# Towards Realistic SUSY Grand Unification for Extended MSSM

Sibo Zheng<EMAIL_ADDRESS>Department of Physics, Chongqing
University, Chongqing 401331, P. R. China

(March 27, 2019)

###### Abstract

Low-energy supersymmetric models such as MSSM, NMSSM and MSSM with vectorlike
fermion are consistent with perturbative unification. While the non-minimal
extensions naturally explain Higgs mass and dark matter in the low energy
region, it is unclear how seriously they are constrained in the ultraviolet
region. Our study shows that $i)$, In the case of embedding MSSM into
$\rm{SU}(5)$, the fit to SM fermion masses requires a singlet $S$, which leads
to unviable embedding of NMSSM into $\rm{SU}(5)$ because such $S$ feeds
singlet $N$ a mass of order unification scale as well. $ii)$, Similar result
holds in the case of embedding NMSSM into $\rm{SO}(10)$, where $S$ is replaced
by some Higgs fields responsible for $\rm{SO}(10)$ breaking. $iii)$, On the
contrary, for the embedding of MSSM with $16$-dimensional vectorlike fermions
into $\rm{SO}(10)$, the Higgs field responsible for the vectorlike mass of
order TeV scale can evade those problems the singlet $N$ encounters because of
an intermediate mass scale in the $126$-dimensional Higgs field.

##  I Introduction

At the frontiers of new physics beyond standard model (SM) natural or TeV-
scale supersymmetry (SUSY) offers us a grand unification (GUT) of SM gauge
coupling constants unif1 ; unif2 ; unif3 ; unif4 . Such natural SUSY hosts a
lot of SUSY particles which can be directly detected at the particle collider
LHC or dark matter direct detection facilities such as Xenon-1T. Meanwhile,
embedding these TeV-scale SUSY models into the ultraviolet completions - SUSY
GUT- may solve the long-standing issues such as SM flavor puzzle and neutrino
masses.

Nowadays experimental data seems to oppose the minimal supersymmetric standard
model (MSSM) either from the bottom or top viewpoint. In the former one, the
$125$ GeV Higgs mass Higgsmass1 ; Higgsmass2  requires either large mixing
effect or soft masses of order $10$ TeV for the stop scalars MSSMHiggs1 ;
MSSMHiggs2 ; MSSMHiggs3 . When the mixing effects among generations are
significant, the constraints from flavor violation tend to require the SUSY
mass order far above the weak scale. Moreover, the direct detection limits of
dark matter MSSMDark1 ; MSSMDark2  impose rather strong pressure on the
scenario of neutralino dark matter. In the later perspective, the minimal SUSY
$\rm{SU}(5)$ referring to MSSM is significantly constrained by the proton
decay protonreivew . It requires the color-triplet Higgs mass of order GUT
scale, which together with unification, leads to the MSSM mass spectrum at
least of order $100$ TeV Murayama ; Hisano ; Bajc .

Therefore, it is of great interest to explore the MSSM with rational
extensions that can resurrect the natural SUSY once again. Along this
direction, there are at least two simple examples -the next-to-minimal
supersymmetric standard model (NMSSM) NMSSM  and the MSSM with vectorlike (VL)
fermions (VMSSM) 0807.3055  \- which are consistent with unification Zheng1 ;
Zheng2 . While these extensions provide natural explanations of Higgs mass and
dark matter in the low energy region, it is unclear what the statuses of them
are in the ultraviolet energy region. This is the main focus of this study.

In this paper, we discuss the embeddings of MSSM, NMSSM and VMSSM into
realistic grand unification (GUT) GG ; SO101 ; SO102 . In each case, both the
SUSY $\rm{SU}(5)$ SUSYSU51 ; SUSYSU52 ; SUSYSU53  and SUSY $\rm{SO}(10)$
SUSYSO10  representations will be explored. In these SUSY GUTs, we discuss the
GUT-scale superpotential

$\displaystyle{}W=W_{Y}+W_{\text{SB}},$ (1)

with following features:

  * •

$W_{Y}$ and $W_{\text{SB}}$ are both renormalizable.

  * •

All of vacuum expectation values (vevs) are dynamically generated from
$W_{\text{SB}}$.

  * •

All of SM matters and extended matters obtain their masses via the Higgs
mechanism in $W_{Y}$.

Since $W_{Y}$ is fixed by the SM and the extra matters such as the singlet $N$
or VL fermions at TeV scale, it is crucial to find suitable content of
$W_{\text{SB}}$ that achieves the breaking of gauge group
$G_{\text{GUT}}\rightarrow G_{\text{SM}}$. In Sec.II, we explore the embedding
of MSSM into realistic SUSY GUTs, where useful conventions and notation will
be introduced. The analysis on the MSSM is of great use to guide us towards
the embeddings of NMSSM and VMSSM. Sec.III and Sec.IV is devoted to study the
embeddings of NMSSM and VMSSM into realistic SUSY GUTs, respectively. Finally,
we conclude in Sec.V.

##  II The Benchmark Model: MSSM

In the minimal $\rm{SU}(5)$ of Standard Model (SM) gauge group, the SM
fermions of each generation are assigned as $1$, $\bar{5}$ ($\psi$) and $10$
($\Phi$) of $\rm{SU}(5)$ for right-hand neutrino $N_{R}$, $L$ and down quark
$\bar{d}$, and $Q$, up quark $\bar{u}$ and $\bar{e}$, respectively; whereas in
the $\rm{SO}(10)$ representation, the SM fermions of each generation are
embedded into a $16$-dimensional representation, which decomposes as
$16=1+\bar{5}+10$ under the $\rm{SU}(5)$.

###  II.1 SU(5)

The Yukawa superpotenail $W_{Y}$ in Eq.(1) contains two parts

$\displaystyle{}W^{\rm{SU}(5)}_{Y}=W^{\rm{SU}(5)}_{Y_{f}}+W^{\rm{SU}(5)}_{Y_{\nu}}$
(2)

which refers to SM fermions without neutrinos and neutrinos, respectively.

