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
Page 6
divergent regime, accompanied by occurrence of a vacuum degeneracy at a
small value of m, is not supported by any symmetry, and is highly unlikely
whenever the inclusion of dynamical fermions does not spoil asymptotic free-
dom and confinement of the original Yang–Mills theory. One may still worry
about the convergence of the expansion3at m = 0, since one could suspect a
priori that infrared singularities would develop invalidating the convergence.
However, also in this respect, the two theories at hand, the one with and
the one without fermions, behave similarly: indeed, there are no massless
particles in the physical spectrum of either pure Yang–Mills or of the theory
with a single fermionic flavor.
Vacuum uniqueness implies no spontaneous breaking of C. Note, that
in the general case the proof of planar equivalence is carried out at finite m
(then it becomes an equivalence between a non-supersymmetric theory and
the one with softly broken supersymmetry [4]). Taking the limit m → 0
at the end requires a special consideration. Infrared singularities due to
massless particles are potentially dangerous; one must check that they do
not contribute in an essential way. To this end the limit N → ∞ must be
taken first.
3 Implications for planar equivalence
According to [11], the absence of spontaneous breaking of C-parity is a
necessary and sufficient condition for the validity of orientifold planar equiva-
lence. The necessity of this condition is absolutely obvious. The very possibil-
ity of a spontaneous breaking of C-parity is related to the possible existence
of two degenerate vacua of the theory transforming into each other under the
action of C parity. In pure Yang–Mills theory the vacuum is unique.4
Since Dirac quarks do not alter the conclusion – charge conjugation can-
not be broken on R4– we can assert that planar equivalence holds when the
theory is formulated on R4. In this section we wish to elucidate this issue
with regards to the “refined” proof of Ref. [8].
Let us recall the main points of [8]. It was shown there that the parti-
tion functions of N = 1 super-Yang–Mills and the orientifold field theory,
after integration over the fermions, coincide at large N. The corresponding
3We thank M.¨Unsal and L. Yaffe for a discussion on this issue.
4For simplicity in this section we set θ = 0.
5
Page 7
derivation is based on the expansion of the partition functions in powers of
lndet(i ?D − m) and then expressing generic terms of the expansion in terms
of Wilson loops5. Then we compare the results, term by term,
?WSYMWSYM...WSYM?conn.= ?WQCD-ORWQCD-OR...WQCD-OR?conn., (4)
at large N. The subscripts above are self-evident.
For example, for a single Wilson loop, Eq. (4) reduces to
?WSYM? = ?WQCD-OR?. (5)
The reason why Eq. (5) holds is that, at large N,
?WSYM? = ?TrU TrU†? = ?TrU??TrU†?,(6)
?WQCD-OR? = ?TrU TrU? = ?TrU??TrU?, (7)
where U is the Wilson loop in the fundamental representation. Under the
assumption that C-parity is not broken,
?TrU? = ?TrU†?,
and Eq. (5) is obviously satisfied. The equality of all connected correlation
functions similarly follows [8] under the assumption that C-parity is not
broken.
A closer look at our proof reveals that in fact we needed to make only
the very mild assumption of no C-parity breaking in pure Yang–Mills theory
(on R4). This is due to the fact that all the above Wilson loop correlation
functions are calculated in the Yang–Mills vacuum, and not in the vacuum
of the theory with quarks. How is the presence of quarks felt here?
Since we expand in fermion loops, starting from pure Yang–Mills theory,
if this expansion is convergent, the number of vacua cannot change compared
to that in pure Yang–Mills theory. The latter has a unique vacuum, and so
does the theory with quarks defined by this expansion. In order to ensure the
convergence of the fermion loop expansion on R4we introduced the infrared
cut-off – a mass term for the fermions. We believe that on R4it is sufficient,
but we have to assume that the limit m → 0 is nonsingular. Practically, it
5It is essential to add a small quark mass, see comments at the end of this section.
6
Page 8
is impossible to think otherwise, since there are no massless particles in the
theories under consideration at m = 0. If the vacuum is unique, certainly
there is no place for the spontaneous breaking of C-parity.
There is another subtle issue which requires a mild assumption in our
proof [8]. We assume that the sum of the correlation functions commutes
with factorization. Lattice simulations can also shed light on this point.
