The naturalness in the BLMSSM and B-LSSM

In order to interpret the Higgs mass and its decays more naturally, we hope to intrude the BLMSSM and B-LSSM. In the both models, the right-handed neutrino superfields are introduced to better explain the neutrino mass problems. In addition, there are other superfields considered to make these models more natural than MSSM. In this paper, the method of $\chi^2$ analyses will be adopted in the BLMSSM and B-LSSM to calculate the Higgs mass, Higgs decays and muon $g-2$. With the fine-tuning in the region $0.67\%-2.5\%$ and $0.67\%-5\%$, we can obtain the reasonable theoretical values that are in accordance with the experimental results respectively in the BLMSSM and B-LSSM. Meanwhile, the best-fitted benchmark points in the BLMSSM and B-LSSM will be acquired at minimal $(\chi^{BL}_{min})^2 = 2.34736$ and $(\chi^{B-L}_{min})^2 = 2.47754$, respectively.


I. INTRODUCTION
The standard model (SM) has been confirmed by many experiments. Especially, the Large Hadron Collider (LHC) have announced a 125.10 GeV SM-like Higgs boson [1][2][3], whose discovery is a great triumph for the SM. Up to now, most of measurements are compatible with the SM predictions at 1 ∼ 2σ level. More than this, there are still some problems that can not be naturally explained by SM, such as the masses of neutrinos [4][5][6][7][8], the hierarchy problem [9], the dark matter(DM) candidates [10,11], flavor physics [12,13] and CP-violating problems [14].... Therefore, it is necessary to extend SM, and it happens that Minimal Supersymmetric SM (MSSM) is a highly motivated one [15][16][17][18].
However, there are still very strong restrictions on supersymmetric parameter space, which will be further explained by the following implications. As we know, the mass of the physical Higgs boson in the MSSM at tree level is less than the Z boson mass , and it can be lifted by the top quark-stop quark loop corrections [19][20][21][22][23]. So we need to acquire a rather large stop masses (around TeV region) to give such a large contribution. However, the Higgs soft mass square is deduced as m 2 Hu ≃ − 3y 2 t 4π 2 m 2 t ln Λ mt ∼ m 2 t (here Λ representing the corresponding new physics(NP) scale while mt corresponding to the scale of the stop mass), and the light stops are good to reproduce the correct scale for electroweak symmetry breaking [22][23][24][25][26]. Therefore, we need to introduce the fine-tuning to obtain relatively light stop mass, which can be easily accommodated by introducing an additional contribution to the Higgs boson mass. Actually, we hope to explain the above problem naturally by extending the MSSM(EMSSM). So far, physicists have proposed many feasible new physical models and in this paper we mainly study the BLMSSM [27][28][29][30][31][32] and B-LSSM [33][34][35][36][37].
The reason why we discuss the BLMSSM is that the baryon(B) and lepton(L) numbers are local gauge symmetries spontaneously broken at the TeV scale. Not only that, broken baryon number can naturally explain the origin of the matter-antimatter asymmetry in the universe. While broken lepton number can explain the neutrino oscillation experiment well by heavy majorana neutrinos contained in the seesaw mechanism inducing the tiny neutrino masses [27][28][29][30][31]. Additionally, there is a natural suppression of flavour violation in the quark and leptonic sectors since the gauge symmetries and particle content forbid tree level flavor changing neutral currents involving the quarks or charged leptons [27,28,30,31]. Other than this, the mass of the physical Higgs boson can be large without assuming a large stop mass [31].
Meanwhile, we also study the B-LSSM where gauge symmetry group SU(3) C ⊗ SU(2) L ⊗ U(1) Y ⊗ U(1) B−L is introduced with B representing baryon number and L standing for lepton number. Besides, the invariance under U(1) B−L gauge group imposes the R-parity conservation which is assumed in the MSSM to avoid proton decay [38]. In the B-LSSM, right-handed neutrinos can naturally be implemented due to the introduction of the righthanded neutrino superfields, which can realize type I seesaw mechanism, thus provide an elegant solution for the existence and smallness of the light left-handed neutrino masses. Furthermore, additional parameter space in the B-LSSM is released from the LEP, Tevatron and LHC constraints through the additional singlet Higgs state and right-handed (s)neutrinos to alleviate the hierarchy problem of the MSSM [39]. Other than this, the model can also provide much more DM candidates comparing that in the MSSM [40][41][42][43].
In this paper, we shall study the natural and realistic EMSSM including both the BLMSSM and B-LSSM by studying the Higgs masses, Higgs decays and muon anomalous magnetic dipole moment(MDM). We first introduce the naturalness conditions specifically in the EMSSM in section II. And the corresponding characteristics for BLMSSM and B-LSSM will be further illustrated in section III. Meanwhile, we derive the concrete theoretical expressions of Higgs decays and muon MDM in both BLMSSM and B-LSSM in section IV.
Considering the χ 2 analyses, the numerical results are discussed in section V to satisfy the phenomenological constraints and the relevant experimental data. Last but not least, we summarize the conclusion in section VI. In appendix A, B and C, we give out the corresponding form factors and couplings used in this paper.

