Mean field quantization of effective string

Journal of High Energy Physics, Jul 2018

Abstract I describe the recently proposed quantization of bosonic string about the meanfield ground state, paying special attention to the differences from the usual quantization about the classical vacuum which turns out to be unstable for d > 2. In particular, the string susceptibility index γstr is 1 in the usual perturbation theory, but equals 1/2 in the mean-field approximation that applies for 2 < d < 26. I show that the total central charge equals zero in the mean-field approximation and argue that fluctuations about the mean field do not spoil conformal invariance.

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Mean field quantization of effective string

Revised: May eld quantization of e ective string Yuri Makeenko 0 1 0 B. Cheremushkinskaya 25 , Moscow, 117218 Russia 1 Institute of Theoretical and Experimental Physics I describe the recently proposed quantization of bosonic string about the meaneld ground state, paying special attention to the di erences from the usual quantization about the classical vacuum which turns out to be unstable for d > 2. In particular, the string susceptibility index str is 1 in the usual perturbation theory, but equals 1/2 in the mean- eld approximation that applies for 2 < d < 26. I show that the total central charge equals zero in the mean- eld approximation and argue that uctuations about the mean eld do not spoil conformal invariance. Long strings; Conformal Field Models in String Theory; Bosonic Strings - Mean 1 Introduction 2 The mean- eld ground state 3 Instability of the classical vacuum 4 Stability of the mean- eld vacuum 5 The string susceptibility index 6 Fluctuations about the mean- eld 7 Computation of the central charge 8 \Semiclassical" correction to the mean eld 8.1 8.2 Correction to Tzz Correction to Se 8.3 Remark on the universality 9 Discussion A.1 T a a A.2 Tzz A Leading-order explicit computations B \Semiclassical" corrections B.1 Contribution to Tzz B.2 Contribution to Se C Mathematica programs C.1 Program for computing diagrams in gure 3 that results in (8.9) C.2 Program for computing diagrams in gure 4 that results in (8.12) 1 Introduction Strings or more generally two-dimensional random surfaces have wide applications in physics: from biological membranes to QCD. However, a nonperturbative theory of quantum strings, which goes back to 1980's, makes sense only if the dimension of target space { 1 { d < 2, where the results both of dynamical triangulation [1{3] and of conformal eld theory [4{6] are consistent and agree. For d > 2 the scaling limit of dynamically triangulated random surfaces is particle-like1 rather than string-like because only the lowest mass scales but the string tension does not scale and tends to in nity in the scaling limit. [8] Analogously, the conformal eld theory approach does not lead to sensible results for 2 < d < 26. [4{6] The conclusion was that quantum string does not exist nonperturbatively for 2 < d < 26, while it beautifully works for d < 2. However, we understand that strings do exist as physical objects in d = 4 space-time dimensions. A potential way out was to adopt the viewpoint that string is not a fundamental object but is rather formed by more fundamental degrees of freedom. This philosophy perfectly applies to QCD string,2 where these are uxes of the gauge eld. String description makes sense only for the distances larger than the con nement scale. For shorter distances the quark-gluon degrees of freedom are more relevant due to asymptotic freedom. This picture is well justi ed both by experiment and by lattice simulations. The string tachyon which is a short-distance phenomenon does not show up in the QCD spectrum. A breakthrough along this line is due to the \e ective string" philosophy [10], which works perturbatively order by order in the inverse string length (for recent advances see [11{ 15]). Then string quantization is consistent even below the critical dimension (d = 26 for the relativistic bosonic string) and a few leading orders reproduce [10, 16, 17] the AlvarezArvis ground-state energy [18{20]. In this Paper I shall pay much attention to this issue. In the recent series of papers [21{24] it has been understood why lattice string formulations resulted in the particle-like continuum limit. A nonperturbative mean- eld solution of the Nambu-Goto string showed that the usual classical vacuum about which string is quantized is unstable for d > 2, while another nonperturbative vacuum is stable for 2 < d < 26, like it happens in the well-known example of the two-dimensional O(N ) sigma-model. For the true ground state the value of the metric at the string worldsheet ( ab = ab in the conformal gauge) becames in nite in the scaling limit. For this reason an in nite amount of stringy modes (which is / =a2 with a being a UV cuto ) can be reached even at the distances of order a. That was in contrast to the usual continuum limit in quantum eld theory, where the amount of degrees of freedom can be in nite only if the correlation length is in nite. The discovered phenomenon is speci c to theories with di eomorphism invariance and was called the Lilliputian continuum limit. The task of this Paper is to analyze properties of the mean- eld vacuum which plays the role of a \classical" string ground state and \quantum" uctuations about it. I put here quotes to emphasize this state is a genuine nonperturbative quantum state from the viewpoint of the usual semiclassical expansion in 0 about the classical vacuum. We thus perform a resummation of this expansion with the leading order given by the sum of bubblelike diagrams. An analogy with the two-dimensional O(N ) sigma-model at large N can be again instructive. I shall pay special attention to a comparison with the KnizhnikPolyakov-Zamolodchikov (KPZ){David-Distler-Kawai (DDK) results [4{6] for the parturbative vacuum, which is applicable for d < 2. 1A detailed description can be found in the book [7]. 