# Explaining the possible 95 GeV excesses in the B-L symmetric SSM

Jin-Lei Yang1,2,3111jlyang@hbu.edu.cn,Ming-Hui Guo1,2,Wen-Hui Zhang1,2,Hai-Bin
Zhang1,2,3222hbzhang@hbu.edu.cn,Tai-Fu<EMAIL_ADDRESS>Department of Physics, Hebei University, Baoding, 071002, China1
Key Laboratory of High-precision Computation and Application of Quantum Field
Theory of Hebei Province, Baoding, 071002, China2
Research Center for Computational Physics of Hebei Province, Baoding, 071002,
China3

###### Abstract

This study investigates the excesses observed in the diphoton and $b\bar{b}$
data around $95\;{\rm GeV}$ within the framework of the $B-L$ supersymmetric
model (B-LSSM). Comparing with the minimal supersymmetric standard model, the
B-LSSM incorporates two singlet chiral Higgs bosons which mix with the SM-like
Higgs boson due to the gauge kinetic mixing effect. The richer Higgs sector
indicates that designating the B-LSSM specific CP-even Higgs state as the
lightest Higgs boson has great potential to explain the excesses at around
$95\;{\rm GeV}$. Considering the two-loop effective potential corrections to
the squared Higgs mass matrix, it is found that the signal strengthes
$\mu(h_{95})_{\gamma\gamma}$, $\mu(h_{95})_{bb}$ can be described
simultaneously in the experimental $1\sigma$ interval. And the B-LSSM specific
parameters $\tan\beta^{\prime}$, $B_{\eta}$, $g_{YB}$, $M_{Z^{\prime}}$ affect
the theoretical predictions on the light Higgs boson masses and signal
strengthes $\mu(h_{95})_{\gamma\gamma}$, $\mu(h_{95})_{bb}$ significantly.

95 GeV excesses, B-LSSM, new Higgs states

##  I Introduction

The discovery of a $125\;{\rm GeV}$ Higgs boson at the Large Hadron Collider
(LHC) in 2012 CMS:2012qbp ; ATLAS:2012yve  is one of the most remarkable
achievements in theoretical physics, and its properties are consistent well
with the standard model (SM) predictions CMS:2022dwd ; ATLAS:2022vkf  which
indicates all of the fundamental particles predicted by the SM are observed.
Then identifying whether the detected Higgs boson is the only fundamental
scalar particle or part of a new physics (NP) theory with extended Higgs
sectors becomes the prime goal of the current LHC programme. Although there is
no new scalar was found at the LHC so far, several intriguing excesses in the
searches for light Higgs bosons below 125 GeV have been observed with
increasing precision in the measurements of Higgs couplings to fermions and
gauge bosons.

The results based on both the CMS Run 1 and the first year of CMS Run 2 data
for Higgs boson searches in the diphoton final state show a $2.8\sigma$ local
excess at the mass about $95\;{\rm GeV}$ CMS:2015ocq ; CMS:2018cyk , which is
compatible with the latest ATLAS result ATLAS:2023CA  based on the previously
reported result utilizing $80\;{\rm fb}^{-1}$ ATLAS:2018xad  in the diphoton
searches

$\displaystyle\mu(\Phi_{95})_{\gamma\gamma}=\frac{\sigma(gg\rightarrow\Phi^{\rm
NP}_{95}){\rm BR}(\Phi^{\rm
NP}_{95}\rightarrow\gamma\gamma)}{\sigma(gg\rightarrow h^{\rm SM}_{95}){\rm
BR}(h^{\rm SM}_{95}\rightarrow\gamma\gamma)}=0.18\pm 0.10.$ (1)

Utilizing the full Run 2 data set, CMS published the results for the Higgs
boson searches in the $\tau^{+}\tau^{-}$ channel which show a local
significance of $3.1\sigma$ at a mass value about $95\;{\rm GeV}$ CMS:2022goy

$\displaystyle\mu(\Phi_{95})_{\tau\tau}=\frac{\sigma(gg\rightarrow\Phi^{\rm
NP}_{95}){\rm BR}(\Phi^{\rm NP}_{95}\rightarrow\tau\tau)}{\sigma(gg\rightarrow
h^{\rm SM}_{95}){\rm BR}(h^{\rm SM}_{95}\rightarrow\tau\tau)}=1.2\pm 0.5.$ (2)

The analysis on the Higgs boson searches in the diphoton final state by CMS
further confirmed the excess at about $95\;{\rm GeV}$, and the result reads
CMS:2023yay

$\displaystyle\mu(\Phi_{95})_{\gamma\gamma}=0.33_{-0.12}^{+0.19}.$ (3)

In addition, the results measured at the Large Electron Positron (LEP) show a
local excess of $2.3\sigma$ at the invariant mass of $b\bar{b}$ around 98
GeV444The excess at LEP is broad due to the hadronic $b\bar{b}$ final state,
so this result is also compatible with a scalar resonance at 95 GeV,
consistent with the diphoton and ditau excesses. We also take the excess for
$b\bar{b}$ final state at about 95 GeV in the following analysis. LEP:2003ing
, which can be expressed in terms of a signal strength as

$\displaystyle\mu(\Phi_{95})_{b\bar{b}}=0.117\pm 0.057.$ (4)

Theoretically, there are numerous discussions on the excesses in NP models.
The analysis carried out in Refs. Moretti:2006sv ; Ellwanger:2010nf ;
Cao:2011pg ; AlbornozVasquez:2012foo ; Ellwanger:2012ke ; Boudjema:2012in ;
Schmidt-Hoberg:2012dba ; Badziak:2013bda ; Badziak:2013gla ; Barbieri:2013nka
; Fan:2013gjf ; Potter:2015wsa ; Ellwanger:2015uaz ; Cao:2016uwt ; Cao:2019ofo
; Cao:2023gkc ; Cao:2024axg  indicates that the diphoton rate may be several
times larger than its SM prediction for the same scalar mass in the next-to-
minimal supersymmetric standard model (NMSSM). In the Two-Higgs doublet model
(2HDM) with an additional real singlet (N2HDM), the possibilities of
explaining the observed excesses were studied in Refs. Biekotter:2019kde ;
Biekotter:2021ovi ; Biekotter:2021qbc ; Heinemeyer:2021msz ; Biekotter:2022jyr
; Biekotter:2022abc ; Biekotter:2023jld ; Azevedo:2023zkg ; Aguilar-
Saavedra:2023vpd . The authors of Ref. Sachdeva:2019hvk  explored the
viability of the radion mixed Higgs to be 125 GeV along with the presence of a
light radion which can account for the CMS diphoton excess well in the Higgs
radion mixing model. Considering the one-loop corrections to the neutral
scalar masses of the $\mu\nu$SSM, the authors of Refs. Biekotter:2017xmf ;
Biekotter:2019gtq  demonstrated how the $\mu\nu$SSM can simultaneously
accommodate two excesses measured at the LEP and LHC at the $1\sigma$ level.
Based on the analysis in Ref. Ashanujjaman:2023etj , extending the scalar
sector with a $SU(2)_{L}$ triplet (the hypercharge $Y=0$) can well provide the
origin of the $95\;{\rm GeV}$ excesses. Whether certain model realizations
could simultaneously explain the two excesses while being in agreement with
all other Higgs boson related limits and measurements was reviewed in Refs.
Azatov:2012bz ; Heinemeyer:2018wzl .

In this work, we investigate the diphoton and $b\bar{b}$ excesses in the $B-L$
supersymmetric model (B-LSSM) Khalil:2008ps ; Elsayed:2011de ; Elsayed:2012ec
; Abdallah:2016vcn ; Khalil:2015naa ; DelleRose:2017uas ; Yang:2020bmh ;
Yang:2021duj ; Abdelalim:2020xfk 555Eq. (2) shows obviously that the ditau
excess has large experimental uncertainties, combined with the analysis in
Ref. Iguro:2022dok  which indicates the CP-even dominated Higgs state has
difficulties in describing the ditau excess, we focus on the diphoton and
$b\bar{b}$ excesses in this work.. Compared with the minimal supersymmetric
standard model (MSSM), the gauge symmetry of the B-LSSM is extended by an
additional $U(1)_{B-L}$ gauge group, i.e. the full gauge structure is
$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{B-L}$. In addition to
the MSSM superfields, two singlet chiral Higgs and three right-handed
neutrinos with nonzero $U(1)_{B-L}$ charge are introduced in the B-LSSM. The
tiny neutrino masses can be acquired naturally through the so-called type I
seesaw mechanism when the $U(1)_{B-L}$ symmetry is broken spontaneously by the
two newly introduced singlet Higgs. The mixings between the SM-like Higgs and
the B-LSSM specific Higgs states are generated through the gauge kinetic
mixing effect which arises in the NP models with two Abelian groups. It
indicates that the rich Higgs sector and Higgs mixing effect provide great
potential to explain the $95\;{\rm GeV}$ excesses. We take one of the B-LSSM
specific CP-even Higgs states as the lightest Higgs to explore whether the
B-LSSM can account for the low-mass diphoton and $b\bar{b}$ excesses. It is
known that the loop corrections from top and stop quarks are crucial to the
SM-like Higgs boson mass in the supersymmetric models, the two-loop effective
potential corrections from top and stop quarks to the CP-even neutral Higgs
boson masses are considered in the calculations.

The paper is organized as follows. The scalar sector of the B-LSSM and the
two-loop effective potential corrections to the CP-even neutral Higgs mass
matrix are given in Sec. II. The numerical results are presented and analyzed
in Sec. III. Finally, a summary is made in Sec. IV. The tedious formulae are
collected in the appendix.

