# Novel $B$-decay signatures of light scalars at high energy facilities

Andrew Blanceab, Mikael Chalaac, Maria Ramosd and Michael Spannowskya
aInstitute of Particle Physics Phenomenology, Physics Department, Durham
University, Durham DH1 3LE, UK
bInstitute for Data Science, Durham University, Durham DH1 3LE, UK
cCAFPE and Departamento de Física Teórica y del Cosmos, Universidad de
Granada, E–18071 Granada, Spain
dLaboratório de Instrumentação e Física Experimental de Partículas,
Departamento de Física da Universidade do Minho, Campus de Gualtar, 4710-057
Braga, Portugal

###### Abstract

We study the phenomenology of light scalars of masses $m_{1}$ and $m_{2}$
coupling to heavy flavour-violating vector bosons of mass $m_{V}$. For
$m_{1,2}\lesssim$ few GeV, this scenario triggers the rare $B$ meson decays
$B_{s}^{0}\to 3\mu^{+}3\mu^{-}$, $B^{0}\to 3\mu^{+}3\mu^{-}$, $B^{+}\to
K^{+}3\mu^{+}3\mu^{-}$ and $B_{s}^{0}\to K^{0*}3\mu^{+}3\mu^{-}$; the last two
being the most important ones for $m_{1}\sim m_{2}$. None of these signals has
been studied experimentally; therefore we propose analyses to test these
channels at the LHCb. We demonstrate that the reach of this facility extends
to branching ratios as small as $6.0\times 10^{-9}$, $1.6\times 10^{-9}$,
$5.9\times 10^{-9}$ and $1.8\times 10^{-8}$ for the aforementioned channels,
respectively. For $m_{1,2}\gg\mathcal{O}(1)$ GeV, we show that slightly
modified versions of current multilepton and multitau searches at the LHC can
probe wide regions of the parameter space of this scenario. Altogether, the
potential of the searches we propose outperform other constraints such as
those from meson mixing.

††preprint: IPPP/19/63

###### Contents

  1. I Introduction
  2. II Framework
  3. III Low mass regime at the LHC$\mathbf{b}$
  4. IV High mass regime at the LHC
  5. V Conclusions
  6. A Concrete composite Higgs model

##  I Introduction

Searches for new physics in final states often considered as “standard
candles”, most notably in searches for supersymmetry (SUSY), have not provided
any evidence of physics beyond the Standard Model (BSM) so far. This fact does
not necessarily disproves low energy SUSY or other popular BSM extensions
Chala:2018pbn , such as composite Higgs models (CHM) Kaplan:1983fs ;
Kaplan:1983sm . However, it supports the search for new physics in radically
new and still unexplored channels.

In this paper we focus on light singlet scalars $a_{1,2}$ that can be produced
in rare decays of $B$ mesons mediated by heavy flavour-violating vector bosons
$V$. This scenario is especially motivated, as it arises naturally in non-
minimal CHMs Gripaios:2009pe ; Vecchi:2013bja ; Sanz:2015sua ; Chala:2016ykx ;
Balkin:2017aep ; DaRold:2019ccj . ($V$ and $a_{1,2}$ can be seen as the
counterparts of the $\rho$ and the pions in QCD.) Likewise, such vector boson
can explain the apparent anomalies observed in tests of lepton flavour
universality Niehoff:2015bfa ; Niehoff:2015iaa ; Carmona:2015ena ;
Megias:2016bde ; GarciaGarcia:2016nvr ; Sannino:2017utc ; Carmona:2017fsn ;
Chala:2018igk . Moreover, the bounds on such vector boson are weakened when it
decays into lighter composite resonances Chala:2018igk , such as the
aforementioned scalars. Finally, also supersymmetric models can trigger
similar decays, mediated by scalar and pseudoscalar sgoldstino particles
Demidov:2011rd .

If, similarly to the Higgs boson, the scalars couple stronger to the muon than
to the electron, processes such as $B_{s}^{0}\to a_{1}a_{2}$ can lead to four
muon final states. To the best of our knowledge, the corresponding signal has
been studied experimentally only at the LHCb Aaij:2016kfs ; the most stringent
limit being $\mathcal{B}(B_{s}^{0}\to 2\mu^{+}2\mu^{-})<2.5\times 10^{-9}$.

However, there are different reasons to consider alternative $B$ meson decay
modes. To start with, the partial width for $a_{2}\to a_{1}a_{1}$ can very
easily dominate over the corresponding leptonic width. In this case, six muon
final states rather than four muon ones are to be studied. And secondly, the
scalars couple to the mediator as a vector current $\sim a_{1}\partial a_{2}$.
When the latter is conserved, namely for $m_{1}\sim m_{2}$ (and in particular
in the massless limit), the $B$ meson decay into such scalars vanishes. In
other words, $\Gamma(B_{s}^{0}\to a_{1}a_{2})\sim(m_{1}^{2}-m_{2}^{2})/m_{B}$.
In this regime, one should rather explore three body decays of $B$ with
emitted mesons. In this work we focus mostly on $B^{+}\to
K^{+}3\mu^{+}3\mu^{-}$. (The inclusion of conjugate modes of charged decays is
implied throughout the paper.)

We also extend previous works on this topic Demidov:2011rd ; Nelson:2013ula ;
Chala:2019vzu  by studying the regime of large scalar masses. In such regime,
$a_{1,2}$ can no longer show up in rare decays of $B$ mesons. However, they
can appear in decays of the vector mediator if it is at the TeV scale and
therefore be produced in $pp$ collisions at the LHC.

This article is organised as follows. In section II, we provide the Lagrangian
relevant for our study and define the region of the parameter space of
phenomenological interest. In section III we focus on the regime
$m_{1,2}\lesssim$ few Gev and provide analyses for the LHCb and estimate the
reach for different $B$ decays. We do not circumscribe to any particular value
of $m_{1,2}$, but rather scan over different values of these. In section IV we
focus instead on the regime $m_{1,2}>$ few GeV and study the corresponding LHC
signatures.

Unless otherwise stated, all limits given in this article stand for 95% CL.

We conclude in section V, while we dedicate Appendix A to building a complete
model that predicts definite values of several of the parameters that we scan
over.

##  II Framework

Let us consider the Lagrangian of the SM extended with a heavy vector $V$, and
two light scalars $a_{1},a_{2}$. The relevant Lagrangian before electroweak
symmetry breaking (EWSB) (in the basis in which up quark and lepton Yukawas
are diagonal) reads

$\displaystyle L=\,$
$\displaystyle\frac{1}{2}m_{V}^{2}V_{\mu}V^{\mu}+\frac{1}{2}m_{1}^{2}a_{1}^{2}+\frac{1}{2}m_{2}^{2}a_{2}^{2}+m_{12}a_{2}a_{1}^{2}+\cdots$
$\displaystyle+V^{\mu}\left[g_{12}a_{1}\overleftrightarrow{\partial_{\mu}}a_{2}+g_{qq}(\overline{q_{L}}\gamma_{\mu}q_{L}+\text{h.c.})\right]~{},$
(1)

with $m_{V}\gg m_{1,2}$. The ellipsis stand for terms not relevant for this
study. Without loss of generality, we assume $m_{2}>m_{1}$. The scalars
$a_{1,2}$ can be more naturally thought of as the real and imaginary
components of a complex field $\Phi$; the Lagrangian being invariant under
$\Phi\to\exp{(i\theta)}\Phi$ up to $\mathcal{O}(1-m_{2}/m1,m_{12})$. In the
Appendix A we match a concrete CHM to the Lagrangian above.

