aainstitutetext: School of Physics, Beijing Institute of Technology, Beijing,
100081, Chinabbinstitutetext: School of Physics Sciences, University of
Chinese Academy of Sciences, Beijing 100039, Chinaccinstitutetext: Center for
Future High Energy Physics, Institute of High Energy Physics, Chinese Academy
of Sciences, Beijing 100039, Chinaddinstitutetext: Dipartimento di Matematica
e Fisica, Università di Roma Tre and
INFN, sezione di Roma Tre, I-00146 Rome, Italyeeinstitutetext: Deutsches
Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany

# Constraining rare B decays by $\mu^{+}\mu^{-}\to tc$ at future lepton
colliders

Sichun Sun b,c Qi-Shu Yan d Xiaoran Zhao c,e Zhijie Zhao<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Motivated by the recent rare B decays measurements, we study the matching
procedure of operators $O_{9},O_{10}$ in the low energy effective Hamiltonian
and operators in the Standard Model effective theory (SMEFT). It is noticed
that there are more related operators in the SMEFT whose coefficients can not
be determined only from the low-energy data from B physics. We demonstrate how
to determine these coefficients with some new physics models, like
$Z^{\prime}$ model and leptoquark models, and then consider how to probe these
operators of SMEFT at high energy by using the process $\mu^{+}\mu^{-}\to tc$
at future muon colliders, which can provide complementary information except
for $\mu^{+}\mu^{-}\to bs$ on the underlying models which lead to rare B decay
processes. We perform a Monte Carlo study (a hadron level analysis) to show
how to separate the signal events from the SM background events and estimate
the sensitivity to the Wilson coefficients for different models.

##  1 Introduction

Searching for new physics is the prime target of both the high energy frontier
and high precision frontier. In the rare decay of B mesons, long-standing
discrepancies were reported between the Standard Model predictions and
experimental measurements, with a hint of non-lepton flavor universality(LFU),
especially in the muon-related final states. These hints are observed in a
$B\rightarrow K\mu^{+}\mu^{-}$, $B_{s}\rightarrow\phi\mu^{+}\mu^{-}$,
$B_{s}\rightarrow\mu^{+}\mu^{-}$ and angular distribution of $B\rightarrow
K^{*}\mu^{+}\mu^{-}$LHCb:2021awg ; LHCb:2021vsc ; LHCb:2017rmj ; ATLAS:2018cur
; CMS:2019bbr . For the LFU violation, the hints were reported by
LHCbLHCb:2021trn ; LHCb:2019hip ; LHCb:2017avl ; LHCb:2015svh ; LHCb:2020lmf ;
LHCb:2020gog  in the ratio

$\displaystyle R_{K}=\frac{BR(B\rightarrow K\mu^{+}\mu^{-})}{BR(B\rightarrow
Ke^{+}e^{-})},R_{K^{*}}=\frac{BR(B\rightarrow
K^{*}\mu^{+}\mu^{-})}{BR(B\rightarrow K^{*}e^{+}e^{-})}$ (1)

Although large hadronic uncertainties can enter in some of these absolute
branching fractions and angular observables for the Standard Model
predictions, $R_{K},R_{K^{*}},B_{s}\rightarrow\mu^{+}\mu^{-}$ are considered
relatively theoretically clean. The deviation in those measurements might lead
to indirect evidence for new physics Allanach:2015gkd ; Hiller:2014ula ;
Altmannshofer:2017fio ; Altmannshofer:2014rta ; Geng:2017svp ;
Ciuchini:2019usw ; Datta:2019zca ; Aebischer:2019mlg ; Ciuchini:2020gvn ;
Jager:2017gal . This picture has suddenly changed with the very recent
experimental updates from CMS collaboration on
BR($B_{(d,s)}\rightarrow\mu^{+}\mu^{-}$) with the full Run 2 data CMS-PAS-
BPH-21-006 , and LHCb analysis of $R_{K}$ and $R_{K*}$ with the full Run 1 and
2 dataset LHCb:2022qnv ; LHCb:2022zom . The newly reported measurements are in
agreement with the Standard Model values, however, the new physics effects can
still come into play due to both theoretical and experimental uncertainties
Ciuchini:2022wbq ; Buras:2023ldz ; Greljo:2022jac ; Das:2022mmh ;
Desai:2023jxh ; Varzielas:2023qlb ; Becirevic:2023aov . While that measurement
provides hints into new physics, the exact mechanics(models) behind those
hints are still unknown, and low-energy measurement cannot fully reveal the
nature behind that.

On the other hand, by scattering high-energy particles, collider experiments
provide unique opportunities to access underlying UV theories. Among various
current and future collidersShiltsev:2019rfl , a multi-TeV muon
colliderAime:2022flm ; MuonCollider:2022xlm  is ideal for such studies. Being
fundamental particles, the entire energy of incoming muons is available to
produce short-distance scattering rather than being spread among partons of
hadrons, and thus a 14 TeV muon collider can be as effective as a 100 TeV
proton-proton colliderDelahaye:2019omf . Such high energy reach strongly
benefits new heavy particles searches, such as minimal dark matter
modelsHan:2020uak ; Bottaro:2021snn  searches, as well as indirect measurement
at high energiesButtazzo:2020uzc . Moreover, vector boson fusion processes are
found to be important at muon collidersCostantini:2020stv , and enable access
to difficult parameters such as the Higgs quartic self-couplingChiesa:2020awd
. More importantly, muon colliders have a special feature: the initial states
are muons, directly related to those low-energy physics. The muon $g-2$
anomaly can be probed directly at muon collidersButtazzo:2020ibd . In the
context of muon $g-2$ anomaly, studies have been performed on testing it under
the SMEFT formalismButtazzo:2020ibd , and model-exhaustive
analysisButtazzo:2020ibd ; Capdevilla:2020qel ; Capdevilla:2021rwo ;
Capdevilla:2021kcf ; Yin:2020afe .

The multi-TeV reach and better precision advantages of muon colliders, make it
a perfect place to probe muon-related B physics in low energy. There are
already proposals studying $\mu^{+}\mu^{-}\to bs$ (here $bs$ denotes both
$b\bar{s}$ and $\bar{b}s$) at multi-TeV scale to discuss the impact of low
energy rare B decays processes Altmannshofer:2022xri ; Huang:2021nkl ;
Huang:2021biu . At the current stage, some rare B decay processes can be
nicely parameterized by effective four-fermion operators at the B physics
scale ($\mu=4.8$ GeV):

$\displaystyle O_{9}$ $\displaystyle=$
$\displaystyle(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\ell}\gamma^{\mu}\ell),$ (2)
$\displaystyle O_{10}$ $\displaystyle=$
$\displaystyle(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\ell}\gamma^{\mu}\gamma_{5}\ell),$
(3) $\displaystyle O^{\prime}_{9}$ $\displaystyle=$
$\displaystyle(\bar{s}\gamma_{\mu}P_{R}b)(\bar{\ell}\gamma^{\mu}\ell),$ (4)
$\displaystyle O^{\prime}_{10}$ $\displaystyle=$
$\displaystyle(\bar{s}\gamma_{\mu}P_{R}b)(\bar{\ell}\gamma^{\mu}\gamma_{5}\ell),$
(5)

For the new physics effects described by these operators, when we go above the
weak scale ($\mu=M_{Z}$ for instance), the Wilson coefficients of such two
operators depend on the specific models (e.g. leptoquark,
scalars,$Z^{\prime}$,etc).

In this work, we adopt the same assumption that the new physics scale is
around $\mu=35$ TeV and it is challenging to discover the new physics
signature at the LHC. We also assume that these new physics above the weak
scale ($\mu=M_{Z}$) can be described by the framework of the standard model
effective field theory (SMEFT). Under appropriate assumptions, it is
noteworthy that there is at least one more operator needed in order to match
the SMEFT to low energy operators $O_{9}$ and $O_{10}$. These three operators
in the SMEFT are given in Eqs. (2.12-2.14). Different new physics models can
lead to different matching conditions across the weak scale.

Different from the proposal presented in Altmannshofer:2022xri , where the
polarization of muon beams and charge tagging of jets in the final state is
assumed, in this work, in order to reveal the nature of new physics related to
the rare B decays, we propose to measure the process $\mu^{+}\mu^{-}\to tc$
(same as $bs$, $tc$ denotes both $t\bar{c}$ and $\bar{t}c$). Same final state
has been studied for $e^{+}e^{-}$ collider in Ref. Bar-Shalom:1997ezk ; Bar-
Shalom:1997htk ; Bar-Shalom:1999dtk . And the $tc\ell\ell$ interaction has
been studied through the $t\to c\ell\ell$ decay in Bause:2020auq . We extend
these works to muon collider, with a detailed Monte-Carlo simulation at hadron
level.

To probe the $tc\ell\ell$ interaction at LHC or future hadron colliders, a
possible way is considering the decay process $t\to c\ell\ell$. From the
latest version of Review of Particle Physics ParticleDataGroup:2022pth , there
is no direct search to this process. Indirect constraints can come from the
measurements of $t\to Zq(q=u,c)$ decay, where $Z$ further decays to leptons.
Currently, the branching ratio of $t\to Zq$ is established at
$\mathcal{O}(10^{-4})$ ATLAS:2015vhj ; CMS:2017wcz ; ATLAS:2018zsq . Probing
the $tc\ell\ell$ interaction is a huge challenge at hadron colliders.

This study shows that the measurement of $tc$ final state can provide
complementary information to the process $\mu^{+}\mu^{-}\to bs$. The four
fermion operators for $\mu^{+}\mu^{-}\to tc$ naturally arise from the operator
matching conditions from the low energy effective field theory to the SMEFT,
since left-handed top and charm quarks form electroweak SU(2) doublets
respectively as the partners of the left-handed bottom and strange quarks in
SMEFT operators. The process $\mu^{+}\mu^{-}\to tc$ can also help to
distinguish different new physics models, which yield different operator
matching conditions. We will consider one $Z^{\prime}$ model and three
leptoquark models.

The leptoquark particles are predicted in the grand unification models and
they can either be scalar or vector bosons. Usually, these particles are
superheavy (e.g. $10^{13}$ GeV), as required by the experimental data of
proton decays. Very light leptoquarks (e.g. 1 TeV or a few 10 TeV) are
consistent with experimental data if their couplings to the first generation
of matter fields are weak. Light leptoquarks are also predicted in the Pati-
Salam model, where lepton numbers are treated as the fourth color quantum
number. These light leptoquarks can be accessible even at the LHC and future
collider projects. Recently, leptoquarks have attracted much attention in
order to interpret the previously claimed B anomalies. A comprehensive review
on the phenomenology of leptoquarks can be found in Dorsner:2016wpm .

Our new findings in this work include 1) The dominant SM background events for
the process $\mu^{+}\mu^{-}\to tc$ are different from the background of the
process $\mu^{+}\mu^{-}\to bs$. Due to the highly boosted top quark in the
final states, jet substructure analysis is crucial to distinguish signal and
background events. 2) In the case that there is no new resonance found in the
TeV muon colliders, measurement of the process $\mu^{+}\mu^{-}\to tc$, can
provide crucial information on the potential nature of the new physics which
leads to the low energy rare B decay processes data. 3) It is noticed that the
final state $W^{\pm}jj$ can have a very large cross section (e,g. 100fb with
collision energy $\sqrt{s}=10$ TeV), and such a final state is mainly from the
weak final state radiation processes. Suppressing the weak final state
radiation might be important for signal findings.

This paper is organized as given below. In section 2, we demonstrate the
relations between the Wilsonian coefficients of the low-energy effective
Hamiltonian and those of the SMEFT. In section 3, we present the values of
Wilson coefficients of the SMEFT derived from different new physics models.
These new physics models can accommodate the rare B decay processes data. In
section 4, we perform a Monte Carlo simulation to explore the sensitivity of
future muon colliders to these new physics scenarios. We end this work with a
few discussions and conclusions. In Appendix, we present the renormalization
group equations of Wilsonian coefficients in the SMEFT and effective
Hamiltonian, respectively.

