$U(1)_{B_3-3L_\mu}$ gauge symmetry as a simple description of $b\to s$ anomalies

TL;DR

This paper proposes a simple gauge extension of the Standard Model with a $U(1)_{B_3-3L_}$ symmetry to explain observed anomalies in B-meson decays, highlighting its minimal particle content and phenomenological implications.

Contribution

It introduces a minimal $U(1)_{B_3-3L_}$ gauge symmetry model that accounts for $R(K)$ and $R(K^*)$ anomalies with specific fermion charge assignments.

Findings

The model explains lepton non-universality in B decays.

It predicts flavor-changing neutral currents involving second and third quark families.

Constraints from current experiments are discussed.

Abstract

We present a simple $U(1)_{B_3-3L_\mu}$ gauge Standard Model extension that can easily account for the anomalies in $R(K)$ and $R(K^*)$ reported by LHCb. The model is economical in its setup and particle content. Among the Standard Model fermions, only the third generation quark family and the second generation leptons transform non-trivially under the new $U(1)_{B_3-3L_\mu}$ symmetry. This leads to lepton non-universality and flavor changing neutral currents involving the second and third quark families. We discuss the relevant experimental constraints and some implications.

Figures

Tables

Table 1. Table 1: Particle content and assignments. The singlet σ 𝜎 \sigma ensures a realistic pattern of neutrino oscillations  Forero:2014bxa .

Fields	$SU{\left(3\right)}_c\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$	$U{\left(1\right)}_{B_3-3L_{\mu }}$	Fields	$SU{\left(3\right)}_c\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$	$U{\left(1\right)}_{B_3-3L_{\mu }}$
$Q_i$	$(3,2,\frac{1}{3})$	$0$	$L_e$	$(1,2,-1)$	$0$
$u_{R_i}$	$(3,1,\frac{4}{3})$	$0$	$e_R$	$(1,1,-2)$	$0$
$d_{R_i}$	$(3,1,-\frac{2}{3})$	$0$	${\nu }_{R_e}$	$(1,1,0)$	$x_1$
$Q_3$	$(3,2,\frac{1}{3})$	$1$	$L_{\mu }$	$(1,2,-1)$	$-3$
$t_R$	$(3,1,\frac{4}{3})$	$1$	${\mu }_R$	$(1,2,-1)$	$-3$
$b_R$	$(3,1,-\frac{2}{3})$	$1$	${\nu }_{R_{\mu }}$	$(1,1,0)$	$x_2$
$Q_L^{\prime }$	$(3,2,\frac{1}{3})$	$1$	$L_{\tau }$	$(1,2,-1)$	$0$
$Q_R^{\prime }$	$(3,2,\frac{1}{3})$	$1$	${\tau }_R$	$(1,1,-2)$	$0$
			${\nu }_{R_{\tau }}$	$(1,1,0)$	$x_3$
${\Phi }_1$	$(1,2,-1)$	$1$	$\chi$	$(1,1,0)$	$y_1$
${\Phi }_2$	$(1,2,1)$	$0$	$\sigma$	$(1,1,0)$	$y_2$

Equations

$$R_{K^{*}}^{\rm expt}=\left\{\begin{array}[]{cc}0.660^{+0.110}_{-0.070}\leavevmode\nobreak\ {\rm(stat)}\pm 0.024\leavevmode\nobreak\ {\rm(syst)}\leavevmode\nobreak\ ,\nobreak\leavevmode\nobreak\leavevmode&0.045\leq q^{2}\leq 1.1\leavevmode\nobreak\ {\rm GeV}^{2}\leavevmode\nobreak\ ,\\ 0.685^{+0.113}_{-0.069}\leavevmode\nobreak\ {\rm(stat)}\pm 0.047\leavevmode\nobreak\ {\rm(syst)}\leavevmode\nobreak\ ,\nobreak\leavevmode\nobreak\leavevmode&1.1\leq q^{2}\leq 6.0\leavevmode\nobreak\ {\rm GeV}^{2}\leavevmode\nobreak\ .\end{array}\right.$$

$$R_K^{\mathrm{expt}}=0.745_{-0.074}^{+0.090}\text{leavevmode}(\mathrm{stat})\pm 0.036\text{leavevmode}(\mathrm{syst})\text{leavevmode},\text{leavevmode}\text{leavevmode}1\le q^2\le 6.0\text{leavevmode}{\mathrm{GeV}}^2\text{leavevmode},$$

$${\mathcal{H}}_{eff}=-\frac{4G_F}{\sqrt{2}}\frac{e^2}{16{\pi }^2}{V}_{tb}{V}_{ts}^{\ast }\sum _i\left({\mathcal{C}}_i(\Lambda )O_i(\Lambda )+{\mathcal{C}}_i^{\prime }(\Lambda )O_i^{\prime }(\Lambda )\right)$$

$$\displaystyle\begin{array}[]{cc}\mathcal{O}_{9}=(\bar{s}\gamma_{\alpha}P_{L}b)(\bar{\ell}\gamma^{\alpha}\ell),&\mathcal{O}_{9}^{\prime}=(\bar{s}\gamma_{\alpha}P_{R}b)(\bar{\ell}\gamma^{\alpha}\ell)\\ \mathcal{O}_{10}=(\bar{s}\gamma_{\alpha}P_{L}b)(\bar{\ell}\gamma^{\alpha}\gamma_{5}\ell),&\mathcal{O}_{10}^{\prime}=(\bar{s}\gamma_{\alpha}P_{R}b)(\bar{\ell}\gamma^{\alpha}\gamma_{5}\ell)\leavevmode\nobreak\ .\end{array}$$

