Glueball Masses in (2+1)-Dimensional Anisotropic Weakly-Coupled
  Yang-Mills Theory
NSF-KITP-07-57
BCCUNY-HEP/07-03
GLUEBALL MASSES IN (2 + 1)-DIMENSIONAL
ANISOTROPIC WEAKLY-COUPLED YANG-MILLS
THEORY
Peter Orlanda.b.c.1
a. Kavli Institute for Theoretical Physics, The University of California, Santa
Barbara, CA 93106, U.S.A.
b. Physics Program, The Graduate School and University Center, The City
University of New York, 365 Fifth Avenue, New York, NY 10016, U.S.A.
c. Department of Natural Sciences, Baruch College, The City University of New
York, 17 Lexington Avenue, New York, NY 10010, U.S.A.
Abstract
The confinement problem has been solved in the anisotropic (2+1)-dimensional SU(N)
Yang-Mills theory at weak coupling. In this paper, we find the low-lying spectrum for
N = 2. The lightest excitations are pairs of fundamental particles of the (1 + 1)-
dimensional SU(2) × SU(2) principal chiral nonlinear sigma model bound in a linear
potential, with a specified matching condition where the particles overlap. This match-
ing condition can be determined from the exactly-known S-matrix for the sigma model.
1giantswing@gursey.baruch.cuny.edu
http://arxiv.org/abs/0704.0940v2
1 Introduction
In recent papers, some new techniques have been developed for calculating quantities in
a (2+1)-dimensional SU(N) gauge theories [1], [2], [3]. These techniques exploit the fact
that in an anisotropic limit of small coupling, the gauge theory becomes a collection of
completely-integrable quantum field theories, namely SU(N)× SU(N) principal chiral
nonlinear sigma models. These integrable systems are decoupled, save for a constraint
which is necessary for complete gauge invariance. In the case of N = 2, is possible to
perturb away from integrability, using exactly-known off-shell matrix elements of the
integrable theory.
Though the gauge theory we consider s not spatially-rotation invariant, it has fea-
tures one expects of real (3+1)-dimensional QCD; it is asymptotically free and confines
quarks at weak coupling. Thus the limit of no regularization is accessible.
One can formally remove the regulator in strong-coupling expansions of (2 + 1)-
dimensional gauge theories; the vacuum state in this expansion yields a string tension
and a mass gap which have formal continuum limits. This can be done in a Hamiltonian
lattice formalism [4], or with an ingenious choice of degrees of freedom and point-
splitting regularization [5]. This leaves open the question of whether these expressions
can be trusted at weak coupling (more discussion of this issue can be found in the
introduction of reference [2]). In particular, one would like to rule out a deconfinement
transition, or different dependence of physical quantities on the coupling (as in compact
QED [6]). There is a proposal for the vacuum state [7], in the formulation of reference
[5] which seems to give correct values for some glueball masses [8], but this proposal
evidently requires more mathematical justification.
In this paper, we will work out the masses of the lightest glueballs for the case
of gauge group SU(2). Our method would also work in principle for SU(N) gauge
theories, and our reason for choosing N = 2 is that the analysis is simplest for that
case.
The basic connection between the gauge theory and integrable systems is most easily
seen in axial gauge [1]. The string tensions in the x1-direction and x2-direction (which
we called the horizontal and vertical string tensions, respectively) for very small g′0,
were found by simple physical arguments. The result for the horizontal string tension
was confirmed for gauge group SU(2), and additional corrections in g′0 were found [2],
through the use of exact form factors for the currents of the sigma model. String
tensions for higher representations can also be worked out, and adjoint sources are not
confined [3].
Careful derivations of the connection between the gauge theory and integrable sys-
tems use the Kogut-Susskind lattice formalism [1], [2]. A shorter derivation was given
in reference [9], which we summarize again here. The formalism is essentially that of
“deconstruction” [10].
The Yang-Mills action is
d3L, where the Lagrangian is L = 1
2e′ 2
TrF 201+
TrF 202−
TrF 212, and where A0,A1 and A1 are SU(N)-Lie-algebra-valued components of the
gauge field, and the field strength is Fµν = ∂µAν − ∂νAµ − i[Aµ, Aν ]. This action
is invariant under the gauge transformation Aµ(x) → ig(x)−1[∂µ − iAµ(x)]g(x), where
g(x) is an SU(N)-valued scalar field. We take e′ 6= e, thereby losing rotation invariance.
