aainstitutetext: Dipartimento di Fisica e Geologia, Università degli Studi di
Perugia, Via A. Pascoli, I-06123, Perugia, Italybbinstitutetext: INFN Sezione
di Perugia, Via A. Pascoli, I-06123, Perugia, Italyccinstitutetext: Institute
for Experimental Particle Physics (ETP), Karlsruhe Institute of Technology
(KIT), Wolfgang-Gaede-Straße 1, 76131 Karlsruhe, Germanyddinstitutetext:
ICRANet, Piazzale della Repubblica, 10-65122, Pescara, Italyeeinstitutetext:
Dipartimento di Fisica, Sapienza-Università di Roma, Piazzale Aldo Moro 5,
00185 Roma, Italyffinstitutetext: ICTP-AP, University of Chinese Academy of
Sciences, Beijing, China

# Constraints on NJL four-fermion effective interactions from neutrinoless
double beta decay

L. Pacioselli  0009-0000-3942-0721 b,1 O. Panella,111Corresponding author.
0000-0003-4262-894X c M. Presilla  0000-0003-2808-7315 b,d,e,f and S.-S. Xue
0000-0002-2702-0859<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the contribution of a heavy right-handed Majorana neutrino to
neutrinoless double beta decay ($0\nu\beta\beta$) via four-fermion effective
interactions of Nambu-Jona-Lasinio (NJL) type. In this physical scenario, the
sterile neutrino contributes to the nuclear transition through gauge, contact,
and mixed interactions. Using the lower limit on the half-life of
$0\nu\beta\beta$ from the KamLAND-Zen experiment, we then constrain the
effective right-handed coupling between the sterile neutrino and the $W$
boson: $\mathcal{G}^{W}_{R}$. Eventually, we show that the obtained bounds are
compatible with those found in the literature, which highlights the
complementarity of this type of phenomenological study with high-energy
experiments.

##  1 Introduction

The Standard Model (SM) for elementary particle physics is the experimentally
confirmed [1, 2, 3] gauge theory that led to the most accurate results in
Physics history. Nevertheless, SM alone cannot answer many questions raised by
experimental evidence: the existence of dark matter, General Relativity, the
baryon asymmetry problem, the hierarchy pattern of fermion masses, etc. For
these reasons, a multitude of Beyond the Standard Model (BSM) theories have
been developed and tested in various experimental facilities, e.g. the
searches for contact interactions at the Large Hadron Collider (LHC) [4, 5]
and the searches for lepton number violation (LNV) at low-energy experiments
[6, 7, 8].

The most interesting physical process for detecting LNV signals is the
neutrinoless double beta decay ($0\nu\beta\beta$) [9, 10, 11]: a rare nuclear
decay strictly forbidden by the SM that, due to the Schechter-Valle theorem
[12], would imply the Majorana nature for neutrinos. The first direct searches
for $0\nu\beta\beta$ decays began in the 1960s [13, 14], but it has never been
observed to date. However, the expected sensitivity gain of the next-
generation experiments looks promising [15, 16, 17, 18]. Hence, from these
experiments are derived only lower limits for the half-life $T^{0\nu}_{1/2}$.
The aim of this paper is to exploit those experimental limits in order to
constrain the parameters of a specific class of BSM models, as extensively
done in the literature [19, 20, 21, 22, 23, 24, 25].

At low energies, the SM with one elementary Higgs boson [26] is rendered, as
an effective field theory (EFT), by the Nambu-Jona-Lasinio (NJL) model [27,
28] of four-fermion interactions, where the Higgs is a composite particle. In
this paper, we consider the latter, in which the effective Lagrangian contains
four-fermion operators responsible, through minimal dynamical symmetry
breaking in a well-defined quantum field theory (QFT) at the high energy
scale, for generating the top quark and Higgs masses [29, 30]. This model has
been used to tackle several open questions of the SM: from the hierarchy
pattern of fermion masses [31] to dark matter particles’ masses [32], and the
recent $W$ boson mass tension [33]. The model not only renders the effective
parity-violating SM at its infrared (IR) fixed point of the electroweak scale
$v=246\text{ GeV}$, but also a parity-preserving theory of massive composite
particles at its ultraviolet (UV) fixed point of the composite scale
$\Lambda\sim\mathcal{O}$(TeV) scales [34]. Hence, the $W$ boson gauge coupling
is no longer purely left but has a non-trivial right coupling from the four-
fermion interactions at high energies. We will study and constrain precisely
this effective right coupling $\mathcal{G}^{W}_{R}$ through this
$0\nu\beta\beta$-driven study, i.e. at low energies ($\sim 100\text{ MeV}$).

The rest of the paper is organized as follows: in Sec. 2 we outline the main
features of the four-fermion interactions of the NJL-type model employed; in
Sec. 3 we use the model interactions contributing to $0\nu\beta\beta$ (gauge,
contact and two mixed) to compute its theoretical half-life; then in Sec. 4 we
exploit the experimental lower limits on $T^{0\nu}_{1/2}$ to extract the
bounds on the new composite scale $\Lambda$ and effective right-handed
coupling $\mathcal{G}^{W}_{R}$ of the model; lastly, we summarize the work
with some remarks in Sec. 5.

##  2 Four-fermion interactions of NJL type

Here we briefly describe the model adopted in this work. As shown in low-
energy experiments, the SM possesses parity-violating (chiral) gauge
symmetries $SU_{c}(3)\times SU_{L}(2)\times U_{Y}(1)$. As a well-defined QFT,
the SM should regularize at the high-energy cutoff $\Lambda_{\rm cut}$, fully
preserving the SM gauge symmetries. A natural UV regularization is provided by
a theory of new physics at $\Lambda_{\rm cut}$, for instance, quantum gravity.
However, the theoretical inconsistency between the SM bilinear Lagrangian and
the natural UV regularization, due to the No-Go theorem [35, 36], implies
quadrilinear four-fermion operators and right-handed neutrinos, which
effectively represent a theory of new physics at the UV cutoff. We adopt the
four-fermion operators of the torsion-free Einstein-Cartan Lagrangian with SM
fermion content and three right-handed neutrinos [31, 34]:

$\mathcal{L}\supset-G_{\rm
cut}\sum_{ff^{\prime}}\big{(}\bar{\psi}^{f}_{L}\psi^{f^{\prime}}_{R}\bar{\psi}^{f^{\prime}}_{R}\psi^{f}_{L}+\bar{\nu}^{fC}_{R}\psi^{f^{\prime}}_{R}\bar{\psi}^{f^{\prime}}_{R}\nu^{fC}_{R}\big{)}+h.c.$
(1)

where $\psi^{f}_{L}$ and $\psi^{f}_{R}$ (for simplicity $\psi^{f}_{R}$ is also
used to represent the sterile neutrinos $\nu^{f}_{R}$) are the two-component
Weyl fields that are gauge doublets and singlets of the symmetry
$SU_{L}(2)\times U_{Y}(1)$, respectively. Fermion family indexes
$f,f^{\prime}=1,2,3$ are summed over for the three lepton families (charge
$q=0,-1$) and for the three quark families (charge $q=2/3,-1/3$). Family
mixing will be duly induced when one makes the unitary transformations $U_{L}$
and $U_{R}$ from gauge to mass eigenstates. Attributing to the new physics at
the cutoff, the effective four-fermion coupling $G_{\rm
cut}\propto\mathcal{O}(\Lambda^{-2}_{\rm cut})$ is assumed to be unique for
all the terms in Eq. (1), and it depends on the running energy scale $\mu$.
Using strong-coupling $G_{\rm cut}\Lambda^{2}_{\rm cut}$ expansion to
calculate two-point Green functions shows the presence of composite bosons
$(\bar{\psi}_{L}\psi_{R})$ and fermions $(\bar{\psi}_{L}\psi_{R})\psi_{L}$
[37]. An EFT for composite particles of masses $M\propto\Lambda$ is realized
in the scaling domain of the stable UV fixed point at the composite scale
$\Lambda$ ($v<\Lambda<\Lambda_{\rm cut}$) [34]. When the running energy scale
$\mu$ decreases below $\Lambda$, the four-fermion interacting dynamics run
into the SM ground state of spontaneous symmetry breaking and the IR fixed
point of the $v$ [29]. In the IR scaling domain, integrating over massive
composite states gives rise to effective four-fermion contact interactions of
coupling $G\propto\mathcal{O}(\Lambda^{-2})$ [38], for example,
$G\big{(}\bar{\psi}^{i}_{La}t^{a}_{R}\big{)}\big{(}\bar{t}^{a}_{R}\psi_{Lia}\big{)}$
of the $\bar{t}t$-condensate model [29] in the third quark family. The one
relevant for the physics case in exam is between quarks and leptons in the
first SM family [39]

