Implications of large CP Violation in B mixing for Supersymmetric Standard Models
IImplications of large CP Violation in B mixing forSupersymmetric Standard Models
S. F. King School of Physics and Astronomy, University of Southampton,Southampton, SO17 1BJ, U.K.
Abstract
Following the anomalous like-sign dimuon charge asymmetry measured by theD0 collaboration at the Tevatron collider we discuss the implications of large CPviolation in B d,s mixing for Supersymmetric (SUSY) Standard Models, focussingon those models which allow a family symmetry and unification. For the MinimalSupersymmetric Standard Model (MSSM) we show that it is only possible toaccount for B s mixing and CP violation at the expense of large squark mixingwhich would require a new approach to family symmetry models. In order todescribe both B s and B d mixing and CP violation we are led to consider SUSYmodels with Higgs fields transforming as triplets under a family symmetry. Wedescribe a realistic such model based on ∆ family symmetry in which tree-levelexchange of the second Higgs family predicts B s and B d mixing and CP violationin good agreement with a recent global fit, while naturally suppressing flavour andCP violation involving the first and second quark and lepton families. E-mail: [email protected] a r X i v : . [ h e p - ph ] S e p Introduction
The Standard Model (SM) has provided a remarkably successful description of quarksand leptons from its inception in the 1960s until the end of the last millennium. In 1998the discovery of neutrino mass and mixing demanded new physics beyond the SM forits explanation (for a review see e.g. [1]). The picture which has emerged in the leptonsector is consistent with three neutrino mass and mixing described by a PMNS matrix U ,although its origin remains unclear and U e is so far unmeasured. By contrast, despiteintense experimental and theoretical scrutiny, there has been no firm evidence of anynew physics in the quark sector, with the CKM picture of CP violation, summarisedby the Unitarity Triangle, becoming ever more precisely determined [2]. Yet despitethis progress, some cracks have begun to appear in B physics which may call for newphysics to describe CP violation beyond the CKM matrix. It is worth recalling thatCP violation is predicted by the SM to be very small in B − B mixing, well below theTevatron sensitivity, due in part to the small phases of the relevant CKM elements. Onthe other hand new physics can compete with the SM box diagrams, in principle withlarge new CP violating phases, giving much larger CP asymmetries in B − B mixingthan predicted by the SM, rendering it observable at the Tevatron.Recently the D0 Collaboration has reported evidence for CP violation in the like-signdimuon charge asymmetry [3] A b sl ≡ N ++ b − N −− b N ++ b + N −− b = − (0 . ± . ± . × − , (1)where N ++ b ( N −− b ) is the number of events with b (¯ b ) containing hadrons decayingsemileptonically into µ + X ( µ − X ) . The D0 result is 3.2 σ away from the standardmodel (SM) prediction ( − . ± . × − [4]. The CDF [5] measurement of A b sl , usingonly 1.6 fb − of data, as compared to the D0 6.1 fb − data set, has a central valuewhich is positive, A b sl = (8 . ± . ± . × − , but is still compatible with the D0measurement at the 1.5 σ level because its uncertainties are 4 times larger than those ofD0. Combining the the D0 and CDF results for A b sl , one finds A b sl (cid:39) − (0 . ± . × − which is still 3 σ away from the SM value.The interpretation of the observed CP asymmetry is in terms of the production of BB meson pairs followed by their subsequent oscillation and semi-leptonic decay wherethe charge of the final state lepton effectively tags whether it is a b or b quark whichdecays. Thus the dilepton asymmetry can be written as A b sl ≡ N ( BB ) − N ( BB ) N ( BB ) + N ( BB ) = P B → B P B → B − P B → B P B → B P B → B P B → B + P B → B P B → B . (2)The measured asymmetry at the Tevatron is interpreted as a linear combination of theasymmetries a d,s sl in B d and B s oscillation and decays [3], A b sl = (0 . ± . a d sl + (0 . ± . a s sl , (3)1here the “wrong charge” asymmetries are a q sl ≡ Γ( ¯ B q → µ + X ) − Γ( B q → µ − X )Γ( ¯ B q → µ + X ) + Γ( B q → µ − X ) . (4)The current experimental values of the separate asymmetries are a d sl = − (0 . ± . × − [6] and a s sl = − (0 . ± . ± . × − [7] which, while being consistent with anegative A b sl of order one per cent, are also consistent with zero as well, and so do notshed much light on which of the two separate asymmetries is responsible, however thereis apparently a mild tendency for both of these asymmetries to be acting together.In addition to these measurements, D0 have reconstructed B s → J/ψφ decays andhave measured the time dependent asymmetry parameter S ψφ = − sin φ B s , where φ B s is the phase of the B s − B s mixing matrix element M s = | M s | e iφ Bs , and finds adiscrepancy with the SM prediction of S ψφ ∼ . σ [8]. A recentpreliminary CDF analysis based on 5.2 fb − of data, finds S ψφ which is consistent withzero but has a central value of S ψφ ∼ . S ψφ in various modelsand the correlations between this and other observables as a means of discriminatingbetween these models has been comprehensively studied [10]. Neglecting the small SMcontribution to S ψφ , the following model independent relation holds between S ψφ and a s sl [11]: a s sl ≈ − | Γ s || M s | S ψφ (5)Following the recent D0 results, it has been shown that it is possible to fit S ψφ and a s sl from the latest data by assuming that new physics contributes significantly to themixing matrix element M s and also by allowing the decay matrix element Γ s to float[12]. More precisely the authors in [12] perform a global fit of all experimental mea-surements, including the recent D0 asymmetry results and the recent CDF preliminaryresults for B s → J/ψφ , allowing the two SM decay matrix elements Γ q to float, whileallowing for new physics to contribute to both mixing matrix elements M q which canbe parameterised as: M q = M ,SMq (1 + h q e iθ q ) . (6)Using the convention where a q sl = Im( − Γ q /M q ) with the dominant real parts of Γ s and M s being positive in the SM, gives, a s sl ≈ | Γ s || M ,SMs | h s sin θ s h s cos θ s + h s , (7)where we have neglected the small phase in the SM matrix element, β s ≈ − .
