#  $J/\psi$ polarization in semi-inclusive DIS at low and high transverse
momentum

Umberto D’Alesio<EMAIL_ADDRESS>Dipartimento di Fisica,
Università di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA),
Italy  INFN Sezione di Cagliari, Cittadella Universitaria, I-09042 Monserrato
(CA), Italy  Luca Maxia<EMAIL_ADDRESS>Dipartimento di Fisica,
Università di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA),
Italy  INFN Sezione di Cagliari, Cittadella Universitaria, I-09042 Monserrato
(CA), Italy  Francesco Murgia<EMAIL_ADDRESS>INFN Sezione di
Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA), Italy  Cristian
Pisano<EMAIL_ADDRESS>Dipartimento di Fisica, Università di
Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA), Italy  INFN
Sezione di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA), Italy
Sangem Rajesh 111Now at INFN, Sezione di Perugia, via A. Pascoli snc, 06123,
Perugia, Italy<EMAIL_ADDRESS>INFN Sezione di Cagliari, Cittadella
Universitaria, I-09042 Monserrato (CA), Italy

###### Abstract

We study the polar and azimuthal decay angular distributions of $J/\psi$
mesons produced in semi-inclusive, deep-inelastic electron-proton scattering.
For the description of the quarkonium formation mechanism, we adopt the
framework of nonrelativistic QCD, with the inclusion of the intermediate
color-octet channels that are suppressed at most by a factor $v^{4}$ in the
velocity parameter $v$ relative to the leading color-singlet channel. We put
forward factorized expressions for the helicity structure functions in terms
of transverse momentum dependent gluon distributions and shape functions,
which are valid when the $J/\psi$ transverse momentum is small with respect to
the hard scale of the process. By requiring that such expressions correctly
match with the collinear factorization results at high transverse momentum, we
determine the perturbative tails of the shape functions and find them to be
independent of the $J/\psi$ polarization. In particular, we focus on the $\cos
2\phi$ azimuthal decay asymmetry, which originates from the distribution of
linearly polarized gluons inside an unpolarized proton. We therefore suggest a
novel experiment for the extraction of this so-far unknown parton density that
could be performed, in principle, at the future Electron-Ion Collider.

##  I Introduction

The $J/\psi$ meson, i.e. the lightest spin-one bound state of a charm quark-
antiquark pair ($c\bar{c}$), has been the subject of considerable attention
since its simultaneous discovery at the Brookhaven National Laboratory
Aubert:1974js  and at the Stanford Linear Accelerator Center Augustin:1974xw
in 1974. From the experimental side, it has the advantage of being abundantly
produced at all high-energy electron-proton and proton-proton colliders due to
its relatively low mass. Moreover, since the $J/\psi$ meson carries the same
total angular-momentum, parity and charge-conjugation quantum numbers of the
photon, $J^{PC}=1^{--}$, it can decay to $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$
pairs with significant branching ratios of about 6% in both channels, thus
making it especially easy to detect. From the theory side, it is an excellent
tool to probe both the perturbative and nonperturbative aspects of QCD. In
fact, since the charm mass $M_{c}$ is much larger than the QCD scale parameter
$\Lambda_{\text{QCD}}$, the $J/\psi$ production mechanism involves the
creation of the $c\overline{c}$ pair, which can be treated perturbatively, and
its subsequent, nonperturbative, transition to the observed quarkonium bound
state. Different frameworks provide different descriptions of the latter
hadronization process. In this paper, we will adopt the factorization approach
based on the effective field theory of nonrelativistic QCD (NRQCD)
Bodwin:1994jh .

According to NRQCD, the formation of a heavy-quarkonium state is described by
means of a double expansion in the strong coupling constant $\alpha_{s}$ and
in the average relative velocity $v$ of the heavy quark-antiquark pair in the
quarkonium rest frame, with $v^{2}\simeq 0.3$ for charmonium and $v^{2}\simeq
0.1$ for bottomonium. The main feature of this approach is that, at short
distances, a heavy quark-antiquark pair can be produced not only directly in a
color-singlet (CS) configuration, but also in a color-octet (CO) state, which
subsequently evolves into the observed quarkonium through soft gluon
radiation. Such nonperturbative hadronization process is encoded in the so-
called long-distance matrix elements (LDMEs), which are expected to be
universal. The CS LDMEs are well known from potential models, lattice
calculations and leptonic decays. Conversely, the CO ones are typically
extracted from fits to data on $J/\psi$ and $\Upsilon$ yields, see for
instance Refs. Butenschoen:2010rq ; Chao:2012iv ; Sharma:2012dy ;
Bodwin:2014gia ; Zhang:2014ybe , but not yet from lattice calculations.

Although NRQCD successfully explains many experimental observations, it still
presents major challenges Brambilla:2010cs ; Andronic:2015wma ;
Lansberg:2019adr . On the one hand, the present knowledge of the CO matrix
elements is not very accurate, because the different sets of their extracted
values are not compatible with each other, even within the large
uncertainties. For this reason, new ways to determine them with better
precision have been recently put forward Qiu:2020xum ; Boer:2021ehu . On the
other hand, phenomenological analyses within the NRQCD framework are not able
to consistently account for all cross sections and polarization measurements
for $J/\psi$ mesons produced both in proton-proton and in electron-proton
collisions.

In this paper, we show how the study of $J/\psi$ polarization in semi-
inclusive, deep-inelastic electron-proton scattering (SIDIS), i.e. $e\,p\to
e^{\prime}\,J/\psi\,X$, can shed light on the still open puzzles of quarkonium
production mechanism and polarization. Such an analysis is complementary to
the one performed in Ref. Boer:2020bbd , where the transverse momentum
spectrum and the azimuthal distribution for the production of unpolarized
$J/\psi$ mesons were considered. A satisfactory theoretical description will
be achieved only if all these observables can be simultaneously described
within a unique framework. Their experimental determination could be reached
at the future Electron-Ion Collider (EIC) AbdulKhalek:2021gbh ;
Accardi:2012qut ; Boer:2011fh , which will be built in the United States.

Along the lines of Ref. Boer:2020bbd , in order to avoid contributions from
photoproduction processes, we only consider the kinematic region where the
virtuality of the photon exchanged in the reaction, $Q$, is equal or greater
than the $J/\psi$ mass, $M_{\psi}$, namely $Q\geq M_{\psi}$. Moreover, we
denote by $q_{\scriptscriptstyle T}$ the transverse momentum of the photon
with respect to the $J/\psi$ and the proton four-momenta. Hence, in the
calculation of the polarization parameters, we have to deal with three
relevant scales: the above-defined transverse momentum $q_{\scriptscriptstyle
T}$, a hard scale $\mu$ (to be identified with $Q$, or $M_{\psi}$, or any
combination of them) and a soft scale (the nonperturbative QCD scale
$\Lambda_{\text{QCD}}$ or, alternatively, the proton mass). Depending on the
value of $q_{\scriptscriptstyle T}$ we can adopt two different factorization
frameworks, both of them allowing for a separation of the short-distance from
the long-distance contributions to the observables under study.

In the high-$q_{\scriptscriptstyle T}$ region, namely $q_{\scriptscriptstyle
T}\gg\Lambda_{\rm QCD}$, in a frame where the $J/\psi$ meson is at rest, the
photon transverse momentum is generated by perturbative radiation. NRQCD and
collinear factorization can be applied and the resulting polarization
parameters will depend on collinear (i.e. integrated over transverse momentum)
parton distribution functions (PDFs) and LDMEs. In the
small-$q_{\scriptscriptstyle T}$ region, $q_{\scriptscriptstyle T}\ll\mu$, the
photon transverse momentum is nonperturbative instead, and transverse momentum
dependent (TMD) factorization Collins:2011zzd ; GarciaEchevarria:2011rb ;
Echevarria:2012js  is expected to be appropriate. Observables should then
depend on TMD PDFs (or TMDs for short) and shape functions Echevarria:2019ynx
; Fleming:2019pzj , which are the generalization of the NRQCD LDMEs.
Alternatively, the latter can be seen as the analog of the TMD fragmentation
functions for light hadrons. We note that in the overlapping region,
$\Lambda_{\text{QCD}}\ll q_{\scriptscriptstyle T}\ll\mu$, both frameworks can
be applied and the results obtained within the two formalisms have to match,
provided they describe the same underlying mechanism Bacchetta:2008xw . This
property has been proven for several observables for which TMD factorization
at the twist-two level has been demonstrated, such as the unpolarized cross
sections, differential in $q_{\scriptscriptstyle T}$ and integrated over the
azimuthal angles of the final particles, for the SIDIS process $e\,p\to
e^{\prime}\,h\,X$, where $h$ is a light hadron, and for Drell-Yan (DY)
dilepton production, $p\,p\to\ell\,\ell^{\prime}\,X$ Collins:1984kg ;
Catani:2000vq .

Although there is not yet a rigorous proof of TMD factorization for the
$J/\psi$ polarization parameters in SIDIS, strong arguments exist in favor of
its validity, if we consider the analogy with $e\,p\to e^{\prime}\,h\,X$, for
which TMD factorization holds at all orders Collins:2011zzd . As already
pointed out in Ref. Boer:2020bbd , these two processes are essentially
equivalent from the point of view of the color flow, which determines the
gauge-link structure of the TMD parton correlators Bacchetta:2018ivt . Since
neither the $J/\psi$ mass nor its spin can affect such structure, we do not
expect any factorization breaking effects due to color entanglement. We
therefore propose reasonable factorized expressions for the $J/\psi$
polarization parameters in terms of twist-two TMDs and shape functions, which
properly match with the collinear results in the intermediate region
$\Lambda_{\text{QCD}}\ll q_{\scriptscriptstyle T}\ll\mu$.

Finally, we point out that the theoretical TMD framework we have devised will
also have important implications for the extraction of gluon TMDs at the EIC.
Indeed, although several proposals for the extraction of these distributions
have been put forward, both in $ep$ Yuan:2008vn ; Bacchetta:2018ivt ;
Mukherjee:2016qxa ; DAlesio:2019qpk  and in $pp$ collisions Yuan:2008vn ;
Dunnen:2014eta ; Lansberg:2017tlc ; Lansberg:2017dzg ; Scarpa:2019fol , as
well as within the more phenomenological generalized parton model approach
Godbole:2013bca ; Godbole:2014tha ; Godbole:2017syo ; Kishore:2018ugo ;
Rajesh:2018qks ; DAlesio:2017rzj ; DAlesio:2018rnv ; DAlesio:2019gnu , they
are still basically unkown. In particular, we find that the so-called $\nu$
parameter, related to the $\cos 2\phi$ azimuthal asymmetry of the leptons from
the $J/\psi$ decay, can give direct access to the distribution of linearly
polarized gluons inside unpolarized protons.

The paper is organized as follows. In Section II we review in detail the main
properties of the cross section using only kinematic considerations and the
symmetries of the strong and electromagnetic interactions, without referring
to any specific model concerning quarkonium formation. In particular, the
well-known result of the angular structure of the cross section, expressed in
terms of four independent helicity structure functions, is obtained along the
lines of the derivation presented in Ref. Lam:1978pu  for the DY process. In
Section III we compute the polarized structure functions within the framework
of collinear factorization and NRQCD at the order $\alpha_{s}^{2}$. Moreover,
we investigate their small-$q_{\scriptscriptstyle T}$ limit and show that the
dominant terms do not depend on the choice of the coordinate axes. Section IV
is devoted to the calculation of the structure functions in the TMD regime at
the order $\alpha_{s}$ and to the study of their matching with the collinear
results in the common region of validity. Summary and conclusions are gathered
in Section V. Details on the reference frames and on the transformations which
connect them can be found in Appendix A. Finally, the explicit expressions for
the partonic helicity structure functions in the Gottfried-Jackson frame are
collected in Appendix B.

##  II Model independent properties of the cross section

We study the process

$e(k)+p(P)\to e(k^{\prime})+J/\psi(P_{\psi})+X(P_{X})\,,$ (1)

with $X$ being an undetected hadronic system, and the subsequent leptonic
decay

$\displaystyle J/\psi(P_{\psi})\to\ell^{+}(l)\,+\,\ell^{-}(l^{\prime})\,,$ (2)

where the four-momenta of the particles are given within brackets. Throughout
this work the mass of the leptons, both in the initial and in the final state,
are neglected, while we denote by $M_{p}$ and $M_{\psi}$ the proton and the
$J/\psi$ masses, respectively. Moreover, we will sum over the lepton and
$J/\psi$ helicities. The virtual photon exchanged in the reaction carries
four-momentum

$\displaystyle q=k-k^{\prime}\,,$ (3)

with $Q^{2}\equiv-q^{2}>0$. In the deep-inelastic limit, where $Q^{2}$,
$P\cdot q$ and $P_{\psi}\cdot q$ are large while the variables

$x_{\scriptscriptstyle B}=\frac{Q^{2}}{2P\cdot q}\,,\qquad y=\frac{P\cdot
q}{P\cdot k}\,,\qquad z=\frac{P\cdot P_{\psi}}{P\cdot q}$ (4)

are finite (with values between zero and one), the square of the invariant
amplitude for this process can be split into a leptonic part and a purely
hadronic part,

$\displaystyle|{\cal M}|^{2}=\frac{1}{Q^{4}}\,L^{\mu\nu}H_{\mu\nu}\,,$ (5)

where the leptonic tensor $L^{\mu\nu}$ reads

$\displaystyle L^{\mu\nu}$
$\displaystyle=e^{2}\left[-g^{\mu\nu}Q^{2}+2({k}^{\mu}{k}^{\prime\nu}+{k}^{\nu}{k}^{\prime\mu})\right]\,,$
(6)

and $H_{\mu\nu}$ is the product of the hadronic current matrix elements,

$\displaystyle H_{\mu\nu}=\langle P|J_{\mu}(0)|P_{X};P_{\psi}\rangle\langle
P_{X};P_{\psi}|J_{\nu}(0)|P\rangle\,.$ (7)

The corresponding cross section reads

$\displaystyle\mathrm{d}\sigma=\frac{1}{2S}\,|{\cal
M}|^{2}\,B_{\ell\ell}\,(2\pi)^{4}\delta^{4}(q+P-l-l^{\prime}-P_{X})\,\frac{\mathrm{d}^{3}P_{X}}{(2\pi)^{3}\,2P^{0}_{X}}\,\frac{\mathrm{d}^{3}k^{\prime}}{(2\pi)^{3}\,2E_{e}^{\prime}}\,\frac{\mathrm{d}^{3}l}{(2\pi)^{3}2l^{0}}\,\frac{\mathrm{d}^{3}l^{\prime}}{(2\pi)^{3}2l^{\prime
0}}\,,$ (8)

where $S=(k+P)^{2}\approx 2\,k\cdot P$ if also the proton mass is neglected,
while $B_{\ell\ell}$ is the branching ratio for the decay process
$J/\psi\to\ell^{+}\ell^{-}$. The integration of $H_{\mu\nu}$ over $P_{X}$
leads to the usual hadronic tensor

$\displaystyle
W_{\mu\nu}=\int\\!\frac{\mathrm{d}^{3}P_{X}}{(2\pi)^{3}\,2P^{0}_{X}}\,\delta^{4}(q+P-P_{\psi}-P_{X})\,H_{\mu\nu}\,.$
(9)

Moreover, it is convenient to express the phase space of the final leptons as
follows

$\displaystyle\frac{\mathrm{d}^{3}k^{\prime}}{(2\pi)^{3}2E^{\prime}_{e}}$
$\displaystyle=\frac{1}{4(2\pi)^{3}}\,Sy\,\mathrm{d}x_{\scriptscriptstyle
B}\,\mathrm{d}y\,\mathrm{d}\phi_{k^{\prime}},$
$\displaystyle\frac{\mathrm{d}^{3}l}{(2\pi)^{3}2l^{0}}\,\frac{\mathrm{d}^{3}l^{\prime}}{(2\pi)^{3}2l^{\prime
0}}$
$\displaystyle=\frac{\mathrm{d}^{4}P_{\psi}}{(2\pi)^{4}}\,\frac{\mathrm{d}\Omega}{32\pi^{2}}\,,$
(10)

where $\phi_{k^{\prime}}$ is the azimuthal angle of the scattered electron,
while $\Omega=(\theta,\varphi)$ refers to the solid angle of lepton $\ell^{+}$
in a reference frame where the system formed by $\ell^{+}$ and $\ell^{-}$ is
at rest. Hence, after integration over $\phi_{k^{\prime}}$, Eq. (8) can be
cast in the form

$\displaystyle\frac{1}{B_{\ell\ell}}\,\frac{\mathrm{d}\sigma}{\mathrm{d}x_{\scriptscriptstyle
B}\,\mathrm{d}y\,\mathrm{d}^{4}P_{\psi}\,\mathrm{d}\Omega}=\frac{1}{4(4\pi)^{4}}\,\frac{y}{Q^{4}}\,L^{\mu\nu}W_{\mu\nu}\,\,.$
(11)

The hadronic tensor $W_{\mu\nu}$ is a function of the four-vectors $q^{\mu}$,
$P^{\mu}$, $P_{\psi}^{\mu}$ and contains the information on the proton
structure as well as on the $J/\psi$ formation process and polarization. It
has to fulfill certain constraints due to electromagnetic gauge invariance,
parity, and hermiticity. More specifically, the gauge-invariance condition,

$\displaystyle q^{\mu}W_{\mu\nu}(q,P,P_{\psi})$
$\displaystyle=q^{\nu}W_{\mu\nu}(q,P,P_{\psi})=0\,,$ (12)

is implied by conservation of the hadronic current $J_{\mu}(0)$ in Eq. (7) and
limits $W_{\mu\nu}$ to be a tensor in the three-dimensional space orthogonal
to $q^{\mu}$. Moreover, as a consequence of the hermiticity condition,

$\displaystyle W_{\mu\nu}(q,P,P_{\psi})$
$\displaystyle=W^{*}_{\nu\mu}(q,P,P_{\psi})\,,$ (13)

the symmetric part of the hadronic tensor in the indices $(\mu,\nu)$ is real,
while its antisymmetric part is imaginary. Since the leptonic tensor in Eq.
(6) is symmetric, only the symmetric part of $W_{\mu\nu}$ contributes to the
cross section. Finally, parity conservation implies

$\displaystyle
W_{\mu\nu}(q,P,P_{\psi})=W_{\mu\nu}(\overline{q},\overline{P},\overline{P}_{\psi})\,,$
(14)

with the definition $\overline{v}^{\mu}\equiv v_{\mu}=(v^{0},-\bm{v})$ for any
vector $v^{\mu}=(v^{0},\bm{v})$. By imposing Eqs. (12) and (14), the hadronic
tensor can be decomposed into a set of four basis tensors multiplied by scalar
functions, the so-called structure functions, which have to be real because of
Eq. (13). To this end, it is useful to introduce the projector operator to the
space orthogonal to $q^{\mu}$,

$\displaystyle\widehat{g}^{\mu\nu}=g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{Q^{2}}\,,$
(15)

satisfying the relation
$\widehat{g}_{\mu\nu}q^{\mu}=\widehat{g}_{\mu\nu}q^{\nu}=0$ and such that,
when contracted with any four-vector, yields a vector orthogonal to $q^{\mu}$.
In particular,

$\displaystyle\widehat{P}^{\mu}=\frac{1}{M_{p}}\,\widehat{g}^{\mu\nu}P_{\nu}=\frac{1}{M_{p}}\,\left(P^{\mu}+\frac{P\cdot
q}{Q^{2}}\,q^{\mu}\right)\,,\qquad\widehat{P}_{\psi}^{\mu}=\frac{1}{M_{\psi}}\,\widehat{g}^{\mu\nu}P_{\psi\nu}=\frac{1}{M_{\psi}}\,\left(P_{\psi}^{\mu}+\frac{P_{\psi}\cdot
q}{Q^{2}}\,q^{\mu}\right)\,,$ (16)

with $\widehat{P}\cdot q=\widehat{P}_{\psi}\cdot q=0$. Hence, the hadronic
tensor can be expressed as

$\displaystyle
W^{\mu\nu}(q,P,P_{\psi})=-W_{1}\,\widehat{g}^{\mu\nu}\,+\,W_{2}\,\widehat{P}^{\mu}\widehat{P}^{\nu}-\frac{1}{2}\,W_{3}\,(\widehat{P}^{\mu}\widehat{P}_{\psi}^{\nu}+\widehat{P}_{\psi}^{\mu}\widehat{P}^{\nu})+W_{4}\,\widehat{P}_{\psi}^{\mu}\widehat{P}_{\psi}^{\nu}\,,$
(17)

in terms of the common Lorentz invariant structure functions
$W_{i}=W_{i}(Q^{2},q\cdot P,q\cdot P_{\psi})$, with $i=1,2,3,4$.

