Stochastic regularization of quantum field theory

ANNALS

OF PHYSICS

140, 247-265

Stochastic

(1982)

Regularization A.

Laboratory

NIEMI

AND

of Quantum

L. C. R.

Field Theory*

WLIEWARDHANA

Center for Theoretical Physics, for Nuclear Science and Department Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

of Physics,

Received January 19, 1982

The equivalence between a D-dimensional classical field theory coupled to an external random source having Gaussian correlations and its D - 2 dimensional quantum counterpart was established. Utilizing this equivalence, a regularization procedure for scalar theories is developed. The regularization amounts to a compactification of the extra two dimensions. The regularization scheme is interpreted in terms of superpropagator modifications.

1.

INTRODUCTION

During recent years [ 1,2] it has been found that a six-dimensional classical scalar field theory, coupled to an external random source having Gaussian correlations,’ and its four-dimensional quantum counterpart are related order by order in perturbation theory. This stochastic quantization has since been generalized to gauge theories [ 31. In the present paper we address the question whether new regularization schemes follow from the pattern of stochastic quantization. We find and study the properties of some new regularization methods for simple scalar theories. In this case we can obtain expressions in closed form for the regulated amplitudes, which make it easier for us to explain and relate our schemes to existing regularization methods. The correlation strength of the external six-dimensional random sources is measured by a parameter that corresponds to Planck’s constant in four-dimensional quantum theories. If we assume that this parameter has a non-trivial space dependence, and, especially if it vanishes for small relative distances, finite Feynman amplitudes emerge. This is the underlying idea in the regularization schemes we study. We start in Section 2 by introducing the six-dimensional stochastic classical field theory. We then explain the approach suggested by Parisi and Sourlas [4], and justify * This work is supported in part through funds provided by the U.S. Department of Energy (DOE) under Contract DE-AC02-76ER0 3069. ’ In stochastic quantization schemes external random sources are introduced in the six-dimensional space. Here we consider only the case when these random sources have Gaussian correlations

247 0003-49 16/82/060247- 19SO5.00/0 Copyright 0 1982 by Academic Press. Inc. All rights of reproduction in any form reserved.

NIEMI AND WIJEWARDHANA

248

it in perturbation theory. In our analysis we carefully include external lines of the Green’s functions, thus generalizing the original proof [ 1,2], that applies to truncated Green’s functions. In Section 3 we study regularization and renormalization within the framework of stochastic quantization. We show how some well-known regularization schemes compare to ours. We then study in detail a regularization schemewhich is convenient in practical calculations. We conclude this section by indicating how renormalization is adopted to stochastically quantized theories. In Section 4 we explain how our regularization schemescan be interpreted in the superfield formalism.

2. STOCHASTIC

QUANTIZATION

OF FIELD THEORIES

Here we consider a classical six-dimensional Euclidean scalar field theory given by the Lagrangian (2.1) Extension of this discussion to more general dimensions and to gauge theories is straightforward, but the extension to fermions has not yet been found. We couple this classical theory to an external source h(x)

and solve perturbatively the classical equation of motion (2.2) where S is the classical action. We denote this solution of Eq. (2.2) by $h(~). We take the random sources h(x) to have Gaussian correlations (2.3a)

(h(x)) = 0,

(h(x) h(y)) = 1’[dhlh(x) 0) exp1-&l

d6xh2(x)l (2.3b)

= d&x - y), (4x,)

hh)

44

hk,)) = (44 + W,) + WJ

hW)Wd

WJ)

h(x,))Wd

WJ)

WJXh(xd

44).

(2.3~)

STOCHASTIC

249

REGULARIZATION

In momentum space (2.3b) reads (l;(k) 6(P))

= (2x)6 A&k + k’).

(2.4)

The parameter A defining the correlation strength has dimensions p2 where p fixes the mass scale. We write A = d;u’, where 2 is a dimensionless number, and later we shall see that d’ corresponds to Plancks constant in the stochastically quantized fourdimensional theory. That is, d’ is equal to 4&z. We now explain the stochastic quantization scheme, invented by Aharony et al. [ 11. First we agree to denote a general six-dimensional point as (x, y), where x is four-dimensional and y is two-dimensional. We then introduce another sixdimensional source J(x, y), of the form 4x9 VI = + 0) where j(x) is arbitrary. form the functional

Following

ZQ[J]=J[dh]exp

[

Ax),

(2.5 1

the suggestions made by Parisi and Sourlas

j’d’xd’g(b,(ry)~(x,p)-~h’(x,p))].

