Muon to electron conversion in the Littlest Higgs model with T-parity
aa r X i v : . [ h e p - ph ] S e p CAFPE-135/10UG-FT-265/10 µ − e conversion in the Littlest Higgs modelwith T-parity F. del ´Aguila, J. I. Illana and M. D. Jenkins
CAFPE and Departamento de F´ısica Te´orica y del Cosmos,Universidad de Granada, E–18071 Granada, Spain [email protected] , [email protected] , [email protected] Abstract
Little Higgs models provide a natural explanation of the little hierarchy betweenthe electroweak scale and a few TeV scale, where new physics is expected. Underthe same inspiring naturalness arguments, this work completes a previous study onlepton flavor-changing processes in the Littlest Higgs model with T-parity exploringthe channel that will eventually turn out to be the most sensitive, µ − e conversionin nuclei. All one-loop contributions are carefully taken into account, results for themost relevant nuclei are provided and a discussion of the influence of the quark mixingis included. The results for the Ti nucleus are in good agreement with earlier work byBlanke et al., where a degenerate mirror quark sector was assumed. The conclusionis that, although this particular model reduces the tension with electroweak precisiontests, if the restrictions on the parameter space derived from lepton flavor violationare taken seriously, the degree of fine tuning necessary to meet these constraints alsodisfavors this model. ontents µ N → e N process 9 Introduction
Lepton flavor is conserved within experimental limits in all known processes, except forneutrino oscillations. This is predicted by the Standard Model (SM) when it is minimallyextended to include Dirac and/or Majorana masses [1]. In such a case any neutral leptonflavor transition is proportional to the neutrino masses m ν ∼ . M is near or above the GeV. Hence, the observationof lepton flavor violation (LFV) in processes like µ → e γ , µ → eee or µ N → e N, or theanalogous τ decays would be a clear signal of new physics beyond the SM.On the other hand, any new physics near the TeV scale involving leptons may inducelarge LFV transitions if it is not aligned with the SM charged lepton Yukawa couplings.In this case the relevant scales in the process are the electroweak scale v ≈
246 GeV andthe new scale f of few TeV, with the corresponding ratio v/f ∼ . m ν /M . ∗ Due to very stringent bounds on LFV processes (as smallas 10 − for µ → e γ ) new physics at the TeV must incorporate a very efficient lepton flavorsuppression mechanism to agree with current and foreseen limits on LFV transitions.Present limits on flavor violating processes set stringent constraints on the possibleextensions of the SM, in particular on those solving the (little) hierarchy problem likesupersymmetry, Little Higgs models and models with large extra dimensions. (See for arecent review [2], and references therein.) These bounds, which in the quark sector and for τ leptons have been improved by BaBar and Belle [3], are also expected to be improvedfor the first two lepton families in the near future. Thus MEG will improve the precisionon µ → e γ and µ → eee by two orders of magnitude [4, 5], whereas PRISM/PRIME couldreduce the bounds on µ N → e N by several orders of magnitude [6]. Super-B factoriesunder study should also improve the precision on τ decays by an order of magnitude [7].Little Higgs models take care of the large top corrections to the Higgs mass by makingthe Higgs a pseudo-Goldstone boson of a larger global symmetry broken at a scale of a fewTeV [8–13]. This appealing solution to the little hierarchy problem encounters difficultieswhen a specific model is defined. In particular, electroweak precision data [14–19] and thelimits for flavor violating processes tend to banish the new physics scale to higher energies.To deal with the electroweak data constraints, T-parity [20–22] was introduced with theLittlest Higgs model with T-parity (LHT) being the most popular solution for a lighternew scale.In this paper we re-evaluate µ N → e N in the Littlest Higgs model with T-parity [23].This complements our previous calculations of µ → e γ and µ → eee in [24] and thosein [23, 25]. The results for the Ti nucleus are in good agreement with previous work [23, 26]where a degenerate mirror quark sector was assumed. Here we have also considered theeffect of a more general quark sector where we include non-degenerate mirror quark masses ∗ Note that amplitudes must be at least suppressed by one of these ratios, and then cross sections areat least quadratically suppressed. µ N → e N is one-loop finite [24, 26, 27].This is apparent for box diagrams but not for Z penguins. In general, this process providesa stronger bound on the new physics scale f , but limits from all three processes are requiredbecause there are accidental suppressions of any of them in certain regions of parameterspace. Obviously, each of them cancel when the heavy lepton Yukawa couplings align withthose of the charged leptons, that is the amplitudes vanish when the corresponding mixingangle θ goes to zero. The amplitudes also scale like f − when the new physics scale goesto infinity with all other parameters finite. Present bounds on these processes are satisfiedfor θ < ∼ .
