Revisiting pseudo-Dirac neutrinos

We study the pseudo-Dirac mixing of left and right-handed neutrinos in the case where the Majorana masses M_L and M_R are small when compared with the Dirac mass, M_D. The light Majorana masses could be generated by a non-renormalizable operator reflecting effects of new physics at some high energy scale. In this context, we obtain a simple model independent closed bound for M_D. A phenomenologically consistent scenario is achieved with M_L,M_R ~ 10^{-7} eV and M_D ~ 10^{-5}-10^{-4} eV. This precludes the possibility of positive mass searches in the planned future experiments like GENIUS or in tritium decay experiments. If on the other hand, GENIUS does observe a positive signal for a Majorana mass \geq 10^{-3} eV, then with very little fine tuning of neutrino parameters, the scale of new physics could be in the TeV range, but pseudo-Dirac scenario in that case is excluded. We briefly discuss the constraints from cosmology when a fraction of the dark matter is composed of nearly degenerate neutrinos.

PACS numbers: 12.15.Ff,12.20.Fv,14.60.Pq,14.60.St I. INTRODUCTION Measurements of the atmospheric neutrino fluxes by the Super-Kamiokande experiment [1] and of the solar neutrino fluxes by several experiments [2] have given a compelling experimental evidence for neutrino masses, mixing and oscillations. The recent results of the SNO experiment [3] favour the existence of neutrino oscillation among active flavours involving ν e from the Sun. Upon inclusion of the LSND result [4], a simultaneous explanation of both the solar and atmospheric results in terms of oscillations would require the existence of at least one sterile neutrino which can oscillate with any of the active flavours. There are many analyses in the literature where various possible active-sterile neutrino oscillation patterns have been studied [5].
In most analyses, the atmospheric anomaly points for its solution towards large angle ν µ → ν τ or ν µ → ν s oscillations, where ν s denotes a sterile neutrino. Results obtained by CHOOZ reactor basedν e disappearance experiment [6] and later by PaloVerde [7] severely constrain ν µ → ν e oscillations for neutrino mass scales relevant for atmospheric neutrinos. This is also in agreement with the flat spectrum observed for the atmospheric e-like events.
In addition, an analysis of the neutral current data disfavours large transitions involving ν e at the atmospheric scale [8]. Recently, the Super-Kamiokande collaboration argued that the oscillations between active-sterile flavours is disfavoured at 3σ level [9]. It should be mentioned, however, that this conclusion may depend on how one analyses the data, and it has been claimed that a maximal ν µ → ν s oscillation solution to the atmospheric neutrino problem is not yet ruled out [10]. Furthermore, it has been argued that the study of neutral current events at Super-Kamiokande, combined with the information obtained from future long baseline experiments, might not even be sufficient to decide between active-active and active-sterile oscillation solutions [11].
The possible role of the active-sterile oscillations in explaining the solar neutrino problem has recently got new light from the first SNO results on the charged current rates. The pre-SNO situation was such that active-sterile large mixing angle (LMA) as well as low mass (LOW) solutions were disfavoured whereas small mixing angle (SMA), vacuum (VAC) and Just-So solutions were well allowed [12]. Upon inclusion of the preliminary SNO results, within the two flavour analysis, it appears that only the VAC solution gives a good fit to the data with best fit point as ∆m 2 ⊙ = 1.4 · 10 −10 eV 2 and tan 2 θ ⊙ = 0.38 [13]. Alternatively, magnetic moment solutions to the solar anomaly are also feasible. Such solutions equally involve large active-sterile oscillations and are currently not ruled out [14].
It may of course be that the solar neutrino oscillations follow in reality a more complicated pattern than an effective two flavour scenario. The SNO and future experiments, especially those which are sensitive to both charged and neutral currents (Borexino and KamLAND), are believed to provide a crucial test of the existence of oscillations to sterile neutrinos of any form. On the other hand, Barger et al. [15] have recently argued that due to the poorly known value of the 8 B flux normalization, even the forthcoming SNO neutral current measurement might not be sufficient to determine the sterile neutrino content in the solar neutrino flux.
Thus, given our current understanding and analyses of the neutrino data, large activesterile oscillations may play some role in solving the solar and atmospheric neutrino anomaly, though it seems to be less probable than active-active solutions. Furthermore, a combined analyses of the neutrino data including the LSND result favours a 2 + 2 spectrum which involves the possibility of large active-sterile oscillations either in the solar or atmospheric sector [16].
All of the above solutions require neutrinos to posses a small but non-vanishing mass.
From the theoretical point of view, the seesaw mechanism [17] offers the simplest and the most natural explanation for small neutrino masses. In this mechanism, one assumes the If, on the other hand, one assumes M D ≫ M R , M L , the situation is quite different.
The resulting mass eigenstates have eigenvalues very close to each other, and they have opposite CP parities. Hence they can form a pseudo-Dirac neutrino [18]. There have been numerous suggestions in the literature for pseudo-Dirac neutrinos as solution to the neutrino anomalies, where the observed flavour suppression is due to a maximal or near to maximal mixing between an active and a sterile neutrino [19].