$$[SU(3{)}_c{]}^{2}U(1{)}_X$$

$${\left[SU(2{)}_L\right]}^{2}U(1{)}_X$$

$${\left[U(1{)}_Y\right]}^{2}U(1{)}_X$$

$$U(1{)}_Y{\left[U(1{)}_X\right]}^2$$

$${\left[U(1{)}_X\right]}^3$$

$${\left[\mathrm{Gravity}\right]}^2\left[U(1{)}_X\right]$$

$$x_1^3+x_2^3+x_3^3$$

$$x_1+x_2+x_3$$

$$x_i=-3,x_j=-x_k.$$

$${\nu }_{R_e}$$

$$\chi$$

$$-{\mathcal{L}}_Y$$

$$\displaystyle\mathcal{M}_{d}=\left(\begin{array}[]{cccc}y^{d}_{11}v_{2}&y^{d}_{12}v_{2}&y^{d}_{13}v_{1}&y_{14}v_{\chi}\\ y^{d}_{21}v_{2}&y^{d}_{22}v_{2}&y^{d}_{23}v_{1}&y_{24}v_{\chi}\\ 0&0&y^{d}_{33}v_{2}&\mu\\ 0&0&y^{d}_{43}v_{2}&M_{Q^{\prime}}\\ \end{array}\right)\ \ \text{and}\ \ \mathcal{M}_{u}=\left(\begin{array}[]{cccc}y^{u}_{11}v_{2}&y^{u}_{12}v_{2}&0&y_{14}v_{\chi}\\ y^{u}_{21}v_{2}&y^{u}_{22}v_{2}&0&y_{24}v_{\chi}\\ y^{u}_{31}v_{1}&y^{u}_{32}v_{1}&y^{u}_{33}v_{2}&\mu\\ y^{u}_{41}v_{1}&y^{u}_{42}v_{1}&y^{u}_{43}v_{2}&M_{Q^{\prime}}\\ \end{array}\right)$$

$$m_{Z^{\prime }}^2\propto g^{\prime 2}{u}^2\text{and}{m}_Z^2\propto (g_1^2+g_2^2)v^2$$

$$-{\mathcal{L}}_{Z^{\prime }}=g^{\prime }{Z}^{\prime \alpha }{J}_{\alpha }^{\prime 0}=g^{\prime }{Z}^{\prime \alpha }\left[-3\bar{\mu }{\gamma }_{\alpha }\mu +\bar{b}{\gamma }_{\alpha }b+\bar{t}{\gamma }_{\alpha }t-3\bar{\nu }{\gamma }_{\alpha }\nu +{\bar{Q}}^{\prime }{\gamma }_{\alpha }{Q}^{\prime }\right].$$

$$\displaystyle J^{\prime 0}_{\alpha}\supset\left(\begin{array}[]{cccc}\bar{d}&\bar{s}&\bar{b}&\bar{Q}^{\prime}\end{array}\right)D_{L}\gamma_{\alpha}\left(\begin{array}[]{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right)D_{L}^{\dagger}\left(\begin{array}[]{c}d\\ s\\ b\\ Q^{\prime}\end{array}\right)+\left(\begin{array}[]{cccc}\bar{d}_{R}&\bar{s}_{R}&\bar{b}_{R}&\bar{Q}^{\prime}\end{array}\right)D_{R}\gamma_{\alpha}\left(\begin{array}[]{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right)D_{R}^{\dagger}\left(\begin{array}[]{c}d_{R}\\ s_{R}\\ b_{R}\\ Q^{\prime}\end{array}\right),$$

$$\displaystyle\mathcal{M}_{d}^{2}\sim\small\left(\begin{array}[]{cccc}\times&0&0&\times\\ 0&\times&0&\times\\ 0&0&\times&\times\\ \times&\times&\times&\times\end{array}\right)\ \ \text{and hence}\ \ D_{L}\sim\small\left(\begin{array}[]{cccc}1&0&0&V_{d4}\\ 0&1&0&V_{s4}\\ 0&0&1&V_{b4}\\ V_{4d}&V_{4s}&V_{4b}&1\end{array}\right)$$

$$-{\mathcal{L}}_{Z^{\prime }}^{{}^{\prime }}\supset g^{\prime }{Z}_{\mu }^{\prime }\text{leavevmode}\left[{\bar{d}}_a^{\prime }{\gamma }^{\mu }(({\Gamma }_L{)}_{ab}{P}_L+({\Gamma }_R{)}_{ab}{P}_R)d_b^{\prime }\right]$$

$$\displaystyle\Gamma_{L}=\left(\small\begin{array}[]{cccc}|V_{4d}|^{2}&V_{4d}V_{4s}^{*}&V_{4d}V_{4b}^{*}&V_{4d}V_{44}^{*}\\ V_{4s}V_{4d}^{*}&|V_{4s}|^{2}&V_{4s}V_{4b}^{*}&V_{4s}V_{44}^{*}\\ V_{4b}V_{4d}^{*}&V_{4b}V_{4s}^{*}&1-|V_{4b}|^{2}&V_{4b}V_{44}^{*}\\ V_{44}V_{4d}^{*}&V_{44}V_{4s}^{*}&V_{44}V_{4b}^{*}&1-|V_{44}|^{2}\end{array}\right)\ \ \text{and}\ \ \Gamma_{R}=\left(\begin{array}[]{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right)$$

$$C_9^{NP}=\left(\frac{8{\pi }^2{v}^2}{e^2{\lambda }_{bs}}\right)\left(\frac{3g^{{}^{\prime }2}{\hat{\lambda }}_{bs}}{m_{Z^{\prime }}^2}\right)=-\frac{3{\hat{\lambda }}_{bs}}{\hat{\alpha }{\lambda }_{bs}}{\left(\frac{g^{\prime }}{m_{Z^{\prime }}}\right)}^2$$

$$\displaystyle\Gamma_{L}\sim\left(\small\begin{array}[]{cccc}0&0&0&0\\ 0&\lambda_{C}^{4}&\lambda_{C}^{2}&\lambda_{C}^{2}\\ 0&\lambda_{C}^{2}&0&1\\ 0&\lambda_{C}^{2}&1&0\end{array}\right).$$

$$C_9^{NP}=-\frac{3}{\hat{\alpha }}{\left(\frac{g^{\prime }}{m_{Z^{\prime }}}\right)}^2.$$