Assuming that $V$ interacts mostly with the third generation quarks, after
EWSB it couples to $\overline{b_{L}}b_{L}$ and $\overline{t_{L}}t_{L}$ as well
as $\overline{b_{L}}s_{L}+\text{h.c.}$ with strengths $\sim g_{qq}$ and

$g_{sb}\equiv g_{qq}V_{ts}^{CKM}V_{tb}^{CKM}\sim 0.04\,g_{qq}~{},$ (2)

respectively.

We distinguish two different regimes depending on the masses of the scalars:
$1\,\text{GeV}\lesssim m_{1,2}\lesssim 4$ GeV (low mass regime) and
$m_{1,2}>4$ GeV (high-mass regime). Likewise, we consider two possible
scenarios for the couplings of $a_{1,2}$ to the fermions. First, we assume
that $a_{1,2}$ are muonphilic. As a second possibility, we assume that they
couple only to the SM leptons and with Higgs-like strength, namely $\sim
g_{1,2}y_{\ell}a_{1,2}\ell^{+}\ell^{-}$, with $y_{\ell}$ the SM Yukawa
couplings and $g_{1,2}$ free dimensionless parameters and lepton independent.

In the low-mass regime, $a_{1}$ decays mostly into muons irrespectively of
whether it is muonphilic or just leptophilic. In the high-mass regime, it
decays mostly into taus unless it is muonphilic.

Regarding the decay of $a_{2}$, if $m_{2}>2m_{1}$, then $a_{2}$ can either
decay into $a_{1}a_{1}$ or into lepton pairs, depending on $m_{12}/g_{2}$:

$\displaystyle\Gamma(a_{2}\to\ell^{+}\ell^{-})$
$\displaystyle=\frac{g_{2}^{2}y_{\ell}^{2}}{8\pi}\left(1-\frac{4m_{\ell}^{2}}{m_{2}^{2}}\right)^{3/2}m_{2}~{},$
(3) $\displaystyle\Gamma(a_{2}\to a_{1}a_{1})$
$\displaystyle=\frac{m_{12}^{2}}{8\pi
m_{2}}\left(1-\frac{4m_{1}^{2}}{m_{2}^{2}}\right)^{1/2}~{}.$ (4)

In what follows, we assume that $m_{12}/m_{2}\gg g_{2}y_{\ell}$ in this
regime, so that $\mathcal{B}(a_{2}\to
a_{1}a_{1})\gg\mathcal{B}(a_{2}\to\ell^{+}\ell^{-})$. Note that this
inequality holds almost trivially, since one expects $m_{12}\sim m_{2}$
whereas the Yukawas are tiny.

If instead $m_{2}<2m_{1}$, $a_{2}$ can either decay into pairs of leptons as
before, or into $a_{1}\ell^{+}\ell^{-}$ with width

$\displaystyle\Gamma(a_{2}\to$ $\displaystyle a_{1}\ell^{+}\ell^{-})\sim$
$\displaystyle\frac{(g_{1}y_{\ell})^{2}}{64\pi^{3}m_{2}^{3}}m_{12}^{2}m_{1}^{2}\bigg{(}1+\frac{m_{2}}{m_{1}}\bigg{)}\bigg{(}\frac{m_{2}}{m_{1}}-1\bigg{)}^{5}~{}.$
(5)

This decay mode dominates if $g_{1}\gtrsim 100g_{2}$. We assume this hierarchy
hereafter. Thus, for example for $g_{1}=3$ and $g_{2}=0.01$, $a_{2}$ decays
always into four leptons mediated by $a_{1}$, which can be either on-shell or
off-shell. Also, they both have widths smaller than $10$ MeV and lifetime
shorter than $10$ fs. As a consequence, both $a_{1,2}$ would seem to have
vanishing experimentally measurable widths and flight distances. Furthermore,
note that the Yukawa suppression helps also avoiding bounds from BaBar and
even the future Belle-II Liu:2018xkx .

Figure -1019: Tree level Feynman diagram for the decays $B_{s}^{0}\to
a_{1}a_{2}$ (left) and $B^{+}\to K^{+}a_{1}a_{2}$ (right).

At low energies, the vector boson $V$ triggers $B$ meson decays into the light
scalars; see Fig. -1019. Depending on the relative size between $m_{B}$ and
$m_{1,2}$ we distinguish two cases:

  * •

If $m_{B}>m_{1}+m_{2}$, we have $B_{s}^{0}\rightarrow a_{1}a_{2}$ .

  * •

If $m_{B}<m_{1}+m_{2}$ and $m_{B}>3m_{1}$, we have instead $B_{s}^{0}\to
a_{1}a_{1}a_{1}$. (Other three body decays, e.g. $B_{s}^{0}\to
a_{1}\mu^{+}\mu^{-}$ are subdominant due to the Yukawa suppression.)

If $m_{B}>m_{1}+m_{2}+m_{K}$, we also have $B^{+}\rightarrow K^{+}a_{1}a_{2}$.
We do not consider any other cases in this paper; see Fig. -1018.

Figure -1018: Dominant decays taking place in the different regions of the
plane $(m_{1},m_{2})$. The gray areas are not considered in this analysis.

The decay width for $B_{s}^{0}\rightarrow a_{1}a_{2}$ reads:

$\Gamma=\frac{f_{B}^{2}}{16\pi
m_{V}^{4}}\left(g_{sb}g_{12}\right)^{2}\frac{\left(m_{1}^{2}-m_{2}^{2}\right)^{2}}{m_{B}}~{}\mathcal{K}\bigg{(}\frac{m_{1}}{m_{B}},\frac{m_{2}}{m_{B}}\bigg{)}$
(6)

with

$\mathcal{K}(x,y)=\bigg{[}x^{4}+(1-y^{2})^{2}-2x^{2}(1+y^{2})\bigg{]}^{1/2}$
(7)

and $f_{B}\sim 0.23$ GeV Cheung:2006tm .

