# Dilepton production in the photodisintegration of the deuteron

Mengchu Cai  School of Physics, Peking University, Beijing 100871, China
Tianbo Liu<EMAIL_ADDRESS>Key Laboratory of Particle Physics and Particle
Irradiation (MOE), Institute of Frontier and Interdisciplinary Science,
Shandong University, Qingdao, Shandong 266237, China  Bo-Qiang Ma
<EMAIL_ADDRESS>School of Physics, Peking University, Beijing 100871, China
Center for High Energy Physics, Peking University, Beijing 100871, China

###### Abstract

We study the lepton pair production in the photodisintegration of the deuteron
process. The complete seven-fold differential cross section is calculated via
the Bethe-Heitler mechanism with final state interactions taken into account.
The deuteron bound state is described by a relativistic covariant deuteron-
nucleon vertex. With numerical results, we find that the differential cross
section has strong dependence on the lepton azimuthal angle in the small polar
angle region and sharp peaks appear in the dependence on the invariant mass of
the produced lepton pair or the two nucleons in the final state. We
demonstrate that such nearly singular feature originates from the collinearity
between the produced lepton or antilepton and the incident photon, and it is
physically regularized by the lepton mass in our calculation. The final state
interaction between the knocked-out nucleon and the recoil nucleon
redistributes the differential cross section over the missing momentum, with a
significant enhancement at large missing momentum and a suppression in the
intermediate region. With a further decomposition of the final state
interaction contribution, it is found that the on-shell term dominates the
near quasi-elastic region while the off-shell term dominates the other end. In
addition, we examine the contribution from the interference between the proton
amplitude and the neutron amplitude, which as expected is found negligible
even if the proton-neutron rescattering is included. The result in this work
can serve as an input for the analysis and background estimation of multiple
exclusive measurements at Jefferson Lab and future electron-ion colliders.

##  I Introduction

The deuteron is the simplest nontrivial nucleus in nature, predominantly
consisting of a proton-neutron bound state, and is considered an ideal place
to learn the structure of cold nuclear matter and the nucleon-nucleon
interaction. It has been studied from both experimental and theoretical
aspects for more than 80 years since the discovery and is nowadays still an
active frontier in nuclear and particle physics. Since the electromagnetic
interaction is better understood, the electron and photon scatterings from the
deuteron with or without disintegration are main processes to probe the
deuteron structure [1, 2, 3, 4, 5]. The near-threshold production of $J/\psi$
from a deuteron target received great interest in recent years because of its
sensitivity to the gluonic field and the nucleon-nucleon short-range
correlations (SRCs) [6, 7, 8]. Furthermore, the final-state interaction in the
incoherent scattering allows to investigate the $J/\psi\,N$ scattering cross
section at low energies [6, 7, 9]. As the $J/\psi$ in the final state is
usually reconstructed from its decay channels to $e^{+}e^{-}$ and
$\mu^{+}\mu^{-}$, the lepton pair production is one of the dominant background
reactions for the precise measurement of the $J/\psi$ production cross
section, especially near the threshold.

Apart from the study of nuclear structures, the deuteron is also utilized as
an effective neutron target because of the lacking of a stable free neutron
target in the exploration of nucleon structures. The deep-inelastic electron
scattering from the deuteron plays an important role in the flavor separation
of parton distribution functions and meanwhile provides valuable information
of the nuclear matter modification of the partonic structure of a nucleon [10,
11]. Beyond the parton distribution functions, which describe the longitudinal
momentum distribution of quarks and gluon in the nucleon, one of the main
goals of current and future nuclear experiments is to have precise measurement
of three-dimensional partonic structures of the nucleon. The generalized
parton distribution (GPD), which encodes the transverse coordinate
distribution of partons, is one of physical quantities to be investigated at
Jefferson Lab and future electron-ion colliders [12, 13, 14]. A golden channel
to access GPDs is the deeply virtual Compton scattering (DVCS) [15], in which
one can practically extract GPDs from the interference term between the DVCS
amplitude and the Bethe-Heitler (BH) amplitude [16, 17, 18, 19], while new
processes with unique sensitivities to the $x$-dependence of GPDs are recently
suggested [20, 21]. Several other processes extended from the DVCS, such as
the timelike Compton scattering (TCS) [22, 23, 24, 25] and the deeply virtual
exclusive meson production (DVMP) [26, 27, 28], are also proposed as
complementary channels to measure GPDs via the similar mechanism. In such
measurements, the Bethe-Heitler process is a dominant background channel,
which should be well understood.

The dilepton production in the photodisintegration of the deuteron through the
BH mechanism shares the same final state particles as those in the TCS process
and the $J/\psi$ production with subsequent decay to a lepton pair from the
deuteron. The BH contribution dominates the total cross section over the
typical TCS region by more than one order of magnitude [22, 23], and is thus a
main source of background preventing the precise measurement of GPDs. For the
$J/\psi$ production, the cross section is small and changes rapidly when the
beam energy is near the threshold, and hence the yield from the BH channel is
one of the key sources of systematic uncertainties in the analysis of the
$J/\psi$ production. Besides, the final state interaction (FSI) also plays an
important role in certain kinematic regions where the missing momentum of the
undetected nucleon is not small [29, 30, 31, 32, 33], and thus can shed light
on the extraction of the $J/\psi\,N$ scattering cross section. All these above
require a careful study of the BH lepton pair production from the
photodisintegration of the deuteron, and it is necessary to include the FSI
effects [6, 8].

In this paper, we perform a complete calculation of the differential cross
section of the dilepton production process in the photodisintegration of the
deuteron through the BH mechanism. The lepton mass is kept finite not only to
differentiate the results of $e^{+}e-$ and $\mu^{+}\mu^{-}$ production but
also to physically regularize the collinear singularities. A relativistic
covariant spectator theory [34, 35, 36, 37] is utilized to describe the
deuteron in terms of the proton and the neutron, which has been proven
successful in describing the elastic and inelastic electron deuteron
scatterings [33, 38, 39] and the deuteron magnetic moment and electromagnetic
form factors [40, 41]. The FSI is taken into account via the nucleon-nucleon
scattering, which is parametrized in a Lorentz covariant form including all
spin-dependent contributions following the formalism in [33], although other
approaches like the Glauber approximation or the generalized eikonal
approximation are also commonly used in previous studies [42, 43, 44, 29, 30,
45, 32]. With numerical results, we further analyze the impact of the FSI in
comparison with the calculation using the plane wave impulse approximation
(PWIA) and its kinematic dependence.

The remaining of this paper is organized as follows. In Sec. II, we present
the theoretical framework of the calculation, including the kinematics of the
process, the covariant deuteron-nucleon vertex, and the nucleon-nucleon
rescattering in the final state. Both the PWIA and the FSI amplitudes are
derived with detailed provided in appendix. In Sec. III, we show the numerical
results for the differential cross section with detailed discussions of the
nearly singular behavior in some particular kinematic regions and the effects
of FSI which is further decomposed into an on-shell term and an off-shell term
to understand the kinematic dependence. Summary and conclusions are given in
Sec. IV.

##  II Theoretical calculation

We consider the process,

$\gamma(k)+d(P)\rightarrow p(p_{1})+n(p_{2})+l^{-}(p_{3})+l^{+}(p_{4}),$ (1)

where the variables in parentheses represent the four-momenta of corresponding
particles. The $l^{-}$ and $l^{+}$ are a pair of charged leptons, such as
$e^{-}e^{+}$ or $\mu^{-}\mu^{+}$. The total four-momentum of the lepton pair
in the final state is $p_{34}=p_{3}+p_{4}$ and the four-momentum transferred
to the hadronic system is $q=k-p_{34}=p_{1}+p_{2}-P$. To describe the angular
distribution of final state particles, it is convenient to specify a reference
frame. Here we choose the deuteron rest frame, or sometimes referred to as the
target rest frame, where

$\displaystyle P$ $\displaystyle=(m_{d},{\bm{0}}),\quad
k=(E_{\gamma},{\bm{k}}),\quad q=(\nu,{\bm{q}}).$ (2)

As illustrated in Fig. 1, we choose the photon momentum ${\bm{k}}$ and the
transferred momentum ${\bm{q}}$ in the $x-z$ plane. The direction of the
proton momentum ${\bm{p}}_{1}$ is labeled by the polar angle $\theta_{1}$ and
the azimuthal angle $\phi_{1}$ with respect to ${\bm{q}}$. The neutron
momentum ${\bm{p}}_{2}$, which is not shown in Fig. 1, is fixed by the
momentum conservation ${\bm{p}}_{2}={\bm{q}}-{\bm{p}}_{1}$, and we denote
$|{\bm{p}}_{2}|=p_{m}$ as the so-called missing momentum. The direction of the
final-state lepton momentum ${\bm{p}}_{3}$ (or ${\bm{p}}_{4}$) is easier to be
defined in the center-of-mass frame of the lepton pair, where we label the
polar angle and the azimuthal angle of $l^{-}$ as $\theta_{l}$ and $\phi_{l}$
with respect to the direction of ${\bm{p}}_{34}$ in the target rest frame.

Figure 1: Kinematics for the dilepton photoproduction on the deuteron target.

With the kinematic variables defined above, we can write the differential
cross section in the target rest frame as

$\displaystyle{\rm d}\sigma$ $\displaystyle=$
$\displaystyle(2\pi)^{4}\delta^{(4)}(k+P-p_{1}-p_{2}-p_{3}-p_{4})$ (3)
$\displaystyle\times$
$\displaystyle\frac{1}{4E_{\gamma}m_{d}}\overline{\sum_{\rm
if}}|\mathcal{M}|^{2}\frac{{\rm{d}^{3}}\bm{p}_{1}}{(2\pi)^{3}2E_{1}}\frac{{\rm{d}^{3}}\bm{p}_{2}}{(2\pi)^{3}2E_{2}}\frac{{\rm{d}^{3}}\bm{p}_{3}}{(2\pi)^{3}2E_{3}}\frac{{\rm{d}^{3}}\bm{p}_{4}}{(2\pi)^{3}2E_{4}}~{},$

where $E_{i}=\sqrt{m_{i}^{2}+\bm{p}_{i}^{2}}$ is the energy of corresponding
particles, $m_{1}=m_{2}=m$ is the nucleon mass, and $m_{3}=m_{4}=m_{l}$ is the
lepton mass. The symbol $\displaystyle{\overline{\sum}_{\rm if}}$ represents
the average over spin states in the initial state and the sum over spin states
in the final state.

