Ward identities in open string field theory

Volume 197, number 1,2 PHYSICS LETTERS B 22 October 1987

WARD IDENTITIES IN OPEN STRING FIELD THEORY

S.P. DE ALWIS l Department of Physics, University of Texas at Austin, Austin, TX 78712, USA

M.T. GRISARU 2 Department of Physics, Brandeis University. Waltham, MA 02254, USA

and

L. MEZINCESCU 3 Department of Physics, University of Miami, Coral Gables, FL 33124, USA

Received 20 July 1987

We use the BRST invariant gauge fixed action for open string field theory to derive the Ward identities of the latter.

It has long been known [ 1 ] that a Lorentz invariant S-matrix for the scattering of massless vector particles (photons) entails the invariance of the amplitude under the linear gauge transformation eu--,eu+2k u where E u, k u are, respectively, the polar- ization and momentum vectors of the photon. In other words the scattering amplitude cUT u involving at least one photon must satisfy a Ward identity k~T~,=O.

In QED this Ward identity is a simple consequence of the gauge invariance of the action under the linear gauge transformation A " ~ A ~'+ Ouq) of the photon field. In Yang-Mills theory this connection is not so obvious since now the theory is gauge invariant only under a non-linear gauge transformation. Actually matters are even more complicated due to the fact that in order to define a perturbation series the gauge has to be fixed. Nevertheless the existence of a BRST symmetry in the gauge fixed quantum theory enables one to derive the Ward identity.

In first quantized string theory the Ward identity

Supported in part by the Robert A. Welch foundation and NSF grant no. PHY-86-05978.

2 Supported in part by NSF grant no. PHY-83-13243. 3 Supported in part by NSF grant no. PHY-87-03390.

is the statement that the S-matrix is invariant under the replacement of any vertex operator V by V+ [ Q, U] where U is an arbitrary vertex operator. This may be proved by conformal field theory methods [2] (up to anomalies arising from degenerate Riemann surfaces). In string field theory [3-5] the statement of the Ward identity may be formulated as follows. One defines an S-matrix generating functional 5e[q~0] whose argument is the asymptotic string field ~o satisfying Qq~0=0. Then 5e[~o] should be invariant under q~o--.q~o+QV'. Now of course given the one to one correspondence between asymptotic string fields and the vertex operators, and the result established by Giddings, Martinec and Witten [6] that the sum of the Feynman diagrams of string field theory at a given order of perturbation theory gives a cover of the moduli space of the Rie- mann surface relevant to that order, the Ward identity follows immediately since equivalence with the first quantized theory is established. However, we believe that it is important to demonstrate the valid- ity of the identity by direct field theory arguments without recourse to the arguments of ref. [ 6 ]. Such arguments should shed some light on the structure of the field theory and they are, at least in some formal

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sense, independent of perturbation theory. Hope- fully they will give some insight into the relation between the algebraic approach of the field theory and the geometric approach of the first quantized theory. Indeed as argued by Polchinski and Cai [7] and independent demonstration of the Ward identity would imply that the field theory Feynman diagrams cover all of moduli space uniformly, though it does not show that it covers it only once.

In this letter we will discuss the derivation of the Ward identity for open string field theory [ 3-5 ]. We will first establish it at tree level. At loop order we can establish it only by being somewhat cavalier about infinities arising from self-contractions of string fields. We will work here with Witten's open string field theory but our arguments can be trivially extended to any other gauge invariant field theory [4,51.

The gauge fixed action (in the Siegel gauge) for Witten's string field theory [8,9] is

1 1 I[ ¢ , B l = ~ f ~ , ,Q~,+sf q ) , q , , ¢+ fB ,bo¢ . (1) In the above

+ c o

q~= E a ,_, /2 (2) - - o o

assembles the string fields A._ 1/2 (with ghost number n - l / 2 ) into one master-field [4] which is Grassmann odd. (1) is invariant under the BRST transformation

sq)=Q~+q),q)-boB, sB=O. (3)

We note that s is nill-potent only on-shell,

s 2 ~ = [Q+ q~, boB] = [Q+ q~, 8I/5 q) l . (4)

To establish the last equality we made use of the Bianchi identity for the curvature associated with ¢ .

The generating functional for string Green's func- tionals is

Z[ J, L, K] =exp( iW[J , L, K])

+if L ,B+i fK.sqb) . (5)

In the above we have explicitly introduced a source for the BRST variation of q~ in order to derive the Slavnov-Taylor identities ~i. Now from (2) and (3)

sA._ 1/2 = QA._ 3/2 + ~ Ap_ 1/2 * A n - p - 3 ~ 2 - boB.+ 1/2 • P

Hence ~2

• s4~= ~ 6~,_,2,sA,_ 1/2 = 0 , (6)

using the Grassmann oddness of A,_ 1/2 and assum- ing that the high frequency modes can be regulated so as to make sense of the integral. Then we have, using ( 6 ) ,

O= f [dcI)] [dB] f h , s ¢ , exp(iI+...)

