TBA, NLO Luscher correction, and double wrapping in twisted AdS/CFT
Changrim Ahn, Zoltan Bajnok, Diego Bombardelli, Rafael I. Nepomechie
aa r X i v : . [ h e p - t h ] O c t TBA, NLO Lüscher correction, and double wrappingin twisted AdS/CFT
August 21, 2018
Changrim Ahn , Zoltan Bajnok , Diego Bombardelli and Rafael I. Nepomechie UMTG-271
Abstract
The ground-state energy of integrably-twisted theories is analyzed in finite volume. We derivethe leading and next-to-leading order (NLO) Lüscher-type corrections for large volumes of thevacuum energy for integrable theories with twisted boundary conditions and twisted S-matrix.We then derive the twisted thermodynamic Bethe ansatz (TBA) equations to describe exactly theground state, from which we obtain an untwisted Y-system. The two approaches are compared byexpanding the TBA equations to NLO, and exact agreement is found. We give explicit results forthe O (4) model and for the three-parameter family of γ -deformed (non-supersymmetric) planarAdS/CFT model, where the ground-state energy can be nontrivial and can acquire finite-sizecorrections. The NLO corrections, which correspond to double-wrapping diagrams, are explicitlyevaluated for the latter model at six loops. Department of Physics and Institute for the Early Universe, Ewha Womans University, DaeHyun 11-1, Seoul120-750, S. Korea; [email protected] Theoretical Physics Research Group, Hungarian Academy of Sciences, 1117 Budapest, Pázmány s. 1/A Hungary;[email protected] Centro de Física do Porto and Departamento de Física e Astronomia Faculdade de Ciências da Universidade doPorto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal; [email protected] Physics Department, P.O. Box 248046, University of Miami, Coral Gables, FL 33124, USA;[email protected] Introduction
The AdS/CFT correspondence in the planar limit can be described by a two-dimensional integrablequantum field theory. The finite-volume energy levels of this integrable theory correspond on oneside to the string energies in the curved
AdS × S background, while to the anomalous dimensionsof gauge-invariant single-trace operators on the other side. Integrability provides tools to solve thefinite-volume spectral problem exactly. (For recent reviews with many references, see [1, 2].)For large volume, L , (long operators of size L ), the asymptotic Bethe ansatz [3, 4] determines thespectrum including all polynomial corrections in L − . In the weak-coupling limit, this result is exactup to L loops; but over L loops, wrapping diagrams start to contribute [5]. In the integrable quantumfield theory, they show up as exponentially small vacuum polarization effects: virtual particles circlingaround the space-time modifies the energy levels [6]. These effects have a systematic expansion whichcounts how many times virtual particles encircle the space-time cylinder (or diagrams wrap around).The leading-order (LO) Lüscher correction corresponds to a single circle or wrapping. Together withthe asymptotic Bethe ansatz, they provide an exact result up to L loops. The next-to-leading (NLO)Lüscher correction corresponds to two circles and double wrapping. Including their contribution de-scribes the energy levels/anomalous dimensions exactly up to L loops.For an exact description, valid for any number of loops, one has to sum up all virtual processes. Forthe ground state, this is done by the thermodynamic Bethe ansatz (TBA), which evaluates the saddlepoint of the partition function for large Euclidean times in the mirror (space-time rotated) description[7, 8, 9, 10, 11, 12, 13, 14]. The TBA provides coupled integral equations for infinitely-many unknownfunctions, whose solutions determine the exact ground-state energy and satisfy the so-called Y-systemrelations, which is characteristic for the model and are the same for all the excited states [15]. Whatis different for the excited states is the analytical structure of these Y-functions [16, 17, 18]. Usingadditional inputs, such as discontinuity relations [19, 20] and analytical structure, the Y-system canbe turned into integral equations for excited states [21, 22], which provide the solution of the finite-volume spectral problem. An ultimate solution would be to replace the infinite Y-system with a finiteT-Q system (see attempts [23, 24, 25, 26, 22] in this direction), which would lead to nonlinear integralequations (NLIE) for only finitely-many unknowns.In the present paper, we would like to analyze the ground state of the three-parameter family of γ -deformed planar AdS/CFT theories [27, 28, 29, 30, 31], for which we refer as γ -deformed theoryfrom now on. Contrary to the undeformed or β -deformed theories, in the most general case, no super-symmetry is preserved, so the ground state is indeed nontrivial and affected by wrapping corrections.The planar gauge theory is nevertheless ultraviolet finite and scale-invariant [32]. This is an ideallaboratory to test ideas directly on the ground state, which actually contains all information aboutthe theory.The γ -deformation can be implemented in several distinct ways: in [33] it was described as anoperatorial twisted boundary condition (the twist depends on the particle number); in [34, 35] as a(c-number) twisted boundary condition and a twisted scattering matrix; finally in [36] the authorsshowed that the untwisted Y-system with twisted asymptotic conditions is consistent with the LOLüscher (single wrapping) correction as calculated on the gauge-theory side. In this paper, based onour previous work [35], we choose twisted boundary condition and twisted S-matrix.We begin by analyzing in Sec 2 the effect of a twisted boundary condition on the ground state ingeneral. We derive exact expressions for the LO and NLO Lüscher corrections valid for any integrabletheory with a twisted boundary condition. The LO correction contains information about the spectrumof the (mirror) theory, while the NLO contains the logarithmic derivative of the scattering matrix. Weshow that a Drinfeld-Reshetikhin type twist [37] of the scattering matrix does not affect the ground-state energy. We then demonstrate the effect of the twist in the TBA equations in general. Theseequations provides the exact description of the ground state for any finite size. By expanding the resultfor large sizes, we must recover the LO and NLO Lüscher corrections. This is explicitly elaborated inthe examples that follow.As a warm up in a simpler case, we analyze in Sec. 3 the O (4) model with twisted boundary condi-1ions. After calculating the LO and NLO Lüscher corrections, we derive the so-called raw (canonical)TBA equations, which contain the twist as chemical potentials. Interestingly, the twist does not showup in the simplified TBA equations except in the asymptotic behavior of the Y-functions. As a con-sequence, the Y-system is the same as the untwisted one. We solve the simplified TBA equations atNLO and compare with the NLO Lüscher correction. We find complete agreement.We turn in Sec. 4 to the γ -deformed AdS/CFT model. We calculate first the LO Lüscher correction.In calculating the NLO correction, we determine the determinant of the two-particle S-matrix S Q Q in all the su (2) L ⊗ su (2) R sectors for the generic Q and Q bound-state case. We then derive theraw TBA equations from first principles by evaluating exactly the chemical potentials originating fromthe twisted boundary condition. (For the untwisted case, the TBA equations were formulated in[10, 11, 12, 13, 14].) The twist disappears from the simplified equations, just as it does in the O(4)case. (See [38] for a general argument on this.) The twist nevertheless reappears in the asymptoticboundary conditions for the Y-functions. Since the simplified equations are not twisted, neither is theY-system, as was anticipated by the authors of [36, 39]. Our derivation confirms their assumption. Wethen expand the TBA equations to NLO and compare with the result of the NLO Lüscher correction.We find complete agreement again.We evaluate in Sec. 5 the weak-coupling expansion of the NLO Lüscher correction, which cor-responds to double-wrapping diagrams. We explicitly compute this correction for L = 3 , therebyobtaining the anomalous dimension of the operator Tr Z in the twisted gauge theory up to six loops.Finally, Sec. 6 contains our conclusion and outlook. In this section we analyze the finite-size corrections for the ground state with a twisted boundarycondition. We consider an integrable (1 + 1) -dimensional quantum field theory that possesses justone multiplet of particles with the same dispersion relation. The particles are labeled by α , and theirinteraction is described by the two-particle scattering matrix S δγαβ ( p , p ) , which does not admit anybound states. We are interested in the ground-state energy of a system of size L with a c -numbertwisted boundary condition in terms of the scattering data. The twisted boundary condition is definedby means of a conserved charge J , which commutes with the scattering matrix [ J, S ] = 0 . The twistsare implemented by introducing a so-called defect line on the circle. It has the effect that, whenever aparticle of type α crosses the defect line from the left to the right, it picks up the transmission phase e iγJ α , where γ is the twist angle supposed to be real. If the particle moves oppositely, then it picks upthe inverse phase e − iγJ α . This ensures that if we formulate the Bethe-Yang equation by moving oneparticle around the circle and scattering with all the other particles and with the defect line in bothdirections, then we obtain equivalent equations.In deriving the finite-size energy of the vacuum with the defect line, E d ( L ) , we analyze the twistedEuclidean torus partition function from two different perspectives, see Figure 1.By compactifying the time-like direction with period R and taking the R → ∞ limit, the ground-stateenergy of the twisted system can be extracted from the twisted partition function as lim R →∞ Z d ( L, R ) = lim R →∞ T r (cid:16) e − H d ( L ) R (cid:17) = e − E d ( L ) R + . . . . (2.1)In the alternative description in which the role of Euclidean time, ˜ x = − it , and space, x , are exchanged,the defect will be localized at a constant imaginary time ˜ t = − ix of the mirror model. It acts as anoperator of the periodic Hilbert space of the mirror model defined by the configurations on a fixed- ˜ t slice. The action of this operator can be calculated from the transmission phase [40]. In the presentcase, the operator is simply e iγJ , and we can evaluate the twisted partition function alternatively as Z d ( L, R ) = Tr ( e − ˜ H ( R ) L e iγJ ) , (2.2) With a view to later applying this formalism to AdS/CFT, we do not assume relativistic invariance; hence, thetwo-particle S-matrix need not be a function of the difference of the particles’ momenta. efect lineL0 xt L 0 defect operatorx~t~
Figure 1: Two possible locations of a defect. On the left it is located in space, and it introduces atwisted boundary condition. On the right it is located in (Euclidean) time, and it acts as an operatoron the periodic Hilbert space.where we use a tilde ˜ to help distinguish quantities in the mirror model. In the first subsection, wesuppose that the volume L is large and expand the partition function at leading and next-to-leadingorders. In this way, we derive the LO and NLO Lüscher-type corrections for the ground state energy ofthe twisted system. Then, in the second subsection, we comment on how one can evaluate the partitionfunction in the saddle-point approximation to obtain the twisted thermodynamic Bethe ansatz (TBA)equations. In this subsection, we evaluate the twisted partition function at LO and NLO for large volumes (i.e., L is large, and R → ∞ ). This means that we keep the first two nontrivial terms in the expansion ofthe twisted partition function lim R →∞ Tr ( e − ˜ H ( R ) L e iγJ ) = 1 + X k,α e iγJ α − ˜ ǫ (˜ p k ) L + X ′ k,l, ( α,β ) e iγJ ( α,β ) − (˜ ǫ (˜ p k )+˜ ǫ (˜ p l )) L + . . . , (2.3)where k, l are the labels of the allowed mirror momenta ˜ p ; α is the color index of the one-particleand ( α, β ) is that of the two-particle state. The sum P ′ is taken over the distinct two-particle states. J is the conserved charge such that J α denotes its eigenvalue on the one particle, while J ( α,β ) is itseigenvalue on the two-particle state. Finally, ˜ ǫ (˜ p ) denotes the energy of the mirror particle. Clearly,the defect does not affect the energy levels, but nevertheless modifies the twisted partition function.Calculations based on the expansion of the partition function for large volumes can be found forboundary entropies in [41], while for the boundary ground state energy in [42]. In evaluating the twisted partition function at LO, we analyze the one-particle contributions. In afinite but large volume, R , the momentum is quantized as e i ˜ p k R = 1 → R π ˜ p k = k ∈ Z , (2.4)which is independent of the color index α = 1 , . . . , N . In the R → ∞ limit, the allowed momentabecome dense, and the summation can be turned into integration. The change from the discrete label k to the continuous momentum variable ˜ p is dictated by the Bethe-Yang equation above as X k → R ˆ d ˜ p π . (2.5)3aking the logarithm of the twisted partition function, the ground-state energy can be obtained E d ( L ) = − lim R →∞ R − log h Tr ( e − ˜ H ( R ) L e iγJ ) i . (2.6)Expanding the log as log(1 + x ) = x + O (cid:0) x (cid:1) and keeping the first term, we obtain E d ( L ) = E (1)0 ( L ) + O ( e − ǫ (0) L ) , E (1)0 ( L ) = − Tr ( e iγJ ) ˆ d ˜ p π e − ˜ ǫ (˜ p ) L , (2.7)where the color summation gives P α e iγJ α = Tr ( e iγJ ) , which is basically the character of the particles’representation. The physical meaning of this formula is clear: The finite-volume vacuum containsvirtual particles, and they modify the vacuum energy by virtual processes. The leading volume-dependent process is when a particle and anti-particle pair appears from the vacuum, and then theparticle travels around the world and annihilates with the anti-particle on the other side. Clearly, inso doing, it crosses the defect line and picks up the phase which, when summed up for the multiplet,results in the character. At the NLO energy correction, we have to expand the logarithm of