Electric dipole moments from spontaneous CP violation in SU(3)-flavoured SUSY
aa r X i v : . [ h e p - ph ] F e b Electric dipole moments from spontaneous CPviolation in SU(3)-flavoured SUSY
J Jones P´erez Departament de F´ısica Te`orica and IFIC, Universitat de Val`encia-CSIC, E-46100, Burjassot,Spain.E-mail: [email protected]
Abstract.
The SUSY flavour problem is deeply related to the origin of flavour and hence tothe origin of the SM Yukawa couplings themselves. Since all CP-violation in the SM is restrictedto the flavour sector, it is possible that the SUSY CP problem is related to the origin of flavouras well. In this work, we present three variations of an SU (3) flavour model with spontaneousCP violation. Such models explain the hierarchy in the fermion masses and mixings, and predictthe structure of the flavoured soft SUSY breaking terms. In such a situation, both SUSY flavourand CP problems do not exist. We use electric dipole moments and lepton flavour violationprocesses to distinguish between these models, and place constraints on the SUSY parameterspace.
1. Introduction
Although Supersymmetry (SUSY) has not been observed yet in nature, it has become one ofthe most popular theories beyond the Standard Model (SM). This popularity is justified by theway it provides solutions to theoretical and cosmological problems that the SM cannot explain.Nonetheless, the Minimal Supersymmetric Standard Model (MSSM), has many problems of itsown. Among these problems, two of the most important are:
The SUSY Flavour Problem:
As we do not have any information about the off-diagonalelements of the soft SUSY-breaking matrices, we could expect them to be all of the sameorder. However, this generally gives too-large contributions to flavour changing neutralcurrents (FCNC) and unobserved processes like Lepton Flavour Violation (LFV). Thus,the off-diagonal elements must be suppressed with respect to the diagonal ones.
The SUSY CP Problem:
Among the many parameters of the MSSM, we have new CPviolating (CPV) phases. It turns out that the phases associated with flavour independentparameters give a too-large contribution to the electron and neutron electric dipole moments(EDMs). This requires these phases to be suppressed if the masses of the SUSY partnersare light.Presented in this way, the suppressions suggest the existance of a naturalness problem withinlow-energy SUSY. Nonetheless, the origin of these problems, and their possible solutions, maybe understood by observing the SM itself. In particular, the SM also presents its own FlavourProblem: if all elements in the flavoured matrices of the Standard Model (the Yukawa matrices)are assumed to be of the same order, one would not be able to reproduce the observed hierarchyn the mass spectrum and mixing angles. Thus, the flavoured matrices in the SM need tohave a structure. It can then be said that the real Flavour Problem lies in the inability tounderstand the generation of structures in flavoured parameters, may these be in the form ofYukawa couplings or soft SUSY-breaking terms.It is interesting to see that the flavoured elements of both the SM and the MSSM needcomplicated structures. This suggests that all flavoured parameters could have a common origin,and that by solving the Flavour Problem of the SM it might be possible to find a way to solvethe Flavour Problem in the MSSM.On the other hand, there is no such analogy for the SUSY CP Problem in the SM.Nevertheless, it is crucial to take into account that, although the CKM phase is large, all CPV inthe SM comes from supressed flavour-dependent terms. In the MSSM, the problematic phasescome from large flavour-independent terms. This might suggest that CPV is not completelyunderstood, and that it should be constrained within the flavour sector.An interesting solution for both of these problems lies in the use of family symmetries [1].The breaking of such a family symmetry can be used to give shape to both Yukawa matricesand soft terms. Moreover, this breaking of the family symmetry can also be associated with thespontaneous breaking of an exact CP symmetry [2], constraining in this way all CPV withinthe flavour sector. Furthermore, since the model predicts a structure for the soft terms, it ispossible to estimate the order of magnitude of the contributions to flavoured observables, beingable to reject the model or not from the low energy phenomenology.In this work, a model based on an SU (3) family symmetry will be presented [2]. Afterdescribing the possible variations to this model, low energy phenomenology in the leptonicsector will be analyzed. In particular, the expectations for LFV and EDM observables shall begiven.