$$\frac{M_{12}}{M_{12}^{\text{SM}}}=1+\frac{g^{{}^{\prime }2}}{m_{Z^{\prime }}^2}{\left(\frac{g_2^2}{16{\pi }^2{v}^2}(V_{ts}{V}_{tb}^{\ast }{)}^2{S}_0\right)}^{-1}$$

$$\displaystyle\frac{\sigma^{\text{NP}}}{\sigma^{\text{SM}}}=\frac{1+(4s_{W}^{2}+18v^{2}g^{{}^{\prime}2}/m_{Z^{\prime}}^{2})^{2}}{1+(1+4s^{2}_{W})^{2}}\leavevmode\nobreak\$$

$$\frac{{\sigma }^{\text{CCFR}}}{{\sigma }^{\text{SM}}}=0.82\pm 0.28\text{leavevmode}.$$

$$\frac{g_{V\mu }}{g_{Ve}}\simeq \frac{g_{A\mu }}{g_{Ae}}\simeq \left|1+\frac{9g^{\prime 2}}{(4\pi {)}^2}\kappa (m_{Z^{\prime }})\right|;$$

$$\frac{g_{V\nu }}{g_{Ae}}=\frac{g_{A\nu }}{g_{Ae}}\simeq \left|1+\frac{1}{3}\frac{9g^{\prime 2}}{(4\pi {)}^2}\kappa (m_{Z^{\prime }})\right|$$

$$R_{\sigma }=\frac{\sigma (pp\to Z^{\prime }+X\to {\mu }^{+}{\mu }^{-}+X)}{\sigma (pp\to Z+X\text{leavevmode}\to {\mu }^{+}{\mu }^{-}+X)}.$$

$$-1.45\le \mathrm{Re}\text{leavevmode}{C}_9^{\mathrm{NP}}\le -\mathrm{0.75.}$$

Full text

I INTRODUCTION

A host of increasingly sophisticated experiments over several decades has been able to thoroughly verify various predictions of the Standard Model of particle physics. These culminated with the discovery of the $125$ GeV Higgs boson at the Large Hadron Collider (LHC) in 2012. Despite its amazing success, there are good reasons to think that the Standard Model may not be the ultimate theory. Apart from many theoretical shortcomings, the existence of neutrino mass suggests the existence of new physics, possibly in the electroweak-TeV range. By making very precise measurements of decay rates and angular observables, the LHCb Collaboration looks for the effect of new particles in various hadronic processes. Of particular interest to LHCb are the processes which are either forbidden or are extremely rare within the Standard Model. Since such processes may be allowed in new physics models, these searches can probe new physics models with good sensitivity, sometimes higher than attainable at the ATLAS and CMS experiments.

LHCb has recently announced anomalous measurements of $b\to s\mu^{+}\mu^{-}$ transitions Aaij:2017vbb . They measured the ratio $R_{K^{*}}\equiv{\cal B}(B^{0}\to K^{*0}\mu^{+}\mu^{-})/{\cal B}(B^{0}\to K^{*0}e^{+}e^{-})$ as

[TABLE]

These measurements involve two ranges of $q^{2}$, the dilepton invariant squared mass. These numbers are very similar to the previous LHCb measurement of the ratio $R_{K}\equiv{\cal B}(B^{+}\to K^{+}\mu^{+}\mu^{-})/{\cal B}(B^{+}\to K^{+}e^{+}e^{-})$ PhysRevLett.113.151601 ,

[TABLE]

These observations are also in tune with the so called $P^{\prime}_{5}$ anomaly observed in the angular variable $P^{\prime}_{5}$ of $B\to K^{*}\mu^{+}\mu^{-}$ decays DescotesGenon:2012zf ; Aaij:2013qta ; Abdesselam:2016llu ; Wehle:2016yoi . In addition to these, LHCb has also observed other several other anomalies all involving $b\to s$ transitions, such as $B_{s}\to\phi\mu^{+}\mu^{-}$ Aaij:2015esa . Specially remarkable is the fact that, although each individual result is not specially significant, all of the anomalies observed in $b\to s\mu^{+}\mu^{-}$ transitions and form a coherent global picture Altmannshofer:2017yso ; Capdevila:2017bsm ; DAmico:2017mtc ; Hiller:2017bzc ; Ciuchini:2017mik ; Geng:2017svp ; Celis:2017doq ; Hurth:2017hxg ; Alok:2017sui .

At a basic level such rare transitions may be described by the effective Hamiltonian,

[TABLE]

where $\mathcal{C}_{i}^{(\prime)}=C_{i}^{(\prime)SM}+C_{i}^{(\prime)NP}$. Here each coefficient is separated into a Standard Model (SM) part and a new physics contribution (NP). The relevant semileptonic operators required to account for the observed $b\to s\mu^{+}\mu^{-}$ anomalies are of the restricted type Descotes-Genon:2013wba ,

[TABLE]

In this paper we propose a consistent gauge model, constructed from first principles, that induces just one of the Wilson operators, $\mathcal{O}_{9}$. Its strength parameter can describe the observed $b\to s\mu^{+}\mu^{-}$ anomalies in agreement with all existing experimental restrictions. It provides a minimal way to account for the $b\to s\mu^{+}\mu^{-}$ discrepancies, while adding as few new particles and symmetries as possible : just a new $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ gauge symmetry. In contrast to other alternative schemes, which typically require several the addition of new charged fermions111 For $Z^{\prime}$ models without need of additional vector like quarks, see Ref. Crivellin:2015lwa ; Bhatia:2017tgo . Altmannshofer:2014cfa ; Altmannshofer:2016jzy ; Crivellin:2015era ; Sierra:2015fma ; Carmona:2015ena ; Ko:2017quv ; Gauld:2013qja ; Kamenik:2017tnu ; Gauld:2013qba ; Crivellin:2015mga ; Hou:2017ozb ; Fuyuto:2015gmk ; Allanach:2015gkd ; Celis:2016ayl ; Boucenna:2016wpr ; Boucenna:2016qad ; Falkowski:2015zwa ; Celis:2015ara ; Belanger:2015nma ; Chiang:2016qov ; Megias:2016bde ; Hou:2018npi or leptoquarks Alonso:2015sja ; Becirevic:2016yqi ; Fajfer:2015ycq ; Becirevic:2017jtw ; Gripaios:2014tna ; Sahoo:2015wya ; Cai:2017wry ; Becirevic:2017jtw ; Crivellin:2017zlb ; Chen:2017hir ; Chen:2016dip ; Hiller:2016kry , all the anomalies induced by our $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ gauge symmetry, including the gravitational ones, cancel without need for adding any new charged states. The only new fermions present are a pair of heavy vector-like quarks $Q^{\prime}_{L,R}$ and right handed neutrinos $\nu_{R}$ required in order to generate small neutrino masses through the type-I seesaw mechanism Valle:2015pba . The only other new particles required are the $Z^{\prime}$ boson associated to the new gauge symmetry and new scalars involved in symmetry breaking, so as to generate mass for the $Z^{\prime}$ boson and neutrinos. The $Z^{\prime}$ boson mediates the anomalous $b\to s\mu^{+}\mu^{-}$ transitions.

