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Confinement and gaussian gluon fields
Physics Letters B 278 (1992) 15-18 North-Holland
PHYSICS LETTERS B
Confinement and gaussian gluon fields T.S. B i r 6 lnstitut j~r TheoretischePhysik, Justus-Liebig-Universitdt Giessen, W-6300 Giessen, FRG Received 29 November 1991; revised manuscript received 8 January 1992
The confinement mechanism is described as an interplay between classical massless longitudinal gluon fields induced by static color charges and transversal quantum fluctuations in the gluon condensate. The nonabelian field energy amounts to a linearly rising effective potential between heavy quarks and antiquarks in this case. The string tension is related to the gluon condensate strength.
W h a t can relativistic heavy-ion e x p e r i m e n t s teach us about Q C D ? The soft Q C D regime can be tested in hot a n d dense m a t t e r over a m u c h larger v o l u m e than would be accessible in i n d i v i d u a l h a d r o n - h a d ron collisions. This is exactly the physical regime, however, where p e r t u r b a t i v e Q C D does not work. This circumstance m o t i v a t e s us to develop theoretical models which i n c o r p o r a t e an a p p r o x i m a t e description o f both the infrared a n d ultraviolet regime o f Q C D ; a nontrivial p r o b l e m for the theoretical phenomenology. Recently an approach to the Q C D v a c u u m state has been proposed, which uses a gaussian wave function ansatz [ 1 ]. This m o d e l describes a gluon condensate, glueball mass, a first o r d e r phase transition and its dependence on the n u m b e r o f light quark flavors [ 2 ]. It was not clear up to now, however, whether the low t e m p e r a t u r e phase o f this m o d e l can be e x t e n d e d to include color confinement. It has been conjectured several times that a chaotic v a c u u m state in Q C D , which describes independently fluctuating gluon fields, leads to an area-law o f electrostatic Wilson-loop expectation values and hence to static c o n f i n e m e n t [ 3 - 5 ]. In the present letter I explore the c o n f i n e m e n t o f static charges in a gaussian fluctuating gluon condensate with variational methods. This description assigns an equally i m p o r t a n t role for b o t h q u a n t u m fluctations and classical fields. Work supported by BMFT, GSI Darmstadt.
Since gauge fields obey constraints there are always only two degrees o f freedom to be quantized for each color. The gaussian wave function ansatz takes this into account: (0o[A ] = e x p ( - 1 ( A q m O T A q m ) ) ,
(1)
where < . > means color and polarization traces and m o d e s u m m a t i o n . Using a h o m o g e n o u s and isotropic QT in this ansatz we arrived at a gap equation d e t e r m i n i n g the mass gap in the v a c u u m state [ 1,2 ]. The vacuum pressure at the variational m i n i m u m was o b t a i n e d as 3M 4 A4as - 167r2 .
(2)
The numerical value was o b t a i n e d using a bag constant Of Abag= 220 MeV, which yielded M = 600 MeV, and an e s t i m a t e d glueball mass o f 2 M = 1200 MeV. Due to the gaussian wave function ansatz a correlation between transversal q u a n t u m fluctuations is o b t a i n e d with a magnitude p r o p o r t i o n a l to this mass scale:
A ~A~ = M--~25abgij J l - A a A b , 6g 2
(3)
where S' denotes the expectation value ,.q= f ~ A ~0*[A] S~o[AI f ~ A ~o*~0
(4)
for any o p e r a t o r S. Although eq. ( 3 ) is not gauge in-
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
15
Volume 278, number 1,2
PHYSICS LETTERSB
variant M 2 itself, being connected to the gluon condensate strength, is. In the presence of static charges classical gluon fields occur due to the Gauss law. We generalize therefore the gaussian wave function ansatz of the pure gluon sector introducing a classical vectorpotential.4 and an electric field/~:
~0[A] = ~Oo[ A - A ] - e x p ( i ( £ A ) ) .
(5)
This gaussian ansatz is centered around the classical field configuration (E,A). In order to satisfy the Gauss law DE+p=0,
(6)
we determine the action of the electric field operator on our ansatz eq. (5):
:~SA
( 1- ~ Q ( A - A . ) + E ) rp.
(8)
where the "dot" means time derivative. The Gauss law can be satisfied on the average assuming [A, .41 = 0 .
