Holographic three-point correlators in the Schrodinger/dipole CFT correspondence

We calculate, for the first time, three-point correlation functions involving “heavy” operators in the Schrodinger/null-dipole CFT correspondence at strong coupling. In particular, we focus on the three-point functions of the dilaton modes and two “heavy” operators. The heavy states are dual to the single spin and dyonic magnon, the single spin and dyonic spike solutions or to two novel string solutions which do not have an undeformed counterpart. Our results provide the leading term of the correlators in the large λ expansion and are in perfect agreement with the form of the correlator dictated by non-relativistic conformal invariance. We also specify the scaling function which can not be fixed by using conformal invariance.


Introduction and conclusions
Any conformal field theory (CFT) is characterized by two pieces of information. The first one is the set of its primary operators and their conformal dimensions which can be read from the two-point correlation functions. The second piece needed is the structure constants coefficients which specify the operator product expansion (OPE) of two primary operators. The knowledge of all two and three point correlators of primary operators is enough to determine through the OPE all higher point correlation functions. Usually, the aforementioned correlation functions are calculated as a series of one or more parameters, the couplings of the theory, that is they are calculated order by order in perturbation theory. It is a rare occasion when one is able to calculate the observables of the theory at large values of the coupling constants or as an exact function of the couplings. One such occasion is that of the maximally supersymmetric gauge theory in four dimensions, N = 4 Super Yang-Mills (SYM), one of the most thoroughly studied CFTs due to its duality with type IIB string theory on AdS 5 × S 5 [1]. Exploiting the integrable structure of the theory an intense activity took place culminating in the determination of its planar spectrum for any value of the 't Hooft coupling λ. A variety of integrability based techniques were used to this end, from the asymptotic Bethe ansatz to the Y-system and the quantum spectral curve (for a review on these techniques see [2]).
Much less is known about the structure constants of the theory. The problem of determining them is much harder than that of the spectrum since the exact form of the

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We should stress that all these operators correspond to point-like strings propagating in the Sch 5 × S 5 background. Extended dyonic giant magnon and spike solutions were found in [37]. Their existence is in agreement with the fact that the theory is integrable. In the same work an exact in the coupling λ expression for the dimensions of the corresponding gauge operators was conjectured. 1 Furthermore, in the large J limit agreement of this expression with the one-loop anomalous dimension of BMN-like operators was found providing further evidence in favor of the correspondence. On the field theory side, it is only the one-loop spectrum of operators belonging in a SL (2) subsector that has been studied [39] finding agreement with the string theory prediction for the anomalous dimension of certain long operators (see also [40]). No higher point correlation functions have been calculated. 2 In the present paper we will use the Schrodinger background in order to calculate, holographically, three-point functions involving two heavy operators and a light one. The light operator will be chosen to be one of the modes of the dilaton while the heavy states will be either generalizations of the single spin and dyonic magnon or the single spin and dyonic spike solutions constructed in [37]. We will also calculate the three point function in the case where the heavy operators are two novel string solutions presented later in this work. As mentioned above the existing results in the literature are at the level of supergravity, in the sense that the corresponding states participating in the correlator are point-like and reduce to BPS states in the limit of zero deformation, i.e. µ = 0. In contradistinction, the heavy states we will be using are extended string solutions which tunnel from one point boundary of Sch 5 to another. These are precisely the points where the dual field theory operators will be situated. Our results provide the leading term of the correlators in the large λ expansion and are in perfect agreement with the form of the correlator dictated by non-relativistic conformal invariance. Unlike the three-point functions in N = 4 SYM conformal invariance of the non-relativistic theory is not enough to fix completely the space-time dependence of the correlator. Instead one is left with an undetermined function, called the scaling function, of the single conformally invariant variable that can be built from three space-time points. Our calculation will determine this functionF (v 12 ) at strong coupling.
A number of extensions and generalizations of the present calculations are possible. One may try to replace the dilaton with another light operator such as the R-current or the energy-momentum tensor T µν . The analogous calculation in the case of the original AdS/CFT scenario can be found in [41]. In order to be able to perform such a calculation one should first find the corresponding bulk-to-boundary propagators in the Schrodinger background. Another possibility would be to focus on the field theory side and try to identify the field theory operators which are dual to the string solutions we are using in this work and calculate their correlation functions at weak coupling possibly employing the integrable structure of the theory. In particular, it would be interesting to focus on operators which can be described by coherent states since in this case one may be able to JHEP09(2018)026 compare the weak and strong coupling results along the lines of [42,43]. Finally, it would be interesting to employ integrability in order to calculate three-point correlation functions involving three heavy operators or one heavy and two light operators both holographically and at weak coupling.

