Low-energy limits on heavy Majorana neutrino masses from the neutrinoless double-beta decay and non-unitary neutrino mixing

In the type-I seesaw mechanism, both the light Majorana neutrinos (\nu_1, \nu_2, \nu_3) and the heavy Majorana neutrinos (N_1, ..., N_n) can mediate the neutrinoless double-beta (0\nu\beta\beta) decay. We point out that the contribution of \nu_i to this 0\nu\beta\beta process is also dependent on the masses M_k and the mixing parameters R_{ek} of N_k as a direct consequence of the exact seesaw relation, and the effective mass term of \nu_i is in most cases dominant over that of N_i. We obtain a new bound |\sum R^2_{ek} M_k|<0.23 eV (or<0.85 eV as a more conservative limit) at the 2\sigma level, which is much stronger than |\sum R^2_{ek} M^{-1}_k|<5 \times 10^{-8} GeV^{-1} used in some literature, from current experimental constraints on the 0\nu\beta\beta decay. Taking the minimal type-I seesaw scenario for example, we illustrate the possibility of determining or constraining two heavy Majorana neutrino masses by using more accurate low-energy data on lepton number violation and non-unitarity of neutrino mixing.


I. INTRODUCTION
Almost undebatable evidence for finite neutrino masses and large neutrino mixing angles has recently been achieved from solar, atmospheric, reactor and accelerator neutrino oscillation experiments [1][2][3][4] . This exciting breakthrough opens a new window to physics beyond the standard model (SM), since the SM itself only contains three massless neutrinos (i.e., ν e , ν µ and ν τ , corresponding to the mass eigenstates ν 1 , ν 2 and ν 3 ). The simplest way to generate non-zero but tiny neutrino masses m i for ν i is to extend the SM by introducing at least two right-handed neutrinos and allowing lepton number violation. In this well-known (type-I) seesaw mechanism [5], the SU(2) L × U(1) Y gauge-invariant mass terms of charged leptons and neutrinos are given by whereH ≡ iσ 2 H * , l L denotes the left-handed lepton doublet, and M R is the mass matrix of right-handed neutrinos. After the spontaneous gauge symmetry breaking, we arrive at the charged-lepton mass matrix M l = Y l v and the Dirac neutrino mass matrix M D = Y ν v, where v ≃ 174 GeV is the vacuum expectation value of the neutral component of the Higgs doublet H and characterizes the Fermi scale of weak interactions. The mass scale of M R (or equivalently the seesaw scale Λ SS ) is crucial, because it is relevant to whether the seesaw mechanism itself is theoretically natural and experimentally testable. Although Λ SS ≪ v is not impossible [6], it is in general expected that Λ SS should be much higher than the Fermi scale. In particular, the conventional seesaw mechanism works at a scale which is not far away from the scale of grand unified theories. Driven by the upcoming running of the Large hadron Collider (LHC), more and more attention has been paid to the TeV scale at which the unnatural gauge hierarchy problem of the SM may be solved or softened by new physics. If the TeV scale is really a fundamental scale, then we are reasonably motivated to speculate that possible new physics existing at this scale and responsible for the electroweak symmetry breaking might also be responsible for the origin of neutrino masses. In this sense, it is meaningful to investigate the TeV seesaw mechanism and balance its theoretical naturalness and experimental testability at the energy frontier set by the LHC [7].
A direct test of the type-I seesaw mechanism demands the experimental discovery of heavy Majorana neutrinos N k (for k = 1, · · · , n) at the LHC, but two prerequisites must be satisfied: (a) their masses M k must be of O(1) TeV or smaller; and (b) their couplings to charged leptons R αk (for α = e, µ, τ and k = 1, · · · , n) must not be too small. The strongest bound on M k and R ek comes from the non-observation of the neutrinoless doublebeta (0νββ) decay [8], as N k can mediate this lepton-number-violating process. Current experimental lower limit on the half-lifetime of the 0νββ decay is usually translated into in some literature [9]. In obtaining Eq. (2), one has ignored the contribution of three light Majorana neutrinos ν i (for i = 1, 2, 3) to the 0νββ decay.
