---

Interaction of d=2 c=1 Discrete States

arXiv:hep-th/9212156v1 1 Jan 1992Interaction of d=2 c=1 Discrete Statesfrom String Field TheoryI.Ya.Aref’eva, ∗P.B.Medvedev †andA.P.Zubarev ‡September 5, 2018AbstractStarting from string field theory for 2d gravity coupled to c=1 matter we analyzethe off-shell tree amplitudes of discrete states.The amplitudes exhibit the polestructure and we use the off-shell calculus to extract the residues and prove thatjust the residues are constrained by the Ward Identities. The residues generate asimple effective action.1IntroductionTwo-dimensional gravity coupled to a c = 1 matter is still a subject of intense study.

Thereason is that it admits non-perturbative investigations as well as it has a reach symmetrystructure. There are two ways to describe the model.

In the first approach one deals witha matrix model and in the second one with the continuum Liouville theory.The characteristic property of the model is an appearance of discrete states (DS)which together with tachyon states exhaust the spectrum of physical states. This factwas firstly recognized in the matrix model description [1].

The appearance of these statesis rather transparent from the Liouville point of view when the model looks like a 2dstring [2, 11]. Since we deal with a string one can expect an infinite number of higherlevel states.

However, since we are dealing wiyh a 2-dimensional gauge invariant theorywe also can expect that all these states can be eliminated by a gauge transformation.In fact due to a nontrivial background not all the states are purely gauge. Namely, thestates with special fixed values of momentum survive in the physical spectrum [2, 6].

Orin other words, there exist non-trivial cohomologies of the corresponding BRST charge[3]-[5].This specific nature of the spectrum provides an enormous symmetry group [7, 9].The discrete states are spin one currents and they generate a W∞current algebra. There∗Steklov Mathematical Institute, Vavilov 42, GSP-1, 117966, Moscow, E-MAIL: Arefeva@qft.mian.su†Institute of Theoretical and Experimental Physics,117259, Moscow‡Samara Polytechnic Institute1

are also spin zero states generating a ground ring [8]. It is natural to conjecture that theexistence of this algebra leads to some Ward Identities (WI) for scattering amplitudes.This problem was addressed in [10] where these identities were derived for amplitudesdefined by formal CFT expressions.

However, one can convince that some of the am-plitudes are ill defined [11]. The origin of the divergences is the specific kinematics ofthe ”particles” living with fixed momenta only.

Two-vectors of energy-momemtum formthe two-dimensional lattice and when summing up a pair of discrete momenta (accordingto energy conservation law with the background charge) one gets the momentum of aDS again. So, all the internal lines of tree-level amplitudes describe a propagation ofreal ”particles”.

Hence, to obtain the amplitude one has to deal with actual poles and ameaning of the WI is rather problematic.A natural way to expose the pole structure of a scattering amplitude is to go off-shell.A consistent off-shell description is provided by string field theory (SFT).SFT for 2d string in a nontrivial background has been proposed in [12]. As usual,the interest in SFT was motivated by the fact that SFT is supposed to be backgroundindependent and to give a framework for discussing nonperturbative effects.Now weobserve that SFT appears to be a suitable tool to investigate the symmetries of correlationfunctions.In this paper we are going to use the machinery of SFT to analyse the specific sin-gularities of Feynman string diagrams of 2d string.

The main advantage of SFT for theproblem in question is the fact that it brings regular way to obtain off-shell amplitudes.A given Feynman tree diagram of SFT defines a meromorphic function of kinematicalinvariants. In the usual case to obtain on-shell tree amplitudes one has to restrict thisoff-shell function on some hyperplane.

In our case, since the energies and momenta ofDS are components of fixed 2d vectors, to go on-shell one has to sit on a fixed point andsome of these points are nothing but the actual poles. Just these poles lead to the abovementioned divergences in the amplitudes of CFT.Our strategy will be the following one.

We start with the general expression for off-shellamplitudes obtained by slightly modifying the off-shell calculation for the critical stringfield theory [13]. We present the explicit formula for the off-shell four-point Feynmandiagrams for arbitrary discrete states.

