# Reduced nuclear helicity amplitudes for deuteron electrodisintegration and
other processes

J. Flores  S. S. Chabysheva  Department of Physics, University of Idaho,
Moscow ID 83844 USA  J. R. Hiller  Department of Physics, University of Idaho,
Moscow ID 83844 USA  Department of Physics and Astronomy, University of
Minnesota-Duluth, Duluth, Minnesota 55812 USA

###### Abstract

We extend the original idea of reduced nuclear amplitudes to capture
individual helicity amplitudes and discuss various applications to exclusive
processes involving the deuteron. Specifically, we consider deuteron form
factors, structure functions, tensor polarization observables,
photodisintegration, and electrodisintegration. The basic premise is that
nuclear processes at high momentum transfer can be approximated by tree graphs
for point-like nucleons supplemented by empirical form factors for each
nucleon. The latter represent the internal structure of the nucleon, and
incorporate nonperturbative physics, which can allow for early onset of
scaling behavior. The nucleon form factors are evaluated at the net momentum
transfer experienced by the given nucleon, with use of $G_{E}$ for a no-flip
contribution and $G_{M}$ for a helicity-flip contribution. Results are
compared with data where available. The deuteron photodisintegration asymmetry
$\Sigma$ is obtained with a value of $\Sigma(90^{\circ})\simeq-0.06$, which is
much closer to experiment than the value of -1 originally expected. The method
also provides an estimate of the momentum transfer values required for scaling
onset. We find that the deuteron structure function $B$ is a good place to
look, above momentum transfers of 10 GeV2.

##  I Introduction

With the advent of the upgraded electron accelerator at the Thomas Jefferson
National Accelerator Facility, scattering experiments with polarized beams and
targets at high energy and high momentum transfer become possible. In the
regime of high momentum transfer to all relevant nucleons, quantum
chromodynamics (QCD) implies that the internal structure of every nucleon is
important. Until ab initio QCD (lattice) calculations for nuclear scattering
processes are available for more than very simple processes, one is led to
consider models that can represent the basic physics.

One such approach is the reduced nuclear amplitude (RNA) analysis pioneered by
Brodsky and Chertok [1]. In addition to their application to a generic
deuteron form factor, the approach has been applied to deuteron disintegration
[2], pion photoproduction [3], and photodisintegration of 3He [4]. As
originally developed, a nuclear process was modeled as a tree-level amplitude
multiplied by a generic form factor for each nucleon, with each form factor
evaluated at the net momentum transferred to that nucleon. In order to model
the behavior of polarization observables [5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15], we extend this approach to a reduced nuclear helicity amplitude (RNHA)
method to combine a tree-level helicity amplitude for point-like nucleons with
the appropriate form factor for each nucleon. When the nucleon does (not) flip
its helicity, we use the electric (magnetic) form factor $G_{EN}$ ($G_{MN}$).
As a check on the procedure, virtual photon absorption by a single nucleon in
the RNHA approach is consistent with the definitions of $G_{EN}$ and $G_{MN}$.

A caveat in applications of the RNA approach is that the normalization is not
determined by the model and is fixed to data at infinite momentum transfer by
the coefficient of the leading power-law behavior. This means that the
normalization cannot be determined in practice; fitting to a data point at
some intermediate kinematics will give the wrong normalization and the wrong
magnitude at higher momentum transfer. Instead, ratios need to be considered,
so that the normalization becomes irrelevant.

The primary criterion for the asymptotic region is in the momentum transfer to
each nucleon. For every nucleon in the process, the momentum transfer must be
above some common threshold, which is at least 1 GeV2. For example, for
deuteron photodisintegration, the momentum transfer to a nucleon is
$-t_{N}=-(p_{N}-p/2)^{2}$, where $p_{N}$ is the final four-momentum of the
nucleon and $p$ is the initial deuteron momentum. When expressed in terms of
the photon energy $E_{\gamma}$ and the final nucleon angle $\theta$, the
constraint to be above 1 GeV2 becomes [16]

$m_{N}E_{\gamma}\left[1-\sqrt{\frac{E_{\gamma}}{m_{N}+E_{\gamma}}}|\cos\theta|\right]\geq
1\,\mbox{GeV}^{2}.$ (1)

This relationship is illustrated in Fig. 1. Notice that away from 90∘, the
lower limit is quite high.

Figure 1:  Angular dependence of the scale for large momentum transfer in
deuteron photodisintegration.

For electrodisintegration, only the most recent data [17, 18] begins to reach
this threshold.

Here we will focus on deuteron processes, including photodisintegration and
electrodisintegration. For recent reviews of deuteron studies at high momentum
transfer, see [19, 20, 16]. Elastic electron scattering data at high momentum
transfer is presented in [21, 22, 23, 24, 25]. Recent photodisintegration data
can be found in [26, 27, 28, 29], and for electrodisintegration data, in [30,
31, 32, 33, 17, 18]. Other analyses of deuteron processes include hidden-color
contributions to deuteron form factors [34], the hard rescattering mechanism
[35], quark-gluon strings [36], the Moscow NN potential [37], and AdS/QCD
models [38, 39].

One recent experiment [17] used the 10.6 GeV electron beam at JLab and the
Hall C spectrometers to measure electron scattering from a liquid deuterium
target. The final electron and the proton were detected, with the kinematics
restricted to the exclusive process $ed\rightarrow e^{\prime}pn$. One
spectrometer measured the final electron at a nominal 12.2∘ degrees from the
beam direction, with a momentum of 8.5-9.1 GeV such that the recorded events
had a distribution of momentum transfer squared reaching 5 GeV2. The events
studied were taken from a bin of 4.5$\pm$0.5 GeV2 in the tail of the
distribution; however, the nominal transfer was 4.2 GeV2, because the majority
of the events were in the lower half of the bin.

A second spectrometer measured the proton momentum at a range of angles to the
beam direction, tuned to select events where the (missing) neutron had an
angle relative to the direction of the momentum transfer that fell within a
chosen bin. In the one-photon exchange approximation, which we assume, the
momentum transferred is, of course, the photon momentum. The published neutron
angles are binned at 35∘, 45∘ and 75∘, with the first two selected to minimize
final-state interactions. For our purposes, the importance of these two angles
is that the momentum transferred to the neutron reaches 1 GeV in a zero-
binding approximation, so that, rather than focus on the internal structure of
the deuteron, we can consider the response to a large momentum transfer to all
the nucleons involved and we can see that experiments may be approaching the
threshold where our model can be applied.

The RNHA model is constructed in detail in Sec. II for two-nucleon processes.
In the remainder of the paper, we consider various processes for the deuteron.
In Sec. III, the form factors,111For discussion specifically in terms of
perturbative QCD, see [34]. structure functions, and tensor polarization
observables of elastic electron scattering from the deuteron are obtained.
Photodisintegration and electrodisintegration are analyzed in Secs. IV and V.
Within the zero-binding approximation, elastic scattering and
photodisintegration live at edges of the kinematic range of
electrodisintegration and are essentially special cases that provide
introductory examples. Section VI contains a summary of the results and
suggestions for additional applications. Many details of the
electrodisintegration helicity amplitudes are left to an appendix.

##  II Construction of the model

The basic process for a two-nucleon system to absorb a photon and exchange
momentum between the nucleons is illustrated in Fig. 2. These diagrams are
modeled on the primitive process of $\gamma^{*}ff\rightarrow ff$, with $f$
representing a point-like nucleon.222In [2], the primitive process was
$\gamma^{*}q\bar{q}\rightarrow q\bar{q}$, with $q$ corresponding to a point-
like proton and $\bar{q}$ to a point-like neutron, and direct interaction of
the photon with the neutron was neglected. Here we amend and extend this, to
retain information about helicity states of the fermions and include photon
absorption by the neutron. The structure of each nucleon is then introduced by
combining the Feynman amplitude for each diagram with the appropriate form
factor for each nucleon, evaluated at the net momentum transfer for that
nucleon. For a deuteron process in the zero-binding limit, the initial
nucleons share the initial deuteron momentum $p$ equally, so that
$p_{p}=p_{n}=p/2$. We also neglect the nucleon mass difference, setting
$m_{p}=m_{n}\equiv m$. The distinction between different photon-absorption
processes is then in the nature of the photon, being either real or virtual,
and in the outcome for the final nucleons, bound as a deuteron or not.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 2:  Tree graphs for deuteron processes that absorb a photon of momentum
$q$ and helicity $\lambda_{\gamma}$. The initial (final) nucleon momentum and
helicity are $p_{N}$ ($p^{\prime}_{N}$) and $\lambda_{N}$
($\lambda^{\prime}_{N}$), with $N=p$ or $n$. The two nucleons exchange
momentum via a vector particle. The four diagrams differ in the nature of the
photon-absorbing nucleon and the order of this absorption and momentum
transfer between nucleons.

