Towards Realistic Susy Spectra and Yukawa Textures from Intersecting Branes
Abstract
ABSTRACT
We study the possible phenomenology of a threefamily PatiSalam model constructed from intersecting D6branes in Type IIA string theory on the orientifold with some desirable semirealistic features. In the model, treelevel gauge coupling unification is achieved automatically at the string scale, and the gauge symmetry may be broken to the Standard Model (SM) close to the string scale. The small number of extra chiral exotic states in the model may be decoupled via the Higgs mechanism and strong dynamics. We calculate the possible supersymmetry breaking soft terms and the corresponding lowenergy supersymmetric particle spectra which may potentially be tested at the Large Hadron Collider (LHC). We find that for the viable regions of the parameter space the lightest CPeven Higgs boson mass usually satisfies GeV, and the observed dark matter density may be generated. Finally, we find that it is possible to obtain correct SM quark masses and mixings, and the tau lepton mass at the unification scale. Additionally, neutrino masses and mixings may be generated via the seesaw mechanism. Mechanisms to stabilize the open and closedstring moduli, which are necessary for the model to be truly viable and to make definite predictions are discussed.
ACT0607, MIFP0729
I Introduction
Although string theory has long teased us with her power to encompass all known physical phenomena in a complete mathematical structure, an actual worked out example is still lacking. Indeed, the major problem of string phenomenology is to construct at least one realistic model with all moduli stabilized, which completely describes known particle physics as well as potentially being predictive of unknown phenomena. With the dawn of the Large Hadron Collider (LHC) era, new discoveries will hopefully be upon us. In particular, supersymmetry is expected to be found as well as the Higgs states required to break the electroweak symmetry. Therefore, it is highly desirable to have complete, concrete models derived from string theory which are able to make predictions for the superpartner spectra, as well as describing currently known particle physics.
In the old days of string phenomenology, model builders were primarily focused on weakly coupled heterotic string theory. However, with the advent of the second string revolution, Dbranes JPEW have created new interest in Type I and II compactifications. In particular, Type IIA orientifolds with intersecting D6branes, where the chiral fermions arise at the intersections of D6branes in the internal space bdl , with Tdual Type IIB description in terms of magnetized Dbranes bachas , have shown great promise during the last few years. Indeed, intersecting Dbrane configurations provide promising setups which may accommodate semirealistic features of lowenergy physics. Given this, it is an interesting question to see how far one can get from a particular string compactification to reproducing the finer details of the Standard Model as a lowenergy effective field theory.
In order to construct globally consistent vacua with intersecting Dbranes, conditions must be imposed which strongly constrain the models. In particular, all RamondRamond (RR) tadpoles must be cancelled and Ktheory Witten9810188 conditions for cancelling the nontrivial anomally also must be imposed. Despite the clear benefits of supersymmetry, there have been many threefamily standardlike models and Grand Unified Theories (GUT) constructed on Type IIA orientifolds Blumenhagen:2000wh ; Angelantonj:2000hi ; Blumenhagen:2005mu which are not supersymmetric. Although these models are globally consistent, they are generally plagued by the gauge hierarchy problem and vacuum instability which arises from uncancelled NeveuSchwarzNeveuSchwarz (NSNS) tadpoles. Later, semirealistic supersymmetric Standardlike, PatiSalam, unflipped as well as flipped models in Type IIA theory on CSU1 ; CSU2 ; Cvetic:2002pj ; CP ; CLL ; Cvetic:2004nk ; Chen:2005ab ; Chen:2005mj and Dudas:2005jx ; Blumenhagen:2005tn ; Chen:2006sd orientifolds were eventually constructed, and some of their phenomenological consequences studied CLS1 ; CLW . Other supersymmetric constructions in Type IIA theory on different orientifold backgrounds have also been discussed ListSUSYOthers . Nonperturbative Dinstanton effects have also been receiving much attention of late, and may play an important role Blumenhagen:2006xt Ibanez:2006da Cvetic:2007ku Ibanez:2007rs .