We discretize the 2-direction, so that the x2 takes on the values x2 = a, 2a, 3a . . . ,
where a is a lattice spacing. All fields are considered functions of x = (x0, x1, x2). We
define the unit vector 2̂ = (0, 0, 1). We replace A2(x) by a field U(x) lying in SU(N),
via U(x) ≈ exp−iaA2(x). There is a natural discrete covariant-derivative operator:
DµU(x) = ∂µU(x) − iAµ(x)U(x) + iU(x)Aµ(x + 2̂a), µ = 0, 1, for any N × N complex
matrix field U(x). The action is S =
x2 a L where
2(g′0)
TrF 201 +
Tr[D0U(x)]†D0U(x)−
Tr[D1U(x)]†D1U(x) , (1.1)
and where g20 = e
0a and (g
2 = e′ 2a. The Lagrangian (1.1) is invariant under the gauge
transformation: Aµ(x) → ig(x)−1[∂µ − iAµ(x)]g(x) and U(x) → g(x)−1U(x)g(x + 2̂a)
where again, g(x) ∈ SU(N) and µ is restricted to 0 or 1. The bare coupling constants g0
and g′0 are dimensionless. We recover from (1.1) the anisotropic continuum action in the
limit a→ 0. The sigma model field is U(x0, x1, x2), and each discrete x2 corresponds to
a different sigma model. The system (1.1) is a collection of parallel (1+1)-dimensional
SU(N) × SU(N) sigma models, each of which couples to the auxiliary fields A0, A1.
The sigma-model self-interaction is the dimensionless number g0.
We feel it worth commenting on the nature of the anisotropic regime and how it
is different from the standard (2 + 1)-dimensional Yang-Mills theory. The point where
the regulator can be removed in the theory is g′0 = g0 = 0. This point can be reached
in our treatment, but only if
(g′0)
2 ≪ 1
e−4π/(g
N) . (1.2)
The left-hand side and ride-hand side are proportional to the two energy scales in the
theory (the latter comes from the two-loop beta function of the sigma model). Thus
our method cannot accommodate fixing the ratio g′0/g0, which is natural in standard
perturbation theory [11]. This is why the mass gap is not of order e, e′ and the string
tension is not of order e2, (e′)2.
We now discuss the Hamiltonian in the axial gauge A1 = 0. The left-handed and
right-handed currents are, jLµ (x)b = iTr tb ∂µU(x)U(x)
† and jRµ (x)b = iTr tb U(x)
†∂µU(x),
respectively, where µ = 0, 1. The Hamiltonian obtained from (1.1) is H0 +H1, where
{[jL0 (x)b]2 + [jL1 (x)b]2} , (1.3)
(g′0)
∂1Φ(x
1, x2)∂1Φ(x
1, x2)
)2 L2−a
jL0 (x
1, x2)Φ(x1, x2)− jR0 (x1, x2)Φ(x1, x2 + a)
+ (g′0)
2qbΦ(u
1, u2)b − (g′0)2q′bΦ(v1, v2)b , (1.4)
where −Φb = A0 b is the temporal gauge field, and where in the last term we have
inserted two color charges - a quark with charge q at site u and an anti-quark with
charge q′ at site v. Some gauge invariance remains after the axial-gauge fixing, namely
that for each x2
jL0 (x
1, x2)b − jR0 (x1, x2 − a)b
− g20Q(x2)b
Ψ = 0 , (1.5)
where Q(x2)b is the total color charge from quarks at x
2 and Ψ is any physical state. To
derive the constraint (1.5) more precisely, we started with open boundary conditions
in the 1-direction and periodic boundary conditions in the 2-direction, meaning that
the two-dimensional space is a cylinder [1], [2].
From (1.4) we see that the left-handed charge of the sigma model at x2 is coupled
to the electrostatic potential Φ, at x2. The right-handed charge of the sigma model
is coupled to the electrostatic potential at x2 + a. The excitations of H0, which we
call Fadeev-Zamoldochikov or FZ particles, behave like solitons, though they do not
correspond to classical configurations. Some of these FZ particles are elementary and
others are bound states of the elementary FZ particles. An elementary FZ particle has
an adjoint charge and mass m1. An elementary one-FZ-particle state is a superposition
of color-dipole states, with a quark (anti-quark) charge at x1, x2 and an anti-quark
(quark) charge at x1, x2 + a. The interaction H1 produces a linear potential between
color charges with the same value of x2. Residual gauge invariance (1.5) requires that
at each value of x2, the total color charge is zero. If there are no quarks with coordinate
x2, the total right-handed charge of FZ particles in the sigma model at x2 − a is equal
to the total left-handed charge of FZ particles in the sigma model at x2.
