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Stochastic Quantization of Fermionic Theories: Renormalization of the Massive Thirring Model

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Stochastic Quantization of Fermionic Theories: Renormalization of the Massive Thirring Model Instituto de Física Teórica IFT Universidade Estadual Paulista October/92 IFT-R043/92 Stochastic Quantization of Fermionic Theories: Renormalization of the Massive Thirring Model J.C.Brunelli Instituto de Física Teórica Universidade Estadual Paulista Rua Pamplona, 145 01405-900 - São Paulo, S.P. Brazil 'This work was supported by CNPq. Instituto de Física Teórica Universidade Estadual Paulista Rua Pamplona, 145 01405 - Sao Paulo, S.P. Brazil Telephone: 55 (11) 288-5643 Telefax: 55(11)36-3449 Telex: 55 (11) 31870 UJMFBR Electronic Address: [email protected] 47553::LIBRARY Stochastic Quantization of Fermionic Theories: 1. Introduction Renormalization of the Massive Thimng Model' The stochastic quantization method of Parisi-Wu1 (for a review see Ref. 2) when applied to fermionic theories usually requires the use of a Langevin system modified by the introduction of a kernel3 J. C. Brunelli (1.1a) InBtituto de Física Teórica (1.16) Universidade Estadual Paulista Rua Pamplona, 145 01405 - São Paulo - SP where BRAZIL l = 2Kah(x,x )8(t - ?). (1.2) Here tj)1 tp and the Gaussian noises rj, rj are independent Grassmann variables. K(xty) is the aforementioned kernel which ensures the proper equilibrium limit configuration for Accepted for publication in the International Journal of Modern Physics A. mas si ess theories. The specific form of the kernel is quite arbitrary but in what follows, we use K(x,y) = Sn(x-y)(-iX + ™)- Abstract In a number of cases, it has been verified that the stochastic quantization procedure does not bring new anomalies and that the equilibrium limit correctly reproduces the basic (jJsfsini g the Langevin approach for stochastic processes we study the renormalizability properties of the models considered4. It should be remarked that, for interacting fermionic of the massive Thirring model. At finite fictitious time, we prove the absence of induced models, a rigorously general proof of the equivalence between the stochastic method and quadrilineax counterterms by verifying the cancellation of the divergencies of graphs with the usual quantization procedures is lacking in contrast with the bosonic case2. It is still four external lines. This implies that the vanishing of the renormalization group beta necessary to investigate specific examples to gain further insights on the peculiarities of function already occurs at finite times. the method. In this sense two dimensional models offer a good laboratory, providing a rich structure to test the general grounds of the theory. In this paper we will focus our attention on the stochastic quantization of the massive Thirring model. One interesting aspect of the field quantized version of the model is the This work was supported by CNPq. absence of infinite coupling constant renormalization5. As it is well known, this implies in the vanishing of the renormalization group beta function to every finite order of perturba- = -(*: jjfe J tion theory. As we shall see, this happens already at finite fictitious time in the stochastic at (2.21) quantization approach. In Sec. 2 we establish the perturbation framework for the massive Thirring model in the stochastic quantization of Paris!-Wu in 2 space-time dimensions. In wh< Sec. 3 the renormalizability of the model is discussed and we derive the renormalization (V(k,tUk',t')} = 2{2nf(-Y+m)S\k - f )i(i - t'). (2-3) group equation. In Sec. 4 we show by explicit calculations, at finite fictitious time and at one loop order, the vanishing of the beta function of the model. Finally, Sec. 5 is devoted Using ^(A,0) — -^(£,0) = 0 as initial conditions the solutions of (2.2) are to the conclusion. l2 t ^ ~^rja(ktr)-{'2g I dr I £ \^ 2. Langevin Equation and Perturb ative Expansion (2.4o) The massive Thirring model, in Euclidian space-time, is defined by the Lag rangi an density 2 *il>(p,T)it>{qiT)7l>b(k - p + i,T)rta(*; t - T) . (2.46) where the relation Çjj^tjf) — (0VO , valid in two dimensions, was used to rewrite the We have introduced the notation current-current interaction in a more convenient form. Throughout this paper we will use ( a+ml)< an antihermitian representation for the 7^ matrices which obey {7pj7v} = —26^» and we r(jfe,i) s G(k%t)(-U + m) = 0(Oe~ * (-/i + rn) (2.5) adopt 70 = xVi, 7J = 1V3 and 75 = 7071 = itr% (ffi, i = 1,2,3 are the Pauli matrices). where G(kjt) is the Green function of the linearized part of (2.2). The Langevin equations for this system were considered previously by Damgaard and Setting g = 0 we can calculate the