The amplitude for $B_{s}^{0}\to a_{1}a_{1}a_{1}$ reads:

$\mathcal{M}=2g_{sb}g_{12}\frac{f_{B}m_{12}}{m_{V}^{2}}\left[\frac{q_{12}^{2}-m_{1}^{2}}{q_{12}^{2}-m_{2}^{2}}+\frac{q_{23}^{2}-m_{1}^{2}}{q_{23}^{2}-m_{2}^{2}}+\frac{q_{13}^{2}-m_{1}^{2}}{q_{13}^{2}-m_{2}^{2}}\right]~{},$
(8)

where we have defined the transferred momenta $q_{12}^{2}=(p_{1}+p_{2})^{2}$,
$q_{23}^{2}=(p_{2}+p_{3})^{2}$ and
$q_{13}^{2}=(p_{1}+p_{3})^{2}=3m_{1}^{2}+m_{B}^{2}-q_{12}^{2}-q_{23}^{2}$.
After integrating over $q_{23}^{2}$, we obtain:

$\frac{d\Gamma}{dq_{12}^{2}}=\frac{\left(g_{sb}g_{12}\right)^{2}m_{2}^{2}}{384\pi^{3}m_{B}^{3}}\left(\frac{f_{B}m_{12}}{m_{V}^{2}}\right)^{2}F\left[\frac{m_{1}}{m_{2}},\frac{m_{B}}{m_{2}},\frac{q_{12}}{m_{2}},\frac{q_{23}}{m_{2}}\right]_{(q_{23}^{2})^{\rm
min}}^{(q_{23}^{2})^{\rm max}}$ (9)

with

$\displaystyle F$
$\displaystyle(x,y,w,v)=\left(1-x^{2}\right)^{2}\left[\frac{1}{1-v^{2}}+\frac{1}{3x^{2}+y^{2}-w^{2}-v^{2}-1}\right]+\frac{v^{2}\left(2+x^{2}-3w^{2}\right)^{2}}{(w^{2}-1)^{2}}+2(x^{2}-1)\times$
$\displaystyle\times\left\\{\frac{3x^{4}+x^{2}(3+y^{2}-9w^{2})+y^{2}(2-3w^{2})+3(w^{4}+w^{2}-1)}{(w^{2}-1)(3x^{2}+y^{2}-w^{2}-2)}\left[\log{\left(v^{2}-1\right)}-\log{\left(1+w^{2}+v^{2}-3x^{2}-y^{2}\right)}\right]\right\\}$
(10)

which should be evaluated at

$\displaystyle\left(q_{23}^{2}\right)^{\rm
max}=\left(E_{2}^{*}+E_{3}^{*}\right)^{2}-\left(\sqrt{E_{2}^{*2}-m_{1}^{2}}-\sqrt{E_{3}^{*2}-m_{1}^{2}}\right)^{2}$
(11) $\displaystyle\left(q_{23}^{2}\right)^{\rm
min}=\left(E_{2}^{*}+E_{3}^{*}\right)^{2}-\left(\sqrt{E_{2}^{*2}-m_{1}^{2}}+\sqrt{E_{3}^{*2}-m_{1}^{2}}\right)^{2}~{},$

where $E_{2}^{*}\equiv q_{12}/2$ and
$E_{3}^{*}\equiv\left(m_{B}^{2}-q_{12}^{2}-m_{1}^{2}\right)/(2q_{12})$. The
final width is obtained integrating over $q_{12}^{2}$ between $4m_{1}^{2}$ and
$(m_{B}-m_{1})^{2}$ .

In the limit $m_{1},m_{2}\rightarrow 0$, the integrated width simplifies to:

$\Gamma\sim\frac{3\left(g_{sb}g_{12}\right)^{2}}{256\pi^{3}}\frac{f_{B}^{2}m_{12}^{2}}{m_{V}^{4}}m_{B}~{}.$
(12)

Finally, the amplitude for $B^{+}\to K^{+}a_{1}a_{2}$ is given by:

$\mathcal{M}=-\frac{g_{sb}g_{12}}{m_{V}^{2}}\left<K(p_{3})|\overline{s}\gamma_{\mu}b|B(p)\right>(p_{2}-p_{1})^{\mu}~{},$
(13)

with

$\displaystyle\left<K(p_{3})|\overline{s}\gamma_{\mu}b|B(p)\right>=$
$\displaystyle
f_{+}(q^{2})\bigg{[}(p+p_{3})_{\mu}-\frac{m_{B}^{2}-m_{K}^{2}}{q^{2}}q_{\mu}\bigg{]}$
$\displaystyle+f_{0}(q^{2})\frac{m_{B}^{2}-m_{K}^{2}}{q^{2}}q_{\mu}$ (14)

and again $q^{2}=(p-p_{3})^{2}$ is the transferred momentum, ranging from
$(m_{1}+m_{2})^{2}<q^{2}<(m_{B}-m_{K})^{2}$. The contraction of this matrix
element with $(p_{2}-p_{1})$ in Eq. 13 simplifies to

$\displaystyle\mathcal{M}$
$\displaystyle=-\frac{g_{sb}g_{12}}{m_{V}^{2}}\bigg{\\{}\frac{\left(m_{B}^{2}-m_{K}^{2}\right)\left(m_{2}^{2}-m_{1}^{2}\right)}{q^{2}}\left[f_{0}(q^{2})-f_{+}(q^{2})\right]$
$\displaystyle+\left[2\left(p_{2}+p_{3}\right)^{2}+q^{2}-m_{1}^{2}-m_{2}^{2}-m_{B}^{2}-m_{K}^{2}\right]f_{+}(q^{2})\bigg{\\}}~{}.$
(15)

For convenience, we trade these variables for $M_{12}^{2}\equiv
m_{2}^{2}-m_{1}^{2}$ and $M_{BK}^{2}\equiv m_{B}^{2}-m_{K}^{2}$, getting

$\frac{d\Gamma}{dq^{2}}=\frac{\left(g_{sb}g_{12}\right)^{2}}{768\pi^{3}m_{V}^{4}m_{B}^{3}}F(q^{2})~{},$
(16)

with

$\displaystyle F(q^{2})$
$\displaystyle=\frac{1}{q^{2}}\left[\frac{(M_{BK}^{2}+q^{2})^{2}}{q^{4}}-4\frac{m_{B}^{2}}{q^{2}}\right]^{1/2}\left[\frac{(M_{12}^{2}+q^{2})^{2}}{q^{4}}-4\frac{m_{2}^{2}}{q^{2}}\right]^{1/2}$
$\displaystyle\times\bigg{\\{}3M_{BK}^{4}M_{12}^{4}|f_{0}(q^{2})|^{2}+\left[q^{4}+2q^{2}\left(M_{BK}^{2}-2m_{B}^{2}\right)+M_{BK}^{4}\right]\left[q^{4}+2q^{2}\left(M_{12}^{2}-2m_{2}^{2}\right)+M_{12}^{4}\right]|f_{+}(q^{2})|^{2}\bigg{\\}}~{}.$
(17)

Following Ref. Ball:2004ye , we parameterize the form factor as

$f_{+}(q^{2})=\frac{r_{1}}{(1-q^{2}/m^{2})}+\frac{r_{2}}{(1-q^{2}/m^{2})^{2}}~{},$
(18)

with $r_{1}=0.162$, $r_{2}=0.173$ and $m^{2}=5.41^{2}$ $\rm{GeV}^{2}$.
Similarly,

$f_{0}(q^{2})=\frac{r_{2}}{(1-q^{2}/m_{\rm fit}^{2})}$ (19)

with $r_{2}=0.330$ and $m_{\rm fit}^{2}=37.46$ $\rm{GeV}^{2}$. Finally, in the
approximation $m_{1},m_{2},m_{K}\rightarrow 0$ and
$f_{0},f_{+}(q^{2})\rightarrow 1$, we obtain:

$\Gamma\sim\frac{\left(g_{sb}g_{12}\right)^{2}}{3072\pi^{3}m_{V}^{4}}m_{B}^{5}~{}.$
(20)

In Fig. -1017, we show the magnitude of three body decays under consideration
and their dependence with $(p_{1}+p_{2})^{2}$.

