- Email: [email protected]
Thermal phase transition in weakly interacting, large NC QCD
18 June 1998
Physics Letters B 429 Ž1998. 232–238
Thermal phase transition in weakly interacting, large NC QCD Joakim Hallin a
a,1
, David Persson
b,2
Institute of Theoretical Physics, Chalmers UniÕersity of Technology and Goteborg UniÕersity, S-412 96 Goteborg, Sweden ¨ ¨ b Department of Physics and Astronomy, UniÕersity of British Columbia, VancouÕer, BC V6T 1Z1, Canada Received 13 March 1998 Editor: W. Haxton
Abstract We consider thermal QCD in the large NC limit, mainly in 1 q 1 dimensions. The gauge coupling is only taken into account to minimal order, by projection onto colour singlets. An expression for the free energy, exact as NC ™ `, is then obtained. A third order phase transition will occur. The critical temperature depends on the ratio NCrL, where L is the Žinfinite. spatial length. In the high temperature limit, the free energy will approach the same value as in the free theory, whereas we have a mesonic like phase at low temperature. Expressions for the quark condensate, ²CC :, are also obtained. q 1998 Published by Elsevier Science B.V. All rights reserved.
Considerable interest has recently been devoted to the assumed deconfinement phase transition in QCD at high temperature andror density. Exact analytical results are incomprehensible sofar, and we have to rely on approximate methods or computer simulations on the lattice. It has been well known for a long time that some insight in the confinement mechanism may be gained by considering the large NC limit, for an SUŽ NC . gauge theory. This limit is particularly fruitful in 1 q 1 dimensions Žsee e.g. Refs. w1–5x.. QCD in 1 q 1 dimensions has also received attention recently due to the remarkable fact that different regularizations seems to lead to different models, even within the same choice of gauge. In his pioneering work, ’t Hooft w1x employed a cutoff around the origin in momentum space, that later was shown to be equivalent to a principle value prescription w2x. Within this framework, no free quarks can propagate, but they are confined into mesons, consisting of quark anti-quark pairs. Wu w3x, on the other hand, suggested another regularization, allowing for a Wick rotation to Euclidean space. The bound state equation is more complicated in this case, and no solution has yet been found. Bassetto and Nardelli with co-workers w6–9x have compared the two regularizations, and computed for example the expectation value of the Wilson loop. The expectation value of the Wilson loop with Wu’s regularization, was recently calculated by Staudacher and Krauth w10x, who showed that confinement is not enforced in this case, unlike using the ’t Hooft regularization. It has been suggested by Chibisov and Zhitnitsky w11x that the models due to different regularizations could show up as different phases, that makes a study of the system at
1 2
Email address: [email protected]. Email address: [email protected].
0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 4 7 8 - X
J. Hallin, D. Perssonr Physics Letters B 429 (1998) 232–238
233
finite temperature intriguing. In 2 dimensional QCD, confinement is obvious as an infinite energy for coloured states. We shall in this letter take the interaction into account only by a projection onto colour singlets, i.e. we neglect the coupling unless it multiplies the infra red divergence. The quark contribution to the free energy in 3 q 1 dimensions is then easily obtained from the free energy presented here in 1 q 1 dimensions through the substitution, LHdprŽ2p . ™ 2VHd 3 prŽ2p . 3. The projected free energy in 3 q 1 dimensions has earlier been considered in the high temperature limit by Skagerstam w12x. The full interacting case in 2 dimensions, that is troubled by infra red divergences, will be more extensively considered in another article. Including the interaction in the strict large NC limit LrNC ™ 0, McLerran and Sen w13x, and employing a different method Hansson and Zahed w14x showed that in the low temperature ’t Hooft meson phase, the partition function ln Z s O Ž L., due to confinement. Using a different resummation scheme for LrNC ™ `, McLerran and Sen w13x also found another phase of almost free quarks, with ln Z s O Ž LNC .. The phase transition itself was not obtainable, and could only take place as T ™ `, i.e. T A NC . By considering finite k s NC rL, we are here able to monitor the phase transition exactly, and for finite k , it will take place at finite T. However, due to the negligence of the interaction we do not have confinement in the strict sense. Our low temperature phase consists of quark–anti-quark pairs, showing up for example in Bose–Einstein distribution functions, but these are not the confined mesons of ’t Hooft’s, since ln Z s O Ž LNC .. ˆ Let us now consider the unnormalized density matrix rˆ s eyb H of free coloured fermions having the Hamiltonian Hˆ s fˆ † Ž p . h Ž p . fˆ Ž p . ,
Ž 1.
