Stringy Hidden Valleys
We study gauge theories where quasi-hidden sectors are added to the MSSM for the sake of string consistency conditions which would otherwise not be satisfied. We focus on quiver gauge theories motivated by weakly coupled type II orientifold compactifications. Model independent features in this class include an anomalous symmetry which protects messenger masses and has strong consequences for superpotential couplings, a rich phenomenology of heavy and light bosons, and axionic couplings required for anomaly cancellation via the Green-Schwarz mechanism. We discuss possibilities for dark matter and supersymmetry breaking in light of these generic features. Dark matter is necessarily non-baryonic, though many dark matter candidates have weak interactions. Most models have a anomaly whose cancellation requires couplings which allow for dark matter annihilation into photons through intermediate axions or anomalous bosons, as in two recently proposed scenarios. There is often an additional non-anomalous symmetry which can give rise to a Fayet-like model of metastable supersymmetry breaking. Breaking of supersymmetry via SQCD can also be realized and flavor masses are often protected. Natural possibilities for mediation include gauge mediation, mediation, and D-instanton mediation, though it is not possible to realize minimal gauge mediation with messengers added for string consistency.
University of Pennsylvania, Philadelphia, PA 19104-6396, USA ††institutetext: Center for Applied Mathematics and Theoretical Physics,
University of Maribor, Maribor, Slovenia ††institutetext: Kavli Institute for Theoretical Physics,
University of California, Santa Barbara, CA 93106-4030, USA
With new experimental data arriving from the Large Hadron Collider and other experiments, recent model building efforts include the study of quasi-hidden gauge sectors which could be discovered in the near future. Such sectors are well motivated from top-down constructions, and it is possible that they interact weakly, but non-trivially, with the standard model. These include hidden valleys Strassler:2006im (), Higgs portals (for example Batell:2011pz ()), dark photons Jaeckel:2010ni (), and a variety of dark matter models Cheung:2010gj (). Hidden valleys, in particular, are strongly motivated by works on string compactifications, such as Cleaver:1998sm (); Cleaver:1998gc (). However, the top-down input into these broad classes of effective field theories is that hidden extensions of the standard model frequently exist, and therefore they should be considered. It would be better, when possible, to have a guiding principle from top-down considerations which lead to precise gauge theoretic structures.
Gauge theories arising in string compactifications are constrained by consistency conditions. For example, the ten dimensional type I superstring is anomaly free only for gauge group . Global consistency conditions can also require the presence of hidden sectors. In the heterotic string, standard model sectors are typically realized in one factor, while the other factor can give rise to rich hidden sector physics which depend on consistency conditions on a holomorphic vector bundles. In type II compactifications, similar constraints are placed on the ranks of gauge groups by Ramond-Ramond tadpole cancellation. These constraints can descend to constraints on the chiral spectrum which are necessary for string consistency. The constraints on the chiral matter spectrum are not always equivalent to the cancellation of four-dimensional gauge anomalies. On one hand, string theory provides mechanisms for the cancellation of certain anomalies, and therefore does not place constraints on the chiral matter spectrum which ensure their absence. On the other hand, there exist constraints on the chiral matter spectrum of a gauge theory which are necessary for string consistency, but for which there is no known field theoretic analog. We refer to such constraints as “stringy” constraints.
It is possible to enumerate all realizations of the MSSM in certain classes of gauge theories motivated by string compactification. Many of these theories do not satisfy conditions necessary for string consistency, and therefore some augmentation is required. We focus exclusively on the possibility of adding hidden sectors, where messenger fields are chosen in a way that remedies visible sector inconsistencies. Doing so can lead to specific gauge theoretic structures which are generic across a broad class of models and have important consequences for low energy physics. Extensions of the MSSM with hidden sectors added for consistency form a proper subset of the broader class of hidden valley models. For brevity, will refer to a model in this class as a “stringy hidden valley.”
In this paper we study a specific class of stringy hidden valleys. In a class of quiver gauge theories motivated by weakly coupled type II orientifold compactifications and their duals, nearly all MSSM quivers do not satisfy the constraints necessary for Ramond-Ramond tadpole cancellation. The violation of these constraints is suggestive of matter fields that should be added to the theory, and we study the possibility that these fields are messengers to a hidden sector. We will find that this class of theories generically has messenger masses protected by symmetry, a rich structure of superpotential couplings, axionic couplings required for anomaly cancellation, and bosons. These generic features give rise to interesting possibilities for supersymmetry breaking and dark matter, including annihilation processes with photons in the final state. We emphasize that these features are non-generic within the broad class of hidden valley models, but are generic in the class we study. We also emphasize that our results do not depend strongly on the number of nodes in the visible sector or possible flux contributions to tadpole cancellation conditions, as we will discuss. For brevity throughout, we will refer to these particular models as stringy hidden valleys, keeping in mind that they are a subset of stringy hidden valleys motivated by type II orientifold compactifications. See figure 1.
Let us briefly describe the broader class of theories. Quiver gauge theories arising in weakly coupled type II orientifold compactifications111Note that the quivers we consider differ slightly from those arising from D3-branes at singularities. There, the singularity structure determines the quiver. Here our approach is from the bottom-up: we build MSSM quivers and hidden sector extensions using the ingredients of type II string theory, independent of any global embedding or realization at a particular singularity. For D3-brane quivers, see for example Aldazabal:2000sa (); Verlinde:2005jr (); Krippendorf:2010hj () and references therein. and their duals have generic properties which affect low energy physics, including the presence of anomalous symmetries which can forbid superpotential couplings in perturbation theory. These terms can be generated with suppression via D-instanton effects Blumenhagen:2006xt (); Ibanez:2006da (); Florea:2006si () or via couplings to singlets. Cancellation of anomalies222This includes cubic abelian, mixed abelian non-abelian, and mixed abelian-gravitational anomalies. via the Green-Schwarz mechanism requires the presence of Chern-Simons couplings of the form and , where the former gives a large Stückelberg mass to the associated gauge boson. Consistent quivers must also satisfy conditions necessary for Ramond-Ramond tadpole cancellation. If they are to realize the MSSM, quivers must satisfy constraints necessary for the hypercharge to remain massless.
