# Catani’s generalization of collinear factorization breaking

Leandro Cieri<EMAIL_ADDRESS>Prasanna K. Dhani<EMAIL_ADDRESS>Germán
Rodrigo<EMAIL_ADDRESS>Instituto de Física Corpuscular, Universitat
de València - Consejo Superior de Investigaciones Científicas, E-46980
Paterna, Valencia, Spain

(February 22, 2024)

###### Abstract

We consider the most general form of soft and collinear factorization for
hard-scattering amplitudes to all orders in perturbative Quantum
Chromodynamics. Specifically, we present the generalization of collinear
factorization to configurations with several collinear directions, where the
most singular behaviour is encoded by generalized collinear splitting
amplitudes that manifestly embed the breaking of strict collinear
factorization in space-like collinear configurations. We also extend the
analysis to the simultaneous soft-collinear factorization with multiple
collinear directions and show how naïve multiplicative factorization do not
hold.

##  I Introduction

Hard scattering amplitudes in perturbative Quantum Chromodynamics (QCD) are
singular if the energy of one or more partons, quarks and gluons, is
negligibly small and/or if two or more parton momenta are parallel to each
other. The singular behavior in these kinematic configurations is factorized
Collins  _et al._ (1989); Ellis  _et al._ (2011) and encoded in process-
independent factors such as collinear splitting amplitudes and soft currents.
Based on the lowest order versions of soft and collinear factorization
Altarelli and Parisi (1977); Bassetto  _et al._ (1983), general algorithms for
predicting jet cross sections at next-to-leading order (NLO) were proposed in
the famous works in Refs. Frixione  _et al._ (1996); Catani and Seymour
(1997); Frixione (1997); Catani  _et al._ (2002). In the last twenty years,
tremendous efforts have been made to advance to next-to-NLO (NNLO) and next-
to-NNLO (N3LO), and there are now a variety of different methodological
approaches (see, e.g., the reviews in Refs. Heinrich (2021); Torres Bobadilla
_et al._ (2021); Agarwal  _et al._ (2023)) capable of achieving high
perturbative orders. Also, resummed calculations to all perturbative orders
have achieved next-to-next-to-next-to-leading logarithmic (N3LL) accuracy in
some specific cases (see, Refs. Camarda  _et al._ (2021); Billis  _et al._
(2021); Neumann and Campbell (2023), and references therein). All these
advances rely on the detailed knowledge of the underlying QCD factorization
properties of hard scattering amplitudes in the soft and collinear limits.

At $\mathcal{O}(\alpha_{\mathrm{S}}^{2})$, soft factorization at tree level is
studied in Refs. Berends and Giele (1989); Campbell and Glover (1998); Catani
and Grazzini (2000a), and at one loop with single soft emission in Refs. Bern
and Chalmers (1995); Bern  _et al._ (1998, 1999); Catani and Grazzini (2000b);
Bierenbaum  _et al._ (2012). At the next perturbative order, i.e.
$\mathcal{O}(\alpha_{\mathrm{S}}^{3})$, the triple-soft limit of hard
scattering amplitudes at tree level is calculated in Refs. Catani  _et al._
(2020); Del Duca  _et al._ (2023); Catani  _et al._ (2023), the double-soft
limit of one-loop amplitudes is computed in Refs. Zhu (2020); Catani and Cieri
(2022); Czakon  _et al._ (2023), and the two-loop case with single soft
emission is analyzed in Refs. Li and Zhu (2013); Duhr and Gehrmann (2013);
Dixon  _et al._ (2020). Furthermore, at $\mathcal{O}(\alpha_{\mathrm{S}}^{4})$
the single soft limit at three loops with two colored particles is recently
known in Refs. Chen  _et al._ (2024); Herzog  _et al._ (2023).

Collinear factorization is considered universal and process independent for
theory predictions at high-energy colliders. Strictly speaking, the singular
factors arising in scattering amplitudes from collinear configurations are
expected to depend only on the momenta and quantum numbers of the external
partons involved in the collinear splitting process, and this aspect is known
in the literature as strict collinear factorization (SCF). At
$\mathcal{O}(\alpha_{\mathrm{S}}^{2})$, tree-level collinear splitting
processes are studied in Refs. Campbell and Glover (1998); Catani and Grazzini
(1999, 2000a); Dhani  _et al._ (2023); Craft  _et al._ (2023), while double-
collinear splitting processes at one loop are analyzed in Refs. Bern  _et al._
(1998); Kosower and Uwer (1999); Bern  _et al._ (1999); Catani  _et al._
(2012); Sborlini  _et al._ (2014a). At the next perturbative order, i.e.
$\mathcal{O}(\alpha_{\mathrm{S}}^{3})$, the quadruple-collinear limit of hard
scattering amplitudes is calculated in Refs. Del Duca  _et al._ (2000);
Birthwright  _et al._ (2005a, b); Del Duca  _et al._ (2020a, b), while the
triple-collinear limit of one-loop amplitudes is discussed in Refs. Catani
_et al._ (2004, 2012); Sborlini  _et al._ (2014b, 2015); Badger  _et al._
(2015); Czakon and Sapeta (2022), and the two-loop case with double-collinear
emission is analyzed in Refs. Bern  _et al._ (2004); Badger and Glover (2004);
Duhr  _et al._ (2015).

A seminal paper Catani  _et al._ (2012) led by Stefano Catani showed, however,
that SCF is not valid beyond the tree level in perturbation theory. In
particular, SCF remains true for configurations where all collinear partons
are produced in the physical final state termed as time-like (TL) collinear
but it is broken for other kinematic collinear configurations generally
denoted as space-like (SL) collinear. This effect has a purely quantum field
theory origin from Feynman loop diagrams and it is related to the intriguing
color coherence properties of QCD. In particular, the soft modes in the loop
couples the collinear partons in the final state to non-collinear ones in the
initial state that are far apart in phase-space kinematics. As a consequence,
in the SL collinear region, QCD scattering amplitudes fulfil a generalized
factorization, in which the singular factors retain certain dependence on the
momenta and quantum numbers of the non-collinear external partons.

The SCF breaking effects have more general implications in the context of
perturbative QCD predictions for jet production at colliders. Starting from
N3LO, these effects endanger the cancellation mechanism of infrared (IR)
divergences that leads to the factorization theorem of mass (collinear)
singularities. Besides challenging the process independent validity of the
age-old mass-singularity factorization theorem, SCF breaking effects also
emerge in the context of factorization of transverse-momentum dependent
distributions (TMDs) Rogers and Mulders (2010); Echevarria  _et al._ (2012);
Echevarría  _et al._ ; Rogers (2013). The resummation of large logarithmic
terms produced by collinear parton evolution is affected by the breaking of
SCF: parton evolution gets tangled up with the color and kinematic structure
of the hard-scattering subprocess, and this leads to the appearance of
entangled large logarithms. Indeed, the physical mechanism that generates the
breaking of SCF is directly related to the mechanism that produces super-
leading logarithms Forshaw  _et al._ (2006, 2008); Keates and Seymour (2009);
Becher  _et al._ (2021) in jet cross sections. A recent study of the
intricacies of the soft photon theorem in QCD can be found in Ref. Ma  _et
al._ (2023).

The purpose of this Letter is to disseminate Stefano Catani’s ideas on the
generalization of soft and collinear factorization that apply to both TL and
SL domains to all orders in QCD perturbation theory. In particular, we cherish
the memory of our last blackboard conversation (Fig. 1) we had with him in
September 2023 at the Department of Physics in Sesto Fiorentino, Firenze,
Italy.

Figure 1: Footage of original handwritten blackboard discussion by Stefano
Catani, September 2023, Sesto Fiorentino, Firenze, Italy.

The outline of the Letter is as follows. In section II we set up the notation
and briefly discuss the scattering process we are interested in. In section
II.1 we discuss the QCD factorization of scattering amplitudes in the soft
limit. We then analyze in detail the factorization of scattering amplitudes in
the collinear limit in section II.2 and in the simultaneous soft-colliner
limit in section II.3. In section III we present the extension of collinear
and soft-collinear factorization to general kinematics. Finally, in section IV
we give our concluding remarks and future outlook.