##  II The B-LSSM and Higgs boson mass with two-loop effective potential
corrections

The gauge group of B-LSSM is extended by an additional $U(1)_{B-L}$ local
gauge group, and a new $Z^{\prime}$ gauge boson is introduced correspondingly.
The chiral scalar superfields in the B-LSSM are

$\displaystyle H_{d}=\left(\begin{array}[]{c}H_{d}^{1}\\\
H_{d}^{2}\end{array}\right)\sim(1,2,-1/2,0),\;H_{u}=\left(\begin{array}[]{c}H_{u}^{1}\\\
H_{u}^{2}\end{array}\right)\sim(1,2,1/2,0),$ (9)
$\displaystyle\eta\sim(1,1,0,-1),\;\;\bar{\eta}\sim(1,1,0,1),$ (10)

where the charges in the brackets correspond to
$SU(3)_{C},\;SU(2)_{L},\;U(1)_{Y},\;U(1)_{B-L}$ respectively. The local gauge
symmetry $SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{B-L}$ breaks down to the
electromagnetic symmetry $U(1)_{\rm em}$ as the Higgs fields receive nonzero
vacuum expectation values (VEV):

$\displaystyle
H_{d}^{1}=\frac{1}{\sqrt{2}}(v_{d}+\phi_{d}+i\sigma_{d}),\qquad\;H_{u}^{2}=\frac{1}{\sqrt{2}}(v_{u}+\phi_{u}+i\sigma_{u}),$
$\displaystyle\eta=\frac{1}{\sqrt{2}}(u_{\eta}+\phi_{\eta}+i\sigma_{\eta}),\qquad\;\quad\;\bar{\eta}=\frac{1}{\sqrt{2}}(u_{\bar{\eta}}+\phi_{\bar{\eta}}+i\sigma_{\bar{\eta}})\;.$
(11)

For convenience, we can define $v^{2}=v_{d}^{2}+v_{u}^{2}$,
$u^{2}=u_{\eta}^{2}+u_{\bar{\eta}}^{2}$, $\tan\beta=v_{u}/v_{d}$,
$\tan\beta^{\prime}=u_{\bar{\eta}}/u_{\eta}$. The tree-level scalar potential
in the B-LSSM can be written as Yang:2018guw

$\displaystyle
V^{(0)}=(m_{H_{d}}^{2}+\mu^{2})H_{d}^{2}+(m_{H_{u}}^{2}+\mu^{2})H_{u}^{2}+(m_{\eta}^{2}+\mu_{\eta}^{2})\eta^{2}+(m_{\bar{\eta}}^{2}+\mu_{\eta}^{2})\bar{\eta}^{2}$
$\displaystyle\qquad\;-2B_{\mu}H_{d}H_{u}-2B_{\eta}\eta\bar{\eta}+\frac{1}{8}(g_{1}^{2}+g_{2}^{2}+g_{YB}^{2})(H_{d}^{2}-H_{u}^{2})^{2}+\frac{1}{2}g_{B}^{2}(\eta^{2}-\bar{\eta}^{2})^{2}$
$\displaystyle\qquad\;+\frac{1}{2}g_{B}g_{YB}(H_{d}^{2}-H_{u}^{2})(\eta^{2}-\bar{\eta}^{2}),$
(12)

where $g_{B}$ is the gauge coupling constant corresponding to the $U(1)_{B-L}$
gauge group, $g_{YB}$ is the coupling constant arising from the kinetic mixing
effect. For the one-loop and two-loop effective potential, the dominant
contributions come from top and stop quarks. In the B-LSSM, the top quark mass
reads

$\displaystyle m_{t}=\frac{1}{\sqrt{2}}Y_{t}v_{u}.$ (13)

The stop quark mass matrix can be written as

$\displaystyle
m_{\tilde{t}}^{2}=\left(\begin{array}[]{cc}m_{\tilde{t}L}^{2},&\frac{Y_{t}}{\sqrt{2}}(v_{u}A_{t}-v_{d}\mu)\\\
\frac{Y_{t}}{\sqrt{2}}(v_{u}A_{t}-v_{d}\mu),&m_{\tilde{t}R}^{2}\end{array}\right),$
(16)

where

$\displaystyle
m_{\tilde{t}L}^{2}=\frac{1}{24}\Big{[}(g_{1}^{2}-3g_{2}^{2}+g_{YB}^{2}+g_{B}g_{YB})(v_{u}^{2}-v_{d}^{2})+2g_{B}(g_{B}+g_{YB})(u_{\bar{\eta}}^{2}-u_{\eta}^{2})\Big{]}$
$\displaystyle\qquad\quad+m_{\tilde{q}}^{2}+\frac{1}{2}v_{u}^{2}Y_{t}^{2},$
$\displaystyle
m_{\tilde{t}R}^{2}=\frac{1}{24}\Big{[}(4g_{1}^{2}+4g_{YB}^{2}+g_{B}g_{YB})(v_{d}^{2}-v_{u}^{2})+2g_{B}(g_{B}+4g_{YB})(u_{\eta}^{2}-u_{\bar{\eta}}^{2})\Big{]}$
$\displaystyle\qquad\quad+m_{\tilde{u}}^{2}+\frac{1}{2}v_{u}^{2}Y_{t}^{2}.$
(17)

On the mass eigenstates, the physical stop masses can be written as

$\displaystyle
M_{\tilde{t}1,2}^{2}=\frac{1}{2}(m_{\tilde{t}L}^{2}+m_{\tilde{t}R}^{2})\pm\Big{[}\frac{1}{4}(m_{\tilde{t}L}^{2}-m_{\tilde{t}R}^{2})^{2}+\frac{1}{2}Y_{t}^{2}(v_{u}A_{t}-v_{d}\mu)^{2}\Big{]}^{1/2}.$
(18)

The effective potential is

$\displaystyle V_{\rm eff}=V^{(0)}+\Delta V=V^{(0)}+V^{(1)}+V^{(2)},$ (19)

with Zhang:1998bm ; Espinosa:1999zm ; Degrassi:2001yf

$\displaystyle
V^{(1)}=-\frac{3}{16\pi^{2}}(m_{t}^{2})^{2}(\log\frac{m_{t}^{2}}{Q^{2}}-\frac{3}{2})+\frac{3}{32\pi^{2}}\sum_{i=1}^{2}(M_{\tilde{t}_{i}}^{2})^{2}(\log\frac{M_{\tilde{t}_{i}}^{2}}{Q^{2}}-\frac{3}{2}),$
$\displaystyle
V^{(2)}=\frac{\alpha_{s}}{16\pi^{3}}\Big{\\{}2J(m_{t}^{2},m_{t}^{2})-4m_{t}^{2}I(m_{t}^{2},m_{t}^{2},0)+\Big{[}2M_{\tilde{t}_{1}}^{2}I(M_{\tilde{t}_{1}}^{2},M_{\tilde{t}_{1}}^{2},0)+2L(M_{\tilde{t}_{1}}^{2},M_{\tilde{g}}^{2},m_{t}^{2})$
$\displaystyle\qquad\quad-4m_{t}M_{\tilde{g}}S_{2\bar{\theta}}I(M_{\tilde{t}_{1}}^{2},M_{\tilde{g}}^{2},m_{t}^{2})+\frac{1}{2}(1+2C_{2\bar{\theta}}^{2})J(M_{\tilde{t}_{1}}^{2},M_{\tilde{t}_{1}}^{2})+\frac{S_{2\bar{\theta}}^{2}}{2}J(M_{\tilde{t}_{1}}^{2},M_{\tilde{t}_{2}}^{2})$
$\displaystyle\qquad\quad+(M_{\tilde{t}_{1}}\leftrightarrow
M_{\tilde{t}_{2}},S_{2\bar{\theta}}\leftrightarrow-
S_{2\bar{\theta}})\Big{]}\Big{\\}},$ (20)

where $V^{(1)}$, $V^{(2)}$ denote the one-loop and two-loop effective
potential respectively. In Eq. (20), $M_{\tilde{g}}$ is the mass of sgluon and

$\displaystyle
S_{2\bar{\theta}}=\frac{2(M_{\tilde{t}})_{12}}{M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2}},$
(21)

where $(M_{\tilde{t}})_{12}=Y_{t}(v_{u}A_{t}-v_{d}\mu)$. For the two-loop
functions $I,\;J$ in Eq. (20) can be found in the appendix D of Ref.
Degrassi:2009yq .

Combining Eq. (12) with Eq. (19), we can obtain the vacuum stability
conditions

$\displaystyle m_{H_{d}}^{2}+\mu^{2}=-\frac{\partial\Delta
V}{v_{d}\partial\phi_{d}}-\frac{1}{4}g_{B}g_{YB}(v_{\eta}^{2}-v_{\bar{\eta}}^{2})+B_{\mu}\tan\beta-\frac{1}{8}(g_{1}^{2}+g_{2}^{2}+g_{YB}^{2})(v_{d}^{2}-v_{u}^{2}),$
$\displaystyle m_{H_{u}}^{2}+\mu^{2}=-\frac{\partial\Delta
V}{v_{u}\partial\phi_{u}}+\frac{1}{4}g_{B}g_{YB}(v_{\eta}^{2}-v_{\bar{\eta}}^{2})+B_{\mu}/\tan\beta+\frac{1}{8}(g_{1}^{2}+g_{2}^{2}+g_{YB}^{2})(v_{d}^{2}-v_{u}^{2}),$
$\displaystyle m_{H_{\eta}}^{2}+\mu_{\eta}^{2}=-\frac{\partial\Delta
V}{v_{\eta}\partial\phi_{\eta}}-\frac{1}{4}g_{B}g_{YB}(v_{d}^{2}-v_{u}^{2})+B_{\eta}\tan\beta^{\prime}-\frac{1}{2}g_{B}^{2}(v_{\eta}^{2}-v_{\bar{\eta}}^{2}),$
$\displaystyle m_{H_{\bar{\eta}}}^{2}+\mu_{\eta}^{2}=-\frac{\partial\Delta
V}{v_{\bar{\eta}}\partial\phi_{\bar{\eta}}}+\frac{1}{4}g_{B}g_{YB}(v_{d}^{2}-v_{u}^{2})+B_{\eta}/\tan\beta^{\prime}+\frac{1}{2}g_{B}^{2}(v_{\eta}^{2}-v_{\bar{\eta}}^{2}).$
(22)