In this study, we consider the lepton pair production via the Bethe-Heitler
mechanism. With one-photon-exchange approximation, we can express the
invariant amplitude square as

$\overline{\sum_{\rm
if}}|\mathcal{M}|^{2}=\frac{e^{6}}{t^{2}}L^{\mu\nu}W_{\mu\nu}~{},$ (4)

where $t=q^{2}$ is the transferred momentum square. The unpolarized leptonic
tensor $L^{\mu\nu}$ is given by

$\displaystyle L^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\operatorname{Tr}\left\\{\left(\not{p}_{3}+m_{l}\right)\left[\gamma^{\alpha}\frac{\not{p}_{3}-\not{k}+m_{l}}{\left(p_{3}-k\right)^{2}-m_{l}^{2}}\gamma^{\mu}\right.+\gamma^{\mu}\frac{\not{k}-\not{p}_{4}+m_{l}}{\left(k-p_{4}\right)^{2}-m_{l}^{2}}\gamma^{\alpha}\right]$
(5) $\displaystyle\times$
$\displaystyle\left.\left(\not{p}_{4}-m_{l}\right)\left[\gamma^{\nu}\frac{\not{p}_{3}-\not{k}+m_{l}}{\left(p_{3}-k\right)^{2}-m_{l}^{2}}\gamma_{\alpha}+\gamma_{\alpha}\frac{\not{k}-\not{p}_{4}+m_{l}}{\left(k-p_{4}\right)^{2}-m_{l}^{2}}\gamma^{\nu}\right]\right\\}~{},$

which is the same as the one for the Bethe-Heitler process [46]. The
information of nuclear structure is contained in the hadronic tensor,

$W_{\mu\nu}=(2m)^{2}\overline{\sum_{\rm{if}}}J_{\mu}(q)J_{\nu}^{\dagger}(q)~{},$
(6)

where
$J_{\mu}(q)=\left\langle\bm{p}_{1}s_{1};\bm{p}_{2}s_{2}\left|\hat{J}_{\mu}(0)\right|\bm{P}\lambda_{d}\right\rangle$
is the nuclear electromagnetic current matrix element in the momentum space,
and $\lambda_{d}$, $s_{1}$, and $s_{2}$ represent the spin states of the
initial deuteron, the final proton, and the final neutron respectively. A
factor $(2m)$ is inserted for each nucleon spinor to be consistent with the
normalization convention in following calculations. The current conservation
requires

$q^{\mu}J_{\mu}(q)=\nu J^{0}(q)-\bm{q}\cdot\bm{J}(q)=0~{}.$ (7)

If taking $\bm{q}$ along the $z$-direction, one can replace the explicit
$J_{z}(q)$ dependence by $\left(\nu/|\bm{q}|\right)J^{0}(q)$ in the
calculation.

Figure 2: The Feynman diagrams for the PWIA in the one-photon-exchange
approximation.

After integrating the four-momentum conservation $\delta$-function and a
global azimuthal angle of final-state particles with respect to the photon
axis, we obtain a seven-fold differential cross section. For convenience, we
adopt the invariant mass square of the proton and the neutron
$s_{pn}=(p_{1}+p_{2})^{2}$ and the invariant mass square of the lepton pair
$s_{ll}=(p_{3}+p_{4})^{2}$ as variables. Then the differential cross section
can be expressed as

${\rm
d}\sigma=\frac{\alpha^{3}}{8(4\pi)^{4}}\frac{|\bm{p}_{1}|^{2}\beta}{m_{d}^{2}E_{\gamma}^{2}t^{2}}\frac{L^{\mu\nu}W_{\mu\nu}}{||\bm{p}_{1}|(m_{d}+\nu)-E_{1}|\bm{q}|\cos\theta_{1}|}{\rm{d}}\Omega_{p}{\rm{d}}\Omega_{ll}{\rm{d}}s_{ll}{\rm{d}}s_{pn}{\rm{d}}t~{},$
(8)

where $\Omega_{p}$ is the solid angle of the proton with ${\rm
d}\Omega_{p}={\rm d}\cos\theta_{1}{\rm d}\phi_{1}$, $\Omega_{ll}$ is the solid
angle of $l^{-}$ in the lepton pair center-of-mass frame with ${\rm
d}\Omega_{ll}={\rm d}\cos\theta_{l}{\rm d}\phi_{l}$, $\beta$ is the lepton
velocity given by

$\beta=\sqrt{1-\frac{4m_{l}^{2}}{s_{ll}}}~{},$ (9)

and $\alpha=e^{2}/(4\pi)$ is the electromagnetic fine structure constant. The
derivation from Eq. (3) to Eq. (8) is provided in Appendix A.

###  II.1 Plane wave contribution

Firstly, we consider the impulse approximation (IA), in which the reaction is
viewed as the Bethe-Heitler process from a quasi-free nucleon in the deuteron.
The mechanism of this plane wave (PW) contribution is depicted by the Feynman
amplitudes in Fig. 2. In this case, electromagnetic current matrix element can
be written as

$\displaystyle J_{\rm{PW}}^{\mu}(q)$ $\displaystyle=$
$\displaystyle\left\langle\bm{p}_{1}s_{1};\bm{p}_{2}s_{2}\left|\hat{J}^{\mu}\right|\bm{P}\lambda_{d}\right\rangle_{\rm{PW}}$
(10) $\displaystyle=$
$\displaystyle\bar{u}\left(\bm{p}_{1},s_{1}\right)\Gamma_{N}^{\mu}(q)G_{0}\left(P-p_{2}\right)\Gamma_{\lambda_{d}}\left(p_{2},P\right)C\bar{u}^{T}\left(\bm{p}_{2},s_{2}\right)~{},$

where the Dirac spinors are normalized as
$u^{\dagger}({\bm{p}},s)u({\bm{p}},s^{\prime})=E/m$,
$C=-i\gamma^{0}\gamma^{2}$ is the charge conjugation operator, and $G_{0}$ is
the bare nucleon propagator given by

$G_{0}(p)=\frac{\not{p}+m}{m^{2}-p^{2}-i\epsilon}~{},$ (11)

where $\epsilon$ is a positive infinitesimal number. The nucleon
electromagnetic current vertex $\Gamma_{N}^{\mu}(q)$ is parametrized as

$\Gamma_{N}^{\mu}(q)=F_{1}(Q^{2})\gamma^{\mu}+\frac{F_{2}(Q^{2})}{2m}i\sigma^{\mu\nu}q_{\nu}~{},$
(12)

where $Q^{2}=-t$ and $F_{1}(Q^{2})$ and $F_{2}(Q^{2})$ are nucleon’s Dirac and
Pauli form factors. In numerical calculations, we use the fit of nucleon
electromagnetic form factors in Ref. [47]. The deuteron-proton-neutron vertex
can be written as

$\displaystyle\Gamma_{\lambda_{d}}(p_{2},P)=\Gamma_{D}^{\mu}(p_{2},P)~{}\xi^{\lambda_{d}}_{\mu}(P),$
(13)

where $\xi_{\mu}^{\lambda_{d}}(P)$ is the deuteron polarization vector and
$\Gamma_{D}^{\mu}(p_{2},P)$ is the covariant vertex function with $P$ and
$p_{2}$ on-shell. According to the Lorentz structure, one can in general
express the vertex function as [34, 36, 37]

$\Gamma_{D}^{\mu}(k_{2},P)\equiv
F\gamma^{\mu}+\frac{G}{m}k^{\mu}-\frac{m-\not{k}_{1}}{m}\left(H\gamma^{\mu}+\frac{I}{m}k^{\mu}\right)~{},$
(14)

where $k=(k_{2}-k_{1})/2$ is the relative four-momentum of the two nucleons
with $k_{1}=P-k_{2}$ being the momentum of the off-shell nucleon. The scalar
functions $F$, $G$, $H$, and $I$ encode the nucleonic structure of the
deuteron and can be evaluated from the deuteron wave functions. Explicit
expressions of these functions are provided in Appendix B and more details can
be found in Refs. [35, 40].

Substituting (10) into (6), we obtain the hadronic tensor in the plane wave
impulse approximation (PWIA) as

$\displaystyle W^{\mu\nu}_{\rm PW}$
$\displaystyle=(2m)^{2}\overline{\sum_{\rm{if}}}J_{\rm{PW}}^{\mu}(q)J^{\nu\,\dagger}_{\rm{PW}}(q)$
$\displaystyle={\rm{Tr}}\left[\Gamma_{N}^{\mu}(q)\frac{\not{P}-\not{p}_{2}+m}{m^{2}-(P-p_{2})^{2}}\Gamma_{D}^{\alpha}\left(p_{2},P\right)\left(\not{p}_{2}-m\right)\widetilde{\Gamma}_{D}^{\beta}\frac{\not{P}-\not{p}_{2}+m}{m^{2}-(P-p_{2})^{2}}\widetilde{\Gamma}_{N}^{\nu}\left(\not{p}_{1}+m\right)\right]$
$\displaystyle\quad\times\frac{1}{3}\sum_{\lambda_{d}=1}^{3}\xi_{\alpha}^{\lambda_{d}}(P)~{}\xi^{*\,\lambda_{d}}_{\beta}(P)~{},$
(15)

where

$\displaystyle\widetilde{\Gamma}_{N}^{\nu}$
$\displaystyle=\gamma^{0}\Gamma_{N}^{\nu\,\dagger}\gamma^{0}=F_{1}(Q^{2})\gamma^{\nu}-\frac{F_{2}(Q^{2})}{2m}i\sigma^{\nu\eta}q_{\eta}~{},$
(16) $\displaystyle\widetilde{\Gamma}^{\beta}_{D}$
$\displaystyle=\gamma^{0}\Gamma^{\beta\,\dagger}_{D}\gamma^{0}=F\gamma^{\beta}+\frac{G}{m}p^{\beta}-\left(H\gamma^{\beta}+\frac{I}{m}p^{\beta}\right)\frac{m-\not{p}_{1}}{m}~{},$
(17)

and $p=\dfrac{1}{2}(p_{2}-p_{1}+q)$. The derivation of Eq. (15) is provided in
Appendix C.

###  II.2 Final state interactions

Figure 3: Final state interactions contribution to the electromagnetic current
matrix element. The fermion line with “$\circ$” represents the particle can be
off-shell and the propagator will be decomposed into positive energy
contribution and negative energy contribution.