An alternative parametrization of the hadronic tensor is provided by four,
frame dependent, helicity structure functions, which, as already pointed out
in Ref. Lam:1978pu  for the DY process, have the advantage of allowing for an
explicit factorization of the leptonic and hadronic variables in the cross
section. In the following, we will show that this is the case also for the
process under study.

As a first step of our derivation, it might be useful to write separately the
contributions coming from transversely and longitudinally polarized virtual
photons. This can be achieved by expressing the leptonic tensor in Eq. (6) in
the form

$\displaystyle
L^{\mu\nu}=e^{2}\,\frac{Q^{2}}{y^{2}}\,\bigg{\\{}-[1+(1-y)^{2}]\,g_{\perp}^{\mu\nu}\,+\,4(1-y)\,\epsilon^{\mu}_{\gamma{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}\epsilon^{\nu}_{\gamma{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}}\bigg{\\}}\,,$ (18)

which is valid upon integration over the azimuthal angle of the electron
scattering plane, taken with respect to the plane formed by the proton and the
virtual photon, and neglecting the proton mass. In Eq. (18), we have
introduced the projector to the space orthogonal to both $q$ and $P$

$\displaystyle g_{\perp}^{\mu\nu}$ $\displaystyle=g^{\mu\nu}-\frac{1}{P\cdot
q}\,(P^{\mu}q^{\nu}+P^{\nu}q^{\mu})-\frac{Q^{2}}{(P\cdot
q)^{2}}\,P^{\mu}P^{\nu}$ (19)

and the longitudinal polarization vector of the exchanged photon,

$\displaystyle\epsilon_{\gamma{{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}}^{\mu}(q)=\frac{1}{Q}\left(q^{\mu}+\frac{Q^{2}}{P\cdot
q}\,P^{\mu}\right)\,,$ (20)

which fulfills the relations $\epsilon^{2}_{\gamma{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}(q)=1$ and $\epsilon_{\gamma{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}}^{\mu}(q)\,q_{\mu}=0$.
Hence we obtain

$\displaystyle
L^{\mu\nu}W_{\mu\nu}=e^{2}\,\frac{Q^{2}}{y^{2}}\,\left\\{[1+(1-y)^{2}]\,{\cal
W}^{\perp}\,+\,(1-y)\,{\cal W}^{{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}}\right\\}\,,$ (21)

with the definitions

$\displaystyle{\cal W}^{\perp}\equiv-
g_{\perp}^{\mu\nu}W_{\mu\nu}=-{W^{\mu}}_{\mu}+\frac{Q^{2}}{(P\cdot
q)^{2}}\,P^{\mu}P^{\nu}\,W_{\mu\nu}\,,\qquad\qquad{\cal W}^{{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}\equiv 4\,\epsilon^{\mu}_{\gamma{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}\epsilon^{\nu}_{\gamma{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}W_{\mu\nu}=\frac{4\,Q^{2}}{(P\cdot
q)^{2}}\,P^{\mu}P^{\nu}\,W_{\mu\nu}\,,$ (22)

which in turn can be expressed in terms of either the invariant structure
functions or the helicity ones.

Figure 1: Reference frames for the process $\gamma^{*}(q)+p(P)\to
J/\psi(P_{\psi})+X$ in which the $J/\psi$ is at rest: Helicity (HX), Target
(TF), Collins-Soper (CS) and Gottfried-Jackson (GJ). They are related to each
other by a rotation around the $Y$-axis. See Appendix A for further details.

It still remains to clarify the dependence of the cross section on the angles
of the decaying lepton $\ell^{+}$. To this end, we write the sum over the
final $J/\psi$ helicities $\lambda,\lambda^{\prime}=-1,0,1$ in an explicit
form,

$\displaystyle{\cal W}^{\cal P}=\sum_{\lambda,\lambda^{\prime}}{\cal W}^{\cal
P}_{\alpha\beta}\,\epsilon^{\alpha}_{\lambda}(P_{\psi})\,\epsilon^{\beta\,*}_{\lambda^{\prime}}(P_{\psi})\,\delta_{\lambda\lambda^{\prime}}=\sum_{\lambda,\lambda^{\prime}}{\cal
W}^{\cal P}_{\lambda\lambda^{\prime}}\,\delta_{\lambda\lambda^{\prime}}\,,$
(23)

where the superscript ${\cal P}=\perp,{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}$ refers to the photon
polarization and we have introduced the frame-dependent helicity structure
functions

$\displaystyle{\cal W}^{\cal
P}_{\lambda\lambda^{\prime}}\equiv\epsilon^{\alpha}_{\lambda}(P_{\psi})\,\epsilon^{\ast\beta}_{\lambda^{\prime}}(P_{\psi})\,{\cal
W}_{\alpha\beta}^{\cal P}\,,$ (24)

with $\epsilon^{\mu}_{\lambda}(P_{\psi})$ being the polarization vectors of
the spin-1 $J/\psi$ meson defined with respect to a covariant set of
coordinate axes $(T^{\mu},X^{\mu},Y^{\mu},Z^{\mu})$ with normalizations
$T^{2}=1$ and $X^{2}=Y^{2}=Z^{2}=-1$. These four-vectors can be defined as
linear combinations of the physical momenta $q^{\mu}$, $P^{\mu}$,
$P_{\psi}^{\mu}$, in such a way that $X^{\mu}$, $Y^{\mu}$, $Z^{\mu}$ become
three-vectors in the quarkonium rest frame Lam:1978pu ; Beneke:1998re . An
illustration of the commonly used frames is given in Fig. 1. In all of them,

$\displaystyle T^{\mu}=\frac{P_{\psi}^{\mu}}{M_{\psi}}\,,\qquad\qquad
Y^{\mu}=\varepsilon^{\mu\nu\alpha\beta}X_{\nu}Z_{\alpha}T_{\beta}\,,$ (25)

with $\varepsilon^{0123}=+1$, while they differ for the choice of $X^{\mu}$
and $Z^{\mu}$, see also Appendix A for details. Once the frame is fixed, the
polarization vectors are given by

$\displaystyle\epsilon^{\mu}_{0}({\bm{P}_{\psi}})=Z^{\mu}=(0,\,0,\,0,\,1)\,,\qquad\epsilon^{\mu}_{\pm
1}(\bm{P}_{\psi})=\frac{1}{\sqrt{2}}(\mp
X^{\mu}-iY^{\mu})=\frac{1}{\sqrt{2}}\,(0,\mp 1,-i,0)\,,$ (26)

and fulfill the orthogonality and completeness relations,

$\displaystyle\epsilon_{\lambda\,\alpha}^{*}(\bm{P}_{\psi})\,\epsilon^{\alpha}_{\lambda^{\prime}}(\bm{P}_{\psi})$
$\displaystyle=-\delta_{\lambda\lambda^{\prime}}\,,$
$\displaystyle\sum_{\lambda=-1,0,1}\,\epsilon_{\lambda}^{\alpha*}(\bm{P}_{\psi})\,\epsilon^{\beta}_{\lambda}(\bm{P}_{\psi})$
$\displaystyle=-g^{\alpha\beta}\,+\,\frac{P^{\alpha}_{\psi}P^{\beta}_{\psi}}{M_{\psi}^{2}}\,.$
(27)

The constraints of parity conservation and hermiticity imposed by QCD on the
hadronic tensor imply the following relations for the helicity structure
functions

$\displaystyle{\cal W}_{\lambda\lambda^{\prime}}^{\cal P}={\cal
W}_{\lambda^{\prime}\lambda}^{\cal P\,*}\,,\qquad\qquad{\cal
W}_{\lambda\lambda^{\prime}}^{\cal P}=(-1)^{\lambda+\lambda^{\prime}}{\cal
W}_{-\lambda-\lambda^{\prime}}^{\cal P}\,.$ (28)

We can therefore decompose ${\cal W}_{\alpha\beta}^{{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}$ and ${\cal W}_{\alpha\beta}^{\perp}$ as Lam:1978pu ; Boer:2006eq

$\displaystyle{\cal W}^{\cal P}_{\alpha\beta}=-({\cal W}^{\cal P}_{T}+{\cal
W}^{\cal P}_{\Delta\Delta})\,({g}_{\alpha\beta}-T_{\alpha}T_{\beta})+({\cal
W}^{\cal P}_{L}-{\cal W}^{\cal P}_{T}-{\cal W}^{\cal
P}_{\Delta\Delta})\,Z_{\alpha}Z_{\beta}\,-\,{\cal W}^{\cal
P}_{\Delta}\,(X_{\alpha}Z_{\beta}+Z_{\alpha}X_{\beta})\,-\,2{\cal W}^{\cal
P}_{\Delta\Delta}\,X_{\alpha}X_{\beta}\,,$ (29)

in terms of the eight independent helicity structure functions

$\displaystyle{\cal W}^{\cal P}_{T}$ $\displaystyle\equiv{\cal W}^{\cal
P}_{11}={\cal W}^{\cal P}_{-1-1}\,,$ $\displaystyle{\cal W}^{\cal P}_{L}$
$\displaystyle\equiv{\cal W}^{\cal P}_{00}\,,$ $\displaystyle{\cal W}^{\cal
P}_{\Delta}$ $\displaystyle\equiv\frac{1}{\sqrt{2}}\,({\cal W}^{\cal
P}_{10}+{\cal W}^{\cal P}_{01})=\sqrt{2}\,\text{Re}\,{\cal W}^{\cal
P}_{10}\,,$ $\displaystyle{\cal W}^{\cal P}_{\Delta\Delta}$
$\displaystyle\equiv{\cal W}^{\cal P}_{1-1}={\cal W}^{\cal P}_{-11}\,,$ (30)

where the subscripts refer to the $J/\psi$ polarization: ${\cal W}^{\cal
P}_{T}$ and ${\cal W}^{\cal P}_{L}$ are respectively the structure functions
for transversely and longitudinally polarized $J/\psi$ mesons, ${\cal W}^{\cal
P}_{\Delta}$ are the single-helicity flip structure functions, while ${\cal
W}^{\cal P}_{\Delta\Delta}$ are the double-helicity flip ones.

A further simplification occurs because the leptonic decay conserves helicity
in the massless limit we are considering, i.e. the lepton spins do not flip in
the coupling with the $J/\psi$ meson. In the quarkonium rest frame, where
$\ell^{+}$ and $\ell^{-}$ are produced back-to-back along the direction
identified by the three-vector

$\displaystyle\bm{l}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)\,,$
(31)

the component along this direction of the total angular momentum of the
dilepton system, and therefore of the $J/\psi$, can be either $+1$ or $-1$,
but not zero. Hence, the sum in Eq. (23) can be performed by choosing a
specific set of polarization vectors $\epsilon^{\alpha}_{\sigma}(P_{\psi})$,
where $\sigma$ is the $J/\psi$ helicity along the direction ${\bm{l}}$,

$\displaystyle{\cal W}^{\cal P}={\cal W}^{\cal
P}_{\alpha\beta}\,\sum_{\sigma=-1,1}\epsilon^{\alpha}_{\sigma}(P_{\psi})\,\epsilon^{\beta\,*}_{\sigma}(P_{\psi})=-{\cal
W}^{\cal P}_{\alpha\beta}\,g_{l\perp}^{\alpha\beta}\,,$ (32)

and the transverse projector with respect to the directions of the four-
vectors $P_{\psi}$ and $l$ is given by

$\displaystyle g_{l\perp}^{\alpha\beta}$
$\displaystyle=g^{\alpha\beta}-\frac{1}{l\cdot
P_{\psi}}\,(l^{\alpha}P_{\psi}^{\beta}\,+\,l^{\beta}P_{\psi}^{\alpha})+\frac{M_{\psi}^{2}}{(l\cdot
P_{\psi})^{2}}\,l^{\alpha}l^{\beta}\,.$ (33)

By contracting the above expression for $g_{l\perp}^{\alpha\beta}$ with the
one for ${\cal W}^{\cal P}_{\alpha\beta}$ in Eq. (29), according to Eq. (32)
we obtain

$\displaystyle{\cal W}^{\cal P}={\cal W}^{\cal
P}_{T}(1+\cos^{2}\theta)\,+\,{\cal W}^{\cal P}_{L}(1-\cos^{2}\theta)\,+\,{\cal
W}^{\cal P}_{\Delta}\sin 2\theta\cos\varphi\,+\,{\cal W}^{\cal
P}_{\Delta\Delta}\sin^{2}\theta\cos 2\varphi\,.$ (34)

Hence, by using this result and the one in Eq. (21), the angular structure of
the cross section in Eq. (11) can be made explicit. We find

$\displaystyle\frac{1}{B_{\ell\ell}}\,\frac{\mathrm{d}\sigma}{\mathrm{d}x_{\scriptscriptstyle
B}\,\mathrm{d}y\,\mathrm{d}^{4}P_{\psi}\,\mathrm{d}\Omega}=\frac{1}{4(4\pi)^{3}}\,\frac{\alpha}{yQ^{2}}\,\left[{\cal
W}_{T}(1+\cos^{2}\theta)\,+\,{\cal W}_{L}(1-\cos^{2}\theta)\,+\,{\cal
W}_{\Delta}\sin 2\theta\cos\varphi\,+\,{\cal
W}_{\Delta\Delta}\sin^{2}\theta\cos 2\varphi\right]\,,$ (35)

with the definitions

$\displaystyle{\cal W}_{\Lambda}\equiv\left[1+(1-y)^{2}\right]{\cal
W}^{\perp}_{\Lambda}\,+\,(1-y){\cal W}^{{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}_{\Lambda}\,\qquad\text{with}\qquad\Lambda=T,L,\Delta,\Delta\Delta\,.$
(36)

We point out that Eq. (35) represents the main result of this section and
shows, as anticipated above, that the helicity structure functions defined in
Eqs. (30) and (36) allow for an explicit factorization of the leptonic and
hadronic variables in the cross section.

It is also convenient to introduce the following ratio of differential cross
sections Boer:2006eq ,

$\displaystyle\frac{\mathrm{d}N}{\mathrm{d}\Omega}\equiv\left(\frac{\mathrm{d}\sigma}{\mathrm{d}x_{\scriptscriptstyle
B}\,\mathrm{d}y\,\mathrm{d}^{4}P_{\psi}}\right)^{-1}\frac{\mathrm{d}\sigma}{\mathrm{d}x_{\scriptscriptstyle
B}\,\mathrm{d}y\,\mathrm{d}^{4}P_{\psi}\,\mathrm{d}\Omega}\,,$ (37)

which is therefore given by

$\displaystyle\frac{\mathrm{d}N}{\mathrm{d}\Omega}=\frac{3}{8\pi}\,\frac{{\cal
W}_{T}(1+\cos^{2}\theta)\,+\,{\cal W}_{L}(1-\cos^{2}\theta)\,+\,{\cal
W}_{\Delta}\sin 2\theta\cos\varphi\,+\,{\cal
W}_{\Delta\Delta}\sin^{2}\theta\cos 2\varphi}{2{\cal W}_{T}+{\cal W}_{L}}\,.$
(38)

Alternatively, this ratio can be expressed in terms of the largely used
polarization parameters $\lambda$, $\mu$, $\nu$,

$\displaystyle\frac{\mathrm{d}N}{\mathrm{d}\Omega}=\frac{3}{4\pi}\,\frac{1}{\lambda+3}\,\left[1+\lambda\cos^{2}\theta+\mu\sin
2\theta\cos\varphi+\frac{1}{2}\,\nu\,\sin^{2}\theta\cos 2\varphi\right]\,,$
(39)

where, by comparison with Eq. (38), we have

$\displaystyle\lambda=\frac{{\cal W}_{T}-{\cal W}_{L}}{{\cal W}_{T}+{\cal
W}_{L}}\,,\qquad\mu=\frac{{\cal W}_{\Delta}}{{\cal W}_{T}+{\cal
W}_{L}}\,,\qquad\nu=\frac{2{\cal W}_{\Delta\Delta}}{{\cal W}_{T}+{\cal
W}_{L}}\,.$ (40)

Notice that Eqs. (38) and (39) describe the typical angular distribution of
the leptons that originate from the decay of a spin-one particle in its rest
frame, for instance a virtual photon produced in the Drell-Yan process
Boer:2011fh  or a $J/\psi$ meson produced in photon-proton collisions
Beneke:1998re .

