Lepton flavor violation (LFV) has been observed in neutrino oscillations. For
charged lepton FV decays only upper limits are known, but sizable branching
ratios are expected in many neutrino mass models. High scale models, such as
the classical supersymmetric seesaw, usually predict that decays $l_i \to 3
l_j$ are roughly a factor $\alpha$ maller than the corresponding decays $l_i
\to l_j \gamma$. Here we demonstrate that the $Z^0$-penguin diagram can give an
enhancement for decays $l_i \to 3 l_j$ in many extensions of the MSSM. We first
discuss why the $Z^0$-penguin is not dominant in the MSSM with seesaw and show
that much larger contributions from the $Z^0$-penguin are expected in general.
We then demonstrate the effect numerically in two example models, namely, the
supersymmetric inverse seesaw and R-parity violating supersymmetry.Comment: 5 pages; v2: minor corrections, final version to appear in PR

Hirsch, M.

Staub, F.

Vicente, A.

English

arXiv

Vicente, Avelino

Open Repository and Bibliography - Liège

Enhancing $l_i  3 l_j$ with the $Z^0$-penguin

M. Hirsch

F. Staub

A. Vicente

Crossref

Physical Review D

Enhancing<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>l</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>→</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mi>l</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math>with the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>Z</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math>-penguin

Lepton flavor violation has been observed in neutrino oscillations. For charged lepton flavor violation decays only upper limits are known, but sizable branching ratios are expected in many neutrino mass models. High-scale models, such as the classical supersymmetric seesaw, usually predict that decays l(i) -> 3l(j) are roughly a factor alpha smaller than the corresponding decays l(i) -> l(j)gamma. Here we demonstrate that the Z(0)-penguin diagram can give an enhancement for decays l(i) -> 3l(j) in many extensions of the minimal supersymmetric standard model (MSSM). We first discuss why the Z(0)-penguin is not dominant in the MSSM with seesaw and show that much larger contributions from the Z(0)-penguin are expected in general. We then demonstrate the effect numerically in two example models, namely, the supersymmetric inverse seesaw and R-parity violating supersymmetry

