Quantisation of the effective string with TBA

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

In presence of a static pair of sources, the spectrum of low-lying states of whatever confining gauge theory in D space-time dimensions is described, at large source separations, by an effective string theory. In the far infrared the latter flows, in the static gauge, to a two-dimensional massless free-field theory. It is known that the Lorentz invariance of the gauge theory fixes uniquely the first few subleading corrections of this free-field limit. We point out that the first allowed correction - a quartic polynomial in the field derivatives - is exactly the composite field ..., built with the chiral components,T and ..., of the energy-momentum tensor. This irrelevant perturbation is quantum integrable and yields, through the thermodynamic Bethe Ansatz (TBA), the energy levels of the string which exactly coincide with the Nambu-Goto spectrum. We obtain this way the results recently found by Dubovsky, Flauger and Gorbenko. This procedure easily generalizes to any two-dimensional CFT. It is known that the leading deviation of the Nambu-Goto spectrum comes from the boundary terms of the string action. We solve the TBA equations on an infinite strip, identify the relevant boundary parameter and verify that it modifies the string spectrum as expected.

Published for SISSA by Springer

Received: May 16, 2013 Accepted: June 19, 2013 Published: July 10, 2013

Michele Caselle,a Davide Fioravanti,b Ferdinando Gliozzia and Roberto Tateoa

aDipartimento di Fisica, Universit di Torino and INFN Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy

bDipartimento di Fisica e Astronomia, Universit di Bologna and INFN Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , [email protected]

Abstract: In presence of a static pair of sources, the spectrum of low-lying states of whatever conning gauge theory in D space-time dimensions is described, at large source separations, by an e ective string theory. In the far infrared the latter ows, in the static gauge, to a two-dimensional massless free-eld theory. It is known that the Lorentz invariance of the gauge theory xes uniquely the rst few subleading corrections of this free-eld limit. We point out that the rst allowed correction - a quartic polynomial in the eld derivatives - is exactly the composite eld T T, built with the chiral components, T and T, of the energy-momentum tensor. This irrelevant perturbation is quantum integrable and yields, through the thermodynamic Bethe Ansatz (TBA), the energy levels of the string which exactly coincide with the Nambu-Goto spectrum. We obtain this way the results recently found by Dubovsky, Flauger and Gorbenko. This procedure easily generalizes to any two-dimensional CFT. It is known that the leading deviation of the Nambu-Goto spectrum comes from the boundary terms of the string action. We solve the TBA equations on an innite strip, identify the relevant boundary parameter and verify that it modies the string spectrum as expected.

Keywords: Wilson, t Hooft and Polyakov loops, Boundary Quantum Field Theory, Exact S-Matrix, Integrable Field Theories

c

Quantisation of the e ective string with TBA

JHEP07(2013)071

[circlecopyrt] SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP07(2013)071

Web End =10.1007/JHEP07(2013)071

Contents

1 Introduction 1

2 The composite perturbation T T and its expectation value 6

3 The exact S matrix for massless ows 7

4 Spectrum degeneracy 12

5 The innite strip 13

6 The boundary TBA 146.1 Basic boundary conditions 146.2 More general boundary conditions 15

7 Conclusions 17

A The ADE general case 18

1 Introduction

Even though a rigorous proof of quark connement in Yang-Mills theories is still missing, numerical experiments and theoretical arguments leave little doubt that this phenomenon is associated to the formation of a thin string-like ux tube, the conning string, which generates, for large quark separations, a linearly rising of conning potential.

The string-like nature of the ux tube is particularly evident in the strong coupling region of lattice gauge theories, where the vacuum expectation value of large Wilson loops is given by a sum over certain lattice surfaces which can be considered as the world-sheet of the underlying conning string. When the coupling constant decreases, this two-dimensional system undergoes a roughening transition [13] where the sum of these surfaces diverges and the colour ux tube of whatever lattice gauge theory undergoes a transition towards a rough phase, which is connected to the continuum limit. It is widely believed that such a phase transition belongs to the Kosterlitz-Thouless universality class [4]. Accordingly, the renormalisation group equations imply that the e ective string action S describing the dynamics of the ux tube in the whole rough phase ows at large scales towards a massless free-eld theory. Thus, for large enough inter-quark separations it is not necessary to know explicitly the specic form of the e ective string action S, but only its infrared limit

S[X] = Scl + S0[X] + . . . , (1.1)

1

JHEP07(2013)071

where the classical action Scl describes the usual perimeter-area term, X denotes the two-dimensional bosonic elds Xi(1, 2), with i = 1, 2, . . . , D 2, describing the transverse

displacements of the string with respect the conguration of minimal energy, 1, 2 are the

coordinates on the world-sheet and S0[X] is the Gaussian action

S0[X] = 2

[integraldisplay]

d2 (@ X [notdef] @ X) . (1.2)

This action is written in the physical or static gauge, where the only degrees of freedom taken into account are the physical ones i.e. the transverse displacements Xi. Even in this infrared approximation the e ective string is highly predictive, indeed it predicts the leading correction to the linear quark-anti-quark potential, known as Lscher term [5, 6]

V (R) = R

d2

b1(@1X [notdef] @1X) + b2(@1@0X [notdef] @1@0X) + b3(@1X [notdef] @1X)2 + . . .

[bracketrightbig]

. (1.5)

Of course the addition of these terms modies the spectrum of the physical states. For instance the interquark potential (1.3) becomes, at rst order in b1 [11],

V (R) = R

(D 2)

24R b1

JHEP07(2013)071

(D 2)

24R + O(1/R2) . (1.3)

Accurate numerical simulations have shown the validity of that expectation [712]. This infrared limit also accounts for the logarithmic broadening of the string width as a function of the inter-quark separation [13]. This phenomenon was rst observed long time ago in the Z2 3D gauge theory [14] and only recently, using the very e cient Lscher-Weisz algorithm [11], has also been observed in a non-abelian Yang-Mills theory [15, 16].

In the last few years there has been a substantial progress in lattice simulations in measuring various properties of the ux tube, in particular the interquark potential and the energy of the excited string states (see e.g. [1728]) which is now sensible to the rst few subleading corrections of the free-eld infrared limit of the e ective action. The latter is the sum of all the terms respecting the symmetry of the system, which in an Euclidean space is SO(2) [notdef] ISO(D 2). The rst few terms are

S = Scl+S0[X]+

[integraldisplay]

hc2(@ X [notdef] @ X)2 + c3(@ X [notdef] @ X)(@ X [notdef] @ X)[bracketrightBig]+Sb+. . . , (1.4)

where Sb is the boundary action characterizing the open string. Indeed quantum eld theories on space-time manifolds with boundaries require, in general, the inclusion in the action of contributions localized at the boundary. If the boundary is a Polyakov line in the 0 direction, on which we assume Dirichlet boundary conditions, the rst few terms are

Sb =

[integraldisplay]

d0

(D 2)

6R2 + O(1/R3) . (1.6)

In 2004 Lscher and Weisz [29] noted that the comparison of the string partition function on a cylinder (Polyakov correlator) with the sum over closed string states in a Lorentz (or rotation) invariant theory yields strong constraints (called open-closed string duality). In particular they showed in this way that b1 = 0. This property was then further generalized

2

in [30]. It was subsequently recognized that an essential ingredient of these constraints is the Lorentz invariance of the bulk space-time [3133]. The conning string action could be regarded as the e ective action obtained from the underlying Yang-Mills theory of the conning vacuum in presence of a large Wilson loop by integrating out all the massive degrees of freedom [33]. This integration does not spoil the original Poincar invariance of the underlying gauge theory, however this symmetry is no longer manifest, being spontaneously broken. As expected, it is realized through non-linear transformations of the Xis. The e ective string action (1.4) should be invariant under the innitesimal Lorentz transformation in the plane ( , j)

Xi = [epsilon1] j ij [epsilon1] jXj@ Xi . (1.7)

For instance, if we apply the transformation (1.7) to the term Sb1 proportional to b1 in the boundary action Sb = Sb1 + Sb2 + . . . we get at once

(Sb1) = b1 [integraldisplay]

[epsilon1]1id0 @1Xi + higher order terms [negationslash]= 0 , (1.8)

thus such a term breaks explicitly Lorentz invariance, hence b1 = 0. In a similar way [33] it is possible to show that b3 = 0 . On the contrary the b2 term is compatible with Lorentz invariance provided we add an innite sequence of terms generated by the non-linearity of the transformation. The associated recursion relations can be easily solved and the nal expression can be written in a closed form [34]

S2 = b2

[integraldisplay]

2, (1.10)

where the integers n, n dene the total energy 2n/R (2n/R) of the left (right) moving massless phonons. Similarly for the open string with xed ends, where n = n, one has

En(R) =

3

JHEP07(2013)071

@1@0X [notdef] @1@0X1 + @1X [notdef] @1X+ (@1@0X [notdef] @1X)2(1 + @1X [notdef] @1X)2[bracketrightbigg]

. (1.9)

It is easy to construct in this way boundary terms of higher order [34]. Actually this procedure was rst applied to the bulk action and it was shown that the requirement of Lorentz invariance of the infrared free-eld limit (1.1) generates the whole Nambu-Goto (NG) action [32, 33, 35]. In the latter reference this method was generalized to the construction of the e ective action of higher dimensional extended objects as D-branes on which other massless modes, besides the Xis, are propagating. It can also be used to construct the allowed bulk corrections to the Nambu-Goto action [3638]; further informations on the bulk corrections of the NG can be found working in a gauge where the Lorentz invariance is manifest [39, 40] (see also a general discussion on this argument in [41]).

A formal light-cone quantisation of the Nambu-Goto action [42] suggested a simple Ansatz for the energy spectrum of the closed string of length R

E(n,n)(R) =

d0

s2R2 + 4

n + n

D 212[parenrightbigg]+

2(n n)

R

s2R2 + 2

n

D 2 24

. (1.11)

We refer to (1.10) and (1.11) as the exact Nambu-Goto spectrum even if we know that this Ansatz is incompatible with Lorentz invariance in D < 26 [43], except maybe in D = 3 [44].