The tree-level amplitudes for the four diagrams in Fig. 2 are

$\displaystyle
M_{a}^{\nu}(\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda_{p},\lambda_{n})$
$\displaystyle=$ $\displaystyle
A_{p}^{\mu\nu}(p/2+q;\lambda^{\prime}_{p},\lambda_{p})\frac{1}{(p^{\prime}_{n}-p/2)^{2}}B_{n\mu}(\lambda^{\prime}_{n},\lambda_{n}),$
(2) $\displaystyle
M_{b}^{\nu}(\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda_{p},\lambda_{n})$
$\displaystyle=$ $\displaystyle
A_{p}^{\nu\mu}(p^{\prime}_{p}-q;\lambda^{\prime}_{p},\lambda_{p})\frac{1}{(p^{\prime}_{n}-p/2)^{2}}B_{n\mu}(\lambda^{\prime}_{n},\lambda_{n}),$
(3) $\displaystyle
M_{c}^{\nu}(\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda_{p},\lambda_{n})$
$\displaystyle=$ $\displaystyle
A_{n}^{\mu\nu}(p/2+q;\lambda^{\prime}_{n},\lambda_{n})\frac{1}{(p^{\prime}_{p}-p/2)^{2}}B_{p\mu}(\lambda^{\prime}_{p},\lambda_{p}),$
(4) $\displaystyle
M_{d}^{\nu}(\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda_{p},\lambda_{n})$
$\displaystyle=$ $\displaystyle
A_{n}^{\nu\mu}(p^{\prime}_{n}-q;\lambda^{\prime}_{n},\lambda_{n})\frac{1}{(p^{\prime}_{p}-p/2)^{2}}B_{p\mu}(\lambda^{\prime}_{p},\lambda_{p}),$
(5)

where

$A_{N}^{\mu\nu}(p;\lambda^{\prime}_{N},\lambda_{N})=\bar{u}^{\prime}_{N}\gamma^{\mu}\frac{{\centernot{p}}+m}{p^{2}-m^{2}}\gamma^{\nu}u_{N},\;\;B_{N}^{\mu}(\lambda^{\prime}_{N},\lambda_{N})=\bar{u}^{\prime}_{N}\gamma^{\mu}u_{N},$
(6)

with $u_{N}$ ($\bar{u}^{\prime}_{N}$) the initial (final) spinor for the
nucleon $N$ with helicity $\lambda_{N}$ ($\lambda^{\prime}_{N}$). The sub-
amplitude $A_{N}$ represents the fermion line that absorbs the photon, and
$B_{N}$ represents the other fermion line. Calculation of these sub-amplitudes
can be checked against the trace theorem for sums over helicities:

$\displaystyle\sum_{\lambda_{N},\lambda^{\prime}_{N}}A_{N}^{\nu^{\prime}\mu^{\prime}*}(p;\lambda^{\prime}_{N},\lambda_{N})A_{N}^{\mu\nu}(p;\lambda^{\prime}_{N},\lambda_{N})$
$\displaystyle=$ $\displaystyle{\rm
Tr}\left[\gamma^{\nu^{\prime}}\frac{{\centernot{p}}+m}{p^{2}-m^{2}}\gamma^{\mu^{\prime}}({\centernot{p}}^{\prime}_{N}+m)\gamma^{\mu}\frac{{\centernot{p}}+m}{p^{2}-m^{2}}\gamma^{\nu}({\centernot{p}}_{N}+m)\right],$
$\displaystyle\sum_{\lambda_{N},\lambda^{\prime}_{N}}B_{N}^{\mu^{\prime}*}(\lambda^{\prime}_{N},\lambda_{N})B_{N}^{\mu}(\lambda^{\prime}_{N},\lambda_{N})$
$\displaystyle=$ $\displaystyle{\rm
Tr}\left[\gamma^{\mu^{\prime}}({\centernot{p}}^{\prime}_{N}+m)\gamma^{\mu}({\centernot{p}}_{N}+m)\right].$
(8)

The full amplitude is constructed from the $M_{X}$ by combining them with form
factors for each nucleon. For a deuteron with initial helicity $\lambda_{d}$,
we have

$M^{\nu}(\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda_{d})=\sum_{\lambda_{p},\lambda_{n}}C_{\lambda_{p}\lambda_{n}}^{\lambda_{d}}\left[\sum_{X=a,b,c,d}M_{X}^{\nu}(\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda_{p},\lambda_{n})\right]G_{p\lambda^{\prime}_{p}\lambda_{p}}(Q^{2}_{p})G_{n\lambda^{\prime}_{n}\lambda_{n}}(Q^{2}_{n}),$
(9)

where $Q^{2}_{N}=-(p^{\prime}_{N}-p_{N})^{2}$,

$C_{\lambda_{p}\lambda_{n}}^{\lambda_{d}}=\left\\{\begin{array}[]{ll}\delta_{\lambda_{p}\pm\frac{1}{2}}\delta_{\lambda_{n}\pm\frac{1}{2}},&\lambda_{d}=\pm
1\\\
\frac{1}{\sqrt{2}}\left(\delta_{\lambda_{p}\frac{1}{2}}\delta_{\lambda_{n}-\frac{1}{2}}+\delta_{\lambda_{p}-\frac{1}{2}}\delta_{\lambda_{n}+\frac{1}{2}}\right),&\lambda_{d}=0,\end{array}\right.$
(10)

and

$G_{N\lambda^{\prime}\lambda}=\left\\{\begin{array}[]{ll}G_{EN},&\lambda^{\prime}=\lambda\\\
G_{MN},&\lambda^{\prime}=-\lambda.\end{array}\right.$ (11)

The form factors $G_{EN}$ and $G_{MN}$ represent the internal structure of the
nucleons. They can be represented by data or empirical fits. For simplicity,
we use the fits [40]

$G_{Ep}\simeq\left(1+\frac{Q_{N}^{2}}{m_{0}^{2}}\right)^{-2},\;\;G_{Mp}\simeq\mu_{p}G_{Ep},\;\;G_{Mn}\simeq\mu_{n}G_{Ep},\;\;G_{En}\simeq-\frac{\mu_{n}\tau}{1+5.6\tau}G_{Ep},$
(12)

where $m_{0}^{2}=0.71$ GeV2, $\tau=\frac{Q_{N}^{2}}{4m^{2}}$, $\mu_{p}=2.79$,
and $\mu_{n}=-1.91$. To limit the analysis to a single mass scale, we take the
parameter $m_{0}$ to be proportional to the nuclear mass, with
$m_{0}^{2}=0.80\,m^{2}$. We do not attempt to compute or assign an overall
normalization to $M^{\nu}$, and the running of the strong coupling constant is
not included.

