6.5pt0.4pt

††thanks: Corresponding author

# The hidden sector variations in the ${\cal N}=1$ supersymmetric three-family
Pati-Salam models with symplectic groups from intersecting D6-branes

Adeel Mansha<EMAIL_ADDRESS>Department of Physics, Zhejiang Normal
University, Jinhua 321004, P. R. China  Tianjun Li<EMAIL_ADDRESS>CAS Key
Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese
Academy of Sciences, Beijing 100190, P. R. China  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing, P. R. China  Lina Wu
<EMAIL_ADDRESS>School of Sciences, Xi’an Technological University, Xi’an
710021, P. R. China

###### Abstract

It is well-known that three-family supersymmetric Pati-Salam models from
intersecting D6-branes, where either one or both of the ${\rm U}(2)$ gauge
factors are replaced by a ${\rm USp}(2)$ group, are quite scarce. In order to
construct all such kind of models with generic additional gauge symmetries, we
fix the observable sectors and study all the possible hidden sectors. Thus, we
are able to completely determine all types of such kind of the inequivalent
models on a $\mathbbm{T}^{6}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})$
orientifold from IIA string theory. We find the gauge coupling relations to be
highly sensitive to the variations in the hidden sector. One of the models
exhibits the gauge coupling unification for a particular solution at the
string scale.

##  I Introduction

Intersecting D6-branes in type IIA string theory have been extremely useful to
geometrically understand the standard model (SM) from a certain Calabi-Yau
compactification of extra dimensions. The gauge structures, the chiral spectra
and the various couplings arise from the D-brane configurations. For example,
the four-dimensional gauge couplings depend on the volume of the cycles
wrapped by the D-branes and the gravitational coupling is determined by their
total internal volume. Similarly, the cubic Yukawa couplings depend
exponentially on the triangular areas of open worldsheet intersections. The
general flavor structure and selection rules for intersecting D-brane models
has been investigated in [1, 2].

The SM fermions belong to the chiral representations of the gauge group ${\rm
SU}(3)_{C}\times{\rm SU}(2)_{W}\times{\rm U}(1)_{Y}$ such that all gauge
anomalies are canceled. Simply placing parallel D-branes in flat space does
not yield chiral fermions. Instead, to realize the chiral fermions we need to
either place D-branes on orbifold singularities [3] or else consider
intersecting D-branes on generalized orbifolds called orientifolds [4, 5]. In
orientifolds, both the discrete internal symmetries of the world-sheet theory
and the products of internal symmetries with world-sheet parity reversal
become gauged.

The ${\cal N}=1$ supersymmetric three-family Pati-Salam models in the IIA
string theory on $\mathbbm{T}^{6}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})$
orientifold with intersecting D6-branes have been constructed in ref. [6],
where the wrapping number is up to 3. Recently, utilizing machine learning
algorithm in ref. [7] several new models with the wrapping number up to 5 were
consrtructed. The phenomenology of models up to the wrapping number of 3 was
first studied in ref. [8] while the phenomenology of a newly found model with
a wrapping number 5 was explored in [9]. In ref. [10, 11], a mathematical
search algorithm was used to possibly obtain the complete landscape of
supersymmetric Pati-Salam models, where again the highest wrapping number
turned out to be 5 as already indicated from the random search in [7].
However, an interesting variation of the Pati-Salam gauge symmetry
$U(4)_{C}\times U(2)_{L}\times U(2)_{R}$ that was not taken into account in
previous searches [6, 7, 10, 11] is the ${\rm U}(4)_{C}\times{\rm
USp}(2)_{L}\times{\rm U}(2)_{R}$ or ${\rm U}(4)_{C}\times{\rm
USp}(2)_{L}\times{\rm USp}(2)_{R}$ where either one or both of the usual
$U(2)$s are replaced by $USp(2)$ groups. The construction of $USp(2)$ groups
can be readily achieved by taking the corresponding stacks of D6-branes
parallel to any of the O6-planes or their $\mathbbm{Z}_{2}$ images as
exemplified in refs. [12, 13]. The choice of $USp(2)$ is simpler to deal with
since there is no associated global anomalous $U(1)$ group which is generic in
the unitary groups.

In this paper, we fix the visible sector stacks $a$, $b$ and $c$ of the
supersymmetric Pati-Salam models with the ${\rm USp(2)}_{L,R}$ group and vary
the hidden sector to search for all possible gauge group factors consistent
with the four-dimensional ${\cal N}=1$ supersymmetry conditions and tadpole
cancellation conditions. This systematic search further pins down the allowed
number of gauge group factors for the supersymmetric Pati-Salam Models from
intersecting D6-branes [14]. Taking the two new models as examples, we show
the string-scale gauge coupling relations can be realized through two-loop
level renormalization group equation (RGE) running by introducing vector-like
particles. In [15], the additional vector-like matters are introduced to push
GUT scale unification up to string scale for the $\mathcal{N}=1$
supersymmetric Pati-Salam models. The unification of three independent gauge
interactions into a single one has been extensively studied previously from
various physical perspectives [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28]. In addition, we point out that the claim in ref. [10, 11] that there are
only 33 independent models with different gauge coupling relations may be
subtle since the gauge groups in the hidden sector might be the unitary
groups. And we shall fix this gap in the future study.

The plan of the paper is as follows. In section II, we briefly review the
model building rules for $\mathcal{N}=1$ supersymmetric Pati-Salam models with
the gauge group ${\rm SU}(4)_{C}\times{\rm USp}(2)_{L}\times{\rm USp}(2)_{R}$
on a $\mathbbm{T}^{6}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})$ orientifold. In
section III we discuss the salient phenomenological features of the newly
engineered models under the variation of the hidden sector gauge group and
list the perturbative particle spectra of the models. Finally, we conclude in
section V.

##  II Pati-Salam model building from
$\mathbbm{T}^{6}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})$ orientifold

The model building rules for supersymmetic Pati-Salam models have been
thoroughly discussed in ref. [6]. We employ similar conventions to construct
supersymmetric Pati-Salam models with symplectic groups [13]. In the
orientifold $\mathbbm{T}^{6}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})$,
$\mathbbm{T}^{6}$ is a product of three 2-tori with the orbifold group
$(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})$ has the generators $\theta$ and
$\omega$ which act on the complex coordinates $z_{i}$ as,

$\displaystyle\theta:\quad(z_{1},z_{2},z_{3})$
$\displaystyle\to(-z_{1},-z_{2},z_{3}),$
$\displaystyle\omega:\quad(z_{1},z_{2},z_{3})$
$\displaystyle\to(z_{1},-z_{2},-z_{3}).$ (1)

Orientifold projection is the gauged $\Omega R$ symmetry, where $\Omega$ is
world-sheet parity that interchanges the left- and right-moving sectors of a
closed string and swaps the two ends of an open string as,

$\displaystyle\textrm{Closed}:$
$\displaystyle\quad\Omega:(\sigma_{1},\sigma_{2})\mapsto(2\pi-\sigma_{1},\sigma_{2}),$
$\displaystyle\textrm{Open}:$
$\displaystyle\quad\Omega:(\tau,\sigma)\mapsto(\tau,\pi-\sigma),$ (2)

and $R$ acts as complex conjugation on coordinates $z_{i}$. This results in
four different kinds of orientifold 6-planes (O6-planes) corresponding to
$\Omega R$, $\Omega R\theta$, $\Omega R\omega$, and $\Omega R\theta\omega$
respectively. These orientifold projections are only consistent with either
the rectangular or the tilted complex structures of the factorized 2-tori. To
have three families of the SM fermions, we need one torus to be tilted, which
is chosen to be the third torus. So we have $\beta_{1}=\beta_{2}=0$ and
$\beta_{3}=1$.

