Tianjun Li,Rui Sun and Chi Zhang

1 CAS Key Laboratory of Theoretical Physics,Institute of Theoretical Physics,Chinese Academy of Sciences,Beijing 100190,China

2 School of Physical Sciences,University of Chinese Academy of Sciences,No.19A Yuquan Road,Beijing 100049,China

3 Korea Institute for Advanced Study,85 Hoegiro,Dongdaemun-Gu,Seoul 02455,Republic of Korea

Abstract We investigate the construction of four-family N=1supersymmetric Pati–Salam models from Type IIAorientifold with intersecting D6-branes.Utilizing the deterministic algorithm introduced in He W,Li T and Sun R(2021 arXiv:hep-th/2112.09632),we obtain 274 types of models with three rectangular tori and distinct gauge coupling relations at string scale,while 6 types of models with two rectangular tori and one tilted torus.In both cases,there exists a class of models with gauge coupling unification at string scale.In particular,for the models with two rectangular tori,one tilted torus and gauge coupling unification,the gaugino condensations are allowed,and thus supersymmetry breaking and moduli stabilization are possible for further phenomenological study.

Keywords:supersymmetric Pati–Salam models,intersecting D6-branes,brane model building

1.Introduction

One of the main motivations of string phenomenology is to find a unifying N=1supersymmetric quantum field theory,a competent framework among various extensions of the fourdimensional Standard Model (SM).D-branes play an important role in constructing interesting models at the phenomenological level,especially in Type I,Type IIA and Type IIB string theories.Chiral fermions appear at

· worldvolume singularities of D-branes [2–8];and

· intersecting loci of D-branes in the internal space [9].

Intersecting D6-branes on Type IIA orientifolds have been used to construct three-family non-supersymmetric models and grand unified models [10–30].Even though these models satisfy the Ramond–Ramond(RR) tadpole cancellation conditions,there are Neveu–Schwarz–Neveu–Schwarz(NS-NS) tadpoles remaining due to their nonsupersymmetric nature.Moreover,the string scale is close to the Planck scale since the intersecting D6-branes are not transversal in the internal space.As a result,there are large Planck scale corrections at the loop level,leading to the gauge hierarchy problem.As a remedy,a large number of supersymmetric standard-like models and grand unified models[31–49]have been constructed,with the gauge hierarchy problem solved.We refer to[50]for a comprehensive review of such non-supersymmetric and supersymmetric models.

Among these supersymmetric models,Pati–Salam models have been a prominent road to the Standard Model,without adding any extra U(1) symmetry around the electroweak level.In[40],Cveti? Liu and one of us(TL)showed how to systematically construct N=1supersymmetric Pati–Salam models from intersecting D6-branes on Type IIA-orientifold.After D-brane splitting and supersymmetric preserving Higgs mechanism applied,the Pati–Salam gauge symmetry SU(4)C×SU(2)L×SU(2)Reventually breaks down to the SM gauge symmetry.Due to the supersymmetry breaking and moduli stabilization triggered by their two confining groups in the hidden sectors,these models do have realistic and phenomenological consequences,as shown in [43,51,52].Until very recently,intriguingly all possible 202752 three-familyN=1supersymmetric Pati–Salam models onhave been found and classified with 33 types of independent models according to gauge coupling relations [1].

Most of these above achievements are based on threefamily model buildings.In fact,the study of the SM with four-families is also worthy of attention due to the flavor democracy hypothesis.In [53],a democratic mass matrix model was introduced to mainly fix the mass gap problem between three families of fermions of the SM,as well as the hierarchy problem of the Yukawa couplings.Allowing three families of fermions,one gets typical SM predictions,such as a low mass of the top quark and the inequality between three neutrino masses [54].If one allows four families of fermions in the democratic mass matrix model,three families of precisely massless neutrinos and a massive neutrino can be realized without the assumption of a larger hierarchy of the Yukawa couplings.Moreover,via a slight breaking of democracy,the three massless neutrinos obtain small masses.Then the flavor problem of the SM can be solved naturally by putting the flavor democracy hypothesis due to the important role the democratic mass matrix model plays.

As a matter of fact,the SM does not make a theoretical prediction on the number of families.The only restriction for the number of SM families comes from the requirement made by the quantum chromodynamics (QCD).The asymptotic freedom of QCD provides an upper bound of 8 for the number of families as discussed in [55].And if we allow four families of fermions for SM,many open issues of SM can be solved in a natural way.For example,introducing a fourth massive generation to SM can alter the crosssection and decay channels of the Higgs particles [56].When the Yukawa couplings of the fourth generation particles are large enough,these particles are natural candidates for electroweak symmetry breaking [57].Motivated by constructing realistic four-family SM,in this paper,we concentrate on building four-family N=1supersymmetric models with gauge symmetry SU(4)C×SU(2)L×SU(2)Ronorientifold with intersecting D6-branes as an extension to [1,40].We obtain four-family Pati–Salam models with gauge coupling unification at string scale or near string scale.In particular,there are models without any filler brane required as well.

The paper is organized as follows.In section 2 we will briefly review the basics of model building from intersecting D6-branes onorientifold.Constraints on D6-brane configuration,such as RR tadpole cancellation condition and supersymmetric condition are also reviewed.In section 3,we present the symmetry-breaking mechanism for U(4)C×U(2)L×U(2)R,i.e.D6-brane splitting and the Higgs mechanism.Four-family of chiral fermion conditions and various symmetry relations,such as T-dualities are also discussed.Section 4 is devoted to the phenomenological features of the four-family models.For each class of model,we list its chiral spectrum for the open string sector.In section 5,we draw conclusions and briefly discuss the limitations of our work.The four-family Pati–Salam models are presented in the appendix.

2.Basics of T6/（Z2×Z2）-orientifolds model construction

To construct realistic and four-family supersymmetric models,we recall the basics of model construction from Type IIA string theory compactified on-orientifold,with D6-branes intersecting at general tilted angles,under similar settings as in [31]and [33].

We begin with the orientifoldwhere D6-branes can be naturally viewed as general 3-cycles.Using the canonical isomorphismwe can easily write down the coordinate chart of the orientifold.Let zi,i=1,2,3 be the complex coordinates of the ith torus,respectively.Let θ and ω be the two generators of the abelian group Z2×Z2.In coordinate (z1,z2,z3),we define orientifold actions θ and ω of Z2×Z2on T6by

In addition,we define actions Ω and R of Z2×Z2on T6,by Ω the parity-reversion on the world-sheet,and R the complex conjugate of T6as a complex manifold.By the very definition of an orientifold,where the world-sheet parity and complex conjugate make no difference,there are exactly four components of the-orientifold,namely the image of T6under the action of ΩR,ΩRθ,ΩRω and ΩRθω.These components are objects bearing RR charges,and thus D6-branes are introduced to cancel their RR charges.