(9)
In this case we arrive at the abelian requirement
ISd3X~l.qm=IgZfd3x[A,.~l 2 ,
It is easy to solve cqs. (8) and (10) in the Fourier decomposition. We get
1 ~ d3x ~,,qm = Im2 f
(15)
Due to earlier studies of the infrared behavior of the gluon propagator it was realized that for the confinement the necessary k-4 contribution is consistent with the Dyson-Schwinger equation [ 6 ]. We investigate next a neutral quark-antiquark system. The Fourier decomposition of the charge distribution is sin(Ik~R~ ) ,
(17)
putting a heavy quark at IRi with color charge qa and an antiquark at - I R i with color charge _qa. The nonabelian contribution to the vacuum energy modification in this case becomes [cf. eq. ( 15 ) ]
g2M2R
oo
sin2t 47~2 j ~ 5 - dt,
f
(18)
0 using the coupling constant g2 for the charge of a single quark:
qaqa=g2.
(19)
The evaluation of the integral yields l zt so we finally get l f d3x ~cl,qm =
I aM2R ,
(20)
(12)
receives an abelian contribution from the classical electric field: 16
k2w-''''"-~,
(16)
(11)
Although this seems to be an enormous reduction of the richness of the nonabelian Gauss law we describe confinement even with this simplification. This implies on the other hand that the nonabelian nature of the Gauss law in QCD may not be essential for the confinement. Now we turn to calculate the modification of the vacuum energy (gluon condensate) due to static charges. The hamiltonian
[EZ(x)+BZ(x)]
d3k Ipa(k) ] 2
(27~)3
where we have used eq. (3) for evaluating ( A - A ) 2 in the gluon condensate background. This energy contribution results in a static quark confinement as long as classical gluon fields.4 are rnassless:
l j d3x °~cl,qrn=
H = / f d3x
(14)
which yields upon the gaussian ansatz
pa(k) =qa2i
(10)
-~pa(k).
(13)
There is furthermore a nonabelian contribution arising as a combined effect of classical and quantum gluon fields
co(k)=k.
£=3,
A~(k) =
l ~ o¢t~cld 3 x = l ~ d3k Ipa(k) 12 (27~)3 k2
(7)
The classical electric field in the Ao = 0 gauge becomes
OA-=p.
19 March 1992
yielding a linear and hence confining potential. The abelian contribution of the qCl system to the energy is recognized as the Coulomb energy
Volume 278, number 1,2
PHYSICS LETTERSB
½f d3x ~ , = o o - ~o~,
(21)
where the UV infinite term is due to considering point charges and is therefore irrelevant for the soft QCD. The total energy of the qdl system in a gluon condensate is //q~=ot(_
1 l 2R). N -~+~M
(22)
19 March 1992
a= ~o~M2 .
(28)
This can be compared to fits to J / ~ experiments carried out by Eichten et al. [ 8 ]. Using the intermediate value of their two sets of parameters which fit experimental data, i.e., a string constant of a = 0.20 GeV 2, we obtain o q = 1.1 in our model. We compare this value to the intermediate value of the experimental fits o~2= 0.40 using renormalisation group scaling 1
1
oq
o~2
4~.
M2
(29)
The present model reproduces not only the linear rise of the effective qdl potential but the absolute confinement of unbalanced color charges too. The simplest case to consider is that of a single quark having the charge density Fourier decomposition
With the values M1 =0.6 GeV (from the gluon condensate) and M2= 3 GeV (J/~¢ mass) we obtain
Ipa(k) 12=g 2 .
which agrees quite well with the experimental finding
(23)
In fact one gets an infrared divergent energy contribution /
f
d 3 x O~el,qm -
g2M2f 47~2
dk = ~ ,
OO~
(24)
0
as long as the classical gluon mass is zero. The mechanism of the deconfinement is therefore cutting this infrared divergence due to a medium generated gluon mass 1
f
d3X~cl,qm -
g2M2 ~
f dk M2 Jk2+m--~2=O~2rn.
(25)
0
The quark-antiquark system in this case yields 1
~ d3x )~clqm'--
°lM2(lm\....1-exp(-mR))mR ,
(26)
while the Coulomb energy remains unchanged. If we assume that at high temperatures both classical and quantum gluon fields can be described by the same medium generated mass, i.e., we obtain a Debye screened effective potential as in lattice QCD [7]
M=m,
Vqc~=°t(m+ exp(-mR)R
2) .
(27)
On the other hand rendering by some mechanism the classical gluon fields massless, this model predicts from the linear rise of the effective qdl potential a string constant of
2--9m M~"
O~]C"led(3 G e V ) = 0 . 4 3 ,
(30)
o f Og2 = 0 . 4 0 .
