We discuss the $\phi^6 $ theory defined in $D=2+1$-dimensional space-time and
assume that the system is in equilibrium with a thermal bath at temperature
$\beta^{-1}$. We use the $ 1/N $ expansion and the method of the composite
operator (CJT) for summing a large set of Feynman graphs.We demonstrate
explicitly the Coleman-Mermin-Wagner theorem at finite temperature.Comment: 12 pages, 1 figure. To be published in Journal Mathematical Physics,
  typos adde

G. N. J. Añaños

Jackiw R.

Lewin L.

English

arXiv

Gino N. J. Ananos

Crossref

Scalar field theory at finite temperature in D = 2 + 1
G. N. J. Añaños 
 
Citation: Journal of Mathematical Physics 47, 012301 (2006); doi: 10.1063/1.2159068 
View online: http://dx.doi.org/10.1063/1.2159068 
View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/47/1?ver=pdfcov 
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JOURNAL OF MATHEMATICAL PHYSICS 47, 012301 2006
0
 This article is copyrightcalar field theory at finite temperature in D=2+1
G. N. J. Añañosa
Instituto de Física Teórica-IFT, Universidade Estadual Paulista, Rua Pamplona 145,
São Paulo, SP 01405-900 Brazil
Received 21 September 2005; accepted 8 November 2005;
published online 17 January 2006
We discuss the 6 theory defined in D=2+1-dimensional space-time and assume
that the system is in equilibrium with a thermal bath at temperature −1. We use the
1/N expansion and the method of the composite operator Cornwall, Jackiw, and
Tomboulis for summing a large set of Feynman graphs. We demonstrate explicitly
the Coleman-Mermin-Wagner theorem at finite temperature. © 2006 American In-
stitute of Physics. DOI: 10.1063/1.2159068
. INTRODUCTION
The conventional perturbation theory in the coupling constant or in , i.e., the loop expansion
an only be used for the study of small quantum corrections to classical results. When discussing
uantum mechanical effects to any given order in such an expansion, one is not usually able to
ustify the neglect of yet higher order. In other words, for theories with a large N dimensional
nternal symmetry group, there exist another perturbation scheme, the 1/N expansion, which
ircumvents this criticism. Each term in the 1/N expansion contains an infinite subset of terms of
he loop expansion. The 1/N expansion has the nice property that the leading-order quantum
orrections are of the same order as the classical quantities. Consequently, the leading order which
dequately characterizes the theory in the large N limit preserves much of the nonlinear structure
f the full theory.
The scalar field 4D=4 theory at finite temperature is of great interest in the field of phase
ransitions in the early universe and heavy ion collisions. When used as a simple model for the
iggs particle in the standard model of electroweak interactions, it may allow the study of sym-
etry breaking phase transitions in the early universe. For N=4 scalar fields, it is also a model of
hiral symmetry breaking in QCD and hence is relevant for the theoretical study of heavy ion
ollisions. Moreover, this theory is an excellent theoretical laboratory in which analytic nonper-
urbative methods can be tested. Now for the case D=3 it has been shown that, in the large N
imit, the 6 theory has a UV fixed point and therefore must have a second IR fixed point1 and for
his we could say that at least for large N the 6D=3 theory is known to be qualitatively different
rom 4D=4 theory. For other ways, theories in less than four space-time dimensions can offer
nteresting and complex behavior as well as tractability, and, for example, the case of three
pace-time dimensions, they can even be directly physical, describing various planar condensed
atter systems. For example, the introduction of the 6 term generates a rich phase diagram, with
he possibility of second order, first order phase transitions or even both transitions occurring
imultaneously. This situation defines the tricritical phenomenon. For example, some systems such
ntiferromagnets in the presence of a strong external field or the He3-He4 mixture exhibits such
ehavior.
Electronic mail: gananos@ift.unesp.br
47, 012301-1022-2488/2006/471/012301/6/$23.00 © 2006 American Institute of Physics
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012301-2 G. N. J. Añaños J. Math. Phys. 47, 012301 2006
 This article is copyrightI. THE EFFECTIVE POTENTIAL
The theory for which we are interested is given by the Lagrangian,
L =
1
2
2 −
1
2
m0
22 −
0
4!N
4 −
0
6!N2
6, 1
s a theory of N real scalar fields with ON symmetry.
For definiteness, we work at zero temperature; however, the finite temperature generalizations
an be easily obtained.2
In this section we are going to use the method of composite operator developed by Cornwall,
ackiw, and Tomboulis CJT3,4 in order to get the effective potential  at leading order in the
/N expansion. The composite operator formalism reduces the problem to summing two particle
rreducible 2PI Feynman graphs by defining a generalized effective action  ,G which is a
unctional not only of ax, but also of the expectation values Gabx ,y of the time ordered
roduct of quantum fields 0Txy0, i.e.,
,G = I +
i
2
Tr Ln G−1 +
i
2
Tr D−1G + 2,G + ¯ , 2
here I=dxD L, G and D are matrices in both the functional and the internal space whose
lements are Gabx ,y, Dab ;x ,y, respectively, and D is defined by
iD−1 =
	2I
	x	y
. 3
he quantity 2 ,G is computed as follows. In the classical action I we must shift the field
by . The new action I+ possesses terms cubic and higher in . This defines an interaction
art Iint , where the vertices depend on . 2 ,G is given by sum of all 2PI vacuum
raphs in a theory with vertices determined by Iint , and the propagators set equal to Gx ,y.
he trace and logarithm in Eq. 2 are functional. After these procedures the interaction Lagrang-
an density becomes
Lint, = −
1
2
	0a
3N
+
0
2a
30N2