For large enough R one can expand (1.10) and (1.11) in powers of 1/R2. Lattice simulations of conning gauge theories in 2+1 and 3+1 dimensions show that the ground state and the rst excitations of the conning string have an energy spectrum very close to that of NG. This suggests that the e ective action constructed along the lines illustrated above can be considered as small perturbations of NG action that can be evaluated using a suitable regularization and a standard expansion in perturbative diagrams [29, 45]. It turns out that only rst order calculations on the parameters ci and bi are practically feasible.

Recently, a new powerful non-perturbative method has been introduced in this context [40]. It is based on the study of the S matrix describing the scattering of the quanta of the string excitations, the phonons, in the word-sheet. Assuming a reasonably simple form for S it turns out that the system is quantum integrable [46], hence one can apply the method of thermodynamic Bethe Ansatz (TBA) to calculate the non-perturbative spectrum of the e ective string. Taking the simplest form of the S matrix and some assumptions on the way of interacting of the phonons with di erent avours (i.e. transverse indices), it turns out the spectrum coincides with the exact NG spectrum of the closed string [46]. This method was also applied to describe some apparent deviations of the spectrum of the closed string [47].

In the present paper we re-derive the NG spectrum starting from the observation that the rst non-Gaussian correction of the string action (1.4), once the coe cients ci assume the values required by Lorentz invariance of the target space, namely c2 = 18 and c3 = 12, coincide with the composite eld T

T, where T and T are the chiral components of the energy momentum tensor T . Thus the e ective string action is, at this perturbative order, a two-dimensional integrable quantum eld theory formed by a conformal eld theory (the infrared Gaussian limit) perturbed by T T. We do not need any further assumption to derive, through the TBA, the NG spectrum.

We also show that a similar spectrum emerges from a general class of CFTs perturbed by T T. The energy levels for the identity primary eld and its descendents coincides with (1.10) and (1.11) once one replaces D 2 with the central charge c. The level

degeneracy di ers in an intriguing way: it is know that the degeneracy of the closed string grows exponentially for large E as exp(E/TH), where TH =

p3/(D 2) -the

Hagedorn temperature- coincides with the inverse of the distance Rc where the ground state of the NG spectrum develops a tachyon, i.e. where the argument of the square root in (1.10) vanishes. A similar relationship between the position of the tachyonic singularity of the ground state and the degeneracy of highly excited states holds for general CFTs. We shall check it in the critical Ising model where this degeneracy can be calculated exactly.

The interplay between quantum integrability and Lorentz invariance in the target space is an intriguing issue: the rst non-Gaussian contribution of the string action is an integrable perturbation only if c2/c3 = 14 as required by Lorentz invariance, however this

does not imply that the generated NG spectrum agrees with that of a Lorentz-invariant string theory. As already mentioned, a pure NG spectrum is compatible with Lorentz invariance in the Minkowski target space only for D = 26 (and perhaps D = 3) and

4

JHEP07(2013)071

perturbative calculations show that the NG spectrum deviates from that of a Lorentz-invariant string already at the order 1/R5 in D > 3 dimensions , and starting at the order 1/R7 non-universal terms are expected to contribute to the energy levels [40, 41, 45]. These contributions could be inserted in the TBA approach by assuming that at shorter distances other perturbations contribute, besides T T.

Actually the leading deviation of the NG spectrum comes from the boundary action (1.9). This is also the most signicant from a phenomenological point of view, being associated with the rst non-NG correction of the interquark potential. We have indeed [29, 45]

V (R) = R

where the third term comes from the c2 and c3 terms in (1.4) [29]. This deviation of the interquark potential has been already observed in lattice gauge theories and the b2 parameter has been evaluated [34, 48].

In this paper we introduce the TBA equations describing these boundary e ects by generalizing the approach of [40] to an innite strip with Dirichlet boundary conditions. We nd that also in this case the equations are solvable explicitly and the energy spectrum is given by (1.11). It is also easy to derive the corrections of the open string NG spectrum due to the boundary constants bi and we recover in particular eq. (1.12).

Even though the non-perturbative method we used is very powerful and the calculations of the level corrections due to these coupling constants could be easily pushed to any perturbative order, we do not think that this formulation of the e ective string is ultra-violet complete. The reason is that the whole spectrum of the theory includes an innite set of negative energy levels: because of the square root in eq. (1.10), the complete energy spectrum is actually [notdef][notdef]E(n,n)(R)[notdef]. The theory can be fermionized and one may assume

that the sea of negative energy levels is completely lled, however we did not succeeded in nding the zero-point energy produced by this sea, because of the huge degeneracy of the string states. As far as this problem is not completely solved, one should regard this formulation as an e ective theory.

In the next section we provide an elementary calculation of the T T nature of the quartic term along with some quantum check. In section 3 we describe in detail the exact S matrix for a critical RG ow of the Ising model in the limit of the massless phonons, following the method of Aliosha Zamoldchikov in the study of the ux between the tricritical Ising model and the critical Ising model and obtain the spectrum of the T T perturbed Ising model in a closed form. In section 4 we study the degeneracy of the spectrum and compare it with the degeneracy of the string. Then in section 5 we put the theory in an innite strip, dene a consistent reection factor and, in section 6, solve the boundary TBA for the open string in the case D = 3 and compare it with the perturbative calculations. Finally, in appendix A, we show that Nambu-Goto like spectra emerge from a large class of T T perturbed CFTs and section 7 contains our conclusions with a summary of the main results.

5

JHEP07(2013)071

(D 2)

24R

1

2R3

(D 2) 24

2 b23(D 2)60R4 + O(1/R5) , (1.12)

2 The composite perturbation T T and its expectation value

The energy-momentum tensor of the free-eld theory (1.2) can be written as

T = @ X [notdef] @ X

1

2 (@ X [notdef] @ X) . (2.1)

Note that this tensor is symmetric, traceless and conserved, as it should. Once we put in eq. (1.4) the values of c2 and c3 prescribed by Lorentz invariance, we have, as anticipated in the Introduction,

S = Scl + S0[X]

[integraldisplay]

d2 T T + Sb + . . . (2.2)

In two-dimensional CFT it is useful to introduce the chiral components Tzz = 12(T11 iT12) and Tzz = 12(T11 + iT12) and use the normalized quantities T = 2Tzz,

T = 2Tzz in

such a way the operator product expansion begins with

T (z)T (w) = D 2

2

1(z w)4

+ . . . (2.3)

and similarly for T. Thus at the end we have

S = Scl + S0[X]

1

22

[integraldisplay]

JHEP07(2013)071

d2 T T + Sb + . . . (2.4)

There is a consistency check based on some general properties of the expectation value of the composite eld T T pointed out by Sasha Zamolodchikov [49] for any two-dimensional quantum eld theory. In the particular case of a CFT on a innite strip we have

hT T[angbracketright] = [angbracketleft]T [angbracketright][angbracketleft]

T[angbracketright] . (2.5)

Under a conformal mapping z ! w = f(z), T transforms as

T (z) =

df dz

2T (w) + c12[notdef]f, z[notdef] , (2.6)

where [notdef]f, z[notdef] = 2pf[prime]

1pf[prime] is the Schwarzian derivative and c the central charge. Starting from the observation that [angbracketleft]T [angbracketright] = 0 in the complex w plane and that the

transformation f(z) = exp(z/R) maps conformally the innite strip into the upper half plane [Ifractur]m(w) 0, we get, as it is well known,

hT [angbracketright]strip = [angbracketleft]

T[angbracketright]strip =

d2 dz2

c12[notdef]f, z[notdef] =

c 24

[parenleftBig]

R

2, (2.7)

therefore in the innite strip limit L ! 1 we should nd

[integraldisplay]

d2 [angbracketleft]T

T[angbracketright] = RL[angbracketleft]T [angbracketright]2strip . (2.8)

On the other hand, the vacuum expectation value of the quartic term of the string action on a cylinder -i.e. the correlator of two Polyakov lines- has been calculated many years ago [50]

6

in the -function regularization and more recently [29] in the dimensional regularization. The result is

1

22

(D 4)E2()2 + 2E4()

[bracketrightbig]

, = i L2R , (2.9)

where L is the circumference of the cylinder. We do not need the explicit expression of the Eisenstein series E2 and E4 because in the innite strip limit L ! 1 they become

E2 = E4 = 1. In this limit we recover eq. (2.8).

We conclude this section with the following remark. The free-eld action (2.2), once perturbed with T T, has a new energy-momentum tensor which is no longer the one dened in (2.1) as it includes a quartic polynomial in the derivatives of Xi. Inserting this new T T perturbation generates a new energy-momentum tensor made with a polynomial of higher degree, and so on. It would be interesting to see whether this kind of recursion generates the same sequence produced by the request of Lorentz invariance in the target space.

3 The exact S matrix for massless ows

More than twenty years ago, Aliosha Zamolodchikov [51] has proposed an interesting variant of the thermodynamic Bethe Ansatz [52] and the exact S matrix approach to two-dimensional quantum eld theory describing interpolating trajectories among pairs of nontrivial CFTs. The simplest instance discussed in [51], concerns the line of second order phase transitions connecting the tricritical Ising model (TIM) identied with the conformal minimal model M4,5 perturbed by 13 to the Ising model (IM). The latter system

corresponds to the CFT M3,4 underlying the infrared xed point of the RG ow.

