Un-oriented Quiver Theories for Majorana Neutrons

In the context of un-oriented open string theories, we identify quivers whereby a Majorana mass for the neutron is indirectly generated by exotic instantons. We discuss two classes of (Susy) Standard Model like quivers, depending on the embedding of SU(2)_W in the Chan-Paton group. In both cases, the main mechanism involves a vector-like pair mixing through a non-perturbative mass term. We also discuss possible relations between the phenomenology of Neutron-Antineutron oscillations and LHC physics in these models. In particular, a vector-like pair of color-triplet scalars or color-triplet fermions could be directly detected at LHC, compatibly with n-\bar{n} limits. Finally we briefly comment on Pati-Salam extensions of our models.


Introduction
Recently we have proposed the possibility that a Majorana mass term for the neutron could be indirectly generated by non-perturbative quantum gravity effects present in string theory: the exotic instantons [1,2] 3  FCNC's. The class of models that we consider seems to meet these requirements.
Exotic instantons propagate the quantum gravity stringy effects to much lower scales, that can be as low as 1000 TeV.
The main purpose of the present paper is to clarify aspects of the mechanism proposed in [1] and to identify quiver theories leading to the interesting phenomenology introduced in [1]. The paper is organized as follows: in Section 2 we briefly review the main features of the models with Majorana mass terms for the neutron; in Section 3, we discuss the phenomenology related to neutron-antineutron oscillations, reviewing and extending our previous considerations and constraining the allowed region in parameter space, with particular attention to possible signatures at LHC; in Section 4 we briefly review the construction of (un-oriented) quiver theories; and identify SM-like (un-oriented) quivers for a Majorana neutron in Section 5 we discuss possible quantum corrections to the Kähler potential and D-terms as well as the role of susy breaking; in Section 6 we present our conclusions and a preliminary discussion of Pati-Salam extensions.
We will also need an Ω-plane for local tadpole cancellation and E2-branes (instantons).
The un-oriented strings between the various stacks account for the minimal super-field content of the MSSM These interact via the super-potential Flavour or family indices are understood unless strictly necessary. Note that W preserves R-parity. The last term violates the continuous R-symmetry and can be generated by E2-branes (instantons) as discussed in [33,34,35] and reviewed later on.
We could also consider some of the possible perturbative R-parity breaking terms (see [36] for a review on the subject): Moreover, soft susy breaking terms can be generated by fluxes or other means, that produce scalar mass terms, Majorana mass terms for gaugini (zino, photino, gluini), trilinear A-terms, bilinear B-terms [37,38].
In the first case, the hypercharge group U (1) Y is a combination of the four anoma- In fact the four U (1)'s can be recombined into U (1) Y , U (1) B−L and two anomalous and Y is a linear combination of 3 charges q 1,1 ,3 .
The presence of anomalous U (1)'s is not a problem in string theory. A generalisation of the Green-Schwarz mechanism disposes of anomalies. In particular in the stringinspired extension of the (MS)SM under consideration, new vector bosons Z appear that get a mass via a Stückelberg mechanism [40] and interact through generalized Chern-Simon (GCS) terms, in such a way as to cancel all anomalies [41,42,22].
If the relevant D-brane stacks intersect four rather than three times, i.e. #U (3) · U (1) = 4, a 4th replica D = D c f =4 of the three MSSM D c f =1,2,3 appears. Moreover, compatibly with tadpole and anomaly cancellation, another chiral super-field C i = 1 2 ijk C jk appears at the intersection of the two images of the U (3) stack of D6-branes, reflecting each other on the Ω-plane 4 .
ijk C jk form a vector-like pair with respect to SU (3). New perturbative Yukawa-like interactions involve D and C and A non-perturbative mixing mass term is generated by non-perturbative E2-instanton effects. The relevant E2-brane (exotic instanton) is transversely invariant under Ω and intersects the physical D6-branes, as discussed in [1]. The non-perturbative mass scale is M 0 ∼ M S e −S E2 with M S the string scale, S E2 the E2 instanton action, depending on the closed string moduli parameterizing the complexified size of the 3-cycle wrapped by the world-volume of

E2.
Integrating out the vector-like pair an effective super-potential of the form is generated.
In this way, one can start with a theory preserving R-parity and have it broken dynamically only through the non-renormalizable R-parity breaking operator (9).
In principle, one can also consider some explicit R-parity breaking terms, including perturbative ones (3), but then one has to carefully study the dangerous effect of these on low-energy processes violating baryon and lepton numbers.