Figure -1017: Differential branching ratios as a function of
$q^{2}=(p_{1}+p_{2})^{2}$. We have fixed $m_{1}=1.2$ GeV, $g_{sb}=g_{12}=1$,
$m_{V}=1$ TeV and $m_{12}=5$ GeV. Due to the different kinematic regions where
these decays take place, we have set $m_{2}=2.5$ GeV and $m_{2}=5$ GeV for the
$B^{+}\rightarrow K^{+}a_{1}a_{2}$ and $B_{0}^{s}\rightarrow a_{1}a_{1}a_{1}$,
respectively.  Figure -1016: Value of $\Gamma\left(B_{s}^{0}\rightarrow
a_{1}a_{2}\right)/\Gamma\left(B^{+}\rightarrow K^{+}a_{1}a_{2}\right)$ in the
plane $(m_{1},m_{2})$. This ratio vanishes along the line $m_{1}=m_{2}$.

In Fig. -1016, we show the ratio of $\Gamma(B_{s}^{0}\to a_{1}a_{2})$ to
$\Gamma(B^{+}\to K^{+}a_{1}a_{2})$. It is very worth noting that it vanishes
in the limit $m_{1}\to m_{2}$; see also Eq. 6. In this regime, searches for
$B_{s}^{0}$ decaying only to muons are irrelevant; extra mesons have to be
tagged instead. There are however no analyses (not even prospects) in this
respect, and this is a gap that we try to overcome in this work.

At high energies, $V$ can be produced on-shell in $pp$ collisions initiated by
bottom quarks, and subsequently decay into third generation quarks and into
$a_{1}a_{2}$ with respective widths:

$\displaystyle\Gamma(V\to
q\overline{q})=\frac{g_{qq}^{2}}{8\pi}\left(1-\frac{m_{q}^{2}}{m_{V}^{2}}\right)\left(1-\frac{4m_{q}^{2}}{m_{V}^{2}}\right)^{1/2}m_{V}~{},$
$\displaystyle\Gamma(V\to a_{1}a_{2})=\frac{g_{12}^{2}}{48\pi}\bigg{[}1$
$\displaystyle-2\frac{m_{1}^{2}+m_{2}^{2}}{m_{V}^{2}}$ (21)
$\displaystyle+\frac{(m_{2}^{2}-m_{1}^{2})^{2}}{m_{V}^{4}}\bigg{]}^{3/2}m_{V}~{},$

with $q=t,b$. Note that the scalar decay mode dominates already for
$g_{12}\gtrsim 3g_{qq}$.

##  III Low mass regime at the LHC$\mathbf{b}$

| $m_{X}\geq m_{1}+m_{2}$ | $m_{X}<m_{1}+m_{2}$
---|---|---
| $m_{2}\geq 2m_{1}$ | $m_{2}<2m_{1}$ | $m_{X}\geq 3m_{1}$
$B_{s}^{0}\rightarrow 3\mu^{+}3\mu^{-}$ | [0.02,0.03] | [0.01,0.02] | [0.02,0.03]
limit ($\times 10^{-9}$) | $~{}[6.7,11.6]$ | $[7.9,18.2]$ | $[6.0,11.9]$
$B^{+}\rightarrow K^{+}3\mu^{+}3\mu^{-}$ | [0.007,0.009] | [0.003,0.009] | four-body
limit ($\times 10^{-9}$) | $~{}[5.9,8.0]$ | $[6.0,16.6]$ | four-body
Table 1: Maximum and minimum efficiencies for selecting signal events in the
channels $B_{s}^{0}\to 3\mu^{+}3\mu^{-}$ ($m_{X}=m_{B_{s}^{0}}$) and $B^{+}\to
K^{+}3\mu^{+}3\mu^{-}$ ($m_{X}=m_{B^{+}}-m_{K^{+}}$) in each kinematic region.
The upper limits ($\times 10^{-9}$) on the corresponding branching ratios for
$3$ fb-1 of data are also shown. We vary $m_{1,2}$ in the coloured region of
Fig. -1018, with $m_{2}<10$ GeV and $m_{1}\geq 1.1$ GeV. (For smaller values
of $m_{1}$ the efficiency is negligible.)

In the low mass regime, the smoking gun signature of the Lagrangian in Eq. II
is rare decays of $B$ mesons into final states containing six muons (and
possibly other lighter mesons). Let us focus first on the channel
$B_{s}^{0}\to 3\mu^{+}3\mu^{-}$. As we have already commented, there are no
searches for this decay mode, and so neither constraints nor any direct way to
estimate the potential of the LHCb to test this process. We therefore suggest
the first analysis in this respect.

We first require events with at least one muon with $p_{T}>1.7$ GeV; this cut
ensures that the events pass the same hardware trigger used at $\sqrt{s}=8$
TeV Aaij:2016kfs . We subsequently require exactly six muons, with vanishing
total charge. We also require all muon tracks to have $p_{T}>0.5$ GeV and
$2.5<\eta<5.0$. Finally, we require all muons tracks to have total momentum
larger than $2.5$ GeV to simulate the threshold for muon identification based
on the penetration power through absorption plates in the detector.

Due to the six muons in the final state, the SM backgrounds are negligible to
very good approximation. They arise mostly from resonant production of
$J/\Psi$ and $\varphi$ with subsequent decays into muons; we completely remove
them by enforcing that no zero charge muon pair has an invariant mass in the
range $[0.95,1.09]\cup[3.0,3.2]$ GeV. (We lose sensitivity to signal events
with $m_{1}$ in that region, though.) Even searches for four muons are
background free Aaij:2016kfs ; Chala:2019vzu , so it is guaranteed that any
observed event in the six lepton final state is due to the signal.

We generate signal $B$ meson events using Pythia v8 Sjostrand:2014zea ; and
MadGraph v5 Alwall:2014hca  with Feynrules v2 Degrande:2011ua  for the decays.
(We have cross checked our event distributions using EvtGen Lange:2001uf .)
Following Ref. Chala:2019vzu , we compare the (mass dependent) efficiencies
for selecting events in the channel $B_{s}^{0}\to 3\mu^{+}3\mu^{-}$ with that
for $B_{s}^{0}\rightarrow 2\mu^{+}2\mu^{-}$. The former is shown in Tab. 1,
while we estimate the latter to be $\varepsilon_{2\mu^{+}2\mu^{-}}\sim 0.14$.
The explanation for the smaller efficiencies for the six muon process is two
fold. First, due to the larger number of final state tracks, there are more
events with no single muon with $p_{T}>1.7$ GeV which therefore do not pass
the trigger; see Fig. -1015. And second, there are more muons with at least
one track with $p_{T}<0.5$ GeV which is therefore not detected; see Fig.
-1014.

