Prospects for the Zee-Babu Model at the LHC and low energy experiments
Miguel Nebot, Josep F. Oliver, David Palao, Arcadi Santamaria
aa r X i v : . [ h e p - ph ] M a r FTUV/07-1103, IFIC/07-70, RM3-TH/07-16
Prospects for the Zee-Babu Model at the LHCand low energy experiments
Miguel Nebot
Centro de F´ısica Te´orica de Part´ıculas (CFTP),Instituto Superior T´ecnico, P-1049-001, Lisboa, Portugal andInstituto Nazionale di Fisica Nucleare (INFN), Sezione di Roma Tre,Dipartamento di Fisica “Edoardo Amaldi”,Universit`a degli Studi Roma Tre, I-00146, Roma, Italy.
Josep F. Oliver, David Palao and Arcadi Santamaria
Departament de F´ısica Te`orica, Universitat de Val`encia andIFIC, Universitat de Val`encia-CSICDr. Moliner 50, E-46100 Burjassot (Val`encia), Spain
Abstract
We analyze the viability of the Zee-Babu model as an explanation of observed neutrino massesand mixings and the possibility that the model is confirmed or discarded in experiments plannedfor the very close future. The allowed parameter space is studied analytically by using someapproximations and partial data. Then, a complete scanning of all parameters and constraints isperformed numerically by using Monte Carlo methods. The cleanest signal of the model will bethe detection of the doubly charged scalar at the LHC and its correlation with measurements ofthe branching ratio of µ → eγ at the MEG experiment. In addition, the model offers interestingpredictions for τ − → µ + µ − µ − experiments, lepton-hadron universality tests, the θ mixing inneutrino oscillations and the h m ν i ee parameter of neutrinoless double beta decay. . INTRODUCTION The hints for neutrino masses accumulated in the last 30 years [1, 2] have been convertedinto a strong evidence in the last 10 years [3, 4, 5, 6, 7, 8, 9, 10]: the only consistentexplanation for solar neutrino data, atmospheric neutrino data, reactor and acceleratorneutrino experiments is based on the hypothesis of massive neutrinos which mix and oscillate.Using this hypothesis all neutrino data can be fitted with just two squared mass differencesand three mixing angles. At present, phases are not needed to explain the data althoughthey can be included in the analyses and hopefully will be tested in the future.This picture, although certainly a big step forward in our understanding of neutrinophysics leaves more open questions than the standard model (SM) picture in which neutrinosare exactly massless because (1) the SM does not contain righthanded neutrinos, thus, Diracneutrinos are not possible, and (2) the renormalizability of the model, with the minimal Higgscontent, enforces the exact conservation of the lepton number which prevents Majoranamass terms. Therefore, to include neutrino masses, we should relax some or several of theabove ingredients of the SM:1. Add righthanded neutrinos.2. Add new fields which could allow for lepton number violation while keeping the renor-malizability of the model.3. Drop the renormalizability of the model.The last possibility is very general and allows for neutrino masses without touching the fieldcontent of the SM. However, its predictivity is very limited and it is only useful when seenas a low energy parametrization of a more complete theory containing heavy non-standardparticles.Allowing for righthanded neutrinos seems the simplest and most economical solution:if neutrinos are like all the other fermions, it is natural to consider righthanded neutrinoscoupled to the lefthanded neutrinos and the Higgs scalar, obtaining in this way a Dirac mass Perturbatively both lepton ( L ) and baryon ( B ) numbers are conserved separately. If nonperturbativeeffects are taken into account only B − L is exactly conserved. Still, L and B violations are tiny at zerotemperature and density. m ν < .Adding other fields to the SM gives many possibilities but obviously, if there are norighthanded neutrinos, all of them require the non-conservation of the lepton number andneutrinos acquire a Majorana mass term. Since supersymmetry provides the best solutionto the hierarchy problem, one natural choice is to use the fields already present in thesupersymmetric extensions of the SM to generate neutrino masses. Very interesting modelsof this type with spontaneous breakdown of R-parity (see for instance [11, 12]), which implieslepton number violation, or explicit R-parity violation (see [13, 14]) have been built. All Although one can check it indirectly through its effects in leptogenesis. µ → eγ , µ → eee experiments in a couple of orders of magnitude (see forinstance [23]). It may be time to explore alternatives for neutrino mass generation thatcould be confirmed or rejected in planned experiments; among them, the natural candidatesare models in which neutrino masses are generated through radiative corrections: as themasses will be suppressed by loop effects the new particles responsible for them could berelatively light and be produced at the LHC, at the International Linear Collider (ILC) andmay have sizable effects in µ → eγ , µ → eee experiments. The simplest model of neutrinomasses is the Zee-Babu (ZB) model [18, 19, 20] which just adds two complex singlet scalarfields to the SM (that is, just 4 new degrees of freedom) with neutrino masses generatedat the two-loop level. Another very interesting model is the Zee model [17] which adds anew scalar doublet and a complex scalar singlet (6 new degrees of freedom), however, thesimplest version of the model gives a too sharp bimaximal prediction for neutrino mixingand has already been excluded [24, 25, 26]. Therefore, we will only consider here theZee-Babu model. In the Zee-Babu model neutrino masses are generated at two loops andare proportional to several Yukawa couplings of the new scalars and inversely proportionalto the square of their masses, therefore the couplings cannot be too small and the scalarmasses cannot be too large otherwise the generated neutrino masses would be too small.This is very interesting because the new scalars may be accessible at the LHC and couldmediate the processes µ → eγ and µ → eee with rates measurable in planned experiments.In this paper we will sharpen the predictions of the model by using both analytical andnumerical methods, specially under the assumption that the new scalars are light enough tobe produced at the LHC. The phenomenology of the Zee-Babu model was recently reviewed The model was first proposed in [18, 19] and studied later in [20]. In the literature, it has been oftenreferred to as the Babu model.
4n [27] where under certain assumptions analytic limits on several couplings and masses wereset. The analysis of [27] is very clever and interesting, however, the calculation of severalprocesses in that paper was taken from older papers with some wrong factors of 2 which,unfortunately, have also propagated to more recent papers. In addition, ref. [27] makes thesimplifying assumption that certain couplings are negligible. Thus, we found interesting toreview the phenomenology of the model by relaxing this assumption. This, unavoidablyrequires a numerical study which will be presented in this paper. We also take into accountnew stronger limits on flavour lepton number violating tau decays from BELLE [29, 30] andBABAR [31]. Thus, in section II we sketch the model and review some of its features: theneutrino masses, contributions to low energy processes like µ → eγ , µ → e , τ → µ andfavoured values for the parameters of the model. In section III we study the production anddecays of the new scalars at the LHC. In section IV we analyze all the relevant constraints onthe parameters of the model and obtain its predictions for lepton flavour violating processesand for neutrino mass parameters ( θ mixing, Majorana and Dirac phases, neutrinolessdouble beta decay parameter h m ν i ee , ...). Section V is devoted to a summary of the results.Finally in section VI we present our conclusions. II. THE ZEE-BABU MODEL
The Zee-Babu model is the minimal extension of the SM providing neutrino masses andmixings compatible with experiment: in addition to the Standard Model field spectrum, itonly contains one singly charged scalar and one doubly charged scalar.In order to fix the notation we will briefly review the Zee-Babu model. We will denote theSM particle content as follows: ℓ will be the lefthanded lepton doublet, e ℓ ≡ iτ ℓ c = iτ C ¯ ℓ T isjust the conjugate lepton doublet used to build Majorana type couplings, e is the righthandedlepton and H is the Higgs boson doublet. We also have, of course, the weak gauge bosons ~W µ , B µ , and the quarks and gluons. As mentioned, the Zee-Babu model contains, in addition,two charged singlet scalar fields h ± , k ±± , (1)with weak hypercharges ± ± Q = See also ref. [28]. + Y and that h and k destroy negatively charged particles, thus h = h − and h † = h + ,while k = k −− and k † = k ++ . Then, the Lagrangian can be split into two parts: L = L SM + L ZB . (2)The first part, L SM , is the Standard Model Lagrangian: L SM = iℓ D ℓ + ie D e + ( ℓY e H + h . c . ) + · · · (3)The dots represent SM gauge boson, Higgs boson and quark kinetic terms, quark Yukawa in-teractions and the SM Higgs potential. Generation and SU (2) indices have been suppressed,and therefore, Y is a completely general 3 × L ZB = D µ h † D µ h + D µ k † D µ k + ˜ ℓf ℓh + + e c g e k ++ + h . c . − V ZB . (4)Since both h and k are SU (2) singlets the covariant derivative only contains couplings to the B gauge boson, which after diagonalization will generate photon and Z -boson couplings withthe scalars, but no W couplings. Due to Fermi statistics the Zee type Yukawa coupling, f ab ,is an antisymmetric matrix in flavour space while the Yukawa coupling, g ab , of the doublycharged scalar k , is a symmetric matrix. The scalar potential V ZB contains all renormalizableinteractions between the scalars h , k and between them and the standard Higgs doublet: V ZB = m ′ h | h | + m ′ k | k | + λ h | h | + λ k | k | + λ hk | h | | k | + λ hH | h | H † H + λ kH | k | H † H + (cid:0) µh k ++ + h . c . (cid:1) . (5)Trilinear terms, like in the Zee model, with two scalar doublets and the h vanish identicallybecause the coupling is antisymmetric in SU (2) indices and, therefore, requires two differentdoublets. The last term is particularly interesting because if µ →