Massless excitations conned on a innite line or a ring naturally separate into right and left movers. In this simple example, only one species of particles is present. The right-right and left-left mover scattering is trivial, while the left-right scattering is described by the amplitude

S(p, q) = 2 + ipq

2 ipq

[integraldisplay]

d2 [angbracketleft]T

T[angbracketright] =

1

22

(D 2)4L 242R3

JHEP07(2013)071

, (3.1)

where the real parameter sets the scale; it plays the role of string tension in the energy spectrum. In (3.1) p is the momentum of the right mover and q the momentum of the left

mover. In the limit ! 1, S(p, q) ! 1, right and left mover excitations decouple and the

scale invariance of the model is fully restored at = 1. Starting from the S matrix (3.1),

Zamolodchikov was then able to derive the thermodynamic Bethe Ansatz equations for the vacuum energy of the theory dened on a innite cylinder with circumference R. The relevant equations are

[epsilon1](p) = Rp

[integraldisplay]

1

0

1

0

dq2 (p, q)

L(q),

[epsilon1](p) = Rp

[integraldisplay]

dq2 (p, q) L(q) , (3.2)

where [epsilon1](p) and

[epsilon1](p) are the pseudoenergies for the right and the left movers, respectively,

(p, q) = i@qln S(p, q) , (3.3)

7

and

L(p) = ln(1 + e[epsilon1](p)) , L(p) = ln(1 + e[epsilon1](p)) . (3.4)

The ground state energy is

E(TBA)(R) =

6Rc(pR) = [integraldisplay]

1

0

dp2 (L(p) +

L(p)) , (3.5)

where c(Rp) is the (owing) e ective central charge with

cUV = cTIM = c(0) = 710 , cIR = cIM = c(1) =

1

2 . (3.6)

The energy levels obtained through the TBA method are automatically dened with respect to the vacuum energy in innite space,

lim

R!1

E(TBA)0(R) = 0 . (3.7)

Within a pure two-dimensional setup, the normalization (3.7) is perfectly acceptable but it di ers from the perturbative denition about the ultraviolet xed point by a bulk vacuum contribution F0R which is not analytic in the perturbing parameter [52]

E(TBA)0(R) =

cUV

6R F0R + regular terms , (R [similarequal] 0) . (3.8) (For the TIM ! IM massless ow, F0 = 2.) The alternative normalisation for the energy

levelsEn(R) = E(TBA)n(R) + F0R , (3.9)

agrees with the denition coming from the short distance expansion, it seems to be a more natural choice in view of a possible embedding in an higher dimensional space and it highlights the similarity between the bulk energy, exactly computable within the TBA scheme, and the linear term in the quark-anti-quark potential (1.3), whose origin traces back to the classical contribution Scl to the e ective string action (1.4)

E0(R) = F0R

cIR

JHEP07(2013)071

6R + . . . , (pR 1) . (3.10)

In the far infrared limit, equations (3.2) lead to an exact asymptotic expansion for

E(TBA)(R) [51, 53]

f(TBA)(t) = 12 RE(TBA)0(R) =

1

24

1 48t

148t2 + [parenleftbigg]

5 192 +

49 4002[parenrightbigg]

t3

+

7192 +491002[parenrightbigg]

t4 +

7128 [phone][parenrightbigg]

t5

11128 +5391602 7238199800 4[parenrightbigg]

t6 + . . . (3.11)

where f(TBA)(t) is the scaling function [51] and t = /(12R2). The rst three terms in (3.11) reproduce the large expansion of the e ective action

S = SIM

1

22

+

[integraldisplay]

d2 T T , (3.12)

8

p

p

C

C

Figure 1. Possible excited state integration contours for the + sector.

where SIM is the Ising model CFT action. Correspondingly, the scaling function on the cylinder admits the perturbative expansion

f(pert)(t) =

cIR

12 +

=

JHEP07(2013)071

cIR 24

2

cIR 24

3 2 + O( 3)

(3.13)

The appearance in (3.11) of coe cients with nonzero trascendentality1 at order O(t3) and greater is a clear signal of contributions from other irrelevant operators [51].2

However, what had not been realised until the work [40] was that certain TBA equations lead to the full Nambu-Goto closed string spectrum. One of the main achievements of the present paper is to generalize the results of [40] to other important families of models by showing that the corresponding TBA spectra are a direct generalization of (1.10) and that the leading part S1(p, q) of the Zamolodchikovs S matrix (3.1) at large

S(p, q) = eipq/i(pq/)3/12+... = S1(p, q)ei(pq/)3/12+... (3.14)

selects precisely the zero trascendentality terms in (3.11) which form the large R expansion of the NG ground state energy. Therefore, following [40], we replace the kernel in (3.2) with

(p, q) = i@qln S1(p, q) = p/ , (3.15)

the resulting TBA equations are

[epsilon1](p) = Rp

p

1

24

1 48t

148t2 + O(t3) , ( = 48t) .

ZC

dq2L(q) = Rp + p

ZC

dq2 q@qL(q) ,

(3.16)

[epsilon1](p) = Rp

p

[integraldisplay]C

dq2 L(q) = Rp +

p

[integraldisplay]C

dq2 q@qL(q) ,

1i.e. with a higher power of .

2It is interesting to observe that this new operator contributes exactly at the same order where the Lorentz-invariant e ective string theory admits new non-NG terms [40, 41, 45].

9

with

L(q) = lnC(1 + [notdef]e[epsilon1](q)) , L(q) = ln C(1 + [notdef]e[epsilon1](q)) . (3.17) The corresponding energy is

E(TBA)(R) = [integraldisplay]C

dp2 L(p)

ZC

dp2L(p) . (3.18)

In (3.17) + = 1 selects the descendents of the identity and energy primary elds, while = 1 selects the conformal family of the spin eld [54]. lnC is the continuous branch

logarithm, C and

C are certain integration contours running from q = 0 to q = 1 on

the real axis for the ground states in each subsector [notdef], but for excited states they circle

around a nite number of poles [notdef]qi[notdef] and [notdef]qi[notdef] of @qL(q) and @q

L(q) (see gure 1):

JHEP07(2013)071

[epsilon1](qj) = i(2nj + (1 [notdef])/2) ,

[epsilon1](qj) = i(2nj + (1 [notdef])/2) , (nj, nj 2 N) . (3.19)

Setting

[epsilon1](p) = Rp ,

[epsilon1](p) = Rp

, (3.20)

we nd the constraints

= 1 + 4

R2 Li2( [notdef],

= 1 + 4

R2 Li2( [notdef], C) . (3.21)

In (3.21) Li2(z, C) denotes the continuous branch dilogarithm (see e.g. [55]):

Li2(z, C) =

[integraldisplay]C

C) ,

dq2 lnC(1 zeq) = Li2(z) + 42m i2nln(z) , (m, n 2 N) (3.22)

with Li2(1) = 212 , Li2(1) = 26 ,

Li2(1, C) =

8

>

>

<

>

>

:

2

6 (cIR 24h(0,0) 24n) , (n = 0, 2, 3, . . . )

2

6 (cIR 24h(1,3) 24n) , (n = 0, 1, 2, . . . )

(3.23)

and

2

6 (cIR 24h(1,2) 24n) , (n = 0, 1, 2, . . . ) (3.24)

where

Li2(1, C) = Li2(1) + 42n =

n =

Xj

nj , n =

Xjnj , (3.25)

and h(0,0) = 0, h(1,2) = 1

16 , h(1,3) = 12 are the conformal dimensions of the primary elds of the minimal model M3,4. Therefore

E(TBA)(n,n)(R) + R = R( +

1)

=

s2R2 +

(Li2( [notdef], C) + Li2( [notdef],C)) + [parenleftbigg]

Li2( [notdef], C) Li2( [notdef], C) 2R

2

=

s2R2 + 4

n + n IR12[parenrightbigg]+

2(n n) R

2, (3.26)

10

with IR = cIR 24h, (h 2 [notdef]h(0,0), h(1,2), h(1,3)[notdef]), or

[notdef](TBA)(n,n)(R) = 2R E(TBA)(n,n)(R) . (3.27)

The result (3.26) reproduces precisely the zero trascendentality coe cients appearing in (3.11)

f(TBA)(t) = 12 RE(TBA)(0,0)(R) =

1

24t

[radicalBigg]

1 (24t)2

cIR

144t

(3.28)

Although, as it will be discussed in greater detail in section 4 below and similarly to the cases studied in [40], this simple model of quantum eld theory does not possess a standard ultraviolet xed point, it still seems reasonable to identify the bulk contribution with the linear term in the short distance expansion

@RE(TBA)(0,0)(R) [similarequal] F0 + [notdef] [notdef] [notdef] = + . . . , (for IR [negationslash]= 0) . (3.29)

Adding F0R = R, a nice match with the Nambu-Goto formula (1.10) at D 2 = IR is

nally obtained:

E(n,n)(R) = E(TBA)(n,n)(R) + R . (3.30)

Naively, we may be tempted to discard completely the negative energy sector[notdef](n,n)(R) =

E(n,n)(R), however, the two branches are not completely disconnected as there is a spectral singularity (exceptional point) at the tachyonic critical point

Rc =

rcIR3 , (3.31)

in the ground state n = n = 0 and, for the other levels, singular points at complex values of R. The appearance of the negative energy sector, absent in the original work [51], is a signal of the somehow pathological nature of the -explicitly solvable- CFT perturbation considered. At the level of the TBA this fact is a direct consequence of the non-localized form of the scattering amplitude S1(p, q) used for the kernel (3.15). Still, even with a range of validity restricted to low energy, the appearance of the NG spectrum in the framework of two-dimensional integrable modes is a very striking result that may have a highly non trivial impact to the study of e ective strings in conning gauge theory.

The results obtained in this section, although discussed from a slightly di erent perspective, heavily rely on [40]. Our model di ers from those studied in [40] in two ways:

In [40] Bose statistics was used for the derivation of the TBA equations. The Ising

model and almost all the known integrable models obey instead an exclusion principle in the momentum space even when they are unmistakably associated to Bose type Lagrangians as, for instance, in the quantum a ne Toda eld models [56]. A well known exception is the theory of a single (non compactied) free Bose eld which

11

=

1

24

1 48t

1 48t2

5 192t3

7 192t4

7 128t5

11128t6 + . . .