01 or f > ∼
10 TeV. Particular cancellations result from amplitude contributionsof different sign, for example between penguins and boxes, at given parameter values. Aswe shall discuss, in the case of µ N → e N these cancellations occur even for degeneratequark masses (or just one quark family).On the other hand, box as well as γ penguin contributions are at most logarithmicallydivergent when the internal fermion masses go to infinity, whereas the Z penguins add aquadratically divergent contribution when these masses are parametrically increased. Thisis similar to what happens in the SM with the top mass m t [28, 29] (and to µ → eee inthese models [24]). In our case the heavy fermion masses are proportional to κf , where κ is the corresponding Yukawa coupling. For a fixed f value, the Z penguins scale like κ , the same as, for instance, the top quark contribution to Zb ¯ b that scales like λ , with m t ∼ λv [30]. In both cases the gauge symmetry, [SU(2) × U(1)] × [SU(2) × U(1)] inthe LHT and SU(2) L × U(1) Y in the SM, is spontaneously broken and the heavy fermionmass splits the fermion multiplet. Of course, these Yukawa couplings can be large and thecorresponding contributions dominant, but they should eventually remain perturbative.In a companion paper we present the evaluation of the three processes µ → e γ , µ → eeeand µ N → e N in the Simplest Little Higgs model [31], finding similar results [32]. Thesecalculations can also be applied to the four-dimensional formulation of extended modelswith large extra dimensions [33, 34]. In this case, however, care must be taken to includethe appropriate number of Kaluza-Klein modes to match the five-dimensional result [35].In Section 2 the LHT model is briefly described with the purpose of introducing notationand conventions, providing the Feynman rules for quarks, needed to perform the calculationof µ N → e N which were not included in [24]. (They agree with those in [36] up to signconventions.) In Section 3 we detail the different contributions, finding complete agreementwith [23] in the total sum. We show in Section 4 that, in general, the conversion rates forboth Ti and Au give a more stringent limit than µ → e γ and µ → eee, with Au beingthe most stringent. Results for Pb are not shown but it was found that it provides alower conversion rate than Ti. We also study the effect of the heavy quark sector on theconversion rate. A degenerate heavy quark sector was previously assumed in the literaturearguing that the quark mixing is a higher order effect. We analyze the degenerate case,and then show the impact of a non-degenerate heavy quark sector including quark mixings.Even assuming degeneracy, the quark mass parameter can entirely cancel the conversionrate if chosen appropriately (in a region where box diagrams cancel penguin diagrams).4lthough the effect of the mixings is found to be less important than that of the quarkmasses, they can have sizable effects on the conversion rate since they can push the valuesof the form factors into regions with cancellations. Finally, Section 5 is devoted to ourconclusions. The Littlest Higgs [13] is a non-linear σ model based on the spontaneous breaking of aglobal SU(5) symmetry into SO(5) by the vacuum expectation value of a five-dimensionaltensor field. The 14 Goldstone fields are parameterized asΣ( x ) = ξ Σ , ξ = e iΠ( x ) /f , Σ = × × × × . (2.1)Only an [SU(2) × U(1)] × [SU(2) × U(1)] subgroup of SU(5) is gauged, with generators Q a = 12 σ a × , Y = 110 diag(3 , , − , − , − , (2.2) Q a = 12 × − σ a ∗ , Y = 110 diag(2 , , , − , − , (2.3)and σ a the three Pauli matrices. This gauge group is broken by Σ down to the SMgroup, whose generators are the combinations { Q a + Q a , Y + Y } . A discrete T-symmetryis introduced [20–22] that exchanges the gauge fields G , of [SU(2) × U(1)] , under theassumption that g ≡ g = g , g ′ ≡ g ′ = g ′ . The T-even combinations remain masslesswhile those T-odd acquire a mass proportional to f . In order that the SM Higgs doubletcontained in Π is T-even and the remaining Goldstone fields T-odd, the T transformationon the scalar fields is defined as follows,Π T −→ − ΩΠΩ , Ω = diag( − , − , , − , −
