A next-to-leading Luescher formula

We propose a next-to-leading Luescher-like formula for the finite-size corrections of the excited states energies in integrable theories. We conjecture the expressions of the corrections for both the energy and the particles' rapidities by interpreting the excited states as momenta-dependent defects. We check the resulting formulas in some simple relativistic model and conjecture those for the AdS5/CFT4 case.


Introduction
This will be done by using the experience with the calculation of double-wrapping corrections in the vacuum of the deformed O(4) σ-model and AdS/CF T [16], and by considering the physical excited states as momenta-dependent defects, which twist the boundary conditions of the vacuum. We shall observe, at least in a diagonal case, that the insertion of such defects modifies the ground-state TBA equations in the same way as resulting from the standard analytical continuation of the ground-state TBA [17,7]. More importantly for our purposes, the twist matrices involved in the NLO Lüscher-like formula for the twisted vacuum [16] will be replaced by the S-matrices describing the scattering between physical and virtual particles. This will give as a result the first proposal of NLO Lüscher-like formulas for the excited states' energies and rapidities, even in a non-diagonal integrable theory.
In this paper, in particular, these formulas will be checked in some simple relativistic model -the Gross-Neveu model, the O(4) σ-model and the sine-Gordon model -against the large volume expansion of their NLIEs, while the generalization for the non-relativistic case of AdS 5 /CF T 4 will be just conjectured. We hope to come back to the actual calculation of the Konishi double-wrapping correction in the near future. Figure 1: On the left the momentum-dependent defect is circling the compactified space. On the right it is located in (Euclidean) time, and it acts as an operator on the periodic Hilbert space of the rotated channel. The red point is a scattering between virtual particles, the blue one between a virtual particle and the defect.

