computing the derivative as $F_{i}^{(0)}(\xi_{N})\approx F_{i}(x_{N})+\delta
x_{N}[dF_{i}(y)/dy]_{y=x_{N}}$, where $\delta
x_{N}=(\xi_{N}-x_{N})\approx(x^{3}M_{p}^{2}/Q^{2})\ll 1$, If we apply this
approximation to the $F_{1}$ ratio, for example, we then obtain:

$\displaystyle\frac{F_{1}^{\rm TMC-Leading}(x_{N})}{F_{1}^{\rm No-
TMC}(x_{N})}\sim\left(1-\varepsilon^{2}\right)\frac{F_{1}^{(0)}(\xi_{N})}{F_{1}^{(0)}(x_{N})}\sim\left(1-\varepsilon^{2}\right)\frac{F_{1}^{(0)}(x_{N})+\mathcal{O}(\varepsilon^{2})}{F_{1}^{(0)}(x_{N})}\sim\left(1-\varepsilon^{2}\right)+\mathcal{O}\left(\varepsilon^{4}\right)\
.$ (7.4)

where we have defined $\varepsilon=(xM/Q)$. Remarkably, we find that the ratio
$F_{1}^{\rm TMC-Leading}(x_{N})/F_{1}^{\rm No-TMC}(x_{N})$ is independent of
$A$ up to $\varepsilon^{2}$. Recall the full TMC ratio $F_{1}^{\rm
TMC}(x_{N})/F_{1}^{\rm No-TMC}(x_{N})$ differs only by the additional
$\\{h_{i},g_{i}\\}$ terms which are suppressed by powers of $(M/Q)^{2}$. This
implies that nuclear TMCs themselves, when defined in terms of averaged
quantities, are essentially universal.

We now explore this approximate universal property of the TMCs, and determine
how we can exploit this property to simplify certain nuclear TMC calculations.

Figure 7.3:  Upper panels: We display the nuclear structure function ratios of
$F_{i}^{\rm TMC}/F_{i}^{\rm TMC-Leading}$ for selected CC and NC processes for
$Q=\\{1.3,1.5,2,3,4,6\\}{\rm~{}GeV}$ (from top to bottom) vs. $x_{N}$. Lower
panels: We display the variation of this ratio compared to the average
variation as computed in Eq. (7.6) for $Q=2$ GeV. In the top row we show
$F_{2}$ results for (a) NC $\gamma$ exchange, (b) CC $W^{+}$ exchange, and (c)
CC $W^{-}$ exchange, and the bottom row shows $F_{3}$ results for the same
processes. In all panels, 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.

#### Comparison of Full vs. Leading-TMCs for Nuclei

In Fig. 7.1 we observed that the “Leading-TMC” yielded a reasonable
approximation to the full “TMC” result, even at low $Q$ and large $x$ values.
Computationally, the “Leading-TMC” result is much simpler to compute as the
full “TMC” result include contributions from the $\\{h_{i},g_{i}\\}$ terms,
each of which includes an integral. Additionally, if we can take advantage of
the approximate $A$ independence observed in the previous section, this may
allow us to greatly simplify the calculation of TMCs for the many different
nuclei present in a typical nPDF fit.

##### Magnitude of Ratios: (Fig. 7.3 Upper panels)

We start by comparing the ratio of the “Leading-TMC” to the full “TMC” result
as shown in the upper panels of Fig. 7.3. The results are shown for selected
CC and NC processes with representative values
$Q=\\{1.3,1.5,2,3,4,6\\}{\rm~{}GeV}$.

We display results for all the nuclei of Table 7.1 In these plots, the
individual nuclear $A$ values are not labeled, but the ratios coalesce into
clearly defined bands for each $Q$ value. This coalescence holds for all
permutations of structure functions and boson exchanges. The different nuclei
are shown as colored solid lines in the figure. We also include results for
the proton (dashed black) and neutron (dotted blue).

Focusing first on the bands of the top panels, we see that for $F_{2}^{A}$ the
differences between the full TMCs and Leading-TMCs are as large as
$\mathcal{O}(10\%-15\%)$ at $x\sim 0.5$ for $Q=1.3{\rm~{}GeV}$, and
$\mathcal{O}(4\%-5\%)$ for $F_{3}^{A}$. For larger $Q$ values we expect these
differences to be reduced as they are proportional to powers of $(M/Q)^{2}$.
At $Q=2$ GeV, which is a typical $Q_{cut}$ value for many global PDF analyses,
the ratio for $F_{2}^{A}$ is $\lesssim 5\%$, and the ratio of $F_{3}^{A}$ is
$\lesssim 2\%$. For $Q\gtrsim 3-4{\rm~{}GeV}$, we find that these differences
reduces to the sub-percent level, and effectively vanish at $Q=6{\rm~{}GeV}$.

##### Nuclear $A$ Dependence of Ratios:

Having discussed the magnitude of the $F_{i}^{\rm TMC}/F_{i}^{\rm TMC-
Leading}$ ratios, we now examine the $A$ dependence of these ratios. A
distinctive feature of Fig. 7.3 is the coalescence of the results into
individual bands, and this suggests that the nuclear $A$ dependence of this
ratio is minimal. [The proton and neutron ratios (shown as dashed and dotted
black lines, respectively), do not necessarily lie within the bands, and we
will discuss these separately.] This apparent universality of the full
TMC/Leading-TMC ratios can be traced back to: i) the fact that this ratio has
only mild dependence on the underlying PDF, and ii) the fact that for nuclei,
in the large $x$ region it is the average $u+d$ (isoscalar) PDF that dominates
this result.

One approach to understand these features is to consider the analytically
computed upper bound for these ratios. If we assume the structure functions
are monotonically decreasing (an entirely reasonable assumption in the large
$x$ region), it is possible to obtain the following constraints on these
ratios [60]:

$\displaystyle\frac{F_{2}^{\rm TMC}}{F_{2}^{\rm TMC-leading}}(x,Q^{2})$
$\displaystyle\ \leq\
1+\left(\frac{M}{Q}\right)^{2}\frac{6x\xi}{r}(1-\xi)+\left(\frac{M}{Q}\right)^{4}\frac{12x^{2}\xi^{2}}{r^{2}}(-\ln\xi-1+\xi)$
(7.5a) $\displaystyle\frac{F_{3}^{\rm TMC}}{F_{3}^{\rm TMC-leading}}(x,Q^{2})$
$\displaystyle\ \leq\
1-\left(\frac{M}{Q}\right)^{2}\frac{2x\xi}{r}\ln\xi\quad.$ (7.5b)

Note, these bounds have absolutely no dependence on the PDF. Here, we also
explicitly see the powers of $(M/Q)^{2}$ which drive the ratios to unity for
large $Q$.

In Fig. 7.4, we plot these bounds as a function of $x$ for selected $Q$
values. Comparing the magnitudes of the bounds with the values of Fig. 7.3, we
see the bounds are quite conservative. Note that these bounds are valid for
any nPDFs in the large $x$ region (where $F_{i}^{A}$ is decreasing
monotonically), and that is the relevant region of interest for the TMC
effects. Consequently, the independence of the bound on the PDF helps us
understand the comparable nuclear $A$ independence of the $F_{i}^{\rm
TMC}/F_{i}^{\rm TMC-Leading}$ ratios observed in Fig. 7.3.

Figure 7.4: We show the upper bounds of $F_{i}^{\rm TMC}/F_{i}^{\rm TMC-
Leading}$ for a) $F_{2}$ and b) $F_{3}$ for selected $Q=\\{1.3,1.5,2,3,4,6\\}$
GeV, (from top to bottom) vs. $x_{N}$ as computed with Eqs. (7.5a) and (7.5b).

##### Nuclear $A$ Variation of Ratios:

The observation that these ratios are relatively insensitive to the nuclear
$A$ value suggests a “short cut” which can be used to efficiently implement
TMCs into numerical calculations.

More specifically, computing the full TMCs for each nucleus $A$ requires a
calculation of the $h_{i}^{A}(\xi_{N},Q^{2})$ and $g_{i}^{A}(\xi_{N},Q^{2})$
terms in Eq. (4.15). Each term requires a separate integration over a nuclear
structure function. Clearly, such steps would require significant CPU time
were they included inside the fitting loop of a global PDF analysis. If it is
true that the size of TMCs to nuclear structure functions are relatively
insensitive to the nuclear $A$ value, as our works suggests, then TMCs can be
estimated through the application of a “universal” correction factor,
$R_{i}^{V}(x,Q^{2})$. Such a factor can be constructed by taking an average
TMC summed over the various $n_{A}$ nuclei:

$R_{i}^{V}(x_{N},Q^{2})\ =\
\frac{1}{n_{A}}\sum_{k}^{n_{A}}\frac{F_{i}^{A_{k},\rm
TMC}(x_{N},Q^{2})}{F_{i}^{A_{k},{\rm TMC-Leading}}(x_{N},Q^{2})}\ =\
\frac{1}{n_{A}}\sum_{k}^{n_{A}}R_{i}^{k,{V}}$ (7.6)

where “$V$” labels the type of the exchanged gauge boson ($\gamma,Z,W^{\pm}$),
and the sum over $k$ excludes the proton and neutron.

The utility of the “universal” correction factor $R_{i}^{V}(x,Q^{2})$ is that
we can compute this once at the beginning of a fit, and then apply this
correction factor inside the fitting loop without the need for recomputing
additional integrals over $h_{i}$ and $g_{i}$ functions.

To determine the potential precision of such an approximation, we show in the
lower panels of Fig. 7.3 a ratio of ratios constructed by taking $(F_{i}^{\rm
TMC}/F_{i}^{\rm TMC-Leading})$ over $R_{i}^{V}$ for $Q=2$ GeV, which is a
typical $Q_{\rm cut}$ value for global fits. It is remarkable how narrow the
band of nuclear PDFs lie in comparison to the universal $R_{i}^{V}$ function.
For $F_{2}$ we see the nuclear bands lie within $0.2\%$ of the universal
curve, while for $F_{3}$ the bands lie within $0.1\%$. For larger $Q$ values,
the precision is further increased. This is clearly sufficient accuracy for
any TMC calculation given the typical nPDF uncertainty in the large $x$
region.

##### Variation of Proton & Neutron Ratios:

In Fig. 7.3 we observed that all the nuclear PDFs were uniformly within a
narrow band. In contrast, the proton and neutron results displayed a wider
variation lying outside these nuclear bands. We will explain the source of
this variation, and understand why the universal correction factor
$R_{i}^{V}(x,Q^{2})$ previously introduced, still works well in the case of
nuclei.

Specifically, in Fig. 7.3 we see that the proton ratios lie above the nuclear
band for the NC and CC $W^{-}$ plots, and below for the CC $W^{+}$ plots;
furthermore, the behavior of the neutron ratios is exactly the opposite of the
proton. We can understand these features by identifying the dominant
contribution for each process. As $x\to 1$, we neglect the gluon and sea
quarks to obtain: $F_{2}(\gamma/Z)\sim\frac{x}{9}[4u(x)+d(x)]$ and
$F_{2}(W^{-})\sim 2xu(x)$, while $F_{2}(W^{+})\sim 2xd(x)$. Recall that for a
proton PDF, the $d/u$ ratio in the large $x$ region can typically be as small
as $\sim 0.1$; that is, the up quark is an order of magnitude larger than the
down quark.

We now understand the pattern of Fig. 7.3. The structure function for the NC
and CC $W^{-}$ processes are driven by the up quark, while the CC $W^{+}$ is
driven by the down quark. Thus, the proton results are above the others for NC
and CC $W^{-}$ but below for CC $W^{+}$ in plots of Fig. 7.3. Since we obtain
the neutron PDFs by isospin symmetry ($u\leftrightarrow d$), so we also
understand why the neutron results are the opposite of the proton.

This exercise also demonstrates why the results for the nuclear PDFs lie
together in a uniform band. Most of the nuclei are closer to the isoscalar
state $(A\sim 2Z)$ than either the proton $(Z=A)$ or neutron state $(Z=0)$.
For an isoscalar nuclei we have $u=d$, and it is effectively the average of
the up and down PDFs $(u+d)/2$ which provides the dominant contributions to
Fig. 7.3. Consequently, we find that the nuclear band is effectively an
average of the proton and neutron result, and thus lies between the two in
Figs. 7.3. The one slight variation of this pattern is the parity-violating
$F_{3}$ NC structure function; this effect must come from the $Z$ boson
contribution because the parity-conserving photon contribution vanishes.

##### Approximate Nuclear $A$ Independence:

We illustrate this further in Fig. 7.5 which shows the ratios for the NC and
CC processes for the a) proton, b) deuteron, and c) 3He. We choose these
examples because the deuteron is an isoscalar, and 3He is one of the most
“non-isoscalar” nuclei (with the exception of hydrogen/proton).

The pattern in Fig. 7.5 is reminiscent of Fig. 7.3. In Fig. 7.5(a) for the
proton, we see the ($u$-quark dominated) NC and CC $W^{-}$ processes lie above
the ($d$-quark dominated) CC $W^{+}$ process. In Fig. 7.5(b) for the isoscalar
deuteron, we see all the curves coincide since $u=d$ for this nuclei. Finally,
in Fig. 7.5(c) for the ${}^{3}\mathrm{He}$, we see the NC and CC $W^{-}$
processes lie slightly above the CC $W^{+}$ process, but the difference is not
as dramatic as for the proton case. Note in these cases, the $u:d$ ratio is
$2:1$ in the proton, $1:1$ in the deuteron, $5:4$ in 3He. For any heavier
nuclei, the $u:d$ ratio will be comparatively closer to unity, and the results
of the individual NC and CC processes will tend to coincide.

(a) (b) (c)

Figure 7.5:  The nuclear structure function ratio $F_{2}^{\rm TMC}/F_{2}^{\rm
TMC-Leading}$ as function of $x_{N}$ for $Q=\\{1.3,1.5,2,3,4,6\\}{\rm~{}GeV}$
for (a) the proton, (b) deuteron, and (c) 3He. For each nucleus, we overlay
the result for NC exchange (solid green), CC $W^{+}$ exchange (dot orange),
and CC $W^{-}$ exchange (dot-dashed blue). Note that for the deuteron in Fig.
(b), the curves for the three CC/NC processes coincide.

The outcome of these findings is that given the typical uncertainty of nuclear
PDFs in the large $x$ region, applying an average, $A$-independent set of TMCs
yields precision which is sufficient for a wide variety of real-world
calculations. We shall exploit this observation in Sec. 7.3 and derive an
approximate parameterization for these TMCs which can be applied to the
general nuclear case.

Setting aside the proton and neutron cases, at $Q=2$ GeV we find the deviation
of individual nuclei from the average to be less than $\sim 0.2\%$ for
$F_{2}$, and less than $\sim 0.1\%$ for $F_{3}$. And of course, these
differences decrease for larger $Q$.

#### Sub-leading TMCs for Nuclei

Figure 7.6: To highlight the effect of the sub-leading terms
$\\{h_{i},g_{i}\\}$, we display the ratios $F_{i}^{{\rm TMC}+h_{2}}/F_{i}^{\rm
TMC-Leading}$ (dotted red) and $F_{i}^{{\rm TMC}}/F_{i}^{\rm TMC-Leading}$
(solid blue), $i=\\{1,2\\}$, for selected $Q=\\{1.3,1.5,2,3,4,6\\}$ GeV (from
top to bottom) as a function of $x_{N}$. We show $F_{1,2}$ for neutral current
(NC) and charged current ($W^{+}$) processes using the nCTEQ15 PDF for
${}^{12}C$. Note $F_{2}^{{\rm TMC}}$ contains both $\\{h_{i},g_{i}\\}$
contributions, $F_{2}^{{\rm TMC}+h_{2}}$ contains only $\\{h_{i}\\}$, and
$F_{2}^{{\rm TMC-Leading}}$ contains no sub-leading contributions.

As a final study, we want to demonstrate the individual size of the sub-
leading $h_{2}^{A}$ and $g_{2}^{A}$ components of the full TMCs in Eq. (4.15).
In Fig. 7.6 we present the results for NC ($\gamma/Z$) and CC ($W^{\pm}$)
processes, and we display a pair of ratios $F_{2}^{\rm TMC+h2}/F_{2}^{\rm TMC-
Leading}$ and $F_{2}^{\rm TMC}/F_{2}^{\rm TMC-Leading}$, for each $Q$ value.

Recall that $F_{2}^{\rm TMC-Leading}$ neglects the sub-leading $h_{2}^{A}$ and
$g_{2}^{A}$ contributions, where $h_{2}^{A}$ is suppressed by $(M/Q)^{2}$, and
$g_{2}^{A}$ is suppressed by $(M/Q)^{4}$. The $F_{2}^{\rm TMC+h2}$ expression
includes the $h_{2}^{A}$ contribution, but neglects $g_{2}^{A}$. Finally, the
full $F_{2}^{\rm TMC}$ expression includes both $h_{2}^{A}$ and $g_{2}^{A}$;
this could equivalently be called $F_{2}^{\rm TMC+h2+g2}$.

Since the $g_{2}^{A}$ term is suppressed by $(M/Q)^{4}$, we expect this will
only contribute for very large $x$ at low $Q^{2}$ values. Therefore, we expect
$F_{2}^{\rm TMC+h2}$ and $F_{2}^{\rm TMC}$ to coincide throughout most of the
parameter space. This expectation is validated in Fig. 7.6 where we do see
that $F_{2}^{\rm TMC+h2}$ and $F_{2}^{\rm TMC}$ differ only for the lowest few
$Q$ values in the large $x$ region. For example, at $Q=2$ GeV the differences
are barely discernible in the figure, and the effect is even smaller as $Q$
increases.

Thus, $h_{2}^{A}$ provides the dominant contribution throughout the kinematic
range, but $g_{2}^{A}$ does contribute for low $Q$ values. The pattern for the
other structure functions $F_{1,3}^{A}$ is similar.

### 7.3 Parameterizing $F_{i}^{\rm TMC}/F_{i}^{\rm Leading-TMC}$

Figure 7.7: We present the nuclear structure function ratio $F_{i}^{\rm
TMC}/F_{i}^{\rm TMC-Leading}$ averaged over the nuclei of Table 7.1 as defined
in Eq. (7.6). The ratios are shown as a function of $x_{N}$ for
$Q=\\{1.3,1.5,2,3,4,6\\}$ GeV (from top to bottom) using the nCTEQ15 nPDFs. We
compare the exact results (green solid) of Eq. (4.15) to the approximate
parameterizations (red dashed) based on Eqs. (7.8) with parameters given in
Table 7.2. For $Q>2$ GeV, we observe the parameterization can match the exact
results to better than 0.2%.

In Sec. 7.2 we demonstrated that the full TMC/Leading-TMC ratios were
effectively insensitive to the nuclear $A$ value. In this section, we provide
a parameterization for these ratios which allows us to efficiently compute the
full TMCs given the Leading-TMC structure functions.

Starting from Eqs. (4.15), we divide $F_{i}^{\rm TMC}(x)$ by the leading term
$F_{i}^{\rm Leading-TMC}(x)$:

$\displaystyle\frac{F_{1}^{\rm TMC}(x)}{F_{1}^{\rm Leading-TMC}(x)}$
$\displaystyle=1+\frac{M^{2}}{Q^{2}}\frac{x\xi}{r}\frac{h_{2}(\xi)}{F_{1}^{(0)}(\xi)}+\frac{M^{4}}{Q^{4}}\frac{2x^{2}\xi}{r^{2}}\frac{g_{2}(\xi)}{F_{1}^{(0)}(\xi)}\
,$ (7.7a) $\displaystyle\frac{F_{2}^{\rm TMC}(x)}{F_{2}^{\rm Leading-TMC}(x)}$
$\displaystyle=1+\frac{M^{2}}{Q^{2}}\frac{6x\xi^{2}}{r}\frac{h_{2}(\xi)}{F_{2}^{(0)}(\xi)}+\frac{M^{4}}{Q^{4}}\frac{12x^{2}\xi^{2}}{r^{2}}\frac{g_{2}(\xi)}{F_{2}^{(0)}(\xi)}\
,$ (7.7b) $\displaystyle\frac{F_{3}^{\rm TMC}(x)}{F_{3}^{\rm Leading-TMC}(x)}$
$\displaystyle=1+\frac{M^{2}}{Q^{2}}\frac{2\xi
x}{r}\frac{h_{3}(\xi)}{F_{3}^{(0)}(\xi)}\quad.$ (7.7c)

The goal of this section is to parameterize these ratios.

As explained in detail in B, if we assume the structure functions vanish at
$x=1$ and expand them in a Taylor series, we can obtain simple expressions for
the ratios in terms of the parameter function $\gamma_{i}(Q)$, which
characterizes the first derivative of the structure function. This allows us
to obtain the approximate expressions for the ratios $h_{i}/F_{j}^{(0)}$ and
$g_{2}/F_{j}^{(0)}$:

$\frac{h_{2}(\xi)}{F_{2}^{(0)}(\xi)}=\frac{1-\xi}{\xi}+\gamma_{2}(Q)\frac{1-\xi}{\xi^{2}}\sum_{j=1}^{j_{\rm
max}}\frac{(-1)^{j}}{j!(j+1)}\,\,_{2}F_{1}\left(2,j+1,j+2,1-\frac{1}{\xi}\right)\
,$ (7.8a)
$\frac{h_{3}(\xi)}{F_{3}^{(0)}(\xi)}=-\ln(\xi)+\gamma_{3}(Q)\frac{1-\xi}{\xi}\sum_{j=1}^{j_{\rm
max}}\frac{(-1)^{j}}{j!(j+1)}\,\,_{2}F_{1}\left(1,j+1,j+2,1-\frac{1}{\xi}\right)\
,$ (7.8b) $\frac{h_{2}(\xi)}{F_{1}^{(0)}(\xi)}=2\xi
r^{2}\left[\frac{1-\xi}{\xi}+\gamma_{1}(Q)\frac{1-\xi}{\xi^{2}}\sum_{j=1}^{j_{\rm
max}}\frac{(-1)^{j}}{j!(j+1)}\,\,_{2}F_{1}\left(2,j+1,j+2,1-\frac{1}{\xi}\right)\right]\
,$ (7.8c)
$\frac{g_{2}(\xi)}{F_{2}^{(0)}(\xi)}=-\ln(\xi)-(1-\xi)+\gamma_{2}(Q)\frac{(1-\xi)^{2}}{\xi^{2}}\sum_{j=1}^{j_{\rm
max}}\frac{(-1)^{j}}{j!(j+2)}\,\,_{2}F_{1}\left(2,j+2,j+3,1-\frac{1}{\xi}\right)\
,$ (7.8d) $\frac{g_{2}(\xi)}{F_{1}^{(0)}(\xi)}=2\xi
r^{2}\left[-\ln(\xi)-(1-\xi)+\gamma_{1}(Q)\frac{(1-\xi)^{2}}{\xi^{2}}\sum_{j=1}^{j_{\rm
max}}\frac{(-1)^{j}}{j!(j+2)}\,\,_{2}F_{1}\left(2,j+2,j+3,1-\frac{1}{\xi}\right)\right]\
,$ (7.8e)

where ${}_{2}F_{1}(a,b,c,z)$ is a hypergeometric function. Although the
summation over the index $j$ can, in principle, go to infinity, we truncate
the series at $j=j_{\rm max}$. The $1/(j!)$ prefactor of the hypergeometric
function helps ensure the series converges quickly, and we find $j_{\rm
max}=4$ yields results that are $\lesssim 1\%$ accuracy.

