# Pushing the precision frontier for
gravitational waves

Michèle Levi111https://sites.google.com/view/levim Niels Bohr Institute,
University of Copenhagen,
Blegdamsvej 17, 2100 Copenhagen, Denmark

(August 6, 2021)

###### Abstract

Pushing the precision frontier for gravitational waves is one of the most
urgent tasks in theoretical physics today, in light of the increasing influx
of data from a rapidly growing worldwide network of gravitational-wave
detectors. Levi will analytically predict gravitational radiation from such
compact binaries and further develop the use of quantum field theory advances
to study gravity. She seeks to uncover profound duality relations between
gauge and gravity.

International Editor Clifford Holt spoke to Levi about her research, the
importance of analytical high-precision predictions for real-world
gravitational-wave data, and potential challenges as she works to develop a
better idea of universal commonalities across classical and quantum field
theories.

Firstly, congratulations on being awarded the Ernest Rutherford Fellowship.
How important are funding mechanisms such as these, and what will the
Fellowship mean for your own research and career?

Thank you! A funding mechanism like this prestigious early-career award from
the STFC (one of the seven UK research councils under UKRI) is very important
and unique. This is the most competitive scheme in the UK for early-career
researchers in my domain, devised to ensure that those 10 carefully-selected
awardees, who are recognised as the very top international talent and
leadership ‘material’ in science early on in their career, will be integrated
into top academic positions in the leading research institutions in the UK.

This will enable the awardees to play a key role in the coming decades in the
scientific and transformative leadership at the national level in the UK and
will also represent the UK at the international level, solidifying its
position as a scientific world-leader for the next generations. As a public
funder, UKRI selects these future leaders to shape strategic UK policies, to
bolster the UK’s scientific place in the world, and to bring about a radical
shift in the current research culture in terms of the way that science is
conducted, namely its standards and its values, but also its role and
commitment within society.

Currently, many academics have sadly forgotten that their raison d’etre is to
serve the public, that they have the privilege of expanding the frontiers of
knowledge mainly thanks to the public money that is funding their research. We
must insist that science that is peer-reviewed and approved for publication
and funding lives up to the standards of FAIR (Findabilty, Accessibilty,
Interoperabilty, Reusabilty) principles, namely of responsible research of
integrity, that is publicly available, widely accessible, and reproducible. We
should also advocate for equality, diversity, and inclusion in Science, which
should properly represent Society, and engage with the public. It is very
clear that UKRI as a public funder has a strong commitment to realising these
aspirations, which are unfortunately still overlooked by the majority of
private funders.

I am excited and honoured to transfer and base my cutting-edge research
programme in the UK, and to be part of making the UK a world-leading force in
the timely domain of gravitational waves with the most innovative theoretical
approaches and computational tools applied to gravity. I am also incredibly
aligned with the aspirations of UKRI to transform the current research
culture, and as I myself belong to an overlap of under-represented and
marginalised groups in science, I am particularly aware to how systemic
inequalities affect the ability of students and researchers to truly realise
their potential and excel, or even to just conform to the common skewed
metrics of academic merit.

I adamantly champion and pursue equality, diversity, and inclusion in every
aspect of my everyday scientific activities. I believe and I see that it is
only through these consistent everyday actions and through my being a role
model that I can ultimately translate my research work to further concrete
scientific and societal impact. I also adamantly advance the broad
communication of science across the research community and through public
outreach, which I regard as a social imperative. I also find it extremely
rewarding to share my knowledge with as many people as possible.

Your project aims to deliver analytical high-precision predictions for real
world gravitational wave data. Can you tell me more about this and why it is
important?

Gravitational radiation is a probing prediction of any complete theory of
gravity. Currently, we still do not have a complete theory of gravity and this
is true for both ends of the scale, since we do not know how to describe
gravity on quantum scales, which are the smallest scales, and we do not have a
theory that captures gravity on the largest scales – cosmological scales,
since we do not know how to explain the puzzle of dark matter or the
accelerated expansion of the Universe that was discovered in 1998.
Understanding how the force of gravity works on these more ‘extreme’ scales is
possibly the deepest and most fundamental longstanding open problem in
physics.

The first Earth-shaking gravitational-wave (GW) detection announced in 2016
launched a new era of gravitational-wave astronomy and precision gravity,
which is only in its infancy, with vast future impacts in astrophysics and
cosmology and, most notably, in fundamental physics. Since this first
milestone detection, ground-based detectors keep multiplying, and many more
are planned for the coming years. They are designed to measure various
frequency ranges, which will enable a richer array of sources and events that
generate gravitational radiation to be captured.

The technology of measurements is constantly upgraded, either in currently-
operating detectors like LIGO or VIRGO, or in upcoming and planned ones, like
KAGRA in Japan, and others in the US, Europe, India, and Korea. Furthermore,
there are also concrete planned space-based experiments from Europe, China,
and Japan that will also target smaller frequencies and other types of
sources. All in all, in the coming years and decades we are headed towards a
growing influx of gravitational-wave data that will have higher and higher
quality.

Let me stress what, I believe, much of the community in Europe and the US does
not fully recognise yet: that Asia is about to emerge as possibly the major
player in gravitational-wave astronomy in the coming years: in Japan, KAGRA is
already operational; in India, IndiGO will become operational in just a few
years; and in Korea, SOGRO is planned to uniquely target mid-range frequencies
from Earth, while China and Japan are also planning space-borne detectors,
TianQin and DECIGO, respectively.

How will your research help to inform the activities of detectors such as
VIRGO and LIGO, as well as perhaps LISA? How could your theoretical work come
to complement future observations/detections?

My research generates theoretical data from which a huge bank of templates is
created to be matched in the detectors by the real data that is measured. The
GW signal is weak and is covered by many sources of noise so, in order to
identify it the matched filtering technique is used for detection, which
requires the most accurate theoretical signal templates possible. When a match
is found, a detection is announced. Yet, lots of real-world data, even if is
great data is just data without interpretation. What transforms data into
knowledge is theory and analysis. This is especially true for the
gravitational-wave signal since it is so weak, so it is really important to
separate the noise from the true signal.

For the most part, the observable differences among the plethora of candidate
theories for gravity suggested along the years have been insanely tiny. In
most cases, all of these theories seem grossly similar within gravitational-
wave measurements. So it is only the theoretical resolution into various
physical effects at a very high-precision, such as those that have been
computed in my research since my graduate studies, that will enable us to
really distinguish among the various suggested theories of gravity, and to
close in on which is the viable one.

Another feature which is rather unique to my own theoretical framework is that
it provides predictions for generic sources of compact binaries, not only for
binaries of two black holes. As the technology in upcoming detectors advances,
and as the sensitivity improves, we are able to observe more and more neutron
stars, for example, in the radiating binaries. Whereas from black holes we can
‘only’ learn about gravity theories, neutron stars also tell us about the
theory of QCD (Quantum Chromodynamics) in extreme conditions that no human-
built lab could produce. With future detectors, we also expect to observe
white dwarfs, and learn more about them.

Have you yet been able to identify any potential challenges that you expect to
have to overcome as you work to develop a better idea of universal
commonalities across classical and quantum field theories?

Right now, we see some analogies between interactions of classical extended
gravitating objects like the huge non-rotating black holes in radiating
binaries, and elementary particles of quantum nature, which we call ‘scalar
particles’ – these are boson particles of zero quantum spin. But the precise
mapping between them, and more importantly its generalisations for all types
of elementary particle, is poorly understood. In particular, my theoretical
work pushed the understanding of spinning gravitating objects, which would
also be analogous to higher-spin particles. Currently, the highest-spin
particle that has been observed in nature is of spin one, and we also believe
there could be particles of spin three/two according to supersymmetric
theories, as well as our theoretically predicted graviton – a massless
particle of spin two. But what about particles with spins larger than two? Do
they exist? Could they be elementary? Or only show up as what we call
‘composite’ particles, which must be composed of elementary ones?

