# A study of the relativistic corrections to tritium $\beta$-decay

Gianluca Cavoto<EMAIL_ADDRESS>Dipartimento di Fisica, Sapienza
Università di Roma, Piazzale Aldo Moro 2, I-00185 Rome, Italy  INFN Sezione di
Roma, Piazzale Aldo Moro 2, I-00185 Rome, Italy  Angelo Esposito
<EMAIL_ADDRESS>Dipartimento di Fisica, Sapienza Università di
Roma, Piazzale Aldo Moro 2, I-00185 Rome, Italy  INFN Sezione di Roma,
Piazzale Aldo Moro 2, I-00185 Rome, Italy  School of Natural Sciences,
Institute for Advanced Study, Princeton, NJ 08540, USA  Guglielmo Papiri
<EMAIL_ADDRESS>Dipartimento di Fisica, Sapienza Università di Roma,
Piazzale Aldo Moro 2, I-00185 Rome, Italy  Department of Physics, Cornell
University, Ithaca, NY 14853, USA  Antonio D. Polosa
<EMAIL_ADDRESS>Dipartimento di Fisica, Sapienza Università
di Roma, Piazzale Aldo Moro 2, I-00185 Rome, Italy  INFN Sezione di Roma,
Piazzale Aldo Moro 2, I-00185 Rome, Italy

###### Abstract

Forthcoming experiments like Project8 and Ptolemy aim at investigating with
high precision the end-point of the tritium $\beta$-decay spectrum sensitive
to the neutrino mass. In light of this, using the standard parametrization in
terms of nuclear polar form factors, we analyze the complete relativistic
expression for the spectrum of the $\beta$-electron emitted by a tritium
nucleus. Given the small parameters in the problem, we discuss the
approximations that can be made, and present the first two corrections to the
standard lowest order formula, improving on the few results available in the
literature.

##  I Introduction

The existence of a non-zero neutrino mass is now well established [1, 2, 3, 4,
5, 6], and provides a compelling evidence for physics beyond the Standard
Model. The differences between the squared masses of the three neutrinos is
precisely measured [7, 8, 9], whereas the value of their absolute mass scale,
their mass ordering, as well as their Dirac or Majorana nature are still
unknown.

To this end, a number of existing and future experiments aim at determining
the lightest neutrino mass. Specifically, this can be done by studying the
spectrum of the electron emitted in $\beta$-decay close to its maximum allowed
energy, the so-called end-point. Among these, the existing KATRIN [10, 11, 12]
and the forthcoming Project 8 [13, 14, 15] and Ptolemy [16, 17, 18]
experiments focus on the decay of tritium:

$\displaystyle{{}^{3}\text{H}}\to{{}^{3}\text{He}^{+}}+e^{-}+\bar{\nu}_{e}\,.$
(1)

The current best laboratory bound on the effective neutrino mass,
$m_{\nu}^{2}\equiv\sum_{i}|U_{ei}|^{2}m_{i}^{2}$, is given by $m_{\nu}\leq
0.8$ eV, as set by KATRIN [11, 12]. Here $m_{i}$ are the neutrino mass
eigenvalues and $U_{ei}$ the entries of the PMNS matrix [e.g., 19]. Ptolemy
aims at the largest event rate, thanks to a high loading of tritium on,
possibly, a flat graphene substrate, as well as an energy resolution as small
as $100$ meV [17, 18]. In light of this, it is important to determine the
expected $\beta$-decay spectrum with a theory uncertainty that is better than
the experimental precision. In this work we analyze the relativistic
corrections affecting the kinematics and nuclear matrix element. Such a
calculation has already been discussed in some previous works [20, 21, 22],
but the corrected expression for the spectrum was never reported completely
and without specific assumptions. Moreover, in light of the increasing
experimental precision, it is crucial to systematically understand the
corrections to the lowest order formula, which is usually employed. To the
best of our knowledge, a discussion of this sort only appeared in [22].
However, we find it to be incorrect and leading to an overestimate of the
corrections. In this work we solve this problem. We also show that, given the
expected event rate in Ptolemy, the statistical uncertainty is far smaller
than the theoretical error associated to the standard lowest order formula,
which thus needs to be improved. Specifically, we argue that, for neutrino
masses $m_{\nu}\gtrsim 100$ meV, the leading and next-to-leading order
formulae have different shapes, potentially distinguishable by an experiment
like Ptolemy.