Main idea
If we consider the physical particles as defects introduced in the compactified space of a cylinder, then we have that the usual defect transmission phase [18] is given by the Smatrix, which describes the scattering of a probe particle against N excitations, that is, in a relativistic case (2.1) Of course, the transmission matrix in this case will depend on the rapidities of the excitations. In this way, we are also ensured that, if the particle moves in the opposite direction, then it picks up the inverse phase: where p = sinh πθ. Furthermore, we have the usual relation between left and right transmission matrices and our transmission matrix satisfies also the defect crossing symmetry [18]: Upon a double Wick rotation, the defect line, which defines, in our case, the asymptotic Bethe equations for the physical theory 2 , becomes a defect operator [18,16] (see Figure 1), 2 At this point a possible reasonable objection could be: the S-matrix describes well the interaction among particles only in the limit of large L. Then, for finite L, the physical excitations should correspond to an which modifies the expression of the mirror 3 partition function: Z(L, R) = Tr(e −H(R)L D) , (2.5) where, introducing the mirror theory rapidityθ = θ + i/2, the defect operator, for a diagonal theory with single species particles, is given by 4 with A, A † being the Zamolodchikov-Faddeev annihilation and creation operators, respectively, in the mirror theory. In particular, the n-particle matrix element of the mirror partition function can be calculated as where Thus, for the large volume expansion of the mirror partition function, also called cluster expansion, we have (α,β) α, β|D|α, β α, β|α, β e −(Ẽ(θ k )+Ẽ(θ l ))L + . . . , (2.9) whereẼ(θ) = cosh πθ. On the other hand, the physical partition function should be dominated at R → ∞ by the ground state energy of the twisted physical theory: Z(L, R) ∼ e −RE d 0 (L) . However, as we shall show in what follows, expression (2.9) will give the finite-size operator describing the exact Bethe equations, with the form of the r.h.s. of (4.3). We found only a posteriori justifications to this fact: the exact Bethe equations would reduce to the asymptotic ones if the probe particle belongs to the rotated theory, or, in other words, particles of physical and rotated theories seem to interact only through the S-matrix, as it is confirmed by the results of this paper. 3 Even though in the relativistic case the space-time rotated theory is the same as the physical one, in view of the generalization to the AdS/CF T case, we already start to use here the terminology related to the non-relativistic case, were the rotated theory is different from the physical one and is called mirror theory, first introduced in the first reference of [5]. 4 One should be careful with the analytical continuation to the mirror theory, since shifting back the integration contour to the real line, poles of the S-matrix, corresponding to bound-states between mirror and physical particles, could be met, giving additional contributions to the energy, called µ-terms. We shall not consider this case through the paper.  corrections to excited states' energies, hence we can say that we are calculating the ground state energy of a theory whose vacuum corresponds actually to some excited state of the original theory. In other words, the insertion into the mirror partition function of the defect operator (2.6), given by the S-matrix related to a particular excited state, selects the contribution of the particular excited state energy to the large R limit of the physical partition function 5 .
In particular, in a generic non-diagonal integrable relativistic theory with particles' flavors labeled by an index a, the r.h.s. of (2.9) can be calculated as follows (see Figure 2) and, for later purposes, we used the crossing symmetry of the defect operator. In general, the expectation value of the non-diagonal defect operator on two generic mirror Bethe states 5 We thank Ryo Suzuki for a comment on this point. is α 1 , α 2 , ..., α n |D|α 1 , α 2 , ..., α n α 1 , α 2 , ..., α n |α 1 , α 2 , ..., α n = (2.14) On the other hand, assuming that the rapidity-dependent defect does not change the mirror Bethe equations, the TBA equations of a diagonal theory result to be modified only by the introduction of a rapidity-dependent chemical potential 15) which enters in the definition of the mirror partition function as follows Following the usual procedure for the derivation of the TBA equations, the mirror free energy is minimized with the constraint of the mirror density equation 17) and the pseudo-energy ε = lnρ/ρ turns out to satisfy the following TBA equation that coincides with the standard excited states TBA equation.
Even in a case where the physical particle was a two-particle bound-state, for instance, we would have the product S(θ 1 −θ +i/2)S(θ 2 −θ +i/2) in the l.h.s. of (2.18), with θ 1,2 being the rapidities of the bound-state's constituents. For example, in the Lee-Yang model, the leading order rapidities would be θ 1,2 = ±i/3 in our normalization, giving the same source term as in [17], once crossing symmetry and unitarity are used.
Unfortunately, we did not find any argument in the particle-defects analogy in order to derive the momentum quantization condition independently from the standard analytic continuation of the TBA. Then the rapidities are not fixed by the TBA equations themselves but somehow in this approach they are put by hand. The only evidence supporting that θ i are the rapidities of the physical excitations is that, as we shall see also in Section 3.3, the equation (2n i + 1)iπ = ε(θ i + i/2) at leading order at large L reduces to the asymptotic Bethe equations.
So, this is just a nice check of the analogy between excited states and defects, at least in the case of a diagonal theory with single species particles. While this analysis can be easily generalized to the case with more particles' species and its extension to non-diagonal theories would be interesting, more importantly we can use this analogy to skip the solution of the excited states TBA equations in a non-diagonal case and to calculate, using equations (2.9) and (2.11), the next-to-leading Lüscher corrections in a generic integrable theory.

Lüscher corrections 3.1 Leading order
Following the analysis of [16], we can easily write now the second term in the r.h.s. of (2.9) in integral form. Then the first correction to the twisted ground-state energy, that corresponds to the energy of some particular excited state in the original model, takes the well known expression of the Lüscher formula: where the trace in the integrand denotes the sum over the mirror states of (2.10) and we recall that a labels the flavor of the particular excited state we want to analyze. In the cases we shall consider in the following sections and for the SU (2) representative of the Konishi multiplet in N = 4 Super-Yang-Mills, the scattering between mirror and physical particles will be diagonal, then the expression above will simplify to 6 Actually, in the supersymmetric case one has to consider the super-trace: S αa αa , with F α = 0, 1 for bosons/fermions respectively.

Next-to-leading order
At the next-to leading order, like in [16], there is a diagonal contribution to the last term in (2.9), given by where the mirror momentap k satisfy the single-particle mirror Bethe equations e ip k R = 1, and whose translation into integral form reads (3.5) The remaining term 1 2 summed over any k, l, α, β, instead needs the solution of the two-particle mirror Bethe equations, that is the diagonalization of the mirror S-matrix. Taking into account the proper Jacobian in the change of variables (k, l) → (p k ,p l ) as in [16], we obtain 7) where δ µ are the eigenvalues' phases of the S-matrix S(θ 1 −θ 2 ) describing the interactions of the two mirror particles, N i T a (θ 1 ,θ 2 , θ i ) denotes the r.h.s. of (2.11), the components of T a are written in (2.12) and we need the expectation values of i T a on the normalized eigenvectors |µ of the mirror S-matrix: An equivalent way to write the sum over µ in the integrand above is as follows where Λ(θ 1 −θ 2 ) is the diagonal matrix of the mirror S-matrix eigenvalues and U is the change of basis matrix, that is needed here differently from the twisted vacuum case [16], since this time the S-matrix does not commute with the defect operator.

