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Neutron Majorana mass from exotic instantons

We show how a Majorana mass for the Neutron could result from non-perturbative quantum gravity effects peculiar to string theory. In particular,"exotic instantons"in un-oriented string compactifications with D-branes extending the (supersymmetric) standard model could indirectly produce an effective operator delta{m} n^t n+h.c. In a specific model with an extra vector-like pair of `quarks', acquiring a large mass proportional to the string mass scale (exponentially suppressed by a function of the string moduli fields), delta{m} can turn out to be as low as 10^{-24}-10^{-25} eV. The induced neutron-antineutron oscillations could take place with a time scale tau_{n\bar{n}}>10^8 s, that could be tested by the next generation of experiments. On the other hand, proton decay and FCNC's are automatically strongly suppressed and are compatible with the current experimental limits. Depending on the number of brane intersections, the model may also lead to the generation of Majorana masses for R-handed neutrini. Our proposal could also suggest neutron-neutralino or neutron-axino oscillations, with implications in UCN, Dark Matter Direct Detection, UHECR and Neutron-Antineutron oscillations. This suggests to improve the limits on neutron-antineutron oscillations, as a possible test of string theory and quantum gravity.


Introduction
Does a Majorana fermion exist in our Universe? This question remains one of the most intriguing for particle physics. When we address this issue, we would immediately think of neutrini. But curiously, Ettore Majorana suggested the neutron as a candidate rather than the neutrino [1]. A Majorana mass term δm n t n+h.c leads to neutron-antineutron oscillations through a non-diagonal mass matrix [2] M eff = m n δm δm * m n with denoting leptons and φ the Higgs doublet. (2) can produce Majorana masses for neutrini m ν ≈ φ 2 /M. For example a simple model generating (2) is the "seesaw mechanism", introducing a heavy RH neutrino N with mass term and Yukawa  For reviews see also [12].
This kind of experiments has an ample margin of improvement. In the near future there is the concrete possibility of increasing the neutron propagation time to ∆t ∼ 1 s and to suppress the magnetic field to B ∼ 10 −6 ÷ 10 −5 Gauss. Thus one could enhance the experimental limit to τ n−n > 10 10 s [13].
Neutron-antineutron transitions for free neutrons at τ ∼ 10 8 s do not lead to dangerous destabilization of nuclei. In the atomic nucleus, one has to consider the presence of nuclear binding energies, that strongly suppress the contribution of any external magnetic fields and of the neutron or antineutron β-decay widths. The effective hamiltonian takes the form where V n and Vn are the binding energies in the nucleus for a neutron and an antineutron. Vn << V n , |Vn − V n | ∼ V n ∼ 10MeV. The neutron in the nucleus is essentially 3 free for a time that can be estimated from the generalized uncertainty principle to be ∆E∆t ∼ 1 −→ t free ∼ 1 E bind ∼ 10 −23 s with E bind the average binding energy of the nucleon in the nucleus. The oscillation probability is given by where τ A is the internuclear transition lifetime, and p A the transition rate. The limits from nuclear stability translated into free-neutron are not so different form the direct search ones. For Oxygen for example it is τ > 2.4 × 10 8 s [14], for Iron τ > 1.3 × 10 8 s [15].
The Majorana mass is induced by effective operators of the form that depend on non-perturbative strong IR dynamics. A complete classification of the matrix elements n|O i |n (for different Lorentz and color structures) can be found in [16]. Using the MIT bag model [16] [17], the calculations involve six-folds integrals of More recent calculations using lattice QCD confirm these estimates [18].

