# Higgs Production and $H^{0}\to c\overline{c}$ at a Future Muon-Ion Collider

Pavan Ahluwalia  Department of Physics, Southern Methodist University, Dallas,
TX 75275, USA  Harmony School of Innovation, Fort Worth, TX 76123, USA
Justine Choi  Department of Physics, Southern Methodist University, Dallas, TX
75275, USA  Highland Park High School, Highland Park, TX 75205, USA  Simba
Masters  Department of Physics, Southern Methodist University, Dallas, TX
75275, USA  Austin Mullins  Department of Physics, Southern Methodist
University, Dallas, TX 75275, USA  Noah Randall  Department of Physics,
Southern Methodist University, Dallas, TX 75275, USA  Stephen J. Sekula
corresponding author<EMAIL_ADDRESS>Department of Physics, Southern
Methodist University, Dallas, TX 75275, USA  SNOLAB, Lively, ON P3Y 1N2,
Canada  Department of Physics, Engineering Physics, and Astronomy, Queen’s
University, Kingston, ON K7L 3N6, Canada

###### Abstract

We investigate the production of Higgs bosons in neutral- and charged-current
interactions at a future Muon-Ion Collider (MuIC) with emphasis on $H^{0}\to
c\overline{c}$ decay. In particular, we focus on the leading production
mechanisms and explore the impact of a range of Parton Distribution Function
(PDF) scenarios, with emphasis on the role of heavy intrinsic quark flavor. We
emphasize estimates of non-resonant di-charm background production and explore
the impact of this background on a possible future measurement of $H^{0}\to
c\overline{c}$ at a future Muon-Ion Collider facility.

††preprint: SMU-HEP-22-11

##  I Introduction

The Electron-Ion Collider at Brookhaven National Laboratory Abdul Khalek  _et
al._ (2022a, b) will open an entirely new window on hadron structure through
the use of high-intensity, high-energy electron and ion collisions. A recent
proposal for later upgrading the EIC to a Muon-Ion Collider (MuIC) has been
presented Acosta and Li (2022) and a case is being built within the community
for such an upgrade path beyond the nominal EIC program Acosta  _et al._
(2022). A MuIC would not only provide a useful step forward toward the
eventual idea of a dedicated muon collider, but itself would facilitate a
novel physics program that complements the High-Luminosity Large Hadron
Collider (HL-LHC) and other future accelerator complexes (c.f. Ref. Shiltsev
and Zimmermann (2021)). In particular, the center-of-mass energy possible in
such a machine would facilitate a program that includes Higgs physics.

In this note, we explore in additional detail the production of Higgs
particles at such a collider. We focus specifically on $H^{0}\to
c\overline{c}$ and di-charm production that would serve as a background to the
Higgs reconstruction. The decay $H^{0}\to c\overline{c}$ is a key part of the
ongoing LHC program (c.f. Refs Aad  _et al._ (2022); CMS Collaboration (2022))
but poses a challenge in the LHC proton-collider environment. Mechanisms for
production of the Higgs boson at a lepton-hadron collider are well-understood
in the context of past such collider studies. A MuIC operated at, for example,
a reasonable center-of-mass energy ($E_{\mu}=960\leavevmode\nobreak\
{\mathrm{\ Ge\kern-1.00006ptV}}$, $E_{p}=275\leavevmode\nobreak\ {\mathrm{\
Ge\kern-1.00006ptV}}$, $\sqrt{s}\approx 1000\leavevmode\nobreak\ {\mathrm{\
Ge\kern-1.00006ptV}}$, achievable without a complete overhaul of the EIC
complex) would allow for single-Higgs production dominated by charged-current
(CC) and neutral-current (NC) deep-inelastic scattering (DIS) involving light-
flavor quarks. This is illustrated in Fig. 1.

The cross-sections for the above processes were estimated in Ref. Acosta  _et
al._ (2022) for a $960{\mathrm{\ Ge\kern-1.00006ptV}}\times 275{\mathrm{\
Ge\kern-1.00006ptV}}$ muon-proton collider. Estimates were made at leading
order (LO) in the matrix element and next-to-leading order (NLO) in the parton
distribution function (PDF). The matrix element was computed using MadGraph
3.3.1 and the sm-no_b_mass model (a "five-flavor scheme" in which the quarks
contributing to the proton are assumed to all be massless except the top
quark). The PDF set used was PDF4LHC15_nlo_mc_pdfas. A number of kinematic
selections are set implicitly by MadGraph; these are disabled (e.g. the
minimum jet $p_{T}$ is set to 0 instead of 20 GeV, etc.). For certain
processes (c.f. Section II.3) we do employ generator-level kinematic criteria.

For zero beam polarization, the authors of Ref. Acosta  _et al._ (2022)
determined the CC (NC) DIS cross-section to be $65\leavevmode\nobreak\
\mathrm{fb}$ ($12\leavevmode\nobreak\ \mathrm{fb}$). In addition, the authors
did a dedicated simulation of a top-associated Higgs production channel that
is strongly influenced by the bottom quark content in the proton and the
modeling of the proton’s gluon content (which can lead to $g^{*}\to
b\overline{b}$ splitting). They estimated the cross-section for this process
(Fig. 2) to be $0.0158\leavevmode\nobreak\ \mbox{\,fb}$ (also at zero
polarization).

The authors of the study also estimated the uncertainties associated with
these production mechanisms owing to the scale and the combined PDF and
$\alpha_{s}$ uncertainties. They estimated the scale uncertainty in the CC and
NC DIS processes to be at the level of 6% and the combined PDF and
$\alpha_{s}$ uncertainty to be at the level of 1%. For the top-Higgs
associated production mechanism, these were estimated to be at the level of
$1-2\%$ and $14\%$, respectively.

In this note, we reproduce some of the key results in Ref. Acosta  _et al._
(2022) and explore in more detail the effect of PDFs on various production
mechanism (especially as regards the modeling of heavy flavor content in the
proton) as well as a dedicated study of the $H\to\overline{c}$ final state.
All simulations for this study were run using MadGraph Alwall  _et al._
(2014); Frederix  _et al._ (2018), Pythia 8.306 Bierlich  _et al._ (2022);
Sjostrand  _et al._ (2006), and Delphes 3 de Favereau  _et al._ (2014).

To facilitate the simulation of lepton-hadron collisions, we implement a
custom calculation of $Q^{2}$ that utilizes the initial- and final-state
leptons to ascertain the momentum transfer-squared to the proton. This is
documented in Appendix B and is based on conversations with some of the
authors of Ref. Acosta  _et al._ (2022). In addition, to facilitate the
hadronization stage we employ a newer model of color flow for the beam remnant
that is implemented in Pythia8, also documented in the appendix.

$\mu^{-}$$\nu_{\mu}$$H^{0}$$q$$p^{+}$$W^{*-}$$W^{*+}$$q^{\prime}$

$\mu^{-}$$\mu^{-}$$H^{0}$$q$$p^{+}$$Z^{*0}$$Z^{*0}$$q$

Figure 1:  Leading-order diagram for the production of a single final-state
Higgs particle in charged-current (left) and neutral-current (right) muon-
proton DIS. In both cases a particle jet is expected both from the proton
remnant and from the scattered quark that participated in the DIS interaction.

