Flavour models with Dirac and fake gluinos
Emilian Dudas, Mark Goodsell, Lucien Heurtier, Pantelis Tziveloglou
CCPHT099.1213
Flavour models with Dirac and fake gluinos
Emilian Dudas a, , Mark Goodsell b, , Lucien Heurtier a, and Pantelis Tziveloglou c,d, a Centre de Physique Th´eorique, ´Ecole Polytechnique, CNRS, 91128 Palaiseau, France b Laboratoire de Physique Th´eorique et Hautes Energies, CNRS, UPMC Universit´e Paris VI,Boite 126, 4 Place Jussieu, 75252 Paris cedex 05, France c Theoretische Natuurkunde and IIHE, Vrije Universiteit Brussel, Pleinlaan 2,B-1050 Brussels, Belgium d International Solvay Institutes, Brussels, Belgium
Abstract
In the context of supersymmetric models where the gauginos may have bothMajorana and Dirac masses we investigate the general constraints from flavour-changing processes on the scalar mass matrices. One finds that the chirality-flipsuppression of flavour-changing effects usually invoked in the pure Dirac caseholds in the mass insertion approximation but not in the general case, and failsin particular for inverted hierarchy models. We quantify the constraints in severalflavour models which correlate fermion and scalar superpartner masses. We alsodiscuss the limit of very large Majorana gaugino masses compared to the chiraladjoint and Dirac masses, where the remaining light eigenstate is the “fake”gaugino, including the consequences of suppressed couplings to quarks beyondflavour constraints. [email protected] [email protected] [email protected] [email protected] a r X i v : . [ h e p - ph ] J un ontents B Models of Flavour 46
B.1 Abelian Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46B.2 Non-abelian extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
C B-meson mixing constraints 51
C.1 Mass insertion approximation . . . . . . . . . . . . . . . . . . . . . . . 52C.2 Decoupled first two generations . . . . . . . . . . . . . . . . . . . . . . 52
D Input 53 Introduction
Supersymmetric extensions of the Standard Model are arguably still the most plausi-ble ways to deal with the various mysteries of the Standard Model. The absence ofa new physics signature at LHC for the time being suggests, however, that we shouldseriously (re)consider non-minimal extensions compared to the minimal supersymmet-ric extension (MSSM) in all its various forms. Furthermore, it has been known sincethe early days of low-energy supersymmetry that flavour-changing processes set severeconstraints on the flavour structure of the superpartner spectrum in the MSSM. Forexample, the simplest models based on a single abelian flavoured gauge group, althoughproviding an approximate alignment mechanism for scalar mass matrices, still requirescalar partners heavier than at least 100 TeV. Both collider and flavour constraintsencourage us to search for non-minimal extensions with suppressed collider boundsand flavour-changing transitions. Supersymmetric extensions with a Dirac gauginosector [1–36] enter precisely into this category.Originally motivated by the preserved R-symmetry, which allows simpler supersym-metry breaking sectors [1, 2], and the possible connection with extra dimensions and N = 2 supersymmetry [6], it was subsequently noticed that Dirac gaugino masses havemany phenomenological advantages over their Majorana counterparts. For example,the Dirac mass is supersoft [5, 37–39], which naturally allows somewhat heavy gluinoscompared to the squarks [40–42]. Furthermore, it was argued later on that in this caseflavour-changing neutral current (FCNC) transitions are suppressed due to protectionfrom the underlying R-symmetry that lead to a chirality flip suppression [8]. It wasalso proved that the collider signatures of superpartner production are suppressed com-pared to the MSSM case due to the heaviness of the Dirac gluino and the absence ofseveral squark decay channels [43–46]. The main goal of this paper is to understandthe most general bounds from flavour physics when we allow Dirac gaugino masses inaddition to Majorana masses.We begin the paper in section 2 by giving the general expressions for the mesonmixing (∆ F = 2, i.e. a change of two units of flavour) FCNC processes in models withboth Dirac and Majorana gluino masses. We also introduce the notation used in the2emainder of the paper.In much of the literature where flavour constraints are discussed, in an attempt toprovide relatively model-independent bounds, scalar mass matrices are treated in theso-called mass insertion approximation, in which scalars are almost degenerate withsmall off-diagonal entries. Indeed, where flavour constraints in Dirac gaugino modelshave been considered, the mass insertion approximation was also used [8, 47]. Hencewe first provide an updated discussion of this case in section 3, with in addition boundsfor differing ratios of Dirac and Majorana gluino masses, with no restrictions providedby the R-symmetry.However, in particular in light of bounds on superpartner masses, the mass inser-tion approximation is actually rather difficult to realise in any flavour model. We aretherefore led to consider general flavour models/scenarios which go beyond this ap-proximation in section 4. An important result is that, surprisingly, we find that thedramatic chirality-flip suppression of [8] is at work only in a small number of cases,whereas in the general case the suppression is much milder and in certain cases theMajorana case is less constrained. Our main working assumption is that the flavoursymmetry explaining the fermion masses and mixings governs simultaneously the su-perpartner spectrum. We find that the simplest single U (1) flavour models do still needheavy scalars. For the case of two U (1)’s we find the unusual feature that, in someregions of parameter space, Dirac models are more constrained than their Majoranacounterparts, due to cancellations occurring in the latter case. We also investigatethe inverted hierarchy case and one example of nonabelian flavour symmetries, discussthe K (and B meson constraints in appendix C) and compare them with their MSSMcounterpart models.As a refreshing aside, in section 5 we consider also the unusual case where thelightest adjoint fermions couple in a suppressed way to the quarks, due to their verylittle gaugino component. This happens when the Majorana gaugino mass is muchbigger than the Dirac and the adjoint fermion masses. This can occur for relativelylight squarks and gluinos or for intermediate scale values. In both cases light adjoint3ermions have suppressed couplings to quarks, a case we refer to as “fake gluino” .The first case can lead to the unusual feature of experimentally accessible squarks, butlong-lived (fake) gluinos. The intermediate scale case is interesting from the viewpointof gauge coupling unification. In this case, radiative corrections lead to heavy scalarsand therefore the scenario is similar in spirit to split supersymmetry [49], but withsuppressed “fake gluino/gaugino” couplings to quarks and to higgs/higgsinos. Sincethe radiative stability of this scenario requires some particular high-energy symmetries,it has specific features distinguishing it from standard split supersymmetry and otherrelated scenarios [6, 50, 51] which we shall discuss.Finally, as a note to the concerned reader, in this paper we largely only discuss∆ F = 2 constraints arising from box diagrams involving gluinos. In principle, thereare also diagrams that contribute at two loops from processes involving the octet scalarpartners of the Dirac gluino, which were discussed in [10] and shown to be small; sim-ilarly we do not include subdominant contributions to the box diagrams coming fromelectroweak gauginos/higgsinos because they do not add qualitatively to the discussion.In addition, there are also constraints coming from ∆ F = 1 processes such as b → sγ , µ → eγ and electric dipole moments. These have been discussed in the context of theMRSSM and the mass insertion approximation [8, 47]. However, with the exception of b → sγ these are all dependent on the Higgs structure of the theory, and not only onthe squark/quark mass matrices, since the Dirac gaugino paradigm allows many pos-sible Higgs sectors [4, 5, 8, 21, 33, 36]. For example, if we insist that the model preservesan exact R-symmetry, then these processes are suppressed so much as to be negligible;but they become relevant if we allow the Higgs sector to break R-symmetry [33]. Thusit is not possible to describe bounds on these in a model-independent way, and we re-frain from attempting to do so. For the case of b → sγ , the constraints are genericallyweaker than the ∆ F = 2 case, and moreover the expressions are the same in boththe Majorana and Dirac cases, since they do not involve a chirality flip; they are thusirrelevant for this paper. We acknowledge K. Benakli and P. Slavich for suggesting the name during collaboration on arelated work [48]. Neutral meson mixing in supersymmetry withDirac gauginos
In recent years, very precise measurements of observables in flavour violation processeshave been made [52] while the Standard Model contribution to some of these processes isnow being known with reasonable accuracy [53]. This results in very strong restrictionson the flavour structure of theories beyond the SM.Some of the strongest constraints arise from neutral meson mixing systems, in par-ticular the neutral K -, B d -, B s - and D - meson systems [54]. An exact theoretical com-putation of these processes is particularly difficult due to unresolved non-perturbative,strong-interaction effects. The general strategy is to compute the amplitude betweenthe valence quarks in the full perturbative theory, then match the amplitude to an ef-fective theory of four-fermion contact interactions. Contact with neutral meson mixingis achieved by estimating the matrix elements between initial and final states, typicallyby use of PCAC [55] and lattice QCD techniques. Within the context of MSSM, the dominant contribution to neutral meson mixingcomes from gluino-squark box diagrams (see e.g. figure 6 for the Kaon system). In thefollowing, we expand the standard computation (see app. A) to include both Majoranaand Dirac gluino masses. In particular
L ⊃ −
12 (
M λ a λ a + M χ χ a χ a + 2 m D χ a λ a + h.c. ) −√ g s (cid:104) ˜ d ∗ Lxi T axy λ aα d Lyiα − ˜ d Rxi T a ∗ xy λ aα d cRyiα (cid:105) + h.c. , (2.1)where λ aα is the Majorana gaugino, χ aα its Dirac partner and T axy , d Li , ˜ d Li are the SU(3)generators, the quarks and the squarks of generation i respectively . The mass matrixis diagonalised by performing an orthogonal transformation and then a phase shift to Our conventions are the ones from [56]. λ a χ a = R ψ a ψ a . (2.2)In basis ψ i , eq. (2.1) becomes L (cid:48) ⊃ −