##  II Review of the calculation

In this work we follow the approach outlined in [22], where the relativistic
matrix element is obtained from an hadronic model for the tritium nucleus
[23]:

$\displaystyle\begin{split}\mathcal{M}={}&\frac{G_{F}V_{ud}}{\sqrt{2}}\,\bar{u}(P_{e})\gamma_{\mu}(1-\gamma_{5})v(P_{\nu})\\\
&\times\bar{u}(P_{f})\bigg{[}G_{V}\big{(}q^{2}\big{)}\gamma^{\mu}+i\frac{G_{M}\big{(}q^{2}\big{)}}{2M_{i}}\sigma^{\mu\nu}q_{\nu}\\\
&\quad-
G_{A}\big{(}q^{2}\big{)}\gamma^{\mu}\gamma_{5}-G_{P}\big{(}q^{2}\big{)}q^{\mu}\gamma_{5}\bigg{]}u(P_{i})\,,\end{split}$
(2)

where $G_{F}$ is the Fermi constant and $V_{ud}$ the entry of the CKM matrix.
Moreover, $P_{i}$ and $P_{f}$ are the 4-momenta of ${{}^{3}\text{H}}$ and
${{}^{3}\text{He}^{+}}$ (with masses $M_{i}$ and $M_{f}$), $P_{e}$ and
$P_{\nu}$ are those of the electron and antineutrino, and $q\equiv
P_{f}-P_{i}$ is the momentum transferred to the nucleus. The form factors are
[22]

$\displaystyle\begin{split}&G_{V}=\frac{g_{V}}{\left(1-q^{2}/M_{V}^{2}\right)^{2}}\,,\quad
G_{M}=\frac{g_{M}}{\left(1-q^{2}/M_{V}^{2}\right)^{2}}\,,\\\
&G_{A}=\frac{g_{A}}{\left(1-q^{2}/M_{A}^{2}\right)^{2}}\,,\quad
G_{P}=\frac{2M_{i}}{m_{\pi}^{2}-q^{2}}G_{A}\,,\end{split}$ (3)

with $m_{\pi}$ the pion mass and $M_{A,V}$ two mass scales of the order of the
GeV.

A version of the full relativistic spectrum has also been discussed in [21].
However, the matrix element employed there is not computed from first
principles, but rather parametrized using Lorentz invariance together with
some considerations regarding which terms should be the leading ones. While
this approach reproduces well the lowest order formula, it is not the most
suitable one for computing higher order corrections.

In the following we neglect the momentum dependence of the form factors. This
approximation is correct up to relative corrections given by

$\displaystyle\begin{split}\frac{q^{2}}{M_{A,V}^{2}}\simeq\frac{m_{e}^{2}}{M_{A,V}^{2}}\sim
10^{-7}\quad&\text{for }G_{V},G_{M}\text{ and }G_{A}\,,\\\
\frac{q^{2}}{m_{\pi}^{2}}\simeq\frac{m_{e}^{2}}{m_{\pi}^{2}}\sim
10^{-2}\quad&\text{for }G_{P}\,.\end{split}$ (4)

Given the above, one can compute the fully relativistic decay rate,
differential in the electron energy, whose form in the lab frame is given by

$\displaystyle\frac{d\Gamma}{dE_{e}}={}$
$\displaystyle\frac{{(G_{F}V_{ud})}^{2}}{2\pi^{3}}F(Z,E_{e})\frac{M_{i}^{2}p_{e}}{M_{12}^{2}}\sqrt{\Delta\left(\Delta-2m_{\nu}\frac{M_{f}}{M_{i}}\right)}$
$\displaystyle\begin{split}&\times\Big{[}g_{V}^{2}\mathcal{W}_{VV}+g_{A}^{2}\mathcal{W}_{AA}+g_{A}g_{V}\mathcal{W}_{AV}\\\
&\quad\;\;+g_{A}^{2}\mathcal{W}_{AP}+g_{A}^{2}\mathcal{W}_{P\\!P}+g_{V}g_{M}\mathcal{W}_{V\\!M}\end{split}$
(5)
$\displaystyle\quad\;\;+g_{A}g_{M}\mathcal{W}_{AM}+g_{M}^{2}\mathcal{W}_{M\\!M}\Big{]}\,,$

