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Non-trivial phase structure of φ4-theory at finite temperature
Physics Letters B 311 ( 1993 ) 207-212 North-Holland
PHYSICS LETTERS B
Non-trivial phase structure of 04-theory at finite temperature * K. Langfeld, L. v. Smekal and H. Reinhardt Institut fiir theoretische Physik, Universitiit Tiibingen, I ~ 7400 Tfibingen, FRG Received 7 May 1993 Editor: P.V. Landshoff
The effective potential for the local composite operator 02 (x) in 2~b4-theory is investigated at finite temperature in an approach based on path-integral linearisation of the 04-interaction. At zero temperature, the perturbative vacuum is unstable, because a non-trivial phase with a scalar condensate (02)0 has lower effective action. Due to field renormalisation, (202)0 is renormalisation group invariant and leads to the correct scale anomaly. At a critical temperature Tc the non-perturbative phase becomes meta-stable implying a first order phase-transition to the perturbative phase. The ratio (202)0/T~2 .~ 62 turns out to be a universal constant.
1. Introduction
It is of interest to study ~b4-theory, because o f its wide field o f applications [ 1 ]. Due to its simple structure it also makes possible an understanding o f basic features of q u a n t u m field theory, e.g., anomalous breaking o f scale invariance [2[, as well as being a suitable test for non-perturbative methods, e.g., a variational approach [3] and numerical simulations on a lattice [4]. Lattice results suggest that q~4-theory is in fact trivial (at best a free theory), due to renormalisation effects [5 ]. On the other hand, many phenomenological applications [6] rely on a non-trivial vacuum structure o f &4-theory. A variational approach to massive ~/)4-theory [31 indicates that besides the trivial phase, a second non-trivial one exists which escapes the triviality bounds from lattice calculations. This phase is known as the precarious phase since the bare coupling constant becomes (infinitesimally) negative during renormalisation. Recently, this phase was also found for the massless case in a modified loop expansion around the mean field [ 7 ]. It was shown that this expansion rapidly converges in zero dimensions and yields renormalisation group invariant results in four dimensions [7]. Thereby it was observed that the so called precarious renormalisation [3] is, in fact, the *
Supported by DFG under contract Re 856/1 - 1.
standard procedure and is hidden in the usual expansion with respect to the coupling strength [7], i.e. expanding the precarious renormalisation condition in powers of the renormalised coupling constant reproduces the standard result o f perturbative renormalisation. In order to gain further insight into the vacuum structure of 2q~4-theory, it is convenient to study the effective potential for the local composite field q52( x ) . This is because the energy density as a function o f the expectation value of q~2 is minimised for (~b2) /: 0, whereas the effective potential of the field q~ itself is minimal at (q~) = 0 (also true in the non-trivial phase [ 7 ] ). In this letter we study the temperature effects of the non-trivial phase o f 2~a-theory. To this end we therefore calculate the effective potential for q~2 in the modified loop-approach presented in [7] for finite temperatures. Since temperature does not influence the renormalisation procedure, the effective potential is renormalisation group invariant for all temperatures, as observed in a perturbative calculation [8]. Due to the need of renormalisation of the operator 4~2, it turns out that (2~ 2) is renormalisation group invariant and its contribution to the energy density gives the correct scale anomaly within our approximation. The vacuum energy density o f the non-trivial phase is considerably lower than the energy density of the
0370-2693/93/$ 06.00 @ 1993-Elsevier Science Publishers B.V. All rights reserved
207
Volume 31 l, number 1,2,3,4
PHYSICS LETTERS B
perturbative vacuum. The thermal expectation value of the energy density of the non-perturbative phase relative to the perturbative phase increases, however, with increasing temperature, and a first order phase transition to the perturbative phase is found at a critical temperature To, in accordance with Landau's theory [9]. The ratio & t h e zero temperature condensate to the critical temperature is found to be a universal number, as is the ratio of the jump in the condensate at the transition point to the critical temperature.
2. The effective potential of 0 2 Our starting point is the Euclidean generating functional for Green's functions, i.e. I/T
29 July 1993
Since the vacuum is isotropic and homogeneous, we have 0z. = Uodz. with Uo = 0oo the vacuum energy density. This implies Uo = ---14 • 2~fl(J,R)(~4).