Hirsch, Martin

Staub, Florian

Vicente Montesinos, Avelino

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

arXiv:1202.1825v2  [hep-ph]  28 Jun 2012Enhancing li → 3lj with the Z0-penguinM. Hirsch,1, ∗ F. Staub,2, † and A. Vicente3, ‡1AHEP Group, Instituto de Física Corpuscular – C.S.I.C./Universitat de ValènciaEdificio de Institutos de Paterna, Apartado 22085, E–46071 València, Spain2Physikalisches Institut der Universität Bonn, 53115 Bonn, Germany3Laboratoire de Physique Théorique, CNRS – UMR 8627, Université de Paris-Sud 11F-91405 Orsay Cedex, FranceLepton flavor violation (LFV) has been observed in neutrino oscillations. For charged lepton FVdecays only upper limits are known, but sizable branching ratios are expected in many neutrinomass models. High scale models, such as the classical supersymmetric seesaw, usually predict thatdecays li → 3lj are roughly a factor α maller than the corresponding decays li → ljγ. Here wedemonstrate that the Z0-penguin diagram can give an enhancement for decays li → 3lj in manyextensions of the Minimal Supersymmetric Standard Model (MSSM). We first discuss why the Z0-penguin is not dominant in the MSSM with seesaw and show that much larger contributions fromthe Z0-penguin are expected in general. We then demonstrate the effect numerically in two examplemodels, namely, the supersymmetric inverse seesaw and R-parity violating supersymmetry.Introduction: Neutrino oscillation experiments [1, 2]have firmly established that lepton flavor is violated inthe neutrino sector, with two of the three measurablemixing angles being surprisingly large. Observation ofthe characteristic “neutrino dip” leaves no doubt thatneutrinos have mass [3, 4] and quite accurate values forthe mass squared differences are known now [5]. In thecharged lepton sector, however, only upper limits on LFVbranching ratios, such as µ→ eγ [6] or µ→ 3e [7], exist.Extending the standard model (SM) only by neu-trino masses does not automatically lead to measurablecharged LFV (CLFV), but sizable branching ratios areexpected in many models. In fact, on quite generalgrounds one expects large CLFV, if physics beyond theSM exists at the TeV scale. A prime example for thisobservation is supersymmetry (SUSY). Here, the massmatrices of the new scalar particles need not (and in gen-eral will not) be aligned with those of the SM fermions.CLFV decays will occur and one can estimate roughlythe branching ratios for radiative decays as [8]Br(li → ljγ) ≃48π3αG2F|(m2f˜)ij |2m8SUSYBr(li → ljνiν¯j) (1)where (m2f˜)ij parameterizes the dominant off-diagonal el-ements of the soft SUSY breaking slepton mass matricesand mSUSY is the typical mass of the SUSY particles,expected to be in the ballpark of O(0.1− 1) TeV. Rathersmall off-diagonal elements are required to satisfy exper-imental bounds [6, 7].In the Constrained Minimal Supersymmetric exten-sion of the SM (CMSSM), on the other hand, CLFV iszero, just as in the SM, simply because neutrinos are as-sumed to be massless. Extending the CMSSM to includeneutrino masses (and mixings), for example by a seesawmechanism, then leads to CLFV decays, because the fla-vor violation necessarily present in the Yukawa couplingsis transmitted to the slepton mass matrices in the RGE(renormalization group equation) running [9]. In suchhigh-scale neutrino mass models, with only MSSM par-ticle content at the electroweak scale, it has been shownthat the photonic penguin diagram gives the dominantcontribution to li → 3lj decays in large regions of pa-rameter space.1 In this case a simple relation can bederived [11]Br(li → 3lj) ≃α3π(log(m2lim2lj)−114)Br(l → l′γ) (2)Thus, usually it is concluded that the decays li → ljγ aremore constraining than the decays li → 3lj.Apart from the photonic penguin, there are also boxdiagrams, Higgs- and Z0-penguin contributing to the de-cays li → 3lj. The latter diagram is not per se smallerthan the photonic penguin. Rather, as we will show be-low, in models with only the MSSM particle content andcouplings, the Z0-penguin is suppressed by a subtle can-cellation among different terms in the amplitude. Sucha cancellation, however, can be easily spoiled if thereare (a) new couplings and/or (b) a larger particle con-tent than in the MSSM. Then, as we will discuss, theZ0-penguin