Classical string solutions
In this section we will write down and analyze the four classical string solutions, which in the following section will be used in the calculation of the three-point correlation functions. In subsection 2.1 we review the dyonic magnon and dyonic spike solutions, in 2.2 we review the single spin giant magnon and in 2.3 we review the single spin spike solutions. These solutions were initially presented and studied in [37]. To be precise, we will consider slight generalizations of the string solutions presented in [37] because these solutions will have a clear interpretation as strings tunneling from one point of the boundary to another and as a result will be of immediate use in the calculation of the three-point correlators. Subsequently, in subsections 2.4 and 2.5 we present two new classical string solutions. The solution of 2.4 is a generalization of the spinning BMN-like string solution that first appeared in [39]. In 2.5 we present a completely new solution, with an oscillating behavior along the holographic direction which does not have an undeformed analogue.
We consider the following consistent truncation of the 10d Sch 5 × S 5 metric 3 that is supplemented by the following B-field

Dyonic giant magnon and dyonic single spike
Here we review and extend the dyonic giant magnon and single spike solutions that were originally presented in [37], 4 since we intend to use them in the three-point function calculation. We consider the following ansatz, for both solutions,

3)
3 With respect to the notation of [37] for the S 3 we have performed the following change of variables More details about the consistent truncation can be found in appendix A. 4 The solutions of [37] have X0 = 0.

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where we have defined the variable y as The explicit expressions for the functions V y (y), θ y (y), Ψ y (y) and Φ y (y) as well as for the constants κ and Z 0 can be found in [37] and it is where the interested reader is referred to.
The new dispersion relation for the giant magnon reads while for the single spike Setting the constant X 0 to zero, i.e. X 0 = 0, we obtain the dispersion relations of [37].
In order to compute the three-point correlation function we need to Euclideanize the world-sheet metric and then rewrite the solution (2.3) in Poincare coordinates. To this end we have used the coordinate transformation relating the global to the Poincare coordinates which can be found in [44]. Namely, where (x + , x − , z, x) parametrize the space in Poincare coordinates. Performing these two steps we arrive at the following solution where the definition for y becomes due to the Wick rotation performed to the world-sheet time. To obtain (2.8) we have also redefined X 0 as X 0 = − i 2 X. The complex solution (2.8) describes a string propagating from the point x(τ = −∞) = − X 2 and x + (τ = −∞) = −i T 2 of the boundary -which is located at z = 0-to another point of the boundary with coordinates x(τ = ∞) = X 2 and x + (τ = ∞) = i T 2 . The two heavy field theory operators of the dual CFT will be inserted at precisely the aforementioned points.
At this point an important comment is in order. Notice that due to the relation x + (τ = ±∞) = ±i T 2 the operator insertion appears to be at points of the boundary where the JHEP09(2018)026 coordinate x + (τ = ±∞) takes imaginary values. However, the dual conformal field theory lives in a spacetime with real coordinates. In order to fix this issue one can analytically continue the quantity as follows T → −iT. A similar analytic continuation was performed for the other separation variable X. Having done this the operator insertion points are now real x + (τ = ±∞) = ± T 2 . This analytic continuation amounts to replacing the conformal invariant ratio v 12 defined in (3.5) with i v 12 in all formulae from section 3.2 on. Finally, let us note that an identical analytic continuation T → −iT should be performed to all solutions presented in the rest of this section.
In what follows we will also need the Lagrangian densities evaluated on the classical solutions. These will be useful in the calculation of the corresponding three-point functions.
Computing the Lagrangians on-shell we obtain the following expressions 5 and where the explicit expression of the function u(y), both for the giant magnon and the spike solutions, can be found in [37].

