JHEP06(2012)018 M Springer May 9, 2012 June 4, 2012 : d,e : March 17, 2012 : Accepted Published Received 10.1007/JHEP06(2012)018 , doi: and Robert Ziegler b,d Published for SISSA by , the universality of soft terms, even if S M Stefan Pokorski [email protected] b,c [email protected] , , Zygmunt Lalak, 1203.1489 a , is radiatively broken. We estimate this effect in a broad class of models. S Rare Decays, Beyond Standard Model, Quark Masses and SM Parameters, The flavour messenger sectors and their impact on the soft SUSY breaking M [email protected] Lichtenbergstr. 2A, D-85748 Garching, Germany Physik Department, Technische Universit¨atM¨unchen, James-Franck-Straße, D-85748 Garching, Germany E-mail: [email protected] Institute of Theoretical Physics, FacultyHo˙za69, of 00-681, Physics, Warsaw, University Poland ofCERN Warsaw, Physics Department, TheoryCH-1211 Division, Geneva 23, Switzerland TUM-IAS, Technische Universit¨atM¨unchen, Max-Planck-Institut f¨urPhysik (Werner-Heisenberg-Institut), F¨ohringerRing 6, D-80805 M¨unchen,Germany b c e d a Open Access ArXiv ePrint: tree-level estimate based on the flavour symmetry. Keywords: Supersymmetric Standard Model Abstract: terms are investigated inis SUSY below flavour the models. SUSYassumed breaking at In mediation the scale In case the when CKM the basis flavour that scale effect gives flavour off-diagonal soft masses comparable to the Lorenzo Calibbi, The messenger sector ofradiative SUSY breaking flavour of models flavour and universality JHEP06(2012)018 18 20 21 22 10 7 ]. The flavour hierarchies are then 1 3 7 9 4 6 – 1 – (1) couplings O 20 21 18 14 19 21 22 10 11 17 0 9 1 16 U(1) F H × A.3.1 Fermion UVC A.3.2 Higgs UVC A.1.1 Fermion UVC A.1.2 Higgs UVC A.2.1 Fermion UVC A.2.2 Higgs UVC A.3 SU(3) A.2 U(1) 4.3 Numerical discussion A.1 U(1) 3.3 Unification 4.1 Tree-level effects 4.2 Radiative effects 2.1 UV completion2.2 with heavy fermions UV completion with heavy scalars 3.1 Perturbativity of3.2 the gauge couplings Perturbativity of the in terms of flavourwhich symmetries. is spontaneously The broken matter by fields thewe vacuum transform expectation will under values call (vevs) the of flavons flavourmalisable scalar in symmetry, fields level the that and following. only ariseof Most from the higher-dimensional Yukawa flavons couplings operators as are involving determined suitable forbidden by powers the at flavour the symmetry renor- [ 1 Introduction A common approach to explain the observed hierarchies in fermion masses and mixings is B Explicit examples of RG induced flavour violation 5 Conclusions A Example models 4 Constraints from SUSY-induced flavour violation 3 Constraints from perturbativity and unification Contents 1 Introduction 2 The structure of the messenger sector JHEP06(2012)018 ] S 6 , 5 M is larger ], fermion ]. 8 M 3 , 2 ], and radiative 9 . We quantitatively estimate S M ], that is, the vanishing of certain entries 4 1 – 2 – , with the flavour messengers included. The latter are M ] from their initial universal values is possible only if 10 but, when the flavour messenger scale is lower than the SUSY breaking medi- 2 . For Gravity Mediation, the flavour physics is then pushed to the Planck scale. S ] where also heavy scalars have been employed. 7 M One can distinguish tree-level effects on the soft mass terms, generated by integrating Independently of the kind of messengers, in general one needs many of them. Since In the UV completion that contains the messenger sector, small fermion masses can More recently, some phenomenological consequences of the messenger sector have been discussedIn in models [ with a low fundamental cutoff one can take these fields down to low scale where they can 1 2 than and in [ give rise to large flavour-changing effects which make such scenarios testable at experiments [ is assumed (or naturally present) asthese the initial effects condition and at showterms that in they soft always masses giveat in comparable tree-level (in the without their super-CKM assuming order universality. basisto Thus of flavour the the off-diagonal magnitude) MFV evolution hypothesis to of [ those the one soft terms would according obtain out the flavour messengers and determinedeffects by the summarised flavour by symmetry the alone [ and RG evolution flavour between messenger theparticularly SUSY scale interesting breaking as mediation they necessarily scale break flavour universality of soft terms, even if it flavour messengers must be verysuppressed, heavy. Direct effectsation of scale, their they exchange are are relevantThis then for is strongly the because flavour in dependence theterms of supersymmetric are the theories sensitive soft we to SUSY are any breaking considering dynamics terms. the that soft couple SUSY to breaking light fields. these fields carry SM quantumcouplings. numbers, they The contribute requirement to of thebound maintaining RG perturbativity on evolution up of the to SM messenger certainperturbative gauge scale masses physics puts as up a a to lower Planck function a scale of very for their high Gravity number. scale Mediation of In or supersymmetry this the breaking work Gauge we mediation, Mediation require the scale. In consequence the light masses arise fromattention small despite vevs its of structure theto is heavy generate scalars. much texture simpler. zeros This inalthough latter Such the allowed case scalar Yukawa matrix by has messengers [ the received are flavour less symmetry. very suitable and show that it cansoft indeed masses have in important the consequences MSSM. both for Yukawa couplings and be thought to ariseheavy from scalars. small In mixing themasses of first correspond case, light to which fields three has either been light with eigenvalues studied heavy of extensively fermions in a or refs. large [ with mass matrix. In the second case, scale. This scale itself remainsscale. undetermined and In can case in it principlenew is be degrees as smaller, of large one freedom, as canmay the the interpret have Planck so-called this important “flavour cutoff impact messengers”.high. scale on as In The low-energy this the dynamics physics work typical of even we mass want if this scale to sector its systematically of analyse characteristic the scale structure is of very the messenger sector explained by small order parameters given by the ratio of flavon vev and the UV cutoff JHEP06(2012)018 I ).  2 (2.1) c j u , spontaneously 1 U,D I,ij F U n G  i ) with the quantum . They couple to the I . The transformation 1 F S φ M U h G + /M  i I (1), so that Yukawa hierar- I φ Y O H,S ∼ are properly chosen, so that ≡ h φ φ φ I + F  U U,D ij n G H y U U n u . The MSSM Yukawa couplings arise from I h U φ Q n -even gauge singlets, respectively (see figure – 3 – Q P d ) with the quantum numbers of the MSSM matter R is the typical scale of the flavour sector dynamics. h D c j i Q d I n reproduce the observed hierarchies of fermion masses i + φ q φ Q h D ij y U,D I,ij & n U,D + . Schematic supergraph for Fermion UVC. M u + h c j φ u i invariant operators involving the flavons q 1 Q, U U ij F Figure 1 Q y + G Q = ). 