order functions for a given logarithmic accuracy.

Equipped with Eqs. (3.15)-(3.16), we can calculate the resummed projected
transverse momentum distributions. In this work, we will present the results
in particular at NLL′, N2LL, and at approximate N2LL′ levels. For NLL′, N2LL,
and strict N2LL′ accurate calculations the required precision for the various
anomalous dimensions and fixed-order functions have been summarised in Table.
1. In particular, for the N2LL′ result, the hard and soft sectors need to be
known at full N2LO accuracy. However, only the logarithmic terms, which are
derived from Eq. (3.2) and Eqs. (3.11)-(3.12), are included in this work. We
thus label our results with this approximation as aN2LL′ in the rest of this
paper.

###  3.3 Observables

In Eq. (3.14), we have presented the master formula of the resummed $q_{\tau}$
spectrum with a general choice of $\vec{\tau}$. In Sec. 4, we will investigate
three observables, $q_{\mathrm{T,out}}$, $q_{\mathrm{T,in}}$, and
$\Delta\Phi_{{t{\bar{t}}}}$. The calculations of the first two observables are
immediate from Eq. (3.14) by choosing the reference vector $\vec{\tau}$ to be
perpendicular or parallel with respect to the top-quark transverse momentum
$\vec{P}_{t}^{\perp}$,

$\displaystyle q_{\tau}\,=\,~{}$ $\displaystyle
q_{\mathrm{T,out}}\,,\qquad\mathrm{if}~{}\vec{\tau}=\pm\;\vec{n}\times\frac{\vec{P}_{t}^{\perp}}{|\vec{P}_{t}^{\perp}|}\,,$
(3.19) $\displaystyle q_{\tau}\,=\,~{}$ $\displaystyle
q_{\mathrm{T,in}}\,,\qquad~{}~{}\\!\mathrm{if}~{}\vec{\tau}=\pm\;\frac{\vec{P}_{t}^{\perp}}{|\vec{P}_{t}^{\perp}|}\,,$
(3.20)

where the unit vector $\vec{n}$ characterises the flight direction of one of
the colliding protons. Since the value of $q_{\tau}$ only concerns the
magnitude of the projected component, the calculation with either $+$ or $-$
sign in Eqs. (3.19)-(3.20) gives the same result. In order to determine the
azimuthal distribution $\Delta\Phi_{{t{\bar{t}}}}$, it is worth noting that in
the vicinity of $\Delta\Phi_{{t{\bar{t}}}}\to\pi$, we are able to perform the
expansion in
$\Delta\phi_{{t{\bar{t}}}}\equiv(\pi-\Delta\Phi_{{t{\bar{t}}}})\sim\lambda_{\tau}$,

$\displaystyle\Delta\Phi_{{t{\bar{t}}}}\equiv\arccos\left[\frac{\vec{P}_{t}^{\perp}\cdot\vec{P}_{\bar{t}}^{\perp}}{|\vec{P}_{t}^{\perp}||\vec{P}_{\bar{t}}^{\perp}|}\right]\sim\pi-\frac{q_{\mathrm{T,out}}}{|\vec{P}_{t}^{\perp}|}+\mathcal{O}(\lambda^{2}_{\tau})\,.$
(3.21)

In this work, only the leading kinematical effects will be taken into account,
such that the azimuthal spectrum can be calculated from the results on
$q_{\mathrm{T,out}}$,

$\displaystyle\frac{{\rm d}{\sigma^{\mathrm{res}}_{{t{\bar{t}}}}}}{{\rm
d}M_{t{\bar{t}}}^{2}\,{\rm d}^{2}\vec{P}_{t}^{\perp}\,{\rm
d}Y_{t{\bar{t}}}\,{\rm d}\Delta\phi_{{t{\bar{t}}}}}\,=$
$\displaystyle\;\left|\vec{P}_{t}^{\perp}\right|\,\frac{{\rm
d}{\sigma^{\mathrm{res}}_{{t{\bar{t}}}}}}{{\rm d}M_{t{\bar{t}}}^{2}\,{\rm
d}^{2}\vec{P}_{t}^{\perp}\,{\rm d}Y_{t{\bar{t}}}\,{\rm
d}q_{\mathrm{T,out}}}+\mathcal{O}(\lambda^{0}_{\tau})\,.$ (3.22)

It merits noting that since the $\Delta\phi_{{t{\bar{t}}}}$ spectrum is
directly proportional to the differential distribution of $q_{\mathrm{T,out}}$
in the leading power approximation, the AATs from the impact space
representation of ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm
d}\Delta\phi_{{t{\bar{t}}}}$ are also tamed due to Eq. (3.5), and thus will
not introduce azimuthally asymmetric singularities to the
$\Delta\phi_{{t{\bar{t}}}}$ spectrum after completing the inverse Fourier
transformation. This behaviour justifies the application of the resummation
schemes in Tab. 1 and also the R(a)GE framework in Eqs. (3.14)-(3.16) onto the
$\Delta\phi_{{t{\bar{t}}}}$ distribution.

###  3.4 Matching to fixed-order QCD

We are now ready to match the resummed predictions derived in the previous
sections to exact fixed-order QCD calculations. As the expansion in
$\lambda_{\tau}$ has been applied in the derivation of the resummed results of
Eq. (3.14), its validation is maintained only within the asymptotic domain. To
continue the resummed spectra to the entire phase space, we match Eq. (3.14)
onto the fixed-order result using a multiplicative scheme [177, 178, 179],

$\begin{split}\cfrac{{\rm d}\sigma_{{t{\bar{t}}}}^{\rm{mat}}}{{\rm
d}\mathcal{Q}}\,\equiv&\;\Bigg{\\{}\left[\cfrac{{\rm
d}\sigma_{{t{\bar{t}}}}^{\rm{res}}}{{\rm d}\mathcal{Q}}-\cfrac{{\rm
d}\sigma_{{t{\bar{t}}}}^{\rm{s}}(\mu_{\mathrm{f.o.}})}{{\rm
d}\mathcal{Q}}\right]\,f_{\rm{tran}}({\mathcal{Q}},c_{\mathrm{m}},r_{\mathrm{m}})+\cfrac{{\rm
d}\sigma_{{t{\bar{t}}}}^{\rm{s}}(\mu_{\mathrm{f.o.}})}{{\rm
d}\mathcal{Q}}\Bigg{\\}}\;\mathcal{R}_{\mathrm{fs}}(\mu_{\mathrm{f.o.}})\\\
=&\;f_{\rm{tran}}({\mathcal{Q}},c_{\mathrm{m}},r_{\mathrm{m}})\,\left(\cfrac{{\rm
d}\sigma_{{t{\bar{t}}}}^{\rm{res}}}{{\rm
d}\mathcal{Q}}\right)\,\mathcal{R}_{\mathrm{fs}}(\mu_{\mathrm{f.o.}})\Bigg{|}_{\mathrm{exp}}+\Big{\\{}1-{f}_{\rm{tran}}({\mathcal{Q}},c_{\mathrm{m}},r_{\mathrm{m}})\Big{\\}}\,\cfrac{{\rm
d}\sigma_{{t{\bar{t}}}}^{\rm{f.o.}}(\mu_{\mathrm{f.o.}})}{{\rm
d}\mathcal{Q}}+\dots\,,\end{split}$ (3.23)

where
$\mathcal{Q}\in\\{q_{\mathrm{T,out}},q_{\mathrm{T,in}},\Delta\phi_{{t{\bar{t}}}}\\}$
represents a general observable of our concern. ${{\rm
d}\sigma_{{t{\bar{t}}}}^{\rm{res}}}/{{\rm d}\mathcal{Q}}$ is the resummed
differential distribution calculated from Eq. (3.14). ${{\rm
d}\sigma_{{t{\bar{t}}}}^{\rm{s}}}/{{\rm d}\mathcal{Q}}$ stands for the
perturbative expansion of ${{\rm d}\sigma_{{t{\bar{t}}}}^{\rm{res}}}/{{\rm
d}\mathcal{Q}}$ at the scale $\mu_{\mathrm{f.o.}}$ and also corresponds to the
leading singular terms in eq. (2.6). $\mu_{\mathrm{f.o.}}$ is the scale of the
fixed-order expansion, and typically identified with $\mu_{R}$ and $\mu_{F}$
in an exact QCD fixed-order calculation. In the numerical study presented in
the next section of this paper, we will take as default choice

$\mu_{\mathrm{f.o.}}^{\mathrm{def}}=M_{t{\bar{t}}}\,.$ (3.24)

Taking the difference between $\sigma_{{t{\bar{t}}}}^{\rm{res}}$ and
$\sigma_{{t{\bar{t}}}}^{\rm{s}}$ yields the pure resummation corrections
beyond the fixed-order accuracy of the calculation that our result is matched
to, as shown in the square brackets of Eq. (3.23). Multiplying this difference
by the transition function $f_{\rm{tran}}$ permits us to regulate the active
range of the resummation through the shape of $f_{\rm{tran}}$, and avoid
double counting at the same time. To accomplish a continuous and progressive
transition towards the exact result, this paper will employ the following
piecewise form of $f_{\rm{tran}}$,

$f_{\rm{tran}}({\mathcal{Q}},c_{\mathrm{m}},r_{\mathrm{m}})=\begin{dcases}1\,,&\mathcal{Q}\leq
c_{\rm{m}}-r_{\mathrm{m}}\,;\\\
1-\frac{(\mathcal{Q}-c_{\rm{m}}+r_{\mathrm{m}})^{2}}{2r_{\mathrm{m}}^{2}}\,,&c_{\rm{m}}-r_{\mathrm{m}}<\mathcal{Q}\leq
c_{\rm{m}}\,;\\\
\frac{(\mathcal{Q}-c_{\rm{m}}-r_{\mathrm{m}})^{2}}{2r_{\mathrm{m}}^{2}}\,,&c_{\rm{m}}<\mathcal{Q}\leq
c_{\rm{m}}+r_{\mathrm{m}}\,;\\\
0\,,&c_{\rm{m}}+r_{\mathrm{m}}\leq\mathcal{Q}\,,\end{dcases}\hskip
18.77567pt\begin{minipage}{140.84256pt} \includegraphics{Diags_Mat/fx.pdf}
\end{minipage}$ (3.25)

where the parameters $c_{\rm{m}}$ and $r_{\mathrm{m}}$ are introduced to
measure the focal point and the transition radius, respectively. In our
calculation, the following parameters will be taken by default,

$\displaystyle c^{\mathrm{def}}_{\rm{m}}$
$\displaystyle=30~{}\mathrm{GeV}\,,r^{\mathrm{def}}_{\mathrm{m}}=20~{}\mathrm{GeV}\,,\qquad\mathrm{if}~{}\mathcal{Q}\in\\{q_{\mathrm{T,out}},q_{\mathrm{T,in}}\\}\,,$
(3.26) $\displaystyle c^{\mathrm{def}}_{\rm{m}}$
$\displaystyle=0.3\,,r^{\mathrm{def}}_{\mathrm{m}}=0.2,~{}\qquad\qquad\qquad\mathrm{if}~{}\mathcal{Q}\in\\{\Delta\phi_{{t{\bar{t}}}}\\}\,.$
(3.27)

Please note, that the focal point of our transition function,
$c_{\mathrm{m}}$, plays the role of a traditional matching scale for the
$q_{\mathrm{T,out}}$ and $q_{\mathrm{T,in}}$ spectra. As the
$\Delta\phi_{{t{\bar{t}}}}$ spectrum, however, is dimensionless,
$c_{\mathrm{m}}$ is dimensionless, and can thus not be directly connected to
an “intrinsic” scale of the scattering process. Instead, we choose it solely
on the basis of the quality of the approximation of the fixed-order expansion
of Eq. (3.14) wrt. the exact QCD result, see Sec. 4.2. Besides the resummed
differential cross sections and its perturbative expansions aforementioned, to
compensate for the power corrections having been truncated during the
asymptotic expansions, Eq. (3.23) also includes the ratio of the exact spectra
to the leading singular terms derived from SCETII and HQET,

$\displaystyle\mathcal{R}_{\mathrm{fs}}(\mu_{\mathrm{f.o.}})\equiv\cfrac{{{\rm
d}\sigma_{{t{\bar{t}}}}^{\mathrm{f.o.}}(\mu_{\mathrm{f.o.}})}/{{\rm
d}\mathcal{Q}}}{{{\rm
d}\sigma_{{t{\bar{t}}}}^{\mathrm{s}}(\mu_{\mathrm{f.o.}})}/{{\rm
d}\mathcal{Q}}}\,.$ (3.28)

Herein, $\sigma_{{t{\bar{t}}}}^{\mathrm{f.o.}}$ denotes the fixed-order QCD
results assessed at the identical scale to that of
$\sigma_{{t{\bar{t}}}}^{\rm{s}}$, and will be evaluated by the program S
HERPA [158, 159, 160]. Starting from N2LO, the fixed-order predictions are not
positive definite on the whole range of $\mathcal{O}$, and indeed must turn
negative close to the fixed-order divergence to be improved by our
resummation. This invariably leads to the vanishing denominators in Eq.
(3.28). We thus expand $\mathcal{R}_{\mathrm{fs}}$ in
$\alpha_{s}(\mu_{\mathrm{f.o.}})$ in the second step of Eq. (3.23) following
the spirit of [179].

##  4 Numerical Results

###  4.1 Parameters and uncertainty estimates

In this part, we will present numeric results for the observables
$q_{\mathrm{T,out}}$, $q_{\mathrm{T,in}}$, and $\Delta\phi_{{t{\bar{t}}}}$,
which are calculated using the master formulae in Eq. (3.14) and Eq. (3.23).
The resummed result of Eq. (3.14) comprises a convolution of the hard, beam,
soft functions. We calculate the NLO amplitudes of all the helicity and color
configurations of the hard sector using program R  ECOLA [129, 130], and then
evolve them by means of the RGE in Eq. (3.2) to derive the logarithmic
contributions at N2LO. We strictly adhere to the on-shell prescription in
renormalizing the top quark mass, and take its value from the Particle Data
Group (PDG) [180]. For computing the beam functions, the package HPOLY [181]
is embedded to calculate the harmonic poly-logarithms participating in the
hard-collinear coefficients in [136, 137, 138]. The partonic content of the
proton is parametrised using the NNPDF31_nnlo_as_0118 [182] parton
distribution function, interfaced through L  HAPDF [183, 184]. To be
consistent, we use the corresponding value of the strong coupling with
$\alpha_{s}(m_{Z})=0.118$. The evaluation of the soft ingredients at NLO
accuracy is straightforward from the analytic expressions presented in Sec.
2.5. To access the N2LO logarithmic terms, we expand the solutions of Eqs.
(3.11)-(3.12) up to $\mathcal{O}(\alpha^{2}_{s})$. In addition to those fixed-
order constituents, Eq. (3.14) also requires the evolution kernels
$\mathcal{D}^{\mathrm{res}}_{[\kappa]}$ and
$\mathcal{V}^{[\kappa]}_{\alpha\beta}$. The analytic result of
$\mathcal{D}^{\mathrm{res}}_{[\kappa]}$ can be found with the approach of
[185]. To appraise $\mathcal{V}^{[\kappa]}_{\alpha\beta}$, we first
diagonalise the one-loop anomalous dimension $\mathbf{\gamma}^{[\kappa]}_{h}$
(see Eq. (3.2)) by means of Diag [186] to reach NLL accuracy. Then, based on
the perturbative scheme proposed in [175, 176, 42], we reinstate the higher
order corrections from $\mathbf{\gamma}^{[\kappa]}_{h}$ to address the N2LL
requirements and beyond.

Combining these evolution kernels and fixed-order functions permits us to
evaluate the differential cross sections. To assess the phase-space and
impact-space integrals therein, the package Cuba is employed to manage the
relevant multidimensional numerical integrations. Over the course, to
circumvent the threshold regime $\beta_{{t{\bar{t}}}}\sim 0$, where the
Coulomb singularity manifests itself and sabotages the factorisation formula
established in Eq. (2.43), the constraint $M_{{t{\bar{t}}}}\geq
400~{}\mathrm{GeV}$, resulting in $\beta_{{t{\bar{t}}}}\gtrsim 0.5$, is
imposed in the phase integral.

With the resummed spectra in hand, we can proceed with the matching procedure
formulated in Eq. (3.23). The leading singular contribution ${\rm
d}\sigma_{{t{\bar{t}}}}^{\mathrm{s}}/{\rm d}\mathcal{Q}$ is obtained by
expanding ${\rm d}\sigma_{{t{\bar{t}}}}^{\mathrm{res}}/{\rm d}\mathcal{Q}$ in
$\alpha_{s}$. To calculate the exact fixed-order QCD differential cross
sections, we use S  HERPA [158, 159, 160] together with R  ECOLA [129, 130]
and R  IVET [187, 188]. In particular, we will restrict the fixed-order
calculations to the domains

$\displaystyle\mathcal{Q}$ $\displaystyle\geq
10^{-1}~{}\mathrm{GeV}\,,\qquad\mathrm{if}~{}\mathcal{Q}\in\\{q_{\mathrm{T,out}},q_{\mathrm{T,in}}\\}\,,$
(4.1) $\displaystyle\mathcal{Q}$ $\displaystyle\geq
10^{-2}\,,~{}\qquad\qquad\mathrm{if}~{}\mathcal{Q}\in\\{\Delta\phi_{{t{\bar{t}}}}\\}$
(4.2)

to avoid numerical inaccuracies. The NLO calculations involve only the tree
level amplitudes444 In this work, the perturbative accuracy is counted with
respect to the Born cross section. Thus, the NLO contributions here correspond
to the tree-level amplitudes of the process $pp\to{t{\bar{t}}}+\mathrm{jet}$.
, which can thus be generated by the built-in tree-level matrix element
generator A  MEGIC [189] and then processed by R  IVET to extract the
observables $q_{\mathrm{T,out}}$, $q_{\mathrm{T,in}}$, and
$\Delta\phi_{{t{\bar{t}}}}$. To access the N2LO results, R  ECOLA is used to
compute the renormalised one-loop virtual amplitudes of the relevant
subprocesses, while the program A  MEGIC calculates the real emission
corrections and performs the dipole subtraction in the Catani-Seymour scheme
[150, 190, 191, 192]. The subsequent event analysis procedures again proceed
through R  IVET as in the NLO case.

