String completion of an $\mathrm{SU(3)_c \otimes SU(3)_L \otimes U(1)_X}$ electroweak model

The extended electroweak $\mathrm{SU(3)_c \otimes SU(3)_L \otimes U(1)_X}$ symmetry framework"explaining"the number of fermion families is revisited. While $331$-based schemes can not easily be unified within the conventional field theory sense, we show how to do it within an approach based on D-branes and (un)oriented open strings, on Calabi-Yau singularities. We show how the theory can be UV-completed in a quiver setup, free of gauge and string anomalies. Lepton and baryon numbers are perturbatively conserved, so neutrinos are Dirac-type, and their lightness results from a novel TeV scale seesaw mechanism. Dynamical violation of baryon number by exotic instantons could induce neutron-antineutron oscillations, with proton decay and other dangerous R-parity violating processes strictly forbidden.


I. INTRODUCTION
Among the open challenges in the standard model we encounter issues like: Why we have three species of fermions? Why the neutrino masses are so small? Why fundamental couplings unify?
How is gravity incorporated in a fundamental way? One of the early extended electroweak models based on the SU(3) c ⊗ SU(3) L ⊗ U(1) X gauge group [1,2] "explains" the number of fermion families from the requirement of anomaly cancellation. Indeed the theory is anomaly free if and only if the number of quark colors is equal to the number of families, i.e. three (species of fermions) is related to quantum consistency [1][2][3][4][5][6][7]. Recently this scenario has been revamped in order to also provide a framework for naturally light neutrinos without invoking superheavy physics [8]. In this scheme these two fundamental issues get related through the embedding of the standard model gauge group in SU(3) c ⊗ SU(3) L ⊗ U(1) X . In the simplest 3-3-1 model considered recently neutrino masses were radiatively generated by one-loop corrections, involving new neutral gauge bosons associated to lepton number violating interactions [8]. Within a simple variant it has been shown that the same physics involved in small neutrino mass generation may also achieve gauge coupling unification [9], alternative to conventional grand unified theories.
A drawback of such SU(3) c ⊗ SU(3) L ⊗ U(1) X based-models is, however, that they cannot be easily embedded in a conventional Grand Unified Theory (GUT). In order to achieve gauge coupling unification the authors in Ref. [9] considered an alternative more complicated version of the model, in which the presence of a neutral sequential lepton octet allowed for the merging of the gauge couplings at high energies in the absence of a bona fide grand-unified structure. A more ambitious theoretical question is whether such a structure could be UV-completed and understood in more fundamental terms.
Here we show that our desire of obtaining a consistent string completion of this type of 3-3-1 theories leads us to the novel variety of seesaw mechanism proposed in [10], in which neutrinos are Dirac particles with masses generated at the tree level. Moreover, we find that neutrino masses vanish in the limit where the up-quark mass vanish. As a result, consistency of the neutrino sector with the observed quark masses suggests that the new dynamics associated with neutrino mass generation must reside near the TeV scale. On this basis we expect that this model can be directly tested at LHC in the next run, through the resonant production of new fields involved in the neutrino mass generation mechanism. In particular, a new Z boson can be produced through the Drell-Yan processes. This boson couples to standard model particles and to the new isosinglet neutral leptons [11]. Another interesting indirect signature predicted by this model is b → sµ + µ − , gauge mediated by the new Z boson [12]. Likewise, there are also lepton flavour violation signals, recently investigated in a simple variant of these models [13]. i) (un)oriented type IIA, with intersecting D6-branes wrapping 3-cycles on the Calabi-Yau compactification CY 3 ; ii) (un)oriented type IIB, with D7-branes and D3-branes wrapping holomorphic divisors in CY 3 . iii) type I, with magnetized D9-branes wrapping a CY 3 . Here we will focus on the first class.
In this case, we can directly calculate low energy interactions for α s → 0, obtaining just an N = 1 supergravity coupled with matter fields. In particular, we will discuss a simple example of a "quiver field theory" embedding of SU(3) c ⊗ SU(3) L ⊗ U(1) X locally free of stringy anomalies or tadpoles.