Neutron Majorana mass from exotic instantons
Henceforth, we would like to show how a Majorana mass for the neutron could indirectly result from non-perturbative effects of quantum gravity type. In particular, we propose a simple un-oriented string theory model with intersecting D-branes, where "exotic stringy instanton effects", perfectly calculable and controllable in the case under consideration, can play this role. Unlike 'gauge' instantons, 'exotic' instantons do not admit an ADHM construction 5 . Though subtly compatible with gauge invariance, thanks to compensating axionic shifts, they elude a natural gauge theory interpretation. In the open-string theory context all instantons, gauge or exotic, admit a simple geometric interpretation: they are nothing but special D-branes, Eucliden D-branes (E-branes) wrapping an internal cycle, that could intersect the 'physical' D-branes. In a restricted class of string compactifications with a (MS)SM-like spectrum, these effects are naturally present and explicitly computable. So, we would like to argue that string theory could produce observable phenomena generated by non-perturbative effects that do not exist in a gauge theory, even without large extra dimensions that would favour a TeV-scale quantum gravity. The second suggestion is that phenomenological aspects of string theory could be simpler to test in rare processes and in particular in neutron physics rather than at colliders.
Obviously, a Majorana mass for the neutron could be generated in other ways, not directly related to string theory, in models that extend the standard model with GUT groups, Left-Right symmetric extensions, R-breaking MSSM or R-breaking NMSSM and so on. For a review of these, see [22]. For example in [23] an SO(10) GUT model without supersymmetry is suggested, that with a more complicated multiplet structure can achieve exact unification, also increasing the life-time of the proton to τ p ∼ 10 34 yr. Assuming that color-sextet scalars survive down to the TeV-scale -so much so that LHC would discover them -diquark couplings of these scalars lead to neutron-antineutron oscillations. A similar model cannot be simply accommodated within open un-oriented string theory 6 .
Alternatively, R-parity breaking MSSM's are consistent with several string inspired models. But if one allows for all R-parity breaking renormalizable terms in the MSSM, neglecting the family structure) then one needs a severe and unnatural fine tuning of the parameters to avoid proton decay with τ p < 10 34 yr 7 . The proton decay constraint does not give much 5 For recent review see [19,20,21] 6 On the other hand, (supersymmetric) SO(10) models can be easily constructed within heterotic string theory [24]- [25]- [26]- [27], or F-theory [28]- [29]- [30], but they are less appealing, less simple and less controllable. 7 R-parity violating operators have unnaturally small couplings but they are allowed by gauge invariance. Other R-parity violating gauge-invariant non-renormalizable effective operators that one could consider are QQQH or LHLH. room for n −n oscillations at δm ∼ (10 8÷10 s) −1 . In general R-parity breaking seems to complicate rather than solve the phenomenological problems of the MSSM. In particular, it introduces 48 extra dangerous parameters wrt the R-parity-conserving MSSM case. As an alternative, one can give up supersymmetry altogether and introduce a sort of "RH-neutron" that via a see-saw mechanism could induce a Majorana mass for the neutron 8 . This last mechanism cannot be embedded -at least in a straightforward fashion -in a string inspired SM-like model or in a supersymmetric GUT.
In the same class of SM-like string inspired model as in the present investigation, but with a more direct mechanism, exotic string instantons can also generate a Majorana mass for the RH neutrino as proposed in [32,33,34]. The Majorana mass for the RH neutrino N is given by M N ∼ M S e −S E , where M S is the string scale and S E measures the (complexified) world-volume of the exotic instanton brane in string units and depends on the moduli fields.
These seem to be the only simple possibilities to generate a neutrino or a neutron mass without Left-Right symmetry or explicit R-parity violating (non-)renormalizable terms. Exotic instantons naturally lead to dynamical R-parity breaking in MSSM, inducing R-violating non-renormalizable effective operators. In particular, as we will see in the next section, with a simple construction one can explain within this paradigm not only why R-parity violating operators are naturally suppressed by high-scale mass powers, but also how one can avoid a proton decay faster than 10 34÷35 yr and n −n oscillations faster than 10 8 −10 10 s 9 . Let us mention en passant that the µ-term problem in the MSSM could also be solved thanks to exotic stringy instantons as proposed in [33].
The string models that one can consider in order to embed (N-MS)SM-like theories, with chiral matter and interesting phenomenology, are divided in three classes: i) type I with magnetized D9-branes wrapping a CY 3 or alike; ii) un-oriented type IIB with space-time filling D3-branes and D7-brane wrapping holomorphic divisors in a CY 3 ; iii) un-oriented type IIA with intersecting D6-branes, wrapping 3-cycles in CY 3 . In the last class of models, the different particle families and tri-linear couplings arise from double and triple intersections, respectively. The interactions can be derived in a direct way from string amplitudes and the low-energy limit can be naturally described by matter coupled N = 1 SUGRA, with chiral and vector multiplets. The remarkable feature that motivates our paper is the presence of non-perturbative stringy effects in the effective action. Gauge instantons, that are point-like configurations in the 4d Minkowski space, in (un-)oriented type IIA, correspond to Euclidean D2 (E2) branes wrapping the same 3-cycle as a stack of "physical" D6-branes. The D6/D2 system has 4 mixed ND directions and the ADHM construction is obtained from open strings. In type I, one has E5 branes in the internal space, with the same magnetization as the D9, that are wrapped on the entire CY 3 . In (un-)oriented IIB one has D-instantons E(-1) or E3 wrapping the same holomorphic divisor as a stack of "physical" D7-branes.
On the other hand, exotic instantons correspond in type IIA to E2 branes, that are still point-like in the 4d Minkowski space but wrap different 3-cycles from the "color" D6 branes. These are not ordinary gauge instanton configurations: there are no ADHM-like constraints, no bosonic moduli in the mixed sectors and the number of mixed ND directions is typically 8. The counterpart in type I are E5 branes wrapping the entire CY 3 , but with different magnetization from the D9's, or E1 wrapping holomorphic cycles. In (un-)oriented type IIB with D3-and D7-branes they are E3 wrapping different holomorphic divisors from the D7's. When the D-branes are space-time filling, Ω-planes are introduced that are necessary for tadpole cancellation [60,61,62,63,64,65] and the consistency of the theory.
Ω-planes combine world-sheet parity with a (non) geometric involution in the target space. As a result Left-and Right-moving modes of the closed strings are identified. Both closed and open strings become un-oriented and more choices for the gauge groups and their representations are allowed [60,61,62]. D-branes come in two different types. There are branes whose images under the orientifold action Ω are different from the initial branes, and also branes that are their own images under the orientifold projection. Stacks of the first type combine with their mirrors and give U (N ) gauge groups. Stacks of the second type give SO(N ) or Sp(2N ) gauge groups. In this context, we could embed realistic gauge groups with chiral matter in a globally consistent model [66,67]. A simple way to construct a local SM-like model with open (un-)oriented strings is to consider a simple intersecting D-brane configuration, with 4-stacks, schematically represented in Fig. 1. This corresponds to a SM extension as [69]. In the next section we will present the basic features for the mechanism generating a Majorana mass for the neutron. Later on we will discuss relevant aspects of the model such as suppressed proton-decay or neutron-neutralino (or neutron-axino) mixings. For the time being, let us stress that E-branes are subject to the Ω-projections very much like the 'physical' D-branes. In particular we will be interested in E2-branes which are 'transversely' invariant under Ω and carry an O(1) gauge group. These and only these carry the minimal number of fermionic zero-modes (two) required for the generation of a dynamical super-potential rather than some higher-derivative F-term.