##  II QCD Factorization in IR limits

Let us consider a generic scattering process with $n$ external partons,
massless quarks and gluons, together with an arbitrary number of non-QCD
particles, particles that are color singlet, such as Higgs bosons, vector
bosons, leptons, photons, etc. The scattering amplitude for this process is
denoted by $\mathcal{M}(p_{1},\ldots,p_{n})$, where $p_{i}$ is the momentum of
the parton with flavor $a_{i}\,(a_{i}=g,q\,{\rm or}\,\bar{q})$, while the
dependence on non-QCD particles is implicitly understood. The external QCD
particles are on-shell $(p_{i}^{2}=0)$ and have physical spin polarizations.
Here, the scattering amplitude is defined in an arbitrary number of space-time
dimensions.

In the context of QCD perturbation theory, the scattering amplitude
$\mathcal{M}(p_{1},\ldots,p_{n})$ is expanded to higher loop orders as follows

$\displaystyle\mathcal{M}(p_{1},\ldots,p_{n})=\sum_{i=0}\mathcal{M}^{(i)}(p_{1},\ldots,p_{n})~{},$
(1)

where $\mathcal{M}^{(0)}$ denotes the lowest order contribution,
$\mathcal{M}^{(1)}$ denotes the one-loop contribution, and so on.

We will consider kinematic configurations where subsets of the external
particles become soft or collinear. To simplify the presentation, we will
denote by $s$ the subset of soft partons and their momenta, by $c_{I}$ a
subset of collinear partons, where the subindex $I$ could label collinear
subsets in different collinear directions, and by $h$ the remaining hard
partons.

###  II.1 SOFT FACTORIZATION

We are interested in the behavior of a hard scattering amplitude $\mathcal{M}$
in the kinematic configuration where $m$ massless external partons become
soft. This limit is formally obtained by first scaling all the momenta of the
soft partons by a factor $\lambda$ and then considering the limit
$\lambda\rightarrow 0$. In this limit, the leading singular behavior is
captured as follows

$\displaystyle\left.\mathcal{M}(\lambda s,h)\right|_{\lambda\to
0}\sim\frac{1}{\lambda^{m}}{\rm mod}\,(\ln^{r}\lambda)+\cdots~{},$ (2)

The power-like behavior $\lambda^{-m}$ on the right-hand side in Eq. (2)
determines the dominant singular terms of $\mathcal{M}$, the logarithmic
factor $\ln^{r}\lambda$ indicates the break in the naïve scaling behavior due
to higher order quantum loop contributions, and the dots denote the
subdominant singular behavior of relative suppression
$\mathcal{O}(\sqrt{\lambda})$ at least.

The emission of soft partons has no influence on the kinematics, spins and
flavors of the hard partons present in the scattering process. However, they
influence the color structure, since the emission of a soft parton always
carries color charges and these do not depend on its softness. As a result, in
QCD, there are color correlations, and the scattering amplitude is not fully
factorized, in contrast to Abelian theories such as Quantum Electrodynamics,
where the contribution of soft photons factorizes from the rest of the
scattering amplitude.

In this multiparticle soft limit, the dominant singular behavior of
$\mathcal{M}(s,h)$ is expressed by the following process-independent
factorization Campbell and Glover (1998); Bern and Chalmers (1995); Bern  _et
al._ (1999); Catani and Grazzini (2000a, b); Feige and Schwartz (2014)

$\displaystyle|\mathcal{M}(s,h)\rangle\simeq{\mbox{\boldmath$J$}}(s;h)|\mathcal{M}(h)\rangle\,,$
(3)

where the factor ${\mbox{\boldmath$J$}}(s;h)$ on the right-hand side of Eq.
(3) is the multiparticle soft current that embodies the dominant singular
behavior, the reduced scattering amplitude $|\mathcal{M}(h)\rangle$ is simply
obtained from the full amplitude on the left-hand side by removing the $m$
external legs with soft parton momenta and the symbol $\simeq$ indicates that
the subdominant singular terms are neglected.

It is important to note that the soft current ${\mbox{\boldmath$J$}}(s;h)$
depends on both soft and hard partons through their momenta and quantum
numbers, such as color. We also note that the soft current is proportional to
the unit operator in the spin subspace of the hard partons. Despite its
dependence on the hard partons, the soft current is universal, i.e., it does
not depend on the specific hard scattering amplitude and the corresponding
specific physical process.

The soft current $\bm{J}$ in Eq. (3) can be expressed in QCD perturbation
theory, by a loop expansion analogous to Eq. (1) as follows

$\displaystyle{\mbox{\boldmath$J$}}(s;h)=\sum_{i=0}{\mbox{\boldmath$J$}}^{(i)}(s;h)~{},$
(4)

where ${\mbox{\boldmath$J$}}^{(0)}$ is the tree-level soft current,
${\mbox{\boldmath$J$}}^{(1)}$ is the one-loop soft current, and so forth.

###  II.2 COLLINEAR FACTORIZATION

Scattering amplitudes in QCD also factorize when two or more external on-shell
momenta become collinear to each other. Collinear factorization and the
corresponding splitting processes are classified into two categories: TL and
SL. If all collinear partons are emitted in the physical final state of the
hard scattering process, we call it TL collinear splitting; otherwise, we
refer to the collinear splitting as SL.

We are interested in the behavior of the scattering amplitude $\mathcal{M}$ in
the kinematic configuration in which $m$ of the $n$ external partons become
parallel to each other. Without loss of generality, we assume that the
collinear momenta are $p_{1},\ldots,p_{m}$ and define their sum as $p_{1\ldots
m}$. Since the decaying parent particle is generally off-shell $(s_{1\ldots
m}\equiv p_{1\ldots m}^{2}\neq 0)$, we define an on-shell vector

$\displaystyle\widetilde{p}^{\mu}=p_{1\ldots m}^{\mu}-\frac{s_{1\ldots
m}}{2n\cdot p_{1\ldots m}}n^{\mu},$ (5)

such that it fulfils $\widetilde{p}^{2}=0$. Then we introduce the Sudakov
parametrization for the collinear momenta Catani and Grazzini (2000a); Catani
_et al._ (2012) as follows

$\displaystyle p_{i}^{\mu}=x_{i}\widetilde{p}^{\mu}+k_{\perp
i}^{\mu}-\frac{k_{\perp
i}^{2}}{x_{i}}\frac{n^{\mu}}{2\,n\cdot\widetilde{p}}\,,~{}i\in
c\equiv\\{1,\ldots,m\\},$ (6)

where the vector $\widetilde{p}$ denotes the collinear direction and the
light-like auxiliary vector $n^{\mu}$ ($n^{2}=0$) is necessary to specify how
the collinear direction is approached or, equivalently, to specify the
transverse components $k_{\perp i}$ ($k_{\perp i}\cdot\widetilde{p}=k_{\perp
i}\cdot n=0$, with $k_{\perp i}^{2}<0$).

Now, the $m$-parton collinear limit is precisely defined by scaling the
transverse momenta $k_{\perp i}^{\mu}$ with an overall factor $\lambda$, and
then setting the limit $\lambda\rightarrow 0$. If we keep the most singular
terms in $\lambda$ and neglect subdominant terms, we obtain the following
amplitude-level factorization Catani  _et al._ (2012)

$\displaystyle|{\mathcal{M}(c,h)}\rangle\simeq{\mbox{\boldmath$Sp$}}(c;\widetilde{p})\,|{\mathcal{M}(\widetilde{p},h)}\rangle~{}.$
(7)

The factorization in Eq. (7) relates the original scattering amplitude with
$n$ partons on the left-hand side to a reduced scattering amplitude with
$n-m+1$ partons on the right-hand side. The latter is obtained by replacing
the $m$ collinear partons $a_{1},\ldots,a_{m}$ by a single parent parton $a$
whose momentum is $\widetilde{p}$ and whose flavor is determined by flavor
conservation.

The process dependence of Eq. (7) is fully taken into account in the
amplitudes on both sides. The factor ${\mbox{\boldmath$Sp$}}(c;\widetilde{p})$
is called the splitting amplitude, which encodes the dominant singular
behavior in the collinear limit and depends only on the momenta and quantum
numbers such as color, flavor and spin of the partons involved in the
collinear splitting process $a\rightarrow a_{1}+\cdots+a_{m}$. This property
of collinear factorization is also referred to in the literature as SCF.

