Explaining 95 (or so) GeV Anomalies in the 2-Higgs Doublet Model Type-I
Akshat Khanna Stefano Moretti Agnivo Sarkar






khanna_akshat@iitgn.ac.in



s.moretti@soton.ac.uk; stefano.moretti@physics.uu.se




agnivosarkar@hri.res.in




    HRI-RECAPP-2024-05



§ ABSTRACT 
We show how the 2-Higgs Doublet Model (2HDM)  Type-I can explain some excesses recently seen at the Large Hadron Collider (LHC) in γγ and τ^+τ^- final states in turn matching Large Electron Positron (LEP) data in bb̅ signatures, all anomalies residing over the 90-100 GeV or so region. The explanation to such anomalous data is found in the aforementioned scenario when in inverted mass hierarchy, in two configurations: i) when the lightest CP-even Higgs state is alone capable of reproducing the excesses; ii) when a combination of such a state and the CP-odd Higgs boson is able to do so. To test further this scenario, we present some Benchmark Points (BPs)  of it amenable to phenomenological investigation.   

Explaining 95 (or so) GeV Anomalies 
in the 2-Higgs Doublet Model Type-I
    Agnivo Sarkar
    September 9, 2024
========================================================================






§ INTRODUCTION



A long-standing anomaly existing in LEP collider data <cit.> is the one hinting at the possibility of  e^+e^-→ Zh events being produced therein, with a Higgs boson state h with a mass of approximately 98 GeV decaying into bb̅ pairs <cit.>. More recently, the CMS collaboration at the LHC has  found an excess near 95 GeV in di-photon events in two separate analyses  <cit.>.
In fact, they also reported an excess in τ^+τ^- pairs, again, around a mass of about 98 GeV. Finally, ATLAS also observed an excess at around 95 GeV in di-photon events, thereby aligning with  CMS although,  especially when including `look elsewhere' effects, their findings are far less significant than the CMS ones.  Altogether, in view of the limited mass resolution of the di-jet invariant mass at LEP, this older anomaly may well be consistent with the  excesses seen by CMS (and, partially) ATLAS in the γγ and, even more so, τ^+τ^- final states (as the mass resolution herein is also rather poor).

As a consequence of this credible mass overlap, many  studies <cit.> have tested the possibility of simultaneously fitting these excesses within Beyond the Standard Model (BSM) frameworks featuring a non-SM Higgs state lighter than 125 GeV, i.e., than the one observed at the LHC in 2012 <cit.>.

A possible route to follow in explaining such events through a companion Higgs state (to the 125 GeV one) is to resort to a  2HDM <cit.>, as done, e.g., in Refs. <cit.>, wherein a Type-III (which allows direct couplings of both Higgs doublets to all SM fermions) with specific fermion textures was invoked successfully as a BSM explanation to the bb̅, γγ and τ^+τ^- excesses seen at LEP and the LHC. Herein, 
both a fully CP-even and a mixed CP-even/odd solution was found, upon refining the 2HDM Type-III Yukawa structure to comply with both  theoretical consistency requirements and experimental measurements of the discovered Higgs mass and couplings (of the 125 GeV Higgs state).

In this study, we show that  solutions of the same kind (i.e., both a fully CP-even and a mixed CP-even/odd one) can also be found in a 2HDM Type-I, wherein only one Higgs doublet gives mass to all SM fermions, again, satisfying the aforementioned  theoretical requirements and experimental constraints.

The  paper is  organised as follows. In the next section, we review the theoretical framework of the 2HDM Type-I. Then we discuss the theoretical and experimental constraints applied to such a BSM scenario in our study, after which we move on to show how the latter can naturally explain the discussed anomalous data in various configurations of its parameters space. Our conclusions  then follow.  