##  III Helicity structure functions within NRQCD and collinear factorization

The helicity structure functions ${\cal W}^{\cal P}_{\lambda\lambda^{\prime}}$
can be calculated within NRQCD and collinear factorization in the kinematic
region $q_{\scriptscriptstyle T}\gg\Lambda_{\text{QCD}}$. One first needs to
evaluate the partonic structure functions ${w}^{{\cal
P}(a)}_{\lambda\lambda^{\prime}}$ for each of the underlying hard scattering
subprocesses, which at the order $\alpha\alpha_{s}^{2}$ are

$\displaystyle\gamma^{*}(q)+a(p_{a})\to
c\overline{c}[n](P_{\psi})+a(p_{a}^{\prime})\,,$ (41)

where $a$ is either a gluon, a quark or an antiquark with helicity
$\lambda_{a}$, and the charm-anticharm quark pair is produced in the
intermediate Fock state $n=\,^{2S+1\\!}L_{J}^{[c]}$, with $S$, $L$, $J$ being
the spin, orbital and total angular momenta of the $c\bar{c}$ pair,
respectively, while $c=1,8$ specifies its color configuration. In addition to
the leading color-singlet production channel with $n=\,^{3}\\!S_{1}^{[1]}$, we
also include the subleading color-octet channels that are relatively
suppressed by at most a factor of $v^{4}$, namely ${}^{1}\\!S_{0}^{[8]}$,
${}^{3}\\!S_{1}^{[8]}$, ${}^{3}\\!P_{J}^{[8]}$, with $J=0,1,2$. The
corresponding Feynman diagrams are shown in Fig. 2. The polarized hadronic
structure functions are then obtained as convolutions of the partonic ones
with suitable parton distribution functions $f_{1}^{a}$. If we denote by
${\cal M}_{\mu\alpha}(\gamma^{*}a\to J/\psi\,X)$ the amplitude for the
processes $\gamma^{*}a\to J/\psi\,X$, where we assume that the $J/\psi$ meson
is produced in a definite helicity state $\lambda$, the partonic structure
functions are given by

$\displaystyle w^{{\cal
P}(a)}_{\lambda\lambda^{\prime}}=\sum_{\lambda_{a}}\,\frac{1}{2}\,\varepsilon^{\mu\nu}_{\cal
P}\,{\cal M}_{\mu\alpha}(\gamma^{*}a\to J/\psi\,X)\,{\cal
M}^{*}_{\nu\beta}(\gamma^{*}a\to
J/\psi\,X)\,\epsilon^{\alpha}_{\lambda}(P_{\psi})\,\epsilon^{*\beta}_{\lambda^{\prime}}(P_{\psi})\,,$
(42)

with ${\cal{P}}=\perp,{\mkern 3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern
2.0mu\vrule depth=0.0pt\mkern 3.0mu}$; the tensors $\varepsilon^{\mu\nu}_{\cal
P}$ being defined as $\varepsilon^{\mu\nu}_{\perp}=-g^{\mu\nu}_{\perp}$ and
$\varepsilon^{\mu\nu}_{\mkern 3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern
2.0mu\vrule depth=0.0pt\mkern 3.0mu}=\epsilon^{\mu}_{\gamma{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}\,\epsilon^{\nu}_{\gamma{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}}$, see also Eqs.
(19)-(20). The $J/\psi$ polarization vectors
$\epsilon_{\lambda}^{\alpha}(P_{\psi})$ are presented in Appendix A in terms
of the four-momenta $q$, $P$, $P_{\psi}$ in the four different frames
considered.

According to NRQCD, the $J/\psi$ polarization state $\lambda$ can be reached
from the $c\bar{c}$ pairs produced in various orbital and spin angular
momentum states in the subprocesses of Eq. (41) Beneke:1998re . Because of
charge and parity conjugation, there are no interference effects between
intermediate states with different orbital angular momentum $L$ and spin $S$.
However, NRQCD does not forbid interferences of different ${}^{3}P_{J}$
states, for which $S=L=1$ Leibovich:1996pa ; Beneke:1995yb ; Beneke:1996tk .
The following decomposition of $w^{{\cal P}(a)}_{\lambda\lambda^{\prime}}$ is
therefore valid,

$\displaystyle w^{{\cal P}(a)}_{\lambda\lambda^{\prime}}=w^{{\cal
P}(a)}_{\lambda\lambda^{\prime}}\left[\,{}^{3}\\!S_{1}^{[1]}\right]+w^{{\cal
P}(a)}_{\lambda\lambda^{\prime}}\left[\,{}^{3}\\!S_{1}^{[8]}\right]\,+\,w^{{\cal
P}(a)}_{\lambda\lambda^{\prime}}\left[\,{}^{1}\\!S_{0}^{[8]}\right]\,+\,w^{{\cal
P}(a)}_{\lambda\lambda^{\prime}}\left[\left\\{L=1,S=1\right\\}^{[8]}\right]\,,$
(43)

which expresses the coherent sum of the partonic structure functions for the
four intermediate $c\bar{c}$ states. Since the ${}^{1}S_{0}^{[8]}$ state is
rotational invariant in the quarkonium rest frame, it leads to a random
orientation of the $J/\psi$ spin. In other words, each $J/\psi$ helicity state
$\lambda$ amounts to one third of the unpolarized cross section. Furthermore,
for the ${}^{3}\\!S_{1}^{[1,8]}$ states, NRQCD spin symmetry implies that the
perturbatively calculable third component of the spin of the $c\bar{c}$ pair,
namely the quantum number $S_{z}$, is not modified in the subsequent
nonperturbative evolution into the observed $J/\psi$ meson. Hence, $S_{z}$ is
equal to the helicity of the $J/\psi$, i.e. $S_{z}=\lambda$. The only
nontrivial term of the sum in Eq. (43) is the last one. This can be calculated
explicitly by projecting the hard scattering amplitude onto states of definite
$S_{z}=\lambda$ and $L_{z}$, squaring the amplitude and summing over $L_{z}$
Beneke:1998re ; Leibovich:1996pa ; Beneke:1995yb . In formulae,

$\displaystyle w^{{\cal
P}(a)}_{\lambda\lambda^{\prime}}\left[\left\\{L=1,S=1\right\\}^{[8]}\right]$
$\displaystyle\propto\sum_{L_{z}}\sum_{\lambda_{a}}\,\frac{1}{2}\,\varepsilon^{\mu\nu}_{\cal
P}\,{\cal M}_{\mu\alpha}\left(\gamma^{*}a\to
c\bar{c}[(1,L_{z};1,\lambda)]\,a\right)\,{\cal
M}^{*}_{\nu\beta}\left(\gamma^{*}\to
c\bar{c}[(1,L_{z};1,\lambda^{\prime})\,a]\right)\epsilon^{\alpha}_{\lambda}\,\epsilon^{*\beta}_{\lambda^{\prime}}$
$\displaystyle\neq\sum_{J=0,1,2}w^{{\cal
P}(a)}_{\lambda\lambda^{\prime}}[\,^{3}P_{J}]\,,$ (44)

where we have denoted the quantum numbers of the $c\bar{c}$ pair by
$(L,L_{z};S,S_{z})$. At the order in $v$ we are considering and by further
exploiting the symmetries of NRQCD, it can be shown that the partonic
structure functions can be written in a factorized form, in which the
nonperturbative contributions, encoded in the same four LDMEs $\langle 0|{\cal
O}(n)|0\rangle$ that describe unpolarized $J/\psi$ production Boer:2020bbd ;
Kniehl:2001tk ; Sun:2017nly ; Sun:2017wxk ; Zhang:2019ecf , are multiplied by
perturbative short-distant coefficients $K^{{\cal P}(a)}_{\alpha\beta}[n]$.
These can be calculated from the hard scattering partonic subprocesses
$\gamma^{*}a\to c\bar{c}[n]\,a$. Hence we can write

$\displaystyle w^{{\cal P}(a)}_{\lambda\lambda^{\prime}}=\sum_{n}w^{{\cal
P}(a)}_{\lambda\lambda^{\prime}}[n]\equiv\sum_{n}K^{{\cal
P}(a)}_{\alpha\beta}[n]\,\epsilon^{\alpha}_{\lambda}(P_{\psi})\,\epsilon^{*\beta}_{\lambda^{\prime}}(P_{\psi})\,\langle
0|{\cal O}(n)|0\rangle\,,$ (45)

where the sums run over the Fock states $n=\,^{3}\\!S_{1}^{[1]}$,
${}^{1}\\!S_{0}^{[8]}$, ${}^{3}\\!S_{1}^{[8]}$, ${}^{3}\\!P_{0}^{[8]}$.

In the actual calculation we neglect the proton mass and any smearing effects
both in the initial and in the final states, hence partons can be taken to be
collinear to the parent proton, $p_{a}^{\mu}=\xi P^{\mu}$. The $w^{{\cal
P}(a)}_{\lambda\lambda^{\prime}}$ can therefore be expressed as functions of
the scalar products of the four-vectors $q$, $p_{a}$, $P_{\psi}$, or,
equivalently, of the usual partonic Mandelstam variables

$\displaystyle\hat{s}=(q+p_{a})^{2}\,,\qquad\hat{t}=(q-P_{\psi})^{2}\,,\qquad\hat{u}=(p_{a}-P_{\psi})^{2}\,.$
(46)

It is then straightforward to obtain the structure functions $w^{{\cal
P}(a)}_{\Lambda}$, with $\Lambda=T,L,\Delta,\Delta\Delta$, from the $w^{{\cal
P}(a)}_{\lambda\lambda^{\prime}}$ in analogy to Eq. (30).

Figure 2: Representative diagrams of the partonic subprocesses contributing to
$J/\psi$ production in SIDIS at the order $\alpha\alpha_{s}^{2}$,
$\gamma^{*}(q)+a(p_{a})\to{J/\psi}(P_{\psi})+a(p_{a}^{\prime})$ with
$a=g,q,\bar{q}$. The six diagrams of type (a) are the only ones corresponding
to the CS production mechanism. Moreover, there are two diagrams for each type
(b), (c), (d).

Their expressions in the GJ frame are collected in Appendix B, while the
corresponding formulae in the other three reference frames considered (TF, CS,
HX) can be obtained by applying the transformations listed in Appendix A. We
note that in the photoproduction limit ($Q\to 0$) our results agree with the
ones in Ref. Beneke:1998re . Furthermore, we are able to reproduce the SIDIS
unpolarized cross section, related to the combination $2W_{T}+W_{L}$,
presented in Ref. Kniehl:2001tk . The structure functions $W_{T}$ and $W_{L}$
for $J/\psi$ production in SIDIS have been calculated also in Ref. Yuan:2000cn
, however we are in disagreement with the relative normalizations of most of
the contributions to the structure functions for longitudinally polarized
virtual photons, $w^{{{\mkern 3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern
2.0mu\vrule depth=0.0pt\mkern 3.0mu}}(g)}_{\Lambda}[n]$, presented in Ref.
Yuan:2000cn . Since $w^{{{\mkern 3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern
2.0mu\vrule depth=0.0pt\mkern 3.0mu}}(g)}_{\Lambda}[n]\to 0$ as $Q\to 0$, this
might explain why the authors of Ref. Yuan:2000cn  recover the photoproduction
results in Ref. Beneke:1998re , but not the unpolarized SIDIS cross section in
Ref. Kniehl:2001tk . Conversely, to the best of our knowledge, the SIDIS
helicity flip structure functions $W_{\Delta}$ and $W_{\Delta\Delta}$ have
been explicitly computed for the first time in the present work.

In order to write down the expressions for the hadronic structure functions,
it is convenient to introduce the scaling variables

$\hat{x}=\frac{Q^{2}}{2p_{a}\cdot q}\,,\qquad\hat{z}=\frac{p_{a}\cdot
P_{\psi}}{p_{a}\cdot q}\,.$ (47)

By comparison with the hadronic variables in Eq. (4), we obtain the relations
$\hat{z}=z$ and $\hat{x}={x_{\scriptscriptstyle B}}/{\xi}$. The latter implies
$\hat{x}p_{a}=x_{\scriptscriptstyle B}P$, with $\hat{x}\geq
x_{\scriptscriptstyle B}$. These variables allow us to perform, in the
$J/\psi$ rest frame, the following Sudakov decomposition of the four-momenta
of the particles involved in the reaction,

$\displaystyle q^{\mu}$
$\displaystyle=\frac{M_{\psi}}{\sqrt{2}\hat{z}}\,n_{+}^{\mu}-\left(1-\frac{\bm{q}_{\scriptscriptstyle
T}^{2}}{Q^{2}}\right)\frac{\hat{z}\,Q^{2}}{\sqrt{2}M_{\psi}}\,n^{\mu}_{-}+q_{\scriptscriptstyle
T}^{\mu}\,,$ $\displaystyle p_{a}^{\mu}$
$\displaystyle=\frac{\hat{z}\,Q^{2}}{\sqrt{2}\,\hat{x}\,M_{\psi}}\,n_{-}^{\mu}\,,$
$\displaystyle P^{\mu}_{\psi}$
$\displaystyle=\frac{M_{\psi}}{\sqrt{2}}\,n^{\mu}_{+}+\frac{M_{\psi}}{\sqrt{2}}\,n^{\mu}_{-}\,,$
(48)

where $n_{+}$ and $n_{-}$ are two light-like vectors such that $n_{+}\cdot
n_{-}=1$ and $\bm{q}_{\scriptscriptstyle T}^{2}=-q_{\scriptscriptstyle
T}^{2}$. Hence the Mandelstam variables in Eq. (46) can be written as

$\displaystyle\hat{s}=\frac{1-\hat{x}}{\hat{x}}\,Q^{2}\,,\qquad\hat{t}=-(1-\hat{z})\left(Q^{2}+\frac{M^{2}_{\psi}}{\hat{z}}\right)-\hat{z}\bm{q}_{\scriptscriptstyle
T}^{2}\,,\qquad\hat{u}=-\frac{\hat{z}}{\hat{x}}\,Q^{2}+M_{\psi}^{2}\,,$ (49)

and the hadronic structure functions are given by the convolutions

$\displaystyle(2\pi)^{3}{\cal W}^{\cal P}_{\lambda\lambda^{\prime}}$
$\displaystyle=\sum_{a}\int_{x_{\scriptscriptstyle B}}^{\hat{x}_{\rm
max}}\frac{\mathrm{d}\hat{x}}{\hat{x}}\int_{z}^{1}\mathrm{d}\hat{z}\,f_{1}^{a}\left(\frac{x_{\scriptscriptstyle
B}}{\hat{x}}\,,\mu^{2}\right)\,w^{{\cal
P}(a)}_{\lambda\lambda^{\prime}}\,\delta\left(\hat{s}+\hat{t}+\hat{u}-M_{\psi}^{2}+Q^{2}\right)\delta(\hat{z}-z)$
$\displaystyle=\sum_{a}\sum_{n}\int_{x_{\scriptscriptstyle B}}^{\hat{x}_{\rm
max}}\frac{\mathrm{d}\hat{x}}{\hat{x}}\int_{z}^{1}\frac{\mathrm{d}\hat{z}}{\hat{z}}\,\frac{1}{Q^{2}}\,f_{1}^{a}\left(\frac{x_{\scriptscriptstyle
B}}{\hat{x}}\,,\mu^{2}\right)\,w^{{\cal P}(a)}_{\lambda\lambda^{\prime}}[n]$
$\displaystyle\qquad\qquad\qquad\times\delta\left(\frac{\bm{q}_{\scriptscriptstyle
T}^{2}}{Q^{2}}+\frac{1-\hat{z}}{\hat{z}^{2}}\,\frac{M_{\psi}^{2}}{Q^{2}}-\frac{(1-\hat{x})(1-\hat{z})}{\hat{x}\hat{z}}\right)\delta(\hat{z}-z)\,,$
(50)

where the final result has been obtained by substituting the expression for
$w^{{\cal P}(a)}_{\lambda\lambda^{\prime}}$ given in Eq. (45), while

$\displaystyle\hat{x}_{\rm max}=\frac{Q^{2}}{Q^{2}+M_{\psi}^{2}}\,.$ (51)

Moreover, in Eq. (50) $\mu^{2}$ is the hard factorization scale, on which also
the partonic structure functions $w^{{\cal
P}(a)}_{\lambda\lambda^{\prime}}[n]$ depend, even if not explicitly shown.