In a highly regarded paper Catani  _et al._ (2012) led by Stefano Catani (see
also Refs. Forshaw  _et al._ (2012); Sterman (2022)), it was found that SCF is
only valid in the so-called TL collinear region and breaks down for loop
amplitudes in the SL collinear region. In the SL collinear region, the
splitting amplitude depends on both the collinear and non-collinear partons
through their momenta and quantum numbers. The modified factorization valid
for both TL and SL regions is therefore as follows Catani  _et al._ (2012)

$\displaystyle|{\mathcal{M}(c,h)}\rangle\simeq{\mbox{\boldmath$Sp$}}(c;\widetilde{p};h)\,|{\mathcal{M}(\widetilde{p},h)}\rangle~{}.$
(8)

Analogous to the soft current in Eq. (4), the splitting amplitude $Sp$ can be
predicted by QCD perturbation theory and its loop expansion is as follows

$\displaystyle{\mbox{\boldmath$Sp$}}(c;\widetilde{p};h)={\mbox{\boldmath$Sp$}}^{(0)}(c;\widetilde{p})+\sum_{i=1}{\mbox{\boldmath$Sp$}}^{(i)}(c;\widetilde{p};h)~{},$
(9)

where ${\mbox{\boldmath$Sp$}}^{(0)}$ is the splitting amplitude at tree level,
${\mbox{\boldmath$Sp$}}^{(1)}$ is the one-loop splitting amplitude, and so on.
The splitting amplitude at tree level, ${\mbox{\boldmath$Sp$}}^{(0)}$, fulfils
SCF and is independent of the hard partons both in the TL and SL regions.

###  II.3 SOFT-COLLINEAR FACTORIZATION

We now analyse the scattering amplitude $\mathcal{M}(s,c,h)$ for the limiting
case in which a subset of partons become soft and at the same time another
subset of partons become collinear to each other. We focus for now on the TL
collinear limit, i.e., all collinear particles are produced in the physical
final state. When examining this soft-collinear limit, we neglect soft
configurations that are uniformly of $\mathcal{O}(p_{i})$ when
$p_{i}\rightarrow 0$, and collinear configurations that are uniformly of
$\mathcal{O}(k_{\perp i})$, when $k_{\perp i}\rightarrow 0$. Their
contributions can be accounted for by augmenting the results of this section
with the lower-order soft and collinear factorization described in sections
II.1 and II.2. Analogous to Eqs. (3) and (7), the singular behavior of the
hard-scattering amplitude is factorized in this multi-partonic soft-collinear
limit and expressed by the following factorization Catani and Grazzini (2000a)

$\displaystyle|\mathcal{M}(s,c,h)\rangle\simeq{\mbox{\boldmath$Sp$}}(c;\widetilde{p})\,{\mbox{\boldmath$J$}}(s;\widetilde{p};h)\,|\mathcal{M}(\widetilde{p},h)\rangle~{},$
(10)

where the perturbative loop expansion for the splitting amplitude $Sp$ is
identical to Eq. (9) without the dependence on the hard non-collinear partons
within the argument. The loop expansion for the soft current
${\mbox{\boldmath$J$}}(s;\widetilde{p};h)$ is analogous to Eq. (4). Note that
the effective interaction of the soft current with the collinear partons
occurs only through the parent parton due to color coherence. In the following
section, we extend the factorization described in sections (II.2-II.3) to the
most general kinematics.

##  III Generalized collinear and soft-collinear factorization

In this section we disseminate Stefano Catani’s new ideas on the
generalization of QCD factorization of hard scattering amplitudes in the
collinear and simultaneous soft-collinear limits. The generalization of
collinear factorization in Eq. (7), to two sets of collinear partons, denoted
by $c_{A}$ and $c_{B}$, with parent momenta $\widetilde{p}_{A}$ and
$\widetilde{p}_{B}$, respectively, is as follows

Figure 2: TL collinear factorization with two collinear directions.  Figure 3:
SL collinear factorization with two collinear directions.

$\displaystyle|{\mathcal{M}(c_{A},c_{B},h)}\rangle\simeq{\mbox{\boldmath$Sp$}}(c_{A};\widetilde{p}_{A})\,{\mbox{\boldmath$Sp$}}(c_{B};\widetilde{p}_{B})\,|{\mathcal{M}(\widetilde{p}_{A},\widetilde{p}_{B},h)}\rangle\,.$
(11)

Strictly speaking, Eq. (11) is valid to all orders in perturbation theory only
in the TL collinear region (see, Fig. 2). However, as discussed in section
II.2, collinear factorization must be modified for a general kinematics that
includes the SL collinear region Catani  _et al._ (2012). The generalization
of collinear factorization to more than one collinear direction, e.g. two
collinear directions, does not correspond to the naïve multiplicative
expectation, i.e., the product of two splitting amplitudes encoding each
collinear splitting according to Eq. (8). Rather, it is given by a single non-
factorizable splitting amplitude acting on the reduced hard scattering
amplitude. The generalized factorization is as follows

$\displaystyle|{\mathcal{M}(c_{A},c_{B},h)}\rangle\simeq{\mbox{\boldmath$Sp$}}(c_{A},c_{B};\widetilde{p}_{A},\widetilde{p}_{B};h)\,|{\mathcal{M}(\widetilde{p}_{A},\widetilde{p}_{B},h)}\rangle~{},$
(12)

and is illustrated in Fig. 3. The collinear factorization in Eq. (12) can now
easily be extended to any number of collinear directions. Analogous to Eq.
(9), the splitting amplitude
${\mbox{\boldmath$Sp$}}(c_{A},c_{B};\widetilde{p}_{A},\widetilde{p}_{B};h)$
has a perturbative loop expansion, where the splitting amplitude at the tree
level fulfills SCF

$\displaystyle{\mbox{\boldmath$Sp$}}^{(0)}(c_{A},c_{B};\widetilde{p}_{A},\widetilde{p}_{B};h)$
$\displaystyle={\mbox{\boldmath$Sp$}}^{(0)}(c_{A};\widetilde{p}_{A}){\mbox{\boldmath$Sp$}}^{(0)}(c_{B};\widetilde{p}_{B})~{},$
(13)

Based on Eq. (10), soft-collinear factorization for two sets of collinear
partons and a set of soft partons in the TL collinear region reads as

$\displaystyle|\mathcal{M}(s,c_{A},c_{B},h)\rangle$
$\displaystyle\simeq{\mbox{\boldmath$Sp$}}(c_{A};\widetilde{p}_{A})\,{\mbox{\boldmath$Sp$}}(c_{B};\widetilde{p}_{B})\,$
$\displaystyle\times{\mbox{\boldmath$J$}}(s;\widetilde{p}_{A},\widetilde{p}_{B};h)|\mathcal{M}(\widetilde{p}_{A},\widetilde{p}_{B},h)\rangle~{},$
(14)

where the soft current effectively interacts with the parent partons of the
two collinear splittings (see, Fig. 4). However, for a general kinematics that
includes the SL collinear region of the phase space, the strict factorization
in Eq. (10) breaks down and the new soft-collinear factorization shown in Fig.
5 is as follows

Figure 4: TL soft-collinear factorization. The soft current effectively
interacts with the collinear parent partons.  Figure 5: SL soft-collinear
factorization. The generalized splitting amplitude embeds the soft
interactions.

