J. Ambj0rna'h* Y. Makeenkoa'€**

a The Niels Bohr Institute, Copenhagen University DK-2100, Copenhagen, Denmark

bIMAPP, Radboud University 6525, A J, Nijmegen, The Ne therlands

'"Institute of Theoretical and Experimental Physics 117218, Moscow, Russia

Received October 20, 2014

We implement a UV regularization of the bosonic string by truncating its mode expansion and keeping the regularized theory "as diffeomorphism invariant as possible". We compute the regularized determinant of the 2d Laplacian for the closed string winding around a compact dimension, obtaining the effective action in this way. The minimization of the effective action reliably determines the energy of the string ground state for a long string and/or for a large number of space-time dimensions. We discuss the possibility of a scaling limit when the cutoff is taken to infinity.

Contribution for the JETP special issue in honor of V. A. Rubakov's 60th birthday

DOI: 10.7868/S0044451015030155


A modern formulation of string theory is based on the Polyakov path integral fl], where the worldsheet metric gab(ui) and the target-space position X'^ui), 11. = 1,... ,d, of the string worldsheet are treated as independent variables. Thanks to the diffeomorphism invariance, the metric can be diagonalized, gab = e^tiab-, by choosing the conformal gauge. The remaining path integration over the so-called Liouville field p decouples on the IIlclSS shell for the bosonic string in d = 26, the critical dimension. Due to this decoupling, the results in d = 26 reproduce those obtained in the early 1970s using the operator formalism. For d ^ 26, the path integral over p has to be taken into account and plays an important role for the consistency of the theory.

The path integral over the target-space string coordinates (and ghosts) is Gaussian and results in a determinant of the 2d Laplace Beltrami operator with proper boundary conditions imposed. For an open

E-mail: ambjorn'fflnbi.dk

E-mail: makeenko'fflnbi.dk

string with fixed ends, these are Dirichlet boundary conditions, for which the determinant was computed in [2,3]. The result is given by the conformal anomaly and determines the effective action for the Liouville field p. The path integral over p can be consistently-treated [4] order by order in the inverse string length and/or in the limit of a large number of space time dimensions d, where the WKB expansion around the saddle points applies. Of special interest in this approach is the ground-state energy as a function of the string length R, which is given by the well-known Alvarez Arvis spectrum [5, 6]. It reveals a tachyonic singularity at distances R < R0. with 1 being the tachyon mass squared. For larger distances, this quantity is well-behaved.

The conformal factor does not appear in the classical string. However, as was pointed out by Polyakov fl], the computation of 2d determinants requires a UV cutoff like A2yfg in the parameter space1). This follows from the diffeomorphism invariance and results in the conformal anomaly, which decouples in the effective action as A —¥ oc. The dependence of the cutoff on the

11 We recall that x/g = e.f in the conformal Range.

motric is of crucial importance for the consideration in this paper.

The emergence of a tachyonic excitation of the string is seen clearly in the zeta-function regularization, where the sum over oscillators (the stringy modes) is computed as

This negative value is the result of an analytic continuation from positive values of the argument of the zeta function, for which the sum is convergent. Of course, the sum of positive numbers in Eq. (1) is infinite and the negative value emerges after the subtraction of an infinity as was illustrated in detail by one of the first calculations [7]. In this paper, we investigate how the sums over the stringy modes (like in Eq. (1)) can be consistently regularized, maximally preserving the dif-feomorphism invariance.

One regularization of this kind is the so-called dynamical triangulation (DT) [8], where the intrinsic geometry of the parameter space (defined by the metric gab(oj)) is approximated by a set of equilateral triangles of side a. The summation over triangulations is done independently of the integration over the targetspace coordinates associated, for instance, with the vertices of the triangles. In this sense, DT discretizes the Polyakov string. DT provides the conceptual foundation for matrix-model solutions of the so-called non-critical string theory. However, for the real bosonic string theory with d > 2, DT also provides an interesting result. In DT, the renormalized mass excitations and the renormalized string tension are by definition nonnegative and it was shown in [9] that if we keep the lowest mass excitation finite as the cutoff a —¥ 0, the string tension scales to infinity. With this otherwise very successful regularization, it thus seems impossible to obtain a bosonic string with a finite tachyonic mass and a finite string tension.

