Dynamical Supersymmetry Breaking in Intersecting Brane Models
RROM2F/2010/11
Dynamical Supersymmetry Breaking in IntersectingBrane Models
F.Fucito , A. Lionetto , J. F. Morales , and
R. Richter
I.N.F.N. Sezione di Roma Tor Vergataand
Dipartimento di Fisica, Universit´a di Roma “Tor Vergata”Via della Ricerca Scientifica, 00133 Roma, Italy
Abstract
In this paper we study dynamical supersymmetry breaking in absence ofgravity with the matter content of the minimal supersymmetric standardmodel. The hidden sector of the theory is a strongly coupled gauge theory,realized in terms of microscopic variables which condensate to form mesons.The supersymmetry breaking scalar potential combines F, D terms with in-stanton generated interactions in the Higgs-mesons sector. We show that fora large region in parameter space the vacuum breaks in addition to super-symmetry also electroweak gauge symmetry. We furthermore present localD-brane configurations that realize these supersymmetry breaking patterns. a r X i v : . [ h e p - t h ] J u l ontents Breaking supersymmetry (SUSY) has always proved to be a challenging anddaunted task. Of all the possible options, dynamical SUSY breaking (DSB)remains one of the most exciting and economical choices of breaking SUSY.In this framework, SUSY is a symmetry of the effective action and it is brokenspontaneously by the choice of the field theory vacuum. As it is well known,the original toy models, in which SUSY was broken by giving a vacuumexpectation value (vev) to the F or D-terms, gave a sparticle spectrum atodds with observations. This is why in the usual SUSY extensions of theStandard Model (MSSM), the breaking is achieved via the introduction ofsuitable terms in the Lagrangian called soft SUSY breaking terms. They donot spoil the divergence properties of the theory and also participate to themechanism for the gauge symmetry breaking of the theory. SUSY breakingis thus intimately connected with all the main features of the MSSM. Thecommon lore wants soft SUSY breaking terms to be generated at higherenergies with respect to the mass of the sparticles in a so called hiddensector. SUSY breaking is then communicated to the visible sector via (gauge1r gravity) messenger particles. Recently many good books on SUSY haveappeared in which these issues are considered at length [1, 2, 3, 4, 5, 6].The progress in the computation of non perturbative effects in gauge andSUSY theories, has opened up a new possibility for SUSY breaking: the clas-sical potential has a trivial vacuum which is corrected by quantum effects.The quantum potential then exhibits DSB. Already at the level of SUSYgauge theories, the existence of condensates, driven by non perturbative ef-fects, does not immediately translates into SUSY breaking. Each case mustbe carefully examined: the standard criterion is to look for a global symme-try, spontaneously broken, in a theory which has no flat directions. Anotherstandard strategy is to look for inconsistencies in the theory i.e. for a clashbetween two different conditions which have to be simultaneously satisfied inorder to preserve SUSY (for example existence of a condensate which thenviolates the Konishi anomaly) [7, 8, 9, 10]. These results were first obtainedfor non chiral QCD like theories. Their extension to chiral models of SUSYGUT type was discussed in [9] and in the framework of gauge mediation in[11, 12, 13].How to incorporate all of this into the framework of string theory isthen a completely different problem. Recently some of the authors of thepresent paper [14] have analysed a quiver model realizing a dynamical SUSYbreaking scenario for a GUT theory. In this paper we want to take a furtherstep and consider string intersecting models with the matter content of theMSSM. There has been much work recently to embed the SM and MSSMusing intersecting branes: . here we discuss SUSY breaking in this scenario.Therefore we consider a model where SUSY is broken spontaneously via anon-perturbatively generated superpotential in a hidden strongly coupledgauge theory. The non-perturbative superpotential will induce spontaneousbreaking of gauge and SUSY via an articulate conspiracy of gauge (D-terms)and Yukawa (F-terms) interactions. Some of the steps we will take resemblethe so called KKLT approach followed in [20] to stabilize some of the moduliof a string theory compactification and more recently in [21, 22, 23]. Howeverin our case gravity plays no role: we just focus on the open string sector.We thus postulate that the details of the compactification are already takencare of, that moduli have been (completely or partially) stabilized and thatthe closed string dynamics is not affecting our results. As we will see, thesepositions are not oversimplifying our model. The open string sector must For recent reviews see [15, 16, 17, 18, 19].