where $M_{12}^{2}\equiv M_{i}^{2}+m_{e}^{2}-2M_{i}E_{e}$, and $F(Z,E_{e})$ is
the Fermi factor accounting for the electromagnetic interaction between the
outgoing electron and the decayed nucleus. Following Primakoff and Rosen [24],
we approximate it as

$\displaystyle F(Z,E_{e})\simeq\frac{2\pi\alpha
ZE_{e}/p_{e}}{1-\exp\left(-2\pi\alpha ZE_{e}/p_{e}\right)}\,,$ (6)

with $Z=2$ the atomic number of ${{}^{3}\text{He}^{+}}$. Finally, we define
$\Delta\equiv E_{e}-E_{e}^{\text{max}}$, with the relativistic end-point given
by111We note a typo in the denominator of Eq. (7) in [22], where $M_{f}$
should be replaced with $M_{i}$.

$\displaystyle
E_{e}^{\text{max}}=\frac{M_{i}^{2}+m_{e}^{2}-\left(M_{f}+m_{\nu}\right)^{2}}{2M_{i}}\,.$
(7)

The $\mathcal{W}$’s appearing in Eq. (II) result from integrating the spin-
averaged squared matrix element over the relativistic 3-body phase space, as
outlined in [22]. Their expressions (again, in the approximation of constant
form factors) are rather cumbersome. We report them fully in Appendix A.

##  III Expansion in the small parameters

The dimensionful quantities entering the $\mathcal{W}$ terms come with
different sizes, resulting in a number of small parameters that one can use to
approximate the full expression, Eq. (II). These parameters are chosen to be

$\displaystyle\begin{split}\frac{m_{e}}{M_{i}}\sim
10^{-4}\,,\quad\frac{Q}{M_{i}}\sim
10^{-5}\,,\quad\frac{|\Delta|}{M_{i}}\sim\frac{m_{\nu}}{M_{i}}\sim
10^{-10}\,,\end{split}$ (8)

where $Q\equiv M_{i}-M_{f}-m_{e}$ is the $Q$-value. Since the spectrum is most
sensitive to the neutrino mass close to the end-point, we also set
$|\Delta|\sim m_{\nu}$, and assumed conservatively $m_{\nu}\sim 1$ eV. For
everything else, we used the numerical values in Table 1.

$M_{i}$ [eV] [25] |  $m_{\pi}$ [eV] [26] |  $m_{e}$ [eV] [26] |  $Q$ [eV] [25] |  $g_{V}$ |  $g_{A}$ [27] |  $g_{M}$ [22]
---|---|---|---|---|---|---
$2.80943249663(19)\times 10^{9}$ | $137.27360(27)\times 10^{6}$ | $0.5109989461(31)\times 10^{6}$ | $18.59201(7)\times 10^{3}$ | $1$ | $1.27641(55)$ | $-6.106$
Table 1: Values used in this work. In order to keep track of all the
uncertainties, we also report the experimental ones. For the pion mass we
report the average between charged and neutral states. The value $g_{V}=1$ is
implied by the conserved vector current hypothesis, while, to the best of our
knowledge, the uncertainty on $g_{M}$ is not found in the literature. The
precise values of $G_{F}$ and $V_{ud}$ only affect the normalization of the
spectrum, which is expected to be fitted from data.

We also report the current errors on the measured parameters. In fact, while
this does not affect the present work, for higher order corrections, the
uncertainties coming from the parameters can compete with the theoretical
uncertainties. This is especially true for the masses, the $Q$-value and the
$g$ couplings. The errors on overall factors are, instead, expected not to
play a role, since a standard experimental analysis would fit the overall
normalization anyway.

