- Email: [email protected]
Numerical calculation of the momentum-space gluon propagator in quenched, reduced QCD3
Volume 173, number 2
PHYSICS LETTERS B
N U M E R I C A L CALCULATION OF THE M O M E N T U M - S P A C E IN Q U E N C H E D , R E D U C E D Q C D 3
5 June 1986
GLUON PROPAGATOR
P.A. A M U N D S E N Phvsik-Department T30, Teehnische Universitfit Mi~nchen, D-8046 Garching near Munich, Fed. Rep. German),
and J. G R E E N S I T E 1 Pt~vsics and Astronoml Department, San Francisco State University, 1600 Hollowav Avenue, San Francisco, CA 94132, USA
Received 22 November 1985
A recently developed method of momentum-space Monte Carlo is applied to compute the momentum-space gluon propagator in quenched, reduced, continuum QCD 3 in axial gauge. There is some evidence that the gluon propagator D~,,(p ) is finite as p ~ O, which might indicate the existence of a non-perturbative gluon mass.
Many attempts to understand the long-range properties o f QCD have centered on the infrared behavior of the gluon propagator. It has been suggested, for example, that the gluon propagator D u v ( p ) goes like 1/p 4 a s p 2 ~ 0 [1], leading to a linear confining force; it has also been suggested that the gluon instead develops a non-perturbative mass, so that Duv(p ) -~ 1/ (p2 + m 2) as p2 -~ 0 [2]. Both proposals are based on an analysis of the truncated Schwinger-Dyson equations, and it is therefore o f some interest to check these ideas directly by a Monte Carlo calculation of the gluon propagator. In this letter we report our numerical results for the momentum-space gluon propagator in planar, continuum QCD3, obtained using a non-standard method that we will refer to as "momentum-space Monte Carlo" (MSMC). MSMC was developed b y us in ref. [3]; it is based on the fact that U(N)-invariant matrix field theories can be reduced, in the N - + oo limit, to matrix models at a single point in s p a c e time [4]. In the "quenched, reduced" version, introduced by Bhanot, Heller, and Neuberger [4], a i Work supported by the US Department of Energy under Contract No. DE-AC03-81ER40009. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
euclidean momentum p a is associated with each matrix index 1 ~ a ~ ~ , s o that the expectation value (in the reduced theory) of a product of matrix elements can be identified with a Green function in the full theory at some particular set of momenta. MSMC simply applies this fact to compute momentum-space Green functions o f planar theories by a Monte Carlo evaluation o f matrix elements in the quenched, reduced version. In ref. [3] the MSMC method was tested b y calculating the propagator in the onedimensional U(N) × U(N) chiral model. The mass gap found in this model, as extracted from the propagator and plotted as a function of the coupling constant, was in excellent agreement with the exact G r o s s - W i t t e n result [5]. An interesting aspect of the quenched reduction technique is that it appears to provide a regularization of continuum QCD with a sharp momentum cutoff. In ordinary (unreduced) QCD, a sharp momentum cutoff would introduce a spurious gluon mass and quadratic divergences in the perturbation expansion. Gross and Kitazawa [4] have verified that these problems do not arise in the quenched, reduced version, at least to one-loop. It is therefore possible to study b o t h lattice and continuum QCD numerically, 179
Volume 173, number 2
PHYSICS LETTERS B
albeit in the planar limit. In this letter we consider only the continuum theory. A study of the positionspace gluon propagator in ordinary lattice QCD, using standard Monte Carlo methods, has also been initiated by Mandula [6]. Quenched, reduced QCD is obtained by applying the Parisi-Gross-Kitazawa quenching prescription [4] to a formulation of QCD proposed by Bars [7]: introduce a set o f D N × N unitary matrix fields Baub(x) (p = 1..... D; a, b -- 1,2 ..... N), where
Bu(x)=Pexp(-ig /
dx'UAu(x')),
5 June 1986
Dii(x ) = (1/N2)(Tr A i(x)A](O)) A/2
1
-
fA
N2-
dpC
[I ~abDal.bexp[i(pau-pb)x u ] /2 v,c -X(7)
where D-1
D~b= 1
l-I dBko
k=l
× (B~iobd p/B/~
-
tt~iO Pi~iO -
pbsba)exp(--SQ)
(8)
(no sum over a, b), and
SQ = (2Tr/A)D(1/2g 2) Tr(PuB 2 uuP~B 2 vv ? Au(x ) = (i/g)B?u(x)OuBu(x) ,
(1)
- PuBuvPvB?udPuBu~PvB~uv) ,
and define gauge-invariant variables
B vv(x) = Bu(x)B?v (x) . In axial gauge B~ b = 6ab, the
(2)
Pu
QCD functional inte-
P'u=
gral is
0
(9) "'.
