A New Higgs Boson with Electron-Muon Flavor-Violating Couplings

Recently, the CMS Collaboration performed a search on a new resonance decaying to $e^\pm\mu^\mp$ in the mass range of 110 GeV to 160 GeV. The search also hints a possible excess at 146 GeV with a $3.8\sigma~(2.8\sigma)$ of local (global) significance. Motivated by that, we try to interpret the results in the context of the type-III two-Higgs-doublet-model. We find that the excess is only moderately constrained by low-energy lepton-flavor-violation processes, in particular the $\mu\to e \gamma$ decay. We also compare the CMS bounds across the entire search region against constraints of $\mu\to e\gamma$ and $\mu\to e$ conversion in nuclei. Our finding indicates that the collider bounds can be superior to those of low-energy processes for the scalar mass between $110 \text{ GeV}$ and $150 \text{ GeV}$, suggesting the importance of this mass range for future searches.


I. INTRODUCTION
As the most recently discovered particle of the Standard Model (SM) [1,2], the 125-GeV Higgs boson, h, is the least studied fundamental particle. After the discovery, immense amount of works on precision measurements of the h properties has been done at the Large Hadron Collider (LHC). It is found that the h properties agree quite well with the SM expectations [3,4]. In particular, since its discovery in the γγ, W + W − and ZZ channels, the fermionic decay channels bb [5,6], τ + τ − [7], and µ + µ − [8] have been established. The consistency of these measurements with the SM predictions is one of the biggest triumphs of the SM. Searches for the remaining predicted decay channels, e.g., e + e − [9, 10] and cc [11,12], are underway. Despite all these, one should not discount other possible decay modes that are not predicted by the SM. Discovering any of these decays will be a clear signal of new physics.
Lepton-flavor-violating (LFV) couplings are prototypical examples of new physics. The LFV couplings of the h lead to LFV decays h → ℓℓ ′ , which are correlated with low-energy LFV decays of charged lepton ℓ → ℓ ′ γ and ℓ → 3ℓ ′ . Additionally, the µ → e conversion in atomic nuclei constrains any processes involving the LFV e-µ coupling. In the case of LFV tau sector, it has been demonstrated that the LHC searches for h → τ ℓ provide more stringent constraints than the low-energy processes [13,14]. This indicates that collider searches for h → τ ℓ, together with their heavy scalar counterparts, are important in constraining new physics parameter space [15][16][17][18][19][20][21][22].
The collider search for h → eµ, on the other hand, has often been overlooked as tools for constraining new physics parameter space. This is due to a conventional wisdom that such a channel is correlated with µ → eγ and µ → e conversion, which provide more stringent constraints [13,14]. In this work, we show that such expectation does not generally apply to a new resonance decaying into e ± µ ∓ . In particular, when the new particle mixes with the h and its mass is less than 160 GeV, its contribution naturally cancels that of the h in the µ → eγ and µ → e conversion, weakening the bounds from these processes.
The strongest bound on h → eµ derived from the full LHC Run II data is set by the CMS with BR(h → e ± µ ∓ ) ≤ 4.4×10 −5 [23]. Beside searching for h → e ± µ ∓ , CMS also looks for the LFV decay of a new resonance, H, for 110 GeV < m H < 160 GeV. The CMS finds an excess at m H ∼ 146 GeV with a 3.8σ (2.8σ) local (global) significance; the best fit cross-section is found to be σ (pp → H → e ± µ ∓ ) = 3.82 +1. 16 −1.09 fb. However, the corresponding ATLAS search [10] does not find any significant excess at 146 GeV. Hence, the nature of the CMS excess remains inconclusive.
The CMS excess will inevitably induce µ → e conversion in nuclei by a tree-level exchange of H. One can estimate the µ → e conversion rate in a simplified model where H is assumed to couple only to e-µ and to gluon via an effective coupling. We find that in this scenario the induced rate for µ → e conversion in the gold nucleus can be as low as 10 −16 , which is safely below current experimental bound. However, in a more complete model, there will be more states that contribute to the process. Hence, it is compelling to investigate this in a more realistic model.
In this work, we compare the CMS search results against the corresponding low-energy bounds in the context of the type-III two-Higgs-doublet model (2HDM). While not advocating the presence of the excess, we try to examine whether it holds up against the current and the future µ → eγ and µ → e conversion bounds. We also highlight the importance of collider searches for LFV decays at a relatively low mass, i.e., between 100 GeV and 150 GeV.