$\mathcal{L}_{q-\ell}\supset
G\big{(}\bar{l}^{i}_{L}N^{e}_{R}\big{)}\big{(}\bar{u}^{a}_{R}\psi_{Lia}\big{)}+{h.c.}$
(2)

Here $l^{i}_{L}=(\nu^{e}_{L},e_{L})$ and $e_{R}$ are lepton doublet and
singlet of $SU(2)_{L}$, respectively, and $\psi_{Lia}=(u_{La},d_{La})$,
$u_{Ra}$, $d_{Ra}$ are the counterparts for the quarks. The color $a$ and the
weak isospin $i$ indexes are summed over. $N^{e}_{R}$ represents the right-
handed component of Majorana neutrino $N^{e}$ of mass $M_{N^{e}}$. On the
basis of mass eigenstates, neglecting the mixing matrix, the contact
interaction (2) becomes

$\mathcal{L}_{q-\ell}\approx
G\big{(}\bar{e}P_{R}N^{e}\big{)}\big{(}\bar{u}^{a}P_{L}d_{a}\big{)}+h.c.\quad,$
(3)

which is the one contributing to $n\to p+N+e$ decay.

On the other hand, the effective four-fermion contact interactions induce
effective $W^{\pm}$ boson right-handed coupling [31]

$\mathcal{L}\supset\mathcal{G}^{W}_{R}\Big{(}\frac{g_{w}}{\sqrt{2}}\Big{)}\big{[}(U^{\ell}_{R})^{\dagger}U^{\nu}_{R}\big{]}^{ll^{\prime}}\bar{l}\gamma^{\mu}P_{R}N^{l^{\prime}}W^{-}_{\mu}+h.c.$
(4)

where ($N^{l}_{R},l_{R}$) are the right-handed leptons in the mass basis,
$U^{\nu}_{R}$ and $U^{\ell}_{R}$ are unitary matrices $3\times 3$
($\ell=e,\mu,\tau$). It’s important to note that the mixing matrix $V_{\ell
N}\equiv\big{[}(U^{\ell}_{R})^{\dagger}U^{\nu}_{R}\big{]}$ is not the PMNS one
$\big{[}(U^{\ell}_{L})^{\dagger}U^{\nu}_{L}\big{]}$. The effective operator of
this type (4) contributes to vector boson fusion (VBF) processes, see the left
Feynman diagram in Figure 1 of Ref. [40]. The mixing $|V_{\ell N}|$ and mass
$M_{N}$ are constrained [40, 41, 42]. In this work, we neglect mixing by
approximating $\big{[}(U^{\ell}_{R})^{\dagger}U^{\nu}_{R}\big{]}\approx 1$.
Hence for the first lepton family, we have

$\mathcal{L}\approx\mathcal{G}^{W}_{R}\Big{(}\frac{g_{w}}{\sqrt{2}}\Big{)}\bar{e}\gamma^{\mu}P_{R}N^{e}W^{-}_{\mu}+h.c.\quad,$
(5)

where $g_{w}$ is the normal $SU(2)_{L}$ coupling. The $\mathcal{G}^{W}_{R}$
parameterizes the effective $W^{\pm}$ right-handed coupling. The upper limit
of its value should be smaller than $\sim 10^{-4}$. It is constrained by the
$W^{\pm}$ decay width and regarding $N^{e}_{R}$ as a dark-matter particle [43,
44].

The effective interactions (3) and (5) are the relevant operators for
computing the theoretical half-life of $0\nu\beta\beta$ decay, which is the
aim of this article.

##  3 Half-Lives Computations

In this section, we calculate the contributions of the effective contact
interaction (3) and gauge interaction (5) to the half-lives of the
neutrinoless double beta decay. The relevant Feynman diagrams for
$0\nu\beta\beta$, shown in Figure 1, with the exchange of a heavy Majorana
neutrino, are four: one for pure gauge interactions (Fig. 1(a)), one for pure
contact interactions (Fig. 1(b)) and two mixed contributions (Figs. 1(c) and
1(d)).

(a) Gauge term.

(b) Contact term.

(c) Mixed term I.

(d) Mixed term II.

Figure 1: Feynman diagrams for $0\nu\beta\beta$, i.e. the nuclear decay of two
neutrons $n$ into two protons $p$, mediated by a Majorana neutrino. The two
vertices connected by the exchanged heavy Majorana neutrino involve, from left
to right, effective $V+A$ gauge interactions (5), four-fermion contact
interactions (3) and one gauge and one contact vertex, taking into account two
permutations.

###  3.1 Gauge Interactions

In a pure gauge transition two neutrons $n$ decay in two protons $p$ emitting
two $W^{-}$ bosons through a SM vertex:

$\frac{g_{w}}{2\sqrt{2}}\cos{\theta_{C}}J^{h}_{\mu}(x)W^{+\mu}(x)$ (6)

where $g_{w}$ is the $SU(2)_{L}$ SM gauge coupling, $\theta_{C}$ is the
Cabibbo mixing angle ($\cos{\theta_{C}}\approx 0.974$), while
$J^{h}_{\mu}(x)=\big{(}\bar{u}\gamma_{\mu}(1-\gamma^{5})d\big{)}(x)$ is the
hadronic weak charged current, where we have factored out $1/2$ from the
chiral projector $P_{L}$ in order to conform with the expressions of nuclear
matrix elements (NME) found in the literature [22, 45, 46, 47]. The two $W$
bosons then decay into a right-handed electron, $e_{R}$, and a heavy Majorana
electron neutrino, $N^{e}_{R}$, through the effective right interaction in Eq.
(5). Summing up these two terms, we can write the gauge interactions
Lagrangian as follows:

$\mathcal{L}_{\text{GI}}(x)=\frac{g_{w}}{\sqrt{2}}\Big{[}\mathcal{G}^{W}_{R}(j^{\mu}_{R}W^{-}_{\mu})(x)+\frac{1}{2}\cos{\theta_{C}}(J^{h}_{\mu}W^{+\mu})(x)\Big{]}+h.c.$
(7)

with $P_{L,R}=\frac{1-\gamma^{5}}{2}$, while
$j^{\mu}_{R}\equiv\bar{e}\gamma^{\mu}P_{R}N^{e}$ is the leptonic current.

For the purpose of this paper, we will consider only nuclear transitions
between $0$-spin and positive parity states, i.e. $0^{+}\to 0^{+}$. The non-
trivial S-matrix element related to Eq. (7) is

$\displaystyle T_{\text{GI}}=$
$\displaystyle-(\mathcal{G}^{W}_{R})^{2}\Big{(}\frac{g_{w}}{\sqrt{2}}\Big{)}^{4}\frac{(\cos{\theta_{C}})^{2}}{8}\Big{(}\frac{1-P_{12}}{\sqrt{2}}\Big{)}\int\frac{d^{4}q}{(2\pi)^{4}}d^{4}xd^{4}ye^{ip_{1}\cdot
x}e^{ip_{2}\cdot y}$ $\displaystyle\times
e^{-iq\cdot(x-y)}\bar{e}_{p_{1}}\gamma^{\mu}P_{R}\frac{i(\not{q}+M_{N})}{q^{2}-M^{2}_{N}+i\epsilon}P_{R}\gamma^{\nu}e^{C}_{p_{2}}$
$\displaystyle\times\frac{(-i\delta^{\lambda}_{\mu})(-i\delta^{\rho}_{\nu})\braket{0^{+}_{F}}{T\\{J^{h}_{\lambda}(y)J^{h}_{\rho}(x)\\}}{0^{+}_{I}}}{\big{(}(q+p_{1})^{2}-M^{2}_{W}+i\epsilon\big{)}\big{(}(q-p_{2})^{2}-M^{2}_{W}+i\epsilon\big{)}}\quad,$
(8)

where $e_{p_{1,2}}$ are the wave functions of the electrons in the final
state, with four-momenta $p_{1}$ and $p_{2}$, $q$ is the four-momentum
transfer, i.e. the Majorana neutrino four-momentum, and
$\frac{1-P_{12}}{\sqrt{2}}$ is the antisymmetric operator due to the
production of two identical fermions. We make the ansatz that the hadronic
current is given by the sum of the nucleonic charged current

