aainstitutetext: School of Sciences, Xi’an Technological University, Xi’an
710021, P. R. Chinabbinstitutetext: CAS Key Laboratory of Theoretical Physics,
Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, P. R. Chinaccinstitutetext:
School of Physical Sciences, University of Chinese Academy of Sciences,
No.19A Yuquan Road, Beijing 100049, P. R. Chinaddinstitutetext: Korea
Institute for Advanced Study,
85 Hoegiro, Dongdaemun-Gu, Seoul 02455, Korea

# String-scale Gauge Coupling Relations in the Supersymmetric Pati-Salam
Models from Intersecting D6-branes

Tianjun Li d Rui Sun ,a and Lina Wu111Corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We have constructed all the three-family ${\cal N}=1$ supersymmetric Pati-
Salam models from intersecting D6-branes, and obtained 33 independent models
in total. But how to realize the string-scale gauge coupling relations in
these models is a big challenge. We first discuss how to decouple the exotic
particles in these models. In addition, we consider the adjoint chiral
mulitplets for $SU(4)_{C}$ and $SU(2)_{L}$ gauge symmetries, the Standard
Model (SM) vector-like particles from D6-brane intersections, as well as the
vector-like particles from the ${\cal N}=2$ subsector. We show that the gauge
coupling relations at string scale can be achieved via two-loop
renormalization group equation running for all these supersymmetric Pati-Salam
models. Therefore, we propose a concrete way to obtain the string-scale gauge
coupling relations for the generic intersecting D-brane models.

##  1 Introduction

The three-family ${\cal N}=1$ supersymmetric Pati-Salam models from
intersecting D6-branes have been studied extensively since we can generate the
Yukawa couplings for the Standard Model (SM) fermions, and break the Pati-
Salam gauge symmetry down to the SM gauge symmetry via the D-brane splitting
and supersymmetry preserving Higgs mechanism Lykken:1998ec ; Cvetic:1999hb ;
Cvetic:2000st ; Aldazabal:2000sa ; Berkooz:1996km ; Cvetic:2004ui ;
Chen:2007ms ; Aldazabal:2000cn ; Cvetic:2001tj ; Blumenhagen:2002gw ;
Chen:2005aba ; Blumenhagen:2005mu ; Aldazabal:2000dg ; Ibanez:2001nd ;
Sabir:2022hko . Realizing the SM in D-brane vacua from Orbifold configuration
and Gepner configurations were also stuided in Honecker:2012jd ;
Honecker:2017air ; Anastasopoulos:2005ba ; Anastasopoulos:2006da ;
Anastasopoulos:2009mr ; Anastasopoulos:2010ca ; Anastasopoulos:2010hu ;
Anastasopoulos:2011zz ; Ecker:2015vea ; Marchesano:2022qbx ; Antoniadis:2004dt
; Antoniadis:2010zze ; Kiritsis:2007zz ; Kiritsis:2003mc . However, deriving
all the possible Pati-Salam models is a challenging topic due to the large
amount of possible wrapping numbers Chen:2007px ; Chen:2007zu ; Cvetic:2003xs
; Cvetic:2002pj ; Cvetic:2002qa ; Cvetic:2002wh ; Cvetic:2003yd ;
Cvetic:2003ch ; Honecker:2003vq ; Chen:2005mj . The full landscape of
intersecting D-brane models have been under investigation via different
perspectives Douglas:2006xy ; Halverson:2019tkf ; Loges:2021hvn ;
Loges:2022mao ; Cvetic:2004nk ; Cvetic:2001nr ; Li:2019nvi ; Li:2021pxo ;
Li:2019miw ; Li:2022fzt . In the Type IIA string theory on
$T^{6}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$ orientifold with intersecting
D6-branes, we propose a systematic method to construct all the possible 202752
${\cal N}=1$ supersymmetric Pati-Salam models He:2021kbj ; He:2021gug . After
modding out the equivalent classes, we obtain 33 physical independent models
with 33 types of gauge coupling relations. The main idea is to solve the
common solutions of three generation conditions, and RR tadpole cancellation
conditions iteratively. Thus, we complete the landscape for one particular
type of intersecting D-brane models for the first time. However, how to
realize the string-scale gauge coupling relations in these models is still a
big challenge.

It is well-known that gauge coupling unification can be achieved in the
Minimal Supersymmetric SM (MSSM) Ellis:1990wk ; Langacker:1991an ;
Amaldi:1991cn , which strongly suggested the Grand Unified Theory (GUT). The
unification scale or say GUT scale is around $2\times 10^{16}$ GeV. On the
other hand, the string scale is about one order larger. For example, in the
weakly coupled heterotic string theory, the string scale $M_{\rm string}$ is
given by Dienes:1996du

$\displaystyle M_{\rm string}=g_{\rm string}\times 5.27\times
10^{17}\leavevmode\nobreak\ {\rm GeV}\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ (1)

where $g_{\rm string}$ is string coupling constant. Because $g_{\rm
string}\sim{\cal O}(1)$ is an order one coupling constant, we obtain

$\displaystyle M_{\rm string}=5\times 10^{17}\leavevmode\nobreak\ {\rm
GeV}\leavevmode\nobreak\ .\leavevmode\nobreak\ \,$ (2)

And then we have a factor about 25 between the GUT scale and string scale.
Thus, how to realize the string-scale gauge coupling unification is an
important question in string phenomenology. In principle, the string scale can
be low due to the large volume of the extra-dimension Antoniadis:2002qm ;
Antoniadis:2011bi . However, recall that the Yang-Mills gauge coupling on the
D6-brane stack $x$ at string scale is given by Dienes:1996du

$(g_{YM}^{x})^{2}=\dfrac{\sqrt{8\pi}M_{\rm string}}{M_{\rm
Pl}}\dfrac{1}{\prod_{i=1}^{3}\sqrt{(n^{i}_{x})^{2}\chi_{i}^{-1}+(2^{-\beta_{i}l^{i}_{x}})^{2}\chi_{i}}},$
(3)

where $M_{\rm Pl}$ is the 4-dimensional Planck scale. To have the proper Yang-
Mills coupling $g_{YM}\sim{\cal O}(1)$ at order one, the string scale is
expected to be closer to the Planck scale $M_{\rm Pl}$ to have proper gauge
couplings. Therefore, we set the string scale at $M_{\rm string}=5\times
10^{17}\leavevmode\nobreak\ {\rm GeV}.$

Gauge coupling unification has been studied extensively previously from
various physics point of view Bachas:1995yt ; Lopez:1996gd ;
Blumenhagen:2003jy ; Barger:2005qy ; Jiang:2006hf ; Barger:2007qb ;
Jiang:2008xrg ; Jiang:2009za ; Kokorelis:2016ckp ; Chen:2017rpn ; Chen:2018ucf
. In general, to achieve the string-scale gauge coupling unification, we need
to introduce the new particles at the intermediate scale, such as the SM
vector-like particles, and the $SU(3)_{C}$/$SU(2)_{L}$ adjoint particles. In
particular, we need to obtain these new particles from string model building
as well. Moreover, there are two kinds of the string-scale gauge coupling
unification: one-step unification Bachas:1995yt , and two-step unification for
flipped $SU(5)\times U(1)_{X}$ models Lopez:1996gd ; Jiang:2006hf ;
Jiang:2008xrg ; Jiang:2009za , where the $SU(3)_{C}\times SU(2)_{L}$ gauge
symmetries are unified around the traditional GUT scale, and then $SU(5)\times
U(1)_{X}$ gauge symmetries are unified at the string scale.

In the three-family ${\cal N}=1$ supersymmetric Pati-Salam models from the
Type IIA string theory on $T^{6}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$
orientifold with intersecting D6-branes He:2021kbj ; He:2021gug , there is one
and only one model with gauge coupling unification at the string scale. For
the rest 32 models, the gauge couplings for $SU(4)_{C}$, $SU(2)_{L}$, and
$SU(2)_{R}$ are not unified at the string scale, and then we have the gauge
coupling relations at the string scale as follows

$k_{3}g_{a}^{2}=k_{2}g_{b}^{2}=k_{Y}g_{Y}^{2}=k_{y}\left(\frac{5}{3}g_{Y}^{2}\right)=g_{\text{U}}^{2}\sim
g_{\text{string}}^{2}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \,$ (4)

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

In the string model building, we generically have exotic particles, and thus
need to study how to decouple the exotic particles first. We explain that the
exotic particles in most of our models can be decoupled except Model 4, 23,
and 32, which have chiral multiplets under $SU(4)_{C}$ symmetric
representation. To realize the string-scale gauge coupling relations, we
assume that the exotic particles in these three models can be decoupled as
well. Moreover, we can have two scenarios to obtain the string-scale gauge
coupling relations: traditional one step, and two steps where the SM gauge
symmetry becomes Pati-Salam gauge symmetry at the intermediate scale, and then
their gauge couplings satisfy the gauge coupling relations at string scale. In
this paper, we only consider the first scenario, and will study the second
scenario in the future. Furthermore, we consider the adjoint chiral mulitplets
for $SU(4)_{C}$ and $SU(2)_{L}$ gauge symmetries, the SM vector-like particles
from D6-brane intersections, as well as the vector-like particles from the
${\cal N}=2$ subsector. We show that the gauge coupling relations at string
scale can be achieved via two-loop renormalization group equation (RGE)
running for all these supersymmetric Pati-Salam models. Therefore, we propose
a concrete way to obtain the string-scale gauge coupling relations for the
generic intersecting D-brane models.

This paper is organized as follows. In Section II we first review the
construction of supersymmetric Pati-Salam models and present the relevant
knowledge. In Section III, we discuss how to decouple the exotic particles. In
Section IV, we perform the two-loop RGEs of gauge couplings as well as one-
loop RGEs of Yukawa couplings. We show that the string-scale gauge coupling
relations are achieved after taking into account the contributions from extra
vector-like particles. In Section V we finally conclude the results and make
an outlook.

##  2 The Supersymmetric Pati-Salam Models from Type IIA
$T^{6}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$ Orientifolds with Intersecting
D6-branes

The supersymmetric Pati-Salam models have been constructed on Type IIA
$T^{6}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$ orientifolds with D6-branes
intersecting at generic angles. The orbifold group
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ results in generators $\theta$ and
$\omega$ and associated with twist vectors $(1/2,-1/2,0)$ and $(0,1/2,-1/2)$
respectively. They act on the complex coordinates $z_{i}$ as Cvetic:2001nr ;
Cvetic:2002pj

$\displaystyle\theta:$
$\displaystyle(z_{1},z_{2},z_{3})\mapsto(-z_{1},-z_{2},z_{3})\leavevmode\nobreak\
,\leavevmode\nobreak\ $ $\displaystyle\omega:$
$\displaystyle(z_{1},z_{2},z_{3})\mapsto(z_{1},-z_{2},-z_{3})\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$

The orientifold projection acts by gauging the $\Omega R$ symmetry, where
$\Omega$ is world-sheet parity and $R$ acts on the complex coordinates as

$R:(z_{1},z_{2},z_{3})\mapsto({\overline{z}}_{1},{\overline{z}}_{2},{\overline{z}}_{3})\leavevmode\nobreak\
.\leavevmode\nobreak\ \,$ (6)

Overall there are four kinds of orientifold 6-planes (O6-planes) contributes
to the actions of $\Omega R$, $\Omega R\theta$, $\Omega R\omega$, and $\Omega
R\theta\omega$ respectively Blumenhagen:2000ea ; Chen:2007zu ; Cvetic:2001nr ;
Cvetic:2002pj . Three stacks of $N_{a}$ D6-branes wraps on the factorized
three-cycles cancelling the RR charges of these O6-planes.

The homology classes of these three cycles are wrapped by the D6-brane stack,
taking the form for a rectangular torus as $n_{a}^{i}[a_{i}]+m_{a}^{i}[b_{i}]$
and for a tilted torus as $n_{a}^{i}[a^{\prime}_{i}]+m_{a}^{i}[b_{i}]$, with
$[a_{i}^{\prime}]=[a_{i}]+\frac{1}{2}[b_{i}]$. We label the generic the
generic one cycle by $(n_{a}^{i},l_{a}^{i})$ in terms of the so-called
wrapping numbers, with $l_{a}^{i}\equiv m_{a}^{i}$ for a rectangular and
$l_{a}^{i}\equiv 2\tilde{m}_{a}^{i}=2m_{a}^{i}+n_{a}^{i}$ for tilted two-torus
respectively. Here $l_{a}^{i}-n_{a}^{i}$ is even for tilted two-tori, yet odd
for rectangular torus. Therefore, the wrapping number for stack $a$ of
D6-branes along the cycle are denoted by $(n_{a}^{i},l_{a}^{i})$, while the
$\Omega R$ images ${a^{\prime}}$ stack of $N_{a}$ D6-branes can be labeled by
wrapping numbers $(n_{a}^{i},-l_{a}^{i})$. The homology three-cycles for $a$
stack of D6-branes and its orientifold image $a^{\prime}$ results in

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

in which $\beta_{i}=0$ when the $i$-th two-torus is rectangular while
$\beta_{i}=1$ for tilted two-torus. Furthermore, the homology three-cycles
wrapped by the four O6-planes leads to

$\displaystyle\Omega R:[\Pi_{\Omega
R}]=2^{3}[a_{1}]\times[a_{2}]\times[a_{3}]\leavevmode\nobreak\
,\leavevmode\nobreak\ \,$ $\displaystyle\Omega R\omega:[\Pi_{\Omega
R\omega}]=-2^{3-\beta_{2}-\beta_{3}}[a_{1}]\times[b_{2}]\times[b_{3}]\leavevmode\nobreak\
,\leavevmode\nobreak\ $ $\displaystyle\Omega R\theta\omega:[\Pi_{\Omega
R\theta\omega}]=-2^{3-\beta_{1}-\beta_{3}}[b_{1}]\times[a_{2}]\times[b_{3}]\leavevmode\nobreak\
,\leavevmode\nobreak\ $ $\displaystyle\Omega R\theta:[\Pi_{\Omega
R}]=-2^{3-\beta_{1}-\beta_{2}}[b_{1}]\times[b_{2}]\times[a_{3}]\leavevmode\nobreak\
.\leavevmode\nobreak\ $ (7)

Based on these, the RR tadpole cancellation condition provides the restriction
rule Cvetic:2001nr ; Aldazabal:2000dg ; Ibanez:2001nd

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

where $2N^{(i)}$ is the number of D6-branes wrapping along the $i$-th O6-plane
(filler branes) and the last term represents the O6-planes with $-4$ RR
charges in D6-brane charge unit. Note that for simplification, we denote
$A_{a}\equiv-n_{a}^{1}n_{a}^{2}n_{a}^{3},\,B_{a}\equiv
n_{a}^{1}l_{a}^{2}l_{a}^{3},\,C_{a}\equiv
l_{a}^{1}n_{a}^{2}l_{a}^{3},\,D_{a}\equiv
l_{a}^{1}l_{a}^{2}n_{a}^{3},\,\tilde{A}_{a}\equiv-
l_{a}^{1}l_{a}^{2}l_{a}^{3},\,\tilde{B}_{a}\equiv
l_{a}^{1}n_{a}^{2}n_{a}^{3},\,\tilde{C}_{a}\equiv
n_{a}^{1}l_{a}^{2}n_{a}^{3},\,\tilde{D}_{a}\equiv
n_{a}^{1}n_{a}^{2}l_{a}^{3}$. These filler branes represent the $USp$ group
carrying the wrapping numbers with one of the O6-planes as shown in table 1.

