Worldsheet S-matrix of beta-deformed SYM

We compute perturbative worldsheet S-matrix of beta-deformed AdS/CFT in the strong and weak `t Hooft coupling limit to compare with exact S-matrix. For the purpose we take near BMN limit of TsT-transformed AdS_5 x S^5 with the twisted boundary condition and compute the S-matrix on worldsheet using light-cone gauge fixed Lagrangian. For the weak coupling side, we compute the S-matrix in the SU(3) sector by applying coordinate Bethe ansatz method to the one-loop dilatation operator obtained from the deformed super Yang-Mills theory. These analysis support the conjectured exact S-matrix in the leading order for both sides of beta-deformed AdS/CFT along with appropriate twisted boundary conditions.


Introduction
The S-matrix plays a key role for studying two-dimensional integrable models. With enough symmetries, the S-matrix can be determined mathematically and can be used to find particle spectrum along with exact dispersion relations and to compute finite-size effects. Based on this philosophy, there have been remarkable developments in applying the integrability methods to the AdS/CFT duality between N = 4 super Yang-Mills theory (SYM) and the type-IIB superstring theory on AdS 5 × S 5 [1]. Exact S-matrix has been proposed [2,3,4] with the dressing phase [5,6], and applied to such tools as Lüscher correction [7] and thermodynamic Bethe asnatz [8].
After these successes, it is natural to extend the utility of the integrable methods to other proposed or conjectured AdS/CFT dualities. These include β-deformed SYM theory [9] which is dual to superstring theory on Lunin-Maldacena background [10] and three-parameter-deformed theory which breaks all the supersymmetry [11]. There are some clues that the deformations still maintain the integrability. First, string sigma models on the deformed backgrounds are classically integrable [12,13]. One-loop dilatation operator on the gauge theory can be mapped to integrable spin chain models with certain twists [14]. All-loop asymptotic Bethe ansatz equations for the deformed theories were conjectured by Beisert and Roiban [15].
Another strong evidence for the integrability has come from the anomalous dimension of Konishi operator computed by twisted Lüscher formula [16] which matches with fourloop perturbative computation [17]. Related computations have been also worked out by the Y-system of the β-deformed SYM [18].
With the assumption of integrability, the S-matrix and associated twisted boundary conditions have been proposed and used to derive the conjectured all-loop asymptotic Bethe ansatz equations [19]. The twisted S-matrix is given bỹ and the corresponding twisted boundary conditions are where A denotes an auxiliary space. Another support of the S-matrix conjecture comes from the strong coupling limit of the twisted AdS/CFT duality. Finite J correction of a classical giant magnon dispersion relation has been computed from the S-matrix element and twisted boundary conditions through Lüscher formula [20] and shown to match with classical sigma model computation for the γ-deformed background [21,22]. While these evidences justify the assumption of integrability, it is desirable to check the S-matrix directly either with the string theory on a deformed background in the strong coupling limit or with the N = 1 supersymmetric or non-supersymmetric gauge theories in the weak coupling limit.
On the other hand, S-matrix in two dimensions which describes localized interaction of two on-shell particle states and the boundary conditions imposed on the particle states which characterize global spatial geometry are inseparable. One can always attribute a part of S-matrix into the boundary conditions and vice versa. Consistency of this ambiguity is guaranteed only at the level of Bethe-Yang equations which determine the energy level spectra of a theory. In the context of the twisted AdS/CFT, it is possible to shift the Drinfeld-Reshetikhin twist F into the boundary condition M as shown in [19]. The resulting theory is described by untwisted S-matrix while the twisted boundary condition is given by This is called "operatorial" boundary condition since it depends not only the particle state which passes through the boundary but also all other states Q's in the "quantum space" which are away from the boundary. This feature is inevitable when one deals with off-diagonal S-matrix. It is shown that this combination of S-matrix and boundary conditions can produce the same "Beisert-Roiban" asymptotic Bethe ansatz equations [23]. However, these operatorial boundary conditions are difficult to realize in the perturbative computations. On string side, Frolov first showed that superstring theory on the T sT -transformed AdS 5 × S 5 with the periodic boundary conditions is equivalent to the undeformed AdS 5 × S 5 with the following twisted boundary conditions [11]: (1.4) Here, γ j = β is a parameter for deformation of scalar field potential in the gauge theory side and three angular momenta are given by J i = dxφ i . Eq.(1.4) is not easy to solve for the multiparticle solutions. For the spin-chain side in the weak coupling limit, the S-matrix can be computed by coordinate Bethe ansatz method. The spin-chain Hamiltonian as a dilatation operator naturally depends on the deformation parameter β in N = 1 SYM. By some nontrivial unitary transformation it can be changed into that of untwisted spin-chain as explained in [14]. However, it generates nontrivial boundary conditions which will be in general nonlocal, i.e. which depends on the states on the quantum space.
For these reasons, we study the S-matrix of the β-deformed SYM at strong and weak 't Hooft coupling regimes which corresponds to (1.1) where the boundary condition (1.2) becomes simply a c-number. For this purpose, we consider string world-sheet action in near BMN limit and with light-cone gauge fixing which is different from Lunin-Maldacena and compute the worldsheet scattering as was done for untwisted case in [24]. In the weak coupling regime, we consider one-loop dilatation operator for three-spin sector. We apply coordinate Bethe ansatz to compute one-loop S-matrix in this sector using the deformed SU(3) spin chain Hamiltonian derived in [14,25] and show that it matches with the exact Drinfeld-Reshetikhin S-matrix (1.1) in this limit.