The initial nucleon spinor, for a deuteron traveling along the negative $z$
direction, is [41]

$u_{N}=\frac{{\centernot{p}}/2+m}{\sqrt{E_{d}/2+m}}\left(\begin{array}[]{c}\phi^{(\lambda_{N})}(-\hat{z})\\\
0\end{array}\right),$ (13)

with

$\phi^{(1/2)}(-\hat{z})=\left(\begin{array}[]{c}0\\\
1\end{array}\right),\;\;\phi^{(-1/2)}(-\hat{z})=\left(\begin{array}[]{c}1\\\
0\end{array}\right).$ (14)

The final nucleon spinor is

$u^{\prime}_{N}=\frac{{\centernot{p}}^{\prime}_{N}+m}{\sqrt{E^{\prime}_{N}+m}}\left(\begin{array}[]{c}\phi^{(\lambda^{\prime}_{N})}(\hat{p}^{\prime}_{N})\\\
0\end{array}\right),$ (15)

with

$\phi^{(1/2)}(\hat{p}^{\prime}_{N})=\left(\begin{array}[]{c}\cos(\theta_{N}/2)\\\
e^{i\phi_{N}}\sin(\theta_{N}/2)\end{array}\right),\;\;\phi^{(-1/2)}(\hat{p}^{\prime}_{N})=\left(\begin{array}[]{c}-e^{-i\phi_{N}}\sin(\theta_{N}/2)\\\
\cos(\theta_{N}/2)\end{array}\right),$ (16)

where $\theta_{N}$ and $\phi_{N}$ are the polar and azimuthal angles of the
outgoing momentum of the particular nucleon.

As discussed in the Introduction, the overall normalization of the RNHA
amplitude is unknown. For comparison with data, we consider quantities which
are themselves ratios or a ratio of the model to data.

##  III Elastic electron scattering

###  III.1 Form factors

The three deuteron form factors, $G_{C}$, $G_{M}$, and $G_{Q}$, are readily
obtained from the hadronic helicity amplitudes of elastic electron-deuteron
scattering in the Breit frame [42]. The kinematics are shown in Fig. 3. The
photon four-momentum is $q=(0,0,0,q_{z})$ and the initial (final) deuteron
four-momentum is $p=(E_{d},0,0,-q_{z}/2)$ ($p^{\prime}=(E_{d},0,0,q_{z}/2)$),
with $q_{z}^{2}=Q^{2}$ and $E_{d}=\sqrt{Q^{2}/4+m_{d}^{2}}$. In the zero-
binding limit,333The difference between the proton and neutron masses is
neglected in addition to the deuteron binding energy, the two being of the
same order. $m_{d}=2m$ and the individual nucleon four-momenta are
$p_{p}=p_{n}=p/2$ and $p^{\prime}_{p}=p^{\prime}_{n}=p^{\prime}/2$. The
hadronic matrix elements are given by

$G_{\lambda^{\prime}_{d},\lambda_{d}}^{\mu}=\sum_{\lambda^{\prime}_{p},\lambda^{\prime}_{n}}C_{\lambda^{\prime}_{p}\lambda^{\prime}_{n}}^{\lambda^{\prime}_{d}}M^{\mu}(\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda_{d}).$
(17)

The initial spinors are as in (13); the final spinors are specified by

$u^{\prime}_{N}=\frac{{\centernot{p}}^{\prime}/2+m}{\sqrt{E_{d}/2+m}}\left(\begin{array}[]{c}\phi^{(\lambda^{\prime}_{N})}(\hat{z})\\\
0\end{array}\right).$ (18)

Figure 3:  Kinematics for elastic electron-deuteron scattering in the Breit
frame. The photon travels along the positive $z$ direction, and the deuteron
comes from the right, along the negative $z$ direction.

The three form factors are then extracted as [42, 43]

$G_{C}=\frac{-1}{2m_{d}\sqrt{1+\eta}}\frac{G_{00}^{+}-2G_{+-}^{+}}{3},\;\;G_{M}=\frac{2}{2m_{d}\sqrt{1+\eta}}\frac{G_{+0}^{x}}{\sqrt{2\eta}},\;\;G_{Q}=\frac{-1}{2m_{d}\sqrt{1+\eta}}\frac{G_{00}^{+}+G_{+-}^{+}}{2\eta},$
(19)

with $\eta\equiv\frac{Q^{2}}{4m_{d}^{2}}$ and the $+$ superscript denoting the
light-front sum of the 0 and $z$ components. For the helicity matrix elements,
the model yields the following $Q^{2}$ dependence:

$\displaystyle G_{00}^{+}$ $\displaystyle=$ $\displaystyle 0.5588{\cal
N}m\left(\frac{m}{Q}\right)^{9}\left[1+129.1\frac{m^{2}}{Q^{2}}+{\cal
O}(\frac{m^{4}}{Q^{4}})\right],$ (20) $\displaystyle G_{+-}^{+}$
$\displaystyle=$ $\displaystyle-69.85{\cal
N}m\left(\frac{m}{Q}\right)^{11}\left[1+4.8\frac{m^{2}}{Q^{2}}+{\cal
O}(\frac{m^{4}}{Q^{4}})\right],$ (21) $\displaystyle G_{+0}^{x}$
$\displaystyle=$ $\displaystyle 8.851{\cal
N}m\left(\frac{m}{Q}\right)^{10}\left[1+4.8\frac{m^{2}}{Q^{2}}+{\cal
O}(\frac{m^{4}}{Q^{4}})\right],$ (22)

with ${\cal N}$ the unknown normalization. The factor of $m/Q$ associated with
each helicity flip [44] is clearly evident. For the form factors, we find

$\displaystyle G_{C}$ $\displaystyle=$
$\displaystyle-\frac{0.5588}{\sqrt{1+\eta}}\frac{{\cal
N}}{12}\left(\frac{m}{Q}\right)^{9}\left[1+379.1\frac{m^{2}}{Q^{2}}+{\cal
O}(\frac{m^{4}}{Q^{4}})\right],$ (23) $\displaystyle G_{M}$ $\displaystyle=$
$\displaystyle\frac{8.851}{\sqrt{\eta(1+\eta)}}\frac{{\cal
N}}{2\sqrt{2}}\left(\frac{m}{Q}\right)^{10}\left[1+4.8\frac{m^{2}}{Q^{2}}+{\cal
O}(\frac{m^{4}}{Q^{4}})\right],$ (24) $\displaystyle G_{Q}$ $\displaystyle=$
$\displaystyle-\frac{0.5588}{\eta\sqrt{1+\eta}}\frac{{\cal
N}}{8}\left(\frac{m}{Q}\right)^{9}\left[1+4.086\frac{m^{2}}{Q^{2}}+{\cal
O}(\frac{m^{4}}{Q^{4}})\right].$ (25)

The leading $\pm$ signs are as expected for large $Q^{2}$.

We have left the kinematic factor $\eta=Q^{2}/16m^{2}$ without substitution,
because there can be three regimes for $Q^{2}$. In addition to $Q^{2}$ large
or small, there can be an intermediate region where $Q^{2}$ is large but
$\eta$ is not. Such an intermediate regime does exist for $G_{M}$ and $G_{Q}$,
where the coefficients of the nonleading terms are small enough for this
correction to be small while $\eta$ is also small. For $G_{C}$, this is not
the case, because the coefficient of the nonleading term is large enough to
require a $Q^{2}$ value for which $\eta$ is also large. In the intermediate
regime, we obtain

$G_{M}\sim\left(\frac{m}{Q}\right)^{11},\;\;G_{Q}\sim\left(\frac{m}{Q}\right)^{11},$
(26)

and for the large-$\eta$ regime

$G_{C}\sim\left(\frac{m}{Q}\right)^{10},\;\;G_{M}\sim\left(\frac{m}{Q}\right)^{12},\;\;G_{Q}\sim\left(\frac{m}{Q}\right)^{12}.$
(27)

Ratios of these form factors at very large $Q^{2}$ can be compared with the
tree-level ratios for a point-like spin-one particle, such as the $W^{+}$,
which are [43]

$\frac{G_{C}}{G_{Q}}=\frac{2}{3}\eta-1,\;\;\frac{G_{M}}{G_{Q}}=-2.$ (28)

Such behavior is immediately reproduced for form factors separated according
to a Drell–Yan frame [45], with the assumption of strict $G^{+}_{00}$
dominance [43]. In terms of our hadronic matrix elements, we have

$\frac{G_{C}}{G_{Q}}=\frac{2}{3}\eta-2\eta\frac{G^{+}_{+-}}{G^{+}_{00}+G^{+}_{+-}},\;\;\frac{G_{M}}{G_{Q}}=-2\sqrt{2\eta}\frac{G^{x}_{+0}}{G^{+}_{00}+G^{+}_{+-}}.$
(29)

As already observed in [43], these Breit-frame ratios cannot both be resolved
by simply assuming $G^{+}_{00}$ dominance. From our model, we obtain

$\frac{G_{C}}{G_{Q}}=\frac{2}{3}\eta+15.6+{\cal
O}(\frac{m^{2}}{Q^{2}}),\;\;\frac{G_{M}}{G_{Q}}=-11.2+{\cal
O}(\frac{m^{2}}{Q^{2}}).$ (30)

The leading $\frac{2}{3}\eta$ is just kinematic. The deviations of 15.6 and
-11.2 from -1 and -2, respectively, are due to nonleading contributions
multiplied by powers of $\eta$. Similar deviations will arise for calculations
done in the Drell–Yan frame, because $\eta$ factors again interfere with
strict $G_{00}^{+}$ dominance. Plots of these ratios are shown in Fig. 4.