Phenomenology: Neutron-Antineutron physics and LHC
An operator like (9) generates neutron-antineutron transitions, violating baryon number with ∆B = 2, as shown in Fig.1 and discussed in [1]. The scale M 5 nn = m 2 g M 2 0 MH in (udd) 2 /M 5 nn is a combination of the gaugino (gluino or zino) mass mg, of the mixing mass term M 0 for the vector-like pair and the Higgsino mass MH. In order to satisfy the present experimental bound M nn > 300 TeV, we can consider different scenarios.
We focus on some of these in the following 5  Another diagram generating n −n transitions is depicted in Fig.2, the analysis of the parameter space is roughly the same as for the first case.
These diagrams respect R-parity at all the vertices, except for the non-perturbative mixing term of the vector-like pair. In fact the super-potential has R(W) = −1 as usual, and one can consistently assign R-charges to C and D , so that their tri-linear Yukawa terms be invariant. Yet their mass term necessarily violates R-parity. Omitting the coupling constants one schematically has where ± indicates the R-parity, φ C,D and ψ C,D are the scalars and the fermions respectively in the superfields C, D , q,q are quarks and squarks, φ H u,d are Higgs bosons, ψ H u,d are the two Higgsini. Note how R-parity is violated only by the last non-perturbative term with mixing mass parameter M 0 not directly connected to the Dirac mass term for D , emerging from its 'standard' coupling to the Higgs.
More precisely, M 0 is replaced by the mass parameter of the lightest mass eigenstate, be it a fermion ψ D ,C as in Fig.1, or a scalar φ D ,C as in Fig.2. The scalars φ D ,C have in general a non-diagonal mass matrix [2] of the form and δµ 2 ∼ mgM 0 as in Fig.2.
The mass eigenvalues of (11) are both doubly degenerate, as manifest in (11). Note that, in the case of m φ D = m φ C = 0 and δµ = 0, one of the mass eigenvalue is negative, i.e. leading to a condensate, On the other hand, we would like to note that Dirac mass terms for fermions ψ D and ψ C are not present at all. For instance, ψ D is like a 4th right-handed down quark without a Left-Handed counterpart. As a result m ± = ±M 0 , where the sign, in fact any phase, can be absorbed into a redefinition of the phases of the fermionic fields.
We can distinguish two branches for LHC and FCNCs phenomenology: i) Normal Susy hierarchy; ii) Inverted Susy hierarchy.
In Normal Susy hierarchy, scalars φ D ,C have the lighest mass eigenstate λ − << |m − |, i.e scalars have lower masses with respect to their supersymmetric fermionic partners. This case is an ordinary hierarchy between fields and their supersymmetric partners. In this case, the relevant contribution for Neutron-Antineutron oscillations is the one in Fig.2. In principle, M 0 mg has to be substituted, in the parameters estimations for n −n oscillations shown above, with the lighest mass λ 2 − . For with mixing angles θ 13 = θ 24 ∼ 10 −6 . So, mixings between X and Y are strongly suppressed in this case, but may be enough for neutron-antineutron transitions: a prefactor of 10 −12 in a n −n scale (M 4 0 µ) 1/5 has to be included. This drastically changes the constraints on the other parameters: for M 0 = 1 − 10 TeV, a light ψ of µ = 1 ÷ 100 GeV would be enough! The phenomenology of (i) is discussed in [2]. In [2], a toy-model was shown in which the so called X , Y are nothing but φ D ,C respectively. There are some subtle differences not allowing a perfect identification X = φ D and Y = φ C . For example, in the main interactions terms, like Yu R d R rather than φ C u L u R , or X d R ψ rather than φ C d LH ; with ψ a sterile Majorana fermion with zero hypercharge rather than an Higgsino. This leads to some subtle differences in hypercharge assignments (reversed in the vector-like pair) and Baryonic number assignment (opposite sign in both), compatible with gauge symmetries. Even so, the phenomenology is very similar to the one discussed in [2], in so far as neutron-antineutron oscillations, LHC signatures, FNCN's, and Post-Sphaleron Baryogenesis are concerned. For instance, the lightest mass eigenstate scalars can have λ − 1 TeV, with possible channels at LHC. In particular, pp → jjE T / is previewed, avoiding stronger constraints from FCNC's processes. Our models also predict pp → 4j (direct bound of 1.2 TeV) or pp → ttjj (direct bound of 900 GeV), but FCNC bounds are stronger than LHC ones, in these cases.
In the Inverted Susy Hierarchy, we consider the opposite scenario in which M 0 << λ − , i.e susy fermions ψ D ,C are lighter than scalars φ D ,C . We also would like to note that, in principle, a scenario in which the fermions ψ D ,C can be at lower masses with respect to their scalar partners φ D ,C is perfectly possible: the second ones can get extra contributions from (non-perturbative) non-supersymmetric closed-string fluxes (NS-NS or R-R), not contributing to fermionic masses 6 . In this case, direct detection of ψ C − ψ D at LHC would be possible. For instance, it is possible to produce these in different processes, having peculiar decay channels like ψ C → qq. We would like to note that in our case one can also generate perturbative Yukawa terms of ψ D with the bottom quark, leading to a decay channel ψ D → Hb. Moreover, an electroweak mixing with the top quark is also possible, that would lead to ψ D → W t. These In order to integrate out the massive super-fields, N , H u , H d , C, D , we have to evaluate the field-dependent mass matrix M IJ (Φ), where Φ collectively denotes the light super-fields, and invert it where Due to the non-trivial dependence on the superfields Q and L, direct inversion inversion of (16) becomes laborious but straight-forward with the result A perturbative approach, alternative but equivalent to the exact inversion (47) is reported in Appendix.
On-shell the F-terms yield Replacing their expressions into W ef f (Φ), we obtain the following extra and potentially dangerous operators (relevant coupling constants are omitted for simplicity): We report also the only one remaining at the 4th order: In the limit of m N → ∞, all the dangerous operators are automatically suppressed. In fact, only QQQQU c /µM 0 remains, but this cannot lead to proton decay, as discussed in [1]. Also combining such operator with other perturbative ones, one can check that all the resulting effective operators are innocuous: there is no operator leading to a final state without at least one susy partner (so, no available phase space for proton decay), without violation of any fundamental symmetry like charge, spin or fermion number.
In fact our models may be tuned not to violate Lepton number, by setting m N = 0, by turning on fluxes or other means that prevent any E2-brane instanton that may generate m N [44]. The price to pay is that a type I see-saw mechanism for the neutrino is not allowed: we cannot generate a Majorana mass without fast proton decay. So, such processes as neutrino-less double-β decay would provide evidence against these class of models. Of course, a Dirac mass for the neutrino W = H α u L α N → L Y = φ α Hu t α ν 0 is allowed if R-H sterile neutrini are present.