Summarizing in the gaussian gluon field model we describe two phases: - confinement with massive transverse vacuum fluctuations and massless longitudinal gluon fields giving infinite energy to single quarks and a linearly rising potential between quark and antiquark; - deconfinement with screened color charges due to an equal medium generated mass for both transverse and longitudinal gluons. As already reported in earlier publications the same model predicts a first order phase transition between these two phases at a temperature of Tc = 230 MeV in the pure gluon sector ( N f = 0 ) and at a temperature of To= 175 MeV including two light quark flavors ( N f = 2 ) . The confinement also occurs at zero temperature at a baryochemical potential of/~B= 1275 MeV [1,2]. It should also be noted that a variational model, which yields a linearly rising static quark potential, does not prove confinement; a real ground state can always have a lower energy than the variational resuit. On the other hand the presently assumed ground state does not yield a less than linearly rising variational potential, it is therefore compatible with our knowledge about color confinement in QCD. Summarizing this letter, the confinement of static charges in the G G F model is represented as an interplay between classical massless longitudinal gluon fields and massive transverse quantum fluctuations in the gluon condensate. The nonabelian field energy 17
Volume 278, number 1,2
PHYSICS LETTERS B
b e c o m e s i n f r a r e d d i v e r g e n t for u n b a l a n c e d color charges while it a m o u n t s to a linearly rising p o t e n t i a l between quarks and antiquarks. T h e e n l i g h t e n i n g d i s c u s s i o n s with Professor B. Miiller, Professor E. S h u r y a k a n d Professor I. Z a h e d are gratefully acknowledged.
18
19 March 1992
References [ 1 ] T.S. Bir6, Ann. Phys. (NY) 191 (1989) 1, [2] T.S. Bir6, Phys. Lett. B 245 (1990) 142. [31 P. Olesen, Nucl. Phys. B 200 (1982) 381. [4] H.G. Dosch, Phys. Lett. B 190 (1987) 177. [ 5 ] H.G. Dosch and Yu.A. Simonov, Z. Phys. C 45 ( 1989 ) 147. [6] M. Baker, J.S. Ball and F. Zachariasen, Nucl. Phys. B 186 (1981) 531,560. [ 7 ] J. Engels, F. Karsch, H. Satz and I. Montvay, Nucl. Phys. B 205 (1982) 545. [8] E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane and T.H. Yan, Phys. Rev. D 21 (1980) 203.
PHYSICS LETTERS B
Confinement and gaussian gluon fields T.S. B i r 6 lnstitut j~r TheoretischePhysik, Justus-Liebig-Universitdt Giessen, W-6300 Giessen, FRG Received 29 November 1991; revised manuscript received 8 January 1992
The confinement mechanism is described as an interplay between classical massless longitudinal gluon fields induced by static color charges and transversal quantum fluctuations in the gluon condensate. The nonabelian field energy amounts to a linearly rising effective potential between heavy quarks and antiquarks in this case. The string tension is related to the gluon condensate strength.
W h a t can relativistic heavy-ion e x p e r i m e n t s teach us about Q C D ? The soft Q C D regime can be tested in hot a n d dense m a t t e r over a m u c h larger v o l u m e than would be accessible in i n d i v i d u a l h a d r o n - h a d ron collisions. This is exactly the physical regime, however, where p e r t u r b a t i v e Q C D does not work. This circumstance m o t i v a t e s us to develop theoretical models which i n c o r p o r a t e an a p p r o x i m a t e description o f both the infrared a n d ultraviolet regime o f Q C D ; a nontrivial p r o b l e m for the theoretical phenomenology. Recently an approach to the Q C D v a c u u m state has been proposed, which uses a gaussian wave function ansatz [ 1 ]. This m o d e l describes a gluon condensate, glueball mass, a first o r d e r phase transition and its dependence on the n u m b e r o f light quark flavors [ 2 ]. It was not clear up to now, however, whether the low t e m p e r a t u r e phase o f this m o d e l can be e x t e n d e d to include color confinement. It has been conjectured several times that a chaotic v a c u u m state in Q C D , which describes independently fluctuating gluon fields, leads to an area-law o f electrostatic Wilson-loop expectation values and hence to static c o n f i n e m e n t [ 3 - 5 ]. In the present letter I explore the c o n f i n e m e n t o f static charges in a gaussian fluctuating gluon condensate with variational methods. This description assigns an equally i m p o r t a n t role for b o t h q u a n t u m fluctations and classical fields. Work supported by BMFT, GSI Darmstadt.