Figure 4:  Ratios $G_{C}/G_{Q}-2\eta/3$ (dashed) and $G_{M}/G_{Q}$ (solid) for
the model deuteron form factors.

###  III.2 Structure functions

Experiments designed to extract these form factors measure cross sections and
polarization observables in elastic electron-deuteron scattering. The
unpolarized cross section

$\frac{d\sigma}{d\Omega}\propto S,\;\;S\equiv
A(Q^{2})+B(Q^{2})\tan^{2}(\theta_{e}/2)$ (31)

depends on the electron scattering angle $\theta_{e}$ and two structure
functions

$\displaystyle A(Q^{2})$ $\displaystyle\equiv$ $\displaystyle
G_{C}^{2}+\frac{8}{9}\eta^{2}G_{Q}^{2}+\frac{2}{3}\eta G_{M}^{2},$ (32)
$\displaystyle B(Q^{2})$ $\displaystyle\equiv$
$\displaystyle\frac{4}{3}\eta(1+\eta)G_{M}^{2}.$ (33)

These have been measured at the highest $Q^{2}$ yet attained at JLab [22, 23,
24], and $A$ has been measured at comparable $Q^{2}$ at SLAC [21]. However,
these do not yet reach the $Q^{2}$ values needed for a definitive comparison.
Figures 5 and 6 show plots of the data divided by the model, including an
arbitrary normalization factor.

Figure 5:  Data for the deuteron structure function $A(Q^{2})$ divided by the
model function, including an arbitrary normalization. Experimental values are
taken from [23] (circles) and [24] (squares).

Figure 6:  Data for the deuteron structure function $B(Q^{2})$ divided by the
model function, including an arbitrary normalization. Experimental values are
taken from [22].

In our model, expansions of these functions in inverse powers of $Q^{2}$ are

$\displaystyle A(Q^{2})$ $\displaystyle=$ $\displaystyle 0.1041{\cal
N}^{2}\left(\frac{m}{Q}\right)^{20}\left[1+1246\frac{m^{2}}{Q^{2}}+{\cal
O}(\frac{m^{4}}{Q^{4}})\right],$ (34) $\displaystyle B(Q^{2})$
$\displaystyle=$ $\displaystyle 13.06{\cal
N}^{2}\left(\frac{m}{Q}\right)^{20}\left[1+9.6\frac{m^{2}}{Q^{2}}+{\cal
O}(\frac{m^{4}}{Q^{4}})\right].$ (35)

Because the expansion for $G_{C}$ is valid only for large $\eta$, we have used
the explicit form of $\eta$ in constructing the expansion for $A$. The
function $B$ is independent of $\eta$; the leading factor of $\eta(1+\eta)$ in
its definition exactly cancels against factors in the relationship of $G_{M}$
to hadronic matrix elements. The expansion for $B$ converges much faster than
the expansion for $A$, and the leading $Q^{2}$ behavior is dominant for
$Q^{2}\gg 10$ GeV2 only for $B$. For $A$, one must wait until impossibly large
$Q^{2}$, which enters a regime where the collective quark substructure is
important, including hidden-color effects [34], and the point-like
approximation used in our model is invalid.

In [22] the large $Q^{2}$ behavior of $B$ is quoted as being $Q^{-24}$ from
perturbative QCD. This faster fall off compared to $A$ is attributed to the
extra suppression of the helicity flip involved in $G_{M}$. However, there are
other compensating factors, and, just as in our model, the behavior of $B$
should be $Q^{-20}$, which is the same as $A$. In Fig. 7 we plot the ratio of
$B$ to $A$ for a large range of $Q^{2}$. This ratio becomes constant at very
large $Q^{2}$. Although the plots begin at low $Q^{2}$, there is nothing in
the model that could reproduce diffractive minima, hence the smooth
appearance.

Figure 7:  Ratio of $B$ to $A$ for the model deuteron structure functions.

###  III.3 Tensor polarization observables

Experiments can also extract tensor polarization observables [20, 25]

$\displaystyle t_{20}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{\sqrt{2}S}\left[\frac{8}{3}\eta
G_{C}G_{Q}+\frac{8}{9}\eta^{2}G_{Q}^{2}+\frac{1}{3}\eta\left(1+2(1+\eta)\tan^{2}(\theta_{e}/2)\right)G_{M}^{2}\right],$
(36) $\displaystyle t_{21}$ $\displaystyle\equiv$
$\displaystyle\frac{2\eta}{\sqrt{3}S\cos(\theta_{e}/2)}\sqrt{\eta+\eta^{2}\sin^{2}(\theta_{e}/2)}G_{M}G_{Q},$
(37) $\displaystyle t_{22}$ $\displaystyle\equiv$
$\displaystyle-\frac{\eta}{2\sqrt{3}S}G_{M}^{2}.$ (38)

The highest $Q^{2}$ measurements of these were also done at JLab [25]. When
$\eta$ is held explicit, expansions in $m/Q$ are

$\displaystyle t_{20}$ $\displaystyle=$
$\displaystyle-\sqrt{2}+1064[1+2(1+\eta)\tan^{2}(\theta_{e}/2)]\left(\frac{m}{Q}\right)^{2}+{\cal
O}(\frac{m^{4}}{Q^{4}}),$ (39) $\displaystyle t_{21}$ $\displaystyle=$
$\displaystyle
38.8\sec(\theta_{e}/2)\sqrt{\eta+\sin^{2}(\theta_{e}/2)}\frac{m}{Q}$
$\displaystyle+\sec(\theta_{e}/2)\sqrt{\eta+\sin^{2}(\theta_{e}/2)}[48606+77869(1+\eta)\tan^{2}(\theta_{e}/2)]\left(\frac{m}{Q}\right)^{3}+{\cal
O}(\frac{m^{4}}{Q^{4}}),$ $\displaystyle t_{22}$ $\displaystyle=$
$\displaystyle-434.5\left(\frac{m}{Q}\right)^{2}+[544703+872133(1+\eta)\tan^{2}(\theta_{e}/2)]\left(\frac{m}{Q}\right)^{4}+{\cal
O}(\frac{m^{6}}{Q^{6}}).$ (41)

While at very large $Q^{2}$, they are

$\displaystyle t_{20}$ $\displaystyle=$
$\displaystyle-\sqrt{2}+133\tan^{2}(\theta_{e}/2)+1064[1+2\tan^{2}(\theta_{e}/2)]\left(\frac{m}{Q}\right)^{2}+{\cal
O}(\frac{m^{4}}{Q^{4}}),$ (42) $\displaystyle t_{21}$ $\displaystyle=$
$\displaystyle
1217\sec(\theta_{e}/2)\sin(\theta_{e}/2)[\tan^{2}(\theta_{e}/2)-0.007972]+\left[\right]\left(\frac{m}{Q}\right)^{2}+{\cal
O}(\frac{m^{4}}{Q^{4}}),$ (43) $\displaystyle t_{22}$ $\displaystyle=$
$\displaystyle-[434.5-54508\tan^{2}(\theta_{e}/2)]\left(\frac{m}{Q}\right)^{2}$
$\displaystyle+[544703+872133\tan^{2}(\theta_{e}/2)]\left(\frac{m}{Q}\right)^{4}+{\cal
O}(\frac{m^{6}}{Q^{6}}).$

The coefficients of nonleading terms are quite large. Thus, very large $Q^{2}$
is required for the leading term to be dominant, well beyond any available
data. The limit of $-\sqrt{2}$ for $t_{20}$ at $\theta_{e}=0^{\circ}$ was an
early prediction of perturbative QCD [46, 44]. However, as argued elsewhere
[43], this value is obtained only at very large $Q^{2}$, and the value is
quite different for small $\eta$. Figures 8 and 9 show plots of these
observables at angles of 0∘ and 30∘, respectively. We also compare with data
[25] in Figs. 10, 11, and 12. At these ‘small’ values of $Q^{2}$, only
$t_{22}$ is consistent with data, something which is likely accidental with
both data and model values near zero.