Hp
where hŽ p . s g 0 Ž g 1 p q m. ,
Ž 2.
n
and Hp s H Ž2dp p. , in n spatial dimensions. We shall here mainly discuss n s 1. In the functional representation the expression for rˆ is w15x n
r Ž h1)h1 ,h 2)h 2 . s Nb exp Ž Ž h1) q h 2) . Vb Ž h1 q h 2 . q h1)h 2 y h 2)h1 . ,
Ž 3.
where
ž
b
ž
Nb s exp 4 NC V ln cosh
Hp
b
Vb s ytanh
ž / 2
v
h
v
2
v
//
,
Ž 4.
1C ,
Ž 5.
(
v Ž p . s p 2 q m2 ,
Ž 6.
and L denotes the spatial length. In the expression for r Žh1)h1 ,h 2)h 2 ., summation over spinor, colour and momentum indices is understood. Moreover NC denotes the number of colours and 1 C is the unit matrix in colour space. We now wish to calculate the partition function for colourless states. This is done by the operator pˆ ,
pˆ s dupˆ u ,
Ž 7.
H
ˆa
that projects onto colour singlets. Here pˆ u s e i a a Q is a general global UŽ NC . gauge transformation. The partition function Z is then Z s tr Ž pr ˆ ˆ. .
Ž 8.
J. Hallin, D. Perssonr Physics Letters B 429 (1998) 232–238
234
With the colour charge operator Qˆ a s fˆ †
Hp
ta 2
fˆ ,
Ž 9.
the global gauge transformation pˆ u is in the functional representation:
pu Ž h1)h1 ,h 2)h 2 . s Nu exp Ž h1) q h 2) . V u Ž h1 q h 2 . q h1)h 2 y h 2)h1 .
Ž 10 .
Here we have defined Nu s exp
1 2
tr ln 14 Ž u q uy1 q 2 . ,
Ž 11 .
and
Vu s
uy1 uq1
1s.
Ž 12 .
Furthermore, tr s LHp tr s tr C denotes the trace over all indices, momentum, spin and colour, while 1 s is the unit matrix in spinor space i.e. a 2 = 2-unit matrix Ž4 = 4 in n s 3.. The group element is u s e i a a . Multiplying rˆ and uˆ one finds ta 2
²h1)h1 < uˆ rˆ
½
Ž h1) q h2) .
Vb q V u 1 q Vb V u
5
Ž h1 q h2 . q h1)h2 y h2)h1 . Ž 13 .
Finally we can then give the expression for the partition function,
½H
5
Z s du tr Ž pˆ u rˆ . s dudet Ž 2 . Nb Nu det Ž 1 q Vb V u . s eyb E 0 duexp L tr C ln Ž 1 q j u . Ž 1 q j uy1 . ,
H
H
H
p
Ž 14 . where E0 s yLNC Hp v is the vacuum energy. We have here defined
j s eyb v ,
Ž 15 .
and e i a j denotes the NC eigenvalues of the UŽ NC . matrix u. On functions of eigenvalues the Haar measure du takes the form NC
H
du s
H is1 Ł d a Ł sin i-j
2
ai y a j
i
2
.
Ž 16 .
This gives NC
Z s ey b E 0
Ł d a i Ł sin2
H is1
i-j
ai y a j 2
NC
Hp Ý ln Ž1 q j e
exp L
iaj
.Ž 1 q j eyi a . j
.
Ž 17 .
js1
In the large NC -limit the integral over UŽ NC . in Ž17. may be calculated by the steepest descent method. We take this limit by letting NC and the spatial length L tend to infinity simultaneously keeping their quotient
ks
NC L
,
Ž 18 .
constant. We solve this by introducing the density of eigenstates, D , in the continuum limit w16,17x, i.e. NC
1
Ý f Ž a j . ™ NCH dtf js1
0
a Ž t . s NC
ac
Hya D Ž a . d a f Ž a . . c
Ž 19 .
