Scalar field theory at finite temperature in D = 2 + 1
G. N. J. AñañosCitation: 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
Scalar field theory at finite temperature in D = 2 + 1
G. N. J. Añañosa兲
Instituto de Física Teórica-IFT, Universidade Estadual Paulista, Rua Pamplona 145, São Paulo, SP 01405-900 Brazil
共Received 21 September 2005; accepted 8 November 2005; published online 17 January 2006兲
We discuss the6 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共Cornwall, Jackiw, and Tomboulis兲 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.关DOI:10.1063/1.2159068兴
I. INTRODUCTION
The conventional perturbation theory in the coupling constant or inប, i.e., the loop expansion can only be used for the study of small quantum corrections to classical results. When discussing quantum mechanical effects to any given order in such an expansion, one is not usually able to justify the neglect of yet higher order. In other words, for theories with a large N dimensional internal symmetry group, there exist another perturbation scheme, the 1 / N expansion, which circumvents this criticism. Each term in the 1 / N expansion contains an infinite subset of terms of the loop expansion. The 1 / N expansion has the nice property that the leading-order quantum corrections are of the same order as the classical quantities. Consequently, the leading order which adequately characterizes the theory in the large N limit preserves much of the nonlinear structure of the full theory.
The scalar field共4兲
D=4 theory at finite temperature is of great interest in the field of phase
transitions in the early universe and heavy ion collisions. When used as a simple model for the Higgs particle in the standard model of electroweak interactions, it may allow the study of sym-metry breaking phase transitions in the early universe. For N = 4 scalar fields, it is also a model of chiral symmetry breaking in QCD and hence is relevant for the theoretical study of heavy ion collisions. Moreover, this theory is an excellent theoretical laboratory in which analytic nonper-turbative methods can be tested. Now for the case D = 3 it has been shown that, in the large N limit, the6theory has a UV fixed point and therefore must have a second IR fixed point1
and for this we could say that at least for large N the共6兲
D=3theory is known to be qualitatively different
from 共4兲
D=4 theory. For other ways, theories in less than four space-time dimensions can offer
interesting and complex behavior as well as tractability, and, for example, the case of three space-time dimensions, they can even be directly physical, describing various planar condensed matter systems. For example, the introduction of the6term generates a rich phase diagram, with the possibility of second order, first order phase transitions or even both transitions occurring simultaneously. This situation defines the tricritical phenomenon. For example, some systems such antiferromagnets in the presence of a strong external field or the He3-He4 mixture exhibits such behavior.
a兲Electronic mail: gananos@ift.unesp.br
47, 012301-1
II. THE EFFECTIVE POTENTIAL
The theory for which we are interested is given by the Lagrangian,
L共兲 =1 2共兲 2−1 2m0 22− 0 4!N 4− 0 6!N2 6, 共1兲
is a theory of N real scalar fields with O共N兲 symmetry.
For definiteness, we work at zero temperature; however, the finite temperature generalizations can be easily obtained.2
In this section we are going to use the method of composite operator developed by Cornwall, Jackiw, and Tomboulis共CJT兲3,4in order to get the effective potential⌫共兲 at leading order in the 1 / N expansion. The composite operator formalism reduces the problem to summing two particle irreducible共2PI兲 Feynman graphs by defining a generalized effective action ⌫共, G兲 which is a functional not only of a共x兲, but also of the expectation values Gab共x,y兲 of the time ordered
product of quantum fields具0兩T共共x兲共y兲兲兩0典, i.e., ⌫共,G兲 = I共兲 + i
2Tr Ln G −1+ i
2Tr D
−1共兲G + ⌫2共,G兲 + ¯ , 共2兲 where I共兲=兰dxDL共兲, G and D are matrices in both the functional and the internal space whose
elements are Gab共x,y兲, Dab共; x , y兲, respectively, and D is defined by iD−1= ␦
2I共兲
␦共x兲␦共y兲. 共3兲
The quantity⌫2共, G兲 is computed as follows. In the classical action I共兲 we must shift the field
by. The new action I共+兲 possesses terms cubic and higher in. This defines an interaction part Iint共,兲 where the vertices depend on . ⌫2共, G兲 is given by sum of all 共2PI兲 vacuum graphs in a theory with vertices determined by Iint共,兲 and the propagators set equal to G共x,y兲. The trace and logarithm in Eq.共2兲 are functional. After these procedures the interaction Lagrang-ian density becomes
Lint共,兲 = − 1 2
冉
0a 3N + 02a 30N2冊
a2−冉
80abc 6N2冊
abc− 1 4!N冉
0+ 02 10N冊
4 −冉
120ab 6!N2冊
ab 2− 1 5!冉
0a N2 a 4冊
− 0 6!N2 6. 共4兲The effective action⌫共兲 is found by solving for Gab共x,y兲 the equation
␦⌫共,G兲
␦Gab共x,y兲
= 0, 共5兲
and substituting the solution in the generalized effective action⌫共, G兲. The vertices in the above equation contain factors of 1 / N or 1 / N2, but a loop gives a factor of N provided the O共N兲 isospin flows around it alone and not into another part of the graph. We usually call such loops bubbles. Then at leading order in 1 / N, the vacuum graphs are bubble trees with two or three bubbles at each vertex. The共2PI兲 graphs are shown in Fig. 1. It is straightforward to obtain
FIG. 1. The 2PI vacuum graphs.
