aainstitutetext: University College London, Gower Street, London WC1E 6BT,
UKbbinstitutetext: Institute of Physics, ELTE Eötvös Loránd University,
Pázmány Péter sétány 1/A, H-1117
Budapest, Hungary

# Sensitivity of Future Tritium
Decay Experiments to New Physics

James A. L. Canning a Frank F. Deppisch a,b Wenna Pei
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Tritium beta-decay is the most promising approach to measure the absolute
masses of active light neutrinos in the laboratory and in a model-independent
fashion. The development of Cyclotron Radiation Emission Spectroscopy
techniques and the use of atomic tritium has the potential to improve the
current limits by an order of magnitude in future experiments. In this paper,
we analyse the potential sensitivity of such future searches to keV-mass
sterile neutrinos and exotic interactions of either the active or sterile
neutrinos. We calculate the relevant decay distributions in both energy and
angle of the emitted electron with respect to a potential polarisation of the
tritium, including the interference with the Standard Model case as well as
incorporating relevant final state corrections for atomic tritium. We present
projected sensitivities on the active-sterile neutrino mixing and effective
coupling constants of exotic currents, demonstrating the potential to probe
New Physics in tritium experiments.

##  1 Introduction

The detection of neutrino oscillations implies that active neutrinos in the
Standard Model (SM) have a small but non-zero mass PhysRevD.98.030001 .
Oscillations are explained through a non-diagonal mixing between flavour and
mass eigenstates and they crucially require that the three active neutrinos do
not have identical (degenerate) masses. Instead, the observed oscillations are
compatible with two different mass splitting scenarios. Oscillation
experiments are insensitive to the absolute values of active neutrino masses
which remain undetermined. The ordering of the neutrino mass eigenstates with
respect to their flavour content is also still unknown. Matter effects in
oscillations of solar neutrinos led to the determination that the so-called
solar mass splitting $\Delta m_{\text{sol}}^{2}>0$, but the order in the
atmospheric splitting $\Delta m_{\text{atm}}^{2}$ is unknown.

The absolute values of eV-scale neutrino masses are being constrained in
single $\beta$-decay111Alternative direct probes are other decays with
neutrinos in the final state such as that of the muon and tau. The resulting
constraints are much weaker and not relevant for our discussion. as well as in
indirect measurements using neutrinoless double $\beta$-decay Agostini:2022zub
; Deppisch:2016rox  and cosmology Ade:2015xua . The process of single
$\beta$-decay, in which $d\to u+e+\bar{\nu}$ at the quark level, will be the
primary focus of this paper. In particular, we have chosen to focus on tritium
$\beta$-decay as future leading experiments are planning to use tritium due to
its short half-life and low $Q$-value. This decay produces helium with the
emission of an electron anti-neutrino and an electron,

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

In order to derive the Lagrangian we start at the quark level which (in terms
of the up and down quarks, $u$ and $d$, respectively) at low energies takes
the form in the SM,

$\displaystyle\mathcal{L}=-\frac{G_{F}}{\sqrt{2}}V_{ud}\left[\bar{e}\gamma^{\mu}(1-\gamma^{5})\nu_{e}\right]\left[\bar{u}\gamma_{\mu}(1-\gamma^{5})d\right]+\text{h.c.},$
(2)

with $V_{ud}$ the CKM mixing matrix element between the up and down quarks and
$G_{F}$ the Fermi constant. This is an effective dimension-6 operator that
serves well as an approximation to the SM because we are looking at
interactions with energies much below the mass of the $W$ boson.

The advantage of single $\beta$-decay as a neutrino mass measurement technique
is that it is model-independent and relies purely upon the kinematics of the
decay. Specifically, it is independent of the Majorana or Dirac nature of
neutrinos and as a laboratory measurement it does not rely on assumptions
beyond SM particle physics, e.g., the exact form of the cosmological model in
astrophysical observations. While the emitted neutrino interacts too weakly to
be consistently detected, the electron is easily observed as a charged
particle. The presence of the neutrino’s mass will affect the emission
spectrum of the electron, most clearly at the endpoint where the electron
takes near the maximum energy that it can kinematically receive, leaving the
neutrino with negligible kinetic energy. The challenge with this method is
that only a very small fraction of the emitted electrons will have an energy
near the endpoint requiring a high number of decay events to reach the
required statistics. Additionally, the electrons’ energies have to be measured
with a very high precision. This is necessary to be sensitive to the spectral
shape near the endpoint. There are many effects such as the energy
uncertainties introduced through excited final states, especially when using
molecular tritium, and the experimental challenges which limit the precision
of such a measurement Otten:2008zz .

Single $\beta$-decay produces an electron anti-neutrino, which, in turn is
described in terms of a weighted sum over its constituent mass eigenstates.
With experiments not able to resolve the individual three thresholds of the
mass eigenstates in the foreseeable future, experimental results are thus
often presented in terms of an effective $\beta$-decay mass Kleesiek2019 ,

$\displaystyle m_{\beta}^{2}=\sum_{i=1}^{3}|U_{ei}|^{2}m_{i}^{2},$ (3)

see the discussion of Eq. (26) and Fig. 2 below. Here, $U_{ei}$ are the
elements of the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) mixing matrix and
$m_{i}$ are the neutrino masses in the respective ordering. The current
strictest bound on $m_{\beta}$ comes from the KATRIN experiment with
$m_{\beta}<800$ meV at 90% confidence level (CL) Aker2022 . KATRIN aims to
reach a sensitivity of $m_{\beta}\sim 200$ meV (90% CL) by the conclusion of
its operation Aker2022 . Using the experimental techniques employed in KATRIN,
it will be very challenging to further improve on this sensitivity. The
minimum value of $m_{\beta}\sim 9$ meV is the ‘worst’-case of normally ordered
neutrino states with a massless lightest neutrino, see Fig. 2.222The lower
limit on $m_{\beta}$ is derived in the SM with three active neutrinos and a
unitary PMNS matrix. It may be further modified under the presence of sterile
neutrinos or other non-unitary effects. To achieve this ultimate goal, new
techniques are required and the most promising approach is based on measuring
the electron energy through the emitted cyclotron radiation in a magnetic
field as championed by the Project 8 collaboration Pettus:2017sxd . Together
with utilising atomic instead of molecular tritium to minimise final state
corrections, the projected sensitivity is $m_{\beta}\sim 40$ meV
Pettus:2017sxd . This will guarantee the observation of $m_{\beta}$ for
inversely ordered neutrinos with a minimum value of $m_{\beta}\sim 50$ meV.
Exploring the potential to achieve even better sensitivities using atomic
tritium and quantum technologies, the Cyclotron Radiation Emission
Spectroscopy Demonstrator Apparatus is being built to develop the necessary
techniques QTNM:2020 .

Probing $m_{\beta}$ at the level of $\mathcal{O}(10)$ meV will require
observing a large number, $N_{1\text{eV}}\approx 10^{5}-10^{6}$, of tritium
decay electrons with energies in the last $\approx$ eV of the endpoint. Such
an experiment, owing to the high count rate, may also be designed to measure
the full tritium decay spectrum. Techniques required to produce atomic tritium
may also allow the partial polarisation of the tritium atom, giving rise to a
potentially non-isotropic angular distribution of the emitted electrons. In
this work, we explore the sensitivity of such experiments to New Physics
specifically, exotic charged currents beyond the SM $V-A$ term in Eq. (2), as
well as the presence of a keV-scale sterile neutrino that is either mixing
with the active neutrinos, is participating in exotic currents beyond $V-A$ or
both.

Such extensions are motivated by the presence of neutrino masses that
necessarily require the existence of a corresponding mechanism of mass
generation. Within the SM, all the fermions exist either within an $SU(2)$
left-handed doublet or a right-handed singlet, with a Yukawa Higgs coupling
the two, generating a Dirac mass. A right-handed neutrino singlet would be
uncharged under all SM gauge interactions and is thus called a sterile
neutrino. The simplest extension to the SM is to analogously generate a Dirac
mass for the neutrino. However, as this state is uncharged, there is nothing
forbidding it to have a Majorana mass as well, thus making it a Majorana
fermion, leading to the seesaw mechanisms for mass generation AKHMEDOV2000215
. This will induce a mixing between the SM-like active and the sterile states.
An extension of the PMNS matrix of dimension $3+n_{s}$, where $n_{s}$ is the
number of sterile states, will describe this mixing. In the context of tritium
decay, this mixing will allow the production of sterile states as long as they
are lighter than the tritium $Q$-value of $18.59$ keV.

The inclusion of right-handed neutrinos also evokes the potential presence of
a fundamental left-right symmetry within nature. We can consider the
possibility that at high energies there exists an equivalent right-handed weak
force which, due to some unknown mechanism, is broken at a much higher scale
than the SM weak force Sirunyan2018 . This would lead to right-handed vector
currents within $\beta$-decay, potentially within either or both of the
hadronic and leptonic currents. Extending this further, we consider the full
basis of exotic operators within the hadronic sector which could couple to
either left or right-handed leptonic currents Ludl:2016ane .

This manuscript is organized as follows. In Sec. 2 we describe the effective
operators and associated model parameters potentially responsible for
$\beta$-decay. In Sec. 3 we then outline the calculation of the tritium decay
rate as well as energy and angular decay distributions. In Sec. 4 we describe
the current and future experiments. In Sec. 5 we demonstrate the limits on the
model parameters that could be reached by an idealized future experiment with
enough statistics to probe $m_{\beta}\approx\mathcal{O}(10)$ meV. Sec. 6
concludes our analysis and the Appendix contains details of final state
corrections to the decay spectrum (Appendix A) and full expressions of the
exotic contributions considered here (Appendix B).

##  2 Effective Operators for Tritium Decay

For the decay of tritium, composed of two neutrons and one proton, a hadronic
Lagrangian must be considered to take account of the internal strong dynamics
Ludl:2016ane ,

$\displaystyle\mathcal{L}_{\text{SM}}=-\frac{G_{F}}{\sqrt{2}}V_{ud}(1+\delta_{\beta})\left[\overline{e}\gamma^{\mu}(1-\gamma^{5})\nu_{e}\right]\left[\overline{{}^{3}\text{He}}\gamma_{\mu}\left(g_{V}-g_{A}\gamma^{5}\right){}^{3}\text{H}\right],$
(4)

with $\delta_{\beta}=0$ within the SM. The coefficients $g_{V}$ and $g_{A}$
parametrize the vector and axial couplings and take the values $g_{V}=1$ and
$g_{A}\approx 1.247$, respectively Simkovic:2007yi . It is from this
Lagrangian that all of the later results denoted ‘SM’ are derived. This form
is analogous to the quark level Lagrangian. This is because the tritium and
helium form an approximate $SU(2)$ isospin doublet equivalent to the quarks
Simkovic:2007yi . Further terms $\mathcal{O}(Q/m_{\text{H}})$ have been
neglected as $Q\sim 20$ keV $\ll m_{\text{H}}\sim 3$ GeV, the tritium mass
Ludl:2016ane . The fermionic fields in the Lagrangian implicitly depend on
their individual spin states. Rather than averaging over all of these spins,
we will consider what happens when the tritium is prepared in a particular
spin state. This will lead to correlations between the electron’s momentum
direction and the tritium’s spin which will break the isotropy of emission.

###  2.1 Exotic Currents

Current | Hadronic | Leptonic
---|---|---
Scalar | $H_{S}=g_{S}~{}\overline{{}^{3}\text{He}}~{}{}^{3}\text{H}$ | $j^{\pm}_{S}=\overline{e}(1\pm\gamma^{5})\nu_{e}$
Pseudoscalar | $H_{P}=g_{P}~{}\overline{{}^{3}\text{He}}~{}\gamma^{5}~{}{}^{3}\text{H}$ | $j^{\pm}_{P}=\overline{e}~{}(1\pm\gamma^{5})\nu_{e}$
Vector/Axial | $H^{\mu}_{V\pm A}=\overline{{}^{3}\text{He}}\gamma^{\mu}(g_{V}\pm g_{A}\gamma^{5})~{}{}^{3}\text{H}$ | $j^{\mu}_{V\pm A}=\overline{e}\gamma^{\mu}(1\pm\gamma^{5})\nu_{e}$
Left-Tensor | $H^{\mu\nu}_{T}=g_{T}~{}\overline{{}^{3}\text{He}}~{}\sigma^{\mu\nu}(1-\gamma^{5})~{}{}^{3}\text{H}$ | $j^{\mu\nu}_{T}=\overline{e}~{}\sigma^{\mu\nu}(1-\gamma^{5})\nu_{e}$
Right-Tensor | $\tilde{H}^{\mu\nu}_{T}=g_{T}~{}\overline{{}^{3}\text{He}}~{}\sigma^{\mu\nu}(1+\gamma^{5})~{}{}^{3}\text{H}$ | $\tilde{j}^{\mu\nu}_{T}=\overline{e}~{}\sigma^{\mu\nu}(1+\gamma^{5})\nu_{e}$
Table 1: Hadronic and leptonic currents expanded in the basis of gamma
matrices with $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$.

We are here working within the regime of effective operators with interactions
taking place much below the electroweak scale. From the SM weak interaction,
we have taken the effective, 4-point Fermi interaction as the basis for the
derivation of the decay rate. However, the vector minus axial (or $V-A$)
coupling of this interaction is not the only Lorentz-invariant operator. We
can also consider the presence of additional currents; these either being of
the right-handed $V+A$ form or of the scalar, pseudoscalar or tensor types. By
considering a basis expansion of the gamma matrices we can consider all
possible contributions to the 4-point interaction Ludl:2016ane . We break down
the Lagrangian as

$\displaystyle\mathcal{L}=\mathcal{L}_{\text{SM}}+\mathcal{L}_{\text{exotic}}+\mathcal{L}^{N}_{\text{exotic}},$
(5)

where

$\displaystyle\mathcal{L}_{\text{exotic}}=-\frac{G_{F}}{\sqrt{2}}V_{ud}\bigg{(}$
$\displaystyle\tilde{\epsilon}_{L}H^{\mu}_{V-A}j_{\mu,V+A}+\epsilon_{R}H^{\mu}_{V+A}j_{\mu,V-A}+\tilde{\epsilon}_{R}H^{\mu}_{V+A}j_{\mu,V+A}$
$\displaystyle+\epsilon_{S}H_{S}j^{-}_{S}+\tilde{\epsilon}_{S}H_{S}j^{+}_{S}-\epsilon_{P}H_{P}j^{-}_{P}-\tilde{\epsilon}_{P}H_{P}j^{+}_{P}$
$\displaystyle+\epsilon_{T}H^{\mu\nu}_{T}j_{T,\mu\nu}+\tilde{\epsilon}_{T}\tilde{H}^{\mu\nu}_{T}\tilde{j}_{T,\mu\nu}\bigg{)},$
(6)

gives all possible contributions involving the hadronic ($H$) and leptonic
currents ($j$) as given in Table 1. The dimensionless constants
$\epsilon_{i}$, $\tilde{\epsilon}_{i}$ give the interaction strength relative
to SM $V-A$ Fermi coupling $G_{F}$. The subscript denotes the hadronic part
and the tilde denotes a right-handed leptonic part. An additional exotic
contribution to the $V-A$ current is also possible, as represented by
$\delta_{\beta}$ in Eq. (4)) but this merely leads to a re-scaling of $G_{F}$
so we have chosen to omit it. The $g_{X}$ in the hadronic currents are form
factors which will, generically, depend upon the momentum exchange $Q^{2}$.
However, the dependence can be well approximated in dipole form as
$g_{X}(Q^{2})=g_{X}(0)/(1-(Q^{2}/M_{X}^{2}))^{2}$ where $M_{X}\sim 1$ GeV $\gg
Q\sim 20$ keV and thus the momentum dependence can be neglected
Simkovic:2007yi .