Given the absence of background, we can estimate the upper limit on the
branching ratio of the new processes at $\sqrt{s}=14$ TeV and luminosity
$\mathcal{L}^{\prime}$ as

$\mathcal{B}_{\text{max}}^{3\mu^{+}3\mu^{-}}\sim\frac{\mathcal{B}_{\text{max}}^{2\mu^{+}2\mu^{-}}\times\varepsilon_{2\mu^{+}2\mu^{-}}}{1.8\times\varepsilon_{3\mu^{+}3\mu^{-}}}\times\frac{\mathcal{L}}{\mathcal{L}^{\prime}}~{},$
(22)

where $\mathcal{B}_{\text{max}}^{2\mu^{+}2\mu^{-}}$ is the upper limit on
$\mathcal{B}\left(B_{s}^{0}\rightarrow 2\mu^{+}2\mu^{-}\right)=2.5\times
10^{-9}$, obtained in Ref. Aaij:2016kfs  with $\mathcal{L}=3~{}\rm{fb}^{-1}$
and $\sqrt{s}=8$ TeV, under the same trigger and reconstruction criteria. The
factor $1.8$ stands for the approximated growth of the $b$ production cross
section from $\sqrt{s}=8$ TeV to $\sqrt{s}=14$ TeV. The prospective bounds on
the branching ratio of this new decay mode are given in Tab. 1.

We also consider the channel $B^{+}\to K^{+}3\mu^{+}3\mu^{-}$. In this case,
on top of the selection criteria proposed before, we require the presence of a
charged kaon which is also required to have $p_{T}>0.5$ GeV and
$2.5<\eta<5.0$. The corresponding efficiencies are shown in Tab. 1. The limit
on the branching ratio can be again obtained as

$\mathcal{B}_{max}^{3\mu^{+}3\mu^{-}K^{+}}\sim\frac{\mathcal{B}_{max}^{2\mu^{+}2\mu^{-}}\times\varepsilon_{2\mu^{+}2\mu^{-}}}{1.8\times
3.7\times\varepsilon_{3\mu^{+}3\mu^{-}K^{+}}}\times\frac{\mathcal{L}}{\mathcal{L}^{\prime}}~{},$
(23)

where the factor $3.7$ stands for the larger $B^{+}$ production cross section
Aaij:2011jp . The bounds obtained this way are also shown in Tab. 1. It is
worth noting that the prospective limits on this channel are comparable or
even more stringent than that on the decay mode without the extra meson (due
mostly to the larger cross section, that compensates the smaller efficiency).
This fact, together with the observation that theoretically this decay mode
dominates for $m_{2}\sim m_{1}$, strongly motivates searches for $B^{+}\to
K^{+}3\mu^{+}3\mu^{-}$.

Figure -1015: Normalized distribution of the transverse momentum of the
hardest muon for $B_{s}^{0}\rightarrow a_{1}a_{2}$ and $B^{+}\rightarrow
K^{+}a_{1}a_{2}$ with $m_{1}=1$ GeV and $m_{2}=2.5$ GeV. These distributions
are compared with the case $B_{0}^{s}\rightarrow 2\mu^{+}2\mu^{-}$.  Figure
-1014: Normalized distribution of the transverse momentum of the softest track
for $B_{s}^{0}\rightarrow a_{1}a_{2}$ and $B^{+}\rightarrow K^{+}a_{1}a_{2}$
with $m_{1}=1$ GeV and $m_{2}=2.5$ GeV. These distributions are compared with
the case $B_{0}^{s}\rightarrow 2\mu^{+}2\mu^{-}$.

For illustration, we translate the expected limits in Tab. 1 to the plane
$(g_{sb},m_{V})$ in Fig. -1013 for definite values of $g_{12}$, $m_{1}$,
$m_{2}$ and $m_{12}$ (when relevant). Prospects for the Upgrade II, defined by
$\mathcal{L}^{\prime}=300$ fb-1, are also shown. It is interesting to see that
with our proposed analyses we can easily test masses larger than 15 TeV,
thereby outperforming constraints obtained from $\Delta M_{s}$ and completely
probing the region in which the anomalies in lepton flavour universality can
be explained.

Likewise, we also translate the aforementioned bounds to the plane
$(m_{1},m_{2})$ in Fig. -1012, fixing $g_{sb}=0.04$ as well as $m_{V}=4$ TeV.
Such values are not yet excluded by measurements of $\Delta M_{s}$; see Refs.
DiLuzio:2017fdq . In both figures, only the weakest limits of Tab. 1 are used.

Figure -1013: Maximum value of $m_{V}$ that can be tested in the searches for
$B_{s}^{0}\to 3\mu^{+}3\mu^{-}$ and $B^{+}\to K^{+}3\mu^{+}3\mu^{-}$ at the
current run of the LHCb (solid lines) and for Upgrade II (dashed lines). The
red dotted line delimits the area excluded by measurements of $\Delta M_{s}$.
In the dash-dotted line the anomalies in $R_{K}$ and $R_{K^{*}}$ can be
explained at the $1\sigma$ level assuming $g_{V\ell\ell}\sim 1$
DiLuzio:2017fdq . We have fixed $g_{12}=0.5$ as well as $m_{1}=1.2$ GeV. We
have set $m_{2}=2.0$ GeV for both $B_{0}^{s}\rightarrow a_{1}a_{2}$ and
$B^{+}\rightarrow K^{+}a_{1}a_{2}$. For $B_{0}^{s}\rightarrow
a_{1}a_{1}a_{1}$, we have fixed instead $m_{2}=m_{12}=5$ GeV. Figure -1012:
Region of the plane $(m_{1},m_{2})$ that can be tested at the current run of
the LHCb (solid) and in Upgrade II (dashed) in searches for $B_{s}^{0}\to
3\mu^{+}3\mu^{-}$ (green) and $B^{+}\to K^{+}3\mu^{+}3\mu^{-}$ (red). We have
fixed $g_{sb}=0.04$, $m_{V}=4$ TeV and $g_{12}=0.5$, as well as $m_{12}=1$ GeV
(only relevant in the upper left region). The sensitivity is negligible in the
slashed region.

We also note that, if a signal is observed in these six-muon channels, the
mass of the scalar particles involved could be reconstructed due to the
outstanding detector resolution of the LHCb. To this aim, we provide two
different algorithms, depending on whether $m_{2}>2m_{1}$ (in which case
$a_{2}\to a_{1}a_{1}$) or rather $m_{2}<2m_{1}$ (and therefore $a_{2}\to
a_{1}\mu^{+}\mu^{-}$).

For the first case, we minimize the difference $|m_{11}^{\rm rec}-m_{12}^{\rm
rec}|+|m_{12}^{\rm rec}-m_{13}^{\rm rec}|$, where $m_{i}^{\rm rec}$ is the
invariant mass of each combination of opposite-sign muons. The two $a_{1}$s
that reconstruct the heavier scalar are those with the minimum $\Delta R$
among themselves; see Fig. -1011 for an example.

Concerning the second case, the muon pairs reconstructing the two $a_{1}$s are
selected as those minimizing the difference $|m_{11}^{\rm rec}-m_{12}^{\rm
rec}|$ among the three pairs of muons. Then, $a_{2}$ is reconstructed from the
two muons not assigned to any $a_{1}$ and the $a_{1}$ that minimizes $\Delta
R(p_{1},p_{\mu\mu})$ (with $p_{1}$ its four-momentum and $p_{\mu\mu}$ the
four-momentum of the aforementioned pair of muons); see Fig. -1010.

Figure -1011: Normalized distribution of the reconstructed $m_{1}$ (solid) and
$m_{2}$ (dashed) for $m_{2}>2m_{1}$. Figure -1010: Normalized distribution of
the reconstructed $m_{1}$ (solid) and $m_{2}$ (dashed) for $m_{2}<2m_{1}$.