The homology cycles for a stack $a$ of $N_{a}$ D6-branes along the cycle
$(n_{a}^{i},l_{a}^{i})$ and their $\Omega R$ images ${a^{\prime}}$ stack of
$N_{a}$ D6-branes with cycles $(n_{a}^{i},-l_{a}^{i})$ are respectively given
as,

$\displaystyle[\Pi_{a}]$
$\displaystyle=\prod_{i=1}^{3}\left(n_{a}^{i}[a_{i}]+2^{-\beta_{i}}l_{a}^{i}[b_{i}]\right),$
$\displaystyle[\Pi_{a^{\prime}}]$
$\displaystyle=\prod_{i=1}^{3}\left(n_{a}^{i}[a_{i}]-2^{-\beta_{i}}l_{a}^{i}[b_{i}]\right),$
(3)

where $\beta_{i}=0$ or $\beta_{i}=1$ for the rectangular or tilted $i^{\rm
th}$ 2-torus, respectively. The homology three-cycles, which are wrapped by
the four O6-planes, are given by

$\displaystyle\Omega R:$ $\displaystyle[\Pi_{\Omega R}]$
$\displaystyle=2^{3}[a_{1}]\times[a_{2}]\times[a_{3}],$ $\displaystyle\Omega
R\omega:$ $\displaystyle[\Pi_{\Omega R\omega}]$
$\displaystyle=-2^{3-\beta_{2}-\beta_{3}}[a_{1}]\times[b_{2}]\times[b_{3}],$
$\displaystyle\Omega R\theta\omega:$ $\displaystyle[\Pi_{\Omega
R\theta\omega}]$
$\displaystyle=-2^{3-\beta_{1}-\beta_{3}}[b_{1}]\times[a_{2}]\times[b_{3}],$
$\displaystyle\Omega R\theta:$ $\displaystyle[\Pi_{\Omega R\theta}]$
$\displaystyle=-2^{3-\beta_{1}-\beta_{2}}[b_{1}]\times[b_{2}]\times[a_{3}].$
(4)

The intersection numbers can be calculated in terms of wrapping numbers as,

$\displaystyle I_{ab}$
$\displaystyle=[\Pi_{a}][\Pi_{b}]=2^{-k}\prod_{i=1}^{3}(n_{a}^{i}l_{b}^{i}-n_{b}^{i}l_{a}^{i}),$
$\displaystyle I_{ab^{\prime}}$
$\displaystyle=[\Pi_{a}]\left[\Pi_{b^{\prime}}\right]=-2^{-k}\prod_{i=1}^{3}(n_{a}^{i}l_{b}^{i}+n_{b}^{i}l_{a}^{i}),$
$\displaystyle I_{aa^{\prime}}$
$\displaystyle=[\Pi_{a}]\left[\Pi_{a^{\prime}}\right]=-2^{3-k}\prod_{i=1}^{3}(n_{a}^{i}l_{a}^{i}),$
$\displaystyle I_{aO6}$
$\displaystyle=[\Pi_{a}][\Pi_{O6}]=2^{3-k}(-l_{a}^{1}l_{a}^{2}l_{a}^{3}+l_{a}^{1}n_{a}^{2}n_{a}^{3}+n_{a}^{1}l_{a}^{2}n_{a}^{3}+n_{a}^{1}n_{a}^{2}l_{a}^{3}),$
(5)

where $k=\sum_{i=1}^{3}\beta_{i}$ and $[\Pi_{O6}]=[\Pi_{\Omega
R}]+[\Pi_{\Omega R\omega}]+[\Pi_{\Omega R\theta\omega}]+[\Pi_{\Omega
R\theta}]$.

###  II.1 Constraints from tadpole cancellation and supersymmetry

D6-branes and O6-planes are the sources of Ramond-Ramond charges that are
constrained by the Gauss’s law in compact space, [29]

$\displaystyle\sum_{a}N_{a}[\Pi_{a}]+\sum_{a}N_{a}\left[\Pi_{a^{\prime}}\right]-4[\Pi_{O6}]=0,$
(6)

where the last terms arise from the O6-planes, which have $-4$ RR charges in
D6-brane charge units. RR tadpole constraint is sufficient to cancel the ${\rm
SU}(N_{a})^{3}$ cubic non-Abelian anomaly while U(1) mixed gauge and
gravitational anomaly or $[{\rm SU}(N_{a})]^{2}{\rm U}(1)$ gauge anomaly can
be cancelled by the Green-Schwarz mechanism, mediated by untwisted RR fields
[30].

Let us define the following products of wrapping numbers,

$\displaystyle A_{a}$ $\displaystyle\equiv-n_{a}^{1}n_{a}^{2}n_{a}^{3},$
$\displaystyle\quad B_{a}$ $\displaystyle\equiv n_{a}^{1}l_{a}^{2}l_{a}^{3},$
$\displaystyle\quad C_{a}$ $\displaystyle\equiv
l_{a}^{1}n_{a}^{2}l_{a}^{3},\quad$ $\displaystyle D_{a}$ $\displaystyle\equiv
l_{a}^{1}l_{a}^{2}n_{a}^{3},$ $\displaystyle\tilde{A}_{a}$
$\displaystyle\equiv-l_{a}^{1}l_{a}^{2}l_{a}^{3},$
$\displaystyle\tilde{B}_{a}$ $\displaystyle\equiv
l_{a}^{1}n_{a}^{2}n_{a}^{3},$ $\displaystyle\tilde{C}_{a}$
$\displaystyle\equiv n_{a}^{1}l_{a}^{2}n_{a}^{3},$
$\displaystyle\tilde{D}_{a}$ $\displaystyle\equiv
n_{a}^{1}n_{a}^{2}l_{a}^{3}.\,$ (7)

Cancellation of RR tadpoles requires introducing a number of orientifold
planes also called “filler branes” that trivially satisfy the four-dimensional
${\cal N}=1$ supersymmetry conditions. The no-tadpole condition is given as,

$\displaystyle-2^{k}N^{(1)}+\sum_{a}N_{a}A_{a}$
$\displaystyle=-2^{k}N^{(2)}+\sum_{a}N_{a}B_{a}=$
$\displaystyle-2^{k}N^{(3)}+\sum_{a}N_{a}C_{a}$
$\displaystyle=-2^{k}N^{(4)}+\sum_{a}N_{a}D_{a}=-16,\,$ (8)

where $2N^{(i)}$ is the number of filler branes wrapping along the $i^{\rm
th}$ O6-plane. The filler branes belong to the hidden sector USp group and
carry the same wrapping numbers as one of the O6-planes as shown in table 1.
USp group is hence referred with respect to the non-zero $A$, $B$, $C$ or
$D$-type.

Table 1: The wrapping numbers for four O6-planes.

$\begin{array}[]{|c|c|c|}\hline\cr\text{Orientifold
Action}&\text{O6-Plane}&(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})\\\
\hline\cr\hline\cr\Omega
R&1&(2^{\beta_{1}},0)\times(2^{\beta_{2}},0)\times(2^{\beta_{3}},0)\\\
\hline\cr\Omega
R\omega&2&(2^{\beta_{1}},0)\times(0,-2^{\beta_{2}})\times(0,2^{\beta_{3}})\\\
\hline\cr\Omega
R\theta\omega&3&(0,-2^{\beta_{1}})\times(2^{\beta_{2}},0)\times(0,2^{\beta_{3}})\\\
\hline\cr\Omega
R\theta&4&(0,-2^{\beta_{1}})\times(0,2^{\beta_{2}})\times(2^{\beta_{3}},0)\\\
\hline\cr\end{array}$

Preserving ${\cal N}=1$ supersymmetry in four dimensions after
compactification from ten-dimensions restricts the rotation angle of any
D6-brane with respect to the orientifold plane to be an element of ${\rm
SU}(3)$, i.e.

$\theta^{a}_{1}+\theta^{a}_{2}+\theta^{a}_{3}=0\mod 2\pi,$ (9)

with $\theta^{a}_{j}=\arctan(2^{-\beta_{j}}\chi_{j}l^{a}_{j}/n^{a}_{j})$.
$\theta_{i}$ is the angle between the $D6$-brane and orientifold-plane in the
$i^{\rm th}$ 2-torus and $\chi_{i}=R^{2}_{i}/R^{1}_{i}$ are the complex
structure moduli for the $i^{\rm th}$ 2-torus. ${\cal N}=1$ supersymmetry
conditions are given as,

$\displaystyle
x_{A}\tilde{A}_{a}+x_{B}\tilde{B}_{a}+x_{C}\tilde{C}_{a}+x_{D}\tilde{D}_{a}=0,$
$\displaystyle\frac{A_{a}}{x_{A}}+\frac{B_{a}}{x_{B}}+\frac{C_{a}}{x_{C}}+\frac{D_{a}}{x_{D}}<0,$
(10)

where
$x_{A}=\lambda,\;x_{B}=2^{\beta_{2}+\beta_{3}}\cdot\lambda/\chi_{2}\chi_{3},\;x_{C}=2^{\beta_{1}+\beta_{3}}\cdot\lambda/\chi_{1}\chi_{3},\;x_{D}=2^{\beta_{1}+\beta_{2}}\cdot\lambda/\chi_{1}\chi_{2}$.