(4)
For small couplings 2R the renormalisation group flfunction behaves as fl(2R) -- f l o ~ + .... with flo a constant. Hence if (~b4) ~: 0, and assuming (¢4) (~b2)2, leads us to regard ,~Rt~2 as a physical quantity instead of 02 . Due to the 04-interaction the path-integral ( I ) cannot be performed exactly, and one has to resort to approximation schemes. One of these is the perturbative expansion with respect to the coupling strength. At one-loop level, the result for the zero-temperature effective potential for ~b is [10]
Ul-loop =
l
~-~c
4
~'2
"l- ~ w c
"~4 /'"
'~'~2 ~m 2--A--~
0
4._ ½m2q52 ..{. 1 2 ¢ 4 _
j(X)O2(X)]) ,
(l)
where 2 is the bare coupling constant and T is the temperature in units of Boltzmann's constant. The functional integration runs over allscalar modes satisfying periodic boundary conditions in the Euclidean time direction, j is an external source which couples linearly to 0 2. The effective action for 0 2 is defined by the Lcgendrc transformation, i.e., £[(/)2c] :~--~6~(x)
-lnZ[j]
"- 6
+ ./d4xO2c(x)j(x),
lnZ[j]
dj(x)
(2)
The effective potential U (4~) is obtained by restricting ~bcto constant classical fields. It is a physical quantity, e.g., its minimum value represents the vacuum energy density [ 10], and should therefore be renormalisation group invariant. Further access to the vacuum energy density is provided by the scale anomaly [2]. Conformal symmetry is broken in the non-perturbative phase by a nonvanishing vacuum expectation value of the trace of the energy-momentum tensor, i.e.,
208
(3)
.
(5)
N o field renormalisation is required to derive (5) implying that ~bc is stillrenormalisation group invariant and a physical field. It would initiallysccm that (5) can account for the scale anomaly, since U1-1oop(Pc) has a m i n i m u m for non-vanishing field.However, this m i n i m u m is an artefact of the one-loop approximation as already pointed out by Coleman and Weinberg, and by Coleman and Gross [I0]. This fact isalso evident from the two-loop effective potential [I l ], since it again has the m i n i m u m at Oc = 0, and significantly differsfrom the one-loop resultfor small classical fields.For large classicalfields,or large masses, the one-loop potential scales according the standard rcnormalisation group fl-functionwhich has an infrared fixed point. Since wc would like to investigate the non-trivial phase of ~b4-theory,non-pcrturbativc methods arc required. W e therefore apply the modified loop expansion [7] and lincariscthe g54-interactionby means of an auxiliary field,i.e.,
Z[j] = / D~73zexp[- / d4x(½0uc~O~ +
26-Z2(x)
+[½m2(0u,u) = -- 2'~fl ('J"R)((/)4) .
2)
iz(x)]~2(x)
-
j(x)(~Z(x))]. 2J
(6)
29 July 1993
PHYSICS LETTERS B
Volume 3l l, number 1,2,3,4
After integrating out the ~b-field, the modified approach [7] is defined by a loop expansion with respect to the field Z around its mean field value. Even the lowest order in this expansion contains quantum fluctuations of the ~-theory, yielding reasonable results for the effective potential of 4~. For the effective potential of 4)2, we obtain in lowest order (i.e., the mean-field approximation)
-InZc[j](T)= fd4x(-3(M-m2+2j)2) +½ Tr l n ( - 0 2 + M ) ,
(7)
-
V4
32;72 Z ,#0
327r2
+-
fds
× e x p { - s ~ 2 + M + (2;TT)2n2]},
(9)
where A is the proper-time cutoff and V the threespace volume. Note that the ultra-violet divergences arise from the behaviour of the integrand in (9) at small s. The integration over the three momentum is easily performed, i.e., V
~
16;73/2
f d
~ ds
e-sM
e-S(2~T)2n2
.
(lO)
n I/A2
/
([3)
~ds e_SX e_,Z/,
(14)
(where e = 3 in (13)). The term in curly brackets in (13) is precisely the loop contribution to the effective potential at zero temperature. The temperature dependence is completely contained in the second term of (13), which is finite. As in the perturbative approach [8], renormalisation is unchanged, if temperature changes. Renormalisation is quite analogous to procedure of the zero-temperature effective potential of q5 [7]. The key observation is that the divergences can be absorbed into the bare parameters 2, m, j by setting
dnf(n) e i2~".
