Higgs boson decays with lepton flavor violation in the $B-L$ symmetric SSM

Recently, the ATLAS and CMS Collaborations reported the latest experimental upper limits on the branching ratios of the lepton flavor violating (LFV) 125 GeV Higgs boson decays, $h\rightarrow e\mu$, $h\rightarrow e\tau$, and $h\rightarrow \mu\tau$. In this paper, we mainly investigate the LFV Higgs boson decays $h\rightarrow e\mu$, $h\rightarrow e\tau$, and $h\rightarrow \mu\tau$ in the minimal supersymmetric extension of the Standard Model with local $\emph{B-L}$ gauge symmetry. At the same time, the corresponding constraints from the LFV rare decays $\mu\rightarrow e\gamma$, $\tau\rightarrow e\gamma$, $\tau\rightarrow \mu\gamma$, and muon $(g-2)$ are considered to analyze the numerical results.

It is interpreted as a signal, eµ means the final state consisting ofēµ and eμ, eτ means the final state consisting ofēτ and eτ , and µτ means the final state consisting ofμτ and µτ .
In our previous work, we studied the lepton flavor violating decays l − j → l − i γ and l − j → l − i l − i l + i in the B-LSSM [18]. The numerical results show that the present experimental limits for the branching ratio of l − j → l − i γ constrain the parameter space of the B-LSSM most strictly. In this work, considering the constraint of the present experimental limits on the branching ratio of l − j → l − i γ, we give the influence of slepton flavor mixing parameters for the branching ratio of h → l i l j and l − j → l − i γ in the B-LSSM. The upper limits on the LFV branching ratio of µ → eγ, τ → eγ, and τ → µγ at 90% C.L. are now given by [154][155][156] Br(µ → eγ) < 4.2 × 10 −13 , Br(τ → eγ) < 3.3 × 10 −8 , Br(τ → µγ) < 4.4 × 10 −8 .
The paper is organized as follows. In Sec. II, we mainly introduce the B-LSSM including its superpotential and the general soft breaking terms. In Sec. III, we give an analytic expression for the branching ratio of the 125 GeV Higgs boson decays with LFV in the B-LSSM. In Sec. IV, we give the numerical analysis, and the summary is given in Sec. V.
Finally, some tedious formulas are collected in the Appendices.

II. THE B-LSSM
Extended the superfields of the MSSM, the B-LSSM [135][136][137][138][139][140] adds two singlet Higgs fieldsη 1 andη 2 and three generations of right-handed neutrinosν c i . The quantum numbers of the gauge symmetry group for the chiral superfields in the B-LSSM can be seen in Table I. Then, the superpotential in the model can be given by whereĤ  Correspondingly, the soft breaking terms of the B-LSSM are generally given as Here the soft breaking terms mainly contain the mass square terms of squarks, sleptons, sneutrinos, and Higgs bosons, the trilinear scalar coupling terms, and the Majorana mass terms. λ B and λ ′ B denote the gauginos of U(1) Y and U(1) B−L , respectively. The SU(2) L ⊗ U(1) Y ⊗ U(1) B−L gauge groups break to SU(3) c ⊗ U(1) em as the Higgs fields receive vacuum expectation values (VEVs), Here, we define u 2 = u 2 1 + u 2 2 , v 2 = v 2 1 + v 2 2 , and tan β ′ = u 2 u 1 in analogy to the ratio of the MSSM VEVs (tan β = v 2 v 1 ). The presence of two Abelian gauge groups gives rise to a new effect absent in any model with only one Abelian gauge group: the gauge kinetic mixing. This mixing couples the B-L sector to the MSSM sector, and even if it is set to zero at the grand unification theory scale, it can be induced via renormalization group equations [157][158][159][160][161][162][163]. In practice, it turns out that it is easier to work with noncanonical covariant derivatives instead of off-diagonal fieldstrength tensors. However, both approaches are equivalent [164]. Hence, in the following, we consider covariant derivatives of the form where As long as the two Abelian gauge groups are unbroken, we have the freedom to perform a change of basis by suitable rotation, and R is the proper way to do it.