0, the complete Lagrangianhas an additional global U (1) symmetry which can be identified with lepton number L (or B − L ). In fact, if we assign lepton number 1 to both, the lepton doublet and the righthandedlepton singlet, we can also assign lepton number − h and k , in sucha way that this quantum number is conserved in all the Lagrangian except in the trilinearcoupling of the scalar potential. Thus, if µ = 0, lepton number is explicitly broken by the µ -coupling in the scalar potential. This is very important because this lepton number violation6ill be transmitted to the fermionic sector and will finally be responsible for the generationof neutrino masses. It is also important to remark that this mechanism for lepton numberviolation requires the simultaneous presence of the four couplings Y , f , g and µ , becauseif any of them vanishes one can always assign quantum numbers in such a way that thereis a global U (1) symmetry. This means that neutrino masses will require the simultaneouspresence of the four couplings.It is also important to note that the gauge-kinetic part of the Lagrangian is invariant underthe following U ( N ) transformations in generation space (for N generations of leptons). ℓ → U ℓ ℓ , e → U e e . (6)Yukawa couplings, however, break this symmetry. This implies that sets of Yukawa couplingsrelated by the following redefinitions are completely equivalent( Y, f, g ) → ( U † ℓ Y U e , U Tℓ f U ℓ , U Te gU e ) , (7)which in turn means that physical observables should transform correctly under these redef-initions. This is important to check the behaviour under flavour transformations of physicalamplitudes, moreover it can also be used to choose a convenient set of parameters in theYukawa sector and count the number of physical parameters following the methods devel-oped in [32]. Thus, using redefinitions of eq. (7) one can choose, without loss of generality, Y diagonal with real and positive elements. One could also choose f real and antisymmetricand leave g as a completely general complex symmetric matrix. In addition, one can useredefinitions of h + and k ++ to set µ real and positive and to remove one of the phases in g .Thus, we finally have
12 moduli (3 from Y , 3 from f and 6 from g ) and 5 phases (all from g ) and the real and positive parameter µ (plus, of course, the rest of the parameters in thescalar potential). However, we will see later that this convention is not compatible with thestandard parametrization of neutrino masses and mixings and it will be more convenientto use a slightly different convention for Yukawa coupling phases: we will also choose Y diagonal with real and positive elements, then we will choose fermion field rephasings toremove 3 phases from the elements of g ab , leaving the elements of f ab complex. Chargedscalar rephasing can be further used to remove the phase of µ and one of the phases of f ab , The counting can be generalized to n generations of leptons. In that case we will have n + n moduli and n − n − − e R k −− ν L e R e L e L ν L h − h H i h H i Figure 1: Diagram contributing to the neutrino Majorana mass at two loops. for instance we can take f µτ real and positive. Of course the counting of parameters is thesame as before: we will have 12 moduli (3 from Y , 3 from f and 6 from g ab ) and 5 phases(3 from g ab and 2 from f ab ) and the real and positive parameter µ .In any of the discussed conventions, Y is directly related to the masses of charged leptons m a = Y aa v , with v ≡ h H i = 174 GeV, the VEV of the standard Higgs doublet. Then thephysical scalar masses are m h = m ′ h + λ hH v , m k = m ′ k + λ kH v . (8) A. The neutrino masses
The first contribution to neutrino masses involving the four relevant couplings appearsat two loops [19, 20] and its Feynman diagram is depicted in fig. 1.The calculation of this diagram gives the following mass matrix for the neutrinos (definedas an effective term in the Lagrangian L ν ≡ − ν cL M ν ν L + h . c . )( M ν ) ab = 16 µf ac m c g ∗ cd I cd m d f bd , (9)with I cd = Z d k (2 π ) Z d q (2 π ) k − m c ) 1( k − m h ) 1( q − m d ) 1( q − m h ) 1( k − q ) − m k . (10) I cd can be calculated analytically [33], however, since m c , m d are the masses of the chargedleptons, necessarily much lighter than the charged scalars, we can neglect them and obtain8 much simpler form I cd ≃ I = 1(16 π ) M π I ( r ) , M ≡ max( m h , m k ) , (11)where ˜ I ( r ) is a function of the ratio of the masses of the scalars r ≡ m k /m h ,˜ I ( r ) = π (log r −
1) for r ≫
11 for r → , (12)which is close to one for a wide range of scalar masses. With this approximation the neutrinomass matrix can be directly written in terms of the Yukawa coupling matrices, f , g , and Y M ν = v µ π M ˜ I f Y g † Y T f T . (13)A very important point is that since f is a 3 × f = 0, andtherefore det M ν = 0. Thus, at least one of the neutrinos is exactly massless at this order .This is a very important result since it excludes the possibility of degenerate neutrino masses.To estimate the value of the largest possible neutrino mass we can take µ ≈ m k ≈ m h ≡ M , then the largest ν mass will be m ν ≈ . × − f g m τ M , (14)which is the typical seesaw formula, suppressed by some additional couplings and loopfactors. Because in this model one of the neutrinos is massless, the heaviest neutrino massis fixed by the atmospheric mass difference, thus m ν ≈ .
05 eV and f g ≈ M m ν m τ > × − , (15)since LEP bounds on charged scalar masses are typically M >
100 GeV. This meansthat f ’s and g ’s cannot be made arbitrarily small and natural values for them can be g, f > ∼ .
01. Then, for these relatively large couplings and scalar masses in a range M ∼
100 GeV −
10 TeV the model will give sizable contributions to low energy processeslike µ → eee , µ → eγ , ..., and scalars that could be produced and detected at the LHC. This result does not change if higher orders in charged leptons masses are taken into account in the loopintegral I ab . However, one expects it will change if higher loops are considered. f, g < M ≈ . × − f g m τ m ν < × TeV , (16)which is out of reach of planned experiments. However, constraints on low energy processesmight require smaller couplings which would lead to much smaller scalar masses. In addition,these estimates are very rough; for example the relevant couplings may be related to muonphysics, and not to tau physics: our estimate on neutrino masses should then be reduced bya factor ( m µ /m τ ) , which of course requires much lighter scalars to match the atmosphericneutrino scale. It is therefore very important to carefully establish the parameters andvalidity of the model. B. Low energy constraints
In order to provide neutrino masses compatible with experiment, the Yukawa couplingsof the charged scalars cannot be too small and their masses cannot be too large. Thisimmediately gives rise to a series of flavour lepton number violating processes, as for instance µ − → e − γ or µ − → e + e − e − , with rates which can be, in some cases, at the verge ofthe present experimental limits. This means that we can use these processes to obtaininformation on the parameters of the model and perhaps to confirm or to exclude the modelin a close future. In this section we will discuss briefly the relevant processes and collect theformulas for our conventions of Yukawa couplings: • ℓ − a → ℓ + b ℓ − c ℓ − d : The interesting observable for these processes is the decay width. Wehave (see for instance [34])R( ℓ − a → ℓ + b ℓ − c ℓ − d ) ≡ Γ( ℓ − a → ℓ + b ℓ − c ℓ − d )Γ( ℓ − a → ℓ − b ν ¯ ν ) = 12(1 + δ cd ) (cid:12)(cid:12)(cid:12)(cid:12) g ab g ∗ cd G F m k (cid:12)(cid:12)(cid:12)(cid:12) . (17)In this expression, the term δ cd takes into account the fact that we may have twoidentical particles in the final state. In the case of τ decays we have to remem-ber that leptonic channels are a small fraction of the decays BR( ℓ − a → ℓ + b ℓ − c ℓ − d ) =R( ℓ − a → ℓ + b ℓ − c ℓ − d )BR( ℓ − a → ℓ − b ν ¯ ν ) (with BR( µ − → e − ν ¯ ν ) ≈ τ − → e − ν ¯ ν ) ≈ .
84% and BR( τ − → µ − ν ¯ ν ) ≈ . rocess Experiment (90% CL) Bound (90% CL) µ − → e + e − e − BR < . × − | g eµ g ∗ ee | < . × − ( m k / TeV) τ − → e + e − e − BR < . × − | g eτ g ∗ ee | < .
010 ( m k / TeV) τ − → e + e − µ − BR < . × − | g eτ g ∗ eµ | < .
006 ( m k / TeV) τ − → e + µ − µ − BR < . × − | g eτ g ∗ µµ | < .
008 ( m k / TeV) τ − → µ + e − e − BR < . × − | g µτ g ∗ ee | < .
008 ( m k / TeV) τ − → µ + e − µ − BR < . × − | g µτ g ∗ eµ | < .
008 ( m k / TeV) τ − → µ + µ − µ − BR < . × − | g µτ g ∗ µµ | < .