According to the product $\bar{5}\times 10=5+45$ and $10\times
10=\bar{5}_{s}+\bar{45}_{a}+\bar{50}_{s}$, where subscript “s” and “a” refers
to symmetric and anti-symmetric, respectively. With the Higgs representations
composed of $5$, $\bar{5}$ and $\bar{45}$ GJ , $W^{\rm{SU}(5)}_{Y_{f}}$ is
given by,

$\displaystyle{}W^{\rm{SU}(5)}_{Y_{f}}$ $\displaystyle=$ $\displaystyle
Y_{d}^{ij}\psi_{ai}(\bar{5})\bar{H}_{b}(\bar{5})\Phi^{ab}_{j}(10)$ (3)
$\displaystyle+$ $\displaystyle
Y_{45}^{ij}\psi_{ai}(\bar{5})\bar{H}^{a}_{bc}(\bar{45})\Phi^{bc}_{j}(10)$
$\displaystyle+$ $\displaystyle
Y_{u}^{ij}\epsilon_{abcde}\Phi^{ab}_{i}(10)H^{e}(5)\Phi^{cd}_{j}(10)$

where $a,b,c$, etc denote the $\rm{SU}(5)$ indexes, $i,j$ are the generation
indexes, and $Y_{u,d,45}$ are Yukawa matrixes. Note, $\Phi^{ab}$ is an anti-
symmetric field and $H^{a}_{bc}=-H^{a}_{cb}$.

The reason to include $\bar{45}$ is clear in the SM fermion mass matrixes as
derived from Eq.(3):

$\displaystyle{}M_{u}$ $\displaystyle=$ $\displaystyle
Y_{u}^{ij}\upsilon^{5}_{u},$ $\displaystyle M_{d}$ $\displaystyle=$
$\displaystyle
Y_{d}^{ij}\upsilon^{\bar{5}}_{d}+Y^{ij}_{45}\upsilon^{\bar{45}}_{d},$
$\displaystyle M_{e}$ $\displaystyle=$ $\displaystyle
Y_{d}^{ij}\upsilon^{\bar{5}}_{d}-3Y^{ij}_{45}\upsilon^{\bar{45}}_{d}.$ (4)

where $\upsilon^{5}_{u}$ and $\upsilon^{\bar{5}}_{d}$ is vev of Higgs doublet
$H_{u}$ and $H_{d}$ in $5$ and $\bar{5}$, respectively; $\upsilon_{45}$ is the
vev of doublet $\sigma_{d}$ in $\bar{45}$, which is defined as
$\left<H^{b5}_{a}\right>=\upsilon_{45}(1,1,1,-3)_{\text{diag}}$ for $a,b=1-4$.
Without the $\bar{45}$, $M_{d}=M_{e}$ at the GUT scale for all of three
generations. While such mass relation is viable for the third generation, they
should be avoided for the first two generations. Adding $\bar{45}$ can tune
the incorrect mass relations to the desired ones,

$\displaystyle{}m_{d}\simeq 3m_{e},~{}m_{s}\simeq m_{\mu}/3,~{}m_{b}\simeq
m_{\tau},$ (5)

The GUT-scale mass relations in Eq.(5) strongly constrain the Yukawa matrixes
$Y_{u,d,45}$. For example, some specific choices on $Y_{u,d,45}$ in the
Georgi-Jarlskog scheme GJ  lead to a stable $b$ quark. In order to solve the
SM flavor issue, we choose the Fritzsch scheme Fritzsch , in which
$Y_{u,d,45}$ take the following forms

$\displaystyle{}Y_{u}^{ij}\upsilon_{u}=\left(\begin{array}[]{ccc}0&A_{u}&0\\\
A_{u}&0&B_{u}\\\ 0&B_{u}&C_{u}\\\ \end{array}\right),$ (9) $\displaystyle
Y_{d}^{ij}\upsilon_{d}=\left(\begin{array}[]{ccc}0&A_{d}&0\\\ A_{d}&0&B_{d}\\\
0&B_{d}&C_{d}\\\ \end{array}\right),$ (13) $\displaystyle
Y_{45}^{ij}\upsilon_{45}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&D_{d}\\\
0&D_{d}&0\\\ \end{array}\right),$ (17)

where there are small mass hierarchies $C_{f}>>B_{f}>>A_{f}$ with
$f=\\{u,d,e\\}$ in Eq.(9) so as to address the SM flavor mass hierarchies.
Substituting Eq.(9) into Eq.(5) implies that there is a fine tuning between
$B_{d}$ and $D_{d}$,

$\displaystyle{}D_{d}=(1\pm\frac{2\sqrt{3}}{3})B_{d}.$ (18)

With this fine-tuning solution, the diagonalizations of matrixes in Eq.(9)
yield the CKM matrix

$\displaystyle{}\mid
V_{\text{CKM}}\mid\simeq\left(\begin{array}[]{ccc}0.974&0.227&0.004\\\
0.227&0.970&0.042\\\ 0.008&0.042&0.999\\\ \end{array}\right),$ (22)

which is in good agreement with experimental data.

For $W_{Y_{\nu}}$ responsible neutrino masses we take a simple form as
follows,

$\displaystyle{}W^{\rm{SU}(5)}_{Y_{\nu}}$ $\displaystyle=$ $\displaystyle
Y_{N}^{ij}N_{Ri}(1)\psi_{aj}(\bar{5})H^{a}(5)$ (23) $\displaystyle+$
$\displaystyle Y_{S}^{ij}S(1)\cdot N_{Ri}(1)N_{Rj}(1)+\text{H.c}$

where $S$ is a singlet of $G_{\text{SM}}=SU(3)_{c}\times SU(2)_{L}\times
U(1)_{Y}$, with $\langle S\rangle$ of order GUT scale. In Eq.(23), one finds
the neutrino Dirac mass $M_{\nu}=Y^{ij}_{N}\upsilon^{5}_{u}$ and the right-
hand neutrino mass $M_{N_{R}}=Y^{ij}_{S}\left<S\right>$, which results in the
left-hand neutrino masses in terms of type-I typeI1 ; typeI2 ; typeI3 ; typeI4
seesaw mechanism,

