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40
Hattori T., Hayashi M., Inagaki T.,1 Kitadono Y.
EFFECTIVE POTENTIAL FOR THEORY AT FINITE TEMPERATURE IN ÄND r®hd^
Department of Physics, Hiroshima University, Higashi-Hiroshima, Hiroshima, 739-8521, Japan PACS numbers: 04,62,+v, 11.iO.Kk, 11.10.Wx, 11.30.Qc
I. Introduction
The idea of spontaneous symmetry breaking has an important role in elementary particle physics. In fact it is understood that the electroweak symmetry is spontaneously broken down through the Higgs mechanism. Grand unified theories (GUT) are constructed on a basis of spontaneous gauge symmetry breaking to yield a theory at low energy scale. It is generally expected that a more fundamental theory with a 'higher symmetry is realized in the early universe. Thus one of the possible environment to test the mechanism of symmetry breaking is found in the early universe where the fundamental symmetry is broken down. In the early stage of universe, especially at GUT era, we can neglect the thermal and curvature effects. The spontaneous symmetry breaking takes place under the influence of the high temperature, strong curvature and non-trivial topology.
Much attention has been paid in the study of the spontaneous symmetry breaking under the circumstance of the high temperature, strong curvature and non-trivial topology. For this purpose finite temperature quantum field theory is investigated in curved space time 11-14]. The effective action of the system is calculated to determine the ground state. Where the ground state of the system breaks a symmetry of the Lagrangian, the symmetry is broken down spontaneously. If the universe is almost static, homogeneous and isotropic, it is quite natural to assume an order parameter of the symmetry dose not depend on the spacetime coordinates. Under this assumption a ground state is found by observing the minimum of the effective potential as a function of an order parameter [6]. Hence, the effective potential of many kinds of quantum field theories is evaluated at finite temperature and/or finite curvature with the spacetime structure fixed (for a review, see Ref. (15) and references therein).
In the present paper we consider a real scalar fields with (j>4 interaction in static homogeneous and isotropic spacetirne. The Ax])4 theory is one of the simplest models where a discrete symmetry is sponta-
neously broken down. To find the ground state of the system we evaluate the effective potential at high temperature and strong curavture. In two, three and four spacetime dimensions the theory is renormalizabie. Beyond five dimensions the theory is nonrenormalizable. We confine ourselves to the spacetirne dimensions greater than or equal to 2 and less than 5 and calculate the renormalized effective potential. The curvature effect to the symmetry breaking comes from a scale factor dependence of the covariant derivative and a coupling between the scalar field and the gravitational field. One-loop and two-loop corrections to scalar field theories in linear curvature approximation were found in Refs. [16] and [17], respectively. This method is general, any renormalizabie theory maybe studied in this way (including backgrounds of non-trivial topology) but at small curvature only.
We assume that the system i; introduce the temperature. The acceptable in a general curved restrict ourselves in the static uni work in the scalar X.f4 theory in Isiv vv lut taiua.-
spacetirne R®Sl>~1 and negative curvature spacetirne R ® H l>~1. An exact expression for scalar and spinor two-point functions is calculated in R ® S1’ "1 and R®Hl)~~l [18-23]. It is one of the fundamental object in dealing with quantum field theories. The vacuum energy density for scalar and spinor fields is obtained in R ® S°~l [24-26]. Thermal effect to the vacuum energy is discussed in [1-4]. The curvature effect for symmetry breaking is studied for the 'hj>4 theory and
the four-fenmion theory in R®SD~1 [27-29]. It has been found that the broken symmetry is restored for a sufficiently small scale factor. Following the imaginary time formalism, we calculate the effective potential at finite temperature in curved spacetime. For £) > 4 a naive coupling expansion is not valid at high temperature. To improve the loop expansion we resume all the ring diagrams [30] and estimate the effective potential for £) > 4,
1 E-mail: inaoaki@hiroshima-u.ac.ip; Also at Information Media Center at Hiroshima University
The paper is organized in the following way. In Sec. II we briefly review a basic formalism to calculate the effective potential for a theory in curved spacetime. In Sec. Ill we calculate the effective potential in R®Sn~' and R®HDA without making any approximation in the spacetime curvature. According to the imaginary time formalism the
temperature is introduced in the theory. The ordinary renormalization procedure in a flat spacetinte is used to obtain the renormalized effective potential. In Sec. IV we numerically calculate the effective potential at finite temperature in R®SD~l and Characteristic behaviors of the effective potential are shown as the scale factor varies in the fixed
temperature. Section V gives the concluding remarks.
