On the Regge limit of Fishnet correlators

We study the Regge trajectories of the Mellin amplitudes of the $0-,1-$ and $2-$ magnon correlators of the Fishnet theory. Since fishnet theory is both integrable and conformal, the correlation functions are known exactly. We find that while for $0$ and $1$ magnon correlators, the Regge poles can be exactly determined as a function of coupling, $2$-magnon correlators can only be dealt with perturbatively. We evaluate the resulting Mellin amplitudes at weak coupling, while for strong coupling we do an order of magnitude calculation.

1 Introduction N = 4 SYM is one of the few most convenient playground for analyzing the scattering amplitudes for a CFT, since in addition to conformal symmetries, it also admits a Lagrangian description. But this has its own technical challenges. A somewhat simpler theory is obtained from the γ−deformed N = 4 SYM in the double scaling limit, called the conformal fishnet theory. In this limit, all the heavier constituents of the N = 4 except the adjoint scalars decouple, giving an effective Lagrangian, where X(Z) are complex traceless N × N adjoint scalars andX(Z) are the conjugates. The reduced coupling ξ is given in the planar limit (N → ∞, g 2 Y M → 0) for specific configuration of the deformation (γ 3 → i∞) by, Due to CPT non-invariance of the interaction term, the theory is inherently non-unitary giving rise to some peculiar features. Owing to integrability and conformal invariance, correlation functions of local and bi-local operators in this theory can be exactly determined as a function of the coupling ξ, by iteratively solving the Bethe-Salpeter equations (reviewed in detail in section 2). Authors of [1,2] analyzed the scattering amplitudes for the fishnet theory in four dimensions. Further, one can analyze the Regge limit of the correlators of the fishnet theory exactly in coupling (ξ) 1 . The exact correlation functions of the local and bi-local operators that we study is given by (from (2.3)), tr (X(x 1 )X(x 2 )) tr X (x 3 )X(x 4 ) = G 0 (x 1 , x 2 |x 3 x 4 ) + G 0 (x 1 , x 2 |x 4 x 3 ), tr (X(x 1 )Z(x 1 )X(x 2 )) tr X (x 3 )X(x 4 )Z(x 4 ) = 1 2 OXZ(x 4 )) = G 2 (x 1 , x 2 |x 3 x 4 ), (1.3) where O XY (z) = tr (XY )(z). These correlation functions are expressed in terms of the n-magnon graphs denoted by G n (x 1 , x 2 |x 3 x 4 ). In [2], the authors study the Regge limit of the 0−magnon four point amplitude in the fishnet theory using standard LSZ reduction techniques in momentum space. An immediate obstruction to generalizing their method to the 1 and 2-magnon cases is the fact that the 1 and 2-magnon states describe a bound state which is off-shell 2 .
We will however discuss the Regge limit of magnon correlators independently following [3]. In [3] the authors showed that for the Mellin amplitude for a CFT correlator, given by [3], where, and P ν,J (s, t) is the Mack polynomial, the Regge limit is defined as s → ∞ and t = fixed. The details of how the Regge limit is obtained will be discussed in the next section. The most important part is basically the spectral weight b J (ν 2 ) which for fishnet CFT can be exactly determined as shown in [4]. In this short note, we achieve a modest goal of determining the Regge trajectories for the 0, 1, 2−magnon correlators in the fishnet CFT using the techniques of [3]. We will also point out various relevant features and subtleties of the computations pertaining to each type of correlators. We now present the main results of our paper.

Results
Our main results can be summarized as follows. We systematically study the Mellin amplitude in the limit of s → ∞ with t held fixed (s and t are the Mellin variables in contrast to Mandelstam invariants) of correlation functions of certain operators in the fishnet theory. We obtain the Regge poles and evaluate the ν integral in the weak and strong coupling limit for these poles. 2 We thank Gregory Korchemsky for pointing this out to us.

1-Magnon correlator
For this case there are two separate Regge trajectories depending upon whether it is even or odd spin.
We have used the following definitions below q = log(s), g = 8π 2 cξ.
Even Spin: The Regge trajectory is, The Mellin amplitudes for strong coupling and weak coupling are as following.
• Weak coupling: where we have considered the limit, s → ∞, g → 0, q = g log s → constant.
Further I n (x) is Modified Bessel function of first kind.
• Strong coupling: Odd Spin: The Regge trajectory is given by, while the Mellin amplitudes are, • Weak coupling: (1. 13) where we have considered the following limit, and J m (x) is Bessel function of first kind.

