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Theoretical Physics 1090/33 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ON A PHASE STRUCTURE OF A THREE-DIMENSIONAL cp4 FIELD THEORY G.V. Efimov INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1990 MIRAMARE-TRIESTE T T r r IC/90/33 INTRODUCTION International Atomic Energy Agency In this paper we will investigate the phase structure of the quantum field models and United Nations Educational Scientific and Cultural Organization 1 (M) INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS - g (1.2) in a space-time of three dimensions R3. The Lagrangians (1.1 and 1.2) describe a one-component scalar field <p( x) and these Lagrangians are invariant under the transformation tp -* —<p. The mass 4 2 ON A PHASE STRUCTURE OF A THREE-DIMENSIONAL tp FIELD THEORY • m in (1.1) is positive, g is coupling constant, and tp0 in (1.2) is a real constant. If the coupling constant is small enough, the Lagrangian (I.I) describes the symmetric interaction, but the Lagrangian (1.2) describes the so-called situation of spontaneous symmetry breaking (SSB) (see, for example. Reft. 1 and 2, containing many earlier references). G.V. Efimov ** Our aim is to investigate the behaviour of the systems (1.1 and 1.2) in the strong cou- International Centre for Theoretical Physics, Trieste, Italy. pling limit, i.e., for 0 —> oo. We will investigate these Lagrangians by the method of canonical transformations. We have used this method for the investigation of the same models in R 2 [2]. By comparison with the Lagrangians in R2 we have the additional divergent diagrams ABSTRACT in R3. Nevertheless, the Lagrangians (1.1 and 1.2) remain to be super-renormalizable and we can write down all necessary counter-terms in the explicit form. We construct the Hamiltonians for Two models of a scalar field <p with the interaction Lagrangians gtp* andg(<p2 - f>l)z Lagrangians (1.1 and 1.2) and introduce operators with counter-terms. We formulate our problem 3 are considered in R . There are phase transitions in these models for a certain g = gc. In the as follows: first model gtp* the symmetry <p —» — <p keeps for any values of p. In the second model where the spontaneous breaking symmetry takes place for small g the symmetry <p —* — <p is restored for What representation of canonical commutative relations is suitable for different values of the dimensionless coupling constant g > gc. The effective perturbation coupling constant is small in both cases. (1.3) and what physical picture corresponds to this representation? 2 M1RAMARE - TRIESTE It turned out that the picture is quite different from the situation in R . Namely, the system (1.1) keeps the symmetry tp —> -tp (or all values of the coupling constant although a kind January 1990 of a phase transition takes place. The symmetry <p —» —ip is restored in the system (1.2) for \ > \c where \c is a critical coupling constant At last we want to stress especially that the effective perturbation coupling constant is small in both cases, i.e., the Lagrangians (I.I and 1.2) describe the systems with weak coupling for any values of g. In other words, the strong coupling regime in the sense that the perturbation expansion has a large coupling constant does not exist in the system s described by the Lagrangians To be submitted for publication. Permanent address: Joint Institute for Nuclear Research, Dubna, USSR. (1.1 and 1.2), T 3 to remove these divergences we should introduce in the interaction Hamiltonian (2.2) an operator 2. HAMILTONIAN v* IN R Rm containing counter-terms which cancel these divergences in perturbation calculations. This In this section we consider the theory of a scalar field <p in the space-time R ' which is operator Rm looks as described by the Lagrangian density 1 fi«= [ dS: \~Amp (x)*Cm *><*) (2.6) iv L2 C(x) = -<p(x)(O -T (2.1) where where gi and g4 are coupling constants. C« = 4! 93 04 Z , We denote coordinates by x = (10,3:1,12) - (zo,£) and momenta by fc • (fco,*i,to> = m Ea = j 4! 9\ Wm - j Vm + ~ 3! (2.7) We shall consider this theory in the Hamiltonian representation. The second-quantized These divergent diagrams can be computed, for example, in the Euclidean metrics (xo Hamiltonian for the Lagrangian (2.1) is written as iky) where the propagator of the field <p(x) is * 1 e"1" H (2.2) Here*2 - *2+Jt|+Jt|.