$J^{h}_{\mu}(x)=\sum_{i}J^{(i)}_{\mu}(x)$ (9)

where the sum runs over the nucleons of the isotope which decays through
$0\nu\beta\beta$. This implies that:

$\braket{0^{+}_{F}}{T\\{J^{h}_{\rho}(x)J^{h}_{\lambda}(y)\\}}{0^{+}_{I}}=e^{i(P_{F}-P_{I})\cdot
y}\braket{0^{+}_{F}}{T\\{J^{h}_{\rho}(x-y)J^{h}_{\lambda}(0)\\}}{0^{+}_{I}}$
(10)

with $P_{I}$ and $P_{F}$ the four-momenta of the nucleus in the initial and
final state, respectively. Now if we define
$G_{\text{eff}}\equiv\mathcal{G}^{W}_{R}G_{F}\cos{\theta_{C}}$ and change the
variables of integration as

$\begin{cases}x=z+\frac{u}{2}\\\
y=z-\frac{u}{2}\end{cases}\quad\text{with}\quad d^{4}xd^{4}y=d^{4}ud^{4}z$
(11)

the integration over $d^{4}z$ gives the energy-momentum conservation, thus we
can recast the matrix element in the form $T_{\text{GI}}\equiv
i(2\pi)^{4}\delta^{4}\big{(}P_{F}+p_{1}+p_{2}-P_{I}\big{)}M_{\text{GI}}$, with

$M_{\text{GI}}=G^{2}_{\text{eff}}M^{4}_{W}M_{N}\frac{1-P_{12}}{\sqrt{2}}\int\frac{d^{4}q}{(2\pi)^{4}}\bar{e}_{p_{1}}\gamma^{\mu}P_{R}\gamma^{\nu}e^{C}_{p_{2}}\frac{W_{\nu\mu}(q)}{\big{(}q^{2}_{0}-\omega^{2}_{N}+i\epsilon\big{)}\big{(}q^{2}_{0}-\omega^{2}_{W}+i\epsilon\big{)}^{2}}$
(12)

in which we have neglected $p_{1}$ and $p_{2}$ with respect to $q$ and defined

$W_{\nu\mu}(q)\equiv\int d^{4}ue^{-iu\cdot
q}\braket{0^{+}_{F}}{T\\{J^{h}_{\nu}(u)J^{h}_{\mu}(0)\\}}{0^{+}_{I}}\quad,$
(13)

where $\omega^{2}_{N(W)}=(\mathbf{q})^{2}+M^{2}_{N(W)}$. Using the identity:

$\displaystyle\frac{1-P_{12}}{\sqrt{2}}\bar{e}_{p_{1}}\gamma^{\mu}P_{R}\gamma^{\nu}e^{C}_{p_{2}}=\frac{1}{\sqrt{2}}\big{(}\bar{e}_{p_{1}}\gamma^{\mu}P_{R}\gamma^{\nu}e^{C}_{p_{2}}-\bar{e}_{p_{2}}\gamma^{\mu}P_{R}\gamma^{\nu}e^{C}_{p_{1}}\big{)}$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}\bar{e}_{p_{1}}\gamma^{\mu}\gamma^{\nu}P_{L}e^{C}_{p_{2}}+\bar{e}_{p_{1}}\gamma^{\nu}\gamma^{\mu}P_{L}e^{C}_{p_{2}}\big{)}=\frac{2}{\sqrt{2}}g^{\mu\nu}\bar{e}_{p_{1}}P_{L}e^{C}_{p_{2}}\quad,$
(14)

we obtain

$M_{\text{GI}}=\frac{2G^{2}_{\text{eff}}}{\sqrt{2}}M^{4}_{W}M_{N}\int\frac{d^{4}q}{(2\pi)^{4}}\bar{e}_{p_{1}}P_{L}e^{C}_{p_{2}}\frac{W^{\mu}_{\mu}(q)}{\big{(}q^{2}_{0}-\omega^{2}_{N}+i\epsilon\big{)}\big{(}q^{2}_{0}-\omega^{2}_{W}+i\epsilon\big{)}^{2}}\quad.$
(15)

As usual for $0\nu\beta\beta$ calculations [23], we insert a complete set of
intermediate nuclear states $\ket{X}$ in $W^{\mu}_{\mu}$:

$W^{\mu}_{\mu}(q)=-i\int
d^{3}\mathbf{u}e^{i\mathbf{u}\cdot\mathbf{q}}\sum_{\ket{X}}\Big{(}\frac{\braket{0^{+}_{F}}{J^{h\mu}(\mathbf{u})}{X}\braket{X}{J^{h}_{\mu}(0)}{0^{+}_{I}}}{q_{0}-E_{F}+E_{X}-i\epsilon}+\frac{\braket{0^{+}_{F}}{J^{h}_{\mu}(0)}{X}\braket{X}{J^{h\mu}(\mathbf{u})}{0^{+}_{I}}}{-q_{0}-E_{I}+E_{X}-i\epsilon}\Big{)}\
.$ (16)

The energy of a state $\ket{X}$ is $E_{X}=E_{CM}(\mathbf{P})+\epsilon_{n}$
where $E_{CM}(\mathbf{P})$ is the translational energy of the center of mass
and $\epsilon_{n}$ is the excitation energy, characterizing state $\ket{X}$.
Note also that the sum over intermediate states includes integration over the
center of mass momentum $\mathbf{P}$ and a sum on the discrete part of the
spectrum, i.e. the excitation level $n$. Now we use the closure approximation,
namely we replace the intermediate state energies with an average value
$\braket{E_{X}}=E_{CM}(\braket{\mathbf{P}})+\bar{\epsilon}_{n}$, where
$\bar{\epsilon}_{n}$ is the average excitation energy of all the intermediate
states $\ket{X}$. Due to the ansatz (9), the center of mass motion
($\mathbf{R}=\frac{1}{A}\sum_{i}\mathbf{u}_{i}$) can be separated out so that
we can rewrite the one body operator matrix elements [48] in Eq. (16) as

$\braket{0^{+}_{F}}{J^{h\mu}(\mathbf{u})}{X}=\sum_{k}\braket{\braket{0^{+}_{F}}{e^{i(\mathbf{P}-\mathbf{P}_{F})\cdot(\mathbf{u}-\mathbf{\xi}_{k})}\tilde{J}^{(k)\mu}(\mathbf{P}-\mathbf{P}_{F})}{X}}{}{hadron}$
(17)

and likewise for the other hadronic current. In (LABEL:hadron)
$\mathbf{\xi}_{i}=\mathbf{u}_{i}-\mathbf{R}$ are the relative coordinates, the
notation $\braket{\bra{}\cdot\ket{}}$ denotes that we are in the space of the
$A-1$ relative coordinates and $\tilde{J}^{(k)\mu}(\mathbf{P}-\mathbf{P}_{F})$
is the nucleon current in momentum space. Hence, for the first term in (16),
we have:

$-i\int\frac{d^{3}\mathbf{P}}{(2\pi)^{3}}d^{3}\mathbf{u}e^{i\mathbf{q}\cdot\mathbf{u}}\sum_{k,\ell}\frac{e^{i(\mathbf{P}-\mathbf{P}_{F})\cdot(\mathbf{u}-\mathbf{\xi}_{k})}e^{i(\mathbf{P}-\mathbf{P}_{I})\cdot\mathbf{\xi}_{\ell}}}{q_{0}+\braket{E_{X}}-E_{F}-i\epsilon}\braket{\braket{0^{+}_{F}}{\tilde{J}^{(k)\mu}(\mathbf{P}-\mathbf{P}_{F})\tilde{J}^{(\ell)}_{\mu}(-\mathbf{P}+\mathbf{P}_{I})}{0^{+}_{I}}}{\quad}{,}$
(18)

integrating over $d^{3}\mathbf{u}$ we obtain the Dirac delta
$\delta^{(3)}(\mathbf{q}+\mathbf{P}-\mathbf{P}_{F})$, which can be used to
integrate over the center of mass motion. Moreover, thanks to this Dirac delta
and that of energy-momentum conservation we have the identities
$\mathbf{P}-\mathbf{P}_{F}=-\mathbf{q}$ and
$-\mathbf{P}+\mathbf{P}_{I}=\mathbf{q}$. We consider nuclei without recoil so
that $E_{F}\approx M_{F}$, $E_{I}\approx M_{I}$ and $M_{F}\approx M_{I}$. By
introducing the so-called closure energy [49]

$\Delta=\braket{E_{X}}-\frac{1}{2}(M_{F}+M_{I})\approx 10\text{ MeV}\quad,$
(19)

we can recast the tensor function $W^{\mu}_{\mu}(q_{0},\mathbf{q})$ in the
form:

$W^{\mu}_{\mu}(q_{0},\mathbf{q})=\sum_{k,\ell}e^{i\mathbf{q}\cdot\mathbf{\xi}_{k\ell}}\braket{\braket{0^{+}_{F}}{\tilde{J}^{(k)\mu}(-\mathbf{q})\tilde{J}^{(\ell)}_{\mu}(\mathbf{q})}{0^{+}_{I}}}{\frac{missing}{missing}}{2i\Delta}{q^{2}_{0}-\Delta^{2}+i\epsilon}$
(20)

where $\mathbf{\xi}_{k\ell}\equiv\mathbf{\xi}_{k}-\mathbf{\xi}_{\ell}$. Now we
can perform the integration over $dq_{0}$ in Eq. (15)

$I(q^{2}_{0})\equiv\int\frac{dq_{0}}{2\pi}\frac{2i\Delta}{(q^{2}_{0}-\Delta^{2})(q^{2}_{0}-\omega^{2}_{N})(q^{2}_{0}-\omega^{2}_{W})^{2}}=\frac{1}{\omega^{2}_{N}\omega^{4}_{W}}\approx\frac{1}{M^{2}_{N}M^{4}_{W}}$
(21)

where we have assumed $\Delta\ll\omega_{W},\omega_{N}$ and that the exchanged
Majorana neutrino is heavy, namely $M_{N}\gg 100\text{ MeV}$ which is its
momentum $|\mathbf{q}|\approx\frac{1}{r_{NN}}\approx 100\text{ MeV}$ with
$r_{NN}\approx 2\text{ fm}$ the average inter-nucleon distance in the nuclei.

Next, we use the non-relativistic limit for the four-vector nucleon currents
[45]:

$\tilde{J}_{0}^{(k)}(\mathbf{q})=\tau^{+}_{k}\Big{(}F_{V}(\mathbf{q}^{2})\mathbb{1}_{2\times
2}+\frac{F_{P^{\prime}}(\mathbf{q}^{2})}{4m^{2}_{p}}q_{0}(\mathbf{\sigma}_{k}\cdot\mathbf{q})+...\Big{)}$
(22)
$\tilde{J}^{(k)}_{i}(\mathbf{q})=\tau^{+}_{k}\Big{(}F_{A}(\mathbf{q}^{2})(\mathbf{\sigma}_{k})_{i}+\frac{F_{V}(\mathbf{q}^{2})+F_{W}(\mathbf{q}^{2})}{2m_{p}}i(\mathbf{\sigma}_{k}\times\mathbf{q})_{i}-\frac{F_{P^{\prime}}(\mathbf{q}^{2})}{4m^{2}_{p}}q_{i}(\mathbf{\sigma}_{k}\cdot\mathbf{q})+...\Big{)}$
(23)

where $m_{p}$ is the proton mass, $\mathbf{\sigma}_{k}$ and $\tau^{+}_{k}$ are
Pauli matrices acting respectively on the spin and isospin space of the $k$-th
nucleon and $F_{o}$ are the nucleon form factors whose parametrizations and
coupling constants $g_{o}$ are fixed as in Ref. [45]. Lastly, taking into
account only $0^{+}\to 0^{+}$ transitions and keeping the first order in the
non-relativistic limit we have

$\displaystyle M_{\text{GI}}$
$\displaystyle=\frac{2G^{2}_{\text{eff}}}{\sqrt{2}M_{N}}\bar{e}_{p_{1}}P_{L}e^{C}_{p_{2}}\frac{m_{e}m_{p}}{4\pi
R_{0}}g^{2}_{A}\Bigg{[}-\mathcal{M}_{GT}+\Big{(}\frac{g^{2}_{V}}{g^{2}_{A}}\Big{)}\mathcal{M}_{F}+\mathcal{M}_{T}\Bigg{]}$
$\displaystyle=\frac{2G^{2}_{\text{eff}}}{\sqrt{2}M_{N}}\bar{e}_{p_{1}}P_{L}e^{C}_{p_{2}}\frac{m_{e}m_{p}}{4\pi
R_{0}}\mathcal{M}_{0\nu}$ (24)

where $R_{0}=1.2A^{1/3}\text{ fm}$ is the mean nuclear radius, $m_{e}$ is the
electron mass and $\mathcal{M}_{o}$ are the standard nuclear matrix elements
(NME) for a $0\nu\beta\beta$ with heavy Majorana neutrino exchange [46, 45].

The $0\nu\beta\beta$ half-life, which is the actual observable of the
experimental searches [6, 8], formula is

$(T^{0\nu}_{1/2})^{-1}_{\text{GI}}=\frac{1}{\ln{2}}\int\frac{d^{3}\mathbf{p}_{1}}{(2\pi)^{3}2E_{1}}\frac{d^{3}\mathbf{p}_{2}}{(2\pi)^{3}2E_{2}}\overline{|M_{\text{GI}}|^{2}}(2\pi)\delta(E_{F}+E_{1}+E_{2}-E_{I})$
(25)

where the probability amplitude squared and summed over the electron spin
polarizations is

$\overline{|M_{\text{GI}}|^{2}}=\frac{G^{4}_{\text{eff}}}{8M^{2}_{N}}\frac{m^{2}_{e}m^{2}_{p}}{\pi^{2}R^{2}_{0}}|\mathcal{M}_{0\nu}|^{2}\sum_{spin}|\bar{e}_{p_{1}}P_{L}e^{C}_{p_{2}}|^{2}\quad.$
(26)

We approximate the electron wave functions in the following way

$e_{p_{1,2}}=\sqrt{F_{0}(Z+2,E_{1,2})}\ u(p_{1,2})$ (27)

where $F_{0}$ is the Fermi function, which describes the distortion of the
electron wave function due to the Coulomb field of the nucleus [50] and
$u(p_{1,2})$ is the actual positive energy Dirac spinor. Now using simple
Dirac algebra we obtain

$\sum_{k,r}|\bar{e}_{r,p_{1}}P_{L}e^{C}_{k,p_{2}}|^{2}=F_{0}(Z+2,E_{1})F_{0}(Z+2,E_{2})2p_{1}\cdot
p_{2}$ (28)

so that, adopting standard notation [47] to express the phase-space
integration

$C=\frac{(G_{F}\cos{\theta_{C}})^{4}m^{2}_{e}}{16\pi^{5}}$ (29) $\displaystyle
G^{(0)}_{11}$ $\displaystyle=\frac{2C}{(\ln{2})4R_{0}^{2}}\int
F_{0}(Z+2,E_{1})F_{0}(Z+2,E_{2})p_{1}p_{2}E_{1}E_{2}\times$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times
dE_{1}dE_{2}\delta(E_{F}+E_{1}+E_{2}-E_{I})$
$\displaystyle\equiv\frac{2C}{(\ln{2})4R_{0}^{2}}\int
f^{(0)}_{11+}p_{1}p_{2}E_{1}E_{2}dE_{1}dE_{2}\delta(E_{F}+E_{1}+E_{2}-E_{I})$
(30)

we finally have the gauge interactions contributing to the $0\nu\beta\beta$
half-life

$\big{(}T^{0\nu}_{1/2}\big{)}^{-1}_{\text{GI}}=(\mathcal{G}^{W}_{R})^{4}\frac{m^{2}_{p}}{M^{2}_{N}}|\mathcal{M}_{0\nu}|^{2}G^{(0)}_{11}$
(31)

###  3.2 Contact Interactions

In this subsection, we go through the same type of computations we just did
but for contact interactions, therefore for brevity’s sake, we focus only on
the biggest differences.