The helicity structure functions discussed so far are expected to describe the
azimuthal decay distributions of $J/\psi$ mesons for large values of the
transverse momentum, namely for $\bm{q}_{\scriptscriptstyle
T}^{2}\gg\Lambda^{2}_{\text{QCD}}$. Their behavior in the
small-$q_{\scriptscriptstyle T}$ region,
$\Lambda^{2}_{\text{QCD}}\ll\bm{q}_{\scriptscriptstyle T}^{2}\ll Q^{2}$, can
be obtained along the lines of Ref. Boer:2020bbd , by replacing the Dirac
delta in Eq. (50) with its expansion in the small-$q_{\scriptscriptstyle T}$
limit,

$\displaystyle\delta\left(\frac{\bm{q}_{\scriptscriptstyle
T}^{2}}{Q^{2}}+\frac{1-\hat{z}}{\hat{z}^{2}}\,\frac{M_{\psi}^{2}}{Q^{2}}-\frac{(1-\hat{x})(1-\hat{z})}{\hat{x}\hat{z}}\right)$
$\displaystyle\approx\hat{x}_{\rm
max}\left\\{\frac{\hat{x}^{\prime}}{(1-\hat{x}^{\prime})_{+}}\,\delta(1-\hat{z})\,+\,\frac{Q^{2}+M^{2}_{\psi}}{Q^{2}+M_{\psi}^{2}/\hat{z}}\,\frac{\hat{z}}{(1-\hat{z})_{+}}\
\delta\left(1-\hat{x}^{\prime}\right)\right.$
$\displaystyle\qquad\left.+\,\delta(1-\hat{x}^{\prime})\delta(1-\hat{z})\ln\bigg{(}\frac{Q^{2}+M_{\psi}^{2}}{\bm{q}_{\scriptscriptstyle
T}^{2}}\bigg{)}\right\\}\,,$ (52)

where $\hat{x}^{\prime}={\hat{x}}/{\hat{x}_{\rm max}}$ . Up to corrections of
the order of ${\cal O}(\Lambda_{\rm QCD}/|\bm{q}_{\scriptscriptstyle T}|)$ and
${\cal O}(|\bm{q}_{\scriptscriptstyle T}|/Q)$, we find the following leading
power behavior of the structure functions for transversely polarized photons

$\displaystyle{\cal W}_{\Lambda}^{\perp}$ $\displaystyle=$
$\displaystyle\widetilde{w}_{\Lambda}^{\perp\,(g)}\,\frac{\alpha_{s}}{2\pi^{2}\bm{q}_{\scriptscriptstyle
T}^{2}}\left[L\left({Q^{2}+M_{\psi}^{2}\over{\bm{q}}_{\scriptscriptstyle
T}^{2}}\right)f_{1}^{g}(x,\mu^{2})+\left(P_{gg}\otimes f_{1}^{g}+P_{gi}\otimes
f_{1}^{i}\right)(x,\mu^{2})\right]\qquad\text{with}\quad\Lambda=T,L,$
$\displaystyle{\cal W}_{\Delta\Delta}^{\perp}$ $\displaystyle=$
$\displaystyle\widetilde{w}_{\Delta\Delta}^{\perp\,(g)}\,\frac{\alpha_{s}}{\pi^{2}\bm{q}_{\scriptscriptstyle
T}^{2}}\left(\delta P_{gg}\otimes f_{1}^{g}+\delta P_{gi}\otimes
f_{1}^{i}\right)(x,\mu^{2})\,,$ (53)

where a sum over $i=q,\bar{q}$ is understood. The single-helicity flip
structure function ${\cal W}_{\Delta}^{\perp}$ is suppressed by a factor
$|\bm{q}_{\scriptscriptstyle T}|/Q$ relatively to the ones in Eq. (53) and
therefore we do not consider it in the following. Similarly, for
longitudinally polarized photons,

$\displaystyle{\cal W}_{L}^{{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}=\widetilde{w}_{L}^{{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}\,(g)}\,\frac{\alpha_{s}}{2\pi^{2}\bm{q}_{\scriptscriptstyle
T}^{2}}\left[L\left({Q^{2}+M_{\psi}^{2}\over{\bm{q}}_{\scriptscriptstyle
T}^{2}}\right)f_{1}^{g}(x,\mu^{2})+\left(P_{gg}\otimes f_{1}^{g}+P_{gi}\otimes
f_{1}^{i}\right)(x,\mu^{2})\right]\,,$ (54)

while all the other structure functions are suppressed. We note that the above
results are independent of the chosen reference frame. The quantities
$\widetilde{w}_{\Lambda}^{{\cal P}\,(g)}$ in Eqs. (53) and (54) are the gluon
helicity structure functions at LO in $\alpha_{s}$, i.e. they refer to the
$2\to 1$ subprocess $\gamma^{*}g\to J/\psi$ in Fig. 3 and read

$\displaystyle\widetilde{w}_{T}^{\perp\,(g)}$
$\displaystyle=2(4\pi)^{2}{\alpha\alpha_{s}e_{c}^{2}\over
M_{\psi}(M_{\psi}^{2}+Q^{2})}\left[{1\over 3}\langle 0|{\cal
O}_{8}(^{1}S_{0})|0\rangle+{4\over
M_{\psi}^{2}}{3M_{\psi}^{4}+Q^{4}\over(M_{\psi}^{2}+Q^{2})^{2}}\langle 0|{\cal
O}_{8}(^{3}P_{0})|0\rangle\right]\delta(1-\hat{z})\,,$
$\displaystyle\widetilde{w}_{L}^{\perp\,(g)}$
$\displaystyle=2(4\pi)^{2}{\alpha\alpha_{s}e_{c}^{2}\over
M_{\psi}(M_{\psi}^{2}+Q^{2})}\left[{1\over 3}\langle 0|{\cal
O}_{8}(^{1}S_{0})|0\rangle+{4\over M_{\psi}^{2}}\langle 0|{\cal
O}_{8}(^{3}P_{0})|0\rangle\right]\delta(1-\hat{z})\,,$
$\displaystyle\widetilde{w}_{\Delta\Delta}^{\perp\,(g)}$
$\displaystyle=-16(4\pi)^{2}{\alpha\alpha_{s}e_{c}^{2}\over
M_{\psi}(M_{\psi}^{2}+Q^{2})^{2}}\langle 0|{\cal
O}_{8}(^{3}P_{0})|0\rangle\,\delta(1-\hat{z})\,,$
$\displaystyle\widetilde{w}_{L}^{{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}\,(g)}$
$\displaystyle={128(4\pi)^{2}}\frac{\alpha\alpha_{s}e_{c}^{2}\,Q^{2}}{M_{\psi}(M_{\psi}^{2}+Q^{2})^{3}}\,\langle
0|{\cal O}_{8}(^{3}P_{0})|0\rangle\,\delta(1-\hat{z})\,,$ (55)

where, as before, the superscripts (subscripts) refer to the photon ($J/\psi$)
polarization states and $e_{c}$ is the electric charge of the charm quark in
units of the proton charge. Furthermore, in Eqs. (53) and (54) we have used
the definitions

$\displaystyle x\equiv\frac{x_{\scriptscriptstyle B}}{\hat{x}_{\rm
max}}=x_{\scriptscriptstyle B}\,\left(1+\frac{M^{2}_{\psi}}{Q^{2}}\right)\,,$
(56)

and

$\displaystyle L\left(\frac{Q^{2}+M^{2}_{\psi}}{\bm{q}_{\scriptscriptstyle
T}^{2}}\right)\equiv
2C_{A}\,\ln\left(\frac{Q^{2}+M^{2}_{\psi}}{\bm{q}_{\scriptscriptstyle
T}^{2}}\right)-\frac{11C_{A}-4n_{f}T_{R}}{6}\,,$ (57)

where $T_{R}=1/2$, $n_{f}$ is the number of active flavors and $C_{A}=N_{c}$,
with $N_{c}$ being the number of colors. The symbol $\otimes$ stands for a
convolution in the longitudinal momentum fractions of the splitting functions
and the parton distributions,

$\displaystyle(P\otimes
f)(x,\mu^{2})=\int_{x}^{1}\frac{\mathrm{d}\hat{x}}{\hat{x}}\,P\left(\hat{x},\mu^{2}\right)f\left(\frac{x}{\hat{x}},\mu^{2}\right)\,.$
(58)

The leading-order (LO) unpolarized splitting functions are explicitly given by

$\displaystyle P_{gg}(\hat{x})$
$\displaystyle=2C_{A}\left[\frac{\hat{x}}{(1-\hat{x})_{+}}+\frac{1-\hat{x}}{\hat{x}}+\hat{x}(1-\hat{x})\right]+\delta(1-\hat{x})\,\frac{11C_{A}-4n_{f}T_{R}}{6}\,,$
$\displaystyle P_{gq}(\hat{x})$
$\displaystyle=P_{g\bar{q}}(\hat{x})=C_{F}\,\frac{1+(1-\hat{x})^{2}}{\hat{x}}\,,$
(59)

with $C_{F}=(N_{c}^{2}-1)/2N_{c}$. As well known, the plus-prescription on the
singular parts of the splitting function $P_{gg}$ is defined so that the
integral of a sufficiently smooth distribution $G$ is given by

$\displaystyle\int_{z}^{1}\mathrm{d}y\,\frac{G(y)}{(1-y)_{+}}=\int_{z}^{1}\mathrm{d}y\,\frac{G(y)-G(1)}{1-y}-G(1)\ln\left(\frac{1}{1-z}\right)\,.$
(60)

Moreover, the splitting functions of an unpolarized parton into a linearly
polarized gluon read Sun:2011iw ; Catani:2010pd

$\displaystyle\delta P_{gg}(\hat{x})$
$\displaystyle=C_{A}\,\frac{1-\hat{x}}{\hat{x}}\,,$ $\displaystyle\delta
P_{gq}(\hat{x})$ $\displaystyle=\delta
P_{g\bar{q}}(\hat{x})=C_{F}\,\frac{1-\hat{x}}{\hat{x}}\,.$ (61)

Finally, we point out that the partonic subprocesses which contribute to the
structure functions at small $q_{\scriptscriptstyle T}$, see Eqs. (53)-(54),
are only the ones in the intermediate Fock states $n=~{}^{1}\\!S_{0}^{[8]}$,
${}^{3}\\!P_{J}^{[8]}$, corresponding to $t$-channel Feynman diagrams of the
(b) and (d) type in Fig. 2. The other partonic channels, i.e. the gluon
induced ${}^{3}\\!S_{1}^{[1,8]}$ subprocesses in Fig. 2 (a) and the quark-
induced subprocesses in Fig. 2 (c), are suppressed and vanish in the limit
$\bm{q}_{\scriptscriptstyle T}^{2}\to 0$. This means that they are not
relevant in our analysis of the matching of the collinear and TMD results.
Hence they will not be considered in the following discussion.

##  IV TMD factorization and matching

Within the TMD framework at the order $\alpha\alpha_{s}$ in the strong
coupling constant and $v^{4}$ in the NRQCD velocity parameter, the underlying
partonic subprocesses are given by

$\displaystyle\gamma^{*}(q)+g(p_{a})\to c\overline{c}[n](P_{\psi})\,,$ (62)

where the $c\bar{c}$ pairs are produced perturbatively in one of the color-
octet bound states $n=\,^{1}S_{0}^{[8]},\,^{3}P_{J}^{[8]}$ with $J=0,1,2$, see
also Fig. 3.

Figure 3: Feynman diagram for the process $\gamma^{*}(q)\,+\,g(p_{a})\to
J/\psi(P_{\psi})$ contributing to $J/\psi$ production in SIDIS at the order
$\alpha\alpha_{s}$. The crossed diagram, in which the directions of the arrows
are reversed, is not shown.

The leading-twist helicity structure functions for incoming transversely
polarized photons read

$\displaystyle{{\cal W}}_{T}^{\perp}$
$\displaystyle={2(4\pi)^{2}{\alpha\alpha_{s}e_{c}^{2}\over
M_{\psi}(M_{\psi}^{2}+Q^{2})}}\left\\{{1\over 3}\,{\cal
C}\big{[}f_{1}^{g}\,\Delta_{T}^{[^{1}S_{0}]}\big{]}(x,\bm{q}_{\scriptscriptstyle
T}^{2};\mu^{2})\,+\,{4\over
M_{\psi}^{2}}{3M_{\psi}^{4}+Q^{4}\over(M_{\psi}^{2}+Q^{2})^{2}}\,{\cal
C}\big{[}f_{1}^{g}\,\Delta_{T}^{[^{3}P_{0}]}\big{]}(x,\bm{q}_{\scriptscriptstyle
T}^{2};\mu^{2})\right\\}\,,$ $\displaystyle{{\cal W}}_{L}^{\perp}$
$\displaystyle={2(4\pi)^{2}\frac{\alpha\alpha_{s}e_{c}^{2}}{M_{\psi}(M_{\psi}^{2}+Q^{2})}}\,\left\\{{1\over
3}\,{\cal
C}\big{[}f_{1}^{g}\,\Delta_{L}^{[^{1}S_{0}]}\big{]}(x,\bm{q}_{\scriptscriptstyle
T}^{2};\mu^{2})\,+{4\over M_{\psi}^{2}}\,{\cal
C}\big{[}f_{1}^{g}\,\Delta_{L}^{[^{3}P_{0}]}\big{]}(x,\bm{q}_{\scriptscriptstyle
T}^{2};\mu^{2})\rangle\right\\}\,,$ $\displaystyle{{\cal
W}}_{\Delta\Delta}^{\perp}$
$\displaystyle=-{16(4\pi)^{2}\frac{\alpha\alpha_{s}e_{c}^{2}}{M_{\psi}(M_{\psi}^{2}+\
Q^{2})^{2}}}\,{\cal C}\big{[}w\,h_{1}^{\perp
g}\,\Delta_{\Delta\Delta}^{[^{3}P_{0}]}\big{]}(x,\bm{q}_{\scriptscriptstyle
T}^{2};\mu^{2})\,,$ (63)

while for incoming longitudinally polarized photons we find

$\displaystyle{{\cal W}}_{L}^{\parallel}$
$\displaystyle={128(4\pi)^{2}}{\alpha\alpha_{s}e_{c}^{2}\,Q^{2}\over
M_{\psi}(M_{\psi}^{2}+Q^{2})^{3}}\,{\cal
C}\big{[}f_{1}^{g}\,\Delta_{L}^{[^{3}P_{0}]}\big{]}(x,\bm{q}_{\scriptscriptstyle
T}^{2};\mu^{2})\,,$ (64)

where we have introduced the transverse momentum convolutions

$\displaystyle{\cal
C}\big{[}f_{1}^{g}\,\Delta_{\Lambda}^{[n]}\big{]}(x,\bm{q}_{\scriptscriptstyle
T}^{2};\mu^{2})$ $\displaystyle=\int\mathrm{d}^{2}\bm{p}_{\scriptscriptstyle
T}\int\mathrm{d}^{2}\bm{k}_{\scriptscriptstyle
T}\,\delta^{2}(\bm{q}_{\scriptscriptstyle T}-\bm{p}_{\scriptscriptstyle
T}-\bm{k}_{\scriptscriptstyle T})\,f_{1}^{g}(x,\bm{p}_{\scriptscriptstyle
T}^{2};\mu^{2})\,\Delta_{\Lambda}^{[n]}(\bm{k}_{\scriptscriptstyle
T}^{2},\mu^{2})\,,$ $\displaystyle{\cal C}\big{[}w\,h_{1}^{\perp
g}\,\Delta_{\Lambda}^{[n]}\big{]}(x,\bm{q}_{\scriptscriptstyle
T}^{2};\mu^{2})$ $\displaystyle=\int\mathrm{d}^{2}\bm{p}_{\scriptscriptstyle
T}\int\mathrm{d}^{2}\bm{k}_{\scriptscriptstyle
T}\,\delta^{2}(\bm{q}_{\scriptscriptstyle T}-\bm{p}_{\scriptscriptstyle
T}-\bm{k}_{\scriptscriptstyle T})\,w(\bm{p}_{\scriptscriptstyle
T},\bm{k}_{\scriptscriptstyle T})\,h_{1}^{\perp
g}(x,\bm{p}_{\scriptscriptstyle
T}^{2};\mu^{2})\,\Delta_{\Lambda}^{[n]}(\bm{k}_{\scriptscriptstyle
T}^{2},\mu^{2})\,.$ (65)

In the above equations, $f_{1}^{g}$ and $h_{1}^{\perp\,g}$ are, respectively,
the unpolarized and linearly polarized gluon TMDs inside an unpolarized proton
Mulders:2000sh ; Meissner:2007rx ; Boer:2016xqr ; Echevarria:2015uaa ;
Gutierrez-Reyes:2019rug ; Luo:2019bmw , while $\Delta_{\Lambda}^{[n]}$ are the
shape functions Echevarria:2019ynx ; Fleming:2019pzj  encoding the smearing
effects in the $c\bar{c}[n]\to J/\psi$ transition. As such, the
$\Delta_{\Lambda}^{[n]}$ could in principle depend on the quantum numbers of
the intermediate $c\bar{c}$ pair, as well as on the polarization $\Lambda$ of
the final $J/\psi$ meson. Moreover, $w(\bm{p}_{\scriptscriptstyle
T},\bm{k}_{\scriptscriptstyle T})$ is the transverse momentum dependent weight
function Boer:2020bbd

$\displaystyle w(\bm{p}_{\scriptscriptstyle T},\bm{k}_{\scriptscriptstyle
T})=\frac{1}{M_{p}^{2}\,\bm{q}_{\scriptscriptstyle
T}^{2}}\,\left[(\bm{p}_{\scriptscriptstyle T}\cdot\bm{q}_{\scriptscriptstyle
T})^{2}-\frac{1}{2}\,\bm{p}_{\scriptscriptstyle
T}^{2}\,\bm{q}_{\scriptscriptstyle T}^{2}\right]\,.$ (66)

In absence of smearing effects, the final $J/\psi$ meson would be collinear to
the $c\bar{c}$ pair originally produced in the hard scattering process and the
shape functions would be given by
$\Delta_{\Lambda}^{[n]}(\bm{k}_{\scriptscriptstyle T}^{2};\mu^{2})=\langle
0|{\cal O}(n)\,|0\rangle\,\delta^{2}(\bm{k}_{\scriptscriptstyle T})$. Hence
the convolutions in Eq. (65) reduce to the products of a LDME $\langle 0|{\cal
O}(n)\,|0\rangle$ with a gluon TMD, namely

$\displaystyle{\cal
C}\big{[}f_{1}^{g}\,\Delta_{\Lambda}^{[n]}\big{]}(x,\bm{q}_{\scriptscriptstyle
T}^{2};\mu^{2})$ $\displaystyle=\langle 0|{\cal
O}(n)\,|0\rangle\,f_{1}^{g}(x,\bm{q}_{\scriptscriptstyle T}^{2};\mu^{2})\,,$
$\displaystyle{\cal C}\big{[}w\,h_{1}^{\perp
g}\,\Delta_{\Lambda}^{[n]}\big{]}(x,\bm{q}_{\scriptscriptstyle
T}^{2};\mu^{2})$ $\displaystyle=\langle 0|{\cal
O}(n)\,|0\rangle\,\frac{\bm{q}_{\scriptscriptstyle
T}^{2}}{2M_{p}^{2}}\,h_{1}^{\perp g}(x,\bm{q}_{\scriptscriptstyle
T}^{2};\mu^{2})\,,\qquad\text{with}~{}~{}\bm{q}_{\scriptscriptstyle
T}=\bm{p}_{\scriptscriptstyle T}\,.$ (67)

TMDs can be calculated perturbatively in the limit
$|\bm{q}_{\scriptscriptstyle T}|\gg\Lambda_{\rm QCD}$, where their soft parts
can be safely neglected. In particular, the unpolarized gluon TMD distribution
at the scale $\mu^{2}$ and order $\alpha_{s}$ can be expressed in terms of
collinear parton distributions as follows Boer:2020bbd ,

$\displaystyle f_{1}^{g}(x,\bm{q}_{\scriptscriptstyle T}^{2};\mu^{2})$
$\displaystyle=\frac{\alpha_{s}}{2\pi^{2}\bm{q}_{\scriptscriptstyle
T}^{2}}\,\,\left[\left(C_{A}\ln\frac{\mu^{2}}{\bm{q}_{\scriptscriptstyle
T}^{2}}-\frac{11C_{A}-2n_{f}}{6}\,\right)f_{1}^{g}(x,\mu^{2})\,+\,(P_{gg}\otimes
f_{1}^{g}\,+\,P_{gi}\otimes f_{1}^{i})(x,\mu^{2})\right]\,,$ (68)

where $i=q,\bar{q}$, the LO splitting functions $P_{gg}$ and $P_{gi}$ are
given in Eq. (59) and the convolution $\otimes$ is defined in Eq. (58).
Similarly, the perturbative tail of the linearly polarized gluon distribution
reads Sun:2011iw

$\displaystyle\frac{\bm{q}_{\scriptscriptstyle
T}^{2}}{2M_{p}^{2}}\,h_{1}^{\perp g}(x,\bm{q}_{\scriptscriptstyle
T}^{2};\mu^{2})$
$\displaystyle=\frac{\alpha_{s}}{\pi^{2}}\,\frac{1}{\bm{q}_{\scriptscriptstyle
T}^{2}}\,(\delta P_{gg}\otimes f_{1}^{g}\,+\,\delta P_{gi}\otimes
f_{1}^{i})(x,\mu^{2})\,,$ (69)

with the LO polarized splitting functions $\delta P_{gg}$ and $\delta P_{gi}$
given in Eq. (61).