Orientifold action | O6-Plane | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$
---|---|---
$\Omega R$ | 1 | $(2^{\beta_{1}},0)\times(2^{\beta_{2}},0)\times(2^{\beta_{3}},0)$
$\Omega R\omega$ | 2 | $(2^{\beta_{1}},0)\times(0,-2^{\beta_{2}})\times(0,2^{\beta_{3}})$
$\Omega R\theta\omega$ | 3 | $(0,-2^{\beta_{1}})\times(2^{\beta_{2}},0)\times(0,2^{\beta_{3}})$
$\Omega R\theta$ | 4 | $(0,-2^{\beta_{1}})\times(0,2^{\beta_{2}})\times(2^{\beta_{3}},0)$
Table 1: The wrapping numbers for four O6-planes.

To have three family of chiral fermions under $SU(4)_{C}\times SU(2)_{L}\times
SU(2)_{R}$ gauge symmetries, further constraints were imposed on the
intersection numbers expressed in terms of the wrapping numbers

$\displaystyle
I_{ab}=[\Pi_{a}][\Pi_{b}]=2^{-k}\prod_{i=1}^{3}(n_{a}^{i}l_{b}^{i}-n_{b}^{i}l_{a}^{i})\leavevmode\nobreak\
,\leavevmode\nobreak\ $ $\displaystyle
I_{ab^{\prime}}=[\Pi_{a}][\Pi_{b^{\prime}}]=-2^{-k}\prod_{i=1}^{3}(n_{a}^{i}l_{b}^{i}+n_{b}^{i}l_{a}^{i})\leavevmode\nobreak\
,\leavevmode\nobreak\ $ $\displaystyle
I_{aa^{\prime}}=[\Pi_{a}][\Pi_{a^{\prime}}]=-2^{3-k}\prod_{i=1}^{3}(n_{a}^{i}l_{a}^{i})\leavevmode\nobreak\
,\leavevmode\nobreak\ $ $\displaystyle
I_{aO6}=[\Pi_{a}][\Pi_{O6}]=2^{3-k}(-l_{a}^{1}l_{a}^{2}l_{a}^{3}+l_{a}^{1}n_{a}^{2}n_{a}^{3}+n_{a}^{1}l_{a}^{2}n_{a}^{3}+n_{a}^{1}n_{a}^{2}l_{a}^{3})$
(9)

where $k=\beta_{1}+\beta_{2}+\beta_{3}$ is the total number of tilted two-
tori, and $[\Pi_{O6}]=[\Pi_{\Omega R}]+[\Pi_{\Omega R\omega}]+[\Pi_{\Omega
R\theta\omega}]+[\Pi_{\Omega R\theta}]$ is the sum of four O6-plane homology
three-cycles. And the intersection numbers shall follow

$\displaystyle\small I_{ab}+I_{ab^{\prime}}=3\leavevmode\nobreak\
,\leavevmode\nobreak\ I_{ac}=-3\leavevmode\nobreak\ ,\leavevmode\nobreak\
I_{ac^{\prime}}=0\leavevmode\nobreak\ ,\leavevmode\nobreak\ \,$ (10)

with $c$ and $c^{\prime}$ exchange as well.

The massless particle spectrum for the supersymmetric Pati-Salam is listed in
table 2, where the gauge symmetry results from
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orbifold projection Blumenhagen:2003jy .
The representations refer to $U(N_{a}/2)$ when the intersecting D6-branes are
of number $N_{a}=8,N_{b}=4,N_{c}=4$. Note that the chiral supermultiplets not
only represents the scalars but also the fermions in this supersymmetric
constructions. Moreover, positive intersection numbers refer to the left-
handed chiral supermultiplets. In our later discussion, some of the introduced
particles for gauge coupling unification can be read off from the spectrum
table as well.

Sector | Representation
---|---
$aa$ |  $U(N_{a}/2)$ vector multiplet
| 3 adjoint chiral multiplets
$ab+ba$ |  $I_{ab}$ $(\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}_{a},\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}_{b})$ fermions
$ab^{\prime}+b^{\prime}a$ |  $I_{ab^{\prime}}$ $(\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}_{a},\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}_{b})$ fermions
$aa^{\prime}+a^{\prime}a$ |  $\frac{1}{2}(I_{aa^{\prime}}-\frac{1}{2}I_{a,O6})\;\;\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\;\;$ fermions
|
$\frac{1}{2}(I_{aa^{\prime}}+\frac{1}{2}I_{a,O6})\;\;\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\;\;$
fermions
Table 2: General massless particle spectrum for intersecting D6-branes at
generic angles.

In addition to the three generation conditions and tadpole cancellation
condition, we further need to require $N=1$ supersymmetry preservation in four
dimension, with the equality and inequality conditions Berkooz:1996km ;
Cvetic:2002pj

$\displaystyle
x_{A}\tilde{A}_{a}+x_{B}\tilde{B}_{a}+x_{C}\tilde{C}_{a}+x_{D}\tilde{D}_{a}=0,$
$\displaystyle A_{a}/x_{A}+B_{a}/x_{B}+C_{a}/x_{C}+D_{a}/x_{D}<0,$ (11)

where $x_{A}=\lambda,\;x_{B}=\lambda 2^{\beta_{2}+\beta
3}/\chi_{2}\chi_{3},\;x_{C}=\lambda 2^{\beta_{1}+\beta
3}/\chi_{1}\chi_{3},\;x_{D}=\lambda 2^{\beta_{1}+\beta 2}/\chi_{1}\chi_{2}$.
Here $\chi_{i}=R^{2}_{i}/R^{1}_{i}$ represent the complex structure moduli of
the $i$-th two-torus, and $\lambda$ is introduced as a positive parameter to
put all the variables $A,\,B,\,C,\,D$ at equal footing.

The supersymmetric Pati-Salam gauge symmetry $SU(4)_{C}\times SU(2)_{L}\times
SU(2)_{R}$ can then be broken down to the SM gauge symmetry via the D-brane
splitting and Higgs mechanism preserving supersymmetry. Concrete
supersymmetric Pati-Salam models have been constructed Li:2019nvi ; Li:2021pxo
, and such D-brane models have also been investigated with powerful
reinforcement machine learning methods Halverson:2019tkf ; Loges:2021hvn . In
He:2021gug ; He:2021kbj , we for the first time propose a systematic method to
construct all the three-family ${\cal N}=1$ supersymmetric Pati-Salam models
and conclude that there are in total 202752 models with 33 different gauge
coupling relations. Among which, there is one class of models with gauge
coupling unification at GUT scale, while the others do not. The absence of
gauge coupling unification at the string scale appear be a generic problem. In
He:2021kbj , we propose that by introducing vector-like particles from ${\cal
N}=2$ subsector the gauge unification problem may be solved. In particular,
the number of these exotic particles are fully determined by the brane
intersection number. These additional particles can be decoupled as discussed
in Chen:2007px , and its gauge coupling relation can be realized at string
scale via two-loop RGE running. This leads to our exploration that whether
this solved for all the supersymmetric Pati-Salam models in general. We will
discuss in details for all the 33 representative models.

##  3 Decoupling of the Exotic Particles

In the string model building, there exist exotic particles in general. So we
need to discuss how to decouple the exotic particles first. In our Pati-Salam
models which are given in Appendix A, we can decouple most of the exotic
particles via Higgs mechanism and instanton effects, etc, except the chiral
multiplets under $SU(4)_{C}$ symmetric representation. The key point is the
gauge anomaly cancellation. First, the chiral multiplets under $SU(4)_{C}$
anti-symmetric representation do not contribute to the gauge anomaly. Their
mass terms are forbidden by the anomalous $U(1)$ gauge symmetries, and can be
generated via the instanton effects Blumenhagen:2006xt ; Haack:2006cy ;
Florea:2006si . Thus, they can be decoupled. Second, for the models without
the chiral multiplets under $SU(4)_{C}$ symmetric representation, all the
exotic particles can be decoupled via Higgs mechanism and instanton effects.
For a concrete example, please see Chen:2007px ; Chen:2007zu . Third, for the
models with the chiral multiplets under $SU(4)_{C}$ symmetric representation,
it seems to us that we cannot decouple the exotic particles. Therefore, we
cannot decouple the exotic particles only in Model 4, 23, and 32 in Appendix
A. To study the gauge coupling unification, we assume the extoic particles in
these models can be decoupled as well.

##  4 String-scale Gauge Coupling Relations

As presented with details in He:2021gug ; He:2021kbj , the full list of
supersymmetric Pati-Salam models have $33$ different gauge coupling relations,
see Appendix A. In these models, $a$ stack of D6-branes gives the $U(4)$ gauge
symmetry, $b$ stack of D6-branes gives the $U(2)_{L}$ gauge symmetry, and $c$
stack of D6-branes gives the $U(2)_{R}$ gauge symmetry. In particular, there
is one class of models with gauge coupling unification at GUT scale. Namely,
the strong, weak and hypercharge gauge couplings $g^{2}_{a},g^{2}_{b}$ and
$\frac{5}{3}g^{2}_{Y}$ satisfy
$g^{2}_{a}=g^{2}_{b}=g^{2}_{c}=(\frac{5}{3}g^{2}_{Y})=4\sqrt{\frac{2}{3}}\pi
e^{\phi^{4}}$, where $\phi^{4}$ represents the dilaton field. However, not all
the supersymmetry Pati-Salam models are with gauge coupling unification at GUT
scale. To have string-scale gauge coupling relation, we utilize the
Renormalization Group Equations (RGEs) evolution Chen:2017rpn ; Chen:2018ucf ;
He:2021kbj .

The RGEs for the gauge couplings at the two-loop level are given by
Barger:2004sf ; Barger:2007qb ; Barger:2005qy ; Gogoladze:2010in

$\frac{d}{d\ln\mu}g_{i}=\frac{b_{i}}{(4\pi)^{2}}g_{i}^{3}+\frac{g_{i}^{3}}{(4\pi)^{4}}\left[\sum_{j=1}^{3}B_{ij}g_{j}^{2}-\sum_{\alpha=u,d,e}d_{i}^{\alpha}{\rm
Tr}\left(h^{\alpha\dagger}h^{\alpha}\right)\right],$ (12)

where $g_{i}(i=1,2,3)$ are the SM gauge couplings and
$h^{\alpha}(\alpha=u,d,e)$ are the Yukawa couplings. The coefficients of beta
functions in SM Machacek:1983tz ; Machacek:1983fi ; Machacek:1984zw ;
Cvetic:1998uw  and supersymmetric models Barger:1992ac ; Barger:1993gh ;
Martin:1993zk  are represented by

$\displaystyle b_{\rm
SM}=\left(\frac{41}{6}\frac{1}{k_{Y}},-\frac{19}{6}\frac{1}{k_{2}},-7\right),\leavevmode\nobreak\
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},\leavevmode\nobreak\
$ (13) $\displaystyle d^{u}_{\rm
SM}=\left(\frac{17}{6}\frac{1}{k_{Y}},\frac{3}{2}\frac{1}{k_{2}},2\right),\leavevmode\nobreak\
d^{d}_{\rm SM}=0,\leavevmode\nobreak\ d^{e}_{\rm SM}=0,\,$ (14) $\displaystyle
b_{\rm
SUSY}=\left(11\frac{1}{k_{Y}},\frac{1}{k_{2}},-3\right),\leavevmode\nobreak\
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},\leavevmode\nobreak\
$ (15) $\displaystyle d^{u}_{\rm
SUSY}=\left(\frac{26}{3}\frac{1}{k_{Y}},6\frac{1}{k_{2}},4\right),\leavevmode\nobreak\
d^{d}_{\rm SUSY}=0,\leavevmode\nobreak\ d^{e}_{\rm SUSY}=0,$ (16)

where $k_{Y}$ and $k_{2}$ are general normalization factors. 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. The general one-loop RGEs for Yukawa
couplings can be found in Gogoladze:2010in . Starting from the electroweak
theory, we run the couplings up from Z boson mass scale $M_{Z}$ to high
energies with the boundary conditions for these equations at $M_{Z}$ as

$g_{1}(M_{Z})=\sqrt{k_{Y}}\frac{g_{em}}{\cos\theta_{W}}\leavevmode\nobreak\
,\leavevmode\nobreak\
g_{2}(M_{Z})=\sqrt{k_{2}}\frac{g_{em}}{\sin\theta_{W}}\leavevmode\nobreak\
,\leavevmode\nobreak\ g_{3}(M_{Z})=\sqrt{4\pi\alpha_{s}}\leavevmode\nobreak\
.$ (17)

From $M_{Z}$ up to a supersymmetry breaking scale $M_{\rm S}$, we consider
only the non-supersymmetric SM spectrum including a top quark pole mass at
$m_{t}=173.34$ GeV. From $M_{\rm S}$ scale, we perform the supersymmetric RGEs
with all the states including the introduced exotic vector-like particles at
$M_{V}$. Moreover, the Z boson mass, the Higgs vacuum expectation value,
strong coupling constant, fine structure constant, and weak mixing angle at
$M_{Z}$ are choosen to be ParticleDataGroup:2018ovx ;
ParticleDataGroup:2020ssz

$\begin{split}&M_{Z}=91.1876{\rm\leavevmode\nobreak\ GeV},\leavevmode\nobreak\
m_{t}=173.34\pm 0.27({\rm{stat}})\pm
0.71({\rm{syst}}){\rm{\leavevmode\nobreak\ GeV}},\leavevmode\nobreak\
v=174.10{\rm{\leavevmode\nobreak\ GeV}},\\\ &\alpha_{s}(M_{Z})=0.1181\pm
0.0011,\leavevmode\nobreak\ \alpha_{em}^{-1}(M_{Z})=128.91\pm
0.02,\leavevmode\nobreak\ \sin^{2}\theta_{W}(M_{Z})=0.23122.\end{split}$ (18)

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.

