aainstitutetext: DAMTP, University of Cambridge, Wilberforce Road, Cambridge,
CB3 0WA, United Kingdombbinstitutetext: Department of Theoretical Physics,
CERN, 1211 Geneva 23, Switzerland

# Flavonstrahlung in the $B_{3}-L_{2}$ $Z^{\prime}$ Model at Current and
Future Colliders

Ben Allanach<EMAIL_ADDRESS>a,1 and Eetu Loisa111Corresponding
author<EMAIL_ADDRESS>
###### Abstract

The $B_{3}-L_{2}$ $Z^{\prime}$ model may explain some gross features of the
fermion mass spectrum as well as $b\rightarrow s\ell\ell$ anomalies. A TeV-
scale physical scalar field associated with gauged $U(1)_{B_{3}-L_{2}}$
spontaneous symmetry breaking, the flavon field $\vartheta$, affects Higgs
phenomenology via mixing. In this paper, we investigate the collider
phenomenology of the flavon field. Higgs and $W$ boson mass data are used to
place bounds upon parameter space. We then examine _flavonstrahlung_
(${Z^{\prime}}\rightarrow Z^{\prime}\vartheta$ production) at colliders as a
means to directly produce and discover flavon particles, which would provide
direct empirical evidence tying the flavon to $U(1)_{B_{3}-L_{2}}$ symmetry
breaking. A 100 TeV FCC-hh or a 10 TeV muon collider would have high
sensitivity to flavonstrahlung, whereas the HL-LHC can observe it only in
extreme corners of parameter space.

###### Keywords:

muon collider, FCC-hh, LHC, HL-LHC

††preprint: CERN-TH-2022-188

##  1 Introduction

The $B_{3}-L_{2}$ model Alonso:2017uky ; Bonilla:2017lsq ; Allanach:2020kss
may be motivated by providing an explanation of some features of the fermion
mass spectrum. It was originally introduced to explain discrepancies between
Standard Model (SM) predictions and experimental measurements of various
observable quantities that involve the $b\rightarrow s\mu^{+}\mu^{-}$ or
$\bar{b}\rightarrow\bar{s}\mu^{+}\mu^{-}$ transition Alguero:2019ptt ;
Alok:2019ufo ; Ciuchini:2019usw ; Aebischer:2019mlg ; Datta:2019zca ;
Kowalska:2019ley ; Arbey:2019duh ; Gubernari:2022hxn ; Alok:2022pjb ;
SinghChundawat:2022zdf .222Discrepancies between predictions and data are
still present when one uses ratios of observables to cancel the dependence
upon CKM matrix elements, whose determination from data is based on $\Delta
M_{s,d}$, $\epsilon_{K}$ and $S_{\psi K_{S}}$ for which we find new physics
contributions that are negligible Buras:2022wpw ; Buras:2022qip . Whilst four
of these measurements (two different $q^{2}$ bins each of $R_{K}$ and
$R_{K^{\ast}}$) have recently returned LHCb:2022qnv ; LHCb:2022zom  to being
compatible with SM predictions, a larger number of other observables are in
tension with them. A $Z^{\prime}$ contribution to the $b\rightarrow
s\mu^{+}\mu^{-}$ process is depicted in the left-hand panel of figure 1. The
$B_{3}-L_{2}$ model is based on an extension of the SM gauge group by a direct
product with an additional spontaneously broken $U(1)_{B_{3}-L_{2}}$ gauge
symmetry, where the charges of the SM fields are proportional to third family
baryon number minus second family lepton number. The model is free of quantum
field theoretic anomalies if one includes one right-handed neutrino Weyl
fermion field per SM family in the field content, a choice which is otherwise
motivated by the fact that it facilitates the see-saw explanation of the light
neutrino masses inferred from empirical data. $U(1)_{B_{3}-L_{2}}$ is broken
near the TeV scale by a _flavon_ field, a SM-singlet complex scalar field
$\theta$ with non-zero $B_{3}-L_{2}$ charge, resulting in a TeV-scale
electrically neutral gauge boson, i.e. a $Z^{\prime}$ state. Much as the Higgs
doublet field of the SM possesses a physical state, the Higgs boson, which has
been discovered in experiments, so the flavon field contains a physical real
scalar state, the flavon particle $\vartheta$.

It is remarkable that a model with spontaneous symmetry breaking at a
relatively low (i.e. TeV) scale can pass the experimental bounds upon it
coming from flavour constraints. Such constraints are notoriously strict when
they involve flavour transitions of the first two generations of electrically
charged fermionic fields, particularly from $K-\overline{K}$ mixing
Charles:2013aka . Moreover, the strength of LHC bounds coming from bump hunts
in the di-muon mass spectrum resulting from $pp\rightarrow
Z^{\prime}\rightarrow\mu^{+}\mu^{-}$ is diminished by the fact that the
$Z^{\prime}$ only couples with an appreciable strength to _third_ family
fermions. Thus, the dominant LHC production process (as depicted in the middle
panel of figure 1) originates from fusing a bottom quark $b$ and an anti-
bottom quark $\bar{b}$ in the initial proton states, providing double
suppression to the $Z^{\prime}$ production cross section from small
(anti-)bottom parton distribution functions Lim:2016wjo . Current direct
searches imply a lower bound upon the $Z^{\prime}$ mass $M_{Z^{\prime}}$ of
around 1-2 TeV Bonilla:2017lsq ; Allanach:2020kss ; Allanach:2021gmj ;
Azatov:2022itm , significantly lower than $Z^{\prime}$ models in which the
$Z^{\prime}$ field couples to quarks universally, where the current lower
bound from ATLAS and CMS is currently around 5 TeV ATLAS:2019erb ; CMS:2021ctt
. A window in the parameter space with 20 GeV $<M_{Z^{\prime}}<$ 300 GeV
Bonilla:2017lsq ; Allanach:2020kss  is all but ruled out at the $95\%$
confidence level Allanach:2021gmj . A recent analysis Azatov:2022itm  found
that although through much of the parameter space the $Z^{\prime}$ can be
discovered at the HL-LHC, either a 3 TeV $\mu^{+}\mu^{-}$ collider, a 10 TeV
$\mu^{+}\mu^{-}$ collider MuonCollider:2022nsa  or a 100 TeV FCC-hh collider
FCC:2018vvp  would cover all of the parameter space compatible with the
model’s explanation of the $b\rightarrow s\mu^{+}\mu^{-}$ anomalies.

(300,75)(-25,-5) [arrow](-25,25)(0,50) [arrow](-50,25)(-25,25)
[arrow](25,50)(0,25) [arrow](0,25)(25,0) (-25,25)(0,25)33
(-12.5,17)[c]$Z^{\prime}$ (-55,25)[r]$b$ (-5,50)[r]$s$ (35,50)[c]$\mu^{+}$
(35,0)[c]$\mu^{-}$

[arrow](80,50)(105,25) [arrow](105,25)(80,0) [arrow](155,50)(130,25)
[arrow](130,25)(155,0) (105,25)(130,25)33 (117.5,33)[c]$Z^{\prime}$
(75,0)[r]$\overline{b}$ (75,50)[r]$b$ (162,50)[l]$\mu^{+}$ (162,0)[l]$\mu^{-}$

[arrow](220,50)(245,25) [arrow](245,25)(220,0) (295,50)(270,25)3
(270,25)(295,0)33 (245,25)(270,25)33 (257.5,33)[c]$Z^{\prime}$
(215,0)[r]$\overline{b},\mu^{+}$ (215,50)[r]$b,\mu^{-}$ (302,50)[l]$\vartheta$
(302,0)[l]$Z^{\prime}$

Figure 1:  Feynman diagram of (left panel) $Z^{\prime}$ mediated contribution
to $b\rightarrow s\mu^{+}\mu^{-}$, (middle panel) the dominant LHC
$Z^{\prime}$ production process, followed by subsequent decay into a di-muon
pair $\mu^{+}\mu^{-}$ and (right panel) flavonstrahlung at a hadron collider
[$b\bar{b}$ initial state] or a muon collider [$\mu^{+}\mu^{-}$ initial
state].

It is remarkable that a TeV-scale $Z^{\prime}$, which generates flavour
changing neutral currents at the tree-level of perturbation theory, is not
only allowed by current data but in fact is motivated by it. Whether or not
the $b\rightarrow s\mu^{+}\mu^{-}$ anomalies persist in collider data, the
$B_{3}-L_{2}$ model is of interest both for this reason and because it may be
pertinent to the fermion mass puzzle (namely why certain hierarchies within
the spectrum of fermion masses and mixings exist). Aside from the
aforementioned $Z^{\prime}$ production, scant collider phenomenology of the
model has been studied in terms of direct searches.

It is our purpose here to study the direct collider phenomenology of the
flavon in the $B_{3}-L_{2}$ model for the first time. The flavon is expected
to have a mass of order $M_{Z^{\prime}}$ (i.e. the TeV scale) and so may be
amenable to direct production and discovery at high energy colliders. We shall
see that the flavon field generically mixes with the Higgs field and is
therefore bounded by some electroweak measurements and Higgs searches. In
detail, we wish to make a first estimate of the current and potential future
collider capabilities of observing the flavon via flavonstrahlung
Allanach:2020kss 333LHC $HZ^{\prime}$ production has been studied in a family
universal $U(1)$ model in refs. Pruna:2011me ; Banerjee:2015hoa ., a process
whose dominant Feynman diagram is depicted in the right-hand panel of figure
1. Observation of a flavon would be an important confirmation of the means of
$U(1)_{B_{3}-L_{2}}$ gauge symmetry breakdown, since $U(1)_{B_{3}-L_{2}}$
breaking could instead be a consequence of the Stueckelberg mechanism
Kors:2005uz , which possesses no explicit flavon field. This then has
implications for the fermion mass puzzle since, for example, the Froggatt-
Nielsen mechanism Froggatt:1978nt , which can explain the hierarchies between
the different families observed in the measured values of the SM fermion
masses, necessarily involves a flavon field.