Generally,Dp-branes are (p+1)-dimensional objects in the spacetime,where strings start from and land on.In our case,viewed from the internal spacethe D6-branes are three-dimensional objects.It is sufficient to identify D6-branes and 3-cycles for physical considerations.The powerful Eilenberg–Zilber theorem tells us that

In the following discussion,we denote [ai],[bi],i=1,2,3 as generators ofH3(T2;Z) for the ith torus respectively.Under the orientifold action Z2×Z2and taking world-sheet into consideration,we find that there are only two patterns f or the latticeH1(T2;Z)?Z2:the rectangular one and the tilted one [11,33,35,45].In the basis [ai],[bi],i=1,2,3,the former is generated by[ai],[bi]overZ,while the latter is generated byoverZ withUnder the basis [a1],[b1],[a2],[b2],[a3],[b3],we can represent a 3-cycle [Πa]of the orientifoldby the coordinatewherenai,laiare all integers,βitakes 0 when the ith torus is rectangular and takes value 1 for the tilted case.Under the action of ΩR,a D6-brane [Πa]=(na1,la1)× (na2,la2)×(na3,la3) becomes [Πa′]=(na1,-la1)× (na2,-la2)× (na3,-la3),with [Πa′]a short-handed notation for the image of[Πa].To sum up,a D6-brane[Πa]and its orientifold image [Πa′]take the form

Table 1.Spectrum of intersecting D6-branes.

Denote [ΠΩR]=23(1,0)×(1,0)×(1,0),then we have

where [ΠΩRω],[ΠΩRθ],[ΠΩRθω]are the images of [ΠΩR]under the action of ω,θ and θω respectively.The coefficient 23comes from identifying 8 O6-branes (±1,0)×(±1,0)×(±1,0),under the quotient of Ω and R actions.The intersection number of D6-branes can be easily computed as

where k=β1+β2+β3.If we denote [ΠO6]=[ΠΩR]+[ΠΩRω]+[ΠΩRθ]+[ΠΩRθω],we have

The spectrum of intersecting D6-branes is given in table 1.

For three-family N=1supersymmetric Pati–Salam model building in Type IIA orientifolds onwith intersecting D6-branes in which the SU(4)C×SU(2)L×SU(2)Rgauge symmetries arise from U(n)branes.To get four families of fermions,we require

The conditionsIab+Iab′=4and Iac=-4 ensure that there are four families of SM fermions,and the conditionIac′=0 means that stack a of D6-branes is parallel to the orientifold image of stack c of D6-branes.Thus there should be open strings stretching between the two stack of D6-branes.Thelight scalar from the NS scalar will obtain masswithZa2c′the minimal squared length of the stretching string.Similarly,the light fermions from the R sector acquire the same masses [13,14,39].These light scalars and fermions form the Higgs fields needed to break the Pati–Salam gauge symmetry to the SM gauge symmetry.

Table 2.Configuration of four O6-planes.

In addition,there are two main constraints for D6-brane configurations and O6-plane configurations,namely the RR Tadpole Cancellation Condition and Supersymmetry Condition.As the sources of RR charges,D6-branes and O6-planes should satisfy the Gauss’ law,for the flux of RR fields through the compact spacewithout boundary should be conserved.This form of Gauss’law is the so-called RR tadpole cancellation condition.To satisfy this condition,stacks of Na,a=1,2,3 D6-branes are often needed to be introduced as the so-called filler brane,with a running through three families of gauge groups.Then the RR tadpole cancellation condition reads

and the coefficient 4 before [ΠO6]comes from the-4 RR charges in the D6-brane charge unit.Other than O6-planes in equation (2.7),we can also introduce D6-branes between the four O6-planes to cancel out the total RR charges,and rewrite equation (2.7) free of O6-planes.To do this,we note

Therefore the tadpole cancellation conditions 2.7 can be expressed in the form

where 2N(i),i=1,2,3,4 are the number of filler branes between the four O6-branes,as shown in table 2.

Furthermore,the four-dimensional N=1 supersymmetric conditions require 1/4 supercharges conserved under (i) orientation reversion of D6-branes;and (ii) group action of Z2×Z2.As shown in [9],the four-dimensional N=1supersymmetry is preserved under the orientation reversion if and only if rotations of D6-branes with respect to O6-planes are elements of SU(3),while their total rotation angles equal to 0.When the four-dimensionalN=1supersymmetry is preserved under orientation reversion,it will be preserved under the Z2×Z2-action manifestly.The supersymmetric condition can be written as [33]

3.Gauge symmetry breaking via brane splittings

To obtain SM or standard-like models via the mechanism of intersecting D6-branes,there should be at least two extra U(1)gauge symmetries for either supersymmetric models or nonsupersymmetric models,as a result of the constraints on the quantum number of the right handed electron [14,31–33].Among these two U(1) gauge symmetries,one is lepton number symmetryU(1)Land anotherU (1)I3Risan analogyfor right-hand weakisospin.Wehave the hyperchargeQYexpressed in the form

The baryonic charge QBarises from U(1)B,via the decomposition U(3)C?SU(3)C×U(1)B.On the other hand,since the U(1)I3Rgauge field should be massless,the gauge group U(1)I3Rmust come from the non-abelian component of U(2)Ror USp symmetry,otherwise the U(1)I3Rwill acquire mass from the B ∧F couplings.To get an anomaly-free U(1)B-L,the U(1)Lsymmetry should come from some non-abelian group for similar reasons.In previous studies of supersymmetric model building [31,32],U(1)I3Rcomes from USp groups.These models indeed have two extra anomaly-free U(1) symmetries,and have at least 8 Higgs doublets.One could in principle break their symmetry groups down to the SM symmetry,but cannot do this without violating the D-flatness and F-flatness,thus the supersymmetry.

In this paper,as introduced in [52],we study a generalized version of the four-family MSSM models.In these models,the b or c stacks are with twice the number of D6-branes.Then the gauge symmetries of these generalized four-family models can be broken to the standard four-family gauge symmetries SU(4)C×SU(2)L×SU(2)Rvia the Higgs mechanism.Taking gauge symmetries U (4)× U (4)L×U (2)Ras example,we consider a U(4)gauge theory with a scalar field in the adjoint representation.By choosing appropriate rotations commuting with the generators of the Lie algebra of SU(4),one can break U(4)to U(2)×U(2),and finally to U(2).We choose the rotation for U(4) scalar field acting on the vacuum expectation value Φ0as

We find that the U(4)gauge symmetry breaks spontaneously to U(2)×U(2),as the matrix in(3.2)lies in the center of the Lie algebra of U(4).What is left to us is to break down U(2)×U(2) to U(2).For each U(2) component in U(2)×U(2),the generators of its Lie algebra are the standard Pauli matrices.If we choose the rotation matrix for U(2)×U(2)as

we find that U(2)×U(2)breaks down to U(2).We also note that a mass of

is acquired through the above process.For models with gaugesymmetry U (4)×U (2)L×U (4)Rand U (4)× U (4)L×U (4)R,one can similarly break the symmetry down toU (4)×U (2)L×U (2)Rfollowing the above procedure.The anomalies of the overall U(1)symmetries are canceled by the generalized Green-Schwarz mechanism[13,14,31],while their fields get massive from the linear B ∧F couplings.

