# Open spin chains and complexity in the high energy limit

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## Abstract

In the high energy limit of scattering amplitudes in Quantum Chromodynamics and supersymmetric theories the dominant Feynman diagrams are characterized by a hidden integrability in the planar limit. A well-known example is that of Odderon exchange, which can be described as a composite state of three reggeized gluons and corresponds to a closed spin chain with periodic boundary conditions. In the \(N=4\) supersymmetric Yang–Mills theory a similar spin chain arises in the multi-Regge asymptotics of the eight-point amplitude in the planar limit. We investigate the associated open spin chain in transverse momentum and rapidity variables solving the corresponding effective Feynman diagrams. We introduce the concept of complexity in the high energy effective field theory and study its emerging scaling laws.

## 1 Introduction

The contribution to a cross section due to Odderon exchange can be calculated by solving an integral equation describing the *t*-channel exchange of three off-shell reggeized gluons, the so-called Bartels–Kwiecinski–Praszalowicz (BKP) equation [8, 9, 10]. Since, in MRK, this equation is invariant under two-dimensional conformal transformations in coordinate representation, one can apply techniques developed in conformal field theories and integrable systems [11, 12, 13, 14, 15] to find the solution to the equation. It is remarkable the existence of a mapping to a closed spin chain (CSC) when working in impact parameter representation [11, 12]. However, due to the different possible choices of normalization conditions for the Odderon wave function, it is difficult to know which analytic solution is relevant for a particular scattering process.

In a recent work [16], we solved the Odderon problem using an orthogonal method to those applied so far in the literature on this subject. Working in transverse momentum and rapidity space we solved exactly the equation governing the Odderon Green function. Our solution was expressed as a set of nested integrations which can be evaluated using numerical Monte Carlo techniques.

*Y*and \(\mathbf{p}_{i=4,5,6}\) a rapidity 0. \(\omega \) is a complex variable, Mellin-conjugate of

*Y*. The reggeized gluon propagators are infrared divergent and need a regulator with mass-dimension \(\lambda \). With this regularization, the gluon Regge trajectory at leading order can be written in the form (with \({\bar{\alpha }}_s = \alpha _s N_c / \pi \))

*s*-channel normal gluon with transverse momentum \(\mathbf{k}\), leaving the third reggeized gluon, with momentum \(\mathbf{p}_k\), as a spectator. This generates pairwise nearest neighbor interactions in the corresponding CSC.

In Ref. [16] we showed how to iterate the BKP ternary kernel acting on the initial condition until we reach convergence for a particular value of the relevant expansion parameter \({\bar{\alpha }}_s Y\). The gluon Green function grows with *Y* for small values of this variable to then rapidly decrease at higher *Y* (keeping \({\bar{\alpha }}_s\) constant). Our solution is compatible with previous approaches where the Odderon intercept has been argued to be of \(\mathcal{O} (1)\) [17, 18] (similar results have been found within the dipole formalism [19]).

In the following section we will explain in some detail our procedure to solve an equation similar to Eq. (3) which has a representation as an integrable open spin chain (OSC) in the planar limit as shown by Lipatov in Ref. [20]. In a nutshell, we first iterate it in the \(\omega \) space to then transform the result to get back to a representation with only transverse momenta and rapidity.

## 2 The equation for the open spin chain

*t*-channel. This is the most complicated contribution to the amplitude stemming from the so-called Mandelstam cuts in the associated partial wave [20, 21, 22, 23]. Since the amplitude carries the quantum numbers of a gluon in that channel, this implies that the contributing effective Feynman diagrams are planar and the gluons with momentum \(\mathbf{p}_1\) and \(\mathbf{p}_3\) cannot be directly connected by the function \(\xi \). This is the reason why we now have an OSC. The corresponding BKP-like integral equation then reads

*Y*dependence using

*t*-channel. Now the corresponding Green functions are IR divergent. Fortunately, the associated \(\lambda \) dependence factorizes in a simple form. Let us go back to the BFKL equation in this new color representation, it now reads

## 3 Solution to the open spin chain

*rungs*following the nomenclature associated to generalised ladder diagrams. We will label a rung between Reggeons (1) and (2) by L (left), a rung between Reggeons (2) and (3) by R (right) and a rung between Reggeons (1) and (3) by M. As an example, in Fig. 2 we can see two six-rung ladder diagrams, one for the OSC case (to the left) and one for the closed case (to the right). In the following, without any loss of generality, we will assume that in the OSC case the Reggeons that cannot interact directly via a gluon exchange are the (1) and (3) (Fig. 3).

