We report on our study of the LFV processes \mu \to e\gamma, \mu\to eee and
\mu \to e conversion in the context of Little Higgs models. Specifically we
examine the Littlest Higgs with T-parity (LHT) and the Simplest Little Higgs
(SLH) as examples of a Product group and Simple group Little Higgs models
respectively. The necessary Feynman rules for both models are obtained in the
't Hooft Feynman Gauge up to order v^2/f^2 and predictions for the branching
ratios and conversion rates of the LFV processes are calculated to leading
order (one-loop level). Comparison with current experimental constraints show
that there is some tension and, in order to be within the limits, one requires
a higher breaking scale f, alignment of the heavy and light lepton sectors or
almost degenerate heavy lepton masses. These constraints are more demanding in
the SLH than in the LHT case.Comment: 6 pages, 3 figures, to appear in Proceedings of the XXXIII Intl.
  Conf. of Theoretical Physics, "Matter to the Deepest", Ustron, Poland,
  September 11-16, 2009; v2: comments and references adde

Illana, Jose Ignacio

Jenkins, Mark D.

Acta Physica Polonica B

English

arXiv

Presented by M.D. Jenkins at the XXXIII International Conference of Theoretical
Physics, “Matter to the Deepest”, Ustroń, Poland, September 11–16, 2009.We report on our study of the LFV processes μ → eγ, μ → eee and μN → eN in the context of Little Higgs models. Specifically, we examine the Littlest Higgs with T-parity (LHT) and the Simplest Little Higgs (SLH) as examples of a Product group and Simple group Little Higgs models, respectively. The necessary Feynman rules for both models are obtained in the ’t Hooft Feynman Gauge up to the order of v2 / f2 and predictions for the branching ratios and conversion rates of the LFV processes are calculated to leading order (one-loop level). Comparison with current experimental constraints show that there is some tension and, in order to be within the limits, one requires a higher breaking scale f, alignment of the heavy and light lepton sectors or almost degenerate heavy lepton masses. These constraints are more demanding in the SLH than in the LHT case

