compat=1.1.0

#  Resolving the Ultracollinear Paradox with Effective Field Theory

Matthew Baumgart  Department of Physics, Arizona State University, Tempe, AZ
85287, USA Panagiotis Christeas  Department of Physics, Arizona State
University, Tempe, AZ 85287, USA

###### Abstract

Naive intuition about scale decoupling breaks down in the presence of fermion
masses. Kinematic enhancements can greatly extend the range where one needs to
keep a finite mass in calculations to obtain a correct result at even the
$\mathcal{O}(1)$ level. Treating a light fermion as massive though, leads to a
known but somewhat obscure paradox, a seeming leading-order sensitivity to an
arbitrarily small mass. We show how a proper formulation in effective field
theory not only resolves the physical conundrum, but answers the very
practical question of when fermion masses are required. This has important
implications for the development of shower Monte Carlo above the weak scale.

##  1 Introduction

Mathematically, the fact that
$\int_{0}^{\infty}\frac{m^{2}}{(k^{2}+m^{2})^{2}}dk^{2}$ = 1 is utterly
unremarkable. The physicist can recognize in this equation though, the seeds
of catastrophe, even paradox. The integrand resembles a fermion propagator
squared, with the numerator projected onto the terms from the “$m$” piece of
$\not{p}+m$. The integration arises easily enough from loop momenta or phase
space. Standard lore tells us to drop the masses of any fermions that are well
below the energy scale of interest, unless one cares about the details of
small power corrections. However, this simple integral reveals a strong
discrepancy between zeroing out $m$ before or after the calculation in
question.

This noncommutation of integration and the limit $m\rightarrow 0$ was
recognized long ago by Smilga and dubbed a “quasiparadox” of massless QED[1].
In fact, the issue plagues general theories of relativistic fermions. One can
push to an absurd limit by imagining a particle with $m=10^{-10^{10^{10}}}$
GeV. That would be significantly below current limits on the photon, and yet
keeping the $m$ piece of its propagator numerator seemingly leads to an
unsuppressed perturbative contribution. By similar logic, one could imagine
using the LHC to determine the Standard Model neutrino masses.

The physical resolution arises from thinking about the production of light
fermions at high energies. In the immediate aftermath of the collision, the
fermion is not in an approximately on-shell, asymptotic state (outside of a
tiny corner of phase space). Instead, it begins life as a wavepacket that
evolves in time, eventually decaying to the physical fermion plus a cloud of
radiation. It is these emissions of other particles that introduce terms of
the form $\frac{m^{2}}{(k^{2}+m^{2})^{2}}$ . Thus, if we want to
experimentally determine the mass then we must observe this radiation.
Sensitivity to the enhanced part of the integrand
$\frac{m^{2}}{(k^{2}+m^{2})^{2}}$, where the mass effects are important,
requires $k_{\perp}^{2}\lesssim m^{2}$. As we will detail in calculations
below, what enters the denominators in our calculations of interest provides a
condition on transverse momentum. However, as Smilga points out, generating
such “ultracollinear” radiation takes time, and for a particle with energy $E$
and flight path of length $L$, there is a lower cutoff on the emission angle
of $\theta\sim 1/\sqrt{E\,L}$. We therefore pick up a contribution

$\int_{k_{\perp{\rm
min}}}\frac{m^{2}}{(k_{\perp}^{2}+m^{2})^{2}}dk_{\perp}^{2}\sim\frac{m^{2}}{E^{2}\theta^{2}}\sim\frac{m^{2}\,L}{E},$
(1)

with $k_{\perp{\rm min}}\sim\sqrt{E/L}\gg m$. This resolves the strong form of
the paradox as sensitivity to particle masses now depends sensibly on the size
of the experiment. Independent of detailed technical considerations, this
places a lower limit on fermion mass measurable by the LHC, $m_{\rm
LHC,\,min}\sim\mathcal{O}(100)$ eV.

We have avoided a deep physical conundrum, and yet the situation remains
unsatisfying. Taken at face value, the result of Eq. 1 implies that we still
must track fermion masses at scales substantially below those of our process’s
center of mass energy. For computations at the LHC, one might worry that every
neglected charged-lepton mass results in a sizable error. It is the purpose of
the present paper to allay these fears, in general, but also to clarify the
energy regime where ultracollinear effects are important, providing a
systematic treatment of how to include them. This will show where masses must
be kept in a practical computation.

We see from Eq. 1 that the relevant question concerns the scale of
$k_{\perp{\rm min}}$. The physical size of the experiment provides an IR
cutoff, but it is not necessarily the IR cutoff for the problem at hand. For
example, light fermions produced well above the weak scale have the ability to
radiate on-shell electroweak bosons. Thus, in this regime, particle
virtualities are regulated in the IR by something at least at the order of
$m_{W}$, which dominates all fermion masses except the top quark. Below this
scale, with weak particles integrated out, but still well above the fermion
mass, $m_{f}$, one must nonetheless account for the vanishingly small phase
space to produce nearly on-shell light particles in high-energy processes.
This is quantified by the ubiquitous Sudakov factors found in observables that
quantify the (non)showering of radiation.

Unlike collisions (or explosions) of macroscopic objects, which tend to spray
debris in all directions, the scattering of high-energy particles produces
well-collimated jets of radiation. One finds an explanation for this
phenomenon in the structure of Feynman diagrams. The internal propagators of
the showering particles provide kinematic enhancement if they are nearly on-
shell, as is typically the case for soft or collinear emissions. This provides
a perturbative origin for jets, but it leaves a further puzzle: why then isn’t
every emission as collinear as possible?. Instead, the process is dominated by
“strong-ordering,” as implemented in shower Monte Carlo like Pythia by either
angular or virtuality ordering, with the widest or most off-shell emissions
coming first [2]. The differential cross section one gets by demanding the
absence of real emissions must still account for kinematically-enhanced
virtual corrections to all orders of perturbation theory. This gives the
overall Sudakov factor, which schematically modifies a Born-level cross
section $d\sigma_{0}\rightarrow
d\sigma_{0}\exp[-C\frac{\alpha}{\pi}\log(Q^{2}/m_{\rm IR}^{2})]$ in the toy
Yukawa theory we study in this work of interest, where $C$ is an
$\mathcal{O}(1)$ number and $m_{\rm IR}$ is parametrically near the scale of
observation, typically a particle mass or kinematic cut. Gauge theories
famously have stronger “double-log” soft-collinear enhancements, giving
$d\sigma_{0}\rightarrow d\sigma_{0}\exp[-C\frac{\alpha}{\pi}\log(Q^{2}/m_{\rm
IR}^{2})^{2}]$. These overall terms thus give the “no-split” probability, as
nicely summarized in [3]. We thus see that for particles with masses $m_{f}\ll
Q$, the rate to produce them from the high-energy collision exactly on-shell
gets highly suppressed.

The Sudakov factors show that even in the absence of unambiguous lower
physical scales like masses, the physics of jets dynamically produces a soft
scale, the “jet mass” ($m_{J}$), that can be parametrically large compared to
$m_{f}$. We propose in this work, a threshold of
$\exp[-C\frac{\alpha}{\pi}\log(Q^{2}/m_{\rm IR}^{2})^{a}]<50\%$, where $a=1,2$
depending on the theory. If the threshold is crossed for $m_{\rm IR}>m_{f}$,
then the particle is more likely to have split than not, and one then works in
an effective field theory (EFT) where the fermion is treated as massless,
which we take to be SCET1 for the appropriate particles (cf. [4, 5, 6, 7] for
the foundational papers on SCET, or Soft-Collinear Effective Theory). Once one
gets to a jet scale where
$\exp(-C\frac{\alpha}{\pi}\log(m_{J}^{2}/m_{f}^{2}))>50\%$, then at $m_{J}$
one matches to a SCET2 theory where that particular fermion is massive.
Lastly, if one drops below $m_{f}$, then one integrates out the particle.111We
leave it for future exploration whether data is best explained by some other
threshold criteria, like having a no-split probability $<32\%$, 5 $\%$, etc.
Regardless of the details, we necessarily have a regime where fermions need to
be included as propagating degrees of freedom, but with their finite masses
accounted for. This approach thus systematically improves the crude
approximation of taking a particle to either be massless at high-energies or
integrated out at low energies. Letting the Sudakov factor decide when to keep
$m_{f}$ in a calculation can greatly modify the naive estimate that once
$m_{f}^{2}/Q^{2}\ll 1$, we are effectively massless. In particular, we see
that the strength of the interaction must be accounted for. For example, if it
is very feeble, then producing a light particle in a high-energy process can
lead directly to on-shell states since the amplitude to radiate anything would
then be highly suppressed.

The natural arena for the correct treatment of ultracollinear physics is the
regime of high-energy multi-particle emissions in colliders and astroparticle
systems. In [8], the authors developed a general-purpose electroweak shower
Monte Carlo, emphasizing the role that ultracollinear corrections play in the
proper handling of both next-to-leading-log (NLL) resummation and next-to-
leading-order (NLO) matching. They found that the (im)proper treatment of
fermion masses affects observables at the the few-% level for
$\mathcal{O}(10\,\text{--}\,100)$ TeV colliders, which is well within the
scope of current theoretical precision. Furthermore, the discrepancy can grow
with energy scale, meaning that astroparticle studies at even higher energies
are especially sensitive. Since the focus of [8] is a practical realistic
shower algorithm, they did not generalize the conceptual issues and questions
that surround ultracollinear enhancement. This is our focus. We even find in a
simple Yukawa theory that finite-mass contributions can even lead to larger,
$\mathcal{O}(1)$ numerical shifts to observables.

The calculation of showering radiation is done by Monte Carlo in practice. It
is a useful question for future study to what extent finite mass effects and
the ultracollinear regime are receiving proper treatment. It is not the case
that practical shower routines have manifestly & universally accounted for
this physics. Its correct implementation would certainly be in keeping though,
with the present sophistication of modern parton showers. For example, recent
efforts of the community include further QCD and EW corrections along with
their resummation [9],[10],[11],[12],[13], non-perturbative physics such as
hadronization, FSR (fragmentation functions) and ISR (parton distribution
functions) modeling [10],[11],[13], and taking into account mass effects of
particles [12],[13],[14]. Additionally, by going beyond next-to-leading-log
(NLL) resummation in observables dominated by soft emissions, [15] fills an
important gap in building general NLL parton showers. When fragmentation
functions are considered, especially in EW theory, the massless case is
strictly considered so far [9, 10]. There is also the matter of better
precision and higher accuracy in the computation which is related to
developing the computational tools and algorithms used for the analysis. More
specialized approaches have been examined that include SCET [14], the complex
mass scheme to treat finite width effects[12],[14], or alternative jet
techniques [14],[16]. Good agreement with experimental results have been
achieved so far [13],[14]. A priori, except for [8], the explicit discussions
of mass in these references are different from our ultracollinear concern.
Along with finite-width effects, we can treat them systematically within EFT,
where fragmentation functions enter as particular operators.

The aim of the present work is to focus on the subtle physics of having
$m_{f}^{2}/(k_{\perp}^{2}+m_{f}^{2})^{2}$ contributions to observables,
disentangling them from the many other technical complications that arise in
properly showering the full Standard Model. To this end, we work with a
“simplest nontrivial example,” a Yukawa theory consisting of one massless
scalar (“higgs”), one massless fermion (“bottom”), and one fermion (“top”)
with mass $M_{t}$. We use these particle names hereafter, with the
understanding that they are not strictly the Standard Model versions. This
allows for three different regimes of jet physics which depend on the energy
scale relative to $M_{t}$ (cf. Fig. 1). We will see that properly accounting
for the mass in the intermediate-regime theory requires inclusion of the top
width, $\Gamma_{t}$, in its propagator. This shifts splitting functions at
$\mathcal{O}(\alpha_{y})$ and is generally more important numerically and more
universal (since this effect occurs for massive particles of any spin) than
the ultracollinear terms with which we motivated our study. However, we
continue to emphasize the ultracollinear regime as its treatment is more
subtle and forces one into very careful thinking about the physics of massive
particles in high-energy processes.

The structure of the paper is as follows, in Section 2 we review the
calculational setup; Appendix A provides further details of the fragmentation
formalism, which allows us to calculate observables in the presence of
multiple emissions. Section 2 also gives the specific implementation of SCET
in our Yukawa theory. This EFT allows for efficient computation of the
fragmentation quantities across a wide variety of scales and the resummation
of large logarithms that results. The treatment of SCET soft modes is
discussed in Appendix B. In Section 3, we sketch some of the calculational
ingredients, leaving details for Appendix C. Having obtained our quantities of
interest, we combine them at leading-log (LL) and a subset of NLL corrections
with NLO matching in Section 4 and Appendix D before presenting numerical
results & concluding in Section 5.

##  2 A Toy Theory for Ultracollinear Physics

We begin by specifying our observable of interest. Thanks to factorization, we
are able to write our normalized differential decay rate as a convolution of
two separate functions, the Wilson Coefficient $C_{i}$ and the Fragmentation
Function $D_{ij}$ where we capture the high-, and low-energy physics,
respectively,

$\displaystyle\dfrac{1}{\Gamma_{0}}\dfrac{d\Gamma_{j+X}}{dz}=\int_{z}^{1}\dfrac{dx}{x}C_{i}\left(x/z\right)D_{ij}\left(z\right).$
(2)

We are thus interested in the differential rate to observe particle $j$,
carrying a fraction, $z$, of its mother’s plus-momentum, where
$p^{+}=E+p^{z}$. Though we have written it as the decay rate of some heavy
particle, it is trivial to replace $\Gamma\rightarrow\sigma$ and discuss
instead a normalized differential cross section. The normalization factor
$\Gamma_{0}$ is typically determined from the total decay rate for a process
of interest or sometimes its tree-level value. It needs specification for a
particular calculation. In our case, $\Gamma_{0}$ is the all-orders two-body
decay rate of the heavy particle.