Following the expansion scheme outlined above, one finds the following leading
order behaviors of the $\mathcal{W}$’s:

$\displaystyle\begin{split}\mathcal{W}_{VV}={}&E_{e}(m_{\nu}-\Delta)-E_{e}m_{\nu}\frac{m_{e}}{M_{i}}\\\
&-E_{e}m_{\nu}\frac{Q}{M_{i}}+\mathcal{O}\\!\big{(}{m_{e}^{2}Q\Delta}/{M_{i}^{2}}\big{)}\,,\end{split}$
(9a)
$\displaystyle\begin{split}\mathcal{W}_{AA}={}&3E_{e}(m_{\nu}-\Delta)-3E_{e}m_{\nu}\frac{m_{e}}{M_{i}}\\\
&+E_{e}(m_{\nu}-4\Delta)\frac{Q}{M_{i}}+\mathcal{O}\\!\left({m_{e}^{2}Q\Delta}/{M_{i}^{2}}\right)\,,\end{split}$
(9b) $\displaystyle\mathcal{W}_{AV}={}$ $\displaystyle
4E_{e}(m_{\nu}-\Delta)\frac{Q}{M_{i}}+\mathcal{O}\\!\left({m_{e}^{2}Q\Delta}/{M_{i}^{2}}\right)\,,$
(9c) $\displaystyle\mathcal{W}_{AM}={}$
$\displaystyle-4E_{e}(m_{\nu}-\Delta)\frac{Q}{M_{i}}+\mathcal{O}\\!\left({m_{e}^{2}Q\Delta}/{M_{i}^{2}}\right)\,,$
(9d) $\displaystyle\mathcal{W}_{AP}={}$
$\displaystyle\mathcal{O}\\!\left({m_{e}^{2}Q^{2}\Delta}/{M_{i}m_{\pi}^{2}}\right)\,,$
(9e) $\displaystyle\mathcal{W}_{P\\!P}={}$
$\displaystyle\mathcal{O}\\!\left({m_{e}^{4}Q\Delta}/{m_{\pi}^{4}}\right)\,,$
(9f) $\displaystyle\mathcal{W}_{V\\!M}={}$
$\displaystyle\mathcal{O}\\!\left({m_{e}^{2}Q\Delta}/{M_{i}^{2}}\right)\,,$
(9g) $\displaystyle\mathcal{W}_{M\\!M}={}$
$\displaystyle\mathcal{O}\\!\left({m_{e}^{3}\Delta}/{M_{i}^{2}}\right)\,.$
(9h)

Before computing the decay rate and its leading corrections, we briefly
comment about the expansion presented in [22]. There, in order to estimate the
relative sizes of the various terms, it is set $M_{f}=M_{i}$. However, this
equality holds true only at lowest order, and receives corrections already at
order $m_{e}/M_{i}$. Because of this, the scalings reported in Eq. (28) of
[22] lead to an _overestimate_ of the corrections (in all cases but one). In
fact, when one keeps track of the fact that $M_{i}\neq M_{f}$, additional
cancellations happen, reducing the size of the correction. We have verified
that, starting from the full expressions reported in Appendix A, the correct
estimates are those in Eqs. (9). Moreover, we report the presence of a wrong
sign in the expression of $\mathcal{W}_{AA,VV,AV}$, given in Eq. (29) of [22],
for the first term in the parentheses proportional to $(g_{V}+g_{A})^{2}$. The
correct expression is given in Appendix A.

Now, in light of all this, the leading order expression for the differential
rate is

$\displaystyle\begin{split}\frac{d\Gamma_{(0)}}{dE_{e}}={}&\frac{\left(G_{F}V_{ud}\right)^{2}}{2\pi^{3}}F(Z,E_{e})\left(g_{V}^{2}+3g_{A}^{2}\right)\\\
&\times
E_{e}p_{e}(m_{\nu}-\Delta)\sqrt{\Delta(\Delta-2m_{\nu})}\,,\end{split}$ (10)

in agreement with the literature [e.g, 21, 22, 17]. The rate up to order
$Q/M_{i}$ instead reads