###  2.2 Sterile Neutrinos

We also consider contributions to tritium decay involving the emission of a
keV-scale sterile neutrino $N$. They can be defined analogously to the above
with the substitution of leptonic currents $j\to J$ that now involve the
sterile state $\nu_{e}\to N$,

$\displaystyle\mathcal{L}^{N}_{\text{exotic}}=-\frac{G_{F}}{\sqrt{2}}V_{ud}\bigg{(}$
$\displaystyle\epsilon^{N}_{L}H^{\mu}_{V-A}J_{\mu,V-A}+\tilde{\epsilon}^{N}_{L}H^{\mu}_{V-A}J_{\mu,V+A}$
$\displaystyle+\epsilon^{N}_{R}H^{\mu}_{V+A}J_{\mu,V-A}+\tilde{\epsilon}^{N}_{R}H^{\mu}_{V+A}J_{\mu,V+A}$
$\displaystyle+\epsilon^{N}_{S}H_{S}J^{-}_{S}+\tilde{\epsilon}^{N}_{S}H_{S}J^{+}_{S}-\epsilon^{N}_{P}H_{P}J^{-}_{P}-\tilde{\epsilon}^{N}_{P}H_{P}J^{+}_{P}$
$\displaystyle+\epsilon^{N}_{T}H^{\mu\nu}_{T}J_{T,\mu\nu}+\tilde{\epsilon}^{N}_{T}\tilde{H}^{\mu\nu}_{T}\tilde{J}_{T,\mu\nu}\bigg{)}.$
(7)

Here, the $\epsilon^{N}_{i}$, $\tilde{\epsilon}^{N}_{i}$ equivalently
parametrize the strength of the exotic currents. This Lagrangian also has an
additional purely left-handed contribution parametrized by $\epsilon^{N}_{L}$.

In calculating the sterile contributions, we will mainly consider a simplified
scenario with one massless, active electron neutrino (the SM contribution) and
one sterile neutrino with mass $m_{N}\lesssim 18$ keV. The active and sterile
states can be expressed as mixtures of the mass eigenstates, $\nu_{1,2}$, with
mixing strength $0\leq V_{eN}\ll 1$ here taken to be real and, without loss of
generality, positive,

$\displaystyle\nu_{e}$
$\displaystyle=\sqrt{1-V_{eN}^{2}}~{}\nu_{1}+V_{eN}~{}\nu_{2},$ $\displaystyle
N$ $\displaystyle=-V_{eN}~{}\nu_{1}+\sqrt{1-V_{eN}^{2}}~{}\nu_{2}.$ (8)

The state $\nu_{1}$ can thus be seen as mostly active whereas $\nu_{2}$ is
mostly sterile when $V_{eN}$ is small.

###  2.3 Existing Constraints

Coupling | $|\operatorname{Re}\epsilon|$ | $|\operatorname{Im}\epsilon|$
---|---|---
$\epsilon_{S}$ | $8\times 10^{-3}$ | $1\times 10^{-2}$
$\tilde{\epsilon}_{S}$ | $1.3\times 10^{-2}$ | $1.3\times 10^{-2}$
$\epsilon_{P}$ | $4.6\times 10^{-7}$ | $2\times 10^{-4}$
$\tilde{\epsilon}_{P}$ | $2\times 10^{-4}$ | $2\times 10^{-4}$
$\epsilon_{T}$ | $1\times 10^{-3}$ | $1\times 10^{-3}$
$\tilde{\epsilon}_{T}$ | $3\times 10^{-3}$ | $3\times 10^{-3}$
$\tilde{\epsilon}_{L}$ | $6\times 10^{-2}$ | -
$\epsilon_{R}$ | $5\times 10^{-4}$ | $5\times 10^{-4}$
$\tilde{\epsilon}_{R}$ | $5\times 10^{-3}$ | $5\times 10^{-3}$
Table 2: Experimental upper bounds on the real and imaginary parts of the
exotic coupling strengths at $90\%$ CL. Adapted from Ludl:2016ane ;
Falkowski:2021  and with experimental sources given in the text.

Experiments have sought to measure or place upper bounds upon the strengths of
exotic currents. The upper bounds quoted in Table 2 mostly come from
$\beta$-decay experiments and from LHC processes, with the stronger of these
bounds being given for each exotic current parameter. The current most
stringent upper bounds on $\epsilon_{P}$ and $\tilde{\epsilon}_{P}$ (both the
real and imaginary parts) come from low-energy experiments involving pion
decay, more precisely the ratio in pion decay of $\pi\to e\nu_{e}$ to
$\pi\to\mu\nu_{\mu}$ which is dependent upon the pseudoscalar current
Cirigliano:2013 ; Falkowski:2021 . The upper limits on
$\operatorname{Re}(\epsilon_{R,L})$ come from $\beta$-decay experiments
through placing bounds on the unitarity of the CKM matrix
cirigliano2010semileptonic . The limit on $\operatorname{Im}(\epsilon_{R})$
has been placed by the emiT collaboration by measuring spin-angle correlations
in $\beta$-decay chupp2012search ; mumm2011new . The largest possible values
of $\operatorname{Re}(\epsilon_{S,T})$ compatible with the excluded regions of
their respective parameter spaces is of the same order of magnitude for both
the low-energy and LHC probes, where the low-energy probe involves $0^{+}\to
0^{+}$ nuclear $\beta$-decays hardy2009superallowed  and radiative pion decay
from the PIBETA collaboration bychkov2009new , for the scalar and tensor
currents, respectively. These bounds are projected to reach a precision level
of $0.1\%$ through measurements of neutron dubbers2008clean ;
alarcon2007precise ; wilburn2009measurement ; povcanic2009nab ;
markisch2009new  and $\operatorname{{}^{6}{He}}$ knecht2011high  decays. The
$\beta$-decay and LHC bounds are also similar in the case of
$\operatorname{Im}(\epsilon_{T})$, where the low-energy limit is based on
spin-angle correlations in $\operatorname{{}^{8}{He}}$. In the cases of
$\operatorname{Im}(\epsilon_{S})$ and $\tilde{\epsilon}_{R,S,T}$ (both the
real and imaginary parts) the strongest bounds come from LHC measurements. In
order for the low-energy probes to match this sensitivity, the measurement of
neutrino angular correlations in Gamow-Teller transitions, $a_{GT}$, needs to
be improved to the level of $\delta a_{GT}/a_{GT}\sim 0.05\%$ Cirigliano:2013
. If sterile neutrinos are Majorana fermions, limits on the operator
coefficients can also be derived from neutrinoless double $\beta$ decay
experiments Tong:2021 .

##  3 Tritium Decay

###  3.1 Differential Decay Rate

Two different $\beta$-decay distributions will be of interest to us: the
energy spectrum as a function of the electron energy and the angular spectrum
as a function of the angle between the electron’s momentum and the tritium’s
spin. Both quantities can be calculated from the fully differential decay
rate,

$\displaystyle
d\Gamma=\frac{1}{2m_{\text{H}}}\frac{d^{3}\boldsymbol{p}_{e}}{(2\pi)^{3}2E_{e}}\frac{d^{3}\boldsymbol{p}_{\nu}}{(2\pi)^{3}2E_{\nu}}\frac{d^{3}\boldsymbol{p}_{\text{He}}}{{(2\pi)}^{3}2E_{\text{He}}}|T|^{2}(2\pi)^{4}\delta^{4}(P_{\text{H}}-P_{\text{He}}-P_{e}-P_{\nu}).$
(9)

where the lower case $\boldsymbol{p_{i}}$ denote the 3-momenta of the
associated particles, whereas $P_{i}$ are the 4-momenta, and $|T|^{2}$ is the
squared matrix element of the process.

Evaluating in the tritium rest frame,
$P_{\text{H}}=(m_{\text{H}},\mathbf{0})$, using the energy-momentum conserving
delta function and expressing in spherical coordinates for the electron and
neutrino momenta gives

$\displaystyle\frac{d\Gamma}{dE_{e}d\Omega_{e}}=\frac{C(E_{e})}{2^{9}\pi^{5}m_{\text{H}}}\int_{\Omega}d\Omega_{\nu}\int_{E_{\nu}^{-}}^{E_{\nu}^{+}}dE_{\nu}|T|^{2}\delta\left(\cos{\theta_{e\nu}}-\frac{{\boldsymbol{p}_{\text{He}}^{2}}-{\boldsymbol{p}_{e}^{2}}-{\boldsymbol{p}_{\nu}^{2}}}{2|\boldsymbol{p}_{e}||\boldsymbol{p}_{\nu}|}\right).$
(10)

Here, $\Omega_{i}$ is the solid angle of emission of the electron or neutrino
(which we choose to be oriented such that the polar angle is with respect to
the spin direction of the tritium nucleus) and $\theta_{e\nu}$ is the angle
between the electron and neutrino three-momenta. The factor $C(E_{e})$
collects the theoretical corrections as described in Sec. 3.2. In the integral
over the neutrino energy, $E^{\pm}_{\nu}(E_{e})$ are the upper and lower
bounds on the neutrino energy for a given electron energy Ludl:2016ane ,

$\displaystyle
E_{\nu}^{\pm}(E_{e})=\frac{(m_{\text{H}}-E_{e})(m_{\text{H}}y+m_{\nu}^{2}+m_{\text{He}}m_{\nu})\pm|\boldsymbol{p}_{e}|\sqrt{(m_{\text{H}}y(m_{\text{H}}y+2m_{\text{He}}m_{\nu})}}{m_{12}^{2}}.$
(11)

Here, we define

$\displaystyle
m_{12}^{2}=(p_{\text{H}}-p_{e})^{2}=m_{\text{H}}^{2}-2m_{\text{H}}E_{e}+m_{e}^{2},$
(12)

and

$\displaystyle y=E_{e}^{\text{max}}-E_{e},$ (13)

where $E_{e}^{\text{max}}$ is the maximum electron energy for tritium decay
with $E_{e}^{\text{max}}-m_{e}\approx 18.59$ keV AUDI2003337 ; Masood:2007rc .
It is worth noting that $E_{e}^{max}$ and hence $y$ have an implicit
dependence on the neutrino mass,

$E_{e}^{\rm{max}}=\frac{m_{\text{H}}^{2}+m_{e}^{2}-(m_{\text{He}}+m_{\nu})^{2}}{2m_{\text{H}}}.$
(14)

In general, the particles will be in particular spin states. Typically these
spins are summed or averaged to derive the decay rate corresponding to the
source having an equal probability of being spin up or down. If we wish to
consider the impact of the spin direction on the emission of the electron
(such as when the ensemble is polarised) then it is more useful to preserve
the choice of spin state. One way to achieve this is by projecting onto the
required spin state whilst still summing over the spins. This is done with the
projection operator $P_{s}=\frac{1}{2}(1+\gamma^{5}\not{S})$ Pal2014  with
$S^{\mu}=(\boldsymbol{p}\cdot\hat{\boldsymbol{s}}/m,\hat{\boldsymbol{s}}+(\boldsymbol{p}\cdot\hat{\boldsymbol{s}})/(m(E+m)))$
where the mass, spin and momentum are for the relevant particle as evaluated
in a given frame. In this case the projector will be applied to the tritium
which in its rest frame gives $S^{\mu}=(0,\hat{\boldsymbol{s}})$ 333The
projector $P_{s}$ is chosen to satisfy
$P_{s}u_{\boldsymbol{p},s^{\prime}}=\delta_{s,s^{\prime}}u_{\boldsymbol{p},s^{\prime}}$
and
$P_{s}v_{\boldsymbol{p},s^{\prime}}=\delta_{s,s^{\prime}}v_{\boldsymbol{p},s^{\prime}}$
where $s,s^{\prime}=\downarrow,\uparrow$ and $u,v$ are the spinors of the
associated particle..

Generally, the tritium ensemble can be partially polarised with a polarisation
factor of

$\displaystyle
f=\frac{N^{\uparrow}-N^{\downarrow}}{N^{\uparrow}+N^{\downarrow}},$ (15)

where $N^{\uparrow(\downarrow)}$ denotes the number of spin up (down) tritium
nuclei or, alternatively, the population probabilities. This arises because
when we are considering the net impact of polarisation across the ensemble we
must sum over all the up and down states with $N^{\uparrow}\to N^{\downarrow}$
corresponding to $\hat{s}\to-\hat{s}\Rightarrow S\to-S$ and thus there is a
cancellation between them. A polarisation of the ensemble will lead to an
angular non-isotropy. When considering the angular distribution we will choose
the maximal value $f=1$, i.e., total polarisation, but any value can be easily
used by taking $S\to fS$ in the matrix element or $\hat{s}\to f\hat{s}$ in the
decay rate. No polarisation corresponds to $f=0$.

When polarised, a correlation arises between the electron’s momentum direction
and the spin of the tritium. As the spin dependence is always linear (due to
the linearity of the singly included projector), the decay rate can be written
as a linear sum of spin-independent and -dependent terms. Thus in general the
double differential distribution with respect to the electron energy $E_{e}$
and its solid angle takes the form

$\displaystyle\frac{d\Gamma}{dE_{e}d\Omega_{e}}=a(E_{e})+b(E_{e})\cos\theta_{e},$
(16)

where $a(E_{e})$, $b(E_{e})$ are functions which will depend on the choice of
Lagrangian and $\theta_{e}$ is the angle between the electron momentum and the
tritium spin.

Because of the $\cos\theta_{e}$ dependence, when integrating over all electron
angles, the contribution from $b(E_{e})$ will vanish leaving the energy
distribution,

$\displaystyle\frac{d\Gamma}{dE_{e}}=4\pi a(E_{e}).$ (17)

The total tritium decay rate is then

$\displaystyle\Gamma=4\pi\int_{m_{e}}^{E_{e}^{\text{max}}}a(E_{e})dE_{e},$
(18)

Alternatively, the double differential distribution can be integrated over the
electron energy to give the angular distribution

$\displaystyle\frac{d\Gamma}{d\cos\theta_{e}}=\frac{\Gamma}{2}(1+k\,\cos\theta_{e}),$
(19)

with the angular correlation factor

$\displaystyle
k=\int_{m_{e}}^{E_{e}^{max}}b(E_{e})dE_{e}\Bigg{/}\int_{m_{e}}^{E_{e}^{max}}a(E_{e})dE_{e}.$
(20)

###  3.2 Theoretical Correction Factors

The factor $C(E_{e})$ in Eq. (10) denotes additional theoretical correction
factors to the spectrum which appear as multiplicative overall factors. See
Appendix A for full expressions and further details on all theoretical
corrections. The most significant of these is the Fermi factor, $F(E_{e})$
which corrects for the electromagnetic interaction between the emitted
electron and the final state nucleus. All of the other multiplicative
correction factors are close to unity for the vast majority of the spectrum.
For atomic tritium, the relevant corrections are: radiative corrections to the
spectrum, $G(E_{e})$; the nuclear screening from the orbital electron,
$S(E_{e})$; finite size nucleus effects, $L(E_{e})B(E_{e})$ and recoil
corrections to the nuclear Coulomb field, $Q(E_{e})$. Thus the overall factor
is

$\displaystyle C(E_{e})=F(E_{e})\times G(E_{e})\times S(E_{e})\times
L(E_{e})\times B(E_{e})\times Q(E_{e}).$ (21)

Figure 1: Fraction of decays to helium energy levels $1s$ (ground state),
$2s$, $3s$, $4s+5s$ of the orbital electron, as well emission of the orbital
electron (continuum) as a function of the $\beta$-decay electron energy below
the endpoint.