$\mu^{-}$$\nu_{\mu}$$H^{0}$$\overline{b}$$p^{+}$$W^{*-}$$W^{*+}$$\overline{t}$

Figure 2:  Leading-order diagram for the production of a single final-state
Higgs particle in top-Higgs associated production via charged-current muon-
proton DIS.

##  II Simulation and Modelling

###  II.1 Cross-Section Estimations

The Snowmass MuIC white paper Acosta  _et al._ (2022) considered Higgs
production using the MadGraph model sm-no_b_mass, which implements a leading-
order (LO) version of the SM Lagrangian and sets the b-quark mass to zero in
the parton-level interactions, effectively implementing a "5-flavor scheme."
The Higgs Effective Field Theory (HEFT) is also provided in MadGraph, and the
FeynRules package provides three alternative SM implementations that are
stated to be NLO in QCD corrections Duhr  _et al._ . The latter models come in
three forms, denoted SM_NLO, SM5F_NLO, and SM4F_NLO for "…QCD corrections with
the top and bottom yukawa [sic] renormalised in the on-shell scheme," "…QCD
corrections in the 5 flavour scheme (massless b but non-vanishing bottom
yukawa [sic] renormalised in the MSbar)," and "…QCD corrections in the 4
flavour scheme (massive b but bottom yukawa [sic] renormalised in the MSbar),"
respectively Duhr  _et al._ .

For each model, 100,000 events were generated. The PDF used for each was
CT18NNLO, unless otherwise specified. In all cases, the proton was redefined
in MadGraph to contain b-quarks as well, e.g. define p = b b~ p. We provide
additional details of the simulation in Appendix B. Results are in Table 1.

Regarding the $tH$ production mechanism it is clear that a MadGraph model
employing a 5-flavor scheme is essential to more accurately estimating the
cross-section in a MuIC. At the level of uncertainty achievable from
systematic on the scale (${}^{+0.083\%}_{-1.7\%}$), central scheme
(${}^{+0.0\%}_{-1.2\%}$), and PDF (${}^{+19\%}_{-11\%}$) there is no clear
difference between sm-no_b_mass and SM5F_NLO-full. It is apparent that the
level of certainty on the b-quark contribution to the proton is a strong
influence on the precision with which this production mechanism can be
estimated in advance, as PDF uncertainty dominates this calculation.

Model Name | Cross-Section [fb]
---|---
| $p^{+}\mu^{-}\to H^{0}\nu_{\mu}+\textrm{jet}$ | $p^{+}\mu^{-}\to H^{0}\mu^{-}+\textrm{jet}$ | $p^{+}\mu^{-}\to tH^{0}\nu_{\mu}$
sm-full | $65.6$ | $11.5$ | $0.000117$
sm-no_b_mass | $65.5$ | $11.7$ | $0.0240$
heft-full | $65.5$ | $11.7$ | N/A
SM_NLO-full | $64.0$ | $11.2$ | N/A
SM4F_NLO-full | $64.0$ | $11.2$ | N/A
SM5F_NLO-full | $64.0$ | $11.4$ | $0.0238$
Table 1: Cross-sections determined using MadGraph (MG5_aMC@NLO) and a range of
available Standard Model implementations available in the framework or from
FeynRules. The PDF set used for all cases is CT18NNLO. In all cases, the pure
statistical uncertainly on the estimated cross-section is at the level of
$5\times 10^{-2}\leavevmode\nobreak\ \mbox{\,fb}$.

###  II.2 Heavy Flavor, PDFs, and Higgs Production

We investigated the effect on production cross-section resulting from the
degree of knowledge of the flavor content of the proton (e.g., at the level of
the strange quark and above). Given the role that the bottom quark plays in
top-Higgs associated production, the inclusion of heavy flavor in both the
underlying model and in the PDF model were of particular interest, should that
mechanism be of future interest to a MuIC program.

We compared (Table 2) the estimated production cross-sections for the three
majors processes — CC DIS, NC DIS, and top-Higgs associated production — using
a range of PDF sets. We begin by simply comparing the estimates obtained when
using the reference PDF set in Ref. Acosta and Li (2022)
(PDF4LHC15_nlo_mc_pdfas) to our baseline PDF set. We find that the we
essentially reproduce the results from that original work. We observe that the
top-Higgs associated production is enhanced when utilizing the CT18NNLO PDF
set, by about 20%. We note that, within the estimated theory uncertainties
discussed earlier that this difference is not significant.

We also compare the production cross-sections using a foundational PDF fit —
CT14 Dulat  _et al._ (2016) — that is varied according to a range of
assumptions about the fitted number of flavors (NF) and the computation of the
PDFs to leading and next-to-leading orders (LO and NLO). We observe no effect
on the CC DIS production process. This is compatible with the expectation that
this mechanism is dominated by the contributions of valence and the lightest
sea quark flavors. This observation is also compatible with the effect on NC
DIS Higgs production, which shows negligible changes across the range of
assumptions.

The most interesting pattern emerges from the variation of the number of heavy
flavors that can contribute to the proton and the $tH$ mechanism. Since this
requires a bottom quark PDF, the variations with NF$<5$ yield no useful
predictions (one cannot model a proton containing a bottom quark when there is
no PDF to accommodate this hypothesis, and one cannot produce the Higgs this
way without the bottom quark). In the case where all six quark flavors are
fitted in the PDF, we observe that increasing the order of the computation of
the PDF decreases the production cross-section. This observation is
statistically valid across the different simulation samples, given the level
of uncertainty on the yield of Monte Carlo simulation events. However, in a
real data sample this difference would not be observable given the current
level of theory uncertainty that contributes to knowledge of this process.
That uncertainty would be a significant systematic uncertainty on the
interpretation of collider data.

Model Name | Production Process Cross-Section [fb]
---|---
| CC DIS | NC DIS | $tH$
CT18NNLO | $63.9$ | $11.0$ | $0.0235$
PDF4LHC15_nlo_mc_pdfas | $63.1$ | $10.9$ | $0.0196$
CT14lo | $63.6$ | $11.2$ | $0.0276$
CT14lo_NF3 | $63.8$ | $11.0$ | N/A
CT14NLO_NF3 | $63.7$ | $10.8$ | N/A
CT14lo_NF4 | $64.0$ | $11.1$ | N/A
CT14NLO_NF4 | $63.9$ | $11.2$ | N/A
CT14lo_NF6 | $63.6$ | $11.2$ | $0.0273$
CT14NLO_NF6 | $63.6$ | $11.4$ | $0.0190$
Table 2: Cross-sections determined using MG5_aMC@NLO, the Standard Model
implementation SM5F_Hcc_NLO available in the framework, and a range of PDF
sets at different orders and with different quark flavor assumptions. In all
cases, the pure statistical uncertainly on the estimated cross-section is at
the level of $2\times 10^{-5}\leavevmode\nobreak\ \mbox{\,fb}$.

###  II.3 Background Processes

It is expected that there will be at least a few key background processes in
the search for $H^{0}\to c\overline{c}$:

  1. 1.

Inclusive dijet production: while heavy flavor will be part of this, given
charm and bottom jet mis-identification rates it is expected that the much
larger production cross-section for inclusive dijets at a MuIC will lead to
contamination of the final sample of $H^{0}\to c\overline{c}$ candidates, even
if light jet suppression is very good for such algorithms. This should be
reducible by increased attention to light-jet rejection performance in any jet
identification algorithm.