Then the elements of the CP-even Higgs squared mass matrix can be written as

$\displaystyle
m_{\phi_{d}\phi_{d}}^{2}=\frac{1}{4}(g_{1}^{2}+g_{2}^{2}+g_{YB}^{2})v_{d}^{2}+B_{\mu}v_{u}/v_{d}+\Big{[}-\frac{1}{\phi_{d}}\frac{\partial\Delta
V}{\partial\phi_{d}}+\frac{\partial^{2}\Delta
V}{\partial\phi_{d}\partial\phi_{d}}\Big{]}\Big{|}_{\rm VEV},$ $\displaystyle
m_{\phi_{d}\phi_{u}}^{2}=m_{\phi_{u}\phi_{d}}^{2}=-B_{\mu}-\frac{1}{4}(g_{1}^{2}+g_{2}^{2}+g_{YB}^{2})v_{d}v_{u}+\Big{[}\frac{\partial^{2}\Delta
V}{\partial\phi_{d}\partial\phi_{u}}\Big{]}\Big{|}_{\rm VEV},$ $\displaystyle
m_{\phi_{d}\phi_{\eta}}^{2}=m_{\phi_{\eta}\phi_{d}}^{2}=\frac{1}{2}g_{B}g_{YB}v_{d}v_{\eta}+\Big{[}\frac{\partial^{2}\Delta
V}{\partial\phi_{d}\partial\phi_{\eta}}\Big{]}\Big{|}_{\rm VEV},$
$\displaystyle
m_{\phi_{d}\phi_{\bar{\eta}}}^{2}=m_{\phi_{\bar{\eta}}\phi_{d}}^{2}=-\frac{1}{2}g_{B}g_{YB}v_{d}v_{\bar{\eta}}+\Big{[}\frac{\partial^{2}\Delta
V}{\partial\phi_{d}\partial\phi_{\bar{\eta}}}\Big{]}\Big{|}_{\rm VEV},$
$\displaystyle
m_{\phi_{u}\phi_{u}}^{2}=\frac{1}{4}(g_{1}^{2}+g_{2}^{2}+g_{YB}^{2})v_{u}^{2}+B_{\mu}v_{d}/v_{u}+\Big{[}-\frac{1}{\phi_{u}}\frac{\partial\Delta
V}{\partial\phi_{u}}+\frac{\partial^{2}\Delta
V}{\partial\phi_{u}\partial\phi_{u}}\Big{]}\Big{|}_{\rm VEV},$ $\displaystyle
m_{\phi_{u}\phi_{\eta}}^{2}=m_{\phi_{\eta}\phi_{u}}^{2}=-\frac{1}{2}g_{B}g_{YB}v_{u}v_{\eta}+\Big{[}\frac{\partial^{2}\Delta
V}{\partial\phi_{u}\partial\phi_{\eta}}\Big{]}\Big{|}_{\rm VEV},$
$\displaystyle
m_{\phi_{u}\phi_{\bar{\eta}}}^{2}=m_{\phi_{\bar{\eta}}\phi_{u}}^{2}=\frac{1}{2}g_{B}g_{YB}v_{u}v_{\bar{\eta}}+\Big{[}\frac{\partial^{2}\Delta
V}{\partial\phi_{u}\partial\phi_{\bar{\eta}}}\Big{]}\Big{|}_{\rm VEV},$
$\displaystyle
m_{\phi_{\eta}\phi_{\eta}}^{2}=g_{B}^{2}v_{\eta}^{2}+B_{\eta}v_{\bar{\eta}}/v_{\eta}+\Big{[}-\frac{1}{\phi_{\eta}}\frac{\partial\Delta
V}{\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial\phi_{\eta}\partial\phi_{\eta}}\Big{]}\Big{|}_{\rm VEV},$
$\displaystyle
m_{\phi_{\eta}\phi_{\bar{\eta}}}^{2}=m_{\phi_{\bar{\eta}}\phi_{\eta}}^{2}=-B_{\eta}-g_{B}^{2}v_{\eta}v_{\bar{\eta}}+\Big{[}\frac{\partial^{2}\Delta
V}{\partial\phi_{\eta}\partial\phi_{\bar{\eta}}}\Big{]}\Big{|}_{\rm VEV},$
$\displaystyle
m_{\phi_{\bar{\eta}}\phi_{\bar{\eta}}}^{2}=g_{B}^{2}v_{\bar{\eta}}^{2}+B_{\eta}v_{\eta}/v_{\bar{\eta}}+\Big{[}-\frac{1}{\phi_{\bar{\eta}}}\frac{\partial\Delta
V}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial\phi_{\bar{\eta}}\partial\phi_{\bar{\eta}}}\Big{]}\Big{|}_{\rm
VEV},$ (23)

where the detailed calculations of the contributions from $\Delta V$ in Eq.
(23) are collected in the appendix. The tree-level squared Higgs mass matrix
in the B-LSSM can be read easily in Eq. (23). And to illustrate clearly that
the rich Higgs sector in the B-LSSM provides the potential to explain the 95
GeV excesses, we calculate the eigenvalues of the tree-level squared Higgs
mass matrix under the approximation $v^{2}\ll u^{2}$, the analytical results
can be written as

$\displaystyle m_{h_{1},{\rm
tree}}^{2}\approx\frac{2B_{\eta}(\tan^{2}\beta^{\prime}-1)^{2}}{1+(\tan^{2}\beta^{\prime}+1)B_{\eta}/(g_{B}^{2}u^{2})},\quad
m_{h_{2},{\rm
tree}}^{2}\approx\frac{(g_{1}^{2}+g_{2}^{2}+g_{YB}^{2})(\tan^{2}\beta-1)^{2}v^{2}}{4(\tan^{2}\beta+1)^{2}},$
$\displaystyle m_{h_{3},{\rm
tree}}^{2}\approx\frac{B_{\mu}}{\tan\beta}(\tan^{2}\beta+1),\quad
m_{h_{4},{\rm tree}}^{2}\approx
2(\frac{g_{B}^{2}u^{2}}{\tan^{2}\beta+1}+B_{\eta}),$ (24)

where the terms proportional to $\mathcal{O}[(\tan^{2}\beta^{\prime}-1)^{3}]$
are neglected for simplicity, $m_{h_{i},{\rm tree}}<m_{h_{j},{\rm tree}}$ for
$i<j$, and $m_{h_{1},{\rm tree}}$, $m_{h_{4},{\rm tree}}$ are the tree-level
masses of the states dominated by the two specific scalar singlets in the
B-LSSM, $m_{h_{2},{\rm tree}}$, $m_{h_{3},{\rm tree}}$ are the tree-level
masses of the states dominated by the two scalar doublets which correspond to
the two scalar doublets in the MSSM. As shown in Eq. (24), $m_{h_{2},{\rm
tree}}$ corresponds to the mass of SM-like Higgs, and $m_{h_{3},{\rm
tree}}\approx m_{A}\gtrsim 380\;{\rm GeV}$ PDG  with $m_{A}$ denoting the CP-
odd Higgs mass in the MSSM, which also indicates the 95 GeV Higgs can not be
achieved in the MSSM. However, the expression of $m_{h_{1},{\rm tree}}$ in Eq.
(24) shows that the 95 GeV Higgs can be obtained in the B-LSSM when
$\tan\beta^{\prime}-1$ approaches to $0$ or $B_{\eta}$ is small. Combined with
the tree-level mixing with the SM-like Higgs as presented in Eq. (23), the
B-LSSM has great potential to explain the observed 95 GeV excesses, while the
excesses can not be explained in the frame work of MSSM Bechtle:2016kui .

On the basis $(\phi_{d},\phi_{u},\phi_{\eta},\phi_{\bar{\eta}})$, the squared
mass matrix of CP-even Higgs can be written as

$\displaystyle
M_{\phi}^{2}=\left(\begin{array}[]{*{20}{c}}m_{\phi_{d}\phi_{d}}^{2}&m_{\phi_{d}\phi_{u}}^{2}&m_{\phi_{d}\phi_{\eta}}^{2}&m_{\phi_{d}\phi_{\bar{\eta}}}^{2}\\\\[6.0pt]
m_{\phi_{d}\phi_{u}}^{2}&m_{\phi_{u}\phi_{u}}^{2}&m_{\phi_{u}\phi_{\eta}}^{2}&m_{\phi_{u}\phi_{\bar{\eta}}}^{2}\\\\[6.0pt]
m_{\phi_{d}\phi_{\eta}}^{2}&m_{\phi_{u}\phi_{\eta}}^{2}&m_{\phi_{\eta}\phi_{\eta}}^{2}&m_{\phi_{\eta}\phi_{\bar{\eta}}}^{2}\\\\[6.0pt]
m_{\phi_{d}\phi_{\bar{\eta}}}^{2}&m_{\phi_{u}\phi_{\bar{\eta}}}^{2}&m_{\phi_{\eta}\phi_{\bar{\eta}}}^{2}&m_{\phi_{\bar{\eta}}\phi_{\bar{\eta}}}^{2}\end{array}\right).$
(29)

In order to present the mass matrices of the Higgs fields in a physical way,
we rotate the fields $H_{d}$, $H_{u}$, $\eta$, $\bar{\eta}$ as

$\displaystyle H_{1}=\cos\beta H_{u}+\sin\beta H_{d},\;\;H_{2}=\cos\beta
H_{d}-\sin\beta H_{u},$ $\displaystyle
H_{3}=\cos\beta^{\prime}\bar{\eta}+\sin\beta^{\prime}\eta,\;\;H_{4}=\cos\beta^{\prime}\eta-\sin\beta^{\prime}\bar{\eta},$
(30)

where $H_{2}$ corresponds to the SM Higgs doublet Cao:2016uwt . Then on the
basis $(\phi_{1},\phi_{2},\phi_{3},\phi_{4})$, the squared mass matrix of CP-
even Higgs can be written as

$\displaystyle M_{h}^{2}=U\cdot M_{\phi}^{2}\cdot U^{\dagger},$ (31)

where

$\displaystyle
U=\left(\begin{array}[]{*{20}{c}}\sin\beta&\cos\beta&0&0\\\\[6.0pt]
\cos\beta&-\sin\beta&0&0\\\\[6.0pt]
0&0&\sin\beta^{\prime}&\cos\beta^{\prime}\\\\[6.0pt]
0&0&\cos\beta^{\prime}&-\sin\beta^{\prime}\end{array}\right),$ (36)
$\displaystyle\phi_{1}=\cos\beta\phi_{u}+\sin\beta\phi_{d},\;\;\phi_{2}=\cos\beta\phi_{d}-\sin\beta\phi_{u},$
$\displaystyle\phi_{3}=\cos\beta^{\prime}\phi_{\bar{\eta}}+\sin\beta^{\prime}\phi_{\eta},\;\;\phi_{4}=\cos\beta^{\prime}\phi_{\eta}-\sin\beta^{\prime}\phi_{\bar{\eta}}.$
(37)

The squared mass matrix $M_{h}^{2}$ can be diagonalized by the unitary matrix
$Z_{h}$ as

$\displaystyle{\rm
diag}(m_{h_{1}}^{2},m_{h_{2}}^{2},m_{h_{3}}^{2},m_{h_{4}}^{2})=Z_{h}\cdot
M_{h}^{2}\cdot Z_{h}^{\dagger},$ (38)

where $h_{i},(i=1,2,3,4)$ denote the mass eigenstates.

