Approaching Minimal Flavour Violation from an SU(5) x S4 x U(1) SUSY GUT

We show how approximate Minimal Flavour Violation (MFV) can emerge from an SU(5) Supersymmetric Grand Unified Theory (SUSY GUT) supplemented by an S4 x U(1) family symmetry, which provides a good description of all quark and lepton (including neutrino) masses, mixings and CP violation. Assuming a SUSY breaking mechanism which respects the family symmetry, we calculate in full explicit detail the low energy mass insertion parameters in the super-CKM basis, including the effects of canonical normalisation and renormalisation group running. We find that the very simple family symmetry S4 x U(1) is sufficient to approximately reproduce the effects of low energy MFV.


Introduction
The mystery of flavour has been with us from the discovery of the muon in 1936 to the discovery of neutrino mass and mixing in 1998. The Standard Model (SM), extended to include neutrino mass, is described by at least 26 parameters, of which no less than 20 are flavour parameters: 10 from the quark sector and at least 10 from the lepton sector. At least two of these parameters are related to CP violation in the quark and lepton sectors, although the latter has not yet been definitively observed.
A lot of effort has been put into trying to understand the flavour structure of the SM (for reviews see e.g. [1]). Its peculiar features include hierarchical charged fermion masses, with the down-type quark and charged lepton masses showing a similar pattern which differs from that of the up-type quarks, while neutrinos are significantly lighter than all other particles. Flavour mixing in the lepton sector has turned out to be much larger than in the quark sector, and the number of generations is not explained.
Following the award of the 2015 Nobel Prize for "the discovery of neutrino oscillations which shows that neutrinos have mass", we still have no more understanding of flavour than back in 1936 when Rabi famously asked of the muon "who ordered that?". Part of the reason for this impasse is the failure of experiment to measure any flavour and CP violation beyond that expected in the SM. The problem is that the SM is not a theory of flavour and, as such, provides no understanding of the origin or nature of flavour.
In the absence of any observed beyond SM flavour and CP violation, a sort of "straw man" ansatz for flavour has emerged known as Minimal Flavour Violation (MFV) [2] in which all flavour and CP-violating transitions are postulated to originate in the SM Yukawa matrices so that they are governed by the CKM matrix. The formulation of MFV in an effective field theory involving a high-energy SU(3) 5 flavour symmetry, broken only by the Yukawa matrices, allows higher-dimensional operators which can contribute considerably to flavour observables [3,4]. Going beyond an effective field theory description, it is possible to implement the idea of MFV in a renormalisable theory by introducing new heavy fermions. In such a setup, the flavour symmetry is broken by scalar fields whose Vacuum Expectation Values (VEVs) are related to the Yukawa matrices in an inverse way [5]. Although this differs from the standard MFV approach, where the fundamental flavour breaking fields are linearly related to the Yukawa matrices, it does reproduce MFV phenomenologically by predicting very SM-like flavour and CP violation, which is of course exactly what is observed.
When considering extensions of the SM, such as Supersymmetry (SUSY) softly broken at the TeV scale, then in general large deviations from SM flavour and CP violation are expected. SUSY models include one-loop diagrams that lead to Flavour Changing Neutral Current (FCNC) processes such as e.g. b → sγ and µ → eγ at rates which are proportional to the size of the off-diagonal elements of the scalar mass matrices, when the latter have been rotated to the super-CKM (SCKM) basis where the Yukawa matrices are diagonal [6]. These SUSY contributions are tamed in the Constrained Minimal Supersymmetric Standard Model (CMSSM) which postulates that, at the high energy scale, the SUSY breaking squark and slepton mass squared matrices are proportional to the unit matrix and the trilinear A-terms are additionally aligned with the Yukawa matrices, resulting in an (approximate) MFV-like structure at low energy [6].
In the framework of Grand Unified Theories (GUTs), the embedding of the SM fermions into GUT multiplets does not allow to implement the SU(3) 5 flavour symmetry of MFV. However, in GUTs based on SU(5) [7] or the Pati-Salam group SU(4) × SU(2) × SU(2) ′ [8], it is possible to introduce an SU(3) 2 flavour symmetry instead, and this has been shown to lead to sufficient suppression of flavour violation [9]. Considering SUSY GUTs, the CMSSM framework always provides a safe haven from unwanted flavour violation, although CP violation in the form of Electric Dipole Moments (EDMs) remains a challenge [6]. However, with SUSY and SUSY GUTs, the real challenge is to justify the assumptions of MFV or the CMSSM, while at the same time providing a realistic explanation of quark and lepton (including neutrino) masses, mixing and CP violation. This non-trivial balancing act is what concerns us in this paper.
The discovery of neutrino mass and mixing has spurred a lot of work aiming to describe flavour in terms of a family symmetry of some kind, in particular discrete non-Abelian family symmetry [1]. It was realised early on that in such models, the idea of spontaneous flavour and CP violation could effectively tame the flavour and CP problems of the SM [10,11] without any ad hoc assumptions about MFV or the CMSSM. The main point is that the same family symmetry introduced to understand the Yukawa sector will also automatically control the flavour structures of the soft SUSY breaking sector. The only requirement is that the SUSY breaking hidden sector must respect the family symmetry, which means that the family (and CP) symmetry breaking scale must be below the mass scale of the messengers which mediate SUSY breaking to the visible sector. SUSY breaking in the framework of supergravity provides one attractive example for such a situation.
The idea of using family symmetry to solve the SUSY favour and CP problems has been fully explored in the framework of an SU(3) family symmetry [11,12], where it was shown that the flavons that spontaneously break family and CP symmetry will perturb the SUSY breaking sector, leading to tell-tale signatures of flavour and CP violation beyond MFV or the CMSSM. Unfortunately, these signatures which were expected to appear in Run1 of the LHC [13] did not in fact materialise, and indeed the allowed parameter space has been much reduced [14].
In the setup discussed in [12], the extra flavour violation can be understood as follows. At leading order, the CMSSM is enforced by the SU (3) family symmetry acting on the squark and slepton mass squared matrices. However the fact that SU (3) is broken by flavons, as it must be to generate the quark and lepton masses, means that flavons appearing in the Kähler potential will give important contributions to the kinetic terms, requiring extra canonical normalisation [15]. Since SUSY breaking also originates from the Kähler potential, the flavons will also modify the couplings of squarks and sleptons to the fields with SUSY breaking F -terms. The resulting corrections to the soft mass squared matrices from unity will be similar to the corrections of the corresponding Kähler metrics, yet both are not aligned due to independent coefficients of the relevant operators. Likewise, the trilinear soft SUSY breaking A-terms will replicate the flavour structure of the Yukawa matrices prior to canonical normalisation, but exact alignment is not realised. All of this occurs at the high scale. Additional flavour violation is generated by renormalisation group (RG) running down to low energy, taking into account the seesaw mechanism [16] which will involve thresholds at an intermediate scale, see e.g. [17,18].
In this paper we show how approximate MFV can emerge from an SU(5) SUSY GUT, supplemented by an S 4 ×U(1) family symmetry [19,20], which provides a good description of all quark and lepton (including neutrino) masses, mixings and CP violation. Assuming that SUSY breaking respects the family symmetry, we calculate in full detail the low energy mass insertion parameters in the SCKM basis. We include the effects of canonical normalisation as well as RG running. Remarkably, due to the peculiar flavour structure of the model, we find that the small family symmetry S 4 × U(1) is sufficient to reproduce the effects of low energy MFV much more accurately than the previous SU(3) family symmetry model. Table 1: The matter, Higgs and flavon superfields of the model in [20] together with their transformation properties under the imposed SU(5) × S 4 × U(1) symmetry.

