CAFPE-109/08 UG-FT-239/08 9 0 Precise limits from lepton flavour violating processes 0 2 on the Littlest Higgs model with T-parity n a J 7 2 F. del A´guila, J. I. Illana and M. D. Jenkins ] h p CAFPE and Departamento de F´ısica Te´orica y del Cosmos, - p Universidad de Granada, E–18071 Granada, Spain e h [ [email protected], [email protected], [email protected] 3 v 1 9 8 2 Abstract . 1 1 8 We recalculate the leading one-loop contributions to µ eγ and µ ee¯e in the → → 0 Littlest Higgs model with T-parity, recovering previous results for the former. When : v all the Goldstone interactions are taken into account, the latter is also ultraviolet Xi finite. The present experimental limits on these processes require a somewhat heavy effective scale 2.5 TeV, or the flavour alignment of the Yukawa couplings of light r a ∼ and heavy leptons at the 10% level, or the splitting of heavy lepton masses to a ∼ similar precision. Present limits on τ decays set no bounds on the corresponding parameters involving the τ lepton. Contents 1 Introduction 3 2 The Littlest Higgs model with T-parity 4 2.1 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Flavour mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 New contributions to LFV processes 10 3.1 µ eγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 → 3.2 µ ee¯e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 → 4 Numerical results 23 5 Conclusions 27 A Physical fields 30 B Feynman rules 31 B.1 SM with massive neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 B.2 LHT model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 C Loop integrals 34 C.1 Two-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 C.2 Three-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 C.3 Four-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 1 Introduction Little Higgs models [1] offer an explanation to the little hierarchy between the Higgs mass M assumed to be near the electroweak scale v = 246 GeV and the new physics (NP) scale h f, whose natural value is expected to be 1 TeV [2]. In contrast with supersymmetry, ∼ where the large one-loop Standard Model (SM) contributions to the Higgs mass are can- celled by the contributions from the corresponding supersymmetric partners with masses 1 TeV and spins differing by 1/2 (see [3] and references therein), Little Higgs (LH) ∼ ± models stabilize M by making the Higgs a pseudo-Goldstone boson of a broken global h symmetry. The cancellation is in this case between particles with the same spin belonging to the same multiplets of this approximate symmetry. Which of these SM extensions, if any, is at work will be hopefully established at the LHC [4,5]. Supersymmetry is linearly realized in the minimal supersymmetric SM (MSSM). This and other simple supersymmetric extensions of the SM have other interesting phenomeno- logicalproperties,like,forinstance, theunificationofgaugecouplingsatveryhighscales[6], which is not the case for LH models. However, they have no built-in low energy mechanism to explain the observed fermion mass hierarchy or flavour conservation. As a matter of fact, it can be argued that supergravity is phenomenologically relevant [3,7] because it can provide the necessary initial conditions to explain the precise fine-tuning required among the many new parameters of the MSSM, which otherwise would result in too large flavour changing processes [8]. This has been historically the problem of many SM extensions [9]. If the NP is near the TeV scale, it faces in general the problem of naturally explaining why it is aligned with the SM Yukawa interactions, as experimentally required. Although the SMcannotexplain thelargehierarchy between fermionmasses, which by thewayisseveral orders of magnitude more demanding than the little hierarchy, it naturally accommodates the absence of flavour changing neutral currents (FCNC) [10]. LH models are not designed to solve the flavour puzzle either, and one must expect stringent constraints on the new parameters involving the heavy sector. The study of FCNC processes in Littlest Higgs models has been addressed