Contents

CPHT-RR065.0911

CERN-PH-TH/2011-243

ICCUB-11-170

Massive wavefunctions, proton decay and FCNCs in local F-theory GUTs

[12mm] Pablo G. Cámara,  Emilian Dudas,  Eran Palti

[5mm] CERN, PH-TH Division, CH-1211 Geneve 23, Switzerland

Departament de Física Fonamental and Institut de Ciències del Cosmos, Universitat de Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain

Centre de Physique Th´eorique, Ecole Polytechnique, CNRS, F-91128 Palaiseau, France.

LPT, UMR du CNRS 8627, Bat 210, Universit´e de Paris-Sud, F-91405 Orsay Cedex, France.

[3mm]

E-mail: pcamara@cern.ch, emilian.dudas@cpht.polytechnique.fr, palti@cpht.polytechnique.fr

[12mm] Abstract

[3mm]

We study the coupling of MSSM fields to heavy modes through cubic superpotential interactions in F-theory SU(5) GUTs. The couplings are calculated by integrating the overlap of two massless and one massive wavefunctions. The overlap integral receives contributions from only a small patch around a point of symmetry enhancement thereby allowing the wavefunctions to be determined locally on flat space, drastically simplifying the calculation. The cubic coupling between two MSSM fields and one of the massive coloured Higgs triplets present in SU(5) GUTs is calculated using a local eight-dimensional SO(12) gauge theory. We find that for the most natural regions of local parameter space the coupling to the triplet is comparable to or stronger than in minimal four-dimensional GUTs thereby, for those regions, reaffirming or strengthening constraints from dimension-five proton decay. We also identify possible regions in local parameter space where the couplings to the lightest generations are substantially suppressed compared to minimal four-dimensional GUTs. We further apply our results and techniques to study other phenomenologically important operators arising from coupling to heavy modes. In particular we calculate within a toy model flavour non-universal soft masses induced by integrating out heavy modes which lead to FCNCs.

1 Introduction

Although string theory is primarily motivated as a fundamental unified theory because of its ultraviolet behaviour, phenomenological model building within string theory often concerns only the infrared spectrum. This is a natural first step given the expected hierarchy between the string and electroweak scales. However, heavy modes play a crucial role in our understanding of much of the physics which is relevant to the Standard Model and extensions of it, for example by inducing higher dimension operators in the infrared. The fact that studying such modes explicitly requires a good understanding of the ultraviolet physics means that this is one of the subjects where string phenomenology can play an important role. Heavy modes are particularly important in the case of Grand Unified Theories (GUTs). For instance, the result of gauge coupling unification at the GUT scale is sensitive to threshold corrections from heavy modes, and one of the classic constraints on GUTs comes from dimension-five proton decay operators that are induced by integrating out heavy modes.

Studying detailed properties of these fields, such as their wavefunction profile, is typically a difficult prospect because of the complicated geometry associated to realistic models of particle physics in string theory. String modes can only be concretely studied in simple geometries where a world-sheet description is available. Kaluza-Klein (KK) modes, and the closely related Landau-levels111These are sometimes referred to as gonions in the intersecting brane literature [1]., are typically difficult to solve for within some complicated Calabi-Yau (CY) geometry. However, in some models, and in particular F-theory (or type IIB) GUTs, many important operators of the theory are associated to only a small patch within the full geometry. The extreme example of this are Yukawa couplings, which are associated to just a single point in the geometry. Analogous to the Yukawa couplings there are triple couplings between heavy modes and massless modes which can be locally studied within a small region around a point. Since locally the complicated global CY geometry is decoupled and essentially we can work on flat space, many properties of heavy modes become accessible. In this paper we use this local approach to study the coupling of heavy modes to massless modes through such a triple coupling operator. This is done by solving for the local form of wavefunctions of massive and massless modes and calculating their triple overlap.

The particular operator that we study, coupling one heavy mode to two massless ones, plays a key role in GUTs. One of the general features of SU(5) GUTs is that associated to the MSSM Higgs doublets there are coloured triplets which complete a GUT representation. These modes have to obtain a mass, leading to the so called doublet-triplet splitting problem. Similarly, associated to the Yukawa couplings there are also triple couplings between one heavy triplet and two MSSM fields. Once the heavy triplets are integrated out these couplings induce dimension-five baryon and lepton number violating operators that lead to proton decay. Thus, understanding such couplings and their flavour structure is of crucial importance for placing constraints on GUT models. In minimal field-theory SU(5) GUTs these couplings are the same as Yukawa couplings and therefore are exactly known. However, in string theory GUTs this is not the case and the couplings can be completely different in nature. This means that without knowledge of how the triplets couple to the matter fields and in particular whether the coupling to the lightest generations is suppressed in a manner similar to the Yukawa couplings it is not possible to use dimension-five proton stability to constrain model building. The primary aim of this paper is to study the nature of the triplet couplings within a realistic string setup thereby performing this crucial step in imposing phenomenological constraints on string theory GUTs.

More generally the paper aims to show that much important physics can be extracted by similar calculations of couplings to heavy modes. Indeed, in section 5 we present a toy model where such a calculation allows to extract Flavour-Changing-Neutral-Current (FCNC) terms which, like proton stability, form one of the important observational constraints on ultraviolet physics. We also discuss how our calculations apply to string theoretic realisations of the Froggatt-Nielsen mechanism for generating flavour structure this being yet another mechanism which relies on higher dimension operators.