II. TYPE-III 2HDM
The type-III 2HDM is conveniently described in the Higgs basis [24], where the two Higgs doublets, H 1,2 , are given by Here v is the vacuum expectation value, and G + and G are the would-be Goldstone bosons. For simplicity, we assume a CP symmetry in the scalar sector so that the two CP -even states, h 1 and h 2 , do not mix with the CPodd state A. Both h 1 and h 2 can mix through where c α (s α ) stands for cos α (sin α). Here we identify h with the 125-GeV Higgs boson. Since the properties of the h agree with the SM predictions [25][26][27], the mixing angle s α is expected to be small. The masses of the extra Higgs bosons H, A, and H + are, in principle, arbitrary.
The Yukawa couplings of H 1 generate fermion masses while the Yukawa couplings of H 2 give rise to potential flavor violations. The most general form of the Yukawa couplings of H 1 and H 2 to the leptons is given by where L denotes the lepton doublet, m i is the ith generation charged lepton mass, and i, j = e, µ, τ indicate lepton generations.
In this work, we will focus on LFV in the e-µ sector, so we take only Y eµ and Y µe in Eq. (3) to be nonzero. The couplings Y eµ and Y µe , in principle, can be complex. However, the imaginary part of Y eµ Y µe is strongly constrained by the electron electric dipole moment measurement [28]. Therefore, for simplicity, we assume that both Y eµ and Y µe are real in this work.
In our minimal scenario, only h and H can be singly produced via gluon fusion and vector boson fusion processes. This makes H the only relevant resonance for CMS and ATLAS LFV searches. The H production cross-sections and non-LFV partial decay widths can be obtained from the would-be SM Higgs boson values by scaling them with a factor of s 2 α . On the other hand, the h production cross-sections and non-LFV partial decay widths are reduced by a factor of c 2 α from their SM values. In our analysis, we use the SM-like Higgs crosssections and decay widths provided by the LHC Higgs Cross Section Working Group [29].

III. LOW-ENERGY LFV CONSTRAINTS
Naturally, the LFV couplings Y eµ and Y µe , together with the scalar mixing s α , will induce µ → eγ, which proceeds via loops. The partial decay width is given by where α em is the electromagnetic fine-structure constant. The Wilson coefficients c L,R arise at one-and two-loop level. The one-loop contributions are given by with s 2α ≡ sin 2α. The main contributions to the Wilson coefficients c L,R are dominated by two-loop photonexchange diagrams involving the W boson and top quark. They are given by with z ab = m 2 a /m 2 b . The loop functions f , g, and h can be found in Ref. [30]. Note the cancellation between the h and the H contributions due to the mixing of Eq. (2). Such a cancellation weakens the µ → eγ constraint as m H approaches m h . It should be noted that the loop functions in Eqs. (5)-(7) are typically O(1). Hence, the one-loop coefficient is parametrically suppressed by In addition to such W and top contributions, we have included contributions from the so-called "set C" diagrams [30]. Numerically, they give a correction of order 10% to the Wilson coefficients. Other contributions, i.e., two-loop Z-exchange diagrams are found to be more suppressed compared to the others.
The Wilson coefficients c L and c R also lead to µ → e conversion in atomic nuclei. In addition, the µ → e conversion also gets tree-level contributions mediated by the CP -even Higgs bosons. In our scenario, the conversion rate is given by [33] (see also [14]) where D and S N are overlap integrals, whose values are given in [33]. The effective coupling g N L is given by where f (q,N ) and m N denote the nucleon form factor and the nucleon mass, respectively. The form factors are given in Refs. [34,35], and in above equation, they are summed over all quark flavors. The effective coupling g N R can be obtained from g N L by replacing Y eµ → Y µe . As in the µ → eγ case, the µ → e conversion constraint also suffers from the same blind spot when m H approaches m h . Currently, the strongest constraint comes from conversion in the gold nucleus, which sets Γ(µ → e)/Γ(captured) < 7 × 10 −13 [36].