⌫2共,G兲 = − 1 4!N
冕
d D x冉
0+0 2 10N冊
关Gaa共x,x兲兴 2− 0 6!N2冕
d D x关Gaa共x,x兲兴3. 共6兲Therefore Eq. 共5兲 becomes
␦⌫共,G兲 ␦Gab共x,y兲 =1 2共G −1兲 ab共x,y兲 + i 2D −1共兲 − 1 12N
冉
0+ 02 10N冊
关␦abGcc共x,x兲兴␦ D共x − y兲 −30 6!N␦ab关Gcc共x,x兲兴 2␦D共x − y兲 = 0. 共7兲Rewriting this equation, we obtain the gap equation 共G−1兲 ab共x,y兲 = Dab−1共;x,y兲 + i 6N
冉
0+ 02 10N冊
关␦abGcc共x,x兲兴␦ D共x − y兲 + i0 5!N2␦ab关Gcc共x,x兲兴 2␦D共x − y兲. 共8兲 Hence i 2Tr D −1G = 1 12N冕
d D x冉
0+ 0 2 10N冊
关Gaa共x,x兲兴 2+ 30 6!N2冕
d D x关Gaa共x,x兲兴3+ cte. 共9兲Using Eqs.共8兲 and 共9兲 in Eq. 共6兲 we find the effective action ⌫共兲 = I共兲 + i 2Tr关Ln G −1兴 + 1 4!N
冕
d D x冉
0+0 2 10N冊
关Gaa共x,x兲兴 2+ 20 6!N2冕
d D x关Gaa共x,x兲兴3, 共10兲 where Gabis given implicitly by Eq.共8兲. The trace in 共10兲 are both the functional and the internalspace. The last two terms on the right-hand side of Eq.共10兲 are the leading contribution to the effective action in the 1 / N expansion. As usual we may simplify the situation by separating Gab
into transverse and longitudinal components, so
Gab=
冉
␦ab−ab
2
冊
g +ab
2 ˜ ,g 共11兲
in this form we can invert Gab,
共G兲ab−1=
冉
␦ab− ab 2冊
g −1+ab 2 ˜g −1. 共12兲Now we can take the trace with respect to the indices of the internal space,
Gaa= Ng + O共1兲, 共G兲aa−1= Ng−1+ O共1兲. 共13兲
From this equation at leading order in 1 / N, Gabis diagonal in a , b. Substituting Eq.共13兲 into Eq.
共10兲 and Eq. 共8兲 and keeping only the leading order one finds that the daisy and superdaisy resumed effective potential for the6 theory is given by
⌫共兲 = I共兲 +iN 2tr共ln g −1兲 + N 4!
冕
d D x冉
0+0 2 10N冊
g 2共x,x兲 +2N0 6!冕
d D xg3共x,x兲 + O共1兲, 共14兲 where the trace is only in the functional space, and the gap equation becomesg−1共x,y兲 = i
冋
씲 + m02+0 6冉
2 N + g共x,x兲冊
+ 0 5!冉
2 N + g共x,x兲冊
2册
␦D共x − y兲 + O冉
1 N冊
. 共15兲It is important to point out that this calculation was done by Townsend.5
III. THE THEORY AT FINITE TEMPERATURE
In order to generalize these results to the case of finite temperature we are going to assume that the system is in equilibrium with a thermal bath a temperature T =−1. Since we are interested in the equilibrium situation it is convenient to use the Matsubara formalism 共imaginary time兲. Consequently it is convenient to continue all momenta to Euclidean values共p0= ip4兲 and take the following Ansatz for g共x,y兲:
g共x,y兲 =
冕
d D p 共2兲D expi共x−y兲p p2+ M2共兲. 共16兲 Substituting Eq.共16兲 in Eq. 共15兲 we get the expression for the gap equation,M2共兲 = m02+0 6
冉
2 N + F共兲冊
+ 0 5!冉
2 N + F共兲冊
2 , 共17兲 where F共兲 is given by F共兲 =冕
d D p 共2兲D 1 p2+ M2共兲, 共18兲 and the effective potential in the D-dimensional Euclidean space can be written asV共兲 = V0共兲 +N 2
冕
dDp 共2兲Dln关p 2+ M2共兲兴 − N 4!冉
0+ 02 10N冊
F共兲 2−2N0F共兲3 6! , 共19兲where V0共兲 is the tree approximation of the potential.