in the literature [11]. In this paper we revise the calculation of the decay rates of the lepton flavour violating (LFV) processes µ eγ [12,13] and → µ ee¯e [13] in the Littlest Higgs model with T-parity (LHT) [14], obtaining an ultravio- → let finite result also for the latter.1 Indeed, when all Goldstone boson interactions of the new leptons are taken into account, the one-loop contributions to the amplitudes are well- defined [16,17], scaling approximately in the two family case like (v2/f2)sin2θ δ, where θ is a measure of the misalignment between the heavy and SM lepton Yukawa couplings and δ is the corresponding heavy lepton mass splitting. As a consequence, the present experimental limits require fine tuning the Yukawa couplings of the new heavy leptons up to 10%, aligning them with their SM counterparts, or making the heavy masses quasi- degenerate. One might also rise the NP scale degrading the motivation of the LH scenario 1See also [15] which appeared when we were preparing this manuscript. There the cancellation of ultraviolet divergences in the LHT model is also shown, flavour violation in the quark sector is explored and the phenomenology of K πνν¯ is analyzed. → 3 itself. In the general case with three families the new contributions must be tuned to a similar precision but the parameter dependence is more involved. The calculation also applies to τ decays, but the corresponding limits are not restrictive at present. Moreover, it can be easily extended to µ e conversion in nuclei [18]. A complete phenomenological − analysis comparing as well different LH models will be presented elsewhere. In LH models the Higgs is a pseudo-Goldstone boson. Thus, M is naturally small as h longasthenewscalef isrelativelylow, becauseoneexpectsthatcancellationsareonlypro- tected to one loop and for the dominant contributions. Hence, 4πf can not be much larger than 10 TeV if we do not wish to invoke some fine tuning again. However, as the model introduces heavy particles the new one-loop contributions to electroweak observables may require rising f significantly above 1 TeV in the absence of model dependent cancellations, in order to be consistent with present electroweak precision data (EWPD) [19]. The LHT is an economical realization of the LH scenario with the further virtue of keeping the new contributions to EWPD small. It incorporates a discrete symmetry under which the new particles areoddandtheSMoneseven. Then allvertices musthave aneven number ofnew particles, if any. Similarly to the R symmetry in supersymmetric models, the T symmetry allows us to weaken the experimental limit on the LH effective scale below the TeV [20]. This symmetry also makes stable the lightest T-odd particle, offering, like R-parity does in the supersymmetric case, an alternative candidate for cold dark matter [21]. Nevertheless, as already emphasized these models are not a priori designed to deal with the flavour problem. Therefore, it is important to investigate the constraints on the model parameters implied by the stringent experimental limits on FCNC. We follow an operational approach and calculate the leading contributions to µ eγ and µ ee¯e → → in the LHT, recovering previous results for the former [12,13] but an ultraviolet finite result for the latter. We focus on these processes because the lepton sector is free from large strong corrections, and the experimental limits are quite demanding. In Section 2 we review the LHT model to introduce the notation and the Feynman rules needed. The one-loop amplitudes of the LFV processes µ eγ and µ ee¯e are discussed in Section 3. → → The calculation is straightforward but cumbersome, requiring a careful bookkeeping of the different terms. In Section 4 we present the numerical results discussing the dependence on the different parameters of the model. Finally, Section 5 is devoted to conclusions, where we also briefly comment on τ decays. 