01. How-ever the corresponding SM matrix element in the B d sector is, M ,SMd = | M ,SMd | e iβ ,with β ≈ .
38, which will contribute significantly to the phase of M d .2he global fit [12] includes the measured B s and B d mass differences ∆ M s and ∆ M d ,the measured time dependent asymmetries S ψφ , S ψK (which determines the unitaritytriangle angle β ), the CP asymmetries a d,s sl as well as the CKM parameters ρ, η , whileallowing Γ q to vary from its SM predicted value. The best fit value for Γ s is abouttwice as large as the SM prediction [4], with similar results obtained in [13] wherethe implications of such a large value are discussed. The decay matrix element Γ s isproportional to the square of a tree-level SM amplitude proportional to V cb arising from W exchange so it is challenging to understand why the best fit value should be so large.The best fit points for new physics contributions to the matrix elements are [12]( h d , θ d ) ∼ (0 . , π ) , ( h s , θ s ) I ∼ (0 . , π ) , ( h s , θ s ) II ∼ (1 . , π ) . (8)Note that there are two different best fit points for ( h s , θ s ), but only a single best fitpoint for ( h d , θ d ) which is only preferred from a zero value at a confidence level of order1.5 σ . All points have the angle θ q in the third quadrant where both sin θ q and cos θ q arenegative, resulting in enhanced negative values of a q sl from Eq.7. Although the precisebest fit points must be regarded as indicative values with rather large error bars, thefit is quite robust with h s = h d = 0 disfavoured at 3.3 σ [12]. There have already beenseveral attempts to explain the recent data [14, 15, 16, 17, 18, 19, 20] (see also [21]).In this paper we consider the implications of large CP violation in B mixing for SUSYStandard Models (for a review see e.g. [22]) focussing on those models which includea family symmetry and allow for unification. We discuss two distinct possibilities. Forthe Minimal Supersymmetric Standard Model (MSSM), discussed in Section 2, we showthat it is only possible to account for B s mixing and CP violation at the expense oflarge squark mixing which would require a new approach to family symmetry models.Moreover this approach cannot account for B d mixing and CP violation. In order todescribe both B s and B d mixing and CP violation, in Section 3 we are led to considerSUSY models with Higgs fields transforming as triplets under a family symmetry, wheretree-level exchange of the second Higgs family may readily account for all the data. Wedescribe a realistic such model based on ∆ family symmetry [23] which predicts B s and B d mixing and CP violation in good agreement with the best fit point I, and naturallyleads to small effects in K mixing and other flavour violating processes. B s mixing in the MSSM As discussed in [24] the MSSM contributions to CP violation in B mixing arise domi-nantly from box diagrams involving down squarks and gluinos. The B q mixing matrixelement can be written as the sum of the SM box diagrams and the SUSY box diagrams, M q = M , SM q + M , SUSY q (9)3here from Eq.6 we identify h q e iθ q = M , SUSY q M , SM q , (10)where q = s, d . Using the matrix elements in [24] with the updated parameters in [17]we find, in the mass insertion approximation (see [25] and references therein), h s e iθ s ≈ (cid:18)
500 GeV m ˜ q (cid:19) (cid:2) (cid:0) ( δ d ) LL + ( δ d ) RR (cid:1) − (cid:0) ( δ d ) LL ( δ d ) RR (cid:1)(cid:3) (11) h d e i ( θ d +2 β ) ≈ (cid:18)
500 GeV m ˜ q (cid:19) (cid:2) (cid:0) ( δ d ) LL + ( δ d ) RR (cid:1) − (cid:0) ( δ d ) LL ( δ d ) RR (cid:1)(cid:3) , where ( δ d i ) LL = ( V D L m Q V † D L ) i /m q and ( δ d i ) RR = ( V D R m D V † D R ) i /m q , with m ˜ q being atypical squark mass (assumed to be degenerate with the gluino ˜ g ) where m Q is the left-handed (L) squark doublet mass squared matrix, m D is the right-handed (R) down-typesquark mass squared matrix, and V D L and V D R are the unitary matrices that diagonalisethe down-type quark mass matrix M d , namely V D L M d V † D R = diag( m d , m s , m b ). This issummarised by the statement that the LL and RR mass mixing between down squarksof different generations in the super CKM basis is the source of the flavour and CPviolation [22]. We have not included the contributions from LR mass mixing which aretightly constrained by b → sγ .Comparing the SUSY predictions in Eq.11 to the best fit points in Eq.8, it is clearthat the values of h s ∼ δ d ) LL ∼ ( δ d ) RR ∼ . − . δ d ) LL (cid:28) ( δ d ) RR ∼ . − . δ d ) RR (cid:28) ( δ d ) LL ∼ . − .