In this paper, we want to make contact with the DT result mentioned using a standard continuum regularization of the bosonic string, namely, truncating the string mode expansion. We consider a closed string winding once around a compact dimension of length R and propagating a (Euclidean) time T. We generically consider a string whose length is larger than the inverse tachyon IIlclSS (if present for the regularized string). We therefore expect a stable ground state and compute its mass as a function of the string length R. This determines the string tension as the energy per a unit length and should provide us with information about the mass

of the lowest state (usually, the tachyon) from the behavior of the energy at small R. We then search for a scaling regime, where the two quantities may or may not remain finite in the limit of an infinite cutoff.

In Sees. 2 and 3, we introduce the string regularization by a truncation of the mode expansion and compute the regularized determinant of the 2d Laplacian for a lot x uin rectangle in the parameter space. We use the Dirichlet boundary condition along the T-axis and periodic boundary conditions along the R-axis. This gives an effective action of the regularized string. We demonstrate how the Luscher term emerges using this regularization. In Sec. 4, we argue that the reparame-terization invariance favors Att = NrT/R for the numbers of modes Nt and Nr along the respective T- and R- axes.

The effective action computed this way depends on the ratio uit/uir- There are two cases where this parameter can be reliably determined by minimizing the effective action: small a1 /R2 and large d. They are considered in Sees. 5 and C. In Sec. 5, we first recall the situation in the classical limit and then analyze the one-loop (semiclassical) correction that determines the renormalization of the string tension. In Sec. 6, we derive the equation which minimizes the effective action at large d. The minimized effective action contains terms of all orders in a'/R2, and we find the effective action in both the large-R. and the small-R limit. We show that, at a finite cutoff, the tachyonic singularity-is present for positive values of the bare string tension A'o, but is absent for a range of negative values of A'0. We find that there exists a critical (negative) value A'» such that if the bare string tension A'0 approaches A'» from above, it is possible to have a renormalized string tension K that stays finite as the cutoff A —¥ oc, but in this case the lowest mass excitation does not scale but goes to infinity. However, there also exists a value Kc, A'» < Kc < 0, such that if A'0 approaches Kc from below, the lowest mass can be kept finite for the cutoff A —¥ oc, but in this case the "renormalized" string tension goes to infinity as A2. This situation seems very similar to what is observed using DT as a regularization.


We consider a closed string winding one time around a compact dimension of length R. We impose Dirichlet boundary condition along the T-axis and periodic boundary condition along the R-axis. We consider

an ujt x u>r rectangle in the paraniotor spaco mapped onto a T x R rectangle in the target space with 0 and R identified along the R-axis.

The one-loop effective action can be computed as the determinant of the 2d Laplacian in the conformal gauge with the above boundary conditions imposed on the lot x ujji rectangle. The Laplacian is


where p = RT/lOrlOt^ and we have


tiTog(—A«2) = ^ ^ log <

m=1 n= — oo v




77111 \ ' LOT J




where we want to think about u = tt/A as a UV latticelike cutoff similar to the lattice cutoff a in DT. For large Ut » uJr, we replace the sum over m by the integral over the "momentum" x = nm/uir-


trlog(—A<r) = — x

^ / dx log ■

/ 2nn\''





To regularize this divergent expression, we integrate over .i: from 0 to A', where the upper limit of the integration is introduced to provide a UV cutoff, which cuts off mode numbers m larger than XlOt/t7■ In Sec. 4, we relate it to the A introduced above. To perform the integral, we use the relation


<i.i;log(.i;2 + y2) = X [log(A'2 + y2) — 2]


+ 2yarctan—. (5) V

where y = 2nn/ujR. As X —¥ oo, in the right-hand side, we recover the term tt|j/| familiar from the zeta-function regularization.

The remaining sum over n can be evaluated by using Plana's summation formula


1 °° r

2/(0) + E /(»)= / <WM-

n=l n


which holds whenever f(z) is analytic for Rej >0.

The first term in the right-hand side of Eq. (6) results in the integral


J dx J dy log(.i:2 + y2) =

0 -Y

= (3A'2+1"2) arctan ^+(A'2+31"2) arctan + A )

+ 2XY [log(A'2 + V2) - 3] - |(A2 + V2). (7)

Here, Y is a UV mode sum cutoff along the lor axis in the same way as X was along the lot axis.


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