A simple model
Let us introduce the minimal superfield content of the MSSM where α = 1, 2 is for SU (2), i = 1, 2, 3 is for SU (3) and the lower index is the U (1) hyper-charge. For simplicity, the family structure is understood. (1) stack the open strings correspond to the weak gauge bosons , L, and a third additional abelian charge I R which acts as the third isospin component of an SU (2) R . The usual electroweak hypercharge is a linear combination of these three U (1) charges: with c 1 = 1/2, c 3 = 1/6, c 4 = −1/2, B = Q B /3 and L = Q L . Alternatively, inverting the above relations, one finds: The chiral particle spectrum from these intersecting branes consists of six sets (labeled by an index i = 1, . . . , 6) of Weyl fermion-antifermion pairs, whose quantum numbers are given in Table II. Note that the combination B − L is anomaly free, while both B and L are anomalous.
As mentioned already, the Q B (gauged baryon number) is anomalous. This anomaly is canceled by the 4D version [70][71][72][73][74] of the Green-Schwarz mechanism [33]. Non anomalous U (1)'s can acquire masses due to effective six-dimensional anomalies associated for instance to sectors preserving N = 2 supersymmetry [75,76]. 3 These two-dimensional 'bulk' masses 3 In fact, also the hypercharge gauge boson of U (1) Y can acquire a mass through this mechanism. In order 7 Figure 1: One usually considers the Baryon and Lepton number preserving renormalisable superpotential together with the soft susy breaking terms: scalar mass terms, Majorana mass terms for gaugini (zino, photino, gluini), trilinear A-terms, bilinear B-terms. The superpotential W preserves R-parity. Models of this kind can be locally embedded in string theory with intersecting or magnetized D-branes. Building global models is more challenging.
In addition, we consider a vector-like pair that we call D c i+2/3 and where 3 * −1/3 are the standard 'quark' charges in the anti-fundamental representation 9 ! ! Figure 2: The diagram inducing neutron-antineutron oscillation: C −2/3 and D +2/3 form the new vector pair, mixing through non-perturbative stringy instanton effects (white crosses). The higgsino in the propagator can connect the two specular parts of the diagram through a Majorana mass term (in general there is an elaborate mixing between higgsini, photino, zino and wino, the mass eigenstates are called neutralini and chargini). Fig. 1 after including Ω-planes. Although it is not our aim to construct a global string theory model with the desired properties, let us mention that several un-oriented string compactifications with intersecting or magnetised D-branes give rise to massless spectra with additional vector-like pairs such as the one we consider here [70,71].

of U (3). This is a minimal extension of the 4-stacks model in
More precisely, one has to keep in mind that the hyper-charge group U (1) Y in this model is in general a combination of 4 U (1)'s in the gauge group As a result Y is a linear combination of 4 charges q 3,2,c,d . In fact the four U (1)'s are recombined into U (1) Y , and other three U (1)s, one of which could be taken to be With these building blocks we can examine the process in Fig. 2 more closely.
It involves a scalar color triplet with baryon charge −2/3 that can come from (9).
These cannot be s-quarks from Q i +1/3 , but the exotic triplets C i −2/3 , resulting from the intersection shown in Fig. 2, can do the job. The second ingredient that we desire for the process in One can integrate out the D i ,C i pair and obtain at E << M 0 the effective operator the flavour structure is understood. At this point in order to complete the diagram in Exotic instantons can generate the desired non-perturbative mass term M 0 ijk D c i C jk , forbidden in perturbation theory by the U (1) factor in U (3), if they carry the correct number of fermionic zero-modes [58,73]. In string theory a term with an antisymmetric tensor can only be generated in a non-perturbative way since it violates the U (1) symmetry under which i 1 ...i N carries charge N , i.e. 3 in our case. Even though one Combined with the terms (11), this dynamically breaks R-parity: it is not possible to identify a consistent transformations under R of C and D and the other superfields in order to preserve the R-parity in all the processes. This way of breaking R-symmetry is more convenient than an explicit way, since it does not generate all the possible renormalizable or non-renormalizable operators.
Integrating out the fermionic modulini one obtains the dynamical super-potential where ijk results from the integration where M S is the string mass scale and S E2 depends on the closed string moduli that parametrize the complexified size of the 3-cycle wrapped by E2.
The superpotential term (11) generates the effective operator withq squarks, q quarks. The conversion of susy particles to SM particles brings in further suppressions. By power counting arguments, up to some adimensional O (1) factor, the 6-fermion effective operator that leads to a Majorana mass for the neutron As mentioned in the introduction, the actual strength of the coupling and the value of δm depend on strong IR dynamics that is beyond the scope of our analysis. Based on phenomenological models and numerical simulations [16] [17] [18] [75] (for interesting astrophysical consequences of TeVscale gravity see [76]). In this case M 0 ∼ 1 − 10 TeV and the vector like pairs would be accessible at LHC. This last possibility leads to higgsini with MH ∼ 10 6÷10 T eV , in contrast with susy at the TeV scale for LHC, or split-supersymmetry [77] with TeV-scale quantum gravity.