A relevant question in the pseudo-Dirac scenario is to explain the unorthodoxy in the hierarchy: M D ≫ M R which is necessary for sterile neutrinos to be light. In the standard model (SM) the Majorana masses M L and M R are non-existing due to the conservation of lepton number. Hence the origin of these mass terms goes beyond the SM and there could be many sources. One possibility is that the masses may be provided at the SM level by non-renormalizable effective operators of the type L 2 φ 2 /M and ν 2 R φ ′2 /M ′ . Here L = (ν L , l L ) is an ordinary lepton doublet, φ and φ ′ are Higgs fields, and M and M ′ are high mass scales derived from some beyond-the-SM theory. The masses M and M ′ are not necessarily connected with the vacuum expectation values of the Higgs fields φ and φ ′ , so it is possible that both M L and M R are much smaller than M D . In any case, it is known that in a viable model M L should be suppressed so that M D ≫ M L . This is required to avoid a contradiction with the accurately determined ρ-parameter. It is conceivable to assume that a similar suppression also happens for M R .
A subsequent question is to understand the smallness of M D . A light Dirac mass can be either (i) due to a small Yukawa coupling in the mass term ν R ν L φ or (ii) just like in the case of M L or M R , a light M D could be generated by a non-renormalizable higher dimensional term [20]. Another possibility is realized in models with large extra spatial dimensions.
In such theories, the Yukawa coupling of the term ν R ν L φ may be suppressed as the righthanded neutrino can be most of the time in the bulk outside our four-dimensional brane [21]. In the following, we assume a small M D relevant for a pseudo-Dirac mixing without addressing to its origin. We examine the mixing of ν L and ν R when the Dirac mass term dominates over the Majorana mass terms, i.e. M D ≫ M L , M R , and discuss the experimental and theoretical bounds one can obtain for the mass parameters. This is illustrated for the case of the electron neutrino.
Our paper is organised as follows. In the next Section, we give the basic formalism for pseudo-Dirac mixing and by a simple exercise we show that the effective electron neutrino mass as probed by neutrinoless double beta decay experiments is exactly M L . In Section III, we set bounds for the masses, M L and M D and derive a closed bound for M D . We also discuss the constraints from cosmology when some fraction of the dark matter is composed of nearly degenerate neutrinos. Finally, in Section IV, we conclude by summarising the main results of this paper.

II. THE PSEUDO-DIRAC SCENARIO
Let us consider the 2 × 2 Dirac-Majorana mass matrix in the (ν L , ν C L ) basis of the form and assume M D ≫ M L , M R . The mixing angle which diagonalises M is easily derived to be We get a pseudo-Dirac neutrino pair with mass eigenvalues For a nonzero M D and M L = M R , this system corresponds to a maximal interlevel mixing of π/4 between the Majorana pair. If M D > 0 is assumed, the neutrino mass-squared difference is If M L = M R , i.e. when the mixing is not maximal, one has In where are normalization factors. In the limit M D ≫ M L and M R , Therefore, the active neutrino component ν L in the mass eigenstates is given by the ampli- implying The effective electron neutrino mass as measured by 0νββ decay experiments is then given to be where in η ± = ±1 are the Majorana phases of the mass eigenstates. It is easy to check that  (4) and (10) a general result follows: where β ≡ M R /M L + 1 > 1. The most stringent experimental upper bound published by the Heidelberg-Moscow experiment in [22] implies M ef f ≤ M exp ef f = 0.2 eV (more recently the experiment has quoted the limit 0.34 eV at 90 % C. L. [23]). Thus, for a given ∆m 2 , to be consistent with 0νββ decay results, the Dirac mass M D must obey the bound A bound for M L from unitarity. A Majorana mass M L of the left-handed neutrino reflects physics beyond SM. In its presence the SM should be considered as an effective theory. It should be replaced by a more fundamental theory at some high energy scale M X , where new physics should enter, since otherwise the processes induced by the Majorana mass term would spoil the unitarity. One can find an upper limit for M X , for example, by studying the high energy behavior of the lepton number violating reactions νν → W W or ZZ, which can occur because of the Majorana mass term. The amplitudes of these reactions increase as proportional to the center of mass energy, leading to a breakdown of the effective theory at high energies. It was recently shown [24] that the most stringent bound for M X is obtained by considering the following linear combination of the zeroth partial wave amplitudes: a 0 ( 1 , where ν ± are helicity components of the mass eigenstate neutrino ν and the final state bosons are longitudinally polarized. This amplitude to obey unitarity, i.e. |a 0 | ≤ 1/2, requires [24] where φ = 174 GeV is the vev of the ordinary Higgs boson. It should be stressed that the Majorana mass M L appears in this formula, not the kinematical mass of the neutrino. At high energies, where neutrinos are ultra-relativistic, the kinematical mass of the neutrino is irrelevant. The condition (13) The smaller the scale of the new physics, the less stringent is the bound. The 0νββ decay to be visible in the planned GENIUS experiment [25], i.e. M L > ∼ 10 −3 eV, would require M X < ∼ 10 17 GeV.
