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. 10) which implies the introduction of (3.8) an arbitrary constant p with the dimension of mass. Now gR is dimensionless and in x (3.4) we have to make the substitution gR —» p gR, where A is an analytic regulator, and Equations (3.6) and (3.7) are the generalization of renormalization group equations to we emphasize that the Z's do not depend on TTIR and tR because the finite parts of the the stochastic approach and show the behaviour of the correlation functions by a shift in counterterms are set to zero. the renormalization point or a momentum reescaüng respectively. The Btudy of these equa- Using (3.4) the 27V-point bare and renormalized correlation functions are related by tions as well as the derivation of the Callan-Symanzik equation, even for non-Markovian

R»mjt,iji,A,/i) (3.5) stochastic process, can be found in Ref. 11. where the bare correlation functions are regularized in a properly manner. The p and t refer to the set of external momenta and fictitious time. 4. Beta Function Calculation The renormalization group equation is derived by the observation that the bare cor- In the equilibrium situation ihep(gR) function for the massive Thirring model vanishes relation functions are independent of the renormalization scale p while the renormalized to all orders in perturbation theory5. Now, in the SQM, we show that the t>ame result is ones depend on p explicitly and implicitly through g m and t . Taking the derivative Ri R R valid out of the equilibrium at least to the one loop order. 0- of (3.5) we come up with The p function, in a minimal subtraction scheme, is given by

din dgR 8 dmR d diR 8 B (3.6) +'1 where we have used (3.8). The relation between the bare and renormalized coupling as the generalization of the renormalization group equation to the stochastic case. By constant (3.4) can be rewritten perturbativaly as dimensional analysis we know that

£^)R (4.2) 8 where the coefficients an are power series in the renormalized coupling constant gH. Taking to each one of them x 2 1 the derivative of gR = fi~ (ZgZ^ )~ g on /* we get (a) = -4

Using the series (4.2) we look for a power series solution for fi(gR,\) and we finally arrive 8 *i r at 2 1 2 = -4g Jdr Jdr J d k\r(px;h ~ r)Y(Pl -p3 -f-*;r- 1 J%*0 = SR^ (4-3) o o

2 (4.56) showing that 0(gH) is totally determined by the residue of the simple poles in À of ZgZ^ . 0 2 To calculate /? in order g we must determine Z# and Zg in order g. However, 2U r as shown in figure 2, the graphs (a) and (b) do not contribute for the wave function (c) = ~Ag jdr Jdr' J d2k re normalization because their loops do not have external momenta dependence. So we can 0 0 write (4.5c) Zf = 1 + O{g2). (4.4)

2 1 2 Figure 2 (d) = 4j /dr Jdr Jd k [v(Pl;Í, -r)D(k ~ 0 0 The stochastic Feynman graphs for the correlation functions with four external lines can be generated at order g2 by substituting $ and ifi given in figure 1 in

2 1 2 e) =45 Jdr Jdr Jd k \r{pi'%ti-r)D{^'9TfU)]M^ 0 0 These graphs can be separated in classes, as the one shown in figure 3, which differ only (4.5c) by the type of lines distribution. For the graph (a), given by (4.5a), we can use (2.5) and (2.6) to obtain Figure 3

The graphs of figure 3 have logarithmic divergencies but we can show a cancellation 0 D 2 # x J d k 8{r - r ) [-fa + ^2 - IS) + m] among them at finite fictitious time. First let us write the analytic expression associated bc T+T' c from now on we shall omit completely the subscript R from the renormalized x I da\ (4.6) parameters. 10 11

Performing the change of variables a = T — T\ b — |(T + r') and a = a/3 the expression Following the same procedure method for the graph (b) we find that the divergent (4.6) assumes the form part cancels exactally with the contribution coming from graph (a) (given by (4.9)). Also, by similar calculations we obtain for the graphs (c), (d) and (e) of figure 3 the following 2 -4

oo x Jd*k [-fr - fl + m] J- (4.7) where we have introduced the notation p = pi +p2- A divergence in the integrand appears, 00 as a pole in a = 0, after the integration of the term jéjé. Isolating this term we obtain \ fdb Ay 0 J da J dftrBl which obviously cancel each other when added. The same occurs for the graphs with four external lines of the other classes. 2 Then, at order g t all divergent terms for the four point correlation functions cancel M,(pa; -a/2 + 6,<3) [(T,)Ô,:(7I.)* at finite fictitious time (note that this result independs of the minimal subtraction scheme where we have regularized analytically the integral in the variable a by the introduction of we have used) and the re normalization constant Zg giveB a complex parameter A, the regulator. By an analytic continuation of the above expression (4.10) for A = 0, the divergence appears as a pole at A = 0°. The pole of expression (4.8) yields12 Finally from (4.3)