We investigate the behaviour of this formula nearthe mass-shell. For any two given external states this point is uniquely defined by thekinematics.

As one expects, this point manifests the potential pole of the amplitude andwe give the rule to calculate the residue. In contrast to the usual case, these residues willbe constants since the value of t is also fixed.

In accordance with the unitarity conditionthis residue must be a product of a pair of two three-point amplitudes. The set of thesethree-point constants can be used to write down an effective action such that the residuescan be obtained from it.

To reflect the fact that we deal with + or −states numberedby pairs of integers (half-integers) m, n the effective action is a function of two sets ofvariables Φ and ¯Φ both of which are numbered by the pairs (n, m). This effective actionalso seems to give the explicit expressions for the residues of the leading poles of n-pointoff-shell tree amplitudes.

This conjecture seems to be natural as the leading singularityis fulfilled by real ”particles” in intermediate states, that brings the expression for theresidue as a product of corresponding three-point amplitudes. Turning back to the issue ofthe WI we realize that the proper objects to be constrained by the WI are just the residuesand not the correlation functions.

These identities are the origin of the symmetries of theeffective action.2

The paper is organised as follows. In section 2 we summarize the main points of themodel.

In section 3 we specify the scheme of SFT off-shell calculations. In section 4 wecalculate the off-shell four-point amplitudes.

In section 5 we speculate on a problem ofeffective action and derive the WI.2The modelIn this section we shall recall the model and briefly discuss some specific features whichare essential for our investigation.The classical action of the model looks like an action of d = 2 string in the Euclideanspace-time with a non-zero background charge for the Liouville mode φS = 18πZd2ξqˆg(ˆgαβ∂αXµ∂βXµ −Q ˆRφ + ghosts )(1)The value of the background charge Q is fixed by the usual requirement of vanishing ofthe conformal anomaly for the total matter Xµ = (x, φ) and ghost (b, c) system:1 + cφ = 26, cφ = 1 + 3Q2, →Q = 2√2. (2)The non-zero Q together with two-dimensional kinematics make the spectrum of the modelsomewhat unusual, as was discovered in [2, 3].

The straightforward way to see it is toanalyze the standard Virasoro constraints (L0 −1)|phys.state⟩= 0, Lm|phys.state⟩= 0for m > 0 in the light-cone parametrizationLm = P +(m)α−m + P −(m)α+m +Xn̸=0,mα+m−nα−n ,L0 = k+k−+ k+ −k−+ ˆN,(3)whereP ±(m) = k± ∓(m + 1),k± = 1√2(k1 ± ik2),α±m = 1√2(αm1 ± iαm2). (4)The constraints Lm = 0 becomes drastically simplified if one analyze the states createdonly by α+m (α−m) modes.

Let us take a state with ˆN = m0|phys. state⟩= (α+−m0 +Xm1+m2=m0Cm1m2α+−m1α+−m2 + ...)|k+, k−⟩,(5)thenLm0|phys.

state⟩= [P +(m0)α−m0 +Xn>m0α+m0−nα−n ]|phys. state⟩== [P +(m0) + oscillator terms]|k+, k−⟩= 0(6)and the necessary condition for a state to be physical for some specific m = m0 isP +(m0) = 0 which fixes the value of k+ to bek+ = m0 + 1(7)3

The value of k−is fixed by the mass-shell condition (L0 −1)|phys.state⟩= 0: k−= −2The example of such a physical state, the ”vector” one, is|phys. state⟩= α+−1|2, −2⟩.

(8)In a more general setting, the discrete states appear as a non-trivial cohomology ofBRST charge QBRST : QBRST |ψ⟩= 0, |ψ⟩̸= QBRST |λ⟩, and they can be classifiedaccording to their ghost number.The nontrivial cohomology can be found for ghostnumber 0, ..., 3 and it can be proved that not only the states with fixed momenta are inthe spectrum but the whole spectrum is exhausted by these states plus the tachyon withnon-discrete momentum. The physical states in the total (matter + ghosts) Fock spacecan be explicitly described in two ways: by using Shur polynomials [9] or in terms ofSU(2) raising and lowering operators [9, 7].