J. Hallin, D. Perssonr Physics Letters B 429 (1998) 232–238
235
We find two distinct phases, depending on the inequality 1y
2
1
H k pe
bv
y1
G 0.
Ž 20 .
As long as Ž20. is satisfied we are in the zero-gap phase, a c s p , where D has support over the whole circle. When Ž20. is not satisfied we are in the one-gap phase. In this case there is a gap in which D lacks support, i.e. an interval where the eigenvalues give a vanishing contribution as NC ™ `. In the zero-gap phase ln Z is given by a sum over quark–antiquark pairs ln Z s yb E0 y
NC2
k2
ln 1 y j Ž p . j Ž q . HH p q
.
Ž 21 .
Even though the states are mesonic-like, we do not have confinement since ln Z A Ž NC rk . 2 s NC Lrk , whereas in the confined phase of ’t Hooft mesons we have Žcf. w14x. ln Z A L. In the one-gap phase we find ln Z s yb E0 q NC2 ln sin
ž
y
NC2
k2
ac 2
/
q
2 NC2
k
ln 1 y x Ž p . x Ž q . HH p q
Hpln ž 1 q j q (j 1 2
,
2
q 2 j cos a c q 1
/ Ž 22 .
where we have defined 1
xs
4j sin
(
2 a 1 q j y j q 2 j cos a c q 1 2 c
2
.
Ž 23 .
2
Furthermore, the critical angle a c is determined from
ž
Hp (j
1qj 2
q 2 j cos a c q 1
/
y1 sk .
Ž 24 .
In Fig. 1, we plot ln ZrŽTNC2 . as a function of the temperature, for k s m. The critical temperature is then found as Tc f 0.8 m. The free energy Ž F s yT ln Z . in the 1-gap phase Žsolid line. grows as T 2 for large T, and is approaching the free energy for free quarks, i.e. without the projection, Ždotted line., as T ™ `. Whereas the free energy in the 0-gap phase Ždashed line., continued above the phase transition would grow like T 3. The high
Fig. 1. The free energy for free quarks Ždotted line., in the 0-gap phase Ždashed line., and in the 1-gap phase Žsolid line., for k s m. All in units of NC2 T 2 r m.
J. Hallin, D. Perssonr Physics Letters B 429 (1998) 232–238
236
temperature behaviour in the 1-gap phase is obtained by an expansion for a c < 1, yielding a c , 4pkrT. As T ™ `, a c ™ 0, i.e. only u s 1 gives a contribution, corresponding to no projection. In addition to the free energy, it is of interest to compare the entropy, and the Žrenormalized. energy density in the two phases. They are given by ss
es
1 E L ET
Ž T ln Z . L ,
Ž 25 .
T2 E
Ž ln Z q b E0 . L , Ž 26 . L ET respectively. Their behaviour is similar to the that of the free energy. In the interacting case it is well known that a nontrivial chiral condensate appears, as shown for m s 0 by Zhitnitsky w18,19x, and in the general case by Burkardt w20x. However, when the interaction only is taken into account through the projection, the vacuum is trivial and no chiral condensate can appear in this case. Nevertheless, the Žrenormalized. expectation value of cc , is an order parameter for the phase transition, and thus of interest. We find E ² cc : s yT Ž ln Z q b E0 . L . Ž 27 . Em In the 0-gap phase it is straightforward to evaluate this from Ž21.. However, in the 1-gap phase we need to define ac E cot T a ' ma, Ž 28 . 2 Em c where y1
as
½
Hp
j Ž1qj .
Ž j 2 q 2 j cos a c q 1 .
3r2
5½
j Ž1yj .
Hp
1
Ž j 2 q 2 j cos a c q 1 .
3r2
v
5
,
Ž 29 .
is obtained from Ž24., using Ž15., and Ž6.. We then find T E m Em
xs
x
(j
2
q 2 j cos a c q 1
Ž1qj . ay Ž1yj .
1
.
v
Ž 30 .
This gives
² cc : s NC2
=
° 1 2 m~y a q H ¢ 2 k 1 q j q (j p
1q
2 y
k2
j q cos a c
(j
2
2
q 2 j cos a c q 1 2 j sin2
j v
q 2 j cos a c q 1
x Ž p. x Ž q.
1
q
(j
2
ac
2 z q 2 j cos a c q 1
Ž1qj . ay Ž1yj .