In a pure contact transition two neutrons $n$ decay simultaneously into two
protons $p$, two electrons $e^{-}$ and exchange a heavy Majorana electron
neutrino $N^{e}$ without mediating bosons

$\mathcal{L}_{\text{CI}}=G\big{(}\bar{e}P_{R}N^{e}\big{)}\big{(}\bar{u}^{a}P_{L}d_{a}\big{)}+h.c.\quad.$
(32)

From the NJL type model [32] we recall that the effective four-fermion
coupling $G$ depends on the new physics energy scale and on the top-quark
Yukawa coupling constant

$G\sim\frac{g^{2}_{t0}}{\Lambda^{2}}$ (33)

so that we can recast the Lagrangian in (32) as follows:

$\displaystyle\mathcal{L}_{\text{\text{CI}}}(x)$
$\displaystyle=\Big{(}\frac{g_{t0}}{\Lambda}\Big{)}^{2}\big{(}\bar{e}P_{R}N^{e}\big{)}(x)\big{(}\bar{u}^{a}P_{L}d_{a}\big{)}(x)+h.c.$
$\displaystyle\equiv\Big{(}\frac{g^{2}_{t0}}{2\Lambda^{2}}\Big{)}j_{R}(x)J^{h}(x)+h.c.\quad,$
(34)

where in the last term we factored out the $1/2$ from the scalar hadronic
current.

The non-trivial $S$-matrix element related to Eq. (34) is

$\displaystyle T_{\text{CI}}=$
$\displaystyle\Big{(}\frac{g^{4}_{t0}}{8\Lambda^{4}}\Big{)}\frac{1-P_{12}}{\sqrt{2}}\int\frac{d^{4}q}{(2\pi)^{4}}d^{4}xd^{4}ye^{ip_{1}\cdot
x}e^{ip_{2}\cdot y}e^{-iq\cdot(x-y)}$
$\displaystyle\times\bar{e}_{p_{1}}P_{R}\frac{i(\not{q}+M_{N})}{q^{2}-M^{2}_{N}+i\epsilon}P_{R}e^{C}_{p_{2}}\braket{0^{+}_{F}}{T\\{J^{h}(x)J^{h}(y)\\}}{0^{+}_{I}}\quad,$
(35)

where the main difference with the pure gauge case is that the leptonic and
hadronic currents are now scalar. Making the same substitutions as in (11),
the ansatz (9), simplifying the leptonic currents with simple Dirac algebra
and defining the scalar function

$W(q)\equiv\int d^{4}ue^{-iq\cdot
u}\braket{0^{+}_{F}}{T\\{J^{h}(u)J^{h}(0)\\}}{0^{+}_{I}}$ (36)

we find:

$M_{\text{CI}}=\Big{(}\frac{g^{4}_{t0}}{4\sqrt{2}\Lambda^{4}}\Big{)}M_{N}\int\frac{d^{4}q}{(2\pi)^{4}}\bar{e}_{p_{1}}P_{R}e^{C}_{p_{2}}\frac{W(q)}{q^{2}_{0}-\omega^{2}_{N}+i\epsilon}\quad.$
(37)

Here too we will compute the function $W(q)$ inserting a complete set of
intermediate nuclear states $\ket{X}$ and adopting the closure approximation
in order to obtain the NMEs. Hence we find:

$W(q_{0},\mathbf{q})=\sum_{k,\ell}e^{i\mathbf{q}\cdot\mathbf{\xi}_{k\ell}}\braket{\braket{0^{+}_{F}}{\tilde{J}^{(k)}(-\mathbf{q})\tilde{J}^{(\ell)}(\mathbf{q})}{0^{+}_{I}}}{\frac{missing}{missing}}{2i\Delta}{q^{2}_{0}-\Delta^{2}+i\epsilon}\quad.$
(38)

We perform the integration over $dq_{0}$ in Eq. (37)

$\int\frac{dq_{0}}{2\pi}\frac{2i\Delta}{(q^{2}_{0}-\Delta^{2})(q^{2}_{0}-\omega_{N}^{2})}=\frac{1}{\omega_{N}(\Delta+\omega_{N})}\approx\frac{1}{M^{2}_{N}}$
(39)

where we have assumed $\Delta\ll\omega_{N}$ and that the exchanged Majorana
neutrino is heavy, i.e. $M_{N}\gg 100\text{ MeV}$.

Next, we use the non-relativistic limit for the scalar nucleon currents [45],
which is the main difference from the pure gauge case:

$\tilde{J}^{(k)}(\mathbf{q})=\tau^{+}_{k}\Big{(}F_{S}(\mathbf{q}^{2})\mathbb{1}_{2\times
2}-\frac{F_{P}(\mathbf{q}^{2})}{2m_{p}}(\mathbf{\sigma}_{k}\cdot\mathbf{q})+...\Big{)}\quad.$
(40)

Taking into account only $0^{+}\to 0^{+}$ transitions and keeping the first
order in the non-relativistic limit we have

$\displaystyle M_{\text{CI}}$
$\displaystyle=\Big{(}\frac{g^{4}_{t0}}{4\sqrt{2}\Lambda^{4}}\Big{)}\frac{1}{M_{N}}\bar{e}_{p_{1}}P_{R}e^{C}_{p_{2}}\frac{m_{e}m_{p}}{4\pi
R_{0}}\Big{[}g^{2}_{V}\mathcal{M}_{F}-\frac{g^{2}_{P^{\prime}}}{12}\big{(}\mathcal{M}^{\prime
PP}_{GT}+\mathcal{M}^{\prime PP}_{T}\big{)}\Big{]}$
$\displaystyle\equiv\Big{(}\frac{g^{4}_{t0}}{4\sqrt{2}\Lambda^{4}}\Big{)}\frac{1}{M_{N}}\bar{e}_{p_{1}}P_{R}e^{C}_{p_{2}}\frac{m_{e}m_{p}}{4\pi
R_{0}}\mathcal{M}_{1}\quad.$ (41)

The $0\nu\beta\beta$ half-life formula is

$\big{(}T^{0\nu}_{1/2}\big{)}^{-1}_{\text{CI}}=\frac{1}{\ln{2}}\int\frac{d^{3}\mathbf{p}_{1}}{(2\pi)^{3}2E_{1}}\frac{d^{3}\mathbf{p}_{2}}{(2\pi)^{3}2E_{2}}\overline{|M_{\text{CI}}|^{2}}(2\pi)\delta(E_{F}+E_{1}+E_{2}-E_{I})$
(42)

where the amplitude squared and summed over the electron spin polarizations is

$\overline{|M_{\text{CI}}|^{2}}=\Big{(}\frac{g_{t0}}{\Lambda}\Big{)}^{8}\frac{m^{2}_{e}m^{2}_{p}}{512\pi^{2}R^{2}_{0}M^{2}_{N}}|\mathcal{M}_{1}|^{2}\sum_{spin}|\bar{e}_{p_{1}}P_{R}e^{C}_{p_{2}}|^{2}\quad.$
(43)

We approximate the electron wave functions with the Fermi function [50] and,
using simple Dirac algebra, we obtain