Utilizing the current precision electroweak data and setting the supersymmetry
breaking scale to be $M_{\text{S}}\simeq 3.0$ TeV, we study the gauge coupling
unification around string scale by solving two-loop RGEs, with

$M_{\text{U}}\sim M_{\text{string}}\simeq 5\times 10^{17}{\leavevmode\nobreak\
\text{GeV}}.$ (19)

Recall that the generic gauge coupling relations at string scale for
supersymmetric Pati-Salam models read

$g_{a}^{2}=k_{2}g_{b}^{2}=k_{Y}g_{Y}^{2}=g_{\text{U}}^{2}\sim
g_{\text{string}}^{2},$ (20)

where $k_{Y}$ and $k_{2}$ are constants for each model and will determine the
value of the mixing angel $\sin\theta_{W}$ at the string scale. The string-
scale gauge coupling relation in the evolution is realized via

$\alpha_{\text{U}}^{-1}\equiv\alpha_{1}^{-1}=(\alpha_{2}^{-1}+\alpha_{3}^{-1})/2$
(21)

with $\alpha_{1}\equiv k_{Y}g_{Y}^{2}/4\pi$, $\alpha_{2}\equiv
k_{2}g_{b}^{2}/4\pi$, $\alpha_{3}\equiv g_{a}^{2}/4\pi$, and the accuracy
$\Delta=|\alpha_{1}^{-1}-\alpha_{2}^{-1}|/\alpha_{1}^{-1}$ is limited to be
less than $1.0\%$. Here, $\alpha_{1}$ and $\alpha_{2}$ are traditional gauge
couplings if and only if $k_{Y}=5/3$ and $k_{2}=1$, respectively.

As discussed in He:2021kbj , Model 1 in table 8 has a canonical hypercharge
normalization $k_{Y}=5/3$, and then the gauge coupling unification is
naturally achieved at most $M_{\rm U}\sim 2\times 10^{16}$ GeV, similar to the
predicted unification scale in the MSSM. However, the unification scale is
about one order of magnitude smaller than string scale, where the discrepancy
can be diminished by the introducing additional vector-like particles
Blumenhagen:2003jy .

Interestingly, we also find that the precise string-scale gauge coupling
relation can be achieved at two-loop level by introducing the extra vector-
like particles from ${\cal N}=2$ subsector or four-dimensional chiral sectors
in other models. The quantum numbers for the vector-like particles are the
same as those of the SM fermions and their Hermitian conjugates. While, the
number $n_{V}$ and the quantum numbers of these particles are highly model
dependent and can be determined by brane construction. Particularly, in our
calculations, multi-pair of extra particles appears naturally, and we only
need to fine tune their masses to unify the gauge coupling near the string
scale $M_{\text{string}}$. From the numerical results, we find that the mass
of these extra particles approach to the string scale as the number of these
particles increase.

The quantum numbers of vector-like particles under $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}$ and their contributions to one-loop level beta
functionsJiang:2006hf ; Barger:2007qb  are

$\displaystyle XQ+{\overline{XQ}}={\mathbf{(3,2,{1\over
6})+({\bar{3}},2,-{1\over 6})}}\,,\quad\Delta b=({1\over 5},3,2)\,;$ (22)
$\displaystyle XU+{\overline{XU}}={\mathbf{({3},1,{2\over
3})+({\bar{3}},1,-{2\over 3})}}\,,\quad\Delta b=({8\over 5},0,1)\,;$ (23)
$\displaystyle XD+{\overline{XD}}={\mathbf{({3},1,-{1\over
3})+({\bar{3}},1,{1\over 3})}}\,,\quad\Delta b=({2\over 5},0,1)\,;$ (24)
$\displaystyle XL+{\overline{XL}}={\mathbf{(1,2,{1\over 2})+({1},2,-{1\over
2})}}\,,\quad\Delta b=({3\over 5},1,0)\,;$ (25) $\displaystyle
XE+{\overline{XE}}={\mathbf{({1},1,{1})+({1},1,-{1})}}\,,\quad\Delta
b=({6\over 5},0,0)\,;$ (26) $\displaystyle
XG={\mathbf{({8},1,0)}}\,,\quad\Delta b=(0,0,3)\,;$ (27) $\displaystyle
XW={\mathbf{({1},3,0)}}\,,\quad\Delta b=(0,2,0)\,.$ (28)

In our numerical calculations, the two-loop level beta functions from these
particles are used in supersymmetric models from Eqs. (B8)-(B11) in
Barger:2007qb . We integrate the renormalization group equations in Eq. (12)
from $M_{Z}$ up to $M_{\rm string}$ with three sections, from $M_{Z}$ to
$M_{\rm S}$, from $M_{\rm S}$ to $M_{V}$, and from $M_{V}$ to $M_{\rm
string}$. The precise energy scale where we realize the string-scale gauge
coupling relations depends on the number of the introduced particle, the
coefficients $b_{i}$, $B_{ij}$ and $d_{i}^{\alpha}$ of the beta-function. The
values of $M_{XG}$, $M_{XW}$ are determined by the intersection energy scale
of section 2(from $M_{\rm S}$ to $M_{V}$), and section 3(from $M_{V}$ to
$M_{\rm string}$) in the plot, where these exotic particles are introduced.
Thus, the mass of the exotic particles are determined by both the number of
the particles, the beta functions and the precise energy scale where we
realize the string-scale gauge coupling relations.

Once these particles are included with mass $M_{V}$, the evolution of
$\alpha_{i}^{-1}(t)$ is depressed if $\Delta b_{i}\neq 0$ and the turning
point is at $\mu=M_{V}$. Taking Model 1 as example, we find that there are two
bending points for the running of gauge couplings in figure 1; the first one
is corresponding to the supersymmetry breaking scale $M_{\rm S}\simeq 3.0$
TeV, while the second one corresponds to the introduced particle $M_{V}$ with
chiral multiplets $XG$ and $XW$ obtain the same mass, i.e.,
$M_{XG}=M_{XW}=M_{V}$. Recall that $\Delta b_{2}(XW)=2$, the plotted lines for
$\alpha_{2}^{-1}$ bend at $M_{XW}$ where the particle $XW$ is introduced. In
the same way, the lines for $\alpha_{3}^{-1}$ bend at the point $M_{XG}$ due
to $\Delta b_{3}(XG)=3$. Moreover, the degree of depression will increase as
$\Delta b_{i}$ increases. That’s to say, the inverse hypercharge coupling
$\alpha_{1}^{-1}$ decreases more rapidly in models with the particles
$XU+\overline{XU}$ than that in models with $XD+\overline{XD}$. And the
depression of the inverse strong coupling $\alpha_{3}^{-1}$ from both pairs of
vector-like particles are the same. Note that when both particles
$XU+\overline{XU}$ and $XD+\overline{XD}$ are added, there will be two bending
points at $M_{\rm S}$ and $M_{V}$ as these particles have the same mass
$M_{V}=M_{XU}=M_{XD}$, while there will be three bending points at $M_{\rm
S}$, $M_{XU}$ and $M_{XD}$ as the mass of these particles are different.

###  4.1 Chiral Multiplets from Adjoint Representations of $SU(4)_{C}$ and
$SU(2)_{L}$

For the Model 1 with $k_{y}=k_{2}=1$, the gauge couplings naturally unifies at
the traditional GUT scale, one order of magnitude smaller than the string
scale. Instead of introducing the vector-like particle from a ${\cal N}=2$
subsector, we consider up to three chiral multiplets in the adjoint
representations of $SU(4)_{C}$ and $SU(2)_{L}$ with masses
$M_{V}(V=XG\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ XW)$. These two
particles $XG$ and $XW$ arise from the $aa$ and $bb$ sectors in the spectrum
table 2, and thus the maximum number is 3. The gauge couplings can be unified
at close to the string scale, $M_{\rm U}=5.0\times 10^{17}$ GeV. The two-loop
RGE running for the guage couplings are shown in figure 1.

Figure 1: Two-loop evolution of gauge couplings for the Model 1 with vector-
like particles. In the model, $k_{Y}=1\times\frac{5}{3}$ and $k_{2}=1$. The
string-scale gauge coupling relations can be achieved by adding $3(XG+XW)$
(a), $2(XG+XW)$ (b) and $(XG+XW)$ (c).

As the number of particles increases, the mass of the particles increases and
approaches the string scale. Moreover, the splitting between the $XW$ and $XG$
masses will be reduced. Thus in the following calculations, we choose to add 3
$(XG+XW)$. The energy scale, number of the particles, and the mass of the
particles for the Model 1-12 are shown in table 3. Above $M_{V}$, the running
of $SU(3)_{C}$ coupling and $SU(2)_{L}$ coupling are reducing due to the non-
zero beta functions $\Delta b_{3}(XG)$ and $\Delta b_{2}(XW)$.

Model No. |  $k_{y}$ |  $k_{2}$ |  $n_{V}$ |  $M_{XG}$(GeV) |  $M_{XW}$(GeV) |  $M_{\rm U}$(GeV)
---|---|---|---|---|---|---
1 |  1 |  1 |  3 |  $1.60\times 10^{16}$ |  $1.60\times 10^{16}$ |  $5.00\times 10^{17}$
2 |  $2.87\times 10^{15}$ |  $2.87\times 10^{15}$ |  $5.00\times 10^{17}$
1 |  $1.65\times 10^{13}$ |  $1.65\times 10^{13}$ |  $5.00\times 10^{17}$
2 |  85/61 |  4/9 |  3 |  $2.76\times 10^{14}$ |  $7.84\times 10^{9}$ |  $5.00\times 10^{17}$
3 |  65/44 |  1/2 |  3 |  $1.28\times 10^{14}$ |  $1.34\times 10^{10}$ |  $5.00\times 10^{17}$
4 |  35/32 |  5/6 |  3 |  $3.50\times 10^{15}$ |  $1.23\times 10^{14}$ |  $5.00\times 10^{17}$
5 |  10/7 |  2/3 |  3 |  $2.87\times 10^{14}$ |  $2.00\times 10^{11}$ |  $5.85\times 10^{17}$
6 |  11/8 |  5/6 |  3 |  $3.13\times 10^{14}$ |  $3.06\times 10^{12}$ |  $5.00\times 10^{17}$
7 |  25/19 |  1 |  3 |  $3.98\times 10^{14}$ |  $1.19\times 10^{14}$ |  $5.00\times 10^{17}$
8 |  10/7 |  1 |  3 |  $1.73\times 10^{14}$ |  $3.17\times 10^{13}$ |  $5.00\times 10^{17}$
9 |  11/8 |  1/6 |  3 |  $2.63\times 10^{14}$ |  $2.06\times 10^{8}$ |  $5.00\times 10^{17}$
10 |  50/47 |  4/9 |  3 |  $5.10\times 10^{15}$ |  $6.90\times 10^{10}$ |  $5.00\times 10^{17}$
11 |  1 |  1/3 |  3 |  $1.21\times 10^{16}$ |  $1.18\times 10^{12}$ |  $5.00\times 10^{17}$
12 |  35/32 |  1/6 |  3 |  $1.22\times 10^{16}$ |  $4.49\times 10^{8}$ |  $5.00\times 10^{17}$
13 |  11/8 |  5/14 |  3 |  $3.18\times 10^{14}$ |  $2.45\times 10^{9}$ |  $5.00\times 10^{17}$
14 |  35/32 |  35/66 |  3 |  $4.62\times 10^{14}$ |  $2.33\times 10^{11}$ |  $5.00\times 10^{17}$
Table 3: String-scale gauge coupling relations achieved with $XG+XW$, from
adjoint representation of $SU(4)$ and $SU(2)$.

For the models with $k_{y}>1$ and $k_{2}<1$, the string-scale gauge coupling
relation can also be achieved by adding $XW+XG$ from adjoint representation of
$SU(4)$ and $SU(2)$. The two-loop RGE running for the gauge couplings of the
Model 2 is plotted in figure 2. Here we include the contributions from
3$(XW+XG)$ to reduce the mass splitting of these added particles. And the two-
loop RGE running of the gauge couplings for Model 3-5 are shown in figures 3,
4, 5, for Model 6-12 in figures 11, 12, 13, 14, 15, 16, 17 in Appendix B, for
Model 13-14 in figures 6 and 7.

Figure 2: Two-loop evolution of gauge couplings for the Model 2 with vector-
like particles. In the model, $k_{Y}=\frac{85}{61}\times\frac{5}{3}$ and
$k_{2}=\frac{4}{9}$. The string-scale gauge coupling relations can be achieved
by adding $3(XW+XG)$ (a) and $5(XQ+\overline{XQ})$ (b).

###  4.2 The Vector-Like Particles from $\mathcal{N}=2$ Subsector with
$k_{y}>1$ and $k_{2}<1$

For the models like Model 2 in table 9, the evolution of
$\alpha_{1}^{-1}(\mu)$ is depressed when $k_{y}>1$ and the evolution of
$\alpha_{2}^{-1}(t)$ is raised when $k_{2}<1$. They induces that the
intersection point of $\alpha_{1}^{-1}(t)$ and $\alpha_{3}^{-1}(t)$ lines is
below the line of $\alpha_{2}^{-1}(t)$, as illustrated in figure 2(a). In this
case, to get an string-scale gauge coupling relation, we need introduce extra
particles $XQ+\overline{XQ}$ or $XW$ as well as $XG$ into the models, which
will mainly modify the evolution of the electroweak and strong couplings
without substantially affecting the U(1) coupling. This is owing to the large
contributions to $\Delta b_{2}$ and $\Delta b_{3}$, rather relatively small
contributions to $\Delta b_{1}$. So as the energy rises from mass scale
$M_{XQ}$, $\alpha_{2}^{-1}$ and $\alpha_{3}^{-1}$ reduce rapidly. Therefore,
string-scale gauge coupling relations are achieved near $M_{\rm U}=3.75\times
10^{17}$ GeV by adding 5 sets of $XQ+\overline{XQ}$ at $M_{XQ}=2.51\times
10^{14}$ GeV. The evolution of gauge couplings for Model 2 with $k_{y}=85/61$
and $k_{2}=4/9$ is shown in figure 2(b) and the energy scale, the number and
mass of the added vector-like particles $XQ+\overline{XQ}$ are list in table
4.