Our focus is on local F-theory GUTs [2, 3, 4, 5, 6, 7, 8]. Within this framework a 7-brane carrying an SU(5) gauge group wraps a 4-dimensional surface inside a CY four-fold. Other 7-branes intersecting this brane are locally modeled by an enhancement of the gauge symmetry over loci in : along complex curves on the group enhances by rank 1, to SU(6) or SO(10), while on the points where complex curves intersect it enhances by at least rank 2, to SU(7), SO(12) or E. We are particularly interested in a point of enhancement to SO(12) as it is there that the down-type Yukawa interaction is localised. To describe the physics near such point we consider an 8-dimensional gauge theory, which is just super Yang-Mills twisted to account for the embedding into the CY four-fold [4, 5, 14], with SO(12) gauge group broken down to SU(5)U(1)U(1) by a spatially varying Higgs field. Matter localises onto complex curves where the Higgs vev vanishes and at the SO(12) enhancement point three such matter curves intersect giving rise to a cubic coupling in the 4-dimensional effective theory. This coupling can be calculated directly from dimensional reduction of the 8-dimensional theory by integrating the overlap of the internal wavefunctions of localised fields. Yukawa couplings are calculated by overlaps of wavefunctions of three massless modes [9] and have been extensively studied in [10, 11, 12, 13, 14, 15, 16] (see also [17, 18, 19, 20] in the context of magnetised D-branes). In this paper we calculate the wavefunctions for massive modes around an SO(12) point. Similar calculations of massive mode wavefunctions for other models were performed in [21, 22, 16]. Once we obtain the wavefunctions for massive modes we can study their overlap with massless wavefunctions, thereby probing the cubic coupling discussed above.

An important property of the dimension-five proton decay operator we are studying is that it is a superpotential operator. Since in type IIB string theory and F-theory the superpotential does not receive corrections, and since integrating out massive string oscillator modes induces corrections, we do not expect the operator to be induced by exchanging massive string oscillator modes. Thus, all the relevant heavy modes which participate in dimension-five proton decay are captured within the effective gauge theory described above.

The calculation of the coupling to massive modes at an SO(12) point is only a part of the full calculation required to understand dimension-five proton stability. It is therefore worth discussing how the present calculation fits within the full picture of dimension-five proton stability in F-theory SU(5) GUTs. We begin by reviewing the constraints on the relevant operators. The effective superpotential couplings take the schematic form

 W⊃YuijHuQiUj+YdijHd(QiDj+LiEj)+^YuijTu(QiQj+UiEj)+^YdijTd(QiLj+UiDj)+MTuTd. (1.1)

Here and denote the up- and down-type Yukawa couplings with the MSSM superfields expressed in standard notation. The coloured triplets are denoted by and and their associated triple couplings are and . In minimal 4-dimensional GUTs we have .222This relation and the above superpotential may be slightly modified by more complicated theories where the cubic couplings arise after fields in non-trivial GUT representations obtain a vev, as proposed for example in [23] to fix the GUT mass relations. The scale is related to the mass of the triplets and is expected to be at or below the GUT scale, . Integrating out the heavy triplets leads to dimension-five operators

 W⊃^Yuij^YdklM(QiQjQkLl+UiEjUkDl). (1.2)

There are a number of diagrams that lead to proton decay and involve these operators.333There are a large number of papers which study nucleon decay in 4-dimensional supersymmetric GUTs. We refer to [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] for a subset. Note that a number of these papers were using old experimental results on the proton lifetime which has since increased by 2 - 3 orders of magnitude. At TeV scale the diagrams involve a 4-point interaction coming from (1.2) which has two fermions and two scalar superpartners, and a loop factor involving wino or Higgsino exchange to turn the scalars into fermions (c.f. figure 1). This results in nucleon decay to kaons primarily (due to the need for a strange quark because of the anti-symmetric colour index). To discuss the constraints on the operators (1.2) let us fix and quote limits on for different generation indices.444Note that using the results of [37] that higher dimension superpotential operators are expected to be suppressed by the winding scale, and those of [38, 39, 40] showing that the winding scale is also the unification scale, implies that is a quite natural suppression scale. Of course the arguments given are simply scaling arguments and should not be taken to hold to significant accuracy. Nevertheless a suppression mass scale larger than seems unlikely given that there are always some heavy modes at or below this scale.

The precise constraints depend on a number of factors such as the soft masses, the size of the -term and tan (with small tan and large soft masses giving generally weaker constraints). Instead of going into the details of these studies, we can concentrate on the most relevant aspect for the study in this paper: the difference between the Higgs and the coloured triplet couplings. In 4-dimensional field-theory analysis these couplings are taken as equal and this leads to an approximate bound . Thus, if coloured triplet couplings are suppressed with respect to Yukawa couplings by a factor larger than the 4-dimensional constraint on , dimension-five proton stability constraints can be satisfied for triplet masses of the order of the GUT scale or above. We can estimate the relevant parameters involving the down-type triplet couplings in the minimal 4-dimensional field-theory analysis,