Our calculations, Eq. (3.14) and Eq. (3.23), involve a set of auxiliary
scales,
$\\{\mu_{i},\nu_{i}\\}\equiv\\{\mu_{h},\mu_{b},\mu_{s},\mu_{\mathrm{f.o.}},\nu_{b},\nu_{s}\\}$
and two shape parameters $\\{c_{m},r_{m}\\}$ of the transition function in the
matching procedure. To estimate the theoretical uncertainties of choosing the
default values of those scales, as presented in Eqs. (3.17) and (3.24), we
appraise the differential cross sections with the scales varied to twice or
half their default values independently. The deviations from the calculation
using the default scales are then combined in the quadrature. The so estimated
error is referred to as $\delta_{\mathrm{scale}}$ hereafter. Moreover, to
investigate the sensitivity to the shape parameters of the transition function
of Eqs. (3.26)-(3.27), the differential spectra are also calculated with the
combinations,

$\displaystyle\\{c_{\rm{m}},r_{\mathrm{m}}\\}$
$\displaystyle=\\{25~{}\mathrm{GeV},15~{}\mathrm{GeV}\\},\\{35~{}\mathrm{GeV},25~{}\mathrm{GeV}\\}\,,\qquad\mathrm{if}~{}\mathcal{Q}\in\\{q_{\mathrm{T,out}},q_{\mathrm{T,in}}\\}\,,$
(4.3) $\displaystyle\\{c_{\rm{m}},r_{\mathrm{m}}\\}$
$\displaystyle=\\{0.25,0.15\\},\\{0.35,0.25\\}\,,\,\qquad\qquad\qquad\qquad\mathrm{if}~{}\mathcal{Q}\in\\{\Delta\phi_{{t{\bar{t}}}}\\}\,,$
(4.4)

which amount to fixing the lower boundaries of $f_{\mathrm{tran}}$ but
adjusting the descending gradients around the central choice in Eqs.
(3.26)-(3.27). Again, the deviations from the central value using the default
choices defines the uncertainty, which we denote by $\delta_{\mathrm{tran}}$.
The total theoretical error is then obtained through
$\delta_{\mathrm{tot}}=\sqrt{\delta^{2}_{\mathrm{scale}}+\delta^{2}_{\mathrm{tran}}}$.

###  4.2 Validation

(a) Numeric results of ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm d}q_{\mathrm{T,in}}$

(b) Numeric results of ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm d}\ln
q_{\mathrm{T,in}}$

Figure 5:  The fixed-order results of the $q_{\mathrm{T,in}}$ and the weighted
$q_{\mathrm{T,in}}$ spectra of the process $pp\to{t{\bar{t}}}+X$ at
$\sqrt{s}=13$ TeV. NLO/N2LO represents the differential cross section
calculated in the full QCD, while NLO${}_{\text{s}}$/N2LO${}_{\text{s}}$
encodes the leading singular behaviour derived from SCETII $+$HQET.

In this part, we will confront the differential cross sections derived from
the factorisation formula of Eq. (2.43) with those evaluated in the full
theory. At this point, it merits reminding that in establishing Eq. (2.43),
the asymptotic expansion has been carried out in $\lambda_{\tau}$ by means of
the region expansion of [69, 70, 71, 72], and only the leading singular
contributions have been taken into account in the approximate outputs. In
order to assess the effectiveness of this approximation, and in turn the
resummation scheme of Eqs. (3.14)-(3.16), it is of the crucial importance for
this work to compare the numeric performances of the exact and approximate
spectra in the asymptotic regime.

Fig. 5 exhibits the calculation of the $q_{\mathrm{T,in}}$ and the weighted
$q_{\mathrm{T,in}}$ spectra at NLO and N2LO precision. In the left panel, we
focus on the distributions ${\rm d}\sigma^{\mathrm{s}}_{{t{\bar{t}}}}/{\rm
d}q_{\mathrm{T,in}}$ and ${\rm d}\sigma^{\mathrm{f.o.}}_{{t{\bar{t}}}}/{\rm
d}q_{\mathrm{T,in}}$, in the notation of Sec. 3.4. Please note that while the
NLO fixed-order expansions are strictly positive throughout, the N2LO ones are
negative at small $q_{\mathrm{T,in}}$ and only return to the positive for
$q_{\mathrm{T,in}}\gtrsim 3\,\text{GeV}$. As shown in the top plot, owing to
the singular contributions from the asymptotic expansion of Eq. (2.6), the
absolute values of the NLO${}_{\text{s}}$ and N2LO${}_{\text{s}}$ results are
both in excellent agreement with the exact results in the low
$q_{\mathrm{T,in}}$ region. As the value of $q_{\mathrm{T,in}}$ grows, the
acuteness of the singular terms becomes gradually alleviated, from which the
overall cross-section is reduced. As a result, the NLO${}_{\text{s}}$ and
N2LO${}_{\text{s}}$ approximations show progressively larger deviations. The
fluctuations visible around $q_{\mathrm{T,in}}=3\,\text{GeV}$ are caused by
the slightly different crossing points from the negative to the positive realm
of the exact N2LO and approximate N2LO${}_{\text{s}}$ results. At some point,
far outside its validity range, the approximate NLO${}_{\text{s}}$ and
N2LO${}_{\text{s}}$ again become unphysical, e.g. at $q_{\mathrm{T,in}}\approx
80\,\text{GeV}$ for the NLO${}_{\text{s}}$ results.

Below the main plot, we show two _ratios_ to highlight the relationship
between the exact and approximate spectra. Independent of the order of the
expansion or the choice on the scale, as indicated by their respective bands,
excellent agreement is observed between ${\rm
d}\sigma^{\mathrm{s}}_{{t{\bar{t}}}}/{\rm d}q_{\mathrm{T,in}}$ and ${\rm
d}\sigma^{\mathrm{f.o.}}_{{t{\bar{t}}}}/{\rm d}q_{\mathrm{T,in}}$ below
$q_{\mathrm{T,in}}=1~{}$GeV. Within a few percents of deviation, this holds up
to $q_{\mathrm{T,in}}\sim 10~{}$GeV for the NLO${}_{\text{s}}$ bands and
$q_{\mathrm{T,in}}\sim 20~{}$GeV in the N2LO${}_{\text{s}}$ results. Further
increase in $q_{\mathrm{T,in}}$ will render the power suppressed terms in Eq.
(2.6) manifestly important, thereby developing appreciable discrepancies
between the QCD and EFT outcomes. For instance, it is seen that in the domain
$q_{\mathrm{T,in}}\sim 30\,\text{GeV}$, the leading singular terms only
account for about $75\%$ of the NLO and $\sim 80\%$ of the N2LO spectra.

In order to further ascertain the asymptotic properties in the
$q_{\mathrm{T,in}}\to 0$ limit, we investigate the
$q_{\mathrm{T,in}}$-weighted differential distributions ${\rm
d}\sigma_{{t{\bar{t}}}}/{\rm d}\ln q_{\mathrm{T,in}}$ in Fig. 5(b). Akin to
Eq. (2.6), the asymptotic expansion can also be applied onto the present case,

$\displaystyle\frac{{\rm d}{\sigma_{{t{\bar{t}}}}}}{{\rm d}\ln
q_{\tau}}\,\sim\;\sigma^{{t{\bar{t}}}}_{\mathrm{B}}\,\sum_{m,n}\,\left[\frac{\alpha_{s}(M_{t{\bar{t}}})}{4\pi}\right]^{m}\,\left[\,\underbrace{c^{(0)}_{m,n}\,{\ln^{n}(\lambda_{\tau})}}_{\mathrm{LP}}+\,\underbrace{c^{(1)}_{m,n}\,\lambda_{\tau}\ln^{n}(\lambda_{\tau})}_{\mathrm{NLP}}+\,\underbrace{c^{(2)}_{m,n}\,\lambda^{2}_{\tau}\ln^{n}(\lambda_{\tau})}_{\mathrm{N}^{2}\mathrm{LP}}+\dots\right]\,.$
(4.5)

Comparing with Eq. (2.6), the power series of Eq. (4.5) exhibits a less
singular behaviour in each order and power in $\lambda_{\tau}$. This
characteristic is reflected by the shallower slopes of the NLO${}_{\text{s}}$
and N2LO${}_{\text{s}}$ curves in the main plot of Fig. 5(b). More explicitly,
whilst the magnitudes of ${\rm
d}\sigma^{\mathrm{s}(\mathrm{f.o.})}_{{t{\bar{t}}}}/{\rm d}\ln
q_{\mathrm{T,in}}$ are nearly 10 times smaller than those of ${\rm
d}\sigma^{\mathrm{s}(\mathrm{f.o.})}_{{t{\bar{t}}}}/{\rm d}q_{\mathrm{T,in}}$
in the vicinity of $q_{\mathrm{T,in}}=0$, they approach each other when
entering the area $q_{\mathrm{T,in}}\sim 10~{}$GeV. On the other hand,
differing from Eq. (2.6), the leading power terms of ${\rm
d}\sigma_{{t{\bar{t}}}}/{\rm d}\ln q_{\mathrm{T,in}}$ are expected to capture
the entire singular behaviour of the spectrum as all subleading power
corrections are finite in the limit $q_{\mathrm{T,in}}\to 0$. To verify this
property and in turn scrutinise our calculations of the leading singular
contributions, the _difference_ between the full theory and the EFT results is
addressed in the middle and bottom graphs, at NLO and N2LO accuracy,
respectively, of Fig. 5(b). For the NLO${}_{\text{s}}$ results, with the
decrease in $q_{\mathrm{T,in}}$, the difference between the full QCD result
and our approximation continuously decreases for all scale choices. In
particular, in the vicinity of $q_{\mathrm{T,in}}\sim 0.1\,\text{GeV}$, the
gaps between the NLO and NLO${}_{\text{s}}$ results shrink to $\sim 0.3~{}$pb,
almost a thousand times smaller than the magnitudes of ${\rm
d}\sigma^{\mathrm{s}(\mathrm{f.o.})}_{{t{\bar{t}}}}/{\rm d}\ln
q_{\mathrm{T,in}}$. We thus conclude that all the singularities of ${\rm
d}\sigma^{\mathrm{f.o.}}_{{t{\bar{t}}}}/{\rm d}\ln q_{\mathrm{T,in}}$ at NLO
have been successfully incorporated in ${\rm
d}\sigma^{\mathrm{s}}_{{t{\bar{t}}}}/{\rm d}\ln q_{\mathrm{T,in}}$. Analogous
behaviors can also be found in the N2LO${}_{\text{s}}$ results from the region
$q_{\mathrm{T,in}}\geq 2~{}$GeV, where the Monte-Carlo integration errors
$\epsilon_{{}_{\mathrm{MC}}}$ for the the difference
$\text{N${}^{2}$LO}-\text{N${}^{2}$LO${}_{\text{s}}$}$ (indicated by the red
bars) are still under control. Further reducing $q_{\mathrm{T,in}}$ leads to
growing numerical uncertainties in the calculation of this small residual
difference of increasingly large individual weighted cross sections.
Nonetheless, numeric zero is still within a few standard deviations
$\epsilon_{{}_{\mathrm{MC}}}$ and, thus, the EFT spectra can still be regarded
as compatible with the exact results. It should again be stressed that since
the absolute values of the N2LO${}_{\text{s}}$ curves are generally above
$\mathcal{O}(10^{3})$ within the asymptotic domain, mitigating their absolute
uncertainties down to the same level as those in the NLO${}_{\text{s}}$
results necessitates the sub-permille relative accuracy in running the
programs, which is rather demanding in time and thus has to be postponed to
future research.

(a) Numeric results of ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm
d}q_{\mathrm{T,out}}$

(b) Numeric results of ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm d}\ln
q_{\mathrm{T,out}}$

Figure 6:  The fixed-order results of the $q_{\mathrm{T,out}}$ and the
weighted $q_{\mathrm{T,out}}$ spectra of the process $pp\to{t{\bar{t}}}+X$ at
$\sqrt{s}=13$ TeV. NLO/N2LO represents the differential cross section
calculated in the full QCD, while NLO${}_{\text{s}}$/N2LO${}_{\text{s}}$
encodes the leading singular behaviour derived from SCETII $+$HQET.

In Fig. 6, we illustrate the fixed-order results of the distributions ${\rm
d}\sigma_{{t{\bar{t}}}}/{\rm d}q_{\mathrm{T,out}}$ and ${\rm
d}\sigma_{{t{\bar{t}}}}/{\rm d}\ln q_{\mathrm{T,out}}$. Within the asymptotic
regime, since the $q_{\mathrm{T,out}}$ observable is subject to the same
factorisation formula as that in the $q_{\mathrm{T,in}}$ case, the general
pattern of the $q_{\mathrm{T,out}}$ and the weighted spectra are similar to
those of Fig. 5. Furthermore, due to the fact that the double-logarithmic
terms, induced by the cusp anomalous dimensions in Eq. (3.2) which dominate
the leading singular behaviour, are independent of the choice of the reference
vector $\vec{\tau}$, the absolute values for ${\rm
d}\sigma_{{t{\bar{t}}}}/{\rm d}q_{\mathrm{T,out}}$ and ${\rm
d}\sigma_{{t{\bar{t}}}}/{\rm d}\ln q_{\mathrm{T,out}}$ are close to those of
$q_{\mathrm{T,in}}$ in Fig. 5. In addition to the top diagrams focusing on the
magnitudes of the respective spectra, the lower two panels of Fig. 6 are again
dedicated to the comparisons of the exact and approximate results. It is seen
that, up to statistical uncertainties, the leading singular terms have
suitably reproduced the asymptotic behaviour of the full theory in the small
$q_{\mathrm{T,out}}$ domain.

(a) Numeric results of ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm
d}\Delta\phi_{{t{\bar{t}}}}$

(b) Numeric results of ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm
d}\ln\Delta\phi_{{t{\bar{t}}}}$

Figure 7:  The fixed-order results of the $\Delta\phi_{{t{\bar{t}}}}$ and the
weighted $\Delta\phi_{{t{\bar{t}}}}$ spectra of the process
$pp\to{t{\bar{t}}}+X$ at $\sqrt{s}=13$ TeV. NLO/N2LO represents the
differential cross section calculated in the full QCD, while
NLO${}_{\text{s}}$/N2LO${}_{\text{s}}$ encodes the leading singular behaviour
derived from SCETII $+$HQET.

Fig. 7 finally depicts the results for the ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm
d}\Delta\phi_{{t{\bar{t}}}}$ and ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm
d}\ln\Delta\phi_{{t{\bar{t}}}}$ spectra. Taking into account the facts that
the azimuthal distributions are in practice derived from the
$q_{\mathrm{T,out}}$ spectrum by means of Eq. (3.22) and also that the
contributions from the region
$\vec{P}_{t}^{\perp}\sim\mathcal{O}(10^{2})\,\text{GeV}$ provide the bulk of
the cross section [18, 19], we find that the ${\rm
d}\sigma_{{t{\bar{t}}}}/{\rm d}\Delta\phi_{{t{\bar{t}}}}$ and ${\rm
d}\sigma_{{t{\bar{t}}}}/{\rm d}\ln\Delta\phi_{{t{\bar{t}}}}$ results in the
interval $\Delta\phi_{{t{\bar{t}}}}\in[0.01,1]$ of Fig. 7 present a
corresponding behaviour to the $q_{\mathrm{T,out}}$ spectra within the
$q_{\mathrm{T,out}}\in[1,100]\,\text{GeV}$ region of Fig. 6. For example, the
NLO${}_{\text{s}}$ curves intercept the N2LO${}_{\text{s}}$ ones around
$\Delta\phi_{{t{\bar{t}}}}=0.01$ and $\Delta\phi_{{t{\bar{t}}}}=0.1$ in Fig.
7, while similar intercepts also take place close to
$q_{\mathrm{T,out}}=1\,\text{GeV}$ and $q_{\mathrm{T,out}}=10\,\text{GeV}$ in
Fig. 6. Finally, as before, Fig. 7 delivers the ratios of the
$\Delta\phi_{{t{\bar{t}}}}$ spectra as well as the differences of the weighted
$\Delta\phi_{{t{\bar{t}}}}$ distributions, to compare the approximate and
exact spectra. As before, except for the calculations crossing from negative
to positive cross sections in the vicinity of $\Delta\phi_{{t{\bar{t}}}}\sim
0.03$ of Fig. 7(a) and the numerical instability observed at
$\Delta\phi_{{t{\bar{t}}}}\sim 0.04$ in Fig. 7(b), we observe excellent
agreement in the asymptotic domain $\Delta\phi_{{t{\bar{t}}}}\lesssim 0.1$.

###  4.3 Resummation improved results

Figure 8:  The resummation improved spectra of $q_{\mathrm{T,in}}$ and
$q_{\mathrm{T,out}}$ at NLL+NLO, N2LL+N2LO, and approximate N2LL′+N2LO
accuracies (aN2LL′+N2LO).

We now match our resummed calculation, validated in the previous section, to
the exact fixed-order result, according to Eq. (3.23). To facilitate our
calculations of the exact spectra, lower cutoffs in the respective observable
have been implemented as shown in Eq. (4.1). They are justified by the above
findings, namely the agreement of the fixed-order expansion of the
approximation and the full QCD calculation, in the asymptotic domain. It is
important to note that the full theory calculation participates in the
matching procedure through the ratio $\mathcal{R}_{\mathrm{fs}}$ defined in
Eq. (3.28). As illustrated in Figs. 5(a), 6(a), and 7(a), this ratio
approaches unity in the asymptotic region for all observables investigated in
this paper. Consequently, for practical evaluations, we are justified to
truncate the impact of the full theory below the boundaries in Eq. (4.1).
Further lowering these boundaries in principle helps to suppress the power
corrections, which however entails much longer run time of our programs.