Trimaximal S 4 × SU (5) model
In this section, we present the basic ingredients of the supersymmetric model of flavour proposed in [20]. It is capable of correctly describing a sizable reactor neutrino mixing angle θ l 13 by generating a neutrino mass matrix of trimaximal form. The model represents a modification of an earlier tri-bimaximal model [19] with only minor changes. Being formulated in a supersymmetric SU(5) grand unified framework, the matter superfields fall into the 10 and5 representations, where the superscript c denotes charge conjugation of the right-handed superfields. Table 1 lists the matter, Higgs and flavon superfields together with their transformation properties under the imposed SU(5) × S 4 × U(1) symmetry. Details of the non-Abelian finite group S 4 are provided in Appendix A. The5-plets, labelled by F , are assigned to a triplet representation of S 4 , while the 10-plets are split into an S 4 doublet T for the first two generations and an S 4 singlet T 3 for the third generation. In addition, righthanded neutrinos N are introduced transforming in the same S 4 triplet representation as F . The SU(5) Higgs fields H 5 , H5 and H4 5 are all S 4 singlets. Note that each of these GUT Higgs representations contains an SU(2) L Higgs doublet. Therefore, the low energy doublet H u originates from H 5 , while H d arises from a linear combination of H5 and H4 5 [6,21]. 1 In addition, we introduce a number of flavon fields Φ f ρ , which are labelled by the corresponding S 4 representation ρ as well as the fermion sector f to which they couple at leading order (LO). Two flavons, Φ u 2 andΦ u 2 , generate the LO up-type quark mass matrix. Three flavon multiplets, Φ d 3 ,Φ d 3 and Φ d 2 , are responsible for the down-type quark and charged lepton mass matrices. Finally, the right-handed neutrino mass matrix is generated from the flavon multiplets Φ ν 3 ′ , Φ ν 2 and Φ ν 1 as well as the flavon η which is responsible for breaking the tri-bimaximal pattern of the neutrino mass matrix to a trimaximal one at subleading order [20]. The additional U(1) symmetry has been introduced in order to control the coupling of the flavon fields to the matter fields in a way which avoids significant perturbations of the LO flavour structure by higher-dimensional operators. We refer the reader to [19] for more details.
The vacuum structure of the flavon fields arises from the F -term alignment mechanism [22]. Introducing a set of so-called driving fields, the corresponding F -term conditions give rise to particular flavon alignments as described in Appendix B. To LO, these are given as [19,20], Φ u 1 As H5 and H4 5 transform differently under U (1), it is clear that the mechanism which spawns the low energy Higgs doublet H d must necessarily break U (1). Although the discussion of any details of the SU (5) GUT symmetry breaking (which, e.g., could even have an extra dimensional origin) are beyond the scope of our paper, we remark that a mixing of H5 and H4 5 could be induced by introducing the pair H ± 24 with U (1) charges ±1 in addition to the standard SU (5) breaking Higgs H 0 24 .
account such shifts as well as all other subleading effects. As our LO results for the mass insertion parameters depend solely on the LO structure of the model, we only report the LO analysis in the main part of this paper. When giving explicit expressions, we therefore limit ourselves to showing the leading contributions, omitting additional higher order corrections. We will indicate such approximations by ≈ throughout the paper. Finally, the VEVs of the two neutral Higgses are: where t β ≡ tan(β) = υu υ d and υ = υ 2 u + υ 2 d = 174 GeV.

