95 GeV excess in a CP -violating µ -from- ν SSM

The CMS and ATLAS have recently reported their results searching for light Higgs boson with mass around 95 GeV, based on the full Run 2 data set. In the framework of the CP-violating (CPV) µν SSM, we discuss a ∼ 2.9 σ (local) excess at 95 GeV in the light Higgs boson search in the diphoton decay mode as reported by ATLAS and CMS, together with a ∼ 2 σ excess (local) in the b ¯ b ﬁnal state at LEP in the same mass range. By introducing CPV phases as well as by mixing CP-even Higgs and CP-odd Higgs, a lighter Higgs boson in the µν SSM can be produced, which can account for the “di-photon excess”


I. INTRODUCTION
In 2012, the 125 GeV Higgs boson discovered by the Large Hadron Collider (LHC) [1,2]; the measured mass of the Higgs boson now is [3] m h = 125.25 ± 0.17 GeV.The discovery of the Higgs boson marked a huge success for the standard model (SM), but it did not stop the search for new physics (NP) at the LHC, and one of them was the search for lighter scalar particles.
As one of the extensions of the SM, the µ-from-ν supersymmetric standard model [39][40][41][42][43][44][45][46] can solve the µ problem [47] of the minimal supersymmetric standard model (MSSM) [48][49][50][51][52], through introducing three singlet right-handed neutrino superfields νc i (i = 1, 2, 3).The neutrino superfields lead the mixing of the neutral components of the Higgs doublets with the right-handed sneutrinos, that is different from the Higgs sector of the MSSM.The mixing can change the Higgs couplings and influence the decay processes of the Higgs bosons.In addition, we also introduce CP violation, and we also get a lighter Higgs at ∼ 95 GeV with a suitable parameter space.
The paper is organized as follows.In Sec.II, we introduce the CP -violating µ-from-ν SSM briefly, about the superpotential and the CPV phases.In Sec.III, we study the excess at 95 GeV in the CPV µ-from-ν SSM.In Secs.IV and V, we show the numerical analysis and the conclusion,respectively.

II. THE MODEL
The superpotential of the µ-from-ν SSM contains Yukawa couplings for neutrinos, two additional types of terms involving the Higgs doublet superfields Ĥu and Ĥd , and the righthanded neutrino superfields νc i [39]: where ĤT u = Ĥ+ u , Ĥ0 u , ĤT d = Ĥ0 d , Ĥ− d , QT i = ûi , di , and LT i = νi , êi (the index T denotes the transposition) represent the MSSM-like doublet Higgs superfields and ûc i , dc i , and êc i are the singlet up-type quark, down-type quark and charged lepton superfields, respectively.In addition, Y u,d,e,ν , λ, and κ are dimensionless matrices, a vector, and a totally symmetric tensor, respectively.a, b = 1, 2 are SU(2) indices with antisymmetric tensor ǫ 12 = 1, and i, j, k = 1, 2, 3 are generation indices.The CP is violated by the parameter λ i , and the CP -violating phase is φ λ i .
In the superpotential, if the scalar potential is such that nonzero vacuum expectative values (VEVs) of the scalar components (ν c i ) of the singlet neutrino superfields νc i are induced, the effective bilinear terms ǫ ab ε i Ĥb once the electroweak symmetry is broken.The last term in Eq. ( 1) generates the effective Majorana masses for neutrinos at the electroweak scale.Therefore, the µ-from-ν SSM can generate three tiny neutrino masses at the tree level through TeV-scale seesaw mechanism [39,[57][58][59][60][61][62][63].
It is worth explaining why the TeV-scale seesaw was chosen.Through a seesaw on the scale of the grand unified theory, one can get Yukawa couplings of the order of one for neutrinos.But we know that the Yukawa coupling of the electron is on the order of 10 −6 , and the Yukawa couplings of neutrinos can also be around on the order of 10 −6 instead of one.In the TeV-scale seesaw, this is sufficient to reproduce the neutrino mass, if the Yukawa coupling of the neutrino is of the same order as the Yukawa coupling of the electron [39].
Here, it is important to note that the VEVs of the left-handed sneutrinos υ ν i are generally small.We know that the Dirac masses for the neutrinos GeV in the TeV-scale seesaw.So we can get an estimate of the VEVs of the left-handed sneutrinos, [39,40].
The general soft supersymmetry-breaking terms of the µ-from-νSSM are given by Here, the first two lines contain mass squared terms of squarks, sleptons, and Higgses.The next two lines consist of the trilinear scalar couplings.In the last line, M 3 , M 2 , and M 1 denote Majorana masses corresponding to SU(3), SU(2), and U(1) gauginos λ3 , λ2 , and λ1 , respectively.In addition to the terms from L sof t , the tree-level scalar potential receives the usual D-and F -term contributions [40,41].
Once the electroweak symmetry is spontaneously broken, the neutral scalars develop in general the VEVs.The CP can violated by the VEVs of the scalar fields: One can define the neutral scalars as and In supersymmetric extensions of the SM, the R parity of a particle is defined as R = (−1) L+3B+2S [48][49][50][51][52]. R parity is violated if either the baryon number (B) or lepton number (L) is not conserved, where S denotes the spin of concerned component field.The last two terms in Eq. ( 1) explicitly violate lepton number and R parity.For example, if one assigns L = 1 to the right-handed neutrino superfields, then the last term 1 3 κ ijk νc i νc j νc k in Eq. ( 2) violates the lepton number by three units contrary to the Li Lj êc k term of the R parity violating MSSM which shows the ∆L = 1 effect.R parity breaking implies that the lightest supersymmetric particle is no longer stable.
Considering the one-loop effective potential, the Higgs potential can be written as we can calculate the minimization conditions of the potential and the Higgs masses in this work.
Here, we note that the Higgs doublets and right-handed sneutrinos are basically decoupled from the left-handed sneutrinos [55], so we did not consider the left-handed sneutrinos in the CP -even and CP -odd scalars part.
The CP -even sector mix with the CP -odd sector, the 10 × 10 mixing matrix is defined by with M 2 S denoting the CP -even neutral scalars, M 2 P is the CP -odd neutral scalars, and M 2 SP represents the mass submatrix for the mixing of CP -even neutral scalars and CP -odd neutral scalars.
The mass squared matrix M 2 h can be diagonalized as with CPV in the CP -even and CP -odd scalar sector the matrix Z H can be complex.
We consider the radiative corrections in mass submatrix M 2 H ; the radiative corrections from the third fermions (f = t, b, τ ) and their superpartners include the two-loop leading-log effects [91][92][93].The CP -even neutral scalars is given as: In detail, the mass submatrix M 2 H is defined by The dominating contributions of radiative corrections ∆ 11 , ∆ 12 and ∆ 22 comes from the fermions (t, b) and their superpartners We did not consider the terms containing coupling Y ν i and υ ν i , because these terms are very small.The radiative corrections from the top quark is given by [53, 85-90] with to save space, the mass matrix and the radiative corrections are given in the Appendix.
In the radiative corrections, the trilinear coupling A t = |A t |e iφ A t can be complex.These seven independent phase have been defined as

