# Lepton flavor violation in the Simplest Little Higgs model

###### Abstract

Little Higgs Models are a possible elegant solution to the hierarchy problem on the Higgs mass. As they predict naturally small deviations with respect to SM results, they are in agreement with all current experimental data. In this contribution, we review lepton flavor violation in the Simplest Little Higgs model focusing on semileptonic lepton flavor violating tau decays (where some new results are presented) and . Within this model, the most promising decay channels for discovering lepton flavor violation are conversion in nuclei, , and .

###### keywords:

Lepton Flavor Violation, Tau decays, Higgs decays, Composite Higgs models## 1 Motivation

I will start recalling the case for searches and studies of lepton flavor violation and also the interest of analysing it within (the Simplest) Little Higgs models.

### 1.1 Lepton Flavor Violation (LFV)

The discovery of neutrino oscillations evidences their non-vanishing mass and makes the charged lepton sector the only fermion subarea where flavor violation has not been unveiled yet. Moreover, the minimal extension of the SM with massive neutrinos predicts non-zero (although undetectable) branching ratios for charged LFV processes, i. e. mu2eg (). Therefore, the discovery of charged LFV would correspond, necessarily, to the effect of new dynamics.

For this reason there is an extensive hunt for new physics in searches for LFV muon, tau, Higgs, decays and conversion in nuclei; on which we had several experimental talks at this workshop exptalks () (see also the corresponding sections of Emilie ()). Without dwelling in more detail into these, let us just highlight the impressive upper limit recently achieved by MEG MEG (), which is a stringent constraint on new physics models. In parallel to this exhaustive experimental activity there is not a corresponding effort on the theory side (but for the easiest decays) and limited activity on the semileptonic LFV tau decays has been carried on TheoryLFV ().

LFV is not intrinsically related to any of the known problems of the Standard Model (SM): dark matter, baryon asymmetry of the universe, dark energy, (little) hierarchy problem, flavor problem, etc. However, it will be hopefully linked to any/some of them, so that its eventual measurement will shed light on any of these issues, helping to find the next standard theory.

### 1.2 Simplest Little Higgs (SLH) model

Little Higgs models arise as an elegant solution to the (little) hierarchy problem on the Higgs mass: since the Higgs boson couples proportionally to others’ particles masses, its mass would get huge quantum loop corrections in the presence of generic heavy new physics. Therefore, GeV would need to result from an extreme fine-tuning among the diverse corrections. A theoretically beautiful solution to this problem is provided by Supersymmetry, but the absence of SUSY particles at a TeV questions that Nature chose this way. Another classical solution to the problem comes from the analog with QCD. Technicolor and its different evolutions again face naturalness problems when confronted to the lack of their imprints on LHC data. Still, the idea of composite Higgs models CompositeHiggs () can be the starting point to formulate a theory in which the Higgs boson is naturally light that accords with all present observations.

Scalar boson masses are not protected by any symmetry, however the pion is so light because it is the pseudo-Nambu-Goldstone boson of chiral symmetry breakdown. The idea of LH models is to justify the small Higgs mass similarly, as a consequence of the breaking of a global symmetry. These models assume a scale of compositeness (above which the new global symmetry is also displayed), which is much smaller than the electroweak vev ( TeV) and the structure of the model is arranged so that the Higgs mass is radiatively generated. There are new ’little’ particles with masses of and the UV completion of the model is expected at some tens of TeVs, where the theory would become strongly coupled ( TeV). Thus we can expand perturbatively our amplitudes in and keep only the leading term.

Among the LH models there are product group and simple group models (). Since the former need and ad-hoc symmetry (T-parity) to solve the hierarchy problem, we will take the simplest of the latter () for our study of LFV tau Lami:2016vrs () and Higgs decays OurLetter () that we present in Sects. 3 and 4 preceded by a short account on the SLH model next (see also PosterSLH ()).