Orientifolds also have discrete D-brane RR charges classified by the
$\mathbbm{Z}_{2}$ K-theory groups [31, 32, 33, 34], which imply [35],

$\displaystyle\sum_{a}\tilde{A}_{a}=\sum_{a}N_{a}\tilde{B}_{a}=\sum_{a}N_{a}\tilde{C}_{a}=\sum_{a}N_{a}\tilde{D}_{a}=0\mod
4~{}.~{}\,$ (11)

In Pati-Salam models, we can avoid the nonvanishing torsion charges by taking
an even number of D-branes, i.e., $N_{a}\in 2\mathbbm{Z}$.

Table 2: General spectrum for intersecting D6-branes at generic angles, where
${\cal M}$ is the multiplicity, and $a_{\yng(2)}$ and $a_{\yng(1,1)}$ denote
respectively the symmetric and antisymmetric representations of ${\rm
U}(N_{a}/2)$. Positive intersection numbers in our convention refer to the
left-handed chiral supermultiplets.

$\begin{array}[]{|c|c|}\hline\cr\text{\bf Sector}&{\bf Representation}\\\
\hline\cr\hline\cr aa&{\rm U}(N_{a}/2)\text{ vector multiplet}\\\ &\text{3
adjoint chiral multiplets}\\\ \hline\cr ab+ba&{\cal
M}(\frac{N_{a}}{2},\frac{\overline{N_{b}}}{2})=I_{ab}(\yng(1)_{a},\overline{\yng(1)}_{b})\\\
\hline\cr ab^{\prime}+b^{\prime}a&{\cal
M}(\frac{N_{a}}{2},\frac{N_{b}}{2})=I_{ab^{\prime}}(\yng(1)_{a},\yng(1)_{b})\\\
\hline\cr aa^{\prime}+a^{\prime}a&{\cal
M}(a_{\yng(2)})=\frac{1}{2}(I_{aa^{\prime}}-\frac{1}{2}I_{aO6})\\\ &{\cal
M}(a_{\yng(1,1)})=\frac{1}{2}(I_{aa^{\prime}}+\frac{1}{2}I_{aO6})\\\
\hline\cr\end{array}$

The general particle representations for intersecting D6-branes models at
angles are shown in table 2. Following the convention of [6] the $N$ number of
D6-brane stacks corresponds to U($N/2$) and USp($N$) respectively. A positive
intersection number in our convention refers to the left-chiral
supermultiplet.

The effective gauge symmetry of supersymmetric Pati-Salam models can be
further broken down to the standard model gauge symmetry via brane splitting.
The relevant details and the formulae of computing gauge kinetic functions and
the gauge coupling relations are given in [12, 9, 13].

##  III Pati-Salam models with symplectic group under the variation of the
hidden sector

Three family supersymmetric Pati-Salam models from symplectic groups where
either one or both of the SU(2) gauge factor is replaced by USp(2) groups are
of special interest because unlike the U(2) stacks the USp brane stacks are
parallel to either of the O6-planes. This fact explains the relative scarcity
of such models as compared to the usual models strictly arising from the
unitary gauge group stacks.

One of the main objectives of our paper is to emphasize that a complete search
of such three family models should not only focus on the observable sector but
also take into account the variations under the hidden sector. Because, it is
possible to find inequivalent models with the same observable sector while a
completely different hidden sector of the D6-brane stacks and O6-planes. In
the following, we discuss the models with one USp(2) group and the models with
two USp(2) groups separately.

###  III.1 Pati-Salam models with the gauge group ${\rm U}(4)_{C}\times{\rm
USp}(2)_{L}\times{\rm SU}(2)_{R}$

Three-family supersymmetric Pati-Salam with the gauge group ${\rm
U}(4)_{C}\times{\rm USp}(2)_{L}\times{\rm SU}(2)_{R}$ have been recently
discussed in ref. [13]. It was found that the number of such models is only 5.
However, by fixing the visible stacks $a$, $b$ and $c$, and searching for all
possible hidden sectors results in two new models. The new models are Model
2.1b and Model 5b. In appendix A we enlist all such three-family
supersymmetric Pati-Salam models with a single symplectic group under the
variation of the hidden sector. We refer readers to consult ref. [13] for the
detailed particle spectra.

###  III.2 Pati-Salam models with the gauge group ${\rm U}(4)_{C}\times{\rm
USp}(2)_{L}\times{\rm USp}(2)_{R}$

Table 3: D6-brane configurations and intersection numbers of Model 1a, and its
gauge coupling relation is
$g^{2}_{a}=F(x_{B})g^{2}_{b}=F(x_{B})g_{c}^{2}=\frac{3(3x_{B}+8)x_{B}+8}{30x_{B}+20}\,\left(\frac{5}{3}\,g^{2}_{Y}\right)=\frac{4\sqrt{2}(x_{B}(3x_{B}+4))^{3/4}}{\sqrt[4]{3}(3x_{B}+2)}\,\pi\,e^{\phi_{4}}$.

$\begin{array}[]{|c||c|c| |r|r| r|r|r| r|r|r|
r|r|r|}\hline\cr\rm{Model~{}\hyperref@@ii[model2USp_1a]{1a}}&\lx@intercol\hfil{\rm
U}(4)\times{\rm USp}(2)_{L}\times{\rm USp}(2)_{R}\times{\rm U}(2)\times{\rm
USp}(8)\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr\rm{stack}&N&(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})&n_{\yng(2)}&n_{\yng(1,1)}&b&c&d&d^{\prime}&1&2&3&4\\\
\hline\cr a&8&(1,-3)\times(0,-1)\times(-3,1)&0&0&3&-3&0&0&0&0&0&0\\\
b&2&(0,1)\times(1,0)\times(0,-2)&0&0&\text{-}&0&6&-6&0&0&0&0\\\
c&2&(1,0)\times(1,0)\times(2,0)&0&0&\text{-}&\text{-}&6&-6&0&0&0&0\\\
d&4&(1,-3)\times(1,-2)\times(3,1)&-16&-56&\text{-}&\text{-}&\text{-}&\text{-}&0&0&0&1\\\
\hline\cr
4&8&(0,-1)\times(0,1)\times(2,0)&\lx@intercol\hfil\beta^{g}_{d}=6,\quad\beta^{g}_{4}=-5,\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil x_{A}=1=x_{C},\quad
x_{D}=9x_{B}+12\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil\chi_{1}=x_{B}\chi_{2}=2/\chi_{3}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\end{array}$

Table 4: The chiral and vector-like superfields, and their quantum numbers
under the gauge symmetry ${\rm SU}(4)\times{\rm USp}(2)_{L}\times{\rm
USp}(2)_{R}\times{\rm SU}(2)\times{\rm USp}(8)$ for the Model 1a.

$\begin{array}[]{|c||c||r|r|r||c|c|c|}\hline\cr\rm{Model~{}\hyperref@@ii[model2USp_1a]{1a}}&\text{Quantum
Number}&Q_{4}&Q_{2L}&Q_{2R}&Q_{em}&B-L&\text{Field}\\\ \hline\cr\hline\cr
ab&3\times(4,\overline{2},1,1,1)&1&-1&0&-\frac{1}{3},\frac{2}{3},-1,0&\frac{1}{3},-1&Q_{L},L_{L}\\\
ac&3\times(\overline{4},1,2,1,1)&-1&0&1&\frac{1}{3},-\frac{2}{3},1,0&-\frac{1}{3},1&Q_{R},L_{R}\\\
bd&6\times(1,2,1,\overline{2},1)&0&1&0&\pm\frac{1}{2}&0&\\\
bd^{\prime}&6\times(1,\overline{2},1,\overline{2},1)&0&-1&0&\mp\frac{1}{2}&0&\\\
cd&6\times(1,1,2,\overline{2},1)&0&0&1&\pm\frac{1}{2}&0&\\\
cd^{\prime}&6\times(1,1,\overline{2},\overline{2},1)&0&0&-1&\mp\frac{1}{2}&0&\\\
d4&1\times(1,1,1,2,\overline{8})&0&0&0&0&0&\\\
d_{\overline{\yng(2)}}&16\times(1,1,1,\overline{3}_{\overline{\yng(2)}},1)&0&0&0&0&0&\\\
d_{\overline{\yng(1,1)}}&56\times(1,1,1,\overline{1}_{\overline{\yng(1,1)}},1)&0&0&0&0&0&\\\
\hline\cr\hline\cr
bc&2\times(1,\overline{2},2,1,1)&0&-1&1&1,0,0,-1&0&H_{u},H_{d}\\\
&2\times(1,2,\overline{2},1,1)&0&1&-1&&&\\\ \hline\cr\end{array}$

Table 5: D6-brane configurations and intersection numbers of Model 1b, and its
gauge coupling relation is
$g_{a}^{2}=\frac{13g_{b}^{2}}{11}=\frac{13g_{c}^{2}}{11}=\frac{61}{55}\frac{5g_{Y}^{2}}{3}=\frac{8}{11}13^{3/4}\,\pi\,e^{\phi_{4}}$.