III. EXCESS AT 95 GEV
The process measured at LEP reported a 2.3σ local excess in the b b final state searches, with the scalar mass at ∼ 96 GeV.The production of a Higgs boson via Higgstrahlung is associated with the Higgs decaying to bottom quarks.Normalized to the SM expectation, the signal strength is defined as The value for µ bb LEP can be found in [29,81] with the method introduced in [82].h 1 is the Higgs which has mass range around ∼ 96 GeV, and h 2 is the 125 GeV Higgs boson in the following.In the framework of the µ-from-ν SSM we use µ bb N P to describe the signal strength; the expression for µ bb N P can be approximated as [34] One can find the SM branching ratios Br SM in Ref. [3], and Γ is the decay widths.The couplings are normalized to the SM prediction of a Higgs boson of the same mass.C h 1 is the coupling of h 1 and gauge boson, and C h 1 uū and C h 1 d d are the couplings of h 1 and upand down-type quarks.The normalized couplings are given as In 2019, the CMS searches for the Higgs boson decaying in the diphoton channel showed a local excess of ∼ 3σ around ∼ 96 GeV [11]; the previous results is that [11,84] Recently, ATLAS reported their new results at 95.4 GeV based on the full Run 2 dataset [15], the "diphoton excess" with a signal strength of Meanwhile, the corresponding CMS result for the "diphoton excess" is given by [12] Neglecting possible correlations one can get a combined signal strength of [13] µ exp γγ = µ AT LAS+CM S γγ = 0.24 +0.09 −0.08 .
In this work, the approximation of the diphoton rate of the h 1 can written as [29,34] The effective coupling C h 1 γγ can be written as [34] |C with The form factors A 1/2 and A 1 are given by [94] By using Eqs.( 23) and ( 29), we calculate the two signal strengths.

IV. NUMERICAL RESULTS
In this section we will discuss the couplings and signal strength of 96 GeV Higgs in CPV µνSSM.The free parameters in our analysis will be We take 3 , and we have defined where i, j = 1, 2, 3.

