A Light Higgs at the LHC and the B-Anomalies

After the Higgs discovery, the LHC has been looking for new resonances, decaying into pairs of Standard Model (SM) particles. Recently, the CMS experiment observed an excess in the di-photon channel, with a di-photon invariant mass of about 96~GeV. This mass range is similar to the one of an excess observed in the search for the associated production of Higgs bosons with the $Z$ neutral gauge boson at LEP, with the Higgs bosons decaying to bottom quark pairs. On the other hand, the LHCb experiment observed a discrepancy with respect to the SM expectations of the ratio of the decay of $B$-mesons to $K$-mesons and a pair of leptons, $R_{K^{(*)}} = BR(B \to K^{(*)} \mu^+\mu^-)/BR(B\to K^{(*)} e^+e^-)$. This observation provides a hint of the violation of lepton-flavor universality in the charged lepton sector and may be explained by the existence of a vector boson originating form a $U(1)_{L_\mu - L_\tau}$ symmetry and heavy quarks that mix with the left-handed down quarks. Since the coupling to heavy quarks could lead to sizable Higgs di-photon rates in the gluon fusion channel, in this article we propose a common origin of these anomalies identifying a Higgs associated with the breakdown of the $U(1)_{L_\mu - L_\tau}$ symmetry and at the same time responsible to the quark mixing, with the one observed at the LHC. We also discuss the constraints on the identification of the same Higgs with the one associated with the bottom quark pair excess observed at LEP.


I. INTRODUCTION
The discovery of a scalar resonance, with properties similar to the ones expected for the Higgs boson in the Standard Model (SM), has provided evidence for the realization of the simplest Higgs mechanism scenario of electroweak symmetry breaking. The couplings of the observed Higgs boson to the SM particles is within a few tens of percent of the ones expected within the SM. Small deviations of these coupling with respect to the SM values are still possible, and are expected in extensions of the Higgs sector that occur in most beyond the SM scenarios. For this reason, since the Higgs discovery, apart from a precise determination of the Higgs couplings, the LHC has been looking for new scalar resonances, with the diphoton channel being one of the most sensitive ones. Recently, the CMS experiment reported a 2.9 σ excess in this channel [1], with a di-photon invariant mass of about 95.3 GeV. This excess was mildly present in the 8 TeV run [2], but became prominent only in the 13 TeV run. While the ATLAS experiment did not observe any significant excess in this mass region in the 8 TeV run [3], it has not yet reported the results of a similar search in the 13 TeV run.

Searches for Higgs boson resonances produced in association with the Z gauge boson, with
Higgs bosons decaying into bottom-quark pairs, were conducted at LEP. The combination of the results of the four experiments, ALEPH, DELPHI, L3 and OPAL, led to the presence of a 2.3 σ local excess at an invariant mass of about 95-100 GeV [4]. The agreement between the invariant mass of the excesses observed at LEP and CMS calls for a possible common origin of these two signatures [5][6][7][8][9][10].
The absence of a coupling to electrons explains the deviation of the above ratios with respect to one, the value expected within the SM. In order to allow the coupling of the new gauge pairs. Therefore, a very natural question is whether such Higgs boson can be identified with the one that is observed by the CMS and LEP experiments. We analyze the signal and existing constraints in this paper and answer this interesting question in the conclusion section.
In this article, we shall describe the simplest scenario that can lead to a realization of this  [21,. The gauge charge L µ − L τ is not flavor universal, but diagonal.
We also need one extra vector-like heavy quark ψ L,R charged under both SM gauge and the chiral components of a vector-like heavy quark, which has the same SM charge as q i L and also charged under U (1) Lµ−Lτ . The charge assignments of φ and ψ under the U (1) Lµ−Lτ symmetry need to satisfy the relation Q φ µ−τ = ±Q ψ µ−τ to allow for the appropriate Yukawa couplings. Without loss of generality, it can be taken to be Q φ µ−τ = −Q ψ µ−τ = 1/2. The anomaly-free condition is satisfied because U (1) Lµ−Lτ is vector-like in the NP particle content. It is also anomaly-free within the SM sector. The vacuum expectation value of the field φ may induce a mixing between the SM quarks and the NP heavy quarks. In the following sections, we will discuss the model in details in the Higgs, gauge boson and quark sectors.