the partition function (2.6) tosecond order: log(1 + x ) = x − x + O (cid:0) x (cid:1) . This will include the square of the one-particle term andthe two-particle term. The former, however, contains a factor R which would lead to a divergence inthe R → ∞ limit, and has to be canceled against a similar part of the two-particle term. We evaluatenow the two-particle contribution and see the needed cancellation. From the remaining terms, weobtain the NLO energy correction.In calculating the two-particle term, we must first determine the allowed momenta. In very largevolume R , the momentum quantization conditions are given by the Bethe-Yang (or, in other terminol-ogy, the asymptotic Bethe ansatz) equations. As the scattering mixes the color indices, we begin bydiagonalizing the two-particle S-matrix: e iR ˜ p k S νµ (˜ p k , ˜ p l ) ψ ν = ψ µ → e iR ˜ p k e iδ µ (˜ p k , ˜ p l ) = 1 . (2.8)The two-particle S-matrix has N eigenvalues, and we denote their phases by δ µ (˜ p k , ˜ p l ) for µ =1 , . . . , N . Unitarity implies δ µ (˜ p k , ˜ p l ) = − δ µ (˜ p l , ˜ p k ) mod 2 π . We assume that the particles arefermionic: S (˜ p, ˜ p ) = − I , thus δ µ (˜ p, ˜ p ) = π . Taking the logarithm of the equations (2.8) for a giveneigenvalue, we arrive at the Bethe-Yang equations R π ˜ p k + 12 π δ µ (˜ p k , ˜ p l ) = k ,R π ˜ p l − π δ µ (˜ p k , ˜ p l ) = l . (2.9)The fermionic nature of the particles excludes k = l ; and in summing up over two-particle states, k > l is understood. In changing to momentum integration, it is better to reorganize the sum as P k>l f ( k, l ) = P k,l f ( k, l ) − P k f ( k, k ) , since the summand f ( k, l ) = e iγJ − (˜ ǫ (˜ p k )+˜ ǫ (˜ p l )) L is sym-metric. The diagonal part, − P k f ( k, k ) , has the one-particle quantization rule (2.4); thus, changingto integration as in (2.5) the contribution to the energy turns out to be: E (2 , ( L ) = 12 Tr ( e iγJ ) ˆ d ˜ p π e − ǫ (˜ p ) L , (2.10)where we used that P ( α,β ) e iγJ ( α,β ) = P µ e iγJ µ = Tr ( e iγJ ) .4e now transform P k,l f ( k, l ) into a double integral. To this end, we compute the Jacobian forthe change of variables ( k, l ) → (˜ p k , ˜ p l ) : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂k∂ ˜ p k ∂k∂ ˜ p l ∂l∂ ˜ p k ∂l∂ ˜ p l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1(2 π ) (cid:12)(cid:12)(cid:12)(cid:12) R + δ µ,k δ µ,l − δ µ,k R − δ µ,l (cid:12)(cid:12)(cid:12)(cid:12) = 1(2 π ) (cid:2) R + R ( δ µ,k − δ µ,l ) (cid:3) , (2.11)where δ µ,k = ∂ ˜ p k δ µ (˜ p k , ˜ p l ) and δ µ,l = ∂ ˜ p l δ µ (˜ p k , ˜ p l ) . As already mentioned, the terms which contributeto the ground-state energy have to be proportional to R . Indeed the dangerous R term R Tr ( e iγJ ) ˆ d ˜ p π ˆ d ˜ p π e − (˜ ǫ (˜ p )+˜ ǫ (˜ p )) L (2.12)will cancel against the − x term of the expansion of the logarithm of the one-particle contribution.The second term of the Jacobi determinant (2.11) is proportional to the volume R , and contributes tothe ground-state energy as E (2 , ( L ) = − ˆ d ˜ p π e − ˜ ǫ (˜ p ) L ˆ d ˜ p π e − ˜ ǫ (˜ p ) L X µ e iγJ µ ∂ ˜ p δ µ (˜ p , ˜ p ) , (2.13)where we have used that δ µ (˜ p , ˜ p ) is antisymmetric in its arguments; and that, as the twist commuteswith the scattering matrix [ e iγJ , S ] = 0 , both can be diagonalized in the same basis. We note that X µ e iγJ µ ∂ ˜ p δ µ (˜ p , ˜ p ) = − i∂ ˜ p Tr ( e iγJ log[ S (˜ p , ˜ p )]) . (2.14)In particular, this implies that if the S-matrix is twisted (à la Drinfeld-Reshetikhin [37]) with anotherconserved charge ˜ S = F SF , such that [ e iγJ , F ] = 0 , then the finite-size correction is the same as inthe undeformed case: ∂ ˜ p Tr ( e iγJ log( ˜ S )) = ∂ ˜ p X α e iγJ α Tr α log( F α S α F α ) = ∂ ˜ p X α e iγJ α log det( F α S α F α )= ∂ ˜ p X α e iγJ α log det S α = ∂ ˜ p Tr ( e iγJ log( S )) , (2.15)where we have denoted by F α ( S α ) the matrix F ( S ) in the subspace where J has eigenvalue J α ,respectively; and we have used the fact that det( F α S α F α ) = det S α det F α .We conclude that the LO and NLO corrections to the finite-volume vacuum energy in the twistedtheory come only from the twisted boundary condition, and are given by E d ( L ) = E (1)0 ( L ) + E (2 , ( L ) + E (2 , ( L )= − Tr ( e iγJ ) ˆ d ˜ p π e − ˜ ǫ (˜ p ) L + 12 Tr ( e iγJ ) ˆ d ˜ p π e − ǫ (˜ p ) L + ˆ d ˜ p π e − ˜ ǫ (˜ p ) L ˆ d ˜ p π e − ˜ ǫ (˜ p ) L i∂ ˜ p Tr ( e iγJ log[ S (˜ p , ˜ p )]) , (2.16)where the omitted terms are of order of O ( e − ǫ (0) L ) , and Tr i () for i = 1 , means that the trace istaken over the one- or two-particle states, respectively. This derivation is an alternative formulationof the virial expansion of the partition function in statistical physics. (See also the result for the O ( n ) case [43].)Our result (2.16) can also be used to make the connection between the scattering description andother descriptions of the theory. Indeed, given an integral equation for the ground-state energy, wecan extract from it the S-matrix by expanding for large volume to NLO.5 .2 Twisted TBA We have so far supposed that the physical volume L is large, and we have calculated the LO and NLOenergy corrections. If the volume is not large and we are interested in the exact description of thevacuum, we have to evaluate the contributions of multiparticle states. This, in the untwisted case, isdone by the TBA; and we shall now see how the derivations are modified in the presence of the twist.The first step in calculating the partition function is the determination of the momentum quan-tization of multiparticle states. This is done by solving the Bethe-Yang equations by means of theasymptotic Bethe ansatz (BA). Here, in addition to the physical momentum-carrying particles, one hasto introduce so-called magnonic particles that take care of the non-diagonal nature of the scattering.They are useful objects, since in terms of them the scattering can be regarded as diagonal. One thenanalyzes the various “diagonal” scattering matrices and looks for bound states: i.e., complex string-likesolutions of the asymptotic BA equations. The scattering matrices of the bound states are determinedfrom the scattering matrices of their constituents. Let us label the particles (momentum-carrying,magnonic and their bound states) by a multilabel n ; and their scattering matrices by S nm ( u n , u m ) ,where u n i i is some generalized rapidity of a particle of type n i . Greek indices such as α will denotemagnons only. The asymptotic BA equations for large particle numbers (thermodynamic limit) takesthe generic form − e i ˜ p n ( u nk ) R Y m Y l S nm ( u nk , u ml ) , (2.17)where the mirror momentum vanishes for magnons ˜ p α ( u α ) = 0 , and S nn ( u nk , u nk ) = − . We notethat not only the momentum, but also the energy vanishes for magnons, ˜ ǫ α = 0 . Thus, the magnonicequations can be inverted, without changing their physical meaning. We have to choose such equationswhich give rise to positive particle densities in the thermodynamic limit. In this limit, the partitionfunction is dominated by finite-density configurations. The density of the particles (holes) of type n canbe introduced as ρ n = ∆ N n R ∆˜ p , ( ¯ ρ n = ∆ ¯ N n R ∆˜ p ), where ∆ N n ( ∆ ¯ N n ) denotes the number of particles (holes)in the interval (˜ p, ˜ p + ∆˜ p ), respectively. In terms of these densities, the energy of the configuration is ˜ E [ ρ ] = R X n ˆ d ˜ p ρ n (˜ p ) ˜ ǫ n (˜ p ) = R X n ˆ du ρ n ( u ) ˜ ǫ n ( u ) , (2.18)while the entropy is S [ ρ, ¯ ρ ] = R X n ˆ du [( ρ n + ¯ ρ n ) log( ρ n + ¯ ρ n ) − ρ n log ρ n − ¯ ρ n log ¯ ρ n ] . (2.19)The particle and the hole densities are not independent, and the derivative of the logarithm of theasymptotic BA (2.17) connects them as ρ n + ¯ ρ n − π ∂ u ˜ p n = ˆ du ′ X m K nm ( u, u ′ ) ρ m ( u ′ ) =: K nm ⋆ ρ m , (2.20)where K nm ( u, u ′ ) = πi ∂ u log S nm ( u, u ′ ) . If we had inverted any of the asymptotic BA equations, thenwe would have obtained the sign-changed kernel here. By choosing the proper signs of the kernelsfor the magnons, we can ensure the positivity of all the densities. If we had started instead with theDrinfeld-Reshetikhin-twisted S-matrix, then S nm in (2.17) would be replaced by ˜ S nm , which differsfrom S nm by constant phases; and these phases would disappear from the kernel K nm . Consequently,the TBA equations are independent of twists of the S-matrix, as is the Lüscher correction (2.15).We have seen that the twist does not change the energy levels of the periodic mirror system, butnevertheless modifies the partition function. Since the twist commutes with the scattering matrix,the particles of the asymptotic BA equations which diagonalize the multiparticle scatterings will havediagonal twist eigenvalues, too. Let us denote the eigenvalue of iγJ on a particle with label n by µ n .6he total contribution of the twist on the multiparticle state is µ [ ρ ] = R X n ˆ du ρ n ( u ) µ n . (2.21)In terms of these quantities, the partition function can be written as Z d ( L, R ) = Tr ( e − ˜ H ( R ) L e iγJ ) = ˆ Y n d [ ρ n , ¯ ρ n ] e S [ ρ, ¯ ρ ]+ µ [ ρ ] − L ˜ E [ ρ ] . (2.22)Evaluating the integrals in the saddle-point approximation, the minimizing condition for the pseudo-energies ǫ n = log ¯ ρ n ρ n turns out to be ǫ n + µ n = ˜ ǫ n L − log(1 + e − ǫ m ) ⋆ K mn . (2.23)Once we have calculated the pseudo-energies, the ground-state energy can be extracted from thesaddle-point value as E d ( L ) = − X n ˆ du π ∂ u ˜ p n log(1 + e − ǫ n ) . (2.24)Clearly the only difference compared with the untwisted case is the appearance in the TBA equations(2.23) of the chemical potential µ n , which is proportional to the charge of the particle. (TBA equationswith chemical potentials have been studied previously; see e.g. [44].)As the determination of the magnons and their charges is model dependent, we work out the detailsin the following for the O (4) model, and then for twisted planar AdS/CFT. O (4) model In this section, as a warm-up, we elaborate explicitly the simpler case of the twisted O (4) model, alsoknown as the su (2) principal chiral model. We calculate the LO and NLO Lüscher corrections, derivethe twisted TBA equations, and compare the two approaches by expanding the TBA equations up tosecond order.The O (4) model is a relativistic theory containing one multiplet of particles with mass m . Thedispersion relation E ( p ) = p m + p can be parameterized in terms of the rapidity as E ( θ ) = m cosh πθ , p ( θ ) = m sinh πθ . (3.1)The particles transform under the bifundamental representation of su (2) . The two-particle S-matrixis the simplest su (2) ⊗ su (2) symmetric, unitary and crossing-invariant scattering matrix [45, 46] S ( θ ) = S ( θ )( θ − i ) ˆ S ( θ ) ⊗ ˆ S ( θ ) , ˆ S ( θ ) = θ I − i P , (3.2)where θ = θ − θ , and the scalar factor S ( θ ) = i Γ( − iθ )Γ( iθ )Γ( + iθ )Γ( − iθ ) (3.3)does not have any poles in the physical strip, showing the absence of physical bound states.We analyze this theory on a circle of size L with a twisted boundary condition. We twist the theorywith independent twist angles γ ∓ for the left and right su (2) factors, respectively: e iγJ = e iγ − J ⊗ I + iγ + I ⊗ J = e iγ − J ⊗ e iγ + J = diag ( ˙ q, ˙ q − ) ⊗ diag ( q, q − ) , (3.4)where J has eigenvalues ± on the two components of the doublet, and ˙ q = e iγ − , q = e iγ + . We couldalso twist the S-matrix, i.e. change S → F SF , but this would have no effect on the ground-stateenergy, as explained in (2.15). 7 .1 Lüscher corrections
We now proceed to evaluate the Lüscher correction for the vacuum (2.16). As the theory is relativis-tically invariant, the mirror dispersion relation is ˜ ǫ (˜ p ) = p m + ˜ p , which we parameterize in termsof the rapidity as above: ˜ p ( θ ) = m sinh πθ . In this parameterization, the leading-order result for theground-state energy is E (1)0 ( L ) = − [2] q [2] ˙ q m ˆ dθ πθ e − mL cosh πθ , (3.5)where we used thatTr ( e iγJ ) = Tr ( e iγ − J ) Tr ( e iγ + J ) = ( ˙ q + ˙ q − )( q + q − ) = [2] q [2] ˙ q . (3.6)It is useful to introduce the q -numbers [ n ] q = q n − q − n q − q − = q n − + q n − + · · · + q − n + q − n , (3.7)for which [ n ] q → n in the untwisted limit q → .In the second-order correction, we have the term without the scattering matrix E (2 , ( L ) = 12 [2] q [2] q m ˆ dθ πθ e − mL cosh πθ . (3.8)In the other term, we have to diagonalize the two-particle S-matrix S ( θ ) = S ( θ ) S ( θ ) ⊗ S ( θ ) , S ( θ ) = S ( θ ) = 1 θ − i ˆ S ( θ ) = θθ − i − iθ − i − iθ − i θθ − i
00 0 0 1 . (3.9)The twist matrix acts on the two-particle states as e iγJ = e iγ − J ⊗ e iγ + J = ˙ A ⊗ A = diag ( ˙ q , , , ˙ q − ) ⊗ diag ( q , , , q − ) , (3.10)and commutes with the scattering matrix. The twist and the S-matrix can be diagonalized in the samebasis, where the S-matrix eigenvalues take the form S = S Λ ⊗ Λ = S diag (1 , , θ + iθ − i , ⊗ diag (1 , , θ + iθ − i , , . (3.11)For the Lüscher correction, we need to calculate Tr ( e iγJ ( − i∂ θ ) log S ) . As the scattering matrix hasthe specific tensor product structure (3.9), we can writeTr ( e iγJ log S ) = Tr (( ˙ A ⊗ A ) (2 log S I ⊗ I + log S ⊗ I + I ⊗ log S )) (3.12) = Tr ( ˙ A ) Tr ( A )2 log S + Tr ( A ) Tr ( ˙ A log S ) + Tr ( ˙ A ) Tr ( A log S )= 2 Tr ( ˙ A ) Tr ( A ) log S + Tr ( A ) X i ˙ A i log Λ i + Tr ( ˙ A ) X i A i log Λ i . In Fourier space, the logarithmic derivatives take a particularly simple form: K ( θ ) = 12 πi ∂ θ log S ( θ ) → ˜ K ( ω ) = 2 tt + t − ,K ( θ ) = 12 πi ∂ θ log θ + iθ − i → ˜ K ( ω ) = − t , (3.13)8here we have indicated the Fourier transform by tilde, and t = e − | ω | . The integrand of the secondorder Lüscher correction is finally π Tr ( e iγJ ( − i∂ θ ) log S ) = [2] q [2] q K + (cid:0) [2] q + [2] q (cid:1) K . (3.14)In terms of these quantities, the second part of the Lüscher correction is E (2 , ( L ) = − [2] q [2] q m ˆ dθ e − mL cosh πθ ˆ dθ cosh πθ e − mL cosh πθ × n K ( θ − θ ) + ([2] − q + [2] − q ) K ( θ − θ ) o . (3.15) Following the general procedure outlined in section 2.2, in order to formulate the twisted TBA equa-tions, we need to classify the particles: momentum-carrying, magnons and their bound states. Wealso have to calculate their scattering matrices; and, additionally to the untwisted case, we also mustidentify the twist charge on all the excitations.