12 ( M ψ a ψ a + M ψ a ψ a + h.c. ) (2.3) − √ g s (cid:104) ˜ d ∗ Lxi T axy ( R ψ aα + R ψ aα ) d Lyiα − ˜ d Rxi T a ∗ xy ( R ψ aα + R ψ aα ) d cRyiα (cid:105) + h.c. The four-fermion effective action is given by [57, 58] H K = (cid:88) i =1 C i Q i + (cid:88) i =1 ˜ C i ˜ Q i , (2.4)where the conventionally chosen basis of the dimension six operators is (now in Diracnotation) Q = d x γ µ P L s x d n γ µ P L s n ,Q = d x P L s x d n P L s n ,Q = d x P L s n d n P L s x ,Q = d x P L s x d n P R s n ,Q = d x P L s n d n P R s x , (2.5)6 Q , , are the R-projection analogues of Q , , and C = ig s W K W L (cid:18) | R r | | R q | ˜ I + 19 M r M q R ∗ r R q I (cid:19) W † K W † L ,C = ig s W K W L I W † K W † L M r M q R r R q ,C = − ig s W K W L I W † K W † L M r M q R r R q ,C = ig s W K W L (cid:18) M r M q R ∗ r R q I − | R r | | R q | ˜ I (cid:19) W † K W † L − ig s W K W L ˜ I W † K W † L | R r | | R q | ,C = ig s W K W L (cid:18) M r M q R ∗ r R q I + 59 | R r | | R q | ˜ I (cid:19) W † K W † L − ig s W K W L ˜ I W † K W † L | R r | | R q | , (2.6)˜ C = ig s W K W L (cid:18) | R r | | R q | ˜ I + 19 M r M q R r R ∗ q I (cid:19) W † K W † L , ˜ C = ig s W K W L I W † K W † L M r M q R ∗ r R ∗ q , ˜ C = − ig s W K W L I W † K W † L M r M q R ∗ r R ∗ q . (2.7)where the Feynman integrals are I = I ( M r , M q , m K , m L ), ˜ I = ˜ I ( M r , M q , m K , m L )and summation over r, q = 1 , K, L = 1 , ... , W IJ is the unitarymatrix that diagonalises the down squark mass-squared matrix m d in a basis wherethe down quark mass matrix is diagonal. Matrix W is given in terms of the squarkdiagonalising matrix Z and the quark diagonalising matrices V L , V R by W = V † L Z LL V † L Z LR V † R Z RL V † R Z RR (2.8)as detailed in appendix A.1.In the simple case that the mass of the gaugino is Dirac-type ( M = M χ = 0),we obtain M = M = m D , R = − iR = √ , so that (cid:80) | R r | | R q | ˜ I = ˜ I and See appendix A.3 for explicit definitions. M r M q R ∗ r R q I = (cid:80) M r M q R r R q I = (cid:80) M r M q R ∗ r R ∗ q I = 0. The effective coeffi-cients simplify to C = ig s W K W L ˜ I W † K W † L , C = 0 , C = 0 ,C = − ig s W K W L ˜ I W † K W † L − ig s W K W L ˜ I W † K W † L ,C = ig s W K W L ˜ I W † K W † L − ig s W K W L ˜ I W † K W † L , ˜ C = ig s W K W L ˜ I W † K W † L , ˜ C = 0 , ˜ C = 0 , (2.9)The derivation of the effective action for the mixing between the other neutralmesons is the same as above. Therefore, the corresponding effective actions are givenby simple substitution: H B d = H K ( s → b, → , → , H B s = H K ( d → s, s → b, → , → , → , → , H D = H K ( d → u, s → c, W → W u ) . (2.10) Flavour violation in the Kaon mixing system is typically parametrised by the real andimaginary part of the mixing amplitude. These two are related to the mass differencebetween K L and K S and the CP violating parameter as∆ m K = 2Re (cid:104) K |H K | K (cid:105) , | (cid:15) K | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Im (cid:104) K |H K | K (cid:105)√ m K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.11)which have both been experimentally measured with great accuracy [52]. Their sizesets strict bounds on the amount of flavour violation allowed by new physics. In orderto compute these observables we need to extract the hadronic matrix elements of theoperators in (2.4). They are first derived in the Vacuum Saturation Approximation8VSA), (cid:104) K | Q | K (cid:105) V SA = 13 m K f K , (cid:104) K | Q | K (cid:105) V SA = − (cid:18) m K m s + m d (cid:19) m K f K , (cid:104) K | Q | K (cid:105) V SA = 124 (cid:18) m K m s + m d (cid:19) m K f K , (cid:104) K | Q | K (cid:105) V SA = (cid:34)
124 + 14 (cid:18) m K m s + m d (cid:19) (cid:35) m K f K , (cid:104) K | Q | K (cid:105) V SA = (cid:34)
18 + 112 (cid:18) m K m s + m d (cid:19) (cid:35) m K f K . (2.12)Since only strong interactions are involved, we get identical expressions for the ‘R-projection’ version of the first three operators. The ratio of the exact over the VSAresult for each of the five operators above is parametrised by the “bag” factors B i , i = 1 , ..., (cid:104) K |H K | K (cid:105) Re (cid:104) K |H SMK | K (cid:105) = C ∆ m K , Im (cid:104) K |H K | K (cid:105) Im (cid:104) K |H SMK | K (cid:105) = C (cid:15) K . (2.13)Flavour violation in B q meson systems is parametrised in a similar way, by themodulus and the phase of the mixing amplitude: (cid:104) B q |H B q | B q (cid:105)(cid:104) B q |H SMB q | B q (cid:105) = C B q e iφ Bq , (2.14)where the B q -meson hadronic matrix elements are obtained by eq. (2.12) by substi-tution ( m K , f K , m s , m d ) → ( m B q , f B q , m b , m q ) and the corresponding bag factors (seeappendix D). Finally, there exists a similar parametrisation of the D-meson mixing CPconserving and violating parameters (as in e.g. [59]) which we do not explicitly describehere, and will be mentioned in the appropriate place in section 4.9 .3 Flavour patterns The stringent experimental bounds on flavour violation processes require that contri-butions from extensions of the Standard Model be highly suppressed. This is typicallyachieved by employing particular patterns for the flavour structure of the BSM theory.In the following we describe how flavour violation is parametrised in the patterns thatwill appear throughout the paper.
Degeneracy - mass insertion approximation
One way to suppress flavour violation is to assume that the masses of the squarks arealmost degenerate, m I = m q + δm I , where m I are the squark mass eigenvalues and δm I are small enough deviations from an “average” squark mass-squared m q , I = 1 , ..., δm I and use of the unitarity of the W matricesdelivers (for I (cid:54) = J , L (cid:54) = N ) W IK W LM I ( m K , m M ) W † KJ W † MN = m IJ m LN I ( m q , m q , m q , m q ) + ... (2.15)where m is the squark squared mass matrix in the basis where the quark mass matrixis diagonal. Flavour violation in this scheme is parametrised by the small ratio of theoff-diagonal elements m IJ over the average squark mass δ L ( R ) L ( R ) ij ≡ m − q m i ( i +3) j ( j +3) . Hierarchy
A slightly different notation is used in the case of hierarchical squark masses where thesquarks of first and second generation are much heavier than those of the third so thattheir contribution to the box diagrams is negligible. Further below we will considersuch flavour patterns, in the simpler case of absent left-right mixing. In this case, onecan parametrise flavour violation processes by ˆ δ Lij ≡ W Li W L † j , ˆ δ Rij ≡ W Ri W R † j , where W Lij and W Rij are the block diagonal matrices of (2.8). The reasoning behind this choicecan be illustrated by the following example [60]. Let us assume that ˜ b L is much lighter10han the other squarks. Then W K W L I ( m K , m L ) W † K W † L (cid:39) (ˆ δ L ) I ( m b L , m b L ) where (ˆ δ L ) = W L W L † . (2.16) Alignment
An alternative to degeneracy or hierarchy for the suppression of flavour violating pro-cesses is to consider that the squark mass-squared matrix is simultaneously diagonalisedwith the quark mass matrix [61]. In this “alignment” flavour pattern, the suppressionappears because W Lij = V L † ik Z LLkj ∼ δ ij and W Rij = V R † ik Z RRkj ∼ δ ij . In this framework,we can take the squark masses to be of the same order m ˜ q but not degenerate. If weignore left-right mixing, we obtain e.g. for the left sector W L i W L j I ( m i , m j ) W L † i W L † j (cid:39) (˜ δ L ) I ( m q , m q ) where ˜ δ L = max k ( W L k W L † k ) (2.17)and similarly for ˜ δ R . In the following we present the bounds for representative points in the gluino parameterspace (
M, m D , M χ ). We focus on near degenerate squarks; hierarchical and alignmentflavour patterns are discussed in section 4. In this approximation, coefficients (2.6) and112.7) of the general effective action for the Kaon mixing system become C = − α s m q (cid:18) | R r | | R q | ˜ f + 19 √ x r x q R ∗ r R q f (cid:19) ( δ LL ) ,C = − α s m q √ x r x q R r R q f ( δ RL ) ,C = 16 α s m q √ x r x q R r R q f ( δ RL ) ,C = − α s m q (cid:104)(cid:16) √ x r x q R ∗ r R q f − | R r | | R q | ˜ f (cid:17) δ LL δ RR − | R r | | R q | ˜ f δ LR δ RL (cid:105) ,C = − α s m q (cid:104)(cid:16) √ x r x q R ∗ r R q f + 59 | R r | | R q | ˜ f (cid:17) δ LL δ RR − | R r | | R q | ˜ f δ LR δ RL (cid:105) , ˜ C = − α s m q (cid:18) | R r | | R q | ˜ f + 19 √ x r x q R r R ∗ q f (cid:19) ( δ RR ) , ˜ C = − α s m q √ x r x q R ∗ r R ∗ q f ( δ LR ) , ˜ C = 16 α s m q √ x r x q R ∗ r R ∗ q f ( δ LR ) , (3.1)while for the B d and B s system we replace δ → δ and δ → δ accordingly. Inthe expressions above, x k = M k /m q with M k the gluino mass eigenstate and we havereplaced, according to appendix A.3 notations with mass scale m q , I ( M r , M q , m q , m q , m q , m q ) = i π m q f ( x r , x q , , ,
1) = if π m q , ˜ I ( M r , M q , m q , m q , m q , m q ) = i π m q ˜ f ( x r , x q , , ,
1) = i ˜ f π m q . (3.2)The bounds on d ↔ s transitions from the Kaon system are proven to be the mostrestrictive and therefore we will focus on them; we discuss the comparison of bounds inappendix C. We allow the SUSY contribution to ∆ m K to be as large as the experimental12ound; however, the contribution to (cid:15) K is restricted by the SM calculation [53]. Ouranalysis takes into account NLO corrections to the effective Hamiltonian [62]; as forthe parameter inputs, they are given in appendix D . In tables 1 and 2, we update the bounds on flavour violation parameters for the MSSMwith a Majorana gluino, for an average gluino mass of 1 . δ ) and c Im( δ ), with c (cid:39)
25. As seen in the tables, the K − K system sets powerful constraints in the size of flavour violation. For example, for m ˜ q = 2 M ˜ g = 3 TeV the best case is √ Re δ (cid:46) √ Im δ is around 25 timessmaller. m ˜ q [GeV] δ LL (cid:54) = 0 δ LL = δ RR (cid:54) = 0 δ LR = δ RL (cid:54) = 0750 0.211 0.002 0.0041500 0.180 0.002 0.0142000 0.157 0.003 0.008Table 1: Majorana gluino bounds for M ˜ g = 1500 GeV. By δ AB we denote (cid:113) | Re ( δ AB ) | and c (cid:113) | Im ( δ AB ) | . m ˜ q [GeV] δ LL (cid:54) = 0 δ LL = δ RR (cid:54) = 0 δ LR = δ RL (cid:54) = 0750 0.192 0.002 0.0051500 0.374 0.003 0.0112000 0.240 0.003 0.019Table 2: Majorana gluino bounds for M ˜ g = 2000 GeV. By δ AB we denote (cid:113) | Re ( δ AB ) | and c (cid:113) | Im ( δ AB ) | . Higher order terms in B and B of (2.12) have been dropped [63]. Saturating the 2 σ deviation in (cid:15) SMK . .2 Dirac gluino As has already been mentioned in the introduction, flavour violation for quasi-degeneratesquarks is suppressed if the gluino is of Dirac type, especially in the large gluino masslimit. This is true both because of the absence of the chirality-flip processes and be-cause we are allowed to increase a Dirac gluino mass over the squark masses withoutaffecting naturalness as much as in the Majorana case. These properties lead to a signif-icant relaxation of the bounds from ∆ m K and (cid:15) K , as seen in figure 1 for representativevalues of δ AB .However, despite the order of magnitude (or better) improvement over the Majoranacase, the bounds on (cid:15) K still require a relatively high flavour degeneracy or that theflavour violating masses in the squark matrix be real. For example, for a 6 TeV gluinoand average squark mass of 1 TeV, (cid:112) | Im ( δ LL ) | can be as high as ∼ (cid:15) K bounds for reasonable values of gluino and squarkmasses. We will also notice that in many other flavour models, Dirac gauginos do notenjoy the suppression of flavour violation with respect to Majorana ones that is seenhere. The mass terms of eq. 2.1 allow for non-standard gluinos, when all M , m D and M χ arenon-zero. One such scenario is when M (cid:29) M χ , m D and corresponds to the interestingcase of a light gluino with a suppressed squark - quark vertex, which we call “fakegluino”. In section 5 we explore this possibility in more detail.In this limit we obtain much lower bounds on flavour violation parameters withrespect to MSSM with Majorana gluino. In order to illustrate the point, we consider m D = M χ = M/
10. Even for an order of magnitude difference between M and m D , M χ , we obtain no restrictions for the size of flavour violation from effective operator Q , where δ LL (cid:54) = 0, δ RR = δ LR = δ RL = 0. For other combinations, we obtain results14 = 0.03 ∆ = 0.1 ∆ = 0.3
500 600 700 800 900 1000200040006000800010000 m q Ž @ GeV D M D @ G e V D ∆ = 0.03 ∆ = 0.1
500 600 700 800 900 1000200040006000800010000 m q Ž @ GeV D M D @ G e V D Figure 1:
Contour plots in parameter space m ˜ q - m D for purely Dirac gluino ( M = M χ =0). Left: δ LL = δ RR = δ , δ LR = δ RL = 0. Right: δ LL = δ RR = δ LR = δ RL = δ .Along the contours ∆ m K = ∆ m exp K (for δ AB = (cid:113) | Re ( δ AB ) | ) and (cid:15) K = (cid:15) exp K (for δ AB = c (cid:113) | Im ( δ AB ) | ). given in tables 3 and 4. m ˜ q [GeV] δ LL = δ RR (cid:54) = 0 δ LR = δ RL (cid:54) = 0750 0.013 0.0281500 0.014 0.0292000 0.014 0.030Table 3: “Fake” gluino bounds for M ˜ g = 1500 GeV. By δ AB we denote (cid:112) | Re ( δ AB ) | and c (cid:112) | Im ( δ AB ) | .In this case, the quark/squark coupling of the fake gluino is suppressed with respectto the standard one by R ∼ m D M = 0 . R for the same lightest gluino mass,leading to bounds reduced by R ∼ .