II. X<j>4 theory in curved spacetime
In this section we summarize how to obtain the effective potential in the Feynmann path-integral
formalism. Here we consider the theory
constructed by a real scalar field with a <|>4 interaction. It is one of the simplest models where the spontaneous symmetry breaking takes place. The theory is defined by the Lagrangian,
1 . , . ili A.
i)
where <j) ¡s a real scalar field, ifi;t corresponds to the bare mass os' the scalar field, kg the bare coupling ca :or the scalar self-interaction, the bare
e< constant between the scalar field and the
gravitational field.
The Lagrangian is invariant under the discrete transformation,
(2)
This Z, symmetry prevents the Lagrangian from having cfd terms. Because of the minus sign of the mass term the symmetric state, (<j>) = 0, is unstable in a classical level. If the non-vanishing expectation value is assigned to the field <f) in the ground state, the discrete symmetry is eventually broken. To see the phase structure of the theory we want to find a ground state.
In quantum theories the ground stale is determined by observing the minimum of the effective potential. Here we briefly review the effective potential for X<j>4 theory. We start with the generating functional of the theory which is given by drm r„ . m)+¥)
where g is the determinant of the metric tensor . In the presence of the source J the classical equation of motion becomes &S[<j>]
= -J(x)
(4)
&5>(x)
where S[#] is the action
S№*jdDxJ^gm. (5)
In curved spacetime the functional derivative is defined
by
S[<|> + 6#] - st<M = ¡dDx.f^-^?4(x). (6)
J of(x)
We divide the field (j) into a classical background <j>j which satisfies the Eq.(4) and a quantum fluctuation <j>, <ji = <{>0 + tiu% (”7)
In terms of and <j> the action S[<|)] is rewritten as
sm=s[^+iir%]
= SIM -
+f \d °x ld 11 y^l~g^ 4~8(y№x)iG~' (x> >')<k.y)
+(J(1/I","(j>3),
where G"’(x,y) is the scalar two-point function,
(8)
iG-'(jc,y)s
8aS[# 1 5(K.r)5<|)(y)
(9)
Therefore the generating functional (3) is expanded to be
~l¥l J J f (.51 ft ]- f (.01
eh _ i
xJz>|e2J
Performing the path-integral,
jfdDxfj0yj-a, >')?.(.«
f D^e1
(10)
(11)
(12)
= [DctfG'1r1/2,
we obtain the generating functional W[J] = St«!»*]~ ¡dDx^J(x)$b{x)
IntDeti G^‘] + 0(*3/2).
2
The effective action n<M is given by the Legendre transform of W[J],
r[<j>c]a= W[J}~ \dDxf^i|)c (x)J(x), (1.3)
where <j>c denotes the expectation value of <j) in the presence of the source J ,
0Щ/] &/ '
(14) R = (D - l)(i) - 2)
1
From Eq, (12) the effective action is expanded to be Щ j = S [фс ] -ylnfDet iG~l ] + 0{hm ). (15)
If there is a translational invariance, фс(х) in the ground state dose not depend on the spacetime coordinates. In such a case it is more convenient to consider the effective potential. The effective potential У(ф) is defined by
= —— На. ф2 + hi + JL in[Der iG~r ] (16)
2 2 4! 20
+0(hm),
where we put a constant value фДх) = ф and Q is the spacetime volume
Q=ldDx^g. (17)
The expectation value of ф in the ground state satisfies
Who,*™ (.*)
./-мі ¿j
This equation is rewritten as
§m.1
§ф.