In Appendix A, we provide a brief recap of the fragmentation formalism and its
implementation in SCET. In particular, the $D_{ij}$ fragmentation functions
are computed as operator matrix elements. When subject to only leading-log
running in the massless case, they give the probability for $i$ partons to
split into $j$ partons. However, we will see that this interpretation breaks
down at higher perturbative order and in the presence of finite particle
width. By matching our calculation at some scale, we determine the Wilson
coefficients, $C_{i}$ for each species of parton. This gives a boundary
condition for renormalization group (RG) evolution. The usual invariance of
the observable, LHS, with change in renormalization scale gives us equations
for $C_{i}$ or $D_{ij}$ that we can use to run to other scales, resumming
large logarithms in the process. This is further detailed in Section 4.

Our toy theory consists of three particles inspired by their Standard Model
namesakes: 1) one massive Dirac fermion, $top$, that is in a doublet of an
SU(2) global symmetry; 2) one massless SU(2)-singlet Dirac fermion, $bottom$;
3) one masslesss SU(2)-doublet complex scalar, $higgs$. They have one flavor
off-diagonal Yukawa interaction,

$\displaystyle\mathcal{L}=\bar{t}\left(i\not{\partial}-M_{t}\right)t+\bar{b}\left(i\not{\partial}\right)b+(\partial^{\mu}h)^{\dagger}(\partial_{\mu}h)+y_{t}\,h\,\bar{t}\,b+h.c.$
(3)

To initiate the fragmentation process, we produce tops at high-energy. Though
the fragmentation functions and splitting functions that evolve them are
agnostic to the production details, for concreteness, we assume a scalar, $X$,
with $M_{X}\gg M_{t}$ that decays exclusively to $t\bar{t}$ via,

$\mathcal{L}_{X}=y_{X}\,X\,\bar{t}\,t.$ (4)

At a qualitative level, because the $X$ decays into a significantly lighter
sector, what it ultimately produces are jets of $t,\,b\,h$ particles. To
accurately describe their contents, we work in four different energy regimes
(cf. Figure 1): 1) A full theory consisting of eqs. 3 & 4, along with $X$
kinetic terms; matching to this determines the high-energy boundary conditions
for $C_{t,b,h}$. 2) Integrating out the $X$, the kinematics of its top-
particle decay products at very high energies is dominated by collinear
splittings. We can easily resum the resulting large-logs by working in SCET1
with all massless particles. 3) As the jets evolve and particle fragmentation
decreases energies, we enter a regime where we can no longer neglect $M_{t}$.
This is described by SCET2 with a soft scale $\sim M_{t}$. 4) Lastly, passing
below $M_{t}$, we integrate out the top, leaving a theory of $b,\,h$ particles
that only interact via irrelevant operators.

Figure 1: Schematic description of our sequence of theories

Our interest is only the collinear sectors of SCET1 and SCET2. We write here
the SCET2 collinear lagrangian, from which one can easily get the SCET1
collinear sector by setting $M_{t}=0$.

$\displaystyle\mathcal{L}_{\text{SCET}}$
$\displaystyle=\bar{t}_{n}\left\\{i\,(n\cdot\partial)+i\not{\partial}_{\perp}\dfrac{1}{\bar{n}\cdot\mathcal{P}}i\not{\partial}_{\perp}\right\\}\dfrac{\not{\bar{n}}}{2}t_{n}-M_{t}^{2}\,\bar{t}_{n}\dfrac{1}{\bar{n}\cdot\mathcal{P}}t_{n}+$
$\displaystyle+\bar{b}_{n}\left\\{i\,(n\cdot\partial)+i\not{\partial}_{\perp}\dfrac{1}{\bar{n}\cdot\mathcal{P}}i\not{\partial}_{\perp}\right\\}\dfrac{\not{\bar{n}}}{2}b_{n}+(\partial_{\mu}h_{n})^{\dagger}(\partial^{\mu}h_{n})$
$\displaystyle+y\,\bar{t}_{n}\left\\{(i\not{\partial}_{\perp}-M_{t})\dfrac{1}{\bar{n}\cdot\mathcal{P}}h_{n}-h_{n}\dfrac{1}{\bar{n}\cdot\mathcal{P}}i\not{\partial}_{\perp}\right\\}\dfrac{\not{\bar{n}}}{2}b_{n}$
$\displaystyle+y\,\bar{b}_{n}\left\\{i\not{\partial}_{\perp}\dfrac{1}{\bar{n}\cdot\mathcal{P}}h^{\dagger}_{n}-h^{\dagger}_{n}\dfrac{1}{\bar{n}\cdot\mathcal{P}}(i\not{\partial}_{\perp}+M_{t})\right\\}\dfrac{\not{\bar{n}}}{2}t_{n}$
$\displaystyle-y^{2}\left\\{h_{n}\bar{t}_{n}\dfrac{1}{\bar{n}\cdot\mathcal{P}}\dfrac{\not{\bar{n}}}{2}t_{n}h^{\dagger}_{n}+h^{\dagger}_{n}\bar{b}_{n}\dfrac{1}{\bar{n}\cdot\mathcal{P}}\dfrac{\not{\bar{n}}}{2}b_{n}h_{n}\right\\}.$
(5)

By the usual convention the label momentum operator $\bar{n}\cdot\mathcal{P}$
only acts on fields to its right. Also by standard SCET practice, the two
lightcone vectors $n^{\mu},\bar{n}^{\mu}$ are defined as $n^{2}=\bar{n}^{2}=0$
and $n\cdot\bar{n}=2$. They form a lightcone basis where we can express any
(momentum) vector as $p^{\mu}=(\bar{n}\cdot p,n\cdot
p,p_{\perp})=(p^{+},p^{-},p_{\perp})$. This lagrangian gives the following
Feynman rules:

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a); \vertex[right=of a] (b); \vertex[above right=of b] (d); \vertex[
below right=of b] (e); \diagram* { (a) -- [double,double
distance=0.5ex,thick,with arrow=0.5,arrow size=0.15em,momentum=$p$](b), (b)--
[fermion,rmomentum=$k$](d), (b)-- [scalar,rmomentum'=$q$](e), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=(+iy)\left[\left(\dfrac{\not{k}_{\perp}}{k^{+}}-\dfrac{\not{p}_{\perp}}{p^{+}}\right)-\dfrac{M_{t}}{p^{+}}\right]\dfrac{\not{\bar{n}}}{2}$
$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a); \vertex[above left=of a] (b); \vertex[below left=of a] (c);
\vertex[above right=of a] (d); \vertex[ below right=of a] (e); \diagram* { (b)
-- [double,double distance=0.5ex,thick,with arrow=0.5,arrow
size=0.15em,momentum=$p$](a), (c)-- [scalar,momentum=$q$](a), (a)--
[double,double distance=0.5ex,thick,with arrow=0.5,arrow
size=0.15em,rmomentum'=$l$](d), (a)-- [scalar,rmomentum'=$k$](e), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=(-iy^{2})\left(\dfrac{1}{p^{+}+k^{+}}\right)\dfrac{\not{\bar{n}}}{2}.$

The double-line indicates a $t$, the single line a $b$, and the dashed line an
$h$. One should note that these Feynman rules assume all incoming momenta. In
practice, we will only need the three-point vertex above. To the order we are
working, the four-point vertex222There is also another four-point vertex that
is the same as the one written above, where the $t$ is substituted by $b$.
could only contribute in a virtual process, but as we show in Section 3.1, it
does not affect the fragmentation functions. One might also worry about the
need for soft interactions. However, as we demonstrate in Appendix B, in
SCET1, where we can neglect $M_{t}$, the soft limit of the collinear
interactions given in Eq. 5 itself reproduces the soft limit of the full-
theory contributions to fragmentation amplitudes333Zero-bin contribution is
equal to the soft contribution.. In SCET2, the soft-collinear vertex that
would contribute is power-suppressed. Thus, we are left with a straightforward
set of all-collinear calculations we detail in Section 3 and Appendix C.

##  3 Finite Mass Effects at LO & Beyond

In Section 2, we discussed the fragmentation function, $D_{ij}$, our main
calculational quantity of interest. We provide here an overview of its
computation in SCET1/SCET2, and how finite particle masses affect results
beyond mere power corrections. In using the fragmentation formalism to compute
particle production at a variety of energy scales, it becomes necessary to run
the operators whose matrix elements give $D_{ij}$. As we will show in Section
4, the anomalous dimension, $\gamma_{ij}$, for $D_{ij}$ is

$\gamma_{ij}=-\dfrac{dD_{ik}}{d\log(\mu)}\,(D)^{-1}_{kj}.$ (6)

Fragmentation Functions are calculated through a fixed order diagram based on
the definitions in Eq. 64. For example, we can take the $D_{th}$ computation
at $\mathcal{O}(\alpha_{y})$:

$\displaystyle D_{th}=\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a)[crossed dot] {}; \vertex[right=of a] (b); \vertex[above right=of b]
(c1); \vertex[ right=of c1] (c2); \vertex[below right=of c2] (d); \vertex[
right=of d] (e)[crossed dot]{}; \diagram* { (a) -- [double,double
distance=0.5ex,thick,with arrow=0.5,arrow size=0.15em,momentum=$p+k$](b) --
[scalar, momentum=$p$](c1), (c2) -- [scalar, momentum=$p$](d)--[double,double
distance=0.5ex,thick,with arrow=0.5,arrow size=0.15em,momentum=$p+k$](e),
(b)-- [fermion,half right,
looseness=1,insertion={[size=10pt]0.5},momentum'=$k$](d) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (7)

By construction, this process consists of an energetic mother parton
fragmenting to a daughter. In the SCET, the mother is created by a collinear
field. We can investigate various contributions for the other states from the
soft and collinear sectors by setting their momenta in these regions. By
energy conservation, one or both of the daughter particles is necessarily
collinear. In SCET2 the soft emission is power suppressed while in SCET1 the
soft contribution is cancelled out by the zero-bin, as shown in B. Therefore,
only the all-collinear process is the leading one of interest. As a
consequence of the power counting in SCET1, we drop $M_{t}$ as a scale below
that of theory’s soft scale. To regulate IR divergences, we impose a fixed
$\mu_{\textrm{I}R}$ cutoff. Our SCET1 splitting functions just reproduce the
standard massless splitting functions [17].

Once the scale is sufficiently close to $M_{t}$ that on-shell top production
is likely, we match to the intermediate-scale theory, SCET2. We can no longer
neglect $M_{t}$, and its presence modifies our results at LO because of its
finite width, and at NLO through the ultracollinear contributions. When the
shower reaches the scale $M_{t}$, we integrate out the top. In this Low-Energy
Theory, there are only trivial fragmentation functions for the remaining
particles, $\sim\delta(1-z)$.

###  3.1 Outline of the FF Computation

The computation of a real-emission fragmentation function will be outlined
with an example of $D_{th}$ in SCET2.444One can obtain the analogous SCET1
$D_{th}$ result by setting $M_{t}=0$ in the SCET2 integrand and imposing a
fixed IR cutoff on perp momentum, $\mu_{\textrm{I}R}$. For notational
simplicity though, we always call the IR scale in the log argument $M_{t}$.
Starting from the definition of the Fragmentation Function given in Eq. 64,

$\displaystyle D_{th}(z)$ $\displaystyle=\frac{z}{4Q}\Tr\int
d\Pi_{X}\,\delta\left(1-\frac{p_{X}^{+}+p^{+}}{Q}\right)\langle
0|\dfrac{\not{\bar{n}}}{2}t(p+p_{X})|h(p)\,X\rangle\langle
h(p)\,X|\bar{t}(p+p_{X})|0\rangle\,\Big{|}_{p_{\perp}=0},$ (8)

where we have used $N_{c}=2$ since the initiating $t$ is a doublet of
SU(2).555The $X$ in Eq. 8 is meant to represent a generic final state of the
SCET2 theory. It is not the heavy $X$ particle we used in the full theory as a
source of $t\bar{t}$. At $\mathcal{O}(\alpha_{y})$ we need to compute the
graph shown at the start of this section. As we can see, the graph is cut on
the observed daughter and recoiling particle lines, which puts these particles
on shell. The unintegrated momentum $p$ is thus on-shell by construction, and
we note that we have rotated to the frame with $p_{\perp}$ = 0. This also
introduces a $\delta$-function setting $k^{2}=0$. After imposing all
$\delta$-function constraints, one straightforward integral is left, for which
we use dimensional regularization to remove its UV divergence,

$\displaystyle
D_{th}(z)=\alpha_{y}\int\dfrac{d^{2-2\epsilon}k_{\perp}}{(2\pi)^{2}}\dfrac{z\,k_{\perp}^{2}+\frac{(1-z)^{2}}{z}M_{t}^{2}}{\left(k_{\perp}^{2}-\left(\frac{1-z}{z}\right)M_{t}^{2}\right)^{2}+\left(\frac{1-z}{z}\right)^{2}M_{t}^{2}\,\Gamma_{t}^{2}}.$
(9)

We recognize that the $M_{t}^{2}$ term in the numerator, combined with the
denominator, provides an ultracollinear correction to the LL result.
Additionally, the kinematics allow the top quark propagator to hit the pole at
$(p+k)^{2}=M_{t}^{2}$. By working in the complex mass scheme[18],[19], and
deforming $M_{t}\rightarrow M_{t}-i\,\Gamma_{t}$, we avoid a divergence, but
do pick up a contribution enhanced by $1/\Gamma_{t}$. At LO,
$\Gamma_{t}=\frac{\alpha_{y}M_{t}}{4}$. The physics of such a large correction
makes sense since we are necessarily in the region of on-shell top production.
In addition to the usual perturbative splitting process that creates $b,\,h$
particles, the top can also decay to them, which it does with a probability
that approaches 100%.666If we were interested in the details of top decay
product distributions near its mass pole, then it may be necessary to resum
the large logs of $\Gamma_{t}/M_{t}$. The Unstable Particle Effective Theory
of [20] is one straightforward formalism for tackling this problem. However,
we take the observables from our processes of interest, jet production from
energetic collider or astroparticle processes, as sufficiently inclusive that
this resummation is beyond our scope. It is also pointed out in Ref. [21] that
including finite widths breaks gauge invariance. Gauge theories are beyond the
scope of this work, but are an important future direction. Upon performing the
integral in Eq. 9 and renormalizing in $\overline{\text{MS}}$, we reach the
final expression,

$\displaystyle
D_{th}(z)=\dfrac{\alpha_{y}}{4\pi}\left[z\,\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)-z\,\log\left(\frac{1-z}{z}\right)-1+\dfrac{\pi\,M_{t}}{\Gamma_{t}}\right],$
(10)

where $\mu$ is the renormalization scale. The NLL terms, with no explicit
$\log(\mu^{2}/M_{t}^{2})$ or $1/\Gamma_{t}$ are due to the presence of finite
$M_{t}$ in both numerator and denominator. These are the ultracollinear
contributions. Lastly, the $1/\Gamma_{t}$ term modifies the running at LO from
the $D^{-1}$ contribution to $\gamma_{ij}$ (cf. Eq. 6).