01. However, we observe from the bounds intables 3 and 4 that the suppression is much less dramatic, of the order 0 .
1. The reason15 ˜ q [GeV] δ LL = δ RR (cid:54) = 0 δ LR = δ RL (cid:54) = 0750 0.017 0.0371500 0.018 0.0382000 0.018 0.039Table 4: “Fake” gluino bounds for M ˜ g = 2000 GeV. By δ AB we denote (cid:112) | Re ( δ AB ) | and c (cid:112) | Im ( δ AB ) | .is that it is not the light but actually the heavy eigenstate that dominates the boxintegral!This can be seen by comparing, for example, the loop integral contribution fromthe chirality-flip process: √ x r x q R ∗ r R q f ( x r , x q ) (cid:39) x f ( x , x ) + x (cid:18) x x (cid:19) f ( x , x ) + 2 √ x x (cid:18) x x (cid:19) f ( x , x )= xy f ( x/y, x/y ) + xy f ( x, x ) + 2 xy √ yf ( x/y, x ) (3.3)where x (cid:39) x , x (cid:39) x with x ≡ M g /m q , y ≡ x x (for the lightest gluino eigenstate)and we have replaced R (cid:39) R (cid:39) x x . Since f ( x/y, x ) ∼ y log y, f ( x/y, x/y ) ∼ y x , the dominant contribution comes from the heavy gluino term x f ( x , x ) and isgiven by √ x r x q R ∗ r R q f ( x r , x q ) (cid:39) x M D M . (3.4)The parametric scaling of the bound on δ AB is then | δ AB Majorana || δ AB fake gluino | ∼ Mm D (3.5)which is much less than the naive scaling of M m D .16 Beyond the mass insertion approximation
Having established in the previous section that the bounds from (cid:15) K do not allow flavour-generic models at LHC-accessible energies even in the case of Dirac gaugino masses, weare led to the conclusion that it is likely that we either require an accidental suppressionof the mixing between the first two generations or we must impose some additionalstructure on the squark mass matrices. It is therefore important to consider flavourmodels. However, in doing so we invariably find that the mass insertion approximationis no longer valid: in fact, it is hard to find any models in which it would actuallyapply. Hence, in this section we shall investigate the consequences - and the generalbounds - when we go beyond the mass insertion approximation in the context of Diracgauginos.One of the most important things that we find in the general case is that the much-vaunted suppression of ∆ F = 2 FCNC processes is in general much less marked; in fact,for certain specific cases the Majorana case is actually less suppressed! We explain thisin section 4.1. In the remainder of the section we then discuss specific flavour modelsto illustrate the different types of behaviour. We shall consider: • The simple case of non-degenerate but same order of magnitude squark masses,where alignment applies. • A simple flavour model realising such a spectrum. • The general case of an inverted hierarchy between the first two squark generationsand the third, `a la reference [60]. In addition to changing the gluino masses toDirac type, we will update the bounds with the latest flavour data and also takeinto account the LHC bounds on squark and gaugino masses. • Models where in addition to the first two generations of squarks, the third gen-eration of right-handed squarks is also heavy. These models provide a minimumof extra coloured particles available to the LHC. • A flavour model realising the above, as given in [64] but with Dirac gaugino17asses. This model highly restricts the allowed flavour violation by imposingadditional symmetries upon the first two generations.In the following, we ignore left - right squark mixing and define W Lij = W ij and W Rij = W i +3 j +3 for i, j (cid:54)
3. We also define˜ f AB = W A i W B j ˜ I ( m D , m Ai , m Bj ) W A † i W B † j , (4.1)where A = L, R . Then the effective action (2.4) can be written as H DiracK = C Q + ˜ C ˜ Q + C Q + C Q (4.2)where the Dirac coefficients (2.9) are written as C = 1136 ig s ˜ f LL , ˜ C = 1136 ig s ˜ f RR , C = − ig s ˜ f LR , C = 59 ig s ˜ f LR (4.3) In reference [8], it was argued that the absence of chirality-flip processes in the caseof Dirac gluinos leads to a suppression in the contribution to the box diagram by afactor x ≡ M g /m q as the Dirac mass becomes larger than the squark masses. In thefollowing we show that this is generally not true beyond mass insertion approximationand even when it is, the flavour bounds are often relaxed by a factor of few rather thanbeing parametrically reduced.This can be immediately seen by taking the large x limit in the loop functionsthat appear in the coefficients (2.6) of the general expression (2.4) for ∆ F = 2 FCNCprocesses. Taking for simplicity equal masses m ˜ q for the squarks in the loop, thesefunctions are (see app. A.3): M g I ( M g , M g , m q , m q ) ≡ i π m q xf ( x ) = i π m q (cid:20) x ( x − − x ( x + 1)ln( x )(1 − x ) (cid:21) , ˜ I ( M g , M g , m q , m q ) ≡ i π m q ˜ f ( x ) = i π m q (cid:20) x − x ln( x ) − − x ) (cid:21) . (4.4)18unction ˜ f ( x ) appears in both Dirac and Majorana cases while xf ( x ) appears onlyin the Majorana case, corresponding to the chirality-flip process. Notice that xf ( x ) isalways positive, and ˜ f ( x ) always negative; moreover they have broadly similar valuesexcept near x = 0; for example f (1) = 1 / , ˜ f (1) = − /
3. As x → ∞ the ratiobetween them tends to − ln( x ) + 2, which is not the aforementioned enhancement by afactor of x .This can be understood in the following way. Following the reasoning of [8], inte-grating out the heavy gluino generates effective operators1 M ˜ g ˜ d ∗ R ˜ s ∗ L d R s L , M g ˜ d L ∂ µ ˜ s ∗ L d L γ µ s L , (4.5)the first of these being the chirality-flip process forbidden in the Dirac case. In themass insertion approximation, the flavour changing loop diagram is then as in figure2(a) and gives ( Q i refers to the four-fermion effective operators of sec. 2) L eff ⊃ Q i ( m ) M g (cid:90) d q q − m q ) ∼ Q i δ M g (4.6)for the chirality-flip case and L eff ⊃ Q i ( m ) M g (cid:90) d q q ( q − m q ) ∼ Q i m q M g δ (4.7)in the same chirality case, in line with the claim in [8]. The insertion of operators ofthe form m ˜ q ∗ ˜ q as effective vertices is of course only valid in the limit m (cid:28) m q ;however, as we shall see below in section 4.2, the above behaviour of the integrandscan also arise in certain cases beyond mass insertion approximation, where there isapproximate unitarity of a submatrix of the squark rotations leading to cancellationsbetween diagrams. However, in all other cases we instead have diagrams like that offigure 2(b), which gives L eff ⊃ Q i W M g (cid:90) M ˜ g d q q − m q ) ∼ Q i W M g ln M g m q (4.8)19igure 2: Loop diagrams in the effective theory where the gaugino has been integratedout. In figure (a) the mass insertions are shown, whereas in figure (b) the mass-insertionapproximation is inappropriate.in the chirality-flip case and L eff ⊃ Q i W M g (cid:90) M ˜ g d q q ( q − m q ) ∼ Q i W M g (4.9)in the same chirality case, where we needed to use the cutoff of M ˜ g in the integrals .This is exactly the behaviour that we find born out in the amplitudes and explainswhy in generic flavour models the Dirac case will not provide a parametric suppressionof the flavour-changing bounds.The logarithmic, instead of a linear suppression for the Dirac amplitude has thenstriking consequences. In the case that the contribution from same-chirality andchirality-flip amplitudes is comparable for reasonable values of x , the flavour bounds onDirac gluinos can be proven more strict than those on Majorana, because in the latterthere can exist cancellations between the same- and flipped- chirality amplitudes. Letus consider the impact that this has on bounds, by taking the ratio between the valueof the Wilson coefficients C i for purely Majorana gauginos C Mi and for purely Dirac C Di . For a given contribution to the integrand (i.e. for the same values of K, L ) inequation (2.6) and taking for simplicity equal masses for the squarks in the loop (while Note that if we define m q K = m q (1 + δ K ), sum the integrals of the above form (4.8) and (4.9)over W K W ∗ K W L W ∗ L we and expand to leading order in δ K we recover (4.6) and (4.7). C M C D =1 + 411 xf ( x, x, f ( x, x,
1) = −
411 ln( x ) + 1911 + O ( x − ln ( x )) ,C M C D =7 ln( x ) −
13 + O ( x − ln ( x )) ,C M C D = −
15 ln( x ) + 75 + O ( x − ln ( x )) . (4.10)For arbitrarily large values of x the Majorana case will have a larger contribution, butfor reasonable values, up to x = O (100), only C is actually enhanced compared tothe Dirac case (for C we would require gluinos about 40 times heavier than squarksto obtain a relative suppression).Finally, we note that the cancellation between the amplitudes can also be relevantwhen the the linear enhancement of the chirality-flip contribution applies, i.e. when f AB and ˜ f AB are proportional to ( ∼ ) I . This is the case when squarks are quasi-degeneratebut also in certain cases beyond the mass insertion approximation for very particularsquark matrix configurations as we shall find below. In this case, for moderate valuesof x the cancellation plays a role: C M C D → xf ( x, x, f ( x, x,
1) = 111 (47 − x −
12 ln( x )) + O ( x − ln ( x )) ,C M C D → x −
62 + 21 ln x + O ( x − ln ( x )) ,C M C D →
110 (28 − x − x ) + O ( x − ln ( x )) (4.11)We observe that the Majorana contribution is smaller than the Dirac for C ( x (cid:46) C ( x (cid:46)
15) while the Dirac is only suppressed by a factor of 10 for C ( x (cid:39) C ( x (cid:39) .2 Alignment In the previous section we examined how flavour constraints in the mass insertionapproximation are affected by a generalised gluino spectrum. However, flavour modelsoften do not lead to a near degeneracy of the squarks’ masses but to different flavourpatterns such as alignment or hierarchy, as mentioned in section 2.3. Moreover, oneexpects non-degeneracy to arise from running: there will always be a split between atleast the first two generations and the third due to the larger Yukawa couplings. Ittherefore makes sense to consider models that can suppress flavour constraints evenwithout requiring degeneracy of the squarks’ masses.
