Stochastic identities in supersymmetric theories

Volume 142B, number 5,6

PHYSICS LETTERS

2 August 1984

STOCHASTIC IDENTITIES IN SUPERSYMMETRIC THEORIES V. de ALFARO Istituto di F~sica Teorica, Universitit di Torino, Turin, Italy and Istituto Nazionale di Fisica Nucleate, sezione di Torino, Turin, Italy

S. FUBINI CERN, Geneva, Switzerland

G. FURLAN Istituto di Fisica Teorica, Universitit di Trieste, Trieste, Italy Istituto Nazionale di Fisica Nucleate, sezione di Trieste, Trieste, Italy and International Center for Theoretical Physics, Trieste, Italy

and G. VENEZIANO CERN, Geneva, Switzerland

Received 19 April 1984

The correspondence between stochastic quantization and supersymmetry is reobtained for the quantum mechanics case by use of "stochastic identities". The method is easily generalized to N = 1 super Yang-Mills theory in the light cone gauge.

1. Supersymmetric theories exhibit many interesting features which make them unique among renormalizable field theories. A very important property o f such theories, first suggested by Nicolai [1], is the fact that, after integration o f the fermionic fields, the resulting functional integral becomes "gaussian" in a new set o f bosonic variables (Nicolai map). Conversely, Parisi and Sourlas [2] have shown that a supersymmetry naturally emerges from considering stochastic equations in real space (i.e. without introducing an extra time). The Nicolai-Parisi-Sourlas approach works perfectly in supersymmetric quantum mechanics and in 2-dimensional field theories, where the new bosonic variables are actually local functions o f the old ones. However, for field theories in more dimensions, and in particular for gauge theories, the standard procedure appears to encounter difficulties. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

The aim o f this note is to introduce a new, simple procedure which can be easily employed in four dimensionatgauge theories. It is based on the existence of certain "stochastic identities". We shall first illustrate our method in the k n o w n case of supersy.mmetric quantum mechanics where we show the equivalence with the standard functional approach. We then apply the procedure to N = 1 gauge supersymmetry in the light cone gauge. 2. The starting point o f the discussion is given by the euclidean supersymmetric action for a system of commuting x ~ ( t ) and anticommuting ~ ( t ) fields, a = 1, ...,N [3]:

I= f ds(½T2 +~)+

(ds=idt),

(1)

where

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Tc~ = dxJds + ~W/Oxa , oa~ + = (f~#

d/ds + a~%w)~= f I / f X ~ .

(2)

mediately. If we take F = )ta(s 1)T3(s2) , then

(3)

f e ()tc~(Sl) T3(s2))

The form ( i ) of the action is equivalent to the usual one up to a partial integration. The action given by eqs. ( I ) - ( 3 ) is invariant under the supersymmetric transformations fxe=X~e,

6X~ = T ~ e ,

6X+ = 0 ,

(4)

from which, using eqs. (2) and (3) one has f T a = ~%e, +

+ = 0. foJ,~

(5)

The Nicolai procedure is well known [ 1 ]. If one integrates over the fermionic variables one obtains the effective action ' fd 2~ , Z e f f = f dg2(T)exp(-~J.g T_a)

(6)

with the functional measure

a,s

\fxc~/ .

(T~(s)) = O,

(8a)

(Tcq(Sl)Ta2(s2)) = 6axa26(Sl - S2)'

(8b)

and in general

(9)

We offer a proof of eqs. (8), (9), which starts from the supersymmetric action (1) without resorting to the Nicolai transformation. Although obvious in the present one dimensional case, this proof is instrumental to the derivation o f the stochastic identities in the case o f N = 1 gauge theory. We start from the fact that, if supersymmetry is not spontaneously broken, then we have (6eF) = O,

(10)

where F is any functional o f x ~ and ha, and 6e is the supersymmetric transformation defined in eqs. (4). Taking F = X~, since 6 e ~ = To,e, eq. (8a) follows ira400

Using the second part of eq. (3) we get (X~(Sl)CO~(s2)) =

-6~#6(s 1 - s2),

(12)

which follows from

fda(X,

Xa(sl) exp(-I)=O ,

(13)

and one is immediately led to eq. (8b). The more general identities (9) can be similarly derived by considering F = X~l (s 1 ) T~2 @2) "" To~n(Sn)" 3. We now considerN= 1 super Yang-Mills theory. The action, written in Minkowski space and in the Wess-Zumino gauge, reads [4,5] * a

+ iXc~]~a~X&},

(14a)

I= f d4x{½F~ +iF+F_ -Xc'c%+} .

