Mean field quantization of effective string

I describe the recently proposed quantization of bosonic string about the mean-field 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 $\gamma_{\rm str}$ 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.


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 d < 2, where the results both of dynamical triangulation [1][2][3] and of conformal field theory [4][5][6] are consistent and agree. For d > 2 the scaling limit of dynamically triangulated random surfaces is particle-like 1 rather than string-like because only the lowest mass scales but the string tension does not scale and tends to infinity in the scaling limit. [8] Analogously, the conformal field theory approach does not lead to sensible results for 2 < d < 26. [4][5][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 fluxes of the gauge field. String description makes sense only for the distances larger than the confinement scale. For shorter distances the quark-gluon degrees of freedom are more relevant due to asymptotic freedom. This picture is well justified 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 "effective string" philosophy [10], which works perturbatively order by order in the inverse string length (for recent advances see [11][12][13][14]). Then string quantization is consistent even below the critical dimension (d = 26 for the relativistic bosonic string) and a few leading orders reproduce [10,15,16] the Alvarez-Arvis ground-state energy [17][18][19]. In this Paper I shall pay much attention to this issue.
In the recent series of papers [20][21][22] it has been understood why lattice string formulations resulted in the particle-like continuum limit. A nonperturbative mean-field 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 (ρ ab =ρδ ab in the conformal gauge) becames infinite in the scaling limit. For this reason an infinite amount of stringy modes (which is ∝ρ/a 2 with a being a UV cutoff) can be reached even at the distances of order a, That was in contrast to the usual continuum limit in quantum field theory, where the amount of degrees of freedom can be infinite only if the correlation length is infinite. The discovered phenomenon is specific to theories with diffeomorphism invariance and was called the Lilliputian continuum limit.
The task of this Paper is to analyze properties of the mean-field vacuum which plays the role of a "classical" string ground state and "quantum" fluctuations 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 α 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 Knizhnik-Polyakov-Zamolodchikov (KPZ) -David-Distler-Kawai (DDK) results [4][5][6] for the parturbative vacuum, which is applicable for d < 2.
It will be shown in the Paper that the total central charge of the system vanishes in the mean-field approximation which is thus consistent in noncritical dimension 2 < d < 26. This is in contrast to the old canonical quantization which works only in the critical di-mension d = 26 and compliments the effective-string approach of Polchinski-Strominger, where the consistency is explicitly demonstrated to a few lower orders of the perturbative expansion [10,15,16]. I then analyze a "semiclassical" correction to the mean-field approximation and show that it does not spoil conformal invariance in spite of logarithmic infrared divergences caused by the propagator of a massless field, which cancel in the sum of diagrams. This Paper is organized as follows. In Sects. 2, 3, 4 I review the results [20,21] which form a background for further investigations. Sect. 5 is devoted to the computation of the mean-field value of the string susceptibility index γ str = 1/2 and its comparison to the perturbative value γ str = 1. In Sect. 6 I formulate a general procedure for expanding about the mean field 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 Sect. 7 I compute the total central charge of the system in the mean-field approximation and show that it vanishes for 2 < d < 26. Sect. 8 is devoted to the "semiclassical" expansion about the mean field. 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 Sect. 9. Some explicit computations are presented in Appendices A, B by using the Mathematica programs from Appendix C.