2. The SU(3) Model An SU (3) flavour symmetry can explain the peculiar structure of the SM Yukawa couplings asthe result of spontaneous symmetry breaking [3]. The three generations of each SM fermionicfield are grouped in an SU (3) triplet , such that the SM Yukawa couplings are not allowed inthe limit of exact symmetry. In order to generate the Yukawas, the symmetry is broken by oneor several scalar vacuum expectation values (vevs) of new fields, called flavons. The flavons arecoupled to the SM fields through non-renormalizable operators, suppressed by a heavy mediatormass, to compensate the SU (3) charges. If the scalar vev is smaller than the mediator scale, thisprovides a small expansion parameter that can be used to explain the hierarchy of the observedYukawa couplings.In the same way, these flavons will couple to the SUSY scalar fields in all possible ways allowedby the symmetry and, after spontaneous symmetry breaking, they will generate a non-trivialflavour structure in the soft-breaking parameters. Therefore, by being generated by insertionsof the same flavon vevs, the structures in the soft-breaking matrices and the Yukawa couplingsare related. The starting point in the analysis of the soft-breaking terms must then necessarilyinvolve an analysis of the texture in the Yukawas, in order to reproduce first the correct massesand mixings.Two assumptions are required to fix the Yukawa couplings: (1) the smallness of CKM mixingangles is due to the smallness of the off-diagonal elements in the Yukawa matrices with respectto the corresponding diagonal elements, and (2) the matrices are symmetric (for simplicity).With these two theoretical assumptions, and using the ratio of masses at a high scale todefine the expansion parameters in the up and down sector as ¯ ε = p m s /m b ≈ .
15 and = p m c /m t ≈ .
05, the Yukawa textures in the quark sector can be fixed to [4]: Y d ∝ x ¯ ε x ¯ ε x ¯ ε ¯ ε x ¯ ε x ¯ ε x ¯ ε , Y u ∝ b ′ ε c ′ ε b ′ ε ε a ′ ε c ′ ε a ′ ε , (1)with x = 1 . x = 0 . x = 1 .
75, and a ′ , b ′ , c ′ being poorly fixed from experimental data.On the other hand, the Yukawa couplings in the leptonic sector can not be determined from theavailable phenomenological data. Within a seesaw scenario, the left-handed neutrino masses andmixings do not give any information to fix the neutrino Yukawa couplings. Therefore only thecharged lepton masses provide useful information on leptonic Yukawas. For simplicity, Yukawaunification at high scales is assumed, a possibility favouring Grand Unified models. In this case,charged lepton and down-quark (and the neutrino and up-quark) Yukawa matrices are the sameexcept for the different vev of an additional Georgi-Jarlskog Higgs field Σ, in the ( B − L + 2 T R )direction, to unify the second and first generation masses.The SU (3) model has several flavon fields, denoted θ , θ (anti-triplets ¯3 ), ¯ θ and ¯ θ (triplets ). These have to be coupled to the matter fields using effective couplings in every possibleway. Unfortunately, given a particular symmetry breaking pattern for the flavons, the SU (3)symmetry is not restrictive enough to reproduce the textures in Eq. (1) and one must imposesome additional symmetries to guarantee the correct power structure and to forbid unwantedterms in the effective superpotential. The basic structure of the Yukawa superpotential (forquarks and leptons) is then given by: W Y = Hψ i ψ cj h θ i θ j + θ i θ j (cid:0) θ θ (cid:1) Σ + ǫ ikl θ ,k θ ,l θ j (cid:0) θ θ (cid:1) + . . . i , (2)where to simplify the notation, the flavon and Σ fields have been normalized to the correspondingmediator mass, i.e., all the flavon fields in this equation should be understood as θ i /M f .After spontaneous breaking of the flavour symmetry (and CP symmetry) the vevs of thedifferent flavon fields are: h θ i = ⊗ (cid:18) a u a d e iχ (cid:19) ; h ¯ θ i = ⊗ (cid:18) a u e iα u a d e iα d (cid:19) ; h θ i = b b e iβ ; h ¯ θ i = b e iβ ′ b e i ( β ′ − β ) ; (3)where a vacuum alignment mechanism is required. The following relations are needed to properlyreproduce the Yukawas in Eq. (1): (cid:18) a u M u (cid:19) = y t , a d M d ! = y b ,b M u = ε, b M d = ¯ ε. (4)This structure is quite general for the different SU (3) models one can build, and for additionaldetails we refer to [2, 3, 5]. It is important to notice that the flavon vevs in Eq. (3) carry phases.This means that if exact CP symmetry is demanded in the original lagrangian, the breakingof SU (3) also breaks CP invariance. By doing this, one can restrict all CPV to lie within theflavour sector, removing the large flavour-independent phases that are the source of the SUSYP Problem. It has also been shown in [2] that it is possible to reproduce