010 ( m k / TeV) µ + e − → µ − e + G M ¯ M < . G F | g ee g ∗ µµ | < . m k / TeV) Table I: Constraints from tree-level lepton flavour violating decays. • µ + e − ←→ µ − e + : The k ++ scalar exchange gives also rise to transitions of the type µ + e − → µ − e + which are well bounded experimentally. The relevant four-fermion effec-tive coupling generated by exchange of the scalar k ++ is (here we use the conventionsfor the effective Hamiltonian and the limits of [35, 36]) G M ¯ M = − √ g ee g ∗ µµ m k . (18)We collect the relevant constraints of this type in table I. • ℓ a → ℓ b ν ¯ ν : These processes receive additional contributions from the exchange ofthe singly charged scalar h + which affect the Fermi muon decay constant but do notmodify the spectrum [21] (cid:18) G µ G µSM (cid:19) ≈ √ G F m h | f eµ | + 12 G F m h (cid:0) | f eµ | + | f eτ | (cid:1) (cid:0) | f eµ | + | f µτ | (cid:1) , (19)where a sum over undetected neutrinos has been performed. The second term is clearlysubdominant if m h ≫
200 GeV, however we have included it in the numerical analysisand have checked that we can neglect it in analytical estimates. In this model thecharged scalar only contributes to lepton decays but does not contribute to hadronicdecays, therefore the effective G β extracted from hadronic decays and G µ are different.However, in the framework of the SM, the equality of G β and G µ has been tested withvery good accuracy once all radiative corrections have been correctly included. In theSM, both | V ud | + | V us | + | V ub | = 1 and G βSM = G µSM are satisfied. Thus, assuming11 βSM = G µSM one can test the unitarity of the CKM matrix, or, conversely, assumingthe unitarity of the CKM matrix one can test the universality of couplings in hadronicand leptonic decays. In the model we are considering this is not true anymore; theCKM matrix is still unitary but, as explained, G β = G βSM = G µSM = G µ . Since theextraction of the experimental values of | V expij | assumes the SM, we will have V expij = G β G µ V ij , (20)where V ij are the truly unitary CKM matrix elements in the model. Therefore | V expud | + | V expus | + | V expub | = G β G µ = G µSM G µ ≈ − √ G F m h | f eµ | , (21)and, since | V expud | + | V expus | + | V expub | = 0 . ± . | f eµ | /m h .On the other hand, the charged scalar contribution will also modify the Fermi couplingextracted from τ decays in the different leptonic channels. After subtracting thedifferent factors from phase space and radiative corrections this is usually expressedin terms of ratios of effective “gauge couplings” g expa for the different leptons which inthe SM are all equal (see for instance [38]). Thus, comparing tau decays to muons andtau decays to electrons we have (since in the SM G a → b ∝ g a g b ) (cid:18) g expµ g expe (cid:19) = (cid:18) G τ → µ G τ → e (cid:19) ≈ √ G F m h (cid:0) | f µτ | − | f eτ | (cid:1) . (22)Similarly (cid:18) g expτ g expµ (cid:19) = (cid:18) G τ → e G µ → e (cid:19) ≈ √ G F m h (cid:0) | f eτ | − | f eµ | (cid:1) , (23) (cid:18) g expτ g expe (cid:19) = (cid:18) G τ → µ G µ → e (cid:19) ≈ √ G F m h (cid:0) | f µτ | − | f eµ | (cid:1) . (24)Universality constraints are summarized in table II , where measured values are translatedinto 90% CL limits. • ℓ − a → ℓ − b γ : In the case of transition amplitudes a = b the interesting observable is thedecay rate. We find (for calculations including singly and doubly charged scalars seefor instance [21, 39, 40])R( ℓ − a → ℓ − b γ ) ≡ Γ( ℓ − a → ℓ − b γ )Γ( ℓ − a → ℓ − b ν ¯ ν ) ≈ α π (cid:12)(cid:12)(cid:12)(cid:12) ( f † f ) ab G F m h (cid:12)(cid:12)(cid:12)(cid:12) + 16 (cid:12)(cid:12)(cid:12)(cid:12) ( g † g ) ab G F m k (cid:12)(cid:12)(cid:12)(cid:12) ! . (25)12 M Test Experiment Bound (90%CL)lept./hadr. univ. P q = d,s,b | V expuq | = 0 . ± . | f eµ | < .
015 ( m h / TeV) µ/e universality g expµ /g expe = 1 . ± . (cid:12)(cid:12) | f µτ | − | f eτ | (cid:12)(cid:12) < .
05 ( m h / TeV) τ /µ universality g expτ /g expµ = 1 . ± . (cid:12)(cid:12) | f eτ | − | f eµ | (cid:12)(cid:12) < .
06 ( m h / TeV) τ /e universality g expτ /g expe = 1 . ± . (cid:12)(cid:12) | f µτ | − | f eµ | (cid:12)(cid:12) < .
06 ( m h / TeV) Table II: Constraints from universality of charged currents.
The factor 16 in front of the doubly charged contribution does not usually appearin the literature [27] and deserves a comment: the Feynman rule for the ke a e b vertexcontains a factor 2 when a = b because there are two identical terms in the Lagrangian,but also the vertex with a = b contains a factor 2 because there are two identical Wickcontractions in this kind of vertices. This factor of 2 for identical particles was missedin [27] which led the authors do define new coupling constants with different factorsof 2 for diagonal and non-diagonal terms . It is also important to remark that thesingly and doubly charged scalar contributions do not interfere because they couple tofermions with different chirality. Again we have to remember that BR( ℓ − a → ℓ − b γ ) =R( ℓ − a → ℓ − b γ )BR( ℓ − a → ℓ − b ν ¯ ν ). • µ − e conversion in nuclei: The new scalars of the model do not couple to quarks and,therefore, do not generate a four-fermion operator that could contribute at tree levelto µ − e conversion. However, radiative corrections, in particular those related with the µ − e − γ vertex, will contribute to the process. It is also clear that those corrections aretightly related to the µ → eγ decay discussed above but are not identical because thephoton in µ − e conversion is not on the mass shell. In fact in ref. [41] if was shown thatin models with doubly charged scalars there is a logarithmic enhancement, log( q /m k ),of the µ − e conversion amplitude with respect to the µ → eγ amplitude. At present thebest limits come from µ − e conversion on T i , σ ( µ − T i → e − T i ) /σ ( e − T i → capture) < . × − [37] which, when translated into limits on the couplings, are slightly worsethan present µ → eγ constraints, but one has to keep in mind that if µ → eγ is relevantand if µ − e conversion limits are improved in the future it will also be relevant. One can also see that those results cannot be right because physical amplitudes should transform correctlyunder the flavour redefinitions of couplings in eq. (7). xperiment Bound (90%CL) δa e = (12 ± × − r (cid:0) | f eµ | + | f eτ | (cid:1) + 4 (cid:0) | g ee | + | g eµ | + | g eτ | (cid:1) < . × ( m k / TeV) δa µ = (21 ± × − r (cid:0) | f eµ | + | f µτ | (cid:1) + 4 (cid:0) | g eµ | + | g µµ | + | g µτ | (cid:1) < . m k / TeV) BR ( µ → eγ ) < . × − r | f ∗ eτ f µτ | + 16 | g ∗ ee g eµ + g ∗ eµ g µµ + g ∗ eτ g µτ | < . × − ( m k / TeV) BR ( τ → eγ ) < . × − r | f ∗ eµ f µτ | + 16 | g ∗ ee g eτ + g ∗ eµ g µτ + g ∗ eτ g ττ | < . m k / TeV) BR ( τ → µγ ) < . × − r | f ∗ eµ f eτ | + 16 | g ∗ eµ g eτ + g ∗ µµ g µτ + g ∗ µτ g ττ | < . m k / TeV) Table III: Constraints from lepton number violating photon interactions. • a = ( g − /
2: For diagonal transitions the model gives additional contributions tothe anomalous magnetic moments of the leptons which are very well measured in thecase of the electron and the muon. We find that the additional contribution to the a a of lepton ℓ a , in this model, is δa a ≡ a expa − a SMa = − m a π (cid:18) ( f † f ) aa m h + 4 ( g † g ) aa m k (cid:19) . (26)It is important to remark that we always find a negative contribution (this is in agree-ment with old and well tested calculations [42, 43]). In the case of the muon a µ , recentanalyses of experimental data and theoretical calculations in the Standard Model sug-gest that the experimental measurement is slightly larger than the SM prediction (fora review see [44]). Several authors have tried to explain this 1 σ to 3 σ effect in differ-ent extensions of the SM. In particular, in [45] the charged scalars of the Zee modelwere used to increase the a µ of the SM. We find this is not possible and instead wewill use the g − g − δa a . We will use [46] δa e = (12 ± × − and δa µ = (21 ± × − . Notice that in both cases the central value is positive, whilethe model gives a negative contribution. To place 90% CL bounds in this situationwe use the Feldman and Cousins prescription [47] which, for the values above, gives | δa e | < . × − and | δa µ | < . × − .Since lepton number is not conserved, another interesting low energy process that could arisein the model is neutrinoless double beta decay (0 ν β ). In this model, the singly and doubly14harged scalars do not couple to hadrons, this means that the (0 ν β ) rate is dominatedby the Majorana neutrino mass exchange and it is proportional to the | ( M ν ) ee | matrixelement, therefore it will be addressed in section IV when we discuss in detail the neutrinomass matrix constraints. C. Perturbativity constraints
Beside requiring that the model produce acceptable (a) neutrino masses and mixings (b)low energy predictions, we also have to address theoretical questions related to the validityof the predictions and the consistency of the model. Indeed to be able to perform anycalculation in this model we have to assume that perturbation theory can be used. Thisimposes strong constraints on the relevant couplings of the model. The Yukawa couplingsof the new scalars receive loop corrections like δf ∼ f (4 π ) , δg ∼ g (4 π ) . (27)If the corrections are going to be much smaller than the couplings, the couplings mustsatisfy f, g ≪ π . Similarly, the trilinear coupling among charged scalars proportional tothe parameter µ induces loop corrections to the charged scalar masses like δm k , δm h ∼ µ (4 π ) . (28)Requiring that the corrections are much smaller than the masses implies µ ≪ πm h , πm k .Since it is difficult to establish the exact values of the couplings for which the perturbativityof the theory breaks down, we will encode this type of constraints in the single parameter κ and will require | f ab | < κ , | g ab | < κ , µ < κ min( m h , m k ) . (29)For the purposes of illustration we will take κ = 1 or κ = 5, the value κ = 5 being ratherconservative (for instance, the strong coupling constant, g s , is considered to become non-perturbative at scales of about 1 GeV, at those scales α s (1 GeV) ∼ . g s (1 GeV) ∼ . µ is important in the generation of neutrino masses, therefore the con-straint µ < κ min( m h , m k ) is important. On the other hand, as seen in tables I, II, III, ifthe scalar masses are relatively light (around 1 TeV or less), low energy processes already15rovide interesting limits on the charged scalar Yukawa couplings. However, if the chargedscalars are heavier, the experimental limits on the new Yukawa couplings are so mild thatmay allow Yukawa couplings large enough to compromise the perturbative validity of thetheory. Then, the perturbativity constraints we just discussed will become relevant. III. THE MODEL AT THE LHC
Extra scalar degrees of freedom arise in many scenarios extending the weak interactionsbeyond the SM. In our case, the scalar sector is enlarged by the addition of two chargedscalars: h and k , which could be produced at the LHC if their masses are low enough. Inparticular, as we will see below, the LHC will be very well suited for searching the doublycharged scalar, k . Studies for searching doubly-charged scalars at future colliders have beendirected in the past [48, 49, 50, 51, 52] . In general this scalar is taken to be one componentof a weak triplet. Such triplets are well motivated on theoretical grounds, specially whenconsidering left-right symmetric models. Our work differs essentially in the gauge charges ofthe scalars. Both, h and k , are charged weak singlets that do not acquire a VEV. This makesthe phenomenology different. More model independent studies have been also considered inthe literature [53, 54].Concerning experimental bounds, LEP searched for these scalars. Their pair production( e + e − → γ ∗ Z ∗ → k ++ k −− ) implies the bound m k >
100 GeV [55, 56, 57]. Single productionvia e + e − → kee as well as contributions to Bhabha scattering have been also studied byLEP [57, 58], but in these cases the bounds depend on the (unknown) values of the Yukawacouplings. Tevatron has also been used to set bounds on this kind of scalars [59, 60, 61].Depending on the details of the model (couplings, decay channels,...) the mass is againfound to be roughly above 100 GeV. A. Collider phenomenology
1. Production
The extra scalars can be pair produced via a Drell-Yan process, fig. 2. Although thisproduction mechanism presents the drawback of having a potentially high threshold due tothe creation of two scalars, it has the important advantage of being proportional to their16 ∗ , Z ∗ qq k −− k ++ Figure 2: Pair production of k gauge charges as well as depending only on one unknown parameter: the mass of the scalar.The partonic cross section at LO reads σ = πα Q β (cid:20) Q q ˆ s − g L + g R ) Q q c w ˆ s − M Z (ˆ s − M Z ) + Γ Z M Z + ( g L + g R ) c w ˆ s (ˆ s − M Z ) + Γ Z M Z (cid:21) , (30)where ˆ s is the energy squared in the center of mass frame (CM) of the quarks, Q stands forelectric charges, g L and g R are given for the quarks by g L = T − s w Q q and g R = − s w Q q and β is the velocity of the produced scalars in this frame β = p − m / ˆ s .Equation (30) shows that pair production is four times more efficient for k than for h dueto their charges (assuming equal masses), which translates into a better discovery potentialfor k . The k pair production cross section, σ kk , at NLO for the LHC and Tevatron is displayedin fig. 3. To compute it, we have used CompHEP [62] with CTEQ6.1L libraries [63] to findthe LO cross section and afterwards we have included a K-factor of 1 .
