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Analytic study of the U(1) LGT with fermion at finite temperature and density
4 September 1997
PHYSICS LElTERS B Physics Letters B 407 (1997) 303-306
Analytic study of the U( 1) LGT with fermion at finite temperature and density Ren Xuezao *, Zhu Yunlun 2, Chen Ying Department
of Physics, Peking University, Beijing 100871, PR China Received 15 May 1997 Editor: H. Georgi
Abstract In this letter, we apply the va~adon~ c~ul~t expansion (VCE) to the U( I ) LGT with fermion at finite density (p # Cl), which has a complex action. The order parameters L and L’ at finite temperature (T # 0) are calculated. Our results are compared with MC results [N. Bilk, I-I. Gausterer and S. Sanielevici, Phys. Lea. B 198 ( 1987) 2351 suggesting new conclusions which are different from that of the mean field approximation. @ 1997 Elsevier Science B.V. PACS: 11.15.Ha; 11.1O.W~ ~w~r~~ Lattice gauge; Fmite temperature; Finite density; V~ation~ cumulant expansion
The standing problems of cosmology and heavyion collisions require taking into account finite temperature and particle density in nonpe~u~ative QCD studies. Therefore the lattice gauge theories with finite chemical potential have attracted considerable attention. However, in the case of finite particle density, the Lagrangian action contains au additional term which is related to the chemical potential CL.This term becomes complex for the gauge groups U( 1) and SU( N) with N 2 3. In this case, numerical simulation encounters serious difficulties. To overcome the difficulties, Engels and Satz [ l] suggested the so-called “partial quenching”. The idea is to neglect the imaginary part of the leading term in the effective action with the hopping-parameter expansion [ 21. But in many cases, the imaginary part of the leading term is important [ 3 1. In the numerical simulation, the only known method ’ On leave from Xichang Normal School, Sichuan 61502 1, China. ’ E-mail: zhuyl~pku.~u.cn.
to treat complex action is the complex Langevin equation [ 41. Unfortunately, there exists no general proof on whether a given complex process converges to the correct ~uilib~um~s~bution [4]. Therefore analytical methods to treat complex actions are necessary. In this paper, we apply the variational cumulant expansion method to calculate the complex action. Now let us look at the U( 1) gauge model with U&(x) = dfferxt E U( 1) on a NzN, (N, = 1) hypercubic lattice with the periodic condition in the timelike direction. The lattice effective action of the U( 1) gauge with a Wilson fermion is given by [ 51
PC7
P,
where so is the Wilson action, BP*and 6)p, are the sums of 0, around a space-like plaquette Pa and a timelike plaquette P,, respectively. Li = Tr l-j:, Ui,T,r+l
0370-2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PI1 SO370-2693(97)00740-S
X. Ren et al./Physics
304
is the Polyakav line, the index i runs over the site of the 3d space lattice. The last two terms in Eq_ ( 1) are the first nonvanishing terms in the hopping-parameter expansion [ 21 for a temporal lattice size such as A$ 5 3. The parameters ht, k2 are defined by h* =~~zK~~re~~~,
(3)
hZ =2(2K)N7e-Nr’M,
(4)
where a denotes the lattice spacing; ~c the chemical potential. The prison function of the system is defined as Z=
71d@ ,s
z;;> * .I-(
(5)
--7F
For p C 0, S is complex, however the p~ition function remains positive-definite. The trial action introduced in VCE takes the following form
1,
f,
where 1, and 1, denote space-like and time-like links, A, B are real variational parameters. Using VCE, the average of an observable quantity Q can be written as
where (. . ->ois the average over the trial system, (- * *}c is the cumulant average over the same system. (Q)m denotes the mth order approximation of (Q}. For an Abelian gauge group, (Q>tn is only related to /3 and B when N, = 1 [ 6f because of the periodic condition for temporal links. It is convenient to use the following diagrammatic notations [ 61:
Letters B 407 (1997) 303-306
Eq. (1) can ~re~tten S=p~cosBp, P,
as
+ (hr + hz - B) ~cosBI, r,
i-i(hr - h2) xsin&,. I,
(12)
Now we can calculate (R) and (I) order by order through the expansion in Eq. (7). Explicitly,
X. Ren et al./Physics 1.0,
Letters B 407 (1997) 303-306
305
1.0,
I
,.-*
_- _______-----
I’
I’
0.8 -
i’ : 0.6 -
h = 4hz = 0.25
:
0.4 t
,’
I’ , ,’
0.2
O.S!o
B
--__
o.i5
015
0.+5
Fig. 1. The VCE results and MC nmdts [51 (p = 0).