1 1 yuk φ Q W i q . These messenger fields are vectorlike and charged under M In the second case one introduces chiralWe are fields now ( going to discuss in more detail these two possibilities, to which we refer In order to UV-complete these models one has to “integrate in” heavy fields at the The structure of our paper is as follows: In section 2 we first discuss the general fields (see figure numbers of the MSSM Higgs fields and as “Fermion UV completion” (FUVC) and “Higgs UV completion” (HUVC). Although we and mixings. scale flavons and the light matter.ter In or the Higgs fundamental fields theory (afterchiral flavour they superfields symmetry mix ( breaking). either with In the MSSM first mat- case one has to introduce where the suppression scale The coefficients of the effectivechies operators arise are exclusively assumed from to theproperties be small of order the parameters MSSMtogether fields with and their the exponents flavons under higher-dimensional in the literature in order to illustrate our general2 discussion. The structure ofWe the want messenger to sector considerbroken models by the with vevs a of general the flavon flavour superfields symmetry group number of messengers thatscale we from use perturbativity. in section Inthe 3 section flavour to structure 4 calculate of we theangles soft discuss bounds valid terms. the on in impact In the large ofappendix particular messenger classes the we we provide messenger of present explicit sector constraints flavour examples on on models. of light UV rotation We completions finally of three conclude popular in flavour section models 5. In the structure of the messengerwith heavy sector, fermions in and heavy particular scalars, the respectively. This two allows possibilities us to of estimate UV the typical completions JHEP06(2012)018 (2.2) (2.3) φ φ , in which 1 1 , S . For instance, one ) α 1 ) with the number of U , i
1 q α 2.2 β S D U + . φ α c i c j + u u U α α 2 − + D Q 1 S β U + Q − β α U Q ... α − I Q ( U φ S α n n u u U + h U S h α − + + H – 4 – U Q n ). α
α ). The diagram dictates the required messenger n c i U ) messengers one wants to use. In particular it is u H U M Q α 2.1 U − ∼ U M U, D U ... + + quantum numbers. In the following, we are going to refer i M − q α 1 . Schematic supergraph for Higgs UVC. F α in eq. ( ∼ Q α Q G Q α U ij Q Q − i Q n is induced by the interactions in eq. ( I q M + (1), since all hierarchies should arise from the symmetry breaking φ Q φ U ij α O M y + 1 Figure 2 Q ⊃ H ) are generated upon integrating out the messengers. This choice can . They have to be chosen appropriately, such that the effective Yukawa W 2.1 F 1 G ij H λ ) and “right-handed” ( Q c j i u q All entries of the light Yukawas can then be generated using the chain diagrams. To be The allowed couplings follow directly from the transformation properties of the mes- can choose the positionhanded” of ( the Higgs insertion,possible which to corresponds use to only the left-handed number or of right-handed messengers. “left- economic one can use the same messengers for different chains, but one has to pay attention Notice that for generatingcan a write Yukawa several entry chain withthe diagrams a possible using given permutations different of number messengers. the of Higgs flavon This and insertions flavon ambiguity one insertions arises in from figure be conveniently carried out bya drawing tree-level given Feynman diagrams, Yukawa entry see figure flavon insertions given by couplings and therefore their to these diagrams as “chain” diagrams, for which we introduce the shorthand notation: that are assumed to be alone (which also implies sengers under couplings in eq. ( with superpotential interactions of the schematic form where we restricted to the up-sector for simplicity. Moreover we have dropped all couplings In order to UV-complete theintroduce effective certain theory numbers above of with vector-like superfields fermionic messengers, with one light fermion has quantum to numbers restrict to the pure cases, aa UV case completion is involving a both kind straightforward of generalisation fields of is the2.1 also following viable. discussion. Such UV completion with heavy fermions JHEP06(2012)018 : M (2.6) (2.4) (2.5) . By ) is to I  2.2 . Usually , one needs ) reduces to min I φ N 2.2 n, s, ), contribute at s = 6= , i.e. a new messenger Q r, n + 6= Q . m, r j ), the determinant of the full m 3 ( u,d = M v ( i I U rs n I y O φ , I for . Y U mn I y n ∝ I X u,d light = m – 5 – c j one clearly obtains a rank 1 matrix (see the dia- min det Qu Q u N + ∝ h j Q β u,d full + M i Qq det φ i α + are some effective couplings involving appropriate powers of ]. In the full theory the role of the messenger couplings in eq. ( 2 n,s QQ β Q ). One can avoid this by introducing a copy of M 3 messengers with and ⊃ min eff m,r N W α in the determinant of the Yukawa matrix. Finally we want to elucidate the necessity of using different messengers for different The above method guarantees that the resulting Yukawa matrices are full-rank, i.e. all Although this argument gives the minimal number of messengers, it is unclear which I  where integrating out the last messenger gram of figure we checked that this is the case for theentries. three example Let models us ofleading assume appendix order A. that to two the Yukawathen determinant entries integrate and out arise all fromthe messengers chains following but that form: this share one. a The messenger. superpotential One of can eq. ( fermionic messengers. In appendix Afor we three explicitly examples construct in the order Fermion messenger to sector illustrate the generalfermion procedure. masses are generated.besides the three In chosen, and general therefore the it correct does mixings, are not generated as ensure well. that However, other Yukawa entries entries using different messengers fornumbers. different entries, In even this if way one they addsthere have a are the total several same number solutions quantum of obtained messengersdiagram. from that permuting is By the precisely counting Higgs the and totalone flavon number insertions can of in easily each these find permutations the for total each of number the of three possible entries UV completions with minimal number of messengers have to becedure included. to derive We minimal thereforethe sets outline three of a entries fermionic simple, of messengers model-independent,to the for pro- the effective each determinant. Yukawa sector. matrix If that Onechoose the gives any first of leading the identifies the order leading summands. determinant order Then is contribution one the constructs sum the of chain diagrams several for terms the one three can chosen That is, the minimum numberof of messengers can be found simply by counting the powers mass matrix must be equal to the determinant of the light mass matrix, up to powers of Because every entry ofat the least full mass matrix involves at most one power of sible correlations of different entries. An(meaning elegant minimal and simple number way of to SM findpointed the representations) out minimal of in number messengers ref. needed [ ingenerate each three sector light eigenvalues was of the fullvectorlike mass messengers. matrix that Since involves the the chiral heavy fields eigenvalues and are the that the resulting Yukawa matrix has full rank, which might not be the case because of pos- JHEP06(2012)018 . M mes- (2.9) (2.8) (2.7) Higgs ∼ 3 H S M messengers ∼ H H n, s,