The plan of the paper is as follows. In Section II we discuss the basics of the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ gauge model including the gauge and gauge-gravity anomaly cancellation conditions. We show that the model is free from anomalies. In Section III we summarize the main properties of the model and show that the flavor changing neutral currents (FCNC) mediated by the $Z^{\prime}$ boson have all the essential features required in order to account for the observed $b\to s\mu^{+}\mu^{-}$ discrepancies. In Section IV we discuss the various constraints on our model coming from flavor, collider and precision physics. We show that, after taking the relevant constraints into account, the model still has enough freedom to account for the observed anomalies in the B-system. Finally, our results are given in Fig. 1 and Sec.V and are summarized in Sec. VI.

II Gauging the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry

In this section we discuss the details of the model. The $\mathrm{SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}}$ and $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ charge assignments for the fermions and scalars are given in Table 1, where $i=1,2$ labels the first two generations of quarks.

Notice that apart from the “Standard Model like” $SU(2)_{L}$ doublet scalar $\Phi_{2}$, we also have added another $SU(2)_{L}$ doublet scalar $\Phi_{1}$ which is also charged under the new $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry. Notice that we have also included two $\mathrm{SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}}$ singlet scalars $\chi$ and $\sigma$ which are also charged under the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry. As we will discuss shortly, the scalar $\chi$ is required to ensure that the model does not have a massless Nambu-Goldstone boson left after symmetry breaking. The scalar $\sigma$ is needed in order to ensure a realistic pattern of neutrino masses and mixing that can describe oscillations. To induce the latter we also include, apart from the Standard Model fermions, three right handed neutrinos $\nu_{R_{i}}$ ($i=e,\mu,\tau$). Finally, we need the vector-like quarks $Q^{\prime}_{L}$ and $Q^{\prime}_{R}$, transforming as $SU(2)_{L}$ doublets, and also carrying $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ charges as shown in Tab. 1.

In order for $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ to be a consistent gauge symmetry it is important to ensure that the model is anomaly free. The $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry can potentially induce the following triangular anomalies:

[TABLE]

In addition we have anomaly conditions involving the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ just with itself and with gravity,

[TABLE]

where $(X)_{i}$ denote the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ fermion charges. Using the assignments in Table. 1, we find that the first four anomalies, i.e. eqs. (6)-(9), cancel, irrespective of the charges of the right handed neutrinos. The anomaly cancellation conditions eqs. (10) and (11) give the following two conditions on the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ charge of the right handed neutrinos

[TABLE]

The only solution for eq. (13) is given by

[TABLE]

This implies that under the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry, one of the right handed neutrinos should transform as $-3$ while the others can carry any arbitrary equal and opposite charge. For definiteness and keeping in mind simple scenarios of neutrino mass generation, we choose to assign the following charges to the right handed neutrinos:

[TABLE]

Also, we like to remark that the vector-like nature of the additional quarks $Q^{\prime}_{L,R}$ implies that the anomaly cancellation conditions do not fix their charges. Thus, they can have any charge under $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry. However, for the sake of simplicity and keeping minimality in mind (see Goldstone boson discussion below) we set their charges to be $1$, the same as the charges of the third generation of quarks.

Once the scalars get vacuum expectation values (vevs) the $\mathrm{SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}}$ $\otimes$ $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry is broken down to $\mathrm{U(1)_{\rm{EM}}}$. Notice that, since only $\Phi_{1}$, $\chi$ and $\sigma$ are charged under the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry, the latter is broken only by the vevs of these scalars. Notice also that both Standard Model doublet scalars $\Phi_{1}$ and $\Phi_{2}$ contribute to the breaking of the $\mathrm{SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}}$ symmetry.

The $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ charges of the scalars are not fixed by anomaly cancellation conditions. But given the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ charges of all the fermions in the model, the charges of scalars can also be restricted by other considerations. The $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ charge of the doublet scalar $\Phi_{1}$ is fixed by requiring an adequate pattern of quark mixing consistent with experiments. The charge of singlet scalar $\chi$ is fixed by the requirement that if the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ charge of $\chi$ is not the same as the charge difference between $\Phi_{1}$ and $\Phi_{2}$, then the scalar potential will have a residual global $U(1)$ symmetry leading to a massless Goldstone boson. This can be avoided by taking

[TABLE]

which provides a term in the scalar potential like $V\supset\kappa\,(\Phi^{\dagger}_{1}\Phi_{2}\,\chi+h.c.)$, where $\kappa$ is a dimensionful parameter. With these assignments for $\nu_{R_{i}}$, $\Phi_{1}$, $\Phi_{2}$ and $\chi$ plus a scalar singlet $\sigma$, transforming as $\sigma\sim 3$, one also has a realistic pattern of neutrino mass and mixing 222We stress that the above choice of $\nu_{R_{i}}$ charges under the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry is just the simplest one consistent with current neutrino oscillation data Forero:2014bxa . A detailed treatment of neutrino properties is outside the scope of this letter and will be addressed separately..

