We analyze the possibility of universality violation in diagonal leptonic
decays of the $Z$ boson, in the context of interfamily "see-saw" models. In a
minimal extension of the Standard Model with right-handed neutrino fields, we
find that universality-breaking effects increase quadratically with the heavy
Majorana neutrino mass and may be observed in the running $LEP$ experiments.Comment: MZ-TH/93-04 #, LaTeX, 14 p. (2 Figs

A. A. Akhundov

A. Axelrod

A. Datta

A. Pilaftsis

C. E. Vayonakis

D. A. Dicus

E. Nardi

F. Halzen

G. Altarelli

J. Bernabéu

J. G. Körner

J. Maalampi

K. Enqvist

K. Hikasa

K. Schilcher

M. Clements

M. E. Peskin

M. Veltman

N. Rius

P. Langacker

S. L. Glashow

T. Appelquist

V. Ganapathi

W. Buch\%müller

W. Buchmüller

W. S. Hou

English

arXiv

We analyze the possibility of universality violation in diagonal leptonic decays of the Z boson, in the context of interfamily 'see-saw' models. In a minimal extension of the Standard Model with right-handed neutrino fields, we find that universality-breaking effects increase quadratically with the heavy Majorana neutrino mass and may be observed in the running LEP experiments

Bernabeu Alberola, José

Korner, Jurgen G.

Pilaftsis, Apostolos

Schilcher, K.