The other major ingredient in our matching procedure is the transition
function $f_{\mathrm{tran}}$ which, together with the shape parameters
$\\{c_{\rm{m}},r_{\mathrm{m}}\\}$, governs the active range of the
resummation. In light of the excellent agreement between QCD and EFT, see
Figs. 5-7, the chosen default values of $\\{c_{\rm{m}},r_{\mathrm{m}}\\}$ are
given in Eq. (3.26) and Eq. (3.27). The transition function ensures that the
resummation of Eq. (3.14) is fully operational in the asymptotic regions. In
the present paper, we choose these regions to be $q_{\mathrm{T,in(out)}}\leq
10\,\text{GeV}$ and $\Delta\phi_{{t{\bar{t}}}}\leq 0.1$. The contribution of
the resummation to the matched result is then continuously reduced, reaching
half its intrinsic value at the focal point $c_{\mathrm{m}}$, 30 GeV in the
$q_{\mathrm{T,in(out)}}$ spectra and 0.3 in the $\Delta\phi_{{t{\bar{t}}}}$
spectrum. The resummation is eventually fully switched off for
$\mathcal{Q}\geq c_{\mathrm{m}}+r_{\mathrm{m}}$, $50\,\text{GeV}$ for
$\mathcal{Q}=q_{\mathrm{T,in(out)}}$ and $0.5$ for
$\mathcal{Q}=\Delta\phi_{{t{\bar{t}}}}$. Alternative settings for
$c_{\mathrm{m}}$ and $r_{\mathrm{m}}$, see Eq. (4.3) and Eq. (4.4), are
assessed to estimate the associated theoretical uncertainty of the matching
processes.

In Fig. 8, we present the resummation improved spectra ${\rm
d}\sigma_{{t{\bar{t}}}}/{\rm d}q_{\mathrm{T,in}}$ and ${\rm
d}\sigma_{{t{\bar{t}}}}/{\rm d}q_{\mathrm{T,out}}$. Within the small
$q_{\mathrm{T,in(out)}}$ regimes, in contrast to the fixed-order results of
Figs. 5(a)-7(a), where substantial N2LO contributions were observed, the
central values here have been stabilised and only receive very small residual
higher-order corrections. More importantly, a reduction in the theoretical
uncertainties, detailed through their respective coloured bands, can be found
with the increase in the logarithmic accuracy. For example, in the limits
$q_{\mathrm{T,in}}\to 0$ and $q_{\mathrm{T,out}}\to 0$, the results on the
aN2LL′+N2LO level, that constitute our best prediction, possess relative
uncertainties of about ${+1\%}$ and ${-5\%}$. They are greatly reduced from
the uncertainties of the N2LL+N2LO calculations, being from $+6\%$ to $-12\%$.
For illustrational purposes we also include the lowest order NLL+NLO
calculation showing uncertainties of $\pm 30\%$. At larger
$q_{\mathrm{T,in(out)}}$ the fixed-order full QCD calculation determines both
spectra, given at N2LO accuracy in both aN2LL′+N2LO and N2LL+N2LO
calculations, and a full ${t{\bar{t}}}+\text{jet}$ two-loop calculation is
needed to improve upon it.

Comparing the spectra of $q_{\mathrm{T,in}}$ with the $q_{\mathrm{T,out}}$
ones, it is interesting to find that their NLL+NLO results are in close
agreement with each other, whereas those at N2LL+N2LO or aN2LL′+N2LO accuracy
start to exhibit larger differences. In particular, the $q_{\mathrm{T,out}}$
spectra are raised by $\approx 1~{}$pb/GeV with regard to the
$q_{\mathrm{T,in}}$ one in the asymptotic regime. To interpret this, it is
beneficial to remind that according to Tab. 1, the NLL resummation only
involves the evolution kernels $\mathcal{D}^{\mathrm{res}}_{[\kappa]}$ and
$\mathcal{V}^{[\kappa]}$ in Eqs. (3.15)-(3.16), both of which, as defined in
Eq. (3.2) and Eq. (B.3), concern only the magnitude of $\vec{b}_{\tau_{\|}}$
and thus fail to differentiate between the $q_{\mathrm{T,out}}$ and
$q_{\mathrm{T,in}}$ observables. However, starting from N2LL, the
$\vec{\tau}$-dependent components, such as the soft function
$\mathcal{S}_{[\kappa]}$ and the gluon beam sector
$\mathcal{B}^{[g_{n(\bar{n})}]}$, come into play, thus including a non-trivial
impact that the projected direction $\vec{\tau}$ can make on the results.

Furthermore, contrary to the azimuthally averaged transverse momentum
distributions calculated in [59, 56], no Sudakov peak is formed in the
projected transverse momentum case, as displayed in Fig. 8. This originates in
part from the kinematic differences of these two observables. In the R(a)GE
framework introduced in Sec. 3.2, the resummed $q_{\mathrm{T}}$ distributions
can be addressed as,

$\displaystyle\frac{{\rm d}\sigma_{{t{\bar{t}}}}}{{\rm
d}q_{\mathrm{T}}}\,\sim\;q_{\mathrm{T}}\,\sum_{\kappa}\,\int{\rm
d}\vec{b}_{\mathrm{T}}\,J_{0}(b_{\mathrm{T}}q_{\mathrm{T}})\,\widetilde{\Sigma}^{\mathrm{res},[\kappa]}_{{t{\bar{t}}}}(\vec{b}_{\mathrm{T}},M_{t{\bar{t}}},\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})\,,$
(4.6)

where $J_{0}$ stands for the zeroth-rank Bessel function. Taking the limit
$q_{\mathrm{T}}\to 0$, while the integrand in Eq. (4.6) approaches a constant
value, the pre-factor $q_{\mathrm{T}}$ linearly suppresses the resummed
spectra, which in turn leads to the formation of a Sudakov peak. Such a linear
factor is absent for $q_{\mathrm{T,in}}$ and $q_{\mathrm{T,out}}$. As
formulated in Eq. (3.14), the situation of the $q_{\tau}$ resummation is

$\displaystyle\frac{{\rm d}\sigma_{{t{\bar{t}}}}}{{\rm
d}q_{\tau}}\,\sim\;\sum_{\kappa}\int{\rm
d}b_{\tau_{\|}}\,\cos(b_{\tau_{\|}}q_{\tau})\,\widetilde{\Sigma}^{\mathrm{res},[\kappa]}_{{t{\bar{t}}}}(b_{\tau_{\|}}\vec{\tau},M_{t{\bar{t}}},\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})\,.$
(4.7)

Thus, the $q_{\tau}$ spectra here receive no kinematical suppressions in the
asymptotic limit, explaining the absence of the Sudakov peaks in Fig. 8.

Figure 9:  The resummation improved azimuthal spectra ${\rm
d}\sigma_{{t{\bar{t}}}}/{\rm d}\Delta\phi_{{t{\bar{t}}}}$ at NLL+NLO,
N2LL+N2LO, and approximate N2LL′+N2LO accuracies (aN2LL′+N2LO).

Finally, in Fig. 9, we show the resummation improved azimuthal separations
between the top and anti-top quarks. Since at the leading power, the
observable $\Delta\phi_{{t{\bar{t}}}}$ exhibits the same factorisation and
resummation patterns as the $q_{\mathrm{T,out}}$ and $q_{\mathrm{T,in}}$
spectra, the curves from Fig. 9 present analogous behaviors to the ones in
Fig. 8, including the converging theoretical uncertainties with the increasing
of the precision of the calculation. Due to the same reasons as before, a
Sudakov peak cannot be observed in the asymptotic domain. In particular, the
latter case appears to be in conflict with the finding in a similar observable
in Drell-Yan production [193] where Sudakov peaks were observed in the
azimuthal spectra. This contradiction is, however, found to be spurious. The
constraints on the transverse momentum that had been imposed on the final
state in [193] through fiducial cuts and the projection of the observable into
multiple transverse momentum slices introduced an effective kinematic damping
factor through the shrinking phase space into Eq. (4.6), restoring a similar
Sudakov peak structure as observed in the azimuthally averaged transverse
momentum distribution. On the contrary, despite requiring $M_{t{\bar{t}}}\geq
400\,\text{GeV}$ in this paper and the prefactor $\vec{P}^{\perp}_{t}$ having
emerged from the expansions in Eq. (3.22), neither of them can collapse the
phase space for $\Delta\phi_{{t{\bar{t}}}}\to 0$ sufficiently for such a
structure to emerge.

##  5 Conclusions

In this paper, we investigated the projected transverse momentum spectrum
$q_{\tau}$, which can be evaluated by integrating out the rejection components
of $\vec{q}_{\mathrm{T}}$ with respect to the reference unit vector
$\vec{\tau}$. We focussed on its singular behaviour in the low $q_{\tau}$
domain in particular.

Using the region expansion methods [69, 70, 71, 72] we resolved the phase
space and loop integrals in terms of the dynamic regions that capture the
asymptotic properties of ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm d}q_{\tau}$ in the
vicinity of $q_{\tau}=0$. It comprises the usual hard, beam-collinear, jet-
collinear, and soft domains. Then, in the context of the effective field
theories, SCETII and HQET, we enumerated the possible configurations
constructed by those momentum modes, and determined the power accuracy in each
case by probing the corresponding factorisation formulae. It is observed that
the leading singular terms of the $q_{\tau}$ distribution are governed by the
hard, beam-collinear, and soft regions, akin to the $q_{\mathrm{T}}$ spectrum
[55, 56, 57, 58, 59, 65], whilst the higher power regular terms entail the
participations of the jet-collinear regime.

In the subsequent resummation, we focussed on the leading asymptotic behaviour
of ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm d}q_{\tau}$. By inspecting the
corresponding factorisation formula in impact space, we illustrated that the
driving factor of the singular behaviour in momentum space is the
characteristic logarithm
$\ln^{k}[{{b}^{2}_{\tau_{\|}}M_{t{\bar{t}}}^{2}}/{b_{0}^{2}}]$, for $k\geq 1$,
which therefore permitted us to apply the framework of RGE and RaGE [87, 88,
89, 90] to accomplish the logarithmic exponentiation and in turn the
resummation. It is worth emphasizing that in presence of the azimuthal
asymmetric terms, this phenomenon is not general on the process
$pp\to{t{\bar{t}}}+X$. The $q_{\tau}$ spectra and the azimuthally averaged
case ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm d}q_{\mathrm{T}}$ are known to be the
only two observables which stem from the double-differential distribution
${\rm d}\sigma_{{t{\bar{t}}}}/{\rm d}q_{\mathrm{T}}$ but are free of the
azimuthal asymmetric divergences. 555 It is interesting to note that in a very
recent calculation [194] on the semi-inclusive deep-inelastic scattering
(SIDIS), the observable $q_{*}$ is proposed and also demonstrated to be
promising to cope with the azimuthal asymmetric divergences. Within the
asymptotic regime, the $q_{*}$ spectrum actually bears a close resemblance to
the $q_{\mathrm{T,out}}$ distribution used in this work, up to rapidity-
dependent factors.

Although the derivation and findings summarised above hold for an arbitrary
choice of reference vector $\vec{\tau}$, we focussed on three specific
observables to demonstrate the validity and usefulness of our framework,
$q_{\mathrm{T,in}}$, $q_{\mathrm{T,out}}$, and $\Delta\phi_{{t{\bar{t}}}}$.
The first two cases are obtained by assigning $\vec{\tau}$ to be parallel and
perpendicular with respect to the transverse momentum of the top quark,
respectively. The azimuthal spectrum ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm
d}\Delta\phi_{{t{\bar{t}}}}$ is extracted from ${\rm
d}\sigma_{{t{\bar{t}}}}/{\rm d}q_{\mathrm{T,out}}$ according to the
kinematical relationship in between. Owing to the independence from
azimuthally asymmetric divergences, we were able to calculate those
observables at NLL+NLO, N2LL+N2LO, and approximate N2LL′+N2LO (aN2LL′+N2LO)
accuracy. From the numerical results, a manifest perturbative convergence is
observed in the asymptotic regime, i.e. the central values at the higher
precisions are generally contained in the uncertainty bands of the lower ones
and the respective uncertainties are systematically reduced down to the
percent level at the aN2LL′+N2LO level. Very recently, the double differential
spectra ${\rm d}\sigma_{{t{\bar{t}}}}/({\rm d}M_{t{\bar{t}}}\,{\rm
d}\Delta\phi_{{t{\bar{t}}}})$ have been measured by the CMS collaboration
[12]. It is expected that after the full release of those experimental data in
the forthcoming future, we can compare our theoretical predictions calculated
using the framework introduced in this paper against these experimental
measurements.

### Acknowledgements

WJ is grateful to Jean-Nicolas Lang for providing detailed instructions on the
usage of R  ECOLA . WJ also would like to thank L. Chen, M. Czakon and R.
Poncelet for the helpful discussions on the N2LO hard function. This work is
supported by a Royal Society through a University Research Fellowship
(URF\R1\180549) and an Enhancement Award (RGF\EA\181033, CEC19\100349, and
RF\ERE\210397).

##  Appendix A The double differential transverse momentum distribution ${\rm
d}^{2}\sigma_{{t{\bar{t}}}}/{\rm d}^{2}\vec{q}_{\mathrm{T}}$

As discussed in Sec. 3.1, in absence of the _azimuthal asymmetric term_ (AAT),
the main task in resumming the singular behaviors on the $q_{\tau}$ spectra is
to exponentiate the characteristic logarithmic terms in impact space.
Nevertheless, in light of the non-trivial contributions of the helicity-
flipping beam functions as well as the soft correlations the resummation of a
general observable is more complicated. Hence, this appendix will calculate
the double differential spectrum as an example to illustrate the effect of the
AATs on the fixed-order expansion and the organisation of the logarithmic
terms. Alternative cases with active AATs contributions can also be found in
[63, 64, 57, 58].

Considering that the contributions to the $\vec{q}_{\mathrm{T}}$ distributions
from the hard jet regions, $N_{\mathrm{J}}\geq 1$, have been demonstrated to
be power suppressed [65, 122], the factorisation formula is now subject to
only the $N_{\mathrm{J}}=0$ configuration at leading power. Therefore, they
can be extracted from Eq. (2.26) after reducing the heavy parton correlation
functions with Eqs. (2.27)-(2.28). It follows that,

$\displaystyle\frac{{\rm d}{\sigma_{{t{\bar{t}}}}}}{{\rm
d}M_{t{\bar{t}}}^{2}\,{\rm d}^{2}\vec{P}_{t}^{\perp}\,{\rm
d}Y_{t{\bar{t}}}\,{\rm
d}^{2}\vec{q}_{\mathrm{T}}}=\sum_{\mathrm{sign}[x_{t}]}\sum_{\kappa}\,\frac{\Theta_{\mathrm{kin}}^{(0)}}{64\pi^{4}\,\beta_{t{\bar{t}}}\,|x_{t}|\,M_{t{\bar{t}}}^{4}\,s}\,\int{\rm
d}^{2}\vec{b}_{\mathrm{T}}\,\exp\left(\vec{q}_{\mathrm{T}}\cdot\vec{b}_{\mathrm{T}}\right)\,\widetilde{\Sigma}_{{t{\bar{t}}}}^{[\kappa]}(\vec{b}_{\mathrm{T}},M_{t{\bar{t}}},\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})\,,$
(A.1)

where
$\kappa\in\\{g_{n}g_{\bar{n}},q^{i}_{n}\bar{q}^{j}_{\bar{n}},q^{i}_{\bar{n}}\bar{q}^{j}_{n}\\}$
again denotes the partonic channel. The constituents of
$\widetilde{\Sigma}_{{t{\bar{t}}}}^{[\kappa]}$ have been specified in Eqs.
(2.29)-(2.30) for each partonic channel. Upon the usage of the identities in
Eqs. (2.27)-(2.28), Eq. (A.1) inherits the problems from Coulomb exchanges as
well, akin to Eq. (2.43). To this end, we will constrain the following
investigation within the domain $M_{t{\bar{t}}}\geq 400\,\text{GeV}$, or
equivalently, $\beta_{t{\bar{t}}}\gtrsim 0.5$, to circumvent the threshold
regime.

Generically, the expressions of $\widetilde{\Sigma}^{[\dots]}_{{t{\bar{t}}}}$
at an arbitrary order in $\alpha_{s}$ can be parameterized as

$\displaystyle\widetilde{\Sigma}^{[\dots]}_{{t{\bar{t}}}}\,\sim\;\sum_{m,n}\,\alpha^{m}_{s}(M_{t{\bar{t}}})\,L_{\mathrm{T}}^{n}\,\Big{\\{}s_{m}(\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})+a_{m}(\widehat{b}_{\mathrm{T}},\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})\Big{\\}}\,,$
(A.2)

where
$L_{\mathrm{T}}\equiv\log\left[{b_{\mathrm{T}}^{2}M_{t{\bar{t}}}^{2}}/{b_{0}^{2}}\right]$
and $\widehat{b}_{\mathrm{T}}\equiv\vec{b}_{\mathrm{T}}/b_{\mathrm{T}}$. The
dimensionless coefficients $s_{m,n}$ and $a_{m,n}$ encode the azimuthal
_symmetric_ and _asymmetric_ contributions, respectively. Substituting Eq.
(A.2) into Eq. (A.1) permits us to evaluate the impact space integrals,
namely,

$\displaystyle\frac{{\rm d}{\sigma_{{t{\bar{t}}}}}}{{\rm
d}M_{t{\bar{t}}}^{2}\,{\rm d}^{2}\vec{P}_{t}^{\perp}\,{\rm
d}Y_{t{\bar{t}}}\,{\rm
d}^{2}\vec{q}_{\mathrm{T}}}\,\sim\;\sum_{m,n}\alpha^{m}_{s}(M_{t{\bar{t}}})\Bigg{\\{}s_{m,n}(\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})\,\mathcal{F}^{(n)}_{\mathrm{T}}(q_{\mathrm{T}},M_{t{\bar{t}}})+\mathcal{A}^{(n)}_{\mathrm{T}}\Big{[}\vec{q}_{\mathrm{T}},M_{t{\bar{t}}},a_{m,n}\Big{]}\Bigg{\\}}\,.$
(A.3)

Herein, since the ASTs are all independent of the impact parameter
$\vec{b}_{\mathrm{T}}$, we can factor them out of the
$\vec{b}_{\mathrm{T}}$-integrals and apply the inverse Fourier transformations
only onto the logarithmic terms, i.e.