Figure 8:  Deuteron tensor polarization observables $t_{20}$ (solid), $t_{21}$
(dashed), and $t_{22}$ (dotted) as computed in the model at an angle of
$\theta_{e}=0^{\circ}$. The asymptotic value of $t_{20}(0^{\circ})$ is
$-\sqrt{2}$, as predicted by perturbative QCD [46, 44].

Figure 9:  Same as Fig. 8 but for an angle of $\theta_{e}=30^{\circ}$.

Figure 10:  Plots of the tensor polarization observable $t_{20}$ of the
deuteron from both data [25] (circles) and the model (squares) considered in
the text. The angle $\theta_{e}$ varies and is as follows in order of
increasing $Q^{2}$: 35.6∘, 33.4∘, 29.8∘, 27.3∘, 23.0∘, and 19.8∘.

Figure 11:  Same as Fig. 10 but for $t_{21}$.

Figure 12:  Same as Fig. 10 but for $t_{22}$.

##  IV Photodisintegration

In the photodisintegration of a deuteron, a real photon is absorbed and the
two constituent nucleons emitted. This process is depicted in Fig. 13. The
initial deuteron and photon four-momenta in the center-of-mass (c.m.) frame
are $p=(E_{d},0,0,-q_{z})$ and $q=(q_{z},0,0,q_{z})$, where the incident
photon is taken along the positive $z$ axis. The final proton and neutron
four-momenta are $p^{\prime}_{p}=(E^{\prime}_{p},\vec{p}^{\,\prime}_{p})$ and
$p^{\prime}_{n}=(E^{\prime}_{n},\vec{p}^{\,\prime}_{n})$, with $\theta_{p}$
and $\phi_{p}$ the polar and azimuthal angles of the final proton. By ignoring
the nucleon mass difference, we have $E^{\prime}_{p}=E^{\prime}_{n}$, because
momentum conservation guarantees
$\vec{p}^{\,\prime}_{n}=-\vec{p}^{\,\prime}_{p}$ in the c.m. frame.

Figure 13:  Kinematics for deuteron photodisintegration in the c.m. frame,
with $\vec{q}$ the photon momentum and $\vec{p}=-\vec{q}$ the deuteron
momentum. The final proton and neutron momenta are $\vec{p}^{\,\prime}_{p}$
and $\vec{p}^{\,\prime}_{n}$. The $\lambda$’s are helicities. Coordinates are
chosen such that the photon enters along the positive $z$ direction and the
azimuthal angle $\phi_{p}$ of the proton is zero.

In terms of the Mandelstam variable $s$, the c.m. energies and momenta are

$E_{d}=(s+4m^{2})/(2\sqrt{s}),\;\;q_{z}=(s-4m^{2})/(2\sqrt{s}),\;\;E^{\prime}_{p}=\sqrt{s}/2,\;\;|\vec{p}^{\,\prime}_{p}|=\sqrt{s-4m^{2}}/2.$
(45)

The photon energy in the lab frame is $E_{\gamma}=(s-4m^{2})/4m$. We will work
at large $s$, so that momentum transfers are large.

The standard definition of helicity amplitudes for photodisintegration is [47]

$F_{i\pm}\equiv\epsilon_{\nu}(\lambda_{\gamma})M^{\nu}(\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda_{d})$
(46)

with $\epsilon$ the polarization vector for a photon with helicity
$\lambda_{\gamma}$ and $M^{\nu}$ given in (9). The index $i$ is associated
with particular helicity combinations as follows:

$\displaystyle F_{1\pm}$ $\displaystyle=$
$\displaystyle\epsilon_{\nu}(1)M^{\nu}(\pm\frac{1}{2},\pm\frac{1}{2},1),\;\;F_{2\pm}=\epsilon_{\nu}(1)M^{\nu}(\pm\frac{1}{2},\pm\frac{1}{2},0),$
(47) $\displaystyle F_{3\pm}$ $\displaystyle=$
$\displaystyle\epsilon_{\nu}(1)M^{\nu}(\pm\frac{1}{2},\pm\frac{1}{2},-1),\;\;F_{4\pm}=\epsilon_{\nu}(1)M^{\nu}(\pm\frac{1}{2},\mp\frac{1}{2},1),$
(48) $\displaystyle F_{5\pm}$ $\displaystyle=$
$\displaystyle\epsilon_{\nu}(1)M^{\nu}(\pm\frac{1}{2},\mp\frac{1}{2},0),\;\;F_{6\pm}=\epsilon_{\nu}(1)M^{\nu}(\pm\frac{1}{2},\mp\frac{1}{2},-1).$
(49)

The other helicity combinations are related to these by parity.

The helicity amplitudes can be used to compute various polarization
observables. The recoil-proton polarization $P_{y}$ measures the asymmetry
parallel/antiparallel to the normal
$\hat{y}\propto\vec{q}\times\vec{p}^{\,\prime}_{p}$ to the scattering plane:

$P_{y}=2{\rm
Im}\sum_{i=1}^{3}[F_{i+}^{\dagger}F_{i+3,-}+F_{i+3,+}^{\dagger}F_{i-}]/f(\theta),$
(50)

where $f(\theta)=\sum_{i=1}^{6}[|F_{i+}|^{2}+|F_{i-}|^{2}]$ is the sum of all
the helicity amplitudes squared. The transferred polarizations
$C_{x^{\prime}}$ and $C_{z^{\prime}}$ measure asymmetries
parallel/antiparallel to the
$\hat{x}^{\prime}\propto\vec{p}^{\,\prime}_{p}\times\hat{y}$ and
$\hat{z}^{\prime}=\hat{p}^{\,\prime}_{p}$ directions:

$\displaystyle C_{x^{\prime}}$ $\displaystyle=$ $\displaystyle 2{\rm
Re}\sum_{i=1}^{3}[F_{i+}^{\dagger}F_{i+3,-}+F_{i+3,+}^{\dagger}F_{i-}]/f(\theta),$
(51) $\displaystyle C_{z^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{6}[|F_{i+}|^{2}-|F_{i-}|^{2}]/f(\theta).$ (52)

The asymmetry $\Sigma$ for linearly polarized photons is given by

$\Sigma=-2{\rm
Re}\left[\sum_{\pm}(F_{1\pm}^{\dagger}F_{3\mp}-F_{4\pm}^{\dagger}F_{6\mp})-F_{2+}^{\dagger}F_{2-}+F_{5+}^{\dagger}F_{5-}\right]/f(\theta).$
(53)

Each observable is formed as a ratio, which sets aside questions of
normalization.