  2. 2.

Charm dijet production: heavy flavor dijet production will, of course,
directly contaminate the $H^{0}\to c\overline{c}$ sample. This should be
reducible by using the kinematics of Higgs decay to discriminate from this
population of events, though of course this population will contain real charm
jets.

  3. 3.

$Z^{0}\to c\overline{c}$ events: production of the Z boson followed by decay
to charm is expected to be a main, semi-irreducible background. It’s reducible
in the sense that the Z dijet mass will be different from the Higgs dijet
mass, but given dijet mass resolution at the level of tens of GeV the key to
separating these classes of events will depend on detector resolution and jet
calibration. A MG5_aMC@NLO simulation of this process yields an expected
inclusive $Z$ production cross-section of $0.331{\rm\,pb}$ ($3.0$) for CC DIS
(NC DIS).

It is possible to simulate aspects of the above backgrounds in MadGraph and
Pythia8. Feynman diagrams for the processes utilized in MG5_aMC@NLO are shown
in Appendices A.4 and A.5. However, it is challenging to simulate all of these
background fully inclusively, as most of the events produced by these
underlying mechanisms (especially the non-resonant dijet backgrounds) tend to
result in events with extremely soft jets or other factors that make them
trivially reducible (e.g. the detector will not see many or most of the
products of the collision).

Figure 3: Event displays of a CC DIS $H^{0}\to c\overline{c}$ event (left) and
an NC DIS di-charm event (right), both of which pass all prototype selection
criteria described in Section III. Although the perspectives are slightly
different, the muon-beam (backward) direction is on the left-most side of each
image. In the signal event, the backward two jets (blue cones) are associated
with the Higgs decay. In the background event, the two most backward jets were
associated with the candidate Higgs when none is present. No reconstructed
muons are present in either event.

We adopt two approaches to simulating the di-charm background. The first is
almost fully inclusive, with only a generator-level restriction that a final
state muon satisfy $|\eta|<6$ (only in NC DIS) and that there be at least one
final-state quark where $p_{q}>5{\mathrm{\ Ge\kern-1.00006ptV}}$ (all
production mechanisms). We refer to this as "inclusive di-charm." A second
sample is generated also requiring that $m_{cc}>50\leavevmode\nobreak\
{\mathrm{\ Ge\kern-1.00006ptV}}$, similar to Ref. Acosta  _et al._ (2022). We
refer to this as "high-mass di-charm." Inclusive di-charm events are essential
in revealing the characteristics in missing energy, muon, and jet kinematics
that can be used as a first approach to rejecting di-charm events. The high-
mass di-charm simulation is essential for a higher statistics sample of this
background more likely to pass final selection criteria. Of course, as we
restrict the phase space at production we incur the cost of larger theoretical
uncertainties. We will return to the challenges of the di-charm background at
the conclusion of this note. We estimate the inclusive (high-mass) di-charm
production cross-section to be $509,000\leavevmode\nobreak\ \mbox{\,fb}$
($97,700\leavevmode\nobreak\ \mbox{\,fb}$). We simulate 200,000 collisions for
each of these samples, a yield generally at or far below what would be
expected for our target integrated luminosities (e.g., $1\mbox{\,fb}^{-1}$ or
$100\mbox{\,fb}^{-1}$).

The systematic uncertainties due to scale, $\alpha_{s}$, and PDFs are
generally larger than these statistical effects (which can be reduced merely
by simulating more events of this type). For example, the scale uncertainty is
about 20% while the central scheme uncertainty is over 100% in both cases
described above and worsens as one further restricts the phase space of the
process. As expected, the more exclusive the process the larger are some
associated theory uncertainties. Restricting the simulation from the fully
inclusive approach (e.g. no limitations on jet $p_{T}$ or $\eta$) at this
level nonetheless retains a sample of events that could be reconstructed by a
detector, albeit in a phase space that is only part of the total inclusive
rate.

The graphs in this note are generally normalized in one of two ways. When
"density" appears on the y-axis, the contents of the figure have been
normalized using the area of distribution and the width of the histogram bins.
Otherwise, we normalize the samples to their production cross-sections (and
branching fractions, where appropriate) and a target luminosity of 100
$\mbox{\,fb}^{-1}$. The density normalization allows for a comparison of shape
differences in samples and populations; the luminosity normalization allows
for a sense of how these populations will look in a future data sample.

We note that, unlike in the case of $H^{0}\to b\overline{b}$ explicitly
studied in Ref. Acosta and Li (2022), $H^{0}\to c\overline{c}$ appears to
suffer from at least two major disadvantages at the level of a generator- or
pre-selection of events:

  1. 1.

The branching fraction for $H^{0}\to c\overline{c}$ is significantly lower
than for $H^{0}\to b\overline{b}$ owing to the much smaller Yukawa coupling to
charm quarks (a factor of about 20);

  2. 2.

The probability of non-resonantly pair producing charm quarks in a high-energy
collider is much greater than that of producing a pair of bottom quarks, due
to the much smaller mass of the charm quark.

These two factors could make the study of $H^{0}\to c\overline{c}$
particularly challenging, even in this kind of collider environment. We
explore the reconstruction and selection of these events to identify the basic
challenges of this process after a modest, baseline selection.

####  II.3.1 Visualization of Events

We show DELPHES event displays of the CC DIS $H^{0}\to c\overline{c}$ process
and for the NC DIS di-charm process in Fig. 3. The events shown here pass all
selection criteria outlined in Section III and are considered extremely
"signal-like." The $H^{0}\to c\overline{c}$ event has two backward jets that
are identified as charm-induced and form a di-jet invariant mass consistent
with the Higgs particle. Other jets are present as a result of the proton beam
remnant, radiation, etc. The background event is induced by a NC DIS process
but no reconstructed muon is present. The two most backward jets are the ones
associated with the Higgs candidate. Both events have significant missing
transverse momentum. The analysis we outline later is very basic, but these
event displays may hint at topological differences in jets that may be
exploited by more advanced methods.

###  II.4 Jet Reconstruction in Detector Model

The detector model employed in this study is based on the "EIC Generic 3T"
model provided in the delphes_eic package Arratia and Sekula (2021). The model
is modified, however, in a few key ways:

  * •

The calorimeter design can be changed to try alternatives, including the
original EIC Generic calorimeter model Abdul Khalek  _et al._ (2022a), a model
similar to that proposed for the ATHENA detector concept Adam  _et al._
(2022), and a model similar to the CMS calorimeter adopted in Ref. Acosta  _et
al._ (2022). These are referred to as the "EIC Generic," "ATHENA," and
"Snowmass" concepts, respectively. We generally adopt the "Snowmass"
configuration in our studies. While it does not appear to provide the best
resolution, it provides a somewhat more conservative baseline. All calorimeter
models include an electromagnetic and hadronic calorimeter component with
identical coverage in pseudorapidity, though not identical cell granularity.

  * •

Jets can be reconstructed using only calorimeter towers ("Calorimeter-only",
or "CaloJets") or using particle flow. We allow for both kinds of jet and
missing energy reconstruction to facilitate studies of the effect of particle
flow algorithms on a future Higgs candidate reconstruction (simply referred to
henceforth as "Jets"). Our baseline, unless otherwise stated, uses particle
flow (c.f. Ref. CMS Collaboration (2009)). All jet reconstruction approaches
use the anti-$k_{T}$ Cacciari  _et al._ (2008) algorithm.