We now prove that the four-dimensional

PI we (2.6)

E-point functions

($qx,)... qqx,))= I’[&I #h(;, 0) ... #$,O) X exp --=

[

1 .

J

d4xd2yh * (x, y ) I

(2.7)

generated by the expression on the right-hand side of Eq. (2.6), are equal to the full E-point Green’s functions in the corresponding four-dimensional quantum theory, order by order in perturbation theory. In Ref. [4] it is suggested to prove this correspondence by employing a hidden supersymmetry of the theory. But here we are interested in regulating ultraviolet divergences, and we find it more convenient to work out the asserted correspondence following the diagrammatic approach by Young 121. The supersymmetry proof is given in [3]. Our proof extends Young’s proof to diagrams with external lines included. Let us consider the E-point function in (2-7) and substitute for 4h(xi, 0), i= 1,..., E, its diagrammatic expansion (Fig. 1). When we take the stochastic average, all lines ending at the sources are joined together in all possible ways by the 6functions of Eqs. (2.3a) to (2.3~). In this way the different diagrams of the classical solutions #h(~i, 0) are joined together and the loops in the diagrams of the E-point function of Eq. (2.7) are formed. We draw bars at the places where the external random sources come together. The diagrams we now have are topologically

250

FIG.

NIEMI

1.

Perturbative

solution

AND

for the field

WIJEWARDHANA

equation

in 4“ theory

up to order

A’ in coupling

constant.

equivalent to the diagrams of the four-dimensional quantum E-point function, save for the bars that appear in the former. As an example we consider the 4” theory. In Fig. 1 we have depicted the diagrams of the perturbative solution to the six-dimensional classical equation of motion with a source term h. In Fig. 2 we show the barred diagrams we get when the four-point function is expanded diagrammatically. Let us now return to the general case of Eq. (2.7). Without loss of generality we consider only the connected barred diagrams. We take such an arbitrary diagram and augment it as follows. We extend the external lines to infinity and identify the points at infinity. In his way we introduce a new vertex to the diagram, and we call this the index point vertex V,. To this vertex we associate the co-ordinates x, ,..., xE, and all other indices attached to the starting points of the external lines. In Fig. 3 we have drawn the augmented diagrams that correspond to the one loop vertex correction of the 4” theory. Due to overall momentum conservation we can interpret the augmented diagram as a vacuum bubble of a hypothetical theory, with momentum space vertex factor 1 at the vertex V,.

< 4 (x,) $(X,)

3-7

4 (XI) 4 (x4) >

=

+ -4 + ‘*-0

FIG. 2. Diagrammatic expansion scheme. The barred lines show where

4

quantization of the four-point function, to order I, ‘, in stochastic two external random sources have been joined to form loops.

251

STOCHASTIC REGULARIZATION

FIG. 3. The augmented diagrams corresponding to the one-loop vertex correction of 4’ theory, both in ordinary quantization and stochastic quantization schemes. The index point vertex is denoted by a circle around the vertex. If the barred lines are removed the remaining lines will form a tree of the augmented diagram.

The number of loops L’, and vertices V’, of the augmented diagram are given by L’=z+E-

v+

(2.8a)

1,

(2.8b)

V’ = v+ 1,

where Z and V are the numbers of internal lines and the vertices of the original diagram. If we remove the L’ barred lines in the augmenteddiagram, we are left with one of its trees which we call t. In fact, if we remove the barred lines from the diagrammatic expansion of the expression in the right-hand side of Eq. (2.7) we are left with all the trees of the four-dimensional quantum diagrams. If we introduce a Feynman parameter ai, i= I,..., Z + E, to every line of the augmented fourdimensional quantum diagram we can characterize its trees, and hence also all topologically equivalent barred diagrams, by the homogeneouspolynomial cP(a, ,..., ar+E)=s

7

Yz(Cf,, .... a,+,)=x

n ai. T

(2.9)

i+Zr

For the one-loop vertex correlation in 4” theory, this polynomial reads ~~(al,a2,a3,a4,a5,a6)=a,a2~3~5+~,a2~3~6