Kähler potential
A characteristic feature of any effective theory is the presence of non-renormalisable operators which are only constrained by the imposed symmetries. In the context of supersymmetry, this is the case for both the superpotential as well as the Kähler potential. The effective coupling of flavon fields to the Kähler potential gives rise to kinetic terms with a non-canonical Kähler metric K = 1, wheref and f are, respectively, the scalar and fermionic components of a generic chiral superfieldf . In order to extract physically meaningful properties of a model, the kinetic terms have to be brought to a canonical form. The required basis transformation is usually referred to as canonical normalisation [15]. In the context of SU (5), we encounter a Kähler metric for each of the three GUT representations containing the matter fields. We denote these by K T , K F and K N , respectively. Using the symmetries of Table 1, the expansions of these 3 × 3 matrices in terms of flavon fields can be obtained from 3) where the c n are order one coefficients which we can assume to be real thanks to the imposed CP symmetry. Products of flavon fields which are allowed to couple in the Kähler potential are collected in the tuples R i , which in turn are unions of tuples S i . These tuples, which contain all possible combinations of up to eight flavons with a minimum contribution of order λ 8 , are defined as (3.9) S 1 and S 2 contain combinations of flavons with U(1) charges that sum up to zero. They can form S 4 invariants when contracted with two doublets or two triplets. Therefore, S 1 and S 2 contribute to K F , K N and the upper-left 2 × 2 block of K T in Eq. (3.2). Moreover, the combinations in S 1 can be contracted to S 4 invariants so that they additionally contribute to the lower-right 1 × 1 block of K T . S 3 gives further contributions to K F and K N but not to K T . Finally, the combinations contained in S 4 have U(1) charges which add up to 5 and allow for S 4 contractions to a doublet. Hence, they contribute to the off-diagonal upper-right block of K T . We remark that the effects of the operators involving the flavon field η are independent of its S 4 transformation properties as a 1 or 1 ′ .
When calculating the Kähler metric from the expressions of Eqs. (3.2-3.4), it is important to take into account all invariant S 4 contractions of two matter fields with a given product of flavons.

Kähler metric with LO corrections
It is straightforward though tedious to determine the matrices K T , K F and K N from Eqs. (3.2-3.4). Keeping only the LO corrections to the unit matrix, we find for the 10 of SU(5) where k i denote real order one coefficients, and θ k i are phases associated with the generally complex flavon VEVs. Here and throughout our paper, the dots in the lower-left corner of the matrix represent the complex conjugates of the corresponding entries in the upperright part of the matrix. The operator T † Φ d 2 Φ d † 2 T /M 2 gives rise to the parameters k 1 and Being associated with T †Φu 2 T /M, the parameter k 2 carries no phase factor becauseφ u 2 ∈ R, cf. Appendix B. Finally, the (13) and (23) 11) where θ d 2 and θ d 3 are the phases of the LO VEVs φ d 2 and φ d 3 , respectively. Analogously, we obtain the matrix K F , 2 where K i ∈ R. The parameters on the diagonal, K 1 and K 2 , originate from different contractions of the term The off-diagonal elements, parametrised by K 3 , are derived from the operator F †Φu 2 F/M and are real due toφ u 2 ∈ R. Hence the LO correction of K F from unity is given by a real matrix.
The corresponding Kähler metric K N for the right-handed neutrinos is identical to K F up to a difference in the order one coefficients of the individual corrections. We thus have 13) where the coefficients K N i are again real.

Canonical normalisation
The expansion of the Kähler potentials in terms of flavon insertions leads to non-canonical kinetic terms. In order to bring the Kähler potential back to its canonical form, a nonunitary transformation has to be applied on the matter superfields. This procedure is known as canonical normalisation (CN) [15], and introduces the 3 × 3 matrices P A which transform the matter superfields A = T, F, N as (3.14) A prescription for deriving the matrices P A can be found in Appendix C.1. To LO, they take the simple form (3.15) In the following sections we study the structure of the Yukawa as well as the soft supersymmetry breaking sectors. The CN transformations of Eq. (3.15) have to be applied to these before aiming at a physical interpretation of the resulting patterns.

Yukawa sector after CN
In this section, we study the fermionic sector of the model, completing the analysis of [19,20] by including the effects of canonical normalisation. Our parametrisation differs slightly from the one used in [19,20] as, in this work, we do not absorb any of the higher order corrections to the mass matrices or the flavon VEVs into the associated leading order terms. See Appendix B for more details.

Up-type quarks
The Yukawa matrix of the up-type quarks can be constructed by considering all the possible combinations of a product of flavons with T T H 5 for the upper-left 2 × 2 block, with T T 3 H 5 for the (i3) elements, and with T 3 T 3 H 5 for the (33) element. The operators which generate a contribution to the Yukawa matrix of order up to and including λ 8 are where the parameters y t and y u i are real order one coefficients. Inserting the flavon VEVs and expanding the S 4 contractions of Eq. (4.1) using the Clebsch-Gordan coefficients given for instance in [19], yields the up-type Yukawa matrix at the GUT scale where the relation to the flavon VEVs, cf. Eqs. (2.2-2.4) as well as Appendix B, is given by Applying the phases of the LO flavon VEVs as given in Eq. (B.2), we moreover have where we have also used the fact that the shift δ u 2,1 of the flavon VEV Φ u 2 in the first component is of order It is worth noting that the (12), (13) and (21), (31) elements of Eq. (4.2) remain zero up to order λ 8 .
Changing to the basis with canonical kinetic terms, we calculate (P −1 T ) T Y u GUT P −1 T . For convenience we also apply an extra phase redefinition on the right-handed superfields, Q u = diag(e iθ y u , e iθ y u , 1). (4.5) As a result we obtain the up-type quark Yukawa matrix in the canonical basis, Compared to Eq. (4.2), the canonical normalisation has significantly modified the offdiagonal entries: the texture zeros are filled in; moreover, the (23) and (32) elements feature a reduced λ-suppression.