At the moment, we cannot even manage to give well-defined predictions of
observables of quantum field theories that capture such higher-spin particles,
and we are certainly not able to write down such quantum field theories
explicitly. Furthermore, we are not certain whether this fundamental issue
exists already at the classical level, i.e. with the type of two-body
interactions that my work is tackling for spinning gravitating objects. The
big challenge for my research programme, therefore, would be to more
rigorously and generally link the classical and quantum sides of these
analogies and to clarify whether there is indeed a fundamental issue with
higher-spin – and, if so, whether it originates in the quantisation of the
theory, or if it already exists at the classical level.

This all relates to possibly the biggest open question in theoretical physics
of how to complete the theory of gravity at high energies, and therefore it
would indeed be a breakthrough for my upcoming research programme to be able
to provide any further insights on that.

What are your short/long term hopes and ambitions?

Since my graduate studies, I have had the privilege of pioneering and leading
innovative analytical work and high-precision computation on the highly
intricate problem of compact binaries that source gravitational radiation. At
the time, I did not believe that I would live to see actual detections of
gravitational waves from these sources, since at that time the detectors had
been operational for more than a decade without any successful detection.

Yet, the detection came in February 2016, and I can still remember it as
though it were yesterday; I was overwhelmed with excitement and wonder for
what was to come when the first observation of a gravitational-wave signal by
LIGO was announced that day. Since then, the scientific prospects of
gravitational-wave science have been continuously improving and exceeding all
expectations. The future truly looks symphonic, filled with gravitational-wave
echoes.

With this increasing influx of data of an ever improving quality, we have a
chance to really push theoretical predictions and to test so much more theory
in gravity and even in QCD. My research programme is exactly aimed at
maximising the potential of what we can learn from gravitational-wave
measurements on the theory of gravity across all scales.

I hope to engage the broader scientific community with this exciting new
domain of research, such that it really crosses over all sub-fields of
theoretical physics, especially since this is a great scientific challenge
that can benefit from so many different specific perspectives. I believe that
every one of these perspectives can bring in something fresh and interesting
to the mix. I also believe that, to some extent, I have already contributed to
the widespread attention that this type of research obtained in the
traditional high energy physics community. In the UK, I hope to engage more
people and institutions with the research in this field as the UK really has
the potential to become a superpower in this domain, especially in the type of
analytical work that I am involved in, which has been traditionally dominated
by France.

I am also a big believer in open code and open data, and I really want to see
individual researchers, big experimental collaborations, scientific journals,
and academic institutions all persistently moving towards this in the coming
years. The scientific challenges at hand are too big, and we need to focus on
the science and make our best joint efforts to make breakthroughs, instead of
yielding to the old zero-sum game mentality that keeps theoretical physics
stagnated.

Simply put, I just hope that, together, we can all increase our knowledge of
the world around us. Isn’t that the greatest privilege of our society?

## Acknowledgements

ML received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant 847523.
Her upcoming research program will be supported by the STFC Rutherford grant
ST/V003895/1 “Harnessing QFT for Gravity”.


#  Relativistic corrections to exclusive decays $\psi(n)\to\rho\pi$ and
possible solution to the “$\rho\pi$-puzzle”

Nikolay Kivel
Physik-Department, Technische Universität München,
James-Franck-Str. 1, 85748 Garching, Germany

###### Abstract

We study relativistic corrections to exclusive $S$-wave charmonium decays into
$\rho\pi$ and $\gamma\pi$ final states. The contribution of relative order
$v^{2}$ and the set of associated higher order corrections are calculated
using NRQCD and collinear factorisation framework. Numerical estimates show
that the dominant effect is provided by the corrections of relative order
$v^{2}$. The numerical values of these contributions are of the same order as
the leading-order ones. These results suggest that the sum of relativistic and
radiative QCD corrections can potentially explain the “$\rho\pi$-puzzle”.

##  1 Introduction

A description of $S$-wave charmonium decays into $\rho\pi$ final state is
already a long-standing problem in QCD phenomenology. The branching ratios for
$J/\psi$ and excited state $\psi(3686)\equiv\psi^{\prime}$ are measured
sufficiently accurately and their ratio is found to be very small [1]

$\displaystyle
Q_{\rho\pi}=\frac{Br[\psi^{\prime}\to\rho\pi]}{Br[J/\psi\to\rho\pi]}\approx
0.20\times 10^{-2}.$ (1)

This corresponds to a strong violation of the “13%-rule”, which suggests that
$Q_{\rho\pi}\approx Q_{e^{+}e^{-}}\simeq 0.13$. The latter is valid only if
the decay amplitudes of $S$-wave charmonium are dominated by the leading-order
contribution in the QCD factorisation framework (pQCD). Therefore the
disagreement between the data and qualitative theoretical expectation
indicates about large dynamical effects, which are not accounted by the
leading-order approximation of pQCD.

The problem has attracted a lot of attention and many different qualitative
ideas and phenomenological models have been proposed in order to understand
the small value of $Q_{\rho\pi}$. Almost all of proposed explanations use
different ideas about long distance QCD dynamics; a comprehensive overview of
the topic can be found in Refs.[2, 3].

The dominant role of some nonperturbative dynamics is related to the fact that
the QCD helicity selection rule suppresses the valence contribution to the
decay amplitude. Therefore, it is necessary to take into account for the one
of outgoing mesons a non-valence component of the wave functions, which is
suppressed by additional power $\Lambda/m_{c}$. However, already long ago in
Refs. [4, 5] it was found that pQCD framework yields a reliable leading-order
estimate for the $J/\psi$ branching ratio. In Ref. [4] the non-valence
contributions are described by the three-particles twist-3 light-cone
distribution amplitudes (LCDAs). These nonperturbative functions are process
independent and the first few moments of these functions can be estimated
using QCD sum rules. At present time corresponding matrix elements were
studied and revised for various mesons, see updates in Refs. [6, 7, 8].
Therefore it is reasonable to believe that pQCD description is a good starting
point in order to develop a systematic description of the process within the
effective field theory framework.

Following this way one faces with the problem in the description of
$\psi^{\prime}\to\rho\pi$, which must be strongly suppressed relative
$J/\psi\to\rho\pi$ in order to get the small ratio (1). There are various
assumptions about possible dynamical origins for this suppression. Often they
are related to the fact that the mass of excited state $\psi^{\prime}$ is
close to the open charm threshold and this can lead to dynamical effects,
which provide the crucial difference between $J/\psi$ and $\psi^{\prime}$
decays. The possible scenarios include: destructive interference of the large
non-valence and valence contributions [4, 9]; suppression of the colour-
singlet $c\bar{c}$-wave function at the origin for $\psi^{\prime}$ and the
dominance of the colour-octet state [10]; cancellation between $c\bar{c}$ and
$D\bar{D}$ components of $\psi^{\prime}$ [11]; cancellation between $S$\- and
$D$-wave components of $\psi^{\prime}$ [12] and others [2].

On the other hand, the potential of the effective field theory framework to
study the problem was not been fully exploited yet. Especially it is
interesting to study the higher order corrections, which are different for
$J/\psi$ and $\psi^{\prime}$. In this way the natural violation of the “13%
rule” can be related to relativistic corrections in NRQCD [13].