1) (2.4)and then Σ T −→ e Σ = ΩΣ Σ † Σ Ω , ξ T −→ Ω ξ † Ω . (2.5)The gauge- and T-invariant Lagrangians for the vector and Goldstone bosons can thenbe written in terms of covariant derivatives. Our sign conventions are those in [24]. After5he electroweak symmetry breaking the photon remains massless, the weak gauge bosonspick up a mass of order v , the T-odd gauge bosons get rotated to their physical states A H , Z H and W ± H , and the scalar fields are also shifted and rotated accordingly. The latterinclude the would-be Goldstone fields η , ω , ω ± eaten by the four heavy gauge bosonsabove.The introduction of fermions in the model is less straightforward. Since the imple-mentation of the lepton sector was described in detail in [24], which was sufficient forthe processes µ → e γ and µ → eee, we present below the relevant Yukawa and gaugeinteraction Lagrangians for the quark sector, also needed to study µ N → e N. As for the lepton sector, one introduces two left-handed fermion doublets in incompleteSU(5) multiplets, one transforming only under SU(2) and the other under SU(2) , foreach SM left-handed quark doublet [22]:Ψ [ ¯5 ]1 = − i σ q L , Ψ [ ]2 = − i σ q L , (2.6)where q iL = u iL d iL ! . An SU(5) transformation V and a T transformation acts as follows:Ψ −→ V ∗ Ψ , Ψ −→ V Ψ , Ψ ←→ ΩΣ Ψ . (2.7)The T-even combination Ψ +ΩΣ Ψ remains massless while the T-odd Ψ − ΩΣ Ψ obtainsa mass when coupled (see below) to a multiplet Ψ R defined asΨ R = ˜ ψ R χ R − i σ q HR , q HR = u HR d HR ! , Ψ R −→ U Ψ R , Ψ R T −→ ΩΨ R , (2.8)so that ξ Ψ R transforms as a representation. The operator U is a non-linear transforma-tion depending on V and the Goldstone fields, that takes values in the Lie algebra of theunbroken SO(5), defined byΣ ≡ ξ Σ −→ V Σ V T ⇒ ξ −→ V ξU † ≡ U ξ Σ V T Σ , (2.9)such that the following Yukawa Lagrangian is SU(5) and T invariant, L Y H = − κf (cid:0) ¯Ψ ξ Ψ R + ¯Ψ Σ ξ † Ψ R (cid:1) + h.c. . (2.10)6efining the T-eigenstates q L = q L − q L √ , q HL = q L + q L √ , q = u, d, (2.11)we see that the left-handed T-odd fields u HL and d HL couple to u HR and d HR , respectively,to get masses of order f . The T-even combinations u L and d L are the SM left-handedquarks. The fields χ R and ˜ ψ R can be assumed to acquire large Dirac masses pairing withadditional fermions [22, 37] so that they decouple from the theory.The left-handed quark fields interact with the gauge bosons through the followingcovariant derivatives: L F L = i ¯Ψ (cid:0)(cid:0) D Ψ + i ¯Ψ (cid:0)(cid:0) D Ψ , (2.12)where D µ = ∂ µ + √ g ( W a µ Q aT + W a µ Q aT ) + √ g ′ ( y Ψ B µ + y Ψ B µ ) , (2.13) D µ = ∂ µ − √ g ( W a µ Q a + W a µ Q a ) + √ g ′ ( y Ψ B µ + y Ψ B µ ) , (2.14)with W ajµ , B jµ the gauge fields of [SU(2) × U(1)] j and y Ψ = y Ψ = 130 , y Ψ = y Ψ = 215 . (2.15)It is necessary to enlarge SU(5) with two extra U(1) factors to introduce these hypercharges[27], that add up to 1 / v/f , but we include them forreference [37]: L F R = iΨ R γ µ (cid:20) ∂ µ + 12 ξ † ( D µ ξ ) + 12 ξ (Σ D ∗ µ Σ ξ † ) (cid:21) Ψ R , (2.16) L ′ F R = i u R γ µ ( ∂ µ + i g ′ y u B µ ) u R + i d R γ µ ( ∂ µ + i g ′ y d B µ ) d R , (2.17)where y u = 2 / y d = − / L Y H = − κ mn f (cid:0) ¯Ψ m ξ Ψ nR + ¯Ψ m Σ ξ † Ψ nR (cid:1) + h.c. , (2.18)with family indices m, n = 1 , ,
3. This is the primary source of heavy flavor mixing in thequark sector, which is completely analogous to the lepton sector. After diagonalization,7iag( κ m ) = V H † L κV HR where V HL ( V HR ) acts on the left (right) handed fields, one finds thatthe masses for the heavy quarks are approximately given by: m d mH = √ κ m f, m u mH = √ κ m f (cid:18) − v f (cid:19) . (2.19)Then the T-odd gauge boson interactions from (2.12) work out as: q HL V H † L (cid:0)(cid:0) G H V uL u L V dL d L ! + h.c. , (2.20)where G H is a T-odd gauge boson ( A H , Z H , W H ) and the matrices V u,dL are rotationsnecessary to make the light sector mass diagonal. The matrices appearing in the couplingcan therefore be defined as [39]: V Hu ≡ V H † L V uL , V Hd ≡ V H † L V dL . (2.21)The two light rotations are then related by V u † L V dL = V CKM and can be obtained from theYukawa Lagrangians for the light sector which we now describe.For the SM quark Yukawa couplings, the first two generations are treated separatelyfrom the third one. This is because of the special structure necessary for the collectivesymmetry breaking mechanism. The third generation requires enlarging Ψ and Ψ intomultiplets of the SU(3) and SU(3) subgroups of SU(5) [39] adding additional T-even andT-odd partners (even and odd combinations of t ′ and t ′ ), usually denoted by T + and T − respectively, Q = (cid:0) i d L , − i u L , t ′ L , , (cid:1) T , Q = (cid:0) , , t ′ L , i d L , − i u L (cid:1) T . (2.22)They have the same transformation properties as Ψ , in (2.7). The quarks masses of thisgeneration originate from the Lagrangian L Y t = − λ √ f ǫ ijk ǫ xy h ¯ Q i Σ jx Σ ky − (cid:0) ¯ Q Σ (cid:1) i ˜Σ jx ˜Σ ky i u R + λ f (¯ t ′ L t ′ R +¯ t ′ L t ′ R )+h.c. , (2.23)where i, j, k = 1 , , x, y = 4 , L Y u = − λ mu √ f ǫ ijk ǫ xy h ¯ Q m i Σ jx Σ ky − (cid:0) ¯ Q m Σ (cid:1) i ˜Σ jx ˜Σ ky i u mR + h.c. ( m = 1 , , (2.24)where i, j, k = 1 , , x, y = 4 , Q m = (i d m L , − i u m L , , , T , Q m = (0 , , , i d m L , − i u m L ) T ( m = 1 , . (2.25)8nlike the up quark case, all three generations of down-type light quarks acquire massthrough the same type of Lagrangian. For simplicity, we have introduced the quark mixing(CKM matrix) in this sector only, choosing the up-type quarks to be directly in the massbasis. This is because of the fact that the three generations of up quarks are implementeddifferently, which explicitly breaks flavor symmetries. Although it is possible to restoreflavor symmetries (see [39] for further details), this is unimportant in our case and it issufficient to assume that all the mixing comes from the rotation in the down sector. Then L Y d = i λ mnd √ f ǫ ij ǫ xyz h ¯Ψ ′ m x Σ iy Σ jz X − (cid:0) ¯Ψ ′ m Σ (cid:1) x ˜Σ iy ˜Σ jz ˜ X i d nR + h.c. ( m, n = 1 , , , (2.26)where i, j = 1 , x, y, z = 3 , , X = (Σ ) − / andΨ ′ m = ( u m L , d m L , , , T , Ψ ′ m = (0 , , , u m L , d m L ) T . (2.27)In this context, the previously defined V uL is set to the identity matrix since there is nofamily mixing in the up sector and V dL would correspond to V CKM and would be one of thetwo matrices necessary to diagonalize λ mnd . The other matrix, V dR would be the rotationfor the right handed sector.The relevant degrees of freedom in V Hu (or V Hd ) are three angles, θ u , θ u and θ u , and three complex phases, as emphasized in [38] where a standard parameterization is proposed. From these Lagrangians one can obtain the Feynman rules for the quark vertices enteringin our calculation. They are in agreement with [36] and are summarized in tables 1 and 2,in our sign conventions. Standard Model Feynman rules are used for vertices that involveonly ordinary particles and they will not be listed. Although the model predicts somecorrections to some of these standard vertices, their contributions are subleading in ourprocess and we can safely ignore them. µ N → e N process The µ N → e N process receives contributions from both penguin and box diagrams likethose shown in figure 1b. Here the quark q is either an up or down quark and is of thesame type before and after the interaction. The gauge bosons exchanged in the penguindiagrams are a Z boson or a photon. The diagrams look very similar to those involved in µ → eee except for the fact that now up and down-type quarks instead of electrons areinvolved on the lower legs. 9FF g L g R A H ¯ u iH u j (cid:18) c W + x H s W v f (cid:19) V ijHu Z H ¯ u iH u j (cid:18) s W − x H c W v f (cid:19) V ijHu W + H ¯ u iH d j √ s W V ijHd A H ¯ d iH d j (cid:18) c W − x H s W v f (cid:19) V ijHd Z H ¯ d iH d j − (cid:18) s W + x H c W v f (cid:19) V ijHd W − H ¯ d iH u j √ s W V ijHu eγ µ ( g L P L + g R P R ) for quarks in the LHT model; x H = 5 gg ′ g − g ′ ) . SFF c L c R η ¯ u iH u j i10 c W m u iH M A H (cid:20) v f (cid:18)
58 + x H s W c W (cid:19)(cid:21) V ijHu − i10 c W m u i M A H V ijHu ω ¯ u iH u j − i2 s W m u iH M Z H (cid:20) v f (cid:18) − x H c W s W (cid:19)(cid:21) V ijHu i2 s W m u i M Z H V ijHu ω + ¯ u iH d j − i √ s W m u iH M W H V ijHd i √ s W m d i M W H V ijHd η ¯ d iH d j i10 c W m d iH M A H (cid:20) − v f (cid:18)
54 + x H s W c W (cid:19)(cid:21) V ijHd − i10 c W m d i M A H V ijHd ω ¯ d iH d j i2 s W m d iH M Z H (cid:20) v f (cid:18) −
14 + x H c W s W (cid:19)(cid:21) V ijHd − i2 s W m d i M Z H V ijHd ω − ¯ d iH u j − i √ s W m d iH M W H (cid:18) − v f (cid:19) V ijHu i √ s W m u i M W H V ijHu Table 2: SFF vertices i e ( c L P L + c R P R ) for quarks in the LHT model; x H = 5 gg ′ g − g ′ ) . jV µ p p qe µ q qe µ q (a) (b)Figure 1: Vertex subdiagrams (a) and topologies (b) involved in µ N → e N.The contributions of the lepton flavor-changing vertex subdiagrams fit into the followingLorentz structure [40–42]: † iΓ µ ( p , p ) = i e (cid:2) γ µ ( F VL P L + F VR P R ) − (i F VM + F VE γ ) σ µν Q ν + (i F VS + F VP γ ) Q µ (cid:3) , (3.1)where V denotes the external vector boson, either Z or γ in our case, and