Further implications
The construction we propose leads to interesting questions and implications that we cannot refrain from commenting on:

Proton decay
Proton decay in our model is more suppressed than in models with explicit R-parity violating terms, depicted in Fig. 5 [78]. Apparently, the proton decay seems to pose a problem also in our case. The effective super-potential operator

Cosmology
Rapid B and L violating interactions induced by RPV operators may wash out any pre-existing baryonic or leptonic asymmetry. Consequently, such processes should be highly suppressed at low temperatures. Since sphalerons, active above the weak-scale, violate B + L, it is typically required that the RPV-induced rates are sufficiently slow above that scale. The bounds on the dRPV operators are similar to those in standard holomorphic RPV. One finds X η 10 −7 and κ eff i < 10 −6 where η stands for any η ijk , η ijk or η ijk [2,22,23]. As we show below, these cosmological bounds typically imply displaced decays at the LHC. Nonetheless these bounds can be easily evaded in several ways (see [2] and references therein). For example, the bounds are irrelevant if the baryon asymmetry is generated at or below the electroweak scale. Conversely, as discussed in [9,23], when a single lepton flavor number is approximately conserved the bounds can be significantly weaker.

LHC PHENOMENOLOGY
The phenomenology of models with dRPV can be very different from those with R-parity conservation and even from those with traditional RPV described by (1). The details depend greatly on the identity of the lightest supersymmetric particle (LSP). Here we briefly comment on three interesting possibilities which crucially differ in their collider phenomenology from standard RPV: stop LSP, gluino LSP and sneutrino LSP, with the first two most relevant for naturalness. Further details on these and other interesting possibilities will be given in [12].
Consider first the stop LSP. In all of the nonholomorphic operators of (2), stop decays are induced from SUSY-conserving interactions in which the stop is Thus the stop LSP case may manifest itself uniquely as four displaced b's, where each pair reconstructs to a single displaced vertex, and the two pairs have a similar invariant mass. The situation is illustrated in Fig. 3. We stress that such decays do not exist in the holomorphic RPV scenario. The collider search for a stop LSP should be significantly altered in order to discover dRPV.
Next consider the case of a sneutrino LSP, where the LSP decay is governed by the η couplings which induce the operators u Li u † Rjν k +d Li u †

Rjẽ †
Lk . Since the 3rd generation couplings are typically least suppressed, the leading decay mode will beν → t L t † R with a decay length cτν 1 mm 10 −2 2 10 −5 2 165 GeV m 2 . Figure 5: Proton decay in R-violating MSSM models [78]. Proton decay strongly constrains the parameters of the operators involved in neutron-antineutron transitions (Fig. 6). These are automatically suppressed in our simple construction that breaks R-parity dynamically.
that previous studies consider exclusively holomorphic RPV, we will study below the case when only the nonholomorphic RPV terms appear. Before analyzing the constraints, let us briefly discuss assumption (III). The inclusion of flavor dynamics implies that the various operators discussed above are suppressed according to their flavor structure. Numerous models that introduce such suppressions exist, including, for example, theories with horizontal symmetries as in FN models [10], or ones with strong interactions [13][14][15]. Consequently, the low energy parameters, η, η , η , κ and κ , are suppressed in a flavor-dependent manner. For example, the η ijk 's can take the form where = O(0.1) is a small parameter and q α are the various charges of the SM fields under the FN symmetry. Similar expressions hold when q α characterize the partial compositeness in the case of an RS-type scenario. While a comprehensive study is beyond the scope of this paper, we stress that all the constraints discussed below are easily satisfied with, for example, a simple choice of FN charges. In particular, a straightforward extension of the alignment model of [16] to the lepton sector allows for a viable dRPV model, without any additional assumption such as the typically needed lepton-number conservation. A complete realization of this scenario will be discussed in an upcoming publication [12].
these bounds are easily satis simple flavor model.