D-1
P
z= f 21DBk0(X) ×exp( fdDx
A Fourier transform of (7) gives an expression for the momentum-space gluon propagator
l-~Tr[au(B,~O~B?u~)]2),
4g 2
(3)
1 ?/2 Di](P)=-~_A/2
dDpC ~ D q .b ~c A D ab tl
with X (2rr)DsD(p -
Bij(x ) = Bio(x)B;o(X) , Ak(X ) = (i/g)B~o(X)OkBko(x) , and DBko is the (functional) Haar
(4)
measure for the unitary-matrix-valued fields. The quenching prescription is simply to make the replacements
ab )-+ exp[i(p a Buv(X
-
b a ]Buy, ab pa)x
pa _ pb),
which is the quantity we wish to calculate. In the numerical evaluation, what we actually compute is the average value of D0.(p ) in each of a large number of momentum subvolumes (or "bins"). Divide momentum space into some large number M (~50) subvolumes Vm (m = 1,2 ..... 114). The average value of Di/(p ) in each subvolume is A/2
Di](P(m)) = fdDx ~
(2n/A) D ,
(5)
where
- a / 2 <.p~ <.A/2
1 ..... D;a= 1..... N)
(11)
where p in subvolume m ,
(6)
is a set of N "quenched" momenta, and each Buy is an x-independent N × N matrix. In this formulation, the gluon propagator (averaged over color indices) is
= 0
otherwise,
(12)
and p belongs to the mth subvolume iff x/DA2(m - 1 ) / M < ]p] Then
180
dDp Am(p)Di](p ) , V m - /2
Am(p) --=1 (p=
(10)
~Dx/f~m/M.
(13)
Volume 173, number 2
1
PHYSICS LETTERS B
5 June 1986
7
dDpc ~ (2rr) D
6
x Z~m(p ~ - pb)D} b .
(14) 5-
The procedure for evaluating (14) is b y a Monte Carlo-within-a-Monte Carlo; i.e. by a sequence of evaluations of eq. (8), with each evaluation using a different set of quenched momenta {p~). In practice it is useful to introduce a fundamental length scale a = 27r/A analogous to a lattice spacing, and rescale all dimensionful parameters in an obvious way:
A ~ 21r/a, p-+'pa -1,
g2 ~'~2aD-4 ,
(15)
so that the quenched action in (9) involves only dimensionless quantities. Each set o f quenched (rescaled) momenta ~p"~} is then generated stochastically with uniform weighting in the D-dimensional p space hypercube -Tr ~< P u ~< 7r,/a = 1, ..., D. Denote the k t h set of quenched momenta by (P~a"'k}'ak~b= 1, ...,K, and let D~b'k denote the value of Di] obtained from a Monte Carlo of (8) using the k t h quenched m o m e n t u m set. Then K
_ 1 1 DiTb,k Oij~(m') -K. k~=l n(m, k) a~b "" X Am~a,k - "~b,k)
(16)
is the average value of the momentum-space gluon propagator Di/~) in the ruth momentum bin, where
n(m, k) = ~ Am~a - "~b) a~b
(17)
is the number o f relative momenta (p~a _ ~ b ) which lie in the m t h m o m e n t u m bin. The numerical calculation was carried out in D = 3 dimensions, at couplings X=g2N=
1,2,5,10,
and N = 15, 20, 25, 30, 35 . For all N, at least 400 update sweeps o f the matrices were made, of which the last 300 contribute to the statistics. The matrices were updated b y Okawa's method [ 8 ] , w i t h at least 800 random SU(2) matrices generated for the updates at each set o f quenched momenta. F o r N = 15 and 20, K = 20 sets o f quench-
~Q'4" 3" 2" 1 0
Fig. l. The inverse two-point function D-1 ('~) versus (dimensionless) p'2at N = 35, D = 3 dimensions, at cou~alingsX = ~'2N = 1, 5, in the low-momentum range 0 < p" < 8. The gluon propagator is averaged over colors and spatial directions, and each data point represents the average value of D-l(~") in a small momentum bin Vm. The straight line represents the best fit to the first 20 data points, and the finite intercept at if2 = 0 impfies a finite gluon mass. ed momenta were used, and for higher N, at least 10 sets were used. The calculational time for the largest jobs (at N = 35) was about 2¼ hours CPU time on a Cyber 875. To improve statistics, the ghion propagator Di] was averaged over two spatial directions O(P'(m)) = 1 [/)11 ('P(m)) + / ) 2 2 ( P ' ( m ) ) ] -