(Jta) + x\ + s|,andfc » y^Jt2 + A22 + fcf We have (23) Here (2.9) Here an appropriate regularization should be introduced. (2.4) We can now say that the theory (2.2 and 2.3) is defined, i.e., the S-matrix The fields p( 1) and ir( z) are canonical variables and obey the canonical commutation relations S-Tcxpi-if* dtH?(t)\ (2.10) [pUKipCr')] = [ir(z),ir(5')]=0 is finite in each order of the perturbation expansion. [?<*),«<« ')]=i6(i~i'). (2.5) 3. CANONICAL TRANSFORMATION The integration in (2.3) is performed over a large "volume" V, here it is a square: Our aim is to realize what is this system (2.1) for different values of the coupling constants j3 and 04. In order to do it we shall use the method of canonical transformations. Let us perform and V = (2 L)2. All operators in (2.3) are taken in the form of normal products. the canonical transformation of our fields This theory contains ultraviolet divergences but it is super-renormalizable, i.e., it has only a finite number of the divergent Feynman diagrams. These diagrams are shown in Fig.l. In order <p(x) B (3.1) where the fields fl (z) and *( i) have the form (2.4) but their mass is equal to Hz= f dx M - m( 1 + f) . (32) where B is a constant field. The explicit form of this transformation (3.1) can be written but we will not S(m,M) =m2 -Af 4! do it here. 2 2 P(m, Af) = tn B + - 3AB) - A) + 4g4B) We now perform this canonical transformation (3.1) for the Hamiltonian (2.2 and 2.3) (3.8) and go over to the normal ordering in the new fields * (a;) and El ( x). We will use the well-known formulas B(m,M) ,Af) + S4(B*-6AB2 + 3A2)+ ^ ) (3.3) In Here: * : denotes the normal ordering of the fields <p(x) and <t> ( x) correspondingly, * is a constant 4! Kim, M) = - and A = A(m,M) = £>m(0) - D«(0) = . 3 Using the formulas (2.9) one can get The canonical transformation of the free Hamiltonian looks as 1 1 , M 1 l 64 ' = Ho1 + I dS: \^-=— + TO2B<I>(SC) + = ©2(i) + L(m, j (3.5) Jv I 2 2 —l(4ir)_3 where (3.9) - M2-m21 Let us put the coefficients of the operator : <t>2 : and * in Hi (3.6) equal to zero Using these formulas we are able to do the canonical transformation of the total Hamil- P(m,AO=0 . (310) tonian (2.2). One can obtain This requirement means that the total Hamiltonian cannot contain the linear term <t> at all and (3.6) can contain only the quadratic term : *2 : in the free Hamiltonian. In other words, the total Hamiltonian should have a "correct" form, i.e., the free Hamiltonian has a standard form and the Here H" is the free Hamiltonian of the fields <t> (x) and n (at) with the mass M. interaction Hamiltonian can contain field operators in degrees more than two. From a physical point of view, it means that our system consists of the scalar particles with the mass M and these noninteracting particles are described by the free Hamiltonian H". , G* =QA . (3.7) It should be noted that Eqs.(3.10) do not define the minimum of the energy E(m, M) in 2 The operator Jtu coincides with (2.6) where we should substitute (3.8) as it was in the <p* -theory in R 13]. It follows from that the counter-terms of the highest perturbation orders are introduced into the Hamiltonian (2.3) in order to remove all divergences m —* M, }]->(?}, jj —» GA from the S-matrix. We want to stress that it is the main reason so we cannot do the variational 5 i estimations for the Hamiltonian (2.2 and 2.3). At the same time the canonical transformations do Let us divide F into two subregions not introduce new divergences in the Hamiltonian. All lenns in the representation (3.6) are finite and the operator R contains all necessary counter-terms (see alto [4]). In other words, we can u where say that Eqs.(3.10) define the ground-state of our system. Introducing the dimensionless variables if if (3.16) ft . b^l^B (311) , x3 Then we shall consider that our system (2.1) is described by the representation (2.2 and 2.3) if (Xj,X«) £ F+ and by the representation (3.14) if e(X3,X«) e F_, So that the conditions (3.16) and using the formulas (3.9) one can get for Eq.(3.10) defines the phase structure of the system under consideration. 1 In addition we should remark that Eqs.(3.12) can have several solutions. Then we have -/ (l + |) + 3X4 (b - f) + 3X36 + 12X? In(l + /) = 0 to choose the solution which leads to the lowest vacuum energy. 6+ 2A (63 - 3/6) (3.12) 4 Thus our direct problem is to solve Eqs.(3.12) and to investigate the behaviour of the function e(X3)X4) (3.13).
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