Ward identities in open string field theory

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Text of Ward identities in open string field theory

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


    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


    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 invar- iant 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 conse- quence 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 invar- iant 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 meth- ods [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 func- tional 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 iden- tity 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

    96 0370-2693/87/$ 03.50 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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

    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 iden- tity would imply that the field theory Feynman dia- grams 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


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

    assembles the string fields A._ 1/2 (with ghost num- ber 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,

    s2~= [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+ifK.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 *An-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] fh , s , exp(iI+...)

    = f [d' l [dBl fs'* Sfi- exp(iI+...)

    =if[d l [aBl(fJ, sa,-fK,[Q+ ,Fl) exp( 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)


    (2=- f J ,~+K, [Q- iS /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--- ( Lo - 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 ].


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


    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)


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

    fc * 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 func- tional 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


    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


    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 ~*Coo[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 *Coo ~ =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 argu- ments 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


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

    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- se