In general, a "quiver" is simply a diagrammatic representation of a gauge theory. A supersymmetric quiver (as in our case) includes all the matter (super)field content, represented with arrows, and their interactions. The corresponding diagrams have the following conventions 2 : i) gauge groups are nodes, which are in correspondence with the gauge superfields; ii) superfields are oriented lines between nodes; iii) superfields in the adjoint representations are arrows going in and out on the same node; those in the bi-fundamental representations (M,P) or (M, P) link two different nodes/gauge groups; iv) the number of arrows on a line corresponds to the multiplicity of the same superfield; v) Closed oriented paths (arrows with the same orientation) like triangles, quadrangles, and so on, represent possible gauge-invariant interaction terms in the superpotential.
In open string theories, quiver diagrams are particularly powerful. This is because D-brane dynamics on Calabi-Yau singularities is described by quiver field theories in the low energy limit α s → 0. In string theory, lines connecting nodes correspond to (un)oriented open strings, while nodes are Dbrane stacks. Intriguingly, we will show how the quiver field theory suggests the existence of novel phenomena characteristically "stringy" in nature. In particular, we will see how the presence of new anomalous massive bosons is inevitably predicted. In gauge theories, anomalous U(1)s lead to quantum inconsistencies, but in string theories these can be cured through a Generalized Green-Schwarz mechanism (GGS) and Generalized Chern Simon terms. As a result, anomaly cancellation implies mixing vertices connecting the γ , Z , Z , X gauge bosons with those of the anomalous U(1)s [22][23][24].
There are other interesting features of quiver field theories related to non-perturbative stringy effects which could manifest at low energies. For example, at the low energy limit, a quiver field theory admits the presence of extra non-perturbative couplings in the effective action, generated by "exotic stringy instantons".
In (un-)oriented type IIA, gauge instantons can be described by Euclidean D2 branes (or E2 branes) wrapping the same 3-cycles of "ordinary physical" D6-branes on the Calabi-Yau CY 3 [25,26]. In In quiver field theories, E-brane instantons are represented as triangles. Open strings attached to one ordinary D-brane and one E-brane are fermionic moduli or modulini, corresponding to 'dotted' arrows. Closed triangles of lines and dotted lines correspond to effective interactions among ordinary fields and modulini. Integrating out moduli, new effective interactions among ordinary fields are generated. As shown in [27][28][29][30][31][32][33][34][35][36][37][38][39], these new interactions can dynamically violate Baryon and Lepton numbers. Indeed, we will discuss how exotic instantons can directly generate ∆B = 2 six quarks transitions generating neutron-antineutron oscillations. However, even if B number is violated in our model, the selection rule ∆B = 2 emerges dynamically, so that proton stability and R parity conservation are ensured.
In this section we describe the basic ingredients of our model, diagrammatically represented as the quiver in Fig. 1. This quiver generates a N = 1 supersymmetric theory with gauge group Here L i (i = 1, 2, 3) accommodates the SU(2) L lepton doublets i = (E L , ν L ) T i together with new neutral components N Li into the anti-triplet representation of U(3) L ; Q i includes the LH doublets q i = (u L , d L ) T i plus extra quark fields u L , s L , b L ; R stands for the right-handed charged lepton multiplets and U , D contain the right-handed quarks u R , d R plus extra three (super)quarks u R , s R , b R . In addition, there are six G 331 singlets denoted by S, i.e. there is a pair of gauge singlets in each generation. Finally, the scalar components of the Higgs superfields Φ 1,2,3 , Φ 1,2,3 are responsible for the G 331 spontaneous symmetry breaking.
The effective trilinear quark and lepton superpotentials, perturbatively generated, are given as Each term corresponds to a closed oriented triangle following the arrows associated to chiral superfields depicted in Fig. 1. Moreover, one can see that R-parity violating terms like LQD are automatically forbidden at the perturbative level. This is related to the quiver orientations: there are no closed oriented triangles corresponding to R-parity violating superpotential terms. As a consequence, Rparity is not imposed ad hoc in our model, but appears as an accidental symmetry.
The first stage of gauge symmetry breaking pattern involves a Stueckelberg mechanism [40], while the latter is induced by scalar vacuum expectation values (VEVs) through the Higgs mechanism. The Note that the quiver also generates perturbatively the µ-terms for the Higgs superfields, required for electroweak breaking These terms correspond to closed circuits involving Higgs superfields in Fig. 1. The proposed quiver can be interpreted as the UV completion of a particular G 331 model with extra neutral leptons in the triplet representation. For recent studies in this class of models see for example Ref. [8] and [10,13].
A remarkable feature of this quiver construction is that neutrinos are of Dirac nature, thanks to the presence of a (sequential) pair of lepton singlets S 3 and to the symmetry structure of the model.