The mean-field ground state
We consider a closed bosonic string in target space with one compactified dimension of circumference β. 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 configuration, if β is larger than a certain value to guarantee that the classical energy of the string dominates over the energy of zero-point fluctuations.
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 µ . To make it quadratic in X µ , we rewrite it, introducing a Lagrange multiplier λ ab and an independent intrinsic metric ρ ab , as 3 (2.1) where K 0 stands for the bare string tension. The equivalence of the two formulations can be proven by path integrating over the functions λ ab (ω) and ρ ab (ω) which take on imaginary and real values, respectively.
It is convenient to choose the world-sheet coordinates ω 1 and ω 2 inside an ω L × ω β rectangle in the parameter space. Then the classical solution X µ cl minimizing the action (2.1) linearly depends on ω The classical value of ρ ab coincides with the classical induced metrics which becomes diagonal for The classical value of λ ab reads λ ab cl = ρ ab cl det ρ cl (2.5) and simplifies to λ ab cl = δ ab if Eq. (2.4) is satisfied. We apply the path-integral quantization to account for quantum fluctuations of the X-fields by splitting X µ = X µ cl + X µ q and then performing the Gaussian path integral over X µ q . We thus obtain the action, governing the fields λ ab and ρ ab , The operator O reproduces the usual two-dimensional Laplacian ∆ for λ ab given by the 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 λ ab and ρ ab with the action (2.8), which runs as is already mentioned over imaginary λ ab (ω) and real ρ ab (ω). The action (2.6) is often called the induced (or emergent) action to be distinguished from the effective action which is usually associated with slowly varying fields in the lowmomentum limit. It is convenient to fix the conformal gauge when ρ ab = ρδ ab , so that √ det ρ = ρ. Then the log of the determinant of the ghost operator [23] O gh 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 [24,25]. The subtleties associated with the boundary conditions will be inessential both for the matter and ghost determinants for L β when only the bulk terms survive. We shall describe in Sect. 6 how to accurately compute the determinants using the Pauli-Villars regularization but let us assume for a moment that λ ab (ω) =λδ ab and ρ ab (ω) =ρδ ab with constantλ andρ. As we shall see these constant values ofλ and ρ are what is needed for the mean-field approximation.
The computation of the matter and ghost determinants for constantλ andρ is presently an exercise in string theory courses with the result Here Λ 2 cuts off eigenvalues of the operators involved. The first and second terms on the right-hand side are classical contributions, while the sign of the third term is negative for d/λ > 2 to comply with positive entropy. Technically, it comes as the product of the eigenvalues divided by Λ, where every multiplier is less than 1. The last term is known as the Lüscher term which is due to the Casimir energy of zero-point fluctuations. Its negative sign is intimately linked to the presence of the tachyon. The next step is to minimize (2.8) overλ,ρ to find the mean-field configuration which describes the string ground state. The difference 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 µ 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 ω β /ω L which plays the role of the modular parameter of the cylinder. This guarantees that ρ ab and λ ab are diagonal as is required by the conformal gauge. We shall return to this issue soon.
The minimum of (2.8) is remarkable simple [20,21] The value of the action (2.8) at the minimum (2.9) is (2.10) 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 infinite set of diagrams of perturbation theory about the classical vacuum. This approach is quite similar to that 4 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 n 2 = 1. After integration over the fields n one obtains an induced action as a functional of u, whose minimum determines the exact vacuum state as N → ∞. For finite N the fluctuations of u about this mean-field 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 n cl = (1, 0, . . . , 0) possesses an O(N − 1) symmetry rather than the O(N ) symmetry as the saddle-point vacuum does and the fields n fluctuate strongly. For our case the number of fields X µ in the sigma model (2.1) is d, so the saddle-point is justified by K 0 ∼ d → ∞. At finite d the saddle-point solution (2.9) is associated with the mean-field approximation.
The minimization of the action (2.8) over ω β /ω L can be now understood as follows. In the mean-field 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-defined if the bare string tension K 0 > K * given by At this critical value of K 0 the square root in (2.9a) vanishes.
• The large-d ground-state energy [17], 5 where an analytic regularization was used, are recovered by Eq. (2.10) for Λ 2 = 0. Analogously, the ground-state energy obtained by the old canonical quantization [18] is reproduced by our mean-field approximation. This is not surprising because fluctuations of ρ are ignored in the canonical quantization.
• The metric (2.9b) becomes infinite when K 0 → K * given by Eq. (2.11). This is crucial for constructing the scaling limit.
At the classical level ρ cl coincides with the induced metric as is displayed in (2.3). In the mean-field approximation it is superseded bȳ where the average is understood in the sense of the path integral over X µ . Equation (2.12) follows from the minimization of the effective action over λ ab . Thus, in the mean-field approximationρ coincides with the averaged induced metric.