IV. COLLIDER SEARCHES FOR LFV DECAYS
The LFV couplings Y eµ and Y µe , together with the mixing angle s α , lead to LFV decays of the h and H. The partial decay width h → e ± µ ∓ is given by The partial decay width for H → e ± µ ∓ can be obtained from Eq. (10) by changing s α → c α and m h → m H .
Both ATLAS and CMS Collaborations have searched for such LFV decay of h without any positive signal [10,37]. Assuming the SM production cross-sections for the h, the ATLAS has set an upper bound BR(h → e ± µ ∓ ) ≤ 6.2 × 10 −5 , while the CMS has set a slightly stronger constraint BR(h → e ± µ ∓ ) ≤ 4.4 × 10 −5 . As mentioned earlier, in our scenario, the h production cross-sections get corrected by c 2 α , so does the bound on the h → e ± µ ∓ branching ratio. Hence, the CMS upper bound now reads c 2 α BR(h → e ± µ ∓ ) ≤ 4.4 × 10 −5 . The collider search for H → e ± µ ∓ was done by the CMS Collaboration [37]. The result was presented in terms of the H production cross-section times the branching ratio into e ± µ ∓ . In our model, it is formulated as where σ SM and Γ SM are the cross-section and the total decay width of the SM-like Higgs with mass m H .

A. The CMS 146-GeV excess
The CMS search for a resonance decaying to e ± µ ∓ has observed an excess of events around 146 GeV with a local significance of 3.8σ [37]. However, the corresponding ATLAS search does not find any excess at 146 GeV. By performing a simple profiled likelihood ratio analysis, we find that the CMS signal is disfavored by the ATLAS measurement at the 2.5σ level.
Taking both CMS and ATLAS results into consideration, we perform a naive combination of the two searches using the counting experiment. The combined tool with asymptotic approximation [38] is used to calculate the best fit cross-section and its significance. The data and background Monte Carlo (MC) from both experiments, as well as the CMS signal MC for m H = 146 GeV, are extracted from the Data-MC plots in Refs. [10,37]. In the case of ATLAS MC, we estimate the signal model using Madgraph5 [39] to generate parton-level productions and H decays, followed by showering and hadronization with Pythia8 [40]. Finally, the detector responses are simulated using Delphes3 [41]. The overall uncertainty is estimated to be 1%, as mentioned in Ref. [10] for the eµ channel. We find that the combination reduces the local significance of the excess to 3.3σ, with the best fit cross section σ × BR(H → e ± µ ∓ ) = 2.92 +0.91 −0.89 fb. The region in s α -Y 2 eµ + Y 2 µe parameter space consistent with the CMS excess, together with the combined CMS and ATLAS data, at 1σ level is shown in Fig. 1. The preferred regions of parameter space are compared against constraints from the h → eµ, µ → eγ, and µ → e conversion searches. Note that the small 1σ region preferred by the CMS excess consistent with the µ → eγ bound is centered around s α = 0.014 and Y 2 eµ + Y 2 µe = 6.6×10 −4 . In that figure, we also show the projected sensitivity of the MEGII experiment. Should the origin of the CMS excess be a new particle, the MEGII experiment will observe the µ → eγ decay.
In addition to the aforementioned constraints, the 146-GeV excess is also constrained by the CMS search for a new resonance decaying into a pair of ZZ * [42] and a pair of tau leptons [43]. In both cases, the bounds are O(100) fb, which correspond to s α ≲ 0.2.
The 146-GeV excess is also constrained by multilepton searches via HA pair production. The A will decay predominantly into eµ, hZ, or HZ if the latter is kinematically open. A significant portion of the H, on the other hand, will decay into eµ for sufficiently small s α . If this is the case, the pair produced HA will result in fourlepton signatures, which are strongly constrained by the CMS multilepton searches [44]. From our analysis, we find that the CMS multilepton constraints imply either m A ≳ 800 GeV or s α ≳ 0.025 provided that A → HZ is kinematically open.

