Decoherence in Neutrino Oscillation between 3D Gaussian Wave Packets

There is renewed attention to whether we can observe the decoherence effect in neutrino oscillation due to the separation of wave packets with different masses in near-future experiments. As a contribution to this endeavor, we extend the existing formulation based on a single 1D Gaussian wave function to an amplitude between two distinct 3D Gaussian wave packets, corresponding to the neutrinos being produced and detected, with different central momenta and spacetime positions and with different widths. We find that the spatial widths-squared for the production and detection appear additively in the (de)coherence length and in the localization factor for governing the propagation of the wave packet, whereas they appear as the reduced one (inverse of the sum of inverse) in the momentum conservation factor. The overall probability is governed by the ratio of the reduced to the sum.

The neutrino oscillation is suppressed by the decoherence effect due to wave-packet separation between different mass eigenstates beyond the so-called (de)coherence length [20,21,22,23,24]; see also Ref. [6] for an earlier ultra-relativistic expression for the coherence length. 1  (In Ref. [26], the decoherence effect is pursued in a non-wavepacket approach, which is further developed recently [27].)In Refs.[28,29,30], decoherence due to wave packet separation is discussed with emphasis on experimental testability.In the latest analysis [30] among them, the density matrix for the produced neutrino state is traced out with respect to the accompanying charged lepton state to find a consistent decoherence effect to that in Ref. [22], with a negative conclusion for the near-future detectability in reactor neutrino experiments such as JUNO.A similar conclusion is made in a talk [31].
So far, all the above analyses are based on a single 1D Gaussian wave function, either in the position or momentum space, taking into account only the effect of the neutrino production process.In this paper, we treat both the in and out neutrino states by 3D Gaussian wave packets to take into account the neutrino detection process too.
This Letter is organized as follows: In Sec. 2, we briefly review the Gaussian wave-packet formalism applied to the neutrino field.In Sec. 3, we show our main result: the amplitude and probability for the neutrino oscillation between two distinct Gaussian wave packets.In Sec. 4, we conclude this Letter.In Appendix A, we present more details on the saddlepoint approximation in deriving the amplitude between the Gaussian wave-packet states.In Appendix B, we show a formula by first square-completing the exponent with respect to T .In Appendix C, we show a concrete two-flavor result.

Neutrino wave packet
In this section, we first show how to treat the neutrino wave packet rigorously, and then present our approximation of neglecting the spin dependence.Readers who are more interested in phenomenological aspects may skip this section.

Plane wave
Let us first expand a free neutrino field in Ith mass eigenstate (after the electroweak symmetry breaking) into the plane waves: where E I (p) is the energy of the Ith mass eigenstate, E I (p) := m2 I + p 2 , and the annihilation and creation operators of the neutrino and antineutrino fields obey If neutrinos are Majorana a c I (p, s) = a I (p, s), the momentum-space plane-wave functions u I (p, s) and v I (p, s) are related by the Majorana condition accordingly. 2Hereafter, we assume the Majorana neutrino for simplicity; switching to a Dirac neutrino and/or inclusion of a sterile neutrino is straightforward.
A momentum eigenbasis of the free neutrino one-particle subspace can be defined by such that the commutator (2) leads to the normalization p, s; I p, In the one-particle subspace, the completeness relation (resolution of identity) is where 1 is the identity operator in the free neutrino one-particle subspace.Finally, for a given momentum p, we write the neutrino velocity in the Ith mass eigenstate as where v I (p) := |v I (p)| and e p := p/p.Its direction does not depend on the mass eigenstates, whereas the magnitude does.

Gaussian wave packet
Here, we review the Gaussian wave-packet formalism, following Appendix A in Ref. [33], to spell out notations in this paper; see also Refs.[34,35] for its application in quantum field theory and Ref. [36] for a historical account of earlier works.
We define the Gaussian basis state |X, P , s; σ, I in the free neutrino one-particle subspace by which is centered at the spacetime point X and the momentum P , with σ being its spatial width-squared. 3 Sandwiching by the momentum eigenbases, it is straightforward to show the completeness (resolution of identity) for the Gaussian basis states: It is important that the completeness relation ( 8) holds for arbitrary fixed σ and X 0 , which are not summed nor integrated.Since the Gaussian wave packets form the complete set, they can be used to expand an arbitrary shape of a wave packet.
It is important that what is mixed by the PKMTYMNS matrix U αI is the neutrino field ν α (x) and is not the momentum eigenbasis |p, s; I .Indeed the neutrino field in an interaction eigenstate, ν α (x), is related to the wave function in a mass eigenstate |p, s; I by In the literature, this point has been neglected so far.Including such a spinor effect is an interesting topic in itself and will be covered in a separate publication [38].For the purpose of the current study, we adhere to the convention of disregarding this effect and sloppily write a wave packet state in an interaction eigenstate ν α as 3 Decoherence from wave-packet separation