the observed CKMphase using the flavon phases. This procedure shall be followed in this work.As mentioned previously, in the context of a supersymmetric theory an unbroken flavoursymmetry would apply equally to the fermion and scalar sectors. This implies that in the limitof exact symmetry the soft-breaking scalar masses and the trilinear couplings must also beinvariant under the flavour symmetry. This has different implications in the case of the scalarmasses and the trilinear couplings, although in the present work only the scalar masses will beanalyzed (we will take A = 0).Scalar flavour-diagonal couplings are invariant under any symmetry, which means thatdiagonal soft-masses are always allowed by the flavour symmetry and will be of the order ofthe SUSY breaking scale. Additionally, for SU (3), the three generations are grouped in a singlemultiplet with a common mass, thus reducing greatly the FCNC problem.After SU (3) breaking, the scalar soft masses deviate from exact universality, and any invariantcombination of flavon fields contribute to the sfermion masses. In this case, the following termswill always contribute to the sfermion mass matrices:( M f ) ij = m (cid:18) δ ij + 1 M f h θ † ,i θ ,j + θ i † θ j + θ † ,i θ ,j + θ i † θ j i + 1 M f ( ǫ ikl θ ,k θ ,l ) † ( ǫ jmn θ ,m θ ,n ) + . . . (cid:19) , (5)where f represents the SU (2) quark and lepton doublets or the up (neutrino) and down (charged-lepton) singlets. Notice there are three different mediator masses, M f = M L , M u , M d , althoughfor simplicity M u is taken equal to M L . This minimal structure shall be denoted as RVV1. Thesoft scalar masses of the slepton sector, in the SCKM basis, take the following structure:( m e cR ) T m = ε y d ¯ ε ¯ ε e − i ( β − χ )13 ¯ ε ε ¯ ε e − i ( β − χ )13 ¯ ε e i ( β − χ ) ¯ ε e i ( β − χ ) y d (6a) m L m = ε y t ε ¯ ε ¯ ε y t e − i ( β − χ )13 ε ¯ ε ε ε y t e − i ( β − χ ) ¯ ε y t e i ( β − χ ) ε y t e i ( β − χ ) y t (6b)where for simplicity we have neglected O (1) constants, which usually have important subleadingphases.It is possible to build other invariant combinations with different flavon fields that can notbe present in the superpotential. This is due to the fact that the superpotential must beholomorphic, i.e. can not include daggered fields, while the soft masses, coming from the K¨ahlerpotential, only need to be real combinations of fields. The type of allowed extra terms dependon the symmetries one imposes to shape the Yukawa matrices. Such symmetries can allow twoexclusive variations without modifying the initial Yukawa structures.The first variation to the minimal structure is achieved by allowing a θ i ¯ θ j term in the softmass matrix (RVV2). When rotated to the SCKM basis, the mass matrices become:( m e cR ) T m = ε y b ¯ ε ¯ ε y . b e − iβ ′ ¯ ε ε ¯ ε y . b e − iβ ′ ¯ ε y . b e iβ ′ ¯ ε y . b e iβ ′ y b (7a) m L m = ε y t ε ¯ ε ε ¯ ε y . t e − i ( χ − β ′ )13 ε ¯ ε ε ε y . t e − i ( χ − β ′ )13 ε ¯ ε y . t e i ( χ − β ′ ) ε y . t e i ( χ − β ′ ) y t (7b)sing mass-insertion notation [6, 7], one can see that the effect of this term in ( m e cR ) T isto exchange one power of ¯ ε by a y . b supression in ( δ e ) RR and ( δ e ) RR . In m L , the sameterms change an ¯ ε by an ε y . t . However, for tan β = 10, and considering that ε ≈ ¯ ε , suchreplacements leave the structure of the mass matrices very similar numerically to the originalone. Nonetheless, it must be remarked that the phase structure of the whole mass matrix ismodified.The second variation to the minimal structure allows a (cid:0) ǫ ikl θ k θ l (cid:1) θ j term (RVV3). The softmatrices, when rotated into the SCKM basis, have the following structure:( M e cR ) T m = ε y b ¯ ε ¯ ε y b e − i ( δ d − χ )13 ¯ ε ε ¯ ε e − i ( β − χ ) ¯ ε y b e i ( δ d − χ ) ¯ ε e i ( β − χ ) y b (8a) M L m = ε y t ε ¯ ε y t e − i (2 χ − β − δ d ) ε y t e − i ( χ − δ d ) ε ¯ ε y t e i (2 χ − β − δ d ) ε ε y t e − i ( β − χ ) ε y t e i ( χ − δ d ) ε y t e i ( β − χ ) y t (8b)where δ d = 2 α d + β ′ + β .This model shows larger deviations from RVV1 in the LL sector. It is important to noticethat ( δ e ) LL is now of order ε ¯ ε instead of ε ¯ ε , which will have considerable consequences inprocesses such as µ → eγ . Likewise, ( δ e ) LL is of order ε y t instead of ¯ ε , so an enhancementin τ → eγ should be expected. Regarding the RR sector, for tan β = 10, the y b suppression at M GUT has roughly the same size as an ¯ ε suppression, so once again the structure of ( m e cR ) T isnumerically similar to RVV1.It is important to emphasize at this point that these deviations from universality in the soft-mass matrices due to flavour symmetry breaking come always through corrections in the