$\displaystyle\mathcal{F}^{(n)}_{\mathrm{T}}$
$\displaystyle(q_{\mathrm{T}},M_{t{\bar{t}}})\,=\;\int{\rm
d}^{2}b_{\mathrm{T}}\,\exp\left(\vec{q}_{\mathrm{T}}\cdot\vec{b}_{\mathrm{T}}\right)\,L^{n}_{\mathrm{T}}\,.$
(A.4)

The calculation of $\mathcal{F}^{(n)}_{\mathrm{T}}$ follows a similar pattern
to Eq. (3.1). The corresponding generating function now reads [93],

$\displaystyle\mathbf{F}_{\mathrm{T}}(\eta,q_{\mathrm{T}},M_{t{\bar{t}}})\,=$
$\displaystyle\;\sum_{n}\frac{\eta^{n}}{n!}\,\mathcal{F}^{(n)}_{\mathrm{T}}(q_{\mathrm{T}},M_{t{\bar{t}}})\,=\;\int{\rm
d}^{2}\vec{b}_{\mathrm{T}}\,\exp\left(\vec{q}_{\mathrm{T}}\cdot\vec{b}_{\mathrm{T}}\right)\,\left(\frac{b_{\mathrm{T}}^{2}M_{t{\bar{t}}}^{2}}{b_{0}^{2}}\right)^{\eta}$
$\displaystyle\,=$ $\displaystyle\;\frac{4\pi
e^{2\gamma_{\mathrm{E}}\eta}}{q_{\mathrm{T}}^{2}}\,\frac{\Gamma[1+\eta]}{\Gamma[-\eta]}\,\left(\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right)^{\eta}\,.$
(A.5)

$\mathcal{F}^{(n)}_{\mathrm{T}}$ can be extracted by expanding
$\mathbf{F}_{\mathrm{T}}(\eta,q_{\mathrm{T}},M_{t{\bar{t}}})$ in $\eta$ as
appropriate. The expressions of the first few ranks are

$\displaystyle\mathcal{F}^{(0)}_{\mathrm{T}}(q_{\mathrm{T}},M_{t{\bar{t}}})$
$\displaystyle=0\,,$ (A.6)
$\displaystyle\mathcal{F}^{(1)}_{\mathrm{T}}(q_{\mathrm{T}},M_{t{\bar{t}}})$
$\displaystyle=-\frac{4\pi}{q_{\mathrm{T}}^{2}}\,,$ (A.7)
$\displaystyle\mathcal{F}^{(2)}_{\mathrm{T}}(q_{\mathrm{T}},M_{t{\bar{t}}})$
$\displaystyle=-\frac{8\pi}{q_{\mathrm{T}}^{2}}\ln\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]\,,$
(A.8) $\displaystyle\;\,\vdots$

From Eq. (A.6) it is seen that the ASTs themselves cannot induce any
asymptotic behaviour in the small $q_{\mathrm{T}}$ domain, whilst the
logarithmic term $L_{\mathrm{T}}^{m}$ $(m\geq 1)$ is able to produce singular
contributions up to
$\ln^{m-1}[M_{t{\bar{t}}}^{2}/q_{\mathrm{T}}^{2}]/q_{\mathrm{T}}^{2}$ after
completing the inverse Fourier transformations. This observation is found by
analogy with Eq. (3.5) and also the circumstances of the azimuthally averaged
observables in [59, 55, 56].

On the other hand, Eq. (A.3) also includes contributions from the AATs
$a_{m,n}$. In light of their explicit dependence on the orientation of
$\vec{b}_{\mathrm{T}}$, the inverse Fourier transformation now comprises both
$a_{m,n}$ and $L^{n}_{\mathrm{T}}$, that is,

$\displaystyle\mathcal{A}^{(n)}_{\mathrm{T}}\Big{[}\vec{q}_{\mathrm{T}},M_{t{\bar{t}}},A\Big{]}\,=\;\int{\rm
d}^{2}b_{\mathrm{T}}\,\exp\left(\vec{q}_{\mathrm{T}}\cdot\vec{b}_{\mathrm{T}}\right)\,L^{n}_{\mathrm{T}}\,A(\phi_{qb})\,,$
(A.9)

where $A(\phi_{qb})$ represents a generic function of the azimuthal difference
$\phi_{qb}$ between the vector $\vec{b}_{\mathrm{T}}$ and
$\vec{q}_{\mathrm{T}}$, which thereafter participates in the functional
$\mathcal{A}^{(n)}_{\mathrm{T}}$ as one of its arguments. Appraising
$\mathcal{A}^{(n)}_{\mathrm{T}}$ will benefit from its generational form as
well,

$\displaystyle\mathbf{A}_{\mathrm{T}}\Big{[}\eta,\vec{q}_{\mathrm{T}},M_{t{\bar{t}}},A\Big{]}\,=$
$\displaystyle\;\sum_{n}\,\frac{\eta^{n}}{n!}\,\mathcal{A}^{(n)}_{\mathrm{T}}\Big{[}\vec{q}_{\mathrm{T}},M_{t{\bar{t}}},A\Big{]}\,=\,\int\,{\rm
d}^{2}\vec{b}_{\mathrm{T}}\,\exp\left(\vec{q}_{\mathrm{T}}\cdot\vec{b}_{\mathrm{T}}\right)\,\left(\frac{b_{\mathrm{T}}^{2}M_{t{\bar{t}}}^{2}}{b_{0}^{2}}\right)^{\eta}\,A(\phi_{qb})\,$
$\displaystyle\,=$
$\displaystyle\;-\frac{\Gamma[2\eta+2]}{q_{\mathrm{T}}^{2}}\,\frac{e^{2\gamma_{\mathrm{E}}\eta}}{4^{\eta}}\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]^{\eta}\,\int^{\frac{\pi}{2}}_{0}\,{\rm
d}\phi\,\Bigg{\\{}c_{\phi}^{-2(1+\eta)}\,e^{\mathrm{i}\pi\eta}\,\bigg{[}A(\phi)-A\Big{(}\frac{\pi}{2}\Big{)}+c_{\phi}A^{\prime}\Big{(}\frac{\pi}{2}\Big{)}\bigg{]}\,$
$\displaystyle\;{}+s_{\phi}^{-2(1+\eta)}\,e^{-\mathrm{i}\pi\eta}\,\bigg{[}A\Big{(}\phi+\frac{\pi}{2}\Big{)}-A\Big{(}\frac{\pi}{2}\Big{)}-s_{\phi}A^{\prime}\Big{(}\frac{\pi}{2}\Big{)}\bigg{]}+c_{\phi}^{-2(1+\eta)}\,e^{-\mathrm{i}\pi\eta}\bigg{[}A\Big{(}\phi+\pi\Big{)}\,$
$\displaystyle\;{}-A\Big{(}\frac{3\pi}{2}\Big{)}+c_{\phi}A^{\prime}\Big{(}\frac{3\pi}{2}\Big{)}\bigg{]}+s_{\phi}^{-2(1+\eta)}\,e^{\mathrm{i}\pi\eta}\,\bigg{[}A\Big{(}\phi+\frac{3\pi}{2}\Big{)}-A\Big{(}\frac{3\pi}{2}\Big{)}-s_{\phi}A^{\prime}\Big{(}\frac{3\pi}{2}\Big{)}\bigg{]}\Bigg{\\}}\,$
$\displaystyle\;{}-\sqrt{\pi}\,\frac{\Gamma[2\eta+2]}{q_{\mathrm{T}}^{2}}\frac{e^{2\gamma_{\mathrm{E}}\eta}}{4^{\eta}}\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]^{\eta}\,\Bigg{\\{}\frac{\cos(\pi\eta)\Gamma[-\frac{1}{2}-\eta]}{\Gamma[-\eta]}\bigg{[}A\Big{(}\frac{\pi}{2}\Big{)}+A\Big{(}\frac{3\pi}{2}\Big{)}\bigg{]}\,$
$\displaystyle\;{}-\frac{\mathrm{i}\sin(\pi\eta)\Gamma[-\eta]}{\Gamma[\frac{1}{2}-\eta]}\bigg{[}A^{\prime}\Big{(}\frac{\pi}{2}\Big{)}-A^{\prime}\Big{(}\frac{3\pi}{2}\Big{)}\bigg{]}\Bigg{\\}}\,,$
(A.10)

where $c_{\phi}=\cos(\phi)$ and $s_{\phi}=\sin(\phi)$. In deriving Eq. (A),
the function $A(\phi_{qb})$ is assumed to be analytically continuous in the
entire domain $\phi_{qb}\in[0,2\pi]$, and the result is expected to be
applicable to any azimuthal asymmetric contributions that stem from the beam
functions, the soft sectors, or the products of these two cases. In the
following discussion, we will focus on scenarios of the beam constituent. The
other two pieces observe the same scaling manner but incur lengthier
expressions.

As defined in Eq. (2.36), the gluon beam function constitutes a $2\times 2$
matrix in the helicity space. With the choice of helicity bases of Eq. (2.37),
while the diagonal entries characterise the ASTs, the off-diagonal ones, which
are always proportional to $e^{\pm\mathrm{i}2\phi_{tb}}$, encode the azimuthal
asymmetric contributions, namely, $A(\phi_{qb})\propto
e^{\pm\mathrm{i}2(\phi_{qb}+\phi_{tq})}$. Here $\phi_{tb}$ ($\phi_{tq}$)
signifies the azimuthal difference between the top quark with respect to the
vector $\vec{b}_{\mathrm{T}}$ ($\vec{q}_{\mathrm{T}}$). Plugging this
expression into Eq. (A) and then performing the expansions in $\eta$, it
yields,

$\displaystyle\mathcal{A}^{(0)}_{\mathrm{T}}\Big{[}\vec{q}_{\mathrm{T}},M_{t{\bar{t}}},e^{\pm\mathrm{i}2(\phi_{qb}+\phi_{tq})}\Big{]}\,=$
$\displaystyle\;-\frac{4\pi}{q_{\mathrm{T}}^{2}}\,e^{\pm\mathrm{i}2\phi_{tq}}\,,$
(A.11)
$\displaystyle\mathcal{A}^{(1)}_{\mathrm{T}}\Big{[}\vec{q}_{\mathrm{T}},M_{t{\bar{t}}},e^{\pm\mathrm{i}2(\phi_{qb}+\phi_{tq})}\Big{]}\,=$
$\displaystyle\;-\frac{4\pi}{q_{\mathrm{T}}^{2}}\,\bigg{\\{}\ln\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]+1\bigg{\\}}\,e^{\pm\mathrm{i}2\phi_{tq}}\,,$
(A.12)
$\displaystyle\mathcal{A}^{(2)}_{\mathrm{T}}\Big{[}\vec{q}_{\mathrm{T}},M_{t{\bar{t}}},e^{\pm\mathrm{i}2(\phi_{qb}+\phi_{tq})}\Big{]}\,=$
$\displaystyle\;-\frac{4\pi}{q_{\mathrm{T}}^{2}}\,\bigg{\\{}\ln^{2}\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]+2\ln\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]\bigg{\\}}\,e^{\pm\mathrm{i}2\phi_{tq}}\,,$
(A.13) $\displaystyle\vdots\;$

It is noted that the situation of $\mathcal{A}^{(m)}_{\mathrm{T}}$ above is
quite different from that for the $\mathcal{F}^{(m)}_{\mathrm{T}}$
aforementioned. Firstly, as opposed to Eqs. (A.6)-(A.8), where the logarithmic
contribution is the only driver for the singularities in the momentum space,
Eq. (A.11) illustrates that the AAT itself can serve as a standalone factor to
induce an asymptotic behavior. Secondly, in comparison with Eqs. (A.7)-(A.8),
the product of $L^{m}_{\mathrm{T}}$ $(m\geq 1)$ and AATs can induce asymptotic
terms up to
$\ln^{m}\left[{M_{t{\bar{t}}}^{2}}/{q_{\mathrm{T}}^{2}}\right]/q_{\mathrm{T}}^{2}$,
which are stronger than the case of ASTs by one power in
$\ln\left[{M_{t{\bar{t}}}^{2}}/{q_{\mathrm{T}}^{2}}\right]$. Thirdly,
differing from Eqs. (A.6)-(A.8) concerning only the absolute value of
$\vec{q}_{\mathrm{T}}$, the expressions for $\mathcal{A}^{(m)}_{\mathrm{T}}$
additionally depend on the orientation of $\vec{q}_{\mathrm{T}}$ through the
phase factor $\exp(\pm 2\phi_{tq})$. It is worth emphasising that even though
the gluon beam function is utilised here as an example to derive the
$\mathcal{A}^{(m)}_{\mathrm{T}}$s above, as a matter of fact, the phase factor
$\exp(\pm k\phi_{tq})$ $(k\in\mathbb{Z})$ can also participate in the soft
sectors at an arbitrary perturbative order upon which the spherical harmonics
expansion was performed [58]. Given those non-trivial impacts from the AATs,
the fixed-order results in the low $q_{\mathrm{T}}$ regime and the subsequent
resummation may show a different behaviour from that solely driven by the
ASTs. The following paragraphs will take the NLO double differential spectrum
${\rm d}\sigma_{{t{\bar{t}}}}/({\rm d}q_{\mathrm{T}}{\rm d}\Delta\Phi_{tX})$
and its required ingredients for a NLL′ resummation as examples to elucidate
this. Here $\Delta\Phi_{tX}$ stands for the azimuthal separation between the
top quark and the recoil. MS: Is this correct? WJ: Sorry that this description
may be too sketchy. Here $\Delta\Phi_{tX}$ stands for the azimuthal separation
between the top quark and the total momentum of the emitted partons.

Figure 10:  Comparison of the NLO double differential spectra ${\rm
d}\sigma_{{t{\bar{t}}}}/({\rm d}q_{\mathrm{T}}{\rm d}\Delta\Phi_{tX})$, shown
as distributions of $\Delta\Phi_{tX}$ in slices of $q_{\mathrm{T}}$. Herein,
NLO represents the QCD result calculated in S  HERPA [158, 159, 160].
$\text{NLO}_{\mathrm{s}}$ labels the azimuthally _symmetric_ and
$\text{NLO}_{\mathrm{s+a}}$ marks the full, i.e. azimuthally _symmetric_ and
_asymmetric_ , contributions derived from SCET${}_{\mathrm{II}}+$HQET,
respectively. The superscript “$\mathrm{Azm.Avg}$” denotes the azimuthally
averaging operation on the exact and approximate spectra.

In Figs. 10-12, we display a comparison of the double differential spectra
${\rm d}\sigma_{{t{\bar{t}}}}/({\rm d}q_{\mathrm{T}}{\rm d}\Delta\Phi_{tX})$
among the exact NLO result, the approximate spectrum NLOs comprising only the
AST contributions, and the full EFT-derived spectrum NLOs+a including both AST
and AAT ingredients. As a first observable, Fig. 10 presents the
$\Delta\Phi_{tX}$ distributions, ranging from the parallel configuration,
$\Delta\Phi_{tX}=0^{\circ}$, to the opposite one,
$\Delta\Phi_{tX}=180^{\circ}$, within three $q_{\mathrm{T}}$ intervals,
$q_{\mathrm{T}}\in[0.1,0.2]$ GeV, $q_{\mathrm{T}}\in[1,2]$ GeV, and
$q_{\mathrm{T}}\in[10,20]$ GeV. As observed in the left panel of Fig. 10,
which focusses on the domain closest to the singularity,
$q_{\mathrm{T}}\in[0.1,0.2]$ GeV, the curves of NLO and NLOs+a are in very
good agreement and exhibit a nearly cosinusoidal behaviour on account of the
phase factors in Eqs. (A.11)-(A.13) and those stemming from the soft
functions. However, if we were to remove the AAT contributions, leaving only
the ASTs as active ingredients of our approximation, governed now by Eqs.
(A.6)-(A.8) which depend only on the magnitude of $\vec{q}_{\mathrm{T}}$, the
resulting curve NLOs is independent of $\Delta\Phi_{tX}$. After taking the
azimuthal averaging operations upon the exact and approximate spectra,
however, it is interesting to note that the result of
NLO${}^{\mathrm{Azm.Avg}}_{\mathrm{s}+\mathrm{a}}$ coincides with that of
NLOs, and both of them reproduce the exact NLOAzm.Avg result well. This
observation demonstrates that the asymptotic terms in Eqs. (A.11)-(A.13) are
indeed able to be eliminated by completing the $\phi_{tq}$ integral, which
confirms the discussions in [58] at least at the NLO level. In the centre and
right plots of Fig. 10, we move on to the slices $q_{\mathrm{T}}\in[1,2]$ GeV
and $q_{\mathrm{T}}\in[10,20]$ GeV. Even though the general patterns herein
follow that in the left panel, it is seen that the power corrections gradually
become more important with the increase in $q_{\mathrm{T}}$. They give rise to
the growing discrepancies between NLO and NLOs+a results, and also lead to the
deviations amid the azimuthally averaged curves.

