Higgs boson decay h → Zγ and muon magnetic dipole moment in the μνSSM

To solve the μ problem and generate three tiny neutrino masses in the MSSM, the μ from ν Supersymmetric Standard Model (μνSSM) introduces three singlet right- handed neutrino superfields, which lead to the mixing of the Higgs doublets with the sneutrinos. The mixing affects the lightest Higgs boson mass and the Higgs couplings. The present observed 95% CL upper limit on signal strength of the 125 GeV Higgs boson decay h → Zγ is 6.6, which still is plenty of space to prove the existence of new physics. In this work, we investigate the signal strength of the 125 GeV Higgs boson decay channel h → Zγ in the μνSSM. Besides, we consider the two-loop electroweak corrections of muon anomalous magnetic dipole moment (MDM) in the model, which also make important contributions compared with one-loop electroweak corrections.

JHEP04(2020)002 will represent a major undertaking of modern particle physics and a probable probe of new physics. Within various theoretical frameworks, the 125 GeV Higgs boson decay h → Zγ has been discussed .
In addition, the current difference between the experimental measurement [64] and SM theoretical prediction of the muon anomalous magnetic dipole moment (MDM) [6], ∆a µ = a exp µ − a SM µ = (26.8 ± 7.7) × 10 −10 , (1.2) represents an interesting but not yet conclusive discrepancy of 3.5 standard deviation, which still stands as a potential indication of the existence of new physics. Up to now, several predictions for the muon anomalous MDM have been discussed in the framework of various SM extensions [65][66][67][68][69][70][71][72][73][74][75][76][77][78][79][80][81]. In near future, the Muon g-2 experiment E989 at Fermilab [82,83] will measure the muon anomalous MDM with unprecedented precision, which may reach a 5σ deviation from the SM, constituting an augury for new physics. In our previous work, we have studied the muon MDM at one-loop level in the µνSSM [62]. To be more precise, here we will consider the two-loop diagrams of the muon anomalous MDM in the framework of the µνSSM. Simultaneously, the accurate theoretical prediction of the muon anomalous MDM can conduce to constrain strictly the parameter space of the model. The paper is organized as follows. In section 2, we introduce the µνSSM briefly, about the superpotential and the soft SUSY-breaking terms. In section 3, we give the decay width and the signal strength of h → Zγ. Section 4 includes the two-loop electroweak corrections of the muon anomalous MDM. Section 5 and section 6 respectively show the numerical analysis and summary. Some formulae are collected in appendix.

The µνSSM
In addition to the MSSM Yukawa couplings for quarks and charged leptons, the superpotential of the µνSSM contains Yukawa couplings for neutrinos, two additional types of terms involving the Higgs doublet superfieldsĤ u andĤ d , and the right-handed neutrino superfieldsν c i , [48]  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. a, b = 1, 2 are SU(2) indices with antisymmetric tensor 12 = 1, and i, j, k = 1, 2, 3 are generation indices. The summation convention is implied on repeating indices in the following.
In the superpotential, if the scalar potential is such that nonzero vacuum expectation values (VEVs) of the scalar components (ν c i ) of the singlet neutrino superfieldsν c i are induced, the effective bilinear terms ab ε iĤ b uL a i and ab µĤ a dĤ b u are generated, with ε i = Y ν ij ν c j and µ = λ i ν c i , once the electroweak symmetry is broken. The last term generates the effective Majorana masses for neutrinos at the electroweak scale. Therefore, the µνSSM can generate three tiny neutrino masses at the tree level through TeV scale seesaw mechanism [49,[84][85][86][87][88][89][90].
In supersymmetric (SUSY) extensions of the standard model, the R-parity of a particle is defined as R = (−1) L+3B+2S [56][57][58][59][60]. R-parity is violated if either the baryon number (B) or lepton number (L) is not conserved, where S denotes the spin of the concerned component field. The last two terms in eq. (2.1) explicitly violate lepton number and Rparity. R-parity breaking implies that the lightest supersymmetric particle (LSP) is no longer stable. In this context, the neutralino or the sneutrino are no longer candidates for the dark matter (DM). However, other SUSY particles such as the gravitino or the axino can still be used as candidates [49,50,87,[91][92][93][94][95][96].
The dark matter candidate must be stable on the cosmic timescale, so that it is still around today [97]. In refs. [91][92][93][94], the authors analyzed the gravitino dark matter candidate in the µνSSM, whose lifetime is long lived compared to the current age of the Universe. The gravitino turns out to be an interesting candidate for DM, which may be searched through gamma-ray observations with Fermi-LAT. Recently, the axino dark matter candidate in the µνSSM also was analyzed [95,96].
The general soft SUSY-breaking terms of the µνSSM 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) gauginosλ 3 ,λ 2 , and λ 1 , respectively. In addition to the terms from L sof t , the tree-level scalar potential receives the usual D-and F -term contributions [49,50].

