# On the flavor/mass dichotomy for mixed neutrinos: a phenomenologically
motivated analysis based on lepton charge conservation in neutron decay

Giuseppe Gaetano Luciano111email<EMAIL_ADDRESS>Applied
Physics Section of Environmental Science Department, Universitat de Lleida,
Av. Jaume II, 69, 25001 Lleida, Spain

###### Abstract

_Flavor/mass dichotomy_ for mixed fields is one of the most controversial
issues in Quantum Field Theory. In this work we approach the problem by
considering mixing of neutrinos and computing the transition amplitude for the
paradigmatic neutron $\beta$-decay
$n\,\rightarrow\,p\,+\,e^{-}\,+\,\bar{\nu}$. Calculations are developed by
utilizing the following different representations of neutrino states: _i_)
Pontecorvo states, _ii_) mass states and _iii_) exact QFT flavor states, which
are defined as eigenstates of the flavor charge. As a guiding principle, we
invoke the conservation of lepton charge in the interaction vertex, which is
fundamentally inbuilt into the Standard Model at tree level. In the short-time
limit, we show that the only fully consistent scenario is that based on QFT
states, whereas the other two frameworks contradict the underlying theory.
This result provides a crucial step towards understanding the rôle of neutrino
mixing in weak decays and contributes to elucidate the flavor/mass controversy
in QFT.

###### pacs:

14.60.Pq, 13.15.+g

##  I Introduction

Neutrino mixing and oscillations are among the most fascinating, but least
understood phenomena in particle physics. Although the idea of neutrino
oscillations was first proposed by Pontecorvo in 1957 Ponte57  by analogy with
kaons, the theoretical framework in its modern form only dates back to the
1970’s Grib ; Frit , after the discovery of the second Maki  and third Third
generation of neutrinos and the later extension to include environmental
effects Wolf ; Mikh . On the experimental side, significant progress was
achieved between 1998 and 2002 with the detection of atmospheric and solar
neutrino oscillations by Super-Kamiokande SuperK ; NP  and SNO SNO ; McD ,
shortly thereafter confirmed by KamLand KamL .

Besides their intrinsic relevance, neutrino oscillations provide to date the
only evidence of physics beyond the Standard Model (SM), thus motivating ever-
new research both at theoretical and experimental level. In this direction,
special focus has been recently placed on the study of
gravitational/acceleration effects on flavor oscillations Stodo ; Burgard ;
Fornengo ; Cardall ; Lambiase ; GiuntiKim ; Mavro ; Paglia ; Vissani ;
Capozziello ; Dvo ; ETG ; BlasSma ; Luc ; Luc2 ; Swami ; Khalifeh , with non-
trivial results in the framework of extended theories Capozziello ; ETG . In
parallel, interesting applications have been found in quantum information
BlasEnt ; Alok ; Kumar ; Roggero ; Li ; Matrella , quantum optics Marini  and
geophysics Geo , where neutrinos are used as a probe to determine the chemical
composition of the inner Earth.

Despite the intensive study in quantum mechanics (QM), comparably less
attention has been devoted to the analysis of neutrino mixing in Quantum Field
Theory (QFT). However, it is the field theoretical investigation that is more
pertinent to high-energy phenomena in particle physics, due to the SM being
built upon QFT. The first consistent treatment of flavor mixing based on the
construction of Fock spaces for quantized neutrino fields was developed in
BV95 , highlighting the shortcomings of the original QM theory. Indeed, while
Pontecorvo mixing transformations act as pure rotations between single-
particle states of neutrinos with definite masses (henceforth “mass states”)
and neutrinos with definite flavors (“flavor states”), their QFT counterparts
exhibit the structure of a rotation nested into a non-commuting Bogoliubov
transformation BV95 ; Garg . This leads to orthogonal vacuum states for mass
and flavor fields BV95 , the latter behaving as a condensate of massive
particle/antiparticle pairs with a complex entangled structure LucPetr ; Cabo
. In turn, the Fock spaces built upon these two vacua are _unitarily_ (i.e.
physically) _inequivalent_ to each other, giving rise to characteristic
effects that lie outside the domain of QM Bare ; CapolDark ; MavDark ;
Blasone:2017nbf ; Jizba ; Quaranta ; LucianoTs  (see SmVit  for a recent
review). We emphasize that similar considerations had been previously outlined
in Berna ; Berna2  in relativistic QM. On mathematical grounds, the
theoretical understanding of QFT mixing has been confirmed by the rigorous
analysis of Hannab .

In recent years renewed interest in the field theoretical analysis of mixing
has been prompted by the study of weak interactions involving neutrinos. In
particular, a fruitful arena where to test the QFT formalism was offered by
the weak decay of accelerated protons Ahluw ; ProtBlas ; ProtMat , whose
occurrence is allowed by the Unruh effect. In this context, by invoking
consistency with the general covariance of QFT, the scalar decay rate for the
proton was computed both in the laboratory frame and in the comoving rest-
frame, with conflicting results on whether to utilize flavor Ahluw ; ProtBlas
or mass states ProtMat  for mixed neutrinos (_flavor/mass dichotomy_ , see
also GiuntiLee ; GiuntiUn ; Akh ; Remarks ; SmaldTur ; GaetanoLucia  for
further discussion on the topic). It must be emphasized that such a
controversy is peculiar to QFT QFTCon , whereas in QM the formal equivalence
between the two representations is warranted by the Stone-von Neumann
uniqueness theorem Stone ; VN . While providing valuable insights on the rôle
of QFT mixing in weak interactions, these studies have posed the problem of
the very nature of asymptotic neutrinos, requiring fresh efforts to sort
things out. In passing, we mention that stimulating analysis of flavor mixing
and weak decays were conducted in CapW  and CYL .

Starting from the above premises, in this work we explore more in-depth the
inherent features of neutrino mixing in QFT. Specifically, we attempt to
elucidate the aforementioned flavor/mass controversy by focusing on the
computation of the transition amplitude for the neutron $\beta$-decay. We
carry out calculations by resorting to three different representations of
neutrino states: _i_) Pontecorvo states, _ii_) mass states and _iii_) exact
QFT flavor states, which are defined as eigenstates of the flavor charge
operators. As a result, we obtain three different expressions for the neutron
decay amplitude, which only overlap in the limit of ultra-relativistic
neutrinos. The question arises as to which of these descriptions provides the
physically correct scenario. Waiting for experimental hints that could point
us toward the solution, here we follow a phenomenologically motivated
approach: by requiring the conservation of lepton charge in the interaction
vertex, which is empirically observed and built into the SM at tree level Cli
, we show that the only fully consistent framework is that founded on QFT
states. On the other hand, both Pontecorvo and mass representations lead to a
violation of lepton charge, contradicting the underlying theory. We remark
that this conclusion is simply reached on the basis of consistency with SM
expectations and does not assume any detector-dependent model for neutrino
oscillations. Such a result should help clarifying the true nature of
asymptotic neutrinos against the variety of exotic claims recently appeared in
the literature.

The remainder of the work is organized as follows: in the next Section we
review the QFT formalism of neutrino mixing. Section III is devoted to the
calculation of the tree-level $\beta$-decay amplitude in the Pontecorvo and
mass representations. Consistency with the conservation of lepton charge is
checked by considering the “short-time limit” of the above amplitudes, i.e.
the limit for small distances from the production vertex. The same analysis is
then developed with exact QFT flavor states in Sec. IV. To avoid unnecessary
technicalities and make the physical insight of our analysis as transparent as
possible, we neglect flavor changing loop induced processes, which however
would not be relevant for our discussion. Conclusions and outlook are
summarized in Sec. V. Throughout the whole manuscript, we use natural units
$\hbar=c=1$.

##  II Neutrino mixing in QFT: mathematical aspects and physical implications

Let us start by reviewing the QFT formalism of neutrino mixing and the main
differences with the QM Pontecorvo treatment (for more details, see BV95 ).
Toward this end, we remind that Pontecorvo mixing transformations for single-
particle flavor states read222In this work we consider a simplified model
involving only two flavors. However, the validity of our result is not spoilt
by the generalization to the third generation.

$\displaystyle|\nu^{r}_{{\bf{k}},e}\rangle_{P}$
$\displaystyle=\cos\theta\,|\nu^{r}_{{\bf{k}},1}\rangle+\sin\theta\,|\nu^{r}_{{\bf{k}},2}\rangle\,,$
(1a) $\displaystyle|\nu^{r}_{{\bf{k}},\mu}\rangle_{P}$
$\displaystyle=-\sin\theta\,|\nu^{r}_{{\bf{k}},1}\rangle+\cos\theta\,|\nu^{r}_{{\bf{k}},2}\rangle\,,$
(1b)

where $|\nu^{r}_{{\bf{k}},i}\rangle$, $i=1,2$, are the states of neutrino with
definite masses $m_{i}$, while $|\nu^{r}_{{\bf{k}},\ell}\rangle_{P}$,
$\ell=e,\mu$, are the states of neutrino with definite flavors $\ell$
(electron and muon, respectively). We are assuming equal momentum ${\bf{k}}$
and helicity $r=1,2$ for neutrinos with different masses. The subscript $P$ is
a reminder for Pontecorvo states.

At level of QFT, the above relations are rewritten in the form

$\displaystyle\nu_{e}(x)$
$\displaystyle=\cos\theta\,\nu_{1}(x)+\sin\theta\,\nu_{2}(x)\,,$ (2a)
$\displaystyle\nu_{\mu}(x)$
$\displaystyle=-\sin\theta\,\nu_{1}(x)+\cos\theta\,\nu_{2}(x)\,,$ (2b)

where $\nu_{\ell}(x)$, $\ell=e,\mu$, are the (interacting) Dirac neutrino
fields with definite flavors, while $\nu_{i}(x)$, $i=1,2$, are the (free)
fields with definite masses. These are expanded according to the usual
Fourier-decomposition BV95

$\nu_{i}(x)\,=\,\frac{1}{\sqrt{V}}\sum_{\textbf{k},r}\left[u_{\textbf{k},i}^{r}\,\alpha^{r}_{\textbf{k},i}(x^{0})\,+\,v_{-\textbf{k},i}^{r}\,\beta^{r\dagger}_{-\textbf{k},i}(x^{0})\right]e^{i\textbf{k}\cdot\textbf{x}}\,,\qquad
i=1,2\,,$ (3)

where
$\alpha^{r}_{\textbf{k},i}(x^{0})=\alpha^{r}_{\textbf{k},i}\,e^{-i\omega_{{\bf{k}},i}x^{0}}$,
$\beta^{r\dagger}_{-\textbf{k},i}(x^{0})=\beta^{r\dagger}_{-\textbf{k},i}\,e^{i\omega_{{\bf{k}},i}x^{0}}$.
The operators $\alpha^{r}_{\textbf{k},i}$ annihilate a field-mode of quantum
numbers ${\bf{k}},r$ and energy
$\omega_{{\bf{k}},i}=\sqrt{m_{i}^{2}+{\bf{k}}^{2}}$. They are defined by

$\alpha^{r}_{\textbf{k},i}|0\rangle_{m}\,=\,\beta^{r}_{\textbf{k},i}|0\rangle_{m}\,=\,0\,,\qquad
i=1,2\,,$ (4)

where $|0\rangle_{m}\equiv|0\rangle_{1}\otimes|0\rangle_{2}$ is the “mass
vacuum”. Similar relations hold for $\beta^{r}_{\textbf{k},i}$.