Figure 11:  Comparison of the NLO double differential spectra ${\rm
d}\sigma_{{t{\bar{t}}}}/({\rm d}q_{\mathrm{T}}{\rm d}\Delta\Phi_{tX})$, shown
as distributions of $q_{\mathrm{T}}$ in slices of $\Delta\Phi_{tX}$. Herein,
NLO represents the QCD result calculated in S  HERPA [158, 159, 160].
$\text{NLO}_{\mathrm{s}}$ labels the azimuthally _symmetric_ and
$\text{NLO}_{\mathrm{s+a}}$ marks the full, i.e. azimuthally _symmetric_ and
_asymmetric_ , contributions derived from SCET${}_{\mathrm{II}}+$HQET,
respectively.

In Fig, 11, we exhibit the $q_{\mathrm{T}}$ distributions in slices of
$\Delta\Phi_{tX}\in[0^{\circ},10^{\circ}]$,
$\Delta\Phi_{tX}\in[80^{\circ},100^{\circ}]$, and
$\Delta\Phi_{tX}\in[170^{\circ},180^{\circ}]$, which correspond to the peak
and trough regions in Fig. 10. We observe that for the majority of the
$q_{\mathrm{T}}$ range, the magnitudes of the exact results are close to the
approximate calculations NLOs and NLOs+a, however, only the NLOs+a reproduces
it well in the region of $q_{\mathrm{T}}\lesssim 1\,$GeV. To interpret this,
it is worth noting that at NLO, the most singular terms generated by the ASTs
and AATs are
$\propto\ln\left[{M_{t{\bar{t}}}^{2}}/{q_{\mathrm{T}}^{2}}\right]/q_{\mathrm{T}}$
and $1/q_{\mathrm{T}}$, respectively. In this regard, the asymptotic behaviors
in the low $q_{\mathrm{T}}$ regime are actually dominated by the AST
contributions, which therefore is the cause for the _nearly_ coinciding curves
in the main plot of Fig. 11. However, it should be stressed that in spite of
the minor roles played by AATs, their participations are still of the crucial
importance in reproducing the correct singular manner of the exact spectra. As
illustrated in the ratio diagrams of Fig. 11, whereas the AST contributions
either underestimate or overshoot the NLO results by $\sim 2\%$ in the
vicinity of $q_{\mathrm{T}}=0.1~{}$GeV, the curves of NLOs+a achieve permille
level agreement with the exact spectra for all three slices. These phenomenona
can also be confirmed by Fig. 12, especially in the centre and bottom diagrams
therein.

Figure 12:  Comparison of the weighted NLO double differential spectra ${\rm
d}\sigma_{{t{\bar{t}}}}/({\rm d}q_{\mathrm{T}}{\rm d}\Delta\Phi_{tX})$, shown
as distributions of $q_{\mathrm{T}}$ in slices of $\Delta\Phi_{tX}$. Herein,
NLO represents the QCD result calculated in S  HERPA [158, 159, 160].
$\text{NLO}_{\mathrm{s}}$ labels the azimuthally _symmetric_ and
$\text{NLO}_{\mathrm{s+a}}$ marks the full, i.e. azimuthally _symmetric_ and
_asymmetric_ , contributions derived from SCET${}_{\mathrm{II}}+$HQET,
respectively.

In light of the non-trivial contributions of the AATs on the leading singular
behaviour, the subsequent task is to work out a consistent approach to embed
their contributions in the resummation. In circumstances where the singular
behaviour is governed solely by the ASTs, the resummation can proceed with
exponentiating the characteristic logarithmic terms in the impact space, as
discussed in Sec. 3. Considering that the AAT emergence still preserves the
factorisation formula and so the according R(a)GEs, we can still apply the
logarithmic exponentiation onto the present situation, which then transforms
Eq. (A.2) into the following expression,

$\displaystyle\widetilde{\Sigma}^{[\dots]}_{{t{\bar{t}}}}\,\sim$
$\displaystyle\;\exp\bigg{[}\;L_{\mathrm{T}}\,f_{0}(\alpha_{s}L_{\mathrm{T}})+f_{1}(\alpha_{s}L_{\mathrm{T}})+{\alpha_{s}\,f_{2}(\alpha_{s}L_{\mathrm{T}})}+{\alpha^{2}_{s}\,f_{3}(\alpha_{s}L_{\mathrm{T}})}+\ldots\;\bigg{]}$
$\displaystyle\times\,\sum_{m}\,\alpha^{m}_{s}\,\Big{\\{}\hat{s}_{m}(\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})+\hat{a}_{m}(\widehat{b}_{\mathrm{T}},\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})\Big{\\}}\,,$
(A.14)

where the coefficient functions $f_{k}(k\geq 0)$ are introduced to collect the
logarithmic terms at the respective accuracies. $\hat{s}_{m}$ and
$\hat{a}_{m}$ stem from the coefficients in Eq. (A.2) capturing the
azimuthally symmetric and asymmetric contributions, respectively. In spite of
the similar appearance of Eq. (A) to the AST-driven result in Eq. (A.2), it is
worth emphasising that the perturbative series in Eq. (A) still possesses the
constituents $\hat{a}_{m}$, which can induce a leading asymptotic behaviour
after completing the $\vec{b}_{\mathrm{T}}$-integral. To this end, it is of
essence to clarify the higher order performance of $\hat{a}_{m}$ so as to
determine the subsequent treatments during the resummation. If, akin to the
logarithmic contributions in Eqs. (A.6)-(A.8), $\hat{a}_{m}$ invokes
deteriorating singularities with the increase in $\alpha_{s}$, a second
exponentiating procedure will be necessary to mitigate the theoretical
uncertainty caused by the perturbative truncation. Otherwise, after suitable
adaptations, we are in principle able to regard the AAT contributions in the
similar manner to the ASTs.

To facilitate the discussion, we note that as for an arbitrary order in
$\alpha_{s}$, the term $\hat{a}_{m}$ exists as a function of $\phi_{tb}$ and
can thus be expanded in terms of the spherical harmonics,

$\displaystyle\hat{a}_{m}(\widehat{b}_{\mathrm{T}},\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})\,=\,\sum_{k}\,\hat{a}_{m}^{(k)}(\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})\;e^{\mathrm{i}k\phi_{tb}}\,,$
(A.15)

where we have utilised the momentum conservation condition
$\vec{v}_{t}=-\vec{v}_{\bar{t}}$ to reduce the dependence on the azimuthal
angle $\phi_{\bar{t}b}$. Substituting this harmonic series into Eq. (A) and
then repeating the derivation of Eqs. (A.11)-(A.13) yields

$\displaystyle e^{\mathrm{i}k\phi_{tb}}\xRightarrow[\text{ }]{\textbf{F.T.
}}\frac{e^{\mathrm{i}k\phi_{tq}}}{q_{\mathrm{T}}^{2}}\,.$ (A.16)

As this consequence holds independent of the rank of the spherical harmonics
or the specific expression of $\hat{a}_{m}$, we can now conclude that for all
perturbative orders, the singular behaviour induced by $\hat{a}_{m}$ is always
$\propto 1/q_{\mathrm{T}}^{2}$. Therefore, we can carry out a perturbative
truncation of the fixed-order functions in Eq. (A)666 This sort of
perturbative truncatablity has also been addressed in [58] in the context of
the CSS resummation framework [91]. . According to Eq. (1.1), there are two
kinds of customary prescriptions in the AST-driven resummation to truncate the
perturbative series, i.e. the NmLL and NmLL′ schemes. In the following
paragraphs, we will implement both methods separately for the resummation of
the double differential distribution ${\rm d}\sigma_{{t{\bar{t}}}}/({\rm
d}q_{\mathrm{T}}{\rm d}\Delta\Phi_{tX})$ at NLL and NLL′ accuracy. Then we
will compare their first few perturbative terms against the solely AST
governed case, e.g. $q_{\tau}$ resummation.

Based on the factorisation in Eq. (A.2) and the exponentiated impact space
cross section in Eq. (A), the resummed $\vec{q}_{\mathrm{T}}$ spectra at NLL
can be organized as follows,

$\displaystyle\frac{{\rm d}\sigma_{{t{\bar{t}}}}}{{\rm d}q_{\mathrm{T}}{\rm
d}\Delta\Phi_{tX}}\Bigg{|}_{\mathrm{NLL}}\,\sim$ $\displaystyle\;\int{\rm
d}^{2}\vec{b}_{\mathrm{T}}\,\exp\left(\vec{q}_{\mathrm{T}}\cdot\vec{b}_{\mathrm{T}}\right)\,\exp\bigg{[}L_{\mathrm{T}}\,f_{0}(\alpha_{s}L_{\mathrm{T}})+f_{1}(\alpha_{s}L_{\mathrm{T}})\bigg{]}$
$\displaystyle\sim$
$\displaystyle\;\alpha_{s}\bigg{\\{}\underbrace{\frac{1}{q_{\mathrm{T}}}\ln\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]}_{\mathrm{complete}}+\underbrace{\frac{1}{q_{\mathrm{T}}}}_{\mathrm{incomplete}}\bigg{\\}}+\alpha_{s}^{2}\bigg{\\{}\underbrace{\frac{1}{q_{\mathrm{T}}}\ln^{3}\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]}_{\mathrm{complete}}+\underbrace{\frac{1}{q_{\mathrm{T}}}\ln^{2}\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]}_{\mathrm{incomplete}}+\underbrace{\frac{1}{q_{\mathrm{T}}}\ln\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]}_{\mathrm{incomplete}}$
$\displaystyle\hskip
170.71652pt{}+\underbrace{\frac{1}{q_{\mathrm{T}}}}_{\mathrm{incomplete}}\bigg{\\}}+\dots\,,$
(A.17)

where in the second step, we expand the resummed result in $\alpha_{s}$ and
keep the first two orders contributing to the low $q_{\mathrm{T}}$ domain. For
brevity, we omit the specific coefficient in front of each singular term. With
the underbraces, we manifest the status of each term. For instance, in
presence of the LL anomalous dimensions, the most singular behaviour at each
perturbative order can be exactly reproduced, whereas, for the lack of the AAT
correction and the NmLL, $m\geq 2$, anomalous dimensions, the terms
$1/q_{\mathrm{T}}^{2}$ at NLO and those of
$\ln^{k}[M_{t{\bar{t}}}^{2}/q_{\mathrm{T}}^{2}]/q_{\mathrm{T}}^{2}$, $k\leq
2$, at N2LO are not complete and thus will receive further correction once the
calculation at the higher accuracy is performed.

A similar analysis can also be applied to the projected transverse momentum
resummation with the help of Eq. (3.14),

$\displaystyle\frac{{\rm d}\sigma_{{t{\bar{t}}}}}{{\rm
d}q_{\tau}}\Bigg{|}_{\mathrm{NLL}}\,\sim$
$\displaystyle\;\int^{\infty}_{-\infty}{\rm
d}{b}_{\tau_{\|}}\,\cos\left({b}_{\tau_{\|}}{q}_{\tau}\right)\;\exp\bigg{[}\;L_{\mathrm{M}}\,f_{0}(\alpha_{s}L_{\mathrm{M}})+f_{1}(\alpha_{s}L_{\mathrm{M}})\bigg{]}$
$\displaystyle\sim$
$\displaystyle\;\alpha_{s}\bigg{\\{}\underbrace{\frac{1}{q_{\tau}}\ln\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\tau}^{2}}\right]}_{\mathrm{complete}}+\underbrace{\frac{1}{q_{\tau}}}_{\mathrm{complete}}\bigg{\\}}+\alpha_{s}^{2}\bigg{\\{}\underbrace{\frac{1}{q_{\tau}}\ln^{3}\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\tau}^{2}}\right]}_{\mathrm{complete}}+\underbrace{\frac{1}{q_{\tau}}\ln^{2}\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\tau}^{2}}\right]}_{\mathrm{complete}}+\underbrace{\frac{1}{q_{\tau}}\ln\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\tau}^{2}}\right]}_{\mathrm{incomplete}}$
$\displaystyle\hskip
170.71652pt{}+\underbrace{\frac{1}{q_{\tau}}}_{\mathrm{incomplete}}\bigg{\\}}+\dots\,,$
(A.18)

where
$L_{\mathrm{M}}\equiv\log\left[{{b}^{2}_{\tau_{\|}}M_{{t{\bar{t}}}}^{2}}/{b_{0}^{2}}\right]$.
Herein, due to the absence of the AAT, the NLO singular terms of the
$q_{\tau}$ spectrum have been completely reproduced by the NLL expansions,
while at N2LO only the terms
$\ln^{k}[M_{t{\bar{t}}}^{2}/q_{\mathrm{T}}^{2}]/q_{\mathrm{T}}^{2}$, $k\geq 2$
can still be finalised by the occurring Sudakov factor. This feature permits
our matching procedure proceeding at the NLL+NLO precision as illustrated in
Sec. 4. This, however, is infeasible for the double differential spectrum
${\rm d}\sigma_{{t{\bar{t}}}}/({\rm d}q_{\mathrm{T}}{\rm d}\Delta\Phi_{tX})$
as their NLO asymptotic behaviour is incomplete, as demonstrated in Eq. (A).

We now continue to explore the resummation at the NLL′ level. In comparison
with the NLL case, the present calculation entails an additional fixed-order
constituent. It follows that

$\displaystyle\frac{{\rm d}\sigma_{{t{\bar{t}}}}}{{\rm d}q_{\mathrm{T}}{\rm
d}\Delta\Phi_{tX}}\Bigg{|}_{\mathrm{NLL}^{\prime}}\,\sim$
$\displaystyle\;\int{\rm
d}^{2}\vec{b}_{\mathrm{T}}\,\exp\left(\vec{q}_{\mathrm{T}}\cdot\vec{b}_{\mathrm{T}}\right)\;\exp\bigg{[}L_{\mathrm{T}}\,f_{0}(\alpha_{s}L_{\mathrm{T}})+f_{1}(\alpha_{s}L_{\mathrm{T}})\bigg{]}$
$\displaystyle\hskip
113.81102pt{}\times\;\Bigg{\\{}1+\alpha_{s}\bigg{[}\hat{s}_{1}(\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})+\hat{a}_{1}(\widehat{b}_{\mathrm{T}},\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})\bigg{]}\Bigg{\\}}$
$\displaystyle\sim$
$\displaystyle\;\alpha_{s}\bigg{\\{}\underbrace{\frac{1}{q_{\mathrm{T}}}\ln\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]}_{\mathrm{complete}}+\underbrace{\frac{1}{q_{\mathrm{T}}}}_{\mathrm{complete}}\bigg{\\}}+\alpha_{s}^{2}\bigg{\\{}\underbrace{\frac{1}{q_{\mathrm{T}}}\ln^{3}\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]}_{\mathrm{complete}}+\underbrace{\frac{1}{q_{\mathrm{T}}}\ln^{2}\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]}_{\mathrm{complete}}+\underbrace{\frac{1}{q_{\mathrm{T}}}\ln\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\mathrm{T}}^{2}}\right]}_{\mathrm{complete}}$
$\displaystyle\hskip
170.71652pt{}+\underbrace{\frac{1}{q_{\mathrm{T}}}}_{\mathrm{incomplete}}\bigg{\\}}+\dots\,,$
(A.19)

and

$\displaystyle\frac{{\rm d}\sigma_{{t{\bar{t}}}}}{{\rm
d}q_{\tau}}\Bigg{|}_{\mathrm{NLL}^{\prime}}\,\sim$
$\displaystyle\;\int^{\infty}_{-\infty}{\rm
d}{b}_{\tau_{\|}}\,\cos\left({b}_{\tau_{\|}}{q}_{\tau}\right)\exp\bigg{[}L_{\mathrm{M}}\,f_{0}(\alpha_{s}L_{\mathrm{M}})+f_{1}(\alpha_{s}L_{\mathrm{M}})\bigg{]}$
$\displaystyle\hskip
113.81102pt{}\times\;\Bigg{\\{}1+\alpha_{s}\bigg{[}\hat{s}_{1}(\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})+\hat{a}_{1}(\mathrm{sign}[b_{\tau_{\|}}],\beta_{t{\bar{t}}},x_{t},Y_{t{\bar{t}}})\bigg{]}\Bigg{\\}}$
$\displaystyle\sim$
$\displaystyle\;\alpha_{s}\bigg{\\{}\underbrace{\frac{1}{q_{\tau}}\ln\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\tau}^{2}}\right]}_{\mathrm{complete}}+\underbrace{\frac{1}{q_{\tau}}}_{\mathrm{complete}}\bigg{\\}}+\alpha_{s}^{2}\bigg{\\{}\underbrace{\frac{1}{q_{\tau}}\ln^{3}\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\tau}^{2}}\right]}_{\mathrm{complete}}+\underbrace{\frac{1}{q_{\tau}}\ln^{2}\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\tau}^{2}}\right]}_{\mathrm{complete}}+\underbrace{\frac{1}{q_{\tau}}\ln\left[\frac{M_{t{\bar{t}}}^{2}}{q_{\tau}^{2}}\right]}_{\mathrm{complete}}$
$\displaystyle\hskip
170.71652pt{}+\underbrace{\frac{1}{q_{\tau}}}_{\mathrm{incomplete}}\bigg{\\}}+\dots\,.$
(A.20)

Confronting Eqs. (A)-(A) with Eqs. (A)-(A), it is seen that the inclusion of
the fixed-order function can make up the missing singular parts at NLO, and
that at N2LO the asymptotic performance of the spectrum ${\rm
d}\sigma_{{t{\bar{t}}}}/({\rm d}q_{\mathrm{T}}{\rm d}\Delta\Phi_{tX})$ is
aligned with the AST-driven case, i.e. ${\rm d}\sigma_{{t{\bar{t}}}}/{\rm
d}q_{\tau}$. It should be stressed that this kind of alignment can also hold
in the perturbative series for other logarithmic accuracies, if one notes that
as to a given precise NkLL′, $k\geq 1$, the missing AAT contributions caused
by the perturbative truncation in practice start from
$\alpha^{k+1}_{s}\hat{a}_{k+1}$, which are mapped onto the singular terms
$\alpha^{k+1}_{s}/q_{\mathrm{T}}^{2}$ through the inverse Fourier
transformation and thus amount to the Nk+1LL′, $k\geq 1$, corrections. To this
end, differing from the AST-driven observables, which are able to switch
between the unprimed and primed schemes flexibly, the double differential
spectra ${\rm d}\sigma_{{t{\bar{t}}}}/({\rm d}q_{\mathrm{T}}{\rm
d}\Delta\Phi_{tX})$ prefers the primed prescriptions in organising the
relevant ingredients in regard to the intact asymptotic series therein and so
the improved matching precision.

