Variations of the Hidden Sector in a Realistic
Intersecting
Brane Model
Abstract
Recently, we discussed the first example of a phenomenologically realistic intersecting D6brane model. In this model, the gauge symmetry in the hidden sector is . However, we find that the gauge symmetry can be replaced by an gauge symmetry, and/or the gauge symmetry can be replaced by an gauge symmetry since the stacks of D6branes contribute to the same RamondRamond tadpoles as those of the stacks. Thus, there are three nonequivalent variations of the hidden sector, and the corresponding gauge symmetries are , , and , respectively. Moreover, we study the hidden sector gauge symmetry breaking, discuss how to decouple the additional exotic particles, and briefly comment on the phenomenological consequences.
pacs:
11.10.Kk, 11.25.Mj, 11.25.w, 12.60.JvI Introduction
The goal of string phenomenology is to construct realistic standardlike string models with all moduli stabilized. In the early days, string model building was mainly concentrated on the weakly coupled heterotic string theory. After the second string revolution, consistent fourdimensional chiral models with nonAbelian gauge symmetry on Type II orientifolds were able to be constructed due to the advent of Dbranes JPEW . In particular, Type II orientifolds with intersecting Dbranes, where the chiral fermions arise from the intersections of Dbranes in the internal space bdl with Tdual description in terms of magnetized Dbranes bachas , have played an important role in string model building during the last few years.
On Type IIA orientifolds with intersecting D6branes, many nonsupersymmetric threefamily standardlike models and Grand Unified Theories (GUTs) were constructed Blumenhagen:2000wh ; Angelantonj:2000hi ; Blumenhagen:2005mu . Although these models were globally consistent, there generically existed uncancelled NeveuSchwarzNeveuSchwarz (NSNS) tadpoles as well as the gauge hierarchy problem. To solve these two problems, semirealistic supersymmetric standardlike models, PatiSalam models, models as well as flipped models have been constructed in Type IIA theory on CSU1 ; CSU2 ; Cvetic:2002pj ; CP ; CLL ; Cvetic:2004nk ; Chen:2005ab ; Chen:2005mj and Blumenhagen:2005tn ; Chen:2006sd orientifolds with intersecting D6branes, and some of their phenomenological consequences have been studied CLS1 ; CLW . Moreover, the supersymmetric constructions in Type IIA theory on other orientifolds were also discussed ListSUSYOthers . There are two main constraints on supersymmetric D6brane model building: RR tadpole cancellation conditions and fourdimensional supersymmetric D6brane configurations. Also, Ktheory conditions provide minor constraints. In addition, to stabilize the closedstring moduli via supergravity fluxes, the flux models on Type II orientifolds have also been constructed Blumenhagen:2003vr ; Cascales:2003zp ; MS ; CL ; Cvetic:2005bn ; Kumar:2005hf ; Chen:2005cf ; Villadoro:2005cu ; Camara:2005dc ; Chen:2006gd ; Chen:2006ip .
It is well known that there are two serious problems in almost all the supersymmetric Dbrane models: no gauge coupling unification at the string scale, and the rank one problem in the Standard Model (SM) fermion Yukawa matrices. Although these problems can be solved in the flux models of Ref. Chen:2006gd where the RR tadpole cancellation conditions are relaxed, these models are in the AdS vacua and the question of how to lift these AdS vacua to the Minkowski vacua or dS vacua correctly is still a big challenge. Recently, we found that there is one and only one intersecting D6brane model on Type IIA orientifold where the above problems can be solved CLL ; Chen:2006gd . Moreover, this model may has a realistic low energy phenomenology Chen:2007px . Although its observable sector has unique phenomological properties, it is possible to have different stacks of the D6branes in the hidden sector.
In this paper, we discuss three nonequivalent variations of the hidden sector where the RR tadpoles are cancelled, the fourdimensional supersymmetry is perserved, and the Ktheory conditions are satisfied. These three variations seem to be the only possibilities. In the original model CLL ; Chen:2006gd , the gauge symmetry in the hidden sector is . Interestingly, we can replace the gauge symmetry by an gauge symmetry, and/or the gauge symmetry by an gauge symmetry since the contributions to the RR tadpoles from the stacks of D6branes are the same as those of the stacks. Thus, there are three nonequivalent variations, and the corresponding gauge symmetries in the hidden sector are , , and , respectively. Moreover, we discuss the hidden sector gauge symmetry breaking, and consider how to decouple the additional exotic particles. Because the observable sector is the same, the discussions on phenomenological consequences, for example, the gauge coupling unification, supersymmetry breaking soft terms, low energy supersymmetric particle spectrum, dark matter density, and the SM fermion masses and mixings, are the same as those in Ref. Chen:2007px ; CLMNL .
This paper is organized as follows. We briefly review the intersecting D6brane model building on Type IIA orientifold in Section II and the realistic intersecting D6brane model in Section III. We study the three variations of the hidden sector in Section IV. Discussion and conclusions are given in Section V.
Ii Intersecting D6Brane Model Building in Type IIA Theory on Orientifold
We briefly review the intersecting D6brane model building in Type IIA theory on orientifold CSU1 ; CSU2 . We consider to be a six torus factorized as whose complex coordinates are , for the th two torus, respectively. The and generators for the orbifold group act on the complex coordinates as following
(1)  
We implement an orientifold projection , where is the worldsheet parity, and acts on the complex coordinates as
(2) 
So, there are four kinds of orientifold 6planes (O6planes) for the actions of , , , and , respectively. Also, we have two kinds of complex structures consistent with orientifold projection for a two torus – rectangular and tilted LUII . If we denote the homology classes of the three cycles wrapped by the D6brane stacks as and with for the rectangular and tilted tori respectively, we can label a generic one cycle by in either case, where in terms of the wrapping numbers for a rectangular two torus and for a tilted two torus. So, the homology threecycles for stack of D6branes and its orientifold image take the form
(3) 
where if the th two torus is rectangular and if it is tilted. Also, we define .
Sector  Representation 