By imposing the canonical anti-commutation relations for the fields,

$\left\\{\nu_{i}^{\alpha}(x),\nu_{j}^{\beta\dagger}(y)\right\\}_{x^{0}=x^{0^{\prime}}}\,=\,\delta^{3}{\left({\bf{x}}-{\bf{y}}\right)}\,\delta_{\alpha\beta}\,\delta_{ij}\,,\qquad\alpha,\beta=1,\dots,4\,,$
(5)

we have

$\left\\{\alpha^{r}_{\textbf{k},i},\alpha^{s\dagger}_{\textbf{q},j}\right\\}\,=\,\delta_{{\bf{k}}{\bf{q}}}\delta_{rs}\delta_{ij}\,,\qquad
i,j=1,2\,,$ (6)

and similarly for $\beta^{r}_{{\bf{k}},i}$, with all other anti-commutators
vanishing.

For more details on the explicit form used for the spinors
$u_{\textbf{k},i}^{r}$, $v_{-\textbf{k},i}^{r}$ and the Dirac
$\gamma$-matrices, we remand to BV95 . Here, it is enough to write down the
following orthonormality and completeness relations

$\displaystyle
u_{{\bf{k}},i}^{r\dagger}u_{{\bf{k}},i}^{s}\,=\,v_{{\bf{k}},i}^{r\dagger}v_{{\bf{k}},i}^{s}\,=\,\delta_{rs}\,,$
(7) $\displaystyle
u_{{\bf{k}},i}^{r\dagger}v_{-{\bf{k}},i}^{s}\,=\,v_{-{\bf{k}},i}^{r\dagger}u_{{\bf{k}},i}^{s}=0\,,$
(8)
$\displaystyle\sum_{r}\left(u_{{\bf{k}},i}^{r}u_{{\bf{k}},i}^{r\dagger}\,+\,v_{-{\bf{k}},i}^{r}v_{-{\bf{k}},i}^{r\dagger}\right)=\mathds{1}\,.$
(9)

For reasons that will appear clear below, in Eq. (3) we have considered the
finite-volume expansion of free fields. Formally, the infinite-volume limit is
obtained by implementing the standard prescription

$\displaystyle\sqrt{V}{\alpha}_{{\bf{k}}}$ $\displaystyle\rightarrow$
$\displaystyle(2\pi)^{{3}/{2}}\,{\alpha}_{\bf{k}}\,,$ (10)
$\displaystyle\frac{1}{V}\sum_{{\bf{k}}}$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{(2\pi)^{3}}\,\int\hskip 0.28453ptd^{3}\textbf{k}\,,$
(11) $\displaystyle\frac{V\delta_{{\bf{k}}{\bf{q}}}}{(2\pi)^{3}}$
$\displaystyle\rightarrow$ $\displaystyle\delta^{3}({\bf{k}}-{\bf{q}})\,.$
(12)

Let us now turn to the expansion of flavor fields. For this purpose, it is
convenient to express the transformations (2) in terms of the mixing generator
BV95

$G_{\theta}(x^{0})\,=\,\exp\left\\{\theta\int
d^{3}{\bf{x}}\,\left[{\nu}^{\dagger}_{1}(x){\nu}_{2}(x)-{\nu}^{\dagger}_{2}(x){\nu}_{1}(x)\right]\right\\}\,,$
(13)

which gives

${\nu}^{\alpha}_{\ell}(x)\,=\,{G}_{\theta}^{-1}(x^{0})\,{\nu}^{\alpha}_{i}(x)\,{G_{\theta}}(x^{0})\,,\qquad(\ell,i)=(e,1),(\mu,2)\,.$
(14)

At finite volume $V$, it is straightforward to see that this is a unitary
operator, i.e.
$G^{-1}_{\theta}(x^{0})=G_{-\theta}(x^{0})=G^{\dagger}_{\theta}(x^{0})$, which
preserves the canonical anitcommutators. Furthermore, it provides us with the
map between the Hilbert space for free fields $\mathcal{H}_{m}$ and that for
mixed fields $\mathcal{H}_{f}$. In particular, the vacuum for flavor fields
(“flavor vacuum”) $|0(x^{0})\rangle_{f}$ is given by

$|0(x^{0})\rangle_{f}=G_{\theta}^{-1}(x^{0})|0\rangle_{m}\,.$ (15)

Following the convention of BV95 , we shall denote by $|0\rangle_{f}$ the
flavor vacuum at $x^{0}=0$.

While being unitary for finite $V$, in the infinite-volume limit (i.e., for
systems with infinite degrees of freedom like fields) $G_{\theta}(x^{0})$
exhibits a non-unitary nature, which results into the orthogonality of flavor
and mass vacua BV95 ; JiMi

${}_{f}\langle 0(x^{0})|0\rangle_{m}=0,\,\,\,\,\forall x^{0}\,.$ (16)

Likewise, flavor vacua at different times are orthogonal to each other Terra ,
i.e. ${}_{f}\langle 0(x^{0^{\prime}})|0(x^{0})\rangle_{f}=0,\forall
x^{0^{\prime}}\neq x^{0}$. We stress that these are distinctive QFT features
that are missing in QM, due to the validity of the Stone-von Neumann theorem
Stone ; VN . To avoid any technical issue, in what follows we shall perform
all computations at finite volume and only at the end we consider the
$V\rightarrow\infty$ limit.

By letting the generator $G_{\theta}(x^{0})$ act on $\nu_{i}(x)$ in Eq. (14),
the flavor fields take the form

$\nu_{\ell}(x)\,=\,\frac{1}{\sqrt{V}}\sum_{\textbf{k},r}\left[u_{\textbf{k},i}^{r}\,\alpha^{r}_{\textbf{k},\nu_{\ell}}(x^{0})\,+\,v_{-\textbf{k},i}^{r}\,\beta^{r\dagger}_{-\textbf{k},\nu_{\ell}}(x^{0})\right]e^{i\textbf{k}\cdot\textbf{x}}\,,$
(17)

with $(\ell,i)=(e,1),(\mu,2)$. The flavor annihilators are given by

$\displaystyle\alpha^{r}_{{\bf{k}},\nu_{e}}(x^{0})$ $\displaystyle=$
$\displaystyle\cos\theta\,\alpha^{r}_{{\bf{k}},1}(x^{0})\,+\,\sin\theta\sum_{s}\left[u^{r\dagger}_{{\bf{k}},1}u^{s}_{{\bf{k}},2}\,\alpha^{s}_{{\bf{k}},2}(x^{0})\,+\,u^{r\dagger}_{{\bf{k}},1}v^{s}_{-{\bf{k}},2}\,\beta^{s\dagger}_{-{\bf{k}},2}(x^{0})\right],$
(18) $\displaystyle\alpha^{r}_{{\bf{k}},\nu_{\mu}}(x^{0})$ $\displaystyle=$
$\displaystyle\cos\theta\,\alpha^{r}_{{\bf{k}},2}(x^{0})\,-\,\sin\theta\sum_{s}\left[u^{r\dagger}_{{\bf{k}},2}u^{s}_{{\bf{k}},1}\,\alpha^{s}_{{\bf{k}},1}(x^{0})\,+\,u^{r\dagger}_{{\bf{k}},2}v^{s}_{-{\bf{k}},1}\,\beta^{s\dagger}_{-{\bf{k}},1}(x^{0})\right],$
(19) $\displaystyle\beta^{r}_{-{\bf{k}},\nu_{e}}(x^{0})$ $\displaystyle=$
$\displaystyle\cos\theta\,\beta^{r}_{-{\bf{k}},1}(x^{0})\,+\,\sin\theta\sum_{s}\left[v^{s\dagger}_{-{\bf{k}},2}v^{r}_{-{\bf{k}},1}\,\beta^{s}_{-{\bf{k}},2}(x^{0})\,+\,u^{s\dagger}_{{\bf{k}},2}v^{r}_{-{\bf{k}},1}\,\alpha^{s\dagger}_{{\bf{k}},2}(x^{0})\right],$
(20) $\displaystyle\beta^{r}_{-{\bf{k}},\nu_{\mu}}(x^{0})$ $\displaystyle=$
$\displaystyle\cos\theta\,\beta^{r}_{-{\bf{k}},2}(x^{0})\,-\,\sin\theta\sum_{s}\left[v^{s\dagger}_{-{\bf{k}},1}v^{r}_{-{\bf{k}},2}\,\beta^{s}_{-{\bf{k}},1}(x^{0})\,+\,u^{s\dagger}_{{\bf{k}},1}v^{r}_{-{\bf{k}},2}\,\alpha^{s\dagger}_{{\bf{k}},1}(x^{0})\right].$
(21)

Notice that these transformations can be inverted by utilizing the symmetry
$\nu_{\ell}(x)\leftrightarrow\nu_{i}(x)$ when $\theta\rightarrow-\theta$. The
time-dependence of the flavor operators (19)-(21) indicates that flavor fields
are interacting fields. It must be noted that this interacting model can be
solved exactly, without needing perturbation expansions Jizba . Furthermore,
we emphasize that the choice of the reference time $x^{0}=0$ at which the
flavor vacuum (15) and the related particle states are defined is not unique.
In principle, any other choice would be possible and equivalent, provided that
the flavor states which are acted upon by the corresponding flavor operators
(19)-(21) are consistently evaluated at the same time as the operators
themselves and the commutators are all considered at equal times BV95 .