The function $\gamma_{a}(Q)$ is given by

$\gamma_{a}(Q)=\lambda_{a}\ln(Q)^{\delta_{a}}\quad.$ (7.9)

The values of $\\{\lambda_{a},\delta_{a}\\}$ are obtained by fitting the
parameterization to the exact results for each $F_{i}^{\rm TMC}$ structure
function. Note that the values of $\\{\lambda_{a},\delta_{a}\\}$ are
independent of the type of the exchanged bosons; they have only a mild
dependence on the specific structure function $F_{i}^{A}$ as displayed in
Table 7.2.

| $F_{1}^{TMC}/F_{1}^{leading}$ | $F_{2}^{TMC}/F_{2}^{leading}$ | $F_{3}^{TMC}/F_{3}^{leading}$
---|---|---|---
nPDFs | $\lambda_{1}$ | $\delta_{1}$ | $\lambda_{2}$ | $\delta_{2}$ | $\lambda_{3}$ | $\delta_{3}$
nCTEQ15 | 2.275 | -0.014 | 2.144 | 0.100 | 2.183 | 0.029
EPPS16 | 2.226 | -0.026 | 2.086 | 0.095 | 2.142 | 0.022
nNNPDF2.0 | 2.226 | -0.025 | 2.086 | 0.103 | 2.197 | 0.023
TUJU19 | 2.290 | -0.0170 | 2.167 | 0.099 | 2.200 | 0.031
Table 7.2: We approximate the $\\{h_{i}(\xi),g_{i}(\xi)\\}$ contributions to
$F_{i}^{\rm TMC}$ using $F_{i}^{{\rm TMC-Leading}}$ together with a
2-parameter function $\gamma_{a}(Q)$. Here, $\gamma_{a}(Q)$ implicitly depends
on the parameters $\\{\lambda_{a},\delta_{a}\\}$ which are fitted for each
structure function $\\{F_{1,2,3}\\}$ and for selected PDFs displayed in the
table. The TMC corrections are accurate to within $\sim 0.3$% for $Q>1.3$ GeV.
The $\\{\lambda_{i},\delta_{i}\\}$ parameters are independent of the exchanged
boson ($\gamma,Z,W^{\pm}$), and are relatively insensitive to the specific
underlying nPDF. Once the $\\{\lambda_{i},\delta_{i}\\}$ parameters are
determined for a particular nPDF, the full TMC corrections can be efficiently
computed (without convolution integrations) to high precision as illustrated
in Fig. 7.7.

Additionally, while we have computed these results assuming an NLO DGLAP
evolution, these results should be similar for an NNLO evolution. This is
because: i) we are working at relatively low $Q$ values where the effects of
the DGLAP evolution from $Q_{0}$ to $Q$ will be minimal, and ii) since this
quantity is a ratio of structure functions, the modification of the ratio is
expected to be less than the impact on the individual structure functions.

We have also computed the results for a variety of different PDF sets to
ensure this parameterization is largely insensitive to the specific PDF. This
also is displayed in Table 7.2, where we observe a mild variation of the
parameters between PDF sets.

In Figure 7.7, we show the comparison between our parameterization with the
exact results obtained using nCTEQ15 nPDFs. We can see that our
parameterization works very well to reproduce the exact results. The
parameterization has the advantage that it can be computed efficiently whereas
the full $F_{i}^{\rm TMC}$ results require additional convolution integrations
for the $\\{h_{i}(\xi),g_{i}(\xi)\\}$ functions. For values of $Q\geq 2$ GeV,
which is a typical cut for the global nPDF fits, we see the parameterization
describes the full TMC results to better than 0.2%.

##### Summary:

In the previous sections we established several results which will facilitate
efficient calculation of TMCs for nuclear structure functions. We briefly
summarize the key observations below.

  * •

TMCs can yield large corrections in the double limit of large $x$ and small
$Q$.

  * •

The “Leading-TMC” result provides an excellent approximation to the full “TMC”
result as the $h_{2,3}$ contributions are suppressed by additional
$(M/Q)^{2}$, and the $g_{2}$ contributions by additional $(M/Q)^{4}$ powers.

  * •

The nuclear dependence of the “TMC” result is approximately independent of
$A$. This observation suggests we can determine an $A$-independent “universal”
correction factor which can be used to quickly calculate the nuclear structure
function TMCs for any nuclei. As illustrated in the bottom panels of Fig. 7.3,
this “universal” approximation holds to ${\lesssim 0.1\%}$ for $Q\geq 2$ GeV.

  * •

As demonstrated in Fig. 7.7, we can parameterize this “universal” correction
factor with Eq. (7.8) into a 2-parameter form that can be easily incorporated
in a efficient nuclear PDF fitting program. This parameterization matches the
exact results to better than 0.2% for $Q\geq 2{\rm~{}GeV}$.

Given the “Leading-TMC” result, the above allows us to efficiently approximate
the full “TMC” result for any nuclear structure function by applying
“universal” correction factor. This avoids the computation of the numerical
integrals contained in the $\\{h_{2},h_{3},g_{2}\\}$ functions.

### 7.4 Reduced Cross Sections with TMCs for Nuclei

When extracting structure functions from DIS data or comparing theoretical
expectations for DIS to data, one often works with the “reduced cross section”
$\sigma_{\rm Red.}$. These quantities are derived from double differential
cross sections scaled by kinematic factors, and provide a direct connection
between DIS kinematics and structure functions.

The general cross section for neutral and charged current DIS can be expressed
as:[75]

$\displaystyle\frac{d^{2}\sigma^{i}}{dx\,dy}$ $\displaystyle=$ $\displaystyle
x(s-M^{2})\,\frac{d^{2}\sigma^{i}}{dxdQ^{2}}=\frac{4\pi\alpha^{2}}{xyQ^{2}}\,\eta^{i}\left\\{\left(1-y-\frac{x^{2}y^{2}M^{2}}{Q^{2}}\right)F_{2}^{i}+\left(y^{2}x\right)F_{1}^{i}\mp\left(y-\frac{y^{2}}{2}\right)xF_{3}^{i}\right\\}\
,$ (7.10)

where i=NC, CC corresponds to neutral-current or charged-current processes.
Additionally, $\eta^{NC}=1$ and $\eta^{CC}=4\eta_{{}_{W}}$, with
$\eta_{{}_{W}}=\frac{1}{2}[\frac{G_{F}M_{W}^{2}}{4\pi\alpha}\frac{Q^{2}}{Q^{2}+M_{W}^{2}}]^{2}$.
In the last term of Eq. (7.10), the “–” sign is taken for an incoming anti-
lepton $\\{e^{+},\bar{\nu}\\}$ and the “+” sign for an incoming lepton
$\\{e^{-},{\nu}\\}$.

##### Neutral Current (NC) Cross Section:

We find it convenient to define the reduced cross section by dividing out the
leading factors.

$\displaystyle\frac{d^{2}\sigma^{\rm NC}}{dx\,dy}$ $\displaystyle=$
$\displaystyle x(s-M^{2})\,\frac{d^{2}\sigma^{\rm
NC}}{dxdQ^{2}}=\frac{4\pi\alpha^{2}}{xyQ^{2}}\ \left[\frac{Y_{+}}{2}\
\sigma_{\rm Red.}^{NC}\right]\ ,$ (7.11) $\displaystyle\sigma_{\rm Red.}^{NC}$
$\displaystyle=$
$\displaystyle\left(1+\frac{2y^{2}\varepsilon^{2}}{Y_{+}}\right)F_{2}^{\rm
NC}\mp\frac{Y_{-}}{Y_{+}}\,xF_{3}^{\rm NC}-\frac{y^{2}}{Y_{+}}F_{L}^{\rm NC}\
,$ (7.12)

with $F_{L}=r^{2}F_{2}-2xF_{1}$, $r=\sqrt{1+4\varepsilon^{2}}$,
$\varepsilon=(xM/Q)$ and $Y_{\pm}=1\pm(1-y)^{2}$; the $r^{2}$ term in $F_{L}$
is essential to obtain the correct sign for the $\varepsilon^{2}$ term of Eq.
(7.12). Note that at LO ($F_{L}\sim 0$) for parity-conserving photon exchange
($F_{3}=0$), so that we have $\sigma_{\rm Red.}^{NC}$ simply reduces to
$F_{2}^{\rm NC}$.

##### Charged Current (CC) Cross Section:

For the charged current process, we have:

$\displaystyle\frac{d^{2}\sigma^{\rm CC}}{dx\,dy}$ $\displaystyle=$
$\displaystyle x(s-M^{2})\,\frac{d^{2}\sigma^{\rm
CC}}{dxdQ^{2}}=\frac{G_{F}^{2}}{2\pi x}\ \frac{Q^{2}}{y}\
\left(\frac{M_{W}^{2}}{Q^{2}+M_{W}^{2}}\right)^{2}\
\left[\rule[-11.38109pt]{0.0pt}{0.0pt}\sigma_{\rm Red.}^{CC}\right]\ ,$ (7.13)
$\displaystyle\sigma_{\rm Red.}^{CC}$ $\displaystyle=$
$\displaystyle\frac{Y_{+}}{2}\
\left(1+\frac{2y^{2}\varepsilon^{2}}{Y_{+}}\right)F_{2}^{\rm
CC}\mp\frac{Y_{-}}{2}\,xF_{3}^{\rm CC}-\frac{y^{2}}{2}F_{L}^{\rm CC}$ (7.14)

Here we have followed the HERA convention[125] where the CC differs from NC by
a factor of $Y_{+}/2$.

For both the NC and CC cases, the factor of $2y^{2}\varepsilon^{2}/Y_{+}$ with
$\varepsilon=(xM/Q)$ multiplying $F_{2}$ arises directly from the DIS
kinematics; however, this is often neglected as it is small for typical HERA
kinematics.

Figure 7.8:  We present reduced cross sections for the following scenarios:
(a) HERA inspired $e^{+}p$ neutral current scattering with $E_{CM}=318$ GeV
for $Q=1.3$ GeV; (b) JLab inspired $e^{-}A$ charged current scattering with
$E_{beam}=24$ GeV for $Q=1.3$ GeV; (b) EIC inspired $e^{-}A$ neutral current
scattering with $E_{CM}=100$ GeV for $Q=1.3$ GeV; (d) LBNF inspired $\nu A$
charged current scattering with $E_{CM}=10$ GeV for $Q=1.3$ GeV. Upper panels:
The upper panels display the full “TMC” results. Lower panels: The lower
panels display the ratio of the full “TMC” to “No-TMC”. 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.

#### TMCs for Nuclear Reduced Cross Sections

As described in the preceding subsections, TMCs can have a large numerical
impact on nuclear structure functions in the low $Q^{2}$ and large $x$ double
limit. To illustrate the TMC impact, in Fig. 7.8 we present results for
experimental scenarios including HERA, JLab, EIC, and LBNF. The upper panels
of Fig. 7.8 display the reduced cross sections, and the lower panels show the
ratio of the full “TMC” to the “No-TMC” result. As before, all the nuclear
isotopes of Table 7.1 are within the colored band, and the proton and neutron
are displayed separately.

In Fig. 7.8-(a) we show the HERA-inspired $e^{+}p$ neutral current
$\sigma_{Red}^{NC}$ at $\sqrt{s}=318{\rm~{}GeV}$ at $Q=1.3$ GeV for the full
“TMC” results. Although HERA ran proton beams, we also show the nuclear
results which may be useful for a future LHeC/FCCeh facility. The TMCs yield a
slight suppression at intermediate $x$ values ($\sim 0.4$), and an enhancement
at larger $x$ values ($\gtrsim 0.6$) which can be significant. The behavior of
$\sigma^{NC}_{Red.}$ strongly resembles that of $F_{2}$ in Fig. 7.1; this is
expected as given our choices of $Q$ and $E_{\rm CM}$ we find $y\ll 1$ and
$Y_{-}\sim 0$, so we see from Eq. (7.12) that the $F_{2}$ contribution
dominates. For the higher $Q$ values (not shown) TMCs are reduced as for
$F_{2}$ in Fig. 7.1, and are generally negligible except in the very large $x$
region.

In Fig. 7.8-(b) we show a JLab-inspired $e^{-}A$ neutral-current
$\sigma_{Red}^{NC}$ with $E_{\rm Beam}=24{\rm~{}GeV}$ at $Q=1.3{\rm~{}GeV}$.
The character of the curves for $E_{\rm Beam}=12{\rm~{}GeV}$ (not shown) is
quite similar to the displayed results. The results here are qualitatively
similar to the $F_{2}$ result of Fig. 7.1 and the HERA case above with a
reduction for intermediate $x$ ($\sim 0.4$), and an enhancement at larger $x$.

In Fig. 7.8-(c) we show an EIC-inspired $e^{-}A$ neutral-current
$\sigma_{Red}^{NC}$ with $\sqrt{s}=100{\rm~{}GeV}$ at $Q=1.3{\rm~{}GeV}$. As
with the above HERA and JLab cases, these results largely follow the $F_{2}$
structure function.

In Fig. 7.8-(d) we show an LBNF-inspired $\nu-A$ charged-current scattering
with $E_{\rm Beam}=10{\rm~{}GeV}$ at $Q=1.3$ GeV. In the low $x$ region, we
see the various nuclear curves separate due to the differing nuclear mass
values.

From this exercise, one can see that for large $\sqrt{s}$, we generally have
small $y$ and $\sigma_{Red}$ closely follows $F_{2}$. For smaller $\sqrt{s}$
values, we have larger $y$ and the $F_{3}$ and $F_{L}$ structure functions can
now contribute.

## 8 Conclusions

To exploit the current and future nuclear measurements and to obtain the
desired accuracy of theory calculations, it is essential to include TMCs to
structure functions in analyses. The present study provides both pedagogical
material and a practical reference for computing and investigating TMCs for
nuclear processes such as at the upcoming Electron Ion Collider (EIC) and Deep
Underground Neutrino Experiment (DUNE). We briefly outline the key components
and results in the following:

  * •

We extend the OPE formalism to the case of interacting nuclei and explicitly
provide the correspondence between TMC expressions for a nucleon and a
nucleus, given in Eq. (3.24). To derive these relations (see Sec. 3.2 and A),
we use only fundamental symmetry principles of the nuclei, and make no
assumptions about the parton or other models.

  * •

We specifically investigate analytically and numerically the magnitude of TMCs
and show it is the quantity $(x_{A}M_{A}/Q)=(x_{N}M_{N}/Q)$ that enters the
master TMC formula, Eq. (4.15). Given that the dominant support of nPDFs is in
the region $x_{N}\in[0,1]$, TMCs are governed by powers of $(M_{N}/Q)$ and not
$(M_{A}/Q)$.

  * •

In Sec. 4.2, we derive the master TMC expression in terms of the rescaled
structure functions. We also investigate the power-suppressed terms
proportional to $h_{i}$ and $g_{i}$, and determine their rescaling properties.

  * •

In terms of the rescaled structure functions, we define the rescaled nPDFs,
which are the ones commonly studied and presented in comparisons between
different nuclear data. In Sec. 5.1, we also discuss the corresponding set of
DGLAP equations and associated sum rules, including isospin symmetry and the
complication of separately extraction $u(x)$ and $d(x)$ nPDFs.

  * •

In particular, we establish the relations between the kinematic variables
including the nuclear scaling variables $\\{x_{A},\xi_{A}\\}$ and the nucleon
scaling variables $\\{x_{N},\xi_{N}\\}$. Additionally, we indicate in Sec. 5.4
some subtle complications we encounter such as defining an appropriate $W^{2}$
cut for the nucleon case.

  * •

We obtain a complete correspondence between the parton model and OPE structure
function results identifying the source of the differences. Additionally, in
Sec. 6 we review the ACOT helicity approach with light-cone kinematics, and
compare with the direct OPE results.

  * •

In Sec. 7, we provide variety of numerical comparisons to demonstrate the size
of the TMCs in various kinematic regions. These comparisons are shown for both
structure functions, and the reduced cross section for some realistic machine
kinematics.

  * •

We also present numerically the dependence of the TMCs on the nuclear $A$
value and the exchanged vector boson $\\{\gamma,Z,W^{\pm}\\}$. We find the
relative magnitude of the TMCs is weakly dependent on the exchanged boson, and
very weakly dependent on the nuclear $A$. This latter observation allow us to
derive an approximate (and computationally expedient) parameterization for the
nuclear TMCs suitable for including inside a global fit function.

Just as Ref. [60] provided the details for the proton case, we hope this
report provides foundational and reference material which can facilitate
incisive investigations of nuclear phenomena.

## Acknowledgments

The authors would like to thank Eric Christy, John, C. Collins, Stefan Krezer,
and Mary Hall Reno for many useful comments and discussions.

A.K. and R.R. acknowledge the support of Narodowe Centrum Nauki under Grant
No. 2019/34/E /ST2/00186. R.R also acknowledges the support of the Polska
Akademia Nauk (grant agreement PAN.BFD.S.BDN. 613. 022. 2021 - PASIFIC 1,
POPSICLE). This work has received funding from the European Union’s Horizon
2020 research and innovation program under the Skłodowska-Curie grant
agreement No. 847639 and as part of the Skłodowska-Curie Innovative Training
Network MCnetITN3 (grant agreement no. 722104), and from the Polish Ministry
of Education and Science. The work at WWU Münster was funded by the DFG
through the Research Training Group 2149 “Strong and Weak Interactions - from
Hadrons to Dark Matter” and the SFB 1225 “Isoquant,” project-id 273811115.
M.K. thanks the School of Physics at the University of New South Wales in
Sydney, Australia for its hospitality and financial support through the Gordon
Godfrey visitors program. The work of A.A. was supported by the U.S.
Department of Energy under Grant No. DE-SC0008791. The work of T.J.H. was
supported by the U.S. Department of Energy under contract DE-AC02-06CH11357.
The research of P. D. was funded as a part of the Center of Excellence in
Quark Matter of the Academy of Finland (project 346326). The work of C.K. was
supported by the U.S. Department of Energy contract DE-AC05-06OR23177, under
which Jefferson Science Associates LLC manages and operates Jefferson Lab.
J.G.M. has been supported by Fermi Research Alliance, LLC under Contract No.
DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science,
Office of High Energy Physics. The work of F.O. was supported by the U.S.
Department of Energy under Grant No. DE-SC0010129. The work of I.S. was
supported by the French CNRS via the IN2P3 project GLUE@NLO.

##  Appendix A Nuclear structure functions with TMCs in the OPE

In this appendix, we build structure functions for massive, unpolarized
nuclear targets following a modified prescription of Ref. [49, 50]. The
purpose of this is two fold. First is to establish that the TMC formulae used
for free nucleons are applicable to nuclei. Second is to provide a explicit
derivation of the TMC master formulae with details beyond what is generally
available in the literature. It is well-documented that constructing TMCs to
structure functions is a tedious endeavor, with many intermediate steps
omitted from the literature. Some notable exceptions exist [49, 50, 54, 81,
60, 126]. For example: Refs. [49, 50] give a largely complete treatment of
$F_{2}$ but implicitly employ derivative, integral, and summation identities.
Some of these intermediate steps are more completely documented in Refs. [81,
126], but other steps may not be fully justified under their assumptions [85].
Similarly, while the structure functions $W_{4}$ and $W_{5}$ (or equivalently
$F_{4}$ and $F_{5}$) are documented in Refs. [54, 60], neither works out the
many permutations of contractions with the momentum factor
$\Pi^{\mu_{1}\dots\mu_{2k}}$ from Eq. (3.19). Keeping track of these
permutations requires care, as demonstrated for $F_{1}$ and $F_{2}$ in Ref.
[126]. Notably, Ref. [60] documents some differences and similarities between
the above treatments. The treatment in this Appendix builds on the steps of
Refs. [49, 50, 85, 54, 81, 64, 63, 126].

We first build a formula for the inclusive DIS cross section in A.1 based on
the notation defined in section 2.1. In A.2, we derive the matrix element,
squared matrix element, and inclusive hadronic tensor in DIS. In A.3, the OPE
is applied to the inclusive hadronic tensor, with the leading-power result
organized according to Wilson coefficients. In A.4, OPE terms are reorganized
by Lorentz structures, resulting in structure-function mixing. Structure
functions for massless and massive nuclear targets are then constructed in
A.5.

