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Physics Letters B 536 (2002) 129–137

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Gauge invariant operators and closed string scattering inopen string field theory

Mohsen Alishahihaa, Mohammad R. Garousib,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-Gaumé

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 brane–anti-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:[email protected] (M. Alishahiha),[email protected] (M.R. Garousi).

On the other hand the most difficult part of theSen’s 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 [6–9].

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

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130 M. Alishahiha, M.R. Garousi / Physics Letters B 536 (2002) 129–137

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(Ψ )= − 1

2α′

(∫Ψ ∗QΨ + 2go

3Ψ ∗Ψ ∗Ψ

),

which is invariant under the gauge transformationδΨ =QΛ+ goΨ ∗Λ− goΛ ∗Ψ . Herego is the openstring coupling,Q is the BRST charge and the stringfield, Ψ , is a ghost number one state in the Hilbertspace of the first-quantized string theory which can beexpanded using the Fock space basis as1

|Ψ 〉 =∫

dp+1k

(φ +Aµα

µ−1 + iαb−1c0

+ i√2Bµα

µ−2 + 1√

2Bµνα

µ−1α

ν−1

+ β0b−2c0 + β1b−1c−1

+ ikµαµ−1b1c0 + · · ·

)c1|k〉.

The gauge invariance of (1) can be fixed can bychoosing Feynman–Siegel gaugeb0|Ψ 〉 = 0. In thisgauge the truncated field up to level two reads

|Ψ 〉 =∫

dp+1k

(φ(k)+Aµ(k)α

µ−1 + i√

2Bµ(k)α

µ−2

+ 1√2Bµν(k)α

µ−1α

ν−1 + β1(k)b−1c−1

)c1|k〉.

The corresponding string vertex is given by

Ψ (0)=∫

dp+1k

[φ(k)c(0)+ iAµ(k)c∂X

µ(0)

− 1√2Bµ(k)c∂

2Xµ(0)

− 1√2Bµν(k)c∂X

µ∂Xν(0)

(2)− 1

2β1(k)∂

2c(0)

]e2ik·X(0).

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

andN matrices for projecting a spacetime field to its componentin the worldvolume and transverse spaces, respectively. So in thisconventionµ,ν = 0,1,2, . . . ,25, andAµα

µ−1 = A · V · α−1 +

A ·N · α−1. Our conventions also setα′ = 2.

M. Alishahiha, M.R. Garousi / Physics Letters B 536 (2002) 129–137 131

language then-string vertex is defined by2∫Ψ ∗Ψ ∗ · · · ∗Ψ= ⟨

f(n)1 ◦Ψ (0) f (n)

2 ◦Ψ (0) · · ·f (n)n ◦Ψ (0)

⟩UHP,

wheref (n)k ◦ Ψ (0) denotes the conformal transforma-

tion of the vertex operatorΨ (0) by the conformal mapf(n)k . Here〈〉UHP denotes correlation function on the

upper-half plane and the conformal mapf (n)k is de-

fined as

f(n)k (zk)= g

(e

2πin(k−1)

(1+ izk

1− izk

)2/n), 1 � k � n,

whereg(ζ )= −iζ−1ζ+1. Therefore the open string action

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

S = −1

4

⟨f(2)2 ◦Ψ (0)f (2)

1 ◦ (QΨ(0))

+ 2go3

f(3)1 ◦Ψ (0) f (3)

2 ◦Ψ (0)f (3)3 ◦Ψ (0)

⟩UHP

.