A straightforward calculation yields
1
~ + T-6-~ 6 . 2-J
(ln A2
3m 2 2
j - m 2 = 0,
~ f(n) = ~ c(v), n~ -oc t/= -~c
c(v) = i
+7
oo
6
The divergence of the discrete sum in (10) can be extracted by applying Poisson's formula, i.e.,
with
ln~- 2 - ~
v¢0
n I/A2
-
(12)
where y = 0.577... is the Euler-Mascheroni constant and the function f, is defined by
L = ½Trln(-02 +M)
L
]'
V4 T4j~ ( ~ T 2 ) 2;72
f~(x) =
Vfd3p
-j3- e-SMexp - 4 - i T
where F is the incomplete F-function, 1/4 the spacetime volume and the limit A -~ oc has been taken in the second term of the right hand side (12), which is finite. All divergences are in the first term, which is in addition, temperature independent. Using the asymptotic expansion for the incomplete F-function the loop contribution becomes
where M is related to the mean field value Z0 by Z0 = ½ i ( M - m 2 + 2j ). The mean field equation for Z0 can be recast into an equation for M, i.e., c ~ l n Z [ j ] _ O. (8) c~M The trace in (7) extends over space-time, and the modes satisfy periodic boundary conditions in the Euclidean time direction. Using Schwinger's proper time regularisation the loop contribution L is
32;72
(11)
F -y+
1 A2 32;72 =
1)
=~ jR
6,
[5)
3m~ ,
16)
2R 17)
where/~ is an arbitrary renormalisation scale. Later, we will check that physical quantities do not depend on/~. Note that the mean field variable M has been introduced in such a way that renormalisation is possible without M picking up divergences. In fact, as 209
Volume 311, number 1,2,3,4
PHYSICS LETTERS B
will be shown later, M is a renormalisation group invariant. In the following we confine ourselves to the massless case (m~ = 0). The bare parameters are uniquely defined, in ( 15)- (17), as functions of the renormalised ones and the regulator. The renormalisation of the coupling constant (15) is essentially the same as in [7], implying a universal renormalisation procedure in the calculation of the effective potential of ¢ and ¢2. Besides coupling constant renormalisation, mass and field renormalisation (16), (17) is also necessary to absorb all divergencies in (7). This is different to the case of the effective potential of ¢ on its own, where the field does not require renormalisation at this level. The situation here is analogous to the Gaussian effective potential approach t o ).~4_ theory (see refs. [3,12] ), where besides the "precarious phase", a second non-trivial phase (called "autonomous phase" in [3] ) would seem to exist, allowing for an infinite shift of the field variable. It was shown, however, that field renormalisation does not truly give a new phase, but rather what is the effective potential of the composite field 2¢ 2 [ 12 ]. Introducing the renormalised quantities in (13), the generating functional (7) becomes
29 July 1993
o.o15
T=0
......
/
T=Tc=0.507.. T = 0 . 6 p.
....
g
,t
,,'t
/
/,,//;'
0.005
"'----_____--~.,'"/"
-ooo5
// / I
/
-0.015 0.0
210
M/,a 2
4.0
6.0
Fig. 1. The effective potential as function of the scalar condensate at various temperatures. A large coupling strength 2R (#) = 1000 was chosen for aesthetic reasons. (
M
1)
32
-2-GM
1 lnZ[j](T)
v.
- l e f t T a f 3 (4-~)M=(RR/6)¢2 ----22R3
M2_
-- 16CtT4j~ ( ~ T 2 )
M jR+ lc~M2 ,
(18,
where c~ = 1/32~ 2. Here M is defined by the mean field equation (8), which in terms of the renormalised quantities reads
3 M - ~---~jR + ctMln --~ + 4~T2f2 (4-~ ) 2R =
0,
(19)
where J~ is defined by (14). In order to perform the Legendre transformation (2) we first obtain the classical field, i.e.,
¢~
10lnZ[jR] _ 6 M
-
v4
ojR
-
;~R
"
(20)
The effective potential (2), is therefore given by 210
c
.
(21)
ln~- 2 We have obtained a finite and renormalisation group invariant result, since a change in the scale Ft can be absorbed by a redefinition of the renormalised coupling constant ~-R according (15). The renormalisation group B-function fl().R) = f1022,
with
fl0:= ~c~
(22)
exhibits an infrared stable fixed point as indicated by a perturbative calculation [10]. Note that since our approach is intrinsically non-perturbative, (22) need not agree with the fl-function from the perturbative calculation. Furthermore, we see that M = -~2n(Oct 2 rather than q52 is renormalisation group invariant, as suggested by the scale anomaly. Due to the need of field renormalisation ¢c2 itself has lost its physical meaning. Fig. 1 shows the effective potential (21) as a function Of 2R¢~ for various temperatures. At zero temper-
Volume 311, n u m b e r 1,2,3,4
P H Y S I C S LETTERS B
29 July 1993
ature it develops a minimum U0 at a non-vanishing value of the condensate M0, i.e.,
-5
0.98
Mo = /z2 exp (flo@R),
Uo=-¼(~p'exp
(4)
fl--~--RR "
(23) o 0.92
This minimum corresponds to a non-trivial vacuum at zero temperature it has lower energy density than the perturbative state (¢2 = 0). The vacuum energy density of the non-perturbative phase is 1 Z = _gerM6
LT0
-~
1
2
0.86
2)2
~4fl(2R)(:qS4:).
(24)
This is precisely the result obtained in [7 ] for U (q~ = 0) in the non-trivial phase, furthermore it gives the correct scale anomaly (4) obtained by general arguments. Note that in the effective ;(-theory, obtained from (6) by integrating out ~b, the trace anomaly already shows up at tree level. It is therefore not so surprising that the trace anomaly survives in the meanfield approximation. Increasing the temperature shifts the scalar condensate to smaller values and lowers the difference of the energy density of perturbative and non-perturbative phase. At a critical temperature T~, the non-trivial phase becomes degenerate with the perturbative one (at q~ = 0). If the temperature is increased further, the non-trivial phase becomes meta-stable and a first order phase transition to the trivial phase at ¢2 = 0 can occur either by quantum or statistical fluctuations. This phase transition implies that the scalar condensate as a function of temperature has a discontinuity at the critical temperature.
