$\displaystyle\dfrac{\alpha_{s}C_{F}}{2\pi}\,\delta(x)\ln\dfrac{\Lambda^{2}_{\text{\tiny{UV}}}}{m^{2}_{q}}+\dfrac{\alpha_{s}C_{F}}{2\pi}\,\bigg{(}\ln\dfrac{\Lambda^{2}_{\text{\tiny{UV}}}}{xm^{2}_{g}}-\dfrac{1-x}{x}\bigg{)}\,.$

The general structure of the result for quasi-PDF $e_{\rm{Q}}(x)$ is given by
Eq. (105). The numerator for the singular and the canonical parts are given by

$\displaystyle N_{e(\rm{s})}$ $\displaystyle=$
$\displaystyle-\dfrac{(1-\epsilon)\,p^{3}}{\big{(}k^{2}_{\perp}+x^{2}p^{2}_{3}+m^{2}_{q}\big{)}^{3/2}}\,,$
(187) $\displaystyle N_{e(\rm{c})}$ $\displaystyle=$ $\displaystyle
2p^{3}\bigg{(}(k^{0})^{2}-x^{2}p^{2}_{3}-k^{2}_{\perp}+m^{2}_{q}+(1-\epsilon)m^{2}_{g}\bigg{)}\,.$
(188)

Table 4 confirms that the moment of $e_{\rm{Q}}(x)$ agrees exactly with that
of $e(x)$ when DR is applied for the UV. This agreement holds true even in the
cut-off scheme.

###  VI.2 Results in Yukawa Model

To derive the renormalized mass of the target in the YM, we follow the
procedure outlined in the previous section. In the $\overline{\mathrm{MS}}$
scheme, the counter-terms are

$\displaystyle\delta_{2}$ $\displaystyle=$ $\displaystyle\dfrac{\alpha_{\rm
Y}}{8\pi}{\cal P}_{\text{\tiny{UV}}}\,,$ (189) $\displaystyle\delta_{\rm{m}}$
$\displaystyle=$ $\displaystyle-\dfrac{3\alpha_{\rm Y}}{8\pi}{\cal
P}_{\text{\tiny{UV}}}\,,$ (190)

and we arrive at the following relation between the renormalized mass of the
target and the bare quark mass,

$\displaystyle m_{\rm{R}}\Big{|}^{\epsilon_{\text{\tiny{UV}}}}$
$\displaystyle=$ $\displaystyle m_{q}\bigg{[}1+\dfrac{\alpha_{\rm
Y}}{4\pi}\bigg{(}-\dfrac{3}{2}\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{m^{2}_{q}}-\dfrac{7}{2}\bigg{)}\bigg{]}\,.$
(191)

In the cut-off scheme, we choose the counter-terms as

$\displaystyle\delta_{2}$ $\displaystyle=$ $\displaystyle\dfrac{\alpha_{\rm
Y}}{8\pi}\ln\dfrac{\Lambda^{2}_{\text{\tiny{UV}}}}{\mu^{2}}\,,$ (192)
$\displaystyle\delta_{\rm{m}}$ $\displaystyle=$
$\displaystyle-\dfrac{3\alpha_{\rm
Y}}{8\pi}\ln\dfrac{\Lambda^{2}_{\text{\tiny{UV}}}}{\mu^{2}}\,,$ (193)

and the relation between the two masses reads

$\displaystyle m_{\rm{R}}\Big{|}^{\Lambda_{\text{\tiny{UV}}}}$
$\displaystyle=$ $\displaystyle m_{q}\bigg{[}1+\dfrac{\alpha_{\rm
Y}}{4\pi}\bigg{(}-\dfrac{3}{2}\ln\dfrac{\mu^{2}}{m^{2}_{q}}-2\bigg{)}\bigg{]}\,.$
(194)

Once again, by repeating the above steps, and not subtracting the counter-
terms, we arrive at the following expressions for the regulated mass and it’s
derivative,