The paper proceeds in section 2 by briefly reiterating some salient points in
the construction of the $B_{3}-L_{2}$ model. The Higgs phenomenology of the
flavon particle is reviewed in section 3 and current collider constraints upon
the parameter space of the $B_{3}-L_{2}$ model are imposed. Then, in section
4, we study the flavonstrahlung process in allowed parts of the parameter
space. We shall see that the cross-section is too small to result in a
realistic measurement at the HL-LHC (except for a small portion of parameter
space in the case that the flavon charge is larger than unity) but that a 10
TeV muon collider of a 100 TeV hadron collider such as the FCC-hh could
facilitate discovery of flavonstrahlung and therefore facilitate discovery of
the flavon field associated with breaking the gauged flavour symmetry. We
summarise and conclude in section 5.

##  2 $B_{3}-L_{2}$ Model

We shall now review some salient points of the construction of the
$B_{3}-L_{2}$ model. For definiteness, we use the simple bottom-up
construction of ref. Allanach:2020kss . The $B_{3}-L_{2}$ model is constructed
by extending the SM gauge group, $SU(3)\times SU(2)_{L}\times U(1)_{Y}$, by an
abelian factor $U(1)_{B_{3}-L_{2}}$ in a direct product. This new symmetry is
gauged, so it comes with an electrically neutral force carrier, the
$Z^{\prime}$ boson. We also introduce the flavon field
$\mathopen{}\mathclose{{}\left(\theta}\right)$, a SM-singlet complex scalar,
which carries charge $q_{\theta}$ under $U(1)_{B_{3}-L_{2}}$. Table 1 displays
the charges of the fields in the model under the new abelian symmetry. Three
right-handed neutrinos $\nu_{R_{1,2,3}}$ are added in order to facilitate
neutrino masses and mixing. With these fields and charges,
$U(1)_{B_{3}-L_{2}}$ acts vectorially on the fermions (i.e. acts in the same
way on left-handed chiral fermions as it does on their right-handed partners),
and gauge anomaly cancellation manifestly takes place. There is an implicit
assumption in the $B_{3}-L_{2}$ model (and other similar models) that they
originate from some more complete ultra-violet model which may be a semi-
simple extension, thus obviating constraints coming from Landau poles
Bause:2021prv .

$Q^{\prime}_{iL}$ | $u^{\prime}_{iR}$ | $d^{\prime}_{iR}$ | $L_{1}^{\prime}$ | $L_{2}^{\prime}$ | $L_{3}^{\prime}$ | $e^{\prime}_{1R}$ | $e^{\prime}_{2R}$ | $e^{\prime}_{3R}$
---|---|---|---|---|---|---|---|---
0 | 0 | 0 | 0 | -3 | 0 | 0 | -3 | 0
$\nu^{\prime}_{1R}$ | $\nu^{\prime}_{2R}$ | $\nu^{\prime}_{3R}$ | $Q^{\prime}_{3L}$ | $u^{\prime}_{3R}$ | $d^{\prime}_{3R}$ | $H$ | $\theta$ |
0 | -3 | 0 | 1 | 1 | 1 | 0 | $q_{\theta}$ |
Table 1: The $U(1)_{B_{3}-L_{2}}$ charge assignments. A prime stands for a
weak eigenstate Weyl fermion and the family index $i$ takes values 1 and 2.
The flavon charge $q_{\theta}$ is a non-zero rational number.

The fermionic couplings of the $Z^{\prime}$ are expressed in the Lagrangian
density

$\mathcal{L}_{Z^{\prime}\psi}=-g_{Z^{\prime}}\mathopen{}\mathclose{{}\left(\overline{Q^{\prime}_{3L}}\not{Z}^{\prime}Q^{\prime}_{3L}+\overline{u^{\prime}_{3R}}\not{Z}^{\prime}u^{\prime}_{3R}+\overline{d^{\prime}_{3R}}\not{Z}^{\prime}d^{\prime}_{3R}-3\overline{L^{\prime}_{2L}}\not{Z}^{\prime}L_{2L}^{\prime}-3\overline{e^{\prime}_{2R}}\not{Z}^{\prime}e^{\prime}_{2R}-3\overline{\nu^{\prime}_{2R}}\not{Z}^{\prime}\nu^{\prime}_{2R}}\right)$
(1)

where $g_{Z^{\prime}}$ is the gauge coupling of the new $U(1)_{B_{3}-L_{2}}$
symmetry. In order to study the phenomenology of the model, however, we must
transform to the mass basis. We denote the 3-component column vectors in
family space by boldface letters,
$\mathbf{Q_{L}}^{\prime}=\mathopen{}\mathclose{{}\left(\mathbf{u_{L}}^{\prime},\mathbf{d_{L}}^{\prime}}\right)$,
$\mathbf{L_{L}}^{\prime}=\mathopen{}\mathclose{{}\left(\mathbf{v_{L}}^{\prime},\mathbf{e_{L}}^{\prime}}\right)$,
$\mathbf{u_{R}}^{\prime}$, $\mathbf{d_{R}}^{\prime}$,
$\mathbf{e_{R}}^{\prime}$ and $\bm{\nu}_{\mathbf{R}}^{\prime}$. The
transformation between the (primed) weak eigenbasis and the (unprimed) mass
eigenbasis written

$\mathbf{P}^{\prime}=V_{I}\mathbf{P}$ (2)

for
$I\in\mathopen{}\mathclose{{}\left\\{u_{L},d_{L},e_{L},\nu_{L},u_{R},d_{R},e_{R},\nu_{R}}\right\\}$.
Encoding the family-dependent couplings of eq. 1 into two 3 by 3 diagonal
matrices

$\Xi:=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&0&1\\\
\end{pmatrix},\quad\Omega:=\begin{pmatrix}0&0&0\\\ 0&1&0\\\ 0&0&0\\\
\end{pmatrix}$ (3)

and defining $\Lambda:=V_{I}^{\dagger}\alpha V_{I}$ where
$I\in\mathopen{}\mathclose{{}\left\\{u_{L},d_{L},e_{L},\nu_{L},u_{R},d_{R},e_{R},\nu_{R}}\right\\},\alpha\in\mathopen{}\mathclose{{}\left\\{\Xi,\Omega}\right\\}$,
we obtain $\mathcal{L}_{Z^{\prime}\psi}$ in the mass eigenbasis:

$\mathcal{L}_{Z^{\prime}\psi}=-g_{Z^{\prime}}\Big{(}\overline{\mathbf{u_{L}}}\Lambda_{\Xi}^{\mathopen{}\mathclose{{}\left(u_{L}}\right)}\not{Z}^{\prime}\mathbf{u_{L}}+\overline{\mathbf{d_{L}}}\Lambda_{\Xi}^{\mathopen{}\mathclose{{}\left(d_{L}}\right)}\not{Z}^{\prime}\mathbf{d_{L}}+\overline{\mathbf{u_{R}}}\Lambda_{\Xi}^{\mathopen{}\mathclose{{}\left(u_{R}}\right)}\not{Z}^{\prime}\mathbf{u_{R}}+\overline{\mathbf{d_{R}}}\Lambda_{\Xi}^{\mathopen{}\mathclose{{}\left(d_{R}}\right)}\not{Z}\mathbf{d_{R}}\\\
-3\overline{\bm{\nu}_{\mathbf{L}}}\Lambda_{\Omega}^{(\nu_{L})}\not{Z}^{\prime}\bm{\nu}_{\mathbf{L}}-3\overline{\mathbf{e_{L}}}\Lambda_{\Omega}^{(e_{L})}\not{Z}^{\prime}\mathbf{e_{L}}-3\overline{\bm{\nu}_{\mathbf{R}}}\Lambda_{\Omega}^{\mathopen{}\mathclose{{}\left(\nu_{R}}\right)}\not{Z}^{\prime}\bm{\nu}_{\mathbf{R}}-3\overline{\mathbf{e_{R}}}\Lambda_{\Omega}^{\mathopen{}\mathclose{{}\left(e_{R}}\right)}\not{Z}^{\prime}\mathbf{e_{R}}\Big{)}.$
(4)

Provided that $\mathopen{}\mathclose{{}\left(V_{d_{L}}}\right)_{23}\neq 0$,
eq. 4 couples the $Z^{\prime}$ to both $(\overline{b}s+\overline{s}b)$ and
$\mu^{+}\mu^{-}$. The $Z^{\prime}$ boson can thus mediate $b\rightarrow
s\mu^{+}\mu^{-}$transitions, thereby influencing $B$-observables.

The kinetic term of the flavon field reads

$\mathcal{L}_{\theta,\text{kin}}=\mathopen{}\mathclose{{}\left(D^{\mu}\theta}\right)^{\ast}\mathopen{}\mathclose{{}\left(D_{\mu}\theta}\right),$
(5)

with its covariant derivative being

$D_{\mu}\theta=\mathopen{}\mathclose{{}\left(\partial_{\mu}-ig_{Z^{\prime}}q_{\theta}Z^{\prime}_{\mu}}\right)\theta.$
(6)

The flavon field spontaneously breaks $U(1)_{B_{3}-L_{2}}$ by developing a
non-zero vacuum expectation value (VEV) $\langle\theta\rangle=v_{\theta}\neq
0$. As a consequence, the $Z^{\prime}$ acquires a mass
$M_{Z^{\prime}}=q_{\theta}g_{Z^{\prime}}v_{\theta}$. In the unitary gauge, the
$Z^{\prime}$ boson eats the massless Goldstone boson associated with the
broken symmetry and obtains a longitudinal polarisation mode. We are left with
three $Z^{\prime}$ degrees of freedom and a single real flavon field,
$\vartheta$.