-0.
(1.9)
?\Ф)
where we have used the relation
ônfc]= ,
(20)
&jK
The equation (19) is called the gap equation. Here we assume that the expectation value of <j> in the ground state is independent of the spacetime coordinate jc . In this case the gap equation (19) reduces to dV(§)\
<іф
= 0.
(21)
»ф-КФ>
(23)
The manifold R®Hl} 1 which is represented by the metric
ds2 -dr1 + o2(dQ2 + sinh2 в^Ш£и.2), (24)
is also a constant curvature spacetime with negative curvature
R = ~{D - 1)(D - 2)
1
(25)
A, Effective potential at I = 0}
The effective, potential (16) is described by the two-point function G(x,y") of the real scalar fields. The effect of the spacetime structure is introduced to the
effective potential through this two-point function. In Euclidean spacetime the effective potential V(f }
reads
m)akRÿ-È.ÿ+h.ÿ
(26)
+— ln[Det (-Cr1)] + 0(fiil2)
2 2 4!
+_A_Tr in(—Cj’-1 ) + 0(tr12).
Ill
The scalar two-point function satisfies the Klein-Gordon equation
j (d^2 +a0.j-^0i? +p.u jG(x}’)
L , ‘ J (27)
The expectation value ((j)) is obtained by solving the gap equation (21).
111. Effective potential in R®S°~l and R®HDA
Here we introduce the curvature and temperature in the theory and calculate the effective potential. First we consider the constant curvature space R<8>S°~l and R®HD~l as Euclidean analog of the static Einstein universe. The manifold R®S°~l is represented by the metric
ds2 =dr2 + a2(dd2 +sin2 QdQ.D_2), (22)
where dilD_2 is the metric on a unit sphere Sü~2 and a is the scale factor. It is a constant curvature spacetime with positive curvature
= —LôB(x,y), yji
where is the Laplacian on. SD"‘. Thus the
effective potential V(0) is rewritten as
(34)2 +d£W -^0R + pô -
+——Trln
20,
+0(tf12)
2-ф2
(28)
40
f
dm
xTr
(34)2 +oB
-m
At the last line we normalize the effective potential so that V(0) = 0 and. neglect the 0(h>n) term. The
integrand in the last line of the Eq. (28) is described by the Green function G(x,x,m),
-i-i
A
Tr
(34) +□,}_! t,0R + )x0 •
-±m2
(29)
= -feDxJgG(x,x;ij) = m).
Thus the effective potential reads
hXn
dm2G(x,x,m).
Substituting Eq, (30) to Eq. (21), we obtain the following expression for the gap equation
(31)
^ Lw
= (#! ^-Ho+^L<<]))2+^3 2 I
On the manifolds J?®5£W the solution of Eq. (27) is given by f 18-23]
= 0.
G(x,x,m) = 'D-2
nr ____
f ius 11 : .
D -
■iav
XjPI
( D....2
• tac
D-
-iaK
D-
— ; cos ; — 2 2a
where o is the geodesic distance between x and y on 50-1 and as is defined by
(xs s xj f(m)ar + (D - i)(l) - 2)i'0 ■ with
(D - 2)2
(33)
a
(4ît)<W)'2 •> 2tc
xcosh
D-2
( 2
2a
F(2aH +1)
(35)
xF
D-2 1 , . .
—-— + aH,- + aH,2aH +l;cosh
where a is the geodesic distance between x and y on HDA and tt„ is defined by
(30) aH s J f(m)a2 - (D -1 )(D - 2)t,0 +
(D-2)2
(36)
On R®HD~l the geodesic distance a is not bounded. The two-point function (35) dose not have any singularities except for the limit cr-> 0.
Substituting Eqs. (32) and (35) to Eq. (30), we obtain the effective potential V (<)>) on R ® Sn~L,
hln
a'
4 (4%)!