a2 − 	80abc6N2 
abc − 14!N	0 + 0
2
10N

4
− 	120ab
6!N2

ab2 − 15!	0aN2 a4
 − 06!N26. 4
he effective action  is found by solving for Gabx ,y the equation
	,G
	Gabx,y
= 0, 5
nd substituting the solution in the generalized effective action  ,G. The vertices in the above
quation contain factors of 1 /N or 1/N2, but a  loop gives a factor of N provided the ON
sospin flows around it alone and not into another part of the graph. We usually call such loops
ubbles. Then at leading order in 1/N, the vacuum graphs are bubble trees with two or three
FIG. 1. The 2PI vacuum graphs.ubbles at each vertex. The 2PI graphs are shown in Fig. 1. It is straightforward to obtain
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012301-3 Scalar field theory at finite temperature in D=2+1 J. Math. Phys. 47, 012301 2006
 This article is copyright2,G =
− 1
4!N
 dDx	0 + 0210N 
Gaax,x2 − 06!N2  dDxGaax,x3. 6
Therefore Eq. 5 becomes
	,G
	Gabx,y
=
1
2
G−1abx,y +
i
2
D−1 −
1
12N
	0 + 0210N 
	abGccx,x	Dx − y
−
30
6!N
	abGccx,x2	Dx − y = 0. 7
ewriting this equation, we obtain the gap equation
G−1abx,y = Dab
−1;x,y +
i
6N
	0 + 0210N 
	abGccx,x	Dx − y
+
i0
5!N2
	abGccx,x2	Dx − y . 8
ence
i
2
Tr D−1G =
1
12N
 dDx	0 + 0210N 
Gaax,x2 + 306!N2  dDxGaax,x3 + cte. 9
sing Eqs. 8 and 9 in Eq. 6 we find the effective action
 = I +
i
2
TrLn G−1 +
1
4!N
 dDx	0 + 0210N 
Gaax,x2 + 206!N2  dDxGaax,x3,
10
here Gab is given implicitly by Eq. 8. The trace in 10 are both the functional and the internal
pace. The last two terms on the right-hand side of Eq. 10 are the leading contribution to the
ffective action in the 1/N expansion. As usual we may simplify the situation by separating Gab
nto transverse and longitudinal components, so
Gab = 		ab − ab
2