+a,a,a,a,+a,a,a,a,+a,a,a,a,+a,a,a,a, + a,a3a5a6 + ala4a5a6 + a2a3a4a5 + a,a,a,a, + a2a3a5a6 + a2a4a5a6y

(2.10)

where we have referred to Fig. 4. One can see that if the corresponding lines are removed, we are indeed left with all the trees of the augmented diagram. Let us denote by ki, i = l,..., L’, the four-dimensional parts of the loop momenta in the hyperplane spanned by the co-ordinates x, ,..., xE, and let qi, i = l,..., L’, be their

252

NIEMI AND WIJEWARDHANA

FIG. 4. Notation used in formula (2.19).

two-dimensional components. Denote by pi, i = l,..., Z + E, the momentum that flows along the line i. Then (2.11)

Pi = 2 ViSkr + qr), i-=1

where the (Z + E) x L’ matrix (vi,) defines the chosen orientations. The sum of the topologically equivalent barred diagrams has the amplitude GE&, ,---y4

= ELI”

. L’ d4k d2q T j ,I=ITo’(2K)l i f’ r=l

5 Arjx, - kj j=l

‘ff ’ I i= 1 (p,’ + nt2)li ’

(2.12)

where Li = 2 if line i is barred and Ai = 1 otherwise. The factor S includes the combinatorial and vertex contributions. The E x L’ matrix (A,) depends on the choice of loops in the augmented diagram. We introduce Feynman parameters in (2-12) by using the generalized Feynman identity,

and get GE(x, ,..., x,)=59-(Z+E+L’)AL’xj. T

fid’k, r=l (2x)

d2dr

02

i f’ ? A,.jx, - kj r=, jy ’ [C:‘f

eta1 9-.ya,+d a,(pi + m2)]‘+E+L’ *

(2.14)

STOCHASTIC

253

REGULARIZATION

We now diagonalize the part corresponding to the variables qi in the denominator. From the change of variables a Jacobian determinant emerges and it cancels the characteristic polynomial (2.9), we have in the numerator, as (2.15) and we end up with GE@, ,..., xE)

.oo L’ X

J0

n r=1

dp,.exp

i f 5 Arjx, . kj 1” ‘ff da,6 I r=l j=l I -0 ,=,

(2.17) where Zi is the four momentum that flows along line i (2.18) But (2.17) equals the four-dimensional quantum amplitude of our diagram save for the factors (~l/47r)~’ and the dimensional factors in S. Before we study these overall factors let us look at the one-loop vertex corrections of 4” theory in Fig. 3. Using the notation of Fig. 4 and denoting the propagator along the lines l,..., 6 by Ill,..., [6], respectively, where, for example, [ 5 1 reads

we get the value of the diagrams as

X exP{i(pi

- XI +

(P2

-PI>

* X2

+

(P3

-P2)

. X3

-P4

* wx4)l

254

NIEMI

AND

WIJEWARDHANA

x ~11~21~31~41~51~61~~11~21~31~51+ [11[21[31[61 + 1I1121141151+ 1I1121141161+ 1I1131141151 + 1I1131141161+ 1I1131151161 + 1I1141151161 + 121131141151 + 121131141161+ 121131151161 (2.19)

+ [21[41[51[6lI.

This corresponds to the general formula (2.12). Applying the generalized Feynman identity (2.13) we get the polynomial (2.10) in the numerator. In the denominator we diagonalize the I,, I,, I,, q, dependent quadratic form and get the Jacobian determinant a1 +a, -a2

0 0

-a2 a2 + a3 + as --a3

0

0

-a3

a5

a3 + a4

0

.O

a5

015+

a6

which cancels the homogeneous polynomial (2.10) we had in the numerator. By evaluating the 1,) I,, I,, q integrals after this diagonalization, we verify the equality of Fig. 3. Now we go back to (2.17) and study the overall factors present there. We write A = 2~~. The power in which 2/47r appears is L’ = I + E - V from Eq. (2.8), which is exactly the power of Planck’s constant in a similar four-dimensional quantum diagram. Hence we can identify 2/47r with Planck’s constant. In S there are several dimensional constants, depending both on the diagram and on the potential V(4). In the general case this potential is a polynomial in the field 4, V[#]

=

f +q r=3 r.

