Neutron Majorana mass from Exotic Instantons in a Pati-Salam model

We show how exotic stringy instantons can generate an effective interaction between color diquark sextets in a Pati-Salam model, inducing a Majorana mass term for the neutron. In particular, we discuss a simple quiver theory for a Pati-Salam like model, as an example in which the calculations of exotic instantons' effects are simple and controllable. We discuss some different possibilities in order to generate $n-\bar{n}$ oscillation testable in the next generation of experiments, Majorana mass matrices for neutrini and a Post-Sphaleron Baryogenesis scenario. Connections with Dark Matter issues and the Higgs mass Hierarchy problem are discussed, in view of implications for LHC and rare processes physics. The model may be viewed as a completion of Left-Right symmetry, alternative to a GUT-inspired scenario. Combined measures in Neutron-Antineutron physics, FCNC, LHC, Dark Matter could rule out the proposed model or uncover aspects of physics at the Planck scale!


Introduction
How can Matter be generated in our Universe? And how are neutrino masses generated? Has the neutron a Majorana mass?
In principle, these three questions could appear unrelated. However, in Left-Right symmetric models with SU (2) L × SU (2) R × SU c (3) × U (1) B−L gauge group, one can find intriguing and elegant connections between these three issues. A Left-Right model 1 E-mail: andrea.addazi@infn.lngs.it 2 E-mail: massimo.bianchi@roma2.infn.it is naturally embedded in a Pati-Salam (P-S) model with G 224 = SU (2) L × SU (2) R × SU (4) c gauge group [1], that in turn can be embedded in an SO(10) GUT.
As originally suggested in [2], new Higgses ∆ R in the (1,3,10) (and ∆ L in the (3, 1, 10 * )) of G 224 can be introduced in PS models in order to spontaneously break Left-Right symmetry, through ∆ R = v R = 0 and ∆ L = 0. This mechanism also produces Majorana masses for Right-Handed neutrinos, that can trigger a seesaw mechanism as suggested in [3] with ∆ c l c l c (1, 3, 1) −2 generating Right-Handed neutrini masses via ∆ c ν c ν c ν c ν c . In GUT SO(10), the (1, 3, 10) of G 224 and its conjugate are contained in the 126 representation 3 . But ∆ c (1, 3, 10) also contains color sextet diquark fields ∆ c q c q c (1, 3, 6) 2/3 , leading to possible new effects. In particular, these sextets can induce Baryon number violating processes beyond the Standard Model (BSM). Color sextets can also play an important role in some post-sphaleron baryogenesis mechanism [4,5,6]. In susy extensions, a quartic superpotential term can appear that, among other terms, produces a term coupling three color sextets ∆ c q c q c and one color singlet ∆ c ν c ν c , as ∆ c u c u c ∆ c d c d c ∆ c d c d c ∆ ν c ν c . When the color singlet ∆ c ν c ν c takes an expectation value, U (1) B−L is spontaneously broken and Right-handed neutrini get a mass [3]. Moreover a Majorana mass for the neutron is generated through the processes shown in Fig. 1-(a)-(b). This can be directly tested in Neutron-Antineutron transition experiments, as firstly proposed in [2]. As shown in [7], constraints from post-sphaleron baryogenesis, and neutrino oscillations imply a precise prediction about neutron-antineutron transitions: an oscillation time τ n−n ≈ 10 10 s accessible to the next generation of experiments! In principle, color sextet scalars could be as light as 1 TeV and they could be directly searched at the LHC, as proposed in [8]: dijet data put constraints on the couplings between colored scalars and quarks. In [9], bounds are shown in comparison with LHC data. On the other hand, FCNC processes could impose stronger constraints on the 3 The complete decomposition reads 126 → (1, 3, 10) + (3, 1, 10 * ) + (2, 2, 15) + (1, 1, 6).  [11]. The transition is induced by color sextets ∆ c u c u c and ∆ c d c d c . b) We show the main diagram for neutron-antineutron oscillation in a supersymmetric SU (4) c × SU (2) L × SU (2) R model [11]. The transition is induced by color sextets ∆ c u c u c and ∆ c d c d c arising from the decomposition of ∆ c (1, 3, 10). The latter participate in a nonperturbative quartic superpotential term. The diagram involves also gauginig (gluini, zino or bino), squarksd c , and susy partners of the color sextets∆ c d c d c .