## 2 A brief sketch of the SLH model

The symmetry structure of the SLH model SLH_bib () is given by , where only the diagonal group is gauged. There are two different symmetry breakdowns (requiring two complex scalar fields, triplets under and , respectively): on the one hand the gauged diagonal subgroup is broken down to the SM electroweak gauge group, yielding 5 Goldstone bosons which give mass to the additional ’little’ gauge bosons (among these only and play a rôle in our study). On the other hand, the global symmetry is broken similarly, with associated Goldstone bosons including the Higgs degrees of freedom.

Every fermion family contains a left-handed triplet (adding to the SM doublets one ’little’ particle) and corresponding singlets. Heavy neutrinos, () are fundamental to our study, since they drive the LFV through their couplings. Though the quark sector is not unambiguously defined, we will follow the anomaly-free embedding for the new quarks. Under reasonable assumptions, only the first generation ’little’ quark, , matters to our discussion delAguila:2011wk ().

## 3 LFV tau decays

We presented our results for (; is short for pseudoscalar meson and for vector resonance) in Ref. Lami:2016vrs () ^{1}^{1}1The SLH preserves lepton universality. As a result,
we obtain the same branching ratios irrespective of .. There we decided to include the effect
of only two heavy neutrinos in our analysis. This corresponds to the case where there is a GIM-like mechanism acting in the mixing matrix among charged leptons and heavy neutrinos which effectively
decouples . In this scenario, also the contributions of and cancel each other partially (according to the similarity of their masses). Here, instead, we will consider the most general
scenario where no particular pattern of this mixing matrix is assumed. Generally, this will increase our predicted LFV observables.

The one-loop diagrams contributing to these decays can be seen in figures 1, 2 and 3 (with the , and box-mediated contributions, respectively). We
have computed them in the unitary gauge, where only physical particles appear. As a result, the cancellation of divergences becomes subtle, and the sum of the divergences of the penguin-like diagrams is
cancelled by that of the box contributions. Parity forbids -mediated contributions to the processes with one pseudoscalar meson. However, since these and the box-mediated contributions turn out to
be of similar magnitude ^{2}^{2}2 and contributions are negligible in all cases. decays are predicted at a comparable rate to the processes. Along our
computation we have kept the leading term in the expansion parameter and set , and to zero ^{3}^{3}3 also sets the largest scale of external momenta, which are then
negligible in the evaluation of the loop integrals..

As a result of the embedding of the SM group into the SLH group, the only couplings entering the amplitudes are the SM and couplings. In addition, the expressions depend on the ratios
and . appears in the change between the flavor and mass bases for
the (light and heavy) neutrinos and turns out to be an important parameter allowing to set the bound TeV delAguila:2011wk () (in the two-heavy neutrino scenario) studying
, and in the SLH model ^{4}^{4}4A thorough discussion of the phenomenological relevance of these decays modes can be found in this reference..

The remaining expressions contain quark bilinear currents which still need to be hadronized to make contact with the experimental searches. This is done in an essentially model-independent way, writing
those fermion bilinears in terms of the QCD quark currents and proceeding to their hadronization guided by chiral symmetry ChPT (), axiomatic field theory properties implemented naturally through
dispersion relations Sergi () and the QCD asymptotics BrodskyLepage (), benefitting as well from the precise data at our disposal on two-meson factors. For the these, we use the expressions
given in Refs. 2mesonFFs () ^{5}^{5}5In the one-meson case, the hadronization is encoded in the corresponding meson decay constant and the mixing parameters Lami:2016vrs ().
The cases can be obtained by considering the dominant channel with the di-meson system around the mass and width of the vector resonance ( for the and for the
)..

For our phenomenological analysis within the SLH model, we have varied randomly the model parameters in the ranges TeV TeV, tan, keeping its product below TeV. In the mixing matrix between charged leptons and heavy neutrinos we have neglected CP violation but kept it general otherwise. We have allowed for a factor of up to ten in the ratio between successive heavy neutrino masses and verified that all LFV low-energy constraints were satisfied before admitting a point in the model’s parameter space. This restriction is needed, as can be seen from fig. 4.