*adjacency matrix*of a given diagram is the square matrix with off-diagonal elements being the number of lines connecting the vertex

*i*with the vertex

*j*. Its diagonal elements are zero. Using a labelling system as described above allows the fast computation of the adjacency matrix and adjacency list for any ladder diagram opening the road for more detailed studies of the diagrammatic topologies that appear after each iteration of the kernel. As an example we now show the adjacency matrices associated to the diagrams in Fig. 2. For the CSC example,

In Ref. [16], we developed a Monte Carlo code to compute the Odderon Green function based on our experience with BFKLex [24, 25, 26, 27, 28, 29, 30]. In principle, a straightforward approach would be to modify as little as possible our existing code for the CSC case and run it for the OSC one. However, we realized at an early stage of this work that we would need to develop a code tailored especially for the OSC case. This was mainly due to the fact that convergence now was not as fast as in the CSC case and therefore we had to optimise our code as much as possible. In the following we will describe some key issues regarding our numerical approach, for more details we refer the reader to Ref. [16].

*Y*. To be specific, we will consider the following values for the transverse momenta (they are shown in polar coordinates, the first entry stands for the modulus of the momentum and the second one for the azimuthal angle):

*Y*from 1 to 5.5 units and in one case up to 6.5 (Fig. 9).

*i*rungs, once iterated, will generate two new diagrams each with \(i+1\) rungs. This leads to a complete binary tree structure: Two of the momenta integrations are trivial since there are three Dirac delta functions to be fulfilled. However, the

*junctions*(see Ref. [16]) in the OSC are only two: \({\mathcal {J}}_{\text {LR}}\), \({\mathcal {J}}_{\text {RL}}\). Lastly, any diagram with no junction, in other words, any diagram \({\mathcal {J}}_{\text {Q}}\) with Q being a sequence of only L or only R is zero. This becomes clear since all \(\mathbf{p}_i\) are chosen to be different from each other and hence none of the Dirac delta functions in the initial condition is fulfilled.

*multiplicity*(in the sense of counting the number of rungs in the diagram) plots, two of them (smaller and larger rapidities) with momentum transfer, \(\mathbf{q} = (4,0)\) (Fig. 5) and two more for \(\mathbf{q} = (31,0)\) (Fig. 7). We compare these to the corresponding plots from the CSC, Figs. 4 and 6 respectively.

We verify anew that the contributions from each iteration for a given *Y* once plotted versus *n*, where *n* is the number of rungs or iterations, follow a Poisson-like distribution, similarly to the CSC. The similarities between the open and CSC do not end there. In both cases we observe a global maximum at some *n* and then a decrease as *n* increases. The Green function is noticeably smaller when the total momentum transfer is larger. Furthermore, the peak of the distribution moves similarly to larger values of *n* for both values of \(\mathbf{q}\) as *Y* increases, whereas its height gets lower and the distributions are much broader. As \(|\mathbf{q}|\) increases the position of the peak shifts to a larger *n*.

There is an interesting qualitative difference between the closed and OSC cases. Let us focus on the bottom plots in Figs. 4 and 5. We observe that in the OSC the decrease in the value of the maximal point of the distribution takes place very slowly as we vary *Y*. This is different to the CSC configuration where the distribution broadens very quickly with *Y*. This sudden broadening of the multiplicity distribution is a distinct signal of the cylinder topology of the contributing effective Feynman diagrams (Fig. 3).