g + ab
2
g̃ , 11
n this form we can invert Gab,
Gab
−1 = 		ab − ab
2

g−1 + ab
2
g̃−1. 12
Now we can take the trace with respect to the indices of the internal space,
Gaa = Ng + O1, Gaa
−1 = Ng−1 + O1 . 13
rom this equation at leading order in 1/N, Gab is diagonal in a ,b. Substituting Eq. 13 into Eq.
10 and Eq. 8 and keeping only the leading order one finds that the daisy and superdaisy
esumed effective potential for the 6 theory is given by
 = I +
iN
2
trln g−1 +
N
4!
 dDx	0 + 0210N 
g2x,x + 2N06!  dDxg3x,x + O1 ,
14here the trace is only in the functional space, and the gap equation becomes
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012301-4 G. N. J. Añaños J. Math. Phys. 47, 012301 2006
 This article is copyrightg−1x,y = i + m02 + 06 	2N + gx,x
 + 05! 	2N + gx,x
2	Dx − y + O	 1N
 . 15
t is important to point out that this calculation was done by Townsend.5
II. THE THEORY AT FINITE TEMPERATURE
In order to generalize these results to the case of finite temperature we are going to assume
hat the system is in equilibrium with a thermal bath a temperature T=−1. Since we are interested
n the equilibrium situation it is convenient to use the Matsubara formalism imaginary time.
onsequently it is convenient to continue all momenta to Euclidean values p0= ip4 and take the
ollowing Ansatz for gx ,y:
gx,y = dDp2
D exp
ix−yp
p2 + M2
. 16
ubstituting Eq. 16 in Eq. 15 we get the expression for the gap equation,
M2 = m0
2 +
0
6
	2
N
+ F
 + 0
5!
	2
N
+ F
2, 17
here F is given by
F = dDp2
D 1p2 + M2 , 18
nd the effective potential in the D-dimensional Euclidean space can be written as
V = V0 +
N
2
 dDp2
D lnp2 + M2 − N4!	0 + 0
2
10N

F2 − 2N0F3
6!
, 19
here V0 is the tree approximation of the potential.
Replacing the continuous four momenta k4 by discrete n and the integration by a summation
=1/T. The effective potential at finite temperature is
V = V0 +
N
2n
  dD−12
D−1 lnn + p2 + M2 − N4!	0 + 0
2
10N

F2 − 2NF36! ,
20
here the expression F is the finite temperature generalization of F, and is given by
F =
1


n=−
  dD−1p2
D−1 1n2 + p2 + M2 . 21
he gap equation for this theory at finite temperature is
M
2 = m0
2 +
0
6
	2
N
+ F
 + 05! 	
2
N
+ F
2. 22
n order to regularize F given by Eq. 21, we use a mixing between dimensional regulariza-
ion and analytic regularization. For this purpose we define the following expression:
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012301-5 Scalar field theory at finite temperature in D=2+1 J. Math. Phys. 47, 012301 2006
 This article is copyrightID,s,m =
1


n=−
  dD−1k2
D−1 1n2 + k2 + m2s . 23
he analytic extension of the inhomogeneous Epstein zeta function can be done and the corre-
ponding analytic extension of ID ,s ,m is
ID,s,m =
mD−2s
2
1/2Ds	s − D2 
 + 4n=1
 	 2
mn

D/2−sKD/2−smn , 24
here Kz is the modified Bessel function of the third kind. Fortunately for D=2+1 the analytic
xtension of the function ID ,s=1,m=M=F is finite and can be expressed in a closed
orm6
F = I3,1,M = −
M
4

	1 + 2 ln1 − e−M
M

 . 25
e note that in D=2+1 we have no pole, at least in this approximation. To proceed to regularize
he second term of Eq. 20, we define
LF =
1


n=1
  dD−1p2
D−1 lnn + p2 + M2 26
hen,
LF
M
= 2M
1


n=1
  dD−1p2
D−1 1n + p2 + M2 27
nd from Eq. 21, we have that
LF
M
= 2MF . 28
or D=2+1, F is finite and is given by Eq. 29 Ref. 6 and integrating the Eq. 28, we
btain
LFR = −
M3
6

−
MLi2e−M

2
−
Li3e−M

3
. 29
The definition of general polylogarithm function Linz can be found in Ref. 7.
The daisy and super daisy resummed effective potential at finite temperature for D=2+1 is
hen given by
V = V0 +
N
2
LFR −
N
4!
	0 + 0210N 
FR2 − 2NFR
2
6!
30
nd the corresponding gap equation see Eq. 22
M
2 = m0
2 +
0
6
	2
N
−
M
4

