Non-Unitary Neutrino Propagation From Neutrino Decay

Neutrino propagation in space-time is not constrained to be unitary if very light states - lighter than the active neutrinos - exist into which neutrinos may decay. If this is the case, neutrino flavor-change is governed by a handful of extra mixing and"oscillation"parameters, including new sources of CP-invariance violation. We compute the transition probabilities in the two- and three-flavor scenarios and discuss the different phenomenological consequences of the new physics. These are qualitatively different from other sources of unitarity violation discussed in the literature.

Neutrino oscillations, first unambiguously observed towards the end of the twentieth century, have proven to be a powerful tool for fundamental physics research. Their observations revolutionized our understanding of neutrinos, revealing that these have tiny but nonzero masses. Moreover, they can be used to reveal new phenomena including the existence of new, weaker-than-weak interactions involving neutrinos and ordinary matter -which lead to nonstandard matter effects [1] -or the existence of light sterile neutrinos or new contributions to the charged-current weak interactions -which lead to different non-unitary 3 × 3 leptonic mixing matrices [2]. Neutrino oscillations also provide powerful testbeds of some of the most basic assumptions of fundamental physics [3], including tests of whether neutrino propagation is Lorentz invariant [4], whether neutrinos obey the CPT-theorem [4,5], whether there are exotic sources of decoherence in the time-evolution of the neutrino states [6,7], etc.
In this letter, we explore the consequences of the hypothesis that the neutrino propagation Hamiltonian is not Hermitian. When applied to the active neutrinos, these include new parameters for neutrino oscillation observables that are not captured by the different scenarios considered in the literature to date. We compute the transition probabilities, concentrating on the case where the non-unitary effects are small, and discuss the different qualitative aspects of the associated phenomenology.
It is important to stress that non-unitary time evolution need not be an extravagant hypothesis: neutrino propagation is non-unitary if the possibility that the three neutrinos interact with other generic light states is taken into account, a generalization of what is known to occur in the neutral kaon system [8]. A similar version of this phenomenon is also realized in resonant leptogenesis [9].
We start by postulating that the neutrino states involved in the production and detection processes are orthonormal. Generically, we refer to these as flavor eigenstates |ν α and our assumption amounts to imposing where α, β are flavor indices. For the case of the light, active neutrinos of the Standard Model (SM), this assumption certainly holds true if both production and detection occur via the standard charged-current weak interactions. In that case, α, β = e, µ, τ . In practice, this implies that as long as the production and detection processes occur through the weak interactions, there are no flavor-changing phenomena in the limit that the baseline is much shorter than the dimensionful parameters that govern propagation, as will become clear momentarily.
Neutrino flavor-evolution in space is governed by the usual Schrödinger-like equation, valid in the limit of ultrarelativistic neutrinos assuming that the initial states are perfectly coherent, where L is the distance traversed by the neutrino. The effective Hamiltonian H eff , which we assume is a generic matrix, can be parameterized as where M and Γ are Hermitian matrices. For Γ = 0, we have standard oscillations. The eigenvalues of M are, as usual, m 2 i /2E, where E is the neutrino energy and m i , i = 1, 2, . . ., are real. Γ also has real eigenvalues and can be diagonalized by a unitary matrix. We explore the most general case where M and Γ cannot be simultaneously diagonalized, i.e., the "mass" eigenstates need not coincide with the "decay" eigenstates. Time evolution governed by a non-Hermitian Hamiltonian is generically expected if there are new interactions that couple the light neutrinos to new, very light many-particle states. We provide a quick description of the formalism, which has been developed for the kaon system [8,10] and can be readily adapted to neutrino propagation. The main difference between the two is that in the neutrino case there are no constraints from CPT invariance which, for the kaons, lead to some simplifications.
Consider a system consisting of light neutrino states |ν i along with new many-particle states |φ k , with the index k understood to run over both discrete and continuous labels required to identify such states. It is convenient for this analysis to work in the mass basis so that |ν i and |φ k are eigenstates of a "free particle" Hamiltonian H 0 : The complete propagation Hamiltonian H of the system is assumed to involve new interactions and can be split into In the context of neutrino oscillations, H 0 describes the standard propagation Hamiltonian for the neutrino mass eigenstates in the absence of new interactions. On the other hand, the new physics piece H ′ is completely general.