*Y*is drawn in Figs. 8 and 9. While for the CSC we see that the energy (\(\ln Y\)) dependence plot of the Green function has its peak at relatively small rapidities (\(Y<3\)) for both \(\mathbf{q} = (4, 0)\) and \(\mathbf{q} = (31, 0)\) and then it starts to decrease noticeably fast, for the OSC on the other hand, we see that for \(\mathbf{q} = (4, 0)\) the curve increases monotonically. If we now increase the momentum transfer, we see that for \(\mathbf{q} = (17, 0)\) (brown dashed line) the curve rises although much slower than for \(\mathbf{q} = (4, 0)\). If we increase further to \(\mathbf{q} = (31, 0)\) (red dashed line) we see that for \(Y>6\) (notice that we have pushed the upper limit of

*Y*here for this plot to 6.5 units) the curve seems to decrease. An even further increase to \(\mathbf{q} = (107, 0)\) (orange dashed line) gives us a plot with a clear maximum at around \(Y\sim 3.4\).

To summarize, we find a similar qualitative behavior between the closed and open Green function for the main features we could assess in our numerical analysis. However, the CSC Green function seems to approach the asymptotia much faster than the OSC one. Moreover, the smaller the momentum transfer, the more amplified this trend is. It will be very interesting to see what happens when one integrates the Green function with impact factors which is a crucial step to construct the full *n*-point amplitudes in the supersymmetric theory (Ref. [31] offers an interesting review on the systematics behind these calculations), but this is a question beyond the scope of this work. Let us now conclude with a section devoted to a study of the graph complexity associated to the Feynman diagrams contributing to the gluon Green function.

## 4 Graph complexity

Let us highlight some aspects of graph theory which we have used in our study of the Reggeon spin chains (some reviews on the subject can be found in [32, 33]). Our graphs \(\mathcal{G}\) consist of a set of vertices *V* and propagators (edges) *P*, \(\mathcal{G}=(V,P)\). All the diagrams are connected since we do not allow for \(\mathbf{p}_1\) to be equal to \(\mathbf{p}_4\), \(\mathbf{p}_2\) to \(\mathbf{p}_5\), or \(\mathbf{p}_3\) to \(\mathbf{p}_6\). Only for connected graphs we can define *spanning trees*, which are those paths within the graph which connect all its vertices without any cycles.

The *complexity* of an undirected connected graph corresponds to the number of all possible spanning trees of the graph \(\mathcal{G}\). The *degree* of a particular vertex *i*, \(d_\mathcal{G} (i)\), is the number of half-edges which are in contact with it. Since we can connect two or three propagators to each node then the vertices can have degree 2 or 3.

*V*| is the number of vertices in \(\mathcal{G}\) then we can define the

*degree matrix*of \(\mathcal{G}\), \(D_\mathcal{G}\), as the diagonal \(|V| \times |V|\) square matrix with diagonal elements \(d_\mathcal{G} (i)\). Its matrix elements are of the form \(D_\mathcal{G} (i,j) = d_\mathcal{G} (i) \delta _{i j}\). As mentioned in a previous section, the adjacency matrix of \(\mathcal{G}\), \(A_\mathcal{G} \), is the \(|V| \times |V|\) square matrix with off-diagonal elements being the number of propagators connecting the vertex

*i*with the vertex

*j*. The

*Laplacian matrix*corresponds to the difference of these two matrices: \(L_\mathcal{G} = D_\mathcal{G} - A_\mathcal{G}\). As we will see, these matrices carry the topological information of the graph. For the sake of clarity, we explicitly write the Laplacian matrices associated to the diagrams in Fig. 2. In the CSC we have

*Matrix-Tree theorem*[34] which is one of most fundamental results in combinatorial theory and states that the complexity of a graph corresponds to the value of the determinant of the Laplacian matrix once we remove one of its rows and one of its columns (the determinant of any of its principal minors). The complexity does not depend on a possible ordering of the vertices. Applying the Matrix-Tree theorem to the two graphs in Fig. 2 we find that the number of possible spanning trees in the CSC example is 532 and in the open case 463. For a fixed number of rungs the number of graphs in the CSC case is much larger than in the OSC configuration. As an example we show the relevant topologies along with their corresponding complexity for four rungs in Fig. 10 for the OSC and in Fig. 11 for the OSC. It is natural to find that the OSC topologies are contained in the CSC possible diagrams. Moreover, the Pomeron ladder topology where all rungs are connecting the same pair of Reggeons appears twice in OSC ({L, L, L, L} and {R, R, R, R}) and thrice in CSC ({L, L, L, L}, {M, M, M, M} and {R, R, R, R}). It is interesting to note that the Pomeron ladder has always the maximal complexity