Q = p − p as in figure 1a. The penguin diagrams for µ → eq¯q can be read from [24] introducing thecorresponding electric charges ( Q q ) and weak couplings ( Z qL,R ) for quarks: M γ peng = − e Q ¯ u ( p ) (cid:2) Q γ µ ( A L P L + A R P R ) + i mσ µν Q ν ( A L P L + A R P R ) (cid:3) u ( p ) × ¯ u q ( p ) Q q γ µ v q ( p ) , (3.2) M Zpeng = e M Z ¯ u ( p ) [ γ µ ( F L P L + F R P R )] u ( p )¯ u q ( p )[ γ µ ( Z qL P L + Z qR P R )] v q ( p ) , (3.3)where m is the muon mass, the electron mass has been neglected, and A L = F γL /Q , A R = F γR /Q , A L = − ( F γM + i F γE ) /m, A R = − ( F γM − i F γE ) /m,F L = − F ZL , F R = − F ZR . (3.4)The form factors F S and F P do not contribute in any case since they are found to beproportional to the ratio of the external lepton and quark masses to the mass of the Wboson and are therefore negligible. The same occurs in the case of F E and F M in the Zboson penguin.Similarly, in the limit of vanishing external momenta the contribution to the amplitude † A minus sign has been extracted from the dipole form factors F M and F E in order to meet thedefinition of the momentum transfer Q in [40], that had opposite sign in [41, 42]. This implies a sign flipin (3.4) accordingly. Z N Z eff F ( q ) Γ capt [GeV] Ti 22 26 17.6 0.54 1 . × − Au 79 118 33.5 0.16 8 . × − Table 3: Relevant input parameters for the nuclei under study. From [44].from any box diagram can be written as [40]: M q box = e B L q [¯ u ( p ) γ µ P L u ( p )] [¯ u q ( p ) γ µ P L v q ( p )]+ e B R q [¯ u ( p ) γ µ P R u ( p )] [¯ u q ( p ) γ µ P R v q ( p )]+ e B L q [¯ u ( p ) γ µ P L u ( p )] [¯ u q ( p ) γ µ P R v q ( p )]+ e B R q [¯ u ( p ) γ µ P R u ( p )] [¯ u q ( p ) γ µ P L v q ( p )]+ e B L q [¯ u ( p ) P L u ( p )] [¯ u q ( p ) P L v q ( p )]+ e B R q [¯ u ( p ) P R u ( p )] [¯ u q ( p ) P R v q ( p )]+ e B L q [¯ u ( p ) σ µν P L u ( p )] [¯ u q ( p ) σ µν P L v q ( p )]+ e B R q [¯ u ( p ) σ µν P R u ( p )] [¯ u q ( p ) σ µν P R v q ( p )] . (3.5)However, we may restrict ourselves to the first term, proportional to B L q , since it is theonly contributing due to the fact that the LHT couplings are primarily left-handed.Finally, in the process µ N → e N only the quark vector current contributes [43]. Theconversion width is then given by [40]:Γ( µ → e) = α Z Z F ( q ) m µ (cid:12)(cid:12) Z ( A L − A R ) − (2 Z + N ) ¯ B L u − ( Z + 2 N ) ¯ B L d (cid:12)(cid:12) , (3.6)where ¯ B L q = B L q + F L M Z ( Z qL + Z qR ) ≡ B L q + F qLL + F qLR . (3.7)The conversion rate is obtained by dividing by the capture rate: R = Γ( µ → e)Γ capt . (3.8)The nuclei we will consider are Ti and
Au, whose relevant parameters are listed intable 3. We now calculate the form factors appearing in the amplitudes in terms of standardloop functions.
For the penguin diagrams, the contributions to the form factors are identical to the µ → eeecase since they only depend on the lepton triangle diagrams. The only difference here comes12rom the couplings to quarks and these are taken into account in the expressions (3.2) and(3.3) for the matrix elements. We therefore just recall our previous results: ‡ A L = α W π M W v f X i V ie ∗ Hℓ V iµHℓ h G (1) W ( y i ) + G (1) Z ( y i ) + 15 G (1) Z ( ay i ) i , (3.9) A R = − α W π M W v f X i V ie ∗ Hℓ V iµHℓ h F W ( y i ) + F Z ( y i ) + 15 F Z ( ay i ) i , (3.10) F L = − α W π s W c W v f X i V ie ∗ Hℓ V iµHℓ y i H W ( y i ) , (3.11)where y i = m Hi M W H , a = M W H M A H = 5 c W s W , (3.12)and the loop functions are given in equations (3.23), (3.29), (3.35), (3.39) and (3.42) of [24]. The boxes do differ considerably and need to be recalculated. The list of diagrams is shownin figure 2. The contributions are split into two form factors depending on the quark thatenters on the lower legs: B L u and B L d . Notice that all diagrams with neutral gauge bosonsinclude several contributions, i.e. diagrams with one or other type of boson and diagramswith mixed types. The first diagram in the second row, for instance, includes four of them:( A H , A H ), ( Z H , Z H ), ( A H , Z H ), ( Z H , A H ).The resulting form factors are as follows: B L u = α W π s W M W v f X ij χ uij " − (cid:18) y i y dj (cid:19) e d ( y i , y dj ) + 2 y i y dj d ( y i , y dj ) − e d ( y i , y uj ) − a e d ( ay i , ay uj ) + 310 e d ( a, ay i , ay uj ) , (3.13) B L d = α W π s W M W v f X ij χ dij " (cid:18) y i y uj (cid:19) e d ( y i , y uj ) − y i y uj d ( y i , y uj ) − e d ( y i , y dj ) − a e