can easily give the dominant contribution toli → 3lj . We will demonstrate this fact numerically withtwo typical example models: (i) a supersymmetric inverseseesaw and (ii) R-parity violating SUSY. The former isan example of a model with extended particle content,while the latter is an example of a model with the MSSMparticle content but new interactions. As we will show,in such models li → 3lj can be more constraining thanli → ljγ. This is the main result of the present paper.Finally, we emphasize that the Z0-penguin can bedominant also in other observables and for other the-1 An exception from this rule is the decay τ → 3µ in the limit oflarge tan β [10].2oretical models, although this fact has not, in general,been discussed before. For example, Z0-dominance canbe found in µ− e conversion in nuclei in supersymmetricmodels with R-parity violation, as can be seen from thenumerical results of reference [12], although the authorsdo not discuss it. Similarly, in the little Higgs modelof [13] one finds numerical results with parameter pointswhere Br(li → 3lj) > Br(li → ljγ), despite the authorsconcluding that both are correlated.Analytical discussion: The total width of the li → 3ljdecay contains contributions from the photon penguin,the Higgs penguin, the Z0-penguin and boxes. Consid-ering only contributions from photon and Z0-penguin,which are the ones of interest to us and numericallythe most important ones, the total width Γ ≡ Γ(l−i →l−j l−j l+j ) can be written as [14]:Γ =e4512π3m5li[ ∣∣AL1 ∣∣2 + ∣∣AR1 ∣∣2 − 2 (AL1AR∗2 +AL2AR∗1 + h.c.)+ (∣∣AL2 ∣∣2 + ∣∣AR2 ∣∣2)(163logmljmli−223)+13{2(|FLL|2+ |FRR|2)+ |FLR|2+ |FRL|2}+ IAF](3)Here, terms denoted A (F ) are due to photon (Z0) ex-change and IAF denotes their interference terms, irrele-vant for the following discussion. Both, photon and Z0penguins have chargino-sneutrino and neutralino-sleptoncontributions. Exact definitions can be found in [11]. Wewill focus on the chargino loops for brevity here, since theeffects we are interested in are most pronounced in theseloops. The photon contributions areA(c)L,Ra =1m2ν˜OL,RAa s(x2) (4)whereas the Z-contributions readFX =1g2 sin2 θWm2ZOL,RFX t(x2) (5)with X = {LL,LR,RL,RR}. In these expressions OL,Rydenote combinations of rotation matrices and couplingconstants and s(x2) and t(x2) are short-hands for thePassarino-Veltman loop functions which depend on x2 =m2χ˜−/m2ν˜. For precise definitions see [11].The scaling A ∼ m−2SUSY and F ∼ m−2Z can be un-derstood, in principle, from simple dimensional analysis.The width of the decay is proportional to m5li , so both Aand F must be A,F ∝ m−2. In this case it is the smallestmass term in the loop which sets the scale, which in Fis mZ . Due to the masslessness of the photon in case ofA the smallest mass scale in the loop is mSUSY . Withm2Z ≪ m2SUSY the Z0 penguin can, in principle, be evenmore important than the photonic one.Numerically, however, it has been found in case of theMSSM that the photonic penguin is dominant [11]. Thiscan be understood as follows. To simplify the discussion,we neglect first FR, since it is always proportional to thecharged lepton Yukawa couplings. Consider then n gen-erations of sneutrinos and neglect the chargino mixing.In this simplified scenario only the wino contributes toF(c)L and it can be written asF(c)L = Mwave +Mp1 +Mp2 (6)withMwave =12g2(gcW − g′sW )ZikV Zij∗V fiwave (7)Mp1 = −g3cWZikV Zij∗V fip1 (8)Mp2 =12g2(gcW + g′sW )ZikV Zij∗V fip2 (9)Summing over the index i is implied. The termsin the sum come from different types of diagrams:wave function diagrams (Mwave), penguins with the Z0-boson attached to the chargino line (Mp1) or the sneu-trino line (Mp2). Moreover, cW = cos θW , sW =sin θW , ZV is a n × n unitary matrix that diag-onalizes the mass matrix of the sneutrinos and weused the abbreviations f iwave = −B1(m2χ˜± ,m2ν˜i), f ip1 =12 C˜0(m2ν˜i,m2χ˜± ,m2χ˜±) − m2χ˜±C0(m2ν˜i,m2χ˜± ,m2χ˜±), fip2 =12 C˜0(m2χ˜± ,m2ν˜i,m2ν˜i). The sum in eq. (6) vanishes exactlyas can be seen by grouping the different termsF(c)L =12g3cWZikV Zij∗V Xi1 +12g2g′sWZikV Zij∗V Xi2 (10)with X i1 = fiwave − 2fip1 + fip2, Xi2 = fip2 − fiwave. Usingthe exact expressions for the loop functions [11] one findsthat the masses cancel out and these combinations