##  Appendix B Evolution kernel for the non-diagonal anomalous dimension

In this part, we will elaborate on the calculation on the evolution kernel
$\mathcal{V}_{\alpha\beta}^{[\kappa]}$ which is induced by the hard non-
diagonal anomalous dimensions $\gamma_{h,\alpha\beta}^{[\kappa]}$ in Eq.
(3.2). Here $\kappa$ runs over
$\\{g_{n}g_{\bar{n}},q^{i}_{n}\bar{q}^{j}_{\bar{n}},q^{i}_{\bar{n}}\bar{q}^{j}_{n}\\}$,
indicating the partonic channel.

We define the perturbative series of $\gamma_{h,\alpha\beta}^{[\kappa]}$ below

$\displaystyle\gamma_{h,\alpha\beta}^{[\kappa]}\,=\,\sum_{m=0}\,\left(\frac{\alpha_{s}}{4\pi}\right)^{m+1}\,\gamma_{h,\alpha\beta}^{[\kappa],(m)}\,,$
(B.1)

where the $\gamma_{h,\alpha\beta}^{[\kappa],(m)}$ are the coefficients in each
order. The N2LO expressions can be found in [128, 161], while progress towards
N3LO precision has been made in [166]. Other than that, the resummation
procedure also entails the QCD-$\beta$ functions,

$\displaystyle\frac{{\rm d}\alpha_{s}}{{\rm
d}\ln\mu}\,=\;-2\alpha_{s}\,\sum_{k=0}\bigg{(}\frac{\alpha_{s}}{4\pi}\bigg{)}^{k+1}\beta_{k}\,$
(B.2)

where the $\beta_{k}$ characterise the anomalous dimensions for the strong
coupling $\alpha_{s}$. By now, they are known to five-loop accuracy [195].

Equipped with these expressions, we are now ready to calculate the kernel
$\mathcal{V}_{\alpha\beta}^{[\kappa]}$. At NLL, according to Tab. 1, the
resummation only concerns the contribution of
$\gamma_{h,\alpha\beta}^{[\kappa],(0)}$ to the kernel
$\mathcal{V}_{\alpha\beta}^{[\kappa]}$, which is independent of either the
scale evolution or the $\alpha_{s}$ running, and thus allows us to solve the
hard RG equations in diagonal space. It yields,

$\displaystyle\mathbf{V}_{h}^{[\kappa]}(\beta_{t{\bar{t}}},x_{t},\mu_{f},\mu_{i})\Bigg{|}_{\mathrm{NLL}(^{\prime})}\,=\;\mathbf{R}^{-1}_{[\kappa]}\,\exp\bigg{\\{}\frac{\mathbf{r}^{[\kappa],(0)}_{h}}{2\beta_{0}}\ln\bigg{[}\frac{\alpha_{s}(\mu_{i})}{\alpha_{s}(\mu_{f})}\bigg{]}\bigg{\\}}\,\mathbf{R}_{[\kappa]}\,,$
(B.3)

where $\mathbf{V}_{h}^{[\kappa]}$ stands for the matrix representation of the
evolution kernel $\mathcal{V}_{\alpha\beta}^{[\kappa]}$.
$\mathbf{r}^{[\kappa],(0)}_{h}$ denotes the resulting diagonal matrix of
$\gamma_{h,\alpha\beta}^{[\kappa],(0)}$ with the invertible transformation
matrix $\mathbf{R}_{[\kappa]}$, namely

$\displaystyle\mathbf{r}^{[\kappa],(0)}_{h}\,=\;\mathbf{R}_{[\kappa]}\gamma_{h}^{[\kappa],(0)}\mathbf{R}^{-1}_{[\kappa]}\,.$
(B.4)

However, the situation on the N2LL′ level is different on the ground that due
to the participation of $\gamma_{h}^{[\kappa],(1)}$, factoring out all the
scale dependent pieces is not straightforward any longer. To this end, we
resort to the perturbative resolution suggested in [175, 176, 42] to introduce
the correction matrices,

$\displaystyle\mathbf{J}^{[\kappa]}_{ij}\,=\;\mathbf{r}^{{[\kappa]},(0)}_{h,ii}\;\delta_{ij}\,\frac{\beta_{1}}{2\beta^{2}_{0}}-\frac{\mathbf{r}^{{[\kappa]},(1)}_{h,ij}}{2\beta_{0}+\mathbf{r}^{{[\kappa]},(0)}_{h,ii}-\mathbf{r}^{{[\kappa]},(0)}_{h,jj}}\,,$
(B.5)

where $\delta_{ij}$ is the Kronecker delta function with the indices $i$ and
$j$ running over the set $\\{1,2\\}$ ($\\{1,2,3\\}$) for the quark (gluon)
channel. $\mathbf{r}^{{[\kappa]},(1)}_{h}$ is defined in the similar way of
Eq. (B.4) except for replacing $\gamma_{h}^{[\kappa],(0)}$ with
$\gamma_{h}^{[\kappa],(1)}$.

Combining the $\mathbf{J}^{[\kappa]}$ matrix above with Eq. (B.3), we then
arrive at the result for N2LL′ precision,

$\displaystyle\mathbf{V}_{h}^{[\kappa]}(\beta_{t{\bar{t}}},x_{t},\mu_{f},\mu_{i})\Bigg{|}_{\mathrm{N}^{2}\mathrm{LL}(^{\prime})}\,=\;\mathbf{R}^{-1}_{[\kappa]}\,\bigg{[}\mathbf{I}+\frac{\alpha_{s}(\mu_{f})}{4\pi}\mathbf{J}^{[\kappa]}\bigg{]}\,\exp\bigg{\\{}\frac{\mathbf{r}^{[\kappa],(0)}_{h}}{2\beta_{0}}\ln\bigg{[}\frac{\alpha_{s}(\mu_{i})}{\alpha_{s}(\mu_{f})}\bigg{]}\bigg{\\}}\,\bigg{[}\mathbf{I}-\frac{\alpha_{s}(\mu_{i})}{4\pi}\mathbf{J}^{[\kappa]}\bigg{]}\,\mathbf{R}_{[\kappa]}\,,$
(B.6)

where $\mathbf{I}$ signifies the unit matrix.