1 + 2 ln1 − e−M
M


+
0
5!
	2
N
−
M
4

1 + 2 ln1 − e−M
M

2. 31
rom this expression we can deduce that there is no possible way to find a solution for M going
o zero, because the terms in Eq. 31 containing the logarithm will not permit, and this situation
8s similar to the scalar theory with ON symmetry in two dimensions 2D at zero temperature.
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012301-6 G. N. J. Añaños J. Math. Phys. 47, 012301 2006
 This article is copyrighthis result is in agreement with the the Coleman-Mermin-Wagner theorem,9 which statement is
elated to the fact that it is impossible to construct a consistent theory of massless scalar in 2D. If
spontaneous breaking of continuous symmetry were to happen at finite T, then one would be
aced with this problem at momentum scales below Tc, i.e., it would be impossible to construct an
ffective 2D theory of the Goldstone bosons zero modes.
V. CONCLUSIONS
In this paper we have found the daisy and super daisy effective potential for the theory 6 in
=2+1-dimensional Euclidean space at finite temperature. The form of effective potential have
een found explicitly using resummation method in the leading order 1 /N approximation
Hartree-Fock approximation. We found that in this approximation there is no symmetry breaking
or any temperature and this is clearly a manifestation of the Coleman-Mermin-Wagner theorem
hich stipulates that the spontaneous symmetry breaking of continuous symmetry cannot happen
n D=2+1 at finite temperature.
CKNOWLEDGMENT
This paper was supported by FAPESP.
1 W. A. Barden, M. Moshe, and M. Bander, Phys. Rev. Lett. 52, 1118 1984.
2 L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 1974.
3 J. M. Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev. D 10, 2428 1974.
4 R. Jackiw, Diverses Topics in Theoretical and Mathematical Physics World Scientific, Singapore, 1995.
5 P. K. Townsend, Phys. Rev. D 12, 2269 1975; Nucl. Phys. B 118, 199 1977.
6 G. N. J. Ananos and N. F. Svaiter, Physica A 241, 627 1997.
7 L. Lewin, Polylogarithms and Associated Functions North-Holland, Amsterdam, 1981; http://mathworld.wolfram.com/
Polylogarithm.html
8 S. R. Coleman, Phys. Rev. D 10, 2491 1974.
9 N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 1966; S. R. Coleman, Commun. Math. Phys. 31, 259 1973.ed as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:
186.217.234.225 On: Tue, 14 Jan 2014 15:18:19


Scalar Field Theory at Finite Temperature in D=2+1

We discuss the phi(6) theory defined in D = 2 + 1-dimensional space-time and assume that the system is in equilibrium with a thermal bath at temperature beta(-1). We use the 1/N expansion and the method of composite operator (CJT) for summing a large set of Feynman graphs. We demonstrate explicitly the Coleman-Mermin-Wagner theorem at finite temperature.Univ Estadual Paulista, Inst Fis Teor, BR-01405900 São Paulo, BrazilUniv Estadual Paulista, Inst Fis Teor, BR-01405900 São Paulo, Brazi

Ananos, G. N. J. [UNESP]