There has been much phenomenologically motivated work studying type II quiver gauge theories without hidden sectors. These studies have been both systematic and example-driven. For example, it has been shown Ibanez:2008my (); Anastasopoulos:2009mr (); Cvetic:2009yh (); Cvetic:2009ez (); Cvetic:2009ng (); Anastasopoulos:2009nk () that D-instantons in MSSM quivers can give rise to realistic Yukawa couplings and neutrino masses while ensuring the absence of R-parity violating operators and dimension five proton decay operators. Alternative mechanisms for realistic neutrino masses have been proposed, including the generation of a realistic Dirac neutrino mass term Cvetic:2008hi () and a Weinberg operator Cvetic:2010mm () by D-instantons. Other issues which have been studied include singlet-extended standard models Cvetic:2010dz (), dynamical supersymmetry breaking Fucito:2010dk (), grand unification Kiritsis:2009sf (); Anastasopoulos:2010hu (), and physics Berenstein:2006pk (); Cvetic:2011iq (); Anchordoqui:2011eg (). Systematics of hypercharge embeddings, bottom-up configurations, and global rational conformal field theory realizations have been carried out in Anastasopoulos:2006da ().
This paper is very similar in spirit to Cvetic:2011iq (), which also utilized string consistency conditions as the guiding principle for physics beyond the standard model. Since it is so closely related to this work, let us briefly review the conclusions. In Cvetic:2011iq (), exotic matter was added to inconsistent type II MSSM quivers for the sake of consistency and in the most general way allowed by the construction, without adding additional quiver nodes. It was shown at the level of standard model representations that some exotics are much more likely than others, with a clear preference for MSSM singlets, triplets without hypercharge, and quasichiral pairs333Quasichiral pairs are vector with respect to the standard model but chiral with respect to some other symmetry in the theory. In this case the symmetry is an anomalous .. All of these possibilities could be relevant for LHC physics, as mass terms for the quasichiral pairs are forbidden by an anomalous symmetry but could be generated at the TeV scale by string instantons or couplings to singlets. We also refer the reader to Cvetic:2011iq () for a more in depth discussion of the constraints we will import on quivers. In this paper, we will study the same class of MSSM quivers but will add hidden sectors for the sake of consistency.
This paper is organized as follows. In section 2 we review the structure of the quiver gauge theories we study, including string consistency conditions, and also the classification of three-node MSSM quivers. We introduce the possibility of adding hidden sectors only for the sake of string consistency and set notation used throughout. In section 3 we show that hidden sectors added for consistency have generic features which affect low energy physics, including an important symmetry , axionic couplings, and bosons. These have important implications for various dark matter and supersymmetry breaking scenarios, which we discuss in detail. In section 4 we present explicit quivers realizing the general ideas of section 3. In section 5 we conclude, briefly stating the main results and discussing possibilities for future work.
2 Basic Setup and Guiding Principles
In this section we will introduce a class of quiver gauge theories, emphasizing how they can arise in weakly coupled type II string compactifications. We discuss constraints on their chiral matter spectrum which are necessary for string consistency and introduce a guiding principle for adding hidden sectors. We will review the classification of three-node MSSM quivers. We will also discuss the applicability of our results to extensions of higher-node MSSM quivers.
2.1 Symmetries, Spectrum, and String Consistency
Let us introduce the class of theories we consider444For a recent in-depth discussion of this class of theories, see Cvetic:2011vz ()., beginning with a discussion of symmetries. A compactification of weakly coupled type II string theory realizes gauge degrees of freedom as open strings ending on D-branes. For specificity, consider type IIa. A D6-brane wrapped on a generic three-cycle exhibits gauge symmetry, though or gauge symmetry can exist if happens to be an orientifold invariant cycle. For generality and simplicity, we consider all gauge factors to be . In the presence of O6-planes, every D6-brane has an associated orientifold-image brane, which will be important for the spectrum. We represent a brane together with its image brane as a node in a quiver, labeled by . The trace of a factor is often anomalous, and the associated abelian and mixed anomalies are automatically canceled in globally consistent string compactifications by four-dimensional terms coming from the dimensional reduction of the Chern-Simons D-brane action. In a bottom-up gauge theory, the necessary Chern-Simons terms can be added by hand (see Anastasopoulos:2006cz (), e.g.). In particular, anomaly cancellation requires terms of the form , which gives a string scale Stückelberg mass to the with associated field strength . Though these degrees of freedom can be integrated out at scales well below the associated mass, the massive ’s impose global selection rules on the low energy effective action, forbidding many terms in the superpotential. We will realize the standard model gauge group via unitary factors . We require the one linear combination is left massless555Throughout, we will use the phrases “massless U(1)” or “light ” to describe a symmetry which is not required to obtain a Stückelberg mass. The terms “massive U(1)” or “heavy ” will be used to denote a symmetry which does not satisfy the constraints (2.1) and therefore obtains a string scale Stückelberg mass. and can be identified as hypercharge. Such a combination is referred to as a hypercharge embedding. In addition, it is possible that another linear combination is left massless and is associated to a light which must obtain a mass via the standard Higgs mechanism.
Let us discuss the spectrum. Chiral matter is localized at the intersection of two D6-branes and the chiral index is given by the topological intersection number of two three-cycles. Given a D6-brane on , another D6-brane on and its image on , strings localized at intersections of with carry representations or under and strings localized at the intersections of and carry representations or . The fundamental and antifundamental representations carry charge under the trace ’s, respectively. Therefore the branes can realize all four combinations of bifundamental representations, which we present as a bidirectional edge between the node and the node. String zero modes localized at the intersection of with come in symmetric or antisymmetric tensor representations under and carry charge under the trace associated to . We label these representations as an arrow from the node to itself with a or to denote the sign of the charge and or to denote symmetric or antisymmetric. See figure 2 for an example. For a given set of nodes and hypercharge embedding, it is straightforward to enumerate all possible realizations of the MSSM spectrum.
We require that the chiral matter spectrum of the quiver satisfies two sets of conditions. The first set are those necessary for tadpole cancellation666Tadpole cancellation is the requirement that the net Ramond-Ramond (D-brane) charge is canceled on the internal space. It is necessary for the consistency of a globally defined string compactification. For example, in type IIa tadpole cancellation places a homological constraint on three-cycles wrapped by D6-branes which can be shown to descend to the weaker conditions (2.1) on chiral matter. These conditions are necessary but not sufficient for D6-brane tadpole cancellation. , which are
for each node, where we have denoted the fundamental and antifundamental by and . A defined by an arbitrary linear combination will obtain a string scale Stückelberg mass unless the masslessness conditions
are satisfied. The second set of conditions we impose is that the hypercharge embedding satisfies these masslessness conditions. In a given quiver, there may also be other linear combinations which satisfy these equations, giving light bosons.
Let us define some terminology that we will use throughout to discuss these conditions. For a gauge node, we refer to the lefthand side of the conditions necessary for tadpole cancellation and a massless hypercharge boson as the “T-charge” and the “M-charge” . In the three-node MSSM quivers we consider with , we will ensure that the T-charges and satisfy (2.1) and the M-charges and satisfy the equations (2.1). In addition, we may refer to the contributions of certain sets of fields to some T-charge or M-charge, where the context will make the content clear. For example, could be the contribution of messenger fields to .