### A.1 Preliminaries and the inclusive DIS cross section formula

To build nuclear structure functions with TMCs in the OPE formalism, we
consider the DIS process as defined in Eq. (2.1) of section 2.1 and depicted
in Fig. 1.1. Given this configuration, the total cross section for a generic
$2\to(n+1)$ scattering process is given by the formula

$\displaystyle\sigma(\ell_{1}+A\to\ell_{2}+X_{n})$ $\displaystyle=\int
dPS_{n+1}\frac{d\sigma}{dPS_{n+1}},\qquad\frac{d\sigma}{dPS_{n+1}}=\frac{1}{\mathcal{F}}\frac{1}{\mathcal{S}_{\ell_{1}}\mathcal{S}_{A}}\sum_{\rm\\{\lambda\\}}\sum_{\rm
dof}|\mathcal{M}|^{2},$ (A.1a) $\displaystyle\mathcal{F}$
$\displaystyle=4\sqrt{(p_{A}\cdot k_{1})^{2}-p_{A}^{2}k_{1}^{2}}~{}\
\overset{k_{1}^{2}=m_{1}^{2}\to 0}{\approx}\ 2(s-M_{A}^{2}).$ (A.1b)

Here, $\mathcal{F}$ is the Moller flux factor and reduces to
$\mathcal{F}\approx 2(s-M_{A}^{2})$ when the mass $k_{1}^{2}=m_{1}^{2}$
vanishes. $d\sigma/dPS_{n+1}$ is the totally differential, nucleus-level cross
section. $\mathcal{S}_{\ell_{1}}$ and $\mathcal{S}_{A}$ are the spin
multiplicities for $\ell_{1}$ and $A$ in unpolarized scattering. The double
summations in the differential cross section run over external helicities
$\\{\lambda\\}$ and other discrete quantum numbers/degrees of freedom (dof).
We introduce the subscript $n$ on $X_{n}$ to make explicitly that it is an
$n$-body state. In addition, $\mathcal{M}$ is the scattering amplitude for the
$2\to(n+1)$ process. The associated $(n+1)$-body phase space measure is given
by

$\displaystyle dPS_{n+1}(k_{1}+p_{A};k_{2},p_{X})=$
$\displaystyle(2\pi)^{4}\delta^{4}\left(k_{1}+p_{A}-k_{2}-\sum_{i=1}^{n}p_{Xi}\right)~{}\frac{d^{3}k_{2}}{(2\pi)^{3}2k^{0}_{2}}\times\prod_{i=1}^{n}\frac{d^{3}p_{Xi}}{(2\pi)^{3}2p^{0}_{Xi}}$
(A.2) $\displaystyle=$ $\displaystyle\ dPS_{n}(q+p_{A};p_{X})\ \times\
\frac{d^{3}k_{2}}{(2\pi)^{3}2k^{0}_{2}}$ (A.3) $\displaystyle=$
$\displaystyle\int d^{4}z\ e^{i(q+p_{A}-p_{X})\cdot z}\
\prod_{i=1}^{n}\frac{d^{3}p_{Xi}}{(2\pi)^{3}2p^{0}_{Xi}}\
\times\frac{d^{3}k_{2}}{(2\pi)^{3}2k^{0}_{2}}\ .$ (A.4)

In the above lines, we employed momentum conservation and $\delta$-function
identities to rewrite the measure in ways that will be used below.
Specifically in the third line, we used

$(2\pi)^{4}\ \delta^{4}(q-p)\ =\ \int d^{4}z\ e^{i(q-p)\cdot z}\ ,$ (A.5)

where $q,p$ and $z$ are 4-vectors. The point of these identities is to
identify the phase space integral over the hadronic matrix element also as the
Fourier transformation over composite operators. The modeling of $X_{n}$’s
phase space is not precisely correct in many presentations of structure
functions in DIS. Often, $p_{X}$ is treated as a one-body configuration
despite being an $n$-body configuration. While this mis-modelling is
ultimately a technicality, it does impact the formal definition of
$W_{\mu\nu}$; see, for instance, Ref. [127]. We reiterate that $X_{n}$ here
(or $X$ in the main text) is an $n$-body state.

Using the above phase space decomposition, we can write an expression for the
DIS cross section that (1) is differential with respect to the kinematics of
$\\{\ell_{2},k_{2}\\}$, and (2) is explicitly inclusive over all hadronic
activity. This expression is given by

$\displaystyle k^{0}_{2}\times\frac{d\sigma}{d^{3}k_{2}}$
$\displaystyle=\frac{1}{(16\pi^{2})}\frac{1}{\pi}\frac{1}{\mathcal{F}}\frac{1}{\mathcal{S}_{\ell_{1}}\mathcal{S}_{A}}~{}\sum_{n=1}^{\infty}\int
d^{4}z~{}e^{i(q+p_{A}-p_{X})\cdot
z}\prod_{i=1}^{n}\frac{d^{3}p_{Xi}}{(2\pi)^{3}2p^{0}_{Xi}}\sum_{\rm\\{\lambda\\}}\sum_{\rm
dof}|\mathcal{M}|^{2}\ .$ (A.6)

Here, $\mathcal{S}_{\ell_{1}}=2$ for an unpolarized $\ell_{1}$, and
$\mathcal{S}_{A}=2$ for an unpolarized, spin-1/2 nuclei $A$. Additionally,
$\mathcal{S}_{\ell_{1}}=1$ when $\ell_{1}$ is an incoming neutrino or
antineutrino because neutrino beams are effectively 100% polarized. More
broadly, for unpolarized nuclear targets with spin $s_{A}>1/2$, the spin-
averaging factor $\mathcal{S}_{A}$ generalizes to
$\mathcal{S}_{A}=(2s_{A}+1)>2$.

In Eq. (A.6), we inserted a sum over the multiplicity $n$ of $X_{n}$.
Conceptually, this is equivalent to the relationship

$\ell_{1}~{}+~{}A~{}\to~{}\ell_{2}~{}+~{}\text{any hadronic
activity}=\sum_{n=1}^{\infty}\left[\ell_{1}+A\to\ell_{2}+X_{n}\right]\ .$
(A.7)

For example, when $n=2$ this sums over the configurations
$pp,pn,p\pi,pK,\pi\pi,\ldots$, which are all included in the inclusive sum.

### A.2 The DIS matrix element and inclusive hadronic tensor

The hadronic tensor $W_{\mu\nu}^{A}$ of Eq. (2.7) is built from the (squared)
matrix element $\mathcal{M}$. For $\ell_{1}A\to\ell_{2}X$ scattering via an
arbitrary electroweak boson $V(q)$, we parameterize this by

$\displaystyle-i\mathcal{M}$ $\displaystyle=\ \langle
X\ell_{2}|J^{\mu}_{\ell_{2}\ell_{1}}(0)\cdot\Delta_{\mu\sigma}^{V}(q)\cdot
J^{\sigma}_{XA}(0)|A\ell_{1}\rangle$ (A.8) $\displaystyle=\
\langle\ell_{2}|J^{\mu}_{\ell_{2}\ell_{1}}(0)|\ell_{1}\rangle\cdot\Delta_{\mu\sigma}^{V}(q)\cdot\langle
X|J^{\sigma}_{XA}(0)|A\rangle,$ (A.9)

where the leptonic part of the matrix element $L^{\mu}$ and current
$J^{\mu}_{\ell_{2}\ell_{1}}$ in terms of 4-component helicity spinors, a
generic coupling normalization $\tilde{g}$, and chiral couplings
$g_{V},g_{A},$ are

$L^{\mu}~{}\equiv~{}\langle\ell_{2}|J^{\mu}_{\ell_{2}\ell_{1}}(0)|\ell_{1}\rangle~{}=~{}-i\tilde{g}~{}\overline{u}(k_{2},\lambda_{2})\left[g_{V}^{\ell}\gamma^{\mu}+g_{A}^{\ell}\gamma^{\mu}\gamma^{5}\right]u(k_{1},\lambda_{1}).$
(A.10)

We assume arbitrary vector and axial-vector couplings between the
$(\ell_{2}\ell_{1})$ system and $V$ in order to accommodate the possibility of
$V$ being a photon or weak boson. The conversion between the different
possibilities is given in Tab. A.1. Similarly, we write the hadronic part of
the matrix element in terms of an arbitrary hadronic current
$J^{\sigma}_{XA}(0)$:

$H^{A\sigma}~{}\equiv~{}\langle
X|J^{\sigma}_{XA}(0)|A\rangle~{}\equiv~{}\overline{u}_{X}(p_{X},\lambda_{X})[\dots]^{\sigma}u(p_{A},\lambda_{A}).$
(A.11)

We assume here and below that $J^{\sigma}_{XA}$ is always a renormalized
object in QCD.

Vertex | Coupling | $g_{R}^{\ell}$ | $g_{L}^{\ell}$ | $g_{V}^{\ell}$ | $g_{A}^{\ell}$
---|---|---|---|---|---
strength
$V-\ell_{1}-\ell_{2}$ | $\tilde{g}$ | $(g_{V}^{\ell}+g_{A}^{\ell})$ | $(g_{V}^{\ell}-g_{A}^{\ell})$ | $\frac{(g_{R}+g_{L})}{2}$ | $\frac{(g_{R}-g_{L})}{2}$
$\gamma-\ell-\ell$ | $eQ^{\ell}$ | $1$ | $1$ | $1$ | $0$
$Z-\ell-\ell$ | $\frac{g_{W}}{\cos\theta_{W}}$ | $-Q^{\ell}\sin^{2}\theta_{W}$ | $(T_{3}^{\ell})_{L}-Q^{\ell}\sin^{2}\theta_{W}$ | $\frac{1}{2}(T_{3}^{\ell})_{L}-Q^{\ell}\sin^{2}\theta_{W}$ | $-\frac{1}{2}(T_{3}^{\ell})_{L}$
$W-\ell_{1}-\ell_{2}$ | $\frac{g_{W}}{\sqrt{2}}$ | $0$ | $1$ | $\frac{1}{2}$ | $-\frac{1}{2}$

Table A.1: Electroweak chiral couplings and coupling strength normalizations
used for fermions $\ell_{1},\ell_{2}$ with weak isospin charge
$(T_{3}^{\ell})_{L}=\pm 1/2$ and electric charge $Q^{\ell}$, with
normalization $Q^{e}=-1$.

In Eq. (A.9), $\Delta_{\mu\sigma}^{V}(q)$ is the propagator of $V$. When we
assume the $\ell_{i}$ are massless, the Dirac equation and SU$(2)_{L}$
invariance ensure that the longitudinal modes of $\Delta_{\mu\sigma}^{V}$ do
not contribute to the scattering process.121212We typically neglect the mass
of the leptons, except for the tau-lepton.  More specifically, the
longitudinal component of $\Delta$ scales as $\Delta_{\mu\sigma}(q)|_{\rm
long}\propto q_{\mu}q_{\sigma}$. Since $\not k_{i}u(k_{i})=m_{i}u(k_{i})=0$
and $q_{\mu}=(k_{1}-k_{2})_{\mu}$, one finds

$\displaystyle L^{\mu}\cdot\Delta_{\mu\sigma}^{V}|_{\rm long}$
$\displaystyle~{}\propto~{}q_{\mu}\cdot\overline{u}(k_{2},\lambda_{2})\left[g_{V}^{\ell}\gamma^{\mu}+g_{A}^{\ell}\gamma^{\mu}\gamma^{5}\right]u(k_{1},\lambda_{1})$
(A.12)
$\displaystyle~{}=~{}\overline{u}(k_{2},\lambda_{2})\left[g_{V}^{\ell}(\not
k_{1}-\not k_{2})+g_{A}^{\ell}(\not k_{1}-\not
k_{2})\gamma^{5}\right]u(k_{1},\lambda_{1})=0.$ (A.13)

The above implies that only the transverse components of
$\Delta_{\mu\sigma}^{V}$ contributes to DIS for massless leptons. For
$V=\gamma,W,Z$ with mass $M_{V}$ (where $M_{\gamma}=0$), one can then write

$-i\mathcal{M}=L^{\mu}(k_{2},k_{1})\Delta_{\mu\sigma}^{V}(q)H^{A\sigma}(p_{X},p_{A})=L^{\mu}\frac{-ig_{\mu\sigma}}{q^{2}-M_{V}^{2}}H^{A\sigma}=\frac{-i}{q^{2}-M_{V}^{2}}L^{\mu}H_{\mu}^{A}.$
(A.14)

After squaring and summing over external helicity states, one obtains

$\sum_{\\{\lambda\\}}|\mathcal{M}|^{2}\ =\ \frac{1}{(q^{2}-M_{V}^{2})^{2}}\
\sum_{\\{\lambda\\}}L^{\mu\nu}\ \cdot\ \sum_{\\{\lambda\\}}H_{\mu\nu}^{A}\ .$
(A.15)

The leptonic tensor $L^{\mu\nu}$ denotes the square of $L^{\mu}$ and can be
solved exactly using trace technology, helicity amplitudes, or other standard
technology. It is given by

$\displaystyle\sum_{\\{\lambda\\}}L^{\mu\nu}=$
$\displaystyle\sum_{\\{\lambda\\}}\left(L^{\dagger}\right)^{\mu}L^{\nu}=\sum_{\\{\lambda\\}}\langle\ell_{1}|\left(J_{\ell_{2}\ell_{1}}^{\dagger}(0)\right)^{\mu}|\ell_{2}\rangle\langle\ell_{2}|J^{\nu}_{\ell_{2}\ell_{1}}(0)|\ell_{1}\rangle$
(A.16) $\displaystyle=$ $\displaystyle
4\tilde{g}^{2}\Big{\\{}(g_{V}^{2}+g_{A}^{2})(k_{1}^{\mu}k_{2}^{\nu}+k_{1}^{\nu}k_{2}^{\mu})-\left[(g_{V}^{2}+g_{A}^{2})(k_{1}\cdot
k_{2})-(g_{V}^{2}-g_{A}^{2})m_{1}m_{2}\right]g^{\mu\nu}$
$\displaystyle\qquad-2i(g_{V}g_{A})k_{1\alpha}k_{2\beta}\epsilon^{\mu\nu\alpha\beta}\Big{\\}}$
(A.17) $\displaystyle\overset{m_{1},m_{2}\to 0}{=}$ $\displaystyle
4\tilde{g}^{2}\Big{\\{}(g_{V}^{2}+g_{A}^{2})(k_{1}^{\mu}k_{2}^{\nu}+k_{1}^{\nu}k_{2}^{\mu})-(g_{V}^{2}+g_{A}^{2})(k_{1}\cdot
k_{2})g^{\mu\nu}$
$\displaystyle\qquad-2i(g_{V}g_{A})k_{1\alpha}k_{2\beta}\epsilon^{\mu\nu\alpha\beta}\Big{\\}}$
(A.18)

For completeness, we provide the expression for
$\sum_{\\{\lambda\\}}L^{\mu\nu}$ with massive leptons. If using this formula,
note that Eq. (A.14) must be modified since $\not
k_{i}u(k_{i})=m_{i}u(k_{i})\neq 0$. For the remainder of this work, we keep
$m_{1},m_{2}=0$. Note that the expression here for the leptonic tensor only
sums over initial- and final-state polarizations; a symmetry factor of
$1/\mathcal{S}_{\ell_{1}}=1/2$ for unpolarized initial-states has not yet been
applied. Following the conventions of Table A.1, for the case of QED, where
$g_{V}=1,~{}g_{A}=0$, and $\tilde{g}=e$, we recover the usual expression:

$\sum_{\\{\lambda\\}}L^{\mu\nu}\Big{|}_{\rm
QED}=4e^{2}\Big{\\{}k_{1}^{\mu}k_{2}^{\nu}+k_{1}^{\nu}k_{2}^{\mu}-(k_{1}\cdot
k_{2})g^{\mu\nu}\Big{\\}}.$ (A.19)

For the case of $W$ boson exchange, where $g_{V}=1/2,~{}g_{A}=-1/2,$ and
$\tilde{g}=g_{W}/\sqrt{2}$, we have

$\sum_{\\{\lambda\\}}L^{\mu\nu}\Big{|}_{W}=g_{W}^{2}\Big{\\{}k_{1}^{\mu}k_{2}^{\nu}+k_{1}^{\nu}k_{2}^{\mu}-(k_{1}\cdot
k_{2})g^{\mu\nu}+ik_{1\alpha}k_{2\beta}\epsilon^{\mu\nu\alpha\beta}\Big{\\}}.$
(A.20)

The exclusive hadronic tensor $H_{\mu\nu}^{A}$ for an $n$-body final state $X$
denotes the square of $H_{\mu}$. For a fixed $n$, it is given by

$\displaystyle\sum_{\\{\lambda\\}}H_{\mu\nu}^{A}=\sum_{\\{\lambda\\}}\left(H^{A\dagger}\right)_{\mu}H_{\nu}^{A}$
$\displaystyle=\sum_{\\{\lambda\\}}\langle
A|\left(J_{XA}^{\dagger}(0)\right)_{\mu}|X\rangle\langle
X|J_{XA\nu}(0)|A\rangle\ .$ (A.21)

From $H_{\mu\nu}^{A}$ we can build the inclusive hadronic tensor
$W_{\mu\nu}^{A}$ that sums over all final states.

### Building the inclusive hadronic tensor

In inclusive DIS, one measures the kinematics of $\ell_{2}$ and remains
totally inclusive regarding the hadronic system $X$. (Though in practice, the
invariant mass of $X$ is often measured.) The inclusiveness criterion means
summing over all possible final-state multiplicities $n$ (where $n$ is same
$n$ in Eq. (A.2)), discrete dof, and continuous dof, i.e., momenta. Performing
this summation allows us to define the inclusive hadronic tensor as

$\displaystyle W_{\mu\nu}^{A}~{}\equiv~{}$
$\displaystyle\frac{1}{4\pi}\sum_{n=1}^{\infty}\int dPS_{n}(q+p_{A};p_{X})\
\sum_{\rm dof}\sum_{\\{\lambda\\}}H_{\mu\nu}^{A}$ (A.22)
$\displaystyle~{}=~{}$ $\displaystyle\frac{1}{4\pi}\int
d^{4}z~{}e^{i(q+p_{A}-p_{X})\cdot z}\
\sum_{n=1}^{\infty}\int\prod_{i=1}^{n}\frac{d^{3}p_{Xi}}{(2\pi)^{3}2p^{0}_{Xi}}$
$\displaystyle\qquad~{}\qquad~{}\qquad~{}\quad\times\sum_{\rm
dof}\sum_{\\{\lambda\\}}\langle A|J_{XA\mu}^{\dagger}(0)|X\rangle\langle
X|J_{XA\nu}(0)|A\rangle$ (A.23) $\displaystyle~{}=~{}$
$\displaystyle\frac{1}{4\pi}\int d^{4}z~{}e^{i(q+p_{A}-p_{X})\cdot z}\
\sumint\langle A|J_{XA\mu}^{\dagger}(0)|X\rangle\langle
X|J_{XA\nu}(0)|A\rangle.$ (A.24)

In the second line, we used the phase space identities of Eq. (A.2). Following
Ref. [127] (and related references), we denote the $3^{n}$-dimensional
integration over momenta and triple summation by $\sumint$. Importantly, the
integration and summation constitute a summation over all possible
configurations spanning the space of $\\{|X\rangle\\}$. Completeness dictates
that

$\sumint|X\rangle\langle
X|=\sum_{n=1}^{\infty}\int\prod_{i=1}^{n}\frac{d^{3}p_{Xi}}{(2\pi)^{3}2p^{0}_{Xi}}~{}\sum_{\rm
dof}\sum_{\\{\lambda\\}}~{}|X\rangle\langle X|=1.$ (A.25)

Before employing Eq. (A.25), we use translation invariance to build a
spacetime separation between $J_{A\mu}^{\dagger}$ and $J_{A\nu}$ that is
constrained by momentum conservation. Such separations in the DIS limit are
restricted to be near the light cone, i.e., with $z^{2}\sim 0$ (c.f., Ref.
[64]). Applying translation invariance to the current $J_{XA\mu}^{\dagger}(0)$
gives

$\displaystyle\langle A|J_{XA\mu}^{\dagger}(0)|X\rangle$
$\displaystyle=\langle A|\left[e^{-i\hat{P}\cdot
z}J_{XA\mu}(z)e^{i\hat{P}\cdot z}\right]^{\dagger}|X\rangle$ (A.26)
$\displaystyle=\langle A|e^{-i\hat{P}\cdot
z}J_{XA\mu}^{\dagger}(z)e^{i\hat{P}\cdot z}|X\rangle=e^{-i(p_{A}-p_{X})\cdot
z}\langle A|J_{XA\mu}^{\dagger}(z)|X\rangle,$ (A.27)

where $\hat{P}^{\alpha}$ is the four-momentum operator that generates the
eigenvalue equation
$\hat{P}^{\alpha}|Y(p_{Y})\rangle=p_{Y}^{\alpha}|Y(p_{Y})\rangle$. Combining
the preceding expressions leads to familiar expression for $W_{\mu\nu}^{A}$:

$\displaystyle W_{\mu\nu}^{A}$ $\displaystyle=\frac{1}{4\pi}\int
d^{4}z~{}e^{iq\cdot z}\langle A|J_{XA\mu}^{\dagger}(z)\ J_{XA\nu}(0)|A\rangle\
.$ (A.28)

In electromagnetism, currents are Hermitian. This leads to the omission of the
$\dagger$ in many texts. In other texts, Eq. (A.28) is written using a
commutator of currents [63, 64]. However, Refs. [85, 47] note that commutators
are not necessary to define $W_{\mu\nu}^{A}$. And indeed, Refs. [63, 64] show
that the commutator-based result reduce to Eq. (A.28) in certain limits.

We are now in position to write the differential cross section for the DIS
process $\ell_{1}+A\to\ell_{2}+$ anything in terms of the inclusive hadronic
tensor $W_{\mu\nu}^{A}$. To do this, we take the differential cross section
formula of Eq. (A.6) and insert the squared matrix element in Eq. (A.15). We
then substitute the summations and integration over the exclusive hadronic
tensor $H_{\mu\nu}$ with the inclusive hadronic tensor $W_{\mu\nu}$ to obtain
the formula,

$\displaystyle k^{0}_{2}\times\frac{d\sigma}{d^{3}k_{2}}$
$\displaystyle=\frac{1}{(16\pi^{2})s}\frac{2}{\mathcal{S}_{\ell_{1}}\mathcal{S}_{A}}~{}\frac{1}{(q^{2}-M_{V}^{2})^{2}}\left(\sum_{\\{\lambda\\}}L^{\mu\nu}\right)\cdot
W_{\mu\nu}^{A}\ .$ (A.29)

This formula holds for arbitrary $A$, including unbound nucleons. Expressions
for the leptonic tensor (summed over all external spins) are found in Eqs.
(A.18)-(A.20). The factor of $2$ in the numerator originates from the adopting
the $(1/4\pi)$ normalization of $W_{\mu\nu}$; when $A$ is a spin-1/2 object,
it cancels the spin-averaging factor $\mathcal{S}_{A}$. The inclusive hadronic
tensor is given by Eq. (A.28), but now we relate it to the time-order matrix
element for virtual Compton scattering, which has a known expansion in the OPE
at leading twist.