§ THE 2HDM TYPE-I


Among the various BSM scenarios, the 2HDM can be considered as a simple extension of the SM. The scalar sector of this model comprises two complex scalar fields ϕ_1 and ϕ_2 which transform as a doublet under the  Electro-Weak (EW) gauge group SU(2)_L× U(1)_Y with hypercharge Y = 1. For a detailed overview of this model interested reader can look into  <cit.>. The most general gauge invariant CP-conserving scalar potential can be written as 

    V(ϕ_1, ϕ_2)     = m_11^2 ϕ_1^†ϕ_1 + m_22^2 ϕ_2^†ϕ_2 - m^2_12[ϕ^†_1ϕ_2 + ϕ^†_2ϕ_1] + λ_1/2(ϕ_1^†ϕ_1)^2 + λ_2/2(ϕ_2^†ϕ_2)^2 
        + λ_3(ϕ_1^†ϕ_1)(ϕ_2^†ϕ_2) + λ_4(ϕ_1^†ϕ_2)(ϕ_2^†ϕ_1) + [  λ_5/2(ϕ_1^†ϕ_2)^2 + h.c. ] + {[ λ_6(ϕ_1^†ϕ_1
        )+ λ_7(ϕ_2^†ϕ_2) ](ϕ^†_1ϕ_2) + h.c.  }.

Given the hermiticity of the scalar potential, all the potential parameters of Eq.(<ref>) must be real. In order to prevent tree level Flavour Changing Neutral Currents (FCNCs), one can postulate an additional discrete 𝒵_2 symmetry under which the scalar fields transform as ϕ_1→ϕ_1, ϕ_2→ -ϕ_2. One can realise that the term proportional to m^2_12, λ_6 and λ_7 in Eq.(<ref>) violates this 𝒵_2 symmetry explicitly. Therefore the potential can be expressed in the following form:

    V(ϕ_1, ϕ_2)     = m_11^2 ϕ_1^†ϕ_1 + m_22^2 ϕ_2^†ϕ_2 + λ_1/2(ϕ_1^†ϕ_1)^2 + λ_2/2(ϕ_2^†ϕ_2)^2 
        + λ_3(ϕ_1^†ϕ_1)(ϕ_2^†ϕ_2) + λ_4(ϕ_1^†ϕ_2)(ϕ_2^†ϕ_1) + {λ_5/2(ϕ_1^†ϕ_2)^2 + h.c. }.


Both these Higgs fields, ϕ_1 and ϕ_2 acquire a non-zero vacuum expectation value
(vev) (i.e., ⟨ϕ_1⟩ = v_1 and ⟨ϕ_2⟩ = v_2) and spontaneously break the EW gauge symmetry down to U(1)_ EM. After the symmetry breaking, the W and Z boson becomes massive and the scalar sector contains five physical Higgs bosons - two CP-even {H, h}, one CP-odd A and a pair of charged states H^± with masses m_H, m_h, m_A  and m_H^±,  respectively. Both these vevs v_1 and v_2 are related to the EW scale v = √(v^2_1 + v^2_2) = 246 GeV. The ratio between these two vevs can be parameterised as tanβ = v_2/v_1. In addition, the mixing angle between the CP-even states {H, h} can be parametrised as α. For present study we consider the inverted hierarchy between the CP-even mass eigenstates. This particular choice alters the usual interpretation of the mass spectrum and the couplings [For example, in the inverted mass hierarchy the alignment limit corresponds to cos(β - α) → 1. However, we will not conform to this limit in the present study.]. Hereafter we align H as to be the SM like Higgs boson and h to be the lighter scalar. In Eq. (<ref>) we present the relations between the potential parameters λ_i's with the physical parameters of the model:  


    λ_1     = c_α^2m_H^2 + s_α^2m_h^2/v^2c_β^2, 
    λ_2     = c_α^2m_h^2 + s_α^2m_H^2/v^2s_β^2, 
    λ_3     = (m_H^2-m_h^2)s_αc_α - (λ_4+λ_5)v^2c_βs_β/v^2c_βs_β, 
    λ_4     = m_A^2 - 2m_H^±^2/v^2, 
    λ_5     = -m_A^2/v^2.

The scalar potential given in Eq.(<ref>) clearly has six independent parameters, and the m_ii^2 terms can be traded off in terms of λ_i using the extremization conditions on the potential. We find relations of the parameters of the potentials given in equation Eq.(<ref>) in terms of physical scalar masses and the angles {β, α} and use those as the input parameters for further analysis.