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Once the electroweak symmetry is spontaneously broken, the neutral scalars develop in general the VEVs: One can define the neutral scalars as and In the µνSSM, the left-and right-handed sneutrino VEVs lead to the mixing of the neutral components of the Higgs doublets with the sneutrinos producing an 8 × 8 CP-even neutral scalar mass matrix, which can be seen in refs. [49,50,53]. The mixing gives a rich phenomenology in the Higgs sector of the µνSSM. In the large m A limit, we give an approximate expression for the lightest Higgs boson mass [63], Here A 2 X 1 comes from the mixing of the neutral components of the Higgs doublets with the right-handed sneutrinos, and m 2 R 1 is the mass squared of the right-handed sneutrino, whose concrete expressions are given by where ∆ 1R , ∆ 2R and ∆ RR are the radiative corrections [63]. Comparing with the MSSM, The radiative corrections m 2 H 1 can be computed more precisely by some public tools, for example, FeynHiggs [98][99][100][101][102][103][104][105], SOFTSUSY [106][107][108], SPheno [109,110], and so on. In the following numerical section, we will use the FeynHiggs-2.13.0 to calculate the radiative corrections for the Higgs boson mass about the MSSM part. Figure 1. The one-loop diagrams contributing to the decay h → Zγ in the µνSSM, where F denotes the fermions and the charginos, W is the W gauge boson, S ± denotes the charged scalars, andf shows the sfermions.

The rare decay h → Zγ
The h → Zγ coupling in the SM is similar to the h → γγ coupling, which is built up by the heavy top quark and W boson loops [15]. In the supersymmetric models of the SM, there are more kinds of particles can make contributions to the LO decay width, W boson, the third-generation fermions (f = t, b, τ ) and the supersymmetric partners [16]. In figure 1, we plot the one-loop diagrams contributing to the decay h → Zγ in the µνSSM, where F denotes the fermions and the charginos, W is the W gauge boson, S ± denotes the charged scalars, andf shows the sfermions. Therefore, the decay width of the loop induced Higgs boson decay h → Zγ in the framework of the µνSSM can be mainly given as The form factors A 0 , A 1/2 and A 1 are showed in appendix A. The concrete expressions of g hf f , g hW W , g hS + α S − α , g hff can be found in ref. [61]. And the expressions of g n Zχ i χ i and g n hχ i χ i are where C Zχ iχi n and C S 1 χ iχi n (h = S 1 ) can be seen in ref. [53]. The decay width of h → Zγ at leading order (LO) in the µνSSM is mediated by charged heavy particle loops built up by W bosons, standard fermions f , charged scalars S ± α , charginos χ i and sfermionsf . When the supersymmetric particles are more heavy, the contributions of supersymmetric particles will be small. The signal strength of Higgs boson decay h → Zγ is a physical quantity that can be observed directly, and it can be written by normalized to the SM values, where ggF stands for gluon-gluon fusion. One can evaluate the Higgs production cross sections 4) and the total decay width of the 125 GeV Higgs boson in the NP is [61] Γ h where we neglected the little contribution which is rare or invisible, and the Γ h SM is the total decay width of the SM Higgs boson. Through eqs.

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where l µ denotes the muon which is on-shell, m µ is the mass of the muon, σ αβ = i 2 [γ α , γ β ], F αβ represents the electromagnetic field strength and muon MDM, a µ = 1 2 (g − 2) µ . Including two-loop electroweak corrections, the muon MDM in the µνSSM can be written by where the one-loop corrections a one−loop µ can be found in ref. [62]. The two-loop diagrams can give important contributions to the muon MDM in a reasonable parameter space. According to refs. [75,80], the main two-loop rainbow diagram (a) and Barr-Zee type diagrams (b,c) contributing to the muon MDM in the µνSSM are shown in figure 2. Here, we ignore some two-loop diagrams which have low contributions, due to the decoupling theorem. In the µνSSM, the two-loop corrections are given as where the terms a W W µ , a W S µ , a γh µ are the contributions corresponding to figure 2 (a-c). Under the assumption m F = m χ β m χ 0 η m W , the concrete expression can be approximately written as where J(x, y, z) = ln x − y ln y − z ln z y − z . (4.7) Here, (· · · ) represents the operation to take the real part of a complex number, the concrete expressions for couplings C can be found in ref. [53].