By substituting Eqs. (67) together with the expressions for
$f_{1}^{g}(x,\bm{q}_{\scriptscriptstyle T}^{2};\mu^{2})$ and $h_{1}^{\perp
g}(x,\bm{q}_{\scriptscriptstyle T}^{2};\mu^{2})$ in Eqs. (68)-(69), evaluated
at the scale $\mu^{2}=Q^{2}+M_{\psi}^{2}$, into Eqs. (63)-(64), we find that,
if smearing is neglected in the final state, only the double-helicity flip
structure function ${\cal W}^{\perp}_{\Delta\Delta}$ exactly matches the
corresponding collinear one in the small-$q_{\scriptscriptstyle T}$ limit
given in Eqs. (53), (55). This is possible without the need of any shape
function because of the absence of a logarithmic term in ${\cal
W}^{T}_{\Delta\Delta}$ at the perturbative order we are considering.
Conversely, in the intermediate region $\Lambda_{\rm
QCD}\ll|\bm{q}_{\scriptscriptstyle T}|\ll Q$,

$\displaystyle\Delta{\cal W}^{\perp}_{\Lambda}\equiv{{\cal
W}^{\perp}_{\Lambda}}{\bigg{|}_{\text{Coll.}}}\,-\,{\cal
W}^{\perp}_{\Lambda}\bigg{|}_{\text{TMD}}$
$\displaystyle=\widetilde{w}^{\perp}_{\Lambda}\,\frac{C_{A}}{2\pi^{2}\bm{q}_{\scriptscriptstyle
T}^{2}}\ln\left(\frac{Q^{2}+M^{2}_{\psi}}{\bm{q}_{\scriptscriptstyle
T}^{2}}\right)\qquad\text{with}\quad\Lambda=T,L,$ $\displaystyle\Delta{\cal
W}^{{\mkern 3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule
depth=0.0pt\mkern 3.0mu}}_{L}\equiv{\cal W}^{{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}_{L}\bigg{|}_{\text{Coll.}}\,-\,{\cal W}^{{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}_{L}\bigg{|}_{\text{TMD}}$ $\displaystyle=\widetilde{w}^{{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}}_{L}\,\frac{C_{A}}{2\pi^{2}\bm{q}_{\scriptscriptstyle
T}^{2}}\,\ln\left(\frac{Q^{2}+M^{2}_{\psi}}{\bm{q}_{\scriptscriptstyle
T}^{2}}\right)\,.$ (70)

This suggests that smearing effects need to be taken into account in those
helicity structure functions which depend on the unpolarized gluon TMD
distribution, through the inclusion of a suitable shape function
Echevarria:2019ynx ; Fleming:2019pzj . Along the lines of the calculation for
unpolarized $J/\psi$ production presented in Ref. Boer:2020bbd , to which we
refer for details, we are able to find the perturbative tails of the shape
functions $\Delta_{T}^{[n]}$ and $\Delta_{L}^{[n]}$ at the order $\alpha_{s}$
by imposing the matching of the TMD and collinear expressions for the helicity
structure functions ${\cal W}^{\perp}_{T}$, ${\cal W}^{\perp}_{L}$, ${\cal
W}^{\mkern 3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule
depth=0.0pt\mkern 3.0mu}_{L}$ in the intermediate region. In other words, the
shape functions are determined by requiring that their contributions to ${\cal
W}^{\perp}_{T}$, ${\cal W}^{\perp}_{L}$, ${\cal W}^{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}_{L}$ exactly cancel against the logarithmic terms on the right-hand
side of Eq. (70). It turns out that $\Delta_{T}^{[n]}=\Delta_{L}^{[n]}$, i.e.
they are independent of the $J/\psi$ polarization and equal to the shape
function $\Delta^{[n]}$ calculated for unpolarized $J/\psi$ production in Ref.
Boer:2020bbd  by applying the same matching procedure,

$\displaystyle\Delta_{T}^{[n]}(\bm{k}_{\scriptscriptstyle
T}^{2},\mu^{2})=\Delta_{L}^{[n]}(\bm{k}_{\scriptscriptstyle
T}^{2},\mu^{2})\equiv\,\Delta^{[n]}(\bm{k}_{\scriptscriptstyle
T}^{2},\mu^{2})=\frac{\alpha_{s}}{2\pi^{2}\bm{k}_{\scriptscriptstyle
T}^{2}}\,C_{A}\,\langle 0|{\cal
O}(n)|0\rangle\,\ln\frac{\mu^{2}}{\bm{k}_{\scriptscriptstyle T}^{2}}\,.$ (71)

This can be checked directly by plugging into the first line of Eq. (65) the
full perturbative expansions of the gluon TMD and the shape function up to the
order $\alpha_{s}$, which are obtained by adding the order-$\alpha_{s}^{0}$
terms to the expressions in Eqs. (68) and (71), respectively. We get

$\displaystyle{\cal
C}\big{[}f_{1}^{g}\,\Delta^{[n]}\big{]}(x,\bm{q}_{\scriptscriptstyle
T}^{2};\mu^{2})$ $\displaystyle=\int\mathrm{d}^{2}\bm{p}_{\scriptscriptstyle
T}\int\mathrm{d}^{2}\bm{k}_{\scriptscriptstyle
T}\,\delta^{2}(\bm{q}_{\scriptscriptstyle T}-\bm{p}_{\scriptscriptstyle
T}-\bm{k}_{\scriptscriptstyle T})\,f_{1}^{g}(x,\bm{p}_{\scriptscriptstyle
T}^{2};\mu^{2})\,\Delta^{[n]}(\bm{k}_{\scriptscriptstyle T}^{2},\mu^{2})$
$\displaystyle=\int\mathrm{d}^{2}\bm{p}_{\scriptscriptstyle
T}\int\mathrm{d}^{2}\bm{k}_{\scriptscriptstyle
T}\,\delta^{2}(\bm{q}_{\scriptscriptstyle T}-\bm{p}_{\scriptscriptstyle
T}-\bm{k}_{\scriptscriptstyle
T})\,\bigg{\\{}\delta^{2}(\bm{p}_{\scriptscriptstyle T})f_{1}^{g}(x,\mu^{2})$
$\displaystyle\qquad+\,\left.\frac{\alpha_{s}}{2\pi^{2}\bm{p}_{\scriptscriptstyle
T}^{2}}\,\left[\left(C_{A}\ln\frac{\mu^{2}}{\bm{p}_{\scriptscriptstyle
T}^{2}}-\frac{11C_{A}-2n_{f}}{6}\,\right)f_{1}^{g}(x,\mu^{2})\,+\,(P_{gg}\otimes
f_{1}^{g}\,+\,P_{gi}\otimes f_{1}^{i})(x,\mu^{2})\right]\right\\}$
$\displaystyle\qquad\qquad\times\left[\delta^{2}(\bm{k}_{\scriptscriptstyle
T})\,+\,\frac{\alpha_{s}}{2\pi^{2}\bm{k}_{\scriptscriptstyle
T}^{2}}\,C_{A}\,\langle 0|{\cal
O}(n)|0\rangle\,\ln\frac{\mu^{2}}{\bm{k}_{\scriptscriptstyle
T}^{2}}\right]\,.$ (72)

Hence, in the limit $\bm{q}_{\scriptscriptstyle
T}^{2}\gg\Lambda_{\text{QCD}}^{2},$

$\displaystyle{\cal
C}\big{[}f_{1}^{g}\,\Delta^{[n]}\big{]}(x,\bm{q}_{\scriptscriptstyle T}^{2})$
$\displaystyle=\frac{\alpha_{s}}{2\pi^{2}\bm{q}_{\scriptscriptstyle
T}^{2}}\,\langle 0|{\cal
O}(n)|0\rangle\,\left[\left(2C_{A}\ln\frac{\mu^{2}}{\bm{q}_{\scriptscriptstyle
T}^{2}}-\frac{11C_{A}-2n_{f}}{6}\,\right)f_{1}^{g}(x,\mu^{2})\,+\,(P_{gg}\otimes
f_{1}^{g}\,+\,P_{gi}\otimes f_{1}^{i})(x,\mu^{2})\,\right]\,.$ (73)

If we substitute this convolution, with the choice
$\mu^{2}=Q^{2}+M_{\psi}^{2}$, into Eqs. (63)-(64), we recover the correct
formulae for ${\cal W}^{\perp}_{T}$, ${\cal W}^{\perp}_{L}$ and ${\cal
W}^{\mkern 3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule
depth=0.0pt\mkern 3.0mu}_{L}$ in Eqs. (53)-(54). In this way we are able to
solve the matching issue of the TMD and collinear results for all the
structure functions in the region
$\Lambda_{\text{QCD}}^{2}\ll\bm{q}_{\scriptscriptstyle T}^{2}\ll Q^{2}$. We
note that, at the perturbative order we are considering, nothing can be said
about the tail of the shape function $\Delta_{\Delta\Delta}^{[n]}$, which
could be in principle different from $\Delta^{[n]}$. Its determination would
require a study of the matching of the helicity structure function ${\cal
W}_{\Delta\Delta}^{\perp}$ at higher order in $\alpha_{s}$, because the
perturbative expansion of $h_{1}^{\perp\,g}$ starts at the order $\alpha_{s}$
and not $\alpha_{s}^{0}$ like for $f_{1}^{g}$. However, a full calculation of
the cross section for $J/\psi$ production in SIDIS at the order
$\alpha^{2}\alpha_{s}^{3}$ within the NRQCD framework, is still missing.

##  V Summary and conclusions

We conclude by summarizing our main results. We have calculated the structure
functions ${\cal W}_{T},{\cal W}_{L},{\cal W}_{\Delta},{\cal
W}_{\Delta\Delta}$, and the corresponding polarization parameters
$\lambda,\mu,\nu$, for $J/\psi$ production in SIDIS. In particular, the
helicity-flip structure functions ${\cal W}_{\Delta}$ and ${\cal
W}_{\Delta\Delta}$, which generate, respectively, $\cos\phi$ and $\cos 2\phi$
azimuthal asymmetries in the angular distribution of the decaying leptons,
have been derived here for the first time. In general, the explicit
expressions depend on the choice of the coordinate axes. We have presented our
results in the Gottfried-Jackson frame and we have shown how to transform them
to the Target, Collins-Soper and Helicity frames.

In the kinematic region where the transverse momentum $q_{\scriptscriptstyle
T}$ of the virtual photon exchanged in the reaction is large,
$q_{\scriptscriptstyle T}\gg\Lambda_{\text{QCD}}$, the theoretical frameworks
of collinear factorization and NRQCD have been adopted. By performing an
analysis up to the order $\alpha_{s}^{2}$ accuracy, with the inclusion of CO
contributions up to the order $v^{4}$ with respect to the CS one, we obtain
the small-$q_{\scriptscriptstyle T}$ behavior of the structure functions
${\cal W}_{T}$, ${\cal W}_{L}$ and ${\cal W}_{\Delta\Delta}$ in terms of the
LO DGLAP splitting functions. Furthermore, ${\cal W}_{T}$ and ${\cal W}_{L}$
receive also large logarithmic corrections, which are absent in ${\cal
W}_{\Delta\Delta}$ and need to be resummed. In contrast, the structure
function ${\cal W}_{\Delta}$ turns out to be suppressed by a factor
$q_{\scriptscriptstyle T}/Q$ with respect to the other ones. Quite remarkably,
these features at small $q_{\scriptscriptstyle T}$ do not depend on the choice
of the reference frame.

The large logarithms of the structure functions at small
$q_{\scriptscriptstyle T}$ can be resummed within the TMD framework. Although
a rigorous proof of TMD factorization only exists for light-hadron production
in SIDIS, it is reasonable to assume its validity also for the production of
heavy quarkonium states, just changing from the fundamental to the adjoint
representation in the gauge-link structure underlying the process
Bacchetta:2018ivt . Hence we propose a factorized expression of the polarized
structure functions in terms of transverse momentum dependent parton
distributions and shape functions, the latter being a generalization of the
NRQCD LDMEs to TMD approach first introduced in Refs. Echevarria:2019ynx ;
Fleming:2019pzj . Our assumption has to fulfill the consistency condition that
both descriptions match in the intermediate region $\Lambda_{\mathrm{QCD}}\ll
q_{\scriptscriptstyle T}\ll\mu$, where $\mu$ is a hard scale typical of the
process. At the order $\alpha_{s}$, we find that a smooth transition from low
$q_{\scriptscriptstyle T}$ to high $q_{\scriptscriptstyle T}$ is possibile if
we choose $\mu=\sqrt{Q^{2}+M^{2}_{\psi}}$. Moreover, while $W_{\Delta\Delta}$
matches without depending on any shape function because of the absence of
logarithmic divergences, from the matching of $W_{L}$ and $W_{T}$ we are able
to deduce the specific form of the color-octet shape functions at large
transverse momentum. Very interestingly, the perturbative tails of the shape
functions turn out to be independent of the $J/\psi$ polarization, as well as
of the quantum numbers of the intermediate color-octet Fock states, except for
their overall magnitude given by the NRQCD LDMEs. These conclusions are in
full agreement with the ones obtained in the analysis of unpolarized $J/\psi$
production in SIDIS Boer:2020bbd . They of course hold for any other
quarkonium state with the same quantum numbers of the $J/\psi$ meson, such as
the $\psi(2S)$ and $\Upsilon(nS)$ states.

Finally, our findings show that a combined analysis of the $J/\psi$
polarization parameters and production yields in the
large-$q_{\scriptscriptstyle T}$ region will be very helpful in testing NRQCD
and will likely improve our knowledge of the CO LDMEs. A similar study in the
small-$q_{T}$ region will provide important information on the shape
functions, especially on their nonperturbative part which we cannot predict.
Moreover, we suggest a novel experiment for the extraction of the distribution
of linearly polarized gluons inside an unpolarized proton, by looking at the
double helicity-flip parameter $\nu$, related to the $\cos 2\phi$ azimuthal
decay asymmetry of the $J/\psi$ meson. The high-luminosity Electron-Ion
Collider to be build in the U.S. would be the ideal facility for carrying out
this proposal.

## Acknowledgments

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement STRONG 2020 - No
824093. U.D. and C.P. also acknowledge financial support by Fondazione di
Sardegna under the project Proton tomography at the LHC, project number
F72F20000220007 (University of Cagliari).