As defined in the introduction sector, the diphoton and $b\bar{b}$ signal
strength are

$\displaystyle\mu(h_{95})_{\gamma\gamma}=\frac{\sigma(gg\rightarrow h^{\rm
NP}_{95}){\rm BR}(h^{\rm
NP}_{95}\rightarrow\gamma\gamma)}{\sigma(gg\rightarrow h^{\rm SM}_{95}){\rm
BR}(h^{\rm SM}_{95}\rightarrow\gamma\gamma)},$
$\displaystyle\mu(h_{95})_{bb}=\frac{\sigma(Z^{*}\rightarrow Zh^{\rm
NP}_{95}){\rm BR}(h^{\rm NP}_{95}\rightarrow
b\bar{b})}{\sigma(Z^{*}\rightarrow Zh^{\rm SM}_{95}){\rm BR}(h^{\rm
SM}_{95}\rightarrow b\bar{b})},$ (39)

where Djouadi:1997yw

$\displaystyle\Gamma^{\rm SM}_{\rm tot}\approx 0.0026\;{\rm GeV},$
$\displaystyle{\rm BR}(h^{\rm SM}_{95}\rightarrow\gamma\gamma)\approx
1.4\times 10^{-3},\;\;{\rm BR}(h^{\rm SM}_{95}\rightarrow b\bar{b})\approx
0.81.$ (40)

Since top quark makes the dominant contributions to the Higgs production, the
cross section for $gg\to h^{NP}_{95}$ and $\sigma(Z^{*}\rightarrow Zh^{\rm
NP}_{95})$ can be estimated by Cao:2016uwt

$\displaystyle\sigma(gg\rightarrow h^{\rm NP}_{95})\approx
C_{h_{95}uu}^{2}\sigma(gg\rightarrow h^{\rm SM}_{95}),$
$\displaystyle\sigma(Z^{*}\rightarrow Zh^{\rm NP}_{95})\approx
C_{h_{95}VV}^{2}\sigma(Z^{*}\rightarrow Zh^{\rm SM}_{95}),$ (41)

where the coefficients $C_{h_{95}uu}$, $C_{h_{95}VV}$ are the normalized
couplings of $95\;{\rm GeV}$ Higgs in NP models with up-quarks and gauge
bosons respectively (in units of the corresponding SM couplings). The branch
ratios for $h^{NP}_{95}\to\gamma\gamma$ or $b\bar{b}$ can be estimated by
Cao:2016uwt

$\displaystyle{\rm BR}(h^{\rm NP}_{95}\rightarrow\gamma\gamma)\approx
C_{h_{95}uu}^{2}{\rm BR}(h^{\rm
SM}_{95}\rightarrow\gamma\gamma)\frac{\Gamma^{\rm SM}_{\rm tot}}{\Gamma^{\rm
NP}_{\rm tot}},$ $\displaystyle{\rm BR}(h^{\rm NP}_{95}\rightarrow
b\bar{b})\approx C_{h_{95}dd}^{2}{\rm BR}(h^{\rm SM}_{95}\rightarrow
b\bar{b})\frac{\Gamma^{\rm SM}_{\rm tot}}{\Gamma^{\rm NP}_{\rm tot}},$ (42)

where the coefficients $C_{h_{95}dd}$ is the normalized couplings of $95\;{\rm
GeV}$ Higgs in NP models with down-quarks, and Djouadi:1997yw

$\displaystyle\Gamma^{\rm NP}_{\rm tot}\approx C_{h_{95}dd}^{2}(\Gamma^{\rm
SM}_{b\bar{b}}+\Gamma^{\rm SM}_{\tau\bar{\tau}})+C_{h_{95}uu}^{2}(\Gamma^{\rm
SM}_{c\bar{c}}+\Gamma^{\rm SM}_{gg})+C_{h_{95}VV}^{2}\Gamma^{\rm
SM}_{WW^{*}},$ $\displaystyle{\rm BR}(h^{\rm
SM}_{95}\rightarrow\tau\tau)\approx 0.082,\;{\rm BR}(h^{\rm
SM}_{95}\rightarrow c\bar{c})\approx 0.037,{\rm BR}(h^{\rm SM}_{95}\rightarrow
gg)\approx 0.058,$ $\displaystyle{\rm BR}(h^{\rm SM}_{95}\rightarrow
WW^{*})\approx 0.012.$ (43)

In the B-LSSM, we have

$\displaystyle
C_{h_{95}uu}=C_{h_{1}uu}=Z_{h,11}\cot\beta+Z_{h,12},\;\;C_{h_{95}dd}=C_{h_{1}dd}=Z_{h,11}\tan\beta+Z_{h,12},$
$\displaystyle C_{h_{95}VV}=C_{h_{1}VV}=Z_{h,12}.$ (44)

##  III Numerical results

Taking the lightest Higgs mass at around $95\;{\rm GeV}$ and the next-to-
lightest Higgs mass at the measured SM-like Higgs mass PDG

$\displaystyle m_{h}=125.09\pm 0.24{\rm GeV},$ (45)

we present the numerical results of the two light Higgs masses, the diphoton
and $b\bar{b}$ signal strengths in this section.

To carry out the numerical evaluations, we take the relevant SM input
parameters as $m_{W}=80.385\;{\rm GeV},\;m_{Z}=90.1876\;{\rm
GeV},\;\alpha_{em}(m_{Z})=1/128.9,\;\alpha_{s}(m_{Z})=0.118$. Due to the
introducing of $U(1)_{B-L}$ gauge group, a new $Z^{\prime}$ gauge boson is
introduced in the B-LSSM. Refs. newZ ; 20 ; 21  give an upper bound on the
ratio between the $Z^{\prime}$ mass $M_{Z^{\prime}}\approx\frac{1}{2}g_{B}u$
and its gauge coupling at $99\%$ CL as

$\displaystyle M_{Z^{\prime}}/g_{{}_{B}}\geq 6\;{\rm TeV}\;.$ (46)

For simplicity, we take $g_{B}=0.5$ in the following analysis, and Eq. (46)
indicates $M_{Z^{\prime}}\geq 3\;{\rm TeV}$ in this case. As analyzed in Refs.
Cacciapaglia:2006pk ; Yang:2023lne , the effects of new $Z^{\prime}$ in the
B-LSSM on the couplings of $Z-$charged leptons which are measured at LEP
precisely PDG , can be eliminated by redefining the neutral currents involving
charged leptons. It indicates that the redefined $Z$ boson in the B-LSSM can
coincide with the one measured at the LEP PDG . The LHC experimental data
constrains $\tan\beta^{{}^{\prime}}<1.5$ 8 . Based on the analysis about
$\bar{B}\rightarrow X_{s}\gamma$ and $B_{s}^{0}\rightarrow\mu^{+}\mu^{-}$ in
our previous work Yang:2018fvw , we take $m_{H^{\pm}}=1.5\;{\rm TeV}$. To
coincide with the constraints from the direct searches of squarks, sgluon at
the LHC ATLAS.PRD ; CMS.JHEP  and the observed Higgs signal analyzed in Ref.
add1 , we take $m_{\tilde{g}}=m_{\tilde{q}}=m_{\tilde{u}}=2.5\;{\rm TeV}$ for
simplicity. It is well known that squarks can make contributions at the one-
loop level to the oblique parameters $S,\;T,\;U$, which suffer strict
constraints from the LEP measurements PDG . Hence, the contributions from
squarks in the B-LSSM to $S,\;T,\;U$ parameters are calculated and the details
are collected in the appendix B. We verify numerically that the $S,\;T,\;U$
parameters can satisfy the LEP measurements for the chosen squark masses.

Firstly, we investigate the masses of the lightest and next-to-lightest Higgs
bosons within the B-LSSM. Taking $g_{YB}=-0.4$, $M_{Z^{\prime}}=4.2\;{\rm
TeV}$, $B_{\eta}=(1\;{\rm TeV})^{2}$, we plot $m_{h_{i}}$ versus
$\tan\beta^{\prime}$ for $A_{t}=0.2\;{\rm TeV}$ in Fig. 1 (a), and $m_{h_{i}}$
versus $A_{t}$ for $\tan\beta^{\prime}=1.075$ in Fig. 1 (b). Where the black,
red lines denote the results for $m_{h_{1}}$, $m_{h_{2}}$ respectively, the
solid, dashed, dotted lines denote the results for $\tan\beta=5,\;15,\;25$
respectively, the green areas denote the range $94\;{\rm
GeV}<m_{h_{i}}<96\;{\rm GeV}$, the gray areas denote the range $124\;{\rm
GeV}<m_{h_{i}}<126\;{\rm GeV}$.

Figure 1: Taking $g_{YB}=-0.4$, $M_{Z^{\prime}}=4.2\;{\rm TeV}$,
$B_{\eta}=(1\;{\rm TeV})^{2}$, $m_{h_{i}}$ versus $\tan\beta^{\prime}$ for
$A_{t}=0.2\;{\rm TeV}$ is plotted in (a), and $m_{h_{i}}$ versus $A_{t}$ for
$\tan\beta^{\prime}=1.075$ is plotted in (b). The black, red lines denote the
results for $m_{h_{1}}$, $m_{h_{2}}$ respectively, the solid, dashed, dotted
lines denote the results for $\tan\beta=5,\;15,\;25$ respectively, the green
areas denote the range $94\;{\rm GeV}<m_{h_{i}}<96\;{\rm GeV}$, the gray areas
denote the range $124\;{\rm GeV}<m_{h_{i}}<126\;{\rm GeV}$.

It can be noted from the picture that $\tan\beta^{\prime}$, $\tan\beta$ and
$A_{t}$ affect the two light Higgs boson masses obviously. Importantly, both
the $95\;{\rm GeV}$ Higgs and the SM-like Higgs with a mass of $125\;{\rm
GeV}$ are attainable within the B-LSSM. Fig. 1 (a) demonstrates that
$\tan\beta^{\prime}$ is constrained strictly for $m_{h_{1}}\approx 95\;{\rm
GeV}$ in our chosen parameter space, and the effects of $\tan\beta^{\prime}$
on $m_{h_{i}}\;(i=1,2)$ are influenced significantly by the value of
$\tan\beta$. This fact can be seen explicitly in Eq. (24). From Fig. 1 (b) we
can see that $A_{t}$ is subject to a strict limitation through the requirement
of setting $m_{h_{2}}\approx 125\;{\rm GeV}$.

As shown in Eq. (24), the lightest Higgs boson is dominated by one of the
B-LSSM specific Higgs states, and $B_{\eta}$, $g_{YB}$, $M_{Z^{\prime}}$ can
affect the theoretical predictions on the two light Higgs boson masses
significantly. Then we take $\tan\beta^{\prime}=1.075$, $\tan\beta=5$,
$A_{t}=0.2\;{\rm TeV}$ and plot $m_{h_{i}}$ versus $B_{\eta}$ for
$M_{Z^{\prime}}=4.2\;{\rm TeV}$ in Fig. 2 (a), and $m_{h_{i}}$ versus
$M_{Z^{\prime}}$ for $B_{\eta}=(1\;{\rm TeV})^{2}$ in Fig. 2 (b). Where the
black, red lines denote the results for $m_{h_{1}}$, $m_{h_{2}}$ respectively,
the solid, dashed, dotted lines denote the results for
$g_{YB}=-0.8,\;-0.4,\;0$ respectively, the green areas denote the range
$94\;{\rm GeV}<m_{h_{i}}<96\;{\rm GeV}$, the gray areas denote the range
$124\;{\rm GeV}<m_{h_{i}}<126\;{\rm GeV}$.