M = J ) has rank 2. φ c s , α , u S c n = 0 u + ). α α S S 2.6 c j m, r j β u S (1) and take = . Note that for generating all ) and ( i r + O 2 ∂W/∂ α α , q j S β = m u H q for β h is a gauge singlet. The superpotential + α α . This gives rise to a new contribution α H S Q H S c j I u + c j i φ u q and 0 Q ∂W/∂S + u Q + – 6 – α u h = u h S α h 0 α j α I QQ β H S H φ S α + + . In particular, it is possible to use only φ i α H M 2 q i 0 . For generating a Yukawa entry in general one can write H α + + ∂W/∂ Q . Diagram corresponding to eq. ( u α i, j φ = h 0 i i H β q α α α as a background value. Solving the F-term equations S + α H i 0 u H H Figure 3 h Q for each 0 h M + c j Q ∂W/∂H 0 u ⊃ i Q q M W ⊃ eff has the SM quantum number of W with the same quantum numbers as 0 ∆ H Q can be inferred from chain diagrams like in figure In the fundamental theory, small fermion masses arise from small vevs of the In non-abelian models these fields can be part of the same multiplet. + 3 0 F sengers. These vevs canthe be MSSM calculated Higgs by vev setting the messenger F-terms to zero and using fields coupling to several diagram with differentHiggs messengers, and corresponding flavon insertion to in theand figure possible avoid gauge permutations singlets. of Explicit examples with HUVC can be found in appendix A. Again we dropped allThe couplings required couplings that and are therefore the assumedG transformation to properties of be the messengers under Yukawa entries, in general corresponding to different charges, one needs different where interactions are of the schematic form 2.2 UV completionWe with now heavy discuss scalars the UVone completion has to of introduce flavour certain models numbers with of heavy vector-like Higgs superfields fields. In general Q to the effective superpotential which ensures that the effective Yukawa matrix for ( JHEP06(2012)018 2 2 α (3.1) remains in the i α of the gauge couplings i b ], where integrating out the 11 , i N vectorlike SU(2) doublet messengers 1). Requiring that 2 + , N 3 0 i b − = i b -function coefficients ) = ( – 7 – β 0 2 + ∆ , b (1) and one has to ensure that they do not blow 0 3 0 i b b O . In the case of HUVC, instead the bound is set by = Z i b M the 1-loop ) up to the cutoff scale Λ provides a lower bound on the π 4 M triplet messengers and c . i α . Above M ]. In appendix A we illustrate this for the case of a U(1) flavour model. 4 are the MSSM coefficients ( vector-like SU(3) 0 i 3 b which is sizable already at N This feature of HUVCs allows to enforce texture zeros in the Yukawa matrices in a very is missing, the entry vanishes in the fundamental theory and remains zero in the low- 3 get modified according to α F i α where perturbative regime ( 3.1 Perturbativity ofWe the want gauge to couplings calculatewith the constraints on theliving at messenger a scale scale for a given UV completion the bounds from perturbativitycontains of many gauge couplings couplings. whichup Similarly, are in the the messenger running Lagrangian messenger between the sector finally messenger arise scale when andcouplings one the in requires cutoff. the that MSSM Further thecomplete constraints is approximate SU(5) on unification not multiplets. the of spoiled. gauge We are This now is going easily to ensured discuss if these the issues messenger in fields more from detail. coloured (as in the case ofof FUVC), the strongest constraint typically arisesand from the is running usually weaker thanreplaced in by heavy the singlets previous with no case.can impact Moreover expect on some running that of heavy models the Higgs with gauge fields couplings. HUVC can will Therefore be be one less constrained than FUVC models regarding As we have seen, the strongly hierarchicalof pattern messengers of with fermion SM masses quantum requires numbers. aof These large the fields number SM have a gauge strong impactcannot couplings, be on so too the that running far below for the a cutoff scale given in messenger order to sector avoid Landau the poles. messenger If mass the scale messengers are G energy effective theory. This elegantoutlined possibility in to [ produce texture zeros has been already 3 Constraints from perturbativity and unification would have to diagonalise huge mass matrices. simple way. Although aone certain can Yukawa entry set would it to be zeroit allowed because is by of the clear particular flavour dynamics that inheavy symmetry, a the Higgs messenger specific is sector. Yukawa present. entry From figure can If only such a arise Higgs if with the the corresponding correct coupling transformation to properties a under This is analogous toheavy the triplet supersymmetric or type-II computingcompute see-saw the the [ vev effective of neutrinomasses its in masses. the scalar fundamental Notice component theoryis that is are formally much the two the easier equivalent interpretation for same ways the of as to Higgs small in UVC, fermion the since the effective calculation theory, while in the fermion messenger case one is then equivalent to integrating out the messengers by their SUSY equations of motion. JHEP06(2012)018 2 b (3.2) (right) shows 2 4 b (left), ∆ . 3 b 1 )). (left), ∆ , 3.2 3 b  for given ∆ Z Λ M min for ∆ M log min 0 i π b 2 M − ) Z 1 M ( i α  i b π – 8 – 2 ∆ −  ). Taking into account both up- and down sector one has = Λ exp 2.5 min M & given by , provided that only right-handed messengers are used in both sectors. M M d min N + . Contour plot of the lower bound on the messenger scale u min N It is interesting to notice that, despite the large number of additional fields charged The UV completions for the example models in the appendix are chosen to fulfil this In the case of a FUVC, one can get, for a given effective model, the minimal number = 3 criterion of minimality, and canmessenger therefore scale be for used typical to flavour illustrate models. the We minimal summarise bounds the onunder results the the in SM table gauge group,with a a theory messenger perturbative sector up living to far the below Planck the scale GUT is scale. still achievable the perturbativity bound assector a as criterion the for solutionleast constrained defining with by the the perturbativity. “minimal” leastminimal This messenger fermionic number notion sectors messenger of can since reduce color only drastically few triplets, solutions the i.e. efficiently degeneracy unify the of up messenger and sector down sectors. of triplets directly fromN eq. ( If also left-handed messengersboth are isospin used, components in general of the the number quark of doublet triplets serve is as larger, messengers. unless In turn, one can use where we considered 1-loopthe running contours and of neglected the theand lower SUSY the bound cutoff thresholds. Λ. on Figure Comparing theif the messenger the two messengers scale panels, are one can coloured. see that the bound is indeed stronger messenger scale Figure 4 (right) requiring perturbativity up to the cutoff scale Λ (obtained using eq. ( JHEP06(2012)018 min (3.4) (3.3) M , that would 2 0 = 10. In this -function that λλ β /M 0) gives an upper . > Planck (Λ) 7 12 12 min M /λ (1), but they cannot be too M . : 2 / M 1 ∼ O HUVC − 2 b  4 6 10 . 1111 10 10 ∆ Λ M . 3 = 8 = 5 = 4 8). log C C C 14 12 , 9 Cλ min 2 5 π M C = , 8 ⇒ ⇒ ⇒ , which are negligible in the large coupling regime we λ λ  2 i φ φ φ – 9 – d g dt FUVC = (4 β β J d = 3 2 b ) 8 10 Q U H C 19 10 12 10 π α α I u ∆ couplings max (4 λ λ U 0 λ Q λ H (1) . 