 ^Yd~tbYdbb=1,^Yd~tsYdbb∼10−1,^Yd~tdYdbb∼^Yd~csYdbb∼10−2,^Yd~cdYdbb∼^Yd~udYdbb∼10−3. (1.3)

This simply comes from using the 4-dimensional field-theory equalities, for example , and the measured quark masses and mixings. We will calculate precisely these ratios for local F-theory SU(5) GUTs and compare to the above values to see if there is enough additional suppression to avoid proton decay or, alternatively, if there is an enhancement thereby making proton decay constraints more severe.555It is important to note that in order to suppress proton decay all the ratios in (1.3) must be suppressed since, probing a superpotential coupling, we are working in a weak eigenstate basis. A single large coupling in the weak basis can lead to several large couplings in the mass eigenstate basis. Thus, to enhance the rate of proton decay it is sufficient that only one of the ratios is larger than in minimal 4-dimensional GUTs. Note also that some of the couplings may vanish through other selection rules such as additional symmetries or the fact that the colour index in the dimension-five operator must be anti-symmetric and so it cannot involve all the same generation. In such cases the ratios involving those operators would not be constrained and it would suffice to suppress only the other ratios.

Calculating the parameters in (1.3) manifestly requires a theory of flavour. The study of flavour structures within F-theory GUTs has been an active research area in the recent years [6, 10, 11, 12, 13, 14, 15, 16, 41, 42, 43, 44, 45, 46, 47, 48]. We make use of the theory of flavour first proposed in [10] and subsequently elucidated in [11, 12, 13, 14, 42, 16].666There are two key motivations for studying this proposal as opposed to say that of [43] which was based on a Froggatt-Nielsen mechanism with additional U(1) symmetries. The first is a practical one: it is not possible to study the flavour structure of [43] locally near an SO(12) point. The second is that within the structure of [43] the relations (1.3) are always at least as strong as in minimal field-theory GUTs since the suppression by the U(1) symmetries acts on the triplets in the same way as on the Higgs doublets. The structure is such that all three generations are localised on a single matter curve and arise from the degeneracy of massless Landau-levels in the presence of flux. This theory of flavour, however, requires ingredients which are not present in our setup. More precisely, it was shown in [13, 14, 42] that to generate non-vanishing Yukawa couplings for anything other than the heaviest generation requires a non-commutative deformation of the theory induced by closed string fluxes or non-perturbative effects. We discuss this in more detail in section 5 but for now it is sufficient to state that this does not affect the calculation we are performing. Unlike Yukawa couplings, the coupling of one massive mode to two massless ones is non-vanishing even in the absence of the required non-commutative deformation. Turning on the additional, necessarily small, such deformation will only perturb slightly our present calculation thereby maintaining its validity. Note that the fact that in the concrete setup we are using the Yukawa couplings are rank one in generation space while the triplet couplings are rank three highlights the fundamental difference between these couplings.

The computation that we perform is a necessary one to understand a number of phenomenological issues. Regarding the particular problem of dimension-five proton stability, it is worth discussing some alternative solutions that have been proposed within the context of F-theory. One way to avoid inducing proton decay is by having a symmetry which forbids it. One such candidate symmetry is a U(1) symmetry, which we label as U(1). Such a (massive) symmetry has been studied in detail in F-theory GUT models, see for example [49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 43, 45]. Although at the GUT level it was found that many models can exhibit such a symmetry, it was shown in a series of papers [53, 54, 43, 45, 58, 59] that the use of hypercharge flux to break the GUT group and induce doublet-triplet splitting is incompatible with such a symmetry.777As pointed out in [62], the same problem arises for Wilson-line GUT breaking. The precise statement is that the presence of a U(1) symmetry necessarily implies the presence of exotic non-MSSM states in the massless spectrum. The mass of the exotic states is set by the scale at which the U(1) symmetry is broken and therefore the constraints coming from dimension-five proton decay translate to constraints on the mass of the exotic states. The phenomenology associated to different masses for the exotic states was studied in detail for a number of models in [45, 59, 60]. Since the exotic states do not form complete GUT multiplets, the most immediate constraints on their masses come from gauge coupling unification. The tension between a large exotics mass to maintain gauge coupling unification and a small mass to preserve an approximate U(1) symmetry implies that it is difficult to practically realise the full suppression necessary for dimension-five proton stability using such a symmetry alone. The suppression due to a U(1) symmetry is additive to that studied in this work and therefore whether we find additional suppression or alternatively an enhancement of coupling to massive modes can allow for or rule out a number of proposed models.

An alternative possibility for suppressing dimension-five proton decay even without a U(1) is keeping the matter curves associated to the up and down Higgs fields in the same homology class but still geometrically separated. The interaction between the up and down triplets may then be suppressed by their small wavefunction overlap, although explicitly studying this would require a calculation of massive wavefunctions similar to that presented in this paper. Apart from the fact that this rather complicated setup has yet to be realised explicitly, there are a number of phenomenological problems with such a setup. The first is that the use of hypercharge flux for doublet-triplet splitting is difficult since it acts in the same way on both the Higgs curves. Another problem is that the theory of flavour of [10] is based on local geometric symmetries which means that in order to correlate the up- and down-type Yukawas, as is required by a realistic CKM matrix, the geometric separation between them should be small. Indeed this is one of the primary motivations presented in [52] for a proposed point of E unification.