3. At the phase-transition We first investigate the change of the scalar condensate with temperature. The condensate M:v = 1 2 g2R~bcImin is obtained by solving the equation of motion (19) for zero external source (jR = 0) which is
0.80
My (~ + c~ln-w- + - J ~ ( x ) = O, X
(25)
0,2 T/M 0
0.3
0.4
Fig. 2. The scalar condensate as function of temperature. where x : = M ~ / 4 T 2. In order to remove the renormalisation point dependence we subtract the corresponding equation for the zero temperature condensate 3//0 given by 3
M0 + (~In--7 p ~ = 0,
(26)
yielding ln--~ + lfz(x)
= 0.
(27)
This equation allows us to calculate My, in terms of its zero temperature value, as function of temperature. A numerical solution of (27) is displayed in fig. 2. It is instructive to relate the jump in the condensate and the critical temperature Tc to a fundamental observable like the scalar condensate at zero temperature M0 = (~,~R~2). At the critical temperature, the energy density of the perturbative thermal ground state U(q~2 = 0) is equal to that of the non-trivial phase U (¢2 mi,)The effective potential at zero field is obtained from (21 ) by a direct calculation, i.e. Uper : =
3
0.1
0.0
U(q~c2 = 0 ) =
= -32(~T4~(4),
16o~T4j~(0)
(28) 211
Volume 311, number 1,2,3,4
PHYSICS LETTERS B
where ( is the R i e m a n n (-function. Equating the energy density of the perturbative thermal state Uper to the one of the non-trivial phase (with non-zero condensate) we obtain
Acknowledgement
-22--~ + ½c~ In ~z2
References
2
- ~-~A(x)
= - ~-~2( ( 4 ) •
(29)
In order to eliminate the renormalisation point dependence we replace/z by M0 by using (23), yielding In M,, 2 M0 = ½ + 7 g [ j ) ( x ) - 2 ( ( 4 ) ] .
(30)
Combining this equation with (27) we finally obtain ½ + ~-~[f3(x
-2~(4)]
+
f 2 ( x ) = 0.
(31)
This relation allows us to calculate the ratio of the j u m p o f condensate to the square of the critical temperature. A numerical solution of (31 yields
Atv (T~) - 4x = 9.30472 .... T?
(32)
Furthermore the critical temperature xs related to the zero temperature condensate by 340 = l n x ln~2
' 2
2 x2 [ f 3 ( x ) - 2 ( ( 4 ) 1 ,
(33)
implying
Mo = 10.29134 .... ;rc2
(34)
A scalar condensate M0 ~ ( 1 0 0 M e V ) 2 at zero temperature would imply a transition temperature o f T~ 31 MeV. We have shown that at a critical temperature the mean-field effective potential F [ ~ 2 ] exhibits a first order phase transition from a non-trivial phase realised at zero temperature to a trivial (perturbative) phase realised at high temperatures. Cosmological consequences of this phase transition will be the subject of further investigations.
212
29 July 1993
We want to thank R.F. Langbein for carefully reading this manuscript and helpful remarks.
[1] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, London, 1972); P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, 1979); E.W. Kolb and M.S. Turner, The Early Universe (Addison-Wesley, Reading, MA, 1990). [2] J.C. Collins, A. Duncan and S.D. Joglekar, Phys. Rev. D 16 (1977) 438. [3] M. Stevenson, Z. Phys. C 24 (1984) 87; Phys. Rev. D 32 (1985) 1389; M. Stevenson and R. Tarrach, Phys. Lett. B 176 (1986) 436. [4 ] C.M. Bender and F. Cooper, Nucl. Phys. B 224 ( 1983 ) 403; M.Liischer, Nucl. Phys. B 318 (1989) 705. [5] K.G. Wilson, Phys. Rev. B 4 (1971) 3184; K.G. Wilson and J. Kogut, Phys. Rep. 12 (1974) 75; M. Aizenman, Phys. Rev. Lett. 47 (1981) 886; B. Freedman, P. Smolensky and D. Weingarten, Phys. Lett. B 113 (1982) 481; J. Fr6hlich, Nucl. Phys. B 200 (1982) 281; C. Arag~o de Caravalho, C.S. Caracciolo and J. Fr6hlich, Nucl. Phys. B 215 (1983) 209. [6] See e.g.T.-P. Cheng and L.-F. Li, Gauge Theory of Elementary Particle Physics (Oxford University Press, New York, 1984). [7] K. Langfeld, L. v. Smekal and H. Reinhardt, Phys. Lett. B 308 (1993) 279. [8] L. Dolan and R. Jackiw, Phys. Rev. D (1974) 9. [9] See e.g.L.D. Landau and E.M. Lifschitz, Statistical Physics (Addison-Wesley, Reading MA, 1958). [I0] S. Coleman and E. Weinberg, Phys. Rev. D 7 (1973) 1888; S. Coleman and D.J. Gross, Phys. Rev. Lett. 31 (1973) 851. [11] R. Jackiw, Phys. Rev. D 9 (1974) 1686. [12] J. Soto, Phys. Lett. B 188 (1987) 340; J.1. Latorre and J. Soto, Phys. Rev. D 34 (1982) 3111.