The equations necessary for tadpole cancellation for are the conditions necessary for the cancellation of cubic anomalies. It is crucial that consistent chiral spectra do not give rise to these anomalies, since the Green-Schwarz mechanism cannot cancel them. The corresponding field theory constraints do not exist for or , however, and we refer to these as “stringy” constraints. We refer the reader to Cvetic:2011iq () for a recent in-depth discussion of these constraints and field theoretic constraints. These constraints are often violated for MSSM quivers. Our guiding principle will be to add hidden sectors so that they are satisfied.
2.2 Classifying Three-node MSSM Quivers
Our results regarding stringy hidden valleys will apply to essentially any MSSM quiver with non-zero T-charge which is canceled by the non-zero T-charge of messengers to a hidden sector. However, three-node MSSM quivers and their extensions provide an excellent example. Let us review their classification.
Consider a quiver with gauge symmetry, which is the minimal number of nodes that can realize the MSSM gauge group and chiral spectrum at low energies. of the standard model arise from the and factors and hypercharge must arise as a linear combination
of the trace ’s. There are only two possible sets which can realize the entire MSSM spectrum utilizing bifundamental, symmetric, and antisymmetric tensor representations. The first is the well-known Madrid embedding Ibanez:2001nd (), given by
and possible MSSM field representations given by777To avoid unnecessary notation throughout, we make the definitions , , , and .
where the unbarred and barred letters represent the fundamental and antifundamental representations of the associated nodes. MSSM singlets can be realized as or , and for this embedding we define the chiral excess of singlets to be . Lacking a better name, the other hypercharge embedding is the non-Madrid embedding, given by
and the possible MSSM field representations are given by
MSSM singlets can be realized as or , and for this embedding we define the chiral excess of singlets to be . Depending on the coupling of these singlets to MSSM fields, they could be right-handed neutrinos or singlets which give rise to a dynamical -term. See Cvetic:2010dz () for singlet-extended MSSM quivers in this class.
Given the possible MSSM field representations in the Madrid and non-Madrid embeddings, (2.2) and (2.2), it is possible to enumerate888Since and carry the same standard model quantum numbers, we treat them as one field with multiplicity four in our counting. all three-node realizations of the exact MSSM spectrum. One can also compute the T-charges and M-charges from equations (2.1) and (2.1). The possible T-charges for the Madrid quivers are given by
and all M-charges are automatically zero. That is, all of the conditions necessary for a massless hypercharge are satisfied and the conditions necessary for tadpole cancellation are only violated on the node. If a chiral excess of singlets are added to the theory, the only difference is . Performing the same analysis for the non-Madrid embedding, one obtains
and all M-charges zero. Therefore, for the non-Madrid embedding the only T-charge or M-charge violation is on the node. If a chiral excess of singlets are added to the theory, the possible T-charges and M-charges remain the same but the multiplicity of quivers changes due to the new fields. For the Madrid embedding with , there are 160 quivers in all, 144 of which violate the condition. For the non-Madrid embedding with , there are 40 quivers in all, 24 of which violate the condition. With regards to anomalies, a simple calculation shows that 144 of the Madrid quivers have a anomaly, and all non-zero anomaly coefficients are non-integral. All quivers with the non-Madrid quivers have a anomaly, and all anomaly coefficients are non-integral. We will utilize these facts later when discussing hidden sectors.
It is remarkable that most three-node MSSM quivers violate the conditions necessary for tadpole cancellation. This is also true of MSSM quivers with a larger number of nodes. Let us briefly discuss possible field theoretic explanations in terms of anomalies. Since the spectrum under of each quiver is the exact MSSM, perhaps with singlets added, there are no cubic non-abelian anomalies and there is no global anomaly Witten:1982fp (). Any mixed anomalies involving abelian symmetries, such as the and anomalies just discussed, can be canceled by the introduction of appropriate Chern-Simons terms999Consistent type II string compactifications provide these terms automatically via dimensional reduction of the Chern-Simons D-brane action. See Anastasopoulos:2006cz (), for example, for a similar discussion purely in field theory., and therefore these quivers are consistent quantum field theories. However, without further modification any quiver which does not satisfy the conditions necessary for tadpole cancellation cannot be embedded into the types of string compactifications we have discussed. One possible solution is to add matter to the theory so that the inconsistent quivers become consistent. This was done for three-node quivers in Cvetic:2011iq (), as discussed in the introduction, where it was shown that string consistency conditions prefer some standard model representations over others. We now turn to another possible solution.
2.3 Adding Hidden Sectors for Consistency
In section 2.2, we showed that most three-node quivers with the exact MSSM spectrum are not consistent with conditions necessary for tadpole cancellation, despite being consistent as quantum field theories after the addition of Chern-Simons terms. Higher node MSSM quivers also exhibit this behavior. We discussed the possibility of adding matter to the visible sector nodes to cancel the T-charge contribution of MSSM fields.
Another possible solution is to add hidden gauge sectors where bifundamental messenger fields with one end on a visible sector node cancel any overshooting in T-charge or M-charge, ensuring that hidden sector is also consistent. This will be our guiding principle for physics beyond the standard model. We call101010As emphasized in the introduction, stringy hidden valleys are potentially a much broader class than the models studied here. More specifically, we are studying stringy hidden valleys in type II quivers. any hidden sector of this type a “stringy hidden valley.” The setup is heuristically depicted in figure 3 and looks similar to common depictions of hidden sectors. We have a visible sector with fields transforming under and associated anomalous ’s, a hidden sector with fields transforming under some hidden sector gauge group , and messenger fields transforming under both the visible sector group and . However, the setups we study will differ from generic hidden sectors in at least two important ways. First, since our hidden sectors are added to cancel some visible sector -charge, the messenger fields to the node will always vector-like with respect to the standard model but chiral111111We will refer to such fields as “quasichiral”. under , and therefore their masses are always protected. Phenomenologically this very important, since pairs which are vector with respect to all symmetries have string scale masses at a generic point in the moduli space of a string compactification. Second, the structure of the hidden sectors will be constrained by string consistency conditions.
In the three-node Madrid and non-Madrid embeddings string consistency requires that messengers are added to the and nodes, respectively, and therefore the symmetry which charges the messengers are the trace and . In discussing , though, we will be able to address aspects of low energy physics in these models which do not depend on the visible sector hypercharge embedding.