The outline of this paper is as follows. In section 2 we introduce the effective theory that we will be using. Following this, in sections 3 and 4 we present the actual calculations of the relevant wavefunctions and their overlaps. In section 5 the results and their phenomenological applications are discussed in detail. We summarise our findings in section 6. In appendix A we present the wavefunctions and overlaps for a more general set of background fluxes, consisting also of oblique fluxes. In appendix B we study in more detail the normalisation of the wavefunctions.

2 The effective theory

We consider F-theory on an elliptically fibered Calabi-Yau 4-fold , with the degrees of freedom of an SU(5) GUT localised on a codimension-2 singularity. In the infrared those are described by a twisted 8-dimensional gauge theory, with gauge group and support on , where is a 4-dimensional Kähler sub-manifold of [4, 5].888If is shrinkable (more formally it has an ample normal bundle), the resulting 4-dimensional gauge coupling can be tuned independently of . However we do not necessarily assume this property in our analysis. In this section we describe the 8-dimensional effective theory and how its massless and massive localised spectrum is calculated. Related computations to the ones that we describe here have also recently appeared in [16].

2.1 The 8-dimensional effective theory

For convenience, we arrange the 8-dimensional fields in adjoint valued, -valued, 4-dimensional multiplets

 A¯m = (A¯m,ψ¯m,G¯m), (2.1) Φmn = (φmn,χmn,Hmn), V = (η,Aμ,D).

The subindices on the fields denote their local differential structure on . Thus, for instance where denotes the space of holomorphic -forms on and is the principal bundle (in the adjoint representation) associated to the gauge group . Here and are chiral multiplets with respective F-terms and . is a vector multiplet with D-term . and are complex scalars while , , and are fermions.

The action for the effective theory was given in [5]. For the bosonic components of the multiplets it reads,999This action can be shown to be equivalent to 8-dimensional super Yang-Mills theory with a non-trivial Higgs bundle [14].

 S8d=M4∗∫R1,2×Sd4x Tr[ω∧ω(12D2−14FμνFμν)−Dμφ∧Dμ¯φ+2iω∧G∧¯¯¯G+H∧¯¯¯¯¯H−F(2,0)∧¯¯¯¯¯H−F(0,2)∧H−¯¯¯G∧∂A¯φ−G∧¯∂Aφ+2(ω∧F(1,1)+i2[φ,¯φ])D−2iω∧F(1,0)Sμ∧F(0,1)μS+…] (2.2)

where is the Kähler form of . Our conventions are such that is dimensionless, and have dimensions of mass and the auxiliary fields , and have dimensions of . denotes the UV cutoff of the theory. In the weakly coupled type IIB limit this is related to the string scale as

 M4∗=(2π)−5α′−2. (2.3)

Above this scale, corrections to eq. (2.2) in the form of higher derivative couplings become important and keeping only the leading term in the expansion, eq. (2.2), is not a valid approximation. Thus, in what follows we shall stick to the regime where, at every point of ,

 ⟨∂A⟩, ⟨∂φ⟩≪M2∗. (2.4)

Ideally we would like to dimensionally reduce the 8-dimensional effective action (2.2) on in order to obtain the spectrum of 4-dimensional fields with masses smaller than . However, such a program would require the precise knowledge of the geometry of , which in general is only available for few highly symmetric spaces such as , or (see for instance [18]). Alternatively, we can solve the equations of motion in a local patch around a particular point of where the energy density of a set of charged modes localises. This approach has been extensively used in recent phenomenological studies of Yukawa couplings in F-theory GUTs (see e.g. [10, 11, 12, 13, 14, 15, 16]). In what follows we describe it in detail.

2.2 Equations of motion for localised fields

Let us first consider 4-dimensional massless fields. Setting the 4-dimensional variations of the fields to zero, the equations of motion that follow for their internal wavefunctions are [5]

 H−F(2,0)=0, (2.5) i[φ,¯φ]+2ω∧F(1,1)+⋆SD=0, (2.6) 2iω∧¯G−¯∂Aφ=0, (2.7) −∂¯H+2ω∧¯∂D+¯G∧¯φ−¯χ∧¯ψ−i2√2ω∧η∧ψ=0, (2.8) ω∧∂Aψ+i2[¯φ,χ]=0, (2.9) ¯∂Aχ−2i√2ω∧∂Aη−[φ,ψ]=0, (2.10) ¯∂Aψ−√2[¯φ,η]=0, (2.11) −√2[¯η,¯χ]−¯∂AG−12[ψ,ψ]=0. (2.12)

where we have also included the equations of motion for the fermionic fluctuations. Eqs. (2.5) and (2.6) are usually dubbed as the F-term and the D-term conditions for the flux, respectively.

Generically, the gauge group is broken by and to a smaller subgroup . We are interested in vacua where and take values in a subgroup belonging to the Cartan of ,

 G→G′×GH (2.13)

with the commutant of and . In that case eq. (2.7) can be simply satisfied by requiring to be holomorphic

 ¯∂A⟨φ⟩=¯∂⟨φ⟩+[A,⟨φ⟩]=¯∂⟨φ⟩=0. (2.14)

Note also that to preserve 4-dimensional Poincaré invariance we must impose

 ⟨χ⟩=⟨ψ⟩=⟨η⟩=0, (2.15)

which, making use of eqs. (2.5)-(2.7), imply that eqs. (2.8) and (2.12) are automatically satisfied.