It is also necessary to obtain the self-fragmentation functions and their
running. These are just the wavefunction renormalization factors for the
fields in our theory.777Since the top is unstable, its pole and residue are
complex valued. Thus, the factor that enters the fragmentation function is
$|\sqrt{Z_{t}}|^{2}$. The imaginary part of $\sqrt{Z_{t}}$ only enters at
$\mathcal{O}(\alpha_{y})$, and as we only work to that order, in practice we
can drop it. Since these computations take a different form, we give an
example here with $D_{hh}$ ($=Z_{h}\,\delta(1-z)$, where $Z_{h}$ is our
$\overline{\text{MS}}$ wavefunction renormalization). For the self-
fragmentation function, the contribution of this graph to the residue to
$\mathcal{O}(\alpha_{y})$ is calculated as

$\displaystyle
Z_{h}=\left(1-\dfrac{1}{p^{+}}\dfrac{d}{dp^{-}}\Sigma(p)\Big{|}_{p^{2}=0}\right)^{-1},$
(12)

where $\Sigma(p)$ is the expression of the loop. We again compute in the SCET2
theory with dimensional regularization and $\overline{\text{MS}}$, obtaining

$\displaystyle
D_{hh}(z)=\left(1-\dfrac{\alpha_{y}}{2\pi}\left[\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)+\frac{1}{2}\right]\right)\delta(1-z)$
(13)

Similar to the real emission graph, the $\log$ coefficient is the leading-log
term. The other $\mathcal{O}(\alpha_{y})$ contribution arises from the finite
$M_{t}$ that we track in the SCET2 theory. The calculations for the other real
& virtual-emission fragmentation functions proceed similarly. In Appendix C,
one can find more calculational details for real and virtual fragmentation
functions. We collect the results for the entire theory in Table 1 below.

In principle, there are other diagrams contributing to $D_{hh}$ and $D_{tt}$
of the following form888There are also two extra graphs, one contributing to
$D_{hh}$ and one to $D_{bb}$, where they are the same but $t$ is substituted
by $b$.:

$\displaystyle D_{hh}:\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a)[crossed dot] {}; \vertex[right=of a] (b); \vertex[ right=of b]
(e)[crossed dot]{}; \vertex[ above left=of b] (e1); \vertex[ above right=of b]
(e2); \diagram* { (a) -- [scalar](b), (b)-- [double,double
distance=0.5ex,thick,with arrow=0.5,arrow size=0.15em](e1), (e2)--
[double,double distance=0.5ex,thick,with arrow=0.5,arrow size=0.15em] (b),
(e1)-- [double,double distance=0.5ex,thick,with arrow=0.5,arrow
size=0.15em,half left, looseness=1](e2), (b)-- [scalar](e) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad
D_{tt}:\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a)[crossed dot] {}; \vertex[right=of a] (b); \vertex[ right=of b]
(e)[crossed dot]{}; \vertex[ above left=of b] (e1); \vertex[ above right=of b]
(e2); \diagram* { (a) -- [double,double distance=0.5ex,thick,with
arrow=0.5,arrow size=0.15em](b), (b)-- [scalar](e1), (b)-- [scalar] (e2),
(e1)-- [scalar,half left, looseness=1](e2), (b)-- [double,double
distance=0.5ex,thick,with arrow=0.5,arrow size=0.15em](e) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (14)

These graphs however, have trivial contribution at $\mathcal{O}(\alpha_{y})$
since $\dfrac{d}{dp^{-}}\Sigma(p)$ would give 0.

###  3.2 Summary of Results

We tabulate here all results that contribute to matching of Wilson
coefficients at $\mathcal{O}(\alpha_{y})$ as we run from scales with $Q\gg
M_{t}$ to energies below $M_{t}$, where the top is integrated out of the
theory. This includes shifts to the LO results from the finite top width,
$\Gamma_{t}$. We also include those parts of the anomalous dimensions that
arise from the finite $M_{t}$ in the SCET2 regime and contribute to running at
NLL order. These are ultracollinear contributions in the sense that their
existence requires the inclusion of nonzero $M_{t}$ in the initial phase of
the calculation, but they persist if one takes $M_{t}\rightarrow 0$ at the end
of it. We dub these terms “NLL-UC”. We stress that we have not performed a
full NLL resummation, as this requires two-loop results in the toy theory that
are beyond our scope. Nonetheless, in presenting numerical results in Section
5, we show the shifts incurred in the Wilson coefficients by including the
NLL-UC terms. This is to demonstrate the size of the contributions that would
be missed under a more naive treatment that keeps the top massless until it is
integrated out.

F.F. Leading-Log Finite $M_{t}$ corrections $D_{tb}\left(z\right)=$
$\dfrac{\alpha_{y}}{4\pi}(1-z)\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)$
$+\dfrac{\alpha_{y}}{4\pi}\left((1-z)\,\log\left(\dfrac{z}{1-z}\right)-1+\pi\dfrac{M_{t}}{\Gamma_{t}}\right)$
$D_{th}\left(z\right)=$
$\dfrac{\alpha_{y}}{4\pi}\,z\,\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)$
$+\dfrac{\alpha_{y}}{4\pi}\left(z\log\left(\dfrac{z}{1-z}\right)-1+\pi\dfrac{M_{t}}{\Gamma_{t}}\right)$
$D_{bt}\left(z\right)=$
$\dfrac{\alpha_{y}}{2\pi}(1-z)\,\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)$
$+\dfrac{\alpha_{y}}{2\pi}\left((1-z)\,\log\left(\dfrac{z^{2}}{1-z}\right)+z\right)$
$D_{bh}\left(z\right)=$
$\dfrac{\alpha_{y}}{2\pi}z\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)$
$+\dfrac{\alpha_{y}}{2\pi}\left(z\log\left(z\right)+1-z\right)$
$D_{hb}\left(z\right)=$
$\dfrac{\alpha_{y}}{2\pi}\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)$
$+\dfrac{\alpha_{y}}{2\pi}\left(\log\left(z\right)-1+z\right)$
$D_{ht}\left(z\right)=$
$\dfrac{\alpha_{y}}{2\pi}\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)$
$+\dfrac{\alpha_{y}}{2\pi}\left(\log\left(\dfrac{z^{2}}{1-z}\right)-z\right)$
$D_{tt}\left(z\right)=$
$\left[1-\dfrac{\alpha_{y}}{8\pi}\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)\right]\delta(1-z)$
0 $D_{hh}\left(z\right)=$
$\left[1-\dfrac{\alpha_{y}}{2\pi}\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)\right]\delta(1-z)$
$-\dfrac{\alpha_{y}}{2\pi}\left(\dfrac{1}{2}\right)\delta(1-z)$
$D_{bb}\left(z\right)=$
$\left[1-\dfrac{\alpha_{y}}{4\pi}\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)\right]\delta(1-z)$
$-\dfrac{\alpha_{y}}{4\pi}\left(\dfrac{3}{2}\right)\delta(1-z)$

Table 1: The fragmentation functions to $\mathcal{O}(\alpha_{y})$. In the
high-scale theory (SCET1), we only get the results in the LL column, and
$M_{t}$ is strictly an IR cutoff on perp momentum. In the intermediate-energy
theory (SCET2), we pickup full $M_{t}$ dependence. Since
$\Gamma_{t}=\frac{\alpha_{y}M_{t}}{4}$, we see that all entries in the “Finite
$M_{t}$ corrections” column survive in the $M_{t}\rightarrow 0$ limit, even
though we need to keep $M_{t}\neq 0$ for their initial determination.

As we will further see in Section 4, it is useful to write the Fragmentation
Functions in matrix form:

$\displaystyle D_{ij}=\left(\begin{matrix}D_{hh}&D_{hb}&D_{ht}\\\
D_{bh}&D_{bb}&D_{bt}\\\ D_{th}&D_{tb}&D_{tt}\end{matrix}\right)$ (15)

In the high-scale SCET1 theory, where all particles are treated as massless,
the LO “fragmentation matrix” is just $\delta(1-z)$ terms on the diagonal. As
discussed above, after passing to the intermediate-scale SCET2 theory with
finite $M_{t}$ & $\Gamma_{t}$, the Fragmentation Functions at LO are modified
to the following expression:

$\displaystyle D^{LO}_{ij}=\left(\begin{matrix}\delta(1-z)&0&0\\\
0&\delta(1-z)&0\\\ 1&1&\delta(1-z)\end{matrix}\right)$ (16)

The off-diagonal, $z$-independent terms arise from the near-certain decay of
the unstable tops. We give the full contributions up to NLO in
$\mathcal{O}(\alpha_{y})$ in Table 1. In Mellin Space, the total contribution
in the intermediate-mass theory (SCET2) to NLO is

$\displaystyle D_{ij}$
$\displaystyle=D^{\text{LO}}_{ij}+D^{M_{t}=0,\,\text{NLO}}_{ij}\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)+D^{M_{t}\neq
0,\,\text{NLO}}_{ij}=$ $\displaystyle=\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ \frac{1}{s}&\frac{1}{s}&1\\\ \end{array}\right)+$ (20)
$\displaystyle+\dfrac{\alpha_{y}}{2\pi}\left(\begin{array}[]{ccc}-1&\frac{1}{s}&\frac{1}{s}\\\\[5.0pt]
\frac{1}{(s+1)}&-\frac{1}{2}&\frac{1}{s(s+1)}\\\\[5.0pt]
\frac{1}{2(s+1)}&\frac{1}{2s(s+1)}&-\frac{1}{4}\\\
\end{array}\right)\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)+$ (24)
$\displaystyle+\dfrac{\alpha_{y}}{2\pi}\left(\begin{array}[]{ccc}-\frac{1}{2}&\frac{-2s-1}{s^{2}(s+1)}&\frac{H(s)}{s}-\frac{2}{s^{2}}-\frac{1}{s+1}\\\\[5.0pt]
\frac{1}{s(s+1)^{2}}&-\frac{3}{4}&\frac{s(s+1)H(s)+s^{3}-4s-2}{s^{2}(s+1)^{2}}\\\\[5.0pt]
\frac{H\left(s+1\right)}{2(s+1)}-\frac{1}{2(s+1)^{2}}-\frac{1}{2s}&\frac{sH(s)-(s+1)^{2}}{2s^{2}(s+1)}&0\\\
\end{array}\right),$ (28)

where $H(s)$ is the generalized harmonic number. One can easily obtain the
high-scale (SCET1) fragmentation matrix from Eq. 28 by dropping the off-
diagonal elements in $D^{\text{LO}}_{ij}$, along with the $D^{M_{t}\neq
0,\,\text{NLO}}_{ij}$ term. As we discuss next, the running and log
resummation necessary for showering particles in the different energy regimes
of our theory is a ‘turn-the-crank’ extension of the work described thus far.

##  4 Running and Matching

The explicit $\mu$-dependence we see in the renormalized fragmentation
functions of Table 1 lets us use RG evolution to run our Wilson coefficients
down from their UV-determined boundary conditions to IR values. In doing so,
we automatically sum the LL contributions to our observable to all orders.
Indeed, one of the primary benefits of calculating in SCET is the
straightforward way that this resummation follows from the RGE of its
operators. Recall from Figure 1 that our calculation has four regimes: full
Yukawa theory, SCET1 with $M_{t}=0$, SCET2 with finite $M_{t}$, and lastly a
theory below $M_{t}$ with the top integrated out. The full description thus
involves three matching steps and two stages of RG running.

The full-theory/SCET1 matching occurs at a high-scale we call $M_{X}$,
thinking of it as the mass of a very particle that decays exlusively to tops.
The SCET1 running is standard and has no top mass effects by construction.
Then, at some scale $M_{\text{intermediate}}$, top particles are more likely
than not to be produced on-shell and we must account for finite $M_{t}$. Since
$M_{t}$ is now comparable to our soft scale, we have a SCET2 theory. The
matching of Wilson coefficients from SCET1 is trivial. However, for all scales
below $M_{\text{intermediate}}$, they run with anomalous dimensions modified
by finite $M_{t}$.999The SCET1/SCET2 transition is most simply handled by
matching our SCET1 of interest to a fictitious SCET2 where the top mass is
tuned by hand to be parametrically below its soft scale, and thus dropped.
Having obtained the Wilson coefficients of this artificial theory, we then
tune $M_{t}$ to be larger until it coincides with the soft scale. This
necessarily modifies operator dimensions, but it cannot change the Wilson
coefficients, which are set by UV physics and are thus decoupled from the soft
scale, including $M_{t}$ effects. Lastly, in matching at $M_{t}$, one
necessarily accounts for the finite-width and ultracollinear contributions due
to $M_{t}$ in the fragmentation functions.