In the PWIA, the spectator nucleon does not participate in the reaction.
Although this approximation is widely utilized in many Monte Carlo
simulations, its validity should be examined quantitatively for precise
measurement. We consider the situation that one of the nucleons is struck by
the virtual photon but interacts with the other nucleon before flying out. The
mechanism of such final-state interaction (FSI) is illustrated in Fig. 3. In
this case, the proton leg and the neutron leg of the deuteron-proton-neutron
vertex can be both off-shell and such kind of vertex function is explored in
Ref. [40], in which the amplitude requires a four-dimensional integration over
the internal momentum $k$. In this study, we simplify the calculation by
applying the covariant spectator theory [34, 35, 36, 37], in which the nucleon
not directly struck by the virtual photon is set on-shell and the four-
dimensional integration is replaced by

$\int\frac{{\rm d}^{3}\bm{k}}{(2\pi)^{3}}\frac{m}{E_{\bm{k}}}~{}.$ (18)

This method has been utilized in the study of the elastic electron-deuteron
scattering [38] and the electrodisintegration of the deuteron [33], and the
results are in good agreement with experimental data.

To evaluate the FSI effect, we need to calculate the rescattering between the
knocked-out nucleon and the recoil nucleon. According to the Dirac matrix
structure, the nucleon-nucleon (NN) scattering matrix can be parametrized as
[48, 49, 33]

$\displaystyle M$
$\displaystyle=F_{S}(s,t)\bm{1}_{1}\bm{1}_{2}+F_{V}(s,t)(\gamma^{\mu})_{1}(\gamma_{\mu})_{2}+F_{T}(s,t)(\sigma^{\mu\nu})_{1}(\sigma_{\mu\nu})_{2}$
(19)
$\displaystyle+F_{P}(s,t)(\gamma^{5})_{1}(\gamma^{5})_{2}+F_{A}(s,t)(\gamma^{5}\gamma^{\mu})_{1}(\gamma^{5}\gamma_{\mu})_{2}~{},$

where the subscripts “$1$” and “$2$” of the $\gamma$-matrices indicate the
spinor space of the corresponding nucleon. $F_{S}$, $F_{V}$, $F_{T}$, $F_{P}$,
and $F_{A}$ are scalar functions of $s$ and $t$ which are the Mandelstam
variables for the NN elastic scattering. Considering the spin states of the
nucleons, one can alternatively expand the NN scattering matrix as [50, 48]

$\displaystyle M\left(\bm{k}_{f},\bm{k}_{i}\right)$
$\displaystyle=\frac{1}{2}\left[(a+b)+(a-b)\left(\bm{\sigma}_{1},\bm{n}\right)\left(\bm{\sigma}_{2},\bm{n}\right)+(c+d)\left(\bm{\sigma}_{1},\bm{m}\right)\left(\bm{\sigma}_{2},\bm{m}\right)\right.$
(20)
$\displaystyle+(c-d)\left.\left(\bm{\sigma}_{1},\bm{l}\right)\left(\bm{\sigma}_{2},\bm{l}\right)+e\left(\bm{\sigma}_{1}+\bm{\sigma}_{2},\bm{n}\right)\right]~{},$

where ${\bm{k}}_{i}$ and ${\bm{k}}_{f}$ are momenta of the nucleon before and
after the scattering in the NN center-of-mass frame, in which the basis
vectors can be constructed as

$\bm{l}=\frac{\bm{k}_{f}+\bm{k}_{i}}{\left|\bm{k}_{f}+\bm{k}_{i}\right|},\quad\bm{m}=\frac{\bm{k}_{f}-\bm{k}_{i}}{\left|\bm{k}_{f}-\bm{k}_{i}\right|},\quad\bm{n}=\frac{\bm{k}_{i}\times\bm{k}_{f}}{\left|\bm{k}_{i}\times\bm{k}_{f}\right|}~{}.$
(21)

The $(\bm{\sigma}_{j},\bm{v})$ represents the projection of the spin Pauli
matrix $\bm{\sigma}$ of the $j$th particle on the direction $\bm{v}={\bm{l}}$,
${\bm{m}}$, and ${\bm{n}}$. The amplitudes $a$, $b$, $c$, $d$, and $e$ are
complex functions of the momentum
$|\bm{k}_{i}|=|{\bm{k}}_{f}|=\sqrt{s-4m^{2}}/2$ and the scattering angle
$\theta$, which can be expressed in terms of Mandelstem variables as

$\displaystyle\cos\theta=1+\frac{2t}{s-4m^{2}}~{}.$ (22)

These two parametrizations of the NN scattering matrix can be related through
the helicity amplitudes as demonstrated in Appendix D. The partial wave
expansion for the amplitudes $a$, $b$, $c$, $d$, and $e$ is presented in Ref.
[51]. In numerical calculations, we use the phase shift analysis in Refs. [52,
53] and the phase shift data are available from the SAID program [54].

Though the recoil nucleon is set on-shell in the covariant spectator theory,
the struck nucleon can be off-shell before the rescattering. One may in
principle include additional terms in the NN scattering matrix with one of the
nucleon off-shell [55]. In this study, we follow the approach in Ref. [33] by
introducing a form factor $F_{N}(p^{2})$ to account for the off-shell effect
and keep the NN scattering matrix parametrization the same form as Eq. (19).
The scalar functions $F_{j}(s,t)$ with $j=S$, $V$, $T$, $P$, and $A$ are then
replaced by

$F_{j}(s,t)\rightarrow F_{j}(s,t,u)F_{N}(s+t+u-3m^{2})~{},$ (23)

where

$F_{N}(p^{2})=\dfrac{(\Lambda_{N}^{2}-m^{2})^{2}}{(p^{2}-m^{2})^{2}+(\Lambda_{N}^{2}-m^{2})^{2}}~{},$
(24)

and $p$ represents the four-momentum of the off-shell nucleon. In this case,
the scattering angle $\theta$ is given by

$\displaystyle\cos\theta=\frac{t-u}{\sqrt{1-\frac{4m^{2}}{s}}\sqrt{(4m^{2}-t-u)^{2}-4m^{2}s}}~{}.$
(25)

When $p^{2}=m^{2}$, the form factor $F_{N}(m^{2})=1$ and the scattering angle
$\theta$ reduces back to the regular expression for the on-shell NN
scattering. $\Lambda_{N}$ is a phenomenological parameter, which we choose to
be $1.2\,\rm{GeV}$ as suggested in Ref. [33].

With Eqs. (18) and (19), the FSI contribution to the current matrix element is
given by

$\displaystyle J_{\rm{FSI}}^{\mu}(q)_{\rm{if}}$ $\displaystyle=$
$\displaystyle\left\langle\bm{p}_{1}s_{1};\bm{p}_{2}s_{2}\left|\hat{J}^{\mu}\right|\bm{P}\lambda_{d}\right\rangle_{\rm{FSI}}$
(26) $\displaystyle=$
$\displaystyle\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\frac{m}{E_{k}}\bar{u}_{\alpha}\left(\bm{p}_{1},s_{1}\right)\bar{u}_{\beta}\left(\bm{p}_{2},s_{2}\right)M_{\alpha\alpha^{\prime};\beta\beta^{\prime}}\left(p_{1},p_{2};k\right)$
$\displaystyle\times$ $\displaystyle
G_{0\,\alpha^{\prime}\eta}\left(P+q-k\right)\Gamma_{N\,\eta\eta^{\prime}}^{\mu}(q)G_{0\,\eta^{\prime}\zeta}\left(P-k\right)(\Gamma_{\lambda_{d}}C)_{\zeta\zeta^{\prime}}\left(k,P\right)\Lambda_{\beta^{\prime}\zeta^{\prime}}^{+}\left(\bm{k}\right)~{}.$

Then we decompose the bare nucleon propagator into positive and negative
energy terms,

$\displaystyle G_{0}(p)$ $\displaystyle=$
$\displaystyle-\frac{m}{E_{p}}\sum_{s}\left[\frac{u(\bm{p},s)\bar{u}(\bm{p},s)}{p^{0}-E_{p}+i\epsilon}+\frac{v(-\bm{p},s)\bar{v}(-\bm{p},s)}{p^{0}+E_{p}-i\epsilon}\right]$
(27) $\displaystyle=$
$\displaystyle-\frac{m}{E_{p}}\left[\frac{\Lambda_{+}(\bm{p})}{p^{0}-E_{p}+i\epsilon}-\frac{\Lambda_{-}(-\bm{p})}{p^{0}+E_{p}-i\epsilon}\right],$

where the projection operators $\Lambda_{\pm}$ are defined as

$\displaystyle\Lambda_{+}(\bm{p})$
$\displaystyle=\frac{\not{\bar{p}}+m}{2m}=\sum_{s}u(\bm{p},s)\bar{u}(\bm{p},s)~{},$
(28) $\displaystyle\Lambda_{-}(\bm{p})$
$\displaystyle=-\frac{\not{\bar{p}}-m}{2m}=-\sum_{s}v(\bm{p},s)\bar{v}(\bm{p},s)~{},$
(29)

corresponding to the positive and negative energy solutions respectively, and
$\bar{p}$ denotes the on-shell four-momentum. Furthermore, we use the relation

$\frac{1}{p^{0}-E_{p}+i\epsilon}=-i\pi\delta\left(p^{0}-E_{p}\right)+\frac{\mathcal{P.V.}}{p^{0}-E_{p}}$
(30)

to write the positive energy term into two parts. Hence, Eq. (26) is
decomposed into three parts,