2 The Littlest Higgs model with T-parity 2.1 The Lagrangian The LHT is a non-linear σ model based on the coset space SU(5)/SO(5), with the SU(5) global symmetry broken by the vacuum expectation value (VEV) of a 5 5 symmetric × 4 tensor, 0 0 1 2×2 2×2 Σ =  0 1 0 . (2.1) 0  12×2 0 02×2      The10unbrokengeneratorsTa,whichleaveinvariantΣ andthensatisfyTaΣ +Σ (Ta)T = 0 0 0 0, expand the SO(5) algebra; whereas the 14 broken generators Xa, which fulfill XaΣ 0 − Σ (Xa)T = 0, expand the Goldstone fields Π = πaXa parameterized as 0 Σ(x) = eiΠ/fΣ eiΠT/f = e2iΠ/fΣ , (2.2) 0 0 where f is the effective NP scale. Only the [SU(2) U(1)] [SU(2) U(1)] subgroup of 1 2 × × × the SU(5) global symmetry is gauged. It is generated by σa 0 0 1 1 Qa =  0 0 0 , Y = diag(3,3, 2, 2, 2), (2.3) 1 2 1 10 − − −  0 0 02×2      0 0 0 2×2 1 1 Qa =  0 0 0 , Y = diag(2,2,2, 3, 3), (2.4) 2 2 2 10 − −  0 0 σa∗   −    with σa the three Pauli matrices. The VEV in Eq. (2.1) breaks this gauge group down to the SM gauge group SU(2) U(1) , generated by the combinations Qa+Qa, Y +Y L× Y { 1 2 1 2} ⊂ Ta . The orthogonal combinations are a subset of the broken generators, Qa Qa, Y { } { 1− 2 1− Y Xa . Thus, the Goldstone fields 2 } ⊂ { } ω0 η ω+ π+ Φ+ i iΦ++ i − 2 − √20 −√2 − √2 − − √2  ω− ω0 η v +h+iπ0 Φ+ iΦ0 +ΦP  i −  −√2 2 − √20 2 − √2 √2     π− v +h iπ0 4 π+ v +h+iπ0  Π =  i − η i  (2.5)  √2 2 5 − √2 2   r   Φ− π− ω0 η ω−   iΦ−− i i   √2 √2 − 2 − √20 −√2     Φ− iΦ0 +ΦP v +h iπ0 ω+ ω0 η   i −   √2 √2 2 −√2 2 − √20      decompose into the SM Higgs doublet ( iπ+/√2,(v + h + iπ0)/2)T, a complex SU(2) L − triplet Φ, and the longitudinal modes of the heavy gauge fields ω±,ω0 and η.2 2 In the following we use for the SM fields and couplings the conventions in Ref. [22]. In particular, φ+ = iπ+, φ0 =π0. − 5 As emphasized in the previous section, we can make the new contributions to elec- troweak precision observables small enough introducing a T-parity under which the SM particles are even and the new particles are odd. An obvious choice for the action of such T-parity on the gauge fields G is the exchange of the gauge subgroups [SU(2) U(1)] and i 1 × [SU(2) U(1)] , 2 × T G G . (2.6) 1 2 ←→ Then, T invariance requires that the gauge couplings associated to both factors are equal. This leaves the following gauge Lagrangian unchanged, 2 1 1 = Tr W Wµν B Bµν , (2.7) LG −2 jµν j − 4 jµν j Xj=1 (cid:20) (cid:16) (cid:17) (cid:21) f f where W = Wa Qa, W = ∂ W ∂ W ig W ,W , B = ∂ B ∂ B .(2.8) jµ jµ j jµν µ jν − ν jµ − jµ jν jµν µ jν − ν jµ h i (Sfummation over ifndex a, wfhich runsfon the cofrrespfonding SU(2) generators, is always assumed when repeated.) The T-even combinations multiplying the unbroken gauge gen- erators correspond to the SM gauge bosons, 1 W3 +W3 B +B W± = [(W1 +W1) i(W2 +W2)], W3 = 1 2 , B = 1 2, (2.9) 2 1 2 ∓ 1 2 √2 √2 whereas the T-odd combinations 1 W3 W3 B B W± = [(W1 W1) i(W2 W2)], W3 = 1 − 2 , B = 1 − 2, (2.10) H 2 1 − 2 ∓ 1 − 2 H √2 H √2 expand the heavy gauge sector. In order to ensure that the SM Higgs doublet is T-even and the remaining Goldstone fields are T-odd, the T action on the scalar fields is defined as follows, T Π ΩΠΩ, Ω = diag( 1, 1,1, 1, 1), (2.11) −→ − − − − − where Ω is an element of the center of the gauge group,3 which commutes with Σ but not 0 with the full global symmetry. Then, Σ T Σ = ΩΣ Σ†Σ Ω, (2.12) 0 0 −→ and the scalar Lagrangian e f2 = Tr (D Σ)†(DµΣ) , (2.13) S µ L 8 (cid:2) (cid:3) 3 Note that we have reversed the sign of Ω as compared to the literature, to make it a group element. 6 with 2 D Σ = ∂ Σ √2i gWa (QaΣ+ΣQaT) g′B (Y Σ+ΣYT) , (2.14) µ µ − jµ j j − jµ j j j=1 X(cid:2) (cid:3) is also gauge and T-invariant. This discrete symmetry must be implemented in the fermion sector too. This is less straightforward. In fact, there is no proposed model fulfilling the three desired conditions: to give masses to all (SM) fermions with Yukawa couplings, preserving a discrete sym- metry under which all new particles are odd and the SM ones even, and keeping the full global symmetry before introducing the symmetry breaking. Although terms explicitly breaking the global symmetries at the Lagrangian level must manifest as badly behaved contributions to physical processes [17], this will not be our case since all the explicit