and SCET+ resummation uncertainties, respectively. As expected, the SCETI
resummation uncertainty dominates in the SCETI region of phase space, and
similarly for SCETII. The SCET+ resummation uncertainty is largest along the
phase-space boundaries, indicating that it is mostly sensitive to variations
of the transition points, i.e., the points where the intrinsic SCET+
resummation is turned off in our matched prediction.

Figure -1002: The double-differential Drell-Yan cross section for fixed
$q_{T}$, as a function of ${\mathcal{T}}$ (left) and for fixed
${\mathcal{T}}$, as a function of $q_{T}$ (right). The solid red curves are
slices of the surface plots shown in the top left panel in figure -1004, up to
Jacobians. The blue dashed (green dotted) curve corresponds to the middle
(bottom) panel of figure -1005. The thin vertical lines indicate the
transition points $a_{i}$ described in section 3.3. The SCETI prediction
(dashed blue) has an unphysical edge at ${\mathcal{T}}=q_{T}$, see figure
-1005, and is not shown beyond ${\mathcal{T}}=0.8\,q_{T}$ to avoid
distraction. See the text for details on the uncertainty bands.

To further highlight the differences between our fully double-differential
resummation and the single-differential resummation at either NNLL${}_{q_{T}}$
or NNLLT, we take slices of the surface plots and overlay them in figure
-1002, keeping $q_{T}$ (left) or ${\mathcal{T}}$ (right) fixed. The solid red
curve corresponds to the matched and fully resummed cross section in eq.
(3.2), with the uncertainty band given by the total perturbative uncertainty
$\Delta_{\mathrm{total}}$, see eq. (3.26). The matched SCETI (dashed blue) and
SCETII (dotted green) predictions correspond to the middle and bottom panel of
figure -1005, respectively. Their uncertainty bands are given by
$\Delta^{\mathrm{I}}_{\mathrm{total}}$ and
$\Delta^{\mathrm{II}}_{\mathrm{total}}$, which only probe a subset of higher-
order terms as predicted by the respective RGE, see eqs. (2.18) and (2.37).
The SCETI prediction features an unphysical sharp edge at
${\mathcal{T}}=q_{T}$, cf. the middle panel of figure -1005, and for this
reason is cut off at ${\mathcal{T}}=0.8\,q_{T}$.

All panels in figure -1002 show that our final prediction smoothly
interpolates between the SCETI and SCETII boundary theories, both for the
central values and for the uncertainties. Specifically, the matched prediction
tends towards SCETI (SCETII) for small (large) values of ${\mathcal{T}}$ and
large (small) values of $q_{T}$. In the left column one clearly sees that
SCETII only captures logarithms of ${\mathcal{T}}$ at fixed order, leading to
a diverging spectrum as ${\mathcal{T}}\to 0$, while the complete NNLL result
features a physical Sudakov peak. Conversely, the SCETI result diverges as
$q_{T}\to 0$, but is rendered physical by the additional $q_{T}$ resummation
at NNLL.

We would like to stress that our fully resummed prediction does not exactly
agree with either boundary theory, even beyond the final transition points
$a_{1}$ and $a_{6}$ where the intrinsic SCET+ resummation is turned off. The
reason for this is that even in these limits, the matched cross section in eq.
(3.2) improves over the matched SCETI and SCETII cross sections in eq. (2.11)
and eq. (2.29) by an additional resummation of power-suppressed terms, cf.
eqs. (3.2) and (3.2). To assess the size of this effect, we again compare both
single-differential resummations (dashed blue and dotted green) to our matched
prediction (solid red) in figure -1001, but for reference include the case
where $\sigma_{+}$ in the matched prediction is evaluated at
$\mu^{\mathrm{I}}$ (solid blue) or $\mu^{\mathrm{II}}$ (solid green) directly.
One can easily verify from e.g. the right panel that for $q_{T}$ above the
right-most vertical line (where $a<a_{1}$), the difference between the solid
blue and the dashed blue curves indeed amounts to a small power-suppressed set
of higher-order terms, while our best prediction (solid red) recovers the
solid blue curve as it must. Similarly, for $q_{T}$ below the left-most
vertical line (where $a>a_{6}$), the difference between the solid green (and
solid red) and dashed green curves can be seen to be a small correction,
reflecting the size of power-suppressed higher-order terms predicted by the
SCETI RGE in this region. The asymptotic limits are interchanged in the left
panel, where $a<a_{1}$ towards the left and $a>a_{6}$ towards the very right
of the plot.

Figure -1001: Slices of the double-differential Drell-Yan cross section at
$q_{T}=15\,\mathrm{GeV}$ (left) and ${\mathcal{T}}=5\,\mathrm{GeV}$ (right).
The solid red, dashed blue, and dotted green curves are identical to the
central results in figure -1002. The solid blue and green curves depict the
SCETI and SCETII limits of our fully resummed result, given in eqs. (3.2) and
(3.2). The thin vertical lines indicate the transition points $a_{i}$
described in section 3.3.

###  4.2 Single-differential spectra with a cut on the other variable

So far we have turned our attention to the cross section differential in both
$q_{T}$ and ${\mathcal{T}}$. In addition to this double spectrum, our setup
also predicts the fully matched and resummed double cumulant cross section,
and the single-differential $q_{T}$ (or ${\mathcal{T}}$) spectrum with a cut
on the other variable; selected results for these observables where already
discussed in section 3.5 from a more technical point of view. In figure -1000,
we show some more detailed results for the single-differential spectra with an
additional cut, where the left panel shows
$\mathrm{d}\sigma({q_{T}^{\mathrm{cut}}})/\mathrm{d}{\mathcal{T}}$ as a
function of ${\mathcal{T}}$ for various values of ${q_{T}^{\mathrm{cut}}}$,
and the right panel shows
$\mathrm{d}\sigma({\mathcal{T}_{\mathrm{cut}}})/\mathrm{d}q_{T}$ as a function
of $q_{T}$ for various values of ${\mathcal{T}_{\mathrm{cut}}}$. By increasing
the value of the cut, they can be seen to approach the inclusive single-
differential spectra (orange solid), with which they must agree when sending
${q_{T}^{\mathrm{cut}}}\to\infty$ or ${\mathcal{T}_{\mathrm{cut}}}\to\infty$,
respectively. This is by construction because we employ cumulant scale setting
as appropriate for this prediction, cf. section 3.5. We observe that cuts on
the other variable shape either spectrum in a very nontrivial way. Tight cuts
$\lesssim 10\,\mathrm{GeV}$ push the peak to lower values and suppress the
tail, where the $q_{T}$ spectrum is somewhat more resilient to cuts on
${\mathcal{T}}$ than vice versa. Intermediate cuts $\sim 10-15\,\mathrm{GeV}$
keep the peak and mostly lead to a suppression in the tail, while the effect
of cuts $\gtrsim 40\,\mathrm{GeV}$ is almost negligible in the $q_{T}$ and
${\mathcal{T}}$ ranges of interest.

Figure -1000: The single-differential ${\mathcal{T}}$ (left) and $q_{T}$
(right) spectrum with a cut on the other variable at NNLL$+$NLO. The different
curves represent different values of the cut. The solid orange lines
correspond to the inclusive single-differential spectrum obtained by lifting
the cut.

##  5 Conclusions

In this paper we calculated the Drell-Yan cross section double-differential in
the transverse momentum $q_{T}$ of the lepton pair and the 0-jettiness
${\mathcal{T}}$. Both ${\mathcal{T}}$ and $q_{T}$ probe the initial state
radiation, leading to Sudakov double logarithms of ${\mathcal{T}}/Q$ and
$q_{T}/Q$ in the cross section. We performed, for the first time, the
simultaneous resummation of both kinds of logarithms, achieving next-to-next-
to-leading logarithmic accuracy and matching the result to next-to-leading
fixed order. We accomplish this resummation by using SCETI and SCETII to
describe the regions ${\mathcal{T}}\ll q_{T}\sim\sqrt{{\mathcal{T}}Q}$ and
${\mathcal{T}}\sim q_{T}\ll\sqrt{{\mathcal{T}}Q}$, respectively, and SCET+ to
describe the bulk of phase space in between these boundaries [51].