(2.20)

and the dimension of the six-dimensional coupling constant is ,LL~-“‘. In general, these are V,. factors of A, in S, and the numbers V, satisfy

We now use the law of conservation of boson ends, E+2Z=

2 TV,., r=3

255

STOCHASTIC REGULARIZATION

and combine the dimensions of the A and S, and get the overall dimensional factor

But in (2S),j(x) has the dimensionsof a four-dimensional source. So the factor pE is cancelled by the factor l/p in (2.5). The remaining ,u’s in (2.17) then give the fourdimensional coupling constants the correct dimensions.We have thus proved that the functional (2.7) generatesthe full E-point functions of the four-dimensional quantum theory. Using the methods of [3] it is straightforward to generalize this proof to gauge theories.

3. STOCHASTIC

REGULARIZATION

AND RENORMALIZATION

Now we set about using the stochastic quantization procedure to regulate fourdimensional quantum theories. One way of achieving this is to interpret the classical theory in a 6 - E dimensional space. Another approach is to modify the correlation strength (2.3) of the external random sources. Here we study the latter. In stochastically quantizing a four-dimensional field theory, we break the O(6) and translational invariances of the six-dimensional classical theory into the corresponding four invariances. If we insist that these be preserved after regularization, we are led to consider correlation modifcations of the form (3.la)

@(P)) = 03 (&lo) 4P’))

= mn(~*~

(3.lb)

434’; a) w + k’),

(41) fi(2) f;(3) L(4)) = c&l) l(2))@(3) G(4)) + W)

&3))$(2)

h(4)) + @(l) Q4)M2)

&3)),

(3.lc)

where k and q are the four- and two-dimensional parts of the six-momentum p. We impose the condition that as a tends to a given value a,, the correlation strength f(k*, q, q’; a) tends to Ad*(q + q’), where A is equal to 47&p*, so as to gain the bare correlations. For a # a, this modified correlation should make the amplitudes finite. As an example we can study correlations of the form C&(P) &P’)) = (27~)~A(P* + m*)” ti*)Y

4~ +P’).

(3.2)

where a is a complex number, and in the limit a + 0 we gain the bare correlation. In this schemewe modify the propagators of the barred lines in the following manner: 1 k*+q’+m*

A

1 1 k2+q2+m-‘k2’+q2+m2

A(k* + q* + m*)” 01*>”

1 k*+q’+m*’

256

NIEMI

AND

WIJEWARDHANA

This resembles the method of analytic modified according to the rule

regularization

k2~m2+(k’+m2)*+d

[6] where

propagators

are

017”

Another scheme can be defined by correlations (l(p)

f;(p’))

= (2~)~ d exp

-$I

that render all diagrams finite. We shall now study regularization form

I

6(p +p’)

schemes that are based on correlations

(l;(k, q) l;(k’, 4’)) = (2ny d@i * - q’) S(p + p’). Here /i is a mass parameter. If infrared divergences by introducing another cutoff mass L and get (Qk, q) Qk’, 4’)) = (27r)6 ii&4*

(3.3)

of the

(3.4)

are present we can modify (3.4)

- 42) byq* - /I’) &p +p’).

(3.5)

In the limit A* + co and A2 -+ 0 we recover the bare correlations (2.3). Since at present we are only interested in curing ultraviolet divergences we stick to the form (3.4). Applying (3.4) to the E-point function (2.14) we get the amplitude GE,,&,

,..., xE)

(3.6) and each choice of the A:(a,,..., a ,+E)‘~ leads to a different regularization scheme. We first show that for finite /i, (3.6) is superficially convergent. We have 4L’ momentum variables in the numerator and 2(1+ E + L’) momentum variables in the denominator. If there are at least two vertices, V’ > 1, and it follows from Eq. (2.8) that the integral on the right-hand side of Eq. (3.6) is superficially convergent. Regularization is not necessary when V’ = 1 since that case corresponds to the free

STOCHASTIC

propagator. So the assertion Later on we shall show how We first study the case independent of the Feynman

257

REGULARIZATION

is proved. The previous proof is not valid for tadpoles. they are regulated. where the regulator masses in (3.6) are all equal and parameters. Then the regulated amplitude reads

GE,,&, ,...,xJ

1

’ [CFI, P, + C:‘f’CF,‘,=l a,r,rv,skr . k + mzl’+E* If we denote the bare amplitude (2.15) by GE(x, ,..., xE, m’) we can write (3.7) as G!i&w.~~;m~)=

6 (-1)” “?O

(;)GE(X,,...,~~;rn*+nn~).