$\displaystyle|\mathcal{M}(s,c,h)\rangle\simeq{\mbox{\boldmath$Sp$}}(c;\widetilde{p};s;h)\,|\mathcal{M}(\widetilde{p},h)\rangle\,,$
(15)

where ${\mbox{\boldmath$Sp$}}(c;\widetilde{p};s;h)$ is the generalized soft-
collinear splitting amplitude, where again only the tree-level compoment of
its perturbative expansion is strictly factorized

$\displaystyle{\mbox{\boldmath$Sp$}}^{(0)}(c;\widetilde{p};s;h)$
$\displaystyle={\mbox{\boldmath$Sp$}}^{(0)}(c;\widetilde{p})\,{\mbox{\boldmath$J$}}^{(0)}(s;\widetilde{p};h)~{}.$
(16)

The soft-collinear factorization in Eq. (15) can now easily be generalized to
any number of collinear directions. For example, in the case of two collinear
directions that we explicitly discussed, we have

$\displaystyle|{\mathcal{M}(s,c_{A},c_{B},h)}\rangle$
$\displaystyle\simeq{\mbox{\boldmath$Sp$}}(c_{A},c_{B};\widetilde{p}_{A},\widetilde{p}_{B};s;h)\,$
$\displaystyle\times|{\mathcal{M}(\widetilde{p}_{A},\widetilde{p}_{B},h)}\rangle~{},$
(17)

where

$\displaystyle{\mbox{\boldmath$Sp$}}^{(0)}(c_{A},c_{B};\widetilde{p}_{A},\widetilde{p}_{B};s;h)$
$\displaystyle={\mbox{\boldmath$Sp$}}^{(0)}(c_{A};\widetilde{p}_{A}){\mbox{\boldmath$Sp$}}^{(0)}(c_{B};\widetilde{p}_{B})$
$\displaystyle\times{\mbox{\boldmath$J$}}^{(0)}(s;\widetilde{p}_{A},\widetilde{p}_{B};h)~{}.$
(18)

##  IV Conclusion

The soft and collinear factorization properties of hard scattering amplitudes
are crucial for understanding the quantum imprints of scattering processes and
for developing the most sophisticated tools to achieve accurate theoretical
predictions for relevant measurable quantities at high-energy colliders. In
the limiting case with multiple soft partons, the scattering amplitude
factorizes into a soft current, which controls the dominant singular behavior,
and the reduced amplitude, which is derived from the original amplitude by
simply removing the soft partons. As expected, due to the long-range
interactions of the soft particles, the soft current depends on the momentum
and color of all partons involved in the scattering process.

In the collinear limit, factorization is generally understood to be universal
in the sense that the most singular factors, the splitting amplitudes, depend
only on the momenta and degrees of freedom of the particles involved in the
specific collinear splitting process. This property is called strict collinear
factorization (SCF). The SCF is valid in the TL-collinear region to all
perturbative orders. However, as shown in the seminal paper Catani  _et al._
(2012), the SCF breaks down for SL-collinear configurations, for which a
generalized factorization has been derived.

In this Letter, we have extended the collinear and simultaneous soft-collinear
factorization to the most general kinematics with multiple collinear
directions. We have shown that the naïve multiplicative factorization does not
hold in a general kinematics and the factorization needs to be further
generalized, including the process dependence on the non-collinear partons. In
the simultaneous soft-collinear limit, the scattering amplitudes do not
factorize into a soft-collinear current and collinear splitting amplitudes.
Instead, the factorization contains a generalized splitting amplitude that
includes the effect of the dominant terms in the soft limit.

The SCF breaking at one loop for two collinear partons is anti-Hermitian
Catani  _et al._ (2012) and therefore has no effect on the squared scattering
amplitude. At higher multiplicities and higher orders, the SCF breaking
effects are partially cancelled also for squared amplitudes Catani  _et al._
(2012); Forshaw  _et al._ (2012). The explicit expressions of the SCF breaking
effects for amplitudes as well as for squared amplitudes with multiple
collinear directions will be presented in a future publication.

###### Acknowledgements.

We are grateful to Stefano Catani for many insightful discussions on the
breaking of strict collinear factorization. We are indebted to the Galileo
Galilei Institute (Firenze) and the organizers of the workshop Theory
Challenges in the Precision Era of the Large Hadron Collider for facilitating
us the very last opportunity to meet our beloved Stefano. We thank Roberto
Bonciani and German Sborlini for useful discussions. This work is supported by
the Spanish Government (Agencia Estatal de Investigació,
MCIN/AEI/10.13039/501100011033) Grant No. PID2020-114473GB-I00, and
Generalitat Valenciana Grant No. PROMETEO/2021/071. LC is supported by
Generalitat Valenciana GenT Excellence Programme (CIDEGENT/2020/011) and CSIC
(ILINK22045). The work of PKD is supported by European Commission MSCA Action
COLLINEAR-FRACTURE, Grant Agreement No. 101108573.