Because we only need to consider photons with helicity +1, the polarization
vector is always $\epsilon=-\frac{1}{\sqrt{2}}(0,1,i,0)$, relative to the
momentum in the positive $z$ direction. The final Dirac spinors are

$u^{\prime}_{N}=\frac{{\centernot{p}}^{\prime}_{N}+m}{\sqrt{E^{\prime}_{N}+m}}\left(\begin{array}[]{c}\phi^{(\lambda^{\prime}_{N})}(\hat{p}^{\prime}_{N})\\\
0\end{array}\right),$ (54)

with $\theta_{n}=\pi-\theta_{p}$, $\phi_{n}=\phi_{p}+\pi=\pi$, and

$\phi^{(1/2)}(\hat{p}^{\prime}_{N})=\left(\begin{array}[]{c}\cos(\theta_{N}/2)\\\
e^{i\phi_{N}}\sin(\theta_{N}/2)\end{array}\right),\;\;\phi^{(-1/2)}(\hat{p}^{\prime}_{N})=\left(\begin{array}[]{c}-e^{-i\phi_{N}}\sin(\theta_{N}/2)\\\
\cos(\theta_{N}/2)\end{array}\right).$ (55)

With these spinors as input, the amplitudes $\epsilon_{\nu}M_{X}^{\nu}$ can be
evaluated in terms of Dirac matrix and spinor products and then combined to
construct the predefined amplitudes $F_{i\pm}$. At large $s$, these RNHA
predictions for the helicity amplitudes reduce to

$\displaystyle F_{1+}$ $\displaystyle\sim$ $\displaystyle
4\frac{\sqrt{2}}{\sqrt{s}}{\rm
csc}^{2}(\frac{\theta_{p}}{2})G_{En}(\theta_{p})G_{Ep}(\theta_{p}),\;\;F_{1-}\sim
0,$ (56) $\displaystyle F_{2+}$ $\displaystyle\sim$ $\displaystyle
2\frac{m}{s}\cot^{3}(\frac{\theta_{p}}{2})[G_{En}(\theta_{p})G_{Mp}(\theta_{p})-G_{Mn}(\theta_{p})G_{Ep}(\theta_{p})],$
(57) $\displaystyle F_{2-}$ $\displaystyle\sim$ $\displaystyle
2\frac{m}{s}\cot(\frac{\theta_{p}}{2})[G_{Mn}(\theta_{p})G_{Ep}(\theta_{p})-G_{En}(\theta_{p})G_{Mp}(\theta_{p})],$
$\displaystyle F_{3+}$ $\displaystyle\sim$ $\displaystyle 0,\;\;F_{3-}\sim 0,$
(58) $\displaystyle F_{4+}$ $\displaystyle\sim$
$\displaystyle-4\sqrt{2}\frac{m}{s}\cot(\frac{\theta_{p}}{2})G_{Mn}(\theta_{p})G_{Ep}(\theta_{p}),\;\;F_{4-}\sim
4\sqrt{2}\frac{m}{s}\cot(\frac{\theta_{p}}{2})G_{En}(\theta_{p})G_{Mp}(\theta_{p}),$
(59) $\displaystyle F_{5+}$ $\displaystyle\sim$
$\displaystyle\frac{2}{\sqrt{s}}\cot^{2}(\frac{\theta_{p}}{2})G_{En}(\theta_{p})G_{Ep}(\theta_{p}),\;\;F_{5-}\sim\frac{2}{\sqrt{s}}G_{En}(\theta_{p})G_{Ep}(\theta_{p}),$
(60) $\displaystyle F_{6+}$ $\displaystyle\sim$ $\displaystyle
4\sqrt{2}\frac{m}{s}\cot^{3}(\frac{\theta_{p}}{2})G_{En}(\theta_{p})G_{Mp}(\theta_{p}),\;\;F_{6-}\sim-4\sqrt{2}\frac{m}{s}\cot(\frac{\theta_{p}}{2})G_{Mn}(\theta_{p})G_{Ep}(\theta_{p}).$
(61)

From these we can calculate the various observables. Plots of the results and
recent data [48, 49, 50] are given in Figs. 14, 15, 16, and 17. Because the
tree-level amplitudes are real, $P_{y}$ is automatically zero. That
$C_{x^{\prime}}$ is of order $m/\sqrt{s}$, rather than zero, is a correction
to hadron helicity conservation [51]. Also, we find the asymmetry
$\Sigma(90^{\circ})$ to be approximately -0.06, rather than the nominal
expectation [52] of -1. In general, the trends with photon energy seem to be
modestly consistent with data.

|
---|---
(a) | (b)
Figure 14:  Recoil proton polarization $P_{y}$ as a function of (a) photon energy $E_{\gamma}$ and (b) proton angle $\theta$. For the latter, the photon energy is 2 GeV. The solid line is the RNHA prediction; the data points are from [48, 49].  |
---|---
(a) | (b)
Figure 15:  Same as Fig. 14 but for the transferred polarization $C_{x^{\prime}}$.  |
---|---
(a) | (b)
Figure 16:  Same as Fig. 15 but for $C_{z^{\prime}}$.  |
---|---
(a) | (b)
Figure 17:  Same as Fig. 14 but for the asymmetry $\Sigma$. The data points
are from [50].

##  V Electrodisintegration

The kinematics of the electrodisintegration process are shown in Fig. 18. The
initial (final) momentum and helicity of the electron are $p_{e}$
($p^{\prime}_{e}$) and $\lambda_{e}$ ($\lambda^{\prime}_{e}$). The
intermediate photon carries four-momentum $q$. The azimuthal angle $\phi_{p}$
of the proton measures the rotation of the hadronic reaction plane relative to
the electron scattering plane.

In the lab frame, with the $z$ axis taken along the photon three-momentum and
the electron mass neglected, the initial and final electron four-momenta are

$p_{e}=(E_{e},E_{e}\sin\theta_{e},0,E_{e}\cos\theta_{e}),\;\;p^{\prime}_{e}=(E^{\prime}_{e},E^{\prime}_{e}\sin\theta^{\prime}_{e},0,E^{\prime}_{e}\cos\theta^{\prime}_{e}),$
(62)

with $E^{\prime}_{e}$ and $\tilde{\theta}=\theta^{\prime}_{e}-\theta_{e}$, the
angle of the scattered electron to the beam direction, being measured. The
photon four-momentum $q=(E_{\gamma},0,0,q_{z})$ is just
$p_{e}-p^{\prime}_{e}$, which yields

$Q^{2}\equiv-q^{2}=2E_{e}E^{\prime}_{e}(1-\cos\tilde{\theta}),\;\;E_{\gamma}=E_{e}-E^{\prime}_{e},\;\;q_{z}=\sqrt{E_{\gamma}^{2}+Q^{2}}.$
(63)

The deuteron four-momentum is $p=(m_{d}=2m,0,0,0)$, and in the zero-binding
limit, the initial proton and neutron four-momenta are
$p_{p}=p_{n}=(m,0,0,0)$. The final nucleon four-momenta are

$p^{\prime}_{p}=(E^{\prime}_{p}=\sqrt{\vec{p}^{\,\prime
2}_{p}+m^{2}},|\vec{p}^{\,\prime}_{p}|\sin\theta_{p}\cos\phi_{p},|\vec{p}^{\,\prime}_{p}|\sin\theta_{p}\sin\phi_{p},|\vec{p}^{\,\prime}_{p}|\cos\theta_{p}),$
(64) $p^{\prime}_{n}=(E^{\prime}_{n}=\sqrt{\vec{p}^{\,\prime
2}_{n}+m^{2}},-|\vec{p}^{\,\prime}_{n}|\sin\theta_{n}\cos\phi_{p},-|\vec{p}^{\,\prime}_{n}|\sin\theta_{n}\sin\phi_{p},|\vec{p}^{\,\prime}_{n}|\cos\theta_{n}).$
(65)

Within the one-photon-exchange approximation, the scattering amplitude is
proportional to

$M_{ed}(\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda^{\prime}_{e};\lambda_{d},\lambda_{e})=\bar{u}^{\prime}_{e}\gamma_{\mu}u_{e}\frac{D_{\mu\nu}}{q^{2}}M^{\nu}(\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda_{d}),$
(66)

with $u_{e}$ ($u^{\prime}_{e}$) the initial (final) spinor of the electron and
$M^{\nu}$ given in (9). The numerator of the photon progator is the sum over
photon polarizations

$D_{\mu\nu}=\sum_{\lambda=-1}^{1}(-1)^{\lambda}\epsilon_{\mu}^{*}(\lambda)\epsilon_{\nu}(\lambda).$
(67)

The polarization four-vectors are444In the hadronic c.m. frame, the
longitudinal polarization vector is
$\epsilon(0)=(q^{\prime}_{z}/Q,0,0,E^{\prime}_{\gamma}/Q)$.

$\epsilon(\pm 1)=\mp\frac{1}{\sqrt{2}}(0,1,\pm
i,0),\;\;\epsilon(0)=(q_{z}/Q,0,0,E_{\gamma}/Q)$ (68)

relative to the photon four-momentum $q=(E_{\gamma},0,0,q_{z})$. Polarization
observables [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] can then be computed from
these helicity amplitudes.