∆B = 2 P
The η term in (2) violates unit. Consequently it is im bounds on ∆B = 2 processes cleon decay, obtained by two obeyed. The simplest way is which will generate a dimen most general flavor index st display the subset necessary ering the left diagram of This leads to n−n oscillation π + π + for i, j, k = 1 and pp → The n −n oscillation time

Neutralino-neutron mixing and more
The non-pertubatively generated effective operator H d QQQ/M 0 curiously implies neutralinoantineutron, antineutralino-neutron, neutralino-neutron mixing (Fig. 9). Higgsini mix with wini, photini and zini. The resulting mass matrix has 6 mass eigenstates: 4 neutralini and 2 chargini. The mass terms for the neutralini read where λ B is the gaugino associated with B µ of U (1) Y , λ 3 the gaugino associated to A 3 µ and Ψ H 1,2 the Higgsini. The mass matrix is given by where M 1 and M 2 are respectively the U (1) Y and SU (2) L soft supersymmetry breaking gaugino mass terms. The eigenstates are usually denoted by χ 0 1,2,3,4 . In general, the mass matrix could be extended when extra U (1)'s appear as in our model by including axiniã [79,80]. On the other hand, one has also to consider the operator H d QQQ/M 0 , this modifies the matrix, giving rise to an effective mixing of neutrons with axini and neutralini. The limits on neutron oscillations in invisible channels are only of τ n−inv > 414 s at 90% CL in suppressed magnetic field [81] [82] 12 . So, there is no phenomenological problem with neutrons oscillating into the stable lightest neutralini or stable axini. Naturally, the transition probabilities will be suppressed if the neutron mass is much smaller or much larger than the neutralini and axini masses.
On the other hand, as shown in Fig. 9, it seems that in this way the transition probabilities χ − n and χ −n could be exactly equal, leading to a rapid transition n −n in 2 − 1000 s. Clearly, if neutralini or axini have masses of m χ,ã >> 10 GeV, 12 These limits are placed in the search for a hint of Mirror Dark Matter. The phenomenology of neutronmirror neutron oscillations are considered in [83] [84]. Currently, there is an anomaly of 5σ (with respect to the null hypothesis) in condition of magnetic field B 0.2 Gauss in Ultra Cold Neutron (UCN) [82]. This remains to be confirmed in future experiments. This could be explained if the Earth itself is the origin of a long range Yukawa type fifth force acting on the neutralini or axini. In this case, the transition probability could be enhanced in condition of strong magnetic field around 0.2 Gauss as a resonance between the experimental magnetic field and the new interaction.
transitions into neutron and antineutron are strongly suppressed and the problem is closed, without any implication for UCN physics. However, the two transition rates could be very different if one considers the full n × n mass matrix mixing neutrons, neutralini, antineutrons, axini. In fact, in general, this matrix can violate CP, because of the Yukawa-like couplings inside the processes and extensions of the two matrix blocks χ − n and χ −n with N axini. In particular, the introduction of N axini introduces new free parameters, as non-diagonal mass terms Uã 1,2,..,N −n , CP-violating phases φ 1,2,..,N and axini masses mã 1,2,...,N . Adjusting the parameters in the model, one can get the interesting case τ χ−n << τ χ−n . This is not so different from the proposal of extending the mass matrix of the neutrini with one or more sterile neutrini and inducing a difference in the processesν i →ν j with respect to ν i → ν j . On the other hand the transitionn − n through oscillations with neutralini and acini becomes an alternative to generate n −n oscillation to be tested in near future experiments.

WIMPs and DAMA
Light neutralini or axini are WIMP's (weakly interacting massive particles) and could be natural Dark Matter candidates or at least account for a fraction thereof. For example, one could imagine the model dependent scenario of axino dark matter, with χ − n fast oscillations and neutralino decaying into an axino and an axion χ →ãa. In this scenario one could assume m χ m n mã (m χ −mã m a << eV ). This situation is also very interesting for UHECR (Ultra High Energy Cosmic Rays), as we will see in the next section. So, our model could connect the ultra-cold neutron phenomenology with underground direct detection experiments.
In the last 10 years or so, significant progress has been made in efforts to directly detect dark matter. The DAMA/NaI [85] and DAMA/Libra [86] experiments have obtained exciting results (see also [87]). In particular, these experiments have observed an annual modulation at 9.3 σ C.L. [88], as expected for a signal from Dark particles. Different anomalies in other direct detection projects, CoGeNT [89], CRESST-II [90] and recently in CDMS-II (CDMS-Si) [91], seem to favor DAMA results. Interestingly, DAMA signal suggests light neutralino candidate in a region of masses 1 − 50 GeV [92] 13 . So light a neutralino is not ruled out at all by LEP, Tevatron and LHC data, the situation is strongly model dependent. The first data from LHC tend to disfavour a TeV-scale MSSM model [96] and the desired m χ 0 ∼ 1 GeV for interesting oscillations is in tension with respect to the neutralino mass lower bound by the Cold DM relic abundance Ω χ h 2 (Ω CDM h 2 ), derived in [97]- [98]: m χ 0 > 7 − 8 GeV.
In contrast to neutralino, the axino mass is unconstrained experimentally. Moreover from the theoretical point of view, one can easily imagine it in the few GeV range [100].
Constraints on a light axino are not so rigidly related to the SUSY scale. Depending on the model, SUSY could be broken at higher scale compatibly with a light axino. For axino, the parameter space is constrained by axion couplings with gluons, photons and fermions (see [101] for a review about axion constraints), but neutron-axino oscillations are not directly related to axion PQ-like scale. So a light axino seems to be favored as a WIMP candidate of 1GeV with respect to neutralino. DAMA collaboration analysis for the neutralino [92] applies directly to the axino.