### A.3 Nuclear structure functions from the OPE I: organization

As summarized in Sec. 3.2, building structure functions in the OPE for
massless and massive hadronic targets is a multi-step process. The advantage,
however, is clear: the OPE, if it holds for QCD, facilitates an all-orders,
operational definitions for parton densities and structure functions.
Moreover, the power counting within the OPE formalism allows one to organize
contributions that are not clearly captured in fixed-order perturbation
theory.

To do this in the manner of Refs. [49, 50], one starts with the inclusive
hadronic tensor $W_{\mu\nu}^{A}$ in the DIS limit (large $Q$, fixed $x_{A}$)
as defined above and relates it to the time-ordered matrix element
$T_{\mu\nu}^{A}$ for virtual Compton
scattering131313$\gamma^{*}(q)A(p_{A})\to\gamma^{*}A$ for electromagnetic
currents, $W^{*}(q)A(p_{A})\to W^{*}A$ for charged currents, etc. in the
short-distance limit (large $Q$, fixed $x_{A}/Q$). The OPE of $T_{\mu\nu}^{A}$
is then organized by Lorentz structures. By comparing the Lorentz structures
of the expansions of $W_{\mu\nu}^{A}$ and $T_{\mu\nu}^{A}$, one can identify
the structure functions of $W_{\mu\nu}^{A}$ in terms of the OPE. When the mass
of a nucleus $A$ is neglected, the OPE of $T_{\mu\nu}^{A}$ simplifies to
familiar expressions. The differences between the fully massive and simplified
massless expressions are the TMCs to the structure functions of
$W_{\mu\nu}^{A}$.

Following this outline, we start by taking the expression for
$W_{\mu\nu}^{A}(p_{A},q)$ in Eq. (A.28) and decomposing it into a sum of
tensor-valued coefficients multiplied by dimensionless, scalar-valued
functions $\tilde{W}_{i}^{A}$. The $\tilde{W}_{i}^{A}$ are the structure
functions. Lorentz symmetry and hermiticity dictate that only certain
combinations of $p_{A}$ and $q$ are allowed as coefficients. The most general
combination allowed by symmetries for unpolarized $A$ is

$\displaystyle W^{A}_{\mu\nu}(p_{A},q)\quad$ $\displaystyle=\frac{1}{4\pi}\int
d^{4}z~{}e^{iq\cdot z}\langle A|J_{\mu}^{\dagger}(z)\ J_{\nu}(0)|A\rangle$
(A.30)
$\displaystyle=-g_{\mu\nu}\tilde{W}_{1}^{A}~{}+~{}\frac{p_{A\mu}p_{A\nu}}{M_{A}^{2}}\tilde{W}_{2}^{A}~{}-~{}i\epsilon_{\mu\nu\rho\sigma}\frac{p_{A}^{\rho}q^{\sigma}}{M_{A}^{2}}\tilde{W}_{3}^{A}~{}+~{}\frac{q_{\mu}q_{\nu}}{M_{A}^{2}}\tilde{W}_{4}^{A}$
$\displaystyle~{}\qquad~{}\quad~{}\quad+\frac{p_{A\mu}q_{\nu}+p_{A\nu}q_{\mu}}{M_{A}^{2}}\tilde{W}_{5}^{A}~{}+~{}\frac{p_{A\mu}q_{\nu}-p_{A\nu}q_{\mu}}{M_{A}^{2}}\tilde{W}_{6}^{A}$
(A.31)
$\displaystyle=-g_{\mu\nu}\tilde{F}_{1}^{A}~{}+~{}\frac{p_{A\mu}p_{A\nu}}{Q^{2}}2x_{A}\tilde{F}_{2}^{A}~{}-~{}i\epsilon_{\mu\nu\alpha\beta}\frac{p^{\alpha}_{A}q^{\beta}}{Q^{2}}{x_{A}}\tilde{F}_{3}^{A}~{}+~{}\frac{q_{\mu}q_{\nu}}{Q^{2}}{2}\tilde{F}_{4}^{A}$
$\displaystyle~{}\qquad~{}\quad~{}\quad+\frac{(p_{A\mu}q_{\nu}+p_{A\nu}q_{\mu})}{Q^{2}}2x_{A}\tilde{F}_{5}^{A}~{}+~{}\frac{(p_{A\mu}q_{\nu}-p_{A\nu}q_{\mu})}{Q^{2}}2x_{A}\tilde{F}_{6}^{A}\
.$ (A.32)

Formally, the normalizations and organization of the $\tilde{W}_{i}^{A}$ and
$\tilde{F}_{i}^{A}$ are conventional. Following Ref. [60], the
$\tilde{W}_{i}^{A}$ are normalized such that each is dimensionless and have at
most a prefactor of $M_{A}^{-2}$. Subsequently, the $\tilde{F}_{i}^{A}$ are
normalized such that

Using $\tilde{F}_{i}^{A}$ instead of $\tilde{W}_{i}^{A}$ factors out known
$Q^{2}$ dependence from the structure functions. This makes scaling with
respect to $x_{A}$ more manifest. Under this normalization,
$\tilde{W}_{i}^{A}$ and $\tilde{F}_{i}^{A}$ are real for $i=1-5$, whereas
$\tilde{W}_{6}^{A}$ and $\tilde{F}_{6}^{A}$ are imaginary [85]. Moreover,
$\tilde{W}_{6}^{A}$ and $\tilde{F}_{6}^{A}$ are only nonzero if charge-parity
symmetry is violated in QCD. However, the coefficient of $\tilde{W}_{6}^{A}$
and $\tilde{F}_{6}^{A}$ will vanish when contracted with the symmetric
$L^{\mu\nu}$ tensor. Choosing $\tilde{W}_{i}^{A}$, $\tilde{F}_{i}^{A}$, or
other normalizations does not change the underlying physics. However,
seemingly innocuous differences in their definitions can impact the final form
of TMCs due to structure function mixing, i.e., the off-diagonal terms in
$A^{i}_{j}$ and $B^{i}_{j}$ in Eq. (3.23).

The next step is to recognize that $\tilde{W}^{A}_{\mu\nu}$ in the DIS limit
can be related to the time-ordered matrix element $T_{\mu\nu}^{A}$ for
$AV^{*}\to AV^{*}$ scattering in short-distance limit. This is given by [40]

$\displaystyle{T}^{A}_{\mu\nu}(p_{A},q)\quad$ $\displaystyle\equiv\int d^{4}z\
e^{iq\cdot z}\ \langle A|\mathcal{T}J_{\mu}^{\dagger}(z)\ J_{\nu}(0)|A\rangle$
(A.34) $\displaystyle=\int d^{4}z\ e^{iq\cdot z}\ \langle
A|J_{\mu}^{\dagger}(z)\ J_{\nu}(0)|A\rangle\ \theta(z^{0}>0)$
$\displaystyle\qquad\ +\int d^{4}z\ e^{iq\cdot z}\ \langle A|J_{\nu}(0)\
J_{\mu}^{\dagger}(z)|A\rangle\ \theta(z^{0}<0)\ $ (A.35) $\displaystyle\equiv
T_{z^{0}>0}(p_{A},q)+T_{z^{0}<0}(p_{A},q),$ (A.36)

where $\theta$ is the usual Heaviside step function normalized to unity. The
matrix elements under the integral can be identified as the (inverse) Fourier
transforms (FT) of $W_{\mu\nu}^{A}$ over $q$:

$\displaystyle{\rm FT}[W^{A}_{\mu\nu}(p_{A},q)](x)$
$\displaystyle=\int\frac{d^{4}q}{(2\pi)^{4}}\ e^{-iq\cdot x}\
W^{A}_{\mu\nu}(p_{A},q)\ =\frac{1}{4\pi}\langle A|J_{\mu}^{\dagger}(x)\
J_{\nu}(0)|A\rangle\ ,$ (A.37a) $\displaystyle{\rm
FT}[\overline{W}^{A}_{\nu\mu}(p_{A},q)](x)$
$\displaystyle=\int\frac{d^{4}q}{(2\pi)^{4}}\ e^{-iq\cdot x}\
\overline{W}^{A}_{\nu\mu}(p_{A},q)\ =\frac{1}{4\pi}\langle A|J_{\nu}(0)\
J_{\mu}^{\dagger}(x)|A\rangle\ .$ (A.37b)

Equation (A.37b) defines the “conjugate” hadronic tensor
$\overline{W}_{\nu\mu}^{A}$. It is related to ${W}_{\mu\nu}^{A}$ by

$\displaystyle\overline{W}^{A}_{\nu\mu}(p_{A},q)\ =\ \frac{1}{4\pi}\int
d^{4}z~{}e^{-i(-q)\cdot z}\langle A|J_{\nu}(0)\ J_{\mu}^{\dagger}(z)|A\rangle\
=\ \left[W^{A}_{\mu\nu}(p_{A},-q)\right]^{\dagger}\ .$ (A.38)

Note that the Hermitian conjugation operator $\dagger$ should be applied to
both the structure functions and the tensor-valued coefficients of
$W^{A}_{\mu\nu}$. For the case of electromagnetic currents, these are
Hermitian and Eq. (A.38) reduces to
$\overline{W}^{A}_{\nu\mu}(p_{A},q)\big{|}_{\rm EM}=W^{A}_{\mu\nu}(p_{A},-q)$.

After exchanging the order of integration, the first term in Eq. (A.36) is

$\displaystyle T_{z^{0}>0}(p_{A},q)$
$\displaystyle=4\pi\int\frac{d^{4}k}{(2\pi)^{4}}\ W^{A}_{\mu\nu}(p_{A},k)\
\int d^{4}z\ e^{i(q-k)\cdot z}\ \theta(z^{0}>0)$ (A.39)
$\displaystyle=4\pi\int\frac{d^{4}k}{(2\pi)^{4}}\ W^{A}_{\mu\nu}(p_{A},k)\
(2\pi)^{3}\delta(\vec{q}-\vec{k})\
\left[\pi\delta(q^{0}-k^{0})+\frac{i}{q^{0}-k^{0}}\right].$ (A.40)

The three-dimensional $\delta(\vec{q}-\vec{k})$ comes from integrating over
all of $\vec{z}$-space and the bracketed factor comes from integrating only
over $z^{0}>0$. Following the argument of Ref. [85], momentum conservation
dictates that $(2p_{A}\cdot q)>Q^{2}$, which implies that ${W}^{A}_{\mu\nu}$
is zero for $(2p_{A}\cdot q)<Q^{2}$, or equivalently that ${W}^{A}_{\mu\nu}$
is zero for $x_{A}>1$ and $x_{A}<0$. Fixing $Q^{2}$ and taking
${W}_{\mu\nu}^{A}$ to be analytic on the plane $\omega_{A}\equiv x_{A}^{-1}$
further implies branch cuts along $|\omega_{A}|\geq 1$ as depicted in Fig.
3.1. Therefore, since the pole in Eq. (A.40) can be decomposed into its
principal $\mathcal{P}$ and singular parts when deformed, i.e.,

$\displaystyle\frac{1}{q^{0}-k^{0}\mp i\varepsilon}\ =\ \mathcal{P}\ \pm\
i\pi\delta(q^{0}-k^{0}),$ (A.41)

the tensor $W^{A}_{\mu\nu}$ can be identified as the discontinuity of
$T_{z^{0}>0}$. Specifically, one finds:

$\displaystyle{\rm disc}\ T_{z^{0}>0}\ $ $\displaystyle=\lim_{\varepsilon\to
0}\
\left[T_{z^{0}>0}(q^{0}+i\varepsilon)-T_{z^{0}>0}(q^{0}-i\varepsilon)\right]$
(A.42) $\displaystyle=4\pi\ \int\frac{d^{4}k}{(2\pi)^{4}}\
W^{A}_{\mu\nu}(p_{A},k)\ (2\pi)^{3}\delta(\vec{q}-\vec{k})\ $
$\displaystyle\qquad\times\
\left\\{\left[i\mathcal{P}-i^{2}\pi\delta(q^{0}-k^{0})\right]-\left[i\mathcal{P}+i^{2}\pi\delta(q^{0}-k^{0})\right]\right\\}$
(A.43) $\displaystyle=4\pi\ W^{A}_{\mu\nu}(p_{A},q)\ .$ (A.44)

A similar result holds for $T_{z^{0}<0}$. Subsequently, one can write in terms
of $\omega_{A}=x_{A}^{-1}$ the relations [40, 85]

$\displaystyle{T}^{A}_{\mu\nu}(p_{A},q)\Big{|}^{\omega_{A}+i\varepsilon}_{\omega_{A}-i\varepsilon}$
$\displaystyle=4\pi\
{W}^{A}_{\mu\nu}(p_{A},q),\quad\text{for}\quad\omega_{A}>0\ ,$ (A.45a)
$\displaystyle{T}^{A}_{\mu\nu}(p_{A},q)\Big{|}^{\omega_{A}-i\varepsilon}_{\omega_{A}+i\varepsilon}$
$\displaystyle=4\pi\
\left[W^{A}_{\mu\nu}(p_{A},-q)\right]^{\dagger},\quad\text{for}\quad\omega_{A}<0\
.$ (A.45b)

and with ${W}^{A}_{\mu\nu}$ and ${W}^{A\dagger}_{\nu\mu}$ vanishing for
$\omega_{A}<1$ and $\omega_{A}>-1$, respectively. Intuitively, Eq. (A.45)
states that ${T}^{A}_{\mu\nu}$ and $W^{A}_{\mu\nu}$, which are defined in
different kinematic limits (the short-distance and DIS limits, respectively),
are nevertheless related through analytic continuation. Therefore, one can
take ${T}^{A}_{\mu\nu}(p_{A},q)$, 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.

### Coefficient functions of $T_{\mu\nu}^{A}$ and moments of structure
functions

The relationship in Eq. (A.45) can be refined by decomposing
${T}^{A}_{\mu\nu}$ in three different ways. The first is according to Lorentz
structures as was done for ${W}^{A}_{\mu\nu}$ in Eq. (A.31):

$\displaystyle{T}^{A}_{\mu\nu}$
$\displaystyle=-g_{\mu\nu}\Delta\tilde{T}^{A}_{1}~{}+~{}\frac{p_{A\mu}p_{A\nu}}{M_{A}^{2}}\Delta\tilde{T}^{A}_{2}~{}-~{}i\epsilon_{\mu\nu\alpha\beta}\frac{p^{\alpha}_{A}q^{\beta}}{M_{A}^{2}}\Delta\tilde{T}^{A}_{3}~{}+~{}\frac{q_{\mu}q_{\nu}}{M_{A}^{2}}\Delta\tilde{T}^{A}_{4}$
$\displaystyle~{}\qquad~{}\quad~{}\qquad+\frac{(p_{A\mu}q_{\nu}+p_{A\nu}q_{\mu})}{M_{A}^{2}}\Delta\tilde{T}^{A}_{5}~{}+~{}\frac{(p_{A\mu}q_{\nu}-p_{A\nu}q_{\mu})}{M_{A}^{2}}\Delta\tilde{T}^{A}_{6}.$
(A.46)

Component-by-component, the dispersion relationship of Eq. (A.45) becomes

$\displaystyle\Delta{\tilde{T}}^{A}_{i}(p_{A},q)\Big{|}^{\omega_{A}+i\varepsilon}_{\omega_{A}-i\varepsilon}$
$\displaystyle=4\pi\
\tilde{W}^{A}_{i}(p_{A},q),\quad\text{for}\quad\omega_{A}>0\ ,$ (A.47a)
$\displaystyle\Delta{\tilde{T}}^{A}_{i}(p_{A},q)\Big{|}^{\omega_{A}-i\varepsilon}_{\omega_{A}+i\varepsilon}$
$\displaystyle=4\pi\ (-1)^{b_{i}}\
{\tilde{W}}^{A}_{i}(p_{A},-q),\quad\text{for}\quad\omega_{A}<0\ ,$ (A.47b)
$\displaystyle\text{and}\quad b_{i}$ $\displaystyle=0\
(1),\quad\text{for}\quad i=1,2,3,4,6\ (5)\ .$ (A.47c)

The factor of $(-1)^{b_{i}}$ comes from applying the $\dagger$ operator in Eq.
(A.45b) to $\tilde{W}_{i}^{A}$ and its prefactor in combination with
propagating the argument $-q$ into the prefactor of $\tilde{W}_{i}^{A}$. We
also take into account whether $\tilde{W}_{i}$ is real or imaginary, as
discussed below Eq. (A.33); recall ${\tilde{W}}^{A}_{6}$ is imaginary.

The second way of decomposing $T_{\mu\nu}^{A}$ is by the OPE (done below) and
gives expressions for $\Delta\tilde{T}_{i}^{A}$.

The third decomposition is by Taylor expansion in the short-distance limit.
The distinction between this 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 $(Q/M_{A})$. Formally, the
short-distance limit corresponds to the limit $\omega_{A}=2(p_{A}\cdot
q)/Q^{2}\to 0$. We therefore assume that we can write

$\displaystyle\zeta_{i}^{A}\ \times\ \omega_{A}^{a_{i}}\ \times\
\Delta\tilde{T}_{i}^{A}(p_{A},q)\ $ $\displaystyle=\ \sum_{k=0}^{\infty}\
t_{i(k)}^{A}(Q^{2})\ \omega_{A}^{k}\ ,\quad\text{where}$ (A.48a)
$\displaystyle a_{i}=0~{}(1)\ \text{for}\ i=1,4~{}(2,3,5,6)\
,\quad\text{and}\quad\zeta_{1}^{A}$ $\displaystyle=1\ ,\quad\
\zeta_{2,4,5,6}^{A}=\left(\frac{Q^{2}}{2M_{A}^{2}}\right)\ ,\quad\
\zeta_{3}^{A}=\left(\frac{Q^{2}}{M_{A}^{2}}\right)\ .$ (A.48b)

Here, $t_{i(k)}^{A}$ is the $k^{th}$ coefficient of the expansion and is only
a function of $Q^{2}$. The factors $\zeta_{i}^{A}$ and $\omega_{A}^{a_{i}}$,
which are fixed in the short distance limit, are the same factors relating the
structure functions $\tilde{W}_{i}$ and $\tilde{F}_{i}$ in Eqs. (2.2) and
(A.33). Following Ref. [85] and using Cauchy’s integral formula

$\displaystyle F(y)=\frac{1}{2\pi i}\int_{\mathcal{C}}\ \frac{dz}{z-y}\ F(z),$
(A.49)

where $\mathcal{C}$ is an appropriately chosen contour, it is possible to
analytically continue $\Delta\tilde{T}_{i}^{A}$:

$\displaystyle\zeta_{i}^{A}\ \times\omega_{A}^{a_{i}}\
\times\Delta\tilde{T}_{i}^{A}(p_{A},q)$
$\displaystyle=\frac{\zeta_{i}^{A}}{2\pi i}\int_{\mathcal{C}}\
\frac{d\omega_{A}^{\prime}}{(\omega_{A}^{\prime}-\omega_{A})}\
\left(\omega_{A}^{\prime}\right)^{a_{i}}\
\Delta\tilde{T}_{i}^{A}(p_{A},q^{\prime})$ (A.50)
$\displaystyle=\frac{\zeta_{i}^{A}}{2\pi i}\int_{\mathcal{C}}\
\frac{d\omega_{A}^{\prime}}{\omega_{A}^{\prime}(1-\frac{\omega_{A}}{\omega_{A}^{\prime}})}\
\left(\omega_{A}^{\prime}\right)^{a_{i}}\
\Delta\tilde{T}_{i}^{A}(p_{A},q^{\prime})$ (A.51)
$\displaystyle=\sum_{k=0}^{\infty}\ \frac{\zeta_{i}^{A}}{2\pi
i}\int_{\mathcal{C}}\ d\omega_{A}^{\prime}\
\left(\frac{\omega_{A}}{\omega_{A}^{\prime}}\right)^{k}\
\left(\omega_{A}^{\prime}\right)^{a_{i}-1}\
\Delta\tilde{T}_{i}^{A}(p_{A},q^{\prime}).$ (A.52)

The geometric series is obtained by fixing the integration variable
$\omega_{A}^{\prime}$, which can be large, but taking the external variable
$\omega_{A}\to 0$.