##  IV High mass regime at the LHC

In the high mass regime, $a_{1,2}$ can no longer be produced in the decay of
$B$ mesons. However, if $V$ is light enough ($m_{V}\lesssim$ few TeV), it can
be produced on shell at colliders, giving rise to $a_{1,2}$ pair production
upon decay. The tree level signal cross section for $g_{qq}=0.5$ and
$g_{12}=1$ ranges between $\sim 0.04$ pb and $\sim 10^{-5}$ pb for $m_{V}$
between $1$ and $5$ TeV.

There are multilepton searches at the LHC which are very sensitive to this
scenario. Most of them rely on substantial missing energy, being therefore not
relevant for our model. In this work, we consider the signal region dubbed
SR0A in the analysis of Ref. Aaboud:2018zeb . The main selection cuts of that
study are (i) at least four isolated leptons; (ii) no hadronic taus; (iii) no
pair of opposite-sign leptons with invariant mass in the range $[81.2,101.2]$
GeV; (iv) $m_{\text{eff}}>600$ GeV, where $m_{\text{eff}}$ stands for the
scalar sum of the $p_{T}$ of all leptons, jets with $p_{T}^{j}>40$ GeV and
missing energy.

Only hadronic tau candidates with $p_{T}^{\tau}>20$ GeV are considered in
(ii); jets are reconstructed using the anti-$k_{t}$ algorithm with $R=0.4$.
The experimental analysis reports the observation of 13 events, while $10.2\pm
2.1$ are predicted in the background-only hypothesis. Using these numbers
including the systematic uncertainty on the SM prediction, we obtain that the
maximum number of allowed signal events is $12$. Scaling the expected number
of background events with the larger luminosity, and assuming the same
uncertainty, the expected maximum number of signal events at the HL-LHC is
$300$.

We recast this analysis using homemade routines based on ROOT v5 Brun:1997pa ,
HepMC v2 Dobbs:2001ck  and FasJet v3 Cacciari:2011ma . We define hadronic taus
as jets with angular separation smaller than $0.2$ from a true hadronic
decayed tau lepton. We establish a flat tau-tagging efficiency of $0.5$. We
consider light leptons to be isolated if the hadronic activity around $\Delta
R=0.2$ of the corresponding lepton is smaller than 10% of its transverse
momentum. On top of the cuts above, we require that the angular separation
between any pair of muons is larger than $0.05$, to simulate their correct
reconstruction at detectors.

We generate signal events for $pp\to V\to a_{1}a_{2}$ with the corresponding
scalar decays with MadGraph v5 Alwall:2014hca  with no parton level cuts. For
the PDFs we use the NNPDF23LO set Ball:2012cx . Signal events are subsequently
passed through Pythia v8 Sjostrand:2014zea  to account for initial and final
state radiation, fragmentation and hadronization effects.

Figure -1009: Region in the plane $(m_{1},m_{2})$ that is excluded by
multilepton searches (solid red) and lepton-tau searches (solid green)
Aaboud:2018zeb . The dashed lines represent the corresponding prospects at the
HL-LHC. We have fixed $g_{qq}=0.5$, $g_{12}=1$. Figure -1008: Minimum value of
$g_{qq}$ that it is excluded by multilepton searches (solid red) and lepton-
tau searches (solid green) Aaboud:2018zeb . The dashed lines represent the
corresponding prospects at the HL-LHC. We have assumed $\mathcal{B}(V\to
a_{1}a_{2})\sim 0.25$.

If the light scalars couple mostly to the tau lepton (second scenario
introduced in Section II), the aforementioned signal region has no
sensitivity. We can rely instead on the signal region SR2 defined in the same
experimental paper of Ref. Aaboud:2018zeb , which requires (i) exactly two
light leptons with invariant mass not in the range $[81.2,101.2]$ GeV; (ii) at
least two hadronic taus with $p_{T}^{\tau}>30$ GeV; $m_{\text{eff}}>650$ GeV.
The experimental collaboration reports the observation of 2 events; the SM
prediction being $2.3\pm 0.8$. Using again the CLs method, we obtain 6 (121)
events as the current (future) maximum allowed signal.

We scan over 20 values of $m_{1}$ and $m_{2}$ in logarithmic scale in the
range $[1,500]$ GeV, with special attention to low masses as well as masses
close to the $Z$ pole.

In Fig. -1009 we depict the region in the $(m_{1},m_{2})$ plane for
$g_{qq}=0.5$ and $g_{12}=1$ that is already excluded in the muonphilic case
and also in the case with couplings to taus. The exclusion prospects for the
HL-LHC, defined by 3 ab-1, are also shown. The tau analysis is much less
constraining (mainly due to the small branching ratio to leptons), and thus we
only show results for $m_{V}=1$ TeV.

The low sensitivity in the small $m_{1}$ region is due to muons being very
collimated. (Decays into taus are furthermore forbidden for $m_{1}\lesssim 4$
GeV.) If it were possible to resolve muons with angular separations as small
as $0.001$, then almost the whole small mass range could be tested in the
muonphilic case.

Likewise, the non excluded region around $m_{1}\sim 100$ GeV results from the
$Z$ veto of the analysis. This region could be covered if the veto on the $Z$
pole is removed and, instead of $m_{\text{eff}}$, the invariant mass of all
final state observable objets (which in our signal, and contrary to the SUSY
targets of the analysis, presents a narrow peak) is used. Such improvement
would also extend the reach to smaller masses. It is therefore desirable that
future updates of the experimental work consider different versions of the cut
on $m_{\text{eff}}$.

In the same vein, in Fig. -1008 we plot the minimum value of $g_{qq}$ that can
be tested for different values of $m_{V}$ and for fixed values of $m_{1,2}$.
We have also fixed $g_{12}$ to the value for which $\mathcal{B}(V\to
a_{1}a_{2})\sim 0.25$.

##  V Conclusions

We have studied the phenomenology of light leptophilic scalars $a_{1,2}$ that
couple to a heavy flavour violating (mostly $b-s$ like) spin-1 resonance $V$.
We have shown that, under very mild conditions, $a_{2}$ decays mostly into
$a_{1}$, which subsequently decays into pairs of leptons. Thus, for scalar
masses $\lesssim$ few GeV, this scenario produces new $B$ meson decays into
six muons, namely $B_{s}^{0}\to 3\mu^{+}3\mu^{-}$ and $B^{+}\to
K^{+}3\mu^{+}3\mu^{-}$. Interestingly, the later dominates over the second
when $m_{1}\sim m_{2}$. None of them has been explored experimentally; we have
therefore proposed dedicated analyses to explore these signals at the LHCb. We
have found that branching ratios as small as $6.0\times 10^{-9}$ ($5.9\times
10^{-9}$) for the first (second) process can be already tested with the
current luminosity. Branching ratios hundred times smaller could be probed at
the Upgrade-II of the LHCb.

For larger scalar masses, $a_{1/2}$ arise rather in the decay of $V$, which
can be produced on shell at $pp$ collisions at the LHC. Current multilepton
searches in final states with muons (taus) constrain most of the parameter
space for $m_{1}\gtrsim 10$ GeV provided that $\sigma(pp\to V\to
a_{1}a_{2})\gtrsim 0.001\,(0.01)$ pb. Smaller masses give rise to very
collimated leptons (or jets) that are difficult to disentangle at detectors.
However, at the HL-LHC, the reach can be extended to $m_{1}\lesssim 5$. And
even further if the current analyses cut on the invariant mass of all visible
objets.