1) for . ) O 2 U(1) M , in GeV requiring perturbativity up to ( one finds: 6 × . λ U(1) 2 2 M SU(3) Model , 9 is given by group theoretical factors that depend on the quantum . U(1) and C 1 1. = (2 & λ max λ . Comparison of HUVC and FUVC for the three example models in appendix A. In general there will be other terms proportional to different Yukawas, of the kind 4 . Let us consider for simplicity only the unavoidable contribution to the general gives rise to additional constraintsunification on is the exactly UV preserved completion. at 1-loop As if is the well additional known, fields MSSM formonly complete make multiplets the of boundare stronger, interested and in, gauge terms, case we find 3.3 Unification One might want to requirenot that spoiled by the the apparent presence unification of of the the messenger MSSM sector living gauge at couplings an is intermediate scale. This in bound on the value of the coupling at the messenger scale One can get very conservative bounds considering a short running with Λ Requiring the absence of a Landau pole below the cutoff (i.e. 1 Here the coefficient numbers of the fields appearingthe in chains the of coupling. figures For some typical couplings that appear in large if the theory shouldthe remain corresponding perturbative bounds considering up the toλ typical the RGE cutoff of Λ. a generic One superpotentialis can coupling proportional easily to estimate the third power of the coupling itself: Table 1 denotes the lower bound on 3.2 Perturbativity of the The couplings in the messenger sector are by construction JHEP06(2012)018 , M GeV. This 10 representations. = 10 ¯ 5 + M 5 is much below the SUSY M and therefore are negligible. All potentially gives rise to additional 6 φ/M become relevant. Notice that the in- M 3 Indeed the embedding of the messengers α 5 instead of – 10 – S φ/M ) via a chain of heavy Higgs triplets has to pass through ]. ]). This is because at the SUSY breaking scale flavour-violating uude 13 5 of SU(5). This leads to a further reduction of the degeneracy of and 5 + , all flavour-violating effects in the soft terms arise dominantly from the qqq` S 5 M . However, in the MSSM even such high scales can have an important impact and M 10 + ]. For HUVC the Higgs messengers can be simply embedded in For FUVC one has now to write the chain diagrams with messengers transforming 12 For sets of fields that do not formFor complete a SU(5) recent multiplets discussion but see still [ maintain MSSM unification, see 10 5 6 messenger sector (see e.g. [ effects are suppressed by powers of ref. [ 4.1 Tree-level effects A common approach in theMediation literature and is apply to a assumematrices spurion high-scale below analysis SUSY the to breaking messenger likebreaking determine scale. Gravity scale the structure If of the the messenger sfermion scale mass determined by the underlying mechanismvery of high scales SUSY as breaking, well. andbreaking This are means sector. usually that generated the In at messengerby particular sector construction, can the interfere and messenger with therefore the sectorwith SUSY can strongly drastic easily consequences violates induce for flavour flavour low-energy universality observables. violation in the sfermion masses In the previous section weto have the shown Planck that scale aimplies flavour requires that theory very all that heavy direct remains messengers, effectspowers perturbative in at of up low the energy order canon of be TeV neglected, scale since physics they due are to suppressed by the presence of light SUSY particles. Their soft masses are as in ordinary SUSYHiggs GUTs. coloured This triplets means live that at the the new GUT triplets scale. can4 be light as long Constraints as from the SUSY-induced flavour violation for the SU(5) compatiblein models these in models the any appendix tree-leveldecay this diagram operators is which ( generates not the thethe dangerous case. ordinary dimension coloured The 5 triplets reason proton decay associated is is that with suppressed the by light Higgs the doublets. mass of Therefore proton these triplets together with small Yukawas precisely for the SU(5) compatible examplesstrong in requirement. the appendix are the unique solutions satisfying this In this case alsotroduction the of perturbativity new bounds Higgscontributions color on to triplets proton decay. at While the in scale general this might lead to new constraints on in SU(5) multiplets is straightforwardSU(5), provided i.e. that the the effective flavour theory symmetry can commutes with be written inas a SU(5) invariant way. messenger sectors that minimise perturbativity constraints. Indeed the UV completions SU(5), independent of the scale where they live. JHEP06(2012)018 ). As (1) . (4.1) 8 φ/M O , the RG , M ) S ] and SU(5) mod- 20 φ/M ( = 0), and therefore O iα + , c  α b ... = β i H a † β H β d + α denotes the SUSY breaking spurion and Q ] the 1-2 sector universality breaking can be † i X q 19 – iα c – 11 – is below the flavour messenger scale 15 + since the interactions of sfermions and messengers S α 7 M Q α † Q α b + i q † i q . While in abelian flavour models this breaking is large, being i 5 F a  2 S X † M . This can be easily realised if the underlying dynamics is flavour-blind, X S ⊃ M (1) couplings in the Lagrangian, in non-abelian models it is due to small flavon K O ]. ]. 22 denote Fermion and Higgs messengers, respectively. , 14 In this section we estimate these constraints for the case of abelian and simple non- The RG effects from the messenger sector destroy universality of sfermion masses in In general integrating out the messenger fields generates the tree-level flavour structure This effect has been discussed for aThis is model perfectly with analogous accidental to flavour the symmetries RG and induced Fermion flavour violation messen- in SUSY seesaw [ 21 7 8 abelian flavour symmetries. The starting point is a universal sfermiongers mass [ matrix at the els [ two ways, by directly generatingon an the off-diagonal diagonal entry and thatwe by will is splitting see, then degenerate the masses convertedclasses second to of contribution flavour an models. is off-diagonal For alwaysfermion such entry larger rotations models in one or depending can on equal the therefore the than obtain mass size constraints the basis. of on first the the one light diagonal for splitting. large strongly violate SU(3) through vevs. Still, in simple non-abeliansizable models (of [ the order of the Cabibbo angle squared) and in 1-3 and 2-3 sector even of for instance in Gaugerunning Mediation. down to low If energies approximately preservesby universality, because the it Yukawa interactions is that broken only aregeneration. small If for instead the the relevant SUSY transitionsspoiled breaking between by scale 1st messenger is and loop above 2nd the corrections, messenger scale, universality is radiative effects as we are going to discuss in4.2 the following. Radiative effects Let us assume aat mechanism the of scale SUSY breaking that generates universal sfermion masses off-diagonal sfermion mass termsmessengers can do mix be with light generated fields inmasses. and this the This mixing flavour typically is generates basis. however off-diagonalversal sfermion not Instead among the Fermion fields case in withstay Gauge same universal Mediation gauge where in quantum sfermion every numbers masses ( basis. are uni- Still, even in such cases flavour universality is broken by Q, H of the soft massesHowever as this expected effective structure bythe also the SUSY depends flavour symmetry on breaking (controlled the mechanism. by details powers of For of the example messenger it sector is and clear that for Higgs messengers no and read schematically where we canonically normalised the fields, soft mass terms for chiral fields and messengers are given in the flavour symmetric limit JHEP06(2012)018 (4.6) (4.2) (4.3) (4.4) (4.5) , 2 q − 1 q  = 2 2 d d + + 1 2 q q .   , , ! ) ∝ , that is the loop corrections 12 2 22 2 0 θ M D D ˜ 22 12 m