Comparing Eq. (A.52) to Eq. (A.48) and using Eqs. (A.47), one obtains

$\displaystyle t_{i(k)}^{A}(Q^{2})$ $\displaystyle=\frac{\zeta_{i}^{A}}{2\pi
i}\int_{\mathcal{C}}\ d\omega_{A}^{\prime}\
\left(\omega_{A}^{\prime}\right)^{a_{i}-k-1}\
\Delta\tilde{T}_{i}^{A}(p_{A},q^{\prime})$ (A.53)
$\displaystyle=\frac{4\pi\zeta_{i}^{A}}{2\pi
i}\int_{1}^{\infty}d\omega_{A}^{\prime}\
\left(\omega_{A}^{\prime}\right)^{a_{i}-k-1}\
\tilde{W}_{i}^{A}(p_{A},q^{\prime})$
$\displaystyle+\frac{4\pi\zeta_{i}^{A}}{2\pi
i}\int_{-\infty}^{-1}d\omega_{A}^{\prime}\
\left(\omega_{A}^{\prime}\right)^{a_{i}-k-1}\ (-1)^{b_{i}}\
{\tilde{W}}_{i}^{A}(p_{A},-q^{\prime})$ (A.54)
$\displaystyle=(-2i)\zeta_{i}^{A}\ \int_{0}^{1}dx_{A}^{\prime}\
\left(x_{A}^{\prime}\right)^{k-1-a_{i}}\
\tilde{W}_{i}^{A}(x_{A}^{\prime},Q^{2})$ $\displaystyle+(-2i)\zeta_{i}^{A}\
\int_{-1}^{0}dx_{A}^{\prime}\ \left(x_{A}^{\prime}\right)^{k-1-a_{i}}\
(-1)^{b_{i}}\ {\tilde{W}}_{i}^{A}(-x_{A}^{\prime},Q^{2})\ .$ (A.55)

The last line is obtained by changing to the variable
$x^{\prime}_{A}=(\omega_{A}^{\prime})^{-1}$. Changing the integration variable
of the second term to $z_{A}=-x_{A}^{\prime}$, collecting factors of $(-1)$,
using $(-1)^{a_{i}}=(-1)^{-a_{i}}$, and relabeling gives

$\displaystyle t_{i(k)}^{A}(Q^{2})$ $\displaystyle=(-2i)\
\int_{0}^{1}dx_{A}^{\prime}\ \left(x_{A}^{\prime}\right)^{k-1}$
$\displaystyle\quad\times\left[\zeta_{i}\
\left(x_{A}^{\prime}\right)^{-a_{i}}\
\tilde{W}_{i}^{A}(x_{A}^{\prime},Q^{2})+\left(-1\right)^{k-1+a_{i}+b_{i}}\zeta_{i}\
\left(x_{A}^{\prime}\right)^{-a_{i}}\
{\tilde{W}}_{i}^{A}(x_{A}^{\prime},Q^{2})\right]$ (A.56) $\displaystyle=(-2i)\
\left[1+(-1)^{k-1+a_{i}+b_{i}}\right]\ \int_{0}^{1}dx_{A}^{\prime}\
\left(x_{A}^{\prime}\right)^{k-1}\ \tilde{F}_{i}^{A}(x_{A}^{\prime},Q^{2}).$
(A.57)

We now denote the $N^{th}$ Mellin moment of the function $M(z)$ by $M^{N}$ and
fix normalizations such that a Mellin transformation and its inverse (over a
path $c$) are:

$M^{N}=\int_{0}^{1}dz~{}z^{N-1}~{}M(z)\quad\text{with}\quad M(z)=\frac{1}{2\pi
i}\int^{c+i\infty}_{c-i\infty}dN~{}z^{-N}~{}M^{N}.$ (A.58)

Under this normalization, the moments of $\tilde{W}^{A}_{i}(x_{A})$ are
related to those of $\tilde{F}^{A}_{i}(x_{A})$ by

Both are related to the $N^{th}$ coefficient function $t_{i(k=N)}^{A}(Q^{2})$
by

$\displaystyle t_{i(N)}^{A}(Q^{2})$ $\displaystyle=-2i\
\left[1+(-1)^{N-1+a_{i}+b_{i}}\right]\ \tilde{F}_{i}^{AN}(Q^{2})$ (A.60a)
$\displaystyle=\left\\{\begin{matrix}0,&N=\text{even}\\\ -4i\
\tilde{F}^{AN}_{i}(Q^{2}),&N=\text{odd}\end{matrix}\right.,\quad\text{for}\quad
i=1,4,5\ ,$ (A.60b) $\displaystyle=\left\\{\begin{matrix}-4i\
\tilde{F}^{AN}_{i}(Q^{2}),&N=\text{even}\\\
0,&N=\text{odd}\end{matrix}\right.,\quad\text{for}\quad i=2,3,6\ .$ (A.60c)

This allows us to rewrite the expansion in Eq. (A.48) as

$\displaystyle\zeta_{i}^{A}\ \times\ \omega_{A}^{a_{i}}\ \times\
\Delta\tilde{T}_{i}^{A}(Q^{2},\omega_{A})\ =\ -4i\ \sum_{N}^{\infty}\
\tilde{F}_{i}^{AN}(Q^{2})\ \omega_{A}^{N}\ ,$ (A.61)

where it is implied that the index $N$ runs only over odd or even integers.

We now make a few brief comments. First, in order for Eq. (A.61) to hold,
$|\omega_{A}|<1$ must be satisfied, i.e., one must be in the DIS limit.
Second, as $N$ increases, the dominant contribution to $\tilde{F}_{i}^{AN}$ is
when the argument of $\tilde{F}_{i}^{A}(z)$ approaches unity since (according
to the definition of Eq. (A.58)) $\tilde{F}_{i}^{AN}$ would otherwise be
suppressed by a small number. Finally, since $\omega_{A}=1$ corresponds to the
elastic limit (see Eq. (2.2c)), $Q^{2}$ must be made increasingly large for
large-$N$ moments to be well-defined and for the invariant mass of the
hadronic system to remain in the perturbative regime.

### Expanding $T_{\mu\nu}^{A}$ with the OPE

Using the OPE, the leading behavior of $T_{\mu\nu}^{A}$ in powers of
$(1/Q^{p})$ can be decomposed into a sum of operators
$\mathcal{O}^{\mu_{1},\dots,\mu_{k}}$ and Wilson coefficients
$c_{\mu\nu\mu_{1}...\mu_{k}}^{\tau,\iota}$. This expansion is given by [40,
49, 50]

$\displaystyle\lim_{z\to 0}\ {T}^{A}_{\mu\nu}(p_{A},q)\quad\overset{\rm
OPE}{=}\quad$ $\displaystyle-2i\
\sum_{k,\iota}~{}c_{\mu\nu\mu_{1}...\mu_{k}}^{\tau=2,\iota}(q)\ \langle
A(p_{A})|\mathcal{O}_{\iota,\tau=2}^{\mu_{1}...\mu_{k}}|A(p_{A})\rangle\
+\mathcal{O}(\tau>2)\ .$ (A.62)

At leading power, the composite operators
$\mathcal{O}_{\iota,\tau}^{\mu_{1}\dots\mu_{k}}$ are symmetric, quark
billinears (or pairs of gluon field strengths) that sandwich uncontracted
covariant derivatives. Tallying up the number of uncontracted Lorentz indicies
implies that $\mathcal{O}_{\iota,\tau}^{\mu_{1}\dots\mu_{k}}$ carries a spin
of $k$. The operators are organized according to their twist $\tau\equiv d-n$,
where $d$ is the dimensionality of
$\mathcal{O}_{\iota,\tau}^{\mu_{1}\dots\mu_{k}}$ in the standard sense of
dimension power counting in an effective field theory. For a fixed spin and
twist, the index $\iota$ catalogs all the Lorentz structures that can possibly
contract with $\mathcal{O}_{\iota,\tau}^{\mu_{1}\dots\mu_{k}}$. Coupling and
renormalization factors are sequestered into the effective Wilson coefficient
$c_{\mu\nu\mu_{1}...\mu_{k}}^{\tau,\iota}(q)$. For fixed spin $k$, a larger
twist $\tau$ corresponds to a larger $(1/Q)$ suppression in the Wilson
coefficient. Schematically, Eq. (A.62) stipulates that in the short-distance
limit, the time-ordered matrix element ${T}_{\mu\nu}^{A}$, which is a function
of $p_{A}$ and $q$, can be expressed in terms of Wilson coefficients, which
are only functions of $q$, and hadronic matrix elements, which are only
functions of $p_{A}$.

The operators $\mathcal{O}^{\mu_{1},\dots,\mu_{k}}$ can be decomposed into
symmetric (traceless) and non-symmetric parts, with
$\mathcal{O}^{\mu_{1},\dots,\mu_{k}}\sim\hat{P}_{A}^{\mu_{1}}\dots\hat{P}_{A}^{\mu_{k}}+{\rm
Tr}$. The trace term captures all contributions proportional to the spacetime
metric and derivatives, which after contractions or applications of equations
of motion give rise to powers of quark masses [49, 50]. Neglecting quark
masses141414In practice, finite quark masses can be incorporated in TMCs
through a rescaling of the Nachtmann scaling variable. See Appendix A.2 of
Ref. [60] and references therein., the matrix elements that follow from acting
on $\mathcal{O}^{\mu_{1},\dots,\mu_{2k}}$ are given by

$\displaystyle\langle
A|\mathcal{O}_{\iota,\tau=2}^{\mu_{1}...\mu_{2k}}|A\rangle$
$\displaystyle=A_{\tau=2}^{2k}\ \times\
\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}},\quad\text{where}$ (A.63a)
$\displaystyle\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}$
$\displaystyle=\sum_{j=0}^{k}(-1)^{j}\ \frac{(2k-j)!}{2^{j}(2k)!}\
\eta(j,2k-2j)\ {\underbrace{\\{g...g\\}}_{\scriptstyle j\
g^{\mu_{n}\mu_{m}}{}^{\prime}s}}\quad{\underbrace{\\{p_{A}...p_{A}\\}}_{\scriptstyle(2k-2j)\
p_{A}^{\mu_{n}}{}^{\prime}s}}(p_{A}^{2})^{j}\ .$ (A.63b)

Note the index change from $k$ to $2k$. The factor $A_{\tau=2}^{2k}$ is the
scalar-valued “reduced” hadronic matrix element and describes long-distance
hadronic dynamics. (The $2k$ in $A_{\tau=2}^{2k}$ is an index, not an
exponent.) The index $j$ sums over all permutations of $p_{A}^{\mu_{n}}$ and
$g^{\mu_{n}\mu_{m}}$, with $\mu_{n},\mu_{m}\in\\{\mu_{1},\dots,\mu_{2k}\\}$,
that contract with a particular Wilson coefficient. For a given $k$ and $j$,
the two $\\{\dots\\}$ brackets denote $j$ spacetime metrics
$g^{\mu_{n}\mu_{m}}$ and $(2k-2j)$ factors of momentum $p_{A}^{\mu_{m}}$. The
permutation multiplicity $\eta$ is

$\eta\left(j~{}\text{metrics},\
2k-2j~{}\text{factors~{}of~{}}p_{A}\right)=\frac{1}{2^{j}}\frac{(2k)!}{j!(2k-2j)!}\
.$ (A.64)

The numerator of $\eta$ is calculated from $(2k)!=[(2k-2j)+(2j)]!$ The
denominator factor of $2^{j}$ accounts for the two-fold symmetry of $j$
symmetric metric tensors, i.e., $g^{\mu_{n}\mu_{m}}=g^{\mu_{m}\mu_{n}}$. If a
metric or momentum factor is pulled from either $\\{\dots\\}$ bracket, $\eta$
is updated accordingly.

In the massless target limit, i.e., when $(M_{A}^{2}/Q^{2})\to 0$, one
neglects $j>0$ terms since they generate powers of
$(p_{A}^{2})^{j}=M_{A}^{2j}$. For this reason, the $j>0$ terms are sometimes
called “kinematical power corrections” [55, 83]. In the $j=0$ limit, Eq.
(A.63) reduces to $2k$ factors of $p_{A}^{\mu_{m}}$:

$\langle
A|\mathcal{O}_{\iota,\tau=2}^{\mu_{1},\dots,\mu_{2k}}|A\rangle\Big{|}_{(M_{A}/Q)^{2}\to
0}=A_{\tau=2}^{2k}(p_{A}^{2})\times\tilde{\Pi}_{\mu_{1}\dots\mu_{2k}}\Big{|}_{j=0}=A_{\tau=2}^{2k}(p_{A}^{2})\times(p_{A}^{\mu_{1}}\dots
p_{A}^{\mu_{2k}})\ .$ (A.65)

At leading power of $\tau$ and for a fixed $2k>2$, the $c$ functions in Eq.
(A.62) can be decomposed as

$\displaystyle c_{\mu\nu\mu_{1},\dots,\mu_{2k}}^{\tau=2,\iota}(q)~{}=~{}$
$\displaystyle\Bigl{[}-2g_{\mu\nu}q_{\mu_{1}}q_{\mu_{2}}C_{1}^{2k}+g_{\mu\mu_{1}}g_{\nu\mu_{2}}Q^{2}C_{2}^{2k}-i\epsilon_{\mu\nu\alpha\beta}g^{\alpha}_{\mu_{1}}q^{\beta}q_{\mu_{2}}C_{3}^{2k}$
$\displaystyle+4\frac{q_{\mu}q_{\nu}}{Q^{2}}q_{\mu_{1}}q_{\mu_{2}}C_{4}^{2k}$
$\displaystyle+2(g_{\mu\mu_{1}}q_{\nu}q_{\mu_{2}}\pm
g_{\nu\mu_{1}}q_{\mu}q_{\mu_{2}})C_{5,6}^{2k}\Bigr{]}\times\frac{2^{2k}}{(Q^{2})^{2k}}\times\left(\prod_{m=3}^{2k}q_{\mu_{m}}\right).$
(A.66)

Here, the $C_{\iota=1,\dots,6}^{2k}$ are scalar-valued coefficients that
parameterize the normalization of each
$c_{\mu\nu\mu_{1},\dots,\mu_{2k}}^{\tau=2,\iota}$. The $C_{\iota}$ are defined
to all orders in QCD but can be identified and matched to quantities in fixed-
order perturbation theory. The tensor part, i.e., the part carrying Lorentz
indices, can then be organized according to Lorentz structures as in Eq.
(A.46). Conventional factors of $2$ in Eq. (A) are pulled from the
$C_{\iota}^{2k}$ in order to simplify later expressions.

Assembling these ingredients and contracting over all $2k>k_{\min}$ indices,
the OPE gives

$\displaystyle\lim_{z\to 0}{T}^{A}_{\mu\nu}(q,p_{A})\ \overset{\rm OPE}{=}\ $
$\displaystyle-2i\ \sum_{k=k_{\min}}^{\infty}\
\Bigl{[}-2g_{\mu\nu}q_{\mu_{1}}q_{\mu_{2}}C_{1}^{2k}+g_{\mu\mu_{1}}g_{\nu\mu_{2}}Q^{2}C_{2}^{2k}-i\epsilon_{\mu\nu\alpha\beta}g^{\alpha}_{\mu_{1}}q^{\beta}q_{\mu_{2}}C_{3}^{2k}$
$\displaystyle+4\frac{q_{\mu}q_{\nu}}{Q^{2}}q_{\mu_{1}}q_{\mu_{2}}C_{4}^{2k}+2(g_{\mu\mu_{1}}q_{\nu}q_{\mu_{2}}\pm
g_{\nu\mu_{1}}q_{\mu}q_{\mu_{2}})C_{5,6}^{2k}\Bigr{]}$
$\displaystyle\times\frac{2^{2k}}{(Q^{2})^{2k}}\times\left(\prod_{m=3}^{2k}q_{\mu_{m}}\right)\times
A_{\tau=2}^{2k}(p_{A}^{2})\times\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}+\mathcal{O}(\tau>2)$
(A.67) $\displaystyle\equiv~{}$
$\displaystyle~{}\Delta\tilde{T}^{A}_{1\mu\nu}+\Delta\tilde{T}^{A}_{2\mu\nu}+\Delta\tilde{T}^{A}_{3\mu\nu}+\Delta\tilde{T}^{A}_{4\mu\nu}+\Delta\tilde{T}^{A}_{5\mu\nu}+\Delta\tilde{T}^{A}_{6\mu\nu}+\mathcal{O}(\tau>2)$
(A.68) $\displaystyle=~{}$
$\displaystyle~{}-g_{\mu\nu}\Delta\tilde{T}^{A}_{1}+\frac{p_{A\mu}p_{A\nu}}{M_{A}^{2}}\Delta\tilde{T}^{A}_{2}-i\epsilon_{\mu\nu\alpha\beta}\frac{p^{\alpha}_{A}q^{\beta}}{M_{A}^{2}}\Delta\tilde{T}^{A}_{3}+\frac{q_{\mu}q_{\nu}}{M_{A}^{2}}\Delta\tilde{T}^{A}_{4}$
$\displaystyle~{}+\frac{(p_{A\mu}q_{\nu}+p_{A\nu}q_{\mu})}{M_{A}^{2}}\Delta\tilde{T}^{A}_{5}+\frac{(p_{A\mu}q_{\nu}-p_{A\nu}q_{\mu})}{M_{A}^{2}}\Delta\tilde{T}^{A}_{6}+\mathcal{O}(\tau>2).$
(A.69)

The starting point for the summation over $k$ depends on the particular Wilson
coefficient; specifically, $k_{\min}=2$ for $C_{2}$ while $k_{\min}=1$ for the
other $C_{i}$ coefficients. In Eq. (A.69), the scalar-valued coefficients
$\Delta\tilde{T}_{i=1,\dots,6}^{A}$, i.e., the quantities without external
Lorentz indices, are related to the hadronic structure functions $W_{i}^{A}$
through the dispersion relationships of Eqs. (A.45) and (A.47), up to $(1/Q)$
corrections. The $\Delta\tilde{T}_{i}$ are given explicitly in terms of
summations over $k$ and $j$ below in Eqs. (A.96)-(A.106).

In the intermediate step Eq. (A.68), each $\Delta\tilde{T}_{\iota=1,\dots,6\
\mu\nu}^{A}$ denotes the collection of contractions that are respectively
proportional to the coefficient $C_{\iota}$. We introduce this step because
$\Delta\tilde{T}_{\iota\ \mu\nu}^{A}$ and $\Delta\tilde{T}_{i}^{A}$ do not
have a one-to-one correspondence when $M_{A}\neq 0$. As a consequence of the
normalizations adopted for $\tilde{W}_{i}^{A}$ in Eq. (A.31), and subsequently
those for $\Delta\tilde{T}_{i}^{A}$ in Eq. (A.46), some
$\Delta\tilde{T}_{i}^{A}$ are sourced by multiple OPE operators when
$M_{A}\neq 0$. This phenomenon of $\tilde{W}_{i}^{A}$ (or
$\Delta\tilde{T}_{i}^{A}$) being sources by two or more $C_{\iota}$ is
sometimes called “structure function mixing,” and is discussed in A.4.

We now derive all six $\Delta\tilde{T}_{\iota\ \mu\nu}^{A}$ from Eq. (A) by
contracting all $\mu_{m}$ indices. After, we reorganize
$\Delta\tilde{T}_{\iota\ \mu\nu}^{A}$ and group terms according to their
Lorentz structure, e.g., collect all terms proportional to $g_{\mu\nu}$ or all
terms proportional to $\epsilon_{\mu\nu\alpha\beta}p^{\alpha}_{A}q^{\beta}$.
We then identify the $\Delta\tilde{T}_{i}^{A}$, which are in terms of hadronic
matrix elements $A_{\tau=2}^{2k}$, Wilson coefficients $C_{\iota}^{2k}$, and
nested summations. In A.5, the summations are first evaluated for the case of
a massless target. The results for the $(M_{A}/Q^{2})\neq 0$ case are then
expressed in terms of structure functions for massless targets.

### Organization of Lorentz contractions in the OPE

Explicit evaluation of all contractions of Lorentz indices in Eq. (A) is an
exacting task. Therefore, as a first step, we organize everything according to
$\Delta\tilde{T}_{\iota\ \mu\nu}^{A}$, and hence $C_{\iota}$. As a
consequence, patterns in $\Delta\tilde{T}_{\iota\ \mu\nu}^{A}$ appear and
simplify the work. The outcome is summarized in A.4.

(i) $\Delta\tilde{T}^{A}_{1\mu\nu}$:

We start with $\Delta\tilde{T}^{A}_{1\mu\nu}$ for $M_{A}\neq 0$ because, like
$\Delta\tilde{T}^{A}_{4\mu\nu}$, the covariant tensor structure for
$\mu_{m=1}\dots\mu_{m=2k}$ consists entirely of momentum factors $q$. That is
to say, all lowered internal indices $\mu_{m}$ are carried by $q_{\mu_{m}}$.
This results in the simplest possible permutations of metrics and momenta in
$\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}$. Combining terms, one gets

$\displaystyle\Delta\tilde{T}^{A}_{1\mu\nu}$ $\displaystyle=-4i\
\sum_{k=1}^{\infty}\left[-g_{\mu\nu}C_{1}^{2k}A_{\tau=2}^{2k}\right]\times\frac{2^{2k}}{(Q^{2})^{2k}}\times\left(\prod_{m=1}^{2k}q_{\mu_{m}}\right)\times\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}+\mathcal{O}(\tau>2),$
(A.70)

where the production factor $(\prod_{m=1}^{2k}q_{\mu_{m}})$ has (re)absorbed
$q_{\mu_{1}}$ and $q_{\mu_{2}}$.

The contraction of $\prod_{m=1}^{2k}q_{\mu_{m}}$ and the momentum factor
$\tilde{\Pi}$ combines all possible permutations of $2k$ instances of
$q_{\mu_{m}}$ with $j$ metrics $g^{\mu_{m}\mu_{n}}$ in $\\{g\dots g\\}$ and
$(2k-2j)$ momenta $p_{A}^{\mu_{m}}$ in $\\{p_{A}\dots p_{A}\\}$. This results
in $(2k-2j)$ products of $(q\cdot p_{A})$. The remaining $(2k-0)-(2k-2j)=2j$
instances of $q_{\mu_{m}}$ then contract with the $2j$ indices from $j$
metrics. As no instance of $g^{\mu_{m}\mu_{n}}$ or $p_{A}^{\mu_{m}}$ have been
pulled from $\tilde{\Pi}$, its permutation factor $\eta$ is the same as in Eq.
(A.64). The outcome is

$\displaystyle\left(\prod_{m=1}^{2k}q_{\mu_{m}}\right)\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}$
$\displaystyle=\sum_{j=0}^{k}(-1)^{j}\ \frac{(2k-j)!}{2^{j}(2k)!}\
\frac{1}{2^{j}}\frac{(2k)!}{j!(2k-2j)!}\ $
$\displaystyle\qquad\times\qquad(q^{2})^{j}(q\cdot
p_{A})^{(2k-2j)}(p_{A}^{2})^{j}$ (A.71)
$\displaystyle=\frac{(Q^{2})^{2k}}{2^{2k}}\ \sum_{j=0}^{k}\
\frac{(2k-j)!}{j!(2k-2j)!}\ \left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\
x_{A}^{-(2k-2j)}\ ,$ (A.72)

where $x_{A}=Q^{2}/(2p_{A}\cdot q)$ is the Bjorken scaling variable and
$p_{A}^{2}=M_{A}^{2}$. Altogether,

$\displaystyle\Delta\tilde{T}^{A}_{1\mu\nu}\quad$ $\displaystyle=\quad-
g_{\mu\nu}\ \times\ (-4i)\
\sum_{k=1}^{\infty}\left[C_{1}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=0}^{k}\
\frac{(2k-j)!}{j!(2k-2j)!}\ \left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\
x_{A}^{-(2k-2j)}\ $ $\displaystyle+\quad\mathcal{O}(\tau>2).$ (A.73)

We stress that that the quantity to the right of $g_{\mu\nu}$ here is not
$\Delta\tilde{T}^{A}_{1}$ in Eq. (A.69). In general, the coefficients
$\Delta\tilde{T}^{A}_{i}$ receive contributions from more than one
$\Delta\tilde{T}^{A}_{\iota\ \mu\nu}$.