$\displaystyle m_{\rm{Reg.}}\Big{|}^{\epsilon_{\text{\tiny{UV}}}}$
$\displaystyle=$ $\displaystyle m_{q}\bigg{[}1+\dfrac{\alpha_{\rm
Y}}{4\pi}\bigg{(}-\dfrac{3}{2}{\cal
P}_{\text{\tiny{UV}}}-\dfrac{3}{2}\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{m^{2}_{q}}-\dfrac{7}{2}\bigg{)}\bigg{]}\,,$
(195)
$\displaystyle\therefore\dfrac{\partial{m_{\rm{Reg.}}}}{\partial{m_{q}}}\Big{|}^{\epsilon_{\text{\tiny{UV}}}}$
$\displaystyle=$ $\displaystyle 1+\dfrac{\alpha_{\rm
Y}}{4\pi}\bigg{(}-\dfrac{3}{2}{\cal
P}_{\text{\tiny{UV}}}-\dfrac{3}{2}\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{m^{2}_{q}}-\dfrac{1}{2}\bigg{)}\,,$
(196) $\displaystyle m_{\rm{Reg.}}\Big{|}^{\Lambda_{\text{\tiny{UV}}}}$
$\displaystyle=$ $\displaystyle m_{q}\bigg{[}1+\dfrac{\alpha_{\rm
Y}}{4\pi}\bigg{(}-\dfrac{3}{2}\ln\dfrac{\Lambda^{2}_{\text{\tiny{UV}}}}{m^{2}_{q}}-2\bigg{)}\bigg{]}\,,$
(197)
$\displaystyle\therefore\dfrac{\partial{m_{\rm{Reg.}}}}{\partial{m_{q}}}\Big{|}^{\Lambda_{\text{\tiny{UV}}}}$
$\displaystyle=$ $\displaystyle 1+\dfrac{\alpha_{\rm
Y}}{4\pi}\bigg{(}-\dfrac{3}{2}\ln\dfrac{\Lambda^{2}_{\text{\tiny{UV}}}}{m^{2}_{q}}+1\bigg{)}\,.$
(198)

Turning now to the results for $e(x)$, the starting expressions for the
singular and canonical terms are

$\displaystyle e^{\rm{(1a)}}_{\rm{(s)}}(x)$ $\displaystyle=$
$\displaystyle\dfrac{g^{2}_{\rm
Y}\mu^{2\epsilon}}{2(2\pi)}\,\delta(x)\int\dfrac{d^{n-2}k_{\perp}}{(2\pi)^{n-2}}\,\dfrac{1}{(k^{2}_{\perp}+m^{2}_{q})}\,,$
$\displaystyle e^{\rm{(1a)}}_{\rm{(c)}}(x)$ $\displaystyle=$
$\displaystyle-\dfrac{g^{2}_{\rm
Y}\mu^{2\epsilon}}{2(2\pi)}\int\dfrac{d^{n-2}k_{\perp}}{(2\pi)^{n-2}}\,\dfrac{2k^{2}_{\perp}-2(1-x^{2})m^{2}_{q}+(1+x)m^{2}_{s}}{\big{(}k^{2}_{\perp}+(1-x)^{2}m^{2}_{q}+xm^{2}_{s}\big{)}^{2}}\,.$
(199)

The result for $e(x)$ in the DR scheme is

$\displaystyle e^{\rm{(1a)}}(x)\Big{|}^{\epsilon_{\text{\tiny{UV}}}}_{m_{q}}$
$\displaystyle=$ $\displaystyle
e^{\rm{(1a)}}_{\rm{(s)}}(x)\Big{|}^{\epsilon_{\text{\tiny{UV}}}}_{m_{q}}+e^{\rm{(1a)}}_{\rm{(c)}}(x)\Big{|}^{\epsilon_{\text{\tiny{UV}}}}_{m_{q}}$
(200) $\displaystyle=$ $\displaystyle\dfrac{\alpha_{\rm
Y}}{4\pi}\,\delta(x)\bigg{(}{\cal
P}_{\text{\tiny{UV}}}+\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{m^{2}_{q}}\bigg{)}+\dfrac{\alpha_{\rm
Y}}{4\pi}\bigg{(}-2\,{\cal
P}_{\text{\tiny{UV}}}-2\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{(1-x)^{2}m^{2}_{q}}+\dfrac{4}{1-x}\bigg{)}\,,$

and we find that the lowest moment of $e(x)$ is

$\displaystyle\int^{1}_{0}dx\,\delta(1-x)+\int\dfrac{dk^{+}}{p^{+}}\,e^{\rm{(1a)}}(k^{+})\Big{|}^{\epsilon_{\text{\tiny{UV}}}}_{m_{q}}+\dfrac{\partial\Sigma}{\partial\not{p}}\Big{|}^{\epsilon_{\text{\tiny{UV}}}}_{m_{q}}$
$\displaystyle=$ $\displaystyle 1+\dfrac{\alpha_{\rm
Y}}{4\pi}\bigg{(}-\dfrac{3}{2}{\cal
P}_{\text{\tiny{UV}}}-\dfrac{3}{2}\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{m^{2}_{q}}-\dfrac{1}{2}\bigg{)}\,,$
(201)

which is in agreement with Eq. (196). Hence the sum rule in Eq. (173) is
exactly satisfied in the DR scheme. The result for $e(x)$ in the cut-off
scheme, with regularization applied to the transverse components, is