6. These represent quite sizeable squark mixing angles,which run into conflict with grand unified theories (GUTs) based on gravity mediatedSUSY breaking. To see this, it is worth bearing in mind that renormalisation group(RG) running from the high energy GUT or Planck scale to low energies tends toincrease the diagonal squark masses m ˜ q by about a factor of 5, while not enhancingthe off-diagonal squark masses [25], so the high energy ( δ d ) LL,RR parameters need tobe 25 times larger than these low energy values which is not possible (they can at mostonly be of order unity). For some gauge mediated SUSY breaking scenario, where themessenger scale is below the GUT scale, the effect of running is reduced so it may bepossible to achieve these low energy values. Another constraint is that, in the frameworkof GUTs, there is the danger of running into conflict with the bound on ( δ e ) LL < . τ → µγ [25] since the slepton masses are only enhanced by about a factor of2 in running from the GUT scale to low energies. In the context of SU (5) GUTsthe low energy ( δ d ) LL parameters are constrained by the high energy requirement that( δ d ) GUT RR = ( δ e ) GUT LL < .
48 [25], which implies the low energy constraint ( δ d ) RR < . SU (3) symmetry models (see [26] and references therein)4mall squark mixing parameters are also expected ( δ d ) RR ∼ − . It seems that thedata is not favoured by conventional SUSY GUTs and family symmetry.Suppose we abandon all pre-conceived prejudices about SUSY breaking, GUTs andfamily symmetry, but continue to assume that the MSSM is the only source of newphysics. Then we can ask if it is possible for the MSSM to describe the observationsand if so then what the data is telling us about squark mixing. For the reasons outlinedabove, from the point of view of the MSSM, it is desirable for the squark mixing to beas small as possible. Therefore we shall consider the smallest mixing describing the datagiven by solution I with ( δ d ) LL ∼ ( δ d ) RR ∼ .
05. Taking into account the RG runningup to the (unknown) SUSY breaking messenger scale, this still suggests a high energytheory capable of giving quite large (2,3) mixing in the squark sector. Following thisreasoning we are led to consider high energy Yukawa matrices in the quark sector whichhave a democratic structure in the (2,3) sectors, Y u (2 , ∼ Y d (2 , ∼ (cid:18) (cid:19) . (12)This ensures that even approximately diagonal squark mass squared matrices wouldgenerate large squark mixing in the super CKM basis. However the small CKM angle θ ∼ | θ u − θ d | would then require an accurate cancellation. To enforce this (ap-proximate) cancellation in a natural way we shall require Y u (2 , ∝ Y d (2 , and rank onesub-matrices to achieve the (2,3) quark mass hierarchies. Both these features could beachieved by an SU (3) family symmetry under which the left and right-handed quarkstransform as triplets Q i , U ci , D ci ∼ φ i with aligned vacuum expectation values (cid:104) φ (cid:105) ∼ (0 , , V , where phases havebeen suppressed (see [26]). Then the (2,3) block could be generated from leading orderoperators of the form, dropping coefficients, H u Q i φ i U cj φ j + H d Q i φ i D cj φ j , (13)which, after the two Higgs doublets of the MSSM H u and H d acquire their VEVs,implies θ ∼ m s (cid:28) m b , m c (cid:28) m t , at leading order. This differs from the usual SU (3) models [26] by the absence of the flavon φ i with VEV (cid:104) φ (cid:105) ∼ (0 , , V . Thisdemocratic (2,3) structure could be plausibly be extended to the charged lepton sectoras well, Y e (2 , ∼ (cid:18) (cid:19) , (14)resulting from the leading order operator, H d L i φ i E cj φ j , (15)which implies maximal charged lepton mixing in the (2,3) sector. In order to achievemaximal (2,3) physical lepton mixing we require the light effective neutrino Majorana5ass matrix to be approximately diagonal. For example this could be due to a typeII see-saw mechanism via a sextet flavon ∆ ij with an approximately diagonal VEV (cid:104) ∆ ij (cid:105) ∼ diag(0 , a, b ) arising from operators of the form [27], H u H u L i L j ∆ ij . (16)This then leads to maximal atmospheric neutrino mixing coming from the charged leptonsector.In the above approach we are essentially saturating the high energy limits with( δ d ) GUT
LL,RR ∼ ( δ e ) GUT
LL,RR ∼ τ → µγ limit ( δ e ) GUT LL < .
48. However there is a very large compensating effectfrom RG running due to the large charged lepton Yukawa couplings which will tend toreduce the magnitude of the slepton masses, including the off-diagonal ones. To leadinglog, the correction will be∆( δ e ) LL ∼ − π ln M GUT M W ∼ − . δ e ) GUT LL ∼
1. Nevertheless, τ → µγ mightbe expected to be not far below its current limit. Such Yukawa induced RG correctionswill also be present for the squark sector, but there the dominant suppression is comingfrom QCD enhancement of the diagonal squark masses.Turning to the less statistically significant evidence for B d mixing, comparing theSUSY predictions in Eq.11 to the best fit points in Eq.8, it is clear that the values of h d ∼ .