$\displaystyle\frac{d\Gamma_{(1)}}{dE_{e}}={}$
$\displaystyle\frac{\left(G_{F}V_{ud}\right)^{2}}{2\pi^{3}}F(Z,E_{e})E_{e}p_{e}\sqrt{\Delta\left(\Delta-2m_{\nu}\frac{M_{f}}{M_{i}}\right)}$
$\displaystyle\times\frac{M_{i}^{2}}{M_{12}^{2}}\bigg{\\{}\left(g_{V}^{2}+3g_{A}^{2}\right)\left(m_{\nu}-\Delta-
m_{\nu}\frac{m_{e}}{M_{i}}\right)$
$\displaystyle\quad+\Big{[}-g_{V}^{2}m_{\nu}+g_{A}^{2}(m_{\nu}-4\Delta)$ (11)
$\displaystyle\qquad-4g_{A}(g_{M}-g_{V})(m_{\nu}-\Delta)\Big{]}\frac{Q}{M_{i}}\bigg{\\}}\,.$

Similarly, the leading and next-to-leading order expressions for the electron
end-point read,

$\displaystyle E_{e,(0)}^{\rm max}={}$ $\displaystyle m_{e}+Q-m_{\nu}\,,$
(12a) $\displaystyle E_{e,(1)}^{\rm max}={}$ $\displaystyle
m_{e}+Q-m_{\nu}-m_{e}\frac{Q}{M_{i}}\,.$ (12b)

##  IV Relevance for Ptolemy

Figure 1: Relative correction to the rate, up to $\mathcal{O}(Q/M_{i})$. For a
sufficiently heavy neutrino, the difference in shape of the two expression is
substantially larger that the expected statistical uncertainty for Ptolemy at
maximum loading. The non-zero values for $\Delta\geq 0$ are due to the
experimental resolution in Eq. (14), which pushes some events above the end-
point.

Thanks to the storage of atomic tritium on a solid state substrate, Ptolemy
aims at employing a large mass of tritium, possibly up to $100$ g [17], which
would also allow to hunt for the cosmic neutrino background. In Figure 1 we
show the relative correction to the leading order rate obtained from Eq.
(III), i.e.

$\displaystyle\Delta\tilde{\Gamma}\equiv\frac{d\tilde{\Gamma}_{(1)}/dE_{e}-d\tilde{\Gamma}_{(0)}/dE_{e}}{d\tilde{\Gamma}_{(0)}/dE_{e}}\,,$
(13)

where $\tilde{\Gamma}$ is the theoretical rate convoluted with the
experimental resolution, take to be a Gaussian of width $\sigma=100$ meV:

$\displaystyle\frac{d\tilde{\Gamma}}{dE_{e}}=\int_{-\infty}^{\infty}dE^{\prime}\,\frac{\text{e}^{-(E_{e}-E^{\prime})/(2\sigma^{2})}}{\sqrt{2\pi}\sigma}\,\frac{d\Gamma}{dE^{\prime}}\,.$
(14)

Importantly, since in a realistic analysis one would also fit the position of
the end-point [17], we use the expression for the exact relativistic
$E_{e}^{\rm max}$, Eq. (7), in both the leading and next-to-leading order
rates.

Considering (conservatively) events in the range $\Delta\in[-2,0]$ eV, Eq.
(10) returns an expectation of $N\sim 10^{12}$ events/year. Assuming a Poisson
distribution, this corresponds to a relative statistical uncertainty of order
$10^{-6}$ after a year of data taking. As one can see from Figure 1, for
sufficiently heavy neutrinos, this is considerably smaller than the relative
difference introduced by the next-to-leading order correction to the rate
reported in Eq. (III). The shape of the latter, in fact, varies by a fraction
of the order of $10^{-4}\div 10^{-5}$ over a large range of energies. Such
corrections must then be included in order to avoid theoretical biases on the
parameter estimation.

##  V Conclusion

In the coming years a host of existing and future experiments should place
increasingly more stringent bounds on the neutrino mass. Among these, Ptolemy
might reach unprecedented rates, and correspondingly low statistical
uncertainties. This improvement on the experimental side demands precise
theoretical prediction.