2-Magnon Correlator
For the 2-magnon case we have evaluated the Regge trajectories as well as the Mellin amplitudes perturbatively in weak coupling and strong coupling limits. The main results for this case are as following, Weak coupling: The Regge trajectories in this case are given by, with γ a n,k being given explicitly in (6.5). The Mellin amplitude in this case is given by, with α 1 = 2π 3 ξ 4 . The explicit integrals are evaluated by (6.24).
Strong coupling : In strong coupling the leading Regge trajectory is given by, and the corresponding Mellin amplitude is given by, The paper is organized as follows. In section 2, we discuss the basics of the fishnet CFT in four dimensions following [6,1,7]. In section 3, we give a brief overview of the "Conformal Regge Theory" following [3,8,9]. Specifically, we elaborate a bit on the pole analysis and the contour prescription associated with the resultant Mellin amplitude in the Regge-limit. In sections 4, 5 and 6, we discuss the application of the Conformal Regge theory to the case of the fishnet correlators. We discuss in details the Regge trajectories associated with the individual types of magnon correlators. For 0 and 1−magnon, we compute the Mellin amplitudes for the leading Regge trajectories both in the weak and strong coupling regimes. For 2−magnon case, we analyze the systematics of the Regge limit separately in the weak and strong coupling regimes. We end the paper with some discussions on what could be the potential issues and further questions. In Appendix A, we give the details of the assumptions specially the pole analysis and contour prescription along the lines of [2] for individual cases. In Appendix B, we provide the details of the integrals. We demonstrate that there is only one integral per case one needs to compute and the subsequent integrals (for the weak coupling systematics) are just finite integrals with respect to one of the Mellin variables. In Appendix C, we provide a separate discussion of the 2−magnon case in the weak and strong coupling regime.

Conformal Fishnet theory in 4d
In this section we review the Bi-scalar fishnet CFT [7] and provide an overview of the basic structure of the correlation functions that can be exactly computed in the planar limit [1]. The Bi-scalar fishnet CFT is obtained as the double scaling limit of the γ deformed N = 4 Super Yang-Mills [7]. The γ-deformation reduces the SU (4) ∼ SO(6) R-symmetry of the theory to U (1) 3 . The double scaling limit is defined as γ i → ∞, g 2 = N c g 2 ym → 0 with ζ 2 j = g 2 e −iγ j held fixed (where i = 1, 2, 3 are the three cartans of SO (6)). Choosing ζ 1 , ζ 2 → 0, all the other fields except two complex scalars decouple and we obtain the classical Lagrangian for the Bi-Scalar CFT.
where X and Z are complex N c × N c matrix fields. At the quantum level, the theory described by this Lagrangian is not conformal and we need suitable double trace counter terms [10,6] with finely tuned couplings. The exact details of these counter terms will not be important for our analysis. The theory with the counter terms is renormalizable and has non-trivial fixed points where the coupling constants of the counter terms can be described as (complex) functions of the coupling constant ξ. The theory at the fixed point is conformal and integrable in the planar limit [11,12,13,14]. The resulting theory is non-unitary and conformal. One can consider correlation functions of the local protected dimension 2 and bi-local operators such as It was shown in [1] that due to the iterative structure of the Feynman graphs that contribute to the unprotected four point functions that can be built out of these operators, they can be computed exactly in the planar limit. These correlation functions exhibit a rich non-perturbative OPE structure. We briefly recall the salient features of their computation. The building blocks for the correlation functions are termed as "n-magnon" correlators, denoted by G n (x 1 , x 2 |x 3 x 4 ), depending on the particle that is being exchanged. The relation between the magnon graphs and actual correlation functions are given below [1].
The 0-1 and 2 magnon graphs have the periodic "fishnet" structure and can be computed using the Bethe-Salpater approach. In terms of the iterative Feynman diagram structure, they can be written down The actual procedure for evaluating these summed diagrams involves expressing these in terms of a graph building operatorĤ. Schematically, the correlator More precisely, sinceĤ commutes with the conformal group, the eigenstate x 1 , x 2 | is basically the three point functions of two scalar operators of dimension ∆ 1 and∆ 2 at position x 1 and x 2 and some spin J operator with ∆ = 2 + iν at x 0 . The eigenvalue equation satisfied byĤ is then given by, where E ∆,J are the eigenvalues of the graph building operator. These eigenfunctions are the conformally invariant three point functions, projected onto a light-like (null) vector n µ . We can then write the graph-building operator as, where the function c 1 (ν, J) in arbitrary dimensions is given by [15], .
The last integral can be put in terms of the familiar conformal block and its shadow viz [16,17,18], and finally from (2.5), [1] G( where p = 1, 2, 1 for n = 0, 1, 2−magnon graphs respectively and c 2 (ν, J) = c 1 (ν, J)/c(ν, J) and is given by [15], This is the starting point of our analysis. For more details about the derivation we refer the reader to [1]. Before going into the characterization of the Regge limit for the individual graphs, we will write down the eigenvalues for the n−magnon graphs. , , where ψ m (x) = d m ψ(x)/dx m and ψ(x) is Digamma function given by d dx (ln Γ(x)). In this notation ψ 0 (x) = ψ(x).
Regge limit correspond to the kinematic regime of large s at fixed t. In Regge limit, the leading part of the amplitude is dominated by Regge poles which are functions of actual physical poles of the amplitude.
In [3] the authors explore an analogy between certain kinematic configurations of conformal correlation functions and Regge limits of flat space scattering amplitudes by studying the correlation functions in the Mellin space. The role of the mandelstam invariants in the scattering is played by the Mellin transform variables s and t. In this section we review Conformal Regge Theory in Mellin space following [3]. The Mellin representation of a four-point conformal correlator is, where M(s, t) is the Mellin amplitude and is the measure with ∆ ij = ∆ i − ∆ j . The Mellin amplitude admits a partial wave decomposition [19], where P ν,J (s, t) is the Mack polynomial; and . (3.5) This will be the focal point of our analysis. We consider the t-channel decomposition with ∆ 1 = ∆ 4 and ∆ 2 = ∆ 3 . In Appendix C of [3], it was shown that the Regge limit of Mellin amplitude matches with the usual momentum space Regge limit 4 . In this work, we are however interested in the conformal Regge limit of the Mellin amplitude, irrespective of the physical implications in the momentum space. For large s and fixed t, the Mack polynomial takes the form [3], The factor a J becomes 1 for general ν and integer J. Using the Sommerfeld-Watson (SW) transform, where ± respectively stands for even and odd spins and, Next we replace J in terms of a complex integral along the contour−C (in fig.(1)), which picks up only integer poles in J. Recalling that + and − signs stand for contributions from even and odd spin respectively we have the following expressions, (3.10) Following [3], we analytically continue J from integer to complex values, i.e. deform the contour C → C (see figure (1)), to pick up the poles J = J(ν) in the complex plane. The poles of J = J(ν) are determined from the spectral function b J (ν) 5 . From [3], we make the correct identification of the spectral function for the fishnet CFT.
where p = 1, 2, 1 for respectively n = 0, 1, 2-magnon graphs. Putting in the normalizations, The leading Regge trajectory is determined by the largest exponent of s J(ν) . and, we get, where, is the phase factor associated with the even and odd parts. We note that for the zero-magnon case, with ∆ i = 1, the t-independent part of the amplitude in (3.14) exactly matches with the momentum space Note that the term e Jπ/2 P ± J takes care of (s → −s) in the SW transform and from now on we will dispense with this term by writing out the (s → −s) term separately. In the following sections we compute the Regge limit of Mellin amplitudes for 0,1 and 2-Magnon correlators.