There are two basic types of nodes in a hidden sector: those connected directly to the visible sector via messengers and those which are not. We utilize lower case latin indices for the first type, calling them nodes, and capital latin indices for the second type, calling them nodes. The most general hidden sector could be composed of connected graphs, each disjoint from one another but connected to the visible sector via messengers. When differentiating between disconnected clusters in the hidden sector, we will utilize a superscripted to describe quantities in the cluster. For example, nodes would be nodes in the cluster which are not directly connected to the visible sector.
2.3.1 “Hidden Hypercharge” Quivers
In adding a hidden sector, it is possible that the hypercharge embedding is modified due to contributions from trace ’s of hidden sector nodes. Before adding hidden sectors, the MSSM quivers had a hypercharge embedding given by a linear combination of trace ’s of visible sector nodes. We previously called this , but henceforth will call it . We write the full hypercharge embedding as a linear combination
where is the contribution from hidden sector nodes. While it may seem strange to discuss hidden sector contributions to hypercharge, it is possible to nevertheless ensure that all hidden sector fields are MSSM singlets, and thus we should consider this possibility. We will argue in a moment that it is actually generic to have non-trivial. We call any quiver with a non-trivial a “hidden hypercharge” quiver. We call its hidden sector a “hypercharged stringy hidden valley”.
Any hidden hypercharge quiver is very constrained. Consider the possibility of a single-cluster hidden sector with non-trivial . In general, we could have
where the sum is over all hidden sector nodes indexed by . For the cluster to be connected and hidden, however, every hidden sector node must be connected to another hidden sector node by a field which is an MSSM singlet. This requires for all and where the type of bifundamental is dictated by the signs of . We take for all without loss of generality, which requires that we use only bifundamentals and and not or . Thus, the non-trivial contribution to hypercharge “propagates” through the entire cluster by the requirement that bifundamental fields are MSSM singlets. For an -cluster hidden sector, then, we have121212We have defined for later convenience, since it is a particularly natural linear combination to consider. We will see it can play a role in both supersymmetry breaking and stabilization of messenger dark matter.
and the first sum is over the clusters while the second is over all nodes in the cluster. The key observation is that all nodes in a given cluster contribute to the hypercharge embedding in the same way. Thus, any cluster is labeled by a rational number which determines its contribution to hypercharge. Physically, has a simple interpretation: is the hypercharge of the messengers to the hidden cluster.
At the level of graph theory, a hidden cluster with is a directed graph: any non-messenger edge is between two hidden nodes and has its two arrows in the same direction, in which case we may draw a single arrow indicating the direction. Any loop in hidden sector edges is directed, in contrast with the general case, and the corresponding fields give a perturbative superpotential coupling composed of MSSM singlets. For example, consider figure 4. The messengers are the fields between nodes of type and node , which has an anomalous that charges the messengers. The hidden sector fields are fields connecting hidden sector nodes. Letting be the singlet from nodes to , one perturbative superpotential coupling in this quiver is given by , represented by the closed loop between those nodes.
Since the notion of a hidden hypercharge quiver may still seem strange, let us make some comments regarding generality. We have argued that the requirement that hidden sector fields are MSSM singlets fixes the hypercharge contribution of the hidden cluster up to a single number . For an cluster hidden sector, the contribution to the hypercharge is determined by the tuple . Only one possibility, given by the tuple , has a trivial contribution to hypercharge. Any other tuple gives a hidden hypercharge quiver. Though hidden hypercharge quivers are not required by string consistency, it would introduce a loss of generality to not consider them. Furthermore, we will argue in section 2.4 that hidden sector contributions to the hypercharge embedding are often required to avoid messenger fields with fractional electric charge.
2.4 Fractionally Charged Massive Particles
In this brief subsection we will address an important aspect of phenomenology which must be taken into account when building quivers with hidden sectors.
Globally consistent string compactifications and quivers in the class we study often exhibit particles with fractional electric charge131313Standard model quarks have fractional electric charge, but this does not pose an issue since they are bound into mesons and baryons, which have integral electric charge.. If they exist, the lightest fractionally charged massive particle is stable and its relic density is subject to strong constraints from primordial nucleosynthesis and the cosmic microwave background. Recent work Langacker:2011db () shows that their existence is essentially ruled out. Therefore, we do not consider quivers which give rise to particles with fractional electric charge. This is an important phenomenological consideration which greatly constrains the quivers we study. For example, in extensions of the Madrid embedding any cluster with a type node must have , since the messengers are necessarily doublets of and otherwise they would have fractional electric charge. In non-Madrid extensions, the the messengers are necessarily singlets of and therefore any cluster with a type node must have . In a cluster with only non-abelian type nodes, confinement can relax these constraints, though others exist which depend on the rank of the non-abelian gauge nodes.
2.5 Comments on Fluxes and Consistency Conditions
In the type IIb string, fluxes are crucial for moduli stabilization and can contribute to tadpole cancellation conditions. This is well known for the D3-tadpole, for example. Since the string consistency conditions (2.1) are necessary for tadpole cancellation, let us consider how the conditions may change in the presence of fluxes.
We will work in type IIa, since the consistency conditions are easily derived there. In the absence of fluxes, D6-brane tadpole cancellation gives a homological constraint on the three-cycles wrapped by D6-branes and O6-planes,
The conditions (2.1) arise from intersecting this equation with each cycle on which a D6-brane stack is wrapped and utilizing the relations between topological intersection numbers and chiral indices. See Cvetic:2011vz () for more details. The equations (2.1) can only be altered if (13) is altered. Schematically141414See Kachru:2004jr (); Camara:2005dc () for a detailed discussion of possible flux contributions to the D6-brane tadpole. this must be of the form
Intersecting with gives a set of constraints similar (2.1), except for an additional possible contribution . If , the constraint on is unchanged. the equations can be altered, though the flux must be tuned if it is to precisely cancel any net T-charge of an MSSM quiver. In the generic case the flux contribution will not exactly cancel the net T-charge of the MSSM quiver, and additional matter must still be added for the sake of consistency.
Therefore, the addition of fluxes will not significantly alter the physical conclusions of this paper, which rely entirely on the fact that there is some net T-charge which is canceled by quasichiral messengers to a hidden sector.
3 General Structure of Low Energy Physics
Equipped with a guiding principle for adding hidden sectors, it is possible to make statements about low energy physics which are true of the stringy hidden valleys we consider, but not of a generic hidden valley. The major conclusions in this section will not require the specification of the visible sector matter content, the hidden sector matter content, or the hypercharge embedding. This is because stringy hidden valleys generically exhibit symmetries and chiral spectra which are non-generic within the class of all hidden valleys. Presentation of concrete quivers will be saved for section 4.