$\displaystyle e^{\rm{(1a)}}(x)\Big{|}^{\Lambda_{\text{\tiny{UV}}}}_{m_{q}}$
$\displaystyle=$ $\displaystyle
e^{\rm{(1a)}}_{\rm{(s)}}(x)\Big{|}^{\Lambda_{\text{\tiny{UV}}}}_{m_{q}}+e^{\rm{(1a)}}_{\rm{(c)}}(x)\Big{|}^{\Lambda_{\text{\tiny{UV}}}}_{m_{q}}$
(202) $\displaystyle=$ $\displaystyle\dfrac{\alpha_{\rm
Y}}{4\pi}\,\delta(x)\ln\dfrac{\Lambda^{2}_{\text{\tiny{UV}}}}{m^{2}_{q}}+\dfrac{\alpha_{\rm
Y}}{4\pi}\bigg{(}-2\ln\dfrac{\Lambda^{2}_{\text{\tiny{UV}}}}{(1-x)^{2}m^{2}_{q}}+\dfrac{4}{1-x}\bigg{)}\,.$

Once again, we find that the sum rule is violated with the above results, but
is satisfied provided $\int e$ and $\frac{\partial\Sigma}{\partial\not{p}}$
are consistently calculated with a regularization to all components of $k$.
The results are

$\displaystyle\int\dfrac{dk^{+}}{p^{+}}\,e^{\rm{(1a)}}(k^{+})\Big{|}^{\Lambda_{\text{\tiny{UV}}}}_{m_{q}}$
$\displaystyle=$ $\displaystyle\dfrac{\alpha_{\rm
Y}}{4\pi}\int^{1}_{0}dy\bigg{(}-2(1-y)\ln\dfrac{\Lambda^{2}_{\text{\tiny{UV}}}}{(1-y)^{2}m^{2}_{q}}+3(1-y)+\dfrac{(1+y)^{2}}{1-y}\bigg{)}\,,$
$\displaystyle\dfrac{\partial\Sigma}{\partial\not{p}}\Big{|}^{\Lambda_{\text{\tiny{UV}}}}_{m_{q}}$
$\displaystyle=$ $\displaystyle\dfrac{\alpha_{\rm
Y}}{4\pi}\int^{1}_{0}dy\bigg{(}-y\ln\dfrac{\Lambda^{2}_{\text{\tiny{UV}}}}{(1-y)^{2}m^{2}_{q}}+y-\dfrac{2y(1+y)}{1-y}\bigg{)}\,,$
$\displaystyle\int^{1}_{0}dx\,\delta(1-x)+\int\dfrac{dk^{+}}{p^{+}}\,e^{\rm{(1a)}}(k^{+})\Big{|}^{\Lambda_{\text{\tiny{UV}}}}_{m_{q}}+\dfrac{\partial\Sigma}{\partial\not{p}}\Big{|}^{\Lambda_{\text{\tiny{UV}}}}_{m_{q}}$
$\displaystyle=$ $\displaystyle 1+\dfrac{\alpha_{\rm
Y}}{4\pi}\bigg{(}-\dfrac{3}{2}\ln\dfrac{\Lambda^{2}_{\text{\tiny{UV}}}}{m^{2}_{q}}+1\bigg{)}\,,$
(203)

which exactly matches with our result in Eq. (198).

For $m_{s}\neq 0$, the singular part of $e(x)$ has two results,

$\displaystyle
e^{\rm{(1a)}}_{\rm{(s)}}(x)\Big{|}^{\epsilon_{\text{\tiny{UV}}}}=\begin{dcases}&e^{\rm(1a)}_{\rm{(s)}}(x)\Big{|}^{\epsilon_{\text{\tiny{UV}}}}_{m_{q}}=\dfrac{\alpha_{\rm
Y}}{4\pi}\,\delta(x)\bigg{(}{\cal
P}_{\text{\tiny{UV}}}+\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{m^{2}_{q}}\bigg{)}\,,\\\\[5.69046pt]
&e^{\rm{(1a)}}_{\rm{(s)}}(x)\Big{|}^{\epsilon_{\text{\tiny{UV}}}}_{\epsilon_{\text{\tiny{IR}}}}=\dfrac{\alpha_{\rm
Y}}{4\pi}\,\delta(x)\bigg{(}{\cal P}_{\text{\tiny{UV}}}-{\cal
P}_{\text{\tiny{IR}}}+\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{\mu^{2}_{\text{\tiny{IR}}}}\bigg{)}\,,\end{dcases}$
(204)

while the canonical part is given by

$\displaystyle
e^{\rm{(1a)}}_{\rm{(c)}}(x)\Big{|}^{\epsilon_{\text{\tiny{UV}}}}_{m_{s}}$
$\displaystyle=$ $\displaystyle-\dfrac{\alpha_{\rm Y}}{4\pi}\bigg{(}2\,{\cal
P}_{\text{\tiny{UV}}}+2\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{xm^{2}_{s}}+\dfrac{1-x}{x}\bigg{)}\,.$
(205)