(18)
Our data f o r / ) - 1 ~ ' ) , for ?t = 1 and 5 at N = 35, is shown in fig. 1. It is clear that the inverse propagator extrapolates to a non-zero value at p2 = 0, which implies the existence o f a non-vanishing gluon mass * 1 The gluon mass is extracted by a linear fit o f / 9 - l ( p ~ ) at low momentum to a simpte one-pole form / ~ - l(p~) ~ (const.) • (~2 + ~/2)
(19)
so that ~ 2 = D - 1 ( 0 ) / ( d D - 1/dff21~. 2 = 0 ) ,
(20)
where r~, like ~'2 and ~ 2 , is dimensionless (the dimensionful mass is obtained by multiplying by a - 1 ). The slope and intercept o f / ) - 1 ( ~ ) is computed from a #1 We have learned that Mandula [6] has also found evidence for a gluon mass, in lattice QCD4, by lattice Monte Carlo in Landau gauge. 181
V o l u m e 173, n u m b e r 2
PHYSICS LETTERS B
5 J u n e 1986
3'
2¸
~E
8 3 2 !
.65
N-1
.i k
Fig, 2. E x t r a p o l a t i o n o f t h e g l u o n mass ~ t o N = *% f r o m
data at N = 15, 20, 25, 30, 35. Data shown is for h = 1 and h= 10. linear best fit of (19) to the first 20 data points shown in fig. 1. Since we work at finite N , it is necessary to extrapolate the result to N = oo. This is done b y computing ~ a t N = 1 5 - 3 5 ,and fitting the data to
= ~** + e / N ,
(21)
as shown in fig. 2 for 3` = 1 and 3` = 10. In fig. 3 we display our values for Wt** at 3` = 1,2, 5, 10. The mass appears to increase linearly with 3`, which is the correct scaling behavior in 3 dimensions; what is unexpected is that ~.~ does n o t seem to extrapolate to 0 at 3` ~ 0. If true, this behavior implies an infinite gluon mass at cutoff A -+ oo (a ~ 0). There are a number of possible explanations for the apparent failure of ~ . . to extrapolate to 0 at 3, -~ 0. First, it may be that high-statistics runs at 3` < 1 will show that the data begins to extrapolate to 0; i.e. 3` > 1 is n o t in the continuum scaling regime. Alternatively, it is possible that the one-pole ansatz in (19) is inadequate for the given momentum range. Although this ansatz worked very well for the propagator of the D = 1 chiral model in ref. [3], the D = 3 gluon propagator may have more structure than the chiral model propagator. So while the fact that D - 1 ( 0 ) 4= 0 implies a non-vanishing gluon mass, it may just be that the numerical values we have extracted for the 182
Fig. 3. The gluon mass r~.o = mooaversus coupling h = gZaN, at h = 1, 2, 5, 10. From the data, it does not appear that r~.o extrapolates to 0 as h ~ 0. mass from a simple linear fit are inaccurate. A third possibility is that the quenched, reduced regularization of continuum QCD somehow introduces a spurious gluon mass. According to Gross and Kitazawa [4] a spurious mass does not arise in this formulation (at least at one loop); nevertheless, it must be considered a possible explanation of our results. A repetition of our calculation for quenched lattice QCD (the quenched Eguchi-Kawai model) may clarify this question. There is another, intriguing, alternative. It may be that the extrapolation of our data to ~oo 4 : 0 at 3` -~ 0 is correct, and that the gluon mass rn.. = "~no./ais truly infinite in the A -~ ~ (or a -~ 0) limit. If so, this may be connected to color confinement by the following argument: Consider continuum QCD in the usual (unreduced) formulation in a physical gauge (Coulomb or axial) defined b y F[A ] = 0. An exact representation of the vacuum wavefunctional in this theory, at time t = 0, is
qz0 [A(x, 0)] = ×exp
f DA(x,t <
(10f ~
--oo
0)6 [F(A)] A [A]
)
dtf dD-axTrf~ .