##  2 Matching and running of different operator bases

Interestingly, these rare B decay processes can be simultaneously explained in
a model-independent way with the effective four-fermion operators. In many
B-physics studies, new physics effects strongly prefer an effective
Hamiltonian with Wilson coefficients of dimension 6 interactions at the
renormalization scale $\mu=4.8$ GeV ,

$\displaystyle\mathcal{H}_{eff}$ $\displaystyle=$
$\displaystyle\mathcal{H}^{SM}_{eff}-\mathcal{N}\sum_{\ell=e,\mu}\sum_{i=9,10}\left(c_{i}O^{bs\ell\ell}_{i}+c^{\prime}_{i}O^{\prime
bs\ell\ell}_{i}\right)+h.c.,$ (6)

where the normalization factor $\mathcal{N}$ is

$\displaystyle\mathcal{N}$ $\displaystyle=$
$\displaystyle\frac{4G_{F}}{\sqrt{2}}V_{tb}V^{*}_{ts}\frac{e^{2}}{16\pi^{2}}.$
(7)

The operators $O_{9}$ and $O_{10}$ are

$\displaystyle O^{bs\ell\ell}_{9}$ $\displaystyle=$
$\displaystyle(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\ell}\gamma^{\mu}\ell),$ (8)
$\displaystyle O^{bs\ell\ell}_{10}$ $\displaystyle=$
$\displaystyle(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\ell}\gamma^{\mu}\gamma_{5}\ell),$
(9)

where $P_{L}=(1-\gamma_{5})/2$ is the left-handed projection operator. For
$O^{\prime}_{i}$, $P_{L}$ is replaced by right-handed projection operator
$P_{R}=(1+\gamma_{5})/2$.

For the purpose of this paper, to study the related operator $O_{9}$ and
$O_{10}$ defined in the low energy B physics scale in higher colliding energy
scale, we need to treat the running and matching of the operators carefully.
Especially the subtleties that arise when the energy runs across the weak
scale. Our study finds that in the energy scale above the weak scale, the
impact of $O_{9}$ and $O_{10}$ operators needs to be reparametrized by three
different operators, rather than two, in the so-called Warsaw basis, known as
the Standard model effective theory (SMEFT). Here we introduce different bases
and coefficient matching as below.

At the scale below the weak scale, potential new physics effects are described
by a low energy effective field theory (LEFT). The LEFT Lagrangian with
dimension 6 operators can be written as

$\displaystyle\mathcal{L}_{LEFT}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{QCD+QED}+\frac{1}{v^{2}}\sum_{i}L_{i}Q_{i},$

where $L_{i}$ are the Wilson Coefficients of LEFT, and $v=246$ GeV is the
vacuum expectation value.

In this paper, we use the convention of Ref. Jenkins:2017jig . The most
relevant operators are

$\displaystyle Q^{V,LL}_{ed}(p,r,s,t)$ $\displaystyle=$
$\displaystyle(\bar{e}_{Lp}\gamma^{\mu}e_{Lr})(\bar{d}_{Ls}\gamma_{\mu}d_{Lt}),$
(10) $\displaystyle Q^{V,LR}_{de}(p,r,s,t)$ $\displaystyle=$
$\displaystyle(\bar{d}_{Lp}\gamma^{\mu}d_{Lr})(\bar{e}_{Rs}\gamma_{\mu}e_{Rt}),$
(11) $\displaystyle Q^{V,LR}_{ed}(p,r,s,t)$ $\displaystyle=$
$\displaystyle(\bar{e}_{Lp}\gamma^{\mu}e_{Lr})(\bar{d}_{Rs}\gamma_{\mu}d_{Rt}),$
(12) $\displaystyle Q^{V,RR}_{ed}(p,r,s,t)$ $\displaystyle=$
$\displaystyle(\bar{e}_{Rp}\gamma^{\mu}e_{Rr})(\bar{d}_{Rs}\gamma_{\mu}d_{Rt}),$
(13)

where $p,r,s,t$ are generation indices of quark or lepton. Since the EW
symmetry is broken, here $e$ and $d$ are the lepton field and down-type quark
field. $L$ and $R$ are the chiral indices of fermions.

One can derive the relations between $Q_{i}$ and $O_{i}$ easily:

$\displaystyle Q^{V,LL}_{ed}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(O_{9}-O_{10}\right),$ (14) $\displaystyle
Q^{V,LR}_{de}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(O_{9}+O_{10}\right),$ (15) $\displaystyle
Q^{V,LR}_{ed}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(O^{\prime}_{9}-O^{\prime}_{10}\right),$ (16)
$\displaystyle Q^{V,RR}_{de}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(O^{\prime}_{9}+O^{\prime}_{10}\right).$ (17)

So we have

$\displaystyle L^{V,LL}_{ed}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{N}v^{2}}{2}(c_{9}-c_{10}),$ (18) $\displaystyle
L^{V,LR}_{de}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{N}v^{2}}{2}(c_{9}+c_{10}),$ (19) $\displaystyle
L^{V,LR}_{ed}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{N}v^{2}}{2}(c^{\prime}_{9}-c^{\prime}_{10}),$
(20) $\displaystyle L^{V,RR}_{de}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{N}v^{2}}{2}(c^{\prime}_{9}+c^{\prime}_{10}).$
(21)

The constraints of $c_{9}$ and $c_{10}$ can be found in Ref.
Altmannshofer:2021qrr .

Assuming the new physics appears at a scale above the weak scale $\Lambda$,
the SMEFT Lagrangian with dimension 6 operators ($\mathcal{O}_{i}$) is defined
as

$\displaystyle\mathcal{L}_{SMEFT}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{SM}+\frac{1}{\Lambda^{2}}\sum_{i}C_{i}\mathcal{O}_{i},$
(22)

where $C_{i}$ are called Wilson Coefficients.

A complete set of non-redundant dimension 6 operators has been derived in Ref.
Grzadkowski:2010es , so-called Warsaw basis. The most relevant operators in
this paper are

$\displaystyle\mathcal{O}^{(1)}_{lq}(p,r,s,t)$ $\displaystyle=$
$\displaystyle(\bar{l}_{p}\gamma^{\mu}l_{r})(\bar{q}_{s}\gamma_{\mu}q_{t}),$
(23) $\displaystyle\mathcal{O}^{(3)}_{lq}(p,r,s,t)$ $\displaystyle=$
$\displaystyle(\bar{l}_{p}\gamma^{\mu}\tau^{I}l_{r})(\bar{q}_{s}\gamma_{\mu}\tau^{I}q_{t}),$
(24) $\displaystyle\mathcal{O}_{qe}(p,r,s,t)$ $\displaystyle=$
$\displaystyle(\bar{q}_{p}\gamma^{\mu}q_{r})(\bar{e}_{s}\gamma_{\mu}e_{t}),$
(25) $\displaystyle\mathcal{O}_{ld}(p,r,s,t)$ $\displaystyle=$
$\displaystyle(\bar{l}_{p}\gamma^{\mu}l_{r})(\bar{d}_{s}\gamma_{\mu}d_{t}),$
(26) $\displaystyle\mathcal{O}_{ed}(p,r,s,t)$ $\displaystyle=$
$\displaystyle(\bar{e}_{p}\gamma^{\mu}e_{r})(\bar{d}_{s}\gamma_{\mu}d_{t}),$
(27)

where $q$, $l$, $e$ are the left-handed quark doublet, left-handed lepton
doublet, and right-handed lepton singlet, respectively. $p,r,s,t$ are
generation indices of quark or lepton.

When the electroweak symmetry breaking occurs, the SM heavy particles (top,
Higgs, $W^{\pm}$, $Z$) are integrated out, and the SMEFT should be matched to
LEFT. The full matching conditions at the tree level have been derived by Ref.
Jenkins:2017jig . In this paper, we only consider the operators with flavor
indices $(p,r,s,t)=(2,2,2,3)$ or $(p,r,s,t)=(2,3,2,2)$, so the matching
conditions are simplified to

$\displaystyle L^{V,LL}_{ed}(2,2,2,3)$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{\Lambda^{2}}\left[C^{(1)}_{lq}(2,2,2,3)+C^{(3)}_{lq}(2,2,2,3)\right],$
(28) $\displaystyle L^{V,LR}_{de}(2,3,2,2)$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{\Lambda^{2}}C_{qe}(2,3,2,2),$ (29) $\displaystyle
L^{V,LR}_{ed}(2,2,2,3)$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{\Lambda^{2}}C_{ld}(2,2,2,3),$ (30) $\displaystyle
L^{V,RR}_{ed}(2,2,2,3)$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{\Lambda^{2}}C_{ed}(2,2,2,3).$ (31)

The related renormalization group equations for the SMEFT and LEFT can be
found in Appendix A and Appendix B, respectively. In this work, we do not
distinguish the left-handed and right-handed contributions. The constraints on
$C^{(1)}_{lq}+C^{(3)}_{lq}$ and $C_{qe}$, which are corresponding to $c_{9}$
and $c_{10}$, can also be applied to $C_{ld}$ and $C_{ed}$, which are
corresponding to $c^{\prime}_{9}$ and $c^{\prime}_{10}$. Especially since
$O^{\prime}_{9}$ and $O^{\prime}_{10}$ are only related to the right-handed
part of down type quarks, they do not contribute to $\mu^{+}\mu^{-}\rightarrow
tc$ directly at higher energy in our main study. From now on, we only consider
three operators: $\mathcal{O}^{(1)}_{lq}$, $\mathcal{O}^{(3)}_{lq}$ and
$\mathcal{O}_{qe}$.

In Table. 1, we list the runnings of three benchmark points: 1) BP1,
$c_{9}=-1.0,c_{10}=-0.1$, as an example of general new physics scenario, 2)
BP2, $c_{9}=c_{10}=0.25$, as an example that $C_{qe}$ is the main contribution
at scale beyond $M_{Z}$, 3) BP3, $c_{9}=-c_{10}=-0.39$, as an example that
$C^{(1)}_{lq}$ and $C^{(3)}_{lq}$ are the main contributions at scale beyond
$M_{Z}$. These benchmark points are allowed in the analysis of Ref.
Ciuchini:2022wbq , which has considered the newest LHCb data. The BP3 is the
best fit in Ref. Altmannshofer:2021qrr . It is observed that the operator
mixings induced by the RGE running have no large effects on the size of Wilson
coefficients of the SMEFT at high energy machines.

| BP1 | BP2 ($c_{9}=c_{10}$) | BP3($c_{9}=-c_{10}$)
---|---|---|---
$c_{9}$ | $-1.00$ | $0.25$ | $-0.39$
$c_{10}$ | $-0.10$ | $0.25$ | $+0.39$
$L^{V,LL}_{ed}(m_{B})$ | $-2.13\times 10^{-5}$ | $0.00$ | $-1.85\times 10^{-5}$
$L^{V,LR}_{de}(m_{B})$ | $-2.60\times 10^{-5}$ | $1.18\times 10^{-5}$ | $0.00$
$L^{V,LL}_{ed}(m_{Z})$ | $-2.16\times 10^{-5}$ | $2.83\times 10^{-8}$ | $-1.86\times 10^{-5}$
$L^{V,LR}_{de}(m_{Z})$ | $-2.60\times 10^{-5}$ | $1.18\times 10^{-5}$ | $-4.41\times 10^{-8}$
$(C^{(1)}_{lq}+C^{(3)}_{lq})(m_{Z})$ | $-3.56\times 10^{-2}$ | $4.67\times 10^{-5}$ | $-3.10\times 10^{-2}$
$C_{qe}(m_{Z})$ | $-4.29\times 10^{-2}$ | $1.95\times 10^{-2}$ | $-7.28\times 10^{-5}$
$(C^{(1)}_{lq}+C^{(3)}_{lq})(\Lambda)$ | $-3.75\times 10^{-2}$ | $1.03\times 10^{-4}$ | $-3.20\times 10^{-2}$
$C_{qe}(\Lambda)$ | $-4.42\times 10^{-2}$ | $2.00\times 10^{-2}$ | $-2.02\times 10^{-4}$
Table 1: The coefficients $L^{V,LL}_{ed}$, $L^{V,LR}_{de}$,
$C^{(1)}_{lq}+C^{(3)}_{lq}$ and $C_{qe}$ at scale $m_{B}=5$ GeV, $m_{Z}=91.19$
GeV and $\Lambda=10$ TeV are listed, with different input of $c_{9}$ and
$c_{10}$. Here, we assume
$C^{(1)}_{lq}=L^{V,LL}_{ed}\times\Lambda^{2}/v^{2},C^{(3)}_{lq}=0$.