In this note we discussed the corrections due to relativistic effects in the
nuclear decay of tritium. Specifically, by systematically keeping track of the
approximations made, we revisit some results available in the literature, and
present a revised compendium of what we identify as the most relevant
corrections to the $\beta$-decay spectrum for the experimental setup as
envisioned by Ptolemy. We show that, given the expected number of events,
these correction are important to avoid theory biases.

This is a first step towards the development of theory predictions which are
accurate enough to face the forthcoming experimental challenges. In
particular, we note that experiments like the ones mentioned above use (or
plan to use) tritium bound either in a molecule or to a solid state substrate.
The presence of a spectator system comes with a number of difficulties and
further correction, which have been discussed in [28, 29, 30, 31, 32, 33, 34,
35]. Consequently, several more effects will then need to be taken into
account to achieve the required precision as, for example, atomic, molecular
and solid state ones.

###### Acknowledgements.

We thank L. E. Marcucci and M. Viviani for discussions on the parametrization
of the nuclear matrix element, and D. del Re, S. Gariazzo and A. Messina for
discussions about statistical analyses and the bounds on the neutrino mass.
For most of the development of this work A.E. has been a Roger Dashen Member
at the Institute for Advanced Study, whose work was also supported by the U.S.
Department of Energy, Office of Science, Office of High Energy Physics under
Award No. DE-SC0009988.

##  Appendix A Full expression for the $\mathcal{W}$ factors

We report the full expression for the factors $\mathcal{W}$ entering the rate
in Eq. (II). When approximating the rate following the scheme described in
Section III, most of them will not contribute to the lowest orders,
considerably simplifying the result. We also notice that, if a precision
larger than $\mathcal{O}\big{(}10^{-7}\big{)}$ is required, it is also
necessary to re-compute the $\mathcal{W}$’s including the momentum dependence
of the form factors in Eq. (2). We first define,

$\displaystyle A_{1}\equiv{}$ $\displaystyle
m_{\nu}\frac{M_{f}+m_{\nu}}{M_{i}}-\Delta\,,\quad A_{2}\equiv
M_{f}\frac{M_{f}+m_{\nu}}{M_{i}}-\Delta\,,$
$\displaystyle\begin{split}A_{3}\equiv{}&m_{\nu}\frac{M_{f}}{M_{i}}-\Delta\,,\quad
B_{1}\equiv\frac{M_{i}E_{e}-m_{e}^{2}}{M_{12}^{2}}\,,\\\
B_{2}\equiv{}&\frac{M_{i}(M_{i}-E_{e})}{M_{12}^{2}}\,,\quad
C\equiv\Delta\left(\Delta-2m_{\nu}\frac{M_{f}}{M_{i}}\right)\,,\end{split}$
(15) $\displaystyle D\equiv{}$
$\displaystyle\frac{M_{i}^{2}\big{(}E_{e}^{2}-m_{e}^{2}\big{)}}{3M_{12}^{4}}\,.$