We will begin with a discussion of the symmetry which charges the messengers and the associated implications for superpotential couplings. Certain classes of couplings are forbidden in perturbation theory; others are always highly suppressed. We will then turn to a discussion of anomalies, phenomenologically relevant axionic couplings necessary for their cancellation, and physics. We will show that these basic building blocks lead to interesting models of dark matter and supersymmetry breaking, realizing mechanisms already present in the literature.
3.1 Light Messengers and Constrained Superpotential Couplings
One type of symmetry plays a distinguished role in all stringy hidden valleys. Messengers are added with a net excess of charge for some visible sector node , in which case they are chiral under . In extensions of three-node quivers with the Madrid or non-Madrid hypercharge embedding the node is the node or the node, respectively, in which case messengers are chiral under or . In the case of multiple visible sector nodes which charge messengers of different type, there will be more than one symmetry . We emphasize that the conclusions of this section are generic for the models we study.
Let us discuss possible superpotential couplings, beginning with couplings present in perturbation theory. Label a generic superpotential coupling of chiral supermultiplets in the visible, messenger, and hidden sectors as , , and , respectively. Depending on the fields present in the coupling, it is possible that and are present in perturbation theory. However, messenger fields carry anomalous charge of the same sign and therefore no operator is present in perturbation theory. This is particularly important for the messenger mass terms , since the anomalous symmetry prevents the associated fields from acquiring a string scale mass and decoupling from low energy physics. On general grounds, therefore, the messenger mass is always protected by symmetry and could be generated at the TeV scale via instantons or couplings to singlet VEVs. This is certainly not required in a generic hidden valley or hidden sector, and it will have important consequences.
Symmetries also dictate the structure of mixed couplings . Since messenger fields are the only fields transforming under both visible sector and hidden sector gauge nodes, there are no closed paths in the quiver corresponding to couplings of the form , and thus these operators are not present in perturbation theory. Couplings of the form are forbidden since they carry always anomalous charge. Thus, the lowest dimension perturbative superpotential coupling involving both visible sector and hidden sector fields is a non-renormalizable term . It is also possible to couple messenger fields only to visible sector fields, and the lowest dimension couplings of this type are . Couplings of messengers to hidden sector fields carry charge and are forbidden. We have exhausted the possibilities for couplings present in perturbation theory.
Let us discuss couplings not present in perturbation theory, due to carrying anomalous charge. These coupling can be generated non-perturbatively via D-instantons, in which case they are exponentially suppressed by a factor , a suppression factor dependent upon the volume of the cycle wrapped by the instanton. A perturbatively forbidden coupling can also be generated from a perturbative couplings to singlets if the singlet has a vacuum expectation value. Compared to the possibility of obtaining directly in perturbation theory, obtaining it via couplings to singlets suppress by a factor of . In either case, the coupling receives a large suppression, which could151515See Cvetic:2008hi (), for example, where instantons were used to generate Dirac neutrino masses of the observed order without the seesaw mechanism. easily be . The results for the minimum suppression of a coupling are presented in table 1, where we have utilized MSSM gauge invariance and the chirality of messengers under to determine the minimum suppression for each coupling. For simplicity, we also require that singlets which generate couplings are not messengers, since then couplings could arise from couplings, for example. Relaxing this assumption would complicate the analysis without significantly changing the structure of couplings.
In summary, the structure of superpotential couplings is strongly constrained by symmetries generically present in stringy hidden valleys. The most important observation compared to a generic quiver is that and couplings are forbidden in perturbation theory, and thus are very suppressed if they are present at all. Suppression of these and other couplings play an important role in ensuring that the hidden sector doesn’t couple strongly to visible sector fields, despite being present in the same connected quiver.
|Coupling Structure||Perturbative||Instanton-induced||Singlet Couplings|
3.2 Anomalies, Required Axionic Couplings, and Bosons
All quivers in the broad class motivated by type II orientifold compactifications have a rich structure of anomalies, axionic couplings required for their cancellation, and physics. In this section we will make statements about them which are generic for stringy hidden valleys, but not for the broader class. We will see that these features can have important consequences for dark matter and supersymmetry breaking.
3.2.1 For All Stringy Hidden Valleys
We have emphasized that all stringy hidden sectors have messengers which are chiral under an anomalous symmetry . Therefore there is a anomaly for the of any node with gauge symmetry. In addition, if , there are mixed anomalies.
These anomalies must be canceled via the Green-Schwarz mechanism through the introduction of Chern-Simons terms. The mixed abelian anomalies are canceled by the introduction of terms of the form
where and are the field strengths of and , is the two-form which gives a Stückelberg mass to , and is the zero-form which is the four-dimensional Hodge dual of , i.e. . If , the mixed abelian non-abelian anomalies require terms of the form
where is the field strength of . These conclusions hold for any stringy hidden valley, and the axionic terms can play an important phenomenological role.
Since the coupling is always present, is always a massive . gauge interactions are suppressed via the large mass. This has strong implications for interactions between messengers and visible sector particles charged under . See section 3.5 for a discussion of masses and low energy physics.
Finally, a non-generic but common possibility is to have a chiral excess of messengers161616By this we simply mean that the net charge of the messengers is non-zero. on the node. In this case there is a anomaly, whose cancellation requires terms of the form and and is a massive . If there is not an excess of chiral messengers on , it is possible that is massless, but this cannot be determined without further specification of the hidden sector spectrum.
3.2.2 For All Hidden Hypercharge Quivers
In this section we discuss further aspects of anomalies, axionic couplings, and physics which are true of any hidden hypercharge quiver. The additional physics is governed by the fact that the hidden sector contributes non-trivially to the hypercharge embedding, which will have interesting implications for dark matter annihilation processes with photons in the final state.
Anomalies and Axionic Couplings
Let us first consider anomalies. Since messengers carry hypercharge and are chiral under , they will always contribute to this anomaly. Unlike anomalies, anomalies can also receive contributions from visible sector fields, in which case it may be possible that visible sector and messenger contributions cancel and there is no anomaly. We will now argue that this is almost never the case for extensions of three-node quivers. The only possible loophole is in the case where confinement relaxes constraints on necessary to ensure the absence of fractionally charged massive particles.