$\begin{array}[]{|c||c|c| |r|r| r|r|r| r|r|r|
r|r|r|}\hline\cr\rm{Model~{}\hyperref@@ii[model2USp_1b]{1b}}&\lx@intercol\hfil{\rm
U}(4)\times{\rm USp}(2)_{L}\times{\rm USp}(2)_{R}\times{\rm U}(1)\times{\rm
U}(1)\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr\rm{stack}&N&(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})&n_{\yng(2)}&n_{\yng(1,1)}&b&c&d&d^{\prime}&e&e^{\prime}\\\
\hline\cr a&8&(1,-3)\times(0,-1)\times(-3,1)&0&0&3&-3&1&2&-3&0\\\
b&2&(0,1)\times(1,0)\times(0,-2)&0&0&0&0&0&0&12&-12\\\
c&2&(1,0)\times(1,0)\times(2,0)&0&0&\text{-}&\text{-}&2&-2&10&-10\\\
d&2&(0,-1)\times(1,-2)\times(-1,1)&1&-1&\text{-}&\text{-}&\text{-}&\text{-}&0&-8\\\
e&2&(2,-5)\times(-1,2)\times(-3,-1)&-85&-155&\text{-}&\text{-}&\text{-}&\text{-}&\text{-}&\text{-}\\\
\hline\cr&&&\lx@intercol\hfil\beta^{g}_{d}=-2,\quad\beta^{g}_{e}=22,\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
1/\chi_{1}=1/(2\chi_{2})=\chi_{3}/2=\sqrt{13}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\end{array}$

Table 6: D6-brane configurations and intersection numbers of Model 1c, and its
gauge coupling relation is
$g_{a}^{2}=\frac{92g_{b}^{2}}{105}=\frac{92g_{c}^{2}}{105}=\frac{162}{175}\frac{5g_{Y}^{2}}{3}=\frac{32}{35}\left(\frac{23}{3}\right)^{3/4}\,\pi\,e^{\phi_{4}}$.

$\begin{array}[]{|c||c|c| |r|r| r|r|r| r|r|r|
r|r|r|}\hline\cr\rm{Model~{}\hyperref@@ii[model2USp_1c]{1c}}&\lx@intercol\hfil{\rm
U}(4)\times{\rm USp}(2)_{L}\times{\rm USp}(2)_{R}\times{\rm U}(1)\times{\rm
U}(1)\times{\rm USp}(2)\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr\rm{stack}&N&(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})&n_{\yng(2)}&n_{\yng(1,1)}&b&c&d&d^{\prime}&e&e^{\prime}&4\\\
\hline\cr a&8&(1,-3)\times(0,-1)\times(-3,1)&0&0&3&-3&0&3&-3&0&0\\\
b&2&(0,1)\times(1,0)\times(0,-2)&0&0&0&0&-3&3&9&-9&0\\\
c&2&(1,0)\times(1,0)\times(2,0)&0&0&\text{-}&\text{-}&-4&4&6&-6&0\\\
d&2&(-1,-4)\times(1,1)\times(-3,1)&6&42&\text{-}&\text{-}&\text{-}&\text{-}&72&0&-1\\\
e&2&(-1,2)\times(1,-3)\times(-3,-1)&-16&-56&\text{-}&\text{-}&\text{-}&\text{-}&\text{-}&\text{-}&1\\\
\hline\cr
4&2&(0,-1)\times(0,1)\times(2,0)&\lx@intercol\hfil\beta^{g}_{d}=73,\quad\beta^{g}_{e}=87,\quad\beta^{g}_{4}=-4,\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
2/\chi_{1}=6\chi_{2}=\chi_{3}=\sqrt{69}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\end{array}$

Table 7: The chiral and vector-like superfields, and their quantum numbers
under the gauge symmetry ${\rm SU}(4)\times{\rm USp}(2)_{L}\times{\rm
USp}(2)_{R}\times{\rm U}(1)\times{\rm U}(1)$ for the Model 1b.

$\begin{array}[]{|c||c||r|r|r||c|c|c|}\hline\cr\rm{Model~{}\hyperref@@ii[model2USp_1b]{1b}}&\text{Quantum
Number}&Q_{4}&Q_{2L}&Q_{2R}&Q_{em}&B-L&\text{Field}\\\ \hline\cr\hline\cr
ab&3\times(4,\overline{2},1,0,0)&1&-1&0&-\frac{1}{3},\frac{2}{3},-1,0&\frac{1}{3},-1&Q_{L},L_{L}\\\
ac&3\times(\overline{4},1,2,0,0)&-1&0&1&\frac{1}{3},-\frac{2}{3},1,0&-\frac{1}{3},1&Q_{R},L_{R}\\\
ad&1\times(4,1,1,-1,0)&1&0&0&-\frac{1}{3},\frac{2}{3},-1,0&0&\\\
ad^{\prime}&2\times(4,1,1,1,0)&1&0&0&-\frac{1}{3},\frac{2}{3},-1,0&0&\\\
ae&3\times(\overline{4},1,1,0,1)&-1&0&0&\frac{1}{3},-\frac{2}{3},1,0&0&\\\
be&12\times(1,2,1,0,1)&0&1&0&\pm\frac{1}{2}&0&\\\
be^{\prime}&12\times(1,\overline{2},1,0,-1)&0&-1&0&\mp\frac{1}{2}&0&\\\
cd&2\times(1,1,2,-1,0)&0&0&1&\pm\frac{1}{2}&0&\\\
cd^{\prime}&2\times(1,1,\overline{2},-1,0)&0&0&-1&\mp\frac{1}{2}&0&\\\
ce&10\times(1,1,2,0,-1)&0&0&1&\pm\frac{1}{2}&0&\\\
ce^{\prime}&10\times(1,1,\overline{2},0,-1)&0&0&-1&\mp\frac{1}{2}&0&\\\
de^{\prime}&8\times(1,1,1,-1,-1)&0&0&0&0&0&\\\
d_{\yng(2)}&1\times(1,1,1,2_{\yng(2)},0)&0&0&0&0&0&\\\
e_{\overline{\yng(2)}}&85\times(1,1,1,0,-2_{\overline{\yng(2)}})&0&0&0&0&0&\\\
\hline\cr\hline\cr
bc&2\times(1,\overline{2},2,1,1)&0&-1&1&1,0,0,-1&0&H_{u},H_{d}\\\
&2\times(1,2,\overline{2},1,1)&0&1&-1&&&\\\ \hline\cr\end{array}$

Table 8: The chiral and vector-like superfields, and their quantum numbers
under the gauge symmetry ${\rm SU}(4)\times{\rm USp}(2)_{L}\times{\rm
USp}(2)_{R}\times{\rm U}(1)\times{\rm U}(1)\times{\rm USp}(2)$ for the Model
1c.