Alignment in the left sector
Alignment is typically obtained in flavour models of additional horizontal U (1) sym-metries [65]. In a minimal representative of such models there is only one horizontal U (1) symmetry, under which the quark superfields are charged with charges X as X [ Q i ] = (3 , , , X [ U i ] = (3 , , , X [ D i ] = (3 , , . (4.12)If we neglect D-term contributions to the squark masses, the order of magnitude struc-ture of the squark mass matrices (before any quark rotations) is m d L ∼ m F (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) , m d R ∼ m F (cid:15) (cid:15)(cid:15) (cid:15) . (4.13)where (cid:15) is a small number, the parameter of U (1) symmetry breaking. Throughoutthis section, (cid:15) = λ , where λ (cid:39) .
22 is the Cabibbo angle. In this flavour model, the In all flavour abelian models in what follows, ∼ means order of magnitude only and not a precisenumber. V dL ∼ (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) , V dR ∼ (cid:15) (cid:15)(cid:15) (cid:15) (4.14)and the squark diagonalising matrices (in the basis where the quarks are diagonal) areapproximated by W L ∼ V d † L and W R ∼ V d † R . Therefore, with this particular choice of U (1) charges, the left-squarks sector exhibits alignment while the right-squarks sectordoes not.We can estimate flavour violation in ∆ m K in the leading order in (cid:15) , by focusing at˜ f LR = (cid:15) (cid:104) ( m R − m R ) (cid:16) ˜ I ( m L , m R , m R ) − ˜ I ( m L , m R , m R ) (cid:17) , +( m L − m L ) ˜ I ( m L , m L , m R ) (cid:105) + O ( (cid:15) ) ∼ i π (cid:15) m q ˜ f ( x ) , (4.15)˜ f LL = (cid:15) ( m L − m L ) ˜ I ( m D , m L , m L ) + O ( (cid:15) ) ∼ i π (cid:15) m q ˜ f ( x ) , ˜ f RR = (cid:15) (cid:104) (cid:88) i ˜ I ( m Ri , m Ri ) − I ( m R , m R ) − I ( m R , m R ) + 2 ˜ I ( m R , m R ) (cid:105) + O ( (cid:15) ) ∼ i π (cid:15) m q ˜ f ( x ) , (4.16)where x = m D /m q and in approximating, we have required that all squark masses areof the same order m ˜ q but not degenerate. In the limit of Dirac gluinos much heavierthan m ˜ q we obtain (cid:104) K | H eff | K (cid:105) ∆ m K (exp) (cid:39) (cid:16) α s . (cid:17) (cid:18)
15 TeV m D (cid:19) e iφ K , (4.17)which is much too large: in order to meet the bounds from (cid:15) K we would need m D ∼O (100) TeV. Here, we might have expected Dirac gaugino masses to soften the bounds23ith respect to Majorana masses. However, this is not the case. Since the strongestconstraint comes from operator ˜ Q , according to equation (4.10) we have a boundabout 5 times stronger for Dirac masses than Majorana ones when x = 100. Alignment in both left and right sectors
As we have seen above, since the constraints are severe for Kaon mixing, models thatsuppress the elements W L and W R are then most attractive (since ˜ f AB obtains largestcontribution from W A W A † ∼ W A and W A W A † ∼ W A ). However, retrieving thecorrect form for the CKM matrix leads to large flavour rotation for the up-quarkmatrix. Therefore, apart from checking that B-meson constraints are satisfied, onemust as well consider constraints from D-meson mixing.Since both down and up squark sectors are involved in the following discussion, werestore the corresponding superscripts in the W matrices, so that W q A ij is the matrix thatdiagonalises the A -handed squarks in the q -type sector, with A = L, R and q = u, d .Defining (cid:104) W qij (cid:105) ≡ (cid:113) W q L ij W q R i,j we can place approximate bounds in this framework W d L , W d R (cid:46) × − , (cid:104) W d (cid:105) (cid:46) × − ,W d L , W d R (cid:46) . , (cid:104) W d (cid:105) (cid:46) . ,W d L , W d R (cid:46) . , (cid:104) W d (cid:105) (cid:46) . ,W u L , W u R (cid:46) . , (cid:104) W u (cid:105) (cid:46) . , (4.18)where all of these should be multiplied by (cid:0) m ˜ q (cid:1) (cid:114)(cid:12)(cid:12)(cid:12) / f ( x ) (cid:12)(cid:12)(cid:12) . The constraints in theleft column of (4.18) come from operators of the type Q , ˜ Q , whereas the ones in theright column come from Q , Q .Of these bounds, it is the D-meson constraint that proves problematic for alignmentmodels, as typically suppressing the W d element will require W u ∼ λ . However, theproblem is not particularly severe: it can either be remedied by having somewhat These approximate bounds include bag factors but no NLO corrections (no magic numbers) (inplots we include all available data including magic numbers). U (1) × U (1) under which the quark superfields have charges [65] Q D U (3 ,
0) ( − ,
2) ( − , ,
1) (4 , −
1) (1 , ,
0) (0 ,
1) (0 ,
0) (4.19)Other examples of models with alignment can be found, e.g., in [66,67]. The symmetrybreaking parameters, coming from flavon fields of charges ( − ,
0) and (0 , − (cid:15) ∼ λ and (cid:15) ∼ λ respectively. The diagonalising matrices are given by W d L ij ∼ λ λ λ λ λ λ , W d R ij ∼ λ λ λ λ λ λ ,W u L ij ∼ λ λ λ λ λ λ , W u R ij ∼ λ λ λ λλ λ , (4.20)which are generically challenged by the bounds given above via D -meson mixing. How-ever, those bounds are derived under the assumption that the amplitude is well domi-nated by a single contribution. We find that, in practice, they are overly conservative.Indeed, in order for this to be the case there has to actually be a substantial hierarchybetween the squark masses, and then since there is a minimum mass for the secondgeneration via LHC bounds we will find that the model will be less constrained thanfeared. Considering this model, the constraint essentially comes from the Q operatorfor D -meson mixing. Moreover, if we were to suppress the amplitude by O ( λ ) thenwe would easily meet the constraints; hence we must only suppress the leading order25ontribution in λ , which we find to be:˜ f LL = λ (cid:20) ˜ I ( m L , m L ) + ˜ I ( m L , m L ) − I ( m L , m L ) (cid:21) + O ( λ )= λ ( m L − m L ) ˜ I ( m L , m L ) + O ( λ ) . (4.21)Clearly if the first two generations are quasi-degenerate then this will vanish sufficientlyto satisfy the constraints. Indeed, particular UV models could have them degenerateup to O ( λ ) [68], which would give a much greater suppression of the FCNC processesthan necessary to avoid current bounds. However, it is actually not necessary to have somuch degeneracy; for example taking m L = 3 m L , m L = 2 m L and taking m D = m L the amplitude is suppressed by a factor of 0 .