0−magnon correlator
In this section we will obtain the Regge limit of the 0-magnon correlator in Mellin space. The Regge limit of the scattering amplitude has already been analyzed in [2]. We perform similar analysis in Mellin space as a warm up for the other magnon graphs. Upto some t-dependent factors, we obtain a match with the Regge amplitude computed in [2]. For the 0−magnon correlator, the external operator dimensions are The Mellin amplitude in the Regge limit is given by (3.14), where, from (2.11) we have the for the 0−Magnon amplitude, Also note that the extra sign in front of the Mellin amplitude stems as discussed follwoing (3.14). Putting ∆ = 2 + iν above we obtain the following expression 7 ,

Regge limit
Solving for the poles of (4.3), we obtain the Regge trajectories, The leading Regge trajectories come from the pole(s) having the most positive real part (in the limit s → ∞). Thus the leading Regge trajectory is obtained from J + 2 [2]. The integral over ν in (4.1) is performed as follows. We first compute residue of the spectral function due to the Regge poles.
Schematically this is given by, where the residue is evaluated at the Regge poles J i = J ± 2 . Evaluating the residue around the Regge poles (for leading Regge trajectories), the Mellin amplitude is given by 8 , where, (4.7) We will now evaluate this integral in weak coupling limit, f → 0 and strong coupling limit, f → ∞.