## References

  * Collins  _et al._ (1989) J. C. Collins, D. E. Soper, and G. F. Sterman, Adv. Ser. Direct. High Energy Phys. 5, 1 (1989), arXiv:hep-ph/0409313 .
  * Ellis  _et al._ (2011) R. K. Ellis, W. J. Stirling, and B. R. Webber,  _QCD and collider physics_, Vol. 8 (Cambridge University Press, 2011).
  * Altarelli and Parisi (1977) G. Altarelli and G. Parisi, Nucl. Phys. B 126, 298 (1977).
  * Bassetto  _et al._ (1983) A. Bassetto, M. Ciafaloni, and G. Marchesini, Phys. Rept. 100, 201 (1983).
  * Frixione  _et al._ (1996) S. Frixione, Z. Kunszt, and A. Signer, Nucl. Phys. B 467, 399 (1996), arXiv:hep-ph/9512328 .
  * Catani and Seymour (1997) S. Catani and M. H. Seymour, Nucl. Phys. B 485, 291 (1997), [Erratum: Nucl.Phys.B 510, 503–504 (1998)], arXiv:hep-ph/9605323 .
  * Frixione (1997) S. Frixione, Nucl. Phys. B 507, 295 (1997), arXiv:hep-ph/9706545 .
  * Catani  _et al._ (2002) S. Catani, S. Dittmaier, M. H. Seymour, and Z. Trocsanyi, Nucl. Phys. B 627, 189 (2002), arXiv:hep-ph/0201036 .
  * Heinrich (2021) G. Heinrich, Phys. Rept. 922, 1 (2021), arXiv:2009.00516 [hep-ph] .
  * Torres Bobadilla  _et al._ (2021) W. J. Torres Bobadilla _et al._ , Eur. Phys. J. C 81, 250 (2021), arXiv:2012.02567 [hep-ph] .
  * Agarwal  _et al._ (2023) N. Agarwal, L. Magnea, C. Signorile-Signorile, and A. Tripathi, Phys. Rept. 994, 1 (2023), arXiv:2112.07099 [hep-ph] .
  * Camarda  _et al._ (2021) S. Camarda, L. Cieri, and G. Ferrera, Phys. Rev. D 104, L111503 (2021), arXiv:2103.04974 [hep-ph] .
  * Billis  _et al._ (2021) G. Billis, B. Dehnadi, M. A. Ebert, J. K. L. Michel, and F. J. Tackmann, Phys. Rev. Lett. 127, 072001 (2021), arXiv:2102.08039 [hep-ph] .
  * Neumann and Campbell (2023) T. Neumann and J. Campbell, Phys. Rev. D 107, L011506 (2023), arXiv:2207.07056 [hep-ph] .
  * Berends and Giele (1989) F. A. Berends and W. T. Giele, Nucl. Phys. B 313, 595 (1989).
  * Campbell and Glover (1998) J. M. Campbell and E. W. N. Glover, Nucl. Phys. B 527, 264 (1998), arXiv:hep-ph/9710255 .
  * Catani and Grazzini (2000a) S. Catani and M. Grazzini, Nucl. Phys. B 570, 287 (2000a), arXiv:hep-ph/9908523 .
  * Bern and Chalmers (1995) Z. Bern and G. Chalmers, Nucl. Phys. B 447, 465 (1995), arXiv:hep-ph/9503236 .
  * Bern  _et al._ (1998) Z. Bern, V. Del Duca, and C. R. Schmidt, Phys. Lett. B 445, 168 (1998), arXiv:hep-ph/9810409 .
  * Bern  _et al._ (1999) Z. Bern, V. Del Duca, W. B. Kilgore, and C. R. Schmidt, Phys. Rev. D 60, 116001 (1999), arXiv:hep-ph/9903516 .
  * Catani and Grazzini (2000b) S. Catani and M. Grazzini, Nucl. Phys. B 591, 435 (2000b), arXiv:hep-ph/0007142 .
  * Bierenbaum  _et al._ (2012) I. Bierenbaum, M. Czakon, and A. Mitov, Nucl. Phys. B 856, 228 (2012), arXiv:1107.4384 [hep-ph] .
  * Catani  _et al._ (2020) S. Catani, D. Colferai, and A. Torrini, JHEP 01, 118 (2020), arXiv:1908.01616 [hep-ph] .
  * Del Duca  _et al._ (2023) V. Del Duca, C. Duhr, R. Haindl, and Z. Liu, JHEP 01, 040 (2023), arXiv:2206.01584 [hep-ph] .
  * Catani  _et al._ (2023) S. Catani, L. Cieri, D. Colferai, and F. Coradeschi, Eur. Phys. J. C 83, 38 (2023), arXiv:2210.09397 [hep-ph] .
  * Zhu (2020) Y. J. Zhu, (2020), arXiv:2009.08919 [hep-ph] .
  * Catani and Cieri (2022) S. Catani and L. Cieri, Eur. Phys. J. C 82, 97 (2022), arXiv:2108.13309 [hep-ph] .
  * Czakon  _et al._ (2023) M. Czakon, F. Eschment, and T. Schellenberger, JHEP 04, 065 (2023), arXiv:2211.06465 [hep-ph] .
  * Li and Zhu (2013) Y. Li and H. X. Zhu, JHEP 11, 080 (2013), arXiv:1309.4391 [hep-ph] .
  * Duhr and Gehrmann (2013) C. Duhr and T. Gehrmann, Phys. Lett. B 727, 452 (2013), arXiv:1309.4393 [hep-ph] .
  * Dixon  _et al._ (2020) L. J. Dixon, E. Herrmann, K. Yan, and H. X. Zhu, JHEP 05, 135 (2020), arXiv:1912.09370 [hep-ph] .
  * Chen  _et al._ (2024) W. Chen, M.-x. Luo, T.-Z. Yang, and H. X. Zhu, JHEP 01, 131 (2024), arXiv:2309.03832 [hep-ph] .
  * Herzog  _et al._ (2023) F. Herzog, Y. Ma, B. Mistlberger, and A. Suresh, JHEP 12, 023 (2023), arXiv:2309.07884 [hep-ph] .
  * Catani and Grazzini (1999) S. Catani and M. Grazzini, Phys. Lett. B 446, 143 (1999), arXiv:hep-ph/9810389 .
  * Dhani  _et al._ (2023) P. K. Dhani, G. Rodrigo, and G. F. R. Sborlini, JHEP 12, 188 (2023), arXiv:2310.05803 [hep-ph] .
  * Craft  _et al._ (2023) E. Craft, M. Gonzalez, K. Lee, B. Mecaj, and I. Moult, (2023), arXiv:2310.06736 [hep-ph] .
  * Kosower and Uwer (1999) D. A. Kosower and P. Uwer, Nucl. Phys. B 563, 477 (1999), arXiv:hep-ph/9903515 .
  * Catani  _et al._ (2012) S. Catani, D. de Florian, and G. Rodrigo, JHEP 07, 026 (2012), arXiv:1112.4405 [hep-ph] .
  * Sborlini  _et al._ (2014a) G. F. R. Sborlini, D. de Florian, and G. Rodrigo, JHEP 01, 018 (2014a), arXiv:1310.6841 [hep-ph] .
  * Del Duca  _et al._ (2000) V. Del Duca, A. Frizzo, and F. Maltoni, Nucl. Phys. B 568, 211 (2000), arXiv:hep-ph/9909464 .
  * Birthwright  _et al._ (2005a) T. G. Birthwright, E. W. N. Glover, V. V. Khoze, and P. Marquard, JHEP 05, 013 (2005a), arXiv:hep-ph/0503063 .
  * Birthwright  _et al._ (2005b) T. G. Birthwright, E. W. N. Glover, V. V. Khoze, and P. Marquard, JHEP 07, 068 (2005b), arXiv:hep-ph/0505219 .
  * Del Duca  _et al._ (2020a) V. Del Duca, C. Duhr, R. Haindl, A. Lazopoulos, and M. Michel, JHEP 02, 189 (2020a), arXiv:1912.06425 [hep-ph] .
  * Del Duca  _et al._ (2020b) V. Del Duca, C. Duhr, R. Haindl, A. Lazopoulos, and M. Michel, JHEP 10, 093 (2020b), arXiv:2007.05345 [hep-ph] .
  * Catani  _et al._ (2004) S. Catani, D. de Florian, and G. Rodrigo, Phys. Lett. B 586, 323 (2004), arXiv:hep-ph/0312067 .
  * Sborlini  _et al._ (2014b) G. F. R. Sborlini, D. de Florian, and G. Rodrigo, JHEP 10, 161 (2014b), arXiv:1408.4821 [hep-ph] .
  * Sborlini  _et al._ (2015) G. F. R. Sborlini, D. de Florian, and G. Rodrigo, JHEP 03, 021 (2015), arXiv:1409.6137 [hep-ph] .
  * Badger  _et al._ (2015) S. Badger, F. Buciuni, and T. Peraro, JHEP 09, 188 (2015), arXiv:1507.05070 [hep-ph] .
  * Czakon and Sapeta (2022) M. Czakon and S. Sapeta, JHEP 07, 052 (2022), arXiv:2204.11801 [hep-ph] .
  * Bern  _et al._ (2004) Z. Bern, L. J. Dixon, and D. A. Kosower, JHEP 08, 012 (2004), arXiv:hep-ph/0404293 .
  * Badger and Glover (2004) S. D. Badger and E. W. N. Glover, JHEP 07, 040 (2004), arXiv:hep-ph/0405236 .
  * Duhr  _et al._ (2015) C. Duhr, T. Gehrmann, and M. Jaquier, JHEP 02, 077 (2015), arXiv:1411.3587 [hep-ph] .
  * Rogers and Mulders (2010) T. C. Rogers and P. J. Mulders, Phys. Rev. D 81, 094006 (2010), arXiv:1001.2977 [hep-ph] .
  * Echevarria  _et al._ (2012) M. G. Echevarria, A. Idilbi, and I. Scimemi, JHEP 07, 002 (2012), arXiv:1111.4996 [hep-ph] .
  * (55) M. G. Echevarría, A. Idilbi, and I. Scimemi, Phys. Lett. B 726, 795, arXiv:1211.1947 [hep-ph] .
  * Rogers (2013) T. C. Rogers, Phys. Rev. D 88, 014002 (2013), arXiv:1304.4251 [hep-ph] .
  * Forshaw  _et al._ (2006) J. R. Forshaw, A. Kyrieleis, and M. H. Seymour, JHEP 08, 059 (2006), arXiv:hep-ph/0604094 .
  * Forshaw  _et al._ (2008) J. R. Forshaw, A. Kyrieleis, and M. H. Seymour, JHEP 09, 128 (2008), arXiv:0808.1269 [hep-ph] .
  * Keates and Seymour (2009) J. Keates and M. H. Seymour, JHEP 04, 040 (2009), arXiv:0902.0477 [hep-ph] .
  * Becher  _et al._ (2021) T. Becher, M. Neubert, and D. Y. Shao, Phys. Rev. Lett. 127, 212002 (2021), arXiv:2107.01212 [hep-ph] .
  * Ma  _et al._ (2023) Y. Ma, G. Sterman, and A. Venkata, (2023), arXiv:2311.06912 [hep-ph] .
  * Feige and Schwartz (2014) I. Feige and M. D. Schwartz, Phys. Rev. D 90, 105020 (2014), arXiv:1403.6472 [hep-ph] .
  * Forshaw  _et al._ (2012) J. R. Forshaw, M. H. Seymour, and A. Siodmok, JHEP 11, 066 (2012), arXiv:1206.6363 [hep-ph] .
  * Sterman (2022) G. Sterman, in  _Snowmass 2021_ (2022) arXiv:2207.06507 [hep-ph] .


#  SpectroLab: An Open Source Matlab Based Toolbox for High Throughput
Spectroscopy Analysis

Christopher Sims  School of Electrical and Computer Engineering, Purdue
University, West Lafayette, Indiana 47907, USA

(August 27, 2024)

###### Abstract

We present an open source software package SpectroLab a Matlab based tool
developed in 2018 for the analysis of spectroscopic data. In this package
there are tools for derivative analysis, stacked energy contours , stacked
plots for theory, 3D volumetric plots, and core level analysis, and derivative
analysis. The package can currently be used for angle-resolved photoemission
spectroscopy (ARPES) data, with the ability to also be used for other
spectroscopic data in the future. We apply this program to the HfP2, ZrSiS,
and Hf2Te2P systems to demonstrate its robustness.