###  4.1 RG Evolution for Wilson Coefficients and Fragmentation Functions

Although it is standard, we review here the evolution equation derivation for
fragmentation functions and their corresponding Wilson coefficients. Our
discussion largely follows that in [22]. We will find that for an unstable
particle, a careful accounting is needed even for a correct treatment at
$\mathcal{O}(1)$. If we take an observable that factorizes like our
differential decay rate of interest (cf. Eq. 2), we can write it as follows,

$\displaystyle\sigma^{\prime}_{j}=C_{i}(\mu)D_{ij}(\mu),$ (29)

where $\sigma^{\prime}_{j}$ is a differential rate involving particle $j$.
Since this quantity must be $\mu$-independent,

$\displaystyle\dfrac{d\sigma^{\prime}_{j}}{d\log(\mu)}=0\quad\Rightarrow\quad\dfrac{dC_{i}(\mu)}{d\log(\mu)}D_{ij}(\mu)+C_{i}(\mu)\dfrac{dD_{ij}(\mu)}{d\log(\mu)}=0$
(30)

It is useful to first examine the fragmentation function; one can write the
bare $D_{0,ij}$ in terms of the renormalized $D_{kj}$,

$\displaystyle D_{0,ij}=Z_{ik}D_{kj},$ (31)

where $Z_{ij}$ are the renormalization factors. This gives us an additional
constraint to setup the RG equation since bare quantities are necessarily
independent of the renormalization scale,

$\displaystyle\dfrac{dD_{0,ij}}{d\log(\mu)}=0\quad\Rightarrow\quad\dfrac{dZ_{ik}}{d\log(\mu)}D_{kj}+Z_{ik}\dfrac{dD_{kj}}{d\log(\mu)}=0$
(32)

Solving for $\dfrac{dD_{kj}}{d\log(\mu)}$ gives

$\displaystyle\dfrac{dD_{mj}}{d\log(\mu)}=-Z^{-1}_{mi}\dfrac{dZ_{ik}}{d\log(\mu)}D_{kj}.$
(33)

We thus recognize the anomalous dimension matrix, $\gamma_{mk}\equiv
Z^{-1}_{mi}\dfrac{dZ_{ik}}{d\log(\mu)}$, which gives

$\displaystyle\dfrac{dD_{mj}}{d\log(\mu)}$
$\displaystyle=-\gamma_{mk}D_{kj}\Rightarrow$
$\displaystyle\dfrac{dD}{d\log(\mu)}$ $\displaystyle=-\gamma\cdot D$ (34)

Since we can compute the renormalized fragmentation functions $D_{ij}$
straightforwardly, it is useful to invert the last equation to get

$\boxed{\gamma=-\dfrac{dD}{d\log(\mu)}D^{-1}}$ (35)

This form shows its utility in deriving the evolution equation for the Wilson
Coefficients. Rewriting the equation for $\mu$-independence of the observable
(cf. Eq. 30), one gets

$\displaystyle\dfrac{dC_{i}}{d\log(\mu)}D_{ij}=-C_{i}\dfrac{dD_{ij}}{d\log(\mu)},$
(36)

which allows us to introduce the anomalous dimension,

$\displaystyle\dfrac{dC_{i}}{d\log(\mu)}D_{ij}=-C_{i}\left(-\gamma_{ik}D_{kj}\right).$
(37)

We can then eliminate $D_{ij}$ to get a flow equation for the Wilson
coefficients phrased purely in terms of $C$ and $\gamma$,

$\displaystyle\dfrac{d\,C_{l}}{d\log(\mu)}$
$\displaystyle=C_{i}\gamma_{il}\Rightarrow$
$\displaystyle\dfrac{d\,C}{d\log(\mu)}$ $\displaystyle=\gamma^{T}\cdot C$ (38)

As mentioned above, we perform our running in two distinct theories. Even
though they have the same particle content, their $\gamma$s differ at the
$\mathcal{O}(1)$ level due to the different scales they inhabit and whether or
not one has a finite $M_{t}$. In “$z$-space” where we work explicitly with
plus-momentum fractions, we construct our observable as a convolution,

$\displaystyle\dfrac{1}{\Gamma_{0}}\dfrac{d\Gamma_{j+X}}{dz}=\int_{z}^{1}\dfrac{dx}{x}C_{i}\left(x/z\right)D_{ij}\left(z\right).$
(39)

The running equations for Wilson Coefficients are then given by

$\displaystyle\dfrac{dC_{l}(z)}{d\log(\mu)}=\int_{z}^{1}\dfrac{dx}{x}C_{i}(x/z)\,\gamma_{il}(z)$
(40)

We can reduce this integro-differential equation to a mere differential
equation by means of a Mellin Transform, defined by the following relations,

$\displaystyle\left\\{\mathcal{M}f\right\\}(s)=\int_{0}^{\infty}z^{s-1}f(z)\qquad(\text{Mellin
Transform})$ (41)
$\displaystyle\left\\{\mathcal{M}^{-1}F\right\\}(z)=\dfrac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}z^{-s}F(s)\qquad(\text{Inverse Mellin
Transform}),$ (42)

with the property needed for simplifying convolution:

$\displaystyle\left\\{\mathcal{M}\int\dfrac{dx}{x}\quad
f(x/z)g(z)\right\\}(s)=F(s)G(s)$ (43)

Thus Eq. 40 transforms into a simpler algebraic equation,

$\displaystyle\dfrac{d\,C_{l}(s)}{d\log(\mu)}=C_{i}(s)\gamma_{il}(s).$ (44)

For simplicity, we use the same notation for functions and their (inverse)
Mellin transforms.

###  4.2 Running & Matching in the Toy Yukawa Theory

We can now apply this general approach to evolving fragmentation functions and
resumming large logs to our specific theory of interest. We already have what
we need in the Mellin space fragmentation function matrices given in Eq. 28.
To compute our low-scale observables, we now just need to apply the formalism
of Section 4.1. We first compute the SCET2 LL anomalous dimensions, which are
just the $\mathcal{O}(\alpha_{y})$ contributions. Using the notation of Eq.
28, we have

$\displaystyle\gamma^{\text{LL}}_{ij}$
$\displaystyle=-\dfrac{dD_{ik}}{d\log(\mu)}D_{kj}^{-1}\Big{|}_{\mathcal{O}(\alpha_{y})}=-2D_{ik}^{m=0,\,\text{NLO}}\left(D_{kj}^{\text{LO}}\right)^{-1}=$
$\displaystyle=\dfrac{\alpha_{y}}{\pi}\left(\begin{matrix}-\delta(1-z)&1&1\\\
z&-\dfrac{1}{2}\delta(1-z)&1-z\\\
\dfrac{z}{2}&\dfrac{(1-z)}{2}&-\dfrac{1}{4}\delta(1-z)\end{matrix}\right)\left(\begin{array}[]{ccc}\delta(1-z)&0&0\\\
0&\delta(1-z)&0\\\ 1&1&\delta(1-z)\\\ \end{array}\right)^{-1}.$ (48)

A simple way to handle the inverse matrix is to convert everything to Mellin
space, and then transform back to $z$ space. It is straightforward to get the
LL SCET1 anomalous dimesnions from Eq. 48. We just need to zero out the off-
diagonal entries in $D_{kj}^{\text{LO}\;-1}$. These terms arise from the
$\Gamma_{t}$ that follows from having finite $M_{t}$. The operator running
thus undergoes an $\mathcal{O}(1)$ shift once our scale is sufficiently low
that we evolve nearly on-shell particles.

Running at NLL just requires us to calculate $\gamma_{ij}$ from Eq. 35 to
$\mathcal{O}(\alpha_{y}^{2})$. However, this result requires a full accounting
of two-loop diagrams (equivalently, three-body phase space for real emissions)
and is beyond our scope. The “ultracollinear” terms in $D_{ij}$, those that
are $M_{t}$ independent, but whose existence depends on keeping $M_{t}$ in the
initial calculation, do contribute to $\gamma_{ij}$ at
$\mathcal{O}(\alpha_{y}^{2})$. When we present results in Section 5 we show
Wilson coefficients that have undergone “NLL-UC” running, which includes just
the ultracollinear contributions at subleading order. This is necessarily an
incomplete result, but it shows the size of the effect that would be missed if
one were to drop fermion masses in a regime where one should account for
them.101010Naively computing $\gamma_{ij}$ to $\mathcal{O}(\alpha_{y}^{2})$
from our one-loop results in Table 1 generates some terms of the form
$\alpha_{y}^{2}\log(\mu)$. In running the Wilson coefficients, these would
integrate to $\alpha_{y}^{2}\log^{2}$ contributions. However, that is an order
that one gets from exponentiating the LL result. If it were to survive, it
would imply that the LL resummation fails. Since that is not the case as it
would violate the $\mu$-independence of our observable, a complete calculation
of $\gamma_{ij}$ to $\mathcal{O}(\alpha_{y}^{2})$ must cancel any
contributions of the form $\alpha_{y}^{2}\log$. Thus, we drop any such terms
in our NLL-UC anomalous dimensions.

The $\mathcal{O}(\alpha_{y}^{2})$ contribution to the anomalous dimension
matrix that captures the ultracollinear terms is:

$\displaystyle\gamma^{\text{NLL-UC}}_{ij}$
$\displaystyle=-\dfrac{dD_{ik}}{d\log(\mu)}D_{kj}^{-1}\Big{|}_{\mathcal{O}(\alpha_{y}^{2}),(\alpha_{y}^{2}\log(\mu))\rightarrow
0}$ $\displaystyle=-\left(\dfrac{d\alpha_{y}}{d\log\mu}\dfrac{\partial
D_{ik}^{m\neq
0,\text{NLO}}}{\partial\alpha_{y}}+2D_{ik}^{m=0,\text{NLO}}\right)\left(D_{ik}^{\text{LO}}+D_{ik}^{m\neq
0,\text{NLO}}\right)^{-1}\Bigg{|}_{\mathcal{O}(\alpha_{y}^{2})}$
$\displaystyle=-\left(\dfrac{d\alpha_{y}}{d\log\mu}\dfrac{\partial
D_{ik}^{m\neq 0,\text{NLO}}}{\partial\alpha_{y}}\cdot
D_{kj}^{-1\,\text{LO}}-2D_{ik}^{m=0,\text{NLO}}\cdot D_{kj}^{m\neq
0,\text{NLO}}\right),$ (49)

where we again classify terms using the terminology of Eq. 28. We recognize in
Eq. 49 factors of the $\beta$-function of our coupling, which is just

$\beta(\alpha_{y})\,=\,\frac{7\alpha_{y}^{2}}{4\pi}.$ (50)

The Mellin transform of the matrix given by Eq. 49 is a complicated expression
that we give in Appendix D. Nonetheless, it is straightforward to construct.

Having obtained what we need to calculate the effects due to running, we can
proceed to consider matching. The basic idea is that our observable,
$\dfrac{1}{\Gamma_{0}}\dfrac{d\Gamma_{j}}{dz}$ is the same in both theories at
the matching scale.

$\displaystyle\dfrac{1}{\Gamma_{0}}\dfrac{d\Gamma_{j}(\mu=\mu_{M}+\epsilon)}{dz}=\dfrac{1}{\Gamma_{0}}\dfrac{d\Gamma_{j}^{\prime}(\mu=\mu_{M}-\epsilon)}{dz}$
(51)

where $j$ labels the observed particle, the LHS represents a higher-scale
theory and the RHS is the lower-scale one. Using the factorized form of our
differential decay rate (cf. 2) and going to Mellin space, we have

$\displaystyle
C_{i}(s,\mu_{M})D_{ij}(s,\mu_{M})=C^{\prime}_{k}(s,\mu_{M})D^{\prime}_{kj}(s,\mu_{M})$
(52)

We note that this equation holds $j$ by $j$, and thus that particle must
necessarily exist in both theories. Solving for $C_{k}^{\prime}$, we get

$\displaystyle
C^{\prime}_{k}(s,\mu_{M})=C_{i}(s,\mu_{M})D_{ij}(s,\mu_{M})\left(D^{\prime}_{jk}(s,\mu_{M})\right)^{-1},$
(53)

matching order by order in $\alpha_{y}$.

In the tower of theories that takes us from the creation of our massive
fermion, the top, to scales below its mass, where we integrate it out, we pass
through three matching scales (cf. Fig. 1). We provide here an overview of the
issues with each of these. In Section 5, we present a specific numerical
implementation.