(3.8)

This resembles the Pauli-Villars scheme [7], where auxiliary heavy mass fields, natively quantized with commutation and anticommutation relations, are added Lagrangian. However, in the present scheme the heavy masses arise as a result compactification of the extra two-dimensions, and hence our scheme is different the Pauli-Villars scheme. The convergence in (3.8) can be improved by iteration. Then we get

GE,,,(x,,...)xE;m2)= + (-1)”

G~EC(~,,...,~E;m2

Ii=0

+

wi2).

alterto the of the from

(3.9)

If L’= 1 in Eq. (3.9), the resulting amplitude can be interpreted as if a similar heavy mass Lagrangian had been subtracted from the original six-dimensional Lagrangian. As an example we apply this scheme to the one-loop vertex correction of 4” theory. The divergence here lies in the integral over k and q (in the notation of Fig. 4). In practice we like to find regulated momentum space amplitudes. Hence we restrict the external momenta to four dimensions and find the regulated truncated amplitude

where

s = (P2 +PA*.

258

NlEMl

AND

WIJEWARDHANA

After doing the p integration we get &j;zdzj;da 1 x [ [z+a(l-a)s+m*]*

- ,z+a(l-a)ls+m’+~*,’

1*

Notice that we have a combination similar to the one in Eq. (3.8), but now .in momentum space. After doing the z integral we get the result a(l-c++m*+/i* a(l-a)s+m*

1 ’

If we apply this scheme to the self-energy correction Fig. 5b) the result will read .I

A* qGjy

!

da,da,da,

S(1 -a,

(3.10)

of 4” theory (shown in

- a, - a3)

[ala2 + a2a3+ v,l*

x [(m2+Ap2)ln(m2+Ap2)-2(m2fI12+Ap2)ln[m2+A2+Ap2] +(m2+U2+Ap2)ln(m2+U2+Ap2)], where A=

aIa2a3 ala2 + a2a3 + a3aI

Here the integral over the Feynman parameters diverge. This happens whenever we have overlapping divergences, and then this scheme must be supplemented with further regularizations. We shall now present another way of choosing the cutoff masses in Eg. (3.6). We first take an arbitrary tree r of the augmented diagram. The corresponding polynomial (2.9) is 9Jal

,..., a,+E) = ai, . --. . ai,,

(a) FIG. 5. (a) Coupling part in $” theory.

constant

renormalization

(b) part

in 4” theory;

(b)

Self-energy

renormalization

STOCHASTIC

259

REGULARIZATION

and the numbers i, ,..., it I are all distinct. Cut off masses are now defined by

Af(a,)...)a,+E)= A*air

(3.11)

and we get the regulated amplitude L’ GE,,&,

,...,

1

’ [~‘;I,~,+C:~f:C:.+sE=,a,~,,lt,~k;k,+m*l’+” In this cutoff procedure ri ,..., iLj are modified as

the four-dimensional 1

1

kf, + m* *kf,+m*-

propagators

(3’12)

along the barred lines

1

kfr+m2+A2’

We shall now show that (3.12) is finite if tadpoles are excluded. The proof is based on Weinberg’s theorem [8], which says that a Feynman diagram is finite if the degrees of divergence of the whole diagram and all of its subdiagrams are negative. Let us choose an arbitrary subdiagram of our diagram. Along every possible loop of this subdiagram there is one barred line on which the propagator is modified by 1

1

k2+m2~k2+m2-

1

k2+m2+A2’

where k is the four momentum flowing along the barred line. It then follows (2.8) that the degree of divergence P of this subdiagram is given by

from

P=2-2v, which is negative if tadpoles are excluded. But the degree of divergence of the whole diagram is also negative and the result follows. Let us consider again the self-energy correction in 4” theory [Fig. 5b]. When we apply the scheme (3.12) to this diagram we introduce the six-dimensional loop

260

NIEMI AND WIJEWARDHANA

-o0 (0)

FIG. 6. (a) Self-energy renormalization four-dimensional (” theory.