Numerical analysis
In the µνSSM, there are many free parameters. We can take some appropriate parameter space, so that we can obtain a transparent numerical results. First, we make the minimal JHEP04(2020)002 flavor violation (MFV) assumptions for some parameters, which assume where i, j, k = 1, 2, 3. m 2 ν c i can be constrained by the minimization conditions of the neutral scalar potential seen in ref. [63]. To agree with experimental observations on quark mixing, one can have where the m u i , m d i and m l i stand for the up-quark, down-quark and charged lepton masses, and we can find the values of the masses from PDG [6]. Through our previous work [90], we have discussed in detail how the neutrino oscillation data constrain neutrino Yukawa couplings Y ν i ∼ O(10 −7 ) and left-handed sneutrino VEVs υ ν i ∼ O(10 −4 GeV) in the µνSSM via the TeV scale seesaw mechanism. Through analysis of the parameter space of the µνSSM in ref. [49], we can take reasonable parameter values to be λ = 0.1, κ = 0.4, A λ = 500 GeV, A κ = −300 GeV and A u 1,2 = A d = A e = 1 TeV for simplicity. Considering the direct search for supersymmetric particles [6], we take mQ 1,2,3 = mũ c 1,2 = mdc 1,2,3 = 2 TeV, mL = mẽc = 1 TeV, M 3 = 2.5 TeV. For simplicity, we will choose the gauginos' Majorana masses M 1 = M 2 .
As key parameters, A u 3 = A t , mũ c 3 and tan β greatly affect the lightest Higgs boson mass. Therefore, the free parameters that affect our next analysis are tan β, υ ν c , M 2 , mũ c 3 , and A t .
In the supersymmetric model, there is a close similarity between the anomalous magnetic dipole moment of muon and the branching ratio ofB → X s γ, in that both get large tan β enhancements from a Higgsino-sfermion-fermion interaction vertex with a downfermion Yukawa coupling [68]. So in the following, we also consider the constraint from the branching ratio ofB → X s γ. The current combined experimental data for the branching ratio ofB → X s γ measured by CLEO [111], BELLE [112,113] and BABAR [114][115][116] give [6] Br(B → X s γ) = (3.49 ± 0.19) × 10 −4 . (5.4) In the next numerical analysis, we use our previous work about the rare decayB → X s γ in the µνSSM [117].

Muon MDM
Firstly, we analyze the muon MDM in the µνSSM. We define the physical quantity In figure 3, we plot the muon anomalous MDM a SUSY µ and the ratio R a varying with the parameter υ ν c , where the gray area denotes the muon MDM at 3.0σ given in eq. (1.2). In figure 3(a), the numerical results show that the muon anomalous magnetic dipole moment a SUSY µ is decoupling with increasing υ ν c , which coincides with the decoupling theorem. We can see that the value of the muon anomalous MDM a SUSY µ in the µνSSM could reach the experimental center value shown in eq. (1.2), when υ ν c is small.
To show the two-loop contributions of the muon MDM, figure 3(b) pictures the ratio R a varying with the parameter υ ν c . Normalized to the one-loop corrections of the muon MDM, the ratio R a can reach around 16% when υ ν c is large. Here, when υ ν c is large, the one-loop corrections of the muon MDM are decoupling quickly than the two-loop corrections. The numerical results also show that the ratio R a can be about 12% when υ ν c is small. Therefore, the two-loop corrections also make important contributions to the muon anomalous MDM in the µνSSM.
To see the difference of two-loop contributions of muon MDM between the µνSSM and the MSSM, we define the physical quantity   Here, (a two−loop µ ) µνSSM and (a two−loop µ ) MSSM respectively denote two-loop contributions of muon MDM of the µνSSM and those of the MSSM, which can be given in section 4. In figure 4, we show that R m varies with υ ν c and tan β. In figure 4(a), we can see that the ratio R m can reach about 27%, when υ ν c is around 3 TeV. When the parameter υ ν c is large, the maximum of the ratio R m is around 20%. In figure 4(b), we can know that when tan β is small, the ratio R m can be more large. Here, compared to the MSSM, the µνSSM has extra right-handed neutrinos which can give new contributions to the muon MDM. Simultaneously, the right-handed neutrino superfields lead to the mixing of right-handed neutrinos with the neutralinos.