Form this expression the kinetic terms up to level twofields read

Squad=∫

dp+1x

(−1

2∂µφ∂

µφ + 1

4φ2

− 1

2∂µAν∂

µAν − 1

2∂µBν∂

µBν − 1

4BµB

µ

− 1

2∂λBµν∂

λBµν − 1

4BµνB

µν

(3)+ 1

2∂µβ1∂

µβ1 + 1

4β2

1

),

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 byO = gc

∫VΨ , where

gc is the closed string coupling andV 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 operatorV , the gauge invariant operatorO can be obtained, using the CFT method, in terms ofthe open string field

(4)

O = gc

∫VΨ = gc

⟨cV(i) c V(−i)f

(1)1 ◦Ψ (0)

⟩UHP ,

wheref (1)1 = 2z

1−z2 andV(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 = −1

4

(∫Ψ ∗QΨ + 2go

3Ψ ∗Ψ ∗Ψ

)(5)+ gc

∫VΨ,

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


these operators among themselves. These correlatorsshould be interpreted as the closed string scatteringamplitude off a D-brane. We shall perform all ofour computations in the level truncation up to leveltwo. In evaluating the correlations (4), one needs thetransformation of the vertex (2) under conformal mapf(1)1 . Using the following propagator

(6)⟨Xµ(z)Xν(w)

⟩= −ηµν ln(z−w),

one finds that the different terms in (2) transform undera general conformal mapf as

f ◦ (ce2ik·X)= f ′2k2−1ce2ik·X,f ◦ (c∂Xµe2ik·X)

= f ′2k2(∂Xµ − ikµf ′′/f ′2)ce2ik·X,

f ◦ (c∂2Xµe2ik·X)= f ′2k2+1

[∂2Xµ + (

f ′′/f ′2)∂Xµ

− ikµ/6(4f ′′′/f ′3 − 3

(f ′′/f ′2)2)]ce2ik·X,

f ◦ (c∂Xµ∂Xνe2ik·X)= f ′2k2+1

[∂Xµ∂Xν − 2i

(f ′′/f ′2)k{µ∂Xν}

− (f ′′/f ′2)2kµkν − 1/12

(2f ′′′/f ′3

− 3(f ′′/f ′2)2)ηµν]ce2ik·X,

f ◦ (∂2ce2ik·X)= f ′2k2+1

[∂2c− (

f ′′/f ′2)∂c(7)−

(f ′′′/f ′3 − 2

(f ′′/f ′2)2)c]e2ik·X.

Note that in the above equations the worldsheet fieldson the right-hand side are functions off (z), alsok2 = k · k. The functionf (1)

1 and its derivatives atpoint z = 0 that should be inserted in the above trans-formations are:f (1)

1 (0)= 0, f ′ (1)1 (0)= 2, f ′′ (1)

1 (0)=0, f ′′′ (1)

1 (0) = 12. The correlation functions over theghost field that left over are〈c(i)c(−i)c(0)〉 = 2i and〈c(i)c(−i)∂2c(0)〉 = 4i.

Plugging the conformal transformation (7) intoEq. (4) and using above correlators for ghost part, weget

O = 2igc

∫dp+1k 22k2

×[(

1

2φ + i

√2B · V · k + 1

2√

2Bµ

µ − 1

2β1

)× ⟨

V(i) V(−i)e2ik·X(0)⟩

+ iAµ

⟨V(i) V(−i)∂Xµe2ik·X(0)

⟩− √

2Bµ

⟨V(i) V(−i)∂2Xµe2ik·X(0)

⟩(8)− √

2Bµν

⟨V(i) V(−i)∂Xµ∂Xνe2ik·X(0)

⟩],

where all correlations should be evaluated on theupper-half plane. Now we have all ingredients we needto compute the open string field theory observable (4).We will do this for both closed string tachyon andmassless states.