sextets with respect to LHC direct searches (see [10] for comparison with experimental limits). For example, the ∆ c dd field couples to two down-type quarks dd, ss, bb: it mediates B 0 d,s −B 0 s,s , K 0 −K 0 oscillations and B mesons decays. On the other hand ∆ c uu mediates D 0 −D 0 oscillations and D-decays like D → Kπ, ππ. These analyses show that for coupling strengths of order 10 −2 , the mass of the color sextets has to exceed the TeV-scale.
In this paper, we propose a (susy-)PS model that is alternative to the SO(10) GUT inspired model mentioned above. We consider an (un)oriented open string model with intersecting D-brane stacks, producing a susy PS like model. Models of this kind have been previously considered e.g. in [12], where an analysis of the mass spectrum and low-energy phenomenology has been carried out. In oriented string theory, a simple way to generate a U (N ) gauge theory is to consider a stack of N D-branes, parallel to each other. In this way the excitations of the open strings stretching between the N Dbranes reproduce at low energy the fields in the adjoint of the U (N ) gauge symmetry.
In type IIA, compactified on a six-dimensional (CY) manifold, one can consider stacks of intersecting D6-branes, filling the 4D ordinary Minkowski spacetime, and wrapping internal 3-cycles. From strings connecting different stacks of branes, we can construct chiral fermions, localised at the four-dimensional intersections of two stacks of D6branes a and b, in the bi-fundamental representation of U (N a ) × U (N b ) [13]. The net (positive -negative) number of intersections of two branes a and b is a topological invariant, representing the number of massless fermions. In the case in which D-branes are space-time filling, Ω-planes have to be introduced in order to cancel tadpoles and irreducible anomalies [14,15,16,17,18,19]. An Ω-plane implements a combination of world-sheet parity and a (non) geometric mirror-like involution in the target space.
As a consequence Left-and Right-moving modes of the closed strings are identified; closed and open strings become un-oriented. More choices for the gauge groups and their representations are allowed [14,15,16]. In this way, one can produce stacks supporting U (N ), SO(N ) or Sp(2N ) gauge groups. This is interesting in order to construct realistic gauge groups, with chiral matter in a globally consistent model [20,21]. The closed strings propagate in the entire ten dimensional space-time: some mediate gravitational interactions, some behave as axions or scalar moduli fields.
In principle, one can construct a PS like gauge group U (4) × Sp R (2) × Sp L (2) or U (4) × U R (2) × U L (2) in terms of intersecting D-brane stacks and Ω-planes. In [12] the case U (4) × U R (2) × U L (2) was analysed in some detail. In the present paper, we focus on the U (4) × Sp R (2) × Sp L (2) case with an Ω + -plane that requires a stack of four Dbranes and its mirror image under Ω, producing U (4), and two stacks of two D-branes each, identified with their own images under Ω, producing Sp L (2) and Sp R (2). This model has extra anomalous U (1)'s that could seem dangerous from a gauge theory point of view. On the other hand, in string theory, Generalized Chern-Simon (GCS) terms appear that cancel anomalies [22,23], in combination with a generalised Green-Schwarz mechanism [24,25]. The extra Z gauge bosons can get a mass through a Stückelberg mechanism [26,27,28,29]. We will return onto phenomenological implications of this in the next section. There is however a real problem in this scenario. It is not possible to represent (1,3,10) in terms of open strings. Perturbative open strings have two ends and can at most carry fundamental charges with respect to two classical gauge groups. On the other hand, the triplet is the adjoint of Sp(2), i.e. the symmetric product of two doublets, and the decaplet of SU (4) is the symmetric product of two tetraplets. States in the (1, 3, 10) (or its conjugate) would correspond to multi-pronged strings with two ends on the U (4) stack and two ends on the Sp(2) stack 4 that do not admit a perturbative description.