The other significant effect is that of the quantum mechanical interactions
between the emitted and orbital electron. Whilst the orbital electron is
initially in a 1s state, it is possible for it to end up in an excited state
or even be emitted into the continuum. This leads to a spectrum of endpoints
as shifted by the helium energy levels. The total decay rate is thus a sum
over the decay rates to different orbital states or the continuum, as shown in
Eq. (A.8). The probability of decay to each state as a function of
$\beta$-electron energy is shown in Fig. 1 with the endpoint shifts and
asymptotic probabilities given in Table 3. The significance of having multiple
endpoints is that instead of having a single kink in the energy spectrum for
where each mass eigenstate becomes kinematically allowed, there are now a
series of kinks of a fixed separation and sharpness. For this paper, we have
assumed an energy resolution of around 100 eV which is larger than the roughly
50 eV separation between energy levels. However, a future sterile neutrino
experiment with a higher resolution could use the multi-kink signature as a
specific signal to increase the search power.

$\text{He}^{+}$ State | Energy [eV] | $|T_{fi}^{(0)}|^{2}$
---|---|---
$1s$ | 0,0 | 70.36%
$2s$ | 40.81 | 24.98%
$3s$ | 48.37 | 1.27%
$4s$ | 51.02 | 0.38%
$5s$ | 52.24 | 0.17%
continuum |  $\geq$54.42 | 2.63%
Table 3: Excitation energy above the ground state associated with final helium
orbital electron states and the corresponding asymptotic branching ratios for
large $\beta$-electron energy.

###  3.3 Standard Model Case

Following the above general discussion on the tritium decay distribution, we
first calculate the SM result for the energy and angular distributions
following the procedure above and using the Lagrangian given by Eq. (4). The
matrix element for this contribution is given by

$\displaystyle|T_{\text{SM}}|^{2}$ $\displaystyle=16G_{F}^{2}|V_{ud}|^{2}$
$\displaystyle\times\left\\{(g_{A}+g_{V})^{2}(P_{e}\cdot
P_{\text{He}})(P_{\nu}\cdot P_{\text{H}})+(g_{A}-g_{V})^{2}(P_{e}\cdot
P_{\text{H}})(P_{\nu}\cdot P_{\text{He}})\right.$
$\displaystyle\quad+(g_{A}^{2}-g_{V}^{2})m_{\text{H}}m_{\text{He}}(P_{e}\cdot
P_{\nu})$
$\displaystyle\quad+(g_{A}^{2}-g_{V}^{2})m_{\text{He}}\left[(P_{\text{H}}\cdot
P_{\nu})(P_{e}\cdot S)-(P_{\text{H}}\cdot P_{e})(P_{\nu}\cdot S)\right]$
$\displaystyle\quad\left.+(g_{A}-g_{V})^{2}m_{\text{H}}(P_{\nu}\cdot
P_{\text{He}})(P_{e}\cdot S)-(g_{A}+g_{V})^{2}m_{\text{H}}(P_{e}\cdot
P_{\text{He}})(P_{\nu}\cdot S)\right\\}.$ (22)

Considering the case where the tritium ensemble is unpolarized ($f=0$) and
using Eq. (10) gives the energy distribution

$\displaystyle 4\pi a_{\text{SM}}(E_{e})$
$\displaystyle=\frac{G_{F}^{2}|V_{ud}|^{2}}{2\pi^{3}}C(E_{e})\sum_{i=1}^{3}|U_{ei}|^{2}\frac{m_{\text{H}}^{2}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\widetilde{y_{i}}\Theta(y_{i})$
$\displaystyle\times\left\\{(g_{V}\\!+\\!g_{A})^{2}\Bigg{[}\frac{m_{\text{H}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}\frac{m_{\text{H}}E_{e}-m_{e}^{2}}{m_{12}^{2}}\left(y_{i}\\!+\\!\mu_{i}m_{\nu,i}\right)\left(y_{i}\\!+\\!\mu_{i}m_{\text{He}}\right)\right.$
$\displaystyle\quad-\frac{m^{2}_{\text{H}}|\boldsymbol{p}_{e}|^{2}}{3m_{12}^{4}}\widetilde{y_{i}}^{2}\Bigg{]}+(g_{V}\\!-\\!g_{A})^{2}E_{e}\left(y_{i}+m_{\nu,i}\frac{m_{\text{He}}}{m_{\text{H}}}\right)$
$\displaystyle\quad+(g_{A}^{2}\\!-\\!g_{V}^{2})m_{\text{He}}\frac{m_{\text{H}}E_{e}-m_{e}^{2}}{m_{12}^{2}}\left(y_{i}+\mu_{i}m_{\nu,i}\right)\Biggr{\\}},$
(23)

where we sum over all the three mass eigenstates of active neutrinos with the
Heaviside function $\Theta(y_{i})$ ensuring energy conservation of each
individual contribution. The quantities $m_{12}^{2}$ and $y_{i}$ are defined
in Eq. (12) and Eq. (13), respectively,
$\mu_{i}=(m_{\nu,i}+m_{\text{He}})/m_{\text{H}}$ and

$\displaystyle\widetilde{y_{i}}=\sqrt{y_{i}\left(y_{i}+m_{\nu,i}\frac{2m_{\text{He}}}{m_{\text{H}}}\right)}.$
(24)

The expression in Eq. (23) matches Simkovic:2007yi  with minor typos
corrected. It can be simplified using the approximations $m_{\text{H}}\approx
m_{\text{He}}\gg E_{e},m_{e},m_{\nu}$ to give

$\displaystyle 4\pi a_{\text{SM}}(E_{e})$
$\displaystyle\approx\frac{G_{F}^{2}|V_{ud}|^{2}}{2\pi^{3}}C(E_{e})(g_{V}^{2}+3g_{A}^{2})m_{\text{H}}m_{\text{He}}\frac{m_{\text{He}}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\frac{m_{\text{H}}E_{e}-m_{e}^{2}}{m_{12}^{2}}$
$\displaystyle\times\sum_{i=1}^{3}|U_{ei}|^{2}\Theta(y_{i})\sqrt{y_{i}\left(y_{i}+m_{\nu,i}\frac{2m_{\text{He}}}{m_{\text{H}}}\right)}\left(y_{i}+m_{\nu,i}\frac{m_{\nu,i}+m_{\text{He}}}{m_{\text{H}}}\right).$
(25)

This is usually expressed in the familiar form shrock1980new ; Huang:2019tdh

$\displaystyle 4\pi
a_{\text{SM}}(E_{e})\approx\frac{G_{F}^{2}|V_{ud}|^{2}}{2\pi^{3}}C(E_{e})(g_{V}^{2}+3g_{A}^{2})m_{\text{H}}m_{\text{He}}\frac{m_{\text{H}}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\frac{m_{\text{H}}E_{e}-m_{e}^{2}}{m_{12}^{2}}y_{0}\sqrt{y_{0}^{2}-m_{\beta}^{2}},$
(26)

using the effective single $\beta$-decay mass
$m_{\beta}^{2}=\sum_{i=1}^{3}|U_{ei}|^{2}m_{i}^{2}$, and where
$y_{0}=y(m_{\nu}=0)$ ensures that all of the neutrino mass dependence has been
combined into $m_{\beta}$.

Figure 2: Kurie plot of the tritium energy distribution as measured from the
zero-mass endpoint for a massless lightest neutrino with NO (red) and IO
(blue) of the three active neutrinos. The red (blue) dashed lines correspond
to the effective $m_{\beta}$ approximation for NO (IO) as given by Eq. (3).
The gray line corresponds to the single massless active neutrino case.

Using the full expression in Eq. (23) with three active neutrinos will result
in three distinct endpoints. Instead, the simplified expression in Eq. (26)
has a single endpoint at $E_{e}=E_{0}-m_{\beta}$. The difference is
illustrated in Fig. 2, showing the Kurie plot of the energy distribution near
the endpoint using the exact expression and the approximation using
$m_{\beta}$, for a massless lightest neutrino in both normally and inversely
ordered scenarios. As described in Huang:2019tdh , the difference in the
spectrum when using this approximated version is negligible for energy
resolutions much larger than the neutrino mass or when looking at points far
from the endpoint. For our calculations we are looking across the whole
spectrum with an energy resolution much larger than the neutrino mass and thus
this difference will be negligible.

The angular correlation term in the SM case is given by

$\displaystyle b_{\text{SM}}(E_{e})$
$\displaystyle=-\frac{G_{F}^{2}|V_{ud}|^{2}}{8\pi^{4}}C(E_{e})\frac{m_{\text{H}}}{m_{12}^{2}}|\boldsymbol{p}_{e}|^{2}\widetilde{y_{i}}$
$\displaystyle\times\left\\{\left[(g_{A}-g_{V})^{2}m_{\text{H}}+(g_{A}^{2}-g_{V}^{2})m_{\text{He}}\frac{m_{\text{H}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}\right.\right.$
$\displaystyle\quad\,\,\,+(g_{A}^{2}-g_{V}^{2})\frac{m_{\text{H}}m_{\text{He}}}{m_{12}^{2}}E_{e}+(g_{A}+g_{V})^{2}\frac{m_{\text{H}}}{m_{12}^{2}}(\alpha-
m_{e}^{2})$
$\displaystyle\quad\,\,\,\left.-(g_{A}+g_{V})^{2}\frac{m_{\text{H}}^{2}}{m_{12}^{2}}\left(y_{i}+\mu_{i}m_{\nu,i}\right)\frac{m_{\text{H}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}\right]\left(y_{i}+\mu_{i}m_{\nu,i}\right)$
$\displaystyle\quad\,\,\,\left.-(g_{A}-g_{V})^{2}m_{\nu,i}^{2}-\frac{1}{3}(g_{A}+g_{V})^{2}\frac{m_{\text{H}}^{3}(m_{\text{H}}-E_{e})}{m_{12}^{4}}\widetilde{y_{i}}^{2}\right\\}.$
(27)

where
$\alpha=m_{\text{H}}E_{e}^{max}+m_{\nu_{i}}^{2}+m_{\text{He}}m_{\nu_{i}}$.
This expression can be greatly simplified under the same approximation, as
before, $m_{\text{H}}\approx m_{\text{He}}\gg E_{e},m_{e},m_{\nu_{i}}$,

$\displaystyle
b_{\text{SM}}(E_{e})\approx-\frac{2g_{A}^{2}-2g_{A}g_{V}}{g_{V}^{2}+3g_{A}^{2}}\frac{|\boldsymbol{p}_{e}|}{E_{e}}a_{\text{SM}}(E_{e})\approx-0.12\frac{|\boldsymbol{p}_{e}|}{E_{e}}a_{\text{SM}}(E_{e}).$
(28)

Using Eqs. (18), (23) and (28) gives a total half-life of
$T_{1/2}=\ln(2)/\Gamma=12.6$ yr and an angular correlation factor of
$k_{\text{SM}}=-0.0154$ in the SM. The total decay rate $\Gamma$ and the
angular correlation factor $k$ can also be obtained from experimental
measurements. The decay rate can be calculated from the half-life by
$\Gamma=\ln(2)/T_{1/2}$. The angular correlation can be found by measuring the
proportion of aligned, i.e., in the same hemisphere as the tritium spin, to
anti-aligned, i.e., in the opposite hemisphere electrons which takes the ratio
$(1+k/2)~{}\mathrel{\mathop{\mathchar 58\relax}}~{}(1-k/2)$.

This angular effect occurs due to the electron’s helicity and chirality. The
emitted electron and anti-neutrino can be either in an anti-aligned spin state
($S=0$), corresponding to the vector hadronic current, or an aligned spin
state ($S=1$), corresponding to the axial hadronic current Pal2014 . The $S=0$
emission is isotropic in spin space. However, the $S=1$ state requires the
spins to not sum within the opposite hemisphere to the tritium’s polarisation
in order to conserve angular momentum. This preferentially places the
electron’s spin within the same hemisphere as the polarisation. As the
electron is left-chiral (and thus preferentially left-helical) it is more
likely to be emitted with momentum opposite to the spin and thus in the
opposite hemisphere. Thus the decay rate to an electron parallel to the
tritium’s polarisation is reduced. Crucially, this effect is proportional to
$\hat{\boldsymbol{p}}_{e}\cdot\hat{\boldsymbol{s}}$ and thus once integrated
over the electron’s solid angle it vanishes. This is to be expected as
polarising the tritium merely orientates it rather than changing the internal
dynamics that lead to decay. Such correlations could also be calculated for
the momentum of the neutrino or the helicity of the electron. Whilst measuring
the electron’s helicity could also be informative, designing an experiment to
simultaneously obverse this and the spectrum is prohibitively challenging so
we will not consider it further here.

###  3.4 Exotic Currents

In addition to the SM contribution we include the effects of the exotic
currents in Eq. (2.1). This will lead to additional contributions to the
energy spectrum and the angular distribution, when polarised. When squaring
the matrix element we will get SM, purely exotic and SM-exotic interference
contributions to the decay rate,

$\displaystyle a(E_{e})$
$\displaystyle=a_{\text{SM}}(E_{e})+\text{Re}(\epsilon_{Y})a_{LL,Y}(E_{e})+|\epsilon_{Y}|^{2}a_{Y}(E_{e}),$
(29) $\displaystyle b(E_{e})$
$\displaystyle=b_{\text{SM}}(E_{e})+\text{Re}(\epsilon_{Y})b_{LL,Y}(E_{e})+|\epsilon_{Y}|^{2}b_{Y}(E_{e}).$
(30)

For all of the currents with an exotic left-handed leptonic current, the
interference term will dominate as it is linear in $\epsilon\ll 1$ with the
potential for cancellation between the linear and quadratic terms when
$\epsilon$ is larger. Conversely, for the right-handed leptonic currents, i.e.
those parametrised by $\tilde{\epsilon}$, a chiral flip is required and thus
the interference term is proportional to the neutrino and electron masses.
These interference terms are thus very small and the quadratic terms will
dominate.

Figure 3: Left: Electron energy distribution, in terms of the electron kinetic
energy, in the SM ($\epsilon_{T}=0$) and with a left-handed tensor current,
$\epsilon_{T}=0.02$. Right: Relative deviation between the two spectra.

We use the form factor values $g_{S}=1.02$, $g_{P}=349$, $g_{T}=1.02$ from
PhysRevLett.112.042501 ; PhysRevLett.115.212002  which are calculated from
phenomenological studies and Lattice QCD. The impact of an additional exotic
current can be seen in Fig. 3, where the presence of a tensor current reduces
the decay rate nearly homogeneously across the energy spectrum. See Appendix B
for full expressions for all exotic currents considered, where we give both
the individual exotic contribution as well as the interference with the SM.