25 for the LHC and1 . k accompanied by two singly charged scalars, fig. 4, or bytwo charged leptons replacing the scalar h ’s. If the k is accompanied by two charged leptonsthe amplitudes are proportional to the Yukawa couplings, whose exact values we ignore andmight be small.It is important to note that the cross section will be dominated by the virtual particlesin the propagators if they could be on-shell. In the case of k being produced with two h , thesingle production will be dominated by the first diagram if ˆ s > m k , because in this case k ∗ can be created on-shell. One might argue that the energy in the center of mass frame of17
00 1000 1500m k (GeV)0.0010.010.1110100 σ (fb) 14 TeV (LHC)2 TeV (Tevatron) Figure 3: Pair production cross section for k . We have used CompHEP (CTEQ6.1L) to obtain theLO and applied a K-factor of 1 .
25 for the LHC and 1 . γ ∗ , Z ∗ k ∗ qq k ++ h − h − γ ∗ , Z ∗ h ∗ qq h − k ++ h − Figure 4: Single production diagrams. the colliding quarks is not fixed, instead it is a fraction of the total energy in the center ofmass frame of the colliding protons, s . However, the cross section involves an integrationover the possible values of ˆ s . If s is large enough to create two k ’s, the integration will bedominated by the real production of two k ’s, thus reducing the single production to pairproduction. Specifically, σ ( k ++ h − h − ) ≈ σ kk Br ( k → hh ). The same reasoning is valid in thecase of single production with leptons. We have performed calculations using CompHEP tocheck this point. Therefore, single production is only important when the available energy, s , is not sufficient to create a pair of k . 18 possible third production mechanism is via the couplings with the Higgs doublet, H .There is little information concerning these couplings because their contribution to lowenergy phenomenology is expected to be negligible in front of the Yukawas f ab and g ab .This is so because the former enter at two loops and the later at tree level. In any case,the amplitudes of these processes are expected to be small because the Higgs couplings toquarks are proportional to their masses.In summary, we find that, from the point of view of production, the best suited channelfor discovery studies is pair production, being four times more efficient for k than for h .
2. Decay
We assume for the moment that the scalars are not long lived, i.e. they decay beforereaching the detector. Different decay channels present different experimental sensitivitiesdepending on the final products. In particular, detectors are much less sensitive to thosechannels containing neutrinos and/or taus in the final states, since neutrinos (includingthose coming from the decay of the taus) will escape undetected. This makes necessary tocompute the branching ratios.The k scalar can always decay to two leptons of the same sign, since m k >
100 GeV. Thewidth reads Γ( k → ℓ a ℓ b ) = | g ab | π (1 + δ ab ) m k . (31)It is worth to stress that k is the only particle in this model that can decay to two like-signleptons, which will be crucial to detect it.If m k > m h , then k can also decay to a pair of h Γ( k → hh ) = 18 π (cid:20) µm k (cid:21) m k s − m h m k . (32)On the other hand, the decay channels of h reduce to those with one lepton and oneneutrino in the final state: Γ( h → ℓ a ν b ) = | f ab | π m h . (33)Since these channels involve always one neutrino it is clear that detecting h will be muchmore complicated than detecting k even if their production rates were similar.19inally we will check that the doubly charged scalar k ++ cannot be long-lived in thismodel once low energy phenomenology and neutrino data are taken into account. Indeedthe k decay width can be written asΓ = m k π (cid:0) | g µµ | + · · · (cid:1) . (34)where the g µµ term takes into account the decay into muons and the dots represent all otherpossible couplings. Then, the long-lived condition translates into | g µµ | < − which cannotbe fulfilled when the limits obtained in tabs. IV-V are used.
3. Detection at colliders
As we have seen the discovery potential of the LHC for k is more promising than for h .On one side because the production cross section of the former is enhanced with respect tothe later and, on the other side, because the experimental sensitivity to the decay channelsof h is smaller. Thus, in the following we will focus on k .After a pair of k ’s is created in the collider, they can decay into a number of final states.The most interesting for us contains four like-sign leptons. From now on we will refer to thischannel as 4 lep . The rest of the possible final states always contain h or τ . These channelsare quite difficult to deal with experimentally because τ and h will decay to neutrinos. Incontrast, detection of electrons and muons is quite efficient. In addition, the decay of a k to a pair of like-sign leptons ( e ± or µ ± ) with high invariant masses constitutes a clear anddistinct signature. This channel has a negligible background coming from SM processes,making it very appropriate for k discovery studies.In order to model the efficiencies and acceptances of the detectors at the LHC we use thecriterion that 10 events of k pair production with subsequent decay to 4 lep lead to discoveryof k . We expect that at least two of such events will be properly detected/identified providingus with four pairs of like-sign leptons which invariant mass will give us the first estimate of m k . This criterion is taken from [50] where the authors perform a study of the discoveryreach at Tevatron for doubly-charged bosons decaying to like-sign leptons in a similar modeland a similar rule can be extracted from [52] where the authors focus on the ATLAS detector By “long-lived” we understand that the scalar can travel a distance of the order cτ > m [61]. k (GeV)110100100010000N Figure 5: Number of events at the LHC in the 4 lep channel for a luminosity L = 300 fb − and √ s = 14 TeV assuming that all produced k pairs decay in this particular channel. at the LHC. A more detailed study of the forthcoming detectors acceptances and efficienciesat the LHC is desirable.To estimate the maximum reach in terms of m k at the LHC we take the most optimisticscenario in which all the k pairs decay to 4 lep . The number of events in this channel isshown in fig. 5 for the optimistic luminosity
300 fb − and CM energy 14 TeV. From thisplot one concludes that the LHC will be able to probe masses up to 1 TeV approximately.In general, the signal in the 4 lep channel will be smaller than the one shown in fig. 5 dueto the presence of the other decay channels. This signal draining will be controlled by thebranching ratio BR lep , which can be expressed in terms of the couplings as BR lep = | g ee | + | g µµ | + 2 | g eµ | | g ee | + | g µµ | + 2 | g eµ | + | g hh | + 2 | g eτ | + 2 | g µτ | + | g ττ | (35)where we have defined the effective coupling of the doubly charged scalar to singly charged The LHC luminosity is expected to be about 100 fb − / year . g hh , as g hh = (cid:20) µm k (cid:21) (cid:18) − m h m k (cid:19) / . (36) IV. ANALYSIS OF THE PARAMETERS OF THE MODEL
As discussed in section II the Yukawa couplings of the model can be written in termsof 12 moduli and 5 phases. Other parameters relevant for the model are the masses ofthe charged scalars, m h and m k , and the coupling µ . Among the 12 moduli, 3 correspondto the 3 charged lepton masses, which are known. Thus, we have 17 additional relevantparameters with respect to the plain SM with massless neutrinos (9 moduli and 5 phasesfrom the new Yukawa couplings, the 3 scalar parameters, m h and m k and the coupling µ ).The remaining parameters in the scalar potential are of no interest for our purposes. Theneutrino mass matrix is rather well known. In our case it contains 2 neutrino masses, 3real mixing angles and 2 phases (1 CKM-type phase and 1 Majorana phase). Thus, therewill still remain 4 moduli and 3 additional phases in the Yukawa couplings (plus m h , m k and µ ). On most of these Yukawa couplings we have some information from section II Bas long as the masses of the charged scalars are not much heavier than 1 TeV. Notice thatthis is the interesting range for scalar masses if they are going to be produced at the LHC.In addition, there are also indirect arguments that suggest that the scalar masses shouldbe relatively light (below 1 TeV) if one likes to avoid strong hierarchy problems , sincethe charged scalar masses will contribute, at one loop, to the mass of the SM Higgs boson.However, the couplings of the SM Higgs boson to the new scalars are unknown and couldbe small. Thus, although the natural range of the masses of the new scalars is about fewTeV or less, they can also be larger. Then, in what follows, we will allow the masses ofthe charged scalar to vary between the LEP lower bound ∼
100 GeV and infinity. We willimmediately see, however, that present information already constrains the charged scalarmasses to be below ∼ TeV. The couplings g ab , f ab and µ must in addition satisfy theperturbativity constraints discussed in section II C.From the previous discussion it is clear that even though we have 17 additional parameterswe also have a lot of information on them both from neutrino oscillations and from low energy Alternatively one could enlarge the model by supersymmetrizing it. f ab ’s, at two loops, the mass matrix determinant is equal to zero,and thus one eigenvalue is zero. Two mass differences are then sufficient to fix the massesboth in the normal hierarchy and in the inverted hierarchy cases (a degenerate spectrumcannot arise in this model). Except for the Dirac phase δ and the Majorana phase φ , wecan almost reconstruct experimentally the neutrino mass matrix by using the informationwe have on the mixing angles and the masses. Without loss of generality we can write theneutrino Majorana mass matrix as M ν = U D ν U T , (37)with U the standard PMNS matrix U = c s − s c c s e − iδ − s e iδ c c s − s c
00 0 1 , (38)while D ν is the diagonal matrix of masses (including the only Majorana phase). Notice thatwriting the mass matrix in this form already implies some phase convention.Since one of the ν masses of the model is zero we only have two possibilities :Normal hierarchy (NH) D NHν = m e iφ
00 0 m , m ≫ m , ∆ S = m ∆ A = m . (39) Here we follow the conventions and results of ref. [65] adapted to our case. D IHν = m m e iφ
00 0 0 , m ≈ m , ∆ S = m − m ∆ A = m . (40)With ∆ S = (7 . ± . × − eV ∆ A = (2 . ± . × − eV , (41) s ≡ sin θ = 0 . ± . s ≡ sin θ = 0 . ± . , s ≡ sin θ ≤ . , . (42)Thus, apart from the poorly known s mixing (we just know it is small) and the phases, δ and φ , the mass matrix can be partially reconstructed in terms of the 2 known massdifferences and the 2 known mixing angles for each of the two cases. In particular, we canimmediately extract the matrix element responsible for (0 ν β ) decays:Normal hierarchy h m NHν i ee = ( M NHν ) ee = p ∆ S c s e iφ + p ∆ A s e − i δ . (43)In this case , given the previous values, it is clear that ( M NHν ) ee < ∼ .