In this work we present two extensions of the MSSM which naturally giverise to SUSY and electroweak gauge symmetry breaking. In both cases theinterplay of F and D-terms is crucial for the SUSY and electroweak gaugesymmetry breaking. As we will see later in Section 5 such extensions of theMSSM can naturally arise from D-brane compactifications.Let us start by stating the gauge symmetry of the setups SU (3) C × SU (2) L × U (1) Y × U (1) × SU (3) H , (1)which in addition to the usual SM gauge symmetry contains an abelian gaugesymmetry and a strongly coupled hidden SU (3) H .The chiral matter sector contains the usual MSSM matter content , namelythe quarks and leptons (cid:126) Φ = { Q L , L, u c , d c , e c , ν c } , (2)which are collected in the vector (cid:126) Φ, where capital letters refer to left-handedsuperfields, while lower case letters denote right-handed fields. In addition tothe MSSM particles we have the Higgs fields H u,d and two quark anti-quarkpairs Q i and (cid:101) Q i , i = 1 ,
2, charged with respect to the hidden SU (3) H andthe additional U (1). Moreover, depending on the considered setup there maybe an additional field Y , which is neutral under the SM gauge groups andonly charged under the U (1).The hidden SU (3) H gauge theory with two quark anti-quark pairs Q and (cid:101) Q will condensate via the generation of a Affleck, Dine and Seibergsuperpotential W non − pert = Λ det ( M ij ) , (3) Note that we include here also the right-handed neutrino ν c . M ij = Q i (cid:101) Q j is the meson matrix. At the scale Λ the gauge theory iseffectively described in terms of the mesons and baryons of SU (3) H whichcan be taken as the microscopic degrees of freedom of the low energy physics.In the following we will denote by Λ M i , the eigenvalues of the meson matrix M ij and will work with the M i as fundamental degrees of freedom. Thus theSUSY - electroweak gauge symmetry breaking matter content, the Higgs-meson sector, is given by (cid:126)X = { H u , H d , M i , Y } , (4)where the presence of Y depends on the choice of the specific setup.As we will see later in Section 5 in D-brane compactifications the hyper-charge is a linear combination of various U (1)’s. Explicitly U (1) Y = 12 U (1) d + 12 U (1) e + ... U (1) = 12 U (1) e − U (1) d , (5)where the dots indicate that there are further contributions for the hyper-charge from other U (1)’s under which the Higgs-meson fields are uncharged.For later convenience we label each U (1) by a subscript that will later specifyits D-brane origin.Below in Table 1 we display the non-trivial charges of the fields (cid:126)X in theHiggs-meson sector with respect to the SU (2) electroweak and U (1) d,e sym-metries. With that choice of charges the generic superpotential containing H u H d M M Y SU (2) U (1) d − − U (1) e − − SU (2) × U (1) d × U (1) e -charges of the Higgs-meson sector only the fields (cid:126)X and respecting the abelian symmetries U (1) d and U (1) e isof the form W = µH u H d M + mM M + Λ M M + (cid:16) µ Y H u H d Y + m Y M Y (cid:17) . (6)In absence of the field Y , clearly the last two terms of (6) are absent.4n order to ensure that the color and electromagnetic gauge symmetriesremain unbroken, we require the vanishing of all the vev’s of the MSSMmatter fields (cid:126) Φ. We look then for solutions with (cid:104) (cid:126) Φ (cid:105) = 0 (cid:104) (cid:126)X (cid:105) = x + θ F x (7)In the following we will try to find SUSY and electroweak gauge symmetrybreaking minima satisfying (7). Thus we extremize the scalar potential onlywith respect to the fields (cid:126)X . We will show in Section 3 that for a wide rangein parameter space, SUSY and the electroweak gauge symmetries are brokenin the (cid:126)X sector. After discussing the general setup above let us now describe in more detailsthe Lagrangians of the models we will consider. For simplicity we take acanonical K¨ahler potential for the Higgs-meson fields (cid:126)X . More precisely wetake the K¨ahler potential to be K ( (cid:126)X, (cid:126)X † , (cid:126) Φ , (cid:126) Φ † ) = (cid:126)X (cid:126)X † + k ij ( (cid:126)X, (cid:126)X † )Φ i Φ † j + ... . (8)Thus the K¨ahler metric for the scalar fields (cid:126)X will take a canonical form aftertaking into account the vanishing vev’s of the MSSM matter fields (cid:126) Φ. Weallow for a general (cid:126)X -dependence of the functions k ij specifying the K¨ahlermetric for the matter fields (cid:126) Φ. When SUSY is broken F X (cid:54) = 0, these K¨ahlerinteractions provide soft symmetry breaking mass terms (of order k (cid:48)(cid:48) ij | F X | )for sparticles. These functions should then satisfy the phenomenologicalrequirement that the masses of sparticles are beyond the observable limits.Let us stress though that the specific type of K¨ahler potential does not affectour conclusions regarding SUSY and gauge symmetry breaking. A similar (cid:126)X -dependence can be introduced in the gauge kinetic functions τ a ( (cid:126)X ) in orderto induce soft symmetry breaking masses (of order τ (cid:48) a F X ) for the gauginos ofthe unbroken symmetries SU (3) C × U (1) em of the MSSM .With these assumptions the Lagrangians