Obtaining reliable numerical predictions required several nontrivial steps:
(1) Matching several factorized cross sections for the different regions of
phase space, for which we use a Venn-diagram method to avoid double counting.
(2) Choosing appropriate profile scales for the various ingredients in the
factorization formulas that respect all relevant canonical scaling relations
and at the same time smoothly interpolate between the different regions of
phase space, and varying these scales to estimate perturbative uncertainties.
This is significantly more involved than in the usual single-differential
case, and is further complicated by the requirement to choose scales in impact
parameter ($b_{T}$) space for SCETII. For example, the rapidity scale for the
collinear-soft function in SCET+ has a canonical scaling that does not
coincide with any scale on the SCETI and SCETII boundaries. (3) Ensuring that
scales and scale variations are still, to the extent possible, inherited from
the single-differential resummation of ${\mathcal{T}}$ and $q_{T}$. This makes
our setup flexible enough to incorporate other procedures for estimating the
uncertainty in the individual resummations. (4) To handle the transition
between SCETI, SCET+ and SCETII, we introduced profile scales in terms of a
regime parameter $a$, designed such that $a=1$ for SCETI and $a=2$ for SCETII.
The precise transition points in $a$ were chosen by comparing the various
singular and nonsingular cross section, and are varied as part of the
uncertainty estimate. (5) We also introduced a new hybrid (i.e., $q_{T}$ and
$b_{T}$ dependent) scale choice for $q_{T}$ resummation, allowing the
resummation to strictly take place in $b_{T}$ space, while turning the
resummation on and off using $q_{T}$.

We demonstrated that our simultaneous resummation of ${\mathcal{T}}$ and
$q_{T}$ yields the correct resummed single-differential cross sections after
integrating over either ${\mathcal{T}}$ or $q_{T}$. This requires choosing
scales at the level of the differential or integrated (cumulative) cross
section as appropriate, which we discussed in detail.

While the predictions obtained here are of some direct phenomenological
interest, as ${\mathcal{T}}$ has been measured in bins of $q_{T}$ [48], our
analysis is also an important step towards precise _and_ differential
predictions for LHC cross sections in general. Specifically, the Monte Carlo
event generator Geneva [49, 50] is based on a NNLL′ resummed prediction for
the cross section differential in ${\mathcal{T}}$, and would benefit from the
simultaneous resummation of $q_{T}$. Indeed, our NNLL results clearly indicate
that only resumming the logarithms of either ${\mathcal{T}}$ or $q_{T}$ gives
a poor description of the double-differential cross section. Our methods apply
at any order and for any color-singlet production process, allowing for a
straightforward extension once the relevant perturbative ingredients become
available. We hope that our analysis can pave the way for going beyond single-
differential resummations in many other contexts as well.

## Acknowledgments

We thank Goutam Das, Markus Diehl, and Markus Ebert for discussions. We would
also like to thank Markus Ebert for his contributions to SCETlib. This work is
supported by the ERC grant ERC-STG-2015-677323 and the D-ITP consortium, a
program of the Netherlands Organization for Scientific Research (NWO) that is
funded by the Dutch Ministry of Education, Culture and Science (OCW). FT and
JM thank Nikhef for hospitality. GL thanks DESY for hospitality and the DESY
theory group thanks GL for an ample supply of Dutch pepernoten.

##  Appendix A Plus distributions and Fourier transform

We use the following standard plus distributions with dimensionless arguments,

$\displaystyle\mathcal{L}_{n}(x)$
$\displaystyle\equiv\biggl{[}\frac{\theta(x)\ln^{n}x}{x}\biggr{]}_{+}$
$\displaystyle=\lim_{\beta\to
0}\biggl{[}\frac{\theta(x-\beta)\ln^{n}x}{x}+\delta(x-\beta)\,\frac{\ln^{n+1}\\!\beta}{n+1}\biggr{]}\,,$
(A.1) $\displaystyle\mathcal{L}^{a}(x)$
$\displaystyle\equiv\biggl{[}\frac{\theta(x)}{x^{1-a}}\biggr{]}_{+}$
$\displaystyle=\lim_{\beta\to
0}\biggl{[}\frac{\theta(x-\beta)}{x^{1-a}}+\delta(x-\beta)\frac{x^{a}-1}{a}\biggr{]}\,.$
(A.2)

In intermediate steps we need a two-dimensional plus distribution originally
defined in ref. [51],

$\displaystyle\mathcal{L}_{\Delta}(x_{1},x_{2})\equiv\lim_{\beta\to
0}\frac{\mathrm{d}}{\mathrm{d}x_{1}}\,\frac{\mathrm{d}}{\mathrm{d}x_{2}}\,\Big{[}$
$\displaystyle\theta(x_{2}-x_{1}^{2})\theta(x_{1}-\beta)\ln x_{1}\,(\ln
x_{2}-\ln x_{1})$
$\displaystyle+\frac{1}{4}\,\theta(x_{1}^{2}-x_{2})\theta(x_{2}-\beta^{2})\ln^{2}x_{2}\Big{]}\,.$
(A.3)

Our shorthands for distributions with dimensionful arguments in one spatial
dimension are

$\displaystyle\mathcal{L}_{n}(k,\mu)\equiv\frac{1}{\mu}\mathcal{L}_{n}\Bigl{(}\frac{k}{\mu}\Bigr{)}\,,\qquad\mathcal{L}^{a}(k,\mu)\equiv\frac{1}{\mu}\mathcal{L}^{a}\Bigl{(}\frac{k}{\mu}\Bigr{)}\,.$
(A.4)

In terms of the second class of (power-like) plus distributions, we further
define

$\displaystyle\mathcal{V}_{a}(x)\equiv\frac{e^{-\gamma_{E}a}}{\Gamma(1+a)}\bigl{[}a\mathcal{L}^{a}(x)+\delta(x)\bigr{]}\,,\qquad\mathcal{V}_{a}(k,\mu)\equiv\frac{1}{\mu}\mathcal{V}_{a}\Bigl{(}\frac{k}{\mu}\Bigr{)}\,,$
(A.5)

which have a group property, assuming identical boundary condition $\mu$,

$\int\\!\mathrm{d}\ell\,\mathcal{V}_{a}(\ell,\mu)\,\mathcal{V}_{b}(k-\ell,\mu)=\mathcal{V}_{a+b}(k,\mu)\,,\qquad\mathcal{V}_{0}(k,\mu)=\delta(k)\,.$
(A.6)

Shifting the boundary condition of $\mathcal{V}_{a}(k,\mu)$ from $\mu$ to
$\mu^{\prime}$ is also straightforward,

$\mathcal{V}_{a}(k,\mu)=\Bigl{(}\frac{\mu^{\prime}}{\mu}\Bigr{)}^{a}\,\mathcal{V}_{a}(k,\mu^{\prime})\,.$
(A.7)

We use the conventions from app. C of ref. [77] for logarithmic plus
distributions in two integer spatial dimensions, with
$k_{T}^{2}\equiv\vec{\kern 0.0ptk}_{T}^{2}\geq 0$,

$\displaystyle\delta(\vec{\kern
0.0ptk}_{T})=\frac{1}{\pi}\delta(k_{T}^{2})\,,\qquad\mathcal{L}_{n}(\vec{\kern
0.0ptk}_{T},\mu)$
$\displaystyle\equiv\frac{1}{\pi\mu^{2}}\biggl{[}\frac{\mu^{2}}{k_{T}^{2}}\ln^{n}\biggl{(}\frac{k_{T}^{2}}{\mu^{2}}\biggr{)}\biggr{]}_{+}^{\mu}\equiv\frac{1}{\pi\mu^{2}}\mathcal{L}_{n}\biggl{(}\frac{k_{T}^{2}}{\mu^{2}}\biggr{)}\,.$
(A.8)

Our convention for the two-dimensional Fourier transform also follows ref.
[77],

$\displaystyle\frac{\mathrm{d}f}{\mathrm{d}\vec{\kern
0.0ptp}_{T}}=\int\\!\frac{\mathrm{d}^{2}\vec{\kern
0.0ptb}_{T}}{(2\pi)^{2}}\,e^{+i\vec{\kern 0.0ptp}_{T}\cdot\vec{\kern
0.0ptb}_{T}}\tilde{f}(\vec{\kern 0.0ptb}_{T})\,,\qquad\tilde{f}(\vec{\kern
0.0ptb}_{T})=\int\\!\mathrm{d}^{2}\vec{\kern 0.0ptp}_{T}\,e^{-i\vec{\kern
0.0ptp}_{T}\cdot\vec{\kern 0.0ptb}_{T}}\frac{\mathrm{d}f}{\mathrm{d}\vec{\kern
0.0ptp}_{T}}\,.$ (A.9)