$\displaystyle{}m_{\nu}=M^{T}_{\nu}M_{N_{R}}^{-1}M_{\nu}=Y_{N}^{T}Y^{-1}_{S}Y_{N}\frac{\upsilon^{5}_{u}}{\langle
S\rangle}.$ (24)

Given $\upsilon^{5}_{u}\sim 10^{2}$ GeV, $\langle S\rangle\sim
10^{15}-10^{16}$ GeV and Yukawa couplings of order unity, the neutrino mass is
of order $\sim 10^{-2}-10^{-3}$ eV. Similar to $Y_{u,d,e}$, $Y_{S}$ and
$Y_{N}$ in Eq.(23) are also constrained by the fit to neutrino mixings as
described by PMNS matrix $U_{\text{PMNS}}=U_{e}^{-1}U_{\nu}^{-1}$, where
$U_{\nu}$ and $U_{e}$ are defined to diagonalize mass matrix $m_{\nu}$ and
$M_{e}$, respectively.

After we have established on a benchmark solution111It is of special interest
to examine whether there is viable solution to the input parameters in the
case that all mass matrixes such as $M_{u,d,e,N_{R}}$ are assigned the
Fritzsch form. to the input parameters at the GUT scale which can explain the
SM flavor issue and neutrino masses, we turn to the structure of
$W_{\text{SB}}$. $1)$. In order to obtain light neutrino masses, $\langle
S\rangle$ of order GUT scale is required. $2)$. In order to break the
$\rm{SU}(5)$ we introduce $75(Z)$. With such $75$, we can add a $50$ and
$\bar{50}$ to achieve the doublet-triplet splitting 0007254  for $5$ and
$\bar{5}$. $3)$. In order to gain nonzero vev $\upsilon^{\bar{45}}_{d}$, we
include another $75(Z^{\prime})$ with a vev of the GUT scale. The reason for
this is that neither $1$ nor $24$ with large vev is favored by the product
$H(5)\times\bar{H}(\bar{5})=1+24$. $4)$, Due to the singlet $S$ there is an
unsafe operator

$\displaystyle{}W_{\text{unsafe}}=SH(5)\bar{H}(\bar{5}),$ (25)

which must be eliminated.

Shown in Table.1 is the $Z_{2}\times Z^{\prime}_{2}$ parity, which can
eliminate the unsafe operator in Eq.(25). Under this parity, $W_{\text{SB}}$
reads as,

$\displaystyle{}W^{\rm{SU}(5)}_{\text{SB}}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}}{2}Z^{2}+\frac{M_{Z^{\prime}}}{2}Z^{\prime
2}+\frac{M_{S}}{2}S^{2}+\frac{M_{S^{\prime}}}{2}S^{\prime 2}$ (26)
$\displaystyle+$
$\displaystyle\frac{\lambda_{Z}}{3}Z^{3}+\frac{\lambda_{S}}{3}S^{3}$
$\displaystyle+$ $\displaystyle\lambda_{1}SZ^{2}+\lambda_{2}SZ^{\prime
2}+\lambda_{3}ZZ^{\prime 2}+\lambda_{4}SS^{\prime 2}$ $\displaystyle+$
$\displaystyle\bar{H}(\bar{45})\left(M_{45}+\lambda_{5}S\right)H(45)$
$\displaystyle+$ $\displaystyle\lambda_{6}S^{\prime}\bar{H}(\bar{50})H(50)$
$\displaystyle+$
$\displaystyle\lambda_{7}Z\bar{H}(\bar{5})H(50)+\lambda_{8}Z^{\prime}H(5)\bar{H}(\bar{50})$
$\displaystyle+$ $\displaystyle\lambda_{9}ZH(45)\bar{H}(\bar{5})+\text{H.c}.$

According to the $F$ terms in Eq.(26), the nonzero singlet vevs
$\langle(1,1,1)_{Z}\rangle=Z$,
$\langle(1,1,1)_{Z^{\prime}}\rangle=Z^{\prime}$, $\langle(1,1,1)_{S}\rangle=S$
and $\langle(1,1,1)_{S^{\prime}}\rangle=S^{\prime}$ are given by,

$\displaystyle{}\frac{S}{M_{Z}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\lambda_{4}}\eta_{3}=-\frac{1}{2\lambda_{2}}\left(-2\lambda_{3}\frac{Z}{M_{Z}}+\eta_{1}\right),$
$\displaystyle\frac{Z^{\prime}}{M_{Z}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\lambda_{S}}}\left(\frac{-Z}{M_{Z}}\right)^{\frac{1}{2}}\left[1+\lambda_{Z}+\frac{\lambda_{1}}{\lambda_{2}}\left(-2\lambda_{3}\frac{Z}{M_{Z}}+\eta_{1}\right)\right]^{\frac{1}{2}}$
$\displaystyle\frac{Z}{M_{Z}}$ $\displaystyle=$
$\displaystyle\frac{-b\pm\sqrt{b^{2}-4ac}}{2a},$ (27)

where

$\displaystyle{}a$ $\displaystyle=$
$\displaystyle\lambda_{1}+\frac{\lambda^{2}_{3}}{\lambda^{2}_{2}}\lambda_{S}-\frac{\lambda_{2}\lambda_{Z}}{\lambda_{S}}+2\frac{\lambda_{1}\lambda_{3}}{\lambda_{S}}$
$\displaystyle b$ $\displaystyle=$
$\displaystyle-\frac{\lambda_{3}}{\lambda_{2}}-\frac{\lambda_{3}\lambda_{S}}{\lambda^{2}_{2}}\eta-\frac{\lambda_{2}}{\lambda_{S}}-\frac{\lambda_{1}}{\lambda_{S}}\eta$
$\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{1}{2\lambda_{2}}\eta_{1}\eta_{2}+\frac{\lambda_{S}}{4\lambda^{2}_{2}}\eta_{1}^{2}+\lambda_{4}\left(\frac{S^{\prime}}{M_{Z}}\right)^{2}$
(28)

and $\eta_{1}=M_{Z^{\prime}}/M_{Z}$, $\eta_{2}=M_{S}/M_{Z}$,
$\eta_{3}=M_{S^{\prime}}/M_{Z}$.