In addition to satisfying the above consistency conditions, all open and closedstring moduli must be stabilized in order to obtain an actual vacuum. Unstabilized moduli are manifest in the lowenergy theory as massless scalar fields, which are clearly in conflict with observations. Given a concrete string model, the lowenergy observables such as particle couplings and resulting masses are functions of the open and closed string moduli. In a fully realistic model, these moduli must therefore be stabilized and given sufficiently large masses to meet the astrophysical/cosmological and collider physics constraints on additional scalar fields. Although satisfying the conditions for supersymmetry in Type IIA (IIB) fixes the complex structure (K\a”ahler) moduli in these models, the K\a”ahler (complex structure) and openstring moduli generally remain unfixed. To stabilize some of these moduli, supergravity threeform fluxes Kachru:Blumen and geometric fluxes Grimm:Zwirner were introduced and flux models on Type II orientifolds have been constructed Cascales:2003zp ; MS ; CL ; Cvetic:2005bn ; Kumar:2005hf ; Chen:2005cf ; Blumenhagen:2006ci ; Camara:2005dc ; Chen:2006gd ; Chen:2006ip . Models where the Dbranes wrap rigid cycles, thus freezing the openstring moduli have also been studied Dudas:2005jx ; Blumenhagen:2005tn ; Chen:2006sd .
Despite substantial progress, there have been other roadblocks in constructing phenomenologically realistic intersecting Dbrane models, besides the usual problem of moduli stabilization. Unlike heterotic models, the gauge couplings are not automatically unified. Additionally, there has been a rank one problem in the Standard Model (SM) fermion Yukawa matrices, preventing the generation of mass for the first two generations of quarks and leptons. For the case of toroidal orientifold compactifications, this can be traced to the fact that not all of the Standard Model fermions are localized at intersections on the same torus. However, one example of an intersecting D6brane model on Type IIA orientifold has recently been discovered where these problems may be solved CLL ; Chen:2006gd . Thus, this particular model may be a step forward to obtaining realistic phenomenology from string theory. Indeed, as we recently discussed Chen:2007px , it is possible within the moduli space of this model to obtain correct quark mass matrices and mixings, the tau lepton mass, and to generate naturally small neutrino masses via the seesaw mechanism. Furthermore, it is possible to generically study the soft supersymmetry breaking terms, from which can be calculated the supersymmetric partner spectra, the Higgs masses, and the resulting neutralino relic density.
This paper is organized as follows. First, we will briefly review the intersecting D6brane model on Type IIA orientifold which we are studying and discuss its basic features. We then discuss the lowenergy effective action, and show that the treelevel gauge couplings are unified near the string scale. We also find that the hidden sector gauge groups will become confining at a high energy scale, thus decoupling chiral exotics present in the model. Next, we study the possible lowenergy superpartner spectra which may arise. We also calculate the Yukawa couplings for quarks and leptons in this model, and show that we may obtain the correct quark masses and mixings and the tau lepton mass for specific choices of the open and closed stringmoduli VEVs. We should emphasize that for the present work, we will not focus on the moduli stabilization problem, as our goal is only to explore the possible phenemological characteristics of the model. However, we do comment on this issue and discuss how it may potentially be solved for this model. We also should note that models with an equivalent observable sector have been constructed in Type IIA and Type IIB theory as Ads and Minkowski flux vacua Chen:2006gd ; Chen:2007af , so that the issue of closedstring moduli stablization has already been addressed to some extent.
Ii A Dbrane Model with Desirable Semirealistic Features
In recent years, intersecting Dbrane models have provided an exciting approach towards constructing semirealistic vacua. To summarize, D6 branes (in Type IIA) fill threedimensional Minkowski space and wrap 3cycles in the compactified manifold, with a stack of branes having a gauge group (or in the case of ) in its world volume. The 3cycles wrapped by the Dbranes will in general intersect multiple times in the internal space, resulting in a chiral fermion in the bifundamental representation localized at the intersection between different stacks. The multiplicity of such fermions is then given by the number of times the 3cycles intersect. Due to orientifolding, for every stack of D6branes we must also introduce its orientifold images. Thus, the D6branes may also have intersections with the images of other stacks, also resulting in fermions in bifundamental representations. Each stack may also intersect its own images, resulting in chiral fermions in the symmetric and antisymmetric representations. The different types of representations that may be obtained for each type of intersection and their multiplicities are shown in Table 1. In addition, there are constraints that must be satisfied for the consistency of the model, namely the requirement for RamondRamond tadpole cancellation and to have a sprectrum with supersymmetry.