Here we make the mass dimension of $\mathrm{d}f/\mathrm{d}\vec{\kern
0.0ptp}_{T}=\mathrm{d}f/(\mathrm{d}p_{x}\,\mathrm{d}p_{y})$ explicit. If $f$
is azimuthally symmetric, i.e., if for $p_{T}\equiv\lvert\vec{\kern
0.0ptp}_{T}\rvert,b_{T}\equiv\lvert\vec{\kern 0.0ptb}_{T}\rvert$,

$\frac{\mathrm{d}f}{\mathrm{d}\vec{\kern 0.0ptp}_{T}}=\frac{1}{2\pi
p_{T}}\frac{\mathrm{d}f}{\mathrm{d}p_{T}}\,,\qquad\tilde{f}(\vec{\kern
0.0ptb}_{T})=\tilde{f}(b_{T})\,,$ (A.10)

the azimuthal integral can be performed, leaving

$\displaystyle\frac{\mathrm{d}f}{\mathrm{d}p_{T}}=p_{T}\int_{0}^{\infty}\\!\mathrm{d}b_{T}\,b_{T}\,J_{0}(b_{T}p_{T})\,\tilde{f}(b_{T})\,,\qquad\tilde{f}(b_{T})=\int_{0}^{\infty}\\!\mathrm{d}p_{T}\,J_{0}(b_{T}p_{T})\frac{\mathrm{d}f}{\mathrm{d}p_{T}}\,,$
(A.11)

where $J_{0}(x)$ is the zeroth-order Bessel function of the first kind.
Integrating the first expression in eq. (A.11) by parts, the cumulant in
$p_{T}$ is given by

$\displaystyle\int_{0}^{p_{T}^{\rm
cut}}\mathrm{d}p_{T}\,\frac{\mathrm{d}f}{\mathrm{d}p_{T}}=p_{T}^{\rm
cut}\int_{0}^{\infty}\\!\mathrm{d}b_{T}\,J_{1}(b_{T}p_{T}^{\rm
cut})\,\tilde{f}(b_{T})\,,$ (A.12)

where $J_{1}(x)$ is the first-order Bessel function of the first kind. Fourier
transforms of $\mathcal{L}_{n}(\vec{\kern 0.0ptp}_{T},\mu)$ can be found in
table 5 of ref. [77], and are most conveniently expressed in terms of

$L_{b}\equiv\ln\frac{b_{T}^{2}\mu^{2}}{b_{0}^{2}}\,,\qquad b_{0}\equiv
2e^{-\gamma_{E}}\approx 1.12291\dots\,.$ (A.13)

##  Appendix B Perturbative ingredients

###  B.1 Anomalous dimensions

We expand the $\beta$ function of QCD as

$\mu\frac{\mathrm{d}\alpha_{s}(\mu)}{\mathrm{d}\mu}=\beta[\alpha_{s}(\mu)]\,,\qquad\beta(\alpha_{s})=-2\alpha_{s}\sum_{n=0}^{\infty}\beta_{n}\left(\frac{\alpha_{s}}{4\pi}\right)^{n+1}\,.$
(B.1)

The coefficients in the $\overline{\text{MS}}$ scheme are, up to three loops
[93, 94],

$\displaystyle\beta_{0}$
$\displaystyle=\frac{11}{3}\,C_{A}-\frac{4}{3}\,T_{F}\,n_{f}\,,\qquad\beta_{1}=\frac{34}{3}\,C_{A}^{2}-\Bigl{(}\frac{20}{3}\,C_{A}\,+4C_{F}\Bigr{)}\,T_{F}\,n_{f}\,,$
(B.2) $\displaystyle\beta_{2}$
$\displaystyle=\frac{2857}{54}\,C_{A}^{3}+\Bigl{(}C_{F}^{2}-\frac{205}{18}\,C_{F}C_{A}-\frac{1415}{54}\,C_{A}^{2}\Bigr{)}\,2T_{F}\,n_{f}+\Bigl{(}\frac{11}{9}\,C_{F}+\frac{79}{54}\,C_{A}\Bigr{)}\,4T_{F}^{2}\,n_{f}^{2}\,.$

We work with $n_{f}=5$ light flavors. The cusp and noncusp anomalous
dimensions are expanded as

$\Gamma^{i}_{\mathrm{cusp}}(\alpha_{s})=\sum_{n=0}^{\infty}\Gamma^{i}_{n}\Bigl{(}\frac{\alpha_{s}}{4\pi}\Bigr{)}^{n+1}\,,\qquad\gamma(\alpha_{s})=\sum_{n=0}^{\infty}\gamma_{n}\Bigl{(}\frac{\alpha_{s}}{4\pi}\Bigr{)}^{n+1}\,.$
(B.3)

The coefficients of the $\overline{\mathrm{MS}}$ cusp anomalous dimension to
three loops are [95, 96, 97]

$\displaystyle\Gamma_{0}^{i}$ $\displaystyle=4C_{i}\,,$
$\displaystyle\Gamma_{1}^{i}$
$\displaystyle=4C_{i}\Bigl{[}C_{A}\Bigl{(}\frac{67}{9}-\frac{\pi^{2}}{3}\Bigr{)}-\frac{20}{9}\,T_{F}\,n_{f}\Bigr{]}=\frac{4}{3}C_{i}\bigl{[}(4-\pi^{2})C_{A}+5\beta_{0}\bigr{]}\,,$
$\displaystyle\Gamma_{2}^{i}$
$\displaystyle=4C_{i}\Bigl{[}C_{A}^{2}\Bigl{(}\frac{245}{6}-\frac{134\pi^{2}}{27}+\frac{11\pi^{4}}{45}+\frac{22\zeta_{3}}{3}\Bigr{)}+C_{A}\,T_{F}\,n_{f}\Bigl{(}-\frac{418}{27}+\frac{40\pi^{2}}{27}-\frac{56\zeta_{3}}{3}\Bigr{)}$
$\displaystyle\qquad+C_{F}\,T_{F}\,n_{f}\Bigl{(}-\frac{55}{3}+16\zeta_{3}\Bigr{)}-\frac{16}{27}\,T_{F}^{2}\,n_{f}^{2}\Bigr{]}\,,$
(B.4)

where here and in the following $C_{i}=C_{F}\,(C_{A})$ for $i=q\,(g)$. The
fixed-order boundary condition of the resummed rapidity anomalous dimension
eq. (2.26) reads, through NNLL,

$\displaystyle\tilde{\gamma}^{i}_{\nu,\mathrm{FO}}(b_{T},\mu)$
$\displaystyle=\frac{\alpha_{s}(\mu)}{4\pi}\Bigl{[}-2\Gamma^{i}_{0}L_{b}\Bigr{]}+\frac{\alpha_{s}^{2}(\mu)}{(4\pi)^{2}}\Bigl{[}-\Gamma^{i}_{0}\beta_{0}L_{b}^{2}-2\Gamma^{i}_{1}L_{b}+\gamma^{i}_{\nu\,1}\Bigr{]}+\mathcal{O}(\alpha_{s}^{3})\,,$
(B.5)

where we have already used that $\gamma^{i}_{\nu\,0}=0$. For our choice of
regulator, the two-loop boundary condition is given by [70]

$\displaystyle\gamma^{i}_{\nu\,1}=C_{i}\left[-C_{A}\left(\frac{128}{9}-56\zeta_{3}\right)-\beta_{0}\frac{112}{9}\right]\,.$
(B.6)

#### Hard function.

The hard function $H_{\kappa}(Q,\mu)$ is proportional to the square of the
hard matching coefficient [see eq. (B.2)]. The hard matching coefficient for
$q\bar{q}\to Z/\gamma^{\ast}$ is renormalized as

$\displaystyle\mu\frac{\mathrm{d}}{\mathrm{d}\mu}C_{q{\bar{q}}}^{V,A}(Q^{2},\mu)$
$\displaystyle=\gamma_{q{\bar{q}}}^{V,A}(Q^{2},\mu)\,C_{q{\bar{q}}}^{V,A}(Q^{2},\mu)\,,$
$\displaystyle\gamma_{q{\bar{q}}}^{V,A}(Q^{2},\mu)$
$\displaystyle=\Gamma^{q}_{\mathrm{cusp}}\bigl{[}\alpha_{s}(\mu)\bigr{]}\ln\frac{-Q^{2}-\mathrm{i}0}{\mu^{2}}+2\gamma_{C}^{q}\bigl{[}\alpha_{s}(\mu)\bigr{]}\,.$
(B.7)

The coefficients of the quark noncusp anomalous dimension up to two loops are

$\displaystyle\gamma^{q}_{C\,0}$ $\displaystyle=-3C_{F}\,,$
$\displaystyle\gamma^{q}_{C\,1}$
$\displaystyle=-C_{F}\biggl{[}C_{A}\Bigl{(}\frac{41}{9}-26\zeta_{3}\Bigr{)}+C_{F}\Bigl{(}\frac{3}{2}-2\pi^{2}+24\zeta_{3}\Bigr{)}+\beta_{0}\Bigl{(}\frac{65}{18}+\frac{\pi^{2}}{2}\Bigr{)}\biggr{]}\,.$
(B.8)

#### Beam functions and PDFs.

In $b_{T}$ space, the TMD beam function is renormalized as

$\displaystyle\mu\frac{\mathrm{d}}{\mathrm{d}\mu}\tilde{B}_{q}(\omega,b_{T},\mu,\nu)$
$\displaystyle=\tilde{\gamma}_{B}^{q}(\omega,\mu,\nu)\,\tilde{B}_{q}(\omega,b_{T},\mu,\nu)\,,$
$\displaystyle\nu\frac{\mathrm{d}}{\mathrm{d}\nu}\tilde{B}_{q}(\omega,b_{T},\mu,\nu)$
$\displaystyle=-\frac{1}{2}\tilde{\gamma}^{q}_{\nu}(b_{T},\mu)\,\tilde{B}_{q}(\omega,b_{T},\mu,\nu)\,,$
$\displaystyle\tilde{\gamma}^{q}_{B}(\omega,\mu,\nu)$
$\displaystyle=2\Gamma_{\mathrm{cusp}}^{q}[\alpha_{s}(\mu)]\ln\frac{\nu}{\omega}+\tilde{\gamma}_{B}^{q}[\alpha_{s}(\mu)]\,.$
(B.9)

We include a tilde to indicate that $\tilde{\gamma}_{B}^{q}$ is related to the
SCET${}_{\text{II}}$ beam function, even though it does not depend on $b_{T}$.
Its coefficients through two loops are [70]

$\displaystyle\tilde{\gamma}_{B\,0}^{q}$ $\displaystyle=6C_{F}\,,$
$\displaystyle\tilde{\gamma}_{B\,1}^{q}$
$\displaystyle=C_{F}\Bigl{[}(2-24\zeta_{3})C_{A}+(3-4\pi^{2}+48\zeta_{3})C_{F}+\Bigl{(}1+\frac{4\pi^{2}}{3}\Bigr{)}\beta_{0}\Bigr{]}\,.$
(B.10)

The resummed rapidity anomalous dimension
$\tilde{\gamma}^{i}_{\nu}(b_{T},\mu)$ is given in eq. (2.26).

The double-differential beam function satisfies the same RGE as the inclusive
SCET${}_{\text{I}}$ beam function,666We note that ref. [51] incorrectly did
not distinguish between $\gamma^{i}_{B}(\alpha_{s})$ and
$\tilde{\gamma}^{i}_{B}(\alpha_{s})$. This lead to the noncusp contribution to
the collinear-soft anomalous dimension being missing in their eq. (3.26), cf.
our corrected eq. (B.1) and the nonvanishing two-loop noncusp coefficient in
our eq. (B.1).

$\displaystyle\mu\frac{\mathrm{d}}{\mathrm{d}\mu}B_{q}(t,x,\vec{\kern
0.0ptk}_{T},\mu)$
$\displaystyle=\int\\!\mathrm{d}t^{\prime}\,\gamma_{B}^{q}(t-t^{\prime},\mu)\,B_{q}(t^{\prime},x,\vec{\kern
0.0ptk}_{T},\mu)\,,$ $\displaystyle\gamma_{B}^{q}(t,\mu)$
$\displaystyle=-2\Gamma^{q}_{\mathrm{cusp}}(\alpha_{s})\,\mathcal{L}_{0}(t,\mu^{2})+\gamma_{B}^{q}[\alpha_{s}(\mu)]\,\delta(t)\,.$
(B.11)

The coefficients of the SCET${}_{\text{I}}$ quark beam anomalous dimension are
[60, 98]

$\displaystyle\gamma_{B\,0}^{q}$ $\displaystyle=6C_{F}\,,$
$\displaystyle\gamma_{B\,1}^{q}$
$\displaystyle=C_{F}\Bigl{[}\Bigl{(}\frac{146}{9}-80\zeta_{3}\Bigr{)}C_{A}+(3-4\pi^{2}+48\zeta_{3})C_{F}+\Bigl{(}\frac{121}{9}+\frac{2\pi^{2}}{3}\Bigr{)}\beta_{0}\Bigr{]}\,.$
(B.12)

We also require the one-loop coefficients of the PDF anomalous dimension,

$\displaystyle\mu\frac{\mathrm{d}}{\mathrm{d}\mu}f_{i}(x,\mu)$
$\displaystyle=\sum_{j}\int_{x}^{1}\\!\frac{\mathrm{d}z}{z}\,2P_{ij}[\alpha_{s}(\mu),z]\,f_{j}\Bigl{(}\frac{x}{z},\mu\Bigr{)}\,,$
$\displaystyle P_{ij}(\alpha_{s},z)$
$\displaystyle=\sum_{n=0}^{\infty}P^{(n)}_{ij}(z)\left(\frac{\alpha_{s}}{4\pi}\right)^{n+1}\,.$
(B.13)

Note that we expand $P_{ij}(\alpha_{s},z)$ in $\alpha_{s}/(4\pi)$. The one-
loop coefficients are

$\displaystyle P^{(0)}_{qq}(z)=2C_{F}\,\theta(z)P_{qq}(z)\,,\quad
P^{(0)}_{qg}(z)=2T_{F}\,\theta(z)P_{qg}(z)\,,$ (B.14)

in terms of the standard color-stripped one-loop QCD splitting functions

$\displaystyle P_{qq}(z)$
$\displaystyle=2\mathcal{L}_{0}(1-z)-\theta(1-z)(1+z)+\frac{3}{2}\delta(1-z)$
$\displaystyle=\Bigl{[}\theta(1-z)\frac{1+z^{2}}{1-z}\Bigr{]}_{+}\,,$
$\displaystyle P_{qg}(z)$
$\displaystyle=\theta(1-z)\bigl{[}1-2z(1-z)\bigr{]}\,.$ (B.15)

#### Soft and collinear-soft functions.

The RGE of the beam thrust soft function reads

$\displaystyle\mu\frac{\mathrm{d}}{\mathrm{d}\mu}S_{i}(k,\mu)$
$\displaystyle=\int\\!\mathrm{d}k^{\prime}\,\gamma_{S}^{i}(k-k^{\prime},\mu)\,S_{i}(k^{\prime},\mu)\,,$
$\displaystyle\gamma_{S}^{i}(k,\mu)$
$\displaystyle=4\Gamma_{\mathrm{cusp}}^{i}(\alpha_{s})\,\mathcal{L}_{0}(k,\mu)+\gamma^{i}_{S}[\alpha_{s}(\mu)]\,\delta(k)\,.$
(B.16)

For the double-differential soft function in $b_{T}$ space we have

$\displaystyle\mu\frac{\mathrm{d}}{\mathrm{d}\mu}\tilde{S}_{i}(k,b_{T},\mu,\nu)$
$\displaystyle=\tilde{\gamma}_{S}^{i}(\mu,\nu)\,\tilde{S}_{i}(k,b_{T},\mu,\nu)\,,$
$\displaystyle\nu\frac{\mathrm{d}}{\mathrm{d}\nu}\tilde{S}_{i}(k,b_{T},\mu,\nu)$
$\displaystyle=\tilde{\gamma}^{i}_{\nu}(b_{T},\mu)\,\tilde{S}_{i}(k,b_{T},\mu,\nu)\,,$
$\displaystyle\tilde{\gamma}^{i}_{S}(\mu,\nu)$
$\displaystyle=4\Gamma_{\mathrm{cusp}}^{i}(\alpha_{s})\,\ln\frac{\mu}{\nu}+\tilde{\gamma}^{i}_{S}[\alpha_{s}(\mu)]\,,$
(B.17)

where we again use a tilde on the $\mu$ anomalous dimension to indicate that
it relates to the SCET${}_{\text{II}}$ soft function. The RGE of the
collinear-soft function in $b_{T}$ space reads