Single spin giant magnon
Here we review and extend the single spin giant magnon solution that can also be obtained from the dyonic one, once we set the conserved momentum J 2 equal to zero, i.e. J 2 = 0. The ansatz we consider is the following 6 where y is defined in (2.4). As in the dyonic case [37], we rewrite the differential equation for θ y (y) in terms of a new function u(y) ≡ cos 2 1 2 θ y as (u ) 2 2 where the two constants β 6 and β 4 have the values For the giant magnon solution and for the new classical string solution we use the notation d = v c.
For the single spike solution we use the notation c = v d. In all cases 0 < v < 1. 6 In all the expressions (for the functions and the constants) we present in the current and in the following subsections, the boundary conditions have been taken into account.

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and the prime denotes differentiation with respect to y. In order for β 4 to be positive 7 we constrain the values of ω as follows The solution of equation (2.13) is known from the analysis in [37] and is given by The equations of motion give the following expressions for V y (y) and Φ y (y) The value of κ is determined by the Virasoro constraint G µν (Ẋ µẊ ν +X µ X ν ) = 0 as follows As a result the dispersion relation of the solutions becomes This dispersion relation can be derived, as usual, by finding the relation among finite combinations of the infinite conserved charges E, J and M . Euclideanizing (2.12) and rewriting it in Poincare coordinates we obtain that where now y is defined in (2.9). Computing the Lagrangian on shell it is straightforward to obtain Notice that setting ω ψ = ω φ − 2 µ m v with ω φ ≡ ω and α = µ 2 m in (2.10) the on-shell Lagrangian of the dyonic magnon becomes that of the single spin giant magnon (2.23).

Single spin single spike
Here we review and extend the single spin single spike solution. This solution can be found by considering the following ansatz where y is defined again by (2.4). After some algebra the equations of motion lead to a differential equation analogous to (2.13) but with the constants taking the values Since β 4 > 0 there is no constraint on the values of ω, contrary to the giant magnon case, and the solution for u(y) is given by (2.17), with β 6 and β 4 defined in equations (2.25) and (2.26) respectively. The expressions for V y (y) and Φ y (y) take the form . (2.28) The dispersion relation of the single spin spike becomes As we did in the giant magnon cases the next step consists of euclideanizing the world-sheet time of the solution (2.24) and rewriting it in Poincare coordinates to get where now y is defined in (2.9). Computing the Lagrangian on shell we have Notice that setting ω φ = α µ v with ω ψ ≡ ω, the on-shell Lagrangians of the dyonic and single spin spikes, namely (2.11) and (2.31), are identified.

Spinning BMN-like strings
In this section, we will present a solution generalizing the BMN-like solution of [39]. Our solution is also winding one of the isometries of the S 5 with the winding number being n ∈ Z. In the limit of zero winding n = 0 and X 0 = 0 we obtain the spinning BMN-like solution that was presented in [39].
To start we consider the following ansatz for the spinning BMN-like solutions In order for Z, in (2.32), to be real the parameter n should satisfy the constraint n < µ m.
The G µν (Ẋ µẊ ν + X µ X ν ) = 0 Virasoro constraint requires that while the parameters of the solution are related to the conserved charges as follows As a result the dispersion relation is given by the following expression Euclideanizing (2.32) and rewriting it in Poincare coordinates we have Subsequently. we compute the Lagrangian on shell, to obtain the following result Note that for the BMN-like solution of [39] n = 0 and as a result the on shell Lagrangian becomes zero.