$\displaystyle
J_{\rm{FSI}}^{\mu}(q)_{\rm{if}}=J_{\rm{FSI}}^{\mu}(q)^{\rm{A}}_{\rm{if}}+J_{\rm{FSI}}^{\mu}(q)^{\rm{B}}_{\rm{if}}+J_{\rm{FSI}}^{\mu}(q)^{\rm{C}}_{\rm{if}}~{},$
(31)

where

$\displaystyle J_{\rm{FSI}}^{\mu}(q)^{\rm{A}}_{\rm{if}}$ $\displaystyle=$
$\displaystyle\sum_{s}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}(i\pi)\frac{m}{E_{k}}\frac{m}{E_{P+q-k}}\delta(P^{0}+q^{0}-E_{k}-E_{P+q-k})$
(32) $\displaystyle\times$
$\displaystyle\bar{u}_{\alpha}\left(\bm{p}_{1},s_{1}\right)\bar{u}_{\beta}\left(\bm{p}_{2},s_{2}\right)M_{\alpha\alpha^{\prime};\beta\beta^{\prime}}\left(p_{1},p_{2};k\right)u_{\beta^{\prime}}\left(\bm{k},s\right)$
$\displaystyle\times$
$\displaystyle\Lambda^{+}_{\alpha^{\prime}\eta}\left(\bm{P}+\bm{q}-\bm{k}\right)\Gamma_{N\,\eta\eta^{\prime}}^{\mu}(q)G_{0\,\eta^{\prime}\zeta}\left(P-k\right)(\Gamma_{\lambda_{d}}C)_{\zeta\zeta^{\prime}}\left(k,P\right)\bar{u}^{T}_{\zeta^{\prime}}\left(\bm{k},s\right)~{}$

is the on-shell positive energy contribution,

$\displaystyle J_{\rm{FSI}}^{\mu}(q)^{\rm{B}}_{\rm{if}}$ $\displaystyle=$
$\displaystyle\sum_{s}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}(-1)\frac{m}{E_{k}}\frac{m}{E_{P+q-k}}\frac{\mathcal{P.V.}}{P^{0}+q^{0}-E_{k}-E_{P+q-k}}$
(33) $\displaystyle\times$
$\displaystyle\bar{u}_{\alpha}\left(\bm{p}_{1},s_{1}\right)\bar{u}_{\beta}\left(\bm{p}_{2},s_{2}\right)M_{\alpha\alpha^{\prime};\beta\beta^{\prime}}\left(p_{1},p_{2};k\right)u_{\beta^{\prime}}\left(\bm{k},s\right)$
$\displaystyle\times$
$\displaystyle\Lambda^{+}_{\alpha^{\prime}\eta}\left(\bm{P}+\bm{q}-\bm{k}\right)\Gamma_{N\,\eta\eta^{\prime}}^{\mu}(q)G_{0\,\eta^{\prime}\zeta}\left(P-k\right)(\Gamma_{\lambda_{d}}C)_{\zeta\zeta^{\prime}}\left(k,P\right)\bar{u}^{T}_{\zeta^{\prime}}\left(\bm{k},s\right)~{}$

is the principal value integral (off-shell) positive energy contribution, and

$\displaystyle J_{\rm{FSI}}^{\mu}(q)^{\rm{C}}_{\rm{if}}$ $\displaystyle=$
$\displaystyle\sum_{s}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\frac{m}{E_{k}}\frac{m}{E_{P+q-k}}\frac{1}{P^{0}+q^{0}-E_{k}+E_{P+q-k}-i\epsilon}$
(34) $\displaystyle\times$
$\displaystyle\bar{u}_{\alpha}\left(\bm{p}_{1},s_{1}\right)\bar{u}_{\beta}\left(\bm{p}_{2},s_{2}\right)M_{\alpha\alpha^{\prime};\beta\beta^{\prime}}\left(p_{1},p_{2};k\right)u_{\beta^{\prime}}\left(\bm{k},s\right)$
$\displaystyle\times$
$\displaystyle\Lambda^{-}_{\alpha^{\prime}\eta}\left(\bm{k}-\bm{P}-\bm{q}\right)\Gamma_{N\,\eta\eta^{\prime}}^{\mu}(q)G_{0\,\eta^{\prime}\zeta}\left(P-k\right)(\Gamma_{\lambda_{d}}C)_{\zeta\zeta^{\prime}}\left(k,P\right)\bar{u}^{T}_{\zeta^{\prime}}\left(\bm{k},s\right)~{}$

is the negative energy contribution.
$J_{\rm{FSI}}^{\mu}(q)^{\rm{A}}_{\rm{if}}+J_{\rm{FSI}}^{\mu}(q)^{\rm{B}}_{\rm{if}}$
is shown in Fig. 4a and $J_{\rm{FSI}}^{\mu}(q)^{\rm{C}}_{\rm{if}}$ is shown in
Fig. 4b. Since the negative energy part (anti-nucleon propagation) is much
smaller than the positive energy part (nucleon propagation), we ignore the
contribution from Eq. (34) in the calculation. This is the same approximation
as adopted in Ref. [33].

Figure 4: The decomposition of the FSI amplitude. The fermion line with
“$\circ^{+}$” represents the positive energy contribution, including the on-
shell term and the off-shell principal value term, and the fermion line with
“$\circ^{-}$” represents the negative energy contribution.

##  III Numerical results and discussions

In this section, we present the numerical results for the unpolarized
differential cross section. We first discuss the angular distribution of the
differential cross section in the PWIA and the singular feature of the cross
section at some kinematics for the $e^{+}e^{-}$ production. Then we discuss
the FSI effect on the differential cross section and the impact respectively
from the on-shell and the off-shell contributions. Besides, we consider the
effect arising from possible interference between the proton and the neutron
amplitudes.

###  III.1 Angular distribution

Since the lepton pair production via the BH process is a dominant background
channel for the measurement of $J/\psi$ production near the threshold, we
choose the photon energy at $8.5\,\rm GeV$, which is just above the $J/\psi$
production threshold from free nucleons and is accessible at Jefferson Lab.
With four particles in the final state, the complete differential cross
section is seven-fold after eliminating an azimuthal symmetric angle for the
scattering from unpolarized deuteron target.

In Fig. 5, we plot the differential cross section as a function of the lepton
azimuthal angle $\phi_{l}$ at several values of $\theta_{l}=10^{\circ}$,
$90^{\circ}$, and $170^{\circ}$ to show the polar angle dependence.
$s_{ll}=9.6\,\rm GeV^{2}$ is set as the mass square of $J/\psi$,
$s_{pn}=5\,\rm GeV^{2}$ and $t=-1.5\,\rm GeV^{2}$ are chosen at some typical
values, and $p_{1}=1.4\,\rm GeV$ and $\phi_{1}=180^{\circ}$ are fixed for the
struck proton. As one can observe from the Fig. 5, the cross section is large
when $\theta_{l}$ is close to $0^{\circ}$ or $180^{\circ}$. At such limit, the
emitted lepton or antilepton is collinear to ${\bm{p}}_{34}$, the lepton pair
momentum. Although the cross section does not have singularities in this
kinematics region, the polar angle of ${\bm{p}}_{34}$ is small for the
particular kinematic choice here. Hence, the enhancement around
$\phi_{l}=180^{\circ}$ or $0^{\circ}$ is actually from the singularity when
one of the lepton pair is collinear with the incident photon as will be
discussed in the next subsection. Since the electron is much lighter than the
muon, the solid curves ($e^{+}e^{-}$) show sharper peaks than the dashed
curves ($\mu^{+}\mu^{-}$). When the polar angle $\theta_{l}$ is at
intermediate values, e.g. $90^{\circ}$ as shown in the figure, the propagator
is always away from the pole and consequently leads to a smooth distribution
in $\phi_{l}$. In such regions, the lepton mass effect is marginal and the
cross sections for $\mu^{+}\mu^{-}$ and $e^{+}e^{-}$ productions are nearly
the same.

Figure 5: The differential cross sections of $e^{+}e^{-}$ production (solid
curves) and $\mu^{+}\mu^{-}$ production (dashed curves) as a function of
$\phi_{l}$ at $E_{\gamma}=8.5~{}\rm{GeV}$, $s_{pn}=5~{}\rm{GeV}^{2}$,
$s_{ll}=9.6~{}\rm{GeV}^{2}$, $t=-1.5~{}\rm{GeV}^{2}$, $p_{1}=1.4~{}\rm{GeV}$,
$\phi_{1}=180^{\circ}$, and
$\theta_{l}=10^{\circ},\,90^{\circ},\,170^{\circ}$, respectively. The label
$d^{7}\sigma$ is a short-handed notation representing the seven-fold
differential cross section ${\rm
d}^{7}\sigma/{\rm{d}}\Omega_{p}{\rm{d}}\Omega_{ll}{\rm{d}}s_{ll}{\rm{d}}s_{pn}{\rm{d}}t$.
The same notation is used in following figures. Figure 6: The differential
cross sections of $e^{+}e^{-}$ production (solid curve) and $\mu^{+}\mu^{-}$
production (dashed curve) as a function of $\phi_{1}$ at
$p_{1}=1.4~{}\rm{GeV}$, $\theta_{l}=30^{\circ}$, $\phi_{l}=180^{\circ}$, and
the same $E_{\gamma}$, $s_{pn}$, $s_{ll}$, and $t$ values as those in Fig. 5.

In Fig. 6, we plot the differential cross section as a function of $\phi_{1}$,
the azimuthal angle of the proton. The $s_{ll}$, $s_{pn}$, and $t$ are set at
the same values as Fig. 5, and $\theta_{l}=30^{\circ}$ and
$\phi_{l}=180^{\circ}$ are fixed for the lepton angle. Since the proton
acquires the large momentum from the virtual photon, which has opposite
transverse momentum to the lepton pair, the struck proton tends to be around
$\phi_{1}=180^{\circ}$. In PWIA, the smearing is caused by the fermi motion,
while FSI will further broaden the distribution.

###  III.2 Singular behavior for $e^{+}e^{-}$ production

For massless leptons, the differential cross section has singularities if the
lepton or the antilepton is collinear to the incident photon. This behavior is
convenient to be understood from the BH amplitude. When the lepton is
collinear to the photon, the lepton propagator in Fig. 2a approaches the pole.
Similarly, the propagator in Fig. 2b reaches the pole when the antilepton is
collinear to the photon. Such singularities are physically regularized by the
lepton mass, which is kept nonzero in our calculation. However, a sharp peak
is still observed for the $e^{+}e^{-}$ production because of the tiny mass of
the electron in comparison with typical scales of the reaction presented here.

According to the physical origin of the singular behavior, it is
straightforward to examine the differential cross section as a function of the
lepton polar angle with respect to the incident photon. Whereas, it is not a
commonly used variable in describing the lepton pair production in deuteron
photodisintegration process. One has to track the singular behavior from the
dependence on other kinematic variables.