A closed bound for M D . As was originally pointed out by Weinberg [26], Majorana masses for the left-handed neutrinos can be generated by higher dimensional operators of the form where i, j, k, l are SU(2) L indices, α, β are flavour indices, and M X is the scale of new physics. This operator breaks the lepton number explicitly, and after spontaneous symmetry breaking it leads to the following Majorana mass (neglecting flavour mixing): where f < ∼ O(1) is a numerical factor. With M X < ∼ M pl this implies Therefore, in this scheme we have where the upper bound is due to the 0νββ decay results.
By using (11) one can infer from (18) Let us now turn to experimental numbers involving the electron neutrino. According to the analysis done in [13] for the solar neutrino problem (that takes into account the recent results of SNO on the ν e charged current rate), the best fit values for pure vacuum solution (ν e ↔ ν s ) are with ∆m 2 = 1.4 · 10 −10 eV 2 and tan 2 θ = 0.38. This does not correspond to a maximal mixing which is the case in the pseudo-Dirac scenario. However, as can be seen from the analysis [13], maximal mixing with θ = π/4 is not completely ruled out even though it is less favoured. To illustrate the situation we set ∆m 2 = 1.4 · 10 −10 eV 2 and β = 2 as reference values which corresponds to maximal active-sterile mixing in the case With these values, (19) gives the numerical range Comparison with (18) shows that for the small ∆m 2 of the vacuum solution, the pseudo- is due to a pure sterile mixing, M L is necessarily so small that the 0νββ decay would stay outside the range that the upcoming GENIUS experiment would be able to probe. On the other hand, the kinematical determination of the electron neutrino mass in tritium decay [27] would also be extremely difficult because of the smallness of M D . Nonetheless, the analysis does predict a nonzero mass value from both these processes and hence the associated scale of new physics § .
TeV scale physics. It follows from (11) and (16) where there are suitable additional scalars. Within the context of nonrenormalizable theories, this is feasible if we consider a higher dimension operator other than the one suggested in (15). To illustrate this, we consider the simplest extension to the SM with an extra scalar doublet, φ ′ . In order to avoid the induced flavour changing neutral currents, we impose a discrete Z 2 symmetry for the field φ ′ . In this case, the lowest possible higher dimensional operator, which can generate a Majorana mass, is of the type A Majorana mass is obtained when the scalars get a vev: If we choose φ / φ ′ ≈ 10, and then set φ ≈ 100 GeV and M X ∼ 10 − 100 TeV, we must problem. For M X = 10 TeV (M X = 100 TeV), f ′ must have unnaturally small values, f ′ < 10 −9 (f ′ < 10 −6 ).
From this example one can conclude that in the pseudo-Dirac scenario the scale of new physics could be very high and that M L and M D are outside laboratory detection at present and also for any future realistic experiments. Naturally, it follows that, if for example GENIUS observes a nonzero signal for the 0νββ decay, pseudo-Dirac scenario is very unlikely.
Cosmological constraints. Here, we discuss the constraints assuming that the new physics arises from an operator of the type L 5 and is consistent with pseudo-Dirac scenario. In the context of cosmology, neutrinos being neutral can be ideal candidates for the hot dark matter. In the non-relativistic limit, the energy density is ρ ν = i m ν i N ν , where N ν is the number density and m ν i are the mass values. In the context of four neutrino flavours, it is expected that there is at least a pair of nearly degenerate neutrinos. It is possible that the splitting between such nearly degenerate pairs could correspond to the solar sector.
It is conceivable that the dark matter is composed of some fraction of such degenerate or nearly degenerate neutrinos with the splitting to be ∼ ∆m 2 ⊙ ≈ 10 −4 − 10 −5 eV; this value of the mass splitting in our case will be close to the Dirac mass. Therefore, for such quasi-degenerate masses m ν ≈ M D , we can relate to the cosmological parameters as [28] α M D ≈ 94 Ω ν eV , where Ω ν is the neutrino density compared with the critical density, and α runs from 1 to n f , where n f is the number of flavors in thermal equilibrium. Using (4) and (10), we can rewrite (23) as The present allowed range is 0.003 < Ω ν < 0.1 [29]. This yields the lower limit Comparing this with the lower limit for M ef f in (17), which was obtained by requiring that the scale M X ≤ M pl , one notices that the bound obtained from cosmology is more stringent only if ∆m 2 n f > ∼ 4.7 · 10 −5 f β eV 2 .
This is not in accordance with the vacuum oscillation solution of the solar deficit problem which requires ∆m 2 ∼ 10 −10 eV 2 . Therefore we conclude that in the limit of the dark matter being composed of some fraction of degenerate neutrinos, cosmology does not give more stringent bounds on ∆m 2 than the oscillation results.

IV. SUMMARY
We have investigated a pseudo-Dirac mixing of left and right-handed neutrinos assuming