Model No. |  $k_{y}$ |  $k_{2}$ |  $n_{XQ}$ |  $M_{XQ}$(GeV) |  |  |  $M_{\rm U}$(GeV)
---|---|---|---|---|---|---|---
2 |  85/61 |  4/9 |  5 |  $2.51\times 10^{14}$ |  |  |  $3.75\times 10^{17}$
3 |  65/44 |  1/2 |  3 |  $1.26\times 10^{12}$ |  |  |  $1.81\times 10^{17}$
4 |  25/32 |  5/6 |  7 |  $1.07\times 10^{16}$ |  |  |  $1.09\times 10^{17}$
Model No. |  $k_{y}$ |  $k_{2}$ |  $n_{XQ}$ |  $M_{XQ}$(GeV) |  $n_{XG}$ |  $M_{XG}$(GeV) |  $M_{\rm U}$(GeV)
5 |  10/7 |  2/3 |  5 |  $1.03\times 10^{15}$ |  3 |  $1.18\times 10^{17}$ |  $5.00\times 10^{17}$
6 |  11/8 |  5/6 |  7 |  $1.51\times 10^{16}$ |  3 |  $3.69\times 10^{16}$ |  $5.00\times 10^{17}$
7 |  25/19 |  1 |  3 |  $1.43\times 10^{14}$ |  3 |  $1.41\times 10^{16}$ |  $5.00\times 10^{17}$
8 |  10/7 |  1 |  3 |  $4.73\times 10^{14}$ |  3 |  $1.50\times 10^{16}$ |  $5.00\times 10^{17}$
Model No. |  $k_{y}$ |  $k_{2}$ |  $n_{XQ}$ |  $M_{XQ}$(GeV) |  $n_{XW}$ |  $M_{XW}$(GeV) |  $M_{\rm U}$(GeV)
9 |  11/8 |  1/6 |  5 |  $3.84\times 10^{14}$ |  3 |  $8.10\times 10^{15}$ |  $5.00\times 10^{17}$
10 |  50/47 |  4/9 |  5 |  $5.13\times 10^{15}$ |  3 |  $4.39\times 10^{16}$ |  $5.00\times 10^{17}$
11 |  1 |  1/3 |  5 |  $1.60\times 10^{16}$ |  3 |  $4.12\times 10^{13}$ |  $5.00\times 10^{17}$
12 |  35/32 |  1/6 |  5 |  $1.66\times 10^{16}$ |  3 |  $1.44\times 10^{12}$ |  $5.00\times 10^{17}$
Table 4: String-scale gauge coupling relations achieved by adding vector-like
particles $XQ+\overline{XQ}$ as well as $XG$ or $XW$, from $\mathcal{N}=2$
subsector. The number of vector-like particles are defined by the intersection
number on a and b stacks, $n_{v}=I_{ab}$.

Similarly, for Model 3 and 4, the non-canonical constants are
$k_{y}=65/44,\leavevmode\nobreak\ k_{2}=1/2$ and
$k_{y}=25/32,\leavevmode\nobreak\ k_{2}=5/6$. The string-scale gauge coupling
relation are also achieved by bringing particles $XQ+\overline{XQ}$ into Model
3 at $1.26\times 10^{12}$ GeV and Model 4 at $1.07\times 10^{16}$ GeV,
respectively. The evolution of gauge couplings are shown in figures 3 and 4,
in which the number of pairs of the new vector-like particles is 3 and 7,
respectively. We note that the mass of the extra particles is related to the
number of pairs of particles. As the number increases, the mass scale is
pushed up to the high energy scale and thus the vector-like particles decay at
high energy scale.

Figure 3: Two-loop evolution of gauge couplings for the Model 3 with vector-
like particles. In the model, $k_{Y}=\frac{65}{44}\times\frac{5}{3}$ and
$k_{2}=\frac{1}{2}$. The string-scale gauge coupling relations can be achieved
by adding $3(XW+XG)$ (a) and $3(XQ+\overline{XQ})$ (b).

Figure 4: Two-loop evolution of gauge couplings for the Model 4 with vector-
like particles. In the model, $k_{Y}=\frac{35}{32}\times\frac{5}{3}$ and
$k_{2}=\frac{5}{6}$. The string-scale gauge coupling relations can be achieved
by adding $3(XW+XG)$ (a) and $7(XQ+\overline{XQ})$ (b).

Of course, the number of these extra vector-like particles is not random, yet
from brane constructions. From Eqs. (22) and (28), the quantum numbers of
these particles under $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ are
$XQ=(3,2,1/6)$, and $\overline{XQ}=(\bar{3},2,-1/6)$. In the supersymmetric
Pati-Salam models, these vector-like particles $(XQ+\overline{XQ})$ arise from
the intersections between $a$ and $b$ stacks of D6-branes or $a$ and
$b^{\prime}$ stacks of D6-brane. The particle $XW$ arise from $bb$ sector of
the adjoint representation of $SU(2)_{L}$.

Base on the brane construction, the number of vector-like particles
$(XQ+\overline{XQ})$ can be determined by the intersection number of the $a$
and $b$ stacks of D6-brane, $I_{ab}$, or $a$ and $b^{\prime}$ stacks of
D6-brane, $I_{ab^{\prime}}$. For example, the intersection number is
$I_{ab^{\prime}}=-2^{0}(4+1)(1)=-5$ in Model 2 (table 9) and the corresponding
number of the additional particles $(XQ+\overline{XQ})$ is 5. If the wrapping
numbers $(n_{a}^{i},l_{a}^{i})$ and $(n_{b}^{i},l_{b}^{i})$ have the same
value, indicating that the D6-branes warpping on the $i-$ torus are parallel
to each other. Therefore, there is no intersection on the $i-$torus, but only
on the other two torus. From table 10, we see that $(n_{a}^{3},l_{a}^{3})$ and
$(n_{b}^{3},l_{b}^{3})$ for Model 3 are the same, thus the intersection number
of the $a$ and $b$ stacks are only calculated on the first two torus, i.e.,
$I_{ab}=2^{0}(-2-5)(1)=7$. Namely, 7 pairs of $XQ+\overline{XQ}$ naturally
arise from brane intersection.

Based on the calculations, we know that the energy scale is pushed up to high
energy scale as $k_{y}$ or $k_{2}$ deceases. Thus, for the models with
$k_{y}>1$, to obtain a string-scale gauge coupling relation, the constant
$k_{2}$ of the model should be smaller than $1/2$. Otherwise, the gauge
couplings are unified at an intermediate energy scale $10^{14}-10^{16}$ GeV,
little smaller than the string scale. For Model 5 with $k_{y}>1$ and
$k_{2}>1/2$, the gauge coupling relation can be achieved at the string scale
$M_{\rm U}\simeq 5\times 10^{17}$ GeV by adding 5 pairs of $XQ+\overline{XQ}$
and 3 $XG$. For Models 5-8, the parameter $k_{y}$ is almost equal. And the
parameter $k_{2}$ in Models 6-8 is larger than that in Model 5. Thus, the
energy scale is smaller than that in Model 5. To increase the energy scale, we
also need to introduce adjoint particle $XG$ for Models 6-8. The evolution of
gauge couplings for Models 5-8 with vector-like particles and $XG$ are
presented in figures 5, 11, 12 and 13, respectively.

Figure 5: Two-loop evolution of gauge couplings for the Model 5 with vector-
like particles. In the model, $k_{Y}=\frac{10}{7}\times\frac{5}{3}$ and
$k_{2}=\frac{2}{3}$. The string-scale gauge coupling relations can be achieved
by adding $3(XW+XG)$ (a) and $5(XQ+\overline{XQ})+3XG$ (b).

However, the parameter $k_{2}$ should not be too small either, as this would
realize the gauge coupling relation beyond Planck level, where we do not know
how to quantize gravity. This thorny issue will arise when we deal with models
9-12. The evolution of gauge couplings for Models 9-12 is shown in figures 14,
15, 16, 17, in Appendix B. Note that the U(1) and strong couplings are unified
below the Planck scale. We find that the electroweak coupling can be unified
with other two couplings below the Planck scale by introducing $XW$, which
only affects the running of electroweak coupling due to $\Delta b_{2}\neq 0$.
Take Model 11 as example, the U(1) and strong couplings are unified at
$1\times 10^{18}$ GeV, while the gauge couplings are unified at same energy
scale after introducing 3 $XW$ at $1.4\times 10^{12}$ GeV. To obtain string-
scale gauge coupling relation for these models, the additional particles are
$XQ+\overline{XQ}$ as well as $XW$. The numbers and the masses of these
particles are given in the gauge revolution figures.

###  4.3 The Vector-Like Particles from Four-dimensional Chiral Sectors

On the other hand, Model 13 in table 20 with $k_{y}=11/8$ and $k_{2}=5/14$
also can achieve string-scale gauge coupling relation while the vector-like
particles $XQ+\overline{XQ}$ are added. The number of these vector-like
particles are defined by the fundamental minus anti-fundamental
representation, with $6-3\equiv 3$. The evolution of gauge couplings for the
model are shown in figure 6. The energy scale is around $8\times 10^{17}$ GeV,
and the new vector-like particles decay to the corresponding SM fermions below
$M_{XQ}\simeq 2\times 10^{12}$ GeV. The number and mass of new vector-like
particles are also shown in the plot. Unlike Model 2-12 discussed above, the
vector-like particles entered in this model come from the four-dimensional
chiral sector of the brane building. Furthermore, Model 14 in table 21 with
$k_{y}=35/32$ and $k_{2}=35/66$, also have a energy scale at $1.8\times
10^{18}$ GeV moderately larger than the string scale. As discussed before, the
contributions of the particle $XW$ are also included in this model. The
particle $XW$ comes from the bb sector of the brane configuration. Comparison
of the masses of vector-like particles with different $XW$ numbers shows that
the mass splitting between $XQ$ and $XW$ increases with the number of $XW$.
When $XW$(s) is (are) added, the $XW$ acquires the same mass as the $XQ$ from
the naturalness point of view. Therefore, the number of $XW$ needs to be
carefully chosen in order to reduce their mass splitting. The evolution of
gauge couplings for the model are shown in figure 7. The string-scale gauge
coupling relations for Model 13-14 are listed in table. 5.

Figure 6: Two-loop evolution of gauge couplings for the Model 13 with vector-
like particles. In the model, $k_{Y}=\frac{11}{8}\times\frac{5}{3}$ and
$k_{2}=\frac{5}{14}$. The string-scale gauge coupling relations can be
achieved by adding $3(XW+XG)$ (a) and $3(XQ+\overline{XQ})$ (b).

Figure 7: Two-loop evolution of gauge couplings for the Model 14 with vector-like particles. In the model, $k_{Y}=\frac{35}{32}\times\frac{5}{3}$ and $k_{2}=\frac{35}{66}$. The string-scale gauge coupling relation can be achieved by adding $3(XW+XG)$ (a) and $3(XQ+\overline{XQ})+XW$ (b).  Model No. |  $k_{y}$ |  $k_{2}$ |  $n_{XQ}$ |  $M_{XQ}$ (GeV) |  $n_{XW}$ |  $M_{XW}$ |  $M_{\rm U}$(GeV)
---|---|---|---|---|---|---|---
13 |  11/8 |  5/14 |  3 |  $1.99\times 10^{12}$ |  |  |  $8.53\times 10^{17}$
14 |  35/32 |  35/66 |  3 |  $1.525\times 10^{14}$ |  1 |  $1.525\times 10^{14}$ |  $5.00\times 10^{17}$
$1.34\times 10^{14}$ |  2 |  $1.30\times 10^{16}$
$1.70\times 10^{14}$ |  3 |  $2.75\times 10^{16}$
Table 5: String-scale gauge coupling relations achieved by adding vector-like
particles $XQ+\overline{XQ}$ from four-dimensional chiral sectors as well as
$XW$ from bb sector.

###  4.4 The Vector-Like Particles from $\mathcal{N}=2$ Subsector with
$k_{y}<1$ and $k_{2}>1$

On the contrary, when $k_{y}<1$ or $k_{2}>1$, the intersection of
$\alpha_{1}^{-1}$ and $\alpha_{3}^{-1}$ lines lies above the line of
$\alpha_{2}^{-1}(t)$, which can be seen from Model 15 without vector-like
particle in figure 8(a). And thus, if we want to push or pull back the energy
scale to string scale $M_{\rm string}$, the extra particles arise from
$\mathcal{N}=2$ subsector, like $(XD+\overline{XD})$ and $(XU+\overline{XU})$,
which will modify the evolution of the $U(1)$ and strong couplings rather than
electroweak coupling. This is due to the non-zero $\Delta b_{1}$ and $\Delta
b_{3}$. As we mentioned earlier, since $\Delta b_{1}(XD)<\Delta b_{1}(XU)$,
the suppression of $\alpha_{1}^{-1}$ is stronger in the models with
$XD+\overline{XD}$ particles than that with $XU+\overline{XU}$ particles. And
thus, the energy scale is higher in the models with $(XD+\overline{XD})$ than
that with $(XU+\overline{XU})$. Of course, in some models with much smaller
$k_{y}$ or much larger $k_{2}$, we need to add both particles to obtain the
string-scale gauge coupling relation. Additional, there is mass splitting
between these two particles. The energy scale, the number and mass of the
added vector-like particles $(XD+\overline{XD})$ and $(XU+\overline{XU})$ are
list in table 6. Another vector-like particle $(XE+\overline{XE})$, from
$\mathcal{N}=2$ subsector, will only modify the running of the $U(1)$ coupling
due to $\Delta b_{1}(XE)\neq 0$. When used with $XG$, from aa sectors, the
results are similar or even better than those of $(XD+\overline{XD})$ and
$(XU+\overline{XU})$. The energy scale, the number and mass of the added
vector-like particles $(XE+\overline{XE})$ and $XG$ are list in table 7.