Figure 18:  Kinematics for deuteron electrodisintegration. The unprimed axes
are defined relative to the electron scattering plane, and the primed axes
relative to the final nucleon momenta. The final proton momentum has polar
angle $\theta_{p}$ and azimuthal angle $\phi_{p}$ relative to the unprimed
frame.

In keeping with the notation of [6, 7] and [15], the differential cross
section for electrodisintegration, summed over the final electron and neutron
helicities in the lab frame, is [14, 15]555In [6], $h$ is $2\lambda_{e}$ but
in [15], $h$ is just $\lambda_{e}$, which leads to additional factors of 2.

$\displaystyle d\sigma$ $\displaystyle\equiv$
$\displaystyle\frac{d\sigma^{5}}{dE^{\prime}d\Omega^{\prime}_{e}d\Omega^{\prime}_{p}}$
$\displaystyle=$
$\displaystyle\frac{m_{p}m_{n}|\vec{p}^{\,\prime}_{p}|}{16\pi^{3}m_{d}}\frac{\sigma_{\rm
Mott}}{f_{\rm
rec}}\left[\nu_{L}R_{L}+\nu_{T}R_{T}+\nu_{TT}R_{TT}+\nu_{LT}R_{LT}+2\lambda_{e}\nu_{LT^{\prime}}T_{LT^{\prime}}+2\lambda_{e}\nu_{T^{\prime}}R_{T^{\prime}}\right],$

where $\Omega^{\prime}_{e}$ ($\Omega^{\prime}_{p}$) is the solid angle of the
scattered electron (proton), $\sigma_{\rm Mott}$ is the Mott cross section,
$f_{\rm
rec}=|1+(E_{\gamma}|\vec{p}^{\,\prime}_{p}|-E^{\prime}_{p}q_{z}\cos\theta_{p})/(m_{d}|\vec{p}^{\,\prime}_{p}|)|$
is the lab recoil factor,

$\displaystyle\nu_{L}$ $\displaystyle=$
$\displaystyle\frac{Q^{4}}{q_{z}^{4}},\;\;\nu_{T}=\frac{Q^{2}}{2q_{z}^{2}}+\tan^{2}\frac{\tilde{\theta}}{2},\;\;\nu_{TT}=\frac{Q^{2}}{2q_{z}^{2}},\;\;\nu_{LT}=\frac{Q^{2}}{\sqrt{2}q_{z}^{2}}\sqrt{\frac{Q^{2}}{q_{z}^{2}}+\tan^{2}\frac{\tilde{\theta}}{2}},$
(70) $\displaystyle\nu_{LT^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{Q^{2}}{\sqrt{2}q_{z}^{2}}\tan\frac{\tilde{\theta}}{2},\;\;\nu_{T^{\prime}}=\tan\frac{\tilde{\theta}}{2}\sqrt{\frac{Q^{2}}{q_{z}^{2}}+\tan^{2}\frac{\tilde{\theta}}{2}},$

and $\tilde{\theta}=\theta^{\prime}_{e}-\theta_{e}$ is the angle between the
incoming and outgoing electron. The response functions $R_{X}$ depend upon the
hadronic helicity amplitudes and the azimuthal angle $\phi_{p}$ of the
hadronic scattering plane. The subscripts refer to the polarization of the
intermediate photon, which enters on substitution of the polarization
expansion (67) for the numerator of the photon propagator in the hadronic
amplitude (66). The amplitude then decomposes into separate leptonic and
hadronic factors

$M_{ed}(\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda^{\prime}_{e};\lambda_{d},\lambda_{e})=-\sum_{\lambda=-1}^{1}\bar{u}^{\prime}_{e}{\centernot{\epsilon}}^{*}(\lambda)u_{e}\frac{(-1)^{\lambda}}{Q^{2}}\epsilon_{\nu}(\lambda)M^{\nu}(\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda_{d}).$
(71)

The leptonic factors give rise to the $\nu_{X}$ coefficients, and the hadronic
factors to the response functions in the square of the amplitude used to
construct the cross section [15]. The subscript L(T) indicates a purely
longitudinal (transverse) contribution, while LT is a cross term between
longitudinal and transverse photon helicities. The TT subscript marks a cross
term between different transverse helicities. A prime indicates a different
combination of transverse helicities.

The response functions are computed from components of the hadronic tensor

$w_{\lambda^{\prime},\lambda}=\frac{2}{3}\sum_{\lambda^{\prime\prime}_{p},\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda^{\prime\prime}_{d},\lambda_{d}}\epsilon_{\nu}^{*}(\lambda^{\prime})M^{\nu*}(\lambda^{\prime\prime}_{p},\lambda^{\prime}_{n},\lambda^{\prime\prime}_{d})\rho^{p}_{\lambda^{\prime\prime}_{p},\lambda^{\prime}_{p}}\epsilon_{\mu}(\lambda)M^{\mu}(\lambda^{\prime}_{p},\lambda^{\prime}_{n},\lambda_{d})\rho^{d}_{\lambda^{\prime\prime}_{d},\lambda_{d}},$
(72)

with $\rho^{p}$($\rho^{d}$) the density matrix for the proton (deuteron)
helicity state. We construct these in the $xyz$ coordinate system of the
electron scattering plane. The particular components are [6]

$\displaystyle R_{L}$ $\displaystyle=$ $\displaystyle
w_{0,0},\;\;R_{T}=w_{1,1}+w_{-1,-1},\;\;R_{T^{\prime}}=w_{1,1}-w_{-1,-1},$
(73) $\displaystyle R_{TT}$ $\displaystyle=$ $\displaystyle 2{\rm
Re}w_{1,-1},\;\;R_{LT}=-2{\rm
Re}\left[w_{0,1}-w_{0,-1}\right],\;\;R_{LT^{\prime}}=-2{\rm
Re}\left[w_{0,1}+w_{0,-1}\right].$

For an unpolarized target, the deuteron density matrix is proportional to the
identity, $\rho^{d}=\frac{1}{3}I$; similarly, if the proton helicity is not
detected, $\rho^{p}=\frac{1}{2}I$. We then have the unpolarized cross section
[6]

$d\sigma_{\rm
unpol}=\frac{m_{p}m_{n}|\vec{p}^{\,\prime}_{p}|}{16\pi^{3}m_{d}}\frac{\sigma_{\rm
Mott}}{f_{\rm
rec}}\sigma_{0},\;\;\sigma_{0}\equiv\nu_{L}R_{L}^{U}+\nu_{T}R_{T}^{U}+\nu_{TT}R_{TT}^{U}+\nu_{LT}R_{LT}^{U},$
(74)

where the $R_{X}^{U}$ are computed with the simple density matrices. These are
then computable in our model, with the basic computation being the evaluation
of $\epsilon(\lambda_{\gamma})_{\mu}M^{\mu}$, which differs from the
photodisintegration calculation in only two ways: $Q^{2}$ is not zero and
$\lambda_{\gamma}$ ranges over all three possibilities.

The unpolarized response functions $R_{LT^{\prime}}^{U}$ and
$R_{T^{\prime}}^{U}$ are identically zero. With $\rho^{d}$ replaced by
$\frac{1}{3}I$ and the form (68) of the polarization vectors taken into
account, ${\rm Re}w(0,1)$ is just the negative of ${\rm Re}w(0,-1)$, and
$w_{1,1}$ is equal to $w_{-1,-1}$. Thus, the inputs to $R_{LT^{\prime}}^{U}$
and $R_{T^{\prime}}^{U}$, as given in (73), immediately cancel.

The recent $ed\rightarrow e^{\prime}pn$ experiment at JLab [17] does not
include polarization but does begin to reach momentum transfers sufficient to
consider the RNHA approach. Once polarization data is available, the
expressions developed here and in the Appendix can be compared.

##  VI Summary

We have extended the reduced nuclear amplitude approach [1, 2] to helicity
amplitudes and applied this model to analysis of elastic electron-deuteron
scattering, deuteron photodisintegration, and deuteron electrodisintegration.
These are just examples of the approach, which is generally applicable to
exclusive nuclear processes. The primary limitation is that, for any process,
the net momentum transfer to every nucleon must be large; therefore, as the
number of nucleons increases, the required beam energy can increase
dramatically. The primary gain is precocious scaling in the dependence on
momentum transfer. What the model (or the original RNA approach) does not
provide, though, is an overall normalization; comparisons must be made in
terms of ratios.