When DR is applied for the IR, we get

$\displaystyle
e^{\rm{(1a)}}(x)\Big{|}^{\epsilon_{\text{\tiny{UV}}}}_{\epsilon_{\text{\tiny{IR}}}}$
$\displaystyle=$ $\displaystyle
e^{\rm{(1a)}}_{\rm{(s)}}(x)\Big{|}^{\epsilon_{\text{\tiny{UV}}}}_{\epsilon_{\text{\tiny{IR}}}}+e^{\rm{(1a)}}_{\rm{(c)}}(x)\Big{|}^{\epsilon_{\text{\tiny{UV}}}}_{\epsilon_{\text{\tiny{IR}}}}$
(206) $\displaystyle=$ $\displaystyle\dfrac{\alpha_{\rm
Y}}{4\pi}\,\delta(x)\bigg{(}{\cal P}_{\text{\tiny{UV}}}-{\cal
P}_{\text{\tiny{IR}}}+\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{\mu^{2}_{\text{\tiny{IR}}}}\bigg{)}-\dfrac{2\alpha_{\rm
Y}}{4\pi}\,\bigg{(}{\cal P}_{\text{\tiny{UV}}}-{\cal
P}_{\text{\tiny{IR}}}+\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{\mu^{2}_{\text{\tiny{IR}}}}\bigg{)}\,.$

Finally, when a cut-off is applied to the transverse direction, the result for
$e(x)$ with $m_{s}\neq 0$ reads

$\displaystyle e^{\rm{(1a)}}(x)\Big{|}^{\Lambda_{\text{\tiny{UV}}}}_{m_{s}}$
$\displaystyle=$ $\displaystyle
e^{\rm{(1a)}}_{\rm{(s)}}(x)\Big{|}^{\Lambda_{\text{\tiny{UV}}}}+e^{\rm{(1a)}}_{\rm{(c)}}(x)\Big{|}^{\Lambda_{\text{\tiny{UV}}}}_{m_{s}}$
(207) $\displaystyle=$ $\displaystyle\dfrac{\alpha_{\rm
Y}}{4\pi}\,\delta(x)\ln\dfrac{\Lambda^{2}_{\text{\tiny{UV}}}}{m^{2}_{q}}-\dfrac{\alpha_{\rm
Y}}{4\pi}\,\bigg{(}2\ln\dfrac{\Lambda^{2}_{\text{\tiny{UV}}}}{xm^{2}_{s}}+\dfrac{1-x}{x}\bigg{)}\,.$

We refer to Section IV.2 for the general structure of the quasi-PDFs in YM.
The numerators for the quasi-PDF $e_{\rm{Q}}(x)$ are given by

$\displaystyle N_{e(\rm{s})}$ $\displaystyle=$
$\displaystyle\dfrac{p^{3}}{\big{(}k^{2}_{\perp}+x^{2}p^{2}_{3}+m^{2}_{q}\big{)}^{3/2}}\,,$
(208) $\displaystyle N_{e(\rm{c})}$ $\displaystyle=$
$\displaystyle\dfrac{p^{3}}{m_{q}}\bigg{(}2m_{q}(k^{0})^{2}-2m_{q}k^{2}_{\perp}-2m_{q}x^{2}p^{2}_{3}+2m^{3}_{q}-m_{q}m^{2}_{s}\bigg{)}\,.$
(209)

We have confirmed numerically that our results for the moment of
$e_{\rm{Q}}(x)$ matches exactly with that of $e(x)$.