In order to derive the mirror nested asymptotic BA equations, we start with an N -particle stateconsisting of down-spin particles only. We label these particles by . They scatter on each other as S ( θ ) = S ( θ ) , (3.16)and they have the dispersion relation ˜ ǫ (˜ p ) = ˜ ǫ (˜ p ) . As the J eigenvalue of the lower component is − on both su (2) sides, the chemical potential is µ = − iγ − − iγ + . We can now introduce up-spins in thesea of down-spins. These are the magnons, which do not change the energy and momentum, ratherdescribe the polarization degrees of freedom. We label them by for the right su (2) factor, and by − for the left su (2) factor. Let us first focus on the positive (right) part, and denote magnon rapiditiesby u . The magnons scatter on the massive particles and on themselves as S ( θ − u ) = θ − u + i θ − u − i , S ( u − u ′ ) = u − u ′ − iu − u ′ + i , (3.17)respectively. The magnons do not have any energy and momentum ˜ ǫ ( u ) = ˜ p ( u ) = 0 , but they dohave chemical potential. Since a magnon swaps a spin from down to up, it changes the charge by : µ = 2 iγ + . This means that a state with m up-spins and N − m down-spins, which contains N type- particles and m type- particles, has J charge − N + 2 m . Inspecting the magnon scatteringmatrices, we can conclude that a magnon and a massive particle cannot form bound states. In contrast,magnons among themselves can bound. Bound states in the thermodynamic limit consist of strings ofany length M ∈ N : u j = u + i M + 1 − j , j = 1 , . . . , M . (3.18)We label this string as M . Clearly, the M = 1 string is the magnon itself. The scattering of the M -string and the massive particle can be calculated from the bootstrap, S M ( θ − u ) = M Y j =1 S ( θ − u j ) = θ − u + i Mθ − u − i M . (3.19)9s S M ( θ − u ) S M ( u − θ ) = 1 , we conclude that S M ( u ) = S M ( u ) . Similarly, the magnon-magnonscatterings are given by S MM ′ ( u − u ′ ) = M Y j =1 M ′ Y j ′ =1 S ( u j − u ′ j ′ ) (3.20) = u − u ′ − i | M − M ′ | u − u ′ + i | M − M ′ | ! u − u ′ − i ( | M − M ′ | + 2) u − u ′ + i ( | M − M ′ | + 2) ! × . . . u − u ′ − i ( M + M ′ − u − u ′ + i ( M + M ′ − ! u − u ′ − i ( M + M ′ ) u − u ′ + i ( M + M ′ ) ! . These bound states have no energy and momentum ˜ ǫ M ( u ) = ˜ p M ( u ) = 0 , while their chemical potentialis the sum of their constituents’: µ M = 2 M iγ + . Similar considerations apply to the left excitations, which are denoted by − M . They scatter only onthemselves and on the massive particle, such that the scattering is independent of the sign of M . Theonly difference is in the chemical potential, as the twists are different on the two sides: µ − M = 2 M iγ − .Summarizing, we have particles for any M ∈ Z . The only massive excitation that has nontrivialenergy and momentum has the label ; all others are magnons. The scattering kernels in Fourier spacehave the form ˜ K = 2 t ( t + t − ) , ˜ K n = ˜ K n = − t n , ˜ K nm = t + t − t − t − ( t n + m − t | n − m | ) − δ nm , (3.21)where t = e − | ω | and n, m > . For the other values, we have K n = K − n , K n = K − n , K − n − m = K n m and K − n m = K n − m = 0 .In the general procedure, one has to invert the magnonic equations before introducing the magnondensities. In so doing, one obtains the “raw” (canonical) twisted TBA equations ǫ + µ = L ˜ ǫ − log(1 + e − ǫ ) ⋆ K + X M =0 log(1 + e − ǫ M ) ⋆ K M , (3.22) ǫ M + µ M = − log(1 + e − ǫ ) ⋆ K M + X M ′ =0 log(1 + e − ǫ M ′ ) ⋆ K M ′ M , M = 0 . (3.23)These equations for the untwisted ( µ = 0 ) case reduce to those in [47], although in slightly differentconvention. Using identities among the kernels, we now bring the TBA equations (3.22), (3.23) to a universal localform. This means that the pseudo-energies can be associated with vertices of a two-dimensional lattice,such that only neighboring sites couple to each other with the following universal kernel s I MN = δ MN − ( K + 1) − MN , s ( θ ) = 12 cosh πθ , (3.24)where I MN = δ M +1 ,N + δ M − ,N and ( K + 1) − MN ⋆ ( K NL + δ NL ) = δ ML . We also have ( K n + δ n ) ⋆ s = − K n , which can be easily seen in Fourier space where ˜ s = t + t − .Let us introduce the Y-functions: Y = e − ǫ , Y M = e ǫ M , M = 0 . (3.25)10e take the equations (3.23) for Y M , act with the operator δ MN − s I MN = ( K + 1) − MN from the right,and use the kernel identity K N ⋆ ( K + 1) − NM = − s δ M, . Since the chemical potentials are annihilatedby the discrete Laplacian µ M ⋆ ( sI MN − δ MN ) = 12 ( µ N − + µ N +1 ) − µ N = 0 , (3.26)they completely disappear from the equations, and we arrive at log Y M = I MM ′ log(1 + Y M ′ ) ⋆ s , M = 0 . (3.27)Finally, we take the equations for M = ± and convolute them with the kernel s . We combine theseequations with the massive equation (3.22). Using the magic property of the kernel K = − s ⋆ K ,and exploiting that µ + ( µ + µ − ) = 0 , we obtain the equation for the massive node log Y + mL cosh πθ = (log(1 + Y ) + log(1 + Y − )) ⋆ s . (3.28)Thus, the twists completely disappear from the “simplified” equations (3.27), (3.28). Nevertheless,they enter in the asymptotics of the Y -functions as lim M →∞ M log Y ± M = − iγ ± , (3.29)since the kernels in (3.23) vanish in this limit. After all, it should not come as a surprise that the Y -system is not twisted, Y + M Y − M = (1 + Y M − )(1 + Y M +1 ) , Y ± ( θ ) = Y ( θ ± i . (3.30)The ground-state energy contains the contribution of the only massive node, E ( L ) = − m ˆ dθ cosh πθ log(1 + Y ) . (3.31) We now make a LO and NLO asymptotic expansion of the simplified TBA equations (3.27), (3.28) for L → ∞ .At leading order, Y is exponentially small and the other Y functions are constant. Let us expandthe Y -functions as Y M = Y M (1 + y M ) + . . . , (3.32)and determine all functions iteratively. The Y-system at leading order will be split into two independentconstant Y -systems. The solutions with the correct initial and asymptotic behaviors will determinethe exponentially small leading-order Y in terms of Y ± . Then, in calculating the NLO y M functions,we can proceed independently for the two parts. Again, the initial condition is provided by Y ,which appears as a multiplicative factor; while uniqueness is provided by the vanishing asymptotics lim M →∞ y M = 0 . The y ± obtained in this way will determine the NLO correction y , which is neededfor the energy correction.Let us now carry out these calculations. Using the fact that s ⋆ f = f if f is constant, we seefrom (3.28) that log Y = − mL cosh πθ + 12 log(1 + Y ) + 12 log(1 + Y − ) , (3.33)where the LO constant Y -functions satisfy the relations ( Y M ) = (1 + Y M − )(1 + Y M +1 ) , M = 0 , (3.34)11s follows from (3.27). The solution with the correct asymptotics (3.29) is Y M = [ M ] q [ M + 2] q , Y − M = [ M ] ˙ q [ M + 2] ˙ q . (3.35)Clearly, the twist dependence reenters through the asymptotic solution. This means that at leadingnon-vanishing order Y ≈ Y = p (1 + Y )(1 + Y − ) e − mL cosh πθ = [2] q [2] ˙ q e − mL cosh πθ , (3.36)which, when substituted back into the energy formula (3.31), reproduces the leading-order Lüschercorrection (3.5). Actually, expanding the log in the energy formula (3.31) to second order log(1 + Y ) = Y − Y reproduces also E (2 , in (3.8). Thus, we need to expand the Y -functions to NLO to obtainthe remaining E (2 , in (3.15).We see from (3.28) and (3.32) that the massive node has the NLO expansion Y = Y (cid:18) s ⋆ (cid:18) Y Y y + Y − Y − y − (cid:19)(cid:19) + . . . . (3.37)We need to calculate y ± . We expand the TBA equations (3.27), keeping only the linear terms in y , y k = s ⋆ (cid:18) Y k +1 Y k +1 y k +1 + Y k − Y k − y k − (cid:19) , k = 0 . (3.38)We solve this equation by Fourier transform ( t + t − )˜ y k = [ k + 1] q [ k + 3] q [ k + 2] q ˜ y k +1 + [ k − q [ k + 1] q [ k ] q ˜ y k − , (3.39)where we have also used the result (3.35) and the identity k − q [ k + 1] q = [ k ] q . Being a second-order difference equation, the generic solution contains two parameters. These parameters can be fixedby demanding that lim k →∞ ˜ y k = 0 and ˜ Y = lim k → Y k ˜ y k . The result is ˜ y k = t k [ k + 1] q [2] q [ k ] q [ k + 2] q ([ k + 2] q − [ k ] q t ) ˜ Y , ˜ y − k = ˜ y k ( q → ˙ q ) , (3.40)which is just the deformed version of the O (4) solution [47]. Thus, for the needed y ± , we have ˜ y = (cid:18) t − t [3] q (cid:19) ˜ Y , ˜ y − = (cid:18) t − t [3] ˙ q (cid:19) ˜ Y . (3.41)Performing inverse Fourier transform, y = − (cid:18) K − K [3] q (cid:19) ⋆ Y , y − = − (cid:18) K − K [3] ˙ q (cid:19) ⋆ Y . (3.42)Substituting back into (3.37), we obtain Y = [2] q [2] ˙ q e − mL cosh πθ (cid:16) s ⋆ h ( K − [3] q K )[2] − q + ( K − [3] ˙ q K )[2] − q i ⋆ [2] q [2] ˙ q e − mL cosh πθ (cid:17) . (3.43)Comparing the double-convolution term with E (2 , in (3.15) in Fourier space, we obtain completeagreement. The twists γ ± have small positive imaginary parts in order to suppress large- M magnonic contributions to thepartition function (2.22). Twisted AdS/CFT
In this section, we apply the previous methodology to the twisted AdS/CFT model. After definingthe model by its scattering matrix, dispersion relation and twist matrix, we derive the LO and NLOLüscher corrections. As the model has infinitely many massive bound states Q ∈ N , in the NLO Lüschercorrection we have a sum of the form P ∞ Q ,Q =1 . We first elaborate the summand Q = Q = 1 indetail, and we then treat the general case, which entails detailed knowledge of all scattering matrices S Q Q . We next derive the twisted TBA equations by evaluating the charges of the magnons and theirbound states in the thermodynamic limit of the mirror asymptotic BA. The twist, just as in the O (4) model, disappears from the universal equations, which lead to the untwisted Y -system. We expandthe TBA equations to NLO and compare to the Lüscher correction, and again find perfect agreement.The AdS/CFT integrable model has an su (2 | ⊗ su (2 | symmetry. The elementary particletransforms under the bifundamental representation of su (2 | . For one copy of su (2 | , Latin indices a = 1 , label the bosonic, while Greek indices α = 3 , label the fermionic components of the four-dimensional representation. We will introduce twist in the bosonic subspace by the generator L ,which has nonvanishing diagonal matrix elements: ( L ) = 1 and ( L ) = − .The symmetry completely determines the left/right scattering matrix, which has the nonvanishingamplitudes S aaaa = S abab + S baab = a = x − − x +1 x +2 − x − s x +2 x − s x − x +1 , S abab − S baab = a , (4.1) S αααα = S αβαβ + S βααβ = a = − , S αβαβ − S βααβ = a , (4.2) S αβab = − ǫ ab ǫ αβ a , S abαβ = − ǫ αβ ǫ ab a , (4.3) S aαaα = a , S αaaα = a , S aααa = a , S αaαa = a , (4.4)where a, b ∈ { , } with a = b ; α, β ∈ { , } with α = β ; and the various coefficients can be extractedfrom [48]. For Q = Q = 1 we shall need explicitly only a , since – as a consequence of someidentities among the various coefficients – we shall be able to express the Lüscher corrections purelyin terms of it. The scattering matrix depends independently on the momenta of the particles p and p via x + x − = e ip , x + + 1 x + − x − − x − = 2 ig , (4.5)where g = √ λ/ (2 π ) and λ = g Y M N is the ’t Hooft coupling. The full scattering matrix has the form S ( p , p ) = S sl (2) ( p , p ) h S su (2 | ( x ± , x ± ) ⊗ S su (2 | ( x ± , x ± ) i − , (4.6)where S sl (2) ( p , p ) is the scalar factor S sl (2) ( u, u ′ ) = u − u ′ + ig u − u ′ − ig Σ − , Σ = 1 − x +1 x − − x − x +2 σ , (4.7)with σ being the dressing factor. We remark that S denotes actually the inverse of the AFZ S -matrix[48], since we are using the relativistic convention e ipL Q j S ( p, p j ) , as in Section 2, instead of e ipL = Q j S ( p, p j ) .The dispersion relation can be easily expressed in terms of x ± as E = − ig (cid:18) x + − x + − x − + 1 x − (cid:19) . (4.8) Indeed, a , . . . , a are given by the coefficients of the ten terms in Eq. (8.7) in [48], respectively.