New classical string solution
In this subsection we will describe a new classical string solution that does not have an undeformed analog. We start by writing the ansatz we will use and is inspired by the giant magnon solution of [37], with the novelty that now the function Z is not a constant but a function of the worldvolume coordinates, that is where y is defined in (2.4). Note that if we set T y = V y = Φ y = X 0 = 0 and Z y = 1 in the solution (2.38) we get the spinning string solution of [39]. As usual the functions T y (y), V y (y), Z y (y) and Φ y (y) are functions of σ and τ and they will be determined through the equations of motion and the Virasoro constraints.
Following the same line of reasoning as in [37] and fixing the integration constants in such a way that if we set Z y = 1 we obtain the spinning string solution of [39], we have the following first order differential equations for the functions T y (y), V y (y) and Φ y (y) From the two Virasoro constraints, we obtain a first order differential equation for Z y (which is actually consistent with the second order differential equation coming from the variation along the direction Z) as well as the following algebraic constraint Since the r.h.s. of (2.42) should be positive, the value of Z y is limited in the following interval As can be seen from (2.42) as soon as we set the deformation parameter to zero, i.e. µ = 0, the only acceptable solution is that the l.h.s. and the r.h.s. of (2.42) vanish independently, i.e. Z y = 1. In that way we are led to the spinning BMN-like string solution of [39].

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In order to solve the equation of motion for Z y (2.42) we need to specify whether Z qrit > 1 or Z qrit < 1. In what follows we will solve the equation in the interval 1 < Z y < Z qrit (2.45) since when Z qrit < 1 the variable y cannot range to ±∞. Integrating (2.42) and imposing we obtain the following solution (2.47) Plotting the above solution it is easy to verify that Z y takes the value 1 at y = −∞ reaching its maximum value Z qrit at y = 0 and Z qrit and goes back to Z y = 1 at y = ∞, with Z y > 0 for y ∈ [−∞, 0] and Z y < 0 for y ∈ [0, +∞]. Now we can calculate the conserved charges for the new classical solution and construct the finite linear combinations of those conserved quantities that will lead us to the dispersion relation. In the following we will also use the following change of variables (2.48) Using the above change of variables it is possible to calculate the three conserved charges E, M and J . 8 All these charges diverge and it is possible to express them in terms of a nonconvergent integral. That integral should be eliminated among the conserved quantities in order to construct the dispersion relation. The non-convergent integral reads (2.49) The conserved charges are given by (2.52) 8 We use the notation E ≡ E/T , M ≡ M/T and J ≡ J/T .

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Using the algebraic constraint (2.43) we eliminate the non-convergent integral and construct the following dispersion relation In order to write the dispersion relation in terms of conserved quantities we introduce the following ratio of the parameters µ, m and ω that we denote by χ and which repeatedly appears in (2.53) χ ≡ µ m ω . (2.54) Using linear combinations of (2.50), (2.51) and (2.52) it is possible to find an expression for χ in terms of E, M and J which reads In terms of χ, the conserved charges and the velocity v of the soliton the dispersion relation becomes finally Euclideanizing (2.38) and rewriting it in Poincare coordinates we have where now y is defined in (2.9). Computing the Lagrangian on shell, we finally have that (2.58)

Three-point correlation functions
In this section we will evaluate the leading term in the semiclassical expansion of the three point function involving the dilaton and two heavy operators. The two heavy operators will be dual to each of the classical solutions presented in the previous section.
In general, as discussed in [26][27][28] the ratio of the three-point correlator of a light state and two heavy ones over the two-point function of the heavy states is given in the semiclassical limit by