###  3.5 Sterile Neutrinos

If we consider the sterile state as only appearing through its mixing with the
active neutrinos, as given by Eq. (2.2), then its additional contribution to
the decay rate and angular distribution is simple. The decay rate expressions
are the same as in Eqs. (23), (27) but with the sum over neutrino masses now
going from $i=1,\dots,3+n_{s}$ with $n_{s}$ sterile states and corresponding
masses $m_{4},\dots,m_{n_{s}+3}$ and mixing matrix elements
$V_{e4},\dots,V_{e{(n_{s}+3)}}$.

Figure 4: Differential decay rate in terms of the electron kinetic energy, for
a massless and various heavy, mostly-sterile neutrinos with a mixing of
$|V_{eN}|^{2}=0.25$.

The impact of a sterile neutrino on the energy distribution in such a case can
be seen in Fig. 4. Within the energy spectrum, the distinctive kink from the
neutrino mass is visible at an energy $E^{\rm{max}}_{e}-m_{N}$. The spectrum
for energies below the kink is reduced as the heavier sterile state reduces
the phase space with heavier masses giving a greater reduction. The spectrum
for energies above the kink is also reduced but is the same for all sterile
masses as it consists only of the active spectrum contribution with sterile
emission being kinematically disallowed. The magnitude of reduction throughout
the entire spectrum is also dependent upon the size of the active-sterile
mixing.

The impact on the angular distribution is depicted in Fig. 5 (left). In the
case of a sterile neutrino only mixing with the active state and thus only
inheriting a $V-A$ current, the effect is small and only arises from the
impact of the larger sterile neutrino mass. For small sterile neutrino masses
the angular correlation factor approaches that of the SM result. For large
sterile neutrino masses the sterile neutrino becomes kinematically impossible
to produce and thus its effect on the spectrum vanishes also. For intermediate
values the heaviness of the sterile neutrino reduces the correlation between
chirality and helicity (the cause of the anisotropy, as explained above) and
reduces the magnitude of $k$, bringing the distribution closer to isotropy.
For comparison, the impact of two exotic currents are also in Fig. 5 (right)
where the deviation from the SM value increases as the parameters increase,
although to differing extent. Note that the effect for the exotic currents is
two orders of magnitude greater than that for active-sterile mixing.

Figure 5: Deviation of the angular correlation factor $k$ from the SM value
$k_{\text{SM}}$ as a function of the sterile neutrino mass for different
values of the active-sterile mixing parameter $V_{eN}$ (left), and as a
function of the exotic couplings $\epsilon_{R}$ and $\tilde{\epsilon}_{L}$
(right).

However, we can also consider the presence of additional exotic currents that
couple to the sterile neutrino directly from the Lagrangian of Eq. (2.2).
These calculations are similar to those in the previous section, however as
the final state is a different neutrino (sterile rather than active) there can
be no interference between the SM result and currents involving a sterile
neutrino,

$\displaystyle a(E_{e})$
$\displaystyle=a_{\text{SM}}(E_{e})+|\epsilon^{N}_{X}|^{2}a_{X}(E_{e}),$ (31)
$\displaystyle b(E_{e})$
$\displaystyle=b_{\text{SM}}(E_{e})+|\epsilon^{N}_{X}|^{2}b_{X}(E_{e}).$ (32)

The functions $a(E_{e})$ and $b(E_{e})$ are the same as for an active neutrino
but using a different value for the neutrino mass and the relevant coupling
$\epsilon^{N}_{X}$ is the one associated with the sterile neutrino.

In this way, we consider two mechanisms in which heavy neutrinos can be
produced: indirectly, through the emission of an electron neutrino which has a
heavy component because of active-sterile mixing, and directly, with an exotic
current emitting a heavy sterile neutrino. Distinguishing between these
scenarios could be difficult with some exotic currents matching the signatures
of active-sterile mixing for particular values of $|\epsilon^{N}_{R}|^{2}$ and
$|V_{eN}|^{2}$. As can be seen in Fig. 6, in the energy spectrum both results
are essentially identical, with a kink in each, due to the exotic current
being approximately equivalent to the SM result. However, the angular
correlation factor has a significantly different dependency on the parameters
with the active-sterile mixing generating a much smaller variation.
Simultaneous measurement of energy and angular spectra can aid in
differentiating between these models as the degeneracies in parameter space
occur at different points for the different scenarios.

Figure 6: Energy and angular distributions showcasing the difference between
an active-sterile mixing and an exotic right-handed leptonic current for
producing a $10$ keV neutrino. Left: Normalised energy distributions with
active-sterile mixing, $|V_{eN}|^{2}=0.25$ (blue, dashed), overlapping with an
exotic sterile $|\epsilon^{N}_{R}|^{2}=0.33$ (orange) contribution. Right:
Deviation of the angular correlation factor $k$ from the SM value as a
function of the active-sterile mixing $|V_{eN}|$ or the right-handed leptonic
current parameter $|\epsilon_{R}^{N}|$.

##  4 Tritium Decay Experiments

The calculations in this paper have been carried out for both the energy
spectrum and the angular distribution of the final state electron. Tritium
$\beta$-decay experiments are mainly interested in measuring the absolute
masses of the active light neutrinos and thus focus on the immediate endpoint
of the energy spectrum.

There are three main experimental measurements of interest for the effects
detailed here: precision measurements of the endpoint of the electron energy
spectrum with the aim of measuring the active neutrino mass; measurements of
the full spectrum, or a bulk segment of it, with sensitivity to any sterile
neutrinos and exotic currents; and measurements of the electron angular
distribution from a polarised source with sensitivity to exotic currents.

###  4.1 KATRIN and TRISTAN

The currently strongest upper bound on the effective neutrino mass is
$m_{\beta}<0.8$ eV at $90\%$ CL Aker2022  which exceeded the previous best
limit of $m_{\beta}<1.1$ eV aker2019improved ; aker2020first . The bound comes
from the KATRIN experiment which uses a high-pass filter to measure the
integrated electron kinetic energy spectrum focusing exclusively on the
electrons within the final 300 eV near the endpoint Aker2022 . The KATRIN
collaboration estimates their ultimate sensitivity to be $m_{\beta}\sim 0.2$
eV Huang:2019tdh , hence there is great need for novel approaches if a
guaranteed measurement of absolute neutrino mass is to be made, especially as
the already large size of the central spectrometer (12 m diameter) makes
scaling up to an even larger experiment - required for going beyond 0.2 eV -
practically infeasible.

The TRISTAN project is a proposed extension of KATRIN with a novel detector
and read-out system for a high-precision keV-scale sterile neutrino search,
utilising KATRIN’s gaseous molecular tritium source. The expected experimental
signature of a sterile neutrino can be located anywhere along the electron
kinetic energy spectrum, depending on the sterile mass. In contrast to this,
the measurement of the effective active neutrino mass only distorts the end-
point of the spectrum. Given that the objective of KATRIN is to constrain the
latter, the apparatus has been designed in a way that accepts only the most
highly energetic electrons, while filtering out the rest of the events.
Therefore, KATRIN in its current state is not equipped to read out and handle
the increased number of events, which would arise as a result of processing
the entirety of the bulk, as opposed to just the end-point. TRISTAN has been
proposed in order to use the KATRIN setup to probe sterile neutrinos by
extending the acceptance range to cover the entirety of the spectrum, while
also making use of upgraded equipment in order to allow for the resulting
higher count rate to be processed. TRISTAN is expected to commence operation
once KATRIN’s data collection campaign has been completed, prospectively in
2025 Houdy:2020vhw .

###  4.2 Project 8

The Project 8 collaboration aims to directly measure the mass of electron
neutrinos making use of Cyclotron Radiation Emission Spectroscopy (CRES)
technology Pettus:2017sxd . When an emitted $\beta$-electron passes through a
magnetic field, its circular acceleration causes it to emit cyclotron
radiation. The frequency $f$ of this radiation depends on the energy of the
electron,

$\displaystyle f=\frac{1}{2\pi}\frac{eB}{E_{e}},$ (33)

where $e$ is the electron charge, $B$ the magnetic field strength and $E_{e}$
the total electron energy. Precise measurements of the frequency (using
technology already developed) thus allows for a high energy resolution,
provided that the magnetic field is sufficiently well known Pettus:2017sxd .
Measurement of the energy spectrum can thus take place with attention being
paid to either the endpoint or the full spectrum. The Project 8 collaboration
proposes to use this technology with the target of placing an upper bound of
$m_{\beta}\lesssim 40$ meV Pettus:2017sxd  but with a target energy resolution
of roughly $100$ eV when looking across the whole spectrum Drewes2017 .
Project 8 have estimated that the limit in sensitivity from a molecular source
of tritium has an irreducible value of $\approx 0.3$ eV monreal2012project  as
a result of the Final State Distribution (FSD) problem, which arises due to
final state excitation of molecular tritium. This is not accurate enough to
guarantee a measurement of the neutrino mass given the ‘worst’-case minimum
value of $m_{\beta}\sim 9$ meV for normal ordering and $m_{\beta}\sim 40$ meV
for inverted ordering doe2013project . A solution to this is to use atomic
instead of molecular tritium, resulting in the achievable sensitivity
increasing by a factor of 2 (or even more, given a large enough effective
volume), which is what allows Project 8 to project their sensitivities probing
as low as 40 meV.

###  4.3 CRESDA

Although the use of atomic tritium as proposed by the Project 8 collaboration
is expected to greatly improve the sensitivity, it is not sufficient if the
neutrino mass realised in nature is smaller than $\approx 40$ meV. Therefore,
other probes are required if a guaranteed measurement is to be made. In an
attempt to increase the energy resolution of measurements exploiting CRES
technology, and hence to push the existing and proposed bounds even further,
the _Quantum Technologies for Neutrino Mass_ (QTNM) consortium, consisting of
several UK institutes has been formed with a proposal to make guaranteed
direct neutrino mass measurements with the help of quantum technologies
QTNM:2020 . First, the aim is to develop the experimental tools and
technologies using hydrogen and deuterium atoms before moving on to tritium -
this is the CRES demonstration apparatus (CRESDA). The main challenges of a
CRES measurement that the current phase of the QTNM proposal aims to address
are the production and confinement of tritium atoms, mapping magnetic fields
in the CRES detection region with high precision, and the realisation of high
sensitivity microwave electronics devices for detection and characterisation
of the cyclotron radiation. Over the 3 year timeline, the goal of CRESDA is
four-fold: to produce and confine large quantities of hydrogen and deuterium
atoms at densities exceeding $10^{12}~{}\text{cm}^{-3}$; use the
hydrogen/deuterium atoms as quantum sensors to implement Rydberg atom
magnetometry techniques to map the magnetic field with a precision on the
order of $\pm 100$ nT; to develop microwave detection and readout electronics
components to operate at frequencies on the order of 10 GHz, and allow a
spectral resolution at the ppm level; and to design a software framework for
simulations, signal processing and sensitivity projections.

###  4.4 Future Sensitivity Goal

Figure 7: Percentage variation in the number of events in the final 1 eV of
the tritium energy distribution as a function of the lightest active neutrino
mass. The red line denotes the normal ordering case and the blue line the
inverted ordering. The gray bands denote the 95$\%$ CL for a detectable change
in the rate as a function of the total number of events in the final eV.

In this work, we aim to determine the sensitivity of future tritium decay
experiments to exotic currents and to keV-scale sterile neutrinos. This
clearly depends on the details of the experiments but in order to estimate the
sensitivity we assume an exposure such that the experiment can in principle
distinguish between the normal order and inverse order of active neutrinos
when measuring the neutrino mass near the end-point. In a $\beta$-decay
experiment of the type considered in this paper, the $\beta$-decay tritium
source produces a number of events $N_{\text{total}}$ over the observation
time. After experimental acceptance and efficiency corrections, collectively
taken into account through a factor $\eta\leq 1$, a number of events is
collected, $N_{\text{events}}=\eta N_{\text{tot}}$ (throughout this work we
take $\eta=1$). In the analysis of, e.g., the light neutrino mass, events are
specifically considered in a window,

$\displaystyle
N_{\text{win}}=\frac{N_{\text{events}}}{\Gamma}\int_{E_{1}}^{E_{2}}\frac{d\Gamma}{dE_{e}}dE_{e},$
(34)

where $E_{1}$ and $E_{2}$ are the upper and lower limits of the energy window
under consideration, respectively. The normalization is with respect to the
total tritium decay rate.

In the determination of the light neutrino mass in future experiments, a focus
will be on the final 1 eV below the endpoint. Fig. 7 shows the relative change
in the number of events in the last 1 eV of the spectrum as a function of the
lightest active neutrino mass, $m_{\text{lightest}}$, for NO and IO cases. In
principle, $m_{\text{lightest}}=0$ is possible, therefore one needs to
consider the worst case scenario when calculating the number of $\beta$-decay
events needed to resolve the two hierarchical cases. One can estimate the
sensitivity via the fluctuation of the number $N_{\text{1eV}}$ of events in
the final eV, $\sigma_{N}=1/\sqrt{N_{\text{1eV}}}$. It can be seen that the
percentage deviation for a massless lightest neutrino in the IO case is
$-0.37\%$. For the 95% CL bound in a one-tailed test requires a minimum number
of events of $N_{\text{1eV}}\approx 2.8\times 10^{5}$. Using Eq. (34), this
corresponds to a total number of events of $N_{\text{tot}}=1.3\times 10^{18}$.
In order to derive our sensitivities, we will assume a default number of total
events of $N_{\text{tot}}=10^{18}$.

The TRISTAN experiment is aiming to reach a sensitivity of
$\mathinner{\\!\left\lvert V_{eN}\right\rvert}^{2}\lesssim 10^{-6}$ on the
active-sterile neutrino mixing in the future, assuming $10^{8}$ counts per
second (cps) over a 3 year data taking period. This corresponds to a total of
$10^{16}$ events. It is estimated that the experiment could reach the
statistical limit with 3 years of data taking and by utilising the maximal
source strength of KATRIN, $10^{10}$ cps, corresponding to $10^{18}$ events.
This matches with our default value for $N_{\text{tot}}$.

###  4.5 Impact of Angular Correlations

In order to perform measurements of the angular distribution the tritium
nucleus must be polarised and the emission direction (or at least its
hemisphere relative to the polarisation) must be measurable. There has already
been much interest in polarising hydrogen nuclei due to their potential use in
nuclear fusion reactions Hupin:2019 . An Atomic Beam Source (ABS) is used for
ANKE at COSY-Jülich to produce a $\sim 20$ K hydrogen beam with a $95\%$
disassociation (i.e. proportion of atomic hydrogen) and $\sim 90\%$
polarisation Mikirtychyants:2012nh . The polarised hydrogen can then be put in
a storage cell with measurements finding that on the order of hours there was
no decay of the polarisation Ciullo:2011 .