(ii) $\Delta\tilde{T}^{A}_{4\mu\nu}$:

The next term we construct is $\Delta\tilde{T}^{A}_{4\mu\nu}$ for $M_{A}\neq
0$, which has an identical tensor structure of contracted covariant indices as
$\Delta\tilde{T}^{A}_{1\mu\nu}$. In other words, like
$\Delta\tilde{T}^{A}_{1\mu\nu}$, all the lowered internal indices $\mu_{m}$
are carried exclusively by factors of $q_{\mu_{m}}$. The result is then,

$\displaystyle\Delta\tilde{T}^{A}_{4\mu\nu}\quad$
$\displaystyle=\quad\frac{q_{\mu}q_{\nu}}{Q^{2}}\ \times\ (-8i)\
\sum_{k=1}^{\infty}\left[C_{4}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=0}^{k}\
\frac{(2k-j)!}{j!(2k-2j)!}\ \left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\
x_{A}^{-(2k-2j)}\ $ $\displaystyle+\quad\mathcal{O}(\tau>2).$ (A.74)

(iii) $\Delta\tilde{T}^{A}_{3\mu\nu}$:

Moving onto $\Delta\tilde{T}^{A}_{3\mu\nu}$ for $M_{A}\neq 0$, exactly one
covariant index from the collection $\mu_{m=1}\dots\mu_{m=2k}$ is carried by
the metric $g_{\mu_{1}}^{\alpha}$. All other covariant instances of $\mu_{m}$
are carried by $(2k-1)$ instances of $q_{\mu_{m}}$. This difference among the
lowered indices introduces a complication that was absent in the previous
cases but nevertheless allows us to write

$\displaystyle\Delta\tilde{T}^{A}_{3\mu\nu}$ $\displaystyle=-2i\
\sum_{k=1}^{\infty}\left[-i\epsilon_{\mu\nu\alpha\beta}\ q^{\beta}\
C_{3}^{2k}A_{\tau=2}^{2k}\right]\ \times\frac{2^{2k}}{(Q^{2})^{2k}}\times
g^{\alpha}_{\mu_{1}}\times\left(\prod_{m=2}^{2k}q_{\mu_{m}}\right)\times\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}\
$ $\displaystyle+\quad\mathcal{O}(\tau>2)\ .$ (A.75)

In principle, $g_{\mu_{1}}^{\alpha}$ can contract with a metric from the
collection $\\{g\dots g\\}$ when it contains the index $\mu_{m=1}$. This
extracts a factor of $q^{\alpha}$ out of the production factor
$(\prod_{m=2}^{2k}q_{\mu_{m}})$, and hence potential permutations, through the
chain $q^{\alpha}=g_{\mu_{1}}^{\alpha}q_{\mu_{n}}g^{\mu_{n}\mu_{1}}$. However,
this generates a term proportional to
$\epsilon_{\mu\nu\alpha\beta}q^{\beta}q^{\alpha}$, which is zero due to the
antisymmetric tensor.

It is also possible for the metric $g_{\mu_{1}}^{\alpha}$ to contract with a
momentum factor from the collection $\\{p_{A}\dots p_{A}\\}$. This extracts a
factor of $p_{A}^{\alpha}=g_{\mu_{1}}^{\alpha}p_{A}^{\mu_{1}}$, and reduces
the possible number of permutations in the momentum factor $\tilde{\Pi}$. The
contractions within $\Delta\tilde{T}^{A}_{3\mu\nu}$ then involve $(2k-1)$
instances of $q_{\mu_{m}}$ contracting with $(2k-2j-1)$ instances of
$p_{A}^{\mu_{m}}$, to make $(2k-2j-1)$ powers of $(q\cdot p_{A})$. The
remaining $(2k-1)-(2k-2j-1)=2j$ instances of $q_{\mu_{m}}$ then contract with
the remaining $j$ metrics. Finally, since the number of available
$p_{A}^{\mu_{m}}$ has shifted (by one unit), the multiplicity factor in Eq.
(A.64) must be updated accordingly. With the updated $\eta$ factor, the
outcome is

$\displaystyle
g_{\mu_{1}}^{\alpha}\left(\prod_{m=2}^{2k}q_{\mu_{m}}\right)\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}\quad=\quad$
$\displaystyle p_{A}^{\alpha}\ \sum_{j=0}^{k-1}(-1)^{j}\
\frac{(2k-j)!}{2^{j}(2k)!}\ \frac{1}{2^{j}}\frac{(2k-1)!}{j!(2k-2j-1)!}\ $
$\displaystyle\qquad\times\ (q^{2})^{j}\ (q\cdot p_{A})^{(2k-2j-1)}\
(p_{A}^{2})^{j}$ (A.76) $\displaystyle=\frac{2p_{A}^{\alpha}}{Q^{2}}\
\frac{(Q^{2})^{2k}}{2^{2k}}\ $ $\displaystyle\sum_{j=0}^{k-1}\
\frac{(2k-j)!(2k-1)!}{(2k)!j!(2k-2j-1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j-1)}\ .$ (A.77)

Note that the summation over $j$ extends only to $j=k-1$ since there are fewer
allowed permutations; this can be surmised from the denominator factor
$1/(2k-2j-1)!$, which would reduce to $1/(-1)!$ were $j=k$. After assembling
the different terms, we obtain

$\displaystyle\Delta\tilde{T}^{A}_{3\mu\nu}\quad$
$\displaystyle=\quad-i\epsilon_{\mu\nu\alpha\beta}\frac{p_{A}^{\alpha}q^{\beta}}{Q^{2}}\
\times\ (-4i)\ \sum_{k=1}^{\infty}\ \left[C_{3}^{2k}A_{\tau=2}^{2k}\right]\ $
$\displaystyle\times\ \sum_{j=0}^{k-1}\
\frac{(2k-j)!(2k-1)!}{(2k)!j!(2k-2j-1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\
x_{A}^{-(2k-2j-1)}\quad+\quad\mathcal{O}(\tau>2)\ .$ (A.78)

Unlike the other $\Delta\tilde{T}^{A}_{i}$ in Eq. (A.69), we can identify the
quantity to the right of the antisymmetric tensor
$\epsilon_{\mu\nu\alpha\beta}$ as $\Delta\tilde{T}^{A}_{3}$ since there is a
one-to-one correspondence.

(iv) $\Delta\tilde{T}^{A}_{5\mu\nu}$:

The steps to determine $\Delta\tilde{T}^{A}_{5\mu\nu}$ (and
$\Delta\tilde{T}^{A}_{6\mu\nu}$) for $M_{A}\neq 0$ closely follow those of
$\Delta\tilde{T}^{A}_{3\mu\nu}$ due to the presence of only one metric
carrying a lowered index from the collection $\mu_{m=1}\dots\mu_{m=2k}$. The
difference, however, is the absence of an antisymmetric tensor. The
contractions of $g_{\mu\mu_{1}}$ and $g_{\nu\mu_{1}}$ with the collections
$\\{g\dots g\\}$ and $\\{p_{A}\dots p_{A}\\}$ will thus generate two sets of
terms each. To obtain both sets of terms, we first write

$\displaystyle\Delta\tilde{T}^{A}_{5\mu\nu}$ $\displaystyle=-4i\
\sum_{k=1}^{\infty}\
\left[(g_{\mu\mu_{1}}q_{\nu}+g_{\nu\mu_{1}}q_{\mu})C_{5}^{2k}A_{\tau=2}^{2k}\right]\
\times\frac{2^{2k}}{(Q^{2})^{2k}}\times\left(\prod_{m=2}^{2k}q_{\mu_{m}}\right)\times\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}\
$ $\displaystyle+\quad\mathcal{O}(\tau>2)\ .$ (A.79)

For the case of the metric $g_{\alpha\mu_{1}}$ contracting with $\\{p_{A}\dots
p_{A}\\}$, we have from Eq. (A.77)

$\displaystyle
g_{\alpha\mu_{1}}\left(\prod_{m=2}^{2k}q_{\mu_{m}}\right)\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}$
$\displaystyle\Bigg{|}_{\mu_{1}\in\\{p_{A}\dots p_{A}\\}}=$
$\displaystyle\frac{2p_{A\alpha}}{Q^{2}}\ \frac{(Q^{2})^{2k}}{2^{2k}}\ $
$\displaystyle\sum_{j=0}^{k-1}\ \frac{(2k-j)!(2k-1)!}{(2k)!j!(2k-2j-1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j-1)}\ .$ (A.80)

For the second case, $g_{\alpha\mu_{1}}$ contracts with a metric in $\\{g\dots
g\\}$ and extracts a factor of
$q_{\alpha}=g_{\alpha\mu_{1}}q_{\mu_{n}}g^{\mu_{n}\mu_{1}}$ out of the
production factor $(\prod_{m=2}^{2k}q_{\mu_{m}})$. The precise factor of
$q_{\mu_{n}}$ that is extracted can be any one of the $(2k-1)$ elements in the
product factor, implying an additional multiplicity factor of this size [54].
The remaining $(2k-2)$ instances of $q_{\mu_{m}}$ then contract with $(2k-2j)$
instances of $p_{A}^{\mu_{m}}$ to make $(2k-2j)$ powers of $(q\cdot p_{A})$.
The remaining $(2k-2)-(2k-2j)=2j-2$ factors of $q_{\mu_{m}}$ then contract
with the $2(j-1)$ indices of the remaining $(j-1)$ metrics. After updating
$\eta$ and noting that the sum now starts at $j=1$, the outcome is

$\displaystyle
g_{\alpha\mu_{1}}\left(\prod_{m=2}^{2k}q_{\mu_{m}}\right)\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}$
$\displaystyle\Bigg{|}_{\mu_{1}\in\\{g\dots g\\}}=q_{\alpha}\
\sum_{j=1}^{k}(-1)^{j}\ \frac{(2k-j)!}{2^{j}(2k)!}\
\frac{1}{2^{j-1}}\frac{(2k-2)!}{(j-1)!(2k-2j)!}\ \times(2k-1)$
$\displaystyle\qquad\times\ (q^{2})^{(j-1)}\ (q\cdot p_{A})^{(2k-2j)}\
(p_{A}^{2})^{j}$ (A.81) $\displaystyle=-\frac{2q_{\alpha}}{Q^{2}}\
\frac{(Q^{2})^{2k}}{2^{2k}}\ \sum_{j=1}^{k}\ $
$\displaystyle\frac{(2k-j)!(2k-1)!}{(2k)!(j-1)!(2k-2j)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j)}\ .$ (A.82)

Assembling all terms for both $g_{\mu\mu_{1}}$ and $g_{\nu\mu_{1}}$, we obtain
the expression

$\displaystyle\Delta\tilde{T}^{A}_{5\mu\nu}$
$\displaystyle=\frac{(p_{A\mu}q_{\nu}+p_{A\nu}q_{\mu})}{Q^{2}}\ $
$\displaystyle\quad\times\ (-8i)\ \sum_{k=1}^{\infty}\
\left[C_{5}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=0}^{k-1}\
\frac{(2k-j)!(2k-1)!}{(2k)!j!(2k-2j-1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j-1)}$
$\displaystyle-\frac{(q_{\mu}q_{\nu}+q_{\nu}q_{\mu})}{Q^{2}}\ $
$\displaystyle\quad\times\ (-8i)\ \sum_{k=1}^{\infty}\
\left[C_{5}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=1}^{k}\
\frac{(2k-j)!(2k-1)!}{(2k)!(j-1)!(2k-2j)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j)}\ $
$\displaystyle+\quad\mathcal{O}(\tau>2)\ .$ (A.83)

Importantly, there are two tensor structures: The first is
$(p_{A\mu}q_{\nu}+p_{A\nu}q_{\mu})$, whose coefficient contributes to
$\Delta\tilde{T}^{A}_{5}$ in Eq. (A.69). The second is $q_{\mu}q_{\nu}$, whose
coefficient contributes to $\Delta\tilde{T}^{A}_{4}$ in Eq. (A.69). Note that
the second term can equally be written as $4(q_{\mu}q_{\nu})$; we do not (yet)
condense the result in order to emphasize that $g_{\mu\mu_{1}}$ and
$g_{\nu\mu_{1}}$ each generates two terms.

(v) $\Delta\tilde{T}^{A}_{6\mu\nu}$:

The next term, $\Delta\tilde{T}^{A}_{6\mu\nu}$ for $M_{A}\neq 0$, can be
obtained directly from $\Delta\tilde{T}^{A}_{5\mu\nu}$ by making the
replacement $q_{\mu}\to-q_{\mu}$ and $C_{5}\to C_{6}$. When making this
replacement, the second part of Eq. (A.83) vanishes since
$(q_{\mu}q_{\nu}+q_{\nu}q_{\mu})\to(q_{\mu}q_{\nu}-q_{\nu}q_{\mu})=0$. The
final result for $\Delta\tilde{T}^{A}_{6\mu\nu}$ is

$\displaystyle\Delta\tilde{T}^{A}_{6\mu\nu}=$
$\displaystyle\frac{(p_{A\mu}q_{\nu}-p_{A\nu}q_{\mu})}{Q^{2}}\
\times(-8i)\times\sum_{k=1}^{\infty}\ \left[C_{6}^{2k}A_{\tau=2}^{2k}\right]\
$ $\displaystyle\times\sum_{j=0}^{k-1}\
\frac{(2k-j)!(2k-1)!}{(2k)!j!(2k-2j-1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\
x_{A}^{-(2k-2j-1)}\quad+\quad\mathcal{O}(\tau>2)\ .$ (A.84)

(vi) $\Delta\tilde{T}^{A}_{2\mu\nu}$:

The final term $\Delta\tilde{T}^{A}_{2\mu\nu}$ for $M_{A}\neq 0$, which is
given by

$\displaystyle\Delta\tilde{T}^{A}_{2\mu\nu}$ $\displaystyle=(-2i)\
\sum_{k=2}^{\infty}\left[Q^{2}C_{2}^{2k}A_{\tau=2}^{2k}\right]\
\times\frac{2^{2k}}{(Q^{2})^{2k}}\times
g_{\mu\mu_{1}}g_{\nu\mu_{2}}\times\left(\prod_{m=3}^{2k}q_{\mu_{m}}\right)\times\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}\
$ $\displaystyle+\quad\mathcal{O}(\tau>2)\ .$ (A.85)

is the most nuanced because the covariant tensor structure involves two
metrics. That is to say, two lowered indices from the collection
$\mu_{m=1}\dots\mu_{m=2k}$ are carried by $g_{\mu\mu_{1}}$ and
$g_{\nu\mu_{2}}$. The contraction of these metrics with the collections
$\\{g\dots g\\}$ and $\\{p_{A}\dots p_{A}\\}$, in turn, generates five terms
with four distinct Lorentz structures. The five possible ways are:

  1. (a)

both $g_{\mu\mu_{1}}$ and $g_{\nu\mu_{2}}$ contract with momenta from
$\\{p_{A}\dots p_{A}\\}$ and extract $p_{A\mu}p_{A\nu}$;

  2. (b)

both $g_{\mu\mu_{1}}$ and $g_{\nu\mu_{2}}$ contract with different metrics
from $\\{g\dots g\\}$ and extract the quantity $q_{\mu}q_{\nu}$ from the
production factor $(\prod_{m=3}^{2k}q_{\mu_{m}})$;

  3. (c)

both $g_{\mu\mu_{1}}$ and $g_{\nu\mu_{2}}$ contract with the same metric from
$\\{g\dots g\\}$ and extract $g_{\mu\nu}$;

  4. (d)

$g_{\mu\mu_{1}}$ contracts with $\\{p_{A}\dots p_{A}\\}$ and $g_{\nu\mu_{2}}$
with $\\{g\dots g\\}$ to extract $(p_{A\mu}q_{\nu})$;

  5. (e)

same as (d) but with the index exchange $\mu\leftrightarrow\nu$ to extract the
quantity $(p_{A\nu}q_{\mu})$.

In case (a), contracting the metrics with the collection $\\{p_{A}\dots
p_{A}\\}$ leaves $(2k-2j-2)$ instances of $p_{A}^{\mu_{m}}$ to contract with
$(2k-2)$ instances of $q_{\mu_{m}}$. The remaining $(2k-2)-(2k-2j-2)=2j$
instances of $q_{\mu_{m}}$ then contracts with the $2j$ indices of the $j$
metrics in the collection $\\{g\dots g\\}$. Updating the permutation factor,
we get

$\displaystyle
g_{\mu\mu_{1}}g_{\nu\mu_{2}}\left(\prod_{m=3}^{2k}q_{\mu_{m}}\right)\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}$
$\displaystyle\Bigg{|}_{\mu_{1},\mu_{2}\in\\{p_{A}\dots
p_{A}\\}}=p_{A\mu}p_{A\nu}\ \sum_{j=0}^{k-1}\ \frac{(2k-j)!}{2^{j}(2k)!}\
\frac{1}{2^{j}}\frac{(2k-2)!}{j!(2k-2j-2)!}\ $ $\displaystyle\qquad\times\
(-1)^{j}(q^{2})^{j}\ (q\cdot p_{A})^{(2k-2j-2)}\ (p_{A}^{2})^{j}$ (A.86)
$\displaystyle=\frac{4p_{A\mu}p_{A\nu}}{Q^{4}}\ \frac{(Q^{2})^{2k}}{2^{2k}}\
\sum_{j=0}^{k-1}\ $ $\displaystyle\frac{(2k-j)!(2k-2)!}{(2k)!j!(2k-2j-2)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j-2)}\ .$ (A.87)

Note that the summation over $j$ extends only to $j=k-1$ since there are fewer
allowed permutations; this can be inferred from the denominator factor
$1/(2k-2j-2)!$.

In case (b), contracting $g_{\mu\mu_{1}}$ and $g_{\nu\mu_{2}}$ with different
metrics from the collection $\\{g\dots g\\}$ leaves $(2k-4)$ instances of
$q_{\mu_{m}}$ to contract with $(2k-2j)$ instances of $p_{A}^{\mu_{m}}$. The
precise $q_{\mu_{m}}$ that are extracted from the product factor
$\left(\prod_{m=3}^{2k}q_{\mu_{m}}\right)$ can be any two of its $(2k-2)$
elements, implying the additional multiplicity factor $(2k-2)\times(2k-3)$.
The remaining $(2k-4)-(2k-2j)=(2j-4)$ factors of $q_{\mu_{m}}$ then contract
with the $(j-2)$ metrics remaining in $\\{g\dots g\\}$. Updating the $\eta$
factor and noting the summation over $j$ starts at $j=2$, we get

$\displaystyle g_{\mu\mu_{1}}g_{\nu\mu_{2}}$
$\displaystyle\left(\prod_{m=3}^{2k}q_{\mu_{m}}\right)\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}\Bigg{|}_{\mu_{1},\mu_{2}\in\\{g\dots
g\\}}$ $\displaystyle=q_{\mu}q_{\nu}\ \sum_{j=2}^{k}\
\frac{(2k-j)!}{2^{j}(2k)!}\ \frac{1}{2^{j-2}}\frac{(2k-4)!}{(j-2)!(2k-2j)!}\
\times\ (2k-3)(2k-2)\ $ $\displaystyle\qquad\times\ (-1)^{j}(q^{2})^{(j-2)}\
(q\cdot p_{A})^{(2k-2j)}\ (p_{A}^{2})^{j}$ (A.88)
$\displaystyle=\frac{4q_{\mu}q_{\nu}}{Q^{4}}\ \frac{(Q^{2})^{2k}}{2^{2k}}\
\sum_{j=2}^{k}\ \frac{(2k-j)!(2k-2)!}{(2k)!(j-2)!(2k-2j)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j)}\ .$ (A.89)

In case (c), contracting $g_{\mu\mu_{1}}$ and $g_{\nu\mu_{2}}$ with the same
metric from the collection $\\{g\dots g\\}$ leaves $(2k-2)$ instances of
$q_{\mu_{m}}$ to contract with all $(2k-2j)$ instances of $p_{A}^{\mu_{m}}$.
The $(2k-2)-(2k-2j)=(2j-2)$ uncontracted instances of $q_{\mu_{m}}$ are then
matched with the $(j-1)$ metrics that are left in $\\{g\dots g\\}$. Updating
$\eta$ and the summation over $j$, which now starts from $j=1$, we get

$\displaystyle g_{\mu\mu_{1}}g_{\nu\mu_{2}}\ $
$\displaystyle\left(\prod_{m=3}^{2k}q_{\mu_{m}}\right)\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}\Bigg{|}_{g^{\mu_{1}\mu_{2}}\in\\{g\dots
g\\}}\ =\ g_{\mu\nu}\ \sum_{j=1}^{k}\ \frac{(2k-j)!}{2^{j}(2k)!}\ $
$\displaystyle\qquad\times\ \frac{1}{2^{j-1}}\frac{(2k-2)!}{(j-1)!(2k-2j)!}\
(-1)^{j}(q^{2})^{(j-1)}\ (q\cdot p_{A})^{(2k-2j)}\ (p_{A}^{2})^{j}$ (A.90)
$\displaystyle=\ -\frac{2g_{\mu\nu}}{Q^{2}}\ \frac{(Q^{2})^{2k}}{2^{2k}}\
\sum_{j=1}^{k}\ \frac{(2k-j)!(2k-2)!}{(2k)!(j-1)!(2k-2j)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j)}\ .$ (A.91)

In case (d), $g_{\mu\mu_{1}}$ contracts with $\\{p_{A}\dots p_{A}\\}$, leaving
$(2k-2j-1)$ instances of $p_{A}^{\mu_{m}}$, while $g_{\nu\mu_{2}}$ contracts
with $\\{g\dots g\\}$, and eventually $(\prod_{\mu=3}^{2k}q_{\mu_{n}})$,
leaving a total of $(2k-3)$ instances of $q_{\mu_{m}}$. The result is
$(2k-2j-1)$ powers of $(q\cdot p_{A})$ and $(2k-3)-(2k-2j-1)=(2j-2)$ factors
of $q_{\mu_{m}}$ to contract with $(j-1)$ metrics. The $q_{\mu_{m}}$ that is
extracted from the product factor can be any one of its initial $(2k-2)$
elements, implying an additional multiplicity factor of this size. Updating
$\eta$ and noting the summation over $j$ runs from $j=1$ to $j=k-1$, we obtain

$\displaystyle
g_{\mu\mu_{1}}g_{\nu\mu_{2}}\left(\prod_{m=3}^{2k}q_{\mu_{m}}\right)\tilde{\Pi}^{\mu_{1}\dots\mu_{2k}}\Bigg{|}^{\mu_{1}\in\\{p_{A}\dots
p_{A}\\}}_{\mu_{2}\in\\{g\dots g\\}}=p_{A\mu}q_{\nu}\ \sum_{j=1}^{k-1}\
\frac{(2k-j)!}{2^{j}(2k)!}\ $ $\displaystyle\qquad\times\
\frac{1}{2^{j-1}}\frac{(2k-3)!}{(j-1)!(2k-2j-1)!}\ \times(2k-2)\
(-1)^{j}(q^{2})^{(j-1)}\ (q\cdot p_{A})^{(2k-2j-1)}\ (p_{A}^{2})^{j}$ (A.92)
$\displaystyle=-\frac{4p_{A\mu}q_{\nu}}{Q^{4}}\ \frac{(Q^{2})^{2k}}{2^{2k}}\
\sum_{j=1}^{k-1}\ \frac{(2k-j)!(2k-2)!}{(2k)!(j-1)!(2k-2j-1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j-1)}\ .$ (A.93)

Case (e) follows from case (d) but with $\mu\leftrightarrow\nu$ exchanged.