Within the SLH model, the correlation between the most restricting low-energy process and the most abundant one- and two-meson LFV tau decays ^{6}^{6}6 and are extremely
correlated, as can be seen in Lami:2016vrs (). This is a result of the hadronization process, as the channel is driven by the -exchange, while the one-pion mode is saturated by the box contribution. is plotted in figures
5 and 6, respectively. In both figures the x-axis is cut at the 90 C.L. upper limit for . The solid line in Fig. 5 indicates the corresponding upper limit for .
A similar line is not shown on figure 6 because if the other low-energy restrictions on LFV processes are fulfilled, is at least four orders of magnitude below
its corresponding upper bound. Thus, in the SLH model, the only semileptonic LFV tau decays that can compete with , and as golden channels for the detection of LFV are
and (which is only a factor smaller than the mode).

Three-dimensional plots that allow to represent the simultaneous dependence of the branching ratios on two model parameters can be found in my talk’s file MyTalk (). This, however, does not yield any new information (provided
the GIM-like suppression is understood) with respect to the most conventional 2-D plots. Then, the dependence of the results on the model parameters is basically the one found for the case with only two effective heavy
neutrinos in the 2-D plots of Ref. Lami:2016vrs (): results depend quite mildly on , tan, max (sin for the GIM-like scenario) and the heavy neutrino
spectroscopy. As expected, BRs:

- decrease with f according to the dependence of the amplitude on ,

- are almost constant for tan, while they exhibit a marked narrow dip around tan, where the BR is reduced by an order of magnitude.

- increase as sin (similarly for max).

- vary smoothly with the neutrino masses hierarchy. In the GIM-like case, the suppression of the BRs gets stronger for .

## 4 LFV Higgs decays

It has not been necessary to update our analyses of OurLetter (). The dependence on the model parameters follows the patters explained in Sect. 3. As it can be seen in fig. 7, even in the case with three
active heavy neutrinos (where the considered LFV Higgs decays BRs are four orders of magnitude larger than in the GIM-like scenario OurLetter ()) the SLH predicts unmeasurable BRs at LHC, provided the low-energy
constraints on LFV processes are satisfied. Similar small BRs for these decays have been found recently within LH models LHresults () ^{7}^{7}7SLH models also predict generally small departures from the SM in Higgs
couplings, which are in good agreement with present measurements Han:2013ic ().. We refer the reader to Ref. OurLetter () for a complete discussion of our results on LFV Higgs decays.

## 5 Conclusions

Little Higgs models (particularly SLH) remain as elegant candidates to alleviate the hierarchy problem on the Higgs mass, respecting all experimental bounds. (S)LH models predict small LFV decay rates which could escape detection at Belle-II and (specially) at LHC. Within SLH, LFV detection should be easier for a general (not GIM-like) pattern of the 3 heavy neutrinos of the model. In that case, the most promising channels for its discovery would be , , and .

## Acknowledgements

The author wants to thank and congratulate the local and international advisory Committees for their organization of the fruitful TAU’16 Workshop. I acknowledge the collaboration with Jorge Portolés and Andrea Lami in the research reported in this proceedings contribution. I am grateful to Jorge Portolés for his critical reading of the manuscript of this text.