##  3 The Matching Conditions of New Physics

At muon colliders, the operators we input are Eq.(23) to Eq. (25). In massless
limit,111For simplicity, in this section we work under the massless limit.
Nevertheless, for the numerical results discussed in later sections, full mass
dependence is always included. the differential cross sections for
$\mu^{+}\mu^{-}\to tc$ and $\mu^{+}\mu^{-}\to bs$ are:

$\displaystyle\frac{\mathrm{d}\sigma(\mu^{+}\mu^{-}\to
X)}{\mathrm{d}\cos\theta}=\frac{3s}{256\pi\Lambda^{4}}\left[|C_{LL}^{X}|^{2}(1+\cos\theta)^{2}+|C_{qe}|^{2}(1-\cos\theta)^{2}\right]$
(32)

where

$\displaystyle
C_{LL}^{bs}=C_{lq}^{(1)}+C_{lq}^{(3)},C_{LL}^{tc}=C_{lq}^{(1)}-C_{lq}^{(3)}$
(33)

We note that the left-handed operators($C_{LL}^{X}$) and right-handed
operator($C_{qe}$) have different $\theta$ dependence, hence we expect that
with flavor tagging and charge identification, they can be distinguished with
differential cross section. For simplicity, in this work, we consider only
inclusive cross section, which can be obtained by integrating over
$\cos\theta$, given by:

$\displaystyle\sigma(\mu^{+}\mu^{-}\to
X)=\frac{1}{32\pi\Lambda^{4}}s(|C_{LL}^{X}|^{2}+|C_{qe}|^{2})$ (34)

In a general new physical model, all three operators $C_{lq}^{(1)}$,
$C_{lq}^{(3)}$ and $C_{qe}$ can be non-zero, and thus both processes
$\mu^{+}\mu^{-}\to tc$ and $\mu^{+}\mu^{-}\to bs$ receives new physical
contribution, to be measured in future muon colliders. We note that for
$\mu^{+}\mu^{-}\to bs$, the corresponding Wilson coefficients $C_{LL}^{bs}$
and $C_{qe}$ are directly in charge of $b\to s\mu^{+}\mu^{-}$ transition in
$B$-physics, and hence constrained by those measurements. On the other hand,
$C_{LL}^{tc}$ is unconstrained. As a general argument, the underlying new
physical which induces $C_{LL}^{bs}$ or equivalently
$C_{lq}^{(1)},C_{lq}^{(3)}$, would induce also $C_{LL}^{tc}$ with size at
similar order.

Therefore, we expect that the cross-section of $\mu^{+}\mu^{-}\to tc$ is
comparable to $\mu^{+}\mu^{-}\to bs$, and as we will show in Section 4, due to
the presence of top quark in the final state of $\mu^{+}\mu^{-}\to tc$, it is
easier to measure than $\mu^{+}\mu^{-}\to bs$. To be more specific, below we
discuss four types of new physics models labeled as Model I-IV, where the
relations between $C_{LL}^{tc}$ and $C_{LL}^{bs}$ are given, and later in
Section 4 we will show how to distinguish them by measuring both
$\mu^{+}\mu^{-}\to tc$ and $\mu^{+}\mu^{-}\to bs$.

  1. Model I

A $Z^{\prime}$ model with flavor symmetry $U(1)_{L_{\mu}-L_{\tau}}$. In this
model, the $L_{\mu}-L_{\tau}$ is promoted into a $U(1)$ gauge symmetry, with a
massive gauge boson $Z^{\prime}$. Originally it was proposed for muon $g-2$
anomaly Baek:2001kca ; Allanach:2015gkd  and neutrino mixing Ma:2001md . Later
it is realized that the coupling to quarks can be described by high
dimensional operators, which can be generated through new heavy quarks
Fox:2011qd . Such interaction can induce flavor violation
Altmannshofer:2014cfa ; Crivellin:2015mga ; Crivellin:2015lwa . Since the
effective interaction between $bs\mu\mu$($tc\mu\mu$) is mediated by an
$s$-channel $Z^{\prime}$, an electroweak singlet, clearly we have
$C_{lq}^{(3)}=0$, hence $C_{LL}^{bs}=C_{LL}^{tc}=C_{lq}^{(1)}$. On the other
hand, in this kind of models $C_{qe}$ is independent from $C_{lq}^{(1)}$.
Regardless of the actual value of $C_{qe}$ and $C_{lq}^{(1)}$, we have
$\sigma(tc)\sim\sigma(bs)$.

  2. Model II

A scalar triplet leptoquark $S_{3}$ model. In this model, the new physics is
mediated by a heavy scalar leptoquark, which belongs to $SU(2)_{L}$ triplet.
The corresponding Lagrangian can be written as Dorsner:2016wpm

$\displaystyle\mathcal{L}_{NP}=\sum_{i,j}\lambda_{ij}\bar{Q}_{i}^{c}(i\tau^{2})\tau^{I}L_{j}S_{3}^{I}+\textrm{h.c.}$
(35)

In this model, both $C_{lq}^{(1)}$ and $C_{lq}^{(3)}$ can be generated at
tree-level, which is given by Gherardi:2020det

$\displaystyle\frac{1}{\Lambda^{2}}[C_{lq}^{(1)}]_{prst}=$ $\displaystyle
3\frac{\lambda_{sp}^{*}\lambda_{tr}}{4M^{2}}$ (36)
$\displaystyle\frac{1}{\Lambda^{2}}[C_{lq}^{(3)}]_{prst}=$
$\displaystyle\frac{\lambda_{sp}^{*}\lambda_{tr}}{4M^{2}}.$ (37)

where $M$ is the mass of the leptoquark $S_{3}$. On the other hand, $C_{qe}$
is zero at tree-level, and it can only be generated through loop corrections,
and hence we expect $C_{qe}\ll C_{lq}^{(1)},C_{lq}^{(3)}$. Therefore, we have
$C_{LL}^{bs}\sim 2C_{LL}^{tc}$, and $\sigma(tc)\sim\frac{1}{4}\sigma(bs)$

  3. Model III

A scalar singlet leptoquark $S_{1}$ model. In this model, new physics is
mediated by a heavy scalar leptoquark, which belongs to $SU(2)_{L}$ singlet.
There are three possible hypercharge assignments, and we consider the case
$Y=\frac{1}{3}$ here. The corresponding Lagrangian is

$\displaystyle\mathcal{L}_{NP}=\sum_{i,j}\lambda_{ij}\bar{Q}^{c}_{i}(i\tau_{2})L_{j}S_{1}+\textrm{h.c.}$
(38)

At the tree level, the relevant Wilson coefficients are

$\displaystyle\frac{1}{\Lambda^{2}}[C_{lq}^{(1),\textrm{tree}}]_{prst}=\frac{\lambda_{sp}^{L*}\lambda_{tr}^{L}}{4M^{2}}$
(39)
$\displaystyle\frac{1}{\Lambda^{2}}[C_{lq}^{(3),\textrm{tree}}]_{prst}=-\frac{\lambda_{sp}^{L*}\lambda_{tr}^{L}}{4M^{2}}$
(40)

Interestingly, we can see that at tree-level $C_{LL}^{bs}=0$. The leading
contribution to $C_{LL}^{bs}$ starts from one-loop level Bauer:2015knc .
Consequently, we expect that $C_{LL}^{bs}\ll C_{LL}^{tc}$, and hence
$\sigma(tc)\gg C_{LL}^{bs}$. For the experiment bounds($C_{9}=-1$ benchmark
point), we get

$\displaystyle\sum_{i}|\lambda_{u_{i}\mu}|^{2}\mathrm{Re}\frac{(\lambda\lambda^{{\dagger}})_{bs}}{V_{tb}V_{ts}^{*}}-1.74|\lambda_{t\mu}|^{2}\sim
12.5\hat{M}^{2}$ (41)

where $\hat{M}$ is the mass of the leptoquark in terms of unit TeV, and from
$B_{s}-\bar{B}_{s}$ mixing we have

$\displaystyle\frac{(\lambda\lambda^{{\dagger}})_{bs}}{V_{tb}V_{ts}^{*}}\sim(1.87+0.45i)\hat{M}$
(42)

For perturbativity, we have $|\lambda_{u_{i}\mu}|^{2}<4\pi$, which yields

$\displaystyle M\lesssim 1.9$\mathrm{T}\mathrm{e}\mathrm{V}$$ (43)

Consequently, we expect that at the energy range of future muon colliders, a
pair of the singlet scalars $S_{1}$ can be produced and be observed directly.

  4. Model IV

A $U_{1}$ vector leptoquark model. It was considered as one possible UV
completion of EFT in Calibbi:2015kma , and examined in more detail in
Barbieri:2015yvd ; Calibbi:2017qbu ; Blanke:2018sro ; Buttazzo:2017ixm . The
Lagrangian is given by

$\displaystyle\mathcal{L}_{NP}=-\frac{1}{2}U_{1,\mu\nu}^{{\dagger}}U^{1,\mu\nu}+M_{U}^{2}U_{1,\mu}^{{\dagger}}U_{1}^{\mu}+g_{U}U_{1,\mu}\lambda_{ij}\bar{Q}_{i}\gamma^{\mu}L_{j}+\textrm{h.c.}$
(44)

The relevant Wilson coefficients are

$\displaystyle\frac{1}{\Lambda^{2}}[C_{lq}^{(1),\textrm{tree}}]_{prst}=g_{U}^{2}\frac{\lambda_{sp}^{L*}\lambda_{tr}^{L}}{2M^{2}}$
(45)
$\displaystyle\frac{1}{\Lambda^{2}}[C_{lq}^{(3),\textrm{tree}}]_{prst}=g_{U}^{2}\frac{\lambda_{sp}^{L*}\lambda_{tr}^{L}}{2M^{2}}$
(46)

Clearly, in this case we have $C_{LL}^{tc}\sim 0,C_{LL}^{bs}\gg C_{LL}^{tc}$.
Moreover, in this minimal setup we have $C_{qe}=0$, though it can be
introduced through adding other particles and/or interactions.

We’d like to note that the above are examples of simplest new physics models,
where only one type of mediator is introduced. The Nature may be more
complicated and contains several types of mediators, e.g. both scalar single
and triple leptoquarks may be presentCrivellin:2017zlb .

##  4 Signatute of $\mu^{+}\mu^{-}\to tc$ at future muon colliders

###  4.1 Cross sections of signal and main background processes

To study the $\mu^{+}\mu^{-}\to tc$ process at future muon collider, we
generate a UFO model with relevant operators by Feynrules Alloul:2013bka , and
import it to Madgraph5_aMC@NLOAlwall:2014hca . We calculate the cross sections
of signals and backgrounds from $E_{cm}=1$ TeV to $30$ TeV, with following
generator cuts: jet transverse momentum $p_{T}(j)>20$ GeV, jet pseudorapidity
$|\eta(j)|<5.0$, and angular distance between jets $\Delta R(j,j)>0.4$. The
energy dependencies of the cross sections are shown in Fig. 1.