(0-11/2
3-D
(37)
dm1 J-
2 it
ri~ + i’cx5jr[ j
(32) anti on R® Hl
M.,
a'
3-D
4 <4jc)!
(/>-l)/2
(38)
_/ D-2 rfCO^'T" *
j
where we have used the relation r(Y)r(Y-a-p) r(Y-a)r(y-p)‘
(34) F(aJ,y,l) =
(39)
The two-point function (32) develops many singularities at a = 2mna where n is an arbitrary integer. This property is a direct consequence of the boundedness of the manifold S°~', In other words the geodesic distance a is bounded in [0,2na). Thus the two-point function (32) satisfies the periodic boundary condition G((y- x)4,o) = G((y -x)4,a + 2ma).
Following the procedure developed in Ref. [22] we can solve Eq.(27) on the manifold R®Ha^ and find the two-point function,
G(x, y\m) =
B. Effective potential at finite temperature
Next we introduce the effect of the finite temperature. Since the manifolds R®SD~l and R ® HD~l have no time evolution, the equilibrium state can be defined. Here we follow the standard procedure of the imaginary time formalism.
At finite temperature Eq.(29) is modified as
Tr
04 )2 +n0.1”^0i? + M-0
= - dx4 jd°~iXy[gG(x,x’,§ = m),
where ß = l/(£ÄjT) with kB the Boltzmann constant
and T the temperature. The fourth component of the coordinate x4 is bounded in [0, ß).
Following the standard procedure of the imaginary time formalism (for a review, see Ref. [30], the scalar two point function at finite temperature is obtained by the replacement,
rd 0)
2 n
(Û —4 (0, =-------71.
" ß
(41)
Thus the effective potential at finite temperature on reads
V(<j>) =—+ ^c()4
111,
4ß (4it)to_1)/ l>’i
4! 3-D
(42)
JD-2 , . )„(D-
F ------+ ia |n
--za<
F| - + ¡ac FI -~ia
{2
with
a,, = Jfimja2 + (I) -1 ){D - '2%0 -and on R®Hl)
2 2 4!
(D-2)
(43)
+
tila
4ß (4n:)iD_i,/2
I>!î
3-D
(44)
D-2
1 —r— + OU
4-D
-a,.
with
aH = Jf((0„ )a2 - (D - !)(£> - 2)£0 +
(D-2)2
(45)
As is known, the ordinary loop expansion is not always valid for Bose fields at finite temperature. Higher order contributions of loop expansion contain terms of the order 0(kT°~2 !§2), At high temperature these terms are not negligible for D > 4. The terms
proportional to TD~2 are described by Ring diagrams. We resum ail the Ring diagrams to improve the loop expansion for D> 4 . The resummed result is obtained by the following replacement in the two-point function [30]
(46)
-|^-»n = -p2
For D = 4 Eq,(46) reduces to the well-known formula,
(47)
2 „ , Kl0T2
We apply the replacement (46) to the as and aH in
Eqs.(42) and (44).
C. Renormalization
The effective potential V(<j>) obtained in the
previous section is divergent in two and four dimensions. To obtain the finite effective potential we must renormalize the theory in two and four spacetime dimensions.
First we introduce the renormalization procedure in a flat spacetime. The effective potential in D-
dimensional flat spacetime is given by
*—f0W, (48)
r0
with /oW = r
2(4tc)
( £> W , , r d"k
F dm2 f-
* j
(49)
Integrating over m2 and k in Eq.(49), we obtain
/’,(*) = r -
D )
-p2 + --
■(....
It is divergent in two- and four-dimensions.
