Analytical Computation of Energy-Energy Correlation at Next-to-Leading Order in QCD
Lance J. Dixon,
1,*
Ming-xing Luo,
2,†
Vladyslav Shtabovenko,
2,‡
Tong-Zhi Yang,
2,§
and Hua Xing Zhu
2,∥
1
SLAC National Accelerator Laboratory, Stanford University, Stanford, California 94039, USA
2
Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang Universi ty, Hangzhou 310027, China
(Received 17 January 2018; published 9 March 2018)
The energy-energy correlation (EEC) between two detectors in e
þ
e
−
annihilation was computed
analytically at leading order in QCD almost 40 years ago, and numerically at next-to-leading order (NLO)
starting in the 1980s. We present the first analytical result for the EEC at NLO, which is remarkably simple,
and facilitates analytical study of the perturbative structure of the EEC. We provide the expansion of the
EEC in the collinear and back-to-back regions through next-to-leading power, information which should
aid resummation in these regions.
DOI: 10.1103/PhysRevLett.120.102001
Introduction.—The energy-energy correlation (EEC) [1]
measures particles detected by two detectors at a fixed
angular separation χ, weighted by the product of the
particles’ energies. The EEC is an infrared-safe characteri-
zation of hadronic energy flow in e
þ
e
−
annihilation. It has
been used for precision tests of quantum chromodynamics
(QCD) and measurement of the strong coupling constant α
s
[2,3]. In perturbative QCD, the EEC is defined by
dΣ
d cos χ
¼
X
i;j
Z
E
i
E
j
Q
2
δð
n
i
·
n
j
− cos χÞdσ; ð1Þ
where i and j run over all the final-state massless partons,
which have four-momenta p
μ
i
and p
μ
j
(including the
case i ¼ j at χ ¼ 0); Q
μ
is the total four-momentum of
the e
þ
e
−
collision and dσ is the differential cross section.
The three-vectors
n
i;j
point along the spatial components
of p
i;j
. The definition Eq. (1) implies the sum rule
1
σ
Z
1
−1
d cos χ
dΣ
d cos χ
¼ 1; ð2Þ
where σ is the total cross section for e
þ
e
−
annihilation to
hadrons.
The leading order (LO) QCD prediction for the EEC has
been available since the 1970s [1]:
1
σ
0
dΣ
d cos χ
¼
α
s
ðμÞ
2π
C
F
3 − 2z
4ð1 − zÞz
5
× ½3zð2 − 3zÞþ2ð2z
2
− 6z þ 3Þlogð1 − zÞ
þ Oðα
2
s
Þ; ð3Þ
where σ
0
is the Born cross section for e
þ
e
−
→ q
¯
q, C
F
is the
quadratic Casimir eigenvalue in the fundamental represen-
tation, and we have introduced z ¼ð1 − cos χÞ/2. The cross
section is strongly peaked at χ ¼ 0 (z ¼ 0) and χ ¼ π
(z ¼ 1), regions that require resummation of logarithms
due to emission of soft and collinear partons. At inter-
mediate angles, higher-order corrections tend to flatten the
distribution.
The EEC was first computed numerically at next-to-
leading order (NLO) in QCD by several groups in the
1980s and 1990s, originally leading to conflicting results.
Different methods were used to handle soft and collinear
singularities from real radiation: phase-space slicing [4–7],
subtraction methods [6,8–14], or hybrid schemes [6,7,15].
Accurate numerical NLO results are available from the
program E
VENT
2, based on dipole subtraction [13,14].
Quite recently, the EEC has been computed at NNLO in
QCD using the CoLoRFulNNLO local subtraction
method [16,17].
In perturbation theory, the EEC is singular in both the
collinear (z → 0) and back-to-back regions (z → 1), as can be
seen explicitly from Eq. (3). The leading-logarithmic collin-
ear behavior can be obtained from the “jet calculus” approach
[18,19], in terms of the anomalous dimension matrix of twist-
two, spin-three operators [11,19]. Resummation of the EEC
in the back-to-back (Sudakov) region has been performed
at next-to-leading-logarithmic (NLL) and NNLL accuracy
[20–22]. Quite recently, a factorization formula for the EEC
has been derived which permits its resummation to N
3
LL
[23]. Possible nonperturbative corrections to the EEC have
also been investigated [24].
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 LETTERS 120, 102001 (2018)
Editors' Suggestion
0031-9007=18=120(10)=102001(6) 102001-1 Published by the American Physical Society
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