#  Target mass corrections in lepton-nucleus DIS:
theory and applications to nuclear PDFs

R. Ruiz  K.F. Muzakka  C. Leger  P. Risse  A. Accardi  P. Duwentäster  T.J.
Hobbs  T. Ježo  C. Keppel  M. Klasen  K. Kovařík  A. Kusina  J.G. Morfín  F.I.
Olness  J.F. Owens,  I. Schienbein  J.Y. Yu.

###### Abstract

Motivated by the wide range of kinematics covered by current and planned deep-
inelastic scattering (DIS) facilities, we revisit the formalism, practical
implementation, and numerical impact of target mass corrections (TMCs) for DIS
on unpolarized nuclear targets. An important aspect is that we only use
nuclear and later partonic degrees of freedom, carefully avoiding a picture of
the nucleus in terms of nucleons. After establishing that formulae used for
individual nucleon targets $(p,n)$, derived in the Operator Product Expansion
(OPE) formalism, are indeed applicable to nuclear targets, we rewrite
expressions for nuclear TMCs in terms of re-scaled (or averaged) kinematic
variables. As a consequence, we find a representation for nuclear TMCs that is
approximately independent of the nuclear target. We go on to construct a
single-parameter fit for all nuclear targets that is in good numerical
agreement with full computations of TMCs. We discuss in detail qualitative and
quantitative differences between nuclear TMCs built in the OPE and the parton
model formalisms, as well as give numerical predictions for current and future
facilities.

††journal: Progress in Particle and Nuclear Physics

## 1 Introduction

Deep-inelastic scattering (DIS) of high-energy leptons off hadrons is a key
process for studying the structure of hadrons in terms of their quark and
gluon degrees of freedom [1, 2, 3, 4]. For both nucleons [5, 6, 7, 8, 9, 10]
and nuclei [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28], present data from electron and neutrino DIS form the backbone of
modern determinations of parton distribution functions (PDFs), which describe
and encode the internal structure of hadrons. This process will again be at
the forefront of the future Electron-Ion Collider (EIC) [29, 30], where DIS
off nucleons and various nuclear targets will be studied with high precision
over a wide kinematic range. Similar endeavors are also in discussion for the
proposed Forward Physics Facility [31]. Additionally, high-precision fixed-
target measurements of JLab can provide complementary information [32]. Beyond
this, DIS also plays an important role at current and future accelerator-based
neutrino oscillation facilities such as DUNE and the Short-Baseline Neutrino
Program at Fermilab [33, 34, 35], where experiments operate at somewhat lower
energies compared to the EIC, and where the (quasi)elastic and resonant
processes must also be taken into account. (Here, precise control over the
contributions from each of these processes is essential, and duality-based
arguments for the scaling of DIS cross sections at higher energies into the
resonance region can play an important role [36, 37].) Due to the relative
weakness of the neutrino-nucleon effective interaction, neutrino experiments
typically use nuclear targets to enhance event rates. Consequently, charged
lepton-nucleus and neutrino-nucleus interactions need to be precisely
understood, particularly the differences from nucleon interactions, to reach
the ambitious physics goals of future experiments.

Along these lines, there are two major theoretical approaches to lepton-
nucleon DIS that are standard material in textbooks and review articles; both
have been employed to include the effects of target nucleons having nonzero
masses. One approach is based on the operator product expansion (OPE) [38, 39,
40]. The other is the QCD-improved parton model rigorously derived from first
principles of QCD in the context of the collinear factorization theorems [41,
42, 43, 44, 45, 46, 47, 48]. The OPE was first employed to derive target mass
corrections (TMCs) to structure functions in DIS at leading order (LO) in QCD
in the seminal paper by Georgi and Politzer in 1976 [49, 50]. In the same year
Barbieri, et al. [51, 52] derived TMCs at LO including effects arising from
non-zero quark masses. Later, the original work of Georgi and Politzer was
extended to next-to-leading order (NLO) in QCD for the structure functions
$W_{1,2}$ in the context of local duality in electro-production and using off-
shell regularization [53]. More recently, Kretzer and Reno presented TMCs for
charged current (CC) and weak neutral current (NC) neutrino-nucleon DIS,
including NLO QCD corrections and heavy quark mass effects using modern
conventions [54].

In the context of the parton model, Ellis, Furmanski, and Petronzio derived
TMCs at LO in QCD in the framework of a non-collinear version of the parton
model that includes the effects of partonic transverse momentum ($k_{T}$)
[55]. Agreement with the OPE-based results [51, 52] was found at leading
power, thereby demonstrating the equivalence of the OPE at LO in QCD to a non-
collinear parton approach, where the incident parton is on-shell but not
collinear with its parent nucleon. We note also that TMC prescriptions based
on the OPE and the factorization approach of Ellis, Furmanski, and Petronzio
were similarly compared for semi-inclusive processes in Ref. [56]. In the
collinear parton model ($k_{T}=0$), TMCs have been accounted for in the
Aivazis-Collins-Olness-Tung (ACOT) formalism [57, 58], which is rigorously
based on the factorization theorem including heavy quark masses [46].
Furthermore, the full QCD framework for the evaluation of tau-neutrino deep-
inelastic charged current cross sections, including NLO corrections, charm
production, tau mass threshold, and target mass effects in the collinear
approximation, has also been presented [59].

An earlier review [60] of TMCs for nucleons in unpolarized DIS culminated in a
set of so-called master formulas [54], which are now updated and presented
below for nuclei in Eq. (4.15). This set of equations is quite remarkable,
organizing the rather complicated expressions in the aforementioned OPE-based
papers into a simple, easy-to-use, and modular form, valid at all orders in
perturbation QCD while still taking into account quark masses. In principle,
TMCs derived for DIS off nucleons [49, 50, 61, 54, 60] apply also to DIS off
nuclear targets since the OPE is (in principle) independent of target states.
However, previous discussions [60] do not address the subtle distinctions
between a nucleon and nuclear target: the fact that established notation does
not consistently distinguish between nucleons and nuclei in their respective
kinematics, the fact that the spin of a generic nucleus can be different from
1/2, and the fact that the master equations in Eq. (4.15) are expressed in
terms of “averaged nucleon mass” has led to the question, why nucleon-like
expressions could possibly be valid for nuclear targets?

With an abundance of recent nuclear data from both electron and neutrino
beams, and with the EIC and 3rd-generation long-baseline oscillations
experiments on the horizon, there is a compelling need to rigorously revisit
the derivation of TMCs to structure functions in DIS with a particular focus
on the nuclear case. One of the main goals for this article is to properly
demonstrate the validity of the TMC master equations for the case of massive
nuclear targets within the framework of the OPE. Compared to past works on the
subject [54, 60], here we present the derivation of TMCs from the OPE in much
greater detail. We hope that this material will be useful for students and
researchers looking for a modern, in-depth discussion of the many
technicalities that are frequently omitted in the literature.

In the following, we discuss and contrast the results for TMCs obtained in
both the OPE and the collinear parton model; throughout this review, we
consider them on equal footing. As discussed above, the collinear parton model
is rigorously based on factorization theorems. These provide field-theoretic
definitions of PDFs and make statements about the error of the factorization
approximation, which is generally inversely proportional to a positive power
of a hard scale of the process.111Such power-suppressed terms in DIS structure
functions are generally called “higher twist” terms. In the context of the OPE
we define the twist $\tau=d-s$ of a field operator $\hat{\mathcal{O}}$ in the
canonical fashion, i.e., the dimension $d$ of $\hat{\mathcal{O}}$ (in the
usual power-counting sense) minus its spin $s$ (the number of un-contracted
Lorentz indices).  It is generally believed that collinear factorization
remains valid in lepton-nucleus and proton-nucleus collisions, possibly with
nuclear-enhanced higher twist terms [62]. However, the literature on
factorization in the nuclear case is sparse and we consider this a working
assumption. Moreover, the non-collinear parton model (where the TMCs were
shown to be equivalent to the OPE results at leading power) is not covered by
the factorization proofs cited above. While the application of the OPE to DIS
is widely accepted, proofs of the OPE have only been presented for simple
scalar models [63, 64]. The question as to whether this result can be extended
to QCD in general remains open to the best of our knowledge. Exploring these
more theoretical questions is interesting and relevant but beyond the scope of
this article. Here, we take the collinear parton model and the OPE for
granted, and explore extensively what happens when we transition from nucleons
to nuclei.

The starting point of our analysis is to consider the full nucleus as our
target and apply only general symmetry principles, e.g., Lorentz invariance,
in deriving nuclear structure functions and their TMCs. This means that until
Sec. 4 there is no reference to (or dependence on) nucleon degrees of freedom.
Furthermore, until Sec. 5 there is no reference to (or dependence on) partonic
degrees of freedom.222Explicitly, until Sec. 5 we consider operators and
matrix elements derived from quark, antiquark, and gluon fields but do not
identify these as partons, nor identify structure functions as combinations of
PDFs, i.e., the parton model. To do this, we first outline in Sec. 2 key
kinematic relations and definitions in DIS of a lepton off a nucleus,
$\ell_{1}(k_{1})+A(p_{A})\to\ell_{2}(k_{2})+X(p_{X})$. As depicted in Fig.
1.1, $A$ is a nuclear target with mass number $A$, $\ell$ denotes either a
charged lepton or neutrino, and $X$ represents all final-state hadrons. In
Sec. 3, we discuss precisely the criteria for light-cone dominance in nuclear
DIS, and then present in Sec. 3.3 a formula for nuclear TMCs. The result is
analogous to the nucleon case [60], but expressed in terms of the nuclear
scaling variable $x_{A}$ and the mass of the nucleus $M_{A}$.

Kinematic variable Description $\nu_{A}=\dfrac{q\cdot
p_{A}}{M_{A}}\;\stackrel{{\scriptstyle lab}}{{=}}\;E_{\ell_{1}}-E_{\ell_{2}}$
Lepton energy loss in the laboratory (nucleon rest) frame $y_{A}=\dfrac{q\cdot
p_{A}}{k_{1}\cdot p_{A}}\;\stackrel{{\scriptstyle
lab}}{{=}}\;\dfrac{\nu_{A}}{E_{\ell_{1}}}$ Inelasticity $y_{A}\in[0,1]$
$Q^{2}=-q^{2}>0$ Boson squared momentum transfer
$x_{A}=\dfrac{Q^{2}}{2p_{A}\cdot q}=\dfrac{Q^{2}}{2M_{A}\nu_{A}}$ Bjorken
$x_{A}$ with $x_{A}\in[0,1]$
$W^{2}=(p_{A}+q)^{2}=M_{A}^{2}+Q^{2}\dfrac{1-x_{A}}{x_{A}}$ Mass squared of
the recoil system
$s=(k_{1}+p_{A})^{2}={\dfrac{Q^{2}}{x_{A}\,y_{A}}}+M_{A}^{2}+m_{\ell}^{2}$
Center of Mass System (CMS) energy squared Figure 1.1: We consider the basic
charged current ($V{=}W^{\pm}$) or neutral current ($V{=}\\{\gamma,Z\\}$)
lepton-nucleus DIS process
$\ell_{1}(k_{1})+A(p_{A})\to\ell_{2}(k_{2})+X(p_{X})$ where the lepton can be
a charged lepton (electron, muon) or a neutrino
$\nu_{e},\nu_{\mu},\nu_{\tau}$. The 4-momentum of the exchange boson is
denoted $q=k_{1}-k_{2}$, and $p_{A}^{2}=M_{A}^{2}$.

In Sec. 4, we go on to perform a rescaling in order to express our formula for
nuclear TMCs to the more familiar averaged nucleon quantities $x_{N}=Ax_{A}$
and $M_{N}=M_{A}/A$. This is the key step that allows us to compare structure
functions across different nuclei, including the proton, in a meaningful way.
As an important consequence of rescaling, we obtain an alternative formula for
nuclear TMCs that is universally applicable to all nuclei. In Sec. 5 we
introduce the QCD-improved parton model, both before and after rescaling. We
show that rescaling at the hadronic level and in the parton model are
consistent and we comprehensively discuss the relationship between nuclear
structure functions and nuclear PDFs (nPDFs). We believe that the discussions
in Secs. 4 and 5, while relatively straight forward, are original and have
never been presented in full detail. It is key for a proper theoretical
definition of nuclear structure functions and nPDFs as they are intuitively
used in the literature.

In Sec. 6, we derive nuclear TMCs in the ACOT light cone formalism, which has
been used in past nCTEQ analyses [21, 19, 20, 17, 16, 65, 66], and compare
them to the OPE results. Having provided rigorous theoretical definitions for
the physical observables, we perform numerical studies in Sec. 7. This
includes a comparison to a selection of data, as well as cross section
predictions for current and future DIS experiments. We provide a
parameterization that is accurate at the sub-percent level and can be used
with any underlying set of nuclear structure functions available in the
massless parton model to obtain the full TMCs for any nucleus in a simple way.
We hope that this parameterization will be useful for the community working on
hadron structure. Finally, in Sec. 8 we highlight the key observations of this
analysis and conclude.

A detailed derivation of TMCs at twist $\tau=2$ for nuclear structure
functions using the OPE is provided in A. Finally, technical details on our
parameterization of the TMCs have been relegated to B.

## 2 Kinematics of lepton-nucleus DIS

Before proceeding with an in-depth treatment of nuclear structure functions,
we layout in this section the definitions, notation, and baseline assumptions
used throughout this work. (However, assumptions related to light-cone
dominance are discussed in Sec. 3.1.) While much of the following is standard
material for modern textbooks, many conventions are used in the literature.
Therefore, this section serves (a) to fix the notation and conventions that we
use, and (b) to make the work both self-consistent and self-contained. We
start with definitions of kinematic variables in Sec. 2.1 and move onto the
definitions of leptonic and hadronic tensors for DIS in Sec. 2.2.

### 2.1 Definitions of kinematic variables

Throughout this work we consider the basic charged-current (CC) or neutral-
current (NC) lepton-nucleus DIS process involving a high-energy lepton
$\ell_{1}(k_{1})$ scattering off an unpolarized nuclear target $A(p_{A})$ of
mass $M_{A}$, with an outgoing lepton $\ell_{2}(k_{2})$ in association with
the inclusive hadronic remnant $X(p_{X})$ of mass $W=\sqrt{p_{X}^{2}}$. As
sketched in Fig. 1.1, this is given by the hadron-level expression

$\displaystyle\ell_{1}(k_{1})+A(p_{A})\to\ell_{2}(k_{2})+X(p_{X})\ .$ (2.1)

We consider that the incoming lepton can be a light charged lepton
$\\{e^{\pm},\mu^{\pm}\\}$, a neutrino $\\{\nu_{e},\nu_{\mu},\nu_{\tau}\\}$, or
an antineutrino
$\\{\overline{\nu_{e}},\overline{\nu_{\mu}},\overline{\nu_{\tau}}\\}$ in the
Standard Model. For clarity, we work in the limit that all incoming leptons
are massless but allow that the outgoing lepton may instead be a $\tau^{\pm}$
lepton. We denote the atomic and mass numbers of nucleus $A$ by $(Z,A)$,
respectively. $X(p_{X})$ represents the fragmentation of $A$. It is an
$n$-body final state with net quantum numbers corresponding to a colorless
state carrying the same spin statistic, QED charge, and weak isospin charge as
$A$, modulo differences between $\ell_{1}$ and $\ell_{2}$. The momentum of $X$
can be parameterized by $p_{X}=\sum^{n}_{i=1}p_{Xi}$.

We work at lowest order in the electroweak theory, in the so-called “one-
boson-exchange” approximation. Under this assumption, the leptonic system
$(\ell_{2}\ell_{1})$ probes the hadronic system via the exchange of a single,
electroweak boson $V$ ($V\in\\{\gamma,Z\\}$ or $V=W^{\pm}$) with time-like
momentum $q=k_{1}-k_{2}=p_{X}-p_{A}$. The kinematic invariants and scaling
variables of DIS are given and defined by

$\displaystyle k_{1}^{2}=m_{\ell_{1}}^{2},\qquad k_{2}^{2}\
=m_{\ell_{2}}^{2},\qquad p_{A}^{2}$ $\displaystyle=M_{A}^{2}>0,\qquad
Q^{2}=-q^{2}>0,\qquad s=(k_{1}+p_{A})^{2},\qquad$ (2.2a) $\displaystyle
x_{A}=\frac{Q^{2}}{2p_{A}\cdot q},\qquad\nu_{A}$
$\displaystyle=\frac{p_{A}\cdot q}{M_{A}},\qquad y_{A}=\frac{p_{A}\cdot
q}{p_{A}\cdot k},$ (2.2b) $\displaystyle W^{2}=p_{X}^{2}=(p_{A}+q)^{2}$
$\displaystyle=M_{A}^{2}-Q^{2}+2\nu_{A}M_{A}=M_{A}^{2}+Q^{2}\frac{1-x_{A}}{x_{A}}\geq
M_{A}^{2}.$ (2.2c)

We generally neglect the lepton masses $\\{m_{\ell_{1}},m_{\ell_{2}}\\}$
except for the tau-lepton. The inelasticity variable $(y_{A})$ and the Bjorken
scaling variable $(x_{A})$ take the values $x_{A},y_{A}\in[0,1]$, with
$x_{A}=1$ corresponding to the elastic limit where $W^{2}=M_{A}^{2}$.

For later use with TMCs to structure functions (Secs. 3.3 and 4), we also
define the quantity

$r_{A}\ =\ \sqrt{1+4x_{A}^{2}M_{A}^{2}/Q^{2}}\ =\
1+\frac{2x_{A}^{2}M_{A}^{2}}{Q^{2}}+\mathcal{O}\left(\frac{4x_{A}^{4}M_{A}^{4}}{Q^{4}}\right)\,,$
(2.3)

as well as the so-called Nachtmann variable [67]

$\xi_{A}\ =\ \frac{2}{1+r_{A}}\,x_{A}\equiv R_{M}x_{A}\ ,\qquad{\rm
where}\qquad R_{M}=\frac{2}{1+r_{A}}\quad.$ (2.4)

$R_{M}$ is the target-mass dependent factor relating the Bjorken scaling
variable $x_{A}$ to the Nachtmann scaling variable $\xi_{A}$. Note that in the
limit that $(M_{A}/Q)\to 0$ or that $x_{A}\to 0$, we have $r_{A}\to 1$,
$R_{M}\to 1$, and $\xi_{A}\to x_{A}$ since
$\xi_{A}=x_{A}(1-x_{A}^{2}M_{A}^{2}/Q^{2})+\mathcal{O}(x_{A}^{4}M_{A}^{4}/Q^{4})$.

The difference between $\xi_{A}$ and $x_{A}$ is illustrated in Fig. 2.1, where
$\xi_{A}$ is plotted as a function of $x_{A}$ for a selection of exchange-
boson virtualities $Q$ with $M_{A}=M_{\rm proton}$. Fig. 2.1(a) displays the
full $\\{\xi_{A},x\\}$ plane. Fig. 2.1(b) highlights the large $x$ region,
where the difference between $\xi_{A}$ and $x$ is more pronounced.
Qualitatively, $x_{A}$ and $\xi_{A}$ are largely indistinguishable for
$x_{A}\lesssim 0.4-0.5$ over a large range of $Q$ and $M_{A}$. Quantitatively,
at around $x_{A}=0.8$, one has approximately $\xi_{A}\approx
0.63~{}(0.71)~{}[0.76]$ for $Q=1.3~{}(2)~{}[3]{\rm~{}GeV}$. For the same $Q$
values, one has approximately $\xi_{A}\approx 0.76~{}(0.78)~{}[0.84]$ at
around $x_{A}=0.9$.

(a) (b)

Figure 2.1: (a) The Nachtmann scaling variable $\xi_{A}(x_{A},Q)$ as a
function of the Bjorken scaling variable $x_{A}$ for a selection of $Q$ values
[GeV], as indicated in the legend, with $M_{A}=M_{\rm proton}$. (b) Same as
(a) but for large $x_{A}$.