In fact, already an order $v^{2}$ nonrelativistic QCD matrix elements
$\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon\
}{\boldsymbol{\nabla}}^{2}\psi\left|\psi(n,\boldsymbol{\epsilon})\right\rangle$
have very different values for $J/\psi$ and $\psi^{\prime}$, that was noticed
already long time ago [14]. Recently, the relativistic corrections to
exclusive $\psi(n)\to p\bar{p}$ decays have been studied in Ref.[15]. It is
found that corrections of relative order $v^{2}$ are large and comparable with
the leading-order contribution. This effect is closely related to the
structure of the integrand in the collinear convolution integral describing
the decay amplitude. This observation holds for both states $J/\psi$ and
$\psi^{\prime}$ but for excited state the absolute effect is larger because
the corresponding matrix element is larger. The similar mechanism may also be
relevant for other hadronic decay channels including $\psi(n)\to\rho\pi$
decays.

Therefore, the main purpose of this paper is to calculate the relativistic
corrections to $\psi(n)\to\rho\pi$ and to study their numerical effect. As a
first step in this direction we will calculate the correction of relative
order $v^{2}$ combining NRQCD expansion with the leading-order collinear
expansion. We will use the NRQCD projection technique developed in Refs.[16,
17, 18, 19], which is also effective for calculations of exclusive amplitudes.
This technique also allows one to resum a part of higher order corrections,
which are related to the corrections to quark-antiquark wave function in the
potential model [19]. Such consideration is also useful providing an estimate
of possible effects from higher order contributions.

##  2 Relativistic corrections to $\psi(n)\to\rho\pi$ and
$\psi(n)\to\gamma\pi$ decays

To describe the $J/\psi(P)\rightarrow\rho(p)\pi(p^{\prime})$ decay amplitude
we use the charmonium rest frame and assume that outgoing momenta are directed
along $z$-axis. The amplitude is defined as

$\left\langle\rho(p)\pi(p^{\prime})\right|iT\left|\psi(n)\right\rangle=i(2\pi)^{4}\delta(p+p^{\prime}-P)i\epsilon_{\alpha\beta\mu\nu}\epsilon^{\alpha}e^{\ast\beta}\frac{p^{\prime\mu}p^{\nu}}{(pp^{\prime})}A_{\rho\pi},$
(2)

where $\epsilon$ and $e^{\ast}$ denotes polarisation vectors of $\psi$ and
$\rho$-meson, respectively. The amplitude $A_{\rho\pi}$ can be described as a
superposition of a hard kernel with nonperturbative matrix elements describing
the long distance coupling with hadronic states. In order to calculate the
hard kernel, we perform an NRQCD matching, which is combined with the
collinear light-cone expansion for the light quarks. This technique allows one
to perform the matching at the amplitude level and to find the hard kernels
for corrections associated with the specific set of higher order NRQCD matrix
elements [19]

$\left\langle{v}^{2n}\right\rangle=\frac{\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon\
}\left(-\frac{i}{2}\overleftrightarrow{\boldsymbol{D}}\right)^{2n}\psi\left|\psi(n,\boldsymbol{\epsilon})\right\rangle}{m_{c}^{2n}\
\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon}\psi\left|\psi(n,\boldsymbol{\epsilon})\right\rangle}\simeq\left\langle{v}^{2}\right\rangle^{n},$
(3)

where the last equality is valid up to corrections $\mathcal{O}(v^{2})$ [20].

The diagrams, which describe the decay amplitude are schematically shown in
Fig.1.

Figure 1:  a) Typical diagrams describing the subprocess $Q\bar{Q}\rightarrow
VP$, where $V=\rho,\gamma$. The blobs denote the light-cone matrix elements,
see explanation in the text. b) An example of diagrams, describing the
contribution with the perturbative photon coupling.

The long distance hadronisation dynamics of outgoing mesons is described by
the twist-2 and twist-3 light-cone distribution amplitudes (LCDAs). Various
properties and models for required LCDAs can be found in Refs.[7, 8]. The
twist-2 light-cone matrix elements read 111For simplicity, we do not
explicitly show the gauge links in the light-cone operators assuming the
appropriate light-cone gauge.

$\left\langle\pi^{+}(p^{\prime})\right|\bar{u}(z_{1+})\hbox to0.0pt{\hbox
to5.00002pt{\hfil$\bar{n}$\hfil}\hss}/\gamma_{5}d(z_{2+})\left|0\right\rangle=-if_{\pi}\left(p^{\prime}\bar{n}\right)\
\int_{0}^{1}du\
e^{iu\left(p^{\prime}\bar{n})(z_{1}n\right)/2+i(1-u)\left(p^{\prime}\bar{n})(z_{2}n\right)/2}\
\phi_{2\pi}(u),$ (4)
$\left\langle\rho^{-}(p)\right|\bar{d}(z_{1-})\gamma_{\bot}^{\mu}\hbox
to0.0pt{\hbox
to6.00235pt{\hfil/\hfil}\hss}nu(z_{2-})\left|0\right\rangle=if_{\rho}^{\bot}e_{\bot}^{\ast\mu}(pn)\
\int_{0}^{1}dy\
e^{iy(pn)(z_{1}\bar{n})/2+i(1-y)\left(pn)(z_{2}\bar{n}\right)/2}\
\phi_{2\rho}^{\bot}(y),$ (5)

where we use auxiliary light-cone vectors

$\displaystyle n=(1,0,0,-1),\ \bar{n}=(1,0,0,1),\
g^{\mu\nu}_{\bot}=g^{\mu\nu}-\frac{1}{2}(n^{\mu}\bar{n}^{\nu}+n^{\nu}\bar{n}^{\mu}),$
(6) $\displaystyle
p^{\prime}=(p^{\prime}\bar{n})\frac{n}{2}+\frac{m^{2}_{\pi}}{(p^{\prime}\bar{n})}\frac{\bar{n}}{2},\
p=(pn)\frac{\bar{n}}{2}+\frac{m^{2}_{\rho}}{(pn)}\frac{n}{2},\
(p^{\prime}\bar{n})\sim(pn)\sim m_{c}.$ (7)

and the short notation for the arguments of quark fields

$q(z_{i+})\equiv q((z_{i}n)\bar{n}/2),\ q(z_{i-})\equiv\
q((z_{i}\bar{n})n/2).$ (8)

The required twist-3 three-particles LCDAs are defined as

$\left\langle\pi^{+}(p^{\prime})\right|\bar{u}(z_{1+})\hbox to0.0pt{\hbox
to5.00002pt{\hfil$\bar{n}$\hfil}\hss}/\gamma_{\bot}^{\mu}\gamma_{5}gG_{\bar{n}\mu}(z_{3+})d(z_{2+})\left|0\right\rangle=-2f_{3\pi}\left(p^{\prime}\bar{n}\right)^{2}\text{FT}\left[\phi_{3\pi}(u_{i})\right],$
(9) $\left\langle\rho^{-}(p,e)\right|\bar{d}(z_{1-})\hbox to0.0pt{\hbox
to6.00235pt{\hfil/\hfil}\hss}ngG_{\mu
n}(z_{2-})u(z_{3-})\left|0\right\rangle=-\
f_{\rho}m_{\rho}(pz)^{2}e_{\bot}^{\ast\mu}\
\text{FT}\left[\phi_{3\rho}(y_{i})\right],$ (10)
$\left\langle\rho^{-}(p,e)\right|\bar{d}(z_{1})\hbox to0.0pt{\hbox
to6.00235pt{\hfil/\hfil}\hss}n\gamma_{5}g\tilde{G}_{\mu n}(\lambda
z)u(-z)\left|0\right\rangle=-if_{\rho}m_{\rho}\zeta_{3}(pz)^{2}e_{\bot}^{\ast\mu}\
\text{FT}\left[\tilde{\phi}_{3\rho}(y_{i})\right],$ (11)

where $G_{\mu n}\equiv G_{\mu\nu}n^{\nu}$ and

$\text{FT}\left[f(u_{i})\right]=\int Du_{i}\
e^{iu_{1}\left(p^{\prime}\bar{n})(z_{1}n\right)/2+iu_{2}\left(p^{\prime}\bar{n})(z_{2}n\right)/2+iu_{3}\left(p^{\prime}\bar{n})(z_{3}n\right)/2}\phi_{3\pi}(u_{1},u_{2},u_{3}),$
(12)

with

$Du_{i}=du_{1}du_{2}du_{3}\delta(1-u_{1}-u_{2}-u_{3}).$ (13)

The FT$\left[\phi_{3\rho}(y_{i})\right]$ is defined analogously but with
$y_{i}(pn)(z_{i}\bar{n})$ in the Fourier transformation. The normalisation
constants $f_{\pi,\rho}$, $\zeta_{3},\ f_{3\pi}$ and models for various LCDAs
will be discussed below.