K¨ahlerpotential. Therefore, these effects will be important only in gravity-mediation SUSY modelswhere the low-energy soft-mass matrices are mainly generated through the K¨ahler potential.In other mediation mechanism, as gauge-mediation or anomaly mediation, where these K¨ahlercontributions to the soft masses are negligible, flavour effects in the soft mass matrices will bebasically absent.The soft mass matrices shown are given at M GUT , which means that no effects coming fromthe running have been included. Such running effects can be sizeable in the LL and LR sectors,due to the presence of heavy RH neutrinos with large Yukawa couplings [8, 9, 10]. Moreover, inthe present case there are new contributions to the running given by the non-universality of thesoft mass matrices. These effects can be important even in the RR sector. Nevertheless, it turnsout that these contributions are, at most, of the same order as the initial matrices. This meansthat the addition of running effects will only change the already unknown O (1) constants, suchthat the low-energy phenomenology can still be understood by analyzing Eqs. (6)-(8).
3. Phenomenology in the Leptonic Sector
Supersymmetric flavour models characterized by unaligned, non-universal scalar masses implythe arising of potentially large mixing among flavours. This has an immediate impact onprocesses with FCNC and LFV, via loop diagrams. This means that the proposed SU (3) familysymmetry can be tested through such processes.In particular, in the lepton sector, the mass insertions are sources of lepton flavour violation,through neutralino or chargino loop diagrams. As a consequence, it is expected to haveconstraints in the allowed SUSY parameter space due to the experimental limits on LFV decayssuch as µ → eγ [11]. igure 1. Current constraints due to µ → eγ (green) and τ → µγ (yellow) in the m - M / planefor tan β = 10 and A = 0. We show constraints for RVV1 (left), RVV2 (center) and RVV3(right). The green dotted region corresponds to the reach of µ → eγ at the MEG experiment,while the yellow hatched region is the reach of τ → µγ at the Super Flavour Factory. DirectLEP bounds and charged LSP constraints are shown in dark brown.By setting A = 0, the branching ratio of µ → eγ decay depends only on the square of the( δ eLL ) and ( δ eRR ) mass insertions. Inspecting the structures shown in Eqs. (6), (7) and (8),it is evident that the µ → eγ branching ratio will be similar for RVV1 and RVV2, while forRVV3 the LL contribution will be enhanced by a factor ¯ ε/ε = 3. Since it is the square of themass insertion what contributes to the branching ratio, we can expect an increase in the LL contribution by an order of magnitude.For τ → µγ , the difference between RVV1 and RVV2 is of a factor ¯ ε/y . b and 3(¯ ε /ε ) y . t for the RR and LL contributions, respectively. Taking the values of the Yukawa couplings at M GUT for tan β = 10, one can check that both ratios are of order 1. Thus, no great differencesshould be expected either between RVV1 and RVV2. RVV3 has got the same sort of insertionsas RVV1, which means that τ → µγ decay is not sensitive to any of the possible variations.The decay τ → eγ has been analyzed in [11], and has been shown not to be sensitive enoughto constrain these kind of models, in comparison with µ → eγ . Thus, by looking at the modulusof the mass-insertions, one can only distinguish RVV3 from RVV1 and RVV2.An analysis of these three variations is shown in Figure 1, for tan β = 10 and A = 0. Thepredictions for RVV1 and RVV2 are similar, as expected. Both models are slightly constrainedby both µ → eγ [12] and τ → µγ [13], where both branching ratios exclude SUSY masseswith m . −
400 GeV, M / . −
300 GeV. RVV3 is much more constrained, as µ → eγ can be an order of magnitude larger. SUSY masses following m . −
900 GeV, M / . −
600 GeV are excluded.The future observation of µ → eγ is crucial for all three models. The proposed increasein sensitivity by two orders of magnitude in the MEG experiment [14] will probe much of theparameter space accesible to the LHC. Thus, if SUSY is found at the LHC and any of thesemodels is at work, it is very likely to observe µ → eγ at MEG. The same can be said for τ → µγ at the Super Flavour Factory [15], with an increase of sensitivity of one order of magnitude.This is also shown in Figure 1, although such constraints are not as strong. Even if µ → eγ is observed at MEG, the question remains on how to distinguish between RVV1and RVV2. The answer to this question lies on the observation of the electron EDM, d e .Since the SU (3) models start with exact CP symmetry, broken by the complex flavon vevs, (GeV) M / ( G e V ) m (GeV) M / ( G e V ) m (GeV) M / ( G e V ) Figure 2.