As remarked in Fuji ; BlasDiMa , the field expansion (17) relies on a special
choice of the bases of spinors, namely those referring to the free field
masses $m_{1}$ and $m_{2}$, respectively. However, it is always possible to
perform a Bogoliubov transformation in order to expand the field operators in
a different basis of spinors, referring to an arbitrarily chosen couple of
mass parameters (for instance, the natural bases corresponding to the couple
of masses $m_{e}$, $m_{\mu}$). The relevant point is that this transformation
leaves measurable quantities (such as flavor charges and oscillation
probabilities) invariant, as it should be. Therefore, in this sense flavor
mixing can be reformulated as a gauge theory BlasDiMa .

Equations (18)-(21) clearly show the non-trivial structure of flavor mixing in
QFT: in fact, besides the standard Pontecorvo rotation, flavor annihilators
also contain a Bogoliubov transformation (the terms in the brackets) arising
from the products of (anti-)spinors with different masses. The presence of
such extra terms lies at the heart of the QFT treatment of mixing, as they
imply that flavor vacuum annihilators do not annihilate mass vacuum, and vice-
versa.

Without loss of generality, we can now select the reference frame such that
${\bf{k}}=\left(0,0,|{\bf{k}}|\right)$. In this setting, only the products of
wave functions with $r=s$ are non-vanishing, which allows us to rewrite Eqs.
(18)-(21) in the simpler form

$\displaystyle\alpha^{r}_{{\bf{k}},\nu_{e}}(x^{0})$ $\displaystyle=$
$\displaystyle\cos\theta\,\alpha^{r}_{{\bf{k}},1}(x^{0})\,+\,\sin\theta\left(|U_{\bf{k}}|\,\alpha^{r}_{{\bf{k}},2}(x^{0})\,+\,\epsilon^{r}|V_{\bf{k}}|\,\beta^{r\dagger}_{-{\bf{k}},2}(x^{0})\right),$
(22) $\displaystyle\alpha^{r}_{{\bf{k}},\nu_{\mu}}(x^{0})$ $\displaystyle=$
$\displaystyle\cos\theta\,\alpha^{r}_{{\bf{k}},2}(x^{0})\,-\,\sin\theta\left(|U_{\bf{k}}|\,\alpha^{r}_{{\bf{k}},1}(x^{0})\,-\,\epsilon^{r}|V_{\bf{k}}|\,\beta^{r\dagger}_{-{\bf{k}},1}(x^{0})\right),$
(23) $\displaystyle\beta^{r}_{-{\bf{k}},\nu_{e}}(x^{0})$ $\displaystyle=$
$\displaystyle\cos\theta\,\beta^{r}_{-{\bf{k}},1}(x^{0})\,+\,\sin\theta\left(|U_{\bf{k}}|\,\beta^{r}_{-{\bf{k}},2}(x^{0})\,-\,\epsilon^{r}|V_{\bf{k}}|\,\alpha^{r\dagger}_{{\bf{k}},2}(x^{0})\right),$
(24) $\displaystyle\beta^{r}_{-{\bf{k}},\nu_{\mu}}(x^{0})$ $\displaystyle=$
$\displaystyle\cos\theta\,\beta^{r}_{-{\bf{k}},2}(x^{0})\,-\,\sin\theta\left(|U_{\bf{k}}|\,\beta^{r}_{-{\bf{k}},1}(x^{0})\,+\,\epsilon^{r}|V_{\bf{k}}|\,\alpha^{r\dagger}_{{\bf{k}},1}(x^{0})\right),$
(25)

where $\epsilon^{r}=(-1)^{r}$. The Bogoliubov coefficients $|U_{\bf{k}}|$ and
$|V_{\bf{k}}|$ are given by

$\displaystyle|U_{\bf{k}}|$ $\displaystyle=$ $\displaystyle
u_{{\bf{k}},i}^{r\dagger}\,u_{{\bf{k}},j}^{r}\,=\,v^{r\dagger}_{-{\bf{k}},i}\,v^{r}_{-{\bf{k}},j}$
(26) $\displaystyle=$
$\displaystyle\frac{|{\bf{k}}|^{2}+\left(\omega_{{\bf{k}},1}+m_{1}\right)\left(\omega_{{\bf{k}},2}+m_{2}\right)}{2\sqrt{\omega_{{\bf{k}},1}\omega_{{\bf{k}},2}\left(\omega_{{\bf{k}},1}+m_{1}\right)\left(\omega_{{\bf{k}},2}+m_{2}\right)}}\,,$
$\displaystyle|V_{\bf{k}}|$ $\displaystyle=$
$\displaystyle\epsilon^{r}\,u_{{\bf{k}},1}^{r\dagger}\,v^{r}_{-{\bf{k}},2}\,=\,-\epsilon^{r}\,u^{r\dagger}_{{\bf{k}},2}v_{-{\bf{k}},1}^{r}$
(27) $\displaystyle=$
$\displaystyle\frac{\left(\omega_{{\bf{k}},1}+m_{1}\right)-\left(\omega_{{\bf{k}},2}+m_{2}\right)}{2\sqrt{\omega_{{\bf{k}},1}\omega_{{\bf{k}},2}\left(\omega_{{\bf{k}},1}+m_{1}\right)\left(\omega_{{\bf{k}},2}+m_{2}\right)}}\,|{\bf{k}}|\,,$

with $i,j=1,2$, $i\neq j$ (similar relations hold for
${\bf{k}}\rightarrow-{\bf{k}}$). It is a matter of calculations to show that

$|U_{{\bf{k}}}|^{2}\,+\,|V_{{\bf{k}}}|^{2}\,=\,1\,,$ (28)

which ensures that flavor ladder operators still satisfy canonical anti-
commutation relations (at equal times).

For our next purposes, we note that the following relations hold true in the
reference frame where ${\bf{k}}=\left(0,0,|{\bf{k}}|\right)$:

$\displaystyle\bar{u}_{{\bf{k}},1}^{r}|U_{\bf{k}}|\,-\,\epsilon^{r}\,\bar{v}^{r}_{-{\bf{k}},1}|V_{{\bf{k}}}|$
$\displaystyle=$ $\displaystyle\bar{u}_{{\bf{k}},2}^{r}\,,$ (29)
$\displaystyle\bar{u}_{{\bf{k}},1}^{r}|V_{\bf{k}}|\,+\,\epsilon^{r}\,\bar{v}^{r}_{-{\bf{k}},1}|U_{{\bf{k}}}|$
$\displaystyle=$ $\displaystyle\epsilon^{r}\,\bar{v}_{-{\bf{k}},2}^{r}\,,$
(30) $\displaystyle
v^{r}_{{\bf{k}},1}|U_{{\bf{k}}}|\,-\,\epsilon^{r}\,u^{r}_{-{\bf{k}},1}|V_{{\bf{k}}}|$
$\displaystyle=$ $\displaystyle v_{{\bf{k}},2}^{r}\,,$ (31) $\displaystyle
v^{r}_{{\bf{k}},1}|V_{{\bf{k}}}|\,+\,\epsilon^{r}\,u^{r}_{-{\bf{k}},1}|U_{{\bf{k}}}|$
$\displaystyle=$ $\displaystyle\epsilon^{r}\,u_{-{\bf{k}},2}^{r}\,,$ (32)

where $\bar{u}_{{\bf{k}},i}^{r}=u_{{\bf{k}},i}^{r\dagger}\,\gamma_{0}$ and
$\bar{v}_{-{\bf{k}},i}^{r}=v_{-{\bf{k}},i}^{r\dagger}\,\gamma_{0}$, $i=1,2$.
Furthermore, we have

$\displaystyle\bar{u}_{{\bf{k}},2}^{r}\,|U_{\bf{k}}|\,+\,\epsilon^{r}\,\bar{v}^{r}_{-{\bf{k}},2}\,|V_{\bf{k}}|$
$\displaystyle=$ $\displaystyle\bar{u}_{{\bf{k}},1}^{r}\,,$ (33)
$\displaystyle
v_{{\bf{k}},2}^{r}\,|U_{\bf{k}}|\,+\,\epsilon^{r}\,u^{r}_{-{\bf{k}},2}\,|V_{\bf{k}}|$
$\displaystyle=$ $\displaystyle v_{{\bf{k}},1}^{r}\,.$ (34)

QFT neutrino states are now defined by the action of flavor creation operators
on the flavor vacuum $|0\rangle_{f}$ at $x^{0}=0$, i.e. BV95

$|\nu^{r}_{{\bf{k}},\ell}\rangle\,\equiv\,\alpha^{r\dagger}_{{\bf{k}},\nu_{\ell}}(0)|0\rangle_{f}\,,\qquad\ell=e,\mu\,.$
(35)

As usual, the time evolution of these states is ruled by
$|\nu^{r}_{{\bf{k}},\ell}(x^{0})\rangle=e^{iH_{0}x^{0}}\alpha^{r\dagger}_{{\bf{k}},\nu_{\ell}}(0)|0\rangle_{f}$.

It should be noticed that the QFT formalism described above reproduces the
standard QM Pontecorvo treatment in the ultra-relativistic limit $\frac{\Delta
m}{|{\bf{k}}|}=\frac{|m_{2}-m_{1}|}{{|{\bf{k}}|}}\ll 1$. To this end, we
observe that the Bogoliubov coefficients $|U_{\bf{k}}|$ and $|V_{{\bf{k}}}|$
at the first order in $\mathcal{O}\left(\frac{\Delta m}{2|{\bf{k}}|}\right)$
take the form

$\displaystyle|U_{\bf{k}}|$ $\displaystyle\,\approx\,1\,,$ (36a)
$\displaystyle|V_{\bf{k}}|$ $\displaystyle\,\approx\,\frac{\Delta
m}{2|{\bf{k}}|}\,,$ (36b)

from which we get $|U_{{\bf{k}}}|\rightarrow 1,|V_{{\bf{k}}}|\rightarrow 0$ as
$\frac{\Delta m}{|{\bf{k}}|}\rightarrow 0$. Inserting these values into the
Bogoliubov transformations (22)-(25), it is straightforward to see that
Pontecorvo rotations for flavor annihilators are trivially recovered, which
entails that flavor and mass vacua are equivalent to each other in this limit.
In turn, this implies that the definition (35) of exact QFT flavor states
reduces to the usual Pontecorvo definition (1).

The above considerations provide us with the basic ingredients to analyze the
effects of neutrino mixing in a self-consistent field theoretical way. In
particular, we shall resort to the field expansions (17) and the Bogoliubov
transformations (22)-(25) to compute the transition amplitude for the neutron
$\beta$-decay.