JHEP07(2013)071

indeed corresponds to infrared limit of the the D = 3 case of [40]. However, even this example does not really represent an exception since an alternative Fermi type TBA with an additional delta-function in the kernels may fully replace the original equations. The change of variable is

[epsilon1]B(p) ! [epsilon1]F (p) + ln(1 + e[epsilon1]F (p)) , ln(1 e[epsilon1]B(p)) ! ln(1 + e[epsilon1]F (p)) . (3.32)

The general case discussed in [40], consists in D 2 species of particles with left-right

mover scattering amplitude Sij(p, q) = S1(p, q), (i, j = 1, 2, . . . D 2). Thus microscopically the D 2 species are actually indistinguishable as their mutual interaction

is totally independent from their avour i and j, a property di cult to understand on physical grounds.

In the following section we study the degeneracy of the string levels and in appendix A, starting from a more general family of exact scattering theories, we shall describe the generalization of these results to more complicated conformal eld theories.

4 Spectrum degeneracy

The degeneracy of the energy levels (3.30) is given by the number of ways of writing n and n in (3.25) i.e. the number of decompositions of n and n into distinct integer summands without regard to the order. This is of course the degeneracy of a free fermionic system on a circle. The generating function is

1

Xn=0'(n)qn =

1

JHEP07(2013)071

Yn=1(1 + qn) = 1

Q1

n=1(1 q2n1)

. (4.1) The asymptotic behaviour of the level degeneracy for large n and n is known to be

'(n)'(n) [similarequal] '(n)2 =

1 16p3n3 e2

pn/3 . (4.2)

For large n [similarequal] n the energy (3.30) is E [similarequal] p8n, so the Ising degeneracy I(n) is

I(n) = 3

TH 3E

3eE/TH = I(E)dE

dn , (4.3)

where

3

cIR , (4.4)

is the Hagedorns temperature TH = sup(T (E)) with T (E) = 1/@ElnI(E) and cIR =

1

2 .

TH =

[radicalbigg]

Comparison with the tachyonic singularity (3.31) at Rc we obtain, as anticipated in the Introduction, RcTH = 1. Notice that the degeneracy of the energy levels of the closed

string in D dimensions is asymptotically

D(n) = 12(D 2)D [parenleftbigg]

TH 3E

D+1eE/TH , (4.5)

where TH di ers from TH by the substitution cIR ! D 2. Further details on this

degeneracy as well as its thermodynamic implications in lattice gauge theory at nite temperature can be found in appendix B of [57].

12

=

Figure 2. R matrix constraint.

5 The innite strip

Consider a single quantum particle conned on a segment of length R. The standard quantization condition for the momentum p of the particle is

ei2piRR (pi)R (pi) = 1 , (5.1)

where (p) = ilnR (p) and (p) = ilnR (p) are the contributions to the total phase

shift from the reections on the left and right boundary, respectively. Although, for an interacting integrable two-dimensional quantum eld theory conned on a space segment of length R equation (5.1) becomes exact only in the asymptotically large R limit, it reveals that one of the main ingredients for the computation of the spectrum is, beside the exact bulk S matrix a consistent reection factor R(p). For massive integrable quantum eld

theories, the basic axiomatic constraints linking the boundary reection factor R(p) to

the two body S matrix amplitude were discussed in [6870]. The generalization of these results to a generic massless perturbed conformal eld theories is a nice and partially open problem that certainly deserves further attention. However a full discussion of this topic would take us too far aeld, and we shall postpone it to the future.

For the time being, we restrict the discussion to the D = 3 case of [46], i.e. the theory of a single Bose eld in two dimensions with left-right scattering amplitude

S1(p, q) = eipq/ . (5.2)

Partially based on TIM ! IM boundary ows discussed in [71], we identify the relevant

constraints to be

R(p)R (p) = 1 , (5.3) which coincides with unitary constraint for R(p), and

R(p)R(p) = S1(p, p) , (5.4) corresponding to the equality between the two scattering diagrams represented in gure 2.

Given the S matrix (5.2), the minimal solution to equations (5.3) and (5.4) is

R0(p) =

pS1(p, p) = eip2/(2) , (5.5)

while multi parameter solutions have the general form

R(p) = R0(p)

13

JHEP07(2013)071

1

Yj=1

R(2j1) j(p) , (5.6)

with

R(m) (p) = e(ip)m , (m odd) . (5.7) The functions (5.7) satisfy the simpler equation

R(m) (p)R(m) (p) = 1 . (5.8)

The signs of the real parameters s appearing (5.6) are not constrained by (5.3) or (5.4). However, as it will clear from the results described in section 6.2 below, for left-right symmetrical boundary conditions, the terms with the largest power 2j 1 3 of p and

j [negationslash]= 0 appearing in (5.6) should have a coe cient j > 0, to ensure a proper convergence of

the TBA integrals and consequently also the validity of the boundary TBA approach itself.

In conclusion, the number of possible reection factors is innite. Since, even for a simple conformal eld theory such as the M3,4 model, the set of all possible boundary condi

tions corresponding to the superposition of Cardys boundary (pure) states [72] is innite dimensional, such a proliferation of free parameter reection factors is not totally surprising.

6 The boundary TBA

The relevant TBA equation for our simple system, obeying Bose type statistics3 and dened on a innite strip of size R and boundary conditions ( , ), is [73, 74]

[epsilon1](p) = 2Rp + (p) + p

ZC

dq2 L(q) , (6.1)

withL(q) = ln C

1 e[epsilon1](q)[parenrightBig]

, (p) = ln (R (ip)/R (ip)) , (6.2)

and vacuum and excited state energies

E(TBA)n(R) =ZC

dp2 L(p) . (6.3)

6.1 Basic boundary conditions

Let us rst consider the basic boundary conditions

(p) = ln (R0(ip)/R0(ip)) = 0 , (6.4) the TBA equation reduces to

[epsilon1](p) = 2Rp + p

ZC

dq2 L(q) . (6.5)

The same logical steps described in section A can be repeated to nd the following simple algebraic constraint

E(TBA)n(R) =

14

JHEP07(2013)071

c( C)

12(2R + E(TBA)n(R)/)

, (6.6)

3This is a conventional choice. The equivalent Fermi statistics TBA, obtained through the change of variable (3.32), would be ne as well.

with c( C) = 1 24n, (n 2 N). Considering only the positive energy solutions, we have

E(TBA)n(R) + R =

n 124[parenrightbigg]

. (6.7)

Equation (6.7) coincides with the NG open string spectrum (1.11) at D = 3. To check the full consistency with the BA equation (5.1), let us consider the single particle excited state

[epsilon1](p) = 2Rp + lnS1(p, ipj) +

or

The result (6.10) nicely ts equation (5.1).

6.2 More general boundary conditions

Let us now consider the case

R(p) = R0(p)R(1) 1(p) = eip2/2ei 1p , (6.11)

since

(p) = ln (R(ip)/R(ip)) = 2p 1 , (6.12) we see that these boundary conditions simply correspond to a shift R ! R + 1, the resulting exact spectrum is

E(TBA)n(R, 1) = E(TBA)n(R + 1) . (6.13)

Note that this case corresponds to the b1 term in the boundary action (1.5). This correction was rst calculated at rst order in b1 in [11] using the function regularization and

in [29] using dimensional regularization. In the latter reference it was also noted that this boundary term corresponds to a shift in R and that this rule actually extends to the next order in b1. In our approach this shift property is valid at any order of 1 and the precise relation between 1 and b1 is 1 = 4b1.

Further, we know that Lorentz symmetry of the target space, as pointed out in (1.8), makes this term inconsistent, yet the TBA approach is perfectly consistent. This clearly shows that the quantum integrability does not imply Lorentz invariance on the target space, as anticipated in the Introduction.

Consider now the case

R(p) = R0(p)R(3) 2(p) = eip2/(2)eip3 2 , (p) = ln (R(ip)/R(ip)) = 2p3 2 . (6.14)

15

s2R2 + 2

p

[integraldisplay]

1

0

dq2 L(q) , (6.8)

where q = ipj is the branch point of L(q) corresponding to [epsilon1](ipj) = i2nj,

(nj = 1, 2, . . . ). At large R, setting [epsilon1](ipj) = i2nj on the l.h.s. of (6.8) and dropping the exponentially subdominant term on the r.h.s.

i2nj [similarequal] i2Rpj + lnS1(ipj, ipj) = i2Rpj + lnS1(pj, pj) , (6.9)

JHEP07(2013)071

ei2RpjS1(pj, pj) = ei2RpjR0(pj)R0(pj) [similarequal] 1 . (6.10)

The corresponding TBA equations are

[epsilon1](p) = 2Rp + 2p3 2 + p

ZC

dq2 L(q) . (6.15)

The solution to (6.15) is of the form

[epsilon1](p) = 2R p + 2p3 , (6.16)

with

+ p2

R = 1 +

p2

R 2 +

1

2R

ZC

dq 2 ln

1 e2R q2q3 [parenrightBig]

. (6.17)

Then it must be exactly = 2 and the TBA yields an exact equation for the energy in terms of = (pR, 2/R3)

= 1 + 1

2R

ZC

dq 2 ln

1 e2R q2q3 2[parenrightBig]

, (6.18)

which can be easily solved numerically for R > 0 and 2 > 0. Yet, in the literature there are interesting, large R, asymptotic corrections to the NG formula which we can match very easily. In fact, we can expand in the form

ln 1 e2R p2p3 2[parenrightBig]= ln 1 e2R p

[parenrightbig]

[parenleftbigg]

JHEP07(2013)071

2p3 2

+ e2 pR 1

[parenrightbigg]

+ . . . (6.19)

obtaining, for 2/R3 small,

= 1 + 1

4R

Li2(1, C)2R +3 24R4 Li4(1,C)[parenrightbigg]+ . . . , (6.20)

which has solution

r1 12R2 Li2(1,C)[parenrightBigg]+ 14R3 24R4 Li4(1,C) + . . . . (6.21)

The continuous branch extension of Li4(1, C) is given by [55]

Li4(1, C) = Li4(1) +

8 34

= 12[parenleftBigg][parenleftBigg]

1 +

Xi(ni)3 = 490 +8 34

Xi(ni)3 , ni 2 N , (6.22)

and the resulting modication to the Nambu-Goto spectrum turns out to be

E(TBA)[notdef]ni[notdef](R, 2) = R(2 2) = E(TBA)n(R) +

23

4R4

160 + 4

Xi(ni)3

[parenrightBigg]

+ . . . (6.23)

with ni = 0, 1, . . . , n =

Pi ni. Equation (6.23) matches the results displayed in the table on page 9 of [45], provided we set D = 3 and identify 2 = 4b2.