PHYSICS LETTERS B
Non-trivial phase structure of 04-theory at finite temperature * K. Langfeld, L. v. Smekal and H. Reinhardt Institut fiir theoretische Physik, Universitiit Tiibingen, I ~ 7400 Tfibingen, FRG Received 7 May 1993 Editor: P.V. Landshoff
The effective potential for the local composite operator 02 (x) in 2~b4-theory is investigated at finite temperature in an approach based on path-integral linearisation of the 04-interaction. At zero temperature, the perturbative vacuum is unstable, because a non-trivial phase with a scalar condensate (02)0 has lower effective action. Due to field renormalisation, (202)0 is renormalisation group invariant and leads to the correct scale anomaly. At a critical temperature Tc the non-perturbative phase becomes meta-stable implying a first order phase-transition to the perturbative phase. The ratio (202)0/T~2 .~ 62 turns out to be a universal constant.
1. Introduction
It is of interest to study ~b4-theory, because o f its wide field o f applications [ 1 ]. Due to its simple structure it also makes possible an understanding o f basic features of q u a n t u m field theory, e.g., anomalous breaking o f scale invariance [2[, as well as being a suitable test for non-perturbative methods, e.g., a variational approach [3] and numerical simulations on a lattice [4]. Lattice results suggest that q~4-theory is in fact trivial (at best a free theory), due to renormalisation effects [5 ]. On the other hand, many phenomenological applications [6] rely on a non-trivial vacuum structure o f &4-theory. A variational approach to massive ~/)4-theory [31 indicates that besides the trivial phase, a second non-trivial one exists which escapes the triviality bounds from lattice calculations. This phase is known as the precarious phase since the bare coupling constant becomes (infinitesimally) negative during renormalisation. Recently, this phase was also found for the massless case in a modified loop expansion around the mean field [ 7 ]. It was shown that this expansion rapidly converges in zero dimensions and yields renormalisation group invariant results in four dimensions [7]. Thereby it was observed that the so called precarious renormalisation [3] is, in fact, the *
Supported by DFG under contract Re 856/1 - 1.
standard procedure and is hidden in the usual expansion with respect to the coupling strength [7], i.e. expanding the precarious renormalisation condition in powers of the renormalised coupling constant reproduces the standard result o f perturbative renormalisation. In order to gain further insight into the vacuum structure of 2q~4-theory, it is convenient to study the effective potential for the local composite field q52( x ) . This is because the energy density as a function o f the expectation value of q~2 is minimised for (~b2) /: 0, whereas the effective potential of the field q~ itself is minimal at (q~) = 0 (also true in the non-trivial phase [ 7 ] ). In this letter we study the temperature effects of the non-trivial phase o f 2~a-theory. To this end we therefore calculate the effective potential for q~2 in the modified loop-approach presented in [7] for finite temperatures. Since temperature does not influence the renormalisation procedure, the effective potential is renormalisation group invariant for all temperatures, as observed in a perturbative calculation [8]. Due to the need of renormalisation of the operator 4~2, it turns out that (2~ 2) is renormalisation group invariant and its contribution to the energy density gives the correct scale anomaly within our approximation. The vacuum energy density o f the non-trivial phase is considerably lower than the energy density of the
0370-2693/93/$ 06.00 @ 1993-Elsevier Science Publishers B.V. All rights reserved
207
Volume 31 l, number 1,2,3,4
PHYSICS LETTERS B
perturbative vacuum. The thermal expectation value of the energy density of the non-perturbative phase relative to the perturbative phase increases, however, with increasing temperature, and a first order phase transition to the perturbative phase is found at a critical temperature To, in accordance with Landau's theory [9]. The ratio & t h e zero temperature condensate to the critical temperature is found to be a universal number, as is the ratio of the jump in the condensate at the transition point to the critical temperature.
2. The effective potential of 0 2 Our starting point is the Euclidean generating functional for Green's functions, i.e. I/T
29 July 1993
Since the vacuum is isotropic and homogeneous, we have 0z. = Uodz. with Uo = 0oo the vacuum energy density. This implies Uo = ---14 • 2~fl(J,R)(~4).