  * •

A muon efficiency model is added to represent the performance of a future,
low-angle muon spectrometer system (intended primarily for beam muon
reconstruction in, for example, neutral-current DIS).

####  II.4.1 Jet Parameter Optimization

LHC experiments employ a range of $R$-parameter choices for anti-$k_{T}$ jets.
For example, standard jets in CMS or ATLAS are typically defined with $R$
parameters of 0.5 or 0.4, respectively Khachatryan  _et al._ (2017); Aad  _et
al._ (2021a). Highly Lorentz-boosted heavy objects decaying to jets will
result in larger radius jet clusters, and the typical $R$ parameter for this
situation is 1.0 Aad  _et al._ (2021b). Substructure within a large-radius jet
can be determined by re-running jet clustering inside the object using charged
particle tracks, and typically this is done using jets with a narrower $R$
parameter (e.g. 0.2) or a sliding scale that adjusts the $R$ parameter
depending on the kinematics of the substructure jet relative to the parent
jet. It is not a priori obvious what is the right or best choice for jet
reconstruction of $H^{0}\to c\overline{c}$ at a future MuIC.

We reconstructed jets using a few different approaches and selected an $R$
parameter that balanced Higgs mass accuracy and resolution. We scanned across
the possibilities $R=0.4-1.0$ in steps of $0.2$. To avoid over-constraining
the choice to just one possible heavy flavor, we looked at the quality of both
di-charm and di-bottom reconstruction. Results are shown in Fig. 4.

Figure 4: Number of bottom (upper) and charm (lower) quarks produced from a
single collision event. All events are derived from either $H^{0}\to
b\overline{b}$ and $H^{0}\to c\overline{c}$ simulation, respectively.

We hypothesized that a baseline means of selecting the best $R$ parameter was
to simply observe the rate at which $H^{0}\to b\overline{b}$ and $H^{0}\to
c\overline{c}$ simulated events were reconstructed as having two heavy flavor
jets. We recognize that the production of charm quarks in the proton remnant
affects this hypothesis, as a fraction (at the level of a few percent) of
$H^{0}\to c\overline{c}$ events will contain at least 3 charm jets.
Nevertheless, most events would be expected to contain only two such heavy
flavor jets.

For iterations $R=0.4$, $R=0.6$, $R=0.8$, and $R=1.0$, events with two b-jets
occurred with a frequency of $87\%$, $90\%$, $91\%$, and $91\%$, respectively.
The statistical uncertainty on these rates is smaller than the reported
precision. Di-charm jet production occurred under the same parameters with
frequencies of $82\%$, $84\%$, $84\%$, and $85\%$. We observe that for both
heavy flavor possibilities there is a clear few percent-level rise in the rate
of di-jet reconstruction as we move from $R=0.4$ to $0.6$. However, after
that, the rise is less dramatic and we generally observe that the
reconstruction rate of heavy flavor di-jets is flatter for larger jet $R$
parameters. We select a point on the trend that is past the early rise and
early in the plateau. Thus, $R=0.6$ was chosen as the most optimal $R$ value
for all future work in this note. A comparison of light and charm jets in CC
DIS $H^{0}\to c\overline{c}$ simulation is shown in Fig. 5.

####  II.4.2 Jet Resolution

In the context of our selected jet reconstruction, we can explore the fast-
simulation-level expected quality of jets used in the reconstruction of Higgs
particles. In particular, the resolution of a jet variable $X$ is defined as
$(X_{reco}-X_{true})/X_{true}$, where "reco" labels the variable determined
purely from a reconstruction-based approach and "true" labels the variable as
determined from a Monte Carlo generator-level set of information. Generator-
level jets are defined as jets reconstructed just as they would be from
calorimeter information, but instead employing all long-lived stable particles
from the simulation before detector-level effects are simulated.

Figure 5: Jet $p_{T}$ spectrum is shown from CC DIS $H^{0}\to c\overline{c}$
simulation, distinguishing true charm-initiated jets and other kinds of jets
("light jets").

The typical jet energy resolution at a modern experiment, such at ATLAS,
varies with the kinematics (e.g. $p_{T}$) of the jet object. ATLAS obtains a
jet energy resolution of about 24% at $p_{T}=20\leavevmode\nobreak\ {\mathrm{\
Ge\kern-1.00006ptV}}$ that improves to about 6% at $300\leavevmode\nobreak\
{\mathrm{\ Ge\kern-1.00006ptV}}$. The CC DIS production of $H^{0}\to
c\overline{c}$ in our simulation results in charm jets with a typical
$p_{T}$in the range of $30-40\leavevmode\nobreak\ {\mathrm{\
Ge\kern-1.00006ptV}}$. The energy resolution obtained in our fast simulation
is shown in Fig. 6. We estimate it for both CaloJets and Jets. We can clearly
see the positive effect of employing particle flow, as this improves
significantly the resolution of the energy and its accuracy to the true value
from the generator-level jet. The root mean-square (RMS) of the particle flow
jet energy resolution is about 21%, consistent with typical LHC jet energy
resolution measurements for a similar $p_{T}$ spectrum.

Figure 6: Jet energy resolution is shown for calorimeter-only jets and
particle flow-based jets.

###  II.5 Higgs Jets and Kinematics

We compare the $p_{T}$, azimuthal angle ($\phi$), and pseudorapidity ($\eta$)
between generator-level jets and reconstructed jets. As a basic check, we
observed that the distribution of $\phi$ was consistently isotropic, as
expected given both a symmetric detector model and a zero crossing angle for
the muon and proton beams.

We observe that the jets in $H^{0}\to c\overline{c}$ CC DIS events are
typically produced in the backward (muon beam) direction, which would be
expected from the asymmetric beam kinematics (Fig. 7). The proton remnant
would be expected to travel in the forward direction; indeed, separating charm
jets from light jets, we observe that the light-jet component of the $\eta$
distribution more often contains a population of jets in the forward
direction. Charm jets, primarily resulting from Higgs decay, tend to travel
forward. The jet density in $\eta$ peaks around $\eta=-1.5$.

Tracking and calorimetry in a future MuIC detector will need to provide
sufficient coverage and pixellation in the backward direction. We show
momentum and polar angle coverage plots in Fig. 8, emphasizing the typical
region populated by generator-level jets in a collider of this configuration
(275x960 ${\mathrm{\ Ge\kern-1.00006ptV}}^{2}$ protons on muons). The majority
of decay products fall at $\eta=[-2.5,0]$. Taking this into account with the
specified jet parameter $R=0.6$, a MuIC collider detector will require
calorimeter and tracking coverage out to at least $|\eta|=4.0$, with emphasis
for jet reconstruction in the direction of the muon beam.

Figure 7: $\eta$ distribution is shown for generator-level jets from $H^{0}\to
c\overline{c}$ simulation subdivided into charm and light-jet populations.

Figure 8: Coverage plot for all generator-level jets (top) and for generator-
level charm jets (bottom) illustrating the relationship between the jet
momentum $p$ as a function of polar angle $\theta$.