couplings entering in the calculation we are interested in can be derived from Lagrangian terms which are symmetric. Following Refs. [23,24] we introduce two left-handed fermion doublets in incomplete SU(5) multiplets, one transforming just under SU(2) and the other 1 under SU(2) , for each SM left-handed lepton doublet: 2 iσ2l 0 1L − Ψ =  0 , Ψ =  0 , (2.15) 1 2  0   iσ2l2L     −      ν iL where l = , i = 1,2, and iL ℓ ! iL Ψ V∗Ψ , Ψ VΨ , (2.16) 1 1 2 2 −→ −→ under an SU(5) transformation V. We define the T-parity action on these fermions T Ψ ΩΣ Ψ . (2.17) 1 0 2 ←→ Then the usual T-even combination Ψ +ΩΣ Ψ remains light and is identified, up to the 1 0 2 proper normalization, with the SM fermion doublet. The T-odd combination Ψ ΩΣ Ψ 1 0 2 − pairs with a right-handed doublet (eigenvector of T), in a complete SO(5) multiplet, · T Ψ =  , Ψ ΩΨ , Ψ UΨ , (2.18) R R R R R · −→ −→  iσ2lHR   −    where U is an SO(5) transformation defined below, to form a heavy Dirac doublet. With this aim in mind, a non-linear Yukawa Lagrangian is introduced, = κf Ψ ξ +Ψ Σ ξ† Ψ +h.c. , (2.19) LYH − 2 1 0 R (cid:0) (cid:1) 7 where ξ = eiΠ/f. This is indeed T-invariant, since Eq. (2.11) implies ξ T Ωξ†Ω, (2.20) −→ and invariant under global transformations, Σ = ξ2Σ VΣVT ξ VξU† UξΣ VTΣ , (2.21) 0 0 0 −→ ⇒ −→ ≡ where V is the global SU(5) tranformation and U a function of V and Π taking values in the Lie algebra of the unbroken SO(5). It must be noted that the gauge singlet χ , R completing the SO(5) representation ˜ ψ R Ψ =  χ  (2.22) R R  iσ2lHR   −    and assumed to be heavy, is T-even.4 On the other hand, the extra doublet ψ˜ , which is R also assumed to be heavy enough to agree with EWPD, is T-odd as desired. We have just introduced all heavy fields we need. However, one important comment is in order. The Yukawa-type Lagrangian fixes the transformation properties of the LYH heavy fermions and then their gauge couplings, in particular the non-linear couplings of the right-handed heavy fermions [25], = iΨ γµD∗Ψ + iΨ γµD Ψ LF 1 µ 1 2 µ 2 1 1 + iΨ γµ ∂ + ξ†(D ξ)+ ξ(Σ D∗Σ ξ†) Ψ (2.23) R µ 2 µ 2 0 µ 0 R (cid:18) (cid:19) with D = ∂ √2ig(Wa Qa +Wa Qa)+√2ig′(Y B +Y B ). (2.24) µ µ − 1µ 1 2µ 2 1 1µ 2 2µ The Lagrangian of Eq. (2.23) includes the proper O(v2/f2) couplings to Goldstone fields, absent in [13,26], that render the one-loop amplitudes ultraviolet finite. Besides, in order to assign the proper SM hypercharge y = 1 to the charged right-handed leptons ℓ , which R − are SU(5) singlets and T-even, one can enlarge SU(5) with two extra U(1) groups, since otherwise their hypercharge would be zero. Then, the corresponding gauge and T invariant Lagrangian reads ′ = iℓ γµ(∂ +ig′yB )ℓ . (2.25) LF R µ µ R 4 If we had defined the T action on the fermions Ψ1 T Σ0Ψ2, ΨR T ΨR and the Yukawa ←→ − −→ − Lagrangianwith Ω’s,LYH =−κf Ψ2ξ+Ψ1Σ0Ωξ†Ω ΨR+h.c., allnew fermionswouldbe T-oddandthe new Lagrangianinvariantunder the new T-parity [24], but not under the full globalsymmetry because Ω (cid:0) (cid:1) does not commute with SU(5) neither with SO(5), although it does commute with the gauge group. We must insist that the explicit couplings entering in our calculation are the same in both cases. 8 For the lepton sector and the calculation we are interested in these are all the necessary Lagrangian terms. However, in order to define what a muon or an electron is, we have to diagonalise the mass matrix (M ) = (λ ) v in the corresponding Yukawa Lagrangian ℓ ij ℓ ij Y L which we assume to have all required properties [25,27]5 (and also to include light neutrino i masses). This gives to leading order a mass term for the charged leptons m ℓ ℓi +h.c., ℓi L R with mℓi′δi′j′ = (VLℓ†)i′i(λℓ)ij(VRℓ)jj′ v (2.26) and Vℓ two unitary matrices.6 L,R Finally, in order to perform the calculation in the mass eigenstate basis we have to diagonalise the full Lagrangian = + + + + ′ + , (2.27) L LG LS LYH LF LF LY and reexpress it in the mass eigenstate basis. The corresponding masses and eigenvectors uptoorderv2/f2 aregiveninAppendixA. TheFeynmanrulesarecollectedinAppendixB. They are obtained expanding