The expression for the amplitude can be written as

$A_{\rho\pi}=\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon}\psi\left|\psi(n,\boldsymbol{\epsilon})\right\rangle\frac{\sqrt{2M_{\psi}}}{2E}\
\frac{1}{4\pi}\int d\Omega\ \text{Tr}\left[\Pi_{1}\hat{A}_{Q}\right],$ (14)

where $\hat{A}_{Q}$ describes subprocess $Q\bar{Q}\rightarrow\rho\pi$ with the
quark-antiquark pair in the initial state. The heavy quark projector on the
triplet spin state $\Pi_{1}$ reads [19]

$\Pi_{1}=\frac{-1}{2\sqrt{2}E(E+m)}\left(\frac{1}{2}\hbox to0.0pt{\hbox
to7.80904pt{\hfil/\hfil}\hss}P+m+\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/\right)\frac{\hbox to0.0pt{\hbox
to7.80904pt{\hfil/\hfil}\hss}P+2E}{4E}\hbox to0.0pt{\hbox
to5.00002pt{\hfil$\epsilon$\hfil}\hss}/\left(\frac{1}{2}\hbox to0.0pt{\hbox
to7.80904pt{\hfil/\hfil}\hss}P-m-\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/\right)\otimes\frac{\mathbf{1}}{\sqrt{N_{c}}},$
(15)

and is normalised as

$\displaystyle\text{Tr}\left[\Pi_{1}\Pi_{1}^{{\dagger}}\right]=4E^{2},$ (16)

where $E$ is the heavy quark energy $p_{Q}=(E,\boldsymbol{q})$,
$p_{\bar{Q}}=(E,-\boldsymbol{q})$ and $E=\sqrt{m_{c}^{2}+\boldsymbol{q}^{2}}$.
The integration $d\Omega$ over the angles of the relative momentum
$\boldsymbol{q}$ in Eq.(14) is used to get the state with $L=0$. Therefore the
relevant amplitude $\hat{A}_{Q}$ is the function of relative momentum square
$\boldsymbol{q}^{2}$ only, which is substituted $\boldsymbol{q}^{2}\rightarrow
m_{c}^{2}\left\langle{v}^{2}\right\rangle$ in the final expression (14),
various technical details concerning NRQCD matching can be found in Refs.[17,
19].

Calculation of the diagrams as in Fig.1 gives

$A_{\rho^{-}\pi^{+}}=-\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon}\psi\left|\psi(^{3}S_{1})\right\rangle\sqrt{2M_{\psi}}\left(\pi\alpha_{s}\right)^{2}\frac{10}{27}\left(1+\frac{m_{c}}{E}\right)\frac{f_{\rho}^{\bot}~{}f_{3\pi}}{\left[4E^{2}\right]^{2}}\left(J_{\pi}+\frac{\
f_{\rho}m_{\rho}\zeta_{3}\ f_{\pi}}{f_{\rho\bot}f_{3\pi}}\ J_{\rho}\right),$
(17)

where the dimensionless collinear convolution integrals $J_{\pi}$ and
$J_{\rho}$ describe contributions with twist-3 $\pi$\- and $\rho$-LCDAs,
respectively. These integrals also depend on the NRQCD parameter
$\left\langle{v}^{2}\right\rangle$. In the leading-order limit
$\left\langle{v}^{2}\right\rangle\rightarrow 0$, $E\rightarrow m_{c}^{2}$
Eq.(17) reproduces the result from Ref.[4]

$A_{\rho^{-}\pi^{+}}^{\text{lo}}=-\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon}\psi\left|\psi(n,\boldsymbol{\epsilon})\right\rangle\sqrt{2M_{\psi}}\left(\pi\alpha_{s}\right)^{2}\frac{20}{27}\frac{f_{\rho}^{\bot}f_{3\pi}}{\left[4m_{c}^{2}\right]^{2}}\left(J_{\pi}^{\text{lo}}+\frac{\
f_{\rho}m_{\rho}\zeta_{3}f_{\pi}}{f_{\rho\bot}f_{3\pi}}\
J_{\rho}^{\text{lo}}\right),$ (18)

where

$\displaystyle J_{\pi}^{\text{lo}}$ $\displaystyle=\int Du_{i}\
\frac{\phi_{3\pi}(u_{i})}{u_{1}u_{2}u_{3}}\int_{0}^{1}dy\
\frac{\phi_{2\rho}^{\bot}(y)}{1-y}\frac{2u_{1}}{\left(y\bar{u}_{2}+u_{2}\bar{y}\right)\left(yu_{1}+\bar{y}\bar{u}_{1}\right)},$
(19) $\displaystyle J_{\rho}^{\text{lo}}$ $\displaystyle=\int_{0}^{1}du\
\frac{\phi_{2\pi}(u)}{1-u}\int Dy_{i}\
\frac{\left(\phi_{3\rho}+\tilde{\phi}_{3\rho}\right)(y_{i})}{y_{1}y_{2}y_{3}}\frac{1}{y_{2}\bar{u}+u\bar{y}_{2}},$
(20)

with $\bar{x}=1-x$. The analytical expressions for the integrals
$J_{\pi,\rho}$ in Eq.(17) are somewhat lengthy and presented in Appendix.

In order to estimate these integrals we use the following models of LCDAs

$\phi_{2\rho}^{\bot}(y)=6y(1-y)\left(1+a_{2\rho}\ C_{2}^{3/2}(2y-1)\right),\ $
(21) $\phi_{2\pi}(u)=6u(1-u)\left(1+a_{2}^{\pi}C_{2}^{3/2}(2u-1)\right),$ (22)
$\phi_{3\rho}(y_{i})=360y_{1}y_{2}y_{3}^{2}(y_{1}-y_{2})\omega_{3\rho},$ (23)
$\tilde{\phi}_{3\rho}(y_{i})=360y_{1}y_{2}y_{3}^{2}\left(1+\frac{\tilde{\omega}_{3\rho}}{\zeta_{3}}\frac{1}{2}\left(7y_{3}-3\right)\right),$
(24) $\ \
\phi_{3\pi}(u_{i})=360u_{1}u_{2}u_{3}^{2}\left(1+\omega_{3\pi}\frac{1}{2}(7\alpha_{3}-3)\right).$
(25)

The different nonperturbative moments, which enter in the definitions (4)-(11)
and (21)-(25), were estimated in Refs.[4, 7, 8]. Their values are summarised
in Table 1. In the numerical estimates we fix for the factorisation scale the
value $\mu=2$ GeV and use $\alpha_{s}\simeq 0.30$.

