Solution to the γγ
→ η
c
puzzle using the principle of maximum conformality
Sheng-Quan Wang,
1,2,‡
Xing-Gang Wu,
2,*
Wen-Long Sang,
3,§
and Stanley J. Brodsky
4,†
1
Department of Physics, Guizhou Minzu University, Guiyang 550025, People’s Republic of China
2
Department of Physics, Chongqing University, Chongqing 401331, People’s Republic of China
3
School of Physical Science and Technology, Southwest University,
Chongqing 400700, People’s Republic of China
4
SLAC National Accelerator Laboratory, Stanford University, Stanford, California 94039, USA
(Received 17 April 2018; published 31 May 2018)
The next-to-next-to-leading-order (NNLO) pQCD prediction for the γγ
→ η
c
form factor was evaluated
in 2015 using nonrelativistic QCD (NRQCD). A strong discrepancy between the NRQCD prediction and
the BABAR measurements was observed. Until now there has been no solution for this puzzle. In this paper,
we present a NNLO analysis by applying the Principle of Maximum Conformality (PMC) to set the
renormalization scale. By carefully dealing with the light-by-light diagrams at the NNLO level, the
resulting high precision PMC prediction agrees with the BABAR measurements within errors, and
the conventional renormalization scale uncertainty is eliminated. The PMC is consistent with all of the
requirements of the renormalization group, including scheme-independence. The application of the PMC
thus provides a rigorous solution for the γγ
→ η
c
form factor puzzle, emphasizing the importance of
correct renormalization scale-setting. The results also support the applicability of NRQCD to hard
exclusive processes involving charmonium.
DOI: 10.1103/PhysRevD.97.094034
Hadronic bound states of the heavy quarks, heavy
quarkonia, with their rich display of dynamics at different
mass scales, provide a unique platform for understanding
quantum chromodynamics (QCD). Nonrelativistic QCD
(NRQCD) [1] has become the standard tool for studying the
properties of heavy quarkonium as an effective field theory.
Most measurements at the Tevatron, the B-factories, the
LHC, etc., agree well with the NRQCD predictions,
although some measurements have exposed serious incon-
sistencies, puzzles, and thus new challenges.
The simplest exclusive charmonium production process,
γ
γ → η
c
, measured in two-photon collisions, provides
such a puzzle. One can define a transition form factor
(TFF) FðQ
2
Þ for charmonium production,
hη
c
ðpÞjJ
μ
EM
jγðk; εÞi ¼ ie
2
ϵ
μνρσ
ε
ν
q
ρ
k
σ
FðQ
2
Þ; ð1Þ
where J
μ
EM
is the electromagnetic current evaluated at the
timelike momentum transfer squared, Q
2
¼ −q
2
¼
−ðp − kÞ
2
> 0. In 2010, the BABAR Collaboration mea-
sured the TFF FðQ
2
Þ over the range of 2 GeV
2
<Q
2
<
50 GeV
2
[2]; the measurements can be parameterized
as jFðQ
2
Þ=Fð0Þj ¼ 1=ð1 þ Q
2
=ΛÞ where Λ ¼ 8.5 0.6
0.7 GeV
2
. The next-to-next-to-leading-order (NNLO)
NRQCD prediction fails to explain the BABAR measure-
ments over a wide range of Q
2
[3]. As a consequence of this
discrepancy, there has even been doubt whether NRQCD is
applicable. However, a careful discussion of the renorm-
alization scale dependence has been missing in the theo-
retical analysis. The renormalization scale was simply set
as μ
r
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Q
2
þ m
2
c
p
, leading to a substantially negative
NNLO contribution and a large value for jFðQ
2
Þ=Fð0Þj,in
disagreement with the data. This disagreement cannot be
explained by taking higher Fock states into consideration
[4,5]. A possible solution is to use a continuum approach to
the two valence-body bound-state problem in relativistic
quantum field theory [6]. In the present paper, we shall
concentrate on how to solve the disagreement within the
NRQCD framework.
The renormalization scale-setting problem is in fact an
important issue for fixed-order pQCD predictions. Guessing
the renormalization scale and setting an arbitrary range for its
value introduces an arbitrary systematic error into pQCD
predictions, which may lead to predictions inconsistent with
the experimental measurements. The conventional treatment
*
Corresponding author.
wuxg@cqu.edu.cn
†
Corresponding author.
sjbth@slac.stanford.edu
‡
sqwang@cqu.edu.cn
§
wlsang@ihep.ac.cn
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 publis hed article’s title, journal citation,
and DOI. Funded by SCOAP
3
.
PHYSICAL REVIEW D 97, 094034 (2018)
2470-0010=2018=97(9)=094034(5) 094034-1 Published by the American Physical Society