A. The Higgs sector and gauge boson sector
In the Higgs sector of this model, we have two extra Higgs S, φ. We can write down the Lagrangian for the Higgs bosons, i = 1, 2, 3 is the SM quark generation index. S and φ are the NP Higgs, and ψ L,R are vector-like heavy quarks. The charge assignment Q φ µ−τ = −Q ψ µ−τ allows for the appropriate Yukawa couplings.
Without loss of generality, it is taken to be 1/2.
The first line in L Higgs are kinetic terms for φ and S, which give mass to Z after symmetry breaking, and others are scalar potential terms. The couplings λ φh , λ Sh and λ Sφ define the Higgs portal terms, which induce the mixing between scalars. The scalar φ, S and SM Higgs H get vevs and can be expanded near the minimum as φ = 1 The gauge boson Z has no mass mixing with the SM gauge bosons. Only the two kinetic terms in the first line of Eq. (3) give the Z mass, which reads In our analysis, we assume the observed 96 GeV di-photon excess observed at CMS is predominantly associated with the CP-even component of φ 0 . This means v D is nearly the same order of the SM Higgs vev v. If there is no another Higgs S, the Z mass is too light and severely constrained by di-jet and di-lepton constraints in Section IV A. We therefore introduce another scalar S with unit charge under U (1) Lµ−Lτ , and that contribute to the Z mass, as described above. Therefore, the Z mass can be as heavy as several TeV with a large value of v S and we treat it as a free parameter thereafter. In the later analysis, we could find out m Z ∼ 4.1 TeV, thus m s 0 should be of the same order. We shall not consider the phenomenology of such a heavy Higgs, as it could easily decouple from the SM Higgs and φ phenomenology. In our analysis, we assume λ Sh and λ Sφ small, so that complemented with the heaviness of s 0 we can neglect its mixing with the light Higgs bosons and only consider the mixing between H and φ. On the other hand, S has different U (1) Lµ−Lτ charge to Ψ L,R and SM quarks. It will not affect the quark sector mixing and we do not need to consider it any more.
For the scalar mixing, after ignore the effect of S, we only consider two scalars φ 0 and h 0 with five free parameters µ φ , µ, λ φ , λ and λ φh . The five parameters can be traded to five physical observables, two vevs v D and v, two masses mh and mφ, and one mixing angle sin α.
h andφ are the mass eigenstates and related to φ 0 and h 0 by The five physical observables are the more useful model parameters. The relations with the old parameters are given in Appendix A.