##  Appendix A Reference frames

In this appendix, we present the explicit expressions of the covariant four-
vectors $(T^{\mu},X^{\mu},Y^{\mu},Z^{\mu})$, which form a set of coordinate
axes in the four dimensional Minkowski space. These vectors are given as
linear combinations of the physical momenta $q^{\mu}$, $P^{\mu}$,
$P_{\psi}^{\mu}$ in four typical reference frames, such that $X^{\mu}$,
$Y^{\mu}$, $Z^{\mu}$ become three-vectors in the quarkonium rest frame
Lam:1978pu ; Beneke:1998re . As already mentioned in Section III, the
normalization conditions are given by $T^{2}=1$ and $X^{2}=Y^{2}=Z^{2}=-1$,
with $T^{\mu}$ and $Y^{\mu}$ fixed by Eq. (25) in all frames, namely
$T^{\mu}={P_{\psi}^{\mu}}/{M_{\psi}}$,
$Y^{\mu}=\varepsilon^{\mu\nu\alpha\beta}X_{\nu}Z_{\alpha}T_{\beta}$. On the
other hand, the definition of $X^{\mu}$ and $Z^{\mu}$ is not unique. For their
determination, we proceed along the lines of Ref. Beneke:1998re , introducing
the following auxiliary four-vectors

$\displaystyle\tilde{q}^{\mu}$ $\displaystyle=q^{\mu}-\frac{q\cdot
P_{\psi}}{M_{\psi}^{2}}\,P_{\psi}^{\mu}\,,$ $\displaystyle\tilde{P}^{\mu}$
$\displaystyle=P^{\mu}-\frac{P\cdot
P_{\psi}}{M_{\psi}^{2}}\,P_{\psi}^{\mu}\,.$ (74)

Since $\tilde{q}\cdot T=\tilde{P}\cdot T=0$, $X^{\mu}$ and $Z^{\mu}$ can be
easily written as a linear combination of $\tilde{q}^{\mu}$ and
$\tilde{P}^{\mu}$. One starts by writing

$\displaystyle Z^{\mu}$
$\displaystyle=A_{z}\,\tilde{q}^{\mu}+B_{z}\,\tilde{P}^{\mu}\,,$ (75)

with one of the two coefficients fixed by the condition $Z^{2}=-1$. Similarly,

$\displaystyle X^{\mu}$
$\displaystyle=A_{x}\,\tilde{q}^{\mu}+B_{x}\,\tilde{P}^{\mu}\,,$ (76)

where $A_{x}$ and $B_{x}$ can be determined by imposing $X^{2}=-1$ and $X\cdot
Z=0$. The remaining sign ambiguity in $A_{x}$ and $B_{x}$ is fixed by
requiring that the three-vector $\bm{Y}$ points in the direction of
$\bm{q}\times(-\bm{P})$ in the $J/\psi$ rest frame.

The four commonly used reference frames described in the following are
specified by the choice of $Z^{\mu}$. The covariant expressions for $X^{\mu}$
and $Z^{\mu}$ are given in terms of the invariants $s=2q\cdot p_{a}$,
$t=M_{\psi}^{2}-Q^{2}-2q\cdot P_{\psi}$, $u=M_{\psi}^{2}+Q^{2}-2p_{a}\cdot
P_{\psi}$, with $p_{a}=\xi P$ in the collinear approach. These invariants are
related to the partonic Mandelstam variables in Eq. (46) as follows

$\displaystyle s=\hat{s}+Q^{2}\,,\qquad t=\hat{t}\,,\qquad u=\hat{u}+Q^{2}\,,$
(77)

and fulfill the equation

$\displaystyle s+t+u=Q^{2}+M_{\psi}^{2}\,.$ (78)

  1. 1.

Target frame. The target frame (TF) is defined in such a way that the vector
$\bm{Z}$ points along the opposite direction of the proton momentum in the
$J/\psi$ rest frame, i.e. $\bm{Z}=-{{\bm{P}}/|{\bm{P}}|}$. Hence the four-
vectors $Z^{\mu}$ and $X^{\mu}$ are given by

$\displaystyle Z^{\mu}_{\text{TF}}$ $\displaystyle=-2{M_{\psi}\over
s+t}p_{a}^{\mu}+{P_{\psi}^{\mu}\over M_{\psi}}\,,$ $\displaystyle
X^{\mu}_{\text{TF}}$
$\displaystyle=-{1\over(s+t)\sqrt{t(Q^{2}t+su)}}\left\\{(s+t)^{2}q^{\mu}+\left[s(M_{\psi}^{2}+Q^{2})+t(2Q^{2}-u)\right]p_{a}^{\mu}-s(s+t)P_{\psi}^{\mu}\right\\}\,.$
(79)

  2. 2.

Collins-Soper frame. In the Collins-Soper frame (CS), the direction of
$\bm{Z}$ is chosen as the bisector of the angle between the proton and the
virtual photon three-momenta in the $J/\psi$ rest frame, $\bm{Z}={{\bf
q}/|{\bm{q}}|}-{{\bm{P}}/|{\bm{P}}|}$. Therefore,

$\displaystyle Z^{\mu}_{\text{CS}}$
$\displaystyle={1\over\sqrt{N_{\text{CS}}}}\left[(s+t)\,q^{\mu}-(s+u-2Q^{2})\,p_{a}^{\mu}\right]\,,$
$\displaystyle X^{\mu}_{\text{CS}}$ $\displaystyle=-{1\over
M_{\psi}}{1\over\sqrt{t(Q^{2}t+su)N_{\text{CS}}}}\left\\{M_{\psi}^{2}s(s+t)\,q^{\mu}+M_{\psi}^{2}\left[s(s+u)+2Q^{2}t\right]\,p_{a}^{\mu}-(s+t)\left[s(M_{\psi}^{2}-t)+Q^{2}t\right]\,P_{\psi}^{\mu}\right\\},$
(80)

with

$N_{\text{CS}}=(s+t)\left[s(M_{\psi}^{2}-t)+Q^{2}t\right]\,.$ (81)

  3. 3.

Gottfried-Jackson frame. In the Gottfried-Jackson frame (GJ), also known as
$u$-channel helicity frame, $\bm{Z}$ is chosen to be along the direction of
the three-momentum of the virtual photon in the $J/\psi$ rest frame. Namely,
$\bm{Z}={{\bm{q}}/|{\bm{q}}|}$ and

$\displaystyle Z^{\mu}_{\text{GJ}}$ $\displaystyle={1\over
M_{\psi}\sqrt{N_{\text{GJ}}}}\left[2M_{\psi}^{2}\,q^{\mu}-(s+u-2Q^{2})\,P_{\psi}^{\mu}\right]\,,$
$\displaystyle X^{\mu}_{\text{GJ}}$
$\displaystyle=-{1\over\sqrt{t(Q^{2}t+su)N_{\text{GJ}}}}\left\\{\left[s(M_{\psi}^{2}+Q^{2})+t(2Q^{2}-u)\right]\,q^{\mu}+\left[4Q^{2}t+(s+u)^{2}\right]\,p_{a}^{\mu}-\left[2Q^{2}t+s(s+u)\right]\,P_{\psi}^{\mu}\right\\},$
(82)

with

$N_{\text{GJ}}=4Q^{2}t+(s+u)^{2}\,.$ (83)

  4. 4.

Helicity frame. In the helicity (HX) or recoil frame, the $Z$-axis is chosen
to be along the direction of the $J/\psi$ three-momentum in the hadronic
center of mass frame, that is
$\bm{Z}=-({\bm{P}}+{\bm{q}})/|{\bm{P}}+{\bm{q}}|$ in the $J/\psi$ rest frame.
Moreover,

$\displaystyle Z^{\mu}_{\text{HX}}$ $\displaystyle=-{1\over
M_{\psi}\sqrt{N_{\text{HX}}}}\left\\{2M_{\psi}^{2}\,(\xi
q^{\mu}+p_{a}^{\mu})-\left[\xi(s+u-2Q^{2})+s+t\right]\,P_{\psi}^{\mu}\right\\}\,,$
$\displaystyle X^{\mu}_{\text{HX}}$
$\displaystyle={1\over\sqrt{t(Q^{2}t+su)N_{\text{HX}}}}\left\\{\left[\xi\left(s(M_{\psi}^{2}+Q^{2})+t(2Q^{2}-u)\right)-(s+t)^{2}\right]\,q^{\mu}\right.$
$\displaystyle+\left[\xi\left((s+u)^{2}+4Q^{2}t\right)-s(M_{\psi}^{2}+Q^{2})+t(u-2Q^{2})\right]\,p_{a}^{\mu}$
$\displaystyle\left.+\left[s(s+t)-\xi\left(s(s+u)+2Q^{2}t\right)\right]\,P_{\psi}^{\mu}\right\\},$
(84)

with

$N_{\text{HX}}=\xi^{2}\left[4Q^{2}t+(s+u)^{2}\right]-2\xi\left[s(M_{\psi}^{2}+Q^{2})+t(2Q^{2}-u)\right]+(s+t)^{2}\,.$
(85)

The transition from one frame to the other is given by a rotation around the
$Y$-axis,

$\begin{pmatrix}X\\\
Z\end{pmatrix}_{\\!F^{\prime}}=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&~{}~{}\cos\theta\end{pmatrix}\begin{pmatrix}X\\\
Z\end{pmatrix}_{\\!F}\,,$ (86)

where, from the GJ to the TF frame:

$\displaystyle\cos\theta_{\text{GJ}\to\text{TF}}$
$\displaystyle=\frac{s(M_{\psi}^{2}+Q^{2})+t(2Q^{2}-u)}{(s+t)\sqrt{N_{\text{GJ}}}}\,,$
$\displaystyle\sin\theta_{\text{GJ}\to\text{TF}}$
$\displaystyle=\frac{2M_{\psi}\sqrt{t(Q^{2}t+su)}}{(s+t)\sqrt{N_{\text{GJ}}}}\,;$
(87)

from the GJ to the CS frame:

$\displaystyle\cos\theta_{\text{GJ}\to\text{CS}}$
$\displaystyle=\frac{M_{\psi}\left[s(s+u)+2Q^{2}t\right]}{\sqrt{N_{\text{CS}}\,N_{\text{GJ}}}}\,,$
$\displaystyle\sin\theta_{\text{GJ}\to\text{CS}}$
$\displaystyle=\frac{(s+u-2Q^{2})\sqrt{t(Q^{2}t+su)}}{\sqrt{N_{\text{CS}}\,N_{\text{GJ}}}}\,;$
(88)

form the GJ to the NX frame:

$\displaystyle\cos\theta_{\text{GJ}\to\text{HX}}$
$\displaystyle=\frac{s(M_{\psi}^{2}+Q^{2})+(2Q^{2}-u)t-\xi\left[4Q^{2}t+(s+u)^{2}\right]}{\sqrt{N_{\text{HX}}\,N_{\text{GJ}}}}\,,$
$\displaystyle\sin\theta_{\text{GJ}\to\text{HX}}$
$\displaystyle=\frac{2M_{\psi}\sqrt{t(Q^{2}t+su)}}{\sqrt{N_{\text{HX}}\,N_{\text{GJ}}}}\,.$
(89)

As a consequence, the hadronic structure functions ${\cal W}_{\Lambda}$ (as
well as the partonic ones) in two different frames $F$ and $F^{\prime}$ are
connected through the linear transformation

$\begin{pmatrix}{\cal W}_{T}\\\ {\cal W}_{L}\\\ {\cal W}_{\Delta}\\\ {\cal
W}_{\Delta\Delta}\end{pmatrix}_{\\!\\!\\!F^{\prime}}=\begin{pmatrix}1-{1\over
2}\sin^{2}\theta&{1\over 2}\sin^{2}\theta&{1\over 2}\sin 2\theta&{1\over
2}\sin^{2}\theta\\\ \sin^{2}\theta&\cos^{2}\theta&-\sin
2\theta&-\sin^{2}\theta\\\ -{1\over 2}\sin 2\theta&{1\over 2}\sin 2\theta&\cos
2\theta&{1\over 2}\sin 2\theta\\\ {1\over 2}\sin^{2}\theta&-{1\over
2}\sin^{2}\theta&-{1\over 2}\sin 2\theta&1-{1\over
2}\sin^{2}\theta\end{pmatrix}\begin{pmatrix}{\cal W}_{T}\\\ {\cal W}_{L}\\\
{\cal W}_{\Delta}\\\ {\cal W}_{\Delta\Delta}\end{pmatrix}_{\\!\\!\\!F}\,.$
(90)

Similarly, for the polarization parameters we have

$\begin{pmatrix}\lambda\\\ \mu\\\
\nu\end{pmatrix}_{\\!\\!\\!F^{\prime}}=\,{1\over
1+\rho}\begin{pmatrix}1-{3\over 2}\sin^{2}\theta&{3\over 2}\sin
2\theta&{3\over 4}\sin^{2}\theta\\\ -{1\over 2}\sin 2\theta&\cos
2\theta&{1\over 4}\sin 2\theta\\\ \sin^{2}\theta&-\sin 2\theta&1-{1\over
2}\sin^{2}\theta\end{pmatrix}\begin{pmatrix}\lambda\\\ \mu\\\
\nu\end{pmatrix}_{\\!\\!\\!F}\ ,$ (91)

with

$\rho={\sin^{2}\theta\over 2}\left(\lambda_{F}-{\nu_{F}\over 2}\right)-\sin
2\theta\,{\mu_{F}\over 2}\,.$ (92)

##  Appendix B Partonic helicity structure functions

We collect here the analytic expressions for the partonic helicity structure
functions in the GJ frame. For each subprocess initiated by a parton $a$, and
for each Fock state $n$, they are given in the form

$\displaystyle w_{\Lambda}^{{\cal
P}\,(a)}[n]=F[n]\,\widehat{w}_{\Lambda}^{{\cal P}\,(a)}[n]\,,$ (93)

where, as already stated, ${\cal P}=\perp,{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}$ refers to the virtual
photon polarization, while $\Lambda=T,L,\Delta,\Delta\Delta$ to the $J/\psi$
one. Moreover, $F[n]$ is a common prefactor independent of the polarizations.
In order to render the formulae more compact, they are given in terms of the
Lorentz invariants $s$, $t$, $u$ in Eq. (77). Moreover, for simplicity we will
drop the superscript $(a)$ on $\widehat{w}_{\Lambda}^{{\cal P}\,(a)}$, as well
as the dependences of $F$ and $\widehat{w}_{\Lambda}^{\cal P}$ on $n$ when not
needed.

  * •

$\gamma^{*}q(\bar{q})\,\to\,c\bar{c}[^{1}S_{0}^{[8]}]\,q(\bar{q})$:

$\displaystyle{F}$ $\displaystyle=-{64\,e_{c}^{2}\alpha\alpha_{s}^{2}\over
9s^{2}t(s+u)^{2}M_{\psi}}\,\langle 0|{\cal O}_{8}(\,^{1}S_{0})|0\rangle,$
$\displaystyle\widehat{w}_{T}^{\perp}$
$\displaystyle=2Q^{2}t\left[Q^{2}t+s(s+u)\right]+s^{2}(s^{2}+u^{2})\,,$
$\displaystyle\widehat{w}_{T}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}$
$\displaystyle=8Q^{2}t\left(Q^{2}t+su\right)\,,$
$\displaystyle\widehat{w}_{L}^{\perp}$
$\displaystyle=\widehat{w}_{T}^{\perp}\,,$
$\displaystyle\widehat{w}_{L}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}$
$\displaystyle=\widehat{w}_{T}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}\,,$
$\displaystyle\widehat{w}^{\perp}_{\Delta}$ $\displaystyle=\widehat{w}^{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}_{\Delta}=\widehat{w}^{\perp}_{\Delta\Delta}=\widehat{w}^{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}_{\Delta\Delta}=0\,.$ (94)

  * •

$\gamma^{*}q(\bar{q})\,\to\,c\bar{c}[^{3}S_{1}^{[8]}]\,q(\bar{q})$:

$\displaystyle F=$ $\displaystyle-{16\,e_{q}^{2}\alpha\alpha_{s}^{2}\over
9s^{2}(Q^{2}-s)^{2}(Q^{2}-u)^{2}\left[4Q^{2}t+(s+u)^{2}\right]M_{\psi}^{3}}\langle
0|{\cal O}_{8}(\,^{3}S_{1})|0\rangle,$ $\displaystyle\widehat{w}_{T}^{\perp}=$
$\displaystyle
4Q^{10}t^{3}+4Q^{8}t^{2}\left[s^{2}+s(t+u)+t(t-u)\right]+2Q^{6}t\left[s^{4}-2s^{3}(t-2u)-s^{2}(7t^{2}-2tu-u^{2})\right.$
$\displaystyle\left.-4st^{2}(t-u)+t^{2}u^{2}\right]-2Q^{4}s\left[s^{4}(t-u)+s^{3}u(5t-2u)-s^{2}(2t^{3}-10t^{2}u-3tu^{2}+u^{3})\right.$
$\displaystyle\left.-st(2t^{3}-4t^{2}u+u^{3})+2t^{3}u^{2}\right]+Q^{2}s^{2}\left[s^{4}(t-2u)+2s^{3}(t^{2}+tu-3u^{2})\right.$
$\displaystyle\left.+2s^{2}(t^{3}+3t^{2}u-2tu^{2}-3u^{3})-2su(-2t^{3}-t^{2}u+tu^{2}+u^{3})+tu^{2}(2t^{2}-2tu-u^{2})\right]$
$\displaystyle+s^{3}u(s^{2}+u^{2})\left[s^{2}+2s(t+u)+2t^{2}+2tu+u^{2}\right]\,,$
$\displaystyle\widehat{w}_{T}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}=$ $\displaystyle
8Q^{2}t(Q^{2}-s)^{2}\left\\{2Q^{4}t^{2}+2Q^{2}t\left[2s^{2}+s(3t+u)+t(t-u)\right]+s^{2}(s+t+u)^{2}+t^{2}u^{2}\right\\}\,,$
$\displaystyle\widehat{w}_{L}^{\perp}=$ $\displaystyle
4M_{\psi}^{2}t\left[2Q^{8}t^{2}+2Q^{6}st(s-2t+u)+Q^{4}s^{2}(s^{2}-2st+2t^{2}-6tu+u^{2})\right.$
$\displaystyle\left.-2Q^{2}s^{3}u(s-2t+u)+2s^{4}u^{2}\right]\,,$
$\displaystyle\widehat{w}_{L}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}=$ $\displaystyle
32Q^{2}M_{\psi}^{2}t^{2}(Q^{2}-s)^{2}(Q^{2}t+su)\,,$
$\displaystyle\widehat{w}^{\perp}_{\Delta}=$ $\displaystyle
2M_{\psi}\sqrt{t(Q^{2}t+su)}\left\\{4Q^{8}t^{2}+2Q^{6}t\left[s^{2}+s(u-3t)-tu\right]\right.$
$\displaystyle-Q^{4}s\left[s^{3}+2s^{2}t+su(6t-u)-4t^{2}u\right]+2Q^{2}s^{2}t\left[s(t+u)+u(u-t)\right]$
$\displaystyle\left.+s^{3}u(s-u)(s+2t+u)\right\\}\,,$
$\displaystyle\widehat{w}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}_{\Delta}=$
$\displaystyle
16Q^{2}M_{\psi}t\sqrt{t(Q^{2}t+su)}(Q^{2}-s)^{2}\left[2Q^{2}t+s^{2}+s(t+u)-ut\right]\,,$
$\displaystyle\widehat{w}^{\perp}_{\Delta\Delta}=$ $\displaystyle
2t(Q^{2}t+su)\left\\{-2Q^{8}t+2Q^{6}\left[s^{2}+3st+t(t+u)\right]-4Q^{4}st(s+t+u)\right.$
$\displaystyle\left.-Q^{2}s^{2}(s^{2}+4su-2t^{2}+u^{2})+2s^{3}u(s+t+u)\right\\}\,,$
$\displaystyle\widehat{w}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}_{\Delta\Delta}=$
$\displaystyle 16Q^{2}M_{\psi}^{2}t^{2}(Q^{2}t+su)(Q^{2}-s)^{2}\,.$ (95)