$\displaystyle\mu\frac{\mathrm{d}}{\mathrm{d}\mu}\tilde{\mathcal{S}}_{i}(k,b_{T},\mu,\nu)$
$\displaystyle=\int\\!\mathrm{d}k^{\prime}\,\gamma_{\mathcal{S}}^{i}(k-k^{\prime},\mu,\nu)\,\tilde{\mathcal{S}}_{i}(k^{\prime},b_{T},\mu,\nu)\,,$
$\displaystyle\nu\frac{\mathrm{d}}{\mathrm{d}\nu}\tilde{\mathcal{S}}_{i}(k,b_{T},\mu,\nu)$
$\displaystyle=\frac{1}{2}\tilde{\gamma}^{i}_{\nu}(b_{T},\mu)\,\tilde{\mathcal{S}}_{i}(k,b_{T},\mu,\nu)\,,$
$\displaystyle\gamma_{\mathcal{S}}^{i}(k,\mu,\nu)$
$\displaystyle=-2\Gamma_{\mathrm{cusp}}^{i}(\alpha_{s})\,\mathcal{L}_{0}\Big{(}k,\frac{\mu^{2}}{\nu}\Big{)}+\gamma^{i}_{\mathcal{S}}[\alpha_{s}(\mu)]\delta(k)\,.$
(B.18)

The soft and collinear-soft noncusp anomalous dimension coefficients are only
nonzero starting at two loops and can be inferred from consistency,

$\displaystyle\gamma^{i}_{S\,0}$
$\displaystyle=\gamma^{i}_{\mathcal{S}\,0}=\tilde{\gamma}^{i}_{S\,0}=0\,,$
$\displaystyle-\gamma^{i}_{S\,1}$
$\displaystyle=\gamma^{i}_{\mathcal{S}\,1}=\tilde{\gamma}^{i}_{S\,1}=C_{i}\left[C_{A}\left(\frac{128}{9}-56\zeta_{3}\right)+\beta_{0}\left(\frac{112}{9}-\frac{2\pi^{2}}{3}\right)\right]\,.$
(B.19)

###  B.2 Fixed-order ingredients

#### Hard process.

The Born cross section for $q{\bar{q}}\to Z/\gamma^{\ast}\to\ell^{+}\ell^{-}$
is given by

$\frac{\mathrm{d}\sigma_{B}^{q}}{\mathrm{d}Q}=\frac{8\pi\alpha_{\text{em}}^{2}}{3N_{c}E_{\mathrm{cm}}^{2}Q}\biggl{[}Q_{q}^{2}+\frac{(v_{q}^{2}+a_{q}^{2})(v_{\ell}^{2}+a_{\ell}^{2})-2Q_{q}v_{q}v_{\ell}(1-m_{Z}^{2}/Q^{2})}{(1-m_{Z}^{2}/Q^{2})^{2}+m_{Z}^{2}\Gamma_{Z}^{2}/Q^{4}}\biggr{]}\,,$
(B.20)

where $Q_{q}$ is the quark charge in units of $\lvert e\rvert$, $v_{\ell,q}$
and $a_{\ell,q}$ are the standard vector and axial couplings of the leptons
and quarks, and $m_{Z}$ and $\Gamma_{Z}$ are the mass and width of the $Z$
boson. The one-loop Wilson coefficient $C_{q{\bar{q}}}^{V,A}(Q^{2},\mu)$ from
matching the quark current in QCD onto SCET was computed in refs. [99, 100].
This leads to the following hard function [46],

$\displaystyle H_{ij}(Q,\mu)$
$\displaystyle=\sum_{q}\frac{\mathrm{d}\sigma_{B}^{q}}{\mathrm{d}Q}\,\bigl{(}\delta_{iq}\delta_{j{\bar{q}}}+\delta_{i{\bar{q}}}\delta_{jq}\bigr{)}\,|C_{q{\bar{q}}}^{V,A}(Q^{2},\mu)|^{2}\,,$
$\displaystyle|C_{q{\bar{q}}}^{V,A}(Q^{2},\mu)|^{2}$
$\displaystyle=1+2\operatorname{Re}\,\biggl{\\{}\frac{\alpha_{s}(\mu)}{4\pi}\,C_{F}\Bigl{[}-\ln^{2}\frac{Q^{2}}{\mu^{2}}+3\ln\frac{Q^{2}}{\mu^{2}}-8+\frac{7\pi^{2}}{6}\Bigr{]}\biggr{\\}}+\mathcal{O}(\alpha_{s}^{2})\,,$
(B.21)

where $\operatorname{Re}$ denotes the real part.

#### Beam functions.

The one-loop matching coefficients for the single-differential
SCET${}_{\text{II}}$ beam function in $b_{T}$ space are given by [101, 70]

$\displaystyle\tilde{\mathcal{I}}_{qj}(\omega,b_{T},z,\mu,\nu)$ (B.22)
$\displaystyle=\delta_{qj}\delta(1-z)+\frac{\alpha_{s}(\mu)}{4\pi}\biggl{[}\delta_{qj}\delta(1-z)\Bigl{(}\Gamma^{q}_{0}\ln\frac{\nu}{\omega}+\frac{\gamma^{q}_{B,0}}{2}\Bigr{)}L_{b}-L_{b}P^{(0)}_{qj}(z)+\tilde{I}_{qj}^{(1)}(z)\biggr{]}+\mathcal{O}(\alpha_{s}^{2})\,,$

where $L_{b}$ was defined in eq. (A.13) and the boundary conditions at
$L_{b}=0,\nu=\omega$ are

$\displaystyle\tilde{I}^{(1)}_{qq}(z)$
$\displaystyle=C_{F}\,\,\theta(z)\theta(1-z)\,2(1-z)\,,$
$\displaystyle\tilde{I}^{(1)}_{qg}(z)$
$\displaystyle=T_{F}\,\theta(z)\theta(1-z)\,4z(1-z)\,.$ (B.23)

As for its anomalous dimension, we use a tilde to indicate that these boundary
conditions are part of the SCET${}_{\text{II}}$ beam function, even though
they do not depend on $b_{T}$.

It is convenient to decompose the matching coefficients for the double-
differential SCET${}_{\text{I}}$ quark beam function as

$\mathcal{I}_{qj}(t,z,\vec{\kern 0.0ptk}_{T},\mu)=\delta(\vec{\kern
0.0ptk}_{T})\,\mathcal{I}_{qj}(t,z,\mu)+\Delta\mathcal{I}_{qj}(t,z,\vec{\kern
0.0ptk}_{T},\mu)\,,$ (B.24)

where $\mathcal{I}_{qj}(t,z,\mu)$ is the matching coefficient for the
inclusive quark beam function [46, 60],

$\displaystyle\mathcal{I}_{qj}(t,z,\mu)$
$\displaystyle=\delta_{qj}\,\delta(t)\delta(1-z)$ (B.25)
$\displaystyle\quad+\frac{\alpha_{s}}{4\pi}\Bigl{[}\Gamma_{0}^{q}\delta_{qj}\,\mathcal{L}_{1}(t,\mu^{2})\,\delta(1-z)+\mathcal{L}_{0}(t,\mu^{2})\,\tilde{P}^{(0)}_{qj}(z)+\delta(t)\,I^{(1)}_{qj}(z)\Bigr{]}+\mathcal{O}(\alpha_{s}^{2})\,,$

with the finite terms in this case given by

$\displaystyle I^{(1)}_{qq}(z)$
$\displaystyle=2C_{F}\,\theta(z)\Bigl{[}\mathcal{L}_{1}(1-z)(1+z^{2})-\frac{\pi^{2}}{6}\delta(1-z)+\theta(1-z)\Bigl{(}1-z-\frac{1+z^{2}}{1-z}\ln
z\Bigr{)}\Bigr{]}\,,$ $\displaystyle I^{(1)}_{qg}(z)$
$\displaystyle=2T_{F}\,\theta(z)\Bigl{[}P_{qg}(z)\Bigl{(}\ln\frac{1-z}{z}-1\Bigr{)}+\theta(1-z)\Bigr{]}\,,$
(B.26)

and using the shorthand

$\displaystyle\tilde{P}^{(0)}_{qj}(z)\equiv
P^{(0)}_{qj}(z)-\delta_{qj}\delta(1-z)\frac{\gamma_{B\,0}^{q}}{2}=\begin{cases}2C_{F}\mathcal{L}_{0}(1-z)(1+z^{2})\,,&j=q\,,\\\
2T_{F}\Bigl{[}(1-z)^{2}+z^{2}\Bigr{]}\,,&j=g\,.\end{cases}$