Strong coupling regime : String worldsheet
The dual gravity solution of N = 1 β-deformed SYM is first constructed by Lunin and Maldacena [10]. This background could be obtained by using sequence of three Here, (φ 1 , φ 2 ) T sT means to take T-dualization along φ 1 , shift φ 2 → φ 2 +γφ 1 and take T-dualization again for φ 1 . As a result of T sTtransformation, all kinds of background fields -metric, B-fields, RR-fields and so on -are deformed or generated. If we use different parametersγ 1,2,3 for each T sT -transformation, LM background could be generalized to three-parameter deformed background which is dual to non-supersymmetric, marginal deformed SYM. The three-parameter deformed AdS 5 × S 5 spacetime metric and antisymmetric B-fields are given by the followings: There is an additional constraint 3 i=1 ρ 2 i = 1 and three tori angles φ 1,2,3 have periodicity under σ → σ + 2π. We will only consider one-parameter deformed theory (γ 1,2,3 =γ) for simplicity but all discussions about string regime in this paper are applicable even for three-parameter deformed theory.

T sT -transformed AdS 5 × S 5 with twisted boundary conditions
We start from AdS 5 × S 5 string with twisted boundary conditions (1.4). The nonlinear sigma model action on usual S 5 is given bỹ with background metricĜ ij and fieldsB ij whose non-zero components arê

Corresponding bosonic string action iŝ
We will use this action to compute worldsheet S-matrix. 1 The above T sT -transformation changes the twisted boundary conditions (1.4) tô where level matching condition is given by P ws = 2π [n 2 + β(J 3 − J 2 )]. This corresponds to "c-number" boundary conditions forφ 2 andφ 3 because they do not depend on J 2 and J 3 .

Gauge fixed Lagrangian
To compute the string worldsheet S-matrix, it is convenient to introduce new variables defined by We also have to remove the redundancy from general coordinate invariance. A standard way is to consider the BMN limit [26] and its curvature corrections: This BMN limit simplifies the metric and B-fields as follows: Although the metric is independent ofγ up to 1/R 2 , worldsheet scattering becomes nontrivial because the B-fields haveγ dependence. 2 The bosonic string Lagrangian becomes now Here, a, b stand for worldsheet coordinates σ and τ . As in usual case, the Hamiltonian is just sum of Lagrange multiplier times constraint. As the epsilon coupled to anti-symmetric B-fields is a non-dynamical field, the variation of the action over worldsheet metric operate on only G-field parts.
To fix the gauge, we can use the first-order formalism which works well for undeformed theory [27,28,29]. First, we define the conjugate momentum

8) and the Hamiltonian
which becomes zero if we impose the Virasoro constraints. Introducing the light-cone gauge X + = τ, P − = const, we can express the Lagrangian L = P µẊ µ − H in terms of ungauged variables L g.f. = P + + P IẊ I = P IẊ I − H L.C. (2.10) where we have imposed the Virasoro constraints. The expression for the light-cone Hamiltonian is given by Here,H is the light-cone Hamiltonian of the undeformed theory. Considering Legendre transformation and solving equations of motion for P I , we finally obtain the gauge fixed bosonic Lagrangian The potential term is To compute worldsheet S-matrix, we need to consider decompactification limit P − → ∞ in which worldsheet parameter space changes from cylinder to plane after rescaling σ → P − √ λ σ. Here, P − has appeared in the integration bound for σ due to light-cone gauge fixing.

Tree-level scattering amplitudes
The string worldsheet S-matrix can be straightforwardly computed from the gauge fixed action (2.12). In the leading order of 1 √ λ , we define T-matrix by We need to compute additional contribution to T fromγ-dependent part of V which contains only Y . In terms of mode expansions [30] Y 11 (σ, τ ) = dp (2.14) Here, ω = p 2 + 1 and the kinematic factor [31] Λ(p, One can notice that the scattering amplitudes depend onγ only in Y Y to Y Y process. Explicitly, only non-zero elements of the Tγ are (2.16) Now, we consider the strong coupling limit of the exact twisted S-matrix to compare with the above tree-level amplitudes. In this limit, we can expand the twisted matrix F for small β =γ/ √ λ 3 with Γ defined in (1.1) as well as the twisted S-matrix where T is the undeformed matrix elements. Because the elements of the twisted Smatrix (1.1) can be written as only amplitudes which are deformed in two-boson to two-boson scatterings arẽ This matches with (2.16).
On the other hand, we can get the twisted boundary conditions for Y from (2.4) which also agree with (1.2) with γ 3 = γ 2 = β and J 1 = J.

Weak coupling regime : Spin-chains
The spin-chain Hamiltonian corresponding to the one-loop dilatation operator of the βdeformed SYM was first studied in [14,32]. Later, more general integrable deformation was investigated in [25]. In this section, we compute S-matrix from the spin-chain Hamiltonian using coordinate Bethe ansatz. For simplicity, we only consider three-state spin-chain which is the simplest sector with nontrivial dependence on the deformation parameter.

Conclusions
In this paper we have computed worldsheet and spin-chain scatterings of the β-deformed SYM in the leading order to check the validity of proposed exact S-matrix and boundary conditions. For the strong 't Hooft coupling regime, we used the light-cone gauge fixed Lagrangian in the T sT -transformed background. We also computed weak coupling Smatrix based on SU(3) spin-chain Hamiltonian. We have shown that these perturbative results match with the exact conjectures.
Here, we have considered only boson to boson scatterings in the leading order. It will be interesting to extend the checks to fermions and the higher-loop order. It will be also interesting to investigate whether our simpler background of the β-deformed theory can be more useful in finding concrete string solutions or higher correlation functions.