LETT]~RE AL NUOVO CIMENTO "COL. 31, ~r 11 11 Lugllo 1981

New Generalized Ward-Takahashi Identities in Gauge Theories.

M. QuIRdS and F. J. D~ URRiE8

Inst i tuto de Estr~ctura de la Materia, C S I C - Serrano 119, Madrid Departamento de l~isica, Universidad de Alcald de ttenares . Madrid

(ricevuto 1'11 Maggio 1981)

When dealing with renormalization in gauge theories, the relevance of the invariance properties of the action is well known. In particular, the Becchi, Rouet and Stora (BRS) transformations (~) leave the action of gauge theories invariant , and that leads to the usual generalized Ward-Takahashi (WT) identities preserving the symmetry of the renormalized theory.

The BRS transformations for the gauge fields are infinitesimal gauge transforma- tions whose parameters are related to the Faddeev-Popov (FP) (2) ghost fields. In this work we present a new invariance property for the action by means of infinitesimal gauge transformations with parameters related to the FP anti-ghost fields. Furthermore, we prove that both the BRS transformations and those presented along this paper can be interpreted as infinitesimal gauge transformations when acting over the FP fields. The new invarianee property leads to new generalized WT identities, as showed below.

In a recent paper (~) we proved that the suitable mathematical structure in order to give a geometrical interpretation to gauge theories, including the FP fields, is a trivial principal bundle (4) with structural group G (the gauge group) and base space /~4 • G • G (locally). ~ being the space-time.

The points of the base space are parametrized by (x, y, z), x being the space-time co-ordinates and (y, z) the parameters of G • The gauge fields A~ are, as usual (5), the coefficients of dx~ in the connection 1-form defined on the fibre bundle. On the other hand the FP ghosts, e ~, and antighosts, ~ , are identified with the terms corre- sponding to the y and z co-ordinates, respectively, in the connection; they are, thus, 1-forms and, hence, antieommuting objects. All the fields A~, v ~ and ~, are dependent on the co-ordinates (x, y, z). Nevertheless, in order to avoid completely the nondesired dependence on y and z of the fields, we take their values at y = z = 0, corresponding

( ' ) C. DECCHI, A. ROUET a n d 1~. STORA: Commun. Math. Phys., 42, 127 ( i975) . (~) L. D . FADDEEV a n d V. N. P o P o v : Phys. Left. B , 25 , 29 (1967). (s) M. Qun~es , F . 5 . DE U R P ~ S , J . HOYOS, M. L. MAZeN a n d E. RODRIGUEZ: J. Math. Phys. (N. Y.), ( to appear). (~) S. KOBAYASHI a n d K . No~IZU: Foundations o] Di]feren~ial Geomeiry, VoL 1 ( N e w Y o r k , N . Y . , 1963). (~) T. T. W U a n d C. N. YANG: Phys. Rev. D, 12, 3845 (1975).

369

3 7 0 ~ . Q U I R 6 S a n d F . J . D]~ U- /~Rf]~$

to the identi ty in G • Thus the forms become fields with values in the exterior algebra (Gaussian algebra) of the tangent space of G • G at the point (e, e), T(~,,)(G • G), the corresponding eommuting/anticommuting character being preserved.

The structure equations of the fibre bundle are proved (a) to be equivalent to the following relations:

(1)

{ d ~ c r 1 a ~ ~ ~, (D#c)a , ~]boC C , d ~ A ~ =

d, e a = --�89 c , d , A ~ = (D~,e) ~ ,

where D~ is the covariant derivative, ]be the structure constants of G and d~, d z the exterior derivative with respect to the variables y and z.

Let ~ be a constant odd (anticommuting) element of the Grassman algebra. We define the transformations /1 = Sd~ and A----Sd~. Multiplying (1) by $, we obtain

(2a) Ae" = - - �89 ~ ] L cb c ~

(2b) AA~, = (D~$c) a ,

1 a - b - c

(2d) ZA~ = (D~,$~) a ,

a b - r (2e) A~a + Jea + $]b~ c e = O .

We have proved, in ref. (a), that A and A are the variations generated by the infini- tesimal gauge transformations of parameters $c ~ and $~, respectively.

Observe that (2e) leaves us the freedom to fix A~ a arbitrarily, provided that Jc ~ is constrained by the same equation.

If we choose

(3) A~ ~ = - - ~ A ~ ,

it is clear, from (2a), (2b) and (3) that ~ is nothing but the BRS transformation, which leaves the Lagraugian

(4) a 2 /~ a 2 - a ~ = - - � 8 8 -- �89 A , ) - - c O~'(Dt, c) a

invariant. As is well known, this Lagrangian corresponds to a pure gauge theory, without matter fields, including the gauge-fixing and FP terms.

I t is worthy pointing out that, in our formalism, the BRS transformations for FP fields arc determined by gauge transformations on the fibre bundle. There is a remaining degree of freedom in (2e) to choose either A~ ~ or Ac a. This freedom enables us to fix eq. (3), which can be interpreted as a condition over the parameter c a of the gauge trans- formation. In the standard formalism the BRS transformations for the FP fields are arbitrarily fixed in order to get the ineariance of the Lagrangian, but without any re- lation with gauge transformations.