The particles of the principal chiral sigma model carry a quantum number r, with
the values r = 1, . . . , N − 1 [21]. Each particle of label r has an antiparticle of the
same mass, with label N − r. The masses are given by
mr = m1
sin rπ
sin π
, m1 = KΛ(g
−1/2e
N + non−universal corrections , (1.6)
where K is a non-universal constant and Λ is the ultraviolet cut-off of the sigma model.
Lorentz invariance in each x0, x1 plane is manifest. For this reason, the linear
potential is not the only effect of H1. The interaction creates and destroys pairs of
elementary FZ particles. This effect is quite small, provided that g′0 is small enough.
Specifically, this means that the square of the 1 + 1 string tension in the x1-direction
coming fromH1 is small compared to the square of the mass of fundamental FZ particle;
this is just the condition (1.2). The effect is important, however, in that it is responsible
for the correction to the horizontal string discussed in the next paragraph in equation
(1.8).
Simple arguments readily show that at leading order in g′0, the vertical and hori-
zontal string tensions are given by
, σH =
(g′0)
CN , (1.7)
respectively, where CN is the smallest eigenvalue of the Casimir of SU(N). These naive
results for the string tension have further corrections in g′0, which were determined for
the horizontal string tension for SU(2) [2]:
0.7296
(g′0)
e4π/g
. (1.8)
The leading term agrees with (1.7). This calculation was done using the exact form
factor for sigma model currents obtained by Karowski and Weisz [12]. The form factor
can also be used to find corrections of order (g′0)
2 to the vertical string tension; this
problem should be solved soon. If the reader is not familiar with form-factor techniques
in relativistic integrable field theories, a self-contained review is in the appendix of
reference [2].
Another recent application of exact form factors to the (2 + 1)-dimensional SU(2)
gauge theory is reference [13], in which form factors of the two-dimensional Ising model
[14] are used to find the profile of the electric string near the high-temperature decon-
fining transition, assuming the Svetitsky-Yaffe conjecture [15].
A rough picture of a gauge-invariant state for the gauge group SU(2) with no quarks
is given in Figure 1. For N > 2, there are more complicated ways in which strings can
join particles. For example, a junction of N strings is possible. Figure 1 is inaccurate
in an important respect; the “ring” of particles held together by horizontal strings is
extremely broad in extent in the x2-direction compared to the x1-direction. This is
because σH ≪ σV.
The lightest states have the smallest number of particles, by virtue of σH ≪ σV.
Thus the lightest glueballs are pairs of FZ particles with the same value of x2. For
small enough g′0, the very lightest state has a mass well-approximated by 2m1. The
purpose of this paper is to find the leading corrections in (g′0)
2 to this result. This
will be done using the S-matrix of the sigma model and the WKB formula. There
are further small corrections, due to the softening of the potential near where particles
overlap, which we do do not determine.
It is clear that the lightest bound states of FZ particles are (1 + 1)-dimensional in
character. If we formulated a gauge theory in which x2 was fixed in U(x0, x1, x2), we
would find the same spectrum, as a function of m1 and σH. In the Kogut-Susskind
lattice formulation, a long row of plaquettes with open boundary conditions is a regular-
ized gauge theory of this type. The only real difference between this (1+1) dimensional
model and that we study is that σH will receive different corrections of order (g
− − − − − − − − − − − −
− − − − − − − − − − − −
− − − − − − − − − − − −
− − − − − − − − − − − −
− − − − − − − − − − − −
− − − − − − − − − − − −
− − − − − − − − − − − −
− − − − − − − − − − − −
Figure 1. A glueball state is a collection of heavy particles, held weakly together
by strings. The horizontal coordinate is x1 and the vertical coordinate is x2.
In the next section we will discuss the wave function of an unbound pair of FZ
particles. We find that this is described by phase shift for the color-singlet sector. In
Section 3, we determine the bound-state spectrum. The problem we solve is very similar
to that of two particle-states of the two-dimensional Ising model with an external
magnetic field [16] (for a good summary of this problem, see reference [17]); the only
genuine difference is the presence of a matching condition where the particles overlap.
This matching condition comes from the phase shift of the scattering problem. We
present our conclusions in Section 4.
2 Scattering states of FZ particles
The lightest glueball state, as discussed above, is simply a pair of FZ particles located
at the points (x1, x2) and (y1, x2) and bound in a linear potential. Residual gauge
invariance (1.5), demands that the state be a color singlet. To begin with, however,
we simply write the form of a free state of two particles.