The gauge symmetry SU(4)C×SU(2)L×SU(2)Rcan be further broken down to SM gauge symmetry by D6-brane splitting and Higgs mechanism.Firstly,one can split stack a of Na=8 D6-branes into stack a1ofNa1=6D6-branes and stack a2Na2=2D6-branes.Then the U(4)Csymmetry breaks down to U(3)×U(1).We denote the numbers of symmetric and antisymmetric representations for SU(4)C,SU(2)Land SU(2)RbyAfter splitting,the symmetric and anti-symmetric representations of SU(4)Cdescend to symmetric representations of SU(3)Cand U(1)B-L,and anti-symmetric representations of SU(3)C.Note that there arenew fields,arising from the intersection of a1stack and a2stack of D6-branes.The anomaly-free gauge symmetry is SU(3)C×U(1)B-L,a subgroup of SU(4)C.

Similarly,the stack c of Nc=4 D6-branes can be broken into stack c1ofNc1=2D6-branes and stack c2Nc2=2D6-branes.Then the U(2)Rsymmetry breaks down to U(1)I3R.The symmetric representations of SU(2)Rdescend to the symmetric representations of U(1)I3Ronly.Also,there arenew fields,arising from the intersection of c1stack and c2stack of D6-branes.The anomaly-free gauge symmetry is U(1)I3R,a subgroup of SU(2)R.After splitting,the gauge symmetry of our model breaks down to SU(3)C×SU(2)L×U(1)B-L×U(1)I3R.

To get just the SM gauge symmetry,we assume the minimal squared distancebetween a2stack and the orientifold image of c1stack of D6-branes to be very small.Then there arechiral multiplets of light fermions,arising from the open string stretching between a2stack of D6-branes and the orientifold image of c1stack of D6-branes.These particles break down SU(3)C×SU(2)L×U(1)B-L×U(1)I3Rto the SM gauge symmetry,playing the same role as the righthanded neutrinos and their complex conjugates.Meanwhile,they preserve the D-flatness and F-flatness,thus the supersymmetry.In conclusion,the whole symmetry breaking chain is

The process of dynamical supersymmetry breaking has been studied in [36]for D6-brane models from Type IIA orientifolds.The kinetic function for a stack a of D6-branes is of the form [43]

where κais a constant with respect to the gauge groups,for instance,κa=1 for SU(Na).We use moduli parameters s and ui,i=1,2,3 in a supergravity basis,which are related to four-dimensional dilation parameter φ4and complex structure moduli parameters Ui,i=1,2,3 as follows

In our present models,the Ui,i=1,2,3 can be computed as in [45]

Moreover,the K?hler potential takes the form of

Note that the three moduli parameters χ1,χ2,χ3are not independent,as they can be expressed in terms of xA,xB,xC,xDand the latter parameters are related by the supersymmetric condition(2.10).Actually,one can determine χ1,χ2,χ3up to an overall coefficient,namely an action of dilation on these parameters.So one has to stabilize this dilation to determine all the moduli parameters.Previous studies [58–60]employ mechanisms like gaugino condensation to stabilize this overall coefficient,dictating that there should be at least two USp groups in the hidden sectors.Moreover,the one-loop beta functions [40]

for each USp(N(i))arising from 2N(i)filler branes are required to be negative.However,in this paper,to include other potential mechanisms,we do not restrict ourselves only to models with at least two USp groups in the hidden sectors.The gauge coupling constant related to stack a of D6-branes is

and the coupling constant of stack b and stack c of D6-branes are determined in the same way.The kinetic function for U(1)Yis a linear combination of those for SU(4)Cand SU(2)R,as shown in [11,43]

The coupling constant gYis determined by

At tree-level,the gauge couplings have the relation

where α,β and γ are ratios between the strong coupling and the weak coupling,and hypercharge coupling,respectively.

3.1.T-duality and its variations

In string theory,if two models are related by T-duality,these models are considered equivalent.By applying T-duality,one can tremendously simplify the process of searching inequivalent models.Before the discussion of T-duality,we first point out two obvious symmetries that relate equivalent models.

(i) Two models are equivalent if they are related by a permutation of three T2;and

(ii) Two D6-models are equivalent if their wrapping numbers on any two2T are in opposite signs,while are the same on the third2T.

The above two symmetries are known as the D6-brane Sign Equivalent Principle (DSEP).Then,follow the convention of[40],we introduce Type I and Type II T-dualities.Type I duality transformation acts on arbitrary two2T,say the jth and kth T2.The wrapping numbers on these tori transform as follows

when Type I T-duality applies.Recall the definitions in equation (2.8),Type I T-duality only makes an exchange betweenMoreover,Type I duality transformation is often combined with the trivial two2T exchange,and we call the combination an extended Type I T-duality.

As for Type II T-duality,it acts on all three different2T.For instance,if we pick the ith,jth and kth2T,the wrapping numbers on these tori transform as

In[40],Type II T-duality often combines with the interchange between b and c stacks of D6-branes

associated to SU(2)Land SU(2)Rgauge groups.

If we composite Type I T-duality and DSEP,we will get a variation of Type II T-duality.Under this symmetry transformation,the wrapping numbers on all three2T change as

It is easy to see that under variation of Type II T-duality,only the signs ofin equation (2.8) change.

However,it is worth mentioning that,the variation(3.18)of Type II T-duality is not an equivalence in our construction of four-family supersymmetric models,if the model is not invariant under SU(2)Land SU(2)Rinterchange.This observation makes sense at least phenomenologically.For a fourfamily supersymmetric model,one can obtain a new model by exchanging the b and c stacks of D6-branes associated to the SU(2)Land SU(2)Rgroups,as the quantum numbers for SU(2)Land SU(2)Rin the particle spectrum will exchange,so will the gauge couplings for these two groups at the string level.We will present with examples of this inequivalence in the next section.

3.2.Supersymmetric 4-family models

Employing the deterministic algorithm in [1],we do not restrict the number of USp groups,and consider two cases,one without any titled torus,and the other with the third torus to be tilted,without loss of generality.Note that four-family models are completely different from the three-family models,thus the argument in [40]to exclude which torus is tilted or cannot be directly applied to our case,due to the even number of generations.

3.2.1.Models without tilted torus.We obtain six classes of 274 supersymmetric four-family models with the deterministic algorithm introduced in [1]with representative models presented in appendix A.The classification is based on the gauge groups,T-equivalences and phenomenological considerations such as gauge coupling relations.

Model 14 is a class of its own,as it is the representative model without tilted torus achieving exact gauge coupling unification at the string scale.The Higgs-like particles in this model arise from the intersection at b andc′ stacks of D6-branes,while the Higgs doublets arise from the massless open string states in a N=2 subsector and form vector-like pairs.

The second class of models has no USp group,which means the tadpole cancellation conditions are satisfied without any filler brane as a rare case.These models are represented by Model 15 and Model 16,which are independent models of T-equivalence.These models are the first four-family examples having no confining USp groups to achieve approximate gauge coupling unification at the string scale.

The third class of models includes Model 17 and Model 18 with negative β function and positive β function respectively.

The fourth class of models includes Model 19–21 with two USp groups.These models are independent under T-duality.

As discussed in[59]there are at least two confining USp groups needed,with negative β function,and thus allow for gaugino condensations,these models would need alternative mechanisms to break the supersymmetry.Furthermore,we can observe from the spectrum tables that Model 15,17 and 20 do not have the proper Higgs doublets with quantum number (1,2,2) under U(4)C×U(2)L×U(2)Rgauge symmetry,from neither the b and c orc′ stacks of brane intersection nor the massless open string states in a N=2 subsector.Thus,we do not have the SM fermion Yukawa couplings at the renormalizable level which are invariant under the global U(1)C×U(1)L×U(1)Rsymmetry in these models.