02 compared to simply taking ˜ f (1), whichis enough to satisfy the bounds for squark at gluino masses of O (2 TeV).To illustrate this, we show plots in figure 3 of the allowed lightest squark mass versusgaugino mass for this model with randomly chosen entries of the above form. In orderto harden the bounds we must introduce a large hierarchy between the squark masses.We take three different hierarchies: m L = 1 . m L = 3 m L , m L = 5 m L = 10 m L and m L = 25 m L = 100 m L (the same hierarchies for both up- and down-type squarks) andcalculate the bounds showing the gluino mass against the lightest squark mass usingNLO corrections and taking into account all ∆ F = 2 constraints. In practice, theD-meson constraint is dominant: we insist that | ∆ m D | is less than the experimentalvalue of 7 . × − GeV (since this is approximately three standard deviations fromzero, and moreover the standard model value is known to much less accuracy).The results of figure 3 agree with our discussion in the end of sec. 4.1. Thecancellation between the chirality-flip and the same chirality process suppresses thecontribution in the Majorana case for moderate x even if the enhancement over theDirac case is linear in x . Since the flavour bounds for m L (cid:38) . ÷ x , a Majorana gluino is less constrained than a Dirac one. Anotherfeature of this model is that, due to the suppression in the unitary rotations, the mainFCNC effects come from the first two generations, even if they are heavier than thethird one. Hence, one should bear in mind that the relevant squark mass for the loopdiagrams is heavier than the m L shown on the abscissa.26
00 400 600 800 1000 1200 14000500010 00015 00020 000 m L3 (cid:72) GeV (cid:76) M g (cid:142) (cid:72) G e V (cid:76) m L1 (cid:61) m L3 m L1 (cid:61) m L3 m L1 (cid:61) m L3 Figure 3: Constraints on the model described in section 4.2. The dashed lines corre-spond to exactly Dirac gauginos, while the solid lines are purely Majorana. We takethe same hierarchies for up- and down-type squarks, with m L = 1 . m L = 3 m L forthe red plots; m L = 5 m L = 10 m L for the green curves and m L = 25 m L = 100 m L for the blue. A particularly attractive scenario in light of the strong LHC bounds on the first twogenerations of squarks and the desire for “natural supersymmetry” is to have an in-verted hierarchy, where the first two generations of squarks are substantially heavierthan the third. This can be simply accommodated in flavour models, as we shall discussbelow.One approach, following [60], is to decouple the first two generations. In this case,the effective action is given by (4.2) with ˜ f AB of (4.1) given by˜ f AB = ˆ δ A ˆ δ B ˜ I ( m D , m A , m B ) (4.22)in the inverted hierarchy limit, as we have described in sec. 2.3. Here m L , m R are the27asses of the ‘left-handed’ and ‘right-handed’ sbottoms. The reader should be carefulwith the “hat” notation however: since ˆ δ A ≡ W A W A we expect the ˆ δ A to be small,coming from two small rotations rather than (in the generic case) one - indeed if therotations come from the squark mass-squared matrices M A ij themselves (rather thanfrom quark rotations) so that W A (cid:39) − M A /m A then we expect ˆ δ A < m A /m A .For m D (cid:29) m L , m R we find (we discuss the limits from B -meson mixing in ap-pendix C) (cid:104) K | H eff | K (cid:105) ∆ m K (exp) = 3 × × (cid:16) α s . (cid:17) (cid:18) m D (cid:19) (cid:16) . δ L ) + 0 . δ R ) − . δ L ˆ δ R (cid:17) (4.23)and hence (cid:113) | Re(ˆ δ L ) | < × − (cid:16) m D (cid:17) , (cid:113) | Im(ˆ δ L ) | < × − (cid:16) m D (cid:17) , (4.24)which are not much weaker than the limits from [60] despite the larger gaugino massand the change from Majorana to Dirac gauginos. The reason is that the flavour datahave been updated and the limits scale only inversely proportional to the gaugino mass,since there is no further suppression of the Dirac case relative to the Majorana case,as described in section 4.1. In fact, since the limits are derived from the constraintson C , ˜ C without the mass insertion approximation, for moderate values of the ratioof gluino to third generation squark masses, the Dirac version of this model is actuallymore constrained than the Majorana one. The above discussion assumed that we could completely decouple the first two gener-ations. However, we know that we cannot make them arbitrarily massive compared tothe third generation without the two-loop RGEs leading either to tachyons or substan-tial fine-tuning to avoid them. Typically a factor of m /m ∼ −
15 is the maximumthat is allowed. Given this, we must still worry about flavour-changing effects from thefirst two generations. 28or example, let us suppose that the heavy eigenstates are not degenerate, buthave masses m (cid:54) m . In the limit where m is much larger than m D , one of thecontributions to ˜ f LR of (4.1) can be written as − π i ˜ f LR ∼ W L W R m . Under the reasonable assumption that there are no accidental cancellations betweenthe different contributions, for m ∼
10 TeV the constraint from (cid:15) K requires W L W R (cid:46) − which is clearly highly restrictive for any flavour model. Therefore we must imposerestrictions upon the heavy squarks.Let us determine the condition for neglecting the contribution from the first twogenerations in the approximation that the first two generations of left- and right-handed squarks are degenerate to leading order with masses m L , m R respectively,with the third generation masses m L , m R . Then, there are corrections δ A m , δ A m , δ A m to the off-diagonal elements of the squark mass-squared matrix, with δ Aij definedsimilar to the mass insertion approximation flavour parameter described in sec. 2.3: δ Aij = m − ( m Aij ) , A = L, R . In this case, eq. (4.1) is expressed as˜ f AB (cid:39) ˆ δ A ˆ δ B ˜ I ( m A , m B )+ δ A ˆ δ B m A ∂∂m A (cid:20) ˜ I ( m A , m B ) − ˜ I ( m A , m B ) (cid:21) + ( A ↔ B )+ δ A δ B m A m B ∂ ˜ I ( m A , m B ) ∂m A ∂m B , (4.25)where we have neglected subleading terms in ˆ δ A,B . If we further take m A = m B = m , m A = m B = m , then this simplifies to − π i ˜ f AB = ˆ δ A ˆ δ B m ˜ f ( m D m ) + δ A δ B m ˜ f ( m D m ) − (cid:20) δ A ˆ δ B m ˜ f ( m D m , m m ) + ( A ↔ B ) (cid:21) , (4.26)29here˜ f ( m D m , m m ) = log m D m + 2 m D − m D m + m (1 + log m m D )( m − m D ) + O ( m − ) . (4.27)Assuming that m D (cid:28) m , in order to neglect the contribution of the first two genera-tions we require δ (cid:46) ˆ δ m m . Since, as explained above, we expect m m (cid:46) ÷
15, wesee that only certain flavour models will actually allow this.
In order to realise a model with heavy first two generations of squarks with sup-pressed mixing between them, we could consider models with a large D-term for anextra abelian gauged flavour symmetry under which only the first two generations arecharged, and obtain a natural supersymmetric spectrum [69]. These D-term contribu-tions were argued to be naturally generated (at least) in effective string models [70],to be positive and, in certain circumstances, to be dominant over the F-term contribu-tions. It is then clear from (B.3) and (B.4) that precisely because the first generationsof fermions are lighter than the third one, the corresponding scalars are predicted tobe heavier . While such models would be one approach to realising the scenario of theprevious subsection, there is currently no extant example that solves the FCNC prob-lem of mixing between the first two generations (owing to the need to have degeneracybetween them).Another class of flavour models adds extra symmetry between the first two gen-erations [71, 72]. In this case, we can effectively take the squark mass matrix to bediagonal, with flavour-changing processes only induced by the quark rotations com-bined with the (possibly small) non-degeneracies in the squark matrix (of course, ifthe squarks were degenerate then the super-GIM mechanism would lead to vanishingof the flavour-changing effects).Taking the model of [64] for m L = m L = m (cid:29) m L , m R = m R = m (cid:39) m R as an illustrative example of this scenario (see appendix B.2 for more details), we have30pproximately ˜ f LR (cid:39) i π W L W L † W R W R † m − m R m ˜ f ( m D m , m L m ) , (4.28)where ˜ f is given in (4.27) and the diagonalising matrices are given in terms of param-eters of the model: W L W L † W R W R † = − s d m d m s | V d | e − i ˜ α , (4.29)with s d and V d that take values s d (cid:39) . V d (cid:39) .
04 in the best fit of one of themodels in [64].The bounds on ∆ m K are easily satisfied by this model, so we focus directly on thebounds on (cid:15) K . We obtain, allowing C (cid:15) K ∈ [0 . , .
73] at 99% confidence level: | ∆ (cid:15) K || (cid:15) K ( SM ) | . (cid:39) . . × . × − m K f K m √ m K (exp) | V d | m d m s s d sin 2 ˜ α | m d R − m b R | m ˜ f =0 . × (cid:18) | V | . (cid:19) (cid:18) s d . (cid:19) (cid:18) sin 2 ˜ α √ / (cid:19) (cid:32) | m d R − m b R | (cid:33) (cid:18)
10 TeV m (cid:19) (cid:32) ˜ f . (cid:33) (4.30)where we have used m D = 2 TeV, m L = m ˜ b L = 1 TeV, m = 10 TeV to evaluate˜ f . The results from [64] are compatible with the 95% confidence level bound C (cid:15) K ∈ [0 . , .
41] and we show the comparison in figure 4.In this model, the Dirac gluino offers an improvement by roughly a factor of fourover the Majorana case. Again, this is in agreement with sec. 4.1 since the dominantcontribution comes from C where the chirality-flip process adds to the same-chiralityone instead of cancelling it. We saw previously that large suppression of FCNC and production of coloured particlescan be obtained in two different ways: 31 (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) m b (cid:142) L (cid:61) m g (cid:142) (cid:72) TeV (cid:76) m d (cid:142) R (cid:72) T e V (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) m g (cid:142) (cid:72) TeV (cid:76) m d (cid:142) R (cid:72) T e V (cid:76) Figure 4: Contour plots for the model of section 4.3.3. Along the contour, | ∆ (cid:15) K | =∆ (cid:15) K ( exp ). The dashed lines correspond to exactly Dirac gauginos, while the solid linesare purely Majorana, as in the original model of [64]. In the left plot, the left-handedsbottom mass is set equal to that of the gluino; in the right plot, the left-handedsbottom is fixed at 1 TeV. The two lines correspond to m d R − m b R = (1 . , .The remaining parameters are chosen as | V d | = 0 . , sin( α ) = 0 . s d = 0 . Large Dirac mass m D (cid:29) M, M χ , due to the underlying R-symmetry, in the massinsertion approximation. • Large Majorana mass M (cid:29) m D , M χ , due to small couplings of the light “fakegaugino” fermion to quarks/squarks.The second case can be realised in two distinct ways.i) We can have a scenario with a very moderate hierarchy and without a see-saw mass:we can take for example M χ ∼ TeV, M ∼