(22)
Volume 173, number 2
PHYSICS LETTERS B
The vacuum is colorless, b u t a colored state in the adjoint representation, with the quantum numbers of a single gluon, can be created by inserting a gluon field at some time t < 0 into the functional integral tIrgluo n [A(x, 0)]
=
f DA(x, t < O) A(-k, -7)
× 6[F(A)]A[A]
×exp(- f dtfdD-lxTrF2u), 0
A(k,
-73: faD- I x A(x, t = - 7 )
exp(ik • x ) .
(23)
ggluon = (q~gluonlHIqZgluon)/(~gluonlq~gluon)
-(~o~1~o) T)A(-k,-T)),
(24)
where ( ) denotes the ordinary euclidean vacuum expectation value
(A(k, T)A(--k, -T)) = f
DAA(k, T)A(-k,
T)
×6[F(A)]A(A)exp(-l f d D x T r F 2 u ) ,
of the gluon propagator , 2 . An infinite gluon mass is one possible behavior consistent with confinement, so that the existence of color confinement in continuum QCD 3 may be the proper interpretation of our data. In conclusion, we have found strong evidence for a non-zero gluon mass in quenched, reduced continuum QCD3, using the momentum-space Monte Carlo method. However, the asymptotic behavior of this mass as cutoff A -+ ~ is unclear; it is even possible that the gluon mass is infinite in that limit. Further high-statistics results at in D = 3 dimensions, and at a range of couplings in D = 4 dimensions, as well as computations in the quenched lattice formulation, may help to resolve this issue. ,2 In D = 2 dimensions the theory can be solved exactly in axial (A 1 = 0) gauge. In that gauge, the gluon does not propagate in time, so that the one-gluon state (23) does not exist at T ~ 0. In temporal (Ao = 0) gauge, Gauss's law is implemented as a constraint on states, and the color adjoint state (23) then lies outside the space of physical states.
Then the energy (above the vacuum) of the onegluon state is easily seen to be [9]
= - ~(d/dT) ln(A(k,
5 June 1986
(25)
and H is the hamiltonian in the physical gauge F[A ] =0. For the free-particle ansatz (19), the energy of such a state is Egluon = (k 2 + m2) 1/2. Color confinement, on the other hand, probably requires that the energy of any color non-singlet state is infinite, which from (24) implies some pathological behavior
References [1] R. Anishetty et al., Phys. Lett. B 86 (1979) 52; U. Bar Gadda, Nucl. Phys. B163 (1980) 312; S. Mandelstam, Phys. Rev. D20 (1979) 3223. [2] J. Cornwall, Phys. Rev. D26 (1982) 1453; J. Smit, Phys. Rev. D10 (1974) 2473. [3] P. Amundsen, J. Greensite and T. Sterling, Phys. Lett. B 149 (1984) 482. [4] T. Eguchi and H. Kawai, Phys. Rev. Lett. 48 (1982) 1063: G. Bhanot, U. Heller and H. Neuberger, Phys. Lett. B 113 (1982) 47; G. Parisi, Phys. Lett. B 112 (1982) 463; D. Gross and Y. Kitazawa, Nucl. Phys. B206 (1982) 440. [5 ] D. Gross and E. Witten, Phys. Rev. D21 (1980) 446. [6] J. Mandula, private communication. [7] I. Bars, Phys. Lett. B 116 (1982) 57. [8] M. Okawa, Phys. Rev. Lett. 49 (1982) 353. [9] J. Greensite and M. Halpern, Berkeley preprint UCBPTH-85/40.