Modes charged under arrange into Landau levels and are localised around points in where . In order to have a description of these modes it therefore suffices to consider a local patch around a localisation point. We can take the Kähler form to be given in the local patch by the expansion

 ω=i2(dz1∧d¯z¯1+dz2∧d¯z¯2)+… (2.16)

where the dots denote higher order terms in the two local complex coordinates and . Our conventions are such that coordinates are dimensionless, denote local vielbein 1-forms and the origin of coordinates is at the localisation point.

Similarly, we expand101010Note that most generally we could have also considered constant terms in these local expansions.

 ⟨A⟩ =−M∗R∥Im(Maijzid¯zj)Qa+… , (2.17) ⟨φ⟩ =M∗R⊥maiziQadz1∧dz2+… , (2.18)

where and are arbitrary numbers related to the quanta of gauge and Higgs fluxes and denote the different Abelian generators of . We have also introduced the standard lengths and in the local patch and its transverse space respectively, measured in units, which for simplicity we have taken to be the same in all directions. Note that we have chosen to parameterise the dimensionful part of with the transverse (“winding”) scale . This is not an arbitrary choice but follows from embedding the 8-dimensional theory into a 10-dimensional theory where the Higgs would correspond to deformations of the 7-brane into the normal directions. In the 8-dimensional theory the Higgs kinetic term arises from the pull-back of the 10-dimensional metric normal to the brane,

 S7−brane⊃M4∗∫d8x g3¯3∂μϕ3∂μ¯ϕ¯3=M4∗∫d8x M2∗R2⊥∂μϕ3∂μ¯ϕ¯3=M4∗∫d4x ∂μφ∧∂μ¯φ (2.19)

where is the metric transverse to the 7-brane and is the complex scalar parameterizing geometric deformations of the 7-brane along the holomorphic normal vector. In order to have a canonically normalised quasi-topological term (c.f. eq. (2.2)), in the last equality we have redefined,

 φ≡M∗R⊥ιϕΩ=M∗R⊥ϕ3dz1∧dz2, (2.20)

with the local holomorphic 3-form of the 3-fold base of . Hence, the factor is the appropriate one for the canonically normalised Higgs field in the 8-dimensional theory.111111The above scalings with and can also be understood from the T-dual setup with magnetised D9-branes. Indeed, T-dualising along the transverse space to the 7-brane, the Higgs and gauge fluxes are mapped respectively to gauge fluxes and on a stack of D9-branes [16]. In a vielbein basis the components of the flux are respectively, T-dualising along the transverse directions, these become the Higgs and gauge fluxes on the 7-brane where , in agreement with (2.17) and (2.18).

Plugging the above local expansions into eqs. (2.9)-(2.11) we obtain that the relevant F-term equations for a given massless 4-dimensional fermionic field read, to leading order in the coordinates,

 DΨ=0, (2.21)

with,

 (2.22)

and

 Di ≡M∗R∥(∂i−12qa(Maji)∗¯zj) D†i ≡M∗R∥(¯∂i+12qaMajizj)i=1,2 (2.23) D3 ≡−M∗R⊥qamai¯zi D†3 ≡M∗R⊥qa(mai)∗zi. (2.24)

In these expressions is the vector of -charges for the localised mode and we have relabeled and to simplify the notation. To obtain a finite set of solutions, we have to supplement these equations with a set of boundary conditions encoding the global obstruction from the topology of and which, in particular, determine the degeneracy of the zero modes.

Massive 4-dimensional fields can similarly be accounted for by the 8-dimensional effective theory. In that case, one obtains the more general set of equations

 D†DΨ=|mλ|2Ψ, (2.25)

where is the mass of the 4-dimensional field and the same definitions above hold. As we explicitly show in next subsection, these are the equations of motion for a set of three complex quantum harmonic oscillators which can be solved by means of standard techniques in quantum mechanics.

2.3 Localised fields and supersymmetric quantum mechanics

Let us first solve the equations of motion for 4-dimensional massless fields, eq. (2.21) or equivalently eq. (2.25) with . For that we closely follow the techniques developed in [21, 22, 16].121212See also [63] for related work.

Different matter representations have different charges under the gauge group generators and therefore different equations governing their wavefunctions. We take to transform in a representation of the gauge group, with the corresponding gauge covariant derivatives defined in eq. (2.23). From eq. (2.22) we observe that the operator which appears in the left-hand-side of eq. (2.25) can be written as

 D†D=−△I+B, (2.26)

where

 △≡∑i=1,2,3D†iDi, (2.27)

and

 B=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝00000[D†2,D2][D2,D†1][D3,D†1]0[D1,D†2][D†1,D1][D3,D†2]0[D1,D†3][D2,D†3][D†2,D2]+[D†1,D1]⎞⎟ ⎟ ⎟ ⎟ ⎟⎠. (2.28)

We have made use of the F-term equations for the background, that we take to preserve supersymmetry in 4-dimensions, in order to simplify the hermitian matrix . Imposing also the D-term condition on the background it is easy to check that is traceless.