d ( ay i , ay dj ) − e d ( a, ay i , ay dj ) . (3.14) ‡ The sign of the dipole form factor A R has been reversed as compared to [24], where the sign of Q wasmistaken. This has no consequences in µ → e γ and a non-significant impact on µ → eee. χ uij = V iµHℓ V ie ∗ Hℓ | V juHu | , χ dij = V iµHℓ V ie ∗ Hℓ | V jdHd | , (3.15)and y ui = m u iH M W H , y di = m d iH M W H . (3.16)The box functions d ( x, y, z ), e d ( x, y, z ), d ( x, y ) and e d ( x, y ) are given in equations (C.26)to (C-29) of [24]. Next we can proceed to compute the conversion rates for different values of the LHTparameters. As in our previous work [24], we restrict ourselves to the case of two leptongenerations. Then, the lepton sector can be fully determined by four parameters: The LHbreaking scale f , the masses of the heavy leptons § m Hi ( i = 1 , θ thatdefines the now 2 × V Hℓ : V Hℓ = V eHℓ V µHℓ V eHℓ V µHℓ ! = cos θ sin θ − sin θ cos θ ! (4.1)We shall again replace the masses m Hi by the parameters δ and ˜ y defined as:˜ y = √ y y , y i = m Hi M W H , i = 1 , , (4.2) δ = m H − m H m H m H . (4.3)Both m Hi and M W H are proportional to f and, therefore, ˜ y and δ are independent of thisscale.All form factors for this process then take the following general form: X i =1 , V ie ∗ Hℓ V iµHℓ F ( y i ) = sin 2 θ F ( y ) − F ( y )] , (4.4)with F a generic function. The dependence of the conversion rate for small δ can beapproximated by: R ∝ (cid:12)(cid:12)(cid:12)(cid:12) v f δ sin 2 θ (cid:12)(cid:12)(cid:12)(cid:12) . (4.5) § We neglect the difference between the masses of the heavy neutrino and the charged lepton of thesame generation, that is of order of ( v/f ) , as for quarks in (2.19). d u jH W H W H ν iH e µ dd u jH ωW H ν iH e µ dd u jH W H ω ν iH e µ dd u jH ωω ν iH e µ uu u jH A H , Z H ℓ iH e µA H , Z H uu u jH η, ω ℓ iH e µA H , Z H uu u jH A H , Z H ℓ iH e µη, ω uu u jH η, ω ℓ iH e µη, ω dd d jH A H , Z H ℓ iH e µA H , Z H dd d jH η, ω ℓ iH e µA H , Z H dd d jH A H , Z H ℓ iH e µη, ω dd d jH η, ω ℓ iH e µη, ω uu ν iH d jH W H µ e W H uu ν iH d jH ωµ e W H uu ν iH d jH W H µ e ω uu ν iH d jH ωµ e ω uu ℓ iH u jH A H , Z H µ e A H , Z H uu ℓ iH u jH η, ω µ e A H , Z H uu ℓ iH u jH A H , Z H µ e η, ω uu ℓ iH u jH η, ω µ e η, ω dd ℓ iH d jH A H , Z H µ e A H , Z H dd ℓ iH d jH η, ω µ e A H , Z H dd ℓ iH d jH A H , Z H µ e η, ω dd ℓ iH d jH η, ω µ e η, ω Figure 2: Box contributions to µ N → e N in the LHT model.15ere the dependence on f and θ is exact. There is, of course, a dependence in ˜ y but thebehaviour with changes in this parameter cannot be expressed as simply.The quark sector requires another set of masses and mixings. The two mixing matricesinvolved in the quark sector, V Hu and V Hd , are related by V Hd = V Hu V CKM so there is,in fact, only one matrix that requires fixing. We will assign masses to the three heavyquarks ignoring the small mass difference (2.19) between up- and down-type quarks of thesame generation ( y ui ≈ y di ). The parameters for this sector are defined analogously to thethree-family lepton sector in [24]:˜ y u = p y u y u , y ui = m u iH M W H , i = 1 , , , (4.6) δ u = m u H − m u H m u H m u H , δ u = m u H − m u H m u H m u H . (4.7) We will firstly assume no flavor mixing in the quark sector, which can be achieved byassuming mass degeneracy of heavy quarks. This is convenient to view the general behaviorof the model clearly. In figure 3 we show the conversion rates divided by the currentexperimental limits ( R ( µ Au → e Au) < × − [45] and R ( µ Ti → e Ti) < . × − [46]). Only values below unity are experimentally allowed. The results of [24] for µ → e γ ( B < . × − [47]) and µ → eee ( B < − [48]) are also plotted using the sameinputs. The masses for the heavy quarks have been chosen to correspond to ˜ y u = 1 and δ u = δ u = 0. Natural values for f = 1 TeV, lepton mixing θ = π/