II. NATURALNESS CRITERIA IN THE EMSSM
As mentioned in Refs [9,44], authors popularized a prescription to quantify fine-tuning by an atypical quantity M Z . That is measuring sensitivity in the Z boson mass to general parameters a i by here, a i control the masses of the various supersymmetric partners of the standard particles.
The reason for ∆ F T taking maximum is that supersymmetry is responsible for stabilizing the weak scale.
In general weak scale supersymmetric theories, the fine-tuning will be introduced more detail in the Higgs potential. In the MSSM, the SM Higgs-like particle h 0 is a linear combination of H u and H d . The Higgs potential for h 0 can be reduced as h 0 is negative and λ h 0 is positive. Minimizing the Higgs potential, we get v 2 ≡ h 0 2 = −2m 2 h 0 /λ h 0 . Then the physical Higgs boson mass can be deduced as m h 0 = −2m 2 h 0 . So the fine-tuning measure can also be defined as [22,45,46] In general, tan β ≥ 2, som 2 h 0 can be given asm 2 where µ is the supersymmetric mass between H u and H d . H 2 u | tree represent the tree-level contributions to the soft supersymmetry breaking mass square for H u , while H 2 u | rad represent radiative ones. Therefore, we obtain the following concrete bounds Thus, the value of µ should be smaller than 400 GeV for 5% fine-tuning. Consequently, the Higgsinos must be light due to the small µ. The dominant contributions to H 2 u | rad arise from stop loop where y t is top quark Yukawa coupling, m 2Q 3 m 2 U 3 and A t represent the corresponding soft parameters, which determine the stop mass mt. Supposing mQ 3 ≃ mt 1 and mŨ 3 ≃ mt 2 , we summarize the concrete bound for Mt ≡ m 2 wheret 1 andt 2 are stop mass eigenstates and satisfy m 2 t 1 + m 2 t 2 = A t /x t , Therefore, we obtain Mt < ∼ 1.2 TeV for 5% fine-tuning.
Above all, the natural EMSSM should possess relatively small (effective) µ term as well as stop masses. In this paper, we shall consider the following natural supersymmetry conditions: 1. The µ term or effective µ term is smaller than 400 √ ∆ F T 5% GeV.

III. THE BLMSSM AND B-LSSM
A. the BLMSSM Extending the local gauge group of the SM to SU In order to cancel the B and L anomalies, vector-like families are needed, which arê Correspondingly, Higgs superfieldsΦ B and ϕ B acquire nonzero vacuum expectation values (VEVs) to break baryon number spontaneously, as well asΦ L andφ L are introduced to break lepton number spontaneously. Other than this, in order to make exotic quarks unstable, the model also introduces superfieldsX andX ′ .X andX ′ mix together, and the lightest mass eigenstate can be a DM candidate.
The superpotential of the BLMSSM is given by [47] where W M SSM represents the MSSM superpotential. λ Q , λ U ..., Y u 4 , Y d 4 ... and µ B , µ L , µ X are the Yukawa couplings presented in the BLMSSM superpotential. The soft breaking terms in the BLMSSM are generally denoted by [47,48] where L M SSM In the BLMSSM, we mainly consider the effects from parameters M BL 0 , A BL 0 , m BL 12 , g BL LB , µ BL and tan β BL for our numerical calculation.
where i, j are generation indices, while Y x,ij and Y ν,ij are the Yukawa couplings in the B-LSSM superpotential. The soft breaking terms presented in the B-LSSM are written as where m 2 η 1 , m 2 η 2 , m 2 ν,ij ... are the concrete soft masses. In the B-LSSM, there are also other parameters .. and tan β, tan β ′ .... To facilitate numerical discussion, we adopt the following assumption: In the B-LSSM, we mainly consider the effects from parameters In the EMSSM, we consider the radiative corrections from exotic fermions and corresponding supersymmetric partners to obtain the physical Higgs mass. The corrections to Higgs masses in the BLMSSM were discussed specifically in Ref. [47], while the ones in the B-LSSM were introduced concretely in Refs. [37,49]. The corresponding parameter constraints in the BLMSSM and B-LSSM are considered respectively in this paper. Then the Higgs decays and (g − 2) µ will be taken over explicitly as follows.