Down-type quarks and charged leptons
The Yukawa matrices of the down-type quarks and the charged leptons can be deduced from the superpotential operators where the y d i are real order one coefficients. For the operators proportional to y d 2 and y d 5 , specific contractions have been chosen as described in [19,20], such that the Gatto-Sartori-Tonin (GST) [26] and Georgi-Jarlskog (GJ) [27] relations are satisfied at LO. For all other operators we do not restrict the contractions to special choices; however, we have checked that in all cases, our LO result can simply be parameterised by an effective coupling constant which is given as a combination of the individual contributions from each contraction. It is worth noting that the operator proportional to y d 4 is only allowed if η transforms as a trivial singlet under S 4 . Separating the contributions of H5 and H4 5 , the S 4 contractions give rise to The parameters in these expressions are related to the flavon VEVs as defined in Eqs. (2.2-2.4) and Appendix B viã Using Eqs. (B.2,B.6), we deduce the following relations for the phases (4.10) The Yukawa matrices of the down-type quarks and the charged leptons are linear combinations of the two structures in Eq. (4.8). Following the construction proposed by Georgi and Jarlskog, we have Y d GUT = Y5 + Y4 5 and Y e GUT = (Y5 − 3Y4 5 ) T , respectively. Performing the canonical normalisation on the Yukawa matrices (P T as well as an additional rephasing of the right-handed superfields by (4.13) We observe that the canonical normalisation modifies the down-type quark and charged lepton Yukawa matrices solely by additional contributions of the same order in the (31), (32) and (13), (23) elements, respectively. Comparing Eq, (4.12) with Eq. (4.6) suggests that the CKM mixing is dominated by the diagonalisation of the down-type quark Yukawa matrix. We will explicitly verify this when calculating the SCKM transformations in Section 6.

Dirac neutrino coupling
Having introduced right-handed neutrinos N in Table 1, their Dirac coupling to the lefthanded SM neutrinos originates from the superpotential terms where y D and y D i are real order one parameters. The corresponding Yukawa matrix is determined as Here, the phase can be deduced from Eq. (B.5).
Applying the CN transformation (P −1 F ) T Y ν P −1 N , the corresponding Yukawa matrix in the basis with canonical kinetic terms takes the form Compared to Eq. (4.15), an additional contribution of the same order arises in the (13), (22) and (31) entries. Moreover, the λ-suppression of the (12), (21) and (33) elements is reduced.

Majorana neutrino mass
The mass matrix of the right-handed neutrinos is obtained from the superpotential terms where w i denote real order one coefficients. This results in a right-handed Majorana neutrino mass matrix M R of the form with (4.20) According to Eqs. (B.2,B.5,B.6), the phases are given by The first matrix of Eq. (4.19) arises from terms involving only Φ ν 1,2,3 ′ . As their VEVs respect the tri-bimaximal (TB) Klein symmetry Z S 2 × Z U 2 ⊂ S 4 , this part is of TB form. The second matrix of Eq. (4.19), proportional to D, is due to the operator w 4 1 M NNΦ d 2 η. As the product of both flavon VEVs involved is not an eigenvector of U, half of the TB Klein symmetry is broken at a relative order of λ. The resulting trimaximal TM 2 [28] structure can accommodate the sizable value of the reactor neutrino mixing angle θ l 13 as explained in [20] in the context of the original model [19].
Performing the CN basis transformation (P −1 N ) T M R P −1 N does not alter the matrix in Eq. (4.19) at the given order, so that M R = M R + O(λ 6 )M.

Effective light neutrino mass matrix
Calculating the effective light neutrino mass matrix which arises via the type I seesaw with a ν , b ν , c ν and d ν being functions of the real parameters A, B, C and D. The deviation from tri-bimaximal neutrino mixing is controlled by d ν ∝ D. Due to the three independent LO input parameters (w 1 ∝ A ,w 2 ∝ B ,w 3 ∝ C), any neutrino mass spectrum can be accommodated in this model. At this order, the canonical normalisation does not modify the effective light neutrino mass matrix as obtained without the CN transformations.
Hence, concerning the results on light neutrino masses and mixing, we can simply refer the reader to the corresponding discussion in [20].

Soft SUSY breaking sector after CN
Having applied the CN basis transformation of the matter superfields to the Yukawa sector, we now turn to the soft SUSY breaking terms. In the context of the general MSSM with R-parity, these are parameterised as [6] − and contain trilinear scalar couplings (A-terms) as well as bilinear scalar masses. A tilde indicates the scalar partnerf of a SM fermion f . Taking into account the SU (5) framework, we construct the effective soft SUSY breaking operators in this section, assuming that the mechanism of SUSY breaking is practically independent of the family symmetry breaking.

Trilinear soft couplings
The flavour structure of the trilinear A-terms is similar to the corresponding Yukawa matrices, as both originate from the same set of superpotential terms. In the case of the soft terms, these are coupled to a hidden sector superfield X with independent real order one coupling constants and suppressed by a mass scale M X . When X develops its SUSY breaking F -term VEV, the scalar components of the Higgs and matter superfields are projected out, thereby generating the trilinear soft terms. There exist in fact extra contributions to the A-terms from superpotential operators involving flavons but no X field. These can be traced back to non-vanishing VEVs for the auxiliary F -components of the flavon fields, which are zero in the SUSY limit but develop a non-trivial value when SUSY breaking terms are included. It turns out that such F -term VEVs are aligned with the LO flavon VEVs in many situations [10,29]. Hence, these extra contributions to the A-terms do not give rise to new flavour structures.
Defining the mass parameters m 0 ≡ F X /M X and A 0 ≡ α 0 m 0 , with α 0 being a real constant, we can obtain the expressions for the trilinear matrices A f GUT /A 0 by copying the Yukawas matrices of Eqs. (4.2,4.8,4.15) with different order-one coefficients and phases: and similarly for A d GUT , A e GUT and A ν . Applying the CN transformation as well as the rephasing of the right-handed superfields proceeds analogously to the Yukawa sector. The resulting trilinear matrices A f GUT /A 0 in the basis of canonical kinetic terms are thus derived from Eqs. (4.6,4.12,4.13,4.17) For example, the up-type quark trilinear matrix takes the form

