An HHL 3-point correlation function in the η-deformed AdS 5 × S 5

Article history: Received 11 January 2015 Accepted 13 February 2015 Available online 18 February 2015 Editor: L. Alvarez-Gaumé We derive the 3-point correlation function between two giant magnons heavy string states and the light dilaton operator with zero momentum in the η-deformed AdS5 × S5 valid for any J1 and η in the semiclassical limit. We show that this result satisfies a consistency relation between the 3-point correlation function and the conformal dimension of the giant magnon. We also provide a leading finite J1 correction explicitly. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

For |x 1 | = |x 2 | = 1, x 3 = 0, the correlation function reduces to Then, the normalized structure constant can be found from (2.1) where c is the normalized constant of the "light" vertex operator. Actually, we are going to compute the normalized structure constant (2.1). For the case under consideration, the "light" state is represented by the dilaton with zero momentum.
According to [9], C 3 for the infinite-size giant magnons and dilaton with zero momentum in the undeformed AdS 5 × S 5 is given by where t = κτ e is the Euclidean AdS time and the term ∂ X K∂ X K is proportional to the string Lagrangian on S 2 computed on the giant magnon solution living in the R t × S 2 subspace.
Since here we are interested in finite-size giant magnons, we have to replace where L gives the size of the giant magnon and θ is the non-isometric angle on the two-sphere [11].
Going to the η-deformed AdS 5 × S 5 case, we have to compute the term ∂ X K∂ X K for this background, which is proportional to the string Lagrangian on S 2 η for finite-size giant magnons: where X K = (φ 1 , θ) are the isometric and non-isometric string coordinates on S 2 η correspondingly. Working in conformal gauge and applying the ansatz where η is related to the deformation parameter η according to [4] and a new variable χ is defined by The prime here and below is a derivative d/dξ . The string tension T for the η deformed case is related to the coupling constant g by T = g 1 +η 2 . (2.5) The first integrals of the equations of motion F 1 and θ can be written as Inserting (2.6), (2.7) into (2.3), we obtain: which leads to Therefore, for the case at hand, the normalized structure constant takes the form One can rewrite Eq. (2.7) as (2.12) Using this, we can express all the results in terms of χ p , χ m by eliminating v, W .
Replacing (2.11) in (2.10) and using (2.12), we can express Cη 3 by The integral can be easily expressed by K and , the complete elliptic integrals of the first and the third kind, respectively, are as follows: (2.14) where we introduced a short notation by . (2.15) Eq. (2.14) is the main result of this paper, which is an exact semiclassical result for the normalized structure constant Cη 3 valid for any value of η and J 1 . Here, χ p and χ m are determined by the angular momentum J 1 and world-sheet momentum p from the following equations 1 : The world-sheet energy of the giant magnon is given by One of nontrivial checks is that the g derivative of = E − J 1 should be proportional to the normalized structure constant Cη 3 since the g derivative of the two-point function inserts the dilaton (Lagrangian) operator into the two-point function of the heavy operators [10]. This can be expressed by as noticed in [11] for the case of undeformed giant magnon. From these, we can obtain the expressions for ∂χ p /∂ g and ∂χ m /∂ g which can be inserted into ∂ /∂ g. The η-deformed case involves much more complicated expressions which can be dealt with Mathematica.
In Appendix A, we provide our Mathematica code which confirms that the structure constant Cη 3 in Eq. (2.14) do satisfy the consistency condition (2.19) exactly.

Leading finite-size effect on Cη 3
It is straightforward to compute a leading finite-size effect on Cη 3 for J 1 g by the limit → 0 in (2.14).
First we expand the parameters χ p , W and v for small as follows: Inserting into Eq. (2.14), we obtain In view of Eqs. (2.12) and (2.15), we can express all the auxiliary parameters in terms of v (or its coefficients v 0 , v 1 , and v 2 ):

This leads to
To fix v 0 , v 1 , and v 2 , one can use the small expansion of the angular difference where we identified the angular difference φ 1 with the magnon momentum p on the worldsheet. The result is [7] v 0 = cot p 2 η 2 + csc 2 p 2 , (3.5) and (3.7) The expansion parameter in the leading order is given by [7] =