We introduce the renormalization procedure by
imposing the renormalization conditions
a<j>2
a4K
aft (51)
l<H) If-M
where M is the renormalization scale. From these conditions we obtain the renormalized mass parameter |ir and the coupling constant Xr ;
2(471)"
(52)
fill
FI 2- — 2(4n) { 2
-W +
—A#2
3 + 6XaM21 D
(53)
+V0M*
D
Replacing the bare parameters, ji,n and Xn, with the renormalized ones, and %r, we ■ obtain the renormalized effective potential
h
2(Anfn
ni
-0-i2rfn
D
+.---~i—2-—+•—M2 I
2-4!(4it) ^ 2j{ r 2 )
3 + 6X,M21 Y - 2 Y -M-Î +—M2
+l:M4 ( — - 2^|f ■— s'ji -\x2r +“M2
(54)
If we take the two-dimensional limit,
renormalized potential reduces to
2 4!
h f , 1 A -h +M*/2 . ji; 0- Jinj i P — 8rev 2 ) [ -ft; )
t »a;#4 [/ ^ ;
1 nrc ' 192re j ;x 2 ;
•-6Vtf3[ j
i \ -3 "" -t-2/l’M4! -p.7 + •— M1 I
V 2 }
(3.5)
Taking the four-dimensional limit, we obtain the renormalize effective potential
W)—f^2+^4
2 4!
, ft 2/ •> , f~|i7; + îir<f/2^
-K
^ }A* *4,
+----In
25670
v. * '
+ J^L_^_.3^r A4
v~M> + IrM2/2/
(56)
128nr ' hll
512îi
AT
128xc2 -p; +lrM2 /2^
M2
786% (-\i;+\rM2/2)'
the effective potential by the renormalization conditions (52) and (53).
To see it we consider the conformai coupled case for simplicity. Under the conformai coupling
\ (57)
0 4(1) -1)
the effective potential (37) on R<2>SD for D = 2 and
£> = 4 reads
v(#)=v0(#)+-lT(|r+~fDm
2a~ 4
with
(58)
2« ^ôF^^r^-\nFÏ2
(59)
| ^m^ïü2--\i2r-*rXrm2 il
and
X,. f?2 , ^ pdm sjüF-Mr + V”2 /2 4lt * 2îC I _ /2
The additional term' /D(c|>) is obviously convergent.
On R®Hn the effective potential (38) for for £> = 2 and £> = 4 reduces to
(61)
Thus the finite expression of the effective potential is
obtained.
In the two- and the four-dimensional manifolds R ® S° and R® HiJ we obtain a finite expression for
There is no divergent terms which depend on the scale factor a. The ultra-violet divergence in the ef' potential (58) and (61) is cancelled out by us same renormalized parameters (52) and (53) o in a flat spacetirne. The renormalized ei potential V(§) is given by replacing V), with the one in Eq. (54),
Extension to the finite temperature case is also trivial. As is well-known, thermal effects do not change the ultra-violet behavior of the theory. Thus no new divergent terms appear at finite temperature (See yr (4>) in the next section). All the divergent terms are cancelled out by applying the renormalization procedure for T - 0 in a flat spacetime.
IV. Numerical calculation
We wish to observe the thermal and curvature effect on the effective potential. For this purpose the effective potential is calculated numerically as a function of the field (¡>.
The expressions (42) and (44) are not useful for a numerical analysis, since the divergent part is not clearly separated. Hence the summation appeared in the expression is not convergent. To obtain the finite expression of the effective potential we use the following trick [29].
At finite temperature in a flat spacetime the effective potential for X<j)4 theory is given by
+
Kk |4J
4p
j.liO.A''
4! dD~lk
2 4!
dm2 J-
(62)
{2n)D~i’—(ù2+k2+Tl + ^m2’
where all the ring diagrams is resummed for D > 4. We replace -p2 to n (46) in the propagator. If the spacetime dimensions are much smaller than four, for example 2 < D < 3, we put II = -p2. We perform the integration over k in Eq.(62) and find
x \Ml)~2k\n
1 - e
l-e'
til,
1
-(W*!+n
3-D
4p (4n)<iW)/2 ^ 2 xf dm2 J] f a>2 + FI + —m2
(65)
Inserting Eq.(65) into Eqs.(42) and (44), we obtain the finite expression for the effective potential on R ® Sl}
vm=v0m+y:m
s>2+^f
hi
1
4p (4rc)
(13-1)/2
3-D
(63)
f dml ¿(03»+n+|m2
(D-3J/2
The divergent term for T = 0 is contained in the infinite summation.
if we perform a summation and integration over m1 and angle variables first in Eq, (62) and leave the integration over k , we obtain the following expression for the effective potential
A.