##  VII Summary

In this paper, we have revisited BC-type sum rules which relate the lowest
moment of certain twist-2 and twist-3 PDFs. While those sum rules have long
been known in the case of light-cone PDFs, we argue that they also hold for
the corresponding quasi-PDFs. We have also scrutinized the sum rules through
model calculations. Specifically, we have calculated the light-cone PDFs
($g_{1}(x),\,$ $g_{T}(x)$) and ($h_{1}(x),\,$ $h_{L}(x)$), and their quasi-PDF
counterparts ($g_{1,\rm{Q}}(x),\,$ $g_{T,\rm{Q}}(x)$) and
($h_{1,\rm{Q}}(x),\,$ $h_{L,\rm{Q}}(x)$) in the QTM and the YM, to lowest
order in perturbation theory. We have regulated the IR divergences in 3
schemes: non-zero gluon mass $m_{g}\neq 0$, non-zero quark mass $m_{q}\neq 0$,
and DR. For the UV divergences, we have made use of 2 schemes: DR, and cut-
off.

Related previous model calculations have focused on the UV-divergent parts of
(the perturbative corrections to) the PDFs. As such, several works in the past
have shown that the BC-type sum rules are valid in cut-off schemes. Here, we
have presented the full results for the PDFs at one-loop order, that is, we
have calculated the UV-divergent and the UV-finite parts of the PDFs. We have
shown that the BC-type sum rules hold for both the UV-divergent and the UV-
finite terms when DR is employed for the UV. However, we have found that these
sum rules are generally violated for the UV-finite terms when a cut-off is
employed. The only exception is the h-sum rule in the QTM, which
“accidentally” remains valid in the cut-off scheme. Violations of the sum
rules can be expected in cut-off schemes because they break rotational/Lorentz
invariance which is the reason why the BC-type sum rules exist in the first
place. We have also shown that working with $m_{g}\neq 0$ at twist-3 can lead
to a violation of the BC-type sum rules. Furthermore, we have clarified two
important issues related to the moments for quasi-PDFs — the moments are
finite if a regulator is applied to the $k_{\perp}$-integral (even though this
integral is finite for quasi-PDFs), and the moments of quasi-PDFs diverge if
calculated after a twist-expansion. Finally, we have calculated the light-cone
PDF $e(x)$, and its corresponding quasi-PDF $e_{\rm{Q}}(x)$ in both the QTM
and the YM. In particular, we have scrutinized the sum rule which relates the
lowest moment of $e(x)$ to the target mass. We repeat that we have not
considered renormalization, which could give rise to additional complications
when trying to establish BC-type and related sum rules — see, for instance,
Ref. Fatma_talk_GHP . Nonetheless, the physics pertaining to the regulated
results, which we have presented in this work, are fundamental for our
concepts.

It is quite likely that there exist more instances in which sum rules or other
relations that are rooted in the Lorentz invariance are spoiled in cut-off
schemes. One potential example are polynomiliaty relations for generalized
parton distributions Ji:1998pc . An important message of our work is that it
is crucial to calculate the perturbative corrections for the various partonic
functions beyond the UV-divergent parts. And if such a calculation suggests a
violation of a certain relation, one must check carefully the cause of the
violation.

###### Acknowledgements.

We express our gratitude to Krzysztof Cichy, Martha Constantinou, Aurora
Scapellato, and Fernanda Steffens for collaboration on topics related to the
present manuscript. This work has been supported by the National Science
Foundation under grant No. PHY-1812359, and by the U.S. Department of Energy,
Office of Science, Office of Nuclear Physics, within the framework of the TMD
Topical Collaboration.

##  Appendix A An interesting point related to DR when applied for IR
divergences

Typically, when DR is applied for both IR and UV divergences, one introduces
an arbitrary scale/cut-off, $\Lambda$, to set the boundaries between the IR
and UV regions ($0<\Lambda<\infty$). In the following, through the example of
($g_{1},\,g_{T}$) in the QTM, we show that the BC sum rule is violated if the
regularization is applied to the transverse dimensions and if
($\epsilon_{\text{\tiny{IR}}},\,\epsilon_{\text{\tiny{UV}}}$) are kept finite.
The reason for this violation can be traced back to the effect of the cut-off
$\Lambda$ which continues to hold if
($\epsilon_{\text{\tiny{IR}}},\,\epsilon_{\text{\tiny{UV}}}$) are kept finite.
As we shall show below, it is only after a Taylor expansion in powers of
($\epsilon_{\text{\tiny{IR}}},\,\epsilon_{\text{\tiny{UV}}}$) $\approx 0$,
that the logarithms in $\Lambda$ drop out (at least for the dominant IR pole
and the finite term, that is,
$\mathcal{O}(\epsilon^{0}_{\text{\tiny{IR}}},\,\epsilon^{0}_{\text{\tiny{UV}}})$),
such that the sum rules are exactly satisfied. This is the case that we have
discussed at stretch throughout our manuscript.