##  III Neutron $\beta$-decay with Pontecorvo and mass states for neutrinos

In this Section we consider the weak decay

$a)\,\,\,n\,\rightarrow\,p\,+\,e^{-}\,+\,\bar{\nu}_{e}\,,$ (37)

as paradigmatic process to check the consistency of neutrino mixing with the
conservation of lepton charge in the (tree-level) interaction
vertex333Although our considerations are specific for the $\beta$-decay
process, results and conclusions are more general and can be extended to all
weak interactions involving neutrinos.. We stress that this symmetry is
familiar to any physics student from the most common muon decay mode
$\mu^{-}\,\rightarrow\,e^{-}\bar{\nu}_{e}\nu_{\mu}$, which would be a direct
conversion $\mu^{-}\,\rightarrow\,e^{-}$ without the introduction of neutrinos
to both balance lepton flavor and explain the $3$-body spectrum of electron
energy.

In parallel to the interaction (37), we also consider the flavor-violating
process

$b)\,\,\,n\,\rightarrow\,p\,+\,e^{-}\,+\,\bar{\nu}_{\mu}\,,$ (38)

which is in principle allowed because of neutrino oscillations. Nevertheless,
for time intervals sufficiently longer than the neutron lifetime $\tau_{n}$,
but much shorter compared to the characteristic oscillation time $T_{osc}$,
the tree-level amplitude for this process is expected to vanish for
consistency with lepton charge conservation. We shall refer to this time scale
as “short-time” limit. Of course, given the experimental values of $\tau_{n}$
and $T_{osc}$, such interval is well defined.

In the scattering theory for finite-range potentials, it is typically assumed
that the interaction Hamiltonian $H_{int}(x)$ can be switched off far before
and after the interaction, i.e.
$\lim_{x^{0}_{in}\rightarrow-\infty}H_{int}(x)=\lim_{x^{0}_{out}\rightarrow+\infty}H_{int}(x)=0$
(adiabatic approximation) Itz , so that initial and final states can be
represented as eigenstates of the free hamiltonian $H_{0}$. However, in the
presence of mixed neutrinos, such an approximation fails, since neutrino
fields cannot be defined as asymptotic operators acting on the mass vacuum. As
discussed in Sec. II, this follows from the fact that Fock spaces for flavor
and mass fields are unitarily inequivalent to each other. The supposed
ambiguity has been successfully addressed in CYL , showing that the long-time
limit of the amplitudes (37) and (38) is actually well-defined when working
with exact QFT states. Furthermore, this limit turns out to be consistent with
SM predictions in the relativistic approximation, which corroborates the
physical rigor of the definition of QFT neutrino states.

For our purposes of computing the short-time limit of the decay amplitudes
(37) and (38), we consider the perturbation theory at the first order. We then
set the integration limits $x^{0}_{in}$, $x^{0}_{out}$ such that $\Delta
x^{0}=x^{0}_{out}-x^{0}_{in}\ll T_{osc}$, as discussed above. In the
interaction picture, denoting the ingoing and outgoing states by
$|\psi_{i}\rangle$ and $|\psi_{f}\rangle$, respectively, the probability
amplitude is given by $\langle
e^{iH_{0}x^{0}}\psi_{f}|U_{I}(x^{0})|\psi_{i}\rangle$, where the time
evolution operator $U_{I}(x^{0})$ to the leading order is

$U_{I}(x^{0})\simeq
1-i\int_{0}^{x^{0}}dx^{0^{\prime}}\,H_{int}(x^{0^{\prime}})\,.$ (39)

Here, $H_{int}(x^{0})=e^{iH_{0}x^{0}}\,H_{int}\,e^{-iH_{0}x^{0}}$ is the
interaction Hamiltonian in the interaction picture.

The effective Lagrangian for the interaction (37) is given by Cli

$\mathscr{L}_{int}(x)\,=\,-\frac{G_{F}}{\sqrt{2}}\left[\bar{e}(x)\hskip
0.56905pt\gamma^{\mu}(\mathds{1}-\gamma^{5})\nu_{e}(x)\right]\left[V_{ud}\,\bar{q}_{u}(x)\gamma_{\mu}\hskip
0.56905pt(f\,-\,\gamma^{5}g)\hskip 0.56905ptq_{d}(x)\right]\,,$ (40)

where $G_{F}$ is the Fermi constant, $V_{ud}$ is the CKM matrix element
describing the coupling between the up and down quarks, while $f,g$ are the
form factors. The electron, neutrino, up and down quark fields have been
denoted by $e(x)$, $\nu_{e}(x)$, $q_{u}(x)$ and $q_{d}(x)$, respectively. As
well known, the projection operator
$P_{L}=1/2\left(\mathds{1}-\gamma_{5}\right)$ appears because charged-current
in weak interactions only couple to left-chiral fermions. Furthermore, we have
omitted the hermitian conjugate in Eq. (40), since it gives a vanishing
contribution to the transition amplitude.

The above expression of $\mathscr{L}_{int}$ is justified within the framework
of (current-current interaction) Fermi theory. Clearly, the actual Lagrangian
is more complicated than Eq. (40) (see Act ). However, such additional
complications will be mostly irrelevant for our following discussion and can
be safely overlooked.

Based on the above considerations, let us compute the amplitude for the decay
channels $a)$ and $b)$. We perform calculations by resorting first to
Pontecorvo states $|\nu_{{\bf{k}},\ell}\rangle_{P}$ in Eqs. (1) and then to
mass states $|\nu_{{\bf{k}},i}\rangle$ as neutrino fundamental representation.
At a later stage, we shall focus on the short-time limit of the resulting
amplitudes, discussing the regime of their validity.

###  III.1 Pontecorvo states

As a first step, we consider the process $a)$. The transition amplitude at
tree level takes the form

$A^{a)}_{P}\,=\,_{P}\langle\bar{\nu}^{r}_{{\bf{k}},e},e^{s}_{{\bf{q}}},\hskip
1.13809ptp^{\sigma_{p}}_{{\bf{p}}_{p}}|\left[-i\int_{x^{0}_{in}}^{x^{0}_{out}}d^{4}x\,\mathscr{L}_{int}(x)\right]|n^{\sigma_{n}}_{{\bf{p}}_{n}}\rangle\,,$
(41)

where we have denoted by ${\bf{p}}_{n}$ ($\sigma_{n}$), ${\bf{p}}_{p}$
($\sigma_{p}$) and ${\bf{q}}$ ($s$) the momentum (helicity) of the neutron,
proton and electron, respectively, and utilized Pontecorvo state for the
outgoing (anti-)neutrino. To simplify the notation, we have omitted the time-
dependence of particle states. Clearly, in the standard scattering theory in
the interaction picture, the ingoing/outgoing states must be evaluated at
$x^{0}=x^{0}_{in}$ and $x^{0}=x^{0}_{out}$, respectively444As observed in CapW
, the amplitude for the case of the exact QFT neutrino states should be
consistently defined as
$\langle\psi_{\ell}(x^{0}_{out})|e^{-iH\left(x_{out}^{0}-x_{in}^{0}\right)}|\psi_{\ell}(x^{0}_{in})\rangle=\langle\psi_{\ell}(x^{0}_{in})|U_{I}(x^{0}_{out},x_{in}^{0})|\psi_{\ell}(x^{0}_{in})\rangle$,
due to the orthogonality of the Hilbert spaces at different times (see Sec.
II). This will be explicitly taken into account in Sec. IV..

Now, by plugging Eq. (40) into (41), the term involving the expectation value
of the quark up and down fields can be easily computed to give

$\displaystyle\langle
p^{\sigma_{p}}_{{\bf{p}}_{p}}|V_{ud}\,\bar{q}_{u}(x)\gamma_{\mu}\hskip
0.56905pt(f\,-\,\gamma^{5}g)\hskip
0.56905ptq_{d}(x)|n^{\sigma_{n}}_{{\bf{p}}_{n}}\rangle$ $\displaystyle\simeq$
$\displaystyle V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}$ (42)
$\displaystyle\times\,e^{-i(\omega_{{\bf{p}}_{p}}x_{out}^{0}-\omega_{{\bf{p}}_{n}}x^{0}_{in})}\,e^{-i\left[\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}\right)\hskip
0.56905ptx^{0}-\left({\bf{p}}_{n}-{\bf{p}}_{p}\right)\cdot{\bf{x}}\right]}\,,$

up to a normalization factor. Here we have denoted by $\omega_{{\bf{p}}_{n}}$
and $\omega_{{\bf{p}}_{p}}$ the energy of the neutron and proton,
respectively.

Concerning the lepton current, it is convenient to compute separately the
terms involving the electron and neutrino fields. For the former we have

$\langle
e^{s}_{\bf{q}}|\bar{e}(x)|0\rangle_{e}\,\simeq\,\bar{u}_{{\bf{q}},e}^{s}\,e^{-i{\bf{q}}\cdot{\bf{x}}}\,e^{-i\omega_{\bf{q}}^{e}(x^{0}_{out}-x^{0})}\,,$
(43)

where $\omega_{\bf{q}}^{e}$ is the electron energy. On the other hand, we get
for the neutrino field

$_{P}\langle\bar{\nu}^{r}_{{\bf{k}},e}|\nu_{e}(x)|0\rangle_{m}\,\,=\,\cos^{2}\theta\,_{P}\langle\bar{\nu}^{r}_{{\bf{k}},1}|\nu_{1}(x)|0\rangle_{m}\,+\,\sin^{2}\theta\,_{P}\langle\bar{\nu}^{r}_{{\bf{k}},2}|\nu_{2}(x)|0\rangle_{m}\,,$
(44)

where $|0\rangle_{m}$ is the vacuum for the massive neutrino fields
$\nu_{i}(x)$, $i=1,2$ (see Eq. (4)). Notice that we have used the mixing
transformations (1a) and (2a) for the neutrino state and field, respectively.

Equation (44) can be further manipulated to give

$_{P}\langle\bar{\nu}^{r}_{{\bf{k}},e}|\nu_{e}(x)|0\rangle_{m}\,\simeq\,e^{-i{\bf{k}}\cdot{\bf{x}}}\left[\cos^{2}\theta\,v^{r}_{{\bf{k}},1}\,e^{-i\omega_{{\bf{k}},1}\left(x^{0}_{out}-x^{0}\right)}\,+\,\sin^{2}\theta\,v^{r}_{{\bf{k}},2}\,e^{-i\omega_{{\bf{k}},2}\left(x^{0}_{out}-x^{0}\right)}\right],$
(45)

where we have still omitted the normalization.