Finally, let us comment how these results would get modified if different
flavour assumptions are made. To start with, if $a_{1,2}$ are not leptophilic
but rather they couple to all SM fermions with Yukawa-like couplings, the
branching ratio of $a_{1}$ into leptons would get reduced by one-to-two orders
of magnitude. In turn, LHCb would be only sensitive to exotic branching ratios
thousand times larger. (Note that such branching ratios are not excluded by
any current measurement, though.) However, LHC searches in multilepton final
states would lose almost all sensitivity in this case.

On the other hand, $V$ might also induce $b-d$ transitions. In that case, we
expect new rare decays such as $B^{0}\to 3\mu^{+}3\mu^{-}$. The production
cross section for $B^{0}$ is $\sim 3.7$ larger than for $B_{s}^{0}$
Aaij:2011jp , from where we estimate that $\mathcal{B}(B^{0}\to
3\mu^{+}3\mu^{-})\gtrsim$ $1.6\times 10^{-9}$ ($\sim 10^{-11}$) can be probed
currently (in the Upgrade-II of the LHCb).

On the theory side, this channel vanishes also at tree level when $m_{1}\sim
m_{2}$. In this regime, we propose searching for $B_{s}^{0}\to
K^{*0}3\mu^{+}3\mu^{-}$, with $K^{*0}\to K^{+}\pi^{-}$; whose branching ratio
is around $2/3$ Tanabashi:2018oca . Upon performing an equivalent analysis to
that described in Sec. III, we obtain efficiencies of about $2$ times smaller,
in comparison to the $B_{s}^{0}\rightarrow 3\mu^{+}3\mu^{-}$ channel.
Consequently, we estimate the LHCb reach to be $\mathcal{B}(B_{s}^{0}\to
K^{*0}3\mu^{+}3\mu^{-})\gtrsim$ $1.8\times 10^{-8}$ currently, and again about
hundred times stronger in the Upgrade-II.

At high scalar masses, the prospects are only slightly better than for $b-s$
transitions, because the production cross section for $V$ at the LHC grows
only by a very small factor. Both low and high energy searches are also more
constraining than bounds on $\Delta M_{d}$ Foldenauer:2016rpi  on a wide
region of the parameter space.

Overall, our study motivates new searches for
$B_{s}^{0}\to(K^{0*})3\mu^{+}3\mu^{-}$ and $B^{+}\to K^{+}3\mu^{+}3\mu^{-}$ at
the LHCb as well as small modifications of current multilepton and multitau
analyses at CMS and ATLAS.

## Acknowledgments

We are grateful to Ulrik Egede for previous collaboration that opened this new
line of research. MC is supported by the Royal Society under the Newton
International Fellowship programme. MR is supported by Fundação para a Ciência
e Tecnologia (FCT) under the grant PD/BD/142773/2018 and also acknowledges
financing from LIP (FCT, COMPETE2020-Portugal2020, FEDER,
POCI-01-0145-FEDER-007334). MR would like to thank the IPPP Durham for
hospitality where the main part of this word was carried out. MS acknowledges
the hospitality of the University of Tuebingen and support of the Humboldt
Society during the completion of parts of this work.

##  Appendix A Concrete composite Higgs model

Figure -1007: Mass of $a_{1}$ as a function of $\theta$ (for $f=1$ TeV),
obtained from the embedding of the left-handed leptons in the symmetric
representation of $SO(7)$.

Non-minimal CHMs is the context where heavy vector bosons and new light
scalars, separated by a large mass gap, arise more naturally. The reason is
that the latter are pseudo Nambu Golstone bosons (pNGBs) from the the
spontaneous breaking of $\mathcal{G}/\mathcal{H}$, at a scale $f\sim$ TeV.

The smallest coset for which the scalar sector consists of the Higgs degrees
of freedom as well as two SM singlets is $SO(7)/SO(6)$ Chala:2016ykx ;
Balkin:2017aep ; DaRold:2019ccj . The corresponding $15$ unbroken and $6$
broken generators, $T$ and $X$ respectively, can be written as:

$\displaystyle T_{ij}^{mn}$
$\displaystyle=-\frac{i}{\sqrt{2}}\left(\delta_{i}^{m}\delta_{j}^{n}-\delta_{i}^{n}\delta_{j}^{m}\right)~{},\quad
m<n\in[1,6]~{};$ (24) $\displaystyle X_{ij}^{mn}$
$\displaystyle=-\frac{i}{\sqrt{2}}\left(\delta_{i}^{m}\delta_{j}^{7}-\delta_{i}^{7}\delta_{j}^{m}\right)~{},\quad
m\in[1,6]~{}.$

Without loss of generality, the pNGB matrix can be written as

$U=\left[\begin{array}[]{cccccc}\mathbf{1}_{3\times 3}&&&\\\\[2.84544pt]
&1-h^{2}/(f^{2}+f^{2}\Sigma)&-ha_{1}/(f^{2}+f^{2}\Sigma)&-ha_{2}/(f^{2}+f^{2}\Sigma)&h/f\\\\[2.84544pt]
&-ha_{1}/(f^{2}+f^{2}\Sigma)&1-a_{1}^{2}/(f^{2}+f^{2}\Sigma)&-a_{1}a_{2}/(f^{2}+f^{2}\Sigma)&a_{1}/f\\\\[2.84544pt]
&-ha_{2}/(f^{2}+f^{2}\Sigma)&-a_{1}a_{2}/(f^{2}+f^{2}\Sigma)&1-a_{2}^{2}/(f^{2}+f^{2}\Sigma)&a_{2}/f\\\\[2.84544pt]
&-h/f&-a_{1}/f&-a_{2}/f&\Sigma\\\ \end{array}\right]~{},$ (25)

with $\Sigma^{2}=1-(h^{2}+a_{1}^{2}+a_{2}^{2})/f^{2}$.

Following the partial compositeness paradigm Kaplan:1991dc , the couplings of
$a_{1,2}$ to the SM fermions, as well as the scalar potential, depend on the
quantum numbers of the composite operators that the SM fermions mix with
breaking the global symmetry. Or equivalently, they depend on how the SM
fermions are embedded in representations of $SO(7)$.

We assume that $q_{L}+u_{R}\sim\mathbf{7}+\mathbf{21}$. Likewise, we assume
that $l_{L}+e_{R}\sim\mathbf{27}+1$. Explicitly,
$L_{L}\equiv\nu_{L}\Lambda^{e}+e_{L}\Lambda^{\nu}$:

$L_{L}=\frac{1}{2}\left(\begin{array}[]{cccc}0_{4\times
4}&\theta\mathbf{v_{1}}^{T}&\gamma\mathbf{v_{2}}^{T}&\mathbf{v_{2}}^{T}\\\
\theta\mathbf{v_{1}}&0&0&0\\\ \gamma\mathbf{v_{2}}&0&0&0\\\
\mathbf{v_{2}}^{T}&0&0&0\end{array}\right)~{},$ (26)

where the vectors read $\mathbf{v_{1}}=(e_{L},-ie_{L},\nu_{L},i\nu_{L})$ and
$\mathbf{v_{2}}=(ie_{L},e_{L},i\nu_{L},-\nu_{L})$ and $\theta$ and $\gamma$
are real parameters. (Note that the different embeddings for quarks and
leptons is primarily justified by the fact that the lepton and quark masses
and mixings are completely different.)