two point function obtaining Tsokos3 and, in momentum space*, they read dt with -£ + m) 1>{k + p - g,t) + jrffc,0 (2.2a) (2.6) 11 we consider that ip and 1/1 have opposite Fourier transforms, i.e. which in the equilibrium limit (taking t = i' —* oo after the integration) reduces to the usual Euclidian fermion propagator. and For g ^ 0 the integral equations (2.4) can be solved by iteration treating g as a small parameter. In figure 1 we illustrate the solution obtained in this way. The Green functions G(ktt) are represented by solid lines while the T(ktt) are represented by broken lines. Replacing Jbt* by Afc; in (3.2) and allowing A —* oo we get for the superficial degree of Noise fields 7](k} t) and rj[k71) are represented respectively by crosses and boxes. The wavy divergence d, for the stochastic diagrams, the expression lines help in visualizing the charge flow by separating the term Tip or ipV from ifii>. In this d = 2L-2(V-l)+NIB-NIyi representation the propagator associated to the wavy line is just the identity. where the contributions — 2(V — 1) and — Njx come from the observation that the replace- Figure 1 ment of the Jfc.-'s in the exponentials (inside the terms Qi and Qv) are equivalent to the replacement of the temporal differences TÍ — TJ by A"2 (77 — TJ) in the integrand and in the integration measure. By construction V = NB, where NB is the total number of internal 3. Renormalization and external broken lines. This relation plus the topological one 4V = NE + 2JVj, with NE and Nj standing for the total number of external and internal Une6 respectively, yields The 2ÍV-point correlation functions are obtained by substituting the N fields ip and N fields tp for its diagrammatical expansion given in figure 1 generating, in this way, a series d = 4 _ ^NEB _ 1NEX (3.3) of stochastic diagrams after the averages over the 77's and 77's are taken. The analytical with NEB and NEX as the total number of external broken and crossed lines respectively. form of these diagrams6, out of the equilibrium, is given by One may observe that for every diagram with loops there is at least one external broken Ntx m ±J J2 L f ' "IB line (that is NEB ^ 0) what leads to the fact that linear divergent diagrams appear only / n ?2^ / n *» n if NEB ~ NEX — 1 and logarithmic divergent ones for NEB = 1 and NEX — 3. where L is the number of loops, V is the number of vertices, NIB is the number of internal These ultraviolet divergencies, once the theory is regularized, can be reabsorbed by broken lines representing Y{k\t — t') (given by (2.5)), iV/x is the number of internal crossed a renormalized Langevin equation obtained from (2.2) and (2.3) through the following lines representing the propagator D(k;i,t') (given by (2.6)) obtained by contracting the redefinition of parameters and field normalization0 lines with noises attached to its ends. In the above expression we have omitted matricial indices and the factors corresponding to external lines. In the £(£')*n broken (crossed) line, joining the vertices with times T|,(TÍ» ) and TÍ (T,^), flows momentum Qi(Qv) which is a. 3 6m2 ,2 (3.4) linear combination of external and loop momentum Jk,-. Using (2.5) and (2.6) the expression (3.1) assumes the form L t = tR b In fact we could employ a renormalization constant Zn in the redefinition of the noise, so that Tj(kti) = Z^^Z^Zl^rj^kyin) (of course Zn and Zt should enter in the redefinition of the others parameters). But as show by Zinn-Justin7 we have the Ward identity Zt = Zjj, for Markovian process, which allow us the use of (3.4). 6 where V"(fc,í)i gt m, J?(fc,i) andi are the bare parameters of the theory. The counterterms, which, by the Euler's theorem for homogeneous functions, satisfies and therefore the renormalization constants Z = {Z^^Za,Sm^Zt} are calculated, order by (3— + m«5— - 27«-=r- + p— - D) CiiN)(«ft A» m*, FA, A,p) « 0 V 03 &m di op J order in perturbation theory, through the renormalization conditions which fix the mass, R R where 3 is a scale for the momenta. We can use this expression to eliminate p%- in (3.6) the physical coupling constant, the normalizations of the field ip and the noise 77. The and we are left with renormalization constants are dimensionless functions of the coupling constant gRt of the dimensional parameters tRt mR and the renormalization point fi. However, in what follows, dm. 6 we use a minimal subtraction scheme where the finite parts of the counterterms are set to N) + 2(7* "7* + l)ii-sH GR (sptgRimRytRjA,/i) = 0 (3.7) zero so that the renormalization constants do not depend on the dimensional parameters. J where In our calculations we apply an analytic regularization0 (for a description of this regularizai ion in the stochastic approach see Ref.
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