### 2.2 Leptonic and hadronic tensors in DIS from experimentally observable
kinematics

Deeply inelastic scattering is a powerful, elucidating probe of the internal
structure of the proton and nuclei, i.e., hadrons. The ability stems from the
fact that under strong but general assumptions one can write the cross section
for any sufficiently inclusive DIS process
$\ell_{1}+A\xrightarrow{V}\ell_{2}+X$ as a combination of: (a) leptonic and
hadronic kinematics, which can be measured, and (b) hadronic structure
functions, which parameterize the internal dynamics of $A$. This discussion is
based solely on kinematics and symmetries, including Lorentz symmetry. The
only approximation of consequence that is made when constructing structure
functions from experimentally observable kinematics is the one where DIS is
mediated by the exchange of only one electroweak boson. (Relaxing this has
been explored elsewhere [68, 69, 70, 71].) Apart from this assumption,
expressions derived from kinematics are completely general: they take into
account all twist contributions, at all orders in the strong coupling
constant, and all hadron- and quark- mass effects. And in particular, the
parton model does not need to be invoked.

Schematically, the differential DIS cross section $d\sigma$ is proportional to
the product of a leptonic tensor $L^{\mu\nu}(k_{1},k_{2})$, which is the
square of the $\ell_{1}\to\ell_{2}V$ splitting matrix element (the matrix
element of the leptonic current), and a hadronic tensor
$\tilde{W}^{A}_{\mu\nu}(p_{A},q)$, which is proportional to the square of the
$AV\to X$ matrix element (the matrix element of the hadronic current).
$d\sigma$ can be written as

$d\sigma~{}\propto~{}\tilde{W}^{A}_{\mu\nu}\ L^{\mu\nu}\quad,$ (2.5)

and is given more completely by Eq. (A.6) in A.1. The tilde $(\sim)$ notation
will be made clear in Sec. 4 when we perform a rescaling of nuclear momentum
to average nucleon momentum.

The leptonic tensor can be constructed from Feynman diagrams with standard
technology. For the case of massless charged leptons exchanging a photon with
the nucleus $A$, the spin-summed leptonic tensor for massless leptons is

$\sum_{\\{\lambda\\}}L^{\mu\nu}\Big{|}_{\rm
QED}=4e^{2}\Big{\\{}k_{1}^{\mu}k_{2}^{\nu}+k_{1}^{\nu}k_{2}^{\mu}-(k_{1}\cdot
k_{2})g^{\mu\nu}\Big{\\}}\quad.$ (2.6)

Here, $\\{\lambda\\}$ represents the sum over spins of external particles;
spin-averaging is not yet performed. Analogous expressions for the exchange of
other bosons can be found in A.2.

In practice, the tensor $\tilde{W}^{A}_{\mu\nu}(p_{A},q)$, which can only be a
function of external momenta $p_{A}$ and $q$ for unpolarized targets, is
decomposed into a sum over tensor-valued coefficient functions multiplied by
dimensionless, scalar-valued functions $\tilde{W}_{i}$. These $\tilde{W}_{i}$
are known in the literature as structure functions. Lorentz symmetry and
Hermiticity dictate that only certain combinations of $p_{A}$ and $q$ are
allowed in the coefficient functions. For the case of an unpolarized (spin-
averaged) nuclear target, the “hadronic” tensor
$\tilde{W}^{A}_{\mu\nu}(p_{A},q)$, in terms of the square of the $AV\to X$
current $J(z)$ in coordinate space, is given by

$\displaystyle\tilde{W}^{A}_{\mu\nu}(p_{A},q)\quad$
$\displaystyle\equiv\frac{1}{4\pi}\sumint d^{4}z\ e^{iq\cdot z}\langle
A(p_{A})|\,J_{\mu}(z)|X(p_{X})\rangle\langle
X(p_{X})|J_{\nu}(0)\,|A(p_{A})\rangle\ $ (2.7)
$\displaystyle=-g_{\mu\nu}\tilde{W}_{1}+\frac{p_{A\mu}p_{A\nu}}{M_{A}^{2}}\tilde{W}_{2}-i\epsilon_{\mu\nu\rho\sigma}\frac{p_{A}^{\rho}q^{\sigma}}{M_{A}^{2}}\tilde{W}_{3}$
$\displaystyle+\frac{q_{\mu}q_{\nu}}{M_{A}^{2}}\tilde{W}_{4}+\frac{p_{A\mu}q_{\nu}+p_{A\nu}q_{\mu}}{M_{A}^{2}}\tilde{W}_{5}+\frac{p_{A\mu}q_{\nu}-p_{A\nu}q_{\mu}}{M_{A}^{2}}\tilde{W}_{6}\
.$ (2.8)

In Eq. (2.7), the normalization factor $1/(4\pi)$ is conventional and the sum
and integral $(\sumint)$ run over all discrete and continuous configurations
of $X$, implying that $\tilde{W}^{A}_{\mu\nu}(p_{A},q)$ is inclusive with
respect to $X$ (see also A.2). We note that, although the targets considered
in this work are nuclei, we refer to quantities such as that given in Eq.
(2.7) as a hadronic tensors, in keeping with convention. For a polarized
target, the decomposition into structure functions takes on a more complicated
structure, see e.g. [61, 72, 29, 73, 74, 75]. In Sec. 3.2 and A.5, we review
the connection between the $\tilde{W}_{i}$ and the internal structure of $A$.

Modern notation calls for using the structure functions $\tilde{F}_{i}$ rather
than $\tilde{W}_{i}$. The mapping between the two sets of dimensionless
structure functions is given by

$\displaystyle\left\\{\tilde{F}_{1},\ \tilde{F}_{2},\ \tilde{F}_{3},\
\tilde{F}_{4},\ \tilde{F}_{5,6}\right\\}\ $ $\displaystyle\qquad~{}\qquad=\
\left\\{\tilde{W}_{1},\ \frac{Q^{2}}{2x_{A}M_{A}^{2}}\tilde{W}_{2},\
\frac{Q^{2}}{x_{A}M_{A}^{2}}\tilde{W}_{3},\
\frac{Q^{2}}{2M_{A}^{2}}\tilde{W}_{4},\
\frac{Q^{2}}{2x_{A}M_{A}^{2}}\tilde{W}_{5,6}\right\\}\ .$ (2.9)

The purpose of using $\tilde{F}_{i}$ is to factor out known dependence on
$Q^{2}$ and make more manifest the phenomena of scaling, i.e., that
$\tilde{F}_{i}$ depend only on $x_{A}$, a dimensionless quantity, up to small,
logarithmic QCD corrections. In the discussion that follows, we focus on
$\tilde{F}_{1},\tilde{F}_{2}$, and $\tilde{F}_{3}$. However, we include a
detailed discussion of $\tilde{F}_{4}$ and $\tilde{F}_{5}$ in the Appendix.
The structure functions $\tilde{F}_{4}$ and $\tilde{F}_{5}$ enter into
differential cross sections, but are suppressed by a factor
$\mathcal{O}(\frac{m_{\ell}^{2}}{M_{A}E_{\ell}})$, where $m_{\ell}^{2}$ is the
lepton mass squared, $M_{A}$ is the hadronic mass, and $E_{\ell}$ is the
energy of one of the external leptons [54]. This suppression is a consequence
of contracting the symmetric leptonic tensor $L^{\mu\nu}$ with $q_{\mu}$ (or
$q_{\nu}$), which subsequently vanishes due to the conservation of weak
currents by massless leptons. Notably, finite lepton-mass effects could be
measured in $\nu_{\tau}$-DIS, such as at the SHIP, FASER, or SND@LHC detectors
at CERN [76, 77, 78, 31]. As for $\tilde{F}_{6}$, which signifies charge-
parity violation, the coefficient vanishes when contracted with Eq. (2.6),
hence it does not contribute to the cross-section.

#### Considerations for Spin 1 and Greater

Contrary to a nucleon target with spin-1/2, nuclei can have spin-1 or greater.
The case of a spin-1 nuclear targets in NC DIS with charged leptons has been
discussed since the 80s[79]. At leading twist, i.e., twist $\tau=2$, the
additional effects of scattering on a polarized spin-1 target reside in a
single new structure function $\tilde{b}_{1}(x)$. This structure function
effectively measures the extent to which a target nucleus deviates from a
trivial bound state of protons and neutrons. For the deuteron, it is expected
that $\tilde{b}_{1}\approx 0$, but for other nuclei one could have
$\tilde{b}_{1}\sim{\cal O}(\tilde{F}_{1})$.

More generally, using gauge invariance and P- and T-invariance for the spin-1
case, the hadronic tensor can be expressed in terms of eight independent
structure functions,
$\\{\tilde{F}_{1},\tilde{F}_{2},\tilde{b}_{1,2,3,4},\tilde{g}_{1},\tilde{g}_{2}\\}$.
Similar results are found for the hadronic tensor of a (space-like) virtual
photon target [80]. The functions $\tilde{F}_{1}$, $\tilde{F}_{2}$,
$\tilde{g}_{1}$, and $\tilde{g}_{2}$ are analogous to the scaling structure
functions of a spin-1/2 target. $\tilde{F}_{1}$ and $\tilde{F}_{2}$ can be
measured in the DIS of unpolarized leptons off an unpolarized target whereas
the measurement of $\tilde{g}_{1}$ and $\tilde{g}_{2}$ requires polarized
lepton beams and a polarized target. The four structure functions
$\tilde{b}_{1,2,3,4}$ are new in the spin-1 case. $\tilde{b}_{1,2}$ are
quantities that appear at leading twist, while $\tilde{b}_{3,4}$ appear at
higher twist. $\tilde{b}_{1,2}$ are connected via
$\tilde{b}_{2}=2x\tilde{b}_{1}$ in a manner analogous to the Callan-Gross
relation, and will receive corrections beyond lowest order in QCD. The
structure functions $\tilde{b}_{1,2,3,4}$ can be measured using an unpolarized
lepton beam but require a polarized spin-1 target. The TMCs for a (spin-1)
deuterium target have been calculated [81] as well as those for a virtual
photon target [82].

In this work we are studying unpolarized DIS off an unpolarized nucleus with
any spin greater or equal to $1/2$. For such circumstances, the hadronic
tensor has the same decomposition into structure functions as for a spin-$1/2$
target. For this reason we will focus only on the $\tilde{F}_{i}$ structure
functions in the remainder of this text; the treatment of additional structure
functions, such as $\tilde{b}_{i}$, can be dealt with in a similar manner, but
we shall not address them explicitly.

## 3 Nuclear structure functions in the OPE

In this section, we discuss nuclear structure functions and their TMCs in the
context of the OPE. We start in Sec. 3.1 with a precise stipulation of the
criteria for light-cone dominance involving nuclear targets, whose masses can
readily exceed $\mathcal{O}(50-100){\rm~{}GeV}$. In Sec. 3.2, we sketch the
construction of nuclear structure functions and their TMCs in the OPE; a
fuller derivation is given in A. We then present in Sec. 3.3 the main formula
for nuclear TMCs in terms of nuclear quantities $x_{A}$ and $M_{A}$.

### 3.1 Light-cone dominance of nuclear DIS

A crucial step in deriving structure functions for nucleons and nuclei
involves employing the OPE to expand the product of currents that comprise the
hadronic tensor of Eq. (2.7). This expansion, however, requires that one is in
the limit of light-cone dominance, i.e., $z^{2}\sim 0$, where $z$ is the
Fourier conjugate of the DIS momentum $q$. While this is well-established in
the case of a nucleon target,333A demonstration of the light cone dominance of
DIS can be found, for example, in the textbook by Muta [63, 64], see (2nd
edition) pages 228-229 and 262.  demonstrating light-cone dominance for
nuclear targets requires care due to the relative sizes of $Q^{2}=-q^{2}>0$
and $M_{A}^{2}$.

In many constructions of PDFs and related quantities in DIS, light cone-
dominance is described as corresponding to the momentum configuration

$Q^{2}\sim|p_{N}\cdot q|\gg M_{N}^{2}\,,$ (3.1)

where $p_{N}$ and $M_{N}$ are respectively the four-momentum and mass of a
single nucleon. Under this condition, naively replacing $p_{N}$ and $M_{N}$ by
the nuclear momentum $p_{A}$ and nuclear mass $M_{A}$ would imply
(incorrectly) that light-cone dominance is only satisfied when

$Q^{2}\sim|p_{A}\cdot q|\gg M_{A}^{2}\,.$ (3.2)

This suggests increasingly large $Q^{2}$ are needed for increasing $A$, and
downplays the validity of both pQCD and the parton model. For example: Eq.
(3.2) implies that $Q\gg 50{\rm~{}GeV}$, which is more than half the $Z$ boson
mass, is needed to describe DIS data for 56Fe, or that $Q\gg 180{\rm~{}GeV}$,
which is beyond the top quark mass, is needed for 197Au.

To resolve this, we apply the arguments of Refs. [63, 64] for light-cone
dominance in DIS with a nucleon to the case of an arbitrary nucleus. Our goal
is to identify the dominant contribution to the Fourier integral in Eq. (2.7)
in the deeply inelastic limit:

$Q^{2}\to\infty,\quad\nu_{A}\to\infty,\quad\text{such
that}\quad\frac{Q^{2}}{\nu_{A}}=2M_{A}x_{A}\quad\text{is fixed}.$ (3.3)

Note that in terms of averaged nuclear quantities,
$Q^{2}/\nu_{A}=Q^{2}/\nu_{N}=2M_{A}x_{A}=2M_{N}x_{N}$. That is to say, a
bigger nuclear mass $M_{A}$ is compensated by a smaller Bjorken variable
$x_{A}$ such that the fixed $Q^{2}/\nu_{A}$ is independent of the atomic
number.

Following Refs. [63, 64], we examine $\tilde{W}^{A}_{\mu\nu}(p_{A},q)$ in Eq.
(2.7), and identify the integration regions that give rise to the dominant
contributions. In the DIS limit of Eq. (3.3), as $|q\cdot z|\to\infty$ the
exponential of the Fourier integral oscillates without bound and thus makes a
vanishing contribution to the integral. Therefore, we need only to consider
the region with finite $|q\cdot z|$ in the deeply inelastic regime.

In the target’s rest frame, $\nu_{A}=q_{0}$. Defining the quantity
$\mathcalligra{r}{}=\vec{q}\cdot\vec{z}/|\vec{q}|$, we have

$q\cdot z\ =\ q_{0}z_{0}-\vec{q}\cdot\vec{z}\ =\
\nu_{A}\left(z_{0}-\frac{|\vec{q}|}{\nu_{A}}\mathcalligra{r}{}\right)\ =\
\nu_{A}\left(z_{0}-\sqrt{1+Q^{2}/\nu_{A}^{2}}\ \mathcalligra{r}{}\right)\ .$
(3.4)

We note again that this equation is independent of $A$.

In the DIS limit, $Q^{2}/\nu_{A}^{2}=(2M_{A}x_{A})/\nu_{A}$ is small and we
can expand the square root:

$q\cdot z\ =\ \nu_{A}(z_{0}-\mathcalligra{r}{})\
-M_{A}x_{A}\mathcalligra{r}{}\ +\ {\cal
O}\left(\frac{M_{A}^{2}\mathcalligra{r}{}}{\nu_{A}}\right)\,.$ (3.5)

Here, the target mass $M_{A}$ appears but only in combination with $x_{A}$ and
always satisfies $M_{A}x_{A}=M_{N}x_{N}$. In order to keep $|q\cdot z|$ finite
in the deeply inelastic limit, each term on the right hand side of Eq. (3.5)
must separately be finite. (Being separately infinite requires the scaling
$(z_{0}-\mathcalligra{r}{})\sim(\mathcalligra{r}{}/\nu_{A})\to\infty$, which
cannot be consistently satisfied.) Since $M_{A}x_{A}$ is fixed,
$\mathcalligra{r}{}$ itself must be finite. Therefore, for some constants
$c>0$ and $d>0$, one has

$|z_{0}-\mathcalligra{r}{}|<c/\nu_{A}\,\quad\text{and}\quad\,|\mathcalligra{r}{}|<d/(x_{A}M_{A})\,.$
(3.6)

The first inequality implies that $|z_{0}|<|\mathcalligra{r}{}|+c/\nu_{A}$.
After squaring and using
$\mathcalligra{r}{}^{2}=\vec{z}^{2}-z_{\perp}^{2}<\vec{z}^{2}$, where
$z_{\perp}$ is the component of $z$ orthogonal to $\vec{q}$, we obtain the
inequality chain

$z_{0}^{2}~{}<~{}(|\mathcalligra{r}{}|+c/\nu_{A})^{2}=\mathcalligra{r}{}^{2}+\frac{2c|\mathcalligra{r}{}|}{\nu_{A}}+{\cal
O}\left(\frac{1}{\nu_{A}^{2}}\right)~{}<~{}\vec{z}^{2}+\frac{2cd}{(x_{A}M_{A}\nu_{A})}+{\cal
O}\left(\frac{1}{\nu_{A}^{2}}\right)\,.$ (3.7)

To obtain the rightmost bound, we used the second inequality in Eq. (3.6).
Using the rightmost equality in Eq (3.3), we obtain the final result:

$z^{2}~{}=~{}z_{0}^{2}-\vec{z}^{2}~{}<~{}\frac{2cd}{(x_{A}M_{A}\nu_{A})}+{\cal
O}\left(\frac{1}{\nu_{A}^{2}}\right)~{}=~{}\frac{4cd}{Q^{2}}+{\cal
O}\left(\frac{1}{\nu_{A}^{2}}\right)\,.$ (3.8)

We therefore find that the dominant region remains $0\leq
z^{2}\leq(2cd/Q^{2})$, and is independent of the target. This assumes, of
course, that no extra $A$ dependence is hidden in the constants $c$ and $d$.
Such an assumption, however, is safe upon inspecting the above derivation. For
example: suppose there were separate (minimal) $c_{A}$ and $d_{A}$ for each
$A$ that satisfied the inequality of Eq. (3.6). Since there are finite many
$A$, we can simply take the largest $c_{A}$ and $d_{A}$, and then define
$c=\max\\{c_{A}\\}$ and $d=\max\\{d_{A}\\}$. This implies that there is a
large enough $c$ and $d$ such that Eq. (3.6) holds independent of $A$.

In light of Eq. (3.8), the requirement of Eq. (3.2) is obviously too harsh.
Instead, we advocate that Eq. (3.1) should be understood as requiring
$Q^{2}\gtrsim$ a few GeV2, independent of the hadronic target, in order to
justify the application of pQCD and the parton model. More precisely, given a
non-perturbative (NP) scale $\Lambda_{\rm NP}\lesssim 1{\rm~{}GeV}$, which
parameterizes the onset of hadronization, light-cone dominance in DIS for a
nucleus $A$ occurs when

$Q^{2}\sim|p_{A}\cdot q|\gg\Lambda_{\rm NP}^{2}.$ (3.9)

Under this criterion, Eq. (3.1) is automatically satisfied since nucleon
masses scale as $M_{N}\sim\Lambda_{\rm NP}$. At smaller $Q^{2}$, but still
satisfying $Q^{2}\gtrsim\Lambda_{\rm NP}^{2}$, perturbative techniques and the
parton model can still be employed as evident by the phenomenon of “precocious
scaling” [50, 49, 53]. This denotes the onset of Bjorken scaling at moderate
energies despite $\alpha_{s}(\mu_{r}=Q)\lesssim\mathcal{O}(1)$ being sizable,
and follows from the fact that light quark masses $m_{q}$ are comparable to
the scale of the QCD Landau pole, $\Lambda_{\rm
QCD}\sim\mathcal{O}(0.1){\rm~{}GeV}$.