###  2.1 Spontaneous symmetry breaking

The introduction of a new SM-singlet scalar field, real or complex, modifies
the scalar potential of the theory. In addition to the mass and quartic self-
interaction terms for the new scalar, the theory also allows for a
renormalisable dimension-4 term connecting the Higgs doublet to the beyond the
SM (BSM) scalar. Thus, the combined scalar potential for the SM-Higgs doublet
$(H)$ and the flavon field $(\vartheta)$ in the $B_{3}-L_{2}$ model reads

$V(H,\theta)=-\mu_{H}^{2}H^{\dagger}H+\lambda_{H}(H^{\dagger}H)^{2}-\mu_{\theta}^{2}\theta^{\ast}\theta+\lambda_{\theta}(\theta^{\ast}\theta)^{2}+\lambda_{\theta
H}\theta^{\ast}\theta H^{\dagger}H.$ (7)

Scalar potentials of this form have received considerable attention in the
past (see for instance Barger:2008jx ; Pruna:2013bma ; Falkowski:2015iwa ),
and we review here some standard steps. To find the lowest energy state of the
potential, we work in the unitary gauge and expand both fields about their
vacuum expectation values (VEVs), denoted by $v_{H}$ and $v_{\theta}$:

$H=\begin{pmatrix}0\\\ \frac{v_{H}+h^{\prime}}{\sqrt{2}}\\\
\end{pmatrix},\qquad\theta=\frac{v_{\theta}+\vartheta^{\prime}}{\sqrt{2}}.$
(8)

Minimising the scalar potential with respect to the VEVs,

$\frac{\partial V}{\partial v_{H}}=0,\quad\frac{\partial V}{\partial
v_{\theta}}=0,$ (9)

gives us the following two expressions:

$\displaystyle v_{H}$
$\displaystyle=\sqrt{\frac{4\mu_{H}^{2}\lambda_{\theta}-2\mu_{\theta}^{2}\lambda_{\theta
H}}{4\lambda_{H}\lambda_{\theta}-\lambda_{\theta H}^{2}}},$ (10)
$\displaystyle v_{\theta}$
$\displaystyle=\sqrt{\frac{4\mu_{\theta}^{2}\lambda_{H}-2\mu_{H}^{2}\lambda_{\theta
H}}{4\lambda_{H}\lambda_{\theta}-\lambda_{\theta H}^{2}}}.$ (11)

The requirements that the extremum be a local minimum and the potential
bounded from below for large field values provide the constraints

$\displaystyle 4\lambda_{H}\lambda_{\theta}-\lambda_{\theta H}^{2}$
$\displaystyle>0,$ (12) $\displaystyle\lambda_{H},\lambda_{\theta}>0,$ (13)

respectively Pruna:2013bma .

When we substitute the VEVs and the expanded scalar fields into eq. 7, terms
bilinear in $h^{\prime}$ and $\vartheta^{\prime}$ appear. Writing the
quadratic part of the potential as

$\mathcal{L}_{\text{quadratic}}=-\frac{1}{2}\begin{pmatrix}h^{\prime}&\vartheta^{\prime}\\\
\end{pmatrix}M^{2}\begin{pmatrix}h^{\prime}\\\ \vartheta^{\prime}\\\
\end{pmatrix}$ (14)

leads to the non-diagonal, symmetric mass matrix

$\displaystyle M^{2}$ $\displaystyle=\begin{pmatrix}\frac{1}{2}\lambda_{\theta
H}v_{\theta}^{2}+3\lambda_{H}v_{H}^{2}-\mu_{H}^{2}&\lambda_{\theta
H}v_{H}v_{\theta}\\\ \lambda_{\theta
H}v_{H}v_{\theta}&\frac{1}{2}\lambda_{\theta
H}v_{H}^{2}+3\lambda_{\theta}v_{\theta}^{2}-\mu_{\theta}^{2}\\\
\end{pmatrix},$
$\displaystyle=\begin{pmatrix}2\lambda_{H}v_{H}^{2}&\lambda_{\theta
H}v_{H}v_{\theta}\\\ \lambda_{\theta
H}v_{H}v_{\theta}&2\lambda_{\theta}v_{\theta}^{2}\\\ \end{pmatrix}.$ (15)

The matrix $M^{2}$ is diagonalised by an orthogonal rotation $P$,
parameterised by an angle $\phi$:

$\displaystyle M^{2}$
$\displaystyle=P^{T}\operatorname{diag}\mathopen{}\mathclose{{}\left(m_{h}^{2},m_{\vartheta}^{2}}\right)P,$
(16) $\displaystyle\begin{pmatrix}2\lambda_{H}v_{H}^{2}&\lambda_{\theta
H}v_{H}v_{\theta}\\\ \lambda_{\theta
H}v_{H}v_{\theta}&2\lambda_{\theta}v_{\theta}^{2}\\\ \end{pmatrix}$
$\displaystyle=\begin{pmatrix}\cos\phi&\sin\phi\\\ -\sin\phi&\cos\phi\\\
\end{pmatrix}\begin{pmatrix}m_{h}^{2}&0\\\ 0&m_{\vartheta}^{2}\\\
\end{pmatrix}\begin{pmatrix}\cos\phi&-\sin\phi\\\ \sin\phi&\cos\phi\\\
\end{pmatrix}.$ (17)

Solving for $\phi$, we find

$\sin 2\phi=\frac{2\lambda_{\theta
H}v_{H}v_{\theta}}{m_{\vartheta}^{2}-m_{h}^{2}}$ (18)

or, via a double angle formula,

$\sin\phi=\sqrt{\frac{1}{2}\mathopen{}\mathclose{{}\left(1-\sqrt{1-\frac{4\lambda_{\theta
H}^{2}v_{H}^{2}v_{\theta}^{2}}{\mathopen{}\mathclose{{}\left(m_{h}^{2}-m_{\vartheta}^{2}}\right)^{2}}}}\right)}.$
(19)

We have thus obtained the field rotation $P$, which transforms the primed
scalar field basis
$\mathopen{}\mathclose{{}\left(h^{\prime},\vartheta^{\prime}}\right)$ into the
(unprimed) mass eigenbasis:

$\begin{pmatrix}h\\\ \vartheta\\\
\end{pmatrix}=\begin{pmatrix}\cos\phi&-\sin\phi\\\ \sin\phi&\cos\phi\\\
\end{pmatrix}\begin{pmatrix}h^{\prime}\\\ \vartheta^{\prime}\\\
\end{pmatrix}.$ (20)

The smallest eigenvalue of $M^{2}$, $m_{h}^{2}$, and the associated eigenstate
$h$ are taken to correspond to the 125 GeV Higgs boson discovered at the LHC
experiments, whereas the larger-mass eigenstate $\vartheta$ is the flavon
boson whose mass is written as $m_{\vartheta}$.

Expanding the scalar fields about their VEVs and rotating into the mass
eigenbasis as described above, the kinetic term
$\mathcal{L}_{\theta,\text{kin}}$ yields an interaction term

$\mathcal{L}_{\theta,\text{kin}}\supset\cos\phi\,g_{Z^{\prime}}^{2}q_{\theta}^{2}v_{\theta}\vartheta
Z^{\prime}_{\mu}{Z^{\prime}}^{\mu}.$ (21)

This term helps give rise to the flavonstrahlung process depicted in the
right-hand panel of figure 1 and it will therefore play a key role in the
present study.

The Higgs-flavon axis of the model can be expressed in terms of three
parameters: $m_{\vartheta},\phi$ and $v_{\theta}$, the last of which can be
decomposed as $v_{\theta}=M_{Z^{\prime}}/({q_{\theta}g_{Z^{\prime}}})$.
Accordingly, the quartic couplings $\lambda_{H}$ and $\lambda_{\theta}$,
together with eq. 18, become

$\displaystyle\lambda_{H}$
$\displaystyle=\frac{m_{h}^{2}\cos^{2}\phi+m_{\vartheta}^{2}\sin^{2}\phi}{2v_{H}^{2}},$
(22) $\displaystyle\lambda_{\theta}$
$\displaystyle=\frac{m_{\vartheta}^{2}\cos^{2}\phi+m_{h}^{2}\sin^{2}\phi}{2v_{\theta}^{2}},$
(23) $\displaystyle\lambda_{\theta H}$
$\displaystyle=\frac{\sin(2\phi)\mathopen{}\mathclose{{}\left(m_{\vartheta}^{2}-m_{h}^{2}}\right)}{2v_{H}v_{\theta}}.$
(24)

###  2.2 Assumptions

We must specify the model further before we can study its phenomenology. In
particular, the $V_{I}$ mixing matrices deserve our attention. With
simplicity, ease of passing flavour bounds and the ability to explain the
neutral current $b\rightarrow s\mu^{+}\mu^{-}$ anomalies as guiding
principles, an example set of mixing matrices $V_{I}$ was proposed in ref.
Allanach:2020kss :

$V_{d_{L}}=\begin{pmatrix}1&0&0\\\ 0&\cos\theta_{sb}&-\sin\theta_{sb}\\\
0&\sin\theta_{sb}&\cos\theta_{sb}\\\ \end{pmatrix},$ (25)

