#  Snowmass 2021 KIAS–P22040
MSSM Under Higgs Factories

Honglei Li  School of Physics and Technology, University of Jinan, Jinan,
Shandong 250022, China  Huayang Song  CAS Key Laboratory of Theoretical
Physics, Institute of Theoretical Physics, Chinese Academy of Sciences,
Beijing 100190, China.  Shufang Su  Department of Physics, University of
Arizona, Tucson, Arizona 85721, USA  Wei Su  Korea Institute for Advanced
Study, Seoul 02455, Korea  Jin Min Yang  CAS Key Laboratory of Theoretical
Physics, Institute of Theoretical Physics, Chinese Academy of Sciences,
Beijing 100190, China.

###### Abstract

The high precision measurements of the Higgs mass and couplings at the future
Higgs factories are sensitive to the parameter space of the Minimal
Supersymmetric Standard Model (MSSM). Focused on the dominant stop sector
contributions, we study the implication of the Higgs precision measurements on
MSSM using multi-variable $\chi^{2}$ fit. The results show nice
complementarity between the indirect searches at Higgs factories and the
direct searches at the current LHC program.

11footnotetext: email address:
sps_lihl@ujn.edu.cn,shufang@arizona.edu,huayangs@itp.ac.cn,weisu<EMAIL_ADDRESS>
Thematic Areas:

$\blacksquare$ (EF02) EW Physics: Higgs Boson as a portal to new physics
$\blacksquare$ (EF01) EW Physics: Higgs Boson properties and couplings
$\blacksquare$ (EF08) BSM: Model specific explorations
$\blacksquare$ (TF07) Collider phenomenology

##  1  Introduction

Direct searches on the new physics beyond the Standard Model (SM) are
extensively explored at the Large Hadron Collider (LHC). Complementary to the
direct searches, the Higgs precision measurements provide an alternative way
to study the new physics effects. There are several proposals to build Higgs
factories in the pursuit of precision Higgs measurements, including the
Circular Electron Positron Collider (CEPC) in China [1], the electron-positron
stage of the Future Circular Collider (FCC-ee) at CERN [2, 3], and the
International Linear Collider (ILC) in Japan [4]. These future Higgs factories
will be the frontier of the precision measurements which will be sensitive to
the new physics beyond the SM.

As a well-motivated model to solve problems like the naturalness problem, and
the origin of dark matter, the MSSM is one of the most promising new physics
scenarios. Each SM particle has its supersymmetric partner with spin differed
by a half. The Higgs sector of the MSSM follows that of the Type-II two Higgs
doublet model with one Higgs doublet couples to the up-type quarks, and the
other doublet couples to the down-type quarks and charged leptons. After the
electroweak symmetry breaking, there are five physical fields labeled as $A$,
$h$, $H$ and $H^{\pm}$, in which $h$ and $H$ are CP-even bosons and $A$ is the
CP-odd one.

##  2 MSSM and Analyses Strategy

The MSSM Higgs sector at the tree level is described by only two input
parameters $m_{A}$ and $\tan\beta$, the mass of the CP-odd Higgs boson and the
ratio of the vacuum expectation of the two Higgs fields. Both the mass and the
couplings of the SM-like light CP-even Higgs $h$ receive radiative
corrections, which are sensitive to the model parameters. The CP-even Higgs
mass matrix is given by

$\cal{M}_{\rm Higgs}\rm=\frac{\sin
2\beta}{2}\left(\begin{array}[]{ll}\cot\beta\ m_{Z}^{2}+\tan\beta\
m_{A}^{2}&~{}~{}~{}~{}-m_{Z}^{2}-m_{A}^{2}\\\ -m_{Z}^{2}-m_{A}^{2}&\tan\beta\
m_{Z}^{2}+\cot\beta\
m_{A}^{2}\end{array}\right)+\left(\begin{array}[]{ll}\Delta_{11}&\Delta_{12}\\\
\Delta_{12}&\Delta_{22}\end{array}\right),$ (1)