Next we study the invariance properties of the Lagraugian under the new trans- formation A.

By partiM integration of the FP term in (4), we get the equivalent Lagrangiau

- - a 2 1 (5) ~ ' = - - �88 ( / ~ ) - - ~ (~A~)~-~ (O~'~)(Duc) a ,

N : E W G]~N~RALIZ]~D W A R D - T A K A H A S H I I D E N T I T I ] ~ S I N CrAUG~ T H e O R I e S 371

which is more appropr ia te than (4) to s tudy the act ion under zi. Using (2c), (2d), (2e) and (3), we get

(6) 3 w = ~(A + B ) ,

where

(7a)

- - [~e --]bo(i~ A~,)A.a c - - ( a o)c~.(~ A~,)--/b+A,,(a A~,)a c ,

n = --�89 ] . ~ A . r ~ " +) +

v ~ . . r , - i . . . . + - ~ . , , o ~lL(o.~+)~ +I,,.I:+A.c o }.

Afte r an ex tens ive used of the J acob i iden t i ty and the an t i - commut ing charac te r of the fields c ~ and ~ , we find

(7b) ~ A = -- O~{(a~a)(0"A~)},

t B = 0 .

We see tha t A ~ , and hence zT.~, is a 4-divergence, so tha t t he act ion is i nva r i an t under ~.

We stress again tha t the t r ans fo rmat ion J is g iven by an inf ini tesimal gauge t rans- fo rmat ion on the fibre bundle. Equa t ions (2e) and (3) fix the va lue of zJe% so that t he invar iance of *he act ion under J resul ts wi thou t any addi t iona l condi t ion e v e r the pa rame te r ~ . I n this way the condi t ion (3) leads to the invar ianee of t he act ion under bo th t ransformat ions A and ~.

Thanks to the new invar ianee p rope r ty for the act ion we shall ob ta in in the fol lowing new general ized W T ident i t ies , to be added to the usual ones coming from the B R S t ransformat ions .

L e t Z be the genera t ing funct ional

(8) Z[j, #, V; u, v, w] = f D A DcD~ exp [iS], d

where the to ta l act ion S is g iven by

~_ 4 .a a_~ a-a ~ a t D ~a l v a l a ~b~e j_ , (9) S=f .~d4~ fdxCjpA, o , j + + ~ 1 6 2 . , , , , - 2 , . -wat : .c '+ +}

t ha t includes sources for the fields A~, e a and ~", as well as addi t ional sources for each nonl inear t e rm (6) in (2c), (2d), (2e).

I t is s t ra igh t forward to p rove t h a t the t e rms corresponding to the sources u and v are invar ian t under A, while

(10)

We make the change

- - a /* b -C �9 gO'~otb~ ~) -- r A~,) c

(11) A~," ---- At~ + - " , , = AA~ e'~ = e ~ + A e a ~,'~ ~ + A~,"

(~) S o u r c e s f o r n o n l i n e a r t e r m s w e r e f i r s t i n t r o d u c e d b y H . KLUBERG-STERN a n d J . B . ZUBER: Phys. Rev. D, 12, 467 (1975).

372 M. QUIRdS and L a. Dr. VR~iES

in the pa th integral (8). The Jacobiau of this t ransformation is uni ty because of the proper ty ~2 = 0.

Using the invariance for the action associated to the Lagrangian (4) as well as the invariance of the u and v terms in (9), the change (11) leads us to the ident i ty

(12) a --(fbc~a b o-c IDADcD~exp[iS]I tOX{j~, ,AA~+ f f c ~ + Z~r/~ + w A )} = 0 j j ~ ~

tha t together with (20), (2d), (2e), (3) and (10) gives

(13) fd.xf,'o{-- o)z , + - - ( - - i ~ ) g a ( O i t ( - - ~ ) g ( - - ~ ) g \ a ] \ aj~,

+ A D c D ~ e x p [ i S %w ]bo(a A . ) c - - 0 ,

where functional derivat ives have been wri t ten for the terms of (12) having a source in the to ta l action (9).

Equat ion (13) is the new generalized WT ident i ty coming from the change (11). The last term is somewhat complicated and cannot be expressed as simple functional derivatives of Z. However, i t is possible to get a more compact expression for (13) by means of the equation of motion for the F P ghost fields, c ~.

As is well known, the equation of motion for the ghosts is obtained through the change c~{x)-*c~(x)+ ac~(x), where ac~(x) are infinitesimal and independent of c~(x). Thus the Jacobian of the t ransformation is unity, so tha t we get

(14) f DA De D~ exp [iS] ~S = 0.