Illana Calero, José Ignacio

Repositorio Institucional Universidad de Granada

Vol. 40 (2009) ACTA PHYSICA POLONICA B No 11LEPTON FLAVOR VIOLATIONIN LITTLE HIGGS MODELS∗José Ignacio Illana, Mark D. JenkinsCAFPE and Departamento de Física Teórica y del CosmosUniversidad de Granada, 18071 Granada, Spain(Received October 26, 2009)We report on our study of the LFV processes µ → eγ, µ → eee andµN → eN in the context of Little Higgs models. Specifically, we examinethe Littlest Higgs with T-parity (LHT) and the Simplest Little Higgs (SLH)as examples of a Product group and Simple group Little Higgs models,respectively. The necessary Feynman rules for both models are obtained inthe ’t Hooft Feynman Gauge up to the order of v2/f2 and predictions for thebranching ratios and conversion rates of the LFV processes are calculatedto leading order (one-loop level). Comparison with current experimentalconstraints show that there is some tension and, in order to be withinthe limits, one requires a higher breaking scale f , alignment of the heavyand light lepton sectors or almost degenerate heavy lepton masses. Theseconstraints are more demanding in the SLH than in the LHT case.PACS numbers: 13.35.Bv, 12.60.Cn, 12.60.Fr, 12.15.Ff1. IntroductionLittle Higgs models [1] introduce a new approximate global symmetryinto the Lagrangian, a subgroup of which is gauged. This symmetry isspontaneously broken at a heavy scale f of the order of 1 TeV. The newgauge group breaks down to the SM gauge group while the SM Higgs doubletthen appears as a pseudo-Goldstone boson of this global symmetry. The lowenergy degrees of freedom are given by a non-linear sigma model which makethis an effective theory valid below the cutoff Λ = 4πf ∼ 10 TeV. Beyondthis cutoff one needs an ultraviolet completion.The new global symmetry is explicitly broken by gauge and Yukawainteractions through a mechanism known as collective symmetry breaking.This allows us to give the Higgs a mass and non-derivative interactions∗ Presented by M.D. Jenkins at the XXXIII International Conference of TheoreticalPhysics, “Matter to the Deepest”, Ustroń, Poland, September 11–16, 2009. Talk alsogiven at the Corfu Summer Institute (2009) by J.I. Illana.(3143)3144 J.I. Illana, M.D. Jenkinsbut preserves the cancellation of the quadratic divergences at the one-looplevel. The dependence on the cutoff reappears at the two-loop level but thissensitivity to a 10 TeV cutoff is not unnatural.The same as in most new physics models, Little Higgs type models allintroduce new gauge bosons and heavy fermions which, in this case, havemasses of the order of the heavy breaking scale f ∼ 1 TeV. These endup giving new contributions to lepton flavor violating processes which wehave analyzed and compared with current experimental limits. We comparetwo different models: the Littlest Higgs with T-parity (LHT) [2] and theSimplest Little Higgs [3] which are examples of Product group and Simplegroup models, respectively.The LHT model introduces, among other particles, heavy partners tothe Standard Model fermions as well as adding four new gauge bosons: W±H ,ZH and AH [4]. The new mixing in the lepton sector comes from the mis-alignment of the heavy and light Yukawa matrices. This is analogous tothe mechanism that produces the quark sector mixing (CKM matrix) in theStandard Model.In the SLH case, each of the lepton families is enlarged to contain anadditional heavy neutrino. Again, there are new heavy gauge bosons inthis model (X±, Z ′, Y 0 and Y 0†). The mixing of the heavy neutrinos withthe light neutrinos in conjunction with a family mixing in the light sectorproduces the new lepton mixing matrix.2. Lepton Flavor Violating processes and their contributionsThe leading order contributions to the LFV processes are all one-loopdiagrams. The topologies contributing to each of the three processes we areinterested in are shown in Fig. 1.The current experimental limits on these processes are B(µ → eγ) <1.2×10−11, B(µ→ eee) < 10−12 and CR(µTi→ eTi) < 4.3× 10−12 [5]. TheStandard Model predictions are much smaller than these limits (order∼ 10−50). This is because the effect is proportional to the neutrino massesdivided by the W± mass to the fourth power. This means that practicallyany LFV signal is a sure sign of physics beyond the Standard Model.3. Results and conclusionsOur procedure to obtain the LFV contributions in the two Little Higgsmodels was as follows [6, 7]:1. We obtained the contributions of all triangle, self-energy and boxdiagrams in terms of generic couplings and standard loop integrals(Passarino–Veltman reduction).Lepton Flavor Violation in Little Higgs Models 3145µ→ eγ:ijVµvertex (triangles)µ→ eee:e2e1µe3+e2e1µe3+(e1 ↔ e2)V -penguins (triangles + SE) e-boxesµN → eN :qeµq+qeµq(q = u, d)V -penguins (triangles+SE) q-boxesFig. 