The subject of this talk was the review of our study of three ($2+1$)
dimensional Quantum Chromodynamics. In our previous works, we showed the
existence of a phase where parity is unbroken and the flavor group $U(2n)$ is
broken to a subgroup $U(n)\times U(n)$. We derived the low energy effective
action for the theory and showed that it has solitonic excitations with Fermi
statistic, to be identified with the three dimensional ``baryon''. Finally, we
studied the current algebra for this effective action and we found a
co-homologically non trivial generalization of Kac-Moody algebras to three
dimensions.Comment: 7 pages, Plain TEX, talk presented by S.G. Rajeev at the XXVI
  INTERNATIONAL CONFERENCE ON HIGH ENERGY PHYSICS, DALLAS TX AUG. 199

Ferretti, G.

Rajeev, S. G.

Yang, Z.

English

arXiv

G. Ferretti

S. G. Rajeev

Z. Yang

Crossref

Three dimensional quantum chromodynamics

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UR-1278
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hep-th/9209129
THREE DIMENSIONAL QUANTUM CHROMODYNAMICS†
G. Ferretti, S.G. Rajeev‡ and Z. Yang
Department of Physics and Astronomy
University of Rochester
Rochester NY 14627
ABSTRACT
The subject of this talk was the review of our study of three (2+1) dimensional Quantum
Chromodynamics. In our previous works, we showed the existence of a phase where parity
is unbroken and the flavor group U(2n) is broken to a subgroup U(n)×U(n). We derived
the low energy effective action for the theory and showed that it has solitonic excitations
with Fermi statistic, to be identified with the three dimensional “baryon”. Finally, we
studied the current algebra for this effective action and we found a co-homologically non
trivial generalization of Kac-Moody algebras to three dimensions.
† This work was supported in part by DOE Grant DE-FG02-91ER40685.
‡ Talk presented by S.G. Rajeev at the XXVI International Conference on High Energy
Physics, Dallas TX Aug. 1992
1
INTRODUCTION
Quantum Chromodynamics (QCD) is the universally accepted theory of strong in-
teractions. In spite of this fact, there are still many unresolved issues that need to be
addressed before we can declare our understanding of QCD complete. Among the least
understood problems are those that cannot be studied by perturbation theory, such as
chiral symmetry breaking and quark confinement. It is therefore natural to look for other
models that retain the basic features of QCD but allow one to study these issues in a
simpler setting. One way to construct such models is to lower the dimensionality of the
system.
Two (1 + 1) dimensional QCD has been extensively studied but it fails to be a good
analogue for some purposes: gauge symmetries in two dimensions are somewhat trivial and
no spontaneous breaking of continuous symmetries can occur. We chose to consider three
dimensional QCD and to study the breaking of the global symmetries in this context. We
find a remarkable similarity between this model and what is known about four dimensional
QCD from the study of the Skyrme model. Working in lower dimensions we have the
extra bonus of obtaining an effective theory that is more tractable. This theory can be
regarded as a limiting case of coset models that are renormalizable in the 1/N expansions.
Furthermore, its current algebra is a nontrivial abelian extension of the naive algebra and
it might give rise to interesting representation theory.
THREE DIMENSIONAL QCD
Let us begin by writing down the Lagrangian density for three dimensional QCD. The
gauge group is SU(Nc), the tensor Fµν is the curvature associated to the gauge field and qi
represent the quark fields. The flavor index i runs from 1 to N and we will always assume
N to be even (N = 2n). Color and spinor indices are suppressed but it should be kept in
mind that, as spinor, qi is a two component complex Grassmann field.
In this notation, the Lagrangian reads
L = − 1
α
tr FµνF
µν +
∑
i
q̄i(γ · ∇+mi)qi (1)
In the massless limit, this Lagrangian possess a global Z2 × U(2n) symmetry. Since the
number of flavor is even, it is possible to make all quarks massive without breaking parity
2
explicitly. This is done by pairing them into doublets (qi, q−i) of equal and opposite masses,
mi = −m−i, i = 1 · · ·n.
Our two main results regarding the spontaneous breaking of the global symmetries
are the following. First, parity is not spontaneously broken if N is even. No Chern-Simons