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Here V L is the vertex operator of the light state and z(τ, σ),x(τ, σ), x − (τ, σ) and X i (τ, σ) with i = 1, . . . 5 denote the ten coordinates of the classical string solution tunneling from the pointx 1 of the boundary -which is at z = 0-to another pointx 2 of the boundary, wherex i = (t i , x i ). This tunneling solution is the string theory dual of the corresponding field theory two point correlator at strong coupling.
The vertex operator of a generic light state will depend on the deformation parameter µ and its form is not known for the Schrodinger background we are using. It would be interesting to construct systematically these vertex operators. However, there is one case where the vertex operator is actually known, even for the deformed background. This is the case of the dilaton since the latter couples universally to all other fields of the theory through the term d 2 σe φ(X) 2 G µν (X)∂X µ∂ X ν + . . . . In this case the ratio of the three to the two-point function takes the form where all quantities appearing in the integrand are evaluated on the classical solution sourced by the two heavy state vertex operators. It is this classical solution that correspond to the two-point correlator. The exponential in (3.2) comes from the fact that the dilaton is expanded as a sum of the eigenfunctions of the mass operator a.k.a. number operator which is a good quantum number in the Schrodinger backgrounds. The relevant for the dilaton expansion is Furthermore, K(x classical (τ, σ);x 3 ) is the bulk to boundary propagator of the specific mode of the dilaton which has momentum M 3 in the x − direction and L classical on shell is the Lagrangian density of the string which is evaluated on the classical two-point solution.
An important comment is in order. Would one like to calculate the three-point correlator not for a single mode characterized by M 3 but for the full dilaton field appearing in the left hand side of (3.3) one should be able to evaluate the three-point correlator for each mode separately and subsequently sum over M 3 . This means that one should evaluate the three-point correlator for values of M 3 that are comparable and bigger than the mass eigenstate M of the heavy state. But in such a situation the approximation M 3 M does not hold any more and the method that we use in this paper to calculate the three-point function is no more applicable since the light state can no longer be considered as a small perturbation and its absorption from the heavy state will considerably alter the latter. As a result, this fact forbids one to check the expression relating the three-point function coefficient to the derivative of the scaling dimension w.r.t. the coupling λ which holds in the original AdS/CFT scenario unless one is able to compute the correlators for arbitrary values of M 3 . We believe that the aforementioned relation should hold in the present case also and it would be interesting to perform the calculation of the three-point function for arbitrary values of M 3 in order to verify this.

Schrodinger symmetry and the propagator for the scalar field
In this subsection, we briefly review the analysis presented in [35,36]. In a quantum field theory possessing Schrodinger symmetry the two-point function of scalar operators is completely determined up to an overall constant and takes the following form wherex i = (t i , x i ) and (M i , ∆ i ) denote the non-relativistic mass and the scaling dimension of the scalar operator. When we have three spacetime points, namelyx 1 ,x 2 andx 3 , it is possible to construct the following kinematic Schrodinger invariant v 12 = − 1 2 Then, the functional form of the three-point function of scalar operators is fixed up to a scaling function F (v 12 ) of this Schrodinger invariant where we have introduced the following notation ∆ ij,k = ∆ i + ∆ j − ∆ k . This is to be contrasted with the case of the original AdS/CF T scenario where the spacetime dependence of a three-point correlator involving scalar fields is fully determined by conformal invariance. We aim in computing the three-point function between two heavy operators and a light scalar operator. The heavy operators will be located at the spacetime pointsx 1 andx 2 while the light operator at the spacetime pointx 3 . In order to simplify the computation we consider the following assumption regarding the time ordering of the operators (3.7) Furthermore for the semiclassical approximation to make sense we impose the condition that the mass eigenvalue of the light should be much less that the corresponding mass eigenvalue of the heavy states, that is Normalizing the three-point function of (3.6) by dividing with the two-point function (3.4) and using (3.7), (3.8) and ∆ 1 = ∆ 2 we arrive to following expression

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where in the last equality we have used the fact that t 3 t 1 < t 2 . Furthermore, ∆ 3 is the conformal dimension of the light operator The strong coupling computation should verify the spacetime structure of (3.9) and predict the expression of the scaling functionF (v 12 ). The bulk to boundary propagator for the scalar field, here the dilaton modes, is given by the following expression where to obtain the second line of (3.12) we have used (3.7). It is exactly that expression of the propagator that we will use in the next subsections to perform the three-point function calculation.