A few comments are in order regarding the parity assignments. Firstly, the
$Z_{2}$ parity eliminates the unsafe operator in Eq.(25). Secondly, without
$Z^{\prime}_{2}$ large $\mu$ term for the doublets in $5$ and $\bar{5}$ would
be induced by the mixings with $45$ and $\bar{45}$. Instead, imposing
$Z^{\prime}_{2}$ forbids the operator $Z^{\prime}H(5)\bar{H}(\bar{45})$, which
then keeps the doublets in $5$ and $\bar{5}$ light. Finally, due to the last
line in Eq.(26) which is consistent with $Z_{2}\times Z^{\prime}_{2}$, the
effective operator $\bar{H}(\bar{5})H(45)Z^{\prime 2}/M_{Z}$ is produced after
integrating $Z$. Thus, the effective superpotential for the doublets in $45$
and $\bar{45}$ at the leading order is given by,

$\displaystyle{}W_{\text{eff}}\sim\left(M_{45}+\lambda_{5}\langle
S\rangle\right)\sigma_{u}\sigma_{d}+\lambda_{3}H_{d}\sigma_{u}Z^{\prime
2}/M_{Z},$ (29)

where corrections due to those mixings among singlets of $Z$, $Z^{\prime}$ and
$S$ have been neglected. From Eq.(29) we obtain the vev

$\displaystyle{}\sigma_{d}\sim\lambda_{3}\frac{Z^{\prime
2}}{(M_{45}+\lambda_{4}\langle S\rangle)M_{Z}}\upsilon_{d}.$ (30)

Given $\lambda_{3}\sim 0.1$, $\upsilon_{d}\sim 10$ GeV, we have
$\sigma_{d}\sim 100$ MeV. A rational Yukawa texture such as
$Y_{45}\sim(0.01,0.1,1)$ for the three generations then reproduces the mass
relations in Eq.(5).

Field |  $N_{R}$ |  $\psi$ |  $\Phi$ |  $5$ |  $\bar{5}$ |  $\bar{45}$ |  $1(S)$ |  $1(S^{\prime})$ |  $75(Z)$ |  $75(Z^{\prime})$ | $45$ |  $50$ | $\bar{50}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$Z_{2}$ | + | + | - | + | - | - | + | + | + | - | - | - | -
$Z^{\prime}_{2}$ | + | + | + | + | + | + | + | - | + | - | + | + | -
Table 1: $Z_{2}\times Z^{\prime}_{2}$ parity assignments in the case of
embedding MSSM into $\rm{SU}(5)$, which are consistent with the
superpotentials in Eq.(2) and Eq.(26).

###  II.2 SO(10)

Unlike the case of $\rm{SU(5)}$, the input parameters in the $\rm{SO(10)}$
which control the SM fermions masses $M_{u,d,e}$ and neutrino masses $m_{\nu}$
are tied to each other. The main reason for this is that the MSSM matters of
each generation are contained in a single $16(\phi)$. Here, we give a brief
review on the embedding of MSSM into $\rm{SO}(10)$.

According to the product $16\times 16=10_{s}+120_{a}+126_{s}$, the Higgs
representation responsible for SM fermion masses can be composed of $10$,
$120$ and $\bar{126}$. By following previous discussions on $\rm{SU}(5)$ GN1 ;
GN2 , the simplest choice is to introduce $10$ and $\bar{126}$ 9209215 , with
Yukawa superpotential given by,

$\displaystyle{}W^{\rm{SO}(10)}_{Y}=\phi_{i}(16)[Y^{ij}_{10}H(10)+Y^{ij}_{126}\bar{H}(\bar{126})]\phi_{j}(16),$
(31)

where matrixes $Y_{10}$ and $Y_{126}$ are both symmetric. From Eq.(31) we have

$\displaystyle{}M_{u}$ $\displaystyle=$ $\displaystyle
Y_{10}^{ij}\upsilon^{10}_{u}+Y_{126}^{ij}\upsilon^{\bar{126}}_{u},$
$\displaystyle M_{d}$ $\displaystyle=$ $\displaystyle
Y_{10}^{ij}\upsilon^{10}_{d}+Y_{126}^{ij}\upsilon^{\bar{126}}_{d},$
$\displaystyle M_{e}$ $\displaystyle=$ $\displaystyle
Y_{10}^{ij}\upsilon^{10}_{d}-3Y_{126}^{ij}\upsilon^{\bar{126}}_{d},$ (32)

for SM quarks and electrons, and

$\displaystyle{}M_{N_{R}}$ $\displaystyle=$ $\displaystyle
Y_{126}^{ij}\upsilon^{\bar{126}}_{s}$ $\displaystyle M_{\nu}$ $\displaystyle=$
$\displaystyle
Y_{10}^{ij}\upsilon^{10}_{u}-3Y_{126}^{ij}\upsilon^{\bar{126}}_{d},$
$\displaystyle M_{L}$ $\displaystyle=$ $\displaystyle
Y_{126}^{ij}\upsilon^{\bar{126}}_{L}$ (33)

for neutrinos, where $\upsilon^{10}_{u,d}$ refer to the doublet vevs in
$H(10)$, and $\upsilon^{\bar{126}}_{s,u,d,L}$ denote the singelt (s), doublet
(u, d) and triplet (L) vevs in $\bar{H}(\bar{126})$, respectively. Therefore,
the neutrino mass arises from both type-I typeI1 ; typeI2 ; typeI3 ; typeI4
and type-II typeII1 ; typeII2 ; typeII3  contributions,

$\displaystyle{}m_{\nu}=M_{L}-M^{T}_{\nu}M_{N_{R}}^{-1}M_{\nu}.$ (34)

Benchmark solutions to the input parameters in Eq.(II.2) and Eq.(II.2) have
been extensively studied 9209215 ; SO10fit1 ; SO10fit2 ; SO10fit3 ; SO10fit4 ,
which demonstrate that the fit to SM flavor masses and mixings is viable. But
the construction of $W_{\text{SB}}$ responsible for the breaking of
$\rm{SO}(10)\rightarrow G_{\text{SM}}$ is challenging SO10Mp ; SO10Mp1 ;
SO10Mp2 ; SO10Mp3 , since some of triplet fields in $126$ and $\bar{126}$
obtain mass of order $\upsilon^{\bar{126}}_{s}\sim 10^{13}$ GeV, which spoils
the perturbative unification. Attempts to solve this issue involve adding $54$
1805.10631  to $W_{\text{SB}}$ or adding $120$ 1805.05776  to $W_{Y}$ in
Eq.(31).