leading to the D and F termscan be written as(cid:32)L D = (cid:16) k ij ( (cid:126)X, (cid:126)X † ) Φ † i e V Φ j + (cid:126)X † e V (cid:126)X + ξ a V a (cid:17) (cid:12)(cid:12)(cid:12) θ ¯ θ + h . c . (cid:32)L F = (cid:104) τ a ( (cid:126)X )Tr( W ( a ) W ( a ) ) + W ( (cid:126)X, (cid:126) Φ) (cid:105) (cid:12)(cid:12)(cid:12) θ + h . c . . (9)5ith k ij ( (cid:126)X, (cid:126)X † ), τ a ( (cid:126)X ) the quark and lepton K¨ahler function and W ( (cid:126)X, (cid:126) Φ)the superpotential. We will split the latter into two terms according to thenumber of quark and lepton superfields involved W ( (cid:126)X, (cid:126) Φ) = W ( (cid:126)X ) + W ( (cid:126)X, (cid:126) Φ) . (10) W determines the field theory vacuum. In addition we would like to breakSUSY and the electroweak gauge symmetry in such a way that W ( (cid:104) (cid:126)X (cid:105) , (cid:126) Φ)gives realistic masses to the quarks and leptons. This implies that the Higgsfields should acquire non-zero vev’s. Moreover, we are interested in non-vanishing vev’s for the F (cid:126)X in order to lift the masses of the sparticles com-pared to their SM partners.The pattern of SUSY and gauge symmetry breaking crucially depends onthe choice of gauge and Yukawa couplings, masses as well as on the Fayet-Iliopoulos terms entering the low energy action. In a string realization ofthis scenario, which will be discussed later, all these couplings and Fayet-Iliopoulos terms will be given by the closed string background in which theD-brane setup is localized and will be input parameters in our analysis. Westress the fact that once closed string dynamics is turned on, what we refer asFayet-Iliopoulos terms here become field dependent functions of the chargedclosed string moduli. In this Section we study the vacuum structure of the field theory modelsfor two simple choices of W of the type displayed in equation (6). In bothcases, given reasonable choices for the parameters of the theories, a vacuumcan be found which breaks SUSY and the electroweak gauge symmetry. Thisbreaking requires an interplay between the F- and D-terms. We start byanalysing a configuration which exhibits the fields H u , H d , M and M in theHiggs-meson sector. Later we will allow for an additional field Y , chargedunder U (1) d and U (1) e . Let us consider the following field content in the Higgs-meson sector (cid:126)X = { H u , H d , M , M } . (11)6he charges of the various fields were displayed in Table 1. For the superpo-tential we take W = µH u H d M + mM M + Λ M M , (12)where the latter term is the non-perturbative ADS superpotential of thehidden SU (3) H . In order to preserve the electromagnetic gauge symmetrywe look for vacuum solutions of the form H u = (cid:18) h u + θ F u (cid:19) H d = (cid:18) h d + θ F d (cid:19) (13) M i = x i + θ F x i V a = θ ¯ θ D a W αa = θ α D a and take τ a = g a for the a = d, e, SU (2) components. In terms of thesevariables the scalar potential can be written as V = | (cid:126)F | + 12 g a (cid:126)D a . (14)Here the F-terms take the form¯ F u = µ x h d ¯ F d = µ x h u ¯ F x = − mx + Λ x x F x = µ h u h d − mx + Λ x x , (15)whereas the D terms are given by D SU (2) = g SU (2) | h d | − | h u | ) D d = g d (cid:0) −| h d | + | x | − | x | + ξ d (cid:1) D e = g e (cid:0) | h u | − | x | + | x | + ξ e (cid:1) . (16)In (16) we included a Fayet-Iliopoulos term for the U (1)’s. We will later dis-cuss under what circumstances SUSY and the electroweak gauge symmetryare broken. 7 upersymmetric solution Let us first consider the case in which SUSY is unbroken. Such solutions canbe found for ξ d + ξ e = 0 and are given by h u = h d = 0 x · x = (cid:18) Λ m (cid:19) | x | − | x | = ξ e . (17)For vanishing FI-terms ξ d = ξ e = 0 the solution takes the simple form h u = h d = 0 x = x = (cid:18) Λ m (cid:19) . (18)Note that these supersymmetric solutions do not break the electroweak gaugesymmetry. Non supersymmetric solution with gauge symmetry unbroken
In the following we analyze the effect of generic Fayet-Iliopoulos terms. Westart by looking for a vacuum in which the SU (2) gauge symmetry is un-broken, i.e. h u = h d = 0. In this case the equation of motion can be easilysolved by taking all the fields to be real and x = (cid:18) Λ m (cid:19) (cid:113) √ ∆ + 1 − ∆ x = (cid:18) Λ m (cid:19) (cid:112) √ ∆ + 1 − ∆ (19)with ∆ = m ( g d ξ d − g e ξ e ) ( g d + g e ) . (20)At the minimum F (cid:126)X = 0 and the potential takes the form V = 12 ( ξ d + ξ e ) g d g e g d + g e . (21)For ξ d + ξ e = 0 the scalar potential is vanishing indicating a SUSY solution.However for ξ d + ξ e (cid:54) = 0 SUSY is broken. It is instructive to look at the sim-plest example of a non-SUSY solution in this class ∆ = 0, i.e. for couplingssatisfying ξ d g d − ξ e g e = 0 . (22)8or this choice the linear terms in ξ d,e in the D-term scalar potential canceleach other at h u,d = 0 and one finds1 g d D d + 1 g e D e = 12 ( ξ d + ξ e ) g d g e g d + g e + 12 ( g e + g d ) (cid:0) | x | − | x | (cid:1) + O ( h )with an extremum at (17) . This is a minimum if the masses of h u,d at theextremum are positive. Expanding to second order in h u,d one finds V (cid:12)(cid:12) ext = 12 ( ξ d + ξ e ) g d g e g d + g e + | h u | ( µ | x | + g e ξ e ) (cid:12)(cid:12)(cid:12) ext + | h d | ( µ | x | − g d ξ d ) (cid:12)(cid:12)(cid:12) ext + O ( h ) . Taking FI terms positive, one finds a positive mass for h d ifΛ > m g d ξ d µ (23)We conclude that for ξ d + ξ e (cid:54) = 0 SUSY is broken if (23) is satisfied.Alternatively one can explicitly compute the eigenvalues of the Hessianmatrix and finds ∂ ij V = diag (cid:32) g d + g e ) (cid:18) Λ m (cid:19) , m ) , m , µ (cid:18) Λ m (cid:19) + 2 g d ξ d , µ (cid:18) Λ m (cid:19) − g d ξ d , , , (cid:33) . (24)The last three entries in this matrix are the goldstone bosons associated tothe breaking of the three U (1) symmetries (with the third U(1) coming fromthe Cartan of SU (2)). The remaining eigenvalues are then positive if (23) issatisfied.Let us display an explicit solution for a concrete choice of parameters g SU (2) = g d = g e = 0 . m = µ = ξ e = ξ d = 1 h u = h d = 0 x = x = 1 V = 0 . . Here the dimensionful quantities m , Λ as well as the vev’s of the fields aregiven in units of, let us say, TeV, whereas the FI-terms ξ d , ξ e are measuredin units of TeV and V in TeV . Notice that for ∆ = 0 the non-supersymmetric solution (19) reduces to (17). on supersymmetric solution with gauge symmetry broken Let us now turn to the second type of solutions in which not only SUSY isbroken but also the electroweak gauge symmetry is. For simplicity we restrictourselves to the case ξ d g d = ξ e g e (25)which breaks gauge and SUSY if0 < Λ < m g d ξ d µ . (26)It is hard to find analytic solutions but a numerical analysis is viable andshows that such configurations lead to a minimum with h u , h d (cid:54) = 0. A repre-sentative of the latter is given by g SU (2) = g d = g e = Λ = 0 . m = µ = ξ e = ξ d = 1 h u = 0 . h d = 0 . x = 0 . x = 0 . V = 0 . , where again the dimensionful quantities m , Λ , ξ d,e as well as the vev’s ofthe fields are given in the same units of our previous solutions. In thissolution, the Higgs fields acquire a non-zero vev triggering the breaking ofthe electroweak gauge symmetry. Now we allow for an additional field Y . Thus we consider the following N = 1chiral field content in the Higgs-meson sector (cid:126)X = { H u , H d , M , M , Y } . (27)In Table 1 we display the charges of the various fields . The superpotentialcontaining the fields (cid:126)X and obeying the gauge symmetries is now given by W = µ M H u H d + µ Y Y H u H d + m M M + m Y Y M + Λ M M , (28)where the last term is due to the non-perturbative ADS superpotential gen-erated in the hidden SU (3) H gauge theory.10n order to preserve SU (3) C and the electromagnetic gauge symmetry weagain look for solutions of the type (13). In addition we parametrize the field Y as follows Y = y + θ F y . Then the F-terms take the form¯ F u,d = h d,u ( µx + µ Y y ) ¯ F Y = µ Y h u h d − m Y x ¯ F x = µ h u h d − mx + Λ x x ¯ F x = − mx − m Y y + Λ x x (29)and the D-terms are given by D SU (2) = g SU (2) (cid:16) | h d | − | h u | (cid:17) D d = g d (cid:0) −| h d | + | x | − | x | + | y | + ξ d (cid:1) D e = g e (cid:0) | h u | − | x | + | x | − | y | + ξ e (cid:1) . (30)In the following we analyze for which values of the parameters, SUSY andthe SU (2) L × U (1) Y gauge symmetry are broken. Supersymmetric solution
Before discussing the broken phase let us again first look for supersymmetricsolutions. Such a solution exists if the parameters satisfy µ Y (cid:54) = 0 and ∆ = mµ Y − µm Y (cid:54) = 0 ξ d = ξ e = 0 . (31)The supersymmetric solution can in that case be written as y = − µx µ Y h u = h d = (cid:18) Λ m Y x µ Y ∆ (cid:19) x = − (cid:18) Λ µ Y x ∆ (cid:19) (32)with x determined by the equation | x | − | x | + | y | = 0 (33)evaluated at (32). Let us point out that for these SUSY solutions the vev’sof the Higgs are non-vanishing and the SU (2) L × U (1) Y symmetry is broken. Non-supersymmetric vacua
Once again switching on Fayet-Iliopoulos terms gives SUSY and electroweakgauge symmetry breaking terms for quite general choices of the parameters.11n analytic solution for the non SUSY minimum is hard to find but theequations can be easily solved numerically for any choice of the gauge andYukawa couplings. Let us display one example for each type of solution, onefor unbroken electroweak gauge symmetry and one for the broken electroweakgauge symmetry. • Gauge symmetry unbroken: g SU (2) = g d = g e = 0 . m = µ = m y = µ y = ξ e = ξ d = 1 h u = h d = 0 x = 0 . x = 1 . y = 0 . V = 1 . • Gauge symmetry broken g SU (2) = g d = g e = 0 . m = µ = m y = − µ y = ξ e = ξ d = 1 h u = 0 . h d = 1 . x = y = 0 . x = 0 . V = 0 . m , m Y , Λ as well as the vevs of the vacuumsolution are measured in units of TeV and the FI-terms are measured in unitsof TeV .The vev’s of h u and h d are different from zero only for the second solutionwhich is thus breaking the electroweak gauge symmetry. In Section 5 we will present some D-brane quivers which mimic the configura-tions discussed above. Before presenting these quivers let us briefly discuss,following [24] (see also [25, 26]) , the various constraints on the transforma-tion properties of the chiral matter fields which arise from string