(4)
For small couplings 2R the renormalisation group flfunction behaves as fl(2R) -- f l o ~ + .... with flo a constant. Hence if (~b4) ~: 0, and assuming (¢4) (~b2)2, leads us to regard ,~Rt~2 as a physical quantity instead of 02 . Due to the 04-interaction the path-integral ( I ) cannot be performed exactly, and one has to resort to approximation schemes. One of these is the perturbative expansion with respect to the coupling strength. At one-loop level, the result for the zero-temperature effective potential for ~b is [10]
Ul-loop =
l
~-~c
4
~'2
"l- ~ w c
"~4 /'"
'~'~2 ~m 2--A--~
0
4._ ½m2q52 ..{. 1 2 ¢ 4 _
j(X)O2(X)]) ,
(l)
where 2 is the bare coupling constant and T is the temperature in units of Boltzmann's constant. The functional integration runs over allscalar modes satisfying periodic boundary conditions in the Euclidean time direction, j is an external source which couples linearly to 0 2. The effective action for 0 2 is defined by the Lcgendrc transformation, i.e., £[(/)2c] :~--~6~(x)
-lnZ[j]
"- 6
+ ./d4xO2c(x)j(x),
lnZ[j]
dj(x)
(2)
The effective potential U (4~) is obtained by restricting ~bcto constant classical fields. It is a physical quantity, e.g., its minimum value represents the vacuum energy density [ 10], and should therefore be renormalisation group invariant. Further access to the vacuum energy density is provided by the scale anomaly [2]. Conformal symmetry is broken in the non-perturbative phase by a nonvanishing vacuum expectation value of the trace of the energy-momentum tensor, i.e.,
208
(3)
.
(5)
N o field renormalisation is required to derive (5) implying that ~bc is stillrenormalisation group invariant and a physical field. It would initiallysccm that (5) can account for the scale anomaly, since U1-1oop(Pc) has a m i n i m u m for non-vanishing field.However, this m i n i m u m is an artefact of the one-loop approximation as already pointed out by Coleman and Weinberg, and by Coleman and Gross [I0]. This fact isalso evident from the two-loop effective potential [I l ], since it again has the m i n i m u m at Oc = 0, and significantly differsfrom the one-loop resultfor small classical fields.For large classicalfields,or large masses, the one-loop potential scales according the standard rcnormalisation group fl-functionwhich has an infrared fixed point. Since wc would like to investigate the non-trivial phase of ~b4-theory,non-pcrturbativc methods arc required. W e therefore apply the modified loop expansion [7] and lincariscthe g54-interactionby means of an auxiliary field,i.e.,
Z[j] = / D~73zexp[- / d4x(½0uc~O~ +
26-Z2(x)
+[½m2(0u,u) = -- 2'~fl ('J"R)((/)4) .
2)
iz(x)]~2(x)
-
j(x)(~Z(x))]. 2J
(6)
29 July 1993
PHYSICS LETTERS B
Volume 3l l, number 1,2,3,4
After integrating out the ~b-field, the modified approach [7] is defined by a loop expansion with respect to the field Z around its mean field value. Even the lowest order in this expansion contains quantum fluctuations of the ~-theory, yielding reasonable results for the effective potential of 4~. For the effective potential of 4)2, we obtain in lowest order (i.e., the mean-field approximation)
-InZc[j](T)= fd4x(-3(M-m2+2j)2) +½ Tr l n ( - 0 2 + M ) ,
(7)
-
V4
32;72 Z ,#0
327r2
+-
fds
× e x p { - s ~ 2 + M + (2;TT)2n2]},
(9)
where A is the proper-time cutoff and V the threespace volume. Note that the ultra-violet divergences arise from the behaviour of the integrand in (9) at small s. The integration over the three momentum is easily performed, i.e., V
~
16;73/2
f d
~ ds
e-sM
e-S(2~T)2n2
.
(lO)
n I/A2
/
([3)
~ds e_SX e_,Z/,
(14)
(where e = 3 in (13)). The term in curly brackets in (13) is precisely the loop contribution to the effective potential at zero temperature. The temperature dependence is completely contained in the second term of (13), which is finite. As in the perturbative approach [8], renormalisation is unchanged, if temperature changes. Renormalisation is quite analogous to procedure of the zero-temperature effective potential of q5 [7]. The key observation is that the divergences can be absorbed into the bare parameters 2, m, j by setting
dnf(n) e i2~".
A straightforward calculation yields
1
~ + T-6-~ 6 . 2-J
(ln A2
3m 2 2
j - m 2 = 0,
~ f(n) = ~ c(v), n~ -oc t/= -~c
c(v) = i
+7
oo
6
The divergence of the discrete sum in (10) can be extracted by applying Poisson's formula, i.e.,
with
ln~- 2 - ~
v¢0
n I/A2
-
(12)
where y = 0.577... is the Euler-Mascheroni constant and the function f, is defined by
L = ½Trln(-02 +M)
L
]'
V4 T4j~ ( ~ T 2 ) 2;72
f~(x) =
Vfd3p
-j3- e-SMexp - 4 - i T
where F is the incomplete F-function, 1/4 the spacetime volume and the limit A -~ oc has been taken in the second term of the right hand side (12), which is finite. All divergences are in the first term, which is in addition, temperature independent. Using the asymptotic expansion for the incomplete F-function the loop contribution becomes
where M is related to the mean field value Z0 by Z0 = ½ i ( M - m 2 + 2j ). The mean field equation for Z0 can be recast into an equation for M, i.e., c ~ l n Z [ j ] _ O. (8) c~M The trace in (7) extends over space-time, and the modes satisfy periodic boundary conditions in the Euclidean time direction. Using Schwinger's proper time regularisation the loop contribution L is
32;72
(11)
F -y+
1 A2 32;72 =
1)
=~ jR
6,
[5)
3m~ ,
16)
2R 17)
where/~ is an arbitrary renormalisation scale. Later, we will check that physical quantities do not depend on/~. Note that the mean field variable M has been introduced in such a way that renormalisation is possible without M picking up divergences. In fact, as 209
Volume 311, number 1,2,3,4
PHYSICS LETTERS B
will be shown later, M is a renormalisation group invariant. In the following we confine ourselves to the massless case (m~ = 0). The bare parameters are uniquely defined, in ( 15)- (17), as functions of the renormalised ones and the regulator. The renormalisation of the coupling constant (15) is essentially the same as in [7], implying a universal renormalisation procedure in the calculation of the effective potential of ¢ and ¢2. Besides coupling constant renormalisation, mass and field renormalisation (16), (17) is also necessary to absorb all divergencies in (7). This is different to the case of the effective potential of ¢ on its own, where the field does not require renormalisation at this level. The situation here is analogous to the Gaussian effective potential approach t o ).~4_ theory (see refs. [3,12] ), where besides the "precarious phase", a second non-trivial phase (called "autonomous phase" in [3] ) would seem to exist, allowing for an infinite shift of the field variable. It was shown, however, that field renormalisation does not truly give a new phase, but rather what is the effective potential of the composite field 2¢ 2 [ 12 ]. Introducing the renormalised quantities in (13), the generating functional (7) becomes
29 July 1993
o.o15
T=0
......