$V_{d_{R}}=1,V_{e_{R}}=1,V_{e_{L}}=1$ and $V_{u_{R}}=1$, where here $1$
denotes the 3 by 3 identity matrix. These imply that
$V_{u_{L}}=V_{d_{L}}V^{\dagger}$ and $V_{\nu_{L}}=U^{\dagger}$, where $V$ and
$U$ are the CKM and PMNS matrices, respectively. Here, we will adhere to the
same set of mixing matrices while keeping in mind that this choice is just
intended to provide an example case for further study. This set of fermion
mixing matrices results in a Lagrangian containing the terms

$\mathcal{L}\supset-
g_{Z^{\prime}}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\frac{1}{2}\sin
2\theta_{sb}\overline{s}\not{Z}^{\prime}P_{L}b+\text{H.c.}}\right)-3\overline{\mu}\not{Z}^{\prime}\mu}\right],$
(26)

where $P_{L}$ is a left-handed spinor helicity projection operator. Once the
$Z^{\prime}$ is integrated out, these terms yield a contribution to the Wilson
coefficient $\mathcal{C}_{9}$ from

${\mathcal{H}}_{\text{WET}}=\ldots+\mathopen{}\mathclose{{}\left[\mathcal{C}_{9}\mathcal{N}(\bar{b}\gamma_{\alpha}P_{L}s)(\bar{\mu}\gamma^{\alpha}\mu)+\text{H.c.}}\right],$
(27)

in the weak effective theory Hamiltonian, which can significantly ameliorate
the $b\rightarrow s\mu^{+}\mu^{-}$ anomalies Alguero:2019ptt ; Alok:2019ufo ;
Ciuchini:2019usw ; Aebischer:2019mlg ; Datta:2019zca ; Kowalska:2019ley ;
Arbey:2019duh ; Gubernari:2022hxn . In eq. 27, $\alpha\in\\{0,1,2,3\\}$ is a
space-time index and

${\mathcal{N}}=\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{*}\frac{\alpha}{4\pi}=1/(36\text{~{}TeV})^{2}$
(28)

is a normalising constant.

We shall set the flavon charge $q_{\theta}$ equal to 1 unless stated
otherwise. The flavonstrahlung cross section is proportional to
$q_{\theta}^{2}$, so it is straightforward to extend most of our results to
different values of the charge.

##  3 Flavon Phenomenology

In this section we first review the phenomenological constraints on the
$U(1)_{B_{3}-L_{2}}$ model obtained in earlier work. These limits apply to the
parameter set $\\{g_{Z^{\prime}},M_{Z^{\prime}},\theta_{23}\\}$, the three
inputs which influence the ability of the model to explain $b\rightarrow
s\mu^{+}\mu^{-}$ anomalies. We then move on to study the flavon sector of the
theory, obtaining an upper bound on the Higgs-flavon mixing angle $\phi$ and
discussing the leading flavon decay channels.

We have updated the FeynRules Alloul:2013bka  implementation of the
$U(1)_{B_{3}-L_{2}}$ model from ref. Allanach:2020kss  by adding the flavon
sector, which was previously neglected.444The model file can be found in the
ancillary information of the arXiv version of this work. We have also used
FeynRules to convert the model into UFO format Degrande:2011ua .

###  3.1 Fit to neutral current $b\rightarrow s\mu^{+}\mu^{-}$ anomalies and
LHC constraints

One may use eq. 26 to match the $U(1)_{B_{3}-L_{2}}$ model to fits of
$b\rightarrow s\mu^{+}\mu^{-}$data, as was done in ref. Allanach:2020kss .
This condition lets us eliminate the mixing angle $\theta_{sb}$ and leaves us
with two free parameters relevant to $B$ decay data: $M_{Z^{\prime}}$ and
$g_{Z^{\prime}}$. Defining the dimensionless quantity $x$ as

$x\coloneqq g_{Z^{\prime}}\frac{1\text{~{}TeV}}{M_{Z^{\prime}}},$ (29)

one obtains

$\theta_{sb}=\frac{1}{2}\sin^{-1}\mathopen{}\mathclose{{}\left(\frac{-5.1\times
10^{-4}\mathcal{C}_{9}}{x^{2}}}\right).$ (30)

In this work, we use $\mathcal{C}_{9}=-0.73\pm 0.15$, which is the best-fit
value obtained in Altmannshofer:2021qrr  prior to the recent LHCb updates
LHCb:2022qnv ; LHCb:2022zom  of the lepton flavour universality ratios $R_{K}$
and $R_{K^{*}}$.

It was found in ref. Allanach:2020kss  that there are both lower and upper
bounds on the value of $x$. The lower bound stems from measurements of
$B_{s}-\overline{B_{s}}$ mixing, which the $Z^{\prime}$ contributes to at
tree-level. The upper bound originates from measurements of the neutrino
trident cross-section,
$\sigma\mathopen{}\mathclose{{}\left(\nu_{\mu}N\rightarrow\nu_{\mu}N\mu^{+}\mu^{-}}\right)$,
which also receives $Z^{\prime}$ contributions at tree-level. We are left with
the constraint $0.04<x<0.67$ Azatov:2022itm  which we will adhere to in this
work. Substituting such values of $x$ into eq. 30 implies that $\theta_{sb}$
is small. For small $\theta_{sb}$, the collider phenomenology that we shall
discuss is not sensitive to its precise value and so should also be valid for
up-to-date fits of $\mathcal{C}_{9}$ incorporating the recent LHCb
results.555See Ciuchini:2022wbq  for a recent fit. This is because
flavonstrahlung production proceeds initially via $Z^{\prime}$ production,
which dominantly proceeds via $b\bar{b}$ fusion and is proportional to
$g_{Z^{\prime}}^{2}\cos^{4}\theta_{sb}\approx
g_{Z^{\prime}}^{2}[1-2\theta_{sb}^{2}+\mathcal{O}(\theta_{sb}^{4})]$.

In ref. Allanach:2021gmj , a set of LHC constraints was recast on the
$M_{Z^{\prime}}-g_{Z^{\prime}}$ plane of the theory. Figure 10 of that paper
shows that the previously allowed light $Z^{\prime}$ region of parameter
space, with $M_{Z^{\prime}}<0.3$ TeV, is all but ruled out, and in the rest of
the paper, we shall assume that $M_{Z^{\prime}}>1$ TeV. A recent recasting of
the CMS high-mass di-lepton searches CMS:2021ctt  eliminated at 95% CL all
parameter points with $M_{Z^{\prime}}\lesssim 2\text{~{}TeV}$ Azatov:2022itm .

###  3.2 Perturbativity of $Z^{\prime}$ couplings

Neglecting fermion massess in comparison with $M_{Z^{\prime}}$, the partial
decay rate of the $Z^{\prime}$ turning into a pair of Weyl fermions reads

$\Gamma\mathopen{}\mathclose{{}\left(Z^{\prime}\rightarrow
f_{i}\bar{f}_{i}}\right)=\frac{C_{i}}{24\pi}(q_{i}g_{Z^{\prime}})^{2}M_{Z^{\prime}},$
(31)

where $C_{i}$ is the number of colour degrees of freedoms of the fermion
$f_{i}$, and $q_{i}$ is its $U(1)_{B_{3}-L_{2}}$ charge666Note that
$\Gamma/M_{Z^{\prime}}$ is independent of rescaling all charges by absorbing
the scaling in $g_{Z^{\prime}}$. in units of $g_{Z^{\prime}}$, as assigned in
Table 1. Summing over all fermion species (except for the right-handed
neutrinos which are assumed to be more massive than the $Z^{\prime}$) yields
the equation

$\frac{\Gamma}{M_{Z^{\prime}}}=\frac{13g_{Z^{\prime}}^{2}}{8\pi}$ (32)

where $\Gamma$ is the total width of the $Z^{\prime}$. We impose the limit
$\Gamma/M_{Z^{\prime}}<1/3$ to ensure that our perturbative cross-section
calculations remain valid. This translates into
$g_{Z^{\prime}}<\sqrt{8\pi/39}~{}=~{}0.80$. The perturbativity condition
together with a fit to $b\rightarrow s\mu^{+}\mu^{-}$ data lead to an upper
bound on the $Z^{\prime}$ mass: $M_{Z^{\prime}}<20$ TeV.

###  3.3 Constraints on the mixing angle $\phi$

There are numerous experimental and theoretical constraints on the Higgs-
flavon mixing angle $\phi$. These are discussed at length in, e.g. refs.
Robens:2015gla ; Robens:2022cun ; Falkowski:2015iwa  in the context of the
real singlet extension of the SM. We expect the constraints to be largely the
same in the $B_{3}-L_{2}$ model, as most constraints are independent of the
presence of the $Z^{\prime}$ and the extra degree of freedom from the
complexity of the flavon field. Because the $B_{3}-L_{2}$ model is seen as a
low energy effective theory, we disregard some theoretical constraints, namely
perturbative unitarity of scattering amplitudes in the high energy limit and
the lack of Landau poles below the Planck scale Bause:2021prv . In this work,
we impose four constraints on the $B_{3}-L_{2}$ model: limits from direct
Higgs searches at hadron colliders, the Higgs signal strength measurements,
agreement with the experimentally measured $W$ boson mass and perturbativity
of the Higgs-quartic couplings at the scale of the effective theory. These
correspond to the four coloured regions in figure 2.

Figure 2: 95% CL limits on the Higgs-flavon mixing angle stemming from direct
Higgs searches at the LHC and Tevatron (obtained using HiggsBounds), ATLAS
signal strength measurements, the $W$ boson mass and perturbativity of
$\lambda_{H}$. The white region is currently allowed. To obtain the
$M_{W}$-bound, we have assumed $g_{Z^{\prime}}=0.15,M_{Z^{\prime}}=3$ TeV, but
because $M_{W}$ is only very weakly dependent on the $Z^{\prime}$ parameters,
we will use this bound for all values of $M_{Z^{\prime}}$ and $g_{Z^{\prime}}$
considered in this work.