Integrat ing by par t s in 8S and taking into account the independence of 8c~(x), we get from (14) the equation of motion for ca(x)

(15) /~ - a a /~ b - v a b - c = f DADcDgexp[ iS]{Oa+ a (Duv) --/bc(O At,)c --]b~w c ) 0

tha t mult ipl ied by w~(x) gives

(16) f DA DcD~ exp [iS] wa]~o(auA~) Oc = wa~,Z + w ~ Ou - - (--ia)z

By substi tut ion of (16) into (13), we obtain the following compact expression for the generalized WT ident i ty in terms of Z:

( - - ia )Z ( - - O ) Z + ou (17) 4x J~ ~u~-- + av ------U- 8]~ aw ~ ]

[ + w~ Z + ~

\ / J 0 .

In terms of the generating functional for connected Green's functions, W = - - i In Z,

N E W GENERALIZED WARD-TAKAHASHI IDENTITIES IN GAUGE THEORIES ~

(17) can be wri t ten as

Iden t i ty (18) can be t ranslated in terms of the generating functional for one-par- ticle irreducible Green's functions F, defined, as usual, by the Legendre t ransformation

(t9) f 4 "a a W[~, #, ~; u, v, w] = P[~, ~, "~; u, v, w] + d z{~,2i + ao~ + ~ r

becoming

(20) +

We can simplify (20) by means of the auxil iar functional

(21)

so tha t (20) can be writ ten as

(22) fd r r' 1= [~Au 8ut, 8v ~ 8"~a ~w a ~6aj o.

Equat ion (22) is the generalized WT ident i ty obtained from the invariance of the action under A. Addit ional information for the functional ~F'/3u~ is obtained from the equation of motion for c~(x), (15), which, in terms of F ' , reads as

$F ' 8F ' { ~a~ . ~ } (23) --8a--g-F O u ~ = ]~c ( wb-? auAt,)O" --*~'~ (Ot~ fli~) .

By derivat ion of (22) with respect to A:~ and ~ , we obtain, in the l imit where all the sources vanish

(24) 0

and, in the same l imit , the derivat ion of (23) with respect to ~c leads to

~ * F ' 8 2 F ' (25) ~08a ~ ~ ~o~u~ o .

From the structure of the to ta l action (9) we deduce the ident i ty

(26) ~1" ~F'

Sw~(x) a(a.~IT,(x) )

374 ~. qU~R6S and F. a. DE Umd~s

that, together with the proper~y ~ S F ' / 8 ( ~ . ~ ) = - 8 F ' / 8 ~ , gives the identi ty

(g7) - - a

The substi tution of (25) and (27) into (24) leads to

~ F ' 82F ' a b

and with the use of the parametrization (8)

(29) - - ~ . 2 ~ ( z - - z ) ~.o,

the transversality of the self-energy for the gauge field associated to /~ ' is easily proved. We have shown that a new invariance for the action of gauge theories does exist

and we have found the related generalized WT identities. We hope that this additional invariance and identities might be of interest in future developments of gauge theories.

We will conclude this note with a few remarks, i) The mathematical structure we have introduced (8) is a tr ivial principal bundle whose base space is again a trivial bundle over P(M, G), M being the space-time manifold (locally R 4) and G the gauge group. The principal fibre bundle P is not trivial, so that instanton solutions are allowed. if) All our conclusions can be trivially enlarged for the general class of Lorentz gauges, - - (1/2~)(~A~) ~. Looking for more general gauges of the type - - (1/2~)(~A~ + Fa(A)) 2, we find that the invariance of the action under A and z~ implies ~a = 0. The point is tha t our geometrical construction determines the strong constraint (2e), so that once we choose A~ to get ~5~ = 0 (this can be done in any gauge), ~c is fixed by (2e) and only in the Lorentz gauge we obtain a 4-divergence for z~s In this way the Lorentz gauge seems to be distinyuished in the sense that the action has a wider symmetry.

General Lagrangians invariant under BRS transformations have been built by CVRCI and F ~ R ~ I (~) and by F~m~ARA, PmU~T and SCtIW~])A (s), using superfield techniques. Recently, BO~ORA and ToNI~ (9) have extended the Ferrara et al. for- malism to find the more general Lagrangian invariant under both transformations A and zT, and OJIMA Q0) has applied the ~ transformation to the case of gravity.

The authors are grateful to Prof. M. ToNI~ for his critical suggestion concerning equations (24) and (28).

(D G. CURCI a n d R. FERRARI: Phys. Leff. B, 63, 91 (1976). These au tho r s h a v e used a partie~flar A t r a n s f o r m a t i o n sa t i s fy ing our e q u a t i o n (2e).- (s) S. FERRARA, O. PIGUET and M. SCHWEDA: NuCL -Phys. B, 119, 493 (1977). (o) L. BO~CORA and M. To,tIN: Phys. Left. B, 98, 48 (1981); L. BONORA, P. PASTI and M. TONIN: (Teomefrie description o/ extended B R S symmetry in super]ield formulation, I F P T 2~/80 (October 1980). (10) I . OJIMA: Prog. Theor. Phys., 64, 625 (1980).