m ( 2 12 y y 2 a 2 11 . where the messengers decouple, all ≈ m ∼ M m ! + ∆ ˜ 2 a ∆ ˜ 2 0 2 0 M ∆ ˜ 12 0 C m m ˜ m − ij 2 0 δ 0 ˜ 2 11 2 22 , θ ˜ m (1) coefficients. In appendix B we illustrate 2 m m

˜ 2 21 ) + m O or SU(3) it is easy to see the second term is ∆ ˜ depends on the RG running that can be esti- m M ∝ 2 – 12 – + ∆ ˜ ( ) = (1) the RG coefficients are in general large. As a ). We neglect the possibility that the two terms ∆ ˜ 2 11 + S 2 0 2 ij 2 11 O m ˜ ˜ 4.5 m 2 12 M m m U(1) ( ∆ ˜ ,

m ∆ ˜ 2 ij ≈ − (1) couplings to the messengers. Instead in non-abelian ∆ ˜ ) ˜ − m which are determined by the RG equations ) = m O 2 22 ≈ 2 22 ( ˜ 2 ij M m ij 2 2 12 m ( m ˜ ˜ m 2 ij ∆ ˜ m ˜ m , 2 ˜ m 2 q − 1 where for simplicity we restrict to 1st and 2nd generation q  S ∝ M denote the U(1) charge of the corresponding superfields. A non-abelian exam- are gaugino soft masses. The 1-2 entry in the super-CKM basis is then approx- i denotes the (complex) rotation angle in the sfermion sector under consideration. 2 12 ). In abelian models there is no extra suppression, because 1st and 2nd generation a , d m 12 i B M q θ ∆ ˜ The diagonal splitting ∆ ˜ pendix sfermions couple with different models 1st and 2ndthe family flavour sfermions group, can which befor implies embedded small that in symmetry their the breaking couplings same effects to representation which messengers under lead are to universal, additional except suppression. Such breaking this issue in an explicit example. mated in the leading-logmessengers. approximation. Since these It couplings dependsconservative estimate are on we take all for the interactions totalthe of RG contribution loop sfermions a factor factor and 10 and into account the (besides logarithm), a typical value one finds in concrete models (see ap- where ple will be discussedare below. of the As same weattention order shall on as see the the later second tree-levelcancel term ˜ since effects. in in general eq. Therefore they ( in involve different the following we restrict our always larger or equal thanexample the in first a one, U(1) provided model the one rotation has angle for does left-handed not down vanish. squarks For imately given by where For simple flavour models like U(1) flavour universal) terms and byneglected: 1st and 2nd family MSSM Yukawa couplings that can be where The final evolution to the soft SUSY breaking scale scale is determined by gauge (thus When this mass matrix isentries evolved receive down corrections to ∆ the ˜ scale SUSY scale JHEP06(2012)018 in 2 jj (4.7) (4.8) ˜ m − F 2 ii ˜ m . ≡ 2 12 ij m , sector is therefore ∆ ˜ β D D 12 22 j & - 4 y y (1) (1) (1) (1) . i  tan S are proportional to two 12 O O O O 5 ∼ θ  M M ) 2 11 12 2 11 . The situation is similar in m . Let us finally illustrate that m jj ∆ ˜ 2 y , θ ∆ ˜ 10 log 2 − √ 2 ˜ − 2 0 m 2 22 π Suppression factor in U(2) ˜ in the 1-2 sector. To estimate the m ) gives a good estimate for the total 2 22 D 22 m 16 y 2 m  12 4.5 F ∼ θ 2 11 ≈ and ∆ ˜ m ], where transitions in 1-3 and 2-3 sector are 12 2 12 θ ∆ ˜ m
18 , – 13 – − 2 11 β β β 17 2 22 m , as displayed in table 4 (1) (1) m ∆ ˜  jj tan tan tan y ∆ ˜ , but possibly charged under additional U(1) factors O O 3 3 5 − . The flavons get hierarchical vevs that induce the quark    F 3 2 22 , G 2 m ˜ m ∆ ˜ D 12 y ≈ is given by flavour models [ . Suppression factor in SU(3) 2 12 2 jj F ˜ ˜ 2 12 m m and flavons as m − 3 2 ii ∆ ˜ ˜ m ]). We can then estimate the off-diagonal sfermion mass at leading log by: U 12 D 13 U 13 U 23 D 23 D 12 ), we then see that also in this case (∆ ˜ ≡ ∆ ∆ ∆ ∆ ∆ ∆ 24 ij , 4.7 . Note that such a suppression can occur only for off-diagonal sfermion masses. D 12 . Additional suppression factors of diagonal sfermion mass splittings ∆ 23 y , Mass splitting 7 ∝ In summary, for the estimation of the mass splitting we consider the case of abelian Note that this estimatesfermion is mass roughly matrix only of constrained thenot by same the vanish, order flavour it symmetry. as gives Ifbasis. one the the would It rotation leading only expect angle contribution depends does for to on a the the tree-level rotation off-diagonal angles, entry whereas in the off-diagonal the entry super-CKM in the flavour flavour symmetries, and keep insuppression. mind that Still in non-abelian thetransforming models abelian as one case can singlets can have(e.g. under additional be [ relevant in non-abelian models for sfermions where the inequality in the firsttypically expression lead accounts to for the a fact furthersize that suppression additional of symmetries soft massesFrom eq. with ( respect to the naively expected the non-abelian symmetry, the generatedflavon ∆ ˜ insertions, like the Yukawa entries. For left-handed down squarks, we then have: the case of some U(2) always unsuppressed. We find thatsplitting the ∆ additional suppression factor foralso the in sfermion the mass non-abelian casecontribution. the Since second we term assume in eq. left-handed ( and right-handed superfields to form triplets of is model dependent andadditional suppression can factor be we consider eventransforming the as of case a of the a order generic SU(3) model with all quarks Table 2 simple non-abelian models. masses. The flavon vevroughly responsible given for by the universality square breaking root in of the the Yukawa coupling JHEP06(2012)018 . L, R (4.9) (4.10) = X and (1) factors, the ij ) O f RR δ ( ij ) can be estimated to be: f LL 2 δ ( ˜ m [Im] [Im] [Im] [Im] [Im] [Im] [Im] [Im] [Im] ] ] ] ] / q 4 4 2 3 2 1 2 1 2 4 3 4 6 , is then given by: ] and can be affected by the ij the low-energy squark/slepton − − − − − − − − − − − − − ) 9 S f 2 22 (1) in the case of sleptons, while i ≡ M 10 10 10 10 10 10 10 10 10 , 10 10 10 10 M m ˜ f ij m m O δ S × × × × × × × × × f × × × × h we collected the quark sector con- 2 11 M log 6 4 0 0 6 0 8 5 2 A 7 6 8 4 M ...... ˜ ( 3 m ˜ v [5 [4 [3 [3 m ≡ p log / f ij 1) in the case of large high-energy gluino ij 3 2 3 5 . R ) 2 12 Y − − − − ˜ ij (0 [Re] 2 [Re] 7 [Re] 6 [Re] 4 [Re] 1 [Re] 6 [Re] 1 [Re] 1 [Re] 1 m f LR is typically 10 a 10 10 10 10 -function coefficients differ by 3 1 3 2 1 2 2 2 3 δ O 2 2 ≡ β – 14 – − − − − − − − − − ], for the following reference values of squark and × × × × R π π 12 1 θ 8 3 8 7 12 10 10 10 10 10 10 10 10 10 27 . . . . 16 16 ). δ – 2 2 1 1 × × × × × × × × × ≈ ≈ 25 4.4 6 0 2 6 8 2 6 9 2 ...... is the EWSB vev and ˜ ij 5 1 6 1 2 4 6 1 9 ab ) 12 v δ f LR (1) down to δ 12 13 12 12 12 12 13 12 12 i i i i ( ) ) ) ) ) ) ) ) ) O U D E D 12 13 12 12 δ δ δ δ E D U D E LL E LR LR LR LR RR U D D XX XX XX h h h h δ δ δ δ δ δ δ δ δ ( ( ( ( ( ( ( ( ( parameterizes the possible suppression due to the gaugino-driven run- R (1) coefficients, . 