13n analogy with the O (4) model, we introduce different twists for the two su (2 | factors, which welabel by α = ± , e iγJ = e iγ − L ⊗ e iγ + L = diag ( ˙ q, ˙ q − , , ⊗ diag ( q, q − , , , (4.9)where again q = e iγ + , ˙ q = e iγ − ; and γ ± are related to the deformation parameters γ i used in [29, 30]by γ ± = ( γ ± γ ) L .The scattering matrix has poles, which signal the existence of bound states. These states transformunder the Q -dimensional totally symmetric representation of su (2 | for any Q ∈ N . The dispersionrelation of the bound states can be obtained from (4.8) by changing the shortening condition to x + + 1 x + − x − − x − = 2 iQg . (4.10)The matrix part of the scattering matrix can be fixed [49] from the Yangian symmetry [50], while thescalar factor can be determined [51] from the bootstrap principle.The mirror model has the analytically-continued scattering matrix: x ± ( p ) → x ± (˜ p ) , where ˜ p = − iE . Since the physical domains of p and ˜ p are different, the bound states are different, too. The mirrorbound states transform under the Q -dimensional totally antisymmetric representation of su (2 | , andthe twist charge acts as e iγ + L = diag ( I Q − , I Q +1 , q I Q , q − I Q ) . (4.11)The scattering matrix of the antisymmetric bound states are related to those of the symmetric ones bychanging the labels ↔ , ↔ and simultaneously flipping x ± ↔ x ∓ inside the matrix part. Com-bining this with the previously mentioned notational differences, we can use the following scatteringmatrices to calculate the Lüscher correction: S = S Q Q sl (2) ( S Q Q su (2 | ⊗ S Q Q su (2 | ) , (4.12)where S Q Q sl (2) ( u , u ) = Q Y j =1 Q Y j =1 S ( u j , u j ) , u nj n = u n + ( Q n + 1 − j n ) ig . (4.13)and S Q Q su (2 | denotes the symmetric-symmetric bound state scattering matrix in the conventions of [49]. The derivation of Section 2 is not general enough to describe the AdS/CFT problem. We have toincorporate two new features: the existence of fermions, and of multiple species of particles that arelabeled by the charge Q . The fermionic nature can be taken into account by changing the trace to thesupertrace. This is equivalent to imposing antiperiodic boundary conditions on the fermions, whichcan be implemented by an e iπF twist, where F is the fermion number operator:STr Q ( e iγJ ) = Tr Q (( − F e iγJ ) = Tr Q ( e i ( πF + γJ ) ) = STr Q ( e iγ − L ) STr Q ( e iγ + L ) = ([2] q − ˙ q − Q . (4.14)Clearly, the supertrace vanishes in the untwisted q → limit. The generalization of the derivation ofSection 2 will contain the scattering matrices S Q Q . They arise from two-particle states with charges Q and Q . As the species are different, we should not constrain the summation on the quantizationnumbers P k 1) log a + log (cid:2) a a (( a + 2 a )( a + 2 a ) − a a ) (cid:3) − q log( a a − a a ) } + ( q ↔ ˙ q ) , (4.25)where ( − F = (1 , , − , − . Using the explicit expressions for the coefficients found in [48], weobserve the following identities a a − a a = a , ( a + 2 a )( a + 2 a ) − a a = − a . (4.26)Substituting these identities into (4.25), we obtain a very simple expression for the matrix part of theNLO Lüscher correction for Q = Q = 1 in terms of only a , (2 − [2] q ) (2 − [2] ˙ q ) (cid:18) [2] ˙ q − [2] ˙ q + [2] q − [2] q (cid:19) ˆ d ˜ p π e − L ˜ ǫ (˜ p ) ˆ d ˜ p π e − L ˜ ǫ (˜ p ) i∂ ˜ p log a (˜ p , ˜ p ) . (4.27) ( Q , Q ) Although the above approach can also be used for the cases ( Q , Q ) = (1 , , (2 , for which theexplicit S-matrices are available [52], it is impractical for higher-dimensional cases. Clearly, a morepowerful approach is needed to treat the general case. Observe from (4.24) that the NLO Lüschercorrection involves the quantity P i ( − F i A i log Λ Q Q i , and a similar quantity with A i replaced by ˙ A i .We exploit the fact that the su (2 | part of the γ + twist e iγJ = I ⊗ e iγ + L involves nontrivially onlythe su (2) R factor in su (2) L ⊗ su (2) R ⊂ su (2 | , as is evident from (4.11). Since su (2) L ⊗ su (2) R is thesymmetry of the scattering matrix, we can perform an expansion in the left ( s L ) and right ( s R ) spins: X i ( − F i A i log Λ Q ,Q i = X ( s L ,s R ) STr[( I ⊗ e iγ + L ) log S Q Q ( s L , s R )]= X ( s L ,s R ) ( − s R (2 s L + 1) [2 s R + 1] q log det S Q Q ( s L , s R ) , (4.28)where S Q Q ( s L , s R ) is the 2-particle S-matrix in the sector with left and right su (2) spins s L and s R ,and we calculated the traces as Tr s L ( I ) = 2 s L + 1 and STr s R ( e iγ + L ) = ( − s R [2 s R + 1] q , respectively.The sum is over all the possible values of s L and s R for the given values of Q and Q .The problem of computing the NLO Lüscher correction therefore reduces to the determinationof det S Q Q ( s L , s R ) for general values of Q and Q , and for all the possible corresponding valuesof s L and s R . This is a formidable technical challenge, since individual S-matrix elements – andparticularly the eigenvalues – are not known explicitly enough in general, and those that are knownexplicitly enough [52, 59] generally have very complicated expressions. Nevertheless, it turns out that– remarkably – these determinants have simple compact expressions, which are constructed from asmall number of elementary building blocks.We propose that, with both particles in symmetric representations and Q , Q > , the deter-minants are given by the expressions in Table 1. In order to save writing, we have introduced thefollowing notation U = x − − x +2 x +1 − x − , U = s x +1 x − , U = s x − x +2 , U = x +1 x +2 − x − x − − , S Q = u − u − iQg u − u + iQg , (4.29) In other words, det S ( s L , s R ) = Q i Λ i ( s L , s R ) , where Λ i ( s L , s R ) are the eigenvalues of the 2-particle S-matrixcorresponding to eigenstates which are also su (2) L ⊗ su (2) R highest-weight states with given values of s L and s R . Forfurther details, see appendix A. As usual, the spins s L , s R are non-negative integers or half-odd integers. R det S Q Q ( s L , s R ) s L U U U S s L +2 S s L +4 . . . S Q + Q − | Q | , ( | Q | + 2) , . . . , ( Q + Q − U U U ( Q + Q − ( U U U ) S s L +1 S s L +3 . . . S Q + Q − ( | Q | + 1) , . . . , ( Q + Q − ( U U U ) (cid:16) U U U (cid:17) δ S | Q | +2 S | Q | +4 . . . S Q + Q − ( | Q | − ≥ 00 1 ( Q + Q )0 ( U U U ) S s L ( Q + Q − U U U ) S s L S s L +2 12 ( Q + Q − U U U ) S s L S s L +2 S s L +4 . . . S Q + Q − ( | Q | + 2) , . . . , ( Q + Q − U U U ) (cid:16) U U U (cid:17) δ S | Q | +2 S | Q | +4 . . . S Q + Q − | Q | 6 = 00 U U U (cid:16) U U U (cid:17) δ S | Q | +2 S | Q | +4 . . . S Q + Q − ( | Q | − ≥ 00 ( U U U ) S S S . . . S Q + Q − Q Table 1: The values of det S Q Q ( s L , s R ) for Q , Q > and for all possible s R and s L , where δ = Q | Q | = ± , and Q ij = Q i − Q j . See (4.29) for further notations.where x ± j are the parameters of the Q j bound-state representation, and u j ± iQ j g = x ± j + x ± j . Notethat there are only three possible values of the right-spin, namely s R = 0 , , , as s R counts thenumber of fermions in the basis of the Hilbert space. If at least one of either Q or Q is 1, then thecorresponding results are collected in Table 2. A brief account of how these results were obtained ispresented in Appendix A.Substituting the results from Tables 1 and 2 into (4.28), and carefully simplifying the resultingexpression, we obtain X i ( − F i A i log Λ Q Q i = [3] q (cid:0) Q Q log U U U + K Q Q (cid:1) (4.30) − [2] q (cid:0) (4 Q Q − Q − Q ) log U + 2 Q (2 Q − 1) log U +2 Q (2 Q − 1) log U + ( Q − Q ) log U + 4 K Q Q (cid:1) +[1] q (cid:0) (5 Q Q − Q − Q ) log U + Q (5 Q − 4) log U + Q (5 Q − 4) log U + 2( Q − Q ) log U + 5 K Q Q (cid:1) , where we have defined K Q Q = Q − X j =0 ( Q − Q + 2 j + 1) Q − j − X k =1 log S Q − Q +2 j +2 k . (4.31)In deriving the result (4.30), we have made use of the fact that log S Q is an antisymmetric functionof Q (i.e., log S − Q = − log S Q , up to an irrelevant additive constant), which in particular implies that K Q Q = K Q Q . We emphasize that (4.30) holds for any Q , Q ∈ N . An analogous result can bederived for P i ( − F i ˙ A i log Λ Q Q i by replacing q → ˙ q in (4.30).17 R det S Q Q ( s L , s R ) s L U U U ( Q + Q − U U U ( Q + Q − ( U U U ) (cid:16) U U U (cid:17) δ ( | Q | − ≥ 00 1 ( Q + Q )0 ( U U U ) (cid:16) U U U (cid:17) δ | Q | 6 = 00 U U U (cid:16) U U U (cid:17) δ ( | Q | − ≥ U U U Q ( Q = Q = 1) Table 2: The values of det S Q Q ( s L , s R ) for all possible s R and s L if either Q or Q is 1.Thus, in order to calculate the Lüscher correction, we have to plug (4.30) into the formula (4.24): E (2 , = ∞ X Q ,Q =1 Q Q ˆ ∞−∞ d ˜ p π e − L ˜ ǫ Q (˜ p ) ˆ ∞−∞ d ˜ p π e − L ˜ ǫ Q (˜ p ) × i∂ ˜ p ( (2 − [2] ˙ q ) h [3] q (cid:0) Q Q log U U U + K Q Q (cid:1) − [2] q (cid:16) (4 Q Q − Q − Q ) log U + 2 Q (2 Q − 1) log U +2 Q (2 Q − 1) log U + ( Q − Q ) log U + 4 K Q Q (cid:17) +[1] q (cid:16) (5 Q Q − Q − Q ) log U + Q (5 Q − 4) log U + Q (5 Q − 4) log U + 2( Q − Q ) log U + 5 K Q Q (cid:17)i +( q ↔ ˙ q )+ Q Q (2 − [2] q ) (2 − [2] ˙ q ) log S Q Q sl (2) (˜ p , ˜ p ) ) . (4.32)We shall compare this result to the TBA output in Section 4.4.4. In [19, 20, 13], the authors derived the TBA equations for the AdS/CFT model with the most generalchemical potentials. Hence, the TBA equations for the γ -deformed theories correspond to some specialcases. However, since we must determine precisely the charges/chemical potentials of the variousexcitations in terms of the deformation parameters, we now briefly sketch the derivation.In order to derive the TBA equations, we have to recall the various types of excitations (bothmassive and magnonic) and their scattering matrices; and we must calculate their twist charges. Welabel the fundamental massive particle as Q = 1 , corresponding to the (3 ˙3) label of the fundamentalrepresentation. The S -matrix of this kind of particles is in fact given by (4.7) and they can form18ound states for any Q with string-like complex roots defined like in (4.13). We label such a massivecomposite particle by Q and the scattering matrix of such particles is (4.13). Since the twist chargeacts trivially in the (3 , subspace, the massive particles are not charged: µ Q = 0 .We now focus on the magnonic excitations. They encode the color su (2 | structure of the scat-tering, and come in independent left and right copies. We first consider the right part. We label amagnon, which introduces label in the sea of massive -particles, by y . It scatters trivially on itself,but nontrivially on the massive particles S y ( u, y ) = x − ( u ) − yx + ( u ) − y s x + ( u ) x − ( u ) , S Qy ( u, y ) = Q Y j =1 S y ( u j , y ) . (4.33)The twist charge of the y particles is µ y = − iγ + .We can also introduce the label in the sea of -particles. These particles are labeled by w . Theyscatter nontrivially only on the y particles and on themselves: S yw ( y, w ) = S − ( v ( y ) − w ) , S ww ( w, w ′ ) = S ( w − w ′ ) , (4.34)where v ( y ) = y + y − , and S n ( u ) is defined as in (4.29), namely S n ( u ) = u − ing u + ing . (4.35)The twist charge of these particles is µ w = 2 iγ + .As the scattering matrix S yw ( y, w ) has a difference form in the variable v ( y ) = y + y − , we mightuse the parameter v instead of y . The