$\sum_{k,r}|\bar{e}_{r,p_{1}}P_{R}e^{C}_{k,p_{2}}|^{2}=F_{0}(Z+2,E_{1})F_{0}(Z+2,E_{2})2p_{1}\cdot
p_{2}$ (44)

so that, adopting standard notation [47] to express the phase-space
integration

$C=\frac{(G_{F}\cos{\theta_{C}})^{4}m^{2}_{e}}{16\pi^{5}}$ (45) $\displaystyle
G^{(0)}_{11}$ $\displaystyle=\frac{2C}{(\ln{2})4R_{0}^{2}}\int
F_{0}(Z+2,E_{1})F_{0}(Z+2,E_{2})p_{1}p_{2}E_{1}E_{2}\times$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times
dE_{1}dE_{2}\delta(E_{F}+E_{1}+E_{2}-E_{I})$
$\displaystyle\equiv\frac{2C}{(\ln{2})4R_{0}^{2}}\int
f^{(0)}_{11+}p_{1}p_{2}E_{1}E_{2}dE_{1}dE_{2}\delta(E_{F}+E_{1}+E_{2}-E_{I})$
(46)

we finally have the contact interactions contribution to the $0\nu\beta\beta$
half-life

$\big{(}T^{0\nu}_{1/2}\big{)}^{-1}_{\text{CI}}=\Big{(}\frac{g^{8}_{t0}}{\Lambda^{8}}\Big{)}\frac{m^{2}_{p}}{64M^{2}_{N}}|\mathcal{M}_{1}|^{2}\frac{G^{(0)}_{11}}{(G_{F}\cos{\theta_{C}})^{4}}\quad.$
(47)

###  3.3 Mixed Interactions

To conclude this section, it is mandatory to consider the case where one
neutron decays through gauge interaction (7) and the other through contact
interaction (34) and vice versa: the so-called mixed diagrams (Figs. 1(c) and
1(d)). Usually, the mixed interactions’ contributions to $0\nu\beta\beta$
half-life cancel each other out or, in general, are negligible compared to the
pure cases [24]. However, with this NJL-type model, we will show that there is
constructive interference between the two mixed terms.

For brevity’s sake, the computations and considerations are only expressed for
$T^{I}_{\text{mix}}$ (Fig. 1(c)) since those for $T^{II}_{\text{mix}}$ (Fig.
1(d)) are perfectly analogous. The non-trivial $S$-matrix element related to
Fig. 1(c) is

$\displaystyle T^{I}_{\text{mix}}=$ $\displaystyle
C_{\text{eff}}\int\frac{d^{4}q}{(2\pi)^{4}}d^{4}xd^{4}ye^{-iq\cdot(x-y)}e^{ip_{1}\cdot
x}e^{ip_{2}\cdot y}\frac{1-P_{12}}{\sqrt{2}}$
$\displaystyle\times\bar{e}_{p_{1}}\gamma^{\mu}P_{R}\frac{i(\not{q}+M_{N})}{q^{2}-M^{2}_{N}+i\epsilon}P_{R}e^{C}_{p_{2}}\frac{(-i\delta)^{\nu}_{\mu}\braket{0^{+}_{F}}{T\\{J^{h}(x)J^{h}_{\nu}(y)\\}}{0^{+}_{I}}}{(q+p_{1})^{2}-M^{2}_{W}+i\epsilon}$
(48)

where
$C_{\text{eff}}\equiv\mathcal{G}^{W}_{R}\cos{\theta_{C}}(\frac{g_{t0}}{\Lambda})^{2}\frac{M^{2}_{W}G_{F}}{2\sqrt{2}}$.
Again, with the same assumptions, substitutions and ansatz of the previous
subsections, we obtain:

$M^{I}_{\text{mix}}=-iC_{\text{eff}}M_{N}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1-P_{12}}{\sqrt{2}}\frac{\bar{e}_{p_{1}}\gamma^{\mu}P_{R}e^{C}_{p_{2}}}{q^{2}_{0}-\omega^{2}_{N}+i\epsilon}\frac{W_{\mu}(q-p_{1})}{q_{0}^{2}-\omega^{2}_{W}+i\epsilon}\quad.$
(49)

The action of the antisymmetric operator over the leptonic states is as
follows:

$\frac{1-P_{12}}{\sqrt{2}}\bar{e}_{p_{1}}\gamma^{\mu}P_{R}e^{C}_{p_{2}}=\frac{1}{\sqrt{2}}\big{(}\bar{e}_{p_{1}}\gamma^{\mu}P_{R}e^{C}_{p_{2}}-\bar{e}_{p_{1}}\gamma^{\mu}P_{L}e^{C}_{p_{2}}\big{)}\quad.$
(50)

Instead for the other $S$-matrix element related to Fig. 1(d) we can obtain

$M^{II}_{\text{mix}}=iC_{\text{eff}}M_{N}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1-P_{12}}{\sqrt{2}}\frac{\bar{e}_{p_{1}}P_{R}\gamma^{\mu}e^{C}_{p_{2}}}{q^{2}_{0}-\omega^{2}_{N}+i\epsilon}\frac{W_{\mu}(q-p_{1})}{q_{0}^{2}-\omega^{2}_{W}+i\epsilon}\quad,$
(51)

where the action of the antisymmetric operator gives

$\frac{1-P_{12}}{\sqrt{2}}\bar{e}_{p_{1}}P_{R}\gamma^{\mu}e^{C}_{p_{2}}=\frac{1}{\sqrt{2}}\big{(}\bar{e}_{p_{1}}P_{R}\gamma^{\mu}e^{C}_{p_{2}}-\bar{e}_{p_{1}}P_{L}\gamma^{\mu}e^{C}_{p_{2}}\big{)}$
(52)

which is identical to the action of the other antisymmetric operator.

Now if we add together the two contributions (49) and (51) we have:

$M_{\text{mix}}\equiv
iC_{\text{eff}}\frac{M_{N}}{\sqrt{2}}\int\frac{d^{4}q}{(2\pi)^{4}}-2\bar{e}_{p_{1}}\gamma^{\mu}\gamma^{5}e^{C}_{p_{2}}\frac{W_{\mu}(q)}{(q^{2}_{0}-\omega^{2}_{N}+i\epsilon)(q_{0}^{2}-\omega^{2}_{W}+i\epsilon)}$
(53)

so we can’t neglect a priori the significance of the mixed diagrams. Let us
briefly show how NMEs are obtained in this case; by applying the closure
approximation we can recast the four-vector function $W_{\mu}(q)$ as follows

$W_{\mu}(q_{0},\mathbf{q})=\sum_{k,\ell}e^{i\mathbf{q}\cdot\mathbf{\xi}_{k\ell}}\braket{\bra{0^{+}_{F}}}{\tilde{J}^{(k)}(-\mathbf{q})\tilde{J}^{(\ell)}_{\mu}(\mathbf{q})}{0^{+}_{I}}\frac{2i\Delta}{q^{2}_{0}-\Delta^{2}+i\epsilon}$
(54)

and performing the integration in $q_{0}$ we obtain

$\int\frac{dq_{0}}{2\pi}\frac{2i\Delta}{(q^{2}_{0}-\Delta^{2})(q^{2}_{0}-\omega^{2}_{N})(q^{2}_{0}-\omega^{2}_{W})}=-\frac{1}{\omega^{2}_{W}\omega^{2}_{N}}\approx-\frac{1}{M^{2}_{W}M^{2}_{N}}\quad,$
(55)

so that we have

$M_{\text{mix}}=i\mathcal{G}^{W}_{R}\Big{(}\frac{g^{2}_{t0}}{\Lambda^{2}}\Big{)}\frac{G_{F}\cos{\theta_{C}}}{2M_{N}}\int\frac{d^{3}\mathbf{q}}{(2\pi)^{3}}\bar{e}_{p_{1}}\gamma^{\mu}\gamma^{5}e^{C}_{p_{2}}\sum_{k,\ell}e^{i\mathbf{q}\cdot\mathbf{\xi}_{k\ell}}\braket{\bra{0^{+}_{F}}}{\tilde{J}^{(k)}(-\mathbf{q})\tilde{J}^{(\ell)}_{\mu}(\mathbf{q})}{0^{+}_{I}}\
.$ (56)

Then we use the already seen non-relativistic expansions for the scalar (40)
and four-vector ((22) and (23)) nucleon currents [45] in order to perform the
integration in $\mathbf{q}$:

$\displaystyle M_{\text{mix}}$
$\displaystyle=i\mathcal{G}^{W}_{R}\Big{(}\frac{g^{2}_{t0}}{\Lambda^{2}}\Big{)}\frac{G_{F}\cos{\theta_{C}}}{2M_{N}}\bar{e}_{p_{1}}\gamma^{0}\gamma^{5}e^{C}_{p_{2}}\frac{m_{e}m_{p}}{4\pi
R_{0}}$
$\displaystyle\qquad\times\Big{[}g_{S}g_{V}\mathcal{M}_{F}+\frac{g_{P}g_{P^{\prime}}}{24}\big{(}\mathcal{M}^{\prime
q_{0}PP^{\prime}}_{GT}+\mathcal{M}^{\prime
q_{0}PP^{\prime}}_{T}\big{)}\Big{]}$ $\displaystyle\equiv
i\mathcal{G}^{W}_{R}\Big{(}\frac{g^{2}_{t0}}{\Lambda^{2}}\Big{)}\frac{G_{F}\cos{\theta_{C}}}{2M_{N}}\bar{e}_{p_{1}}\gamma^{0}\gamma^{5}e^{C}_{p_{2}}\frac{m_{e}m_{p}}{4\pi
R_{0}}\mathcal{M}_{5}$ (57)

where, from reference [47], $\mathcal{M}^{\prime q_{0}PP^{\prime}}_{GT}\approx
10^{-2}\mathcal{M}^{\prime P^{\prime}P^{\prime}}_{GT}$ and
$\mathcal{M}^{\prime q_{0}PP^{\prime}}_{T}\approx 10^{-2}\mathcal{M}^{\prime
P^{\prime}P^{\prime}}_{T}$, which would allow mixed contributions to be
neglected. However the coupling constants $g_{P}$ and $g_{P^{\prime}}$ are of
the order of $10^{2}$ whereas the others are all of the order of $1$, so mixed
diagrams give a contribution of the same order as gauge and contact ones. It
is important to point out in Eq. (57) that only the $\gamma^{0}$ matrix
survived integration in $\mathbf{q}$ because it is the only one that
contributes to the mix of scalar and four-vector nuclear currents to the first
perturbative order.

The $0\nu\beta\beta$ half-life formula is

$\big{(}T^{0\nu}_{1/2}\big{)}^{-1}_{\text{mix}}=\frac{1}{\ln{2}}\int\frac{d^{3}\mathbf{p}_{1}}{(2\pi)^{3}2E_{1}}\frac{d^{3}\mathbf{p}_{2}}{(2\pi)^{3}2E_{2}}\overline{|M_{\text{mix}}|^{2}}(2\pi)\delta(E_{F}+E_{1}+E_{2}-E_{I})$
(58)

where the amplitude squared and summed over the electron spin polarizations is

$\overline{|M_{\text{mix}}|^{2}}=\big{(}\mathcal{G}^{W}_{R}\big{)}^{2}\Big{(}\frac{g_{t0}}{\Lambda}\Big{)}^{4}\frac{(G_{F}\cos{\theta_{C}})^{2}}{4M^{2}_{N}}\frac{m^{2}_{e}m^{2}_{p}}{(4\pi
R_{0})^{2}}|\mathcal{M}_{5}|^{2}\sum_{spin}|\bar{e}_{p_{1}}\gamma^{0}\gamma^{5}e^{C}_{p_{2}}|^{2}\quad.$
(59)

Lastly, we approximate the electron wave functions with the Fermi function
[50] and with simple Dirac algebra we obtain

$\sum_{k,r}|\bar{e}_{p_{1}}\gamma^{0}\gamma^{5}e^{C}_{p_{2}}|^{2}=F_{0}(Z+2,E_{1})F_{0}(Z+2,E_{2})4(E_{1}E_{2}+m^{2}_{e}+\mathbf{p}_{1}\cdot\mathbf{p}_{2})$
(60)

so that, with standard notation [47] to express the phase-space integration

$C=\frac{(G_{F}\cos{\theta_{C}})^{4}m^{2}_{e}}{16\pi^{5}}$ (61) $\displaystyle
G^{(0)}_{66}$ $\displaystyle=\frac{2C}{(\ln{2})4R_{0}^{2}}\int
F_{0}(Z+2,E_{1})F_{0}(Z+2,E_{2})\frac{(2E_{1}E_{2}+2m^{2}_{e})}{4E_{1}E_{2}}$
$\displaystyle\qquad\qquad\qquad\qquad\quad\times
p_{1}p_{2}E_{1}E_{2}dE_{1}dE_{2}\delta(E_{F}+E_{1}+E_{2}-E_{I})$
$\displaystyle\equiv\frac{2C}{(\ln{2})4R_{0}^{2}}\int\frac{1}{16}f^{(0)}_{66}p_{1}p_{2}E_{1}E_{2}dE_{1}dE_{2}\delta(E_{F}+E_{1}+E_{2}-E_{I})$
(62)

we have the non-zero mixed interactions contribution to the $0\nu\beta\beta$
half-life

$\big{(}T^{0\nu}_{1/2}\big{)}^{-1}_{\text{mix}}=\big{(}\mathcal{G}^{W}_{R}\big{)}^{2}\Big{(}\frac{g^{4}_{t0}}{\Lambda^{4}}\Big{)}\frac{m^{2}_{p}}{2M^{2}_{N}}|\mathcal{M}_{5}|^{2}\frac{G^{(0)}_{66}}{(G_{F}\cos{\theta_{C}})^{2}}\quad.$
(63)

##  4 Bounds on the Model Parameters

Finally, here we will put together the results of the last section to obtain
the constraints on the BSM model’s parameters. Assuming that the three
contributions (31), (47) and (63) are dominant to $0\nu\beta\beta$ half-life,
we have

$T^{0\nu}_{1/2}\equiv(T^{0\nu}_{1/2})_{\text{GI}}+(T^{0\nu}_{1/2})_{\text{CI}}+(T^{0\nu}_{1/2})_{\text{mix}}\quad,$
(64)

and by imposing the experimental lower bounds on $0\nu\beta\beta$, due to the
non-observation of this rare decay, Eq. (64) becomes the following inequality:

$\displaystyle(T^{0\nu}_{1/2})_{\text{exp}}\leq$ $\displaystyle\
\frac{M^{2}_{N}}{(\mathcal{G}^{W}_{R})^{4}}\Bigg{(}\frac{1}{m^{2}_{p}|\mathcal{M}_{0\nu}|^{2}G^{(0)}_{11}}\Bigg{)}+\frac{\Lambda^{4}M^{2}_{N}}{(\mathcal{G}^{W}_{R})^{2}}$
$\displaystyle\times\Bigg{(}\frac{2(G_{F}\cos{\theta_{C}})^{2}}{g^{4}_{t0}m^{2}_{p}|\mathcal{M}_{5}|^{2}G^{(0)}_{66}}\Bigg{)}+\Lambda^{8}M^{2}_{N}\Bigg{(}\frac{64(G_{F}\cos{\theta_{C}})^{4}}{g^{8}_{t0}m^{2}_{p}|\mathcal{M}_{1}|^{2}G^{(0)}_{11}}\Bigg{)}$
(65)

which is the main analytical result of the paper. To obtain the explicit
constraint we insert the appropriate values for the NMEs
($\mathcal{M}_{0\nu}$, $\mathcal{M}_{1}$, $\mathcal{M}_{5}$ from Tables V and
VI of reference [47], with IBM model), the Phase Space Factors, PSFs,
($G^{(0)}_{11}$, $G^{(0)}_{66}$ from Table VII of reference [47]) and for the
other quantities. For the top-quark Yukawa coupling constant
$g_{t0}\sim\mathcal{O}(1)$ as suggested in reference [32].