The right handed up/down quarks and lepton fields are also charged under the aforementioned Z_2 symmetry and transform as u^i_R→ -u^i_R, d^i_R→ -d^i_R and ℓ^i_R→ -ℓ^i_R, respectively. From these charge assignment one realises that all charged fermions exclusively couple to the Φ_2 field, leading to the traditional 2HDM Type-I scenario. In Eq.(<ref>) we write down the Yukawa part of the Lagrangian in the mass eigenstate basis.  


    - ℒ_Yukawa    = +∑_f = u,d,ℓ[ m_fff̅ + (m_f/vκ^f_hf̅fh + m_f/vκ^f_Hf̅fH - im_f/vκ^f_Af̅γ_5fA)] 
         + √(2)/vu̅(m_uVκ^u_H^+P_L + Vm_dκ^d_H^+P_R)dH^+ + √(2)m_ℓκ^ℓ_H^+/vν̅_Lℓ_RH^+ + h.c.

Here m_f is the fermion mass, V is the CKM matrix and P_L/R = 1 ±γ_5/2 are the projection operators. The explicit form of the scaling function κ_i are detailed in Table  <ref>.





§ CONSTRAINTS

In this section, we describe different theoretical and experimental constraints which are required to restrict the parameter space of the Type-I 2HDM. 





 §.§ Theoretical Constraints


    
  * Vacuum Stability: 
    The vacuum stability conditions ensure that the potential must be bounded from below in all possible field direction. To achieve these, the λ_i's parameters must follow certain relationship such that the quartic terms in the potential must dominate in large field values. Here we list down the  conditions on λ_i's which is needed to meet the stability criteria, that prevents the potential from becoming infinitely negative,  <cit.>
    
    λ_1  > 0,      λ_2  > 0,       λ_3 + √(λ_1 λ_2)  > 0,       λ_3 + λ_4 - |λ_5| + √(λ_1 λ_2)  > 0.

    
    
  * Unitarity: The unitarity constraints are necessary to ensure that the theory remains predictive at high energies. At tree level, unitarity imposes specific conditions on the energy growth of all possible 2 → 2 scattering processes. Ref<cit.> derives the unitarity conditions for the 2HDM model explicitly. According to the unitarity constraint, the following relations should be obeyed:
    
    |u_i| ≤ 8π ,

    where
     
    u_1     = 1/2(λ_1 + λ_2 ±√((λ_1 - λ_2)^2 + 4|λ_5|^2)), 
     
        u_2     = 1/2(λ_1 + λ_2 ±√((λ_1 - λ_2)^2 + 4λ_4^2)), 
    
        u_3     = 1/2(3(λ_1 + λ_2) ±√(9(λ_1 - λ_2)^2 + 4(2λ_3+λ_4)^2)), 
    
        u_4     = λ_3 + 2λ_4 ± 3|λ_5|, 
    
        u_5     = λ_3 ± |λ_5|, 
    
        u_6     = λ_3 ±λ_4.

    
  * Perturbativity: The perturbativity condition on the parameters of the scalar potential, which imposes an upper limit on all the quartic couplings, demands that, for all i values, one has λ_i ≤ |4π|.






 §.§ Experimental Constraints



    
  * EW Precision Tests We evaluated the EW precision constraints by computing the S, T and U parameters using the SPheno package <cit.>, with the model files written in SARAH  <cit.>. These so-called `oblique parameters' provide stringent constraints on new physics, thereby demanding that any extension to the SM Higgs sector should conform to high precision data from LEP (primarily). The numerical values of these observables are <cit.>


    S = -0.02 ± 0.10,         T = 0.03 ± 0.12,         U  = 0.01 ± 0.11.


    
  * BSM Higgs Boson Exclusion We assessed the exclusion limits from direct searches for the BSM scalars at the LHC, LEP and the Tevatron. These exclusion limits were evaluated at the 95 % Confidence Level (C.L.) using the HiggsBounds-6 <cit.> module via the HiggsTools <cit.> package. In our analysis we have also demanded that our lighter Higgs must comply with the results of  <cit.>, where the scalars are produced in association with a massive vector boson or a top anti-quark pair and further decays via leptonic modes.