003 eV and thereforedifficult to see in (0 ν β ) decay experiments.Inverted hierarchy h m IHν i ee = ( M IHν ) ee = p ∆ A + ∆ S c s e iφ + p ∆ A c c . (44)In this case, unless a cancellation occurs between the two terms for e iφ = −
1, ( M IHν ) ee isnaturally of order 0 .
05 eV and, therefore, observable in planned (0 ν β ) decay experiments.Equation (13) gives the mass matrix M ν in terms of the parameters of the model – theYukawa couplings, the scalar masses and the trilinear coupling –, we can thus try to fix someparameters by matching the M ν , obtained from the neutrino oscillation parameters, to thecalculated one. Since the mass matrix is symmetric, in principle this gives 6 equations.However, one of them is trivially satisfied because, by construction, both matrices already Notice the dependence on the Dirac phase δ . This is a consequence of our convention for Majorana phases.One could redefine phases and make this quantity independent on δ , but this will not affect predictionsor constraints on observables. M ν ) = 0. To choose the remaining 5 equations we will use that the eigenvectorcorresponding to the 0 eigenvalue is very simple; as det f = 0, there is an eigenvector a of f with zero eigenvalue f · a = 0, a = ( f µτ , − f eτ , f eµ ). Obviously, a will also be an eigenvectorof M ν with zero eigenvalue when expressed in terms of masses and mixings and, therefore, U D ν U T a = 0 or, since U is unitary, D ν U T a = 0. This gives us three equations, one ofwhich is satisfied trivially because one of the diagonal values of D ν is zero. The other twoequations will allow us to express the ratios of couplings f ij just in terms of mixing anglesand phases . Thus, in the NH case we have ( D NHν ) = 0 and (cid:0) U T a (cid:1) = 0 ⇒ f eτ f µτ = tan θ cos θ cos θ + tan θ sin θ e − iδ , (cid:0) U T a (cid:1) = 0 ⇒ f eµ f µτ = tan θ sin θ cos θ − tan θ cos θ e − iδ . (45)These equations immediately tell us that the standard PMNS convention of phases is notcompatible with all f ab being real. However, we can take a phase convention in which f µτ isreal and positive and leave f eτ and f eµ complex with phases fixed by eq. (45).With values like s ∼ . s ∼ . s < .
02, the first term on the right hand sideof eqs. (45) dominates and we get f eτ ≃ f µτ / ≃ f eµ . With this relation we can go backto the low energy bounds in table II and table III and find that the strongest constraintson the f ij couplings come from µ → eγ (which strongly bounds | f eτ f µτ | ) and tell us that | f ei | < ∼ . m h / TeV) and | f µτ | < ∼ . m h / TeV).The equations corresponding to the inverted hierarchy case, ( D IHν ) = 0, are (cid:0) U T a (cid:1) = 0 ⇒ f eτ f µτ = − sin θ tan θ e − iδ , (cid:0) U T a (cid:1) = 0 ⇒ f eµ f µτ = cos θ tan θ e − iδ . (46)In this case, it is clear that f eτ /f eµ = − tan θ ≈ − | f eµ | > | f µτ | , | f eτ | > | f µτ | . Nowwe can use these relations in the low energy bounds in table II and table III and find that thestrongest constraints on the f ij couplings come from lepton-hadron universality (see tableII) , 5 | f µτ | < ∼ | f ei | < ∼ . m h / TeV).We still have 3 additional equations we can use to fix the parameters of the model. In Therefore, the decay branching ratios of the scalar h to the different leptons are fixed by the mixing angles.This can probably be exploited [28] at the ILC . and following [27] we will take thethree elements m , m and m in the equalities m ij ≡ ( M ν ) ij = ζ f ia m a g ∗ ab m b f jb with the m ij defined by eq. (37) and ζ = µ π M ˜ I . Thus, if ω ab = m a g ∗ ab m b we have m = ζ ( f µτ ω ττ − f eµ f µτ ω eτ + f eµ ω ee ) ,m = ζ ( − f µτ ω µτ − f µτ f eτ ω eτ + f µτ f eµ ω eµ + f eτ f eµ ω ee ) ,m = ζ ( f µτ ω µµ + 2 f µτ f eτ ω eµ + f eτ ω ee ) . (47)Because of the hierarchy among the charged lepton masses, it is natural to assume thatthose ω ab containing the electron mass, ω ee , ω eµ , ω eτ , are much smaller than ω µµ , ω µτ , ω ττ ,in that case we can neglect them (we will check later the goodness of this approximationwithin the numerical analysis), and we have m ≃ ζ f µτ ω ττ , m ≃ − ζ f µτ ω µτ , m ≃ ζ f µτ ω µµ . (48)In the normal hierarchy case this gives ( s ij ≡ sin θ ij , c ij ≡ cos θ ij ) ζ f µτ ω ττ ≃ m c s + m e iφ ( c c − e iδ s s s ) ,ζ f µτ ω µτ ≃ − m c c s + m e iφ ( c s + e iδ c s s )( c c − e iδ s s s ) ,ζ f µτ ω µµ ≃ m c c + m e iφ ( c s + e iδ c s s ) . (49)With m ≃ .
05 eV and m ≃ .
009 eV, | ω ττ | ≃ | ω µτ | ≃ | ω µµ | ≃ .
05 eV2 ζ | f µτ | , (50)setting a definite hierarchy among the g ab couplings: g ττ : g µτ : g µµ ∼ m µ /m τ : m µ /m τ : 1 . (51)In the inverted hierarchy case, eqs. (47) give ζ f µτ ω ττ ≃ m ( c s + e iδ c s s ) + m e iφ ( c c − e iδ s s s ) ,ζ f µτ ω µτ ≃ m ( s s − e iδ c c s )( c s + e iδ c s s )+ m e iφ ( c s + e iδ c s s )( c c − e iδ s s s ) , (52) ζ f µτ ω µµ ≃ m ( s s − e iδ c c s ) + m e iφ ( c s + e iδ c s s ) , Except that they cannot be in the same column or the same row of M ν , because in that case the equationsare related by M ν a = 0. m ≃ m ≃ .
05 eV, also yielding for e iφ = 1 | ω ττ | ≃ | ω µτ | ≃ | ω µµ | ≃ .
05 eV2 ζ | f µτ | , (53)and the hierarchy of couplings in eq. (51). However, in the IH case there is a strong cancel-lation for Majorana phases close to π , and one can obtain a smaller value for ω µµ , thus wecan only write | ω µµ | > .