Such a separation of scales implies the inequality

$Q^{2}\gtrsim\Lambda_{\rm NP}^{2}\gg\Lambda_{\rm QCD}^{2}\sim m_{q}^{2}\ ,$
(3.10)

where we take $m_{q}$ to be the light-quark constituent mass at soft quark
momenta. This demonstrates that, up to corrections of order
$\mathcal{O}(\Lambda_{\rm QCD}^{2}/Q^{2})$, light quarks can be approximated
as free and massless partons, even for low $Q^{2}$. However, when such $Q^{2}$
is probing a nuclear target $A$ at large $x_{A}$, it is paramount to stress
that $x_{A}^{2}M_{A}^{2}/Q^{2}$ is not guaranteed to be small or below unity.
In this case, TMCs are important.

### 3.2 Structure functions in the OPE

In this section, we briefly outline the key elements of the OPE required to
obtain expressions for structure functions with TMCs at leading twist $\tau$.
Formally, TMCs are $\mathcal{O}(M_{A}^{2}/Q^{2})$ corrections to structure
functions that are first generated by the same operators that define the
structure functions themselves at leading power. For this reason they are
sometimes called “kinematical” power corrections [83, 55]. Such power
corrections can be isolated from “dynamical” power corrections, which are
proportional to the QCD coupling constant, by setting the coupling constant to
zero. We generally follow the early literature of Refs. [67, 49, 50, 53, 84,
85], which shed light on the underlying dynamics of hadrons, but note that
TMCs can also be derived using diagrammatic methods and finite parton $k_{T}$
[83, 55]. While we only outline the derivation here, a fuller derivation of
nuclear structure functions and their TMCs at leading power in the OPE is
provided in A.

Constructing TMCs for structure functions in the OPE provides a systematic
organization of short and long distance physics that does not manifestly rely
on the perturbativity of QCD. (This means that it is possible to obtain all-
orders results.) Inspired by the program of Refs. [49, 50], the construction
of structure functions follows by first recognizing that the inclusive
hadronic tensor describing $\gamma^{*}A\to X$ scattering in the deeply
inelastic limit,444The subscripts $XA$ on the vector currents
$J_{\mu},J_{\nu}$ are simply labels to denote the hadronic current that takes
$A$ to $X$. The current is independent of states $A,X$ but depends on whether
the current is electromagnetic or weak.

$\tilde{W}^{A}_{\mu\nu}(p_{A},q)\ =\ \frac{1}{4\pi}\int d^{4}z~{}e^{iq\cdot
z}\langle A|J_{XA\mu}^{\dagger}(z)\ J_{XA\nu}(0)|A\rangle\ ,$ (3.11)

which is equivalent to the expression in Eq. (2.7) of Sec. 2.2, and the time-
ordered amplitude for virtual Compton scattering process
$\gamma^{*}A\to\gamma^{*}A$ in the short-distance limit

$\displaystyle\tilde{T}^{A}_{\mu\nu}(p_{A},q)\ \equiv$ $\displaystyle\int
d^{4}z\ e^{iq\cdot z}\ \langle A|\mathcal{T}J_{A\mu}^{\dagger}(z)\
J_{A\nu}(0)|A\rangle\ ,$ (3.12) $\displaystyle=$ $\displaystyle-
g_{\mu\nu}\Delta\tilde{T}^{A}_{1}+\frac{p_{A\mu}p_{A\nu}}{M_{A}^{2}}\Delta\tilde{T}^{A}_{2}-i\epsilon_{\mu\nu\alpha\beta}\frac{p^{\alpha}_{A}q^{\beta}}{M_{A}^{2}}\Delta\tilde{T}^{A}_{3}$
$\displaystyle+\frac{q_{\mu}q_{\nu}}{M_{A}^{2}}\Delta\tilde{T}^{A}_{4}+\frac{(p_{A\mu}q_{\nu}\pm
p_{A\nu}q_{\mu})}{M_{A}^{2}}\Delta\tilde{T}^{A}_{5,6}\ ,$ (3.13)

are related by the following dispersion relationship [40, 85]

$\displaystyle\tilde{T}^{A}_{\mu\nu}(p_{A},q)\Big{|}^{(1/x_{A})+i\varepsilon}_{(1/x_{A})-i\varepsilon}$
$\displaystyle=4\pi\ \tilde{W}^{A}_{\mu\nu}(p_{A},q),\quad\text{for}\quad
x_{A}>0\ ,$ (3.14a)
$\displaystyle\tilde{T}^{A}_{\mu\nu}(p_{A},q)\Big{|}^{(1/x_{A})-i\varepsilon}_{(1/x_{A})+i\varepsilon}$
$\displaystyle=4\pi\
\left[\tilde{W}^{A}_{\mu\nu}(p_{A},-q)\right]^{\dagger},\quad\text{for}\quad
x_{A}<0\ .$ (3.14b)

In Eq. (3.13), $\tilde{T}^{A}_{\mu\nu}(p_{A},q)$ is decomposed into
coefficients of Lorentz structures in the same manner as
$\tilde{W}^{A}_{\mu\nu}(p_{A},q)$ in Eq. (2.8). As
$\tilde{T}^{A}_{\mu\nu}(p_{A},q)$ is time ordered, the first (second) relation
in Eq. (3.14) is for DIS with (anti)particles. We assume here and below that
the hadronic currents $J_{A}^{\mu}$ and $J_{A}^{\nu}$ are always renormalized
objects in QCD.

Figure 3.1:  (Left) A contour (circle) along the radius of convergence of the
time-ordered matrix element $\tilde{T}^{A}_{\mu\nu}(p_{A},q)$ in the complex
$(1/x_{A})$ plane for $\gamma^{*}A\to\gamma^{*}A$ scattering. Branch cuts
(sawtooth lines) are along the real axis at $x_{A}^{-1}<1$ and $x_{A}^{-1}>1$
lines. (Right) A deformation of the contour along the branch cuts with
vanishing arcs.

The distinction between the short-distance limit and the DIS limit is
important: In the DIS limit, $(Q^{2}/M_{A}^{2})\to\infty$ while
$x_{A}=(Q^{2}/2p_{A}\cdot q)$ is fixed. In the short-distance limit,
$(Q^{2}/M_{A}^{2})\to\infty$ while $x_{A}/Q$ is fixed, meaning that $x_{A}$
grows with $(Q/M_{A})$; equivalently, the quantity $(1/x_{A})$ is small with
increasing $(Q/M_{A})$ in the short-distance limit. Formally,
$\tilde{T}^{A}_{\mu\nu}(p_{A},q)$ and $\tilde{W}^{A}_{\mu\nu}(p_{A},q)$ are
defined in two different limits but can be related through analytic
continuation. When analytically continued to complex values of $x_{A}$,
$\tilde{T}^{A}_{\mu\nu}$ has desirable analytic properties for $|(1/x_{A})|<1$
(the short-distance limit), whereas $\tilde{W}^{A}_{\mu\nu}$ is meaningful for
$|(1/x_{A})|>1$ (the DIS limit). Hence the discontinuities (branch cuts) of
$\tilde{T}^{A}_{\mu\nu}$ correspond to the kinematics of DIS. Therefore, one
can take $\tilde{T}^{A}_{\mu\nu}(p_{A},q)$, decompose it into a contour
integral over $x_{A}^{-1}$ using Cauchy’s integral formula, and deform the
contour around the discontinuities at $x_{A}^{-1}<1$ and $x_{A}^{-1}>1$ as
shown in Fig. 3.1. (The sawtooth lines along $x_{A}^{-1}>1$ are the branch
cuts of $\tilde{T}^{A}_{\mu\nu}$.) Taking the difference of ${T}^{A}_{\mu\nu}$
when evaluated just below and above the branch cut, i.e., at the points
$x_{A}^{-1}\pm i\varepsilon$ for $x_{A}^{-1}>1$, gives
$\tilde{W}^{A}_{\mu\nu}(p_{A},q)$. (For a fuller derivation, see the steps
before Eq. (A.45) in A.3.)

An important consequence of Eq. (3.14) is establishing a link between
individual $\Delta\tilde{T}^{A}_{i}$ in Eq. (3.13) and the Mellin moments of
the structure functions $\tilde{F}_{i}^{A}$ defined in Eq. (2.2). Following
the steps below Eq. (A.49) in A.3, one can write for the case555See A.3 for
the treatment of all six $\tilde{F}_{i}^{A}$. of $\tilde{F}_{2}$ [85]

$\displaystyle\left(\frac{Q^{2}}{2x_{A}M_{A}^{2}}\right)\
\Delta\tilde{T}_{2}^{A}(Q^{2},x_{A})$ $\displaystyle=-4i\ \sum_{N=0}^{\infty}\
\tilde{F}_{2}^{AN}(Q^{2})\ x_{A}^{-N}\ .$ (3.15a)

Here, $\tilde{F}_{i}^{AN}(Q^{2})$ is the $N^{th}$ Mellin moment of the
structure function $\tilde{F}_{i}^{A}(Q^{2},x_{A})$. The normalizations of the
Mellin transformation and its inverse are set by the definitions

$F^{N}=\int_{0}^{1}dz~{}z^{N-1}~{}F(z)\quad\text{with}\quad F(z)=\frac{1}{2\pi
i}\int^{c+i\infty}_{c-i\infty}dN~{}z^{-N}~{}F^{N}\ ,$ (3.16)

where $c$ has to be to the right of any poles of $F^{N}$ in the complex
$N$-plane. As a brief comment, the Mellin transform is an integral over the
positive reals, while the inverse transform is an integral over the complex
plane. We therefore assume that structure functions can be analytically
continued. (In practice, Mellin and inverse-Mellin transforms are implemented
in the many computer codes that perform the DGLAP evolution equations in
Mellin space, see e.g. Ref. [86].)

The next step in deriving TMCs in the OPE formalism is to take the operator-
product expansion of $\tilde{T}^{A}_{\mu\nu}$ itself in the short-distance
limit. After considerable reorganization (see A.3), one obtains expressions
(see A.4) for $\Delta\tilde{T}_{i}^{A}(Q^{2},x_{A})$ at leading power in the
OPE. More specifically, one constructs a power series with respect to powers
of $x_{A}^{-1}$, and therefore obtains expressions for
$\tilde{F}_{i}^{AN}(Q^{2})$, at leading power.

In more detail: the OPE makes use of basic symmetries, e.g., Lorentz
invariance and unitarity, to expand hadronic matrix elements into a complete
set of local operators $\mathcal{O}$ [87]. As described in Sec. 3.1, this
hinges on field operators being along the light cone, i.e., when
$z^{2}\rightarrow 0$. For forward Compton scattering, the OPE of
$\tilde{T}^{A}_{\mu\nu}$ is given by [40]:

$\displaystyle\lim_{z^{2}\to 0}\ {T}^{A}_{\mu\nu}(p_{A},q)\ \overset{\rm
OPE}{=}$
$\displaystyle-2i\sum_{\iota,n}~{}c_{\mu\nu\mu_{1}...\mu_{n}}^{\tau=2,\iota}(q)\
\langle A|\mathcal{O}_{\iota,\tau=2}^{\mu_{1}...\mu_{n}}|A\rangle$
$\displaystyle\qquad+\ \text{power corrections at }\ \mathcal{O}(\tau>2)$
(3.17) $\displaystyle=\ $
$\displaystyle-2i\sum_{k=k_{\min}}^{\infty}\Bigl{[}-g^{\mu\nu}q_{\mu_{1}}q_{\mu_{2}}C_{1}^{2k}+g^{\mu}_{\mu_{1}}g^{\nu}_{\mu_{2}}Q^{2}C_{2}^{2k}-i\epsilon^{\mu\nu\alpha\beta}g_{\alpha\mu_{1}}q_{\beta}q_{\mu_{2}}C_{3}^{2k}$
$\displaystyle\quad+\frac{q^{\mu}q^{\nu}}{Q^{2}}q_{\mu_{1}}q_{\mu_{2}}C_{4}^{2k}+(g^{\mu}_{\mu_{1}}q^{\nu}q_{\mu_{2}}\pm
g^{\nu}_{\mu_{1}}q^{\mu}q_{\mu_{2}})C_{5,6}^{2k}\Bigr{]}\cdot\left.\begin{cases}1\,;&k=1\\\
\prod^{2k}_{m=3}q_{\mu_{m}}\,;&k>1\end{cases}\right\\}$
$\displaystyle\quad\times\frac{2^{2k}}{Q^{4k}}A^{2k}_{\tau=2}\tilde{\Pi}^{\mu_{1}...\mu_{2k}}\
\quad+\quad\mathcal{O}(\tau>2).$ (3.18)

In Eq. (3.18), the summation over $n$ is changed to $n=2k$. The starting point
for the summation over $k$ depends on the particular Wilson coefficient;
specifically, $k_{\min}=2$ for $C_{2}^{2k}$. while $k_{\min}=1$ for all other
$C_{\iota}^{2k}$’s. In Eq. (3.17), the
$\mathcal{O}^{\mu_{1}\dots\mu_{n}}_{\iota,\tau}$ are composite local
operators, consisting of quark fields, QCD covariant derivatives, and/or gluon
field strengths. Operators with $n$ un-contracted Lorentz indices carry a spin
of $n$. Operators and their coefficients are organized according to their
twist $\tau=d-n$, where $d$ is the dimensionality of
$\mathcal{O}^{\mu_{1}\dots\mu_{n}}_{\iota,\tau}$ in the standard sense of
dimension power counting in an effective field theory. In this study, we focus
exclusively on the OPE at leading twist, i.e., $\tau=2$. We neglect power
corrections at $\tau>2$ but attempt to keep track of their presence. The index
$\iota$ catalogs all operators $\mathcal{O}$ at a fixed spin and twist. All
numerical prefactors are sequestered into effective Wilson coefficients
$c_{\mu\nu\mu_{1}...\mu_{n}}^{\tau,\iota}(q)$, which are only functions of the
external momentum $q$.

The Wilson coefficients666In principle, the Wilson coefficients
$C_{\iota}^{2k}$ also carry a species index $f$, which runs over the gluon and
quark flavors. In this case, $C_{\iota}^{2k}$ should be replaced by
$\sum_{f=u,g,\dots}C_{if}^{2k}$, but this dependence is neglected here for
simplicity.  $C_{\iota}^{2k}$, which are the same for all possible nuclei, can
be identified and matched to quantities calculated in perturbative QCD. The
quantity $A^{2k}_{\tau=2}$ is the scalar-valued coefficient of the expectation
value $\langle
A|\mathcal{O}_{\iota,\tau=2}^{\mu_{1}...\mu_{2k}}|A\rangle=A^{2k}_{\tau=2}\times\tilde{\Pi}^{\mu_{1}...\mu_{2k}}$.
$A^{2k}_{\tau=2}$ is sometimes labeled the reduced matrix element of $\langle
A|\mathcal{O}_{\iota,\tau=2}^{\mu_{1}...\mu_{2k}}|A\rangle$. Importantly, the
Lorentz structure of this expectation value
$(\tilde{\Pi}^{\mu_{1}...\mu_{2k}})$ can only be a function of $p_{A}$ as the
$q$ dependence is sequestered elsewhere. Consequentially,
$\tilde{\Pi}^{\mu_{1}...\mu_{2k}}$ can be organized according to permutations
and powers of momenta $p_{A}^{\mu_{m}}$ and metrics $g^{\mu_{m}\mu_{n}}$, and
scales as [49, 50]

$\displaystyle\tilde{\Pi}^{\mu_{1}...\mu_{2k}}$
$\displaystyle=\sum_{j=0}^{k}(-1)^{j}\ \frac{(2k-j)!}{2^{j}(2k)!}\
{\underbrace{\\{g...g\\}}_{\scriptstyle j\
g^{\mu_{n}\mu_{m}}{}^{\prime}s}}\quad{\underbrace{\\{p_{A}...p_{A}\\}}_{\scriptstyle(2k-2j)\
p_{A}^{\mu_{n}}{}^{\prime}s}}(p_{A}^{2})^{j}\ +\ \text{power corrections}$
(3.19) $\displaystyle\sim{\underbrace{\\{p_{A}...p_{A}\\}}_{\scriptstyle 2k\
p_{A}^{\mu_{m}}{}}}~{}+~{}{\underbrace{\\{p_{A}...p_{A}\\}}_{\scriptstyle(2k-2)\
p_{A}^{\mu_{m}}{}}}{\underbrace{\\{g...g\\}}_{~{}\scriptstyle~{}1\
g^{\mu_{m}\mu_{n}}{}}}(p_{A}^{2})^{1}~{}+~{}{\underbrace{\\{p_{A}...p_{A}\\}}_{\scriptstyle(2k-4)\
p_{A}^{\mu_{m}}{}}}{\underbrace{\\{g...g\\}}_{~{}\scriptstyle~{}2\
g^{\mu_{m}\mu_{n}}{}}}(p_{A}^{2})^{2}~{}+~{}\dots\,.$ (3.20)

Focusing on Eq. (3.20), one sees that the momentum factor
$\tilde{\Pi}^{\mu_{1}...\mu_{2k}}$ scales as a tower of momentum and mass
terms. For a fixed index $k$, the series ranges from the product of $2k$
instances of the nucleus’ momentum $p_{A}^{\mu_{m}}$ and no instance of the
nucleus’ mass $p_{A}^{2}=M_{A}^{2}$ (as shown in the leftmost term), to no
instances of $p_{A}^{\mu_{m}}$ but one instance of
$(p_{A}^{2})^{k}=(M_{A}^{2})^{k}$. The brackets $\\{...\\}$ represent the
collection of all possible permutations of Lorentz indices. The dimension of
each term is [mass]2k. Again, we neglect higher-order power corrections.

Whether one retains $(M_{A}^{2})^{j}$ terms or neglects them outright is the
distinction between obtaining structure functions with TMCs or obtaining
structure functions without TMCs (which we shall later label as “No-TMC”). For
example: after contracting all Lorentz indices in Eq. (3.18), one can neglect
all factors of $(M_{A}^{2}/Q^{2})$. This is equivalent to truncating the
summation in Eq. (3.19) at $j=0$. Doing so and comparing the result to Eq.
(3.15) recovers (see A.5 for details):

$\displaystyle\tilde{F}_{i}^{AN}=\int_{0}^{1}dy~{}y^{N-1}~{}\tilde{F}_{i}^{A}(y,Q^{2})$
$\displaystyle=C_{i}^{N}(Q^{2})A^{N}_{\tau=2}+\ \text{power
corrections},\quad\text{for}\quad i=1,3,4,5,$ (3.21a)
$\displaystyle\tilde{F}_{2}^{A(N-1)}=\int_{0}^{1}dy~{}y^{N-2}~{}\tilde{F}^{A}_{2}(y,Q^{2})$
$\displaystyle=C_{2}^{N}(Q^{2})A^{N}_{\tau=2}+\ \text{power corrections}\ .$
(3.21b)

Non-trivially, Eq. (3.21) states that one can identify the product of the
Wilson coefficient and reduced hadronic matrix element,
$(C_{\iota}^{2k}A^{2k}_{\tau=2})$, as integer Mellin moments of structure
functions, up to power corrections. (We note that footnote 6 is still
applicable.)