The first purpose of this paper is to point out that the constraint in Eq. (2) is not always useful for the type-I seesaw mechanism either at a superhigh-energy scale or at the electroweak or TeV scale. The reason is simply that the contribution of ν i to the 0νββ decay is in most cases dominant over the contribution of N k to the same process, leading to a much stronger bound on M k and R ek through the exact seesaw relation: at the 2σ level, which is equivalent to m ee < 0.23 eV (or < 0.85 eV as a more conservative bound) [10,11] for the effective mass of the 0νββ decay mediated by ν i . The second purpose of this paper is to look at whether the future measurements of lepton number violation and non-unitarity of neutrino flavor mixing are possible to shed light on M k . Taking the minimal type-I seesaw scenario [12] for example, we shall illustrate the possibility of determining or constraining two heavy Majorana neutrino masses by using more accurate low-energy data on the 0νββ decay and non-unitary neutrino mixing and CP violation.

II. STRONGER BOUND ON THE 0νββ DECAY
After the spontaneous gauge symmetry breaking (i.e., SU(2) L × U(1) Y → U(1) em ), the mass terms in Eq. (1) turn out to be where E and ν L stand respectively for the column vectors of (e, µ, τ ) and (ν e , ν µ , ν τ ) L . Without loss of generality, one may take M l = Diag{m e , m µ , m τ }. The overall (3 + n) × (3 + n) neutrino mass matrix in Eq. (4) can be diagonalized by a unitary transformation; i.e., where M ν = Diag{m 1 , m 2 , m 3 } and M N = Diag{M 1 , · · · , M 3 }. After this diagonalization, the flavor states of three light neutrinos (ν e , ν µ , ν τ ) can be expressed in terms of the (3 + n) mass states of light and heavy neutrinos (ν 1 , ν 2 , ν 3 and N 1 , · · · , N n ), and thus the standard charged-current interactions between ν α and α (for α = e, µ, τ ) can be written as in the basis of mass states. So V is just the neutrino mixing matrix responsible for neutrino oscillations, while R describes the strength of charged-current interactions between (e, µ, τ ) and (N 1 , · · · , N n ). V and R are correlated with each other through V V † +RR † = 1. Hence V itself is not exactly unitary in the type-I seesaw mechanism and its deviation from unitarity is simply characterized by non-vanishing R [13].
Note that V and R are also correlated with each other through the exact seesaw relation which can directly be derived from Eq. (5). Taking the (ee)-elements for both terms on the left-hand side of Eq. (7), we immediately arrive at This simple but interesting result implies that the effective mass of three light Majorana neutrinos in the 0νββ decay is directly associated with the masses, mixing angles and CPviolating phases of heavy Majorana neutrinos in the type-I seesaw mechanism: Note that both light Majorana neutrinos ν i and heavy Majorana neutrinos N k can mediate the 0νββ decay, as shown in Fig. 1. When the contribution of N k is least suppressed [14], the overall decay width of the 0νββ process in the type-I seesaw scenario can approximately be expressed as 1 where A is the atomic number, F (A, M k ) ≃ 0.1 depending mildly on the decaying nucleus, and M A ≃ 900 MeV [14]. Given M k > ∼ 10 2 GeV, the second term in the square brackets of Eq. (10) turns out to be < ∼ 8.1 × 10 −6 . Hence this term is negligible in most cases, unless the contribution of ν i is vanishing or vanishingly small due to a contrived cancellation among three different V 2 ei m i terms (or equivalently, among n different R 2 ek M k terms), which is in principle not impossible. Let us consider two limits in which the contributions of light and heavy Majorana neutrinos to Γ 0νββ are decoupled.
• In the limit of Current experimental data on the 0νββ decay yield an upper bound on this effective mass term, m ee < 0.23 eV at the 2σ level [11], which has extensively been used to constrain the masses, flavor mixing angles and Majorana CP-violating phases of three light neutrinos in the unitary limit of V . However, it should be kept in mind that this upper bound corresponds to some "favorable" values of the relevant nuclear matrix elements [8]. If their "unfavorable" values are used, one may also arrive at m ee < 0.85 eV at the 2σ level [11].