Table 1:  The values of the moments, which parametrise the hadronic LCDAs. All values are given at the scale $\mu=2$ GeV. For the pion moments, the values are taken from Ref.[7], for the $\rho$-meson from Ref.[8].  $f_{\pi},$MeV |  $f_{\rho}$, MeV |  $f_{\rho}^{\bot}$, MeV | $a_{2\pi}$ | $a_{2\rho}$ |  $f_{3\pi},$ GeV2 | $\zeta_{3\rho}$ | $\omega_{3\rho}$ | $\tilde{\omega}_{3\rho}$ | $\omega_{3\pi}$
---|---|---|---|---|---|---|---|---|---
$131$ | $216$ | $143$ | $0.19$ | $0.11$ | $0.31\times 10^{-2}$ | $0.02$ | $0.09$ | $-0.04$ | $-1.1$

All the convolution integrals calculated with the models (21)-(25) are well
defined, which confirms that collinear factorisation is also valid beyond the
leading-order approximation.

As a first step of the numerical analysis let us consider the the leading-
order estimate for the branching ratio of $J/\psi$. For that purpose we use
the estimates for the NRQCD matrix element obtained in Ref.[18]

$\left|\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon}\psi\left|J/\psi\right\rangle\right|^{2}\simeq
0.440.$ (26)

For the various masses in Eq.(18) we use $M_{\psi}=3.1\ $GeV, $m_{\rho}=775\
$MeV, for the pole $c$-quark mass $m_{c}=1.4\ $GeV and for the total width
$\Gamma_{J/\psi}=93\ $KeV [1]. Then for the sum of all final states
$\rho^{\pm}\pi^{\mp}$ and $\rho^{0}\pi^{0}$ we obtain

$\text{Br}[J/\psi\rightarrow\rho\pi]_{\text{lo}}\simeq 1.0\%,$ (27)

which is somewhat smaller then the corresponding experimental value
$1.69(15)\%.$ This updated result confirm the conclusion of Ref.[4], that the
LO NRQCD approximation works sufficiently well for the $J/\psi$ decay.222We
assume that the difference about factor two is not a large discrepancy taking
into account various uncertainties from scale setting, pole mass $m_{c}$,
etc., which we do not consider now.  On the other hand this approximation can
not describe branching ratio $\psi^{\prime}\to\rho\pi$.

Consider now the effect provided by the relativistic corrections in Eq.(17).
The one part is provided by the resummation of relativistic corrections in the
factor $E=m_{c}^{2}\sqrt{1+\left\langle{v}^{2}\right\rangle}$ in Eq.(17). This
effect can be understood as transition from the scale $4m_{c}^{2}$ to the
scale $M_{\psi}^{2}\simeq 4m_{c}^{2}(1+\left\langle{v}^{2}\right\rangle)$.
These corrections reduce the ratio $Q_{\rho\pi}$ due to the factor
$(1+\left\langle{v}^{2}\right\rangle_{J/\psi})^{2}/(1+\left\langle{v}^{2}\right\rangle_{\psi^{\prime}})^{2}\sim
M_{\psi}^{4}/M_{\psi^{\prime}}^{4}\sim 0.50$. However, this can not explain
the very small value $Q_{\rho\pi}$ in Eq.(1).

The second effect of the relativistic corrections is associated with the
modification of the hard kernels in the convolution integrals $J_{\rho,\pi}$.
Because these integrals depend on meson LCDAs, the resulting effect of the
relativistic corrections is also sensitive to hadronic nonperturbative
structure.

For the numerical calculation we use for $J/\psi$ the estimate from Ref.[18]

$\left\langle{v}^{2}\right\rangle_{J/\psi}\approx 0.225,$ (28)

and for the excite state $\psi^{\prime}$ we apply the following estimate

$\left\langle{v}^{2}\right\rangle_{\psi^{\prime}}=\frac{M_{\psi^{\prime}}-M_{J/\psi}+E_{1}}{m_{c}}\approx
0.64,$ (29)

where $E_{1}=\left\langle{v}^{2}\right\rangle_{J/\psi}m_{c}\simeq 315\ $MeV is
the binding energy for $J/\psi$. The resulting value of
$\left\langle{v}^{2}\right\rangle_{\psi^{\prime}}$ is much larger than
$\left\langle{v}^{2}\right\rangle_{J/\psi}$, which can have a significant
numerical effect and, therefore, affect the value of $Q_{\rho\pi}$.

The given calculation of the relativistic corrections is complete at the
relative order $v^{2}$ only. The resummation of higher orders
$\left\langle{v}^{2}\right\rangle^{n}$ with $n>2$ describes the part of the
relativistic corrections associated with the quark-antiquark wave function
only [19]. We use this approximation in order to study a possible effect from
higher-order contributions. Therefore, for the comparison, we present the
values of the integrals in Eq.(17) obtained in the leading-order approximation
$J^{\text{lo}}$ ($\left\langle{v}^{2}\right\rangle\rightarrow 0$), in the
next-to-leading approximation $J^{\text{nlo}}$, which takes into accont the
next-to-leading correction
$J^{\text{nlo}}=J^{\text{lo}}+\left\langle{v}^{2}\right\rangle J^{(1)}$, and
the integral $J$, which includes all powers
$\left\langle{v}^{2}\right\rangle^{n}$ :
$J=J^{\text{lo}}+\sum\left\langle{v}^{2}\right\rangle^{n}J^{(n)}$.

The total integral in Eq.(17) is described by the sum of two contributions
with the different LCDAs

$J_{\rho\pi}=J_{\pi}+\frac{\ f_{\rho}m_{\rho}\zeta_{3}\
f_{\pi}}{f_{\rho\bot}f_{3\pi}}\ J_{\rho},$ (30)

where, schematically, $J_{\pi}=\phi_{3\pi}\ast T_{\pi}\ast\phi_{2\rho}^{\bot}$
and $J_{\rho}=\phi_{3\rho}\ast
T_{\rho}\ast\phi_{2\pi}+\tilde{\phi}_{3\rho}\ast\tilde{T}_{\rho}\ast\phi_{2\pi}$
(the asterisk denotes the convolution integrals, $T_{\pi,\rho}$ are the hard
kernels). Using parameters from Table 1 one finds

$\frac{\ f_{\rho}m_{\rho}\zeta_{3}\ f_{\pi}}{f_{\rho\bot}f_{3\pi}}\approx
0.99.$ (31)

Therefore the normalisation couplings in the definitions (4)-(11) do not
provide any numerical difference between the two terms in Eq.(30). The results
for convolution integrals (30) are presented in Table 2

Table 2:  Numerical result for the convolution integrals $J_{\rho\pi}$ | $J_{\rho\pi}^{\text{lo}}$ | $J_{\rho\pi}^{\text{nlo}}/J_{\rho\pi}^{\text{lo}}$ | $\ J_{\rho\pi}/J_{\rho\pi}^{\text{lo}}$
---|---|---|---
$J/\psi$ | $630$ | $0.53$ | $0.45$
$\psi^{\prime}$ | $630$ | $-0.46$ | $-0.65$

The effect of the relativistic corrections is negative and the values of the
LO integrals are substantially reduced. Notice that neglecting the higher-
order corrections in $v^{2}$ in the square of the integral, one gets in case
of $J/\psi$ the strong cancellation

$\displaystyle|J_{\rho\pi}^{\text{nlo}}|^{2}=(J_{\rho\pi}^{\text{lo}})^{2}(2J_{\rho\pi}^{\text{nlo}}/J_{\rho\pi}^{\text{lo}}-1)+\mathcal{O}(v^{4})\simeq
0.06(J_{\rho\pi}^{\text{lo}})^{2}.$ (32)

Therefore we assume that it is better to take the large NLO correction
exactly, i.e. do not expanding the square of the integral in powers of
$v^{2}$. At the same time the numerical effect from other higher order
corrections is already much smaller.