(b) part in six-dimensional 4’ theory. (b) Self-energy tadpole in

momenta to the divergent part only, and keep the external momenta four-dimensional. Then the integrand of the truncated part of this diagram reads 1

1

1

kt + m2 -k:+m2+A2

1

)( kg + m2 -k:+m2+A2

1 I( (l-k,

- k,)’ + m2 ’

Now the k,, k, integrals are finite. Let us study another example to illustrate the difficulties that can arise. Consider the perturbative #3 quantum theory in six-dimensions. The classical lagrangian used in stochastic quantization is now defined in eight-dimensions. The most ultraviolet divergent diagram is the one loop self-energy correction [Fig. 6a] which is quadratically divergent. We again interpret the scheme (3.12) so that we introduce eight-dimensional loop momentum in the divergent part only and keep the momenta of the external lines in six-dimensions. The integrand then reads 1

1

1

k2 + m,2 - k2+m2+A2

(p-k)2+m2

and the integral is logarithmically divergent. We then apply (3.9) with L= 1 to the divergent part and end up with a finite result. As a further example consider the one-loop tadpole depicted in Fig. 6b. The integrand is simply 1

k2 +m2 and the Feynman amplitude becomes finite if we apply (3.12) and (3.9) in tandem. Iteration of (3.9) twice would also render the amplitude finite. We can iterate (3.12) further by introducing more two-dimensional momentum

STOCHASTIC

261

REGULARIZATION

variables in the denominator. If we introduce a two-dimensional for each line in the diagram, (3.12) will read

momentum variable

GE,,&, ,...,

and we end up with subtraction

the standard 1

regularization

procedure

1

k2+m2~k2+m2-

which

amounts to the

1

k*$m*+A*

for all lines. This also amounts to the introduction of higher derivatives to the Lagrangian. We shall now outline how 4” theory is renormalized consistently in the stochastic quantization scheme. The discussion will be sketchy as the technical details are well known. We start with the classical six-dimensional Lagrangian, which is a functional of the bare field #,, and the bare coupling constant A,,. We couple this to the random source h. If we solve

(a+m;)m,++h for #,, perturbatively and form the functional (2.6) we generate the four-dimensional bare quantum theory, with coupling constant p*A,. dimensional quantum theory we have a coupling constant g and defined at some convenient renormalization point. We now reorder the quantum theory in powers of g. For this purpose we expand powers of g

(3.14) diagrams of the But in the foura mass m, both the diagrams of ,u2&, and m, in

(3.16)

NIEMl AND WIJEWARDHANA

262

We also define the field renormalization constant z=1+:

,e2

C” 2 g”. ( )

Using the renormalization conditions we determine a,, b,. and c, order by order in perturbation theory in g in the usual manner.

4. SUPERSPACE FORMULATION

OF STOCHASTIC

REGULARIZATION

We shall finally study how we can interpret our regularization schemeswithin the superlield formalism [4]. The starting point is the functional (2-6),

-%IJl=J [dhlexp /Jd6x ($,(x)J(x)-&P(X)) Our regularization schemeswere based on modifying the correlations (2.3a-2.3c) to non-Gaussian ones as in (3.1). The regulated diagrams can be generated by the functional

(4.1) where we have assumed that the bilocal operator D-‘(x invariant. It is related to the correlation (3.1) by (W&J’))

= @Y ml4

-y)

is translational

&P +P’>*

(4.2)

We generalize (4.1) to Z;EG(J) = j [dh] [d#] S([-0

+ m’] 9 + V’(4) + h)

x det (-0 + mz + V”(d)} exp J’ d”x(#(x) J(x) -- ’ j d6xd6yh(x) D-‘(x 24

-v)

h(y)) 1 ,

(4.3)

where the integration over d now picks up all possible solutions to the classical field equations (2.2). In perturbation theory the functionals (4.1) and (4.3) are of course the same. We write the b-functional in Fourier representation and the determinant as

STOCHASTIC

263

REGULARIZATION

an integral over anticommuting fields q, fi, and do the Gaussian functional over h. In this way we end up with the form ZREG[J] = {det(D-I)}-“* Q x

exp

integral

1 [d#][dw][dij][d~] d6xd6y + o(x) D(x - y) w(y)