-—I
4 !
h_ 1 __
h 2
rU"
I 2
P (4it)(/)',)/2 T(D-l
V 2
(64)
i\kdL
A; In
l-e
^k2+Sl+lifl2
l-e
= V0«f>) + Vr(<f>),
where V0 ((j)) is the effective potential (43) for T = 0 in a flat spacetime. The divergent term at T = 0 is included in V'(O). The divergence is cancelled out after the renormalization discussed in the previous section. V7.(<|)) is finite in the spacetime dimensions 2 < D < 5.
Comparing Eq.(63) with Eq.(64), we find the relationship
JLJL. rL£.
2 (4%f V 2
n+-02| -(Ilf
h H—
p (4ic)t/>_,î/a T(D-l
h
+-
1
p (4k){D~1)I2 (D-1
tfk^dk In
I — e
-$*Jk2+n+\$r /
1 - e
hX
40 (4jt)(,)~1)/2 '
-ÿÎk2 + ll 3-D
dm2 X
•5 _ "k ->
coi +I1 +—m‘
(D-3)i2
r( £>- 2 , \ / D — 2 . 1
n —...nas.JI | —.....ias j
T7r—X7T—T~........
I ( —+ i<xs jl ! -.ta< |
\ L
and on R ® H
I
<.jkl)'2dkln
K 2
~p^jk " +TI+X4>” i -
Î - e
•¡n
hX
4(3
(fdml X
3-D)
2 *~r k j
m: + 1I +—m
0-3)/2
D-2
F —-—f-otH
4-JO
F —■--------i-a.
(66)
(67)
-a
rf ~
. 2
The renormalized effective potential V(<js) is given by replacing V0 with the one in Eq.(54). It should be noted that we replace -|,r with II in the second line of Eq. (54) for D> 4 to improve the loop expansion at high temperature. By using the expressions (66) and
(68) we show some characteristic behaviors of the effective potential near the critical point where the phase transition takes place. The effective potential develops a non-vanishing imaginary part in the present renormalization conditions. Below we draw the real part of the effective potential We take the natural unit and put h = 1. All the mass scale is normalized by the renormalization scale M.
0.0001
T=1.45M
T=1.35M
<u
EC
-0.0001
T=1.4M
-0.0002
0
0.1
0.2
f/M
0.3
0.4
Fig. 1. Behavîor of the effective potential for ¡i2 = 0.1M1, X ~ M "~D and D ~ 3.5 in a fiat spaceüme as T varies.
mow
(54
VOili
T = <
neoi
ure is no less
■ um •/ for
Mini ■ ,
ï sy li.
:mpe
due T.., the
minimum of the effective potential locates at f = 0. The ground state keeps the Z2 symmetry under the transformation (2). There is a second order phase transition, as T is increased for D ~ 3.5.
In Fig. 2 and 3 a D dependence of the effective potential is presented with the temperature fixed at T = 2.35M . To draw the figures we use the resumed propagator. The dependence becomes larger near the four spacetime dimensions. It seems that the critical temperature has a gap at an integer dimensions.
We observe the behavior of the effective potential in R®SD~l at finite temperature. It is found in Ref, [27] that a positive curvature suppress the symmetry breaking for 7 = 0 in the Einstein universe. To see the curvature effect at finite temperature, we take a temperature lower than the critical one and calculate the effective potential in R%S°~l. The behaviors of the effective is illustrated for a conformally coupled scalar field, £ = (D - 2)!(AD - 4), and for a minimally coupled one, £ = 0 in Figs. 4 and 5, respectively. As is seen in the figures, a positive curvature restores the symmetry. The ground state is always symmetric under
the transformation (2) for T>TC in R®SD~l. For p. = 0 the spontaneous symmetry breaking is induced
Fig, 2. Behavior of the effective potential for T -135M in a flat spacetime at D - 3.3,3.5,3.7,3.9 for ji2 =0.1 M2 and
x=m*-ü.