    
  * SM-Like Higgs Boson Discovery We examined the compatibility of our 125 GeV Higgs boson with the discovered SM-like Higgs boson using a goodness-of-fit test. Specifically, we calculated the χ-square value with HiggsSignals-3 <cit.> via HiggsTools, comparing the predicted signal strengths of our Higgs boson to those observed experimentally. We retained the parameter spaces that satisfies the condition χ_125^2 < 189.42, corresponding to a 95 % C.L. with 159 degrees of freedom.

    
  * Flavour Physics We incorporated constraints from B-physics observables, which are sensitive to potential new physics contributions in loop mediated FCNC processes. Specifically, we tested the most stringent bound on the Branching Ratio (ℬℛ) of the B→ X_s γ decay using Next-to-Leading Order (NLO) calculations in QCD as discussed in <cit.>. 

    
    ℬℛ(B → X_s γ) = Γ (B → X_s γ)/Γ_SLℬℛ_SL


    where, ℬℛ_ SL is the semi-leptonic branching ratio and Γ_SL is the semi-leptonic decay width.
    








    We took our input parameters from the most recent values from the Particle Data Group (PDG) compilation <cit.>, as follows:

    
    α_s(M_Z)     = 0.1179 ± 0.0010 ,               m_t = 172.76 ± 0.3, 
    m_b/m_c    = 4.58 ± 0.01,               α  = 1/137.036, 
     BR_ SL    = 0.1049 ± 0.0046,      |V_ts^*V_tb/V_cb|^2 = 0.95 ± 0.02, 
    
             m_b( MS)     = 4.18 ± 0.03,               m_c = 1.27 ± 0.02, 
    
            m_Z     = 91.1876 ± 0.0021,           m_W = 86.377 ± 0.012.

    
    The following restriction has been imposed, which represents the 3 σ experimental limit: 

    
    2.87 × 10^-4 < ℬℛ(B → X_s γ) < 3.77 × 10^-4.


    Other B-physics observables, like ℬℛ(B^+ →τ^+ ν_τ), ℬℛ(D_s →τν_τ), ℬℛ(B_s →μ^+ μ^-) and  ℬℛ(B^0 →μ^+ μ^-) have been taken care of by using the FlavorKit tool <cit.> provided by SPheno package  <cit.>. Our calculated b → s γ results were also found to be consistent with the FlavorKit tool.









§ EXPLAINING THE ANOMALIES



The primary objective of this paper is to investigate whether the 2HDM Type-I can explain the excesses observed at the LHC and LEP data over the 92-98 GeV or so mass range. To do so, we need to define the signal strength corresponding to these three excesses. The signal strength is formulated as a ratio between the observed number of events to the expected number of events for a hypothetical SM Higgs of mass 95 GeV. Assuming the Narrow Width Approximation (NWA), the signal strength for the τ^+ τ^-, γγ and b b channels can be parameterised as cross section (σ) times the branching ratio (ℬℛ),



    μ_τ^+ τ^-(ϕ)     = σ_ 2HDM(gg→ϕ)/σ_ SM(gg→ h_ SM)×ℬℛ_ 2HDM(ϕ→τ^+ τ^-)/ℬℛ_ SM(h_ SM→τ^+ τ^-), 
    μ_γγ(ϕ)     = σ_ 2HDM(gg→ϕ)/σ_ SM(gg→ h_ SM)×ℬℛ_ 2HDM(ϕ→γγ)/ℬℛ_ SM(h_ SM→γγ), 
    μ_b b(ϕ)     = σ_ 2HDM(e^+e^-→ Z ϕ)/σ_ SM(e^+e^-→ Z h_ SM)×ℬℛ_ 2HDM(ϕ→ b b)/ℬℛ_ SM(h_ SM→ b b).

Here, h_ SM corresponds to a SM like Higgs Boson with a mass of 95 GeV while ϕ is a 2HDM Type-I Higgs state with the same mass. The experimental measurements for these three signal strengths around 95 GeV are expressed as 



    μ_γγ^ exp    = μ_γγ^ ATLAS+CMS = 0.24^+0.09_-0.08,  <cit.>
    μ_τ^+ τ^-^ exp    = 1.2 ± 0.5,  <cit.>
    μ_b b^ exp    = 0.117 ± 0.057.  <cit.>







Although the ditau excess is most prominent around 100 GeV and the b b̅ excess near 98 GeV, a search around 95 GeV could provide a unified explanation for all these three anomalies. This is because the mass resolution in the ditau final states is substantially large, and the LEP excess, associated with the b b̅ final states, is also broad. Therefore, a common origin for these excesses may plausibly reside around 95 GeV.