Here g 1 corresponds to the measured hypercharge coupling, which is modified in the B-LSSM and given together with g B and g Y B in Ref. [165]. Next, we can redefine the U(1) gauge fields.
Since we will be dealing with the decay of Higgs bosons later, we want to give the mass square matrix of scalar Higgs bosons by gauge kinetic mixing as follows: where g 2 = g 2 1 + g 2 2 + g 2 Y B , T = 1 + tan 2 β 1 + tan 2 β ′ , n 2 = ReBµ u 2 and N 2 = ReBµ ′ u 2 , respectively. The mass matrices in the B-LSSM can be obtained with the help of SARAH [166][167][168][169][170].
The mass of the SM-like Higgs boson in the B-LSSM can be written as where m 0 h 1 is the lightest tree-level Higgs boson mass. ∆m 2 h is the radiative correction for which an approximate expression including two-loop leading-log radiative corrections can be given as [171][172][173] where 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 = T u,33 being the trilinear Higgs stop coupling and µ denoting the Higgsino mass parameter.
The Higgs boson's existence was confirmed by the ATLAS and CMS Collaborations [174,175] at CERN in 2012 based on colliding experiments at the LHC. The measured mass of the SM-like Higgs boson now is [176] m h = 125.10 ± 0.14 GeV.
In the subsequent numerical analysis, we use Eqs. (15) and (16) to calculate the mass of the SM-like Higgs boson and select the relevant parameters that satisfy the constraint of the SM-like Higgs boson measured mass in 3σ. and The formulas for neutral fermion loop contributions F (a,b)ij L,R are as follows: where x i = m 2 i /m 2 W , G i is loop function and C is coupling constant which can be found in the Appendices. The formulas for the self-energy diagrams contribution F (B)ij L,R are as follows: Σ is loop contribution of self-energy diagrams, where B 0,1 (p 2 , m 2 0 , m 2 1 ) is the two-point function [177][178][179][180][181][182][183]. Then, we can obtain the decay width of h →l i l j [5,10], the decay width of h → l i l j is and the branching ratio of h → l i l j is Here

IV. NUMERICAL ANALYSIS
First of all, in order to obtain reasonable numerical results, we need to study some sensitive parameters and important mass matrices. Then, to show the numerical results clearly, we will discuss the processes of h → eµ, h → eτ and h → µτ in three subsections.
We have to be clear that we are looking at 125 GeV Higgs boson decays with lepton flavor violation in the B-LSSM, so we need to consider the constraint of the SM-like Higgs boson mass. The remaining key parameters that affect the Higgs boson mass are tan β, tan β ′ , g B , and g Y B . By constantly adjusting the parameters, the final numerical analysis strictly conforms to the constraint of the SM-like Higgs boson measured mass in 3σ. Finally, we know that the B-LSSM contains all the ingredients to induce nonzero neutrino mass, but since the neutrino mass is very small, it makes little contribution to the problem we study. So, here we choose that neutrino masses are zero in the numerical analysis. The other one is about the neutrino Yukawa sector. Although the B-LSSM contains LFV sources in the neutrino Yukawa sector, such as the Y ν matrix, the neutrino oscillation causes Y ν ∼ O(10 −6 ), which contributes very little to the problem we study, so we approximately ignore the influence of the neutrino Yukawa sector in the numerical analysis.