After assembling all five cases and regrouping, we find

$\displaystyle\Delta\tilde{T}^{A}_{2\mu\nu}$
$\displaystyle=\frac{2p_{A\mu}p_{A\nu}}{Q^{2}}\ \times(-4i)\
\sum_{k=1}^{\infty}\left[C_{2}^{2k}A_{\tau=2}^{2k}\right]\ $
$\displaystyle\quad~{}\times\sum_{j=0}^{k-1}\
\frac{(2k-j)!(2k-2)!}{(2k)!j!(2k-2j-2)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j-2)}$ $\displaystyle-
g_{\mu\nu}\ $ $\displaystyle\times(-4i)\
\sum_{k=1}^{\infty}\left[C_{2}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=1}^{k}\
\frac{(2k-j)!(2k-2)!}{(2k)!(j-1)!(2k-2j)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j)}$
$\displaystyle+\frac{2q_{\mu}q_{\nu}}{Q^{2}}\ $ $\displaystyle\times(-4i)\
\sum_{k=2}^{\infty}\left[C_{2}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=2}^{k}\
\frac{(2k-j)!(2k-2)!}{(2k)!(j-2)!(2k-2j)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j)}$
$\displaystyle-\frac{2(p_{A\mu}q_{\nu}+p_{A\nu}q_{\mu})}{Q^{2}}\ \times(-4i)\
\sum_{k=2}^{\infty}\left[C_{2}^{2k}A_{\tau=2}^{2k}\right]\ $
$\displaystyle\quad~{}\times\sum_{j=1}^{k-1}\
\frac{(2k-j)!(2k-2)!}{(2k)!(j-1)!(2k-2j-1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j-1)}\ +\
\mathcal{O}(\tau>2)\ .$ (A.94)

The result implies that $\Delta\tilde{T}_{2\mu\nu}^{A}$, and hence the Wilson
coefficient $C_{2}^{2k}$, contributes to $\Delta T_{1}^{A}$, $\Delta
T_{2}^{A}$, $\Delta T_{4}^{A}$, and $\Delta T_{5}^{A}$ in Eq. (A.69). The
absence of contributions to $\Delta T_{3}^{A}$ and $\Delta T_{6}^{A}$, which
are asymmetric under $\mu\leftrightarrow\nu$ exchange, can be traced to the
tensor coefficient multiplying $C_{2}^{2k}$ in
$c_{\mu\nu\mu_{1},\dots,\mu_{2k}}^{\tau=2,\iota}$, which is symmetric under
$\mu\leftrightarrow\nu$ exchange.

### A.4 Nuclear structure functions from the OPE II: structure-function mixing

Using the expressions for $\Delta\tilde{T}_{\iota\ \mu\nu}^{A}$, we finally
obtain the quantities $\Delta\tilde{T}_{i}^{A}$ in Eq. (A.69). For
$\Delta\tilde{T}^{A}_{1}$ and $M_{A}\neq 0$, we collect like-terms from
$\Delta\tilde{T}^{A}_{1\mu\nu}$ and $\Delta\tilde{T}^{A}_{2\mu\nu}$ to obtain

$\displaystyle\Delta\tilde{T}^{A}_{1}=(-4i)\ $ $\displaystyle\
\sum_{k=1}^{\infty}\ \left[C_{1}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=0}^{k}\
\frac{(2k-j)!(2k)!}{(2k)!j!(2k-2j)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j)}\ $
$\displaystyle+(-4i)\ $ $\displaystyle\
\sum_{k=1}^{\infty}\left[C_{2}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=1}^{k}\
\frac{(2k-j)!(2k-2)!}{(2k)!(j-1)!(2k-2j)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j)}\ +\
\mathcal{O}(\tau>2)\ $ (A.95) $\displaystyle=(-4i)\ $ $\displaystyle\
\sum_{l=0}^{\infty}\ x_{A}^{-(2l+1)}\ \sum_{j=0}^{\infty}\
\frac{(2l+j+1)!}{j!(2l+1)!}\ \left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ $
$\displaystyle\ \quad\ \times\
\left[\left(C_{1}^{2l+2j+1}A_{\tau=2}^{2l+2j+1}\right)+\frac{j\left(C_{2}^{2l+2j+1}A_{\tau=2}^{2l+2j+1}\right)}{(2l+2j+1)(2l+2j)}\right]\
+\ \mathcal{O}(\tau>2)\ .$ (A.96)

To reach Eq. (A.96), a factor of $1=(2k)(2k-1)j/(2k)(2k-1)j$ is introduced
into the $C_{2}$ term to complete the factorials. The summation in the $C_{2}$
term can then be extended to include $j=0$ since the contribution is zero. We
also re-index the summation over $k$ into a summation over $2l=2k-2j-1$. After
this shift it becomes manifest that the contribution from the $C_{2}$ Wilson
coefficient vanishes when the summation over $j$ is truncated at $j=0$. It
also becomes clear that structure-function mixing is due to TMCs since
neglecting terms where $j>0$ is equivalent to taking the $(M_{A}^{2}/Q^{2})\to
0$ limit.

The coefficient $\Delta\tilde{T}^{A}_{2}$ is sourced entirely from
$\Delta\tilde{T}^{A}_{\mu\nu 2}$, and is given by

$\displaystyle\Delta\tilde{T}^{A}_{2}$ $\displaystyle=(-4i)\
\frac{2M_{A}^{2}}{Q^{2}}\
\sum_{k=1}^{\infty}\left[C_{2}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=0}^{k-1}\
\frac{(2k-j)!(2k-2)!}{(2k)!j!(2k-2j-2)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j-2)}\ $
$\displaystyle+\quad\mathcal{O}(\tau>2)\ $ (A.97) $\displaystyle=(-4i)\
\frac{2M_{A}^{2}}{Q^{2}}\ \sum_{l=0}^{\infty}\ x_{A}^{-(2l-1)}\
\sum_{j=0}^{\infty}\ \frac{(2l+j+1)!}{j!(2l+1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}(2l+1)(2l)\
\frac{\left(C_{2}^{(2l+2j+1)}A_{\tau=2}^{(2l+2j+1)}\right)}{(2l+2j+1)(2l+2j)}\
$ $\displaystyle+\quad\mathcal{O}(\tau>2)\ .$ (A.98)

In reaching Eq. (A.98) we again completed the factorials and set $2l=2k-2j-1$.

Similarly, $\Delta\tilde{T}^{A}_{3}$ for $M_{A}\neq 0$ is sourced entirely
from $\Delta\tilde{T}^{A}_{\mu\nu 3}$. It is given by

$\displaystyle\Delta\tilde{T}^{A}_{3}=$ $\displaystyle(-4i)\
\frac{M_{A}^{2}}{Q^{2}}\ \sum_{k=1}^{\infty}\
\left[C_{3}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=0}^{k-1}\
\frac{(2k-j)!(2k-1)!}{(2k)!j!(2k-2j-1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j-1)}\ $
$\displaystyle+\quad\mathcal{O}(\tau>2)\ $ (A.99) $\displaystyle=$
$\displaystyle(-4i)\ \frac{M_{A}^{2}}{Q^{2}}\ \sum_{l=0}^{\infty}\
x_{A}^{-(2l-1)}\ \sum_{j=0}^{\infty}\ \frac{(2l+j)!}{j!(2l)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ (2l)\
\frac{\left(C_{3}^{(2l+2j)}A_{\tau=2}^{(2l+2j)}\right)}{(2l+2j)}\ $
$\displaystyle+\quad\mathcal{O}(\tau>2)\ ,$ (A.100)

where the index $l=k-j$ was used after completing the factorials.

For $\Delta\tilde{T}^{A}_{4}$ and $M_{A}\neq 0$, we collect terms from
$\Delta\tilde{T}^{A}_{\mu\nu 4}$, $\Delta\tilde{T}^{A}_{\mu\nu 2}$, and
$\Delta\tilde{T}^{A}_{\mu\nu 6}$. The result is

$\displaystyle\Delta\tilde{T}^{A}_{4}\quad=(-4i)\ \frac{2M_{A}^{2}}{Q^{2}}\
\sum_{k=1}^{\infty}\ \left[C_{4}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=0}^{k}\
\frac{(2k-j)!(2k)!}{(2k)!j!(2k-2j)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j)}\ .$
$\displaystyle+$ $\displaystyle(-4i)\ \frac{2M_{A}^{2}}{Q^{2}}\
\sum_{k=2}^{\infty}\left[C_{2}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=2}^{k}\
\frac{(2k-j)!{(2k-2)!}}{(2k)!(j-2)!(2k-2j)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j)}\ $ $\displaystyle-$
$\displaystyle(-4i)\ \frac{4M_{A}^{2}}{Q^{2}}\ \sum_{k=1}^{\infty}\
\left[C_{5}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=1}^{k}\
\frac{(2k-j)!{(2k-1)!}}{(2k)!(j-1)!(2k-2j)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j)}\ $
$\displaystyle+\mathcal{O}(\tau>2)\ $ (A.101) $\displaystyle=$
$\displaystyle(-4i)\ \frac{2M_{A}^{2}}{Q^{2}}\
\sum_{l=0}^{\infty}x_{A}^{-(2l+1)}\
\sum_{j=0}^{\infty}\frac{(2l+j+1)!}{j!(2l+1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ $
$\displaystyle\times\left[\left(C_{4}^{(2l+2j+1)}A_{\tau=2}^{(2l+2j+1)}\right)+\
\frac{j\ (j-1)\
\left(C_{2}^{(2l+2j+1)}A_{\tau=2}^{(2l+2j+1)}\right)}{(2l+2j+1)(2l+2j)}-\
\frac{2j\ \left(C_{5}^{(2l+2j+1)}A_{\tau=2}^{(2l+2j+1)}\right)}{(2l+2j+1)}\
\right]\ $ $\displaystyle+\mathcal{O}(\tau>2)\ .$ (A.102)

We complete the factorials using factors of $1=(2k)j/(2j)j$, etc., and use
$2l=2k-2j-1$ to re-index the summation over $k$. The summations over $j$ can
also be extended to $j=0$ since the additional terms are proportional to $j$
or $j(j-1)$, i.e., are zero at $j=0$ and/or $j=1$, and hence do not
contribute.

The coefficient $\Delta\tilde{T}^{A}_{5}$ for $M_{A}\neq 0$ is generated by
$\Delta\tilde{T}^{A}_{2\mu\nu}$ and $\Delta\tilde{T}^{A}_{5\mu\nu}$. The
result is

$\displaystyle\Delta\tilde{T}^{A}_{5}$ $\displaystyle=(-4i)\
\frac{2M_{A}^{2}}{Q^{2}}\ \sum_{k=1}^{\infty}\
\left[C_{5}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=0}^{k-1}\
\frac{(2k-j)!(2k-1)!}{(2k)!j!(2k-2j-1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j-1)}$
$\displaystyle-(-4i)\ \frac{2M_{A}^{2}}{Q^{2}}\
\sum_{k=2}^{\infty}\left[C_{2}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=1}^{k-1}\
\frac{(2k-j)!{(2k-2)!}}{(2k)!(j-1)!(2k-2j-1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j-1)}\ $
$\displaystyle+\quad\mathcal{O}(\tau>2)\ $ (A.103) $\displaystyle=(-4i)\
\frac{2M_{A}^{2}}{Q^{2}}\ \sum_{l=0}^{\infty}x_{A}^{-2l}\
\sum_{j=0}^{\infty}\frac{(2l+j+1)!}{j!(2l+1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ $
$\displaystyle\times\left[\frac{(2l+1)\
\left(C_{5}^{(2l+2j+1)}A_{\tau=2}^{(2l+2j+1)}\right)}{(2l+2j+1)}\ -\frac{j\
(2l+1)\
\left(C_{2}^{(2l+2j+1)}A_{\tau=2}^{(2l+2j+1)}\right)}{(2l+2j+1)(2l+2j)}\
\right]$ $\displaystyle\ +\mathcal{O}(\tau>2)\ ,$ (A.104)

where the index reassignment is $2l=2k-2j-1$.

Finally, the coefficient $\Delta\tilde{T}^{A}_{6}$ is generated solely by
$\Delta\tilde{T}^{A}_{6\mu\nu}$. It is given by

$\displaystyle\Delta\tilde{T}^{A}_{6}=$ $\displaystyle(-4i)\
\frac{2M_{A}^{2}}{Q^{2}}\ \sum_{k=1}^{\infty}\
\left[C_{6}^{2k}A_{\tau=2}^{2k}\right]\ \sum_{j=0}^{k-1}\
\frac{(2k-j)!(2k-1)!}{(2k)!j!(2k-2j-1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ x_{A}^{-(2k-2j-1)}\ $
$\displaystyle+\quad\mathcal{O}(\tau>2)\ $ (A.105) $\displaystyle=$
$\displaystyle(-4i)\ \frac{2M_{A}^{2}}{Q^{2}}\ \sum_{l=0}^{\infty}\
x_{A}^{-(2l-1)}\ \sum_{j=0}^{\infty}\ \frac{(2l+j)!}{j!(2l)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ (2l)\
\frac{\left(C_{6}^{(2l+2j)}A_{\tau=2}^{(2l+2j)}\right)}{(2l+2j)}\ $
$\displaystyle+\quad\mathcal{O}(\tau>2)\ ,$ (A.106)

where the index $l=k-j$ was used after completing the factorials.

We briefly stress that the $(M_{A}^{2}/Q^{2})$ prefactor in all
$\Delta\tilde{T}^{A}_{i}$ is conventional. They exist to match the
normalization of $\tilde{T}^{A}_{i}$ in Eq. (A.69) and structure functions
$\tilde{W}^{A}_{i}$ in Eq. (A.31). The massless target limit should be
understood as truncating the summation over $j$ at $j=0$. In addition, the re-
indexing from $k$ to $l$ in the above expressions is not random. Different
choices of relabeling can be found in the literature and depend on the precise
interpretation of the quantities to the right of the $x_{A}^{-N}$ factor. Our
choices are motivated by the attempt to align the ratio of factorials into
binomial distributions, e.g., $\binom{k+j}{j}=(k+j)!/j!k!$.

### A.5 Nuclear structure functions from the OPE III: massless and massive
targets

### Massless nuclear targets

As a final preliminary step to obtaining TMCs at leading twist, we establish
the connection between the effective Wilson coefficients $C_{\iota}$, the
reduced matrix element $A_{\tau=2}$, and structure functions in the limit that
$(M_{A}^{2}/Q^{2})\to 0$; in this limit, the TMCs will not be present, and we
label this result as “No TMC” in Eq.(A.107). For each
$\Delta\tilde{T}^{A}_{i}$ above, truncating terms with $j>0$ and using the
Taylor expansion of $\Delta\tilde{T}^{A}_{i}$ from Eq. (A.61) gives the
following:

$\displaystyle\Delta\tilde{T}^{A}_{1}\Bigg{|}_{j=0}$
$\displaystyle=-4i\sum_{l=0}^{\infty}\
x_{A}^{-(2l+1)}\left(C_{1}^{2l+1}A_{\tau=2}^{2l+1}\right)+\mathcal{O}(\tau>2)=\sum_{l=0}^{\infty}\
\omega_{A}^{2l+1}\tilde{F}_{1}^{A(2l+1)}\Bigg{|}_{\rm No~{}TMC},$ (A.107a)
$\displaystyle\left(\frac{Q^{2}}{2x_{A}M_{A}^{2}}\right)\
\Delta\tilde{T}^{A}_{2}\Bigg{|}_{j=0}$ $\displaystyle=-4i\sum_{l=0}^{\infty}\
x_{A}^{-2l}\ \left(C_{2}^{2l+1}A_{\tau=2}^{2l+1}\right)\
+\mathcal{O}(\tau>2)=\sum_{l=0}^{\infty}\ \omega_{A}^{2l}\
\tilde{F}_{2}^{A(2l)}\Bigg{|}_{\rm No~{}TMC},$ (A.107b)
$\displaystyle\left(\frac{Q^{2}}{x_{A}M_{A}^{2}}\right)\
\Delta\tilde{T}^{A}_{3}\Bigg{|}_{j=0}$ $\displaystyle=-4i\sum_{l=0}^{\infty}\
x_{A}^{-2l}\
\left(C_{3}^{2l}A_{\tau=2}^{2l}\right)+\mathcal{O}(\tau>2)=\sum_{l=0}^{\infty}\
\omega_{A}^{2l}\ \tilde{F}_{3}^{A(2l)}\Bigg{|}_{\rm No~{}TMC},$ (A.107c)
$\displaystyle\left(\frac{Q^{2}}{2M_{A}^{2}}\right)\Delta\tilde{T}^{A}_{4}\Bigg{|}_{j=0}$
$\displaystyle=-4i\sum_{l=0}^{\infty}\
x_{A}^{-(2l+1)}\left(C_{4}^{2l+1}A_{\tau=2}^{2l+1}\right)+\mathcal{O}(\tau>2)=\sum_{l=0}^{\infty}\
\omega_{A}^{2l+1}\tilde{F}_{4}^{A(2l+1)}\Bigg{|}_{\rm No~{}TMC},$ (A.107d)
$\displaystyle\left(\frac{Q^{2}}{2x_{A}M_{A}^{2}}\right)\
\Delta\tilde{T}^{A}_{5}\Bigg{|}_{j=0}$ $\displaystyle=-4i\sum_{l=0}^{\infty}\
x_{A}^{-(2l+1)}\left(C_{5}^{2l+1}A_{\tau=2}^{2l+1}\right)+\mathcal{O}(\tau>2)=\sum_{l=0}^{\infty}\
\omega_{A}^{2l+1}\tilde{F}_{5}^{A(2l+1)}\Bigg{|}_{\rm No~{}TMC},$ (A.107e)
$\displaystyle\left(\frac{Q^{2}}{2x_{A}M_{A}^{2}}\right)\
\Delta\tilde{T}^{A}_{6}\Bigg{|}_{j=0}$ $\displaystyle=-4i\sum_{l=0}^{\infty}\
x_{A}^{-2l}\ \left(C_{6}^{2l}A_{\tau=2}^{2l}\right)\
+\mathcal{O}(\tau>2)=\sum_{l=0}^{\infty}\ \omega_{A}^{2l}\
\tilde{F}_{6}^{A(2l)}\Bigg{|}_{\rm No~{}TMC}\ .$ (A.107f)

The conclusion is that, in the massless target limit, one can identify the
leading power of the product of the $N^{\rm th}$ Wilson coefficient
$C_{i}^{N}$ (note the index shift from $\iota$ to $i$) and hadronic matrix
element $A_{\tau=2}^{N}$ as a Mellin moment of structure functions themselves.
This is summarized by

$\displaystyle\tilde{F}_{i}^{AN}\Bigg{|}_{\rm No~{}TMC}$
$\displaystyle=C_{i}^{N}A_{\tau=2}^{N}\ +\
\mathcal{O}(\tau>2)\quad\text{for}\quad i=1,3-6\ ,$ (A.108a)
$\displaystyle\tilde{F}_{2}^{A(N-1)}\Bigg{|}_{\rm No~{}TMC}$
$\displaystyle=C_{2}^{N}A_{\tau=2}^{N}\ +\ \mathcal{O}(\tau>2)\ .$ (A.108b)

### Massive nuclear targets

Given the relationship between the moments of $\tilde{F}_{i}^{A}\big{|}_{\rm
No~{}TMC}$ and the product $(C_{i}A_{\tau=2})$ in the massless target limit,
we are now in position to bootstrap expressions for $\tilde{F}_{i}^{A}$ with
target mass corrections. The key to this is comparing the expansion of
$\Delta\tilde{T}^{A}_{i}$ in Eq. (A.61) with the full expressions for
$\tilde{T}^{A}_{i}$ in Eqs. (A.96)-(A.106). One can identify the moments of
structure functions with TMCs as everything encapsulated by the summation over
$j$, i.e., the coefficients multiplying the $x_{A}^{-2l(\pm 1)}$ factor. Using
Eq. (A.108), the unknown/non-perturbative $(C_{i}A_{\tau=2})$ can then be
replaced by the measurable $\tilde{F}_{i}^{A}\big{|}_{\rm No~{}TMC}$.
Expressions in $x_{A}$-space are obtained by inverse Mellin transformations.
The transformations and summations over $j$ are solvable using various
identities.