Combining Eqs. (42), (43) and (45), the amplitude (41) becomes

$\displaystyle A^{a)}_{P}$ $\displaystyle=$ $\displaystyle
i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)e^{-i\omega_{\bf{q}}^{e}x^{0}_{out}}\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})$
(46)
$\displaystyle\times\int_{x_{in}^{0}}^{x_{out}^{0}}\,dx^{0}\left[\cos^{2}\theta\,v^{r}_{{\bf{k}},1}\,e^{-i\omega_{{\bf{k}},1}x^{0}_{out}}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},1}\right)x^{0}}\right.$
$\displaystyle\left.+\,\sin^{2}\theta\,v^{r}_{{\bf{k}},2}\,e^{-i\omega_{{\bf{k}},2}x^{0}_{out}}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},2}\right)x^{0}}\right]$
$\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,e^{-i(\omega_{{\bf{p}}_{p}}x_{out}^{0}-\omega_{{\bf{p}}_{n}}x^{0}_{in})}\,,$

where we have encompassed all the factors omitted above in the normalization
$\mathcal{N}=\left[{\sqrt{2}\left(2\pi\right)^{3}}\right]^{-1}$.

In a similar fashion, if we consider the decay channel $b)$ in Eq. (38), the
amplitude reads

$A^{b)}_{P}\,=\,_{P}\langle\bar{\nu}^{r}_{{\bf{k}},\mu},e^{s}_{{\bf{q}}},\hskip
1.13809ptp^{\sigma_{p}}_{{\bf{p}}_{p}}|\left[-i\int_{x^{0}_{in}}^{x^{0}_{out}}d^{4}x\,\mathscr{L}_{int}(x)\right]|n^{\sigma_{n}}_{{\bf{p}}_{n}}\rangle\,.$
(47)

In this case, the expectation value involving the neutrino field is

$_{P}\langle\bar{\nu}^{r}_{{\bf{k}},\mu}|\nu_{e}(x)|0\rangle_{m}\,\simeq\,e^{-i{\bf{k}}\cdot{\bf{x}}}\,\sin\theta\cos\theta\left[v^{r}_{{\bf{k}},2}\,e^{-i\omega_{{\bf{k}},2}\left(x^{0}_{out}-x^{0}\right)}\,-\,v^{r}_{{\bf{k}},1}\,e^{-i\omega_{{\bf{k}},1}\left(x^{0}_{out}-x^{0}\right)}\right],$
(48)

where we have resorted to the transformation (1b) for the state
$|\bar{\nu}_{{\bf{k}},\mu}\rangle_{P}$.

From Eq. (48), we get

$\displaystyle A^{b)}_{P}$ $\displaystyle=$ $\displaystyle
i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\sin\theta\cos\theta\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)e^{-i\omega_{\bf{q}}^{e}x^{0}_{out}}\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})$
(49)
$\displaystyle\times\int_{x_{in}^{0}}^{x_{out}^{0}}\,dx^{0}\,\left[v^{r}_{{\bf{k}},2}\,e^{-i\omega_{{\bf{k}},2}x^{0}_{out}}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},2}\right)x^{0}}\right.$
$\displaystyle\left.-\,v^{r}_{{\bf{k}},1}\,e^{-i\omega_{{\bf{k}},1}x^{0}_{out}}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},1}\right)x^{0}}\right]$
$\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,e^{-i(\omega_{{\bf{p}}_{p}}x_{out}^{0}-\omega_{{\bf{p}}_{n}}x^{0}_{in})}\,.$

For vanishing mixing (i.e. $\theta\rightarrow 0$), this expression is
identically zero, as expected in the absence of flavor oscillations.

###  III.2 Short-Time Limit

Let us now use Eqs. (46) and (49) to analyze lepton charge conservation in the
tree-level interaction vertex for the processes
$n\,\rightarrow\,p\,+\,e^{-}\,+\,\bar{\nu}_{e}$ and
$n\,\rightarrow\,p\,+\,e^{-}\,+\,\bar{\nu}_{\mu}$. For this purpose, we study
the structure of $A_{P}^{a)}$ and $A_{P}^{b)}$ in the short-time
approximation. We set the limits of integration as $x^{0}_{in}=-\Delta t/2$
and $x^{0}_{out}=\Delta t/2$, such that $\Delta t$ is sufficiently longer than
the neutron lifetime $\tau_{n}$ to ensure (on average) the neutron decay, but
shorter than the characteristic neutrino oscillation time $T_{osc}$, in
compliance with the discussion below Eq. (38). Notice that a similar
assumption has been considered in Nishi  in the context of the pion decay.

With reference to the process $a)$, the transition amplitude (46) becomes

$\displaystyle A^{a)}_{P}$ $\displaystyle=$ $\displaystyle
i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)e^{-i\omega_{\bf{q}}^{e}\Delta
t/2}\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})$ (50)
$\displaystyle\times\int_{-\Delta t/2}^{\Delta
t/2}\,dx^{0}\left[\cos^{2}\theta\,v^{r}_{{\bf{k}},1}\,e^{-i\omega_{{\bf{k}},1}\Delta
t/2}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},1}\right)x^{0}}\right.$
$\displaystyle\left.+\,\sin^{2}\theta\,v^{r}_{{\bf{k}},2}\,e^{-i\omega_{{\bf{k}},2}\Delta
t/2}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},2}\right)x^{0}}\right]$
$\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,e^{-i(\omega_{{\bf{p}}_{p}}+\omega_{{\bf{p}}_{n}})\Delta
t/2}\,.$

By integrating over $x^{0}$, we then get

$\displaystyle A^{a)}_{P}$ $\displaystyle=$ $\displaystyle
2i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)e^{-i\omega_{\bf{q}}^{e}\Delta
t/2}\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})$ (51)
$\displaystyle\times\left\\{\cos^{2}\theta\,v^{r}_{{\bf{k}},1}\,e^{-i\omega_{{\bf{k}},1}\Delta
t/2}\
\frac{\sin\left[\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},1}\right)\Delta
t/2\right]}{\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},1}}\right.$
$\displaystyle\left.\,+\sin^{2}\theta\,v^{r}_{{\bf{k}},2}\,e^{-i\omega_{{\bf{k}},2}\Delta
t/2}\,\frac{\sin\left[\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},2}\right)\Delta
t/2\right]}{\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},2}}\right\\}$
$\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,e^{-i(\omega_{{\bf{p}}_{p}}+\omega_{{\bf{p}}_{n}})\Delta
t/2}\,.$

A comment is now in order: in the limit of large $\Delta t$, the
$\sin(\tilde{\omega}\Delta t)/\tilde{\omega}$ factors in the above relation
become the usual energy-conserving Dirac $\delta$ functions. Here we have used
the shorthand notation
$\tilde{\omega}_{i}=\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},i}$,
$i=1,2$. On the other hand, the apparent energy fluctuations for finite time
intervals can be understood in terms of (and are constrained by) the time-
energy uncertainty relation for neutrino oscillations, with $\Delta t$ being
the interval defined above. This scenario has been rigorously analyzed in both
perturbative QM BilPhys  and QFT SmaldFl , where it has been shown that the
Mandelstam-Tamm version of time-energy uncertainty relations for neutrino
oscillations can be cast in the form of flavor-energy uncertainty relations.
In other terms, neutrino flavor charges obtained via Noether’s theorem and
energy turn out to be incompatible observables in the usual quantum mechanical
sense. Flavor-energy uncertainty relations have been recently studied also in
stationary curved spacetime BlasSma .

Let us now consider Eq. (51) in the short-time limit. Clearly, the dominant
contributions in this regime are those for which $\tilde{\omega}_{i}\approx
0$. At the leading order, we obtain

$\displaystyle A^{a)}_{P}$ $\displaystyle\simeq$ $\displaystyle
i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)\Delta
t\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})\,\left(\cos^{2}\theta\,v^{r}_{{\bf{k}},1}\,+\sin^{2}\theta\,v^{r}_{{\bf{k}},2}\right)$
(52) $\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,.$

Since observed neutrinos are relativistic, it is convenient to study the
result (52) in the relativistic limit. This approximation will also be useful
for a more direct comparison with the analysis of the next Section. In this
regard, we use the identity (31) and rewrite $A^{a)}_{P}$ in the form

$\displaystyle A^{a)}_{P}$ $\displaystyle\simeq$ $\displaystyle
i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)\Delta
t\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})\,\left\\{v^{r}_{{\bf{k}},1}\left[1-\sin^{2}\theta\left(1-|U_{\bf{k}}|\right)\right]-\sin^{2}\theta\,\epsilon^{r}\,u^{r}_{-{\bf{k}},1}\,|V_{\bf{k}}|\right\\}$
(53) $\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,.$

By means of Eqs. (36), we obtain to the first order in
$\mathcal{O}\left(\frac{\Delta m}{2|{\bf{k}}|}\right)$

$\displaystyle A^{a)}_{P}$ $\displaystyle\simeq$ $\displaystyle
i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)\Delta
t\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})\,\left(v^{r}_{{\bf{k}},1}\,-\,\sin^{2}\theta\,\frac{\Delta
m}{2|{\bf{k}}|}\epsilon^{r}\,u^{r}_{-{\bf{k}},1}\right)$ (54)
$\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,,$

which becomes in the ultarelativistic limit (i.e. for $\frac{\Delta
m}{|{\bf{k}}|}\rightarrow 0)$

$\displaystyle A^{a)}_{P}$ $\displaystyle\simeq$ $\displaystyle
i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)\Delta
t\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})\,v^{r}_{{\bf{k}},1}$
(55) $\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,.$

Notice that this expression exhibits the same structure as the amplitude for
the emission of a free (anti-)neutrino with mass $m_{1}$. Further discussion
on the meaning of this result will be given below.

We now consider the short-time approximation of the amplitude (49). Following
the same steps as above, we are led to

$\displaystyle A^{b)}_{P}$ $\displaystyle\simeq$ $\displaystyle
i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\sin\theta\cos\theta\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)\Delta
t\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})\left(v_{{\bf{k}},2}^{r}\,-\,v^{r}_{{\bf{k}},1}\right)$
(56) $\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,,$

which is evidently non-vanishing, since $v_{{\bf{k}},2}^{r}\neq
v^{r}_{{\bf{k}},1}$ for $m_{2}\neq m_{1}$.