inverse of the relation, however, is not unique. Defining y − ( v ) = ( v − i √ − v ) with the branch cuts running from ±∞ to ± , we can describe any y with ℑ m ( y ) < for v ∈ [ − , . Clearly y + ( v ) = y − ( v ) − describes the other ℑ m ( y ) > case; and in thescattering matrices S y which depend on y , and not on v , we have to specify which root is taken. Asa consequence, we have two types of y particles y | δ with δ = ± ; and the scattering matrices split as S y ( u, y ) → S y | δ ( u, v ) := S y ( u, y δ ( v )) . Clearly, the y | δ magnons scatter on the momentum boundstates as S Q y | δ ( u, v ) = Q j S y | δ ( u j , v ) .Let us now focus on the magnonic bound states. Detailed investigation showed [53] that v and w particles can form bound states for any positive integer M . It consist of M v -particles v ± ( M +2 − j ) = v ± ( M + 2 − j ) ig for j = 1 , . . . , M with y j = ( y − − j ) ∗ , and M w -particles with synchronized parameters w j = v + ( M + 1 − j ) ig for j = 1 , . . . , M . The scattering matrix of the v | M particle with all otherparticles is simply the product of the scatterings of each of its individual constituents S v | M i ( v, q ) = M +1 Y j =1 S y |− i ( v M +2 − j , q ) M Y j S wi ( w j , q ) M − Y j =1 S y | + i ( v M − j , q ) . (4.36)The twist charge of the bound state simply sums up to µ v | M = 2 M ( − iγ + ) + M iγ + = 0 .The w -type particles can form bound states among themselves: an N -string of w -particles can beformed as w j = w + ( N + 1 − j ) ig . The scattering of the N -string with any other particle is S w | N i ( w, q ) = N Y j =1 S w i ( w j , q ) , (4.37)while the twist charge is µ w | N = 2 N iγ + .We summarize the various scattering matrices and chemical potentials in Table 3.Once we know the chemical potentials, we can calculate the kernels K jj ′ ( u, u ′ ) = 12 πi ∂ u log S jj ′ ( u, u ′ ) , (4.38)19 v | M w | N y | δ µQ S Q Q S Q v | M S Q y | δ v | M S v | M Q S v | M v | M S v | M y | δ w | N S w | N w | N S w | N y | δ N iγy | δ S y | δ Q S y | δ v | M S y | δ w | N − iγ Table 3: Scattering matrices of the various particles and their chemical potentials for any of the two su (2 | wings.and write the TBA equations one by one. To ensure positive particle densities, we have to invert theequations for v | M and for y |− . The equation for the massive nodes then read as ǫ Q = L ˜ ǫ Q − log(1+ e − ǫ Q ) ⋆K Q Q + X α = ± log(1+ e − ǫ αv | M ) ⋆K v | M Q − δ log(1+ e − ǫ αy | δ ) ⋆K y | δ Q . (4.39)Note that for particles of type v | M and y | δ , we must include contributions of the two su (2 | ) copies,which we denote by α = ± . The remaining equations are valid for the two su (2 | factors separately,so we omit the α index: ǫ v | M = − log(1 + e − ǫ Q ) ⋆ K Q v | M + log(1 + e − ǫ v | M ′ ) ⋆ K v | M ′ v | M − δ log(1 + e − ǫ y | δ ) ⋆ K y | δ v | M ,ǫ w | N = − µ w | N − log(1 + e − ǫ w | N ′ ) ⋆ K w | N ′ w | N − δ log(1 + e − ǫ y | δ ) ⋆ K y | δ w | N , (4.40) ǫ y | δ = iπ − µ y | δ − log(1 + e − ǫ Q ) ⋆ K Q y | δ + log(1 + e − ǫ v | M ) ⋆ K v | M y | δ − log(1 + e − ǫ w | N ) ⋆ K w | N y | δ . Once these equations are solved, the ground-state energy can be obtained as E ( L ) = − ∞ X Q =1 ˆ du π ∂ u ˜ p Q log(1 + e − ǫ Q ) . (4.41)In [19, 20] the authors analyzed the TBA equations with generic chemical potentials, and formu-lated the requirement under which the Y-system remains unchanged. Our chemical potentials, whichcorrespond to γ -deformations, satisfy their requirement. The TBA equations can usually be brought into a local form. As already remarked, this means thatthe pseudo-energies can be drawn on a two-dimensional lattice, such that only neighboring sites coupleto each other with the universal kernel s I MN = δ MN − ( K + 1) − MN , s ( u ) = g gπu , (4.42)where I MN = δ M +1 ,N + δ M − ,N and ( K + 1) − MN ⋆ ( K NL + δ NL ) = δ ML . To simplify the notation, weintroduce the following Y -functions Y Q = e − ǫ Q , Y v | M = e ǫ v | M , Y w | N = e ǫ w | N , Y δ = − e ǫ y | δ . (4.43)Clearly, we have two copies for the magnonic Y -functions: Y αv | M , Y αw | N , Y αδ where α = ± refers to thetwo su (2 | copies. Acting with the operator (4.42) on the TBA equations (4.39), (4.40), and usingkernel identities such as ( K + 1) − MN ⋆ K N = s δ M, as well as the special properties of the chemical To compare with [13, 14], we note that Y w | N = Y AFN | w , Y v | M = Y AFM | vw and K Q Q vx = K Q Q AFvwx . Also, K n ( u ) = πi ddu log S n ( u ) , where S n ( u ) is defined in (4.35); its Fourier transform is ˜ K n = sign( n ) t | n | , t = e − | ω | g . µ w | N − − µ w | N + µ w | N +1 = 0 and µ w | = − µ y , we arrive at their simplified form [14]. Forlater purposes, we write the simplified equations for v | M and w | N magnons, and a useful combination(hybrid) of the un-simplified equations for Q and y particles [16] log Y Q = − L ˜ ǫ Q + log(1 + Y Q ) ⋆ (cid:16) K Q Q sl (2) + 2 s ⋆ K Q − ,Q vx (cid:17) + X α = ± (cid:20) log (cid:18) Y αv | (cid:19) ⋆ s ˆ ⋆K yQ + log(1 + Y αv | Q − ) ⋆ s − log 1 − Y α − − Y α + ˆ ⋆s ⋆ K Q vx + 12 log 1 − Y α − − Y α + ˆ ⋆K Q + 12 log(1 − Y α − )(1 − Y α + )ˆ ⋆K yQ (cid:21) , (4.44) log Y α − Y α + = − log(1 + Y Q ) ⋆ K Q + 2 log(1 + Y Q ) ⋆ K Q xv ⋆ s + 2 log 1 + Y αv | Y αw | ⋆ s , (4.45) log Y α + Y α − = log(1 + Y Q ) ⋆ K Q y , (4.46) log Y αv | M = − log(1 + Y M +1 ) ⋆ s + I MN log(1 + Y αv | N ) ⋆ s + δ M log 1 − Y α − − Y α + ˆ ⋆s , (4.47) log Y αw | M = I MN log(1 + Y αw | N ) ⋆ s + δ M log 1 − Y α − − Y α + ˆ ⋆s , (4.48)where in the convolution ˆ ⋆ we integrate over the interval [ − , only. To conform with part of theliterature, we have renamed some kernels K MQvx = K v | M Q , K QMxv = K Q v | M , K yQ = K y |− Q + K y | + Q , K Qy = K Q y |− − K Q y | + . The ground-state energy is given by summing the contributions of the massivenodes only: E ( L ) = − ∞ X Q =1 ˆ du π ∂ u ˜ p Q log(1 + Y Q ) . (4.49)Evidently, as in the case of the O (4) model, the chemical potentials and so the twists completelydisappear from the simplified equations: They show up only in the asymptotics of the Y w | N functions,as lim N →∞ log Y w | N = − µ w | N . It follows that the Y -system relations are not modified by the twists,as was supposed in [36]. Equations (4.48)-(4.49) together with the asymptotic prescription give thecomplete solution for the finite-size energy of the twisted AdS/CFT model for any coupling g . We nowcheck this solution against LO and NLO Lüscher corrections. We now expand the simplified TBA equations to leading and next-to-leading order. We expand any Y -functions as Y = Y (1 + y + . . . ) . (4.50)We solve iteratively these equations similarly to the O (4) case: At leading order, all the massivenodes Y Q are exponentially small, which splits the Y -system into two independent subsystems whichhave constant asymptotic solutions. These constant values then determine the LO exponentially smallexpressions for Y Q . At NLO, we obtain linear integral equations for the y corrections of the twosubsystems, whose initial values are provided by the asymptotic Y Q functions. The solution of thelinearized equations determine the NLO correction for the massive nodes y Q , which provides the NLOenergy correction. At LO, the massive Y Q functions are exponentially small, and we can neglect the convolutions involvingall log(1 + Y Q ) . The magnonic Y α ± , Y αv | M , Y αw | N functions are constants. From (4.46), we see that21 α + = Y α − . It then follows from (4.47) and (4.48) that the equations for Y αv | M and Y αw | N are the sameas those for one of the wings of the O (4) model (3.27). From the asymptotic behavior, we see that thesolution for v | M is the same as in the undeformed model, while the solution for w | N is that of thedeformed model: Y αv | M = M ( M + 2) , Y + w | N = [ N ] q [ N + 2] q , Y − w | N = [ N ] ˙ q [ N + 2] ˙ q . (4.51)Since ⋆ s = , the equations (4.45) for Y α ± can be solved as Y α + = Y α − = s Y αv | Y αw | = 2[2] α , (4.52)where we have further streamlined the notation by defining [ n ] + = [ n ] q , [ n ] − = [ n ] ˙ q . (4.53)The sign in (4.52) can be fixed by the last equation in (4.40), and is consistent with the vanishing ofthe ground-state energy (4.56) in the undeformed ( q, ˙ q → ) limit. We now use that ⋆K yQ = 1 (see(6.12) in [13]) to write log Y Q = − L ˜ ǫ Q + 12 X α = ± (cid:20) log (cid:18) Y αv | (cid:19) + log(1 + Y αv | Q − ) + log (cid:18) − Y α − (cid:19)(cid:18) − Y α + (cid:19)(cid:21) . (4.54)Using the asymptotic solution (4.51), (4.52), we obtain the leading-order result for Y Q Y Q = (2 − [2] q )(2 − [2] ˙ q ) Q e − L ˜ ǫ Q (˜ p ) . (4.55)Substituting back into the energy formula (4.49), the LO correction reads as E ( L ) ≃ E (1)0 ( L ) = − ∞ X Q =1 ˆ d ˜ p π Y Q = − (2 − [2] q )(2 − [2] ˙ q ) ∞ X Q =1 Q ˆ d ˜ p π e − L ˜ ǫ Q (˜ p ) , (4.56)which agrees with the result (4.19) that we obtained from the Lüscher calculation. Expanding the energy formula (4.49) to NLO, we obtain E ( L ) = − ∞ X Q =1 ˆ d ˜ p π log(1 + Y Q ) ≃ − ∞ X Q =1 ˆ d ˜ p π Y Q (1 + y Q ) + ∞ X Q =1 ˆ d ˜ p π Y Q , (4.57)The quadratic term nicely reproduces our previous result (4.20) for E (2 , , since using again (4.55)gives E (2 , ( L ) = ∞ X Q =1 ˆ d ˜ p π Y Q = 12 (2 − [2] q ) (2 − [2] ˙ q ) ∞ X Q =1 Q ˆ d ˜ p π e − L ˜ ǫ Q (˜ p ) . (4.58)In order to evaluate E (2 , ( L ) = − ∞ X Q =1 ˆ d ˜ p π Y Q y Q , (4.59)22e must first calculate y Q . This will be given by the solution of the following linearized set of TBAequations: y Q = Y Q ⋆ (cid:16) K Q Q sl (2) + 2 s ⋆ K Q − ,Q vx (cid:17) + X α = ± h A αv | y αv | ⋆ s ˆ ⋆K yQ + A αv | Q − y αv | Q − ⋆ s − y α − − y α + − Y α + ˆ ⋆s ⋆ K Q vx + y α − − y α + Y α + − 1) ˆ ⋆K Q + y α − + y α + Y α + − 1) ˆ ⋆K yQ , (4.60) y α + + y α − = 2 (cid:16) A αv | y αv | − A αw | y αw | (cid:17) ⋆ s − Y Q ⋆ K Q + 2 Y Q ⋆ K Q xv ⋆ s , (4.61) y α + − y α − = Y Q ⋆ K Q y , (4.62) y αv | M = (cid:16) A αv | M − y αv | M − + A αv | M +1 y αv | M +1 (cid:17) ⋆ s − Y M +1 ⋆ s + δ M y α − − y α + − Y α + ˆ ⋆s , (4.63) y αw | N = (cid:16) A αw | N − y αw | N − + A αw | N +1 y αw | N +1 (cid:17) ⋆ s + δ N y α + − y α − − Y α + ˆ ⋆s , (4.64)where A αv | M = Y αv | M Y αv | M = M ( M + 2)( M + 1) , A αw | N = Y αw | N Y αw | N = [ N ] α [ N + 2] α [ N + 1] α . (4.65)We start with the equation (4.64) for y αw | N . The difference between α = + and α = − is only inthe asymptotics (4.51), (4.52). Since one equation can be obtained from the other by interchanging q ↔ ˙ q , we do not write out explicitly the α index. Replacing y + − y − in (4.64) with the contributionsfrom the massive nodes (4.62), and using the explicit form of the asymptotic solution, we obtain anequation similar to the one for the O (4) case: y w | N = (cid:18) [ N − N + 1][ N ] y w | N − + [ N + 1][ N + 3][ N + 2] y w | N +1 (cid:19) ⋆ s + δ N c ⋆ s , (4.66)where c = [2][2] − Y Q ⋆ ˆ K Qy , ˆ K Qy ( u, v ) = K Qy ( u, v ) (Θ( v + 2) − Θ( v − , (4.67)and Θ( v ) is the standard unit step function. We solve the difference equation in Fourier space. Weuse that ˜ s = (2 cosh ωg ) − = ( t + t − ) − where t ≡ e − | ω | g . The solution which decreases for large N (torespect the asymptotics of Y w | N ) and is compatible with the δ N, term is ˜ y w | N = ˜ c t [2] (cid:18) [ N + 1][ N ] t N − − [ N + 1][ N + 2] t N +1 (cid:19) . (4.68)We now analyze the equation (4.63) for y αv | M . This difference equation is not the same as for theundeformed O (4) model, as it has inhomogeneous terms, y v | M = (cid:18) ( M − M + 1) M y v | M − + ( M + 1)( M + 3)( M + 2) y v | M +1 (cid:19) ⋆ s − Y M +1 ⋆ s + δ M c ⋆ s , (4.69)where c = 2[2] − Y Q ⋆ ˆ K Qy . (4.70)Taking the Fourier transform, we obtain the difference equation ( t + t − )˜ y v | M = ( M − M + 1) M ˜ y v | M − + ( M + 1)( M + 3)( M + 2) ˜ y v | M +1 − ˜ Y M +1 + δ M ˜ c . (4.71) We note that in [17] there is an erroneous term in Eq. (2.7): − Y Q ⋆ s should be instead − Y Q ⋆ K Q , as in (4.61). A and A reads as ˜ y v | M = (cid:18) M + 1 M t M − − M + 1 M + 2 t M +1 (cid:19) A − M X k =1 ˜ Y k +1 t − k − (cid:0) t − k − k − (cid:1) ( t − − ( k + 1) ! + (cid:18) M + 1 M t − M − M + 1 M + 2 t − − M (cid:19) A − M X k =1 ˜ Y k +1 t k − (cid:0) t − ( k + 2) − k (cid:1) ( t − − ( k + 1) ! . (4.72)The parameters can be fixed from lim M →∞ ˜ y v | M = 0 and from the M = 1 term as A = t (cid:18) t − c − A (cid:19) , A = ∞ X k =1 ˜ Y k +1 t k − (cid:0) t − ( k + 2) − k (cid:1) ( t − − ( k + 1) . (4.73)The NLO hybrid equation for y Q is (4.60); we plug into it the equations (4.61) and (4.62), andobtain y Q = Y Q ⋆ (cid:16) K Q Q sl (2) + 2 s ⋆ K Q − ,Q vx (cid:17) + X α =1 , " A αv | − Y α + y αv | ⋆ s ˆ ⋆K yQ − A αw | Y α + − y αw | ⋆ s ˆ ⋆K yQ + Y Q ⋆ K Q xv Y α + − ⋆ s ˆ ⋆K yQ + Y Q ⋆ K Q y − Y α ± ˆ ⋆s ⋆ K Q vx − Y Q ⋆ K Q y Y α ± − 1) ˆ ⋆K Q − Y Q ⋆ K Q Y α + − 1) ˆ ⋆K yQ + A αv | Q − y αv | Q − ⋆ s i , (4.74)Since y v | and y w | can be expressed in terms of Y Q , we see that the solution for y Q has the generalform y Q = Y Q ⋆ K Q Q sl (2) + Y Q ⋆ M Q Q . (4.75)Consider the first term Y Q ⋆ K Q Q sl (2) . It is easy to see that its contribution to the integrand in theenergy formula (4.59) Y Q y Q = Y Q ( Y Q ⋆ K Q Q sl (2) ) , (4.76)with Y Q given by (4.55), matches with the “scalar part” of the integrand of the Lüscher correction E (2 , in (4.24). We now proceed to analyze the remaining contribution in (4.75), and show that itgives the “matrix part” of the integrand of the Lüscher correction. Q = Q = 1 To warm up, let us evaluate the NLO correction for the Q = Q = 1 case; thus, we calculate M .In so doing, we can freely put Y Q = 0 for Q > . The corresponding solutions read as y w | = [2][2] − Y ⋆ K y ˆ ⋆ (cid:18) K − K (cid:19) , (4.77) y v | = 2[2] − Y ⋆ K y ˆ ⋆ (cid:18) K − K (cid:19) ,y + − y − = Y ⋆ K y ,y + + y − = 2[2] − Y ⋆ K y ˆ ⋆ (cid:18) (cid:18) − [3][2] (cid:19) K − (cid:18) − (cid:19) K (cid:19) ⋆ s − Y ⋆ K + 2 Y ⋆ K xv ⋆ s . It is convenient to substitute these solutions directly into (4.60), i.e., y = Y ⋆ K sl (2) + X α = ± " A v | y αv | ⋆ s ˆ ⋆K y − y α − − y α + − Y α + ˆ ⋆s ⋆ K vx + y α − − y α + Y α + − 1) ˆ ⋆K + y α − + y α + Y α + − 1) ˆ ⋆K y . (4.78)24sing the explicit form of the asymptotic solutions, one can see that the terms involving the convolutionwith K completely cancel. Exploiting further that K y ˆ ⋆K = K xv (which can be shown using relationsfrom Sec. 6 in [13]), we arrive at y = Y ⋆ K sl (2) + X α = ± (cid:20) [2] α α − (cid:0) Y ⋆ K y ˆ ⋆K + Y ⋆ K ˆ ⋆K y − Y ⋆ K xv ⋆ s ˆ ⋆K y (cid:1) + [3] α − α − Y ⋆ K xv ⋆ s ˆ ⋆K y − α − Y ⋆ K y ˆ ⋆s ⋆ K vx (cid:21) . (4.79)This expression further simplifies to y = Y ⋆K sl (2) + X α = ± (cid:20) [2] α α − Y ⋆ ( K y ˆ ⋆K + K ˆ ⋆K y )+ 2[2] α − Y ⋆ (cid:0) K xv ⋆ s ˆ ⋆K y − K y ˆ ⋆s ⋆ K vx (cid:1)(cid:21) . (4.80)In the second term, using K y ˆ ⋆K = K xv and K vx = K ˆ ⋆K y , we can write K xv ⋆ s ˆ ⋆K y − K y ˆ ⋆s ⋆ K vx = K y ˆ ⋆ ( K ⋆ s − s ⋆ K )ˆ ⋆K y = 0 , (4.81)as both s and K depend on the differences of their arguments, and therefore their convolution iscommutative. In the previous term in (4.80), we can obtain K y ˆ ⋆K + K ˆ ⋆K y = 12 πi ∂ u log (cid:18) x − − x +2 x − − /x +2 x +1 − /x − x +1 − x − u − u − i/gu − u + 2 i/g (cid:19) = 1 πi ∂ u log x − − x +2 x +1 − x − s x +1 x − ! = − πi ∂ u log a ( u , u ) , (4.82)where we have used identities from Sec. 6 in [13] and Eq. (3.7) in [14], and we have recalled thedefinition in (4.1) of a . The final expression for the Q = Q ′ = 1 contribution to the energy (4.59) istherefore given by y = Y ⋆ K sl (2) + Y ⋆ M , M = 12 πi ∂ u log a ( u , u ) X α [2] α − [2] α , (4.83)which completely reproduces the result (4.27) obtained directly from the Lüscher correction. Q , Q We now consider the general case. Let us recall the result (4.74) for y Q y Q = Y Q ⋆ (cid:16) K Q Q sl (2) + 2 s ⋆ K Q − ,Q vx (cid:17) + X α =1 , " A αv | − Y α + y αv | ⋆ s ˆ ⋆K yQ − A αw | Y α + − y αw | ⋆ s ˆ ⋆K yQ + Y Q ⋆ K Q xv Y α + − ⋆ s ˆ ⋆K yQ + Y Q ⋆ K Q y − Y α ± ˆ ⋆s ⋆ K Q vx − Y Q ⋆ K Q y Y α ± − 1) ˆ ⋆K Q − Y Q ⋆ K Q Y α + − 1) ˆ ⋆K yQ + A αv | Q − y αv | Q − ⋆ s i , (4.84)and analyze it term by term. Since we have already checked in Section (4.4.2) the matching of thefirst term with the scalar part of the Lüscher result, we start by considering the second term of (4.84),which can be rewritten as Y Q ⋆ s ⋆ K Q − ,Q vx = 2 Y Q ⋆ K Q − ⋆ s ˆ ⋆K yQ + 2 Y Q ⋆ Q − X j =0 K Q − Q +2 j +1 ⋆ s , (4.85)25here we used the property s ⋆ K Q = K Q ⋆ s , valid for any Q . Now we consider the terms in the squarebrackets of (4.84), again suppressing the index α . Using the solution (4.72) for M = 1 and taking itsinverse Fourier transform, we can express the first term as A v | − Y + y v | ⋆ s ˆ ⋆K yQ = 32 − [2] ( Y Q ⋆ K Q y − [2] ˆ ⋆ (cid:18) K − K (cid:19) + 13 Y Q Q ⋆ [( Q − K Q +1 − ( Q + 1) K Q − ] ) ⋆ s ˆ ⋆K yQ , (4.86)where the term in the second line can be rewritten, by using the identity ( K n +1 + K n − + nδ n, ± δ ) ⋆s = K n , as − − [2] Y Q ⋆ K Q − ⋆ s ˆ ⋆K yQ + Q − Q (2 − [2]) Y Q ⋆ K Q ˆ ⋆K yQ . The K contribution in the first line of (4.86) cancels, as we have already seen in the Q = Q = 1 case, with the successive term in (4.84) − A w | Y + − y w | ⋆ s ˆ ⋆K yQ = − − [2]) Y Q ⋆ K Q y ˆ ⋆ ( K − [3] K ) ⋆ s ˆ ⋆K yQ , (4.87)while the terms with K give [3] − − [2]) Y Q ⋆ K Q y ˆ ⋆K ⋆ s ˆ ⋆K yQ . (4.88)Summing this contribution to the first two terms in the second line of (4.84), we obtain − [2] ( Y Q ⋆ K Q y ˆ ⋆s ⋆ K ˆ ⋆K yQ − Y Q ⋆ K Q y ˆ ⋆K ⋆ s ˆ ⋆K yQ )+ [2]2 − [2] Y Q ⋆ K Q − ⋆ s ˆ ⋆K yQ + 22 − [2] Y Q ⋆ K Q y ˆ ⋆s ⋆ K Q − , (4.89)where we used the identities K Qvx = K ˆ ⋆K yQ + K Q − and K Q xv = K Qy ˆ ⋆K + K Q − . As alreadynoticed for the case Q = Q = 1 , the first line in the expression above vanishes because K ⋆s = s⋆K .The successive two terms in the second line of (4.84) give − [2]2 π (2 − [2]) Y Q ⋆ i∂ u log a Q Q ( u , u ) , (4.90)where we used the identity (4.82) generalized for any Q , Q , K Q y ˆ ⋆K Q + K Q ˆ ⋆K yQ = 1 πi ∂ u log x − Q − x + Q x + Q − x − Q s x + Q x − Q ! = − πi ∂ u log a Q Q ( u , u ) . (4.91)Let us turn to the last and most complicated term. Using the inverse Fourier transform of (4.72) for M = Q − , we can write it as follows A v | Q − y v | Q − ⋆ s = Y Q Q Q ⋆ Q − X k =0 k ( k − Q ) [( Q + 1) K Q − Q +2 k − − ( Q − K Q − Q +2 k +1 ] ⋆ s + Y Q ⋆ K Q y Q (2 − [2]) ˆ ⋆ [( Q − K Q +1 − ( Q + 1) K Q − ] ⋆ s , (4.92) The latter identity is reported in footnote 4 of [16]; the former can be derived analogously using equations (6.19)and (6.39) in [13]. The same equations, together with (6.14), can also be used to obtain (4.85). − − [2] Y Q ⋆ K Q y ˆ ⋆s ⋆ K Q − + Q − Q (2 − [2]) Y Q ⋆ K Q y ˆ ⋆K Q . (4.93)Now, taking into account that in summing over α the first term in (4.92) gets a factor 2 and the otherterms get similar coefficients with q → ˙ q , we can sum all the contributions above to get the solutionfor y Q for generic values of Q , Q : y Q = Y Q ⋆ ( K Q Q sl (2) + 2 Q − X j =0 K Q − Q +2 j +1 ⋆ s − ∂ u log a Q Q ( u , u ) πi − X α = ± " ∂ u log a Q Q ( u , u )2 πiQ (2 − [2] α ) + ∂ u log a Q Q ( u , u ) ⋆ πiQ (2 − [2] α ) + 2 Q Q Q − X k =0 k ( k − Q ) [( Q + 1) K Q − Q +2 k − − ( Q − K Q − Q +2 k +1 ] ⋆ s ) , (4.94)where we used the following identity K Q y ˆ ∗ K Q = 12 πi ∂ u log x − Q − x + Q x + Q − x − Q x + Q − /x + Q x − Q − /x − Q ! ≡ πi ∂ u log a Q Q ( u , u ) , (4.95)its hermitian conjugate (recall that x ( u ) ∗ = 1 /x ( u ∗ ) in the mirror kinematics) K Q ˆ ∗ K yQ = 12 πi ∂ u log x − Q − x + Q x + Q − x − Q x − Q x − Q − x + Q x + Q − x + Q x − Q x − Q x + Q ! = 12 πi ∂ u log a Q Q ( u , u ) ∗ , (4.96)and a Q Q ( u , u ) a Q Q ( u , u ) ∗ = h a Q Q ( u , u ) i − . Moreover, we can write the sum of the twoconvolutions involving the universal kernel s ( u ) in (4.94) as Q Q Q − X k =0 k ( k − Q ) [( Q + 1) K Q − Q +2 k − − ( Q − K Q − Q +2 k +1 ] ⋆ s + Q − X j =0 K Q − Q +2 j +1 ⋆ s = 12 πiQ Q ∂ u K Q Q , (4.97)where we used the definition (4.31) of K Q Q . Remarkably, despite the long computation, the finalexpression for y Q is quite simple and reads y Q = Y Q ⋆ πi ∂ u ( log S Q Q sl (2) + 2 Q Q K Q Q − a Q Q ( u , u ) − X α = ± − [2] α ) (cid:20) Q log a Q Q ( u , u ) + 1 Q log a Q Q ( u , u ) ⋆ (cid:21) ) . (4.98) Actually, identities (4.82), (4.91), and (4.96) are valid up to vanishing derivatives ∂ u log r x − x +2 , ∂ u log s x − Q x + Q and ∂ u log x + Q x − Q , respectively. Y Q , into the formula (4.59) for the energycorrection, we obtain E (2 , = ∞ X Q ,Q =1 Q Q ˆ ∞−∞ d ˜ p π e − L ˜ ǫ Q (˜ p ) ˆ ∞−∞ d ˜ p π e − L ˜ ǫ Q (˜ p ) × i∂ ˜ p ( (2 − [2] ˙ q ) h [3] q (cid:16) − Q Q log a Q Q + K Q Q (cid:17) − [2] q (cid:16) − Q Q log a Q Q − Q log a Q Q − Q log a Q Q ∗ + 4 K Q Q (cid:17) + [1] q (cid:16) − Q Q log a Q Q − Q log a Q Q − Q log a Q Q ∗ + 5 K Q Q (cid:17)i +( q ↔ ˙ q )+ Q Q (2 − [2] q ) (2 − [2] ˙ q ) log S Q Q sl (2) (˜ p , ˜ p ) ) . (4.99)Finally, through the following identifications a Q Q = ( U U U ) − , a Q Q = U U U , a Q Q ∗ = U U U − , (4.100)we find full agreement with the result (4.32) from the Lüscher computation. In this section we calculate the weak-coupling expansion of the ground-state energy of the twistedAdS/CFT model. In order to perform the weak-coupling expansion, we use the parameterization x ± (˜ p ) = (˜ p − iQ )2 g s g Q + ˜ p ∓ ! , (5.1)which follows from (4.10) and (4.18). At leading order in g , and so at weak coupling, we have x − = ˜ p − iQg + O ( g ) , x + = g ˜ p + iQ + O ( g ) . (5.2) The LO correction can be calculated from (4.56) by using the expansion of the exponential termappearing in Y Q : e − L ˜ ǫ Q (˜ p ) = ∞ X j =0 c j g L + j ) (˜ p + Q ) L + j . (5.3)In particular c = 1 , while the higher-order terms can be easily generated with Mathematica. Usingthe fact that n ! f ( n ) ( z ) = ¸ dw πi f ( w )( w − z ) n +1 , we perform the integral in (4.56) by residues ˆ ∞−∞ d ˜ p π p + Q ) k = (cid:18) k − k − (cid:19) (2 Q ) − k . (5.4)The summation over Q gives rise to a series of ζ -functions: E (1)0 ( L ) = − (2 − [2] q )(2 − [2] ˙ q ) ∞ X j =0 c j − L + j ) (cid:18) L + j ) − L + j − (cid:19) ζ L + j ) − g L + j ) . (5.5)This result is exact up to g L where the NLO Lüscher correction starts to play a role. We evaluatethe leading g L -order contribution of the NLO Lüscher correction in the next subsection.28 .2 NLO contribution, double wrapping The simplest term of the NLO correction comes from (4.58) and contains Y Q . Its contribution at order g