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

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<ul><li><p>Physics Letters B 536 (2002)</p><p>Gauge invariant operators and closed string scattering inopen string field theory</p><p>Mohsen Alishahiha a, Mohammad R. Garousi b,a,c</p><p>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</p><p>Received 6 February 2002; accepted 18 April 2002</p><p>Editor: L. Alvarez-Gaum</p><p>Abstract</p><p>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.</p><p>1. Introduction</p><p>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.</p><p>E-mail addresses: (M. Alishahiha), (M.R. Garousi).</p><p>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].</p><p>In an other attempt but related to the closed stringstates in the open string field theory, the gauge invari-</p><p>0370-2693/02/$ see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(02)0 18 06 -3</p></li><li><p>130 M. Alishahiha, M.R. Garousi / Physics Letters B 536 (2002) 129137</p><p>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.</p><p>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.</p><p>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.</p><p>2. Open string field theory</p><p>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.</p><p>2.1. Cubic string field theory action</p><p>The cubic open string field theory action is givenby</p><p>(1)S( )= 12</p><p>( Q + 2go</p><p>3 </p><p>),</p><p>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</p><p>| =</p><p>dp+1k( +A1 + ib1c0</p><p>+ i2B</p><p>2 +</p><p>12B</p><p>1</p><p>1+ 0b2c0 + 1b1c1+ ik1b1c0 + </p><p>)c1|k.</p><p>The gauge invariance of (1) can be fixed can bychoosing FeynmanSiegel gauge b0| = 0. In thisgauge the truncated field up to level two reads</p><p>| =</p><p>dp+1k((k)+A(k)1 +</p><p>i2B(k)</p><p>2</p><p>+ 12B(k)</p><p>1</p><p>1 + 1(k)b1c1</p><p>)c1|k.</p><p>The corresponding string vertex is given by</p><p> (0)=</p><p>dp+1k[(k)c(0)+ iA(k)cX(0)</p><p> 12B(k)c</p><p>2X(0)</p><p> 12B(k)cX</p><p>X(0)</p><p>(2) 121(k)</p><p>2c(0)]e2ikX(0).</p><p>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).</p><p>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</p><p>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.</p></li><li><p>M. Alishahiha, M.R. Garousi / Physics Letters B 536 (2002) 129137 131</p><p>language the n-string vertex is defined by2 = f (n)1 (0) f (n)2 (0) f (n)n (0) UHP,</p><p>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</p><p>upper-half plane and the conformal map f (n)k is de-fined as</p><p>f(n)k (zk)= g</p><p>(e</p><p>2in(k1)</p><p>(1+ izk1 izk</p><p>)2/n), 1 k n,</p><p>where g( )=i 1+1 . Therefore the open string action</p><p>can be calculated as following in terms of correlationfunctions of the CFT on the UHP</p><p>S =14</p><p>f(2)2 (0)f (2)1 </p><p>(Q(0)</p><p>)+ 2go</p><p>3f(3)1 (0) f (3)2 (0)f (3)3 (0)</p><p>UHP</p><p>.</p><p>Form this expression the kinetic terms up to level twofields read</p><p>Squad =</p><p>dp+1x(1</p><p>2</p><p> + 142</p><p> 12A</p><p>A 12B</p><p>B 14BB</p><p> 12B</p><p>B 14BB</p><p>(3)+ 121</p><p>1 + 1421</p><p>),</p><p>which can be used to write the spacetime propagatorsof the corresponding fields.</p><p>2.2. Gauge invariant operator</p><p>The gauge invariant operators in string field theoryhave been constructed in [10,11]. The general formof these operators are given by O = gc</p><p>V , where</p><p>gc is the closed string coupling and V is an on-shellclosed string vertex operator with dimension (0,0). In</p><p>2 We assume that there is a normal order sign between fields atdifferent points in the correlation functions.</p><p>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</p><p>(4)O = gc</p><p>V = gc</p><p>cV(i) cV(i)f (1)1 (0)</p><p>UHP ,</p><p>where f (1)1 = 2z1z2 and V(z)V(z) is the matter part ofthe closed string vertex operator.</p><p>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</p><p>S =14</p><p>( Q + 2go</p><p>3 </p><p>)(5)+ gc</p><p>V,</p><p>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].