We employ the solution of modifying $W_{\text{SB}}$ 1805.10631 , where the
Higgs fields are composed of $210(Y)$, $H(126)$, $\bar{H}(\bar{126})$ and
$54(X)$, and $W_{\text{SB}}$ takes the form222We follow the notation in
0405300  for Yukawa coupling constants.

$\displaystyle{}W^{\text{SO}(10)}_{\text{SB}}$ $\displaystyle=$
$\displaystyle\frac{M_{Y}}{2}Y^{2}+\frac{M_{X}}{2}X^{2}+M_{126}H(126)\bar{H}(\bar{126})$
(35) $\displaystyle+$
$\displaystyle\lambda_{1}Y^{3}+\lambda_{2}YH(126)\bar{H}(\bar{126})$
$\displaystyle+$ $\displaystyle
Y\left[\lambda_{3}H(126)H(10)+\lambda_{4}\bar{H}(\bar{126})H(10)\right]$
$\displaystyle+$ $\displaystyle
X\left[\lambda_{8}X^{2}+\lambda_{10}Y^{2}+\lambda_{11}H^{2}(126)\right.\ $
$\displaystyle+$ $\displaystyle\left.\
\lambda_{12}\bar{H}^{2}(\bar{126})+\lambda_{13}H^{2}(10)\right]+\text{H.c}.$

Here are a few comments about $W_{\text{SB}}$ in order. $1)$. Under the
notation of $SU(4)\times SU(2)_{L}\times SU(2)_{R}$ the SM singlet vevs
$Y_{1}=\langle(1,1,1)_{Y}\rangle$, $Y_{2}=\langle(15,1,1)_{Y}\rangle$ and
$Y_{3}=\langle(15,1,3)_{Y}\rangle$ in $Y$ and $X=\langle(1,1,1)_{X}\rangle$ in
$X$ are responsible for the breaking $\rm{SO}(10)\rightarrow
G_{\text{SM}}\times U(1)_{B-L}$. $2)$. The SM singlet vevs
$\upsilon^{126}_{s}=\langle(\overline{10},1,3)_{126}\rangle$ and
$\upsilon^{\bar{126}}_{s}=\langle(10,1,3)_{\bar{126}}\rangle$ result in the
breaking of $G_{\text{SM}}\times U(1)_{B-L}\rightarrow G_{\text{SM}}$.

In the limit $\upsilon^{126}_{s}=\upsilon^{\bar{126}}_{s}<<Y_{i},X$ the vevs
are given by 1805.10631 ,

$\displaystyle{}\frac{Y_{1}}{Y_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{3}}\eta^{2},$ $\displaystyle\frac{M_{Y}}{Y_{2}}$
$\displaystyle=$
$\displaystyle-\frac{\lambda_{1}}{5\sqrt{2}}\left(3+\eta^{2}\right),$
$\displaystyle\frac{X}{Y_{2}}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{1}}{\lambda_{10}\sqrt{30}}\left(-2+\eta^{2}\right),$
$\displaystyle\frac{M_{126}}{Y_{2}}$ $\displaystyle=$
$\displaystyle-\frac{\lambda_{2}}{120}\left(6\sqrt{2}+12\eta+\sqrt{2}\eta^{2}\right),$
$\displaystyle\frac{M_{X}}{Y_{2}}$ $\displaystyle=$
$\displaystyle\frac{-1}{20\sqrt{2}\lambda_{1}\lambda_{10}}\left[\left(5\lambda_{10}^{3}+2\lambda_{1}^{2}\lambda_{8}\right)\eta^{2}+20\lambda_{10}^{3}-4\lambda_{1}^{2}\lambda_{8}\right]$

with $\eta=Y_{3}/Y_{2}$.

##  III NMSSM

With the embedding of MSSM into realistic SUSY GUTs as a benchmark, in this
section we analyze the NMSSM. According to the starting points in the
Introduction, a viable embedding should satisfy two constraints:

  * •

The mass of $N$ should be of order TeV scale.

  * •

The vev of $N$ should be of order TeV scale.

Both of them may be spoiled by a few dangerous mixings between $N$ and Higgs
fields which contain singlet vev of order GUT scale. The key point is whether
there is suitable symmetry to avoid such mixings.

###  III.1 SU(5)

In this situation, $W_{Y}$ in Eq.(2) should be extended to include
$NH_{u}H_{d}+\frac{\kappa}{3}N^{3}+\text{H.c}$ in the NNSSM, which means
that333Operator $NH(5)\bar{H}(\bar{5})$ contributes to Yukawa interaction
$NH_{c}\bar{H}_{c}$ beyond MSSM, with $H_{c}$ and $\bar{H}_{c}$ being the
color-triplet Higgs fields. However, it does not affect proton decay at all,
as the singlet $N$ mass is always far larger than proton mass.,

$\displaystyle{}\delta W_{Y}=\lambda
NH(5)\bar{H}(\bar{5})+\frac{\kappa}{3}N^{3}.$ (37)

Eq.(37) do not affect the fit to SM flavor masses and mixings in Sec.IIA.
Nevertheless, compared to MSSM, $W_{\text{SB}}$ is allowed to contain
superpotential terms

$\displaystyle{}W_{\text{unsafe}}=NZ^{2}+NZ^{\prime 2}+NS^{2}+N^{2}S.$ (38)

These new terms in Eq.(38) yield corrections to the $F$-terms of $Z$,
$Z^{\prime}$, $S$ such as $F_{S}=F^{\text{MSSM}}_{S}+2\lambda_{N}SN$, which
can be adjusted to the case of MSSM by e.g. $\left<N\right>=0$. Even so, the
singlet vevs $\left<Z\right>$, $\left<Z^{\prime}\right>$ and $\left<S\right>$
still lead to either large $N$ mass or large mixing.