The fifth class of models no less than three USp groups,and includes Models 19–25.Among these models,Model 24 and Model 25 are related by T-dualities with b and c stacks of branes swapping.To see this,we show how Model 24 and Model 25 are related.To begin with,a stacks of D6-branes in Model 24 and Model 25 are related by the DSEP:

And the b stack of D6-branes in Model 24 are related to the c stack of D6-branes in Model 25,by Type I T-duality (3.15),the DSEP,and the interchange (3.17) of b and c stacks:

Table 3.Spectrum of chiral particles of Model 14.

Table 4.Spectrum of chiral particles of Model 15.

Table 5.Spectrum of chiral particles of Model 17.

Table 6.Spectrum of chiral particles of Model 20.

Table 7.Spectrum of chiral particles of Model 24.

Table 8.The composite particle spectrum for Model 14.

Table 9.The composite particle spectrum for Model 20.

Table 10.The composite particle spectrum for Model 25.

Table 11.Spectrum of chiral particles of Model 36.

Table 12.Spectrum of chiral particles of Model 37.

Table 13.Spectrum of chiral particles of Model 39.

While the c stack of D6-branes in Model 24 and the b stack of D6-branes in Model 25 are related by Type II T-duality,DSEP and the b ?c exchange:

Even though these two models are related by the generalized T-duality as above,they are not phenomenologically equivalent.Model 24 achieves U(4) and U(2)Runification,while Model 25 has U(4)and U(2)Lunification due to b and c stacks swapping.The Higgs particles of Model 24 come from the intersection of b andc′ stacks of D6-branes.Since all the beta functions of models within this class are negative,we may break the supersymmetry and stabilize the moduli via gaugino condensations.

The sixth class of models are with large wrapping numbers 5,6,7,8,9,10,11,13,15,17,represented by Models 26–35 which did not appear in the former search.Three-family models with large wrapping number 5 have been found in [51],but it is the first time to find four-family models with wrapping numbers at this scale.

3.2.2.Models with one tilted torus.Employing the deterministic algorithm,we obtain in total 6 types of gauge coupling relations with represented models presented in appendix B.

Model 36 is a class of its own,as it is the only type of model with one tilted torus achieving exact gauge coupling unification at the string scale.The Higgs particles in this model arise from the massless open string states in a N=2 subsector and form vector-like pairs because the b stack branes for these models are parallel to c stack brane images on the third two-tori.Since all the beta functions are negative in this model,we can break the supersymmetry and stabilize the moduli via gaugino condensations.

The second class of models includes Model 37 and Model 38,and has no USp group.These models are related by type II T-duality.More specifically,we will below show how they are related by T-dualities explicitly.It is easy to find that the a stacks of both models are related by DSEP and Type II T-duality

Note that the b stack of wrapping numbers of Model 38 are obtained by applying the variation of Type II T-duality and DSEP on the b stack of wrapping numbers of Model 37:

The c stacks of Model 37 and Model 38 are related by applying DSEP twice:

Models 39–42 with one USp group in the hidden sector are related by T-dualities in a similar way.The Higgs particles in Model 37 again come from the massless open string states in a N=2 subsector and form vector-like pairs because the b stack branes for these models are parallel to c stack brane on the third two-tori.Since there are no USp groups in the hidden sectors,gaugino condensations do not work in this case.One needs to stabilize the modulus and break the supersymmetry via a different mechanism.For Model 39,there are four Higgs doublets arising from N=2 subsectors due to the parallel of b stack branes and c stack brane on the third two-tori.In addition,there are models 43 and 44 with distinct gauge coupling relations at string scale.

4.Phenomenological analysis

4.1.Models without tilted torus

We begin with Model 14.The gauge group of Model 14 is U (4)×U (2)L×U (2)R× USp (2)×USp (4).We tabulate the full spectrum of chiral particles of Models 14 and 15 in tables 3 and 4 respectively.Interestingly both Model 15 and Model 16 do not have any USp group,and then they do not have any exotic particles charged under USp groups as well.We tabulate the full spectrum of chiral particles of Model 15 below as a representative for this class.

Model 17 and Model 18 are constructed with one USp group.We tabulate the full spectrum of chiral particles of Model 17 in table 5 as a representative of this class of models.

Models 19–21 are built with two USp groups.The gauge groups for Model 19,Model 20 and Model 21 areU(4)×U (4)L×U (2)R×USp (2)×USp (4),U (4)× U (2)L×U (2)R×USp (4)2andU (4)×U (4)L×U (2)R× USp (4)×USp (12),respectively.We tabulate the full spectrum of chiral particles in Model 20 in table 6.

Models 22–25 are built with at least three USp groups.Their gauge groups areU (4)×U (2)L×U (2)R× USp (2)2×USp (4),U (4)× U (2)L× U (2)R× USp (2)3×USp (8),U (4)×U (2)L×U (2)R× USp (2)3×USp (4) andU (4)×U (2)L×U (2)R× USp (2)3×USp (4),respectively.Note that Model 24 and Model 25 are related by T-duality,but are not phenomenologically equivalent.This can be easily seen from the fact that Model 24 has U(4) and U(2)Rgauge coupling unification at the string scale,while Model 25 has U(4) and U(2)Lgauge coupling unification.We represent the full spectrum of chiral particles in Model 24 in table 7.

The exotic particles charged by USp groups may form bound states and composite particles at some intermediate energy scale,as the strong coupling dynamics of the USp groups require.The composite particles are consistent with anomaly cancellation conditions,as in the QCD case.These composite particles thus are charged only under the SM gauge symmetry [34].There are essentially two kinds of neutral bound states.The first one comes from decomposing the rank 2 anti-symmetric representation of the USp groups into two fundamental representations and then taking the pseudo inner product of the fundamental representations.The second one comes from the rank 2N anti-symmetric representation of USp(2N)for N≥2.The first bound state is similar to a meson that is the inner product of a fundamental representation and an anti-fundamental representation of SU(3)Cin QCD.The second bound state is a USp(2N) singlet,which is an analog to a baryon being a rank 3 anti-symmetric representation of SU(3)C.Models 14,20 and 25 contain the second kind of bound state.Now we take Models 14,20 and 25 as examples to show explicitly how bound states are formed.

The composite particle spectrum for Model 14 is listed in table 8.The confining group is USp(4),with two charged intersections.The mixing of intersection a4 and c4 results in the chiral supermultiplets (4,1,2,1,1).

Moreover,the confined particle spectrum for Model 20 in table 9.The confining group is USp(4),and has three charged intersections.Besides self-confinement,it is also viable to form mixed confinement between sections within the same confining group.The chiral supermultiplets(4,2,1,1,1),(1,2,2,1,1) and(4,1,2,1,1) are yielded by the mixed-confinement between intersections a4,b4 and c4.