25 could either be achieved by ( δ d ) LL ∼ ( δ d ) RR ∼ . × − or ( δ d ) LL (cid:28) ( δ d ) RR ∼ × − or ( δ d ) RR (cid:28) ( δ d ) LL ∼ × − . From the point of view ofconventional family SU (3) symmetry models the above squark mixings are much largerthan the expected values ( δ d ) RR ∼ − [26]. Although these low energy squark mixingslook more modest, they must originate from high scale squark mixings which are manytimes larger than these values (25 times if the high scale is the GUT scale). Againwe consider the smallest mixing case corresponding to ( δ d ) LL ∼ ( δ d ) RR ∼ . × − corresponding to high scale values perhaps of order λ ∼ .
2. This suggests a modelwith (1,3) quark mixing angles of order λ in the up and down sectors, θ u ∼ θ d ∼ λ ascompared to θ ∼ λ , which again demands some natural cancellation mechanism. Inorder to achieve this we may extend the SU (3) approach above, by introducing usingthe anti-triplet flavon φ i with aligned VEV (cid:104) φ (cid:105) ∼ (1 , , V , where phases havebeen suppressed (see [26]), which introduces the additional operators H u Q i φ i U cj φ j + H u Q i φ i U cj φ j + H d Q i φ i D cj φ j + H d Q i φ i D cj φ j . (18)The combined effect of the operators in Eqs.13,18 is to yield the quark Yukawa couplings,assuming V /V ∼ λ , ignoring phases, Y u ∼ Y d ∼ λ λλ λ λλ λ λ . (19)6fter the maximal angle (2,3) rotations, the remaining matrices are diagonalised byseparate and equal (1,3) rotations of order λ in both the up and down sectors whichcancel as required. However this results in first family eigenvalues of order λ in boththe up and down sectors leading to the relations m u /m t = m d /m b ∼ λ which arein strong disagreement with their observed values. This result could in principle beevaded when other operators or corrections are included as required to describe thesecond family quark masses m c , m s , but apparently only at the expense of an unnaturalcancellation of the leading order first family result above. Therefore we are not able toaccommodate large and natural (1,3) mixing of order λ in the up and down sectors assuggested by the B d mixing fits. In general we do not see how to achieve this in a naturalway in the MSSM including family symmetry and GUTs. We conclude that either theweaker requirement of non-standard B d mixing be discarded, which as remarked in theIntroduction is only required at a confidence level of order 1.5 σ , or one must abandonthe MSSM as a natural explanation for CP violation in both B s and B d mixing withinthe framework of family symmetries. B s and B d mixing withthree families of SUSY Higgs It is obviously easier to account for large CP violation in B s and B d mixing by thetree-level exchange of some new heavy particles than if they only appear in a one-loopdiagram, as is the case for squarks and gluinos in the MSSM which only appear in theSUSY box diagram considered in the previous section. Indeed many of the approachessuggested in [14, 15, 16, 17] are based the tree-level exchanges of some new particles.A good example of this approach is the suggestion [15] that CP violation in B s mixingmay be due to the tree-level exchange of a new neutral spin-0 boson H = ( H + iA ) / √ H does not get a VEV atleading order so that H and A have approximately the same mass M H = M A . TheYukawa couplings of H to b, s quarks in the mass eigenstate basis are given [15] , − H ( y bs b R s L + y sb s R b L ) + H.c. (20)Tree-level H exchange gives rise to the following operator which contributes to B s mixing, y bs y ∗ sb M H ( b R s L )( b L s R ) . (21)One may readily extend this idea to allow for couplings of H to b, d quarks in themass eigenstate basis, − H ( y bd b R d L + y db d R b L ) + H.c. (22)7ree-level H exchange then gives rise to the following operator which contributes to B d mixing, y bd y ∗ db M H ( b R d L )( b L d R ) . (23)Following the approach of the previous section, the B q mixing matrix element can bewritten as the sum of the SM box diagrams and the Higgs tree-level exchange diagrams, M q = M , SM q + M , Higgs q (24)where from Eq.6 we identify h q e iθ q = M , Higgs q M , SM q , (25)where q = s, d . Using the matrix elements in [17] we find, h s e iθ s ≈ − . × y bs y ∗ sb (cid:18)
200 GeV M H (cid:19) h d e i ( θ d +2 β ) ≈ − . × y bd y ∗ db (cid:18)
200 GeV M H (cid:19) . (26)Comparing the Higgs predictions in Eq.26 to the best fit points in Eq.8, it is clear thatthe values of h s ∼ y bs ∼ y sb ∼ − , depending on the phases. Similarly, h d ∼ .