A. Mass and coupling
In Fig. 1 (a), we can see that, when h 1 is near 96 GeV, h 2 is closer to 125 GeV with the increase of tan β.Although both h 1 and h 2 can conform to the experimental mass range in our parameter space, if we assume a theory uncertainty of up to 3 GeV [34], the parameter range will be larger.One can clear see that 1 (b); the LHC measurements of the SM-like Higgs bosons couplings to fermions and massive gauge bosons are still not very precise [34,95].If in the future some collider can measure these couplings to the percent level, then we can choose a more reasonable parameter space.
. The input parameters are in Table I.I.
In Fig. 2, left column, we can see that m h 1 is very sensitive to υ ν c .As υ ν c becomes larger, m h 1 will rapidly become smaller; on the contrary, m h 2 will slowly become larger.In order for m h 1 to be around 95.4 GeV, the value of υ ν c will not have a wide range.Similar to υ ν c , m h 1 is also very sensitive to κ, and with the increase of λ, a larger κ value can be taken to keep m h 1 near 95.4 GeV.On the contrary, κ has little effect on m h 2 ; especially when λ=0.085 or 0.087, m h 2 will be very stable.In Fig. 3, we have showed the component of h 1 and h 2 ; for h 1 , CP -odd H u component and CP -even right-handed sneutrinos component are the main components, and with the increase of λ, CP -odd H d component will become smaller, while the CP -even right-handed sneutrino component will become larger.While for h 2 , CP -even H u is the main component, as λ increases, the CP -even H u component will increase, the CP -even H d component will gradually decrease, and the CP -odd H d component will gradually increase.
Then, in Fig. 4, we analyze the correlation between the Higgs masses and the CPV phases.For the first row, we take λ = 0.09, tan β = 6, and κ = 0.315, for the second row, λ = 0.095, tan β = 4.5, and κ = 0.315, for the third row, λ = 0.098, tan β = 4.3, and κ = 0.32, for the last row, λ = 0.085, tan β = 3, and κ = 0.04.Other parameters remain the same as Table I.We should remark that there are many CPV phase values that can be constrained by the Higgs masses in Fig. 7 (a), and here we select only one of them.The signal strengths and the Higgs masses do not increase or decrease all the time, but oscillate like a sine function.In Fig. 4, the first row, when φ λ grows from 4 to 5, the mass range of the "diphoton excess" can fit the experimental constraints, and the SM-like Higgs mass can be kept around 125 GeV.
Similar to the first row in Fig. 4, the second and third rows show the variation of h 1 and h 2 with the phases φ vu and φ v d .m h 1 is very sensitive to φ vu and φ v d .When φ vu is 0.7π − π, We take λ = 0.09 and other parameters as in Table I. and υ ν c ; with the increase of the value of υ ν c , the values of µ bb N P and µ γγ N P will increase.And if we set υ ν c = 1000, the value of µ bb N P of tan β=3.5 will be larger than that of tan β=3, because we conclude from Fig. 1   the increase of tan β.In the second row in Fig. 5, as the growth of κ, the value of µ bb N P will slowly increase, while the value of µ γγ N P will slowly decrease.However, either tan β = 3 or tan β = 3.5 ensures that µ bb N P and µ γγ N P are both within their respective 1σ experimental error.
The effect of CPV phases is also very obvious.Here, we take φ At as an example, we take φ At from −π to π.In the first row in Fig. 6, we can see that the minimum value of h 1 appears to be around 96.2 GeV and varies periodically with φ At .The maximum value of h 1 does not exceed 1 GeV larger than the minimum value.Meanwhile, the highest point of h 2 is at 125.3 GeV, with a maximum value 1.5GeV larger than the minimum value.
The impact of φ At on the normalized couplings is relatively small.In the second row in Fig. 6, we can see that the peak of |C h 1 V V | is close to 0.313, and the highest point of |C h 2 V V | is slightly less than 0.856.This implies that, if the "95.4 excess" is a new particle, then its coupling with gauge bosons should be much smaller than the coupling of the SM-like Higgs with gauge bosons.
In the last row in Fig. 6, we found that φ At also has a slight effect on the two signal strengths, µ γγ N P is always slightly bigger than the central value of µ γγ exp , but does not exceed 1σ experimental error.At the same time, µ bb N P is lower than the central value of µ bb exp , but it also does not exceed 1 σ experimental error.
Let us remark that, in the random scan plots, we must first ensure that the mass of h 2 can conform to the experimental constraints, because the mass determination for SM-like Higgs is already very accurate.In Fig. 7 (a), we chose 1 GeV theory uncertainty for the SM-like Higgs.For h 1 , we can choose the theory uncertainty up to 3 GeV, because CMS shows only that there is a 2.8 σ excess at 95.3 GeV [9,13,17], which is not exactly observed.I.
Table II, most points can explain the dphoton excess and the b b excess.

V. CONCLUSION
In this paper we introduced CPV in the µ-from-νSSM, which leads to CP -even Higgs sector mixed with CP -odd Higgs sector.We also analyzed an excess in the diphoton decay In the numerical part, we find a suitable parameter space, based on which we show the properties of the lightest and next-to-lightest Higgs boson.We found it very easy get both µ bb N P and µ γγ N P to reach the central values of their respective experiments at the same time.We also analyze the influence of relevant parameters and CPV phases on the signal strengths or Higgs masses, and improve the signal strengths as much as possible while ensuring that the SM-like Higgs meets the experimental constraints and the "diphoton excess" is around 95 GeV.
No. 2022GXNSFDA035068, and the youth top-notch talent support program of the Hebei Province.