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:hep-ph/9307295v1  19 Jul 1993MZ-TH/93-04FTUV/93-24May 1993UNIVERSALITY-BREAKING EFFECTSIN LEPTONIC Z DECAYSJ. Bernabe´u (a), J.G. Ko¨rner (b), A. Pilaftsis (b), ∗, K. Schilcher (b)(a) Departament de Fisica Teo`rica, Univ. de Valencia,and IFIC, Univ. de Vale`ncia–CSIC,E-46100 Burjassot (Vale`ncia), SPAIN(b) Institut fu¨r Physik, Johannes-Gutenberg Universita¨t, Staudinger Weg 7,W-6500 Mainz, FRGABSTRACTWe analyze the possibility of universality violation in diagonal leptonic decays ofthe Z boson, in the context of interfamily ”see-saw” models. In a minimal extension ofthe Standard Model with right-handed neutrino fields, we find that universality-breakingeffects increase quadratically with the heavy Majorana neutrino mass and may be observedin the running LEP experiments.PACS numbers: 12.15.J, 13.38, 14.80.E∗Address after 1, Oct. 1993, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon,ENGLAND. E-mail address: pilaftsis@vipmza.physik.uni-mainz.de1Heavy Majorana neutrinos with masses of few TeV have entered the domain of cos-mology and astrophysics, as possible candidates to account for the net baryon numberof the universe through lepton (L)-number violating processes [1]. On the other hand,if such heavy neutral leptons are realized in nature, their existence may be discoveredthrough their production and the L-violating decay at present or future collider ma-chines [2]. Another place of looking for new physics originating from heavy neutrinos isthe rare leptonic decays of H0 [3] and Z particle [4]. Since such decays are forbiddenin the minimal Standard Model (SM), they constitute an interesting framework to con-strain new physics beyond the SM . Such rare decays [3, 5] have recently been analyzedin a ”see-saw”-type model [6] with intergenerational mixings [7]. An interesting aspectof this minimal scenario is that the Appelquist-Carazzone theorem [8] is not operativeand vertex-correction diagrams with intermediate heavy Majorana neutrinos (Ni) showa quadratic mass dependence, i.e. m2Ni/M2W . This mass dependence is a common featurefor all theories based on the spontaneous-symmetry breaking mechanism. For example, inthe Feynman–’t Hooft gauge this mass dependence can be seen to arise from the couplingof the would-be charged Goldstone bosons to heavy fermions. Similar non-decoupling ef-fects in the quark sector have been extensively studied in the past for the flavor changingdecays Z → bs¯ [9] and the diagonal Z → bb¯ [10].In this note we study universality-breaking effects induced by heavy Majorana neu-trinos in leptonic Z decays. Actually, we will analytically calculate the following physicalobservable:Ubr =Γ(Z → τ+τ−) − Γ(Z → l+l−, l = e or µ)Γ(Z → τ+τ−) + Γ(Z → l+l−, l = e or µ). (1)Ubr is a measure of universality violation in the leptonic sector provided lepton masses canbe neglected and is experimentally constrained to be [11] |Ubr| ≤ 1.5 10−2. The model weare concerned with extends the SM by one right-handed neutrino field for each family.The renormalizable form of all relevant interactions and details of the notation we will2use here can be found in [7, 5].In the SM the universality-violating parameter Ubr has a value different from zerodue to τ lepton mass effects. However, pure phase-space (PS) corrections turn out tobe rather small, i.e. |U(PS)br | ≃ 3 [1 + (1 − 4 sin2 θW )2]−1 m2τ/M2Z ≃ 1.1 10−3. Thisstandard source for a non-zero value of Ubr is beyond the sensitivity of the present LEPexperiments. Since one can expect to analyze about 105 leptonic decays of the Z bosonat LEP per year, one may reach an accuracy for |Ubr| at the level of 3. 