Base on the brane construction, the number of vector-like particles
$(XD+\overline{XD})$, $(XU+\overline{XU})$ and $(XE+\overline{XE})$ can be
determined by the intersection number of the $a$ and $c$ stacks $I_{ac}$. For
example, the intersection number is $I_{ac}=2^{0}(1)(-2-5)=-7$ in Model 15
(table 22) and the corresponding number of the vector-like particles
$(XD+\overline{XD})$ is 7. In this model, the wrapping numbers
$(n_{a}^{3},l_{a}^{3})$ and $(n_{c}^{3},l_{c}^{3})$ have the same value yet
with opposite sign, indicating that the D6-branes warpping on the third torus
are parallel to each other. Therefore, the intersection number of the $a$ and
$c$ stacks are only calculated on the first two torus.

Figure 8: Two-loop evolution of gauge couplings for the Model 15 with vector-like particles. In the model, $k_{Y}=\frac{25}{28}\times\frac{5}{3}$ and $k_{2}=\frac{7}{6}$. The string-scale gauge coupling relation can be achieved by adding $7(XD+\overline{XD})$ (a) as well as $7(XU+\overline{XE})$ (b). Model No. |  $k_{y}$ |  $k_{2}$ |  $n_{v}$ |  $M_{XD}$(GeV) |  $M_{XU}$(GeV) |  $M_{\rm U}$(GeV)
---|---|---|---|---|---|---
15 |  25/28 |  7/6 |  7 |  $5.60\times 10^{16}$ |  |  $7.22\times 10^{17}$
16 |  10/7 |  2 |  3 |  $6.9\times 10^{4}$ |  |  $1.2\times 10^{17}$
17 |  1/4 |  11/6 |  3 |  |  $3.70\times 10^{10}$ |  $1.43\times 10^{17}$
18 |  10/7 |  18/5 |  3 |  $8.90\times 10^{3}$ |  $1.60\times 10^{14}$ |  $3.25\times 10^{17}$
19 |  1 |  5/3 |  3 |  $2.00\times 10^{8}$ |  $1.40\times 10^{17}$ |  $4.90\times 10^{17}$
20 |  1 |  2 |  3 |  $3.71\times 10^{7}$ |  $1.05\times 10^{14}$ |  $5.00\times 10^{17}$
21 |  1 |  54/19 |  3 |  $2.13\times 10^{6}$ |  $2.43\times 10^{13}$ |  $5.00\times 10^{17}$
22 |  1 |  9/5 |  3 |  $1.80\times 10^{8}$ |  $6.20\times 10^{15}$ |  $5.50\times 10^{17}$
23 |  5/8 |  13/6 |  3 |  $8.20\times 10^{10}$ |  $8.20\times 10^{10}$ |  $3.60\times 10^{17}$
24 |  10/13 |  2 |  5 |  $9.00\times 10^{12}$ |  $8.00\times 10^{14}$ |  $5.58\times 10^{17}$
25 |  4/7 |  17/9 |  5 |  $1.08\times 10^{14}$ |  $1.08\times 10^{14}$ |  $6.61\times 10^{17}$
26 |  25/28 |  11/6 |  7 |  $8.71\times 10^{13}$ |  $2.75\times 10^{15}$ |  $3.43\times 10^{17}$
Table 6: String-scale gauge coupling relations achieved by adding vector-like
particles $XD+\overline{XD}$ or/and $XU+\overline{XU}$ from $\mathcal{N}=2$
subsector. The number of vector-like particles are defined by the intersection
number on a and c stacks, $n_{v}=I_{ac}$.

For Model 15, the parameters $k_{y}=25/28$ and $k_{2}=7/6$ slightly deviate
from 1, the string-scale gauge coupling relation can be achieved by
introducing vector-like particles, _e.g._ , a string-scale gauge coupling
relation obtained at $M_{\rm U}=7.22\times 10^{17}$ GeV by adding 7 pair of
$XD+\overline{XD}$. During the evolution of gauge couplings, the extra
particles are introduced around $5.6\times 10^{15}$ GeV. While, when the same
number of vector-like particles $XU+\overline{XU}$ are added at $7\times
10^{14}$ GeV, a GUT scale gauge coupling relation is obtained at $1.8\times
10^{16}$ GeV. Furthermore, adding both particles and fine-tuning their masses,
$M_{\rm U}=5.0\times 10^{17}$ GeV can be achieved and the accuracy is as small
as $0.39\%$. The evolution of gauge couplings for the model with vector-like
particles are show in figure 8. Models 16 is similar and the evolution of
their gauge coupling is shown in figure 18.

Figure 9: Two-loop evolution of gauge couplings for the Model 17 with vector-
like particles. In the model, $k_{Y}=\frac{1}{4}\times\frac{5}{3}$ and
$k_{2}=\frac{11}{6}$. The string-scale gauge coupling relations can be
achieved by adding $5(XU+\overline{XU})$ (a) as well as $5(XD+\overline{XD})$
(b).

For Model 17, the parameters $k_{y}=1/4$ and $k_{2}=11/6$ deviate
significantly from $1$, the evolution for U(1) coupling $\alpha_{1}^{-1}$
cannot intersect the other two couplings. In order to get a string-scale gauge
coupling relation, we choose to introduce 5 pairs of $XU+\overline{XU}$ at
$3.7\times 10^{10}$ GeV. The evolution of gauge couplings for this model
without and with vector-like particles are shown in figure 9. Note that the
energy scale is pushed above the Planck scale by adding $XD+\overline{XD}$.
This is common for models with parameters $k_{y}\ll 1$ and $k_{2}\gg 1$. In
such case, we introduce additional $XU+\overline{XU}$ particles to pull the
energy scale to intermediate scales $10^{14}-10^{15}$ GeV. Therefore, a
string-scale gauge coupling relation can be achieved by adding both
$XD+\overline{XD}$ and $XU+\overline{XU}$ and fine-tuning their masses. The
masses for these vector-like particles $M_{V}$ and the energy scales for
Models 18-26 are listed in table 6, and the corresponding evolution of gauge
couplings are illustrated in figures 19, 20 and 21, in Appendix B.

Figure 10: Two-loop evolution of gauge couplings for the Model 27 with vector-
like particles. In this model, $k_{Y}=\frac{4}{9}\times\frac{5}{3}$ and
$k_{2}=\frac{10}{9}$. The string-scale gauge coupling relations can be
achieved by adding $5(XE+\overline{XE})+XG$ around $10^{13}$ GeV.

However, if one and only one of parameters $k_{y}$ and $k_{2}$ deviates
significantly from $1$, the situations are more complicated. When $k_{2}\sim
1$ and $k_{y}\ll 1$, by adding $(XD+\overline{XD})$ and $(XU+\overline{XU})$
the string-scale gauge coupling realtion is pushed too high and above the
Planck scale. Another kind of vector-like particles $(XE+\overline{XE})$ from
$\mathcal{N}=2$ subsector as well as $XG$ from aa sector are added in these
models to obtain the string-scale gauge coupling relations. For Model 27 with
$k_{y}=4/9$ and $k_{2}=10/9$, the gauge coupling relation can be realized at
the string scale $M_{\rm U}=5.00\times 10^{17}$ GeV by adding
$5(XE+\overline{XE})+XG$ at $10^{13}$ GeV. The details for Model 27, and 28-29
are listed in table 7 and the corresponding running of gauge couplings are
plotted in figures 10, and 22.

Model No. |  $k_{y}$ |  $k_{2}$ |  $n_{XE}$ |  $M_{XE}$(GeV) |  $n_{XG}$ |  $M_{XG}$(GeV) |  $M_{\rm U}$(GeV)
---|---|---|---|---|---|---|---
27 |  4/9 |  10/9 |  5 |  $3.40\times 10^{13}$ |  1 |  $6.20\times 10^{13}$ |  $5.00\times 10^{17}$
28 |  5/11 |  1 |  5 |  $5.50\times 10^{12}$ |  1 |  $3.80\times 10^{15}$ |  $5.00\times 10^{17}$
29 |  1/4 |  7/6 |  5 |  $7.80\times 10^{9}$ |  1 |  $1.70\times 10^{12}$ |  $5.00\times 10^{17}$
30 |  10/7 |  27/11 |  3 |  $1.51\times 10^{12}$ |  3 |  $3.11\times 10^{12}$ |  $5.00\times 10^{17}$
31 |  5/3 |  13/5 |  3 |  $1.36\times 10^{13}$ |  3 |  $4.98\times 10^{12}$ |  $5.00\times 10^{17}$
32 |  7/4 |  21/10 |  3 |  $5.72\times 10^{16}$ |  3 |  $1.94\times 10^{13}$ |  $5.00\times 10^{17}$
33 |  2 |  26/5 |  3 |  $3.44\times 10^{8}$ |  2 |  $4.92\times 10^{7}$ |  $5.00\times 10^{17}$
Table 7: String-scale gauge coupling relations achieved by adding vector-like
particles $XE+\overline{XE}$ from $\mathcal{N}=2$ subsector and $XG$ from the
adjiont sector. The number of $XE+\overline{XE}$ is defined by the
intersection number on a and c stacks, $n_{XE}=I_{ac}$.

For Model 30-33, even the parameter $k_{y}$ is greater than 1, the
intersection of $\alpha_{1}^{-1}$ and $\alpha_{3}^{-1}$ is still above the
line of $\alpha_{2}^{-1}$ because $k_{2}$ is too large. Thus, to get a string-
scale gauge coupling relation, the vector-like particles added are
$(XE+\overline{XE})$ from $\mathcal{N}=2$ subsector as well as $XG$ from aa
sectors. For Model 30, a string-scale gauge coupling relation is achieved at
$5.00\times 10^{17}$ GeV by fine-tuning the masses of $XE$ and $XG$ as well as
the number of $XG$. The appropriate number of $XG$ are chosen to reduce the
mass splitting. For Model 30-33, the energy scales, number and mass of these
additional particles are listed in table 7 and the corresponding evolution are
figured in figure 23 in Appendix B.

##  5 Discussion and Conclusions

In He:2021gug , we have constructed all the three-family ${\cal N}=1$
supersymmetric Pati-Salam models in the Type IIA string theory on
$T^{6}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$ orientifold with intersecting
D6-branes, and obtained all the possible 33 independent models in total.
However, how to realize the string-scale gauge coupling relations for these
models is still a big challenge. In this paper, we systematically studied the
string-scale gauge coupling relations for all these models. First, we
discussed how to decouple the exotic particles in these models. Second,
utilizing the two-loop RGEs revolutions, we obtained string-scale gauge
coupling relations by introducing additional particles from the adjoint
representations of $SU(4)_{C}$ and $SU(2)_{L}$ gauge symmetries, SM vector-
like particles from four-dimensional chiral sectors, as well as vector-like
particles from $\mathcal{N}=2$ subsector. Although most of these
supersymmetric Pati-Salam models do not directly have traditional gauge
coupling unification at string scale, their gauge coupling relations can
indeed be realized at string scale. Therefore, we solved the string-scale
gauge coupling relation problems for the generic intersecting D6-brane models.
It seems to us that this systematic method can be applied to the other
intersecting D-brane model building as well.

###### Acknowledgements.

TL 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. RS is supported by KIAS Individual
Grant PG080701 and PG080704.

##  Appendix A Supersymmetric Pati-Salam Models

In this Appendix, we present the supersymmetric Pati-Salam models with $33$
types of allowed gauge coupling relations on the landscape of supersymmetric
Pati-Salam model building.