183
PHYSICS LETTERS B
N U M E R I C A L CALCULATION OF THE M O M E N T U M - S P A C E IN Q U E N C H E D , R E D U C E D Q C D 3
5 June 1986
GLUON PROPAGATOR
P.A. A M U N D S E N Phvsik-Department T30, Teehnische Universitfit Mi~nchen, D-8046 Garching near Munich, Fed. Rep. German),
and J. G R E E N S I T E 1 Pt~vsics and Astronoml Department, San Francisco State University, 1600 Hollowav Avenue, San Francisco, CA 94132, USA
Received 22 November 1985
A recently developed method of momentum-space Monte Carlo is applied to compute the momentum-space gluon propagator in quenched, reduced, continuum QCD 3 in axial gauge. There is some evidence that the gluon propagator D~,,(p ) is finite as p ~ O, which might indicate the existence of a non-perturbative gluon mass.
Many attempts to understand the long-range properties o f QCD have centered on the infrared behavior of the gluon propagator. It has been suggested, for example, that the gluon propagator D u v ( p ) goes like 1/p 4 a s p 2 ~ 0 [1], leading to a linear confining force; it has also been suggested that the gluon instead develops a non-perturbative mass, so that Duv(p ) -~ 1/ (p2 + m 2) as p2 -~ 0 [2]. Both proposals are based on an analysis of the truncated Schwinger-Dyson equations, and it is therefore o f some interest to check these ideas directly by a Monte Carlo calculation of the gluon propagator. In this letter we report our numerical results for the momentum-space gluon propagator in planar, continuum QCD3, obtained using a non-standard method that we will refer to as "momentum-space Monte Carlo" (MSMC). MSMC was developed b y us in ref. [3]; it is based on the fact that U(N)-invariant matrix field theories can be reduced, in the N - + oo limit, to matrix models at a single point in s p a c e time [4]. In the "quenched, reduced" version, introduced by Bhanot, Heller, and Neuberger [4], a i Work supported by the US Department of Energy under Contract No. DE-AC03-81ER40009. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
euclidean momentum p a is associated with each matrix index 1 ~ a ~ ~ , s o that the expectation value (in the reduced theory) of a product of matrix elements can be identified with a Green function in the full theory at some particular set of momenta. MSMC simply applies this fact to compute momentum-space Green functions o f planar theories by a Monte Carlo evaluation o f matrix elements in the quenched, reduced version. In ref. [3] the MSMC method was tested b y calculating the propagator in the onedimensional U(N) × U(N) chiral model. The mass gap found in this model, as extracted from the propagator and plotted as a function of the coupling constant, was in excellent agreement with the exact G r o s s - W i t t e n result [5]. An interesting aspect of the quenched reduction technique is that it appears to provide a regularization of continuum QCD with a sharp momentum cutoff. In ordinary (unreduced) QCD, a sharp momentum cutoff would introduce a spurious gluon mass and quadratic divergences in the perturbation expansion. Gross and Kitazawa [4] have verified that these problems do not arise in the quenched, reduced version, at least to one-loop. It is therefore possible to study b o t h lattice and continuum QCD numerically, 179
Volume 173, number 2
PHYSICS LETTERS B
albeit in the planar limit. In this letter we consider only the continuum theory. A study of the positionspace gluon propagator in ordinary lattice QCD, using standard Monte Carlo methods, has also been initiated by Mandula [6]. Quenched, reduced QCD is obtained by applying the Parisi-Gross-Kitazawa quenching prescription [4] to a formulation of QCD proposed by Bars [7]: introduce a set o f D N × N unitary matrix fields Baub(x) (p = 1..... D; a, b -- 1,2 ..... N), where
Bu(x)=Pexp(-ig /
dx'UAu(x')),
5 June 1986
Dii(x ) = (1/N2)(Tr A i(x)A](O)) A/2
1
-
fA
N2-
dpC
[I ~abDal.bexp[i(pau-pb)x u ] /2 v,c -X(7)
where D-1
D~b= 1
l-I dBko
k=l
× (B~iobd p/B/~
-
tt~iO Pi~iO -
pbsba)exp(--SQ)
(8)
(no sum over a, b), and
SQ = (2Tr/A)D(1/2g 2) Tr(PuB 2 uuP~B 2 vv ? Au(x ) = (i/g)B?u(x)OuBu(x) ,
(1)
- PuBuvPvB?udPuBu~PvB~uv) ,
and define gauge-invariant variables
B vv(x) = Bu(x)B?v (x) . In axial gauge B~ b = 6ab, the
(2)
Pu
QCD functional inte-
P'u=
gral is
0
(9) "'.