For our purposes the second description ismore convenient.The universal formula for on-shell physical states in terms of conformal fields Yr reads[8]Y ±J,n = cW ±J,n,(9)whereW ±J,J−n =vuut(2J −n)!n!(2J)! [H−, ...[H−|{z}n, W ±J,J]...],(10)andH−=Idz2πie−i√2X(z), W ±J,J = ei√2JXe√2(1∓J)φ.

(11)J is a positive integer or a half integer. For the sake of simplicity we shall assume the”space” dimension x to be compactified, in this case the tachyon state will have a discretemomentum also.

The momenta of discrete states read [4]- [6]:kµ = (p, −iε) =√2(n, −i(1 ∓J))(12)whereJ = 0, 12, 1, 32, 2, ...;n = −J, −J + 1, ..., J.3The Scheme of the Off-shell CalculationsThe strategy and all the necessary tools for off-shell calculations in SFT for usual criticalstrings were developed in a series of papers by Samuel et al. The interested reader canfind the details in the original manuscripts [13].

Here we adopt this calculus for the caseof the critical string in a non-trivial background.The action for 2d SFT is taken to be of the usual Witten type and the amplitudesare computed using perturbation theory. With each Feynman graph is associated a stringconfiguration Rτ.

External strings are semi-infinite rectangular strips of width π. Thei-th internal string propagator is a strip of length τi and width π. The interaction gluesthe strips in pairwise manner.The external states located at points wi are represented by vertex operators Yr = cWr.For any fixed set of the off-shell states Yr-s it is possible to obtain the the amplitude but,to treat the states of a general form by using off-shell conformal methods one has to4

guarantee that Yr-s are conformal fields of definite conformal dimension ∆r. So we haveto specify the proper rule for going off-shell.

Taking YJ,n in the formY ±J,n = cVJ,ne√2(1∓J)φ(13)we can relax the mass-shell condition by simply substituting:√2(1 ∓J)φ →εφ for someparameter ε. The fields Y εJ,n = cVJ,neεφ will be conformal fields with conformal dimension∆(ε) = −1 + J2 −12ε2 +√2ε(14)which goes to zero on-shell (ε =√2(1 ∓J))The contribution to a particular Feynman graph isAN = (N−3Yi=1Z ∞0dτi) < Y1(w1)Y2(w2)Idw′1b(w′1) .

. .

YN(wN) >Rτ,(15)where the correlation function is taken on string configuration.Although the world-sheet action for the first-quantized string is quadratic the compli-cated geometry of the string configuration makes the computation non-trivial. The keyidea is to conformaly map the string configuration to the upper half-plane.

The map ρ(z)which takes the half-plane into the string configuration for the N-point tree diagram wasgiven by Giddings and Martinec [14].The correlation function is a product of two factors ⟨...⟩Rτ = ⟨...⟩x,φRτ ⟨...⟩bcRτ which canbe calculated separately. The on-shell condition is irrelevant for the b −c factor, henceone obtains the standard result< ... >bcRτN−3Yi=1dτi = (z1 −z2)(z1 −zN)(z2 −zN)NYr=1(dρdze−ρ)−1zrN−3Yi=1dzi,(16)where zr are the asymptotic positions on the real axis of N external states.

The x −φfactor transforms under the mapping ρ into:< ... >x,φRτ =NYr=1(dρdze−ρ)−∆r+1zr⟨NYr=1Wr(zr)⟩. (17)Note, that for on-shell states all the ∆-s are equal to zero.

Collecting the factors (16) and(17) together one achieves for the amplitude (15)AN = (N−3Yi=1Zdzi) < Y1(z1)Y2(z2)W3(z3) . .

. WN−1(zN−1)YN(zN) >z−plane (NYr=1eNrr00 ∆r).