vector multiplet and 3 adjoint chiral multiplets  
; 
For a stack of D6branes that do not lie on the top of any O6plane, we obtain the gauge symmetry with three adjoint chiral superfields due to the orbifold projections. While for a stack of D6branes on the top of an O6plane, we obtain the gauge symmetry with three antisymmetric chiral superfields. The bifundamental chiral superfields arise from the intersections of two different stacks of D6branes or one stack of D6branes and its image CSU1 ; CSU2 . In short, the general spectrum for intersecting D6branes at generic angles, which is valid for both rectangular and tilted two tori, is given in Table 1. Moreover, a model may contain additional nonchiral (vectorlike) multiplet pairs from , , and sectors if two stacks of the corresponding Dbranes are parallel and on the top of each other on one two torus. The multiplicity of the nonchiral multiplet pairs is given by the product of the intersection numbers on the other two twotori.
Before further discussions, let us define the products of wrapping numbers
(4) 
The fourdimensional supersymmetric models from Type IIA
orientifolds with intersecting D6branes are mainly constrained by
the RR tadpole cancellation conditions and the fourdimensional
supersymmetric D6brane configurations, and also
constrained by the Ktheory conditions:
(1) RR Tadpole Cancellation Conditions
The total RR charges of D6branes and O6planes must vanish since the RR field flux lines are conserved. And then we obtain the RR tadpole cancellation conditions as follows
(5) 
where are the number of D6branes wrapping along
the th O6plane which is defined in Table 2.
Orientifold Action  O6Plane  

1  
2  
3  
4 
(2) FourDimensional Supersymmetric D6Brane Configurations
The fourdimensional supersymmetry can be preserved by the orientation projection if and only if the rotation angle of any D6brane with respect to the O6plane is an element of bdl , or in other words, mod , where is the angle between the D6brane and the O6plane in the th two torus. This supersymmetry conditions can be rewritten as Cvetic:2002pj
(6) 
where , and are the
complex structure parameters. The positive parameter has
been introduced to put all the variables
on an
equal footing.
(3) Ktheory Conditions
The discrete Dbrane RR charges classified by the Ktheory groups in the presence of orientifolds, which are subtle and invisible by the ordinary homology MS ; Witten9810188 , should also be taken into account Cascales:2003zp . The Ktheory conditions are
(7) 
Iii The Realistic Intersecting D6Brane Model
There may be one and only one intersecting D6brane model in Type IIA theory on orientifold with a realistic phenomenology CLL ; Chen:2006gd ; Chen:2007px . Let us briefly review it. We present the D6brane configurations and intersection numbers in Table 3, and its spectrum in Table 4. We put the , , and stacks of D6branes on the top of each other on the third two torus, and then we have the additional vectorlike particles from subsectors.
stack 
(,)  (,)  (,)  A  S  

8  ( 0,1)  ( 1, 1)  ( 1, 1)  0  0  3  0(3)  3  0(3)  1  1  0  0 

4  ( 3, 1)  ( 1, 0)  ( 1,1)  2  2      0(6)  0(1)  0  1  0  3 

4  ( 3,1)  ( 0, 1)  ( 1,1)  2  2          1  0  3  0 

2  ( 1, 0)  ( 1, 0)  ( 2, 0)                     