Soft scalar masses
The scalar mass terms of the soft supersymmetry breaking Lagrangian originate from the Kähler potential. Non-renormalisable couplings of the matter superfields to the square X † X/M 2 X of the SUSY breaking field X generate soft masses when the F -term of X develops a VEV. The structure of the soft mass matrices is therefore similar to the Kähler metric K of the corresponding GUT multiplet. As for the trilinear soft terms, all order one coefficients are independent of those appearing in K. The scalar masses before canonical normalisation are then obtained from K T , K F and K N of Eqs. (3.10,3.12,3.13) Moreover, the ones on the diagonal of K have to be rescaled by a new factor of order one. In the case of the 10 of SU (5), the 2+1 structure requires the introduction of two extra parameters, b 01 and b 02 . Explicitly, we get Performing the transformations to the basis of canonical kinetic terms results in soft scalar mass matrices of the form For convenience, we will absorb the order one parameter B 0 into the soft SUSY breaking mass m 0 , so that the leading contribution on the diagonal of M 2 F GUT /m 2 0 is nothing but unity. For the right-handed fields contained in the GUT multiplets, an additional rephasing has to be applied. We will come back to this when calculating the soft terms in the SCKM basis in Section 6.2. Notice that we have dropped all λ-suppressed corrections of the diagonal elements. This simplification is justified as FCNC processes are induced by loop diagrams involving the off-diagonal entries of the sfermion mass matrices. The simplification of the diagonal elements in Eqs. (5.6,5.7) does not affect these off-diagonals in our LO analysis, even when going to the SCKM basis.

SCKM basis
Predictions relating a theoretical model with its phenomenological implications are typically given in the basis in which the Yukawa matrices are diagonal and positive, corresponding to the physical quark and lepton mass eigenstates. The so-called SCKM basis is the analogue in a supersymmetric framework. Changing to the SCKM basis, all canonically normalised quantities undergo a unitary transformation of the superfields which diagonalises the effective Yukawa couplings in the superpotential. In this basis it is convenient to define a set of dimensionless parameters, known as the "mass insertion parameters", which directly enter the expressions of phenomenological flavour observables.
In principle, the SCKM transformation should be performed after electroweak symmetry breaking. The canonically normalised Yukawa, trilinear and soft mass matrices should be evolved from the GUT scale M GUT to the weak scale M W using the correspond-ing renormalisation group equations (RGEs). Only at that point, the diagonalisation of the Yukawa matrices should take place, leading to the definition of a SCKM basis. Following this procedure, there is obviously no notion of mass insertion parameters at the scale M GUT as there is no proper definition of the SCKM basis.
An alternative approach which is commonly used consists in diagonalising the Yukawa matrices at (or rather just below) the GUT scale. The so-obtained basis is approximately identical to the SCKM basis provided the RGE contributions to the off-diagonal elements of the Yukawa matrices remain negligible. 3 This is the case as long as the RGE effects can be absorbed into a redefinition of the (unknown) order one coefficients. It is then possible to introduce mass insertion parameters already at M GUT . Their low energy values have to be determined from the corresponding RG evolution. In this work, we will adopt the latter approach as it allows for a semi-analytical study of the relations between the high and low energy parameters by means of a perturbative λ-expansion.

SCKM transformations
The SCKM transformations are applied on the matter superfieldsf L, where we use the tilde to denote the SCKM basis. The derivation and the explicit form of the unitary transformations can be found in Appendix C.2. Applying this change of basis to the Yukawa matrices yields These results, which are valid at the high scale, agree with the LO results derived in [19,20]. This shows that the canonical normalisation does not affect the LO expressions of the quark and charged lepton masses. Up to phase convention, the CKM matrix is given by V CKM GUT = (U u L ) T U d * L (see Appendix C.2 for explicit expressions). Extracting the mixing angles shows that the LO CKM mixing arises purely from the down-type quark sector, incorporating the GST relation [26] θ q 12 ≈ m d /m s , and agrees with the results obtained in [19,20]. Concerning the CP violation, we find the Jarlskog invariant [31] to be The PMNS matrix is dominated by the trimaximal TM 2 neutrino mixing V ν which diagonalises the effective light neutrino mass matrix of Eq. (4.22). Including the charged lepton corrections, we have U PMNS GUT = (U e L ) T V * ν with mixing angles given as and a leptonic Jarlskog invariant of the form

Soft terms in the SCKM basis
In order to obtain the flavour structure of the soft SUSY breaking terms in a basis which is suitable for physical interpretations, we have to apply the SCKM transformations on the canonical trilinear soft couplings and soft scalar masses, cf. Section 5. The action of the U f L,R matrices on the A-terms is identical to the transformation of the Yukawa matrices: However, due to different order one coefficients, the A-terms remain non-diagonal in the SCKM basis. The soft masses of Eqs. (5.6,5.7) are transformed differently for different components of the SU (5) multiplets. Moreover, we have to associate the mass matrices of the effective soft Lagrangian in Eq. (5.1) with M 2 T GUT and M 2 F GUT and take into account the additional rephasing transformations of the right-handed superfields, see Eqs. (4.5,4.11), that were performed after CN. Then, the soft masses in the SCKM basis are We find the following leading order expressions, where the order one coefficients are defined in Eqs. (D.4,D.5). Note that we have absorbed the order one coefficient B 0 into m 0 , cf. Eq. (5.7), so that (m 2 d ) RR GUT /m 2 0 and (m 2 e ) LL GUT /m 2 0 have 1s on the diagonal.

Mass insertion parameters
In supersymmetry, flavour changing processes are induced by the mismatch of fermion and sfermion mass eigenstates. Having changed the basis of the superfields to the SCKM basis, the Yukawa matrices are diagonal. Thus, the off-diagonal entries of the scalar mass matrices determine the size of the resulting FCNCs. As both the left-and the right-handed fermions have their own scalar partners, there are three types of scalar mass matrices 1) where µ is the higgsino mass which we take to be real. In Eq. (7.1), the first contribution on the right-hand sides originates from the soft breaking Lagrangian, while the second term is the supersymmetric F -term contribution to the scalar masses. We note that it is formally possible to define m 2 From the model building perspective, a convenient measure of flavour violation is provided by a set of dimensionless parameters, known as the mass insertion parameters. These are defined as [32] ( where the average masses in the denominators are

Mass insertion parameters δ at the GUT scale
Inserting the results of Section 6, it is straightforward to calculate the mass insertion parameters at the GUT scale. The full LO expressions are given in Appendix D. In the following we only report the flavour structure of the various δs in terms of their λ-suppression.