## References

  * [1] ATLAS collaboration, _Measurement of the $t\bar{t}$ production cross-section using dilepton events in $pp$ collisions at $\sqrt{s}=5.02$ TeV with the ATLAS detector_.
  * [2] A. M. Sirunyan et al., CMS collaboration, _Measurement of the inclusive $\mathrm{t}\overline{\mathrm{t}}$ cross section in pp collisions at $\sqrt{s}=5.02$ TeV using final states with at least one charged lepton_, JHEP 03 (2018), 115, [arXiv:1711.03143 [hep-ex]].
  * [3] A. Tumasyan et al., CMS collaboration, _Measurement of the inclusive $\mathrm{t}\overline{\mathrm{t}}$ production cross section in proton-proton collisions at $\sqrt{s}$ = 5.02 TeV_, JHEP 04 (2022), 144, [arXiv:2112.09114 [hep-ex]].
  * [4] G. Aad et al., ATLAS collaboration, _Measurement of the $t\bar{t}$ production cross-section and lepton differential distributions in $e\mu$ dilepton events from $pp$ collisions at $\sqrt{s}=13\,\text{TeV}$ with the ATLAS detector_, Eur. Phys. J. C 80 (2020), no. 6, 528, [arXiv:1910.08819 [hep-ex]].
  * [5] G. Aad et al., ATLAS collaboration, _Measurement of the $t\bar{t}$ production cross-section in the lepton+jets channel at $\sqrt{s}=13$ TeV with the ATLAS experiment_, Phys. Lett. B 810 (2020), 135797, [arXiv:2006.13076 [hep-ex]].
  * [6] A. M. Sirunyan et al., CMS collaboration, _Measurement of the top quark pair production cross section in dilepton final states containing one $\tau$ lepton in pp collisions at $\sqrt{s}=$ 13 TeV_, JHEP 02 (2020), 191, [arXiv:1911.13204 [hep-ex]].
  * [7] A. M. Sirunyan et al., CMS collaboration, _Measurement of $\mathrm{t\bar{t}}$ normalised multi-differential cross sections in pp collisions at $\sqrt{s}=13$ TeV, and simultaneous determination of the strong coupling strength, top quark pole mass, and parton distribution functions_, Eur. Phys. J. C 80 (2020), no. 7, 658, [arXiv:1904.05237 [hep-ex]].
  * [8] G. Aad et al., ATLAS collaboration, _Measurements of top-quark pair single- and double-differential cross-sections in the all-hadronic channel in $pp$ collisions at $\sqrt{s}=13~{}\textrm{TeV}$ using the ATLAS detector_, JHEP 01 (2021), 033, [arXiv:2006.09274 [hep-ex]].
  * [9] A. Tumasyan et al., CMS collaboration, _Measurement of differential $t\bar{t}$ production cross sections in the full kinematic range using lepton+jets events from proton-proton collisions at $\sqrt{s}$ = 13 TeV_, Phys. Rev. D 104 (2021), no. 9, 092013, [arXiv:2108.02803 [hep-ex]].
  * [10] G. Aad et al., ATLAS collaboration, _Measurements of differential cross-sections in top-quark pair events with a high transverse momentum top quark and limits on beyond the Standard Model contributions to top-quark pair production with the ATLAS detector at $\sqrt{s}$ = 13 TeV_, JHEP 06 (2022), 063, [arXiv:2202.12134 [hep-ex]].
  * [11] A. M. Sirunyan et al., CMS collaboration, _Measurement of differential $\mathrm{t\bar{t}}$ production cross sections using top quarks at large transverse momenta in $pp$ collisions at $\sqrt{s}=$ 13 TeV_, Phys. Rev. D 103 (2021), no. 5, 052008, [arXiv:2008.07860 [hep-ex]].
  * [12] CMS collaboration, _Measurement of differential cross sections for the production of top quark pairs and of additional jets in pp collisions at $\sqrt{s}=13~{}\mathrm{TeV}$_.
  * [13] P. Nason, S. Dawson and R. K. Ellis, _The Total Cross-Section for the Production of Heavy Quarks in Hadronic Collisions_ , Nucl. Phys. B 303 (1988), 607–633.
  * [14] W. Beenakker, H. Kuijf, W. L. van Neerven and J. Smith, _QCD Corrections to Heavy Quark Production in p anti-p Collisions_ , Phys. Rev. D 40 (1989), 54–82.
  * [15] W. Beenakker, W. L. van Neerven, R. Meng, G. A. Schuler and J. Smith, _QCD corrections to heavy quark production in hadron hadron collisions_ , Nucl. Phys. B 351 (1991), 507–560.
  * [16] M. L. Mangano, P. Nason and G. Ridolfi, _Heavy quark correlations in hadron collisions at next-to-leading order_ , Nucl. Phys. B 373 (1992), 295–345.
  * [17] M. Czakon, P. Fiedler and A. Mitov, _Total Top-Quark Pair-Production Cross Section at Hadron Colliders Through $O(\alpha^{4}_{S})$_, Phys. Rev. Lett. 110 (2013), 252004, [arXiv:1303.6254 [hep-ph]].
  * [18] M. Czakon, D. Heymes and A. Mitov, _High-precision differential predictions for top-quark pairs at the LHC_ , Phys. Rev. Lett. 116 (2016), no. 8, 082003, [arXiv:1511.00549 [hep-ph]].
  * [19] M. Czakon, P. Fiedler, D. Heymes and A. Mitov, _NNLO QCD predictions for fully-differential top-quark pair production at the Tevatron_ , JHEP 05 (2016), 034, [arXiv:1601.05375 [hep-ph]].
  * [20] M. Czakon, D. Heymes and A. Mitov, _fastNLO tables for NNLO top-quark pair differential distributions_ , arXiv:1704.08551 [hep-ph].
  * [21] M. Czakon, D. Heymes, A. Mitov, D. Pagani, I. Tsinikos and M. Zaro, _Top-pair production at the LHC through NNLO QCD and NLO EW_ , JHEP 10 (2017), 186, [arXiv:1705.04105 [hep-ph]].
  * [22] S. Catani, S. Devoto, M. Grazzini, S. Kallweit, J. Mazzitelli and H. Sargsyan, _Top-quark pair hadroproduction at next-to-next-to-leading order in QCD_ , Phys. Rev. D 99 (2019), no. 5, 051501, [arXiv:1901.04005 [hep-ph]].
  * [23] S. Catani, S. Devoto, M. Grazzini, S. Kallweit and J. Mazzitelli, _Top-quark pair hadroproduction at NNLO: differential predictions with the $\overline{MS}$ mass_, JHEP 08 (2020), no. 08, 027, [arXiv:2005.00557 [hep-ph]].
  * [24] J. Gao, C. S. Li and H. X. Zhu, _Top Quark Decay at Next-to-Next-to Leading Order in QCD_ , Phys. Rev. Lett. 110 (2013), no. 4, 042001, [arXiv:1210.2808 [hep-ph]].
  * [25] M. Brucherseifer, F. Caola and K. Melnikov, _$\mathcal{O}(\alpha_{s}^{2})$ corrections to fully-differential top quark decays_, JHEP 04 (2013), 059, [arXiv:1301.7133 [hep-ph]].
  * [26] S. Catani, S. Devoto, M. Grazzini, S. Kallweit and J. Mazzitelli, _Top-quark pair production at the LHC: Fully differential QCD predictions at NNLO_ , JHEP 07 (2019), 100, [arXiv:1906.06535 [hep-ph]].
  * [27] A. Behring, M. Czakon, A. Mitov, A. S. Papanastasiou and R. Poncelet, _Higher order corrections to spin correlations in top quark pair production at the LHC_ , Phys. Rev. Lett. 123 (2019), no. 8, 082001, [arXiv:1901.05407 [hep-ph]].
  * [28] M. Czakon, A. Mitov and R. Poncelet, _NNLO QCD corrections to leptonic observables in top-quark pair production and decay_ , JHEP 05 (2021), 212, [arXiv:2008.11133 [hep-ph]].
  * [29] W. Bernreuther and Z.-G. Si, _Distributions and correlations for top quark pair production and decay at the Tevatron and LHC._ , Nucl. Phys. B 837 (2010), 90–121, [arXiv:1003.3926 [hep-ph]].
  * [30] J. H. Kuhn, A. Scharf and P. Uwer, _Electroweak effects in top-quark pair production at hadron colliders_ , Eur. Phys. J. C 51 (2007), 37–53, [hep-ph/0610335].
  * [31] W. Bernreuther, M. Fuecker and Z.-G. Si, _Weak interaction corrections to hadronic top quark pair production_ , Phys. Rev. D 74 (2006), 113005, [hep-ph/0610334].
  * [32] J. H. Kühn, A. Scharf and P. Uwer, _Weak Interactions in Top-Quark Pair Production at Hadron Colliders: An Update_ , Phys. Rev. D 91 (2015), no. 1, 014020, [arXiv:1305.5773 [hep-ph]].
  * [33] W. Hollik and D. Pagani, _The electroweak contribution to the top quark forward-backward asymmetry at the Tevatron_ , Phys. Rev. D 84 (2011), 093003, [arXiv:1107.2606 [hep-ph]].
  * [34] D. Pagani, I. Tsinikos and M. Zaro, _The impact of the photon PDF and electroweak corrections on $t\bar{t}$ distributions_, Eur. Phys. J. C 76 (2016), no. 9, 479, [arXiv:1606.01915 [hep-ph]].
  * [35] C. Gütschow, J. M. Lindert and M. Schönherr, _Multi-jet merged top-pair production including electroweak corrections_ , Eur. Phys. J. C 78 (2018), no. 4, 317, [arXiv:1803.00950 [hep-ph]].
  * [36] A. Denner and M. Pellen, _NLO electroweak corrections to off-shell top-antitop production with leptonic decays at the LHC_ , JHEP 08 (2016), 155, [arXiv:1607.05571 [hep-ph]].
  * [37] N. Kidonakis, _NNNLO soft-gluon corrections for the top-quark $p_{T}$ and rapidity distributions_, Phys. Rev. D 91 (2015), no. 3, 031501, [arXiv:1411.2633 [hep-ph]].
  * [38] N. Kidonakis, _Next-to-next-to-leading soft-gluon corrections for the top quark cross section and transverse momentum distribution_ , Phys. Rev. D 82 (2010), 114030, [arXiv:1009.4935 [hep-ph]].
  * [39] N. Kidonakis, _Two-loop soft anomalous dimensions and NNLL resummation for heavy quark production_ , Phys. Rev. Lett. 102 (2009), 232003, [arXiv:0903.2561 [hep-ph]].
  * [40] N. Kidonakis, _NNNLO soft-gluon corrections for the top-antitop pair production cross section_ , Phys. Rev. D 90 (2014), no. 1, 014006, [arXiv:1405.7046 [hep-ph]].
  * [41] N. Kidonakis, _Top-quark double-differential distributions at approximate N 3LO_, Phys. Rev. D 101 (2020), no. 7, 074006, [arXiv:1912.10362 [hep-ph]].
  * [42] V. Ahrens, A. Ferroglia, M. Neubert, B. D. Pecjak and L. L. Yang, _Renormalization-Group Improved Predictions for Top-Quark Pair Production at Hadron Colliders_ , JHEP 09 (2010), 097, [arXiv:1003.5827 [hep-ph]].
  * [43] A. Ferroglia, B. D. Pecjak and L. L. Yang, _Soft-gluon resummation for boosted top-quark production at hadron colliders_ , Phys. Rev. D 86 (2012), 034010, [arXiv:1205.3662 [hep-ph]].
  * [44] A. Ferroglia, S. Marzani, B. D. Pecjak and L. L. Yang, _Boosted top production: factorization and resummation for single-particle inclusive distributions_ , JHEP 01 (2014), 028, [arXiv:1310.3836 [hep-ph]].
  * [45] B. D. Pecjak, D. J. Scott, X. Wang and L. L. Yang, _Resummed differential cross sections for top-quark pairs at the LHC_ , Phys. Rev. Lett. 116 (2016), no. 20, 202001, [arXiv:1601.07020 [hep-ph]].
  * [46] M. Czakon, A. Ferroglia, D. Heymes, A. Mitov, B. D. Pecjak, D. J. Scott, X. Wang and L. L. Yang, _Resummation for (boosted) top-quark pair production at NNLO+NNLL’ in QCD_ , JHEP 05 (2018), 149, [arXiv:1803.07623 [hep-ph]].
  * [47] P. Hinderer, F. Ringer, G. F. Sterman and W. Vogelsang, _Toward NNLL Threshold Resummation for Hadron Pair Production in Hadronic Collisions_ , Phys. Rev. D 91 (2015), no. 1, 014016, [arXiv:1411.3149 [hep-ph]].
  * [48] M. Beneke, M. Czakon, P. Falgari, A. Mitov and C. Schwinn, _Threshold expansion of the $gg(q\bar{q})$ $\to\overline{QQ}+X$ cross section at $O(\alpha^{4}_{s})$_, Phys. Lett. B 690 (2010), 483–490, [arXiv:0911.5166 [hep-ph]], [Erratum: Phys.Lett.B 778, 464–464 (2018)].
  * [49] M. Beneke, P. Falgari and C. Schwinn, _Threshold resummation for pair production of coloured heavy (s)particles at hadron colliders_ , Nucl. Phys. B 842 (2011), 414–474, [arXiv:1007.5414 [hep-ph]].
  * [50] M. Beneke, P. Falgari, S. Klein and C. Schwinn, _Hadronic top-quark pair production with NNLL threshold resummation_ , Nucl. Phys. B 855 (2012), 695–741, [arXiv:1109.1536 [hep-ph]].
  * [51] M. Cacciari, M. Czakon, M. Mangano, A. Mitov and P. Nason, _Top-pair production at hadron colliders with next-to-next-to-leading logarithmic soft-gluon resummation_ , Phys. Lett. B 710 (2012), 612–622, [arXiv:1111.5869 [hep-ph]].
  * [52] J. Piclum and C. Schwinn, _Soft-gluon and Coulomb corrections to hadronic top-quark pair production beyond NNLO_ , JHEP 03 (2018), 164, [arXiv:1801.05788 [hep-ph]].
  * [53] W.-L. Ju, G. Wang, X. Wang, X. Xu, Y. Xu and L. L. Yang, _Top quark pair production near threshold: single/double distributions and mass determination_ , JHEP 06 (2020), 158, [arXiv:2004.03088 [hep-ph]].
  * [54] W.-L. Ju, G. Wang, X. Wang, X. Xu, Y. Xu and L. L. Yang, _Invariant-mass distribution of top-quark pairs and top-quark mass determination_ , Chin. Phys. C 44 (2020), no. 9, 091001, [arXiv:1908.02179 [hep-ph]].
  * [55] H. X. Zhu, C. S. Li, H. T. Li, D. Y. Shao and L. L. Yang, _Transverse-momentum resummation for top-quark pairs at hadron colliders_ , Phys. Rev. Lett. 110 (2013), no. 8, 082001, [arXiv:1208.5774 [hep-ph]].
  * [56] H. T. Li, C. S. Li, D. Y. Shao, L. L. Yang and H. X. Zhu, _Top quark pair production at small transverse momentum in hadronic collisions_ , Phys. Rev. D 88 (2013), 074004, [arXiv:1307.2464 [hep-ph]].
  * [57] S. Catani, M. Grazzini and A. Torre, _Transverse-momentum resummation for heavy-quark hadroproduction_ , Nucl. Phys. B 890 (2014), 518–538, [arXiv:1408.4564 [hep-ph]].
  * [58] S. Catani, M. Grazzini and H. Sargsyan, _Azimuthal asymmetries in QCD hard scattering: infrared safe but divergent_ , JHEP 06 (2017), 017, [arXiv:1703.08468 [hep-ph]].
  * [59] S. Catani, M. Grazzini and H. Sargsyan, _Transverse-momentum resummation for top-quark pair production at the LHC_ , JHEP 11 (2018), 061, [arXiv:1806.01601 [hep-ph]].
  * [60] S. Alioli, A. Broggio and M. A. Lim, _Zero-jettiness resummation for top-quark pair production at the LHC_ , JHEP 01 (2022), 066, [arXiv:2111.03632 [hep-ph]].
  * [61] J. Mazzitelli, P. F. Monni, P. Nason, E. Re, M. Wiesemann and G. Zanderighi, _Next-to-Next-to-Leading Order Event Generation for Top-Quark Pair Production_ , Phys. Rev. Lett. 127 (2021), no. 6, 062001, [arXiv:2012.14267 [hep-ph]].
  * [62] J. Mazzitelli, P. F. Monni, P. Nason, E. Re, M. Wiesemann and G. Zanderighi, _Top-pair production at the LHC with MINNLO PS_, JHEP 04 (2022), 079, [arXiv:2112.12135 [hep-ph]].
  * [63] P. M. Nadolsky, C. Balazs, E. L. Berger and C. P. Yuan, _Gluon-gluon contributions to the production of continuum diphoton pairs at hadron colliders_ , Phys. Rev. D 76 (2007), 013008, [hep-ph/0702003].
  * [64] S. Catani and M. Grazzini, _QCD transverse-momentum resummation in gluon fusion processes_ , Nucl. Phys. B 845 (2011), 297–323, [arXiv:1011.3918 [hep-ph]].
  * [65] J. C. Collins, _INTRINSIC TRANSVERSE MOMENTUM. 1. NONGAUGE THEORIES_ , Phys. Rev. D 21 (1980), 2962.
  * [66] G. F. Sterman, _Mass Divergences in Annihilation Processes. 1. Origin and Nature of Divergences in Cut Vacuum Polarization Diagrams_ , Phys. Rev. D 17 (1978), 2773.
  * [67] L. D. Landau, _On analytic properties of vertex parts in quantum field theory_ , Nucl. Phys. 13 (1959), no. 1, 181–192.
  * [68] S. Coleman and R. E. Norton, _Singularities in the physical region_ , Nuovo Cim. 38 (1965), 438–442.
  * [69] M. Beneke and V. A. Smirnov, _Asymptotic expansion of Feynman integrals near threshold_ , Nucl. Phys. B 522 (1998), 321–344, [hep-ph/9711391].
  * [70] V. A. Smirnov, _Applied asymptotic expansions in momenta and masses_ , Springer Tracts Mod. Phys. 177 (2002), 1–262.
  * [71] V. A. Smirnov, _Analytic tools for Feynman integrals_ , vol. 250, 2012.
  * [72] B. Jantzen, _Foundation and generalization of the expansion by regions_ , JHEP 12 (2011), 076, [arXiv:1111.2589 [hep-ph]].
  * [73] C. W. Bauer, D. Pirjol and I. W. Stewart, _Soft collinear factorization in effective field theory_ , Phys. Rev. D 65 (2002), 054022, [hep-ph/0109045].
  * [74] C. W. Bauer and I. W. Stewart, _Invariant operators in collinear effective theory_ , Phys. Lett. B 516 (2001), 134–142, [hep-ph/0107001].
  * [75] C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, _An Effective field theory for collinear and soft gluons: Heavy to light decays_ , Phys. Rev. D 63 (2001), 114020, [hep-ph/0011336].
  * [76] C. W. Bauer, S. Fleming and M. E. Luke, _Summing Sudakov logarithms in $B\to X_{s}\gamma$in effective field theory._, Phys. Rev. D 63 (2000), 014006, [hep-ph/0005275].
  * [77] C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein and I. W. Stewart, _Hard scattering factorization from effective field theory_ , Phys. Rev. D 66 (2002), 014017, [hep-ph/0202088].
  * [78] M. Beneke, A. P. Chapovsky, M. Diehl and T. Feldmann, _Soft collinear effective theory and heavy to light currents beyond leading power_ , Nucl. Phys. B 643 (2002), 431–476, [hep-ph/0206152].
  * [79] M. Beneke and T. Feldmann, _Multipole expanded soft collinear effective theory with nonAbelian gauge symmetry_ , Phys. Lett. B 553 (2003), 267–276, [hep-ph/0211358].
  * [80] C. W. Bauer, D. Pirjol and I. W. Stewart, _Factorization and endpoint singularities in heavy to light decays_ , Phys. Rev. D 67 (2003), 071502, [hep-ph/0211069].
  * [81] B. O. Lange and M. Neubert, _Factorization and the soft overlap contribution to heavy to light form-factors_ , Nucl. Phys. B 690 (2004), 249–278, [hep-ph/0311345], [Erratum: Nucl.Phys.B 723, 201–202 (2005)].
  * [82] M. Beneke and T. Feldmann, _Factorization of heavy to light form-factors in soft collinear effective theory_ , Nucl. Phys. B 685 (2004), 249–296, [hep-ph/0311335].
  * [83] E. Eichten and B. R. Hill, _An Effective Field Theory for the Calculation of Matrix Elements Involving Heavy Quarks_ , Phys. Lett. B 234 (1990), 511–516.
  * [84] H. Georgi, _An Effective Field Theory for Heavy Quarks at Low-energies_ , Phys. Lett. B 240 (1990), 447–450.
  * [85] B. Grinstein, _The Static Quark Effective Theory_ , Nucl. Phys. B 339 (1990), 253–268.
  * [86] M. Neubert, _Heavy quark symmetry_ , Phys. Rept. 245 (1994), 259–396, [hep-ph/9306320].
  * [87] J.-y. Chiu, A. Jain, D. Neill and I. Z. Rothstein, _The Rapidity Renormalization Group_ , Phys. Rev. Lett. 108 (2012), 151601, [arXiv:1104.0881 [hep-ph]].
  * [88] J.-Y. Chiu, A. Jain, D. Neill and I. Z. Rothstein, _A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory_ , JHEP 05 (2012), 084, [arXiv:1202.0814 [hep-ph]].
  * [89] Y. Li, D. Neill and H. X. Zhu, _An exponential regulator for rapidity divergences_ , Nucl. Phys. B 960 (2020), 115193, [arXiv:1604.00392 [hep-ph]].
  * [90] Y. Li and H. X. Zhu, _Bootstrapping Rapidity Anomalous Dimensions for Transverse-Momentum Resummation_ , Phys. Rev. Lett. 118 (2017), no. 2, 022004, [arXiv:1604.01404 [hep-ph]].
  * [91] J. C. Collins, D. E. Soper and G. F. Sterman, _Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production_ , Nucl. Phys. B 250 (1985), 199–224.
  * [92] S. Catani, D. de Florian and M. Grazzini, _Universality of nonleading logarithmic contributions in transverse momentum distributions_ , Nucl. Phys. B 596 (2001), 299–312, [hep-ph/0008184].
  * [93] G. Bozzi, S. Catani, D. de Florian and M. Grazzini, _Transverse-momentum resummation and the spectrum of the Higgs boson at the LHC_ , Nucl. Phys. B 737 (2006), 73–120, [hep-ph/0508068].
  * [94] G. Bozzi, S. Catani, D. de Florian and M. Grazzini, _Higgs boson production at the LHC: Transverse-momentum resummation and rapidity dependence_ , Nucl. Phys. B 791 (2008), 1–19, [arXiv:0705.3887 [hep-ph]].