##  I Introduction

In recent years, spectroscopic techniques such as angle resolved photoemission
spectroscopy (ARPES)[1, 2], scanning tunneling microscopy (STM) [3, 4, 5],
neutron scattering [6, 7, 8], etc. have improved to the point where the
scientific communities use these tools extensively. These techniques reveal
features in momentum and energy space that can give deeper understanding of
the interactions of exotic quasi-particles that have taken interest in recent
years such as topological insulators[9, 10], Weyl semimetals[11, 12, 13],
nodal-line semimetals[14, 15, 16], and High order Topological insulators
(HOTI)[17, 18]. Furthermore, these techniques have been expanded to measure
other dimensions such as spin, time, magnetism, etc. These techniques are
always prone to noise which will decrease the ability to interpret the data
correctly. Previously, researchers have employed proprietary programs in order
to conduct the data analysis. These programs limit the ability for users to
tune the way results are presented and create figures and graphs with a high
degree of control. Here, we report a Matlab based program that is written in
an easy to read way that allows for the analysis of ARPES data. Using Matlab’s
built in tools, we have design a program that gives users a high control in
the way they present their data.

The data analysis of these materials is critical in order to understand
complex interactions that are difficult to resolve by eye. By importing the
data into Matlab, we provide a way for researchers to use the vast packages
included in this program that are nontrivial to code in proprietary data
analysis programs such as edge detection or neural networks. We present a
suite of functions that allow for easy analysis and presentation of
spectroscopic data in the Matlab suite called spectrolab. The suite needs
minimal programming experience since all of the tools included can be done in
one click. As oppose to other data analysis techniques, our package is all
inclusive and can be used with minimal need to modify the code in anyway. The
only need for the program to work is an _.ibw_ file that is formatted
correctly. We present ARPES data in from the HfP2 [19], ZrSiS [20, 21], and
Hf2Te2P [22] systems in order to show that this program is applicable to
different types of ARPES data. This paper includes a detailed description of
the theory behind how the main functions work in section 2. In section 3, we
outline the functionalities of each of the functions in the initial release.
In section 4, we go over how to install and use SpectroLab. Finally in section
5, we go over a few examples of using the software package in real systems.

##  II Methods

As the data is imported into SpectroLab it is automatically interpreted into
$k$-space from the raw data using the following equations[23]:

$k_{x}=\frac{1}{\hbar}\sqrt{2mE_{kin}}sin\vartheta cos\varphi$ (1)
$k_{y}=\frac{1}{\hbar}\sqrt{2mE_{kin}}sin\vartheta sin\varphi$ (2)

In the program these are interpreted by the following approximation

$k_{x}=0.5112*\sqrt{(E-4.5)}*(\pi*X*0.046/180)$ (3)
$k_{y}=0.5112*\sqrt{(E-4.5)}*(\pi*Z/180)$ (4)

In addition to this all supporting variables are automatically calculated
within the setvariables() function.

⬇

KY_Angle_Max = 0.5112*sqrt(E -4.5)*(pi*Zmax/180);

KY_Angle_Min = 0.5112*sqrt(E -4.5)*(pi*Zmin/180);

KX_Angle_Max = 0.5112*sqrt(E-4.5)*(pi*Xmax*0.046/180) -1.68;

KX_Angle_Min = 0.5112*sqrt(E-4.5)*(pi*Xmin*0.046/180) - 1.68;

###  II.1 Sobel filter (2nd Derivative)

The common edge detection algorithms are the Sobel, Canny, Prewitt, Roberts,
and fuzzy logic methods[24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. In this
paper, we use a modified version of the Sobel filter in order to conduct
second derivative calculations. While it is possible to use a convolution
neural network (CNN) for feature extraction[34, 35], these networks require a
large amount of computational resources and are only applicable to the data
set they are trained on. Generative adversarial neural networks (GANs)[36] are
a much better alternative to create a neural network feature extraction since
it can work on an infinite amount of data sets once it is properly trained.
The downside is that a large amount of data sets ($>$1000) are required in
order to properly train the neural network. Therefore, we forgo the inclusion
of a neural network in our program. The Sobel filter is a 3$\times$3 matrix
that calculates the second derivative locally in an image file (where A is the
image). There are two versions, the horizontal and the vertical which can be
used, in the distributed program. The horizontal $F_{x}$ is convoluted with
the image to form the second derivative of the image.

$F_{x}=\begin{bmatrix}-1&0&+1\\\ -2&0&+2\\\
-1&0&-1\end{bmatrix}*A,F_{y}=\begin{bmatrix}-1&-2&-1\\\ 0&0&0\\\
+1&+2&+1\end{bmatrix}*A$ (5)

In order to detect edges after the Sobel filter is applied, one would find the
square of the absolute value of the resultant $|F_{x}|^{2}$. By not conducting
this last step we can resolve weak signals of ARPES. In the case of sharp
data, the edge detection feature can be turned on with an additional line of
code which we have included. All edge detection algorithms can be accesses via
Matlab’s image toolbox.

⬇

Data2D = squeeze(Mapping(:,EnergyLevel,:))’;

h_Sobel = fspecial(’sobel’);

f_y = imfilter(double(Data2D), h_Sobel,’replicate’);

imagesc(f_y)

###  II.2 Constant Energy Contours

An energy contour is a slice of a 3D band mapping in ARPES
($k_{x}$,energy,$k_{x}$) where at a constant energy, the 2D $k_{x}$,$k_{y}$
surface is revealed. For constant energy contours we create a function that
automatically interprets the energy levels and recreates them in an energy
contour. For labeling we included a rounding function that rounds to the
nearest significant figure. However, the program can interpret the exact value
for publication.

⬇

Energy_Limit = 200; %specified by user

Energy_Cut = 1:int16(Energy_Limit/6):Energy_Limit;

Energy_Contour = ypos - Energy_Cut % check limits;

hold on

subaxis(2,6,1,’sv’,vertical_space,’sh’,0.01);

f = squeeze(B(:,ypos,:))’;

imshow(imrotate(f,0),[], ’XData’, [KX_Angle_Min+ + offsetx KX_Angle_Max+
offsetx]...

, ’YData’, [KY_Angle_Min + offsety KY_Angle_Max + offsety], ’colormap’,
setmap);

for ii = 2:6

subaxis(2,6,ii,’sv’,0.01,’sh’,0.01);

f = squeeze(B(:,Energy_Contour(ii),:))’;

imshow(imrotate(flipud(f),0),[], ’XData’, [KX_Angle_Min KX_Angle_Max]...

, ’YData’, [KY_Angle_Min KY_Angle_Max], ’colormap’, ’gray’ );

end

###  II.3 Volumetric plot

The volumetric plot function takes advantage of how data is formatted for
ARPES. Typically data is encoded in an (X,Y,Z,C) format. Where X,Y,and Z are
$k_{x}$, $k_{y}$, and energy, while C is the interpretation of the intensity
at that location. By using an alpha map, we can make weak signals transparent
and reveal the 3D band structure for visualization. This has the ability to be
a very powerful tool for investigating anisotropy in bands.

⬇

%VolumetricPlot()

% edit this to get a good mapping

alphamap([0 linspace(0.0005, 0.4, 255)]);

###  II.4 Momentum cuts

We have included several methods to aid the taking $k_{x}$ and $k_{y}$
momentum cuts from a 3D band mapping. We take these cuts by selecting a value
of $k_{x}$ and $k_{y}$ and resolving the cut in a 2D image.

⬇

f = squeeze(B(Kxlocation,:,:));

f = imresize(f,5);

imshow(f’,[], ’XData’, ...

[KX_Angle_Min KX_Angle_Max], ’YData’,...

[abs(Ymin) - Fermi_Level, abs(Ymax)- Fermi_Level], ’colormap’, setmap);

###  II.5 Energy & momentum distribution curves

Energy & momentum distribution curves (EDC & MDC) are used for the data
analysis of spectroscopic data. In a set amount of intervals a line of data is
taken out of the image and then plotted in order to enable researchers to
resolve difficult to resolve images from ARPES or other spectroscopic data. We
use the _waterfall()_ function in order to reconstruct the EDC and MDC plots
seen in literature [22]. The plot also have the benefit of being three
dimensional with the ability to add a color map to further define intensity
differences.