For the $\mu^{+}\mu^{-}\to tc$ and $bs$ processes, we consider the BP3 as an
example where $c_{9}=-c_{10}=-0.39$. The cross section of the main signal
process $\mu^{+}\mu^{-}\to tc$ derived from the 4-fermion interactions are
proportional to collision $s^{2}$, and it increases with the increase of the
collision energy, as shown in Fig. 1. For Wilson coefficients $C^{1}_{lq}$ and
$C^{3}_{lq}$, we consider their relations in Model I and II. As we expect,
their cross sections have relation $\sigma(tc)\sim\sigma(bs)$ for Model I, and
$\sigma(tc)\sim\frac{1}{4}\sigma(bs)$ for Model II.

The main background processes include $b\bar{b}$, $c\bar{c}$, $q\bar{q}$
($q=u,d,s$), $W^{+}W^{-}$, $ZZ$, $t\bar{t}$, and $Wjj$ ($j=u,d,s,c$). These
processes are from S-channel and decrease with the increase of the collision
energy. It is noteworthy that the cross sections of diboson processes
$W^{+}W^{-}$ can be much larger than those of di-quarks final states. The
process $W^{\pm}jj$ is also included, which describes the weak final state
radiation. A real $W$ boson is radiated from di-quark process, which
introduces a $\ln^{2}(s/M^{2}_{W})$ term to the cross section Bell:2010gi .
$W^{\pm}jj$ cross section is around $55$ fb and remains constant even when the
collision energy increases. It is remarkable that the cross section of
$W^{\pm}jj$ is much larger than those of other processes. It actually shows
the impact of electroweak radiation. A more clever way should be doing
analysis with full next-to-leading order corrections, but it is a huge
challenge. On one hand, due to the non-abelian nature of electroweak symmetry,
the vitual and real corrections do not fully cancel Ciafaloni:2000df ;
Ciafaloni:2000rp . On the other hand, the W/Z mass suggests an IR scale $\sim
100$ GeV, and it is physical. Electroweak shower always ends at this scale and
the electroweak “partons” do not “hadronize”. Pythia8 only provides limited
features to the electroweak shower Christiansen:2014kba ; Christiansen:2015jpa
. A summary of these challenges can be found in Ref. Accettura:2023ked . As a
tree-level analysis, we have to treat it as a separated process. Therefore,
suppressing the background events of $\mu^{+}\mu^{-}\to W^{\pm}jj$ will be
crucial and necessary.

Near the threshold, the cross section of all background processes is huge
compared to the signal process. With the increase of collision energy, the
signal cross section can even be larger than those of background processes if
the collision energy is large enough (say $\sqrt{s}=30$ TeV) for some new
physics models. Nonetheless, when $\sqrt{s}=10$ TeV, it might be still
challenging to discover the signal events since the cross section of
background processes is several orders larger than that of the signals
processes.

(a)

(b)

Figure 1: The energy dependencies of the cross sections of $\mu^{+}\mu^{-}\to
X$ are displayed, with generator cuts: $p_{T}(j)>20$ GeV, $|\eta(j)|<5.0$ and
$\Delta R(j,j)>0.4$. For the signal, $X=tc/bs$. In other cases, $X$ is the
final state of the background. The BP3 with $c_{9}=-c_{10}=-0.39$ is shown
here. Their values are evaluated below the cutoff $\Lambda=10$ TeV by our RGEs
and matching conditions in the Appendix. Two models are considered: (a) Model
I, $C^{(1)}_{lq}=L^{V,LL}_{ed}\times\Lambda^{2}/v^{2}$ and $C^{(3)}_{lq}=0$,
and (b) Model II,
$C^{(1)}_{lq}=3C^{(3)}_{lq}=\frac{3}{4}L^{V,LL}_{ed}\times\Lambda^{2}/v^{2}$.

(a)

(b)

Figure 2: The runnings of (a) $C^{(1)}_{lq}+C^{(3)}_{lq}$ and (b)
$C^{(1)}_{lq}-C^{(3)}_{lq}$ from EW scale to $\Lambda=10$ TeV are displayed.
Two NP scenarios are considered: 1) Model I,
$C^{(1)}_{lq}=L^{V,LL}_{ed}\Lambda^{2}/v^{2}$ and $C^{(3)}_{lq}=0$, and 2)
Model II,
$C^{(1)}_{lq}=3C^{(3)}_{lq}=\frac{3}{4}L^{V,LL}_{ed}\times\Lambda^{2}/v^{2}$.

As we show in Fig. 1, the process $\mu^{+}\mu^{-}\to tc$ can only be observed
when $E_{cm}>10$ TeV. At such high energy, we can expect two energetic jets to
be observed at the detector for either signal or background events.

In Fig. 1, the BP3 with $c_{9}=-c_{10}=-0.39$ given in Table 1 is considered.
Their values are evaluated to the EW scale by our RGEs Eq. 60 and Eq. 61. And
then they are matched to SMEFT by Eq. 28 and Eq. 29. From Eq. 28, we also know
that the constraint from B physics can be only applied to the sum of
$C^{(1)}_{lq}+C^{(3)}_{lq}$ 222The generation indices are always
$(p,r,s,t)=(2,2,2,3)$ for $C^{(1)}_{lq}$ and $C^{(3)}_{lq}$ and
$(p,r,s,t)=(2,3,2,2)$ for $C_{qe}$ in this paper.. So any new physics
scenarios with $C^{(1)}_{lq}$ and $C^{(3)}_{lq}$ as free parameters cannot be
constrained by the data from B physics experiments. In Fig. 2, we show the
running of $C^{(1)}_{lq}+C^{(3)}_{lq}$ and $C^{(1)}_{lq}-C^{(3)}_{lq}$ from EW
scale to new physics scale at $\Lambda=10$ TeV for the BP3 case with
$c_{9}=-c_{10}=-0.39$. We have considered two simple cases at EW scale
($M_{Z}$): 1) Model I with
$C^{(1)}_{lq}=L^{V,LL}_{ed}\frac{\Lambda^{2}}{v^{2}}$ and $C^{(3)}_{lq}=0$; 2)
Model II with
$C^{(1)}_{lq}=3C^{(3)}_{lq}=\frac{3}{4}L^{V,LL}_{ed}\frac{\Lambda^{2}}{v^{2}}$.
Considering the experimental uncertainties, it is challenging to distinguish
these two cases from the effects of RGE running of only
$C^{(1)}_{lq}+C^{(3)}_{lq}$.

It is noteworthy that although these two NP cases, i.e. Model I and Model II,
produce the same $c_{9}$ and $c_{10}$ at $\mu=M_{B}$, their high energy
behaviors are different from each other.

###  4.2 Jet Level Analysis

It should be mentioned that for a muon collider, the collision environment is
relatively clean and there are no serious pileups and underlying events that
occurred at a hadron collider, but there exist beam-induced backgrounds which
arise from the muon decay. By using the jet grooming techniques, it is
possible to suppress such beam-induced backgrounds Collamati:2021sbv . We will
neglect the beam-induced backgrounds in our study.

Meanwhile, since it is more and more realistic to use the particle flow method
to measure the jet energy, which can reduce the uncertainty of jet energy down
to $5\%-20\%$ Nachman:2022emq . The particle flow method can also help us to
distinguish a light jet, a W/Z boson jet, and a top jet: a W/Z/top jet can
have more charged tracks in a detector compared with a light jet.

To further investigate the kinematic features of jets from signal processes,
we input events generated with BP3 and Model I to PYTHIA8 Bierlich:2022pfr
for parton shower and hadronization, and FastJet Cacciari:2011ma  to
reconstruct jets. It is expected that jet algorithms developed for electron-
positron colliders can also be applied well to muon colliders. Therefore, we
use the generalized $k_{t}$ algorithm for $e^{+}e^{-}$ collisions, which is
extended from a simple $k_{t}$ algorithm Catani:1991hj . This algorithm
defines two distances:

$\displaystyle d_{ij}$ $\displaystyle=$
$\displaystyle\min(E^{2p}_{i},E^{2p}_{j})\frac{1-\cos\theta_{ij}}{1-\cos{R}},$
(47) $\displaystyle d_{iB}$ $\displaystyle=$ $\displaystyle E^{2p}_{i},$ (48)

where $p$ and $R$ are inputs from user. If a $d_{ij}$ is smallest then
particle $i$ and $j$ are recombined, while $d_{iB}$ is smallest then $i$ is
called an “inclusive jet”. In this context, we choose $p=1$.

It should be pointed out that the denominator $(1-\cos{R})$ is replaced by
$(3+\cos{R})$ while choosing $\pi<R<3\pi$ in FastJet. In this case, $d_{iB}$
is always larger than $d_{ij}$ so that only one inclusive jet can be found. If
we also choose $p=1$, the generalized $k_{t}$ algorithm is identical to the
original $k_{t}$ algorithm Catani:1991hj , which only has a single distance:

$\displaystyle d_{ij}$ $\displaystyle=$
$\displaystyle\min(E^{2}_{i},E^{2}_{j})\left(1-\cos\theta_{ij}\right),$ (49)

and one can extract “exclusive jet” only. For the high energy muon collider,
muon beams may radiate energetic particles which can have large angles to the
beams. Such kinds of particles are included in the exclusive jets. So the
original algorithm is not sufficient in our context.

Below we will demonstrate a case study with the collision energy $\sqrt{s}=10$
TeV at the jet level analysis, with the BP3 and Model I as theoretical input.
We demand all signal and background events have hadronic decays and neglect
those semi-leptonic and pure leptonic final states.

Since the dominant background processes are $WW(ZZ)$, $tt$, and $jj$ which can
have more than two energetic jets in the final state, in order to avoid the
combinatoric issues, it is better to have less number of jets in the final
state. For example, when $R=0.05$ and $E_{j}>100$ GeV, the number of jets for
a signal event can be much more than $10$, which is difficult to analyze. Then
appropriate jet parameters, like cone parameter and energy cut, are crucial
for the jet numbers and the preselection of signal events. Due to the fact
that both $W/Z$ bosons and top quarks are highly boosted in the final state
(each of them carries an energy 5 TeV), it is crucial to set the value of jet
parameter so as to capture the whole decay products of $W/Z$ bosons and top
quarks. Therefore, in this study, we will focus on the boosted events in our
analysis. Considering that the future ECAL (HCAL) sub-detectors can have a
granularity good enough, there is no doubt that a future detector of a muon
collider is able to resolve the substructure of such a highly boosted top jet.
In principle, a future detector of a muon collider should be very similar to
the detectors of CEPC and ILC.

The optimization of parameters $p$ and $R$ is a complicated task. As a first
attempt, we choose $p=1$ so it works similarly to a $k_{t}$ algorithm for
hadron collider Catani:1993hr ; Ellis:1993tq . For the $t\bar{c}(\bar{t}c)$
final state, we hope to find a $R$ parameter to reconstruct a heavy jet around
top mass and a light jet. We have scanned the $R$ parameter from $0.05$ to
$0.15$ and have found that $R=0.10$ can satisfy this requirement.

In Fig. 3(a), we display the energy distribution of the leading three jets in
$t\bar{c}(\bar{t}c)$ final states with $R=0.10$ and $p=1$. These jets are
sorted by energy. Obviously, the first two jets have energy around $E_{cm}/2$,
which means that they probably originated from a hard process. We can also
observe that the energy of the 3rd jet can reach several hundred GeV even TeV
levels. In such a high-energy collider, parton shower can radiate some high-
energy particles and can be detected as a hard jet. To reduce these
radiations, we implement a cut $E(j)>500$ GeV to jets for each event. After
this cut, we plot the number of jets in Fig. 3(b). As we can see, the peak is
$N_{jets}=2$ for two final state processes, and the peak for $W^{\pm}jj$
background is 3. Therefore to demand the number of jets $N_{j}=2$ for each
event can heavily suppress the background events of $\mu^{+}\mu^{-}\to
W^{\pm}jj$.

(a)

(b)

Figure 3: (a) The jet energy distributions of process $\mu^{+}\mu^{-}\to tc$
are displayed, where jets are sorted by energy. (b) The number of jets after
implementing cut $E(j)>500$ GeV.