The state of the SU(2) × SU(2) ≃ O(4) nonlinear sigma model with a particles of
momenta p1 and p2 and quantum numbers j1 and j2 (which take the values 1, 2, 3, 4)
is described by the wave function
ψp1p2(x
1, y1)j1,j2 =
eip1x
1+ip2y
Aj1,j2 , x
1 < y1
eip2x
1+ip1y
k1,k2=1
Sk1k2j1j2 (p1, p2)Ak2,k1 , x
1 > y1
, (2.1)
where Aj1j2 is an arbitrary set of complex numbers and S
(p1, p2) is the two-particle
S-matrix. We have not yet imposed (1.5).
The wave function (2.1) is written in a form where the O(4) symmetry is manifest.
It is straightforward to write it in a form where the left SU(2)L and the right SU(2)R
symmetries are manifest, by writing
ψp1p2(x
1, y1)
j1,j2
(δj14ac − iσj1ac)
bd − iσ
∗ ψp1p2(x
1, y1)j1,j2 (2.2)
describing a pair of color dipoles, one with quantum numbers a, b̄ and the other with
quantum numbers c̄, d, where σj , j = 1, 2, 3 denotes the Pauli matrices.
We impose the physical state condition (1.5) on (2.2) by requiring that a = b and
c = d and summing over these colors. The projected wave function is, up to an overall
constant,
ψp1p2(x
1, y1) =
eip1x
1+ip2y
, x1 < y1
eip2x
1+ip1y
S0(p1, p2) , x
1 > y1
, (2.3)
where S0(p1, p2) is the singlet projection of the O(4) S-matrix. This S-matrix was first
obtained by Zamolodchikov and Zamolodchikov [18]. A useful form is given in reference
[12]:
S0(p1, p2) = S0(θ) = −
π − iθ
π + iθ
exp i
1− e−ξ
1 + eξ
, (2.4)
where the relative rapidity θ is given by θ = θ2− θ1, p1 = m sinh θ1, p2 = m sinh θ2 and
where we denote the particle mass m1, given by (1.6), by m (because there is only one
mass for the case of N = 2). A derivation of (2.4) is in the appendix of reference [2].
The singlet S-matrix is just given by a phase shift φ(θ): S0(θ) = exp iφ(θ). The
phase shift has a simple form in the low-energy, non-relativistic limit, |p1 − p2| ≪ m.
In this limit, θ ≈ |p1− p2|/m. The integral on the right-hand side of (2.4) can be done
by Taylor expanding in |p1 − p2|/m yielding
φ(θ) = φ(p1, p2) = π −
3− 2 ln 2
|p1 − p2| +O
|p− r|2
. (2.5)
3 The low-lying glueball spectrum
Let us now consider the states of a bound pair of FZ particles in the potential V (x1, y1) =
2σH|x1 − y1| (the reason for the factor of two is simply that the particles are joined by
a pair of strings). We use the non-relativistic approximation, used to find (2.5). For
our problem, the horizontal string tension times the size of a typical bound state is
small compared to the mass, by (1.2). This justifies the non-relativistic approximation
for low-lying states. The mass of a low-lying glueball is given by
M = 2m+ E ,
where E is the energy eigenstate of the two-particle problem.
Let us introduce center-of-mass coordinates, X = (x1+y1)/2 and x = y1−x1. The
reduced mass of the system is m/2. We factor out the phase depending on X , leaving
us only with a wave function depending on x. The Schrödinger equation we consider
+ 2σH|x|ψ = Eψ
with a matching condition at x = 0 between the wave function ψ(x) at x > 0 and the
wave function at x < 0. There is actually a further complication, which we do not
consider here; the potential changes slightly in the region where x ≈ 0. This is due
to the fact that the color charge is slightly smeared out. This smearing out can be
calculated from the form factor [12].
Our results (2.3), (2.5) for the unbound two-particle state, tell us that for x1 ≈ y1,
where the effect of the potential can be ignored, the bound-state wave function in the
center-of-mass frame will be of the form
ψ(x) =
cos(px+ ω) , x < 0
cos[−px+ ω − φ(p)] , x > 0 , (3.1)
for some angle ω, where p = p1 − p2 and φ(p) = π− 3−2 ln 2πm |p|+O(|p|
2/m2). The value
of p near x = 0 is given by p = (mE)1/2, where E is the energy eigenvalue of the state.
This is the matching condition between the wave function for x > 0 and for x < 0.