= f [d' l [dBl fs¢'* Sfi- exp(iI+...)

=if[d l [aBl(fJ, sa,-fK,[Q+ ,Fl) × e x p ( i I + . . . ) . (7)

In the last line we have used the BRST invariance of I and eq. (4). (Here F=8I/8~). Thus we have the Slavnov-Taylor identity

£2Z[ J, L, K] = 0 , (8)

where

(2=- f J , ~ + K , [ Q - i S / S J , F] ,

with (/) --, ir /S J, B--,iS/SL in F. We note that

= - J J , [ Q - i8/8 J, F] + .... (9) /2 2

The ellipses represent terms which are 0 (K× 51/8 q~). Now let q~c be a classical string field and q)o be a

classical solution of the free field equation £oq)o--- ( L o - 1) q~o=0, where Lo is the world sheet hamiltonian for coordinates plus ghosts. The S-matrix generating functional may then be defined by [ 10]

5°[40] =exp(Zo))Z[J, O, 0] 1~2o ~ ° , (10)

~ For a recent review see ref. [ 10]. ~2 Eq. (6) is true in our case because of the sum over all n. For

an argument that the second equality is true term by term in n see ref. [ 11 ].

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where

2~c = fq~ c - 8 * Co Lo~-), (1 1)

when ~o is also taken to satisfy the gauge condition bo~o= 0, Q~ o= 0. (Here 0 is the piece independent of ghost zero-modes in the BRST charge Q=co£ +boM+Q.) From (7) and (10) we have

[S, g2]= f q)*CoLO~K. (12)

Hence

[S, g2]Z(J, L, K) = i f [dq)] [dB]

×f¢c * coLo( QdP + qb . q5 ' boB) exp (iI) .

Let us for the time being ignore wave function renor- malization, so that the present discussion is ony valid at tree level. In the limit q~c--'~o the expression fq~c*co(...) would be zero were it not for the one string poles coming from the linear term in ¢ inside the parentheses. Using also the fact that [Lo. Q] = 0 and the gauge condition boq~ =0 inside the functional integral we have

[S@c , £2]Z[J, K, L] @c~@o J,K,L=O

=X~0,oZ[j, . . . . a , ~ o (13) L,, 1~1 ]J,K,L,=O

where

So~c = O ~ • (14)

Let us now show the decoupling of (on-shell asymptotic) spurious states, i.e. the invariance of the S-matrix under q)o--,q%+SokU. From (9) we have

5~[ q% + So 71]

=exp(27,o~) exp(Z',~o)Z[J , K, L] Io

=exp[Z~ , f21 exp(22~o)Z[J, L, g l Io

=~e[q,o]

1 + ~ l~[Z~, , g2]" exp(X~o)Z[J,L,K] [0 • (15)

In the second equation above we have used (13) with q~o-O W. Now the task is to show that the infinite series in the last equation is zero. First we observe

from (8) and (13) with Oq%=0

.., g2 exp(Z~o)Zlo = 0 . (16)

In the above the dots represent products of g227~, and Z~,t2. Also the terms of the form

12 ... exp(27~o)Z I o = 0 , (17)

since £2 vanishes for J=K=L=O. All other terms contain at least one factor of ~2 and vanish when J = K = 0 and when the fact that F=8I/8~ is used inside the functional integral. Let us look at this in more detail. Consider [2~,, £2]nexp(27~o)Z. For n/> 2 this is

[£, £2] 2 [Z, £2] ~-2 exp(X~o) ZI 0

× [X~,, £2] n-2 exp(27~o)ZIo.

The second and third terms in parentheses are zero (see(17)) and the fourth is

--SkL'~'~2S~tt (Sso ~t~) n-2 exp(Z~o)Z[ o

= - f [ d ~ ] [dB] f ~*Co£o[Q+~,8I/8~l

x f ,CoLo f OY ,CoLo @ 2

×exp(Z~ o) exp(iI+. . .)]o • (18)

Using also the fact that

= (818 q)) f ~ o *Co£o ~ =0

because of the on-shell conditions on qbo and ~ , we find that the above term is zero. Thus were are left with

X~E2,S~,f2[Z~,, E2] n-2 exp(X~o)Z] o •

Obviously the process can be repeated with arguments similar to those given above being used to show the vanishing of ff~2 terms until we are left with the LHS of (16) which vanishes.