Fig. 4 shows the invariant masses of the two most energetic jets of signal and
background events. Here, we label the heavier jet as HJ in Fig. 4(a) and the
lighter one as LJ in Fig. 4(b). As we expect, HJ has a peak around the top
mass and the distribution of LJ is flat for the signal. For the backgrounds
with heavy particles ($t$, $W$, and $Z$), we also observe peaks around their
masses. It is because these particles are highly boosted in such a high energy
machine. The reconstruction of massive jets at ATLAS can be found in Ref.
ATLAS:2020gwe . It is shown that the $W$ and top jets mass resolution can
reach $10\%$, with suitable reconstruction scheme. We can expect that this
resolution can reach $1\sim 2\%$ at future muon collider, so applying cuts on
jet invariant mass is helpful to extract the $tc$ signal.

(a)

(b)

Figure 4: The invariant mass of (a) the heavier jet and (b) the lighter jet in
signal and backgrounds are displayed.

B-tagging and C-tagging techniques can help with signal and background
separation. As shown in ATLAS:2019bwq , the larger the transverse momentum,
the better the B tagging efficiency. Therefore we expect a higher B-tagging
efficiency for a 10 TeV collision. Although B mesons and D mesons have similar
proper times at $10^{-12}$ seconds, their masses are different. If C-tagging
techniques ATLAS:2021cxe  can be applied in our analysis, we expect better
results can be obtained.

Since we only consider hadronic decay, the top quark should decay to $b$ quark
and $W$ boson, and $W$ further decays to light quarks. So the HJ of a signal
event should also be tagged as a $b$-jet in a future detector. To consider the
$b$-tagging effects, we track all decayed products of $b$-hadrons after the
hadronization. If the constituents of a jet have a b-hadron, we can label this
jet as a true $b$-jet. The same procedure can be applied to labeling a
$c$-jet.

Fig. 5(a) and Fig. 5(b) show the number of true $b$-jets and $c$-jets for
signals and backgrounds with quarks. Obviously, a $b$-jet and a $c$-jet are
found in the $t\bar{c}(\bar{t}c)$ signal. For backgrounds with two b quarks
($t\bar{t}$ and $b\bar{b}$), two $b$-jets can be found in most events. For the
$c\bar{c}$ backgrounds, most events include two $c$-jets. Most of the
$q\bar{q}$ events do not have both $b$\- and $c$-jet, but a small fraction of
such events can have $b$\- or $c$-jets in the final states. For example, in
the parton shower, a quark has a certain probability to radiate a gluon, and
subsequently, this gluon may split to heavy quarks like $b\bar{b}$ and
$c\bar{c}$. Such a kind of process is easier to occur for an energetic light
quark, which may lead to an increase in the mistagging rate of light quarks.
For backgrounds with gauge bosons, as we plot in Fig. 5(c) and Fig. 5(d),
$ZZ$, $W^{+}W^{-}$ or $W^{\pm}jj$ can decay to $b$ or $c$ quark, so we also
observe some $b$\- or $c$-jets in these events.

With this flavor tagging information, we can implement a $b$-tagging cut. In
our analysis, the $b$-tagging rate is $\epsilon_{b}=0.7$, while the mistagging
rate is $\epsilon_{c}=0.1$ and $\epsilon_{q}=0.01$ for $c$-jets and light
jets, respectively.

(a)

(b)

(c)

(d)

Figure 5: The number of (a) true $b$-jets and (b) true $c$-jets in signals and
backgrounds with quarks are displayed. As a reference, the number of (c) true
$b$-jet and (d) true $c$-jet in the background with gauge boson are displayed.

Below we introduce some simple cuts to separate signal and background events:

  * •

Cut1: $N_{jet}=2$, $N_{lepton}=0$ and $M_{jj}>8$ TeV,

  * •

Cut2: $150$ GeV$<M(HJ)<200$ GeV,

  * •

Cut3: $M(LJ)<75$ GeV,

  * •

Cut4: the heavy jet is b-tagged and the light jet is not tagged.

The results of the cut flows are listed in Table. 2. As we can see, the cuts
for the invariant masses of heavier and lighter jets are efficient to reduce
backgrounds with heavy particles. The $b$-tagging cut is efficient to reduce
backgrounds with $W$ boson. After these cuts, the huge backgrounds are reduced
successfully, and the signal events can be observed with remarkable
significance.

From Table 2, it is interesting to note that the $ZZ$ background is zero after
Cut2, but few $WW$ events survive. It is because Pythia8 has different
treatments to the $ZZ\to q_{1}\bar{q}_{1}q_{2}\bar{q}_{2}$ and $W^{+}W^{-}\to
q_{1}\bar{q}_{2}q_{3}\bar{q}_{4}$. For the $ZZ$ events, there is no confusion
that $q_{1}\bar{q}_{1}$ and $q_{2}\bar{q}_{2}$ are combined to a color
singlet, respectively. For the $W^{+}W^{-}$ events, color reconnection can
lead to the formation of alternative “flipped” singlets $q_{1}\bar{q}_{4}$ and
$q_{3}\bar{q}_{2}$. In this case, few $W^{\pm}$ propagators are forced off the
mass shell Sjostrand:1993hi . In this work, we use the default tune of
Pythia8, which is called Monash 2013 Skands:2014pea . The default model for
color reconnection is so-called MPI-based model Sjostrand:1987su . Pythia8
also provides models intended for $e^{+}e^{-}\to W^{+}W^{-}$ Sjostrand:1993hi
; Sjostrand:1993rb . The choice of color reconnection model has some effects
to our analysis, but do not change our final conclusion.

| No Cuts | Cut1 | Cut2 | Cut3 | Cut4 | $S/B$ | $\sigma$
---|---|---|---|---|---|---|---
$\mu^{+}\mu^{-}\to tc$ | $847$ | $389$ | $247$ | $166$ | $101$ | 0.38 | 5.26
$\mu^{+}\mu^{-}\to t\bar{t}$ | $7.15\times{10^{3}}$ | $3323$ | $2534$ | $90$ | $35$
$\mu^{+}\mu^{-}\to q\bar{q}$ | $3.56\times{10^{4}}$ | $1.65\times{10^{4}}$ | $1696$ | $1167$ | $75$
$\mu^{+}\mu^{-}\to c\bar{c}$ | $1.73\times{10^{4}}$ | $7663$ | $788$ | $559$ | $70$
$\mu^{+}\mu^{-}\to b\bar{b}$ | $9137$ | $3744$ | $386$ | $276$ | $53$
$\mu^{+}\mu^{-}\to W^{+}W^{-}$ | $2.51\times{10^{5}}$ | $2.39\times{10^{5}}$ | $1181$ | $186$ | $0$
$\mu^{+}\mu^{-}\to Wjj$ | $3.55\times{10^{5}}$ | $1.11\times{10^{4}}$ | $976$ | $603$ | $35$
$\mu^{+}\mu^{-}\to ZZ$ | $1.52\times{10^{4}}$ | $1.50\times{10^{4}}$ | $0$ | $0$ | $0$
Table 2: The number of events before and after each cut are listed, where
$\sigma$ is defined as $\sigma=S/\sqrt{S+B}$. We use BP3 and Model I as
theoretical inputs for the signal. We assume the luminosity is $10$ ab-1 at 10
TeV muon collider. The details of these cuts are described in the text.

For a purpose of comparison and contrast, we also perform an analysis of the
process $\mu^{+}\mu^{-}\to b\bar{s}(\bar{b}s)$. We introduce the following
cuts to separate signal and background events

  * •

Cut1: $N_{jet}=2$, $N_{lepton}=0$, and $M_{jj}>8$ TeV,

  * •

Cut2: $M(HJ)<75$ GeV,

  * •

Cut3: One jet is b-tagged.

The results of cut flows are listed in Table. 3.

As we discussed above, there should be no massive jets in such signal events,
so we can veto the heavier jets by demanding the heavier jet should not have
too much larger jet mass. The jet mass cut of the heavier jets can work to
reject backgrounds like $t\bar{t}$, $WW$, and $ZZ$. The remained backgrounds
with $W$ boson can be further reduced by applying a b-tagging cut. After these
cuts, it is observed that heavy flavor final states will be the dominant
background which leads to a small $S/B$.

| No Cuts | Cut1 | Cut2 | Cut3 | $S/B$ | $\sigma$
---|---|---|---|---|---|---
$\mu^{+}\mu^{-}\to bs$ | $1533$ | $693$ | $309$ | $214$ | 0.08 | 3.99
$\mu^{+}\mu^{-}\to t\bar{t}$ | $7152$ | $3323$ | $1$ | $1$
$\mu^{+}\mu^{-}\to q\bar{q}$ | $3.56\times{10^{4}}$ | $1.65\times{10^{4}}$ | $7128$ | $356$
$\mu^{+}\mu^{-}\to c\bar{c}$ | $1.73\times{10^{4}}$ | $7663$ | $3368$ | $735$
$\mu^{+}\mu^{-}\to b\bar{b}$ | $9137$ | $3744$ | $1662$ | $673$
$\mu^{+}\mu^{-}\to W^{+}W^{-}$ | $2.51\times{10^{5}}$ | $2.39\times{10^{5}}$ | $4151$ | $462$
$\mu^{+}\mu^{-}\to Wjj$ | $3.55\times{10^{5}}$ | $1.11\times{10^{4}}$ | $3957$ | $373$
$\mu^{+}\mu^{-}\to ZZ$ | $1.52\times{10^{4}}$ | $1.50\times{10^{4}}$ | $149$ | $63$
Table 3: The number of events before and after each cut is listed. We use BP3
and Model I as theoretical inputs for the signal. We assume the luminosity is
$10$ ab-1 at 10 TeV muon collider. The details of these cuts are described in
the text.

In Fig. 6, we demonstrate the $2\sigma$ and $3\sigma$ bounds on $C^{X}_{LL}$
from the measurement of total cross sections. Based on our analysis,
$C^{tc}_{LL}$ can be constrained to $[-1.8\times 10^{-2},+1.8\times 10^{-2}]$
($[-2.2\times 10^{-2},+2.2\times 10^{-2}]$) at $2\sigma$ ($3\sigma$) limit at
10 TeV muon collider with luminosity $\mathcal{L}=10$ ab-1, while
$C^{bs}_{LL}$ can be constrained to $[-2.3\times 10^{-2},+2.3\times 10^{-2}]$
($[-2.8\times 10^{-2},+2.8\times 10^{-2}]$) at $2\sigma$ ($3\sigma$) limit.

(a)

(b)

Figure 6: The $2\sigma$ and $3\sigma$ bounds of (a) $C^{tc}_{LL}$ and (b)
$C^{bs}_{LL}$ at 10 TeV muon collider with luminosity $\mathcal{L}=10$ ab-1
are shown.

Similarly, we implement the same cuts to analyze the case when only $C_{qe}$
is switched on. In Fig. 7, we show the $2\sigma$ and $3\sigma$ bounds on
$C_{qe}$. There is no wonder that these bounds are close to that we obtain for
$C^{X}_{LL}$ when only the information of the total cross section is utilized.

(a)

(b)

Figure 7: The $2\sigma$ and $3\sigma$ bounds of $C_{qe}$ obtained by measuring
(a) $tc$ and (b) $bs$ final state at 10 TeV muon collider with luminosity
$\mathcal{L}=10$ ab-1 are shown.

Figure 8: The 2-D 95% CL bounds are shown for the case of $C_{qe}=0$(top
left), $C_{\ell q}^{(3)}=0$(top right), $C_{\ell q}^{(1)}=3C_{\ell
q}^{(3)}$(bottom).