In Fig. 7, we plot the differential cross section as a function of $s_{ll}$,
with other kinematic variables set at $s_{pn}=4.9\,\rm GeV^{2}$, $t=-1.5\,\rm
GeV^{2}$, $p_{1}=1.3\,\rm GeV$, $\phi_{1}=180^{\circ}$,
$\theta_{l}=15^{\circ}$, and $\phi_{l}=180^{\circ}$. One can observe that the
cross section for $e^{+}e^{-}$ production sharply peaks around
$s_{ll}=9.15\,\rm{GeV}^{2}$, where the electron is collinear to the incident
photon. To clearly see this correspondence, we first examine the polar angle
$\theta_{34}$ of the lepton pair momentum $\bm{p}_{34}$. At fixed $s_{pn}$ and
$t$, $\theta_{34}$ can be expressed as a function of $s_{ll}$. Their relation
at the chosen kinematics is drawn in Fig. 8. Then we consider the direction of
the electron in the lab frame. The lepton-photon collinear configuration
happens when $\phi_{e}=180^{\circ}$ and $\theta_{e}^{\rm Lab}=\theta_{34}$,
where $\theta_{e}^{\rm Lab}$ is the relative angle between the electron
momentum ${\bm{p}}_{3}$ and the lepton pair momentum ${\bm{p}}_{34}$ in the
laboratory frame. From the relation between $\theta_{e}^{\rm Lab}$ and
$s_{ll}$ drawn in Fig. 8, one can find the intersecting point, which indicates
$\theta_{e}^{\rm Lab}=\theta_{34}$. The corresponding $s_{ll}$ value is just
at $9.15\,\rm GeV^{2}$. According to the definition of angles in Fig. 1, the
positron-photon collinear configuration will happen at $\phi_{l}=0^{\circ}$
and can be analyzed via the same procedure. Since muon is much heavier, the
cross section of $\mu^{+}\mu^{-}$ production does not show a sharp peak.

Figure 7: Differential cross sections for $e^{+}e^{-}$ production (solid line)
and $\mu^{+}\mu^{-}$ production (dashed line) as a function of $s_{ll}$ at
$E_{\gamma}=8.5~{}\rm{GeV}$, $s_{pn}=4.9~{}\rm{GeV}^{2}$,
$t=-1.5~{}\rm{GeV}^{2}$, $p_{1}=1.3~{}\rm{GeV}$, $\theta_{l}=15^{\circ}$ and
$\phi_{l}=\phi_{1}=180^{\circ}$. Figure 8: $\theta_{34}$ (solid line) and
$\theta_{e}^{\rm{Lab}}$ (dashed line) vary with $s_{ll}$ at
$E_{\gamma}=8.5~{}\rm{GeV}$, $s_{pn}=4.9~{}\rm{GeV}^{2}$,
$t=-1.5~{}\rm{GeV}^{2}$.

Similarly, one can also observe the sharp peak behavior in the dependence on
other variables. In Fig. 9, we plot the differential cross section as a
function of $s_{pn}$, the invariant mass of the final proton-neutron system,
while the other variables are set at $s_{ll}=9\,\rm{GeV^{2}}$,
$t=-2\,\rm{GeV}^{2}$, $p_{1}=1.6\,\rm{GeV}$, $\theta_{l}=15^{\circ}$,
$\phi_{l}=180^{\circ}$, and $\phi_{1}=180^{\circ}$. The cross section for
$e^{+}e^{-}$ productions peaks around $s_{pn}=5.7\,\rm GeV^{2}$.

Figure 9: Differential cross sections for $e^{+}e^{-}$ production (dashed
line) and $\mu^{+}\mu^{-}$ production (dashed line) as a function of $s_{pn}$
at $E_{\gamma}=8.5~{}\rm{GeV}$, $s_{ll}=9~{}\rm{GeV^{2}}$,
$t=-2~{}\rm{GeV}^{2}$, $p_{1}=1.6~{}\rm{GeV}$, $\theta_{l}=15^{\circ}$ and
$\phi_{l}=\phi_{1}=180^{\circ}$.

We note that the peak position in $s_{ll}$ or $s_{pn}$ shifts to smaller
values with increasing $\theta_{l}$. When $\theta_{l}$ is large enough, the
momenta of the lepton and antilepton are always away direction of the incident
photon, and thus the differential cross section has smooth behavior as a
function of $s_{pn}$ or $s_{ll}$ as shown in Figs. 14 and 15.

###  III.3 Final state interactions effect

In this subsection, we discuss the FSI contribution to the differential cross
section. Because the FSI effects on the $\mu^{+}\mu^{-}$ and $e^{+}e^{-}$
productions are similar, we only present the results for the $e^{+}e^{-}$
production.

In the PWIA, the reaction is approximated as the scattering from a quasi-free
nucleon. As a general feature, the differential cross section decreases with
increasing $|t|$. To explore the deviation from the quasi-free picture, it is
convenient to introduce the Bjorken variable $x$, which goes to $1$ for
elastic scattering from a free nucleon. In Fig. 10, we show the phase space in
the $s_{pn}-t$ plane together with lines indicating corresponding $x$ values.
With fixed $s_{pn}$, the $x$ becomes closer to $1$ as $|t|$ decreases, which
means closer to the quasi-free kinematics.

In Fig. 11, we plot the differential cross section as a function of $t$ with
other variables fixed at $s_{pn}=5.1\,\rm{GeV}^{2}$, $s_{ll}=9\,\rm{GeV}^{2}$,
$p_{1}=1.45\,\rm{GeV}$, $\theta_{e}=60^{\circ}$, $\phi_{e}=90^{\circ}$, and
$\phi_{1}=180^{\circ}$. Instead of showing the cross section in the full $t$
region, we only present the result in $-3.5\,{\rm GeV}<t<-1.5\,{\rm GeV}$,
which is viewed as a reasonable range for the approach dealing with the FSI in
the current calculation. When $|t|$ is very small, e.g. $|t|<1\,\rm GeV^{2}$,
one may need to take into account the contribution from meson exchange current
[56]. For very large $|t|$, e.g. $|t|>5\,\rm GeV^{2}$, the color coherence
phenomena will complicate the situation [56, 57, 58]. The dashed curve
represents the result from PWIA, the solid curve represents the result
including the on-shell FSI contribution, and the dot-dashed curve represents
the result including both the on-shell and the off-shell FSI contributions. As
can be observed in Fig. 11, the full FSI curve and the on-shell FSI curve have
little difference when $|t|$ is relatively small. This can be understood from
Fig. 10, where one may find that the $x$ becomes closer to $1$ as $|t|$
decreases with fixed $s_{pn}$. At this limit, the kinematics is close to the
quasi-free scattering and thus the off-shell FSI has little impact on the
cross section.

Figure 10: The $s_{pn}-t$ phase space region at $E_{\gamma}=8.5~{}\rm{GeV}$
and $s_{ll}=9~{}\rm{GeV}^{2}$. The allowed kinematic range for $t$ is larger
than that plotted in the figure. Dashed lines show the Bjorken $x$ at several
fixed values. Figure 11: Differential cross sections for $e^{+}e^{-}$
production as a function of $t$ at $E_{\gamma}=8.5~{}\rm{GeV}$,
$s_{pn}=5.1~{}\rm{GeV}^{2}$, $s_{ll}=9~{}\rm{GeV}^{2}$,
$p_{1}=1.45~{}\rm{GeV}$, $\theta_{e}=60^{\circ}$, $\phi_{e}=90^{\circ}$, and
$\phi_{1}=180^{\circ}$ in three cases: in the PWIA (gray dashed curve),
including the on-shell FSI (black solid curve), and including both the on-
shell and the off-shell FSIs (red dot-dashed curve).

In Fig. 12, we plot the differential cross section as a function of the
missing momentum $p_{m}$, which is the amplitude of the momentum of the
undetected nucleon. In the PWIA, the proton is knocked out by the virtual
photon and acquires large momentum, while the spectator neutron carries
relatively low momentum. Once the FSI is taken into account, the undetected
neutron may received momentum transfer via the NN scattering. Therefore, one
may expect that the FSI effect is more significant in the region of relatively
large $p_{m}$. This intuitive picture is confirmed by the numerical results in
Fig. 12. The three curves are almost indistinguishable at small $p_{m}$, e.g.
when $p_{m}<0.1\,\rm GeV$. While in the large $p_{m}$ region, e.g.
$p_{m}>0.4\,\rm GeV$, significant enhancement is caused by the FSI effects. It
is also interesting to find that in the medium $p_{m}$ region, $0.1\,{\rm
GeV}<p_{m}<0.3\,\rm GeV$, the cross section results including the FSI are less
than the PWIA result, in which the suppression is from the interference
between the PWIA amplitude and the FSI amplitude. Besides, one may find that
the full FSI curve and the on-shell FSI curve have little difference in Fig.
12. It is due to the particular kinematics, which gives $x$ close to 1, i.e.
the quasi-free region, where the off-shell FSI contribution is considerably
small as discussed above.

Figure 12: Differential cross sections for $e^{+}e^{-}$ production as a
function of the missing momentum $p_{m}$ at $E_{\gamma}=8.5~{}\rm{GeV}$,
$s_{pn}=5.5~{}\rm{GeV}^{2}$, $t=-2~{}\rm{GeV}^{2}$, $s_{ll}=m_{J/\psi}^{2}$,
$\theta_{e}=5^{\circ}$, and $\phi_{e}=\phi_{1}=180^{\circ}$. The gray dashed
curve represents the result with PWIA, the black solid curve represents the
result including the on-shell FSI, and the red dot-dashed curve represents the
result including both the on-shell and the off-shell FSIs. Figure 13:
Differential cross sections for $e^{+}e^{-}$ production as a function of
$\phi_{1}$ at $E_{\gamma}=8.5~{}\rm{GeV}$, $s_{pn}=5.5~{}\rm{GeV}^{2}$,
$t=-2~{}\rm{GeV}^{2}$, $s_{ll}=m_{J/\psi}^{2}$, $p_{1}=1.75~{}\rm{GeV}$,
$\phi_{e}=180^{\circ}$, and $\theta_{e}=45^{\circ}$. The gray dashed curve
represents the result with PWIA, the black solid curve represents the result
including the on-shell FSI, and the red dot-dashed curve represents the result
including both the on-shell and the off-shell FSIs.

In Fig. 13, we plot the differential cross section as a function of $\phi_{1}$
with other variables fixed at $s_{pn}=5.5\,\rm{GeV}^{2}$, $t=-2\,{\rm
GeV}^{2}$, $s_{ll}=m_{J/\psi}^{2}$, $p_{1}=1.75\,\rm{GeV}$,
$\phi_{e}=180^{\circ}$, and $\theta_{e}=45^{\circ}$. The final-state proton
momentum $p_{1}=1.75~{}\rm{GeV}$ corresponds to the missing momentum
$p_{m}=0.152\,\rm{GeV}$, which is in the region where the FSI effects suppress
the cross section. The on-shell FSI curve is slightly above the full FSI curve
in the whole $\phi_{1}$ range.