Intersecting Dbrane configurations provide promising setups which may accommodate semirealistic features of lowenergy physics. Given this, it is an interesting question to see how far one can get from a particular string compactification to reproducing the finer details of the Standard Model as a lowenergy effective field theory. There have been many consistent models studied, but only a small number have the proper structures to produce an acceptable phenomenology. A good candidate for a realistic model which may possess the proper structures was discussed in CLL ; Chen:2006gd ; Chen:2007px in Type IIA theory on the orientifold. This background has been extensively studied and we refer the reader to CSU1 ; CSU2 for reviews of the basic model building rules. We present the D6brane configurations and intersection numbers of this model in Table 2, and the resulting spectrum which is essentially that of a threefamily PatiSalam in Table 3 CLL ; Chen:2006gd . We put the , , and stacks of D6branes on the top of each other on the third two torus, and as a result there are additional vectorlike particles from subsectors.
Sector  Representation 

vector multiplet and 3 adjoint chiral multiplets  
; 
The anomalies from three global s of , and are cancelled by the GreenSchwarz mechanism, and the gauge fields of these s obtain masses via the linear couplings. Thus, the effective gauge symmetry is . In order to break the gauge symmetry, on the first torus, we split the stack of D6branes into and stacks with 6 and 2 D6branes, respectively, and split the stack of D6branes into and stacks with two D6branes for each one, as shown in Figure 1. In this way, the gauge symmetry is further broken to . Moreover, the gauge symmetry may be broken to by giving vacuum expectation values (VEVs) to the vectorlike particles with the quantum numbers and under the gauge symmetry from intersections CLL ; Chen:2006gd .
1  2  3  4  

8  0  0  3  0  3  0  1  1  0  0  

4  2  2      0  0  0  1  0  3  

4  2  2          1  0  3  0  
1 
2  
2 
2  
3 
2  
4 
2  

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 
Since the gauge couplings in the Minimal Supersymmetric Standard Model (MSSM) are unified at the GUT scale GeV, the additional exotic particles present in the model must necessarily become superheavy. To accomplish this it is first assumed that the and stacks of D6branes lie on the top of each other on the first torus, so we have two pairs of vectorlike particles with quantum numbers . These particles can break down to the diagonal near the string scale by obtaining VEVs, and then states arising from intersections and may obtain vectorlike masses close to the string scale from superpotential terms of the form
(1) 
where we neglect the couplings of order one. Moreover, we assume that the and obtain VEVs near the string scale, and their VEVs satisfy the Dflatness of . We also assume that there exist various suitable highdimensional operators in the effective theory, and thus the adjoint chiral superfields may obtain GUTscale masses via these operators. With and , we can give GUTscale masses to the particles from the intersections , , and via the supepotential:
(2) 
The beta function for is and the gauge coupling for will become strongly coupled around GeV, and then we can give GeV scale VEVs to and preserve the Dflatness of . The remaining states may also obtain intermediate scale masses via the operators
(3) 
To have one pair of light Higgs doublets, it is necessary to finetune the mixing parameters of the Higgs doublets. In particular, the term and the righthanded neutrino masses may be generated via the following highdimensional operators
(4) 
where and are Yukawa couplings, and is the string scale. Thus, the term is TeV scale and the righthanded neutrino masses can be in the range GeV for and .
Iii The Lowenergy Effective Action
In building a concrete string model which may be testable, it is not enough to simply reproduce the matter and gauge symmetry of the known lowenergy particle states in the Standard Model. It is also necessary to make predictions regarding the superpartner spectra and Higgs masses. If supersymmetry exists as expected and is softy broken, then it is possible to calculate the soft SUSY breaking terms, which determine the low energy sparticle spectra. Furthermore, if the neutralino is the lightest supersymmetric particle (LSP), then it is expected to make up a large fraction of the observed dark matter density, at Bennett:2003bz ; Spergel:2003cb , and this is calculable from the soft terms. Ideally, one would also like to be able to calculate the Yukawa couplings for the known quarks and leptons, and be able to reproduce their masses and mixings.
To discuss the lowenergy phenomenology we start from the lowenergy effective action. From the effective scalar potential it is possible to study the stability Blumenhagen:2001te , the treelevel gauge couplings CLS1 ; Shiu:1998pa ; Cremades:2002te , gauge threshold corrections Lust:2003ky , and gauge coupling unification Antoniadis:Blumen . The effective Yukawa couplings Cremades:2003qj ; Cvetic:2003ch , matter field Kähler metric and softSUSY breaking terms have also been investigated Kors:2003wf . A more detailed discussion of the Kähler metric and string scattering of gauge, matter, and moduli fields has been performed in Lust:2004cx . Although turning on Type IIB 3form fluxes can break supersymmetry from the closed string sector Cascales:2003zp ; MS ; CL ; Cvetic:2005bn ; Kumar:2005hf ; Chen:2005cf , there are additional terms in the superpotential generated by the fluxes and there is currently no satisfactory model which incorporates this. Thus, we do not consider this option in the present work. In principle, it should be possible to specify the exact mechanism by which supersymmetry is broken, and thus to make very specific predictions. However, for the present work, we will adopt a parametrization of the SUSY breaking so that we can study it generically.