2  ( 1, 0)  ( 0,1)  ( 0, 2)                     

2  ( 0, 1)  ( 1, 0)  ( 0, 2)                     

2  ( 0, 1)  ( 0, 1)  ( 2, 0)                     
Quantum Number  Field  
1  1  0  
1  0  
0  0  
1  0  0  
0  1  0  
0  1  0  
0  0  1  
0  0  1  
0  2  0  
0  2  0  
0  0  2  
0  0  2  
1  1  0  
1  1  0  
1  1  
1  0  1  
0  1  1  ,  
0  1  1 
We have shown that the gauge symmetry in the observable sector can be broken down to the SM gauge symmetry via the GreenSchwarz mechanism, D6brane splittings and supersymmtry preserving Higgs mechanism. The gauge couplings for , and are unified at the string scale, and the additional exotic particles may be decoupled around the string scale. Also, we calculated the supersymmetry breaking soft terms, and the corresponding low energy supersymmetric particle spectrum that can be tested at the Large Hadron Collider (LHC). The observed dark matter density can also be generated. In addition, we can explain the SM quark masses and mixings, and the tau lepton mass. The neutrino masses and mixings may be generated via seesaw mechanism as well. Similar to the GUTs Nanopoulos:1982zm , we have roughly the wrong fermion mass relation , and the correct electron and muon masses can be generated via highdimensional operators CLMNL . Furthermore, all the gauge symmetries will become strong around the string scale CLMNL .
Iv Three variations of the Hidden Sector
In the realistic intersecting D6brane model CLL ; Chen:2006gd , the observable sector is unique. Interestingly, we find three nonequivalent variations of the hidden sector where we can cancel the RR tadpoles, preserve the fourdimensional supersymmetry, and satisfy the Ktheory conditions. And it seems to us that there is no other variation. In the original model CLL ; Chen:2006gd , the gauge symmetry in the hidden sector is . We notice that the gauge symmetry can be replaced by an gauge symmetry, and/or the gauge symmetry by an gauge symmetry because the contributions to the RR tadpoles from the stacks of D6branes are the same as those of the stacks. Thus, there are three nonequivalent variations, and the corresponding gauge symmetries in the hidden sector are , , and , respectively. Let us present them one by one in the following subsections.
iv.1 Hidden Sector
stack 
(,)  (,)  (,)  A  S  

8  ( 0,1)  ( 1, 1)  ( 1, 1)  0  0  3  0(3)  3  0(3)  0(2)  0(1)  0  0 

4  ( 3, 1)  ( 1, 0)  ( 1,1)  2  2      0(6)  0(1)  1  0(1)  0  3 

4  ( 3,1)  ( 0, 1)  ( 1,1)  2  2          1  0(1)  3  0 

4  ( 1, 0)  ( 1,1)  ( 1, 1)  0  0              1  1 

2  ( 0, 1)  ( 1, 0)  ( 0, 2)                     

2  ( 0, 1)  ( 0, 1)  ( 2, 0)                     

Representation  Field  


0  1  0  1  

0  0  1  1  

0  0  0  1  

0  0  0  1  

In the first variation of the hidden sector, we replace the gauge symmetry by an gauge symmetry. We present the D6brane configurations and intersection numbers in Table 5. Moreover, the particle spectrum has two parts: (1) the spectrum for old particles is given in Table 4 by removing all the particles that are charged under ; (2) the spectrum for the new particles is given in Table 6.
The anomalies from the global of are cancelled by the GreenSchwarz mechanism, and its gauge field obtains mass via the linear couplings. Then, the effective gauge symmetry is . The gauge symmetry can be broken down to via D6brane splitting. Interestingly, we do not have any additional chiral exotic particles that are charged under . The simple way to give masses to the extra exotic particles and is instanton effects Blumenhagen:2006xt ; Ibanez:2006da ; Cvetic:2007ku ; Ibanez:2007rs . However, we do not have the suitable threecycles wrapped by E2 instantons ^{1}^{1}1Note that the E2 branes must also wrap rigid cycles., and thus the instanton effects are not available. Similar results hold for the next two subsections. In addition, the and will become strong at about the string scale CLMNL , and then we will have some composite particles in the antisymmetric and symmetric representations, and from , and and from , respectively. So we can break the by giving suitable stringscale vacuum expectation values (VEVs) to and , and we can give the stringscale VEVs to and . Note that we give the TeVscale VEVs to and the stringscale VEVs to Chen:2007px , we can give the GUTscale masses to and and the TeVscale masses to the and via the highdimensional operators CLMNL . Furthermore, if we could give the stringscale masses to the three adjoint chiral superfields and we do not break the via D6brane splitting, the gauge symmetry will become strong around the string scale. Then we can have the singlet composite field in the antisymmetric representation with charge under from . And we can give the stringscale VEVs to and while keeping the Dflatness of . Therefore, we may also give the GUTscale masses to the and via the highdimensional operators CLMNL .
iv.2 Hidden Sector
stack 
(,)  (,)  (,)  A  S  