1
v j 2 q 2 j cos a c q 1
HH p q 1 yx Ž p. x Ž q. (
¶ •. ß
0 Ž 31 .
We have been able to show that ² cc : is continuous over the phase transition, whereas higher derivatives give cumbersome expressions.
J. Hallin, D. Perssonr Physics Letters B 429 (1998) 232–238
237
However, in the limit of infinite massive quarks analytical results are obtained. As m ™ `, we must also let Tc ™ `, such that their quotient
z s mrTc ,
Ž 32 .
is kept fixed, in order to keep a finite particle density. Furthermore, we define t according to T s Tct . The momentum integrals are now independent of p, so we write
Hp™ L .
Ž 33 .
The critical temperature is then determined from
k
1 s
2
Hp e
v r Tc
y1
™L
1 z
e y1
.
Ž 34 .
.
Ž 35 .
In the 0-gap phase we then find ² cc : s 2 Nc2
L
ž / k
2
1 e
2 z rt
y1
In the 1-gap phase the critical angle is now determined from
(1 q 2 e
z rt
cos a c q e
2 z rt
e z rt q 1 s
,
1 q krL
Ž 36 .
that after some simplifications give ² cc : s Nc2
L
1
½ ž /ž q
2
k
k 2q
L/
1 e
z rt
q1
5
.
Ž 37 .
Notice here the Bose–Einstein distribution in the low temperature 0-gap phase, as opposed to the Fermi–Dirac distribution in the 1-gap phase at high temperature. It is now a straightforward exercise to show that E 2rE t 2 ² cc : is discontinuous, so that it is a third order phase transition. This is in agreement with the study of Gattringer et al. w21x concerning static quarks on a line. Furthermore, in the corresponding lattice model Gross and Witten w16x found a third order phase transition, so we conjecture that the phase transition most likely is third order also for arbitrary mass.
Acknowledgements We are grateful to Ariel Zhitnitsky for proposing the subject, as well as stimulating discussions; and to Hans Hansson for interesting remarks, and pointing out w14x. J.H.’s research was funded by NFR Žthe Swedish Natural Science Research Council., and D.P.’s by STINT Žthe Swedish Foundation for Cooperation in Research and Higher Education..
References w1x w2x w3x w4x w5x
G. ’tHooft, Nucl. Phys. B 75 Ž1974. 461. C.G. Callan, N. Coote, D. Gross, Phys. Rev. D 13 Ž1976. 1649. T.T. Wu, Phys. Lett. B 71 Ž1977. 142. M.B. Einhorn, Phys. Rev. D 14 Ž1976. 3451. R. Brower, W.L. Spence, Phys. Rev. D 19 Ž1979. 3024.
238 w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x
J. Hallin, D. Perssonr Physics Letters B 429 (1998) 232–238 A. Bassetto, L. Griguolo, Phys. Rev. D 53 Ž1996. 7385. A. Bassetto, G. Nardelli, Int. J. Mod. Phys. A 12 Ž1997. 1075. A. Bassetto, G. Nardelli, A. Shuvaev, Nucl. Phys. B 495 Ž1997. 451. A. Bassetto, D. Colferai, G. Nardelli, Nucl. Phys. B 501 Ž1997. 227. M. Staudacher, W. Krauth, hep-thr9709101, 1997. B. Chibisov, A.R. Zhitnitsky, Phys. Lett. B 362 Ž1995. 105. B.S. Skagerstam, Z. Phys. C 24 Ž1984. 97. L.D. McLerran, A. Sen, Phys. Rev. D 32 Ž1985. 2794. T.H. Hansson, I. Zahed, Phys. Lett. Ž1993.. J. Hallin, P. Liljenberg, Phys. Rev. D 52 Ž1995. 1150. D.J. Gross, E. Witten, Phys. Rev. D 21 Ž1980. 446. G. Mandal, Mod. Phys. Lett. A 5 Ž1990. 1147. A.R. Zhitnitsky, Phys. Lett. B 165 Ž1985. 405. A.R. Zhitnitsky, Sov. J. Nucl. Phys. 44 Ž1986. 139. M. Burkardt, Phys. Rev. D 53 Ž1996. 933. C.R. Gattringer, L.D. Paniak, G.W. Semenoff, Annals Phys. 256 Ž1997. 74.