The wave function for x < 0 an Airy function. So is the wave function for x > 0.
We therefore obtain the approximate WKB form
ψ(x) =
C(x+ E
)−1/4 cos
(2mσH)
1/2(x+ E
)3/2 − π
, x < 0
C ′( E
− x)−1/4 cos
(2mσH)
1/2( E
− x)3/2 + π
, x > 0
, (3.2)
for some constants C and C ′. The expression (3.2) can be made to agree with (3.1) for
small x, provided the generalization of the Bohr-Sommerfeld quantization condition
2(m)1/2
E3/2n +
3− 2 ln 2
πm1/2
E1/2n −
π = 0 , n = 0, 1, 2, . . . , (3.3)
is satisfied by E = En. The only new feature in this semi-classical formula is the
second term, produced by the phase shift. Absorbing the horizontal string tension in
the energy, by defining un = Enσ
H , this cubic equation becomes
2(m)1/2
u3/2n +
3− 2 ln 2
πm1/2
π = 0 .
The second term can be ignored for sufficiently small σH, i.e. sufficiently small g
There is a unique real solution of the cubic equation (3.3) for a given integer n ≥ 0,
because 3− 2 ln 2 = 1.613706 > 0. The low-lying glueball masses are given by
Mn = 2m+ En = 2m+
ǫ1/3n −
3(3− 2 ln 2)σH
ǫ−1/3n
, (3.4)
where
3πσH(n+
4m1/2
4m1/2(n + 1
3(3− 2 ln 2)σH
. (3.5)
4 Conclusions
We have identified the low-lying glueballs of the anisotropic Yang-Mills theory in (2 +
1) dimensions as bound pairs of the fundamental massive particles of the principal
chiral nonlinear sigma model. We found a matching condition for the bound-state
wave function at the origin, which when combined with elementary methods yields the
spectrum of the lightest states.
There are other aspects of the two-particle bound-state problem we have not con-
sidered here. First, the potential is not precisely linear in the region where the two
particles are close together. The corrections to the potential can be determined us-
ing form factors. This will slightly modify (3.3). A completely different issue is that
there are small corrections to the form factors themselves, coming from the presence
of bound states. This, in turn, will give a further correction to the horizontal string
tension found in [2]. Such corrections to form factors in theories close to integrability
were first discussed by Delfino, Mussardo and Simonetti [19]. The bound-state energies
proliferate between 2m and 4m, as g′0 → 0. Our method breaks down as the bound-
state mass reaches 4m, because the bound state develops an instability towards fission
into a pair of two-particle bound states. This is analogous to the situation for the Ising
model in a field [16], [17] as we stated earlier. It should be worthwhile to understand
the relativistic corrections to the bound-state formula, along the lines of the work of
Fonseca and Zamolodchikov [20].
A similar calculation is possible for SU(N). The exact S-matrix of the principal
chiral nonlinear sigma model is known for N > 2 [21]. An interesting feature is that the
phase shift should vanish as N → ∞, with g20N fixed, meaning that the wave function
would be continuous where FZ particles overlap.
It would be interesting to study the scattering of a glueball by an external particle.
If the scattering is sufficiently short range, the FZ particles could be liberated from the
glueball, after which hadronization would ensue.
The results of this paper and of references [1] and [2] may be extendable to the
standard (2 + 1)-dimensional isotropic Yang-Mills theory with g′0 = g0. The strat-
egy we have in mind is an anisotropic renormalization procedure. At the start is a
standard field theory with an isotropic cut-off. By anisotropically integrating out high-
momentum degrees of freedom, the isotropic theory will flow to an anisotropic theory
with a small momentum cut-off in the x2-direction and a large momentum cut-off in the
x1 direction. If the renormalized couplings satisfy the condition (1.2), we could apply
our techniques. A check of such a method would be approximate rotational invariance
of the string tension. This would give an analytic first-principles method of solving
the isotropic gauge theory with fixed dimensionful coupling constant e, and no cut-off.
The only other analytic weak-coupling argument for a mass gap and confinement in
(2 + 1)-dimensions, namely that of orbit-space distance estimates, discussed by Feyn-
man [22], by Karabali and Nair in the second of references [5], and by Semenoff and
the author [23] is suggestive, but has not yielded definite results yet1.
Acknowledgments
This research was supported in part by the National Science Foundation under Grant
No. PHY05-51164 and by a grant from the PSC-CUNY.
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	Introduction
	Scattering states of FZ particles
	The low-lying glueball spectrum
	Conclusions