B. Bounds for other masses
While the excess at 146 GeV seen by CMS could be a result of a new particle, we should not draw a definite conclusion on its nature without more data. Thus, we also study the possibility that the CMS search finds no excess anywhere in the search region 110 GeV ≤ m H ≤ 160 GeV. In this scenario, we take the 95% confidence level upper limits on σ × BR(H → eµ) reported by CMS and compare them against constraints from µ → eγ and µ → e conversion.
The bounds from collider search, µ → eγ, and µ → e conversion experiments are functions of the mixing angle s α and the LFV couplings |Y eµ | 2 + |Y µe | 2 . While the µ → eγ and µ → e conversion constraints depend on the product s 2α |Y eµ | 2 + |Y µe | 2 , the collider constraint is a more complicated function. For illustrative purposes, the collider bound, for certain value of H mass, would trace out a curve in the s α -|Y eµ | 2 + |Y µe | 2 plane similar to the preferred region for the 146-GeV excess in Fig. 1. In order to compare these two types of constraints on the entire CMS search region, we project the constraints onto the s 2α |Y eµ | 2 + |Y µe | 2 line. Such a projection maps the µ → eγ and µ → e conversion constraints to a point. On the other hand, the bound from CMS search gets mapped into a bounded-from-below interval. If the minimum of such an interval is smaller than the µ → eγ (µ → e conversion) projection, it is said that the collider bound is more constraining than the µ → eγ (µ → e con- parameter space. In Fig. 2 we show the comparison between the collider bounds and the µ → eγ and µ → e conversion constraints for 110 GeV ≤ m H ≤160 GeV. Note that, in Fig. 2, only the minimum of the collider constraint projection is shown for each m H value. As can be seen from the plot, for m H ≲ 140 GeV, the collider search can be more constraining than even the projected MEGII bound. Hence, we advocate both the ATLAS and CMS Collaborations to continue searching for H → e ± µ ∓ in that low-m H region.

V. CONCLUSION AND DISCUSSION
In this work, we have analyzed the LHC searches for a new resonance decaying to e ± µ ∓ in the context of the type-III 2HDM. This analysis is motivated by a recent CMS search [37] that explores the LFV decay of a new scalar boson between 110 GeV < m H < 160 GeV. Within the search region, the CMS finds a possible excess at m H = 146 GeV with a local (global) significance of 3.8σ (2.8σ). A simplistic combination of the CMS and ATLAS searches reduces the local significance to 3.3σ, with σ × BR(H → e ± µ ∓ ) = 2.92 +0.91 −0.89 fb. In the type-III 2HDM context, the 146-GeV excess is only moderately constrained by the current µ → eγ data, while the future MEGII search will probe the whole parameter region preferred by the excess.
It is interesting to note that the 146-GeV excess could be related to another excess from the so-called "multilepton anomalies" [45]. We leave this investigation for possible future work.
In the event that the excess is due to an upward fluctuation in the data, we analyze the bounds on σ × BR(H → e ± µ ∓ ) provided by CMS over the whole search region. The comparison between the CMS bounds and the low-energy counterparts is shown in Fig. 2. From the plot, we see that for 110 GeV ≤ m H ≲ 140 GeV, the current CMS bound is better than the current and even the projected MEGII bounds for µ → eγ. Thus, we encourage both CMS and ATLAS Collaborations to continue searching for LFV decays of a new resonance in this low-mass region.
Lastly, we note that even though we take Y ee and Y µµ to be zero in this work, it does not mean we need a large hierarchy between the flavor-violating and the flavor-conserving couplings. In fact, Y ee/µµ can be com-parable to Y eµ/µe and still be consistent with low-energy constraints. We set them to zero in the spirit of minimal scenario.