We have observed that the jet energy resolution for charm jets is consistent,
in this fast simulation, with typical resolutions at existing experiments. We
investigate also the mass reconstruction of the dijet system. This is expected
to be a key discriminant in any Higgs reconstruction. When comparing jets and
CaloJets (Fig. 9) it is clear that particle flow provides a more accurate and
precise mass reconstruction.

Figure 9: Comparison of Particle Flow and Calorimeter Jets for $H\to c\bar{c}$
Higgs reconstruction.

Our baseline configuration results in a median (peak) dijet mass of
$111\leavevmode\nobreak\ {\mathrm{\ Ge\kern-1.00006ptV}}$
($117\leavevmode\nobreak\ {\mathrm{\ Ge\kern-1.00006ptV}}$ for truth-matched
charm jets in the CC DIS $H^{0}\to c\overline{c}$ simulation. The standard
deviation of the mass peak is about $21\leavevmode\nobreak\ {\mathrm{\
Ge\kern-1.00006ptV}}$. We would expect to thus be able to separate the
$H^{0}\to c\overline{c}$ mass peak from a $Z\to c\overline{c}$ background mass
peak at the level of about $1\leavevmode\nobreak\ \sigma$; however, this is
without employing standard methods for improving the calibration of jets, such
as jet-energy scale and muon-in-jet corrections (which compensate for jet mis-
reconstruction and energy loss due to semi-leptonic heavy quark decays). These
results are consistent with those obtained using $H\to b\bar{b}$ decay in Ref.
Acosta  _et al._ (2022).

Figure 10: Dijet invariant mass distribution for CCDIS $H\to c\bar{c}$
compared between alternative calorimeter models.

We also compare the mass reconstruction using the different calorimeter models
described earlier. We see no particular advantage in this particular quantity
in choosing one calorimeter model over the others (Fig. 9). The importance of
calorimeter granularity and coverage will be more apparant in the study of
missing momentum.

We adopt a charm-tagging performance for jets similar to that in Ref. Aad  _et
al._ (2022), with 27%, 8%, and 1.6% c-jet, b-jet, and light-jet efficiency,
respectively. This is taken as independent of the jet momentum, though in
reality we recognize that there would be angular and momentum effects that
would need to be folded into the development and performance assessment of any
charm-tagging algorithm.

###  II.6 Muon Identification

The production of di-charm final states, from either resonant or non-resonant
mechanisms, will occur more often in NC DIS than CC DIS. However, Higgs
production is dominated by the CC DIS mechanism. Rejecting the significant
production of charm through NC DIS will require attention to muon and missing
momentum reconstruction.

A backward muon with high energy will be a potential hallmark of the NC DIS
process. At the generator level, muon momentum and pseudorapidity both show
the characteristics expected of these "tagging muons" — those scattered from
the original beam-beam interaction (Fig. 11). Tagging muons from NC DIS
interactions are characteristically in the direction of the muon beam
($\eta<-3$) and very high in momentum, typically with more than 10% of the
lepton beam’s original energy. Final-state muons, resulting from the decay of
heavy particles, have a much softer momentum spectrum and tend to be much more
central in the detector (following more the angular distribution of jets).

Figure 11: Generator-level characteristics of muons from the beam-beam
interaction (tagging) or from final-state decays of heavy particles (FS).

Identifying these muons will require the deployment of a far-backward muon
system that can detect high-momentum minimum ionizing particles all the way
back to $\eta\approx 6$ or lower. LHCb is an excellent example of an
experiment with a low-angle muon spectrometer that could serve as a model for
a future MuIC detector subsystem with this capability Alves  _et al._ (2013);
Anderlini  _et al._ (2020). We assume a conservative LHCb-like performance in
the region $\eta=[-5.0,4.0]$ LHCb Collaboration . This results in 97% (3%)
muon (pion) efficiency throughout the region of acceptance for muons.

###  II.7 Missing Momentum and Energy in CC DIS

Missing momentum and energy are expected to be key discriminators in the study
of $H^{0}\to c\overline{c}$ decay, especially in distinguishing CC and NC DIS
production. Missing energy reconstruction in DELPHES is performed using the
energy flow objects (tracks and clusters). Missing transverse four-momentum is
computed simply using

$\vec{p}^{miss}_{T}=-\sum_{i}\vec{p}_{i}$ (1)

where $\vec{p}_{i}$ is the momentum of an energy flow object. An example of
the baseline resolution
($(p^{miss,reco}_{T}-p^{miss,true}_{T})/p^{miss,true}_{T}$ is shown in Fig.
12, comparing $\mathrm{E_{T}^{miss}}$ calculated using raw calorimeter objects
and using particle flow objects. As expected, particle flow-based
reconstruction provides a superior resolution and fidelity to the true missing
transverse momentum.

Figure 12: MET Resolution for calorimeter only and particle flow. Events are
from CC DIS signal simulation.

As an assessment for future consideration of different calorimeter granularity
designs, we also calculate the $\mathrm{E_{T}^{miss}}$ for the alternative
calorimeter models. A comparison in CC DIS signal simulation is shown in Fig.
13. These include the ATHENA-like model, a baseline EIC detector model, and
what we refer to as the Snowmass calorimeter (our baseline for this note).
Only particle flow is utilized. We observe that the Snowmass granularity and
coverage offers the overall largest resolution of the three designs. For
example, in this fast simulation the Snowmass particle-flow-based
$\mathrm{E_{T}^{miss}}$ resolution, determined by the standard deviation of
the distribution, is about 33%. In comparison, the resolution in the EIC and
ATHENA models (which are similar in granularity) is about 29%. This suggests
both that our results from this note will be conservative and that designs
with more granularity would benefit future experiments that need stronger
$\mathrm{E_{T}^{miss}}$ resolution.

Figure 13: Comparison of the MET Resolution for the different calorimeter
models in $H^{0}\to c\overline{c}$ deay.

###  II.8 Missing Momentum and Energy in NC DIS

Our fast simulations of CC DIS signal and di-charm background suggests there
is little benefit in separating these two populations based solely on the
shape of the $\mathrm{E_{T}^{miss}}$ distribution. However, the distinction
between CC DIS signal and NC DIS di-charm background is far more significant.
NC DIS events maintain the beam muon and produce little or no real
$\mathrm{E_{T}^{miss}}$, except from final-state physics involving one or more
low-energy neutrinos or from mis-reconstruction of the event. The
$\mathrm{E_{T}^{miss}}$ in these two populations is compared in Fig. 14. As
expected, $\mathrm{E_{T}^{miss}}$ peaks more sharply toward zero in the NC DIS
di-charm background, while it extends to typically much larger values in CC
DIS signal simulation.

Figure 14: Comparison of MET between di-charm background and $H^{0}\to
c\overline{c}$ signal in both CC DIS and NC DIS production modes.

We hypothesized that there might be additional benefits in looking at the
angular distribution of the MET relative to the di-jet system that forms the
Higgs candidate. We recognize that in a collision where only the Higgs and the
recoiling neutrino are produced, the $\mathrm{E_{T}^{miss}}$ is entirely
defined as being opposite the sum of the jet particles and thus is 100%
correlated with the di-jet system. However, in collisions where part or all of
the beam remnant is present, or where additional radiation or other particles
appear in the detector, the correlation will be weakened. Naively, we
anticipated that in CC DIS Higgs signal events we should expect to see a
strong peak in the azimuthal angle difference between the
$\mathrm{E_{T}^{miss}}$ and the Higgs candidate around $\pm\pi$. In NC DIS di-
charm events, where there is no real significant source of
$\mathrm{E_{T}^{miss}}$ and the possibility of a muon in the detector, we
anticipated that there would be no or a much weaker correlation.