$\begin{array}[]{|c||c||r|r|r||c|c|c|}\hline\cr\rm{Model~{}\hyperref@@ii[model2USp_1c]{1c}}&\text{Quantum
Number}&Q_{4}&Q_{2L}&Q_{2R}&Q_{em}&B-L&\text{Field}\\\ \hline\cr\hline\cr
ab&3\times(4,\overline{2},1,0,0,1)&1&-1&0&-\frac{1}{3},\frac{2}{3},-1,0&\frac{1}{3},-1&Q_{L},L_{L}\\\
ac&3\times(\overline{4},1,2,0,0,1)&-1&0&1&\frac{1}{3},-\frac{2}{3},1,0&-\frac{1}{3},1&Q_{R},L_{R}\\\
ad^{\prime}&3\times(4,1,1,1,0,1)&1&0&0&-\frac{1}{3},\frac{2}{3},-1,0&0&\\\
ae&3\times(\overline{4},1,1,0,1,1)&-1&0&0&\frac{1}{3},-\frac{2}{3},1,0&0&\\\
bd&3\times(1,\overline{2},1,1,0,1)&0&-1&0&\mp\frac{1}{2}&0&\\\
bd^{\prime}&3\times(1,2,1,1,0,1)&0&1&0&\pm\frac{1}{2}&0&\\\
be&9\times(1,2,1,0,-1,1)&0&1&0&\pm\frac{1}{2}&0&\\\
be^{\prime}&9\times(1,\overline{2},1,0,-1,1)&0&-1&0&\mp\frac{1}{2}&0&\\\
cd&4\times(1,1,\overline{2},1,0,1)&0&0&-1&\mp\frac{1}{2}&0&\\\
cd^{\prime}&4\times(1,1,2,1,0,1)&0&0&1&\pm\frac{1}{2}&0&\\\
ce&6\times(1,1,2,0,-1,1)&0&0&1&\pm\frac{1}{2}&0&\\\
ce^{\prime}&6\times(1,1,\overline{2},0,-1,1)&0&0&-1&\mp\frac{1}{2}&0&\\\
de&72\times(1,1,1,1,-1,1)&0&0&0&0&0&\\\ d4&1\times(1,1,1,-1,0,2)&0&0&0&0&0&\\\
e4&1\times(1,1,1,0,1,2)&0&0&0&0&0&\\\
d_{\yng(2)}&6\times(1,1,1,2_{\yng(2)},0,1)&0&0&0&0&0&\\\
e_{\overline{\yng(2)}}&16\times(1,1,1,0,-2_{\overline{\yng(2)}},1)&0&0&0&0&0&\\\
\hline\cr\hline\cr
bc&2\times(1,\overline{2},2,1,1)&0&-1&1&1,0,0,-1&0&H_{u},H_{d}\\\
&2\times(1,2,\overline{2},1,1)&0&1&-1&&&\\\ \hline\cr\end{array}$

Three-family supersymmetric Pati-Salam with the gauge group ${\rm
U}(4)_{C}\times{\rm USp}(2)_{L}\times{\rm SU}(2)_{R}$ were first constructed
in ref. [12]. The authors of ref. [12] (one of us, TL) only obtained a single
consistent model with ${\cal N}=1$ supersymmetry with three generations. We
reproduce the model in table 3 in its T-dual form [6] and its detailed
particle spectrum is presented in table 4.

$\displaystyle g^{2}_{a}=F(x_{B})g^{2}_{b}=F(x_{B})g_{c}^{2}$
$\displaystyle=\frac{5(3x_{B}+4)}{2(3x_{B}^{2}+13x_{B}+6)}\,\left(\frac{5}{3}\,g^{2}_{Y}\right)=\frac{4\sqrt{2}(x_{B}(3x_{B}+4))^{3/4}}{\sqrt[4]{3}(3x_{B}+2)}\,\pi\,e^{\phi_{4}},$
where $\displaystyle F(x_{B})$
$\displaystyle=\frac{x_{B}(3x_{B}+4)}{2(3x_{B}+2)}$ (12)

Choosing $F(x_{B})=1$ by setting the value of free parameter
$x_{B}=\frac{1}{3}\left(\sqrt{13}+1\right)$ the tree-level MSSM gauge
couplings are unified at the string scale,

$\displaystyle
g^{2}_{a}=g^{2}_{b}=g^{2}_{c}=\left(\frac{5}{3}\,g^{2}_{Y}\right)=4\
2^{3/4}\sqrt[4]{\frac{1}{3}\left(\sqrt{13}-3\right)}\,\pi\,e^{\phi_{4}}.$ (13)

Even though the gauge couplings are unified, this does not fix the actual
value of the couplings as these still depend upon the value taken by the four-
dimensional dilaton $\phi_{4}$. In order for the gauge couplings to have the
value observed for the MSSM ($g^{2}_{\rm unification}\approx 0.511$), we must
choose $\phi_{4}=-3.32221$ such that $e^{-\phi_{4}}\approx 27.7216$, where the
string scale is given by

$M_{St}=\pi^{1/2}e^{\phi_{4}}M_{Pl}\approx 1.55688\times
10^{17}~{}\mbox{GeV},$ (14)

where $M_{Pl}$ is the reduced Planck scale. Obviously, there are one-loop
threshold corrections arising from the ${\cal N}=1$ and ${\cal N}=2$ open
string sectors [36]. Additionally, there are exotic particles charged under
both observable and hidden sector gauge groups, which are expected to pick up
large masses, but could still affect the running of the gauge couplings.

Fixing the observable sectors $a$, $b$ and $c$ of Model 1a and varying the
hidden sector we have found two additional models that are consistent with the
supersymmetry conditions and the no-tadpole constraints. These additional new
models are listed as Model 1b and Model 1c in tables 5 and 6 respectively. The
respective particle spectra of the models are also given in table 7 and table
8.

In particular as can be noted from the descriptions of tables 5 and 6 that
unlike Model 1a, there is no free parameter in the gauge coupling relations of
the models 1b and 1c. However the gauge couplings still exhibit approximate
gauge coupling unification at the string scale.

##  IV String-scale Gauge coupling relations

In this section, we will discuss the RGE running for the gauge couplings in
the Model 1b and Model 1c. The RGEs for the gauge couplings at the two-loop
level are given by [37, 21, 19, 38, 25, 26]

$\frac{d}{d\ln\mu}g_{i}=\frac{b_{i}}{(4\pi)^{2}}g_{i}^{3}+\frac{g_{i}^{3}}{(4\pi)^{4}}\left[\sum_{j=1}^{3}B_{ij}g_{j}^{2}-\sum_{\alpha=u,d,e}d_{i}^{\alpha}{\rm
Tr}\left(h^{\alpha\dagger}h^{\alpha}\right)\right],$ (15)

where $\mu$ is the running mass scale, $g_{i}(i=1,2,3)$ are the SM gauge
couplings and $h^{\alpha}(\alpha=u,d,e)$ are the Yukawa couplings. By solving
the two-loop RGEs for SM gauge couplings, we perform numerically calculations
including the one-loop RGEs for Yukawa couplings and taking into account the
new physics contributions and threshold. And the whole integral is divided
into three segments. The first one is from the electroweak scale, i.e. Z boson
mass scale $M_{Z}=91.1876{\rm~{}GeV}$, up to the supersymmetry breaking scale
$M_{S}$, where we consider only the non-supersymmetric SM spectrum including a
top quark pole mass at $m_{t}=173.34$ GeV and the corresponding gauge
couplings at the scale $M_{Z}$ are

$\displaystyle g_{1}(M_{Z})=\sqrt{k_{Y}}\frac{g_{em}}{\cos\theta_{W}}\,,\quad
g_{2}(M_{Z})=\sqrt{k_{2}}\frac{g_{em}}{\sin\theta_{W}}\,,\quad
g_{3}(M_{Z})=\sqrt{4\pi\alpha_{s}}\,.$ (16)

The Higgs vacuum expectation value, strong coupling constant, fine structure
constant, and weak mixing angle at $M_{Z}$ are choosen to be [39, 40]

$\displaystyle
v=174.10{\rm{~{}GeV}}\,,\quad\sin^{2}\theta_{W}(M_{Z})=0.23122\,,\quad$
$\displaystyle\alpha_{s}(M_{Z})=0.1181\pm
0.0011\,,\quad\alpha_{em}^{-1}(M_{Z})=128.91\pm 0.02\,.$ (17)

Next, we perform the supersymmetric RGEs from $M_{\rm S}$ scale, and the
contributions form the introduced exotic vector-like particles are included
from $M_{V}$ up to string scale. Thus, in the running of the gauge couplings,
there can be two bending point corresponding to $M_{S}$ and $M_{V}$. Based on
the experimental lower limits of supersymmetry and gauge hierarchy
preservation, we have the supersymmetry breaking scale $M_{\rm S}$ at TeV
scale, such as $2.5$ TeV or $3.0$ TeV. The difference has effect less than
$5\%$ on the scale of unification $M_{\rm U}$ while the larger value for
$M_{\rm S}$ reduces the unification scale. Thus in the following calculations,
we set the supersymmetry breaking scale at $3.0$ TeV. At last, we get the
gauge coupling relations at the string scale as

$g_{a}^{2}=k_{2}g_{b}^{2}=k_{Y}g_{Y}^{2}=g_{\text{U}}^{2}\sim
g_{\text{string}}^{2}~{},~{}\,$ (18)

where $g_{a}$, $g_{b}$, and $g_{Y}$ are respectively the gauge couplings for
${\rm SU}(3)_{C}$, ${\rm SU}(2)_{L}$, and ${\rm U}(1)_{Y}$. $k_{2}$ and
$k_{Y}$ are rational numbers. The canonical normalization in ${\rm SU}(5)$ and
${\rm SO}(10)$ models give $k_{2}=1$ and $k_{Y}=5/3$.