term for the color gauge field can arise and the theory should be in the confining phase.
The proof of this statement is an adaptation of the argument given by Vafa and Witten for
four dimensional QCD2. It relies on the possibility of defining a positive fermionic measure
for the path integral. For completeness, let us remark that this argument fails when the
number of flavors is odd3, indicating that the theory is in a parity violating phase and
therefore it is not a good analogue for QCD.
Second, in the massless limit, the flavor symmetry U(2n) is spontaneously broken to
U(n)×U(n). By arguments similar to those given by Coleman and Witten4, one can show
that if the symmetry is broken, then it must break to this specific subgroup. One is then
left with the task of finding at least one correlation function of the theory that breaks the
U(2n) symmetry in the limit of vanishing quark masses. This function can be chosen to
be the two point correlation function of the flavor currents:
< J iµ(k)J
k
ν (−k) >=
iNc
4π
ǫikǫµνρk
ρ +O(k2). (2)
In the massless limit, the leading term of this correlation function is proportional to
ǫik = ±δik, the plus sign being present for i = k > 0, the minus sign being present in the op-
posite case. The coefficient in (2) can be calculated exactly from some no-renormalization
theorems5.
EFFECTIVE LAGRANGIAN
Having understood the symmetry breaking pattern, we can now write down the ef-
fective Lagrangian1 for the light particles of the theory, i.e. the Goldstone bosons “pions”
associated to the broken generators. These are described by a field Φ valued in the Grass-
mannian manifold Gr2n,n = U(2n)/U(n) × U(n). Earlier, a different coset model with
a non trivial topological term had been proposed by Rabinovici et al.6, where the target
manifold was not the complex Grassmannian but the projective quaternionic space.
By the standard properties of the Grassmannian, Φ can be regarded as a 2n × 2n
3
traceless hermitian matrix satisfying the condition Φ2 = 1. The odd dimensional analogue
of the Wess-Zumino-Witten term (WZW) can also be found by studying the generators
of H4(Gr2n,n). As in the four dimensional case, the WZW term cannot be written as the
integral of a local density but one can regard space-time M3 as the boundary of a four
dimensional manifold with boundary M4.
The form of the effective action is, in differential geometry notation,
S =
Fπ
2
∫
M3
dΦ ∗ dΦ+ Nc
64π
∫
M4
tr Φ(dΦ)4. (3)
The constant Fπ has the dimension of a mass in natural units and can be regarded as the
“pion decay constant”. The particular form for the WZW term (second term in eq. (3)),
has been chosen so that the equation of motion derived from it possess the same discrete
symmetry as three dimensional QCD. In particular, the equation of motion must violate
both internal ((t, x, y) → (t,−x, y)) and external (Φ → −Φ) parity but it must preserve
the combination of both. The coefficient Nc/64π in front of the WZW term is fixed by
requiring exp(−S) to be independent on the extension to M4 and by comparison with (2).
One can relax the assumption that led to eq. (3) and include the vector mesons as
dynamical degrees of freedom into the Lagrangian. It turns out that this is the correct way
to study the solitonic excitations (“baryons”) of the theory. The vector mesons provide
the short range repulsion needed to stabilize the soliton.
In our theory, a Chern-Simons term for these vector mesons arises from the WZW
term. This is the Chern-Simons term for the unbroken subgroup U(n) × U(n), not to be
confused with the (unexisting) color Chern-Simons term. Each U(n) factor in the unbroken
subgroup gives rise to its own Chern-Simons term. The relative coefficients of the two terms
must be chosen to be equal and opposite in order to preserve parity as a symmetry of the
effective theory. By writing Φ = χǫχ†, χ ∈ U(2n), ǫ as in (2) and denoting by A1 and A2
the vector fields associated to each U(n), the new effective action becomes
S̃ =
Fπ
2
∫
tr (∇µχ†∇µχ+m2πǫχǫχ†)dx3 +
k
4π
2
∑
i=1
(−1)i
∫
tr (AidAi +
2
3
A3i ) (4)
The two models (3) and (4) coincide in the limit of infinite vector meson mass.
4
BARYONS AS SOLITONS
The “baryons ” of three dimensional QCD are described by the static solitonic solu-
tions of eq (4) with winding number one7. (Recall that π2(Gr2n,n) = Z, the baryon number
B is always identified with the winding number.) The low lying baryons transform under