Physics Letters B 429 Ž1998. 232–238
Thermal phase transition in weakly interacting, large NC QCD Joakim Hallin a
a,1
, David Persson
b,2
Institute of Theoretical Physics, Chalmers UniÕersity of Technology and Goteborg UniÕersity, S-412 96 Goteborg, Sweden ¨ ¨ b Department of Physics and Astronomy, UniÕersity of British Columbia, VancouÕer, BC V6T 1Z1, Canada Received 13 March 1998 Editor: W. Haxton
Abstract We consider thermal QCD in the large NC limit, mainly in 1 q 1 dimensions. The gauge coupling is only taken into account to minimal order, by projection onto colour singlets. An expression for the free energy, exact as NC ™ `, is then obtained. A third order phase transition will occur. The critical temperature depends on the ratio NCrL, where L is the Žinfinite. spatial length. In the high temperature limit, the free energy will approach the same value as in the free theory, whereas we have a mesonic like phase at low temperature. Expressions for the quark condensate, ²CC :, are also obtained. q 1998 Published by Elsevier Science B.V. All rights reserved.
Considerable interest has recently been devoted to the assumed deconfinement phase transition in QCD at high temperature andror density. Exact analytical results are incomprehensible sofar, and we have to rely on approximate methods or computer simulations on the lattice. It has been well known for a long time that some insight in the confinement mechanism may be gained by considering the large NC limit, for an SUŽ NC . gauge theory. This limit is particularly fruitful in 1 q 1 dimensions Žsee e.g. Refs. w1–5x.. QCD in 1 q 1 dimensions has also received attention recently due to the remarkable fact that different regularizations seems to lead to different models, even within the same choice of gauge. In his pioneering work, ’t Hooft w1x employed a cutoff around the origin in momentum space, that later was shown to be equivalent to a principle value prescription w2x. Within this framework, no free quarks can propagate, but they are confined into mesons, consisting of quark anti-quark pairs. Wu w3x, on the other hand, suggested another regularization, allowing for a Wick rotation to Euclidean space. The bound state equation is more complicated in this case, and no solution has yet been found. Bassetto and Nardelli with co-workers w6–9x have compared the two regularizations, and computed for example the expectation value of the Wilson loop. The expectation value of the Wilson loop with Wu’s regularization, was recently calculated by Staudacher and Krauth w10x, who showed that confinement is not enforced in this case, unlike using the ’t Hooft regularization. It has been suggested by Chibisov and Zhitnitsky w11x that the models due to different regularizations could show up as different phases, that makes a study of the system at
1 2
Email address: [email protected]. Email address: [email protected].
0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 4 7 8 - X
J. Hallin, D. Perssonr Physics Letters B 429 (1998) 232–238
233
finite temperature intriguing. In 2 dimensional QCD, confinement is obvious as an infinite energy for coloured states. We shall in this letter take the interaction into account only by a projection onto colour singlets, i.e. we neglect the coupling unless it multiplies the infra red divergence. The quark contribution to the free energy in 3 q 1 dimensions is then easily obtained from the free energy presented here in 1 q 1 dimensions through the substitution, LHdprŽ2p . ™ 2VHd 3 prŽ2p . 3. The projected free energy in 3 q 1 dimensions has earlier been considered in the high temperature limit by Skagerstam w12x. The full interacting case in 2 dimensions, that is troubled by infra red divergences, will be more extensively considered in another article. Including the interaction in the strict large NC limit LrNC ™ 0, McLerran and Sen w13x, and employing a different method Hansson and Zahed w14x showed that in the low temperature ’t Hooft meson phase, the partition function ln Z s O Ž L., due to confinement. Using a different resummation scheme for LrNC ™ `, McLerran and Sen w13x also found another phase of almost free quarks, with ln Z s O Ž LNC .. The phase transition itself was not obtainable, and could only take place as T ™ `, i.e. T A NC . By considering finite k s NC rL, we are here able to monitor the phase transition exactly, and for finite k , it will take place at finite T. However, due to the negligence of the interaction we do not have confinement in the strict sense. Our low temperature phase consists of quark–anti-quark pairs, showing up for example in Bose–Einstein distribution functions, but these are not the confined mesons of ’t Hooft’s, since ln Z s O Ž LNC .. ˆ Let us now consider the unnormalized density matrix rˆ s eyb H of free coloured fermions having the Hamiltonian Hˆ s fˆ † Ž p . h Ž p . fˆ Ž p . ,
Ž 1.