12=3(/...+2i(ht X
=i(h
-
115
1.75
13
)
______ ___---,_*I ,,,*
(pl+(hI+h2-B)q f,
c I+
A5
Fig. 2. The VCE results (pa = In 2, K = $
-h2)
‘.O-
1,
x
110
vi (L’)
0.8
*. -)c
I
~2>Wr(~P),
+
(h
+
h2 -
B)(Wl
>,I9
,’
!’ 1’
,’ !’
0.6 -
h, = 2hz =
+‘?I6
= -ii( hl - h2) (/?r.& + hi + h:! - B)& ...
where
5” = hz(K)I~o(m7 I = 2d - 2,
Ri = 2d - 3 - i.
In this paper, we approximate the calculation to the 4th order. The variational parameter B can be determined as follows [ 61. In the approximation to nth order, we take (Q) N (Q), = (Q)&+, where the Bh are the intersection points of (Q), and (Q)o, i.e. the solution of the equation:
(Q)m - (Q>o =
5 n=I
;(Q(S
- So)“)c= o,
.
The order parameters are (L) and (L*): (L) = (R) + i(l),
(L*) = (R) - i(Z).
The results and discussion are summarized as follows: - (A) In the case of hl = h:! = 0, there are no fermions. The system is reduced to a pure U( 1)
o%oI
0.25 I
0.5 I
0.75 /
1.0
1.25 I
1.5 /
1.:75
11
Fig. 3. The VCE results (pa = In d,
K = & ).
model. The resultant phase diagram is shown in Fig. 1. The second-order phase transition is explicit. The critical point & = 0.375 is comparable with the MC result 0.41 [ 51, while PC obtained in the mean field approximation is 0.338. (B) In the case of ,u = 0, there are fermions, but no chemical potential. The fermion part is like an external field [ 71. The global U( 1) symmetry of the system has been broken. The deconfiment phase transition disappears for any nonzero K, because (L) = (L.*) is always larger than zero.
X. Ren et al./ Physics L.etters B 407 ff9971303-306
306
- (C) In the case of ,U # 0 , the phase diagram is given in Figs. 2 and 3. The expectation values of (L) and (L*) are real, but they are no longer equal and (L)/(L*) < 1. Let n+, n- denote the number density of particle and anti-particle, respectively, then according to Ref. I81 n=n+-L
= ;y
=
(h,L
-
h2L").
(131
Therefore, n does not equal zero in the case of ht Z h2 (,x # 0). It means that the density of particles is greater than that of anti-particles. However, if we use the mean field approximation [ 51, {&P/(~*)MF
= h2/%,
n is identically zero.
Acknowledgements This work is partially supported by the National Natural Science Foundation of China. References [ 11 J. Engles and H. Sat& Phys. Rev. Lett. 55 (1985) 2242; whys. L&t. B 1.59 (1985) 151.
[2J A, Hasenfratz and P. Hasenfratz, Phys. L&t. B iO4 (1981) 489. [3] M.A. Stephanov, Phys. Rev. Lea. 76 (1996) 4472. [4] PH. Damgaard and H. Hiiffel, Phys. Rep. 152 (1987) 227. [5] N. Bilic, H. Gaustem and S. Sauielevici, Phys. I&t. B 198 (1987) 235. [6] X.T. Zheng, Y.L. Li and C.H. Lei, Chin. Phys. Lett. 9 (1992) 573. [7] E Green and E Katwh, Nuci. Phys. B 238 (1984) 297. [8] J. Kogut, H. Matsuoka, M. Stone and H.W. Wyld, Nucl. Phys. B 225 (1983) 93.
