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

TL;DR

This paper explores new light scalar decay signatures in B mesons at high-energy colliders, proposing novel experimental analyses at LHCb to detect these rare processes and outperform existing constraints.

Contribution

It introduces new B decay channels involving light scalars and proposes experimental strategies at LHCb to search for these signals, extending the reach beyond current bounds.

Findings

LHCb can probe branching ratios as low as 6.0×10^{-9} for certain channels

Proposed analyses can test scalar masses below a few GeV

Searches outperform existing meson mixing constraints

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.

Tables

Table 1. Table 1: Maximum and minimum efficiencies for selecting signal events in the channels B s 0 → 3 ​ μ + ​ 3 ​ μ − → superscript subscript 𝐵 𝑠 0 3 superscript 𝜇 3 superscript 𝜇 B_{s}^{0}\to 3\mu^{+}3\mu^{-} ( m X = m B s 0 subscript 𝑚 𝑋 subscript 𝑚 superscript subscript 𝐵 𝑠 0 m_{X}=m_{B_{s}^{0}} ) and B + → K + ​ 3 ​ μ + ​ 3 ​ μ − → superscript 𝐵 superscript 𝐾 3 superscript 𝜇 3 superscript 𝜇 B^{+}\to K^{+}3\mu^{+}3\mu^{-} ( m X = m B + − m K + subscript 𝑚 𝑋 subscript 𝑚 superscript 𝐵 subscript 𝑚 superscript 𝐾 m_{X}=m_{B^{+}}-m_{K^{+}} ) in each kinematic region. The upper limits ( × 10 − 9 absent superscript 10 9 \times 10^{-9} ) on the corresponding branching ratios for 3 3 3 fb -1 of data are also shown. We vary m 1 , 2 subscript 𝑚 1 2 m_{1,2} in the coloured region of Fig.  -898 , with m 2 < 10 subscript 𝑚 2 10 m_{2}<10 GeV and m 1 ≥ 1.1 subscript 𝑚 1 1.1 m_{1}\geq 1.1 GeV. (For smaller values of m 1 subscript 𝑚 1 m_{1} the efficiency is negligible.)

	$m_X\ge m_1+m_2$
	$m_2\ge 2m_1$		$m_X\ge 3m_1$
$B_s^0\to 3{\mu }^{+}3{\mu }^{-}$	[0.02,0.03]	[0.01,0.02]	[0.02,0.03]
limit ($\times {10}^{-9}$)	$[6.7,11.6]$	$[7.9,18.2]$	$[6.0,11.9]$
$B^{+}\to K^{+}3{\mu }^{+}3{\mu }^{-}$	[0.007,0.009]	[0.003,0.009]	four-body
limit ($\times {10}^{-9}$)	$[5.9,8.0]$	$[6.0,16.6]$	four-body

Equations

$$L=$$

$$+V^{\mu }\left[g_{12}{a}_1\overleftrightarrow{{\partial }_{\mu }}{a}_2+g_{qq}(\overline{q_L}{\gamma }_{\mu }{q}_L+\text{h.c.})\right],$$

$$g_{sb}\equiv g_{qq}{V}_{ts}^{CKM}{V}_{tb}^{CKM}\sim 0.04g_{qq},$$

$$\Gamma (a_2\to {\ell }^{+}{\ell }^{-})$$

$$\Gamma (a_2\to a_1{a}_1)$$

$$\Gamma (a_2\to$$

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

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

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

$$\mathcal{M}=2g_{sb}{g}_{12}\frac{f_B{m}_{12}}{m_V^2}\left[\frac{q_{12}^2-m_1^2}{q_{12}^2-m_2^2}+\frac{q_{23}^2-m_1^2}{q_{23}^2-m_2^2}+\frac{q_{13}^2-m_1^2}{q_{13}^2-m_2^2}\right],$$

$$\frac{d\Gamma }{d{q}_{12}^2}=\frac{{\left(g_{sb}{g}_{12}\right)}^2{m}_2^2}{384{\pi }^3{m}_B^3}{\left(\frac{f_B{m}_{12}}{m_V^2}\right)}^{2}F{\left[\frac{m_1}{m_2},\frac{m_B}{m_2},\frac{q_{12}}{m_2},\frac{q_{23}}{m_2}\right]}_{(q_{23}^2{)}^{\min }}^{(q_{23}^2{)}^{\max }}$$

$$F$$

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

$${\left(q_{23}^2\right)}^{\max }={\left(E_2^{\ast }+E_3^{\ast }\right)}^2-{\left(\sqrt{E_2^{\ast 2}-m_1^2}-\sqrt{E_3^{\ast 2}-m_1^2}\right)}^2$$

$${\left(q_{23}^2\right)}^{\min }={\left(E_2^{\ast }+E_3^{\ast }\right)}^2-{\left(\sqrt{E_2^{\ast 2}-m_1^2}+\sqrt{E_3^{\ast 2}-m_1^2}\right)}^2,$$