By considering helicity amplitudes, many more quantities can be studied,
including polarization dependence. All three of the deuteron’s electromagnetic
form factors can be calculated and from there various elastic scattering
observables can be constructed. In Sec. III we considered the standard
structure functions $A$ and $B$ as well as the tensor polarizations $t_{2m}$.
Generally, the model implies the need for momentum transfers larger than one
would have hoped for seeing simple perturbative QCD scaling. However, our
results do imply that the deuteron structure function $B$ is a good place to
look, above a transfer of 10 GeV2.

The RNHA results for polarization observables in deuteron photodisintegration,
considered in Sec. IV, are somewhat consistent with experiment. In particular,
our result for the asymmetry $\Sigma$, with a value of
$\Sigma(90^{\circ})\simeq-0.06$, is much better than the value of -1
originally expected [52]. Higher photon energies would, of course, be useful.

We have also constructed the RNHA framework for analysis of deuteron
electrodisintegration, in Sec. V. This stands ready for comparison with
experiment when data is available at sufficient energies. One aspect that does
remain is to consider polarization of the outgoing proton, in addition to
polarization of the beam and target.

Other processes that one might consider include deeply virtual Compton
scattering on the deuteron, pion photoproduction on the deuteron [3], and
photodisintegration of 3He [4]. In each case, our approach can provide not
only information about helicity amplitudes but also an analysis of nonleading
momentum transfer dependence with respect to the onset of perturbative QCD
scaling. We look forward to experiments at larger momentum transfers for all
of these processes.

###### Acknowledgements.

This work began in conversations with S.J. Brodsky and D.-S. Hwang. Some
calculations were checked by W. Miller and C. Salveson. Diagrams were drawn
with use of JaxoDraw [53].

##  Appendix A Electrodisintegration with polarization

If we consider polarization for the beam and the target,666For discussion of a
polarized outgoing proton, see [7] and [15]. the proton density matrix is
still just $\rho^{p}=\frac{1}{2}I$, but the deuteron density matrix in the
$xyz$ frame is [6]

$\rho^{d}=\frac{1}{3}\left(\begin{array}[]{ccc}1+\sqrt{\frac{3}{2}}T_{10}+\frac{1}{\sqrt{2}}T_{20}&-\sqrt{\frac{3}{2}}(T_{11}^{*}+T_{21}^{*})&\sqrt{3}T_{22}^{*}\\\
-\sqrt{\frac{3}{2}}(T_{11}+T_{21})&1-\sqrt{2}T_{20}&-\sqrt{\frac{3}{2}}(T_{11}^{*}-T_{21}^{*})\\\
\sqrt{3}T_{22}&-\sqrt{\frac{3}{2}}(T_{11}-T_{21})&1-\sqrt{\frac{3}{2}}T_{10}+\frac{1}{\sqrt{2}}T_{20}\end{array}\right).$
(75)

For a target polarization defined relative to the beam direction, rather than
the $xyz$ system used above, the tensor polarization coefficients $T_{JM}$ are
related to the coefficients $\tilde{T}_{JM}$ defined relative to the beam [6].
If only $\tilde{T}_{10}$ and $\tilde{T}_{20}$ are nonzero,777The spherical
tensor moments are related to the Cartesian tensor moments as
$\tilde{T}_{10}=\sqrt{\frac{3}{2}}P_{z}$ and
$\tilde{T}_{20}=\frac{1}{\sqrt{2}}P_{zz}$. the nonzero $T_{JM}$ are

$\displaystyle T_{10}$ $\displaystyle=$
$\displaystyle\cos\tilde{\theta}\,\tilde{T}_{10},\;\;T_{11}=-\frac{1}{\sqrt{2}}\sin\tilde{\theta}\,\tilde{T}_{10},$
(76) $\displaystyle T_{20}$ $\displaystyle=$ $\displaystyle\frac{1}{4}(1+3\cos
2\tilde{\theta})\tilde{T}_{20},\;\;T_{21}=-\sqrt{\frac{3}{8}}\sin
2\tilde{\theta}\,\tilde{T}_{20},\;\;T_{22}=\sqrt{\frac{3}{32}}(1-\cos
2\tilde{\theta})\tilde{T}_{20}.$

The density matrix can then be written as

$\rho^{d}=\left(\frac{1}{3}I+\tilde{T}_{10}\rho^{dV}+\tilde{T}_{20}\rho^{dT}\right),$
(77)

where

$\rho^{dV}=\frac{1}{3}\left(\begin{array}[]{ccc}\sqrt{\frac{3}{2}}\cos\tilde{\theta}&\frac{\sqrt{3}}{2}\sin\tilde{\theta}&0\\\
\frac{\sqrt{3}}{2}\sin\tilde{\theta}&0&\frac{\sqrt{3}}{2}\sin\tilde{\theta}\\\
0&\frac{\sqrt{3}}{2}\sin\tilde{\theta}&-\sqrt{\frac{3}{2}}\cos\tilde{\theta}\end{array}\right)$
(78)

and

$\rho^{dT}=\frac{1}{3}\left(\begin{array}[]{ccc}\frac{1}{4\sqrt{2}}(1+3\cos
2\tilde{\theta})&\frac{3}{4}\sin 2\tilde{\theta}&\frac{3}{\sqrt{32}}(1-\cos
2\tilde{\theta})\\\ \frac{3}{4}\sin
2\tilde{\theta}&-\frac{1}{2\sqrt{2}}(1+3\cos 2\tilde{\theta})&-\frac{3}{4}\sin
2\tilde{\theta}\\\ \frac{3}{\sqrt{32}}(1-\cos
2\tilde{\theta})&-\frac{3}{4}\sin 2\tilde{\theta}&\frac{1}{4\sqrt{2}}(1+3\cos
2\tilde{\theta})\end{array}\right).$ (79)

The response functions can then be separated into unpolarized, vector, and
tensor contributions as
$R_{X}=R_{X}^{U}+\tilde{T}_{10}R_{X}^{V}+\tilde{T}_{20}R_{X}^{T}$, with
$R_{X}^{U}$, $R_{X}^{V}$, and $R_{X}^{T}$ computed with $\rho^{d}$ replaced by
$\frac{1}{3}I$, $\rho^{dV}$, and $\rho^{dT}$, respectively.

With $d\sigma_{\rm unpol}$ defined as the unpolarized cross section, given in
(74), the full cross section can be written as

$d\sigma=\left[1+\tilde{T}_{10}\left(A_{d}^{V}+2\lambda_{e}A_{ed}^{V}\right)+\tilde{T}_{20}\left(A_{d}^{T}+2\lambda_{e}A_{ed}^{T}\right)\right]d\sigma_{\rm
unpol},$ (80)

in terms of the single and double asymmetries

$\displaystyle A_{d}^{V}$ $\displaystyle=$
$\displaystyle\left[\nu_{L}R_{L}^{V}+\nu_{T}R_{T}^{V}+\nu_{TT}R_{TT}^{V}+\nu_{LT}R_{LT}^{V}\right]/\sigma_{0},$
(81) $\displaystyle A_{ed}^{V}$ $\displaystyle=$
$\displaystyle\left[\nu_{LT^{\prime}}R_{LT^{\prime}}^{V}+\nu_{T^{\prime}}R_{T^{\prime}}^{V}\right]/\sigma_{0},$
(82) $\displaystyle A_{d}^{T}$ $\displaystyle=$
$\displaystyle\left[\nu_{L}R_{L}^{T}+\nu_{T}R_{T}^{T}+\nu_{TT}R_{TT}^{T}+\nu_{LT}R_{LT}^{T}\right]/\sigma_{0},$
(83) $\displaystyle A_{ed}^{T}$ $\displaystyle=$
$\displaystyle\left[\nu_{LT^{\prime}}R_{LT^{\prime}}^{T}+\nu_{T^{\prime}}R_{T^{\prime}}^{T}\right]/\sigma_{0}.$
(84)

For a recent summary of data, see [54].

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