The coefficients of beta functions in Eq. (15) in SM [41, 42, 43, 44] and
supersymmetric models [45, 46, 47] are represented by

$\displaystyle b_{\rm SM}$
$\displaystyle=\left(\frac{41}{6}\frac{1}{k_{Y}},-\frac{19}{6}\frac{1}{k_{2}},-7\right),\quad
B_{\rm
SM}=\begin{pmatrix}\frac{199}{18}\frac{1}{k_{Y}^{2}}&\frac{27}{6}\frac{1}{k_{Y}k_{2}}&\frac{44}{3}\frac{1}{k_{Y}}\cr\frac{3}{2}\frac{1}{k_{Y}k_{2}}&\frac{35}{6}\frac{1}{k_{2}^{2}}&12\frac{1}{k_{2}}\cr\frac{11}{6}\frac{1}{k_{Y}}&\frac{9}{2}\frac{1}{k_{2}}&-26\end{pmatrix},\quad$
(19) $\displaystyle d^{u}_{\rm SM}$
$\displaystyle=\left(\frac{17}{6}\frac{1}{k_{Y}},\frac{3}{2}\frac{1}{k_{2}},2\right),\quad
d^{d}_{\rm SM}=0,\quad d^{e}_{\rm SM}=0,\,$ (20) $\displaystyle b_{\rm SUSY}$
$\displaystyle=\left(11\frac{1}{k_{Y}},\frac{1}{k_{2}},-3\right),\quad B_{\rm
SUSY}=\begin{pmatrix}\frac{199}{9}\frac{1}{k_{Y}^{2}}&9\frac{1}{k_{Y}k_{2}}&\frac{88}{3}\frac{1}{k_{Y}}\cr
3\frac{1}{k_{Y}k_{2}}&25\frac{1}{k_{2}^{2}}&24\frac{1}{k_{2}}\cr\frac{11}{3}\frac{1}{k_{Y}}&9\frac{1}{k_{2}}&14\end{pmatrix},~{}$
(21) $\displaystyle d^{u}_{\rm SUSY}$
$\displaystyle=\left(\frac{26}{3}\frac{1}{k_{Y}},6\frac{1}{k_{2}},4\right),\quad
d^{d}_{\rm SUSY}=0,\quad d^{e}_{\rm SUSY}=0,$ (22)

where $k_{Y}$ and $k_{2}$ are general normalization factors. The general one-
loop RGEs for Yukawa couplings can be found in [38].

Figure 1: Two-loop evolution of gauge couplings for the Model 1b without
(left) and with (right) $12(XA+\overline{XA})$ at $1.81\times 10^{17}$ GeV and
$3XG$ at $3.8\times 10^{16}$ GeV.

Figure 2: Two-loop evolution of gauge couplings for the Model 1c without
(left) and with (right) $9(XA+\overline{XA})$ at $2.93\times 10^{16}$ GeV and
$3XG$ at $3.75\times 10^{16}$ GeV.

Defining the parameters as

$\alpha_{1}\equiv k_{Y}g_{Y}^{2}/4\pi,\quad\alpha_{2}\equiv
k_{2}g_{b}^{2}/4\pi,\quad\alpha_{3}\equiv g_{a}^{2}/4\pi,$ (23)

the two-loop evolution of gauge couplings are shown in Figs.1 and 2 for the
models 1b and 1c, where the string-scale gauge coupling relation is achieved
by setting
$\alpha_{\text{U}}^{-1}\equiv\alpha_{1}^{-1}=(\alpha_{2}^{-1}+\alpha_{3}^{-1})/2$
and the accuracy $\Delta=|\alpha_{1}^{-1}-\alpha_{2}^{-1}|/\alpha_{1}^{-1}$ is
limited to be less than $0.1\%$. The string-scale gauge coupling relations are
realized by adding the vector-like particles $XA+{\overline{XA}}$ from the
fundamental representation of $SU(4)$ gauge group and the particle $XG$ from
the adjoint representation of $SU(2)$. The quantum numbers of these exotic
particles under $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ are
$XA+{\overline{XA}}={\mathbf{(1,2,{0})+(1,{\bar{2}},0)}}$ and
$XG=\mathbf{(8,1,0)}$. For the particles $XA+{\overline{XA}}$, the non-zero
coefficients of one- and two-loop beta functions in the supersymmetric models
are $\Delta b_{2}=1$ and $\Delta B_{22}=7$, which only modify the evolution of
the electroweak coupling; while the non-zero contributions from particle $XG$
are $\Delta b_{3}=3$ and $\Delta B_{33}=54$, which alter the evolution of the
strong coupling. Thus, the adding of these particles bend the values of
$\alpha_{2}^{-1}$ and $\alpha_{3}^{-1}$ to achieve the unification on right
panel of Figs. 1 and 2. Based on the intersection of D-branes, $12$ and $9$
pairs of $XA+\overline{XA}$, from the $be+be^{\prime}$ sectors in the spectrum
Tables 7 and 8, are added in the models 1b and 1c, respectively. While, the
particle $XG$ arises from the $aa$ sectors, and in our calculations we
introduce the particle $XG$ with the maximum number 3.

##  V Conclusion

We have found new three-family supersymmetric Pati-Salam models where either
one or both of the ${\rm SU}(2)_{L,R}$ gauge factors arise from an original
${\rm USp}(2)_{L,R}$ group on a
$\mathbbm{T}^{6}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})$ orientifold from
intersecting D6-branes at angles from IIA string theory without fluxes. Unlike
the special unitary group, the symplectic group is a simpler choice as there
is no associated global anomalous U(1) group. The unitary symplectic group
arises by placing the D6-brane stack parallel to any of the four O6-planes.

By fixing the observable sector while varying the hidden sector we are able to
ascertain the all possible gauge group factors consistent with the no-tadpole
constraint. We have displayed the perturbative particle spectra of these
representative models where the exotic particles are found to be decoupled in
some of the models. This seemingly minor change from SU(2) to USp(2) group is
quite restrictive and drastically reduces the number of consistent three-
family models. Therefore, while the number of three-family models with ${\rm
SU}(4)_{C}\times{\rm SU}(2)_{L}\times{\rm SU}(2)_{R}$ groups is found to be
202,752 [11, 10], the similar models by replacing either one of the SU(2)
factors with a USp(2) is only 5 [13]. Interestingly, three-family
supersymmetric Pati-Salam model with the specific gauge group ${\rm
SU}(4)_{C}\times{\rm USp}(2)_{L}\times{\rm USp}(2)_{R}$ are even more
constrained and we have only found two such models.

Finally, we also point out a discrepancy in ref. [10] where a complete search
of the landscape of supersymmetric Pati-Salam models constructed from Type IIA
string theory on $\mathbbm{T}^{6}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})$
orientifolds was claimed. Here, we have demonstrated that in order to chart
out the complete landscape the inequivalent models with similar visible
sectors but having a different hidden sector should also be taken into account
as it can not only change the number of consistent models but also can affect
the gauge coupling relations.