25 could be achieved,assuming Higgs masses of about 200 GeV, by y bd ∼ y db ∼ − , depending on the phases.In both cases the required Yukawa couplings are proportional to the Higgs mass.Of course such tree-level exchanges must be kept under control so that they don’tinduce too much flavour changing in other places where the constraints are more severe,especially involving the first two quark and lepton families as is the case for example with (cid:15) K . To overcome these challenges, it has been suggested that the hypothesis of MinimalFlavour Violation (MFV) could be extended to the two (or more) Higgs doublet model[17]. Here we shall follow a different approach, namely to use the idea of family symmetryto control the magnitude of the flavour changing Yukawa couplings in the framework ofa model with three families of SUSY Higgs doublets.The basic idea we shall discuss is very simple, namely that there are three SUSYHiggs families which form triplets under some family symmetry group, just like thethree families of quarks and leptons. The three SUSY Higgs families can be writtenas H ui , H di where the index i = 1 , , H u , H d . The idea is that the three families of quarks Q i , U ci , D ci and Higgs H ui , H di all transform as triplets ∼ family symmetry and leads to a successful description of all quark and lepton massesand mixing, including tri-bimaximal lepton mixing [28]. However only the dominantthird Higgs family couplings were considered [23] and flavour and CP violation arisingfrom the subdominant first and second family Higgs couplings were not considered. Inthe following we shall revisit this model, focussing on the quark sector, assuming exactlythe same particle content and symmetries as in [23], but including the effects of differentfield orderings and contractions not previously considered.In the considered model [23] the quarks and Higgs transform as triplets under a∆ family symmetry, which is broken by anti-triplet flavons φ , φ h , φ , φ which de-velop aligned VEVs, dropping phases in the following discussion for simplicity (they arerecovered in Appendix A), (cid:104) φ (cid:105) , (cid:104) φ h (cid:105) ∝ (cid:0) (cid:1) T , (cid:104) φ (cid:105) ∝ (cid:0) (cid:1) T , (cid:104) φ (cid:105) ∝ (cid:0) (cid:1) T . (27)The leading order down-type quark Yukawa couplings result from the following flavoncouplings, suppressing coupling constants,1 M d M h [ ( Qφ ) ( D c φ ) ( H d φ h ) + ( Qφ ) ( D c φ ) ( H d φ h ) + ( Qφ ) ( D c φ ) ( H d φ h ) + ( Qφ ) ( D c φ ) ( H d φ h ) ] , (28)with similar couplings generating the up-type and lepton Yukawa couplings. The vacuumalignments in Eq.27 then imply that only the third Higgs family couples to quarks andleptons with the leading order Yukawa matrix, defined by Y ij d Q i D cj H d , given as, Y ij d ∼ (cid:15) d (cid:15) d (cid:15) d (cid:15) d (cid:15) d (cid:15) d (cid:15) d (29)where the expansion parameter (cid:15) d ≈ .
15, where we have assumed [23] , (cid:104) φ h (cid:105) M h ≈ (cid:104) φ (cid:105) M d ∼ , (cid:104) φ (cid:105) M d ≈ (cid:15) d , (cid:104) φ (cid:105) M d ≈ (cid:15) d . (30) More precisely (cid:104) φ h (cid:105) M h ≈ (cid:104) φ (cid:105) M d ≈ . (cid:15) u ≈ .
05. When supplemented by additional corrections [23], such Yukawamatrices, after the third Higgs family develop their VEVs, have been shown to providea successful description of quark masses and mixing [29].The couplings in Eq.28 are controlled by the additional symmetries of [23] and aregenerated by the exchange of heavy messenger particles of mass M d and M h , as indicatedin Fig.1. The quark singlet messengers with masses M d are assumed to be lighter thanthe quark doublet messengers of mass M Q , and hence give the dominant contributionwith M u ≈ M d being responsible for (cid:15) d ≈ (cid:15) u . The assumed messenger sector alsoensures that φ h couples directly to the Higgs doublets with the group theory contractionsgiving the ∆ singlet as indicated by the subscript in ( H d φ h ) . This implies that onlythe third Higgs family couples to quarks.In Fig.2 we display other messenger diagrams, not considered in [23], allowed byall the symmetries of the model and involving the same flavon fields but contracteddifferently. These operators will contribute at a suppressed level due to the (assumed)heavier primed messenger masses. The diagram on the left in Fig.2 will lead to thefollowing additional operators, suppressed by two primed messenger masses,1 M d M (cid:48) d M (cid:48) h [ ( Qφ h ) ( D c φ ) ( H d φ ) + ( Qφ h ) ( D c φ ) ( H d φ ) + ( Qφ h ) ( D c φ ) ( H d φ ) + ( Qφ h ) ( D c φ ) ( H d φ ) ] , (31)with similar couplings for up-type quarks and leptons. The diagram on the right in Fig.2will lead to the following additional operators, suppressed by three primed messengermasses, 1 M (cid:48) d M (cid:48) h [ ( Qφ ) ( D c φ h ) ( H d φ ) + ( Qφ ) ( D c φ h ) ( H d φ ) + ( Qφ ) ( D c φ h ) ( H d φ ) + ( Qφ ) ( D c φ h ) ( H d φ ) ] , (32)with similar couplings for up-type quarks and leptons. Other messenger diagrams withtwo flavons