8  ( 0,1)  ( 1, 1)  ( 1, 1)  0  0  3  0(3)  3  0(3)  0(2)  0(0)  1  1 

4  ( 3, 1)  ( 1, 0)  ( 1,1)  2  2      0(6)  0(1)  0(3)  3  0  1 

4  ( 3,1)  ( 0, 1)  ( 1,1)  2  2          0(3)  3  1  0 

4  ( 0, 1)  (1, 1)  (1, 1)  0  0              1  1 

2  ( 1, 0)  ( 1, 0)  ( 2, 0)                     

2  ( 1, 0)  ( 0,1)  ( 0, 2)                     


Representation  Field  


0  1  0  1  

0  0  1  1  

0  0  0  1  

0  0  0  1  

In the second variation of the hidden sector, we replace the gauge symmetry by an gauge symmetry. We present the D6brane configurations and intersection numbers in Table 7. The particle spectrum also has two parts: (1) the spectrum for old particles is given in Table 4 by removing all the particles that are charged under ; (2) the spectrum for the new particles is given in Table 8.
Note that the wrapping numbers for the stack of D6branes are equivalent to those of the stack by T duality and orientifold action, we can think that we have an gauge symmetry in the begining. Only the global of is anomalous symmetry, and its gauge field obtains mass via the linear couplings. After we put four D6branes on the place with equivalent wrapping numbers (just like the D6brane splittings), we break the down to the where the generator in is
(8) 
Thus, the lefthanded and righthanded SM fermions have charges and , respectively. In order to keep the gauge coupling unification, we have to break the so that it will not become part of the . In short, we have to break completely.
Because the and will become strong at about the string scale CLMNL , we will have some composite particles in the antisymmetric and symmetric representations, and from , and and from , respectively. So we can break the completely by giving suitable stringscale VEVs to , , , and . Moreover, we can have the singlet composite particle in the antisymmetric representation with charge under from . And then we can give the stringscale VEVs to and while keeping the Dflatness of . Note that also have stringscale VEVs, we may give the GUTscale masses to , , , and via the highdimensional operators CLMNL . Moreover, and may form the vectorlike particles if we break the and down to the diagonal Chen:2007px .
iv.3 Hidden Sector
stack 
(,)  (,)  (,)  A  S  

8  ( 0,1)  ( 1, 1)  ( 1, 1)  0  0  3  0(3)  3  0(3)  0(2)  0(1)  0(2)  0(0) 

4  ( 3, 1)  ( 1, 0)  ( 1,1)  2  2      0(6)  0(1)  1  0(1)  0(3)  3 

4  ( 3,1)  ( 0, 1)  ( 1,1)  2  2          1  0(1)  0(3)  3 

4  ( 1, 0)  ( 1,1)  ( 1, 1)  0  0              0(1)  0(2) 

4  ( 0, 1)  (1, 1)  (1, 1)  0  0                 


Representation  Field  

0  1  0  1  0  

0  1  0  0  1  

0  0  1  1  0  

0  0  1  0  1  

0  0  0  1  1  
0  0  0  1  1  

0  0  0  1  1  
0  0  0  1  1  

In the third variation of the hidden sector, we replace the gauge symmetry by , and replace the gauge symmetry by . We present the D6brane configurations and intersection numbers in Table 9. The particle spectrum also has two parts: (1) the spectrum for old particles is given in Table 4 by removing all the particles that are charged under ; (2) the spectrum for the new particles is given in Table 10.
As discussed in above subsections, we can break the down to the gauge symmetry via GreenSchwarz mechanism and D6brane splitting, and we have to break the gauge symmetry completely. In order to break the and gauge symmetries, we put the and stacks of D6branes on the top of each other on the second two torus, and put the and stacks on the top of each other on the third two torus. Then, we have additional vectorlike particles , , , and , as given in Table 10. And there exist the following Yukawa couplings
(9) 
where and are Yukawa couplings. If we give the diagonal stringscale VEVs to and