A suitable approach to obtain the zero mode wavefunctions is therefore to make a change of basis which diagonalises and to solve the equations of motion in that basis. Let be the matrix which diagonalises and has canonically normalised column vectors,

 J−1⋅B⋅J=(M∗R∥)2diag(0,λ1,λ2,λ3), (2.29)

where . The dimensionless eigenvalues are given by the three roots of the characteristic polynomial of the non-trivial part of , which is a depressed cubic equation. We can rotate the operator to the diagonal basis by taking,

 ~D≡(J−1)∗⋅D⋅J. (2.30)

Notice that has again the same structure as in (2.22) but in the new basis covariant derivatives are given by

 ~Dp=3∑k=1JkpDk=1||ξp||3∑k=1ξp,kDk, (2.31)

with the -th eigenvector of and its norm. In particular, it is simple to check that the only non-vanishing commutators of the rotated covariant derivatives are the diagonal ones,

 [~D†p,~Dp]=−(M∗R∥)2λp ,p=1,2,3. (2.32)

As we have advanced, this is the algebra for the ladder operators of a set of three quantum harmonic oscillators.

Generically we can distinguish four towers of solutions to eq. (2.25), one per eigenvector of . These four towers can be identified with the four complex fermions of a (broken) supermultiplet. In particular, if the three non-trivial eigenvalues are different from zero, there is a massless chiral supermultiplet in the 4-dimensional spectrum, which corresponds to the ground state of one of the above four towers. To see this more explicitly, let us assume for a while that with two positive and one negative real eigenvalues. We take to be the negative eigenvalue. The localised normalisable solution satisfies the equations

 ~D⋅⎛⎜ ⎜ ⎜⎝0φ00⎞⎟ ⎟ ⎟⎠=0⇔⎧⎪ ⎪⎨⎪ ⎪⎩~D1φ=0~D†2φ=0~D†3φ=0. (2.33)

We can identify the raising and lowering operators as,

 ^a1 ≡i~D1, ^a2 ≡i~D†2, ^a3 ≡i~D†3, (2.34) ^a†1 ≡i~D†1, ^a†2 ≡i~D2, ^a†3 ≡i~D3,

so that the function in eq. (2.33) is annihilated by the three operators. More generically, for fields transforming in the representation , we have in the diagonal basis

 R :~D†~D=∑i=1,2,3^a†i^aiI+(M∗R∥)2diag(−λ1,0,λ2−λ1,λ3−λ1). (2.35)

Since the ground state in each of the four towers of fermions is by definition annihilated by all lowering operators , the four entries in the last term correspond to the masses of these ground states. Their wavefunctions are given in terms of the function as,

 Ψp=ξpNφ(z1,z2,¯z1,¯z2) ,p=0,1,2,3 (2.36)

where is a normalisation constant. Similarly, wavefunctions for the heavier modes in each tower are obtained by acting on the corresponding ground state wavefunction with the raising operators. We can label these fields by three quantum numbers, , and , according to

 Ψp,(n,m,l)=(R∥/M∗)n+l+m√m!n!l!(−λ1)n/2λm/22λl/23(~D†1)n(~D2)m(~D3)lΨp, (2.37)

where the particular pre-factor ensures the correct normalisation. Thus, calculating massive wavefunctions is a simple task of applying differential operators to functions. The mass of the resulting 4-dimensional fields is given by

 M2Ψ0,(n,m,l) =(M∗R∥)2[−(n+1)λ1+mλ2+lλ3], (2.38) M2Ψ1,(n,m,l) =(M∗R∥)2(−nλ1+mλ2+lλ3), M2Ψ2,(n,m,l) =(M∗R∥)2[−(n+1)λ1+(m+1)λ2+lλ3], M2Ψ3,(n,m,l) =(M∗R∥)2[−(n+1)λ1+mλ2+(l+1)λ3].

In particular, denotes the wavefunction for the massless chiral fermion transforming in the representation.

Whereas this description is complete for massless chiral fields, massive fields contain both chiralities and the above wavefunctions only represent half of their degrees of freedom, namely those transforming in the representation of the gauge group. Wavefunctions for the components of the massive fields can be worked out following the same procedure, taking care of the change of sign in the charges. One may easily check that the analogous operator to (2.35) for fields transforming in the representation is

 ¯R :~D†~D=∑i=1,2,3^a†i^aiI+(M∗R∥)2diag(−λ1,−2λ1,λ3,λ2). (2.39)

Wavefunctions are therefore given again by the same functions , with , but the corresponding masses are shifted with respect to eq. (2.38). Massive components transforming in the and the representations pair up non-trivially. For instance, the first excited states of the massless mode in the representation pair up with same-mass ground states in the representation.131313This non-trivial pairing has its origin in the fact that the 4-dimensional mass term comes from the 8-dimensional kinetic term, and the latter contains a operator acting non-trivially on . Note also that there is no massless fermion transforming in the representation, as expected.

Wavefunctions for the scalar fields can be worked out in a similar way and, in particular, they can be shown to be identical to the ones of their corresponding fermionic superpartners, as a consequence of supersymmetry and flatness of the local patch.