The supergravity action depends upon three functions, the holomorphic gauge kinetic function, , K\a”ahler potential , and the superpotential . Each of these will in turn depend upon the moduli fields which describe the background upon which the model is constructed. The holomorphic gauge kinetic function for a D6brane wrapping a calibrated threecyce is given by Blumenhagen:2006ci
(5) 
In terms of the threecycle wrapped by the stack of branes, we have
(6) 
from which it follows that
(7) 
where for and for or gauge groups and where we use the and moduli in the supergravity basis. In the string theory basis, we have the dilaton , three Kähler moduli , and three complex structure moduli Lust:2004cx . These are related to the corresponding moduli in the supergravity basis by
(8) 
and is the fourdimensional dilaton. To second order in the string matter fields, the K\a”ahler potential is given by
(9)  
The untwisted moduli , are light, nonchiral scalars from the field theory point of view, associated with the Dbrane positions and Wilson lines. These fields are not observed in the MSSM, and if they were present in the low energy spectra may disrupt the gauge coupling unification. Clearly, these fields must get a large mass through some mechanism. One way to accomplish this is to require the Dbranes to wrap rigid cycles, which freezes the open string moduli Blumenhagen:2005tn . However, there are no rigid cycles available on without discrete torsion, thus we will assume that the openstring moduli become massive via highdimensional operators.
For twisted moduli arising from strings stretching between stacks and , we have , where is the angle between the cycles wrapped by the stacks of branes and on the torus respectively. Then, for the K\a”ahler metric in Type IIA theory we find the following two cases:

, ,
(10) 
, ,
(11)
For branes which are parallel on at least one torus, giving rise to nonchiral matter in bifundamental representations (for example, the Higgs doublets), the K\a”ahler metric is
(12) 
The superpotential is given by
(13) 
while the minimum of the F part of the treelevel supergravity scalar potential is given by
(14) 
where and , is inverse of , and the auxiliary fields are given by
(15) 
Supersymmetry is broken when some of the Fterms of the hidden sector fields acquire VEVs. This then results in soft terms being generated in the observable sector. For simplicity, it is assumed in this analysis that the term does not contribute (see Kawamura:1996ex ) to the SUSY breaking. Then the goldstino is included by the gravitino via the superHiggs effect. The gravitino then obtains a mass
(16) 
which we will take to be TeV in the following. The normalized gaugino mass parameters, scalar masssquared parameters, and trilinear parameters respectively may be given in terms of the K\a”ahler potential, the gauge kinetic function, and the superpotential as
(17)  
where is the K\a”ahler metric appropriate for branes which are parallel on at least one torus, i.e. involving nonchiral matter.
The above formulas for the soft terms depend on the Yukawa couplings, via the superpotential. An important consideration is whether or not this should cause any modification to the lowenergy spectrum. However, this turns out not to be the case since the Yukawas in the soft term formulas are not the same as the physical Yukawas, which arise from worldsheet instantons and are proportional to , where is the worldsheet area of the triangles formed by a triplet of intersections at which the Standard Model fields are localized. As we shall see in a later section, the physical Yukawa couplings in Type IIA depend on the K\a”ahler moduli and the openstring moduli. This ensures that the Yukawa couplings present in the soft terms do not depend on either the complexstructure moduli or dilaton (in the supergravity basis). Thus, the Yukawa couplings will not affect the lowenergy spectrum in the case of moduli dominant and mixed and dominant supersymmetry breaking.
To determine the SUSY soft breaking parameters, and therefore the spectra of the models, we introduce the VEVs of the auxiliary fields Eq. (15) for the dilaton, complex and Kähler moduli Brignole:1993dj :
(18) 
The factors and are the CP violating phases of the moduli, while the constant is given by
(19) 
The goldstino is included in the gravitino by in field space, and parameterize the goldstino direction in space, where . The goldstino angle determines the degree to which SUSY breaking is being dominated by the dilaton and/or complex structure () and Kähler () moduli. As suggested earlier, we will not consider the case of moduli dominant supersymmetry breaking as in this case, the soft terms are not independent of the Yukawa couplings.