Hp
where hŽ p . s g 0 Ž g 1 p q m. ,
Ž 2.
n
and Hp s H Ž2dp p. , in n spatial dimensions. We shall here mainly discuss n s 1. In the functional representation the expression for rˆ is w15x n
r Ž h1)h1 ,h 2)h 2 . s Nb exp Ž Ž h1) q h 2) . Vb Ž h1 q h 2 . q h1)h 2 y h 2)h1 . ,
Ž 3.
where
ž
b
ž
Nb s exp 4 NC V ln cosh
Hp
b
Vb s ytanh
ž / 2
v
h
v
2
v
//
,
Ž 4.
1C ,
Ž 5.
(
v Ž p . s p 2 q m2 ,
Ž 6.
and L denotes the spatial length. In the expression for r Žh1)h1 ,h 2)h 2 ., summation over spinor, colour and momentum indices is understood. Moreover NC denotes the number of colours and 1 C is the unit matrix in colour space. We now wish to calculate the partition function for colourless states. This is done by the operator pˆ ,
pˆ s dupˆ u ,
Ž 7.
H
ˆa
that projects onto colour singlets. Here pˆ u s e i a a Q is a general global UŽ NC . gauge transformation. The partition function Z is then Z s tr Ž pr ˆ ˆ. .
Ž 8.
J. Hallin, D. Perssonr Physics Letters B 429 (1998) 232–238
234
With the colour charge operator Qˆ a s fˆ †
Hp
ta 2
fˆ ,
Ž 9.
the global gauge transformation pˆ u is in the functional representation:
pu Ž h1)h1 ,h 2)h 2 . s Nu exp Ž h1) q h 2) . V u Ž h1 q h 2 . q h1)h 2 y h 2)h1 .
Ž 10 .
Here we have defined Nu s exp
1 2
tr ln 14 Ž u q uy1 q 2 . ,
Ž 11 .
and
Vu s
uy1 uq1
1s.
Ž 12 .
Furthermore, tr s LHp tr s tr C denotes the trace over all indices, momentum, spin and colour, while 1 s is the unit matrix in spinor space i.e. a 2 = 2-unit matrix Ž4 = 4 in n s 3.. The group element is u s e i a a . Multiplying rˆ and uˆ one finds ta 2
²h1)h1 < uˆ rˆ
½
Ž h1) q h2) .
Vb q V u 1 q Vb V u
5
Ž h1 q h2 . q h1)h2 y h2)h1 . Ž 13 .
Finally we can then give the expression for the partition function,
½H
5
Z s du tr Ž pˆ u rˆ . s dudet Ž 2 . Nb Nu det Ž 1 q Vb V u . s eyb E 0 duexp L tr C ln Ž 1 q j u . Ž 1 q j uy1 . ,
H
H
H
p
Ž 14 . where E0 s yLNC Hp v is the vacuum energy. We have here defined
j s eyb v ,
Ž 15 .
and e i a j denotes the NC eigenvalues of the UŽ NC . matrix u. On functions of eigenvalues the Haar measure du takes the form NC
H
du s
H is1 Ł d a Ł sin i-j
2
ai y a j
i
2
.
Ž 16 .
This gives NC
Z s ey b E 0
Ł d a i Ł sin2
H is1
i-j
ai y a j 2
NC
Hp Ý ln Ž1 q j e
exp L
iaj
.Ž 1 q j eyi a . j
.
Ž 17 .
js1
In the large NC -limit the integral over UŽ NC . in Ž17. may be calculated by the steepest descent method. We take this limit by letting NC and the spatial length L tend to infinity simultaneously keeping their quotient
ks
NC L
,
Ž 18 .
constant. We solve this by introducing the density of eigenstates, D , in the continuum limit w16,17x, i.e. NC
1
Ý f Ž a j . ™ NCH dtf js1
0
a Ž t . s NC
ac
Hya D Ž a . d a f Ž a . . c
Ž 19 .
J. Hallin, D. Perssonr Physics Letters B 429 (1998) 232–238
235
We find two distinct phases, depending on the inequality 1y
2
1
H k pe
bv
y1
G 0.