####  3.3.1 Higgs signal strength

The rotation into the mass eigenbasis of the scalar fields in eq. 20 modifies
all SM Higgs couplings by a factor of $\cos\phi$. The Higgs signal strength,
defined as the production cross-section times branching ratio (BR) normalised
to the SM prediction,
$\mu:=(\sigma\times\text{BR})_{\text{obs}}/(\sigma\times\text{BR})_{\text{SM}}$
for a given Higgs production and decay mode, is then predicted to be
$\mu=\cos^{2}\phi$ irrespective of the mode. The ATLAS and CMS Run 2
combination results for the global signal strength read
$\mu_{\text{ATLAS}}>0.92,\,\mu_{\text{CMS}}>0.90$ at 95 % CL Workman:2022ynf .
The more stringent of the two, $\mu_{\text{ATLAS}}$, yields for the Higgs-
flavon mixing angle: $\mathopen{}\mathclose{{}\left|\sin\phi}\right|<0.28$.
This limit on $\sin\phi$ is independent of the flavon mass.

####  3.3.2 Direct searches

We utilise the public code HiggsBounds 5.3.2beta Bechtle:2012lvg ;
Bechtle:2013wla ; Bechtle:2008jh ; Bechtle:2011sb ; Bechtle:2020pkv ;
Bechtle:2015pma  to obtain 95% CL direct search limits on an extra scalar
field from the LHC and Tevatron. This bound is stronger at lower flavon masses
and starts to wane for $m_{\vartheta}\gtrsim$ 750 GeV; the constraint is
visible on the upper left-hand side of figure 2.

####  3.3.3 Perturbativity of quartic couplings

For the theory to remain perturbative, we will enforce the conditions
$\mathopen{}\mathclose{{}\left|\lambda_{H},\lambda_{\theta},\lambda_{\theta
H}}\right|<4\pi$ on the quartic couplings. For most values of the ratio
$x=g_{Z^{\prime}}\text{~{}TeV}/M_{Z^{\prime}}$ in its allowed region
$x\in[0.04,0.67]$ the quartic Higgs coupling $\lambda_{H}$, which is
independent of $x$, places a more stringent bound on $\sin\phi$ than either
$\lambda_{\theta H}\propto x$ or $\lambda_{\theta}\propto x^{2}$. The
constraint arising from perturbativity of $\lambda_{H}$ corresponds to the
orange region in figure 2. At the end of the allowed interval, where $x\sim
0.6$, the bound from $\lambda_{H}$ is superseded by $\lambda_{\theta H}$.
However, for $m_{\vartheta}\lesssim$ 5 TeV, neither of these limits is
competitive against the bound coming from $M_{W}$ measurements. For flavon
masses much more massive than this, the perturbativity of the couplings
becomes the tightest constraint on the mixing angle. This can be phrased in
another way: for a given mixing angle, there is an upper bound on the flavon
mass coming from perturbativity of the three quartic couplings. The strictest
bound may depend on $x$ and the flavon charge $q_{\theta}$, but $\lambda_{H}$
will always provide an upper limit independent of $x$ and $q_{\theta}$.

####  3.3.4 $W$ boson mass

We now investigate the prediction of the $W$ boson mass in the $B_{3}-L_{2}$
model. In the real singlet extension of the SM, for an extra scalar field more
massive than $\sim$ 300 GeV, agreement between the experimentally measured W
boson mass $M_{W}$ and the model prediction at the one-loop level places a
bound more austere than that arising from the oblique $S$, $T$ and $U$
parameters Lopez-Val:2014jva . The $Z^{\prime}$ boson cannot influence the
oblique parameters in the $B_{3}-L_{2}$ model, but it does affect $M_{W}$ via
$Z^{\prime}$-induced vertex corrections, as we will see. We thus posit that
$M_{W}$ will provide the stricter of the two limits in the $B_{3}-L_{2}$
model, too, and confirm the assertion by calculating the $W$ boson mass in the
model.

Predicting the value of $M_{W}$ is based on matching the 4-Fermi theory muon
lifetime with the 1-loop calculation using the full Lagrangian of the theory
(see Sirlin:1980nh ; Hollik:1988ii ; Denner:1991kt ; Wells:2005vk  for more
detailed accounts). The matching yields an expression connecting the
experimentally measured Fermi coupling constant $G_{F}$ to the parameters of
the $B_{3}-L_{2}$ model:

$\frac{G_{F}}{\sqrt{2}}=\frac{e^{2}}{8M_{W}^{2}\sin^{2}\theta_{W}}\mathopen{}\mathclose{{}\left(1+\Delta
r}\right),$ (33)

where $\Delta r$ contains all of the loop corrections to the decay process in
the full theory. Taking $M_{Z}$, $G_{F}$ and $\alpha$ as experimental inputs
and working with the on-shell definition of the weak mixing angle where
$\sin^{2}\theta_{W}=1-M_{W}^{2}/M_{Z}^{2}$ to all orders in perturbation
theory, we can rearrange the above equation to obtain a prediction for the
$W$-boson mass:

$M_{W}^{2}=\frac{1}{2}M_{Z}^{2}\mathopen{}\mathclose{{}\left[1+\sqrt{1-\frac{4\pi\alpha}{\sqrt{2}G_{F}M_{Z}^{2}}\mathopen{}\mathclose{{}\left[1+\Delta
r(M_{W}^{2})}\right]}\,}\right].$ (34)

Given a small perturbation $\delta(\Delta r)$, resulting from BSM physics, the
$W$ boson mass changes by

$\Delta
M_{W}\simeq-\frac{1}{2}M_{W}\frac{\sin^{2}\theta_{W}}{\sin^{2}\theta_{W}-\cos^{2}\theta_{W}}\delta(\Delta
r).$ (35)

At the one-loop level in the $U(1)_{B_{3}-L_{2}}$ model, there are two kinds
of BSM contributions to $\Delta r$:777We neglect loops containing $\ell\ell
h$\- and $\ell\ell\vartheta$-vertices, for $\ell=e,\mu$, as their
contributions are of order $m_{\ell}^{2}/v_{H}^{2}~{}\ll~{}1$. those arising
from $Z^{\prime}$-vertices (figure 3) and those arising from Higgs-flavon
mixing (figure 4), which were evaluated in ref. Lopez-Val:2014jva  and will
always act to make the $W$ boson lighter. Each of the two sets of diagrams
depends on different parameters: the size of the $Z^{\prime}$ vertex
contributions is a function of $\\{M_{Z^{\prime}},g_{Z^{\prime}}\\}$ whereas
Higgs-flavon mixing hinges on $\\{\sin\phi,m_{\vartheta}\\}$.

We employ the FeynArts, FormCalc and LoopTools packages (versions 3.11, 9.9
and 2.16, respectively) Hahn:2000kx ; Hahn:1998yk  to aid with the calculation
and to evaluate the results numerically. Ignoring terms of order
$m_{\mu}^{2}/M_{W}^{2}$ and $m_{\mu}^{2}/M_{Z^{\prime}}^{2}$, the Lorentz
structure of the $U(1)_{B_{3}-L_{2}}$ amplitude is identical to that of the
4-Fermi theory and we can match the the two theories at the amplitude level.
In doing so, we find that $\delta(\Delta r)$ is dominated by the oblique
corrections to the vector boson propagators, overwhelming the
$Z^{\prime}$-induced effects by several orders of magnitude. The reason is
that all of the $Z^{\prime}$-effects are of order
$g_{Z^{\prime}}^{2}m_{\mu}^{2}/M_{Z^{\prime}}^{2}$ and thus become negligible
for a TeV scale $Z^{\prime}$. Thus, the resulting constraint on $\sin\phi$ is
essentially independent of $M_{Z^{\prime}}$ and $g_{Z^{\prime}}$.

We proceed by comparing the theoretical prediction with the experimentally
determined $W$ boson mass and insist on agreement at the 2$\sigma$ level. The
empirical and SM-predicted values of $M_{W}$ are obtained from the Particle
Data Group Workman:2022ynf . For the experimentally measured $M_{W}$, we use
the current world average (prior to the 2022 CDF measurement888If the 2022 CDF
measurement is considered, all non-zero mixing angles are ruled out at the
$2\sigma$ level. The $U(1)_{B_{3}-L_{2}}$ model can only make the $W$ boson
lighter than predicted by the SM, thus _increasing_ the tension between
experiment and theory.), $M_{W}^{\text{exp}}=80.377\pm 0.012$ GeV, whereas the
SM prediction stands at $M_{W}^{\text{th}}=80.356\pm 0.006$ GeV. Combining the
two errors in quadrature, we use eq. 35 to obtain a $U(1)_{B_{3}-L_{2}}$ model
prediction for the mass of the $W$ boson and require that the predicted and
measured values disagree by less than $2\sigma$. This constraint corresponds
to the yellow region in figure 2 and is the most stringent for
$0.5<m_{\vartheta}/\text{~{}TeV}<5$.

Figure 3: The $Z^{\prime}$-induced vertex and self-energy corrections
contributing to the $\Delta r$ parameter. There are no contributions from
similar diagrams but with the flavon Goldstone replacing the $Z^{\prime}$ in
the loop because the $U(1)_{B_{3}-L_{2}}$ symmetry is vectorial and does not
come with a Yukawa sector.