Rapidities' corrections
As stated above, we cannot derive the momentum quantization condition from the particledefects analogy. So, to understand what are the finite-size corrections for the rapidities of the physical particles, we consider the exact Bethe equations as from the standard analytic continuation of the TBA, written in terms of the pseudo-energy in a diagonal, single species case (2n i + 1)iπ = ε(θ i + i/2) . (3.10) Thus, we expand at large volume, up to order e −2LẼ , the r.h.s. of (3.10): where φ(θ) = 1 2πi d dθ ln S(θ). The first line corresponds to the asymptotic Bethe equations, while the second line is the leading order correction. In the non-diagonal case, this corresponds to the formula proposed in [4] δΦ (1) Now, with the help of the formulae for the energy corrections (3.5) and (3.7), and the three last lines of (3.12), we can guess for the non-diagonal case the following next-to-leading order rapidities' corrections: and δΦ (2,2) i,a = i 2 Equations (3.5), (3.7), (3.13) and (3.14) are the main results of this paper. Now we shall check them against the large volume expansion of some relativistic models' well known NLIEs.

Gross-Neveu model
The S-matrix of the chiral SU (2) Gross-Neveu model is where the scalar factor is .

(4.2)
In the U (1) sector, the DdV equation can be written as [19] g where K 0 (x) = 1 2πi ∂ x log S 0 (x) and f ± (x) = f (x ± i/2) 7 . Actually, since in the formula for the energy one needs g + and its complex conjugate 1/ḡ − , we find convenient to define A ≡ g + and to use the following non-linear integral equation (NLIE): where one has to take the principal value of the convolution involving K ++ 0 and the last term in the exponential is due to its (−1/2) residue in y = x. Now, in order to calculate the leading energy correction, we consider the leading order of A(x) in the large L expansion, 7 We are using through all this Section the notations of [19].
which gives the well known Lüscher term At the next-to-leading order, the function A(x) is that implies the following energy correction where S ± (θ) ≡ N j=1 S 0 (θ − θ j ± i/2). We checked that the integrands of (4.8) are the same of equations (3.5) and (3.7) with the S-matrix given by (4.1) and (4.2), with flavor label a = 1, for any values of the physical rapidities. The agreement between (3.7) and the second integral of (4.8) is not automatic: in order to show it, one has to use the identities . (4.9) Thus, at least for the states belonging to the U (1) sector of the Gross-Neveu model we have full agreement with our formulas (3.5), (3.7).
On the other hand, the exact Bethe equations are given by (4.10) Then, at the zeroth order at large L, it gives the ABA (4.11) At the first order, (4.10) gives the Bajnok-Janik formula [4] At the second order, we obtain (4.14) where the NLO corrections result to be δΦ (2,1) . (4.16) While in order to show the agreement between (4.15) and (3.13), in the case of a = 1 for instance, it is enough to use the unitarity of S 0 , the matching between (4.16) and (3.14) requires the use of identities (4.9). Still, the integrands of (3.14) and (4.16) differ by a term that is, for example in the one-particle case, equal to where θ 1 is the rapidity single physical excitation. This term is antisymmetric under the exchangeθ 1 ↔θ 2 , then we have complete agreement upon the double integration inθ 1 ,θ 2 .

Sine-Gordon and O(4) σ-model
We checked our formulas also in the cases of sine-Gordon, whose limit ν → ∞ is the Gross-Neveu model, and the O(4) σ-model (SU (2) principal chiral model), which is basically the tensor product of two Gross-Neveu models.
In the first case, it is enough to replace the S-matrix (4.1) by 18) and the scalar factor by All the other quantities follow from these in both sides, Lüscher and NLIE, of the comparison.
In the case of the O(4) σ-model, instead, one needs to use 20) where the scalar factor S 0 is the same as that for the Gross-Neveu model (4.2). In more detail, the main formula for the double-wrapping correction (3.7) becomes where T SU (2) a , U and Λ SU (2) are the same quantities involved in the SU (2) chiral Gross-Neveu model, while the NLIE for the U (1) sector changes to [19] A (4.22) In both cases, we obtained exact agreement between our NLO Lüscher-like formulas and the expansions of the well known NLIEs [20,19], at least for excitations belonging to the U (1) sectors, in the same way as for the Gross-Neveu model. In the sine-Gordon case we needed to use a generalized version of the identities (4.9): .
(4.23) Clearly, it would be interesting to perform the same check for other sectors, and for other integrable relativistic theories. In the case of complex solutions, bound-states etc. there can be additional contributions which are not taken into account in our analysis.