The decay h → Zγ
In this subsection, we present the numerical results of the signal strength for h → Zγ. We plot figure 5 and figure 6 through scanning the parameter space shown in table 2, where the green dots are the corresponding physical quantity's values of the remaining parameters after being constrained by the experimental constraints above. In figure 5(a)   comparing with the MSSM. Thus, the lightest Higgs boson in the µνSSM can easily account for the mass around 125 GeV, especially for small tan β.
In figure 5(b), we also picture the signal strength µ ggF γγ varying with tan β. We can see that the signal strength µ ggF γγ almost is around 1, which is consistent with the experimental value in the error range. Here, the relatively large stop mass and stau mass reduce the signal strength µ ggF γγ . In ref. [61], the signals of the Higgs boson decay channels h → γγ, h → V V * (V = Z, W ), and h → ff (f = b, τ ) in the µνSSM have been investigated. When the lightest stop mass mt 1 700 GeV and the lightest stau mass mτ 1 300 GeV, the signal strengths of these Higgs boson decay channels in the µνSSM are in agreement with those in the SM.

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Through figure 5(a) and figure 5(b), the numerical results show that the signal strength µ ggF Zγ in the µνSSM still has a large deviation from 1, even though the signal strength µ ggF γγ in the µνSSM is in keeping with that in the SM. We plot the signal strength µ ggF Zγ versus the parameter υ ν c in figure 6(a). The numerical results present that the signal strength µ ggF Zγ can have a large deviation from 1, when the value of the parameter υ ν c is small. The parameter υ ν c directly affects the mass of chargino. The small chargino mass gives a large contribution to the signal strength µ ggF Zγ . In addition, the parameter υ ν c leads to the mixing of the neutral components of the Higgs doublets with the sneutrinos. The mixing affects the lightest Higgs boson mass and the Higgs couplings, which is different from the MSSM.
To see more clearly, we also plot the ratio R Zγ/γγ ≡ Γ NP (h → Zγ)/Γ NP (h → γγ) versus the parameter υ ν c in figure 6(b). We can see that 0.55 R Zγ/γγ 0.71, when υ ν c is small. Here, small value of the parameter υ ν c can give more large contributions to the decay width Γ NP (h → Zγ) than Γ NP (h → γγ). Thus, the signal strength µ ggF Zγ in the µνSSM has a large deviation from 1, through small value of the parameter υ ν c which affects the mass of chargino and leads to the mixing of the neutral components of the Higgs doublets with the sneutrinos.

Summary
In the framework of the µνSSM, the three singlet right-handed neutrino superfieldsν c i are introduced to solve the µ problem of the MSSM and generate three tiny Majorana neutrino masses at the tree level through the seesaw mechanism. The gravitino or the axino in the µνSSM also can be a dark mater candidate. The right-handed sneutrino VEVs lead to the mixing of the neutral components of the Higgs doublets with the sneutrinos. Therefore, the mixing would affect the lightest Higgs boson mass and the Higgs couplings, which gives a rich phenomenology in the Higgs sector of the µνSSM, being different from the MSSM.
In this paper, we analyze the signal strength of the Higgs boson decay h → Zγ in the µνSSM. Even though the signal strength of h → γγ in the µνSSM is in accord with that in the SM, the signal strength of h → Zγ in the µνSSM still has a large deviation from 1, due to the small mass of chargino and the mixing of the neutral components of the Higgs doublets with the sneutrinos. The present observed 95% CL upper limit on the signal strength of the h → Zγ decay still is 6.6 [9]. However, high luminosity or high energy large collider [11][12][13] built in the future will detect the Higgs boson decay h → Zγ, which may see the indication of new physics.
Here, we also consider the two-loop corrections of the muon anomalous MDM in the µνSSM. Normalized to the one-loop corrections of the muon MDM, the two-loop corrections in the µνSSM can be around 16%. Compared to the MSSM, the µνSSM has extra right-handed neutrinos which can give new contributions to the muon anomalous MDM. Therefore, the two-loop corrections also make important contributions to the muon anomalous MDM in the µνSSM. In near future, the Muon g-2 experiment E989 at Fermilab [82, 83] will measure the muon anomalous magnetic dipole moment with unprecedented precision, which may reach a 5σ deviation from the SM, constituting an augury for new physics beyond the SM.

A Form factors
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