If one does not truncate the momentum factor
$\tilde{\Pi}^{\mu_{1}...\mu_{2k}}$ in Eq. (3.19) at $j=0$ and instead retains
the sums over all $j$, and hence retains the sum over powers of
$p_{A}^{2}=M_{A}^{2}$, then one obtains expressions that are analogous to Eq.
(3.21). For example: one can identify the Mellin transformation of the
structure function $F_{2}$ with a nonzero target mass as the product
$(C_{\iota=2}^{2k}A^{2k}_{\tau=2})$ with a coefficient proportional to
$(M_{A}^{2}/Q^{2})^{j}$:

$\int_{0}^{1}dx_{A}\,x_{A}^{N-2}\tilde{F}_{2}^{A,{\rm
TMC}}(x_{A},Q^{2})=\sum_{j=0}^{\infty}\Biggl{(}\frac{M_{A}^{2}}{Q^{2}}\Biggr{)}^{j}\frac{(N+j)!}{j!\
(N-2)!}\ \frac{C_{2}^{N+2j}{A}^{N+2j}_{\tau=2}}{(N+2j)(N+2j-1)}\ .$ (3.22)

Similar expressions can be found in A.5 for the other structure functions.
Intuitively, the right hand side of Eq. (3.22) states that structure functions
with TMCs can be thought simply as the product in moment space of a structure
function for a massless target, i.e., the $(C_{i}A_{\tau=2})$ factor, and a
kinematical factor, i.e., everything else. For this reason, the TMCs under
discussion are sometimes called “purely kinematical” [55, 83, 88]. It is not
obvious but, with the use of generating functions, the inverse-Mellin
transform of Eq. (3.22) has a closed form [49, 50, 81]. That is to say, one
obtains expressions for structure functions with TMCs in terms of structure
functions for a massless target in $x$-space. These results are summarized in
the following section for $F_{1}^{A},F_{2}^{A}$, and $F_{3}^{A}$, and in A.5
for all $F_{i}$.

### 3.3 Master formula for structure functions with TMCs in $\ell A$ DIS

Following the procedure outlined in Sec. 3.2, we obtain a set of master
formulas for twist-2 target mass corrections to structure functions for nuclei
that are similar to those in Eq. (23) in Ref. [60] for nucleons. Using the
notation of Refs. [54, 60], the general master formula for target mass-
corrected structure functions $F_{j}^{A,\rm TMC}$ for $j=1,\ldots,6$ reads:

$\displaystyle\tilde{F}_{j}^{{A,{\rm
TMC}}}(x_{A},Q^{2})=\sum_{i=1}^{6}A_{j}^{i}\tilde{F}_{i}^{A,(0)}(\xi_{A},Q^{2})+B_{j}^{i}\tilde{h}_{i}^{A}(\xi_{A},Q^{2})+C_{j}\tilde{g}_{2}^{A}(\xi_{A},Q^{2})\,.$
(3.23)

On the left-hand side, the $F_{j}^{A,\rm TMC}$ take as arguments the Bjorken
scaling variable $x_{A}$ and scale $Q$; on the right-hand side, individual
terms are given in terms of structure functions for massless nuclei
$\tilde{F}_{i}^{A,(0)}$, which take as arguments the Nachtmann variable
$\xi_{A}$ and scale $Q$. The coefficients $A_{j}^{i}$, $B_{j}^{i}$, $C_{j}$
are derived in A, and are the same that are given in Tables I, II, III in [54]
for $j=1,\dots,5$.

Suppressing the $Q^{2}$ dependence for brevity, one finds at twist $\tau=2$
the following:

$\displaystyle\tilde{F}_{1}^{A,{\rm TMC}}(x_{A})$
$\displaystyle=\left(\frac{x_{A}}{\xi_{A}r_{A}}\right)\tilde{F}_{1}^{A,(0)}(\xi_{A})+\left(\frac{M_{A}^{2}x_{A}^{2}}{Q^{2}r_{A}^{2}}\right)\,\tilde{h}_{2}^{A}(\xi_{A})+\left(\frac{2M_{A}^{4}x_{A}^{3}}{Q^{4}r_{A}^{3}}\right)\,\tilde{g}_{2}^{A}(\xi_{A})\,,$
(3.24a) $\displaystyle\tilde{F}_{2}^{A,{\rm TMC}}(x_{A})$
$\displaystyle=\left(\frac{x_{A}^{2}}{\xi_{A}^{2}r_{A}^{3}}\right)\tilde{F}_{2}^{A,(0)}(\xi_{A})+\left(\frac{6M_{A}^{2}x_{A}^{3}}{Q^{2}r_{A}^{4}}\right)\,\tilde{h}_{2}^{A}(\xi_{A})+\left(\frac{12M_{A}^{4}x_{A}^{4}}{Q^{4}r_{A}^{5}}\right)\,\tilde{g}_{2}^{A}(\xi_{A})\,,$
(3.24b) $\displaystyle\tilde{F}_{3}^{A,{\rm TMC}}(x_{A})$
$\displaystyle=\left(\frac{x_{A}}{\xi_{A}r_{A}^{2}}\right)\tilde{F}_{3}^{A,(0)}(\xi_{A})+\left(\frac{2M_{A}^{2}x_{A}^{2}}{Q^{2}r_{A}^{3}}\right)\,\tilde{h}_{3}^{A}(\xi_{A})\,,$
(3.24c) $\displaystyle\tilde{F}_{4}^{A,{\rm TMC}}(x_{A})$
$\displaystyle=\left(\frac{x_{A}}{\xi_{A}r}\right)\
\tilde{F}_{4}^{A,(0)}(\xi_{A})\Bigg{|}_{\rm No~{}TMCs}\ -\
\left(\frac{2M_{A}^{2}x_{A}^{2}}{Q^{2}r^{2}}\right)\
\tilde{F}_{5}^{A,(0)}(\xi_{A})\ +\
\left(\frac{M_{A}^{4}x_{A}^{3}}{Q^{4}r^{3}}\right)\
\tilde{F}_{2}^{A,(0)}(\xi_{A})\ $ $\displaystyle\qquad\ +\
\left(\frac{M_{A}^{2}x_{A}^{2}}{Q^{2}r^{3}}\right)\
\tilde{h}_{5}^{A}(\xi_{A})\ -\
\left(\frac{2M_{A}^{4}x_{A}^{4}}{Q^{4}r^{4}}\right)\
\left(2-\xi^{2}M_{A}^{2}/Q^{2}\right)\ \tilde{h}_{2}^{A}(\xi_{A})\ $
$\displaystyle\qquad\ +\ \left(\frac{2M_{A}^{4}x_{A}^{3}}{Q^{4}r^{5}}\right)\
\left(1-2x_{A}^{2}M_{A}^{2}/Q^{2}\right)\ \tilde{g}_{2}^{A}(\xi_{A})\ ,$
(3.24d) $\displaystyle\tilde{F}_{5}^{A,{\rm TMC}}(x_{A})$
$\displaystyle=\left(\frac{x_{A}}{\xi_{A}r^{2}}\right)\
\tilde{F}_{5}^{A,(0)}(\xi_{A})\ -\
\left(\frac{M_{A}^{2}x_{A}^{2}}{Q^{2}r^{3}\xi_{A}}\right)\
\tilde{F}_{2}^{A,(0)}(\xi_{A})\ $ $\displaystyle\qquad\ +\
\left(\frac{M_{A}^{2}x_{A}^{2}}{Q^{2}r^{3}}\right)\
\tilde{h}_{5}^{A}(\xi_{A})\ -\
\left(\frac{2M_{A}^{2}x_{A}^{2}}{Q^{2}r^{4}}\right)\ \left(1-x\xi
M_{A}^{2}/Q^{2}\right)\ \tilde{h}_{2}^{A}(\xi_{A})\ $ $\displaystyle\qquad\ +\
\left(\frac{6M_{A}^{4}x_{A}^{3}}{Q^{4}r^{5}}\right)\
\tilde{g}_{2}^{A}(\xi_{A})\ ,$ (3.24e) $\displaystyle\tilde{F}_{6}^{A,{\rm
TMC}}(x_{A})$ $\displaystyle=\left(\frac{x_{A}}{\xi_{A}r_{A}^{2}}\right)\
\tilde{F}_{6}^{A,(0)}(\xi_{A})\ +\
\left(\frac{2M_{A}^{2}x_{A}^{2}}{Q^{2}r^{3}}\right)\ \tilde{h}_{6}(\xi_{A})\
.$ (3.24f)

Here, the functions $\tilde{h}_{i}^{A}(\xi_{A},Q^{2})$ and
$\tilde{g}_{2}^{A}(\xi_{A},Q^{2})$ are given by the integrals

$\displaystyle\tilde{h}_{2}^{A}(\xi_{A},Q^{2})$
$\displaystyle=\int_{\xi_{A}}^{1}du_{A}\
\frac{\tilde{F}_{2}^{A,(0)}(u_{A},Q^{2})}{u_{A}^{2}},$ (3.25a)
$\displaystyle\tilde{h}_{3}^{A}(\xi_{A},Q^{2})$
$\displaystyle=\int_{\xi_{A}}^{1}du_{A}\
\frac{\tilde{F}_{3}^{A,(0)}(u_{A},Q^{2})}{u_{A}},$ (3.25b)
$\displaystyle\tilde{h}_{5}^{A}(\xi_{A},Q^{2})$
$\displaystyle=\int_{\xi_{A}}^{1}du_{A}\
\frac{2\tilde{F}_{5}^{A,(0)}(u_{A},Q^{2})}{u_{A}},$ (3.25c)
$\displaystyle\tilde{h}_{6}^{A}(\xi_{A},Q^{2})$
$\displaystyle=\int_{\xi_{A}}^{1}du_{A}\
\frac{\tilde{F}_{6}^{A,(0)}(u_{A},Q^{2})}{u_{A}},$ (3.25d)
$\displaystyle\tilde{g}_{2}^{A}(\xi_{A},Q^{2})$
$\displaystyle=\int_{\xi_{A}}^{1}du_{A}\ \tilde{h}_{2}^{A}(u_{A},Q^{2}).$
(3.25e)

The equation Eq. (3.23) does not assume or imply any Callan-Gross relation
between $F_{1}$ and $F_{2}$. Therefore, one can compute the longitudinal
structure function and obtain:

$\displaystyle\tilde{F}_{L}^{{A,{\rm TMC}}}(x_{A})$
$\displaystyle=r_{A}^{2}\tilde{F}_{2}^{{A,{\rm
TMC}}}(x_{A})-2x_{A}\tilde{F}_{1}^{{A,{\rm TMC}}}(x_{A})$
$\displaystyle=\frac{x_{A}^{2}}{\xi_{A}^{2}r_{A}}[\tilde{F}_{2}^{A,(0)}(\xi_{A})-2\xi_{A}\tilde{F}_{1}^{A,(0)}(\xi_{A})]+\frac{4M_{A}^{2}x_{A}^{3}}{Q^{2}r_{A}^{2}}\,\tilde{h}_{2}^{A}(\xi_{A})+\frac{8M_{A}^{4}x_{A}^{4}}{Q^{4}r_{A}^{3}}\,\tilde{g}_{2}^{A}(\xi_{A})$
$\displaystyle=\frac{x_{A}^{2}}{\xi_{A}^{2}r_{A}}\tilde{F}_{L}^{A,(0)}(\xi_{A})+\frac{4M_{A}^{2}x_{A}^{3}}{Q^{2}r_{A}^{2}}\,\tilde{h}_{2}^{A}(\xi_{A})+\frac{8M_{A}^{4}x_{A}^{4}}{Q^{4}r_{A}^{3}}\,\tilde{g}_{2}^{A}(\xi_{A})\
.$ (3.26)

This general result is non-zero and thus violates the Callan-Gross relation,
as it should. The leading term $\tilde{F}_{L}^{A,(0)}$ will be non-zero for
finite quark masses, and the sub-leading terms
$\tilde{h}_{i}^{A},\tilde{g}^{A}_{2}$ contribute for finite hadron mass
$M_{A}$.

Note that effects from heavy quark masses are taken into account in the
$\tilde{F}_{i}^{A,(0)}$ as outlined in Sec. 4 and Ref. [60]. This is not done
by generalizing the Nachtmann variable $\xi_{A}$ directly in Eq. (3.23).

### Remarks

We note that the formula for TMCs in Eq. (3.23) was derived without invoking
perturbative QCD. In principle, one can start from matrix elements that
describe collinear parton splitting at some fixed order in QCD, construct
structure functions, compare to $\tilde{T}_{\mu\nu}^{A}$ in Eq. (3.18), and
obtain expressions for $C_{\iota}^{2k}$ to some power in the strong coupling
constant $\alpha_{s}$. However, by constructing structure functions in the
OPE, the $C_{\iota}^{2k}$ in Eq. (3.18) are defined at any order $\alpha_{s}$.
Ultimately, this implies that all the $\alpha_{s}$ dependence of
$\tilde{F}_{i}^{A,\rm TMC}$ is contained in $\tilde{F}_{i}^{A,(0)}$. This also
means that the coefficients $A_{j}^{i}$, $B_{j}^{i}$ and $C_{j}$, and the
variable $\xi_{A}$ are the same irrespective to the order (LO, NLO, NNLO, …)
at which the structure functions $\tilde{F}_{i}^{A,(0)}$ are considered.

In the above expressions for $F_{i}^{{{\rm TMC}}}$, we emphasize that it is
$(x_{A},Q^{2})$ and not $(\xi_{A},Q^{2})$ that is the correct point in phase
space to evaluate structure functions. While on the surface it may appear
strange to have the left-hand side of Eq. (3.23) to be a function of $x_{A}$
while the right-hand side be a function of $\xi_{A}$, this difference arises
naturally in the calculation. In particular, final-state kinematics in $\ell
A$ DIS are constrained by momentum conservation, with
$\delta^{4}(q+P_{A}-P_{X})\sim\delta(x_{A}-\xi_{A})$. Thus, we can write
schematically

$F_{i}^{{\rm TMC}}(x_{A},Q^{2})\sim
F_{i}^{(0)}(x_{A},Q^{2})\,\delta(x_{A}-\xi_{A})\sim
F_{i}^{(0)}(\xi_{A},Q^{2}).$ (3.27)

Note that it would be incorrect to write $F_{i}^{{\rm TMC}}(\xi_{A},Q^{2})\sim
F_{i}^{(0)}(\xi_{A},Q^{2})$, as the Mellin and inverse-Mellin transformations
in Sec. 3.2 are defined with respect to $x_{A}$, not $\xi_{A}$.

Another feature of Eq. (3.23) is that $h_{2}$ and $g_{2}$ appear in the
formulas for both $F_{1}^{{\rm TMC}}$ and $F_{2}^{{\rm TMC}}$. This follows
directly from the forms of $\tilde{T}_{\mu\nu}^{A}$ and the momentum factor
$\tilde{\Pi}$ in Eq. (3.18), in the massive target limit. For example: the
terms proportional to $C_{1}^{2k}$ and to $C_{2}^{2k}$ both contribute to
$\Delta\tilde{T}^{A,1}$, which multiplies the Lorentz structure $-g_{\mu\nu}$
in Eq. (3.18). As shown in Eq. (3.20), $\tilde{\Pi}$ in the massive-target
limit is proportional to terms that (a) consist exclusively of
$p_{A}^{\mu_{n}}$ factors as well as to terms that (b) include factors of
momentum $p_{A}^{\mu_{n}}$, metric $g^{\mu_{m}\mu_{n}}$, and mass $M_{A}$.
Among the many possible contractions in Eq. (3.18) are the “(a)” terms
contracting with the $C_{1}^{2k}$ term and the “(b)” terms contracting with
the $C_{2}^{2k}$ term. Both sets of contractions generate a term proportional
to $-g_{\mu\nu}$. However, the “(b)” set of contractions only enters
$\Delta\tilde{T}^{A,1}$ as a TMC, i.e., it vanishes in the
$(M_{A}^{2}/Q^{2})\to 0$ limit. Arguably, one can think of $C_{1}^{2k}$
feeding into $F_{1}^{{\rm TMC}}$ as a helicity-conserving contribution, which
survives in the massless limit, whereas the contribution from $C_{2}^{2k}$ is
a helicity-inverting contribution, which vanishes in the massless limit.

## 4 Rescaling

Summary table of key relations:
Nucleus A Nucleon N $M_{A}=A\,M_{N}$ $M_{N}=M_{A}/A$ $p_{A}=A\,p_{N}$
$p_{N}=p_{A}/A$ $x_{A}=\frac{Q^{2}}{2p_{A}\cdot q}\equiv{x_{N}}/{A}$
$x_{N}=\frac{Q^{2}}{2p_{N}\cdot q}\equiv A\,x_{A}$ $x_{A}\in[0,1]$
$x_{N}\in[0,A]$ $W_{A}^{2}=(p_{A}+q)^{2}$ $W_{N}^{2}=(p_{N}+q)^{2}$
$\nu_{A}=(q\cdot p_{A})/M_{A}\equiv\nu_{N}$ $\nu_{N}=(q\cdot
p_{N})/M_{N}\equiv\nu_{A}$ $y_{A}=\nu_{A}/E\equiv y_{N}$
$y_{N}=\nu_{N}/E\equiv y_{A}$ Nachtmann Variable & Hadronic Mass
$r_{A}=\sqrt{1+\frac{4x_{A}^{2}M_{A}^{2}}{Q^{2}}}\equiv r_{N}$
$r_{N}=\sqrt{1+\frac{4x_{N}^{2}M_{N}^{2}}{Q^{2}}}\equiv r_{A}$
$\xi_{A}=R_{M}x_{A}\equiv\xi_{N}/A$ $\xi_{N}=R_{M}x_{N}\equiv A\,\xi_{A}$
$\xi_{A}\in[0,1]$ $\xi_{N}\in[0,A]$ Since $r_{A}=r_{N}\equiv r$, then
$R_{M}=\frac{2}{1+r_{A,N}}$ Also,
$\xi_{A}/x_{A}=\xi_{N}/x_{N}=R_{M}=\frac{2}{1+r}$ We define
$\varepsilon=(xM/Q)$. Table 4.1:  We summarize the key relations for a nucleus
($A$) and nucleon ($N$). We also find it convenient to define
$\varepsilon=(xM/Q)$; we omit the subscripts on $\varepsilon$ for brevity as
$(x_{A}M_{A}/Q)=(x_{N}M_{N}/Q)$. We caution that $W_{A}$ and $W_{N}$ are not
simply related, c.f., Sec. 5.4. Note, the target mass modifies the scaling
variable via $\xi_{A}=R_{M}\,x_{A}$. Additionally, we introduce the shorthand
notation $r=r_{A}=r_{N}$.

In this section we introduce a rescaling between the nuclear and averaged
nucleon kinematics. Many of the key relations are summarized in Table 4.1. We
typically identify the nuclear variables with an “$A$” subscript
$(x_{A},\xi_{A},r_{A})$, and those of the nucleon with an “$N$” subscript
$(x_{N},\xi_{N},r_{N})$. Similarly, the original nuclear structure functions
are identified with a tilde $(\tilde{W},\tilde{F})$, while those re-scaled to
the kinematics of an averaged nucleon are without $(W,F)$.

### 4.1 Nuclear and Nucleon Kinematics

As can be seen from Eqs. (3.24) and (3.26), the $M_{A}^{2j}$ terms are always
accompanied by $x_{A}^{2j}$ factors in $x$-space. Furthermore, the structure
functions $\tilde{F}_{2}^{A,(0)}(x_{A})$ have their main support at777Note
that we have introduced the nucleon scaling variable $x_{N}=Ax_{A}$; this will
be discussed in detail throughout this section. Although $x_{N}$ can in
principle extend to $x_{N}=A$, the dominant range of the kinematics is $0\leq
x_{N}\lesssim 1$. Discussion on the $x_{N}>1$ region are given in Sec. 5.5.
$x_{A}=x_{N}/A\sim 1/A$. Therefore, we have effectively

$\frac{M_{A}^{2j}x_{A}^{2j}}{Q^{2j}}=\frac{(M_{N}^{2j}A^{2j})\,x_{A}^{2j}}{Q^{2j}}=\frac{M_{N}^{2j}x_{N}^{2j}}{Q^{2j}}\sim\frac{M_{N}^{2j}}{Q^{2j}}\,,$
(4.1)

where $M_{N}=M_{A}/A$ is the (average) nucleon mass such that the terms with
$j>0$ in Eq. (3.20) are suppressed for $Q^{2}>M_{N}^{2}$ independent of the
nuclear target.

