Flavour Physics in the Littlest Higgs Model with T-Parity: Effects in the K, B_d/s and D systems
aa r X i v : . [ h e p - ph ] A ug Flavour Physics in the Littlest Higgs Model with T -Parity:Effects in the K , B d/s and D systems ∗ Stefan Recksiegel
Technische Universit¨at M¨unchen, Physikdepartment, T31The Littlest Higgs Model with T parity (LHT) is an interesting alter-native model for New Physics at the TeV scale. Although Flavour Physicswas not the reason for creating the LHT model, significant effects (such aslarge CP violation where not predicted by the SM) can be created withoutviolating existing experimental bounds. We study the B -, K - and especiallythe D -sector.PACS numbers: 12.60.Cn, 12.60.Fr, 13.25.Ft, 13.25.Hw, 13.20.Eb
1. Introduction: Gauge hierarchy in the SM
A major problem in the Standard Model (SM) is theGauge Hierarchy problem, Top-loop corrections make the Higgsmass unstable, ∆ m H = −| λ t | / π (cid:2) Λ UV + . . . (cid:3) . To prevent m H → m Planck , we need incredible fine-tuning. One possible solu-tion is SUSY, where the top-loop is cancelled with a stop-loop, ∆ m H =2 | λ s | / π (cid:2) Λ UV + . . . (cid:3) . It is also possible to lower the Planck mass withextra dimensions, another possible solution to the Gauge Hierarchy problemis the Little Higgs mechanism.
2. The Little(st) Higgs Model (with T parity)
In the Little Higgs class of models [1], the Higgs Boson is a pseudo-Goldstone boson of a spontaneously broken global symmetry. Gauge andYukawa couplings break the symmetry explicitely, but every single couplingconserves enough of the symmetry to keep the Higgs massless. This way, theradiative corrections to the Higgs mass are only logarithmically divergentat one loop (and not quadratically as in the SM). ∗ Presented at the Flavianet Workshop on Low energy constraints on extensions of theStandard Model, 23-27 July 2009, Kazimierz, Poland (1) proceedings printed on December 13, 2018
One popular implementation of the Little Higgs mechanism is the Lit-tlest Higgs Model [2], where the Higgs boson is a pseudo-Goldstone bosonfrom breaking a global SU (5) symmetry to a global SO (5) at the scale f ∼ O (TeV). The exact mechanism for symmetry breaking is unspeci-fied, therefore the Littlest Higgs model is an effective theory valid up toΛ ∼ πf .There are 14 Nambu-Goldstone bosons from symmetry breaking: theSM Higgs, new heavy gauge bosons W ± H , Z H , A H , a scalar triplet Φ, and aheavy partner for the top quark, T . In the original Littlest Higgs, custodial SU (2) is broken already at tree level, then electroweak precision (EWP)observables demand f & − − T + ) are odd, all SMparticles are even. There are therefore no contributions by T odd particlesat the tree level, but the cancellation of divergences still works since it is aloop effect. This allows lowering the scale f to ∼ T − in addition to the T even T + . (Just like R parity in SUSY, T parity canalso produce a candidate for Dark Matter.)The new parameters in the LHT model are f , the NP scale which alsofixes M W H etc. The mixing between t and T is described by x L . Thereare three mirror quark masses: m H , m H and m H (the model is MinimalFlavour Violating (MFV) if these are degenerate) and a mirror quark mixingmatrix V Hd containing three angles and three [4] phases. The up-type mirrorfermion mixing matrix is given by V † Hu V Hd = V CKM . (There are also 9 mirrorlepton parameters, but these are not of interest in the context of this study.)