Again, it is interesting to consider the relativistic limit. By using Eqs.
(31) and (36), we obtain after some algebra

$\displaystyle A^{b)}_{P}$ $\displaystyle\simeq$
$\displaystyle-i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\sin\theta\cos\theta\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)\Delta
t\,\frac{\Delta
m}{2|{\bf{k}}|}\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})\,\epsilon^{r}\,u^{r}_{-{\bf{k}},1}\,,$
(57) $\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,.$

It must be emphasized that Eq. (56) (or, equivalently, Eq. (57)) signals a
clear violation of lepton charge in the tree-level interaction vertex, as it
states that the flavor-violating process
$n\,\rightarrow\,p\,+\,e^{-}\,+\,\bar{\nu}_{\mu}$ has a nonzero probability
even for intervals much shorter than the characteristic neutrino oscillation
time. In the next Section, we will show that the origin of this flaw can be
traced back to the incorrect definition of Pontecorvo states as eigenstates of
the flavor charge in QFT. As further evidence of this, from Eq. (57) we see
that the correct vanishing amplitude is recovered in the ultra-relativistic
limit $\frac{\Delta m}{|{\bf{k}}|}\rightarrow 0$, where, in fact, Pontecorvo
states well-approximate exact QFT states (see the discussion below Eqs. (35)).

###  III.3 Mass states

Proceeding in the same way as done in the previous subsection, we now compute
the neutron $\beta$-decay amplitude by using the mass eigenstates
$|\nu^{r}_{{\bf{k}},i}\rangle$, $i=1,2$, as fundamental representation for
neutrinos. This amounts to consider the two processes

$n\,\rightarrow\,p\,+\,e^{-}\,+\,\bar{\nu}_{i}\,,\qquad i=1,2\,.$ (58)

In this context, we observe that Eq. (41) (or, equivalently, Eq. (47)) must be
rewritten as

$A_{i}=\langle\bar{\nu}^{r}_{{\bf{k}},i},e^{s}_{{\bf{q}}},\hskip
1.13809ptp^{\sigma_{p}}_{{\bf{p}}_{p}}|\left[-i\int_{x^{0}_{in}}^{x^{0}_{out}}d^{4}x\,\mathscr{L}_{int}(x)\right]|n^{\sigma_{n}}_{{\bf{p}}_{n}}\rangle\,,\qquad
i=1,2\,,$ (59)

where the subscript $i$ refers to the $i^{th}$ mass state.

The amplitude (59) can be straightforwardly derived from Eq. (46) by using the
transformation (1) to counter-rotate the expectation value (45) in the mass
basis. In doing so, we obtain

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)e^{-i\left(\omega_{\bf{q}}^{e}\,+\,\omega_{{\bf{k}},1}\right)x^{0}_{out}}\cos\theta\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})$
(60)
$\displaystyle\times\int_{x_{in}^{0}}^{x_{out}^{0}}\,dx^{0}\,v^{r}_{{\bf{k}},1}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},1}\right)x^{0}}$
$\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,e^{-i(\omega_{{\bf{p}}_{p}}x_{out}^{0}-\omega_{{\bf{p}}_{n}}x^{0}_{in})}\,,$

which becomes in the short-time limit

$\displaystyle A_{1}$ $\displaystyle\simeq$ $\displaystyle
i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)\Delta
t\,\cos\theta\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})\,v^{r}_{{\bf{k}},1}$
(61) $\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,.$

This is predictably consistent with Eq. (55) derived in the ultra-relativistic
limit, where the effects of the mass difference can in fact be neglected.

Similarly, we infer the following expression for $A_{2}$

$\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)e^{-i\left(\omega_{\bf{q}}^{e}\,+\,\omega_{{\bf{k}},2}\right)x^{0}_{out}}\sin\theta\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})$
(62)
$\displaystyle\times\int_{x_{in}^{0}}^{x_{out}^{0}}\,dx^{0}\,v^{r}_{{\bf{k}},2}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},2}\right)x^{0}}$
$\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,e^{-i(\omega_{{\bf{p}}_{p}}x_{out}^{0}-\omega_{{\bf{p}}_{n}}x^{0}_{in})}\,.$

In the short-time approximation this yields

$\displaystyle A_{2}$ $\displaystyle\simeq$ $\displaystyle
i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)\Delta
t\,\sin\theta\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})\,v^{r}_{{\bf{k}},2}$
(63) $\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,.$

To make a comparison with existing literature, it is convenient to consider
for a while the decay rates for the two channels in Eq. (58), which are
obtained by squaring the amplitudes $A_{i}$, $i=1,2$. Then, the approach is to
compute the total decay rate for the processes (37) and (38) through an
incoherent sum of the contributions from the different mass eigenstates
GiuntiKim ; suminc , each averaged by $\cos^{2}\theta$ or $\sin^{2}\theta$,
depending on whether one refers to the emission of an electron or muon
(anti-)neutrino555Notice that neutrinos can be produced incoherently if their
mass differences are larger than the energy uncertainty in the production
process. This is quite plausible for heavy neutrinos ($m\gtrsim
100\,\mathrm{KeV}$ for typical energy of $10\,\mathrm{MeV}$). However, the
present analysis is presumed to have a more general applicability than this
specific case.. Following this recipe and resorting to Eqs. (61) and (63), it
is immediate to see that also in this framework the decay probability for the
process (38) is non-vanishing in the short-time limit, since it is given by
the sum of two positive contributions. As already discussed for Pontecorvo
states, this outcome is in contrast with what expected at tree level in the
SM, which means that mass states are not eligible to be the fundamental
representation for mixed neutrinos, if we want to preserve the internal
consistency of the theory.

##  IV Neutron $\beta$-decay with exact QFT flavor states

Let us now go through the above calculations by using exact QFT neutrino
states. Toward this end, we remind that these states are defined as
eigenstates of flavor charges operators BV95 . Indeed, by using the definition
(35), it is straightforward to check that BV95

$\displaystyle::Q_{\nu_{e}}(0)::|\nu^{r}_{\;{\bf
k},e}\rangle\,=\,|\nu^{r}_{\;{{\bf{k}}},e}\rangle\,,\qquad::Q_{\nu_{\mu}}(0)::|\nu^{r}_{\;{\bf
k},\mu}\rangle\,=\,|\nu^{r}_{\;{{\bf{k}}},\mu}\rangle\,,$ (64)
$\displaystyle::Q_{\nu_{e}}(0)::|\nu^{r}_{\;{\bf
k},\mu}\rangle\,=\;::Q_{\nu_{\mu}}(0)::|\nu^{r}_{\;{{\bf{k}}},e}\rangle\,=\,0\,,\qquad::Q_{\nu_{\ell}}(0)::|0\rangle_{f}\,=\,0\,,$
(65)

where

$Q_{\nu_{\ell}}(x^{0})\,\equiv\,\int
d^{3}{{\bf{x}}}\,\nu_{\ell}^{\dagger}(x)\hskip
0.85358pt\nu_{\ell}(x)\,,\qquad\ell=e,\mu\,,$ (66)

are the flavor charge operators and $::Q_{\nu_{\ell}}::$ denotes the normal
ordering respect to the flavor vacuum. By contrast, it must be emphasized that
Pontecorvo states (and, a fortiori, mass states) are not eigenstates of the
flavor charges Nucl .

In this framework, the decay amplitude (41) becomes

$A^{a)}\,=\,\langle\bar{\nu}^{r}_{{\bf{k}},e},e^{s}_{{\bf{q}}},\hskip
1.13809ptp^{\sigma_{p}}_{{\bf{p}}_{p}}|\left[-i\int_{x^{0}_{in}}^{x^{0}_{out}}d^{4}x\,\mathscr{L}_{int}(x)\right]|n^{\sigma_{n}}_{{\bf{p}}_{n}}\rangle\,.$
(67)

The term involving the expectation value of the neutrino field given in Eq.
(44) is now replaced by

$\displaystyle\langle\bar{\nu}^{r}_{{\bf{k}},e}(x^{0}_{in})|\nu_{e}(x)|0(x^{0}_{in})\rangle_{f}$
$\displaystyle\simeq$ $\displaystyle
e^{-i{\bf{k}}\cdot{\bf{x}}}\left\\{\cos^{2}\theta\,v_{{\bf{k}},1}^{r}\,e^{i\omega_{{\bf{k}},1}(x^{0}-x^{0}_{in})}\right.$
(68)
$\displaystyle\,+\sin^{2}\theta\left[\left(v^{r}_{{\bf{k}},1}\,|U_{\bf{k}}|\,-\,\epsilon^{r}\,u^{r}_{-{\bf{k}},1}\,|V_{\bf{k}}|\right)|U_{\bf{k}}|\,e^{i\omega_{{\bf{k}},2}(x^{0}-x^{0}_{in})}\right.$
$\displaystyle\left.\left.+\left(v^{r}_{{\bf{k}},1}\,|V_{{\bf{k}}}|\,+\,\epsilon^{r}\,u^{r}_{-{\bf{k}},1}\,|U_{{\bf{k}}}|\right)|V_{{\bf{k}}}|\,e^{-i\omega_{{\bf{k}},2}(x^{0}-x^{0}_{in})}\right]\right\\},$

which is defined consistently with the orthogonality of the Hilbert spaces at
different times, see footnote 3 (notice the presence of the flavor vacuum and
the Bogoliubov coefficients $U_{\bf{k}}$ and $V_{\bf{k}}$). Here we have used
the field expansion (17) and the explicit form of the flavor
annihilation/creation operators given in Eqs. (22)-(25). As above,
normalization will be taken into account only at the end of calculations.