We introduce the average nucleon momentum $p_{N}:=p_{A}/A$ inside the nucleus
and define a nucleon scaling variable

$x_{N}=\frac{Q^{2}}{2p_{N}\cdot q}=A\,x_{A}\ ,\qquad{\rm where}\quad
x_{N}\in[0,A]\ ,$ (4.2)

in contrast to the original Bjorken variable

$x_{A}=\frac{Q^{2}}{2p_{A}\cdot q}=x_{N}/A\ ,\qquad{\rm where}\quad
x_{A}\in[0,1]\ .$ (4.3)

Note that the original $x_{A}$ variable can be constructed directly from the
external momenta of the particles whereas the “averaged quantities” $x_{N}$
(and $p_{N}$) are not directly observable for nuclear targets with $A>1$.

Using $M_{A}=AM_{N}$, we find for the Nachtmann variable

$\xi_{A}=\frac{2x_{A}}{1+\sqrt{1+4x_{A}^{2}M_{A}^{2}/Q^{2}}}=\frac{1}{A}\xi_{N}$
(4.4)

with

$\xi_{N}=\frac{2x_{N}}{1+\sqrt{1+4x_{N}^{2}M_{N}^{2}/Q^{2}}}\,.$ (4.5)

Conversely, the quantity $R_{M}$, which relates the Bjorken $x$ variable to
the scaling variable $\xi$ via $\xi=R_{M}x$, is the same for both the nucleus
and nucleon case:

$R_{M}=\frac{2}{1+r_{A}}=\frac{2}{1+r_{N}}\ .$ (4.6)

This is because the quantity $r_{A}$ appearing in the master formula is also
the same:

$r_{A}=\sqrt{1+4x_{A}^{2}M_{A}^{2}/Q^{2}}=\sqrt{1+4x_{N}^{2}M_{N}^{2}/Q^{2}}=r_{N}\,.$
(4.7)

Hence, we have $\xi_{A}=R_{M}x_{A}$ and $\xi_{N}=R_{M}x_{N}$, where $R_{M}$
takes the same value in both equations.

### 4.2 Rescaled Structure Functions

Turning to the structure functions, we define a rescaled hadronic tensor via
the relation

$A\,W^{A}_{\mu\nu}(p_{N},q):=\tilde{W}^{A}_{\mu\nu}(p_{A},q)\ ,$ (4.8)

expressed in terms of rescaled structure functions:

$F_{2}^{A}(x_{N},Q^{2}):=\tilde{F}_{2}^{A}(x_{A},Q^{2})\ ,\qquad
x_{N}F_{1,3}^{A}(x_{N},Q^{2}):=x_{A}\tilde{F}_{1,3}^{A}(x_{A},Q^{2})\,.$ (4.9)

The general relations in Eq. (4.9) hold also for $\tilde{F}_{4,5,6}$, both
with and without TMCs, and for any value of the twist. As we show below, they
are consistent with the master formula for the target-mass-corrected structure
functions. Hence, the pattern of the master equations will be consistent
between the nucleus and nucleon.

Indeed, one easily finds from Eq. (3.25):

$\tilde{h}_{2}^{A}(\xi_{A})=\int_{\xi_{A}}^{1}du_{A}\frac{\tilde{F}_{2}^{A(0)}(u_{A})}{u_{A}^{2}}=A\int_{\xi_{N}}^{A}du_{N}\frac{F_{2}^{A(0)}(u_{N})}{u_{N}^{2}}=:Ah_{2}^{A}(\xi_{N})\,,$
(4.10)
$\tilde{h}_{3}^{A}(\xi_{A})=\int_{\xi_{A}}^{1}du_{A}\frac{\tilde{F}_{3}^{A(0)}(u_{A})}{u_{A}}=A\int_{\xi_{N}}^{A}du_{N}\frac{F_{3}^{A(0)}(u_{N})}{u_{N}}=:Ah_{3}^{A}(\xi_{N})\,,$
(4.11)
$\tilde{h}_{5}^{A}(\xi_{A})=\int_{\xi_{A}}^{1}du_{A}\frac{2\tilde{F}_{5}^{A(0)}(u_{A})}{u_{A}}=A\int_{\xi_{N}}^{A}du_{N}\frac{2F_{5}^{A(0)}(u_{N})}{u_{N}}=:Ah_{5}^{A}(\xi_{N})\,,$
(4.12)
$\tilde{h}_{6}^{A}(\xi_{A})=\int_{\xi_{A}}^{1}du_{A}\frac{\tilde{F}_{6}^{A(0)}(u_{A})}{u_{A}}=A\int_{\xi_{N}}^{A}du_{N}\frac{F_{6}^{A(0)}(u_{N})}{u_{N}}=:Ah_{6}^{A}(\xi_{N})\,,$
(4.13)
$\tilde{g}_{2}^{A}(\xi_{A})=\int_{\xi_{A}}^{1}du_{A}\tilde{h}_{2}^{A}(u_{A})=\frac{1}{A}\int_{\xi_{N}}^{A}du_{N}Ah_{2}^{A}(u_{N})=:g_{2}^{A}(\xi_{N})\,.$
(4.14)

With these expressions, and using $x_{N}{=}Ax_{A}$, $\xi_{N}{=}A\xi_{A}$,
$r_{A}{=}r_{N}$, $M_{A}{=}AM_{N}$, we obtain the master TMC formula for the
rescaled nuclear structure functions:

$\displaystyle F_{1}^{A,{\rm TMC}}(x_{N},Q^{2})$
$\displaystyle=\left(\frac{x_{N}}{\xi_{N}r_{N}}\right)F_{1}^{A,(0)}(\xi_{N},Q^{2})+\left(\frac{M_{N}^{2}x_{N}^{2}}{Q^{2}r_{N}^{2}}\right)\,h_{2}^{A}(\xi_{N},Q^{2})+\left(\frac{2M_{N}^{4}x_{N}^{3}}{Q^{4}r_{N}^{3}}\right)\,g_{2}^{A}(\xi_{N},Q^{2})\,,$
$\displaystyle F_{2}^{A,{\rm TMC}}(x_{N},Q^{2})$
$\displaystyle=\left(\frac{x_{N}^{2}}{\xi_{N}^{2}r_{N}^{3}}\right)F_{2}^{A,(0)}(\xi_{N},Q^{2})+\left(\frac{6M_{N}^{2}x_{N}^{3}}{Q^{2}r_{N}^{4}}\right)\,h_{2}^{A}(\xi_{N},Q^{2})+\left(\frac{12M_{N}^{4}x_{N}^{4}}{Q^{4}r_{N}^{5}}\right)\,g_{2}^{A}(\xi_{N},Q^{2})\,,$
$\displaystyle F_{3}^{A,{\rm TMC}}(x_{N},Q^{2})$
$\displaystyle=\left(\frac{x_{N}}{\xi_{N}r_{N}^{2}}\right)F_{3}^{A,(0)}(\xi_{N},Q^{2})+\left(\frac{2M_{N}^{2}x_{N}^{2}}{Q^{2}r_{N}^{3}}\right)\,h_{3}^{A}(\xi_{N},Q^{2})\
,$ (4.15c) $\displaystyle{F}_{4}^{A,{\rm TMC}}(x_{A},Q^{2})$
$\displaystyle=\left(\frac{x_{A}}{\xi_{A}r}\right)\
{F}_{4}^{A,(0)}(\xi_{A},Q^{2})\ -\
\left(\frac{2M_{A}^{2}x_{A}^{2}}{Q^{2}r^{2}}\right)\
{F}_{5}^{A,(0)}(\xi_{A},Q^{2})\ +\
\left(\frac{M_{A}^{4}x_{A}^{3}}{Q^{4}r^{3}}\right)\
{F}_{2}^{A,(0)}(\xi_{A},Q^{2})\ $ $\displaystyle\qquad\ +\
\left(\frac{M_{A}^{2}x_{A}^{2}}{Q^{2}r^{3}}\right)\ h_{5}^{A}(\xi_{A},Q^{2})\
-\ \left(\frac{2M_{A}^{4}x_{A}^{4}}{Q^{4}r^{4}}\right)\
\left(2-\xi^{2}M_{A}^{2}/Q^{2}\right)\ h_{2}^{A}(\xi_{A},Q^{2})\ $
$\displaystyle\qquad\ +\ \left(\frac{2M_{A}^{4}x_{A}^{3}}{Q^{4}r^{5}}\right)\
\left(1-2x_{A}^{2}M_{A}^{2}/Q^{2}\right)\ g_{2}^{A}(\xi_{A},Q^{2})\ ,$ (4.15d)
$\displaystyle{F}_{5}^{A,{\rm TMC}}(x_{A},Q^{2})$
$\displaystyle=\left(\frac{x_{A}}{\xi_{A}r^{2}}\right)\
{F}_{5}^{A,(0)}(\xi_{A},Q^{2})\ -\
\left(\frac{M_{A}^{2}x_{A}^{2}}{Q^{2}r^{3}\xi_{A}}\right)\
{F}_{2}^{A,(0)}(\xi_{A},Q^{2})\ $ $\displaystyle\qquad\ +\
\left(\frac{M_{A}^{2}x_{A}^{2}}{Q^{2}r^{3}}\right)\ h_{5}^{A}(\xi_{A},Q^{2})\
-\ \left(\frac{2M_{A}^{2}x_{A}^{2}}{Q^{2}r^{4}}\right)\ \left(1-x\xi
M_{A}^{2}/Q^{2}\right)\ h_{2}^{A}(\xi_{A},Q^{2})\ $ $\displaystyle\qquad\ +\
\left(\frac{6M_{A}^{4}x_{A}^{3}}{Q^{4}r^{5}}\right)\ g_{2}^{A}(\xi_{A},Q^{2})\
,$ (4.15e) $\displaystyle{F}_{6}^{A,{\rm TMC}}(x_{A},Q^{2})$
$\displaystyle=\left(\frac{x_{A}}{\xi_{A}r_{A}^{2}}\right)\
{F}_{6}^{A,(0)}(\xi_{A},Q^{2})\ +\
\left(\frac{2M_{A}^{2}x_{A}^{2}}{Q^{2}r^{3}}\right)\ h_{6}(\xi_{A},Q^{2})\ .$
(4.15f)

The master equations take the same form as before, but are entirely expressed
in terms of averaged nucleon variables. This is one of the main results of
this rescaling demonstration.

### 4.3 Impact of Quark Masses

Partonic Masses
---
$\bar{\xi}_{A}=R_{ij}\xi_{A}=R_{ij}R_{M}x_{A}$ |  | $\bar{\xi}_{N}=R_{ij}\xi_{N}=R_{ij}R_{M}x_{N}$
For $q_{i}+V\to q_{j}$
$R_{ij}=\frac{1}{\displaystyle
2Q^{2}}[Q^{2}-m_{i}^{2}+m_{j}^{2}+\Delta(Q^{2},m_{i}^{2},m_{j}^{2})]$
$\Delta(a,b,c)=\sqrt{a^{2}+b^{2}+c^{2}-2(ab+bc+ca)}$
For $g+V\to q_{j}+q_{k}$
$R_{jk}=1+\frac{\displaystyle(m_{j}+m_{k})^{2}}{\displaystyle Q^{2}}$
Table 4.2:  Summary table of partonic mass relations. Note, the target mass
modifies the scaling variable via $\xi_{A}=R_{M}\,x_{A}$. This is further
modified by the parton masses
$\overline{\xi}_{A}=R_{ij}\xi_{A}=R_{ij}R_{M}x_{A}$; hence these
multiplicative corrections factorize between the hadron and parton masses
[60]. In the limit the initial-state parton mass $m_{i}\to 0$, we obtain the
“slow rescaling” result: $R_{ij}=\left(1+m_{j}^{2}/Q^{2}\right)$.

Notably, we stress that all of the above derivations have made no mention of
the parton model. This can be understood from the fact that the quark and
gluon sub-structure of nuclei enters only implicitly in intermediate steps of
the derivation of the master formula since the hadronic currents appearing in
the hadronic tensor are composed of quark and gluon fields and we have relied
on the validity of the OPE for the products of these currents. Thus, due to
the nature of the factorization, all detail related to quark masses is
contained entirely inside the $F_{i}^{A,(0)}$ structure functions, and the
forms of the master formulas in Eq. (4.15) are unchanged.

The treatment of the quark masses was presented in detail in Ref. [60], but it
is instructive to briefly mention how the quark masses effect the rescaling of
the kinematic variables as summarized in Table 4.2. Just as the hadron mass
modifies the scaling variable via the relation $\xi=R_{M}x$, the quark masses
further modify the scaling variable via the relation
$\bar{\xi}=R_{ij}\xi=R_{ij}R_{M}\xi$. The $R_{M}$ factor depends on the hadron
mass, and the $R_{ij}$ factor depends on the incoming and outgoing quark
masses $\\{m_{i},m_{j}\\}$, respectively. Note the hadronic and partonic
correction factors factorize, and are applied multiplicatively. Additionally,
observe that in the limit the initial-state quark mass vanishes $m_{i}=0$, we
have the familiar “slow-rescaling” limit $R_{ij}=(1+m_{j}^{2}/Q^{2})$.

## 5 Parton model

We now turn to a discussion of structure functions in the context of the
parton model originally proposed by Feynman in 1969 [89] and applied to
electron-proton DIS by Bjorken and Paschos [90]. In the years following these
works, partons were identified with quarks and gluons, and the heuristic
parton model including QCD effects was rigorously derived from first
principles of QCD in the context of factorization theorems [41, 42, 43, 44,
45, 46, 47, 48]. In this section we summarize the main elements of the
conventional “massless” QCD-improved parton model, also known as Zero Mass
Variable Flavor Number Scheme (ZM-VFNS), where quark-mass effects in the hard
scattering cross sections and target-mass effects are neglected. There is an
extensive body of literature on computations with heavy partons; Collins
extended the factorization proof [42] to the case of heavy partons in Ref.
[45], and implementations can be found in Refs. [58, 91, 92, 93, 94]. A
discussion of the parton model including such mass effects will be given in
Sec. 6.

In the QCD-improved parton model the nuclear structure functions
$({\cal\tilde{F}}^{A}_{k})$ are given as convolutions of target-independent,
short-distance, Wilson coefficients ($C_{k,i}$) with universal nuclear parton
distribution functions ($\tilde{f}_{i}^{A}$):

${\cal\tilde{F}}^{A}_{k}(x_{A},Q^{2})=\int_{x_{A}}^{1}\frac{dy_{A}}{y_{A}}\
\tilde{f}_{i}^{A}(y_{A},Q^{2})\ C_{k,i}(x_{A}/y_{A})\ +\ldots$ (5.1)

where the ellipses represent higher twist contributions
${\cal\tilde{F}}_{k}^{A,\tau\geq 4}(x_{A},Q^{2})$, and a sum over $i=q,g$
species is understood. As in the previous section, we use the tilde notation
to identify the nuclear PDFs $\tilde{f}_{i}^{A}(x_{A},Q^{2})$, and the
rescaled PDFs are without the tilde $f_{i}^{A}(x_{N},Q^{2})$. Here, we use the
shorthand

${\cal\tilde{F}}^{A}_{k}(x_{A})=[\tilde{F}_{1}(x_{A}),\tilde{F}_{2}(x_{A})/x_{A},\tilde{F}_{3}(x_{A})]\qquad(k=1,2,3)\,.$
(5.2)

The nuclear PDFs are defined as Fourier transforms of matrix elements of
local, twist-2 operators $({\cal O})$ composed of quark and gluon fields:

$\tilde{f}_{i}^{A}(x_{A},Q^{2})\sim\langle A(p_{A})|{\cal
O}_{i}|A(p_{A})\rangle\,,$ (5.3)

where $x_{A}\in[0,1]$ and the factorization and renormalization scales have
been identified with the exchange boson virtuality $\mu_{F}=\mu_{R}=Q$.

For a spin $j$ target there are $2(2j+1)$ independent helicity-dependent quark
distributions $\tilde{q}_{+}^{m}(x)$ and $\tilde{q}_{-}^{m}(x)$ with $-j\leq
m\leq j$. Here $\tilde{q}_{+}^{m}(x)$ ($\tilde{q}_{-}^{m}(x)$) describes the
number density for finding a quark with momentum fraction $x$ and spin $m$
parallel (anti-parallel) to the the $z$-axis of a hadron moving with infinite
momentum along the $z$-axis. Due to parity invariance of the strong
interaction, we have a relation between the up (parallel) and down (anti-
parallel) spin ($\tilde{q}_{+}^{m}=\tilde{q}_{-}^{-m}$) such that there remain
only $2j+1$ independent distributions. However, this symmetry is broken once
weak interaction effects are taken into account.

In this paper we focus on unpolarized DIS. However, the application to
polarized DIS are relatively straightforward. For a spin-$j$ target, the
unpolarized quark densities are related to the helicity-dependent
distributions as follows:

$\tilde{q}_{i}^{A}(x_{A},Q^{2}):=\frac{1}{2j+1}\sum_{m=-j}^{j}(\tilde{q}^{m}_{+}+\tilde{q}^{m}_{-})\,.$
(5.4)

For a spin-$1/2$ target (and using
$\tilde{q}_{+}^{1/2}=\tilde{q}_{-}^{-1/2}$), this reduces to the well-known
expression:

$\tilde{q}_{i}^{A}(x_{A},Q^{2}):=\tilde{q}^{1/2}_{+}+\tilde{q}^{1/2}_{-}\,.$
(5.5)

Similar definitions can be written down for the unpolarized gluon
distribution.

### 5.1 Nuclear DGLAP Evolution

The unpolarized nuclear PDFs (for any target spin) satisfy the usual
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [95,
96, 97]

$\displaystyle\frac{d\tilde{f}_{i}^{A}(x_{A},Q^{2})}{d\ln Q^{2}}$
$\displaystyle=$
$\displaystyle\frac{\alpha_{s}(Q^{2})}{2\pi}\int_{x_{A}}^{1}\frac{dy_{A}}{y_{A}}\,P_{ij}(y_{A})\,\tilde{f}_{j}^{A}(x_{A}/y_{A},Q^{2})$
(5.6) $\displaystyle\equiv$
$\displaystyle\frac{\alpha_{s}(Q^{2})}{2\pi}\int_{x_{A}}^{1}\frac{dy_{A}}{y_{A}}\,P_{ij}(x_{A}/y_{A})\,\tilde{f}_{j}^{A}(y_{A},Q^{2})\quad.$
(5.7)

Furthermore, the following sum rules due to charge, baryon number, and
momentum conservation are satisfied:

$\displaystyle\int_{0}^{1}$ $\displaystyle dx_{A}\
\tilde{u}_{v}^{A}(x_{A},Q^{2})=2Z+N\,,$ (5.8a) $\displaystyle\int_{0}^{1}$
$\displaystyle dx_{A}\ \tilde{d}_{v}^{A}(x_{A},Q^{2})=Z+2N\,,$ (5.8b)
$\displaystyle\int_{0}^{1}$ $\displaystyle dx_{A}\
x_{A}\sum_{i}\tilde{f}_{i}^{A}(x_{A},Q^{2})=1\,,$ (5.8c)

where $Z$ is the electric charge of the nucleus with baryon number $A=Z+N$.

### 5.2 Relation to the OPE

A Mellin transformation can turn a convolution integral $(A\otimes
B)(x)=\int_{x}^{1}dy/yA(y)B(x/y)$ into an ordinary product of Mellin moments:

$\int_{0}^{1}dxx^{n-1}(A\otimes B)(x)\equiv(A\otimes B)_{n}=A_{n}\cdot B_{n}\
,$ (5.9)

where

$A_{n}\equiv\int_{0}^{1}dxx^{n-1}A(x)\,\qquad\text{and}\qquad
B_{n}\equiv\int_{0}^{1}dxx^{n-1}B(x)\ .$ (5.10)

Applying this to structure functions we obtain:

$\int_{0}^{1}dx_{A}x_{A}^{n-1}\,{\cal\tilde{F}}^{A}_{k}(x_{A},Q^{2})=\sum_{n}\tilde{f}_{i}^{A,n}(Q^{2})\,C_{k,i}^{n}\,.$
(5.11)

where on the RHS the Mellin transformation has yielded a simple sum of Mellin
moments. (Note, we indicate the Mellin moment by the “$n$” superscript, as the
tilde-notation has already been used for the pre-scaled PDFs as we discuss in
the following section.)