UHECR and GZK effect
Other implications of neutron oscillations with a sterile partner like a neutralino or an axino could come for Ultra High Energy Cosmic Rays (UHECR) phenomenology. A possible effect of n −χ 0 or n − χ 0 or n −ã oscillations on the Greisen-Zatsepin-Kuzmin (GZK) cutoff 14 shape in UHECR could be detected 15 . In fact proton can collide with CMB photons, producing protons and π 0 , or neutron and π + , with practically the same probability P pp,pn 1/2 and a mean free path l mf p ∼ 5 Mpc. Then the produced neutrons could oscillate in a time interval τ ∼ 1 − 500 s into neutralini or/and axini, which can propagate in the CMB without interactions. An example of an interesting, but model dependent, scenario may be as the following. Consider the case of a neutralino with m χ m n and an axino with a mass smaller then m χ and dispersion tensor and rotation of the galactic halo. Finally the possible contributions of non-thermalized Dark Matter components to the galactic halo, such as the SagDEG stream, or other kinds of streams as those arising from caustic halo models, are discussed in [94] [95]. These could change the local DM speed distribution and the local density. 14 UHE nucleons interact with the CMB radiation field [102] [103], there are two signatures that can be related to these: lepton pair-production p+γCMB → e + e − p [104] [105], and pion photo-production pγCMB → π 0 p, π + n called Greisen-Zatsepin-Kuzmin (GZK) cutoff [106] [107]. So, the position of GZK cutoff is approximately defined by the energy where lepton pair-production and the pion photo-production rates. The energy losses become practically equal at EGZK 50 EeV [108]. 15 This effect is similar to the neutron-mirror neutron oscillations discussed in [109]. However there is an important difference: the mirror neutrons in the the mirror sector β-decay into mirror protons. Then in the Mirror scenario we have also to consider the interactions of the mirror protons with mirror CMB. From BBN limits, Mirror CMB temperature must be less then the ordinary CMB one. On the other hand, in our case neutralini or axini have not other relevant interactions with matter to consider, if they are assumed as WIMP-like particles. So the resulting effect on the GZK shape could be very different.
assume that neutron-neutralino transition rate is much faster than neutron-axino one, this last much faster then axino-neutralino transition. This corresponds just to an effective mass matrix with a non-diagonal mixing terms constrained by the hierarchy: µ χ−n >> µã −n >> µ χ−ã . Then one can have a decay of χ into one axion and one axino through the coupling photino-axino-axion. On the other hand, one can assume χ to be stable against other decays unrelated with this interaction. Assuming the rate for χ → aã to be much slower than forn → χ, such as τ χ→aã > 1000 s, one could imagine a chain of processes as the one represented in Fig. 9 that would involve: i) pγ CM B → nπ + , ii) n − χ oscillations in 1 − 500 s iii) χ → aã in more than 1000 s; iv)ã → n after a length of l mpr >> 1Mpc (also considering the very high Lorentz factor); v) neutron β-decays into protons. This chain leads to a very efficient propagation of protons and to a modification of the spectrum above the GZK cut-off. We would like to stress that this particular model is also connected with UCN and Dark Matter Underground Direct Detection experiments.
The total effect could be a modification of the spectrum at energy above the GZK.
In [110] Auger's data, the GZK seems to appear shifted below in energy wrt theoret- On the other hand, looking at the Telescope Array (TA) data [111] [112] the experimental GZK cutoff seems to be above theoretical expectations, apparently in contradiction with Auger data. However, Auger and Telescope Array spectra are consistent within the systematic uncertainties (see [113] for analysis in common between the collaborations).
Another unclear situation comes from the determination of the nuclei fractions, which are controversial and affected by a lot of uncertainties. Auger atmospheric depth data X M ax [g/cm 2 ], an indicator of the UHECR chemical composition, seem to suggest that the larger part of higher energy points are nuclei: protons seem to be suppressed at energy around 10 19 eV, smaller than the GZK cutoff energy scale, also considering the large uncertainties of the energy scale mentioned above. In particular, Auger Collaboration claims the presence of nuclei in UHECR, with a gradual transition from light to heavy composition between 10 18 eV and 5 × 10 19 eV [114]. If these results were confirmed, the n − χ 0 and/or n −ã and/or χ −ã oscillations would not affect the GZK cutoff shape. But these estimates are very model-dependent since it is necessary to extrapolate models of hadronic interactions to energies much higher than those at which they were tuned, i.e. the TeVscale (LHC). On the contrary, HiRes [115] [116] and TA [111] [112] show that the chemical composition is dominated by protons from 10 18 eV to 10 20 eV. But they use a different data analysis and they have much less statistics with respect to Auger, therefore it is still not known if the disagreement is real or not (see [113]).
The observations that only 30% of UHECR are within cones of few degrees from some known astrophysical source, like AGN, Blazars, Supernovae etc, seems to disadvantage the hypothesis that only protons compose UHECR at E > 10 19 eV due to the basic fact that for a proton of this energy the trajectory cannot be curved more than few degrees by an average intergalactic magnetic field. A nucleus with atomic number Z is Z times easier to accelerate and its trajectory to be curved with a magnetic field (see [117]). However, the propagation of UHE protons with E > 10 19 eV could be more and more efficient because of neutron-neutralino and/or neutron-axino oscillations, they could come from unknown sources at cosmological distances (depending on model considered) not contained in the visible horizon. In this last case the angular correlation analysis could not be conclusive.
The mechanism proposed is independent from the proton sources, which could be distant Blazars, or exotic new physics processes like superheavy particle decays (for a review se [119]), monopole-antimonopole annihilations, cosmic strings or other topological defects (for a review see [118]), scalaron oscillations in f (R) modifications of gravity [120], and so on.
Naturally, a hybrid scenario can explain UHECR with E > 10 19 eV : a fraction could be UHE nuclei coming from AGNs or other astrophysical known sources, and a part could be protons coming from unknown sources.
So it would seem that the status of UHECR is still completely open. In future, with more statistics, error bars on the individual points will shrink a bit. Room for some improvement will come from better measurements of air fluorescence and so on 16  the observatory project JEM-EUSO will be sent on the International Space Station, with the opportunity to collect much more statistics, alas with poorer resolution [121].
For the moment, it seems more reliable to test neutron exotic oscillations in UCN experiments or in neutron base-lines. In particular the oscillations n −n − χ 0 or with axini could be studied in future neutron-antineutron experiments.
On the other hand the limits on proton-charginos oscillations are more stringent (the limits are the same as for proton decay), but this is not necessarily connected with n −n or χ 0 − n diagrams in the parameter space under consideration, including MSSM parameters, extra U (1)'s, M 0 etc.