After studying lepton flavor violating processes, we consider the off-diagonal terms for the soft breaking slepton mass matrices m 2 L,ē and the trilinear coupling matrix T e , which are defined by [188][189][190][191][192][193] Here, the definition of the slepton flavor mixing is more detailed than that of in our previous paper [18] which studied the lepton flavor violating decays l − j → l − i γ and l − j → l − i l − i l + i in the B-LSSM. In our previous paper [18], we just considered the off-diagonal terms for the trilinear coupling matrix T e . In the subsequent numerical analysis, we will show that the branching ratios of h → eµ, h → eτ , and h → µτ in the B-LSSM depend on the slepton mixing parameters δ XX 12 , δ XX 13 , and δ XX 23 (X = L, R), respectively, constrained by the present experimental limits on the branching ratio of l − j → l − i γ. We also impose a constraint on the NP contribution to the muon anomalous magnetic dipole moment (MDM), a µ = (g −2) µ , in the B-LSSM [18,194]. The new experimental average for the difference between the experimental measurement and SM theoretical prediction of a µ now is given by [195] ∆a µ = a exp µ − a SM µ = (25.1 ± 5.9) × 10 −10 .
Therefore, the NP contribution to the muon anomalous MDM in the B-LSSM, ∆a N P µ , should be constrained as 13.3 × 10 −10 ≤ ∆a N P µ ≤ 36.9 × 10 −10 , where we consider 2σ experimental error.
We all know that LFV processes are flavor dependent, just as LFV rate for e → µ transitions depends on the slepton mixing parameters δ XX 12 (X = L, R), which can be confirmed by Fig. 2. In the following, we choose tan β = 11, tan β ′ = 1. In Fig. 2, we can clearly see that when the slepton mixing parameters close δ XX 12 (X=L, R)= 0, Br(h → eµ) can be approximately O(10 −16 ), which is too small to be detected experimentally. However, with the slepton mixing parameters, δ XX 12 (X = L, R) increases, both Br(h → eµ) and Br(µ → eγ) will grow rapidly, and Br(µ → eγ) will soon exceed the experimental limit. Although Br(h → eµ) does not exceed the experimental limit, it will also be very close to the experimental limit. Especially in Fig. 2(e), Br(h → eµ) will reach the experimental upper limit with the increase of δ LR 12 . As we can see from Br(µ → eγ) versus slepton mixing parameters δ LL 12 (b), δ RR 12 (d), and δ LR 12 (f), where the dashed line denotes the present limit of Br(µ → eγ) at 90% C.L. as shown in Eq.(4). Here, the red solid line is ruled out by the present limit of Br(µ → eγ), and the black solid line is consistent with the present limit of Br(µ → eγ).
on the slepton mixing parameters δ XX 12 (X = L, R). When we consider the constraint of Br(µ → eγ) to Br(h → eµ), Br(h → eµ) can be up to O(10 −12 ). Therefore, it can be seen that the limit of Br(µ → eγ) to Br(h → eµ) is very strict, which makes Br(h → eµ) difficult to reach the experimental upper limit.
The dashed line in Fig. 3  In order to clearly see the constraints of Br(µ → eγ) and ∆a µ on Br(h → eµ), we scan the parameter space shown in Table II   the gray area denotes the ∆a µ at 2σ given in Eq. (31). Here, the red triangles are excluded by the present limit of Br(µ → eγ), the green squares are eliminated by ∆a µ at 2σ, and the black circles simultaneously conform to the present limit of Br(µ → eγ) and the ∆a µ at 2σ.
In Fig. 4, the dashed line represents the upper limit of the experiment, the red triangles are excluded by the present limit of Br(µ → eγ), the green squares are eliminated by the ∆a µ at 2σ, and the black circles simultaneously conform to the present limit of Br(µ → eγ) and the ∆a µ at 2σ. It can be intuitively seen that Br(µ → eγ) and ∆a µ have a strict limitation on Br(h → eµ). Under the constraints of Br(µ → eγ) and ∆a µ , Br(h → eµ) can reach O(10 −11 ). That means that Br(h → eµ) now is very hard to get to the upper limit of the experiment. singlet Higgs fields and three generations of right-handed neutrinos, which give new sources for lepton flavor violation and make a contribution to search for NP.