Figure 4: Feynman diagrams corresponding to the three $W$ boson self-energy
contributions involving the Higgs boson. We work in the Feynman gauge
($\xi=1$) and $G$ stands for the charged SM Goldstone boson. Each SM
$h$-vertex is suppressed by a factor of $\cos\phi$ after the Higgs field mixes
with the flavon field, but this is complemented by a set of identical diagrams
with the physical flavon field running in the loop. Though not drawn here, the
$Z$-boson self-energies are modified in a similar manner.

###  3.4 Flavon decay channels

As the flavon couples to SM fields only through mixing with the Higgs field,
its tree-level decay channels resemble those of the SM Higgs whenever
$\sin\phi\neq 0$. A key difference, however, is that the flavon is assumed to
be considerably heavier than the Higgs, which allows decays into on-shell
$W^{-}W^{+},ZZ$ and $t\overline{t}$ final states. Assuming a TeV scale flavon
and $Z^{\prime}$, and that $m_{\vartheta}<2M_{Z^{\prime}}$ (which covers most
of the parameter space studied in this work), there are three channels that
dominate the flavon decay rate. The leading channel is $\vartheta\rightarrow
WW$ with tree-level partial width

$\Gamma_{\vartheta\rightarrow WW}=\frac{m_{\vartheta}^{3}\sin^{2}{\phi}}{16\pi
v_{H}^{2}}+\mathcal{O}\mathopen{}\mathclose{{}\left(\frac{M_{W}^{2}}{m_{\vartheta}^{2}}}\right).$
(36)

This is followed by $\vartheta\rightarrow hh$ (obtained with FeynRules):

$\Gamma_{\vartheta\rightarrow
hh}=\frac{m_{\vartheta}^{3}\cos^{4}\phi\sin^{2}\phi}{32\pi
v_{H}^{2}}+\mathcal{O}\mathopen{}\mathclose{{}\left(\frac{M_{W}^{2}}{m_{\vartheta}^{2}}}\right)+\mathcal{O}\mathopen{}\mathclose{{}\left(\frac{v_{H}}{v_{\theta}}}\right)$
(37)

and $\vartheta\rightarrow ZZ$:

$\Gamma_{\vartheta\rightarrow ZZ}=\frac{m_{\vartheta}^{3}\sin^{2}{\phi}}{32\pi
v_{H}^{2}}+\mathcal{O}\mathopen{}\mathclose{{}\left(\frac{M_{Z}^{2}}{m_{\vartheta}^{2}}}\right).$
(38)

These expressions lead to the relation $\Gamma_{\vartheta\rightarrow
WW}/\Gamma_{\vartheta\rightarrow hh}\approx
2\approx\Gamma_{\vartheta\rightarrow WW}/\Gamma_{\vartheta\rightarrow ZZ}$,
which is clearly demonstrated in figure 5. The leading fermionic final state
is a $t\overline{t}$ pair because the top quark Yukawa coupling is the largest
Yukawa coupling in the SM. We have evaluated the BRs numerically using
MadWidth Alwall:2014bza .

Figure 5: Tree-level flavon BRs for the case $m_{\vartheta}<2M_{Z^{\prime}}$
so that the flavon is unable to decay into a pair of on-shell $Z^{\prime}$
bosons. The three leading final states, $WW$, $ZZ$ and $hh$, come in the
approximate ratio 2:1:1. The upper panel is an enlargened version of the
shaded region of the lower panel.

We may also study the case with $m_{\vartheta}>2M_{Z^{\prime}}$, although,
owing to the constraint $M_{Z^{\prime}}\gtrsim$ 2 TeV, this necessarily takes
us to multi-TeV flavon masses. If one allows for a flavon mass of order 10
TeV, the decay $\vartheta\rightarrow Z^{\prime}Z^{\prime}$ can become one of
the leading channels. This is shown in figure 6 where the BR into a pair of
$Z^{\prime}$s keeps increasing rapidly as more phase space is made available
by lowering $M_{Z^{\prime}}/m_{\vartheta}$. To obtain the figure, we have
arbitrarily picked $m_{\vartheta}=12$ TeV, $\sin\phi=0.05$ and
$g_{Z^{\prime}}=0.5$. The pink region is excluded at the 95% CL by LHC data,
as shown in figure 10 of ref. Azatov:2022itm . This constraint, which is a
function of $g_{Z^{\prime}}/M_{Z^{\prime}}$, is included for completeness only
— the primary purpose of the figure is to show how the mass ratio influences
the BRs.

Figure 6: Flavon BRs as a function of the mass ratio
$M_{Z^{\prime}}/m_{\vartheta}$ when the flavon is much heavier than the
$Z^{\prime}$ and the $U(1)_{B_{3}-L_{2}}$ gauge coupling is order one. As the
mass ratio is lowered, we see that the $Z^{\prime}Z^{\prime}$ final state
becomes increasingly important. The region excluded at the 95% CL by LHC data,
for this particular choice of parameters, is shown in pink. The green and blue
lines are overlap to such a degree that they are indistinguishable by eye.

##  4 Flavonstrahlung at Colliders

We now proceed to study the tree-level flavonstrahlung cross-section at hadron
and muon colliders of varying centre-of-mass energies (the dominant Feynman
diagram is shown again in figure 7). Flavonstrahlung would likely not be the
first detectable direct BSM signal in the $U(1)_{B_{3}-L_{2}}$ model, as it is
more probable that exclusive $Z^{\prime}$ production would be observed at
lower energies and luminosities. As to whether flavonstrahlung would be the
first sign of the flavon particle depends primarily on the value of the Higgs-
flavon mixing angle. For sizeable mixing angles, we may discover the flavon
through the conventional SM Higgs production processes before reaching the
energies and luminosities required for flavonstrahlung. However, observations
of $Z^{\prime}$ and flavon resonances alone would not tell us whether the two
particles interact with each other and whether the scalar field is involved in
generating the $Z^{\prime}$ mass. Flavonstrahlung is unique in that it
combines the $Z^{\prime}$ and flavon in a single process. The subsequent
decays of the $Z^{\prime}$ and $\vartheta$ via their leading channels,
$Z^{\prime}\rightarrow\mu^{-}\mu^{+}$ and $\vartheta\rightarrow W^{-}W^{+}$,
yield the final state $W^{+}W^{-}\mu^{+}\mu^{-}$ with a $WW$ resonance at the
flavon mass $m_{\vartheta}$ and a di-muon resonance at $M_{Z^{\prime}}$. As we
have seen, the flavon may also decay to $ZZ$ or $HH$ with sizeable BRs, and we
leave it for future work to determine which channel is best for
flavonstrahlung hunting.

1.4 (300,75)(100,0) [arrow](220,50)(245,25) [arrow](245,25)(220,0)
(295,50)(270,25)3 (270,25)(295,0)33 (245,25)(270,25)33
(260.5,33)[c]${Z^{\prime}}$ (215,0)[r]$\overline{b},\mu^{+}$
(215,50)[r]$b,\mu^{-}$ (302,50)[l]$\vartheta$ (302,0)[l]$Z^{\prime}$

Figure 7: Flavonstrahlung at a hadron collider or a muon collider.

It should be noted that there are two resonant contributions to the
flavonstrahlung cross-section: in addition to
$pp\rightarrow{Z^{\prime}}^{\ast}\rightarrow Z^{\prime}\vartheta$ (case 1),
discussed above, the cross-section also picks up a contribution from
$pp\rightarrow Z^{\prime}\rightarrow{Z^{\prime}}^{\ast}\vartheta$ (case 2),
where the intermediate $Z^{\prime}$ is on-shell. Choosing for concreteness the
leading decay channels $Z^{\prime}\rightarrow\mu^{-}\mu^{+}$ and
$\vartheta\rightarrow W^{-}W^{+}$, case 2 yields a resonance peak in the
invariant mass of the 4-particle final state at
$q^{2}~{}=~{}M_{Z^{\prime}}^{2}$ as opposed to the di-muon resonance in case
1. We use the event generator MadGraph5_aMC@NLO v.3.4.1 Alwall:2014hca
(abbreviated “MG5_aMC” in the following) to illustrate the two resonances
schematically in figure 8 for the final state $\mu^{+}\mu^{-}W^{+}W^{-}$ at a
hadron collider. The orange and red lines corresponds to cases 1 and 2,
respectively. We have used the condition $|M^{\ast}-M|<5\Gamma$ as the
definition of a propagator being on-shell, with $M^{\ast}$ the invariant mass
of the four-momentum carried by the propagator and $M$ and $\Gamma$ the pole
mass and width of the particle. This is achieved using the
$\text{BW}_{\text{cut}}$ parameter in MG5_aMC.

Figure 8: Illustration of the two resonant contributions to the
flavonstrahlung cross-section for the leading $\mu^{+}\mu^{-}W^{+}W^{-}$ final
state. The blue region shows the final state invariant mass distribution in
the absence of SM backgrounds at the 14 TeV HL-LHC. The orange curve
delineates the contribution arising from case 1, where the second $Z^{\prime}$
in figure 7 is on-shell, whereas the red curve corresponds to case 2, where
the first $Z^{\prime}$ is on-shell.

As to which resonance contributes more to the cross-section depends on the
partonic energy and the masses of the flavon and $Z^{\prime}$. Because case 2
entails lower centre-of-mass energies than case 1, it is more prevalent at
lower energy colliders such as the HL-LHC or a 3 TeV muon collider which may
lack high enough partonic energies to put both a TeV scale $Z^{\prime}$ and
$\vartheta$ on-shell simultaneously. Case 1, on the other hand, is favoured at
higher energy colliders such as the FCC-hh or a 10 TeV muon collider where the
sum of the $Z^{\prime}$ and $\vartheta$ masses is less than the partonic
centre of mass energy for a substantial fraction of collisions.