(18)where Nrr00 are the coefficients of the Neuman functions.4Four-Point Off-Shell Amplitude.As it has been mentioned in the Introduction, because of the specific nature of allowedmomenta one falls into a puzzling situation as soon as the calculation of amplitudes for5

DS is performed. Namely, let us consider, for example, an s-channel four point amplitudewith the following values of particle’s momenta k1 = (√2, 0), k2 = (−√2, 0), k3 =(√2, 0), k4 = (−√2, −i2√2) which satisfy the shifted energy-momentum conservationlaw in presence of the background charge:P pi = 0,P εi = Q.

It appears to be natural todefine the invariant variable s in the presence of the background charge by the followingformulas = (k1 + k2)2 −(ε1 + ε2)2 + 2√2(ε1 + ε2)(19)Hence for this configuration the value of s is also fixed to be s = 0. We are sitting on thepole, the amplitude is obviously divergent and this is the typical case for the theory.In this section we derive the expression for the residues of the intermediate state polesfor general s-channel four-point amplitude:A4 =Zdz3⟨Y1(z1)Y2(z2)W3(z3)Y4(z4)⟩(4Yr=1eNrr00 ∆r).

(20)Here we have omitted the r, n indices for Y -s and labelled them by the numbers of states.The positions of the points zi on the real axis are fixed by the parameter α of the Giddingsmapping ρ(z) as follows z2 = −z3 = α z1 = −z4 = 1/αTo convert the variable α to the Koba-Nielsen variable x one maps the upper half-planeinto itself by using SL(2, R) mappingx = α2 −1α2 + 1αz + 1αz −1(21)The three points z1, z2, z4 are mapped to ∞, 1, 0 as it must be and the final result ofstraightforward but tedious calculations, readsAs4 = limx1→∞Z 11/2 dx⟨Y1(x1)Y2(1)W3(x)Y4(0)⟩(22)x2∆1x12(∆3+∆4−∆1−∆2)(1 −x)∆2+∆3[κ(x)2 ]P4r=1 ∆r.Here we exploited the fact that N1100 = N4400 = ln κ/α,N2200 = N3300 = ln κα, where κ is aregular function in the region 1/2 ≤x ≤1, (√2−1 ≥α ≥0), nonzero at the point x = 1.To analyze the pole structure of (22) note that the poles correspond to the divergencesof the integral on the upper limit or, in other words, to terms in the integrand havingfactors of 1 −x to negative powers. Such factors come from the explicit factor of (1 −x)∆2+∆3 and from possible contractions of the pair Y2(1)W3(x) .

One can present theOPE for these two operators in the formY2(1)W3(x) ∼(1 −x)2n2n3−ε2ε3(1 −x)R,(23)where the first factor originates from the product of the exponents and the second one- from the contraction of the Shur polynomials. It is important to stress that R is aninteger.

Hence, we can present the amplitude in the formAs4 =Z 11/2 dx(1 −x)2n2n3−ε2ε3+∆2+∆3−RF(x),(24)where F(x) = F(x; sr, nr, εr) is a regular function at x ∼1.6

Moreover, according to the adopted definition of the invariant variable s (19) we have2n2n3 −ε2ε3 + ∆2 + ∆3 = 12s −2,(25)so, expanding F(x) in the Taylor seriesF(x) =∞Xi=1Fi(1 −x)iwe get the final expression for A(s)4A(s)4=∞Xi=1112s −R −1 + i(12)12s−n−1+iFi. (26)Now, we turn to extract some information from eq.(26).

Note, that for every givenon-shell states 2 and 3 s is an even number:s = 2[(n2 + n3)2 −(J2 + J3)2 + 2(J2 + J3) + 4] = s0(27)Hence, there is one and only one pole for i0 = R + 1 −12s0 in A(s)4and the F on−shelli0is theresidue (maybe equal to zero) in this pole. Moreover, to find the residue there is no needto develop the off-shell calculations, it is sufficient to extract from the on-shell correlationfunction⟨Y1(∞)Y2(1)W3(x)Y4(0)⟩(28)the coefficient of the term ∼(1 −x)−1.