2For a brief introduction see e.g. [9]. { 2 { where the consistency is explicitly demonstrated to a few lower orders of the perturbative expansion [10, 16, 17]. I then analyze a \semiclassical" correction to the mean- eld approximation and show that it does not spoil conformal invariance in spite of logarithmic infrared divergences caused by the propagator of a massless eld, which cancel in the sum of diagrams. This Paper is organized as follows. In sections 2, 3, 4 I review the results [21{23] which form a background for further investigations. Section 5 is devoted to the computation of the mean- eld value of the string susceptibility index str = 1=2 and its comparison to the perturbative value str = 1. In section 6 I formulate a general procedure for expanding about the mean eld and describe Pauli-Villars' regulation for computing the energy-momentum tensor and its trace anomaly, which does not rely on approximating the involved determinants by (the exponential of) the conformal anomaly. It section 7 I compute the total central charge of the system in the mean- eld approximation and show that it vanishes for 2 < d < 26. Section 8 is devoted to the \semiclassical" expansion about the mean eld. I show that logarithmic infrared divergences which might spoil conformal invariance are mutually canceled. The results obtained and tasks for the future are discussed in section 9. Some explicit computations are presented in appendices A, B by using the Mathematica programs from appendix C. 2 The mean- eld ground state We consider a closed bosonic string in target space with one compacti ed dimension of circumference . The string wraps once around this compact dimension and propagates through the distance L. The string world-sheet has thus topology of a cylinder. There is no tachyon for such a string con guration, if is larger than a certain value to guarantee that the classical energy of the string dominates over the energy of zero-point uctuations. The Nambu-Goto string action is given by the area of the surface embedded in target space. It is highly nonlinear in the embedding-space coordinate X . To make it quadratic in X , we rewrite it, introducing a Lagrange multiplier ab and an independent intrinsic metric ab, as3 S = K0 d2! pdet + K0 Z 2 where K0 stands for the bare string tension. The equivalence of the two formulations can be proven by path integrating over the functions ab(!) and ab(!) which take on imaginary and real values, respectively. It is convenient to choose the world-sheet coordinates !1 and !2 inside an !L ! rectangle in the parameter space. Then the classical solution Xcl minimizing the action (2.1) 3We denote det = det ab and det = det ab. ab) ; (2.1) { 3 { linearly depends on ! The classical value of ab coincides with the classical induced metrics L2 !L2 ; !2 2 ! which becomes diagonal for The classical value of ab reads ! = L !L: calb = calbpdet cl and simpli es to calb = ab if eq. (2.4) is satis ed. We apply the path-integral quantization to account for quantum uctuations of the Xq . We thus obtain the action, governing the elds ab and ab, X- elds by splitting X = Xcl + Xq and then performing the Gaussian path integral over Z 1 O = pdet K0 Z 2 Sind = K0 d2! pdet + d 2 ab) + tr log( O); The operator O reproduces the usual two-dimensional Laplacian right-hand side of eq. (2.5). Its determinant is to be computed with the Dirichlet boundary condition imposed. Quantum observables are determined by the path integral over and ab with the action (2.8), which runs as is already mentioned over imaginary and real ab(!). The action (2.6) is often called the induced (or emergent ) action to be distinguished from the e ective action which is usually associated with slowly varying elds for ab given by the in the low-momentum limit. It is convenient to x the conformal gauge when ab = the log of the determinant of the ghost operator [25] ab, so that pdet = . Then Ogh ba = a b 1 2 ( ba log ) is to be added to the induced action (2.6) [or (4.11) below]. The operator (2.7) acts on twodimensional vector functions whose one component obeys the Dirichlet boundary condition and the other obeys the Robin boundary condition [26, 27]. The subtleties associated with the boundary conditions will be inessential both for the matter and ghost determinants for when only the bulk terms survive. We shall describe in section 6 how to accurately compute the determinants using the Pauli-Villars regularization but let us assume for a moment that what is needed for the mean- eld approximation. an exercise in string theory courses with the result Se = K0 guarantees that ab and return to this issue soon. for L . Here 2 cuts o eigenvalues of the operators involved. The rst and second terms on the right-hand side are classical contributions, while the sign of the third term is negative for d= > 2 to comply with positive entropy. Technically, it comes as the product of the eigenvalues divided by , where every multiplier is less than 1. The last term is to nd the mean- eld con guration which describes the string ground state. The di erence from the classical ground state (2.2), (2.3), (2.5) is that we now minimize the action, taking into account the determinants coming from X and ghosts, while the classical (perturbative) ground state minimizes the classical action. Additionally, similarly to the classical case we have to minimize (2.8) over the ratio ! =!L which plays the role of the modular parameter of the cylinder. This ab are diagonal as is required by the conformal gauge. We shall The minimum of (2.8) is remarkable simple [21{23] (d 3K0 2) : s 2 { 5 { The value of the action (2.8) at the minimum (2.9) is The meaning of the above minimization procedure is clear: we have constructed a saddle-point approximation to the path integral, which takes into account an in nite set of diagrams of perturbation theory about the classical vacuum. This approach is quite similar to that4 in the two-dimensional O(N ) sigma-model, where one sums up bubble diagrams of the 1=N -expansion by introducing the Lagrange multiplier u to resolve the constraint ~n2 = 1. After integration over the elds ~n one obtains an induced action as a functional 4See e.g. the book [28]. = = ! = 1 2 + 2 !L L s 2 2K0 !2 2 s + (d 2) 6K0 1 4 r 1 + uctuations of u about this mean- eld vacuum have to be included, but they are small even at N = 3 because, roughly speaking, there is only one u while the induced action is of order N , i.e. large as is needed for a saddle point. Alternatively, the perturbative vacuum ~ncl = (1; 0; : : : ; 0) possesses an O(N 1) symmetry rather than the O(N ) symmetry as the saddle-point vacuum does and the elds ~n uctuate strongly. For our case the number of elds X in the sigma model (2.1) is d, so the saddle-point is justi ed by K0 d ! At nite d the saddle-point solution (2.9) is associated with the mean- eld approximation. The minimization of the action (2.8) over ! =!L can be now understood as follows. In the mean- eld approximation we consider the action to be large, doing all integrals by the saddle point, including the integral over the modular parameter, which is present for the cylinder topology. A few comments concerning the solution (2.9) are in order: Equation (2.9a) is well-de ned if the bare string tension K0 > K given by K = d 1 + pd2 2d 2 The classical vacuum (2.2), (2.3), (2.5) is recovered by (2.9) as K0 ! 