## References

- (1) S. T. Petcov, Sov. J. Nucl. Phys. 25 (1977) 340 [Yad. Fiz. 25 (1977) 641] Erratum: [Sov. J. Nucl. Phys. 25 (1977) 698] Erratum: [Yad. Fiz. 25 (1977) 1336]. W. J. Marciano, A. I. Sanda, Phys. Lett. B67 (2016), 303-305. S. M. Bilenky, S. T. Petcov and B. Pontecorvo, Phys. Lett. B67 (2016), 309. T-P. Cheng, L-F. Li Phys. Rev. D16 (1977), 1425.
- (2) Talks given by A. Nehrkorn, M. Liu, K. De Bruyn, S. Eidelman, A. Bravar and C. Wu at the TAU’16 Workshop, these proceedings.
- (3) E. Passemar, introductory talk at the TAU’16 Workshop, these proceedings. A. Lusiani, summary talk of the TAU’16 Workshop, these proceedings.
- (4) A. M. Baldini et al. [MEG Collaboration], Eur. Phys. J. C 76 (2016) no.8, 434.
- (5) A. Brignole, and A. Rossi, Nucl. Phys. B701 (2004), 3-53. T. Fukuyama, A. Ilakovac, and T. Kikuchi, Eur. Phys. J. C56 (2008), 125-146. E. Arganda, M. J. Herrero and J. Portolés, JHEP (06)2008, 079. M. J. Herrero, J. Portolés and A. M. Rodríguez-Sánchez, Phys. Rev. D80 (2009), 015023. W. Liu, C-X. Yue, and J. Zhang, Eur. Phys. J. C68 (2010), 197-207. T. Goto, Y. Okada and Y. Yamamoto, Phys. Rev. D83 (2011), 053011. T. Hua, and C-X. Yue, Commun. Theor. Phys. 62 (2014), no. 3, 388-392. A. Celis, V. Cirigliano and E. Passemar, Phys. Rev. D89 (2014), 013008.
- (6) H. Georgi, D. B. Kaplan, Phys. Lett. B145 (1984), 216. M. J. Dugan, H. Georgi and D. Kaplan, Nucl. Phys. B254 (1985), 299.
- (7) A. Lami, J. Portolés and P. Roig, Phys. Rev. D 93 (2016) no.7, 076008.
- (8) A. Lami and P. Roig, Phys. Rev. D 94 (2016) no.5, 056001.
- (9) Poster presented by M. A. Arroyo-Ureña at TAU’16, these proceedings.
- (10) D. E. Kaplan and M. Schmaltz, JHEP 0310 (2003) 039. M. Schmaltz, JHEP 0408 (2004) 056.
- (11) F. del Águila, J. I. Illana and M. D. Jenkins, JHEP 1103 (2011) 080.
- (12) S. Weinberg, Physica A 96 (1979) 327. J. Gasser and H. Leutwyler, Annals Phys. 158 (1984) 142; Nucl. Phys. B 250 (1985) 465. G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B 321 (1989) 311. G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafael, Phys. Lett. B 223 (1989) 425.
- (13) Talks given by Sergi Gonzàlez-Solís (one replacing Rafel Escribano, who unfortunately could not finally attend the conference) at the TAU’16 Workshop, these proceedings.
- (14) S. J. Brodsky and G. R. Farrar, Phys. Rev. Lett. 31 (1973) 1153. G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22 (1980) 2157.
- (15) D. Gómez Dumm and P. Roig, Eur. Phys. J. C (2013) 73: 2528. O. Shekhovtsova, T. Przedzinski, P. Roig and Z. Was, Phys. Rev. D 86 (2012) 113008.
- (16) It can be found at http://indico.ihep.ac.cn/event/5221/other-view?view=standardall.detailed .
- (17) B. Yang, J. Han and N. Liu, arXiv:1605.09248 [hep-ph]. S. Chamorro, A. Moyotl and M. Pérez, to be published in Journal of Physics Conference Series (IOP). Joint Proceedings of the XV Mexican Workshop on Particles and Fields the XXX Annual Meeting of the Division of Particles and Fields of the Mexican Physical Society. See also A. Moyotl’s poster at TAU’16, these proceedings.
- (18) X.-F. Han, L. Wang, J. M. Yang and J. Zhu, Phys. Rev. D87 (2013), no. 5, 055004.