3.1. Tachyon amplitude

The matter part of the vertex operator of closedstring tachyon inserted at the midpoint of open stringwith momentumpµ (p · p = 2) is given by

V(i) V(−i)= eip.X(i)eip.D.X(−i),

where 2V = η + D, and we have used the doublingtrick [12]. Plugging this operator into Eq. (8) andusing the standard propagator (6), one can evaluate thecorrelators in (8). The result is

O(pµ)= igc

8(2π)p+1e4 ln2p·V ·p

×(φ + 4i p ·N ·A+ 2i

√2p · V ·B

− 8√

2p ·N ·B ·N · p + 1√2Bµ

µ − β1

),

here the spacetime fields are functions of−p · V .Fourier-transforming to the position space, e.g.,

φ(k)=∫

dp+1x

(2π)p+1φ(x) e−ik.x,

the operatorO becomes

O(pµ)= igc

8

∫dp+1x eip.x

×(φ(x)+ 4i p ·N · A(x)

+ 2i√

2p · V · B(x)− 8

√2p ·N · B(x) ·N · p

(9)+ 1√2Bµ

µ(x)− β1(x)

),


where the tilde sign over fields means, e.g.,φ(x) =e−4 ln2∂2

φ(x). According to the proposed action (5)the expression (9) is spacetime action representing thecoupling of the closed string tachyon with the openstring fields.

Having a gauge invariant operator one would pro-ceed to compute the correlation function of this gaugeinvariant operator. Since this operator is supposed tobe state corresponding to the on-shell closed string,therefore the correlation function of this operatorshould give the scattering amplitude of the closedstring off a D-brane. Now we are going to computeexplicitly this correlation function to see if we can re-produce the corresponding amplitudes in the bosonicstring theory. We shall do this up to level two trunca-tion in open string field.

Consider the following two point function

(10)⟨O(p1)O(p2)

⟩,

where〈〉 denotes the correlation function in the stringfield theory. According to what we have said, thisshould be interpreted as theS-matrix elements of twoclosed string states. In order to evaluate the abovecorrelation, one needs the propagator of the spacetimeopen string fields which can be obtained from thekinetic term of the string field theory action in (3), i.e.,

〈φ(x)φ(y)〉 = −i

∫dp+1k

eik·(x−y)

k2 − 12

,

⟨Aµ(x)Aν(y)

⟩= −i

∫dp+1k

ηµνeik·(x−y)

k2 ,

〈β1(x)β1(y)〉 = i

∫dp+1k

eik·(x−y)

k2 + 12

,

⟨Bµ(x)Bν(y)

⟩= −i

∫dp+1k

ηµνeik·(x−y)

k2 + 12

,⟨Bµν(x)Bλρ(y)

⟩(11)= − i

2

∫dp+1k

(ηµληνρ + ηµρηνλ)eik·(x−y)

k2 + 12

.

Now inserting (9) into (10) and using above propaga-tors, one finds

〈O(p1)O(p2)〉= ig2

c

2(2π)p+1δp+1(p1 + p2)

×{e8(s− 1

2 ) ln2

2s − 1− e8s ln2p1 ·N · p2

2s

+ e8(s+ 12 ) ln2[1

2(p1 ·N · p2)2 + 3

32(s + 12)− 1

8]2s + 1

},

wheres = p1 · V · p1. In the above expression, thoseterms in each pole which are proportional to thedenominator give contact terms in which we are notinterested. Hence, the pole structure of the amplitudeis

〈O(p1)O(p2)〉= ig2

c

2(2π)p+1δp+1(p1 + p2)

×{

1

2s − 1− p1 ·N · p2

2s

(12)+12(p1 ·N · p2)

2 − 18

2s + 1+ · · ·

},

where dots represent some contact terms. We shallshow that the above poles exactly reproduce thes-channel poles of the corresponding amplitude in thebosonic string theory.

3.2. Graviton amplitude

As an other example, let us to consider the masslessstates scattering off a D-brane in the framework ofthe string field theory. To do this we need to findobservables corresponding to the on-shell masslessclosed string states which could be dilaton, gravitonor Kalb–Ramond field. In other words we need tocompute the operator (8) for the corresponding vertexoperators. The matter part of these vertex operatorsinserted at the midpoint are given by

V(i) V(−i)

= (ε ·D)µν∂Xµ(i)eip·X(i)∂Xν(−i)eip·D·X(−i)

with pµpµ = 0 = pµεµν = εµνp

ν . For graviton wehaveεµν = ενµ andεµµ = 0 [12].