On the other hand, we will show that a spontaneously breaking pattern to the SM, giving masses to the neutrini, can be recovered in this model. In fact, we will see that φ RR (1, 3, 1) and φ LL (1, 1, 3) appear as excitations of open strings with both ends attached to Sp(2) R or Sp(2) L , while ∆(1, 1, 10) and its conjugate ∆ c (1, 1, 10 * ) appear from open strings joining U (4) and U (4) identified which one other under Ω. As a Similarly to the case of ∆(1, 3, 10), color sextets are contained in ∆(1, 1, 10).
Our main suggestion is that the super-potential can be generated by non-perturbative quantum gravity effects peculiar to string theory, called "exotic instantons". These are associated to Euclidean branes (E2-branes in our case), wrapping internal 3-cycles, that could directly produce such interactions, in a calculable and controllable way in models like type IIA (un)oriented strings. We would like to stress that this class of instantons exists in string theory only, not in gauge theories. The resulting superpotential term is suppressed by the scale where M S is the string scale and e +S E2 depends on the 'size' of the 3-cycles wrapped by the relevant E2-brane. We would like to remark that the suppression scale is higher (in principle also much higher) than the string scale. This is a peculiarity of the non-renormalizable nature of such a non-perturbative term in the string effective action. As a consequence, the hierarchy depends on the particular model: e +S E2 can be approximately 1 for a 'small' 3-cycles or e +S E2 >> 1 for a 'large' 3-cycles. So, depending on the String scale, assumed to be larger than some TeV's at least, and the size of the 3-cycle, it is possible to generate such an operator near the LHC scale or at a much higher scale. This leads to two very different branches for phenomenology. In particular, for M 0 10 13 GeV, color sextets appear near the TeV scale, with potential implications in meson physics and at LHC, as mentioned above. On the other hand, for M 0 M S 10 TeV, a post-sphaleron scenario is possible and testable at the next generation of experiments on neutron-antineutron oscillations, with heavy color sextets, at a scale m 6 >> T eV that can be generated by closed-string fluxes, as shown in [30] for quiver theories and reviewed below. In this case, there is no possibility to produce  (10) scenario, as remarked in [7]. Our string-inspired scenario also naturally provides several candidates of WIMP dark matter as we will see.
We would like to mention that such an operator as (2) [47,48].
The paper is organized as follows: in Section 2, we briefly review what are stringy instantons and quivers. In Section 3 we propose a simple and consistent quiver for a Pati-Salam model generating a Majorana mass for the neutron through exotic instantons. In Section 4 we discuss some phenomenology resulting from this model. In Section 5 we present our conclusions and final remarks.

Exotic Instantons and Quivers
In this section, we briefly review D-brane instantons and unoriented quiver theories.

Instantons
In 4-dimensional gauge theory, instantons are point-like configurations, that extremize the Euclidean action for a given topological charge. In string theory, instantons admit a simple geometric interpretation: they are special Euclidean branes wrapping some (internal) cycle. In theories with (unoriented) open strings, these are Euclidean Dbranes (E-branes) that can intersect the 'physical' D-branes 6 . In (un-)oriented type IIA, gauge instantons can be classified as Euclidean D2 (E2) branes wrapping the same 3-cycle as a stack of "physical" D6-branes. In (un-)oriented IIB, instantons are E(-1) or E3 wrapping the same holomorphic divisor as a stack of "physical" D7-branes. In type I, instantons are E5 branes in the internal space, with the same magnetization as the D9, wrapping the entire CY 3 .

Quivers
The effective low energy description of the dynamics of D-branes at Calabi-Yau singularities is captured by a quiver field theory. Usually, the (supersymmetric) quiver wrapping non-compact cycles so much so that g 2 Y M,Dp = g s (α ) p+1/2 /V p+1 → 0. These simple rules allow one to subsume the system of D-branes' stacks and open strings with a simple diagram. In this notation, perturbative interaction terms involving the matter super-fields correspond to closed oriented polygons, starting with triangles. On the other hand, interactions between standard super-fields and modulini also correspond to closed oriented polygons involving solid and dashed lines 7 .