/
T=Tc=0.507.. T = 0 . 6 p.
....
g
,t
,,'t
/
/,,//;'
0.005
"'----_____--~.,'"/"
-ooo5
// / I
/
-0.015 0.0
210
M/,a 2
4.0
6.0
Fig. 1. The effective potential as function of the scalar condensate at various temperatures. A large coupling strength 2R (#) = 1000 was chosen for aesthetic reasons. (
M
1)
32
-2-GM
1 lnZ[j](T)
v.
- l e f t T a f 3 (4-~)M=(RR/6)¢2 ----22R3
M2_
-- 16CtT4j~ ( ~ T 2 )
M jR+ lc~M2 ,
(18,
where c~ = 1/32~ 2. Here M is defined by the mean field equation (8), which in terms of the renormalised quantities reads
3 M - ~---~jR + ctMln --~ + 4~T2f2 (4-~ ) 2R =
0,
(19)
where J~ is defined by (14). In order to perform the Legendre transformation (2) we first obtain the classical field, i.e.,
¢~
10lnZ[jR] _ 6 M
-
v4
ojR
-
;~R
"
(20)
The effective potential (2), is therefore given by 210
c
.
(21)
ln~- 2 We have obtained a finite and renormalisation group invariant result, since a change in the scale Ft can be absorbed by a redefinition of the renormalised coupling constant ~-R according (15). The renormalisation group B-function fl().R) = f1022,
with
fl0:= ~c~
(22)
exhibits an infrared stable fixed point as indicated by a perturbative calculation [10]. Note that since our approach is intrinsically non-perturbative, (22) need not agree with the fl-function from the perturbative calculation. Furthermore, we see that M = -~2n(Oct 2 rather than q52 is renormalisation group invariant, as suggested by the scale anomaly. Due to the need of field renormalisation ¢c2 itself has lost its physical meaning. Fig. 1 shows the effective potential (21) as a function Of 2R¢~ for various temperatures. At zero temper-
Volume 311, n u m b e r 1,2,3,4
P H Y S I C S LETTERS B
29 July 1993
ature it develops a minimum U0 at a non-vanishing value of the condensate M0, i.e.,
-5
0.98
Mo = /z2 exp (flo@R),
Uo=-¼(~p'exp
(4)
fl--~--RR "
(23) o 0.92
This minimum corresponds to a non-trivial vacuum at zero temperature it has lower energy density than the perturbative state (¢2 = 0). The vacuum energy density of the non-perturbative phase is 1 Z = _gerM6
LT0
-~
1
2
0.86
2)2
~4fl(2R)(:qS4:).
(24)
This is precisely the result obtained in [7 ] for U (q~ = 0) in the non-trivial phase, furthermore it gives the correct scale anomaly (4) obtained by general arguments. Note that in the effective ;(-theory, obtained from (6) by integrating out ~b, the trace anomaly already shows up at tree level. It is therefore not so surprising that the trace anomaly survives in the meanfield approximation. Increasing the temperature shifts the scalar condensate to smaller values and lowers the difference of the energy density of perturbative and non-perturbative phase. At a critical temperature T~, the non-trivial phase becomes degenerate with the perturbative one (at q~ = 0). If the temperature is increased further, the non-trivial phase becomes meta-stable and a first order phase transition to the trivial phase at ¢2 = 0 can occur either by quantum or statistical fluctuations. This phase transition implies that the scalar condensate as a function of temperature has a discontinuity at the critical temperature.
