Predictions for Flavour Observables in a RS Model with Custodial Symmetry
aa r X i v : . [ h e p - ph ] A ug Predictions for Flavour Observables in a RS Model withCustodial Symmetry ∗ Bjoern Duling
Physik Department, Technische Universit¨at M¨unchen, 85748 Garching, GermanyIn these proceedings we present the main results for particle-antiparticlemixing and rare decays in the Randall-Sundrum (RS) model with custodialsymmetry. To investigate the strong bound on the Kaluza-Klein (KK) massscale M KK & O (20) TeV implied by the measurement of ǫ K we perform afine-tuning analysis that on the one hand confirms the quoted bound onthe KK mass scale but on the other hand reveals that consistence withexperiment can still be achieved for small or moderate fine-tuning. In ouranalysis of rare decays of K and B mesons we find that due to the custodialsymmetry the coupling of the Z boson to right handed quarks yields thedominant contribution. This feature of the model leads to distinct patternsand correlations that allow to distinguish the model from other frameworksof physics beyond the Standard Model (SM).PACS numbers: 11.10.Kk, 12.15.Ji, 12.60.-i, 13.20.Eb, 13.20.He
1. Introduction
The extradimensional models first proposed by Randall and Sundrum[1] are based on a slice of
AdS that is bounded by the so-called UVand IR (3-)branes and is characterised by a 5-dimensional bulk metric ds = e − ky η µν dx µ dx ν − dy . Effective energy scales in this setup dependon the position along the extra dimension via the warp factor e − ky , whichaddresses the gauge hierarchy problem if the Higgs field is localised close toor on the IR brane and the geometric model parameters are chosen prop-erly. The only free parameter coming from space-time geometry then isgiven by the mass scale of the lightest KK excitations present in the model.Allowing the SM fields except for the Higgs boson to propagate into thebulk [2, 3, 4] in addition allows to address the flavour problem by naturallygenerating the hierarchies in the fermion spectrum and mixing angles [2, 4]. ∗ Presented at the Flavianet topical workshop
Low energy constraints on extensions ofthe Standard Model (1)
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This is achieved by a non-universal localisation of fermions with differentflavours along the extra dimension. In this setup the fermion masses andmixing angles arise from the overlap of the fermion profiles with the IRbrane. Since the localisation of the fermion profiles depends exponentiallyon flavour dependent O (1) bulk mass parameters large hierarchies in thefermion spectrum can be achieved in a natural way. On the other hand thenon-universal localisation of fermions leads to flavour changing neutral cur-rent (FCNC) interactions of the SM fermions and KK gauge bosons, albeitsuppressed by the so-called RS-GIM mechanism [5, 6] that is an intrinsicfeature of the model.The particular model we will analyse is based on an enlarged bulk gaugegroup G bulk = SU (3) c × SU (2) L × SU (2) R × U (1) X × P LR . The fermions inthis model are embedded into representations of G bulk in such a way thatthe T parameter [7, 8] and the Zb L ¯ b L coupling [9] are protected from toolarge corrections. This allows for KK masses of order M KK ≃ (2 −
3) TeVwhich are in the reach of the LHC [10, 11, 12, 13, 14]. A more detaileddescription of the model can be found e.g. in [15].In the following section we present our main results for ∆ F = 2 and∆ F = 1 processes. A detailed analysis of these processes can be found in[16, 17]. In Sec. 3 we present our conclusions.
2. Analysis of particle-antiparticle mixing and rare decays
The mixing of neutral mesons imposes very stringent constraints onthe KK mass scale of the RS setup [18]. In particular ǫ K , the measureof indirect CP violation in K − ¯ K mixing, sets a lower bound M KK & O (20) TeV on the KK mass scale if textureless, anarchic Yukawa matricesare assumed. This has two reasons: In ǫ K the accidental suppression ofthe SM prediction by the very small phase of (cid:0) M K (cid:1) SM meets the presenceof effective operators that are induced by the tree level exchange of heavyKK gluons and that are chirally and QCD enhanced by a total factor ofapproximately 140. Both effects taken together are too large to be overcomeby the RS-GIM mechanism; however, all other observables for which eitherthe strong enhancement of the RS contribution or the suppression of theSM prediction is absent are adequately protected.To further investigate this matter, in [16] (see also [19]) we perform afine-tuning analysis for ǫ K . Our result is given in the left panel of Fig. 1where we show the fine-tuning in ǫ K as a density plot for a KK mass scale M KK ≈ .