Repositório Institucional UNESP

Scalar field theory at finite temperature in D=2+1

Scalar field theory at finite temperature in D = 2 + 1G. N. J. Añaños  Citation: Journal of Mathematical Physics 47, 012301 (2006); doi: 10.1063/1.2159068 View online: http://dx.doi.org/10.1063/1.2159068 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/47/1?ver=pdfcov Published by the AIP Publishing                      This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:186.217.234.225 On: Tue, 14 Jan 2014 15:18:19SIcqjictcaotHmcctltfismtsabaJOURNAL OF MATHEMATICAL PHYSICS 47, 012301 20060 This article is copyrightcalar field theory at finite temperature in D=2+1G. N. J. AñañosaInstituto de Física Teórica-IFT, Universidade Estadual Paulista, Rua Pamplona 145,São Paulo, SP 01405-900 BrazilReceived 21 September 2005; accepted 8 November 2005;published online 17 January 2006We discuss the 6 theory defined in D=2+1-dimensional space-time and assumethat the system is in equilibrium with a thermal bath at temperature −1. We use the1/N expansion and the method of the composite operator Cornwall, Jackiw, andTomboulis for summing a large set of Feynman graphs. We demonstrate explicitlythe Coleman-Mermin-Wagner theorem at finite temperature. © 2006 American In-stitute of Physics. DOI: 10.1063/1.2159068. INTRODUCTIONThe conventional perturbation theory in the coupling constant or in , i.e., the loop expansionan only be used for the study of small quantum corrections to classical results. When discussinguantum mechanical effects to any given order in such an expansion, one is not usually able toustify the neglect of yet higher order. In other words, for theories with a large N dimensionalnternal symmetry group, there exist another perturbation scheme, the 1/N expansion, whichircumvents this criticism. Each term in the 1/N expansion contains an infinite subset of terms ofhe loop expansion. The 1/N expansion has the nice property that the leading-order quantumorrections are of the same order as the classical quantities. Consequently, the leading order whichdequately characterizes the theory in the large N limit preserves much of the nonlinear structuref the full theory.The scalar field 4D=4 theory at finite temperature is of great interest in the field of phaseransitions in the early universe and heavy ion collisions. When used as a simple model for theiggs particle in the standard model of electroweak interactions, it may allow the study of sym-etry breaking phase transitions in the early universe. For N=4 scalar fields, it is also a model ofhiral symmetry breaking in QCD and hence is relevant for the theoretical study of heavy ionollisions. Moreover, this theory is an excellent theoretical laboratory in which analytic nonper-urbative methods can be tested. Now for the case D=3 it has been shown that, in the large Nimit, the 6 theory has a UV fixed point and therefore must have a second IR fixed point1 and forhis we could say that at least for large N the 6D=3 theory is known to be qualitatively differentrom 4D=4 theory. For other ways, theories in less than four space-time dimensions can offernteresting and complex behavior as well as tractability, and, for example, the case of threepace-time dimensions, they can even be directly physical, describing various planar condensedatter systems. For example, the introduction of the 6 term generates a rich phase diagram, withhe possibility of second order, first order phase transitions or even both transitions occurringimultaneously. This situation defines the tricritical phenomenon. For example, some systems suchntiferromagnets in the presence of a strong external field or the He3-He4 mixture exhibits suchehavior.Electronic mail: gananos@ift.unesp.br47, 012301-1022-2488/2006/471/012301/6/$23.00 © 2006 American Institute of Physicsed as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:186.217.234.225 On: Tue, 14 Jan 2014 15:18:19IicJ1ifpweTpgTiTaeibb012301-2 G. N. J. Añaños J. Math. Phys. 47, 012301 2006 This article is copyrightI. THE EFFECTIVE POTENTIALThe theory for which we are interested is given by the Lagrangian,L =122 −12m022 −04!N4 −06!N26, 1s a theory of N real scalar fields with ON symmetry.For definiteness, we work at zero temperature; however, the finite temperature generalizationsan be easily obtained.2In this section we are going to use the method of composite operator developed by Cornwall,ackiw, and Tomboulis CJT3,4 in order to get the effective potential  at leading order in the/N expansion. The composite operator formalism reduces the problem to summing two particlerreducible 2PI Feynman graphs by defining a generalized effective action  ,G which is aunctional not only of ax, but also of the expectation values Gabx ,y of the time orderedroduct of quantum fields 0Txy0, i.e.,,G = I +i2Tr Ln G−1 +i2Tr D−1G + 2,G + ¯ , 2here I=dxD L, G and D are matrices in both the functional and the internal space whoselements are Gabx ,y, Dab ;x ,y, respectively, and D is defined byiD−1 =	2I	x	y. 3he quantity 2 ,G is computed as follows. In the classical action I we must shift the fieldby . The new action I+ possesses terms cubic and higher in . This defines an interactionart Iint , where the vertices depend on . 