along the Higgs line will not introduce new operator structures but willonly change the overall coefficient of the operators in Eq.31,32 with one power of M (cid:48) d being replaced by M (cid:48) h . Additional operator structures are also present in which thesinglet contractions (represented by the subscripts) are replaced by the one dimensionalrepresentations (cid:48) , (cid:48)(cid:48) present in ∆ , as discussed in Appendix A. However we assumethat messengers in other one dimensional or higher dimensional representations areabsent or very heavy.The new couplings in Eqs.31,32, together with the vacuum alignments in Eq.27, allowthe first and second Higgs families to couple to quarks and leptons with the leading order10 D c H d φ h φ φM h M d M d Figure 1: The messenger diagram for the down-type quark sector as assumed in [23] where φ representsany of the flavons φ , φ , φ allowed by the symmetry. Analogous diagrams are present in the up-typequark and lepton sectors. Q D c H d φ h φ φM d M h M d Q D c H d φ h φ φM h M d M d Figure 2: Other possible messenger diagrams involving the same flavons as in Fig.1. Additional dia-grams with two flavons along the Higgs line are also possible but we assume that diagrams with threeflavons along the same line are suppressed due to higher dimensional messenger representations beingabsent or very heavy. The primed messenger masses are assumed to larger than the unprimed ones,leading to the diagram on the left being suppressed compared to Fig.1, and the diagram on the rightbeing even more suppressed. Y ijkd Q i D cj H dk , given as, Y ij d ∼ (cid:15) d α d (cid:15) d α d (cid:15) d (cid:15) d (cid:15) d α d α h , Y ij d ∼ (cid:15) d α d (cid:15) d (cid:15) d α d α h , (33)where we have used the expansion parameters in Eq.30, together with the suppressionfactors, α d ≈ M d M (cid:48) d , α h ≈ M h M (cid:48) h . (34)Eq.33 shows that the second family Higgs H d couples more strongly to quarks thanthe first family Higgs H d whose effects can be approximately ignored. Assuming as aleading order approximation that only the third Higgs family develops VEVs, whichis reasonable since the VEVs are radiatively generated as a result of the large thirdfamily Yukawa coupling in Eq.29 of order unity, then we may diagonalise the quarkmass matrix resulting from Eq.29 by small angle rotations θ d ∼ (cid:15) d , θ d ∼ (cid:15) d , θ d ∼ (cid:15) d ,to go to the diagonal down-type quark mass basis d, s, b . The neutral component ofthe second family Higgs H d then has a Yukawa coupling matrix (the first equation inEq.33) which, when rotated to the down quark mass basis, takes the leading order form, Y ij d mass ∼ (cid:15) d (cid:15) d (cid:15) d α d (cid:15) d (cid:15) d (cid:15) d α d (cid:15) d (cid:15) d (cid:15) d α d α h , (35)in the convention where the rows correspond to d L , s L , b L and the columns correspondto d R , s R , b R . Clearly the flavour violating couplings of H d involving b R in Eq.35 arerelatively suppressed by a factor of α d compared to those involving b L , as can be under-stood from Fig.2. Note that there are no cancellations of the couplings in Eq.35 in the d, s, b basis since each operator in Eqs.28,31,32 has an independent order unity coeffi-cient which has been suppressed for clarity. The flavour violating couplings in Eqs.20,22can then be read off from Eq.35, where we identify H ≡ H d , y bs ∼ (cid:15) d α d α h , y sb ∼ (cid:15) d α d α h , y bd ∼ (cid:15) d α d α h , y db ∼ (cid:15) d α d α h . (36)With the phases included, as discussed in Appendix A, the couplings in Eq.36 leadto CP violation with, | y bd y ∗ db || y bs y ∗ sb | ∼ (cid:15) d ∼ × − , (37)which leads to the prediction, using Eq.26, h d h s ∼ , (38)12n good agreement with the best fit point I in Eq.8. This prediction is independent ofthe messenger masses and hence is equally valid for other messenger diagrams with twoflavons along the Higgs line.Assuming best fit point I the required Higgs mass may be related to α d , α h usingEq.26 , (cid:18) M H
200 GeV (cid:19) ≈ α d α h . (39)For example α d ∼ α h ∼ / M H ∼
400 GeV. This choice of couplingswould then imply, y sb ∼ (cid:15) d α d α h ∼ . × − , y db ∼ (cid:15) d α d α h ∼ . × − . (40)The hierarchical structure of flavour changing couplings in Eq.35 suppresses thecontribution of second family Higgs exchange to K mixing, which is described by avery small Yukawa coupling y sd ∼ y ds ∼ (cid:15) d α d α h ∼ × − , (41)where we have assumed α d ∼ α h ∼ /
3. The charged leptons are expected to haveflavour violating couplings with a similar structure to Eq.35, with the hierarchical struc-ture again leading to suppressed lepton flavour violating processes with y µe ∼ y eµ ∼ y sd ∼ y ds ∼ × − . (42)The diagonal coupling of the second family Higgs to muons is also quite suppressed, y µµ ∼ (cid:15) d α d α h ∼ × − , (43)leading to a negligible contribution to Br ( B s → µ + µ − ). For example, assuming (cid:15) d ≈ . α d ∼ α h ∼ / M H ∼