45 TeV. It turns out that the RS contribution to ǫ K for the chosenKK mass scale is on average by two orders of magnitude too large. Still,as can also be seen from Fig. 1, there is a non-vanishing number of points lavianet˙Topical˙Workshop˙Kazimierz printed on January 8, 2019 BG ( ǫ K ) plotted against ǫ K normalised to its exper-imental value. For visualisation the blue line gives the average fine-tuning. Right:The same for ∆ M K . in parameter space that reproduce ǫ K with merely small or moderate fine-tuning. This discovery does not remove the tension pointed out in [18] - ourdetailed numerical calculation in fact roughly confirms the bound on the KKmass scale derived therein. However, since due to the reasons outlined above ǫ K is the only critical ∆ F = 2 observable in the RS model (for comparison,in the right panel of Fig. 1 we also show the fine-tuning in ∆ M K which issmall or moderate for all points in parameter space) we proceed with thesubset of phenomenologically valid parameter points and investigate theirimpact on rare decays of K and B mesons. K and B decays Using only parameter points that reproduce the experimentally mea-sured ∆ F = 2 observables with small or moderate fine-tuning we will nowhave a look at the impact on rare decays [17] (see also [20]). Also theseprocesses receive tree level contributions this time dominantly from the ex-change of Z , Z H , Z ′ gauge bosons with masses m Z , M KK , M KK , respec-tively. Z , Z H , Z ′ are the mass eigenstates of the system containing all SU (2) L × SU (2) R × U (1) X gauge bosons with masses up to M KK1 . Allthree of those mass eigenstates comprise admixtures of KK gauge bosonsand hence have flavour violating couplings to SM quarks. Of these the Zd iL ¯ d jL and Z ′ d iL ¯ d jL couplings are suppressed by virtue of the custodial sym-metry by factors of O (10 ) and O (10), respectively. In an effective theoryanalysis [21] (see also [22]) it is shown analytically that the mixing of SMquarks with heavy KK quarks after EWSB yields no significant contributionto the suppressed couplings. Higher KK excitations contribute only subleading effects and are neglected in thepresent analysis.
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Fig. 2. Left: Br ( K L → π ν ¯ ν ) plotted against Br ( K + → π + ν ¯ ν ). The shadedarea represents the experimental 1 σ -range, the Grossman-Nir bound is displayedby the dashed line. The black point represents the SM prediction. Right: Theshort-distance contribution to Br ( K L → µ + µ − ) plotted against Br ( B s → µ + µ − ). Considering the impact of the custodial symmetry on the quarks cou-plings of Z and Z ′ , the different masses of the three gauge bosons and thevolume suppression in the leptonic coupling of Z ′ and Z H one finds that thecoupling of the Z boson to right-handed down quarks dominates the RS con-tribution to rare K and B decays. This surprising result suggests a patternin rare decays that is very different from the one expected in the minimal RSmodel without custodial protection. In particular, the rather soft hierarchyin the right-handed Z couplings to different quark flavours can only partlycompensate the hierarchy in the SM CKM factors λ Kt ≪ λ dt ≪ λ st , resultingin much larger relative effects in rare K decays than in decays of B d or B s mesons. This is in contrast to the minimal RS model where the relative NPeffects are expected to be similar in size in the K , B d and B s systems. Inthe following, the consequences of this finding will be highlighted by meansof showing numerical results for the rare decays K → πν ¯ ν , B s → µ + µ − and K L → µ + µ − .In the theoretically very clean but experimentally challenging K → πν ¯ ν system we find the branching ratios Br ( K L → π ν ¯ ν ) and Br ( K + → π + ν ¯ ν )to be potentially enhanced by factors of three and two, respectively (leftpanel of Fig. 2). These enhancements are consistent with the existing mea-surement of the K + → π + ν ¯ ν decay for most parameter points and allow inparticular to reach the central experimental value for the branching ratio ofthis decay.Comparing B and K branching ratios such as Br ( B → µ + µ − ) and theshort-distance contribution to Br ( K L → µ + µ − ) as done in the right panelof Fig. 2 one sees that, as outlined above, the effects in the K mode canreceive much stronger enhancements than in the B decay mode. For thisparticular B decay we find the enhancement of the branching ratio to betypically below 15%. lavianet˙Topical˙Workshop˙Kazimierz printed on January 8, 2019 Br ( K L → µ + µ − ) plotted against Br ( K + → π + ν ¯ ν ). Right: Br ( K L → π ν ¯ ν ) plotted against S ψφ . The dominance of the right-handed Z coupling leads to the RS contri-butions entering Br ( K L → µ + µ − ) SD and Br ( K + → π + ν ¯ ν ) to enter withopposite sign. This in turn results in an inverse correlation between thesetwo quantities that is shown in the left panel of Fig. 3. This picture is inclear contrast to models where NP enters exclusively or dominantly throughleft-handed couplings as is the case in the Littlest Higgs model with T-parity(LHT) (for a recent flavour analysis see [23]). In such models the NP con-tribution enters both quantities with the same sign and the correlation isfound to be roughly linear.Finally, in the right panel of Fig. 3 we contrast the CP violating ∆ F = 2and ∆ F = 1 observables S ψφ and Br ( K L → π ν ¯ ν ). Both observables canbe enhanced significantly beyond their SM values. However it turns outthat a simultaneous enhancement of both observables is very unlikely whichoffers interesting experimental prospects once S ψφ has been measured at theLHCb experiment.
3. Conclusions
The main results of our analysis of particle-antiparticle mixing [16] andrare decays [17] can be summarised as follows: • A detailed fine-tuning analysis of ǫ K confirms the stringent genericbounds on M KK but at the same time shows that the rich flavourstructure of the model can accommodate ǫ K for mass scales as low as M KK ≃ .
45 TeV with small or moderate fine-tuning • Large effects are possible in B − ¯ B mixing. In particular the CPviolating observable S ψφ can take values − ≤ S ψφ ≤ S ψφ ≃ . • For rare decays of K mesons enhancements by factors up to threeare possible while the effects in rare B decays are much smaller andtypically below 15% Flavianet˙Topical˙Workshop˙Kazimierz printed on January 8, 2019 • Simultaneous large enhancements of S ψφ and rare K decays are veryunlikely. This possibly allows to test the RS model with custodialsymmetry experimentally in the foreseeable future. Acknowledgements