flavor symmetry just as predicted by the quark model. The only important difference is
in the size of the baryon. The size predicted by the sigma model is larger by a factor
log(Fπ/Ncmπ) than the prediction from the naive quark model. This seems to indicate
that three is the lowest critical dimension for the applicability of the Skyrme model.
There are also higher baryon number solutions. Particularly pleasing is the existence of
a cylindrically symmetric B = 2 “di-baryon” solution. The existence of these solutions, in
particular the stability of the di-baryon solution has been proven by numerical methods,
since the equation of motion that arises for the cylindrically symmetric ansatz are not
exactly solvable. By use of relaxation methods we have obtained the radial profile for the
baryon density in both the B = 1 and the B = 2 case7.
CURRENT ALGEBRA
The effective action (3) admits an interesting current algebra. The canonical formula-
tion of the WZW model yields the Kac-Moody algebra in two dimension but fails to give
a Lie algebra in four. In our intermediate case we find that the current algebra (more
precisely the current-field algebra) is still a Lie algebra8. Care must be taken in choosing
the correct expression for the current J in a way that makes the Poisson brackets linear
in Φ and J . If the two dimensional space-like surface Σ is a torus, we can write these
relations in a plane wave basis:
{Φam,Φbn} = 0, {Jam,Φbn} = fabcΦcm+n
{Jam, Jbn} = fabcJcm+n −
k
16π
dabcǫijminjΦ
c
m+n
(5)
Above, m,n are two-dimensional vectors with integer components. Also, dabc is the usual
symmetric cubic invariant of U(2n) and fabc the structure constants.
The canonical formulation is completed by two first class constraints
Φ2 − 1 = 0 [J,Φ]
+
+
k
16π
ǫij(∂iΦ∂jΦ) = 0, (6)
5
and by the Hamiltonian function
H =
1
2
∫
Σ
tr
(
− 1
Fπ
√
g
(
J +
k
32π
ǫij∂iΦ∂jΦ
)2
+
Fπ
√
g
4
gij∂iΦ∂jΦ
)
d2x. (7)
It should be stressed that the quantities J and Φ are invariant under the action of
“gauge transformations” generated by the constraints. The constraints themselves express
the fact that the co-adjoint orbit of this algebra is the cotangent bundle of the Grassman-
nian.
We could quantize the action (3) if we could find a unitary, highest weight repre-
sentation for this algebra. At first sight this seems rather unphysical because (3) is not
perturbatively renormalizable. However, (3) belongs to a class of models that are renor-
malizable in the 1/N expansion. This is more clear if we consider a less symmetrical
Grassmannian, where Φ takes values in GrN,nU(N)/U(N − n) × U(n). For n = 1 this
is just the usual CPN−1 model, known to be renormalizable in the large N limit. For
n > 1, this model is still renormalizable in the limit N → ∞, n finite. Its current algebra
is still given by (5) and everything is left unchanged, except that now tr Φ = N − 2n 6= 0.
What fails in this case is the connection with three dimensional QCD because for these
“asymmetrical” models it is not possible to preserve parity. We expect these models to
yield a good quantum theory and this would be very exciting by itself. The original model
on Gr2n,n can probably be approached as limiting case.
6
REFERENCES
[1.] G. Ferretti, S.G. Rajeev and Z. Yang, “The Effective Lagrangian of Three Dimensional
Quantum Chromodynamics” UR1255 Int. J. Mod. Phys. to appear, (1992).
[2.] C. Vafa and E. Witten, “Parity Conservation in Quantum Chromodynamics” Phys.
Rev. Lett. 53 535-536 (1984).
[3.] A.N. Redlich, “Gauge Noninvariance and Parity Nonconservation
of Three-Dimensional Fermions” Phys. Rev. Lett. 52 18-21 (1984).
[4.] S. Coleman and E. Witten, “Chiral-Symmetry Breakdown in Large-N Chromodynam-
ics” Phys. Rev. Lett. 45 100-102 (1980).
[5.] S. Coleman and B. Hill, “No More Corrections to the Topological Mass Term in
QED3” Phys. Lett. B159 184-188 (1985).
G. Ferretti and S.G. Rajeev, “CPN−1 Model with a Chern-Simons Term” Mod. Phys.
Lett. A23 2087-2094 (1992).
[6.] E. Rabinovici, A. Schwimmer and S. Yankielowicz, “Quantization in the Presence of
Wess-Zumino Terms” Nucl. Phys. B248 523-535 (1984).
[7.] G. Ferretti, S.G. Rajeev and Z. Yang, “Baryons as Solitons in Three Dimensional
Quantum Chromodynamics” UR1256 Int. J. Mod. Phys. to appear, (1992).
[8.] G. Ferretti and S.G. Rajeev, “Current Algebra in Three Dimensions” Phys. Rev. Lett.
to appear, (1992).
7


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