The $\Delta\mathcal{I}_{qj}$ piece can be interpreted as a correction over the
limit $t\ll k_{T}^{2}$, where recoil from collinear radiation is power
suppressed and the double-differential beam function becomes proportional to
$\delta(\vec{\kern 0.0ptk}_{T})$. Specifically, it scales as

$\Delta\mathcal{I}_{qj}(t,z,\vec{\kern
0.0ptk}_{T},\mu)\sim\biggl{[}\frac{1}{t}\biggr{]}_{+}\biggl{[}\frac{1}{k_{T}^{2}}\biggr{]}_{+}\times\mathcal{O}\Bigl{(}\frac{t}{k_{T}^{2}}\Bigr{)}\quad\text{for}\quad
t\ll k_{T}^{2}\,,$ (B.27)

and by construction satisfies

$\int\,\mathrm{d}^{2}\vec{\kern
0.0ptk}_{T}\,\Delta\mathcal{I}_{qj}(t,z,\vec{\kern 0.0ptk}_{T},\mu)=0\,.$
(B.28)

At one loop it can be extracted from the full calculation of
$\mathcal{I}_{qj}(t,z,\vec{\kern 0.0ptk}_{T},\mu)$ [102, 59] and has the
compact form

$\displaystyle\Delta\mathcal{I}_{qj}(t,z,\vec{\kern 0.0ptk}_{T},\mu)$
$\displaystyle=\frac{\alpha_{s}(\mu)}{4\pi}\Delta I_{qj}^{(1)}(t,z,\vec{\kern
0.0ptk}_{T})+\mathcal{O}(\alpha_{s}^{2})\,,$ $\displaystyle\Delta
I_{qj}^{(1)}(t,z,\vec{\kern 0.0ptk}_{T})$
$\displaystyle=\frac{\theta(t)}{t}\tilde{P}^{(0)}_{qj}(z)\biggl{[}\frac{1}{\pi}\delta\Bigl{(}k_{T}^{2}-\frac{1-z}{z}t\Bigr{)}-\delta(\vec{\kern
0.0ptk}_{T})\biggr{]}\,.$ (B.29)

The second line is regular in $t$ because the term in square brackets vanishes
as $t\to 0$. After accumulating over the transverse plane up to
$q_{T}^{\mathrm{cut}}>0$, we have

$\int\,\mathrm{d}^{2}\vec{\kern
0.0ptk}_{T}\,\theta(q_{T}^{\mathrm{cut}}-\lvert\vec{\kern
0.0ptk}_{T}\rvert)\,\Delta I_{qj}^{(1)}(t,z,\vec{\kern
0.0ptk}_{T})=-\frac{\theta(t)}{t}\tilde{P}^{(0)}_{qj}(z)\,\theta\Bigl{[}(q_{T}^{\mathrm{cut}})^{2}<\frac{1-z}{z}t\Bigr{]}\,.$
(B.30)

#### Soft and collinear-soft functions.

The (beam) thrust soft function is [103, 104, 46]

$\displaystyle S_{i}(k,\mu)$
$\displaystyle=\delta(k)+\frac{\alpha_{s}(\mu)}{4\pi}\Bigl{[}-4\Gamma_{0}^{i}\,\mathcal{L}_{1}(k,\mu)+\frac{\pi^{2}}{3}C_{i}\,\delta(k)\Bigr{]}+\mathcal{O}(\alpha_{s}^{2})\,.$
(B.31)

The one-loop collinear-soft function in $b_{T}$ space is [51]

$\displaystyle\tilde{\mathcal{S}}_{i}(k,b_{T},\mu,\nu)=\delta(k)$
$\displaystyle+\frac{\alpha_{s}(\mu)}{4\pi}\biggl{\\{}-\Gamma_{0}^{i}L_{b}\,\mathcal{L}_{0}(k,\mu)+\Gamma_{0}^{i}\Bigl{[}-\frac{1}{2}L_{b}^{2}-L_{b}\ln\frac{\nu}{\mu}\Bigr{]}\,\delta(k)-\frac{\pi^{2}}{3}C_{i}\,\delta(k)\biggr{\\}}$
$\displaystyle+\mathcal{O}(\alpha_{s}^{2})\,.$ (B.32)

It is again convenient to decompose the double-differential soft function
computed in ref. [51] into separate pieces with distinct power counting,

$S_{i}(k,\vec{\kern 0.0ptk}_{T},\mu,\nu)=\delta(k)\,S_{i}(\vec{\kern
0.0ptk}_{T},\mu,\nu)+\Delta S_{i}(k,\vec{\kern 0.0ptk}_{T},\mu,\nu)\,.$ (B.33)

Here $S_{i}(\vec{\kern 0.0ptk}_{T},\mu,\nu)$ is the standard single-
differential $q_{T}$ soft function, which in $b_{T}$ space at one loop is
given by [69]

$\tilde{S}_{i}(b_{T},\mu,\nu)=1+\frac{\alpha_{s}(\mu)}{4\pi}\Bigl{[}-\frac{\Gamma^{i}_{0}}{2}L_{b}^{2}+2\Gamma^{i}_{0}L_{b}\ln\frac{\mu}{\nu}-\frac{\pi^{2}}{3}C_{i}\Bigr{]}\,.$
(B.34)

The second term in eq. (B.33) can again be interpreted as a correction, in
this case over the limit $k\gg\vec{\kern 0.0ptk}_{T}$ where the contribution
of soft radiation to the ${\mathcal{T}}=k$ measurement becomes power
suppressed. In momentum space this term satisfies

$\displaystyle\int\\!\mathrm{d}k\,\Delta S_{i}(k,\vec{\kern
0.0ptk}_{T},\mu,\nu)$ $\displaystyle=0\,,$ $\displaystyle\Delta
S_{i}(k,\vec{\kern 0.0ptk}_{T},\mu,\nu)$
$\displaystyle\sim\biggl{[}\frac{1}{k}\biggr{]}_{+}\biggl{[}\frac{1}{k_{T}^{2}}\biggr{]}_{+}\times\mathcal{O}\Bigl{(}\frac{k_{T}^{2}}{k^{2}}\Bigr{)}\quad\text{for}\quad
k_{T}^{2}\ll k^{2}\,.$ (B.35)

Equivalently, in position space we have

$\displaystyle\int\\!\mathrm{d}k\,\Delta\tilde{S}_{i}(k,b_{T},\mu,\nu)$
$\displaystyle=0\,,$ $\displaystyle\Delta\tilde{S}_{i}(k,b_{T},\mu,\nu)$
$\displaystyle\sim\biggl{[}\frac{1}{k}\biggr{]}_{+}\times\mathcal{O}\Bigl{(}\frac{1}{b_{T}^{2}k^{2}}\Bigr{)}\quad\text{for}\quad\frac{1}{b_{T}^{2}}\ll
k^{2}\,.$ (B.36)

At one loop, $\Delta S_{i}$ is given by

$\displaystyle\Delta S_{i}(k,\vec{\kern 0.0ptk}_{T},\mu,\nu)$
$\displaystyle=\frac{\alpha_{s}(\mu)}{4\pi}\Delta S_{i,1}(k,\vec{\kern
0.0ptk}_{T})+\mathcal{O}(\alpha_{s}^{2})\,,$ $\displaystyle\Delta
S_{i,1}(k,\vec{\kern 0.0ptk}_{T})$
$\displaystyle=4C_{i}\biggl{[}\frac{2}{\pi\mu^{3}}\mathcal{L}_{\Delta}\Bigl{(}\frac{k}{\mu},\frac{k_{T}^{2}}{\mu^{2}}\Bigr{)}-\delta(k)\mathcal{L}_{1}(\vec{\kern
0.0ptk}_{T},\mu)\biggr{]}\,.$ (B.37)

The second line is not yet manifestly independent of $\mu$, but can be
simplified noting that