4 Discussion and conclusions
The result of Ref. [11], and of the previous investigation [19], can be inter-
preted as follows. The expansion in fermion loops does not exist in the ori-
entifold field theory on R3×S1with periodic boundary conditions, provided
the S1radius is small enough. Inclusion of fermions drastically changes the
vacuum structure compared to that of pure Yang–Mills theory. Technically,
the spontaneous breaking of C-parity in the theory on R3×S1, demonstrated
in [11], is associated with an order parameter that breaks P invariance. In-
deed, an expectation value of the Wilson line in the compactified direction
is generated. This is equivalent to the expectation value of a component of
a vector. Needless to say, this is an artifact of working with a Lorentz-non-
invariant theory. There are no lessons to be drawn from this analysis for
Lorentz-invariant theories.
Our work was motivated by the potential importance of the claimed non-
perturbative planar (large N) equivalence between supersymmetric gluody-
namics and the orientifold field theory (one-flavor QCD at N = 3). Such an
equivalence, once firmly established, will provide a unique analytic tool in
studying both theories nonperturbatively. Although numerical methods are
— or will become — available for light dynamical fermions, the importance
of making reliable analytic predictions can hardly be overestimated.
Acknowledgments
We are very grateful to Mithat¨Unsal and Larry Yaffe who pointed out a
loophole in an argument presented in the draft version of this paper. A.A.
thanks S. Elitzur, S. Hands, T. Hollowood, C. Hoyos, P. Kumar, A. Patella
and B. Svetitsky for discussions. M.S. is grateful to T. Cohen for useful
7
Page 9
exchange of opinions. G.V. would like to thank L. Giusti and E. Rabinovici
for interesting discussions.
A.A. is supported by the PPARC advanced fellowship award. The work
of M.S. is supported in part by DOE grant DE-FG02-94ER408.
References
[1] M. J. Strassler, On methods for extracting exact nonperturbative results in
nonsupersymmetric gauge theories, hep-th/0104032.
[2] A. Armoni, M. Shifman and G. Veneziano, Nucl. Phys. B 667, 170 (2003)
[hep-th/0302163].
[3] A. Armoni, M. Shifman and G. Veneziano, Phys. Rev. Lett. 91, 191601 (2003)
[hep-th/0307097].
[4] A. Armoni, M. Shifman and G. Veneziano, From super-Yang–Mills theory to
QCD: Planar equivalence and its implications, in From Fields to Strings: Cir-
cumnavigating Theoretical Physics, Eds. M. Shifman, A. Vainshtein, and J.
Wheater, (World Scientific, Singapore, 2005), Vol. 1, p. 353, hep-th/0403071.
[5] A. Armoni, M. Shifman and G. Veneziano, Phys. Lett. B 579, 384 (2004)
[hep-th/0309013].
[6] T. DeGrand, R. Hoffmann, S. Schaefer and Z. Liu, Phys. Rev. D 74, 054501
(2006) [hep-th/0605147].
[7] P. Keith-Hynes and H. B. Thacker, Double hairpin diagrams and the pla-
nar equivalence of N = 1 supersymmetric Yang–Mills theory and one-flavor
QCD, hep-lat/0610045; Relics of supersymmetry in ordinary one-flavor QCD:
Hairpin diagrams and scalar-pseudoscalar degeneracy, hep-th/0701136.
[8] A. Armoni, M. Shifman and G. Veneziano, Phys. Rev. D 71, 045015 (2005)
[hep-th/0412203].
[9] A. Patella, Phys. Rev. D 74, 034506 (2006) [hep-lat/0511037].
[10] J. L. F. Barbon and C. Hoyos, JHEP 0601, 114 (2006) [hep-th/0507267].
[11] M.¨Unsal and L. G. Yaffe, Phys. Rev. D 74, 105019 (2006) [hep-th/0608180].
8
Page 10
[12] P. Kovtun, M.¨Unsal and L. G. Yaffe, JHEP 0312, 034 (2003) [hep-
th/0311098]; JHEP 0507, 008 (2005) [hep-th/0411177].
[13] F. Sannino, Phys. Rev. D 72, 125006 (2005) [hep-th/0507251].
[14] T. J. Hollowood and A. Naqvi, Phase transitions of orientifold gauge theories
at large-N in finite volume, hep-th/0609203.
[15] T. DeGrand and R. Hoffmann, QCD with one compact spatial dimension,
hep-lat/0612012.
[16] C. Vafa and E. Witten, Phys. Rev. Lett. 53, 535 (1984).
[17] E. Witten, Phys. Rev. Lett. 81, 2862 (1998) [hep-th/9807109].
[18] M. A. Shifman and A. I. Vainshtein, Nucl. Phys. B 296, 445 (1988).
[19] T. D. Cohen, Phys. Rev. D 64, 047704 (2001) [hep-th/0101197].
9
Download full-text