Such a beam could not directly be the source for a Project 8 like experiment
which requires a fraction $T_{2}/T\leq 10^{-6}$ and temperature $130-170$ mK
in order to perform its precision endpoint measurements Project8:2017nal . The
Project 8 experiment expects to use a strong magnetic trap filled with 4He to
both prevent molecular recombination and cool the tritium. At the low
pressures of the experiment, virtually all of the recombination will occur at
the walls; using a strong magnetic field will trap the T, using its magnetic
moment, whilst allowing any T2, which lacks a magnetic moment, to escape. The
gas of 4He, also lacking a magnetic moment, could be used to maintain contact
with the walls of the vessel and cool the tritium Project8:2017nal ;
Clark:2014 .

Measuring both the energy and emission direction of the electron is expected
to be challenging. However, the measurements suggested herein require only
knowledge of the hemisphere of emission relative to the tritium spin. Thus if
the tritium is moving sufficiently rapidly at the time of its decay, with its
momentum parallel to its spin, the emission hemisphere of the electron could
be inferred from its kinetic energy provided the boost from being emitted
parallel to the tritium motion is larger than the energy provided by the
decay. A cut could be performed in the data to remove any emissions
ambiguously on the boundary between the two hemispheres. Alternatively, the
detection antenna could be placed in the direction of the spin and whether the
emitted electron was headed towards or away from the antenna determined.
Crucially, the direction of nuclear polarisation and the external magnetic
field are independent and thus so is the electron ‘pitch angle’ and
‘polarisation angle’. Hence there is freedom in choosing the polarisation
direction relative to the magnetic field to maximise the sensitivity to the
electron emission hemisphere.

Overall, designing an experiment that can perform all three measurements (end-
point, full spectrum and emission angle) is expected to be very challenging
due to the high precision required for setting new limits on the light
neutrino mass. However, performing full and end-point spectrum measurements is
within the target design of the Project 8 experiment and existing polarisation
technologies could be implemented within a similar setup to measure the
angular and energy spectra. Tritium is not well suited to probe the angular
spectrum, as the $Q$ value is low leading to a small kinetic energy compared
to the electron mass, leading to a washout of deviations. We nevertheless
discuss the prospect of probing it as tritium searches such as Project 8 are
expected to become crucial for neutrino physics.

##  5 Future Sensitivity to New Physics

###  5.1 Statistical Analysis

So far the presence of sterile neutrinos and exotic currents has gone
unobserved. This lack of observation has placed bounds on the parameters that
quantify these effects, $V_{eN}$, $m_{N}$ and the $\epsilon_{X}$. Sensitivity
to New Physics can be calculated using a minimal-$\chi^{2}$ test with binning
for either the energy spectrum or the angular distribution. For the angular
distribution it is appropriate to use two bins representing the ‘aligned’ and
‘anti-aligned’ hemispheres. For the energy spectrum the number of bins is
merely limited by the energy resolution of the experiment provided that the
bin width is larger than said resolution Huang:2019tdh .

In order to reflect the uncertainty in the overall normalisation of the energy
spectrum, a nuisance parameter, $A$, on the overall expected rate is
introduced. Thus our test statistic is calculated using

$\displaystyle
t=\text{min}_{A}\left[\sum_{i=1}^{N_{\text{bins}}}\frac{(N_{\text{BSM}}^{(i)}-(1+A)N_{\text{SM}}^{(i)})^{2}}{N_{\text{SM}}^{(i)}}+\left(\frac{A}{\sigma_{A}}\right)^{2}\right],$
(35)

where the subscript ‘BSM’ denotes the contribution associated with New
Physics, with the corresponding parameter of interest, such as
$\mathinner{\\!\left\lvert V_{eN}\right\rvert}^{2}$, $m_{N}$ or any of the
exotic currents considered, $\epsilon_{X}$. The uncertainty on the nuisance
parameter $\sigma_{A}$ is set to $\sigma_{A}=2$ (conservatively large but with
little impact). In Eq. (35), $t$ denotes the test-statistic and is the value
we use to measure the deviation from the SM. Theoretically, the expected
number of events needs to be the median of the expected events over a large
number of runs, as every run will produce a slightly different number of
observed events, leading to statistical fluctuations. This can be simulated by
running a series of Monte Carlo tests, which is usually a computationally
rather expensive task. Instead, one can use the Asimov data-set, which allows
for the median values over many runs to be replaced by their expectation
values gcowan2011 .

Throughout this paper the number of bins for the energy spectrum is set to
twenty (apart from Fig. 11, where the bin width is 300 eV, comparable with the
choice by TRISTAN). In the asymptotic limit, which is assumed in this paper
due to the very large number of events, the number of events in each bin can
be replaced by the expectation values, as outlined above. Thus, the content of
each bin is a fixed number, rather than a randomly fluctuating distribution,
which removes the degrees of freedom associated with the bins themselves. As a
result, the log-likelihood tends to a $\chi^{2}$ distribution with one degree
of freedom per strength parameter (i.e., $|V_{eN}|^{2}$ or one of the
$\epsilon$). At the $95\%$ CL, a $\chi^{2}$ in excess of $3.841$ for one
degree of freedom and $5.99$ for two degrees of freedom is significant. Limits
on the corresponding parameters within the ‘observed’ model can thus be placed
at the 95% CL if deviations from the SM are not observed. The $\chi^{2}$ has
been calculated with an overall number of events of $N_{\text{tot}}=10^{18}$
across all bins which corresponds to a total of $\approx 2.8\times 10^{5}$
events in the final eV below the endpoint.

###  5.2 Exotic Currents

In future $\beta$-decay experiments, measurements of the energy or angular
spectrum will allow one to either obverse or place limits upon exotic
currents. Here we summarise the upper bounds that can be placed on the
$\epsilon$ parameters in the absence of a deviation from the SM expectation.
These are for an ideal experiment, without systematic uncertainties, with a
total expected number of $10^{18}$ events.

Figure 8: $\chi^{2}$ deviation for exotic current contributions as a function
of the relevant coupling constant, using the tritium energy distribution. The
curves are plotted in solid below the coupling constant’s upper bound (cf.
Table 2) and dashed above. The horizontal black line is the 95$\%$ CL
sensitivity for $N_{\text{tot}}=10^{18}$ events, while the $N_{\text{tot}}$
scale on the right-hand axis shows the number of events required to reach a
95% CL sensitivity. The labels denote the relevant coupling constant
contributing at a time.

Fig. 8 gives the $\chi^{2}$ deviation when individual exotic currents are
included as a function of their corresponding parameter. As can be seen, the
sensitivity to scalar and tensor currents is the greatest and much more
significant than that to pseudoscalar currents. This is due to the functional
form of the additional terms with the pseudoscalar current being functionally
suppressed. For the right-handed leptonic currents, the linear term is
proportional to the neutrino mass, and thus negligible, meaning that the
quadratic term is the main contribution. This gives these currents a different
gradient to the others as the $\chi^{2}$ for these is dependent upon
$\epsilon^{4}$ unlike the others which go like $\epsilon^{2}$. The energy
sensitivity to the vector currents is much less as the additional terms are
nearly identical to the SM expression (see Appendix B) and thus sensitivity to
them is obscured by the uncertainty in the overall normalisation (i.e., the
minimisation over $A$). The dip in the right-handed hadronic current comes
from the point at which the linear and quadratic terms are roughly comparable
in magnitude leading to partial cancellation. For values of $\epsilon$ above
this, the quadratic term dominates and the right-handed hadronic term tends
towards being the same as the right-handed leptonic term.

For the angular distribution, the dependencies are largely the same but with
some currents experiencing much greater sensitivity. This of course assumes an
experiment that uses perfectly polarised tritium and that can probe the
angular distribution with no loss of efficiency. For the scalar and tensor
currents the primary variation comes from the energy-dependent change in the
energy spectrum (and thus the overall rate) which is not obscured by the
overall normalisation. This is the primary effect measured in both the energy
and angular distributions thus giving them the same sensitivity. However, for
the pseudoscalar current the energy dependent change is highly suppressed so
for the angular distribution the additional angular term is most significant.
For all of the right-handed terms the normalisation uncertainty removes most
of the energy sensitivity but cannot simultaneously remove the angular
sensitivities which have a different dependence thus they are much more
significant. On the right-hand scales in the two figures we note the required
number of events to give a $95\%$ CL sensitivity. This follows inversely
proportional to the $\chi^{2}$.

Figure 9: As Fig. 8 but using the angular distribution.

Figure 10: 95$\%$ CL regions for BSM contributions driven by two exotic
currents for $N_{\text{tot}}=10^{18}$ events. The region based on the energy
(angular) distribution is in orange (blue). The combinations of currents are:
scalar and tensor (top left), scalar and pseudoscalar (top right), right-
handed hadronic and right-handed leptonic (bottom, left for energy and right
for angular).

In actual New Physics scenarios it is unlikely that exotic currents are
triggered individually as they all potentially contribute to the energy and
angular distributions. A result that is compatible with the SM in these
measurements is, for example, also compatible with the simultaneous presence
of scalar and tensor currents. This can be seen in Fig. 10 where 95$\%$ CL
exclusion contours are shown for pairs of parameters. In the scalar,
pseudoscalar and tensor cases the additional linear terms dominate so the
regions appear to be straight lines of cancellation. For the scalar versus
tensor plot the difference between energy and angular distribution is
negligible. However, the additional sensitivity of the angular pseudoscalar
distribution leads to different gradient lines in the scalar vs pseudoscalar
plot. For the right-handed leptonic current the dominant contribution is
quadratic unlike the linear dominance for the right-handed hadronic. This
leads to parabolic regions for small values but which become more complicated
for larger values where the quadratic part of the right-handed hadronic
current becomes relevant.

###  5.3 Sterile Neutrinos

We can apply the same treatment as above to look at the sensitivity to
production of a sterile neutrino. This neutrino could either mix with the
active states or be produced directly through exotic currents. Thus we can
place limits on the mixing, $V_{eN}$ or the parameters $\epsilon^{N}$ which
will in general be dependent upon the mass of the (mostly) sterile state.

Figure 11: Projected sensitivities to the active-sterile mixing $|V_{eN}|^{2}$
as a function of sterile neutrino mass $m_{N}$ at 95% CL, for a total number
of events $N_{\text{tot}}=10^{16}$ (red solid) and $N_{\text{tot}}=10^{18}$
(black solid), alongside corresponding expected sensitivities of TRISTAN
(dashed red and dashed black). The gray shaded band corresponds to the
$1\sigma$ variation of the $95\%$ CL for a large number of simulations. The
shaded regions are excluded by 3H (blue) and 63Ni (yellow) searches together
with future experimental constraints expected from KATRIN (cyan dot-dashed)
and HUNTER (orange dot-dashed). The dotted lines show current astrophysical
constraints from: X-ray data (pink) ng2019new , CMB+BAO+$H_{0}$ observations
(green) vincent2015revisiting  and supernova data (purple) shi1993type .

Considering first, the case of a one-active + one-sterile neutrino gives a
sensitivity to mass and mixing as shown in Fig. 11. The impact of the sterile
neutrino is most felt through the ‘kink’ in the energy spectrum which occurs
at an energy equal to the sterile mass below the endpoint. As can be seen, the
greatest sensitivity to mixing is at sterile masses around half of the maximum
kinetic energy of the electron. For low sterile masses the sensitivity is
reduced because the sterile spectrum looks nearly identical to the active
spectrum. For high sterile masses the sterile spectrum increasingly vanishes
as the neutrino cannot be kinematically generated, this merely leaves a
reduced active spectrum but with the overall rate reduction hidden by the
uncertainty in the normalisation of the SM spectrum (i.e. the minimisation
over $A$ within the $\chi^{2}$, see Eq. (35)).

In Fig. 11, the tritium sensitivity is compared to existing bounds (shaded
regions) and projected sensitivities from other experiments as well as (model-
dependent) astrophysical bounds. As can be seen, the projected limits are
clearly stronger than any existing constraints from direct $\beta$-decay
measurements or even the projected sensitivity from KATRIN mertens2019novel
over the considered sterile mass range. The solid black (red) curve
corresponds to the projected sensitivity based on our calculations, for a
total number of events $N_{\text{tot}}=10^{18}$ ($N_{\text{tot}}=10^{16}$).
For comparison, the corresponding projected sensitivity of TRISTAN
mertens2019novel  is given by the dashed black and red curves.

Astrophysical and cosmological considerations can be used to constrain the
presence of sterile neutrinos in the universe. Sterile neutrinos are viable
dark matter candidates, which could have been produced in the early universe
by mixing with the active neutrinos. If they exist, the heavy states can decay
into an active neutrino alongside a mono-energetic photon, producing a
distinct line in the X-ray domain. Data from NuSTAR is thus able to constrain
the mass and mixing of a new heavy sterile state ng2019new . A heavy neutrino
state which decouples between Big Bang Nucleosynthesis (BBN) and recombination
can make the Universe appear to be younger and hence result in a larger Hubble
parameter. This effect can be balanced out by the decay of the heavy states
into neutrinos. The CMB and the sound horizon from Baryon Acoustic
Oscillations (BAO) which are remnants from the recombination era can be used
to put bounds on the mass and mixing of such a heavy state
vincent2015revisiting . Active-sterile neutrino mixing can have profound
effects in supernova explosions. A rather large mixing may disable a supernova
from exploding, while in the case of a proto-neutron star, the oscillation can
lead to a cooling effect and thus decrease the cooling time, which has been
observed to be of the order of several seconds. These potential effects
constrain the allowed active-sterile mixing shi1993type .

Another future search, HUNTER, will utilise the decay of cesium following
electron capture to measure the masses of active and sterile neutrinos. In
Phase 1, which is currently undergoing construction, the source will comprise
of $10^{8}$ atoms, resulting in $2.1\times 10^{5}$ events per 360 days of
active running. However, this will only cover a mass range of approximately 50
keV to several 100 keV, and hence lies outside of the relevant parameter space
for this work. In Phases 2 and 3, HUNTER will operate with $4\times 10^{9}$
and $3\times 10^{11}$ source atoms, resulting in $4.3\times 10^{7}$ and
$8.7\times 10^{10}$ events per 360 days of active running, respectively. In
terms of its sensitivity to the active-sterile mixing angle, HUNTER is
projected to surpass $10^{-5}$ in Phase 2, and is aiming to reach $10^{-9}$ in
Phase 3 martoff2021hunter . HUNTER Phase 2 is expected to be significantly
less sensitive than TRISTAN or a new future search based on our calculations,
while Phase 3 (as well as Phase 2) will not cover the mass range below
approximately 5 keV. Therefore, despite the promising projections for HUNTER
Phase 3, a separate search is required in the 0-5 keV range. The constraints
from X-ray, cosmological and supernovae data are in places more sensitive but
are significantly more model-dependent.

Figure 12: Sensitivity at $95\%$ CL to tensor and right-handed leptonic exotic
currents as a function of the sterile neutrino mass using the energy (solid)
and angular (dashed) distributions for currents parametrized by
$\epsilon_{T}^{N}$ (black) and $\widetilde{\epsilon}_{L}^{N}$ (blue). The
active-sterile mixing is set to zero, $\mathinner{\\!\left\lvert
V_{eN}\right\rvert}^{2}=0$.