  * •

Matching between the Full Theory and SCET1 at $M_{X}$:

The main nontrivialites at this stage are the complications that arise when
passing from a conventional quantum field theory to SCET. However, this is now
a standard procedure involving a well-established effective field theory [4,
5, 6, 7]. One simplification relative to the original QCD SCET is that we have
no issues with gauge invariance in a pure Yukawa theory. By construction, the
kinematically-enhanced fragmentation region of a collinear splitting of the
top falls out of the leading-power SCET Feynman rule. It is this matching into
SCET that sets the UV boundary conditions for our Wilson coefficients that
will undergo further running. At LO, only tops arise from the $X$ decay.
However, at NLO it becomes possible to produce bottom or higgs as a hard,
three-body decay, a production mechanism not captured the fragmentation of a
top with the SCET vertex. Thus, matching to our observable,
$(1/\Gamma_{0})\,d\Gamma/dz$, we find

$\displaystyle C_{t}(z,\mu=M_{X})$ $\displaystyle=\delta(1-z)$ $\displaystyle
C_{b}(z,\mu=M_{X})$ $\displaystyle=\alpha_{y}\left(\frac{3z-2}{8\pi}\right)$
$\displaystyle C_{h}(z,\mu=M_{X})$
$\displaystyle=\alpha_{y}\frac{z\left(-3z^{2}+z+2\right)+2\log(1-z)}{8\pi
z^{2}}.$ (54)

  * •

Matching between SCET1 and SCET2 at $M_{\text{Intermediate}}$:

In principle, there are many shifts that can occur in passing from a SCET1 to
a SCET2 theory. The former has a hierarchy between the virtualities of its
soft and collinear sectors, and the latter does not. Additionally, the change
in power counting of the soft sector also shuffles the (ir)relevance of
interactions. Furthermore, the SCET2 theory is sensitive to the the finite
value of $M_{t}$ by construction. Nevertheless, the $C_{i}$ Wilson
coefficients of our fragmentation operators enjoy many simplifications. As
discussed above, the UV physics that sets $C_{i}$ is decoupled from $M_{t}$.
In practice, we match to a fictitious SCET2 with $M_{t}=0$, and then tune
$M_{t}$ up to the SCET2 soft scale, which can have no effect on the $C_{i}$ we
obtained. Additionally, our fragmentation functions are computed from
collinear fields alone. The SCET1 soft contributions are removed by zero-bin
subtraction [23], while the SCET2 soft vertex is power-suppressed. This means
that we are matching $D_{ij}$ computed in the collinear regime of both
theories with $M_{t}=0$. Thus, we have

$\displaystyle
C_{i}(s,M_{\text{Intermediate}})D_{ij}(s,M_{\text{Intermediate}})=C^{\prime}_{k}(s,M_{\text{Intermediate}})D^{\prime}_{kj}(s,M_{\text{Intermediate}}),$
(55)

with the same $D$-matrices on both sides. Thus, we get $C_{i}=C_{i}^{\prime}$,
i.e. a trivial matching. It is the case though, that the NLL running in the
SCET2 theory will be immediately different from that in SCET1 due to the
finite $M_{t}$ effects.

  * •

Matching between SCET2 and the Low-Scale Theory at $M_{t}$:

In some sense, this last matching is quite conventional; we integrate out a
particle below its mass. In the context of fragmentation, our matching must
account for the fact that the $t$ had been a propagating source of $h,b$
particles, and now is gone from the theory. This is where the finite-width
contributions to $D_{th}$ and $D_{tb}$ come in (cf. Table 1). In matching to
the Low Scale Theory with no $t$, they give one last, $\mathcal{O}(1)$
injection of $h,b$ as we account for every $t$ particle’s decay into them.
Once in the low-scale theory (LST), interactions only proceed through
irrelevant operators. We thus get no leading-power contributions to
fragmentation. Thus, we have a simple fragmentation matrix,

$\displaystyle D_{ij\,\text{LST}}(s,\mu=M_{t})=\left(\begin{matrix}1&0\\\
0&1\end{matrix}\right).$ (56)

Since we only need to match $h,b$ final states, we only need the SCET2
fragmentation matrix with the $h,b$ columns. To NLO in Mellin space, these
are:

$\displaystyle D_{ij,2}(z)(s,\mu=M_{t})=\left(\begin{matrix}1&0\\\ 0&1\\\
\frac{1}{s}&\frac{1}{s}\end{matrix}\right)+\dfrac{\alpha_{y}}{8\pi}\left(\begin{matrix}-2&-4\frac{2s+1}{s^{3}+s^{2}}\\\
\frac{4}{s(s+1)^{2}}&-3\\\
2\left(\frac{H\left(s+1\right)}{s+1}-\frac{1}{s}-\frac{1}{(s+1)^{2}}\right)&2\left(\frac{\frac{sH(s)}{s+1}-s-1}{s^{2}}\right)\end{matrix}\right),$

The triviality of $D_{ij\,\text{LST}}$ lets us easily compute the
$C_{i\,\text{LST}}$ from Eq. 52.

##  5 Numerical Results & Conclusions

We have seen the subtle and important physics of tracking a fermion mass as
one propagates it down in scale from its production to its eventual decay. It
is nevertheless useful to provide a numerical example to see how these effects
shift results quantitatively. In keeping with our Standard Model inspiration,
we take $M_{t}$ = 173 GeV and we set $y_{t}(M_{t})=1$. We make the $b$ and $h$
massless. For describing fragmentation, at scales immediately above $M_{t}$,
we work in SCET2.

We can ask though, at what scale we can stop tracking $M_{t}$ and instead pass
to the SCET1 theory with $M_{t}=0$. As follows from our discussion in Section
1, our condition to account for the fermion mass is the scale below which its
“no-split” probability, i.e. the odds that we have produced a physical, on-
shell particle are greater than 50%. We compute the Sudakov (no-split) factor
for our $t$ following the presentation in [3]. That reference defines the
splitting function, $P_{ij}$, as the $z$-space anomalous dimension with
$\alpha_{y}/\pi$ stripped off. Inverse Mellin transforming
$\gamma^{\text{LL}}_{th}$ from Table 3, we get $P_{th}=1/4+z/2$. Thus, the
Sudakov factor is

$\Delta(Q,M_{t})=\exp\left(-\int_{M_{t}^{2}}^{Q^{2}}\frac{\alpha_{y}(\sqrt{t})}{4\pi}\frac{1}{t}dt\right).$
(58)

For the above values of $y_{t},M_{t}$, we find that the value of $Q$ for which
$\Delta(Q,M_{t})=0.5$ is $Q=1.5\times 10^{8}$ TeV. We thus take this scale as
the upper limit of our SCET2 theory. This may seem like a shockingly high
scale at which to still be tracking finite $M_{t}$. After all, we are nine
orders of magnitude larger and $(M_{t}/Q)^{2}$ effects are beyond negligible.
However, the lesson of the ultracollinear terms is that naive power
suppression breaks down as kinematic enhancements can provide compensating
factors. Said differently, in the limit that $\alpha_{y}\rightarrow 0$, even
if $M_{X}\gg M_{t}$, the $t\bar{t}$ pair in the $X$ decays into will
necessarily be on-shell as they will have no interactions that allow them to
split and shower into other particles. Furthermore, since our Yukawa theory
has only collinear, rather than soft & collinear divergences like gauge
theories, the Sudakov factor is only single-log enhanced. This causes a
relative suppression in the exponential argument, causing an extension of the
regime where we need to track $M_{t}$.

For the case of real QCD, if we use the double-log $q\rightarrow gq$ Sudakov
in [3] and plug in the observed values for top and bottom quark masses
($m_{t},m_{b}$) along with $\alpha_{s}$, we find that one should keep bottom
quark masses in calculations up to $Q\approx$ 235 GeV and top quark masses up
to $Q\approx$ 43 TeV. Returning to our toy theory, we summarize the parameters
used in our example in Table 2. The interesting mass effects occur in the
SCET2 theory and its matching to the low-scale theory. Thus, to simplify our
example we take the SCET1 regime to run over a negligible range of scales. In
effect, we match directly from the full theory to the SCET2 theory, since the
Wilson coefficients we obtain are independent of $M_{t}$.111111The boundary
conditions we set in Eq. 59 are thus taken as the initial boundary conditions
for the SCET2 running, i.e. $C_{i}(M_{X})=C_{i}(M_{\text{Intermediate}})$. We
do not need to specify a numerical value for $M_{X}$, but we take it to be
just a small factor greater than the $M_{\text{Intermediate}}=1.5\times
10^{8}$ TeV, determined by our Sudakov condition.

As discussed in Section 4.2, matching to the full theory at NLO gives
$\mathcal{O}(\alpha_{y})$ contributions to $C_{h,b}$ since it is possible to
produce $h,b$ particles from a hard three-body decay of the $X$. By
construction, these are the portions of the $X$ decay rate that do not follow
from producing a $t$, which subsequently fragments into the other fields
following the SCET Feynman rules. We give the $z$-space boundary conditions in
Eq. 54, but in practice we run & match in Mellin space. Those boundary
conditions are:

$\displaystyle C_{t}(s,\mu=M_{X})$ $\displaystyle=1$ $\displaystyle
C_{b}(s,\mu=M_{X})$ $\displaystyle=\alpha_{y}\left(\frac{s-2}{8\pi
s(s+1)}\right)$ $\displaystyle C_{h}(s,\mu=M_{X})$
$\displaystyle=\alpha_{y}\left(-\frac{H(s-2)}{4\pi(s-2)}+\frac{1-5s}{8\pi\left(s-s^{3}\right)}\right).$
(59)

Presenting results is more physically transparent back in $z$ or plus
momentum-fraction space. Thus, after numerically solving our RG equation to
run and/or match our Wilson coefficients at the scale $\mu=M_{t}$, we perform
an inverse Mellin transform. We set our contour as
$s(t)=1+t\cdot\exp(i\,\phi)$, where $\phi=3\pi/4$, and integrate numerically
over $t$,

$\displaystyle
C_{i}(z)=\text{Im}\left\\{\dfrac{1}{\pi}\int_{0}^{\infty}e^{i\phi}z^{-s(t)}C_{i}\left(s(t)\right)dt\right\\},$
(60)

We tested robustness of the results in shifting the definition of
$s(t)=a+t\cdot\exp(i\,\phi)$, varying $a$ and $\phi$ by $\mathcal{O}(1)$
amounts. Solving the RG equation necessitates running the coupling,
$\alpha_{y}$, as well,

$\displaystyle\beta(\alpha_{y})=\dfrac{7}{4\pi}\alpha_{y}^{2};\qquad\alpha_{y}(\mu)=\dfrac{4\pi}{16\pi^{2}-7\log(\dfrac{\mu}{M_{t}})}$
(61)

where we use $\alpha_{y}(M_{t})=1/4\pi$ as a boundary condition for
$\alpha_{y}(\mu)$.

Energy Scale EFT Boundary Conditions $\gtrsim$ Few $\times 10^{8}$ TeV Full
Theory Few-1.5 $\times 10^{8}$ TeV SCET1 At $M_{X}=$ Few $\times 10^{8}$ TeV;
$C_{t}=\delta(1-z)$; $C_{b,h}=$cf. Eq. 54 1.5 $\times 10^{8}$
TeV-$M_{t}(=173\;\text{GeV})$ SCET2 At $M_{\text{Intermediate}}=1.5\times
10^{8}$ TeV; Trivial from matching SCET1 onto SCET2 Below $M_{t}$ Low-Scale
Theory (No running) At $M_{t}$; Matching Conditions that remove $t$

Table 2: Numerical inputs & operational steps for computing with our toy
theory.

###  5.1 Effect of Mass on Running

In Figure 2 we show the numerical significance of the finite $M_{t}$
contributions by comparing three different versions of running the Wilson
Coefficients. Firstly, we have the massless case where no mass contributions
are considered throughout our running. That is to say that we include neither
finite-width ($\Gamma_{t}$) nor ultracollinear contributions. This is just LL
conventional running with the usual splitting functions as obtainable from
[17]. Said differently, this is what the running would be if we had a massless
SCET1 theory from $M_{\text{Intermediate}}$ to $M_{t}$. In the second case, we
include the finite width effect, which enters the LL anomalous dimension (cf.
Eq. 48 and Table 3). This is the case spelled out in Table 2 with a SCET2
theory involving a finite $M_{t}$ running over our scales of interest. We see
that the effect is numerically large, as we expect since the finite width
shifts the $\mathcal{O}(\alpha_{y})$ anomalous dimension at this same level.
This does not involve however, any of the ultracollinear contributions. To see
their effect we plot a third curve, which includes a subset of the
$\mathcal{O}(\alpha_{y}^{2})$ anomalous dimension corrections (cf. Tables
4-6). These give our “NLL-UC” running, which is not a complete accounting of
NLL, as discussed above. Nonetheless, we see the ultracollinear terms on their
own give a non-negligible shift to the $C_{i}$.

Figure 2: Running of $C_{t}(z),C_{h}(z),C_{b}(z)$ at massless LL (dashed red),
at LL including the finite width (dotted dashed blue), and at NLL-UC (solid
yellow).

###  5.2 Effect of Mass on Matching

If we were generating particles in this $t,b,h$ theory originating from some
high-energy process (which we take to be the decay of some heavy $X$ for
concreteness), running down in scale would eventually drop below $M_{t}$. This
is the final step of Table 2, matching to a theory without the $t$. In Fig. 3,
we plot the result for the two remaining Wilson coefficients, $C_{b},C_{h}$.
Once again, we show three variations. In all cases, we assume that the UV
theory in this matching is SCET2 with finite $M_{t}$. Thus, the LL running
does account for finite $\Gamma_{t}$. By matching at LO, we again include the
finite width, which is unavoidable even in a leading treatment with a massive
particle. At NLO, we include the ultracollinear terms in the fragmentation
functions (cf. Table 1). We see in Fig. 4, this just provides a percent-level
shift. However, in both Figs. 3 and 4, we see that running at NLL-UC provides
a qualitatively larger shift. It is an interesting question for future work if
this survives in the complete NLL result or gets substantially cancelled by
the other $\mathcal{O}(\alpha_{y}^{2})$ contributions to $\gamma_{ij}$.

Figure 3: Results of $C_{h}(z),C_{b}(z)$ in Running + Matching at LL+LO
(dashed blue), LL+NLO (solid orange), NLL-UC+NLO (solid green).

Figure 4:  Ratio of
$\dfrac{|C_{\text{NLO}}^{\text{LL}}-C_{\text{LO}}^{\text{LL}}|}{C_{\text{LO}}^{\text{LL}}}$
(orange) and $\dfrac{|C_{\text{NLO}}^{\text{NLL-
UC}}-C_{\text{LO}}^{\text{LL}}|}{C_{\text{LO}}^{\text{LL}}}$ (green) for
$C_{b},C_{h}$.