400 GeV using the results in [15] we estimate that thecontribution from second family Higgs exchange to the branching ratio is ∆ Br ( B s → µ + µ − ) ≈ × − which is negligible compared to the SM prediction of about 3 . × − .This contrasts with other non-standard Higgs models [15, 17] which tend to predict Br ( B s → µ + µ − ) larger than the SM value and close to the current experimental limitof about 5 . × − .Similarly the diagonal coupling of the second family Higgs to taus is, y ττ ∼ (cid:15) d α d α h ∼ . × − , (44)leading to a new contribution, assuming the same parameters as above, ∆ Br ( B s → τ + τ − ) ≈ . × − , which is also somewhat below the SM prediction of about 2 . × − , including a phase space suppression factor of 0.75 in both cases. However, a13ifferent choice of parameters could enhance the new physics contribution and make itcompetetive with the SM contribution.It is worth recalling that in the SM the decay matrix element Γ s is proportionalto the square of a tree-level amplitude proportional to V cb ∼ × − arising from W exchange. As remarked in the Introduction, the best fit value for Γ s is about twice aslarge as the SM prediction and it is challenging to understand this. For example, in thepresent model, second family Higgs exchange with mass M H ∼
400 GeV and couplingsin Eqs.40,41,42,43,44 give a contribution to Γ s which is completely negligible comparedto the SM W exchange contribution. The corresponding charged Higgs exchange con-tributions are also expected to be suppressed compared to the SM. For example, theinteraction y cb H +d c R b L involves a coupling, y cb ∼ y sb ∼ (cid:15) d α d α h ∼ . × − , (45)which is again smaller than V cb ∼ × − , with all charged Higgs couplings involvingat least this suppression, and the charged Higgs mass being heavier than the W mass.Finally we remark that the model in [23], as developed above, is based on ∆ familysymmetry combined with the E SSM [32] which predicts three complete SUSY Higgsfamilies as part of three complete 27 dimensional SUSY matter representations at theTeV scale (minus three right-handed neutrinos which get high see-saw scale masses sincethey carry no charges in this model). In addition there is a pair of SUSY doublets
L, L in conjugate representations which form a TeV scale Dirac mass as required for GUTscale unification. These may be absent if the requirement of GUT scale unification isrelaxed [33]. The E SSM can also be tested via its prediction of a Z (cid:48) N gauge boson withflavour conserving couplings [32, 33]. Following the anomalous like-sign dimuon charge asymmetry measured by the D0 collab-oration at the Tevatron collider we have discussed the implications of large CP violationin B d,s mixing for Supersymmetric (SUSY) Standard Models, focussing on those mod-els which allow a family symmetry and unification. For the Minimal SupersymmetricStandard Model (MSSM) we have seen from Eq.11 that it is only possible to account for B s mixing and CP violation at the expense of large squark mixing which would requirea new approach to family symmetry models. However, assuming such a framework, itseems very difficult to account for a significant amount of B d mixing and CP violation.In order to describe both B s and B d mixing and CP violation, as suggested by arecent global fit, we were led to consider SUSY models with Higgs fields transformingas triplets under a family symmetry. We have described a realistic such model basedon ∆ family symmetry combined with the E SSM in which tree-level exchange of thesecond Higgs family predicts B s and B d mixing and CP violation in the ratio h d /h s ∼ / (cid:15) K and the lepton sector, and is distinguished from otherHiggs models by predicting Br ( B s → µ + µ − ) consistent with the SM prediction. Acknowledgements
We would like to thank J. Flynn, D. King, C. Luhn and R. Zwicky for useful discus-sions. We acknowledge the support of a Royal Society Leverhulme Trust Senior ResearchFellowship and the STFC Rolling Grant ST/G000557/1.
AppendixA The origin of phases
In this Appendix we discuss the question of the origin of the phases in the flavourviolating Yukawa couplings in Eq.36. Following the approach in [26], we shall assumethat CP is preserved in the high energy theory but is spontaneously broken by the flavonVEVs in Eq.27 whose phases can be restored as follows, (cid:104) φ (cid:105) ∝ (cid:0) e iω (cid:1) T , (cid:104) φ h (cid:105) ∝ (cid:0) e i ( ω + φ h ) (cid:1) T , (cid:104) φ (cid:105) ∝ (cid:0) e iω e i ( ω + φ ) (cid:1) T , (cid:104) φ (cid:105) ∝ (cid:0) e iω e i ( ω + φ ) e i ( ω + φ ) (cid:1) T , (46)where the phases ω i can be removed by SU (3) transformations but not in the ∆ the-ory. Another difference between SU (3) and ∆ is that the discrete symmetry allowsnine distinct one dimensional representations [30], which, depending on the messengerrepresentations, allows many more new operators than those given in Eqs.31, 32, corre-sponding to the different singlet contractions ( × ) r where r = 1 , . . .
9. Here we restrictourselves to A type messengers in the first three one dimensional representations whichcan be obtained from the products × as follows, = 11 + 22 + 33 , (cid:48) = 11 + ω