In particular, it can induce transitions between the |ν i and the |φ k . At any time t, the state of the system |ψ(t) can be written as a linear combination of the light neutrino eigenstates |ν i and the |φ k as The time evolution of |ψ(t) is governed by the Schrödinger equation where c(t) and C(t) are column vectors formed by the coefficients c i (t) and C k (t) respectively. Eq. (7) is exact.
Because H is Hermitian, the evolution of the complete system is unitary. Any neutrino produced at time zero satisfies i |c i (0)| 2 = 1, while the probability that it remains a neutrino at some time t is P ν→ν = i |c i (t)| 2 . It is clear that for all t. * For the case in which the processes of production and detection involve only linear combinations of neutrino states |ν i , it has proved useful to devise a way to reduce Eq. (7) to a differential equation only for the the vector c(t). This can be accomplished under the Weisskopf-Wigner approximation (WW). WW assumes that the spectrum of accessible |φ k modes is very broad and that the matrix elements of the Hamiltonian with respect to the new states φ j |H ′ |φ k can be neglected. Under these conditions, [10,11] i d dt where M and Γ are Hermitian matrices with matrix elements given by [10] andĒ is the average energy of the neutrino beam. Within WW, Eq.
(3) appears naturally as a result of "integrating out" the new states, taking into account that the new states may be on-shell. Moreover, it is easy to see that Γ is positive definite. That is, WW only yields neutrino states that "decay" into the new states but never the other way around. Furthermore, off-diagonal Γ ij occur when different H 0 eigenstates can access the same φ k state, i.e., the different neutrino "mass" eigenstates can "decay" into the same final state.
In this work we assume that neutrino evolution is dictated by Eq. (9), but we are also be interested in violations of WW that may invalidate the constraint that Γ is positive definite. In particular, these could happen if the matrix elements φ j |H ′ |φ k cannot be neglected. In other words, we assume that there are conditions under which Eq. (9) is a good description of neutrino propagation physics while the restriction that Γ is positive definite need not apply. On the other hand, Eq. (8) is a consequence of the more general hypothesis that the neutrinos mix with new, unidentified degrees of freedom, so we pay special attention to what these constraints imply. Of course, if one wishes to simply explore how well neutrino oscillations are governed by the standard laws of quantum mechanics, no constraints on Γ, other than those imposed by experimental data, need apply.
Solving Eq. (2) is straightforward. Let N be a generic matrix such that where h i are complex numbers, and define the Hamiltonian eigenstates as The matrix N is not uniquely defined by Eq. (13); rescalings of the eigenvalues are still possible. In order to define it completely, we further impose In general, the states |ν i and |ν j , i = j, are not orthogonal. We define where I is the identify matrix, H and δ are Hermitian matrices, and δ ii = 0, ∀i. It is convenient to express N in terms of δ. Eq. (1) and (13) imply hence where V is a unitary matrix. When δ = 0, N is a unitary matrix and the Hamiltonian eigenstates form an orthonormal basis in spite of the fact that the h i are, in general, complex. This special case is the one usually considered when one addresses neutrino decay (see, for example, [12,13]; for a detailed discussion see [14]). It is equivalent to postulating that M and Γ can be simultaneously diagonalized, and the eigenvalues of Γ are proportional to the lifetimes of the neutrino mass eigenstates. Γ ≪ M implies δ ij ≪ 1 and We will restrict our discussions to this case, unless otherwise noted. The solution to Eq. (2), assuming that the neutrino is in state |ν α at L = 0, is and the oscillation amplitudes are trivially related to the oscillation "probabilities," P αβ ≡ |A αβ | 2 , which are the observables directly accessible to neutrino oscillation experiments.
Eq. (21) leads to unitarity-violating effects that are qualitatively different from postulating the existence of new oscillation lengths (i.e., light sterile neutrinos), or postulating that the weak-interaction eigenstates are not orthogonal [2]. Some of the differences are easy to spot. For instance Eq. (21) does not allow for any flavor change in the limit L → 0 (N N −1 ≡ I even if N is not unitary!), unlike the effects discussed in [2]. Also, Eq. (21) does not contain any new oscillation lengths: the new dimensionful parameters lead to exponential decay (or growth) of P αβ , as will be discussed in more detail in the next paragraphs.