PHYSICS LElTERS B Physics Letters B 407 (1997) 303-306
Analytic study of the U( 1) LGT with fermion at finite temperature and density Ren Xuezao *, Zhu Yunlun 2, Chen Ying Department
of Physics, Peking University, Beijing 100871, PR China Received 15 May 1997 Editor: H. Georgi
Abstract In this letter, we apply the va~adon~ c~ul~t expansion (VCE) to the U( I ) LGT with fermion at finite density (p # Cl), which has a complex action. The order parameters L and L’ at finite temperature (T # 0) are calculated. Our results are compared with MC results [N. Bilk, I-I. Gausterer and S. Sanielevici, Phys. Lea. B 198 ( 1987) 2351 suggesting new conclusions which are different from that of the mean field approximation. @ 1997 Elsevier Science B.V. PACS: 11.15.Ha; 11.1O.W~ ~w~r~~ Lattice gauge; Fmite temperature; Finite density; V~ation~ cumulant expansion
The standing problems of cosmology and heavyion collisions require taking into account finite temperature and particle density in nonpe~u~ative QCD studies. Therefore the lattice gauge theories with finite chemical potential have attracted considerable attention. However, in the case of finite particle density, the Lagrangian action contains au additional term which is related to the chemical potential CL.This term becomes complex for the gauge groups U( 1) and SU( N) with N 2 3. In this case, numerical simulation encounters serious difficulties. To overcome the difficulties, Engels and Satz [ l] suggested the so-called “partial quenching”. The idea is to neglect the imaginary part of the leading term in the effective action with the hopping-parameter expansion [ 21. But in many cases, the imaginary part of the leading term is important [ 3 1. In the numerical simulation, the only known method ’ On leave from Xichang Normal School, Sichuan 61502 1, China. ’ E-mail: zhuyl~pku.~u.cn.
to treat complex action is the complex Langevin equation [ 41. Unfortunately, there exists no general proof on whether a given complex process converges to the correct ~uilib~um~s~bution [4]. Therefore analytical methods to treat complex actions are necessary. In this paper, we apply the variational cumulant expansion method to calculate the complex action. Now let us look at the U( 1) gauge model with U&(x) = dfferxt E U( 1) on a NzN, (N, = 1) hypercubic lattice with the periodic condition in the timelike direction. The lattice effective action of the U( 1) gauge with a Wilson fermion is given by [ 51
PC7
P,
where so is the Wilson action, BP*and 6)p, are the sums of 0, around a space-like plaquette Pa and a timelike plaquette P,, respectively. Li = Tr l-j:, Ui,T,r+l
0370-2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PI1 SO370-2693(97)00740-S
X. Ren et al./Physics
304
is the Polyakav line, the index i runs over the site of the 3d space lattice. The last two terms in Eq_ ( 1) are the first nonvanishing terms in the hopping-parameter expansion [ 21 for a temporal lattice size such as A$ 5 3. The parameters ht, k2 are defined by h* =~~zK~~re~~~,
(3)
hZ =2(2K)N7e-Nr’M,
(4)
where a denotes the lattice spacing; ~c the chemical potential. The prison function of the system is defined as Z=
71d@ ,s
z;;> * .I-(
(5)
--7F
For p C 0, S is complex, however the p~ition function remains positive-definite. The trial action introduced in VCE takes the following form
1,
f,
where 1, and 1, denote space-like and time-like links, A, B are real variational parameters. Using VCE, the average of an observable quantity Q can be written as
where (. . ->ois the average over the trial system, (- * *}c is the cumulant average over the same system. (Q)m denotes the mth order approximation of (Q}. For an Abelian gauge group, (Q>tn is only related to /3 and B when N, = 1 [ 6f because of the periodic condition for temporal links. It is convenient to use the following diagrammatic notations [ 61:
Letters B 407 (1997) 303-306
Eq. (1) can ~re~tten S=p~cosBp, P,
as
+ (hr + hz - B) ~cosBI, r,
i-i(hr - h2) xsin&,. I,
(12)
Now we can calculate (R) and (I) order by order through the expansion in Eq. (7). Explicitly,
X. Ren et al./Physics 1.0,
Letters B 407 (1997) 303-306
305
1.0,
I
,.-*
_- _______-----
I’
I’
0.8 -
i’ : 0.6 -
h = 4hz = 0.25
:
0.4 t
,’
I’ , ,’
0.2
O.S!o
B
--__
o.i5
015
0.+5
Fig. 1. The VCE results and MC nmdts [51 (p = 0).