One can also look at the sensitivity to exotic currents of sterile neutrino
beyond $V-A$. The sensitivity on these currents is dependent upon the mass of
the sterile neutrino as can be seen in Fig. 12. For the same reason as above,
the energy spectrum is less sensitive for low and high masses. This does not
apply however to the angular spectrum for low masses. Unlike with the active-
sterile mixing, there is no multiplier of the active spectrum suppressing it
as the sterile spectrum increases. This means that the additional currents
contribute more significantly to changes in the rate. The normalisation
uncertainty can hide this effect but not simultaneously to the angular change
which varies differently and thus the angular spectrum does not lose
sensitivity from this uncertainty. However, for heavier masses the reduced
sensitivity instead comes from the sterile neutrino becoming kinematically
impossible to produce and thus the impact of its presence is lost.

Parameter | Energy | Angular
---|---|---
$\epsilon^{N}_{S}$ | $2\times 10^{-4}$ | $2\times 10^{-3}$
$\tilde{\epsilon}_{S}^{N}$ | $2\times 10^{-4}$ | $2\times 10^{-3}$
$\epsilon_{P}^{N}$ | $5\times 10^{-2}$ | $0.4$
$\tilde{\epsilon}_{P}^{N}$ | $5\times 10^{-2}$ | $0.4$
$\epsilon_{T}^{N}$ | $3\times 10^{-5}$ | $1\times 10^{-4}$
$\tilde{\epsilon}_{T}^{N}$ | $3\times 10^{-5}$ | $1\times 10^{-4}$
$\epsilon_{L}^{N}$ | $8\times 10^{-5}$ | $2\times 10^{-3}$
$\tilde{\epsilon}_{L}^{N}$ | $8\times 10^{-5}$ | $2\times 10^{-4}$
$\epsilon_{R}^{N}$ | $8\times 10^{-5}$ | $3\times 10^{-4}$
$\tilde{\epsilon}_{R}^{N}$ | $8\times 10^{-5}$ | $5\times 10^{-4}$
Table 4: Projected sensitivity at 95% CL on the coupling constants of exotic
currents of a sterile neutrino with mass $m_{N}=10$ keV. The sensitivity
assumes $10^{18}$ events in either the energy or angular distribution.

Similarly, we consider the sensitivity to different currents with sterile
neutrinos when considered individually with the sterile mass fixed to $10$
keV, i.e., the mass where the strongest sensitivity is approximately reached,
see Fig. 12. Unlike for the active case, there is no linear term in the
coefficients as the sterile neutrino cannot interfere with the SM term meaning
that turning on each current individually only leaves their quadratic terms.
Furthermore, many of these terms actually have exactly or approximately
identical quadratic terms. As in the active-sterile mixing case but unlike the
active neutrino exotic currents, for a $10$ keV neutrino the energy
distribution is more sensitive than the angular distribution. This is because
the energy distribution contains the distinctive kink signature coming from
the kinematic exclusion of heavy neutrinos above their threshold which is
absent from the angular distribution by integrating over the entire spectrum.
Table 4 gives the $95\%$ CL upper bounds on the relevant coupling constants,
for a 10 keV sterile neutrino. In general, the right-handed leptonic currents
are more sensitive than their active counterparts whilst the left-handed
currents are less sensitive. This is because the lack of a linear term for the
left-handed currents makes the dependence quadratic and thus the current
sensitivity is reduced; for the right-handed currents the active counterparts
lacked a linear term anyway so instead the distinct spectrum from the sterile
neutrino gives a greater sensitivity than the nearly identical extra terms in
the active case. Again these parameters are taken to be real and positive, if
seen as complex then the limits are equivalently placed upon
$|\epsilon^{N}_{X}|$.

Figure 13: Projected sensitivity at $95\%$ CL on the active-sterile mixing
$V_{eN}$ and the right-handed current of sterile neutrinos,
$\tilde{\epsilon}^{N}_{L}$ for $m_{N}=10$ keV and $10^{18}$ events. The region
derived from the energy (angular) distribution is given in orange (blue).

Finally, we consider the presence of both an exotic current and the active-
sterile mixing simultaneously, assuming a 10 keV sterile neutrino. The
projected sensitivity at $95\%$ CL on $|V_{eN}|^{2}$ and the right-handed
coupling $\tilde{\epsilon}^{N}_{L}$ can be seen in Fig. 13. Using the energy
distribution, the sensitivity to either is practically identical, leading to a
circular contour, because the primary sensitivity is to the kink which affects
both equally. However, using the angular distribution, the sensitivity to the
right-handed current is much higher because the corresponding contribution has
an overall factor of $2g_{A}^{2}+2g_{A}g_{V}$, unlike the active neutrino case
(which the active-sterile mixing will enhance), which is proportional to
$2g_{A}^{2}-2g_{A}g_{V}$. Overall the angular sensitivity is weaker for
aforementioned reasons.

##  6 Conclusions

Tritium $\beta$-decay is the method of choice for measuring the masses of
active neutrinos model-independently. Its sensitivity is not affected by
whether the neutrinos are of Dirac or Majorana nature, and as a laboratory
measurement it is independent of astrophysical and cosmological
considerations. Achieving sensitivities of order
$\mathcal{O}(10~{}\text{meV})$, as would be required in the most pessimistic
scenario of normally ordered neutrino states with a massless lightest
neutrino, is nevertheless extremely challenging. The next generation of
experiments, such as Project 8 and CRESDA, promise high resolutions near the
endpoint with the potential for measurements across the entire
$\beta$-spectrum. Their attempts to measure the neutrino masses also opens up
the possibility to search for New Physics such as sterile neutrinos and exotic
currents, complementing dedicated probes of the overall tritium spectrum
(e.g., the proposed TRISTAN extension of the KATRIN experiment) and in other
$\beta$-decaying or electron-capture isotopes, such as the proposed HUNTER
experiment.

Current $\beta$-decay limits on the active-sterile mixing are of the order
$|V_{eN}|^{2}\sim 10^{-2}-10^{-3}$ coming from experiments with tritium or
nickel. In this work we have analysed the prospect of future tritium
experiments to probe the existence of sterile keV-scale neutrinos as well as
exotic currents beyond $V-A$. The signature of emission of sterile neutrinos
through active-sterile mixing is a kink in the spectrum with a reduced overall
rate compared to the SM prediction. Assuming a statistics with
$N_{\text{tot}}=10^{18}$ events, corresponding to a sensitivity to the
inversely ordered active neutrino scenario in the final eV below the endpoint,
future experiments could reach sensitivities of $|V_{eN}|^{2}\sim
10^{-7}-10^{-8}$ within the mass range $1~{}\text{keV}\lesssim m_{N}\lesssim
18~{}\text{keV}$.

Exotic charged currents beyond the Fermi interaction and its $V-A$ structure
can also be probed in tritium decay, leaving an imprint on both the energy and
angular distributions. The latter requires polarisation of the source and
tritium is generally not ideal due to the low $Q$ value and correspondingly
small kinetic energy of the electron. We nevertheless analyse it to assess the
potential, e.g. in CRES type experiments. We have calculated the double
differential decay rate in terms of both the electron energy and the electron
momentum direction with respect to the nuclear spin of the tritium nucleus.
The potential sensitivity to exotic currents varies significantly depending on
the functional form of their contribution to the spectrum, with a suitable
future tritium decay experiment expected to improve on some of the current
constraints. For sterile neutrinos, which may also participate in exotic
currents (maybe more likely so), the sensitivity depends on their mass. The
strongest sensitivity is typically reached for $m_{N}\approx 10~{}\text{keV}$.
Our analysis shows that future tritium $\beta$-decay experiments have the
potential for many uses beyond measuring the active neutrino masses. Designing
an experiment to observe the endpoint and entire spectrum with sufficient
resolution as well as polarising the tritium and measuring the electron
direction may be challenging but our work is intended as motivation to explore
the prospects further.

###### Acknowledgements.

The authors acknowledge support from the Science and Technology Facilities
Council, part of U.K. Research and Innovation, Grant No. ST/T000880/1. The
authors would like to thank the members of the QTNM collaboration for useful
comments. The work of F.F.D. was performed in part at the Aspen Center for
Physics, which is supported by National Science Foundation grant PHY-1607611.
This work was partially supported by a grant from the Simons Foundation.

##  Appendix A Correction Factors to the Beta-Decay Spectrum

In deriving the $\beta$-decay spectrum in Eq. (23), the factor $C(E_{e})$ was
added to account for additional theoretical corrections to the distribution of
the electron kinetic energy. These corrections come from many different
effects and are reproduced here from Kleesiek2019  and Mertens:2014nha . We
give a brief description of their origin along with their functional form for
the sake of convenience. The impact of each correction on the bulk region of
the distribution is discussed. They are approximately ordered by the magnitude
of their effect upon the spectrum. The corrections are presented using the
following conventions. Natural units ($\hbar=c=1$) are used throughout unless
otherwise stated. Energies and momenta are often given in units of the
electron mass $m_{e}$, namely, $W=E_{e}/m_{e}$ is the electron energy in units
of $m_{e}$, $W_{0}=(E_{e}^{\text{max}}-V_{f})/m_{e}$ is the tritium endpoint
energy in units of $m_{e}$, with $V_{f}$ describing a change in the endpoint
for excited states, and $k=\sqrt{W^{2}-1}$ is the magnitude of the electron
spatial momentum in units of $m_{e}$. The electron speed in units of $c$ is
$\beta=|\boldsymbol{p}_{e}|/E_{e}$. The atomic charge of the daughter nucleus,
i.e., helium in our case, is denoted as $Z=2$. The fine structure constant is
$\alpha=1/137.036$ and $\lambda_{t}=|g_{A}/g_{V}|=1.247$ is the ratio of
nuclear axial vector and vector coupling constants. The nuclear radius is
denoted as $R_{n}\approx 2.88\times 10^{-3}/m_{e}$ Kleesiek2019 .

Figure 14: Plots showing the effect of theoretical corrections on the
differential spectrum. Upper left: radiative corrections. Upper right: orbital
electron shielding. Lower left: scaling of the electric field and convolution
of electron and neutrino wave functions, both within the nucleus. Lower right:
recoiling nuclear charge.

###  A.1 Fermi function

The Fermi function corrects for the interaction between the charges of the
emitted electron and the daughter nucleus. The fully relativistic expression
of the Fermi function, $F(Z,E_{e})$, can be derived from the relativistic
eigenfunctions for hydrogen-like nuclei under the assumption that the electric
field within the nucleus is very similar to that at its surface Fermi1934 .
The functional form for $F(Z,E_{e})$ is given by Kleesiek2019

$\displaystyle
F(Z,E_{e})=4(2|\boldsymbol{p}_{e}|R_{n})^{-2(1-\gamma)}e^{\pi\eta}\frac{|\Gamma(\gamma+i\eta)|^{2}}{\Gamma(2\gamma+1)^{2}}\approx\frac{2\pi\eta(1.002037-0.001427|\boldsymbol{p}_{e}|/E_{e})}{1-e^{-2\pi\eta}},$
(36)

where $\eta=\alpha ZE_{e}/|\boldsymbol{p}_{e}|$ is known as the Sommerfeld
parameter and $\gamma=(1-(\alpha Z)^{2})^{\frac{1}{2}}$. The approximate form
(known as the Simpson approximation PhysRevD.23.649 ) applies in the limit
$\gamma\approx 1$, i.e., small $Z$ as is the case for tritium.

###  A.2 Radiative Corrections

Radiative corrections come from loop photon diagrams with interactions between
the electron and either the initial neutron, final proton or intermediary $W$
boson. First order calculations lead to an infrared divergence near the
endpoint due to the emission of soft (i.e., low energy), real photons. These
can be corrected by considering soft photon emission to all orders. Taking
this into consideration gives

$\displaystyle G(E_{e})$
$\displaystyle=\left(\frac{E_{e}^{\text{max}}-E_{e}}{m_{e}}\right)^{(2\alpha/\pi)t(\beta)}\left\\{1+\frac{2\alpha}{\pi}\left[t(\beta)\left(\ln
2-\frac{3}{2}+\frac{E_{e}^{\text{max}}-E_{e}}{E_{e}}\right)\right.\right.$
$\displaystyle+\frac{1}{4}(t(\beta)+1)\left(2(1+\beta^{2})-2\ln\left(\frac{2}{1-\beta}\right)+\frac{(E_{e}^{\text{max}}-E_{e})^{2}}{6E_{e}^{2}}\right)$
$\displaystyle\left.\left.+\frac{1}{2\beta}\left(L(\beta)-L(-\beta)+L\left(\frac{2\beta}{1+\beta}\right)+\frac{1}{2}L\left(\frac{1-\beta}{2}\right)-\frac{1}{2}L\left(\frac{1+\beta}{2}\right)\right)\right]\right\\},$
(37)

where $t(\beta)=\text{arctanh}(\beta)/\beta-1$ and $L(x)$ is Spence’s
function, $L(x)=\int_{0}^{x}\frac{\ln(1-t)}{t}dt$ Repko:1984cs .

###  A.3 Screening

The Coulomb interaction between the emitted electron and the daughter nucleus
assumes that the nuclear charge is bare. Therefore, an extra correction factor
$S(Z,E_{e})$ needs to be considered in order to take into account the
screening effect of the bound atomic electron on the Coulomb field. This
correction is of the form Behrens1982-zb

$\displaystyle
S(Z,W)=\frac{\overline{W}}{W}\left(\frac{\overline{k}}{k}\right)^{-1+2\gamma}\frac{\mathinner{\\!\left\lvert\Gamma(\gamma+i\overline{\eta})^{2}\right\rvert}}{\mathinner{\\!\left\lvert\Gamma(\gamma+i\eta)^{2}\right\rvert}}e^{\pi(\overline{\eta}-\eta)},$
(38)

where $\overline{W}=W-V_{0}/m_{e}$, $\overline{k}=\sqrt{\overline{W}^{2}-1}$
and $\overline{\eta}=\alpha Z\overline{W}/\overline{k}$. Here, $V_{0}=(76\pm
10)$ eV is the nuclear screening potential of a $1s$ helium orbital electron
PhysRevC.60.034608 .

###  A.4 Recoil Effects

Recoil effects arise from the weak magnetism and $V-A$ terms, and the three-
body phase space. These terms are all dependent upon the $Q$ value and thus
vanish in the zero recoil case. To take them into account, usually a
correction term $R(W,W_{0})$ is applied to the energy distribution,

$\displaystyle R(W,W_{0})=1+\frac{1}{C}\left(AW-\frac{B}{W}\right),$ (39)

where

$\displaystyle A$
$\displaystyle=2(5\lambda_{t}^{2}+\lambda_{t}\mu+1)/m_{\text{He}},$
$\displaystyle B$ $\displaystyle=2\lambda_{t}(\mu+\lambda_{t})/m_{\text{He}},$
(40) $\displaystyle C$ $\displaystyle=1+3\lambda_{t}^{2}-bW_{0},$

and $\mu=5.107$ is the difference in bare triton and helion magnetic moments.
In this work the differential decay rates are calculated from the full matrix
element, and hence include all significant $V-A$ terms and also take into
account 3-body effects. Therefore, this correction term is not part of the
overall correction factor, but is nevertheless included here for completeness.