10−3. Since we areinterested in values of |Ubr| much larger than |U(PS)br |, we will neglect lepton-mass effectsin the calculation of quantum corrections.Decomposing now the transition matrix element T (Z → ll¯) into two parts (with thesuperscripts (0), (1) denoting the electroweak loop order), i.e. T (Z → ll¯) = T (0)(Z →ll¯) + T (1)(Z → ll¯) , withT (0)(Z → ll¯) =igW4 cos θWεµZ u¯lγµ(1− 4 sin2 θW − γ5)vl¯ ,and defining ∆T (1) = T (1)(Z → τ+τ−) − T (1)(Z → l+l−, l 6= τ) , we find that∆T (1) =igWαW8pi cos θWεµZ u¯lγµ(1− γ5)vl¯ ∆BijFZ(λi, λj) . (2)Then, the universality-breaking parameter Ubr takes the simple formUbr =Re(T (0)∆T (1))|T (0)|2=αWpi1− 2 sin2 θW1 + (1− 4 sin2 θW )2∆BijFZ(λi, λj) , (3)whereλi =m2niM2W, ∆Bij = BτiB∗τj −BliB∗lj , l 6= τ . (4)In Eq. (4), Bli is a Cabbibo-Kobayashi-Maskawa-type nG × 2nG matrix appearing in theleptonic charged-current interaction, andmni indicates the masses of all neutral leptons niin our minimal scenario (i.e. i = 1, 2, . . . , 2nG, with nG denoting the number of families).The function FZ(λi, λj) originates from the one-loop graphs depicted in Fig. 1 and contains3all the non-decoupling physics mediated by heavy Majorana neutrinos. Its explicit formwill be discussed below.Before proceeding to give the analytical form of FZ , we remark that the universality-violating parameter Ubr does not involve any infra-red (IR) singularities, as they do notdepend on neutrino masses. Therefore, soft-photon emmision graphs need not to beconsidered here. Also, the Z − l − l vertex should be renormalized at the one loop level.In fact, a large number of renormalization constants should not appear in the universality-breaking parameter Ubr. If we adopt the on-shell renormalization scheme [12, 13], wherethe input renormalization parameters are the electric charge e, MW , MZ , the Higgs massMH and all fermion masses contained in the model, the counterterm Lagrangian LCintrelevant for the renormalization of the Z − l − l¯ vertex is then given by [14]LCint = ie Z1/2AZ l¯γµl Zµ +ie4sW cW[1 +δee+1− 2c2W2s2Wδρ + δZ1/2ZZ+ δZ lL]l¯γµ(1− γ5)l Zµ −ie sWcWl¯γµ[1 +δee+12s2Wδρ + δZ1/2ZZ+ δZ lL1− γ52+ δZ lR1 + γ52]l Zµ, (5)where δρ = δM2Z/M2Z − δM2W/M2W , cW = cos θW = MW/MZ and sW = sin θW . Due toGIM-type cancellation [15] the only non-vanishing contribution to the function FZ comesfrom the wave-function renormalization constants of the left- and right-handed leptons,i.e. δZ lL and δZlR. In fact, one has to calculate the difference of the self-energy derivativesgiven by ∆Z lL = δZτL − δZl 6=τL = ∆ ∂Σ(p/)/∂p/|p/=ml ,mτ→0. The corresponding constant forthe right-handed leptons ∆Z lR vanishes in the limit ml, mτ → 0. It is easy to see that onlythe neutrino-mass dependent self-energy graphs mediated by W± and χ± are of interesthere. The individual contributions to FZ(λi, λj) arising from the diagrams 1(a)–1(f) andthose from the wave-function renormalization constant ∆Z lL are given byF(a)Z =12[Cij(L2(λi, λj) − λZ[K1(λi, λj) − K2(λi, λj) + K˜(λi, λj)] )4+ C∗ij√λiλjK1(λi, λj)], (6)F(b)Z = −14[CijλiλjK1(λi, λj) + C∗ij√λiλj(12CUV −12+ λZK˜(λi, λj)− L2(λi, λj))], (7)F(c)Z = − δij[λZ I˜(λi) + 3c2WL1(λi)], (8)F(d)Z =18δij(1− 2s2W )λi ( CUV − 2L1(λi) ) , (9)F(e)Z + F(f)Z = − δijs2WcWλiI(λi) , (10)F(∆ZlL)Z = −18δij(1− 2s2W )λi(CUV +32−31− λi−(λi + 2)λi lnλi(1− λi)2), (11)whereCij =nG∑k=1B∗lkiBlkj , λZ =M2ZM2W,CUV =1ε− γE + ln 4pi − lnM2Wµ2. (12)The functions I, I˜, L1, K1, K2, K˜ and L2 involved