Model 1 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{4}$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 2 | 3 | 4
$a$ | 8 | $(1,1)\times(1,0)\times(1,-1)$ | 0 | 0 | 3 | 0 | 0 | -3 | 0 | 1 | 0 | -1
$b$ | 4 | $(-1,0)\times(-1,3)\times(1,1)$ | 2 | -2 | - | - | 0 | 0 | 0 | 0 | -3 | 1
$c$ | 4 | $(0,1)\times(-1,3)\times(-1,1)$ | 2 | -2 | - | - | - | - | -3 | 1 | 0 | 0
1 | 2 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=\frac{1}{3}x_{B}=x_{C}=\frac{1}{3}x_{D}$
2 | 2 | $(1,0)\times(0,-1)\times(0,2)$ | $\beta^{g}_{1}=\beta^{g}_{2}=\beta^{g}_{3}=\beta^{g}_{4}=-3$
3 | 2 | $(0,-1)\times(1,0)\times(0,2)$ |  $\chi_{1}=1$, $\chi_{2}=\frac{1}{3}$, $\chi_{3}=2$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |
Table 8: D6-brane configurations and intersection numbers of Model 1, and its gauge coupling relation is $g^{2}_{a}=g^{2}_{b}=g^{2}_{c}=(\frac{5}{3}g^{2}_{Y})=4\sqrt{\frac{2}{3}}\pi e^{\phi^{4}}$. Model 2 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{3}$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 2 | 3 | 4
$a$ | 8 | $(1,-1)\times(1,0)\times(1,1)$ | 0 | 0 | 3 | 0 | 0 | -3 | -1 | 0 | 1
$b$ | 4 | $(-1,4)\times(0,1)\times(-1,1)$ | 3 | -3 | - | - | -7 | 0 | 0 | 1 | 0
$c$ | 4 | $(-2,1)\times(-1,1)\times(1,1)$ | -2 | -6 | - | - | - | - | -1 | -2 | 2
2 | 2 | $(1,0)\times(0,-1)\times(0,2)$ | $x_{A}=\frac{1}{9}x_{B}=\frac{1}{4}x_{C}=\frac{1}{9}x_{D}$
3 | 2 | $(0,-1)\times(1,0)\times(0,2)$ |  $\beta^{g}_{2}=-3$, $\beta^{g}_{3}=-3$, $\beta^{g}_{4}=-2$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\chi_{1}=\frac{1}{2}$, $\chi_{2}=\frac{2}{9}$, $\chi_{3}=1$
Table 9: D6-brane configurations and intersection numbers of Model 2, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{4}{9}g^{2}_{b}=\frac{17}{9}g^{2}_{c}=\frac{85}{61}(\frac{5}{3}g^{2}_{Y})=\frac{32\pi e^{\phi^{4}}}{15}$. Model 3 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{2}$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 4
$a$ | 8 | $(-1,-1)\times(1,1)\times(1,1)$ | 0 | -4 | 0 | 3 | 0 | -3 | 1 | -1
$b$ | 4 | $(-5,2)\times(-1,0)\times(1,1)$ | -3 | 3 | - | - | 0 | 1 | 0 | 5
$c$ | 4 | $(-2,-1)\times(0,1)\times(1,1)$ | 1 | -1 | - | - | - | - | 1 | 0
1 | 2 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=\frac{14}{5}x_{B}=2x_{C}=7x_{D}$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\beta^{g}_{1}=-3$, $\beta^{g}_{4}=1$
|  |  |  $\chi_{1}=\sqrt{5}$, $\chi_{2}=\frac{7}{\sqrt{5}}$, $\chi_{3}=\frac{4}{\sqrt{5}}$
Table 10: D6-brane configurations and intersection numbers of Model 3, and its gauge coupling relation is $g^{2}_{a}=\frac{5}{6}g^{2}_{b}=\frac{7}{6}g^{2}_{c}=\frac{35}{32}(\frac{5}{3}g^{2}_{Y})=\frac{8}{27}5^{3/4}\sqrt{7}\pi e^{\phi^{4}}$. Model 4 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(4)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 4
$a$ | 8 | $(2,1)\times(1,0)\times(1,-1)$ | 1 | -1 | 3 | 0 | 0 | -3 | -2
$b$ | 4 | $(1,0)\times(1,-3)\times(1,1)$ | 2 | -2 | - | - | -4 | 0 | 1
$c$ | 4 | $(-1,2)\times(-1,1)\times(-1,1)$ | 2 | 6 | - | - | - | - | 1
4 | 4 | $(0,-1)\times(0,1)\times(2,0)$ | $x_{A}=\frac{4}{3}x_{B}=8x_{C}=\frac{8}{3}x_{D}$
|  |  | $\beta^{g}_{4}=0$
|  |  |  $\chi_{1}=4$, $\chi_{2}=\frac{2}{3}$, $\chi_{3}=4$
Table 11: D6-brane configurations and intersection numbers of Model 4, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{1}{2}g^{2}_{b}=\frac{13}{6}g^{2}_{c}=\frac{65}{44}(\frac{5}{3}g^{2}_{Y})=\frac{16}{5}\sqrt{\frac{2}{3}}\pi e^{\phi^{4}}$. Model 5 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(4)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 3
$a$ | 8 | $(-1,1)\times(-1,0)\times(1,1)$ | 0 | 0 | 3 | 0 | 0 | -3 | 0
$b$ | 4 | $(-1,4)\times(0,1)\times(-1,1)$ | 3 | -3 | - | - | -8 | 0 | 1
$c$ | 4 | $(1,0)\times(2,-3)\times(1,1)$ | 1 | -1 | - | - | - | - | -3
3 | 4 | $(0,-1)\times(1,0)\times(0,2)$ | $x_{A}=\frac{1}{6}x_{B}=\frac{1}{4}x_{C}=\frac{1}{6}x_{D}$
|  |  | $\beta^{g}_{3}=-2$
|  |  |  $\chi_{1}=\frac{1}{2}$, $\chi_{2}=\frac{1}{3}$, $\chi_{3}=1$
Table 12: D6-brane configurations and intersection numbers of Model 5, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{2}{3}g^{2}_{b}=2g^{2}_{c}=\frac{10}{7}(\frac{5}{3}g^{2}_{Y})=\frac{16}{5}\sqrt{\frac{2}{3}}\pi e^{\phi^{4}}$. Model 6 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)$
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stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 3
$a$ | 8 | $(1,-1)\times(1,0)\times(1,1)$ | 0 | 0 | 3 | 0 | 0 | -3 | 0
$b$ | 4 | $(-2,5)\times(0,1)\times(-1,1)$ | 3 | -3 | - | - | -8 | 0 | 2
$c$ | 4 | $(-2,1)\times(-1,1)\times(1,1)$ | -2 | -6 | - | - | - | - | -2
3 | 2 | $(0,-1)\times(1,0)\times(0,2)$ | $x_{A}=\frac{1}{6}x_{B}=\frac{2}{5}x_{C}=\frac{1}{6}x_{D}$
|  |  | $\beta^{g}_{3}=-2$
|  |  |  $\chi_{1}=\sqrt{\frac{2}{5}}$, $\chi_{2}=\frac{\sqrt{\frac{5}{2}}}{6}$, $\chi_{3}=2\sqrt{\frac{2}{5}}$
Table 13: D6-brane configurations and intersection numbers of Model 6, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{5}{6}g^{2}_{b}=\frac{11}{6}g^{2}_{c}=\frac{11}{8}(\frac{5}{3}g^{2}_{Y})=\frac{8\sqrt[4]{2}5^{3/4}\pi e^{\phi^{4}}}{7\sqrt{3}}$. Model 7 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(4)^{2}$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 2 | 4
$a$ | 8 | $(1,1)\times(1,0)\times(1,-1)$ | 0 | 0 | 3 | 0 | 0 | -3 | 1 | -1
$b$ | 4 | $(1,0)\times(1,-3)\times(1,1)$ | 2 | -2 | - | - | -4 | 0 | 0 | 1
$c$ | 4 | $(1,2)\times(-1,1)\times(-1,1)$ | -2 | -6 | - | - | - | - | 2 | -1
2 | 4 | $(1,0)\times(0,-1)\times(0,2)$ | $x_{A}=\frac{1}{3}x_{B}=x_{C}=\frac{1}{3}x_{D}$
4 | 4 | $(0,-1)\times(0,1)\times(2,0)$ |  $\beta^{g}_{2}=-2$, $\beta^{g}_{4}=-2$
|  |  |  $\chi_{1}=1$, $\chi_{2}=\frac{1}{3}$, $\chi_{3}=2$
Table 14: D6-brane configurations and intersection numbers of Model 7, and its MSSM gauge coupling relation is $g^{2}_{a}=g^{2}_{b}=\frac{5}{3}g^{2}_{c}=\frac{25}{19}(\frac{5}{3}g^{2}_{Y})=4\sqrt{\frac{2}{3}}\pi e^{\phi^{4}}$. Model 8 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{3}$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 2 | 4
$a$ | 8 | $(1,1)\times(1,0)\times(1,-1)$ | 0 | 0 | 3 | 0 | 0 | -3 | 0 | 1 | -1
$b$ | 4 | $(-1,0)\times(-1,3)\times(1,1)$ | 2 | -2 | - | - | -3 | 0 | 0 | 0 | 1
$c$ | 4 | $(0,1)\times(-2,3)\times(-1,1)$ | 1 | -1 | - | - | - | - | -3 | 2 | 0
1 | 2 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=\frac{2}{3}x_{B}=2x_{C}=\frac{2}{3}x_{D}$
2 | 2 | $(1,0)\times(0,-1)\times(0,2)$ |  $\beta^{g}_{1}=-3$, $\beta^{g}_{2}=-2$, $\beta^{g}_{4}=-3$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\chi_{1}=\sqrt{2}$, $\chi_{2}=\frac{\sqrt{2}}{3}$, $\chi_{3}=2\sqrt{2}$
Table 15: D6-brane configurations and intersection numbers of Model 8, and its MSSM gauge coupling relation is $g^{2}_{a}=g^{2}_{b}=2g^{2}_{c}=\frac{10}{7}(\frac{5}{3}g^{2}_{Y})=\frac{16\sqrt[4]{2}\pi e^{\phi^{4}}}{3\sqrt{3}}$. Model 9 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{2}$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 4
$a$ | 8 | $(1,-1)\times(-1,1)\times(1,-1)$ | 0 | 4 | 3 | 0 | 0 | -3 | -1 | 1
$b$ | 4 | $(1,-4)\times(1,0)\times(1,1)$ | 3 | -3 | - | - | -7 | 0 | 0 | 1
$c$ | 4 | $(-2,1)\times(2,1)\times(-1,1)$ | -3 | -13 | - | - | - | - | -1 | -4
1 | 2 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=28x_{B}=\frac{28}{23}x_{C}=7x_{D}$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\beta^{g}_{1}=-3$, $\beta^{g}_{4}=1$
|  |  |  $\chi_{1}=\sqrt{\frac{7}{23}}$, $\chi_{2}=\sqrt{161}$, $\chi_{3}=8\sqrt{\frac{7}{23}}$
Table 16: D6-brane configurations and intersection numbers of Model 9, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{1}{6}g^{2}_{b}=\frac{11}{6}g^{2}_{c}=\frac{11}{8}(\frac{5}{3}g^{2}_{Y})=\frac{8}{405}161^{3/4}\sqrt{2}\pi e^{\phi^{4}}$. Model 10 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{2}\times USp(4)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 3 | 4
$a$ | 8 | $(-1,-1)\times(1,1)\times(1,1)$ | 0 | -4 | 0 | 3 | 0 | -3 | 1 | -1 | -1
$b$ | 4 | $(-4,1)\times(-1,0)\times(1,1)$ | -3 | 3 | - | - | 0 | 2 | 0 | 0 | 4
$c$ | 4 | $(-2,-1)\times(0,1)\times(1,1)$ | 1 | -1 | - | - | - | - | 1 | -2 | 0
1 | 4 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=\frac{5}{2}x_{B}=2x_{C}=10x_{D}$
3 | 2 | $(0,-1)\times(1,0)\times(0,2)$ |  $\beta^{g}_{1}=-3$, $\beta^{g}_{3}=-2$, $\beta^{g}_{4}=0$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\chi_{1}=2\sqrt{2}$, $\chi_{2}=\frac{5}{\sqrt{2}}$, $\chi_{3}=\sqrt{2}$
Table 17: D6-brane configurations and intersection numbers of Model 10, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{4}{9}g^{2}_{b}=\frac{10}{9}g^{2}_{c}=\frac{50}{47}(\frac{5}{3}g^{2}_{Y})=\frac{16}{27}2^{3/4}\sqrt{5}\pi e^{\phi^{4}}$. Model 11 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)\times USp(4)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 3
$a$ | 8 | $(-1,1)\times(-1,0)\times(1,1)$ | 0 | 0 | 3 | 0 | 0 | -3 | 0 | 0
$b$ | 4 | $(-1,4)\times(0,1)\times(-1,1)$ | 3 | -3 | - | - | -4 | 0 | -4 | 1
$c$ | 4 | $(1,0)\times(1,-3)\times(1,1)$ | 2 | -2 | - | - | - | - | 0 | -3
1 | 2 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=\frac{1}{12}x_{B}=\frac{1}{4}x_{C}=\frac{1}{12}x_{D}$
3 | 4 | $(0,-1)\times(1,0)\times(0,2)$ |  $\beta^{g}_{1}=-2$, $\beta^{g}_{3}=-2$
|  |  |  $\chi_{1}=\frac{1}{2}$, $\chi_{2}=\frac{1}{6}$, $\chi_{3}=1$
Table 18: D6-brane configurations and intersection numbers of Model 11, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{1}{3}g^{2}_{b}=g^{2}_{c}=(\frac{5}{3}g^{2}_{Y})=\frac{16\pi e^{\phi^{4}}}{5\sqrt{3}}$. Model 12 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)\times USp(4)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 4
$a$ | 8 | $(-1,-1)\times(2,1)\times(1,1)$ | 0 | -8 | 0 | 3 | 0 | -3 | 1 | -2
$b$ | 4 | $(4,-1)\times(1,0)\times(1,1)$ | -3 | 3 | - | - | 0 | 2 | 0 | 4
$c$ | 4 | $(-2,-1)\times(-1,1)\times(1,1)$ | 2 | 6 | - | - | - | - | 1 | 2
1 | 4 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=\frac{13}{2}x_{B}=\frac{13}{8}x_{C}=26x_{D}$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\beta^{g}_{1}=-3$, $\beta^{g}_{4}=4$
|  |  |  $\chi_{1}=\sqrt{\frac{13}{2}}$, $\chi_{2}=2\sqrt{26}$, $\chi_{3}=\frac{\sqrt{\frac{13}{2}}}{2}$
Table 19: D6-brane configurations and intersection numbers of Model 12, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{1}{6}g^{2}_{b}=\frac{7}{6}g^{2}_{c}=\frac{35}{32}(\frac{5}{3}g^{2}_{Y})=\frac{16}{135}26^{3/4}\pi e^{\phi^{4}}$. Model 13 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{2}$
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stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 4
$a$ | 8 | $(-1,-1)\times(1,1)\times(1,1)$ | 0 | -4 | 6 | -3 | 0 | -3 | 1 | -1
$b$ | 4 | $(-1,2)\times(-1,0)\times(5,1)$ | 9 | -9 | - | - | -10 | -9 | 0 | 1
$c$ | 4 | $(-2,-1)\times(0,1)\times(1,1)$ | 1 | -1 | - | - | - | - | 1 | 0
1 | 2 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=22x_{B}=2x_{C}=\frac{11}{5}x_{D}$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\beta^{g}_{1}=-3$, $\beta^{g}_{4}=-3$
|  |  |  $\chi_{1}=\frac{1}{\sqrt{5}}$, $\chi_{2}=\frac{11}{\sqrt{5}}$, $\chi_{3}=4\sqrt{5}$
Table 20: D6-brane configurations and intersection numbers of Model 13, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{5}{14}g^{2}_{b}=\frac{11}{6}g^{2}_{c}=\frac{11}{8}(\frac{5}{3}g^{2}_{Y})=\frac{8}{63}5^{3/4}\sqrt{11}\pi e^{\phi^{4}}$. Model 14 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)$
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stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 3
$a$ | 8 | $(1,-1)\times(1,0)\times(1,1)$ | 0 | 0 | -3 | 6 | 0 | -3 | 0
$b$ | 4 | $(-2,1)\times(0,1)\times(-5,1)$ | -9 | 9 | - | - | 0 | 8 | 10
$c$ | 4 | $(-2,1)\times(-1,1)\times(1,1)$ | -2 | -6 | - | - | - | - | -2
3 | 2 | $(0,-1)\times(1,0)\times(0,2)$ | $x_{A}=\frac{5}{6}x_{B}=10x_{C}=\frac{5}{6}x_{D}$
|  |  | $\beta^{g}_{3}=6$
|  |  |  $\chi_{1}=\sqrt{10}$, $\chi_{2}=\frac{\sqrt{\frac{5}{2}}}{6}$, $\chi_{3}=2\sqrt{10}$
Table 21: D6-brane configurations and intersection numbers of Model 14, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{35}{66}g^{2}_{b}=\frac{7}{6}g^{2}_{c}=\frac{35}{32}(\frac{5}{3}g^{2}_{Y})=\frac{8\sqrt[4]{2}5^{3/4}\pi e^{\phi^{4}}}{11\sqrt{3}}$. Model 15 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{2}$
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stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 4
$a$ | 8 | $(1,-1)\times(-1,1)\times(1,-1)$ | 0 | 4 | 0 | 3 | 0 | -3 | -1 | 1
$b$ | 4 | $(0,1)\times(-2,1)\times(-1,1)$ | -1 | 1 | - | - | 0 | -1 | -1 | 0
$c$ | 4 | $(-1,0)\times(5,2)\times(-1,1)$ | 3 | -3 | - | - | - | - | 0 | -5
1 | 2 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=2x_{B}=\frac{14}{5}x_{C}=7x_{D}$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\beta^{g}_{1}=-3$, $\beta^{g}_{4}=1$
|  |  |  $\chi_{1}=\frac{7}{\sqrt{5}}$, $\chi_{2}=\sqrt{5}$, $\chi_{3}=\frac{4}{\sqrt{5}}$
Table 22: D6-brane configurations and intersection numbers of Model 15, and its gauge coupling relation is $g^{2}_{a}=\frac{7}{6}g^{2}_{b}=\frac{5}{6}g^{2}_{c}=\frac{25}{28}(\frac{5}{3}g^{2}_{Y})=\frac{8}{27}5^{3/4}\sqrt{7}\pi e^{\phi^{4}}$. Model 16 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{2}$
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stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 2 | 4
$a$ | 8 | $(1,1)\times(1,0)\times(1,-1)$ | 0 | 0 | 2 | 1 | 0 | -3 | 1 | -1
$b$ | 4 | $(-1,0)\times(-2,1)\times(1,3)$ | -5 | 5 | - | - | 8 | 8 | 0 | 6
$c$ | 4 | $(0,1)\times(-2,3)\times(-1,1)$ | 1 | -1 | - | - | - | - | 2 | 0
2 | 2 | $(1,0)\times(0,-1)\times(0,2)$ | $x_{A}=\frac{2}{3}x_{B}=\frac{1}{9}x_{C}=\frac{2}{3}x_{D}$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\beta^{g}_{2}=-2$, $\beta^{g}_{4}=2$
|  |  |  $\chi_{1}=\frac{1}{3}$, $\chi_{2}=2$, $\chi_{3}=\frac{2}{3}$
Table 23: D6-brane configurations and intersection numbers of Model 16, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{18}{5}g^{2}_{b}=2g^{2}_{c}=\frac{10}{7}(\frac{5}{3}g^{2}_{Y})=\frac{24\pi e^{\phi^{4}}}{5}$. Model 17 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{2}$
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stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 2 | 4
$a$ | 8 | $(1,1)\times(1,0)\times(1,-1)$ | 0 | 0 | 3 | 0 | 0 | -3 | 1 | -1
$b$ | 4 | $(-1,0)\times(-2,3)\times(1,1)$ | 1 | -1 | - | - | 0 | 0 | 0 | 2