Figure 14: Differential cross sections for $e^{+}e^{-}$ production as a
function of $s_{pn}$ at $E_{\gamma}=8.5~{}\rm{GeV}$, $t=-1.7~{}\rm{GeV}^{2}$,
$s_{ll}=9~{}\rm{GeV}^{2}$, $p_{m}=0.45~{}\rm{GeV}$, $\phi_{1}=180^{\circ}$,
$\theta_{e}=60^{\circ}$, and $\phi_{e}=90^{\circ}$ in the PWIA (gray dashed
curve), including the interference between the PWIA and the FSI amplitudes
(blue dot-dashed curve), and including both the interference term and the FSI
amplitude square (black solid curve). The $\theta_{m}$ axis is nonlinear.
Figure 15: Differential cross sections for $e^{+}e^{-}$ production as a
function of $s_{ll}$ at $E_{\gamma}=8.5~{}\rm{GeV}$,
$s_{pn}=4.9~{}\rm{GeV}^{2}$, $t=-1.5~{}\rm{GeV}^{2}$, $p_{1}=1.3~{}\rm{GeV}$,
$\theta_{e}=40^{\circ}$, and $\phi_{1}=\phi_{e}=180^{\circ}$. The gray dashed
curve represents the result with PWIA, the black solid curve represents the
result including the on-shell FSI, and the red dot-dashed curve represents the
result including both the on-shell and the off-shell FSIs.

To clearly see the contribution from the PWIA and FSI interference term and
that from the FSI amplitude square term separately, we plot the differential
cross section as a function of $s_{pn}$ with other variables fixed at
$t=-1.7~{}\rm{GeV}^{2}$, $s_{ll}=9~{}\rm{GeV}^{2}$, $\phi_{1}=180^{\circ}$,
$\theta_{e}=60^{\circ}$, and $\phi_{e}=90^{\circ}$ in Fig. 14, where the
dashed curve represent the result with PWIA, the dot-dashed curve includes the
interference term between the PWIA and FSI amplitudes, and the solid curve
includes both the interference term and the FSI amplitude square term. As can
be observed, the interference term generally suppress the cross section. For a
chosen missing momentum, e.g. $p_{m}=0.45~{}\rm{GeV}$, there exists a relation
between the polar angle of the missing momentum with respect to the virtual
photon, $\theta_{m}$, and the $s_{pn}$, as marked on the top and bottom axes
in Fig. 14. At forward angles of the missing momentum, the interference term
dominants FSI effects and the cross section is suppressed in comparison with
the PWIA result even if both the interference term and the FSI amplitude
square term are included. However, when the $\theta_{m}$ is large, which is
more likely to be caused by the NN rescattering effects, the differential
cross section is enhanced by the FSI and reaches a peak around $\theta_{m}\sim
65^{\circ}$ for the particular kinematic choice in Fig. 14. This is consistent
with the typical FSI effects as mentioned in Refs. [42, 56, 30, 33] that the
recoil nucleon receives transverse momentum through the rescattering while the
knocked-out nucleon tends to eject away from the forward direction.

In Fig. 15, we plot the cross section as a function of $s_{ll}$ with other
variables fixed at $s_{pn}=4.9~{}\rm{GeV}^{2}$, $t=-1.5~{}\rm{GeV}^{2}$,
$p_{1}=1.3~{}\rm{GeV}$, $\theta_{e}=40^{\circ}$, and
$\phi_{1}=\phi_{e}=180^{\circ}$. As has been discussed in Sec. III.2, at large
$\theta_{e}$, e.g. $\theta_{e}=40^{\circ}$, the lepton momentum is away from
the photon collinear direction and thus there is not a sharp peak in the
$s_{ll}$ dependence for this particular kinematic choice. The FSI enhances the
differential cross section but does not distort the trend of the curve in this
case.

###  III.4 Interference between the proton and the neutron

Figure 16: The differential cross sections with and without the interference
term between the proton and the neutron amplitudes. The results are drawn at
$E_{\gamma}=8.5~{}\rm{GeV}$, $s_{pn}=5.5~{}\rm{GeV}^{2}$,
$t=-2~{}\rm{GeV}^{2}$, $s_{ll}=m_{J/\psi}^{2}$, $\theta_{e}=5^{\circ}$, and
$\phi_{e}=\phi_{1}=180^{\circ}$.

For the reaction of deuteron disintegration, as shown in Fig. 4, the virtual
photon can either couple to the proton or couple to the neutron. In principle,
one need to add the two possibilities at the amplitude level and then there is
an interference term between the proton amplitude and the neutron amplitude.
However, such contribution is highly suppressed at certain kinematics as
considered in this study.

In the PWIA, the nucleon struck by the virtual photon acquires large momentum,
while the spectator nucleon inherits the fermi motion momentum which is
unlikely to be greater than several hundred MeV. If in the final state a high
momentum nucleon is detected, supposed to be the proton, it is likely the one
coupled to the virtual photon. Once the FSI is taken into account, one may
imagine that the detected high momentum nucleon is not directly coupled to the
virtual photon but receives the large momentum from the NN scattering. In this
case we need to demonstrate whether the interference contribution is still
negligible. Assuming the isospin symmetry, the electromagnetic current
operator matrix element is given by

$J^{\mu}(q)_{\rm{if}}=\frac{1}{\sqrt{2}}\left(J_{p}^{\mu}(q)_{\rm{if}}-J_{n}^{\mu}(q)_{\rm{if}}\right)~{},$
(35)

where the subscript $N=p,n$ indicates the nucleon which absorbs the virtual
photon. Then the interference term for the nuclear tensor is

$W_{\rm{interf}}^{\mu\nu}=-(2m)^{2}\overline{\sum_{\rm{if}}}\frac{1}{2}\left(J_{p}^{\mu}(q)_{\rm{if}}J_{n}^{\nu\,\dagger}(q)_{\rm{if}}+J_{n}^{\mu}(q)_{\rm{if}}J_{p}^{\nu\,\dagger}(q)_{\rm{if}}\right)~{}.$
(36)

In Fig. 16, we compare the differential cross section with and without the
interference term between the proton amplitude and the neutron amplitude in
the range of the missing momentum from $0$ to $0.8\,\rm GeV$. The gray dashed
and blue dotted curves represent the results with and without the interference
term in the PWIA, respectively. The black solid and the red dashed curves
represent the results with and without the interference term including FSI
effects, respectively. No visible difference is observed in either case.

##  IV Summary and conclusions

In this paper, we study the lepton pair production in the photodisintegration
of the deuteron. We perform a calculation of the complete seven-fold
differential cross section with the lepton pair produced through the Bethe-
Heitler mechanism. The deuteron bound state is described by relativistic
covariant deuteron-nucleon vertex, in which the form factors are given by
linear combinations of deuteron wave functions that are the solution of the
Gross equation [37]. Apart from the calculation with PWIA, we take into
account the FSI, which arises from the rescattering between the knock-out
nucleon and the spectator nucleon. The NN scattering amplitude is
parameterized in a general Lorentz covariant form including all spin-dependent
contributions, and the parametrization functions are derived from the partial
wave expansion of the Saclay formalism.

We also carry out numerical calculations. The differential cross section can
vary by orders of magnitude in dependence on the lepton azimuthal angle
$\phi_{l}$ when the polar angle $\theta_{l}$ is approaching $0^{\circ}$ or
$180^{\circ}$, while for the middle polar angle the azimuthal angle dependence
is mild. In the $s_{ll}$ and $s_{pn}$ dependence, sharp peaks are also
observed. All these nearly singular behaviors arise from the collinear
singularity for massless leptons when the lepton or the antilepton is moving
along the incidence photon direction. At the certain kinematics investigated
in this study, the produced lepton pair is moving in forward angle, and thus
the sharp peak does not appear when the lepton polar angle is near vertical
angle. This singularity is physically regularized by keeping the lepton mass
in our calculation and numerically we observe that the differential cross
section for $\mu^{+}\mu^{-}$ production is much smoother than that for
$e^{+}e^{-}$ due to the heavier mass.

The FSI effects have strong dependence on the missing momentum $p_{m}$,
because the recoil nucleon can receive energy-momentum transfer in the
rescattering. When the missing momentum is relatively small, the FSI
contribution is negligible and the PWIA provides a good approximation to the
full result. While the missing momentum is large, the FSI leads to significant
enhancement of the differential cross section. In the intermediate missing
momentum region, we observe a suppression of the cross section due to the
interference between the PWIA amplitude and the FSI amplitude. Separating the
FSI contribution into the on-shell part and the off-shell part, we find that
the on-shell term dominates at relatively small $|t|$ region in which the
Bjorken $x$ goes close to 1, the quasi-elastic limit. When $x$ is away from
$1$, corresponding to large $|t|$ region, the off-shell term dominates the FSI
effect. Furthermore, we investigate the contribution from the interference
between the proton amplitude and the neutron amplitude. With explicit
numerical calculation, we demonstrate that this contribution is negligible in
the kinematics region considered in this study, even if the FSI effects are
taken into account.

The dilepton production in deuteron photodisintegration process can not only
provide us valuable information to understand the deuteron structure but also
serve as an important input in the analysis of $J/\psi$ production, deeply
virtual meson production, and other exclusive measurements using deuteron
target or beam. Therefore, it is a very relevant process for the physics
program at JLab and future electron-ion colliders.

###### Acknowledgements.

We are grateful to Ron L. Workman and Igor I. Strakovsky for sending us the
SAID manual that contains information on the nucleon-nucleon amplitudes. We
thank Haiyan Gao, Gregory Matousek, Richard Tyson, and Zhiwen Zhao for helpful
discussions. This work was supported by the National Natural Science
Foundation of China under contract No. 12075003 and No. 12175117.