Instability of the classical vacuum
The usual semiclassical (or one-loop) correction to the classical ground-state energy due to zero-point fluctuations [28] is described in textbooks. The sum of the two reads The original computation [17] used the Nambu-Goto string. How the same result can be obtained for the Polyakov string is shown in [27].
To make the bulk part of (3.1) finite, it is usually introduced the renormalized string tension which is kept finite as Λ → ∞. Then it is assumed that it works order by order of the perturbative expansion about the classical vacuum, so that K R can be made finite by fine tuning K 0 . 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 K 0 . 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 δλ toλ. However, the sum of the two linear inρ terms in Eq. (2.8) vanishes forλ given by Eq. (2.9a), so the action does not depend onρ 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 and defining the field Minimizing the action with the source term added for constant j ab = jδ ab , we find [21] λ(j) = 1 in the mean-field approximation for ω L = L and ω β = β 1 √ K 0 . Inverting Eq. (3.6), we obtain To understand the properties of the vacuum, we compute an "effective potential" by performing the Legendre transformation like in the studies of symmetry breaking in quantum field theory.
In the mean-field approximation we then obtain with j(ρ) given by Eq. (3.7) as it should. Near the classical vacuum we have 0 <ρ − 1 1 and the potential (3.9) decreases with increasingρ because the second term on the right-hand side has the negative sign, demonstrating an instability of the classical vacuum. If K 0 > K * given by Eq. (2.11), the potential (3.9) linearly increases withρ for largeρ and thus has a (stable) minimum at which is the same as (2.9b) for The coefficient in front of the quadratic term is positive for K 0 > K * which explicitly demonstrates the (global) stability of the mean-field minimum (2.9b). The situation is different for d < 2, where quantum corrections increase the vacuum energy. For this reason the classical vacuum is energetically favorable to the mean-field one. It is explicitly seen for d < 0 from Eq. (3.9) whereρ − 1 has to be negative. The function Γ(ρ) then increases with decreasingρ nearρ = 1 and the mean-field 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-field vacuum is energetically favorable. This reminds spontaneous generation of ρ in quantum field theory. The situation is opposite for d < 2, where the classical vacuum has lower energy than the mean-field vacuum.

Stability of the mean-field vacuum
Let us now consider stability of the mean-field vacuum under wavy fluctuations, when with ω-dependent δρ and δλ. The divergent part of the effective action reads [21] For constant λ ab =λδ ab and ρ =ρ this reproduces the divergent part of Eq. (2.8) above.
Expanding to quadratic order in fluctuations we find from (4.2) The first term on the right-hand side of Eq. (4.4) plays a very important role for dynamics of quadratic fluctuations. Because the path integral over λ ab goes parallel to imaginary axis, i.e. δλ ab is pure imaginary, the first term is always positive. Moreover, its exponential plays the role of a (functional) delta-function as Λ → ∞, forcing δλ ab = δλ δ ab . The same is true for a constant part of δλ ab .
For the effective action to the second order in fluctuations we then find the following quadratic form: Here c is a regularization-dependent constant.
In the scaling limit, where [20,21] as Λ → ∞ keeping the renormalized string tension K R fixed, we have so only A λλ diverges as Λ 2 . Therefore, typical δλ ∼ 1/Λ so that λ ab is localized at the valueλ ab =λδ ab . This is quite similar to what is shown in the book [29] for the fluctuations about the classical vacuum. Thus only ρ fluctuates.
Equation (4.9) holds in the conformal gauge, where ρ ab √ det ρ = δ ab . In the general case the field λ ab (ω) is localized at the valuē λ ab =λρ ab det ρ, (4.10) whereλ 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 [23] forλ = 1. As shown in [20] the action (4.11) is consistent only for a certain value ofλ which is regularization-dependent. One hasλ = 1 for the zeta-function regularization butλ < 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 µ and ρ do not interact in the action (4.11) since in the conformal gauge ρ ab =ĝ ab ρ. Hereĝ ab is a fiducial metric which we can setĝ ab = δ ab without loss of generality. But the dependence of the determinants on ρ appears because the world-sheet regu- depends on ρ owing to diffeomorphism invariance. For smooth ρ the determinants are given by the usual conformal anomaly [23]. 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 Sect. 6. Integrating over the matter and ghost fields, we arrive forĝ ab = δ ab to the induced action where the ghost operator is displayed in Eq. (2.7). Evaluating the determinants, we find for smooth ρ which forλ = 1 reproduces the usual result.
We see from Eq. (4.15) (as well as from Eq. (4.5) with δλ = 0) that the action, describing fluctuations 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 fluctuations is concern, it is the same about the mean-field vacuum as about the usual classical vacuum. This instability is probably linked to the presence of negative-norm states for d > 26 [30,31].