D-1
P
z= f 21DBk0(X) ×exp( fdDx
A Fourier transform of (7) gives an expression for the momentum-space gluon propagator
l-~Tr[au(B,~O~B?u~)]2),
4g 2
(3)
1 ?/2 Di](P)=-~_A/2
dDpC ~ D q .b ~c A D ab tl
with X (2rr)DsD(p -
Bij(x ) = Bio(x)B;o(X) , Ak(X ) = (i/g)B~o(X)OkBko(x) , and DBko is the (functional) Haar
(4)
measure for the unitary-matrix-valued fields. The quenching prescription is simply to make the replacements
ab )-+ exp[i(p a Buv(X
-
b a ]Buy, ab pa)x
pa _ pb),
which is the quantity we wish to calculate. In the numerical evaluation, what we actually compute is the average value of D0.(p ) in each of a large number of momentum subvolumes (or "bins"). Divide momentum space into some large number M (~50) subvolumes Vm (m = 1,2 ..... 114). The average value of Di/(p ) in each subvolume is A/2
Di](P(m)) = fdDx ~
(2n/A) D ,
(5)
where
- a / 2 <.p~ <.A/2
1 ..... D;a= 1..... N)
(11)
where p in subvolume m ,
(6)
is a set of N "quenched" momenta, and each Buy is an x-independent N × N matrix. In this formulation, the gluon propagator (averaged over color indices) is
= 0
otherwise,
(12)
and p belongs to the mth subvolume iff x/DA2(m - 1 ) / M < ]p] Then
180
dDp Am(p)Di](p ) , V m - /2
Am(p) --=1 (p=
(10)
~Dx/f~m/M.
(13)
Volume 173, number 2
1
PHYSICS LETTERS B
5 June 1986
7
dDpc ~ (2rr) D
6
x Z~m(p ~ - pb)D} b .
(14) 5-
The procedure for evaluating (14) is b y a Monte Carlo-within-a-Monte Carlo; i.e. by a sequence of evaluations of eq. (8), with each evaluation using a different set of quenched momenta {p~). In practice it is useful to introduce a fundamental length scale a = 27r/A analogous to a lattice spacing, and rescale all dimensionful parameters in an obvious way:
A ~ 21r/a, p-+'pa -1,
g2 ~'~2aD-4 ,
(15)
so that the quenched action in (9) involves only dimensionless quantities. Each set o f quenched (rescaled) momenta ~p"~} is then generated stochastically with uniform weighting in the D-dimensional p space hypercube -Tr ~< P u ~< 7r,/a = 1, ..., D. Denote the k t h set of quenched momenta by (P~a"'k}'ak~b= 1, ...,K, and let D~b'k denote the value of Di] obtained from a Monte Carlo of (8) using the k t h quenched m o m e n t u m set. Then K
_ 1 1 DiTb,k Oij~(m') -K. k~=l n(m, k) a~b "" X Am~a,k - "~b,k)
(16)
is the average value of the momentum-space gluon propagator Di/~) in the ruth momentum bin, where
n(m, k) = ~ Am~a - "~b) a~b
(17)
is the number o f relative momenta (p~a _ ~ b ) which lie in the m t h m o m e n t u m bin. The numerical calculation was carried out in D = 3 dimensions, at couplings X=g2N=
1,2,5,10,
and N = 15, 20, 25, 30, 35 . For all N, at least 400 update sweeps o f the matrices were made, of which the last 300 contribute to the statistics. The matrices were updated b y Okawa's method [ 8 ] , w i t h at least 800 random SU(2) matrices generated for the updates at each set o f quenched momenta. F o r N = 15 and 20, K = 20 sets o f quench-
~Q'4" 3" 2" 1 0
Fig. l. The inverse two-point function D-1 ('~) versus (dimensionless) p'2at N = 35, D = 3 dimensions, at cou~alingsX = ~'2N = 1, 5, in the low-momentum range 0 < p" < 8. The gluon propagator is averaged over colors and spatial directions, and each data point represents the average value of D-l(~") in a small momentum bin Vm. The straight line represents the best fit to the first 20 data points, and the finite intercept at if2 = 0 impfies a finite gluon mass. ed momenta were used, and for higher N, at least 10 sets were used. The calculational time for the largest jobs (at N = 35) was about 2¼ hours CPU time on a Cyber 875. To improve statistics, the ghion propagator Di] was averaged over two spatial directions O(P'(m)) = 1 [/)11 ('P(m)) + / ) 2 2 ( P ' ( m ) ) ] -
