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CCNY-HEP-96/3
February 1996





On the 0++ Glueball Mass




Stuart Samuel


Department of Physics
The City College of New York
New York, New York, 10031 USA




ABSTRACT

An approximate vacuum wave functional 0 is proposed for 2+1-
dimensional Yang-Mills theories. Using
 7 May 96 0, one can compute the 0++
glueball mass MG in terms of the string tension. By using the idea of
dimensional reduction, a prediction for MG can be made in 3+1
dimensions. One finds MG 1.5 GeV.


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I. Introduction


Yang-Mills theories without quarks are expected to produce one or

more bound states or glueballs. When quarks are present, such pure glue

states should mix with qq
- states but some meson might have a dominant
gluon content. Possible candidates for such states are f0(1300), f0(1590),
fJ(1710) and (1440).[1] The '(958) is expected to obtain a large fraction
of its mass due to fluctuations in gluonic topological charge.[2-4]

Initial lattice studies suggested that the lightest 0++ glueball might
have a mass MG of ~1 GeV. Such lattice glueball computations have been
among the most numerically demanding due to a poor signal to noise ratio.

However, due to advances in algorithms and powerful computers, much
progress has been made. The most recent lattice results give values of MG of
1550  50 MeV [5] and 1740  71 MeV [6], suggesting that the fJ(1710) or
f0(1590) might be the lightest 0++ glueball state. The fJ(1710) is favored
since it has decay widths consistent with numerical simulations.[7]

The purpose of this letter is to obtain an approximate analytic
computation of the 0++ glueball mass in terms of the string tension . The

calculation is carried out in D=2+1 dimensions and extended by dimensional

reduction to D=3+1 dimensions. Analytic studies often provide more

physical insight than numerical simulations. In addition, the interplay

between numerical and analytic approaches can assist either approach in

obtaining new methods and results.

Our method consists in postulating the form of the ground state wave
functional 0. The proposed 0 is adjusted to agree with weak and strong
coupling limits. Support for our 0 comes from the previous work [8,9]. In
particular, numerical and analytic studies [10-17] of the lattice Hamiltonian
formulation [18] give rise to vacuum functionals similar to our 0.

The Hamiltonian for the D=d+1 dimensional Yang-Mills theory is


g2 1
H = ddx Ea Ea(x) + ddx Ba
2 i i 2g2 ij Baij(x) , (1.1)



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1
where Ea a a b c
i (x) = , Ba - + fabcA A , g is the gauge coupling
i Aa ij = iAj jAi i j
i (x)
constant and fabc are the structure constants: [a,b] = ifabcc, where the Lie-
algebra generators a are normalized so that Tr(ab) = ab/2.

In 2+1 dimensions, our key result is

8
MG , (1.2)
g2 Cf
where Cf is the value of casimir operator for the fundamental representation:
aa = Cf I. For SU(N), Cf = (N2-1)/(2N).

II. The Approximate Vacuum Wave Functional

Express the ground state 0 as

0 = exp(-f(B)) , (2.1)
where f is a functional of B. As g goes to zero, the perturbative fpt(B) is

1 d a a
f
pt(B) = d x
B (x) 1 B (x)
4g2 ij
- 2 ij

1 d d a a
= d x
d y
B (x) 1
(x,y)B (y) , (2.2)
4g2 ij
- 2 ij
-1/2
where the kernel (-2) (x,y) is

d
1
p ( ( ))
(x,y) = d
exp ip x - y

- 2 ( p p . (2.3)
2 ) d
In the abelian case for which there are no interactions, perturbation theory

and Eq.(2.2) are exact. For the non-abelian case, Eq.(2.2) is not gauge-

invariant but the violations of gauge invariance are of order g2.