B. The quark sector
After the scalar sector, we are going to specify the mixings between the SM quarks and heavy vector-like quarks, which are responsible to induce the flavor violating couplings needed for the explanation of the B-anomalies. The most general interactions in the quark sector are given by: where q L are left-handed SM quarks, u R and d R are up-type and down-type right-handed SM quarks. ψ is the heavy vector-like quark, H is the SM Higgs and φ is the NP Higgs. i, j are the generation index, y u , y d are the SM Yukawa couplings.H is defined asH ≡ iσ 2 H * .
The SU (2) L doublet ψ can be written as components (ψ u , ψ d ), which are mass degenerate.
After the scalars obtained vevs, the mass and interaction terms for quarks are, where the sum over q = u, d goes over the components of SU (2) L doublet. Since SM quarks are not charged under U (1) Lµ−Lτ , their Yukawa couplings are SM-like with 9 free parameters.
For simplicity, we include only the third and second generation of SM quarks together with the heavy quark ψ in Eq. (7), to illustrate the essence of NP phenomenology. To further simplify the calculation, SM Yukawa couplings are chosen to be flavor diagonal. We shall relegate to the Appendix the discussion of the obtention of the CKM matrix elements, which is not relevant for our analysis. Thus, the mass matrix M q and interaction matrix Λ q are Note for the up-type quarks u 3 = t and u 2 = c and for down-quark d 3 = b and d 2 = s.
In the matrices M q and Λ q , only the diagonal mass elements are modified when changing from u to d, while the terms proportional to λ 2,3 stay the same. Since ψ is heavy vector-like quark, the masses have the sequence m ψ m q 3 > m q 2 , where as we will show λ 2 v D > m ψ is necessary to obtain a large enough mixing between heavy quarks and SM 2nd generation quarks. For the up sector, since the top quark mass is of the order of the weak scale, m q 3 can be comparable with m ψ . To avoid large mixing between the top and the 3rd generation, and hence large Yukawas to obtain the proper top mass, we further assume λ 2 λ 3 which suggests the heavy quark mostly mixes with 2nd generation left-handed SM quarks. After diagonalizing M q , we get the mass of the quark mass-eigenstates, where in the last equality we only keep the leading term and we define tan . Following the mass and coupling assumption, tan θ 3 1. As explained above, to fit the B-meson decay anomalies, λ 2 v D needs to be larger than m ψ , which suggests sin θ 2 and cos θ 2 are of O(1). Note that the mass ofψ u andψ d are almost degenerate, only different by small 2nd generation quark mass m q 2 and coupling λ 3 .
The mixing matrices U q L,R connect flavor and mass eigenstates as where the states with tilde are the mass eigenstates for the quarks. The mixing matrices at leading order are where s θ 2 , c θ 2 , t θ 2 are abbreviations for sin θ 2 , cos θ 2 and tan θ 2 .
We see that U q R is close to identity matrix with off-diagonal terms suppressed by small SM quark mass m q 2 ,q 3 or tan θ 3 . Because the heavy vector-like quark has the same gauge SM charge as the left-handed SM quarks, thus the mixing dominantly happens between the left-handed quarks. For the left-handed mixing U q L in Eq. (13), the leading term are the same for up-type an down-type quarks, if neglecting the SM quark masses.
The SM Cabibbo-Kobayashi-Maskawa (CKM) matrix elements should be obtained from However, since we did not include off-diagonal matrix elements for the SMquarks, all non-vanishing elements will be proportional to rations of the SM-quark masses and the heavy fermion masses. In reality, the successful generation of CKM matrix needs to include the 1st generation SM quark mass and the off-diagonal terms in SM Yukawa couplings. In principle, with 9 free mass parameters in SM quark sector, there should be no problem to generate CKM matrix. We will demonstrate the generation of all the V CKM terms in the Appendix B, and show that it does not affect the NP phenomenology we are interested here.
After the discussion of the quark sector, it is worth to mention that the charged leptons can easily get mass with diagonal SM Yukawa terms. For neutrino mass and Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix under gauge interaction U (1) Lµ−Lτ , it is generally easy to accommodate. The reason is that L µ − L τ gauge satisfies µ ↔ τ permutation which leads to mixing angle θ 23 close to maximal (45 • ) while having a vanishing mixing angle θ 13 at leading order (see review [51]). Since neutrino mass is not relevant of the NP phenomenology we are interested in, we do not modify the lepton sector further.

C. Light and heavy quark interactions
In this subsection, the flavor eigenstate scalars and quarks are rotated into mass eigenstates. In the mass eigenstate, the Yukawa interactions for scalars and quarks are, where the diagonal interactions with scalars are given in Eq. (15). The dots in the last line are for the omitted off-diagonal interactions, which dominantly opens the decay ofψ into 2nd generation SM quarks. The decay width of processψ q →φq 2 is proportional to while the other decaysψ q →φq 3 orψ q →hq 2 are suppressed by either t 2 θ 3 or s 2 α . The off-diagonal interactions will not contribute to scalarh,φ couplings to gluon-gluon and photon-photon, thus they are irrelevant for the NP phenomenology.
The gauge interactions between Z and quark mass eigenstates at leading order are given below: where the first two lines are diagonal interactions, and the lines from three to five are off- The lepton interactions of Z are Due to the charge assignments, in the absence of mixing, the Z coupling to leptons would be 2 times larger than the one to heavy quarks. We calculate the decay branching ratios (BR) for Z into heavy vector-like quarkψψ, 2nd generation SM quarksss andcc, 2nd and 3rd generation SM charged leptons µ + µ − , τ + τ − and neutrinosν µ ν µ ,ν τ ν τ . The width of the decay to 3rd generation quarks are suppressed by t 4 θ 3 . The only significant flavor off-diagonal decays for Z are to 2nd generation quark and heavy vector-like quarkssψ d ,ψ ds andcψ u ,ψ uc . The flavor off-diagonal decays for Z to 3rd generation are suppressed by t 2 θ 3 . The decay BRs for Z are given in Fig. 2.
Before closing this subsection, we calculate the leading order couplings between the SM Z boson and SM quarks, which are modified due to mixing effects,