1, while the expansion in 1=K0 makes sense of the semiclassical (perturbative) expansion about this vacuum. The usual one-loop results are recovered to order 1=K0. The large-d ground-state energy [ 18 ],5 where an analytic regularization was used, are recovered by eq. (2.10) for 2 = 0. Analogously, the ground-state energy obtained by the old canonical quantization [ 19 ] is reproduced by our mean- eld approximation. This is not surprising because uctuations of are ignored in the canonical quantization. squared. Equation (2.10) is well-de ned for larger than p (d 2)=3K0 1= , but becomes imaginary otherwise. The singularity was linked [18{20] to the tachyon mass The metric (2.9b) becomes in nite when K0 ! K given by eq. (2.11). This is crucial for constructing the scaling limit. At the classical level cl coincides with the induced metric as is displayed in (2.3). In the mean- eld approximation it is superseded by (2.12) where the average is understood in the sense of the path integral over X . Equation (2.12) follows from the minimization of the e ective action over approximation coincides with the averaged induced metric. ab. Thus, in the mean- eld 5The original computation [ 18 ] used the Nambu-Goto string. How the same result can be obtained for the Polyakov string is shown in [29]. { 6 { Instability of the classical vacuum The usual semiclassical (or one-loop) correction to the classical ground-state energy due to zero-point uctuations [30] is described in textbooks. The sum of the two reads To make the bulk part of (3.1) nite, it is usually introduced the renormalized string S1l = K0 (d 2 2) 2 L (d 2) L in the mean- eld approximation for !L = L and ! = 1pK0. Inverting eq. (3.6), we (j) = 1 2 + r 1 + j + K0 2 1 + j + K0 ! 1. Then it is assumed that it works order by order of the perturbative expansion about the classical vacuum, so that KR can be made nite by ne We see however from eq. (2.10) how it may not be case. The right-hand side of eq. (2.10) never vanishes with changing K0. The point of view on eq. (3.1) should be that for d > 2 the one-loop correction simply lowers the energy of the classical ground state which therefore may be unstable. As we show in the next section, the action (2.8) indeed increases if we add a constant imaginary addition to . However, the sum of the two linear in terms in eq. (2.8) vanishes for given by eq. (2.9a), so the action does not depend on at the minimum. This reminds a valley in the problem of spontaneous symmetry breaking. To investigate it, we proceed in the standard way, adding to the action the source term { 7 { and de ning the eld Ssrc = K0 Z 2 d2! jab ab ab(j) = 2 K0 jab log Z: Minimizing the action with the source term added for constant jab = j ab, we nd [23] 1 2 (j) = 1 + j + 2 K0 + s 1 4 1 + j + like in the studies of symmetry breaking in quantum eld theory. In the mean- eld approximation we then obtain 1 K0L 2 K0 s 2d 2 K0 1): (3.9) 1 and the potential (3.9) decreases with increasing because the second term on the right-hand side has the negative sign, demonstrating an instability of the classical vacuum. If K0 > K given by eq. (2.11), the potential (3.9) linearly increases with for large and thus has a (stable) minimum at (0) = + 1 2 2 r which is the same as (2.9b) for ! = 1=pK0. Near the minimum we have " ( ) = 1 + The coe cient in front of the quadratic term is positive for K0 > K which explicitly demonstrates the (global) stability of the mean- eld minimum (2.9b). The situation is di erent for d < 2, where quantum corrections increase the vacuum energy. For this reason the classical vacuum is energetically favorable to the mean- eld one. It is explicitly seen for d < 0 from eq. (3.9) where 1 has to be negative. The function ( ) then increases with decreasing near = 1 and the mean- eld solution is a maximum, not a minimum. The conclusion of this section is that the classical vacuum is not stable for d > 2 where the mean- eld vacuum is energetically favorable. This reminds spontaneous generation of in quantum eld theory. The situation is opposite for d < 2, where the classical vacuum has lower energy than the mean- eld vacuum. { 8 { Stability of the mean- eld vacuum Let us now consider stability of the mean- eld vacuum under wavy uctuations, when (!) = + (!); and . The divergent part of the e ective action reads [23] For constant ab = this reproduces the divergent part of eq. (2.8) above. Expanding to quadratic order in uctuations q det( ab + ab) = + 1 2 aa 1 8 2 = 1 1 aa 2 2pdet + 2 ; where S2 = Z d2p (2 )2 A d2! 2 K0 d 2 Z The rst term on the right-hand side of eq. (4.4) plays a very important role for dynamics of quadratic uctuations. Because the path integral over imaginary axis, i.e. ab is pure imaginary, the rst term is always positive. Moreover, its ab goes parallel to exponential plays the role of a (functional) delta-function as The same is true for a constant part of ab. ! 1, forcing ab = ab. For the e ective action to the second order in uctuations we then nd the following (p) ( p) + 2A (p) ( p) + A (p) ( p) 2 ; Here c is a regularization-dependent constant. A A A 2 = = = (26 96 1 2 K0 d 2 { 9 { In the scaling limit, where [21{23] as ! 1 keeping the renormalized string tension KR xed, we have K0 ! K + KR2 1 + r 1 2 d ! ; so only A value diverges as 2 . Therefore, typical 1= so that ab is localized at the This is quite similar to what is shown in the book [31] for the uctuations about the classical vacuum. Thus only uctuates. case the eld ab(!) is localized at the value Equation (4.9) holds in the conformal gauge, where abpdet = ab. In the general where is constant for the world-sheet parametrization in use. We can therefore rewrite the right-hand side of eq. (2.1) in the scaling limit as which reproduces the Polyakov string formulation [25] for = 1. As shown in [ 21, 22 ] the action (4.11) is consistent only for a certain value of which is regularization-dependent. One has = 1 for the zeta-function regularization but < 1 for the proper-time regularization or the Pauli-Villars regularization. A subtlety with the computation of the determinants in the conformal gauge is that X and do not interact in the action (4.11) since in the conformal gauge ab = g^ab . Here g^ab is a ducial metric which we can set g^ab = ab without loss of generality. larization But the dependence of the determinants on appears because the world-sheet reguab = ab: ab = abpdet ; " = 1 2pdet = 1 2 depends on owing to di eomorphism invariance. For smooth the determinants are given by the usual conformal anomaly [25]. An advantage of using the Pauli-Villars regularization in the conformal gauge is that the implicit dependence on the metric becomes explicit as is described in section 6. Integrating over the matter and ghost elds, we arrive for g^ab = ab to the induced Sind = where the ghost operator is displayed in eq. (2.7). Evaluating the determinants, we nd which for = 1 reproduces the usual result. We see from eq. (4.15) (as well as from eq. (4.5) with = 0) that the action, describing uctuations of the metric, is positive only for d < 26 and becomes negative if d > 26. Thus, as far as the local stability of the action under wavy uctuations is concern, it is the same about the mean- eld vacuum as about the usual classical vacuum. This instability is probably linked to the presence of negative-norm states for d > 26 [32, 33]. ) Z d2! a reg 1 2 tr log + The string susceptibility index A very important characteristics of the string dynamics is the string susceptibility index str which characterizes the string entropy and is determined from the preexponential in the number of surfaces of xed area A by e F (A) Z d2z / A!1 A str 2 eCA; where C is