It is instructive to discuss the case of two neutrino flavors in detail in order to illustrate the phenomena described by Eq. (21). In this case, we define where ǫ, ζ are real and positive, ζ ∈ [0, 2π). Further defining h 1,2 = a 1,2 − ib 1,2 and parameterizing the 2 × 2 unitary matrix V with the mixing angle θ, in the usual way, † we find keeping in mind that ǫ ≪ 1. Setting α, β = e, µ for the sake of definiteness we find the oscillation probabilities where ∆ = a 2 − a 1 plays the role of ∆m 2 /2E in the standard case and can be chosen positive. The expressions above ignore terms of O(ǫ 2 ), an approximation that is not appropriate in the limit θ → 0. The two new dimensionful parameters b 1,2 lead to the exponential decay/growth of all oscillations probabilities. Lorentz invariance dictates that b i ∝ d i /E, where d 1,2 are constants with dimensions of mass-squared. In the limit ǫ → 0 we recover the well-known expressions for neutrino oscillations under the assumption that the neutrino mass eigenstates have a finite lifetime.
In the more general case where the "mass" eigenstates do not coincide with the "decay" eigenstates, ǫ = 0 and the non-unitarity of the propagation leads to new "mixing" parameters, ǫ and ζ.
As discussed earlier, the physics responsible for Γ = 0 imposes constraints on the different parameters. In the two-flavor case, in the basis where M is diagonal with diagonal elements a 1 and a 2 (chosen positive), Γ ii = b i in the limit Γ ij ≪ a 1 , a 2 , ∀i, j. In the same basis, defining Γ 12 = Γ * 21 = b, In the context of the Weisskopf-Wigner approximation, Γ is constrained to be positive-definite: In turn, these imply that ǫ ≤ 2 √ b 1 b 2 /∆. The less stringent constraint β P αβ ≤ 1, for all α, translates into b 1 , b 2 ≥ 0 and where C ζ , C ′ ζ are O(1) non-illuminating functions of ζ such that 1 < C ζ 2. This constraint allows for ǫ = 0 as long as both b 1 , b 2 = 0. Importantly and opposed to the case in which WW is assumed, we find that for small mixing (sin 2θ ≪ 1), ǫ values larger than b 1,2 /∆ are allowed. This is potentially relevant for the "1 − 3 sector," as well as for the application of this formalism to sterile neutrinos. † It is straightforward to show that, like in the standard unitary-case, potential "Majorana phases" in V play no role in neutrino oscillations, even for δ = 0.
Note that these are only good approximations in scenarios where β P αβ > 1, in which case P αβ need to be carefully reinterpreted as they cannot stand, mathematically speaking, for probabilities. Nonetheless, the above expressions are easy to explore -only two new dimensionless parameters appear -and are useful in order to illustrate the consequences of ǫ, ζ = 0, Some interesting features are worthy of note. Even when the decay effects are "turned off," unitarity is violated -P ee + P eµ = 1 -along with, in the case ζ = π/2, 3π/2, time-reversal invariance -P eµ = P µe . The oscillation length is the same for all P αβ , L osc = 2π/∆, but the survival probabilities are "out of phase", i.e., maxima and minima do not correspond to L = nL osc , for natural n. The amplitudes of the oscillations -differences between the smallest and largest P αβ -for appearance and disappearance are also different. For example A µe = sin 2 2θ 1 + 2ǫ cos ζ sin 2θ .
As expected, CPT-invariance is preserved, i.e., P αβ = Pβᾱ, while CP-invariance is not unless ζ = π/2, 3π/2. Nonunitary propagation leads to new CP-invariance-violating phenomena, even in the two-flavor case. CP-invariance and T-invariance violation are also present, as long as ǫ cos ζ = 0, in the regime where the oscillatory terms average out, i.e., ∆L ≫ 1. In this case, P ee = e −2b1L cos 4 θ + e −2b2L sin 4 θ, (38) P µµ = e −2b1L sin 4 θ + e −2b2L cos 4 θ, Pēē = P ee , while P eµ − P µe = P eµ − Pēμ ∝ ǫ cos ζ. All expressions above ignore terms O(ǫ 2 ) and are not good approximations in the limit sin 2θ ≪ ǫ. In the limit θ → 0, it is easy to compute the oscillation probabilities. Ignoring O(ǫ 3 ) terms ‡ while P µe = P eµ . CP-invariance is preserved -none of the expressions depend of ζ, so P αβ = Pᾱβ, ∀α, β, but nonzero ǫ implies flavor change as long as the neutrino masses are different (∆ = 0). In the θ → 0 limit, the constraint α P αβ < 1 translates into b 1 , b 2 ≥ 0 and where min(b 1 , b 2 ) indicates the smaller between b 1 and b 2 , so nonzero ǫ requires both b 1 , b 2 nonzero. Furthermore, α P αβ < 1 also implies that oscillatory effects cannot dominate over the exponential decay. It is easy to extend the discussion to the three-flavor case. We define where ǫ i ≥ 0 and ζ i ∈ [0, 2π). Thus, in the limit ǫ i ≪ 1, N is given by Eq. (19) where where s ij = sin θ ij , c ij = cos θ ij and δ CP is the "Dirac" CP-odd phase. As in the two-flavor case, neutrino oscillations are not sensitive to "Majorana phases" in V . Using Eq. (21), the transition probabilities can be written as where h i = a i − ib i . As before, ∆ ij = (a i − a j ) play the role of oscillation frequencies, ∆m 2 ij /2E, while the different b i ∝ d i /E, i = 1, 2, 3 lead to exponential decay (or growth).