III. LIGHT HIGGS AND THE CMS EXCESS
In this section, we first discuss the light Higgsφ with mass 96 GeV and how it can resolve the excess in CMS data. Later, we use the Z in the same model to fit the R ( * ) K excess. The dominantly production channels at LHC are gluon-gluon fusion (ggF), associate production withtt, vector boson fusion (VBF) and associate production (VH). In the section II C, the coupling betweenφ and quarks are given. For mφ = 96 GeV, the coupling strength κ is calculated, which is the coupling forφ devided by the coupling of SM-like Higgs with the same mass.
where V = W, Z. For κ ggφ ef f , the first term is the contribution from SM top quark, and the second term contains the contributions from heavy vector-like quarksψ u,d , therefore there is a factor of 2. For κ γγφ ef f , the third term is again contribution from heavy vector-like quark, with 5/4 fromψ u,d electricmagnetic charge square comparing to top quark.
The ggF, and the inclusive VBF and VH production cross-sections at 13 TeV LHC are given below [52] The dominant decay channels ofφ have the following decay widths [53] Γ(φ →bb) = 1. 9 MeV × κbbφ ef f The branching ratio forφ tobb and γγ are approximately To fit the 13 TeV CMS di-photon excess [1] , one needs We show the parameter space {sin α, sin θ 2 } which fits the CMS excess in Fig. 3, with sin θ 2 ≈ λ 2 v D /mψ. The cyan solid line provides 80% of CMS excess in Eq. (28), while the two dashed line are for 60% and 100% of the excess respectively. The benchmark point is denoted by a red star in the plot, and its parameters are also listed in Table II. For the benchmark point, we list the 96 GeV Higgs branching ratios in Table III Table II.

A. Constraints from SM Higgs measurements
In this subsection, we are going to check the limits from SM Higgs measurements. For convenience, we list the coupling strength κ for the SM-like Higgsh below, This CMS analysis includes ggF, VBF, VH and ttH productions, and various SM Higgs decay modes. For ATLAS, the production modes are the same, but only H → γγ and H → ZZ → 4 decay modes are included in the κ analysis [55], It is clear that κ from CMS and ATLAS are in agreement with each other. We took the κ measurements as constraints and plot the corresponding contours at 90% confidence level (C.L.) in Fig. 3, with CMS in the left panel and ATLAS in the right panel.
Since in our signal modelφ is produced from ggF and decay in di-photon channel, it is appropriate to pay special attention in the SM Higgs di-photon channel measurement as well. The most recent 13 TeV (36 fb −1 ) measurements for SM Higgs property in the di-photon decay channel, from CMS [56] and ATLAS [57] give the following signal strengths  Table II. for different production modes and the combined result,

IV. Z AND THE B-ANOMALIES
After discussing the possibility ofφ fitting the CMS excess, we turn to the B-anomalies.
Integrating out the heavy gauge boson Z , there is an effective flavor violating operator with down-type quarks [13], This can be related to the C N P 9 operator considered in Ref. [59][60][61][62][63][64][65][66][67]: therefore, the coefficient C N P 9 can be rewritten as Recent global fits include more data from experiments, e.g. angular observables from Belle [68], and have found that the significance of NP contributions has increased [65,66]. If one restricts the analysis to lepton flavor universality violation process, the value C N P 9 = −1.56 quoted above was obtained by a fit to the data by the authors of Ref. [65], with a significance of 4.1 σ, while the authors of Ref. [66] obtained a slightly different result, namely, C N P 9 = −1.76, with a significance of 3.9 σ. However, extending the analysis to a more complete set of observables, namely all those included in Ref. [66], a best fit value of C N P 9 = −1.11 is obtained, and the significance increases to 5.8 σ. In our analysis, we shall consider the values obtained from the fit in Ref. [65]. The alternative values of C N P 9 obtained in Ref. [66] do not affect our phenomenological analysis in any significant way, since they can be easily accommodated by few tens of percent changes in the mass of the gauge boson Z . After plugging in the SM parameters (α em = 1/137, |V ts | ∼ 0.04), we obtain a requirement on NP parameters, TeV di-lepton search [72]. The brown area is excluded by 7 TeV ATLAS search on di-leptons [73].  Table II, and the red star denotes our benchmark point.
For the explanation of the R K and R * K anomalies, we have chosen a benchmark with t θ 3 = 0.1 and s θ 2 = 0.89, which are consistent with the parameters shown by the red star in Fig. 3. For such a benchmark, one obtains a relationship between g D and m Z given by and the di-photon signal cross-section σ ggF BR(φ → γγ) at the 13 TeV LHC. We see that di-photon signal at CMS requires a large value of sin θ 2 , while the explanation of the B-anomalies requires a moderate value of sin θ 2 since −C N P 9 is proportional to s θ 2 c θ 2 .