Contours of | d e | = 1 × − e cm (light red), | d e | = 5 × − e cm (orange)and | d e | = 1 × − e cm (yellow) in the m - M / plane for tan β = 10 and A = 0. We showpredictions for RVV1 (left), RVV2 (center) and RVV3 (right). Current EDM bound (1 . × − )is shown in dark red. Current LFV bounds are also shown in green.the problematic flavour-independent phases are gone. Nonetheless, d e can be still generated bycombinations of phases in flavour violating terms. The three RVV models predict their ownphase structure along with their flavour structure, so it is plausible to consider d e as a suitableobservable for distinguishing between the models.In [11] it was shown that the most important contribution to d e , when A = 0, comes froma bino-mediated diagram proportional to ℑ m [( δ e ) LL ( δ e ) LR ( δ e ) RR ], which is enhanced by m τ tan β . Due to the initial CP symmetry, the ( δ e ) LR term is real, so one has to analyze thephases in the flavour-violating terms.For RVV1, one can see in Eq. (6) that the leading phases cancel. This means that d e will comefrom phases in subleading terms, contained within the O (1) parameters, and thus will be highlysupressed. The most important phase within these subleading terms depends on 2( χ − β ).RVV2, even though having roughly the same order of magnitude for tan β = 10, has a non-vanishing leading phase. We can thus expect d e in RVV2 to be somewhat larger than in RVV1.Nonetheless, the largest d e comes from RVV3, which not only has a larger flavour structure, butalso a leading non-vanishing phase.Inspection of Eqs (6)-(8) leads to the following predictions:( d e ) RVV1 ≪ ¯ ε y t d e ) RVV2 ∼ ε ¯ ε ( y b y t )9 . sin( χ − β ′ ) (10)( d e ) RVV3 ∼ ε ¯ εy b y t sin(2( χ − δ d )) (11)Figure 2 shows the sensitivity of current [16] and future [17] d e experiments, for the threemodels. In order to be able to compare them consistently, all phases have been set to zero,except χ = π/
4. This maximizes d e in RVV1 and RVV3, and gives a large contribution toRVV2.In the Figure, both RVV1 and RVV2 can be seen to survive the current constraints, but atthe same time are large enough to be probed at future EDM experiments. In particular, forRVV1, the observation of a d e ∼ − would favour light SUSY masses. On the other hand,RVV2 predicts a value of d e of about one order of magnitude larger than RVV1 for any particularvalue of m and M / . This means that by reaching d e ∼ − one can probe a much largerpart of the evaluated parameter space, with m . M / . d e bounds. Thismeans that, for SUSY masses with m < M / . χ must besupressed, or δ d must have a value that cancels χ . Such constraints seem unnatural and againstthe whole motivation for these kind of models. This leads one to conclude that both EDM andLFV data greatly disfavour RVV3 in most of the parameter space to be probed by the LHC, incontrast to RVV1 and RVV2.
4. Conclusions
It has been shown that it is possible to relate the SM and SUSY Flavour Problems using an SU (3) family symmetry, and by doing so, both problems can be solved simultaneously. Thebreaking of SU (3) can also lead to the breaking of exact CP, constraining all phases within theflavour sector, and solving the SUSY CP Problem.Three possible exclusive variations of this model have been analyzed, and the observation ofLFV processes, as well as EDMs, has proven to be essential to distinguish them. Of the threemodels, RVV1 and RVV2 survive the current LFV and EDM bounds even with light SUSYmasses. If SUSY is observed at the LHC and one of these two models is at work, it would bevery likely to observe µ → eγ and τ → µγ decays in the upcoming experiments, as well as anelectron EDM, if the phases are not small. RVV3 turns out to be highly disfavoured by thelow-energy observables in most of the evaluated parameter space.Thus, the SU (3) models turn out to have a testeable phenomenology in the short term. It isinteresting to notice that, as the flavon phases in the lepton sector are the same as those of thequark sector, correlations between both sectors are expected to appear [18]. This would givemore hints on the correct way of building the model, reducing in this way the ambiguity in thesymmetries required for building the Yukawa and soft matrices. Acknowledgments
I acknowledge support from the Spanish MCYT FPA2005-01678, and would like to thankO. Vives and L. Calibbi, who collaborated in the original work this talk is based on.
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