Oscillation probability of neutrino wave packet
Now we compute the oscillation probability amplitude between interaction eigenstates of a neutrino ν α at a source and ν β at a detector.We take the initial and final states as Gaussian wave packets with spatial widths-squared σ S and σ D .These wave packets are centered at spacetime points X S and X D in the position space, as well as at P S and P D in the momentum space.The amplitude reads where we have used the diagonality of the mass eigenstates ( 4) and (7).
After expanding the amplitude ( 13) by the complete basis (5), we may compute it using the saddle-point approximation: where in which P := P ; see Appendix A for more details on the saddle-point approximation in deriving this amplitude.The following relation is also useful: where vI := |v I | = P /E I P ; see Eq. ( 6).In physical terms, σ sum and σ red represent the summed and reduced width-squared, respectively.Additionally, T denotes the elapsed time, while L indicates the displacement from the source to the detector.The weighted average of the initial and final momenta is symbolized by P .Finally, ĒI and vI correspond to the on-shell energy and velocity of the Ith mass eigenstate for the weight-averaged momentum, respectively.
In the denominator in Eq. ( 14), the factors and √ • • • correspond to the broadening with time of the wave packet in the transverse and longitudinal spatial directions, respectively, as observed in the Gaussian wave function within the position space [39,33], where only the transverse directions broaden with time in the ultra-relativistic limit. 4Hereafter, we will write them The oscillation probability now reads We emphasize that what we have computed in Eq. ( 14) is the quantum mechanical amplitude between normalizable states whose absolute square directly becomes the probability (19).The second-last line in Eq. ( 19) is nothing but the ordinary plane-wave oscillation probability.Physically, each term in the I, J summation can be regarded as the interference between the Ith and Jth mass eigenstates, both acting as mediators between the initial interaction eigenstate ν α and the final ν β .The last line exhibits the wave-packet effects: The first term in the exponent shows how the deviation from the momentum conservation is exponentially suppressed.In a plane-wave limit σ red → ∞, this part is reduced to the momentum delta function: Meanwhile, the second and last terms exhibit the localization of the wave packets at L = vI T and vJ T for the Ith and Jth mass eigenstates, respectively.

Decoherence effect
Here we show how the decoherence effect appears in two different approaches.Firstly, we square-complete the real part of the exponent, which represents the wave-packet effects, with respect to L: where we have used the parallelism of the velocities (18) and have defined5 The last term in the exponent (in the last line) can be regarded as the decoherence effect due to the separation of the wave packets.That is, the interference between Ith and Jth mass eigenstates is exponentially suppressed when the elapsed time exceeds a certain value, T T coh IJ , where Note that the elapsed time T corresponds to the average length of the propagations of the Ith and Jth neutrino wave packets because of the second-last term in the exponent, which localizes the probability around where ≈ denotes the ultra-relativistic limit, which gives Let us turn to the second approach.The exact timings of the neutrino emission and detection at the source and detector, respectively, are hardly measured in actual experiments, particularly in the case of reactor neutrinos.Therefore, for developing physical insight, it would be helpful to square-complete T first.The result is Several comments are in order: • From the last term in the exponent in Eq. ( 26), we again observe the emergence of a coherence length L coh IJ : That is, the interference between Ith and Jth mass eigenstates is exponentially suppressed when L L coh IJ .We see that the first and second approaches agree: • The second-last term effectively chooses configurations around This backs up the equivalence of two approaches (28).
• The third-last term shows that the interference between Ith and Jth mass eigenstates is exponentially suppressed when the size of perpendicular displacement |L ⊥ | exceeds the square-summed wave packet size √ σ sum .

Ultra-relativistic limit
For completeness, we also write down the ultra-relativistic limit of Eq. ( 21) and (26), where the dots denote terms of O P −3 :

Conclusion
In this Letter, we computed the amplitude and probability for the neutrino oscillation between two distinct Gaussian wave packets that have different centers of position and momentum as well as different widths.
We find that the spatial widths-squared for the production and detection appear additively in the (de)coherence length as well as in the localization factor governing the propagation of the wave packet, whereas they appear as the reduced one (inverse of the sum of inverse) in the momentum conservation factor.The overall probability is governed by the ratio of the reduced to the sum.We have obtained the (de)coherence lengths in two ways, which coincide with each other in the ultra-relativistic limit. 1 − i where is the exponent before performing the p-integration; p ⋆ is the solution to the saddle-point equation ∂G I (p ⋆ ) /∂p i = 0; and, around the saddle point p ⋆ , the exponent is approximated as usual: Iteratively solving p ⋆ = ∞ n=0 p (n) with p (n) = O(σ −n sum ), we may obtain arbitrary higher order terms.For example, the next-leading order term of the saddle point is B Complete square with respect to T As discussed in the main text, T is hardly measured.It would be useful to obtain an expression of square-completed T from all the exponents in Eq. ( 19) in order to integrate out T . 6The result is 6 Our analysis in this paper is within the level of quantum mechanics.In quantum field theory, an example of the emergence of such integration over T from the final-state phase-space integral can be seen in e.g.Sec. 4 in Ref. [33].
where we have defined a complex parameter: Its further analysis will be presented elsewhere; see footnote 6.
In a heuristic ultra-relativistic limit leaving only characteristic leading contributions in each part, this reduces to with

C Two-flavor example
It might be instructive to present a concrete two-flavor example between interaction eigenstates, say, ν α and ν β , with the PKMTYMNS matrix The resultant probability of non-oscillation from Eq. ( 21) is