be-come just numerical constants: X i1 = −34 and Xi2 = −14 .Therefore, one is left with F(c)L ∝∑i ZikV Zij∗V , which van-ishes due to unitarity of the ZV matrix2.2 In reference [15], where the authors study B → Xsl+l− in su-persymmetry, the Z-penguin contributions are found to be sub-dominant due to the same type of cancellation that is found inour work.3This cancellation can be spoiled by two effects, ei-ther (i) the sneutrinos mix with other particles which arenot SU(2)L doublets so that the factorization no longerholds, or (ii) the charginos are not pure wino and higgsinostates. The last effect is of course present in the MSSMand therefore this cancellation is not exact. Nevertheless,the Z0-contributions are suppressed due to their propor-tionality to the square of the chargino mixing angle (twowino-higgsino insertions are necessary since there is noH˜± − ν˜L − lL coupling). We neglected so far Higgsinointeractions because in many models these couplings arevery small in comparison to the gauge interactions (forexample, a SUSY scale type-I seesaw model would haveYν ∼ 10−6). However, in models where the Higgsino canhave much larger Yukawa interactions, a large enhance-ment of the Z0-contributions can be expected. This willbe addressed numerically in the next section.Before turning to a numerical discussion, we considerfor simplicity a toy model consisting of two generationsof left-handed sneutrinos which can mix with one genera-tion of right-handed sneutrinos. A 3× 3 rotation matrixis in general parametrized by 3 angles, but we will as-sume here for simplification that two of them vanish andcall the third one Ψ. In addition, we introduce a newinteraction for the Higgsinos κνcH˜u l˜L. We give in Fig. 1the computed F(c)L for arbitrarily chosen sneutrino andchargino masses as a function of Ψ for different values ofκ. The red dotted line shows the case for Ψ = κ = 0.As clearly seen, F(c)L depends on the left-right mixing al-ready for small values of κ. However, increasing κ, F(c)Lbecomes totally dominated by the new κ interactions andenhances Br(li → 3lj).0.0 0.5 1.0 1.5 2.0 2.5 3.0-8-7-6-5-4-3-2-1YlogF LHcLFIG. 1: F(c)L for our toy model as a function of the sneutrinoleft-right mixing angle Ψ and for different values of κ: 10−4(blue), 10−2 (black dashed), 0.1 (blue dashed) and 1.0 (black).Numerical examples: We turn to the full fledged nu-merical study of two examples: an inverse seesaw modeland the MSSM with R-parity violation. For this pur-pose, we have created for both models SPheno modules[16, 17] using the Mathematica package SARAH [18–20].These modules calculate the low-energy observables ex-actly including all possible diagrams [20].Inverse Seesaw: In inverse seesaw the MSSM particlecontent is extended by three generations of right-handedneutrino superfields νˆc and of gauge singlets NˆS whichcarry lepton number [21, 22]. The superpotential readsWIS =WMSSM+Yν νˆcLˆHˆu+MR νˆcNˆS+µN2NˆSNˆS (11)After electroweak symmetry breaking (EWSB) the effec-tive mass matrix for the light neutrinos is approximatelymν ≃v2u2 Yν(MTR )−1µNM−1Y Tν . Since µN can be ofO(10−1) keV or even smaller while MR is of O(mSUSY ),the neutrino Yukawa couplings have to be much largerthan for a standard weak-scale seesaw to explain neu-trino data.Due to the extended particle content, new contribu-tions for Br(li → 3lj) are expected in the inverse see-saw. For example, the Higgs mediated contributionswere recently studied in [23]. In Fig. 2 (top) we showthe different contributions to Br(µ → 3e) for a varia-tion of the SUSY masses. To disentangle RGE effectswe have calculated once the spectrum for a CMSSMinput (m0 = 500 GeV, M1/2 = 1 TeV, tan(β) = 10,A0 = −300 GeV) and rescaled all dimensionful param-eters at the SUSY scale. This changes the sfermionmasses but not the mixing matrices. Yν has been cho-sen to explain neutrino data for diag(µN ) = 10−1 keVand MR = 1 TeV. Clearly, the Z0-penguins dominateand are nearly independent of the SUSY scale. Onlyin the limit mSUSY → mZ the other contributions cancompete. In Fig. 2 (bottom) the branching ratios forµ→ eγ and µ→ 3e and the current experimental boundsof 2.4 · 10−12 and 1.0 · 10−12 are depicted [6, 7]. WhileBr(µ→ eγ) would be in conflict with experiment only formν˜1 < 1.2 TeV, Br(µ → 