4, mass splitting δ = 1and ˜ y = 4 are taken as a reference, here and in the following unless stated otherwise. Noticethat the reference value for ˜ y has been shifted from the one used in [24]. The reason for thiscan be seen in the top right hand graph of figure 3 which shows the dependence on ˜ y . Thevalue ˜ y = 1 chosen in [24] falls very close to a point where penguin and box diagrams canceleach other out. This artificially reduces the value of the conversion rate to values belowthe values expected for µ → eee and µ → e γ even though in almost any other region the µ N → e N process dominates. These cancellations also occur elsewhere in the parameterspace and in the other processes. This only means that the process in question does notset a bound on the parameters in that area. The key observation is that the rates for thethree processes, µ → e γ , µ → eee and µ N → e N, in general do not simultaneously vanishfor any of the values of the parameters making it impossible for the model to fit into thoseareas. Scanning over parameter space has been discussed in some detail previously [23, 26],finding large correlations for non-vanishing cross-sections between different processes. Herewe prefer to look for the parameter regions where all processes get small, fulfilling allexperimental constraints. This is so when the heavy Yukawa couplings effectively alignwith those of the charged leptons, i.e. small θ or δ . Obviously, all new contributions alsocancel for a large new physics scale, since the corresponding cross-sections scale like f − .16 Au → e Au µ Ti → e Ti µ → ee¯e µ → e γ R a t i o t o c u rr e n t li m i t . f [TeV] 2015105010 − − . ˜ y − − − − − R a t i o t o c u rr e n t li m i t . sin 2 θ − − . δ − − Figure 3: Ratios of LHT predictions to current limits, assuming ˜ y u = 1 and degenerateheavy quark masses, as functions of the different parameters keeping the others at theirreference values. 17 → e γ µ → eee µ Au → e Au µ Ti → e TiLimit 1 . × − − − − × − . × − − f / TeV > θ < < − | δ | < < − Table 4: Constraints on LHT parameters from present and future experimental exclusionlimits, assuming ˜ y u = 1 and degenerate heavy quark masses.Barring the possibility of these cancellations, it is clear from figure 3 that the mostrestrictive process is µ Au → e Au, whose constraints are somewhat more demanding thanthose previously obtained in [24] for µ → e γ and µ → eee and than the ones for µ Ti → e Ti. However, this last process is expected to have the greatest improvements in futureexperiments [4–6]. We compare the bounds on the parameters derived from µ → e γ and µ → eee with the new ones from µ N → e N in table 4 for current and future measurements.The bounds on each of the parameters come from keeping all the others at the referencevalues and finding the region where the conversion rate (or branching ratio) is within theexperimental limit. Notice that these bounds depend strongly on the choice of input valuesso these bounds are not strict.A comment about the large ˜ y behavior is in order. Figure 4 shows the contributions ofphoton penguins, Z penguins, boxes and their coherent sum to the total µ − e conversionrate in Au. Notice that only the Z penguins are responsible for the non-decoupling effect,similar to that of the top quark in the SM [28–30]. They saturate the µ − e conversionrate for large ˜ y , analogously as for µ → eee in figure 3. For such large values the maincontributions results from (3.11), which grows like κ κ for large heavy Yukawa coupling κ i , ˜ y = √ y y ∝ κ κ .Figure 5 shows exclusion contours in the (sin 2 θ, δ ) plane for the present experimentalconstraints for µ Au → e Au. The case of ˜ y = 1 shows large allowed areas due to the factthat this is close to a cancellation point. In the other cases, the mixing angle and the masssplitting are correlated if we are to remain within the experimental bound, as they werefor µ → e γ and µ → eee. The bound is relaxed for higher values of the scale f .In figure 6 we show the dependence on the heavy quark mass parameter ˜ y u while keepingall quark masses degenerate. We observe that the location of the cancellations dependsstrongly on the value of the lepton mass parameter ˜ y . Because of these regions, even adegenerate heavy quark sector can be tuned to suppress the lepton flavor changing effect.We therefore conclude that the values of the quark masses can have large effects on theconversion rate. The regions themselves, however, are relatively narrow concentrating onvery definite ˜ y u values. 18 otalboxes only γ -penguin onlyZ-penguin only .. R a t i o t o c u rr e n t li m i t ˜ y − − − − − Figure 4: Contributions from photon penguins, Z penguins, boxes and their coherent sumto the total µ − e conversion rate in Au, assuming ˜ y u = 1, degenerate heavy quark massesand reference values for the remaining parameters.Figure 5: Contours of R ( µ Au → e Au) = 4 . × − for values of f = 0 .
5, 1, 2, 3, 4 TeV(from bottom up) and ˜ y = 0 . ,
1, 4 (left to right), assuming ˜ y u = 1 and degenerate heavyquark masses. 19 y = 0 .