An Unoriented Quiver for a Pati-Salam model
In this section, we construct a simple quiver for a Pati-Salam model inducing a Majorana mass for the neutron. We propose a simple quiver in Fig. 2, leading to an N = 1 The perturbative super-potential that we obtain from the quiver reads  On the other hand, the non-perturbative superpotential term: can be generated by an E2-brane instanton that intersect twice with the U (4) stack of D6-branes so as to produce a four-∆ c (as well as a four-∆) interaction. The fermionic modulini τ i α , with i = 1, ..., 4 and α, β = 1, 2 interact with the super-fields ∆'s via 8 Mass deformed quivers and dimers have been recently investigated in [30].
These interactions are induced by mixed disk amplitudes as the one in Fig. 4, that emerge at the intersections between two D6-brane stacks and one E2 instanton 9 . In our case the two D6-branes are actually the D6-branes of the U (4) stack and their images. Integrating out the fermionic modulini produces two ijkl so that The suppression scale M 0 is related to the string scale M S by M 0 ∼ M S e +S E2 , where S E2 depends on the closed string moduli that parametrize the (complex) size of the 3-cycles, wrapped by the E2-instantons 10 .
We denote by ∆ 6 the 'diquark' super-field in the 6, T 3 in the triplet 3 and S the singlet 1 and find The complete super-potential after symmetry breaking SU (4) For similar calculations in related contexts, see [66,67,68,69,70,71]. 10 In general, the calculations could be much more complicated, in the presence of bulk fluxes, that can also induce soft susy breaking mass terms for the susy partners. For example gaugino mass terms with M λ ∼ Ω ijk 0,3 τ H ijk + iF ijk can be generated in Type IIB contexts by internal 3-form fluxes. In the presence of fluxes, one has to verify that physical branes and instantons are not lifted, i.e. the cycles they wrap and their intersections are not eliminated. With the introduction of bulk fluxes, one also has to consider the back-reactions on the exotic instantons, that could change then number of zero modes. This could modify our present analysis. In fact, when S takes an expectation value, a cubic interaction term is generated. In the next section, we will discuss the consequences, for Neutron-Antineutron physics and for LHC phenomenology, in more in details.

Neutron-Antineutron oscillation through color diquark sextets
In a susy PS-like model SU (4) × SU (2) R × SU (2) L , one can construct a diagram like the one in Fig. 1 for Neutron-Antineutron oscillation, through the 'exotic' interaction ∆ c 10 ∆ c 10 ∆ c 10 ∆ c 10 /M 0 , containing (with S c ν c ν c ≡ ∆ c ν c ν c ). The operator (11) induces a neutron-antineutron transition depicted in Fig. 2, as a result of the super-potential termf 11 v R Q c Q c ∆ c /M 2 , whose com- The process in Fig. 1-(b) produces an effective operator G n−n (udd) 2 with We can now discuss different choices of the parameters leading to very different branches for phenomenology. The motivation of such a variety of possibilities is related to the fact that in (12) one can produce a scale of 300 − 1000 TeV, testable in the next generation of experiments, with different choices of the other parameters.
The cases with M 0 10 19 GeV and M 0 10 13 GeV are equivalent to the GUT SO(10) inspired scenario, discussed in [11] in Fig. 1-(b). In these cases M ∆ c u c u c ∼ 1 TeV.
Both cases are well compatible with the mechanism proposed in the previous section.
In fact a scenario in which M 0 M S can be envisaged, if e S E 2 ∼ 1 i.e. small 3-cycles wrapped by E 2 in CY . A priori, the string scale can be considered as a free parameter, it can be as high as 10 19 TeV as low as a few TeV's. For instance, if M S = 1 ÷ 10 TeV, the hierarchy problem of the Higgs mass is automatically solved, and M 0 can be as high as 10 13 GeV (or more) if e S E 2 = 10 10 i.e. for an E 2 wrapping a large 3-cycle in the CY .