0 O ˜ m are  3 ij . Bounds on flavour-violating mass-insertions. Here a M Similarly, radiative effects given by messenger loops will induce flavour-violating entries The corresponding mass insertion 4.3 Numerical discussion The expressions aboveobtained can from be FCNC compared andstraints to LFV from the the processes. existing literature various [ In bounds table on the mass insertions radiatively generated LR mass-insertions, ( where mass. for squarks it rangesmass, from in the A-term matrices inaligned the to super-CKM the basis, correspondingand even Yukawas. the if Yukawas are Given they the vanish that same, at while the their high flavour energy structures or of are the A-terms where the factor ning of the diagonal entries, cf. eq. ( Values in [ ] denote expected future bounds. Seebasis the text depends for details. directlymessenger on sector. the specific flavour symmetry [ Table 3 JHEP06(2012)018 ) ) E 12 2. . θ eγ 4.9 0 L, R GeV, ≈ [Im] [Im] [Im] [Im] [Im] → , which ] ] ] 10 ) 2 3 1 2 1 8 3 2 3  µ − − − − − ( − − − 10 1000] GeV, 10 10 10 10 10 O , 10 10 10 ≈ = 10 × × × × × × × × 2 8 1 7 2 9 2 6 M , which includes ...... /M = 10 [3 [3 [2 = [100 S - 2 M > M /M 2 1 2 M S S − − − [Re] 8 [Re] 1 [Re] 4 [Re] 2 [Re] 1 M M 10 10 10 GeV and 1 2 1 2 1 . For the leptonic angles − − − − − × × × 17 DR 9 6 2 12 10 10 10 10 10 . . . θ 10 = 10 and 1 1 1 × × × × × = 1 TeV). The angles without 500] GeV, DL 12 ≈ ]). Doing like that, we neglected , 3 3 9 3 9 θ ˜ q . . . . . /M ] to update the existing bounds on 6 1 6 4 2 29 p S m M 28 M = [50 i ≡ 1 D 12 [Im] [Im] [Im] [Im] [Im] [Im] θ M ] ] ] h 2 4 2 3 1 2 4 3 4 − − − − − − − − − 10 10 10 10 10 10 10 10 10 – 15 – 8 × × × × × × ) depends only on the rotation angle and the ratio × × × 0 2 1 4 1 5 9 9 3 ...... 4.9 = 1. For the leptonic sector we used the new exclu- = 10 (corresponding to [4 [3 [3 1000] GeV, ˜ q 15]; then, we have taken as bound for a given mass- , 3 , , for the case of an abelian flavour symmetry. Bounds for 3 2 3 /m /M ˜ 4 − − − g S must account for the Cabibbo angle, i.e. must be [Re] 2 [Re] 5 [Re] 3 [Re] 5 [Re] 1 [Re] 1 m 10 10 10 . We performed a random variation of the SUSY parameters = [5 M 3 2 3 1 2 2 = [100 DL 12 12 − − − − − − × × × ˜ ` θ β ) . 4 0 5 m 10 10 10 10 10 10 . . . or 13 E LR 2 2 1 δ − from the MEG collaboration [ × × × × × × 6 6 3 4 6 9 UL 12 ...... = 1 TeV, θ . eγ 1 8 5 2 3 7 ˜ q 2 and ( → m (1) coefficients) in a given SUSY breaking scenario. They are unavoidable 12 µ ) O UR DR DR 12 13 12 E 2000] GeV, tan i i i i RR ) down to 10 , δ D U D E 12 12 13 12 , θ , θ , θ eγ EL 12 ER . Constraints on rotation angles in abelian flavour models, obtained using eq. ( 12 θ θ θ θ = 1 and the bounds of table θ ,( θ h h h h UL 12 DL DL → 13 12 12 R θ θ θ ) This table can be used to estimate the constraints on the Yukawa matrix (valid up Since the estimated effect in eq. ( µ = [100 rotation angle E LL δ whenever the SUSY breaking scalemSUGRA. is As above one the can messengergeneral see, flavour scale models. these They bounds putsince are abelian at least flavour quite either models strong in mSUGRAIn although scenarios realistic they in models trouble, hold the for constraints pretty are even stronger, since typically they are compatible results are summarised in table non-abelian models can be obtainedprovided taking in into table account the additional suppression factors to unknown of SUSY and messenger scale,imaginary for a part given of ratio the onescales rotation obtains for an angle. upper which This bound we on bound considerin the depends the Gravity real two logarithmically and Mediation extreme on corresponds cases the the to latter ratio representing messengers of the at typical minimal value satisfying perturbativity constraints. The in the following ranges: µ insertion the value for(which which has 90% of been the computedcancellations points using among of different the the contributions expressions larger scan in than are [ roughly excluded 10%. by BR( gluino masses: sion limit on ( we show in brackets alsoBR( the expected future bound provided that MEG will improve the limit on Table 4 with specification are the geometric mean of both, e.g. JHEP06(2012)018 ]. 31 , 1) from . 30 scale like (0 4 O = 9 CP violation and R ¯ . Still, non-abelian K 2. For the non-abelian R . in the 1-2 sector), but − θ 0 2  K ≈ ∼ in the reach of the MEG  ∼ L θ eγ ) ) ) ) ) → ) ) ) ) 6 6 3 3 3 3 3 3 3      . The constraints are less severe in     µ . Comparing to the bounds given . For the leptonic rotation angles ( ( ( ( ( ( ( ( ( 2 2 3 3 3     5 SU(3) EL      DL θ θ ≈ ≈ 3 2          DR U(1) ER θ θ , , – 16 – ER DR θ θ DR UR ER 2 TeV (notice that the bounds in table 12 12 12 i i i i ≈ ≈ D U D E 12 12 13 12 , θ , θ , θ DL 13 DR 13 ' θ θ θ θ θ θ h h h h DL ˜ q UL EL 12 12 EL DL 12 θ θ θ θ θ m , and small CPV phases, the bounds can be evaded only for rotation angle UR S θ mixing, as well as a rate for M ≈ D UL − θ D , K − K , we see that the U(1) model (as any abelian model) is seriously challenged by 4 . Rotation angles for example models of the appendix A with . 2 As an illustration, we compare the obtained bounds with the rotation angles predicted (1 TeV)). With this setup, the U(1) model should still exhibit sizable deviation from The phenomenological implications of some SU(3) models have been recently discussed in [ / 9 ˜ q models. We have analysedmodels, in which detail contains the vector-like superfieldslight structure Higgs that of fields. mix the In either messenger theYukawa with matrices, sector latter just light of case, by fermions it this removing or certain is kind with messengers particularly of from simple the to theory, obtain without texture modifying zeros in the LFV are still possible, provided that the SUSY masses are5 not too heavy. Conclusions In this paper, we have discussed the general features of the UV completions of SUSY flavour a quite heavy SUSYm spectrum, the SM in experiment. On the other hand,the bounds. the non-abelian Notice example however that is large still effects perfectly in compatible SU(3) with models for in the U(1) andwe SU(3) assumed models the of SU(5) appendix relations in A table in table the 1-2 sector constraints.vanishing scalar Even masses considering at an additional suppression with SU(5), implying non-abelian models (that gives at leastthe an small additional suppression breaking of universality isfermion partially mass compensated matrices by the aremodels fact typically are that symmetric definitively in preferred and these from models therefore what concerns the effect that we have discussed here. Table 5 case we included intable parentheses the total effective angle using the additional suppression factor in JHEP06(2012)018 ]. We thank the Theory 33 , we have provided bounds on 32 4 and SU(3). 0 U(1) × – 17 – GeV for the example models we have considered. This implies that the mes-