In Fig. 8, we show the 2D constraints from $\mu^{+}\mu^{-}\to tc$(blue area),
$\mu^{+}\mu^{-}\to bs$(red area), and their combination(magenta area). In the
top left panel of Fig. 8, we show the constraints under the assumption that
$C_{qe}=0$. We note that only BP3 satisfies this assumption in our benchmark
points, while the underlying models are unknown. Constraints from
$\mu^{+}\mu^{-}\to bs$ can probe whether the four-fermion operators are in
charge of the beyond the standard model rare B decay processes, while
$\mu^{+}\mu^{-}\to tc$ are essential to distinguish different underlying
models. In the top right panel of Fig. 8, we show the constraints under the
assumption that $C_{lq}^{(3)}=0$, i.e. in Model I. We can see that in this
case both $\mu^{+}\mu^{-}\to tc$ and $\mu^{+}\mu^{-}\to bs$ have similar
dependence on $C_{qe}$ and $C_{lq}^{(1)}$, though the former provides stronger
constraints. Combining both channels we can distinguish BP1, BP2, and BP3. In
the bottom panel of Fig. 8, we show the constraints under the assumption that
$C_{lq}^{(1)}=3C_{lq}^{(3)}$, i.e. Model II. We can see that in this case
$\mu^{+}\mu^{-}\to tc$ is more sensitive to $C_{qe}$, while $\mu^{+}\mu^{-}\to
bs$ is more sensitive to $C_{lq}^{(1)}$. Both channels are complementary and
combining them we can distinguish different benchmark points.

As we have explained in section 2, the constraints on
$C^{(1)}_{lq}+C^{(3)}_{lq}$ and $C_{qe}$ can also be applied to $C_{ld}$ and
$C_{ed}$. It is difficult to distinguish the left-handed and right-handed
contributions with unpolarized muon beams. If the B decays data are in favor
of the operators $O^{\prime}_{9}$ and $O^{\prime}_{10}$, correspondingly we
should observe deviations in $\mu^{+}\mu^{-}\to bs$ process, but not in
$\mu^{+}\mu^{-}\to tc$ process. To further distinguish the contributions of
left-handed or right-handed, the measurement of cross sections with polarized
muon beams might be necessary.

##  5 Summary and Discussion

For the signal processes $\mu^{+}\mu^{-}\to bs$ Altmannshofer:2022xri , it has
been revealed that b tagging is crucial to reject background events from $jj$
final states. The dominant background events are $jj$. In order to extract the
information on the Wilson coefficients $C_{9}$ and $C_{10}$, it is found that
the polarized muon beams and the measurement of forward-backward asymmetry of
the final state are needed.

In this work, instead of assuming polarized muon beams and charge tagging of a
final state, we propose to measure the signal processes $\mu^{+}\mu^{-}\to
tc$. Then it is found that the major task is to reject $t\bar{t}$ and $WW$
events, which might be easier to pick out signal events. It is also found that
the weak radiation $Wjj$ is large and might be a relevant background.

It is also found that in order to resolve the signal events, detectors with
high granularity are needed since top quarks and W bosons in the hadronic
final states are around 5 TeV and they are highly boosted objects. In order to
capture the substructure of these massive jets, the cone parameter should be
set as around $0.09-0.1$ when the collision energy is assumed to be
$\sqrt{s}=10$ TeV, which is much smaller compared with the cone parameter
adopted as $R=0.4$ or $0.5$ at the LHC. To extract signal events from large
background events, a refined analysis of the TeV region jet substructure
Butterworth:2008iy ; Kaplan:2008ie ; Kilian:2021whd  should be applied to
achieve much better performance. It is expected that a modern top tagger
technique can improve the top jet identification and reject the W/Z jets.

To distinguish new physics models, it is found that the measurement of
$\mu^{+}\mu^{-}\to tc$ can pinpoint Model I and Model II. Model III can be
discovered due to the resonance near a few TeV regions. Model IV could be
favored if there is no signal of the process $\mu^{+}\mu^{-}\to tc$ are
measured.

Finally, we have some comments on the systematic uncertainties. In real data
analysis, the systematic uncertainties should include uncertainties of
collision energy, uncertainties of luminosity of incoming muon beams,
statistical uncertainty, theoretical uncertainties, etc. In this work, the
statistical uncertainty is assumed to be dominant. There are other extra
theoretical sources which might also contribute to the systematic
uncertainties:

  * •

The first one is the theoretical uncertainty from higher order corrections. In
particular, the electroweak corrections will become important at a multi-TeV
muon collider, so we have to consider background like $Wjj$. The automation
with full electroweak corrections is still challenged and under development.
Our tree-level results are preliminary.

  * •

The second one is the uncertainty from parton shower and hadronization. As we
have seen, the color reconnection effect lead to some interesting behaviour of
$W^{+}W^{-}$ background. The description of such kind of effects is strongly
depending on the tuning of MC generators.

  * •

In our study, the initial state radiation (ISR) is switched off. One can
expect that the ISR of multi-TeV muon beam will be important, and the
radiation of massive particles (W, Z and Higgs) should be considered. However,
our MC tools are incomplete to give an exact description.

To explore these theoretical uncertainties are also the challenges for the
future muon collider. We still have many works to do.

###### Acknowledgements.

Z.J. Zhao has been partially supported by a China and Germany Postdoctoral
Exchange Program between the Office of China Postdoctoral Council (OCPC) and
DESY, , and partially supported by the Natural Science Foundation of China
under the grant No. 11875260. S.C. Sun is supported by the National Natural
Science Foundation of China, No.12105013. Q.S. Yan is supported by the Natural
Science Foundation of China under the grant No. 11475180 and No. 11875260.
X.R. Zhao is supported by the Italian Ministry of Research (MUR) under grand
PRIN 20172LNEEZ.

##  Appendix A SMEFT Renormalization Group Equation

The RGE of SMEFT Wilson coefficients can be written as

$\displaystyle\frac{dC_{i}}{d\ln{\mu}}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\beta_{i}.$ (50)

All 1-loop $\beta$ functions of operators in Warsaw basis have been derived in
Ref. Celis:2017hod .

The $\beta$ functions of operators 23$\sim$27 are 333We only consider the
terms are non-zero when $(p,r,s,t)=(2,2,2,3)$ ($(2,3,2,2)$ for $C_{qe}$). This
is the same for the LEFT case.

$\displaystyle\left[\beta^{(1)}_{lq}\right]_{prst}$ $\displaystyle=$
$\displaystyle\frac{2}{3}{g^{\prime}}^{2}\left([C^{(1)}_{lq}]_{wwst}+[C_{qe}]_{stww}\right)\delta_{pr}-{g^{\prime}}^{2}[C^{(1)}_{lq}]_{prst}$
(51)
$\displaystyle+9g^{2}[C^{(3)}_{lq}]_{prst}+\frac{1}{2}[\Gamma_{u}^{\dagger}\Gamma_{u}]_{vt}[C^{(1)}_{lq}]_{prsv},$
$\displaystyle\left[\beta^{(3)}_{lq}\right]_{prst}$ $\displaystyle=$
$\displaystyle\frac{2}{3}g^{2}[C^{(3)}_{lq}]_{wwst}\delta_{pr}+3g^{2}[C^{(1)}_{lq}]_{prst}-(6g^{2}+{g^{\prime}}^{2})[C^{(3)}_{lq}]_{prst}$
(52)
$\displaystyle+\frac{1}{2}[\Gamma_{u}^{\dagger}\Gamma_{u}]_{vt}[C^{(3)}_{lq}]_{prsv},$
$\displaystyle\left[\beta_{qe}\right]_{prst}$ $\displaystyle=$
$\displaystyle\frac{4}{3}{g^{\prime}}^{2}\left([C^{(1)}_{lq}]_{wwpr}+[C_{qe}]_{prww}\right)\delta_{st}+2{g^{\prime}}^{2}[C_{qe}]_{prst}$
(53)
$\displaystyle+\frac{1}{2}[\Gamma_{u}^{\dagger}\Gamma_{u}]_{vr}[C^{(1)}_{lq}]_{pvst},$
$\displaystyle\left[\beta_{ld}\right]_{prst}$ $\displaystyle=$
$\displaystyle\frac{2}{3}{g^{\prime}}^{2}[C_{ld}]_{wwst}\delta_{pr}+\frac{2}{3}{g^{\prime}}^{2}[C_{ed}]_{wwst}\delta_{pr}-2{g^{\prime}}^{2}[C_{ld}]_{prst}$
(54) $\displaystyle\left[\beta_{ed}\right]_{prst}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}{g^{\prime}}^{2}\left(2[-C_{ld}-C_{ed}]_{wwst}\right)\delta_{pr}+4{g^{\prime}}^{2}[C_{ed}]_{prst}$
(55)

where $g$ and $g^{\prime}$ are the gauge coupling of $SU(2)$ and $U(1)$,
respectively. $\Gamma_{u}$ is the $3\times 3$ Yukawa mass matrix of u-type
quarks. For simplicity, we only consider the top quark is massive, and only
the element $\Gamma_{u}(3,3)=1$ is non-zero. Note the Eq. 51$\sim$53 only
contains the most important contributions and mixing of the corresponding
operators. The mixing of the full set of operators is beyond the scope of this
work.

The 1-loop RGE running of SM parameters is given by these $\beta$ functions:

$\displaystyle\beta_{g}$ $\displaystyle=$ $\displaystyle-\frac{19}{6}g^{3},$
(56) $\displaystyle\beta_{g^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{41}{6}{g^{\prime}}^{3},$ (57) $\displaystyle\beta_{g_{s}}$
$\displaystyle=$ $\displaystyle-7g^{2}_{s},$ (58)
$\displaystyle\left[\beta_{\Gamma_{u}}\right]_{33}$ $\displaystyle=$
$\displaystyle\frac{9}{4}g^{2}\Gamma_{u}(3,3)-\frac{17}{12}{g^{\prime}}^{2}\Gamma_{u}(3,3)-8g^{2}_{s}\Gamma_{u}(3,3)+\frac{9}{2}\Gamma^{3}_{u}(3,3).$
(59)

Our simplified RGE running has been compared with two tools: DSixTools
Celis:2017hod ; Fuentes-Martin:2020zaz  and Wilson Aebischer:2018bkb . The
differences between our results and these tools are below $1\%$.

##  Appendix B LEFT Renormalization Group Equation

The definition of RGE of LEFT is the same as Eq. 50, but $C_{i}$ are replaced
by $L_{i}$. The $\beta$ functions of operator 10$\sim$13 are

$\displaystyle\left[\beta^{V,LL}_{ed}\right]_{prst}$ $\displaystyle=$
$\displaystyle\frac{4}{3}e^{2}q^{2}_{e}\delta_{pr}\left([L^{V,LL}_{ed}]_{wwst}+[L^{V,LR}_{de}]_{stww}\right)+12e^{2}q_{e}q_{d}[L^{V,LL}_{ed}]_{prst},$
(60) $\displaystyle\left[\beta^{V,LR}_{de}\right]_{prst}$ $\displaystyle=$
$\displaystyle\frac{4}{3}e^{2}q^{2}_{e}\delta_{st}\left([L^{V,LL}_{ed}]_{wwpr}+[L^{V,LR}_{de}]_{prww}\right)-12e^{2}q_{e}q_{d}[L^{V,LR}_{de}]_{prst},$
(61) $\displaystyle\left[\beta^{V,LR}_{ed}\right]_{prst}$ $\displaystyle=$
$\displaystyle\frac{4}{3}e^{2}q^{2}_{e}\delta_{pr}\left([L^{V,LR}_{ed}]_{wwst}+[L^{V,RR}_{ed}]_{wwst}\right)-12e^{2}q_{e}q_{d}[L^{V,LR}_{ed}]_{prst},$
(62) $\displaystyle\left[\beta^{V,RR}_{ed}\right]_{prst}$ $\displaystyle=$
$\displaystyle\frac{4}{3}e^{2}q^{2}_{e}\delta_{pr}\left([L^{V,LR}_{ed}]_{wwst}+[L^{V,RR}_{ed}]_{wwst}\right)+12e^{2}q_{e}q_{d}[L^{V,RR}_{ed}]_{prst},$
(63)

where $e$ is the coupling constant of QED. $q_{e}=-1$ and $q_{d}=-1/3$ are the
charges of lepton and d-type quark, respectively.