Figure 2: Taking $\tan\beta^{\prime}=1.075$, $\tan\beta=5$, $A_{t}=0.2\;{\rm
TeV}$, $m_{h_{i}}$ versus $B_{\eta}$ for $M_{Z^{\prime}}=4.2\;{\rm TeV}$ is
plotted in (a), and $m_{h_{i}}$ versus $M_{Z^{\prime}}$ for $B_{\eta}=(1\;{\rm
TeV})^{2}$ is plotted in (b). The black, red lines denote the results for
$m_{h_{1}}$, $m_{h_{2}}$ respectively, the solid, dashed, dotted lines denote
the results for $g_{YB}=-0.8,\;-0.4,\;0$ respectively, the green areas denote
the range $94\;{\rm GeV}<m_{h_{i}}<96\;{\rm GeV}$, the gray areas denote the
range $124\;{\rm GeV}<m_{h_{i}}<126\;{\rm GeV}$.

Fig. 2 (a) shows that both $m_{h_{1}}$ and $m_{h_{2}}$ increase as $B_{\eta}$
increases, and the dotted lines intersect. At $g_{YB}=0$, there is no mixing
effects between the SM-like Higgs with the B-LSSM specific Higgs states at the
tree level (the fact can be seen obviously in Eq. (23)). In this scenario,
$h_{2}$ is primarily governed by the SM-like Higgs state and $m_{h_{2}}$ does
not depend on the value of $B_{\eta}$. Conversely, $m_{h_{1}}$ is dominated by
the B-LSSM specific Higgs state and rises with increasing $B_{\eta}$. As
$B_{\eta}$ increases, the mass of the B-LSSM specific Higgs state surpasses
the mass of the SM-like Higgs state, causing $h_{1}$ to be dominated by the
SM-like Higgs state and $h_{2}$ to be dominated by the B-LSSM specific Higgs
state. This situation results in the intersection of the two dotted lines in
Fig. 2 (a). The three red lines in Fig. 2 (b) show that $M_{Z^{\prime}}$
affects $m_{h_{2}}$ obviously when $g_{YB}=-0.8$, because the mixing effects
between the SM-like Higgs with the B-LSSM specific Higgs states are large for
large $g_{YB}$ and play important roles on SM-like Higgs boson masses.

Based on the preceding analysis, it is evident that both the $95\;{\rm GeV}$
Higgs boson and the $125\;{\rm GeV}$ SM-like Higgs boson are attainable in the
B-LSSM. Due to the masses of the two light Higgs bosons are influenced
significantly in a complex manner by $\tan\beta$, $\tan\beta^{\prime}$,
$A_{t}$, $B_{\eta}$, $g_{YB}$, $M_{Z^{\prime}}$, we scan the following
parameter space

$\displaystyle\tan\beta=(2,40),\;\tan\beta^{\prime}=(1,1.5),\;A_{t}=(-1,1)\;{\rm
TeV},$ $\displaystyle B_{\eta}=(0.1^{2},2^{2})\;{\rm
TeV}^{2},\;g_{YB}=(-1,0),\;M_{Z^{\prime}}=(3,10)\;{\rm TeV},$ (47)

to comprehensively explore the collective influences of $\tan\beta$,
$\tan\beta^{\prime}$, $A_{t}$, $B_{\eta}$, $g_{YB}$, $M_{Z^{\prime}}$ on the
signal strengthes of diphoton and $b\bar{b}$ events, and keep $94\;{\rm
GeV}<m_{h_{1}}<96\;{\rm GeV}$, $124\;{\rm GeV}<m_{h_{2}}<126\;{\rm GeV}$ in
the scanning. It can be noted in Eq. (47) that we take $\tan\beta$ in the
range $2<\tan\beta<40$, and the neutral and charged Higgs are limited to be
heavier for larger $\tan\beta$ by the LHC searches PDG . For $\tan\beta=40$,
the neutral Higgs mass and charged Higgs mass are required to be larger than
about $1.3\;{\rm TeV}$, $0.9\;{\rm TeV}$ respectively PDG . In our chosen
parameter space, the charged Higgs mass is taken as $m_{H^{\pm}}=1.5\;{\rm
TeV}$ for simplicity as mentioned above, and the two neutral Higgs masses are
larger than $1.4\;{\rm TeV}$, which indicate that the chosen parameter space
well coincide with the LHC searches.

To explore the best fit describing 125 GeV Higgs mass and the diphoton,
$b\bar{b}$ excesses, a $\chi^{2}$ test is performed. The $\chi^{2}$ function
can be constructed as

$\displaystyle\chi^{2}=\sum_{1}\Big{(}\frac{O_{i}^{\rm th}-O_{i}^{\rm
exp}}{\sigma_{i}^{\rm exp}}\Big{)}^{2},$ (48)

where $O_{i}^{\rm th}$ denotes the $i-$th observable computed theoretically,
$O_{i}^{\rm exp}$ is the corresponding experimental value and $\sigma_{i}^{\rm
exp}$ is the uncertainty in $O_{i}^{\rm exp}$. Then the allowed ranges of
$A_{t}-\tan\beta$, $B_{\eta}-\tan\beta^{\prime}$, $M_{Z^{\prime}}-g_{YB}$ are
plotted in Fig. 3 (a), Fig. 3 (b), Fig. 3 (c) respectively, and the obtained
$\mu(h_{95})_{\gamma\gamma}-\mu(h_{95})_{bb}$ is plotted in Fig. 3 (d). In
Fig. 3, the ‘black stars’ denote the best fit corresponding to
$\chi^{2}=0.058$ with the diphoton excess in Eq. (3), $b\bar{b}$ excess in Eq.
(4) and 125 GeV Higgs mass in Eq. (45), the gray points denote the results
excluded by considering the $\mu(h_{95})_{\gamma\gamma},\;\mu(h_{95})_{bb}$ in
the experimental $2\sigma$ intervals, and the green points, orange points
denote the results with $\mu(h_{95})_{\gamma\gamma},\;\mu(h_{95})_{bb}$ in the
experimental $1\sigma,\;2\sigma$ intervals respectively.

Figure 3: Scanning the parameter space in Eq. (47) and keeping $94\;{\rm
GeV}<m_{h_{1}}<96\;{\rm GeV}$, $124\;{\rm GeV}<m_{h_{2}}<126\;{\rm GeV}$ in
the scanning, the allowed ranges of $A_{t}-\tan\beta$ (a),
$B_{\eta}-\tan\beta^{\prime}$ (b), $M_{Z^{\prime}}-g_{YB}$ (c) and the
obtained $\mu(h_{95})_{\gamma\gamma}-\mu(h_{95})_{bb}$ (d) are plotted. The
‘black stars’ denote the best fit corresponding to $\chi^{2}=0.058$ with the
diphoton excess in Eq. (3), $b\bar{b}$ excess in Eq. (4) and 125 GeV Higgs
mass in Eq. (45), the gray points denote the results excluded by considering
the $\mu(h_{95})_{\gamma\gamma},\;\mu(h_{95})_{bb}$ in the experimental
$2\sigma$ intervals, and the green, orange points denote the results with
$\mu(h_{95})_{\gamma\gamma},\;\mu(h_{95})_{bb}$ in the experimental
$1\sigma,\;2\sigma$ intervals respectively.

The green points in Fig. 3 indicates obviously that the B-LSSM can reproduce
the diphoton and $b\bar{b}$ excesses in the experimental $1\sigma$ intervals
simultaneously. Fig. 3 (a) shows that negative $A_{t}$ is excluded by
considering the two excesses in the experimental $1\sigma$ interval, while the
negative $A_{t}$ is allowed for the case of considering the experimental
$2\sigma$ interval, and most of $A_{t}$ are excluded by the conditions
$94\;{\rm GeV}<m_{h_{1}}<96\;{\rm GeV}$, $124\;{\rm GeV}<m_{h_{2}}<126\;{\rm
GeV}$. As shown in Fig. 3 (b), the two light Higgs masses prefer large
$B_{\eta}$ for small $\tan\beta^{\prime}$ and $B_{\eta}$ is limited strictly
for given $\tan\beta^{\prime}$, because $B_{\eta}$, $\tan\beta^{\prime}$
affect the two light Higgs boson mass significantly as shown in Eq. (24). In
addition, $\tan\beta^{\prime}$ is limited in the range
$1.04\lesssim\tan\beta^{\prime}\lesssim 1.16$ by considering the two excesses
in the experimental $1\sigma$ interval. Fig. 3 (c) shows that $g_{YB}$ is
limited in the range $g_{YB}\lesssim-0.3$ and $M_{Z^{\prime}}/|g_{YB}|\lesssim
11\;{\rm TeV}$ by considering the two excesses in the experimental $1\sigma$
interval, which provides a new bound on the ratio of $Z^{\prime}$ boson mass
with the kinetic mixing parameter $g_{YB}$ if the diphoton and $b\bar{b}$
excesses are verified in future.

Figure 4: Scanning the parameter space in Eq. (47) and keeping $94\;{\rm
GeV}<m_{h_{1}}<96\;{\rm GeV}$, $124\;{\rm GeV}<m_{h_{2}}<126\;{\rm GeV}$ in
the scanning, the results of $|Z_{h,11}|^{2}-|Z_{h,12}|^{2}$ (a),
$|Z_{h,13}|^{2}-|Z_{h,14}|^{2}$ (b) and ${\rm Br}(h^{\rm
NP}_{95}\to\gamma\gamma)-{\rm Br}(h^{\rm NP}_{95}\to bb)$ (c) are plotted,
where the ‘black stars’ denote the best fit corresponding to $\chi^{2}=0.058$
with the diphoton excess in Eq. (3), $b\bar{b}$ excess in Eq. (4) and 125 GeV
Higgs mass in Eq. (45), the gray points denote the results excluded by
considering the two excesses in the experimental $2\sigma$ intervals, and the
green, orange points denote the results with
$\mu(h_{95})_{\gamma\gamma},\;\mu(h_{95})_{bb}$ in the experimental
$1\sigma,\;2\sigma$ intervals respectively.