a nonuniversal constant. F (A) on the left-hand side has the meaning of the Helmholtz free energy of a canonical ensemble at xed area A. Introducing the Lagrange multiplier, we rewrite (5.1) as Z d2z A = dj ej(R d2z A) ; where the integral over j runs parallel to the imaginary axis. This j is the same as introduced in section 3 except for the integral over j. Let us rst consider the integrand. The saddle-point solution is given by eq. (3.6). Then the integrand in (5.2) has an extremum at j(A) given by eq. (3.7) with substituted by A=Amin, Amin = L . Expanding about the extremum, we nd [24] jA K0Amin (A) = s 2d 2 s + K0 r 2K0 Amin A Amin A Amin A Amin 1 1 A Amin 3=2 ( j)2 : 1 + 2 K0 The integral over j = j j(A) goes along the imaginary axis and thus converges. For F (A) we obtain F (A) = p2d 2K0pA(A Amin) A(K0 + 2) + log [A(A Amin)] + const: (5.4) According to the de nition (5.1) of the string susceptibility index, we expect F (A) = regular + (2 str) log 3 4 A Amin for A Amin. Comparing with (5.4), this determines str = 1=2. It can be shown [24] that the one-loop correction contributes only to the regular part of F (A) and does not change the singular part that gives str = 1=2. This value can be exact because it is linked only to the emergence of the square-root singularity which is not changed by higher orders. We are to compare the mean- eld result for str with the one-loop computation of (5.1) about the classical vacuum which is almost trivially done by changing ! A and gives (5.1) / A 1 e(d 2) 2A=2 resulting in str = 1. We can compare it with the formula of the d ! 1 expansion [34] generalized to an arbitrary genus in [ 35, 36 ]. Since we deal with the worldsheet having topology of a cylinder which has two boundaries, its Euler character equals 0 like for a torus. This explains why there is no d-dependence of str. We have got str = 1 rather than str = 2 as in [ 35, 36 ] because we deal with an open rather than a closed string. The discrepancy between the obtained mean- eld value str = 1=2 and the perturbative value str = 1 is due to the fact that the vacua are di erent. The former applies for 2 < d < 26, while the latter applies for d < 2. 6 Fluctuations about the mean- eld The instability of the e ective action for d > 26 implies that we cannot straightforwardly make a systematic 1=d expansion as d ! +1. This is in contrast to the d ! 1 limit which comes along with the usual perturbative expansion because the vacuum is then just classical. The usual semiclassical expansion as d ! 1 cannot be extended to d > 2 because the vacuum states are di erent for d < 2 and d > 2. To go beyond the mean eld for 2 < d < 26, we de ne the partition function Z[h] = Z D e Sind=h (5.5) (5.6) (6.1) with Sind given by eq. (4.14). Here we have introduced an additional parameter h to control the \semiclassical" expansion about the mean eld which plays the role of a \classical" vacuum. This procedure makes sense of the change d ! d=h for the number of the X- elds and simultaneously 2 ! 2=h for the number of the ghost elds. Diagrammatically, this mean eld corresponds to summing up bubbles of both matter and ghosts. The mean- eld approximation is associated with h ! 0, while the expansion about the mean eld goes so that is convergent. Here M equations by For the proper-time regularization we have instead det( O) det( O + 2M 2) det( O + M 2)2 ; det( O) reg ! 1 is the regulator mass which is related to in the above 2 = 1 are then proportional to hl. In reality h = 1 but we can expect that the actual expansion parameter is 6h=(26 d) like in the usual semiclassical expansion as d ! 1. Then the expansion can make sense for d = 4. The action in eq. (6.1) is given by (4.14). For the Pauli-Villars regularization the determinants are regularized by the ratio of massless to massive determinants [24] (6.2) (6.3) (6.4) (6.5) (6.6) (6.7) # : (6.8) (6.9) the type to get The path integral over the regulator elds generates the propagator XM (k)XM ( k) = K0( k2 + M 2 ) We have added in (6.2) the ratio of the determinants for the masses p 2M and M to cancel the logarithmic divergence at small , because the Seeley expansion D ! e O ! E = 4 1 1 pdet + R 24 + : : : starts with the term 1= . This is speci c to the two-dimensional case. The massive determinants in eq. (6.2) can also be represented as path integrals of det d=2 Z DXM e 2 over the elds XM (!) with normal statistics or YM (!) with ghost statistics and the double number of components. We can explicitly add these regulator elds to the action (4.12) ) Z d2! + K0 Z 2 d2! " and the triple vertex of the XM XM interaction ( p)XM (k + p)XM ( k) truncated = K0M 2 The latter vanishes for M = 0 as it should owing to conformal invariance, but explicitly breaks it at nonzero M . Notice that path integration over all matter elds (both X and the regulators) runs with a simple nonregularized measure. This makes it very convenient to derive (regularized) Noether's currents and to calculate their anomalies. An instructive exercise is how to compute the usual anomaly in the trace of the energymomentum tensor 2 [Taa]mat = 4 K0 41 + 0 1 13 HJEP07(218)4 M 2(YM(i))2 + 2M 2Xp22M A5 : (6.11) Averaging (6.11) over the regulator elds, we obtain the diagrams depicted in gure 1, where the solid line corresponds to the propagator of the regulator elds Xp2M or YM while the wavy line corresponds to . We have explicitly in momentum space d Z h d2k (2 )2 2M 2 k2 + 2M 2 2M 2 k2 + M 2 d h = M 2 log 2; (6.12) Figure 1b = 2 d Z h d2k (2 )2 4M 4 ( k2 + 2M 2 )( (k p)2 + 2M 2 ) 2M 4 ( k2 + M 2 )( (k p)2 + M 2 ) d d Figure 1a = 2 reproducing eq. (6.4), and where and for brevity we denoted det a = det (6.13) (6.14) (6.15) (6.16) : (6.17) G(p) = 12 B 0 4m4 arctanh ppp2+8m2 p2+4m2 2m4 arctanh ppp2+4m2 1 p2+2m2 p(p2 + 8m2) p(p2 + 4m2) C A p m 2 = p m2 = M 2 : Ogh b a det The e ect of the diagram in gure 1c and the next orders is to complete the result to scalar curvature R as is discussed in appendix A. Adding all diagrams and using eq. (6.4), we obtain for the contribution from matter h[Taa]mati = 4 K0(1 ) 2 + d 12 R: It still remains to compute the contribution of the ghost determinant which we also regularize by the Pauli-Villars regularization d 2 det Ogh ba + 2M 2 a b Ogh ba + M 2 a 2 b The computation of the contribution from ghosts is pretty much similar to the one [25{27] for the perturbative vacuum and adding it with (6.16) we obtain for the trace of the total energy-momentum tensor (matter plus ghosts) hTaai D [Taa]mat + [Taa]gh E = 4 K0(1 ) d 2 1 2 + d 26 12 R; (6.18) which is the same as = acting on (4.15). The average in this formula is over the matter and ghost elds but not over which plays the role of an external eld. For given by eq. (2.9a) the divergent term vanishes, so we reproduce the usual conformal anomaly. The reason is that we have essentially made a one-loop calculation for the Polyakov-like action (4.12) with a constant ducial metric ^ab = ab and the result coincides with the one about the classical vacuum because of the background independence. 