It has been conjectured that, in strong coupling, the ground state wave
functional is governed by fsc(B) with [9]


fsc(B) = ddx Ba
2 ij Baij(x) , (2.4)



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where  is a parameter with dimensions of length(4-d). There is much

evidence in support of Eq.(2.4).[9-17] In particular, lattice theory gives

Eq.(2.4) as the leading strong coupling result.[15]

In a confining theory in space-time dimensions D with 2 < D 4,
Eq.(2.4) can be derived by the following argument. Work in a basis of

generalized closed Wilson loops of arbitrary shape and number and in

arbitrary representations. This gauge-invariant set forms a complete set of

variables. Expand the vacuum functional in such a basis. Using constructive

field theory, the vacuum state can be obtained by doing a functional integral

weighted by exp(-S) in which one integrates over half of space-time

corresponding to t<0 and uses free boundary conditions at t=0. The

functional at t=0, obtained in this way, is the exact ground state wave

functional. In a confining theory, Wilson loops of large area are suppressed.

Due to the kinetic energy term g2 ddx Ea a
i Ei (x)/2 in H, one sees that the

there is also a contribution per unit length to the energy. Hence, Wilson

loops of large area or large perimeter are suppressed. One concludes that,

in a confining theory, f(B) in Eq.(2.1) is a sum over arbitrary numbers of

Wilson loops in arbitrary representations for loops which are of small size.

As a consequence, f(B) acts like a localized field theory. In computing

vacuum expectation values, a confining theory in D dimensions becomes a

localized field theory in d=D-1 dimensions governed by an action equal to
2f(B) (the factor of two arises because |o|2 enters in computations). Now,
use the idea behind the renormalization group. To compute the behavior of

a large Wilson loop or long-distance correlation function, one may integrate

out short-distance degrees of freedom. Integrate out to a scale slightly

larger that the confinement distance. Then, the resulting field theory will

be dominated by the local gauge-invariant operator of lowest dimension.
This operator is Baij Baij. In space-time dimensions D with D 4, strong
coupling corresponds to large distances. Hence, the effective vacuum

functional in the strong coupling limit is given by Eq.(2.4).

From the above discussion, it is clear that the true vacuum functional

is quite complicated. However, a simplified functional, which interpolates


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between the weak coupling (Eq.(2.2)) and strong coupling (Eq.(2.4)) forms,

might produce good results for computations. Consider the approximate

vacuum functional governed by

1 d
f(B) = d x
TrB (x) - k
D D + m 2
( )-1/2(x,y)B (y) , (2.5)
2 g2 ij k 0 ij

where Bij(x) = aBaij(x), Tr stands for trace, Dk is the covariant derivative in
the adjoint representation and m0 is a mass parameter.

At short distances and small coupling, the derivative term in f(B)

dominates and Eq.(2.5) reduces to Eq.(2.2). At large distances, the mass

term dominates and f(B) in Eq.(2.5) reduces to Eq.(2.4) with

1
 = 2m g2 . (2.6)
Here, we have replaced the parameter m0 by a renormalized parameter m.
In integrating out short-distance degrees of freedom, one expects m0 to be
renormalized. In particular, m contains contributions related to the energy

per unit length of the Wilson lines which enter Eq.(2.5), as we now explain.

An explicit formula for the kernel in Eq.(2.5) for SU(N) is

-
1 /2
- k
D D + m2
( ) (x,y ) = d

k 0 
s t ;s t 2
1 1 2 2 0

. 2
DX()


( ) 2
( )
d
X + m
exp - 12 0  (2.7)
X(0) = y 0

X( ) = x


. i
0 . i

exp i d
A X () exp i d
A X () ,
i
i
xy 0 t1s2 yx t2s
1

where denotes the path-order product along X() and Ai = aAai (X()).
The measure DX in Eq.(2.7) is the Feynman one [19] for a particle of unit
mass in d dimensions. The subscripts s1, t1, s2, t2, which run from 1 to N
for SU(N), are matrix indices:


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Tr B k 2
( )-1/2
ij(x) - D D + m (x,y ) B =
k 0 ij(y)
-1/2
k 2
Bs1t1 ( D + m ) (x,y) s2t2
ij (x) - D k 0 Bij (y) . (2.8)
s t ;s t
1 1 2 2


Eq.(2.7) shows that - k
D D + m2
( )-1/2(x,y) involves a sum over paths of
k 0


a Wilson line in the adjoint representation which goes from the space-point

y to the space-point x. Each path contributes with a particular probability.
As m0 is increased, paths of smaller size are weighted comparatively more
than paths of larger size. The Wilson lines in Eq.(2.7) render Eq.(2.5) gauge

invariant. When the kinetic term g2ddx Ea a
i Ei (x)/2 in H acts on the Wilson

line, a contribution to the energy proportional to the length of the non-back-

tracking part of the path is produced. Hence, it is energetically favorable to
have m0 be non-zero. In the case of a U(1) group, the matrix factor in curly
brackets in Eq.(2.7) is replaced by 1 and no such Wilson line contribution
arises and it is energetically favorable for m0 to be zero.

Besides having the correct strong and weak coupling limits, there is
numerical evidence from lattice simulations that 0 in Eqs.(2.1) and (2.5) is
a reasonable approximation for the 2+1 dimensional Yang-Mills theory.
Consider expanding f(B) in inverse powers of m0. The result is

2
f(B) =  2 k
( )
0 d x
Tr(B(x)B(x)) + 2 d x
Tr D B(x)D B(x) + ... , (2.9)
k

where

 1 1
0 = . (2.10)
2m0 g2 , 2 = - 4m30 g2
Ref.[13] has performed Monte Carlo simulations of the ground state

functional at intermediate couplings for SU(2) and found that the form in
Eq.(2.9) fits the data well with 0=(0.91  0.02)/g4 and 2= - (0.19  0.05)/g8.
Analytic strong coupling lattice computations in the Hamilonian formulation
lead to similar results.[16] As predicted from Eq.(2.5), 2 should be negative
and this is borne out in simulations. The Monte Carlo data implies
m0 1.55 g2 for SU(2).


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III. Consequences of The Approximate Vacuum Functional


Let us first consider D=2+1 space-time dimensions. In this case, the

strong-coupling functional in Eq.(2.5) leads to confinement because the

dimensionally reduced effective theory is similar to a localized Yang-Mills
theory in two dimensions. The string tension is obtained from the vacuum

expectation value of a spatially oriented Wilson loop. The result is

g2mCf
= . (3.1)
4
The 0++ glueball mass MG is obtained as the coefficient of the
1 1
exponential falloff of the correlation function <2 Baij Baij(0,0) 2 Bakl Bakl(0,t)> ~
c(t) exp(-MGt) for large t. By exploiting Lorentz invariance, MG can be
extracted from the vacuum functional via

1 1
<0|2 Baij Baij(x) 2 Bakl Bakl(y)|0> ~ c(r) exp(-MG r) , (3.2)
where r = |x - y|. In two space dimensions and at large distances, the
propagator for Ba12 in the effective field theory governed by |0|2 is
(-2 + m2)1/2. One finds, using perturbation theory in this effective theory,

that Eq.(3.2) holds with

MG 2m . (3.3)
Eq.(3.3) has a physical interpretation. The parameter m can be thought of

as a constituent mass for a gluon. More precisely, m is the effective mass of

an adjoint-representation configuration of gluons inside a bound state. Since

two adjoint representations are needed to make a singlet, the glueball mass

is approximately 2m.

Combining (3.1) and (3.3), one arrives at the main result in Eq.(1.2).