(14b)

Eq. (14a) is the standard action, with gauge indices suppressed. As in the quantum mechanical case, we shall rather employ the form (14b) [equivalent to (14a) up to topological terms ,2] where we have introduced (a i are the Pauli matrices and auv are defined as in ref. [5])

F~ v(°~ v)~ = (oi) ~Fi '

ran(Sn))

= ~ farafi(Sr - Ss) ( iI-Ir,s T~i(si)) .

(l i)

(7)

Our point o f view is to consider as fundamental the stochastic identities that follow from this treatment. These are

(Tal(s 1) ...

= {Ta(s 1)TO(s2) + ~(Sl)CO;(s2)}e .

1 = - f d 4 x {r~F2v

d~2(T) = I-I dTa(s) = I-I d x a ( s ) D e t (fiT#)

c~,s

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Fi= -(Foi + ½ieijk Fik)= -(Ei+ iHi) ,

(15)

o.)~+ _ i ~ a X'~ Defining for later convenience the components of the vector potential A u as A L=A o-A3,

A R = A o + A 3, A+_=A l-+iA2,(16)

and using the matrix representation, we find immediately the relation between the infinitesimal variations of the F ' s and of the A's to be 8F+ = 6(F 1 + iF2) = - D L f A + + D + f A L , 26F 3 = - D _ f A ÷ +

(17)

DR6A L - DLfA R + D + f A _ ,

,1 Our convention however is a ~= (1 ,~) and eo123 = 1. ,2 For possible non-perturbative difficulties, see ref. [6].

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8F_ = 5(F 1 - iF2) = D _ f A R - D R 6 A _ . (17 cont'd)

(18)

It is however well known that gauge fixing and accompanying Faddeev-Popov terms break supersymmetry. This is o f course a complication with respect to the quantum mechanical case. Also, as suggested by Nicolai, we have three "gaussian" variables as compared to f o u r A u ' s [6]. This latter fact suggests taking an axial gauge nUA u = 0, which eliminates directly one component o f A u and does not require an interacting Faddeev-Popov ghost [7]. In such class o f gauges we can still save part o f the supersymmetry (18) if the equation 8(n • A) = ienUouY~ = 0 ,

(19)

has solutions other than @ = 0. This is the case if and only ifdet (n" o) = n 2 = 0, i.e. in light cone gauges * 3. In particular, if we take n -A = A 0 - A 3 = 0, eq. (19) will hold provided ea = ~ = 8a2 e. We shall call 8 s the restricted transformation associated with such ca. It is straightforward to write down the restricted transformations for the components o f A u and ~ using the definition o f the restricted variations and eqs. (18): they are 8sA 2 = 6sA += 0, 8sA R=2ie~, 1 , 6sA_ = 2ieY~2,

6 s h l = - F 3 e, 6sh2=-F+e,

6s~=Ss6O+=0.

0 = (5 s [Xa(x)FiCy)] )

= -(eF]Cx)(o.i)lFt{y))+ (ha(x)~sFt{y)).

The action I is invariant under the supersymmetric transformations

6A u = iea(au)a&Y~&, ~ho~=Fi(oi)~e~, 8Y~= 0 .

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(20)

(22)

Using (21) we see that 6sF i is either 0 or proportional to some ~ a+ = 8 I / 6 h c' so that the last term in eq. (22) is either 0 or a a-function depending on the values o f a and i. Writing out all the constraints following from (22) and solving, we find:

(F3(x)F3(Y) ) = ~ (F+(x)F_(y))= 64(x - y ) ,

(23)

(F3(x)F+(y) ) = (F3(x)F_ (y) )= (F +(x) F+(y) ) = O, which are our "stochastic identities", since all the correlations in (23) coincide with those o f the free theory. In order to get a handle on the single missing correlation function (F_(x)F_(y)) we use the identity fiI

e

6A+(x) e + 6s(O_ hi(x)) = OLF_ ~-,

(24)

which can be easily checked from eqs. (14) and (20). We then write:

(F.v(x)½ eBLF_(Y)) = (F z.(x) [e 6I/6A + + 6 s(D _ h I (x))]) = (-e6Fr_/6A +) - (SsF~.(x)D_hl(X)).