(18)
Our data f o r / ) - 1 ~ ' ) , for ?t = 1 and 5 at N = 35, is shown in fig. 1. It is clear that the inverse propagator extrapolates to a non-zero value at p2 = 0, which implies the existence o f a non-vanishing gluon mass * 1 The gluon mass is extracted by a linear fit o f / 9 - l ( p ~ ) at low momentum to a simpte one-pole form / ~ - l(p~) ~ (const.) • (~2 + ~/2)
(19)
so that ~ 2 = D - 1 ( 0 ) / ( d D - 1/dff21~. 2 = 0 ) ,
(20)
where r~, like ~'2 and ~ 2 , is dimensionless (the dimensionful mass is obtained by multiplying by a - 1 ). The slope and intercept o f / ) - 1 ( ~ ) is computed from a #1 We have learned that Mandula [6] has also found evidence for a gluon mass, in lattice QCD4, by lattice Monte Carlo in Landau gauge. 181
V o l u m e 173, n u m b e r 2
PHYSICS LETTERS B
5 J u n e 1986
3'
2¸
~E
8 3 2 !
.65
N-1
.i k
Fig, 2. E x t r a p o l a t i o n o f t h e g l u o n mass ~ t o N = *% f r o m
data at N = 15, 20, 25, 30, 35. Data shown is for h = 1 and h= 10. linear best fit of (19) to the first 20 data points shown in fig. 1. Since we work at finite N , it is necessary to extrapolate the result to N = oo. This is done b y computing ~ a t N = 1 5 - 3 5 ,and fitting the data to
= ~** + e / N ,
(21)
as shown in fig. 2 for 3` = 1 and 3` = 10. In fig. 3 we display our values for Wt** at 3` = 1,2, 5, 10. The mass appears to increase linearly with 3`, which is the correct scaling behavior in 3 dimensions; what is unexpected is that ~.~ does n o t seem to extrapolate to 0 at 3` ~ 0. If true, this behavior implies an infinite gluon mass at cutoff A -+ oo (a ~ 0). There are a number of possible explanations for the apparent failure of ~ . . to extrapolate to 0 at 3, -~ 0. First, it may be that high-statistics runs at 3` < 1 will show that the data begins to extrapolate to 0; i.e. 3` > 1 is n o t in the continuum scaling regime. Alternatively, it is possible that the one-pole ansatz in (19) is inadequate for the given momentum range. Although this ansatz worked very well for the propagator of the D = 1 chiral model in ref. [3], the D = 3 gluon propagator may have more structure than the chiral model propagator. So while the fact that D - 1 ( 0 ) 4= 0 implies a non-vanishing gluon mass, it may just be that the numerical values we have extracted for the 182
Fig. 3. The gluon mass r~.o = mooaversus coupling h = gZaN, at h = 1, 2, 5, 10. From the data, it does not appear that r~.o extrapolates to 0 as h ~ 0. mass from a simple linear fit are inaccurate. A third possibility is that the quenched, reduced regularization of continuum QCD somehow introduces a spurious gluon mass. According to Gross and Kitazawa [4] a spurious mass does not arise in this formulation (at least at one loop); nevertheless, it must be considered a possible explanation of our results. A repetition of our calculation for quenched lattice QCD (the quenched Eguchi-Kawai model) may clarify this question. There is another, intriguing, alternative. It may be that the extrapolation of our data to ~oo 4 : 0 at 3` -~ 0 is correct, and that the gluon mass rn.. = "~no./ais truly infinite in the A -~ ~ (or a -~ 0) limit. If so, this may be connected to color confinement by the following argument: Consider continuum QCD in the usual (unreduced) formulation in a physical gauge (Coulomb or axial) defined b y F[A ] = 0. An exact representation of the vacuum wavefunctional in this theory, at time t = 0, is
qz0 [A(x, 0)] = ×exp
f DA(x,t <
(10f ~
--oo
0)6 [F(A)] A [A]
)
dtf dD-axTrf~ .