Three-point function of dilaton with giant magnons
Before presenting the actual computation, we collect the essential components for the calculation of the three-point function of two giant magnons, either dyonic or single spin, and the operator dual to the dilaton. In particular, we rewrite in Poincare coordinates the dyonic and single spin giant magnon solutions of (2.8) and (2.22) respectively by using ratios of conserved quantities 9 (3.13) where the function V y (y), 10 is given by the following expression In the last line of (3.14) we have used the fact that

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to write the quantity under the square root as a function of conserved quantities. Note that J 2 is a finite quantity. Setting J 2 = 0 (or equivalently ω φ = ω ψ + 2 µ m v) in (3.15) fixes δ = 1 and from (3.14) we obtain the expression for the function V y (y) in the case of the single spin giant magnon solution. Furthermore, we have used the fact that t 3 → −∞ to write X 2 2 T as v 12 (see (3.5) and keep in mind that t 21 = T and x 2 12 = X 2 ). At this point it should be apparent why we had to generalize the solutions of [37]. Those solutions have X = 0 and this will make the invariant v 12 = 0. Had we sticked to the solutions of [37] all the dependence of the three-point functions on v 12 would have been lost. Now we are in the position to write the expression for the normalized three-point function of the dyonic giant magnon by inserting the appropriate solution in the general expression for the three-point correlator (3.2) to get where I GM is given by the following double integral 11 The two integrals can be performed independently and we can write and The integral in (3.19) can be evaluated analytically using the approximation tanh y ≈ y

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Combining (3.16) with (3.18), (3.20) and (3.21) we arrive to the following expression for the three-point coupling for the dyonic giant magnon casẽ whereF is the scaling function defined in (3.11). The Schrodinger invariant ratio v 12 appears in several places in (3.23). It appears in the beta function through its dependence on Ξ, as well as in the part of the prefactor ( E M + v 12 2 ) ∆ 3 2 . This is the string prediction for the leading term in the large λ expansion of the scaling functionF . Notice that the spacetime dependence in (3.16) is the one dictated by non-relativistic conformal invariance (3.9). In order to express the ratio µ m ω φ as a function of conserved quantities we have used the following expression 12 Finally, notice that in the dyonic giant magnon solution there are two finite conserved quantities, namely p and J 2 , and we have expressed the three-point coupling in terms of those quantities, together with the finite ratio of E and M . Setting δ = 1 (or equivalently J 2 = 0) we obtain the three-point coupling for the single spin giant magnon casẽ Before closing this section lets us make the following comment. As pointed out in [45], in order to extract the correlation function of energy eigenstates of the dilatation operator in the strong coupling regime of the correspondence one should integrate the light vertex operator not only over a single classical string worldsheet but also over the moduli space of classical string solutions, that corresponds to the heavy states under consideration. However as we will argue, in our case this is completely equivalent to ignoring the integration over the moduli space since the latter integration trivializes, as it happens with the first two examples presented in [45]. To see this consider the following modification/deformation of JHEP09(2018)026 our classical string solution (2.8) where the variable y is shifted universally by τ 3 as follows so the equations of motion and the Virasoro constraints are satisfied. In the new modified solution (3.26) and (3.27) there are in total 4 moduli parameters, namely τ 0 , τ 1 , Since (as pointed out in [45]) it is equivalent to realize the moduli space of saddle points by shifting either the Schrodinger coordinates or those of S 5 , we choose the latter case. The reason behind that choice is that the vertex operators for heavy states in a Schrodinger spacetime are completely unknown. Notice that we have also allowed for an independent parameter τ 0 in the linear term of the x − coordinate.
As a consequence the moduli space of our solutions is parametrized by τ 0 , τ 1 , τ 2 & τ 3 over which one should also integrate. Thus in the ratio of the three to the two-point function in (3.2) we insert the solution (3.26) and (3.27), with an extra integration over the moduli parameters. Notice that the contribution from the shifts by τ 1 and τ 2 will cancel in the ratio of the three-point to the two-point correlator, as it happens with the first two examples of [45]. This is actually a consequence of the fact that the dilaton vertex operator does not carry any R-charges.
However, there is another modification needed. Namely, one should take into account that the vertex operators of the heavy states are not identical for the in and for the out state. Moreover, since there is a moduli space of saddle points (parametrized by τ 0 , τ 1 , τ 2 & τ 3 ) instead of a single one, the contribution of the vertex operators for heavy states in the ratio (3.2) will not be just an overall factor that cancels completely with the contribution from the two-point functions. Although the vertex operators for heavy states are not known it is plausible to assume that they are proportional to e i M x − times derivatives of x − and other coordinates. This implies that the modification coming from the vertex operators is But this contribution will precisely cancel the e (i M 3 )(−i α τ 0 ) coming from the numerator of (3.17) leaving the double integral independent of τ 0 . In that way the integral I 1 of (3.19) remains untouched and in the integral I 2 of (3.20) we should replace y withỹ. Consequently, the averaging over all the moduli parameters will trivialize lim T →∞ and the result will be identical to what it was before and independent of the moduli parameters. We conclude that the averaging over the moduli space trivializes and is not relevant in our calculations. We should mention that the same argument applies also to all the other three-point correlation functions considered in the subsequent sections.