theory andthat not always have an analogue in the field theory context.For concreteness we focus on type IIA string theory in which the basicbuilding blocks are D6-branes which fill the four dimensional space-time andwrap a three-cycle in the internal compactification manifold. The gauge sym-metry living on the worldvolume of a stack of N D6-branes transforms undera U ( N ) group. Chiral matter appears at the intersection between two stacks For analogous work see [27, 28, 29, 30, 31, 32, 33, 34, 35]. First local (bottom-up)constructions were discussed in [36, 37, 38]. n ( R ) a ( π a ◦ π (cid:48) a + π a ◦ π O6 ) a ( π a ◦ π (cid:48) a − π a ◦ π O6 )( a , b ) π a ◦ π b ( a , b ) π a ◦ π (cid:48) b Table 2: Chiral spectrum for intersecting D6-branes.of D6 branes, a and b , and transforms as a bifundamental representationunder U ( N a ) × U ( N b ).In order to obtain N = 1 in the four-dimensional spacetime one intro-duces an orientifold action, which implies the presence of O U ( N a ).The multiplicities of the chiral matter fields are given in terms of the inter-section numbers of the three-cycle π a , π (cid:48) a and π O , where π a is the three-cyclewrapped by the stack a , π (cid:48) a its orientifold image and π O is the whole classof orientifold invariant three-cycles in the internal manifold.The field content of an intersecting brane model is determined by theintersection numbers according to Table 2. Here we denote by ∆ n ( R ) thenet number of chiral fields in a representation R , i.e.∆ n ( R ) = n ( R ) − n ( ¯R ) . (34)Note that although the antisymmetric representation of U (1) does not existthe corresponding intersection number in the Table 2 may be not vanishing.In the following, we will refer to this intersection number as ∆ n ( a ) evenfor U (1) a gauge groups where a does not exist. This intersection numberwill enter in the consistency conditions constraining the string model. Thisnotation will then allow us to write the various string consistency conditionsin a unifying compact way independently of the gauge group rank.Even in a local set up the tadpole cancellation condition constrains thespectrum of the string model. While for non-abelian gauge groups theseconstraints boil down to the usual anomaly cancellation conditions in fieldtheory, in presence of U (1) symmetries they further constraint the spectrum13f U (1) charges in the string model.Generically, anomalous U (1) acquire a mass via the Green-Schwarz mech-anism of anomaly cancellation. Non-anomalous U (1) gauge bosons can alsobecome massive via non-trivial Chern-Simons (CS) couplings with RR fields.The massive U (1)’s are generically not part of the low energy effective gaugesymmetry but remain as unbroken global symmetries at the perturbativelevel and thus may forbid various desired couplings. Since the standardmodel gauge symmetry contains the abelian subgroup U (1) Y , we requirethat a linear combination U (1) Y = (cid:88) x q x U (1) x , (35)remains massless. This happens if the CS coupling (cid:82) D C ∧ F U (1) vanishesfor all the D6 branes. The condition for the presence of a massless U (1) Y translates then into a condition on the cycles wrapped by the D-branes, whichtogether with the tadpole cancellation condition constrains the charges of thechiral matter field content. In the following we will discuss both constraintsin more detail. Tadpole condition
The tadpole condition is a constraint on the cycles wrapped by the D-branes.In the framework of intersecting D6-brane models the tadpole conditions read (cid:88) x N x ( π x + π (cid:48) x ) − π O = 0 . Here x denotes the different D-brane stacks present in the model. Thiscondition ensures that the total RR charge carried by the D6-branes (andtheir images) exactly cancels that of the O6 planes. Multiplying this equationwith the homology class of the cycle that is wrapped by a stack a gives, aftera few manipulations and using the relations displayed in Table 2∆ n ( a ) + ( N a − n ( a ) + ( N a + 4)∆ n ( a ) = 0 , (36)where ∆ n ( a ) = (cid:88) x N x (cid:104) ∆ n ( a , x ) + ∆ n ( a , x ) (cid:105) (37)14s the total number of U ( N a ) fundamentals (minus antifundamentals). Equa-tion (36) is nothing else than the anomaly cancellation for non-abelian gaugetheories for SU ( N a ) gauge groups with rank N a >
2. We stress that also for N a = 1 , Massless U(1)’s
Since the SM contains the U (1) Y hypercharge as a gauge symmetry we requirethe presence of a massless U (1) which can be identified with U (1) Y . Only forspecific choices of the coefficients in (35) the matter particles have the properhypercharge. In addition, once the coefficients q x are given, the correspondinglinear combination of the U (1) (cid:48) s remains massless if the condition [39] (cid:88) x q x N x ( π x − π (cid:48) x ) = 0 (38)is satisfied. This condition ensures that the CS coupling (cid:82) C ∧ F U (1) Y cancels.Analogously to the analysis performed for the tadpole constraints wemultiply (38) with the homology class π a of the cycle wrapped by the D-brane stack a . After reinterpreting the intersection numbers in terms of themultiplicities according to the Table 2 we obtain (cid:88) x (cid:54) = a q x N x [∆ n ( a , x ) − ∆ n ( a , x )] = q a N a (cid:16) ∆ n ( a ) + ∆ n ( a ) (cid:17) . (39)We remark that this equation gives a constraint for every D-brane stackpresent in the model. Thus for the six-stack model (which we will considerlater) we expect six additional constraints due to the presence of a masslesshypercharge. Now we have all the ingredients to engineer some consistent string modelsbased on intersecting D6 branes and O6 planes for which the field contentcontains the MSSM particles and the dynamical SUSY breaking Higgs-mesonsector of the types we discussed in Section 3. We present string realizationsbased on six stacks of D6-branes which lead to the gauge symmetry G = U (3) a × SU (2) b × U (1) c × U (1) d × U (1) e × U (3) f . (40)15he stack b is on top of an O6 plane, i.e. b = b (cid:48) , thus leading to a Sp (1) ∼ SU (2) gauge group. The hypercharge is of the form [40] U (1) Y = (cid:88) q x U (1) x = 16 U (1) a + 12 [ U (1) c + U (1) d + U (1) e + U (1) f ] . (41)We consider models with one and three generations of MSSM particles and aHiggs-meson sector of the form discussed in Section 3 which satisfies the tad-pole and massless U (1) Y conditions. The Higgs-meson superpotential is ofthe types (12) and (28), respectively. As explained in Section 3 these configu-rations break SUSY and the electroweak gauge symmetry after condensationof the hidden SU (3) H gauge theory. The D-brane quivers have the sameYukawa interactions of the MSSM and satisfy some basic phenomenologicalrequirements. The latter include the absence of R-parity violating couplingsand of some dimension five operators which could lead to a disastrous shortproton lifetime.The field content of the discussed models is summarized in Table 3. Here A = a, b, c, . . . denotes the various brane stacks. Their images with respectto the O6 orientifold plane are denoted by A (cid:48) = a (cid:48) , b (cid:48) , c (cid:48) . . . . The first columnof the table displays the string origin of each state with AB labelling an openstring originating from the brane stack A and ending on the brane stack B .Different models will be distinguished by different choices of the multiplicities n ’s in the last column of the table.The field content of the various models can be alternatively summarizedby gauge quiver diagrams. In the latter, each node stands for a D branestack carrying a definite gauge groups and each arrow for an open stringconnecting two stacks and thus giving a chiral matter multiplet. Two arrowsconnecting two different brane stacks stand for a chiral matter multiplettransforming in the bifundamental representation of the gauge groups, whilethose arrows connecting a brane stack and its image account for chiral mattermultiplet transforming in the symmetric, or antisymmetric, representation ofthe gauge groups. For simplicity we will present only string models with nosymmetric or antisymmetric matter but more general choices with a similarphenomenology are allowed.The tadpole condition translates into the requirement that the numberof arrows arriving at a node is equal to the number of arrows leaving it. Toperform this counting, the flavour multiplicity of each arrow must be keptinto account. For an arrow entering a certain node, its flavour group is givenby the number of branes in the stack at the opposite end of the arrow. Finally16ector matter U (3) a SU (2) b U (1) c U (1) d U (1) e U (1) Y numberab Q L n e’a u c ¯ − − n u d’a u c ¯ − − n − n u da d c ¯ n d ea d c ¯ n − n d bc L − − n cd’ e c n − n e ce’ e c n e ce ν c − n − n ν cd ν c − n ν bd H d − − H u Y − n Y e’ffd’ M − M − Models with SUSY breaking. The multiplicities in the last column de-pend on the specific model. n is the number of generations and n Y = 1 , n, n u , n d , n e , n ν ) are(1 , , , ,
1) or (3 , , , ,
2) for the models without the Y field and (1 , , , ,
0) or(3 , , , ,
3) for the models with a Y field and one or three generations respectively. image nodes contribute with an opposite sign while an extra ± U (1) Y -condition (38) leads to the same countingweighted by the U (1) Y charge q x of the flavor node x . In this case there isno extra contributions from the arrows crossing the O6 plane since the O6plane does not contribute to the CS coupling. In Figures 1 and 2 we display a one and three generation configuration whichexhibits a SUSY breaking superpotential for the Higgs-meson sector of thetype (12) discussed in Section 3.1. 17 ne generation quiver
Let us start by analysing a one generation quiver which mimics the SUSYbreaking configuration discussed in Section 3.1. The choice of the multiplicity Q Q ~ LQu dHHQ Q ~ u d e ν a d’ f e’ c b d e Figure 1:
One generation quiver leading to SUSY breaking. This diagram issymmetric with respect to the orientifold O6 plane located on the b stack of branes.Certain mirror nodes are omitted for the sake of simplicity. numbers in Table 3 is n = 1 n u = 1 n d = 1 n e = 1 n ν = 1 n Y = 0 , which satisfies the constraints arising from tadpole cancellation as well as themasslessness of the hypercharge U (1) Y discussed in Section 4. That choiceleads for the MSSM spectrum to Q L = ( a, b ) u c = (¯ a, ¯ e ) d c = (¯ a, d ) L = ( b, ¯ c ) e c = ( c, e ) ν c = ( c, ¯ d )and for the Higgs-meson sector to H u = ( b, e ) H d = ( b, ¯ d ) M = ( d, ¯ e ) M = ( ¯ d, e ) . (42)Here the mesons arise after condensation and will be given in terms of Q and (cid:101) Q M = (cid:101) Q Q and M = (cid:101) Q Q . (43)18he perturbative superpotential is given by all closed loops in the quiverdiagram, where one can jump from a node x to its orientifold image changingthe orientation of the loop. Let us perform the analysis concretely for thesuperpotential in the Higgs-meson sector, which is given by W = µ M H u H d + m M M + Λ M M . (44)From Figure 1 one can easily see that the term M M indeed represents aclosed loop (from e’ to d’ and back) in the quiver diagram. The superpotentialterm M H u H d requires a little bit more work. Let us start at node b andgo to node e (cid:48) , which describes the H u field. From node e (cid:48) we go via node f to node d (cid:48) and pick up the meson M . To close the loop we jump to thenode which is the orientifold image of d (cid:48) , namely node d but keeping in mindthat such a jump implies reversing the orientation of the loop. Then onecan close the loop with the inclusion of H d . The last term in (44) is thenon-perturbative ADS superpotential term . Note that the superpotentialin the Higgs-meson sector is of the type (12) discussed in Section 3.1. Therewe showed that for a particular choice in parameter space the vacuum breaksSUSY as well as the electroweak gauge symmetry.Analogously to the Higgs-meson sector, one can determine the superpo-tential containing the chiral MSSM superfields. Again that corresponds tofinding all closed loops involving two quark and lepton fields in the quiverdiagram 1. One obtains for the quiver potential W = Q L H u u c + Q L H d d c + 1Λ M LH d e c + 1Λ M LH u ν c . (45)For simplicity we omit dimensionless couplings. This superpotential gener-ates mass terms for quark and leptons. Interestingly, the origin of masses forquark and leptons in (45) are very different. Masses for the quarks arise fromthe familiar Yukawa couplings with the MSSM Higgs fields while those forthe leptons arise from the quartic couplings involving a meson field. Moreprecisely, at a SUSY breaking vacuum of the type we discussed above thissuperpotential generates masses for the quarks of order h u,d and for the lep-tons of order x h u,d . In addition soft SUSY breaking masses for the sparticlesfollow from non-trivial vevs of the F-fields (cid:126)F X . For a recent derivation of the ADS superpotential in the context of intersecting D6-branes see [41]. hree generation quiver The three generation quiver is displayed in Figure 2 and given by the multi-plicity choice n = 3 n u = 2 n d = 2 n e = 2 n ν = 2 n Y = 0which gives the MSSM spectrum Q L = 3 × ( a, b ) u c , = 2 × (¯ a, ¯ e ) u c = 1 × (¯ a, ¯ d ) d c , = 2 × (¯ a, d ) d c = 1 × (¯ a, e ) L = 3 × ( b, ¯ c ) e c = 1 × ( c, d ) e c , = 2 × ( c, e ) ν c = 1 × ( c, ¯ e ) ν c , = 2 × ( c, ¯ d ) . Let us stress that the choice of multiplicities do pass all the string consistencyconstraints derived in Section 4. The Higgs-meson sector is the same as forthe one generation configuration, see equation (42). Thus it exhibits the samesuperpotential (44) as for the one generation quiver and therefore as shownin Section 3.1 leads to simultaneous SUSY and electroweak gauge symmetrybreaking. Q ~ Q QQ Q ~ uu dee dL HH u d ν ν a d’ f e’ b d e c Figure 2:
Three generation quiver leading to SUSY breaking.
Finding all closed loops containing exactly two fields (cid:126)X gives the MSSMsuperpotential W = Q L H u u c , + Q L H d d c , + L H d e c + L H u ν c (46)+ M Λ (cid:16) Q L H u u c + Q L H d d c + L H d e c , + L H u ν c , (cid:17) . M acquire non-zero vev’s this superpotential givesa mass to all the SM fields. In (46) the Yukawa couplings for different familieshave different coefficients and this might be of use to account for the observedmass hierarchies in the standard model. Note also that this quiver doesnot contain any R-parity violating couplings or dangerous dimension fiveoperators.We remark that the perturbative presence of a Dirac neutrino mass term(of the order of the SM mass scale), in conjunction with the presence ofa large Majorana mass term for the right-handed neutrinos, induced by aD-instanton, might lead to a neutrino mass of the right size via the seesawmechanism [42, 43, 44, 45, 46, 47, 48, 49].Finally let us comment on the mass of U (1) d and U (1) e . Only their sumsatisfies the constraints for a massless U (1). That implies that both U (1) d and U (1) e become massive via the CS-coupling. This mass crucially dependson the details of the compactification (string coupling, string mass, volumeof the cycles the D-branes wrap as well as on the gauge flux living on them,etc.) [50, 51, 52, 53]. Here we assume that both masses are below the scaleΛ, such that at the energy scale Λ one has to treat them as an abelian gaugesymmetry. Figures 3 and 4 display the quiver diagrams for a one and three generationconfiguration, which exhibit a superpotential of the type (28). Let us startagain with the one generation quiver.