A. the Higgs decays
The LHC produces the Higgs chiefly from the gluon fusion. Meanwhile, the leading order(LO) contributions for h 0 → gg originate from the one-loop diagrams, which can be modified through virtual top quark in the SM. In the EMSSM, the LO contributions need to be added by the Higgs-new particle couplings, whose effects are significant. So the decay width of h 0 → gg can be shown as [47,[50][51][52][53] with x a = m 2 h 0 /(4m 2 a ). q andq denote the concrete quarks and squarks in the EMSSM. The LO contributions for decay h 0 → γγ also originate from one-loop diagrams. In the SM, the concrete contributions are mainly derived from top quark and charged gauge boson W ± . Due to the Higgs-new particle couplings in the EMSSM, the decay width of h 0 → γγ can be expressed as [47,[50][51][52][53][54][55] The decay width for h 0 → ZZ, W W are given by [56,57] With the Born approximation, the decay width of the physical Higgs into fermion pairs h 0 → ff is written as [58] Γ where the form factors A 1/2 (x), A 0 (x), A 1 (x) and F (x) are summarized in the appendix A.
In the BLMSSM, the concrete expressions for g h 0 W W and g h 0 ZZ have been discussed in Ref. [47]. The relevant expressions that present in the B-LSSM are specifically discussed in the following appendix B.
The signal strengths for the Higgs decay channels are quantified by the following ratios [59] µ ggF γγ, where ggF and VBF stand for gluon-gluon fusion and vector boson fusion respectively.
Meanwhile, µ γγ,V V * are mainly affected by gluon-gluon fusion while µ ff is more likely to be influenced by vector boson fusion. The Higgs production cross sections can be further . Therefore, the ratios of the signal strengths from the Higgs decay channels are reduced as here, The effective Lagrangian for the muon MDM can be actually summarized as follows where σ αβ = i[γ α , γ β ]/2, F αβ is the electromagnetic field strength. Other than this, l µ denotes the muon fermion, m µ represents the corresponding muon mass and a µ is the muon MDM.
Generally, we obtain the muon MDM through the effective Lagrangian method [18,60,61] a where Q f = −1, C ± 2,6 represent the Wilson coefficients of the corresponding operators O ∓ with D µ = ∂ µ + ieA µ and ω ∓ = ( In the EMSSM, the muon MDM corresponding to FIG. 1(a) can be formulated as where y i denote y) are the one-loop functions and given out in appendix A. Similarly, the muon MDM for FIG.1(b) is deduced as follows √ y F y mµ B 1 (y S , y F ) In the BLMSSM, the concrete expressions for (A 1 ) I , (A 2 ) I , (C 1 ) I and (C 2 ) I can be found in Ref. [62]. The corresponding expressions that present in the B-LSSM will be specifically discussed in the following appendix C.
First of all, we analyze the numerical results in the BLMSSM. With ∆ F T in the region of The black triangle shows the best-fitted benchmark point with minimal (χ BL min ) 2 = 2.34736. The green, blue, and black regions are respectively 90%, 95%, and 99% confidence levels with χ 2 < (χ BL min ) 2 +10.65, (χ BL min ) 2 + 12.59 and (χ BL min ) 2 + 16.81. It is clear to see that g BL LB is changing from 0.1 to 0.9 while tan β BL is in the region 4 ∼ 40. Not only that, µ ggF γγ and µ ggF V V are both around 1.0 ∼ 1.3 and µ V BF f f is fixed in the range of 0.9 to 1.2, whose parameter spaces for χ 2 < (χ BL min ) 2 + 10.65 are obviously smaller than that for χ 2 < (χ BL min ) 2 + 12.59 and (χ BL min ) 2 + 16.81. ∆a µ can be limited to 1.0 × 10 −9 ∼ 3.0 × 10 −9 with the fine-tuning in the region 0.67% − 2.5%. So µ ggF γγ , µ ggF V V , µ ggF V V and ∆a µ which agree well with the concrete experimental results can naturally be explained in the BLMSSM.
Other than this, the B-LSSM numerical analyses are also taken over. We present the

VI. CONCLUSION
In this paper, we adopt the method of χ 2 analyses in the BLMSSM and B-LSSM to calculate the Higgs mass, Higgs decays and muon g − 2, which will be better than MSSM.
After scanning the parameter space, we point out some sensitive parameters in the BLMSSM and B-LSSM. In the BLMSSM, g BL LB is changing from 0.1 to 0.9 while tan β BL is limited in 4 ∼ 40 as the fine-tuning in the region 0.67% − 2.5%. As well as, we observe that 0.2TeV < µ B−L < 1.0TeV, 10 < ∼ tan β B−L < ∼ 40 and −0.5 < ∼ g B−L Y B < 0 in the B-LSSM with fine-tuning fluctuating between 0.67% and 5%. With the constraints µ < ∼ 400 √ ∆ F T 5% GeV, The form factors are defined as The concrete expressions that present in the B-LSSM are specifically discussed in the following(in this part i = 1): 1. CP-even Higgs-charge Higgs-charge Higgs contribution 2. CP-even Higgs-slepton-slepton contribution 3. CP-even Higgs-up squark-up squark contribution 4. CP-even Higgs-down squark-down squark contribution concrete expressions for (A 1 ) I , (A 2 ) I , (C 1 ) I and (C 2 ) I that present in the B-LSSM can be specifically discussed as Y * e,ab U e R,ja Z E kb N i3 ; (C1) U e, * R,ia Y e,ab ; U e, * R,ia Y e,ab ;