##  Appendix A Differential cross section

The four body phase space is

$\int\mathrm{d}\Pi=\int(2\pi)^{4}\delta^{(4)}(k+P-p_{1}-p_{2}-p_{3}-p_{4})\frac{\mathrm{d}^{3}\bm{p}_{1}}{(2\pi)^{3}2E_{1}}\frac{\mathrm{d}^{3}\bm{p}_{2}}{(2\pi)^{3}2E_{2}}\frac{\mathrm{d}^{3}\bm{p}_{3}}{(2\pi)^{3}2E_{3}}\frac{\mathrm{d}^{3}\bm{p}_{4}}{(2\pi)^{3}2E_{4}}~{}.$
(37)

To separate the phase space for the lepton pair part and the nuclear part, the
following delta function integration is inserted,

$\displaystyle\int\frac{\mathrm{d}^{4}p_{34}}{(2\pi)^{4}}\,(2\pi)^{4}\,\delta^{(4)}(p_{34}-p_{3}-p_{4})\int\frac{\mathrm{d}s_{ll}}{2\pi}\,2\pi\,\delta(s_{ll}-p_{34}^{2})$
$\displaystyle=\int\frac{\mathrm{d}s_{ll}}{2\pi}\int\frac{\mathrm{d}^{3}\bm{p}_{34}}{(2\pi)^{3}2E_{34}}\,(2\pi)^{4}\,\delta^{(4)}(p_{34}-p_{3}-p_{4})~{}.$
(38)

Where $p^{0}_{34}=E_{34}=\sqrt{|\bm{p}_{34}|^{2}+s_{ll}}$. Then, the phase
space is

$\displaystyle\int\mathrm{d}\Pi$ $\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}s_{ll}}{2\pi}(2\pi)^{4}\delta^{(4)}(k+P-p_{1}-p_{2}-p_{34})\frac{\mathrm{d}^{3}\bm{p}_{1}}{(2\pi)^{3}2E_{1}}\frac{\mathrm{d}^{3}\bm{p}_{2}}{(2\pi)^{3}2E_{2}}\frac{\mathrm{d}^{3}\bm{p}_{34}}{(2\pi)^{3}2E_{34}}$
(39) $\displaystyle\times$
$\displaystyle(2\pi)^{4}\,\delta^{(4)}(p_{34}-p_{3}-p_{4})\frac{\mathrm{d}^{3}\bm{p}_{3}}{(2\pi)^{3}2E_{3}}\frac{\mathrm{d}^{3}\bm{p}_{4}}{(2\pi)^{3}2E_{4}}~{}.$

The phase space for lepton pair is calculated in the lepton pair center of
mass frame,

$\displaystyle\mathrm{d}\Pi_{ll}$ $\displaystyle=$
$\displaystyle\int(2\pi)^{4}\,\delta^{(4)}(p_{34}-p_{3}-p_{4})\frac{\mathrm{d}^{3}\bm{p}_{3}}{(2\pi)^{3}2E_{3}}\frac{\mathrm{d}^{3}\bm{p}_{4}}{(2\pi)^{3}2E_{4}}$
(40) $\displaystyle=$
$\displaystyle\int\frac{|\bm{p}_{c}|}{\sqrt{s_{ll}}}\frac{\mathrm{d}\Omega_{ll}}{(4\pi)^{2}}~{},$

in which $\bm{p}_{c}$ is the momenta of the lepton in the center of mass
frame. The phase space for the proton and the neutron is calculated in the
laboratory system, in which $z$ axis is along $\bm{q}$ direction,

$\displaystyle\mathrm{d}\Pi_{pn}$ $\displaystyle=$
$\displaystyle\int(2\pi)^{4}\delta^{(4)}(k+P-p_{1}-p_{2}-p_{34})\frac{\mathrm{d}^{3}\bm{p}_{1}}{(2\pi)^{3}2E_{1}}\frac{\mathrm{d}^{3}\bm{p}_{2}}{(2\pi)^{3}2E_{2}}\frac{\mathrm{d}^{3}\bm{p}_{34}}{(2\pi)^{3}2E_{34}}$
(41) $\displaystyle=$
$\displaystyle\int\frac{1}{(2\pi)^{5}}\frac{|\bm{p}_{1}|^{2}}{8E_{34}}\frac{\mathrm{d}^{3}|\bm{p}_{34}|\mathrm{d}\Omega_{1}}{||\bm{p}_{1}|(m_{d}+\nu)-E_{1}|\bm{q}|\cos\theta_{1}|}~{}.$

where $\bm{p}_{1}$ is determined by the equation

$m_{d}+\nu=\sqrt{m^{2}+|\bm{p}_{1}|^{2}}+\sqrt{m^{2}+|\bm{p}_{1}|^{2}+|\bm{q}|^{2}-2|\bm{q}||\bm{p}_{1}|\cos\theta_{1}}~{}.$
(42)

Using the following relations,

$\displaystyle s_{pn}$ $\displaystyle=$ $\displaystyle
t+2m_{d}\nu+m_{d}^{2}~{},$ $\displaystyle|\bm{p}_{34}|$ $\displaystyle=$
$\displaystyle\sqrt{(E_{\gamma}-\nu)^{2}-s_{ll}}~{},$
$\displaystyle\cos\theta_{34}$ $\displaystyle=$
$\displaystyle\frac{E_{\gamma}^{2}+(E_{\gamma}-\nu)^{2}-s_{ll}-(\nu^{2}-t)}{2E_{\gamma}\sqrt{(E_{\gamma}-\nu)^{2}-s_{ll}}}~{},$
(43)

one can make variables transformation from $s_{ll}$, $|\bm{p}_{34}|$, and
$\theta_{34}$ to $s_{ll}$, $s_{pn}$, and $t$. With Eqs. (3) and (4), the
differential cross section is expressed by

$\displaystyle{\rm
d}\sigma=\frac{\alpha^{3}}{8(4\pi)^{4}}\frac{|\bm{p}_{1}|^{2}\beta}{m_{d}^{2}E_{\gamma}^{2}t^{2}}\frac{L^{\mu\nu}W_{\mu\nu}}{||\bm{p}_{1}|(m_{d}+\nu)-E_{1}|\bm{q}|\cos\theta_{1}|}{\rm{d}}\Omega_{p}{\rm{d}}\Omega_{ll}{\rm{d}}s_{ll}{\rm{d}}s_{pn}{\rm{d}}t~{}.$
(44)

##  Appendix B Relations between scalar invariant functions and deuteron wave
functions

Realtions between four scalar invariant functions and deuteron wave functions
in the deuteron rest frame are given by

$\displaystyle F(\bm{p})$ $\displaystyle=$
$\displaystyle\pi\sqrt{2m_{d}}\left(2E_{p}-m_{d}\right)\left[u(\bm{p})-\frac{1}{\sqrt{2}}w(\bm{p})+\sqrt{\frac{3}{2}}\frac{m}{|\bm{p}|}v_{t}(\bm{p})\right]~{},$
(45) $\displaystyle G(\bm{p})$ $\displaystyle=$ $\displaystyle\pi
m\sqrt{2m_{d}}\left(2E_{p}-m_{d}\right)\left[\frac{u(\bm{p})}{E_{p}+m}+\frac{\left(2E_{p}+m\right)}{\sqrt{2}|\bm{p}|^{2}}w(\bm{p})+\sqrt{\frac{3}{2}}\frac{1}{|\bm{p}|}v_{t}(\bm{p})\right]~{},$
(46) $\displaystyle H(\bm{p})$ $\displaystyle=$
$\displaystyle\pi\sqrt{3m_{d}}\frac{E_{p}m}{|\bm{p}|}v_{t}(\bm{p})~{},$ (47)
$\displaystyle I(\bm{p})$ $\displaystyle=$
$\displaystyle-\pi\frac{\sqrt{2}m^{2}}{\sqrt{m_{d}}}\left[\left(2E_{p}-m_{d}\right)\left(\frac{u(\bm{p})}{E_{p}+m}-\frac{E_{p}+2m}{\sqrt{2}|\bm{p}|^{2}}w(\bm{p})\right)+\sqrt{3}\frac{m_{d}}{|\bm{p}|}v_{s}(\bm{p})\right]~{},$
(48)

where $|\bm{p}|=\sqrt{\dfrac{(P\cdot k_{2})^{2}}{P^{2}}-k_{2}^{2}}$, $P$ is
the four-momentum of the deuteron and $k_{2}$ is the four-momentum of the on-
shell nucleon. The derivation details can be found in Refs. [35, 40]. The
normalization of wave functions is

$\int_{0}^{\infty}\left[u^{2}(\bm{p})+w^{2}(\bm{p})+v_{t}^{2}(\bm{p})+v_{s}^{2}(\bm{p})\right]|\bm{p}|^{2}{\rm
d}|\bm{p}|=1~{}.$ (49)

In our calculation, numerical solutions for the deuteron wave functions
$u(\bm{p})$, $w(\bm{p})$, $v_{t}(\bm{p})$ and $v_{s}(\bm{p})$ are from Ref.
[37].

##  Appendix C Plane wave contribution to the nuclear tensor

we show construction details for the Dirac trace in Eq. (15) in this appendix.
With Eq. (10), $W^{\mu\nu}$ is written in the form

$\displaystyle W^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{(2m)^{2}}{3}\sum_{\lambda_{d},s_{1},s_{2}}\left\langle\bm{p}_{1}s_{1};\bm{p}_{2}s_{2}\left|\hat{J}^{\mu}\right|\bm{P}\lambda_{d}\right\rangle_{\rm{PW}}\left\langle\bm{P}\lambda_{d}\left|\hat{J}^{\nu\,\dagger}\right|\bm{p}_{1}s_{1};\bm{p}_{2}s_{2}\right\rangle_{\rm{PW}}$
(50) $\displaystyle=$
$\displaystyle\frac{(2m)^{2}}{3}\sum_{\lambda_{d},s_{1},s_{2}}\bar{u}\left(\bm{p}_{1},s_{1}\right)\Gamma_{N}^{\mu}(q)\frac{\not{P}-\not{p}_{2}+m}{m^{2}-(P-p_{2})^{2}}\Gamma_{\lambda_{d}}\left(p_{2},P\right)C\gamma^{0}u^{*}\left(\bm{p}_{2},s_{2}\right)$
$\displaystyle\times$
$\displaystyle\bar{u}^{*}\left(\bm{p}_{2},s_{2}\right)C^{\dagger}\gamma^{0}\tilde{\Gamma}_{\lambda_{d}}\frac{\not{P}-\not{p}_{2}+m}{m^{2}-(P-p_{2})^{2}}\tilde{\Gamma}_{N}^{\nu}u\left(\bm{p}_{1},s_{1}\right)~{}.$

Note that $\gamma^{0}\gamma^{\mu\,\dagger}\gamma^{0}=\gamma^{\mu}$,
$\gamma^{0}\gamma^{\mu\,*}\gamma^{0}=\gamma^{\mu\,T}$,
$\gamma^{0}C^{\dagger}\gamma^{0}=-C^{-1}$, and
$C\gamma^{\mu}C^{-1}=-\gamma^{\mu\,T}$,

$\displaystyle\sum_{s_{2}}C\gamma^{0}u^{*}\left(\bm{p}_{2},s_{2}\right)\bar{u}^{*}\left(\bm{p}_{2},s_{2}\right)C^{\dagger}\gamma^{0}=\frac{\not{p}_{2}-m}{2m}~{}.$
(51)