Finally, in order to show explicitly the components of the lightest 95 GeV
Higgs and the results of ${\rm Br}(h^{\rm NP}_{95}\to\gamma\gamma)$, ${\rm
Br}(h^{\rm NP}_{95}\to bb)$ analyzed in Fig. 3, we present the results of
$|Z_{h,11}|^{2}-|Z_{h,12}|^{2}$, $|Z_{h,13}|^{2}-|Z_{h,14}|^{2}$, ${\rm
Br}(h^{\rm NP}_{95}\to\gamma\gamma)-{\rm Br}(h^{\rm NP}_{95}\to bb)$ in Fig. 4
(a), Fig. 4 (b), Fig. 4 (c) respectively. The definitions of the ‘black
stars’, gray points, green points, orange points are same to the ones of Fig.
3. It is obvious in Fig. 4 (a), (b) that the lightest $95\;{\rm GeV}$ Higgs is
dominated by the new scalar singlets introduced in the B-LSSM, and the mixing
effects between scalar doublets with B-LSSM specific scalar singlets provide
the potential to explain diphoton and $b\bar{b}$ excesses at about $95\;{\rm
GeV}$. It indicates that considering the two loop effective potential
corrections is valuable in calculating the two light Higgs boson masses and
the scalar mixing effects. In addition, Fig. 4 (c) shows that considering
$\mu(h_{95})_{\gamma\gamma},\;\mu(h_{95})_{bb}$ in the experimental $2\sigma$
intervals prefers ${\rm Br}(h^{\rm NP}_{95}\to\gamma\gamma)$ and ${\rm
Br}(h^{\rm NP}_{95}\to bb)$ in the ranges $(0.17\sim 0.28)\%$ and $(70.3\sim
77.5)\%$ respectively, which are different from the corresponding branching
ratios of $h^{\rm SM}_{95}$.

##  IV Summary

Motivated by the diphoton and $b\bar{b}$ excesses at a mass about $95\;{\rm
GeV}$ in the CMS and LEP data, and the B-LSSM can produce a $95\;{\rm GeV}$
light Higgs naturally due to the introducing of two specific singlet states,
our investigation is centered on the analysis of the two light Higgs boson
masses along with the corresponding diphoton and $b\bar{b}$ signal strengths
within the B-LSSM framework. The two loop effective potential corrections to
the squared Higgs mass matrix are considered in the calculations, because the
loop corrections play crucial roles in determining the SM-like Higgs boson
mass within the supersymmetric models. Considering the lightest Higgs boson
mass at about $95\;{\rm GeV}$ and the next-to-lightest Higgs boson mass at
about $125\;{\rm GeV}$, it is found that the B-LSSM specific Higgs states can
account for the observed diphoton and $b\bar{b}$ excesses around $95\;{\rm
GeV}$ simultaneously. As we consider the B-LSSM specific Higgs state as the
lightest Higgs boson, the specific parameters $\tan\beta^{\prime}$,
$B_{\eta}$, $g_{YB}$ and $M_{Z^{\prime}}$ in the B-LSSM affect the theoretical
predictions on the two light Higgs boson masses significantly. Fitting
$\mu(h_{95})_{\gamma\gamma}$, $\mu(h_{95})_{bb}$ in the experimental $1\sigma$
interval will impose stringent constraints on the parameter space of the
B-LSSM, and a new bound on the ratio of $Z^{\prime}$ boson mass with the
kinetic mixing parameter $g_{YB}$ can be provided if the diphoton and
$b\bar{b}$ excesses at about $95\;{\rm GeV}$ are verified in future. In
addition, considering the two loop effective potential corrections is valuable
in calculating the two light Higgs boson masses and the scalar mixing effects,
which play important roles in explaining the excesses at about $95\;{\rm
GeV}$.

##  Appendix A The two-loop effective potential corrections to the CP-even
neutral Higgs mass amtrix.

Generally, the corrections from the effective potential $\Delta V$ to the CP-
even neutral Higgs boson masses can be written as

$\displaystyle\Big{[}-\frac{1}{\phi_{d}}\frac{\partial\Delta
V}{\partial\phi_{d}}+\frac{\partial^{2}\Delta
V}{\partial\phi_{d}\partial\phi_{d}}\Big{]}\Big{|}_{VEV}=-\frac{1}{v_{d}}\Big{[}\frac{\partial\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}}+\frac{\partial\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}\Big{]}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{1}}^{2}}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}})^{2}+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}\partial\phi_{d}}+2\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{2}}^{2}}(\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}})^{2}+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}\partial\phi_{d}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}(\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}})^{2}$
$\displaystyle\qquad+\frac{\partial\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}\partial\phi_{d}}+2\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}+2\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}},$ (49)
$\displaystyle\Big{[}-\frac{1}{\phi_{u}}\frac{\partial\Delta
V}{\partial\phi_{u}}+\frac{\partial^{2}\Delta
V}{\partial\phi_{u}\partial\phi_{u}}\Big{]}\Big{|}_{VEV}=-\frac{1}{v_{u}}\Big{[}\frac{\partial\Delta
V}{\partial m_{t}^{2}}\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}}$
$\displaystyle\qquad+\frac{\partial\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}}\Big{]}+\frac{\partial^{2}\Delta
V}{\partial m_{t}^{2}\partial m_{t}^{2}}(\frac{\partial
m_{t}^{2}}{\partial\phi_{u}})^{2}+\frac{\partial\Delta V}{\partial
m_{t}^{2}}\frac{\partial^{2}m_{t}^{2}}{\partial\phi_{u}\partial\phi_{u}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial
M_{\tilde{t}_{1}}^{2}}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}})^{2}$
$\displaystyle\qquad+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}\partial\phi_{u}}+2\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}}+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{2}}^{2}}(\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}})^{2}$
$\displaystyle\qquad+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}\partial\phi_{u}}+2\frac{\partial^{2}\Delta
V}{\partial m_{t}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}+2\frac{\partial^{2}\Delta V}{\partial
m_{t}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}\partial C_{2\bar{\theta}_{t}}^{2}}(\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}})^{2}+\frac{\partial\Delta
V}{\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}\partial\phi_{u}}+2\frac{\partial^{2}\Delta
V}{\partial m_{t}^{2}\partial C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}}$
$\displaystyle\qquad+2\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}\partial C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}}+2\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}},$ (50)
$\displaystyle\Big{[}-\frac{1}{\phi_{\eta}}\frac{\partial\Delta
V}{\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial\phi_{\eta}\partial\phi_{\eta}}\Big{]}\Big{|}_{VEV}=-\frac{1}{v_{\eta}}\Big{[}\frac{\partial\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}}+\frac{\partial\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}\Big{]}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{1}}^{2}}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}})^{2}+\frac{\partial\Delta
V}{\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}\partial\phi_{\eta}}+2\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{2}}^{2}}(\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}})^{2}+\frac{\partial\Delta
V}{\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}(\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}})^{2}$
$\displaystyle\qquad+\frac{\partial\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}\partial\phi_{\eta}}+2\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}+2\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}},$ (51)
$\displaystyle\Big{[}-\frac{1}{\phi_{\bar{\eta}}}\frac{\partial\Delta
V}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial\phi_{\bar{\eta}}\partial\phi_{\bar{\eta}}}\Big{]}\Big{|}_{VEV}=-\frac{1}{v_{\bar{\eta}}}\Big{[}\frac{\partial\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}\Big{]}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{1}}^{2}}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}})^{2}+\frac{\partial\Delta
V}{\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}\partial\phi_{\bar{\eta}}}+2\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{2}}^{2}}(\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}})^{2}+\frac{\partial\Delta
V}{\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}(\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}})^{2}$
$\displaystyle\qquad+\frac{\partial\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}\partial\phi_{\bar{\eta}}}+2\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}+2\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}},$ (52)
$\displaystyle\Big{[}\frac{\partial^{2}\Delta
V}{\partial\phi_{d}\partial\phi_{u}}\Big{]}\Big{|}_{VEV}=\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}\partial\phi_{u}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial m_{t}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}$ $\displaystyle\qquad+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}}+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}\partial C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}}+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}\partial\phi_{u}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}\partial m_{t}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}}\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}$ $\displaystyle\qquad+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}\partial C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}\partial C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}}+\frac{\partial\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}\partial\phi_{u}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial m_{t}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}$ $\displaystyle\qquad+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}+\frac{\partial^{2}\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}},$ (53)
$\displaystyle\Big{[}\frac{\partial^{2}\Delta
V}{\partial\phi_{d}\partial\phi_{\eta}}\Big{]}\Big{|}_{VEV}=\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}\partial C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}}+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}\partial\phi_{\eta}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}$
$\displaystyle\qquad+\frac{\partial\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}},$ (54)
$\displaystyle\Big{[}\frac{\partial^{2}\Delta
V}{\partial\phi_{d}\partial\phi_{\bar{\eta}}}\Big{]}\Big{|}_{VEV}=\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial\Delta
V}{\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}\partial C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial\Delta
V}{\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}\partial\phi_{\bar{\eta}}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}$
$\displaystyle\qquad+\frac{\partial\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}},$ (55)
$\displaystyle\Big{[}\frac{\partial^{2}\Delta
V}{\partial\phi_{u}\partial\phi_{\eta}}\Big{]}\Big{|}_{VEV}=\frac{\partial^{2}\Delta
V}{\partial m_{t}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial m_{t}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial m_{t}^{2}\partial C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}\partial C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}}+\frac{\partial\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}\partial\phi_{\eta}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}$
$\displaystyle\qquad+\frac{\partial\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}},$ (56)
$\displaystyle\Big{[}\frac{\partial^{2}\Delta
V}{\partial\phi_{u}\partial\phi_{\bar{\eta}}}\Big{]}\Big{|}_{VEV}=\frac{\partial^{2}\Delta
V}{\partial m_{t}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}+\frac{\partial^{2}\Delta
V}{\partial m_{t}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial m_{t}^{2}\partial C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial\Delta
V}{\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}\partial C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial\Delta
V}{\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}\partial\phi_{\bar{\eta}}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}$
$\displaystyle\qquad+\frac{\partial\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}},$ (57)
$\displaystyle\Big{[}\frac{\partial^{2}\Delta
V}{\partial\phi_{\eta}\partial\phi_{\bar{\eta}}}\Big{]}\Big{|}_{VEV}=\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial\Delta
V}{\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{1}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{1}}^{2}\partial C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{2}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial\Delta
V}{\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}\partial\phi_{\bar{\eta}}}$
$\displaystyle\qquad+\frac{\partial^{2}\Delta V}{\partial
M_{\tilde{t}_{2}}^{2}\partial M_{\tilde{t}_{1}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial M_{\tilde{t}_{2}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}$
$\displaystyle\qquad+\frac{\partial\Delta V}{\partial
C_{2\bar{\theta}_{t}}^{2}}\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{1}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}+\frac{\partial^{2}\Delta
V}{\partial C_{2\bar{\theta}_{t}}^{2}\partial
M_{\tilde{t}_{2}}^{2}}\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}},$ (58)

where

$\displaystyle
C_{2\bar{\theta}_{t}}^{2}=1-\frac{2|m_{\tilde{t}_{12}}^{2}|^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{2}}.$
(59)