The string susceptibility index
A very important characteristics of the string dynamics is the string susceptibility index γ str which characterizes the string entropy and is determined from the preexponential in the number of surfaces of fixed area A by 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 fixed area A. Introducing the Lagrange multiplier, we rewrite (5.1) as where the integral over j runs parallel to the imaginary axis. This j is the same as introduced in Sect. 3 except for the integral over j.
Let us first 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) withρ substituted by A/A min , A min = Lβ. Expanding about the extremum, we find [22] jA The integral over ∆j = j − j(A) goes along the imaginary axis and thus converges. For F (A) we obtain According to the definition (5.1) of the string susceptibility index, we expect for A A min . Comparing with (5.4), this determines γ str = 1/2. It can be shown [22] that the one-loop correction contributes only to the regular part of F (A) and does not change the singular part that gives γ 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-field result for γ str with the one-loop computation of (5.1) about the classical vacuum which is almost trivially done by changing ρ → Aρ and gives resulting in γ str = 1. We can compare it with the formula of the d → −∞ expansion [32] generalized to an arbitrary genus in [33,34]. 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 γ str . We have got γ str = 1 rather than γ str = 2 as in [33,34] because we deal with an open rather than a closed string. The discrepancy between the obtained mean-field value γ str = 1/2 and the perturbative value γ str = 1 is due to the fact that the vacua are different. The former applies for 2 < d < 26, while the latter applies for d < 2.

Fluctuations about the mean-field
The instability of the effective action for d > 26 implies that we cannot straightforwardly make a systematic 1/d expansion as d → +∞. This is in contrast to the d → −∞ limit which comes along with the usual perturbative expansion because the vacuum is then just classical. The usual semiclassical expansion as d → −∞ cannot be extended to d > 2 because the vacuum states are different for d < 2 and d > 2.
To go beyond the mean field for 2 < d < 26, we define the partition function with S ind given by Eq. (4.14). Here we have introduced an additional parameter h to control the "semiclassical" expansion about the mean field which plays the role of a "classical" vacuum. This procedure makes sense of the change d → d/h for the number of the X-fields and simultaneously 2 → 2/h for the number of the ghost fields. Diagrammatically, this mean field corresponds to summing up bubbles of both matter and ghosts. The mean-field approximation is associated with h → 0, while the expansion about the mean field goes in h. Diagrams with l loops are then proportional to h l . 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 → −∞. 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 [22] is convergent. Here M → ∞ is the regulator mass which is related to Λ in the above equations by For the proper-time regularization we have instead We have added in (6.2) the ratio of the determinants for the masses √ 2M and M to cancel the logarithmic divergence at small τ , because the Seeley expansion starts with the term 1/τ . This is specific to the two-dimensional case. The massive determinants in Eq. (6.2) can also be represented as path integrals of the type over the fields X M (ω) with normal statistics or Y M (ω) with ghost statistics and the double number of components. We can explicitly add these regulator fields to the action (4.12) to get The path integral over the regulator fields generates the propagator and the triple vertex of the δρX µ M X ν M interaction 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 fields (both X µ 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 Averaging (6.11) over the regulator fields, we obtain the diagrams depicted in Fig. 1, where e) c) a) b) d) Figure 1. Diagrams contributing to the average of (6.11) or (7.2) over the regulator fields.
the solid line corresponds to the propagator of the regulator fields X √ 2M or Y M while the wavy line corresponds to δρ. We have explicitly in momentum space reproducing Eq. (6.4), and and for brevity we denoted The effect of the diagram in Fig. 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 It still remains to compute the contribution of the ghost determinant which we also regularize by the Pauli-Villars regularization The computation of the contribution from ghosts is pretty much similar to the one [23][24][25] for the perturbative vacuum and adding it with (6.16) we obtain for the trace of the total energy-momentum tensor (matter plus ghosts) which is the same as δ/δρ acting on (4.15). The average in this formula is over the matter and ghost fields but not over ρ which plays the role of an external field. Forλ 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 fiducial metricρ ab =ρδ ab and the result coincides with the one about the classical vacuum because of the background independence. For this purpose let us compute the correlator of the two zz-components of the energymomentum tensor T zz ≡ T (z) = T mat (z) + T gh (z). (7.1)