in Eqs. (6)-(11) are given in Ap-pendix A. It is straightforward to see that the ultraviolet (UV ) divergences (i.e. CUV )cancel in the summation of all FZ terms. To be precise, the UV pole in F(d)Z cancelsagainst the UV one of the wave-function renormalization F(∆ZlL)Z and the UV constant inEq. (7) vanishes due to the identity [3]:2nG∑i=1mniBliC∗ij = 0 .For definiteness, we will consider a interfamily-mixing model with two familiesonly. Employing relations between the mixing matrix Bli and heavy Majorana neutrinomasses [5] together with Eqs. (3) and (6)–(11), we arrive at the simple result|Ubr| ≃αW8pi(sντL )4 − (sνlL )4(1 + x−1/2)2λN1[1 +12ln x −ln x1− x(1 + 2x1/2)], (13)where x = m2N2/m2N1. We also assume N2 to be heavier than N1, i.e. x ≥ 1. In Eq. (13)sνlL is the usual neutrino-mixing angle between heavy Majorana neutrinos and the charged5lepton l and is generally defined as [16] (sνlL )2 =∑Ni|BlNi |2 . The neutrino-mixing angle(sντL )2 turns out to be severely constrained by the recent LEP data on τ decays. Infact, a global analysis allowing mixing of exotic particles gives an upper bound of about7. 10−2 [17]. The e- and µ-family is much more constrained, i.e. (sνe(µ)L )2 < 0.01. Dueto this fact we have to deal with universality-breaking effects in the heaviest lepton fam-ily only. It is important to notice that the non-decoupling terms (i.e. proportional tom2N/M2W ) come from the ”seemingly” suppressed (sντL )4 terms. Table 1 shows the dra-matic non-decoupling behaviour of the loop function FZ . For comparison, we also showthe corresponding values for a calculation where terms proportional to (sντL )4 have beenneglected. In our numerical estimates we have assumed that there is no large mass dif-ference between the two heavy Majorana neutrinos, i.e. x ≃ 1. We have varied the heavyneutrino mass mN (∼ mN1 ∼ mN2) up to its perturbative unitarity bound. Such an upperbound may be imposed by requiring that the total width of N1 and N2, denoted by ΓNi,satisfies the condition ΓNi/mNi ≤ 1/2 [5]. This leads to the constraint on the mass of thelightest heavy Majorana neutrino N1m2N1 ≤2M2WαW (sντL )21 + x−1/2x1/2. (14)Taking the above upper bound into account, we find that the universality-violating pa-rameter |Ubr| can be up to 10 times larger than the naive value obtained by consideringonly terms proportional to the mixing (sντL )2. This enhancement factor (i.e. the crucialm2N/M2W dependence) results from the coupling of the would-be charged Goldstone bosonχ+ to the heavy Majorana neutrinos in the diagram 1(b). If we assume very large massdifferences between the heavy neutrinos, |Ubr| smoothly decreases to negligibly small val-ues. The reason is that one effectively recovers the one generation ”see-saw” model insuch a case, i.e. mN1 → 0 as x → ∞ in Eq. (14). In Fig. (2) we present exclusion plotsfor LEP experiments. We see, for example, that possible universality-breaking effects ofthe order of 10−2 can easily be understood within our minimal model.6Non-SM contributions, due to heavy neutrinos, could also be observed by compar-ing the prediction of the SM for the leptonic rates Γ(Z → l+l−) with those epxected whenone includes heavy neutral lepton effects. Besides the universality-violating phenomenadiscussed in this paper, one could have non-decoupling physics introduced by these heavyneutral leptons which can be constrained by analyzing the oblique electroweak parametersS, T , U or ε1, ε2, ε3 as defined in [18]. These contributions occur both through vacuumpolarization terms and in vertex corrections