3. Flavour effects from LHT u jH W H u iH W H sd ds Although the LHT model does not introduce new operators inaddition to the SM ones, it is not
MFV because of the mirror quarkmixing. New particles contribute to Flavour Changing Neutral Cur-rent (FCNC) processes as shown in the figure. A detailed discussionof Flavour Physics in the LHT model is given in [5].The LHT amplitudes can be written as (e.g. K sector) P i = u,c,t λ Ki F i ( m i , m T + , . . . ) + ξ Ki G i ( m iH , M W H , . . . ), where the firstterm is the T even contribution and the second term is the T oddcontribution. This way the Inami-Lim functions become X K = X SM + X even + ξ Ki /λ Kt X odd , with the CKM factors λ Kt = V ∗ ts V td and the mirror roceedings printed on December 13, 2018 quark mixing ξ Ki = V ∗ isHd V idHd . Because of the CKM hierarchy 1 /λ Kt ≫ /λ B d t ≫ /λ B s t , we expect the largest effects in K physics, but suitable ξ ji can produce large effects also in B d , B s .It has to be checked very carefully whether the LHT effects do notviolate existing experimental FCNC constraints. We studied [6] the con-straints on ∆ M K and ǫ K from the K system, the mass differences in the B system ∆ M B d and ∆ M B s , as well as the CP asymmetry in B d decays S J/ψK S . (Constraints from b → sγ are not a problem, the effects from LHTin this channel are very moderate.)We generated random points in the LHT parameter space, checkedthese constraints and kept only points that fulfill all constraints. Theinput parameters were evenly distributed over their respective 1 σ ranges.Although a lot of points in pa-rameter space have to be triedto find one that does not vi-olate any of the experimentalconstraints, fine tuning is notreally a problem: Typically, ǫ K as generated by arbitrary modelparameters is one or two or-ders of magnitude too large,but there are also many pointsthat generate correct ǫ K without large fine tuning ∆ BG ( O ) = max j (cid:12)(cid:12)(cid:12) p j O ∂O∂p j (cid:12)(cid:12)(cid:12) [7]. Some of the most spectacular points need no fine tuning at all.
4. General results from LHT flavour study
The decays K + → π + ν ¯ ν and especially K L → π ν ¯ ν areexcellent probes of new physicsbecause they can be calculatedvery cleanly. In the LHTmodel, K L → π ν ¯ ν can beenhanced significantly over theSM value (black dot) up to afactor of 3-5, and also K + → π + ν ¯ ν can easily be enhanced to the central value (dashed line) of the cur-rent experimental range. Most data points lie on two axes: One of constant K L → π ν ¯ ν and one parallel to the Grossmann-Nir bound, this is due tothe specific operator structure of the LHT model and distinguishes theexperimental signature from other models. proceedings printed on December 13, 2018 The CP-asymmetry S ψφ of the decay B s → ψφ is much smaller in the SMthan S J/ψK S because the corresponding CKM angle β s is only about − K L → π ν ¯ ν and S ψφ , though possible, seemunlikely. This is very different from the situation between Br ( B s → µ + µ − )and S ψφ , here simultaneous significant effects are rather likely because bothobservables profit from a modified b → s penguin. The enhancement of Br ( B s → µ + µ − ) of up to 30% over the SM result is, however, rather mod-erate compared to e.g. SUSY.Another interesting signa-ture of the LHT model is thecorrelation between the Br’s of K L → µ + µ − SD and K + → π + ν ¯ ν , which is very differentfrom e.g. the RS model withcustodial protection (c.f. con-tribution by B¨orn Duling inthis volume). Correlations likethese might prove instrumentalin distinguishing different models of NP in the experiment. D ¯ D Oscillations (in the LHT model) (This section is based on [8, 9].) D ¯ D is more complicatedthan K ¯ K and B ¯ B mixing: K ¯ K and B ¯ B mixing is dominated byshort-distance physics, i.e. charm/top loops (c.g. figure). D ¯ D has almost no short-distance contribution: The correspondingCKM factors are small and the down-type quarks in the loops too light.Therefore the SM contribution to D ¯ D mixing is long-distance and thereforedifficult to estimate. In our analysis, we vary the SM contribution in areasonable range and use theoretical estimates only to bound the values.The D mass eigenstates are | D / i = 1 / p | p | + | q | (cid:0) p | D i ± q | ¯ D i (cid:1) ,the observables are the normalised mass and width differences, x D ≡ roceedings printed on December 13, 2018 ∆ M D / Γ , y