###### Acknowledgements.

We would like to thank Mudassar Sabir very much for collaboration in the
initial stage of the project and helpful discussions. This research is
supported in part by the National Key Research and Development Program of
China Grant No. 2020YFC2201504, by the Projects No. 11875062, No. 11947302,
No. 12047503, and No. 12275333 supported by the National Natural Science
Foundation of China, by the Key Research Program of the Chinese Academy of
Sciences, Grant No. XDPB15, by the Scientific Instrument Developing Project of
the Chinese Academy of Sciences, Grant No. YJKYYQ20190049, and by the
International Partnership Program of Chinese Academy of Sciences for Grand
Challenges, Grant No. 112311KYSB20210012.

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##  Appendix A Pati-Salam models with a symplectic group under the variations
of the hidden sector

In this appendix, we list all representative three-family supersymmetric Pati-
Salam models with a symplectic group under the variations of the hidden sector
obtained from random scanning method. $a,b,c,d$ in the first column in every
table represent the four stacks of D6-branes, respectively. Similarly,
$1,2,3,4$ in the first columns is a short-handed notation for the filler
branes along the $\Omega R$, $\Omega R\omega$, $\Omega R\theta\omega$ and
$\Omega R\theta$ O6-planes, respectively. The second column in each table
lists the numbers of D6-branes in the respective stack. In the third column we
record the wrapping numbers of each D6-brane configuration. The rest of the
columns record the intersection numbers between various stacks. For instance,
in the $b$ column of table 9, from top to bottom, the numbers represent
intersection numbers $I_{ab},I_{bc},I_{bd}$, etc. As usual, $b^{\prime}$ and
$c^{\prime}$ are the orientifold $\Omega R$ image of $b$ and $c$ stacks of
D6-branes. We also list the relation between $x_{A},x_{B},x_{C},x_{D}$, which
are determined by the supersymmetry conditions (10), as well as the relation
between the moduli parameter $\chi_{1},\,\chi_{2},\,\chi_{3}$. The one loop
beta functions $\beta^{g}_{i}$ for each of the hidden sector stack is also
listed. The gauge coupling relations are given in the caption of each table.

Table 9: D6-brane configurations and intersection numbers of Model 1, and its
gauge coupling relation is
$g^{2}_{a}=\frac{8}{7}\,g^{2}_{b}=\frac{4}{3}\,g_{c}^{2}=\frac{6}{5}\,\left(\frac{5}{3}\,g^{2}_{Y}\right)=\frac{32}{7}\sqrt[4]{\frac{2}{3}}\,\pi\,e^{\phi_{4}}$.

$\begin{array}[]{|c|c|c |r|r|r| r|r|r| r|r|r|
r|r|r|}\hline\cr\vrule\lx@intercol\hfil\text{Model~{}\hyperref@@ii[model0]{1}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil{\rm
U}(4)\times{\rm USp}(2)_{L}\times{\rm U}(2)_{R}\times{\rm
U}(2)\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr\rm{stack}&N&(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})&n_{\yng(2)}&n_{\yng(1,1)}&b&c&c^{\prime}&d&d^{\prime}&1&2&3&4\\\
\hline\cr a&8&(-1,-3)\times(0,-1)\times(-1,-1)&-2&2&3&-3&0&4&4&0&0&0&0\\\
b&2&(1,0)\times(1,0)\times(2,0)&0&0&\text{-}&4&-4&2&-2&0&0&0&0\\\
c&4&(0,1)\times(3,-4)\times(1,-1)&1&-1&\text{-}&\text{-}&\text{-}&4&-10&0&0&0&0\\\
d&4&(-1,1)\times(1,-2)\times(-3,-1)&-2&-22&\text{-}&\text{-}&\text{-}&\text{-}&\text{-}&0&0&0&0\\\
\hline\cr&&&\lx@intercol\hfil\beta^{g}_{d}=8,\quad
x_{A}=\frac{3}{4}x_{B}=\frac{1}{3}x_{C}=\frac{1}{24}x_{D}\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
18\chi_{1}=8\chi_{2}=\chi_{3}/2=\sqrt{6}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\end{array}$

Table 10: D6-brane configurations and intersection numbers of Model 2a, and
its gauge coupling relation is
$g^{2}_{a}=\frac{11}{36}\,g^{2}_{b}=\frac{1}{3}\,g_{c}^{2}=\frac{3}{5}\,\left(\frac{5}{3}\,g^{2}_{Y}\right)=\frac{2\sqrt{2}}{9}11^{3/4}\,\pi\,e^{\phi_{4}}$.

$\begin{array}[]{|c|c|c |r|r|r| r|r|r| r|r|r|
r|r|r|}\hline\cr\vrule\lx@intercol\hfil\rm{Model~{}\hyperref@@ii[model2.1a]{2a}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil{\rm
U}(4)\times{\rm USp}(2)_{L}\times{\rm U}(2)_{R}\times{\rm U}(2)\times{\rm
USp}(2)\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr\rm{stack}&N&(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})&n_{\yng(2)}&n_{\yng(1,1)}&b&c&c^{\prime}&d&d^{\prime}&1&2&3&4\\\
\hline\cr a&8&(-1,-3)\times(-1,0)\times(1,-1)&-2&2&3&0&-3&4&4&0&0&0&0\\\
b&2&(1,0)\times(0,1)\times(0,-2)&0&0&\text{-}&-1&1&1&-1&0&0&0&0\\\
c&4&(0,1)\times(1,-3)\times(1,-1)&2&-2&\text{-}&\text{-}&\text{-}&4&-4&-3&0&0&0\\\
d&4&(-1,1)\times(-1,-1)\times(1,-3)&0&-12&\text{-}&\text{-}&\text{-}&\text{-}&\text{-}&-3&0&0&0\\\
\hline\cr
1&2&(1,0)\times(1,0)\times(2,0)&\lx@intercol\hfil\beta^{g}_{d}=7,~{}\beta^{g}_{1}=0,\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
x_{A}=\frac{1}{3}x_{B}=\frac{1}{33}x_{C}=\frac{1}{9}x_{D}\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
33\chi_{1}=3\chi_{2}=\frac{11}{2}\chi_{3}=\sqrt{11}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\end{array}$

Table 11: D6-brane configurations and intersection numbers of Model 2b, and
its gauge coupling relation is
$g^{2}_{a}=\frac{11}{36}\,g^{2}_{b}=\frac{1}{3}\,g_{c}^{2}=\frac{3}{5}\,\left(\frac{5}{3}\,g^{2}_{Y}\right)=\frac{2\sqrt{2}}{9}11^{3/4}\,\pi\,e^{\phi_{4}}$.

$\begin{array}[]{|c|c|c |r|r|r| r|r|r| r|r|r|
r|r|r|}\hline\cr\vrule\lx@intercol\hfil\rm{Model~{}\hyperref@@ii[model2.1b]{2b}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil{\rm
U}(4)\times{\rm USp}(2)_{L}\times{\rm U}(2)_{R}\times{\rm U}(2)\times{\rm
USp}(6)\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr\rm{stack}&N&(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})&n_{\yng(2)}&n_{\yng(1,1)}&b&c&c^{\prime}&d&d^{\prime}&1&2&3&4\\\
\hline\cr a&8&(-1,-3)\times(-1,0)\times(1,-1)&-2&2&3&0&-3&4&4&0&0&0&0\\\
b&2&(1,0)\times(0,1)\times(0,-2)&0&0&\text{-}&-1&1&-2&2&0&0&0&0\\\
c&4&(0,1)\times(1,-3)\times(1,-1)&2&-2&\text{-}&\text{-}&\text{-}&-10&7&0&0&0&0\\\
d&4&(1,1)\times(2,-1)\times(-1,3)&2&22&\text{-}&\text{-}&\text{-}&\text{-}&\text{-}&0&0&0&-1\\\
\hline\cr
3&6&(0,-1)\times(1,0)\times(0,2)&\lx@intercol\hfil\beta^{g}_{d}=14,~{}\beta^{g}_{3}=-5,\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
x_{A}=\frac{1}{3}x_{B}=\frac{1}{63}x_{C}=\frac{1}{9}x_{D}\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
63\chi_{1}=3\chi_{2}=\frac{21}{2}\chi_{3}=\sqrt{21}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\end{array}$

Table 12: D6-brane configurations and intersection numbers of Model 3, and its
gauge coupling relation is
$g^{2}_{a}=\frac{5}{18}\,g^{2}_{b}=\frac{1}{3}\,g_{c}^{2}=\frac{3}{5}\,\left(\frac{5}{3}\,g^{2}_{Y}\right)=\frac{4\sqrt{2}}{9}5^{3/4}\,\pi\,e^{\phi_{4}}$.