Let us begin with the Madrid embedding where , taking messengers. As discussed in section 2.2 all but sixteen of the three-node MSSM quivers have a anomaly, and any non-zero anomaly coefficient is non-integral. The messengers are doublets and give a net contribution to the anomaly of . We must have with for either the absence of fractionally charged massive particles171717In a cluster with a node. As mentioned in section 2.4 this constraint can be relaxed if all nodes in the cluster are non-abelian. or the existence of an electrically neutral messenger component, and in this case, since cancellation of charge requires even, the messengers give an integral contribution to the anomaly coefficient. Since the contribution of all possible visible sectors to the anomaly coefficient is non-integral, all stringy hidden valley extensions of three-node MSSM quivers with the Madrid embedding and exhibit a anomaly. For non-Madrid extensions, we will often require to ensure the absence of fractionally charged massive particles, in which case messengers will give an integral contribution to the anomaly which will not cancel the non-integral contribution from visible sector fields. Thus, all non-Madrid quivers of this type have a anomaly. Summarizing, any extensions of a Madrid (non-Madrid) quiver with () has a () anomaly.
We have just argued that anomalies are extremely common in hidden hypercharge quivers, being very precise with the case of extensions of visible sectors with three-nodes. For any quiver with such an anomaly, anomaly cancellation via the Green-Schwarz mechanism requires the presence of a term
where is the field strength of hypercharge. In section 3.3 will see that these terms play a crucial role in recent models for dark matter annihilation processes with photons in the final state.
The Physics of Hidden Hypercharge Quivers
bosons appear in many top-down constructions and can greatly impact low energy physics, as reviewed in Langacker:2008yv (). The models we have proposed have a rich structure of physics. In hidden hypercharge quivers there is a “natural” which is usually massless, and is always massless for any extension of a three-node quiver. Recall from section 2.3 that we write the hypercharge embedding as
where the two terms are the contributions of the visible sector and the hidden sector to the hypercharge linear combination, respectively. For all quivers, we require that the linear combination satisfies the linear equations (2.1). If independently satisfies these linear equations, as it does for all extensions of three-node MSSM quivers181818 is just the Madrid or non-Madrid linear combination. These linear combinations are massless for all three-node MSSM quivers, and it is easy to see that adding a hidden sector will not cause these conditions to be violated for ., then will also satisfy these equations. Therefore any such quiver will give rise to a light boson. We will think of this as coming from and henceforth call it , though we could equivalently consider . couples to any messengers which have hypercharge, but never to hidden sector fields. is closely related to . If there is a single cluster, is just rescaled by .
There are further interesting statements that one can make about physics in hidden hypercharge quivers. To do so, it is useful to consider two possibilities for the cluster : the case where cluster has a chiral excess of messengers on some node , and the case where it has no such chiral excess for any node.
Let us first consider the possibility where there is no chiral excess of messengers on any node, and examine the linear combination as in equation (12). From (2.1) the conditions on a node necessary for to remain massless are
where the sum is over hidden sector nodes in cluster and we remind the reader that messengers are chiral under . This is equivalent to the condition on nodes necessary for a massless , and therefore they are satisfied since is massless. Similar statements apply for nodes. The only condition left to satisfy is the condition on the node, given by
This is stronger than the node condition for a necessary , but it is satisfied since we are considering the case where there is no chiral excess of messengers on any node, so that each term in square brackets is zero. Therefore is also massless and there is yet another light . If there are many such clusters, there can be many light bosons.
Let us consider the other case, where there is a chiral excess of messengers on some node . From (19) it is clear there is not necessarily a light corresponding to . However, the chiral excess induces mixed abelian anomaly since the messengers ending on node also carry hypercharge. Such an anomaly is canceled by Chern-Simons terms of the form and where and are the field strengths of and and and are a two-form and its Hodge dual zero-form. The Chern-Simons terms introduced to cancel anomalies of this type can play an important role in dark matter annihilation, as we will now discuss.
3.3 Dark Matter and a Possible Monochromatic -ray Line
It has been known for many years that string consistency often requires the presence of hidden sectors which can give rise to interesting dark matter candidates. In the heterotic string, the standard model spectrum is typically constructed from one of the two factors. The other factor generically gives rise to another gauge sector which interacts with the visible sector only gravitationally. In weakly coupled type II orientifold compactifications and F-theory, “filler branes” which do not intersect the standard model branes are often required for tadpole cancellation. See, for example, Cvetic:2001nr (). These interact gravitationally with the standard mode, but not via gauge interactions.
It is also possible that nature contains a dark matter sector which couples weakly to the standard model, but nevertheless can exhibit dark matter annihilation into standard model particles via gauge interactions or suppressed couplings to visible sector particles. In the last six months many models of this type have been explored, due in part to the possible experimental observation of a -ray line from dark matter annihilation near the galactic center Weniger:2012tx (); Tempel:2012ey (); Boyarsky:2012ca (); Su:2012ft (). Regardless of whether this signal survives further scrutiny, particularly by the Fermi LAT collaboration itself, it is important to discuss whether dark matter candidates in our theories can annihilate via processes with visible sector particles in the final state, particularly photons.
3.3.1 Annihilation via Axionic Couplings and Vertices
We showed in section 3.2 that the stringy hidden valleys we study generically have a rich structure of physics and axionic couplings. These can have important consequences for dark matter annihilation.
Let us briefly review two ideas in the literature which are very common in our models and give rise to dark matter annihilation processes with photons in the final state. The first utilizes an intermediate anomalous boson to give the dark matter annihilation process . This was proposed a few years ago in Dudas:2009uq () and more recently in Dudas:2012pb () after the possible observation of the -line. The key feature is an anomalous symmetry under which dark matter is charged. Anomaly cancellation via the Green-Schwarz mechanism requires the presence of axionic couplings which give an effective vertex that makes the annihilation process possible. One difficulty is that the annihilation cross section is suppressed by the mass, which is typically very large. See section 3.5.
This possibility is extremely common in our models. Structurally, all that is needed is dark matter charged under some symmetry and a anomaly. In our models there are many symmetries which may play this role and this possibility could be checked on a quiver by quiver basis. However, is a distinguished symmetry in all of our quivers. As we have argued in section 3.2, hidden hypercharge quivers always have messengers which contribute to the anomaly coefficient and the quiver exhibit a anomaly unless the contribution from the visible sector precisely cancels those of the messengers. We have argued that this never happens for extensions of three-node MSSM quivers, and therefore a anomaly is generic in those models. In addition, even if the hidden sector is not hypercharged there is almost always a anomaly just from the visible sector contribution. Thus, dark matter charged under can nearly always realize the scenario of Dudas:2009uq (), at least structurally. By the definition of , such dark matter is messenger dark matter, which we will discuss.