For Model 25,the composite particle spectrum is given in table 10.There are two confining groups USp(4) and USp(2),each with two charged intersections.The mixed-confinement between intersections b2,c2 yields the chiral supermultiplet (1,2,2,1,1,1,1),while the mixed-confinement between intersections a4,c4 yields the chiral supermultiplet(4,1,2,1,1,1,1).Note that when there is only one charged intersection,mixed-confinement will not be formed,thus only the tensor representations from self-confinement are left.Checking from the composite particle spectra,one finds that no new anomaly is introduced to the remaining gauge symmetry.Thus our models are free of anomalies.The above analysis for composite particles applies to all our models except Model 15 and Model 16 without any confining group.U (4)×U (2)L×U (2)R×USp (2)2with two confining

4.2.Models with one tilted torus

In this section,we show basic phenomenological properties of models with one tilted torus.Model 36 represents the models with exact gauge coupling unification at the string level so far.The gauge symmetries therein are U (4)×U (4)L×U (2)R×USp (2).We note that Model 39 groups in the hidden sector.The full spectrum of this model is shown in table 11.

Model 37 and Model 38 have no USp group.The gauge group for the two models is U(4)C×U(2)L×U(2)R.Since there is no USp group,the gaugino condensation mechanism will not work.Thus,one needs to find other mechanisms for supersymmetry breaking.Also,there are no exotic particles in these two models,as exotic particles are charged under USp groups.We show the full chiral spectrum in the open string sectors for Model 37 in table 12.

Models 39,40,41,42 have only one USp(2) group in the hidden sector.Their gauge symmetry is all and Model 40 are T-dual to each other,as well as Model 41 and Model 42.But they are clearly not equivalent models at the phenomenological level due to b and c stacks of brane swapping.In Model 39,we have SU(3)Cand U(1)Ygauge coupling unification at the string level,while in Model 40 this gauge unification gets swapped to SU(3)Cand SU(2)Lgauge coupling unification at the string level.Similarly,this b and c stacks of brane swapping appear between Model 41 and Model 42 as well.The whole spectrum of chiral particles of Model 39 in table 13.

For the models with one tilted torus,the models represented by Model 36 are the only class with gauge coupling unification and carry two confining USp groups.Thus gaugino condensation can trigger supersymmetry breaking and moduli stabilization [36].

5.Discussions and conclusions

Utilizing the deterministic algorithm,we obtain various classes of four-family supersymmetric models from intersecting D6-branes onorientifold,with and without tilted torus.In total,there are 274 physical independent four-family supersymmetric models without tilted torus,and 6 physical independent four-family supersymmetric models with the third torus tilted,without loss of generality.

For models without tilted torus,Model 14 represents the model with gauge coupling unification at the string scale.Models 15 and 16 are the rare models without any USp group in the hidden sectors,with tadpole cancellation conditions satisfied.Models 22–26 are with at least two confining USp groups.Thus gaugino condensation can be triggered to break the supersymmetry and stabilize the moduli.Moreover,there are Models 27–35 with wrapping numbers absolutely larger than 5 which was not reached for three-family supersymmetric Pati–Salam models as discussed in [1].

For models with one tilted torus,Models 37 and 38 satisfy the tadpole cancellation conditions without any filler branes.Models such as Models 39 and 40 are related with b and c stacks of branes swapping.Model 36 represents the models with exact gauge coupling unification at the string scale,with two confining USp groups allowing gaugino condensation as well.This class of models would be ideal for further phenomenology model buildings(such as in [61]) as gaugino condensation can be triggered to break the supersymmetry and stabilize the moduli,while gauge coupling unified at string scale.

Acknowledgments

TL and CZ are supported by the National Key Research and Development Program of China Grant No.2020YFC2201504,by the Projects No.11875062,No.11947302,and No.12047 503 supported by the National Natural Science Foundation of China,as well as by the Key Research Program of the Chinese Academy of Sciences,Grant NO.XDPB15.RS is supported by KIAS Individual Grant PG080701.RS would like to thank Weikun He for their useful discussions.CZ would like to thank Lina Wu for their helpful discussions.

Appendix A.Four-family standard models from intersecting D6-branes without tilted tori

In the appendix,we list all representative four-family models obtained from our random scanning method.In the first columns for each table,a,b,c represent three stacks of D6-branes,respectively.Also in the first columns,1,2,3,4 is a short-handed notation for the filler branes along the ΩR,ΩRω,ΩRθω and ΩRθ O6-planes,respectively.The second columns for each table list the numbers of D6-branes in every stack,respectively.In the third column of each table,we record wrapping numbers of each D6-brane configuration and designate the third T2to be tilted.The rest columns of each table record intersection numbers between stacks.For instance,in the b column of table 14–44,from top to bottom,the numbers represent intersection numbers Iab,Ibb,Icb,etc..As usual,b′ andc′ are the orientifold ΩR image of b and c stacks of D6-branes.We also list the relation between xA,xB,xC,xD,which are determined by the supersymmetry condition equation (2.10),as well as the relation between the moduli parameter χ1,χ2,χ3.The one loop β functionsβigare also listed.To have a clearer sight of gauge couplings,we list them up in the caption of each table,which makes it easier to check whether they are unified.

Table 14.D6-brane configurations and intersection numbers of Model 14,and its gauge coupling relation is 2 2 2 5 3 a b c Y2 8 3 4 4====φ g g g g 2 e.π Model 14××××U U U USp USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 2 4 L R b b′ c c′ 1 4 a 8 (-1,0)×(-1,1)×(1,2)-1 1 4 0-4 0 0 2 b 4 (0,1)×(-1,2)×(-1,2) 3-3 – – 0-8-4 0 c 4 (1,1)×(-1,0)×(-1,2) 1-1 – – – – 0-2 1 2 (1,0)×(1,0)×(1,0)x x x x===1 2 14 14 A B C D 4 4 (0,1)×(0,1)×(1,0) β=-2 1,β g=0 g 4 χ=1 12,χ=2 12,χ=3 12

Table 15.D6-brane configurations and intersection numbers of Model 15,and its gauge coupling relation is 3 4 4====π φ g g g g 2 7 9 2 7 3 2 35 23 53 a b c Y 2 4 5 e 3 3.Model 15××U U U stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 L R b b′ c c′a 8 (-1,0)×(-1,1)×(1,1) 0 0 8-4-4 0 b 4 (-1,2)×(0,1)×(-1,3) 5-5 – –-16 0 c 4 (1,2)×(-2,1)×(-1,1)-7-9 – – – –15 16 x x x x===1 6 A B C D χ=5 1 6,χ=2 15,χ=3 25

Table 16.D6-brane configurations and intersection numbers of Model 16,and its gauge coupling relation is 2 10 9 2 10 3 2 50 29 53 a b c Y2 2 3 23 3 4 4====φ g g g g 11 e.π Model 16××U U U stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 4 4 L R b b′ c c′a 8 (-1,0)×(1,1)×(-1,1) 0 0 8-4-4 0 b 8 (-1,1)×(-1,3)×(0,1) 4-4 – – 0-8 c 8 (-1,1)×(1,2)×(-1,-1) 8 24 – – – –1 x x x x===A B C D 11 13 13 χ=1 2 1 11 3,χ=2 11,χ=3 11