Summarizing, we have shown that at each localisation point in there are four towers of fields with equal gauge charges, corresponding to the degrees of freedom of a broken supermultiplet. In these conventions, for there is a localised massless chiral supermultiplet transforming in the representation of the gauge group. The degeneracy of this field is only globally determined. It is also easy to check that for the roles of and are exchanged and there is instead a massless chiral supermultiplet transforming in the representation. These two possibilities are separated by a wall of marginal stability at . At this wall at least one of the three eigenvalues vanishes and 4-dimensional fields arrange into supermultiplets (or supermultiplets if all eigenvalues are zero) with conserved Kaluza-Klein momentum. In that case, the wavefunction of the fields is no-longer localised along the matter curve, and their mass is determined by the particular topology of the curve. We present in section 3 an example of this type.

2.4 Validity of the local approach

The wavefunctions and consequently, through their overlaps, the cubic couplings in the 4-dimensional theory depend on the parameters of the 8-dimensional theory such as the fluxes and the local scales and . It is therefore important to quantify the possible range of these parameters which is consistent with the local effective theory being used.

The first constraint we must impose is which ensures that higher derivative corrections to the 8-dimensional effective action are negligible. Using the expressions (2.17) and (2.18) this gives

 MaijR2∥≪1 ,R⊥R∥mai≪1. (2.40)

These amount to small intersection angles and small flux densities. The flux parameters and would be integer quantised in a homogenous setup but in the local setup need not be so. However, generically they are expected to be of order one and we shall therefore take them as so while keeping in mind that the local freedom to adjust the fluxes allows for some flexibility in satisfying the consistency constraints. Taking the fluxes as such we can rephrase (2.40) in terms of geometric constraints. We define

 R≡R∥R⊥,ε≡R⊥R∥, (2.41)

using which we can write

 ε≪1,εR≪1. (2.42)

We can also consider these as constraints on mass scales: it is simple to check that for large the eigenvalues scale as,

 λ1, λ2∼R ,λ3∼1, (2.43)

and are independent of . Therefore, from eqs. (2.38) we observe two types of massive modes with masses scaling as or . The mass of these modes should be kept below the cutoff scale of the theory which is consistent with the constraints (2.42). For generic order one fluxes the two above constraints can be simultaneously satisfied by taking

 R∥≫R⊥ ,R∥≫1. (2.44)

In this limit there is a large number of 4-dimensional massive fields below the cutoff scale . Note that, although not necessarily required, these constraints also allow for length scales . The stability of the 8-dimensional effective theory against corrections for such small values of depends on the particular connection between the local and global scales, which we now discuss.

The relation between the local scales and and the global ones is model dependent and generically too complicated to be computed explicitly in given models. Whereas the limit (2.44) can always be taken in the local setup, once we begin to relate local scales to global ones new phenomenological constraints are expected to arise from the observed values of and . For instance, if is completely homogenous then we have approximately

 R∥∼α−14GUT. (2.45)

Since we observe this implies that cannot be too large. The relation is approximate and the space is in general not homogenous, but nonetheless it is difficult to conceive a departure of the local scale too far from (2.45). We must therefore keep in mind that although formally our calculations can be made very precise by taking the limit (2.44), in a phenomenologically viable setup there will be corrections (essentially corrections) that are not hugely suppressed. Taking this into account, and that the constraints are only approximate up to order one factors, in what follows we allow ourselves to take in the range , with also in the range but chosen appropriately such that for each value of eqs. (2.42) are satisfied. Note that the most natural values are towards the lower end of the range, however, due to the strong model dependence of the relation between local and global scales this range is only an approximate one and some flexibility should be allowed.

Similarly, we expect the local scale to be related to the global ones and in particular to . The particular relation strongly depends on the geometry of the CY base . In the case of a torus we have

 R⊥∼gsMPlanckα1/2GUTM∗. (2.46)

It is therefore important to note that in a torus, and more generally in a near homogeneous setup, the observed values of and are not compatible with the constraints (2.42) and we expect higher derivative corrections to the effective 8-dimensional theory coming from large brane intersection angles. At a deeper level this can be taken as motivation for local models based on contractible cycles as then the scaling with respect to the Planck scale is expected to be modified to the schematic form . More generally the geometry can lead to differences between and the global scales either coming from inhomogeneities of the divisors or from the geometry allowing a decoupling of the intersecting brane setup from the overall volume. Given this in general we do not attempt to relate the local scales with the global ones.

3 The SO(12) enhancement point

3.1 The SO(12) point and background fluxes

We now apply the procedure described in the previous section to the point in where the down-type Yukawa coupling localises. At that locus there is an SO(12) enhancement of the gauge symmetry which can be seen by decomposing the adjoint

 SO(12) ⊃ SU(5)×U(1)1×U(1)2, (3.1) 66 → 24(0,0)⊕1(0,0)⊕1(0,0)⊕(5(−1,0)⊕5(1,1)⊕10(0,1) ⊕ c.c.).

The spontaneous breaking of the SO(12) symmetry away from the enhancement point can be obtained by turning on a background for the Higgs scalar,

 ⟨φ⟩=M∗R⊥(z1v1Q1+z2v2Q2), (3.2)

where and are dimensionless parameters. The generators and are those corresponding to the U(1) factors in the decomposition of SO(12). This Higgs background describes three sets of intersecting 7-branes with localised matter on their intersection curves

 10M : z2=0, ¯5M : z1=0, ¯5H : v2z1+v1z2=0. (3.3)

We will be solving for the wavefunctions of modes localised along these curves.