Iv Gauge Coupling Unification
The MSSM predicts the unification of the three gauge couplings at an energy GeV. In intersecting Dbrane models, the gauge groups arise from different stacks of branes, and so they will not generally have the same volume in the compactified space. Thus, the gauge couplings are not automatically unified, in contrast to heterotic models. For branes wrapping cycles not invariant under , the holomorphic gauge kinetic function for a D6 brane stack is given by Eq. (7). where and are the complex structure moduli and dilaton in the supergravity basis.
The gauge coupling constant associated with a stack P is given by
(20) 
Thus, for the model under study the holomorphic gauge function is identified with stack and the holomorphic gauge function with stack . The holomorphic gauge function is then given by taking a linear combination of the holomorphic gauge functions from all the stacks. Note that we have absorbed a factor of in the definition of so that the electric charge is given by . In this way, it is found Blumenhagen:2003jy that
(21) 
Recalling that the complex structure moduli are obtained from the supersymmetry conditions, we have for the present model
(22) 
Thus, we find that the treelevel MSSM gauge couplings will be automatically unified at the string scale
(23) 
Even though the gauge couplings are unified, this does not fix the actual value of the couplings as these still depend upon the value taken by the fourdimensional dilaton . In order for the gauge couplings to have the value observed for the MSSM (), we must choose such that , which fixes the string scale as
(24) 
where is the reduced Planck scale.
It should be kept in mind that values given for the gauge couplings at the string scale are only the treelevel results. There are oneloop threshold corrections arising from the and open string sectors Lust:2003ky which may alter these results. In addition, there is exotic matter charged under both observable and hidden sector gauge groups, which are expected to pick up large masses, but could still affect the running of the gauge couplings.
V Confinement of the Hidden Sector Fields
In addition to the matter content of the MSSM, there is also matter charged under the hidden sector gauge groups. These states will generally have fractional electric charges, similar to the socalled ‘cryptons’ Ellis:1990iu ; Benakli:1998ut ; Ellis:2004cj ; Ellis:2005jc . Obviously, no such matter is observed in the lowenergy spectrum so these exotic states must receive a large mass. Such a mass may arise if the hidden sector gauge couplings are asymptotically free and become confining at some high energy. Indeed, in the present case we find that the functions for the groups are all negative CLL ,
(25) 
where we consider all of the chiral exotic particles present even though it is expected that these states will decouple as discussed previously. From the holomorphic gauge kinetic function, the gauge couplings are found to take the values
(26)  
at the string scale. We may then straightforwardly run these couplings to lowenergy energy via the oneloop RGE equations,
(27) 
where we find that the couplings for the and hidden sector groups will become strong at a scale GeV, while the couplings for the and groups will become strong around GeV as shown in Figure 2.
We should note that it is also possible to decouple the chiral exotic states in the manner discussed in section II.
Vi Soft Terms and Superpartner Spectra
Next, we turn to our attention to the soft supersymmetry breaking terms at the GUT scale defined in Eq. (III). In the present analysis, not all the Fterms of the moduli get VEVs for simplicity, as in Font:2004cx ; Kane:2004hm . As discussed earlier, we will assume that so that the soft terms have no dependence on the physical Yukawa couplings. Thus, we consider two cases:

The moduli dominated SUSY breaking where both the cosmological constant and the goldstino angle are set to zero, such that .

The and moduli SUSY breaking where the cosmological constant and .
vi.1 SUSY breaking with moduli dominance
For this case we take so that the terms are parameterized by the expression
(28) 
where and with . With this parametrization, the gaugino mass terms for a stack may be written as
(29) 
The Bino mass parameter is a linear combination of the gaugino mass for each stack,
(30) 
where the the coefficients correspond to the linear combination of factors which define the hypercharge, .
For the trilinear parameters, we have
(31)  
where ,, and label the stacks of branes whose mutual intersections define the fields present in the corresponding trilinear coupling and the angle differences are defined as
(32) 
We must be careful when dealing with cases where the angle difference is negative. Note for the present model, there is always either one or two of the which are negative. Let us define the parameter
(33) 
such that indicates that only one of the angle differences are negative while indicates that two of the angle differences are negative.
Finally, the squark and slepton (1/4 BPS) scalar masssquared parameters are given by
(34) 
The functions in the above formulas defined for are
(35)  
and for are
(36)  
The function is just the derivative
(37) 
and and are defined Kane:2004hm as
(38) 
(39) 
Note that the only explicit dependence of the soft terms on the