22 + ω , (cid:48)(cid:48) = 11 + ω
22 + ω , (47)which are familiar from A [31] where ω = exp(2 πi/ , (cid:48) , (cid:48)(cid:48) representations permits new operators corresponding to one messenger in eachof the allowed one dimensional representations, where the invariant singlet is given by = (cid:48) × (cid:48)(cid:48) . Thus the operators in Eqs.31, 32 need to be augmented by others of the moregeneral form, ( · · · ) ( · · · ) (cid:48) ( · · · ) (cid:48)(cid:48) appearing in all possible combinations. Assumingthe messengers in the , (cid:48) , (cid:48)(cid:48) representations all have the same mass, the expansionparameters and predictions given previously will not change. In particular the flavourviolating couplings will have their magnitudes unchanged from the values quoted in15q.36, but their phases will all be different from each other in a complicated way whichdepends on the order unity couplings which control the precise linear combinations ofthe different operators which contribute to these couplings. In the limit that only singletoperators are permitted ( · · · ) ( · · · ) ( · · · ) it is easy to show that arg( y bs ) = arg( y sb )and arg( y bd ) = arg( y db ), even with the most general flavon VEVs in Eq.46, so the extraoperators of the form ( · · · ) ( · · · ) (cid:48) ( · · · ) (cid:48)(cid:48) are in fact necessary in order to allow newsources of CP violation in B s and B d mixing. References [1] R. N. Mohapatra et al. , Rept. Prog. Phys. (2007) 1757 [arXiv:hep-ph/0510213].[2] C. Amsler et al. [Particle Data Group], Phys. Lett. B (2008) 1.[3] V. M. Abazov et al. [The D0 Collaboration], arXiv:1005.2757 [hep-ex].[4] A. Lenz and U. Nierste, JHEP (2007) 072 [arXiv:hep-ph/0612167].[5] CDF Collaboration, Note 9015, Oct. 2007.[6] E. Barberio et al. [Heavy Flavor Averaging Group], arXiv:0808.1297 [hep-ex].[7] V. M. Abazov et al. [D0 Collaboration], arXiv:0904.3907 [hep-ex].[8] V. M. Abazov et al. [D0 Collaboration], Phys. Rev. Lett. (2008) 241801[arXiv:0802.2255 [hep-ex]].[9] L. Oakes (CDF Collaboration), talk at FPCP 2010,May 25-29, Torino, Italy.[10] W. Altmannshofer, A. J. Buras, S. Gori, P. Paradisi and D. M. Straub, Nucl. Phys.B (2010) 17 [arXiv:0909.1333 [hep-ph]]; A. J. Buras, PoS E PS-HEP2009 (2009) 024 [arXiv:0910.1032 [hep-ph]].[11] Y. Grossman, Y. Nir and G. Perez, Phys. Rev. Lett. (2009) 071602[arXiv:0904.0305 [hep-ph]].[12] Z. Ligeti, M. Papucci, G. Perez and J. Zupan, arXiv:1006.0432 [hep-ph].[13] C. W. Bauer and N. D. Dunn, arXiv:1006.1629 [Unknown].[14] A. Dighe, A. Kundu and S. Nandi, arXiv:1005.4051 [Unknown].[15] B. A. Dobrescu, P. J. Fox and A. Martin, arXiv:1005.4238 [Unknown].[16] C. H. Chen and G. Faisel, arXiv:1005.4582 [Unknown].1617] A. J. Buras, M. V. Carlucci, S. Gori and G. Isidori, arXiv:1005.5310 [Unknown].[18] C. H. Chen, C. Q. Geng and W. Wang, arXiv:1006.5216 [Unknown].[19] J. K. Parry, arXiv:1006.5331 [Unknown].[20] P. Ko and J. h. Park, arXiv:1006.5821 [Unknown].[21] K. Kawashima, J. Kubo and A. Lenz, Phys. Lett. B (2009) 60 [arXiv:0907.2302[hep-ph]]; A. Lenz, Phys. Rev. D (2007) 065006 [arXiv:0707.1535 [hep-ph]];O. Eberhardt, A. Lenz and J. Rohrwild, arXiv:1005.3505 [Unknown].[22] D. J. H. Chung, L. L. Everett, G. L. Kane, S. F. King, J. D. Lykken and L. T. Wang,Phys. Rept. (2005) 1 [arXiv:hep-ph/0312378].[23] R. Howl and S. F. King, Phys. Lett. B (2010) 355 [arXiv:0908.2067 [hep-ph]].[24] L. Randall and S. f. Su, Nucl. Phys. B (1999) 37 [arXiv:hep-ph/9807377].[25] M. Ciuchini, A. Masiero, P. Paradisi, L. Silvestrini, S. K. Vempati and O. Vives,Nucl. Phys. B (2007) 112 [arXiv:hep-ph/0702144].[26] S. Antusch, S. F. King and M. Malinsky, JHEP (2008) 068 [arXiv:0708.1282[hep-ph]]; S. Antusch, S. F. King, M. Malinsky and G. G. Ross, Phys. Lett. B (2009) 383 [arXiv:0807.5047 [hep-ph]]; L. Calibbi, J. Jones-Perez, A. Masiero,J. h. Park, W. Porod and O. Vives, Nucl. Phys. B (2010) 26 [arXiv:0907.4069[hep-ph]].[27] S. F. King and C. Luhn, arXiv:0912.1344 [Unknown]; S. F. King and C. Luhn,Nucl. Phys. B (2009) 269 [arXiv:0905.1686 [hep-ph]].[28] I. de Medeiros Varzielas, S. F. King and G. G. Ross, Phys. Lett. B (2007) 201[arXiv:hep-ph/0607045].[29] G. Ross and M. Serna, Phys. Lett. B (2008) 97 [arXiv:0704.1248 [hep-ph]].[30] E. Ma, Mod. Phys. Lett. A (2006) 1917 [arXiv:hep-ph/0607056].[31] E. Ma and G. Rajasekaran, Phys. Rev. D (2001) 113012 [arXiv:hep-ph/0106291].[32] S. F. King, S. Moretti and R. Nevzorov, Phys. Rev. D (2006) 035009 [arXiv:hep-ph/0510419]; S. F. King, S. Moretti and R. Nevzorov, Phys. Lett. B (2006) 278[arXiv:hep-ph/0511256]; S. F. King, S. Moretti and R. Nevzorov, Phys. Lett. B (2007) 57 [arXiv:hep-ph/0701064]; P. Athron, S. F. King, D. J. . Miller, S. Moretti,R. Nevzorov and R. Nevzorov, arXiv:0901.1192 [hep-ph]; P. Athron, S. F. King,D. J. Miller, S. Moretti and R. Nevzorov, arXiv:0904.2169 [hep-ph]; J. P. Hall andS. F. King, JHEP (2009) 088 [arXiv:0905.2696 [hep-ph]].1733] R. Howl and S. F. King, JHEP (2008) 030 [arXiv:0708.1451 [hep-ph]]; R. Howland S. F. King, Phys. Lett. B652