A. Constraints on Z
In Eq. (16), we have already presented the interactions of Z and in Fig. 2 we show the branching ratios of Z . We find out that, depending on the Z mass, the dominant decay channel of Z are Z →ss (cc), Z →ψ uψu (ψ dψd ), Z →μµ (τ τ ) and Z → ν µνµ (ν τντ ).
We now consider the constraints from Z searches at LHC. The search for new neutral gauge bosons decaying to di-lepton [69][70][71][72][73] and di-jet [74] put strong constraints on a possible Z . After consider the Z production and decay branching ratios, in in the left panel of Fig. 4, we show the constraints in g D vs. m Z 2D plane with benchmark parameters tan θ 3 = 0.1, sin θ 2 = 0.89 and mψ = 800 GeV. All the colored regions are excluded. The limits from di-jet are much weaker than di-lepton, because Z couples to SM quark only through heavy vector-like quark mixing while leptons are directly charged. Thus di-jet constraint is not shown in Fig. 4. The solution to B-anomalies requires m Z /g D 4.1 TeV. It is plotted as a solid blue line and denoted as "R(K)/R(K * )" in Fig. 4. To evade the constraints, Z must be heavier than about 4.1 TeV, while g D must be larger than about 0.8.
For U (1) Lµ−Lτ gauge boson, it is known that neutrino trident production ν µ N → ν µ N µ + µ − , where N is nuclei, gives very stringent constraint [13,75]. The recent results are from CHARM-II [76], CCFR [77] and NuTeV [78], which leads to constraint m Z /g D 540 GeV for m Z 10 GeV [79]. It is easy to see that our benchmark point for model is safe from neutrino trident constraint. Observe that a simultaneous explanation of the B-anomalies and the LEP bottom forward-backward asymmetry with a single Z is not possible, since the latter demands a light Z , with mass comparable to the Z mass and significant couplings to the right-handed bottom quarks [80], what would lead to inconsistencies with the above constraints.

A. Constraints on the heavy quarks
The heavy vector-like quark is subject to constraints from LHC searches. Its mass is mψ λ 2 2 v 2 D + m 2 ψ , and it is about mψ 800 GeV for the benchmark point. At the LHC, such a quark would be produced by QCD processes and will predominantly decay into theψ →φs/c channels, followed byφ →cc,ss,bb. There are other possible decay channels, e.g.ψ q →φq 3 andψ q →hq 2 however suppressed by either t 2 θ 3 or s 2 α , andψ → Zq suppressed by small quark mass or t θ 3 (see Eq. (18)). The decayψ → Z s/c is also possible but is kinematically forbidden by the benchmark setup m Z > mψ. Therefore, the dominant constraints come from searches for heavy fermions which decay into three quarks. A recent relevant search is from ATLAS 13 TeV (36 fb −1 ) looking for R-Parity violating (RPV) gluino at the LHC, which decays to three quarks [81]. It sets 95% C.L. upper limit on the cross section times branching ratio varies between 0.80 fb at gluino mass 900 GeV and 0.011 fb at gluino mass 1800 GeV. At LHC Run-I, ATLAS [82] and CMS [83] [83] has also set limit if it decays to qqb, and the constraint is slightly better than qqq decay channel. The ATLAS 8 TeV search [82] has looked for bbb decay channel, and the constraint is roughly the same as qqq channel. The heavy vector-like quarkψ can decay to qbb, ifφ →bb. Such decay does not match to three jets invariant mass reconstruction for either qqb or bbb, therefore more possible combinatorial errors will reduce the signal efficiency. Moreover, it will pay double suppression from branching ratioφ →bb. As a result,ψ decays into three light flavor jets are more general and useful. In summary, our benchmark model is not excluded by the three jet resonance searches at LHC.