In order to avoid all of mixing terms in Eq.(38), we need to impose new
parity. The first observation is that an odd $N$ under a $Z_{2}$ parity as
shown in Table.1 excludes the first three terms in Eq.(38). But the last term
therein still remains 444A economic solution to keep light $N$ is adding
another singlet $S^{\prime}=1$ which is even under the $Z_{2}$. With such
$S^{\prime}$ the $W_{\text{SB}}$ is further extended by,
$\displaystyle{}\delta
W_{\text{SB}}(S^{\prime})=\frac{1}{2}M_{S^{\prime}}S^{\prime
2}+m_{SS^{\prime}}SS^{\prime}+N^{2}S^{\prime}.$ (39) from which
$F_{S^{\prime}}=M_{S^{\prime}}S^{\prime}+m_{SS^{\prime}}S+N^{2}$, and $\langle
S^{\prime}\rangle=-\langle S\rangle$ if $M_{S^{\prime}}=m_{SS^{\prime}}$. The
two different contributions to $N$ mass cancel each other, leaving us a light
$N$. Unfortunately, neither Abelian or $Z_{N}$ parity can ensure
$M_{S^{\prime}}=m_{SS^{\prime}}$.. Similar result holds for $Z_{N}$ or an
Abelian symmetry.

###  III.2 SO(10)

Similar to the embedding of NMSSM into $\rm{SU}(5)$, $W_{Y}$ in Eq.(31) is
modified by,

$\displaystyle{}\delta W_{Y}=\lambda NH^{2}(10)+\frac{\kappa}{3}N^{3}.$ (40)

Instead of Eq.(38), $W_{\text{SB}}$ in Eq.(35) is allowed by gauge invariance
to contain

$\displaystyle{}W_{\text{unsafe}}$ $\displaystyle=$ $\displaystyle
NY^{2}+NH(126)\bar{H}(\bar{126})+NX^{2}+XH^{2}(10),$

in which $N$ mixes with the SM singlets of $Y$, $126$, $\bar{126}$ and $X$.
Thus, all of Yukawa couplings in Eq.(III.2) have to be extremely small.

What kind of parity allows Yukawa superpotential in Eq.(40) but eliminates
that in Eq.(III.2) simultaneously? The first observation is that a $Z_{2}$
parity does not work. Since Eq.(III.2) would imply $N$ an odd field, which
contradicts with the Yukawa superpotential in Eq.(40). Similar results hold
for any $Z_{N}$ parity. Because the rational assignment $n_{Y}=0$ as required
by successful symmetry breaking, demonstrates that in order to allow the
Yukawa superpotential in Eq.(40), $n_{10}=n_{\bar{126}}=n_{126}=N/2$.
Accordingly, $n_{N}=0$ from $N10(H)10(H)$, which implies that some of terms in
Eq.(III.2) are still allowed. To conclude, in our setup embedding NMSSM into
the minimal $\rm{SO}(10)$ is not viable.

##  IV VMSSM

Let us proceed to discuss the embedding of VMSSM into SUSY $\rm{SU(5)}$ and
$\rm{SO}(10)$. The VL fermions with mass of order TeV scale can be composed of
$5$ with $\bar{5}$, $10$ with $\bar{10}$ in the $\rm{SU}(5)$ or $16$ with
$\bar{16}$ in the $\rm{SO}(10)$ respectively Zheng1 ; Zheng2 . A realistic
embedding should satisfy the following constraints.

  * •

The vev of Higgs field $\rho$ responsible for the VL fermion masses should be
of order TeV scale.

  * •

The VL fermions are prevented from directly coupling to the Higgs fields
triggering high-scale gauge symmetry breaking.

Violating the first constraint is likely to occur because either
$\rho=\\{1,24,75\\}$ or $\rho=\\{1,45,210\\}$ may directly couple to
$S$,$S^{\prime}$ $Z$, $Z^{\prime}$ in the case of $\rm{SU}(5)$ or $X$ and $Y$
in the case of $\rm{SO}(10)$, respectively, which tends to yield
$\langle\rho\rangle$ of GUT scale. In contrast, $\langle\rho\rangle$ of order
TeV scale can be only realized by the effective operator such as

$\displaystyle{}W_{\text{eff}}(\rho)=\frac{M_{\rho}}{2}\rho^{2}+\frac{\rho\cdot
A\cdot B\cdots}{M^{n}_{\text{U}}}$ (42)

where $A$, $B$, $\cdots$ refer to $H(5)$, $\bar{H}(\bar{5})$, $S$,
$S^{\prime}$, $Z$, $Z^{\prime}$ in the $\rm{SU}(5)$, or $H(126)$,
$\bar{H}(\bar{126})$, $X$ and $Y$ in the $\rm{SO}(10)$, respectively, with
$M_{U}$ denoting the GUT scale.

###  IV.1 SU(5)

For the VL fermions of $5(\Sigma)$ and $\bar{5}(\bar{\Sigma})$, $W_{Y}$ in
Eq.(2) is extended by,

$\displaystyle{}\delta W_{Y}=\rho\bar{\Sigma}(\bar{5})\Sigma(5)+\text{H.c},$
(43)

where $\rho=\\{1,24\\}$ of $\rm{SU}(5)$. The reason for adding $\rho$ is that
either singlet vev $S$ or $S^{\prime}$ in the Sec.IIA is too large to provide
a VL mass of order TeV.