$c$ | 4 | $(0,1)\times(-2,3)\times(-1,1)$ | 1 | -1 | - | - | - | - | 2 | 0
2 | 2 | $(1,0)\times(0,-1)\times(0,2)$ | $x_{A}=\frac{2}{3}x_{B}=x_{C}=\frac{2}{3}x_{D}$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\beta^{g}_{2}=-2$, $\beta^{g}_{4}=-2$
|  |  |  $\chi_{1}=1$, $\chi_{2}=\frac{2}{3}$, $\chi_{3}=2$
Table 24: D6-brane configurations and intersection numbers of Model 17, and its MSSM gauge coupling relation is $g^{2}_{a}=2g^{2}_{b}=2g^{2}_{c}=\frac{10}{7}(\frac{5}{3}g^{2}_{Y})=\frac{8\pi e^{\phi^{4}}}{\sqrt{3}}$. Model 18 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{2}$
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stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 4
$a$ | 8 | $(-1,-1)\times(1,1)\times(1,1)$ | 0 | -4 | 3 | 0 | 0 | -3 | 1 | -1
$b$ | 4 | $(-2,1)\times(2,1)\times(-1,1)$ | -3 | -13 | - | - | 7 | 0 | -1 | -4
$c$ | 4 | $(1,-4)\times(1,0)\times(1,1)$ | 3 | -3 | - | - | - | - | 0 | 1
1 | 2 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=28x_{B}=\frac{28}{23}x_{C}=7x_{D}$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\beta^{g}_{1}=-3$, $\beta^{g}_{4}=1$
|  |  |  $\chi_{1}=\sqrt{\frac{7}{23}}$, $\chi_{2}=\sqrt{161}$, $\chi_{3}=8\sqrt{\frac{7}{23}}$
Table 25: D6-brane configurations and intersection numbers of Model 18, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{11}{6}g^{2}_{b}=\frac{1}{6}g^{2}_{c}=\frac{1}{4}(\frac{5}{3}g^{2}_{Y})=\frac{8}{405}\sqrt{2}161^{3/4}\pi e^{\phi^{4}}$. Model 19 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(4)^{2}$
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stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 2 | 4
$a$ | 8 | $(1,1)\times(1,0)\times(1,-1)$ | 0 | 0 | 3 | 0 | 0 | -3 | 1 | -1
$b$ | 4 | $(-2,1)\times(-1,1)\times(1,1)$ | -2 | -6 | - | - | 4 | 0 | -1 | 2
$c$ | 4 | $(0,1)\times(-1,3)\times(-1,1)$ | 2 | -2 | - | - | - | - | 1 | 0
2 | 4 | $(1,0)\times(0,-1)\times(0,2)$ | $x_{A}=\frac{1}{3}x_{B}=x_{C}=\frac{1}{3}x_{D}$
4 | 4 | $(0,-1)\times(0,1)\times(2,0)$ |  $\beta^{g}_{2}=-2$, $\beta^{g}_{4}=-2$
|  |  |  $\chi_{1}=1$, $\chi_{2}=\frac{1}{3}$, $\chi_{3}=2$
Table 26: D6-brane configurations and intersection numbers of Model 19, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{5}{3}g^{2}_{b}=g^{2}_{c}=(\frac{5}{3}g^{2}_{Y})=4\sqrt{\frac{2}{3}}\pi e^{\phi^{4}}$. Model 20 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{3}$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 2 | 3 | 4
$a$ | 8 | $(1,1)\times(1,0)\times(1,-1)$ | 0 | 0 | 3 | 0 | 0 | -3 | 1 | 0 | -1
$b$ | 4 | $(-1,0)\times(-2,3)\times(1,1)$ | 1 | -1 | - | - | 3 | 0 | 0 | -3 | 2
$c$ | 4 | $(0,1)\times(-1,3)\times(-1,1)$ | 2 | -2 | - | - | - | - | 1 | 0 | 0
2 | 2 | $(1,0)\times(0,-1)\times(0,2)$ | $x_{A}=\frac{1}{3}x_{B}=\frac{1}{2}x_{C}=\frac{1}{3}x_{D}$
3 | 2 | $(0,-1)\times(1,0)\times(0,2)$ |  $\beta^{g}_{2}=-3$, $\beta^{g}_{3}=-3$, $\beta^{g}_{4}=-2$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\chi_{1}=\frac{1}{\sqrt{2}}$, $\chi_{2}=\frac{\sqrt{2}}{3}$, $\chi_{3}=\sqrt{2}$
Table 27: D6-brane configurations and intersection numbers of Model 20, and its MSSM gauge coupling relation is $g^{2}_{a}=2g^{2}_{b}=g^{2}_{c}=(\frac{5}{3}g^{2}_{Y})=\frac{16\sqrt[4]{2}\pi e^{\phi^{4}}}{3\sqrt{3}}$. Model 21 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{3}$
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stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 2 | 3 | 4
$a$ | 8 | $(1,1)\times(1,0)\times(1,-1)$ | 0 | 0 | 2 | 1 | 0 | -3 | 1 | 0 | -1
$b$ | 4 | $(-1,0)\times(-2,1)\times(1,3)$ | -5 | 5 | - | - | 10 | 7 | 0 | -1 | 6
$c$ | 4 | $(0,1)\times(-1,3)\times(-1,1)$ | 2 | -2 | - | - | - | - | 1 | 0 | 0
2 | 2 | $(1,0)\times(0,-1)\times(0,2)$ | $x_{A}=\frac{1}{3}x_{B}=\frac{1}{18}x_{C}=\frac{1}{3}x_{D}$
3 | 2 | $(0,-1)\times(1,0)\times(0,2)$ |  $\beta^{g}_{2}=-3$, $\beta^{g}_{3}=-5$, $\beta^{g}_{4}=2$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\chi_{1}=\frac{1}{3\sqrt{2}}$, $\chi_{2}=\sqrt{2}$, $\chi_{3}=\frac{\sqrt{2}}{3}$
Table 28: D6-brane configurations and intersection numbers of Model 21, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{54}{19}g^{2}_{b}=g^{2}_{c}=(\frac{5}{3}g^{2}_{Y})=\frac{48}{19}\sqrt[4]{2}\pi e^{\phi^{4}}$. Model 22 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{4}$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 2 | 3 | 4
$a$ | 8 | $(1,1)\times(1,0)\times(1,-1)$ | 0 | 0 | 2 | 1 | 0 | -3 | 0 | 1 | 0 | -1
$b$ | 4 | $(-1,0)\times(-1,1)\times(1,3)$ | -2 | 2 | - | - | 4 | 4 | 0 | 0 | -1 | 3
$c$ | 4 | $(0,1)\times(-1,3)\times(-1,1)$ | 2 | -2 | - | - | - | - | -3 | 1 | 0 | 0
1 | 2 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=\frac{1}{3}x_{B}=\frac{1}{9}x_{C}=\frac{1}{3}x_{D}$
2 | 2 | $(1,0)\times(0,-1)\times(0,2)$ |  $\beta^{g}_{1}=-3$, $\beta^{g}_{2}=-3$, $\beta^{g}_{3}=-5$, $\beta^{g}_{4}=-1$
3 | 2 | $(0,-1)\times(1,0)\times(0,2)$ |  $\chi_{1}=\frac{1}{3}$, $\chi_{2}=1$, $\chi_{3}=\frac{2}{3}$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |
Table 29: D6-brane configurations and intersection numbers of Model 22, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{9}{5}g^{2}_{b}=g^{2}_{c}=(\frac{5}{3}g^{2}_{Y})=\frac{12}{5}\sqrt{2}\pi e^{\phi^{4}}$. Model 23 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(4)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 4
$a$ | 8 | $(-2,1)\times(-1,0)\times(1,1)$ | -1 | 1 | 3 | 0 | 0 | -3 | 2
$b$ | 4 | $(-1,2)\times(-1,1)\times(-1,1)$ | 2 | 6 | - | - | 4 | 0 | 1
$c$ | 4 | $(1,0)\times(1,-3)\times(1,1)$ | 2 | -2 | - | - | - | - | 1
4 | 4 | $(0,-1)\times(0,1)\times(2,0)$ | $x_{A}=\frac{4}{3}x_{B}=8x_{C}=\frac{8}{3}x_{D}$
|  |  | $\beta^{g}_{4}=0$
|  |  |  $\chi_{1}=4$, $\chi_{2}=\frac{2}{3}$, $\chi_{3}=4$
Table 30: D6-brane configurations and intersection numbers of Model 23, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{13}{6}g^{2}_{b}=\frac{1}{2}g^{2}_{c}=\frac{5}{8}(\frac{5}{3}g^{2}_{Y})=\frac{16}{5}\sqrt{\frac{2}{3}}\pi e^{\phi^{4}}$. Model 24 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(4)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 3
$a$ | 8 | $(1,1)\times(-1,0)\times(-1,1)$ | 0 | 0 | 3 | 0 | 0 | -3 | 0
$b$ | 4 | $(1,0)\times(2,-3)\times(1,1)$ | 1 | -1 | - | - | 8 | 0 | -3
$c$ | 4 | $(-1,4)\times(0,1)\times(-1,1)$ | 3 | -3 | - | - | - | - | 1
3 | 4 | $(0,-1)\times(1,0)\times(0,2)$ | $x_{A}=\frac{1}{6}x_{B}=\frac{1}{4}x_{C}=\frac{1}{6}x_{D}$
|  |  | $\beta^{g}_{3}=-2$
|  |  |  $\chi_{1}=\frac{1}{2}$, $\chi_{2}=\frac{1}{3}$, $\chi_{3}=1$
Table 31: D6-brane configurations and intersection numbers of Model 24, and its MSSM gauge coupling relation is $g^{2}_{a}=2g^{2}_{b}=\frac{2}{3}g^{2}_{c}=\frac{10}{13}(\frac{5}{3}g^{2}_{Y})=\frac{16}{5}\sqrt{\frac{2}{3}}\pi e^{\phi^{4}}$. Model 25 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{3}$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 2 | 3 | 4
$a$ | 8 | $(1,1)\times(1,0)\times(1,-1)$ | 0 | 0 | 3 | 0 | 0 | -3 | 1 | 0 | -1
$b$ | 4 | $(2,-1)\times(1,-1)\times(1,1)$ | -2 | -6 | - | - | 7 | 0 | -1 | -2 | 2
$c$ | 4 | $(-1,4)\times(0,1)\times(-1,1)$ | 3 | -3 | - | - | - | - | 0 | 1 | 0
2 | 2 | $(1,0)\times(0,-1)\times(0,2)$ | $x_{A}=\frac{1}{9}x_{B}=\frac{1}{4}x_{C}=\frac{1}{9}x_{D}$
3 | 2 | $(0,-1)\times(1,0)\times(0,2)$ |  $\beta^{g}_{2}=-3$, $\beta^{g}_{3}=-3$, $\beta^{g}_{4}=-2$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\chi_{1}=\frac{1}{2}$, $\chi_{2}=\frac{2}{9}$, $\chi_{3}=1$
Table 32: D6-brane configurations and intersection numbers of Model 25, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{17}{9}g^{2}_{b}=\frac{4}{9}g^{2}_{c}=\frac{4}{7}(\frac{5}{3}g^{2}_{Y})=\frac{32\pi e^{\phi^{4}}}{15}$. Model 26 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 3
$a$ | 8 | $(1,1)\times(1,0)\times(1,-1)$ | 0 | 0 | 3 | 0 | 0 | -3 | 0
$b$ | 4 | $(2,-1)\times(1,-1)\times(1,1)$ | -2 | -6 | - | - | 8 | 0 | -2
$c$ | 4 | $(-2,5)\times(0,1)\times(-1,1)$ | 3 | -3 | - | - | - | - | 2
3 | 2 | $(0,-1)\times(1,0)\times(0,2)$ | $x_{A}=\frac{1}{6}x_{B}=\frac{2}{5}x_{C}=\frac{1}{6}x_{D}$
|  |  | $\beta^{g}_{3}=-2$
|  |  |  $\chi_{1}=\sqrt{\frac{2}{5}}$, $\chi_{2}=\frac{\sqrt{\frac{5}{2}}}{6}$, $\chi_{3}=2\sqrt{\frac{2}{5}}$
Table 33: D6-brane configurations and intersection numbers of Model 26, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{11}{6}g^{2}_{b}=\frac{5}{6}g^{2}_{c}=\frac{25}{28}(\frac{5}{3}g^{2}_{Y})=\frac{8\sqrt[4]{2}5^{3/4}\pi e^{\phi^{4}}}{7\sqrt{3}}$. Model 27 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{2}\times USp(4)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 2 | 4
$a$ | 8 | $(1,-1)\times(-1,1)\times(1,-1)$ | 0 | 4 | 0 | 3 | 0 | -3 | -1 | 1 | 1
$b$ | 4 | $(0,1)\times(-2,1)\times(-1,1)$ | -1 | 1 | - | - | 0 | -2 | -1 | 2 | 0
$c$ | 4 | $(-1,0)\times(4,1)\times(-1,1)$ | 3 | -3 | - | - | - | - | 0 | 0 | -4
1 | 4 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=2x_{B}=\frac{5}{2}x_{C}=10x_{D}$
2 | 2 | $(1,0)\times(0,-1)\times(0,2)$ |  $\beta^{g}_{1}=-3$, $\beta^{g}_{2}=-2$, $\beta^{g}_{4}=0$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\chi_{1}=\frac{5}{\sqrt{2}}$, $\chi_{2}=2\sqrt{2}$, $\chi_{3}=\sqrt{2}$
Table 34: D6-brane configurations and intersection numbers of Model 27, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{10}{9}g^{2}_{b}=\frac{4}{9}g^{2}_{c}=\frac{4}{7}(\frac{5}{3}g^{2}_{Y})=\frac{16}{27}2^{3/4}\sqrt{5}\pi e^{\phi^{4}}$. Model 28 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)\times USp(4)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 3
$a$ | 8 | $(1,1)\times(-1,0)\times(-1,1)$ | 0 | 0 | 3 | 0 | 0 | -3 | 0 | 0
$b$ | 4 | $(1,0)\times(1,-3)\times(1,1)$ | 2 | -2 | - | - | 4 | 0 | 0 | -3
$c$ | 4 | $(-1,4)\times(0,1)\times(-1,1)$ | 3 | -3 | - | - | - | - | -4 | 1
1 | 2 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=\frac{1}{12}x_{B}=\frac{1}{4}x_{C}=\frac{1}{12}x_{D}$
3 | 4 | $(0,-1)\times(1,0)\times(0,2)$ |  $\beta^{g}_{1}=-2$, $\beta^{g}_{3}=-2$
|  |  |  $\chi_{1}=\frac{1}{2}$, $\chi_{2}=\frac{1}{6}$, $\chi_{3}=1$
Table 35: D6-brane configurations and intersection numbers of Model 28, and its MSSM gauge coupling relation is $g^{2}_{a}=g^{2}_{b}=\frac{1}{3}g^{2}_{c}=\frac{5}{11}(\frac{5}{3}g^{2}_{Y})=\frac{16\pi e^{\phi^{4}}}{5\sqrt{3}}$. Model 29 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)\times USp(4)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 4
$a$ | 8 | $(1,-1)\times(-2,1)\times(1,-1)$ | 0 | 8 | 0 | 3 | 0 | -3 | -1 | 2
$b$ | 4 | $(-2,1)\times(1,1)\times(-1,1)$ | -2 | -6 | - | - | 0 | -2 | -1 | -2
$c$ | 4 | $(-4,-1)\times(1,0)\times(-1,1)$ | 3 | -3 | - | - | - | - | 0 | -4
1 | 4 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=\frac{13}{2}x_{B}=\frac{13}{8}x_{C}=26x_{D}$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\beta^{g}_{1}=-3$, $\beta^{g}_{4}=4$
|  |  |  $\chi_{1}=\sqrt{\frac{13}{2}}$, $\chi_{2}=2\sqrt{26}$, $\chi_{3}=\frac{\sqrt{\frac{13}{2}}}{2}$
Table 36: D6-brane configurations and intersection numbers of Model 29, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{7}{6}g^{2}_{b}=\frac{1}{6}g^{2}_{c}=\frac{1}{4}(\frac{5}{3}g^{2}_{Y})=\frac{16}{135}26^{3/4}\pi e^{\phi^{4}}$. Model 30 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{3}$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 2 | 4
$a$ | 8 | $(1,1)\times(1,0)\times(1,-1)$ | 0 | 0 | 2 | 1 | 0 | -3 | 0 | 1 | -1
$b$ | 4 | $(-1,0)\times(-1,1)\times(1,3)$ | -2 | 2 | - | - | 2 | 5 | 0 | 0 | 3
$c$ | 4 | $(0,1)\times(-2,3)\times(-1,1)$ | 1 | -1 | - | - | - | - | -3 | 2 | 0
1 | 2 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=\frac{2}{3}x_{B}=\frac{2}{9}x_{C}=\frac{2}{3}x_{D}$
2 | 2 | $(1,0)\times(0,-1)\times(0,2)$ |  $\beta^{g}_{1}=-3$, $\beta^{g}_{2}=-2$, $\beta^{g}_{4}=-1$
4 | 2 | $(0,-1)\times(0,1)\times(2,0)$ |  $\chi_{1}=\frac{\sqrt{2}}{3}$, $\chi_{2}=\sqrt{2}$, $\chi_{3}=\frac{2\sqrt{2}}{3}$
Table 37: D6-brane configurations and intersection numbers of Model 30, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{27}{11}g^{2}_{b}=2g^{2}_{c}=\frac{10}{7}(\frac{5}{3}g^{2}_{Y})=\frac{48}{11}\sqrt[4]{2}\pi e^{\phi^{4}}$. Model 31 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)\times USp(4)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1 | 3
$a$ | 8 | $(1,1)\times(-1,0)\times(-1,1)$ | 0 | 0 | 2 | 1 | 0 | -3 | 0 | 0
$b$ | 4 | $(1,0)\times(1,-1)\times(1,3)$ | -2 | 2 | - | - | 8 | 4 | 0 | -1
$c$ | 4 | $(-1,4)\times(0,1)\times(-1,1)$ | 3 | -3 | - | - | - | - | -4 | 1
1 | 2 | $(1,0)\times(1,0)\times(2,0)$ | $x_{A}=\frac{3}{4}x_{B}=\frac{1}{4}x_{C}=\frac{3}{4}x_{D}$
3 | 4 | $(0,-1)\times(1,0)\times(0,2)$ |  $\beta^{g}_{1}=-2$, $\beta^{g}_{3}=-4$
|  |  |  $\chi_{1}=\frac{1}{2}$, $\chi_{2}=\frac{3}{2}$, $\chi_{3}=1$
Table 38: D6-brane configurations and intersection numbers of Model 31, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{13}{5}g^{2}_{b}=3g^{2}_{c}=\frac{5}{3}(\frac{5}{3}g^{2}_{Y})=\frac{16}{5}\sqrt{3}\pi e^{\phi^{4}}$. Model 32 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(4)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 4
$a$ | 8 | $(2,1)\times(1,0)\times(1,-1)$ | 1 | -1 | 2 | 1 | 0 | -3 | -2
$b$ | 4 | $(1,0)\times(1,-1)\times(1,3)$ | -2 | 2 | - | - | 0 | 4 | 3
$c$ | 4 | $(-1,2)\times(-1,1)\times(-1,1)$ | 2 | 6 | - | - | - | - | 1
4 | 4 | $(0,-1)\times(0,1)\times(2,0)$ | $x_{A}=2x_{B}=\frac{4}{3}x_{C}=4x_{D}$
|  |  | $\beta^{g}_{4}=2$
|  |  |  $\chi_{1}=2\sqrt{\frac{2}{3}}$, $\chi_{2}=\sqrt{6}$, $\chi_{3}=2\sqrt{\frac{2}{3}}$
Table 39: D6-brane configurations and intersection numbers of Model 32, and its MSSM gauge coupling relation is $g^{2}_{a}=\frac{21}{10}g^{2}_{b}=\frac{7}{2}g^{2}_{c}=\frac{7}{4}(\frac{5}{3}g^{2}_{Y})=\frac{8}{5}6^{3/4}\pi e^{\phi^{4}}$. Model 33 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(4)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 3
$a$ | 8 | $(1,1)\times(-1,0)\times(-1,1)$ | 0 | 0 | 2 | 1 | 0 | -3 | 0
$b$ | 4 | $(1,0)\times(2,-1)\times(1,3)$ | -5 | 5 | - | - | 16 | 8 | -1
$c$ | 4 | $(-1,4)\times(0,1)\times(-1,1)$ | 3 | -3 | - | - | - | - | 1
3 | 4 | $(0,-1)\times(1,0)\times(0,2)$ | $x_{A}=\frac{3}{2}x_{B}=\frac{1}{4}x_{C}=\frac{3}{2}x_{D}$
|  |  | $\beta^{g}_{3}=-4$
|  |  |  $\chi_{1}=\frac{1}{2}$, $\chi_{2}=3$, $\chi_{3}=1$
Table 40: D6-brane configurations and intersection numbers of Model 33, and
its MSSM gauge coupling relation is
$g^{2}_{a}=\frac{26}{5}g^{2}_{b}=6g^{2}_{c}=2(\frac{5}{3}g^{2}_{Y})=\frac{16}{5}\sqrt{6}\pi
e^{\phi^{4}}$.