In order to capture contributions from both resonances in our cross-section
computations, we shall study the process $pp\rightarrow
Z^{\prime}\rightarrow\vartheta Z^{\prime}\rightarrow\vartheta\mu^{+}\mu^{-}$,
where the rightmost $Z^{\prime}$ propagator in the Feynman diagram of figure 7
splits into a di-muon pair.999All flavonstrahlung cross-sections reported
hereafter are computed for the $\vartheta\mu^{+}\mu^{-}$ final state, even
when not explicitly stated. We require only that the final state flavon and
muons are on-shell, thus allowing both case 1 and case 2 from above to
contribute to the total amplitude. Concentrating on the di-muon final state is
well-motivated by its clean experimental signature, as well as the large
$Z^{\prime}\rightarrow\mu^{+}\mu^{-}$ branching ratio, making it the most
promising mode for observing flavonstrahlung.

In the presence of non-zero Higgs-flavon mixing, a $Z^{\prime}\vartheta$ final
state may also be produced at tree-level via a $t$-channel bottom quark or
muon exchange. These processes are shown in figure 9. The associated matrix
elements are suppressed because the $\vartheta bb$\- and
$\vartheta\mu\mu$-vertices come with couplings
$\sin\phi\mathopen{}\mathclose{{}\left(m_{b}/v_{H}}\right)$ and
$\sin\phi\mathopen{}\mathclose{{}\left(m_{\mu}/v_{H}}\right)$, respectively.
Assuming $\sin\phi\sim 0.1$, the inclusion of the $t$-channel exchange
typically changes the $\vartheta Z^{\prime}$ production cross-sections by
approximately $0.1\%$ and never by more than around $3\%$. We thus neglect
contributions arising from the $t$-channel fermion exchange in this work.

(98,86) (175,-95) Black
[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=3,arrowinset=0.2](128,-10)(176,-26)
[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=3,arrowinset=0.2](176,-26)(176,-74)
[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=3,arrowinset=0.2](176,-74)(130,-90)
(176,-74)(224,-90)53 [dash,dashsize=6.2](177,-26)(224,-10) (193,-50)[c]$b,\mu$
(130,-7)[r]$b,\mu^{-}$ (128,-90)[r]$\overline{b},\mu^{+}$
(240,-8)[r]$\vartheta$ (228,-92)[l]$Z^{\prime}$ (98,86) (75,-95) Black
[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=3,arrowinset=0.2](128,-10)(176,-26)
[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=3,arrowinset=0.2](176,-26)(176,-74)
[arrow,arrowpos=0.5,arrowlength=5,arrowwidth=3,arrowinset=0.2](176,-74)(130,-90)
(176,-26)(224,-10)53 [dash,dashsize=6.2](176,-74)(224,-90) (193,-50)[c]$b,\mu$
(130,-7)[r]$b,\mu^{-}$ (128,-90)[r]$\overline{b},\mu^{+}$
(241,-8)[r]$Z^{\prime}$ (228,-92)[l]$\vartheta$

Figure 9: Flavon production with an associated $Z^{\prime}$ via a $t$-channel
fermion exchange. The bottom quark exchange corresponds to hadron colliders,
whereas muon exchange is possible at muon colliders. For the regions of
parameter space considered in this work, the contribution to the $\vartheta
Z^{\prime}$ production cross-section from this channel is typically of order
$0.1\%$ or less.

###  4.1 Flavonstrahlung at Hadron Colliders

We import the UFO model file into MG5_aMC and use it to calculate leading-
order flavonstrahlung cross-sections for proton-proton ($pp$) collisions. The
largest partonic contribution to $Z^{\prime}$ production comes from the
$b\overline{b}$ initial state. We thus use the five-flavour parton
distribution function (PDF) NNPDF2.3LO where the $b$ quark is absorbed into
the proton and jet definitions and treated as massless. There are also
negligible contributions to the cross-sections from $s\overline{b}$,
$b\overline{s}$ and $s\overline{s}$ initial states, which are nevertheless
included in our numerical estimates.

We apply the default MG5_aMC cuts on the phase space of the di-muon pair
throughout the computations. Placing cuts according to the specifications of
each detector studied in this work would have a negligible impact on our
results, and the designs of future detectors are not yet fixed anyway. We thus
require that the final-state muon isolation satisfy $\Delta R>0.4$ and that
transverse momenta and pseudo-rapidities of the muons fulfil the conditions
$p_{T}>10$ GeV and $\mathopen{}\mathclose{{}\left|\eta}\right|<2.5$. No cuts
are applied on the final state flavon.

To study the process, we select currently allowed combinations of
$\\{M_{Z^{\prime}},g_{Z^{\prime}}\\}$, and compute the flavonstrahlung cross-
section as a function of the flavon mass. These combinations are illustrated
in the left-hand panel of figure 10 which is adapted from figure 10 of ref.
Azatov:2022itm . The region above the solid black line has been excluded at
the 95% CL by the CMS high-mass Drell-Yan searches. The dashed blue lines mark
the condition $0.04<x<0.67$, whereas the green lines delineate perturbativity
conditions. We have added five coloured stars representing example parameter
combinations to the plot for which we compute benchmark cross-sections. A
representative value of the Higgs-flavon mixing angle, $\sin\phi=0.15$, is
chosen in all simulations, keeping in mind that the flavonstrahlung cross-
section is proportional to $\cos^{2}\phi$. We also keep the flavon charge
$q_{\theta}$ at unity for now.

We first consider the cross-section
$\sigma\mathopen{}\mathclose{{}\left(pp\rightarrow\vartheta
Z^{\prime}\rightarrow\vartheta\mu^{+}\mu^{-}}\right)$ at a centre-of-mass
energy $\sqrt{s}=$ 14 TeV, representing the HL-LHC. The cross-sections for the
five example points are shown in the right-hand panel of figure 10, where the
colours of the lines correspond to the colours of the stars in the left-hand
panel. The exact choices of $\\{M_{Z^{\prime}},g_{Z^{\prime}}\\}$ are listed
in the legend. Assuming HL-LHC integrated luminosity of 3000 $\text{fb}^{-1}$,
the plot suggests we expect to produce less than $\mathcal{O}(1)$
flavonstrahlung events. Thus, the flavonstrahlung cross-sections are too small
for discovery at the HL-LHC.

Figure 10: The left-hand panel, based on figure 9 of ref. Azatov:2022itm ,
shows the $g_{Z^{\prime}}-M_{Z^{\prime}}$ plane of the parameter space.
Everything above the solid black line is excluded at the 95% CL by the LHC
whereas the dashed black line indicates the projected 95% CL sensitivity of
the HL-LHC. The dashed and solid green lines indicate the
$\Gamma/M_{Z^{\prime}}=1/3$ and $\Gamma/M_{Z^{\prime}}=1$ bounds, above which
perturbative computations become inaccurate. The region between the blue
dashed lines is the region allowed by the fits discussed in section 3.1. Were
another fit including the new $R_{K}$ and $R_{K^{\ast}}$ measurements
LHCb:2022qnv ; LHCb:2022zom  to be performed, the position of the lower blue
line would be revised downward. Coloured stars have been superposed on the
figure, with each star labelling a benchmark point in the parameter plane. The
right-hand panel shows tree-level flavonstrahlung cross-sections for 14 TeV
$pp$ collisions with the flavon charge $q_{\theta}$ set to unity. Each
coloured line corresponds to a parameter space point labelled by a star of the
same colour.

Figure 11 shows how the picture changes if the flavon charge $q_{\theta}$,
which can be any rational number, is varied. We have selected the two
parameter combinations from figure 10 which yield the largest and third
largest cross-sections and let $q_{\theta}$ take values 3, 5 and 10. We find
that for flavon charges $q_{\theta}\gtrsim 5$ and for $Z^{\prime}$ masses and
couplings near the current exclusion limits, the HL-LHC may be able to
discover flavonstrahlung up to around 1 TeV flavon masses, but a detailed
study would be necessary to confirm this. Either way, the above relies on a
finely tuned selection of input parameters and does not change the overall
conclusion the HL-LHC lacks sufficient partonic energies and luminosity to
look for flavonstrahlung in more than a corner of the currently available
parameter space. For the rest of the present paper, we return to flavon
charges of unity.

Figure 11: Flavonstrahlung cross-sections for 14 TeV $pp$ collisions but with
a variable flavon charge. The charge $q_{\theta}$ takes on values 3, 5 and 10,
represented by solid, dashed and dash-dotted lines, respectively. The line
colours represent different points in the $M_{Z^{\prime}}$ – $g_{Z^{\prime}}$
plane and are congruent with the colours of the stars in figure 10. Figure 12:
Tree-level flavonstrahlung cross-sections for 100 TeV $pp$ collisions for
$q_{\theta}=1$. Each coloured line corresponds to a parameter space point
labelled by a star of the same colour in figure 10.

We now examine whether a 100 TeV hadron collider, such as the FCC-hh with an
integrated luminosity of 20–30 $\text{ab}^{-1}$, would be capable of
discovering flavonstrahlung. The simulated cross-sections for the the five
parameter space points indicated by the coloured stars in figure 10 are shown
in figure 12. The resulting cross-sections are greatly enhanced compared to
HL-LHC and the larger partonic energies come with the benefit that the cross-
sections are not as sensitive to the flavon and $Z^{\prime}$ masses.