Effects of RG running
Having calculated the GUT scale mass insertion parameters, it is now necessary to consider their evolution down to the electroweak scale. Only then are we able to compare the predictions of the model to experimental measurements of flavour observables. This evolution is described by the RG equations which are given explicitly in Appendix E in the SCKM basis. Technically, we perform the RG running in two stages, first from M GUT to M R where the right-handed neutrinos are integrated out, and then from M R to M SUSY ∼ M W . In order to derive analytical results, we estimate the effects of the running using the leading logarithmic approximation. As the Yukawa matrices themselves are also affected by the running, it is necessary to apply further basis transformations on the superfields which diagonalise the low energy Yukawas matrices. Details of the various steps involved in calculating the low energy mass insertion parameters can be found in Appendix F. For the down-type squarks and the charged sleptons, the resulting effects can simply be absorbed into new order one coefficients. It is interesting to see that this is not the case for the up-type squarks, where the order of the (13) and (23) elements of δ u LR gets modified. For completeness, we present the flavour structure of the low energy δs in terms of their λ-suppression, which should be compared to Eqs. (7.4-7.6).

Conclusion
Despite its tremendous success, the Standard Model of particle physics is widely viewed as the low energy limit of a more fundamental theory. In order to understand the nature of flavour in such extensions of the SM it is necessary to answer the following three questions. However, the concept of MFV is not a theory of flavour as such. Moreover, it does not seem to provide a framework in which the first two questions of the flavour puzzle can be addressed in a satisfactory way.
In this paper, we have investigated the issue of flavour violation within a supersymmetric GUT model of flavour which is based on the simple family symmetry S 4 ×U(1) [20]. The existence of three families of quarks and leptons is related to the non-Abelian factor of the family symmetry whose triplets are the only faithful irreducible representations. The structure of the Yukawa matrices arises from the breaking of the family symmetry. This aspect was thoroughly studied in [19,20] where it was shown to provide a good description of all quark and lepton masses, mixings and CP violation.
Applying the family symmetry on the soft SUSY breaking sector, we have worked out the mass insertion parameters which describe the sources of flavour violation beyond the SM. Our calculation relies on the assumption that the SUSY breaking mechanism respects the family symmetry. Working in an expansion in powers of the Wolfenstein parameter λ, we take into account the effect of canonical normalisation as well as renormalisation group running. Our results for the low energy mass insertion parameters are summarised in Eqs. (7.7-7.9), with the explicit expressions given in Appendix F.3. We find that δ f LL and δ f RR are approximately equal to the identity with only small off-diagonal entries. Considering the parameters δ f LR we observe that the diagonal elements feature the same hierarchies as the corresponding diagonal Yukawa matricesỸ f , while the offdiagonal elements are strongly suppressed. This shows that our S 4 × U(1) SUSY GUT approximately reproduces the effects of low energy MFV, where one would simply impose δ f LL = δ f RR = 1 and δ f LR ∝Ỹ f . The phenomenological implications of the deviations form MFV will be discussed quantitatively in a dedicated paper [33], where we will present and discuss the predictions of our model of flavour with respect to a number of different flavour observables in detail.

A S 4 and CP symmetry
The non-Abelian finite group S 4 can be defined in terms of the presentation where S, T and U denote the generators of the group. Explicit matrix representations are basis dependent. In this work we apply the basis where the T generator is diagonal and complex for the doublet and triplet representations. Defining ω = e 2πi/3 , we have The corresponding Clebsch-Gordan coefficients are all real and can be found e.g. in [19]. In addition to the flavour symmetry S 4 , we impose the canonical CP symmetry in our theory. As has been discussed in the literature, see e.g. [34,35], the consistent combination of a flavour and a CP symmetry requires certain conditions to be fulfilled; in particular that the subsequent application of a CP, a flavour and a further CP transformation leads to a transformation belonging to the flavour group. The possibility to combine the group S 4 with CP has been explored previously, see e.g. [24,34]. Here we are interested in combining S 4 symmetry, defined in the above basis, with the canonical CP transformation, i.e. the CP transformation that acts trivially in flavour space with X r = 1 for all representations r of S 4 . Note that this particular CP transformation X r fulfils the constraints of being a unitary and symmetric matrix. Moreover, it represents a consistent choice for a CP transformation (see e.g. [24]), which corresponds to the involutionary automorphism that maps the generators S, T and U in the following way since S and U are represented by real matrices in our chosen basis, while the generator T is given as a diagonal complex matrix in the two-and three-dimensional representations.
As with all automorphisms of S 4 , this is an inner one. In particular, one can check that the automorphism of Eq. (A.1) is "class-inverting" [36], i.e. it maps the group element g into the class which includes g −1 . This is true, since all automorphisms are inner ones and all classes of S 4 are ambivalent, i.e. the elements g and g −1 are in the same class.
With only real Clebsch-Gordan coefficients, a canonical CP symmetry imposed on the theory entails that all coefficients in the (super-)potential are real. Moreover, we observe that the residual symmetry in the neutrino sector at LO comprises the CP symmetry if all three neutrino flavons share the same phase factor. Following the comments of Footnote 5 of Appendix B, this is the case in our setup, cf. also Eqs. (B.1,B.2), so that the common phase can be factored out of the neutrino mass matrix, leading to an effective LO result which conserves CP. Furthermore, the canonical CP transformation X r = 1 commutes with the Klein group generated by S and U and thus at LO the residual symmetry is given by the direct product Z 2 × Z 2 × CP.