with the first term being the tree-level contributions and
$\Delta_{11},\Delta_{12},\Delta_{22}$ in the second term are the loop-induced
Higgs mass corrections [5, 6, 7]. In particular, the dominant contribution to
the SM-like Higgs mass comes from the stop sector. To capture the dominant
effects, we mainly consider four parameters $\tan\beta,m_{A},m_{\rm
SUSY},X_{t}$, with $m_{\rm SUSY}$ being the soft SUSY breaking parameter for
stop masses, and $X_{t}$ being the left-right mixing in the stop mass matrix:
$X_{t}=A_{t}-\mu\cot\beta$. In our study, we use the effective mixing angle
$\alpha_{eff}$ to calculate the Higgs couplings following the Type-II 2HDM
case [8], which takes the form of

$\displaystyle\tan\alpha_{eff}$ $\displaystyle=$
$\displaystyle\frac{-(m_{A}^{2}+m_{Z}^{2})\sin\beta\cos\beta+\Delta_{12}}{m_{Z}^{2}\cos^{2}\beta+m_{A}^{2}\sin^{2}\beta+\Delta_{11}-m_{h^{0},eff}^{2}}.$
(2)

To study the sensitivity of MSSM parameters to the Higgs precision
measurements at Higgs factories, we use a multi-variable $\chi^{2}$ fit:

$\chi^{2}_{total}=\chi^{2}_{m_{h}}+\chi^{2}_{\mu}=\frac{(m_{h}^{\rm
MSSM}-m_{h}^{\rm obs})^{2}}{(\Delta
m_{h})^{2}}+\sum_{i=f,V..}\frac{(\mu_{i}^{\rm MSSM}-\mu_{i}^{\rm
obs})^{2}}{(\Delta\mu_{i})^{2}},$ (3)

here $\chi^{2}_{m_{h}}$ and $\chi^{2}_{\mu}$ are the contributions to the
overall $\chi^{2}$ from the Higgs mass and signal strength
$\mu_{i}^{\rm{MSSM}}=(\sigma\times\textrm{Br}_{i})_{\rm{MSSM}}/(\sigma\times\textrm{Br}_{i})_{\rm{SM}}$
measurements, respectively. For $\chi^{2}_{m_{h}}$, given the small
experimental uncertainties, we set $\Delta m_{h}$ to be 3 GeV which mainly
comes from theoretical uncertainty of unknown higher order radiative
corrections [9, 10, 11, 12]. In our analyses, we determine the allowed
parameter region at the $95\%$ Confidence Level (C.L.) by a multi-variable fit
to the Higgs decay signal strengths of various channels and Higgs mass.

##  3 $\chi^{2}$ Fit Results

Using the $\chi^{2}$ analysis, we study the effects of Higgs precision
measurements from Higgs factories on the MSSM parameter space [8]. We study
the individual constraint from the Higgs mass, loop induced
$h\gamma\gamma+hgg$ channels, as well the channels of Higgs decays to a pair
of fermions or gauge bosons. We found CP-odd Higgs mass $m_{A}$ is sensitive
to the precisions of Higgs decay channels, while $m_{\rm SUSY}$, $X_{t}$ and
$\tan\beta$ are sensitive to the precision of Higgs mass determination. For
large $\tan\beta$, $m_{\rm SUSY}$ and $X_{t}$ are also sensitive the
precisions of fermion and vector gauge boson couplings. For the max-mixing
scenario, the loop-induced $hgg$ and $h\gamma\gamma$ couplings are the main
restrictions on $m_{\rm SUSY}$ when $\tan\beta>7$.

Putting all contributions together, we project the constraints onto the two-
dimensional planes. Detailed results can be found in Ref. [8]. Here we only
show parts of $m_{A}-\tan\beta$ and $m_{A}-m_{\rm SUSY}$. We also compare the
sensitivity of indirect search via Higgs precision measurements with direct
search limits at current and future LHC runs.