universal for all charged lepton flavors. How-ever, a first estimate for the δρ parameter [19] (i.e. δρ = αemT ) provides a weaker boundthan that obtained by Eq. (14). In particular, the dominant non-SM contribution (de-noted below as δρ∗) to δρ, apart from heavy-top and Higgs-particle effects, comes fromthe νiNj , NiNj intermediate states of the Z self-energy graph. In this way, one getsδρ∗|q2=0 ≃αW16pi(sντL )4 m2NM2W, for x = 1 . (15)We can readily compare Eq. (15) with constraints on the masses of the heavy neutrinosthat are derived on the basis of perturbative unitarity. For example, if (sντL )2 = 0.1, themaximal value that δρ∗ can take is 0.8 10−2, which is still in accordance with phenomeno-logical contraints of a possible mass shift of the W boson [20]. However, to complete theanalysis a global consideration of all electroweak oblique corrections is required.In conclusion, we have explicitly shown that heavy neutral leptons introduce aquadratic mass dependence (i.e. αW m2N/M2W ) in the leptonic vertex function Z − l − l¯.This situation is not peculiar for the minimal model considered in this work, but a gen-eral feature for all theories based on the spontaneous symmetry-breaking mechanism, e.g.similar effects will be present in the model described in [4]. In general, we have found thatthe mass of possible non-decoupling neutral particles can be constrained by the alreadyexisting or future LEP data, as a function of their mixing to the ordinary charged leptons.AP wishes to thank University of Vale`ncia for the kind hospitality. Helpful dis-7cussions and criticisms of F.J. Botella, B. Kniehl, A. Santamaria and J.W.F. Valle aregratefully acknowledged. The work of JB has been supported by CICYT, Spain, underGrant No. AEN 90-0040, the work of JGK and KS by the BMFT, FRG, under thecontract No. 06MZ730, and the work of AP by a grant from the Postdoctoral GraduateCollege of Mainz, FRG.A The one-loop integralsIt is useful first to define the functions B1(λi) and B2(λi, λj) which are given by thefollowing expressions:B1(λi) = (1− y)λi + y[1− λZ yx(1− x)] , (A1)B2(λi, λj) = 1− y + y[xλi + (1− x)λj − λZ yx(1− x)] , (A2)where x and y are Feynman parameters. The loop functions I, I˜, L1, K1, K2, K˜ and L2can then be written in terms of the following integrals:I(λi) =∫ dxdy yB1(λi), (A3)I˜(λi) =∫dxdy y2B1(λi)[1− yx(1− x)] , (A4)L1(λi) =∫dxdy y lnB1(λi) , (A5)K1(λi, λj) =∫ dxdy yB2(λi, λj), (A6)K2(λi, λj) =∫dxdy y2B2(λi, λj), (A7)K˜(λi, λj) , =∫dxdy y3x(1− x)B2(λi, λj), (A8)L2(λi, λj) =∫dxdy y lnB2(λi, λj) , (A9)where the integration variables x and y are constrained to the interval [0, 1]. The aboveintegrals can be best performed numerically.8References[1] C.E. Vayonakis, Phys. Lett. B286 (1992) 92; K. Enqvist, I. Vilja, Phys. Lett. B299(1993) 281; W. Buchmu¨ller, T. Yanagida, Phys. Lett. B302 (1993) 240.[2] For recent articles on the subject see e.g., D. A. Dicus, P. Roy, Phys. Rev. D44(1991) 1593; W. Buchmu¨ller, C. Greub, Nucl. Phys. B363 (1991) 345; J. Maalampi,K. Mursula, R. Vuopionpera¨, Nucl. Phys. B363 (1992) 23; A. Datta, A. Pilaftsis,Phys. Lett. B278 (1992) 162.[3] A. Pilaftsis, Phys. Lett. B285 (1992) 68; J.G. Ko¨rner, A. Pilaftsis, K. Schilcher,Phys. Rev. D47 (1993) 1080.[4] J. Bernabe´u etal., Phys. Lett. B187 (1987) 303; N. Rius, J.W.F. Valle,Phys. Lett. B246 (1990) 249.[5] J.G. Ko¨rner, A. Pilaftsis, K. Schilcher, Phys. Lett. B300 (1993) 381.[6] T. Yanagida, Proc. of Workshop on Unified Theory and Baryon Number of the Uni-verse, eds O. Swada and A. Sugamoto, (KEK, 1979) p. 95; M. Gell-Mann, P. Ramondand R. Slansky, Supergravity, eds P. van Nieuwenhuizen and D. Friedman (North-Holland, Amsterdam, 1979) p. 315.