$\mathcal{L}_{\Delta}(x_{1},x_{2})-\delta(x_{1})\mathcal{L}_{1}(x_{2})=\frac{\mathrm{d}}{\mathrm{d}x_{1}}\frac{\mathrm{d}}{\mathrm{d}x_{2}}\theta(x_{2}-x_{1}^{2})\Bigl{[}-\frac{1}{2}\ln^{2}\frac{x_{1}^{2}}{x_{2}}\Bigr{]}\,.$
(B.38)

It is straightforward to show this by writing all three distributions in terms
of $\theta(x_{1}-\beta)$ and $\theta(x_{2}-\beta^{2})$ for infinitesimal
$\beta$, collecting terms, and noting that the result is finite as $\beta\to
0$. From eq. (B.38) we can immediately read off the fixed-order double
cumulant of $\Delta S_{i,1}$ for ${\mathcal{T}}_{\mathrm{cut}}>0$,
$q_{T}^{\mathrm{cut}}>0$,

$\displaystyle\int^{{\mathcal{T}}_{\mathrm{cut}}}\\!\mathrm{d}k\,\int\\!\mathrm{d}^{2}\vec{\kern
0.0ptk}_{T}\,\theta(q_{T}^{\mathrm{cut}}-\lvert\vec{\kern
0.0ptk}_{T}\rvert)\,\Delta S_{i,1}(k,\vec{\kern
0.0ptk}_{T})=4C_{F}\,\theta(q_{T}^{\mathrm{cut}}-{\mathcal{T}}_{\mathrm{cut}})\Bigl{[}-2\ln^{2}\frac{{\mathcal{T}}_{\mathrm{cut}}}{q_{T}^{\mathrm{cut}}}\Bigr{]}\,,$
(B.39)

where the dependence on $\mu$ drops out as expected. Inserting eq. (B.38) and
integrating by parts also yields the cumulant up to
${\mathcal{T}}_{\mathrm{cut}}>0$ in position space,

$\displaystyle\int^{{\mathcal{T}}_{\mathrm{cut}}}\\!\mathrm{d}k\,\Delta\tilde{S}_{i,1}(k,b_{T})=4C_{F}\biggl{[}\frac{1}{4}x^{2}\phantom{}_{3}F_{4}\Bigl{(}1,1,1;2,2,2,2;-\frac{x^{2}}{4}\Bigr{)}-2\ln^{2}\frac{xe^{\gamma_{E}}}{2}\biggr{]}\,,$
(B.40)

where $x\equiv b_{T}{\mathcal{T}}_{\mathrm{cut}}$ and
$\phantom{}{}_{i}F_{j}(x_{1},\ldots,x_{i};y_{1},\ldots,y_{j};z)$ is the
generalized hypergeometric function. The right hand side of eq. (B.40)
asymptotes to $1/x^{2}$ as $x\to\infty$, as required by the scaling law in eq.
(B.2). We also need the spectrum of $\Delta\tilde{S}_{i,1}$ at
${\mathcal{T}}>0$ in position space,

$\displaystyle\Delta\tilde{S}_{i,1}({\mathcal{T}},b_{T})=4C_{F}\,\frac{1}{{\mathcal{T}}}\biggl{[}\frac{1}{2}x^{2}\phantom{}_{2}F_{3}\Bigl{(}1,1;2,2,2;-\frac{x^{2}}{4}\Bigr{)}-4\ln\frac{xe^{\gamma_{E}}}{2}\biggr{]}\,,$
(B.41)

where this time $x\equiv b_{T}{\mathcal{T}}$ and the term in square brackets
again asymptotes to $1/x^{2}$ as $x\to\infty$.

###  B.3 Renormalization-group evolution

#### SCETI.

The closed-form solution of eq. (B.1) is [104, 64]

$\displaystyle B_{q}(t,x,\vec{\kern 0.0ptk}_{T},\mu)$
$\displaystyle=\exp\Bigl{[}4K^{q}_{\Gamma}(\mu_{B},\mu)+K^{q}_{\gamma_{B}}(\mu_{B},\mu)\Bigr{]}$
$\displaystyle\quad\times\int\\!\mathrm{d}t^{\prime}\,\mathcal{V}_{-2\eta^{q}_{\Gamma}(\mu_{B},\,\mu)}(t-t^{\prime},\mu_{B}^{2})\,B_{q}(t^{\prime},x,\vec{\kern
0.0ptk}_{T},\mu_{B})\,,$ (B.42)

where $\mathcal{V}_{\eta}$ was defined in eq. (A.5). Similarly, the solution
of eq. (B.1) is

$\displaystyle S_{i}(k,\mu)$
$\displaystyle=\exp\Bigl{[}-4K^{i}_{\Gamma}(\mu_{S},\mu)+K^{i}_{\gamma_{S}}(\mu_{S},\mu)\Bigr{]}\int\\!\mathrm{d}k^{\prime}\,\mathcal{V}_{4\eta^{i}_{\Gamma}(\mu_{S},\,\mu)}(k-k^{\prime},\mu_{S})\,S_{i}(k^{\prime},\mu_{S})\,.$
(B.43)

The definitions of $\eta^{i}_{\Gamma}$, $K^{i}_{\Gamma}$, and $K^{i}_{\gamma}$
and their approximate analytical form at NNLL are given for example in app.
A.4 of ref. [105]. The solution of the hard RGE in eq. (B.1) in our notation
is also found there.

#### SCETII.

Evolving first in $\nu$ and then in $\mu$ (from right to left), the solution
of eq. (B.1) is

$\displaystyle\tilde{B}_{q}(\omega,b_{T},\mu,\nu)$
$\displaystyle=\exp\Bigl{[}2\eta^{q}_{\Gamma}(\mu_{B},\mu)\ln\frac{\nu}{\omega}+K^{q}_{\tilde{\gamma}_{B}}(\mu_{B},\mu)\Bigr{]}$
$\displaystyle\quad\times\exp\Bigl{[}-\frac{1}{2}\ln\frac{\nu}{\nu_{B}}\,\tilde{\gamma}^{q}_{\nu}(b_{T},\mu_{B})\Bigr{]}\,\tilde{B}_{q}(\omega,b_{T},\mu_{B},\nu_{B})\,,$
(B.44)

where the resummed rapidity anomalous dimension
$\tilde{\gamma}^{q}_{\nu}(b_{T},\mu_{B})$ is given by eq. (2.26). For the
double-differential soft function renormalized as in eq. (B.1), we have

$\displaystyle\tilde{S}_{i}(k,b_{T},\mu,\nu)$
$\displaystyle=\exp\Bigl{[}-4\eta^{i}_{\Gamma}(\mu_{S},\mu)\ln\frac{\nu}{\mu_{S}}+4K^{i}_{\Gamma}(\mu_{S},\mu)+K^{i}_{\tilde{\gamma}_{S}}(\mu_{S},\mu)\Bigr{]}$
$\displaystyle\quad\times\exp\Bigl{[}\ln\frac{\nu}{\nu_{S}}\,\tilde{\gamma}^{i}_{\nu}(b_{T},\mu_{S})\Bigr{]}\,\tilde{S}_{i}(k,b_{T},\mu_{S},\nu_{S})\,.$
(B.45)

#### SCET+.

Again evolving first in $\nu$ and then in $\mu$, the solution of the
collinear-soft RGE in eq. (B.1) in $b_{T}$ space is given by

$\displaystyle\tilde{\mathcal{S}}_{i}(k,b_{T},\mu,\nu)$
$\displaystyle=\exp\Bigl{[}4K^{i}_{\Gamma}(\mu_{\mathcal{S}},\mu)+K^{i}_{\gamma_{\mathcal{S}}}(\mu_{\mathcal{S}},\mu)\Bigr{]}\int\\!\mathrm{d}k^{\prime}\,\mathcal{V}_{-2\eta^{i}_{\Gamma}(\mu_{\mathcal{S}},\,\mu)}\Bigl{(}k-k^{\prime},\frac{\mu_{\mathcal{S}}^{2}}{\nu}\Bigr{)}$
$\displaystyle\quad\times\exp\Bigl{[}\frac{1}{2}\ln\frac{\nu}{\nu_{\mathcal{S}}}\,\tilde{\gamma}^{i}_{\nu}(b_{T},\mu_{\mathcal{S}})\Bigr{]}\,\tilde{\mathcal{S}}_{i}(k^{\prime},b_{T},\mu_{\mathcal{S}},\nu_{\mathcal{S}})\,.$
(B.46)

The rapidity evolution factor on the second line does not depend on
$k^{\prime}$ and thus may be taken out of the convolution integral.