By means of the identities (31) and (32), Eq. (68) can be rewritten as

$\displaystyle\langle\bar{\nu}^{r}_{{\bf{k}},e}(x^{0}_{in})|\nu_{e}(x)|0(x^{0}_{in})\rangle_{f}$
$\displaystyle\simeq$ $\displaystyle
e^{-i{\bf{k}}\cdot{\bf{x}}}\left\\{\cos^{2}\theta\,v_{{\bf{k}},1}^{r}\,e^{i\omega_{{\bf{k}},1}(x^{0}-x^{0}_{in})}\right.$
(69)
$\displaystyle\left.+\sin^{2}\theta\left[v^{r}_{{\bf{k}},2}\,|U_{\bf{k}}|\,e^{i\omega_{{\bf{k}},2}(x^{0}-x^{0}_{in})}\,+\,\epsilon^{r}\,u^{r}_{-{\bf{k}},2}\,|V_{{\bf{k}}}|\,\,e^{-i\omega_{{\bf{k}},2}(x^{0}-x^{0}_{in})}\right]\right\\}.$

Combining Eqs. (42), (43) and (69), the decay amplitude (67) takes the form

$\displaystyle A^{a)}$ $\displaystyle=$ $\displaystyle i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)e^{-i\omega_{\bf{q}}^{e}x^{0}_{out}}\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})$
(70)
$\displaystyle\times\int_{x^{0}_{in}}^{x^{0}_{out}}\,dx^{0}\left\\{\cos^{2}\theta\,v^{r}_{{\bf{k}},1}\,e^{-i\omega_{{\bf{k}},1}x^{0}_{in}}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},1}\right)x^{0}}\right.$
$\displaystyle\,+\sin^{2}\theta\left[v^{r}_{{\bf{k}},2}\,|U_{{\bf{k}}}|\,\,e^{-i\omega_{{\bf{k}},2}x^{0}_{in}}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},2}\right)x^{0}}\right.$
$\displaystyle\left.\left.+\,\epsilon^{r}\,u^{r}_{-{\bf{k}},2}\,|V_{{\bf{k}}}|\,e^{i\omega_{{\bf{k}},2}x^{0}_{in}}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}+\omega_{{\bf{k}},2}\right)x^{0}}\right]\right\\}$
$\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,e^{-i(\omega_{{\bf{p}}_{p}}x_{out}^{0}-\omega_{{\bf{p}}_{n}}x^{0}_{in})}\,.$

The structure of this amplitude is clearly different from the one in Eq. (46),
due to the presence of the Bogoliubov coefficients and the extra contribute of
the anti-particle degrees of freedom (the term multiplying the $u$-neutrino
mode)666The negative neutrino energy $\omega_{{\bf{k}},2}$ in the fourth line
of Eq. (70) should not be misleading, since it is associated to the “hole”
contributions in the flavor vacuum condensate. Of course, this is a richness
of the QFT treatment of mixing, which arises from the peculiar structure of
flavor vacuum (see the discussion in the Introduction).. Regardless of
irrelevant phase factors, Eq. (46) is however recovered in the ultra-
relativistic limit, where $|U_{\bf{k}}|\rightarrow 1$,
$|V_{\bf{k}}|\rightarrow 0$ and exact flavor states reduce to the standard
Pontecorvo states. We show below that the presence of such additional term in
Eq. (70) is essential to restore the conservation of lepton charge in the
short-time limit and validate the use of exact QFT flavor states as correct
neutrino representation.

On the other hand, if we consider the decay channel in Eq. (38), the amplitude
(47) becomes

$A^{b)}\,=\,\langle\bar{\nu}^{r}_{{\bf{k}},\mu},e^{s}_{{\bf{q}}},\hskip
1.13809ptp^{\sigma_{p}}_{{\bf{p}}_{p}}|\left[-i\int_{x^{0}_{in}}^{x^{0}_{out}}d^{4}x\,\mathscr{L}_{int}(x)\right]|n^{\sigma_{n}}_{{\bf{p}}_{n}}\rangle\,,$
(71)

where the expectation value involving the neutrino field is now given by

$\displaystyle\langle\bar{\nu}^{r}_{{\bf{k}},\mu}(x^{0}_{in})|\nu_{e}(x)|0(x^{0}_{in})\rangle_{f}$
$\displaystyle\simeq$ $\displaystyle
e^{-i{\bf{k}}\cdot{\bf{x}}}\,\sin\theta\cos\theta\left[\left(v^{r}_{{\bf{k}},1}\,|U_{\bf{k}}|\,-\,\epsilon^{r}\,u^{r}_{-{\bf{k}},1}\,|V_{\bf{k}}|\right)\,e^{i\omega_{{\bf{k}},2}(x^{0}-x^{0}_{in})}\right.$
(72)
$\displaystyle\left.-\,v^{r}_{{\bf{k}},1}\,|U_{{\bf{k}}}|\,e^{i\omega_{{\bf{k}},1}(x^{0}-x^{0}_{in})}\,+\,\epsilon^{r}\,u^{r}_{-{\bf{k}},1}\,|V_{{\bf{k}}}|\,e^{-i\omega_{{\bf{k}},1}(x^{0}-x^{0}_{in})}\right].$

This can be simplified by use of the relation (31) to give

$\displaystyle\langle\bar{\nu}^{r}_{{\bf{k}},\mu}(x^{0}_{in})|\nu_{e}(x)|0(x^{0}_{in})\rangle_{f}$
$\displaystyle\simeq$ $\displaystyle
e^{-i{\bf{k}}\cdot{\bf{x}}}\,\sin\theta\cos\theta\left[v^{r}_{{\bf{k}},2}\,e^{i\omega_{{\bf{k}},2}(x^{0}-x^{0}_{in})}\right.$
(73)
$\displaystyle\left.-\,v^{r}_{{\bf{k}},1}\,|U_{{\bf{k}}}|\,e^{i\omega_{{\bf{k}},1}(x^{0}-x^{0}_{in})}\,+\,\epsilon^{r}\,u^{r}_{-{\bf{k}},1}\,|V_{{\bf{k}}}|\,e^{-i\omega_{{\bf{k}},1}(x^{0}-x^{0}_{in})}\right].$

The amplitude (71) is then equal to

$\displaystyle A^{b)}$ $\displaystyle=$ $\displaystyle i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\sin\theta\cos\theta\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)e^{-i\omega_{\bf{q}}^{e}x^{0}_{out}}\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})$
(74)
$\displaystyle\times\int_{x^{0}_{in}}^{x^{0}_{out}}\,dx^{0}\,\left[v^{r}_{{\bf{k}},2}\,e^{-i\omega_{{\bf{k}},2}x^{0}_{in}}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},2}\right)x^{0}}\right.$
$\displaystyle-\,v^{r}_{{\bf{k}},1}\,|U_{{\bf{k}}}|\,e^{-i\omega_{{\bf{k}},1}x^{0}_{in}}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},1}\right)x^{0}}$
$\displaystyle\left.+\,\epsilon^{r}\,u^{r}_{-{\bf{k}},1}\,|V_{{\bf{k}}}|\,e^{i\omega_{{\bf{k}},1}x^{0}_{in}}\,e^{-i\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}+\omega_{{\bf{k}},1}\right)x^{0}}\right]$
$\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,e^{-i(\omega_{{\bf{p}}_{p}}x_{out}^{0}-\omega_{{\bf{p}}_{n}}x^{0}_{in})}\,.$

Again, the ultra-relativistic limit gives back Eq. (49) up to an irrelevant
phase factor.

###  IV.1 Short-Time Limit

We now consider the short-time limit of the amplitudes (70) and (74). Toward
this end, we follow the same recipe as in Sec. III.2 and set
$x^{0}_{in}=-\Delta t/2$ and $x^{0}_{out}=\Delta t/2$. By integrating over
$x^{0}$, we get for $A^{a)}$

$\displaystyle A^{a)}$ $\displaystyle=$ $\displaystyle 2i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)e^{-i\omega_{\bf{q}}^{e}\,\Delta
t/2}\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})$ (75)
$\displaystyle\times\left\\{\cos^{2}\theta\,v^{r}_{{\bf{k}},1}\,e^{i\omega_{{\bf{k}},1}\Delta
t/2}\
\frac{\sin\left[\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},1}\right)\Delta
t/2\right]}{\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},1}}\right.$
$\displaystyle+\,\sin^{2}\theta\left[v^{r}_{{\bf{k}},2}\,|U_{\bf{k}}|\,e^{i\omega_{{\bf{k}},2}\Delta
t/2}\
\frac{\sin\left[\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},2}\right)\Delta
t/2\right]}{\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}-\omega_{{\bf{k}},2}}\right.$
$\displaystyle\left.\left.+\,\epsilon^{r}\,u^{r}_{-{\bf{k}},2}\,|V_{{\bf{k}}}|\,e^{-i\omega_{{\bf{k}},2}\Delta
t/2}\
\frac{\sin\left[\left(\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}+\omega_{{\bf{k}},2}\right)\Delta
t/2\right]}{\omega_{{\bf{p}}_{n}}-\omega_{{\bf{p}}_{p}}-\omega_{{\bf{q}}}^{e}+\omega_{{\bf{k}},2}}\right]\right\\}$
$\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,e^{-i(\omega_{{\bf{p}}_{p}}+\omega_{{\bf{p}}_{n}})\Delta
t/2}\,.$

In the short-time approximation, this yields to the first order

$\displaystyle A^{a)}$ $\displaystyle\simeq$ $\displaystyle
i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)\Delta
t\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})\,\left\\{\cos^{2}\theta\,v^{r}_{{\bf{k}},1}\,+\,\sin^{2}\theta\left[v^{r}_{{\bf{k}},2}\,|U_{\bf{k}}|\,+\,\epsilon^{r}\,u^{r}_{-{\bf{k}},2}\,|V_{{\bf{k}}}|\right]\right\\}$
(76) $\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,,$

which can be further manipulated by means of the identity (34) to give

$\displaystyle A^{a)}$ $\displaystyle\simeq$ $\displaystyle
i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)\Delta
t\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})\,v^{r}_{{\bf{k}},1}$
(77) $\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,.$

As remarked above, this correctly matches the amplitude (55) calculated with
Pontecorvo states in the ultra-relativistic approximation.

Let us now turn to the process (38). In the short-time limit, the amplitude
(74) takes the form

$\displaystyle A^{b)}$ $\displaystyle\simeq$ $\displaystyle
i\,\mathcal{N}\hskip
0.28453ptG_{F}\,\sin\theta\cos\theta\,\delta^{3}\left({\bf{p}}_{n}-{\bf{p}}_{p}-{\bf{q}}-{\bf{k}}\right)\Delta
t\,\bar{u}^{s}_{{\bf{q}},e}\,\gamma^{\mu}(\mathds{1}-\gamma^{5})\left[v^{r}_{{\bf{k}},2}\,-\,v^{r}_{{\bf{k}},1}\,|U_{{\bf{k}}}|\,+\,\epsilon^{r}\,u^{r}_{-{\bf{k}},1}\,|V_{{\bf{k}}}|\right]$
(78) $\displaystyle\times\,V_{ud}\,\bar{u}_{{\bf{p}}_{p}}^{\sigma_{p}}\hskip
0.85358pt\gamma_{\mu}(f\,-\,\gamma^{5}g)\hskip
0.85358ptu_{{\bf{p}}_{n}}^{\sigma_{n}}\,.$

By comparison with Eq. (56), we see that the main difference is now the
splitting of the mode with mass $m_{1}$ into two contributions multiplying the
Bogoliubov coefficients $|U_{\bf{k}}|$ and $|V_{\bf{k}}|$, respectively. While
the former survives in the ultra-relativistic limit (returning Eq. (56)), the
latter has no counterpart in Pontecorvo framework, as it vanishes for
$|V_{\bf{k}}|\rightarrow 0$. However, it is exactly this term which allows to
get consistency with expectations from the SM at tree level. Indeed, due to
the identity (31), it is immediate to realize that the quantity in the square
brackets is identically zero, which gives

$A^{b)}=0\,.$ (79)

This proves that the usage of the exact flavor states leads to the
conservation of lepton charge in the production vertex.