0.002
> "0.002
®
cc
D=4.1
-0,004
-0,008
D=4,3 D=4.5,/ V "5=4.7.■
»=4.9
0
0.2
0.4
f/M
0.6
Fig. 3. Behavior of the effective potential for T = 1.35M in a flat spacetimeat D = 4.1,4.3,4.5,4.7,4.9 for p.2 =0AM1 and
X^M4^’.
by only a radiative correction [31]. The broken symmetry is restored by a larger scale factor, in other words, a smaller curvature.
The curvature effect to the symmetry breaking cornes from a curvature dependence of the covariant derivative and a coupling t, between the scalar field and the gravitational field. We clearly observe in Fig. 5, that the curvature effect can restore the broken symmetry without a coupling £,. The curvature effect for Ç = 0 is smaller than the one for a conformally coupled case.
Next we study the curvature effect in the negative curvature space R®HD~l at finite temperature, we fix the temperature above the critical one and see whether the Z, symmetry is broken down in an environment of the small scale factor a. In Figs. 6 and 7 we plot the
typical behavior of the effective potential at D- 3.5 as the scale factor a varies. As is seen in the figures, we 10-11
”7 1 T
/
a=290/M / a=300/M
05
cc
a=310/M /
-2x10
-11
0 0.004 0.008
(p/M
(a) ¡i2 =0, X = M^, T = 0.01 M <Tc
0.4 f/M
{b) n2 = 0 AM2, I = M *~D, T = 1.0M < T1;
Fig. 4. Behavior of the effective potential for a conformally coupled scalar field ç = (D~2)/(4i)-4) in R®S2'5 as a varies.
10
r11
a= 170/M \ a=175/M
©
oc
-10'
rii
a=180/M /
-2x10
■11
0
0.004 0.008
(p/M
(a) M-2 = 0, X = M*~D, T = 0.01 M <Tc
0.002
S. -0.001
-0.002
-0.003
(b) p2 = 0. IM2, X = M *~D, T = 1.0 M < Tc
Fig. 5. Behavior of the effective potential for a minimally coupled scalar field £, = 0 in R®S2S as a varies, observe that there is a second order phase transition and the Z2 symmetry is broken down as a is decreased. In the negative curvature case a larger scale factor breaks the symmetry for )i = 0. We see in Fig. 7, a negative curvature breaks the symmetry without a coupling E,.
to*11
5x10'1Z 0
■5x10'12 -10'11 -1.5x10'11
o_
>
DC
a=220/y /
a=2f()/M
a=200/M
0 0,002 0,004 0.006 0.008
f/M
(a) |i2 = 0, X = M4~D, T = 0.02M > Tc
V. Concluding Remarks
We have investigated the behavior of the effective potential at finite temperature and curvature in arbitrary dimensions 2 < D < 5 . We discuss the curvature and thermal effects to the effective potential
0.004
û 0
£E
-0.004
-0.008
^ Г r ~7 1 / / a=9/M / /
\ a=6/M
\ \
\ \ \ \ a=3/M /
- 1 1 _J . L
0.2
0.6
0.8
0.4 (p/M
(b) |i2 =().1M2, X--M ,T = 1.5 M > Tc Fig. 6. Behavior of the effective potential for a conformally coupled scalar field £ = (£>--2)/(4£>-4) in R®H2S as a varies,
10-ti —-T_
T
5x10'1:2 (-
o L
■ -5x10“12 (--10-1’ h 1.5x10'1 ! j-
a=115/M /
a=t 10/M
a= 105/ад
0,002 0.004 0.006 0.008
cp/M
(а) дг = 0, 'к~ M D'4, 7" = 0,02 M :>
at T = 0 and a —» » , we calculate the renormalized effective potential for finite T and a, The broken Z, symmetry is restored at a certain critical temperature and scale factor in R®SD~\ Above the critical temperature the restored symmetry is broken down again at a certain critical scale factor in R ®H°~l. The phase transition from the broken phase to the symmetric phase is of the second order for D = 3.5. The critical temperature and scale factor depend on the parameters of the theory and D.