Returning to the positivity issue, briey mentioned at the end of section 5, notice that the integral appearing in equation (6.15) is divergent for 2 < 0. It is therefore important to check the sign of the latter coe cient obtained from numerical simulations.

16

The n = 0 state of the spectrum (6.23) was compared with numerics in the case of the three dimensional SU(2) lattice gauge theory in [48]: ()3/2b2 [similarequal] 0.015(6)(6). b2 was also

evaluated in [33] for a class of holographic conning gauge theories and also in this case, with Dirichlets boundary conditions, the b2 coe cient is negative. Thus, in both cases, the TBA equation (6.15) should provide a qualitatively good description of the deviation of the Nambu-Goto spectrum (1.11), caused by the presence of b2-perturbed boundaries, over a wide range of 1/R2 about the infrared xed point.

Further, an high precision numerical simulation of the three dimensional Ising gauge model was reported very recently in [34]. The agreement with the theoretical prediction turned out to be very good allowing a precise estimate of the boundary parameter b2. The

numerical outcome, ()3/2b2 [similarequal] 0.032(2), leads now to a negative sign for 2. We interpret

this result as a clear signal of the presence of additional (infrared subleading) contributions associated to extra terms with higher powers of p in the boundary factor (p) and/or in the TBA convolution kernel. We shall postpone a more complete discussion on this issue and a comparison between TBA and Monte Carlo results to the future.

7 Conclusions

In this paper we pointed out that the e ective string theory describing the conning colour ux tube which joins a static quark-antiquark pair can be seen as a two-dimensional CFT of central charge D2 perturbed by the composite eld T

T made with the energy momentum tensor T . This perturbation is quantum integrable and the spectrum can be calculated with the TBA, as rst noted in [46]. We generalized this result to a large class of conformal models. In the case of periodic boundary conditions the energy levels E(n,n)(R) are labeled by two integers n and n which depend on the monodromy of the dilogarithm in the complex plane of the momentum p. In a generic ADE system these energies can be parametrised in the form E(n,n)(R) = R + E +

, where the two quantities E and

obey the following

consistency conditions

E =

(IR 24n) 12(R +/)

(IR 24n) 12(R + E/)

JHEP07(2013)071

, (7.1)

where IR is the e ective central charge. The solution of these two algebraic equations is exactly the NG spectrum. We then discussed the degeneracy of these states which is growing exponentially for large n and n. Similar conclusions can be drawn for the open string case, where we wrote and solved the boundary TBA. We found that the reection factor may depend on a set of arbitrary parameters which are associated to the coupling constants of the boundary string action. The deviation of the NG spectrum due to these terms can be easily calculated, at least at the rst order in these coupling constants, and it turns out that the results coincide with those of the standard perturbative calculations. These results give a novel perspective on the TBA, and will hopefully lead to a new way to study the interquark potential by means of nonlinear integral equations, exact S matrices and form-factors for correlation functions.

One of the most interesting open questions of the present approach is tied to the fact that the above equations for each pair n, n admit two solutions [notdef]E(n,n)(R). Thus this

17

, =

theory has an innite set of negative energy levels. Even if the theory can be fermionized and we may assume that the sea of negative energy states is completely lled we did not succeed in evaluating the zero-point energy associated to it and its possible e ect on the NG spectrum. One possible way out is to assume that at a given perturbative order in the 1/R2 expansion of the NG spectrum some other irrelevant operator starts to contribute. This is in particular what happens in the massless ow from the tricritical Ising model to the critical one, which was the starting point of our analysis.

As a nal remark we notice that TBA equations both with and without boundaries have emerged in the context of N = 4 super Yang-Mills for the study of the quark-

antiquark potential [75, 76] and gluon scattering amplitudes [77]. The latter quantities are equivalent to light-like polygonal Wilson loops and thus correspond to the area of minimal surfaces in AdS5 in the classical string theory limit. Although there are some similarities between the current setup and those of [75, 76] and [77], there are also important di erences in the underlying physics and the analytic properties of the corresponding TBA equations and we are currently unable to identify a precise link between the results of [40], further developed here, and these important preceding works on AdS5/CF T4.

Acknowledgments

This project was partially supported by INFN grants IS FI11, P14, PI11, the Italian MIURPRIN contract 2009KHZKRX-007 Symmetries of the Universe and of the Fundamental Interactions, the UniTo-SanPaolo research grant Nr TO-Call3-2012-0088 Modern Applications of String Theory (MAST), the ESF Network Holographic methods for strongly coupled systems (HoloGrav) (09-RNP-092 (PESC)) and MPNS-COST Action MP1210 The String Theory Universe.

A The ADE general case

The analysis described in section 3 can be immediately generalised to any perturbed conformal eld theory with known exact S matrix description, as for example, the massive theories represented by the diagonal reectionless ADE-related scattering models of [52, 58]. The TBA equations are

[epsilon1]i(p) = Rei(p)

N

Xj=1

JHEP07(2013)071

[integraldisplay]Cj

dq2 ij(p, q) Lj(q) ,

[epsilon1]i(p) = Rei(p)

N

Xj=1

ZCjdq2 ij(p, q)Lj(q) ,

(A.1)

where ei(p) =

qp2 + m2i is the dispersion relation of the i-th particle and N is the rank of the corresponding ADE algebra. The kernels are

ij(p) = i@q ln Sij(p, q) , (A.2)

where Sij(p, q) are the S matrix amplitudes of [56, 5961] parametrised using the momenta p and q of the two particles involved in the scattering. In the ultraviolet regime miR ! 0

18

the TBA equations (A.1) show the decoupling of the pseudoenergies for right movers from the left mover ones

[epsilon1]i(p) = R mip

N

Xj=1

[integraldisplay]Cj

dq2 ij(p, q) Lj(q) ,

[epsilon1]i(p) = R mip

N

Xj=1

ZCjdq2 ij(p, q)Lj(q) .

(A.3)

In this limit, the energy can be found exactly [52]. The set of pure numbers mi mi/m1 and i = 1, . . . , N x a relative scale among the particle species. They cannot be arbitrary numbers, but must be proportional to the components of the Perron-Frobeniuns eigenvector of the corresponding Cartan matrix. For the vacuum states all this has been analysed in [52, 58], furthermore here we wish to take into consideration excited states as well [54, 62, 63] by introducing complex contours Ci and

Ci for the continuous branch dilogarithm and

fugacities [notdef] i[notdef] inside the statistical functions

Li(p) = lnCi(1 + ie[epsilon1]i(p)) , Li(p) = ln Ci(1 + ie[epsilon1]i(p)) . (A.4)

The fugacities are all equal to unity for sectors related to the CFT identity operator, while they may assume di erent values for conformal families of other primary elds [54, 64, 65]. In the ultraviolet limit the energy is

E(TBA)(n,n)(R) = E +

, E =

12Rc(C) ,

JHEP07(2013)071

=

12Rc(

C) , (A.5)

where the constants

c(C) =

12

[integraldisplay]Ci

dp2 Li(p) , c(

N

Xi=1R mi

N

Xi=1R mi

C) =

12

ZCidp2Li(p) , (A.6)

can be written in terms of the solutions to (A.1) and computed exactly using the dilogarithm trick [52, 66]. Besides, they are easily related to the conformal central charge cIR and the

conformal weights (h, h) of the primary elds

c(C) = IR 24n , c(

C) = IR 24n , (n, n 2 N) , (A.7)

and IR = cIR 24(h +

h). The central charge and the conformal dimensions are those for the coset models [67]

1 [notdef]

1

2

, G 2 AN, DN, EN . (A.8)

Let us now come to the announced generalisation of the analysis described in section 3, and introduce the following variant of massless TBA equation for ADE systems

[epsilon1]i(p) = Rp mi p

mi

N

Xj=1 mj

ZCjdq2Lj(q)

N

Xj=1

[integraldisplay]Cj

dq2 ij(p, q) Lj(q) , (A.9)

[epsilon1]i(p) = Rp mi p

mi

N

Xj=1 mj

[integraldisplay]Cj

dq2 Lj(q)

N

Xj=1

ZCjdq2 ij(p, q)Lj(q) . (A.10)

19

The elegant denition of e ective length

r = R +/ , r = R + E/ , (A.11) allows to recast equations (A.9) and (A.10) in the form (A.3), with

rE =

c(C)

12 =

N

Xi=1r mi

[integraldisplay]Ci

dp2 Li(p) , r

=

c( C)

12 =

N

Xi=1 rmi

ZCidp2Li(p) , (A.12)

C) coincide with those already introduced in (A.7) and

computable via equations (A.3) and (A.6). Finally, the following self-consistent constraints must hold

E =

(IR 24n) 12(R +/)

where, importantly, the c(C) and c(

JHEP07(2013)071

, =

(IR 24n) 12(R + E/)

. (A.13)

The latter pair of algebraic equations for E and

can be easily solved, giving

E(n,n)(R) = E +

+ R = [notdef][radicalBigg]2R2 + 4 [parenleftbigg]

n + n

IR

12

+

2(n n) R

2. (A.14)

In conclusion, we have shown that the spectrum of Nambu-Goto with D 2 = IR emerges

from a wide class of TBA models. Actually, we can generalize this analysis to many other interesting models, including innite families of perturbed CFT theories described by non-diagonal S matrices and we suspect that (A.14) can be obtained for any CFT.