007 eV2 ζ | f µτ | . (54)In both cases one expects g µµ to be the largest coupling among the three considered. Ofcourse, g ee , g eµ and g eτ can also be large and are only constrained by low energy processes andperturbativity constraints. One should notice, however, that in the inverted hierarchy case,the approximation made in going from eqs. (47) to eqs. (48) may be a priori less justifiablethan in the normal hierarchy case when θ →
0, as the eigenvector corresponding to the zeroeigenvalue, ( f µτ , − f eτ , f eµ ) is proportional to ( e iδ tan θ , sin θ , cos θ ), i.e. f µτ ∝ tan θ ,and since the terms retained in eq. (47) are proportional to f µτ , it is not obvious that theterms proportional to ω ei can be neglected.Assuming then that | g µτ | ≈ | g µµ | ( m µ /m τ ) and | g ττ | ≈ | g µµ | ( m µ /m τ ) , we can go back totables I and III to find the relevant constraints on the couplings. The best constraint comesfrom τ − → µ + µ − µ − , which tells us that | g µµ | < ∼ . m k / TeV), | g µτ | < ∼ . m k / TeV), | g ττ | < ∼ . m k / TeV).We can use all this information to set analytical bounds on the relevant parameters ofthe model in the line discussed at the beginning of section II A.
A. Analytical constraints
1. NH case
First, just from the neutrino mass formula, we have | g µµ || f µτ | ≥ − max( m k , m h )˜ I TeV max( m k , m h ) µ . (55)Now we can show that due to the logarithmic growth of ˜ I for m k ≫ m h and the fact that˜ I ≤ m k < m h , 27ax( m k , m h ) e Im h ≥ . (56)Thus | g µµ || f µτ | ≥ − m h TeV max( m k , m h ) µ . (57)Now we use the perturbativity bound on µ , µ < κ min( m h , m k ) κ | g µµ || f µτ | ≥ − m h TeV max( m k , m h )min( m k , m h ) , (58)which can be rewritten as (use that m h m k = max( m k , m h ) min( m k , m h )) (cid:18) max( m k , m h )TeV (cid:19) ≤ κ | g µµ || f µτ | m k TeV (59)We can use that m k ≤ max( m k , m h ) and the perturbative constraints | g µµ | < κ , | f µτ | < κ to find an upper limit on the masses of the charged scalars m h , m k ≤ max( m k , m h ) < κ TeV . (60)On the other hand, if we use | g µµ | < ∼ . m k / TeV), coming from τ → µ and | f µτ | < ∼ . m h / TeV), coming from µ → eγ we immediately obtain a lower bound on the masses ofthe scalars m k , m h > min( m k, m h ) > . √ κ TeV . (61)If we only use the τ → µ constraint in (58) we find a bound on the | f µτ | coupling κ | f µτ | ≥ . × − (cid:18) m h min( m k , m h ) (cid:19) , (62)and using that m h > min( m k , m h ) we find an absolute limit on the coupling | f µτ | > . √ κ . (63)Thus, using either the experimental bounds and/or the perturbativity bounds, we can alsoset upper and lower limits on the different couplings | g µµ | , µ and the interesting observables BR ( µ → eγ ) and BR ( τ → µ ). As discussed in section III the LHC will be able to find thedoubly charged scalar of the model k ++ as long as it is lighter than about 1 TeV, thus it isinteresting to know what are the constraints on the parameters of the model if m k < eneral case m k < . √ κ TeV ≤ m h , m k < κ TeV 0 . √ κ TeV ≤ m h , m k < κ / TeV0 . κ TeV < µ < κ TeV 0 .
26 TeV < µ < κ
TeV0 . √ κ < | f µτ | < κ , . κ ≤ | g µµ | ≤ κ . √ κ < | f µτ | < κ , . κ ≤ | g µµ | ≤ κBR ( µ → eγ ) ≥ . × − /κ BR ( µ → eγ ) ≥ × − /κ BR ( τ → µ ) ≥ . × − /κ BR ( τ → µ ) ≥ × − /κ Table IV: Normal Hierarchy analytical constraints: we assume ω ie ≃ κ and explicitly displayed. the model adding this additional constraint. We collect all the limits we obtain in table IV.It is important to remark the assumptions we use to obtain these bounds: we assume thatbecause the small electron mass, as compared with the tau lepton and muon masses, ω ie ≃ κ and explicitlydisplayed.
2. IH case
The same kind of bounds can be obtained for the IH case with a few remarks. In the IHcase f µτ is not the largest coupling among the f ′ s , since | f µτ | ≈ √ s | f eµ | with s small.Thus perturbativity bounds should be applied to f eµ . In addition the best experimentallimit is also on f eµ , | f eµ | < . m h / TeV). Then, it is convenient to write the main equationsin terms of f eµ instead of f µτ . Finally in the IH hierarchy case there is the possibility of29 eneral case m k < . √ κ TeV < m k , m h < κ TeV 0 . √ κ TeV ≤ m h , m k < κ / TeV0 . κ TeV < µ < κ TeV 0 . < µ < κ TeV0 . √ κ < | f eµ | < κ , . κ ≤ | g µµ | ≤ κ . √ κ < | f eµ | < κ , . κ ≤ | g µµ | ≤ κBR ( µ → eγ ) ≥ × − /κ BR ( µ → eγ ) ≥ × − /κ BR ( τ → µ ) ≥ × − /κ BR ( τ → µ ) ≥ × − /κ . /κ < s < .
02 0 . /κ < s < . cancellations for φ = π which allow for a slightly smaller ω µµ . We have in this case s | g µµ || f eµ | ≥ . × − m h TeV max( m k , m h ) µ . (64)Then we can repeat essentially the same arguments used for the NH, together with theupper limit on s , s < .
02, to obtain lower and upper limits on the masses of the scalars, m h , m k , on the coupling | f eµ | , which is related to | f eτ | and | f µτ | , on the coupling | g µµ | ,related to | g µτ | and | g ττ | , and on the trilinear coupling µ . In addition, since in the IH casethere is a strong dependence on s we can also set a lower bound on it. As in the NH casewe also give the corresponding limits one would find under the assumption that the doublecharged scalar k ++ is found at the LHC and, therefore, has a mass smaller than 1 TeV. Wesummarize all the limits in table V. B. Numerical analysis
The information we obtained above is very useful; however, to obtain it we have madeuse of different approximations: 30 ) We assumed that the ω ee , ω eµ , ω eτ can be neglected in front of the other couplings. Thisapproximation is reasonable because these ω ’s are proportional to the electron mass, ω ei = m e g ei m i , which is much smaller than the other two lepton masses. However, itcould happen that, for some reason, the g ei couplings are much larger than the others.It is therefore important to perform a complete analysis without this assumption. b) We took central values for the measured oscillation parameters. c) In the analytical limits we only used data from neutrino oscillations and bounds from τ → µ and µ → eγ (or lepton/hadron universality in the IH case) together withthe perturbativity constraints. As discussed in section II B there are many moreexperimental constraints that can affect the results and should be taken into account.It is clear that the only way to analyze the model without those approximations is by meansof an exhaustive numerical exploration of the parameter space of the model. The basic toolto achieve this goal will be the use of Monte Carlo (MC) techniques; however, because of thelarge number of independent parameters and their diverse relevance, straightforward appli-cation of MC techniques is not sufficiently efficient and thus some additional considerationsand refinements will be required.The crudest MC exploration of the available parameter space would involve random gen-eration of complete sets of 17 independent basic parameters, calculation of the correspondingpredictions for the observables and finally an acceptation/rejection process in terms of theagreement between those predictions and the appropriate experimental constraints. Be-side the large number of parameters to be considered, the relations among them previouslydiscussed render such a crude approach almost hopeless.Realizing that not all observables play an equal role, that is, some of them are muchmore informative or constraining than others, we can go one further step in the use ofsimple MC techniques: instead of the simplest MC outlined above, we can construct aMC process devised to automatically produce mass matrices in agreement with neutrinooscillation experiments.Knowing the masses and mixing angles, if we were to reconstruct the mass matrix M ν using experimental input, the only missing ingredients would be the Dirac phase δ , theMajorana phase φ (see eqs. (38–40)) and the poorly known mixing θ , for which we onlyknow it is small and ignore its exact value or even if it is zero. Equation (13) gives the mass31atrix M ν in terms of the new parameters – the Yukawa couplings, the scalar masses andthe trilinear coupling –, we can thus try to fix some parameters by matching the extracted M ν from oscillation data and the calculated one. This procedure achieves two goals: itguarantees that neutrino oscillations are adequately produced and it reduces the freedomin parameter space entering numerical study by trading some of the couplings by measuredneutrino oscillation parameters. For each set { θ , θ , θ , ∆ A , ∆ S , δ, φ } we thus obtain nu-merical values for the entries in M ν . We will then use eqs. (45) or eqs. (46) to fix f eµ and f eτ in terms of f µτ and the generated mixing angles. Then come eqs. (47); these three complexrelations involve the six complex couplings g ab , the trilinear coupling and the scalar masses.Together with m k , m h and µ , knowing three independent g ab ’s in eqs. (47) will be sufficientto fix the remaining ones; effectively this means that we will generate g ee , g eµ and g eτ , andthus automatically fix g µµ , g µτ and g ττ . Notice that this phase convention is compatiblewith the standard choice for the neutrino mass matrix, eqs. (37-38).To summarize, by generating five quantities – ∆ A , ∆ S , θ ij – according to experimental knowl-edge, two phases – δ and φ –, one real coupling f µτ , two masses – m h and m k –, the trilinearcoupling µ and three complex g ab , we are spanning the 12 moduli and 5 phases needed to de-scribe the model. That is, instead of the crude and utterly inefficient Monte Carlo procedurein terms of { m k , m h , µ, f ab , g ab } , we can use { s ij , ∆ A , ∆ S , δ, φ, f µτ , m h , m k , µ, g ee , g eµ g eτ } toexplore the whole parameter space and guarantee the agreement with neutrino oscillationsresults prior to the use of the remaining experimental constraints, which constitute the nextstep, as they are then applied to accept/reject “candidate points”. Notice that we have notspecified the generation process of the different quantities involved: some discussion will beaddressed below, the details of the numerical generation are summarized in table VI.Despite being operative and useful, this refined MC procedure is not the last word as onecan do better. For this purpose we resort to the use of Markov Chain driven Monte Carlo(MCMC) processes of the Metropolis type.We have discussed the benefits of a refined simple MC procedure with respect to thecrudest one: the next (and final) step to complete the numerical toolkit we use is therather straightforward conversion of this refined MC into a Metropolis-like simulation whichprovides the results to be discussed. This is largely beneficial as (1) the efficiency of theMCMC process is sufficient to produce a reliable and smooth output for the different subcasesunder study, (2) the refined MC gives a helpful check of the consistency of the whole process.32 arameter Value Shape Parameter Value Shape∆ S (7 . ± . × − eV Flat ∆ A (2 . ± . × − eV Flatsin θ . ± .