3. At the phase-transition We first investigate the change of the scalar condensate with temperature. The condensate M:v = 1 2 g2R~bcImin is obtained by solving the equation of motion (19) for zero external source (jR = 0) which is
0.80
My (~ + c~ln-w- + - J ~ ( x ) = O, X
(25)
0,2 T/M 0
0.3
0.4
Fig. 2. The scalar condensate as function of temperature. where x : = M ~ / 4 T 2. In order to remove the renormalisation point dependence we subtract the corresponding equation for the zero temperature condensate 3//0 given by 3
M0 + (~In--7 p ~ = 0,
(26)
yielding ln--~ + lfz(x)
= 0.
(27)
This equation allows us to calculate My, in terms of its zero temperature value, as function of temperature. A numerical solution of (27) is displayed in fig. 2. It is instructive to relate the jump in the condensate and the critical temperature Tc to a fundamental observable like the scalar condensate at zero temperature M0 = (~,~R~2). At the critical temperature, the energy density of the perturbative thermal ground state U(q~2 = 0) is equal to that of the non-trivial phase U (¢2 mi,)The effective potential at zero field is obtained from (21 ) by a direct calculation, i.e. Uper : =
3
0.1
0.0
U(q~c2 = 0 ) =
= -32(~T4~(4),
16o~T4j~(0)
(28) 211
Volume 311, number 1,2,3,4
PHYSICS LETTERS B
where ( is the R i e m a n n (-function. Equating the energy density of the perturbative thermal state Uper to the one of the non-trivial phase (with non-zero condensate) we obtain
Acknowledgement
-22--~ + ½c~ In ~z2
References
2
- ~-~A(x)
= - ~-~2( ( 4 ) •
(29)
In order to eliminate the renormalisation point dependence we replace/z by M0 by using (23), yielding In M,, 2 M0 = ½ + 7 g [ j ) ( x ) - 2 ( ( 4 ) ] .
(30)
Combining this equation with (27) we finally obtain ½ + ~-~[f3(x
-2~(4)]
+
f 2 ( x ) = 0.
(31)
This relation allows us to calculate the ratio of the j u m p o f condensate to the square of the critical temperature. A numerical solution of (31 yields
Atv (T~) - 4x = 9.30472 .... T?
(32)
Furthermore the critical temperature xs related to the zero temperature condensate by 340 = l n x ln~2
' 2
2 x2 [ f 3 ( x ) - 2 ( ( 4 ) 1 ,
(33)
implying
Mo = 10.29134 .... ;rc2
(34)
A scalar condensate M0 ~ ( 1 0 0 M e V ) 2 at zero temperature would imply a transition temperature o f T~ 31 MeV. We have shown that at a critical temperature the mean-field effective potential F [ ~ 2 ] exhibits a first order phase transition from a non-trivial phase realised at zero temperature to a trivial (perturbative) phase realised at high temperatures. Cosmological consequences of this phase transition will be the subject of further investigations.
212
29 July 1993
We want to thank R.F. Langbein for carefully reading this manuscript and helpful remarks.
[1] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, London, 1972); P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, 1979); E.W. Kolb and M.S. Turner, The Early Universe (Addison-Wesley, Reading, MA, 1990). [2] J.C. Collins, A. Duncan and S.D. Joglekar, Phys. Rev. D 16 (1977) 438. [3] M. Stevenson, Z. Phys. C 24 (1984) 87; Phys. Rev. D 32 (1985) 1389; M. Stevenson and R. Tarrach, Phys. Lett. B 176 (1986) 436. [4 ] C.M. Bender and F. Cooper, Nucl. Phys. B 224 ( 1983 ) 403; M.Liischer, Nucl. Phys. B 318 (1989) 705. [5] K.G. Wilson, Phys. Rev. B 4 (1971) 3184; K.G. Wilson and J. Kogut, Phys. Rep. 12 (1974) 75; M. Aizenman, Phys. Rev. Lett. 47 (1981) 886; B. Freedman, P. Smolensky and D. Weingarten, Phys. Lett. B 113 (1982) 481; J. Fr6hlich, Nucl. Phys. B 200 (1982) 281; C. Arag~o de Caravalho, C.S. Caracciolo and J. Fr6hlich, Nucl. Phys. B 215 (1983) 209. [6] See e.g.T.-P. Cheng and L.-F. Li, Gauge Theory of Elementary Particle Physics (Oxford University Press, New York, 1984). [7] K. Langfeld, L. v. Smekal and H. Reinhardt, Phys. Lett. B 308 (1993) 279. [8] L. Dolan and R. Jackiw, Phys. Rev. D (1974) 9. [9] See e.g.L.D. Landau and E.M. Lifschitz, Statistical Physics (Addison-Wesley, Reading MA, 1958). [I0] S. Coleman and E. Weinberg, Phys. Rev. D 7 (1973) 1888; S. Coleman and D.J. Gross, Phys. Rev. Lett. 31 (1973) 851. [11] R. Jackiw, Phys. Rev. D 9 (1974) 1686. [12] J. Soto, Phys. Lett. B 188 (1987) 340; J.1. Latorre and J. Soto, Phys. Rev. D 34 (1982) 3111.