III The model

Having satisfied the anomaly cancellation conditions we now turn to the scalar and Yukawa sectors of the model. As shown in Table 1, except for the Standard-Model-like Higgs scalar $\Phi_{2}$, all other scalars are charged under the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry. Here our main focus is on explaining the anomaly, so we will skip the details of the scalar sector (fairly standard) in this work, and focus on the Lagrangian characterizing the Yukawa sector, which can be written as follows

[TABLE]

where $\tilde{\Phi}=i\tau_{2}\Phi^{*}$ and $i,j=1,2$ represent the first two families. After spontaneous symmetry breaking the scalars acquire vevs $\left\langle\Phi_{i}\right\rangle=v_{i}$; $i=1,2$, $\left\langle\chi\right\rangle=v_{\chi}$ and $\left\langle\sigma\right\rangle=v_{\sigma}$. The resulting quark mass matrices in the basis $\left(\bar{Q}_{1},\bar{Q}_{2},\bar{Q}_{3},\bar{Q}^{\prime}_{L}\right)^{T}$ and $(q_{1},q_{2},q_{3},Q^{\prime}_{R})$ are given by

[TABLE]

The resulting charged current weak interactions of quarks and leptons can be easily generated taking into account the mixing of Standard Model quarks with the new vector-like quarks. The mass matrices of (26) have enough freedom to be able to generate the required Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix of charged current interactions. However, the presence of vector-like quarks charged under the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry leads to interesting implications for the neutral currents. Hence we focus here on the weak neutral current, mediated both by the Standard Model $Z$ boson as well as the new $Z^{\prime}$. Since the scalars $\Phi_{1}$, $\chi$ and $\sigma$ are all charged under $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$, their vevs break the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry and contribute to the $Z^{\prime}$ mass. Thus, in the limit in which $\left\langle\chi\right\rangle,\left\langle\sigma\right\rangle\gg v_{1}$ and taking negligible $Z-Z^{\prime}$ mixing, the neutral gauge boson masses are given by,

[TABLE]

where $g^{\prime}$, $g_{1}$ and $g_{2}$ are the coupling constants of the $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$, U(1)Y and SU(2)L symmetries, respectively. The vevs of the doublet scalars $\left\langle\Phi_{i}\right\rangle=v_{i}$ ($i=1,2$) satisfy $v^{2}\equiv(174\,\text{GeV})^{2}=v_{1}^{2}+v_{2}^{2}$, whereas $u^{2}\equiv v_{\chi}^{2}+9v_{\sigma}^{2}+v^{2}_{1}$.

The key part of the theory in order to account for the B-decay anomaly is the neutral current. One can easily see that the Standard Model part of the neutral current, involving only $Z$ boson interactions, is canonical, obeying the Glashow-Iliopoulos-Maiani mechanism. In contrast however, by construction, the neutral current Lagrangian associated with the $Z^{\prime}$ boson will give rise to flavor changing transitions, that is

[TABLE]

From Eq. (28) it is clear that in our model, in the gauge basis, the $Z^{\prime}$ couples vectorially to the third generation Standard Model quarks and the second generation leptons. A chiral variant of this feature can also be obtained as in Cline:2017lvv . After spontaneous symmetry breaking the part associated to down-type quarks becomes

[TABLE]

where $D_{L}$ is the rotation matrix that diagonalizes $\mathcal{M}_{d}^{2}=\mathcal{M}_{d}\mathcal{M}_{d}^{\dagger}$ (namely, $\text{diag}(m_{d}^{2},m_{s}^{2},m_{b}^{2},M_{Q^{\prime}}^{2})=D_{L}\mathcal{M}_{d}^{2}D_{L}^{\dagger}$) and $D_{R}=\mathbb{I}$. The squared mass matrix for down-type quarks can be taken to the form,

[TABLE]

As a result, the interactions between $Z^{\prime}$ and the down-type quarks in the mass basis can be rewritten as follows

[TABLE]

where $d^{\prime}=(d,s,b,Q^{\prime})$,

[TABLE]

One can see that our model implies that the $R(K)$ and $R(K^{*})$ get modified only by the operator $\mathcal{O}_{9}$. Hence, $C_{9}^{{}^{\prime}NP}=C_{10}^{NP}=C_{10}^{{}^{\prime}NP}=0$. The associated strength parameter is given as

[TABLE]

where $\lambda_{bs}\equiv V_{tb}V_{ts}^{\ast}$, $\hat{\lambda}_{bs}\equiv V_{4b}V_{4s}^{\ast}$ and $\hat{\alpha}=e^{2}/(8\pi^{2}v^{2})\approx 1.9\times 10^{-8}\,\text{GeV}^{-2}$. In what follows we show that our model not only qualitatively satisfies all requirements to explain the observed anomalies, but can also quantitatively explain them, satisfying all relevant experimental constraints. In our numerical computations in the next section we will work in the limit $|\hat{\lambda}_{bs}|\simeq|\lambda_{bs}|\sim\lambda_{C}^{2}$. Also, for a concrete benchmark we take $V_{44},V_{4b}\sim 1$ and $V_{4s}\sim\lambda_{C}^{2}$. Furthermore, in order to avoid generating unacceptable $K^{0}-\bar{K}^{0}$ mixing we take $V_{4d}\sim 0$. In this limit we have that eq. (66) becomes

[TABLE]

In this approximation the Wilson coefficient turns out to be

[TABLE]

Notice that the above choice constitutes just a simple benchmark of our model. The result in the last equation serves to illustrate in a simple manner that our model provides a successful way to account for the b-anomalies from first principles.

IV Experimental constraints

Having shown that the flavor changing neutral current (FCNC) mediated by the $Z^{\prime}$ boson has the right form, we now turn to the experimental constraints on the two parameters, the $Z^{\prime}$ mass and coupling strength, relevant for describing in our model the anomalies recently observed in rare B decays.