We end this section with a further observation. It was noticed in [49] that for any two-dimensional quantum eld theory the expectation values of the composite eld T T admits an exact representation in terms of the expectation value of the energy-momentum tensor itself. Using the standard CFT convention and setting

T = 2Tzz ,

T = 2Tzz , = 2Tzz = 2Tzz , (A.15) the result of [49], written using the double integer labeling introduced in the previous sections, on the cylinder is

hn, n[notdef]T

T[notdef]n, n[angbracketright] = [angbracketleft]n, n[notdef]T [notdef]n, n[angbracketright][angbracketleft]n, n[notdef]

T[notdef]n, n[angbracketright] [angbracketleft]n, n[notdef] [notdef]n, n[angbracketright]2 . (A.16)

With the help of the following relations linking the expectation values of the energy-tensor components with the energy eigenvalues E(n,n)(R) and the total momentum of the state

P(n,n)(R) = 2(n n)/R

hn, n[notdef]Tyy[notdef]n, n[angbracketright] =

1RE(n,n)(R) , [angbracketleft]n, n[notdef]Txx[notdef]n, n[angbracketright] = @RE(n,n)(R) ,

(A.17)

hn, n[notdef]Txy[notdef]n, n[angbracketright] =

i RP(n,n)(R) ,

we have [49]

2R

2 [angbracketleft]n, n[notdef]T

@R(E2(n,n)(R) P 2(n,n)(R)) =

T[notdef]n, n[angbracketright] . (A.18)

20

Here, we would like to remark that inserting espression (A.14) for the energy levels in (A.18) leads to

hn, n[notdef]T

T[notdef]n, n[angbracketright] = 22, (A.19) exactly and independently from the particular state (n, n) under consideration. Finally, using (A.19) in (A.16) gives

hn, n[notdef] [notdef]n, n[angbracketright] =

q22 + [angbracketleft]n, n[notdef]T [notdef]n, n[angbracketright][angbracketleft]n, n[notdef]T[notdef]n, n[angbracketright] . (A.20)

Since, = 0 corresponds to a conformal invariant theory, the eld can be identied with the CFT perturbing operator, thus the exact result (A.20) should contain fundamental information on the further contributions needed in (2.4) and (3.12), or in an arbitray T T perturbed CFT, to build the full action associated to the Nambu-Goto like spectrum (A.14).

References

[1] C. Itzykson, M.E. Peskin and J.-B. Zuber, Roughening of Wilsons surface, http://dx.doi.org/10.1016/0370-2693(80)90483-9

Web End =Phys. Lett. B 95 http://dx.doi.org/10.1016/0370-2693(80)90483-9

Web End =(1980) 259 [http://inspirehep.net/search?p=find+J+Phys.Lett.,B95,259

Web End =INSPIRE ].

[2] J.B. Kogut and J. Shigemitsu, Crossover from weak to strong coupling in SU(N) lattice gauge theories, http://dx.doi.org/10.1103/PhysRevLett.45.410

Web End =Phys. Rev. Lett. 45 (1980) 410 [Erratum ibid. 45 (1980) 1217] [http://inspirehep.net/search?p=find+J+Phys.Rev.Lett.,45,410

Web End =INSPIRE ].

[3] A. Hasenfratz, E. Hasenfratz and P. Hasenfratz, Generalized roughening transition and its e ect on the string tension, http://dx.doi.org/10.1016/0550-3213(81)90426-0

Web End =Nucl. Phys. B 180 (1981) 353 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B180,353

Web End =INSPIRE ].

[4] J. Kosterlitz and D. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6 (1973) 1181 [http://inspirehep.net/search?p=find+J+J.Phys.,C6,1181

Web End =INSPIRE ].

[5] M. Lscher, K. Symanzik and P. Weisz, Anomalies of the free loop wave equation in the WKB approximation, http://dx.doi.org/10.1016/0550-3213(80)90009-7

Web End =Nucl. Phys. B 173 (1980) 365 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B173,365

Web End =INSPIRE ].

[6] M. Lscher, Symmetry breaking aspects of the roughening transition in gauge theories, http://dx.doi.org/10.1016/0550-3213(81)90423-5

Web End =Nucl. http://dx.doi.org/10.1016/0550-3213(81)90423-5

Web End =Phys. B 180 (1981) 317 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B180,317

Web End =INSPIRE ].

[7] M. Hasenbusch and K. Pinn, Surface tension, surface sti ness and surface width of the three-dimensional Ising model on a cubic lattice, Physica A 192 (1993) 342 [http://arxiv.org/abs/hep-lat/9209013

Web End =hep-lat/9209013 ] [http://inspirehep.net/search?p=find+EPRINT+hep-lat/9209013

Web End =INSPIRE ].

[8] M. Caselle, R. Fiore, F. Gliozzi, M. Hasenbusch and P. Provero, String e ects in the Wilson loop: a high precision numerical test, http://dx.doi.org/10.1016/S0550-3213(96)00672-4

Web End =Nucl. Phys. B 486 (1997) 245 [http://arxiv.org/abs/hep-lat/9609041

Web End =hep-lat/9609041 ] [http://inspirehep.net/search?p=find+EPRINT+hep-lat/9609041

Web End =INSPIRE ].

[9] K.J. Juge, J. Kuti and C. Morningstar, Fine structure of the QCD string spectrum, http://dx.doi.org/10.1103/PhysRevLett.90.161601

Web End =Phys. http://dx.doi.org/10.1103/PhysRevLett.90.161601

Web End =Rev. Lett. 90 (2003) 161601 [http://arxiv.org/abs/hep-lat/0207004

Web End =hep-lat/0207004 ] [http://inspirehep.net/search?p=find+EPRINT+hep-lat/0207004

Web End =INSPIRE ].

[10] B. Lucini and M. Teper, Conning strings in SU(N) gauge theories, http://dx.doi.org/10.1103/PhysRevD.64.105019

Web End =Phys. Rev. D 64 (2001) http://dx.doi.org/10.1103/PhysRevD.64.105019

Web End =105019 [http://arxiv.org/abs/hep-lat/0107007

Web End =hep-lat/0107007 ] [http://inspirehep.net/search?p=find+EPRINT+hep-lat/0107007

Web End =INSPIRE ].

[11] M. Lscher and P. Weisz, Quark connement and the bosonic string, http://dx.doi.org/10.1088/1126-6708/2002/07/049

Web End =JHEP 07 (2002) 049 [http://arxiv.org/abs/hep-lat/0207003

Web End =hep-lat/0207003 ] [http://inspirehep.net/search?p=find+EPRINT+hep-lat/0207003

Web End =INSPIRE ].

[12] M. Caselle, M. Pepe and A. Rago, Static quark potential and e ective string corrections in the (2 + 1)-d SU(2) Yang-Mills theory, http://dx.doi.org/10.1088/1126-6708/2004/10/005

Web End =JHEP 10 (2004) 005 [http://arxiv.org/abs/hep-lat/0406008

Web End =hep-lat/0406008 ] [http://inspirehep.net/search?p=find+EPRINT+hep-lat/0406008

Web End =INSPIRE ].

[13] M. Lscher, G. Munster and P. Weisz, How thick are chromoelectric ux tubes?, http://dx.doi.org/10.1016/0550-3213(81)90151-6

Web End =Nucl. Phys. http://dx.doi.org/10.1016/0550-3213(81)90151-6

Web End =B 180 (1981) 1 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B180,1

Web End =INSPIRE ].

21

JHEP07(2013)071

[14] M. Caselle, F. Gliozzi, U. Magnea and S. Vinti, Width of long color ux tubes in lattice gauge systems, http://dx.doi.org/10.1016/0550-3213(95)00639-7

Web End =Nucl. Phys. B 460 (1996) 397 [http://arxiv.org/abs/hep-lat/9510019

Web End =hep-lat/9510019 ] [http://inspirehep.net/search?p=find+EPRINT+hep-lat/9510019

Web End =INSPIRE ].

[15] F. Gliozzi, M. Pepe and U.-J. Wiese, The width of the conning string in Yang-Mills theory, http://dx.doi.org/10.1103/PhysRevLett.104.232001

Web End =Phys. Rev. Lett. 104 (2010) 232001 [http://arxiv.org/abs/1002.4888

Web End =arXiv:1002.4888 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1002.4888

Web End =INSPIRE ].

[16] F. Gliozzi, M. Pepe and U.-J. Wiese, The width of the color ux tube at 2-loop order, http://dx.doi.org/10.1007/JHEP11(2010)053

Web End =JHEP http://dx.doi.org/10.1007/JHEP11(2010)053

Web End =11 (2010) 053 [http://arxiv.org/abs/1006.2252

Web End =arXiv:1006.2252 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1006.2252

Web End =INSPIRE ].

[17] M. Caselle et al., Rough interfaces beyond the Gaussian approximation, http://dx.doi.org/10.1016/0550-3213(94)90035-3

Web End =Nucl. Phys. B 432 http://dx.doi.org/10.1016/0550-3213(94)90035-3

Web End =(1994) 590 [http://arxiv.org/abs/hep-lat/9407002

Web End =hep-lat/9407002 ] [http://inspirehep.net/search?p=find+EPRINT+hep-lat/9407002

Web End =INSPIRE ].

[18] M. Caselle, M. Hasenbusch and M. Panero, Comparing the Nambu-Goto string with LGT results, http://dx.doi.org/10.1088/1126-6708/2005/03/026

Web End =JHEP 03 (2005) 026 [http://arxiv.org/abs/hep-lat/0501027

Web End =hep-lat/0501027 ] [http://inspirehep.net/search?p=find+EPRINT+hep-lat/0501027

Web End =INSPIRE ].

[19] M. Caselle, M. Hasenbusch and M. Panero, High precision Monte Carlo simulations of interfaces in the three-dimensional Ising model: a comparison with the Nambu-Goto e ective string model, http://dx.doi.org/10.1088/1126-6708/2006/03/084

Web End =JHEP 03 (2006) 084 [http://arxiv.org/abs/hep-lat/0601023

Web End =hep-lat/0601023 ] [http://inspirehep.net/search?p=find+EPRINT+hep-lat/0601023

Web End =INSPIRE ].