Plugging above closed string vertex operator intoEq. (8) and using the worldsheet propagator (6), onefinds

O(ε,p)= igc

8(2π)p+1e4 ln2p·V ·p

× (φ a + 4iAµb

µ + 2i√

2Bµcµ

− 8√

2Bµνdµν − β1a

),


here the spacetime fields are functions of−p · V . Thekinematic factorsa, bµ, cµ, anddµν are

a = Tr(ε ·D)− p ·D · ε ·D · p,bµ = a p ·Nµ + p ·D · ε ·Dµ − εµ ·D · p,cµ = a p · V µ − 4p ·D · ε ·Dµ − 4εµ ·D · p,dµν = a

(p ·Nµp ·Nν − 1

16ηµν

)+ 2(ε ·D){µν}

+ 2p ·D · ε ·D{µp ·Nν} − 2ε{µ ·D · pp ·Nν}.

Fourier-transforming the open string fields to theposition space, we get

O(ε,p)= igc

8

∫dp+1x

(φ(x)a + 4iAµ(x)b

µ

+ 2i√

2Bµ(x)cµ − 8

√2Bµν(x)d

µν

− β1(x)a),

where the tilded fields are defined the same as thosein (9). Inserting above operator in (10) and usingthe spacetime propagators (11), one can evaluate thescattering amplitude of two massless states from D-brane, that is

〈O(ε1,p1)O(ε2,p2)〉= ig2

c

2(2π)p+1δp+1(p1 + p2)

×{e8(s− 1

2 ) ln2a1a2

2s − 1− e8s ln2b1 · b2

2s

+ e8(s+ 12) ln2[− 1

32c1 · c2 + 12 Tr(d1 · d2)− 1

256a1a2]2s + 1

}= ig2

c

2(2π)p+1δp+1(p1 + p2)

×{

a1a2

2s − 1− b1 · b2

2s

(13)

+ − 132c1 · c2 + 1

2 Tr(d1 · d2)− 1256a1a2

2s + 1+ · · ·

},

where dots represent some contact terms.So far we have obtained theS-matrix elements of

two closed string states in the framework of the openstring field theory. Although we have only been ableto find thes-channel poles using level truncated openstring field, we will see that thet-channel can bealso obtained by taking into account the infinite terms

coming from the all level in the open string field. Wewill back to this point later in the conclusion. Nowwe are going to compare these results with the stringtheory computation.

4. Scattering amplitudes in string theory

In this section we study the scattering amplitude oftwo closed string states from D-brane in the bosonicstring theory.

4.1. Tachyon amplitude

Scattering amplitude of two closed string tachyonsfrom D-brane in the bosonic string theory is given bythe following correlation function:

A∼∫

d2z1 d2z2

× ⟨eip1·X(z1) eip1·D·X(z1) eip2·X(z2) eip2·D·X(z2)

⟩.

Using the propagator (6) it is straightforward calcula-tions to evaluate the above correlation and show thatthe resulting integrand isSL(2,R) invariant. Fixingthis symmetry by choosingz1 = iy and z2 = i, onearrives at

A= ig2c

2(2π)p+1δp+1(p1 +p2)

(14)×B(−1− t/2,−1+ 2s),

wheret = −(p1 + p2)2 ands = p1 · V · p1.

Consider the following definition of the beta func-tion:

(15)B(α,β)=∞∑n=0

1

α + n

(−1)n

n! (β − 1) · · · (β − n).

From this expression we see that the beta function hassimple poles atα,β = 0,−1,−2,−3, . . . . Each termof the above expansion has one simple pole in theα-channel, however, the poles in theβ-channel appearwhen adding all infinite poles of theα-channel in theabove expansion.

By making use of this expression for the betafunction the amplitude (14) reads

A= ig2c

2(2π)p+1δp+1(p1 +p2)


×{

1

2s − 1− (−2− t/2)

2s

+12! (−2− t/2)(−3− t/2)

2s + 1+ · · ·

}.