Appendix A: mass matrices
The CP -even neutral scalars have the composition S T = (h d , h u , (ν c i ) Re ), and one can write the mass matrix M 2 S as The radiative corrections from the top quark and bottom quark and their corresponding supersymmetric partners, the corrections in the mass matrix, can be expressed as In the same way, the CP -odd neutral scalars mass matrix is that M 2 σuσu = Re(e iφ λ e −iφv u e iφv d e Im(e iφ λ e iφ A t e iφv u e −iφv d e Im(e iφ λ e iφ A t e iφv u e −iφv d e

5 FIG. 1 :
FIG. 1: Values for m h 2 versus m h 2 in (a) and the normalized couplings|C h 2 V V | versus |C h 1 V V | in

FIG. 2 :
FIG. 2: Values for m h 1 versus the parameters υ ν c in (a) and κ in (c) and values for m h 2 versus the parameters υ ν c in (b) and κ in (d).We take tan β = 5 and other parameters in TableI.

( 2 FIG. 3 :
FIG. 3: The component of h 1 (a) and h 2 (b).The input parameters are in Table I, and we take tan β = 4.31.

FIG. 4 :
FIG. 4: The correlation of h 1 and the CPV phases in (a) (c) (e) (g); the gray area represents the experimental error of 3 GeV.The correlation of h 2 and the CPV phases in (b) (d) (f) (h); the gray area is the theory uncertainty of 1 GeV.

FIG. 5 :
FIG.5:The left column shows that the signal strength µ bb N P varies with υ ν c (a) and κ (c); the right column shows that the signal strength µ γγ N P varies with υ ν c (b) and κ (d).We take λ = 0.09 and

h 1
can be kept within the experimental error of 3 GeV.When φ vu continues to increase or decrease, m h 1 will drop very quickly.Similarly, when φ v d is [−0.75π,−0.45π], m h 1 can conform to the experimental error.For SM-like Higgs, φ vu and φ v d can provide a wide range to keep m h 2 near 125 GeV.Different from other CPV phases, it is difficult to determine the value of φ υ c ν , and only a narrow range allows h 1 and h 2 to be simultaneously constrained by their respective experiments.B. Signal strengths First, in Figs.5 (a) and (b), we can see the correlation of the signal strengths µ bb N P , µ γγ N P

InFIG. 6 :
FIG. 6: The Higgs masses vary with the phase φ At in (a)-(b); the correlation of couplings and the phase φ At in (c)-(d); and (e)-(f) are the correlation of signal strengths and the phase φ At .Here, we take tan β=4.5, λ=0.086, and φ At = [−π, π], and other parameters are the same as in TableI.

FIG. 7 :
FIG. 7:The left plot (a) shows the mass range of h 1 and h 2 ; for h 2 , we take 1 GeV theory uncertainty, for h 1 , CMS gives a mass of around 95.4 GeV, and we take 3 GeV experimental error.Correlation of these two signal strengths in (b); the gray area is the 1σ experimental error.The values of parameters are in TableII.

TABLE I :
Input parameters to fit the LEP and the CMS excesses.All dimensionful parameters are given in GeV.
that the Higgs masses and couplings will increase with

TABLE II :
The parameter space of the random scan plot.All dimensionful parameters are given in GeV.
(−Im(e iφ λ e iφ A t e iφv u e −iφv d e Im(e iφ λ e iφ A t e iφv u e −iφv d e iφ λ e iφ A t e iφv u e −iφv d e (Im(e iφ λ e iφ A t e iφv u e −iφv d e 2 sin 2 β |µ| 2 cot β − Re(e iφ λ e iφ A t e iφv u e −iφv d e iφ λ e iφ A t e iφv u e −iφv d e sin 2 β −Im(e iφ λ e iφ A t e iφv u e −iφv d e |A t | 2 − Re(e iφ λ e iφ A t e iφv u e −iφv d e sin 2 β −Im(e iφ λ e iφ A t e iφv u e −iφv d e (λ i λ * j + λ * i λ j ) − Re(e iφ λ e iφ A t e iφv u e −iφv d e Im(e iφ λ e iφ A t e iφv u e −iφv d e (λ i λ * j + λ * i λ j ) − Re(e iφ λ e iφ A t e iφv u e −iφv d e sin 2 β −Im(e iφ λ e iφ A t e iφv u e −iφv d e (λ i λ * j + λ * i λ j ) − Re(e iφ λ e iφ A t e iφv u e −iφv d e