Flavour changing neutral currents
Extra contributions to FCNC's may appear in our models, mediated by φ C , in normal susy hieararchy, as cited above. But these can be sufficiently suppressed, compatible with n −n limits.
Other possible contributions, directly connected to n −n transitions, are strongly suppressed in our model, as discussed in [1] (See Fig. 11-12

Standard Model like quivers generating a Majorana Neutron
Our aim, in this section, is to identify possible (un)oriented quiver field theories for the models introduced above, thus generating a neutron Majorana mass. As discussed above the ingredients are un-oriented strings stretched between D6-branes stacks. We will also need E2-branes, wrapping some 3-cycles in CY 3 . Thanks to the local CY condition, the resulting theory preserves N = 1 supersymmetry.

What is a quiver field theory?
In general, a quiver, a collection of arrows, represents a gauge theory, with its matter anomalies. An interesting example, studied in [54], is the case of C 3 /Z n singularities, in the presence of non-compact D7-branes, fractional D3-branes, and Ω-planes.

Explicit examples
An example of a simple quiver theory generating a Majorana mass for the neutron is shown in Fig. 3. This consists in: one stack of three D6-branes producing the U   (with the hat, we denote the images in the right side of the Ω-plane in Fig. 1). As a consequence, a perturbative Yukawa term C ij Q i Q j is generated. On the other hand, D c QH d is generated exactly as the corresponding standard one D c QH d .
Finally, the relevant exotic O(1) instanton E2, generating the non-perturbative mixing between D and C, is also represented in Fig. 3. As dashed lines we also denote the modulini living at the intersections between E 2 and the U (3) and U (1) stacks of D6-branes. The hypercharge in this model is the combination of 3 charges coming from U (1) 3 ,U (1),U (1): We can fix the coefficients from the conditions that arise in order to recover the standard hypercharges: leading to the result For the quiver in Fig. 3, it is possible to generate a mixing mass term for the We should remark that the quiver represented in Fig. 3 could generate extra Rparity breaking terms λ LQD c in (3) or λ LQD c , leading to a mixing of quarks and leptons. These dangerous operators can be tuned to zero, since not all closed triangles or more generally polygons necessarily correspond to interaction terms. As an alternative, we can consider the quiver in Fig. 4, consistent with the hypercharges: leading to the result We get all the standard Yukawa terms in this case, too. We may also have the Rparity breaking term µ a L a H u with ∆L = 1. We will assume that such a term is absent at the perturbative level and is not generated non-perturbatively. Since other R-parity violating terms in (3) are automatically disallowed at the perturbative level, our model is R-parity invariant to start with. Taking into account the non-perturbative term QQQH indirectly generated through exotic intantons, and the µ-term µH u H d possibly generated by exotic instantons, too, similarly to the other one discussed above.
Other R-parity breaking contributions may arise from higher order vertices, corresponding to closed polygons with more than 3 sides, not present in the other case: Clearly these operators are dangerous. For instance, combining W V >3 with the nonperturbative operator (9) with (44), yields The first term can lead to neutron-antineutron transitions and di-nucleon decays pp → π + π + , K + K + , the second term to proton decay p → π 0 e + . The ratio of the proton life-time to the neutron-antineutron transition time is This hierarchy is much higher than the present limit on D − C vector-like pairs at colliders. In fact, with τ p−decay 10 34÷35 yr and τ nn 3 yr, M 0 10 −35 M S << M 0 | exp , where M 0 | exp 0.5 ÷ 1TeV is the direct bound from colliders discussed above.
For di-nucleon decay the situation is better, but also in this case the required tuning is extremely delicate, considering that τ di−decay 10 32 yr [55]. So, we conclude that a tuning of the coupling constants y LH d D c Q , y U c Q c H d D c to zero would be necessary in this case.