Comparing it with the corresponding amplitude inthe open string field theory (12), one finds that thetachonic pole is exactly the same. For the masslesspole we write−2 − t/2 = p1 · N · p2 − s. However,the term proportional tos in the massless pole givescontact term. So we get, up to some contact terms,exact agreement in two cases. Similarly, for the firstmassive pole one may rewrite it as

1

2

(−2− t

2

)(−3− t

2

)= 1

2(p1 ·N · p2)

2 − (p1 ·N · p2)

(s + 1

2

)

+ 1

2

(s + 1

2

)2

− 1

8.

Again terms that are proportional to(2s + 1) givecontact term. After dropping these terms, one findsexactly the massive pole as (12).

4.2. Graviton amplitude

The scattering amplitude of two massless closedstring states from D-brane in the bosonic string theoryis evaluated in [16]. The result is

A= ig2c

4(2π)p+1δp+1(p1 + p2)

× {e1B(−t/2,1 + 2s)+ e2B(−t/2,2s)

− e3B(1− t/2,2s)+ e4B(1− t/2,1+ 2s)

+ e5B(−1− t/2,1+ 2s)

+ e6B(1− t/2,−1+ 2s)

+ e7B(−1− t/2,−1+ 2s)

− e8B(−t/2,−1+ 2s)

− e9B(2− t/2,−1+ 2s)

(16)+ e10B(3 − t/2,−1+ 2s)},

where the kinematic factorse1, . . . , e10 are3

e1 = 1

2p2 · εT1 · ε2 · p1 + 1

2p2 · ε1 · εT2 · p1

+ p2 · ε1 ·D · ε2 · p2

+ p2 · εT1 · ε2 ·D · p2

+ p2 · ε1 · εT2 ·D · p2 + (1 ↔ 2),

e2 = −Tr(ε1 ·D)p1 · ε2 · p1 + (1↔ 2),

e3 = −p1 ·D · ε1 ·D · ε2 ·D · p2

+ 1

2p1 ·D · εT1 · ε2 ·D · p2

+ 1

2p1 ·D · ε1 · εT2 ·D · p2

− Tr(ε1 ·D)p1 ·D · ε2 ·D · p1 + (1↔ 2),

e4 = Tr(ε1 ·D · ε2 ·D)−p1 ·D · ε2 ·D · ε1 ·D · p2

− p2 ·D · ε1 ·D · ε2 ·D · p1,

e5 = Tr(ε1 · εT2

)− p1 · εT2 · ε1 · p2 − p2 · ε1 · εT2 · p1,

e6 = 1

2Tr(ε1 ·D)Tr(ε2 ·D)

− Tr(ε1 ·D)p2 ·D · ε2 ·D · p2

+ (p2 · ε1 · p2)(p1 ·D · ε2 ·D · p1)

+ 1

2(p2 · ε1 ·D · p2)(p1 · ε2 ·D · p1)

+ (p2 · ε1 ·D · p2)(p1 ·D · ε2 · p1)

+ 1

2(p2 ·D · ε1 · p2)(p1 ·D · ε2 · p1)

+ (1 ↔ 2),

e7 = (p1 · ε2 · p1)(p2 · ε1 · p2),

e8 = −(p2 · ε1 · p2)(p1 · ε2 ·D · p1 + p1 ·D · ε2 · p1)

+ (1 ↔ 2),

e9 = −(p2 ·D · ε1 ·D · p2)(p1 · ε2 ·D · p1

+ p1 ·D · ε2 · p1)+ (1 ↔ 2),

e10 = (p1 ·D · ε2 ·D · p1)(p2 ·D · ε1 ·D · p2).