Conjectures for AdS/CFT
On the basis of the previous results, it is not so difficult to guess the following formula as the usual quadratic contribution to the next-to-leading energy corrections for excited states in AdS 5 /CF T 4 is the mirror dispersion relation depending on the mirror momentump, and sTr denotes the super-trace, that in the simple case of diagonal mirror-physical scattering, like in the SU (2) sector for instance, becomes S Q,1 is the S-matrix describing the scattering between a mirror bound-state with charge Q and a single physical particle [21], where S is the AdS 5 /CF T 4 S-matrix [22]: So, (5.1) can be written in terms of the SU (2|2)-invariant S-matrix as It matches also the prediction from the large volume expansion of the excited states TBA where Y (0) Q is the asymptotic solution [6] for the Y Q -functions of the excited states TBA equations [7]. The more complicated contribution reads δE (2,2) a (L) = 2i where is the diagonalized SU (2|2)-invariant S-matrix for the scattering between two generic mirror bound-states [23], and U Q 1 ,Q 2 SU (2|2) is its change of basis matrix. In order to calculate the finite-size correction to the 8-loop anomalous dimension of the Konishi operator one needs just formulas (5.5) and (5.7), with a = 1, j = 1, 2, p 1 = −p 2 = 2π/3, together with the leading order formulas of [4] for energy and rapidities' corrections, expanded to g 16 . A preliminary check of (5.7) can be done by isolating the scalar factor from Λ SU (2|2) and choosing a = 1; then one gets that matches the result from the TBA δE (2,2) where one considers the only contribution in the NLO solution Y (1) Q 2 given by the dressing kernel Y (1) (5.11) It will be also possible to go further in the number of loops -up to 11 in particular, since at 12 loops triple-wrapping effects appear -by considering the following conjectures for the corrections of the rapidities δΦ (2,1) δΦ (2,2) i = −2 Q 1 ,Q 2 dp 1 2π e −Ẽ Q 1 (p 1 )L dp 2 2π e −Ẽ Q 2 (p 2 )L sTr ∂p 1 N i T Q 1 ,Q 2 a (p 1 ,p 2 , p i ) × sTr N i T Q 1 ,Q 2 a (p 1 ,p 2 , p i )U Q 1 ,Q 2 SU (2|2) ∂p 1 log Λ Q 1 ,Q 2 SU (2|2) (p 1 ,p 2 ) (U Q 1 ,Q 2 SU (2|2) ) −1 . (5.13) We do not consider here the contributions from possible µ-terms, which could appear at this or higher loops.

Summary and Outlook
We proposed next-to-leading Lüscher-like formulas, involving just the S-matrix and the dispersion relation of the theory, for generic integrable theories, also for the finite-size corrections to the rapidities of the physical particles/excited states.
In doing this, we were helped by a similar derivation for the twisted ground-states [16], and by the idea that the excited states could be seen as defects, at least from the point of view of the interaction with the virtual/mirror particles.
We found full agreement with the large volume expansion of the NLIE in the cases of few relativistic theories and conjectured a generalization of our formulae for the case of AdS 5 /CF T 4 . Due to the complication of the calculations, we leave the check against the known results for the Konishi operator, the attempt to calculate its anomalous dimension up to 11 loops and the analysis of possible µ-terms for future work.
A simpler check in the AdS/CF T case could be a comparison between the results from our formulas at strong coupling, with a single physical particle and at most two mirror single particles (Q 1 = Q 2 = 1) and a mirror bound-state with Q = 2, against the F-term already calculated at any order for generic string solutions moving in AdS 3 × S 1 [24] or the giant magnon [3,25].
Other checks could be also performed in theories different form those analyzed in this paper or in other sectors of the excited states' spectrum.
Finally, it would be absolutely interesting to study possible generalizations of this approach to higher orders in the large volume expansion, in order to investigate possible simplifications or regular patterns, which could guide us to the exact solution of the spectra in particular sectors of interesting integrable theories, such as various examples of AdS d+1 /CF T d .