In this case, the unsafe superpotential at least includes

$\displaystyle{}W_{\text{unsafe}}\supset S\Sigma\bar{\Sigma},$ (44)

which can be excluded by imposing the $Z_{2}$ parity assignments as shown in
Table.2. $W_{\text{unsafe}}$ in Eq.(44) can also contain the following terms
depending on $\rho$,

$\displaystyle{}W_{\text{unsafe}}=\left\\{\begin{array}[]{lcl}\rho Z^{2}+\rho
Z^{\prime 2}+S\rho Z^{\prime}+\rho^{2}S,~{}\rho=24\\\ \rho Z^{2}+\rho
Z^{\prime 2}+\rho^{2}S+\rho S^{2},~{}~{}~{}\rho=1\\\ \end{array}\right.$ (47)

Field | $N_{R}$ |  $\psi$ |  $\Phi$ | $\Sigma$ | $\bar{\Sigma}$ |  $5$ |  $\bar{5}$ |  $\bar{45}$ | $\rho$ |  $1(S)$ | $1(S^{\prime})$ |  $75(Z)$ | $75(Z^{\prime})$ | $45$ |  $50$ | $\bar{50}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$Z_{2}$ | + | + | - | + | - | + | - | - | - | + | + | + | - | - | - | -
Table 2: $Z_{2}$ parity assignments for embedding NMSSM into $\rm{SU}(5)$,
which is consistent with the superpotential in Eq.(43). Yet, it is unable to
exclude all the unsafe structures in Eq.(47).

Besides the unsafe operators in Eq.(47), there are also no suitable Feynman
graphs to generate the desired effective operator with correct mass order in
Eq.(42). In principle, the form of effective operator in Eq.(42) can be
divided as follows:

  * •

It is of type $\rho AB\cdots/M^{2}_{\text{U}}$, with $A$ and $B$ referring to
$S$, $Z$ or $Z^{\prime}$. In this situation, one obtains
$\langle\rho\rangle\sim$ GUT scale given all of $A$, $B$, $\cdots$ are of
order GUT scale.

  * •

It looks like $\rho AB\cdots H(5)/M_{\text{U}}$ or $\rho
AB\cdots\bar{H}(\bar{5})/M_{\text{U}}$, which contain only a doublet vev. Such
operator contributes to $\langle\rho\rangle\sim$ TeV for $M_{\rho}\sim$ GUT
scale. Unfortunately, there is no suitable intermediator.

  * •

It is of form $\rho AB\cdots H(5)\bar{H}(\bar{5})\cdots/M^{2}_{\text{U}}$ or
$\rho AB\cdots\bar{H}(\bar{5})H(45)\cdots/M^{2}_{\text{U}}$, where there are
at least two doublet vevs. In this case, $\langle\rho\rangle$ is less than
$\mathcal{O}(1)$ eV.

In summary, since there is no appropriate vev scale, the embedding of VMSSM
into $\rm{SU}(5)$ is not viable. Similar result can be found for VL $10$ and
$\bar{10}$, where $\rho=\\{1,24,75\\}$ of $\rm{SU}(5)$.

###  IV.2 SO(10)

As shown in Zheng2 , the non-minimal extension through the $16$-dimensional VL
fermions remains consistent with the $\rm{SO}(10)$ unification. In this model,
$W_{Y}$ is modified by,

$\displaystyle{}\delta W_{Y}=\rho\Delta(16)\bar{\Delta}(\bar{16}),$ (48)

where $\rho=\\{1,45,210\\}$ of $\rm{SO}(10)$. Similar to Sec.IVA, the main
challenge to the embedding is the generation of singlet vev
$\langle\rho\rangle\sim$ TeV scale for a $\rho$ mass of GUT scale. What
differs from the discussions in Sec.IVA is the existence of the intermediate
mass scales $\upsilon^{126}_{s}$ and $\upsilon^{\bar{126}}_{s}$ in the
$\rm{SO}(10)$, which is critical to solve the problem.

Figure 1: Super Feynman graph for the generation of higher-dimensional
effective operator in the case of $\rho=210$.

We firstly consider $\rho=210$. We add Higgs fields $54(V)$ and $54(U)$ to
$W_{\text{SB}}$, with the $Z_{2}$ parity assignments as shown in Table.3. The
$Z_{2}$ parity excludes the unsafe operator

$\displaystyle{}W_{\text{unsafe}}=Y\Delta(16)\bar{\Delta}(\bar{16}),$ (49)

and simultaneously allows Yukawa interactions

$\displaystyle{}\delta W_{\text{SB}}$ $\displaystyle=$
$\displaystyle\frac{M_{\rho}}{2}\rho^{2}+\frac{M_{V}}{2}V^{2}+\frac{M_{U}}{2}U^{2}$
(50) $\displaystyle+$ $\displaystyle\lambda^{\prime}_{10}\rho
UY+\lambda^{\prime\prime}_{10}\rho VY$ $\displaystyle+$
$\displaystyle\lambda^{\prime}_{8}UVX+\lambda^{\prime\prime}_{8}U^{2}X+\lambda^{\prime\prime\prime}_{8}V^{2}X.$

In terms of Eq.(50) and Eq.(35) the effective operator for $\rho$ is given by,

$\displaystyle{}W_{\text{eff}}$ $\displaystyle\simeq$
$\displaystyle\frac{M_{\rho}}{2}\rho^{2}+\rho(\lambda^{\prime}_{10}U+\lambda^{\prime\prime}_{10}V)\left[\frac{H(126)\bar{H}(\bar{126})}{M_{Y}}+\cdots\right]$
(51) $\displaystyle+$
$\displaystyle\rho\frac{\lambda^{\prime}_{10}UH^{3}(126)\bar{H}(\bar{126})}{M_{Y}M_{X}M_{V}}+\rho\frac{\lambda^{\prime\prime}_{10}VH^{3}(126)\bar{H}(\bar{126})}{M_{Y}M_{X}M_{U}}$
$\displaystyle+$ $\displaystyle\mathcal{O}(\rho^{3})$

For calculating an effective superpotential in the infrared region from those
in the ultraviolet region, integrating out heavy chiral superfields in the
Feynman graphs is equivalent to solving the nonlinear equations of $F$ terms
related to these heavy chiral superfileds. The leading-order terms with
coefficient $\lambda^{\prime}_{10}U+\lambda^{\prime\prime}_{10}V=\delta F_{Y}$
in Eq.(51) are obtained after integrating out superfield $Y$ for $\lambda_{1}$
and $\lambda_{10}$ less than unity. Similarly, the next-leading order
operators therein are induced by further integrating out $X$, referring to
which the Feynman graph is shown in Fig.1. Note, we have used the mass term in
Eq.(51) representing those quadratic terms, and neglected the higher-order
terms.