Since gauge fields obey constraints there are always only two degrees o f freedom to be quantized for each color. The gaussian wave function ansatz takes this into account: (0o[A ] = e x p ( - 1 ( A q m O T A q m ) ) ,
(1)
where < . > means color and polarization traces and m o d e s u m m a t i o n . Using a h o m o g e n o u s and isotropic QT in this ansatz we arrived at a gap equation d e t e r m i n i n g the mass gap in the v a c u u m state [ 1,2 ]. The vacuum pressure at the variational m i n i m u m was o b t a i n e d as 3M 4 A4as - 167r2 .
(2)
The numerical value was o b t a i n e d using a bag constant Of Abag= 220 MeV, which yielded M = 600 MeV, and an e s t i m a t e d glueball mass o f 2 M = 1200 MeV. Due to the gaussian wave function ansatz a correlation between transversal q u a n t u m fluctuations is o b t a i n e d with a magnitude p r o p o r t i o n a l to this mass scale:
A ~A~ = M--~25abgij J l - A a A b , 6g 2
(3)
where S' denotes the expectation value ,.q= f ~ A ~0*[A] S~o[AI f ~ A ~o*~0
(4)
for any o p e r a t o r S. Although eq. ( 3 ) is not gauge in-
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
15
Volume 278, number 1,2
PHYSICS LETTERSB
variant M 2 itself, being connected to the gluon condensate strength, is. In the presence of static charges classical gluon fields occur due to the Gauss law. We generalize therefore the gaussian wave function ansatz of the pure gluon sector introducing a classical vectorpotential.4 and an electric field/~:
~0[A] = ~Oo[ A - A ] - e x p ( i ( £ A ) ) .
(5)
This gaussian ansatz is centered around the classical field configuration (E,A). In order to satisfy the Gauss law DE+p=0,
(6)
we determine the action of the electric field operator on our ansatz eq. (5):
:~SA
( 1- ~ Q ( A - A . ) + E ) rp.
(8)
where the "dot" means time derivative. The Gauss law can be satisfied on the average assuming [A, .41 = 0 .
(9)
In this case we arrive at the abelian requirement
ISd3X~l.qm=IgZfd3x[A,.~l 2 ,
It is easy to solve cqs. (8) and (10) in the Fourier decomposition. We get
1 ~ d3x ~,,qm = Im2 f
(15)
Due to earlier studies of the infrared behavior of the gluon propagator it was realized that for the confinement the necessary k-4 contribution is consistent with the Dyson-Schwinger equation [ 6 ]. We investigate next a neutral quark-antiquark system. The Fourier decomposition of the charge distribution is sin(Ik~R~ ) ,
(17)
putting a heavy quark at IRi with color charge qa and an antiquark at - I R i with color charge _qa. The nonabelian contribution to the vacuum energy modification in this case becomes [cf. eq. ( 15 ) ]
g2M2R
oo
sin2t 47~2 j ~ 5 - dt,
f
(18)
0 using the coupling constant g2 for the charge of a single quark:
qaqa=g2.
(19)
The evaluation of the integral yields l zt so we finally get l f d3x ~cl,qm =
I aM2R ,
(20)
(12)
receives an abelian contribution from the classical electric field: 16
k2w-''''"-~,
(16)
(11)
Although this seems to be an enormous reduction of the richness of the nonabelian Gauss law we describe confinement even with this simplification. This implies on the other hand that the nonabelian nature of the Gauss law in QCD may not be essential for the confinement. Now we turn to calculate the modification of the vacuum energy (gluon condensate) due to static charges. The hamiltonian
[EZ(x)+BZ(x)]
d3k Ipa(k) ] 2
(27~)3
where we have used eq. (3) for evaluating ( A - A ) 2 in the gluon condensate background. This energy contribution results in a static quark confinement as long as classical gluon fields.4 are rnassless:
l j d3x °~cl,qrn=
H = / f d3x
(14)
which yields upon the gaussian ansatz
pa(k) =qa2i
(10)
-~pa(k).
(13)
There is furthermore a nonabelian contribution arising as a combined effect of classical and quantum gluon fields
co(k)=k.
£=3,
A~(k) =
l ~ o¢t~cld 3 x = l ~ d3k Ipa(k) 12 (27~)3 k2
(7)
The classical electric field in the Ao = 0 gauge becomes
OA-=p.
19 March 1992
yielding a linear and hence confining potential. The abelian contribution of the qCl system to the energy is recognized as the Coulomb energy
Volume 278, number 1,2
PHYSICS LETTERSB
½f d3x ~ , = o o - ~o~,
(21)
where the UV infinite term is due to considering point charges and is therefore irrelevant for the soft QCD. The total energy of the qdl system in a gluon condensate is //q~=ot(_
1 l 2R). N -~+~M
(22)
19 March 1992
a= ~o~M2 .