03 Flat sin θ . ± .
08 Flatsin θ [10 − ; 2 × − ] Log flat δ [0; 2 π [ Flat φ [0; 2 π [ Flat m h [10 ; 10 ] GeV Log Flat m k [10 ; 10 ] GeV Log Flat f µτ [10 − ; κ ] Log flat µ [1; 10 ] GeV Log flat | g ee | [10 − ; κ ] Log flat arg( g ee ) [0; 2 π [ Flat | g eµ | [10 − ; κ ] Log flat arg( g eµ ) [0; 2 π [ Flat | g eτ | [10 − ; κ ] Log flat arg( g eτ ) [0; 2 π [ FlatTable VI: Numerical values. Let us now discuss the remaining details concerning the simulations; notice that, even ifin the following we refer to the generation of parameters, which is appropriate for the MCprocess, the corresponding feature when dealing with MCMC processes is not generation but in fact how they enter the stepwise acceptance function, however, to avoid essentiallyduplicated discussions we will just mention what concerns the plain MC case. The mainidea that drives our election of shapes and ranges of the different parameters is the need toperform an adequate exploration of the available parameter space, in particular one has toensure that the regions which can yield interesting signals like the production of scalars atthe LHC or branching ratios of exotic processes close to present experimental bounds areproperly studied. In particular: • Neutrino oscillations results, i.e. the squared masses differences ∆ A , ∆ S and the mixingparameters sin θ , sin θ , are generated with flat distributions within a ± . σ range around the quoted experimental value (this corresponds to 90% confidence levelor probability region for a gaussian-distributed uncertainty of the measurement). Forsin θ , however, we only have an upper bound: to span a reasonable range of valuesit is generated through a logarithmically flat distribution from the upper bound downto very small values, cut off at 10 − . • The Dirac and Majorana phases, δ and φ , are generated according to flat distributions33panning the whole available range [0; 2 π [. • Concerning the independent Yukawa couplings f ab and g ab , moduli are generatedthrough distributions logarithmically flat to explore values that could potentially spanseveral orders of magnitude. The applied upper bounds correspond to the differentnaturalness/perturbative cases under consideration. The arguments, as the Dirac andMajorana phases, are generated through flat distributions over the complete [0; 2 π [range. • The masses of the new scalar fields m k and m h are generated with logarithmically flatdistributions reaching up to 10 TeV and bounded below at ∼
100 GeV to incorporateLEP-motivated constraints. In any case, the precise upper bound is irrelevant as faras it is beyond the analytic bounds presented in tables IV and V. • The trilinear coupling µ is also generated with a distribution flat in its logarithm andlimited by the perturbativity requirement. • We apply the remaining experimental constraints presented in section II B in a sharp(straightforward acceptance or rejection) way: the only acceptable predictions are theones within the quoted 90% CL ranges/bounds. • The simulation described in the previous points allows a very wide range of scalarmasses. However, as discussed, the most interesting case is when m k < k ++ can be discovered at the LHC. Thus, we have performed an inde-pendent simulation requiring m k < • All the simulations are done for both the NH and IH cases and for two values of theperturbativity constraint κ = 1 and κ = 5.The arbitrariness in the choice of priors and their impact in the final results is always aconcern in this type of analyses. Because of this we have used several priors. In the caseof the neutrino oscillation parameters we have repeated the analysis fixing the parametersat the central values, taking gaussian distributions around central values and using the flatdistributions we have finally presented here. The differences are marginal and we chose topresent results for flat distributions because the results are slightly more conservative. Forother parameters we also tried plain flat priors, but, specially for parameters that range34n several orders of magnitude, logarithmically flat distributions span more efficiently theparameter space. We checked that the distributions obtained for the observables considered,for which we found analytical lower and upper limits, do not depend too much on the choice.At this point the machinery used to perform the announced numerical studies has beencompletely presented, however some comments on the nature and interpretation of the out-put it produces are in order.The model under study naturally “lives” in a parametric space of high dimensionality.The standard statistical arsenal offers two different approaches to reduce this high dimen-sional information and produce tolerably low dimensional – usually one or two dimensional– output: the frequentist and the bayesian frameworks. Very schematically: • Frequentists assign confidence levels to the marginalized output through the best fitachievable with the remaining parametric freedom. • Bayesians assign probability densities to the marginalized output through the integra-tion over the remaining parametric freedom of the likelihood (of data for the givenparameters) times the prior distribution/weight of parameters (this is just Bayes con-ditional probability inversion formula at work).Beside the long standing quarrel existing among practitioners of one or the other approach,both, with their information reduction schemes, unavoidably present some drawbacks to-gether with their statistical merits. As we want sensitivity to the parameter space availablefor the model to work, the procedure we have followed might look quite bayesian. Beingaware of the dependence on prior election and the imprecise nature of the details behindmany constraints , we do not intend at all to try and produce would-be highly orthodoxstatistical results neither interpret them as if they were so, and thus we have chosen thenumerical details of the simulations – that is both experimental constraints and priors – asstated above without any further qualm. Usually the available information is just a 90% CL range and little additional knowledge on the distributionoriginating this range is given. Moreover the perturbativity constraints, as clearly seen in tables IV and Vare determinant and, like all theoretical constraints, no obvious confidence levels or statistical significancecan be assigned to them. . RESULTS In this section we collect the main results of the paper.First we would like to study the impact of the different assumptions and experimentaldata in the analysis.To illustrate the impact of low energy constraints ( µ → eγ , τ → µ , ...) we performan independent simulation with only neutrino data and another simulation including allconstraints (in the case of IH and κ = 5) and represent the resulting distribution of m k forthe two simulations. We represent with a dashed line the results of the simulation only withneutrino data and with a solid line the simulation with all present experiments included. oscillationsFull simulation ν m k (GeV) IH, κ = 5, ν oscillations & full simulation Figure 6: Impact of low energy constraints ( µ → eγ, τ → µ , ...): m k distribution when onlyneutrino data is included (dashed) as compared with the case in which all experiments are included(solid). Displayed data correspond to the IH case and κ = 5. It is clear from the figure that only neutrino data allow (even prefer) relatively low massesof the order of 1 TeV or below. However, when low energy experimental data is includedthe lower limit on the m k is pushed to larger values. We have to remark that the shape ofthe curves basically reflects the volume of the parameter space (from the other parameters) In the following we obtain the distributions as five million point samples from a MCMC exploration ofthe parameter space as described in section IV B. m k < m k < m h distribution from a simulation with κ = 1 and another one with κ = 5 (both in IHcase with all experimental information included). = 5 = 1 κκ m h (GeV) IH full simulation, κ = 5 & κ = 1 Figure 7: Impact of the perturbativity constraints: m h distribution for κ = 1 (dashed) and κ = 5(solid). IH case with all experimental data included. We confirm with this figure the scaling of the bounds with the perturbativity assumptions,encoded in the parameter κ , obtained analytically. Thus, for smaller values of κ the allowedrange of m h is much smaller. For κ = 1, the preferred region of m h is in the range ∼ −
100 TeV (although with long tails) while for κ = 5 this range is enlarged to 1 − κ as shown analytically (see table V).From figure 6 it is clear that all present data allow a wide range of k ++ masses however,the k ++ , as discussed in III, can only be discovered at the LHC if m k < κ = 5) allpresent constraints but assuming, in addition, that the k ++ has been discovered at the LHCand therefore has a mass m k < BR ( µ → eγ ) distribution inthe two cases, general case and m k < BR ( µ → eγ )37f the discovery of the k ++ at the LHC. While present data allow branching ratios in therange 10 − − − , if the k ++ is discovered at the LHC then BR ( µ → eγ ) ∼ − − − and, therefore, will be probed at the MEG experiment. -17 -16 -15 -14 -13 -12 -11 Full simulation 1 TeV m k < BR( µ → eγ ) IH κ = 5, full simulation & m k < Figure 8: Impact of the discovery of the k ++ at the LHC ( m k < BR ( µ → eγ ) distributionfor the general case, IH and κ = 5 with all present experimental results, (dashed) and requiring inaddition that k ++ has been seen at the LHC ( m k < κ = 5 (solid). Until now we have presented results only for the IH case. In general, as also seen inour approximate analytical results, we expect roughly similar results in the NH and the IHcase, except for a few parameters and/or observables. In particular we mentioned that inthe IH case there is a preference for the Majorana phase around φ = π because in that casethere is a cancellation in the neutrino mass formulas. This is confirmed by the numericalcalculation: in fig. 9 we represent the distribution of φ for both, the NH case (dashed line)and the IH case (solid line). The data is taken from a simulation with κ = 5, including allpresent experimental constraints and requiring that m k < φ = π .We also expect large differences in the NH and IH cases for the parameter sin θ . Infig. 10 we represent the sin θ distribution for the two cases, NH and IH ( κ = 5, full dataand m k < θ > ∼ × − , which is not so far from the present upper limit, sin θ < × − .38 NHIH φ (Majorana) κ = 5, m k < Figure 9: Differences between the NH and IH cases: distribution of the Majorana phase φ : dashedin the NH case and solid in the IH case. All present experimental data included and κ = 5. -3
5 10 -2 -2 NHIH sin θ κ = 5, m k < × Figure 10: Differences between the NH and IH cases: distribution of sin θ as in fig. 9. Finally, to illustrate another interesting difference between the two cases, NH and IH,we have represented in fig. 11 the distribution of h m ν i ee , the relevant matrix element in theneutrinoless double beta decay experiments. As before we assume κ = 5 and m k < h m ν i ee is a function of only the neutrino masses and the mixing angles. Thus, theshape of the curves and their position is just a consequence of the fact that the model39redicts a massless neutrino. In any case, from the figure it is clear that the model predicts h m ν i ee ∼ . − .