###  5.3 Conclusions

From a rather ordinary mathematical problem arises a very interesting physical
paradox that requires nontrivial conceptual labor to resolve. As Smilga
noticed long ago [1], small fermion mass terms are not obeying commutation of
the massless limit, and thus it is unsafe to simply toss them aside. The use
of Effective Field Theory and its systematic formalism of operators and Wilson
coefficients reveals when one must track a fermion mass. The answer boils down
to when one expects to produce an on-shell particle, as quantified by the
Sudakov factor. It is not hard to cook up an example, as we have done here,
where finite mass effects endure long beyond the naive intution of
$(m_{\text{light}}/m_{\text{heavy}})^{2}$ suppression.

We see furthermore that the numerical stakes of keeping finite particle masses
(or not) can be significant. There are $\mathcal{O}(\text{few})$ shifts in the
Wilson coefficients, $C_{i}$, depending on whether one tracks the finite width
in running, with further possible $\mathcal{O}(10\text{s}\%)$ corrections from
the ultracollinear contributions to the anomalous dimensions (cf. Fig. 2).
Similar scale shifts from these terms persist in our final matching step,
where we integrate out the massive fermion (cf. Figs. 3 & 4).

Now that we have studied and understood the subtleties of finite fermion mass
effects in a controlled setting, it is a natural follow-up to study more
realistic theories. Our quick exercise with QCD Sudakovs in Section 5 shows
that even with double-log running, one may need to track $b$ and $t$ quark
masses about an order of magnitude higher than one might expect. For showers
used for astroparticle physics, like HDMSpectra [9], the range of scales under
consideration is enormous, stretching from weak to Planck scale. One could
imagine all manner of BSM physics in this range. For each particle, we must
answer the question of when to keep its mass. By extending this work, one can
arrive at a systematic answer.

Acknowledgments: This work has benefitted from useful discussions with Benoit
Assi, Sean Fleming, Stefan Hoeche, Aneesh Manohar, Thomas Mehen, Davison
Soper, & Ubirajara van Kolck. We especially thank Varun Vaidya for clarifying
subtle points in SCET. We were grateful to have Ozan Erdoğan’s collaboration
during the early stages of this project. MB and PC are supported by the DOE
(HEP) Award DE-SC0019470.

##  Appendix A Fragmentation Formalism

We review here the fragmentation formalism and how our calculational
quantities of interest arise. From SCET, we take the operator definitions in
[24, 25], which follow from the classic treatment in perturbative QCD given in
[26]. As originally formulated, the physical concern was observing hadrons in
jets initiated by perturbative partons. Our situation here is qualitatively
different, as the theory is fully perturbative. Nonetheless, we can still work
with fragmentation functions, as in [9], which provide a means to compute the
rate to observe a particle $j$ that radiated from a possibly different
particle $i$.

It is natural to conceive of the fragmentation function as a kind of
probability distribution. In its original QCD framing, this would give the
probability to observe a hadron with a particular energy fraction of its
mother parton. In our perturbative theory, this becomes the probability of
observing a “parton” with a particular plus-momentum fraction of its mother
parton. The probabilistic interpretation breaks down beyond the Leading-Log
level. One can get explicitly negative results, in fact. While inconvenient,
this not a fundamental pathology. Only observable amplitudes need to respect
the strict unitarity of quantum mechanics. Nonetheless, having even an
approximately probabilistic quantity is useful for guiding its construction.
Also, the need to conserve probability at LL provides a useful sum rule with
which to check the calculation, as detailed below.

Following Collins, as reviewed in [27], constructing a function which gives
the number density of particles with a particular fraction of their mother’s
$E$ or $p^{+}$ starts us in a frame where the mother has zero transverse
momentum, ($k_{\perp}$ = 0), and we integrate over the daughter’s
$\vec{p}_{\perp}$. We also have that $p^{+}=z\,k^{+}$ and
$p^{-}=(\vec{p}_{\perp}^{2}+m_{d}^{2})/p^{+}$, where in our case the daughter
mass $m_{d}=M_{t}$ if it is a top and zero otherwise. In this frame, if the
initiating particle is a fermion, the fragmentation function to observe
particle $j$ emitted from it is

$\displaystyle D_{ij}(z)=\frac{1}{z}\int
d^{2}p_{\perp}\int\frac{dx^{-}d^{2}x_{\perp}}{2(2\pi)^{3}}e^{ik^{+}x^{-}/2}\dfrac{1}{2N_{c,i}}\Tr\sum_{X}\langle
0|\dfrac{\not{\bar{n}}}{2}\chi_{i}(0,x^{-},x_{\perp})|j(p)\,X\rangle\langle
j(p)\,X|\chi_{i}^{\dagger}(0)|0\rangle\,\Big{|}_{p^{Xj}_{\perp}=0}.$ (62)

We average over spin and color of the initiating particle (in our case $N_{c}$
= 2 for top and higgs, $N_{c}$ = 1 for the bottom). Naively, $D_{ij}(z)$
depends on the plus-momentum of both mother ($k^{+}$) and daughter ($p^{+}$).
However, we have boost invariance in directions orthogonal to the $\perp$
direction. Thus, it can only depend on $z(\equiv p^{+}/k^{+})$.

The difficulty with the form of Eq. 62 is that it fixes an unobservable
quantity, the mother’s perp momentum, $k_{\perp}=0$. A better operational
definition fixes the axis of the observed daughter’s momentum. Thus, we rotate
and boost to a $p_{\perp}=0$ frame with $p^{+}$ unchanged. The mother perp
momentum becomes $\vec{k}_{\perp}=-\vec{p}_{\perp}/z$. Doing this gives

$\displaystyle
D_{ij}(z)=z\int\frac{dx^{-}}{4\pi}e^{ik^{+}x^{-}/2}\dfrac{1}{2N_{c,i}}\Tr\sum_{X}\langle
0|\dfrac{\not{\bar{n}}}{2}\chi_{i}(0,x^{-},0_{\perp})|j(p)\,X\rangle\langle
j(p)\,X|\chi_{i}^{\dagger}(0)|0\rangle\,\Big{|}_{p^{j}_{\perp}=0}.$ (63)

In practice, we compute purely in momentum space and thus calculate the
following quantities,

$\displaystyle D^{\text{fermion}}_{ij}(z)$
$\displaystyle=\frac{z}{Q}\dfrac{1}{2N_{c,i}}\Tr\int\Pi_{X}\,\delta\left(1-\frac{p_{X}^{+}+p^{+}}{Q}\right)\langle
0|\dfrac{\not{\bar{n}}}{2}\chi_{i}(k)|j(p)\,X\rangle\langle
j(p)\,X|\chi_{i}^{\dagger}(k)|0\rangle\,\Big{|}_{p^{j}_{\perp}=0}$
$\displaystyle D^{\text{scalar}}_{ij}(z)$
$\displaystyle=z\dfrac{1}{N_{c,i}}\Tr\int\Pi_{X}\,\delta\left(1-\frac{p_{X}^{+}+p^{+}}{Q}\right)\langle
0|\phi_{i}(k)|j(p)\,X\rangle\langle
j(p)\,X|\phi_{i}^{\dagger}(k)|0\rangle\,\Big{|}_{p^{j}_{\perp}=0}.$ (64)

We get a very useful set of constraints from the set of sum rules that
massless fragmentation functions satisfy up to order
$\alpha_{y}\log(\mu^{2}/m_{t}^{2})$, which follow from the conservations of
probability and momentum:

$\displaystyle\frac{1}{2}\sum_{j}\int dz\,D^{R}_{ij}(z)+\int
dz\,D^{V}_{ii}(z)=1$ $\displaystyle\sum_{j}\int dz\,z\,D^{R}_{ij}(z)+\int
dz\,z\,D^{V}_{ii}(z)=1$ (65)

where $R$ stands for fragmentation with real-emission while $V$ indicates
self-fragmentation. Including the effects of finite $M_{t}$, the probabilistic
interpretation of $D_{ij}$ breaks down. Contributions due to the finite width
$\Gamma_{t}$ give positive contributions to $D_{tb},\,D_{th}$. However, in our
$\overline{\text{MS}}$ scheme, there is no compensating negative term in
$D_{tt}$. Furthermore, the $\mathcal{O}(\alpha_{y})$ corrections (without log
enhancement) arising from the ultracollinear contributions contain strictly
negative terms (cf. Table 1) allowing $D_{ij}$ to take negative values for
some range of $\mu$ and $z$.

##  Appendix B Soft $\&$ Zero-Bin Contribution

Throughout the following calculations of this Appendix we note the different
notation used to label the daughter and mother momenta with respect to the one
used in the definition of the fragmentation functions in A.

The calculation of the soft contribution to a graph can come from the Full
Theory calculation, followed by taking the appropriate soft and collinear
limits for the graph’s momenta. We recall that it is only in the SCET1 theory
that a vertex with a soft field is leading-power. Thus, we can work in the
simpler $M_{t}$ = 0 case. As an example we look at the bare $b\rightarrow h$
fragmentation function,

$\displaystyle D_{bh}:\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a)[crossed dot] {}; \vertex[right=of a] (b); \vertex[above right=of b]
(c1); \vertex[ right=of c1] (c2); \vertex[below right=of c2] (d); \vertex[
right=of d] (e)[crossed dot]{}; \diagram* { (a) -- [with arrow=0.5,arrow
size=0.15em,momentum=$p+k$](b) -- [scalar, momentum=$p$](c1), (c2) -- [scalar,
momentum=$p$](d)--[with arrow=0.5,arrow size=0.15em,momentum=$p+k$](e), (b)--
[double,double distance=0.5ex,thick,with arrow=0.5,arrow size=0.15em,half
right, looseness=1,insertion={[size=10pt]0.5},momentum'=$k$](d) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (66) $\displaystyle
D_{0,bh}=\dfrac{z}{2}y^{2}\int\dfrac{d^{4}k}{(2\pi)^{4}}$
$\displaystyle\Tr{\dfrac{\not{\bar{n}}}{2}\not{p}\not{k}\not{p}}2\pi\delta\left(k^{2}\right)\dfrac{\delta\left(Q-p^{+}-k^{+}\right)}{\left((p+k)^{2}\right)^{2}},$
(67)

with $p^{-}=0$, $p_{\perp}=0$ and computing the trace:

$\displaystyle D_{0,bh}=\dfrac{z}{2}y^{2}\int\dfrac{d^{4}k}{(2\pi)^{4}}$
$\displaystyle(p^{+}k^{-}p^{+})2\pi\delta\left(k^{2}\right)\dfrac{\delta\left(Q-p^{+}-k^{+}\right)}{\left(p^{+}k^{-}-k^{2}_{\perp}\right)^{2}}.$
(68)

We now take $k$ to have the scaling of a soft momentum $k$, where
$(k^{+},k^{-},k_{\perp})\sim(\lambda^{2},\lambda^{2},\lambda^{2})$, compared
to $(p^{+},p^{-},p_{\perp})\sim(1,\lambda^{2},\lambda)$, where $\lambda$ is
the SCET bookkeeping power counting parameter:

$\displaystyle
D_{0,bh}^{\text{soft}}=\dfrac{z}{2}y^{2}\int\dfrac{d^{4}k}{(2\pi)^{4}}$
$\displaystyle(p^{+}k^{-}p^{+})2\pi\delta\left(k^{2}\right)\dfrac{\delta\left(Q-p^{+}\right)}{\left(p^{+}k^{-}\right)^{2}}.$
(69)

Using the on-shell condition of $k$, integrating over $k^{-}$:

$\displaystyle
D_{0,bh}^{\text{soft}}=\dfrac{z}{2}y^{2}\int\dfrac{d^{2}k_{\perp}dk^{+}}{2(2\pi)^{3}}$
$\displaystyle\dfrac{(p^{+})^{2}}{(k^{+})^{2}}k^{2}_{\perp}\dfrac{(k^{+})^{2}}{(p^{+})^{2}}\dfrac{\delta\left(Q-p^{+}\right)}{\left(k^{2}_{\perp}\right)^{2}}.$
(70)

Thus, our soft contribution will be:

$\displaystyle
D_{0,bh}^{\text{soft}}=\dfrac{z}{2Q}y^{2}\int\dfrac{d^{2}k_{\perp}dk^{+}}{2(2\pi)^{3}}$
$\displaystyle
k^{2}_{\perp}\dfrac{\delta\left(1-z\right)}{\left(k^{2}_{\perp}\right)^{2}}.$
(71)

For the zero-bin contribution of the same graph, we compute it in the
collinear sector of SCET1,

$\displaystyle
D_{0,bh}^{\text{zero}}=\dfrac{z}{2}y^{2}\int\dfrac{d^{4}k}{(2\pi)^{4}}$
$\displaystyle\text{Tr}\left\\{\dfrac{\not{\bar{n}}}{2}\left(p^{+}+k^{+}\right)\dfrac{\not{n}}{2}\left(\dfrac{\not{k}_{\perp}}{k^{+}}-\dfrac{\not{p}_{\perp}+\not{k}_{\perp}}{p^{+}+k^{+}}\right)\dfrac{\not{\bar{n}}}{2}\dfrac{\not{n}}{2}k^{+}\right.$
$\displaystyle\qquad\left.\left(\dfrac{\not{k}_{\perp}}{k^{+}}-\dfrac{\not{p}_{\perp}+\not{k}_{\perp}}{p^{+}+k^{+}}\right)\dfrac{\not{\bar{n}}}{2}\left(p^{+}+k^{+}\right)\dfrac{\not{n}}{2}\right\\}2\pi\delta\left(k^{2}\right)\dfrac{\delta\left(Q-p^{+}-k^{+}\right)}{\left((p+k)^{2}\right)^{2}},$
(72)

and then take the soft limit $k$. We also impose on-shellness for $p$ with
$p_{\perp}=0$, $p^{-}=0$,