  * •

$\gamma^{*}q(\bar{q})\,\to\,c\bar{c}[^{3}P_{J}^{[8]}]\,q(\bar{q})$:

$\displaystyle F=$ $\displaystyle-{256\,e_{c}^{2}\alpha\alpha_{s}^{2}\over
3s^{2}t(s+u)^{4}\left[4Q^{2}t+(s+u)^{2}\right]M_{\psi}^{3}}\,\langle 0|{\cal
O}_{8}(\,^{3}P_{0})|0\rangle,$ $\displaystyle\widehat{w}_{T}^{\perp}=$
$\displaystyle-16Q^{10}t^{3}-8Q^{8}t^{2}\left[3s^{2}+2su-u(4t+u)\right]+4Q^{6}t\left[2s^{4}+s^{3}(5t-2u)\right.$
$\displaystyle\left.+s^{2}(10t^{2}+7tu+2u^{2})+s(8t^{3}+4t^{2}u+7tu^{2}+2u^{3})+t(4t^{3}-6tu^{2}-3u^{3})\right]$
$\displaystyle+2Q^{4}\left[2s^{6}+s^{5}(4u-6t)-s^{4}(t^{2}+8tu-4u^{2})+2s^{3}(3t^{3}+8t^{2}u-2tu^{2}+2u^{3})\right.$
$\displaystyle\left.+2s^{2}(2t^{4}+9t^{3}u+3t^{2}u^{2}-4tu^{3}+u^{4})+2stu(4t^{3}+t^{2}u-4tu^{2}-3u^{3})+t^{2}u^{2}(4t^{2}+6tu+3u^{2})\right]$
$\displaystyle+2Q^{2}s\left[-3s^{6}+s^{5}(t-9u)+s^{4}(6t^{2}+11tu-12u^{2})+2s^{3}(2t^{3}+12t^{2}u+9tu^{2}-6u^{3})\right.$
$\displaystyle\left.+s^{2}u(12t^{3}+20t^{2}u+10tu^{2}-9u^{3})+su^{2}(4t^{3}+8t^{2}u+5tu^{2}-3u^{3})+tu^{3}(4t^{2}+6tu+3u^{2})\right]$
$\displaystyle+s^{2}(s+u)^{2}(s^{2}+u^{2})\left[3s^{2}+6s(t+u)+4t^{2}+6tu+3u^{2}\right]\,,$
$\displaystyle\widehat{w}_{T}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}=$ $\displaystyle
8Q^{2}t\left\\{-8Q^{8}t^{2}+4Q^{6}t\left[s^{2}-2su+u(4t+u)\right]\right.-2Q^{4}\left[s^{3}(3t-2u)+s^{2}t(2t+u)\right.$
$\displaystyle\left.-s(8t^{3}+4t^{2}u+7tu^{2}+2u^{3})-t(4t^{3}-6tu^{2}-3u^{3})\right]-Q^{2}\left[s^{4}(t+6u)-2s^{3}(7t^{2}+4tu-u^{2})\right.$
$\displaystyle\left.-2s^{2}(6t^{3}+13t^{2}u+3tu^{2}-u^{3})-2su(4t^{3}+t^{2}u-4tu^{2}-3u^{3})-tu^{2}(4t^{2}+6tu+3u^{2})\right]$
$\displaystyle\left.+s(s+u)\left[s^{3}(4t+3u)+s^{2}(4t^{2}+6tu+u^{2})+su^{3}+u^{2}(4t^{2}+6tu+3u^{2})\right]\right\\}\,,$
$\displaystyle\widehat{w}_{L}^{\perp}$
$\displaystyle=32Q^{10}t^{3}-16Q^{8}t^{2}\left[s^{2}+2s(t-u)+4t^{2}+6tu+u^{2}\right]+8Q^{6}t\left[2s^{3}(3t-u)+s^{2}t(7t-2u)\right.$
$\displaystyle\left.+2s(2t^{3}-3t^{2}u-7tu^{2}-u^{3})+t(4t^{3}+12t^{2}u+11tu^{2}+2u^{3})\right]+2Q^{4}t\left[s^{4}(24u-7t)\right.$
$\displaystyle\left.-4s^{3}(4t^{2}+tu-2u^{2})+2s^{2}u(16t^{2}+15tu-4u^{2})+4su(8t^{3}+20t^{2}u+15tu^{2}+2u^{3})+tu^{4}\right]$
$\displaystyle+2Q^{2}st\left[5s^{5}+s^{4}(8t-11u)+2s^{3}(4t^{2}+4tu-9u^{2})+2s^{2}u(8t^{2}+20tu+9u^{2})\right.$
$\displaystyle\left.+su^{2}(24t^{2}+40tu+21u^{2})+u^{5}\right]+s^{2}(s+u)^{2}\left[s^{4}+2s^{3}u+2s^{2}u(8t+u)\right.$
$\displaystyle\left.+2su(8t^{2}+8tu+u^{2})+u^{4}\right]\,,$
$\displaystyle\widehat{w}_{L}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}=$ $\displaystyle
8Q^{2}\left\\{16Q^{8}t^{3}+8Q^{6}t^{2}\left[s^{2}-2s(t-u)-4t^{2}-6tu-u^{2}\right]\right.-4Q^{4}t\left[s^{4}+2s^{3}t\right.$
$\displaystyle\left.+s^{2}(5t^{2}+10tu+u^{2})-2s(2t^{3}-3t^{2}u-7tu^{2}-u^{3})-t(4t^{3}+12t^{2}u+11tu^{2}+2u^{3})\right]$
$\displaystyle+Q^{2}\left[-2s^{6}+4s^{5}(t-u)+s^{4}(5t^{2}+4tu-4u^{2})+4s^{3}(4t^{3}+t^{2}u-3tu^{2}-u^{3})\right.$
$\displaystyle\left.+2s^{2}(8t^{4}+32t^{3}u+21t^{2}u^{2}-2tu^{3}-u^{4})+4stu(8t^{3}+20t^{2}u+15tu^{2}+2u^{3})+t^{2}u^{4}\right]$
$\displaystyle+s(s+u)\left[2s^{5}+2s^{4}(t+2u)+s^{3}u(3t+4u)+s^{2}u(16t^{2}+17tu+4u^{2})\right.$
$\displaystyle\left.\left.+su(16t^{3}+32t^{2}u+17tu^{2}+2u^{3})+tu^{4}\right]\right\\}\,,$
$\displaystyle\widehat{w}^{\perp}_{\Delta}=$ $\displaystyle
2M_{\psi}\sqrt{t(Q^{2}t+su)}\left\\{-16Q^{8}t^{2}+8Q^{6}t\left[2t^{2}+4tu+u(u-s)\right]+2Q^{4}\left[2s^{4}+s^{3}(2u-3t)\right.\right.$
$\displaystyle\left.+s^{2}(4t^{2}+3tu+2u^{2})+s(4t^{3}+12t^{2}u+11tu^{2}+2u^{3})-tu(4t^{2}+8tu+3u^{2})\right]$
$\displaystyle-Q^{2}s\left[7s^{4}+4s^{3}(t+2u)-2s^{2}u(6t+u)-4stu(2t+u)+u^{2}(8t^{2}+12tu+3u^{2})\right]$
$\displaystyle\left.+s^{2}(s-u)(s+u)^{2}(3s+4t+3u)\right\\}\,,$
$\displaystyle\widehat{w}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}_{\Delta}=$
$\displaystyle-8Q^{2}M_{\psi}\sqrt{t(Q^{2}t+su)}\left\\{16Q^{6}t^{2}-8Q^{4}t\left[2t^{2}+4tu+u(u-s)\right]\right.-2Q^{2}\left[2s^{4}+s^{3}(2u-3t)\right.$
$\displaystyle\left.+s^{2}(4t^{2}+3tu+2u^{2})+s(4t^{3}+12t^{2}u+11tu^{2}+2u^{3})-tu(4t^{2}+8tu+3u^{2})\right]$
$\displaystyle\left.+s(s+u)\left[3s^{3}+s^{2}(u-4t)+s(u^{2}-8t^{2}-8tu)+u(8t^{2}+12tu+3u^{2})\right]\right\\}\,,$
$\displaystyle\widehat{w}^{\perp}_{\Delta\Delta}=$ $\displaystyle
4M_{\psi}^{2}(Q^{2}t+su)\left\\{-4Q^{6}t^{2}+2Q^{4}t\left[(s-t)^{2}+(t+u)^{2}\right]-Q^{2}t\left[s^{3}-s^{2}u-su(4t+u)+u^{3}\right]\right.$
$\displaystyle\left.+s^{2}(s+u)^{2}(s+2t+u)\right\\}\,,$
$\displaystyle\widehat{w}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}_{\Delta\Delta}=$
$\displaystyle-16Q^{2}M_{\psi}^{2}t(Q^{2}t+su)\left\\{4Q^{4}t-2Q^{2}(s^{2}-2st+2t^{2}+2tu+u^{2})\right.$
$\displaystyle\left.+(s+u)\left[s^{2}+u^{2}-2s(2t+u)\right]\right\\}\,.$ (96)

  * •

$\gamma^{*}g\,\to\,c\bar{c}[^{1}S_{0}^{[8]}]\,g$:

$\displaystyle{F}=$ $\displaystyle{32\,e_{c}^{2}\alpha\alpha_{s}^{2}\over
s^{2}t(s+t)^{2}(s+u)^{2}(t+u)^{2}M_{\psi}}\,\langle 0|{\cal
O}_{8}(\,^{1}S_{0})|0\rangle,$ $\displaystyle\widehat{w}_{T}^{\perp}=$
$\displaystyle
2Q^{6}t^{3}u^{2}+2Q^{4}st^{2}u\left[s(t+u)+t^{2}+tu+2u^{2}\right]+Q^{2}s^{2}t\left[s^{4}+2s^{3}(t+u)+3s^{2}(t+u)^{2}\right.$
$\displaystyle\left.+2s(t^{3}+3t^{2}u+4tu^{2}+2u^{3})+t^{4}+2t^{3}u+5t^{2}u^{2}+4tu^{3}+3u^{4}\right]$
$\displaystyle+s^{3}u\left[s^{4}+2s^{3}(t+u)+3s^{2}(t+u)^{2}+2s(t+u)^{3}+(t^{2}+tu+u^{2})^{2}\right]\,,$
$\displaystyle\widehat{w}_{T}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}=$ $\displaystyle
4Q^{2}t\left\\{2Q^{4}t^{2}u^{2}+2Q^{2}stu\left[t^{2}+tu+2u^{2}+s(t+u)\right]\right.+s^{2}\left[s^{2}(t+u)^{2}\right.$
$\displaystyle\left.\left.+2s(t^{3}+2t^{2}u+2tu^{2}+u^{3})+t^{4}+2t^{3}u+3t^{2}u^{2}+2tu^{3}+2u^{4}\right]\right\\}\,,$
$\displaystyle\widehat{w}_{L}^{\perp}=$
$\displaystyle\widehat{w}_{T}^{\perp}\,,$
$\displaystyle\widehat{w}_{L}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}=$
$\displaystyle\widehat{w}_{T}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}\,,$
$\displaystyle\widehat{w}^{\perp}_{\Delta}=$ $\displaystyle\widehat{w}^{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}_{\Delta}=\widehat{w}^{\perp}_{\Delta\Delta}=\widehat{w}^{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern
3.0mu}_{\Delta\Delta}=0\,.$ (97)

  * •

$\gamma^{*}g\,\to\,c\bar{c}[^{3}S_{1}^{[1]}]\,g$:

$\displaystyle F=$ $\displaystyle{256\,e_{c}^{2}\alpha\alpha_{s}^{2}\over
27s^{2}(s+t)^{2}(s+u)^{2}(t+u)^{2}\left[4Q^{2}t+(s+u)^{2}\right]M_{\psi}}\,\langle
0|{\cal O}_{1}(\,^{3}S_{1})|0\rangle,$ $\displaystyle\widehat{w}_{T}^{\perp}=$
$\displaystyle
M_{\psi}^{2}\left\\{4Q^{8}t^{4}+4Q^{6}t^{3}\left[2s(s+u)+t(t-u)\right]+2Q^{4}t^{2}\left[s^{4}-s^{3}(t-6u)+2s^{2}(t^{2}+2u^{2})\right.\right.$
$\displaystyle\left.+3st(t-u)u+t^{2}u^{2}\right]+Q^{2}st\left[s^{4}(2u-t)+s^{3}(3t^{2}+tu+8u^{2})+s^{2}u(8t^{2}+3tu+4u^{2})\right.$
$\displaystyle\left.+st(5t-3u)u^{2}+2t^{2}u^{3}\right]+s^{2}\left[s^{4}(t^{2}+u^{2})+2s^{3}u(t^{2}+tu+u^{2})+s^{2}u^{2}(4t^{2}+2tu+u^{2})\right.$
$\displaystyle\left.\left.+2st^{2}u^{3}+t^{2}u^{4}\right]\right\\}\,,$
$\displaystyle\widehat{w}_{T}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}=$ $\displaystyle
8Q^{2}M_{\psi}^{2}t(Q^{2}t+su)\left\\{t^{2}\left[s^{2}+2Q^{2}(Q^{2}+t)\right]+tu\left[s(s+t)+2Q^{2}(s-t)\right]+u^{2}\left[s^{2}-st+t^{2}\right]\right\\}\,,$
$\displaystyle\widehat{w}_{L}^{\perp}=$ $\displaystyle
2t\left\\{4Q^{10}t^{3}-4Q^{8}t^{2}\left[s(t-2u)+2t(t+u)\right]+2Q^{6}t\left[s^{3}t+2t^{2}(t+u)^{2}+st(2t^{2}-5tu-7u^{2})\right.\right.$
$\displaystyle\left.+s^{2}(3t^{2}-2tu+3u^{2})\right]+2Q^{4}s\left[3t^{2}u(t+u)^{2}+s^{3}t(t+2u)+s^{2}(5t^{2}u+u^{3})\right.$
$\displaystyle\left.-st(t^{3}-2t^{2}u+tu^{2}+4u^{3})\right]+Q^{2}s^{2}\left[s^{4}t-2s(t-u)^{2}u(t+u)+2s^{3}(t+u)^{2}+3tu^{2}(t+u)^{2}\right.$
$\displaystyle\left.\left.+s^{2}t(t^{2}+2tu+6u^{2})\right]+s^{3}u(s+t+u)^{2}(s^{2}+u^{2})\right\\}\,,$
$\displaystyle\widehat{w}_{L}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}=$ $\displaystyle
4Q^{2}\left\\{8Q^{8}t^{4}+8Q^{6}t^{3}\left[s^{2}-s(t-2u)-2t(t+u)\right]\right.+2Q^{4}t^{2}\left[s^{4}+4t^{2}(t+u)^{2}+s^{3}(6u-4t)\right.$
$\displaystyle\left.+2st(2t^{2}-5tu-7u^{2})-s^{2}(4t^{2}+10tu-7u^{2})\right]+2Q^{2}st\left[s^{4}(u-t)+6t^{2}u(t+u)^{2}\right.$
$\displaystyle\left.+s^{3}(t^{2}-3tu+4u^{2})+st(2t^{3}+8t^{2}u-3tu^{2}-9u^{3})+s^{2}(4t^{3}-7tu^{2}+3u^{3})\right]$
$\displaystyle+s^{2}\left[s^{4}(t^{2}+u^{2})+2s^{3}(t^{3}+2t^{2}u+u^{3})+s^{2}(t^{4}+6t^{3}u+2t^{2}u^{2}-4tu^{3}+u^{4})\right.$
$\displaystyle\left.\left.+2stu(t^{3}+3t^{2}u-2u^{3})+5t^{2}u^{2}(t+u)^{2}\right]\right\\}\,,$
$\displaystyle\widehat{w}^{\perp}_{\Delta}=$ $\displaystyle
M_{\psi}\sqrt{t(Q^{2}t+su)}\left\\{-8Q^{8}t^{3}-4Q^{6}t^{2}(s^{2}-st-2t^{2}+3su-3tu)+2Q^{4}t\left[s^{3}(t-2u)\right.\right.$
$\displaystyle\left.+2s^{2}(t-2u)u+stu(4t+7u)-2t^{2}u(t+u)\right]-Q^{2}s\left[s^{4}t+2s^{3}(t^{2}+tu+u^{2})\right.$
$\displaystyle\left.+s^{2}(3t^{3}+3t^{2}u-tu^{2}+2u^{3})-2stu(t^{2}+2tu+3u^{2})+3t^{2}u^{2}(t+u)\right]$
$\displaystyle\left.+s^{2}(s-u)\left[s^{3}(t-u)-su^{3}+tu^{2}(t+u)+s^{2}(t^{2}-2u^{2})\right]\right\\}\,,$
$\displaystyle\widehat{w}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}_{\Delta}=$
$\displaystyle-4Q^{2}M_{\psi}\sqrt{t(Q^{2}t+su)}\left\\{8Q^{6}t^{3}+4Q^{4}t^{2}\left[s(s-t+3u)-t(2t+3u)\right]-2Q^{2}t\left[s^{3}(t-2u)\right.\right.$
$\displaystyle\left.+2s^{2}u(t-2u)+stu(4t+7u)-2t^{2}u(t+u)\right]+s\left[s^{3}(t^{2}+tu+2u^{2})+s^{2}(t^{3}+3t^{2}u+2u^{3})\right.$
$\displaystyle\left.\left.-stu^{2}(3t+5u)+3t^{2}u^{2}(t+u)\right]\right\\}\,,$
$\displaystyle\widehat{w}^{\perp}_{\Delta\Delta}=$ $\displaystyle
M_{\psi}^{2}t(Q^{2}t+su)\left\\{-4Q^{6}t^{2}-4Q^{4}t\left[su-t(t+u)\right]+2Q^{2}s\left[s^{2}t-s(t^{2}+u^{2})+tu(t+u)\right]\right.$
$\displaystyle\left.+s^{2}(s+t+u)(s^{2}+u^{2})\right\\}\,,$
$\displaystyle\widehat{w}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}_{\Delta\Delta}=$
$\displaystyle-8Q^{2}M_{\psi}^{2}t(Q^{2}t+su)\left[2Q^{4}t^{2}-2Q^{2}t(t^{2}-su+tu)+s(s-t)u(t+u)\right]\,.$
(98)

  * •

$\gamma^{*}g\,\to\,c\bar{c}[^{3}S_{1}^{[8]}]\,g$:

$\displaystyle\widehat{w}_{\Lambda}^{\cal P}[\,^{3}S_{1}^{[8]}]=$
$\displaystyle{15\over 8}\widehat{w}_{\Lambda}^{\cal P}[\,^{3}S_{1}^{[1]}]\,.$
(99)

  * •

$\gamma^{*}g\,\to\,c\bar{c}[^{3}P_{J}^{[8]}]\,g$:

The formulae for the $P$-wave components of $\widehat{w}^{\cal P}_{T}$ and
$\widehat{w}^{\cal P}_{L}$ are too lengthy to be shown here. They are
available upon request as a Wolfram Mathematica notebook file. Only
$\widehat{w}^{\cal P}_{\Delta}$ and $\widehat{w}^{\cal P}_{\Delta\Delta}$ are
given below in an analytical form.

