125 GeV Higgs decay with lepton flavor violation in the $\mu\nu$SSM

Recently, the CMS and ATLAS Collaborations have reported direct searches for the 125 GeV Higgs decay with lepton flavor violation, $h\rightarrow \mu \tau$. In this work, we analyze the signal of the lepton flavour violating (LFV) Higgs decay $h\rightarrow \mu \tau$ in the $\mu$ from $\nu$ Supersymmetric Standard Model ($\mu\nu$SSM) with slepton flavor mixing. Simultaneously, we consider the constraints from the LFV decay $\tau \rightarrow \mu \gamma$, the muon anomalous magnetic dipole moment and the lightest Higgs mass around 125 GeV.


I. INTRODUCTION
The discovery of the Higgs boson by the ATLAS and CMS Collaborations [1,2] is a great success of the Large Hadron Collider (LHC). Combining the updated data of the ATLAS and CMS Collaborations, the measured mass of the Higgs boson now is [3] m h = 125.09 ± 0.24 GeV. (1) The next step is focusing on searching for its properties. In the Standard Model (SM), which is renormalizable, lepton flavour violating (LFV) Higgs decays are forbidden [4]. But recently, a direct search for the 125 GeV Higgs decay with lepton flavor violation, h → µτ , has been described by the CMS Collaboration [5,6]. The upper limit on the branching ratio of h → µτ at 95% confidence level (CL) is [6] Br(h → µτ ) < 1.20 × 10 −2 .
Here, interpreted as a signal, µτ means the final state consisting ofμτ and µτ .
The ATLAS and CMS experiments do not currently show a significant deviation from the SM. Therefore, the experiments still need to make more precise measurements in the future.
Within the µνSSM, we have studied some LFV processes, l − j → l − i γ, l − j → l − i l − i l + i , muon conversion to electrons in nuclei and Z → l ± i l ∓ j in our previous work [120][121][122]. The numerical results show that the LFV rates for l j − l i transitions in the µνSSM depend on the slepton flavor mixing, and the present experimental limits for the branching ratio of l − j → l − i γ constrain the slepton mixing parameters most strictly [122]. In this work, considering the constraint of τ → µγ, we continue to analyze the LFV Higgs decay h → µτ in the µνSSM with slepton flavor mixing.
The paper is organized as follows. In Section 2, we briefly present the µνSSM, including its superpotential and the general soft SUSY-breaking terms. Section 3 contains the analytical expressions of the 125 GeV Higgs decay with lepton flavor violation in the µνSSM. The numerical analysis and the summary are given in Section 4 and Section 5, respectively. Some formulae are collected in Appendix A and Appendix B.

II. THE µνSSM
In addition to the superfields of the MSSM, the µνSSM introduces right-handed neutrino superfieldsν c i (i = 1, 2, 3). Besides 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- , andê c i denote the singlet up-type quark, down-type quark and charged lepton superfields, respectively. Here, Y , λ, and κ are dimensionless matrices, a vector, and a totally symmetric tensor. i, j, k = 1, 2, 3 are the generation indices, a, b = 1, 2 are the SU(2) indices with antisymmetric tensor ǫ 12 = 1. In the superpotential, the last two terms explicitly violate lepton number and R-parity. Note that the summation convention is implied on repeated indices in this paper.
where in the unrotated basis The concrete expressions for the independent coefficients of mass matrices M 2 S , M 2 P , M 2 S ± , M n and M c can be found in Ref. [121]. Using 8 ×8 unitary matrices R S , R P and R S ± , the unrotated basises S ′ , P ′ and S ± ′ can be respectively rotated to the mass eigenvectors S, P and S ± : Through the unitary matrices Z n , Z − and Z + , neutral and charged fermions can also be rotated to the mass eigenvectors χ 0 and χ, respectively.