2 ,G is given by sum of all 2PI vacuumraphs in a theory with vertices determined by Iint , and the propagators set equal to Gx ,y.he trace and logarithm in Eq. 2 are functional. After these procedures the interaction Lagrang-an density becomesLint, = −12	0a3N+02a30N2a2 − 	80abc6N2 abc − 14!N	0 + 0210N4− 	120ab6!N2ab2 − 15!	0aN2 a4 − 06!N26. 4he effective action  is found by solving for Gabx ,y the equation	,G	Gabx,y= 0, 5nd substituting the solution in the generalized effective action  ,G. The vertices in the abovequation contain factors of 1 /N or 1/N2, but a  loop gives a factor of N provided the ONsospin flows around it alone and not into another part of the graph. We usually call such loopsubbles. Then at leading order in 1/N, the vacuum graphs are bubble trees with two or threeFIG. 1. The 2PI vacuum graphs.ubbles at each vertex. The 2PI graphs are shown in Fig. 1. It is straightforward to obtained as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:186.217.234.225 On: Tue, 14 Jan 2014 15:18:19RHUwseiiFrw012301-3 Scalar field theory at finite temperature in D=2+1 J. Math. Phys. 47, 012301 2006 This article is copyright2,G =− 14!N dDx	0 + 0210N Gaax,x2 − 06!N2  dDxGaax,x3. 6Therefore Eq. 5 becomes	,G	Gabx,y=12G−1abx,y +i2D−1 −112N	0 + 0210N 	abGccx,x	Dx − y−306!N	abGccx,x2	Dx − y = 0. 7ewriting this equation, we obtain the gap equationG−1abx,y = Dab−1;x,y +i6N	0 + 0210N 	abGccx,x	Dx − y+i05!N2	abGccx,x2	Dx − y . 8encei2Tr D−1G =112N dDx	0 + 0210N Gaax,x2 + 306!N2  dDxGaax,x3 + cte. 9sing Eqs. 8 and 9 in Eq. 6 we find the effective action = I +i2TrLn G−1 +14!N dDx	0 + 0210N Gaax,x2 + 206!N2  dDxGaax,x3,10here Gab is given implicitly by Eq. 8. The trace in 10 are both the functional and the internalpace. The last two terms on the right-hand side of Eq. 10 are the leading contribution to theffective action in the 1/N expansion. As usual we may simplify the situation by separating Gabnto transverse and longitudinal components, soGab = 		ab − ab2g + ab2g̃ , 11n this form we can invert Gab,Gab−1 = 		ab − ab2g−1 + ab2g̃−1. 12Now we can take the trace with respect to the indices of the internal space,Gaa = Ng + O1, Gaa−1 = Ng−1 + O1 . 13rom this equation at leading order in 1/N, Gab is diagonal in a ,b. Substituting Eq. 13 into Eq.10 and Eq. 8 and keeping only the leading order one finds that the daisy and superdaisyesumed effective potential for the 6 theory is given by = I +iN2trln g−1 +N4! dDx	0 + 0210N g2x,x + 2N06!  dDxg3x,x + O1 ,14here the trace is only in the functional space, and the gap equation becomesed as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:186.217.234.225 On: Tue, 14 Jan 2014 15:18:19IItiCfSwawwTIt012301-4 G. N. J. Añaños J. Math. Phys. 47, 012301 2006 This article is copyrightg−1x,y = i + m02 + 06 	2N + gx,x + 05! 	2N + gx,x2	Dx − y + O	 1N . 15t is important to point out that this calculation was done by Townsend.5II. THE THEORY AT FINITE TEMPERATUREIn order to generalize these results to the case of finite temperature we are going to assumehat the system is in equilibrium with a thermal bath a temperature T=−1. Since we are interestedn the equilibrium situation it is convenient to use the Matsubara formalism imaginary time.onsequently it is convenient to continue all momenta to Euclidean values p0= ip4 and take theollowing Ansatz for gx ,y:gx,y = dDp2D expix−ypp2 + M2. 16ubstituting Eq. 16 in Eq. 15 we get the expression for the gap equation,M2 = m02 +06	2N+ F + 05!	2N+ F2, 17here F is given byF = dDp2D 1p2 + M2 , 18nd the effective potential in the D-dimensional Euclidean space can be written asV = V0 +N2 dDp2D lnp2 + M2 − N4!	0 + 0210NF2 − 2N0F36!, 19here V0 is the tree approximation of the potential.Replacing the continuous four momenta k4 by discrete n and the integration by a summation=1/T. The effective potential at finite temperature isV = V0 +N2n  dD−12D−1 lnn + p2 + M2 − N4!	