$\displaystyle F=$
$\displaystyle-{192\,\sqrt{t(Q^{2}t+su)}\,e_{c}^{2}\alpha\alpha_{s}^{2}\over
s^{2}t^{2}(s+t)^{3}(s+u)^{4}(t+u)^{3}\left[4Q^{2}t+(s+u)^{2}\right]M_{\psi}^{2}}\,\langle
0|{\cal O}_{8}(\,^{3}P_{0})|0\rangle,$
$\displaystyle\widehat{w}^{\perp}_{\Delta}=$ $\displaystyle
t\left\\{64Q^{10}t^{3}(s^{3}t-st^{2}u-s^{2}u^{2}+tu^{3})-16Q^{8}t^{2}\left[2s^{5}u+s^{4}(6t^{2}-2tu+6u^{2})\right.\right.$
$\displaystyle+s^{3}(5t^{3}+11t^{2}u+10u^{3})-s^{2}(t^{4}+7t^{3}u+6tu^{3}-4u^{4})-tu^{2}(t^{3}-5t^{2}u-10tu^{2}-2u^{3})$
$\displaystyle\left.-su(6t^{4}+11t^{3}u-5t^{2}u^{2}+2tu^{3}-2u^{4})\right]-8Q^{6}t\left[2s^{7}(t+u)+2s^{6}(t^{2}+tu+5u^{2})\right.$
$\displaystyle-s^{5}(5t^{3}-6t^{2}u+6tu^{2}-20u^{3})-s^{4}(8t^{4}+8t^{3}u-18t^{2}u^{2}+5tu^{3}-24u^{4})$
$\displaystyle-s^{3}(t^{5}+9t^{4}u+14t^{3}u^{2}-19t^{2}u^{3}+9tu^{4}-14u^{5})+t^{2}u^{2}(2t^{4}+3t^{3}u-7t^{2}u^{2}-12tu^{3}-3u^{4})$
$\displaystyle+s^{2}(2t^{6}+15t^{5}u+23t^{4}u^{2}+16t^{3}u^{3}+43t^{2}u^{4}+5tu^{5}+6u^{6})$
$\displaystyle\left.+stu(4t^{5}+23t^{4}u+29t^{3}u^{2}+23t^{2}u^{3}+27tu^{4}+3u^{5})\right]+4Q^{4}\left[s^{8}(6t^{2}-4u^{2})\right.$
$\displaystyle+s^{7}(15t^{3}+23t^{2}u+12tu^{2}-12u^{3})+s^{6}(17t^{4}+37t^{3}u+32t^{2}u^{2}+32tu^{3}-20u^{4})$
$\displaystyle+s^{5}(6t^{5}+28t^{4}u+40t^{3}u^{2}+21t^{2}u^{3}+39tu^{4}-20u^{5})$
$\displaystyle+s^{4}(4t^{6}+10t^{5}u+33t^{4}u^{2}+56t^{3}u^{3}+6t^{2}u^{4}+31tu^{5}-12u^{6})$
$\displaystyle+s^{3}(2t^{7}+6t^{6}u-3t^{5}u^{2}-4t^{4}u^{3}+17t^{3}u^{4}-25t^{2}u^{5}+9tu^{6}-4u^{7})$
$\displaystyle+s^{2}tu(6t^{6}+20t^{5}u+13t^{4}u^{2}+11t^{3}u^{3}+31t^{2}u^{4}+5u^{6})+t^{4}u^{3}(2t^{3}+4t^{2}u+tu^{2}-u^{3})$
$\displaystyle\left.+st^{2}u^{2}(6t^{5}+22t^{4}u+37t^{3}u^{2}+52t^{2}u^{3}+44tu^{4}+9u^{5})\right]+2Q^{2}s\left[s^{8}(9u^{2}+12tu-5t^{2})\right.$
$\displaystyle-s^{7}(13t^{3}-9t^{2}u-43tu^{2}-29u^{3})-s^{6}(19t^{4}-2t^{3}u-35t^{2}u^{2}-64tu^{3}-52u^{4})$
$\displaystyle-s^{5}(9t^{5}-t^{4}u-53t^{3}u^{2}-83t^{2}u^{3}-88tu^{4}-62u^{5})$
$\displaystyle+s^{4}u(16t^{5}+60t^{4}u+110t^{3}u^{2}+145t^{2}u^{3}+80tu^{4}+45u^{5})$
$\displaystyle+s^{3}u(6t^{6}+34t^{5}u+68t^{4}u^{2}+105t^{3}u^{3}+127t^{2}u^{4}+47tu^{5}+21u^{6})$
$\displaystyle+s^{2}u^{2}(14t^{6}+52t^{5}u+95t^{4}u^{2}+114t^{3}u^{3}+105t^{2}u^{4}+28tu^{5}+6u^{6})$
$\displaystyle\left.+stu^{3}(10t^{5}+47t^{4}u+83t^{3}u^{2}+79t^{2}u^{3}+45tu^{4}+6u^{5})+2t^{3}u^{4}(t^{3}+2t^{2}u-u^{3})\right]$
$\displaystyle-s^{2}(s^{2}-u^{2})\left[s^{6}(3u^{2}+10tu-t^{2})+s^{5}(t^{3}+28t^{2}u+33tu^{2}+6u^{3})\right.$
$\displaystyle+s^{4}(3t^{4}+45t^{3}u+49t^{2}u^{2}+41tu^{3}+6u^{4})+s^{3}(t^{5}+30t^{4}u+46t^{3}u^{2}+40t^{2}u^{3}+41tu^{4}+6u^{5})$
$\displaystyle+s^{2}u(7t^{5}+22t^{4}u+46t^{3}u^{2}+49t^{2}u^{3}+33tu^{4}+3u^{5})$
$\displaystyle\left.\left.+stu^{2}(7t^{4}+30t^{3}u+45t^{2}u^{2}+28tu^{3}+10u^{4})+t^{2}u^{3}(t^{3}+3t^{2}u+tu^{2}-u^{3})\right]\right\\}\,,$
$\displaystyle\widehat{w}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}_{\Delta}=$
$\displaystyle
8Q^{2}t\left\\{32Q^{8}t^{3}(s^{3}t-st^{2}u-s^{2}u^{2}+tu^{3})-8Q^{6}t^{2}\left[2s^{5}u+s^{4}(6t^{2}-2tu+6u^{2})\right.\right.$
$\displaystyle+s^{3}(5t^{3}+11t^{2}u+10u^{3})-s^{2}(t^{4}+7t^{3}u+6tu^{3}-4u^{4})$
$\displaystyle\left.-su(6t^{4}+11t^{3}u-5t^{2}u^{2}+2tu^{3}-2u^{4})+tu^{2}(2u^{3}+10tu^{2}+5t^{2}u-t^{3})\right]$
$\displaystyle-4Q^{4}t\left[2s^{7}(t+u)+2s^{6}(2t^{2}+2tu+5u^{2})+s^{5}(20u^{3}-4tu^{2}+10t^{2}u-5t^{3})\right.$
$\displaystyle-s^{4}(8t^{4}+8t^{3}u-18t^{2}u^{2}+7tu^{3}-24u^{4})-s^{3}(t^{5}+9t^{4}u+14t^{3}u^{2}-15t^{2}u^{3}+11tu^{4}-14u^{5})$
$\displaystyle+s^{2}(2t^{6}+15t^{5}u+23t^{4}u^{2}+16t^{3}u^{3}+41t^{2}u^{4}+5tu^{5}+6u^{6})$
$\displaystyle+stu(4t^{5}+23t^{4}u+29t^{3}u^{2}+23t^{2}u^{3}+27tu^{4}+3u^{5})$
$\displaystyle\left.+t^{2}u^{2}(2t^{4}+3t^{3}u-7t^{2}u^{2}-12tu^{3}-3u^{4})\right]+2Q^{2}\left[2s^{8}(t^{2}-tu-2u^{2})\right.$
$\displaystyle+s^{7}(7t^{3}+9t^{2}u+6tu^{2}-12u^{3})+s^{6}(3t^{4}+27t^{3}u+20t^{2}u^{2}+28tu^{3}-20u^{4})$
$\displaystyle+s^{5}u(16t^{4}+42t^{3}u+21t^{2}u^{2}+43tu^{3}-20u^{4})$
$\displaystyle+s^{4}(4t^{6}+8t^{5}u+33t^{4}u^{2}+54t^{3}u^{3}+18t^{2}u^{4}+37tu^{5}-12u^{6})$
$\displaystyle+s^{3}(2t^{7}+6t^{6}u-t^{5}u^{2}+8t^{4}u^{3}+27t^{3}u^{4}-11t^{2}u^{5}+11tu^{6}-4u^{7})$
$\displaystyle+s^{2}tu(6t^{6}+20t^{5}u+19t^{4}u^{2}+25t^{3}u^{3}+39t^{2}u^{4}+4tu^{5}+5u^{6})+t^{4}u^{3}(2t^{3}+4t^{2}u+tu^{2}-u^{3})$
$\displaystyle\left.+st^{2}u^{2}(6t^{5}+22t^{4}u+37t^{3}u^{2}+52t^{2}u^{3}+44tu^{4}+9u^{5})\right]+s(s+u)\left[s^{7}(3t^{2}+10tu+7u^{2})\right.$
$\displaystyle+s^{6}(8t^{3}+26t^{2}u+30tu^{2}+20u^{3})+s^{5}(7t^{4}+20t^{3}u+43t^{2}u^{2}+42tu^{3}+32u^{4})$
$\displaystyle+2s^{4}(t^{5}+9t^{4}u+27t^{3}u^{2}+35t^{2}u^{3}+23tu^{4}+15u^{5})$
$\displaystyle+s^{3}u(20t^{5}+48t^{4}u+56t^{3}u^{2}+45t^{2}u^{3}+26tu^{4}+17u^{5})$
$\displaystyle+2s^{2}u(3t^{6}+7t^{5}u+7t^{4}u^{2}+14t^{3}u^{3}+24t^{2}u^{4}+12tu^{5}+3u^{6})$
$\displaystyle\left.\left.+stu^{2}(8t^{5}+32t^{4}u+57t^{3}u^{2}+60t^{2}u^{3}+37tu^{4}+6u^{5})+2t^{3}u^{3}(t^{3}+2t^{2}u-u^{3})\right]\right\\}\,,$
$\displaystyle\widehat{w}^{\perp}_{\Delta\Delta}=$ $\displaystyle
2M_{\psi}\sqrt{t(Q^{2}t+su)}\left\\{16Q^{8}t^{3}(s^{3}t-s^{2}u^{2}-st^{2}u+tu^{3})-4Q^{6}t^{2}\left[2s^{5}(t+u)+4s^{4}(t^{2}+u^{2})\right.\right.$
$\displaystyle+s^{3}(5t^{3}+9t^{2}u+2tu^{2}+8u^{3})-s^{2}(t^{4}+3t^{3}u+4tu^{3}-4u^{4})-tu^{2}(t^{3}-5t^{2}u-6tu^{2}-2u^{3})$
$\displaystyle\left.-su(6t^{4}+7t^{3}u-t^{2}u^{2}+2tu^{3}-2u^{4})\right]-4Q^{4}t\left[s^{6}(t^{2}+tu+2u^{2})\right.$
$\displaystyle-s^{5}(2t^{3}-3t^{2}u+2tu^{2}-4u^{3})-s^{4}(t^{4}-t^{3}u-8t^{2}u^{2}+2tu^{3}-6u^{4})$
$\displaystyle+s^{3}u(t^{4}+t^{3}u+6t^{2}u^{2}-3tu^{3}+4u^{4})+s^{2}(t^{6}+8t^{5}u+9t^{4}u^{2}+8t^{3}u^{3}+10t^{2}u^{4}+tu^{5}+2u^{6})$
$\displaystyle\left.+stu(2t^{5}+10t^{4}u+11t^{3}u^{2}+13t^{2}u^{3}+7tu^{4}+u^{5})+t^{2}u^{2}(t^{4}+2t^{3}u-tu^{3}-u^{4})\right]$
$\displaystyle-2Q^{2}st\left[2s^{7}t+s^{6}(3t^{2}+3tu-5u^{2})+2s^{5}(2t^{3}-t^{2}u-tu^{2}-7u^{3})\right.$
$\displaystyle+s^{4}(5t^{4}+5t^{3}u-12t^{2}u^{2}-6tu^{3}-18u^{4})+s^{3}(5t^{4}u-16t^{2}u^{3}-5tu^{4}-12u^{5})$
$\displaystyle+s^{2}u^{2}(11t^{4}+9t^{3}u+t^{2}u^{2}+3tu^{3}-5u^{4})+su^{3}(11t^{4}+10t^{3}u+6t^{2}u^{2}-tu^{3}-2u^{4})$
$\displaystyle\left.-2tu^{5}(2t^{2}+2tu+u^{2})\right]-s^{2}(s+u)\left[s^{6}t(u-t)-2s^{5}(t-u)^{2}(t+u)\right.$
$\displaystyle+s^{4}(t^{4}+9t^{3}u-3t^{2}u^{2}-3tu^{3}-4u^{4})+s^{3}u(12t^{4}+3t^{3}u-12t^{2}u^{2}-3tu^{3}-2u^{4})$
$\displaystyle+s^{2}tu(2t^{4}+6t^{3}u+3t^{2}u^{2}-3tu^{3}+2u^{4})+stu^{2}(2t^{4}+12t^{3}u+9t^{2}u^{2}+2tu^{3}+u^{4})$
$\displaystyle\left.\left.+t^{2}u^{4}(t^{2}-2tu-u^{2})\right]\right\\}\,,$
$\displaystyle\widehat{w}^{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 2.0mu\vrule depth=0.0pt\mkern 3.0mu}_{\Delta\Delta}=$
$\displaystyle
16Q^{2}M_{\psi}t\sqrt{t(Q^{2}t+su)}\left\\{8Q^{6}t^{2}(s^{3}t-s^{2}u^{2}-st^{2}u+tu^{3})-2Q^{4}t\left[2s^{5}(t+u)+4s^{4}(t^{2}+u^{2})\right.\right.$
$\displaystyle+s^{3}(5t^{3}+9t^{2}u+2tu^{2}+8u^{3})-s^{2}(t^{4}+3t^{3}u+4tu^{3}-4u^{4})$
$\displaystyle\left.-su(6t^{4}+7t^{3}u-t^{2}u^{2}+2tu^{3}-2u^{4})-tu^{2}(t^{3}-5t^{2}u-6tu^{2}-2u^{3})\right]$
$\displaystyle-2Q^{2}\left[2s^{6}u(t+u)+s^{5}u(t^{2}+tu+4u^{2})-s^{4}(t^{4}-t^{3}u-4t^{2}u^{2}-tu^{3}-6u^{4})\right.$
$\displaystyle-s^{3}u(t^{4}-t^{3}u-4t^{2}u^{2}+2tu^{3}-4u^{4})+s^{2}(t^{6}+8t^{5}u+9t^{4}u^{2}+10t^{3}u^{3}+9t^{2}u^{4}+tu^{5}+2u^{6})$
$\displaystyle\left.+stu(2t^{5}+10t^{4}u+11t^{3}u^{2}+13t^{2}u^{3}+7tu^{4}+u^{5})+t^{2}u^{2}(t^{4}+2t^{3}u-tu^{3}-u^{4})\right]$
$\displaystyle+s(s+u)\left[3s^{5}(t+u)^{2}+s^{4}(4t^{3}+9t^{2}u+10tu^{2}+5u^{3})+s^{3}u(6t^{3}+15t^{2}u+14tu^{2}+7u^{3})\right.$
$\displaystyle-s^{2}(t^{5}-2t^{4}u-12t^{3}u^{2}-13t^{2}u^{3}-5tu^{4}-3u^{5})$
$\displaystyle\left.\left.-su(t^{5}+6t^{4}u+6t^{3}u^{2}+4t^{2}u^{3}-tu^{4}-2u^{5})+2tu^{4}(2t^{2}+2tu+u^{2})\right]\right\\}\,.$
(100)

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