10 In general, the messenger sector contains many new fields charged under the SM 10 A Example models In this appendix we explicitlymessengers construct for the three UV example completions models: both U(1), with U(1) fermion and Higgs grateful to the Institutepitality of and Theoretical financial Physics of supportsupport the during of University the their of TUM-IAS stays Warsaw fundedpartially for in by kind the supported Warsaw. German hos- by Excellence S.P. Initiative.scientific the and research This R.Z. contract grant work acknowledge N202 PITN-GA-2009-237920 has 103838 been UNILHC (2010-2012). and the MNiSzW Acknowledgments We would like to thankThe C. Feynman Hagedorn, diagrams A. have Romanino beenDivision and drawn G. of using G. CERN JaxoDraw Ross [ for for hospitality useful discussions. during several stages of this work. L.C. and R.Z. are ence of flavour messengers affecting theels RG the running. sfermion mass-splittings Not can surprisingly, beflavon in non-abelian suppressed vevs with mod- and respect ease toand the the LFV abelian bounds observables case are by on still small the possible, rotation provided that angles. the SUSY However, spectrum large is effects not for too heavy. Kaon be easily extended to simpleconstrained non-abelian and models. hence it We find isTeV that difficult scale. abelian to models Even marry though them are it to strongly effects, is Gravity still Mediation well-known we that with find abelian SUSY it models atof remarkable the induce universal that soft large this masses flavour at remains changing the true SUSY even breaking under mediation the scale, as strong a assumption consequence of the pres- off-diagonal sfermion masses asflavour expected constraints by cannot the beThese flavour evaded constraints symmetry. even depend Therefore under only on the thetherefore the strong assumption apply diagonal to of mass large universal splittingthe classes and soft rotation the of terms. angles rotation flavour of angles, models. the and light In fermions that table are valid in any abelian flavour model and can O sengers have no directRG impact running on of low-energy observables. theof sfermion Still, sfermion masses. their mass-splittings presence andsymmetry We affects breaking misaligned have the scale emphasised A-terms is lower that isWe than the have unavoidable the quantitatively SUSY whenever radiative estimated breaking the generation this scale, RG like flavour in effect Gravity and Mediation. found it comparable to the tree-level senger sector does not relyscenarios. on SUSY and can be easily generalised togauge non-supersymmetric group. As aenergy consequence, scales forces requiring the the masses theory of to the remain messengers perturbative to up be to far high- above the TeV scale, typically the transformation properties of the SM fields. Our discussion on the structure of the mes- JHEP06(2012)018 (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.1) (A.2) 1. This − , 9 ) =  5 = 35) φ ( × = ) H is present already i d . ) = 9 messengers in ( 2 = 48 possibilities) U 33 i H y ), one needs in total d ×     ( 4 3 4 )+   i H 2.5 q × that is 7 ( 4 3     H )+ 10 i i 5 4 2 c 1 q    P ( u  c 1     H d − i , ∼ 2 ∼ c 2 − d U d u 1 P c 2 ) are Y Y − d − D , 1 0 1 23. − − . U A.2 0) 0) 1) det U . 0 0 0 0 , , , c 3 − 2 2 1 − D d D 0 0 , , , ∼ 0 − U − −  U 0 0 0 00 0 . According to eq. ( − i − Q Q 0 D f 0 0 : (3 : (3 : (2 – 18 – − Q − − Q 0 1 3 3 3 1 3 flavour symmetry. We take for the charges of the , , , − − q 2 2 2 Q , , , Q 1 10 0 1 1 c 1 c 1 H  − Q q − d u Q 2 2 − = q − Q 2     ) 2 i 3 2 q 1 Q − u   ( 1 − 5 4 2 q H 1    q )+ 6 5 3 i    q (     H i ]: ∼ P  u 34 Y ∼ ) = 10 messengers in the up-sector and u i Y u ( H ) is the charge of the fermion H i det f ( )+ i H q ( H In the simple U(1) case, this multiplicity arises from permuting the position of the Higgs insertion in i 10 In total we have usedwhich 10 is messengers exactly in the the minimal up number sector required. and 9 messengers ineach of the the down three sector, chain diagrams. where the messengers are labelledon with the their renormalisable U(1) level. charges. For the Note down that sector we choose (from 6 the up sector we choose (from the total number of possibilities where P the down-sector. All terms contributingtrue to in the every determinant U(1) are model). ofentries. We the can These same choose three the order term (which entries that is are is then the generated product from of the the three chain diagonal diagrams. For the chains in A.1.1 Fermion UVC The determinants of the Yukawas matrices in eq. ( The correct hierarchy of fermion massesparameter and mixing of can the be order achieved choosing of the the expansion Cabibbo angle, the Higgs fields are neutralgives and rise a to single the flavon effective is Yukawa introduced matrices: with charge The first example isMSSM based superfields on [ a U(1) A.1 U(1) JHEP06(2012)018 , to  2 (5) u − (A.9) , (A.14) (A.12) (A.13) (A.11) 3 H − , (4) d 4 − H , 4 5 − − which implies S α ij + (A.10) λ + + )) j . 2 , . d ) 3 ( ··· 5 , φ 4 , H , , 2 4 + 5 , the resulting Yukawa S , )+ S     i 3 q 3 4 (2) u , ( + M   2 . to zero. But at the same 2 H H  0) 0) , ( d d , ,     − ∼ 0 1 2 3 3 2 Y (1) , S H d α   − − x 3 1 ( ( c j 3) 2) H d 33 , , S d α α − . 1 M i u 33 , (1 (1 x q − 13 d 23 d + 2 λ α α α α λ λ d ij 3 − u 13 u 23 λ − , together with its conjugate partner 3 4 , +  λ   λ λ 3 S ,...,Y + 0) (4) 4 d , 0) 0) 5 4 2 − , , (1) u 1 S    2 3 , )) +1 H from the theory, so that also the 1-1 entry j − 4 H − 5) 4) 2) − ( Y , , , u + ( ( d ( − α (1 (1 (1 4 α α h X,X (5) , u H 0 − α α α d 32 5 = 12 d 22 d H α – 19 – )+ S λ x λ λ u 12 u 22 u 23 i − 5 q , λ λ λ ( (4) + d 4 2 5 S ≡ 6    ( H 6 5 3 u ) ( −    0) 0) 0) φ h , , , φH − u 6) 5) 3) 4 3 1 1) , , , = X,Y + − − − H ( − (1 (1 (1 ( ( ( 5 ( x d c j α , Higgs UV completions easily allow for the presence of α α α α α α − u H i S u 11 u 12 u 13 d 11 d 21 d 31 1 ) q x λ λ λ λ λ λ α 2.2 ( d (5) u ij u         λ + H H d = = + + h would not take a vev. In order to restore it, one has to introduce ··· ) d u ) x x Y Y ( d ⊃ ( + u (5) d H H are the MSSM Higgs fields. Taking 5) ) W H x − ( d u ( u h u H H h x 6) − M ( and d H x u 6 , giving 6 messengers in the up sector and 5 in the down sector which are necessarily X h α 2  . This results in setting the 1-2, 1-3 and 2-1 entries in As an example, we consider the above U(1) model where we now remove the heavy As explained in section ⊃ φ (4) u + W time this also removes thewould coupling be zero as singlet messengers. Specifically, onethe must add theory. vector-like Besides singlets mass terms, the new allowed couplings are removal of a certain Higgs mightgap interrupt has the then chain needed to for be other fixed Yukawa entries. by This using additionaldown-Higgs singlet messenger messengers. with chargeH -4, that is messengers, since in this casefull every rank. entry comes with a different coupling texture zeros in the Yukawas, simplythe by removing theory the (which corresponding Higgs however messenger sets from all Yukawas which the same charge to zero). But the with the shorthand notation Notice that in contrast to the fermion messenger case one does not have to add copies of where couplings for the up and the down sector are given by The relevant part of the superpotential is given by The charges of the heavy Higgsfigure messengers can be easily inferreddistinct. from the Thus chain one diagram of hasthe in charge of total 11 Higgs messengers (plus conjugates) that we denote by A.1.2 Higgs UVC JHEP06(2012)018 , 5 0), − , S (A.21) (A.17) (A.18) (A.19) (A.20) (A.15) (A.16) and 0), (1 4 , 1 − S − ( { , . 