Ž 20 .
As long as Ž20. is satisfied we are in the zero-gap phase, a c s p , where D has support over the whole circle. When Ž20. is not satisfied we are in the one-gap phase. In this case there is a gap in which D lacks support, i.e. an interval where the eigenvalues give a vanishing contribution as NC ™ `. In the zero-gap phase ln Z is given by a sum over quark–antiquark pairs ln Z s yb E0 y
NC2
k2
ln 1 y j Ž p . j Ž q . HH p q
.
Ž 21 .
Even though the states are mesonic-like, we do not have confinement since ln Z A Ž NC rk . 2 s NC Lrk , whereas in the confined phase of ’t Hooft mesons we have Žcf. w14x. ln Z A L. In the one-gap phase we find ln Z s yb E0 q NC2 ln sin
ž
y
NC2
k2
ac 2
/
q
2 NC2
k
ln 1 y x Ž p . x Ž q . HH p q
Hpln ž 1 q j q (j 1 2
,
2
q 2 j cos a c q 1
/ Ž 22 .
where we have defined 1
xs
4j sin
(
2 a 1 q j y j q 2 j cos a c q 1 2 c
2
.
Ž 23 .
2
Furthermore, the critical angle a c is determined from
ž
Hp (j
1qj 2
q 2 j cos a c q 1
/
y1 sk .
Ž 24 .
In Fig. 1, we plot ln ZrŽTNC2 . as a function of the temperature, for k s m. The critical temperature is then found as Tc f 0.8 m. The free energy Ž F s yT ln Z . in the 1-gap phase Žsolid line. grows as T 2 for large T, and is approaching the free energy for free quarks, i.e. without the projection, Ždotted line., as T ™ `. Whereas the free energy in the 0-gap phase Ždashed line., continued above the phase transition would grow like T 3. The high
Fig. 1. The free energy for free quarks Ždotted line., in the 0-gap phase Ždashed line., and in the 1-gap phase Žsolid line., for k s m. All in units of NC2 T 2 r m.
J. Hallin, D. Perssonr Physics Letters B 429 (1998) 232–238
236
temperature behaviour in the 1-gap phase is obtained by an expansion for a c < 1, yielding a c , 4pkrT. As T ™ `, a c ™ 0, i.e. only u s 1 gives a contribution, corresponding to no projection. In addition to the free energy, it is of interest to compare the entropy, and the Žrenormalized. energy density in the two phases. They are given by ss
es
1 E L ET
Ž T ln Z . L ,
Ž 25 .
T2 E
Ž ln Z q b E0 . L , Ž 26 . L ET respectively. Their behaviour is similar to the that of the free energy. In the interacting case it is well known that a nontrivial chiral condensate appears, as shown for m s 0 by Zhitnitsky w18,19x, and in the general case by Burkardt w20x. However, when the interaction only is taken into account through the projection, the vacuum is trivial and no chiral condensate can appear in this case. Nevertheless, the Žrenormalized. expectation value of cc , is an order parameter for the phase transition, and thus of interest. We find E ² cc : s yT Ž ln Z q b E0 . L . Ž 27 . Em In the 0-gap phase it is straightforward to evaluate this from Ž21.. However, in the 1-gap phase we need to define ac E cot T a ' ma, Ž 28 . 2 Em c where y1
as
½
Hp
j Ž1qj .
Ž j 2 q 2 j cos a c q 1 .
3r2
5½
j Ž1yj .
Hp
1
Ž j 2 q 2 j cos a c q 1 .
3r2
v
5
,
Ž 29 .
is obtained from Ž24., using Ž15., and Ž6.. We then find T E m Em
xs
x
(j
2
q 2 j cos a c q 1
Ž1qj . ay Ž1yj .
1
.
v
Ž 30 .
This gives
² cc : s NC2
=
° 1 2 m~y a q H ¢ 2 k 1 q j q (j p
1q
2 y
k2
j q cos a c
(j
2
2
q 2 j cos a c q 1 2 j sin2
j v
q 2 j cos a c q 1
x Ž p. x Ž q.
1
q
(j
2
ac
2 z q 2 j cos a c q 1
Ž1qj . ay Ž1yj .