We first calculate $g_{1}(x)$:

$\displaystyle g^{\rm{(1a)}}_{1}(x)$ $\displaystyle=$ $\displaystyle
2\,\alpha_{s}C_{F}\,(1-x)\,\left\\{\mu^{2\epsilon}(1-\epsilon)\,\int\dfrac{d^{n-2}k_{\perp}}{(2\pi)^{n-2}}\dfrac{1}{k^{2}_{\perp}}\right\\}$
(210) $\displaystyle=$ $\displaystyle
2\,\alpha_{s}C_{F}\,(1-x)\,\left\\{-\dfrac{(4\pi)^{-1+\epsilon_{\text{\tiny{IR}}}}(-1+\epsilon_{\text{\tiny{IR}}})\bigg{(}\dfrac{\Lambda}{\mu_{\text{\tiny{IR}}}}\bigg{)}^{-2\epsilon_{\text{\tiny{IR}}}}}{\epsilon^{2}_{\text{\tiny{IR}}}\Gamma(-\epsilon_{\text{\tiny{IR}}})}+\dfrac{(4\pi)^{-1+\epsilon_{\text{\tiny{UV}}}}(-1+\epsilon_{\text{\tiny{UV}}})\bigg{(}\dfrac{\Lambda}{\mu_{\text{\tiny{UV}}}}\bigg{)}^{-2\epsilon_{\text{\tiny{UV}}}}}{\epsilon^{2}_{\text{\tiny{UV}}}\Gamma(-\epsilon_{\text{\tiny{UV}}})}\right\\}\,.$

On the other hand, the singular and the canonical parts for $g_{T}(x)$ are

$\displaystyle g^{\rm{(1a)}}_{T(\rm{s})}(x)$ $\displaystyle=$
$\displaystyle-2\,\alpha_{s}C_{F}\,\delta(x)\,\left\\{\mu^{2\epsilon}\epsilon\,\int\dfrac{d^{n-2}k_{\perp}}{(2\pi)^{n-2}}\dfrac{1}{k^{2}_{\perp}}\right\\}$
(211) $\displaystyle=$ $\displaystyle
2\,\alpha_{s}C_{F}\,\delta(x)\,\left\\{\dfrac{(4\pi)^{-1+\epsilon_{\text{\tiny{IR}}}}\bigg{(}\dfrac{\Lambda}{\mu_{\text{\tiny{IR}}}}\bigg{)}^{-2\epsilon_{\text{\tiny{IR}}}}}{\Gamma(1-\epsilon_{\text{\tiny{IR}}})}-\dfrac{(4\pi)^{-1+\epsilon_{\text{\tiny{UV}}}}\bigg{(}\dfrac{\Lambda}{\mu_{\text{\tiny{UV}}}}\bigg{)}^{-2\epsilon_{\text{\tiny{UV}}}}}{\Gamma(1-\epsilon_{\text{\tiny{UV}}})}\right\\}\,,$
$\displaystyle g^{\rm{(1a)}}_{T(\rm{c})}(x)$ $\displaystyle=$ $\displaystyle
2\,\alpha_{s}C_{F}x\,\left\\{\mu^{2\epsilon}\,\int\dfrac{d^{n-2}k_{\perp}}{(2\pi)^{n-2}}\dfrac{1}{k^{2}_{\perp}}\right\\}$
(212) $\displaystyle=$ $\displaystyle
2\,\alpha_{s}C_{F}x\,\left\\{\dfrac{(4\pi)^{-1+\epsilon_{\text{\tiny{IR}}}}\bigg{(}\dfrac{\Lambda}{\mu_{\text{\tiny{IR}}}}\bigg{)}^{-2\epsilon_{\text{\tiny{IR}}}}}{\epsilon^{2}_{\text{\tiny{IR}}}\Gamma(-\epsilon_{\text{\tiny{IR}}})}-\dfrac{(4\pi)^{-1+\epsilon_{\text{\tiny{UV}}}}\bigg{(}\dfrac{\Lambda}{\mu_{\text{\tiny{UV}}}}\bigg{)}^{-2\epsilon_{\text{\tiny{UV}}}}}{\epsilon^{2}_{\text{\tiny{UV}}}\Gamma(-\epsilon_{\text{\tiny{UV}}})}\right\\}\,.$

Therefore,

$\displaystyle\int dx\,g_{1}^{\rm{(1a)}}(x)\neq\int
dx\,g_{T}^{\rm{(1a)}}(x)\,.$ (213)

However, in the limit of
($\epsilon_{\text{\tiny{IR}}},\,\epsilon_{\text{\tiny{UV}}}$) $\rightarrow 0$,
the $\Lambda$-dependence drops out:

$\displaystyle g_{1}^{\rm{(1a)}}(x)$ $\displaystyle=$ $\displaystyle
2\,\alpha_{s}C_{F}\,(1-x)\,\left\\{\dfrac{1}{4\pi}\bigg{(}\dfrac{1}{\epsilon_{\text{\tiny{UV}}}}-\dfrac{1}{\epsilon_{\text{\tiny{IR}}}}\bigg{)}+\dfrac{1}{2\pi}\bigg{(}\ln\dfrac{\Lambda}{\mu_{\text{\tiny{IR}}}}-\ln\dfrac{\Lambda}{\mu_{\text{\tiny{UV}}}}\bigg{)}\right\\}\,,$
(214) $\displaystyle g^{\rm{(1a)}}_{T(\rm{s})}(x)$ $\displaystyle=$
$\displaystyle 0\,,$ (215) $\displaystyle g^{\rm{(1a)}}_{T(\rm{c})}(x)$
$\displaystyle=$ $\displaystyle
2\,\alpha_{s}C_{F}\,x\,\left\\{\dfrac{1}{4\pi}\bigg{(}\dfrac{1}{\epsilon_{\text{\tiny{UV}}}}-\dfrac{1}{\epsilon_{\text{\tiny{IR}}}}\bigg{)}+\dfrac{1}{2\pi}\bigg{(}\ln\dfrac{\Lambda}{\mu_{\text{\tiny{IR}}}}-\ln\dfrac{\Lambda}{\mu_{\text{\tiny{UV}}}}\bigg{)}\right\\}\,.$
(216)

Clearly,

$\displaystyle\int dx\,g_{1}^{\rm{(1a)}}(x)=\int dx\,g_{T}^{\rm{(1a)}}(x)\,.$
(217)

This point is very interesting, and the reasoning for such a result is rather
simple: sum rules do not hold for finite values of
($\epsilon_{\text{\tiny{IR}}},\,\epsilon_{\text{\tiny{UV}}}$) because of the
effect of the cut-off $\Lambda$, which has been applied to the transverse
dimensions to demarcate the IR and UV regions. Therefore, obviously, in the
process of creating boundaries between IR and UV regions, we ended up breaking
rotational invariance. We note in passing that due to this reason, we did not
quote numerical results for the moments with finite
($\epsilon_{\text{\tiny{IR}}},\,\epsilon_{\text{\tiny{UV}}}$), as we did in
other instances with nonzero parton mass regulators.

It is also interesting to check if the above observation changes when DR, for
both IR and UV, is applied to all components of the loop momenta. Such a case
would require us to create boundaries between IR and UV by introducing
$\Lambda$ on all components of the loop momenta. Our starting point for
$g^{\rm{(1a)}}_{1}$ is

$\displaystyle\int\dfrac{dk^{+}}{p^{+}}\lambda\,g^{\rm{(1a)}}_{1}(k^{+})$
$\displaystyle=$
$\displaystyle-\dfrac{ig^{2}C_{F}\mu^{2\epsilon}}{4p^{+}}\,\int\dfrac{d^{n}k}{(2\pi)^{n}}\int^{1}_{0}dy\dfrac{2(1-y)}{(k^{2}-Q^{2})^{3}}N_{g1}(k)\,,$
(218)

where

$\displaystyle N_{g1}(k)=4\lambda p^{+}(n-2)k^{2}-8m_{q}(n-2)k^{+}(k\cdot
s)\,,$ (219)

and $Q^{2}=0$. Using

$\displaystyle\mu^{2\epsilon}(n-2)\int\dfrac{d^{n}k}{(2\pi)^{n}}\,\dfrac{1}{k^{4}}$
$\displaystyle=$
$\displaystyle\dfrac{i}{8\pi^{2}}\bigg{(}\dfrac{1}{\epsilon_{\text{\tiny{UV}}}}-\dfrac{1}{\epsilon_{\text{\tiny{IR}}}}+\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{\mu^{2}_{\text{\tiny{IR}}}}\bigg{)}\,,$
(220)
$\displaystyle\mu^{2\epsilon}(n-2)\int\dfrac{d^{n}k}{(2\pi)^{n}}\,\dfrac{k^{+}(k\cdot
s)}{k^{6}}$ $\displaystyle=$
$\displaystyle\mu^{2\epsilon}s^{+}\dfrac{(n-2)}{n}\int\dfrac{d^{n}k}{(2\pi)^{n}}\,\dfrac{k^{2}}{k^{6}}=\dfrac{is^{+}}{32\pi^{2}}\bigg{(}\dfrac{1}{\epsilon_{\text{\tiny{UV}}}}-\dfrac{1}{\epsilon_{\text{\tiny{IR}}}}+\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{\mu^{2}_{\text{\tiny{IR}}}}\bigg{)}\,,$
(221)