B. Tadpole cancellation and U(1)X conditions
The quiver in Fig.1 preserves a linear combination U(1) X = a C a U(1) a , with a = 3 c , 3 L , 1, 1 (1) , that can be obtained from the following system: Here we have adopted the convention + for outgoing arrows and − for incoming ones. The solution corresponds to the defining symmetry of the SU(3) c ⊗ SU(3) L ⊗ U(1) X model [10].
In order to describe a consistent model, the quiver must be free of chiral gauge and gravitational anomalies, with U(1) X unbroken at the string level. These requirements are related to the fulfillment of two more stringent conditions. The first one is local tadpole cancellation [42][43][44]: a N a (π a + π a ) = 4π Ω , where π a denotes 3-cycles wrapped by "ordinary" D6-branes, πâ stands for the corresponding 3-cycles wrapped by the "Ω-image" D6-branes, and π Ω is the contribution of the Ω-plane. More conveniently, Eq. (6) can be expressed in terms of superfields as with The second important condition observed by the quiver field theory reads a C a N a (π a − π a ) = 0 , and guarantees the existence of a massless vector boson associated with the unbroken U(1) X = a C a U(1) a symmetry [42][43][44]. Again, in terms of field representations, Eq. (9) can be written as and is satisfied by U(1) X accordingly: We conclude this section pointing out that the remaining Abelian and mixed anomalies can be canceled by a Generalized Green-Schwarz mechanism with Stueckelberg, axionic and generalized Chern-Simons couplings, following the lines of [22][23][24]. This mechanism introduces non trivial interactions among the various gauge bosons of the model and provides potentially interesting phenomenological implications.

A. Quark sector
Fermion masses are obtained perturbatively, after spontaneous breaking of the gauge symmetry, from the Yukawa interactions present in Eq. (2). For the quarks one has the following mass matrices where we have assumed that the scalar fields φ Here n ( ) 1,2,3 characterizes the SU(3) L breaking and k ( ) 1,2,3 the subsequent SU(2) L breakdown. One can verify that a realistic pattern of quark masses and interactions can be obtained from the above mass matrices, though its detailed study is beyond the scope of the present paper. One characteristic feature which we can comment is the existence of heavy exotic quarks which, in general, mix with those of the standard model leading to an effective violation of unitarity of the CKM matrix [45,46]. Furthermore, the presence of heavy exotic quarks may lead to a number of phenomenological implications, such as accommodating the recent diphoton anomaly [47,48].

B. Neutrino masses
After spontaneous symmetry breaking, one obtains a Dirac neutrino mass [10] −L mass = 1 √ 2 ν LNL y 2 k 2 + y 3 k 3ỹ2 k 2 +ỹ 3 k 3 y 2 n 2 + y 3 n 3ỹ2 n 2 +ỹ 3 n 3 where y 2,3 andỹ 2,3 are 3 × 3 Yukawa matrices and we have denoted S = S 1···3 andS = S 4···6 . The light neutrino masses can be readily estimated in the one family approximation which, as usual, is diagonalized by a bi-unitary transformation M diag = U † ν MU S . As can be seen, the effective light neutrino mass vanishes as the scalar vacuum expectation values n 2,3 become large with respect to |k 2 n 3 − k 3 n 2 |, very much as expected in the conventional Majorana neutrino seesaw mechanism.
Another feature is that the light neutrino become massless in the limit where the dynamical alignment parameter [10] k 2 n 3 − k 3 n 2 approaches zero. The same holds for the up quark. Hence, in the present formulation, the same alignment yields a massless u quark, implying a tension between small neutrino masses and a realistic u quark. However, this tension is still comparable with e.g. the Yukawa hierarchy between the electron and the top quark in the SU (3) where f and f stand for the flavor indices of the corresponding fields and i is the U(3) c index.
Integrating over the modulini space associated to the D6-E2 intersections, we obtain with the flavor matrix Y f1f2f3f4f5f6 ≡ K f6 . As the coefficients K (1,2) parametrize particular homologies of the mixed disk amplitudes, we treat them as free parameters, since our model is local. Thus, the superpotential (17) leads to an effective dimension 9 six-quark operator O nn = (u c d c d c ) 2 /M 5 responsible for neutron-antineutron oscillations. The related new physics scale M can be written as M 5 = y −1 1 e +S E2 M 3 S m 2 g , where mg is determined by gaugino-mediated quark-squark SUSY reductions (see [49] for example), and y 1 ≡ Y 111111 .
In terms of M, the n −n transition time (in vacuum) reads τ nn M 5 /Λ 6 QCD . The current bounds on the n −n transition time are τ nn 3 yrs, constraining the new physics scale to M 300 TeV.