Gauge invariant operators and closed string scattering in open string field theory

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  • Physics Letters B 536 (2002) 129137www.elsevier.com/locate/npe

    Gauge invariant operators and closed string scattering inopen string field theory

    Mohsen Alishahiha a, Mohammad R. Garousi b,a,c

    a Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5531, Tehran, Iranb Department of Physics, Ferdowsi University, Mashhad, Iranc Department of Physics, University of Birjand, Birjand, Iran

    Received 6 February 2002; accepted 18 April 2002

    Editor: L. Alvarez-Gaum

    Abstract

    Using the recent proposal for the observables in open string field theory, we explicitly compute the coupling of closed stringtachyon and massless states with the open string states up to level two. Using these couplings, we then calculate the tree levelS-matrix elements of two closed string tachyons or two massless states in the open string field theory. Up to some contact terms,the results reproduce exactly the corresponding amplitudes in the bosonic string theory. 2002 Elsevier Science B.V. All rightsreserved.

    1. Introduction

    The open string tachyon condensation has attractedmuch interest recently. Regarding the recent devel-opment in string field theory (for example, see [1]and [2] and their references), it is believed that theopen string field theory [3] might provide a direct ap-proach to study the physics of string theory tachyonand could give striking evidence for the tachyon con-densation conjecture regarding the decay of unstableD-branes or the annihilation of braneanti-brane sys-tem [4]. Therefore it would be very interesting to studyand develop the structure of the open string field the-ory itself.

    E-mail addresses: alishah@theory.ipm.ac.ir (M. Alishahiha),garousi@theory.ipm.ac.ir (M.R. Garousi).

    On the other hand the most difficult part of theSens conjecture for open string tachyon is the waythe closed string emerges in the tachyonic vacuum.So it would be a natural question to ask that how onecan see the closed string states in the open string fieldtheory. In fact it has been shown that the off-shellclosed strings arise because certain one-loop openstring diagrams can be cut in a manner that produces aclosed string pole [5]. Therefore unitarity implies thatthey should also appear as asymptotic states. Of courseone cannot remedy this by adding an explicit closed-string field to the theory. This would just double theresidue of the pole. They cannot be also considered asa bound states, since they appear in the perturbationtheory. Closed string in open string field theory hasbeen studied in several papers including [69].

    In an other attempt but related to the closed stringstates in the open string field theory, the gauge invari-

    0370-2693/02/$ see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(02)0 18 06 -3

  • 130 M. Alishahiha, M.R. Garousi / Physics Letters B 536 (2002) 129137

    ant operators in open string field theory have beenconsidered in [10,11]. These gauge invariant opera-tors could also provide us the on-shell closed string inthe open string field theory. In fact these operators areparameterized by on-shell closed string vertex opera-tors and can arise from an open/closed transition ver-tex that emerged in one-loop open string theory. Actu-ally this open/closed vertex was studied in [9] whereit was shown that supplemented with open string ver-tex it would generate a cover of the moduli spaces ofsurfaces involving open and closed string punctures.

    It has also been suggested in [10,11] that the cor-relation function of these gauge invariant operatorscould be interpreted as the on-shell scattering ampli-tude of the closed strings from D-brane. This is theaim of this Letter to study this correspondence in moredetail. We shall study the scattering amplitude of twoclosed string states off a D-brane in the framework ofthe open string field theory by making use of thesegauge invariant operators.

    The Letter is organized as follows. In Section 2, weshall review the open string field theory action as wellas the gauge invariant operators introduced in [10,11].In Section 3 we will evaluate the scattering amplitudeof the closed strings in the framework of string fieldtheory. In Section 4 the same scattering amplitudeswill be obtained in the bosonic string theory where wewill show that up to some contact terms, the results arein agreement with the open string field theory results.The Section 5 is devoted to the discussion and somecomments.

    2. Open string field theory

    In this section we shall review the open stringfield theory and the structure of the gauge invariantoperators which could provide observables of the openstring field theory.

    2.1. Cubic string field theory action

    The cubic open string field theory action is givenby

    (1)S( )= 12

    ( Q + 2go

    3

    ),

    which is invariant under the gauge transformation =Q+ go go . Here go is the openstring coupling, Q is the BRST charge and the stringfield, , is a ghost number one state in the Hilbertspace of the first-quantized string theory which can beexpanded using the Fock space basis as1

    | =

    dp+1k( +A1 + ib1c0

    + i2B

    2 +

    12B

    1

    1+ 0b2c0 + 1b1c1+ ik1b1c0 +

    )c1|k.