IV.1 B meson mixing

The existence of a non-zero $Z^{\prime}bs$ coupling induces $B_{s}-\bar{B}_{s}$ mixing at the tree-level, resulting in very stringent limits on $g^{\prime}$ and $m_{Z^{\prime}}$. Such a tree level $Z^{\prime}$ exchange modifies the $B_{s}-\bar{B_{s}}$ meson mixing amplitude $M_{12}$, which can be quantified as Buras:2014yna

[TABLE]

where $g_{2}$ is the $SU(2)_{L}$ gauge coupling and $S_{0}$ is Inami-Lim function with value $\simeq 2.3$ Lenz:2010gu ; Buras:2012jb . The mixing amplitude $M_{12}$, related to the mass difference by $\Delta m_{B^{0}_{s}}=2|M_{12}|$ is measured precisely at per mill level Olive:2016xmw , while the calculation of $M_{12}$ suffers from several uncertainties. The two dominant sources of uncertainties are hadronic matrix element and CKM factor; the former is estimated by FLAG 2016 Aoki:2016frl as $\sim 12\%$, while the latter is $\sim 5\%$ Charles:2015gya ; Bona:2006sa . However, a recent accurate estimate provided by the Fermilab Lattice and MILC collaborations Bazavov:2016nty has improved the hadronic uncertainty and pushed the FLAG average down to $\sim 6$%. Including such sources of uncertainties, CKMfitter constrains $\left|\frac{M_{12}}{M_{12}^{\text{SM}}}\right|<1.32$ Charles:2015gya while UTfit $\left|\frac{M_{12}}{M_{12}^{\text{SM}}}\right|<1.28$ Bona:2006sa at 2-standard-deviation ($2\sigma$). Note that the summer 2016 result  Bona:2006sa of UTfit includes the result of Ref. Bazavov:2016nty . In order to constrain the the parameter space of $g^{\prime}$ and $m_{Z^{\prime}}$, we allow new physics contribution in $\left|\frac{M_{12}}{M_{12}^{\text{SM}}}\right|$ up to 30% and 15%, which are displayed by the purple shaded region and purple dashed line in Fig. 1 respectively 333See Ref. Kohda:2018xbc for a more detailed discussion on the new physics contribution in $\left|\frac{M_{12}}{M_{12}^{\text{SM}}}\right|$..

IV.2 Neutrino trident production

The production of a $\mu^{+}\mu^{-}$ pair in the scattering of a muon neutrino in the Coulomb field of a heavy nucleus, i.e. $\nu_{\mu}N\to\nu_{\mu}N\mu^{+}\mu^{-}$, provides a sensitive probe for $g^{\prime}$. In our model the correction to the trident cross section can be expressed as Altmannshofer:2014cfa

[TABLE]

where $v=246$ GeV and $s_{W}=\sin\theta_{W}$. The measurement of the trident cross section by the CCFR collaboration is Mishra:1991bv

[TABLE]

Utilizing Eqs. (74) and allowing $2\sigma$ error in $\frac{\sigma^{\text{CCFR}}}{\sigma^{\text{SM}}}$, we find the excluded region shown by the blue solid line in Fig. 1.

IV.3 Lepton Flavor Universality in $Z$ boson decay

The presence of $Z^{\prime}\mu\mu$ and $Z^{\prime}\nu_{\mu}\nu_{\mu}$ couplings will break lepton flavor universality (LFU) in $Z$ boson decay. This is manifest in the $Z$ boson couplings to muons and muon neutrinos through loop effects. The corrections to the vector and axial vector couplings of $Z\mu\mu$ coupling relative to the Standard-Model-like $Zee$ can be expressed as Altmannshofer:2014cfa ; Altmannshofer:2016brv

[TABLE]

and similarly for $Z\nu\nu$

[TABLE]

where $\kappa(m_{Z^{\prime}})$ is the loop factor associated with the $Z^{\prime}$ loop Haisch:2011up , whose real part is taken to match the convention of Ref. ALEPH:2005ab . The factor 1/3 in Eq.(77) accounts for the fact that out of three neutrino flavor only $Z\to\nu_{\mu}\bar{\nu}_{\mu}$ is affected by the $Z^{\prime}$. The vector and axial vector couplings of $Z$ boson in Eqs. (76) and (77) can be found from the average of 14 electroweak measurements in Ref. ALEPH:2005ab . The relevant ones are: $g_{Ve}=-0.03816\pm 0.00047$, $g_{Ae}=-0.50111\pm 0.00035$, $g_{V\mu}=-0.0367\pm 0.0023$, $g_{A\mu}=-0.50120\pm 0.00054$ and $g_{V\nu}=g_{A\nu}=0.5003\pm 0.0012$. We find that $g_{A\mu}/g_{Ae}=1.00018\pm 0.00128$ provides the most stringent constraint, where the uncertainties are added in quadrature. The resulting $2\sigma$ upper limit on $g^{\prime}$ is shown by the red dashed line in Fig. 1.

IV.4 Constraint from $Z\to 4\ell$

The ATLAS TheATLAScollaboration:2013nha and CMS Sirunyan:2017zjc collaborations both have set upper limits on the branching ratio of the $Z$ boson decay to four charged leptons. The ATLAS TheATLAScollaboration:2013nha analysis was performed with Run 1 data (7 TeV + 8 TeV), while CMS Sirunyan:2017zjc utilized 13 TeV 35.9 fb*-1* data to set the limit on $\mathcal{B}r(Z\to 4\ell)$. In particular, the observed value reported by Ref. Sirunyan:2017zjc is $\mathcal{B}r(Z\to 4\mu)=(4.83^{+0.23}_{-0.22}(\mbox{stat})^{+0.32}_{-0.29}(\mbox{syst})\pm 0.08(\mbox{theo})\pm 0.12(\mbox{lumi}))\times 10^{-6}$, while the SM prediction is $(4.37\pm 0.03)\times 10^{-6}$  Sirunyan:2017zjc .

In our model the $Z\to 4\ell$ decay will receive contributions from processes involving the $Z^{\prime}$ as the intermediate state, such as contribution from $Z\to\mu^{+}\mu^{-}Z^{\prime}$ followed by $Z^{\prime}\to\mu^{+}\mu^{-}$, resulting stringent bound on $g^{\prime}$ for $m_{Z^{\prime}}<m_{Z}$.