One generation quiver
For the one generation model the multiplicity numbers in Table 3 are chosen n = 1 n u = 1 n d = 0 n e = 0 n ν = 0 n Y = 1and lead to the MSSM spectrum Q L = ( a, b ) u c = (¯ a, ¯ e ) d c = (¯ a, e ) L = ( b, ¯ c ) e c = ( c, d ) ν c = ( c, ¯ e )and to the Higgs-meson spectrum H u = ( b, e ) H d = ( b, ¯ d ) Y = ( d, ¯ e ) M = ( d, ¯ e ) M = ( ¯ d, e ) (47)21 Q ~ HHQ Q ~ u d u dLe Q ν Y a d’ f e’ b c d e Figure 3:
One generation quiver with additional an Y leading to SUSY breaking. where again M and M denote the mesons after the condensation of SU (3) H .The superpotential is given by all possible loops in the quiver and takes,in the Higgs-meson sector, the form W = µ M H u H d + µ Y Y H u H d + m M M + m Y Y M + Λ M M . (48)This is exactly the superpotential analysed in Section 3.2, where it has beenshown that for some choice of parameters there exist a SUSY and electroweakgauge symmetry breaking vacuum.One moreover obtains the desired MSSM Yukawa couplings W = Q L H u u c + 1Λ Q L H d (cid:16) M + Y (cid:17) d c + L H d e c + L H u ν c (49)which gives masses to all the MSSM matter fields after H u , H d , M and Y acquire a non-zero vev. Three-generation quiver
An extension of the above discussed configuration to three families is givenby the following choice for the multiplicities in Table 3 n = 3 n u = 2 n d = 2 n e = 2 n ν = 3 n Y = 1 , U (1) Y . With that choiceone obtains the MSSM spectrum Q L = 3 × ( a, b ) u c , = 2 × (¯ a, ¯ e ) u c = 1 × (¯ a, ¯ d ) d c , = 2 × (¯ a, d ) d c = 1 × (¯ a, e ) L = 3 × ( b, ¯ c ) (50) e c = 1 × ( c, d ) e c , = 2 × ( c, e ) ν c , , = 3 × ( c, ¯ d ) , while the Higgs-meson sector is the same as for the one generation configu-ration, see equation (47).The corresponding quiver is displayed in Figure 4 and the superpotentialfor the Higgs-meson sector is again given by (48). Thus as shown in Section3.2 there exists a vacuum which breaks SUSY and also SU (2) × U (1) Y gaugesymmetry. Q ~ Q Q HHQ Q ~ u d uu de dLe ν Y a d’ f e’ b d e c Figure 4:
Three generation quiver with additional an Y leading to SUSY breaking. The MSSM superpotential is given by W = Q L H u u c , + Q L H d d c , + L H d e c (51)+ 1Λ (cid:16) M + Y (cid:17)(cid:16) Q L H u u c + Q L H d d c + L H d e c , + L H u ν c , , (cid:17) . After the Higgs-meson fields acquire a vev all SM fields get a mass. As in theprevious example, the fact that one of the families of the right-handed quarkshas a different string origin with respect to the other two generations may23ccount for the observed mass hierarchies in the MSSM. As before the quiverdoes not contain any R-parity violating couplings or dangerous dimension fiveoperators. Moreover, D-instantons can generate large Majorana masses forthe right-handed neutrinos. Then the small neutrino masses can be explainedvia the seesaw mechanism.As before we assume that the masses of the U (1) d and U (1) e inducedby the CS-couplings are below the scale Λ. That forces us to treat thesymmetries U (1) d and U (1) e as gauge symmetries at the energy scale Λ. We discussed two extensions of the MSSM which lead, for a large region inparameter space, to SUSY and electroweak gauge symmetry breaking. Boththese extensions contain a hidden SU (3) H that condensates via the genera-tion of an ADS superpotential. The condensates (mesons) couple to the Higgssector which mediates the SUSY and electroweak gauge symmetry breakingto MSSM matter content. In Section 3 we show explicitly that, dependingon the region in parameter space, these extensions can give rise to differentvacua, that do or do not break SUSY and/or electroweak gauge symmetry.There are large region in parameter space for which both symmetries arebroken.Later in Section 5 we present local D-brane configurations which satisfysevere string consistency constraints and mimic the previously discussed fieldtheory setups. They exhibit the required superpotential to break SUSY andelectroweak gauge symmetry. Moreover, all MSSM Yukawa couplings arerealized and these configurations can naturally explain some of the observedmass hierarchies of the MSSM.In this work we assumed that the closed string sector is stabilized thus weignore all closed string dynamics. Specifically all Yukawa, gauge couplingsand Fayet-Iliopoulos terms are input parameters. It would be nice to find aglobal realization of these local D-brane configurations in which one can studywhether moduli stabilization indeed give the Yukawa and gauge couplings,as well as Fayet-Iliopoulos terms in a range that eventually leads to SUSYand gauge symmetry breaking. 24 cknowledgments The authors wish to thank P. Anastasopoulos, M. Bianchi, E. Dudas andM. Serone for useful discussions. This work was partially supported by theERC Advanced Grant n.226455 “Superfields” , by the Italian MIUR-PRINcontract 20075ATT78 and by the NATO grant PST.CLG.978785.
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