The effective potential develops a non-vanishing imaginary part. If we take the other renormaliztion conditions,
Э1У±
Эф2
s-H,
эх
Эф4
(68)
ф/М
(b) = Q.1M2, 1 = M °-4, T = 1.5 A# > Tc
Pig, 7, Behavior of the effective potential for a minimally coupled scalar field % = 0 in R ® H2 S as a varies,
for both a minimally coupled and conformally coupled scalar fields.
Starting from the theory with broken Z2 symmetry
we can define a real effective potential for D>4.
Although the present work is restricted to the calculation of the effective potential, we are interested in applying our result to the full analysis of the phase structure and physical problems. A consequence of symmetry breaking may be found to study critical phenomena in the early stage of universe. It gives rise to a possibility that some cosmological observable show sings of symmetry breaking. The ph ■ • ■ ■ '• i
the evolution of the spacetir-'■ .ne structure depends on the gr - •
/ • ■ trough the expectation value ' ■ ;
1 dependence may change the i 3 potenti 1 7 ■ the spaeetii ;
■ the syr 1 breaking, 17 ■ ' of this ■ action we rrr :■ ■ ration and the gap equation si: 1,1
L 5 ” J*
The thermal effect may be significantly stronger in most of realistic situations at the early stage of the universe. The curvature effect may cause a nonstatic field configurations. Decreasing the temperature, spontaneous symmetry breaking may occur from the negative curvature place. There is a possibility to observe the combined effect of the temperature and curvature is a fluctuation of some fields. However, we cannot deal with the nonstatic configurations in the effective potential approach. For summing up contributions from different region we will need a new idea.
Acknowledgments
The authors would like to thank T, Fujihara and D. Kimura for useful discussions. We also thank the Yukawa Institute for Theoretical Physics at Kyoto University. Discussions during the YITP workshop YITP-W-04-07 on “Thermal Quantum Field Theories and Their Applications” were useful to complete this work.
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Moretti V,,1 Pinamonti NJ ., .
HOLOGRAPHY AND CONFORMAL SYMMETRY i- 4CÍÍ HOLE HORIZONS
Department of Mathematics, Faculty of Science, University of Trento, Istituto Nazionale di Alta Matematica “F.Severi", unitä locale di Trento & Istituto Nazionale di Fisica Nucleate, Gruppo Collegato di Trento, via Sommarive 14,1-38050
Rovo (TN), Italy
Maldacena [4] conjectured that the quantum field theory in a, asymptotically AdS, d + 1 dimensional spacetime (the “bulk”) is in correspondence with a. conformal theory in a d dimensional manifold (the (conformal) “boundary” at spacelike infinity). Notice that the d dimensional conformal group on the boundary acts as the asymptotic isometry group on the bulk. Afterwards, Witten [5] showed that that correspondence can be reset in terms of observables of the two theories. More recently Rehren [6.7] proved rigorously some holographic theorems concerning boundary and bulk observables in AdS background,
1. Introduction
In the last fifty years much work was done in order to understand the statistical origin of black-hole entropy. The Holographic principle, proposed for the first time by't Hooft and Susskind [1,2,3] is one of the most promising idea to deal with that problem. In few words the quantum theory responsible for the statistical black hole entropy should be suited on the event horizon, moreover it has to describe the events that take place in the spacetime. In some sense as a photograph describe a landscape. Starting from these ideas and using the machinery of string theory,
1 E-mail: moretti@science.unitn.it
2 E-mail: pinamont@science.unitn.it