</p><p>3. Scattering amplitudes in string field theory</p><p>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</p></li><li><p>132 M. Alishahiha, M.R. Garousi / Physics Letters B 536 (2002) 129137</p><p>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</p><p>(6)X(z)X(w)= ln(zw),one finds that the different terms in (2) transform undera general conformal map f as</p><p>f (ce2ikX)= f 2k21ce2ikX,f (cXe2ikX)</p><p>= f 2k2(X ikf /f 2)ce2ikX,f (c2Xe2ikX)</p><p>= f 2k2+1[2X + (f /f 2)X</p><p> ik/6(</p><p>4f /f 3 3(f /f 2)2)]ce2ikX,f (cXXe2ikX)</p><p>= f 2k2+1[XX 2i(f /f 2)k{X}</p><p> (f /f 2)2kk 1/12(2f /f 3 3(f /f 2)2)]ce2ikX,</p><p>f (2ce2ikX)= f 2k2+1</p><p>[2c (f /f 2)c</p><p>(7)(f /f 3 2(f /f 2)2)c]e2ikX.</p><p>Note that in the above equations the worldsheet fieldson the right-hand side are functions of f (z), alsok2 = k k. The function f (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 andc(i)c(i)2c(0) = 4i .</p><p>Plugging the conformal transformation (7) intoEq. (4) and using above correlators for ghost part, weget</p><p>O= 2igc</p><p>dp+1k 22k2</p><p>[(</p><p>12 + i2B V k + 1</p><p>2</p><p>2B</p><p> 121</p><p>) V(i)V(i)e2ikX(0)+ iA</p><p>V(i)V(i)Xe2ikX(0)2B</p><p>V(i)V(i)2Xe2ikX(0)(8)2B</p><p>V(i)V(i)XXe2ikX(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.</p><p>3.1. Tachyon amplitude</p><p>The matter part of the vertex operator of closedstring tachyon inserted at the midpoint of open stringwith momentum p (p p = 2) is given byV(i)V(i)= eip.X(i)eip.D.X(i),where 2V = + 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</p><p>O(p)= igc8(2)p+1e4 ln2pV p</p><p>( + 4i p N A+ 2i2p V B</p><p> 82p N B N p+ 12B</p><p> 1),</p><p>here the spacetime fields are functions of p V .Fourier-transforming to the position space, e.g.,</p><p>(k)=</p><p>dp+1x(2)p+1</p><p>(x) eik.x,</p><p>the operatorO becomes</p><p>O(p)= igc8</p><p>dp+1x eip.x</p><p>((x)+ 4i p N A(x)</p><p>+ 2i2p V B(x) 82p N B(x) N p</p><p>(9)+ 12B</p><p>(x) 1(x)),</p></li><li><p>M. Alishahiha, M.R. Garousi / Physics Letters B 536 (2002) 129137 133</p><p>where the tilde sign over fields means, e.g., (x) =e4 ln2 2(x). According to the proposed action (5)the expression (9) is spacetime action representing thecoupling of the closed string tachyon with the openstring fields.</p><p>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.</p><p>Consider the following two point function</p><p>(10)O(p1)O(p2),where denotes the correlation function in the stringfield theory. According to what we have said, thisshould be interpreted as the S-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.,</p><p>(x)(y) = i</p><p>dp+1k eik(xy)</p><p>k2 12,</p><p>A(x)A(y)</p><p>=i dp+1k eik(xy)k2</p><p>,</p><p>1(x)1(y) = i</p><p>dp+1k eik(xy)</p><p>k2 + 12,</p><p>B(x)B(y)</p><p>=i dp+1k eik(xy)k2 + 12</p><p>,B(x)B(y)</p><p>(11)= i</p><p>2</p><p>dp+1k (</p><p> + )eik(xy)k2 + 12</p><p>.</p><p>Now inserting (9) into (10) and using above propaga-tors, one finds</p><p>O(p1)O(p2)= ig</p><p>2c</p><p>2(2)p+1p+1(p1 + p2)</p><p>{e8(s 12 ) ln2</p><p>2s 1 e8s ln2p1 N p2</p><p>2s</p><p>+ e8(s+ 12 ) ln2[ 12 (p1 N p2)2 + 332 (s + 12 ) 18 ]</p><p>2s + 1},</p><p>where s = 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</p><p>O(p1)O(p2)= ig</p><p>2c</p><p>2(2)p+1p+1(p1 + p2)</p><p>{</p><p>12s 1 </p><p>p1 N p22s</p><p>(12)+12 (p1 N p2)2 18</p><p>2s + 1 + },</p><p>where dots represent some contact terms. We shallshow that the above poles exactly reproduce the s-channel poles of the corresponding amplitude in thebosonic string theory.</p><p>3.2. Graviton amplitude</p><p>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 KalbRamond 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</p><p>V(i)V(i)= ( D)X(i)eipX(i)X(i)eipDX(i)</p><p>with pp = 0 = p = p . For graviton wehave = and = 0 [12].</p><p>Plugging above closed string vertex operator intoEq. (8) and using the worldsheet propagator (6), onefinds</p><p>O(,p)= igc8(2)p+1e4 ln2pV p</p><p> ( a + 4iAb + 2i2Bc 82Bd 1a</p><p>),</p></li><li><p>134 M. Alishahiha, M.R. Garousi / Physics Letters B 536 (2002) 129137</p><p>here the spacetime fields are functions of p V . Thekinematic factors a, b, c, and d are</p><p>a = Tr( D) p D D p,b = a p N + p D D D p,c = a p V 4p D D 4 D p,d = a</p><p>(p Np N 1</p><p>16</p><p>)+ 2( D){}</p><p>+ 2p D D{p N} 2{ D pp N}.Fourier-transforming the open string fields to theposition space, we get</p><p>O(,p)= igc8</p><p>dp+1x</p><p>((x)a+ 4iA(x)b</p><p>+ 2i2 B(x)c...</p></li></ul>


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