From the concrete expression of $\Delta V$ defined in Eqs. (19, 20), we can
obtain

$\displaystyle\frac{\partial V^{(1)}}{\partial
m_{t}^{2}}=-\frac{3m_{t}^{2}}{8\pi^{2}}(\log\frac{m_{t}^{2}}{Q^{2}}-1),\;\;\frac{\partial^{2}V^{(1)}}{\partial
m_{t}^{2}\partial m_{t}^{2}}=-\frac{3}{8\pi^{2}}\log\frac{m_{t}^{2}}{Q^{2}},$
(60) $\displaystyle\frac{\partial V^{(1)}}{\partial
M_{\tilde{t}_{1}}^{2}}=\frac{3M_{\tilde{t}_{1}}^{2}}{16\pi^{2}}(\log\frac{M_{\tilde{t}_{1}}^{2}}{Q^{2}}-1),\;\;\frac{\partial^{2}V^{(1)}}{\partial
M_{\tilde{t}_{1}}^{2}\partial
M_{\tilde{t}_{1}}^{2}}=\frac{3}{16\pi^{2}}\log\frac{M_{\tilde{t}_{1}}^{2}}{Q^{2}},$
(61) $\displaystyle\frac{\partial V^{(1)}}{\partial
M_{\tilde{t}_{2}}^{2}}=\frac{3M_{\tilde{t}_{2}}^{2}}{16\pi^{2}}(\log\frac{M_{\tilde{t}_{2}}^{2}}{Q^{2}}-1),\;\;\frac{\partial^{2}V^{(1)}}{\partial
M_{\tilde{t}_{2}}^{2}\partial
M_{\tilde{t}_{2}}^{2}}=\frac{3}{16\pi^{2}}\log\frac{M_{\tilde{t}_{2}}^{2}}{Q^{2}},$
(62)

and the results of $\partial V^{(2)}/\partial M^{2}$,
$\partial^{2}V^{(2)}/(\partial M^{2}\partial M^{2})$ with
$M=m_{t},\;M_{\tilde{t}_{1}},\;M_{\tilde{t}_{2}}$ can be found in the appendix
of Ref. Degrassi:2009yq . The derivatives of
$m_{t}^{2},\;M_{\tilde{t}_{1}}^{2},\;M_{\tilde{t}_{2}}^{2},\;C_{2\bar{\theta}_{t}}^{2}$
over $\phi_{i},\;(i=d,u,\eta,\bar{\eta})$ can be written as

$\displaystyle\frac{\partial
m_{t}^{2}}{\partial\phi_{u}}=Y_{t}^{2}v_{u},\;\;\frac{\partial^{2}m_{t}^{2}}{\partial\phi_{u}\partial\phi_{u}}=Y_{t}^{2},$
(63) $\displaystyle\frac{\partial
m^{2}_{\tilde{t}1,2}}{\partial\phi_{d}}=\frac{1}{8}G^{2}\phi_{d}\pm\frac{\frac{1}{24}G_{1}^{2}\phi_{d}M_{\tilde{t}LR}^{2}+Y_{t}\mu
m_{\tilde{t}12}^{2}}{\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{1/2}},$
$\displaystyle\frac{\partial^{2}m^{2}_{\tilde{t}1,2}}{\partial\phi_{d}\partial\phi_{d}}=\frac{1}{8}G^{2}\mp\frac{2\Big{\\{}-\frac{1}{24}G_{1}^{2}v_{d}M_{\tilde{t}LR}^{2}+Y_{t}\mu
m_{\tilde{t}12}^{2}\Big{\\}}^{2}}{\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{3/2}}\mp\frac{\frac{1}{24}G_{1}^{2}M_{\tilde{t}LR}^{2}-\frac{2}{24^{2}}G_{1}^{4}v_{d}^{2}-(\mu
Y_{t})^{2}}{\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{1/2}},$
(64) $\displaystyle\frac{\partial
m^{2}_{\tilde{t}1,2}}{\partial\phi_{u}}=(Y_{t}^{2}-\frac{1}{8}G^{2})v_{u}\pm\frac{\frac{1}{24}G_{1}^{2}v_{u}M_{\tilde{t}LR}^{2}-Y_{t}A_{t}m_{\tilde{t}12}^{2}}{\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{1/2}},$
$\displaystyle\frac{\partial^{2}m^{2}_{\tilde{t}1,2}}{\partial\phi_{u}\partial\phi_{u}}=Y_{t}^{2}-\frac{1}{8}G^{2}\mp\frac{2\Big{\\{}\frac{1}{24}G_{1}^{2}v_{u}M_{\tilde{t}LR}^{2}-Y_{t}A_{t}m_{\tilde{t}12}^{2}\Big{\\}}^{2}}{\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{3/2}}\pm\frac{\frac{1}{24}G_{1}^{2}M_{\tilde{t}LR}^{2}+\frac{2}{24^{2}}G_{1}^{4}v_{u}^{2}+(A_{t}Y_{t})^{2}}{\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{1/2}},$
(65) $\displaystyle\frac{\partial
m^{2}_{\tilde{t}1,2}}{\partial\phi_{\eta}}=\frac{1}{4}g_{B}g_{YB}v_{\eta}\mp\frac{\frac{1}{24}G_{2}^{2}v_{\eta}M_{\tilde{t}LR}^{2}}{\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{1/2}},$
$\displaystyle\frac{\partial^{2}m^{2}_{\tilde{t}1,2}}{\partial\phi_{\eta}\partial\phi_{\eta}}=\frac{1}{4}g_{B}g_{YB}\pm\frac{(\frac{1}{12}G_{2}^{2}m_{\tilde{t}12}^{2}v_{\eta})^{2}}{\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{3/2}}\mp\frac{\frac{1}{24}G_{2}^{2}M_{\tilde{t}LR}^{2}}{\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{1/2}},$
(66) $\displaystyle\frac{\partial
m^{2}_{\tilde{t}1,2}}{\partial\phi_{\bar{\eta}}}=-\frac{1}{4}g_{B}g_{YB}v_{\bar{\eta}}\pm\frac{\frac{1}{24}G_{2}^{2}v_{\bar{\eta}}M_{\tilde{t}LR}^{2}}{\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{1/2}},$
$\displaystyle\frac{\partial^{2}m^{2}_{\tilde{t}1,2}}{\partial\phi_{\bar{\eta}}\partial\phi_{\bar{\eta}}}=-\frac{1}{4}g_{B}g_{YB}\pm\frac{(\frac{1}{12}G_{2}^{2}m_{\tilde{t}12}^{2}v_{\bar{\eta}})^{2}}{\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{3/2}}\pm\frac{\frac{1}{24}G_{2}^{2}M_{\tilde{t}LR}^{2}}{\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{1/2}},$
(67)
$\displaystyle\frac{\partial^{2}m^{2}_{\tilde{t}1,2}}{\partial\phi_{d}\partial\phi_{u}}=\mp(Y_{t}A_{t}M_{\tilde{t}LR}^{2}+\frac{1}{12}G_{1}^{2}v_{u}m_{\tilde{t}12}^{2})(Y_{t}\mu
M_{\tilde{t}LR}^{2}+\frac{1}{12}G_{1}^{2}v_{d}m_{\tilde{t}12}^{2})/(M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4})^{3/2},$
(68)
$\displaystyle\frac{\partial^{2}m^{2}_{\tilde{t}1,2}}{\partial\phi_{d}\partial\phi_{\eta}}=\pm\frac{1}{12}G_{2}^{2}v_{\eta}m_{\tilde{t}12}^{2}\Big{\\{}Y_{t}\mu
M_{\tilde{t}LR}^{2}+\frac{1}{12}G_{1}^{2}v_{d}m_{\tilde{t}12}^{2}\Big{\\}}/\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{3/2},$
(69)
$\displaystyle\frac{\partial^{2}m^{2}_{\tilde{t}1,2}}{\partial\phi_{d}\partial\phi_{\bar{\eta}}}=\mp\frac{1}{12}G_{2}^{2}v_{\bar{\eta}}m_{\tilde{t}12}^{2}\Big{\\{}Y_{t}\mu
M_{\tilde{t}LR}^{2}+\frac{1}{12}G_{1}^{2}v_{d}m_{\tilde{t}12}^{2}\Big{\\}}/\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{3/2},$
(70)
$\displaystyle\frac{\partial^{2}m^{2}_{\tilde{t}1,2}}{\partial\phi_{u}\partial\phi_{\eta}}=\mp\frac{1}{12}G_{2}^{2}v_{\eta}m_{\tilde{t}12}^{2}\Big{\\{}Y_{t}A_{t}M_{\tilde{t}LR}^{2}+\frac{1}{12}G_{1}^{2}v_{u}m_{\tilde{t}12}^{2}\Big{\\}}/\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{3/2},$
(71)
$\displaystyle\frac{\partial^{2}m^{2}_{\tilde{t}1,2}}{\partial\phi_{u}\partial\phi_{\bar{\eta}}}=\pm\frac{1}{12}G_{2}^{2}v_{\bar{\eta}}m_{\tilde{t}12}^{2}\Big{\\{}Y_{t}A_{t}M_{\tilde{t}LR}^{2}+\frac{1}{12}G_{1}^{2}v_{u}m_{\tilde{t}12}^{2}\Big{\\}}/\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{3/2},$
(72)
$\displaystyle\frac{\partial^{2}m^{2}_{\tilde{t}1,2}}{\partial\phi_{\eta}\partial\phi_{\bar{\eta}}}=\mp\frac{(\frac{1}{12}G_{2}^{2}m_{\tilde{t}12}^{2})^{2}v_{\eta}v_{\bar{\eta}}}{\Big{\\{}M_{\tilde{t}LR}^{4}+2m_{\tilde{t}12}^{4}\Big{\\}}^{3/2}},$
(73) $\displaystyle\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}}=\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}})-(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{d}}\Big{]},$
$\displaystyle\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}\partial\phi_{d}}=\frac{4}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{4}}\Big{[}(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{d}}-3m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}})\Big{]}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}$
$\displaystyle\qquad\quad\;\;-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}})-(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{d}}\Big{]}+\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}\partial\phi_{d}}-\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}\partial\phi_{d}})\Big{]},$
(74) $\displaystyle\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}}=\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}})-(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{u}}\Big{]},$
$\displaystyle\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}\partial\phi_{u}}=\frac{4}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{4}}\Big{[}(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{u}}-3m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}})\Big{]}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}$
$\displaystyle\qquad\quad\;\;-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}})-(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{u}}\Big{]}+\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}\partial\phi_{u}}-\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}\partial\phi_{u}})\Big{]},$
(75) $\displaystyle\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}}=\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}})\Big{]},$
$\displaystyle\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}\partial\phi_{\eta}}=\frac{4}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{4}}\Big{[}-3m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}})\Big{]}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}})\Big{]}$
$\displaystyle\qquad\quad\;\;+\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}\partial\phi_{\eta}}-\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}\partial\phi_{\eta}})\Big{]},$
(76) $\displaystyle\frac{\partial
C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}}=\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}})\Big{]},$
$\displaystyle\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\bar{\eta}}\partial\phi_{\bar{\eta}}}=\frac{4}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{4}}\Big{[}-3m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}})\Big{]}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}})\Big{]}$
$\displaystyle\qquad\quad\;\;+\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}\partial\phi_{\bar{\eta}}}-\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}\partial\phi_{\bar{\eta}}})\Big{]},$
(77)
$\displaystyle\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}\partial\phi_{u}}=\frac{4}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{4}}\Big{[}(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{u}}-3m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}})\Big{]}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}$
$\displaystyle\qquad\quad\;\;-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}})-(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{d}}\Big{]}+\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}\partial\phi_{u}}-\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}\partial\phi_{u}})$
$\displaystyle\qquad\quad\;\;+\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{u}}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}})-\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{d}}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}})\Big{]},$ (78)
$\displaystyle\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}\partial\phi_{\eta}}=\frac{4}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{4}}\Big{[}-3m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}})\Big{]}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}})$
$\displaystyle\qquad\quad\;\;-(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{d}}\Big{]}+\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}\partial\phi_{\eta}}-\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}\partial\phi_{\eta}})$
$\displaystyle\qquad\quad\;\;-\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{d}}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}})\Big{]},$ (79)
$\displaystyle\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{d}\partial\phi_{\bar{\eta}}}=\frac{4}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{4}}\Big{[}-3m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}})\Big{]}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}})$
$\displaystyle\qquad\quad\;\;-(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{d}}\Big{]}+\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{d}\partial\phi_{\bar{\eta}}}-\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{d}\partial\phi_{\bar{\eta}}})$
$\displaystyle\qquad\quad\;\;-\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{d}}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}})\Big{]},$ (80)
$\displaystyle\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}\partial\phi_{\eta}}=\frac{4}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{4}}\Big{[}-3m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}})\Big{]}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}})$
$\displaystyle\qquad\quad\;\;-(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{u}}\Big{]}+\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}\partial\phi_{\eta}}-\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}\partial\phi_{\eta}})$
$\displaystyle\qquad\quad\;\;-\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{u}}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}})\Big{]},$ (81)
$\displaystyle\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{u}\partial\phi_{\bar{\eta}}}=\frac{4}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{4}}\Big{[}-3m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}})\Big{]}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}})$
$\displaystyle\qquad\quad\;\;-(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{u}}\Big{]}+\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{u}\partial\phi_{\bar{\eta}}}-\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{u}\partial\phi_{\bar{\eta}}})$
$\displaystyle\qquad\quad\;\;-\frac{\partial
m_{\tilde{t}_{12}}^{2}}{\partial\phi_{u}}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}})\Big{]},$ (82)
$\displaystyle\frac{\partial^{2}C_{2\bar{\theta}_{t}}^{2}}{\partial\phi_{\eta}\partial\phi_{\bar{\eta}}}=\frac{4}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{4}}\Big{[}-3m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\bar{\eta}}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\bar{\eta}}})\Big{]}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial
M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}}-\frac{\partial
M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}})\Big{]}$
$\displaystyle\qquad\quad\;\;+\frac{4m_{\tilde{t}_{12}}^{2}}{(M_{\tilde{t}_{1}}^{2}-M_{\tilde{t}_{2}}^{2})^{3}}\Big{[}m_{\tilde{t}_{12}}^{2}(\frac{\partial^{2}M_{\tilde{t}_{1}}^{2}}{\partial\phi_{\eta}\partial\phi_{\bar{\eta}}}-\frac{\partial^{2}M_{\tilde{t}_{2}}^{2}}{\partial\phi_{\eta}\partial\phi_{\bar{\eta}}})\Big{]},$
(83)