In order to determine the upper limit on $g^{\prime}$ from $\mathcal{B}r(Z\to 4\mu)$, we utilized the $\mathrm{SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}}$ prediction and observed value of Ref. Sirunyan:2017zjc , with the errors in the latter symmetrized and added in quadrature. The cross sections are generated in the Monte Carlo event generator MadGraph5_aMC@NLO Alwall:2014hca , interfaced to PYTHIA 6.4 Sjostrand:2003wg for hadronization and showering and finally fed into fast detector simulator Delphes 3.3.3 deFavereau:2013fsa so as to incorporate detector effects. In our analysis we adopt the PDF set NN23LO1 PDF Ball:2013hta . The effective Lagrangians written in eq. (28) and eq. (57) are implemented in FeynRules 2.0 Alloul:2013bka . Following the analysis of Ref. Sirunyan:2017zjc , we select events with four isolated muons with two opposite sign same flavor dimuon pairs. The muons in the quadruplet are required to be separated by $\Delta R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}>0.02$, with each of them having maximum pseudo-rapidity $\left|\eta\right|<2.5$. The two leading muons in an event should have transverse momenta $p_{T}$ $>$ $20$ GeV and $15$ GeV respectively, while the other two muons are required to have $p_{T}>5$ GeV. The four muons will constitute two same flavor oppositely charged muon pairs in an event. The pair closest to the $Z$ boson mass should have invariant mass $>40$ GeV and both pairs should have invariant mass $<120$ GeV. All pairs of oppositely charged muons must have invariant mass $>4$ GeV. We finally impose an invariant mass cut $80\leavevmode\nobreak\ \mbox{GeV}<m_{4\mu}<100\leavevmode\nobreak\ \mbox{GeV}$ on the four muons in the event. Finally, demanding the SM plus NP contribution i.e. the total contribution from $Z^{\prime}\to\mu^{+}\mu^{-}$ not to exceed the $2\sigma$ error, we overlay the solid red line ($2\sigma$ upper limit) in the left panel of Fig. 1. The limits from Ref. TheATLAScollaboration:2013nha are weaker than those of Ref. Sirunyan:2017zjc , which we do not show in Fig. 1.

IV.5 Constraint from $pp\to Z^{\prime}+X\to\mu^{+}\mu^{-}+X$

The $Z^{\prime}$ boson will be produced at LHC predominantly by the flavor conserving $b\bar{b}\to Z^{\prime}$ processes with a correction from flavor violating $s\bar{b}\to Z^{\prime}$ (and its conjugate process). Hence, the search for heavy resonances in the dimuon final state by the ATLAS and CMS collaborations will constrain he parameter space of our model. In particular ATLAS Aaboud:2017buh has set a $95\%$ CL (confidence level) upper limit on $\sigma(pp\to Z^{\prime}+X)\mathcal{B}r(Z^{\prime}\to\mu^{+}\mu^{-})$ in the 150 GeV $\lesssim m_{Z^{\prime}}\lesssim$ 5 TeV mass range with the 13 TeV and 36.1 fb*-1* dataset, where $X$ conforms inclusive activity. CMS Sirunyan:2018exx has also searched for heavy resonances decaying to dimuon pair in the mass range 200 GeV $\lesssim m_{Z^{\prime}}\lesssim$ 5.5 TeV with 13 TeV, also with $\sim$ 36 fb*-1* dataset, setting a $95\%$ CL upper limit on $R_{\sigma}$ defined as:

[TABLE]

We reinterpret $R_{\sigma}$ and extract $\sigma(pp\to Z^{\prime}+X)\mathcal{B}r(Z^{\prime}\to\mu^{+}\mu^{-})$ by multiplying with the Standard Model prediction of $\sigma(pp\to Z+X)\mathcal{B}r(Z\to\mu^{+}\mu^{-})=1928.0$ pb CMS:2015nhc .

In order to determine the upper limit, we generate matrix element (ME) of the $pp\to Z^{\prime}$ process up to two additional jets in the final state to include inclusive contributions. The ME is then merged and matched with parton shower (PS) following the MLM Alwall:2007fs matching prescription. We restrict ourselves up to two additional jets due to computational limitations. It should be noted that we have not used any $K$ factor in our analysis. We finally convert the observed ATLAS (CMS) $95\%$ CL upper limit on $\sigma(pp\to Z^{\prime}+X)\mathcal{B}r(Z^{\prime}\to\mu^{+}\mu^{-})$ to constraint $g^{\prime}$ and $m_{Z^{\prime}}$ in the mass range 150 GeV $<m_{Z^{\prime}}<$ 5 TeV (200 GeV $<m_{Z^{\prime}}<$ 5.5 TeV) which is shown by the black (light-blue) shaded region in Fig. 1.

V Results

We now discuss the constraints on the coupling $g^{\prime}$ and mass $m_{Z^{\prime}}$ in our model, which is summarized in Fig. 1.

The region of coupling-mass ($g^{\prime}-m_{Z^{\prime}}$) allowed by the various flavour and collider physics constraints are divided into two ranges indicated in Fig. 1. The left panel corresponds to the case of light ($m_{Z^{\prime}}<1$ TeV) masses, while the heavy mass range ($1\,\rm{TeV}<m_{Z^{\prime}}<10$ TeV) $Z^{\prime}$ is shown in the right panel. In order to find the allowed parameter space which describes the observed anomalies in $b\to s\ell\ell$ transition, we follow the global-fit analysis presented in Ref. Capdevila:2017bsm . The analysis uses all available $b\to s\ell\ell$ data from LHCb, ATLAS, CMS and Belle, with the best fit value ${\rm Re}\leavevmode\nobreak\ C^{\rm NP}_{9}=-1.11$, and $2\sigma$ range

[TABLE]