7 Computation of the central charge If ab is considered as a classical background metric, only matter and ghosts contribute to the central charge of the Virasoro algebra which equals d 26 like in eq. (6.18). Then the conformal anomaly vanishes only in d = 26 (the critical dimension) which reproduces the result of the old canonical quantization. We shall now see how this is modi ed when quantum uctuations of ab are taken into account in the mean- eld approximation. For this purpose let us compute the correlator of the two zz-components of the energymomentum tensor Tzz T (z) = Tmat(z) + Tgh(z): Classically, the X- eld does not interact, as is already pointed out, with the metric in the conformal gauge because of conformal invariance. Like in the previous section we shall make use of the Pauli-Villars regularization, where Tmat(z) explicitly depends on the regularizing elds as (7.1) (7.2) Tmat(z) = 2 K0 The diagrams contributing to the correlator hT (z)T (0)i in the mean- eld approximation are depicted in gure 2, where the solid line corresponds to the propagator of the eld X (and its regulators Xp2M and YM ) or the ghosts (and their regulators), while the wavy line corresponds to the propagator of HJEP07(218)4 h ( k) (k)i = To each closed line there is associated a factor of (d 26)=h coming from summation over the matter and ghosts like in eq. (6.18). The diagram in gure 2a (which have a combinatorial factor of 2) gives the usual result hT (z)T (0)ia) = d 26 2hz4 associated with the central charges of free elds: d for matter and 26 for ghosts, whose di erence vanishes only in the critical dimension d = 26. Only massless elds contribute to the most singular as z ! 0 part of the correlator shown in eq. (7.4) via the propagator Xq (z)Xq (0) = log(zz): 1 The diagram in gure 2b is usually associated with the next order of the perturbative expansion about the classical vacuum because it has two loops, but in the mean- eld approximation it has to be considered together with the diagram in gure 2a since both are of the same order in h. We shall return soon to the discussion of this issue. Every of the two closed loops in the diagram in gure 2b involves the momentum-space integral 2 where we have absorbed the ratio = into M 2 for simplicity. The power counting predicts a quadratically divergent term like M 2 zz in the integral (7.6), but it vanishes in the conformal gauge. Each of the two closed lines is associated ether with matter (the factor of d) of ghosts (the fector of 26). Multiplying the contribution of the two loops by the propagator (7.3), we nd for the diagram in gure 2b hT (z)T (0)ib) = (d 26) 12h (d 26) 6 12h (26 d) 12h z4 = d 26 2hz4 : (7.4) (7.5) = p 2 z 12 ; (7.6) (7.7) Notice this result is pure anomalous: it comes entirely from the regulator elds but M has canceled. Both diagrams in gure 2 give a \classical" (i.e. saddle-point) contribution from the viewpoint of the mean eld. Adding (7.4) and (7.7), we obtain zero value of the total central charge in the mean- eld approximation. The fact that the total central charge of the bosonic string is always zero in the meaneld approximation, independently on the number of the target-space dimensions d, is remarkable. Thus it always reminds the string in the critical dimension d = 26. A very similar situation occurs in the Polchinski-Strominger approach [10] to the e ective string theory, where the Alvarez-Arvis ground-state energy (same as (2.10) for = 1) was obtained from the requirement of vanishing the central charge at large to order 1= [10], 1= 3 [16] and 1= 5 [17]. The mean- eld approximation we used apparently sums up bubble graphs to all orders in 1= and explicitly results in the Alvarez-Arvis formula. 8 \Semiclassical" correction to the mean eld Let us consider a \semiclassical" correction to the mean- eld approximation which comes from averaging over uctuations of about . Integrating over the matter and ghost elds (including their regulators), we obtain the following induced action for the eld to quadratic order in : Se = Z ) Si(n2d) = (26 d) Z 96 h d2p (2 )2 ( p)G(p) (p) with G(p) given by eq. (6.14). This is not the end of the story because there are diagrams with three, four, etc. 's in (4.14), whose contributions we denote as Si(n3d), Si(n4d), etc. As is explicitly demonstrated in appendix A, it is convenient to introduce instead of another variable ' by (z) = e'(z); (z) = e'(z) 1 and to expand in '. Then the terms higher than quadratic order in ' are mutually canceled in the sum Sind = Si(n2d) + Si(n3d) + : : : = (26 d) Z d2p (2 )2 '( p)G(p)'(p) + O('3) large, pi2 M 2 . There is no reason to expect the cancellation in this case. in the IR limit where all variables pi's obey pipj M 2 , so the induced action (8.3) reproduces the usual e ective action for smooth '(z). However, we consider below explicitly the case of four ''s, where two momenta are small, pi2 M 2 , but two other momenta are The e ective action describes \slow" uctuations of ' with p2 M 2 and emerges after averaging over \fast" uctuations with p2 M 2 . The quadratic part of the e ective action gets then contribution from averaging higher terms in the induced action (which are generically nonlocal), so we write it in the spirit of DDK (a good review is [ 37 ]) as (8.1) (8.2) (8.3) (8.4) a) d) b) e) c) HJEP07(218)4 with a certain constant b2. The di erence between the induced action (8.3) and the e ective action (8.4) will show up when virtual momenta of the propagator h'( p)'(p)i in diagrams, which emerge after averaging over ', are large: p2 & M 2 . Hence the higher order in ' terms in eq. (8.3) may and will, as we see below, then play an important role. The reason why they survive is, roughly speaking, a quadratic divergence of the involved integrals. These terms are however subordinated in h because h'( p)'(p)i / h owing to eq. (7.3). It is instructive to give yet another explanation why the higher terms can emerge. Let us consider Tzz given by eqs. (7.1) and average (7.2) over the regulator elds with ' playing again the role of an external eld. The result is given by the diagrams in gure 1 whose analytic expressions are listed in eqs. (A.7){(A.11) of appedix A, where it is explicitly shown the cancellation of higher than quadratic terms when all momenta squared of external lines are small (i.e. pi2 M 2 ). I do not see again any reason to expect such a cancellation for momenta of the order of M 2 , so counterparts of the higher terms in eq. (8.3) may emerge. The result of the averaging over the regulators will not be yet the energy-momentum (pseudo)tensor because the averages in the path-integral language are associated with T products in the operator language. To obtain a genuine Tzz, we have to normal order the operators ' which produces additional terms like the diagrams in gure 3 coming from normal ordering in '4. We thus write T z'z = 1 2 1 + O('3) again in the spirit of DDK. nonlinear One more source of the nonlinearity is the well-known fact that the norm of ' is jj 'jj2 = Z d2z (z)( '(z))2: We can adopt the philosophy of DDK and replace the path integral over ' with the non(8.5) (8.6) linear norm (8.6) by the path integral over the eld '0 with a linear one by introducing the Jacobian for the transformation from ' to '0. It has again the form of (the exponential of) the action (8.3) and simply changes its coe cients. We shall therefore replace in eq. (8.3) 26 d 6h ) b2 0 1 ; 1 b 0 The di erence between this b20 and b2 in eq. (8.4) comes to order h from the diagrams with one propagator which are computed below. My last comment before proceeding with the computations is that the propagator h'( p)'(p)i behaves as 1=p2 for small p2, so one might expect therefore logarithmic IR divergences, associated with this behavior, which would spoil conformal invariance. However, the low-momentum e ective action is quadratic in the variable ' as is already mentioned (and demonstrated by explicit computations in appendix A), so the divergences are expected to cancel each other because the induced action coincides with the e ective action in the IR domain. We shall see in explicit computations this is indeed the case. The remaining contribution to be calculated will come from virtual momenta squared of the order of the cuto : k2 M 2 . I believe this is a heuristic proof of the theorem about the cancellation of the IR divergences. 8.1 Correction to Tzz The diagrams of the next to the leading order in h which describe \quantum" corrections to the mean- eld approximation for Tzz are depicted in gure 3. Their analytic expressions are listed in eqs. (B.1){(B.5) of appendix B. Every individual diagram has an IR divergence coming from the '-' propagator, but it has indeed canceled in the sum as anticipated. Actually the cancellation happens for the sum a) + 2b) 2c) because d) = 12 e) so only the diagrams in gure 3a, 3b and 3c contribute resulting in It is instructive to present the result in the DDK form Multiplying (8.9) by the normalizations of the propagator (7.3) and of the integrals and summing with the leading-order diagrams in gure 1, we obtain for the coe cient Q in eq. (8.5) where b20 is de ned in eq. (8.8). Q = q0 b 2 0 13 6 + O(h); (8.7) (8.8) HJEP07(218)4 (8.9) (8.10) (8.11) a) d) b) e) Z 1 d 0 An analogous direct computation of the quadratic in ' term in eq. (8.5) is a bit more tedious and involves 12 diagrams: 7 of which are new, while the contribution of the sum of remaining 5 diagrams is like 12 Q@2'2. The integrals involve two external momenta, which complicates their computation. The diagrams of the next to the leading order in h which describe \quantum" corrections to the mean- eld approximation for Se are depicted in gure 4. Their analytic expressions are listed in eqs. (B.12){(B.17) of appendix B. Every individual diagram has again an IR divergence coming from the '-' propagator, but it has canceled in the sum as anticipated. Actually the cancellation happens for the sum a) + 2b) 4c) + f ) because d) = e); so only the diagrams in gure 4a, 4b, 4c and 4f contribute. Accounting for combinatorial factors, we obtain Multiplying (8.12) by the normalization of the propagator (7.3) and of the integrals, accounting for ghosts and summing with the mean- eld result, we obtain for the coupling constant in the e ective action (8.4) 1 b2 = 1 b 2 0 5 + O(h): 8.3 Remark on the universality In the above computations of Tzz and Se we substituted G(p) in eq. (8.3) by p2 because otherwise the computation is hopeless. A question arises as to whether this a ects the results because characteristic virtual momenta squared in the diagrams are M 2 . It is possible to verify that by changing the regularization procedure (6.3) to ization (8.14) involves N Pauli-Villars regulators with masses pnM (n = 1; : : : ; N ) which complicates the computation. It can be shown however that the results (8.11), (8.13) do not change which is an argument in favor of their universality. 9 Discussion The main result of this Paper is that a quantization of the e ective string about the mean- eld ground state works in 2 < d < 26. The mean- eld approximation corresponds to conformal eld theory with the central charge vanishing for any d, resulting in the Alvarez-Arvis ground-state energy and complimenting the Polchinski-Strominger approach. A \semiclassical" expansion about the mean eld can be treated adopting the philosophy of DDK. The di erence from DDK is that our ' is massless as a consequence of the minimization at the mean- eld saddle point. The massless ' is thus a consequence of the nonperturbative mean- eld ground state for 2 < d < 26 in contrast to the usual perturbative one for d < 2. This may lead to infrared logarithms which would spoil conformal invariance when accounting for uctuations about the mean eld, but we argued they have to cancel because the low-momentum (or e ective) action is quadratic in '. This cancellation is explicitly shown to the lowest order of the \semiclassical" expansion about the mean eld. Thus, we expect that conformal invariance should be maintained order by order of the expansion. The explicit computation shows the (induced) action governing uctuations about the mean eld is however not quadratic in ', while only its low-momentum limit | the e ective action | is quadratic. The reason for that is, roughly speaking, quadratic divergences of the involved integrals. Using the Pauli-Villars regularization, I have shown how to systematically treat the induced action (4.14) and to deal with these higher order terms without assuming that ' is smooth and the determinants are approximated by the conformal anomaly. Their emergence may in uence the results and deserve further investigation. The most interesting question is what would be the spectrum of the Nambu-Goto string beyond the mean- eld approximation. In particular, whether the universal correction to the Alvarez-Arvis spectrum at the order 1= 5 (see [12] and references therein) is reproduced in the \semiclassical" expansion about the mean eld at one loop. This issue will be considered elsewhere. Acknowledgments I am grateful to Jan Ambj rn for sharing his insight into Strings. A Leading-order explicit computations To the leading order in h we consider (or ') as an external eld over which we shall average to next orders. The contribution of diagrams in gure 1 to Taa from the regulators read 2M 2 a) = b) = c) = d) = e) = Z Z Z Z Z d2k d2k d2k d2k (2 )2 p p 2M 8 60 p [(k (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) HJEP07(218)4 M=!1 2p2 + 3q2 + 3r2 + 2t2 + 3pq + 2pr + pt + 4qr + 2qt + 3rt : p 32M 10 p 2M 10 q q p r q r Multiplying each wavy line by , passing to the coordinate space and summing up the contributions of the diagrams in gure 1 with these of ghosts, we nd hTaai = = 26 26 12h (4 12h d d + ' 2 ) 6 where we have used (8.2) and expanded in '. We have thus reproduced eq. (6.18) to this order. In eq. (A.6) we simply subtracted 26 from d to account for the ghost contribution because the contribution of diagrams which emerge from ghost and the regulators of ghosts