Table 17.D6-brane configurations and intersection numbers of Model 17,and its gauge coupling relation is 2 5 6 2 5 2 2 25 16 53 a b c Y2 4 9 3 4 4====φ g g g g 14 e.π Model 17×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 2 L R b b′ c c′ 2 a 8 (-1,0)×(-1,1)×(1,1) 0 0 6-2-4 0 0 b 4 (-1,2)×(0,1)×(-1,2) 3-3 – –-8 0 0 c 4 (1,2)×(-2,1)×(-1,1)-7-9 – – – – 4 2 2 (1,0)×(0,1)×(0,1)x x x x===27 A B C D 14 14 β=-2 g2 χ=72 1 4,χ=2 27,χ=2 3 27

Table 18.D6-brane configurations and intersection numbers of Model 18,and its gauge coupling relation is 2 2 2 5 3====φ g g g g 3 5 3 a b c Y2 8 3 3 4 4.2 e π Model 18×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 4 4 4 L R b b′ c c′ 3 a 8 (-1,0)×(-1,2)×(1,1) 1-1 4 0-4 0-2 b 8 (-1,1)×(0,1)×(-1,1) 0 0 – – 0 0 2 c 8 (1,1)×(-1,1)×(-1,1)-4-12 – – – –-2 3 4 (0,1)×(1,0)×(0,1)1 4 x x x x===A B C D β g=2 12 3 χ=2 1,χ=1 2 22,χ=2 3

Table 19.D6-brane configurations and intersection numbers of Model 19,and its gauge coupling relation is 2 2 2 5 3====φ g g g g 2 a b c Y2 8 3 3 4 4.2 e π Model 19××××U U U USp USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 4 2 2 4 L R b b′ c c′ 1 4 a 8 (-1,0)×(-1,1)×(1,2)-1 1 0 4-4 0 0 2 b 8 (0,1)×(-1,1)×(-1,1) 0 0 – – 2-6-2 0 c 4 (1,1)×(-1,0)×(-1,2) 1-1 – – – – 0-2 1 2 (1,0)×(1,0)×(1,0) xA=xB=xC=2xD 4 4 (0,1)×(0,1)×(1,0) β=-4 1,β g=0 g 4 χ=2 1,χ=2 2,χ=2 3

Table 20.D6-brane configurations and intersection numbers of Model 20,and its gauge coupling relation is 2 21 22 2 7 2 2 7 4 53 a b c Y2 4 11 3 4 4====φ g g g g 3 10 e.π Model 20×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 4 L R 2 b b′ c c′ 2 4 a 8 (-1,0)×(1,1)×(-1,2) 1-1 6-2-4 0 0-2 b 4 (-3,2)×(-1,2)×(0,1) 1-1 – –-4 0 0 3 c 4 (-2,1)×(1,2)×(-1,-2) 5 27 – – – –-1 4 2 4 (1,0)×(0,1)×(0,1)1 x x x x===A B C D 20 38 34 4 4 (0,1)×(0,1)×(1,0) β=-5 g2,β g=5 4 χ=3 5 2 1 2,χ=2 1 10,χ=3 1 10

Table 21.D6-brane configurations and intersection numbers of Model 21,and its gauge coupling relation is 2 2 2 20 11====φ g g g g 4 5 3 a b c Y 2 32 9 4 4.2 e π Model 21××××U U U USp USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 4 2 4 12 L R b b′ c c′ 2 4 a 8 (-1,0)×(1,1)×(-1,2) 1-1 4 0-4 0 0-2 b 8 (-1,1)×(-1,1)×(0,1) 0 0 – –-6 6 0 2 c 4 (-2,1)×(1,2)×(-1,-2) 5 27 – – – –-1 4 2 12 (1,0)×(0,1)×(0,1)1 x x x x===1 2 A B C D2,β g=4 4 g 16 4 (0,1)×(0,1)×(1,0) β=-5 4 χ=2 2 1,χ=1 1 2 22,χ=3 2 2

Table 22.D6-brane configurations and intersection numbers of Model 22,and its gauge coupling relation is 2 2 2 10 7.2 5 3 2 4 4====φ g g g g a b c Y 2 2 3 e π 2 Model 22××××U U U USp USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 2 4 L R b b′ c c′ 2 3 4 a 8 (-1,0)×(-1,1)×(1,1) 0 0 4 0-4 0 0-1 1 b 4 (0,1)×(-1,3)×(-1,1) 2-2 – – 0-12 1 0 0 c 4 (1,1)×(-1,0)×(-2,2) 0 0 – – – – 2 0-2 2 4 (1,0)×(0,1)×(0,1)1 3 x x x x===A B C D 13 13 3 g 2 (0,1)×(1,0)×(0,1) β=-3 g2,β=-4 3,β=-2 g4 4 2 (0,1)×(0,1)×(1,0)χ=1 13,χ=2 13,χ=3 23

Table 23.D6-brane configurations and intersection numbers of Model 23,and its gauge coupling relation is 2 2 5 2 2 25 16 53 a b c Y2 2 3 3 4 4====φ g g g g 11 e.π 3 Model 23××××U U U USp USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 2 8 L R b b′ c c′ 1 2 3 4 a 8 (-1,0)×(1,1)×(-1,1) 0 0 4 0-4 0 0 0 1-1 b 4 (-1,2)×(-1,1)×(0,1) 1-1 – –-4 6-2 0 0 1 c 4 (-1,1)×(1,3)×(-1,-1) 0 12 – – – – 3-1 3 1 1 2 (1,0)×(1,0)×(1,0)1 x x x x===A B C D 12 g g 11 12 2 8 (1,0)×(0,1)×(0,1) β=-1 1,β=-5 g2,β=-1 g 4 3,β=-2 3 2 (0,1)×(1,0)×(0,1)χ=1 2 1 11 2,χ=2 11,χ=3 11 4 2 (0,1)×(0,1)×(1,0)

Table 24.D6-brane configurations and intersection numbers of Model 24,and its gauge coupling relation is 2 3 2 2 2 5 3====φ g g g g 2 3 e 2 a b c Y.3 4 4 π 3 Model 24××××U U U USp USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 2 4 L R b b′ c c′ 1 2 3 4 a 8 (-1,0)×(-1,1)×(1,1) 0 0 0 4-4 0 0 0-1 1 b 4 (0,1)×(-1,1)×(-3,1)-2 2 – – 2-4-1 3 0 0 c 4 (1,2)×(-1,0)×(-1,1)-1 1 – – – – 0 2 0-1 1 2 (1,0)×(1,0)×(1,0)x x x x===3 A B C D 32 32 2 g 4 (1,0)×(0,1)×(0,1) β=-5 g1,β=-1 g2,β=-4 g 4 3,β=-3 3 2 (0,1)×(1,0)×(0,1)χ=3 1 2,χ=3 2,χ=2 3 3 4 2 (0,1)×(0,1)×(1,0)