Utilizing Eq.(72), the $2\sigma$ allowed region of ${\rm Re}\leavevmode\nobreak\ C_{9}$ is translated into $g^{\prime}$ vs $m_{Z^{\prime}}$ plane, and shown by the green shaded region in Fig. 1, while the central value ${\rm Re}\leavevmode\nobreak\ C^{\rm NP}_{9}=-1.11$ is shown by black dot-dashed line. Note that the authors of Ref. Capdevila:2017bsm also present results taking into account only lepton-flavor universality observables, such as $R_{K^{(*)}}$ etc., which we do not display in Fig. 1 for simplicity. Although in our analysis we have only utilized the global fit values from Ref. Capdevila:2017bsm , there exist other global-fit analyses based on the effective Hamiltonian formalism (see e.g. Refs. Altmannshofer:2017yso ; DAmico:2017mtc ; Hiller:2017bzc ; Geng:2017svp ; Ciuchini:2017mik ; Celis:2017doq ; Hurth:2017hxg ), which can also be used with similar results as displayed in Fig. 1. The resulting allowed regions are indeed very similar, with a significant overlap region 444There are other observables like $P^{\prime}_{5}$, derived in a model independent way from $B\to K^{*}\mu^{+}\mu^{-}$ decay Mandal:2015bsa , which indicate new physics contribution in the right handed current..

We have also studied all relevant constraints from flavor physics as well as from LEP precision observables ALEPH:2005ab and direct searches at the LHC Aaboud:2017buh ; Sirunyan:2018exx ; Sirunyan:2017zjc . The most relevant LEP constraint from these decays is shown by red dashed line in Fig.1. The region above the red dashed line is excluded by LFU in $Z$ boson decay. The ATLAS collaboration TheATLAScollaboration:2013nha has also looked for the decay $Z\to 4\mu$ which in our model is mediated by the $Z^{\prime}$. This places a constraint on our model parameter space, shown by the solid red line (the region above the line is ruled out). The most stringent limits for $m_{Z^{\prime}}<150$ GeV come from $B_{s}$ mixing Amhis:2014hma ; Charles:2015gya , which is shown by the purple shaded region in both figures; except for $25\leavevmode\nobreak\ \mbox{GeV}\lesssim m_{Z^{\prime}}\lesssim 40$ GeV where constraint from $Z\to 4\mu$ becomes strongest.

In the region $150\leavevmode\nobreak\ \mbox{GeV}\lesssim m_{Z^{\prime}}\lesssim 2.2$ TeV the constraint from $B_{s}$ mixing is superseded by the search for heavy $Z^{\prime}$ boson in the dimuon final state by the ATLAS Aaboud:2017buh and CMS collaborations Sirunyan:2018exx . The black shaded region in Fig.1 is excluded by ATLAS, while the light-blue shaded region is ruled out by CMS. In general, limits from CMS Sirunyan:2018exx are a bit stronger than those of ATLAS Aaboud:2017buh ; except for $150\leavevmode\nobreak\ \mbox{GeV}\lesssim m_{Z^{\prime}}\lesssim 200$ GeV, where CMS provides no result. One sees that the direct search limits from ATLAS and CMS rule out a simultaneous explanation of all $b\to s$ transition anomalies for $Z^{\prime}$ masses in the range from $150\leavevmode\nobreak\ \mbox{GeV}\lesssim m_{Z^{\prime}}\lesssim 2.2$ TeV, leaving however the possibility for discovery in the range $m_{Z^{\prime}}\lesssim 150$ GeV and $m_{Z^{\prime}}\gtrsim 2.2$ TeV  555See Ref. Kohda:2018xbc for the details on how a $Z^{\prime}$, if it were confirmed in the near future, could be associated to $b\to s\ell\ell$ anomalies.. We also mention that low-energy experiments also set severe constraints on the model. For example, the constraint from neutrino trident production is shown by the solid blue line, with region above it excluded Altmannshofer:2014cfa ; Mishra:1991bv . In addition, the decay $B\to K^{(*)}\nu\bar{\nu}$ Buras:2014fpa can also constrain the our model parameters, though we find that the limits on $g^{\prime}$ are weaker and we do not display in Fig.1. Finally, another potentially strong constraint can come from $D^{0}$-${\bar{D}}^{0}$ mixing Kumar:2018kmr . However, we have checked that for our choice of benchmark point the limits coming from this constraint are weaker and hence not displayed in Fig.1.

VI Conclusions

In this letter we have presented a rather simple anomaly free $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ Standard Model extension that can account for the recent anomalies in $b\to s\ell\ell$ transitions reported by the LHCb collaboration. The model is minimalistic both in its setup as well as particle content. Amongst the Standard Model quarks and leptons, only the third generation quark family and the second generation leptons transform non-trivially under our new postulated $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry. This leads to a very simple pattern for lepton non-universality and flavor changing neutral currents involving the second and third families which reproduces the LHCb findings in a way consistent with all the relevant experimental constraints, except for the range from 150 GeV up to $\sim 2.2$ TeV, where the understanding of the $B\to K^{(*)}$ anomaly reported by LHCb would clash with the direct searches in the di-muon channel by the ATLAS and CMS collaborations. One should also stress, as seen in left panel of Figs. 1, that the $Z^{\prime}$ associated to our $\mathrm{U(1)_{B_{3}-3L_{\mu}}}$ symmetry can be as light as 10 GeV, in contrast to $Z^{\prime}$s associated to other gauge extensions based on B-L Das:2012ii ; Ma:2015mjd or 331 theories Deppisch:2013cya ; Queiroz:2016gif . As a last comment, we mention that, throughout the paper, we have assumed dominance of the vector boson mediated neutral current contribution, neglecting all the scalars. We checked that, indeed, there is a realistic limit in parameters space where this can be achieved.

Acknowledgements.

We thank Avelino Vicente, Masaya Kohda and Amol Dighe for useful comments and discussions. Work supported by the Spanish grants SEV-2014-0398 and FPA2017-85216-P (AEI/FEDER, UE), PROMETEO/2018/165 (Generalitat Valenciana) and the Spanish Red Consolider MultiDark FPA2017‐90566‐REDC. CB acknowledges financial support from the Collaborative Research Center SFB1258. The work of TM is supported by MOST-104-2112-M-002-017-MY2.