(22)
Volume 173, number 2
PHYSICS LETTERS B
The vacuum is colorless, b u t a colored state in the adjoint representation, with the quantum numbers of a single gluon, can be created by inserting a gluon field at some time t < 0 into the functional integral tIrgluo n [A(x, 0)]
=
f DA(x, t < O) A(-k, -7)
× 6[F(A)]A[A]
×exp(- f dtfdD-lxTrF2u), 0
A(k,
-73: faD- I x A(x, t = - 7 )
exp(ik • x ) .
(23)
ggluon = (q~gluonlHIqZgluon)/(~gluonlq~gluon)
-(~o~1~o) T)A(-k,-T)),
(24)
where ( ) denotes the ordinary euclidean vacuum expectation value
(A(k, T)A(--k, -T)) = f
DAA(k, T)A(-k,
T)
×6[F(A)]A(A)exp(-l f d D x T r F 2 u ) ,
of the gluon propagator , 2 . An infinite gluon mass is one possible behavior consistent with confinement, so that the existence of color confinement in continuum QCD 3 may be the proper interpretation of our data. In conclusion, we have found strong evidence for a non-zero gluon mass in quenched, reduced continuum QCD3, using the momentum-space Monte Carlo method. However, the asymptotic behavior of this mass as cutoff A -+ ~ is unclear; it is even possible that the gluon mass is infinite in that limit. Further high-statistics results at in D = 3 dimensions, and at a range of couplings in D = 4 dimensions, as well as computations in the quenched lattice formulation, may help to resolve this issue. ,2 In D = 2 dimensions the theory can be solved exactly in axial (A 1 = 0) gauge. In that gauge, the gluon does not propagate in time, so that the one-gluon state (23) does not exist at T ~ 0. In temporal (Ao = 0) gauge, Gauss's law is implemented as a constraint on states, and the color adjoint state (23) then lies outside the space of physical states.
Then the energy (above the vacuum) of the onegluon state is easily seen to be [9]
= - ~(d/dT) ln(A(k,
5 June 1986
(25)
and H is the hamiltonian in the physical gauge F[A ] =0. For the free-particle ansatz (19), the energy of such a state is Egluon = (k 2 + m2) 1/2. Color confinement, on the other hand, probably requires that the energy of any color non-singlet state is infinite, which from (24) implies some pathological behavior
References [1] R. Anishetty et al., Phys. Lett. B 86 (1979) 52; U. Bar Gadda, Nucl. Phys. B163 (1980) 312; S. Mandelstam, Phys. Rev. D20 (1979) 3223. [2] J. Cornwall, Phys. Rev. D26 (1982) 1453; J. Smit, Phys. Rev. D10 (1974) 2473. [3] P. Amundsen, J. Greensite and T. Sterling, Phys. Lett. B 149 (1984) 482. [4] T. Eguchi and H. Kawai, Phys. Rev. Lett. 48 (1982) 1063: G. Bhanot, U. Heller and H. Neuberger, Phys. Lett. B 113 (1982) 47; G. Parisi, Phys. Lett. B 112 (1982) 463; D. Gross and Y. Kitazawa, Nucl. Phys. B206 (1982) 440. [5 ] D. Gross and E. Witten, Phys. Rev. D21 (1980) 446. [6] J. Mandula, private communication. [7] I. Bars, Phys. Lett. B 116 (1982) 57. [8] M. Okawa, Phys. Rev. Lett. 49 (1982) 353. [9] J. Greensite and M. Halpern, Berkeley preprint UCBPTH-85/40.
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