B. Constraints from flavor physics
We start the discussion from b → sγ constraints. The SM branching ratio for b → sγ is given by [86]: where F µν , G µν are the photon and gluon field strengths. The SM CKM matrix values are given by [87]: For the formula for A γ,g from the SM W -loop, we refer to the appendix of Ref. [86]. In our model, the contributions to b → sγ decay come from Z , Z flavor off-diagonal couplings in Eq. (16) and Eq. (18). We are not going to calculate the contributions directly, but only make an estimation for the order of magnitude comparing with the contribution from W -loop. It is not difficult to estimate their contribution to the Wilson coefficients A γ , A g : From the equation above, we can see that the contribution is either suppressed by mbms/m 2 ψ or m 2 W /m 2 Z , which leads to negligible contribution for the b → sγ decay. To be more explicit, we find that for m Z ∼ 4.1 TeV, g D ∼ 1, mψ ∼ 800 GeV and t θ 3 1, we have: which are very small and do not induce any relevant modification to the b → sγ decay rate.
Note that the operator which explains the lepton-flavor university violation in Eq. (39) will not contribute to B s → µ + µ − decay, as our Z couplings to muon pair is vector-like [88].
Apart from the operatorb L γ µs Lμ γ µ µ, there are also ∆F = 2 flavor changing operators generated: Both operators will contribute toB s − B s mixing, but we expect that the left-handed one dominates, as the right-handed one is suppressed by the masses of bottom and strange quarks, i.e.: The bound on the Wilson coefficient of the first operator is given by (see Table 1.1 of Ref. [89] and [90]): Combined with Eq. (41), we can derive the bounds on the combination of the mixing angles or the factor g 2 D /m 2 Z : which is satisfied by our benchmark point (s θ 2 c θ 2 t θ 3 0.041, m Z /g D 4.1 TeV).
Next, we consider the searches for tZ flavor changing neutral current in the top quark decays at LHC. The relevant interactions are which will contribute to the decay channelt → Zc with branching ratio estimated as: The most stringent bound on such branching ratio comes from ATLAS search at 13 TeV (36 fb −1 ) [91]. The results read which sets a lower bound on our parameters, It is easy to see that such bound is satisfied in the region of parameters consistent with our benchmark point.
Finally, we discuss the constraints from flavor physics in the lepton sector. The gauge boson Z under U (1) Lµ−Lτ can contribute to tau lepton decay τ → µν τνµ via 1-loop box diagram [13]. It excludes Z mass lower than 650 GeV for g D = 1, and the benchmark point with a 4 TeV Z is quite safe with this limit. In this work, we have discussed the possibility that such Higgs boson could be associated with the recent excess in the di-photon channel observed at the CMS experiment at the 13 TeV LHC run. We have shown that this is possible and provided the constraints on the quark-mixing parameters that could lead to such a possibility. An additional, naturally heavy, Higgs boson, contributing to the breakdown of the U (1) Lµ−Lτ symmetry, but not to the quark mixing, is necessary in order to give the Z gauge boson a sufficiently high mass to avoid the LHC constraints. A simultaneous explanation of these observables with the Higgs boson excess observed at LEP may be achieved in a way that is consistent with most observations, although it is in tension with the rate of the SM-like Higgs boson decaying to di-photons in the gluon fusion production channel measured by the ATLAS experiment in the 13 TeV LHC run. (A1)

Appendix B: Limits from CKM matrix
The most general interactions for U (1) µ−τ within the quark sector are given by: The mass matrix to generate CKM matrix in the quark sector are written in the basis of (q 3 , q 2 , q 1 , ψ). For up-type and down-type quarks, the mass matrices are In principle, all the matrix elements except the fourth line can be non-zero. Here we consider the simple matrices in Eq. (B2) to generate correct CKM matrix, with only three non-zero parameters in the off-diagonal terms for SM quark mass. To avoid modifying our existing results, we can set λ 1 = 0. Our notation is M diagonal = U † L M U R , thus q flavor