- I‘ d%w[(-•

+ m’) 4 + v’(g)]

+ ii[-0 + m2+ v”(g)]?I>. I

(4.4)

We can forget the determinant here as it is a overall factor. Following [4] we introduce the eight-dimensional superspace by adding to the original six-dimensional space two anticommuting variables 19,r% e2 = P = {S, e, = 0. We define the integration

in this superspace by

and define the superlield

@(x,e,gj= $(x)+ v(x)e+ srgx>+ w(x)six The exponential

in (4.4) then reads

1

%,[@I+~~@=+ [-a-AD&-g+m2 @+V[@]+.P@, where the supercurrent

(4.5 >

J(x, g 19) is defined to be 2-(x,

8, e> = J(x) ee.

The super-Lagrangian (4.5) is of the same functional form as the original sixdimensional classical Lagrangian, save for the factor D, and the generating functional in superspace is of the conventional path integral form. We conclude that our regularizations amount to propagator modifications in superspace, 1 k2+q2+AEa+m2

1 + k2 + q2 + AD(k, q) Sra + m2 *

264

NlEMl

AND

WIJEWARDHANA

As an example we regulate the one-loop vertex superspace formalism the truncated amplitude reads

correction

in Fig. 5a. In the

1 (k’ + q2 + AL%a + m2) 1 ’ ((k +p2 +I$)~

+ q2 + d&511 + m’) ’

where p2 and p3 are the independent external four-momenta. Notice that when we take the external lines to carry physical four-momentum, there are no incoming twodimensional momenta and incoming anticommuting momenta. Performing the a, ~5 integrals we get 1 r”‘jmm

d4k

+

d=q

Ad

1 (k +p2

1 k2 + q2 + m2 A~ [(k+p,+p,;2+q2+m2,2

+ pJ2

+ q* + m2

1 ’

By choosing d(k, q) = @I= - q=) we end up with our previous result (3.10).

CONCLUSIONS

We have studied the stochastic quantization scheme for scalar theories. Following the suggestion made in [4] we have extended the earlier proofs to include external lines of Green’s functions; and using the ideas in [3] our proof can be straightforwardly generalized to gauge theories. We have suggested some regularization schemes that arise from the stochastic quantization and shown how scalar theories are regulated. We have also shown how renormalization is adapted to stochastic quantization. Finally, we showed how our regularization schemes could be interpreted as propagator modifications in the superspace formulation of field theories. Only the anticommuting part of the superpropagator is modified, and as the ordinary part remains intact, we hope that new gauge invariant regularization schemes could be found. We are presently studying this possibility.

STOCHASTIC REGULARIZATION

265

ACKNOWLEDGMENTS We thank Professor R. Jackiw for numerous and valuable suggestions and criticisms.

REFERENCES 1. A. AHARONY, Y. IMRY AND S. K. MA, Phys. Rev. Left. 37 (1976), 1364. The relation is here proved for one loop diagrams in 4” theory. 2. A. YOUNG, J. Phys. C 10 (1977), L257. The proof given in Ref. [l] is generalized to all orders in perturbation theory. 3. B. MCCLAIN, A. NIEMI AND C. TAYLOR, to appear. A supersymmetric proof is given following the suggestions made in Ref. [4] that applies to Yang-Mills theories and to ordinary scalar theories. 4. G. PARISI AND N. SOURLAS,Phys. Rev. Left. 43 (1979), 744. 5. N. NAKANISKI. “Graph Theory and Feynman Integrals,” Gordon Breach, New York. 1970: J. BJORKEN AND S. DRELL, “Relativistic Quantum Fields,” McGraw-Hill, New York. 1965; C. ITZYKSON AND J. B. ZUBER, “Quantum Field Theory,” McGraw-Hill, New York, 1980. 6. E. R. SPEER in “Renormalization Theory” (G. Velo and A. S. Wightman, Eds.), Erice Summer School 1975, Reidel, Dordrecht, 1976, and references cited therein. 7. C. ITZYKSON AND J. B. ZUBER, “Quantum Field Theory,” McGraw-Hill, New York, 1980. 8. S. WEINBERG, Phys. Rev. 118 (1960), 838.