In order to better understand the above results, we observe that the
amplitudes $A^{a)}$ and $A^{b)}$ in the short-time limit provide information
on the decay processes in proximity of the interaction vertex. We can think of
associating a wave function $v^{r}_{{\bf{k}},\nu_{e}}$ to the emitted electron
(anti-)neutrino in Eqs. (52) and (77) (and, similarly, in Eqs. (56) and (78)).
For the case of exact flavor states, Eq. (77), one can identify
$v^{r}_{{\bf{k}},\nu_{e}}=v^{r}_{{\bf{k}},1}$, with
$v^{r\dagger}_{{\bf{k}},1}\,v^{r}_{{\bf{k}},1}=1$ being properly normalized
(see Eq. (7)). By contrast, Eq. (52) suggests that
$v^{r}_{{\bf{k}}\,\nu_{e}}=\cos^{2}\theta\,v^{r}_{{\bf{k}},1}\,+\,\sin^{2}\theta\,v^{r}_{{\bf{k}},2}$
for Pontecorvo states. However, by using Eqs. (7) and (26), simple
calculations show that

$\displaystyle v^{r\dagger}_{{\bf{k}}\,\nu_{e}}\,v^{r}_{{\bf{k}}\,\nu_{e}}$
$\displaystyle=$
$\displaystyle(\cos^{2}\theta\,v^{r\dagger}_{{\bf{k}},1}\,+\,\sin^{2}\theta\,v^{r\dagger}_{{\bf{k}},2})\,(\cos^{2}\theta\,v^{r}_{{\bf{k}},1}\,+\,\sin^{2}\theta\,v^{r}_{{\bf{k}},2})$
(80) $\displaystyle=$
$\displaystyle\cos^{4}\theta\,+\,\sin^{4}\theta\,+\,2\sin^{2}\theta\,\cos^{2}\theta\,|U_{\bf{k}}|\,,$

from which we infer that the above wave function is not normalized, since
$|U_{{\bf{k}}}|<1$ for $m_{1}\neq m_{2}$.

On the other hand, we notice that Eq. (56) includes the wave function
$v^{r}_{{\bf{k}},\Delta_{\nu_{e}}}=\sin\theta\cos\theta(v^{r}_{{\bf{k}},2}-v^{r}_{{\bf{k}},1})$.
Not even this combination is normalized to unity, since

$\displaystyle
v^{r\dagger}_{{\bf{k}},\Delta_{\nu_{e}}}v^{r}_{{\bf{k}},\Delta_{\nu_{e}}}$
$\displaystyle=$
$\displaystyle\sin^{2}\theta\cos^{2}\theta(v^{r\dagger}_{{\bf{k}},2}-v^{r\dagger}_{{\bf{k}},1})\,(v^{r}_{{\bf{k}},2}-v^{r}_{{\bf{k}},1})$
(81) $\displaystyle=$ $\displaystyle
2\sin^{2}\theta\cos^{2}\theta\left(1-|U_{{\bf{k}}}|\right)\,.$

However, this is exactly the missing contribution to restore the normalization
of the wave function (80). Indeed, we have

$v^{r\dagger}_{{\bf{k}}\,\nu_{e}}\,v^{r}_{{\bf{k}}\,\nu_{e}}\,+\,v^{r\dagger}_{{\bf{k}},\Delta_{\nu_{e}}}v^{r}_{{\bf{k}},\Delta_{\nu_{e}}}\,=\,\cos^{4}\theta\,+\,\sin^{4}\theta\,+\,2\sin^{2}\theta\,\cos^{2}\theta\,|U_{\bf{k}}|\,+\,2\sin^{2}\theta\cos^{2}\theta\left(1-|U_{{\bf{k}}}|\right)\,=\,1.$
(82)

A similar reasoning can be developed for mass states. In this case
$v^{r}_{{\bf{k}}\,\nu_{e}}=\cos\theta\,v^{r}_{{\bf{k}}\,1}$ or
$v^{r}_{{\bf{k}}\,\nu_{e}}=\sin\theta\,v^{r}_{{\bf{k}}\,2}$, depending on
whether one considers Eq. (61) or (63). Again, such wave functions are not
normalized if taken separately. Nevertheless, the following condition holds

$\cos\theta^{2}\,v^{r\dagger}_{{\bf{k}}\,1}v^{r}_{{\bf{k}}\,1}\,+\,\sin\theta^{2}\,v^{r\dagger}_{{\bf{k}}\,2}v^{r}_{{\bf{k}}\,2}\,=\,1\,.$
(83)

The above arguments clearly show that neither Pontecorvo nor mass states
provide a self-consistent description of the neutron decay with neutrino
mixing, as they are not exact eigenstates of the flavor charge. However, the
reason why the sum over the two flavors (Eq. (82)) or the two masses (Eq.
(83)) allows to recover the correct result can be understood by considering
the time-dependent flavor charges (66) and the corresponding conserved charges
for neutrinos with definite masses

$Q_{\nu_{i}}\,\equiv\,\int d^{3}{{\bf{x}}}\,\nu_{i}^{\dagger}(x)\hskip
0.85358pt\nu_{i}(x)\,\,,\qquad i=1,2\,.$ (84)

In BJV  it has been proved that

$\sum_{i}Q_{i}\,=\,\sum_{\ell}Q_{\ell}(t)\,=\,Q\,,$ (85)

where $Q$ represents the total charge of the system. The above equality can be
interpreted as the conservation of the _total lepton number_ : when there is
no mixing, this number is given by the sum of two separately conserved family
lepton charges (left-hand side). On the other hand, in the presence of mixing
the same conserved number is obtained by adding time-dependent flavor charges,
which are indeed associated to neutrino oscillations (central term).

In conclusion, we remark that our analysis is focused on the short-time limit
of the $\beta$-decay amplitude. Clearly, a comprehensive study should also
take into account the opposite $\Delta t\rightarrow\infty$ limit. This has
been investigated in CYL , showing that the usage of exact QFT states leads to
predictions which are consistent with the SM model for the case of
relativistic neutrinos.

##  V Discussion and Conclusions

In this work we have analyzed the flavor/mass dichotomy for mixed neutrino
fields. We have considered the neutron $\beta$-decay
$n\,\rightarrow\,p\,+\,e^{-}\,+\,\bar{\nu}_{e}$ and the corresponding flavor-
violating channel $n\,\rightarrow\,p\,+\,e^{-}\,+\,\bar{\nu}_{\mu}$ as a test
bench. For these processes, we have explicitly computed the transition
amplitude by using the scattering theory at tree level. Special focus has been
devoted to the study of the short-time limit, i.e. the limit for very small
distances from the interaction vertex. We have developed calculations by
resorting to the following different representations of neutrino states: _i_)
Pontecorvo states, _ii_) mass states and _iii_) exact QFT flavor states, which
are defined as eigenstates of the flavor charge. We have found that the usage
of the latter representation leads to results consistent with the conservation
of lepton charge in the vertex, whereas Pontecorvo and mass states fail. This
provides a solid argument in favor of exact QFT states as neutrino fundamental
representation. We stress that such a result has been achieved based on the
sole requirement of consistency with SM at tree level. In other terms, our
formalism does not require any detector-dependent model for the emission and
absorption of neutrinos, nor a particular empirical description of flavor
oscillations MatsasDet .

In the above treatment, we have not calculated explicitly the neutron decay
width. This should be taken into account for a more direct comparison of our
results with existing literature. Nevertheless, the fact that the amplitude
$A_{n\rightarrow p+e^{-}+\bar{\nu}_{\mu}}$ computed with the exact QFT flavor
states identically vanishes in the short-time limit is enough to infer that
the decay width vanishes as well. By contrast, this does not happen when
working with Pontecorvo states or, a fortiori, with mass states, for which the
amplitude is in general nonzero. This outcome reveals unambiguously that both
Pontecorvo and mass representations are at odds with SM predictions.

Further aspects remain to be addressed in order to improve the above analysis.
Strictly speaking, we should have carried out calculations with three neutrino
generations, including the possibility of $CP$ violation. Likewise, a
description based on wave-packets would be more appropriate to take into
account finite spatial localization of neutrino production and detection.
However, we expect that such generalizations do not affect the overall
validity of our results. Furthermore, we have shown that the amplitudes
computed with exact QFT and Pontecorvo states are in good agreement in the
ultra-relativistic limit. Since observed neutrinos are essentially ultra-
relativistic, it is quite difficult to find experimental evidences that help
to discern between the two representations at present. In spite of this,
valuable hints could be provided by the PTOLEMY experiment Betts , which aims
at detecting neutrinos of the cosmic background (CNB) through capture on
tritium. Given that the temperature of the CNB is estimated around $T\approx
2\,\mathrm{K}$, it is reasonable to expect that these particles are mostly
non-relativistic, thus offering a promising phenomenological window on the
issue.

Finally, relevant advances in our understanding of neutrino mixing and
oscillations in QFT might be achieved via the study of these phenomena in non-
trivial spacetime. In this context, an emblematic research line is represented
by the analysis of the accelerated proton decay with in Rindler metric Ahluw ;
ProtBlas ; ProtMat ; GaetanoLucia . Work along this direction is presently
under active investigation and will be elaborated in future works.

###### Acknowledgements.

The author is grateful to Massimo Blasone for useful discussions. He also
acknowledges the Spanish “Ministerio de Universidades” for the awarded Maria
Zambrano fellowship and funding received from the European Union -
NextGenerationEU. He is grateful for participation in the COST Association
Action CA18108 “Quantum Gravity Phenomenology in the Multimessenger Approach”
and LISA Cosmology Working group.

## References

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