For $\psi^{\prime}\to\rho\pi$ the numerical effect is bigger because
$\left\langle\boldsymbol{v}^{2}\right\rangle_{\psi^{\prime}}$ is larger. One
can also see that the dominant part of the correction is also provided by the
contribution of relative order $v^{2}$, which is obtained exactly in this
calculation. The numerical dominance of this correction can be explained by
the numerical enhancement of the corresponding convolution integrals in the
same way as for the baryon decays [15].

Let us assume that the relativistic correction of order $v^{2}$ provides the
dominant numerical effect for $J/\psi$ and $\psi^{\prime}$ states. Then, this
allows one to suggest a possible explanation of the small
$\psi^{\prime}\to\rho\pi$ width, which explains the ”$\rho\pi$-puzzle”. The
NRQCD description of decay amplitudes also involves the
$\mathcal{O}(\alpha_{s})$ NLO QCD radiative correction, which can also provide
substantial numerical effect. Usually this contribution is considered to be of
the same order as relativistic corrections of relative order $v^{2}$. The
value of radiative corrections for $J/\psi$ and $\psi^{\prime}$ states is the
same except the NRQCD matrix element. Therefore, if the radiative
$\mathcal{O}(\alpha_{s})$ correction is positive and large enough in order to
compensate the negative contribution $J^{\text{nlo}}$ for $\psi^{\prime}$ then
this naturally explains the small width for $\psi^{\prime}$. On the other
hand, such positive contribution will improve the description of
$J/\psi\to\rho\pi$ increasing the value of the convolution integral, i.e. in
this case the $\mathcal{O}(\alpha_{s})$ also compensates the negative effect
of the relativistic correction. This is in agreement with the observation that
the leading-order description $J/\psi\to\rho\pi$ provides qualitatively good
estimate.

Potentially this analysis can also be applied for other meson decays. The good
feature of the collinear factorisation is that the hadronic nonperturbative
content is described in terms of universal process independent LCDAs. Many of
these functions were already studied in the literature. Even if the hard
kernels are the same the differences in the models for LCDAs can affect the
numerical balance and change the value $Q_{hh^{\prime}}$. Consider, for
example, the decay of S-wave charmonia into $\gamma\pi^{0}$ final state. In
this case the decay amplitude is described by the same diagrams as in Fig.1(a)
but with the photon LCDAs instead of $\rho$-meson. These diagrams describe the
photon as a hadron, i.e. such contributions are sensitive to the
nonperturbative components of the photon wave function. The contribution with
the perturbative photon coupling appear from the diagrams Fig.1(b) only and
therefore they are suppressed by electromagnetic coupling $\alpha$ or by
additional QCD coupling $\alpha_{s}$. In such situation the contributions with
nonperturbative photon can provide a sizeable impact, see e.g. discussion in
Ref.[21].

The data for the branching fractions $\psi(n)\to\gamma\pi$ are known [1]

$\text{Br}[J/\psi\rightarrow\gamma\pi^{0}]=3.56(17)\times 10^{-5},\ \
\text{Br}[\psi^{\prime}\rightarrow\gamma\pi^{0}]=1.04(22)\times 10^{-6},$ (33)

which yields

$Q_{\gamma\pi}\simeq 0.03.$ (34)

The width $\Gamma[J/\psi\rightarrow\gamma\pi^{0}]$ can be well estimated using
data for $\Gamma[J/\psi\rightarrow\rho\pi^{0}]$ and VDM model [4].333 We
assume that the contribution of $J/\psi\to\gamma^{*}\to\gamma\pi$ is
subleading, in contrast to the analysis in Ref.[4]. We guess that this
contribution is overestimated in [4] due to the specific model of the pion
LCDA .. This indirectly support the picture with the dominant contribution
from the non-perturbative photon coupling. However, the ratio $Q_{\gamma\pi}$
is about an order of magnitude larger than $Q_{\rho\pi}$.

We will use the models for the photon LCDAs from Ref.[21]. The twist-2 light-
cone matrix element reads

$\left\langle\gamma(q,e)\right|\bar{q}(z_{1-})\hbox to0.0pt{\hbox
to5.00002pt{\hfil$\bar{n}$\hfil}\hss}/\gamma_{\bot}^{\mu}q(z_{2-})\left|0\right\rangle=ie_{q}e\
f_{\gamma}e_{\bot}^{\ast\mu}(qn)\ \int_{0}^{1}dy\
e^{iy(pn)(z_{1}\bar{n})/2+i(1-y)\left(pn)(z_{2}\bar{n}\right)/2}\
\phi_{2\gamma}^{\bot}(y),$ (35)

where $e_{u}=2/3$, $e_{d}=-1/3$, electric charge $e=\sqrt{4\pi\alpha}$. The
model for $\phi_{2\gamma}^{\bot}$ reads [21]

$\phi_{2\gamma}^{\bot}(y)\simeq 6y(1-y),\ \ \ \
f_{\gamma}(2\text{GeV})\simeq-47\ \text{MeV}.$ (36)

Twist-3 DAs matrix elements are defined as

$\left\langle\gamma(p)\right|\bar{q}(z_{1-})\hbox to0.0pt{\hbox
to6.00235pt{\hfil/\hfil}\hss}ngG_{\mu
n}(z_{3-})q(z_{2-})\left|0\right\rangle=-\ e_{q}e\
f_{3\gamma}(qn)^{2}\varepsilon_{\bot}^{\ast\mu}\
\text{FT}\left[\phi_{3\gamma}(y_{i})\right],$ (37)
$\left\langle\gamma(p)\right|q(z_{1-})\hbox to0.0pt{\hbox
to6.00235pt{\hfil/\hfil}\hss}n\gamma_{5}g\tilde{G}_{\mu
n}(z_{2-})q(z_{3-})\left|0\right\rangle=-ie_{q}e\
f_{3\gamma}(qn)^{2}\varepsilon_{\bot}^{\ast\mu}\
\text{FT}\left[\tilde{\phi}_{3\gamma}(y_{i})\right],$ (38)

where Fourier transformation is the same as in Eq.(12). The models for twist-3
LCDAs read [21]

$\phi_{3\gamma}(y_{i})=360y_{1}y_{2}y_{3}^{2}\left(y_{1}-y_{2}\right)\
\omega_{3\gamma},\ \ $ (39)
$\tilde{\phi}_{3\gamma}(y_{i})=360y_{1}y_{2}y_{3}^{2}\left(1+\tilde{\omega}_{3\gamma}\frac{1}{2}\left(7y_{3}-3\right)\right),\
\ \tilde{\omega}_{3\gamma}\approx\tilde{\omega}_{3\rho}/\zeta_{3}.$ (40)

where

$f_{3\gamma}(2\text{GeV})=-0.32\times 10^{-2}\text{GeV}^{2},\ \
\omega_{3\gamma}\approx\omega_{3\rho},\ \ \ \
\tilde{\omega}_{3\gamma}\approx\tilde{\omega}_{3\rho}/\zeta_{3}.$ (41)

The $\gamma\pi$-decay amplitude can be obtained from Eq.(17) substituting
photon LCDAs instead of $\rho$-meson ones. The LO amplitude reads

$A_{\gamma\pi^{0}}^{\text{lo}}=-\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon}\psi\left|\psi(n,\boldsymbol{\epsilon})\right\rangle\sqrt{2M_{\psi}}\left(\pi\alpha_{s}\right)^{2}\sqrt{2\pi\alpha}\frac{20}{27}\frac{f_{\gamma}~{}f_{3\pi}}{\left[4m_{c}^{2}\right]^{2}}\left(J_{\gamma\pi}^{\text{lo}}+\frac{\
f_{3\gamma}\ f_{\pi}}{f_{3\pi}f_{\gamma}}\ J_{\gamma\pi}^{\text{lo}}\right).$
(42)

The ratio of the normalisation couplings in Eq.(42) yields ($\mu=2$GeV)

$\frac{\ f_{3\gamma}\ f_{\pi}}{f_{3\pi}f_{\gamma}}\approx 2.92,$ (43)

which is different from the analogous ratio for the $\rho$-meson (31). The
leading-order numerical estimates gives

$\text{Br}[J/\psi\rightarrow\gamma\pi^{0}]_{\text{lo}}\simeq 3.82\times
10^{-5},$ (44)

which very well agrees with the data. The results for the total convolution
integral $J_{\gamma\pi}$ are presented in Table 3.