The relation between the structure functions in the parton model and the OPE
can now be easily seen by comparing the Mellin-moment of (5.1) with the
$j{=}0$–term in (3.22):

${\cal\tilde{F}}_{k}^{(0),n}=C_{k,i}^{n}\,\tilde{A}_{i}^{n}=C_{k,i}^{n}\,\tilde{f}_{i}^{A,n}(Q^{2})\
,\ (k=1,2,3)\,,$ (5.12)

where a sum over $i=q,g$ is understood. From this equation we see that the
matrix elements $\tilde{A}^{n}$ of Eq. (3.22) are simply the Mellin moments of
the parton distribution functions:

$\tilde{A}_{i}^{n}(Q^{2})=\tilde{f}_{i}^{A,n}(Q^{2})\equiv\int_{0}^{1}dx_{A}x_{A}^{n-1}\,\tilde{f}_{i}(x_{A},Q^{2})\
.$ (5.13)

Similarly, the Wilson coefficients $C_{k,i}^{n}$ of (5.12) are directly
identified with those of Eq. (3.22).

### 5.3 Rescaling

The parton distributions $\tilde{f}_{i}^{A}(x_{A})$ are number densities
defined on the interval $x_{A}\in[0,1]$ such that
$\tilde{f}_{i}^{A}(x_{A})dx_{A}$ can be interpreted as the number of partons
“$i$” carrying a fraction of the parent hadron momentum in the interval
$[x_{A},x_{A}+dx_{A}]$. We define rescaled parton densities
$f_{i}^{A}(x_{N},Q^{2})$ in the variable $x_{N}\in[0,A]$ with $x_{N}=Ax_{A}$
by imposing that the number of partons remains unchanged in the corresponding
momentum intervals:

$f_{i}^{A}(x_{N},Q^{2})dx_{N}:=\tilde{f}_{i}^{A}(x_{A},Q^{2})dx_{A}\ .$ (5.14)

This equation may appear a bit odd as $x_{N}\in[0,A]$ on the LHS, while
$x_{A}\in[0,1]$ on the RHS; we will provide additional clarification in the
following discussion. Additionally, it should be noted that the rescaled
nuclear PDF $f_{i}^{A}(x_{N},Q^{2})$ are the ones usually used in the
literature as $f_{i}^{A}(x_{N},Q^{2})$ can be meaningfully compared across
different nuclei, in contrast to $\tilde{f}_{i}^{A}(x_{A},Q^{2})$.

It can be easily verified that the rescaling defined here at the level of the
PDFs is consistent with the rescaling introduced above at the hadronic level.

Correspondingly, the rescaled objects satisfy analogous DGLAP evolution
equations:

$\displaystyle\frac{df_{i}^{A}(x_{N},Q^{2})}{d\ln Q^{2}}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}(Q^{2})}{2\pi}\int_{x_{N}}^{A}\frac{dy_{N}}{y_{N}}\
P_{ij}\left(\frac{x_{N}}{y_{N}}\right)\ f_{j}^{A}(y_{N},Q^{2})\,.$ (5.15)

Furthermore, the sum rules take the following form after the rescaling:

$\displaystyle\int_{0}^{A}$ $\displaystyle dx_{N}$ $\displaystyle
u_{v}^{A}(x_{N},Q^{2})=2Z+N\,,$ (5.16a) $\displaystyle\int_{0}^{A}$
$\displaystyle dx_{N}$ $\displaystyle d_{v}^{A}(x_{N},Q^{2})=Z+2N\,,$ (5.16b)
$\displaystyle\int_{0}^{A}$ $\displaystyle dx_{N}$ $\displaystyle
x_{N}\sum_{i}f_{i}^{A}(x_{N},Q^{2})=A\,.$ (5.16c)

Note the above integrations extend to $A$ since $x_{N}\in[0,A]$.

While the rescaled nuclear PDFs are formally defined on the interval
$x_{N}\in[0,A]$, it is important to note how the bulk of the PDFs are
distributed. For a proton with three valence quarks, we generally expect the
valence PDFs to be peaked in the region of $x_{A}=x_{N}\sim 1/3$. In a similar
manner, for nuclei of mass $A$, we generally expect the valence PDFs to be
peaked in the region of $x_{A}\sim 1/(3A)$, or equivalently $x_{N}\sim 1/3$,
which is independent of $A$ and thus facilitates a meaningful comparison
between different nuclei. Therefore, even though $x_{N}$ can in principle span
the range $[0,A]$, we expect the dominant support of the PDFs to be in the
region $x_{A}\leq 1/A$, or $x_{N}\leq 1$. For $x_{N}$ to exceed unity, one
parton alone would essentially need to acquire more momentum than an average
nucleon; this is highly unlikely. Thus, it is common to assume the probability
for $x_{N}>1$ to be very small, and global analyses of nPDFs typically
implement the condition that $f_{i}^{A}(x_{N},Q^{2})=0$ for $x_{N}>1$. This
induces a very small error, and we discuss the possibility of $x_{N}>1$
further in Sec. 5.5.

Setting $f_{i}^{A}(x_{N},Q_{0}^{2})=0$ for $x_{N}\geq 1$ at the initial scale
$Q_{0}$ yields an important computational benefit since the DGLAP evolution
framework consistently leads to $f_{i}^{A}(x_{N},Q^{2})=0$ for $x_{N}\geq 1$
at all larger scales $Q$. Consequently, the advantage of setting
$f_{i}^{A}(x_{N}{\geq}1,Q^{2})=0$ is that the same evolution equations can be
used for all nuclei, including nucleons:

$\frac{df_{i}^{A}(x_{N},Q^{2})}{d\ln
Q^{2}}=\begin{cases}\quad\frac{\alpha_{s}(Q^{2})}{2\pi}\int_{x_{N}}^{1}\frac{dy_{N}}{y_{N}}\
P_{ij}\left(\frac{x_{N}}{y_{N}}\right)\
f_{j}^{A}(y_{N},Q^{2})\,\quad&:0<x_{N}\leq 1\ ,\\\\[5.0pt] \qquad
0&:1<x_{N}\leq A\,.\end{cases}$ (5.17)

Figure 5.1: We display the nCTEQ15 nPDFs [17] as a function of $x$ at $Q=10$
GeV for selected nuclear $A$ values as indicated in the legend. These are the
scaled nPDFs of Eq. (5.18), and this feature is most evident when comparing
the $xu(x,Q)$ and $xd(x,Q)$ of the proton ($A=1$).

With the above framework for the nuclear PDFs, we can perform the DGLAP
evolution on the interval $x\in[0,1]$ in a manner similar to the proton
evolution. However, the sum rules of Eq. (5.16a), (5.16b), and (5.16c) differ
from the proton case. Thus, it is common (but not necessary) to further
decompose the nuclear PDF $f_{i}^{A}$ as

$f_{i}^{A}(x,Q)=\frac{Z}{A}f_{i}^{p/A}(x,Q)+\frac{A-Z}{A}f_{i}^{n/A}(x,Q)\,,$
(5.18)

where $f_{i}^{p/A}$ and $f_{i}^{n/A}$ represent effective “bound proton and
neutron” PDFs for finding a parton “$i$” inside a nucleon. The “bound proton”
PDF $f_{i}^{p/A}$ satisfies identical sum rules as the (free) proton PDF, so
this quantity can be computed using standard DGLAP programs. The “bound
neutron” PDF is commonly obtained from the “bound proton” one by assuming
isospin symmetry.888While isospin symmetry is typically used to relate
$f_{i}^{n/A}$ to $f_{i}^{p/A}$, in principle one could add isospin violating
contributions. The recent Marathon data [98] suggests there may be isospin
violations in the very large $x$ region [99].

For comparison of nPDFs across different $A$ nuclei, it is typically the
scaled nPDF $f_{i}^{(A,Z)}(x,Q)$ of Eq. (5.18) which can be meaningfully
compared, as we illustrate in Fig. 5.1. This quantity is defined on
$x\in[0,1]$ and normalized so that the integrated momentum is unity.
Conversely, although the effective “bound proton and neutron” PDFs of Eq.
(5.18) are convenient for computing the DGLAP evolution, these are not
physically measurable objects. Caution is required when comparing them.

In particular, if we try to invert the relations of Eq. (5.18) for the up and
down quarks, we find:

$\displaystyle u^{p/A}$ $\displaystyle=$
$\displaystyle\frac{Z}{2Z-A}\,u^{A}-\frac{A-Z}{2Z-A}\,d^{A}\qquad{\rm
and}\qquad d^{p/A}=\frac{Z}{2Z-A}\,d^{A}-\frac{A-Z}{2Z-A}\,u^{A}\quad.$ (5.19)

This relation has the expected limits that for a proton with $A=p$ and
$\\{A,Z\\}=\\{1,1\\}$ we find $\\{u^{p/A}{=}u^{A=p};\ d^{p/A}{=}d^{A=p}\\}$.
Also, for a neutron with $A=n$ and $\\{A,Z\\}=\\{1,0\\}$ we find
$\\{d^{p/A}{=}u^{A=n};\ u^{p/A}{=}d^{A=n}\\}$. However, for an isoscalar
target $(A=2Z)$, Eq. (5.19) is singular because in the isoscalar limit it is
impossible to differentiate the $u^{A}$ and $d^{A}$ distributions and the
decomposition into $u^{p/A}$ and $d^{p/A}$ becomes arbitrary. Since the
majority of the nuclear data is taken on nuclear targets which lie along the
nuclear stability line in the $\\{A,Z\\}$ plane where $A\sim 2Z$ (except for
very heavy nuclei), our ability to separately determine the nuclear $u^{A}$
and $d^{A}$ distributions is limited.

### 5.4 Kinematic $\boldsymbol{W}$ Cut

Determinations of PDFs are performed by global analyses using a wide array of
data sets that can be described by the parton model within the factorization
framework. It is therefore important to exclude data in the kinematic region
that are not reliably predicted within this model.

In lepton-nucleon scattering, there are different types of scattering
scenarios depending on the value of the hadronic invariant mass,
$W^{2}=(p+q)^{2}$, and the value of the virtuality of the exchanged boson $Q$
[100]. We can divide these cases into:

  1. 1.

elastic scattering ($W=M_{N}$),

  2. 2.

shallowly inelastic scattering (SIS) ($W\leq 2$ GeV),

  3. 3.

soft DIS ($W>2$ GeV, $Q<1$ GeV), and

  4. 4.

DIS ($W>2$ GeV and $Q\geq 1$ GeV).

Among these scenarios, the DIS and soft DIS (dominated by non-resonant pion
production) cases are the most reliably described by the parton model.
Therefore, a $W$ cut is traditionally imposed on DIS data to extract PDFs.

While imposing a $W$ cut is straightforward for the case of lepton-proton
scattering, it is more subtle for lepton-nucleus scattering when we compute in
terms of the rescaled variables. For $\ell_{1}+A\rightarrow\ell_{2}+X$
scattering, we have:

$W_{A}^{2}=(p_{A}+q)^{2}=M_{A}^{2}+Q^{2}\
\frac{1-x_{A}}{x_{A}}=A^{2}M_{N}^{2}+Q^{2}\ \frac{1-x_{N}/A}{x_{N}/A}\quad.$
(5.20)

The average $W_{A}$ per nucleon, $W_{\rm aver}=W_{A}/A$, is then given by

$\displaystyle W_{\rm aver}^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{W_{A}}{A}\right)^{2}=M_{N}^{2}+Q^{2}\
\frac{A-x_{N}}{A^{2}x_{N}}\quad.$ (5.21)

Additionally, we observe that $W_{\rm aver}^{2}$ and $W_{N}^{2}$ are not the
same:

$\displaystyle W_{N}^{2}$ $\displaystyle=$
$\displaystyle(p_{N}+q)^{2}=M_{N}^{2}+Q^{2}\ \frac{1-x_{N}}{x_{N}}\quad.$
(5.22)

Note, that for $x_{N}\in[0,A]$, we find $W_{\rm aver}^{2}$ is always positive
while $W_{N}^{2}$ can be negative for $x_{N}>1$. On the other hand, $W_{N}$ is
independent of $A$ and can be used to compare different nuclei as long as
$x_{N}$ is restricted to values below unity.

If we impose a lower bound on $W_{A}$, this translates to the well-known upper
bound on $x_{A}$ of

$0\leq x_{A}\leq\left[1+\frac{W_{A}^{2}-M_{A}^{2}}{Q^{2}}\right]^{-1}\leq
1\,.$ (5.23)

In terms of the rescaled variable $x_{N}=Ax_{A}\ \in[0,A]$ this translates to

$0\leq x_{N}\leq A\left[1+\frac{W_{A}^{2}-M_{A}^{2}}{Q^{2}}\right]^{-1}\leq
A\,.$ (5.24)

Consequently, when we impose a $W$ kinematic cut for the nuclear case ($A>1$),
we need to be cognizant of these issues. One could try to limit the resonance
region with a mass cut ($m_{\rm cut}$) by setting $W_{N}^{2}=M_{N}^{2}+m_{\rm
cut}^{2}/A^{2}$. For $A>1$, the resulting high-$x$ cut turns out to be more
relaxed than the usual $W$ cut. Thus, the use of traditional $W$ cut here is
much safer and hence, despite its pathological nature at $x>1$, this cut can
still be used. While cutting on $W_{A}$ or $(W_{A}/A)$ is more natural, the
value of $W_{N}$, which is $A$ dependent, needs further investigations.

Figure 5.2:  We display the neutron structure function $F_{2}^{n}$ as measured
by the BONuS data [101] at selected $Q^{2}$ values. The open (filled) circles
represent data for a beam energy of E = 4.223 (5.262) GeV. This is compared
with ABKM PDF predictions [94] including higher twist effects and target mass
corrections. We observe the resonance structure at low $W^{2}$ values. Figure
taken from Ref. [101].

### 5.5 nPDFs for $x_{N}\geq 1$

The rescaled PDFs allow us to compare distributions from different nuclei in a
meaningful manner as illustrated in Fig. 5.1. Operationally, all current
global nuclear PDF analyses work on the interval $x_{N}\in[0,1]$ even though
the technically allowed range is $x_{N}\in[0,A]$. Although the details of the
$x_{N}>1$ region are beyond the scope of this study, we do want to briefly
mention experimental measurements that provide insights in this extreme
kinematic region.

#### The Fermi Region

The effects of nuclear binding can modify the partonic momentum, and thus
dramatically impact the resulting structure functions. Fig. 5.3 shows the
characteristic form of the nuclear correction factor $F_{2}^{Fe}/F_{2}^{D}$
with shadowing at small $x_{N}$ ($x_{N}<0.1)$, anti-shadowing ($x_{N}\sim
0.1)$, the EMC region for intermediate $x$, and the Fermi region for large
$x$. The nuclear binding effects are especially evident at larger momentum
fractions (large $x$) in the Fermi region where the PDFs are steeply falling.
There are a variety of theoretical approaches to describe this region, as well
as experimental measurements of these nuclear binding effects. We provide only
a brief overview below and refer to the reference for additional details.

Figure 5.3: We display a characteristic nuclear correction ratio
$F_{2}^{Fe}/F_{2}^{D}$ for iron over deuterium. We observe the shadowing
region ($x\lesssim 0.1$), the anti-shadowing region ($x\sim 0.2$), the EMC
region ($x\sim 0.5$), and the Fermi region ($x\gtrsim 0.8$). The figure is
taken from Ref. [65] which contains the details and references for the data
sets.

#### Challenges of the Fermi Region

The limitations of constraining the PDFs in the $x_{N}\sim 1$ region are
evident when examining the nuclear ratio $F_{2}^{A}/F_{2}^{p}$ in the large
$x_{N}$ region. As $x$ increases, the ${\cal R}=F_{2}^{A}/F_{2}^{p}$ ratio
transitions from ${\cal R}\lesssim 1$ in the EMC region, to ${\cal R}\gtrsim
1$ as $x_{N}\to 1$ in the Fermi region.

We find that $F_{2}^{p}(x)$ with $A=1$ for the proton must vanish at
$x_{N}=1$, whereas $F_{2}^{A}(x)$ can be finite since $x_{N}\in[0,A]$. This
means that the denominator of ${\cal R}$ is vanishing while the numerator is
finite. Hence, ${\cal R}$ will rapidly increase at $x\sim 1$, and this is
consistent with experimental measurements. More generally, we expect
$F_{2}^{A_{1}}>F_{2}^{A_{2}}$ for $A_{1}>A_{2}$, and this is the case for
$F_{2}^{{}^{12}C}/F_{2}^{D}$ as shown in Fig. 5.4.

#### Theoretical Implementations

Figure 5.4: We display the ratio $F_{2}^{A}/F_{2}^{d}$ for selected nuclei
data sets in the large $x$ region. The different curves show the impact of the
$\varepsilon$ parameter for the rescaling $x^{\prime}_{N}\to[x_{N}-\varepsilon
x_{N}^{\kappa}\log_{10}A]$ with $\kappa=10$. This figure is taken from Ref.
[19].

On the theoretical side, there are a number of approaches that have been
proposed to model the nPDFs into the $x_{N}>1$ region, and we point to a
selection of references [102, 103, 104, 105, 106, 107, 37, 108, 109, 11, 110].
Additionally, there is a LO solution of DGLAP equation for quarks taking into
account the full support (including $x_{N}>1$ region) which is detailed in
Ref. [111].

One technique to describe the $x_{N}>1$ region is to use a convolution of the
nucleon PDFs $f(x,Q)$ defined on the interval $x\in[0,1]$ with a smearing
kernel $S_{A}$ which will shift the PDFs out to larger $x$ values:

$f_{A}(x,Q)=\int_{x}^{A}\frac{dy}{y}S_{A}(y,Q)\,f(x/y,Q)\quad.$ (5.25)

Here, the smearing kernel $S_{A}$ is typically a Gaussian with an
$A$-dependent width; larger nuclei have a larger smearing width, and hence a
larger proportion of the $f_{A}$ distribution will populate the $x>1$ region.
The behavior of the PDF convolution can also be approximated using an
$A$-dependent rescaling: $x^{\prime}_{N}\to[x_{N}-\varepsilon
x_{N}^{\kappa}\log_{10}A]$, c.f., Ref. [19]. This mimics the PDF convolution
method described above, but the rescaling avoids the convolution integral, so
it is computationally fast to evaluate.

In Fig. 5.4 we display $F_{2}^{A}/F_{2}^{d}$ for selected nuclei and data
sets, and indicate various levels of rescaling (controlled by the
$\varepsilon$ parameter). The data points are described in Ref. [19], and the
solid curve (blue) with $\varepsilon{=}0$ is the default theory with no
$x$-rescaling. The solid points are within the kinematic cuts of the global
fit ($Q>1.3$ GeV, $W_{N}>1.7$ GeV), and the hollow points are outside these
cuts. As we increase the $\varepsilon$ parameter, we see this increases the
ratio in the large $x$ region, and can improve the comparison of the data and
theory.

Judging from this sample of data, it appears that a detailed modeling of the
Fermi region can improve the description of the data in this extreme kinematic
region (low $W_{N}$, low $Q$, large $x_{N}$). These corrections could be a
combination of TMCs, $x_{N}\,{>}\,1$ effects, higher twist, or other non-
perturbative corrections. Thus, a complete analysis must include and balance
all these contributions.

#### Experimental Investigations

There are a number of studies that have explored the structure functions in
the Fermi region at very large $x_{N}$, and we mention a few examples below.