Weak Coupling: f → 0
Following [2], we manipulate the integral in (4.6) into a form that is valid for primarily weak coupling and then we evaluate the integral in the weak coupling limit. This integral can be effectively reduced to 7 Note that the authors of [2,4] use ∆ = 2 + 2iν. 8 We are just looking at the even spin hence considering M + (0) . The odd spin case i.e, M − (0) can be tackled in a same fashion by putting proper signs as delineated in the discussion following (3.14). an integral over the interval −f 2 ≤ ν ≤ f 2 so that, with, where the approximate sign denotes that this equality is valid modulo terms of order O(s −1 ) which vanish in the limit s → ∞ 9 . It is convenient to perform a change of variables ν = f 2 √ 1 − x 2 and define, (4.10) and in this notation, Introducing, q = f 2 log(s/4), we can finally write, where in the last line, we have performed a change of variables x → −x to combine the two regions of integration. Now we will analyze the Regge amplitude in the weak coupling limit. More precisely, we take the following set of limits.
Expanding the integrand in the weak coupling limit, we write first few terms, This integral can be done with the help of integrals described in Appendix B and specifically, the integrals that go into the final evaluation are those in (B.7). Upto a few orders of expansion in f we have the following result, 9 We thank Gregory Korchemsky for sharing his notes on this manipulation with us. Interested readers will find the details of this manipulation in Appendix A.1 This is the main result in the weak coupling limit of 0−magnon correlator. Apart from the t−dependent factors, the integrand arranges itself into the same structure as that of [2] 10 . We are computing Regge amplitudes from the Conformal Regge theory (CRT) point of view, independent of the LSZ approach in [2]. The CRT also aids to compute the Regge limit of the 1-magnon correlators with off-shell states.

Strong Coupling: f → ∞
For strong coupling, we perform an order of magnitude analysis [2]. The procedure is as follows, we first look at the behavior of the Regge poles J ± 2 and J ± 4 as a function of ν. We observe that the dominant contribution comes from J + 2 near ν = 0 11 , (4.14) We clearly see that in the Regge limit, J + 2 dominates over J − 2 which is exponentially suppressed. Also note that this is true for any coupling. Let us define, Hence around ν = 0 from (4.6) we have for M + (0) , Since the dominant contribution to the Regge amplitude ( i.e in the limit s → ∞) comes from the region ν ∼ 0, the ν integral effectively reduces to, .
(4.17) 10 We have additional t-dependent terms, and we have defined our ∆ = 2 + iν whereas [2] had used the definition ∆ = 2 + 2iν. 11 All other poles are subleading near ν = 0 and all the poles including J + 2 are subleading in the limit ν → ∞ Further around f → ∞, Collecting everything, we obtain the Regge amplitude in the strong coupling to be, A similar order of magnitude analysis can be done for the weak coupling also. We find that the leading behavior matches one obtained from (4.11).
where g = 2πc √ χ 1 = 8π 2 cξ. We now determine the Regge poles for this spectral function. Replacing ∆ = 2 + iν in (5.3), we can see that, the above has four sets of poles at, There are a few observations in order. The leading trajectory clearly comes from J = −1 + g 2 − ν 2 .
For g = 0, the first and second set above collide to give double poles.

Regge limit: even spin
In this section we compute the Regge amplitude for the even spin. For even spin we need to consider Also we evaluate Res. .
So that, after the J−integral the Mellin amplitude can be written as (where we have dispensed with the factor (−1) J P ± J as in the 0-Magnon analysis ), Again it is shown in the Appendix A.2, that ν integral in (5.6) reduces effectively to an integral over the range [−g, g] as in the 0-magnon case, (5.8) We will use this expression to investigate the weak coupling g → 0 limit.

Weak Coupling: g → 0
In order to evaluate (5.8), we use the following transformation of variables, and rewrite the Mellin amplitude as, with q = g log s and Analogous to the 0-magnon case, we evaluate the integral (5.10) in the limit The integrand now takes the form, (5.13) Again as in the case of zero magnon weak coupling, we can do the integration term by term. Effectively it gets reduced to evaluating the following integral, Further details of the integrals are explained in Appendix B, especially the integrals that go into this evaluation are effectively those in (B.3). However here we write the explicit expression upto a few orders of expansion in g taking into account ζ(∆ i , t), This is the main result for weak coupling of the 1−magnon correlator.

Strong Coupling: g → ∞
Similarly, for the strong coupling limit, the contribution to the integral occurs around ν ∼ 0 and is Leading contribution to exponent of s thus comes from the vicinity of ν = 0, whereby the integrand of (5.6) becomes, 16c 4 ν 2 πg 2 sin(πJ R ) So the approximate integral is , from which it follows that, Further the strong coupling limit g → ∞ we have the asymptotic relation ∼ g 2 , and, (5.20) The contribution from J − e is exponentially suppressed compared to the above.

Regge limit: odd spin
In this section we study the Regge amplitude corresponding to Regge poles associated with the SW transform for odd spins (see (5.4)). Further we set J ± o = −1 ± i g 2 + ν 2 and we evaluate, .