Computation of the central charge
Classically, the X-field does not interact, as is already pointed out, with the metric ρ in the conformal gauge because of conformal invariance. Like in the previous Section we shall make use of the Pauli-Villars regularization, where T mat (z) explicitly depends on the regularizing fields as The diagrams contributing to the correlator T (z)T (0) in the mean-field approximation are depicted in Fig. 2, where the solid line corresponds to the propagator of the field X (and its regulators X √ 2M and Y M ) or the ghosts (and their regulators), while the wavy line corresponds to the propagator of δρ 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 Fig. 2a (which have a combinatorial factor of 2) gives the usual result associated with the central charges of free fields: d for matter and 26 for ghosts, whose difference vanishes only in the critical dimension d = 26. Only massless fields contribute to the most singular as z → 0 part of the correlator shown in Eq. (7.4) via the propagator The diagram in Fig. 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-field approximation it has to be considered together with the diagram in Fig. 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 Fig. 2b involves the momentum-space integral (7.6) where we have absorbed the ratioρ/λ into M 2 for simplicity. The power counting predicts a quadratically divergent term like M 2 ρ 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 find for the diagram in Fig. 2b T Notice this result is pure anomalous: it comes entirely from the regulator fields but M has canceled. Both diagrams in Fig. 2 give a "classical" (i.e. saddle-point) contribution from the viewpoint of the mean field. Adding (7.4) and (7.7), we obtain zero value of the total central charge in the mean-field approximation. The fact that the total central charge of the bosonic string is always zero in the meanfield 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 effective string theory, where the Alvarez-Arvis ground-state energy (same as (2.10) forλ = 1) was obtained from the requirement of vanishing the central charge at large β to order 1/β [10], 1/β 3 [15] and 1/β 5 [16]. The mean-field approximation we used apparently sums up bubble graphs to all orders in 1/β and explicitly results in the Alvarez-Arvis formula.

"Semiclassical" correction to the mean field
Let us consider a "semiclassical" correction to the mean-field approximation which comes from averaging over fluctuations of ρ aboutρ.
Integrating over the matter and ghost fields (including their regulators), we obtain the following induced action for the field δρ to quadratic order in δρ: with G(p) given by Eq. (6.14). This is not the end of the story because there are diagrams with three, four, etc. δρ's in (4.14), whose contributions we denote as S ind , etc. As is explicitly demonstrated in Appendix A, it is convenient to introduce instead of δρ another variable ϕ by ρ(z) =ρ e ϕ(z) , δρ(z) =ρ e ϕ(z) − 1 (8.2) and to expand in ϕ. Then the terms higher than quadratic order in ϕ are mutually canceled in the sum in the IR limit where all variables p i 's obey p i p j M 2ρ , so the induced action (8.3) reproduces the usual effective action for smooth ϕ(z). However, we consider below explicitly the case of four ϕ's, where two momenta are small, p 2 i M 2ρ , but two other momenta are large, p 2 i ∼ M 2ρ . There is no reason to expect the cancellation in this case. The effective action describes "slow" fluctuations of ϕ with p 2 M 2ρ and emerges after averaging over "fast" fluctuations with p 2 ∼ M 2ρ . The quadratic part of the effective 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 [35]) as with a certain constant b 2 . The difference between the induced action (8.3) and the effective action (8.4) will show up when virtual momenta of the propagator ϕ(−p)ϕ(p) in diagrams, which emerge after averaging over ϕ, are large: p 2 > ∼ M 2ρ . Hence the higher order in ϕ 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 ϕ(−p)ϕ(p) ∝ h owing to Eq. (7.3). It is instructive to give yet another explanation why the higher terms can emerge. Let us consider T zz given by Eqs. (7.1) and average (7.2) over the regulator fields with ϕ playing again the role of an external field. The result is given by the diagrams in 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 Tproducts in the operator language. To obtain a genuine T zz , we have to normal order the operators ϕ which produces additional terms like the diagrams in Fig. 3 coming from normal ordering in ϕ 4 . We thus write again in the spirit of DDK. One more source of the nonlinearity is the well-known fact that the norm of ϕ is nonlinear We can adopt the philosophy of DDK and replace the path integral over ϕ with the nonlinear norm (8.6) by the path integral over the field ϕ 0 with a linear one by introducing the Jacobian for the transformation from ϕ to ϕ 0 . It has again the form of (the exponential of) the action (8.3) and simply changes its coefficients. We shall therefore replace in Eq.
The difference between this b 2 0 and b 2 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 ϕ(−p)ϕ(p) behaves as 1/p 2 for small p 2 , so one might expect therefore logarithmic IR divergences, associated with this behavior, which would spoil conformal invariance. However, the low-momentum effective action is quadratic in the variable ϕ 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 effective 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 cutoff: k 2 ∼ M 2ρ . I believe this is a heuristic proof of the theorem about the cancellation of the IR divergences.