Extended quivers and CY singularities
The quivers, proposed in Fig.3

Kähler potential, D-terms and perturbative corrections
So far we have focussed on the super-potential interactions, both perturbative and nonperturbative ones. We have argued that barring explicit R-parity violating terms in the Lagrangian, R-parity is broken dynamically by non-perturbative exotic instanton effects. This implies that it is preserved in perturbation theory, at least in so far as we keep supersymmetry unbroken. Since supersymmetry has to be broken by 'soft terms' one may be worried about proton decay and other undesired effects. However, even before addressing the issues related to soft SUSY breaking, one may wonder whether Dterms and corrections to the Kähler potential may affect our analysis significantly. Although little is known about quantum corrections to the Kahler potential and D-terms in the intersecting D-brane models, some progress has been made in [59,60,61,62].
The main idea is to use in a sense the locally supersymmetric version of the exact Novikov-Shifman-Vainstein-Zakharov β function in order to derive an exact (perturbative) relation between corrections to K(Φ, Φ † ) and thus anomalous dimensions γ, related to wave-function renormalisation Z Φ , and running of g Y M and thus β function.
Except for theories or sectors with at least N = 2 susy, whereby the K'ahler potential for the vector multiplet is directly related to the holomorphic pre-potential and thus to the gauge couplings i.e. the gauge kinetic function and can be computed, when susy is minimal i.e. N = 1, the relation is much looser. In principle K and the D-terms in general, can get any sort of perturbative corrections. However these are to be compatible with the 'classical' symmetries, which include R-parity, baryon number B and Lepton number L. It is also known that standard 'gauge' instantons can only generate terms violating 'anomalous' symmetries, while 'exotic' instanton can violate non-anomalous ones, such as B − L in the (MS)SM. Depending on the number of fermionic zero-modes both gauge and exotic instantons may correct the gauge kinetic function(s), D-terms and the Kahler potential. It is rather reasonable to assume that such non-perturbative corrections be absent or very small in the quiver models in our classes, even when the string scale is close to -but smaller than -the Planck scale so much so that the full super-gravity structure should be taken into account.
In summary, the only 'seed' of R-parity breaking and Baryon (and/or Lepton) number violation seems to be the super-potential.
When supersymmetry gets broken, say in a hidden (strongly coupled) sector and then communicated to the visible sector, the situation gets more intricate. The structure of the low-energy Lagrangian, though constrained by the original supersymmetry, allows for dangerous mixings. In the quiver models we consider, proton stability, as previously discussed, largely relies on Lepton number conservation or on the fact that the final states should contain at least one susy partner. In Pati-Salam like models, it's built in via the selection rule ∆B = 2.

Conclusions
We have produced two examples of consistent quiver fields theories, indirectly generating a Majorana mass term for the neutron by means of exotic instantons. These are free of local tadpoles and thus irreducible anomalies. The phenomenology exposed by this class of models is interesting both for neutron-antineutron physics and LHC or other colliders, where a new vector-like pair could be detected. On the other hand, the models we suggest can be tuned to suppress FCNC's. However, in order to prevent fast proton decay, Lepton number is not to be violated. An alternative is to consider SO(10) GUT models or Pati-Salam like models in string theory that can lead to ∆B = 2 processes but no ∆B = 1 [63,64,65,66,67,68]. Although perturbative un-oriented strings do not admit spinor representations of orthogonal groups, P-S like models are easy to embed in this context [70]. In Appendix: Integrating out massive super-fields We find it more intuitive to apply a perturbative approach that we report in the following for pedagogical purposes.
the inverse mass matrix can be calculated as a perturbative series In our case, combining (48) and the inverse of (47) one gets the first perturbation then the second perturbation is the third perturbation is At the fourth order, we recover the exact result cited above in the paper.