Replacing the continuous four momenta k4by discretenand the integration by a summation
共= 1 / T兲. The effective potential at finite temperature is
V共兲 = V0共兲 + N 2
兺
n ⬁冕
dD−1 共2兲D−1ln关n+ p2+ M2共兲兴 − N 4!冉
0+ 02 10N冊
F共兲 2−2NF共兲 3 6! , 共20兲 where the expression F共兲 is the finite temperature generalization of F共兲, and is given byF共兲 =1 n=−
兺
⬁ ⬁冕
dD−1p 共2兲D−1 1 n 2 + p2+ M2共兲. 共21兲 The gap equation for this theory at finite temperature isM2共兲 = m02+0 6
冉
2 N + F共兲冊
+ 0 5!冉
2 N + F共兲冊
2 . 共22兲In order to regularize F共兲 given by Eq. 共21兲, we use a mixing between dimensional regulariza-tion and analytic regularizaregulariza-tion. For this purpose we define the following expression:
I共D,s,m兲 = 1 n=−
兺
⬁ ⬁冕
dD−1k 共2兲D−1 1 共n 2 + k2+ m2兲s. 共23兲The analytic extension of the inhomogeneous Epstein zeta function can be done and the corre-sponding analytic extension of I共D,s,m兲 is
I共D,s,m兲 = m D−2s 共21/2兲D⌫共s兲
冋
⌫冉
s − D 2冊
+ 4兺
n=1 ⬁冉
2 mn冊
D/2−s KD/2−s共mn兲册
, 共24兲where K共z兲 is the modified Bessel function of the third kind. Fortunately for D=2+1 the analytic extension of the function I共D,s=1,m=M共兲兲=F共兲 is finite and can be expressed in a closed form6
F共兲 = I共3,1,M共兲兲 = −M共兲
4
冉
1 +2 ln共1 − e−M共兲兲
M共兲
冊
. 共25兲We note that in D = 2 + 1 we have no pole, at least in this approximation. To proceed to regularize the second term of Eq.共20兲, we define
LF共兲 =1 
兺
n=1 ⬁冕
dD−1p 共2兲D−1ln关n+ p2+ M2共兲兴 共26兲 then, LF共兲 M =共2M兲 1 兺
n=1 ⬁冕
dD−1p 共2兲D−1 1 n+ p2+ M2共兲 共27兲 and from Eq.共21兲, we have thatLF共兲
M =共2M兲F共兲. 共28兲
For D = 2 + 1, F共兲 is finite and is given by Eq. 共29兲 共Ref. 6兲 and integrating the Eq. 共28兲, we obtain LF共兲R= − M共兲3 6 − M共兲Li2共e−M共兲兲 2 − Li3共e−M共兲兲 3 . 共29兲
The definition of general polylogarithm function Lin共z兲 can be found in Ref. 7.
The daisy and super daisy resummed effective potential at finite temperature for D = 2 + 1 is then given by V共兲 = V0共兲 +N 2LF共兲R− N 4!
冉
0+ 02 10N冊
共F共兲R兲 2−2N共F共兲R兲2 6! 共30兲and the corresponding gap equation关see Eq. 共22兲兴
M2共兲 = m02+0 6
冉
2 N − M共兲 4冋
1 + 2 ln共1 − e−M共兲兲 M共兲册
冊
+0 5!冉
2 N − M共兲 4冋
1 + 2 ln共1 − e−M共兲兲 M共兲册
冊
2 . 共31兲From this expression we can deduce that there is no possible way to find a solution for M going to zero, because the terms in Eq.共31兲 containing the logarithm will not permit, and this situation is similar to the scalar theory with O共N兲 symmetry in two dimensions 共2D兲 at zero temperature.8
This result is in agreement with the the Coleman-Mermin-Wagner theorem,9 which statement is related to the fact that it is impossible to construct a consistent theory of massless scalar in 2D. If a spontaneous breaking of continuous symmetry were to happen at finite T, then one would be faced with this problem at momentum scales below Tc, i.e., it would be impossible to construct an
effective 2D theory of the Goldstone bosons zero modes.
IV. CONCLUSIONS
In this paper we have found the daisy and super daisy effective potential for the theory6in
D = 2 + 1-dimensional Euclidean space at finite temperature. The form of effective potential have
been found explicitly using resummation method in the leading order 1 / N approximation 共Hartree-Fock approximation兲. We found that in this approximation there is no symmetry breaking for any temperature and this is clearly a manifestation of the Coleman-Mermin-Wagner theorem which stipulates that the spontaneous symmetry breaking of continuous symmetry cannot happen in D = 2 + 1 at finite temperature.
ACKNOWLEDGMENT
This paper was supported by FAPESP.
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