It will be precisely the F on−shelli0.As it is known the total s,t-channel amplitude is a sum of two Feyman diagrams ofSFT. The first diagram is presented below and the second one is obtained by the cyclicpermutation of the external states: (1234) →(4123).

The residue of the correspondingt-pole obviously will be given by the corresponding term in the expansion of eq. (28) withpermuted labels of vertex operators.5The Effective LagrangianAs it was proposed by Klebanov and Polyakov [7] the model can be described by itseffective action.

In this section we are going to connect the concept of effective actionwith the scattering amplitudes.The above discussion of the four-point amplitude (see also [11]) shows that it is naturalto associate an effective action with the leading singularities of the amplitudes. Indeed,using the OPE for Y2(1) and W3(x)Y +J2,n2(1)W +J3,n3(x) =11 −xf J2+J3−1,n2+n3J2n2,J3n3Y +J2+J3−1,n2+n3(1) + regular terms(29)the expression for the residue can be presented in the form˜A4 = ResA4|s=s0 = f J′n′J2n2,J3n3f J1n1J′n′,J2n2(30)the structure constants of OPE were calculated in [8, 7]f J3n3J1n1,J2n2 = δJ3,J1+J2−1δn3,n1+n2 ˜fJ1n1,J4n4 =7

δJ3,J1+J2−1δn3,n1+n2˜N(J3m3)˜N(J1m1) ˜N(J2m2)(J2m1 −J1m2). (31)Here ˜N(Jm) is a normalisation factor.This result (eq.

(30)) can be reproduced by a simple effective field theory which de-scribes an interaction of two independent fields ΦJ,n and ¯ΦJ,n with indices J ≥0, −J ≤n ≤J having a trivial propagator. The Lagrangian has the formL(Φ, ¯Φ) =Xa¯ΦaΦa + gXa,b,cΦaΦb ¯Φcf bab.

(32)where Φa = Φs,n and ¯Φa = ¯Φs,−n andf cab =< Y +a Y +b Y −c >(33)It seems natural that the Lagrangian (32) correctly reproduces the coefficients of the lead-ing singularities for all the n-point tree amplitudes by a simple one to one correspondencebetween graphs of the SFT and the ones of the effective theory. There is no rigorousproof at our disposal but there are some indirect arguments in favour of this hypothesis.In the previous paper [11] it was shown that if a special regularization is adopted thenthe resedues of the leading poles (with respect to the regularization parameter) are givenby the effective Lagrangian (32).

Following this interpretation of the effective theory weconclud that there are no reasons to add to the effective action the quartic and higherorder terms if we deal with the tree amplitudes.Now, let us discuss the issue of Ward Identities. The symmetry properties of on-shellamplitudes were considered in [10].

The identities were derived by using the usual contourdeformation trick. Namely, inserting the zero-dimensional chargeQ±s,m =Idz2πiW ±s,m(z),(34)in the correlation function, changing the contour of integration and using the OPE’s ofW +s,m and Y ±si,mi one gets the WI.

However this consideration is rather formal, since someof amplitudes are ill defined, as it was explained above. The well defined objects areoff-shell amplitudes and it seems natural to search for some WI for these amplitudes.However there is an essential obstacle to do this as it is impossible to apply the contourdeformation argument to off-shell amplitudes.

This originates from the fact that it isimpossible to define an action of the non-local operator (34) on conformal fields of non-integer or non-half-integer dimension. So, WI for off-shell amplitudes of SFT cannot bederived in this way.To solve this puzzle let us recall that we have the expression for residues of amplitudesin terms of on-shell correlations functions in eq.(28).