1. Topologies contributing to LFV processes.2. We worked with the Little Higgs Lagrangians of both the LHT andSLH models in order to obtain the relevant Feynman rules for ourprocesses in the ’t Hooft–Feynman gauge. This calculation is done upto the order of v2/f2 for all relevant couplings.3. We substitute the couplings in our generic expressions and simplify asmuch as possible to obtain the form factors for each process analytically.4. Finally the form factors are introduced in to the expressions for theamplitudes and evaluated.In our calculation we find that all contributions are ultraviolet finite in bothmodels for all three processes.To do our numerical analysis we made the simplification of using onlytwo lepton generations. This leaves us with four free parameters that wecan vary. The idea is to set three of the parameters to natural values andmove the fourth parameter. The four parameters are:• The breaking scale f .• The mixing angle θ appearing in the (now 2× 2) mixing matrix V `H .• An average mass of the two heavy leptons given by ỹ = mH1mH2/M2Gwhere MG is the mass of the charged heavy gauge boson (X± in SLHand W±H in LHT) and mHi is the mass of the heavy lepton.• A relative splitting between the heavy lepton masses δ = m2H2−m2H1mH1mH2.3146 J.I. Illana, M.D. JenkinsThe natural values for these parameters are f ∼ 1 TeV, θ ∼ π/4, ỹ ∼ 1 andδ ∼ 1. Other parameters have also been set to typical values. Specifically,the heavy quark masses appearing in µN → eN are all set tomQ = 500 GeVand we have assumed minimal mixing in the quark sector. In the SLH modelthere is an extra parameter β that has been set to tanβ = 1.Figures 2 and 3 show the ratio of the Little Higgs predictions to thepresent experimental limits for the three lepton flavor violating processesunder study. A complete analysis can be found in [6, 7]...LHTδRatio0.20.150.10.050-0.05-0.1-0.15-0.2102101110−110−2µ→ e conv.µ→ eeeµ→ eγ..LHTfRatio20181614121086420102101110−110−2Fig. 2. Ratio of predictions to current experimental limits for the LHT model...SLHδRatio0.20.150.10.050-0.05-0.1-0.15-0.2102101110−110−2µ→ e conv. (Univ.)µ→ e conv. (A.F.)µ→ eeeµ→ eγ..SLHfRatio20181614121086420102101110−110−2Fig. 3. Ratio of predictions to current experimental limits for the SLH model.Lepton Flavor Violation in Little Higgs Models 3147The form factors and amplitudes have an approximate behavior given byA ∝ v2f2sin 2θ δ . (1)The dependence on ỹ is more involved and cannot be expressed as simply.In the LHT model we see that, in general, the process that gives thestrongest restrictions on the parameter space is µTi→ eTi. We see that inorder to get the prediction within the limit we have several options. Onecan make the breaking scale f somewhat heavier (f & 4.7 TeV) or onecan make either the mixing or mass splitting small, i.e., sin 2θ . 0.05 orδ . 5%. These considerations make the model somewhat unnatural. On theone hand, if you raise the scale you defeat the basic purpose of the modeland, on the other, if you reduce the angles and splittings you are forcinga fine-tuning of the light and heavy mass matrices to within a few percent.One would need some additional symmetry in the model to justify this.In the SLH model the results are similar. Note that here there aretwo possible embeddings for the quark sector: the Universal embedding andthe Anomaly Free embedding. This produces different results in the caseof µN → eN (see Fig. 3). The heavier restrictions again come from theµ→ e conversion process and they are much more extreme in the Universalembedding. To fulfill the experimental constraints we need a heavy scalef & 8 TeV or very small mixing angles or splittings, i.e., sin 2θ . 0.01 orδ . 1%. The same considerations on naturalness that apply in the LHTmodel, apply in the SLH model. However, the restrictions are more de-manding in this case.Certain very small regions of the parameter space seem to be alloweddue to incidental cancellations of loop functions. However, these regions areruled out when all processes are taken into account. In other words, thesecancellations never happen in all three processes simultaneously.REFERENCES[1] N. Arkani-Hamed, A.G. Cohen, H. Georgi, Phys. Rev. Lett. 86, 4757 (2001).[2] N. Arkani-Hamed, A.G. Cohen, E. Katz, A.E. Nelson, J. High Energy Phys.0207, 034 (2002).[3] D.E. Kaplan, M. Schmaltz, J. High Energy Phys. 0310, 039 (2003).[4] M. Blanke et al., J. High Energy Phys. 0705, 013 (2007).[5] C. Amsler et al. [Particle Data Group], Phys. Lett. B667, 1 (2008).[6] F. del Águila, J.I. Illana, M.D. Jenkins, J. High Energy Phys. 0901, 080 (2009).[7] F. del Águila, J.I. Illana, M.D. Jenkins, work in preparation.

Lepton Flavor Violation in Little Higgs Models

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