It then follows that

$\displaystyle W^{\mu\nu}$ $\displaystyle=$
$\displaystyle{\rm{Tr}}\left[\Gamma_{N}^{\mu}(q)\frac{\not{P}-\not{p}_{2}+m}{m^{2}-(P-p_{2})^{2}}\Gamma_{D}^{\alpha}\left(p_{2},P\right)\left(\not{p}_{2}-m\right)\widetilde{\Gamma}_{D}^{\beta}\frac{\not{P}-\not{p}_{2}+m}{m^{2}-(P-p_{2})^{2}}\widetilde{\Gamma}_{N}^{\nu}\left(\not{p}_{1}+m\right)\right]$
(52) $\displaystyle\times$
$\displaystyle\frac{1}{3}\sum_{\lambda_{d}=1}^{3}\xi_{\alpha}^{\lambda_{d}}(P)~{}\xi^{*\,\lambda_{d}}_{\beta}(P)~{}.$

##  Appendix D Parametrization for NN elastic scattering matrix

Two ways of parameterization in Eqs. (19) and (20) for the nucleon nucleon
scattering matrix can be related by the helicity amplitudes. Helicity
amplitudes are defined by scattering amplitudes in which initial states and
final states have the specific helicity $\lambda_{i}$ and
$\lambda^{\prime}_{i}$,

$\mathcal{M}_{\lambda_{1}^{\prime}\lambda_{2}^{\prime};\lambda_{1}\lambda_{2}}=\bar{u}_{\alpha^{\prime}}\left(\bm{p}_{1}^{\prime},\lambda_{1}^{\prime}\right)\bar{u}_{\beta^{\prime}}\left(\bm{p}_{2}^{\prime},\lambda_{2}^{\prime}\right)M_{\alpha^{\prime}\alpha,\beta^{\prime}\beta}\,u_{\alpha}\left(\bm{p}_{1},\lambda_{1}\right)u_{\beta}\left(\bm{p}_{2},\lambda_{2}\right)~{}.$
(53)

where $\alpha$, $\alpha^{\prime}$ and $\beta$, $\beta^{\prime}$ are the Dirac
index for the first and the second particle, respectively.
$u\left(\bm{p}_{i},\lambda_{i}\right)$ is the Dirac spinor with momentum
$\bm{p}_{i}$ and helicity $\lambda_{i}$ and is defined by

$u(\bm{p}_{i},\lambda)=N\begin{pmatrix}1\\\
2\lambda\tilde{p}_{i}\end{pmatrix}|\lambda\rangle_{i}~{},$ (54)

where $\tilde{p}=\dfrac{|\bm{p}|}{E_{p}+m}$, $N=\sqrt{\dfrac{E_{p}+m}{2m}}$
and $E_{p}=\sqrt{m^{2}+\bm{p}^{2}}$. $|\lambda\rangle_{i}$ in the center of
mass frame is listed in Table 1. We use the convention in Refs. [59, 36, 37].

Table 1: $|\lambda\rangle_{i}$ in Eq. (54)  | Initial state |  |  | Final state |
---|---|---|---|---|---
| $\lambda=\frac{1}{2}$ | $\lambda=-\frac{1}{2}$ |  | $\lambda=\frac{1}{2}$ | $\lambda=-\frac{1}{2}$
$|\lambda\rangle_{1}$ | $\begin{pmatrix}1\\\ 0\end{pmatrix}$ | $\begin{pmatrix}0\\\ 1\end{pmatrix}$ | $|\lambda\rangle_{1}$ | $\begin{pmatrix}\cos\frac{1}{2}\theta\\\ \sin\frac{1}{2}\theta\end{pmatrix}$ | $\begin{pmatrix}-\sin\frac{1}{2}\theta\\\ \cos\frac{1}{2}\theta\end{pmatrix}$
$|\lambda\rangle_{2}$ | $\begin{pmatrix}0\\\ 1\end{pmatrix}$ | $\begin{pmatrix}1\\\ 0\end{pmatrix}$ | $|\lambda\rangle_{2}$ | $\begin{pmatrix}-\sin\frac{1}{2}\theta\\\ \cos\frac{1}{2}\theta\end{pmatrix}$ | $\begin{pmatrix}\cos\frac{1}{2}\theta\\\ \sin\frac{1}{2}\theta\end{pmatrix}$

Considering parity conservation, time reversal invariance and the Pauli
principle, there are five independent helicity amplitudes [50],

$\displaystyle M_{1}$ $\displaystyle\equiv$
$\displaystyle\mathcal{M}_{\frac{1}{2}\frac{1}{2};\frac{1}{2}\frac{1}{2}}~{},$
$\displaystyle M_{2}$ $\displaystyle\equiv$
$\displaystyle\mathcal{M}_{\frac{1}{2}\frac{1}{2};-\frac{1}{2}-\frac{1}{2}}~{},$
$\displaystyle M_{3}$ $\displaystyle\equiv$
$\displaystyle\mathcal{M}_{\frac{1}{2}-\frac{1}{2};\frac{1}{2}-\frac{1}{2}}~{},$
$\displaystyle M_{4}$ $\displaystyle\equiv$
$\displaystyle\mathcal{M}_{\frac{1}{2}-\frac{1}{2};-\frac{1}{2}\frac{1}{2}}~{},$
$\displaystyle M_{5}$ $\displaystyle\equiv$
$\displaystyle\mathcal{M}_{\frac{1}{2}\frac{1}{2};\frac{1}{2}-\frac{1}{2}}~{}.$
(55)

Substituting Dirac spinors with specific helicities from Eq. (54) and the
parameterization Eq. (19) of $M_{\alpha^{\prime}\alpha,\beta^{\prime}\beta}$
into Eq. (55) to calculate independent helicity amplitudes, we obtain the
relations between $M_{1}$, $M_{2}$, $M_{3}$, $M_{4}$, $M_{5}$ and
$F_{S}(s,t)$, $F_{V}(s,t)$, $F_{T}(s,t)$, $F_{P}(s,t)$, $F_{A}(s,t)$,

$\displaystyle M_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2m^{2}}\left(m^{2}(\cos\theta+1)F_{S}+(m^{2}+m^{2}\cos\theta+4\bm{p}^{2})F_{V}\right.$
$\displaystyle+$
$\displaystyle\left.2m^{2}(\cos\theta-3)F_{T}+(-m^{2}\cos\theta+3m^{2}+4\bm{p}^{2})F_{A}\right)~{},$
$\displaystyle M_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2m^{2}}\left((\cos\theta-1)E_{p}^{2}F_{S}+m^{2}(\cos\theta-1)F_{V}\right.$
$\displaystyle+$
$\displaystyle\left.2(\cos\theta+3)(m^{2}+2\bm{p}^{2})F_{T}+\bm{p}^{2}(\cos\theta-1)F_{P}-m^{2}(\cos\theta+3)F_{A}\right)~{},$
$\displaystyle M_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{m^{2}}\left(m^{2}\cos^{2}\frac{\theta}{2}F_{S}+(m^{2}+2\bm{p}^{2})\cos^{2}\frac{\theta}{2}F_{V}\right.$
$\displaystyle+$
$\displaystyle\left.2m^{2}\cos^{2}\frac{\theta}{2}F_{T}-(m^{2}+2\bm{p}^{2})\cos^{2}\frac{\theta}{2}F_{A}\right)~{},$
$\displaystyle M_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{m^{2}}\left(\sin^{2}\frac{\theta}{2}E_{p}^{2}F_{S}+m^{2}\sin^{2}\frac{\theta}{2}F_{V}\right.$
$\displaystyle+$
$\displaystyle\left.2m^{2}\sin^{2}\frac{\theta}{2}F_{T}-\bm{p}^{2}\sin^{2}\frac{\theta}{2}F_{P}-m^{2}\sin^{2}\frac{\theta}{2}F_{A}\right)~{},$
$\displaystyle M_{5}$ $\displaystyle=$
$\displaystyle\frac{E_{p}}{2m}\left(-\sin\theta F_{S}-\sin\theta
F_{V}-2\sin\theta F_{T}+\sin\theta F_{A}\right)~{}.$ (56)

Relations between $a$, $b$, $c$, $d$, $e$ and helicity amplitudes $M_{i}^{\rm
SAID}$ are given in Ref. [50] and listed here for convenience,

$\displaystyle M^{\rm SAID}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(a\cos\theta+b-c+d+ie\sin\theta)~{},$ $\displaystyle
M^{\rm SAID}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(a\cos\theta-b+c+d+ie\sin\theta)~{},$ $\displaystyle
M^{\rm SAID}_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(a\cos\theta+b+c-d+ie\sin\theta)~{},$ $\displaystyle
M^{\rm SAID}_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-a\cos\theta+b+c+d-ie\sin\theta)~{},$ $\displaystyle
M^{\rm SAID}_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-a\sin\theta+ie\cos\theta)~{}.$ (57)

The partial wave expansion for amplitudes $a$, $b$, $c$, $d$ and $e$ is
discussed in Ref. [51]. Using the phase shift analysis of Refs. [52, 53] and
phase shift data available from SAID program [54], we obtain the value of
amplitudes at the specific momentum $|\bm{p}|$ and scattering angle $\theta$.
The unpolarized differential cross section in our convention is given by

$\frac{{\rm d}\sigma}{{\rm
d}\Omega}=\frac{1}{4}\sum_{\lambda_{1}\lambda_{1}^{\prime}\lambda_{2}\lambda_{2}^{\prime}}|\mathcal{M}_{\lambda_{1}^{\prime}\lambda_{2}^{\prime};\lambda_{1}\lambda_{2}}|^{2}\frac{(2m)^{4}}{(2\pi)^{2}}\frac{1}{16s}~{}.$
(58)

In SAID program’s convention,

$\frac{{\rm d}\sigma}{{\rm
d}\Omega}=\frac{1}{4}\sum_{\lambda_{1}\lambda_{1}^{\prime}\lambda_{2}\lambda_{2}^{\prime}}|\mathcal{M}^{\rm
SAID}_{\lambda_{1}^{\prime}\lambda_{2}^{\prime};\lambda_{1}\lambda_{2}}|^{2}~{}.$
(59)

An additional factor is needed,

$M_{i}=\frac{2\pi\sqrt{s}}{m^{2}}M^{\rm SAID}_{i}~{}.$ (60)

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