*t*(

*n*) of all the diagram topologies for any given number

*n*of rungs. For example, in Figs. 10 and 11 \(t(n) = 56\) for the Pomeron ladder topologies (see top left diagrams in both figures). The complexity of the Pomeron ladder is equal to the number of spanning trees in a \(2 \times n\) grid which is given by

We have studied what is the contribution to the gluon Green function from different graph complexities. Of course each effective Feynman diagram carries a certain statistical weight in our Monte Carlo method to generate the solution to the BKP equation which is related to the particular number of reggeized gluon propagators and squared Lipatov’s vertices present in a graph with *n* rungs. It is clear that the average complexity of the graphs grows with *n*. We have found an interesting scaling behavior obtained by the following method. Let us consider all those diagrams with the same complexity for a fixed number of rungs. Then we evaluate the average weight of their contribution to the gluon Green function as a complexity class. All of this is calculated for a fixed value of the strong coupling and rapidity. We show a characteristic sample of our results in Fig. 12 for the CSC and in Fig. 13 for the OSC.

Per complexity value, those corresponding to a lower number of rungs have a bigger mean weight but their number is smaller. The amount of possible complexity values grows very fast with *n*. This is not surprising since the number of nodes in the graphs is proportional to the number of rungs. What we find very remarkable is that for the larger values of complexity in each set with the same *n* all different complexities contribute with a very similar weight to the solution of the BKP equation. This trend is independent of having a closed or open Reggeon graph and for different values of the available parameters. We suspect that this “complexity democracy” is likely to be related to the underlying integrability found by Lipatov. To establish the definite link is the subject of some of our current investigations.

## 5 Conclusions and outlook

More recently, Lipatov found a new integrable spin chain, in this case open, in the context of the calculation of scattering amplitudes in the \(N=4\) supersymmetric Yang–Mills theory. This new open spin chain structure is important since it is the most complicated piece to evaluate when dealing with the multi-Regge limit of scattering amplitudes in Mandelstam regions for amplitudes with a large number of legs. It is very important to understand it in detail in order to advance in our knowledge of the all-orders structure of amplitudes in a theory which allows for a smooth matching between the weak and the strong coupling limits [39, 40, 41, 42, 43, 44, 45, 46, 47].

In the present work we have solved the open spin chain problem exactly again using Monte Carlo integration. The infrared divergencies present in the calculation have been shown to factorize and we have investigated the infrared finite part of the gluon Green function. We have shown that it decreases with energy, as in the closed spin chain case although this behavior is delayed as the momentum transfer is reduced. Our results will allow to fix some of the uncertainties present when evaluating the eight-point amplitude in exact kinematics. For this it is still needed to integrate our results over impact factors and this will be the subject of our future work. Our techniques are valid for any number of reggeized gluons and apply also at next-to-leading and higher orders [48]. This implies that they will be important for the evaluation of the general *n*-point amplitudes in exact kinematics [49, 50, 51].

As a by-product of our work, we have found an intriguing scaling behavior of what we can call weighted complexity of the Feynman graphs contributing to the gluon Green function. The complexity of a diagram is a well-defined quantity in graph theory. We have evaluated the average weight per topology in the sense of its total contribution to the gluon Green function and found that it is approximately constant for a fixed number of rungs of the class of effective Feynman diagrams. This “complexity democracy” is very likely related to the integrability found by Lipatov. It will be interesting to find the precise link between both concepts.

## Notes

### Acknowledgements

We acknowledge support from the Spanish Government Grants FPA2015-65480-P, FPA2016-78022-P and Spanish MINECO Centro de Excelencia Severo Ochoa Programme (SEV-2016-0597).

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