[20] N. Hari Dass and P. Majumdar, String-like behaviour of 4 D SU(3) Yang-Mills ux tubes,

http://dx.doi.org/10.1088/1126-6708/2006/10/020

Web End =JHEP 10 (2006) 020 [http://arxiv.org/abs/hep-lat/0608024

Web End =hep-lat/0608024 ] [http://inspirehep.net/search?p=find+EPRINT+hep-lat/0608024

Web End =INSPIRE ].

[21] B. Bringoltz and M. Teper, Closed k-strings in SU(N) gauge theories: 2 + 1 dimensions, http://dx.doi.org/10.1016/j.physletb.2008.04.052

Web End =Phys. Lett. B 663 (2008) 429 [http://arxiv.org/abs/0802.1490

Web End =arXiv:0802.1490 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0802.1490

Web End =INSPIRE ].

[22] A. Athenodorou, B. Bringoltz and M. Teper, The spectrum of closed loops of fundamental ux in D = 3 + 1 SU(N) gauge theories, http://pos.sissa.it/cgi-bin/reader/contribution.cgi?id=PoS(LAT2009)223

Web End =PoS(LAT2009)223 [http://arxiv.org/abs/0912.3238

Web End =arXiv:0912.3238 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0912.3238

Web End =INSPIRE ].

[23] A. Athenodorou, B. Bringoltz and M. Teper, Closed ux tubes and their string description in D = 2 + 1 SU(N) gauge theories, http://dx.doi.org/10.1007/JHEP05(2011)042

Web End =JHEP 05 (2011) 042 [http://arxiv.org/abs/1103.5854

Web End =arXiv:1103.5854 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1103.5854

Web End =INSPIRE ].

[24] A. Athenodorou, B. Bringoltz and M. Teper, Closed ux tubes and their string description in D = 3 + 1 SU(N) gauge theories, http://dx.doi.org/10.1007/JHEP02(2011)030

Web End =JHEP 02 (2011) 030 [http://arxiv.org/abs/1007.4720

Web End =arXiv:1007.4720 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1007.4720

Web End =INSPIRE ].

[25] M. Caselle and M. Zago, A new approach to the study of e ective string corrections in LGTs, http://dx.doi.org/10.1140/epjc/s10052-011-1658-6

Web End =Eur. Phys. J. C 71 (2011) 1658 [http://arxiv.org/abs/1012.1254

Web End =arXiv:1012.1254 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.1254

Web End =INSPIRE ].

[26] M. Bill, M. Caselle and R. Pellegrini, New numerical results and novel e ective string predictions for Wilson loops, http://dx.doi.org/10.1007/JHEP01(2012)104

Web End =JHEP 01 (2012) 104 [Erratum ibid. 1304 (2013) 097] [http://arxiv.org/abs/1107.4356

Web End =arXiv:1107.4356 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1107.4356

Web End =INSPIRE ].

[27] A. Mykkanen, The static quark potential from a multilevel algorithm for the improved gauge action, http://dx.doi.org/10.1007/JHEP12(2012)069

Web End =JHEP 12 (2012) 069 [http://arxiv.org/abs/1209.2372

Web End =arXiv:1209.2372 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1209.2372

Web End =INSPIRE ].

[28] A. Athenodorou and M. Teper, Closed ux tubes in higher representations and their string description in D = 2 + 1 SU(N) gauge theories, http://dx.doi.org/10.1007/JHEP06(2013)053

Web End =JHEP 06 (2013) 053 [http://arxiv.org/abs/1303.5946

Web End =arXiv:1303.5946 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1303.5946

Web End =INSPIRE ].

[29] M. Lscher and P. Weisz, String excitation energies in SU(N) gauge theories beyond the free-string approximation, http://dx.doi.org/10.1088/1126-6708/2004/07/014

Web End =JHEP 07 (2004) 014 [http://arxiv.org/abs/hep-th/0406205

Web End =hep-th/0406205 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0406205

Web End =INSPIRE ].

[30] O. Aharony and E. Karzbrun, On the e ective action of conning strings, http://dx.doi.org/10.1088/1126-6708/2009/06/012

Web End =JHEP 06 (2009) http://dx.doi.org/10.1088/1126-6708/2009/06/012

Web End =012 [http://arxiv.org/abs/0903.1927

Web End =arXiv:0903.1927 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0903.1927

Web End =INSPIRE ].

[31] H.B. Meyer, Poincar invariance in e ective string theories, http://dx.doi.org/10.1088/1126-6708/2006/05/066

Web End =JHEP 05 (2006) 066 [http://arxiv.org/abs/hep-th/0602281

Web End =hep-th/0602281 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0602281

Web End =INSPIRE ].

[32] O. Aharony, Z. Komargodski and A. Schwimmer, The e ective action on the conning string, talk given at String 2009, June 2226, Roma, Italy (2009).

22

JHEP07(2013)071

[33] O. Aharony and M. Field, On the e ective theory of long open strings, http://dx.doi.org/10.1007/JHEP01(2011)065

Web End =JHEP 01 (2011) 065 [http://arxiv.org/abs/1008.2636

Web End =arXiv:1008.2636 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1008.2636

Web End =INSPIRE ].

[34] M. Bill, M. Caselle, F. Gliozzi, M. Meineri and R. Pellegrini, The Lorentz-invariant boundary action of the conning string and its universal contribution to the inter-quark potential, http://dx.doi.org/10.1007/JHEP05(2012)130

Web End =JHEP 05 (2012) 130 [http://arxiv.org/abs/1202.1984

Web End =arXiv:1202.1984 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1202.1984

Web End =INSPIRE ].

[35] F. Gliozzi, Dirac-Born-Infeld action from spontaneous breakdown of Lorentz symmetry in brane-world scenarios, http://dx.doi.org/10.1103/PhysRevD.84.027702

Web End =Phys. Rev. D 84 (2011) 027702 [http://arxiv.org/abs/1103.5377

Web End =arXiv:1103.5377 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1103.5377

Web End =INSPIRE ].

[36] O. Aharony and M. Dodelson, E ective string theory and nonlinear Lorentz invariance, http://dx.doi.org/10.1007/JHEP02(2012)008

Web End =JHEP 02 (2012) 008 [http://arxiv.org/abs/1111.5758

Web End =arXiv:1111.5758 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1111.5758

Web End =INSPIRE ].

[37] F. Gliozzi and M. Meineri, Lorentz completion of e ective string (and p-brane) action, http://dx.doi.org/10.1007/JHEP08(2012)056

Web End =JHEP http://dx.doi.org/10.1007/JHEP08(2012)056

Web End =08 (2012) 056 [http://arxiv.org/abs/1207.2912

Web End =arXiv:1207.2912 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.2912

Web End =INSPIRE ].

[38] M. Meineri, Lorentz completion of e ective string action, http://pos.sissa.it/cgi-bin/reader/contribution.cgi?id=PoS(Confinement X)041

Web End =PoS(Confinement X)041 [http://arxiv.org/abs/1301.3437

Web End =arXiv:1301.3437 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1301.3437

Web End =INSPIRE ].

[39] O. Aharony, M. Field and N. Klingho er, The e ective string spectrum in the orthogonal gauge, http://dx.doi.org/10.1007/JHEP04(2012)048

Web End =JHEP 04 (2012) 048 [http://arxiv.org/abs/1111.5757

Web End =arXiv:1111.5757 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1111.5757

Web End =INSPIRE ].

[40] S. Dubovsky, R. Flauger and V. Gorbenko, E ective string theory revisited, http://dx.doi.org/10.1007/JHEP09(2012)044

Web End =JHEP 09 (2012) http://dx.doi.org/10.1007/JHEP09(2012)044

Web End =044 [http://arxiv.org/abs/1203.1054

Web End =arXiv:1203.1054 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1203.1054

Web End =INSPIRE ].

[41] O. Aharony and Z. Komargodski, The e ective theory of long strings, http://dx.doi.org/10.1007/JHEP05(2013)118

Web End =JHEP 05 (2013) 118 [http://arxiv.org/abs/1302.6257

Web End =arXiv:1302.6257 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1302.6257

Web End =INSPIRE ].

[42] J. Arvis, The exact qq potential in Nambu string theory, http://dx.doi.org/10.1016/0370-2693(83)91640-4

Web End =Phys. Lett. B 127 (1983) 106 [ http://inspirehep.net/search?p=find+J+Phys.Lett.,B127,106

Web End =INSPIRE ].

[43] P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Quantum dynamics of a massless relativistic string, http://dx.doi.org/10.1016/0550-3213(73)90223-X

Web End =Nucl. Phys. B 56 (1973) 109 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B56,109

Web End =INSPIRE ].

[44] L. Mezincescu and P.K. Townsend, 3D strings and other anyonic things, http://dx.doi.org/10.1002/prop.201200001

Web End =Fortsch. Phys. 60 http://dx.doi.org/10.1002/prop.201200001

Web End =(2012) 1076 [http://arxiv.org/abs/1111.3384

Web End =arXiv:1111.3384 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1111.3384

Web End =INSPIRE ].

[45] O. Aharony and N. Klingho er, Corrections to Nambu-Goto energy levels from the e ective string action, http://dx.doi.org/10.1007/JHEP12(2010)058

Web End =JHEP 12 (2010) 058 [http://arxiv.org/abs/1008.2648

Web End =arXiv:1008.2648 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1008.2648

Web End =INSPIRE ].

[46] S. Dubovsky, R. Flauger and V. Gorbenko, Solving the simplest theory of quantum gravity, http://dx.doi.org/10.1007/JHEP09(2012)133

Web End =JHEP 09 (2012) 133 [http://arxiv.org/abs/1205.6805

Web End =arXiv:1205.6805 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1205.6805

Web End =INSPIRE ].

[47] S. Dubovsky, R. Flauger and V. Gorbenko, Evidence for a new particle on the worldsheet of the QCD ux tube, http://arxiv.org/abs/1301.2325

Web End =arXiv:1301.2325 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1301.2325

Web End =INSPIRE ].