Since in [16] the authors were interested in the lowenergy contact terms of the amplitude from which thelow energy effective action can be found, the aboveamplitude was expanded in a limit in whichs, t → 0.On the other hand, since in our point of view we are

3 Note that our convention for matrixD, V , N is differentfrom those in [16], i.e.,Dhere= −Vthere, Nhere= Dthere, Vhere=Nthere.


not looking for the low energy expansion, instead,we expand the amplitude in thes-channel using theexpansion (15) for the beta functions, that is

A= ig2c

4(2π)p+1δp+1(p1 + p2)

×{

1

2s − 1(e6 + e7 − e8 − e9 + e10)

+ 1

2s

[e2 − e3 + 2e7 − e8 + e9 − 2e10

+ t

2(e6 + e7 − e8 − e9 + e10)

]+ 1

2s + 1

[e1 + e2 + e4 + e5 + 3e7 − e8 + e10

+ t2

8(e6 + e7 − e8 − e9 + e10)

+ t

2

(e2 − e3 + 1

2e6 + 5

2e7

− 3

2e8 + 1

2e9 − 3

2e10

)]+ · · ·

},

where dots represent thes-channel poles for highermassive modes. Using the conservation of momentumon the worldvolume of the D-brane, we recognize thatthe aboves-channel poles, up to some contact terms,are exactly identical to the poles appear in the stringfield theory amplitude (13). This ends our illustrationof the equality of the scattering amplitude of twoclosed string states from D-brane in the open stringfield theory and bosonic string theory.

5. Conclusion

In this Letter we studied the closed string scatteringamplitude off a D-brane in the framework of the openstring field theory. Using the CFT method we havebeen able to show, explicitly, that the gauge invariantobservable in open string field theory introduced in[10,11] reproduces the correct pole structure of thescattering amplitude in the bosonic string theory. Wehave checked this in the level truncation method upto level two. We note, however, that although thes-channel poles can be directly obtained from the leveltruncation computation, to see thet-channel poles weneed to evaluate the scattering amplitude to all levelin the open string field. This can be seen from the

form of the beta function (15). In fact each term in thisexpansion has a pole ins, but to see a pole int we needto take into account infinite terms in the summation.In principle one can compute the scattering amplitudeto all level and then resume the expansion findingthe exact expression for theS-matrix as (16) up tosome contact terms. Therefore we conclude that thescattering amplitude of the closed string states in theopen string field theory framework is the same as onein the bosonic string theory framework up to somecontact terms. This contact terms might be related to afield redefinition.

As an other example of the scattering amplitude onecan also compute the interaction of one closed stringand two open string fields. In the open string fieldtheory, this corresponds to〈OΨ ∗Ψ ∗ Ψ 〉. The resultshould be the same as the corresponding amplitude inthe bosonic string theory up to some contact terms. Weleft the detail computation of this case for the futurework.

Probably more interesting problem is to find thegauge invariant operators in the open supersting fieldtheory. Although the superstring field theory actionis not as simple as the bosonic one, but since in thegauge invariant operator only the string field and on-shell closed string are involved one might suspect thatthe gauge invariant operator has the same form as thebosonic one. If this were that case one would proceedto compute the scattering amplitude in the superstringcase.

Finally we note that there are two approaches instring field theory: (1) open string field theory whichwe have used throughout this Letter and (2) boundarystring field theory originally introduced in [17,18](for recent discussions see [19]). It is believed thatthese two string field theories are equivalent. Moreprecisely there should be a field redefinition, thoughsingular, which maps open string field theory to theboundary string field theory. Therefore an interestingquestion one might ask is what is the gauge invariantoperators in the boundary string field theory? Thisgauge invariant operator would have the same role asone in the open string field theory, namely it shouldcorrespond to the on-shell closed string states.4

4 The closed string in the BSFT has been studied in [20].


Acknowledgements

We would like to thank H. Arfaei and A. Ghodsifor discussions and S.L. Shatashvili for comments anddiscussions. M.A. would also like to thank Institute fortheoretical Physics (ITFA), Amsterdam, for hospital-ity.

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