The $\mathcal{W}$’s are then given by

$\displaystyle\mathcal{W}_{VV}={}$ $\displaystyle
A_{1}B_{1}A_{2}B_{2}-M_{f}A_{1}B_{1}+E_{e}A_{3}-CD\,,$ (16a)
$\displaystyle\mathcal{W}_{AA}={}$ $\displaystyle
A_{1}B_{1}A_{2}B_{2}+M_{f}A_{1}B_{1}+E_{e}A_{3}-CD\,,$ (16b)
$\displaystyle\mathcal{W}_{AV}={}$ $\displaystyle
2\left(A_{1}B_{1}A_{2}B_{2}-E_{e}A_{3}-CD\right)\,,$ (16c)
$\displaystyle\mathcal{W}_{AP}={}$ $\displaystyle
2\left(m_{e}^{2}{M_{f}}A_{1}B_{2}+m_{\nu}^{2}{M_{f}}E_{e}-m_{e}^{2}{M_{i}}A_{3}-m_{\nu}^{2}{M_{i}}A_{2}B_{1}\right)/m_{\pi}^{2}\,,$
(16d)
$\displaystyle\begin{split}\mathcal{W}_{P\\!P}={}&2\big{[}\left(m_{e}^{2}+m_{\nu}^{2}\right){M_{i}^{2}}(A_{1}B_{1}A_{2}B_{2}-CD)+2m_{\nu}^{2}m_{e}^{2}{M_{i}}A_{2}B_{2}-\big{(}m_{e}^{2}+m_{\nu}^{2}\big{)}{M_{i}^{2}M_{f}}A_{1}B_{1}\\\
&\quad-2m_{\nu}^{2}m_{e}^{2}{M_{i}M_{f}}\big{]}/{m_{\pi}^{4}}\,,\end{split}$
(16e)
$\displaystyle\begin{split}\mathcal{W}_{V\\!M}={}&\frac{M_{i}+M_{f}}{M_{i}}(CD-
A_{1}B_{1}A_{2}B_{2})+\frac{M_{f}(M_{i}+M_{f})}{2M_{i}}A_{1}B_{1}-\frac{M_{i}+M_{f}}{2M_{i}}E_{e}A_{3}+\frac{M_{f}}{M_{i}}E_{e}A_{1}B_{2}\\\
&+\frac{M_{i}}{M_{12}^{2}}(A_{1}A_{2}+C)A_{2}B_{1}\,,\end{split}$ (16f)
$\displaystyle\mathcal{W}_{AM}={}$ $\displaystyle(M_{i}+M_{f})(CD-
A_{1}B_{1}A_{2}B_{2}+E_{e}A_{3})/{M_{i}}\,,$ (16g)
$\displaystyle\mathcal{W}_{M\\!M}={}$ $\displaystyle-
m_{e}^{2}m_{\nu}^{2}\frac{M_{f}}{4M_{i}^{3}}-\big{(}m_{e}^{2}+m_{\nu}^{2}\big{)}\frac{3M_{f}}{8M_{i}^{2}}A_{1}B_{1}-\frac{m_{e}^{2}m_{\nu}^{2}}{4M_{i}^{3}}A_{2}B_{2}-\frac{m_{e}^{2}+m_{\nu}^{2}}{4M_{i}^{2}}(A_{1}B_{1}A_{2}B_{2}-CD)+\frac{m_{e}^{2}}{6M_{i}}E_{e}\frac{C}{M_{12}^{2}}$
$\displaystyle\begin{split}&-\frac{M_{f}}{2M_{i}^{3}}\left[\frac{m_{e}^{2}}{3}\left(\frac{m_{\nu}^{2}}{2}+\frac{M_{i}^{2}A_{1}^{2}}{M_{12}^{2}}\right)+\left(\frac{4M_{i}^{2}A_{1}^{2}}{M_{12}^{2}}-m_{\nu}^{2}\right)\frac{\big{(}M_{i}E_{e}-m_{e}^{2}\big{)}^{2}}{3M_{12}^{2}}\right]+\frac{m_{\nu}^{2}}{2M_{i}^{2}}E_{e}A_{2}B_{1}\\\
&+\frac{1}{4M_{i}}E_{e}\left(\frac{3A_{1}A_{2}}{4}+C\right)\frac{\big{(}M_{i}E_{e}-m_{e}^{2}\big{)}^{2}}{M_{12}^{4}}+\frac{m_{e}^{2}}{2}(A_{1}A_{2}+C)A_{1}\frac{M_{i}-E_{e}}{M_{12}^{4}}\end{split}$
(16h)
$\displaystyle+\frac{1}{4M_{i}^{2}}(A_{1}A_{2}+C)\left[\frac{M_{i}E_{e}}{3M_{12}^{2}}\left(m_{\nu}^{2}-\frac{M_{i}^{2}A_{1}^{2}}{M_{12}^{2}}\right)+\left(\frac{m_{\nu}^{2}}{3}+\frac{M_{i}^{2}A_{1}^{2}}{M_{12}^{2}}\right)\frac{M_{i}\big{(}M_{i}-E_{e}\big{)}\big{(}M_{i}E_{e}-m_{e}^{2}\big{)}}{M_{12}^{4}}\right]\,.$

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