L can be calculated using eq. (5.4) to be E (2 , ( L ) = (2 − [2] q ) (2 − [2] ˙ q ) − L (cid:18) L − L − (cid:19) ζ L − g L . (5.6)The most complicated term is E (2 , ( L ) . We have to evaluate (4.59) based on the solution given in(4.98). The twist dependence comes in two distinct ways as: E (2 , ( L ) = (2 − [2] q ) (2 − [2] ˙ q ) (cid:20) A ( L ) + B ( L ) (cid:18) q − ˙ q − (cid:19)(cid:21) g L . (5.7)We first calculate B ( L ) for any value of L . The weak-coupling expansion of the functions a Q Q and a Q Q ∗ are given by ∂ ˜ p log a Q Q (˜ p , ˜ p ) = O ( g ) , ∂ ˜ p log a Q Q (˜ p , ˜ p ) ∗ = 2 iQ ˜ p + Q + O ( g ) . (5.8)We substitute these results into (4.98) and then into (4.59), we perform the integrals as in (5.4), andsum up the independent terms to obtain: B ( L ) = − − L (cid:18) L − L − (cid:19) (cid:18) LL (cid:19) ζ L − ζ L − . (5.9)This gives the complete answer for the given (2 − [2] q )(2 − [2] ˙ q )(4 − [2] q − [2] ˙ q ) dependence of thedouble-wrapping correction at leading nonvanishing order for any L .We now proceed to calculate A ( L ) . It acquires contributions from the first line of (4.98), which wedenote by A sl (2) , A K and A , respectively, A ( L ) = A sl (2) ( L ) + A K ( L ) + A ( L ) , (5.10)where A sl (2) ( L ) = X Q ,Q Q Q ˆ d ˜ p π e − Lǫ Q (˜ p ) ˆ d ˜ p π e − Lǫ Q (˜ p ) i∂ ˜ p log S Q Q sl (2) (˜ p , ˜ p ) , (5.11) A K ( L ) = 2 X Q ,Q Q Q ˆ d ˜ p π e − Lǫ Q (˜ p ) ˆ d ˜ p π e − Lǫ Q (˜ p ) i∂ ˜ p K Q Q (˜ p , ˜ p ) , (5.12) A ( L ) = − X Q ,Q Q Q ˆ d ˜ p π e − Lǫ Q (˜ p ) ˆ d ˜ p π e − Lǫ Q (˜ p ) i∂ ˜ p log a Q Q (˜ p , ˜ p ) . (5.13)In order to compute A , we expand a Q Q to leading order in g : ∂ ˜ p log a Q Q (˜ p , ˜ p ) = − iQ ˜ p + Q + O ( g ) . (5.14)Substituting the result back into (5.13) gives A ( L ) = − − L (cid:18) L − L − (cid:19) (cid:18) LL (cid:19) ζ L − ζ L − . (5.15)Observe that the transcendentality of A ( L ) and B ( L ) are different. It seems the deformation − [2] carries transcendentality . A similar effect was observed already in [54, 38].29o calculate A sl (2) , we have to expand the logarithm of the dressing factor log S Q Q sl (2) (˜ p , ˜ p ) in themirror-mirror kinematics. According to [13], it has the structure log S Q Q sl (2) (˜ p , ˜ p ) = − log S Q Q su (2) (˜ p , ˜ p ) − Q Q (˜ p , ˜ p ) . Hence, we can write πi ∂ ˜ p log S Q Q sl (2) (˜ p , ˜ p ) = − K Q Q − πi ∂ ˜ p log Σ Q Q (˜ p , ˜ p ) . (5.16)Explicitly performing the weak-coupling expansion of (6.14) in [51], we obtain (see (B.4)) i∂ ˜ p log Σ Q Q (˜ p , ˜ p ) = − (cid:20) ψ (1 − i p + iQ )) − ψ (1 + 12 ( i (˜ p − ˜ p ) + Q + Q )) + c.c (cid:21) , (5.17)where ψ ( x ) = ∂ x (log Γ( x )) is the polygamma function. The su (2) scalar factor results in K Q Q su (2) = K Q Q = − π (cid:20) ψ ( 12 ( i (˜ p − ˜ p ) − Q + Q )) + ψ (1 + 12 ( i (˜ p − ˜ p ) − Q + Q )) (5.18) − ψ ( 12 ( i (˜ p − ˜ p ) + Q + Q )) − ψ (1 + 12 ( i (˜ p − ˜ p ) + Q + Q )) + c.c (cid:21) . Finally, i∂ ˜ p K Q Q = − (cid:20) Q − Q + (( Q − Q ) + (˜ p − ˜ p ) ) × (5.19) (cid:0) ψ (1 + 12 ( i (˜ p − ˜ p ) − Q + Q )) − ψ ( 12 ( i (˜ p − ˜ p ) + Q + Q )) (cid:1) + c.c. (cid:21) . Denoting the contributions to A sl (2) by A Σ and A su (2) , we have that A sl (2) ( L ) = A Σ ( L ) + A su (2) ( L ) , (5.20)where A Σ ( L ) = − X Q ,Q Q Q ˆ d ˜ p π e − Lǫ Q (˜ p ) ˆ d ˜ p π e − Lǫ Q (˜ p ) i∂ ˜ p log Σ Q Q (˜ p , ˜ p ) , (5.21) A su (2) ( L ) = X Q ,Q Q Q ˆ d ˜ p π e − Lǫ Q (˜ p ) ˆ d ˜ p π e − Lǫ Q (˜ p ) πK Q Q (˜ p , ˜ p ) . (5.22)Using methods explained in Appendix B, we evaluated the integrals by residues. To demonstratethe structure of the result, we write out explicitly A Σ (see (B.10) and (B.13)): A Σ ( L ) = − − L (cid:18) L − L − (cid:19) ζ L − X Q L − X j =0 (cid:18) L + j − j (cid:19) − L +1 ( L − − j )! ( − Q ) − L − j ψ ( L − j − ( Q + 1) − X Q ,Q L − X j ,j =0 (cid:18) L + j − j (cid:19) − L +2 ( L − − j )! ( − Q ) − L − j × (cid:18) L + j − j (cid:19) − L +1 ( L − − j )! ( − Q ) − L − j ψ (2 L − j − j − ( Q + Q + 1) . (5.23)These terms can be expressed in terms of multiple zeta values (MZV) by rewriting ψ ( n ) ( Q + 1) = ( − n +1 n !( ζ ( n + 1) − Q X j =1 j − n − ) , (5.24) For n = 0 , one has to replace ζ (1) with γ E . L . The integrals can be evaluated similarly for A su (2) and A K with a similar structural final result, although some care must be taken to the Q − Q dependentterm for Q = Q . In the next subsection, we present explicit results for the smallest nontrivial length: L = 3 . L = 3 The LO wrapping correction (5.5) for L = 2 is divergent, as we have for j = 0 the term ζ L − = ζ .Similar observations were made in [55, 38]. We therefore focus now on the first nontrivial case, namely L = 3 . The LO correction for this case goes as follows: E (1)0 (3) = − (2 − [2] q )(2 − [2] ˙ q ) (cid:18) ζ g − ζ g + 945256 ζ g − ζ g + . . . (cid:19) . (5.25)The simple double-wrapping contribution (5.6) at leading order is E (2 , (3) = (2 − [2] q ) (2 − [2] ˙ q ) ζ g . (5.26)In calculating the term E (2 , (3) , we recall from (5.7) that E (2 , (3) = (2 − [2] q ) (2 − [2] ˙ q ) (cid:20) A (3) + B (3) (cid:18) q − ˙ q − (cid:19)(cid:21) g . (5.27)From (5.9), we have B (3) = − ζ ζ . (5.28)We calculated the contributions to A (3) one by one. The simplest is A (3) = − ζ ζ , (5.29)as follows from (5.15). In the more complicated terms, we calculated the integrals by residues asexplained in Appendix B. Then, in summing up the expressions, we employed the following strategies: • We performed the sums analytically by replacing the polygamma functions with harmonic sumsusing (5.24), and then rearranging all the sums into MZVs. These MZVs could then be expressedin terms of elementary ones, which contained only products of simple zetas with transcendentalityless than or equal to . • Alternatively, for terms involving polygamma functions depending on Q + Q , we replaced thepolygamma functions with their integral representations ψ ( n ) ( z ) = ˆ ∞ (cid:18) δ n, e − t t − ( − n t n e − tz − e − t (cid:19) dt , (5.30)and performed the summations P ∞ Q ,Q =1 explicitly. The remaining integral over t could beevaluated numerically with very high precision (100 digits), and the result could be expressed interms of products of zeta functions (and the Euler constant γ E ) with the help of the online MZVcalculator, EZ-Face. • Finally, for polygamma functions depending on Q − Q , we evaluated the sums numericallyas P ∞ Q ,Q =1 = 2 P ∞ Q P Q − Q =1 + P ∞ ( Q = Q )=1 , and again expressed the result in terms of zetafunctions using EZ-Face. EZ-Face is documented in [56], and can be accessed at http://oldweb.cecm.sfu.ca/projects/EZFace/index.html 31e found the following results: A Σ (3) = 811024 ζ ζ + 21512 ζ ζ − ζ ,A su (2) (3) = − ζ ζ + 3154096 ζ ,A K (3) = − ζ − ζ ζ − ζ ζ + 63512 ζ . (5.31)By summing up, we obtain the total A contribution A (3) = A Σ (3) + A su (2) (3) + A K (3) + A (3) = − ζ − ζ . (5.32)Thus, the total anomalous dimension is E (3) = E (1)0 (3) + E (2 , (3) + E (2 , (3) + . . . = − (2 − [2] q )(2 − [2] ˙ q ) (cid:18) ζ g − ζ g + 945256 ζ g − ζ g + . . . (cid:19) − (2 − [2] q )(2 − [2] ˙ q ) ([2] q + [2] ˙ q − ζ ζ g + . . . +(2 − [2] q ) (2 − [2] ˙ q ) (cid:18) − ζ + 1894096 ζ (cid:19) g + . . . , (5.33)where we recall that − [2] q = 4 sin( γ + ) and − [2] ˙ q = 4 sin( γ − ) in terms of the deformationparameters γ ± = ( γ ± γ ) , as in our case L = 3 .The result (5.33) is indeed the total anomalous dimension, since the vacuum energy does not receiveany contributions from the asymptotic Bethe ansatz. Remarkably, even though at intermediate stagesof the computation there appear terms involving even zeta functions and Euler’s constant γ E , all suchterms finally cancel. We have computed the NLO finite-volume correction to the vacuum energy in twisted AdS/CFTby two apparently independent approaches: Lüscher (4.32) and TBA (4.99). The fact that bothapproaches yield identical results provides a strong consistency check on the AdS/CFT S-matrices andTBA equations that have been developed in the literature, as well as on the final result. This result isexpressed in terms of a double infinite sum of contributions from the infinitely-many types of massivemirror bound states. Our computations check the complete (both horizontal and vertical parts of the) Y -system, and go beyond the five-loop calculations presented in [17, 57, 58], which checked at thesingle wrapping order only the vertical part.Our result is valid for any value of the coupling constant. However, by making a weak-couplingexpansion, we have obtained a prediction (5.33) for the anomalous dimension of the operator Tr Z in the twisted gauge theory up to six loops. It should be possible to check this prediction directly inperturbation theory by taking into account both single-wrapping and double-wrapping diagrams. Toour knowledge, this is the first complete computation of double wrapping in the literature. It may beinteresting to investigate also the strong-coupling limit.The key results needed for the NLO Lüscher computation were the determinants of the (untwisted)AdS/CFT S-matrices in all the su (2) L ⊗ su (2) R sectors, presented in Tables 1 and 2. The simplicity ofthese results suggests that they may have some group-theoretical formulation. In particular, it shouldbe possible to find a general proof, presumably based on su (2 | Yangian symmetry.It would be interesting to extend our analysis of finite-size corrections in twisted AdS/CFT, whichhas so far been restricted to the ground state, to excited states beyond the LO result of [38]. It would32lso be interesting to understand the origin of the divergence of the LO and NLO results for L = 2 ,which was already noticed in similar contexts in [55, 38]. Finally, one can now begin to contemplatetriple and higher wrapping. Acknowledgments We thank Orlando Alvarez, Gleb Arutyunov, János Balog, Sergey Frolov, Árpád Hegedűs, Mariusde Leeuw, Christoph Sieg and Stijn van Tongeren for useful discussions and/or correspondence; andthe referees for their valuable comments. CA, DB and RN are grateful for the warm hospitalityextended to them at ELTE and at the Perimeter Institute during the course of this work. This workwas supported in part by WCU Grant No. R32-2008-000-101300 (CA), OTKA 81461 (ZB), the FCTfellowship SFRH/BPD/69813/2010 and the network UNIFY for travel financial support (DB), and bythe National Science Foundation under Grant PHY-0854366 and a Cooper fellowship (RN). A Determinants of S-matrices in the su (2) L ⊗ su (2) R sectors We describe here how we obtained the results in Tables 1 and 2 for det S ( Q ,Q ) ( s L , s R ) , the determi-nants of the AdS/CFT S-matrices in the su (2) L ⊗ su (2) R sectors, which enter into the NLO Lüschercomputation. Our straightforward approach was to explicitly compute these determinants for smallvalues of Q and Q (up to 8), and then infer the general pattern.For the cases ( Q , Q ) = (1 , , (1 , , (2 , , we used the explicit S-matrices from [52] to directlycompute the eigenvalues. For the cases ( Q , Q ) = (1 , Q ) , we used results from [59]: from Eq. (56)there, it follows that (up to the overall factors), det S (1 ,Q ) ( Q − , 1) = a , det S (1 ,Q ) ( Q , 12 ) = 1 Q det (cid:18) a a a a (cid:19) , det S (1 ,Q ) ( Q − , 12 ) = Q det (cid:18) a a a a (cid:19) , det S (1 ,Q ) ( Q + 12 , 0) = a = 1 , det S (1 ,Q ) ( Q − , 0) = Q + 1 Q − a a a a a a a a a , det S (1 ,Q ) ( Q − , 0) = Q − a . (A.1)One can verify using the explicit values of a ji that a = 1 Q det (cid:18) a a a a (cid:19) = U U U ,Q det (cid:18) a a a a (cid:19) = Q + 1 Q − a a a a a a a a a = U U U U ,Q − a = U U U , (A.2) We note a couple of typos in appendix B of [59]: a should not have the factor x + z + in the denominator; and a ismissing an overall minus sign. ( Q , Q ) , we made use of the formalism developed in [49]. As an example,let us consider the case ( Q , Q ) = (2 , . Since the state of a single Q -particle (the Q -dimensionaltotally symmetric representation of su (2 | ) has the su (2) L ⊗ su (2) R decomposition V Q × V + V Q − × V + V Q − × V , (A.3)the decomposition of the corresponding 2-particle states can be obtained from the tensor product (cid:16) V × V + V × V + V × V (cid:17) ⊗ (cid:16) V × V + V × V + V × V (cid:17) , (A.4)where in this appendix we denote by × the tensor product of the su (2) L and su (2) R representations.For concreteness, let us focus on the computation of det S (2 , (1 , ) . The tensor product in (A.4)can be decomposed, by the Clebsch-Gordan theorem, into a sum of irreducible representations of su (2) L ⊗ su (2) R . In this decomposition, there appear four representations with ( s L , s R ) = (1 , ) ,which are the relevant ones for computing this determinant. These four representations come from thefollowing channels: (cid:0) V × V (cid:1) ⊗ (cid:16) V × V (cid:17) (cid:16) V × V (cid:17) ⊗ (cid:16) V × V (cid:17) (cid:0) V × V (cid:1) ⊗ (cid:16) V × V (cid:17) (cid:16) V × V (cid:17) ⊗ (cid:16) V × V (cid:17) . (A.5)The corresponding highest-weight states | ψ ( Q ,Q ) I ( s L , s R ) i with s L = m L = 1 and s R = m R = aregiven (up to an overall normalization factor) by | ψ (2 , (1 , 12 ) i ∝ | , i II2 − | , i II2 , | ψ (2 , (1 , 12 ) i ∝ | , i II3 , | ψ (2 , (1 , 12 ) i ∝ | , i II4 , | ψ (2 , (1 , 12 ) i ∝ | , i II1 − | , i II1 , (A.6)respectively, where the states | k, l i II i are defined in [49]. It is convenient to introduce a basis | e i i ofthese so-called type-II states with N ≡ k + l = 1 : | e i = | , i II1 , | e i = | , i II2 , | e i = | , i II3 , | e i = | , i II1 , | e i = | , i II2 , | e i = | , i II4 . (A.7)Although these states are orthogonal, they are not normalized. Indeed, defining n i ≡ h e i | e i i , (A.8) We are grateful to M. de Leeuw for pointing this out to us. 34t readily follows from the definitions of the states [49] that here n i = (2 , , , , , . An orthonormalbasis | ˜ e i i is therefore given by | ˜ e i i ≡ √ n i | e i i , h ˜ e i | ˜ e j i = δ ij . (A.9)The S-matrix acts as S | e i i = X j | e j i U ji . (A.10)Numerical values for the coefficients U ji can be computed using formulas in [49], for given numericalvalues of momenta p , p , coupling constant g , and representations Q , Q . Hence, we can obtainthe corresponding coefficients ˜ U ji in the normalized basis ˜ U ji ≡ h ˜ e j | S | ˜ e i i = r n j n i U ji = X k,l M jk U kl M − li , (A.11)where we have introduced the diagonal matrix M ij ≡ √ n i δ ij . A useful check is that the matrix ˜ U ji (unlike U ji ) is unitary.We use (A.6) to express the highest-weight states | ψ ( Q ,Q ) I ( s L , s R ) i in terms of the normalizedbasis | ψ ( Q ,Q ) I ( s L , s R ) i = X i | ˜ e i i c iI , c iI ≡ h ˜ e i | ψ ( Q ,Q ) I ( s L , s R ) i , (A.12)where the states themselves are normalized, h ψ ( Q ,Q ) I ( s L , s R ) | ψ ( Q ,Q ) J ( s L , s R ) i = δ IJ . (A.13)We can finally construct the S -matrix in the ( s L , s R ) sector, S ( Q ,Q ) IJ ( s L , s R ) ≡ h ψ ( Q ,Q ) I ( s L , s R ) | S | ψ ( Q ,Q ) J ( s L , s R ) i = X i,j c ∗ iI ˜ U ij c jJ . (A.14)Another useful check is that the matrix S ( Q ,Q ) IJ ( s L , s R ) is also unitary. Computing numerically thedeterminant of this matrix det S ( Q ,Q ) ( s L , s R ) ≡ det (cid:16) S ( Q ,Q ) IJ ( s L , s R ) (cid:17) , (A.15)we find for the case in question (namely, ( Q , Q ) = (2 , and ( s L , s R ) = (1 , ) ) that the resultcoincides with ( U U U ) S , in agreement with Table 1. Other cases ( Q , Q ) and other sectors ( s L , s R ) can be treated in a similar way. Note that sectors with s R = 1 , , are constructed withstates of type I, II, III, respectively. After some effort to accumulate results for sufficiently many cases,the general pattern summarized in Tables 1 and 2 became evident.Before closing this section, it may be worthwhile to frame the problem that we have addressedhere in a more general context. Consider an S-matrix (solution of the Yang-Baxter equation) thatis invariant under a group G , which here is su (2) L ⊗ su (2) R . As is well known (see e.g. [60, 61]), We note that version 1 in the arXiv of [49] contains a number of typos, most of which are corrected in the journal.However, some typos remain in the latter. In particular, in (5.14): ¯ Q ij = b i d j − b j d i . Also, in A − in (5.17): in the (2,2)element of the big matrix, c − should be instead c +1 ; and in the (2,1) matrix element, the sign in front of [ M +( l − l ) / should be plus instead of minus. Finally, in (A.8), the formulas for b , . . . , b should have sign plus instead of minus; andthe formulas for d and d should not have i in the denominator. We are grateful to G. Arutyunov and M. de Leeuw forcorrespondence on these points. We use the convention that the determinant of a number (i.e., a × matrix) is the number itself. S ab defined in the tensor product of two vector spaces V a ⊗ V b in whichrepresentations Π a and Π b of G act, [Π a ( g ) ⊗ Π b ( g ) , S ab ] = 0 , g ∈ G . (A.16)The representation space decomposes into a sum of irreducible representations of G parameterized byhighest weights Λ k , which here are ( s L , s R ) , V a ⊗ V b = X k V (Λ k ) . (A.17)Since the S-matrix is G -invariant (A.16), it has the corresponding spectral resolution S ab = X k ρ k P Λ k , (A.18)where P Λ k is a projector onto the irreducible subspace V (Λ k ) .In the seminal work [60] on the construction of rational S -matrices, it was essential to assume thatthe Clebsch-Gordan series (A.17) is multiplicity free (i.e., a given irreducible representation appears atmost once), in which case ρ k in (A.18) is a scalar. For AdS/CFT, the decomposition (A.17) is unfor-tunately not multiplicity free: the Clebsch-Gordan series contains multiple irreducible representations,as we have seen in the example (A.5). Hence, ρ k becomes an r × r matrix, where r is the multiplicityof the corresponding irreducible representation with highest weight Λ k . In the AdS/CFT case, ρ k isthe matrix that we have defined in (A.14). The problem of explicitly determining this matrix can bequite complicated even for rational S -matrices, see e.g. [61]. In the present work, we have restrictedto the problem of computing its determinant. B Details of the weak-coupling expansion B.1 Weak coupling expansion of the dressing phase The dressing phase in the mirror-mirror kinematics is given by [51] − i log Σ Q Q ( y , y ) = Φ( y +1 , y +2 ) − Φ( y +1 , y − ) − Φ( y − , y +2 ) + Φ( y − , y − )+ 12 (cid:2) − Ψ( y +1 , y +2 ) + Ψ( y +1 , y − ) − Ψ( y − , y +2 ) + Ψ( y − , y − ) (cid:3) − (cid:2) − Ψ( y +2 , y +1 ) + Ψ( y +2 , y − ) − Ψ( y − , y +1 ) + Ψ( y − , y − ) (cid:3) (B.1) + 1 i log i Q Γ( Q − i g ( y +1 + y +1 − y +2 − y +2 )) i Q Γ( Q + i g ( y +1 + y +1 − y +2 − y +2 )) − y +1 y − − y − y +2 s y +1 y − y − y +2 , where Ψ( x , x ) = i ˛ C dw πi w − x log Γ(1 + i g ( x + x − − w − w − ))Γ(1 − i g ( x + x − − w − w − )) , (B.2)and for Φ( x , x ) we just note that it starts in any kinematics at least with g . We calculate theO(1) expansion of the phase (B.2). Using the property Ψ( x , x ) = Ψ( x , − Ψ( x , x − ) , beingvalid if | x | 6 = 1 , and that for | x | > it starts at g , it is easy to see that we need to calculate Ψ( x , x ) ≡ Ψ( x , for | x | < , i.e. for x +2 . Since we are interested in the derivative of the expanded The representations Π a and Π b need not be irreducible representations of G . Indeed, in the AdS/CFT case, theyare sums of irreducible representations, as in (A.3). ∂ , we need to expand − (Ψ( y +1 , 0) + Ψ( y − , only.Rescaling the integration variable w by g and evaluating the leading residue for small g , we obtain Ψ( y +1 , 0) = i log Γ(1 + i (˜ p + iQ ))Γ(1 − i (˜ p + iQ )) + . . . , Ψ( y − , 0) = i log Γ(1 + i (˜ p − iQ ))Γ(1 − i (˜ p − iQ )) + . . . (B.3)The logarithmic derivative of the whole dressing phase is then − πi ∂ ˜ p log Σ Q Q (˜ p , ˜ p ) = 12 π (cid:20) − ψ (1 − i p + iQ )) + ψ (1 + 12 ( i (˜ p − ˜ p ) + Q + Q )) + c.c (cid:21) , (B.4)where c.c. denotes complex conjugate, and we used that ψ ( − i (˜ p − iQ )) + c.c = ψ (1 − i (˜ p + iQ )) + c.c for integer Q . B.2 Performing the integrals by residues We demonstrate here how we performed the integrals by evaluating A Σ (5.21). In view of the result(B.4), we start by evaluating the term with ψ (1 − i (˜ p + iQ )) + c.c. . Its contribution factorizes forthe indices , into a product of two factors. The more complicated factor is X Q Q ˆ ∞−∞ d ˜ p π p + Q ) L (cid:20) ψ (1 − i p + iQ )) + ψ (1 + i p − iQ )) (cid:21) . (B.5)Let us analyze the pole structure of the integrand. Additionally to the two “kinematical” poles at ˜ p = ± iQ , the polygamma function has poles for ψ ( − n ) if n ≥ . These poles are located on the lowerhalf plane (LHP) for the first and on the upper half plane (UHP) for second polygamma function: 12 ( Q + 2 ∓ i ˜ p ) = − n −→ ˜ p = ∓ i (2( n + 1) + Q ) . (B.6)We now use the trick in [62] of exploiting the reality of the integrand to rewrite the integral as ℜ e X Q Q ˆ ∞−∞ d ˜ p π p + Q ) L (cid:20) ψ (1 − i p + iQ )) (cid:21) , (B.7)and close the contour on the UHP. In so doing, we have to pick up the residue at ˜ p = iQ only: i X Q Q ∂ L − p ( L − ψ (1 − i (˜ p + iQ ))(˜ p + iQ ) L | ˜ p = iQ = − X Q L − X j =0 (cid:18) L + j − j (cid:19) − L +2 ( L − − j )! ( − Q ) − L − j ψ ( L − j − ( Q + 1) . (B.8)We now note that the Q -dependent terms give ∞ X Q =1 Q ˆ ∞−∞ d ˜ p π p + Q ) L = ∞ X Q =1 (cid:18) L − L − (cid:19) Q − L − L = 2 − L (cid:18) L − L − (cid:19) ζ L − . (B.9)Hence, the factorizing contribution to A Σ , which we denote by A (1)Σ , is given by A (1)Σ = − − L (cid:18) L − L − (cid:19) ζ L − X Q L − X j =0 (cid:18) L + j − j (cid:19) − L +2 ( L − − j )! ( − Q ) − L − j ψ ( L − j − ( Q + 1) . (B.10)37et us concentrate now on the nonfactorizing contributions, which we denote by A (2)Σ . Using againthe reality trick, we can write A (2)Σ = − X Q ,Q Q Q ˆ d ˜ p π ˆ d ˜ p π p + Q ) L (˜ p + Q ) L ψ (cid:18) Q + Q − i (˜ p − ˜ p )) (cid:19) , (B.11)and close the ˜ p integration contour on the UHP. By picking up the only residue at ˜ p = iQ , theresult is − i X Q Q ∂ L − p ( L − ψ (1 + ( Q + Q − i (˜ p − ˜ p )))(˜ p + iQ ) L | ˜ p = iQ = X Q L − X j =0 (cid:18) L + j − j (cid:19) − L +2 ( L − − j )! ( − Q ) − L − j ψ ( L − j − ( Q + 1 + 12 ( Q + i ˜ p )) . (B.12)The next integral we close on the lower half plane and pick up the residue at − iQ : A (2)Σ = − X Q ,Q L − X j ,j =0 (cid:18) L + j − j (cid:19) − L +2 ( L − − j )! 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