Energy-energy correlation in N = 4 super Yang-Mills theory at
next-to-next-to-leading order
J. M. Henn,
1
E. Sokatchev,
2
K. Yan,
1
and A. Zhiboedov
3
1
Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, 80805 München, Germany
2
LAPTh, Universit´e Savoie Mont Blanc, CNRS, B.P. 110, F-74941 Annecy-le-Vieux, France
3
Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland
(Received 3 April 2019; published 26 August 2019)
We develop further an approach to computing energy-energy correlations (EEC) directly from finite
correlation functions. In this way, one completely avoids infrared divergences. In maximally super-
symmetric Yang-Mills theory (N ¼ 4 sYM), we derive a new, extremely simple formula relating the EEC to
a triple discontinuity of a four-point correlation function. We use this formula to compute the EEC in N ¼ 4
sYM at next-to-next-to-leading order in perturbation theory. Our result is given by a twofold integral
representation that is straightforwardly evaluated numerically. We find that some of the integration kernels
are equivalent to those appearing in sunrise Feynman integrals, which evaluate to elliptic functions. Finally,
we use the new formula to provide the expansion of the EEC in the back-to-back and collinear limits.
DOI: 10.1103/PhysRevD.100.036010
I. INTRODUCTION
The energy-energy correlation (EEC) [1] measures the
energy flow through a pair of detectors separated by an
angle χ; see Fig. 1. The EEC has several nice properties and
may help to understand better the nature of jets in quantum
field theory. It is an infrared-safe observable [2,3] that
can be computed perturbatively. Moreover, it has simple
factorization properties in the back-to-back (χ → π) [4–8]
and collinear (χ → 0) [9,10] limit. This knowledge can be
used to match fixed-order predictions to resummation
calculations [6–8]. On the other hand, the EEC is exper-
imentally measurable, and in particular has been used for
precision tests of QCD and measurement of the strong
coupling constant α
s
[11–13].
The EEC at leading order (LO) is known since [1], while
the next-to-leading order (NLO) and next-to-next-to-lead-
ing order (NNLO) results were obtained numerically in
Refs. [14–21] and [22,23], respectively. Only very recently,
the NLO result was computed analytically [24]. Analytic
results are important both conceptually and practically, for
example to test numerical codes and to control numerical
instabilities in special kinematic regions.
Although the EEC is infrared finite, the standard
approach to computing involves infrared divergent scatter-
ing amplitudes and phase-space integrals [24,25]. On the
other hand, it can be defined starting from correlation
functions, which are infrared finite [26 –30]. For example,
for e
þ
þ e
−
→ γ
→ X, the main ingredient is a four-point
correlation function of two energy-momentum tensors
(representing the two detectors), and two electromagnetic
currents, which create the electron-positron pair from the
vacuum.
To the best of our knowledge, this approach has not yet
been implemented in QCD. On the other hand, these ideas
were applied in N ¼ 4 super Yang-Mills (sYM) [30–32],
culminating in the first analytic calculation of and EEC at
NLO [33]. The structure of this result, and in particular the
types of polylogarithmic functions appearing in it, foreshad-
o wed the structures later found in QCD [24]. Therefore, any
analytic information at NNLO is very desirable.
We show that for an analog of the electromagnetic
current in N ¼ 4 sYM, the EEC is computed by a new,
extremely simple formula, given by a twofold integral of a
particular triple discontinuity of the four-point correlation
function; see Eq. (7) below.
FIG. 1. Graphical representation of the energy-energy correla-
tion: particles produced out of the vacuum by the source are
captured by the two detectors located at spatial infinity in the
directions of the unit vectors
n
1
and
n
2
.
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI. Funded by SCOAP
3
.
PHYSICAL REVIEW D 100, 036010 (2019)
2470-0010=2019=100(3)=036010(7) 036010-1 Published by the American Physical Society