we obtain

$\displaystyle\int\dfrac{dk^{+}}{p^{+}}g^{\rm{(1a)}}_{1}(k^{+})\Big{|}^{\epsilon_{\text{\tiny{UV}}}}_{\epsilon_{\text{\tiny{IR}}}}$
$\displaystyle=$
$\displaystyle\dfrac{\alpha_{s}C_{F}}{2\pi}\dfrac{1}{2}\bigg{(}\dfrac{1}{\epsilon_{\text{\tiny{UV}}}}-\dfrac{1}{\epsilon_{\text{\tiny{IR}}}}+\ln\dfrac{\mu^{2}_{\text{\tiny{UV}}}}{\mu^{2}_{\text{\tiny{IR}}}}\bigg{)}\,,$
(222)

which agrees with Eq. (42). Our starting point for $g^{\rm{(1a)}}_{T}$ is
given by

$\displaystyle\int\dfrac{dk^{+}}{p^{+}}\dfrac{m_{q}s^{i}_{\perp}}{p^{+}}\,g^{\rm{(1a)}}_{T}(k^{+})$
$\displaystyle=$
$\displaystyle-\dfrac{ig^{2}C_{F}\mu^{2\epsilon}}{4p^{+}}\,\int\dfrac{d^{n}k}{(2\pi)^{n}}\int^{1}_{0}dy\dfrac{2(1-y)}{(k^{2}-Q^{2})^{3}}N_{gT}(k)\,,$
(223)

where

$\displaystyle
N_{gT}(k)=4m_{q}s^{i}_{\perp}(n-2)k^{2}-8m_{q}(n-2)k^{i}_{\perp}(k\cdot s)\,.$
(224)

We see that the structure of the individual terms in $N_{gT}(k)$ and
$N_{g1}(k)$ exactly agree. Therefore, prior to carrying out the integrals
explicitly, one can already see that the BC sum rule will be satisfied. A
direct consequence of this term-by-term equivalence is that the sum rule
continues to hold also for finite values of
$(\epsilon_{\text{\tiny{IR}}},\epsilon_{\text{\tiny{UV}}})$. Specifically, we
find the following equality,

$\displaystyle\int\dfrac{dk^{+}}{p^{+}}g^{\rm{(1a)}}_{1}(k^{+})$
$\displaystyle=$
$\displaystyle\dfrac{\alpha_{s}C_{F}}{2\pi}\,\bigg{\\{}(4\pi)^{\epsilon_{\text{\tiny{IR}}}}\bigg{(}\dfrac{\Lambda}{\mu_{\text{\tiny{IR}}}}\bigg{)}^{-2\epsilon_{\text{\tiny{IR}}}}\bigg{(}\dfrac{1}{\epsilon^{2}_{\text{\tiny{IR}}}\Gamma(-\epsilon_{\text{\tiny{IR}}})}-\dfrac{(-1+\epsilon_{\text{\tiny{IR}}})}{\epsilon_{\text{\tiny{IR}}}\Gamma(3-\epsilon_{\text{\tiny{IR}}})}\bigg{)}$
(225)
$\displaystyle\phantom{Thistextisinvisible.}-(4\pi)^{\epsilon_{\text{\tiny{UV}}}}\bigg{(}\dfrac{\Lambda}{\mu_{\text{\tiny{UV}}}}\bigg{)}^{-2\epsilon_{\text{\tiny{UV}}}}\bigg{(}\dfrac{1}{\epsilon^{2}_{\text{\tiny{UV}}}\Gamma(-\epsilon_{\text{\tiny{UV}}})}-\dfrac{(-1+\epsilon_{\text{\tiny{UV}}})}{\epsilon_{\text{\tiny{UV}}}\Gamma(3-\epsilon_{\text{\tiny{UV}}})}\bigg{)}\bigg{\\}}$
$\displaystyle=$
$\displaystyle\int\dfrac{dk^{+}}{p^{+}}g^{\rm{(1a)}}_{T}(k^{+})\,,$

which holds for arbitrary values of the cut-off $\Lambda$. Ultimately, all of
these observations arise from the very same situation, namely, whether or not
we are applying a cut-off in a rotationally invariant manner. While our
observation here is very important and fundamental, we believe it is not
widely known and, in fact, we are not aware of a paper which discusses this
point.

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