Three-point function of dilaton with single spikes
The calculation of the three-point function for the single spike solution (either dyonic (2.8) or single spin (2.30)) has to be performed with special care. Integration of the on-shell Lagrangian (either in (2.11) or in (2.31)) will lead to an infinity and for this reason a proper normalization has to be implemented. The most natural/physical one is to subtract in both cases the Lagrangian for a classical configuration that moves very close to the equator, i.e. θ = π ⇒ u = 0 which represents a hoop winding an infinite number of times the equator of S 5 . The remaining on-shell Lagrangians for the three-point function calculation become and L 1s-norm on shell = i After properly normalizing the Lagrangians, the computation of the three-point function, where instead of giant magnons we have single spikes, is similar to the calculation we presented in the previous subsection. Now the function V y (y) is given by Setting J 2 = 0 (or equivalently ω φ = α µ v ) we obtain the expression for the function V y (y) in the case of the single spin single spike solution. Substituting all the ingredients in (3.2) we arrive to a double integral (in τ and y) for the three-point function, similar to that of (3.18). In the dyonic spike case the calculation of those integrals give the following results and The three-point coupling for the dyonic single spike case is given by the following expressioñ (3.36)

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where the expressions of T y and V y as functions of Z y can be obtained by integrating (2.39) and (2.40) using also (2.42). Here we quote those results, initially for the function T y and subsequently for the function V y The connection between the quantities X 0 and v 12 is v 12 = − X 2 0 . Notice that in the new classical string solution all the conserved quantities, namely E in (2.50), M in (2.51) and J in (2.52) are infinite and in the expression of the three-point coupling only ratios of them may appear.
Plugging all the necessary ingredients in (3.2), we obtain the normalized three-point function as follows where we have changed variables, from σ to Z y according to (2.48). The integral with respect to τ in the expression above can be calculated analytically using the approximation tanh y ≈ y and the result is Contrary to the cases of the previous subsections, the integral with respect to Z y cannot be performed analytically, since the expression for Ξ N S depends also on the integration variable Z y . The three-point coupling for the new classical string solution is written in terms of that integral as follows and obtain the three-point coupling as a function of the deformation parameter µ. Since we can vary the deformation parameter in a continuous way it is possible to obtain plots for the real and the imaginary value of the three-point coupling as a function of µ. These plots can are presented in figure 1.

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.