Table 25.D6-brane configurations and intersection numbers of Model 25,and its gauge coupling relation is 2 2 3 2 2 5 4 53====φ g g g g 2 3 e a b c Y 3 4 4.2 π 3 Model 25××××U U U USp USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 2 4 L R b b′ c c′ 1 2 3 4 a 8 (1,0)×(1,1)×(1,-1) 0 0 4 0-4 0 0 0 1-1 b 4 (-1,2)×(-1,1)×(0,1) 1-1 – – 4 2-2 0 0 1 c 4 (0,1)×(1,3)×(-1,-1)-2 2 – – – – 3-1 0 0 1 4 (1,0)×(1,0)×(1,0)x x x x===A B C D g g 13 12 12 2 2 (1,0)×(0,1)×(0,1) β=-1 1,β=-5 2,β=-4 g 4 g3,β=-3 3 2 (0,1)×(1,0)×(0,1)χ=3 1 2,χ=2 13,χ=3 23 4 2 (0,1)×(0,1)×(1,0)

Table 26.D6-brane configurations and intersection numbers of Model 26,and its gauge coupling relation is 3 4 4====π φ g g g g 2 11 10 2 11 5 2 55 37 53 a b c Y 2 8 13 e 7 5.Model 26×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 2 L R 3 b b′ c c′ 2 3 4 a 8 (1,-1)×(1,0)×(1,1) 0 0-3 7-4 0-1 0 1 b 4 (-5,2)×(1,1)×(-1,0)-3 3 – –-2 0-2 5 0 c 4 (-3,1)×(-1,1)×(-1,1) 0 12 – – – – 1 3 3 2 2 (1,0)×(0,1)×(0,1)26 5 x x x x===13 A B C D 26 5 3 3,βg4=-1 2 (0,1)×(1,0)×(0,1) βg2=-1,β g=2 4 2 (0,1)×(0,1)×(1,0) χ=13 1,χ=2 13 2 5,χ=2 13 3

Table 27.D6-brane configurations and intersection numbers of Model 27,and its gauge coupling relation is 2 6 7 2 2 7 2 2 5 53 a b c Y2 6 7 4 4====φ g g g g 2 3 e.π Model 27××U U U stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 L R b b′ c c′a 8 (-2,-1)×(1,1)×(1,1) 0-8 12-8-4 0 b 4 (-1,1)×(6,2)×(-1,0) 4-4 – – 8 0 c 4 (1,1)×(-1,0)×(-2,2) 0 0 – – – –xA=9xB=3xC=9xD χ=3 1,χ=3 3 2,χ=2 3 3

Table 28.D6-brane configurations and intersection numbers of Model 28,and its gauge coupling relation is====φ g g g g 23 e a2 47 112 2 18 7 2 30 19 53 b c Y2 1 3 27 3 4 4.π Model 28××××U U U USp USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 2 4 L R b b′ c c′ 1 2 a 8 (1,-1)×(1,0)×(2,1) 1-1-5 9-4 0 0-2 b 4 (-7,2)×(1,1)×(-1,0)-5 5 – – 9 11 0-2 c 4 (-2,1)×(-2,1)×(-2,1) 5 27 – – – –-1 4 1 2 (1,0)×(1,0)×(1,0)92 7 x x x x===4646 7 A B C D 2 1,βg2=4 χ=23 g 4 (1,0)×(0,1)×(0,1) β=-5 1,χ=2 23 2 7,χ=4 23 3

Table 29.D6-brane configurations and intersection numbers of Model 29,and its gauge coupling relation is 2 5 24 2 19 8 2 95 62 53 a b c Y2 2 45 3 4 4====φ g g g g 86 e.π Model 29×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 2 L R 2 b b′ c c′ 2 4 a 8 (2,-1)×(1,0)×(2,1) 0 0-6 10-4 0-2 2 b 4 (-8,1)×(1,1)×(-1,0)-7 7 – – 15 11-1 0 c 4 (-3,1)×(-2,1)×(-2,1) 9 39 – – – – 4 6 2 2 (1,0)×(0,1)×(0,1)43 4 x x x x===8643 4 A B C D 4 2 (0,1)×(0,1)×(1,0) β g=3 2,β g=4 4 43 χ=86 1,χ=2,χ=2 86 2 4 3

Table 30.D6-brane configurations and intersection numbers of Model 30,and its gauge coupling relation is 4====φ g g g g a 2 760 31 2 2 15 7 9 5 3 b c Y2 48 31.3 4 4 2 3 7 e π Model 30×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 2 L R 2 b b′ c c′ 1 3 a 8 (-1,1)×(-1,0)×(1,1) 0 0-5 9-4 0 0 0 b 4 (-2,1)×(-1,1)×(-4,1) 3 29 – – 33-35-1 8 c 4 (1,0)×(-9,-2)×(-1,1) 7-7 – – – – 0 2 1 2 (1,0)×(1,0)×(1,0)x x x x===42 42 1,β g=4 A B C D 28 3 3 g 2 (0,1)×(1,0)×(0,1) β=-5 3 χ=2 1 73,χ2,χ=4=3 21 3 73

Table 31.D6-brane configurations and intersection numbers of Model 31,and its gauge coupling relation is 4====φ g g g g a 2 1309 47 2 2 25 13 5 5 3 b c Y2 8 47.3 4 4 2 185 e π Model 31××U U U stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 L R b b′ c c′a 8 (0,1)×(-1,-2)×(2,1) 0 0 10-6-4 0 b 4 (1,-2)×(1,-3)×(4,1)-23-73 – – 48-24 c 4 (1,-10)×(0,-1)×(-2,1) 8-8 – – – –15 x x x x===1 A B C D 74 χ=1 1 370,χ=2 5 74,χ=2 3 74 5

Table 32.D6-brane configurations and intersection numbers of Model 32,and its gauge coupling relation is====φ g g g g a 2 1248 41 2 2 11 5 53 b c Y2 64 123 3 4 4 11 77 e.π Model 32×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 14 L R b b′ c c′ 1 a 8 (-1,1)×(-1,0)×(1,1) 0 0-5 9-4 0 0 b 4 (-2,1)×(-2,1)×(-4,1) 13 51 – – 45-35-1 c 4 (1,0)×(-11,-2)×(-1,1) 9-9 – – – – 0 1 14 (1,0)×(1,0)×(1,0)x x x x===56 56 A B C D g1 112 11 β=-5 χ=4 1 7 11,χ=2 77 2,χ=8 3 7 11

Table 33.D6-brane configurations and intersection numbers of Model 33,and its gauge coupling relation is 4====φ g g g g a 2 1680 47 2 2 65 29 53 b c Y 2 256 141 3 4 4 13 2 13 e.π Model 33×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 10 L R b b′ c c′ 1 a 8 (-1,1)×(-1,0)×(1,1) 0 0-5 9-4 0 0 b 4 (-2,1)×(-2,1)×(-4,1) 13 51 – – 51-45-1 c 4 (1,0)×(-13,-2)×(-1,1) 11-11 – – – – 0 1 10 (1,0)×(1,0)×(1,0)128 13 x x x x===64 64 A B C D βg1=-5 χ=8 1 2 13,χ=4 26 2,χ=16 3 2 13

Table 34.D6-brane configurations and intersection numbers of Model 34,and its gauge coupling relation is 4====φ g g g g a 2 2176 53 2 2 25 11 53 b c Y 2 192 53 3 4 4 15 3 5 e.π Model 34×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 6 L R b b′ c c′ 1 a 8 (-1,1)×(-1,0)×(1,1) 0 0-5 9-4 0 0 b 4 (-2,1)×(-2,1)×(-4,1) 13 51 – – 57-55-1 c 4 (1,0)×(-15,-2)×(-1,1) 13-13 – – – – 0 1 6 (1,0)×(1,0)×(1,0)48 5 x x x x===72 72 A B C D βg1=-5 χ=4 1 35,χ=6 15 2,χ=8 3 35