Table 3: Numerical result for the convolution integrals $J_{\gamma\pi}$ | $J_{\gamma\pi}^{\text{lo}}$ | $\ J_{\gamma\pi}^{\text{nlo}}/J_{\gamma\pi}^{\text{lo}}$ | $\ J_{\gamma\pi}/J_{\gamma\pi}^{\text{lo}}$
---|---|---|---
$J/\psi$ | $932$ | $0.62$ | $0.45$
$\psi^{\prime}$ | $932$ | $-0.49$ | $-0.74$

Comparing with the analogous results for the $\rho\pi$-channel one finds that
the both descriptions are qualitatively similar despite the different ratio
(43) and the differences between the LCDAs $\phi_{2\gamma}$ and
$\phi_{2\rho}^{\bot}$. Therefore one can again assume that
$\mathcal{O}(\alpha_{s})$ radiative corrections also play a crucial role in
the understanding of the value of the decay width $\gamma\pi$. Moreover, the
contributions with a perturbative photon, as in Fig. 1(b) can probably explain
the larger value of $Q_{\gamma\pi}$.

##  3 Conclusions

In conclusion, we calculated and investigated relativistic corrections to the
decay amplitudes $\psi(n)\to\rho\pi$ and $\psi(n)\to\gamma\pi$ within the pQCD
(NRQCD and collinear factorisation) framework. This calculation includes the
exact correction of relative order $v^{2}$ and subset of the higher order
corrections associated with the quark-antiquarks wave function. Numerical
estimates show that an order $v^{2}$ correction is large and give the dominant
numerical effect, which can be related to the structure of the collinear
integrals. If this observation is not affected by other higher order
relativistic corrections, then one has to consider the relative
$v^{2}$-contribution as a special case. The obtained relativistic corrections
are negative and large. In case $\psi^{\prime}\to\rho\pi$ the relative $v^{2}$
contribution is much larger than the leading-order one. The different effects
of relativistic corrections for $J/\psi\to\rho\pi$ and
$\psi^{\prime}\to\rho\pi$ suggest a possible explanation for the
$\rho\pi$-puzzle. If the QCD radiative correction is positive and large enough
then it interferes destructively with the relativistic correction for
$\psi^{\prime}\to\rho\pi$ giving the small branching fraction. At the same
time such radiative correction will improve the description of
$J/\psi\to\rho\pi$ reducing the negative effect of the relativistic
correction. Therefore, we believe that further investigation of relative order
$v^{4}$ corrections and QCD radiative corrections can help to clarify this
scenario.

We also expect that the same approach can be be used for an analysis other
hadronic decay channels. As a simplest example, the decay
$\psi(n)\to\gamma\pi$ was also considered. We studied the contribution, which
is given by similar diagrams but with nonperturbative photon instead of
$\rho$-meson. Despite the difference between twist-2 LCDAs for photon and
$\rho-$meson, the qualitative effect from the relativistic corrections is
quite similar to $\psi\to\rho\pi$, they are also large and negative. Therefore
one can guess that the similar scenario with radiative corrections is also
applicable here. The only difference with $\psi(n)\to\rho\pi$ is provided by
the contributions with perturbative photon coupling, which are suppressed by
$\mathcal{O}(\alpha_{s})$ or $\mathcal{O}(\alpha)$. Therefore, it can be that
these expects are responsible for the larger value of $Q_{\gamma\pi}$.

##  4 Appendix

Here we provide the analytical expressions for the intergrals $J_{\pi}$ and
$J_{\rho}$ introduced in Eq.(17). In order to simplify notations we use

$\left\langle v^{2}\right\rangle\equiv\boldsymbol{v}^{2},\ \
\delta=1-1/\sqrt{1+\boldsymbol{v}^{2}}.\ $ (45)

The first integral in Eq.(17) reads

$J_{\pi}\left(\boldsymbol{v}^{2}\right)=\int Du_{i}\
\frac{\phi_{3\pi}(u_{i})}{u_{1}u_{2}u_{3}}\int_{0}^{1}dy\
\frac{\phi_{2}^{\bot}(y)}{y\bar{y}}\left(\frac{2A_{\pi}}{D_{1}D_{3}}+\frac{B_{\pi}}{D_{1}D_{2}}\right),\
\ \bar{y}=1-y,$ (46)

where

$D_{i}=\delta_{i1}\
\left(y_{1}\bar{u}_{2}+\bar{y}_{1}u_{2}\right)+\delta_{i2}\
\left(y_{2}\bar{u}_{1}+\bar{y}_{2}u_{1}\right)+\delta_{i3}\ u_{3},$ (47)

with

$y_{1}=y,\ y_{2}=\bar{y}.$ (48)

The symbol $\delta_{ik}$ denotes the Kronecker delta.

The numerators $A_{\pi}$ and $B_{\pi}~{}$are given by the sums

$A_{\pi}=\sum_{k=0}^{4}f_{k}^{A}I_{k}[13],\ \ \
B_{\pi}=\sum_{k=0}^{4}f_{k}^{B}I_{k}[12],$ (49)

where

$I_{k}[ij]=\frac{1}{2}\int_{-1}^{1}d\eta\
\frac{\boldsymbol{v}^{k}\eta^{k}}{\left(1+\boldsymbol{v}\eta\
a_{i}\right)\left(1-\boldsymbol{v}\eta\
a_{j}\right)}=\frac{\boldsymbol{v}^{k}}{a_{i}+a_{j}}\sum_{n=0}^{\infty}\boldsymbol{v}^{n}\frac{a_{j}^{n+1}+(-1)^{n}a_{i}^{n+1}}{n+1+k}\frac{1}{2}\left[1+(-1)^{n+k}\right].\
\ $ (50)

with

$a_{j}=\delta_{1j}\left(1-\delta\right)\frac{y_{1}-u_{2}}{y_{1}\bar{u}_{2}+\bar{y}_{1}u_{2}}+\delta_{2j}\left(1-\delta\right)\frac{y_{2}-u_{1}}{y_{2}\bar{u}_{1}+\bar{y}_{2}u_{1}}-\delta_{3j}\left(1-\delta\right).$
(51)