(5.21)
Therefore after the J-integral, the Mellin amplitude is cast into the following form, with, We evaluate this integral in both the strong coupling and weak coupling limit.

Weak Coupling: g → 0
We study the weak coupling limit of (5.22). To begin with, we define ν 2 + g 2 =ŷ 2 so that we have With this change of variable (5.22)can be written as, with, F (J ± o (ŷ)) = 16c 4 ŷ 2 − g 2 sinh π ŷ 2 − g 2 csc(±iπŷ) where we have used the definition in (5.12) and, csc (π(−1 ± ix)) = − csc(±iπx). (5.26) Now we consider the transformation of the variableŷ = g/z. We then have the integral as , with, (5.28) Next we observe in above that, With this observation if we define, we can write (4.47) as with q = g log s. We evaluate this integral in the limit As before s → ∞ has already been taken into account in the Mellin amplitude. The weak coupling limit gives us the following expansion (upto order O(g 3 )), Thus the problem is effectively reduced to the following integral with n ∈ Z + , Thus we have the expression for Mellin amplitude,

Strong Coupling: g → ∞
Analogous to the even spin case, maximal contribution again comes from ν = 0 12 . We write, Note that the coupling dependence gives a phase. And so we have to consider contribution from both J o + and J o − . For brevity, we set J 0± = −1 ± ig. Then, And consequently, .

Weak Coupling
For the weak coupling regime (ξ 1), we can either have ν O(ξ) and ν ∼ O(ξ) which leads to two completely different perturbative solutions for the Regge poles. In the next two subsections, we will consider each of these sub-regimes in the weak coupling limit. We will elaborate on different solutions for the Regge poles and the schematics of the ν integral briefly in the following.
) . This transformation is the so called "shadow transformation" and these second series of poles are what are called "shadow poles". We can focus only on one set of poles for the evaluation of Mellin amplitude and we will consider (6.4). justification for this is given in Appendix C. Now while evaluating the J-integral in (6.1), we make a transformation of the variable from J to α n,1 or β n,1 depending on which family we are looking at. Consequently, our spectral functions (6.3) takes the form,  with poles at α n,1 and β n,1 respectively. Note that the poles of α n,1 and β n,1 lead to the perturbative solution. The corresponding Jacobian of transformation is, Θ(ν) = ∂J n (ν) ∂γ n,1 , where γ n,1 = α n,1 or β n,1 . (6.7) Explicitly, the Jacobian of transformation is given by, Θ(ν) = ξ 2 1 + ξ 2 ∂γ n,2 ∂γ n,1 + ξ 4 ∂γ n,3 ∂γ n,1 + . . . .

(6.8)
Note that if |ν| ∼ O(1), then the perturbative expansion in (6.4) converges for 0 < ξ < 1. However, for |ν| < 1, we expect a new perturbativ expansion even at weak coupling regardless of the explicit dependence of ν on the coupling. In the next section, we will derive the perturbative expansion when |ν| < 1.

ν ∼ O(ξ)
From the previous analysis, it seems that perturbative expansion breaks down when ν ∼ O(ξ). Strictly speaking the perturbation expansion breaks down when ν ∼ O(ξ 4 ) 13 . More generally however, we need to consider when ν and ξ are comparable. In order to make this more precise, we consider the following regimes: 1. For ν = xξ 4 and x = O(1), we can write,  while, E For this case, the Jacobian of transformation is, which starts with O(1) in the coupling and α 1 = 2π 3 ξ 4 . Note that this is includes the perturbative regime (6.9).

Strong Coupling
We will now investigate the strong coupling regime ξ 1. There are similar two regions of interest ν ξ and ν ∼ O(ξ). We will analyze these two cases separately below.

ν O(ξ)
For, ν O(ξ), we consider an expansion around ξ → ∞ keeping ν fixed. For strong coupling ξ → ∞, the denominator of (6.3) can be written in a summation representation as, (6.25) Since E 2 ∼ 1/J 4 for large J, we can make an ansatz, Putting this ansatz in (6.3), we obtain that the solutions for a 1 , We neglect a 1 = −2 4 √ 2 since the exponent of s for this root is extremely subleading compared to the others. The coefficients are obtained recursively as (we evaluate explicitly upto order ξ −3 ), For each value of a 1 , we obtain separate solutions for J. For example, a 1 = 2 4 √ 2 gives , Note that the expansions above (including (6.29), (6.30)) are valid for |ν| < 1 but breaks down for |ν| ∼ O(ξ).

