#### Open spin chains and complexity in the high energy limit

Eur. Phys. J. C
Open spin chains and complexity in the high energy limit
Grigorios Chachamis 0
Agustín Sabio Vera 0
0 Instituto de Física Teórica UAM/CSIC & Universidad Autónoma de Madrid , Nicolás Cabrera 15, 28049 Madrid , Spain
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 high energy regime in Quantum Chromodynamics
(QCD) has attracted a lot of attention for many years due to
the rich structure of the theory in this limit. Novel effective
degrees of freedom, reggeized quarks and gluons, arise when
evaluating scattering amplitudes in the multi-Regge
kinematical region (MRK). They can form composite states with
different quantum numbers among which the best known
are the Pomeron and the Odderon [
1–3
]. The (perturbative)
hard Pomeron (even under charge and parity conjugation)
is best described within the Balitsky–Fadin–Kuraev–Lipatov
(BFKL) formalism [
4–6
]. It corresponds to the dominant
contribution to cross sections at very large energies and, together
with the Odderon (odd under charge and parity
conjugation), in perturbation theory they correspond, respectively,
to a composite state of two and three reggeized gluons (see
Fig. 1). A large amount of effort has been expended to study
the properties of the Pomeron while for the Odderon the
understanding, both theoretically and phenomenologically,
is more limited [
7
].
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 [
8–10
]. 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 [
11–15
] 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.
The solution to the BKP equation for three Reggeons
corresponds to a six-point amplitude with off-shells gluons
fω p1, p2, p3; p4, p5, p6 (which we write as fω
(p1, p2, p3)). pi are Euclidean two-dimensional transverse
vectors where pi=1,2,3 have rapidity Y and pi=4,5,6 a
rapidity 0. ω is a complex variable, Mellin-conjugate of Y . The
reggeized gluon propagators are infrared divergent and need
a regulator with mass-dimension λ. With this regularization,
the gluon Regge trajectory at leading order can be written in
the form (with α¯ s = αs Nc/π )
α¯ s ln p2
ω(p) = − 2 λ2
When the square of a Lipatov’s emission vertex,
α¯ s θ (k2 − λ2)
ξ pi , p j , pk , k = 4 π k2
(1)
(pi + k)2p2j − (pi + p j )2k2
pi2(k − p2)2
,
(2)
is integrated over the transverse momentum of the gluon with
momentum k it generates a further λ dependence which
cancels that of the gluon trajectory. The function ξ in Eq. (2)
couples two reggeized gluons with momenta pi and p j via a
s-channel normal gluon with transverse momentum k,
leaving the third reggeized gluon, with momentum pk , as a
spectator. This generates pairwise nearest neighbor interactions
in the corresponding CSC.
The BKP equation in the Odderon case combines all these
elements making use of a ternary kernel:
+
d2k ξ (p2, p3, p1, k) fω (p1, p2 + k, p3 − k)
d2k ξ (p1, p3, p2, k) fω(p1 + k, p2, p3 − k). (3)
The initial condition for evolution in rapidity corresponds to
three reggeized-gluon propagators normalized in the form of
three two-dimensional Dirac delta functions δ(2) (p1 − p4)
δ(2) (p2 − p5) δ(2) (p3 − p6).
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 α¯ s Y . The gluon Green function grows with Y for small
values of this variable to then rapidly decrease at higher Y
(keeping α¯ s constant). Our solution is compatible with
previous approaches where the Odderon intercept has been argued
to be of 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 ω 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
When evaluating the eight-gluon amplitude in the N = 4
supersymmetric theory in MRK and in certain physical
regions we encounter a contribution with three Reggeized
gluons exchanged in the t -channel. This is the most
complicated contribution to the amplitude stemming from the
socalled Mandelstam cuts in the associated partial wave [
20–
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 p1 and p3 cannot be directly connected by the
function ξ . This is the reason why we now have an OSC. The
corresponding BKP-like integral equation then reads
d2k ξ (p2, p3, p1, k) fω(p1, p2 + k, p3 − k). (4)
Following closely our approach for the CSC, we can solve
this equation by iteration. With this in mind, it is convenient
to use the operator notation
O(k) ⊗ f (p1, p2, p3)
≡ ξ (p1, p2, p3, k) f (p1 + k, p2 − k, p3)
+ ξ (p2, p3, p1, k) f (p1, p2 + k, p3 − k),
to write the equation in the form
(ω − ω(p1) − ω(p2) − ω(p3)) fω (p1, p2, p3)
= δ(2) (p1 − p4) δ(2) (p2 − p5) δ(2) (p3 − p6)
d2ki e(ω(p1)+ω(p2)+ω(p3))(yi−1−yi )O(ki ) ⊗
× e(ω(p1)+ω(p2)+ω(p3))yn δ(2) (p1 − p4)
× δ(2) (p2 − p5) δ(2)(p3 − p6).