+ OO +OO (4.11) 2 4TT5 J da J dtrai*(pii<1-o/2

5. Conclusions xD« (P3;-a/2 + ft,<3)[(7p)»e(7p)íe]^í(a) J d/3- 3 2(1 + py We have shown explicitly, at order of one loop, the cancellation of the divergencies in where, after performing the integration in a, the integration in j8 is trivial. So, the final the stochastic four point function for the massive Thirring model at finite fictitious time. result for the divergent piece of graph (a) of figure 3 is given by This result implies in the vanishing of the re normalization group beta function at finite oo times. The generalization of the result (4.11), for all orders in g% depends on an analysis -VT fdb [rfaji! -ò)7jlJ?(p4;M4)l í^(p2;i2,6)7pD(p3;6,i3)l (4.9) which shows explicitly the cancellation mechanism of divergencies among the graphs in 12 each order, as happens with the ones in figure 3. However, the asymptotic conservation of References vectorial and chiral currents in the massless limit are the key elements responsible for the vanishing of the beta function in the equilibrium situation5. The generalization of these 1. G. Parisi and Y. Wu, Sei. Sinica 24, 483(1981). proprieties for the stochastic approach can be used, on all order perturbation calculation, 2. P. H. Damgaard and H. Huffel, Phys. Rep. 152, 227(1987). to demonstrate the maintenance of the asymptotic scale invari&nce of the massive Thirring model. These results will be published elsewhere. 3. T. Pukai, H. Nakazato, I. Ohba, K. Okano and Y. Yamanaka, Prog. Theor. Phys. 69, 1600(1983); B. Sakita/ Stochastic Quantization, in Proc. 7"* Johns Hopkins Workshop (Ed. Domokos), World Scientific, Singapore (1983); J. D. Breil, S. Gupta and A. Zaks, Nucl. Phys. B233, 61(1984); P. H. Damgaard and K. Tsokos, Nucl Acknowledgement Phys. B235[FS11], 75(1984); K. Ishikawa, iW. Phys. B241,589(1984); R. Horsley and W. Schoenmaker, Phys. Rev. D31, 822(1985). I wish to thank M. Gomes for useful discussions during the course of this work. 1 also thank E. Abdalla and D. Menezes for reading the manuscript. 4. M. B. Gavela and N. Parga, Phys. Lett. 174B, 319(1986); It. Kirechner, E. R. Nissimov and S. J. Pacheva, Phys. Lett. 174B, 324(1986); M. B. Gavela and N. Parga, Nucl Phys. B275[FS17], 546(1986); R. Tzani, Phys. Rev. D33,1146(1986); E. S. Egoryan, E. R. Nissimov and S. Pacheva, Theor. Math. Phys. 73f 1274(1987); M. Reuter, Phys. Rev. D35, 3076(1987).

5. M. Gomes and J. H. Lowenstein, Nucl. Phys. B45, 252(1972); A. H. Mueller and T. L. Trueman, Phys. Rev. D4,1635(1971). 6. J. Alfaro, Nucl. Phys. B253, 464(1985); J. Alfaro, NonrenormaUxable theories do not have a stochastic interpretation, Preprint LPTENS 84/8(1984); S. Chaturvedi, A. K. Kapoor and V. Srinavasan, Phys. Rev. D34, 3846(1986); A. Gonz ales-Arroyo, Lecture Notes in Physics 216,171(1985). 7. J. Zinn-Justin, Nucl Phys. B275[FS17], 135(1986).

8. G. 't Hooft, Nucl Phys. B61, 455(1973); J. C. Collins and A. J. Macfarlane, Phys. Rev. D10,1201(1974); J. C. Collins, Nucl. Phys. B80, 341(1974).

9. E. R. Speer, Seminars on Renormalisation Theory, Volume 1: Lectures on Analytic Renormalisaiion - University of Maryland - Technical Report 73-067 (1972); E. R. Speer, Dimensional and Analytic Renormalization, in Rer.ormalization Theory, eds. G. Velo e A. S. Wightman, D. Reidel Pub. Co. (1976).

10. S. Chaturvedi, A. K. Kapoor and V. Srinavasan, Int. J. Mod. Phys. A3, 163(1988).

11. S. Pugnetti, Nucl. Phys. B300[FS22], 143(1988). 12. I. M. Gel'fand and G. E. Shilov, Generalized Functions Vol. /.Academic Press, New York (1964). 14 15 Figure Captions 4>(k,t) Figure 1 — Perturbative expansion of the fermion fields. Broken lines represent F, solid lines G ?nd wavy lines emphasize the separation of the terms Tip or ifiT from iprfr. The r noises 7} and rj are represented by crosses and boxes, respectively.

Figure 2 - The 2-point function to one-loop order. •-H-L

Figure 3 - A class of stochastic diagrams contribuing to 4-point function to one-loop order.

Figure 1

a—

(a) (b)

Figure 2 (e)

Figure 3