12=3(/...+2i(ht X
=i(h
-
115
1.75
13
)
______ ___---,_*I ,,,*
(pl+(hI+h2-B)q f,
c I+
A5
Fig. 2. The VCE results (pa = In 2, K = $
-h2)
‘.O-
1,
x
110
vi (L’)
0.8
*. -)c
I
~2>Wr(~P),
+
(h
+
h2 -
B)(Wl
>,I9
,’
!’ 1’
,’ !’
0.6 -
h, = 2hz =
+‘?I6
= -ii( hl - h2) (/?r.& + hi + h:! - B)& ...
where
5” = hz(K)I~o(m7 I = 2d - 2,
Ri = 2d - 3 - i.
In this paper, we approximate the calculation to the 4th order. The variational parameter B can be determined as follows [ 61. In the approximation to nth order, we take (Q) N (Q), = (Q)&+, where the Bh are the intersection points of (Q), and (Q)o, i.e. the solution of the equation:
(Q)m - (Q>o =
5 n=I
;(Q(S
- So)“)c= o,
.
The order parameters are (L) and (L*): (L) = (R) + i(l),
(L*) = (R) - i(Z).
The results and discussion are summarized as follows: - (A) In the case of hl = h:! = 0, there are no fermions. The system is reduced to a pure U( 1)
o%oI
0.25 I
0.5 I
0.75 /
1.0
1.25 I
1.5 /
1.:75
11
Fig. 3. The VCE results (pa = In d,
K = & ).
model. The resultant phase diagram is shown in Fig. 1. The second-order phase transition is explicit. The critical point & = 0.375 is comparable with the MC result 0.41 [ 51, while PC obtained in the mean field approximation is 0.338. (B) In the case of ,u = 0, there are fermions, but no chemical potential. The fermion part is like an external field [ 71. The global U( 1) symmetry of the system has been broken. The deconfiment phase transition disappears for any nonzero K, because (L) = (L.*) is always larger than zero.
X. Ren et al./ Physics L.etters B 407 ff9971303-306
306
- (C) In the case of ,U # 0 , the phase diagram is given in Figs. 2 and 3. The expectation values of (L) and (L*) are real, but they are no longer equal and (L)/(L*) < 1. Let n+, n- denote the number density of particle and anti-particle, respectively, then according to Ref. I81 n=n+-L
= ;y
=
(h,L
-
h2L").
(131
Therefore, n does not equal zero in the case of ht Z h2 (,x # 0). It means that the density of particles is greater than that of anti-particles. However, if we use the mean field approximation [ 51, {&P/(~*)MF
= h2/%,
n is identically zero.
Acknowledgements This work is partially supported by the National Natural Science Foundation of China. References [ 11 J. Engles and H. Sat& Phys. Rev. Lett. 55 (1985) 2242; whys. L&t. B 1.59 (1985) 151.
[2J A, Hasenfratz and P. Hasenfratz, Phys. L&t. B iO4 (1981) 489. [3] M.A. Stephanov, Phys. Rev. Lea. 76 (1996) 4472. [4] PH. Damgaard and H. Hiiffel, Phys. Rep. 152 (1987) 227. [5] N. Bilic, H. Gaustem and S. Sauielevici, Phys. I&t. B 198 (1987) 235. [6] X.T. Zheng, Y.L. Li and C.H. Lei, Chin. Phys. Lett. 9 (1992) 573. [7] E Green and E Katwh, Nuci. Phys. B 238 (1984) 297. [8] J. Kogut, H. Matsuoka, M. Stone and H.W. Wyld, Nucl. Phys. B 225 (1983) 93.