ν ∼ O(ξ)
For large |ν| and large ξ the two are loosely related by ν ∼ O(ξ) 15 . In this case , wee will consider a different expansion for the Regge poles. It is a double expansion, The solutions for a 1 are,

Evaluation of the Mellin amplitude
For strong coupling, one can take |ξ| 1 for all practical purposes. Since the ν−integral (from (−∞, ∞)), has distinct regions for ν ∼ ξ, ν ξ and ν > ξ, can be subdivided according to these regimes. For ν ξ, (6.28) becomes, with the Jacobian of transformation, Now define J R = −1 + a 1 ξ, so that an expansion in 1/ξ about J R in the limit ξ → ∞ gives, Thus, to the leading order, However, as for the exponent of (s/4) we will consider at least upto order 1/ξ. This is because since this is in exponent, the variation over ξ is stronger than that in M.
Further, E with poles at a 1 = 2 4 √ 2 , ±i2 4 √ 2 where as argued below (6.27), we are not considering the negative real pole. We would like to emphasize here that given (6.36), (6.40) is in fact exact coupling. We can further this argument by expanding around ν = 0 (assuming that the ν−integral is peaked around the origin 16 and neglecting the effect of the poles) 17 , This log s dependence is crucial and matches with the strong coupling analysis of 0, 1−magnon cases.
The rest of the power law analysis can be obtained in a straightforward manner by simply picking out the residues of the a 1 −integral. Observe that the dominant contribution will be given by a 1 = 2 4 √ 2, while the others give a phase. Therefore in ξ → ∞ limit, 16 This is true for all practical purposes. 17 Note that, here we have written the expression excluding all the overall sign factors and s → −s factor. Before concluding, we will compare between the leading Regge trajectories for the 0, 1, 2−magnon correlators for the fishnet theory. The leading Regge trajectories, for the magnon correlators are characterized by, • 0−magnon: For the purpose of sheer comparison with N = 4 SYM we provide with weak coupling(f → 0) and strong coupling(f → ∞) expansions of the Regge trajectory. In weak coupling, and in strong coupling , • 1−magnon: The growth of Regge spin for both 0 and 1 magnon correlators with ν has been plotted in figure 2.
Note that the plots are given in terms of the reduced coupling ξ which are related to the relevant couplings for 0− and 1−magnon by f = 4 √ 2cπ 2 ξ and g = 8π 2 cξ respectively. Observe the obvious shift in the intercept which is clear from the weak coupling expressions of the respective Regge trajectories.

Discussions
We present the salient observations of our exercise in the following.
• We have considered the Regge limit of the 0, 1, 2−magnon correlators in the four dimensional conformal fishnet theory. The techniques of Conformal Regge Theory in Mellin space as expounded in [3] has been deployed along the lines of [2] in order to derive the weak coupling expansion for the fishnet correlators. For 0, 1−magnon correlators, we find exact Regge trajectories and compute the Regge limit of the Mellin amplitude in the weak coupling. For the strong coupling limit we do an order of magnitude computation with regards to the leading behavior For 0-magnon correlator we obtain a match with the analysis of [2] in both regimes of coupling.
• For the 2−magnon case, solving the spectral function for any finite value of the coupling seems a formidable task in contrast with the 0, 1−magnon correlators. However, a systematic expansion in the weak/strong coupling limit is still possible. We have analyzed the weak coupling limit in detail while for the strong coupling we have naively compared the leading power law singularity in the Regge limit (along the lines of [2]).
• In comparison with [2], we would like to point out one subtle difference. [2] used the LSZ-type prescription to analyze the on-shell scattering amplitude. For the 0−magnon case, every exchange including the external operators are on-shell. For the 1, 2−magnon case, some or all of the external operators are off-shell 18 . Though we have analyzed the Regge limit of the correlators themselves using the techniques of [3] thereby bypassing the LSZ-type analysis in [2], it is worth of investigating whether we can devise systematic perturbative methods in terms of Feynman Diagrams for the 1, 2−magnon case.
• Another possible direction is to compare the strong coupling results (the order to magnitude of the leading term) with the holographic counterpart i.e. the quantum holographic fishchain model recently discussed in [20,21].
• It would be nice to correlate these Regge trajectories of the "n"-magnon correlators to known results for N = 4 SYM 19 .

Acknowledgements
We thank Gregory Korchemsky for useful discussions during initial stages of the work. We also thank

A Details of Pole Analysis
In this Appendix, we review the contour manipulation that the authors of [2] use to compute the Regge limit at weak coupling and extend it to our analysis of Mellin amplitudes.