Figure 1:  95% C.L. allowed region in $\tan\beta$ vs. $m_{A}$ for $X_{t}=0$
(zero-mixing, left panel) and $X_{t}=2m_{\rm SUSY}$ (max-mixing, right panel),
with the CEPC (solid), the FCC-ee (dotted) and the ILC (dashed) precision. The
LHC Run-II direct search limits based on $A/H\to\tau\tau$ [13] are shown in
the grey shaded region.

Figure 2: 95% C.L. allowed region in $m_{\rm SUSY}$ vs. $m_{A}$ plane with
CEPC precisions for $X_{t}=0$ (zero-mixing, left panel) and $X_{t}=2m_{\rm
SUSY}$ (max-mixing, right panel). For each panel, different colored curve
corresponds to different values of $\Delta m_{h}=1,\ 2,\ 3$ GeV, with region
to the right of the curve allowed.

As shown in Fig. 1, We obtained the 95% C.L. allowed region given the Higgs
factory precisions, and presented the result in the parameter space of $m_{A}$
vs. $\tan\beta$ for various Higgs factories. In general, $m_{\rm SUSY}<900$
GeV are excluded for both the zero-mixing and max-mixing cases. For the zero-
mixing case, when $m_{\rm SUSY}<1$ TeV, the parameter space of $\tan\beta<40$
is excluded. Limits on $\tan\beta$ get lower for larger values of $m_{\rm
SUSY}$, which is sensitive in particular for $1\ {\rm TeV}<m_{\rm SUSY}<1.5$
TeV. For the max-mixing case, limits on $\tan\beta$ is much lower for $m_{\rm
SUSY}=1$ TeV. Those features are mainly due to the Higgs mass constraint. It
also tells that FCC-ee is of stronger probe ability because of the higher
luminosity. The LHC Run-II direct search limits based on $A/H\to\tau\tau$ [13]
are shown in the grey shaded region, which is complementary to the indirect
limits from Higgs precision measurements.

In Fig. 2, we show the 95% C.L. allowed region in $m_{A}$ versus $m_{SUSY}$
plane for $\Delta m_{h}=3$ GeV (solid curve) 2 GeV (dashed curve) and 1 GeV
(dotted curve). The lower limit on $m_{\rm SUSY}$ for the zero-mixing case,
and the upper limit on $m_{\rm SUSY}$ for the max-mixing case depend
sensitively on the values of $\Delta m_{h}$. Therefore, it is crucial to
improve the precision in the $m_{h}$ calculation in the MSSM, which allows us
to obtain tight constraints on the SUSY mass scale, in particular, on the stop
sector, once Higgs precision measurements are available at future Higgs
factories.

##  4 Conclusion

As a summary, we performed a multi-variable fit to both the signal strength
for various Higgs decay channels at Higgs factories and the Higgs mass. We
obtained the 95% C.L. allowed region given the Higgs factory precisions, and
presented the result in the parameter space of $m_{A}$ vs. $m_{\rm SUSY}$, and
$m_{A}$ vs. $\tan\beta$. We found that the lower limits on $\tan\beta$ depends
sensitively on the values of $m_{\rm SUSY}$ and $m_{A}$, in particular, for
$m_{\rm SUSY}<1.5$ TeV and $m_{A}<2$ TeV. Limits on $m_{\rm SUSY}$ also depend
sensitively on $\Delta m_{h}$, indicating the importance of a precise
determination of the Higgs mass in the MSSM. For $\tan\beta=50$ of the max-
mixing scenario, $m_{\rm SUSY}\in(0.8,1.2)$ TeV when $\Delta m_{h}=1$ GeV. We
also compared the reach of the CEPC, the FCC-ee and the ILC. We found that the
reach of the CEPC is similar to that of the FCC-ee, while the reach of the ILC
is typically better, given the slight better precision in the Higgs WBF
measurements.

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