In order to see the influence of other basic parameters on the numerical results, we again set appropriate numerical values for slepton flavor mixing parameters such as δ LL 13 = 0.6, δ RR 13 = 0.4, and δ LR 13 = 0.8. Then, we look at the influence of basic parameters M E , tan β, tan β ′ , and g Y B on Br(h → eτ ) and Br(τ → eγ), respectively, which can be intuitively seen at 90% C.L. as shown in Eq. (5). ∆a N P µ versus slepton flavor mixing parameters δ XX 13 (c), where the gray area denotes the ∆a µ at 2σ given in Eq. (31). Here, the red triangles are excluded by the present limit of Br(τ → eγ), the green squares are eliminated by the ∆a µ at 2σ, and the black circles simultaneously conform to the present limit of Br(τ → eγ) and the ∆a µ at 2σ.
in Fig. 6. Br(h → eτ ) and Br(τ → eγ) decrease with the increase of M E or g Y B , and the branching ratios for these processes h → eτ and τ → eγ increase with the increase of tan β or tan β ′ . M E , tan β, tan β ′ , and g Y B affect the numerical results mainly through the new mass matrix of sleptons, Higgs bosons, and neutralino. Since the two decay processes contain different coupling vertices, the basic parameters M E , tan β, tan β ′ , and g Y B have different effects on the two processes.
By scanning the parameter space shown in Table II with δ LL 13 = δ RR 13 = δ LR 13 = δ XX 13 , we can clearly see the constraints of Br(τ → eγ) and ∆a µ on Br(h → eτ ) from Fig. 7. Although there are many points close to the experimental upper limit in Fig. 7(a), only the black circles satisfy the constraints of Br(τ → eγ) and ∆a µ . By looking at the black circles in Fig. 7(a), Br(h → eτ ) can approach O(10 −5 ), which is about only 2 orders of magnitude away from the experimental upper limit. Perhaps in the near future, the accuracy of the experimental upper limit will be further improved, and the 125 GeV Higgs boson decays with LFV may be detected.

C. 125 GeV Higgs boson decays with lepton flavor violation h → µτ
In the last, we analyze the process 125 GeV Higgs boson decays with LFV h → µτ in the B-LSSM. We still consider the influence of slepton flavor mixing parameters δ XX 23 (X = L, R) on Br(h → µτ ) and Br(τ → µγ) first, and then we consider the restriction of the experimental upper limit of rare process Br(τ → µγ) on process Br(h → µτ ).
In Fig. 8, we plot the influence of slepton flavor mixing parameters δ XX 23 (X = L, R) on Br(h → µτ ) and Br(τ → µγ), where the dashed line still represents the experimental upper limit of Br(h → µτ ) and Br(τ → µγ). The red solid line is ruled out by the present limit of Br(τ → µγ), and the black solid line is consistent with the present limit of Br(τ → µγ). (c), where the gray area denotes the ∆a µ at 2σ given in Eq. (31). Here, the red triangles are excluded by the present limit of Br(τ → µγ), the green squares are eliminated by the ∆a µ at 2σ, and the black circles simultaneously conform to the present limit of Br(τ → µγ) and the ∆a µ at 2σ.
Br(h → µτ ) in the B-LSSM can reach O(10 −4 ), which can be easily seen in Figs. 9(a)-9(g). By scanning the parameter space shown in Table II, we also get the related images of The couplings of charged scalars, neutral fermions and charged fermions are written as Y * e,ab Z E, * n(3+a) U e L,ib N η3 + √ 2 3 a=1 Z E, * na U e L,ia (g 1 N η1 + g 2 N η2 Y * e,ab U e R,ja Z E mb N η3 . (B5) The couplings of neutral scalars and neutral fermions can be written by The couplings of neutral scalars and charged fermions are written as Y * e,ab U e R,ia U e L,ib Z H k1 .
The matrices Z, N, U, and V above can be found in the version of the B-LSSM that is encoded in SARAH [166][167][168][169][170].