Constraining right-handed neutrinos

Several models of neutrino masses predict the existence of neutral heavy leptons. Here, we review current constraints on heavy neutrinos and apply a new formalism separating new physics from Standard Model. We discuss also the indirect effect of extra heavy neutrinos in oscillation experiments.


I. INTRODUCTION
In the Standard Model (SM), neutrinos are massless particles contradicting the experimental observation of neutrino oscillations, hence physics beyond the SM is required.
An effective lepton number violation dimensionfive operator O 5 ∝ LLΦΦ, can be added to the SU(3) c ⊗ SU(2) L ⊗ U(1) Y model in order to introduce neutrino masses [1,2], where L is one of the three lepton doublets and Φ is the SM scalar doublet. Unfortunately we cannot say too much more about this operator. We do not have any clue about its mechanism, nor its mass scale, nor its flavour structure.
A common possibility is to assume that O 5 is induced, at the tree level, by the exchange of heavy messenger particles. In this way, seesaw models postulate neutral heavy states act as messenger particles to induce neutrino mass. For instance, a right-handed neutrino could be included associated to each of the three isodoublet neutrinos (Type I seesaw) [3]: The existence of processes with ∆L = 2, such as neutrinoless double beta decay, or lepton flavour violation processes (LFV) as µ → eγ , would give hints on the possible existence of these heavy Majorana neutrino messengers. Hence we could find signatures of heavy neutrinos and their mixings by studying this kind of processes.

II. THE METHOD
As we said in the previous section, heavy neutrinos are introduced in several extensions of the SM such as linear and inverse seesaw models [6][7][8], leading to a rich structure in the lepton mixing matrix. In order to work with this kind of models we will use a symmetric parameterization, consistent with the general formalism [4], neatly arXiv:1505.01097v1 [hep-ph] 5 May 2015 separating "new physics (NP)" and "Standard Model physics (SM)".
The mixing matrix U can be decomposed in a new physics part and its Standard Model part R N P = ω n−1 n ω n−2 n . . . ω 1 n . . . ω 3 4 ω 2 4 ω 1 4 (5) and it can be divided in four blocks where N is the block corresponding to the standard three neutrinos, including their mixings between them and the extra neutrinos.
At the same time, the matrix N can be also decomposed as where U SM is the usual mixing matrix of the Standard Model and the matrix N N P includes all the new physics information through the α ij parameters (a more complete discussion is given in [25]) In summary, by choosing a convenient order for the products of the rotation matrices, ω ij , we can obtain a parameterization which puts all the information in a convenient form.
A. Simplest extension of SM: 3 +1 neutrinos The formalism for the simplest extension of the SM includes one extra right handed singlet with the mixing relations between the gauge and mass eigenstates given as [10] The unitary mixing matrix U 4×4 can be written as 12) where N 3×3 is again the sub-matrix related with the standard neutrinos Eq.(8) It is important to notice that N 3×3 is not unitary whereas U 4×4 is unitary because includes all neutrinos in the model.
Comparing the terms in N 3×3 with the terms of U SM we obtain the following expressions for the α factors α 32 =s 34s24 , α 31 =s 34 c 24s14 .

B. Aplication to 3 +3 model
Usually, more than one extra neutrino is introduced in the theory, as in sequential-type seesaw mechanisms where 3 (Type I) or 6 (Inverse and Linear) extra singlets are included with the SU (2) L SM doublets.
For such models our parameterization becomes with these expressions for α the parameters

III. OSCILLATION CONSTRAINTS
The general expression for the survival and conversion neutrino probability is given by [11] where δ αβ appears due to the unitarity of the mixing matrix. However, in a model with extra heavy neutrinos, this equation will change because the mixing matrix describing the three standard neutrinos (N 3×3 in our symmetric notation) will not be unitary and the effective probability will not be normalized to 1. In this case, the probability includes the W terms from the truncated matrix N 3×3 For the electron (anti) neutrino survival probability, we get the expression and using Eq.(11) and Eq.(12) we obtain [25] P ee = α 4 11 1 − cos 4 θ 13 sin 2 2θ 12 sin 2 ∆m 2 12 L 4E − sin 2 2θ 13 sin 2 ∆m 2 13 L 4E (19) Note that the effect of the extra neutrinos is totally included in the α 4 11 factor, illustrating the utility of our symmetric formalism.
Considering now only one extra fourth neutrino, Eq.(18) would change to One could be tempted to study whether the presence of an extra light singlet leptons (sterile neutrinos) could play some role in the reported neutrino anomalies (as MiniBooNE [12]). Unfortunately this is not the case and we stick to the (natural) assumption that the extra states are heavy and do not take part in oscillation effects. In this case one can get a constraint from some reported combined analysis [13] sin 2 θ 14 = |S e4 | 2 < 0.04 (90% C.L.) (21) α 4 11 = cos 2 θ 14 < 0.96 (22)

IV. FUTURE OSCILLATION EXPERIMENTS
As we can see from the previous section, it is not possible with current experiments to obtain a very strong constraint on the new physics α parameters so we consider future experimental proposals such as LENA [14].
LENA is a future neutrino experiment which will use a 51 Cr artificial neutrino source with 5 MCi intensity, producing a total of 1.9 × where n e is the number of electron targets, φ Cr is the neutrino flux coming from the source, ∆t is 28 days which corresponds to the half-life of the source, and R (T, T ) is the resolution function where T is the recoil energy, T is the true energy and σ = 0.075 √ T IM eV is the expected energy resolution. An estimate of the expected sensitivity as a function of the total percent error of the experiment can be performed in advance, being shown in the Fig. 2.

V. OTHER CONSTRAINTS
The effects of heavy neutrinos would show up as peaks in the leptonic decays of pions and kaons or from their direct production at higher collider energies [5]. One can perform an analysis of all these data, and combine the corresponding restrictions on heavy neutrinos parameters. Fig. 3 compiles the bounds on the heavy neutrino mixing with electron neutrinos at 90 % C.L. It agglutinates bounds coming from peak searches at lepton decays, as π → e ν [15,16] and K → e ν [17,18]; meson decays at PS191 [19], NA3 [20] and CHARM [21] and Z 0 decays at DELPHI [22] and L3 [23]. In Fig. 3 we show also the excluded region from neutrinoless double beta decay experiments [24], valid only if the heavy neutrinos are Majorana particles. For completeness we show the bounds on the mixing with muon and tau neutrinos in Fig. 4.
From a more complete combined analysis of these data we can get constraints on our new physics parameters associated to the presence of the heavy neutrinos [25].

VI. CONCLUSIONS
Extra neutral heavy leptons are motivated in order to introduce neutrino mass but no positive evidence of these particles has been found so far.
Signatures of these heavy neutrinos arising from their mixings with the light ones could be searched at laboratory experiments. The study of these bounds would be useful to shed light upon the mass generation mechanism of neutrinos and probe the scale of new physics.
Models beyond SM, such as seesaw models, imply a very large number of parameters. The symmetric parameterizatio of the neutrino mixing matrix, describing the charged current, provides a very useful way to separate new physics from SM effects, concentrating the information and making easier to work with it. A more detailed account of our work will be described elsewhere [25].

VII. ACKNOWLEDGMENTS
FJE wants to thank the organizing committee for giving him the opportunity to talk at ICHEP 14 and to his collaborators DVF, OGM, MAT and JWFV for their advices. Likewise he thanks the AHEP group at Valencia for hospitality. This work has been supported by CONACyT grant 166639 (Mexico) and by the Spanish grants FPA2014-58183-P and Multidark CSD2009-00064 (MINECO), and PROMETEOII/2014/084 (Gen. Valenciana).