• In the limit of n k=1 R 2 ek M k → 0, which is rather contrived, Eq. (10) can be simplified to as a rough approximation. Imposing the bound m ′ ee < 0.23 eV and inputting M A ≃ 900 MeV and F (A, M k ) ≃ 0.1 [14], for example, we obtain which is a bit stronger than the upper bound shown in Eq. (2). If the more conservative bound m ′ ee < 0.85 eV is taken, one will arrive at R 2 ek M −1 k < 1.0 × 10 −8 GeV −1 , much closer to the result given in Eq. (2). Such rough bounds have been used by a number of authors in their preliminary studies of possible collider signatures of heavy Majorana neutrinos [9].
Note again that we have ignored the mild dependence of F (A, M k ) on the decaying nuclei in the above discussions. Otherwise, different 0νββ decays should be separately analyzed.
Below Eq. (10), we have pointed out that the contribution of three light Majorana neutrinos ν i to Γ 0νββ is dominant in most cases. This observation is especially true for the conventional type-I seesaw mechanism with superhigh M k (e.g., max (M k ) ∼ 10 15 GeV) and extremely small R αk (e.g., |R αk | ∼ 10 −13 ) [15]. When the seesaw mechanism is realized at the electroweak or TeV scale to generate experimentally accessible signatures of heavy Majorana neutrinos N k at the LHC, however, one usually has to require max (M k ) < ∼ O(1) TeV and |R αk | > ∼ 10 −3 up to O(0.1) [7], which imply a terrible cancellation in every term of Eq. (8) so as to give rise to tiny masses of ν i . Because such a terrible cancellation in m ee does not necessarily mean the same cancellation in m ′ ee , it is possible to get m ′ ee ≫ m ee as a special case, as already discussed in Eqs. (12) and (13). But the situation might become quite subtle if the masses of heavy Majorana neutrinos are degenerate [16]. Since the function F (A, M k ) depends both on the atomic number A and the heavy Majorana neutrino masses M k , m ′ ee might be exceedingly small for one decaying nucleus in the m ee → 0 limit but not for another in the same limit. A careful analysis of the relative magnitudes of m ee and m ′ ee for different 0νββ decays is nevertheless beyond the scope of the present paper and will be done elsewhere.
For n = 3, R can be parametrized in terms of nine rotation angles θ ij and nine phase angles δ ij (for i = 1, 2, 3 and j = 4, 5, 6) [13]. In this representation, where s ij ≡ sin θ ij and c ij ≡ cos θ ij . The approximation made in Eq. (14) is very reasonable because |RR † | is at most of O(10 −2 ) [17] and thus all the mixing angles of R must be very small (at most at the O(0.1) level). Given m ee < ∼ 1 eV and M 1 ≈ M 2 ≈ M 3 ∼ v or O(1) TeV, for instance, the constraint in Eq. (14) implies that two phase differences δ 14 − δ 15 and δ 14 −δ 16 should be very close to ±π/2 in order to assure significant cancellations among three terms. Fixing M i ∼ 10 2 GeV (for i = 1, 2, 3) as an example, we find that the level of finetuning is at least of O(10 −9 ) with s 2 1j ∼ 10 −2 or O(10 −7 ) with s 2 1j ∼ 10 −4 (for j = 4, 5, 6). Such unnatural cancellations seem to be unavoidable in the type-I seesaw models at the electroweak or TeV scale, unless the relevant mixing angles are extremely small. Current experimental data can only provide us with a rough bound s 2 14 + s 2 15 + s 2 16 < ∼ 1.1 × 10 −2 [17], unfortunately. In a specific type-I seesaw model with the fine-tuning conditions ( , which in turn leads to m ee = 0. Then non-zero but tiny M ν and m ee can be achieved by introducing a small perturbation to the texture of M D . If the magnitudes of M k are too big or those of R αk are too tiny, of course, there will be no hope to produce and detect heavy Majorana neutrinos and test the seeasw mechanism at the LHC [19].