25 .. R a t i o t o c u rr e n t li m i t Au ˜ y u − − − − − .. Ti ˜ y u − − − − − Figure 6: Dependence of the µ N → e N conversion rate on the heavy quark mass parameter˜ y u for degenerate heavy quarks and several values of the lepton mass parameter ˜ y . To study the effect on the conversion rate of the mixing matrix as it compares to theeffect of the quark masses, we must choose the quark mass parameters in an area freeof accidental suppressions. One such choice is the quark mass parameter ˜ y u = 1 and thelepton mass parameter ˜ y = 4. With these values, in figure 7 we make a scatter plot showingthe conversion rate as a function of the quark mass splittings. The quark mixing anglesand phases in V Hu ( V Hd = V Hu V CKM ) are uniformly distributed in their full range. Thesolid lines correspond to minimal heavy quark mixing, that is, alignment of heavy and lightquark flavors ( V Hu = I and V Hd = V CKM ).From the right panel of figure 7 we can conclude that the mixing angles, althoughnumerically important, do not alter the result as much as the mass parameters ˜ y u , δ u and δ u , as long as we are far from a cancellation area. However, taking ˜ y = 1 the rightpanel shows that one can obtain virtually any value for the conversion rate by choosingthe mixing angles appropriately. As previous studies have demonstrated, LHT models are heavily constrained by flavor.In this paper we complete our study presented in [24] with the third basic LFV process, µ N → e N. We have complemented other calculations [23, 25] of lepton flavor violatingeffects for this process and compared with predictions for µ → e γ and µ → eee.20 iAu .. R a t i o t o c u rr e n t li m i t δ u = δ u TiAu .. δ u = δ u − − Figure 7: Ratio of conversion rate to current limit as a function of the heavy quarkmass splittings for random values of mixing angles. The solid lines correspond to V Hd = V Hu V CKM = V CKM . The average quark mass is fixed to ˜ y u = 1 and the leptonmass parameter to ˜ y = 4 (left) and ˜ y = 1 (right). Red (green) points are for Au (Ti).We have first briefly reviewed the quark sector Lagrangians for the LHT and findagreement with Feynman rules calculated in [36]. Generic limits on the LHT parameterscan be found in table 4. In earlier works only conversion in Ti was considered but we findthat, currently, conversion in Au gives the most stringent limits on the LHT parametersin general, producing normalized conversion rates (i.e. the conversion rate divided by thecurrent experimental limit) of up to an order of magnitude larger than in the Ti case andrequiring a Little Higgs breaking scale of almost 10 TeV or fine tuning at the percent levelof the mixing angles and mass splittings in the lepton sector, which must be correlated inorder to fulfill current experimental limits. However, future limits on the Ti conversionrate will clearly be the most restrictive.Nevertheless, certain values of the different parameters can suppress the conversionrate by causing accidental cancellations among the different contributions. This meansthat, in some small regions, these processes do not actually provide any restrictions on themodel, and only those derived from the other processes remain. It is important to note,however, that the cancellation regions for the various processes do not in general overlapand therefore do not allow for the model to survive within them, since there is always atleast one process that is above the experimental limits. For µ N → e N, the origin of thesecancellations is typically a sign difference between the penguin contributions and boxes,which allows them to cancel each other.We have also considered the effect of a more general quark sector where we includenon-degenerate quark masses and arbitrary mixings, whereas previous studies assumed21 degenerate quark sector. We find that even in the degenerate case, there is a sizableinfluence of the heavy quark mass parameter and, in some cases, it can completely cancelthe conversion rate by pushing the prediction into one of the aforementioned suppressionregions. We also observe that the remaining quark sector parameters, namely mass splittingand mixing angles, can also shift the predictions somewhat although, as may be expected,the effect is smaller (less than a factor 10 in the normalized conversion rate) as long as weare far from a suppression region. Otherwise, moving the angles and splittings can againaccidentally reduce the conversion rate and change the value considerably or even cancelit completely. Acknowledgments
Work supported by the Spanish MICINN (FPA2006-05294), Junta de Andaluc´ıa (FQM 101,FQM 03048) and European Community’s Marie-Curie Research Training Network undercontract MRTN-CT-2006-035505 “Tools and Precision Calculations for Physics Discoveriesat Colliders”.
A Comparison with previous calculations
We present below the relation of the form factors for µ → e γ , µ → eee and µ N → e N usedin this and our previous work [24], which follow from [40, 49], with those introduced byBuras et al. [23, 36] corrected in [26]: G F √ π ¯ D ′ µ ee , odd = 2 A R , (A.1) G F √ π ¯ Z µ eodd = −
12 ( A L + F LR ) , (A.2) G F √ π ¯ Y µ e e, odd = s W B L + 2 F LL − F LR ) , (A.3) G F √ π ¯ X µ eodd = − s W ( B L u + F uLL − F uLR ) , (A.4) G F √ π ¯ Y µ eodd = s W ( B L d + F dLL − F dLR ) . (A.5)In the comparison with their results for the final expressions we have found agreement.Nevertheless, left and right hand sides of (A.3–A.5) differ by a common function of theheavy fermion mass involved in the line with no flavor change between external legs, G F √ π X ij χ ij (cid:20) − y j + y j (8 − y j ) ln y j (1 − y j ) (cid:21) , (A.6)22hat vanishes due to the unitarity of the mixing matrices, X i χ ij = 0 . (A.7) References [1] For a pedagogical introduction, see: R. N. Mohapatra and P. B. Pal,
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