We show the results of a basic study of this in Fig. 15. As predicted, the
difference in the missing energy angle for CC DIS signal peaked at $\pm\pi$,
while the NC DIS di-charm background was somewhat more diffuse. Nevertheless,
we still observe strong $\pm\pi$ values for this angular difference in the
transverse plane. We do not exploit this possible distinction in angular
correlations for the remainder of this note, but we point out that such a
variable (with additional refinement) may be of future use in an analysis that
targets extreme separation between CC DIS signal and NC DIS di-charm
background.

Figure 15: A comparison of $\Delta\Phi_{\mathrm{MET,jj}}$ vs
$\mathrm{E_{T}^{miss}}$ for (top) CC DIS signal and (bottom) NC DIS di-charm
background simulation.

##  III A Prototype Data Analysis for $H^{0}\to c\overline{c}$

We present here the outcome of a very basic event selection and candidate
refinement. This is meant to provide a conservative view of the potential for
selecting $H^{0}\to c\overline{c}$ candidates, with emphasis on CC DIS
production (the leading cross-section). We apply a basic optimization
procedure to set criteria at various stages of the analysis, employing the
method of varying a criterion to maximize the following figure of merit Cowan
_et al._ (2011)

$\sqrt{2\left((s+b)\cdot\ln(1+s/b)-s\right)}$ (2)

where $s$ ($b$) is the number of events in the signal (background) class. This
quantity is the asymptotic formula for significance in the limit of large
backgrounds, which is a reasonable assumption in this study. The class
definitions vary depending on the goal of the selection. We conservatively
round any criteria determined from this procedure to avoid over-interpretation
of the optimization results on current simulation approaches and statistical
limitations (e.g., $66{\mathrm{\ Ge\kern-1.00006ptV}}$ becomes $70{\mathrm{\
Ge\kern-1.00006ptV}}$ on an energy criterion).

The most basic event characteristics are that the muon-proton collisions,
through the CC DIS process, will produce a significant and real (e.g. due to
at least one high-momentum neutrino) quantity of missing momentum (Fig. 16).
In addition, the event should not contain evidence of a high-momentum muon at
low angle in the backward direction; that signature is a hallmark of the NC
DIS process (Fig. 17). Given the lower cross-section for Higgs production in
NC DIS and the substantially increased rate of production of non-resonant di-
charm and $Z^{0}\to c\overline{c}$ in NC DIS, our goal is to minimize
contamination from NC DIS events. We define the signal class as containing
luminosity-weighted CC DIS $H^{0}\to c\overline{c}$ events and the background
class to contain NC DIS $Z^{0}\to c\overline{c}$ and inclusive di-charm
events. We find that $\mathrm{E_{T}^{miss}}>50\leavevmode\nobreak\ {\mathrm{\
Ge\kern-1.00006ptV}}$ and rejecting events with a backward (bwd) muon
satisfying $p^{bwd}_{\mu}>60{\mathrm{\ Ge\kern-1.00006ptV}}$ optimizes our
criteria. The optimizations are not performed accounting for correlations in
the two variables and are only one-sided in each variable (a minimum in
$\mathrm{E_{T}^{miss}}$ and maximum in $p_{\mu}$). We refer to the selection
on MET and the muon veto as the "Event Tag."

Figure 16: $\mathrm{E_{T}^{miss}}$ with no other selection criteria applied for a target integrated luminosity of $100\mbox{\,fb}^{-1}$. Figure 17: Muon momentum with no other selection criteria applied for a target integrated luminosity of $100\mbox{\,fb}^{-1}$. |  | Yield ($100\leavevmode\nobreak\ \mbox{\,fb}^{-1}$)/Total Efficiency [%]/Relative Efficiency [%]
---|---|---
Selection | Criteria | $H^{0}\to c\overline{c}$ |  $Z^{0}\to c\overline{c}$ (CC DIS) |  $Z^{0}\to c\overline{c}$ (NC DIS) | Incl. $c\overline{c}$ (CC DIS) | Incl. $c\overline{c}$ (NC DIS)
No Selection | None | 188/-/- | 2800/-/- | 4500/-/- | 15000/-/- |  $5\times 10^{7}$/-/-
Event Tag |  $\mathrm{E_{T}^{miss}}>50\leavevmode\nobreak\ {\mathrm{\ Ge\kern-1.00006ptV}}$ $p_{\mu}^{bwd}<60\leavevmode\nobreak\ {\mathrm{\ Ge\kern-1.00006ptV}}$ | 79/42/42 | 1,900/68/68 | 25/0.6/0.6 | 7,000/45/45 | 51,000/0.1/0.1
Jets |  $|\eta_{j}|<4.0$ $p_{T}^{j}>20\leavevmode\nobreak\ {\mathrm{\ Ge\kern-1.00006ptV}}$ | 76/40/96 | 1,800/65/96 | 23/0.5/93 | 3,700/24/53 | 46,000/0.09/91
Charm Dijet |  $\eta_{j}<0$ 2 Charm Tagged | 4/2/5 | 96/3.5/5 | 0.9/0.02/4 | 56/0.4/2 | 1500/0.003/3
Table 3: Selection stages, criteria, and efficiency on various samples. For
each sample, and for each selection category, we report the number of events
(scaled to 100$\mbox{\,fb}^{-1}$ of collision data), the cumulative efficiency
(in percent) of the selection, and the relative efficiency (in percent)
compared to the previous selection category. Given the limited statistics
available in the NC DIS dicharm background, we note that only 6 unweighted
events survive all selection criteria; the resulting prediction for that
sample therefore has a 41% MC simulation statistical uncertainty.

Candidate events should then contain at least two jets within the fiducial
tracking volume and with a sufficient minimum $p_{T}$ to be compatible with
the decay of a heavy resonance. This is referred to as the "Jets Selection."
Among those jets, there should be at least two charm-tagged jets. These jets
should be located primarily in the backward direction. Constraints that
emphasize the selection of events consistent with these expectations are
referred to as "Charm Dijet Selection."

Finally, we expect the dijet system to form an invariant mass consistent with
the Higgs particle. Constraints on the event that emphasize specifically the
Higgs signature are referred to as the "Higgs Selection."

The selection criteria and their efficiency on the three main samples —
$H^{0}\to c\overline{c}$, $Z^{0}\to c\overline{c}$, and di-charm — are shown
in Table 3. The samples are weighted to a target luminosity of
$100\leavevmode\nobreak\ \mbox{\,fb}^{-1}$. We note that the NC DIS di-charm
background was the most difficult to simulate in sufficient quantity for this
basic study. Whether using the more inclusive sample or the sample where
$m_{cc}>50{\mathrm{\ Ge\kern-1.00006ptV}}$ at generator-level, we obtain
almost the same number of raw events passing all selection criteria: about 6.
This means any weighted prediction using these samples have about a 40% MC
simulation statistical uncertainty and can only be used to guide thinking at
this stage.