    The gauge invariance of (1) can be fixed can bychoosing FeynmanSiegel gauge b0| = 0. In thisgauge the truncated field up to level two reads

    | =

    dp+1k((k)+A(k)1 +

    i2B(k)

    2

    + 12B(k)

    1

    1 + 1(k)b1c1

    )c1|k.

    The corresponding string vertex is given by

    (0)=

    dp+1k[(k)c(0)+ iA(k)cX(0)

    12B(k)c

    2X(0)

    12B(k)cX

    X(0)

    (2) 121(k)

    2c(0)]e2ikX(0).

    In writing the above vertex, we have used the doublingtrick [12]. Hence, the worldsheet field X(z) in aboveequation is only holomorphic part of X(z, z).

    To make sense out of the abstract form of the openstring field theory action, one can use CFT method.In this method we usually use the conformal mappingand calculation of the correlation function of a CFTon a disk or upper-half plane [13,14]. In the CFT

    1 Here, we use the convention fixed in [12] that uses the Vand N matrices for projecting a spacetime field to its componentin the worldvolume and transverse spaces, respectively. So in thisconvention , = 0,1,2, . . . ,25, and A1 = A V 1 +A N 1. Our conventions also set = 2.

  • M. Alishahiha, M.R. Garousi / Physics Letters B 536 (2002) 129137 131

    language the n-string vertex is defined by2 = f (n)1 (0) f (n)2 (0) f (n)n (0) UHP,

    where f (n)k (0) denotes the conformal transforma-tion of the vertex operator (0) by the conformal mapf(n)k . Here UHP denotes correlation function on the

    upper-half plane and the conformal map f (n)k is de-fined as

    f(n)k (zk)= g

    (e

    2in(k1)

    (1+ izk1 izk

    )2/n), 1 k n,

    where g( )=i 1+1 . Therefore the open string action

    can be calculated as following in terms of correlationfunctions of the CFT on the UHP

    S =14

    f(2)2 (0)f (2)1

    (Q(0)

    )+ 2go

    3f(3)1 (0) f (3)2 (0)f (3)3 (0)

    UHP

    .

    Form this expression the kinetic terms up to level twofields read

    Squad =

    dp+1x(1

    2

    + 142

    12A

    A 12B

    B 14BB

    12B

    B 14BB

    (3)+ 121

    1 + 1421

    ),

    which can be used to write the spacetime propagatorsof the corresponding fields.

    2.2. Gauge invariant operator

    The gauge invariant operators in string field theoryhave been constructed in [10,11]. The general formof these operators are given by O = gc

    V , where

    gc is the closed string coupling and V is an on-shellclosed string vertex operator with dimension (0,0). In

    2 We assume that there is a normal order sign between fields atdifferent points in the correlation functions.

    order to be gauge invariant, the closed string vertexoperator has to be inserted at the midpoint of openstring. From open string point of view,V is an operatorwhich acts on a string field. Given any on-shell closedstring vertex operator V , the gauge invariant operatorO can be obtained, using the CFT method, in terms ofthe open string field

    (4)O = gc

    V = gc

    cV(i) cV(i)f (1)1 (0)

    UHP ,

    where f (1)1 = 2z1z2 and V(z)V(z) is the matter part ofthe closed string vertex operator.

    This form of the gauge invariant operator can beunderstood from the closed/open vertex studied in [9],where is was shown that the extended open string fieldtheory with the action

    S =14

    ( Q + 2go

    3

    )(5)+ gc

    V,

    with V being an on-shell closed string vertex definedat the midpoint of the open string, would provide a the-ory which covers the full moduli space of the scatter-ing amplitudes of open and closed string with a bound-ary. We note, however, that scattering amplitudes ofopen and closed string with a boundary are actually theclosed string scattering off a D-brane. We should thenbe able to reproduce the closed string scattering ampli-tudes in the framework of the open string field theory.In the next section we are going to write down the ex-plicit form of the gauge invariant operator as well astheir correlation function among themselves to see towhat extent we can reproduce the known results of theclosed string scattering amplitudes from a D-brane inthe bosonic string theory [15,16].

    3. Scattering amplitudes in string field theory

    In this section we will consider the gauge invariantoperators in the string field theory. Using CFT methodwe shall compute the explicit form of the operatorsin terms of spacetime open string fields. According tothe proposed action (5) the result can be thought as anspacetime action representing the closed/open vertex.We shall also compute the correlation function of