[48] B.B. Brandt, Probing boundary-corrections to Nambu-Goto open string energy levels in 3D SU(2) gauge theory, http://dx.doi.org/10.1007/JHEP02(2011)040

Web End =JHEP 02 (2011) 040 [http://arxiv.org/abs/1010.3625

Web End =arXiv:1010.3625 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1010.3625

Web End =INSPIRE ].

[49] A.B. Zamolodchikov, Expectation value of composite eld T T in two-dimensional quantum eld theory, http://arxiv.org/abs/hep-th/0401146

Web End =hep-th/0401146 [http://inspirehep.net/search?p=find+EPRINT+hep-th/0401146

Web End =INSPIRE ].

[50] K. Dietz and T. Filk, On the renormalization of string functionals, http://dx.doi.org/10.1103/PhysRevD.27.2944

Web End =Phys. Rev. D 27 (1983) 2944.

[51] A. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, http://dx.doi.org/10.1016/0550-3213(91)90423-U

Web End =Nucl. Phys. B 358 (1991) 524 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B358,524

Web End =INSPIRE ].

[52] A. Zamolodchikov, Thermodynamic Bethe Ansatz in relativistic models. Scaling three state Potts and Lee-Yang models, http://dx.doi.org/10.1016/0550-3213(90)90333-9

Web End =Nucl. Phys. B 342 (1990) 695 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B342,695

Web End =INSPIRE ].

23

JHEP07(2013)071

[53] T.R. Klassen and E. Melzer, Spectral ow between conformal eld theories in (1 + 1)-dimensions, http://dx.doi.org/10.1016/0550-3213(92)90422-8

Web End =Nucl. Phys. B 370 (1992) 511 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B370,511

Web End =INSPIRE ].

[54] P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, http://dx.doi.org/10.1016/S0550-3213(96)00516-0

Web End =Nucl. http://dx.doi.org/10.1016/S0550-3213(96)00516-0

Web End =Phys. B 482 (1996) 639 [http://arxiv.org/abs/hep-th/9607167

Web End =hep-th/9607167 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9607167

Web End =INSPIRE ].

[55] L. Vepstas, An e cient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions, http://dx.doi.org/10.1007/s11075-007-9153-8

Web End =Numer. Alg.s 47 (2008) 211 [http://arxiv.org/abs/math/0702243

Web End =math/0702243 ].

[56] H. Braden, E. Corrigan, P. Dorey and R. Sasaki, A ne Toda eld theory and exact S matrices, http://dx.doi.org/10.1016/0550-3213(90)90648-W

Web End =Nucl. Phys. B 338 (1990) 689 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B338,689

Web End =INSPIRE ].

[57] M. Caselle, L. Castagnini, A. Feo, F. Gliozzi and M. Panero, Thermodynamics of SU(N) Yang-Mills theories in 2 + 1 dimensions I The conning phase, http://dx.doi.org/10.1007/JHEP06(2011)142

Web End =JHEP 06 (2011) 142 [http://arxiv.org/abs/1105.0359

Web End =arXiv:1105.0359 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1105.0359

Web End =INSPIRE ].

[58] T.R. Klassen and E. Melzer, The Thermodynamics of purely elastic scattering theories and conformal perturbation theory, http://dx.doi.org/10.1016/0550-3213(91)90159-U

Web End =Nucl. Phys. B 350 (1991) 635 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B350,635

Web End =INSPIRE ].

[59] T.R. Klassen and E. Melzer, Purely elastic scattering theories and their ultraviolet limits, http://dx.doi.org/10.1016/0550-3213(90)90643-R

Web End =Nucl. Phys. B 338 (1990) 485 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B338,485

Web End =INSPIRE ].

[60] V. Fateev and A. Zamolodchikov, Conformal eld theory and purely elastic S matrices, http://dx.doi.org/10.1142/S0217751X90000477

Web End =Int. http://dx.doi.org/10.1142/S0217751X90000477

Web End =J. Mod. Phys. A 5 (1990) 1025 [http://inspirehep.net/search?p=find+J+Int.J.Mod.Phys.,A5,1025

Web End =INSPIRE ].

[61] P. Christe and G. Mussardo, Integrable systems away from criticality: the Toda eld theory and S matrix of the tricritical Ising model, http://dx.doi.org/10.1016/0550-3213(90)90119-X

Web End =Nucl. Phys. B 330 (1990) 465 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B330,465

Web End =INSPIRE ].

[62] P. Dorey and R. Tateo, Excited states in some simple perturbed conformal eld theories, http://dx.doi.org/10.1016/S0550-3213(97)00838-9

Web End =Nucl. Phys. B 515 (1998) 575 [http://arxiv.org/abs/hep-th/9706140

Web End =hep-th/9706140 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9706140

Web End =INSPIRE ].

[63] V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable quantum eld theories in nite volume: excited state energies, http://dx.doi.org/10.1016/S0550-3213(97)00022-9

Web End =Nucl. Phys. B 489 (1997) 487 [http://arxiv.org/abs/hep-th/9607099

Web End =hep-th/9607099 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9607099

Web End =INSPIRE ].

[64] M.J. Martins, Complex excitations in the thermodynamic Bethe ansatz approach, http://dx.doi.org/10.1103/PhysRevLett.67.419

Web End =Phys. Rev. http://dx.doi.org/10.1103/PhysRevLett.67.419

Web End =Lett. 67 (1991) 419 [http://inspirehep.net/search?p=find+J+Phys.Rev.Lett.,67,419

Web End =INSPIRE ].

[65] P. Fendley, Excited state thermodynamics, http://dx.doi.org/10.1016/0550-3213(92)90404-Y

Web End =Nucl. Phys. B 374 (1992) 667 [http://arxiv.org/abs/hep-th/9109021

Web End =hep-th/9109021 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9109021

Web End =INSPIRE ].

[66] A. Kuniba and T. Nakanishi, Spectra in conformal eld theories from the Rogers dilogarithm, http://dx.doi.org/10.1142/S0217732392002895

Web End =Mod. Phys. Lett. A 7 (1992) 3487 [http://arxiv.org/abs/hep-th/9206034

Web End =hep-th/9206034 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9206034

Web End =INSPIRE ].

[67] P. Goddard, A. Kent and D.I. Olive, Virasoro algebras and coset space models, http://dx.doi.org/10.1016/0370-2693(85)91145-1

Web End =Phys. Lett. B http://dx.doi.org/10.1016/0370-2693(85)91145-1

Web End =152 (1985) 88 [http://inspirehep.net/search?p=find+J+Phys.Lett.,B152,88

Web End =INSPIRE ].

[68] A. Fring and R. Koberle, Factorized scattering in the presence of reecting boundaries, http://dx.doi.org/10.1016/0550-3213(94)90229-1

Web End =Nucl. http://dx.doi.org/10.1016/0550-3213(94)90229-1

Web End =Phys. B 421 (1994) 159 [http://arxiv.org/abs/hep-th/9304141

Web End =hep-th/9304141 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9304141

Web End =INSPIRE ].

[69] S. Ghoshal and A.B. Zamolodchikov, Boundary S matrix and boundary state in two-dimensional integrable quantum eld theory, http://dx.doi.org/10.1142/S0217751X94001552

Web End =Int. J. Mod. Phys. A 9 (1994) 3841 [Erratum ibid. A 9 (1994) 4353] [http://arxiv.org/abs/hep-th/9306002

Web End =hep-th/9306002 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9306002

Web End =INSPIRE ].

[70] P. Dorey, R. Tateo and G. Watts, Generalizations of the Coleman-Thun mechanism and boundary reection factors, http://dx.doi.org/10.1016/S0370-2693(99)00004-0

Web End =Phys. Lett. B 448 (1999) 249 [http://arxiv.org/abs/hep-th/9810098

Web End =hep-th/9810098 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9810098

Web End =INSPIRE ].

[71] P. Dorey, C. Rim and R. Tateo, Exact g-function ow between conformal eld theories, http://dx.doi.org/10.1016/j.nuclphysb.2010.03.010

Web End =Nucl. http://dx.doi.org/10.1016/j.nuclphysb.2010.03.010

Web End =Phys. B 834 (2010) 485 [http://arxiv.org/abs/0911.4969

Web End =arXiv:0911.4969 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0911.4969

Web End =INSPIRE ].

24

JHEP07(2013)071

[72] J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, http://dx.doi.org/10.1016/0550-3213(89)90521-X

Web End =Nucl. Phys. B 324 http://dx.doi.org/10.1016/0550-3213(89)90521-X

Web End =(1989) 581 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B324,581

Web End =INSPIRE ].

[73] A. LeClair, G. Mussardo, H. Saleur and S. Skorik, Boundary energy and boundary states in integrable quantum eld theories, http://dx.doi.org/10.1016/0550-3213(95)00435-U

Web End =Nucl. Phys. B 453 (1995) 581 [http://arxiv.org/abs/hep-th/9503227

Web End =hep-th/9503227 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9503227

Web End =INSPIRE ].

[74] P. Dorey, A. Pocklington, R. Tateo and G. Watts, TBA and TCSA with boundaries and excited states, http://dx.doi.org/10.1016/S0550-3213(98)00339-3

Web End =Nucl. Phys. B 525 (1998) 641 [http://arxiv.org/abs/hep-th/9712197

Web End =hep-th/9712197 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9712197

Web End =INSPIRE ].

[75] N. Drukker, Integrable Wilson loops, http://arxiv.org/abs/1203.1617

Web End =arXiv:1203.1617 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1203.1617

Web End =INSPIRE ].

[76] D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation, http://dx.doi.org/10.1007/JHEP08(2012)134

Web End =JHEP 08 (2012) 134 [http://arxiv.org/abs/1203.1913

Web End =arXiv:1203.1913 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1203.1913

Web End =INSPIRE ].

[77] L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for scattering amplitudes, http://dx.doi.org/10.1088/1751-8113/43/48/485401

Web End =J. http://dx.doi.org/10.1088/1751-8113/43/48/485401

Web End =Phys. A 43 (2010) 485401 [http://arxiv.org/abs/1002.2459

Web End =arXiv:1002.2459 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1002.2459

Web End =INSPIRE ].

JHEP07(2013)071

25