A.1 Details of 0-Magnon Analysis
In this subsection we will explain the details of how do we reach the equation (4.8). We follow essentially [2] and review the method for our case. We start with 18 We thank Gregory Korchemsky for enlightening us on this issue. 19 We thank Nikolay Gromov for suggesting this to us. with and , First for brevity we define, We split the integration region in (A.1) as following, The key step is to show that at large s, where the second relation follows from first one upon replacing ν → −ν and taking into account that Φ ± (−ν) = Φ ± (ν). If we now substitute (A.6) into (A.5) then we obtain, To prove (A.6) first we introduce the change variable ν 2 − φ 4 = x 2 so that, With this change of variable (A.6) becomes, Here we took into account that J + 2 and J − 2 are conjugate to each other for real x such that 1−f 4 −x 2 > 0. In a similar fashion, the integral on the right-hand side in the first line of (A.6) we find upon changing Now to match (A.10) into (A.9) we will rotate the integration contour in the integral (A.9). Before that we need to understand the contour prescription of the integral in (A.9) a bit. To get a hold of in which way we need to close the contour we observe that in the large x limit we have, This suggests that we would like to close the x-contour in the lower half of complex x−plane in (A.9) .
The contour that we will use is as below,

Figure 4: Contour Prescription for (A.9)
Now, referred to the above contour prescription, we have ∞ 0 x dx Res. x Observe that we have closed the contour in the lower half plane to ensure that the integral over C, which is a semi-circular arc of infinite radius, vanish. Also note that the residue sum comes with an overall negative sign because we have closed the contour in the clockwise sense.
Now it is very clear from the above representation that only those poles which lie in the lower half-plane , as shown in the figure, i.e, the poles with negative imaginary parts can contribute to the residue sum i.e, the poles that contribute have the generic structure, Next we observe that at these poles, the residue give negative exponents of s at weak coupling. We would like to point this out specifically that this is only the case unanimously in the weak coupling regime, around f → 0. At strong coupling things are not so. Therefore the following reasoning that we are going to present will go through in the weak coupling limit 20 .
20 But we took advantage of (4.8) in weak coupling anyway. So we are not bothered here about strong coupling! Now with the above in place , these contributions are exponentially suppressed compared to the line integral in the Regge limit s → ∞ i.e, the Regge limit. These are the O(s −1 ) terms we wrote explicitly in (A.7) and we are going to neglect these terms in Regge limit. Hence forth while writing we will not write these pole contributions, if any, explicitly and any equality will be understood modulo contributions coming from these poles.
Now we introduce the "Wick Rotation" x = −ix E and finally obtain from (A.9), 14) The integrand has two square-root branch cuts [−∞, −f 2 ) and [f 2 , ∞) and deforming the contour we should not cross the cut.
Next we split up (A.14), To proceed further, we use a crucial observation about the "physical spectrum of t". The vital information is that the physical spectrum for t consists of real values only. And henceforth we will base our analysis on the physical spectrum of t. With this piece of information we observe that the collections of Gamma with suitable values for p, q Since we have (this can be proved for instance using the Euler integral representation of Gamma function) Hence, Φ − (x E ) is real over the entire interval y ∈ [0, ∞). However the factor is purely real for y E ∈ [0, f 2 ] but is purely imaginary for y E ∈ [f 2 , ∞). Thus the piece of integral in (A.15) over the interval [f 2 , ∞) vanishes identically and we have the left-hand side of (A.9) and (A. 10) coincide upto corrections that vanish in s → ∞.
Hence we have the desired relation (A.7).

A.2 Details of 1-Magnon Analysis
In this subsection we will deliver the details of the manipulation leading to the equation (5.8).We start with looking into the following integral, Because the integrand is even under (ν → −ν), we have Under the transformation of variable ν 2 − g 2 = y 2 , where, J + ± = −1 ± iy. The analysis that follows now will actually mimic that done in the previous subsection for zero magnon.
But anyway we give the details step by step. What we do is to convert the above integral effectively into a complex contour integral as shown in the following figure. This is actually a "Wick rotation" which we explain below. For further analysis we refer to the following figure.
(y) (y) C y P where, {y P } are the poles of F (J − e (y)) in y. Note that we have closed the contour in the lower half plane to ensure that the integral over C, which is a semi-circular arc of infinite radius, vanish. Also note that the residue sum comes with an overall negative sign because we have closed the contour in the clockwise sense. Now it is very clear from the above representation that only those poles which lie in the lower half-plane , as shown in the figure, i.e, the poles with negative imaginary parts can contribute to the residue sum i.e, the poles that contribute have the generic structure, y P = (y P ) − i¯ (y P ),¯ (y P ) > 0 (A.24) And since each pole contributes a factor of the form towards the residue, it is immediately clear that these contributions, if any, have the form, Clearly these contributions are exponentially suppressed compared to the line integral in the limit s → ∞ i.e, the Regge limit. Hence forth while writing we will not write these pole contributions, if any, explicitly and any equality will be understood modulo contributions coming from these poles. Now we introduce y = −iy E (this is the "Wick rotation" 21 we mentioned above) and finally obtain, Now, let us look at the wick rotated part, with suitable values for p, q (there are precisely three such combinations in the expression (A.28)). Since we have (this can be proved for instance using the Euler integral representation of Gamma function) Hence, F (J − e (y E )) is real over the entire interval y ∈ [0, ∞). However the factor is purely real for y E ∈ [0, g] but is purely imaginary for y E ∈ [g, ∞). Thus the piece of integral in (A.6) over the interval [g, ∞) vanishes identically and we have therefore, On the other hand, now consider the integral Under the transformation −ν 2 + g 2 =ỹ 2 , we have J e − = −1 −ỹ and , Now note that F (J + − (ỹ)) above is same as F (J + − (y E )) in (A.28) with the replacement y E →ỹ. Thus we have the relation, Equipped with this we have the following identities, Finally, we add them together to arrive at , (A.39)