2 2  3 1 (5) d  . H , this a vev for ∼     3     have charges 5 4 2 3 2 d −    1   , 0) Y } S , 5 4 2 2 8 3 5 0) 0)       φ , , c 1 c 1 det 6 5 3 6 5 3 , u d    ,    1 , − − φ     c 2     1) (0     0) , u 3 0) , , , 0) (0 0) (1 ∼  1 1 ∼ . The effective Yukawas are then given by , , (0 − ]. The charges of the MSSM superfields are (0 φ } 3 −   2 0 0 U     3 { ) are  0)     D , 2  2 1 1 −  1 2 1 1 5   − 0 0   1  − ∼ 1) (  2 2 2 1) – 20 – 1) (1 1) ( 1) (1 , A.17       − 2 2 U , 2 , , ,   1 1 (0  2 1 1   (0 − , ∼ , this one a vev for   1 D 1 2 2  U 2 d  q  2 2 2 − − 2 Y −   1  : (0 : (0 : (0 S 2 2 2 1  = 1   1 q q 3 3 3  model in ref. [ 1 , , ,      2 2 2 0 3 2 , , ,     ,  1 c 1 c 1 ∼ 2 q d u  ∼ u ∼ : U(1) Y d u ∼ d Y × Y Y 1  { det 0 U(1) × ), whilst the 2-3 and 3-2 entries cannot be set to zero without removing the 2-2 and take vevs (2) d } H 1) Let us notice that with a similar procedure one can eliminate the 3-1 entry (by drop- − , while in the down sector we take (out of 18 possibilities) so that one needs 3 messengerssectors in the the diagonal up entries, and and 5only. in build In the the down the associated sector. up chains We sector with again we right-handed choose take messengers in both (out of 6 possibilities) the chain A.2.1 Fermion UVC The determinants of the Yukawas in eq. ( the Higgs are(0 neutral and the three flavons A.2 U(1) Next we consider the U(1) that has the interesting feature that the Cabibbo angle arises entirely from the up sector. which finally gives together with the first interaction aping vev to and the 3-3 entry,the respectively. following texture In for this U(1) example, one can therefore enforce at most The last coupling induces a vev for JHEP06(2012)018 ). L − (A.27) (A.25) (A.26) (A.22) (A.23) (A.24) B ( . Let us 21 ∝ Y σ , = and 123 is assumed to be  1) . 12 /M  − × i Y ,     (Σ) = 2. The MSSM , Σ 1) h (1 33 H , (1) 3 u,d 2 u,d , . 1 ), we see that the leading Y   , ). O 0) 0) , , (1 15. In order to differentiate 1 1 . 3 u,d 2 u,d 2 u,d A.25 0 ∼    A.17 transform as triplets of SU(3). ) = 3, − − ( ( i c ∼ , , 123 d 0 3 u,d 3 u,d c 2 ¯ d   123 φ M 1) 1)  , d ( ¯ φ     h − − − H and 3 c , , . 2 123 ∼ c 3  0) 1 u 05, c , . (0 d , u − 2 23 0 − 0 (0     ( ,  2 − , 23 2 3 D  q 23 , ∼ U  0) 0) σ ) is  , ) = 1, , 2 3 − × 1) − u 00 2 2 23 (0 ∼ 23  2 0) 123 , − ¯ φ – 21 – − D 1)  1 , Q ( ( A.25 u,d − − − ( (0 , − H Y 23 , 3 σ  σ  (1). The expansion parameter 3  1 q D , 1) q , 2 23 2 23 O − 2) −   det 123 (0 2 ,  − q 1 = , ∼ ) = 2, 23 23 − 3 3 (1 (   i  ¯ φ 0 , , ( 23 123 123 M ¯ φ   H 2) 2) h and − −     4. The above set-up gives rise to Yukawas of the form , , 2 , ∼ −  ) 3 : (0 : (0 = , with = u,d ,  H d u Y 0 H , ]. All the MSSM superfields, H H 123  (0 19 F ) , ∼  11 i 3 = ¯ φ M h 23  Some of these possibilities might require additional fields in order to write the theory in an SU(3) 11 We choose to buildpossibilities the three chains with the Higgs in the middle (in totalinvariant we way: have actually only 27 three possibilities have the minimal number of fields and are SU(3) invariant. so that 6 messengers incontribution both to sectors are the required. determinantconstruct From is eq. a given ( minimal by the set Yukawa of entries messengers for the up sector (the down sector is analogous). A.3.1 Fermion UVC The determinant of the Yukawas in eq. ( Unwanted operators are forbidden with additionalcharged symmetries, while under the which the MSSM flavonstake are superfields a are single neutral. U(1) Higgs fields For have our purposes, here we can simply with different in the uplepton and and down down-quark mass sector, matrices, with a field Σ is introduced, with A.3 SU(3) Finally we provide anmodel example in with ref. a [ non-abelianThree flavour anti-triplets symmetry. flavons are We introduced take with the vevs SU(3) of the form By introducing heavy Higgs messengers with charges (plus conjugates): one can generate all entries in the Yukawa matrices of eq. ( In total we use 3is messengers the for minimal the up number sector needed. and 5 messengersA.2.2 for the down sector, Higgs which UVC JHEP06(2012)018 12 (B.6) (B.1) (B.2) (B.3) (B.4) (B.5) (A.28) (A.29) (A.30) (A.31) charge. , H S M , M 1: S − . We consider M log M = i 4 22 log H ) d  2, λ  † − 1 d ∗ − λ = ( α  . − , d 32 H . 3 λ 11 , − ∗ ) DR 3, 12 DL 12 d 31 S d α θ ): θ λ − , λ M i M  d 12 d 22 † 1) d + = λ λ 11 11 − λ ) 3 ) ( A.12 3 ˜ log 2 d ˜ , 2 q H ∗ − ≈ c 2 , = 4: c 1 i +( α m u m u ( ( (4) ,H 6 22 H 22 d 22 DL − 12 ) − ) 2) 1 − d λ − 3 u ∗ − ,S λ U λ ( U d 21 3 † 22 22 † d λ ) = 2, − ) in the case of HUVC. The starting point is (2) u − 6 ˜ , which couples directly to the SM fermions. ˜ 2 d 2 q λ 3 1 λ ( – 22 – + H ,H ( H m Q m Q  , θ ,S 4 ( ( 3) − − A.1 6 − h  ∗ − − 3 − 1 ( 3 2 11 α − H 11 q = = q ) ) α d 12 u d λ λ ,H λ d 22 d 21 † ∗ † 6 λ λ d 11 d u ROT 12 ROT 12 ). As an effect of the flavour symmetry breaking, Higgs ) ) H λ λ λ ˜ ˜ 2 q 2 d ( ( ≈  would lead to antisymmetric Yukawa matrices. h h 2 0 m m 3 2 0 2 0 ( ( A.10 ˜ m H DR 12 ˜ ˜ m m θ 2 2 2 π π π 12 6 12 16 16 16 ≈ ≈ ≈ 11 11 ) ) RG 12 ˜ ˜ ) 2 q 2 d ˜ 2 d m m ( ( m ( − − 22 22 ) ) ˜ ˜ under SU(3) and neutral under U(1) 2 q 2 d A SU(3) triplet messenger m m 6 ( ( 12 As mentioned above there is also aby contribution the to the running, off-diagonal that entries directly at generated leading log reads: The mixing angles can be easily estimated from eq. ( where we have neglectedfollowing contributions the to contribution the off-diagonal of entries the in the A-terms. super-CKM basis: These splittings induce the derived in the messenger mass eigenbasis.for the Concentrating on mass the splittings down in 1-2 leading sector, we log obtain approximation: We now show anthe explicit U(1) example model of describedthe the superpotential in RG effect in appendix discussed eq.messengers in ( mix section among themselves. The RGEs for the soft masses can be conveniently plus the corresponding conjugate fields. B Explicit examples of RG induced flavour violation as In addition one needs three anti-triplets Higgs with or alternatively two sexplet singlets with where all 6 messengers are SU(3) singlets and are labelledA.3.2 with the additional U(1) Higgs UVC One needs to introduce (both in the up and in the down sector) a heavy Higgs transforming JHEP06(2012)018 ˜ 2 q , m non- (B.7) (B.8) (B.9) (1994) 1) − ( 3 H ]. B 406 B 420 are just sup- , 12 and (1993) 319 ) S ... ˜ SPIRE 2 d M 2) M + IN m − j ( 3 ][ 1) log H B 398 − Nucl. Phys. 1  ... ( Nucl. Phys. 3 , 10 we considered for our 3. In the running for and ( ∗ − ,  , H + ∼ α ∗ 3 12 i 23 α d 23 ) 1) ), since they involve different ˜ φ 6= 2 2 q λ − d 23 ) ( ∗ 3 i m λ ij d 13 ( ∗ 6 B.4 H λ u 13 ¯ i Nucl. Phys. H λ couple with + , 23 ), ( 23 ): M 3 j c φ + arXiv:1007.1532 α h d ∗ − ∗ = 0 for 5 [ i B.3 α q + α 23 i between the first and second generation A.30 j d 22 α d 22 i 23 2 d λ 2) Supersymmetric musings on the predictivity of  (1) between the third generation mass and λ φ ∗ + h ∗ − d 12 ( 3 u 12 ∼ 2) λ λ H − ∼ O ( i 3 3 2 23 – 23 – + Stitching the Yukawa quilt + ].  φ 2 3 4 H ) ∗ 6  ]. 2 23 ∗ − i (2010) 085022 2 3 ij acquires small components of the other messengers: α 3 ( α 6 α 6= 3 and α 2 d φ Mass matrix models 6 M d 21 Mass matrix models: the sequel ¯ 2 d ), we see that i H d 21 h SPIRE λ λ λ 3 H 3 λ ∗ ]. IN SPIRE ∗ α u 11 D 82 α Hierarchy of quark masses, Cabibbo angles and CP-violation ∝ B.8 d 11 [ ∝ IN λ λ + + ]. + ][ indices and we omitted the messenger mass terms and other  ) 6 2 0 ij = 0 for SPIRE F ( factor with respect to the abelian case discussed above and the 6 H ˜ i IN m 2 d i 3 H 2  ∼ c j ][ SPIRE φ π d h 6 0 6 (1979) 277 Phys. Rev. i IN q 16 , H ][ d λ ≈ This article is distributed under the terms of the Creative Commons ⊃ B 147 RG 12 hep-ph/9303320 ) denote SU(3) W ˜ 2 q [ j m ( hep-ph/9310320 and [ , this induces a splitting ˜ i 2 d hep-ph/9212278 468 (1993) 19 family symmetries Nucl. Phys. [ m Let us now discuss the same effect for the SU(3) model with HUVC. We start from P. Ramond, R. Roberts and G.G. Ross, K. Kadota, J. Kersten and L. Velasco-Sevilla, C. Froggatt and H.B. Nielsen, M. Leurer, Y. Nir and N. Seiberg, M. Leurer, Y. Nir and N. Seiberg, (1) coefficients. 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