III. 125 GEV HIGGS DECAY WITH LEPTON FLAVOR VIOLATION
The corresponding effective amplitude for 125 GeV Higgs decay with lepton flavor violation h →l i l j can be written as with where F (V )ij L,R denotes the contributions from the vertex diagrams in Fig. 1, and F (S)ij L,R stands for the contributions from the self-energy diagrams in Fig. 2, respectively.
The one-loop vertex diagrams for h →l i l j in the µνSSM are depicted by Fig. 1. Then, we can have where F and charged fermion χ β,ζ loops, respectively. After integrating the heavy freedoms out, we (N = S, P ) and charged fermion χ β,ζ loops. formulate the neutral fermion loop contributions F (a,b)ij L,R as follows: Here, the concrete expressions for couplings C (and below) can be found in Appendix A and Ref. [123], x = m 2 /m 2 W , m is the mass for the corresponding particle, and the loop functions G i are given as In a similar way, the charged fermion loop contributions F In Fig. 2, we show the self-energy diagrams contributing to h →l i l j in the µνSSM. The contributions from the self-energy diagrams F (S)ij L,R can be given as with The Σ of the self-energy diagrams in Fig. 2(c,d) can be obtained below Here, B 0,1 (p 2 , m 2 0 , m 2 1 ) are two-point functions [124][125][126][127][128][129][130].
Then, we can obtain the decay width of h →l i l j [9,14] Γ If interpreted as a signal, the decay width of h → l i l j is and the branching ratio of h → l i l j is where Γ h ≃ 4.1 × 10 −3 GeV [131] denotes the total decay width of the 125 GeV Higgs boson.