At the low energy scale, the 1-loop running of QCD and QED coupling is given
by the following $\beta$ functions:

$\displaystyle\beta_{g_{s}}$ $\displaystyle=$
$\displaystyle-\frac{23}{3}g^{3}_{s},$ (64) $\displaystyle\beta_{e}$
$\displaystyle=$ $\displaystyle\frac{80}{9}e^{3},$ (65)

## References

  * (1) LHCb collaboration, _Measurement of the $B^{0}_{s}\to\mu^{+}\mu^{-}$ decay properties and search for the $B^{0}\to\mu^{+}\mu^{-}$ and $B^{0}_{s}\to\mu^{+}\mu^{-}\gamma$ decays_, _Phys. Rev. D_ 105 (2022) 012010 [2108.09283].
  * (2) LHCb collaboration, _Analysis of Neutral B-Meson Decays into Two Muons_ , _Phys. Rev. Lett._ 128 (2022) 041801 [2108.09284].
  * (3) LHCb collaboration, _Measurement of the $B^{0}_{s}\to\mu^{+}\mu^{-}$ branching fraction and effective lifetime and search for $B^{0}\to\mu^{+}\mu^{-}$ decays_, _Phys. Rev. Lett._ 118 (2017) 191801 [1703.05747].
  * (4) ATLAS collaboration, _Study of the rare decays of $B^{0}_{s}$ and $B^{0}$ mesons into muon pairs using data collected during 2015 and 2016 with the ATLAS detector_, _JHEP_ 04 (2019) 098 [1812.03017].
  * (5) CMS collaboration, _Measurement of properties of B ${}^{0}_{\mathrm{s}}\to\mu^{+}\mu^{-}$ decays and search for B${}^{0}\to\mu^{+}\mu^{-}$ with the CMS experiment_, _JHEP_ 04 (2020) 188 [1910.12127].
  * (6) LHCb collaboration, _Test of lepton universality in beauty-quark decays_ , _Nature Phys._ 18 (2022) 277 [2103.11769].
  * (7) LHCb collaboration, _Search for lepton-universality violation in $B^{+}\to K^{+}\ell^{+}\ell^{-}$ decays_, _Phys. Rev. Lett._ 122 (2019) 191801 [1903.09252].
  * (8) LHCb collaboration, _Test of lepton universality with $B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-}$ decays_, _JHEP_ 08 (2017) 055 [1705.05802].
  * (9) LHCb collaboration, _Angular analysis of the $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay using 3 fb-1 of integrated luminosity_, _JHEP_ 02 (2016) 104 [1512.04442].
  * (10) LHCb collaboration, _Measurement of $CP$-Averaged Observables in the $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ Decay_, _Phys. Rev. Lett._ 125 (2020) 011802 [2003.04831].
  * (11) LHCb collaboration, _Angular Analysis of the $B^{+}\rightarrow K^{\ast+}\mu^{+}\mu^{-}$ Decay_, _Phys. Rev. Lett._ 126 (2021) 161802 [2012.13241].
  * (12) B. Allanach, F. S. Queiroz, A. Strumia and S. Sun, _$Z^{\prime}$ models for the LHCb and $g-2$ muon anomalies_, _Phys. Rev. D_ 93 (2016) 055045 [1511.07447].
  * (13) G. Hiller and M. Schmaltz, _Diagnosing lepton-nonuniversality in $b\to s\ell\ell$_, _JHEP_ 02 (2015) 055 [1411.4773].
  * (14) W. Altmannshofer, C. Niehoff, P. Stangl and D. M. Straub, _Status of the $B\rightarrow K^{*}\mu^{+}\mu^{-}$ anomaly after Moriond 2017_, _Eur. Phys. J. C_ 77 (2017) 377 [1703.09189].
  * (15) W. Altmannshofer and D. M. Straub, _New physics in $b\rightarrow s$ transitions after LHC run 1_, _Eur. Phys. J. C_ 75 (2015) 382 [1411.3161].
  * (16) L.-S. Geng, B. Grinstein, S. Jäger, J. Martin Camalich, X.-L. Ren and R.-X. Shi, _Towards the discovery of new physics with lepton-universality ratios of $b\to s\ell\ell$ decays_, _Phys. Rev. D_ 96 (2017) 093006 [1704.05446].
  * (17) M. Ciuchini, A. M. Coutinho, M. Fedele, E. Franco, A. Paul, L. Silvestrini et al., _New Physics in $b\to s\ell^{+}\ell^{-}$ confronts new data on Lepton Universality_, _Eur. Phys. J. C_ 79 (2019) 719 [1903.09632].
  * (18) A. Datta, J. Kumar and D. London, _The $B$ anomalies and new physics in $b\to se^{+}e^{-}$_, _Phys. Lett. B_ 797 (2019) 134858 [1903.10086].
  * (19) J. Aebischer, W. Altmannshofer, D. Guadagnoli, M. Reboud, P. Stangl and D. M. Straub, _$B$ -decay discrepancies after Moriond 2019_, _Eur. Phys. J. C_ 80 (2020) 252 [1903.10434].
  * (20) M. Ciuchini, M. Fedele, E. Franco, A. Paul, L. Silvestrini and M. Valli, _Lessons from the $B^{0,+}\to K^{*0,+}\mu^{+}\mu^{-}$ angular analyses_, _Phys. Rev. D_ 103 (2021) 015030 [2011.01212].
  * (21) S. Jäger, M. Kirk, A. Lenz and K. Leslie, _Charming new physics in rare B-decays and mixing?_ , _Phys. Rev. D_ 97 (2018) 015021 [1701.09183].
  * (22) CMS collaboration, _Measurement of ${\rm B^{0}_{s}}\to\mu^{+}\mu^{-}$ decay properties and search for the ${\rm B}^{0}\to\mu\mu$ decay in proton-proton collisions at $\sqrt{s}=13~{}\rm{TeV}$_, tech. rep., CERN, Geneva, 2022.
  * (23) LHCb collaboration, _Test of lepton universality in $b\rightarrow s\ell^{+}\ell^{-}$ decays_, 2212.09152.
  * (24) LHCb collaboration, _Measurement of lepton universality parameters in $B^{+}\to K^{+}\ell^{+}\ell^{-}$ and $B^{0}\to K^{*0}\ell^{+}\ell^{-}$ decays_, 2212.09153.
  * (25) M. Ciuchini, M. Fedele, E. Franco, A. Paul, L. Silvestrini and M. Valli, _Constraints on Lepton Universality Violation from Rare $B$ Decays_, 2212.10516.
  * (26) A. J. Buras and F. De Fazio, _331 Model Predictions for Rare $B$ and $K$ Decays, and $\Delta F=2$ Processes: an Update_, 2301.02649.
  * (27) A. Greljo, J. Salko, A. Smolkovič and P. Stangl, _Rare $b$ decays meet high-mass Drell-Yan_, 2212.10497.
  * (28) A. Das, T. Nomura and T. Shimomura, _Multi muon/anti-muon signals via productions of gauge and scalar bosons in a $U(1)_{L_{\mu}-L_{\tau}}$ model at muonic colliders_, 2212.11674.
  * (29) N. Desai and A. Sengupta, _Status of leptoquark models after LHC Run-2 and discovery prospects at future colliders_ , 2301.01754.
  * (30) I. d. M. Varzielas and A. Sengupta, _Constraining flavoured leptoquarks with LHC and LFV_ , 2301.04119.
  * (31) D. Bečirević, G. Piazza and O. Sumensari, _Revisiting $B\to K^{(\ast)}\nu\bar{\nu}$ decays in the Standard Model and beyond_, 2301.06990.
  * (32) V. Shiltsev and F. Zimmermann, _Modern and Future Colliders_ , _Rev. Mod. Phys._ 93 (2021) 015006 [2003.09084].
  * (33) C. Aime et al., _Muon Collider Physics Summary_ , 2203.07256.
  * (34) Muon Collider collaboration, _The physics case of a 3 TeV muon collider stage_ , 2203.07261.
  * (35) J. P. Delahaye, M. Diemoz, K. Long, B. Mansoulié, N. Pastrone, L. Rivkin et al., _Muon Colliders_ , 1901.06150.
  * (36) T. Han, Z. Liu, L.-T. Wang and X. Wang, _WIMPs at High Energy Muon Colliders_ , _Phys. Rev. D_ 103 (2021) 075004 [2009.11287].
  * (37) S. Bottaro, D. Buttazzo, M. Costa, R. Franceschini, P. Panci, D. Redigolo et al., _Closing the window on WIMP Dark Matter_ , _Eur. Phys. J. C_ 82 (2022) 31 [2107.09688].
  * (38) D. Buttazzo, R. Franceschini and A. Wulzer, _Two Paths Towards Precision at a Very High Energy Lepton Collider_ , _JHEP_ 05 (2021) 219 [2012.11555].
  * (39) A. Costantini, F. De Lillo, F. Maltoni, L. Mantani, O. Mattelaer, R. Ruiz et al., _Vector boson fusion at multi-TeV muon colliders_ , _JHEP_ 09 (2020) 080 [2005.10289].
  * (40) M. Chiesa, F. Maltoni, L. Mantani, B. Mele, F. Piccinini and X. Zhao, _Measuring the quartic Higgs self-coupling at a multi-TeV muon collider_ , _JHEP_ 09 (2020) 098 [2003.13628].
  * (41) D. Buttazzo and P. Paradisi, _Probing the muon $g-2$ anomaly with the Higgs boson at a muon collider_, _Phys. Rev. D_ 104 (2021) 075021 [2012.02769].
  * (42) R. Capdevilla, D. Curtin, Y. Kahn and G. Krnjaic, _Discovering the physics of $(g-2)_{\mu}$ at future muon colliders_, _Phys. Rev. D_ 103 (2021) 075028 [2006.16277].
  * (43) R. Capdevilla, D. Curtin, Y. Kahn and G. Krnjaic, _No-lose theorem for discovering the new physics of (g-2) $\mu$ at muon colliders_, _Phys. Rev. D_ 105 (2022) 015028 [2101.10334].
  * (44) R. Capdevilla, D. Curtin, Y. Kahn and G. Krnjaic, _Systematically testing singlet models for (g $-$ 2)μ_, _JHEP_ 04 (2022) 129 [2112.08377].
  * (45) W. Yin and M. Yamaguchi, _Muon g-2 at a multi-TeV muon collider_ , _Phys. Rev. D_ 106 (2022) 033007 [2012.03928].
  * (46) W. Altmannshofer, S. A. Gadam and S. Profumo, _Snowmass White Paper: Probing New Physics with $\mu^{+}\mu^{-}\to bs$ at a Muon Collider_, in _2022 Snowmass Summer Study_ , 3, 2022, 2203.07495.
  * (47) G.-y. Huang, F. S. Queiroz and W. Rodejohann, _Gauged $L_{\mu}{-}L_{\tau}$ at a muon collider_, _Phys. Rev. D_ 103 (2021) 095005 [2101.04956].
  * (48) G.-y. Huang, S. Jana, F. S. Queiroz and W. Rodejohann, _Probing the RK(*) anomaly at a muon collider_ , _Phys. Rev. D_ 105 (2022) 015013 [2103.01617].
  * (49) S. Bar-Shalom, G. Eilam, A. Soni and J. Wudka, _Probing the flavor changing tc vertex via tree level processes: e+ e- — $>$ t anti-c electron-neutrino anti-electron-neutrino, t anti-c e+ e- and t —$>$ c W+ W-_, _Phys. Rev. Lett._ 79 (1997) 1217 [hep-ph/9703221].
  * (50) S. Bar-Shalom, G. Eilam, A. Soni and J. Wudka, _Implications of a W+ W- (Z Z) - Higgs - t anti-c interaction for e+ e- — $>$ t anti-c electron-neutrino anti-electron-neutrino, t anti-c e+ e-, t anti-c Z and for t —$>$ c W+ W-, c Z Z in a two Higgs doublet model_, _Phys. Rev. D_ 57 (1998) 2957 [hep-ph/9708358].