In order to get chiral matter, U(1) flux must be turned on along the generators and . Moreover, to break the GUT group and induce doublet-triplet splitting also flux must be turned on along the hypercharge direction of SU(5) so that,

 SU(5) → SU(3)×SU(2)L×U(1)Y, ¯5 → (¯3,1)1/3 ⊕ (1,2)−1/2, 10 → (¯3,1)−2/3 ⊕ (3,2)1/6 ⊕ (1,1)1.

The geometric properties of these fluxes are dependent not only on but also on the full CY four-fold . Let us recall some of the defining global properties of the fluxes. First, the hypercharge flux must be turned on along a cycle which is homologically non-trivial when pulled back to the GUT divisor , but trivial in the full CY so that U(1) is massless [64, 4, 6, 7]. Secondly, the fluxes must be turned on such that they induce the correct chiral matter spectrum, which is determined as

 n(3,1)−1/3−n(¯3,1)1/3 = M5, n(1,2)1/2−n(1,2)−1/2 = M5+N, (3.4)

for the curves and

 n(3,2)1/6−n(¯3,2)−1/6 = M10, n(¯3,1)−2/3−n(3,1)2/3 = M10−N, n(1,1)1−n(1,1)−1 = M10+N, (3.5)

for the curves, where denotes the number of massless 4-dimensional fields transforming in the representation of SU(3)SU(2)U(1). Here the fluxes , and are specified by fractional line-bundles , and such that , and . To maintain complete representations on the matter curves we want for those curves, while to induce doublet-triplet splitting we want and for the Higgs curve.

Working locally near a point on we are not sensitive to the full global structure of the fluxes. Indeed, all the geometry of the cycles locally reduces to the four possible components of the flux along , , and . These components are constrained by the local D-term condition

 ω∧F(1,1)=0, (3.6)

whereas the F-term condition simply requires .

We assume that locally the flux takes a constant profile in , neglecting a possible spatial dependence of the flux. This can be expected to be a decent approximation if the curvature around the enhancement point is small, as in that case it can be thought as the leading term of a Taylor expansion in the local coordinates, as we have argued in the previous section. Taking varying flux into account would lead to technical difficulties, in particular D-terms would generally not be solved by a flat-space profile, which would imply having to work in curved space [14]. Moreover, for simplicity, in the main part of the paper we do not turn on flux along the oblique components and . Oblique fluxes turn out to not affect the physics of the wavefunction in a qualitatively important way. We relegate a treatment of the more general flux including such components to appendix A. A general suitable choice for the U(1) flux is therefore

 F(1,1)=2iM2∗R2∥(dz1∧d¯z1−dz2∧d¯z2)(−M1Q1+M2Q2+γQY), (3.7)

where , and are dimensionless real constants and the generator is along the hypercharge direction in SU(5). The gauge potential associated with this flux reads

 A=iM∗R∥(z1d¯z1−¯z1dz1−z2d¯z2+¯z2dz2)(−M1Q1+M2Q2+γQY). (3.8)

The relation between the local values of the flux , and and the global integrated values in (3.4) and (3.5) is subtle. The connection is that the local fluxes , and determine the chirality of the localised fields, which generically arrange in supermultiplets as we have described in previous section. At a given localisation point, eq. (2.21) has an infinite number of solutions. Consistency with the topological data of the flux and however selects a finite subset of size given by the global integrated values in (3.4) and (3.5), in accordance with standard index theorems.

Using the flux (3.7) the chirality for a given localised mode is determined by the analogous local expressions to eqs. (3.4) and (3.5). That is, for the curve,

 δ(3,1)−1/3 = sign[−M1+13γ], δ(1,2)1/2 = sign[(−M1+13γ)−56γ], (3.9)

for the curve,

 δ(3,2)1/6 = sign[M2+16γ], δ(¯3,1)−2/3 = sign[(M2+16γ)−56γ], δ(1,1)1 = sign[(M2+16γ)+56γ], (3.10)

and for the curve,

 δ(3,1)−1/3 = sign[M1−M2+13γ], δ(1,2)1/2 = sign[(M1−M2+13γ)−56γ]. (3.11)

where () means that the corresponding set of localised massless 4-dimensional fields transforms in the () representation of SU(3)SU(2)U(1) and we have taken without loss of generality.141414The relative signs between the components in expressions (3.9), (3.10) and (3.11) are determined from the group theory charges of the states while the overall sign for each curve is determined by studying the form of the wavefunctions in section 3.2, such that given the sign of the fluxes the correct state localises. In this regard, it is also worth stressing that if the local flux vanishes along the matter curve for a given representation the wavefunction does not localise, as we have already commented in the previous section.

With these relations we see that there are some constraints on the local fluxes that one has to satisfy in order to properly model the massless spectrum. In particular, we shall require that the expressions in (3.9) are all negative and that the expressions in (3.10) are all positive. This can be implemented by taking and positive and much larger than . On the Higgs curve we require that the second expression of (3.11) is negative. The sign of the first expression of (3.11) determines whether locally there is a massless triplet, a massless anti-triplet or a vector-like pair, with the mass of the latter depending on the particular topology of the matter curve,

 M1−M2+13γ > 0,Massless(3,1)−1/3 (3.12) M1−M2+13γ < 0,Massless(¯3,1)1/3 M1−M2+13