$$\Gamma \sim \frac{3{\left(g_{sb}{g}_{12}\right)}^2}{256{\pi }^3}\frac{f_B^2{m}_{12}^2}{m_V^4}{m}_B.$$

$$\mathcal{M}=-\frac{g_{sb}{g}_{12}}{m_V^2}⟨K(p_3)|\overline{s}{\gamma }_{\mu }b|B(p)⟩(p_2-p_1{)}^{\mu },$$

$$⟨K(p_3)|\overline{s}{\gamma }_{\mu }b|B(p)⟩=$$

$$+f_0(q^2)\frac{m_B^2-m_K^2}{q^2}{q}_{\mu }$$

$$\mathcal{M}$$

$$\displaystyle+\left[2\left(p_{2}+p_{3}\right)^{2}+q^{2}-m_{1}^{2}-m_{2}^{2}-m_{B}^{2}-m_{K}^{2}\right]f_{+}(q^{2})\bigg{\}}~{}.$$

$$\frac{d\Gamma }{d{q}^2}=\frac{{\left(g_{sb}{g}_{12}\right)}^2}{768{\pi }^3{m}_V^4{m}_B^3}F(q^2),$$

$$F(q^2)$$

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

$$f_{+}(q^2)=\frac{r_1}{(1-q^2/m^2)}+\frac{r_2}{(1-q^2/m^2{)}^2},$$

$$f_0(q^2)=\frac{r_2}{(1-q^2/m_{\mathrm{fit}}^2)}$$

$$\Gamma \sim \frac{{\left(g_{sb}{g}_{12}\right)}^2}{3072{\pi }^3{m}_V^4}{m}_B^5.$$

$$\Gamma (V\to q\overline{q})=\frac{g_{qq}^2}{8\pi }\left(1-\frac{m_q^2}{m_V^2}\right){\left(1-\frac{4m_q^2}{m_V^2}\right)}^{1/2}{m}_V,$$

$$\displaystyle\Gamma(V\to a_{1}a_{2})=\frac{g_{12}^{2}}{48\pi}\bigg{[}1$$

$$\displaystyle+\frac{(m_{2}^{2}-m_{1}^{2})^{2}}{m_{V}^{4}}\bigg{]}^{3/2}m_{V}~{},$$

$${\mathcal{B}}_{\text{max}}^{3{\mu }^{+}3{\mu }^{-}}\sim \frac{{\mathcal{B}}_{\text{max}}^{2{\mu }^{+}2{\mu }^{-}}\times {\epsilon }_{2{\mu }^{+}2{\mu }^{-}}}{1.8\times {\epsilon }_{3{\mu }^{+}3{\mu }^{-}}}\times \frac{\mathcal{L}}{{\mathcal{L}}^{\prime }},$$

$${\mathcal{B}}_{max}^{3{\mu }^{+}3{\mu }^{-}{K}^{+}}\sim \frac{{\mathcal{B}}_{max}^{2{\mu }^{+}2{\mu }^{-}}\times {\epsilon }_{2{\mu }^{+}2{\mu }^{-}}}{1.8\times 3.7\times {\epsilon }_{3{\mu }^{+}3{\mu }^{-}{K}^{+}}}\times \frac{\mathcal{L}}{{\mathcal{L}}^{\prime }},$$

$$T_{ij}^{mn}$$

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

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

Full text

††preprint: IPPP/19/63

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

[TABLE]

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

[TABLE]

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 $m_{12}/g_{2}$:

[TABLE]

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

[TABLE]

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 .

At low energies, the vector boson $V$ triggers $B$ meson decays into the light scalars; see Fig. -899. 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. -898.

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

[TABLE]

with

[TABLE]

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

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

[TABLE]

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:

[TABLE]

with

[TABLE]

which should be evaluated at

[TABLE]

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:

[TABLE]

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

[TABLE]

with

[TABLE]

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

[TABLE]

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

[TABLE]

with

[TABLE]

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

[TABLE]

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

[TABLE]

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:

[TABLE]

In Fig. -897, we show the magnitude of three body decays under consideration and their dependence with $(p_{1}+p_{2})^{2}$.

In Fig. -896, 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:

[TABLE]

[TABLE]

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$\mathbf{b}$

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. -895. And second, there are more muons with at least one track with $p_{T}<0.5$ GeV which is therefore not detected; see Fig. -894.

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

[TABLE]

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

[TABLE]

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 $B^{+}\to K^{+}3\mu^{+}3\mu^{-}$.

For illustration, we translate the expected limits in Tab. 1 to the plane $(g_{sb},m_{V})$ in Fig. -893 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. -892, 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.

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. -891 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. -890.

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.

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. -889 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. -888 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

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:

[TABLE]

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

[TABLE]

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, $L_{L}\equiv\nu_{L}\Lambda^{e}+e_{L}\Lambda^{\nu}$:

[TABLE]

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:

[TABLE]

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:

[TABLE]

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

[TABLE]

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. -887.

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

[TABLE]

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:

[TABLE]

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

[TABLE]

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

[TABLE]

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 $\mathbf{7}+\mathbf{7}$.