  * (51) S. Bar-Shalom and J. Wudka, _Flavor changing single top quark production channels at e+ e- colliders in the effective Lagrangian description_ , _Phys. Rev. D_ 60 (1999) 094016 [hep-ph/9905407].
  * (52) R. Bause, H. Gisbert, M. Golz and G. Hiller, _Lepton universality and lepton flavor conservation tests with dineutrino modes_ , _Eur. Phys. J. C_ 82 (2022) 164 [2007.05001].
  * (53) Particle Data Group collaboration, _Review of Particle Physics_ , _PTEP_ 2022 (2022) 083C01.
  * (54) ATLAS collaboration, _Search for flavour-changing neutral current top-quark decays to $qZ$ in $pp$ collision data collected with the ATLAS detector at $\sqrt{s}=8$ TeV_, _Eur. Phys. J. C_ 76 (2016) 12 [1508.05796].
  * (55) CMS collaboration, _Search for associated production of a Z boson with a single top quark and for tZ flavour-changing interactions in pp collisions at $\sqrt{s}=8$ TeV_, _JHEP_ 07 (2017) 003 [1702.01404].
  * (56) ATLAS collaboration, _Search for flavour-changing neutral current top-quark decays $t\to qZ$ in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector_, _JHEP_ 07 (2018) 176 [1803.09923].
  * (57) I. Doršner, S. Fajfer, A. Greljo, J. F. Kamenik and N. Košnik, _Physics of leptoquarks in precision experiments and at particle colliders_ , _Phys. Rept._ 641 (2016) 1 [1603.04993].
  * (58) E. E. Jenkins, A. V. Manohar and P. Stoffer, _Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching_ , _JHEP_ 03 (2018) 016 [1709.04486].
  * (59) W. Altmannshofer and P. Stangl, _New physics in rare B decays after Moriond 2021_ , _Eur. Phys. J. C_ 81 (2021) 952 [2103.13370].
  * (60) B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, _Dimension-Six Terms in the Standard Model Lagrangian_ , _JHEP_ 10 (2010) 085 [1008.4884].
  * (61) S. Baek, N. G. Deshpande, X. G. He and P. Ko, _Muon anomalous g-2 and gauged L(muon) - L(tau) models_ , _Phys. Rev. D_ 64 (2001) 055006 [hep-ph/0104141].
  * (62) E. Ma, D. P. Roy and S. Roy, _Gauged L(mu) - L(tau) with large muon anomalous magnetic moment and the bimaximal mixing of neutrinos_ , _Phys. Lett. B_ 525 (2002) 101 [hep-ph/0110146].
  * (63) P. J. Fox, J. Liu, D. Tucker-Smith and N. Weiner, _An Effective Z’_ , _Phys. Rev. D_ 84 (2011) 115006 [1104.4127].
  * (64) W. Altmannshofer, S. Gori, M. Pospelov and I. Yavin, _Quark flavor transitions in $L_{\mu}-L_{\tau}$ models_, _Phys. Rev. D_ 89 (2014) 095033 [1403.1269].
  * (65) A. Crivellin, G. D’Ambrosio and J. Heeck, _Explaining $h\to\mu^{\pm}\tau^{\mp}$, $B\to K^{*}\mu^{+}\mu^{-}$ and $B\to K\mu^{+}\mu^{-}/B\to Ke^{+}e^{-}$ in a two-Higgs-doublet model with gauged $L_{\mu}-L_{\tau}$_, _Phys. Rev. Lett._ 114 (2015) 151801 [1501.00993].
  * (66) A. Crivellin, G. D’Ambrosio and J. Heeck, _Addressing the LHC flavor anomalies with horizontal gauge symmetries_ , _Phys. Rev. D_ 91 (2015) 075006 [1503.03477].
  * (67) V. Gherardi, D. Marzocca and E. Venturini, _Matching scalar leptoquarks to the SMEFT at one loop_ , _JHEP_ 07 (2020) 225 [2003.12525].
  * (68) M. Bauer and M. Neubert, _Minimal Leptoquark Explanation for the $R_{D^{(*)}}$ , $R_{K}$ , and $(g-2)_{\mu}$ Anomalies_, _Phys. Rev. Lett._ 116 (2016) 141802 [1511.01900].
  * (69) L. Calibbi, A. Crivellin and T. Ota, _Effective Field Theory Approach to $b\to s\ell\ell^{(^{\prime})}$, $B\to K^{(*)}\nu\overline{\nu}$ and $B\to D^{(*)}\tau\nu$ with Third Generation Couplings_, _Phys. Rev. Lett._ 115 (2015) 181801 [1506.02661].
  * (70) R. Barbieri, G. Isidori, A. Pattori and F. Senia, _Anomalies in $B$-decays and $U(2)$ flavour symmetry_, _Eur. Phys. J. C_ 76 (2016) 67 [1512.01560].
  * (71) L. Calibbi, A. Crivellin and T. Li, _Model of vector leptoquarks in view of the $B$-physics anomalies_, _Phys. Rev. D_ 98 (2018) 115002 [1709.00692].
  * (72) M. Blanke and A. Crivellin, _$B$ Meson Anomalies in a Pati-Salam Model within the Randall-Sundrum Background_, _Phys. Rev. Lett._ 121 (2018) 011801 [1801.07256].
  * (73) D. Buttazzo, A. Greljo, G. Isidori and D. Marzocca, _B-physics anomalies: a guide to combined explanations_ , _JHEP_ 11 (2017) 044 [1706.07808].
  * (74) A. Crivellin, D. Müller and T. Ota, _Simultaneous explanation of R(D(∗)) and b→s$\mu$+ $\mu$-: the last scalar leptoquarks standing_, _JHEP_ 09 (2017) 040 [1703.09226].
  * (75) A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, _FeynRules 2.0 - A complete toolbox for tree-level phenomenology_ , _Comput. Phys. Commun._ 185 (2014) 2250 [1310.1921].
  * (76) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al., _The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations_ , _JHEP_ 07 (2014) 079 [1405.0301].
  * (77) G. Bell, J. H. Kuhn and J. Rittinger, _Electroweak Sudakov Logarithms and Real Gauge-Boson Radiation in the TeV Region_ , _Eur. Phys. J. C_ 70 (2010) 659 [1004.4117].
  * (78) M. Ciafaloni, P. Ciafaloni and D. Comelli, _Bloch-Nordsieck violating electroweak corrections to inclusive TeV scale hard processes_ , _Phys. Rev. Lett._ 84 (2000) 4810 [hep-ph/0001142].
  * (79) M. Ciafaloni, P. Ciafaloni and D. Comelli, _Electroweak Bloch-Nordsieck violation at the TeV scale: ’Strong’ weak interactions?_ , _Nucl. Phys. B_ 589 (2000) 359 [hep-ph/0004071].
  * (80) J. R. Christiansen and T. Sjöstrand, _Weak Gauge Boson Radiation in Parton Showers_ , _JHEP_ 04 (2014) 115 [1401.5238].
  * (81) J. R. Christiansen and S. Prestel, _Merging weak and QCD showers with matrix elements_ , _Eur. Phys. J. C_ 76 (2016) 39 [1510.01517].
  * (82) C. Accettura et al., _Towards a Muon Collider_ , 2303.08533.
  * (83) F. Collamati, C. Curatolo, D. Lucchesi, A. Mereghetti, N. Mokhov, M. Palmer et al., _Advanced assessment of beam-induced background at a muon collider_ , _JINST_ 16 (2021) P11009 [2105.09116].
  * (84) B. Nachman et al., _Jets and Jet Substructure at Future Colliders_ , _Front. in Phys._ 10 (2022) 897719 [2203.07462].
  * (85) C. Bierlich et al., _A comprehensive guide to the physics and usage of PYTHIA 8.3_ , 2203.11601.
  * (86) M. Cacciari, G. P. Salam and G. Soyez, _FastJet User Manual_ , _Eur. Phys. J. C_ 72 (2012) 1896 [1111.6097].
  * (87) S. Catani, Y. L. Dokshitzer, M. Olsson, G. Turnock and B. R. Webber, _New clustering algorithm for multi - jet cross-sections in e+ e- annihilation_ , _Phys. Lett. B_ 269 (1991) 432.
  * (88) S. Catani, Y. L. Dokshitzer, M. H. Seymour and B. R. Webber, _Longitudinally invariant $K_{t}$ clustering algorithms for hadron hadron collisions_, _Nucl. Phys. B_ 406 (1993) 187.
  * (89) S. D. Ellis and D. E. Soper, _Successive combination jet algorithm for hadron collisions_ , _Phys. Rev. D_ 48 (1993) 3160 [hep-ph/9305266].
  * (90) ATLAS collaboration, _Optimisation of large-radius jet reconstruction for the ATLAS detector in 13 TeV proton–proton collisions_ , _Eur. Phys. J. C_ 81 (2021) 334 [2009.04986].
  * (91) ATLAS collaboration, _ATLAS b-jet identification performance and efficiency measurement with $t{\bar{t}}$ events in pp collisions at $\sqrt{s}=13$ TeV_, _Eur. Phys. J. C_ 79 (2019) 970 [1907.05120].
  * (92) ATLAS collaboration, _Measurement of the c-jet mistagging efficiency in $t\bar{t}$ events using pp collision data at $\sqrt{s}=13$ TeV collected with the ATLAS detector_, _Eur. Phys. J. C_ 82 (2022) 95 [2109.10627].
  * (93) T. Sjostrand and V. A. Khoze, _On Color rearrangement in hadronic W+ W- events_ , _Z. Phys. C_ 62 (1994) 281 [hep-ph/9310242].
  * (94) P. Skands, S. Carrazza and J. Rojo, _Tuning PYTHIA 8.1: the Monash 2013 Tune_ , _Eur. Phys. J. C_ 74 (2014) 3024 [1404.5630].
  * (95) T. Sjostrand and M. van Zijl, _A Multiple Interaction Model for the Event Structure in Hadron Collisions_ , _Phys. Rev. D_ 36 (1987) 2019.
  * (96) T. Sjostrand and V. A. Khoze, _Does the W mass reconstruction survive QCD effects?_ , _Phys. Rev. Lett._ 72 (1994) 28 [hep-ph/9310276].
  * (97) J. M. Butterworth, A. R. Davison, M. Rubin and G. P. Salam, _Jet substructure as a new Higgs search channel at the LHC_ , _Phys. Rev. Lett._ 100 (2008) 242001 [0802.2470].
  * (98) D. E. Kaplan, K. Rehermann, M. D. Schwartz and B. Tweedie, _Top Tagging: A Method for Identifying Boosted Hadronically Decaying Top Quarks_ , _Phys. Rev. Lett._ 101 (2008) 142001 [0806.0848].
  * (99) W. Kilian, S. Sun, Q.-S. Yan, X. Zhao and Z. Zhao, _Highly Boosted Higgs Bosons and Unitarity in Vector-Boson Fusion at Future Hadron Colliders_ , _JHEP_ 05 (2021) 198 [2101.12537].
  * (100) A. Celis, J. Fuentes-Martin, A. Vicente and J. Virto, _DsixTools: The Standard Model Effective Field Theory Toolkit_ , _Eur. Phys. J. C_ 77 (2017) 405 [1704.04504].
  * (101) J. Fuentes-Martin, P. Ruiz-Femenia, A. Vicente and J. Virto, _DsixTools 2.0: The Effective Field Theory Toolkit_ , _Eur. Phys. J. C_ 81 (2021) 167 [2010.16341].
  * (102) J. Aebischer, J. Kumar and D. M. Straub, _Wilson: a Python package for the running and matching of Wilson coefficients above and below the electroweak scale_ , _Eur. Phys. J. C_ 78 (2018) 1026 [1804.05033].