$\displaystyle
D_{0,bh}^{\text{zero}}=\dfrac{z}{2}y^{2}\int\dfrac{d^{4}k}{(2\pi)^{4}}$
$\displaystyle\Tr{\dfrac{\not{\bar{n}}}{2}\left(p^{+}\right)\dfrac{\not{n}}{2}\left(\dfrac{\not{k}_{\perp}}{k^{+}}\right)\dfrac{\not{\bar{n}}}{2}\dfrac{\not{n}}{2}k^{+}\left(\dfrac{\not{k}_{\perp}}{k^{+}}\right)\dfrac{\not{\bar{n}}}{2}\left(p^{+}\right)\dfrac{\not{n}}{2}}2\pi\delta\left(k^{2}\right)\dfrac{\delta\left(Q-p^{+}\right)}{\left(p^{+}k^{-}\right)^{2}}.$
(73)

Computing the trace:

$\displaystyle
D_{0,bh}^{\text{zero}}=\dfrac{z}{2}y^{2}\int\dfrac{d^{4}k}{(2\pi)^{4}}\dfrac{(p^{+})^{2}}{k^{+}}k^{2}_{\perp}2\pi\delta\left(k^{2}\right)\dfrac{\delta\left(Q-p^{+}\right)}{\left(p^{+}k^{-}\right)^{2}}.$
(74)

Integrating over $k^{-}$:

$\displaystyle
D_{0,bh}^{\text{zero}}=\dfrac{z}{2}y^{2}\int\dfrac{d^{2}k_{\perp}dk^{+}}{(2\pi)^{3}}\dfrac{\delta\left(Q-p^{+}\right)}{k_{\perp}^{2}},$
(75)

and defining $z=p^{+}/Q$:

$\displaystyle
D_{0,bh}^{\text{zero}}=\dfrac{z}{2Q}y^{2}\int\dfrac{d^{2}k_{\perp}dk^{+}}{(2\pi)^{3}}\dfrac{\delta\left(1-z\right)}{k_{\perp}^{2}}.$
(76)

We thus see from equations 71 and 76, that
$\boxed{D_{0,bh}^{\text{zero}}\equiv D_{0,bh}^{\text{soft}}}$. Thus the soft
modes do not affect our results as they are cancelled by the zero-bin modes.
The same result holds for the other anomalous dimensions, as well.

##  Appendix C Calculational Details

Throughout the following calculations of this Appendix we note the different
notation used to label the daughter and mother momenta with respect to the one
used in the definition of the fragmentation functions in A.

###  C.1 Calculation of $D_{th}$ in Intermediate Theory (SCET2 with $M_{t}\neq
0$)

The diagram for $D_{th}$ calculation is the following:

$\displaystyle D_{th}:\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a)[crossed dot] {}; \vertex[right=of a] (b); \vertex[above right=of b]
(c1); \vertex[ right=of c1] (c2); \vertex[below right=of c2] (d); \vertex[
right=of d] (e)[crossed dot]{}; \diagram* { (a) -- [double,double
distance=0.5ex,thick,with arrow=0.5,arrow size=0.15em,momentum=$p+k$](b) --
[scalar, momentum=$p$](c1), (c2) -- [scalar, momentum=$p$](d)--[double,double
distance=0.5ex,thick,with arrow=0.5,arrow size=0.15em,momentum=$p+k$](e),
(b)-- [half right, looseness=1,insertion={[size=10pt]0.5},momentum'=$k$](d) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (77)

Using the operator definition of the fragmentation function as pointed out in
A, along with the SCET2 Feynman rules, as presented in Section 2, in the
complex mass scheme, we get:

$\displaystyle D_{0,th}=\dfrac{z}{2}y^{2}\int\dfrac{d^{4}k}{(2\pi)^{4}}$
$\displaystyle\text{Tr}\left\\{\dfrac{\not{\bar{n}}}{2}\left(p^{+}+k^{+}\right)\dfrac{\not{n}}{2}\left(\dfrac{\not{k}_{\perp}}{k^{+}}-\dfrac{\not{p}_{\perp}+\not{k}_{\perp}}{p^{+}+k^{+}}-\dfrac{\sqrt{M^{2}_{t}-iM_{t}\Gamma_{t}}}{p^{+}+k^{+}}\right)\dfrac{\not{\bar{n}}}{2}\dfrac{\not{n}}{2}k^{+}\right.$
$\displaystyle\qquad\left.\left(\dfrac{\not{k}_{\perp}}{k^{+}}-\dfrac{\not{p}_{\perp}+\not{k}_{\perp}}{p^{+}+k^{+}}+\dfrac{\sqrt{M^{2}_{t}+iM_{t}\Gamma_{t}}}{p^{+}+k^{+}}\right)\dfrac{\not{\bar{n}}}{2}\left(p^{+}+k^{+}\right)\dfrac{\not{n}}{2}\right\\}2\pi\delta\left(k^{2}\right)\dfrac{\delta\left(Q-p^{+}-k^{+}\right)}{\left((p+k)^{2}-M_{t}^{2}\right)^{2}+M_{t}^{2}\Gamma^{2}},$
(78)

where the “0” subscript denotes that this is the bare fragmentation function.
We follow the steps outlined in 3.1, performing the calculation in the
reference frame, discussed in A where daughter’s perp momentum is zero
($p_{\perp}=0$):

$\displaystyle D_{0,th}$
$\displaystyle=\dfrac{z}{2}y^{2}\int\dfrac{d^{4}k}{(2\pi)^{4}}$
$\displaystyle\text{Tr}\left\\{\dfrac{\not{\bar{n}}}{2}\left(p^{+}+k^{+}\right)\dfrac{\not{n}}{2}\left(\dfrac{p^{+}\not{k}_{\perp}}{(p^{+}+k^{+})k^{+}}-\dfrac{\sqrt{M^{2}_{t}-iM_{t}\Gamma_{t}}}{p^{+}+k^{+}}\right)\dfrac{\not{\bar{n}}}{2}\dfrac{\not{n}}{2}k^{+}\left(\dfrac{p^{+}\not{k}_{\perp}}{(p^{+}+k^{+})k^{+}}+\dfrac{\sqrt{M^{2}_{t}+iM_{t}\Gamma_{t}}}{p^{+}+k^{+}}\right)\right.$
$\displaystyle\qquad\left.\dfrac{\not{\bar{n}}}{2}\left(p^{+}+k^{+}\right)\dfrac{\not{n}}{2}\right\\}2\pi\delta\left(k^{2}\right)\dfrac{\delta\left(Q-p^{+}-k^{+}\right)}{\left((p+k)^{+}(p+k)^{-}-k_{\perp}^{2}-M_{t}^{2}\right)^{2}+M_{t}^{2}\Gamma^{2}}.$
(79)

The next step is to calculate the trace and simplify the Dirac structure:

$\displaystyle
D_{0,th}=\dfrac{z}{2}y^{2}\int\dfrac{d^{4}k}{(2\pi)^{3}}\left(p^{+}+k^{+}\right)^{2}k^{+}\left(\dfrac{k_{\perp}^{2}p_{+}^{2}+M_{t}^{2}k_{+}^{2}-M_{t}\Gamma
k_{+}^{2}}{k^{2}_{+}(p_{+}+k_{+})^{2}}\right)\dfrac{\delta\left(k^{2}\right)\delta\left(Q-p^{+}-k^{+}\right)}{\left((p+k)^{+}(p+k)^{-}-k_{\perp}^{2}-M_{t}^{2}\right)^{2}+M_{t}^{2}\Gamma^{2}}.$
(80)

We integrating $k^{-}$ with $k$ on-shell and also $p^{-}=0$ due to $p$ on-
shellness since we are in the daughter’s reference frame:

$\displaystyle
D_{0,th}=\dfrac{z}{2}\dfrac{y^{2}}{2}\int\dfrac{d^{2}k_{\perp}dk^{+}}{(2\pi)^{3}}\left(k_{\perp}^{2}\dfrac{p_{+}^{2}}{k^{2}_{+}}+M_{t}^{2}\right)\dfrac{\delta\left(Q-p^{+}-k^{+}\right)}{\left((p+k)^{+}\dfrac{k_{\perp}^{2}}{k^{+}}-k_{\perp}^{2}-M_{t}^{2}\right)^{2}+M_{t}^{2}\Gamma^{2}}+\mathcal{O}\left(\alpha^{2}_{Y}\right).$
(81)

Here in this step we focused only on the terms that are leading order in
$\alpha_{y}$, any terms of $\mathcal{O}\left(\alpha^{2}_{y}\right)$ are
subsequently dropped. Defining $z\equiv\dfrac{p^{+}}{Q}$ and integrating over
$k^{+}$ with the remaining delta function, $\delta\left(Q-k^{+}-p^{+}\right)$:

$\displaystyle
D_{0,th}=\dfrac{z}{2}\dfrac{y^{2}}{2}\int\dfrac{d^{2}k_{\perp}}{(2\pi)^{3}}\left(k_{\perp}^{2}\dfrac{z^{2}}{(1-z)^{2}}+M_{t}^{2}\right)\dfrac{\dfrac{(1-z)^{2}}{z^{2}}}{\left(k_{\perp}^{2}-M_{t}^{2}\dfrac{1-z}{z}\right)^{2}+M_{t}^{2}\Gamma^{2}\dfrac{(1-z)^{2}}{z^{2}}}.$
(82)

Regularizing the UV divergence through Dimensional Regularization in
$d=2-2\varepsilon$:

$\displaystyle
D_{0,th}=\dfrac{z}{2}\dfrac{y^{2}}{2}\mu^{2\varepsilon}\int\dfrac{d^{d}k_{\perp}}{(2\pi)^{3}}\dfrac{\left(k_{\perp}^{2}+M_{t}^{2}\dfrac{(1-z)^{2}}{z^{2}}\right)}{\left(k_{\perp}^{2}-M_{t}^{2}\dfrac{1-z}{z}\right)^{2}+M_{t}^{2}\Gamma^{2}\dfrac{(1-z)^{2}}{z^{2}}},$
(83)

Integrating $k_{\perp}$ from $[0,\infty)$ and expanding at $\varepsilon=0$:

$\displaystyle
D_{0,th}=\dfrac{\alpha_{y}}{4\pi}\left(z\dfrac{1}{\varepsilon}+z\log\left(\dfrac{\mu^{2}}{M_{t}^{2}\dfrac{1-z}{z}\sqrt{1+\dfrac{\Gamma^{2}}{M_{t}^{2}}}}\right)-\dfrac{M_{t}}{\Gamma_{t}}\arctan\left(\dfrac{\Gamma_{t}}{M_{t}}\right)+\pi\dfrac{M_{t}}{\Gamma_{t}}\right).$
(84)

We note that the $M_{t}^{2}$ term in the numerator of Eq. 83 contributes the
$z\log[z/(1-z)]$ in $D_{0,th}$ that one can spot in Eq. 84. This is precisely
the ultracollinear contribution. It relies on keeping finite $M_{t}$ in the
calculation, but it persists after the fact even if we take $M_{t}\rightarrow
0$. The final form of the fragmentation function is given after keeping the LO
terms of $\dfrac{\Gamma_{t}}{M_{t}}\rightarrow 0$ (to
$\mathcal{O}(\alpha_{y})$, $\Gamma_{t}=\alpha_{y}M_{t}/4$) and subtracting the
$\dfrac{1}{\varepsilon}$ term. Therefore, we have the renormalized
fragmentation function, $D_{th}$:

$\displaystyle\boxed{D_{th}=\dfrac{\alpha_{y}}{4\pi}\left(z\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)-z\log\left(\dfrac{1-z}{z}\right)-1+\pi\dfrac{M_{t}}{\Gamma_{t}}\right)}$
(85)

###  C.2 Calculation of $D_{hh}$ in Intermediate Theory with $M_{t}\neq 0$

The diagram for $D_{hh}$ calculation is the following:

$\displaystyle D_{hh}:\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a)[crossed dot] {}; \vertex[right=of a] (b); \vertex[right=of b] (d);
\vertex[ right=of d] (e)[crossed dot]{}; \diagram* { (a) --
[scalar,momentum=$p$](b), (b)-- [fermion,half right,
looseness=1,momentum'=$p-k$](d), (b)-- [double,double
distance=0.5ex,thick,with arrow=0.5,arrow size=0.15em,half left,
looseness=1,momentum=$k$](d), (d)-- [scalar,momentum=$p$](e) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (86)

We follow the steps outlined in 3.1 and we calculate the residue of the
expression represented by this graph:

$\displaystyle\text{Res}\left[\Sigma\left(p\right)\right]=\lim_{p^{2}\rightarrow
0}\dfrac{d}{dp^{2}}\left\\{\left(p^{2}\right)^{2}\cdot\dfrac{1}{p^{2}}\Sigma\left(p\right)\cdot\dfrac{1}{p^{2}}\right\\}=\lim_{p^{-}\rightarrow
0}\dfrac{dp^{-}}{dp^{2}}\dfrac{d}{dp^{-}}\Sigma\left(p\right)=\lim_{p^{-}\rightarrow
0}\dfrac{1}{p^{+}}\dfrac{d}{dp^{-}}\Sigma\left(p\right),$ (87)

where $\Sigma\left(p\right)$ is given by:

$\displaystyle\Sigma\left(p\right)=(-1)y^{2}\int\dfrac{d^{4}k}{(2\pi)^{4}}$
$\displaystyle\text{Tr}\left\\{\left(\dfrac{-\not{k}_{\perp}}{p^{+}-k^{+}}-\dfrac{\not{k}_{\perp}}{k^{+}}-\dfrac{M_{t}}{k^{+}}\right)\dfrac{\not{\bar{n}}}{2}\dfrac{\not{n}}{2}\left(\dfrac{-\not{k}_{\perp}}{p^{+}-k^{+}}-\dfrac{\not{k}_{\perp}}{k^{+}}+\dfrac{M_{t}}{k^{+}}\right)\dfrac{\not{\bar{n}}}{2}\dfrac{\not{n}}{2}\right\\}$
$\displaystyle\dfrac{k^{+}}{k^{+}k^{-}-k^{2}_{\perp}-M_{t}^{2}}\left(\dfrac{(p-k)^{+}}{(p-k)^{+}(p-k)^{-}-k^{2}_{\perp}}\right),$
(88)

and the $(-1)$ comes from the fermion loop factor. The derivative gives:

$\displaystyle\text{Res}\left[\Sigma\left(p\right)\right]=\lim_{p^{-}\rightarrow
0}\dfrac{1}{p^{+}}\dfrac{d}{dp^{-}}\Sigma\left(p\right)$
$\displaystyle=-y^{2}2\int\dfrac{d^{4}k}{(2\pi)^{4}}\dfrac{1}{p^{+}}\left(\dfrac{p_{+}^{2}k^{2}_{\perp}}{k^{+}(p^{+}-k^{+})}-\dfrac{M_{t}^{2}(p^{+}-k^{+})}{k^{+}}\right)$
$\displaystyle\dfrac{(k-p)^{+}}{k^{+}k^{-}-k^{2}_{\perp}-M^{2}_{t}}\left(\dfrac{1}{(p-k)^{+}(-k^{-})-k^{2}_{\perp}}\right)^{2}$
$\displaystyle=-y^{2}2\int\dfrac{d^{4}k}{(2\pi)^{4}}\dfrac{1}{p^{+}}\left(\dfrac{p_{+}^{2}k^{2}_{\perp}}{k^{+}(p^{+}-k^{+})}-\dfrac{M_{t}^{2}(p^{+}-k^{+})}{k^{+}}\right)$
$\displaystyle\dfrac{1}{k^{+}k^{-}-k^{2}_{\perp}-M^{2}_{t}}\dfrac{1}{(k^{+}-p^{+})}\left(\dfrac{1}{k^{-}-\dfrac{k^{2}_{\perp}}{(k^{+}-p^{+})}}\right)^{2}.$
(89)

Above, the trace is
$\text{Tr}\left\\{\dfrac{\not{\bar{n}}}{2}\dfrac{\not{n}}{2}\dfrac{\not{\bar{n}}}{2}\dfrac{\not{n}}{2}\right\\}=2$.
Using the residue theorem, the integral over $k^{-}$ can be calculated by
expanding the integrand around the pole,
$k^{-}=(k_{\perp}^{2}+M_{t}^{2})/k^{+}$. The integral is nonzero if
$p^{+}>k^{+}>0$. Thus,

$\displaystyle\text{Res}\left[\Sigma\left(p\right)\right]=$
$\displaystyle-\dfrac{y^{2}}{2}2\int\dfrac{d^{2}k_{\perp}dk^{+}}{(2\pi)^{3}}\dfrac{1}{p^{+}}\left(\dfrac{p_{+}^{2}k^{2}_{\perp}}{k^{+}(p^{+}-k^{+})}-\dfrac{M_{t}^{2}(p^{+}-k^{+})}{k^{+}}\right)\dfrac{1}{(k^{+}-p^{+})}\dfrac{1}{k^{+}}\left(\dfrac{1}{\dfrac{k^{2}_{\perp}}{k^{+}}+\dfrac{M_{t}^{2}}{k^{+}}-\dfrac{k^{2}_{\perp}}{(k^{+}-p^{+})}}\right)^{2}$
$\displaystyle=-\dfrac{y^{2}}{2}2\int\dfrac{d^{2}k_{\perp}dk^{+}}{(2\pi)^{3}}\dfrac{1}{p^{+}}\left(\dfrac{p_{+}^{2}k^{2}_{\perp}}{k^{+}(p^{+}-k^{+})}-\dfrac{M_{t}^{2}(p^{+}-k^{+})}{k^{+}}\right)\dfrac{1}{(k^{+}-p^{+})k^{+}}\left(\dfrac{1}{\dfrac{M_{t}^{2}}{k^{+}}-\dfrac{k^{2}_{\perp}p^{+}}{k^{+}(k^{+}-p^{+})}}\right)^{2}$
$\displaystyle=-\dfrac{y^{2}}{2}2\int\dfrac{d^{2}k_{\perp}dk^{+}}{(2\pi)^{3}}\dfrac{1}{p^{+}}\left(-k^{2}_{\perp}+M_{t}^{2}\dfrac{(k^{+}-p^{+})^{2}}{(p^{+})^{2}}\right)\left(\dfrac{1}{k^{2}_{\perp}-\dfrac{M_{t}^{2}(k^{+}-p^{+})}{p^{+}}}\right)^{2}.$
(90)

The $k_{\perp}$ in the numerator of expression above is not Euclidean. Thus we
need to rewrite the expression as follows:

$\displaystyle\text{Res}\left[\Sigma\left(p\right)\right]=-\dfrac{y^{2}}{2}2\int\dfrac{d^{2}k_{\perp}dk^{+}}{(2\pi)^{3}}\dfrac{1}{p^{+}}\left(k^{2}_{\perp}+M_{t}^{2}\dfrac{(k^{+}-p^{+})^{2}}{(p^{+})^{2}}\right)\left(\dfrac{1}{k^{2}_{\perp}-\dfrac{M_{t}^{2}(k^{+}-p^{+})}{p^{+}}}\right)^{2}.$
(91)

Integrating first over $k_{\perp}$ on $\left[0,\infty\right)$ using
Dimensional Regularization:

$\displaystyle\text{Res}\left[\Sigma\left(p\right)\right]=-\dfrac{y^{2}}{2}2\int_{0}^{p^{+}}\dfrac{dk^{+}}{8(\pi)^{2}}\frac{\pi(k^{+}-p^{+})\mu^{2\epsilon}\csc(\pi\epsilon)(k^{+}\epsilon-2p^{+})\left(\frac{p^{+}}{M_{t}^{2}(p^{+}-k^{+})}\right)^{\epsilon}}{(p^{+})^{2}(\epsilon-2)}.$
(92)

Integrating over $k^{+}$ and expanding to $\mathcal{O}(\epsilon^{0})$, we get
the 1-loop contribution as:

$\displaystyle\text{Res}\left[\Sigma\left(p\right)\right]=\dfrac{\alpha_{y}}{2\pi}\left[-\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)-\dfrac{1}{\epsilon}-\dfrac{1}{2}\right].$
(93)

Working toward the fragmentation function, we restore the z-dependence,
through a $\delta(1-z)$ as we have in the self-fragmentation:

$\displaystyle
D_{0,hh}=\dfrac{\alpha_{y}}{2\pi}\left[-\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)-\dfrac{1}{\epsilon}-\dfrac{1}{2}\right]\delta(1-z).$
(94)

The final form of the renormalized fragmentation function, after subtracting
the $\dfrac{1}{\varepsilon}$, is

$\displaystyle\boxed{D_{hh}(z)=\dfrac{\alpha_{y}}{2\pi}\left[-\log\left(\dfrac{\mu^{2}}{M_{t}^{2}}\right)-\dfrac{1}{2}\right]\delta\left(1-z\right).}$
(95)

The -1/2 term is the ultracollinear contribution in this case.

##  Appendix D Mellin Space Anomalous Dimensions

As we saw in 4.2, performing the running requires us to calculate the
anomalous dimension matrix in Mellin space. This can be done at LL and NLL-UC,
as dictated in equations 48, 49, using the matrix elements for fragmentation
functions in Mellin space from 28. We list the expressions for anomalous
dimension matrix elements at LL and NLL-UC in the tables below:

|  $\gamma^{\text{LL}}_{bi}$ for $i\in\left\\{h,b,t\right\\}$
---|---
$\gamma^{\text{LL}}_{bh}$ | $\dfrac{\alpha_{y}}{\pi}\dfrac{(s-1)}{s^{2}}$
$\gamma^{\text{LL}}_{bb}$ | $-\dfrac{\alpha_{y}}{\pi}\dfrac{\left(s^{3}+s^{2}+2\right)}{2s^{2}(s+1)}$
$\gamma^{\text{LL}}_{bt}$ | $\dfrac{\alpha_{y}}{\pi}\dfrac{1}{\left(s^{2}+s\right)}$

|  $\gamma^{\text{LL}}_{hi}$ for $i\in\left\\{h,b,t\right\\}$
---|---
$\gamma^{\text{LL}}_{hh}$ | $-\dfrac{\alpha_{y}}{\pi}\dfrac{\left(s^{2}+1\right)}{s^{2}}$
$\gamma^{\text{LL}}_{hb}$ | $\dfrac{\alpha_{y}}{\pi}\dfrac{(s-1)}{s^{2}}$
$\gamma^{\text{LL}}_{ht}$ | $\dfrac{\alpha_{y}}{\pi}\dfrac{1}{s}$

|  $\gamma^{\text{LL}}_{ti}$ for $i\in\left\\{h,b,t\right\\}$
---|---
$\gamma^{\text{LL}}_{th}$ | $\dfrac{\alpha_{y}}{\pi}\dfrac{1+3s}{4(s^{2}+s)}$
$\gamma^{\text{LL}}_{tb}$ | $\dfrac{\alpha_{y}}{\pi}\dfrac{3+s}{4(s^{2}+s)}$
$\gamma^{\text{LL}}_{tt}$ | $-\dfrac{\alpha_{y}}{\pi}\dfrac{1}{4}$

Table 3: Mellin Transforms of the Anomalous Dimension at LL.

|  $\gamma^{\text{NLL-UC}}_{bi}$ for $i\in\left\\{h,b,t\right\\}$
---|---
$\gamma^{\text{NLL-UC}}_{bh}$ | $\alpha_{y}^{2}\dfrac{\left(-2(s+1)s^{4}H(s+1)-\left(5s^{4}+10s^{3}+9s^{2}+12s+8\right)sH(s)+2s^{7}-5s^{6}-2s^{5}+35s^{4}+38s^{3}+26s^{2}+32s+16\right)}{8\pi^{2}(s+1)^{3}s^{5}}$
$\gamma^{\text{NLL-UC}}_{bb}$ | $-\alpha^{2}_{y}\dfrac{\left(4\left(5s^{4}+12s^{3}+11s^{2}+12s+8\right)sH(s)+27s^{8}+81s^{7}+117s^{6}+51s^{5}-144s^{4}-148s^{3}-48s^{2}-112s-64\right)}{32\pi^{2}s^{5}(s+1)^{3}}$
$\gamma^{\text{NLL-UC}}_{bt}$ | $\alpha^{2}_{y}\dfrac{\left(\left(5s^{4}+10s^{3}+9s^{2}+12s+8\right)sH(s)+\left(9s^{6}+13s^{5}-24s^{4}-38s^{3}-22s^{2}-32s-16\right)\right)}{8\pi^{2}s^{4}(s+1)^{3}}$

Table 4: Mellin Transforms of the Anomalous Dimension for
$b\rightarrow\left\\{h,b,t\right\\}$ at NLL-UC.

|  $\gamma^{\text{NLL-UC}}_{hi}$ for $i\in\left\\{h,b,t\right\\}$
---|---
$\gamma^{\text{NLL-UC}}_{hh}$ | $-\alpha_{y}^{2}\dfrac{\left(4(s+1)s^{4}H(s+1)+2\left(11s^{4}+22s^{3}+11s^{2}+8s+8\right)sH(s)+11s^{7}+22s^{6}-11s^{5}-78s^{4}-80s^{3}-44s^{2}-48s-32\right)}{16\pi^{2}s^{5}(s+1)^{2}}$
$\gamma^{\text{NLL-UC}}_{hb}$ | $\alpha^{2}_{y}\dfrac{\left(-\left(11s^{4}+24s^{3}+13s^{2}+8s+8\right)sH(s)+\left(3s^{6}-6s^{5}+7s^{4}+28s^{3}+8s^{2}+20s+16\right)\right)}{8\pi^{2}s^{5}(s+1)^{2}}$
$\gamma^{\text{NLL-UC}}_{ht}$ | $\alpha^{2}_{y}\dfrac{\left(\left(11s^{4}+22s^{3}+11s^{2}+8s+8\right)sH(s)-\left(11s^{5}+37s^{4}+44s^{3}+18s^{2}+24s+16\right)\right)}{8\pi^{2}s^{4}(s+1)^{2}}$

Table 5: Mellin Transforms of the Anomalous Dimension for
$h\rightarrow\left\\{h,b,t\right\\}$ at NLL-UC.

|  $\gamma^{\text{NLL-UC}}_{ti}$ for $i\in\left\\{h,b,t\right\\}$
---|---
$\gamma^{\text{NLL-UC}}_{th}$ | $\alpha^{2}_{y}\dfrac{\left(2s\left(3s^{2}+5s+4\right)H(s)+(s+1)\left(8s^{4}H(s+1)-\left(5s^{4}+15s^{3}-4s^{2}+16s+16\right)\right)\right)}{16\pi^{2}s^{4}(s+1)^{2}}$
$\gamma^{\text{NLL-UC}}_{tb}$ | $\alpha^{2}_{y}\dfrac{\left(4\left(4s^{4}+11s^{3}+12s^{2}+9s+4\right)sH(s)-\left(13s^{6}+49s^{5}+59s^{4}+39s^{3}+68s^{2}+92s+32\right)\right)}{32\pi^{2}s^{4}(s+1)^{3}}$
$\gamma^{\text{NLL-UC}}_{tt}$ | $-\alpha^{2}_{y}\dfrac{\left(s\left(3s^{3}+8s^{2}+9s+4\right)H(s)-\left(2s^{4}+7s^{3}+19s^{2}+24s+8\right)\right)}{8\pi^{2}s^{3}(s+1)^{3}}$

Table 6: Mellin Transforms of the Anomalous Dimension for
$t\rightarrow\left\\{h,b,t\right\\}$ at NLL-UC.

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