The coefficients $f_{k}^{A,B}\equiv f_{k}^{A,B}(u_{i},y;\delta)$ in Eq.(49)
read

$f_{0}^{A}=\frac{\delta}{2}\left(3u_{3}-2-\delta\right),\ \ \ \
f_{1}^{A}=\frac{\delta}{2}\frac{\left(1-\delta\right)}{\left(2-\delta\right)}u_{3},$
(52)
$f_{2}^{A}=\frac{1}{2}\frac{\left(1-\delta\right)^{2}}{\left(2-\delta\right)^{2}}\left(4-3(2-\delta)u_{3}+2\delta(1-\delta)\right),$
(53)
$f_{3}^{A}=-\frac{1}{2}\frac{\left(1-\delta\right)^{3}}{\left(2-\delta\right)^{2}}u_{3},\
\
f_{4}^{A}=-\frac{1}{2}\frac{\left(1-\delta\right)^{4}}{\left(2-\delta\right)^{2}}.$
(54) $\displaystyle f_{0}^{B}$
$\displaystyle=u_{1}y_{1}+u_{1}y_{2}-\frac{\delta}{2}(u_{1}+u_{2}+y_{1}+y_{2}-\delta)$
(55)
$\displaystyle+\frac{\delta}{2-\delta}\left(u_{1}y_{1}+u_{1}y_{2}-(2-\delta)\left(u_{1}+u_{2}+y_{1}+y_{2}\right)+(2-\delta)^{2}\right),$
(56)
$f_{1}^{B}=\frac{1}{2}\frac{\left(1-\delta\right)}{\left(2-\delta\right)}\left\\{4(u_{1}y_{1}-u_{2}y_{2})+\delta(u_{1}+y_{1}-u_{2}-y_{2})\right\\},$
(57) $\displaystyle f_{2}^{B}$
$\displaystyle=\frac{1}{2}\frac{\left(1-\delta\right)^{2}}{\left(2-\delta\right)^{2}}\left\\{6\left(u_{1}+u_{2}+y_{1}+y_{2}\right)-2\left(u_{1}y_{1}+u_{1}y_{2}\right)-8\right.$
(58)
$\displaystyle\left.+\delta\left(4-3\left(u_{1}+u_{2}+y_{1}+y_{2}\right)\right)+2\delta(2-\delta)\right\\},$
(59)
$f_{3}^{B}=-\frac{1}{2}\frac{\left(1-\delta\right)^{3}}{\left(2-\delta\right)^{2}}\left(u_{1}+y_{1}-u_{2}-y_{2}\right),\
\
f_{4}^{B}=-\frac{1}{2}\frac{\left(1-\delta\right)^{4}}{\left(2-\delta\right)^{2}}.$
(60)

The $\rho$-meson integral in Eq.(17) reads

$J_{\rho}=\int_{0}^{1}du\ \frac{\phi_{2\pi}(u)}{u\bar{u}}\int Dy_{i}\ \
\frac{1}{y_{1}y_{2}y_{3}}\left(\frac{2A_{\rho}}{y_{3}D_{2}}+\frac{B_{\rho}}{D_{1}D_{2}}\right).$
(61)

The numerators $A_{\rho}$ and $B_{\rho}$ can be written as

$A_{\rho}=\phi_{3\rho}(y_{i})\sum_{k=0}^{4}\left[f_{k}^{A}\right]I_{k}[23]+\tilde{\phi}_{3\rho}(y_{i})\sum_{k=0}^{4}\left[\tilde{f}_{k}^{A}\right]I_{k}[23],$
(62)
$B_{\rho}=\phi_{3\rho}(y_{i})\sum_{k=0}^{4}\left[f_{k}^{B}\right]I_{k}[12]+\tilde{\phi}_{3\rho}(y_{i})\sum_{k=0}^{4}\left[\tilde{f}_{k}^{B}\right]I_{k}[12],$
(63)

where the integrals $I_{k}$ are defined in Eq.(50) with a bit different
combination $a_{j}$

$a_{j}=\delta_{1j}\left(1-\delta\right)\frac{y_{1}-u_{2}}{y_{1}\bar{u}_{2}+\bar{y}_{1}u_{2}}+\delta_{2j}\left(1-\delta\right)\frac{y_{2}-u_{1}}{y_{2}\bar{u}_{1}+\bar{y}_{2}u_{1}}+\delta_{3j}\left(1-\delta\right)\
,$ (64)

and we again use for the two-particle LCDA $\ u_{1}=u$, $u_{2}=1-u$.

The coefficients $f_{k}^{A,B}$ and $\tilde{f}_{k}^{A,B}$ defined in Eqs.(62)
and (63) read

$\displaystyle f_{0}^{A}$
$\displaystyle=\frac{1}{4}\left(u_{1}(2y_{3}-\delta)+\delta\left(\delta-
y_{2}-y_{3}\right)\right)+\frac{\delta^{2}}{2}$ (65) $\displaystyle\ \ \ \ \ \
\ \ \ \
+\frac{1}{4}\frac{\delta}{(2-\delta)}\left(u_{1}\left(6+4y_{3}-3\delta\right)+\left(2-\delta\right)\left(3y_{2}+2y_{3}-2-3\delta\right)\right),$
(66) $\displaystyle f_{1}^{A}$
$\displaystyle=-\frac{1}{4}\frac{1-\delta}{2-\delta}\left(4u_{1}+\delta\right)y_{3},$
(67)
$f_{2}^{A}=-\frac{1}{4}\frac{(1-\delta)^{2}}{(2-\delta)^{2}}\left(2u_{1}y_{3}+(2-\delta)\left(2u_{1}+2y_{2}+y_{3}\right)-2(1-\delta)(2-\delta)\right),$
(68) $f_{3}^{A}=\frac{1}{4}\frac{(1-\delta)^{3}}{(2-\delta)^{2}}y_{3},\ \
f_{4}^{A}=0.$ (69) $\displaystyle f_{0}^{B}$
$\displaystyle=-f_{2}^{B}=\frac{\delta}{4}\left(u_{1}-u_{2}-y_{1}+y_{2}\right),\
$ (70) $\displaystyle\ \ f_{1}^{B}$
$\displaystyle=-f_{3}^{B}=-\frac{\delta}{4}(1-\delta)\left(u_{1}+u_{2}-y_{1}-y_{2}\right),\
\ \ f_{4}^{B}=0.$ (71) $\displaystyle\tilde{f}_{0}^{A}$
$\displaystyle=\frac{1}{4}\left(u_{1}(2y_{3}-\delta)+\delta\left(\delta-
y_{2}-y_{3}\right)\right)$ (72)
$\displaystyle+\frac{1}{4}\frac{\delta}{(2-\delta)}\left(u_{1}\left(2+4y_{3}-\delta\right)+\left(2-\delta\right)\left(2+y_{2}-2y_{3}-3\delta\right)\right),$
(73)
$\tilde{f}_{1}^{A}=-\frac{1}{4}\frac{1-\delta}{2-\delta}\left(4u_{1}y_{3}+\delta\left(2u_{1}-2y_{2}+y_{3}\right)\right),$
(74)
$\tilde{f}_{2}^{A}=-\frac{1}{4}\frac{(1-\delta)^{2}}{(2-\delta)^{2}}\left(2u_{1}y_{3}-3y_{3}(2-\delta)+\delta(2-\delta)\right),$
(75)
$\tilde{f}_{3}^{A}=\frac{1}{4}\frac{(1-\delta)^{3}}{(2-\delta)^{2}}\left(2u_{1}-2y_{2}+y_{3}\right),\
\ \ \tilde{f}_{4}^{A}=-\frac{1}{2}\frac{(1-\delta)^{4}}{(2-\delta)^{2}}.$ (76)
$\tilde{f}_{0}^{B}=\frac{\delta}{4}\left(3\left(u_{1}+u_{2}+y_{1}+y_{2}\right)-4-2\delta\right),\
\ \
\tilde{f}_{1}^{B}=-\frac{\delta}{4}\frac{1-\delta}{2-\delta}\left(u_{1}-u_{2}+y_{1}-y_{2}\right),$
(77)
$\tilde{f}_{2}^{B}=\frac{1}{4}\frac{(1-\delta)^{2}}{(2-\delta)^{2}}\left(8-3\left(2-\delta\right)\left(u_{1}+u_{2}+y_{1}+y_{2}\right)+4\delta(1-\delta)\right),$
(78)
$\tilde{f}_{3}^{B}=\frac{1}{4}\frac{(1-\delta)^{3}}{(2-\delta)^{2}}\left(u_{1}-u_{2}+y_{1}-y_{2}\right),\
\ \ \tilde{f}_{4}^{B}=-\frac{1}{2}\frac{(1-\delta)^{4}}{(2-\delta)^{2}}.$ (79)

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