B Details of various integrals
We note that in zero magnon and one magnon weak coupling case we finally are left with evaluation of the integrals of the form Now we can generate all such integrals from the basic integral by repeated applications of derivative (for non-negative n) or anti derivative (for negative n) with respect to L of the the following basic integral, where I µ (L) is Modified Bessel function of first kind.
For non-negative n, we have the following differential relation, with n = 0 corresponds to no differentiation.
For example, On the other hand we note that for n < 0 the integrand is singular at x = 0. So in this case the integral as such does not exist. However the integral can still be given meaning in the sense of Cauchy Principal value. Thus we have the following integral under consideration, We can get this integral from I 0 (L) by repeated anti derivative operation i.e, repeated indefinite integral w.r.t L. Thus if we define, then,Ĩ n (P ) = L n I 0 (P ) = P dP n Pn dP n−1 . . .

C Details of 2-Magnon Analysis
In this section we give detailed account of the claim that in Regge limit s → ∞, the integral A 1 in (C.1) is highly suppressed compared to the integral A 2 in (6.23) so that in the Regge limit we can dispense with the former and focus on the latter.

(C.3)
Now we have for the integral A 1 ,after doing the J intgral, where, F(J a n (ν), ν) = ν sinh πν Θ a n (ν) sin πJ a n (ν) Putting all the expressions one obtains the following perturbative expressions, with L = (s/4). We modify our integral as follows, In order to do each of the above integrals we will resort to contour integral. Basically we will consider the following contour integral, And therefore we focus our attention towards doing the contour integral for which we will do pole analysis for each integrand I a m in order to take advantage of the Residue theorem, This pole analysis thus shows is that each term of the integral (C.2) is exponentially suppressed compared to the integral A 2 in the Regge limit s → ∞. We now look at the remaining integral, F(J a n (ν), ν) (C.18) First we do the analysis for the integral, where we have defined, λ = log L. Note that as L → ∞ so does λ i.e, λ → ∞. So as far as the integral is concerned we are dealing with a "Stationary Phase" type of configuration. Before proceeding further we would like to mention again , as in the previous analyzes, that the physically relevant values of t are given by the poles of the prefactor Γ(2m − t/2 + 2) above and so the relevant values are t q = 4m + 2q + 4 with q being non-negative integer. While doing many details of the calculation we have to keep this in mind. Now let us focus on the integral. Note that this integral can be written as , Now it really does not solve the integration completely, however for our purpose we really don't need to evaluate the integral. All we need is whether we can find an upper bound for the "Remainder Term", 1 iλ  Note that while we have done this analysis with explicitly the O(ξ 0 ) expression the same conclusion will hold true for higher orders because in higher order basically we will encounter higher order Polygamma functions with the argument, however, unchanged. Also observe that, here we have not considered the contribution coming from the "shadow poles"(see the discussion following (6.5)). The reason for this is that while closing the contour in the upper half-plane the residue contributions from shadow poles are 22 one has to be a bit careful while analyzing at ν = 0 and instead of taking just value we have to consider limiting value.
not picked up. The same are picked up when the contour is closed in lower half plane. So in some sense the Regge poles are divided into two mutually exclusive sets as far as the integral over ν is concerned.
The same conclusions as delineated in this section follow for the shadow poles as well.
Therefore assembling this whole analysis we finally reach the conclusion that in the Regge Limit s → ∞ or equivalently L → ∞ the integral A 1 is highly suppressed to A 2 and this is true in every order of ξ-expansion.