1
v j 2 q 2 j cos a c q 1
HH p q 1 yx Ž p. x Ž q. (
¶ •. ß
0 Ž 31 .
We have been able to show that ² cc : is continuous over the phase transition, whereas higher derivatives give cumbersome expressions.
J. Hallin, D. Perssonr Physics Letters B 429 (1998) 232–238
237
However, in the limit of infinite massive quarks analytical results are obtained. As m ™ `, we must also let Tc ™ `, such that their quotient
z s mrTc ,
Ž 32 .
is kept fixed, in order to keep a finite particle density. Furthermore, we define t according to T s Tct . The momentum integrals are now independent of p, so we write
Hp™ L .
Ž 33 .
The critical temperature is then determined from
k
1 s
2
Hp e
v r Tc
y1
™L
1 z
e y1
.
Ž 34 .
.
Ž 35 .
In the 0-gap phase we then find ² cc : s 2 Nc2
L
ž / k
2
1 e
2 z rt
y1
In the 1-gap phase the critical angle is now determined from
(1 q 2 e
z rt
cos a c q e
2 z rt
e z rt q 1 s
,
1 q krL
Ž 36 .
that after some simplifications give ² cc : s Nc2
L
1
½ ž /ž q
2
k
k 2q
L/
1 e
z rt
q1
5
.
Ž 37 .
Notice here the Bose–Einstein distribution in the low temperature 0-gap phase, as opposed to the Fermi–Dirac distribution in the 1-gap phase at high temperature. It is now a straightforward exercise to show that E 2rE t 2 ² cc : is discontinuous, so that it is a third order phase transition. This is in agreement with the study of Gattringer et al. w21x concerning static quarks on a line. Furthermore, in the corresponding lattice model Gross and Witten w16x found a third order phase transition, so we conjecture that the phase transition most likely is third order also for arbitrary mass.
Acknowledgements We are grateful to Ariel Zhitnitsky for proposing the subject, as well as stimulating discussions; and to Hans Hansson for interesting remarks, and pointing out w14x. J.H.’s research was funded by NFR Žthe Swedish Natural Science Research Council., and D.P.’s by STINT Žthe Swedish Foundation for Cooperation in Research and Higher Education..
References w1x w2x w3x w4x w5x
G. ’tHooft, Nucl. Phys. B 75 Ž1974. 461. C.G. Callan, N. Coote, D. Gross, Phys. Rev. D 13 Ž1976. 1649. T.T. Wu, Phys. Lett. B 71 Ž1977. 142. M.B. Einhorn, Phys. Rev. D 14 Ž1976. 3451. R. Brower, W.L. Spence, Phys. Rev. D 19 Ž1979. 3024.
238 w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x
J. Hallin, D. Perssonr Physics Letters B 429 (1998) 232–238 A. Bassetto, L. Griguolo, Phys. Rev. D 53 Ž1996. 7385. A. Bassetto, G. Nardelli, Int. J. Mod. Phys. A 12 Ž1997. 1075. A. Bassetto, G. Nardelli, A. Shuvaev, Nucl. Phys. B 495 Ž1997. 451. A. Bassetto, D. Colferai, G. Nardelli, Nucl. Phys. B 501 Ž1997. 227. M. Staudacher, W. Krauth, hep-thr9709101, 1997. B. Chibisov, A.R. Zhitnitsky, Phys. Lett. B 362 Ž1995. 105. B.S. Skagerstam, Z. Phys. C 24 Ž1984. 97. L.D. McLerran, A. Sen, Phys. Rev. D 32 Ž1985. 2794. T.H. Hansson, I. Zahed, Phys. Lett. Ž1993.. J. Hallin, P. Liljenberg, Phys. Rev. D 52 Ž1995. 1150. D.J. Gross, E. Witten, Phys. Rev. D 21 Ž1980. 446. G. Mandal, Mod. Phys. Lett. A 5 Ž1990. 1147. A.R. Zhitnitsky, Phys. Lett. B 165 Ž1985. 405. A.R. Zhitnitsky, Sov. J. Nucl. Phys. 44 Ž1986. 139. M. Burkardt, Phys. Rev. D 53 Ž1996. 933. C.R. Gattringer, L.D. Paniak, G.W. Semenoff, Annals Phys. 256 Ž1997. 74.