(25)

In the case o f F _ it is easy to verify that both terms vanish giving

3L(F_(x)F_(y)} = 0 ,

(26)

Eqs. (17) take a particularly simple form, to be used later, in the A L = 0 gauge. For the moment we just use it to derive that

while forF+ we pick up OL64(x - y ) from 6F+/6A+ and recover the derivative o f one of eqs. (23). Eq. (26) is consistent with (and suggests) the conclusion that the missing relation also holds:

6sF + = 0,

(F_(x)F_(y)) = 0 ,

28sF 3 = 2w~e,

8sF_ = -2~l'e ,

(21)

Invariance o f the action (14b) under (20), (21) can be easily checked. Within the (standard) assumption that supersymmerry suffers no quantum level anomalies, we can now derive some particularly simple Ward identities that follow from 8sI = 0. These take the form

, 3 This might explain the problems met by Nicolai [6] in the Ao = 0 gauge. Similar difficulties would also occur in covariant gauges.

(27)

i.e. that the integration "constant" of (26) is zero, as it is for the (F+F+) case. This conclusion may be unwarranted and we could be left with a non trivial function of XL, x _+ for (F F }. On the other hand we should remember that in the A L = 0 gauge we have still the possibility o f performing xR-independent gauge transformations and it is quite possible thaf (F_F_) can be set to 0 after completely fixing the gauge For footnote see next page.

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I f e q . (27) and its n-point generalizations [cf. eq. (22)] hold true, as we believe to be the case, that would mean that the bosonic action is gaussian in the variables F i. There is another way to arrive at the same conclusion: it simply consists o f looking at the Nicolai transformation (17) in the light cone gauge (6A L = 0) and computing its jacobian: J = Det (6F/6A) = Det ( - aL) DET ( ] ~ & ) .

(28)

It is easy to see that this Jacobian coincides with the product o f the ghost determinant (which is a number) and the fermion determinant. The only subtle point about this approach is that the functional integration over the F i is over a complex domain (but not the whole complex plane). In order to have real F ' s we would have to work in euclidean space but, in such case, we cannot choose the light cone gauge. Again we are not sure whether these are anything more than purely formal difficulties. The "stochastic identities" method avoids this problem by working all the time in Minkowski space. 4. Our conclusion is that the correspondence between stochastic quantization and supersymmetry can indeed be extended from elementary models to a gauge theory in the physical 3 + 1 s p a c e - t i m e . The main result is that the "stochastic identities" can be directly derived from supersymmetry. The use o f the new procedure is crucial in gauge theories where it allows to cope with the new features introduced b y gauge fixing. Indeed the fact that only light cone gauges are invariant under a subgroup o f supersymmetric transformations tells us that only in those gauges the stochastic structure o f supersymmetric theories becomes clear and apparent. Work is in progress to apply our treatment to ex-

,4 This is related to fixing an ie prescription around PL = 0. As

discussed by Mandelstam [ 8 ], one particular prescription ("modified light cone") ensures both good ultraviolet behaviour and a regular P L ~ 0 limit. Thus eq. (27) should follow at least in that gauge.

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tended supersymmetric gauge theories. The situation looks particularly promising in the N = 2 case where some stochastic identities can indeed be derived. The question o f course arises whether also in this case the full Nicolai transformation can be applied or whether one should be satisfied with a subset o f stochastic identities which would not imply the full gaussian structure of the theory. The very fact that even in the N = 1 case all the stochastic identities are not on the same ground might hint to the possibility that our procedure is not always equivalent to that o f Nicolai. The physical interest o f the stochastic identities will be discussed in the future; o f course their presence is related to the special cancellation o f infinities appearing in supersymmetric gauge theory. This is also implied by the fact that our stochastic identities are valid in the light cone gauge where the cancellations of infinities are particularly transparent [8]. We wish to express our gratitude to D. Amati, S. Ferrara and H. Nicolai for their interest in this work and their kind advice and criticism. We acknowledge illuminating discussions with P. Di Vecchia, R. F10me, G. Parisi and G.C. Rossi.

References [1] H. Nicolai, Phys. Lett. 89B (1980) 341; Nucl. Phys. B 176 (1980) 419. [2] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43 (1979) 244; Nucl. Phys. B 206 (1982) 321. [3] E. Witten, Nucl. Phys. B 188 (1981) 513; B 207 (1982) 253; S. Cecotti and L. Girardello, Ann. Phys. (NY) 145 (1983) 81. [4] A. Salam and J. Strathdee, Phys. Lett. 51B (1974) 353; S. Ferrara and B. Zumino, Nuel. Phys. B 79 (1974) 413. [5 ] For a review see: J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton University Press, Princeton, NJ, 1983). [6] H. Nicolai, Phys. Lett. l17B (1982) 408.

[7] W. Kummer, Acta Phys. Austr. 41 (1975) 315. [8] S. Mandelstam, Nucl. Phys. B 213 (1983) 149; L. Brink, O. Lindgren and B. Nilsson, Phys. Lett. 123B (1983) 323.