Apart from the $F$-term contributions in Eq.(51), the potential for singlet
component in $\rho$ also contains soft SUSY-breaking terms such as
$A_{\rho}\rho^{3}+\text{H.c}$. The scale of $A_{\rho}$ depends on the details
of SUSY breaking. It can be neglected in gauge mediation, but be of order
$\sim 1$ TeV in the other scenarios. In the later case, it is easily to verify
that the $F$-term contribution still dominates over those soft terms. In what
follows, we simply ignore those soft terms for the estimate of
$\langle\rho\rangle$.

The leading-order contribution in Eq.(51) has to be eliminated if one wants to
obtain singlet vev $\langle\rho\rangle\sim$ TeV scale. Alternatively, the
coefficient $\delta F_{Y}$ must be suppressed without much fine tuning. There
is a dynamical realization for this purpose. From Eq.(50), one finds that the
SUSY vacuum described by the vevs in Eq.(II.2) remains only if the following
constraints

$\displaystyle{}0$ $\displaystyle=$
$\displaystyle\lambda^{\prime}_{8}UV+\lambda^{\prime\prime}_{8}U^{2}+\lambda^{\prime\prime\prime}_{8}V^{2},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\lambda^{\prime}_{10}U+\lambda^{\prime\prime}_{10}V,$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
M_{V}V+\frac{X}{2\sqrt{15}}\left(\lambda^{\prime}_{8}U+2\lambda^{\prime\prime\prime}_{8}V\right),$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
M_{U}U+\frac{X}{2\sqrt{15}}\left(\lambda^{\prime}_{8}V+2\lambda^{\prime\prime}_{8}U\right),$
(52)

are satisfied. Given GUT-scale mass parameters $M_{U}$, $M_{V}$, $\langle
U\rangle$ and $\langle V\rangle$, there are indeed rational solutions to
Eq.(IV.2), under which the leading-order operator in Eq.(51) vanishes due to
$\delta F_{Y}=0$ (the second formula in Eq.(IV.2)). Therefore, the next
leading-order contribution in Eq.(51) dominates the effective superpotential
for $\rho$ below GUT scale as long as $M_{U}\neq M_{V}$,

$\displaystyle{}W_{\text{eff}}\sim
M_{\rho}\rho^{2}+\rho\frac{VH^{3}(126)\bar{H}(\bar{126})}{M^{3}_{U}},$ (53)

which contributes to a nonzero vev:

$\displaystyle{}\langle\rho\rangle\sim\frac{\langle
V\rangle}{M_{\rho}}\frac{\left(\upsilon^{126}_{s}\right)^{4}}{M^{3}_{U}}.$
(54)

Given singelt vev $\upsilon^{126}_{s}\simeq\upsilon^{\bar{126}}_{s}\sim
10^{13}$ GeV fixed by fit to SM flavor masses 1805.10631 , and $M_{\rho}\sim
M_{U}\sim\langle V\rangle\sim 10^{16}$ GeV, we have $\langle\rho\rangle\sim
1-10$ TeV.

Field |  $\phi$ |  $\Delta$ |  $\bar{\Delta}$ |  $10$ | $\bar{126}$ | $\rho$ |  $X$ |  $126$ |  $54(V)$ | $54(U)$ | $Y$
---|---|---|---|---|---|---|---|---|---|---|---
$Z_{2}$ | + | + | - | + | + | - | + | + | - | - | +
Table 3: $Z_{2}$ parity assignments in the VMSSM model with $\rho=210$, which
are consistent with the superpotentials in Eq.(31),Eq.(35), Eq.(48) and
Eq.(50), and simultaneously avoid the unsafe operator in Eq.(49).

The analysis for $\rho=210$ can shed light on other cases such as
$\rho=\\{1,45\\}$. For $\rho=1$ we can naively choose $Z_{2}$ odd fields
$U=V=210$ in Fig.1. However, an unsafe operator
$V(U)\Delta(16)\bar{\Delta}(\bar{16})$ appears again. For $\rho=45$, one may
choose $U=45$ and $V=45$ or $54$, which is unfavored by an operator similar to
$\rho=1$.

##  V Conclusion

According to the observed Higgs mass at the LHC and the dark matter direct
detection limits, the conventional MSSM - the simplest natural SUSY that is
consistent with unification - is under more pressure than ever. Such stress
can be greatly relaxed in the extended MSSM models such as NMSSM and VMSSM
which retain the unification and are still simple. In this paper, following
the assumptions that $W_{Y}$ is fixed by the SM matter content and its TeV-
scale extension, and they receive their masses from $W_{\text{SB}}$ through
Higgs mechanism, we have studied the embeddings of these three models into
SUSY GUTs.

First of all, we discussed the MSSM, where the realistic $\rm{SU}(5)$ and
$\rm{SO}(10)$ realizations serve as benchmark solutions to the SM flavor issue
and neutrino masses. Then, we utilize the benchmark MSSM as guidance to the
embedding of NMSSM and VMSSM. We have found that the embedding of NMSSM is not
viable due to a large mount of mixings between the singlet $N$ and the Higgs
fields responsible for the GUT symmetry breaking. But the problem can be
evaded in the VMSSM, because the Higgs field $\rho$ which provides
$16$-dimensional VL mass of order TeV scale can avoid the same problems the
singlet $N$ encounters due to the intermediate mass scale in the
$126$-dimensional Higgs.

Acknowledgments. The author thanks the anonymous referee for useful
suggestions. This work is supported by the National Natural Science Foundation
of China under Grant No.11775039 and the Fundamental Research Funds for the
Central Universities with project No. cqu2017hbrc1B05 at Chongqing University.

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