The scalar potential can be written as
$V(h,a_{1,2})=V_{q}(h,a_{1,2})+V_{l}(h,a_{1,2})$, where the first and second
contributions of the RHS come from loops of quarks and leptons, respectively.
It can be also shown that the quark sector respects a symmetry $a_{1,2}\to-
a_{1/2}$, as well as the shift symmetry of the singlets. Consequently,
$V_{q}(h,a_{1,2})=V_{q}(h)$. It is completely fixed by the measurements of the
Higgs mass and its VEV.

The only model dependence come from $V_{l}(h,a_{1,2})$, which to leading order
in the expansion in the global symmetry breaking parameters reads:

$\displaystyle V_{l}$ $\displaystyle\sim
c_{1}f^{4}\bigg{[}\left(\Lambda_{D}^{\mathbf{1*}}\right)^{\alpha}\left(\Lambda_{D}^{\mathbf{1}}\right)_{\alpha}\bigg{]}+c_{2}f^{4}\bigg{[}\left(\Lambda_{D}^{\mathbf{6*}}\right)^{\alpha}_{i}\left(\Lambda_{D}^{\mathbf{6}}\right)^{i}_{\alpha}\bigg{]}~{},$

where the dressed spurion reads $\Lambda_{D}^{\alpha}\equiv
U^{T}\Lambda^{\alpha}U$ with $\alpha=e,\nu$. (The indices $\mathbf{1}$ and
$\mathbf{6}$ indicate the projection into the singlet and the sextuplet in the
decomposition $\mathbf{27}=1+\mathbf{6}+\mathbf{20}$ from $SO(7)$ to $SO(6)$.)
The constants $c_{1}$ and $c_{2}$ are free parameters encoding the (unknown)
details on the strongly coupled UV. Writing explicitly the one-loop induced
potential, we find:

$\displaystyle V_{l}$ $\displaystyle=4f^{3}c_{2}\gamma\Sigma
a_{2}+2f\left(c_{1}-2c_{2}\right)\gamma\Sigma a_{2}h^{2}$
$\displaystyle+\frac{1}{2}c_{2}f^{2}\bigg{[}\left(\gamma^{2}+\theta^{2}-7+2\frac{c_{1}}{c_{2}}\right)h^{2}$
$\displaystyle+4\left(\theta^{2}-1\right)a_{1}^{2}+4\left(\gamma^{2}-1\right)a_{2}^{2}\bigg{]}$
$\displaystyle+\left(c_{1}-2c_{2}\right)\bigg{[}\left(\theta^{2}-1\right)a_{1}^{2}+\left(\gamma^{2}-1\right)a_{2}^{2}-h^{2}\bigg{]}h^{2}~{}.$
(27)

We further expand this expression in powers of $1/f$, and keep only terms up
to dimension four:

$\displaystyle V_{l}$ $\displaystyle\sim 4f^{3}c_{2}\gamma
a_{2}+2f^{2}c_{2}\bigg{[}\left(\gamma^{2}-1\right)a_{2}^{2}+\left(\theta^{2}-1\right)a_{1}^{2}\bigg{]}$
$\displaystyle+2f\gamma\bigg{[}\left(c_{1}-3c_{2}\right)a_{2}h^{2}-c_{2}\left(a_{1}^{2}+a_{2}^{2}\right)a_{2}\bigg{]}$
$\displaystyle+(c_{1}-2c_{2})\bigg{[}\left(\theta^{2}-1\right)a_{1}^{2}+\left(\gamma^{2}-1\right)a_{2}^{2}\bigg{]}h^{2}+...$
(28)

where the three dots encode terms involving the Higgs boson solely.

The requirements $c_{1}\sim 3c_{2}$ and $\gamma\sim 1$ make the interactions
between $a_{2}$ and the Higgs (in particular mixings) very small. In order to
avoid bounds from Higgs searches, we restrict to this case hereafter. The
tadpole can then be removed by the field redefinition $a_{2}\rightarrow
a_{2}+\sqrt{2/3}f$.

Let us also fix $f=1$ TeV, as well as $c_{2}\sim
g_{*}^{2}y_{l}^{2}/(4\pi)^{2}\sim 10^{-6}$. The latter is the value expected
from SILH power counting Giudice:2007fh  for $g_{*}\sim 3$, with $g_{*}$ the
typical strong coupling between composite resonances. This choice fixes both
$m_{2}$ and $m_{12}$ to $\sim 3.1$ GeV and $\sim 0.002$ GeV, respectively;
while $m_{1}$ depends solely on $\theta$. We compute numerically this
dependence and it is depicted in Fig. -1007.

On another front, the Yukawa Lagrangian to dimension five reads:

$\displaystyle L_{Y}$
$\displaystyle=yf\overline{l_{L}}^{\alpha}\left(\Lambda_{D}^{\alpha}\right)^{\dagger}_{77}e_{R}+\text{h.c.}$
$\displaystyle=y_{\ell}\overline{l_{L}}He_{R}\left[1+\frac{1}{f}(a_{2}-i\theta
a_{1})+\cdots\right]~{}.$ (29)

The vector resonance associated to the generator $T^{56}$ is the only one that
couples to $a_{1,2}$. We identify it with $V$. The interaction between $V$ and
the pNGBs is entirely determined by the CCWZ formalism Coleman:1969sm ;
Callan:1969sn ; Panico:2015jxa , reading:

$L=\frac{1}{2g_{*}^{2}}m_{V}^{2}\left(g_{*}V_{\mu}^{a}-e_{\mu}^{a}\right)^{2}~{},$
(30)

where $e_{\mu}$ is the trace of the Maurer-Cartan form $\omega$ along the
unbroken generators:

$\omega_{\mu}=-iU^{\dagger}\partial_{\mu}U=d_{\mu}^{a}X_{a}+e_{\mu}^{a}T_{a}~{}.$
(31)

We expect $m_{V}\sim g_{*}f$. Therefore, the interaction between the vector
resonance and the light scalars reads:

$L_{int}=\sqrt{2}\,g_{*}\frac{V_{\mu}a_{2}\overleftrightarrow{\partial^{\mu}}a_{1}}{1+\Sigma}\sim\frac{g_{*}}{\sqrt{2}}V_{\mu}a_{2}\overleftrightarrow{\partial^{\mu}}a_{1}~{}.$
(32)

Finally, the vector resonance can not couple directly to the left-handed
quarks. The coupling $g_{qq}$ is therefore suppressed by $v^{2}/f^{2}\lesssim
0.1$.

Altogether, this model matches into the parameterization in Eq. II. For
example, let us take $\theta=1.2$. We obtain: $m_{1}\sim 1.3$ GeV, $m_{2}\sim
3.1$ GeV, $m_{12}\sim 0.002$ GeV, $g_{qq}\sim 0.1$, $g_{12}\sim 2$, $g_{2}\sim
0.17$, $g_{1}\sim 0.22$.

These numbers are also obtained if the leptons are embedded in
$\mathbf{7}+\mathbf{7}$.

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