III. THE MINIMAL SEESAW SCENARIO
The exact seesaw relation in Eq. (7) allows us to determine M k in terms of m i and the mixing parameters of V and R. To illustrate this point, let us focus on the minimal type-I seesaw scenario which contains only two heavy Majorana neutrinos [12]. In this simpler case, it is easy to obtain two real and linear equations of M 1 and M 2 from Eq. (8) Note that either m 1 = 0 or m 3 = 0 must hold in the minimal type-I seesaw model [12], and thus the non-vanishing neutrino masses can be determined from current experimental data on two independent neutrino mass-squared differences ∆m 2 21 ≡ m 2 2 −m 2 1 and ∆m 2 32 ≡ m 2 3 −m 2 2 corresponding to solar and atmospheric neutrino oscillations. After a simple calculation, we arrive at Using the exact and convenient parametrization of V and R advocated in Ref. [13], we have for the minimal type-I seesaw scenario, where c 1i ≡ cos θ 1i and s 1i ≡ sin θ 1i (for i = 2, · · · , 5).
There are at least two phenomenological merits of this parametrization for our present discussions: (1) it can automatically reproduce the standard (unitary) parametrization of the light neutrino mixing matrix [21] when the non-unitary mixing angles of R are switched off; and (2) it can lead to a very simple result of m ee , which is equal to the standard (unitary) expression of m ee multiplied by a factor c 2 14 c 2 15 [13]. Substituting Eq. (17) into Eq. (16), we obtain the explicit results of M 1 and M 2 for two different patterns of the light neutrino mass spectrum.
It is worth remarking that the non-unitarity of V is signified by non-vanishing R, whose mixing angles and CP-violating phases are quite possible to have some nontrivial observable effects. For example, an appreciable CP-violating asymmetry up to the percent level is expected to show up between ν µ → ν τ and ν µ → ν τ oscillations in some medium-or longbaseline experiments at a neutrino factory [23], just as a consequence of non-vanishing R.
A neutrino telescope could also be a useful tool to probe the non-unitary effect in ultrahighenergy cosmic neutrino oscillations [24].

IV. SUMMARY
We have carefully examined the contributions of both light Majorana neutrinos ν i with masses m i (for i = 1, 2, 3) and heavy Majorana neutrinos N k with masses M k (for k = 1, · · · , n) to the 0νββ decay in the type-I seesaw mechanism, in which the light neutrino mixing matrix V is non-unitary due to the non-vanishing coupling matrix R between N k and charged leptons. The exact seesaw relation allows us to establish a straightforward relationship between (m i , V αi ) and (M k , R αk ). We have pointed out that the constraint | R 2 ek M −1 k | < 5 × 10 −8 GeV −1 used in some literature is in most cases too loose for a type-I seesaw mechanism either at a superhigh-energy scale or at the electroweak or TeV scale, because the contribution of ν i to the 0νββ decay is in most cases dominant over the contribution of N k to the same process. Such an observation leads us to a much stronger bound on M k and R ek ; i.e., | R 2 ek M k | < 0.23 eV (or < 0.85 eV) at the 2σ level, extracted from the present experimental upper bound on the 0νββ decay.
We have also looked at whether the future measurements of lepton number violation and non-unitarity of neutrino flavor mixing at low energies are possible to shed light on the masses of heavy Majorana neutrinos. Taking the minimal type-I seesaw scenario for example, we have illustrated the possibility of determining or constraining two heavy Majorana neutrino masses by using more accurate low-energy data on the 0νββ decay and non-unitary neutrino mixing and CP violation. Such an analysis can simply be extended to the more general cases of the type-I seesaw mechanism with three or more heavy Majorana neutrinos.
As stressed in Ref. [13], testing the unitarity of the light neutrino mixing matrix V in neutrino oscillations and searching for the signatures of heavy Majorana neutrinos N k at TeV-scale colliders can be complementary to each other, both qualitatively and quantitatively, in order to deeply understand the intrinsic properties of Majorana particles. We optimistically expect that some experimental breakthrough in this aspect will pave the way towards the true theory of neutrino mass generation and flavor mixing.
I am indebted to M. Chaichian for warm hospitality during my visiting stay in Helsinki, where this work was started, and to T. Ohlsson for warm hospitality during the Nordita scientific program "Astroparticle physics -A Pathfinder to New Physics" in Stockholm, where this work was finalized. I am also grateful to W. Chao, H. Zhang and S. Zhou for useful discussions and comments. This research was supported in part by the National Natural Science Foundation of China under grant No. 10425522 and No. 10875131.