10 TeV, 1 TeV (cid:46) m ˜ q (cid:46) R ˜ g ∼ m D /M ,so the mixing between the gauginos and the fake gaugino could be almost arbitrarilysmall.ii) A second way is by having a large, intermediate scale gluino mass. A theoreticalmotivation for this case is gauge coupling unification. According to [73], MSSM withadditional adjoint chiral fields leads to a good unification of couplings at the stringscale for adjoint masses around 10 GeV. In the case they considered, the low-energyeffective theory is just the MSSM. From the gauge unification viewpoint however, wecan switch the masses of the gauginos/gluinos with those of the chiral adjoint fermions,keeping the scalar adjoint masses heavy. This switch will not affect gauge couplingunification at one-loop, whereas it will significantly change phenomenology, which wecall “fake split SUSY” for obvious reasons in what follows. In this section we thereforeconsider in more detail this scenario and comment on its qualitative phenomenologicalconsequences.Case i) is clearly easy to justify. Before discussing phenomenological implications,let elaborate more about case ii). The obvious question is the stability of the hierarchy M (cid:29) m D , M χ under radiative corrections. For this, we need to consider the effectivetheory when we integrate out the gauginos and the sfermions. The adjoint fermion33 (the “fake gaugino”) has no tree-level renormalisable couplings to the squarks andsleptons, but it does couple via the gauge current to the gaugino λ and the adjointscalar Σ: the relevant terms are L ⊃ − (cid:18) M λ a λ a + M χ χ a χ a + 12 B Σ Σ a Σ a + i √ gf abc Σ a λ b χ c + h.c. (cid:19) − m Σ a Σ a − ( m D Σ a + m D Σ a ) − ( m D λ a χ a + c.c. ) . (5.1)On the second line we included the terms coming from the Dirac gaugino mass term,which necessarily also generates the term ( m D Σ a + m D Σ a ) . We do not absorb theseinto m Σ , B Σ because these corrections are RGE invariant and therefore apply at anyrenormalisation scale [37–39]. Instead we defineˆ B Σ ≡ B Σ + 2 m D , ˆ m ≡ m + 2 | m D | . (5.2)Since we are making the logical assumption that the adjoint scalars are at least asmassive as the other scalars in the theory, we can integrate them out along with thegaugino λ : at one loop we generate a term M χ of M χ = 2 g C ( G ) ˆ B Σ M (cid:90) d p (2 π ) p (( p + m D ) + M p )(( p + ˆ m ) − ˆ B ) , (5.3)which gives to leading order in B Σ /m , m D /MM χ = 2 C ( G ) g π × ˆ B Σ M (cid:18) − log M ˆ m (cid:19) M (cid:29) ˆ m Σ , ˆ B Σ ˆ m M ˆ m Σ (cid:38) M. (5.4)This clearly prevents an arbitrary hierarchy between M and M χ . We might considersimply ignoring ˆ B Σ ; however, it will always have a D-term contribution from the Diracmass, so that without tuning we can say | ˆ B Σ | (cid:38) | m D | . More honestly, we should lookif there can be a symmetry preventing the generation of such a term. Indeed this isthe case: If we rotate the adjoint field Σ then this prevents both M χ and B Σ , but alsoprevents the Dirac mass m D . However, if we break this symmetry with the vev of a34eld φ such that φ/M high ≡ (cid:15) then we generate m D ∼ (cid:15)M , M χ ∼ (cid:15) M , B Σ ∼ (cid:15) M ∼ m D (5.5)and thus the above contribution is irrelevant: the see-saw (and direct) masses for the“fake” gluino are of order m D /M where the scale is controlled by the parameter (cid:15) . Wealso note that since this hierarchy is protected by the approximate symmetry, it is notaffected by renormalisation group running from above the SUSY-breaking scale δM χ ∼ (cid:15) g s π M , (5.6) δB Σ ∼ M M χ g π log (cid:18) Λ M (cid:19) ∼ (cid:15) g π M log (cid:18) Λ M (cid:19) . Taking M ∼ m ˜ q ∼ m Σ ∼ GeV and assuming that the “fake” gluino mass isof order M χ ∼ (cid:15) to be of order 10 − (so that we couldtake (cid:104) φ (cid:105) ∼ M, M high ∼ M GUT ). Hence we get the following masses M ∼ GeV (cid:38) m Σ (cid:29) m D , (cid:112) B Σ ∼ GeV (cid:29) M χ ∼ ↔ fake wino/bino, theresulting low-energy effective theory in this case is different compared to standard splitSUSY. Indeed, we should consider whether there are any light higgsinos remaining inthe spectrum: in split SUSY, there is an R-symmetry that protects the mass of thehiggsinos, whereas we have broken this, and we would expect the higgsinos to obtaina mass through diagrams similar to the one considered above: µ (cid:39)
14 2 g Y M B µ ˜ I ( m h , m H , M B , M B ) + 34 2 g M B µ ˜ I ( m h , m H , M W , M W ) , (5.8)where M ˜ B i , M ˜ W i with i = 1 , m h ( m H ) are the light (heavy) In terms of the parameter ˆ B Σ defined in (5.2), we find δ ˆ B Σ ∼ ( M M χ − m D ) g π log (cid:0) Λ M (cid:1) . m h (cid:39) m h u m h d − B µ m h u + m h d , m H (cid:39) m h u + m h d . (5.9)In writing (5.8) we neglected M χ in the loop. In this case, a more compact form forthe integrals is, for example˜ I ( m h , m H , M W , M W ) = (cid:90) d p (2 π ) p ( p + m h )( p + m H )[( p + m D ) + M W p ] . (5.10)Whereas the general expression is rather involved, in the limit M (cid:29) m H and (forsimplicity) with equal gaugino mass parameters for SU (2) and U (1) factors M ˜ W = M ˜ B (cid:39) M , it simplifies to µ (cid:39) g Y + 3 g π B µ M − m H /M log m H M . (5.11)However, this can be repaired in a similar fashion: we can suppose that the Higgs fieldsare charged under the same U (1) symmetry that the adjoints are charged under. Thiswould suppress the µ and B µ terms, and also prevent any superpotential couplingsbetween the adjoints and the higgsinos. We would have µ ∼ (cid:15) M, B µ ∼ (cid:15) M so wewould have B µ (cid:29) | µ | and the heavy Higgs scalars would be parametrically heavierthan the electroweak scale. In this scenario we effectively take infinite tan β and requirethe down-quark and lepton Yukawa couplings non-holomorphic and generated in thehigh-energy theory (see e.g. [20, 74]). Then, in split SUSY the effective lagrangian contains higgs/higgsino/gaugino cou-plings L eff ⊃ − H † √ g u σ a ˜ W a + ˜ g (cid:48) u ˜ B ) ˜ H u − H T (cid:15) √ − ˜ g d σ a ˜ W a + ˜ g (cid:48) d ˜ B ) ˜ H d . (5.12)In usual split SUSY, ˜ g u = g sin β, ˜ g d = g cos β, ˜ g (cid:48) u = g (cid:48) sin β, ˜ g (cid:48) d = g (cid:48) cos β ; however, inour case these couplings will be strongly suppressed by the fake gaugino/bino compo- There is another solution, where instead we extend the Higgs sector by another pair of doublets.These could be consistent with unification at any scale ; this is being explored in another work [48]. R , R (cid:48) . If the adjoint superpotential couplings W ⊃ λ S H d SH u + 2 λ T H d T H u had not been suppressed, then they would have provided couplings of the same form.Instead, the absence of such couplings at low-energy could be therefore a signature ofa remote N = 2 supersymmetric sector, instead of a more conventional split SUSYspectrum.Finally, in the absence of couplings λ S,T , the model has difficulties to accommodatea good dark matter candidate, due to the small couplings of the fake electroweakinosto quarks and leptons.
In the context of split SUSY, where squarks are very heavy compared to the gluino,one striking experimental signature is the long lifetime of the gluino and associateddisplaced vertices or (for even heavier squarks) gluino stability. Indeed the lifetime ofthe gluino could be sufficiently long to propagate on macroscopic distances in detectors[75–77]. This lifetime, in the standard split SUSY context, can be estimated in anapproximate manner according to [77] as follows τ ˜ g = 4 sec N × (cid:16) m ˜ q GeV (cid:17) × (cid:18) M (cid:19) , (5.13)where N is a quantity varying with M and m ˜ q but of order one for our range of masses.As we saw in the previous sections, the fake gluino couplings are altered by thediagonalisation of the gluino mass matrix and contain a tiny contribution of the originalgluino gauge coupling, proportional to R ˜ g ∼ m D /M . In case i) above, the mixingbetween the gauginos and the fake gaugino could be almost arbitrarily small by having m D (cid:28) TeV, meaning that the fake gluino could still have displaced vertices withoutrequiring large mass scales. Particularly interesting is the case where the usual gluinosare not accessible (they are heavier than say 5 TeV), whereas some of the squarksare. Displaced vertices /long lifetime for the fake gluino with light squarks would bea direct probe of a high-energy N = 2 supersymmetric spectrum. Pair productionof faked gluinos in this case lead to displaced vertices, since although some squarks37ould be light, their small couplings to the fake gluino suppresses such processes. Onthe other hand, direct squark production is possible, but subsequent squark decays toquarks/neutralinos go dominantly through the Higgsino components and correspondingYukawas couplings. They are therefore unsuppressed only for third generation squarks( and eventually third generation sleptons if similar arguments are applied to the othergauginos). Of course, the heavier the usual gluino, the bigger the fine-tuning needed inorder to keep a squark to be light. Some fine-tuning, moderate for gluino mass below10 TeV or so, is unavoidable for such a scenario to be realised in nature. However, itsvery different phenomenological implications could be worth further study.In case ii) above, the fake gluino couplings to quarks/squarks are proportional to g s R (cid:39) g s m D M ∼ (cid:15) and encodes the small gluino composition of the lightest fermionoctet. According to our numerical choice of masses we get R ∼ (cid:15) ∼ − . This affectstherefore the fake gluino lifetime, which has to be modified according to τ χ = 4 × sec N × (cid:18) − R ˜ g (cid:19) (cid:32) − R χ (cid:33) × (cid:16) m ˜ q GeV (cid:17) × (cid:18) M χ (cid:19) ∼ years(5.14)where we define R ˜ g and R χ to be the rotation matrices for the gluino and neutralinorespectively. For the scales given, this lifetime is hence longer than the age of theuniverse, and so we should make sure that fake gluinos are not produced in the earlyuniverse .We could also consider different moderate hierarchies with interesting low-energyimplications. For example, let us suppose that M χ ∼ m D ∼ TeV and gluino andsquark masses M ∼ m ˜ q ∼
100 TeV, while the higgsinos remain light; in split SUSYgluino decays are prompt inside the detector, but in our “fake split SUSY” case, now R ˜ g ∼ − and we can take R χ ∼
1. The gluino propagation length is increased bya factor of 10 and the vertex starts to become displaced. Although the squarks arestill very heavy, they could produce testable CP violating FCNC effects in the Kaonsystem ( (cid:15) K ). For more discussion of this issue we refer the reader to [48] Conclusions
Flavour physics sets severe constraints on supersymmetric models of flavour. In modelsin which the scale of mediation of supersymmetry breaking is similar or higher thanthe scale of flavour symmetry breaking, fermion masses and mixing hierarchies are cor-related with the flavour structure of superpartners. In the MSSM constructing a fullysuccessful flavour model of this type is difficult and usually requires the simultaneouspresence of several ingredients like abelian and non-abelian symmetries. At first sight,flavour models with Dirac gauginos are simpler to build, due to the flavour suppres-sion argued in the literature in their R-symmetric pure Dirac limit, for gluinos heavierthan squarks. In this paper, we found that this suppression is only strong in the near-degeneracy (mass insertion approximation) limit, whereas in most flavour models thisapproximation is not valid.We analysed the simplest Dirac flavour models with abelian symmetries realisingvarious degrees of alignment of fermion and scalar mass matrices and for non-abeliansymmetries realising a natural supersymmetric spectrum with heavy first two gener-ations. We found only a moderate improvement in the flavour constraints over theMSSM case. We also showed in an explicit example in section 4.2 that due to cancel-lations in the Majorana case, it is even possible that a Dirac model is for some partsof the parameter space more constrained than its MSSM cousin.We also considered generalised Lagrangians with both Majorana and Dirac masses,by not imposing an R-symmetry in the UV. We considered, in particular, the case inwhich the gluino Majorana mass is very large compared to that of the chiral octetfermion and the Dirac mass M (cid:29) M χ , m D . This led to the scenario dubbed “fakegluino” in which the light adjoint fermions are not the N = 1 partners of the gaugefields, but the other fermions in the N = 2 gauge multiplets. In this case, couplingsof the light “fake gluino” to quarks are suppressed parametrically by the ratio m D /M .This leads to a potentially new exotic phenomenology in which squarks can be lightand accessible experimentally, while the light adjoint fermions can be long-lived andgenerate displaced vertices or escape detection. Experimental discovery of a squarkand simultaneously of long-lived light gluinos would be spectacular evidence of such a39pectrum. An extreme case with heavy gluinos and light adjoint fermions is obtained bypushing a Majorana gluino mass and squark masses to an intermediate scale M ∼ GeV, which leads to good gauge coupling unification. The outcome is similar in spiritto split supersymmetry, with however light adjoint fermion couplings to quarks and(for electroweakinos) to higgs/higgsino which are highly suppressed compared to splitsupersymmetry.
Acknowledgements
We would like to thank Robert Ziegler for interesting discussions. M. D. G. wouldlike to thank Karim Benakli, Luc Darm´e and Pietro Slavich for useful discussionsand collaboration on the related work [48]. We thank the Galileo Galilei Institute forTheoretical Physics for the hospitality and the INFN for partial support in the initialstage of this work. This work was supported in part by the European ERC AdvancedGrant 226371 MassTeV, the French ANR TAPDMS ANR-09-JCJC-0146, the contractPITN-GA-2009-237920 UNILHC and the Marie-Curie contract no. PIEF-GA-2012-330060 BMM@3F. P. T. is supported in part by Vrije Universiteit Brussel through theStrategic Research Program “High-Energy Physics”, in part by the Belgian FederalScience Policy Office through the Inter-university Attraction Pole P7/37 and in partby FWO-Vlaanderen through project G011410N.