to the required order. The coupling overlooked in [13] is L the v2/f2 correction to the right-handed coupling g of the Zν¯i νj vertex, resulting from R H H the expansion of the last two terms of in Eq. (2.23). F L 2.2 Flavour mixing The new contributions to charged LFV processes must be proportional to the ratio of the electroweak and the LHT breaking scales v2/f2 and to a combination of the matrix elements describing the misalignment of the heavy and charged lepton Yukawa couplings. Let us then set our conventions for the description of the heavy-light mixing relevant to our analysis, and in particular to the Feynman rules discussed above and collected in 5 Right-handed leptons, as the other right-handedSM fermions, are usually taken to be singlets under the non-abelian symmetries, transforming only under the gauge abelian subgroup. We must note that this may be a too strong assumption. If we want to couple them to their left-handed counterpart, one may be inspired by the following observation. There is only one SU(5) singlet in the decomposition of the product of two Σ’s and one left-handed fermion multiplet, 5αi=1ǫα1α2α3α4α5[(Σ)α1α2(Σ)α3α4Ψ2α5 + (Σ†)α1α2(Σ†)α3α4Ψ1α5], where ǫα1α2α3α4α5 is the totally antisyPmmetric tensor and the second term is the T transformed of the first one. Alternatively, one could multiply three Σ’s and the other left-handed fermionmultiplet, 5αi=1ǫα1α2α3α4α5δα6α7[(Σ)α1α2(Σ)α3α4(Σ)α5α6Ψ1α7+(Σ†)α1α2(Σ†)α3α4(Σ†)α5α6Ψ2α7], with δα6α7 the Kronecker delta. In both cases, we get the wrong Higgs coupling. This is so because this prodPuct is an SU(5) singlet and then the Higgs coupling reads iπ+l− + (v + h + iπ0)ν/√2. (In these expressions there are neither Ω’s nor Σ0’s because the determinant of Ω is 1 and ǫα1α2α3α4α5(Σ0)α1β1(Σ0)α2β2(Σ0)α3β3(Σ0)α4β4(Σ0)α5β5 = ǫβ1β2β3β4β5.) Then, getting the correct coupling (v+h iπ0)l−/√2+iπ−ν requires the explicit breaking of SU(5). If ξ is introduced in the game, one − eventually has to break SO(5) as well. 6 We denote the mass eigenstates with primes when necessary to distinguish them from the current eigenstates. 9 Appendix B. The SM interaction and mass eigenstates are related by the unitary matrices in Eq. (2.26), ℓ = Vℓℓ′ , ℓ = Vℓℓ′ . (2.28) L L L R R R Then the SM charged current Lagrangian reads g g SM = ν W/ †ℓ +h.c. = ν′ Vν†VℓW/ †ℓ′ +h.c., (2.29) LCC −√2 L L −√2 L L L L where we have also introduced the corresponding rotation for the neutrinos. Thus, only the combination V† = Vν†Vℓ is observable. It must be noted, however, that the PMNS L L neutrino contributions to LFV processes are negligible in the SM because so are their masses. Hence, Vν can be assumed to be unity. Similarly we can also diagonalise the L heavy Yukawa couplings in Eq. (2.19), mli′δi′j′ = (VLH†)i′iκij(VRH)jj′ √2f, (2.30) H where VH acts on the left-handed fields and VH acts on the right-handed fields. Note that L R there is no distinction between up- and down-type leptons. The T-odd gauge boson inter- actions arising from the corresponding kinetic terms for left-handed leptons in Eq. (2.23) are proportional to Vνν l G/ l +h.c. = l VH†G/ L L +h.c. (2.31) L− − L+ HL L H Vℓℓ ! L L where G and l are the heavy, T-odd gauge bosons and fermions and l are the SM, − L− L+ T-even fermions in the interaction basis, whereas G = A ,Z ,W ; l = (ν ,ℓ )T; H H H H H H H and ν and ℓ are the corresponding mass eigenstates. Then, in analogy with the PMNS L L matrix, the observable rotations are now V VH†Vν, V VH†Vℓ. (2.32) Hν ≡ L L Hℓ ≡ L L Note that both matrices are related, V† V = V† [28], but this relation can not Hν Hℓ PMNS be tested unless V can be measured. The new contributions to the LFV amplitudes Hν describing a muon decay to an electron are then proportional to Vie∗Viµ, with i counting Hℓ Hℓ the heavy lepton doublets. 3 New contributions to LFV processes As noted above, the SM contributions to the LFV processes µ eγ and µ ee¯e are → → negligible forthey areproportionaltotheobserved neutrino masses. Ontheother hand the new LHT contributions can be a priori large. In particular, one expects that the dominant 10