B Vacuum alignment
The vacuum alignment of the flavon fields is achieved by coupling them to a set of so-called driving fields and requiring the F -terms of the latter to vanish. These driving fields, whose transformation properties under the family symmetry are shown in Table 2, are SM gauge singlets and carry a charge of +2 under a continuous R-symmetry. The flavons and the GUT Higgs fields are uncharged under this U(1) R , whereas the supermultiplets containing the SM fermions (or right-handed neutrinos) have charge +1. As the superpotential must have a U(1) R charge of +2, the driving fields can only appear linearly and cannot have any direct interactions with the SM fermions or the right-handed neutrinos. (5) 1 1 1 1 1 1 1 1 1 1 [20], which serve to align the flavon VEVs.
The LO alignment of the flavon fields, see Eq. (2.2), has been thoroughly discussed in [19,20]. The particular setup also provides correlations amongst the VEVs. As described in Appendix D of [19] and in Section 4 of [20], 4 the vanishing of the F -terms of the driving fields X new Denoting the phase of each flavon VEV φ f ρ by θ f ρ , Eq. (B.1) correlates the LO phases as 6 leaving as free variables only the two phases θ d 2 , θ d 3 , which correspond to the LO VEVs of the two flat superpotential directions: Φ d 2,1 and Φ d 3,2 respectively. In order to find the higher order terms of the flavon VEVs, we start by writing each one of them as a series expansion in λ, up to and including order λ 12 . For example, the leading operators of the superpotential fix Φ u 2,1 /M to be zero up to λ 4 , while Φ u 2,2 /M 4 The introduction of the new flavon field η in [20] favours the exchange of the S 4 doublet driving field V 2 , which was introduced in [19], by the S 4 singlet field V 1 . Furthermore, the field V η , transforming in the same representation of S 4 as η, is introduced in order to relate the new flavon field to an explicit mass scale. 5 The proportionality constant between φ ν 3 ′ and φ ν 2 is a square root of an order one real number, which we assume to be positive, such that φ ν 3 ′ and φ ν 2 have the same phases. 6 Here and in Eq. (B.6), a possible phase shift by π has been ignored as real coefficients can generally be positive or negative. has to be φ u 2 λ 4 [19]. When considering the subleading operators, the VEVs of Φ u 2,1 and Φ u 2,2 receive corrections (shifts) which we parametrise as All flavon VEVs are parametrised in a similar manner. The aim is to find the order of λ at which each shift δ has to be non-zero. The computation consists of taking into account all possible operators and solving the F -term conditions resulting from the set of driving field order by order in λ, up to λ 12 . Each vanishing expression is solved for the lowest order shift involved. At the end, all shifts can be expressed in terms of the LO flavon VEVs. We find Note that the shifts presented in Eq. (B.4) are the first non-trivial ones. However, in our calculations of the mass matrices we take into account all shifts up to O(λ 8 ). It should be pointed out that the alignment of Φ ν 3 ′ is not perturbed up to order λ 8 , so that it preserves the S symmetry to that level. On the other hand, the alignment of Φ ν 2 is already perturbed at order λ 5 which, however, does not break the S generator as it is nothing but the identity for the doublet representation. Taking into account also CN effects, one can show that m eff ν has the form of Eq. (4.22) up to O(λ 7 ). Eq. (B.4) is in agreement with the discussion presented in Section 4 of [20], barring the absorptions of δ u 2,2 λ,δ u 2,2 λ,δ d 3,2 (4) λ, δ ν 3 ′ λ, δ ν 1 λ and δ η λ into the corresponding LO VEVs. Being interested in the CP transformation properties of the fields, such absorptions must not be made in the current work, as the phases of shifts and LO VEVs are generally different. In particular, we find the following relations between the shifts and the LO VEVs, .

(B.5)
Similar relations also hold for higher order shifts. Although such shifts have to be taken into account when performing a systematical λ-expansion, their explicit expressions are irrelevant for our phenomenological study. The phases of the LO shifts can be deduced straightforwardly from Eq. (B.5). Denoting the phase of δ f ρ,i by θ f ρ,i we obtain C Basis transformations

C.1 Canonical normalisation
In order to find the transformations which map the Kähler potential into its canonical form, we express the hermitian matrix K A as in Eq. (3.14), i.e. P † A P A = K A . Note that the matrix P A is not unique since P A → Q A P A with unitary Q A will satisfy Eq. (3.14) just as well. Moreover, K A can always be decomposed as where D A is the diagonalised form of K A . Therefore it is sufficient to find a hermitian matrix P A which satisfies Eq. (3.14), i.e. P † A P A = P A P A = K A . Expanding K A and P A in powers of λ, with k n , p n being matrices, allows one to calculate P A iteratively. With k 0 = 1, the result reads

C.2 SCKM transformations
The SCKM rotation matrices that diagonalise the Yukawas are found through the singular value decomposition. In particular, if We fix the phases of the matrices Ω f L by requiring that the CKM and PMNS mixing matrices are given in the standard phase convention, while the phases of Ω f R are fixed by demanding real and positive charged fermion masses. To LO, we find the following structure of the SCKM transformation matrices in terms of their λ-suppression.
With these SCKM transformations, it is straightforward to calculate the CKM mixing to leading order, The associated measure of CP violation is given by the Jarlskog invariant J q CP GUT and can be calculated from the imaginary part of V CKM GUT 21 GUT 31 . The explicit result can be found in Eq. (6.5).

D Mass insertion parameters at the GUT scale
In the following we present the explicit expression for the various LO mass insertion parameters at the GUT scale whose λ-suppressions have been stated in Eqs. (7.4-7.6). Using the definitions of Eqs. (7.2,7.3), we obtain These δ parameters are expressed in terms of the coefficients of the soft mass matrices in Eqs. (6.13-6.21), where we have defined , a e 23 = 9 The phases θ y u,c,s,b , θ z u,d i , θx 2 can be expressed in terms of the flavon phases θ d 2 , θ d 3 according to Eqs. (4.4,4.10). This has been done in Eq. (D.4), but we refrain from doing so in Eq. (D.5) in order to highlight the fact that allã f ij become real in the limit where the contributions of the auxiliary components of the flavon superfields to the A-terms are neglected such that the relation θ a f = θ y f holds.