Monte Carlo simulations of the 2+1 dimensional SU(2) Yang-Mills

theory have accurately [20] determined the string tension to be
MC = (0.112  0.002) g4. Using this value for in Eq.(1.2), we obtain
MG 1.2 g2. The Monte Carlo calculations of ref.[21] give MG

= (1.59  0.01) g2. Thus, we find MG/
3.6, while numerical simulations
[21] give MG/
= 4.77  0.05. Using the difference between our results and


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those of the lattice as a means of estimating systematic uncertainty, our

method for computing the 0++ glueball mass is accurate to about 25%. For a

summary of computations of MG/
, see the references in ref.[22].
For SU(3), Monte Carlo simulations give MC = (0.307  0.004) g4.[23]
This value of in conjunction with Eq.(1.2) leads to the SU(3) results

MG 1.84 g2 , MG/
3.3 . (3.4)
The result in Eq.(3.4) is in reasonably agreement with strong coupling

Hamiltonian computations [24].

Recently, another analytic approach, based on finding exact
eigenstates of the kinetic energy term in H, gives MG = N/ g2 for SU(N).[25]
For SU(2), one obtains MG 0.64 g2, which is considerably smaller than the
Monte Carlo result, suggesting that corrections involving the potential

energy term d2x Ba12 Ba12(x)/2g2 may be important. For SU(3), the approach

gives MG 0.95 g2, which is about half the value in Eq.(3.4).

IV. Extrapolation to D=4


The result in Eq.(1.2) does not directly apply to D=3+1 dimensions.

However, one can appeal to the idea of dimensional reduction.[26] If a 3+1

dimensional gauge theory is confining, then the computation of spatial

correlation functions and Wilson loops involves only those degrees of
freedom within the order of Lc in the time direction, where Lc is the
confinement length. Hence, it should be possible to approximate a 3+1

dimensional confining theory by a 2+1 dimensional theory. This idea has

been used several times in past work[15,27] and is embodied in the result
that fsc(B) can be used for the purposes of computing large spatial Wilson
loops. Due to the beta function being negative,  is expected to be

exponentially large in 1/g2. In short, assuming four-dimensional

confinement, results for MG/
should be approximately the same in 2+1
and 3+1 dimensions. In fact, Monte Carlo simulations of the SU(2) gauge


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theory indicate that the glueball spectra, in units of
, in 2+1 and 3+1
dimensions agree to about 15%.[21]

Assuming the validity of dimensional reduction, Eq.(3.4) with
= 440 MeV gives a value of the 0++ glueball mass MG in D=3+1
dimensions for the SU(3) gauge theory of

MG 1.5 GeV . (4.1)
V. Summary


In summary, by using an approximate vacuum functional which

interpolates between strong and weak coupling forms, we have obtained the

relation in Eq.(1.2) between the lightest 0++ glueball mass and the string

tension in 2+1 dimensions. Using dimensional reduction, we obtain a value
of about 1.5 GeV for MG in 3+1 dimensions.

Our value of 1.5 GeV for MG is much larger than many of the natural

scales in Yang-Mills theory:
is 440 MeV, the deconfining phase transition
temperature is ~250 MeV, the mass of the color-singlet gluon cloud around

a quark is ~300 MeV assuming that the contribution to the constituent mass

of a quark in a bound state comes from such a gluon cloud, and the
topological susceptibility <2> which enters the ' mass is <2>1/4 180 MeV

[28-33]. On this basis one might expect approximate analytic computations
of MG to yield values less than 1 GeV. For example, if the continuum strong
coupling 2+1 dimensional Hamiltonian result [25] is assumed to extrapolate
to D=3+1 by dimensional reduction, one would obtain MG 760 MeV.

Since, as mentioned above, the error in our results in 2+1 dimensions

is estimated to be about 25% and that the error in extrapolating to 3+1
dimensions is about 15%, the total error in MG is around 30%. Our value of
1.5 GeV for MG is about 15% less than the most accurate current lattice
value of 1740 MeV [6].


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Acknowledgments

We thank V. P. Nair for discussions and Columbia University for hospitality.

This work is supported in part by the United States Department of Energy

under the grant number DE-FG02-92ER40698.





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