Table 35.D6-brane configurations and intersection numbers of Model 35,and its gauge coupling relation is====φ g g g g a 2 2736 59 2 2 85 37 53 b c Y2 64 177 3 4 4 17 170 e.π Model 35×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 2 L R b b′ c c′ 1 a 8 (-1,1)×(-1,0)×(1,1) 0 0-5 9-4 0 0 b 4 (-2,1)×(-2,1)×(-4,1) 13 51 – – 63-65-1 c 4 (1,0)×(-17,-2)×(-1,1) 15-15 – – – – 0 1 2 (1,0)×(1,0)×(1,0)160 17 x x x x===80 80 A B C D βg1=-5 χ=4 1 10 17,χ=2 170 2,χ=8 3 10 17

Appendix B.Four-family standard models from intersecting D6-branes with one tilted torus

Table 36.D6-brane configurations and intersection numbers of Model 36,and its gauge coupling relation is 2 2 2 5 3 4 2====φ g g g g 2 2 e a b c Y.π Model 36×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 2 L R 2 b b′ c c′ 2 4 a 8 (1,1)×(0,-1)×(1,1) 0 0 0 4 0-4 0 0 b 4 (1,0)×(-4,1)×(-1,-1)-3 3 – – 0 0 0 4 c 4 (0,-1)×(4,1)×(1,1) 3-3 – – – – 4 0 2 2 (1,0)×(0,1)×(0,-2) xA=4xB=xC=4xD 4 2 (0,1)×(0,1)×(-2,0) βg2=-2,βg4=-2 χ1=1,χ2=4,χ3=2

Table 37.D6-brane configurations and intersection numbers of Model 37,and its gauge coupling relation is 2 2 2 10 7 53====φ g g g g 2 2 4 e 4 2 a b c Y.π Model 37××U U U stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 4 4 L R b b′ c c′a 8 (1,-1)×(-1,0)×(-1,-1) 0 0 0 4 0-4 b 8 (0,1)×(-1,-2)×(1,1)-2 2 – – 0 0 c 8 (1,0)×(1,-2)×(1,1) 2-2 – – – –12===1 2 x x x x A B C D χ1=1,χ=2 12,χ3=2

Table 38.D6-brane configurations and intersection numbers of Model 38,and its gauge coupling relation is 2 2 2 10 7 53====φ g g g g 2 2 4 e 4 2 a b c Y.π Model 38××U U U stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 4 4 L R b b′ c c′a 8 (1,0)×(-1,1)×(-1,-1) 0 0 4 0 0-4 b 8 (1,-2)×(0,-1)×(-1,1) 2-2 – – 0 0 c 8 (1,-2)×(-1,0)×(-1,-1) 2-2 – – – –12===1 2 x x x x A B C D χ=1 12,χ2=1,χ3=2

Table 39.D6-brane configurations and intersection numbers of Model 39,and its gauge coupling relation is 2 2 2 5 3====φ g g g g 2 4 4 a b c Y2 8 3.2 e π Model 39×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 4 2 2 L R b b′ c c′ 1 a 8 (1,-1)×(1,0)×(1,1) 0 0 4 0-4 0 0 b 8 (0,1)×(-1,2)×(-1,1) 2-2 – – 0 4 4 c 4 (-1,0)×(-1,-4)×(1,-1)-3 3 – – – – 0 1 2 (1,0)×(1,0)×(-2,0)1 2 x x x x===2 A B C D βg1=-2 χ=2 12 1,χ=1 2 22,χ=2 2 3

Table 40.D6-brane configurations and intersection numbers of Model 40,and its gauge coupling relation is 2 2 2 10 7====φ g g g g 2 5 3 a b c Y2 8 3 4 4.2 e π Model 40×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 4 2 L R b b′ c c′ 2 a 8 (-1,-1)×(0,-1)×(-1,-1) 0 0 4 0-4 0 0 b 4 (-1,0)×(4,1)×(-1,1) 3-3 – – 0-4 0 c 8 (0,-1)×(2,-1)×(-1,1)-2 2 – – – – 4 2 2 (1,0)×(0,1)×(0,-2) xA=2xB=xC=4xD βg2=-2 χ=2 1,χ=2 2 2,χ=2 3

Table 41.D6-brane configurations and intersection numbers of Model 41,and its gauge coupling relation is 2 2 2 5 3====φ g g g g 2 a b c Y2 8 3 4 4.2 e π Model 41×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 4 2 2 L R b b′ c c′ 2 a 8 (-1,0)×(1,1)×(-1,1) 0 0 0 4 0-4 0 b 8 (1,2)×(1,0)×(1,-1)-2 2 – – 0-4 4 c 4 (-1,4)×(0,-1)×(1,-1) 3-3 – – – – 0 2 2 (1,0)×(0,1)×(0,-2)x x x x===12 14 14 A B C D β=-2 g2 χ=1 1 22,χ=2 12,χ=2 3

Table 42.D6-brane configurations and intersection numbers of Model 42,and its gauge coupling relation is 2 2 2 10 7====φ g g g g 2 5 3 a b c Y2 8 3 4 4.2 e π Model 42×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 4 2 L R b b′ c c′ 1 a 8 (-1,0)×(-1,-1)×(1,-1) 0 0 0 4-4 0 0 b 4 (-1,-4)×(-1,0)×(1,-1)-3 3 – – 4 0 0 c 8 (-1,-2)×(0,1)×(1,1)-2 2 – – – – 4 1 2 (1,0)×(1,0)×(-2,0)x x x x===2 A B C D 12 12 βg1=-2 χ=1 1 22,χ=2 2,χ=2 2 3

Table 43.D6-brane configurations and intersection numbers of Model 43,and its gauge coupling relation is 2 8 3 2 2 5 3 a b c Y 2 16 9 3 4 4====φ g g g g 2 e.π Model 43×××U U U USp stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 2 L R b b′ c c′ 3 a 8 (0,1)×(1,-1)×(1,-1) 0 0 1 3-4 0 0 b 4 (-1,1)×(0,1)×(-2,4) 2-2 – –-9-5 2 c 4 (4,-1)×(1,0)×(1,1)-3 3 – – – – 0 3 2 (0,1)×(1,0)×(0,2)1 2 x x x 4x===A B C D βg3=-4 χ=2 1,χ=2 2 2,χ=3 12

Table 44.D6-brane configurations and intersection numbers of Model 44,and its gauge coupling relation is 2 16 5 2 2 10 7====φ g g g g 2 5 3 a b c Y 2 16 5.2 e π 4 Model 44×××U U U stack N (n1,l1)×(n2,l2)×(n3,l3) n□□4 2 2 L R b b′ c c′a 8 (0,1)×(1,-1)×(1,-1) 0 0 1 3-4 0 b 4 (-1,1)×(0,1)×(-2,4) 2-2 – –-6-6 c 4 (2,-1)×(1,0)×(2,2)-2 2 – – – –12 x x x 2x===A B C D χ1=1,χ2=2,χ3=1