In our analysis, we have combined the di-photon measurements from the ATLAS and CMS experiments, denoted as μ_γγ^ ATLAS and μ_γγ^ CMS, respectively. The ATLAS measurement yields a central value of 0.18 ± 0.1  <cit.> while the CMS measurement yields a central value of 0.33^+0.19_-0.12  <cit.>. The combined measurement, denoted by μ_γγ^ ATLAS+CMS is determined by taking the average of these two central values, assuming both these measurements are uncorrelated. The corresponding combined uncertainty is calculated by adding corresponding uncertainties in quadrature.  To determine if the observed excess can be explained through our model or otherwise, we perform a χ^2 analysis using the central values μ^ exp and the 1 σ uncertainties Δμ^ exp associated with the signal related to the excesses as defined in Eq. (<ref>). The contribution to the χ^2 for each of the channel is calculated using the equation


    χ^2_γγ, τ^+ τ^-, b b = (μ_γγ, τ^+ τ^-, b b(ϕ) - μ_γγ, τ^+ τ^-, b b^ exp)^2/(Δμ_γγ, τ^+ τ^-, b b^ exp)^2.


Hence, the resulting χ^2 which we will use to determine if the excesses are explained by the 2HDM Type-I, or otherwise, is the following:


    χ^2_γγ, τ^+ τ^-, b b = χ^2_γγ + χ^2_ τ^+ τ^- + χ^2_b b.



We test this BSM scenario into two cases: firstly, we consider both the CP-even and CP-odd (i.e., ϕ=h+A. except for bb̅ where ϕ=h) Higgs states simultaneously in explaining the anomaly  and, secondly, we only exploit the CP-even Higgs (i.e., ϕ=h) in order to explain it. Hence, we align our H state (recall that we have m_h<m_H) with the SM Higgs boson, so that  m_H=125 GeV,  and start a Monte Carlo (MC) sampling of the various input parameters.



 §.§ The Overlapping Solution

In case of overlapping solution, the signal strength corresponding to the τ^+ τ^- and γγ channels receive substantial contribution from both the CP-even and CP-odd states simultaneously[Here we have considered CP-conserving potential. As a result, the h and A states would not interfere with each other.]. In contrast, for the b b̅ mode only the CP-even state contribute as the trilinear coupling AZZ is zero at tree level. As a result, the signal strength can be expressed in the following manner -

    μ_τ^+ τ^-(h+A)  = μ_τ^+ τ^-(h) + μ_τ^+ τ^-(A),   μ_γγ(h+A)  = μ_γγ(h) + μ_γγ(A),   μ_bb̅ (h).






We generated MC samples in the scan range as described in Table  <ref>. After testing them against various theoretical and experimental constraints, the allowed parameter space is illustrated in Figure <ref>. The region of the parameter space that pass the theoretical constraints are depicted by the blue points while the region that pass the experimental constraints are depicted by red points. The plot illustrates that a nearly degenerate solution with both the CP-even and CP-odd Higgs in the 92-98 GeV mass range is viable for the charged Higgs mass ranging around 160 < m_H^± < 195. We will use the overlapping region of the two coloured point distributions to test the aforementioned anomalies.



Figure  <ref> illustrates the total chi-square distribution for points that are compatible with all three anomalies and the best fit point is marked by a star, corresponding to χ^2_ min. The experimentally observed signal strength with their 1 σ band is also superimposed in the plot to test them against this model's best fit point. The chi-square fit reveals two distinct branches, differentiated by the sign of sin(β - α). The main branch, characterized by positive values, is densely populated, while the second branch, which has negative values, is sparsely populated due to the elimination of many points by various constraints. The figure  <ref> also illustrates the sign of the electroweak coupling parameter. The best fit point indicated by a red star has a large positive sin (β - α) value, while the vector boson coupling with the SM Higgs boson, indicated by the color bar, is weakened in that case. 