For example: Refs. [112, 113] examined SLAC DIS data from a variety of nuclear
targets and found that in the larger $x_{N}$ region ($x_{N}\,{\gtrsim}\,0.5$)
the Fermi motion effects are similar to those in the deuteron, and increase
for heavier nuclei. Measurements in this region are particularly challenging,
and are subject to a variety of uncertainties as outlined in the Appendix of
Ref. [112].

For the CC neutrino DIS process, CCFR [114] measured ${\sim}2000$ events in
the region $x_{N}>0.75$, and used this to study $F_{2}$ at very large $x_{N}$.
They fit the high $x_{N}$ behavior to a decaying exponential
$\propto\exp(-sx_{N})$ normalized at $x_{N}=0.65$, and find a slope of
$s=8.3\pm 0.7$. They note this compares favorably with E133 at SLAC [115],
which observed $s\,{\sim}\,[7,8]$.

Hall C at Jefferson Lab measured the NC structure function $F_{2}$ on a
variety of nuclei ranging from 2H and 3He up to 197Au in the large $x_{N}$
region [116, 117] out to $x_{N}\sim 1.4$. This work extends an earlier study
of Ref. [118], and related measurements performed at SLAC [119]. Taking
account of the kinematic pre-factors and the $\\{h_{2},g_{2}\\}$
contributions, they extract $F_{2}^{(0)}(\xi_{N},Q^{2})$, where $\xi_{N}$ is
the Nachtmann variable, and fit this to a decaying exponential of the form
$\exp(-s\xi_{N})$ in the region $\xi_{N}\geq 0.75$. The large $\xi_{N}$ fits
finds the slope is in the general range $s\sim 15\pm 2$. The slope has a mild
increase with increasing $Q^{2}$, and a mild decrease with increasing nuclear
$A$. These results also compare favorably with the BCDMS muon scattering data
[120] which finds $s=16.5\pm 0.6$ for $x_{N}\,{\sim}\,[0.75,1.05]$.

#### Parton Momentum in the Fermi Region

Based on the above large $x_{N}$ measurements, we can ask the question: How
much parton momentum is in the region $x_{N}>1$?

The experimental measurements discussed above parameterize $F_{2}$ at large
$x_{N}$ as $\exp[-sx_{N}]$, where $s$ can range from about ${\sim}8.3$ to
${\sim}14$. With $s$=8.3, the fraction of $F_{2}$ in the range
$x_{N}\in[1,\infty]$ is less than ${\sim}6\%$ that in the range
$x_{N}\in[0.65,1]$. Given that the average momentum fraction in the interval
$x_{N}\in[0.65,1]$ is $\lesssim 2\%$ of the total nucleon momentum, this gives
a rough estimate of the potential momentum fraction beyond $x_{N}>1$ of
$6\%\times 2\%\lesssim 0.1\%$ of the total momentum fraction. For $s\sim 14$,
the decrease at large $x_{N}$ is even steeper; here, the $F_{2}$ ratio of
$x_{N}\in[0.65,1]$ compared to $x_{N}\in[1,\infty]$ drops to $\lesssim 0.1\%$,
and the potential momentum fraction beyond $x_{N}>1$ estimate becomes
$1\%\times 2\%\lesssim 0.02\%$ of the total.

In both of the above cases, the integrated parton momentum is very small in
the $x_{N}>1$ region, but the impact on the $F_{2}^{A_{1}}/F_{2}^{A_{2}}$
ratio still yields the characteristic increase in the $x_{N}\sim 1$ Fermi
region.

### 5.6 Threshold problem and higher twist contributions

On the basis of the correspondence between the limits $x_{N}\to 1$ and
$W_{N}^{2}\to M_{A}^{2}$, the very high-$x_{N}$ behavior of DIS structure
functions at fixed $Q^{2}$ is sensitive to the presence of power-suppressed
corrections beyond leading twist (twist-4 for unpolarized processes). It has
been conjectured that the problematic threshold behavior in target-mass
corrected structure functions in the free nucleon case are at least partly due
to the absence of higher-twist contributions. As an approximate test of the
inclusion of twist-4 contributions, Ref. [121] proposed a $1/Q^{2}$ expansion
of the OPE mass-corrected structure functions, which indeed suppresses the
problematic threshold behavior. These calculations were done for $F_{2}$, and
later, for other tensor structure function in Ref. [122]. While this was done
for the free nucleon, a similar exercise might be carried out for nuclear
structure functions as another TMC prescription.

In particular, Kulagin and Petti [121] showed that by expanding the target
mass corrected structure functions to leading order in $1/Q^{2}$, the
resulting functions have the correct $x_{N}\to 1$ limits. While avoiding the
threshold problem, this prescription, however, raises the question of whether
the $1/Q^{2}$ approximation is sufficiently accurate for structure functions
near $x_{N}\approx 1$ at moderate $Q^{2}$.

To test the convergence of the $1/Q^{2}$ expansion at large $x$, Ref. [122]
further expands the OPE result(s) to include ${\cal O}(1/Q^{4})$ corrections.
In fact, one can demonstrate that for a structure function that behaves at
large $x$ as $(1-x)^{n}$, the target mass corrected result will vanish in the
$x_{N}\to 1$ limit up to order $1/Q^{2n-2}$ in the expansion. For $n\approx
3$, as is typical for nucleon structure functions, the threshold problem will
therefore appear only at order $1/Q^{6}$.

## 6 The parton model with quark and hadron masses

We now turn to a discussion of the parton model, including the effects of
quark and hadron mass. Such mass effects can be rigorously included in the
QCD-improved parton model, as shown in a general factorization proof with
quark masses by Collins [46]. The underlying scheme is also known in the
literature as the Aivazis-Collins-Olness-Tung (ACOT) scheme [57, 58, 123],
which is an example of what is called a General Mass Variable Flavor Number
Scheme (GM-VFNS) in the modern literature [58, 91, 92, 124, 93, 94]. In the
following subsections we summarize DIS structure functions in the ACOT scheme,
including quark and hadron (nuclear) mass effects, and compare and contrast
these expressions to the structure functions in the OPE. As discussed in the
introduction (Sec. 1), we consider the theoretical status of both approaches,
the OPE and the collinear-factorization-based QCD-improved parton model, on
equal footing.

In the naïve, massless parton model, we have a hadron of momentum $p_{A}$ that
emits a collinear parton of momentum $k=x_{A}\,p_{A}$. If we try and extend
this simple picture to the case of massive partons, the presence of the parton
mass $m_{i}$ violates the collinear kinematics and we encounter ambiguities
proportional to the parton mass. We work in the ACOT formalism [57, 58, 123]
to compute the structure functions in the helicity basis. This has the
advantage that the polarization vectors are boost invariant between the hadron
and parton frame, and provides advantages for incorporating both the hadron
and parton masses, as we shall illustrate below. We observed in Sec. 4 that
the hadron mass modifies the scaling variable via the relation
$\xi_{A}~=~R_{M}~x_{A}$, where the combination $R_{M}x_{A}$ is the Nachtmann
variable. Within the context of the OPE and the parton model, the effect of
the quark masses factorizes from the hadron modification. The rescaling
variable is modified by an additional factor $R_{ij}$, which is a function of
the quark masses $\\{m_{i},m_{j}~\\}$ and is defined in Table 4.2. Thus, we
have $\xi_{A}~=R_{ij}~R_{M}~x_{A}$.

In the following, we also review the structure functions in the light-front
formalism, and compare with the TMC expressions obtained from the OPE. There
has been extensive discussion in the literature about computations with
massive partons [58, 91, 92, 93, 94]. The present discussion is focused on the
detailed organization of the massive parton contributions in the formalism
outlined by Collins [46]. This work extended the factorization proof to the
case of massive partons and demonstrated that all contributions proportional
to the parton mass can be fully factorized. Thus, we include all terms of
order ${\cal O}(m_{i}/Q)^{2}$, and the non-factorizable terms are suppressed
by powers of $(\Lambda_{\rm NP}/Q)^{2}$.

### 6.1 Helicity Formulation of the Parton Model

When we generalize the massless QCD parton model to the massive case, there
are potential ambiguities proportional to the parton mass $m_{i}$, which may
enter if we are not careful about organizing our perturbative expansion. This
organization is more easily dealt with in the helicity basis as we outline.

The original ACOT formalism for massive DIS was computed in the helicity
basis, and this provides some advantages in working with both hadron and
parton masses. For example: in the helicity formalism, we can define
polarization vectors (Sec. 6.2) in a boost-invariant manner so that they are
the same for both the hadron and parton reference frames (Sec. 6.3). This has
the advantage that there is no mixing of terms between the partonic and
hadronic structure functions in the helicity basis. Additionally, by working
with light-front momenta, $\\{p^{\pm},k^{\pm}\\}$, we can take advantage of
the fact that $p^{+}\gg p^{-}$ and $k^{+}\gg k^{-}$, and define our scaling
variable $\xi$ as $k^{+}=\xi p^{+}$, which is valid for both massless and
massive partons.

For the present discussion, we will focus on the target mass corrections which
enter the hadronic structure functions. We will find the helicity formalism
matches the OPE master formula for the leading terms. We compare these result
both analytically in Sec. 6.4 and numerically in Sec. 6.5 (and Sec. 7).

### 6.2 Helicity Decomposition

In analogy to what was done in Sec. 2, we decompose the hadronic tensor
$W_{\mu\nu}$ by projecting out Lorentz tensors to obtain the six structure
functions $F_{i}$. We can also project out the helicity components using
polarization vectors $\epsilon_{n}^{\nu}(p,q)$:

$F_{\lambda}\equiv
F_{mn}=\epsilon_{m}^{\mu\,*}(p,q)W_{\mu\nu}(p,q)\epsilon_{n}^{\nu}(p,q)\ .$
(6.1)

Here, helicity indices $m,n$ span the helicity components plus ($+$), minus
($-$), longitudinal ($0$), and scalar ($s$). Angular momentum conservation
reduces these to six combination $\\{++,--,00,ss,0s,s0\\}$, just as we have
six hadronic structure functions in the tensor basis $F_{i}$ with
$\\{i=1...6\\}$. Additionally, the three combinations
$\\{F_{ss},F_{s0},F_{0s}\\}$ are suppressed by the lepton mass in a manner
analogous to $\\{F_{4},F_{5},F_{6}\\}$, so typically these are ignored. The
remaining combinations are $\\{F_{++},F_{00},F_{--}\\}$, which we also
indicate with the short-hand notation $F_{\lambda}$ with
$\lambda=\\{+,0,-\\}$.

We obtain the hadron-level cross section by contracting the hadronic and
leptonic tensors:

$\displaystyle d\sigma$ $\displaystyle\sim$ $\displaystyle L^{\mu\nu}(q,k)\
W_{\mu\nu}(p,q)$ (6.2) $\displaystyle\equiv$ $\displaystyle
L^{m^{\prime}n^{\prime}}(q,k)\ d_{m^{\prime}}^{m}(\psi)^{-1}\
d_{n^{\prime}}^{n}(\psi)\ F_{mn}(p,q)\ ,$

where we show the corresponding relations in both the tensor and helicity
basis. Here the Wigner rotation matrix $d_{n^{\prime}}^{n}(\psi)$ is the
$SO(2,1)$ analog of the familiar $3\times 3$ “rotation” matrix (since the
exchanged particle is spin-1 with 3 polarization states), and $\psi$ is the
boost angle that transforms between the hadronic Breit (“brick wall”) frame,
where $p\|q$, and the leptonic frame where $k\|k^{\prime}$, with

$\cosh\psi=\frac{2p\cdot(k+k^{\prime})}{\Delta[-Q^{2},p^{2},p_{x}^{2}]}=\frac{\xi^{2}M^{2}-Q^{2}+2\xi(s-M^{2})}{\xi^{2}M^{2}+Q^{2}}\quad\xrightarrow[M\to
0]{}\quad\frac{2-y}{y}\quad.$ (6.3)

Here, $\Delta^{2}[a,b,c]=a^{2}+b^{2}+c^{2}-2(ab+bc+ca)$. An elegant feature of
Eq. (6.2) is that the leptonic ($L^{mn}$) and hadronic ($F_{mn}$) helicity
structure functions are simple and all the kinematic complications due to the
target mass are contained in the $d(\psi)$ rotation matrices. This simplicity
is a consequence of the underlying group-theoretic approach to the factorized
structure.999While the helicity formalism elegantly exhibits the underlying
symmetries of the interaction, as we go to higher-order the computation of the
Wigner rotations for multiple intermediate particles can become more complex
[58].

### 6.3 Boost Invariant Polarizations

If we work in light-front coordinates $\\{x^{+},\vec{x},x^{-}\\}$ with
$x^{\pm}=(x^{0}\pm x^{3})/\sqrt{2}$ and $\vec{x}=\\{x^{1},x^{2}\\}$, then

$\displaystyle p^{\mu}$ $\displaystyle=$
$\displaystyle\left\\{p^{+},\vec{0},\frac{M^{2}}{2p^{+}}\right\\}\qquad\text{and}\qquad
q^{\mu}=\left\\{-\xi p^{+},\vec{0},\frac{Q^{2}}{2\xi p^{+}}\right\\}\ ,$ (6.4)

such that $p^{2}=M^{2}$ and $q^{2}=-Q^{2}$. Additionally,
$2p{\cdot}q=Q^{2}/x=Q^{2}/\xi\,{-}\xi M^{2}$. We now choose to define the
parton momentum for a massive state as

$\displaystyle k^{\mu}$ $\displaystyle=$ $\displaystyle\left\\{\xi
p^{+},\vec{0},\frac{m^{2}}{2\xi p^{+}}\right\\}\ ,$ (6.5)

such that $k^{2}=m^{2}$ and $k^{+}=\xi p^{+}$. This scaling relation holds for
both massless and massive partons.

We require two reference vectors to define the polarizations. For the hadron
polarizations $\epsilon_{n}^{\nu}(p,q)$ we choose the vectors $\\{p,q\\}$,
while for the parton polarizations $\epsilon_{n}^{\nu}(k,q)$ we choose the
vectors $\\{k,q\\}$. For both the hadron and parton polarizations, the scalar
polarization is given by $\epsilon_{0}^{\mu}=q^{\mu}/Q$, and the transverse
polarizations are defined as usual to be $\epsilon^{\pm}=(0,\mp
1,-i,0)/\sqrt{2}$. The longitudinal polarizations are constructed from the
reference vectors using a completeness relation [57]. The polarization vectors
only depend on the reference vectors $p^{\mu}$ and $k^{\mu}$ to the extent
that they define the $t-z$ plane in conjunction with $q^{\mu}$; see Ref. [57]
for details.

Thus, the polarization vectors are invariant between the hadron and parton
reference frame. Additionally, there is a direct relation between the hadronic
helicity structure functions $F_{\lambda}$ and the partonic helicity structure
functions $\omega_{\lambda}$:

$\displaystyle F_{\lambda}$ $\displaystyle=$
$\displaystyle\delta^{\lambda\lambda^{\prime}}f\otimes\omega_{\lambda^{\prime}}\quad.$
(6.6)

Here, $f$ is the PDF, $\otimes$ represents a convolution,
$\delta^{\lambda\lambda^{\prime}}$ is the usual Kronecker $\delta$-function,
and $\lambda$ sums over the six helicity configurations. Specifically, we note
there is no mixing as indicated by the $\delta^{\lambda\lambda^{\prime}}$
function. This is in contrast to the structure functions in the tensor basis:

$\displaystyle F_{i}$ $\displaystyle=$ $\displaystyle
c^{ij}\,f\otimes\omega_{j}$ (6.7)

where the mixing coefficients $c^{ij}$ contain off-diagonal elements, with
indicies $i,j$ summing over the six tensor structure functions [57].

| TMC | ACOT-TMC | $\frac{\mathrm{ACOT-TMC}}{\mathrm{TMC}}$
---|---|---|---
$F_{1}$ | $\frac{x}{\xi\,r}$ | $1$ | $\frac{\xi\,r}{x}\sim 1+\varepsilon^{2}$
$F_{2}$ | $\frac{x^{2}}{\xi^{2}\,r^{3}}$ | $\frac{x}{\xi\,r^{2}}$ | $\frac{\xi\,r}{x}\sim 1+\varepsilon^{2}$
$F_{3}$ | $\frac{x}{\xi\,r^{2}}$ | $\frac{1}{r}$ | $\frac{\xi\,r}{x}\sim 1+\varepsilon^{2}$
Table 6.1: Tabulation of prefactors from the master formulas for the TMC
(obtained using the OPE) of Eq. (3.24) and ACOT-TMC light-front derivations of
Eq. (6.10), where $\varepsilon=(xM/Q)$, and
$r=r_{A}=r_{N}=\sqrt{1+4\varepsilon^{2}}$. While the $F_{i}$’s are generally
evaluated at $x_{N}$, using Table 4.1 we have $x_{N}/\xi_{N}=x_{A}/\xi_{A}$
and these ratios are equivalent. The TMC and ACOT-TMC prefactors match to
${\cal O}(\varepsilon^{2})$, and this is plotted in Fig. 6.1. The source of
this factor is discussed in the text.

### 6.4 Relation to Master Formulas

Finally, coming back to the hadronic level, we can relate the hadronic
helicity structure functions $F_{\lambda}$ to the tensor structure functions
$F_{i}$ using Eq. (6.1) and obtain:101010Note that while the $F_{i}$ structure
functions of Eq. (4.15) are evaluated at $x_{N}$, the relations of this
section hold in general; hence, we omit the subscripts on $x_{N}$ here.

$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{F_{+}+F_{-}\right\\}\ ,\qquad
F_{2}=\frac{x}{r^{2}}\left\\{F_{+}+F_{-}+2F_{0}\right\\}\ ,\qquad
F_{3}=\frac{1}{r}\left\\{-F_{+}+F_{-}\right\\}\ ,$ (6.8a) and the inverse
relations $\displaystyle F_{\pm}$ $\displaystyle=$ $\displaystyle
F_{1}\mp\frac{r}{2}F_{3}\ ,\qquad F_{0}=-F_{1}+\frac{r^{2}}{2x}F_{2}\ ,$
(6.8b)

where $r=r_{A}=r_{N}=\sqrt{1+4\varepsilon^{2}}$ and $\varepsilon=(xM/Q)$. The
$F_{0}$ helicity structure function is related to the longitudinal structure
$F_{L}$ of Eq. (3.26) via

$\displaystyle F_{L}$ $\displaystyle=$ $\displaystyle
2x\,F_{0}=r^{2}F_{2}-2xF_{1}\ .$ (6.9)

Translating the above relations into the master formula for the TMCs for the
light-front approach yields:

$\displaystyle F_{1}^{\mathrm{ACOT-TMC}}(x)$ $\displaystyle=$
$\displaystyle\,F_{1}^{(0)}(\xi)$ (6.10a) $\displaystyle F_{2}^{\mathrm{ACOT-
TMC}}(x)$ $\displaystyle=$ $\displaystyle\ \frac{x}{\xi\,r^{2}}$
$\displaystyle\,F_{2}^{(0)}(\xi)$ (6.10b) $\displaystyle F_{3}^{\mathrm{ACOT-
TMC}}(x)$ $\displaystyle=$ $\displaystyle\frac{1}{r}$
$\displaystyle\,F_{3}^{(0)}(\xi)\quad,$ (6.10c)

and $F_{L}^{\mathrm{ACOT-TMC}}(x)$ can be constructed using Eq. (3.26). In the
above formula, we can incorporate the parton masses by using the full
rescaling variable $\xi_{A}~=R_{ij}~R_{M}~x_{A}$, and computing
$F_{i}^{(0)}(\xi)$ including the quark mass effects. The proof of
factorization for heavy quarks is detailed in Ref. [46].