To gain some insight into the reach of the collider, we discard as
undiscoverable those parameter space points where less than 10 flavonstrahlung
events are expected to be produced. Parameter space points passing this very
rough criterion are not automatically within the reach of the collider, and we
leave for a future work the detailed study of the SM backgrounds and detector
effects which would allow for a more precise estimate of the collider
sensitivity. Applying this crude method, we find that for all but the smallest
allowed values of the gauge coupling, $g_{Z^{\prime}}\gtrsim 0.4$, the
collider can explore the parameter space up to $\sim$ 5 TeV flavon and
$Z^{\prime}$ masses. For $g_{Z^{\prime}}\lesssim 0.4$, the mass reach will
likely be more limited. The blue line in figure 12, corresponding to the
smallest allowed coupling for a 2.5 TeV $Z^{\prime}$ boson, demonstrates that
even in this case we may have sensitivity up to a flavon mass of around 2 TeV.

###  4.2 Flavonstrahlung at Muon Colliders

We may also simulate flavonstrahlung at 3 TeV and 10 TeV $\mu^{+}\mu^{-}$
colliders assumed to reach integrated luminosities of 1 $\text{ab}^{-1}$ and
10 $\text{ab}^{-1}$, respectively. Despite the centre-of-mass energies being
lower than that of the FCC-hh, the fact that nearly all of the beam energy is
typically carried by the colliding muon pair may allow these colliders to have
high sensitivity to flavonstrahlung. We once again focus on the five example
points from figure 10 and use MG5_aMC to perform the simulations. The results
for the 3 TeV and 10 TeV colliders are shown in figures 13 and 14,
respectively.

The cross-sections do not include initial state radiation (ISR) effects, as
this feature is not yet implemented in MG5_aMC. Due to the fact that lepton
PDFs peak at momentum fraction $x=1$, the inclusion of ISR effects is not
expected to change the cross-sections dramatically. In a similar fashion to
Bao:2022onq , we estimate the magnitude of ISR by allowing for a single
collinearly emitted photon in the final state with kinematic parameters such
that the photon falls outside the acceptance of the detector. To this end, we
enforce that the pseudorapidity $\eta$ of the photon be in the domain
$2.5<|\eta|<1000$ and require that its transverse momentum $p_{T}$ satisfy
$0.001\text{ GeV}<p_{T}<0.1\text{ GeV}$. The inclusive collinear photon is
found to increase the cross-sections by $\sim 30\%$ in the case of 10 TeV muon
collisions. The cross-sections of figures 13 and 14 do not include the
collinearly emitted photon, and one should take them as lower estimates of the
real cross-section.

Figure 13: Tree-level flavonstrahlung cross-sections for 3 TeV
$\mu^{+}\mu^{-}$ collisions for $q_{\theta}=1$. Each coloured line corresponds
to a parameter point labelled by a star of the same colour in figure 10. The
cross-sections do not include ISR effects.

Using the same discoverability criterion as before, we observe in figure 13
that the cross-sections at the 3 TeV collider are large enough to explore
regions of the parameter space satisfying $M_{Z^{\prime}}\lesssim$ 5 TeV and
$m_{\vartheta}\lesssim 2.5$ TeV with good sensitivity. Barring very small
flavon charges $q_{\theta}\ll 1$, the 3 TeV collider would likely be able to
discover or rule out flavonstrahlung in this region of $m_{\vartheta}$ and
$M_{Z^{\prime}}$. For $Z^{\prime}$ masses larger than $\sim 5$ TeV, the
amplitudes are increasingly suppressed by the off-shell $Z^{\prime}$
propagators, whereas flavon masses of order 3 TeV and greater are unreachable
because the collider would be unable to produce an on-shell flavon in the
final state.

Figure 14: Tree-level flavonstrahlung cross-sections for 10 TeV
$\mu^{+}\mu^{-}$ collisions for $q_{\theta}=1$. Each coloured line corresponds
to a parameter point labelled by a star of the same colour in figure 10. The
cross-sections do not include ISR effects.

Similarly, the 10 TeV muon collider has an excellent reach in the parameter
space region where $M_{Z^{\prime}}\lesssim 15$ TeV and $m_{\vartheta}\lesssim
8$ TeV. As figure 14 shows, the flavonstrahlung cross-sections are very
sensitive to the size of the coupling $g_{Z^{\prime}}$ (they scale
as101010Note that the cross-section scales as
$g_{Z^{\prime}}^{6}v_{\vartheta}^{2}$, but at a fixed value of
$M_{Z^{\prime}}$, $v_{\vartheta}\propto 1/g_{Z^{\prime}}$.
$g_{Z^{\prime}}^{4}$) compared with sensitivity to $m_{\vartheta}$ and
$M_{Z^{\prime}}$. The cross-sections are greater than around $10^{-1}$ fb for
even the smallest allowed values of the coupling, meaning that the 10 TeV
collider would likely be able to cover all of the parameter space , at least
when $q_{\theta}$ is of order one or more.

###  4.3 Summary of future collider prospects

We see that the flavonstrahlung cross-sections are likely too small for the
process to be discovered at the HL-LHC, but a 100 TeV hadron collider and a 3
or 10 TeV muon collider could have excellent discovery prospects for multi-TeV
mass flavons and $Z^{\prime}$ bosons. At a qualitative level, this is very
similar to the pure $Z^{\prime}$ search prospects in the $B_{3}-L_{2}$ model,
as shown in ref. Azatov:2022itm .

Comparing the 100 TeV hadron collider with the 10 TeV muon collider, we notice
that for a given parameter space point, the flavonstrahlung cross-sections are
two or three orders of magnitude greater at the muon collider. This is not
surprising, considering that flavonstrahlung at a hadron collider occurs
primarily through a $bb$ partonic initial state, whereas the muon collider can
harness almost the entire beam for the production of flavonstrahlung. The muon
collider is limited by the kinematical threshold $m_{\vartheta}\lesssim$ 8 TeV
coming from the smaller beam energy, but even for the largest allowed
couplings, the reach of the 100 TeV hadron collider is stopped in the same
neighbourhood due to diminishing cross-sections.

To truly estimate discovery prospects, one of course should calculate
backgrounds. However, it seems very likely that these can be controlled highly
efficiently: with invariant mass cuts upon the reconstructed $Z^{\prime}$ from
its decay products and also from the reconstructed flavon particle, from its
decay products.

##  5 Conclusions

The $B_{3}-L_{2}$ model is well motivated, being pertinent to the fermion mass
puzzle111111In particular, the model describes why the CKM quark mixing matrix
elements $|V_{ub}|\ll 1$, $|V_{ts}|\ll 1$, $|V_{td}|\ll 1$ and $|V_{cb}|\ll
1$. Alonso:2017uky ; Bonilla:2017lsq ; Allanach:2020kss  as well as the
$b\rightarrow s\mu^{+}\mu^{-}$ anomalies, and it is directly testable at
colliders. Collider signatures remain currently relatively unstudied aside
from a few $Z^{\prime}$ bump-hunts in di-fermion invariant masses. Even these
had to be reinterpreted from the experimental papers because most
interpretations of various bump-hunts in di-fermion invariant masses assumed
family universality. The $B_{3}-L_{2}$ model is far from the family universal
limit; for studies of $Z^{\prime}$ production in the $B_{3}-L_{2}$ model, see
refs. Bonilla:2017lsq ; Allanach:2020kss ; Allanach:2021gmj ; Azatov:2022itm .

We have instead studied the prospects for flavonstrahlung, where the flavon is
produced along with an associated $Z^{\prime}$. Our results indicate that the
flavon will not be directly discovered at the HL-LHC because its production
cross-section is too small, but in a future 3 or 10 TeV muon collider or a 100
TeV FCC, flavonstrahlung discovery prospects are good. Flavons could also be
produced at hadron colliders by traditional SM Higgs production processes,
provided that they mix with the Higgs (i.e. provided that $\phi\neq 0$). Thus,
they could be produced via gluon-gluon fusion, via weak boson fusion or via
associated production with a di-top fermion. The advantage of the
flavonstrahlung process is that it is present even in the limit of zero mixing
with the SM Higgs field $\phi\rightarrow 0$. In fact, it becomes negligible
for maximal mixing with the SM Higgs field, but the model must be far from
this limit because of current bounds from Higgs measurements, as shown in
figure 2.

Several other similar bottom-up models possess the flavonstrahlung signature,
for example the Third Family Hypercharge Model Allanach:2018lvl  and its
variants Allanach:2019iiy , gauged muon minus tau lepton number He:1990pn  and
several other gauged $U(1)$ family non-universal models Greljo:2021npi . In
the future, it may be of interest to compute the cross-sections for these
models too in order to see how they compare.

One may ask what the first hints would be in collider data from beyond the SM
effects of the $B_{3}-L_{2}$ model. The answer, possibly, is the $b\rightarrow
s\mu^{+}\mu^{-}$ anomalies that are currently being investigated
Alonso:2017uky ; Bonilla:2017lsq ; Allanach:2020kss . One can also obtain
deviations in $M_{W}$, as figure 2 implies. This figure also reminds us that
(since we have applied the current bounds, the sensitivity of which will
increase with increased integrated luminosity) the LHC may also observe
deviations from the SM limit in the signal strength of various Higgs cross-
sections, should the flavon and Higgs fields mix. The first direct evidence
for the model, however, would likely be from the classic
$Z^{\prime}\rightarrow\mu^{+}\mu^{-}$ resonance search, followed possibly by
other $Z^{\prime}$ decay modes. Flavon particle production (via the usual
Higgs production modes, but suppressed by powers of $\sin\phi$) could also be
observed. Flavonstrahlung may eventually be observed at future colliders, with
a resonant $\mu^{+}\mu^{-}$ produced in association with a flavon boson, which
would primarily decay to $W^{+}W^{-}$, $ZZ$ or $HH$, as displayed on figure 5.

## Acknowledgements

This work has been partially supported by STFC consolidated grant
ST/T000694/1. We thank the Cambridge Pheno Working Group for helpful
discussions.

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