[7] A. Pilaftsis, Z. Phys. C55 (1992) 275.[8] T. Appelquist, J. Carazzone, Phys. Rev. D11 (1975) 2856.[9] A. Axelrod, Nucl. Phys. B209 (1982) 349; M. Clements etal., Phys. Rev. D27 (1983)570; V. Ganapathi etal., Phys. Rev. D27 (1983) 579; W.S. Hou, N. Deshpande,G. Eilam, A. Soni, Phys. Rev. Lett. 57 (1986) 1406; J. Bernabe´u, A. Santamaria,M.B. Gavela, Phys. Rev. Lett. 57 (1986) 1514.9[10] A.A. Akhundov, D.Y. Bardin, T. Rieman, Nucl. Phys. B276 (1986) 1; J. Bernabe´u,A. Pich, A. Santamaria, Phys. Lett. B200 (1988) 569; Nucl. Phys. B363 (1991) 326.[11] Review of Particle Properties, Phys. Rev. D45 (1992) S1.[12] A. Sirlin, Phys. Rev. D22 (1980) 971.[13] W. Hollik, Fort. Phys. 38 (1990) 165.[14] K. Aoki etal., Supl. of the Progr. of Theor. Phys. 73 (1982) 1.[15] S.L. Glashow, J. Iliopoulos, L. Maiani, Phys. Rev. D2 (1970) 1285.[16] P. Langacker, D. London, Phys. Rev. D38 (1988) 886.[17] E. Nardi, E. Roulet, D. Tommasini, Nucl. Phys. B386 (1992) 239.[18] M.E. Peskin, T. Takeuchi, Phys. Rev. Lett. 65 (1990) 964; Phys. Rev. D46 (1992)381; G. Altarelli, R. Barbieri, Phys. Lett. B253 (1990) 161; G. Altarelli, R. Barbieri,S. Jadach, Nucl. Phys. B369 (1992) 3.[19] M. Veltman, Nucl. Phys. B123 (1977) 89.[20] See for example, F. Halzen, B.A. Kniehl, M.L. Stong, Z. Phys. C58 (1993) 119; whohave recently performed a global analysis of all electroweak parameters in the SMat two-loop level.10Figure and Table CaptionsFig. 1: One-loop irreducible vertex graphs contributing to the non-universality pa-rameter Ubr in the Feynman–’t Hooft gauge.Fig. 2: Exclusion plots for LEP experiments. The areas lying to the right of thecurves are excluded due to the following conditions: (i) The validity ofperturbative unitarity (solid line) – see also text, (ii) |Ubr| ≤ 1. 10−2 (dashedline), (iii) |Ubr| ≤ 7. 10−3 (dash-dotted line), (iv) |Ubr| ≤ 3. 10−3 (dottedline).Tab. 1: Numerical estimates for the universality-violating quantity |Ubr| within theperturbatively allowed parameter space. The values in parentheses areobtained by neglecting the ”seemingly” suppressed terms proportional to(sντL )4.11Table 1(sντL )2mN 0.06 0.05 0.04 0.03 0.02 0.01 0.005[TeV]0.3 1.7 10−4 1.3 10−4 9.8 10−5 6.7 10−5 4.0 10−5 1.8 10−5 8.4 10−6(9.4 10−5) (7.8 10−5) (6.3 10−5) (4.7 10−5) (3.1 10−5) (1.6 10−5) (7.8 10−6)0.5 4.9 10−4 3.8 10−4 2.8 10−4 1.9 10−4 1.2 10−4 5.2 10−5 2.5 10−5(2.8 10−4) (2.3 10−4) (1.8 10−4) (1.4 10−4) (9.2 10−5) (4.6 10−5) (2.3 10−5)0.7 8.4 10−4 6.4 10−4 4.7 10−4 3.2 10−4 1.9 10−4 8.2 10−5 3.8 10−5(4.2 10−4) (3.5 10−4) (2.8 10−4) (2.1 10−4) (1.4 10−4) (7.0 10−5) (3.5 10−5)1.0 1.4 10−3 1.1 10−3 7.7 10−4 5.1 10−4 2.9 10−4 1.2 10−4 5.5 10−5(5.9 10−4) (4.9 10−4) (3.9 10−4) (2.9 10−4) (1.9 10−4) (9.8 10−5) (4.9 10−5)2.0 4.3 10−3 3.1 10−3 2.1 10−3 1.3 10−3 6.9 10−4 2.5 10−4 1.0 10−4(9.4 10−4) (7.8 10−4) (6.3 10−4) (4.7 10−4) (3.1 10−4) (1.5 10−4) (7.8 10−5)3.0 8.7 10−3 6.2 10−3 4.1 10−3 2.5 10−3 1.2 10−3 4.0 10−4 1.5 10−4(1.1 10−3) (9.6 10−4) (7.7 10−4) (5.7 10−4) (3.8 10−4) (1.9 10−4) (9.6 10−5)4.0 1.5 10−2 1.0 10−2 6.8 10−3 4.0 10−3 1.9 10−3 5.9 10−4 2.0 10−4(1.3 10−3) (1.1 10−3) (8.7 10−4) (6.5 10−4) (4.3 10−4) (2.2 10−4) (1.1 10−4)6.0 − − − 8.3 10−3 3.9 10−3 1.1 10−3 3.3 10−4(7.5 10−4) (5.0 10−4) (2.5 10−4) (1.2 10−4)8.0 − − − − − 1.8 10−3 5.1 10−4(2.7 10−4) (1.4 10−4)10.0 − − − − − − 7.3 10−4(1.4 10−4)12

Universality-Breaking Effects in Leptonic Z Decays

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