D ≡ ∆Γ D / , as well as q/p ≡ q ( M D ∗ − i Γ D ∗ ) / ( M D − i Γ D ).Obviously CP is violated when | q/p | 6 = 1.Rather recently, D ¯ D oscillations have been observed [10], a measurementreceived with great interest by the commmunity: x D = 0 . +0 . − . , y D =0 . +0 . − . , | q/p | = 0 . +0 . − . . Although this establishes oscillation, CP violation has not (yet) been observed, | q/p | is consistent with 1. In theSM, no significant CP violation is expected.To establish whether the LHTmodel can produce a significant CP violation in the D system, we de-termine ( M D ) SM and (Γ D ) SM so thattogether with the LHT contribution, x D and y D coincide with experiment.This approach is reasonable, becauseeven the expected relative sign of( M D ) SM and (Γ D ) SM [11] does not match the values necessary to repro-duce the measured values of x D and y D with the SM contributions, i.e. verylittle is known about these quantities from the theoretical side. We obtaintwo solutions for each LHT parameter point as shown in the figure.Essentially all LHT parameter points are consistent with expectationsfor the magnitude of SM contributions. In some cases, ( M D ) SM / (Γ D ) SM can be rather large, but these are not our most spectacular/interesting datapoints.Obviously, requiring x D and y D tocoincide with experiment restricts theallowed points to a rather narrow re-gion in the Abs / Arg M D plane. Since V † Hu V Hd = V CKM and the CKM-matrixis rather close to the unity matrix, theexperimental constraints on ǫ K ex-clude points with large Arg M D (lightblue/grey triangles). Even without these points, i.e. ob-serving all experimental constraints,very large (for the D system) CPasymmetries of several percent arepossible. The LHT model couldeven generate asymmetries of ± D → Kφ , but this would cor-respond to semileptonic asymmetries a D SL close to unity. Such large values proceedings printed on December 13, 2018 of a D SL are already excluded by the measurements of | q/p | exp = 0 . +0 . − . because a D SL = ( | q | −| p | ) / ( | q | + | p | ). We can therefore conclude that theLHT model can easily saturate the CP violation in the D system that isstill allowed by current measurements.Let us last look at the correlationbetween the D system and the B s sys-tem: We find that simulataneous largeNP effects in both systems are possible,but unlikely, just as we found that simul-tanous large effects in the K and the B system are unlikely in the LHT model.Again, it is easier to produce large NPeffects that do not violate existing experimental constraints in one sectorthan in two.
6. Conclusions
The LHT model is an interesting, economical alternative to SUSY etc. insolving the Little Hierarchy problem. There are rather few parameters, themodel passes the EW precision tests and (surprisingly, because this is notwhat the model was created for) there are interesting, sometimes spectaculareffects on Flavour observables. For example, large CP violation in D ¯ D oscillations is possible. We hope that in the near future, experimental resultswill show us whether nature has chosen anything like the LHT model forphysics at the TeV scale. REFERENCES [1] N. Arkani-Hamed, A. G. Cohen and H. Georgi, Phys. Rev. Lett. (2001)4757 [hep-th/0104005]; Phys. Lett. B (2001) 232 [hep-ph/0105239].[2] N. Arkani-Hamed, A. G. Cohen, E. Katz and A. E. Nelson, JHEP (2002)034 [hep-ph/0206021].[3] H. C. Cheng and I. Low, JHEP (2003) 051 [hep-ph/0308199].[4] M. Blanke, A. J. Buras, A. Poschenrieder, S. Recksiegel, C. Tarantino, S. Uhligand A. Weiler, Phys. Lett. B , 253 (2007) [hep-ph/0609284].[5] M. Blanke, A. J. Buras, A. Poschenrieder, S. Recksiegel, C. Tarantino, S. Uhligand A. Weiler, JHEP , 066 (2007) [hep-ph/0610298].[6] M. Blanke, A. J. Buras, B. Duling, S. Recksiegel and C. Tarantino,arXiv:0906.5454 [hep-ph]; M. Blanke, A. J. Buras, S. Recksiegel andC. Tarantino, arXiv:0805.4393 [hep-ph].[7] R. Barbieri and G. F. Giudice, Nucl. Phys. B (1988) 63. roceedings printed on December 13, 2018 , 81 (2007) [hep-ph/0703254].[9] I. I. Bigi, M. Blanke, A. J. Buras and S. Recksiegel, JHEP , 097 (2009)[arXiv:0904.1545].[10] B. Aubert et al. , Phys. Rev. Lett. (2007) 211802 [hep-ex/0703020];M. Staric et al. , Phys. Rev. Lett. (2007) 211803 [hep-ex/0703036].[11] I. I. Y. Bigi and N. G. Uraltsev, Nucl. Phys. B (2001) 92 [hep-ph/0005089];A. F. Falk, Y. Grossman, Z. Ligeti, Y. Nir and A. A. Petrov, Phys. Rev. D69