Inserting the operator (34) in eq. (28) we get the relation between corresponding correlations functions˜fsm,s1m1⟨Y +s1+s−1,m1+m(∞)Y +s2,m2(1)W +s3,m3(x)Y −s4,m4(0)⟩±˜fsm,s2m2⟨Y +s1,m1(∞)Y +s2+s−1,m2+m(1)W +s3,m3(x)Y −s4,m4(0)⟩±˜fsm,s3m3⟨Y +s1,m1(∞)Y +s2,m2(1)W +s3+s−1,m3+m(x)Y −s4,m4(0)⟩±˜fsm,s4m4⟨Y +s1,m1(∞)Y +s2,m2(1)W +s3,m3(x)Y −s4+s−1,m4+m(0)⟩= 0(35)8

It is evident that this relation is also true for the terms proportional to (1 −x)−1 inexpansion of eq. (35) which define the corresponding residues.

So, the proper objects tobe constrained by the WI are the residues and the identities read˜fsm,s1m1 ˜A((s1 + s + 1, m1 + m)+(s2, m2)+, (s3, m3)+(s4, m4)−)±± ˜fsm,s2m2 ˜A((s1, m1)+(s2 + s + 1, m2 + m)+, (s3, m3)+(s4, m4)−)±± ˜fsm,s3m3 ˜A((s1, m1)+(s2, m2)+, (s3 + s + 1, m3 + m)+(s4, m4)−)±± ˜fsm,s4m4 ˜A((s1, m1)+(s2, m2)+, (s3, m3)+(s4 −s + 1, m4 + m)−) = 0. (36)Coming back to the effective action (32) it is evident that relations (36) should rep-resent a symmetry of the effective Lagrangian.

This symmetry does exist. Namely, theLagrangian (32) is invariant under infinite number of infinitesimal transformationsδcΦa = facb Φb,δc ¯Φa = fbac ¯Φb,(37)6Concluding RemarksIn summary, we have argued that off-shell scattering amplitudes of 2d open string discretestates exhibit the pole structure and the residues are described by the effective Lagrangianwith rich symmetry structure.

We have presented the rigourous proof of these statementsfor four-point open string tree amplitudes. The N-point case needs more detaled analysis.Our effective action contains only cubic terms.

This does not agree the with Klebanovand Polyakov conjecture [7] that the effective action is nonpolynomial. However, it ispossible that loops will destroy its cubic character.References[1]D.Gross, I.Klebanov and M.J.Newman,Nucl.Phys.

B350 (1991) 621 D.Gross andI.Klebanov,Nucl. Phys.

B352 (1991) 671. [2]A.Polyakov, Mod.Phys.Lett.

A6 (1991) 635. [3]B.Lian and G.Zukerman, Phys.

Lett. 254B (1991) 417; Phys.

Lett. 266B (1991) 21.

[4]P.Bouwknegt, J.McCarthy and K.Pilch, CERN preprints, CERN-TH.6162/91 (1991);CERN-TH.6279/91 (1991). [5]K.Itoh and N.Ohta, Preprint FERMILAB-PUB-91/228-T.[6]I.Ya.Aref’eva and A.V.Zubarev, Non-critical NSR string field theory.

Preprint MIRAN,1992. [7]L.Klebanov and A.M.Polyakov, Mod.Phys.Lett.A6(1991)3273.

[8]E.Witten, Ground ring of two dimensional string theory, preprint IASSNS-HEP-91/51. [9]E.Witten and B.Zwiebach, preprint IASSNS-HEP-92/4 (1992)[10]E.Verlinde, preprint IASSNS-HEP-92/5 (1992)9

[11]I.Ya.Aref’eva and A.V.Zubarev, Divergences of Discrete States Amplitudes and EffectiveLagrangian in 2D String Theory. Preprint MIRAN, 1992.

[12]I.Ya.Aref’eva and A.V.Zubarev, Mod. Phys.Lett.A2 (1992)677.

[13]S.Samuel, Nucl. Phys.

B308 (1988) 285, 317;R.Bluhm and S.Samuel, Nucl. Phys.

B323 (1989) 337. [14]S.Giddings and E.Martinec, Nucl.

Phys. B278 (1986) 91.10

---

출처: arXiv:9212.156 • 원문 보기