Correction to T zz
The diagrams of the next to the leading order in h which describe "quantum" corrections to the mean-field approximation for T zz are depicted in Fig Every individual diagram has an IR divergence coming from the ϕ-ϕ propagator, but it has indeed canceled in the sum as anticipated. Actually the cancellation happens for the sum a) + 2b) − 2c) because d) = 1 2 e) so only the diagrams in Fig. 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 Fig. 1, we obtain for the coefficient Q in Eq. (8.5) where b 2 0 is defined in Eq. (8.8). An analogous direct computation of the quadratic in ϕ 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 1 2 Q∂ 2 ϕ 2 . The integrals involve two external momenta, which complicates their computation.

Correction to S eff
The diagrams of the next to the leading order in h which describe "quantum" corrections to the mean-field approximation for S eff are depicted in Fig. 4

Remark on the Universality
In the above computations of T zz and S eff we substituted G(p) in Eq. (8.3) by p 2 because otherwise the computation is hopeless. A question arises as to whether this affects the results because characteristic virtual momenta squared in the diagrams are ∼ M 2ρ . It is possible to verify th by changing the regularization procedure (6.3) to Such a modification of the Pauli-Villars regularization is discussed in [22]. The regularization (8.14) involves N Pauli-Villars regulators with masses √ nM (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.

Discussion
The main result of this Paper is that a quantization of the effective string about the mean-field ground state works in 2 < d < 26. The mean-field approximation corresponds to conformal field 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 field can be treated adopting the philosophy of DDK.
The difference from DDK is that our ϕ is massless as a consequence of the minimization at the mean-field saddle point. The massless ϕ is thus a consequence of the nonperturbative mean-field 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 fluctuations about the mean field, but we argued they have to cancel because the low-momentum (or effective) action is quadratic in ϕ. This cancellation is explicitly shown to the lowest order of the "semiclassical" expansion about the mean field. Thus,we expect that conformal invariance should be maintained order by order of the expansion.
The explicit computation shows the (induced) action governing fluctuations about the mean field is however not quadratic in ϕ, while only its low-momentum limit -the effective 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 ϕ is smooth and the determinants are approximated by the conformal anomaly. Their emergence may influence the results and deserve further investigation.
The most interesting question is what would be the spectrum of the Nambu-Goto string beyond the mean-field approximation. In particular, whether the universal correction to the Alvarez-Arvis spectrum at the order 1/β 5 (see [12] and references therein) is reproduced in the "semiclassical" expansion about the mean field at one loop. This issue will be considered elsewhere.

Acknowledgment
I am grateful to Jan Ambjørn for sharing his insight into Strings.

A Leading-order explicit computations
To the leading order in h we consider δρ (or ϕ) as an external field over which we shall average to next orders.
The contribution of diagrams in Fig. 1 to T a a from the regulators read 2p 2 + 3q 2 + 3r 2 + 2t 2 + 3pq + 2pr + pt + 4qr + 2qt + 3rt 60π . (A.5) Multiplying each wavy line by −δρ, passing to the coordinate space and summing up the contributions of the diagrams in Fig. 1 with these of ghosts, we find T a a = 26 − d 12h ∂ 2 δρ − (2δρ ∂ 2 δρ + ∂ a δρ ∂ a δρ) + 3(δρ 2 ∂ 2 δρ + δρ ∂ a δρ ∂ a δρ) −(4δρ 3 ∂ 2 δρ + 6δρ 2 ∂ a δρ ∂ a δρ) + O(δρ 5 ) where we have used (8.2) and expanded in ϕ. 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-field and perturbative vacua just as it is for the matter fields and regulators. The same applies below for T zz .
, (B.5) For the computation of integrals it is convenient to multiply a generic integral Notice that d) = 1 2 e) so only the diagrams in Fig. 3a, 3b and 3c contribute. The infrared divergence coming from the ϕ-ϕ 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 Fig. 1 and Fig. 3, we obtain

C Mathematica programs
This Appendix can be downloaded as an ancillary file [36].