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 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. 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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 ��4�D=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 ��6�D=3 theory is known to be qualitatively different rom ��4�D=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, 012301-1022-2488/2006/47�1�/012301/6/$23.00 © 2006 American Institute of Physics 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 http://dx.doi.org/10.1063/1.2159068 http://dx.doi.org/10.1063/1.2159068 I i c J 1 i f p w e T � p g T i T a e i b b 012301-2 G. N. J. Añaños J. Math. Phys. 47, 012301 �2006� This article is copyright I. THE EFFECTIVE POTENTIAL The theory for which we are interested is given by the Lagrangian, L��� = 1 2 �����2 − 1 2 m0 2�2 − �0 4!N �4 − �0 6!N2�6, �1� s a theory of N real scalar fields with O�N� 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 �CJT�3,4 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 �a�x�, but also of the expectation values Gab�x ,y� of the time ordered roduct of quantum fields �0�T���x���y���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 Gab�x ,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 G�x ,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 �a�2 − 8�0�a�b�c 6N2 �a�b�c − 1 4!N �0 + �0�2 10N �4 − 12�0�a�b 6!N2 �a�b�2 − 1 5! �0�a N2 �a�4 − �0 6!N2�6. �4� he effective action ���� is found by solving for Gab�x ,y� the equation ���,G� Gab�x,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 O�N� 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 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 R H U w s e i i F � r w 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�2 10N �Gaa�x,x��2 − �0 6!N2 � dDx�Gaa�x,x��3. �6� Therefore Eq. �5� becomes ���,G� Gab�x,y� = 1 2 �G−1�ab�x,y� + i 2 D−1��� − 1 12N �0 + �0�2 10N � abGcc�x,x�� D�x − y� − 3�0 6!N ab�Gcc�x,x��2 D�x − y� = 0. �7� ewriting this equation, we obtain the gap equation �G−1�ab�x,y� = Dab −1��;x,y� + i 6N �0 + �0�2 10N � abGcc�x,x�� D�x − y� + i�0 5!N2 ab�Gcc�x,x��2 D�x − y� . �8� ence i 2 Tr D−1G = 1 12N � dDx �0 + �0�2 10N �Gaa�x,x��2 + 3�0 6!N2 � dDx�Gaa�x,x��3 + cte. �9� sing Eqs. �8� and �9� in Eq. �6� we find the effective action ���� = I��� + i 2 Tr�Ln G−1� + 1 4!N � dDx �0 + �0�2 10N �Gaa�x,x��2 + 2�0 6!N2 � dDx�Gaa�x,x��3, �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, �G�ab −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 + O�1�, �G�aa −1 = Ng−1 + O�1� . �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 tr�ln g−1� + N 4! � dDx �0 + �0�2 10N g2�x,x� + 2N�0 6! � dDxg3�x,x� + O�1� , �14� here the trace is only in the functional space, and the gap equation becomes 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 I I t i C f S w a w � w T I t 012301-4 G. N. J. Añaños J. Math. Phys. 47, 012301 �2006� This article is copyright g−1�x,y� = i�� + m0 2 + �0 6 �2 N + g�x,x� + �0 5! �2 N + g�x,x� 2 D�x − y� + O 1 N . �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 g�x ,y�: g�x,y� =� dDp �2 �D expi�x−y�p 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��� =� dDp �2 �D 1 p2 + M2��� , �18� nd the effective potential in the D-dimensional Euclidean space can be written as V��� = V0��� + N 2 � dDp �2 �D ln�p2 + M2���� − N 4! �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−1 �2 �D−1 ln��n + p2 + M� 2���� − N 4! �0 + �0�2 10N F����2 − 2N�F����3 6! , �20� here the expression F���� is the finite temperature generalization of F���, and is given by F���� = 1 � � n=−� � � dD−1p �2 �D−1 1 �n 2 + p2 + M� 2��� . �21� he gap equation for this theory at finite temperature is M� 2��� = m0 2 + �0 6 �2 N + F���� + �0 5! �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: 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 T s w e f W t t a F o t a F t i 012301-5 Scalar field theory at finite temperature in D=2+1 J. Math. Phys. 47, 012301 �2006� This article is copyright I��D,s,m� = 1 � � n=−� � � dD−1k �2 �D−1 1 ��n 2 + k2 + m2�s . �23� he analytic extension of the inhomogeneous Epstein zeta function can be done and the corre- ponding analytic extension of I��D ,s ,m� is I��D,s,m� = mD−2s �2 1/2�D��s��� s − D 2 + 4� n=1 � 2 mn� D/2−s KD/2−s�mn�� , �24� here K��z� is the modified Bessel function of the third kind. Fortunately for D=2+1 the analytic xtension of the function I��D ,s=1,m=M�����=F���� is finite and can be expressed in a closed orm6 F���� = I��3,1,M����� = − M���� 4 1 + 2 ln�1 − 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−1p �2 �D−1 ln��n + p2 + M� 2���� �26� hen, �LF���� �M� = �2M�� 1 � � n=1 � � dD−1p �2 �D−1 1 �n + p2 + M� 2��� �27� nd from Eq. �21�, we have that �LF���� �M� = �2M��F���� . �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����Li2�e−M������ �2 − Li3�e−M������ �3 . �29� The definition of general polylogarithm function Lin�z� 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�2 10N �F����R�2 − 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 ln�1 − e−M������ M����� + �0 5! �2 N − M���� 4 �1 + 2 ln�1 − 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 8 s similar to the scalar theory with O�N� 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 15:18:19 T r a f e I D b � f w i A 012301-6 G. N. J. Añaños J. Math. Phys. 47, 012301 �2006� This article is copyright his 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