##  Appendix B The Evolution for The Gauge Couplings

Figure 11: Two-loop evolution of gauge couplings for the Model 6 with vector-
like particles. In the model, $k_{Y}=1\times\frac{11}{8}$ and
$k_{2}=\frac{5}{6}$. The string-scale gauge coupling relations can be achieved
by adding $3(XG+XW)$ (a) and $7(XQ+\overline{XQ})+3XG$ (b).

Figure 12: Two-loop evolution of gauge couplings for the Model 7 with vector-
like particles. In the model, $k_{Y}=\frac{25}{19}\times\frac{5}{3}$ and
$k_{2}=1$. The string-scale gauge coupling relations can be achieved by adding
$3(XG+XW)$ (a) and $3(XQ+\overline{XQ})+3XG$(b).

Figure 13: Two-loop evolution of gauge couplings for the Model 8 with vector-
like particles. In the model, $k_{Y}=\frac{10}{7}\times\frac{5}{3}$ and
$k_{2}=1$. The string-scale gauge coupling relations can be achieved by adding
$3(XG+XW)$ (a) and $3(XQ+\overline{XQ})+3XG$(b).

Figure 14: Two-loop evolution of gauge couplings for the Model 9 with vector-
like particles. In the model, $k_{Y}=\frac{11}{8}\times\frac{5}{3}$ and
$k_{2}=\frac{1}{6}$. The string-scale gauge coupling relation can be achieved
by adding $3(XG+XW)$ (a) and $5(XQ+\overline{XQ})+3XW$(b).

Figure 15: Two-loop evolution of gauge couplings for the Model 10 with vector-
like particles. In the model, $k_{Y}=\frac{50}{47}\times\frac{5}{3}$ and
$k_{2}=\frac{4}{9}$. The string-scale gauge coupling relation can be achieved
by adding $3(XG+XW)$ (a) and $5(XQ+\overline{XQ})+3XW$ (b).

Figure 16: Two-loop evolution of gauge couplings for the Model 11 without and
with vector-like particles. In the model, $k_{Y}=1\times\frac{5}{3}$ and
$k_{2}=\frac{1}{3}$. The string-scale gauge coupling relation can be achieved
by adding $3(XG+XW)$ (a) and $5(XQ+\overline{XQ})+3XW$ (b).

Figure 17: Two-loop evolution of gauge couplings for the Model 12 with vector-
like particles. In the model, $k_{Y}=\frac{35}{32}\times\frac{5}{3}$ and
$k_{2}=\frac{1}{6}$. The string-scale gauge coupling relation can be achieved
by adding $3(XG+XW)$ (a) and $5(XQ+\overline{XQ})+3XW$ (b).

Figure 18: Two-loop evolution of gauge couplings for the Model 16 with vector-
like particles. In this model, $k_{Y}=\frac{10}{7}\times\frac{5}{3}$ and
$k_{2}=2$. The string-scale gauge coupling relation can be achieved by adding
$3(XD+\overline{XD})$ (a) as well as $3(XU+\overline{XU})$ (b).

Figure 19: Two-loop evolution of gauge couplings for the Model 18 (a), Model
19 (b), Model 20 (c), Model 21 (d), Model 22 (e) and Model 23 (f) with vector-
like particles. The string-scale gauge coupling relation can be achieved by
adding $3(XD+\overline{XD})+3(XU+\overline{XU})$.

Figure 20: Two-loop evolution of gauge couplings for the Model 24 (a) and
Model 25 (b) with vector-like particles. In this model,
$k_{Y}=\frac{10}{13}\times\frac{5}{3}$ and $k_{2}=2$. The string-scale gauge
coupling relation can be achieved by adding
$5(XD+\overline{XD})+5(XU+\overline{XU})$.  Figure 21: Two-loop evolution of
gauge couplings for the Model 26 with vector-like particles. In this model,
$k_{Y}=\frac{25}{28}\times\frac{5}{3}$ and $k_{2}=\frac{11}{6}$. The string-
scale gauge coupling relation can be achieved by adding
$7(XD+\overline{XD})+7(XU+\overline{XU})$.

Figure 22: Two-loop evolution of gauge couplings for the Model 28 (a) and
Model29 (b) with vector-like particles. The string-scale gauge coupling
relation can be achieved by adding $5(XE+\overline{XE})+XG$.

Figure 23: Two-loop evolution of gauge couplings for the Model 30 (a), Model
31 (b), Model 32 (c) and Model 33 (d) with vector-like particles. The string-
scale gauge coupling relation can be achieved by adding
$3(XE+\overline{XE})+3XG$.

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