Another possibility was recently proposed Fan:2012gr () which utilized similar axionic couplings. The theory has a hidden sector with a non-anomalous and an gauge factor with quarks carrying appropriate charge to give rise to neutral or -charged hidden sector pions. There are axionic couplings of the form and where and are the hypercharge and field strengths, respectively. The -charged pions are stable due to being the lightest charged particles and are identified as dark matter. They can annihilate to -neutral hidden sector pions which can then decay to photons via the axionic couplings. See section 4.1.3 for a concrete realization similar to this possibility in a stringy hidden valley.
Our models frequently realize axionic couplings similar to these. In certain cases it is possible to add these axionic couplings by hand, as in Fan:2012gr (). The more interesting case, however, is when they are required for anomaly cancellation. As argued in section 3.2, there is a anomaly for any node and also a if is non-abelian, requiring the presence of couplings and . The key coupling allowing annihilation to photons is the axionic coupling to the hypercharge field strength, here . This is necessary for the cancellation of a anomaly, which nearly always exists. Therefore our models typically have the couplings utilized in necessary to explain dark matter annihilation via the mechanism of Fan:2012gr (), or a similar mechanism. In a given quiver, there may be anomalous ’s other than which could play this role.
3.3.2 Messenger Dark Matter and Anomalies
Since all quivers we study have messenger fields to hidden gauge nodes, one simple possibility is that dark matter is comprised of messengers fields and . Since they are quasichiral, the messenger mass is always protected by symmetry and can therefore be light, perhaps or . We see from table 1 that any perturbative superpotential coupling of messengers to a standard model field is string suppressed, and that similar couplings obtained via instanton effects or couplings to singlets are also very suppressed. Messenger dark matter in stringy hidden valleys will always be non-baryonic, since string consistency does not require the addition of messengers charged under when extending MSSM quivers.
Let us discuss possibilities under which messenger dark matter is stable against decay. A simple possibility is that a symmetry ensures stability, which is certainly possible if there is a natural symmetry under which only messengers are charged. As shown in section 3.2, quivers with a hypercharged stringy hidden sector very frequently191919Always, for extensions of three-node quivers. have a massless which charges only the messengers and could protect messenger dark matter candidates from decay. In addition, any hidden hypercharge quiver and many others will have symmetries, perhaps anomalous, which charge only the messengers to the hidden cluster. In concrete quivers, there could be other massless symmetries which charge the messengers, or massive symmetries. Therefore, symmetries which could protect messenger dark matter from decay are very common.
Let us discuss possible annihilation processes for messenger dark matter in generality. always charges both the messengers and some set of standard model fields, allowing for dark matter annihilation via for standard model fermions . In addition, unless visible sector contributions to the anomaly coefficient exactly cancel the messenger contributions, dark matter can annihilate to photons via as discussed in section 3.3.1. However, is heavy and dark matter annihilation cross sections are suppressed. Purely in a low energy effective theory, though, one can treat the mass of as a parameter and constrain the phenomenologically allowed parameter space, as in Dudas:2012pb (). See section 3.5 for a discussion of anomalous masses. In addition, any stringy hidden valley necessarily gives rise to couplings and . Since messengers end on nodes, the axionic couplings could give rise to dark matter annihilation processes with photons in the final state, similar to Fan:2012gr ().
Let us discuss more specific possibilities which depend on the visible sector hypercharge embedding. For messenger dark matter to have any hope of being realistic in an extension of the Madrid embedding, it must be a messenger to a cluster with , which is required for the charged messenger to have an electrically neutral component . Such a particle is a natural WIMP candidate. For the Madrid embedding, , and dark matter can annihilate into an anomalous . Since messengers are doublets of , annihilation to via the process will dominate over the process involving an intermediate . In an extension of the non-Madrid embedding, messenger fields must end on a cluster with for field to have an electrically neutral component and . Dark matter can annihilate to via an intermediate anomalous . The messengers do not carry hypercharge, but in the case where the standard model fields generate a anomaly, dark matter can nevertheless decay as . This is possible for any extension of a three-node quiver, since there is always a anomaly, as argued in section 3.2.
3.3.3 Hidden Sector Dark Matter
Another possibility is that dark matter is comprised of fields transforming only under hidden sector nodes. As such, they necessarily standard model singlets. Since hidden sector fields are much less constrained than messenger fields, there are more possibilities and we will therefore be brief. Symmetries ensuring stability are similar to the messenger dark matter case, except that hidden sector dark matter is not charged under , the distinguished massless common in hidden hypercharge quivers.
Since hidden sector dark matter does not carry charge, it cannot decay via a vertex. However, as argued in section 3.2 there are broad classes of quivers which exhibit a anomaly, which introduces a vertex into the theory, allowing for dark matter annihilation into photons via . In such a case dark matter is necessarily charged under and could annihilate to photons via the axionic couplings and as suggested in Fan:2012gr (). This mechanism does not rely on the propagation of a heavy . Finally, there are never anomalies, since this would require hidden sector fields which carry hypercharge. Therefore the vertex is not required to exist in the low energy theory and it is unlikely202020In the absence of couplings there could be and no anomaly. that hidden sector dark matter ending only on nodes will decay into photons.
3.4 Spontaneous Global Supersymmetry Breaking
In a globally consistent string compactification, the proper framework for discussing supersymmetry and its breaking is supergravity, where the dynamics and stabilization of closed string moduli play an important role in determining possible supersymmetry breaking and mediation scenarios. As discussed, string consistency often requires the presence strongly coupled gauge sectors which interact only gravitationally with the standard model. It is possible that supersymmetry is broken in this sector and gravity mediation ensues. Such analyses require the specification of a global string compactification with moduli stabilized and is outside the realm of the quiver gauge theories we study. However, in the limit it is natural to study the possibility of global supersymmetry breaking. Though an embedding into supergravity may spoil212121For example, in a string compactification the Fayet-Iliopoulos term depends on closed string moduli and may dynamically relax to zero, restoring supersymmetry in the Fayet models we will discuss. Realizing this model in supergravity would require stabilization at a point in moduli space with non-zero . the global supersymmetry conclusions gained via studying a quiver gauge theory, this is the best one can do at the quiver level and the conclusions may nevertheless hold in supergravity embeddings. In this section we will discussed global supersymmetry breaking scenarios in stringy hidden valleys.
One way to break supersymmetry is to embed a non-abelian gauge theory into the low energy spectrum which exhibits strong gauge dynamics that break supersymmetry Affleck:1984xz (). A prototype which has been studied extensively is supersymmetric QCD with gauge symmetry and vector-like flavors Affleck:1983mk (). Metastable supersymmetry breaking Intriligator:2006dd (); Intriligator:2007py () is a common and intriguing possibility, in SQCD and in general. In addition, classic supersymmetry breaking models which do not utilize strong gauge dynamics have been realized in simple D-brane quivers Florea:2006si (); Aharony:2007db (), where D-instantons play a crucial role in determining scales in the model. Global realizations include Cvetic:2007qj (); Cvetic:2008mh (). We find that supersymmetry breaking via SQCD and a retrofitted Fayet model similar to those of Aharony:2007db () can appear naturally in the models we study.