###  B.4 Beam function convolutions with RG kernels

In SCETI the combined beam and soft renormalization group running in momentum
space has the functional form $\mathcal{V}_{\eta}(k,\mu)$ [see eq. (A.5)],
where the convolution variable $k$ is ultimately fixed by the overall
${\mathcal{T}}$ measurement. To evaluate the resummed SCETI cross section as a
function of ${\mathcal{T}}$ (${\mathcal{T}}_{\mathrm{cut}}$) we require
convolutions of $\mathcal{V}_{\eta}$ with the double-differential beam
function at finite ${\mathcal{T}}$ (integrated up to
${\mathcal{T}}_{\mathrm{cut}}$). Convolutions of $\mathcal{V}_{\eta}$ with the
inclusive beam function reduce to a linear combination of

$Q_{i}\\!\int\\!\mathrm{d}k^{\prime}\,\mathcal{V}_{\eta}(k-k^{\prime},\mu)\,\mathcal{L}_{n}(Q_{i}k^{\prime},\mu^{2})=\int\\!\mathrm{d}k^{\prime}\,\mathcal{V}_{\eta}(k-k^{\prime},\mu)\,\mathcal{L}_{n}\Bigl{(}k^{\prime},\frac{\mu^{2}}{Q_{i}}\Bigr{)}\,,$
(B.47)

with $Q_{i}$ as in eq. (2.1), and are straightforward to evaluate using
results in app. B of ref. [64]. The same is true for convolutions of the
evolution kernel with the single-differential ${\mathcal{T}}$ soft function
and, in SCET+, with the collinear-soft function. (Cross terms between beam,
csoft, or soft functions are absent at one loop.) The only nontrivial
ingredient are convolutions of $\mathcal{V}_{\eta}(k,\mu)$ with the one-loop
$\Delta I_{qj}^{(1)}$ piece defined in eq. (B.2). Depending on the
measurement, we distinguish the following four cases for the resulting Mellin
kernel:

  1. 1.

cumulant up to ${\mathcal{T}}_{\mathrm{cut}}>0$, cumulant up to
$q_{T}^{\mathrm{cut}}>0$,
$r\equiv(q_{T}^{\mathrm{cut}})^{2}/(Q_{i}{\mathcal{T}}_{\mathrm{cut}})$:

$\displaystyle Q_{i}\\!\int\\!\mathrm{d}k\,\int\\!\mathrm{d}^{2}\vec{\kern
0.0ptk}_{T}\,\theta({\mathcal{T}}_{\mathrm{cut}}-k)\,\theta(q_{T}^{\mathrm{cut}}-\lvert\vec{\kern
0.0ptk}_{T}\rvert)\,\int\\!\mathrm{d}k^{\prime}\,\mathcal{V}_{\eta}(k-k^{\prime},{\mathcal{T}_{\mathrm{cut}}})\,\Delta
I_{qj}^{(1)}(Q_{i}k^{\prime},z,\vec{\kern 0.0ptk}_{T})$
$\displaystyle=\theta\Bigl{(}r>\frac{1-z}{z}\Bigr{)}P_{qj}^{(0)}(z)\,\frac{e^{-\gamma_{E}\eta}}{\Gamma(1+\eta)}\biggl{[}-B_{1-rz/(1-z)}(1+\eta,0)\biggr{]}\,,$
(B.48)

where $B_{x}(a,b)$ is the incomplete Beta function,

$B_{x}(a,b)=\int_{0}^{x}\\!\mathrm{d}t\,t^{a-1}(1-t)^{b-1}\,.$ (B.49)

  2. 2.

spectrum at ${\mathcal{T}}>0$, cumulant up to $q_{T}^{\mathrm{cut}}>0$,
$r\equiv(q_{T}^{\mathrm{cut}})^{2}/(Q_{i}{\mathcal{T}})$:

$\displaystyle Q_{i}\\!\int\\!\mathrm{d}k\,\int\\!\mathrm{d}^{2}\vec{\kern
0.0ptk}_{T}\,\delta({\mathcal{T}}-k)\,\theta(q_{T}^{\mathrm{cut}}-\lvert\vec{\kern
0.0ptk}_{T}\rvert)\,\int\\!\mathrm{d}k^{\prime}\,\mathcal{V}_{\eta}(k-k^{\prime},{\mathcal{T}})\,\Delta
I_{qj}^{(1)}(Q_{i}k^{\prime},z,\vec{\kern 0.0ptk}_{T})$ (B.50)
$\displaystyle=\frac{1}{{\mathcal{T}}}\,\theta\Bigl{(}r>\frac{1-z}{z}\Bigr{)}P_{qj}^{(0)}(z)\,\frac{e^{-\gamma_{E}\eta}}{\Gamma(1+\eta)}\biggl{[}-\eta
B_{1-rz/(1-z)}(1+\eta,0)-\Bigl{(}1-\frac{rz}{1-z}\Bigr{)}^{\eta}\biggr{]}\,.$

  3. 3.

cumulant up to ${\mathcal{T}}_{\mathrm{cut}}$, spectrum at $q_{T}>0$,
$r\equiv(q_{T})^{2}/(Q_{i}{\mathcal{T}}_{\mathrm{cut}})$:

$\displaystyle Q_{i}\\!\int\\!\mathrm{d}k\,\int\\!\mathrm{d}^{2}\vec{\kern
0.0ptk}_{T}\,\theta({\mathcal{T}}_{\mathrm{cut}}-k)\,\delta(q_{T}-\lvert\vec{\kern
0.0ptk}_{T}\rvert)\,\int\\!\mathrm{d}k^{\prime}\,\mathcal{V}_{\eta}(k-k^{\prime},{\mathcal{T}_{\mathrm{cut}}})\,\Delta
I_{qj}^{(1)}(Q_{i}k^{\prime},z,\vec{\kern 0.0ptk}_{T})$
$\displaystyle=\frac{2}{q_{T}}\,\theta\Bigl{(}r>\frac{1-z}{z}\Bigr{)}P_{qj}^{(0)}(z)\,\frac{e^{-\gamma_{E}\eta}}{\Gamma(1+\eta)}\Bigl{(}1-\frac{rz}{1-z}\Bigr{)}^{\eta}\,.$
(B.51)

  4. 4.

spectrum at ${\mathcal{T}}>0$, spectrum at $q_{T}>0$,
$r\equiv(q_{T})^{2}/(Q_{i}{\mathcal{T}})$:

$\displaystyle Q_{i}\\!\int\\!\mathrm{d}k\,\int\\!\mathrm{d}^{2}\vec{\kern
0.0ptk}_{T}\,\delta({\mathcal{T}}-k)\,\delta(q_{T}-\lvert\vec{\kern
0.0ptk}_{T}\rvert)\,\int\\!\mathrm{d}k^{\prime}\,\mathcal{V}_{\eta}(k-k^{\prime},{\mathcal{T}})\,\Delta
I_{qj}^{(1)}(Q_{i}k^{\prime},z,\vec{\kern 0.0ptk}_{T})$
$\displaystyle=\frac{2}{q_{T}}\frac{1}{{\mathcal{T}}}\,P_{qj}^{(0)}(z)\,\mathcal{V}_{\eta}\Bigl{(}1-\frac{rz}{1-z}\Bigr{)}$
(B.52)

In the first three cases the overall $\theta$ function cuts off the final PDF
integral at

$z<z_{\mathrm{cut}}\equiv\frac{1}{1+r}\,,$ (B.53)

and the Mellin kernel is regular up to and including $z_{\mathrm{cut}}$. In
the last case we instead find a singularity at $z=z_{\mathrm{cut}}$, i.e., the
subtraction from $\mathcal{V}_{\eta}$ now acts directly on the PDF integral.
In either case we have exploited that terms proportional to $\delta(1-z)$ are
cut off since $r>0$, so we could replace $\tilde{P}_{qj}^{(0)}$ by
$P_{qj}^{(0)}$ throughout. We have also set
$\mu={\mathcal{T}}\,({\mathcal{T}_{\mathrm{cut}}}$) in the boundary condition
of $\mathcal{V}_{\eta}$ on the left-hand side, which can always be achieved by
a shift in $\mu$. This ensures that the right-hand side depends only on the
dimensionless parameters $r$ and $\eta$, up to an overall dimensionful
prefactor. It is straightforward to check that for $\eta\to 0$ (at fixed
order), the above results reduce to cumulants (spectra) of $\Delta
I^{(1)}_{qj}$ itself.

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