E Renormalisation group equations in SCKM basis
The renormalisation group equations for the parameters of the superpotential as well as the soft breaking terms are usually given in the gauge flavour basis, see e.g. [6], with the transformation to the SCKM basis being defined only at the electroweak scale. As already discussed in Section 6, we find it useful to diagonalise the Yukawa matrices already at the high scale. In such a high scale SCKM basis, the RGEs will explicitly depend on the CKM mixing matrix. Here we define for convenience Introducing the parameter t = ln(µ/M x ), with µ being the renormalisation scale and M x the high energy scale, we have for the Yukawas and the trilinear A-parameters, The running of the soft scalar masses in the SCKM basis is given by For completeness, we also show the evolution of the µ parameter, i.e. the coupling of the bilinear superpotential term H u H d , where g i , M i , i = 1, 2, 3 are the gaugino couplings and masses respectively.

F Renormalisation group running
In this appendix, we provide analytical expression for the RG evolved Yukawa couplings, soft terms and mass insertion parameters. We estimate the effects of RG running using the leading logarithmic approximation. In order to formulate the two-stage running (i) from M GUT to M R , where the right-handed neutrinos are integrated out, and (ii) from M R to For M GUT ≈ 2 × 10 16 GeV, M R ≈ 10 14 GeV and M low ≈ 10 3 GeV, η ≈ 0.19 is of the order of our expansion parameter λ ≈ 0.22 and η N ≈ 0.03.

F.1 Low energy Yukawas
The SCKM transformations, discussed in Section 6, diagonalise the Yukawa matrices at high scales. RG running to low energies re-introduces off-diagonal elements in the low energy Yukawa matrices. These off-diagonal entries inỸ u low andỸ d low are proportional to the quark masses and the V CKM elements. As the CKM matrix features only a mild running, the RG corrections can be treated as a perturbation. InỸ e low , the off-diagonal terms are proportional to the charged lepton masses and the elements of Y ν . The corresponding RG equations are provided explicitly in Eq. (E.2) for convenience. To LO in λ, we find, where g U ≈ √ 0.52 is the universal gauge coupling constant at the GUT scale.

F.2 Low energy soft terms
Similar to the Yukawa matrices, the parameters of the soft terms have to be run down to low energies. Moreover, it is mandatory to perform further transformations to the "new" SCKM basis which renderỸ f low diagonal again. The running of the trilinear terms is similar to the one of the corresponding Yukawas. To LO in λ, η and η N , we derive the following expressions in the "new" SCKM basis. (F.14) The first terms in Eqs. (F. 8-F.10) are analogous to the first terms in Eqs. (F.2 -F.4); they are usually ignored. The second terms contain the universal gaugino mass M 1/2 contributions, which generate non-zero diagonal trilinear couplings through the running, even for A 0 → 0. The sources of the off-diagonal entries in the Yukawa couplings are also present for the trilinear soft terms. We see that the (13) element inÃ u low , which was zero inÃ u GUT , is now filled in, and there is an O(λ 6 ) contribution (but additionally suppressed by a factor of η) to the (23) element, which was of order λ 7 inÃ u GUT . The (32) element inÃ u low , withã u 23 given in Eq. (D.5), is of the same order in λ as the one that is already present inÃ u GUT . All the off-diagonal elements generated by the running inÃ d low and iñ A e low are of the same order in λ as the ones that were already present at the high scale.
Analogously to the trilinear A-terms, we find for the soft scalar mass, where we have introduced the ratio x = M 2 1/2 /m 2 0 and R q = (2b 02 + c Hu ) y 2 t + α 2 0 a 2 t , (F.21) with c Hu = m 2 Hu GUT /m 2 0 . Furthermore, the small quantities T f L,R are defined as The contributions T f L,R to the running soft masses are usually ignored, and it is common practice to set them to zero in a numerical scan. In our study, we will therefore not consider them any further.
The off-diagonal entries in the soft scalar masses which are induced by the running are of the same order in λ as the high scale ones, with an additional suppression by η. Only for the LL masses of the down-squarks and charged sleptons, the contributions due to R q and R ( ′ ) l can be relatively large as those factors take values up to ∼ 35 in a numerical scan. Generally, however, the main effect of the RG evolution on the scalar masses is the change of the diagonal elements. The masses of the first two generations of (m 2 u ) LL low , (m 2 u ) RR low , (m 2 d ) LL low and all three generations of (m 2 d ) RR low , (m 2 e ) RR low are increased at low energy scales due to the second terms in Eqs. (F. 15-F.20). The (33) elements of (m 2 u ) LL low , (m 2 u ) RR Low and (m 2 d ) LL low can still remain relatively light, as they also feel the effect of R q , defined in Eq. (F.21), entering with a negative sign. Similarly, the enhancement of all three diagonal entries of (m 2 e ) LL low is reduced due to the term −2η N R l which encodes seesaw effects.

F.3 Low energy mass insertion parameters
With these preparations, we can now formulate the mass insertion parameters at the low energy scale.
Up-type quark sector: , (1 + R y t − 2η y 2 t )e i(3θ d  (22) and (23) elements of the down-type Yukawa and soft trilinear structures, respectively. Originating from the second term of Eq. (4.7), z d 4 e iθ z d 4 = y d 2δ d 3,2 (4) φ d 2 , so that θ z d 4 = 6θ d 2 +4θ d 3 . We see that the term proportional to η λ 6 , which was generated inÃ u low 23 via th RG evolution, is the source of the associated term in (δ u LR ) 23 , which was of order λ 7 at the GUT scale. In Eqs. (F.31-F.43) we have defined the factors p u L 1G = b 01 + 6.5 x, p u L 3G = b 02 + 6.5 x − 2ηR q + which are related to the full sfermion mass matrices by