A K and B meson mixing in supersymmetry
A.1 From the Lagrangian to Feynman rules
In MSSM, the dominant contribution to K and B meson mixing comes from a boxdiagram with squarks and gluinos propagating in the loop. Starting from the superfield40agrangian, we have L MSSM ⊃ (cid:90) d θ Q † e g s V a T a Q + D † e − g s V a T a ∗ D ⊃ −√ g s (cid:104) ˜ d ∗ Lxi T axy λ aα d Lyiα − ˜ d Rxi T a ∗ xy λ aα d cRyiα (cid:105) + h.c. , (A.1)where g s is the strong coupling constant, i = 1 , , T axy are the SU (3) generators and λ aα is the gluino Weyl fermion. Also, the fermion in the chiralsuperfield D is denoted by d cRxiα = ( d Rxi ) cα and describes the charge conjugate of theright-handed down quark field. Its scalar superpartner is ˜ d ∗ Rxi .After adopting four-component notation d = d Lα d cR ˙ α ; ˜ g a = λ aα λ a ˙ α , (A.2)and using identities Ψ ci Γ I Ψ cj = − ( − A g IJ Ψ j Γ J Ψ i (A.3)where Γ I = { I , γ , γ µ γ , γ µ , Σ µν , Σ µν γ } , g IJ = diag(1 , , , − , − , −
1) and ( − A =+1 ( −
1) for a commuting (anticommuting) Ψ i , (A.1) can be written as − √ g s T axy (cid:104) ˜ d ∗ Lxi ˜ g a P L d yi − ˜ d ∗ Rxi ˜ g a P R d yi (cid:105) + h.c. (A.4)This last expression can also be written using charge conjugated fields − √ g s T axy (cid:104) ˜ g a P R d cxi ˜ d Lyi − ˜ g a P L d cxi ˜ d Ryi (cid:105) + h.c. . (A.5)Before writing down the Feynman rules for couplings (A.4) and (A.5), we switch to thesquark mass eigenstate basis.Going first to the basis where the down quark mass matrix and the gluino - squark- quark coupling are diagonal, one can write d L → V L d L , d cR → V R d cR . (A.6)41he down squark mass matrix is now denoted by m d L m ˜ d = − ˜ d † m d ˜ d , ˜ d = ˜ d Li ˜ d Ri (A.7)and can be diagonalised by the unitary matrix Z IJ ˜ d Li = Z iI ˜ D I , ˜ d Ri = Z i +3 I ˜ D I (A.8)such that L m ˜ d = − ˜ D ∗ I m I ˜ D I , m = Z † m d Z (A.9)where ˜ D I with I = 1 , . . . , i = 1 , , m = diag( m I ) is the diagonal matrix of the mass eigenstates. Then, (A.4) and(A.5) are written as (we denote ( W † ) IJ ≡ Z † IJ ) − √ g s T axy (cid:104) ˜ D ∗ Ix W † Ii ˜ g a P L d yi − ˜ D ∗ Ix W † Ii +3 ˜ g a P R d yi (cid:105) + h.c. (A.10)and − √ g s T axy (cid:104) ˜ g a P R d cxi W iI ˜ D Iy − ˜ g a P L d cxi W i +3 I ˜ D Iy (cid:105) + h.c. , (A.11)where W is defined by W = V † L Z LL V † L Z LR V † R Z RL V † R Z RR . (A.12)The corresponding Feynman rules for the vertices are (see figure 5) V axy = − i √ g s T axy ( W † Ii P L − W † Ii +3 P R ) , G axy = − i √ g s T axy ( W iI P R − W i +3 I P L ) (A.13)42 x ˜ gd y V axy ˜ gD x d cy V axy ˜ gG axy D y d x ˜ gG axy D y d cx Figure 5:
Feynman rules for gluino - squark - quark vertices. s w V ayw ˜ g a G alm d l D Iy D mJ ˜ g b d x G bxy V bmn s n s w V bmw D mI G alm d l ˜ g b ˜ g a D yJ d x G bxy V ayn s n s w V ayw ˜ g a G blm d cl D Iy D mJ ˜ g b d x G bxy V amn s cn s w V ayw ˜ g a G bly d l D Iy D mJ ˜ g b d cx G bxm V amn s cn Figure 6:
The four box diagrams (denoted M , , , from top left to bottom right) thatcontribute to K - K mixing. The fields s w , d x , d l and s n have 4-momenta p , , , . A.2 From the amplitude to the effective action
The amplitudes of the diagrams in figure 6 are (we neglect external momenta) [57] iM = − (cid:90) d p (2 π ) ip − m I ip − m J d x G bxy i (p + M ˜ g ) p − M g V bmn s n d l G alm i (p + M ˜ g ) p − M g V ayw s w iM = (cid:90) d p (2 π ) ip − m I ip − m J d x G bxy i (p + M ˜ g ) p − M g V bmw s w d l G alm i (p + M ˜ g ) p − M g V ayn s n iM = (cid:90) d p (2 π ) ip − m I ip − m J d x G bxy i (p + M ˜ g ) p − M g G blm d cl s cn V amn i (p + M ˜ g ) p − M g V ayw s w iM = − (cid:90) d p (2 π ) ip − m I ip − m J d l G bly i (p + M ˜ g ) p − M g G bxm d cx s cn V amn i (p + M ˜ g ) p − M g V ayw s w d x , s w etc. now denote commuting spinors. The total amplitude is simplified byusing SU (3) generator identities T axy T amn T blm T byw = 136 ( δ xw δ nl + 21 δ xn δ lw ) ; T axy T anm T bml T byw = 136 (10 δ xw δ nl − δ xl δ nw )(A.14)as well as the Fierz identities such asΨ P L Ψ Ψ P R Ψ = 12 Ψ P L γ µ Ψ Ψ P R γ µ Ψ Ψ P L γ µ Ψ Ψ P L γ µ Ψ = − Ψ P L γ µ Ψ Ψ P L γ µ Ψ (same for P R )Ψ P R Ψ Ψ P R Ψ = 12 Ψ P R Ψ Ψ P R Ψ −
18 Ψ Σ µν P R Ψ Ψ Σ µν Ψ . (A.15)We can identify an effective Lagrangian that delivers this total amplitude. In our casewe use [58] ig s L KKeff = W K W L (cid:18) I + 19 M g I (cid:19) W † K W † L d x γ µ P L s x d n γ µ P L s n + W K W L (cid:18) I + 19 M g I (cid:19) W † K W † L d x γ µ P R s x d n γ µ P R s n + W K W L (cid:32) M g I − ˜ I d x P L s x d n P R s n + M g I +5 ˜ I d x P L s n d n P R s x (cid:33) W † K W † L + M g W K W L I W † K W † L (cid:18) d x P R s x d n P R s n − d x P R s n d n P R s x (cid:19) + M g W K W L I W † K W † L (cid:18) d x P L s x d n P L s n − d x P L s n d n P L s x (cid:19) + W K W L ˜ I W † K W † L (cid:18) − d x P L s x d n P R s n − d x P L s n d n P R s x (cid:19) . (A.16)44 .3 Loop Integrals The following loop functions are being used throughout the main part of this work. I n ( m , ..., m n − , m n ) ≡ (cid:90) d p (2 π ) p − m )( p − m ) ... ( p − m n − )( p − m n ) ≡ i π m n − n f n ( x , x , ..., x n − )˜ I n ( m , ..., m n − , m n ) ≡ (cid:90) d p (2 π ) p ( p − m )( p − m ) ... ( p − m n − )( p − m n ) ≡ i π m n − n ˜ f n ( x , x , ..., x n − )with x i ≡ m i m n . Here we collect useful relations related to functions I , , and ˜ I , , : ( ∼ ) I ( M , M , m , m ) = i ( ∼ ) f ( x, x, π m ∼ ) I ( M , M , m , m , m ) = i ( ∼ ) f ( x, x, , π m ∼ ) I ( M , M , m , m , m , m ) = i ( ∼ ) f ( x, x, , , π m where f ( x, x,
1) = 2 x − − ( x + 1) ln( x )(1 − x ) ˜ f ( x, x,
1) = x − − x ln( x )(1 − x ) f ( x, x, ,
1) = − x − x + 5 + 2(1 + 2 x ) ln( x )2(1 − x ) ˜ f ( x, x, ,
1) = − x + 4 x + 1 + 2 x (2 + x ) ln( x )2(1 − x ) f ( x, x, , ,
1) = − x + 9 x + 9 x − − x + 1) ln( x )6(1 − x ) ˜ f ( x, x, , ,
1) = x + 9 x − x − − x (1 + x ) ln( x )3(1 − x ) x → x → ∞ arelim x → f ( x, x,
1) = − ln( x ) − O ( x ln( x )) , lim x →∞ f ( x, x,
1) = ln( x ) x − x + O ( x − ln( x ))lim x → ˜ f ( x, x,
1) = − − x ln( x ) + O ( x ) , lim x →∞ ˜ f ( x, x,
1) = − x + O ( x − ln( x ))lim x → f ( x, x, ,
1) = ln( x ) + 52 + O ( x ln( x )) , lim x →∞ f ( x, x, ,
1) = − x + O ( x − ln( x ))lim x → ˜ f ( x, x, ,
1) = 12 + 2 x ln( x ) + O ( x ) , lim x →∞ ˜ f ( x, x, ,
1) = ln( x ) x − x + O ( x − ln( x ))lim x → f ( x, x, , ,
1) = − ln( x ) −
176 + O ( x ln( x )) , lim x →∞ f ( x, x, , ,
1) = 16 x + O ( x − )lim x → ˜ f ( x, x, , ,
1) = − − x ln( x ) + O ( x ) , lim x →∞ ˜ f ( x, x, , ,
1) = − x + O ( x − ln( x ))and f ( x = 1) = 16 , ˜ f ( x = 1) = − ,f ( x = 1) = − , ˜ f ( x = 1) = 112 ,f ( x = 1) = 120 , ˜ f ( x = 1) = − . B Models of Flavour
B.1 Abelian Models
An inverted hierarchy was invoked some time ago in the literature [69,78,79] in order toease the FCNC and CP constraints in supersymmetric models. To our knowledge, thefirst class of models in which the inverted hierarchy is really predicted are supersymmet-ric generalisations of abelian flavour models of the Froggatt-Nielsen type [80]. Thesemodels contain an additional abelian gauge symmetry U (1) X under which the threefermion generations have different charges (therefore the name horizontal or flavoursymmetry), spontaneously broken at a high energy scale by the vev of (at least) onescalar field Φ, such that (cid:15) = (cid:104) Φ (cid:105) / Λ (cid:28) h Uij ∼ (cid:15) q i + u j + h u , h Dij ∼ (cid:15) q i + d j + h d , (B.1)where q i ( u i , d i , h u , h d ) denote the U (1) X charges of the left-handed quarks (right-handed up-quarks, right-handed down-quarks, H u and H d , respectively). Quark massesand mixings in the simplest models are given as m u m t ∼ (cid:15) q + u , m c m t ∼ (cid:15) q + u , m d m b ∼ (cid:15) q + d , m s m b ∼ (cid:15) q + d , sin θ ∼ (cid:15) q , sin θ ∼ (cid:15) q , sin θ ∼ (cid:15) q . (B.2)A successful fit of the experimental data requires larger charges for the lighter genera-tions q > q > q , u > u > u , d > d > d , (B.3)one simple example being for example [81] q = 3 , q = 2 , q = 0 , u = 5 , u = 2 , u = 0 , d = 1 , d = 0 , d = 0 . (B.4)Scalar soft masses in abelian flavour models are typically of the form m ij = X i δ ij (cid:104) D (cid:105) + c ij (cid:15) | q i − q j | ( ˜ m F ) , (B.5)where X i (cid:104) D (cid:105) are D-term contribution for the scalar of charge X i , whereas the secondterms proportional to ( ˜ m F ) describe F-term contributions. In the case where D-termsare smaller or at most of the same order than the F-term contributions, the order ormagnitude estimate of the FCNC in the mass insertion approximation is completelydetermined by U (1) charges to be( δ u,dij ) LL ∼ (cid:15) | q i − q j | , ( δ dij ) RR ∼ (cid:15) | d i − d j | , ( δ uij ) RR ∼ (cid:15) | u i − u j | . (B.6)If two charges are equal (this is the case for right-handed d quarks above d = d ),mass insertion approximation is however not valid anymore.47 .2 Non-abelian extension We present here in some details the model used in Section 4.3.3. The model was pro-posed in [64] and is a flavour model based on a G × U (1) local horizontal symmetry,where G is a discrete nonabelian subgroup of SU (2) global . Whereas the discrete non-abelian symmetry is preferable over the continuous SU (2) global for theoretical reasons,for low-energy flavour physics it was argued in [64] that there is no major differencebetween the discrete and the continuous case.The simplest choice for the flavour charges is to consider an SU (5) invariant pattern X and X , with Higgses uncharged. We need a minimum number of two flavons, anSU(2) doublet φ with charge X φ and an SU (2) singlet χ with charge −