###  II.6 Stacked theory

In order to import theory, we create a 2D surface in a 3D box using Matlab’s
_meshgrid()_ function to import an image of the theoretical plot. To insure
that the images that we import are of the same size we use the largest image
dimensions as the map for all of the subsequent images. We do this because the
3D plot will not work unless we use the same size map. We then offset the
images by one for each image. Now, one can scale the distance between image
layers using the _daspect()_ function of Matlab to scale the Z (Energy) in
order to resolve the layers.

⬇

%# plot each slice as a texture-mapped surface (stacked along the Z-dimension)

for k=1:size(im)

surface(’XData’,X-0.5, ’YData’,Y, ’ZData’, Z.*-k, ...

’CData’,im{k}, ’CDataMapping’,’direct’, ...

’EdgeColor’,’none’, ’FaceColor’,’texturemap’)

end

##  III Analysis tools of SpectroLab

Here we included an overview of the uses of SpectroLab. The fermi level can be
found with the _fermisurface()_ [Fig. 2(a)] and the _ypos_ variable

Table 1: Main tools of SpectroLab Program | Description
---|---
Kx Cuts | Creates $k_{x}$ momentum slices
Ky Cuts | Creates $k_{y}$ momentum slices
Plot_KX_KY | For quick check of data analysis
Fermi Surface | Draws Fermi surface from 3D map
Momentum 2D | 2D ARPES cuts
Stacked plot | Slices and stacks 3D momentum cuts
Stacked theory | Stack theory similar to stacked plot
Density Curves | EDC and MDC
Energy Contour plot | ECP in exact steps
Volumetric plot | 3D volumetric plot
Dispersion | photon energy dependence
Inner core | Analysis of inner core spectra

###  III.1 Graphical User Interface (GUI)

We include a graphical user interface (GUI) [Fig. 1] that can be used as a
general tool, more advance data selection and analysis will need to be done in
code.

###  III.2 Constant Energy Contours

We include a function to extract constant energy contours (ECs) from a 3D band
mapping. StackedECP() [Fig. 2(c,d) is a feature that calculates ECs and stacks
them on top of each other in order to visualize the bands dispersing in 3D
space. The energy contour functions do the same, however the ECs are presented
in a 2D format.

###  III.3 Volumetric plot

In the volumetric plot function, a 3D band mapping is interpreted with an
alpha map (transparency map). This allows for all bands to be mapped in a 3D
space. In addition, users can rotate and zoom in to further examine the
electronic structure.

###  III.4 Momentum cuts

We have included several methods to aid the taking $k_{x}$ and $k_{y}$
momentum cuts from a 3D band mapping. Plot_KX_KY, Ky cuts (Fig. 3(a)), and Kx
cuts (Fig. 3(b)) are all features that allow for one to construct these plots.
Plot_KX_KY is a debug tool but can still be used to accomplish this feature.

###  III.5 Energy & momentum distribution curves

Energy distribution curves (EDCs) and momentum distribution curves (MDCs) are
important tools for the analysis of ARPES data. Here, we construct programs
that can create 3D EDC [Fig. 4(a)] and MDC [Fig. 4(b)] plots similar to those
found in previous literature. The advantage of these types of plots is that
they are able to give a 3D interpretation of the EDC and MDC which will allow
for better understanding of the intensity at certain points (such as
discerning if there is a band gap or weak feature in the band gap).

###  III.6 Photon energy dependence

Dispersion allows for one to interpret photon energy dependent cuts (not a
mapping of photon energy) to include for interpretation or publication. The
Dispersion map function can interpret an arbitrary amount of cuts specified by
the user. [Fig. 5]

###  III.7 Stacked theory

Stacked theory is a function that allows one to import theoretical DFT energy
contours and plot them in a way to compare to the _stackedECP()_ function
[Fig. 2(b)]. There is also a feature that allows for plotting theory in one of
the energy contour functions.

##  IV Installation and usage

In this section, we present how to install and use SpectroLab.

###  IV.1 Get SpectroLab

SpectroLab is an open source software package distributed under the MIT
License. The code can be downloaded directly from the public code repository:
github.com/christophersims/SpectroLab.

###  IV.2 Installation and running the code

The code requires no installation, however, the folders that contain the code
must be added to the path in order for them to run correctly (or the user
could place all of the .m files in one directory). The user must have a Matlab
version year of at least 2018 with the image toolbox add-on installed.

The code can be run by simply specifying the file path of the data one wishes
to analyze. If there are no formatting errors the code should run without
errors.

###  IV.3 File formats

For ARPES there are three major file formats used (from the Igor suite), .txt,
.ibw, and .pxt. Important data is lost in the .txt format and the .pxt format
is compressed and cannot be decompressed outside of the Igor program.
Therefore, this program only uses the Igor Binary Wave format (.ibw). In
addition, the program only accepts files in the format
($k_{x}$,Energy,$k_{y}$), however, the file can be reformatted within the
program.

##  V Examples

###  V.1 Incorrectly formatted input

If the input of the code in not formatted correctly (in the
($k_{x}$,Energy,$k_{y}$) format) it is possible to still run the code by
utilizing Matlab’s permute() function. This function allows for one to swap
the matrices in order for the program to run correctly. In addition to using
the permute the variables in variables() must also be changed in order to
match the code. In figure , we present data that was formatted incorrectly but
was able to run in the program and produce good results. Here we present the
data of Hf2Te2P (Fig. 6) and ZrSiS (Fig. 7) which were imported using the
permute function.

###  V.2 Plotting theory

Our implementation of plotting theory is currently an image importer, however,
we have the code that allows for the input if the surface states and Fermi
surface generated with the iterative surface green functions[37] imported from
Wanniertools[38]. The calculations are not included in this package and must
be calculated with tools such as Quantum Espresso[39], VASP[40], Abinit[41],
Wannier90[42], etc. In figure 2(b), we show the capability so show stacked
theory plots in order to coincide with the stackedECP() function. In addition,
we have included a ECP function that also allows for the inclusion of
theoretical calculation figures (see supplementary).

###  V.3 Volumetric plot

Here we present a volumetric plot of both a 3D band mapping correctly
formatted Fig. 8(a) and an incorrectly formatted (corrected in code) in Fig.
8(b). This method is a powerful tool that will enable researchers to visualize
bands in a 3D space.

##  VI Conclusion

In conclusion, we have designed a Matlab based program that can be used to
interpret spectroscopic data. Included in this tool are multiple functions
which can be used for the analysis of ARPES data. To demonstrate the
robustness of this package we show ARPES analysis of several data sets. This
package can be used as a standalone with no other dependency on other
programs. The features of this analysis suite is extensive with the addition
of new analysis techniques such as the 3D volumetric plot. In the future, we
aim to integrate the analysis of XRD and STM data into the SpectroLab suite.

## Supplemetary Material

See Supplementary material for further figures of spectrolab’s output in the
HfP2, Hf2Te2P, and ZrSiS systems.

## Acknowledgments

The authors would like to thank the Matlab community which provided several
methods and programs used in SpectroLab.

C.S. acknowledges the generous support of the Purdue ECE ASIRE fellowship and
the National GEM consortium fellowship

Correspondence and requests for additional functionality should be addressed
to C.S.
(Email: Sims58@purdue.edu).