005 eV in the NH and h m ν i ee ∼ . − .
06 eV in the IH case. -3 -2 -1 NHIH h m ν i ee (eV) κ = 5, m k < Figure 11: Predictions for h m ν i ee in the NH (dashed) and the IH (solid) cases. All present experi-mental data included and κ = 5. Now, for the most interesting observables, those which give some interesting constraintsor good perspectives in future tests, we present two-dimensional contour plots of the cor-responding distributions. The density of points has been calculated using 50 bins in alogarithmic scale for each axis. Then, 10 contour lines equally spaced, ranging from themaximum density to 1 / m k < m k < κ = 5). Scaling for more restrictive assumptions can be inferredfrom tables IV and V. We discuss the relevant plots for both the NH and IH cases.40 . Normal Hierarchy Here we consider correlations among observables in the normal hierarchy case. k )VeG( m (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) ) ✁ e ✂ ( R B k )VeG( m (cid:0) (cid:0) (cid:0) (cid:0) ) ✁ e ✂ ( R B k 10 TeV and BR ( µ → eγ ) above 10 − , but a large region ofvalues is not excluded m k ∼ − GeV and BR ( µ → eγ ) ∼ − − − , howeverif the doubly charged scalar, k ++ , is discovered at the LHC ( m k < m k > ∼ 600 GeV and BR ( µ → eγ ) > ∼ − and that k ++ masses below 200 GeV and BR ( µ → eγ ) below 10 − are very difficult to obtain in the model.Since BR ( µ → eγ ) depends more explicitly on m h than on m k it is interesting to study thecorrelation between BR ( µ → eγ ) and m h . In fig. 13 we depict the allowed region in the plane BR ( µ → eγ )– m h , on the left for the general case and on the right for the case m k < m h can be in a very wide range of values m h ∼ − GeV butthe preferred values are m h ∼ 40 TeV. On the other hand, if m k < For specific numbers we take 3 contours in the plots. m h is much smaller, m h ∼ − GeV, and is shifted to the lower edge. It still allowsa large range of masses not accessible at the LHC. h )VeG( m (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) ) ✁ e ✂ ( R B h )VeG( m (cid:0) (cid:0) (cid:0) (cid:0) ) ✁ e ✂ ( R B k 50 TeV and BR ( µ → eγ ) ∼ − . However,if nature chooses a k ++ light enough to be produced at the LHC the model is much moreconstrained: it predicts that m k is relatively large (masses below 400 GeV are only marginallyallowed and the preferred masses are above 800 GeV). In addition BR ( µ → eγ ) > − and the preferred range is above 2 × − .Figure 16 is also similar to fig. 13 but slightly more restrictive. In the general case we find m h ∼ − GeV and preferred values m h ∼ 40 TeV. For m k < m h is m h ∼ 500 GeV − 70 TeV which will make its detection at the LHC problematic.The constraints on BR ( τ → µ ) are also stronger in the IH case, fig. 16, than in the NHcase. The allowed regions are similar but more restrictive. Thus we find that in the generalcase the preferred values of BR ( τ → µ ) are in the 10 − range, although values as small as10 − are allowed. In the m k < BR ( τ → µ ) > ∼ × − (to be compared with present limits BR ( τ → µ ) < . × − ), but values like 10 − arenot completely excluded.Finally, as has been shown analytically, in the IH case there is a lower bound on sin θ .43 h )VeG( m (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) ) ✁ e ✂ ( R B h )VeG( m (cid:0) (cid:0) (cid:0) (cid:0) ) ✁ e ✂ ( R B k 01 region although values below 10 − do not seem completelyexcluded (the present upper limit is sin θ < . θ require larger values of BR ( µ → eγ ). If we also require that the k ++ can be discovered atthe LHC we find the preferred values of the model are constrained to a region sin θ > ∼ . BR ( µ → eγ ) > ∼ − . We also see that values of sin θ below 0 . 005 and BR ( µ → eγ )below 10 − are very unlikely in this case. Notice that mixings as small as sin θ ∼ . BR ( µ → eγ ) at the level of 10 − .44 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) ) e ( RB (cid:0) (cid:0) (cid:0) ✁ n i s (cid:0) (cid:0) (cid:0) (cid:0) ) e ( RB (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) ✁ n i s k In this section we would like to highlight some of the most relevant conclusions drawnfrom the previous analysis.The explanation of observed neutrino mixings in the model sets very strong constraintson the structure of the couplings of the singly charged scalars to fermions, in particular, inthe NH case the couplings must satisfy f eτ ≃ f µτ / ≃ f eµ . To fix the absolute value we needanother observable, in the NH case we find that the strongest bound comes from µ → eγ and tells us that | f ei | < ∼ . m h / TeV) and | f µτ | < ∼ . m h / TeV). In the case of IH thecouplings must satisfy f eτ ≃ − f eµ and | f eτ | > ∼ | f µτ | . Then, from lepton-hadron universalitywe find | f ei | < ∼ . m h / TeV), and | f µτ | < ∼ . m h / TeV), bounds which are similar to thebounds obtained from µ → eγ .The structure of the couplings of the doubly charged scalar is also very constrained byneutrino masses and mixings. In the case of NH they must satisfy, to a good degree ofprecision, that | g µτ | ≈ | g µµ | ( m µ /m τ ) and | g ττ | ≈ | g µµ | ( m µ /m τ ) . In the case of IH thisrelation does not need to be satisfied exactly because the electron couplings g ei can berelevant. However, we have seen that in a large region of the parameter space this relationis also required. Then, the best constraint comes from τ − → µ + µ − µ − which tells us that | g µµ | < ∼ . m k / TeV), | g µτ | < ∼ . m k / TeV), | g ττ | < ∼ . m k / TeV). The g ei couplingsare not constrained by neutrino data but are constrained by low-energy processes which aresummarized in tables I and III.We find that the neutrinoless double beta decay parameter h m ν i ee is strongly constrained45n the model. We find 0 . eV < |h m ν i ee | < . 004 eV in the NH case and 0 . eV < |h m ν i ee | < . 06 eV in the IH case. This is just a consequence of the measured neutrinomasses and mixings and the particular structure of neutrino masses of the model whichpredicts a massless neutrino.If this model is the right explanation for neutrino masses and if m k < lep channel (see section III) . There could besome dilution of the signal because the k can also decay into tau leptons or into two singlycharged scalars but in the case of NH this can only be relevant for m k > m k > m h . In the IH case the dilution of the 4 lep signal is a bit larger, still, most of the parameter space with m k < lep channel as long as the 2 h channel is not open.If more than 10 events are produced at the LHC in the 4 lep channel we find that BR ( µ → eγ ) > − in both the NH and the IH cases. These values are precisely the sensitivityexpected in the MEG experiment at the Paul Scherrer Institute (PSI) which will start torun soon [23]. In fact one goal of the experiment is to obtain a significant result before thestart of the LHC experiments. If this goal is achieved and nothing is seen we can reverse theargument and claim that it will be very difficult to find the charged scalars of this model atthe LHC. We find that if BR ( µ → eγ ) < − , then, m k > 900 GeV and m h > 600 GeV inthe NH case and that both scalar masses will be above the TeV in the case of IH.It is also important to remark that the photonic vector form factor in muon-electronconversion in nuclei is enhanced with respect to the tensor form factor due to logarithmiccorrections of loops with doubly charged scalars. Thus, if the current precision in muon-electron conversion experiments is increased in the next years, there will be additional testson the model.If the doubly charged scalars are light enough to be produced at the LHC there are alsointeresting contributions to rare tau decays. For instance, in the IH case we find that mostof the parameter space lies in the region BR ( τ → µ ) > ∼ × − , The NH case allows forslightly smaller branching ratios BR ( τ → µ ) > ∼ − . These results have to be comparedwith the the present limit, BR ( τ → µ ) < . × − or the ranges that might be explored We assume that, once efficiencies are taken into account, this corresponds to 2 reconstructed events[50, 52]. 46n SuperB factories ∼ − –10 − [67, 68].The model gives a negative contribution to the a µ = ( g µ − / a µ set interesting constraints on the parameters of the model.In general it is much more difficult to satisfy all the constraints in the IH case than inthe NH case. This can be seen in the MC acceptance rate which is much lower in the IHcase than in the NH case. In fact, we have seen that satisfying all the neutrino mass datain the IH case requires certain cancellations in the neutrino mass formulas which imply thatthe neutrino Majorana phase φ cannot be zero, actually, most of the parameter space liesin the region e iφ ≃ − θ mixing. We find that, even in the general case, most of the allowed parameter spacerequires sin θ > ∼ . − do not seem completely excluded. Inaddition, smaller values of sin θ require larger values of BR ( µ → eγ ). If we also requirethat the k ++ is seen at the LHC through the 4 lep channel we find that values of sin θ below 0 . 005 are very unlikely (the present upper limit is sin θ < . 02 and mixings as smallas sin θ ∼ . 005 will be tested in a near future [66] ).In short, the requirement that the model is able to explain the observed pattern ofneutrino masses and mixings places very strong limits on the parameters of the model. Thus,if the doubly charged scalar of the model is seen at the LHC through the 4 lep channel, themodel predicts large contributions to several low energy processes, µ → eγ , τ → µ , hadron-lepton universality tests, which should be within reach of the next round of experiments. 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