It is important to comment now on the infrared finiteness of
this expression. The original non-forward BFKL equation is
projected on a color singlet representation and reads
(ω − ω(p1) − ω(p2))) fω (p1, p2) = δ(2) (p1 − p3)
+ 2
d2k ξ (p1, p2, k) fω(p1 + k, p2 − k).
The solution to this equation is λ independent in the limit
λ → 0 because each of the logarithmic divergencies
generated upon integration over one of the gluons with momentum
k is compensated by the same logarithmic λ dependence in
two of the gluon Regge trajectories. The original BKP
equation (for any number of exchanged reggeons) projected on
a singlet is IR finite for the same reason. For example, the
divergence generated by
(11)
(5)
(6)
(7)
(8)
(9)
(10)
in Eq. (3) is cancelled against the one present in (ω(p1)
+ ω(p2))/2.
The situation is different when we project on the adjoint
representation in the t -channel. Now the corresponding
Green functions are IR divergent. Fortunately, the associated
λ dependence factorizes in a simple form. Let us go back to
the BFKL equation in this new color representation, it now
reads
(ω − ω(p1) − ω(p2))) fω (p1, p2) = δ(2) (p1 − p3)
+
d2k ξ (p1, p2, k) fω(p1 + k, p2 − k).
(12)
We can see that there is a divergent λ dependence generated
by half of the trajectories contribution (ω(p1) + ω(p2)) /2
not compensated by the integration of ξ . In Refs. [
24,25
] we
have shown that in this case the gluon Green function can be
written as
⎛
fBaFdjKoLint (p1, p2, Y) = ⎝
2 ⎞ α¯s2Y
λ
where fBaFdKjoLint is finite when λ → 0. As we mentioned
before, the usual BKP equation for three reggeized gluons
is finite while the open BKP-like equation we are studying
lacks the link term ξ between p1 and p3. This means that
the combination of trajectories (ω(p1) + ω(p3)) /2
generates divergencies which are not cancelled. Similarly to the
adjoint BFKL case these divergencies factorize at the level
of the Green function in the form
fBaKdjPoint (p1, p2, p3, Y)
= ⎝
⎛
2 ⎞ α¯s2Y
λ
p2p2 ⎠
1 3
fBaKdPjoint(p1, p2, p3, Y).
(14)
It is the infrared finite function fBKP the one we will study
in depth in this work. Let us now indicate that the λ
independence can be achieved order-by-order in a coupling
expansion but this is not very useful for our purposes. We work
instead with an effective field theory where each Feynman
diagram is not infrared finite. Finiteness is achieved only after
summing up all contributing effective Feynman diagrams to
all those orders being numerically significant. We explained
this method in detail in Refs. [
24,25
]. We describe the main
features emerging from our solution in the following section.
3 Solution to the open spin chain
In this section we evaluate the solution to the OSC and
compare it to that in the closed case. It is instructive to discuss
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first a few key points regarding the type of diagrams one
encounters in the two cases.
We will always have three Reggeons labeled (1), (2) and
(3) vertically aligned and ordered such that (1) is to the left
and (3) to the right (see the plot at the right hand side of Fig. 1).
In the CSC case each Reggeon is allowed to interact with any
of the other two via gluon exchange. We will call these gluon
exchanges 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).
From Fig. 2, it is evident that an ordered list of the labels
that correspond to the rungs of a ladder diagram defines it
unambiguously. In our diagrams we only have two types of
nodes: those with two attached legs and those with three.
There are only six of the first type which correspond to
threepoint vertices where we have removed one propagator
carrying one of the external momenta pi . The 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,
BFKLex [
24–30
]. 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
].
We want to compute Eq. (9) in order to obtain f (p1, p2,
p3, Y ) for some given momenta configuration p1, p2, p3, p4,
p5, p6 and for varying rapidity 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):
p1 = (10, 0);
p2 = (20, π );
p3 = q − p1 − p2;
p4 = (20, 0);
p5 = (25, π );
p6 = q − p4 − p5.