## References

  * Damascelli [2004] A. Damascelli, Phys. Scr. T109, 61 (2004).
  * Lu  _et al._ [2012] D. Lu, I. M. Vishik, M. Yi, Y. Chen, R. G. Moore, and Z.-X. Shen, Annual Review of Condensed Matter Physics 3, 129 (2012).
  * Binnig  _et al._ [1982] G. Binnig, H. Rohrer, C. Gerber, and E. Weibel, Phys. Rev. Lett. 49, 57 (1982).
  * Tersoff and Hamann [1983] J. Tersoff and D. R. Hamann, Phys. Rev. Lett. 50, 1998 (1983).
  * Binnig and Rohrer [1987] G. Binnig and H. Rohrer, Rev. Mod. Phys. 59, 615 (1987).
  * Wilkinson, Wollan, and Koehler [1961] M. K. Wilkinson, E. O. Wollan, and W. C. Koehler, Annu. Rev. Nucl. Sci. 11, 303 (1961).
  * Prince [2004] E. Prince, Appl. Spectrosc. Rev. 34, 159 (2004).
  * Strobl  _et al._ [2017] M. Strobl, R. Harti, C. Gruenzweig, R. Woracek, and J. Plomp, Journal of Imaging 3, 64 (2017).
  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Xia  _et al._ [2009] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nat. Phys. 5, 398 (2009).
  * Xu  _et al._ [2015] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, Science 349, 613 (2015).
  * Lv  _et al._ [2015] B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, Phys. Rev. X 5, 031013 (2015).
  * Huang  _et al._ [2015] S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, S. Jia, A. Bansil, H. Lin, and M. Z. Hasan, Nat. Commun. 6, 7373 (2015).
  * Burkov, Hook, and Balents [2011] A. A. Burkov, M. D. Hook, and L. Balents, Phys. Rev. B 84, 235126 (2011).
  * Fang  _et al._ [2015] C. Fang, Y. Chen, H.-Y. Kee, and L. Fu, Phys. Rev. B 92, 081201 (2015).
  * Fang  _et al._ [2016] C. Fang, H. Weng, X. Dai, and Z. Fang, Chin. Phys. B 25, 117106 (2016).
  * [17] S. Regmi, M. M. Hosen, B. Ghosh, B. Singh, G. Dhakal, C. Sims, B. Wang, F. Kabir, K. Dimitri, Y. Liu, A. Agarwal, H. Lin, D. Kaczorowski, A. Bansil, and M. Neupane, http://arxiv.org/abs/1911.03703v1 .
  * Schindler  _et al._ [2018] F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Sci. Adv. 4, eaat0346 (2018).
  * Sims  _et al._ [2019] C. Sims, M. M. Hosen, H. Aramberri, C.-Y. Huang, G. Dhakal, K. Dimitri, F. Kabir, S. Regmi, X. Zhou, T.-R. Chang, H. Lin, D. Kaczorowski, N. Kioussis, and M. Neupane, arXiv:1906.09642 (2019).
  * Neupane  _et al._ [2016] M. Neupane, I. Belopolski, M. M. Hosen, D. S. Sanchez, R. Sankar, M. Szlawska, S.-Y. Xu, K. Dimitri, N. Dhakal, P. Maldonado, P. M. Oppeneer, D. Kaczorowski, F. Chou, M. Z. Hasan, and T. Durakiewicz, Phys. Rev. B 93 (2016).
  * Hosen  _et al._ [2017] M. M. Hosen, K. Dimitri, I. Belopolski, P. Maldonado, R. Sankar, N. Dhakal, G. Dhakal, T. Cole, P. M. Oppeneer, D. Kaczorowski, F. Chou, M. Z. Hasan, T. Durakiewicz, and M. Neupane, Phys. Rev. B 95 (2017).
  * Hosen  _et al._ [2018] M. M. Hosen, K. Dimitri, A. K. Nandy, A. Aperis, R. Sankar, G. Dhakal, P. Maldonado, F. Kabir, C. Sims, F. Chou, D. Kaczorowski, T. Durakiewicz, P. M. Oppeneer, and M. Neupane, Nat. Commun. 9 (2018).
  * Comin and Damascelli [2015] R. Comin and A. Damascelli,  _ARPES: A Probe of Electronic Correlations_ (2015) pp. 31–71.
  * Senthilkumaran and Rajesh [2009] N. Senthilkumaran and R. Rajesh, in  _2009 International Conference on Advances in Recent Technologies in Communication and Computing_ (IEEE, 2009).
  * Ma  _et al._ [2010] C. Ma, W. Gao, L. Yang, and Z. Liu, in  _2010 2nd International Conference on Mechanical and Electronics Engineering_ (IEEE, 2010).
  * Gao  _et al._ [2010] W. Gao, X. Zhang, L. Yang, and H. Liu, in  _2010 3rd International Conference on Computer Science and Information Technology_ (IEEE, 2010).
  * Seif, Salut, and Marsono [2010] A. Seif, M. M. Salut, and M. N. Marsono, in  _2010 IEEE Conference on Sustainable Utilization and Development in Engineering and Technology_ (IEEE, 2010).
  * Mintz [1994] D. Mintz, CVGIP: Image Understanding 59, 137 (1994).
  * Kanopoulos, Vasanthavada, and Baker [1988] N. Kanopoulos, N. Vasanthavada, and R. Baker, IEEE J. Solid-State Circuits 23, 358 (1988).
  * Chen, Tsang, and Xu [2012] L. Chen, I. W. Tsang, and D. Xu, IEEE Trans. Neural Netw. Learn. Syst. 23, 902 (2012).
  * Canny [1986] J. Canny, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 679 (1986).
  * Huertas and Medioni [1986] A. Huertas and G. Medioni, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 651 (1986).
  * Shrivakshan and Chandrasekar [2012] G. T. Shrivakshan and C. Chandrasekar, in  _IJCSI International Journal of Computer Science Issues_ (2012).
  * Bishop [1996] C. M. Bishop, “Neural networks: A pattern recognition perspective,” (1996).
  * [35] H. Peng, X. Gao, Y. He, Y. Li, Y. Ji, C. Liu, S. A. Ekahana, D. Pei, Z. Liu, Z. Shen, and Y. Chen, http://arxiv.org/abs/2001.10908v1 .
  * Goodfellow  _et al._ [2014] I. J. Goodfellow, J. Pouget-abadie, M. Mirza, B. Xu, D. Warde-farley, S. Ozair, A. Courville, and Y. Bengio, “Generative adversarial nets,” (2014).
  * Sancho  _et al._ [1985] M. P. L. Sancho, J. M. L. Sancho, J. M. L. Sancho, and J. Rubio, J. Phys. F: Met. Phys. 15, 851 (1985).
  * Wu  _et al._ [2018] Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov, Comput. Phys. Commun. 224, 405 (2018).
  * Giannozzi  _et al._ [2009] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcovitch, J. Phys.: Condens. Matter 21, 395502 (2009).
  * Kresse and Hafner [1993] G. Kresse and J. Hafner, Phys. Rev. B 48, 13115 (1993).
  * Gonze  _et al._ [2009] X. Gonze, B. Amadon, P.-M. Anglade, J.-M. Beuken, F. Bottin, P. Boulanger, F. Bruneval, D. Caliste, R. Caracas, M. Côté, T. Deutsch, L. Genovese, P. Ghosez, M. Giantomassi, S. Goedecker, D. Hamann, P. Hermet, F. Jollet, G. Jomard, S. Leroux, M. Mancini, S. Mazevet, M. Oliveira, G. Onida, Y. Pouillon, T. Rangel, G.-M. Rignanese, D. Sangalli, R. Shaltaf, M. Torrent, M. Verstraete, G. Zerah, and J. Zwanziger, Comput. Phys. Commun. 180, 2582 (2009).
  * Mostofi  _et al._ [2008] A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Comput. Phys. Commun. 178, 685 (2008).

Figure 1: Graphical User interface (a) The main page the GUI figure with the
main release of SpectroLab. Figure 2: Example of functions (a) Result of the
fermisurface() function. (b) Result of the stacked_theory() program. (c)
Output of the stackedECP() function in the Hfp2 system. (d) Output of the
stackedECP() function in another dataset in the HfP2 system. Figure 3: $k_{x}$
and $k_{y}$ momentum cuts (a) Fermi surface (rightmost panel) and $k_{x}$
momentum cuts (cut1-cut5) taken from a 3D mapping by using the kxcuts
function. (b) same as (a) but for $k_{y}$ momentum cuts Figure 4: Energy &
Momentum Distribution Curves (a) EDC curve (b) MDC curve Figure 5: Dispersion
map A dispersion map of HfP2 resolved with the dispersion() function Figure 6:
Hf2Te2P (a) Normal energy contour (b) 2nd derivative energy contour Figure 7:
ZrSiS Energy Contour (a) Normal energy contour (b) 2nd derivative energy
contour Figure 8: Volumetric plot (a) volumetric plot of correctly formatted
code. (b) Volumetric plot of incorrectly formatted input. There is no errors
in the incorrectly formatted 3D band mapping contour