Complete expressions for the different P αβ are rather cumbersome and not particularly illuminating. They depend on all decay parameters as well as the new mixing parameters ǫ 1,2,3 and the new CP-odd phases ζ 1,2,3 . As in the two-flavor case, the transitions for antineutrinos are governed by the same expressions, except for N → (N −1 ) T , or V → V * , ∆ → −∆. In terms of the parameterization introduced here, this translates into δ CP → −δ CP , ζ i → π−ζ i . As expected, CPT-invariance is preserved, P αβ = Pβᾱ, but CP-invariance and T-invariance are violated unless ǫ i cos ζ i = 0 for all i. This is true even when three-flavor effects are "turned-off" (some mixing angles and some mass-squared differences vanish) and persists in the limit ∆ 31 L, ∆ 21 L ≫ 1.
For the case of standard neutrinos, bounds on some of the new physics parameters have been discussed in the literature, mostly in the context of "standard" neutrino decay (δ ij = 0 in Eq. (15)). However, the bounds on the ǫ i we will discuss are indirect, all of them deriving from the bounds on the b i , and relatively weak if WW is abandonned. Here we summarize qualitatively some of these bounds. A quantitative discussion on how non-unitary neutrino propagation is constrained by current data and the potential reach of next-generation experiments is currently being pursued in the follow-up to this letter [15].
In what follows, it will prove convenient to define the Hamiltonian eigenstates as where the d i 's have dimensions of mass-squared. Below, we assume that m 2 i agree with the results obtained by analyzing the world's neutrino data under the assumption that neutrino propagation is unitary, and use the standard definition for the ordering of the masses [3].
A finite neutrino lifetime can dramatically impact all the indirect information we have on primordial neutrinos. The nature of the decay products and the interactions, however, plays a role in determining if and how decaying neutrinos impact the cosmic microwave background, structure formation, etc, so bounds on decay parameters are very model-dependent (see, for example, [16,17]), ranging from very stringent to non-existent. A future observation of nonzero neutrino mass effects in cosmic surveys may change the picture dramatically [18].
The recent observation of ultra-high energy neutrinos from potentially extra-galactic sources [19] implies that at least one of the d i 's is tiny. Much more information can be obtained -especially if the d i 's are not zero -with more statistics and flavor information [20] (see also, for example, [21][22][23][24][25]). A similarly very stringent bound -at least one of the d i is zero for all practical purposes -comes from the observation of neutrinos from SN1987A [12].
If Γ is positive definite, ǫ i d j d k /∆m 2 and the bounds above imply that at least two of the ǫ i are tiny given the constraints from solar neutrino data (at least ǫ 10 −5 ), while the third one might be of order a few percent ( 10 −5 eV 2 /|∆m 2 13 |). The less stringent requirement α P αβ ≤ 1 allows for more ǫ i of order several percent, potentially enhanced by the fact that θ 13 is small. A detailed discussion will be presented in [15]. If no restriction is imposed on Γ or α P αβ , joint analyses of nonzero d i and ǫ i are required in order to establish the currently allowed values of these phenomenological parameters.
In summary, non-unitary neutrino propagation can be realized if neutrinos couple to new very light states which can interact among themselves. This scenario is qualitatively different from other hypotheses regarding new physics in the neutrino sector and leads to new phenomena -including new sources of CP-invariance violation and new mass-scales -that can only be probed in oscillation experiments. A lot of work remains to be done, including a quantitative discussion of current bounds on non-unitary propagation, three-flavor phenomenology and prospects for next-generation oscillation experiments, potential applications to the short-baseline anomalies and other searches for new neutrino oscillation lengths, etc. We also need to address matter effects, especially when it comes to addressing their impact on the non-unitarity parameters -especially the ǫ parameters -and some of the constraints we have discussed here.