We also simulated NC DIS $H^{0}\to c\overline{c}$ production (not shown in
Table 3), but as the analysis was designed to eliminate NC DIS production this
resulted in a negligible contribution to the overall signal yield (about 1% of
the $H^{0}\to c\overline{c}$ yield). We note that where we expect consistent
performance in selection criteria between samples, we do observe it, giving
confidence to the underlying simulation models. For example, we expect the
event tag criteria to substantially eliminate NC DIS; they do, maintaining CC
DIS in any sample at a high level of efficiency while suppressing NC DIS-based
samples. We expect that charm identification will perform with about the same
efficiency on the $H$ and $Z$ simulation samples, since each contains high-
momentum real charm-initiated jets. Indeed, we see about the same performance
in both samples (a 5% di-charm efficiency).

Based on our simulations, the most promising target for a future MuIC
$H^{0}\to c\overline{c}$ search would be to emphasize the CC DIS production
mechanism. This conclusion is based solely on the challenge posed in rejection
NC DIS di-charm background from this analysis. Missing momentum and muon
identification will be essential ingredients in this effort; the performance
on these two physical observables, in turn, is defined by the quality of the
calorimeter (and tracking, for particle flow) as well as the coverage of a
backward muon system dedicated to beam-muon tagging. Our approach has utilized
a calorimeter of reduced granularity compared to alternative models emphasized
for the EIC detectors and is therefore expected to provide a much more
conservative performance on MET than we think is achievable using the kind of
calorimeter designs anticipated for EIC experiments.

Proposed EIC detectors do not emphasize muon reconstruction with low-angle
coverage, so the deployment of such a dedicated muon system will be necessary
for the MuIC. We see from Table 3 that the NC DIS di-charm background is of
particular concern owning to its substantial cross-section in the jet fiducial
region of interest. Reducing this background will rely heavily on efficient
event tagging with a strong separation between NC and CC DIS candidate events.

We note that even at early stages of the selection the limited statistics of
the di-charm sample are apparent. It was not possible for this basic study to
simulate the target luminosity’s worth of inclusive di-charm events, forcing
us to apply large weights to these events. The effect of limited statistics
becomes greater at increasingly restrictive selection. In our figures, for any
distribution beyond the event tag stage we we utilize the high-mass di-charm
simulation. Even here, for the present study we have limited statistics
compared to the target luminosity.

Figure 18: Charm-tagged di-jet mass with all selection criteria applied for a
target integrated luminosity of $100\mbox{\,fb}^{-1}$. The NC DIS di-charm
background is not reliable owning to the low statistics of the population
surviving all selections.

We show in Fig. 18 a distribution of the charm-tagged di-jet mass. We
recognize first and foremost that it is difficult to draw conclusions about
the NC DIS di-charm background; nevertheless, the earlier stages of event
selection suggest that even at this most restricted stage one should expect
this population to dominate in the current prototype analysis. Additional
resources will be needed to simulate sufficient samples of this background and
additional approaches, such as more information and/or a correlated
multivariate analysis, will be needed to improve selection of signal and
rejection of this overwhelming background.

Beyond the NC DIS di-charm background it is clear that work will also be
needed to improve the ability to separate $Z$-boson-induced backgrounds from
the $H^{0}\to c\overline{c}$ signal. Separation of these topologies at the LHC
has made substantial progress in the past decade, and no doubt a refined
approached can be brought to the question of the viability of a future MuIC
search for $H^{0}\to c\overline{c}$. In many ways, this measurement at the
MuIC may be as challenging as was the original measurement of $H^{0}\to
b\overline{b}$ at the LHC. The accomplishment of that task within a decade of
starting the LHC program provides lessons for this future-looking issue.

##  IV Conclusions

We have studied the production of the $H^{0}\to c\overline{c}$ final state at
a future MuIC facility using a conservative baseline detector inspired by
earlier work for the Snomass community planning exercise. We have assumed
rather modest (27%) charm jet-tagging efficiency, which is generally
considered to be the minimum level of performance a modern algorithm can
manage. We have assumed a more coarse calorimeter than those generally planned
for EIC experiments, but in addition to baseline EIC-like detector
configurations we have added a low-angle LHCb-like muon system to a fast
simulation of an experiment.

We identify NC DIS non-resonant di-charm background as the leading challenge
in any MuIC search for $H^{0}\to c\overline{c}$. The second-leading background
is expected to arise from $Z^{0}\to c\overline{c}$ production. We have not
explored in this note the reality that the $b\to c$ jet mis-identification
rate, typically at the level of a 10% or so, will mean an additional
background both from $Z^{0}\to b\overline{b}$ and, more importantly, $H^{0}\to
b\overline{b}$ events. Given existing approaches to simultaneous fits of di-
jet samples for both $b$ and $c$ final states, we have confidence that should
other backgrounds be reduced this one can be resolved through the use of a
simultaneous fit of a Higgs candidate sample for both final states, including
mis-identification rates assessed on independent samples.

We comment on an element that is missing from this study and that would
necessarily be a target for future work. The charm-tagging mis-identification
rate for light jets in this study was 1.6%. Naively, the light di-jet mis-
identification rate would be than about $2.6\times 10^{-4}$. However, the
light di-jet production cross-section from NC DIS or CC DIS will be much
larger than the cross sections studied for background processes in this note.
We expect a significant challenge from light di-jet background in a future
$H^{0}\to c\overline{c}$ analysis, one that will also be especially difficult
to simulate in sufficient quantity and that might benefit from entirely data-
driven approaches in a real data analysis. We leave this as a comment for
future experimental physicists.

We conclude from this preliminary work that any future MuIC experiment must
have good calorimeter granularity and readout capability, good tracking
coverage for the purpose of both particle-flow reconstruction and jet
acceptance (especially for charm-jet tagging performance at lower angles), and
muon identification capability at low angle. Nevertheless, these are necessary
but not sufficient ingredients for ultimately selecting $H^{0}\to
c\overline{c}$ candidates in data. The challenge of background reduction,
rejection, and characterization will provide daunting if such a search is
undertaken. We caution the use of $H^{0}\to c\overline{c}$ as a motivation for
a future MuIC program. However, we advocate for future consideration of this
channel as part of a healthy MuIC physics analysis portfolio.

##  V Acknowledgements

We are extremely grateful to Darin Acosta, Osvaldo Miguel Colin, and Xunwu Zuo
for valuable conversations and insights about Monte Carlo simulation and
underlying physics issues in the production processes for the Higgs. We are
extremely grateful to Fred Olness for valuable discussions about PDFs in the
study of proton structure and Higgs production. We also acknowledge the
support of the Department of Physics at SMU especially for providing
facilities to support the research activities of the authors. J. Choi
expresses gratitude toward Highland Park High School for its encouragement of
academic excellence and especially for its Science and Technology Festival,
which resulted in becoming a part of this project. P. Ahluwahlia expresses
gratitude toward the Harmony School of Innovation for its encouragement of
academic excellence and independent research. We are grateful to the SMU
Office of Information Technology and the Center for Research Computation at
SMU for access to and support on the ManeFrame II supercomputer cluster.

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##  Appendix A MG5_aMC@NLO Feynman Diagrams

This appendix contains the Feynman diagrams that are employed by MG5_aMC@NLO
in the calculation of the matrix element, and thus the cross-section, for
Higgs production and background processes of interest to the effort described
in this note.