0 + 0210NF2 − 2NF36! ,20here the expression F is the finite temperature generalization of F, and is given byF =1n=−  dD−1p2D−1 1n2 + p2 + M2 . 21he gap equation for this theory at finite temperature isM2 = m02 +06	2N+ F + 05! 	2N+ F2. 22n order to regularize F given by Eq. 21, we use a mixing between dimensional regulariza-ion and analytic regularization. For this purpose we define the following expression:ed as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:186.217.234.225 On: Tue, 14 Jan 2014 15:18:19TswefWttaFotaFti012301-5 Scalar field theory at finite temperature in D=2+1 J. Math. Phys. 47, 012301 2006 This article is copyrightID,s,m =1n=−  dD−1k2D−1 1n2 + k2 + m2s . 23he analytic extension of the inhomogeneous Epstein zeta function can be done and the corre-ponding analytic extension of ID ,s ,m isID,s,m =mD−2s21/2Ds	s − D2  + 4n=1 	 2mnD/2−sKD/2−smn , 24here Kz is the modified Bessel function of the third kind. Fortunately for D=2+1 the analyticxtension of the function ID ,s=1,m=M=F is finite and can be expressed in a closedorm6F = I3,1,M = −M4	1 + 2 ln1 − e−MM . 25e note that in D=2+1 we have no pole, at least in this approximation. To proceed to regularizehe second term of Eq. 20, we defineLF =1n=1  dD−1p2D−1 lnn + p2 + M2 26hen,LFM= 2M1n=1  dD−1p2D−1 1n + p2 + M2 27nd from Eq. 21, we have thatLFM= 2MF . 28or D=2+1, F is finite and is given by Eq. 29 Ref. 6 and integrating the Eq. 28, webtainLFR = −M36−MLi2e−M2−Li3e−M3. 29The definition of general polylogarithm function Linz can be found in Ref. 7.The daisy and super daisy resummed effective potential at finite temperature for D=2+1 ishen given byV = V0 +N2LFR −N4!	0 + 0210N FR2 − 2NFR26!30nd the corresponding gap equation see Eq. 22M2 = m02 +06	2N−M41 + 2 ln1 − e−MM+05!	2N−M41 + 2 ln1 − e−MM2. 31rom this expression we can deduce that there is no possible way to find a solution for M goingo zero, because the terms in Eq. 31 containing the logarithm will not permit, and this situation8s similar to the scalar theory with ON symmetry in two dimensions 2D at zero temperature.ed as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:186.217.234.225 On: Tue, 14 Jan 2014 15:18:19TrafeIDbfwiA012301-6 G. N. J. Añaños J. Math. Phys. 47, 012301 2006 This article is copyrighthis result is in agreement with the the Coleman-Mermin-Wagner theorem,9 which statement iselated to the fact that it is impossible to construct a consistent theory of massless scalar in 2D. Ifspontaneous breaking of continuous symmetry were to happen at finite T, then one would beaced with this problem at momentum scales below Tc, i.e., it would be impossible to construct anffective 2D theory of the Goldstone bosons zero modes.V. CONCLUSIONSIn this paper we have found the daisy and super daisy effective potential for the theory 6 in=2+1-dimensional Euclidean space at finite temperature. The form of effective potential haveeen found explicitly using resummation method in the leading order 1 /N approximationHartree-Fock approximation. We found that in this approximation there is no symmetry breakingor any temperature and this is clearly a manifestation of the Coleman-Mermin-Wagner theoremhich stipulates that the spontaneous symmetry breaking of continuous symmetry cannot happenn D=2+1 at finite temperature.CKNOWLEDGMENTThis paper was supported by FAPESP.1 W. A. Barden, M. Moshe, and M. Bander, Phys. Rev. Lett. 52, 1118 1984.2 L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 1974.3 J. M. Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev. D 10, 2428 1974.4 R. Jackiw, Diverses Topics in Theoretical and Mathematical Physics World Scientific, Singapore, 1995.5 P. K. Townsend, Phys. Rev. D 12, 2269 1975; Nucl. Phys. B 118, 199 1977.6 G. N. J. Ananos and N. F. Svaiter, Physica A 241, 627 1997.7 L. Lewin, Polylogarithms and Associated Functions North-Holland, Amsterdam, 1981; http://mathworld.wolfram.com/Polylogarithm.html8 S. R. Coleman, Phys. Rev. D 10, 2491 1974.9 N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 1966; S. R. Coleman, Commun. Math. Phys. 31, 259 1973.ed as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:186.217.234.225 On: Tue, 14 Jan 2014 15:18:19

We discuss the phi(6) theory defined in D = 2 + 1-dimensional space-time and assume that the system is in equilibrium with a thermal bath at temperature beta(-1). We use the 1/N expansion and the method of composite operator (CJT) for summing a large set of Feynman graphs. We demonstrate explicitly the Coleman-Mermin-Wagner theorem at finite temperature

Scalar Field Theory at Finite Temperature in D=2+1

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