where

$\displaystyle
G^{2}=g_{1}^{2}+g_{2}^{2}+g_{YB}^{2},\;G_{1}^{2}=5g_{1}^{2}-3g_{2}^{2}+5g_{YB}^{2}+2g_{B}g_{YB},\;G_{2}^{2}=4g_{B}^{2}+10g_{B}g_{YB},$
$\displaystyle m_{\tilde{t}12}^{2}=Y_{t}(\mu v_{d}-A_{t}v_{u}),$
$\displaystyle
M_{\tilde{t}LR}^{2}=\frac{1}{24}[24(M_{Q}^{2}-M_{U}^{2})+G_{1}^{2}(v_{u}^{2}-v_{d}^{2})+G_{2}^{2}(v_{\bar{\eta}}^{2}-v_{\eta}^{2})].$
(84)

##  Appendix B The contributions from squarks to $S,\;T,\;U$ parameters in the
B-LSSM.

The one-loop diagrams contributing to the $\gamma,\;W,\;Z$ self-energy and
$\gamma-Z$ mixing are plotted in Fig. 5, where the loop particles are squarks.

Figure 5: The Feynman diagrams contributing to the $\gamma,\;W,\;Z$ self-
energy and $\gamma-Z$ mixing, where the loop particles are squarks.

Since the $Z-Z^{\prime}$ mixing is limited to be very small ($\lesssim
10^{-4}$) experimentally, we can neglect the $Z-Z^{\prime}$ mixing
contributions safely in the calculation. Then the results can be written as

$\displaystyle\Pi_{ZZ}=\frac{1}{16\pi^{2}}(C_{Z\tilde{Q}_{i}\tilde{Q}_{j}})^{2}\Big{[}I_{1}(M_{\tilde{Q}_{i}}^{2},M_{\tilde{Q}_{j}}^{2})+I_{2}(M_{\tilde{Q}_{i}}^{2},M_{\tilde{Q}_{j}}^{2})q^{2}\Big{]}-$
$\displaystyle\qquad\quad\;\frac{1}{16\pi^{2}}(C_{ZZ\tilde{Q}_{i}\tilde{Q}_{i}})I_{3}(M_{\tilde{Q}_{i}}^{2}),$
(85)
$\displaystyle\Pi_{WW}=\frac{1}{16\pi^{2}}(C_{W\tilde{U}_{i}\tilde{D}_{j}})^{2}\Big{[}I_{1}(M_{\tilde{U}_{i}}^{2},M_{\tilde{D}_{j}}^{2})+I_{2}(M_{\tilde{U}_{i}}^{2},M_{\tilde{D}_{j}}^{2})q^{2}\Big{]}-$
$\displaystyle\qquad\quad\;\;\frac{1}{16\pi^{2}}(C_{WW\tilde{Q}_{i}\tilde{Q}_{i}})I_{3}(M_{\tilde{Q}_{i}}^{2}),$
(86)
$\displaystyle\Pi_{\gamma\gamma}=\frac{1}{16\pi^{2}}(C_{\gamma\tilde{Q}_{i}\tilde{Q}_{j}})^{2}I_{2}(M_{\tilde{Q}_{i}}^{2},M_{\tilde{Q}_{j}}^{2})q^{2},$
(87) $\displaystyle\Pi_{\gamma
Z}=\frac{1}{16\pi^{2}}(C_{\gamma\tilde{Q}_{i}\tilde{Q}_{j}}C_{Z\tilde{Q}_{i}\tilde{Q}_{j}})I_{2}(M_{\tilde{Q}_{i}}^{2},M_{\tilde{Q}_{j}}^{2})q^{2},$
(88)

where $C_{abc}$ and $C_{abcd}$ denote the coupling constants of fields $abc$,
$abcd$ respectively, $M_{\tilde{Q}_{i}}$ denotes the squark mass,
$M_{\tilde{U}_{i}}$ denotes the up-squark mass, $M_{\tilde{D}_{i}}$ denotes
the down-squark mass, and

$\displaystyle I_{1}(x,y)=x({\rm div}-\ln x)+y({\rm div}-\ln y)+f(x,y),$ (89)
$\displaystyle I_{2}(x,y)=\frac{1}{18(x-y)^{2}}\Big{[}(x+y)^{2}+8xy+6x^{2}\ln
x+6y^{2}\ln y$ $\displaystyle\qquad\qquad\;-\frac{12xy}{x-y}(x\ln x-y\ln
y)\Big{]},$ (90) $\displaystyle I_{3}(x)=x({\rm div}-\ln x),$ (91)

where

$\displaystyle{\rm div}\equiv\frac{2}{4-d}-\gamma+1+\ln(4\pi).$ (92)

Then the $S,\;T,\;U$ parameters Peskin:1991sw ; Yang:2022gvz  at one-loop
level in the B-LSSM can be written as

$\displaystyle\hat{S}=-\frac{c_{W}}{s_{W}}(c_{W}^{2}\Pi^{\prime}_{\gamma
Z}-s_{W}^{2}\Pi^{\prime}_{\gamma
Z}+c_{W}s_{W}\Pi^{\prime}_{\gamma\gamma}-c_{W}s_{W}\Pi^{\prime}_{ZZ}),$
$\displaystyle\hat{T}=\frac{\Pi_{33}-\Pi_{WW}}{M_{W}^{2}}=-\frac{1}{M_{W}^{2}}(c_{W}^{2}\Pi_{ZZ}+s_{W}^{2}\Pi_{\gamma\gamma}+2s_{W}c_{W}\Pi_{\gamma
Z}-\Pi_{WW}),$
$\displaystyle\hat{U}=c_{W}^{2}\Pi^{\prime}_{ZZ}+s_{W}^{2}\Pi^{\prime}_{\gamma\gamma}+2s_{W}c_{W}\Pi^{\prime}_{\gamma
Z}-\Pi^{\prime}_{WW},$ (93)

where $\Pi_{V_{1}V_{2}}^{\prime}$ denotes $\partial\Pi_{V_{1}V_{2}}/\partial
q^{2}$ with $V_{1}V_{2}$ denoting $\gamma\gamma,\;ZZ,\;WW,\;\gamma Z$.

###### Acknowledgements.

The work has been supported by the National Natural Science Foundation of
China (NNSFC) with Grants No. 12075074, No. 12235008, Hebei Natural Science
Foundation for Distinguished Young Scholars with Grant No. A2022201017, No.
A2023201041, Natural Science Foundation of Guangxi Autonomous Region with
Grant No. 2022GXNSFDA035068, and the youth top-notch talent support program of
the Hebei Province.

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