Meson physics and FCNC's
A natural question for phenomenology is if our model is predictive for meson physics in K, D, B, B s decay channels or in The answer is positive, the present model can generate these processes, but they are strongly suppressed, as shown in Fig. 10 and Fig. 11.
Another delicate question that we cannot by-pass is about FCNCs in quark sectors: are they generated in our simple model? The answer is again positive, but they are highly suppressed. Essentially, the relevant diagrams come from the variant in Fig. 11, Figure 9: Example of a mechanism for UHECR protons propagation, involving rapid oscillations between neutron and neutralino τ n−χ 1 − 500 s; neutralino decay into axion and axino with τ χ→aã > (5 ÷ 10)τ n−χ and finally the transition of the axino into the neutron with τã −n >> τ n−χ .
closing one more quark-antiquark line. The 4-loop suppression is beyond any observable effects.

Running coupling
The

Event Reconstruction
An importante piece in the event reconstruction is the lateral distribution of the signal, as recorded in the Cherenkov stations. A lateral distribution function, based on shower simulations and previous data from the Auger Observatory, is fitted to the experimental curve, in order to obtain the value of signal at 1000 m from shower core (see figure 2). A universal calibration curve using this information and those from fluorescence detector, is used to obtain the primary energy, within a systematic error of 22%, whereas the statistical error is ∼ 15 %. The calibration curve is then used for all showers, also for non-hybrid detection. Figure 3 shows a high-precision energy spectrum, at energies above 10 18 eV. The operation in hybrid mode optimizes event reconstruction: the timing information from telescope pixels together with the time and signal structure from surface stations, as cited above, gives a resolution for core location within 50 m and an angular resolution for the arrival direction of primary of 0.6 o [8]. Non hybrid events have angular resolution about 1.5 o .

Mass Composition
The atmospheric profile of an extensive air shower, i.e, the number of particles as a function of depth, carries important information about the mechanisms of particle production and absorption. The depth at which the shower reaches its maximum, called X max in the literature gives important clue about the mass of the primary particle [6]. The change of < X max > per decade of energy, the so-called Elongation Rate and its shower-to-shower fluctuations, RMS (X max ) are sensitive to changes in mass composition with energy. These variables have different behaviors, if the primary is a proton or an iron nucleus, as predicted by different hadronic interaction models. Figure 5 shows the experimental behavior of both variables with energy, together with the predictions of the models EPOS 1.99, SIBYLL 2.1, QGSJET01C, QGSJETII [6]. We find a tendency in the composition: if the models describe correctly the interactions, when the energy gets higher, the composition departs from that expected for protons and tends (L, L ) could play also an important role in lepto-genesis. Finally, if their mass were around 1 − 10 TeV, they could be detectable at LHC, for example in decay-channels of the singlets.