are identical for the mean- eld and perturbative vacua just as it is for the matter elds and regulators. The same applies below for Tzz. (k2 + M 2) = 0; pz) p q p q = 26 12h d d 3 2 + O('5); 16M 8kz(kz qz q qz rz p tz) tz) q p r q r M=!1 2pz2 + 3qz2 + 3rz2 + 2tz2 + 6pzqz + 8pzrz + 10pztz + 7qzrz + 8qztz + 6rztz : (A.11) Using (8.2), we analogously to eq. (A.6) obtain for Tzz a) = b) = c) = d) = e) = Z Z Z Z Z d2k d2k d2k d2k d2k M=!1 3pz2 + 4qz2 + 3rz2 + 9pzqz + 12pzrz + 9qzrz 240 (A.7) (A.8) (A.9) (A.10) (A.12) The analogous contribution of diagrams in gure 1 to Tzz read i.e. the free energy-momentum tensor to this order. The reason why I presented in this appendix the explicit computations of Taa and Tzz is to emphasize that numerical factors are most important to get the free-theory results. The cancellation would no longer take place if these factors were changed due to induced interactions, as we shall immediately see in the next appendix. ; pz qz p rz ' + 3 \Semiclassical" corrections Contribution to Tzz The contributions of diagrams in gure 3 to Tzz involve a) = b) = c) = d) = e) = (k q)2[(k ; pz) pz) q)2[(k p)2 + 2M 2](k2 + 2M 2)2(q2 + 2M 2) pz) q)2[(k p)2 + 2M 2](k2 + 2M 2)(q2 + 2M 2) pz) pz) pz) p)2 + 2M 2](k2 + 2M 2)2 q2[(k pz) pz) p)2 + 2M 2](k2 + 2M 2) q2[(k For the computation of integrals it is convenient to multiply a generic integral by the projector Z d2k (2 )2 ka(kb pb)f (k2; p2; kp) = f1(p2)gab + f2(p2)papb P ab = 2 papb p2 gab Z d2k (2 )2 Z 2 (2 )2 kz(kz kp k 2 f (k2; p2; kp) = f2(p2)p2: pz)f (k2; p2; kp) = f2(p2)pz2: This trick is implemented in the Mathematica program of appendix C, where the integrals are computed by rst integrating over the two relative angles and then by the two absolute values of the virtual momenta. ing for combinatorial factors, we obtain Performing the computation by the Mathematica program in appendix C and account2c) d) + e) = 1 2 13pz2 288 : (B.1) (B.2) (B.3) ; (B.4) : (B.5) (B.6) (B.7) (B.8) (B.9) (B.10) Notice that d) = 12 e) so only the diagrams in gure 3a, 3b and 3c contribute. The infrared divergence coming from the '-' propagator has indeed canceled in the sum, as anticipated. Multiplying (B.10) by the normalization of the propagator (7.3) and of the integrals and summing the diagrams in gure 1 and gure 3, we obtain 1 B.2 Contribution to Se The contributions of diagrams in gure 4 to Se involve q)2[(k 2M 8 ; q)2[(k p)2 + 2M 2](k2 + 2M 2)2(q2 + 2M 2) q)2[(k p)2 + 2M 2](q2 + 2M 2) (k q)2[(k p)2 + M 2](q2 + M 2) Performing the computation by the Mathematica program of appendix C and accounting for combinatorial factors, we obtain 4c) d) + e) + f ) = 5p2 48 : Notice that again d) = e) so only the diagrams in gure 4a, 4b, 4c and 4f contribute. The infrared divergence coming from the '-' propagator has indeed canceled in the sum, as anticipated. a) = b) = c) = d) = e) = f ) = 2M 6 8M 6 4M 4 4M 4 2M 4 16M 8 8M 6 16M 8 : (k q)2[(k p)2 + M 2](k2 + M 2)2(q2 + M 2) q)2[(k p)2 + 2M 2](k2 + 2M 2)(q2 + 2M 2) (k q)2[(k p)2 + M 2](k2 + M 2)(q2 + M 2) p)2 + 2M 2](k2 + 2M 2)2 q2[(k p)2 + 2M 2](k2 + 2M 2) q2[(k 2M 6 2M 4 (B.11) (B.12) (B.13) (B.14) ; (B.15) ; (B.16) (B.17) (B.18) Multiplying (B.18) by the normalization of the propagator (7.3) and of the integrals, accounting for ghosts and summing with the mean- eld result, we obtain Se = (B.19) C Mathematica programs C.1 Program for computing diagrams in gure 3 that results in (8.9) (* Computation of T_zz by integration over angles and x=k^2, y=q^2 *) HJEP07(218)4 x =. y =. M =. al = 1 ii1 = Normal[ ii2 = Normal[ M = 1 Resa1 = ResaM M =. ii1 = Normal[ ii2 = Normal[ M = Sqrt[2] Resb2 = ResbM Normal[Series[(2 k^2 Cos[bet]^2 - al k p Cos[bet] - k^2) M^6/(k^2 + q^2 2 k q Cos[the])/(k^2 + al^2 p^2 - 2 al k p Cos[bet] + M^2)/(q^2 + al^2 p^2 2 al p q Cos[bet + the] + M^2)/(k^2 + M^2)/(q^2 + M^2), {p, 0, 2}]]; ixyan = Integrate[%, {bet, 0, 2 Pi}] iixlay = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions -> {x > y > 0}] Series[Integrate[iixlay, {y, 0, x - mu^2}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]] iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions -> {y > x > 0}] Series[Integrate[iiylax, {y, x + mu^2, Infinity}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]] ResaM = Integrate[FullSimplify[ii1 + ii2], {x, 0, Infinity}, Assumptions -> {M > mu > 0}] tResa = FullSimplify[(Resa2 - 2 Resa1)/16/Pi^2] Normal[Series[(2 k^2 Cos[bet]^2 - al k p Cos[bet] k^2) M^6/(k^2 + q^2 - 2 k q Cos[the])/(k^2 + al^2 p^2 2 al k p Cos[bet] + M^2)/(k^2 + M^2)^2/(q^2 + M^2), {p, 0, 2}]]; ixyan = Integrate[%, {bet, 0, 2 Pi}] iixlay = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions -> {x > y > 0}] Series[Integrate[iixlay, {y, 0, x - mu^2}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]] iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions -> {y > x > 0}] Series[Integrate[iiylax, {y, x + mu^2, Infinity}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]] Assumptions -> {M > mu > 0}] tResb = FullSimplify[(Resb2 - 2 Resb1)/16/Pi^2] Normal[Series[(2 k^2 Cos[bet]^2 - al k p Cos[bet] k^2) M^4/(k^2 + q^2 - 2 k q Cos[the])/(k^2 + al^2 p^2 2 al k p Cos[bet] + M^2)/(k^2 + M^2)/(q^2 + M^2), {p, 0, 2}]]; ixyan = Integrate[%, {bet, 0, 2 Pi}] iixlay = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions -> {x > y > 0}] Series[Integrate[iixlay, {y, 0, x - mu^2}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]] iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions -> {y > x > 0}] Series[Integrate[iiylax, {y, x + mu^2, Infinity}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]] RescM = Integrate[FullSimplify[ii1 + ii2], {x, 0, Infinity}, Assumptions -> {M > mu > 0}] tResc = FullSimplify[(Resc2 - 2 Resc1)/16/Pi^2] FullSimplify[tResa + 2 tResb - 2 tResc, Assumptions -> {mu > 0}] (* gives -13 p^2/288 *) C.2 Program for computing diagrams in gure 4 that results in (8.12) (* Computation of S_eff by integration over angles and x=k^2, y=q^2 *) M = 1 Resc1 = RescM M =. x =. y =. M =. k = Sqrt[x] q = Sqrt[y] al = 1 Normal[Series[M^8/(k^2 + q^2 - 2 k q Cos[the])/(k^2 + al^2 p^2 2 al k p Cos[bet] + M^2)/(q^2 + al^2 p^2 2 al p q Cos[bet + the] + M^2)/(k^2 + M^2)/(q^2 + M^2), {p, 0, 2}]]; ixyan = Integrate[%, {bet, 0, 2 Pi}]; iixlay = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions -> {x > y > 0}] ii1 = Normal[Series[Integrate[iixlay, {y, 0, x - mu^2}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]; iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions -> {y > x > 0}] ii2 = Normal[Series[Integrate[iiylax, {y, x + mu^2, Infinity}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]; ResaM = Integrate[FullSimplify[ii1 + ii2], {x, 0, Infinity}, Assumptions -> {M > mu > 0}] M = Sqrt[2]; Resa2 = ResaM; M = 1; M =. SResa = FullSimplify[(Resa2 - 2 Resa1)/16/Pi^2] Normal[Series[M^8/(k^2 + q^2 - 2 k q Cos[the])/(k^2 + al^2 p^2 2 al k p Cos[bet] + M^2)/(k^2 + M^2)^2/(q^2 + M^2), {p, 0, 2}]]; ixyan = Integrate[%, {bet, 0, 2 Pi}]; ii1 = Normal[Series[Integrate[iixlay, {y, 0, x - mu^2}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]; iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions -> {y > x > 0}] ii2 = Normal[Series[Integrate[iiylax, {y, x + mu^2, Infinity}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]; ResbM = Integrate[FullSimplify[ii1 + ii2], {x, 0, Infinity}, Assumptions -> {M > mu > 0}] M = Sqrt[2]; Resb2 = ResbM; M = 1; Resb1 = ResbM; M =. SResb = FullSimplify[(Resb2 - 2 Resb1)/16/Pi^2] Normal[Series[M^6/(k^2 + q^2 - 2 k q Cos[the])/(k^2 + al^2 p^2 2 al k p Cos[bet] + M^2)/(k^2 + M^2)/(q^2 + M^2), {p, 0, 2}]]; ixyan = Integrate[%, {bet, 0, 2 Pi}]; iixlay = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions -> {x > y > 0}] ii1 = Normal[Series[Integrate[iixlay, {y, 0, x - mu^2}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]; iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions -> {y > x > 0}] ii2 = Normal[Series[Integrate[iiylax, {y, x + mu^2, Infinity}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]; RescM = Integrate[FullSimplify[ii1 + ii2], {x, 0, Infinity}, Assumptions -> {M > mu > 0}] M = Sqrt[2]; Resc2 = RescM; M = 1; Resc1 = RescM; M =. SResc = FullSimplify[(Resc2 - 2 Resc1)/16/Pi^2] Normal[Series[M^4/(k^2 + q^2 - 2 k q Cos[the])/(k^2 + al^2 p^2 2 al k p Cos[bet] + M^2)/(q^2 + M^2), {p, 0, 2}]]; ixyan = Integrate[%, {bet, 0, 2 Pi}]; iixlay = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions -> {x > y > 0}] ii1 = Normal[Series[Integrate[iixlay, {y, 0, x - mu^2}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]; iiylax = Integrate[ixyan, {the, 0, 2 Pi}, Assumptions -> {y > x > 0}] ii2 = Normal[Series[Integrate[iiylax, {y, x + mu^2, Infinity}, Assumptions -> {M^2 > x > mu^2 > 0}], {mu, 0, 0}]]; ResfM = Integrate[FullSimplify[ii1 + ii2], {x, 0, Infinity}, Assumptions -> {M > mu > 0}] Resf2 = ResfM; SResf = FullSimplify[(Resf2 - 2 Resf1)/16/Pi^2] FullSimplify[SResa + 2 SResb - 4 SResc + SResf, Assumptions -> {mu > 0}] FullSimplify[% 48 Pi/(2 Pi)^2, Assumptions -> {mu > 0}] (* gives -5 p^2/48 *) Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Phys. Lett. A 3 (1988) 1651 [INSPIRE]. 321 (1989) 509 [INSPIRE]. [1] V.A. Kazakov, A.A. Migdal and I.K. Kostov, Critical Properties of Randomly Triangulated Planar Random Surfaces, Phys. Lett. B 157 (1985) 295 [INSPIRE]. [2] F. David, Planar Diagrams, Two-Dimensional Lattice Gravity and Surface Models, Nucl. Phys. B 257 (1985) 45 [INSPIRE]. [3] J. Ambj rn, B. Durhuus and J. Frohlich, Diseases of Triangulated Random Surface Models and Possible Cures, Nucl. Phys. B 257 (1985) 433 [INSPIRE]. Gravity, Mod. Phys. Lett. A 3 (1988) 819 [INSPIRE]. [4] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Fractal Structure of 2D Quantum [5] F. David, Conformal Field Theories Coupled to 2D Gravity in the Conformal Gauge, Mod. [6] J. Distler and H. Kawai, Conformal Field Theory and 2D Quantum Gravity, Nucl. Phys. B [7] J. Ambj rn, B. Durhuus and T. Jonsson, Quantum geometry. A statistical eld theory approach, Cambridge University Press, Cambridge U.K. (1997). [8] J. Ambj rn and B. Durhuus, Regularized bosonic strings need extrinsic curvature, Phys. Lett. B 188 (1987) 253 [INSPIRE]. [9] Y. Makeenko, QCD String as an E ective String, in proceedings of the Low dimensional World Scienti c (2012), pp. 211{222 [arXiv:1206.0922] [INSPIRE]. [10] J. Polchinski and A. Strominger, E ective string theory, Phys. Rev. Lett. 67 (1991) 1681 118 [arXiv:1302.6257] [INSPIRE]. [11] S. Dubovsky, R. Flauger and V. Gorbenko, E ective String Theory Revisited, JHEP 09 [12] O. Aharony and Z. Komargodski, The E ective Theory of Long Strings, JHEP 05 (2013) [13] S. Dubovsky, R. Flauger and V. Gorbenko, Flux Tube Spectra from Approximate Integrability at Low Energies, J. Exp. Theor. Phys. 120 (2015) 399 [arXiv:1404.0037] [INSPIRE]. [INSPIRE]. [INSPIRE]. [14] S. Hellerman, S. Maeda, J. Maltz and I. Swanson, E ective String Theory Simpli ed, JHEP 09 (2014) 183 [arXiv:1405.6197] [INSPIRE]. Phys. A 31 (2016) 1643001 [arXiv:1603.06969] [INSPIRE]. [15] B.B. Brandt and M. Meineri, E ective string description of con ning ux tubes, Int. J. Mod. [16] J.M. Drummond, Universal subleading spectrum of e ective string theory, hep-th/0411017 [17] O. Aharony, M. Field and N. Klingho er, The e ective string spectrum in the orthogonal gauge, JHEP 04 (2012) 048 [arXiv:1111.5757] [INSPIRE]. HJEP07(218)4 142 [arXiv:1601.00540] [INSPIRE]. 93 (2016) 066007 [arXiv:1510.03390] [INSPIRE]. Lett. B 770 (2017) 352 [arXiv:1703.05382] [INSPIRE]. [22] J. Ambj rn and Y. Makeenko, Scaling behavior of regularized bosonic strings, Phys. Rev. D [23] J. Ambj rn and Y. Makeenko, Stability of the nonperturbative bosonic string vacuum, Phys. [24] J. Ambj rn and Y. Makeenko, The use of Pauli-Villars regularization in string theory, Int. J. Mod. Phys. A 32 (2017) 1750187 [arXiv:1709.00995] [INSPIRE]. [25] A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett. B 103 (1981) 207 [26] B. Durhuus, P. Olesen and J.L. Petersen, Polyakov's Quantized String With Boundary Terms, Nucl. Phys. B 198 (1982) 157 [INSPIRE]. Geometry, Nucl. Phys. B 216 (1983) 125 [INSPIRE]. Cambridge U.K. (2002), pp. 208{210. [27] O. Alvarez, Theory of Strings with Boundaries: Fluctuations, Topology and Quantum [28] Y. Makeenko, Methods of contemporary gauge theory, Cambridge University Press, (1987), pp. 173{174. Rev. D 6 (1972) 1655 [INSPIRE]. [29] Y. Makeenko, An interplay between static potential and Reggeon trajectory for QCD string, Phys. Lett. B 699 (2011) 199 [arXiv:1103.2269] [INSPIRE]. [30] L. Brink and H.B. Nielsen, A Simple Physical Interpretation of the Critical Dimension of Space-Time in Dual Models, Phys. Lett. B 45 (1973) 332 [INSPIRE]. [31] A.M. Polyakov, Gauge elds and strings, Harwood Academic Publishers, Reading U.K. [32] R.C. Brower, Spectrum generating algebra and no ghost theorem for the dual model, Phys. [33] P. Goddard and C.B. Thorn, Compatibility of the Dual Pomeron with Unitarity and the Absence of Ghosts in the Dual Resonance Model, Phys. Lett. B 40 (1972) 235 [INSPIRE]. [34] A.B. Zamolodchikov, On the entropy of random surfaces, Phys. Lett. B 117 (1982) 87 [INSPIRE]. (2008) and online pdf version at http://qft.itp.ac.ru/ZZ.pdf. [18] O. Alvarez , The Static Potential in String Models, Phys. Rev. D 24 ( 1981 ) 440 [INSPIRE]. [19] J.F. Arvis , The Exact qq Potential in Nambu String Theory, Phys . Lett. B 127 ( 1983 ) 106 [20] P. Olesen , Strings and QCD , Phys. Lett. B 160 ( 1985 ) 144 [INSPIRE]. [21] J. Ambj rn and Y. Makeenko , String theory as a Lilliputian world , Phys. Lett. B 756 ( 2016 ) [35] S. Chaudhuri , H. Kawai and S.H.H. Tye , Path Integral Formulation of Closed Strings, Phys. Rev. D 36 ( 1987 ) 1148 [INSPIRE] . Phys. Lett. B 187 ( 1987 ) 149 [INSPIRE]. [36] I.K. Kostov and A. Krzywicki , On the Entropy of Random Surfaces With Arbitrary Genus, [37] A. Zamolodchikov and A. Zamolodchikov, Lectures on Liouville theory and matrix models,


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Yuri Makeenko. Mean field quantization of effective string, Journal of High Energy Physics, 2018, 104, DOI: 10.1007/JHEP07(2018)104