$\tilde{F}_{2}^{A}$:

To build $\tilde{F}_{2}^{A}$ for a non-zero target mass, we start from
$\Delta\tilde{T}^{A}_{2}$ in Eq. (A.98). Using Eq. (A.61) (or Eq. (A.107)), we
identify everything to the right of $x_{A}^{-(2l-1)}$ in Eq. (A.98) as the
$(2l)^{th}$ moment of $\tilde{F}_{2}^{A}(x_{A},Q^{2})$ with TMCs. This allows
us to write

$\displaystyle\tilde{F}_{2}^{A(2l)}$ $\displaystyle=\sum_{j=0}^{\infty}\
\frac{(2l+j+1)!}{j!(2l+1)!}\ \left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\
(2l+1)(2l)\
\frac{\left(C_{2}^{(2l+2j+1)}A_{\tau=2}^{(2l+2j+1)}\right)}{(2l+2j+1)(2l+2j)}\
.$ (A.109)

For an arbitrary integrable function $B(y)$ over $y\in[0,1]$ and $m\geq 0$, we
have the identities

$\displaystyle\frac{1}{(m+1)}\ \int_{0}^{1}dy\ y^{m+1}\ B(y)$
$\displaystyle=\int_{0}^{1}dy\ y^{m}\ H(y),\ $ (A.110a)
$\displaystyle\frac{1}{(m+2)(m+1)}\ \int_{0}^{1}dy\ y^{(m+2)}\ B(y)$
$\displaystyle=\int_{0}^{1}dy\ y^{m}\ G(y),\ \ \text{where}$ (A.110b)
$\displaystyle H(y)$ $\displaystyle=\int_{y}^{1}dy^{\prime}\ B(y^{\prime}),\
\quad\text{and}\quad$ (A.110c) $\displaystyle G(y)=\int_{y}^{1}dy^{\prime}\
\int_{y^{\prime}}^{1}dy^{\prime\prime}\ B(y^{\prime\prime})$
$\displaystyle=\int_{y}^{1}dy^{\prime}\ (y^{\prime}-y)\ B(y^{\prime})\ $
(A.110d)

These identities will be used repeatedly for the different structure
functions. Schematically, the identities show that each factor of “$1/m$,”
which originates from completing the (numerator) factorials in
$\Delta\tilde{T}^{A}_{i}$, corresponds to an additional integral over a
structure function. Using the second identity and Eq. (A.108), we have

$\displaystyle\frac{C_{2}^{(2l+2j+1)}A_{\tau=2}^{(2l+2j+1)}}{(2l+2j+1)(2l+2j)}$
$\displaystyle=\frac{1}{(2l+2j+1)(2l+2j)}\ \int_{0}^{1}dy\ y^{2l+2j+1}\cdot
y^{-2}\tilde{F}_{2}^{A}(y)\Bigg{|}_{\rm No~{}TMCs}$ (A.111)
$\displaystyle=\int_{0}^{1}dy\ y^{2l+2j-1}\
\tilde{g}_{2}(y),\quad\text{where}$ (A.112) $\displaystyle\tilde{g}_{2}(y)$
$\displaystyle\equiv\int_{y}^{1}dy^{\prime}\int_{y^{\prime}}^{1}dy^{\prime\prime}\
(y^{\prime\prime})^{-2}\ \tilde{F}_{2}^{A}(y^{\prime\prime})\Bigg{|}_{\rm
No~{}TMCs}\ .$ (A.113)

This allows allows us to evaluate the summation over $j$:

$\displaystyle\tilde{F}_{2}^{A(2l)}$ $\displaystyle=\int_{0}^{1}dy\ y^{2l-1}\
\tilde{g}_{2}(y)\ (2l+1)(2l)\ \sum_{j=0}^{\infty}\
\frac{(2l+j+1)!}{j!(2l+1)!}\ \left(\frac{y^{2}M_{A}^{2}}{Q^{2}}\right)^{j}\ $
(A.114) $\displaystyle=\int_{0}^{1}dy\ y^{-1}\ \tilde{g}_{2}(y)\
\frac{(2l+1)(2l)\ y^{2l}}{(1-y^{2}M_{A}^{2}/Q^{2})^{2l+2}}\ ,$ (A.115)

where we used the first of the following identities, which also hold for
$z>1$:

$\displaystyle\sum_{j=0}^{\infty}\ \frac{(n+j)!}{j!\ n!}\ z^{j}$
$\displaystyle=\frac{1}{(1-z)^{(n+1)}}\ ,$ (A.116a)
$\displaystyle\sum_{j=1}^{\infty}\ j\ \frac{(n+j)!}{j!\ n!}\ z^{j}$
$\displaystyle=\frac{(n+1)\ z}{(1-z)^{(n+2)}}\ ,$ (A.116b)
$\displaystyle\sum_{j=2}^{\infty}\ j\ (j-1)\ \frac{(n+j)!}{j!\ n!}\ z^{j}$
$\displaystyle=\frac{(n+1)\ (n+2)\ z^{2}}{(1-z)^{(n+3)}}\ .$ (A.116c)

The first identity is derived by Taylor expanding the right-hand side and
completing the factorial. The second follows from taking the derivative of the
first identity and multiplying by $z$; similarly, the third by taking two
derivatives and scaling by $z^{2}$.

After relabeling $N=2l$, and using the second of the following identities

$\displaystyle(n+a)z^{-n}$ $\displaystyle=-z^{(1+a)}\ \frac{d}{dz}\
z^{-(n+a)}\ ,$ (A.117a) $\displaystyle(n+a)(n+a+1)z^{-n}$ $\displaystyle=\
z^{(2+a)}\frac{d^{2}}{dz^{2}}\ z^{-(n+a)}\ ,$ (A.117b)

we can take the inverse Mellin transformation of $\tilde{F}_{2}^{AN}$. We
remark that these derivative identities will be used repeatedly for the
different structure functions. Schematically, they show that each factor of
“$n$” or “$(n+1)$,” which originates from completing the factorials in
$\Delta\tilde{T}^{A}_{i}$, corresponds to a derivative over the generating
functions $G(y)$ and $H(y)$.

Inserting the derivative identity and taking the inverse Mellin transformation
gives

$\displaystyle\tilde{F}_{2}^{A}(x_{A})$ $\displaystyle=\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}dN\ x_{A}^{-N}\ \tilde{F}_{2}^{AN}$ (A.118)
$\displaystyle=\frac{x_{A}^{2}}{2\pi i}\frac{d^{2}}{dx_{A}^{2}}\int_{0}^{1}dy\
\frac{\tilde{g}_{2}(y)}{y(1-y^{2}M_{A}^{2}/Q^{2})^{2}}\
\int_{c-i\infty}^{c+i\infty}dN\
\frac{(y/x_{A})^{N}}{(1-y^{2}M_{A}^{2}/Q^{2})^{N}}\ $ (A.119)
$\displaystyle=x_{A}^{2}\ \frac{d^{2}}{dx_{A}^{2}}\int_{0}^{1}dy\
\frac{\tilde{g}_{2}(y)}{y(1-y^{2}M_{A}^{2}/Q^{2})^{2}}\
\delta\left[\log\left(\frac{(y/x_{A})}{(1-y^{2}M_{A}^{2}/Q^{2})}\right)\right]\
,$ (A.120)

where we employed the contour integral

$\displaystyle\int_{c-i\infty}^{c+i\infty}dN\ z^{N}=(2\pi i)\
\delta\left[\log(z)\right]\ .$ (A.121)

To evaluate the $\delta$ function, we use the decomposition formula

$\displaystyle\delta\left[g(z)\right]=\sum_{z_{k}}\ \frac{1}{|g^{\prime}(z)|}\
\delta(z-z_{k}),\quad\text{such~{}that}\quad g(z=z_{k})=0.$ (A.122)

This allows us to express the $\delta$ function in $\tilde{F}_{2}^{A}(x_{A})$
in terms of the zeros of the logarithm, i.e., when
$(y/x)=(1-y^{2}M_{A}^{2}/Q^{2})$. While the logarithm vanishes for two values
of $y$, only one is physical. The unphysical solution $y_{2}$ lies outside the
domain $y\in[0,1]$, meaning the contribution from $\delta(y-y_{2})$ is always
zero. The physical solution is the Nachtmann variable [67]:

$\displaystyle\xi_{A}=\frac{2x_{A}}{1+r}\ ,\quad\text{where}\quad
r\equiv\sqrt{1+4x_{A}^{2}M_{A}^{2}/Q^{2}}=\frac{(1+\xi_{A}^{2}M_{A}^{2}/Q^{2})}{(1-\xi_{A}^{2}M_{A}^{2}/Q^{2})}\
,$ (A.123)

and corresponds to the following solution for the $\delta$ function:

$\displaystyle\delta\left[\log\left(\frac{(y/x_{A})}{(1-y^{2}M_{A}^{2}/Q^{2})}\right)\right]\
=\ \left[\frac{y(1-y^{2}M_{A}^{2}/Q^{2})}{(1+y^{2}M_{A}^{2}/Q^{2})}\right]\
\delta(\xi_{A}-y)\ .$ (A.124)

Some identities that are useful for mapping between results in the literature
include

$\displaystyle\frac{x_{A}}{r\xi_{A}}=\frac{1}{(1+\xi_{A}^{2}M_{A}^{2}/Q^{2})}\quad\text{and}\quad\frac{x_{A}}{\xi_{A}}=\frac{1}{(1-\xi_{A}^{2}M_{A}^{2}/Q^{2})}\
.$ (A.125)

Replacing the $\delta$ function with Eq. (A.124) allows us to finally write

$\displaystyle\tilde{F}_{2}^{A}(x_{A})$ $\displaystyle=x_{A}^{2}\
\frac{d^{2}}{dx_{A}^{2}}\int_{0}^{1}dy\
\frac{\tilde{g}_{2}(y)}{y(1-y^{2}M_{A}^{2}/Q^{2})^{2}}\
\left[\frac{y(1-y^{2}M_{A}^{2}/Q^{2})}{(1+y^{2}M_{A}^{2}/Q^{2})}\right]\
\delta(\xi-y)\ $ (A.126) $\displaystyle\tilde{F}_{2}^{A}(x_{A})$
$\displaystyle=\ x_{A}^{2}\ \frac{d^{2}}{dx_{A}^{2}}\
\left[\frac{(1+r)^{2}}{4r}\ \tilde{g}_{2}(\xi_{A})\right]\ \ .$ (A.127)

Allowing the derivative to act on the quantities inside the bracket recovers
Eq. (3.24b). The final expression is summarized in Eq. (A.161).

$\tilde{F}_{3}^{A}$:

To build $\tilde{F}_{3}^{A}$, we similarly use Eq. (A.61) and identify
everything to the right of $x_{A}^{-(2l-1)}$ in Eq. (A.100) as the $2l^{th}$
moment of $\tilde{F}_{3}^{A}(x_{A},Q^{2})$ with TMCs. We then write

$\displaystyle\tilde{F}_{3}^{A(2l)}$ $\displaystyle=\sum_{j=0}^{\infty}\
\frac{(2l+j)!}{j!(2l)!}\ \left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ (2l)\
\frac{\left(C_{3}^{(2l+2j)}A_{\tau=2}^{(2l+2j)}\right)}{(2l+2j)}\ .$ (A.128)

Using the integral identity of Eq. (A.110a) and the relationship in Eq.
(A.108) between the moment of $F_{3}^{A}$ and the product $(C_{3}A_{\tau=2})$
when $(M_{A}^{2}/Q^{2})\to 0$, we have

$\displaystyle\frac{C_{3}^{(2l+2j)}A_{\tau=2}^{(2l+2j)}}{(2l+2j)}$
$\displaystyle=\frac{1}{(2l+2j)}\ \int_{0}^{1}dy\ y^{2l+2j}\cdot
y^{-1}\tilde{F}_{3}^{A}(y)\Bigg{|}_{\rm No~{}TMCs}\ $ (A.129)
$\displaystyle=\int_{0}^{1}dy\ y^{2l+2j-1}\
\tilde{h}_{3}(y),\quad\text{where}$ (A.130) $\displaystyle\tilde{h}_{3}(y)$
$\displaystyle\equiv\int_{y}^{1}dy^{\prime}\ (y^{\prime})^{-1}\
\tilde{F}_{3}^{A}(y^{\prime})\Bigg{|}_{\rm No~{}TMCs}\ .$ (A.131)

Using this, the summation over $j$ is then given by

$\displaystyle\tilde{F}_{3}^{A(2l)}$ $\displaystyle=\int_{0}^{1}dy\ y^{2l-1}\
\tilde{h}_{3}(y)\ (2l)\ \sum_{j=0}^{\infty}\ \frac{(2l+j)!}{j!(2l)!}\
\left(\frac{y^{2}M_{A}^{2}}{Q^{2}}\right)^{j}\ $ (A.132)
$\displaystyle=\int_{0}^{1}dy\ y^{-1}\ \tilde{h}_{3}(y)\ \frac{(2l)\
y^{2l}}{(1-y^{2}M_{A}^{2}/Q^{2})^{2l+1}}\ .$ (A.133)

After relabeling $N=2l$, using the first derivative identity in Eq. (A.117),
the contour integral of Eq. (A.121), the $\delta$ function decomposition of
Eq. (A.124), and the identities for $\xi_{A}$, we get as the inverse Mellin
transformation of $\tilde{F}_{3}^{A}$

$\displaystyle\tilde{F}_{3}^{A}(x_{A})$ $\displaystyle=\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}dN\ x_{A}^{-N}\ \tilde{F}_{3}^{AN}$ (A.134)
$\displaystyle=\frac{-x_{A}}{2\pi i}\frac{d}{dx_{A}}\ \int_{0}^{1}dy\
\frac{\tilde{h}_{3}(y)}{y(1-y^{2}M_{A}^{2}/Q^{2})}\
\int_{c-i\infty}^{c+i\infty}dN\
\frac{(y/x_{A})^{N}}{(1-y^{2}M_{A}^{2}/Q^{2})^{N}}\ $ (A.135)
$\displaystyle=-x_{A}\frac{d}{dx_{A}}\ \int_{0}^{1}dy\
\frac{\tilde{h}_{3}(y)}{y(1-y^{2}M_{A}^{2}/Q^{2})}\
\left[\frac{y(1-y^{2}M_{A}^{2}/Q^{2})}{(1+y^{2}M_{A}^{2}/Q^{2})}\right]\
\delta(\xi_{A}-y)\ ,$ (A.136) $\displaystyle\tilde{F}_{3}^{A}(x_{A})$
$\displaystyle=\ -x_{A}\frac{d}{dx_{A}}\ \left[\frac{(1+r)}{2r}\
\tilde{h}_{3}(\xi_{A})\right]\ \ .$ (A.137)

Allowing the derivative to act on the quantities inside the bracket recovers
Eq. (3.24c). The final expression is summarized in Eq. (A.161).

$\tilde{F}_{6}^{A}$:

Building $\tilde{F}_{6}^{A}$ with TMCs proceeds identically to
$\tilde{F}_{3}^{A}$. We first identify everything to the right of
$x_{A}^{-(2l-1)}$ in Eq. (A.106) as the $2l^{th}$ moment of
$\tilde{F}_{6}^{A}(x_{A},Q^{2})$. Explicitly,

$\displaystyle\tilde{F}_{3}^{A(2l)}$ $\displaystyle=\sum_{j=0}^{\infty}\
\frac{(2l+j)!}{j!(2l)!}\ \left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ (2l)\
\frac{\left(C_{6}^{(2l+2j)}A_{\tau=2}^{(2l+2j)}\right)}{(2l+2j)}\ .$ (A.138)

The procedure for this case is similar to $\tilde{F}_{3}^{A}$. Using the
relationship in Eq. (A.108), we obtain

$\displaystyle\tilde{F}_{6}^{A}(x_{A})$ $\displaystyle=-x\frac{d}{dx_{A}}\
\left[\frac{(1+r)}{2r}\ \tilde{h}_{6}(y)\right]\
,\quad\text{where}\quad\tilde{h}_{6}(y)\equiv\int_{y}^{1}dy^{\prime}y^{\prime-1}\
\tilde{F}_{6}^{A}(y^{\prime})\Bigg{|}_{\rm No~{}TMCs}\ .$ (A.139)

The final expression is summarized in Eq. (A.161). To our knowledge, the TMCs
to the $\tilde{F_{6}}$ structure function have not been previously reported in
the literature.

$\tilde{F}_{1}^{A}$:

In principle, building $\tilde{F}_{1}^{A}$ follows the same procedure as the
previous cases. However, the nuance here is that there are contributions from
$(C_{1}A_{\tau=2})$ and $(C_{2}A_{\tau=2})$ when $j>0$. The solution is to
decompose these into the moments of $\tilde{F}_{1}^{A}$ and
$\tilde{F}_{2}^{A}$, respectively. Using Eqs. (A.61) and (A.107), we identify
everything to the right of $x_{A}^{-(2l+1)}$ in Eq. (A.96) as the
$(2l+1)^{st}$ moment of $\tilde{F}_{1}^{A}(x_{A},Q^{2})$ with TMCs.
Symbolically, this is given by

$\displaystyle\tilde{F}_{1}^{A(2l+1)}$ $\displaystyle=\sum_{j=0}^{\infty}\
\frac{(2l+j+1)!}{j!\ (2l+1)!}\ \left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\
\left[\left(C_{1}^{2l+2j+1}A_{\tau=2}^{2l+2j+1}\right)+\frac{j\left(C_{2}^{2l+2j+1}A_{\tau=2}^{2l+2j+1}\right)}{(2l+2j+1)(2l+2j)}\right]\
$ (A.140) $\displaystyle=$ $\displaystyle\int_{0}^{1}dy\
\sum_{j=0}^{\infty}\frac{(2l+j+1)!}{j!\ (2l+1)!}\
\left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\left[y^{2l+2j}\tilde{F}_{1}^{A}(y)\Big{|}_{\rm
No~{}TMCs}+j\ y^{2l+2j-1}\tilde{g}_{2}(y)\right],$ (A.141)

where we expressed $(C_{2}A_{\tau=2})$ as an integral over $\tilde{g}_{2}(y)$
according to Eq. (A.111). Using Eq. (A.116), we evaluate the summation of each
term separately, giving

$\displaystyle\tilde{F}_{1}^{A(2l+1)}=$ $\displaystyle\int_{0}^{1}dy\
\frac{\tilde{F}_{1}^{A}(y)|_{\rm No~{}TMCs}\ }{y(1-y^{2}M_{A}^{2}/Q^{2})}\
\left[\frac{y^{2l+1}}{(1-y^{2}M_{A}^{2}/Q^{2})^{2l+1}}\right]$
$\displaystyle+\left(\frac{M_{A}^{2}}{Q^{2}}\right)\ \int_{0}^{1}dy\
\frac{\tilde{g}_{2}(y)}{(1-y^{2}M_{A}^{2}/Q^{2})^{2}}\
\left[\frac{(2l+2)y^{2l+1}}{(1-y^{2}M_{A}^{2}/Q^{2})^{2l+1}}\right].$ (A.142)

We now set $N=(2l+1)$, take the inverse Mellin transformation with respect to
$N$, and use the first derivative identity in Eq. (A.117) to remove the
$(N+1)$. The result is

$\displaystyle\tilde{F}_{1}^{A}(x_{A})=$ $\displaystyle\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}dN\ x_{A}^{-N}\ \tilde{F}_{1}^{AN}$ (A.143)
$\displaystyle=$ $\displaystyle\frac{1}{2\pi i}\int_{0}^{1}dy\
\frac{\tilde{F}_{1}^{A}(y)|_{\rm No~{}TMCs}\ }{y(1-y^{2}M_{A}^{2}/Q^{2})}\
\int_{c-i\infty}^{c+i\infty}dN\
\left[\frac{(y/x_{A})^{N}}{(1-y^{2}M_{A}^{2}/Q^{2})^{N}}\right]$
$\displaystyle-\left(\frac{M_{A}^{2}x_{A}^{2}}{2\pi iQ^{2}}\right)\ $
$\displaystyle\frac{d}{dx_{A}}\int_{0}^{1}dy\
\frac{\tilde{g}_{2}(y)}{(1-y^{2}M_{A}^{2}/Q^{2})^{2}}\
\int_{c-i\infty}^{c+i\infty}dN\ \left[\frac{(y/x_{A})^{N}\
x_{A}^{-1}}{(1-y^{2}M_{A}^{2}/Q^{2})^{N}}\right]$ (A.144) $\displaystyle=$
$\displaystyle\left[\frac{\tilde{F}_{1}^{A}(y)|_{\rm No~{}TMCs}\
}{(1+y^{2}M_{A}^{2}/Q^{2})}\right]_{y=\xi_{A}}\ $
$\displaystyle-\left(\frac{M_{A}^{2}x_{A}^{2}}{Q^{2}}\right)\
\frac{d}{dx_{A}}\ \left[\frac{\tilde{g}_{2}(y)\ x_{A}^{-1}\
y}{(1-y^{2}M_{A}^{2}/Q^{2})(1+y^{2}M_{A}^{2}/Q^{2})}\right]_{y=\xi_{A}}\ .$
(A.145)

To reach the last line, we used the contour integral of Eq. (A.121) and the
$\delta$ function decomposition of Eq. (A.124). Using the $\xi_{A}$
identities, the final expressions is

$\displaystyle\tilde{F}_{1}^{A}(x_{A})=$ $\displaystyle\frac{x_{A}}{r\xi_{A}}\
\tilde{F}_{1}^{A}(\xi_{A})\Bigg{|}_{\rm No~{}TMCs}\
-\left(\frac{M_{A}^{2}x_{A}^{2}}{Q^{2}}\right)\ \frac{d}{dx_{A}}\
\left[\frac{(1+r)}{{2}r}\ \tilde{g}_{2}(\xi_{A})\right]\quad.$ (A.146)

Applying the derivative to the quantities inside the bracket recovers Eq.
(3.24a). Note that the factor of $1/2$ in the square brackets is absent in Eq.
(14) of Ref. [60]. This appears to be a typo in Ref. [60] as the final result
matches Eq. (3.24a). Furthermore, after employing $\xi_{A}$ identities, we
find agreement with Eq. (3.1) of Ref. [54]. The final expression is summarized
in Eq. (A.161).

$\tilde{F}_{5}^{A}$:

Building $\tilde{F}_{5}^{A}$ with TMCs, which involves mixing with
$\tilde{F}_{2}^{A}$, is similar to $\tilde{F}_{1}^{A}$. Subsequently, we
identify everything to the right of $x_{A}^{-(2l)}$ in Eq. (A.104) as the
$(2l+1)^{th}$ moment of $\tilde{F}_{5}^{A}(x_{A},Q^{2})$:

$\displaystyle\tilde{F}_{5}^{A(2l+1)}=\sum_{j=0}^{\infty}\frac{(2l+j+1)!}{j!\
(2l+1)!}\ \left(\frac{M_{A}^{2}}{Q^{2}}\right)^{j}\ $ $\displaystyle\times$
$\displaystyle\left[\frac{(2l+1)\
\left(C_{5}^{(2l+2j+1)}A_{\tau=2}^{(2l+2j+1)}\right)}{(2l+2j+1)}-\frac{j\
(2l+1)\
\left(C_{2}^{(2l+2j+1)}A_{\tau=2}^{(2l+2j+1)}\right)}{(2l+2j+1)(2l+2j)}\