The correspondence between the ACOT-TMC and TMC expressions are summarized in
Table 6.1. In the right-most column, we display the relative conversion factor
($\xi r/x$), which is the same for all three $F_{i}^{\mathrm{ACOT-TMC}}(x)$
results. While we do not compute $F_{4,5,6}$, we expect these structure
functions also follow this pattern. For $F_{i}$, we see that the TMC and ACOT-
TMC approach align to ${\cal O}(\varepsilon^{2})$, where $\varepsilon=(xM/Q)$
vanishes outside the dominant TMC region of large $x$ and small $Q$. Note,
these comparisons are for the leading term $F_{i}^{\mathrm{Leading-TMC}}$ only
(see below and Sec. 7), as the higher order terms in $(M/Q)^{2}$ with the
$h_{i}$ and $g_{i}$ functions are not included in the ACOT expressions.

Figure 6.1: We display the relative factor $(\xi r/x)$ between the ACOT and
leading OPE terms as tabulated in Table 6.1 for $Q=\\{1.3,1.5,2,3,4,6\\}$ GeV
as a function of $x$ with $M=M_{proton}$. The lower two $Q$ values
$\\{1.3,1.5\\}$ are dashed to indicate these values are below typical
kinematic cuts ($Q_{cut}\sim 2$ GeV) for global PDF fits.

### 6.5 Relation of Parton Model to OPE

The parton model ACOT-TMC results differ from the TMC results obtained with
OPE in two respects: i) there is an overall factor of $(\xi r/x)$, and ii) the
OPE result contains additional $(M/Q)^{2}$ and $(M/Q)^{4}$ terms. We will
discuss the origin of these differences below.

##### Correspondence of the leading factor $(\xi r/x)$:

The TMC (OPE) results of Eq. (4.15) contain an overall factor of $(\xi r/x)$
as compared with the ACOT-TMC results of Eq. (6.10). The source of this factor
comes from fully including the hadron mass in the calculation as outlined in
A.5 which computes both the massless and massive results. The relative factor
is most obvious in the case of $F_{1}$, where the ACOT-TMC result is simply
given by $F_{1}^{(0)}(\xi)$.

In contrast, the massive OPE results of Eq. (A.146) include the additional
$(\xi r/x)$ factor which arises from the delta function of Eq. (A.124). This
delta function enforces the relationship between the Bjorken and Nachtmann
variables, and is a manifestation of momentum conservation. The same factor is
present in all the OPE results for $F_{i}$ as computed in A.5, and is
summarized in Table 6.1 for $F_{1,2,3}$. Additionally, this factor is also
present in Ref. [83]; c.f., Eq. (1.24) where, matching notation,
$(1+\mu\xi^{2})=(r\xi/x)$ with $\mu=(M/Q)^{2}$.

In Fig. 6.1 we show the size of this relative factor as a function of $x$ for
selected $Q$ values. For small $x$ and large $Q$ this factor is close to
unity, but grows for large $x$ and small $Q$ values where the TMCs are
typically substantial.

##### Correspondence of the $M/Q$ Powers:

The OPE master equation, Eq. (3.24), includes higher powers of $(M^{2}/Q^{2})$
which are formally twist-2. The source of these terms comes from the intrinsic
transverse momentum of the parton [55, 83]. The partonic cross sections are
evaluated with parton momenta on-shell, but not collinear with the parent
hadron. The details are provided in Refs. [55, 83], which include the
transverse momentum in the description of the partons, [c.f., $f(x,k_{T}^{2})$
in Eq. (1.22) of Ref. [83]]. As the transverse momentum is bounded to be of
order $M^{2}$, these contributions give rise to the twist-2 $(M^{2}/Q^{2})$
and $(M^{4}/Q^{4})$ terms of Eq. (3.24). Thus, if the transverse momentum is
accounted for in the parton model, one matches the OPE result.

##### Recap:

We have derived the structure function results for the parton model using the
ACOT formalism in a helicity basis as summarized in Eq. (6.10). We can obtain
a complete correspondence between the parton model and OPE structure function
results if i) we fully account for the hadron mass in the delta function
(c.f., Eq. (A.124)), which ensures the relationship between the Bjorken and
Nachtmann variables, and ii) we account for the transverse momentum of the
parton (c.f., Ref. [83]).

## 7 Numerical results

Having discussed various theoretical and phenomenological aspects of TMCs for
nuclei from an analytic perspective, we now turn our focus to the quantitative
aspect of TMCs. In this section, we explore the numerical impact of TMCs on
first the proton structure functions (Sec. 7.1), and then the nuclear
structure functions (Sec. 7.2). We will examine both the leading-TMC and sub-
leading-TMC contributions, and explore the variation across the nuclear $A$
range. These observations will allow us to design a simple parameterization of
the TMCs which are computationally efficient (Sec. 7.3). Finally, we will
compute the impact of the TMCs for DIS reduced cross sections with sample
kinematics from HERA, JLab, EIC, and LBNF (Sec. 7.4).

In order to build intuition for the numerical results throughout this section,
it is useful to recall several relations between structure functions and quark
and antiquark PDFs, but which hold only at LO in the absence of CKM mixing and
masses. In particular, for neutrino and antineutrino DIS on an arbitrary
nuclear target $(A)$, where $A$ may also be just a proton, the charged current
structure functions $(F^{W^{\pm}}_{i})$ and PDFs are related by:

$\displaystyle F_{1}^{\nu A}$ $\displaystyle=$
$\displaystyle(d+s+\bar{u}+\bar{c}),\qquad$ $\displaystyle F_{1}^{\bar{\nu}A}$
$\displaystyle=$ $\displaystyle\quad(u+c+\bar{d}+\bar{s}),$ (7.1a)
$\displaystyle F_{2}^{\nu A}$ $\displaystyle=$ $\displaystyle
2x\left(d+s+\bar{u}+\bar{c}\right),\qquad$ $\displaystyle F_{2}^{\bar{\nu}A}$
$\displaystyle=$ $\displaystyle 2x\left(u+c+\bar{d}+\bar{s}\right)$ , (7.1b)
$\displaystyle F_{3}^{\nu A}$ $\displaystyle=$
$\displaystyle+2\left(d+s-\bar{u}-\bar{c}\right),\qquad$ $\displaystyle
F_{3}^{\bar{\nu}A}$ $\displaystyle=$
$\displaystyle{-}2\left(u+c-\bar{d}-\bar{s}\right)$ $\displaystyle\ ,$ (7.1c)

in the limit of four quarks. Here $\\{u,d,...\\}$ are the PDFs of a full
nucleus $A$. Likewise, when the exchange of $Z$ bosons can be neglected in
charged lepton DIS, the neutral current structure functions $(F^{\gamma}_{i})$
are related to PDFs at LO by

$F_{2}^{l^{\pm}A}=x\frac{1}{9}\left[4(u+\bar{u})+(d+\bar{d})+4(c+\bar{c})+(s+\bar{s})\right].$
(7.2)

While the LO relations shown above are intuitively useful, our calculations
are performed at full NLO in QCD including the quark mass contributions;
specifically, we use the S-ACOT($\chi$) scheme.

Since we will compare separate components of the TMC contributions, to reduce
ambiguity we use the following nomenclature throughout our presentation.

  * TMC:

We identify the full set of contributions to the structure function, given by
the OPE master equation of Eq. (4.15), as the “TMC” result; this label is
without qualifiers.

  * No-TMC:

As we take the $(M^{2}/Q^{2})\to 0$ limit, the pre-factors of $F_{i}^{A,(0)}$
become unity, and the higher-order $(M/Q)^{2}$ and $(M/Q)^{4}$ terms
containing the $\\{h_{i},g_{i}\\}$ functions vanish.111111Note, we always
retain the full quark mass dependence in all calculations as this is
factorized from the hadron-level kinematics. C.f., Ref. [60] for details.
Additionally, we have $\xi\to x$ so that the expressions in Eq. (4.15) reduce
to $F_{i}^{A,(0)}(x)$ alone. We refer to this as the “No-TMC” result.

  * Leading-TMC:

The “Leading-TMC” structure functions are obtained from Eq. (4.15) by only
keeping the first term on the RHS which is proportional to $(M/Q)^{0}$.
Specifically, we are neglecting the terms proportional to $(M/Q)^{2}$ and
$(M/Q)^{4}$ which contain the $\\{h_{i},g_{i}\\}$ functions.

  * ACOT-TMC:

We obtain the “ACOT-TMC” structure functions from the ACOT master equation of
Eq. (6.10). This result is similar to the “Leading-TMC,” but the pre-factors
differ by $(\xi r/x)$ as detailed in Table 6.1. Importantly, the “ACOT-TMC”
result does not include the higher-order $(M/Q)^{2}$ and $(M/Q)^{4}$ terms
containing the $\\{h_{i},g_{i}\\}$ functions.

### 7.1 Proton Structure Functions with TMC

In comparison to structure functions for massless protons, structure functions
with TMCs for nuclei contain two additional layers of complexity. The first,
of course, is the larger nucleon content; the second are the TMCs themselves.
Therefore, in order to establish a baseline intuition of TMCs for nuclear
structure functions, we consider briefly TMCs for proton structure functions.
For more comprehensive studies of TMCs for protons, see Ref. [60] and
references therein.

We begin with Fig. 7.1, where in the upper panel we plot the absolute
structure functions $F_{i}$ for charged current $(W^{-})$ and neutral current
$(\gamma,Z)$ exchange showing “No-TMC” (solid), full “TMC” (short dash),
“Leading-TMC” (dash-dot), and “ACOT-TMC” (long dash-dot), at
$Q=1.5{\rm~{}GeV}$ and $10{\rm~{}GeV}$ as a function of the Bjorken scaling
variable $x$. In the lower panel we show the ratio of the full “TMC” to the
“No-TMC” results. The inset panel displays the ratio of the “Leading-TMC” to
the “ACOT-TMC” which is simply a function of the relative $(\xi r/x)$ factor
discussed in Sec. 6.

#####  $F_{i}$ Structure Functions: (Fig. 7.1 Upper Panels)

We focus first on the upper panels of Fig. 7.1 which shows the proton
structure functions vs. $x.$ All the structure functions are steeply
decreasing for large $x$. This reflects the underlying PDF structure of Eqs.
(7.1) and (7.2): PDFs, and hence structure functions, must decrease at large
$x$ due to momentum conservation.

For both $Q=1.5$ GeV and $Q=10$ GeV, we observe that the full “TMC,” “Leading-
TMC” and “ACOT-TMC” results roughly coincide. The good agreement between the
full “TMC” and “Leading-TMC” allows us to infer that the sub-leading
contributions containing the $\\{h_{i},g_{i}\\}$ terms are small; this is to
be expected as they are suppressed by powers of $(M/Q)^{2}$ and $(M/Q)^{4}$.
We will further examine the impact of these sub-leading terms in Sec. 7.2.

The approximate agreement of the “Leading-TMC” and the “ACOT-TMC” reflects the
impact of the relative factor of $(\xi r/x)$ discussed in Sec. 6.4. To better
display this difference, in the inset panels we plot the ratio of these two
structure functions for the lower $Q=1.5$ GeV value; at $Q=10$ GeV, the
difference is minimal as is evident in Fig. 6.1.

In contrast, at $Q=1.5$ GeV, the “No-TMC” results are dramatically different
from the other three curves. The “No-TMC” expression is most similar to the
“Leading-TMC” as illustrated below for the case of $F_{1}$:

$\displaystyle F_{1}^{\rm TMC-leading}(x_{N})\ $ $\displaystyle=$
$\displaystyle\ \frac{x_{N}}{\xi_{N}r_{N}}$ $\displaystyle
F_{1}^{(0)}(\xi_{N})\ ,$ (7.3a) $\displaystyle F_{1}^{\rm No-TMC}(x_{N})$
$\displaystyle=$ $\displaystyle F_{1}^{(0)}(x_{N})\quad,$ (7.3b)

where $r_{N}=\sqrt{1+4\varepsilon^{2}}$. There are two differences between
these expressions: i) for “No-TMC” the pre-factor is set to unity, and ii) the
argument of $F_{i}^{(0)}$ is set to $x$ instead of $\xi$. Given that
$\xi=2x/(1+r)\simeq x(1-x^{2}M^{2}/Q^{2})$, the expression for $F_{1}^{\rm No-
TMC}$ is neglecting the $(M/Q)$ mass effects.

We can separately explore the impact of these two components and determine
which is dominant. For $F_{1}$ the relative factor $x/(\xi r)$ is precisely
the inverse of what is displayed in Fig. 6.1. Thus, we observe that for
$Q=1.5$ GeV and $x=0.9$, this factor yields a shift of ${\sim}20\%$. Comparing
with Fig. 7.2, we see this is clearly not sufficient to explain the observed
large difference between the “No-TMC” and “Leading-TMC” results. While $F_{2}$
and $F_{3}$ have slightly different prefactors, we obtain the same conclusion
that these prefactors do not generate the sizeable differences observed
between the “No-TMC” and the other results.

Thus, the dominant variation of $F_{i}^{\rm No-TMC}$ and $F_{i}^{\rm TMC-
Leading}$ must be due to the different arguments: $x$ vs. $\xi$. This
correspondence of $x$ and $\xi$ is displayed in Fig. 2.1 where we observe that
$x\sim\xi$ for small $x$ and large $Q$, but for large $x$ and small $Q$ we
find $\xi<x$. Because the PDFs are steeply falling at large $x$ (where $x$ and
$\xi$ have the largest deviation), it is the difference of $\xi$ vs. $x$ in
the arguments on the RHS of Eq. (7.3) that is driving the variation observed
in Fig. 7.2.

While we have detailed the comparison of $F_{1}$ in Fig. 7.2(a), the same
rationale applies to the other structure functions displayed.

In contrast to the sizeable differences between the “No-TMC” result and the
other three curves for the lower $Q$ value ($1.5$ GeV), the “No-TMC” result is
roughly comparable to the others for $Q=10$ GeV In total, these comparisons
affirm that TMCs are negligible for small-$x$ or for intermediate to large
$Q$.

(a) (b)

(c) (d)

Figure 7.1:  Upper panels: Proton structure functions for charged current
$W^{-}$ (a) $F_{1}$, (b) $F_{2}$, and (c) $F_{3}$ and (d) neutral current
$\gamma/Z$ $F_{2}$ as a function of momentum fraction $x_{N}$. We display
results for the full “TMC” (solid blue), “Leading-TMC” (dashed green), “No-
TMC” (dot-dash orange, and “ACOT-TMC” (dotted red) at $Q=1.5{\rm~{}GeV}$ and
$10{\rm~{}GeV}$. Lower panels: Ratio of $F_{i}$ with full “TMC” to “No-TMC” at
$Q=1.5{\rm~{}GeV}$ and $10{\rm~{}GeV}$. Insets: Ratio of $F_{i}$ with
“Leading-TMC” to “ACOT-TMC” at $Q=1.5{\rm~{}GeV}$.

#####  $F_{i}$ Structure Function Ratios: (Fig. 7.1 Lower Panels)

To highlight the importance of including the TMCs, we now examine the lower
panels of Fig. 7.1 which display the ratio of $F_{i}^{\rm TMC}$ to $F_{i}^{\rm
No-TMC}$. The ratio plots accentuate subtle features that are not evident on
the log-scale plots of the upper panels.

If we first focus on the $Q=1.5$ GeV results, we observe the deviation between
$F_{i}^{\rm TMC}$ and $F_{i}^{\rm No-TMC}$ differs markedly for the separate
$F_{1,2,3}$ structure functions. These differences arise from: i) the
different prefactors, see e.g. Eq. (4.15), ii) the different argument of
$F_{i}^{(0)}$ (as noted previously), and also iii) the sub-leading
$\\{h_{i},g_{i}\\}$ contributions.

It is the combination of these components that cause the variation among the
Fig. 7.1 sub-figures. For example, we see the $F_{1}$ curves deviate at
smaller $x$ values as compared with the $F_{2}$ and $F_{3}$ results.
Additionally, the $F_{2}$ and $F_{3}$ ratios dip below unity for intermediate
$x$ values ($x\sim 0.3-0.5$) while the $F_{1}$ ratio is above unity.

Symbol | $A$ | $Z$ | Symbol | $A$ | $Z$ | Symbol | $A$ | $Z$ | Symbol | $A$ | $Z$
---|---|---|---|---|---|---|---|---|---|---|---
H | 1 | 1 | Be | 9 | 4 | Ca | 40 | 20 | Xe | 131 | 54
D | 2 | 1 | C | 12 | 6 | Fe | 56 | 26 | W | 184 | 74
3He | 3 | 2 | N | 14 | 7 |  Cu${}_{\rm\,iso}$ | 64 | 32 | Au | 197 | 79
He | 4 | 2 | Ne | 20 | 10 |  Kr${}_{\rm\,iso}$ | 84 | 42 |  Au${}_{\rm\,iso}$ | 197 | 98.5
Li | 6 | 3 | Al | 27 | 13 |  Ag${}_{\rm\,iso}$ | 108 | 54 |  Pb${}_{\rm\,iso}$ | 207 | 103.5
Li | 7 | 3 | Ar | 40 | 18 |  Sn${}_{\rm\,iso}$ | 119 | 59.5 | Pb | 208 | 82
Table 7.1: List of nuclear PDFs considered in this work. We use the nCTEQ15
nPDF set [17]. The nuclei indicated with “iso” subscript are isoscalar
corrected; thus, the $Z$ value can be half-integer. We will also show results
with a neutron PDF computed using isospin symmetry.

(a) (b)

(c) (d)

Figure 7.2:  Ratio of nuclear structure functions $F_{i}$ for the full “TMC”
over “No-TMC” vs. $x_{N}$ at $Q=1.5{\rm~{}GeV}$. We display ratios for charged
current $W^{-}$ (a) $F_{1}$, (b) $F_{2}$, and (c) $F_{3}$ and (d) neutral
current $\gamma/Z$ $F_{2}$ for the nuclei listed in Table 7.1. The proton is
indicated with the black dashed line, and the neutron with the blue dotted
line. The narrow band of colored lines shows the various nuclear results.

### 7.2 Nuclear Structure Functions with TMCs

Having examined the proton structure functions in Fig. 7.1, we now consider
the corresponding nuclear structure functions of Fig. 7.2. The list of nuclear
isotopes that we consider are summarized in Tab. 7.1.

#### Full TMCs for Nuclei

In Fig. 7.2 we plot the ratio of the full “TMC” structure functions
$F_{i}^{\rm TMC}$ to the “No-TMC” structure functions $F_{i}^{\rm No-TMC}$ for
charged current $(W^{-})$ and neutral current $(\gamma,Z)$ exchange for the
nuclei of Table 7.1. This ratio highlights the impact of the TMCs.

We observe the general behaviour of the nuclear ratios displayed in Fig. 7.2
are very similar to the corresponding proton results in the lower panels of
Fig. 7.1. Because the nuclear and proton results are so similar, we only plot
a single $Q$ value ($1.5~{}$GeV) in Fig. 7.2 as other $Q$ values (e.g., $Q=10$
GeV) will be similar to the proton results. We will study the detailed $Q$
dependence in the following.

The results of Fig. 7.2 clearly demonstrate that the TMCs for the nuclei are
effectively independent of $A$. We attribute this curious finding to the re-
scaling property ($x_{A}M_{A}=x_{N}M_{N}$) as shown in Sec. 4. This re-scaling
allows us to identically rewrite the TMCs for structure functions in Eq.
(3.23), which are functions of nuclear-level quantities $x_{A}$ and $M_{A}$,
in terms of nucleon-level quantities $x_{N}$ and $M_{N}$. As a result, the
variable nucleon content of nuclei, which is the defining characteristic of
nuclei, is averaged out, resulting in near universal behavior.

To illustrate the emergence of this universality, let us Taylor-expand the
nuclear structure function ratio $F_{1}^{\rm TMC-Leading}/F_{1}^{\rm No-TMC}$
to show this is independent of the nuclear $A$ value up to corrections
$\varepsilon^{2}=(xM/Q)^{2}$. We use the expressions of Eq. (7.3) and