IV. NUMERICAL ANALYSIS
In order to obtain transparent numerical results in the µνSSM, we take the minimal 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. [121]. To agree with experimental observations on quark mixing, one can have and V = V u L V d † L denotes the CKM matrix. For the trilinear coupling matrix (A e Y e ) and soft breaking slepton mass matrices m 2 L,ẽ c , we will take into account the off-diagonal terms for the matrices, which are named the slepton flavor mixings and are defined by [132][133][134][135][136][137] The following numerical results will show that the branching ratio of h → µτ depends on the slepton mixing parameters δ XX 23 (X = L, R). At first, the constraints from some experiments should be considered. Through our previous work [119], 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) via the seesaw mechanism. Here, due to the neutrino sector only weakly affecting h → µτ , we can take no account of the constraints from neutrino experiment data.
The neutral Higgs with mass around 125 GeV reported by ATLAS and CMS contributes a strict constraint on the relevant parameters of the µνSSM. For a large mass of the pseudoscalar M A and moderate tan β, the SM-like Higgs mass of the µνSSM is approximately written as [107,138] Compared with the MSSM, the µνSSM gets an additional term, Thus, the SM-like Higgs in the µνSSM can easily account for the mass around 125 GeV, especially for small tan β. Including two-loop leading-log effects, the main radiative corrections △m 2 h can be given as [139][140][141] where υ = 174 GeV, α 3 is the strong coupling constant, M S = √ mt 1 mt 2 with mt 1,2 denoting the stop masses,Ã t = A t − µ cot β with A t = A u 3 being the trilinear Higgs-stop coupling and µ = 3λυ ν c denoting the Higgsino mass parameter.
We also impose a constraint on the SUSY contribution to the muon magnetic dipole moment a µ in the µνSSM, which is given in Appendix B for convenience. The difference between experiment and the SM prediction on a µ is [142][143][144] ∆a µ = a exp µ − a SM µ = (24.8 ± 7.9) × 10 −10 , with all errors combining in quadrature. Therefore, the SUSY contribution to a µ in the  In the scan, we keep the chargino masses m χ β > 200 GeV (β = 1, 2), the neutral fermion masses m χ 0 η > 200 GeV (η = 1, · · · , 7), and the scalar masses m Sα,Pα,S ± α > 500 GeV (η = 2, · · · , 8), to avoid the range ruled out by the experiments [142]. The results are also constrained by the muon anomalous magnetic dipole moment 1.1 ×10 −10 ≤ ∆a µ ≤ 48.5 ×10 −10 , where a 3σ experimental error is considered. In Ref. [123], we have investigated 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. 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 are in agreement with the SM. Therefore, the scanning results in this paper coincide with the experimental data of these Higgs boson decay channels.
Note that, when the calculation program is scanning one of the slepton mixing parameters δ XX 23 (X = L, R), the other two slepton mixing parameters δ XX 23 (X = L, R) are set to zero. So, we can see the contribution of every slepton mixing parameter alone. Then in Fig. 3 Here, the red triangles are ruled out by the present limit of Br(τ → µγ), and the black circles are consistent with the present limit of Br(τ → µγ).
In Fig. 3, when slepton mixing parameters δ Br(h → µτ ) and Br(τ → µγ) grow quickly. With increasing δ XX 23 (X = L, R), Br(τ → µγ) can easily go beyond the present experimental limit of Br(τ → µγ), shown in the plot as the red triangles. Although Br(h → µτ ) cannot reach the present experimental upper limit of Br(h → µτ ), Br(h → µτ ) becomes larger and approaches the present experimental limit with increasing δ XX 23 (X = L, R). Especially in Fig. 3(a), considering nonzero slepton mixing parameters δ LR 23 , Br(h → µτ ) can achieve O(10 −4 ), which is below the present experimental limit by just two orders of magnitude. Compared to the MSSM, exotic singlet righthanded neutrino superfields in the µνSSM induce new sources for lepton-flavor violation, considering that the righthanded neutrino and sneutrinos can mix and couple with the other particles seen in Eq. (8) and Appendix A. In Fig. 3(a,c,e), the red triangles overlap with the black circles, because some parameters strongly affect Br(τ → µγ) but do not affect Br(h → µτ ).
We will research this further in the following.
To see how other parameters affect the results, we appropriately fix δ LR 23 = 0.02 and δ LL 23 = δ RR 23 = 0.2. Then, we scan the parameter space shown in Table II, where µ = M 2 = m L = m E ≡ M SUSY . In the scanning, we also keep the chargino masses m χ β > 200 GeV (β = 1, 2), the neutral fermion masses m χ 0 η > 200 GeV (η = 1, · · · , 7), and the scalar masses m Sα,Pα,S ± α > 500 GeV (η = 2, · · · , 8), to avoid the range ruled out by the experiments [142]. Then in Fig. 4, we plot Br(h → µτ ) respectively versus tan β (a) and M SUSY (b), where the dashed line stands for the upper limit on Br(h → µτ ) at 95% CL shown in Eq. (2). We show Br(τ → µγ) varying with tan β (c) and M SUSY (d) respectively, where the dashed line denotes the present limit of Br(τ → µγ) which can be seen in Eq. (31). We also picture the muon anomalous magnetic dipole moment ∆a µ versus tan β (e) and M SUSY (f) respectively, where the gray area denotes the ∆a µ at 3.0σ given in Eq. (30). Here, the red triangles are excluded by the present limit of Br(τ → µγ), the green squares are eliminated by the ∆a µ at 3.0σ, and the black circles conform to both the present limit of Br(τ → µγ) and the ∆a µ in Eq. (30). Here, the red triangles are excluded by the present limit of Br(τ → µγ), the green squares are eliminated by the ∆a µ at 3.0σ, and the black circles simultaneously conform to the present limit of Br(τ → µγ) and the ∆a µ at 3.0σ. at 3.0σ.
In Fig. 4(d,f), the numerical results show that Br(τ → µγ) and the muon anomalous magnetic dipole moment ∆a µ are decoupling with increasing M SUSY . For large M SUSY , it is hard to give large contribution to ∆a µ . So, the large M SUSY are easily excluded by the ∆a µ at 3.0σ given in Eq. (30), which can be seen in the graph as the green squares. For small M SUSY , there can be a large contribution to Br(τ → µγ). Therefore, the small M SUSY are easily ruled out by the present experimental limit of Br(τ → µγ), shown as the red triangles. In Fig. 4(b), Br(h → µτ ) is non-decoupling with increasing M SUSY , which is in agreement with the research in the MSSM [44,67]. Due to the introduction of slepton mixing parameters, the non-decoupling behaviour of Br(h → µτ ) tends to O((m h /M SU SY ) 0 ), which is somewhat different from the Appelquist-Carazzone decoupling theorem [150]. (As a side note, in Ref. [151], a non-decoupling behaviour in computation of the Higgs mass showed that it was linked to an ambiguity in the treatment of tan β, which is a renormalization scheme dependent parameter.) We can also see that the red triangles overlap with the black circles in Fig. 4(b), because the parameter tan β does not affect Br(h → µτ ) visibly in this parameter space. In Fig. 4(a,c, The couplings between CP-even neutral scalars and the other CP-even (or CP-odd) neutral scalars are formulated as with where the unitary matrices R S , R P (and Z n , Z − , Z + below) can be found in Ref. [121], and the small terms containing Y ν i ∼ O(10 −7 ) and υ ν i ∼ O(10 −4 GeV) are ignored.
The interaction Lagrangian between CP-even neutral scalars and neutral fermions is formulated as where and The interaction Lagrangian of neutral scalars and charged fermions can be written as where the coefficients are The interaction Lagrangian of charged scalars, charged fermions, and neutral fermions can be similarly written by 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 . Adopting the effective Lagrangian approach, the MDM of the muon can be written by [152][153][154] where ℜ(· · ·) denotes the operation to take the real part of the complex number, and C L,R . (B4) The effective coefficients C L,R(n) 2,6 denote the contributions from the neutralinos χ 0 η and the charged scalars S − α loops Here, the loop functions I i (x 1 , x 2 ) are given as I 4 (x 1 , x 2 ) = 1 96π 2 11 + 6 ln x 2