Up to date, the most stringent experimental bounds on $0\nu\beta\beta$ are:

$(T^{0\nu}_{1/2})_{\text{exp}}=\begin{cases}1.8\times 10^{26}\text{
yr}\quad(^{76}\text{Ge~{}\cite[cite]{[\@@bibref{Number}{GERDA:2020xhi}{}{}]})}\\\
2.3\times 10^{26}\text{
yr}\quad(^{136}\text{Xe~{}\cite[cite]{[\@@bibref{Number}{KamLAND-
Zen:2022tow}{}{}]}})\end{cases}\quad,$ (66)

(a)

(b)

Figure 2: Bounds on $\mathcal{G}^{W}_{R}$ as a function of the heavy Majorana
neutrino mass $M_{N}$, for fixed values of $\Lambda$. The regions excluded by
Eq. (65) are those colored above the three curves. Note that the plots start
from the reference value $M_{N}=10\text{ GeV ($0.01$ TeV)}$, because of the
heavy Majorana neutrino approximation. (a) Constraints obtained with
$0\nu\beta\beta$ lower bound $(T^{0\nu}_{1/2})=2.3\times 10^{26}\text{ yr}$
from KamLAND-Zen experiment. (b) Constraints obtained with expected
$0\nu\beta\beta$ lower bound $(T^{0\nu}_{1/2})\sim 10^{28}\text{ yr}$ from
next generation experiments.

so that the analysis of (65) will be done with 76Ge and 136Xe isotopes.
Nevertheless, the constraints obtained with 76Ge will be less stringent than
with 136Xe, so we ignore the former.

By assigning reference values to the energy scale $\Lambda$ ($\Lambda=1$ TeV,
$\Lambda=5.1$ TeV as suggested in [32], $\Lambda=10$ TeV), we can extract from
(65) an upper bound for the effective coupling $\mathcal{G}^{W}_{R}$ as a
function of the heavy Majorana neutrino mass $M_{N}$, as shown in Fig. 2(a)
with colored areas representing the excluded regions. The constraint imposed
by the non-observation of the $0\nu\beta\beta$ decay when $\Lambda=10\text{
TeV}$ (green curve) becomes very weak before reaching $M_{N}=1\text{ TeV}$,
while for the other values of $\Lambda=1,5.1\text{ TeV}$ (blue and orange
curves, respectively) the limits obtained for the effective coupling constant
are $\mathcal{G}^{W}_{R}\lesssim 2.6\times 10^{-3}$ and
$\mathcal{G}^{W}_{R}\lesssim 2.3\times 10^{-3}$ at $M_{N}=1\text{ TeV}$.

In a similar way, it is possible to foresee the bounds on the effective right
coupling $\mathcal{G}^{W}_{R}$ coming from future $0\nu\beta\beta$ experiments
[15, 16, 18]. The next generation of $0\nu\beta\beta$ experiments aims at
sensitivities for the half-life of the order of
$(T^{0\nu}_{1/2})_{\text{exp}}\sim 10^{28}\text{ yr}\ (^{136}\text{Xe
projection~{}\cite[cite]{[\@@bibref{Number}{nEXO:2021ujk}{}{}]}})$. The future
exclusion limits are given in Fig. 2(b). The bounds are much stronger and will
constrain $\mathcal{G}^{W}_{R}$ below $10^{-3}$ for the whole parameter space
considered, with an improvement of a factor $\sim 2.5$ at the benchmark point
$M_{N}=\Lambda=1\text{ TeV}$.

The results presented in this manuscript are valid only for Majorana neutrino
masses heavier than the typical momentum transfer for $0\nu\beta\beta$
reactions, i.e. $M_{N}\gg 100\text{ MeV}$. Therefore, limit curves in Fig. 2
are presented for the reference value of $M_{N}=10\text{ GeV}\gg 100\text{
MeV}$. We do not address the light neutrinos regime ($M_{N}<100\text{ MeV}$)
since the computation would rely on different approximations, such as the use
of NMEs dependent on the mass of the neutrinos. We focus on a mass range up to
$M_{N}=1$ TeV since it implies small effective right coupling constant
$\mathcal{G}^{W}_{R}$ values, which are preferred in this model [32]. In
addition, the chosen mass range is particularly appealing as it is often
associated with high-energy searches for heavy neutrinos at colliders.

Finally, the same effective right coupling $\mathcal{G}^{W}_{R}$ has been
studied in reference [44] from stellar cooling data [51]. We find that our
constraints are, in general, stronger.

##  5 Summary and Remarks

In this paper, we studied the constraints on the parameters of a BSM model of
NJL type arising from the non-observation of the rare $0\nu\beta\beta$ decay.
We presented the NJL type EFT [31, 39] with its distinctive, quark-quark and
quark-lepton, four-fermion operators which we have shown can contribute to the
half-life of $0\nu\beta\beta$ [13, 14, 52]. After that, we computed these
gauge, contact, and mixed contributions by using the ansatz and approximations
typical of double beta decay calculations [50]. Hence we exploited the non-
observation of $0\nu\beta\beta$ from state-of-the-art experiments and
translated the newly calculated theoretical half-life into a constraint in the
3D parameter space ($M_{N}$, $\Lambda$, $\mathcal{G}^{W}_{R}$) of the examined
BSM model.

By setting some values, of the order of magnitude suggested by the reference
[32], for the New Physics energy scale $\Lambda$, we converted the 3D
constraint into an upper bound, dependent on the mass of the heavy Majorana
neutrino $M_{N}$, for the effective coupling constant $\mathcal{G}^{W}_{R}$.
This right coupling constant is that of vertexes of the SM gauge boson $W$ and
right-handed currents, which means that in our physical scenario, they are
non-vanishing and parity symmetry could be restored at the energy scale
$\Lambda\sim\mathcal{O}(\text{TeV})$. Furthermore, we used the expected half-
life limits from the next-generation experiments to foresee the future
projected upper bound on $0\nu\beta\beta$. Finally, we confirmed the
consistency of our constraints through observed stellar cooling data.

If we include the flavor mixing $V_{\ell
N}\equiv\big{[}(U^{\ell}_{R})^{\dagger}U^{\nu}_{R}\big{]}$ (2) between three
SM families, the final results will receive the contributions of Majorana
neutrinos $N^{\mu}$ ($M_{N^{\mu}}$) and $N^{\tau}$ ($M_{N^{\tau}}$) from the
second and third families. Namely, effective right-handed interacting vertexes
$\sim\mathcal{G}^{W}_{R}V_{eN^{\mu,\tau}}W_{\mu}\bar{e}\gamma^{\mu}P_{R}N^{\mu,\tau}$
are present in Figs. 1(a), 1(c) and 1(d). In these cases, the effective
coupling becomes
$(\mathcal{G}^{W}_{R})^{2}|V_{eN^{\mu,\tau}}|^{2}<(\mathcal{G}^{W}_{R})^{2}$
because of $|V_{eN^{\mu,\tau}}|^{2}<1$. We do not know the mixing $V_{\ell
N}$, masses $M_{N^{e}}$, $M_{N^{\mu}}$ and $M_{N^{\tau}}$ of sterile Majorana
neutrinos $N^{e}$, $N^{\mu}$ and $N^{\tau}$. In this study of $0\nu\beta\beta$
process, Figures 2(a) and 2(b) show constraints on the general parameter space
of the effective right-handed coupling
$\mathcal{G}^{W}_{R}|V_{eN^{e,\mu,\tau}}|$ and sterile Majorana neutrinos’
masses $M_{N^{e,\mu,\tau}}$. The results represent actually contributions from
all sterile Majorana neutrinos $N^{e}$, $N^{\mu}$, and $N^{\tau}$, because of
nontrivial mixing $V_{\ell N}\not=0$. $N^{e}$ could be light and a candidate
for warm DM particles in the present Universe. While massive $N^{\mu}$ and
$N^{\tau}$ could have already decayed to SM particles in the early Universe.
However, they could dominantly contribute to the $0\nu\beta\beta$ process via
Feynman diagrams of Figure 1. To disentangle contributions from sterile
Majorana neutrinos of three SM families requires necessarily different
experiments and observations for different lepton final states. Therefore, it
is worthwhile to study the phenomenology of the effective operators (3) and
(2), relating to the experiments of Probing Heavy Majorana Neutrinos and the
Weinberg Operator through Vector Boson Fusion Processes in Proton-Proton
Collisions [40], the experiment of MiniBooNE Neutrino Oscillation [53, 54],
and the measurement of the Positive Muon Anomalous Magnetic Moment [55].

###### Acknowledgements.

The work of M. P. was supported by the Alexander von Humboldt-Stiftung.

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