In TeV-scale gravity scenari, one can also consider an alternative scenario in which intriguing implications for LHC. A recent anomaly in pp → l 1 l 2 jj with significance near 3σ, compatible with Left-Right symmetry, was seen by CMS [74]. In a Left-Right model, this is interpreted as sequential W R and N R production as [75,76] pp → W R → l 1 N R → l 1 l 2 W * R → l 1 l 2 jj However, as mentioned above, exotic intantons can dynamically break U (1) B−L . For instance, a Majorana mass matrix for RH neutrini can be generated by exotic instantons rather than by S, as cited above. In this case, S could also be a light particle, if a residual discrete symmetry of U (1) B−L stabilizes it. In other words, S can behave as a Majoron, but it is not exactly a Majoron [77]. We can call it an exoticon. We suppose that the exoticon interacts with the three color sextets with a coupling µ S . So, in this case, we have to replace v B−L with µ S in (12). The exoticon carries B − L = −2, so n →nS does not violate B −L. We also note another important difference with respect to Majorons: the Majoron mass m φ = y L v L , with v L vev of a global U (1) L (and y L coupling), is related to its interaction with neutrini, as g φνν = m ν /v L ; such a relation, in general, is not satisfied by exoticons. A massive exoticon cannot be emitted in a n−n transition, in the vacuum: CPT symmetry protects neutron by transitions n →n + S, i.e m n = mn. However, in a nuclear environment, such a transition is allowed! Such a transition is followed by annihilation of the antineutron with another neutron in the nuclear environment, as (Z, A) → (Z, A − 2) + 3π. We can roughly estimate the corresponding decay width to be Γ (δm/µ S ) 2 ∆E , where ∆E 10 ÷ 100 MeV is the average energy in the nuclear environment. Limits on n −n oscillation in the nuclei are Γ −1 nn ∼ 10 −32 yr [78], corresponding to µ S > 10 30 δm keV. Another spectacular signature of an exoticon could be a nucleon-nucleon disappearances as nn → S, ∆B = 2. This could be detected as a nuclear transition (Z, A) → (Z, A − 2) + missing energy.
We can easily estimate the rate of such a transition as Γ ∼ κ np (M nn ) −3 m 14 N G 2 n−n GeV, where κ np ∼ 10 −6 approximately accounts for the hadronic non-perturbative correction.
Such an estimate leads us to conclude that such a process is very suppressed, roughly as 10 40 yr. Finally, an exoticon can also be detected in a neutrinoless double-beta-decay, as a Majoron. However, there are several important differences with respect to the Majoron: a 0ν2β + S does not violate lepton number, it is an apparent violation. For a 0ν2β + S process, limits on the exoticon production imply (m ν /µ S ) < 10 −5 [79,80] that corresponds to a bound µ S > 10 keV. Limits from supernovae cooling processes ν → ν c S or νν → SS are competitive (for electronic neutrini m ν /µ S < 10 −5 ) [81]. B → l + l − l + l − K suppressed by the GCS couplings with respect to B → l + l − K.
Extra Z from anomalous symmetries are different with respect to Z R (Z-boson of the SU (2) R ), and kinetic mixings Z − Z or Z − Z R can be envisaged. Z R can also interact with Z, γ, Z through G.C.S. A complete study of the resulting cascade processes is 12 We would like to stress that GCS terms generate UV divergent triangles that are cured by considering UV completions with KK states or string excitations. For issues in scattering amplitudes and collider physics see [89], for recent discussion about string theory and causality see [87,88]. For non-local (string inspired) quantum field theories see [90,91,92] beyond the purpose of this paper. Concerning the hierarchy problem of the Higgs mass, our model is compatible with TeV-scale supersymmetry, but this model has more undetermined parameters than the MSSM, i.e. it could be more elusive and more difficult to constrain.

Conclusions and remarks
In this paper, we have shown how to generate a Majorana mass for the neutron,