1. The totalfield content is summarised in Table 5. The zero U (1) charge of the 3rd generation10 a ¯5 a ¯5 H u H d φ a χSU (2) 2 1 2 1 1 1 2 1 U (1) X X ¯5 X X φ − X is left free,in order to accommodate different values of tan β .The relevant part of the superpotential is given by W = h u H u Q U + h u Q a U H u φ a Λ (cid:16) χ Λ (cid:17) X + X φ + h u Q U a H u φ a Λ (cid:16) χ Λ (cid:17) X + X φ + h u H u Q a U b (cid:15) ab (cid:16) χ Λ (cid:17) X + h u Q a U b H u φ a Λ φ b Λ (cid:16) χ Λ (cid:17) X +2 X φ + h d H d Q D (cid:16) χ Λ (cid:17) X + h d Q a D H d φ a Λ (cid:16) χ Λ (cid:17) X + X + X φ + h d Q D a H d φ a Λ (cid:16) χ Λ (cid:17) X ¯5 + X φ + h d H d Q a D b (cid:15) ab (cid:16) χ Λ (cid:17) X + X ¯5 + h d Q a D b H d φ a Λ φ b Λ (cid:16) χ Λ (cid:17) X + X ¯5 +2 X φ . (B.7)We have imposed here that all exponents are non-negative X ≥ , X ≥ , X + X φ ≥ , X ¯5 + X φ ≥ , X + X ¯5 ≥ . (B.8)The h ’s are complex O (1) coefficients, Λ is a high flavour scale and a, b are the SU (2)48ndices. In the leading order in small parameters, the structure of the K¨ahler potentialdoes not affect the predictions in the fermion sector. Using the flavon vevs (cid:104) φ a (cid:105) = (cid:15) φ Λ , (cid:104) χ (cid:105) = (cid:15) χ Λ , (B.9)one can calculate masses and mixings in terms of the original parameters.The Yukawa matrices turn out to be given by Y u = h u (cid:15) (cid:48) u − h u (cid:15) (cid:48) u h u (cid:15) u h u (cid:15) u h u (cid:15) u h u , (B.10) Y d = h d (cid:15) (cid:48) u (cid:15) d /(cid:15) u − h d (cid:15) (cid:48) u (cid:15) d /(cid:15) u h d (cid:15) u (cid:15) d h d (cid:15) (cid:15) u h d (cid:15) d h d (cid:15) , (B.11)with (cid:15) u ≡ (cid:15) φ (cid:15) X + X φ χ , (cid:15) d ≡ (cid:15) φ (cid:15) X ¯5 + X φ χ , (cid:15) (cid:48) u ≡ (cid:15) X χ , (cid:15) ≡ (cid:15) X χ . (B.12)Imposing that the charges are integers then gives a series of possibilities. A par-ticularly simple possibility, which turns out to be the most successful from the flavourprotection viewpoint is for (cid:15) χ ∼ (cid:15) φ ∼ . , X = X ¯5 = X = − X φ = 1 , tan β = 5 . (B.13)The main features of the model are as follows: • The model has U (1) X D-term contributions which are dominant over the F-termones (cid:104) D (cid:105) (cid:29) m F . • The squark mass matrices are almost diagonal in the flavour basis, with rotationmatrices Z which are very close to the identity, compared to the analogous ones49or the quarks U . In this case, the matrices appearing in the gluino couplings aredetermined by quark rotations W (cid:39) U † . • Due to the SU (2) original symmetry only broken by the small parameter (cid:15) φ , thefirst two generation squarks, both left and right-handed, are essentially degener-ate with mass given by m L , m L = (cid:104) D (cid:105) , with non-degeneracies (provided by theflavour breaking) which are negligible. • The main splitting is between the first two and the third generation. For leftsquarks, there is an hierarchy m L (cid:29) m L since the third generation is unchargedunder U (1) X and therefore gets only F-term contributions m L ∼ m F . This isalso true for the right-handed up-type squarks. • The right-handed down-type squarks are charged and get D-term contributions.In the simplest example we consider here, the third generation is almost degen-erate with the first two, m R = m Rh + δm R , where δm R ∼ m F . The most constraining operator is as usual Q , from (cid:15) K . For models of the typedescribed above, the corresponding coefficient in the leading approximation is given by C = α s V L ¯ V L V R ¯ V R m R − m m ˜ f ( m L m , m D m ) . (B.14)The relevant rotations are given in the leading approximation by V L ∼ (cid:15) u , ¯ V L ∼ (cid:114) m d m s (cid:15) u ,V R ∼ sin θ d , ¯ V R ∼ (cid:114) m d m s sin θ d , (B.15)where tan θ d ≡ | h d | (cid:15) d | h d | (cid:15) (B.16)is a free parameter of order one fixed to tan θ d = 0 . V ub /V cb . The product of rotations is therefore given at the leading order in the flavonparameters by V L ¯ V L V R ¯ V R ∼ m d m d (cid:15) u sin θ d , (B.17)50otice that the right-handed rotations in (B.15) are large. Because of this lack ofsuppression, right-handed sbottom has to be heavy. The charge assignment (B.13)is then the most advantageous one and realises the minimal implementation of thenatural SUSY spectrum. C B-meson mixing constraints
As discussed in section 2, the bounds from B -meson mixing can be calculated in thesame way as Kaon mixing, using the translations in equation (2.10) but taking thevalues in table 8. These give us equations of the form Ce iφ =1 + ( x + iy ) e − iβ . (C.1)We have limits on C, φ although they are correlated and it is difficult to use thatinformation. Hence the most conservative bounds that we can set are simply to makesure that
C, φ always lie within their 3 σ ranges. These lead to | x d | < . , | y d | < . , | x s | < . , | y s | < . , (C.2)where x q ≡ (cid:104) B q |H B q | B q (cid:105) ∆ m B q ( SM ) , y q ≡ (cid:104) B q |H B q | B q (cid:105) ∆ m B q ( SM ) (C.3)These limits are unlikely to change substantially over the next 20 years: the projectedimprovement in sensitivity from SuperKEKB with 50ab − is from ± . ± .
15 in C B d [82] (more or less the same as the current UTFIT value), from ± . ± .
03 in φ B d [82] (an improvement of about 2 over the UTFIT present value) while LHCb with50fb − will improve the uncertainty on φ B s to ± .
007 [83] – a factor of 5 improvement.We typically find that the bounds from B-meson mixing are subdominant to thosefrom Kaon mixing; we shall explore this in the mass-insertion approximation and heavy-first-two-generations scenarios below. In this section we shall specialise for clarity to51he exactly Dirac gaugino case.
C.1 Mass insertion approximation
In the mass insertion approximation, defining A ≡ (cid:16) α s . (cid:17) (cid:18) GeV m D (cid:19) (cid:32) ˜ f (1) − / (cid:33) (C.4)we find x d + iy d =42 × A × (cid:20) . δ LL δ LL + δ RR δ RR ] − . δ LR δ RL − . δ LL δ RR (cid:21) x s + iy s =2 × A × (cid:20) . δ LL δ LL + δ RR δ RR ] − . δ LR δ RL − . δ LL δ RR (cid:21) . (C.5)These can be simply translated into bounds using equation (C.2). However, if wecompare with the bounds from Kaon mixing we have∆ M K (SUSY)∆ M K (exp) =280 × A × (cid:20) . δ LL δ LL + δ RR δ RR ] − δ LR δ RL − . δ LL δ RR (cid:21) | (cid:15) K (SUSY) || (cid:15) K (SM) | .
73 =6 . × × A × (cid:12)(cid:12)(cid:12)(cid:12) Im (cid:18) . δ LL δ LL + δ RR δ RR ] − δ LR δ RL − . δ LL δ RR (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (C.6)We see clearly that the bounds from ∆ M K and, in particular, (cid:15) K are much morestringent than those from the B meson oscillations. C.2 Decoupled first two generations
We expect that the B-meson mixing bounds should be most relevant in the limit thatthe first two generations are heavy; here we shall consider that case. For these purposeswe can ignore mixing between the first two generations `a la [60] and thus have m D α s L dirac =ˆ δ LLi ˆ δ LLi Q + δ RRi δ RRi Q + ˆ δ LLi ˆ δ RRi (cid:16) Q − Q (cid:17) (C.7)52here i = 1 for B d , 2 for B s .These lead to (taking the bag factors into account from table 9) x d + iy d =2260 × (cid:16) α s . (cid:17) (cid:18) GeV m D (cid:19) (cid:20) . δ LLi ˆ δ LLi + ˆ δ RRi ˆ δ RRi ] − . δ LLi ˆ δ RRi (cid:21) x s + iy s =95 × (cid:16) α s . (cid:17) (cid:18) GeV m D (cid:19) (cid:20) . δ LLi ˆ δ LLi + ˆ δ RRi ˆ δ RRi ] − . δ LLi ˆ δ RRi (cid:21) . (C.8)These lead to bounds | Re(ˆ δ LL ˆ δ LL ) | < . × − , | Im(ˆ δ LL ˆ δ LL ) | < . × − , | Re(ˆ δ LL ˆ δ LL ) | < . × − , | Im(ˆ δ LL ˆ δ LL ) | < . × − . (C.9)Hence the stronger B-meson bounds come from the B d data rather than B s , but (cid:15) K still provides the strongest constraint on the model parameter space, given in equation(4.24). These bounds are much weaker than the those from [60], presumably due to theDirac mass and the factor of 10 increase in the gaugino mass that we are now requiredto take. Note that, since there is no square root here, changing the gaugino mass by afactor of ten weakens the bound by a factor of a hundred; whereas in the (cid:15) K case it isonly a factor of ten (even for Dirac gauginos). Hence as we make the gauginos heavierwe further weaken the relevance of the B -mixing compared to (cid:15) K . D Input
Here we collect the Bag factors and B-meson mixing data that we have used in settingbounds. In addition we use bag factors and magic numbers given in [53, 62, 84] that wehave not reproduced here. 53arameter Value Ref. α s ( M Z ) 0.1184 [52] f K . m K m s (2 GeV) 0 .
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17 MeV m B s . ± .
24 MeV m b . ± .
03 GeV (
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