Thus we have shown the required Ward identity at tree level. Before we investigate what happens in

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higher orders, it is perhaps instructive to illustrate how the Ward identity is obtained diagrammatically at the level of the four-point tree amplitude. This is given by the s, t, and u channel diagrams. The explicit expressions for the s-channel diagrams for instance is given by

12boL5 I ( A ] 2 ( A I 5(01Vt,21510) s - - I

X 3(AI 4(AI V[3415 10>345 •

The labels 1, 2 etc. refer to the different string Fock spaces. The antisymmetrization of 1, 2 and 3, 4 is a result of the Grassmann character of the string state (A I- Let us see what happens when one of the string states say 1 (AI is replaced by t (21QI. Using the con- servation of the BRST charge at the vertices, the physical state conditions ( A [ Q = 0 and the anti- commutator {bo, Q} =£o, it is easy to see [ 10] that one gets a contact term

~(A] 2(A] 3(A] 4(AI 5(01 Vtl2]~

~V[3415 10)5 10)1234 •

Similar contact terms are generated from the t and u channel diagrams so that the Ward identity is sat- isfied if the following relation is true:

V[12][34] + V[3111241 --t- V[231[141 ---.~ 0 , (19)

where VEo]t~.q=m(0[ V[ijl,nV[kllmlO)m. The relative signs in (19) are fixed by the total antisymmetry of the amplitude under interchange of any pair of strings. Now the associativity of the star algebra is equivalent to the statement that Vo~ z is cyclically symmetric. In fact Gross and Jevicki [ 12] proved associativity by showing the latter. The identity (19) is then easily seen to be a trivial consequence of this fact.

Let us now discuss (albeit in somewhat heuristic fashion) the extension to arbitrary loop order in string perturbation theory. First we define an effec- tive action F in the usual way,

F[~c, B c K ] = W [ J , L , K ] - f J * ~ o - f L , B ~ , (20)

with

cI2~=SW/SJ, B~=SW/SK, J=-SFI8~c,

L = - S F / g B ~ , 8F/gKIq~¢,Bm~)W/6KIj, L. (21)

8Z/SK=i(8 W/SK) exp(iW)

=ftd dB] (Q~+~,~ -boB) exp(i/+. . . )

= i [ Qq)c + q~ * q~c - boBs] exp( iW) ,

ignoring a term 8 q~c [ 1 ]/8 J[ 1 ] = [ 821"/ 8qb[1]gqb[1]] - l , which should be zero since the quantity in the square bracket (being of the form a(0)...) is infinite. So we have (using (21))

8F/SK= Q ~ + ~¢ • ~c - boBc . (22)

We replace (10) by

a 8F 8 Z~c= ~s[118~[118~tc[2] oa j t [2] ' (23)

where we have expanded the string fields in a basis q~=E&ls) (see e.g. ref. [8]) and repeated indices (arguments) are summed (integrated). Putting 3]t2=SeF/~c[118(b~[2] Io and taking ~b o to be a solution of A12~bo[2] =0, we have

[Z,~c, t2] = f ~ ' 3 ' 8 ~

A 8J[3] 0 8 - + f ~ c [ 1 1 . ~ , 2 . ~ 8J[31 •

In the above, the dot product represents the sums over modes whose indices have been suppressed, the arrows imply taking the limit ~¢-+~b o when the expression is acting on Z, and the result amounts to keeping ony the single string pole contribution in the limit as ~c-+~bo. From (21) and (22),

8J[2] o - 8 K [ 8 2 / " o 8K[11 116q~[21 = Q a [ 1 - 2 ] .

Then we have the result (13) with 27~ c now defined by (23). The rest of the argument goes through as before. Therefore, modulo questions of regularization the Ward identity re[ 4 0 + 0 g j] = 6P[ ~bo] holds to all orders in perturbation theory and indeed inde- pendently of perturbation theory. Presumably a more careful treatment of the question of regularization would clarify issues related to BRST anomalies introduced by the boundaries of moduli space.

Now we have

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We wish to thank A. S t rominger , C. T h o r n and B.

Zwiebach for discussions.

References

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(1986) 93. [3] E. Witten, Nucl. Phys. B 268 (1986) 253. [4] W. Siegel, Phys. Lett. B 149 (1984) 157, 162;B 151 (1985)

391,396; H. Hata, K. Itoh, T. Kugo, H. Kunimoto and K. Ogawa, Phys. Lett. B 172 (1986) 186; Phys. Rev. D 34 (1986) 2360.

[ 5] A. Neveu and P.C. West, Phys. Lett. B 168 (1986) 192; Nuel. Phys. B 278 (1986) 601.

[6] S. Giddings, E. Martinec and E. Witten, Phys. Lett. B 176 (1986) 362.

[ 7 ] J. Polehinski and Y. Cai, Texas preprint UTTGH (1987). [8] C.B. Thorn, Nucl. Phys. B 287 (1987) 61. [9] M. Bochicchio, Phys. Lett. B 188 (1987) 332; B 193 (1987)

31. [ 10 ] H. Osborn, Lectures on gauge theory methods, Cambridge

preprint DAMTP/86-14. [ 11 ] A.R. Bogojevic, preprint BROWN-HET-615 (1987). [ 12 ] D. Gross and A. Jevicki, Princeton preprints ( 1986 ).

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Ward identities in open string field theory