###  A.5 Finite Nuclear Size

In many cases it is sufficient to consider the nucleus as a point-like
structure. However, the finite structure of the daughter nucleus can be
included by using the corrections $L(Z,W)$ and $B(Z,W)$
wilkinson1990evaluation . The function $L(Z,W)$ accounts for the scaling of
the Coulomb field and is given by

$\displaystyle L(Z,W)=1+\frac{13}{60}(\alpha Z)^{2}-\frac{WR_{n}\alpha
Z}{15}\frac{41-26\gamma}{2\gamma-1}-\frac{\alpha
ZR_{n}\gamma}{30W}\frac{17-2\gamma}{2\gamma-1}.$ (41)

The function $B(Z,W)$, on the other hand, takes into account the convolution
of the electron and neutrino wave functions with that of the nucleus for the
whole nuclear volume,

$\displaystyle B(Z,W)=1+B_{0}+B_{1}W+B_{2}W^{2},$ (42)

with

$\displaystyle B_{0}$ $\displaystyle=-\frac{233}{630}(\alpha
Z)^{2}-\frac{1}{5}(W_{0}R_{n})^{2}+\frac{2}{35}(W_{0}R_{n}\alpha Z),$
$\displaystyle B_{1}$ $\displaystyle=-\frac{21}{35}R_{n}\alpha
Z+\frac{4}{9}W_{0}R_{n}^{2},$ (43) $\displaystyle B_{2}$
$\displaystyle=-\frac{4}{9}R_{n}^{2}.$

###  A.6 Recoiling Coulomb Field

The Fermi function treats the nucleus as infinitely heavy. In reality, it is
not and as a result the nucleus recoils upon the emission of the electron and
neutrino. This in turn means the origin of the Coulomb field is in fact a
moving source - an effect captured by the function wilkinson1982analysis

$\displaystyle Q(Z,W,W_{0})=1-\frac{\pi\alpha
Z}{m_{\text{He}}k}\left(1+\frac{1-\lambda_{t}^{2}}{1+3\lambda_{t}^{2}}\frac{W_{0}-W}{3W}\right).$
(44)

###  A.7 Orbital Electron Interactions

The bound orbital electron will undergo a sudden change of state from a
hydrogen energy level to a helium energy level. It may also interact with the
emitted $\beta$-electron, with the possibility that the emitted electron may
knock the atomic electron out of the atom, and become bound, while the
original orbital-electron is ejected and is observed as the $\beta$-particle.
Considering both of these scenarios for decay to the $1s$ helium state induces
the correction Haxton:1985cz

$I(Z,W)=1+\frac{729}{256}a^{2}(\tau)+\frac{27}{16}a(\tau),$ (45)

with

$\displaystyle
a(\tau)=\exp\left[2\tau\arctan\left(-\frac{2}{\tau}\right)\right]\left(\frac{\tau^{2}}{1+\frac{1}{4}\tau^{2}}\right)^{2},$
(46)

and $\tau=-2\alpha/k$. This case only accounts for exchange with the $1s$
electron and thus $I(Z,W)$ is not included in our calculation, as the more
exact continuous emission correction described in the next section is applied
instead.

###  A.8 Excited States and Continuous Emission of The Atomic Electron

The tritium orbital electron is in principle indistinguishable from the
emitted electron after $\beta$-decay. Thus it is important to consider what
happens to this electron which may either end up in a bound helium state or in
the continuum. Included in this must also be the possibility of quantum
mechanical exchange with the emitted electron becoming bound and vice versa.
The energy of the orbital electron determines the energy available to the
$\beta$-electron as it may give off or take energy from the decay process.
Table 3, based on PhysRevC.27.1815 , summarises the energy gap between the
possible excitation states of the atomic electron left behind, and also states
the transition probabilities, i.e., branching ratio, for each.

The probability of population of each excited final state is shown in the
zeroth-order sudden approximation (third column) and includes the interaction
between the $\beta$ and atomic electron. The sudden approximation, which is
well supported in this 4-point interaction approximation, assumes that a rapid
change in the Hamiltonian allows a transition amplitude of the form
PhysRevC.27.1815  $\bra{b}U(t_{2},t_{1})\ket{a}$, to be simply approximated by
the overlap $\bra{b}\ket{a}$, as $t_{2}-t_{1}\sim 0$, where $U(t_{2},t_{1})$
is some transition function depending on time $t_{1}$ and a later time
$t_{2}$. Therefore, the transition probability is
$|\bra{b}U(t_{2},t_{1})\ket{a}|^{2}\sim|\bra{b}\ket{a}|^{2}$.

The total decay rate is thus expressed as a sum over decays to different
energy levels. The decay rate to an individual energy level is identical as
described above, however modified by a shift in the endpoint. This is done by
shifting the effective masses for the tritium and helium by including the
electron and its binding or kinetic energy,

$\displaystyle
E_{e}^{\text{max}}\to\begin{cases}E_{e}^{\text{max}}(n),\;m_{\text{H}}\to
m_{\text{H}}-R_{h},\;m_{\text{He}}\to
m_{\text{He}}+m_{e}-4R_{h}/n^{2},&n=1,2,3,\dots,\\\
E_{e}^{\text{max}}(\tau),\;m_{\text{H}}\to
m_{\text{H}}-R_{h},\;m_{\text{He}}\to
m_{\text{He}}+m_{e}+4R_{h}/\tau^{2},&\tau=-2\alpha/k,\end{cases}$ (47)

where $R_{h}=13.61$ eV is the Rydberg energy PhysRevD.98.030001  and with the
top case being to a helium bound state and the bottom being to the continuum.
Combining these terms with the appropriate weightings gives the full decay
rate over all possible bound and continuum helium final states Haxton:1985cz ,

$\displaystyle\frac{d\Gamma_{\text{full}}}{dE_{e}}$
$\displaystyle=\sum_{n=1}^{\infty}2\frac{d\Gamma}{dE_{e}}(E_{e}^{\text{max}}(n))\left[256n^{5}\frac{(n-2)^{2n-4}}{(n+2)^{2n+4}}+\frac{\alpha^{2}(\tau)}{n^{3}}-16n\frac{(n-2)^{n-2}}{(n+2)^{n+2}}\alpha(\tau)\right]$
$\displaystyle+\frac{1}{\pi}\int_{-\infty}^{\tau}\frac{d\tau^{\prime}}{{\tau^{\prime}}^{4}}\frac{d\Gamma}{dE_{e}}(E_{e}^{\text{max}}(\tau^{\prime}))\frac{2\pi\tau^{\prime}}{e^{2\pi\tau^{\prime}}-1}\left[\alpha^{2}(\tau)-\alpha(\tau)\alpha(\tau^{\prime})+\alpha^{2}(\tau^{\prime})\right],$
(48)

Here, $a(\tau)$ is given in Eq. (46) and $(n-2)^{n-2}\to 1$ as $n\to 2$ for
the sake of evaluation. The first line involves summing over decays to
discrete bound energy levels while the second is an integral over all possible
continuum energies for the non-measured electron. This leads to an energy
dependence for the probability of decay to a particular energy level as can be
seen in Fig. 1.

##  Appendix B Full Expressions for Tritium Differential Decay Rate

The $\beta$-decay rate can be generally written in the form

$\displaystyle\frac{d\Gamma}{dE_{e}d\Omega_{e}}$
$\displaystyle=a(E_{e})+b(E_{e})\cos\theta_{e}$ (49)

If two effective operators $\mathcal{O}_{X}$, $\mathcal{O}_{Y}$ in Eq. (5)
contribute to the decay for a given neutrino mass eigenstate, the transition
matrix element may be composed as

$\displaystyle T=\epsilon_{X}T_{X}+\epsilon_{Y}T_{Y},$ (50)

where $T_{X}$ and $T_{Y}$ are the individual matrix elements and
$\epsilon_{X}$, $\epsilon_{Y}$ are the associated effective coupling
constants. The terms in the differential decay rate Eq. (49) can then be
written as

$\displaystyle a(E_{e})$
$\displaystyle=|\epsilon_{X}|^{2}a_{\text{X}}(E_{e})+\text{Re}(\epsilon_{X}\epsilon_{Y}^{*})a_{X,Y}(E_{e})+|\epsilon_{Y}|^{2}a_{Y}(E_{e}),$
(51) $\displaystyle b(E_{e})$
$\displaystyle=|\epsilon_{X}|^{2}b_{\text{X}}(E_{e})+\text{Re}(\epsilon_{X}\epsilon_{Y}^{*})b_{X,Y}(E_{e})+|\epsilon_{Y}|^{2}b_{Y}(E_{e}).$
(52)

Most importantly, as the SM contribution with both left-handed lepton and
quark vector currents is always present, we consider its interference with an
exotic operator. Thus, $X=LL$, $\epsilon_{X}=1$ (for a single active neutrino)
and $Y$ is one of the exotic operators in Eq. (2.1) or Eq. (2.2).

Below are presented the exact expressions for some of these terms. We
concentrate on the individual contribution of each operator and the
interference of exotic operators with that in the SM. The expressions are
written in terms of a single generic neutrino mass eigenstate $\nu$ with mass
$m_{\nu}$, which can be either one of the active neutrinos, $\nu_{1,2,3}$, or
a sterile neutrino $\nu_{N}$. In the latter case, there is no interference
with any contributions from active neutrinos. For simplification, we define

$\displaystyle\widetilde{C}(E_{e})$
$\displaystyle=\frac{1}{4\pi}\frac{G_{F}^{2}|V_{ud}|^{2}}{2\pi^{3}}C(E_{e})\Theta(E_{e}^{\text{max}}-E_{e}-m_{\nu}),$
(53) $\displaystyle m_{12}^{2}$
$\displaystyle=(p_{\text{H}}-p_{e})^{2}=m_{\text{H}}^{2}-2m_{\text{H}}E_{e}+m_{e}^{2},$
(54) $\displaystyle|\boldsymbol{p}_{e}|$
$\displaystyle=\sqrt{E_{e}^{2}-m_{e}^{2}},$ (55) $\displaystyle\mu$
$\displaystyle=(m_{\nu}+m_{\text{He}})/m_{\text{H}},$ (56) $\displaystyle y$
$\displaystyle=E_{e}^{\text{max}}-E_{e},$ (57) $\displaystyle\widetilde{y}$
$\displaystyle=\sqrt{y\left(y+m_{\nu}\frac{2m_{\text{He}}}{m_{\text{H}}}\right)}$
(58) $\displaystyle\alpha$
$\displaystyle=m_{\text{H}}E_{e}^{\rm{max}}+m_{\nu}^{2}+m_{\text{He}}m_{\nu}.$
(59)

###  B.1 Individual SM and Exotic Contributions

Vector Currents $LL$, $RR$, $RL$, $LR$

The SM contribution $(LL)$ is related to that of $\tilde{\epsilon}_{R}$ $(RR)$
by using $\gamma_{5}\to-\gamma_{5}$ (or equivalently $S\to-S$). This results
in the same energy distributions and reverse angular correlations,

$\displaystyle a_{LL}(E_{e})$ $\displaystyle=a_{RR}(E_{e})$
$\displaystyle=\widetilde{C}(E_{e})\frac{m_{\text{H}}^{2}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\widetilde{y}$
$\displaystyle\times\left\\{(g_{V}\\!+\\!g_{A})^{2}\left[\frac{m_{\text{H}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}\frac{m_{\text{H}}E_{e}-m_{e}^{2}}{m_{12}^{2}}(y\\!+\\!\mu
m_{\nu})(y\\!+\\!\mu
m_{\text{He}})\right.-\frac{m^{2}_{\text{H}}|\boldsymbol{p}_{e}|^{2}}{3m_{12}^{4}}\widetilde{y}^{2}\right]$
$\displaystyle\quad+(g_{V}\\!-\\!g_{A})^{2}E_{e}\left(y+m_{\nu}\frac{m_{\text{He}}}{m_{\text{H}}}\right)+(g_{A}^{2}\\!-\\!g_{V}^{2})m_{\text{He}}\frac{m_{\text{H}}E_{e}-m_{e}^{2}}{m_{12}^{2}}(y+\mu
m_{\nu})\Biggr{\\}},$ (60) $\displaystyle b_{LL}(E_{e})$
$\displaystyle=-b_{RR}(E_{e})$
$\displaystyle=-\widetilde{C}(E_{e})\frac{m_{\text{H}}}{m_{12}^{2}}|\boldsymbol{p}_{e}|^{2}\widetilde{y}$
$\displaystyle\times\left\\{\left[(g_{A}-g_{V})^{2}m_{\text{H}}+(g_{A}^{2}-g_{V}^{2})m_{\text{He}}\frac{m_{\text{H}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}\right.\right.$
$\displaystyle\quad\,\,\,+(g_{A}^{2}-g_{V}^{2})\frac{m_{\text{H}}m_{\text{He}}}{m_{12}^{2}}E_{e}+(g_{A}+g_{V})^{2}\frac{m_{\text{H}}}{m_{12}^{2}}(\alpha-
m_{e}^{2})$
$\displaystyle\quad\,\,\,\left.-(g_{A}+g_{V})^{2}\frac{m_{\text{H}}^{2}}{m_{12}^{2}}(y+\mu
m_{\nu})\frac{m_{\text{H}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}\right](y+\mu
m_{\nu})$
$\displaystyle\quad\,\,\,\left.-(g_{A}-g_{V})^{2}m_{\nu}^{2}-\frac{1}{3}(g_{A}+g_{V})^{2}\frac{m_{\text{H}}^{3}(m_{\text{H}}-E_{e})}{m_{12}^{4}}\widetilde{y}^{2}\right\\}.$
(61)

The contributions related to $\tilde{\epsilon}_{L}$ $(LR)$ and $\epsilon_{R}$
$(RL)$ can be obtained from the SM $(LL)$ terms by applying $g_{A}\to-g_{A}$
with an additional $S\to-S$ for the right-handed lepton term,

$\displaystyle a_{LR}(E_{e})$
$\displaystyle=a_{RL}(E_{e})=a_{LL}(E_{e})(g_{A}\to-g_{A}),$ (62)
$\displaystyle b_{LR}(E_{e})$
$\displaystyle=-b_{RL}(E_{e})=-b_{LL}(E_{e})(g_{A}\to-g_{A}).$ (63)

This makes a negligible difference in form.