$\begin{array}[]{|c|c|c |r|r|r| r|r|r| r|r|r|
r|r|r|}\hline\cr\vrule\lx@intercol\hfil\rm{Model~{}\hyperref@@ii[model2.2]{3}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil{\rm
U}(4)\times{\rm USp}(2)_{L}\times{\rm U}(2)_{R}\times{\rm U}(2)\times{\rm
USp}(2)\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr\rm{stack}&N&(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})&n_{\yng(2)}&n_{\yng(1,1)}&b&c&c^{\prime}&d&d^{\prime}&1&2&3&4\\\
\hline\cr a&8&(-1,0)\times(-1,-3)\times(1,-3)&0&0&3&-3&0&0&0&0&0&0&0\\\
b&2&(0,1)\times(1,0)\times(0,-2)&0&0&\text{-}&1&-1&6&-6&0&0&0&0\\\
c&4&(1,1)\times(0,1)\times(-1,-3)&-2&2&\text{-}&\text{-}&\text{-}&3&0&0&0&0&0\\\
d&4&(2,1)\times(1,-3)\times(1,-3)&-16&-56&\text{-}&\text{-}&\text{-}&\text{-}&\text{-}&0&1&0&0\\\
\hline\cr
2&2&(1,0)\times(0,-1)\times(0,2)&\lx@intercol\hfil\beta^{g}_{d}=3,~{}\beta^{g}_{2}=-5,\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
x_{A}=\frac{1}{45}x_{B}=\frac{1}{3}x_{C}=\frac{1}{3}x_{D}\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil\chi_{1}=15\chi_{2}=\frac{15}{2}\chi_{3}=\sqrt{5}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\end{array}$

Table 13: D6-brane configurations and intersection numbers of Model 4, and its
gauge coupling relation is
$g^{2}_{a}=\frac{11}{18}\,g^{2}_{b}=\frac{2}{3}\,g_{c}^{2}=\frac{4}{5}\,\left(\frac{5}{3}\,g^{2}_{Y}\right)=\frac{4}{9}\,11^{3/4}\,\pi\,e^{\phi_{4}}$.

$\begin{array}[]{|c|c|c |r|r|r| r|r|r| r|r|r|
r|r|r|}\hline\cr\vrule\lx@intercol\hfil\rm{Model~{}\hyperref@@ii[model4]{4}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil{\rm
U}(4)\times{\rm USp}(2)_{L}\times{\rm U}(2)_{R}\times{\rm U}(2)\times{\rm
USp}(4)\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr\rm{stack}&N&(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})&n_{\yng(2)}&n_{\yng(1,1)}&b&c&c^{\prime}&d&d^{\prime}&1&2&3&4\\\
\hline\cr a&8&(-1,0)\times(-1,-3)\times(1,-1)&-2&2&3&-3&0&4&4&0&0&0&0\\\
b&2&(0,1)\times(1,0)\times(0,-2)&0&0&\text{-}&2&-2&2&-2&0&0&0&0\\\
c&4&(2,3)\times(0,1)\times(-1,-1)&-1&1&\text{-}&\text{-}&\text{-}&8&-8&0&0&0&0\\\
d&4&(2,1)\times(1,-1)\times(1,-3)&-2&-22&\text{-}&\text{-}&\text{-}&\text{-}&\text{-}&0&1&0&0\\\
\hline\cr
2&4&(1,0)\times(0,-1)\times(0,2)&\lx@intercol\hfil\beta^{g}_{d}=12,~{}\beta^{g}_{2}=-5,\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
x_{A}=\frac{1}{33}x_{B}=\frac{2}{3}x_{C}=\frac{2}{9}x_{D}\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
3\chi_{1}=33\chi_{2}=\frac{11}{2}\chi_{3}=\sqrt{11}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\end{array}$

Table 14: D6-brane configurations and intersection numbers of Model 5a, and
its gauge coupling relation is
$g^{2}_{a}=\frac{7}{22}\,g^{2}_{b}=\frac{1}{3}\,g_{c}^{2}=\frac{3}{5}\,\left(\frac{5}{3}\,g^{2}_{Y}\right)=\frac{4\sqrt{2}}{11}\frac{7^{3/4}}{\sqrt[4]{3}}\,\pi\,e^{\phi_{4}}$.

$\begin{array}[]{|c|c|c |r|r|r| r|r|r| r|r|r|
r|r|r|}\hline\cr\vrule\lx@intercol\hfil\rm{Model~{}\hyperref@@ii[model6a]{5a}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil{\rm
U}(4)\times{\rm USp}(2)_{L}\times{\rm U}(2)_{R}\times{\rm U}(2)\times{\rm
USp}(6)\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr\rm{stack}&N&(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})&n_{\yng(2)}&n_{\yng(1,1)}&b&c&c^{\prime}&d&d^{\prime}&1&2&3&4\\\
\hline\cr a&8&(1,3)\times(1,0)\times(1,-1)&-2&2&3&0&-3&4&4&0&0&0&0\\\
b&2&(1,0)\times(0,1)\times(0,-2)&0&0&\text{-}&-1&1&2&-2&0&0&0&0\\\
c&4&(0,-1)\times(-1,3)\times(1,-1)&2&-2&\text{-}&\text{-}&\text{-}&7&-10&0&0&0&0\\\
d&4&(1,-1)\times(2,1)\times(1,-3)&-2&-22&\text{-}&\text{-}&\text{-}&\text{-}&\text{-}&0&0&1&0\\\
\hline\cr
3&6&(0,-1)\times(1,0)\times(0,2)&\lx@intercol\hfil\beta^{g}_{d}=11,~{}\beta^{g}_{3}=-5,\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
x_{A}=\frac{1}{3}x_{B}=\frac{1}{63}x_{C}=\frac{1}{9}x_{D}\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
63\chi_{1}=3\chi_{2}=\frac{21}{2}\chi_{3}=\sqrt{2}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\end{array}$

Table 15: D6-brane configurations and intersection numbers of Model 5b, and
its gauge coupling relation is
$g^{2}_{a}=\frac{7}{22}\,g^{2}_{b}=\frac{1}{3}\,g_{c}^{2}=\frac{3}{5}\,\left(\frac{5}{3}\,g^{2}_{Y}\right)=\frac{4\sqrt{2}}{11}\frac{7^{3/4}}{\sqrt[4]{3}}\,\pi\,e^{\phi_{4}}$.

$\begin{array}[]{|c|c|c |r|r|r| r|r|r| r|r|r|
r|r|r|}\hline\cr\vrule\lx@intercol\hfil\rm{Model~{}\hyperref@@ii[model6b]{5b}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil{\rm
U}(4)\times{\rm USp}(2)_{L}\times{\rm U}(2)_{R}\times{\rm U}(2)\times{\rm
USp}(2)\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr\rm{stack}&N&(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})&n_{\yng(2)}&n_{\yng(1,1)}&b&c&c^{\prime}&d&d^{\prime}&1&2&3&4\\\
\hline\cr a&8&(1,3)\times(1,0)\times(1,-1)&-2&2&3&0&-3&4&4&0&0&0&0\\\
b&2&(1,0)\times(0,1)\times(0,-2)&0&0&\text{-}&-1&1&-1&1&0&0&0&0\\\
c&4&(0,-1)\times(-1,3)\times(1,-1)&2&-2&\text{-}&\text{-}&\text{-}&-4&4&-3&0&0&0\\\
d&4&(1,1)\times(-1,1)\times(-1,-3)&0&12&\text{-}&\text{-}&\text{-}&\text{-}&\text{-}&3&0&0&0\\\
\hline\cr
1&2&(1,0)\times(1,0)\times(2,0)&\lx@intercol\hfil\beta^{g}_{d}=7,~{}\beta^{g}_{1}=0,\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
x_{A}=\frac{1}{3}x_{B}=\frac{1}{33}x_{C}=\frac{1}{9}x_{D}\hfil\lx@intercol\vrule\lx@intercol\\\
&&&\lx@intercol\hfil
33\chi_{1}=3\chi_{2}=\frac{11}{2}\chi_{3}=\sqrt{11}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\end{array}$