The following is the key for the particles in the diagrams:

  * •

Quarks and gluons are represented by their usual lower-case letters (u, d, c,
s, b, t, g);

  * •

Anti-quarks are denoted by the placement of a tilde next to the quark letter;

  * •

Photons are denoted by $a$;

  * •

Z and W bosons are denoted by lower-case versions of their letter names.

###  A.1 Charged Current DIS Vector-Boson Fusion (SM5F_NLO)

Figure 19: Diagrams used in the calculation of the CC DIS VBF process
(SM5F_NLO).

Figure 20: Diagrams used in the calculation of the CC DIS VBF process
(SM5F_NLO).

###  A.2 Neutral Current DIS Vector-Boson Fusion (SM5F_NLO)

Figure 21: Diagrams used in the calculation of the NC DIS VBF process
(SM5F_NLO).

Figure 22: Diagrams used in the calculation of the CC DIS VBF process
(SM5F_NLO).

###  A.3 top-Higgs Vector-Boson Fusion (SM5F_NLO)

Figure 23: Diagrams used in the calculation of the top-Higgs DIS VBF process
(SM5F_NLO).

###  A.4 Neutral-Current DIS Di-Charm Production (No Higgs Resonance)

Figure 24: Diagrams used in the calculation of the NC DIS di-charm (non-
resonant) process (SM5F_NLO). The processes emphasized here are induced by up
quarks in the proton. Figure 25: Diagrams used in the calculation of the NC
DIS di-charm (non-resonant) process (SM5F_NLO). The processes emphasized here
are induced by up quarks in the proton. Figure 26: Diagrams used in the
calculation of the NC DIS di-charm (non-resonant) process (SM5F_NLO). The
processes emphasized here are induced by down quarks in the proton. Figure 27:
Diagrams used in the calculation of the NC DIS di-charm (non-resonant) process
(SM5F_NLO). The processes emphasized here are induced by down quarks in the
proton. Figure 28: Diagrams used in the calculation of the NC DIS di-charm
(non-resonant) process (SM5F_NLO). The processes emphasized here are induced
by charm quarks in the proton. Figure 29: Diagrams used in the calculation of
the NC DIS di-charm (non-resonant) process (SM5F_NLO). The processes
emphasized here are induced by charm quarks in the proton. Figure 30: Diagrams
used in the calculation of the NC DIS di-charm (non-resonant) process
(SM5F_NLO). The processes emphasized here are induced by gluons in the proton.

###  A.5 Charged-Current DIS Di-Charm Production (No Higgs Resonance)

Figure 31: Diagrams used in the calculation of the CC DIS di-charm (non-
resonant) process (SM5F_NLO). The processes emphasized here are induced by up
quarks in the proton. Figure 32: Diagrams used in the calculation of the CC
DIS di-charm (non-resonant) process (SM5F_NLO). The processes emphasized here
are induced by up quarks in the proton. Figure 33: Diagrams used in the
calculation of the CC DIS di-charm (non-resonant) process (SM5F_NLO). The
processes emphasized here are induced by down quarks in the proton. Figure 34:
Diagrams used in the calculation of the CC DIS di-charm (non-resonant) process
(SM5F_NLO). The processes emphasized here are induced by down quarks in the
proton. Figure 35: Diagrams used in the calculation of the CC DIS di-charm
(non-resonant) process (SM5F_NLO). The processes emphasized here are induced
by strange quarks in the proton. Figure 36: Diagrams used in the calculation
of the CC DIS di-charm (non-resonant) process (SM5F_NLO). The processes
emphasized here are induced by strange quarks in the proton. Figure 37:
Diagrams used in the calculation of the CC DIS di-charm (non-resonant) process
(SM5F_NLO). The processes emphasized here are induced by charm quarks in the
proton. Figure 38: Diagrams used in the calculation of the CC DIS di-charm
(non-resonant) process (SM5F_NLO). The processes emphasized here are induced
by charm quarks in the proton.

##  Appendix B MG5_aMC@NLO and Pythia8 Configurations

We here document, for the benefit of others who may wish to conduct similar
simulations, the configuration of the matrix-element-level and hadronization-
level simulation tools. All of these are achieved through the use of the
MG5_aMC@NLO interface, and the following code examples are provided assuming
all simulation is run through this interface.

###  B.1 Generator-level configuration

The following code block illustrates the configuration of a proton-muon
collider with the goal of producing the Higgs particle through CC DIS.

⬇

set group_subprocesses False

define p = p b b~

define j = j b b~

import model SM5F_NLO

generate p mu- > j vm h

output pmu_275x960_ccdis_vbf_h-SM5F_NLO

The order of the initial- and final-state particles is important to the user-
defined $Q^{2}$ calculation, defined below, as the calculation expects the
initial lepton and the final lepton to have fixed positions in the generator
string (e.g. the second and fourth positions).

The above configuration separates the subprocesses that result in the same
final state; this is not observed to be necessary during signal simulation,
but essential to the stability of the background simulation, which involves
many more processes. We follow the procedure of Ref. Acosta  _et al._ (2022)
and add bottom quark flavor to the proton definition, allowing it also to be
produced in any explicit jet in the final state (e.g. through a flavor-
changing or flavor-preserving process). This allows bottom quarks to be
controlled by the PDF.

The follow generate strings illustrate how we produced the Higgs also through
NC DIS, top-Higgs production, and how we produced di-charm background.

⬇

# NC DIS Higgs Production

generate p mu- > j mu- h

# CC DIS di-charm Production, no Higgs

generate p mu- > j vm c c~ /h

# NC DIS di-charm Production, no Higgs

generate p mu- > j mu- c c~ /h

###  B.2 Production-level configuration

The following code block illustrates the production-level configuration that
would be executed to make a larger number of simulated events, having defined
the generator-level process. We employ the CT18NNLO PDF, which is LHAPDF ID
14000.

⬇

launch pmu_275x960_ccdis_vbf_h-SM5F_NLO

shower=Pythia8

0

set nevents 20000

set ebeam1 275

set ebeam2 960

set polbeam1 0

set polbeam2 0

set pdlabel1 lhapdf

set lhaid 14000

set ptj 5

set ptl 0

set ptjmax -1

set ptlmax -1

set pt_min_pdg {}

set pt_max_pdg {}

set etaj -1

set etal -1

set etalmin 0

set eta_min_pdg {}

set eta_max_pdg {}

set mxx_min_pdg {}

set drjl 0

set drjlmax -1

set ptheavy 0

set maxjetflavor 5

set dynamical_scale_choice 0

#set systematics_program none

The dynamical scale choice enforces the user-level definition we provide
below. The systematic error computation can be disabled if all one needs are
generated events (e.g. if one is not interested in a production cycle in
knowing the theory uncertainties on the cross-section).

###  B.3 $Q^{2}$ definition

We modify the file in the output folder for each process named
SubProcesses/setscales.f and define the following user-defined scale:

⬇

rscale = max(0d0,-sumdot(p(0,2),p(0,6),-1d0))

This assumes the the intial and final-state particle are the second and the

###  B.4 QCD color flow model in Pythia8

We also modify the color flow model for the beam remnant since the lepton-
hadron collisions have been observed the pose a challenge to the default
model. Specifically, we modify the MG5_aMC@NLO pythia8_card_default.dat for
each simulation process and add the following lines at the end:

⬇

BeamRemnants:remnantMode = 1

ColourReconnection:mode = 1