Given that the di-photon excess is most pronounced around 95 GeV, we also plot the allowed points within the 94-96 GeV range over the CMS and ATLAS results for the signal strength in the γγ channel, as shown in Figure  <ref>. The expected and observed CMS limits are shown by the black dashed and solid lines. The green and yellow bands indicate the 1 σ and 2 σ uncertainties and the plot is overlaid by the ATLAS observed 95 % confidence level limits on the signal strength with the red dashed and solid lines. The combined signal strength (CMS+ATLAS) at 95.4 GeV with its error bar is also shown using a red dot. The points explaining the anomaly at the 1 σ level, for 3 degrees of freedom, corresponding to the γγ, ττ and b b channels as shown in equation  <ref>, which requires χ^2 < 3.53, are plotted in dark red while the points explaining it at the 3 σ level, demanding χ^2 < 7.8147 are shown in peru color (Less likely points are given in sky blue). The best fit point in the 94-96 range is also indicated using midnight blue color. It can be clearly seen that the parameter points are suited to explain the excesses observed. The details of the best fit points as marked in the two figures are shown in Table <ref>.




 




 §.§ The Single Solution







In this case, only the h state is responsible for explaining all the three  anomalies. We sampled points for the scan range described in Table  <ref>, and the points that pass the various constraints are depicted in  Figure <ref>, wherein the blue shaded region indicates the points that pass the various theoretical constraints while the red shaded region indicates region of parameter space that survives after imposing different experimental bounds. The plots clearly represent the allowed parameter space for the CP-odd and Charged Higgs masses, given that we fix our CP-even mass to lie in the range 92-97 GeV. Though the charged Higgs mass is bounded at around 160 GeV, the CP-odd pseudoscalar covers almost the entire scan range. Note that the allowed CP-odd scalar mass also lie in the 90-100 GeV mass window, hinting to the overlapping solution that we discussed in the previous section. We move ahead with testing the overlapping points with the observed anomalies. 



The total χ^2 fit for the points passing all the constraints is displayed in Figure <ref>, wherein the best fit point (i.e., again, the χ^2_ min one) is indicated with a star. We have again overlaid the plot with the experimentally observed data with the 1 σ band. The plot clearly depicts that the best fit point lies within the experimental 1 σ boundary. The two bands here again are differentiated with the sign of sin (β - α), wherein the densely populated region is positive while the sparsely populated one negative. This is also depicted in Figure <ref>, where the best fit point is depicted by a red star, which has a large sin(β - α) value, while the vector boson coupling with the SM Higgs boson is weakened, which is indicated by the color bar.
The mass region lying between 94-96 GeV is also displayed on top of the CMS and ATLAS results for the signal strength in the γγ channel in Figure <ref> (with the same color coding as that in Figure <ref>). The figure clearly depicts the best fit point in that particular region to be close to the mean experimentally observed value. Finally the best fit BPs in these cases are shown in Table <ref>.











§ CONCLUSIONS


In summary, we have proven that somewhat anomalous data produced at LEP and the LHC in bb̅ and τ^+τ^- as well as γγ final states, respectively, all clustering in a 10 GeV or so mass window around 95 GeV, are consistent with the possibility of the 2HDM-Type I in inverted mass hierarchy explaining these. Specifically, two configurations are possible: one where both the (degenerate) h and A states cooperate to explain the aforementioned anomalies and another where only the h states does so. This is an intriguing result, as such Higgs states can well be probed in collateral signatures specific to the 2HDM Type-I in inverted mass hierarchy, as emphasised in various previous literature
<cit.>. To aid testing this theoretical hypothesis, we have produced two pairs of BPs amenable to further phenomenological analysis,   each pair corresponding to one of the above two solutions, wherein the parameter space point giving the best fit to all anomalies. 



§ ACKNOWLEDGMENTS

The work of S.M. is supported in part through the NExT Institute and the STFC Consolidated Grant No. ST/ L000296/1. A.S acknowledges the support from SERB-National Postdoctoral fellowship (Ref No: PDF/2023/002572). A.K. acknowledges the support from Director's Fellowship at IIT Gandhinagar. All authors thank Tanmoy Mondal and Prasenjit Sanyal for their help in discovering a mistake in their calculations.