One important feature that we must consider with either SQCD or Fayet breaking is that messenger fields often play a crucial role. In such a case supersymmetry breaking can give vacuum expectation values to the scalar components of the messengers, breaking the MSSM gauge group in the common case of non-singlet messengers. In particular, in extensions of the Madrid embedding the messengers carry charge and supersymmetry breaking involving messengers VEVs would trigger electroweak symmetry breaking. For simplicity we will avoid this possibility, when necessary, in the examples of section 4.
3.4.1 Breaking Supersymmetry with SQCD
Since we take hidden sector gauge group , realizations of supersymmetry breaking with strong gauge dynamics necessarily require an gauge group. In a generic hidden sector there could be many such factors with rich gauge dynamics, but for simplicity we will restrict our attention to the possibility of a single non-abelian factor with gauge group with flavors which are vector-like with respect to . All flavors are necessarily bifundamentals, and for simplicity we also require that they have one end on a common node which is not the node. Given these restrictions, it is natural to classify the possibilities according to whether the flavors are messengers or hidden sector fields. We refer to these scenarios as “messenger SQCD” and “hidden sector SQCD”, respectively. Of course, hybrid scenarios are also possible if is an node.
Over time it has been shown that SQCD can break supersymmetry for many values of and , originally in the confined regime in Affleck:1983mk (). More recently it has been shown Intriligator:2006dd (); Intriligator:2007py () that SQCD can give rise to metastable supersymmetry breaking in the free magnetic range . For a recent discussion of these ideas and their history, see Intriligator:2007cp ().
If the node is an node, the SQCD flavors can end on a visible sector node with gauge symmetry and . The flavors are what we have been calling “messenger” fields, where this should not necessarily be confused with messengers of gauge mediated supersymmetry breaking. The quiver takes the form shown in figure 5
and the field content beyond the standard model is copies of and , and the flavors are chiral with respect to the trace of . To avoid detailed analyses of supersymmetry breaking scenarios for different values of and , we will utilize facts about SQCD despite the fact that our gauge group is . This is certainly a valid assumption at scales below the mass of the non-anomalous boson associated to the trace of . We will give a concrete example of these models in section 4.2. Let us discuss some generic features here.
An important feature of these realizations of SQCD is that the mass term is protected by symmetry but can be generated at a low scale via D-brane instantons or couplings to singlets. In the absence of this symmetry, the flavors will typically obtain a large mass far above the confinement scale , giving a pure SQCD theory at low energies which does not break supersymmetry. We view this as an advantage of these models and assume that the masses of the flavors is far below the confinement scale. A natural concern in this theory is that it may be difficult to realize the Affleck-Dine-Seiberg non-perturbative superpotential which plays an important role in supersymmetry breaking, since explicitly appears and is forbidden by symmetry. However, it is known Haack:2006cy () that a gauge invariant222222In these constructions the non-gauge invariance of is compensated for by the non-gauge invariance of a closed-string modulus appearing in the correction. ADS superpotential can be generated even in the case where carries net anomalous charge. We have argued in section 3.2 that is always anomalous in models with stringy hidden sectors. Given these arguments, one can apply standard techniques of supersymmetry breaking via SQCD with various various of and .
For SQCD with messenger flavors in our models, the allowed values of and are constrained by the fact that messengers are added to cancel some non-zero T-charge, and the T-charges are concretely determined by possible visible sector realizations of the MSSM. For example, in extensions of the three-node Madrid hypercharge embedding the only possible non-zero T-charge is for , as discussed in section 2.2, which constrains the allowed values of and via the equation
We have assumed a single SQCD node of type. Due to the condition for nodes in equation (2.1), SQCD extensions of the non-Madrid embedding must have which is a not a multiple of , as must any stringy hidden sector with an SQCD node attached to a visible sector node. It is also possible to write down the allowed values of and for extensions of higher-node MSSM quivers. There are allowed values of and which break supersymmetry via the ADS superpotential.
Finally, for SQCD supersymmetry breaking with messenger flavors it is possible that the messengers fill out non-trivial standard model representations, in which case the ADS superpotential Higgses . The only possibility for the messenger flavors to be standard model singlets is an in extension of the non-Madrid embedding with . See section 4.2 for an example.
Hidden Sector Flavors
The other possibility is that the gauge theory which breaks supersymmetry is realized on an type node, in which case the flavors cannot be messenger fields. In this case there is no constraint on the allowed values of and since the flavors are hidden and they are not required to cancel a T-charge. Hidden sector fields are not required to be quasichiral and therefore in this case it is possible to realize vanilla SQCD with vector-like flavors. However, such flavors do not have masses protected by symmetry and are very heavy at a generic point in the moduli space of a string compactification. If so, the flavors can be integrated out, giving pure glue SQCD at low energies which does not break supersymmetry.
In clusters with , the hidden sector SQCD flavors could also be quasichiral bifundamentals with protected masses, giving rise to a scenario very similar to that of the messenger flavor case. However, compared to the messenger flavor case the structure superpotential couplings is different, according to table 1, and the possibilities are not as constrained.
3.4.2 Breaking Supersymmetry via a Retrofitted Fayet Model
In Aharony:2007db () a retrofitted Fayet model which broke supersymmetry was presented in a simple quiver. We remind the reader that a Fayet model generically contains a symmetry with a non-zero Fayet-Iliopoulos term and some number of fields charged under the . Since the F-term and D-term equations cannot be simultaneously satisfied, supersymmetry is broken. Given the many symmetries in our hidden sectors, it seems natural that this model of supersymmetry breaking could be realized.
We would like to realize the Fayet model without needing to specify a concrete spectrum or hypercharge embedding. There are typically many heavy anomalous bosons in a given quiver, but as emphasized in Intriligator:2005aw () the corresponding D-term equations should not be imposed since the bosons can be integrated out of the low energy theory. Therefore, successful Fayet models should utilize massless symmetries. Fortunately, in hidden hypercharge quivers there is typically a light corresponding to the gauge symmetry , as discussed in section 3.2. We will study the possibility of a single cluster hidden sector, though the arguments we present can be trivially generalized to the case of multiple cluster hidden sectors. Given that the hypercharged stringy hidden sector has a single sector, we will rescale by to give a symmetry for simplicity. This allows the discussion to proceed without reference to the value of .
Let us discuss how