The units of the moduli in the two-vectors above are
expressed in GeV. The momentum transfer q satisfies
p1 + p2 + p3 = q = p3 + p4 + p6
and takes the following values: q = {(4, 0), (17, 0), (31, 0),
(107, 0)}. Finally, we vary the rapidity Y from 1 to 5.5 units
and in one case up to 6.5 (Fig. 9).
As in the CSC case, non-zero contributions to f (p1, p2,
p3, Y ) will appear only after we consider two rungs. Any
given diagram with i rungs, once iterated, will generate two
new diagrams each with i + 1 rungs. This leads to a complete
binary tree structure:
0
L
R
LL
LR
RL
RR
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: JLR, JRL.
Lastly, any diagram with no junction, in other words, any
diagram JQ with Q being a sequence of only L or only R is
(17)
(18)
zero. This becomes clear since all pi are chosen to be
different from each other and hence none of the Dirac delta
functions in the initial condition is fulfilled.
In the following we are presenting four multiplicity (in the
sense of counting the number of rungs in the diagram) plots,
two of them (smaller and larger rapidities) with momentum
transfer, q = (4, 0) (Fig. 5) and two more for 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 q as Y increases,
whereas its height gets lower and the distributions are much
broader. As |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).
The six-point gluon Green functions correspond to the
areas under the distributions just described. Their
dependence with 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 q = (4, 0) and q = (31, 0) and then it starts to
decrease noticeably fast, for the OSC on the other hand, we
see that for q = (4, 0) the curve increases monotonically.
If we now increase the momentum transfer, we see that for
q = (17, 0) (brown dashed line) the curve rises although
much slower than for q = (4, 0). If we increase further to
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 q = (107, 0) (orange dashed line) gives us a plot
with a clear maximum at around Y ∼ 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 G consist
of a set of vertices V and propagators (edges) P, G = (V , P).
All the diagrams are connected since we do not allow for p1
to be equal to p4, p2 to p5, or p3 to p6. Only for connected
graphs we can define spanning trees, which are those paths
within the graph which connect all its vertices without any
cycles.
If |V | is the number of vertices in G then we can define
the degree matrix of G, DG , as the diagonal |V | × |V | square
matrix with diagonal elements d (i ). Its matrix elements are
G
of the form DG (i, j ) = dG (i )δi j . As mentioned in a previous
section, the adjacency matrix of G, A , is the |V | × |V |
G
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: LG = DG − AG . 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
is the well-known 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 × n
grid which is given by
t (n) = 4 t (n − 1) − t (n − 2),
with t(0) = 0, t(1) = 1 or, equivalently,
t (n) =
(2 +
√3)n − (2 − √3)n
2√3
.
(21)
(22)
It will be relevant to investigate whether similar relations
hold for the sub-leading in complexity topologies in both the
OSC and CSC cases.
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
The high energy limit of scattering amplitudes in field
theory and gravity [
35–38
] has a very rich structure. In QCD an
integrable structure arises when the center-of-mass energy is
much larger than any Mandelstam invariant in the scattering
process. This integrability, realized in coordinate
representation, is related to the mapping to a Heisenberg ferromagnet
with non-compact spins and periodic boundary conditions
(closed spin chain). Recently, we have shown how to solve
this system in momentum and rapidity space making use of
Monte Carlo integration techniques. Our solution calculates
the gluon Green function exactly and is general, it does not
rely on any choice of normalization of the composite state
wave function. For the three reggeon case it corresponds to
the solution to the perturbative Odderon.
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 [
39–47
].
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
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Fig. 10 Open spin chain topologies with 4 rungs. The labels of the
topologies from left to right and top to bottom are {L, L, L, L}, {L,
L, L, R}, {L, L, R, L}, {L, L, R, R}, {L, R, L, L}, {L, R, L, R}, {L,
R, R, L}, {L, R, R, R}, {R, L, L, L}, {R, L, L, R}, {R, L, R, L}, {R,
L, R, R}, {R, R, L, L}, {R, R, L, R}, {R, R, R, L}, {R, R, R, R}. The
corresponding complexity for each graph is shown
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1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
316 4 323 4 319 4 3235 64 3165 64 3195 64 3515 64 3515 64 3556 64
5 6 5 6 5 6
7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8
Fig. 11 Closedspinchaintopologieswithn=4rungs.Thereareintotal3n =81topologies.Trivially,thesetoftopologiesfoundintheOSC
(Fig.10)iscontainedasasubsethere.Thecorrespondingcomplexityforeachgraphisshown
123
10-7
q=4, Y=1
q=4, Y=1
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 [
49–51
].
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.
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-20160597).
Open Access This article is distributed under the terms of the Creative
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