#### Scalar Currents

$\displaystyle a_{S}(E_{e})$
$\displaystyle=\frac{1}{2}g_{S}^{2}\widetilde{C}(E_{e})\frac{m_{\text{H}}^{2}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\widetilde{y}$
$\displaystyle\times\Biggr{\\{}(m_{\text{He}}+m_{\text{H}}-E_{e})\frac{m_{\text{H}}E_{e}-m_{e}^{2}}{m_{12}^{2}}(y\\!+\\!\mu
m_{\nu})-\frac{1}{3}\frac{m_{\text{H}}^{2}|\boldsymbol{p}_{e}|^{2}}{m_{12}^{4}}\widetilde{y}^{2}$
$\displaystyle-\frac{m_{\text{H}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}\frac{m_{\text{H}}E_{e}-m_{e}^{2}}{m_{12}^{2}}(y\\!+\\!\mu
m_{\nu})^{2}\Biggr{\\}},$ (64) $\displaystyle b_{S}(E_{e})$ $\displaystyle=0.$
(65)

$\displaystyle a_{\tilde{S}}(E_{e})$ $\displaystyle=a_{S}(E_{e}),$ (66)
$\displaystyle b_{\tilde{S}}(E_{e})$ $\displaystyle=0.$ (67)

#### Pseudoscalar Currents

$\displaystyle a_{P}(E_{e})$ $\displaystyle=a_{S}(E_{e})(g_{S}\to g_{P}),$
(68) $\displaystyle b_{P}(E_{e})$ $\displaystyle=0.$ (69)

$\displaystyle a_{\tilde{P}}(E_{e})$ $\displaystyle=a_{S}(E_{e})(g_{S}\to
g_{P}),$ (70) $\displaystyle b_{\tilde{P}}(E_{e})$ $\displaystyle=0.$ (71)

#### Tensor Currents

$\displaystyle a_{T}(E_{e})$ $\displaystyle=a_{\tilde{T}}(E_{e})=$
$\displaystyle=16\widetilde{C}(E_{e})g_{T}^{2}\frac{m_{\text{H}}^{2}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\widetilde{y}\Biggr{[}\frac{m_{\text{H}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}\frac{m_{\text{H}}E_{e}-m_{e}^{2}}{m_{12}^{2}}(y\\!+\\!\mu
m_{\nu})(3y+2\mu m_{\text{He}}+\mu m_{\nu})$
$\displaystyle-\frac{1}{3}\frac{m_{\text{H}}^{2}|\boldsymbol{p}_{e}|^{2}}{m_{12}^{4}}\widetilde{y}^{2}+2E_{e}\left(y+m_{\nu}\frac{m_{\text{He}}}{m_{\text{H}}}\right)-2(m_{\text{H}}-E_{e})(y\\!+\\!\mu
m_{\nu})\frac{m_{\text{H}}E_{e}-m_{e}^{2}}{m_{12}^{2}}\Biggr{]},$ (72)
$\displaystyle b_{T}(E_{e})$ $\displaystyle=-b_{\tilde{T}}(E_{e})$
$\displaystyle=16\widetilde{C}(E_{e})g_{T}^{2}\frac{m_{\text{H}}^{2}}{m_{12}^{2}}|\boldsymbol{p}_{e}|^{2}\widetilde{y}\left[\frac{1}{3}\frac{m_{\text{H}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}\frac{m_{\text{H}}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\widetilde{y}\right.$
$\displaystyle\qquad\left.+\left(2+\frac{m_{\text{H}}E_{e}-m_{e}^{2}}{m_{12}^{4}}m_{\text{H}}(y+\mu
m_{\nu})-\frac{m_{\text{H}}E_{e}}{m_{12}^{2}}-\frac{2m_{\nu}^{2}}{m_{\text{H}}|\boldsymbol{p}_{e}|}\right)(y+\mu
m_{\nu})\right].$ (73)

###  B.2 Interference Terms

#### Vector Currents: $(LL)$ with $(LR)$

$\displaystyle a_{LL,LR}(E_{e})$
$\displaystyle=\widetilde{C}(E_{e})m_{e}m_{\nu}\frac{m_{\text{H}}^{2}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\widetilde{y}\Bigg{[}2(g_{A}^{2}-g_{V}^{2})m_{\text{He}}+(g_{A}^{2}\\!+\\!g_{V}^{2})(m_{\text{H}}-E_{e})$
$\displaystyle\qquad\qquad\qquad\left.-(g_{A}^{2}+g_{V}^{2})\frac{m_{\text{H}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}(y+\mu
m_{\nu})\right],$ (74) $\displaystyle b_{LL,LR}(E_{e})$
$\displaystyle=-\widetilde{C}(E_{e})g_{A}g_{V}m_{e}m_{\nu}|\boldsymbol{p}_{e}|^{2}\frac{m_{\text{H}}}{m_{12}^{2}}\widetilde{y}\left[1-\frac{m_{\text{H}}}{m_{12}^{2}}(y\\!+\\!\mu
m_{\nu})\right].$ (75)

All other interference terms with differing leptonic handedness can be related
by $\gamma^{5}\to-\gamma^{5}$ (with $S\to-S$) or $g_{A}\to-g_{A}$ ,

$\displaystyle a_{LR,LL}(E_{e})$
$\displaystyle=a_{RR,RL}(E_{e})=a_{RL,RR}(E_{e})=a_{LL,LR}(E_{e}),$ (76)
$\displaystyle b_{LR,LL}(E_{e})$
$\displaystyle=b_{RR,RL}(E_{e})=-b_{RL,RR}(E_{e})=-b_{LL,LR}(E_{e}).$ (77)

#### Vector Currents: $(LL)$ with $(RL)$

$\displaystyle a_{LL,RL}(E_{e})$
$\displaystyle=-\widetilde{C}(E_{e})\frac{m_{\text{H}}^{2}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\widetilde{y}$
$\displaystyle\times\left\\{(g_{A}^{2}\\!-\\!g_{V}^{2})\left[\frac{m_{\text{H}}(m_{\text{H}}\\!-\\!E_{e})}{m_{12}^{2}}\frac{m_{\text{H}}E_{e}\\!-\\!m_{e}^{2}}{m_{12}^{2}}(y\\!+\\!\mu
m_{\nu})(y\\!+\\!\mu
m_{\text{He}})\\!-\\!\frac{m_{\text{H}}^{2}|\boldsymbol{p}_{e}|^{2}}{3m_{12}^{4}}\widetilde{y}^{2}\right]\right.$
$\displaystyle\quad\,\left.+(g_{A}^{2}\\!-\\!g_{V}^{2})E_{e}\left(ym_{\nu}\frac{m_{\text{He}}}{m_{\text{H}}}\right)\\!+\\!(g_{A}^{2}\\!+\\!g_{V}^{2})m_{\text{He}}\left(\frac{m_{\text{H}}E_{e}\\!-\\!m_{e}^{2}}{m_{12}^{2}}(y\\!+\\!\mu
m_{\nu})\right)\right\\},$ (78) $\displaystyle b_{LL,RL}(E_{e})$
$\displaystyle=-\widetilde{C}(E_{e})\frac{m_{\text{H}}}{m_{12}^{2}}|\boldsymbol{p}_{e}|^{2}\widetilde{y}$
$\displaystyle\times\left\\{\left[(g_{A}^{2}-g_{V}^{2})m_{\text{H}}+(g_{A}^{2}+g_{V}^{2})m_{\text{He}}\frac{m_{\text{H}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}+(g_{A}^{2}+g_{V}^{2})\frac{m_{\text{H}}m_{\text{He}}}{m_{12}^{2}}E_{e}\right.\right.$
$\displaystyle\quad\,\,\,\left.+(g_{A}^{2}\\!-\\!g_{V}^{2})\frac{m_{\text{H}}}{m_{12}^{2}}(\alpha\\!-\\!m_{e}^{2})\\!-\\!(g_{A}^{2}\\!-\\!g_{V}^{2})\frac{m_{\text{H}}^{2}}{m_{12}^{2}}(y\\!+\\!\mu
m_{\nu})\frac{m_{\text{H}}(m_{\text{H}}\\!-\\!E_{e})}{m_{12}^{2}}\right](y\\!+\\!\mu
m_{\nu})$
$\displaystyle\quad\,\,\,\left.-(g_{A}^{2}-g_{V}^{2})m_{\nu}^{2}-\frac{1}{3}(g_{A}^{2}-g_{V}^{2})\frac{m_{\text{H}}^{3}(m_{\text{H}}-E_{e})}{m_{12}^{4}}\widetilde{y}^{2}\right\\}.$
(79)

All other interference terms with differing hadronic handedness can be related
by $\gamma^{5}\to-\gamma^{5}$ (with $S\to-S$) or $g_{A}\to-g_{A}$ ,

$\displaystyle a_{LR,RR}(E_{e})$
$\displaystyle=a_{RR,LR}(E_{e})=a_{RL,LL}(E_{e})=a_{LL,RL}(E_{e}),$ (80)
$\displaystyle b_{LR,RR}(E_{e})$
$\displaystyle=b_{RR,LR}(E_{e})=-b_{RL,LL}(E_{e})=-b_{LL,RL}(E_{e}).$ (81)

#### Vector Currents: $(LL)$ with $(RR)$

$\displaystyle a_{LL,RR}(E_{e})$
$\displaystyle=-\widetilde{C}(E_{e})m_{e}m_{\nu}\frac{m_{\text{H}}^{2}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\widetilde{y}\Bigg{[}(2(g_{A}^{2}+g_{V}^{2})m_{\text{He}}+(g_{A}^{2}-g_{V}^{2})(m_{\text{H}}-E_{e})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\,\,\,\,\left.-(g_{A}^{2}-g_{V}^{2})\frac{m_{\text{H}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}(y+\mu
m_{\nu})\right],$ (82) $\displaystyle b_{LL,RR}(E_{e})$ $\displaystyle=0.$
(83)

All other interference terms differing in handedness for both can be related
by $\gamma^{5}\to-\gamma^{5}$ (with $S\to-S$) or $g_{A}\to-g_{A}$ ,

$\displaystyle a_{LR,RL}(E_{e})$
$\displaystyle=a_{LL,RR}(E_{e})=a_{RL,LR}(E_{e})=a_{RR,LL}(E_{e}),$ (84)
$\displaystyle b_{LR,RL}(E_{e})$
$\displaystyle=b_{LL,RR}(E_{e})=b_{RL,LR}(E_{e})=b_{RR,LL}(E_{e})=0.$ (85)

####  $LL$ with Left-handed Scalar Current

$\displaystyle a_{LL,S}(E_{e})$
$\displaystyle=\widetilde{C}(E_{e})g_{S}g_{V}m_{e}\frac{m_{\text{H}}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\widetilde{y}$
$\displaystyle\times\left[m_{\text{H}}\left(1+\frac{m_{\text{He}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}\right)\left(y+\mu
m_{\nu}\right)-m_{\nu}^{2}\right],$ (86) $\displaystyle b_{LL,S}(E_{e})$
$\displaystyle=-\widetilde{C}(E_{e})g_{S}g_{A}m_{e}m_{\text{He}}\frac{m_{\text{H}}^{2}}{m_{12}^{2}}\frac{|\boldsymbol{p}_{e}|^{2}}{m_{12}^{2}}\widetilde{y}(y+\mu
m_{\nu}).$ (87)

####  $LL$ with Right-handed Scalar Current

$\displaystyle a_{LL,\tilde{S}}(E_{e})$
$\displaystyle=-\widetilde{C}(E_{e})g_{S}g_{V}m_{\nu}\frac{m_{\text{H}}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\widetilde{y}$
$\displaystyle\times\left[(m_{\text{H}}+m_{\text{He}})E_{e}-m_{e}^{2}-m_{\text{H}}\frac{m_{\text{H}}E_{e}-m_{e}^{2}}{m_{12}^{2}}\left(y+\mu
m_{\nu}\right)\right],$ (88) $\displaystyle b_{LL,\tilde{S}}(E_{e})$
$\displaystyle=-\widetilde{C}(E_{e})g_{S}g_{A}m_{\nu}M\frac{|\boldsymbol{p}_{e}|^{2}}{m_{12}^{2}}\widetilde{y}\left[\frac{m_{\text{H}}^{2}}{m_{12}^{2}}(y+\mu
m_{\nu})-(m_{\text{H}}+m_{\text{He}})\right].$ (89)

####  $LL$ with Left-handed Pseudoscalar Current

$\displaystyle a_{LL,P}(E_{e})$
$\displaystyle=\widetilde{C}(E_{e})g_{P}g_{A}m_{e}\frac{m_{\text{H}}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\widetilde{y}$
$\displaystyle\times\left[m_{\text{H}}\left(\frac{m_{\text{He}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}-1\right)\left(y+\mu
m_{\nu}\right)+m_{\nu}^{2}\right],$ (90) $\displaystyle b_{LL,P}(E_{e})$
$\displaystyle=-\widetilde{C}(E_{e})g_{P}g_{V}m_{e}m_{\text{He}}\frac{m_{\text{H}}^{2}}{m_{12}^{2}}\frac{|\boldsymbol{p}_{e}|^{2}}{m_{12}^{2}}\widetilde{y}(y+\mu
m_{\nu}).$ (91)

####  $LL$ with Right-handed Pseudoscalar Current

$\displaystyle a_{LL,\tilde{P}}(E_{e})$
$\displaystyle=\widetilde{C}(E_{e})g_{P}g_{A}m_{\nu}\frac{m_{\text{H}}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\widetilde{y}$
$\displaystyle\times\left[(m_{\text{H}}-m_{\text{He}})E_{e}-m_{e}^{2}-m_{\text{H}}\frac{m_{\text{He}}E_{e}-m_{e}^{2}}{m_{12}^{2}}(y+\mu
m_{\nu})\right],$ (92) $\displaystyle b_{LL,\tilde{P}}(E_{e})$
$\displaystyle=\widetilde{C}(E_{e})g_{P}g_{V}m_{\nu}m_{\text{H}}\frac{|\boldsymbol{p}_{e}|^{2}}{m_{12}^{2}}\widetilde{y}\left[\frac{m_{\text{H}}^{2}}{m_{12}^{2}}(y+\mu
m_{\nu})-(m_{\text{H}}-m_{\text{He}})\right].$ (93)

####  $LL$ with Left-handed Tensor Current

$\displaystyle a_{LL,T}(E_{e})$
$\displaystyle=-12\widetilde{C}(E_{e})g_{T}m_{e}\frac{m_{\text{H}}|\boldsymbol{p}_{e}|}{m_{12}^{2}}\widetilde{y}$
$\displaystyle\times\left[m_{\text{H}}\left((g_{A}+g_{V})\frac{m_{\text{He}}(m_{\text{H}}-E_{e})}{m_{12}^{2}}+(g_{A}-g_{V})\right)(y+\mu
m_{\nu})-(g_{A}-g_{V})m_{\nu}^{2}\right],$ (94) $\displaystyle
b_{LL,T}(E_{e})$
$\displaystyle=12\widetilde{C}(E_{e})g_{T}(g_{A}+g_{V})m_{e}m_{\text{He}}\frac{m_{\text{H}}^{2}}{m_{12}^{2}}\frac{|\boldsymbol{p}_{e}|^{2}}{m_{12}^{2}}\widetilde{y}(y+\mu
m_{\nu}).$ (95)

####  $LL$ with Right-handed Tensor Current

$\displaystyle a_{LL,\tilde{T}}(E_{e})$
$\displaystyle=12\widetilde{C}(E_{e})g_{T}m_{\nu}\frac{M|\vec{\boldsymbol{p}}_{e}|}{m_{12}^{2}}\widetilde{y}$
$\displaystyle\times\left[(g_{A}\\!+\\!g_{V})\left(m_{\text{H}}E_{e}-m_{e}^{2}-m_{\text{H}}\frac{m_{\text{H}}E_{e}-m_{e}^{2}}{m_{12}^{2}}(y+\mu
m_{\nu})\right)\\!+\\!(g_{A}\\!-\\!g_{V})m_{\text{He}}E_{e}\right],$ (96)
$\displaystyle b_{LL,\tilde{T}}(E_{e})$
$\displaystyle=4\widetilde{C}(E_{e})g_{T}m_{\nu}m_{\text{H}}\frac{|\boldsymbol{p}_{e}|^{2}}{m_{12}^{2}}\widetilde{y}$
$\displaystyle\times\left[3(g_{A}-g_{V})m_{\text{He}}+(g_{A}+g_{V})m_{\text{H}}-(g_{A}+g_{V})\frac{m_{\text{H}}^{2}}{m_{12}^{2}}(y+\mu
m_{\nu})\right].$ (97)

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