(28)
This can be compared to fits to J / ~ experiments carried out by Eichten et al. [ 8 ]. Using the intermediate value of their two sets of parameters which fit experimental data, i.e., a string constant of a = 0.20 GeV 2, we obtain o q = 1.1 in our model. We compare this value to the intermediate value of the experimental fits o~2= 0.40 using renormalisation group scaling 1
1
oq
o~2
4~.
M2
(29)
The present model reproduces not only the linear rise of the effective qdl potential but the absolute confinement of unbalanced color charges too. The simplest case to consider is that of a single quark having the charge density Fourier decomposition
With the values M1 =0.6 GeV (from the gluon condensate) and M2= 3 GeV (J/~¢ mass) we obtain
Ipa(k) 12=g 2 .
which agrees quite well with the experimental finding
(23)
In fact one gets an infrared divergent energy contribution /
f
d 3 x O~el,qm -
g2M2f 47~2
dk = ~ ,
OO~
(24)
0
as long as the classical gluon mass is zero. The mechanism of the deconfinement is therefore cutting this infrared divergence due to a medium generated gluon mass 1
f
d3X~cl,qm -
g2M2 ~
f dk M2 Jk2+m--~2=O~2rn.
(25)
0
The quark-antiquark system in this case yields 1
~ d3x )~clqm'--
°lM2(lm\....1-exp(-mR))mR ,
(26)
while the Coulomb energy remains unchanged. If we assume that at high temperatures both classical and quantum gluon fields can be described by the same medium generated mass, i.e., we obtain a Debye screened effective potential as in lattice QCD [7]
M=m,
Vqc~=°t(m+ exp(-mR)R
2) .
(27)
On the other hand rendering by some mechanism the classical gluon fields massless, this model predicts from the linear rise of the effective qdl potential a string constant of
2--9m M~"
O~]C"led(3 G e V ) = 0 . 4 3 ,
(30)
o f Og2 = 0 . 4 0 .
Summarizing in the gaussian gluon field model we describe two phases: - confinement with massive transverse vacuum fluctuations and massless longitudinal gluon fields giving infinite energy to single quarks and a linearly rising potential between quark and antiquark; - deconfinement with screened color charges due to an equal medium generated mass for both transverse and longitudinal gluons. As already reported in earlier publications the same model predicts a first order phase transition between these two phases at a temperature of Tc = 230 MeV in the pure gluon sector ( N f = 0 ) and at a temperature of To= 175 MeV including two light quark flavors ( N f = 2 ) . The confinement also occurs at zero temperature at a baryochemical potential of/~B= 1275 MeV [1,2]. It should also be noted that a variational model, which yields a linearly rising static quark potential, does not prove confinement; a real ground state can always have a lower energy than the variational resuit. On the other hand the presently assumed ground state does not yield a less than linearly rising variational potential, it is therefore compatible with our knowledge about color confinement in QCD. Summarizing this letter, the confinement of static charges in the G G F model is represented as an interplay between classical massless longitudinal gluon fields and massive transverse quantum fluctuations in the gluon condensate. The nonabelian field energy 17
Volume 278, number 1,2
PHYSICS LETTERS B
b e c o m e s i n f r a r e d d i v e r g e n t for u n b a l a n c e d color charges while it a m o u n t s to a linearly rising p o t e n t i a l between quarks and antiquarks. T h e e n l i g h t e n i n g d i s c u s s i o n s with Professor B. Miiller, Professor E. S h u r y a k a n d Professor I. Z a h e d are gratefully acknowledged.
18
19 March 1992
References [ 1 ] T.S. Bir6, Ann. Phys. (NY) 191 (1989) 1, [2] T.S. Bir6, Phys. Lett. B 245 (1990) 142. [31 P. Olesen, Nucl. Phys. B 200 (1982) 381. [4] H.G. Dosch, Phys. Lett. B 190 (1987) 177. [ 5 ] H.G. Dosch and Yu.A. Simonov, Z. Phys. C 45 ( 1989 ) 147. [6] M. Baker, J.S. Ball and F. Zachariasen, Nucl. Phys. B 186 (1981) 531,560. [ 7 ] J. Engels, F. Karsch, H. Satz and I. Montvay, Nucl. Phys. B 205 (1982) 545. [8] E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane and T.H. Yan, Phys. Rev. D 21 (1980) 203.