  * [95] M. A. Ebert and F. J. Tackmann, _Resummation of Transverse Momentum Distributions in Distribution Space_ , JHEP 02 (2017), 110, [arXiv:1611.08610 [hep-ph]].
  * [96] P. F. Monni, E. Re and P. Torrielli, _Higgs Transverse-Momentum Resummation in Direct Space_ , Phys. Rev. Lett. 116 (2016), no. 24, 242001, [arXiv:1604.02191 [hep-ph]].
  * [97] W. Bizon, P. F. Monni, E. Re, L. Rottoli and P. Torrielli, _Momentum-space resummation for transverse observables and the Higgs p ⟂ at N3LL+NNLO_, JHEP 02 (2018), 108, [arXiv:1705.09127 [hep-ph]].
  * [98] T. Becher and M. Neubert, _Drell-Yan Production at Small $q_{T}$, Transverse Parton Distributions and the Collinear Anomaly_, Eur. Phys. J. C 71 (2011), 1665, [arXiv:1007.4005 [hep-ph]].
  * [99] M. G. Echevarria, A. Idilbi and I. Scimemi, _Factorization Theorem For Drell-Yan At Low $q_{T}$ And Transverse Momentum Distributions On-The-Light-Cone_, JHEP 07 (2012), 002, [arXiv:1111.4996 [hep-ph]].
  * [100] T. Becher and G. Bell, _Analytic Regularization in Soft-Collinear Effective Theory_ , Phys. Lett. B 713 (2012), 41–46, [arXiv:1112.3907 [hep-ph]].
  * [101] J. C. Collins, D. E. Soper and G. F. Sterman, _Factorization of Hard Processes in QCD_ , Adv. Ser. Direct. High Energy Phys. 5 (1989), 1–91, [hep-ph/0409313].
  * [102] A. V. Manohar and I. W. Stewart, _The Zero-Bin and Mode Factorization in Quantum Field Theory_ , Phys. Rev. D 76 (2007), 074002, [hep-ph/0605001].
  * [103] B. Ananthanarayan, A. Pal, S. Ramanan and R. Sarkar, _Unveiling Regions in multi-scale Feynman Integrals using Singularities and Power Geometry_ , Eur. Phys. J. C 79 (2019), no. 1, 57, [arXiv:1810.06270 [hep-ph]].
  * [104] J. Plenter and G. Rodrigo, _Asymptotic expansions through the loop-tree duality_ , Eur. Phys. J. C 81 (2021), no. 4, 320, [arXiv:2005.02119 [hep-ph]].
  * [105] G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, F. Langer, V. Magerya, A. Pöldaru, J. Schlenk and E. Villa, _Expansion by regions with pySecDec_ , Comput. Phys. Commun. 273 (2022), 108267, [arXiv:2108.10807 [hep-ph]].
  * [106] J. C. Collins and G. F. Sterman, _Soft Partons in QCD_ , Nucl. Phys. B 185 (1981), 172–188.
  * [107] J. C. Collins, D. E. Soper and G. F. Sterman, _Factorization for Short Distance Hadron - Hadron Scattering_ , Nucl. Phys. B 261 (1985), 104–142.
  * [108] J. C. Collins, _Proof of factorization for diffractive hard scattering_ , Phys. Rev. D 57 (1998), 3051–3056, [hep-ph/9709499], [Erratum: Phys.Rev.D 61, 019902 (2000)].
  * [109] J. C. Collins and A. Metz, _Universality of soft and collinear factors in hard-scattering factorization_ , Phys. Rev. Lett. 93 (2004), 252001, [hep-ph/0408249].
  * [110] J. R. Gaunt, _Glauber Gluons and Multiple Parton Interactions_ , JHEP 07 (2014), 110, [arXiv:1405.2080 [hep-ph]].
  * [111] M. D. Schwartz, K. Yan and H. X. Zhu, _Factorization Violation and Scale Invariance_ , Phys. Rev. D 97 (2018), no. 9, 096017, [arXiv:1801.01138 [hep-ph]].
  * [112] S.-J. Chang and S.-K. Ma, _Feynman rules and quantum electrodynamics at infinite momentum_ , Phys. Rev. 180 (1969), 1506–1513.
  * [113] G. F. Sterman, _Mass Divergences in Annihilation Processes. 2. Cancellation of Divergences in Cut Vacuum Polarization Diagrams_ , Phys. Rev. D 17 (1978), 2789.
  * [114] A. Mitov and G. Sterman, _Final state interactions in single- and multi-particle inclusive cross sections for hadronic collisions_ , Phys. Rev. D 86 (2012), 114038, [arXiv:1209.5798 [hep-ph]].
  * [115] I. Z. Rothstein and I. W. Stewart, _An Effective Field Theory for Forward Scattering and Factorization Violation_ , JHEP 08 (2016), 025, [arXiv:1601.04695 [hep-ph]].
  * [116] T. C. Rogers and P. J. Mulders, _No Generalized TMD-Factorization in Hadro-Production of High Transverse Momentum Hadrons_ , Phys. Rev. D 81 (2010), 094006, [arXiv:1001.2977 [hep-ph]].
  * [117] J. F. Donoghue and D. Wyler, _On Regge kinematics in SCET_ , Phys. Rev. D 81 (2010), 114023, [arXiv:0908.4559 [hep-ph]].
  * [118] C. W. Bauer, B. O. Lange and G. Ovanesyan, _On Glauber modes in Soft-Collinear Effective Theory_ , JHEP 07 (2011), 077, [arXiv:1010.1027 [hep-ph]].
  * [119] S. Fleming, _The role of Glauber exchange in soft collinear effective theory and the Balitsky–Fadin–Kuraev–Lipatov Equation_ , Phys. Lett. B 735 (2014), 266–271, [arXiv:1404.5672 [hep-ph]].
  * [120] I. Moult, S. Raman, G. Ridgway and I. W. Stewart, _Anomalous Dimensions from Soft Regge Constants_ , arXiv:2207.02859 [hep-ph].
  * [121] R. Bonciani, S. Catani, M. Grazzini, H. Sargsyan and A. Torre, _The $q_{T}$ subtraction method for top quark production at hadron colliders_, Eur. Phys. J. C 75 (2015), no. 12, 581, [arXiv:1508.03585 [hep-ph]].
  * [122] J. C. Collins and D. E. Soper, _Back-To-Back Jets in QCD_ , Nucl. Phys. B 193 (1981), 381, [Erratum: Nucl.Phys.B 213, 545 (1983)].
  * [123] R. J. Hill and M. Neubert, _Spectator interactions in soft collinear effective theory_ , Nucl. Phys. B 657 (2003), 229–256, [hep-ph/0211018].
  * [124] A. Pineda and J. Soto, _The Renormalization group improvement of the QCD static potentials_ , Phys. Lett. B 495 (2000), 323–328, [hep-ph/0007197].
  * [125] N. Brambilla, X. Garcia i Tormo, J. Soto and A. Vairo, _The Logarithmic contribution to the QCD static energy at N**4 LO_ , Phys. Lett. B 647 (2007), 185–193, [hep-ph/0610143].
  * [126] P. Bärnreuther, M. Czakon and A. Mitov, _Percent Level Precision Physics at the Tevatron: First Genuine NNLO QCD Corrections to $q\bar{q}\to t\bar{t}+X$_, Phys. Rev. Lett. 109 (2012), 132001, [arXiv:1204.5201 [hep-ph]].
  * [127] M. Beneke, P. Falgari and C. Schwinn, _Soft radiation in heavy-particle pair production: All-order colour structure and two-loop anomalous dimension_ , Nucl. Phys. B 828 (2010), 69–101, [arXiv:0907.1443 [hep-ph]].
  * [128] A. Ferroglia, M. Neubert, B. D. Pecjak and L. L. Yang, _Two-loop divergences of massive scattering amplitudes in non-abelian gauge theories_ , JHEP 11 (2009), 062, [arXiv:0908.3676 [hep-ph]].
  * [129] S. Actis, A. Denner, L. Hofer, A. Scharf and S. Uccirati, _Recursive generation of one-loop amplitudes in the Standard Model_ , JHEP 04 (2013), 037, [arXiv:1211.6316 [hep-ph]].
  * [130] S. Actis, A. Denner, L. Hofer, J.-N. Lang, A. Scharf and S. Uccirati, _RECOLA: REcursive Computation of One-Loop Amplitudes_ , Comput. Phys. Commun. 214 (2017), 140–173, [arXiv:1605.01090 [hep-ph]].
  * [131] L. Chen, M. Czakon and R. Poncelet, _Polarized double-virtual amplitudes for heavy-quark pair production_ , JHEP 03 (2018), 085, [arXiv:1712.08075 [hep-ph]].
  * [132] S. Di Vita, T. Gehrmann, S. Laporta, P. Mastrolia, A. Primo and U. Schubert, _Master integrals for the NNLO virtual corrections to $q\overline{q}\to t\overline{t}$ scattering in QCD: the non-planar graphs_, JHEP 06 (2019), 117, [arXiv:1904.10964 [hep-ph]].
  * [133] S. Badger, E. Chaubey, H. B. Hartanto and R. Marzucca, _Two-loop leading colour QCD helicity amplitudes for top quark pair production in the gluon fusion channel_ , JHEP 06 (2021), 163, [arXiv:2102.13450 [hep-ph]].
  * [134] M. K. Mandal, P. Mastrolia, J. Ronca and W. J. Bobadilla Torres, _Two-loop scattering amplitude for heavy-quark pair production through light-quark annihilation in QCD_ , JHEP 09 (2022), 129, [arXiv:2204.03466 [hep-ph]].
  * [135] M.-X. Luo, X. Wang, X. Xu, L. L. Yang, T.-Z. Yang and H. X. Zhu, _Transverse Parton Distribution and Fragmentation Functions at NNLO: the Quark Case_ , JHEP 10 (2019), 083, [arXiv:1908.03831 [hep-ph]].
  * [136] M.-X. Luo, T.-Z. Yang, H. X. Zhu and Y. J. Zhu, _Transverse Parton Distribution and Fragmentation Functions at NNLO: the Gluon Case_ , JHEP 01 (2020), 040, [arXiv:1909.13820 [hep-ph]].
  * [137] M.-x. Luo, T.-Z. Yang, H. X. Zhu and Y. J. Zhu, _Unpolarized quark and gluon TMD PDFs and FFs at N 3LO_, JHEP 06 (2021), 115, [arXiv:2012.03256 [hep-ph]].
  * [138] M.-x. Luo, T.-Z. Yang, H. X. Zhu and Y. J. Zhu, _Quark Transverse Parton Distribution at the Next-to-Next-to-Next-to-Leading Order_ , Phys. Rev. Lett. 124 (2020), no. 9, 092001, [arXiv:1912.05778 [hep-ph]].
  * [139] D. Gutierrez-Reyes, S. Leal-Gomez, I. Scimemi and A. Vladimirov, _Linearly polarized gluons at next-to-next-to leading order and the Higgs transverse momentum distribution_ , JHEP 11 (2019), 121, [arXiv:1907.03780 [hep-ph]].
  * [140] S. Catani and P. K. Dhani, _Collinear functions for QCD resummations_ , arXiv:2208.05840 [hep-ph].
  * [141] R. Angeles-Martinez, M. Czakon and S. Sapeta, _NNLO soft function for top quark pair production at small transverse momentum_ , JHEP 10 (2018), 201, [arXiv:1809.01459 [hep-ph]].
  * [142] S. Fleming and A. K. Leibovich, _The Resummed Photon Spectrum in Radiative Upsilon Decays_ , Phys. Rev. Lett. 90 (2003), 032001, [hep-ph/0211303].
  * [143] S. Fleming and A. K. Leibovich, _The Photon Spectrum in Upsilon Decays_ , Phys. Rev. D 67 (2003), 074035, [hep-ph/0212094].
  * [144] T. Becher and G. Bell, _The gluon jet function at two-loop order_ , Phys. Lett. B 695 (2011), 252–258, [arXiv:1008.1936 [hep-ph]].
  * [145] P. Banerjee, P. K. Dhani and V. Ravindran, _Gluon jet function at three loops in QCD_ , Phys. Rev. D 98 (2018), no. 9, 094016, [arXiv:1805.02637 [hep-ph]].
  * [146] T. Becher and M. Neubert, _Toward a NNLO calculation of the anti-B — $>$ X(s) gamma decay rate with a cut on photon energy. II. Two-loop result for the jet function_, Phys. Lett. B 637 (2006), 251–259, [hep-ph/0603140].
  * [147] R. Brüser, Z. L. Liu and M. Stahlhofen, _Three-Loop Quark Jet Function_ , Phys. Rev. Lett. 121 (2018), no. 7, 072003, [arXiv:1804.09722 [hep-ph]].
  * [148] T. Becher, M. Neubert, L. Rothen and D. Y. Shao, _Factorization and Resummation for Jet Processes_ , JHEP 11 (2016), 019, [arXiv:1605.02737 [hep-ph]], [Erratum: JHEP 05, 154 (2017)].
  * [149] M. Balsiger, T. Becher and A. Ferroglia, _Resummation of non-global logarithms in cross sections with massive particles_ , JHEP 09 (2020), 029, [arXiv:2006.00014 [hep-ph]].
  * [150] S. Catani and M. H. Seymour, _A General algorithm for calculating jet cross-sections in NLO QCD_ , Nucl. Phys. B 485 (1997), 291–419, [hep-ph/9605323], [Erratum: Nucl.Phys.B 510, 503–504 (1998)].
  * [151] V. A. Smirnov, _Analytical result for dimensionally regularized massless on shell double box_ , Phys. Lett. B 460 (1999), 397–404, [hep-ph/9905323].
  * [152] J. B. Tausk, _Nonplanar massless two loop Feynman diagrams with four on-shell legs_ , Phys. Lett. B 469 (1999), 225–234, [hep-ph/9909506].
  * [153] E. Bejdakic, _Feynman integrals, hypergeometric functions and nested sums_ , Other thesis, 10 2009.
  * [154] M. Czakon, _Automatized analytic continuation of Mellin-Barnes integrals_ , Comput. Phys. Commun. 175 (2006), 559–571, [hep-ph/0511200].
  * [155] M. Czakon, _MBasymptotics_ , https://mbtools.hepforge.org.
  * [156] M. Ochman and T. Riemann, _MBsums - a Mathematica package for the representation of Mellin-Barnes integrals by multiple sums_ , Acta Phys. Polon. B 46 (2015), no. 11, 2117, [arXiv:1511.01323 [hep-ph]].
  * [157] S. Catani, I. Fabre, M. Grazzini and S. Kallweit, _${t{{\bar{t}}}H}$ production at NNLO: the flavour off-diagonal channels_, Eur. Phys. J. C 81 (2021), no. 6, 491, [arXiv:2102.03256 [hep-ph]].
  * [158] T. Gleisberg, S. Hoeche, F. Krauss, A. Schalicke, S. Schumann and J.-C. Winter, _SHERPA 1. alpha: A Proof of concept version_ , JHEP 02 (2004), 056, [hep-ph/0311263].
  * [159] T. Gleisberg, S. Hoeche, F. Krauss, M. Schonherr, S. Schumann, F. Siegert and J. Winter, _Event generation with SHERPA 1.1_ , JHEP 02 (2009), 007, [arXiv:0811.4622 [hep-ph]].
  * [160] E. Bothmann et al., Sherpa collaboration, _Event Generation with Sherpa 2.2_ , SciPost Phys. 7 (2019), no. 3, 034, [arXiv:1905.09127 [hep-ph]].
  * [161] A. Ferroglia, M. Neubert, B. D. Pecjak and L. L. Yang, _Two-loop divergences of scattering amplitudes with massive partons_ , Phys. Rev. Lett. 103 (2009), 201601, [arXiv:0907.4791 [hep-ph]].
  * [162] S. Moch, J. A. M. Vermaseren and A. Vogt, _The Three loop splitting functions in QCD: The Nonsinglet case_ , Nucl. Phys. B 688 (2004), 101–134, [hep-ph/0403192].
  * [163] J. M. Henn, G. P. Korchemsky and B. Mistlberger, _The full four-loop cusp anomalous dimension in $\mathcal{N}=4$ super Yang-Mills and QCD_, JHEP 04 (2020), 018, [arXiv:1911.10174 [hep-th]].
  * [164] A. von Manteuffel, E. Panzer and R. M. Schabinger, _Cusp and collinear anomalous dimensions in four-loop QCD from form factors_ , Phys. Rev. Lett. 124 (2020), no. 16, 162001, [arXiv:2002.04617 [hep-ph]].
  * [165] F. Herzog, S. Moch, B. Ruijl, T. Ueda, J. A. M. Vermaseren and A. Vogt, _Five-loop contributions to low-N non-singlet anomalous dimensions in QCD_ , Phys. Lett. B 790 (2019), 436–443, [arXiv:1812.11818 [hep-ph]].
  * [166] Z. L. Liu and N. Schalch, _Infrared singularities of multi-leg QCD amplitudes with a massive parton at three loops_ , arXiv:2207.02864 [hep-ph].
  * [167] A. A. Vladimirov, _Correspondence between Soft and Rapidity Anomalous Dimensions_ , Phys. Rev. Lett. 118 (2017), no. 6, 062001, [arXiv:1610.05791 [hep-ph]].
  * [168] M. A. Ebert, B. Mistlberger and G. Vita, _Transverse momentum dependent PDFs at N 3LO_, JHEP 09 (2020), 146, [arXiv:2006.05329 [hep-ph]].
  * [169] G. Das, S.-O. Moch and A. Vogt, _Soft corrections to inclusive deep-inelastic scattering at four loops and beyond_ , JHEP 03 (2020), 116, [arXiv:1912.12920 [hep-ph]].
  * [170] C. Duhr, B. Mistlberger and G. Vita, _Soft integrals and soft anomalous dimensions at N 3LO and beyond_, JHEP 09 (2022), 155, [arXiv:2205.04493 [hep-ph]].
  * [171] C. Duhr, B. Mistlberger and G. Vita, _Four-Loop Rapidity Anomalous Dimension and Event Shapes to Fourth Logarithmic Order_ , Phys. Rev. Lett. 129 (2022), no. 16, 162001, [arXiv:2205.02242 [hep-ph]].
  * [172] I. Moult, H. X. Zhu and Y. J. Zhu, _The four loop QCD rapidity anomalous dimension_ , JHEP 08 (2022), 280, [arXiv:2205.02249 [hep-ph]].
  * [173] D. Neill, I. Z. Rothstein and V. Vaidya, _The Higgs Transverse Momentum Distribution at NNLL and its Theoretical Errors_ , JHEP 12 (2015), 097, [arXiv:1503.00005 [hep-ph]].
  * [174] D. Kang, C. Lee and V. Vaidya, _A fast and accurate method for perturbative resummation of transverse momentum-dependent observables_ , JHEP 04 (2018), 149, [arXiv:1710.00078 [hep-ph]].
  * [175] A. J. Buras, M. Jamin, M. E. Lautenbacher and P. H. Weisz, _Effective Hamiltonians for $\Delta S=1$ and $\Delta B=1$ nonleptonic decays beyond the leading logarithmic approximation_, Nucl. Phys. B 370 (1992), 69–104, [Addendum: Nucl.Phys.B 375, 501 (1992)].
  * [176] G. Buchalla, A. J. Buras and M. E. Lautenbacher, _Weak decays beyond leading logarithms_ , Rev. Mod. Phys. 68 (1996), 1125–1144, [hep-ph/9512380].
  * [177] A. Banfi, P. F. Monni, G. P. Salam and G. Zanderighi, _Higgs and Z-boson production with a jet veto_ , Phys. Rev. Lett. 109 (2012), 202001, [arXiv:1206.4998 [hep-ph]].
  * [178] A. Banfi, G. P. Salam and G. Zanderighi, _NLL+NNLO predictions for jet-veto efficiencies in Higgs-boson and Drell-Yan production_ , JHEP 06 (2012), 159, [arXiv:1203.5773 [hep-ph]].
  * [179] W. Bizoń, X. Chen, A. Gehrmann-De Ridder, T. Gehrmann, N. Glover, A. Huss, P. F. Monni, E. Re, L. Rottoli and P. Torrielli, _Fiducial distributions in Higgs and Drell-Yan production at N 3LL+NNLO_, JHEP 12 (2018), 132, [arXiv:1805.05916 [hep-ph]].
  * [180] R. L. Workman et al., Particle Data Group collaboration, _Review of Particle Physics_ , PTEP 2022 (2022), 083C01.
  * [181] J. Ablinger, J. Blümlein, M. Round and C. Schneider, _Numerical Implementation of Harmonic Polylogarithms to Weight w = 8_ , Comput. Phys. Commun. 240 (2019), 189–201, [arXiv:1809.07084 [hep-ph]].
  * [182] R. D. Ball et al., NNPDF collaboration, _Parton distributions from high-precision collider data_ , Eur. Phys. J. C 77 (2017), no. 10, 663, [arXiv:1706.00428 [hep-ph]].
  * [183] A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht, M. Schönherr and G. Watt, _LHAPDF6: parton density access in the LHC precision era_ , Eur. Phys. J. C 75 (2015), 132, [arXiv:1412.7420 [hep-ph]].
  * [184] E. Bothmann, A. Buckley, I. A. Christidi, C. Gütschow, S. Höche, M. Knobbe, T. Martin and M. Schönherr, _Accelerating LHC event generation with simplified pilot runs and fast PDFs_ , arXiv:2209.00843 [hep-ph].
  * [185] T. Becher, M. Neubert and B. D. Pecjak, _Factorization and Momentum-Space Resummation in Deep-Inelastic Scattering_ , JHEP 01 (2007), 076, [hep-ph/0607228].
  * [186] T. Hahn, _Routines for the diagonalization of complex matrices_ , physics/0607103.
  * [187] A. Buckley, J. Butterworth, D. Grellscheid, H. Hoeth, L. Lonnblad, J. Monk, H. Schulz and F. Siegert, _Rivet user manual_ , Comput. Phys. Commun. 184 (2013), 2803–2819, [arXiv:1003.0694 [hep-ph]].
  * [188] C. Bierlich et al., _Robust Independent Validation of Experiment and Theory: Rivet version 3_ , SciPost Phys. 8 (2020), 026, [arXiv:1912.05451 [hep-ph]].
  * [189] F. Krauss, R. Kuhn and G. Soff, _AMEGIC++ 1.0: A Matrix element generator in C++_ , JHEP 02 (2002), 044, [hep-ph/0109036].
  * [190] S. Catani, S. Dittmaier, M. H. Seymour and Z. Trocsanyi, _The Dipole formalism for next-to-leading order QCD calculations with massive partons_ , Nucl. Phys. B 627 (2002), 189–265, [hep-ph/0201036].
  * [191] T. Gleisberg and F. Krauss, _Automating dipole subtraction for QCD NLO calculations_ , Eur. Phys. J. C 53 (2008), 501–523, [arXiv:0709.2881 [hep-ph]].
  * [192] M. Schönherr, _An automated subtraction of NLO EW infrared divergences_ , Eur. Phys. J. C 78 (2018), no. 2, 119, [arXiv:1712.07975 [hep-ph]].
  * [193] W.-L. Ju and M. Schönherr, _The q T and $\Delta$$\phi$ spectra in W and Z production at the LHC at N3LL’+N2LO_, JHEP 10 (2021), 088, [arXiv:2106.11260 [hep-ph]].
  * [194] A. Gao, J. K. L. Michel, I. W. Stewart and Z. Sun, _A Better Angle on Hadron Transverse Momentum Distributions at the EIC_ , arXiv:2209.11211 [hep-ph].
  * [195] P. A. Baikov, K. G. Chetyrkin and J. H. Kühn, _Five-Loop Running of the QCD coupling constant_ , Phys. Rev. Lett. 118 (2017), no. 8, 082002, [arXiv:1606.08659 [hep-ph]].