied direction.1It was concluded that planar equivalence does not hold on

R3× S1, at least for sufficiently small radii.2

The above result was advertised (see e.g. the title of [11]) as raising doubts

concerning the validity of planar equivalence on R4. Shortly after, the phase

1In a revised version of the paper [11], it was argued that not only C-parity is broken,

but CPT as well.

2In fact, a more careful reading of [11] implies that C parity conservation and planar

equivalence are invalid only if periodic boundary conditions are imposed on the fermions.

With antiperiodic boundary conditions, both C parity and planar equivalence do hold, in

contrast to claims otherwise [13]. The sensitivity to the boundary conditions is a clear-cut

indication that this is a finite-size effect. The effects found in [11] are just Casimir-like

effects that vanish as the theory decompactifies, i.e. as R → ∞.

1

Page 3

structure of the orientifold field theory on S3× S1was analyzed [14]. It was

shown that, if the S3radius is sufficiently small so that perturbation theory

can be trusted, the theory undergoes a phase transition as the S1radius

increases.At large radius charge conjugation is restored.

analysis of [14] certainly cannot be trusted in the domain of large S3radii, it

could still be considered as pointing in the opposite direction, i.e. that planar

equivalence holds on R4. Another indication that planar equivalecnce holds

on R4is provided by a recent lattice simulation [15], showing that QCD on

a circle undergoes a phase transition from a C-parity violating phase to a

C-parity preserving phase above a critical radius.

The purpose of this paper is to argue that C-parity does not break spon-

taneously in any vector-like gauge theory on R4. Although we will not be

able to give a rigorous mathematical proof of the type known for spatial

parity [16], we will present several convincing physical arguments that seem

impossible to overcome.

In addition, we clarify certain aspects of our proof [8] and point out which

particular aspects of the set-up of Ref. [11] are to blame for the failure of

C-parity and planar equivalence on R3× S1.

Although the

2

C-parity in pure Yang–Mills and

vector-like gauge theories

In this section we will first argue that C-parity is not spontaneously broken

in Yang–Mills theories on R4. We will then argue that, if C is not broken in

pure Yang–Mills theory, it cannot be spontaneously broken if we add fermions

in vector-like (real) representations.

The impossibility of spontaneous breaking of P-parity at θ = 0 was proven

long ago [16]. This proof is in essence nondynamical and is based only on

certain general features of Yang–Mills theories. Unlike spatial parity, the

issue of spontaneous breaking of C invariance depends on the dynamics.

This is the reason why we cannot prove our assertion at the same level of

rigor as that of Ref. [16]. Instead, we rely on a number of independent

physical arguments which exploit known features of the gauge dynamics.

Consider first pure Yang–Mills theory on the cylinder R3× S1with a

large radius, and the suspected order parameter for the spontaneous C-parity

2

Page 4

breaking, the Polyakov line in the compact direction (let us call it t),

P = Tr exp

?

i

?

A0dt

?

. (1)

If C is spontaneously broken, the vacuum expectation value (VEV) of the

Polyakov line will acquire an imaginary part, corresponding to two degenerate

vacua with Im?P? = ±K with K a non-vanishing constant.

However, such a nonvanishing VEV would contradict color confinement in

pure Yang–Mills theory. Indeed, the order parameter (1) is simultaneously

an order parameter for the group’s center. It must vanish in the confine-

ment phase: and it does at low temperatures, a well-established fact. There

is a critical temperature below which color confinement is recovered, and,

correspondingly, ?P? = 0.

There is a subtle point in this argument. One can say that, as the ra-

dius of the cylinder grows and eventually crosses the critical value beyond

which ?P? = 0, the Polyakov line no longer represents an appropriate order

parameter for the spontaneous breaking of C invariance of the theory.

Therefore, let us look at this problem from a more general perspective.

The Yang–Mills Lagrangian is C invariant. Therefore, for the spontaneous

breaking to take place in pure Yang–Mills theory its vacuum structure must

be nontrivial. There should exist two (more generally, an even number of)

degenerate vacua with opposite C parities.

A certain amount of knowledge has been accumulated regarding the vac-

uum structure of pure Yang–Mills theory. From Witten’s work [17] we know

that the vacuum of pure Yang–Mills theory is nondegenerate at θ = 0. The

vacuum energy is given by the partition function

Z=

?

DAµexp

?

−

?

R4

d4x

?

−1

2g2TrF2

µν+ i

θ

16π2TrF˜F

??

≡ exp(−V E(θ)) .(2)

The vacuum at θ = 0 is the absolute minimum of energy with respect to

other θ vacua since it is a sum of positive contributions (in the Euclidean

formulation). What is of relevance for us is that it is also nondegenerate. The

latter property can be seen explicitly at large N by considering the string

dual to pure Yang–Mills [17].

3

Page 5

The uniqueness (nondegeneracy) of the pure Yang–Mills vacuum can be

argued from a different side, starting from a slightly broken supersymmetric

gluodynamics (i.e. N = 1 supersymmetric pure Yang–Mills theory). This

theory has N degenerate vacua due to the spontaneous breaking of the Z2N

chiral symmetry down to Z2. The order parameter is the gluino condensate

which was exactly calculated in [18].

When a small mass m/Λ ≪ 1 is given to the gluino field, the vacuum

degeneracy is lifted resulting in a theory with a nondegenerate vacuum. Pure

Yang–Mills theory is recovered by taking the limit m/Λ → ∞. If the vacua of

pure Yang–Mills theory were degenerate, in order to end up with degenerate

vacua starting from a nondegenerate one, one would have a phase transition

to occur at an intermediate value of m. This is certainly impossible if the

expansion in fermion loops (see below) is convergent.

Finally, let us note that the uniqueness of the Yang–Mills vacuum is

supported by lattice simulations.

Let us now include fermions. In vector-like theories C invariance is pre-

served at the Lagrangian level. As in the previous case of pure Yang–Mills,

spontaneous breaking requires degenerate vacua.

uniqueness of the vacuum state of pure Yang–Mills (at θ = 0), we may ask

what should happen in order that the inclusion of dynamical fermions intro-

duces vacuum degeneracy.

Since fermions appear in the action bilinearly, it is easy to formally inte-

grate them out. The resulting theory is defined by the partition function

Taking for granted the

Z =

?

DAµexp{−SYM+ ln det(?D + m)} , (3)

where m is a mass parameter (m ?= 0, which can be viewed either as physical

or as an infrared regulator to be sent to zero at the end), and the expression

(3) is given in Euclidean space. Because of the fermion determinant (which

is taken in the appropriate representation of SU(N)color) the new effective

action is nonlocal, but this is irrelevant for our purposes.

The expansion in fermion loops is just the expansion of the above (func-

tional) integrand in powers of ln det(?D + m) with the leading term corre-

sponding to the pure Yang–Mills theory. If such an expansion is convergent,

the unique nondegenerate vacuum of pure Yang–Mills is inherited by the

theory with dynamical fermions. It is certainly the case for sufficiently large

m, as a result of the Appelquist-Carazzone theorem. A sudden switch to a

4