3e) rules out the entire range.In this example, we have assumed µN and MR to bediagonal and all flavor violation comes from Yν , as isusually done in literature. However, neutrino oscillationdata could equally well be fitted with the flavor violationcoming from µN andMR. In that case CLFV observableswould be much smaller and consistent with experimentaldata. However, the relative order between 2- and 3-bodydecays won’t change, i.e. Br(li → 3lj) will be most likelyobserved before Br(li → ljγ) if inverse seesaw is realizedin nature.R-parity violation: As second example, we take theMSSM particle content but extend the superpotential bythe lepton number violating terms [25]W/R =WMSSM +12λijkLˆiLˆjEˆck + λ′ijkLˆiQˆjDˆck + ǫiLˆiHˆu(12)While the ǫ-parameters are highly constrained by neu-trino data [24], the bounds for the tri-linear couplingsare much weaker and some entries can be of O(1) [26].In the following, all entries of λ and λ′are set to zero4✺   ✶    ✶✺   ✷   ✲✷✁✲✷✂✲✷✷✲✷ ✲✶✄✲✶✁♠⑦✗☎❧♦❣✭P❋✆✮✱❧♦❣✭P❇✆✮✱❧♦❣✭P❆✆✮✺   ✶    ✶✺   ✷   ✲✶✁✲✶✂✲✶✷✲✶ ✲✽✲✁✲✂♠⑦✗✄❧♦❣✭❇r✭✖✦❡✌✮✮❀❧♦❣✭❇r✭✖✦✸❡✮✮FIG. 2: Top: Different contributions to Br(µ → 3e) asfunction of the lightest sneutrino mass: Z0-penguins (reddotted), photonic penguins (blue dashed), combined Higgs-penguins/box diagrams (green). Bottom: Br(µ → 3e) (black)and Br(µ → eγ) (blue) and the current experimental bounds(dashed lines). The dips are an effect of a mass crossing be-tween charginos and sneutrinos.but λ132 and λ232. We give in Fig. 3 the results forBr(µ → 3e) and Br(µ → eγ) for a mixed bi- and tri-linear as well as for the pure tri-linear scenario varying|λ∗132 · λ232|. In the mixed case ǫi and the vacuum ex-pectation values of the sneutrinos, viL, have been chosento be consistent with neutrino data and a moderate fla-vor violation in the sneutrino sector has been induced bym2l˜iHd= (45 GeV)2. It can be seen that in the mixedcase Br(µ → 3e) > Br(µ → eγ) holds when |λ∗132 · λ232|crosses 2.5·10−5, while for the pure tri-linear case withoutany flavor violation at tree level in the sneutrino sectorthe three body decays dominate even for much smallervalues.In both cases we get an upper limit for |λ∗132 · λ232| of2.5 · 10−3 from the bounds on Br(µ → 3e) for sneutrinomasses of 730 GeV. So far, in the literature just the limitsfor mν˜ = 100 GeV from the photonic penguins [27] havebeen published. These are much weaker, after rescalingthe bound ∼ 7.1 · 10−5(730 GeV100 GeV)4≃ 0.2.Summary: We have shown in this letter that the Z0-penguin can give the dominant contribution in leptonflavor violating three body decays in many models. The✲✾ ✲✽✲✼✲✻ ✲✺ ✲✹ ✲✸ ✲✷✲✷✷✲✷ ✲✶✽✲✶✻✲✶✹✲✶✷✲✶ ❧♦❣✭❥✕✄✁✂☎✆ ✕☎✂☎❥✮✝✞✟✠❇r✠✖✦❡✌✡✡❀✝✞✟✠❇r✠✖✦☛❡✡✡FIG. 3: Br(µ → 3e) (black) and Br(µ → eγ) (blue) varying|λ∗132 ·λ232| in a mixed bi- and trilinear (solid lines) and a puretri-linear (dotted lines) RpV scenario. The dashed lines showthe experimental limits.importance of the Z0-penguin increases with increasingSUSY particles masses. As numerical examples, we havebriefly discussed the supersymmetric inverse seesaw andthe MSSM with R-parity violation.AcknowledgementsWe thank Werner Porod, Debottam Das and DanielE. Lopez-Fogliani for fruitful discussions. A.V. ac-knowledges support from the ANR project CPV-LFV-LHC NT09-508531. This work was supported by theSpanish MICINN under grants FPA2008-00319/FPAand FPA2011-22975 by the MULTIDARK ConsoliderCSD2009-00064, by the Generalitat Valenciana grantPrometeo/2009/091 and by the EU grant UNILHCPITN-GA-2009-237920.∗ Electronic address: mahirsch@ific.uv.es† Electronic address: fnstaub@physik.uni-bonn.de‡ Electronic address: avelino.vicente@th.u-psud.fr[1] Y. Fukuda et al. [Super-Kamiokande Collaboration],Phys. Rev. Lett. 81, 1562 (1998)[2] SNO, Q. R. Ahmad et al., Phys. Rev. Lett. 89, 011301(2002), [nucl-ex/0204008].[3] S. Abe et al. [KamLAND Collaboration], Phys. Rev.Lett. 100, 221803 (2008) [arXiv:0801.4589 [hep-ex]].[4] Y. Ashie et al. [Super-Kamiokande Collaboration], Phys.Rev. Lett. 93, 101801 (2004) [hep-ex/0404034].[5] T. Schwetz, M. Tortola and J. W. F. Valle, New J. Phys.13, 109401 (2011) [arXiv:1108.1376 [hep-ph]].[6] J. Adam et al. [MEG Collaboration], Phys. Rev. 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Enhancing l(i) -> 3l(j) with the Z(0)-penguin