Different Ω's and fluxes
Instead of an Ω − -plane one can consider an Ω + -plane in Fig. 3. This construction generates color sextets that could induce n −n oscillation in a different way, similar to the scalar color sextet of Babu-Mohapatra SO(10) model [23]. They could play an interesting role in the baryo-genesis 18 . In this case the (L, L ) are not antisymmetric singlets but symmetric triplets.
The soft susy breaking terms could also be induced by bulk fluxes. For example, gaugino masses could be generated by bulk fluxes such as NS-NS H ijk or R-R F ijk 3-form fluxes, from an interaction λ t Γ ijk τ H ijk + iF ijk λ ∼ M λ λ t λ. So in more complicated situations, one has to consider the back-reaction of the fluxes on the "exotic" instantons [129,130]. These could modify the simple analysis proposed in this paper 19 .

Majorana mass for RH neutrini
In our model, a Majorana mass terms for the RH neutrini N i can be generated that induces the observed small neutrino masses thanks to the see-saw mechanism. Majorana mass terms for RH neutrini are forbidden in perturbation theory by U (1) symmetries such as U (1) B−L . However they can be generated by non-perturbative stringy instanton effects. In unoriented type IIA string models, the pseudo-scalars needed to make the U (1)s massive correspond to the R-R 3-form integrated over 3-cycles.

Extra (anomalous) U (1)'s and Z'
In any string-inspired extension of the (MS)SM of Fig. 1, new vector bosons Z 20 appear that get a mass by a Stückelberg mechanism [140]. In addition, Generalized Chern-Simon (GCS) terms are introduced in order to cancel anomalies [141], in combination with a generalised Green-Schwarz mechanism [142]. If one assumes the string mass scale to be at M S = 1 − 10 TeV, even in our model, processes such as Z → ZZ or Z → Zγ could produce interesting signals at the LHC, as already discussed in the literature [143]- [79]. 20 For discussions about the existence of additional massive neutral gauge bosons see [131]- [139] 24

Conclusions and Remarks
We have shown how exotic instantons can indirectly generate a Majorana mass for the neutron. The crucial ingredients are a local intersecting D6-brane configuration with Ω6-planes giving rise to the MSSM super-fields plus a vector-like pair of 'quark' super-fields D , C. An O(1) instanton (E2-brane) singly intersecting the relevant D6branes generates a dynamical super potential mass term for D , C. Integrating these out, while taking into account their interactions with the standard MSSM super-fields, produces an effective Baryon number violating term that in turn leads to the desired highly-suppressed Majorana mass for the neutron.
We have then discussed phenomenological implications and commented on potential drawbackks of the proposed mechanism. Proton decay and FCNC are highly suppressed while several signals of neutron-antineutron or neutron-neutralino/axino oscillations can give rise to interesting signatures in DM, UCN and UHECR experiments. This shows how interesting string theory could be for near future experiments, with its peculiar non-perturbative stringy instantons effects, not admitting a natural gauge theory interpretation. In particular, these could generate Majorana masses for neutrini and for neutrons. As a consequence, the next generation of experiments on neutrinoless-double-beta decays and neutron-antineutron oscillations could test quantum gravity non-perturbative effects. In particular, limits on n −n oscillations are quite mild with respect to limits on proton decay: τ n−n > 10 8 s ∼ 10 −33 τ p→πe,Kν,etc. .
The stringy instantons effects are completely calculable in some string models containing the Standard Model, as the one we have considered in the present paper. In more complicated string models as heterotic strings or in the presence of fluxes, stringy instantons effects becomes more difficult to calculate, but their existence is a quite general feature.
We have also seen how these effects could interplay with large extra-dimensions, with a rich phenomenology for LHC. However, large extra dimensions are not necessary to generate interesting rare processes like n −n oscillations with non-perturbative stringy instantons effects. We would like to stress that our mechanism can be compatible with highly suppressed proton decay. This is a crucial feature: if future experiments on proton decay would enhance the limits, the most interesting models for neutron-antineutron phenomenology would become models of the present kind that naturally avoid too fast a proton decay, contrary to L-R symmetric or R-violating (renormalizable) extensions of the MSSM. In fact, for these last two classes of models, an improvement on proton decay limits (for example at 10 35 − 10 37 yr) would strongly constrain n −n oscillation at M ≈ 300 − 1000 TeV.
We conclude that string theory could be experimentally testable in some of its Standard Model like versions, as a consequence of its better known non-perturbative aspects. Further theoretical discovery about non-perturbative aspects of string theory could show up as absolutely unique and interesting for experimental physics, in unexplored ways that one cannot imagine at present. Future experiments on rare processes as n −n could help us to clarify our understanding of the Universe and disclose its hidden Beauty.