# Interference Effects for $0\nu\beta\beta$ Decay in the Left-Right Symmetric
Model

Fahim Ahmed<EMAIL_ADDRESS>Mihai Horoi<EMAIL_ADDRESS>Department
of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA

###### Abstract

Various mechanisms may contribute to neutrinoless double beta decay in the
left-right symmetric model. The interference between these mechanisms also
contribute to the overall decay rate. The analysis of the contributions of
these interference terms is important for disentangling different mechanisms.
In the present paper we study interference effects contributing to the decay
rate for neutrinoless double-$\beta$ decay in the left-right symmetric model.
The numerical values for maximum interference for several nuclides are
calculated. It is observed that, for most of the interference terms, the
contribution is smaller than 20$\%$ for all the nuclei considered in the
study. However, the interference between the mass-mechanisms (light and heavy)
and $\eta$ mechanism is observed to be in the range 30$\%$-50$\%$. The
variation of the interference effect with the $Q$ values is also studied.

##  I Introduction

The lepton number violating (LNV) rare nuclear process of neutrinoless
double-$\beta$ decay ($0\nu\beta\beta$),

$\displaystyle\prescript{A}{Z}{\mathrm{X}}\rightarrow\prescript{A}{Z+2}{\mathrm{X}}+2e^{-},$
(1)

could be an important low-energy manifestation of physics beyond the Standard
Model (BSM). In contrast with the two neutrino double- $\beta$ decay
($2\nu\beta\beta$), where two antineutrinos are also emitted, in
$0\nu\beta\beta$ the lepton number is violated by two units ($\Delta L=2$).
Experimental observation of $0\nu\beta\beta$ would indicate BSM physics since
lepton number is conserved in the Stardard Model (SM). In addition,
$0\nu\beta\beta$ would prove the Majorana nature of neutrinos Schechter and
Valle (1982). Apart from the extensively studied ”standard mass-mechanism” of
light left-handed (LH) neutrino exchange Horoi (2013), several BSM mechanisms
are proposed to contribute to the $0\nu\beta\beta$ decay Horoi (2013);
Vergados et al. (2012).

The left-right symmetric model (LRSM) is a natural extension of the SM where
the parity is assumed to be restored at energies higher than the electroweak
scale. Actively investigated at the LHC Khachatryan et al. (2014), in the LRSM
scenario several competing mechanisms contribute to $0\nu\beta\beta$ due to
the presence of the right-handed (RH) fields Barry and Rodejohann (2013).
Additionally, LRSM provides a natural framework for type-I Mohapatra and
Senjanovic (1980) and type-II Mohapatra and Senjanovic (1981) seesaw
mechanisms generating small neutrino masses. Moreover, the seesaw mechanism
requires the existence of heavy, sterile neutrinos Mohapatra and Senjanovic
(1981). Neutrino mixing schemes would then naturally incorporate heavy-mass
eigenstates for both LH and RH neutrinos (see Sec. II for details).

The study of the $0\nu\beta\beta$ decay rate allows us to extract the new
neutrino physics parameters resulting from such BSM physics scenarios.
However, the neutrino oscillation experiments alone can not determine the
absolute masses of the neutrinos. Moreover, if the regular ”mass-mechanism”
dominates, then $0\nu\beta\beta$ decay will allow us to determine the absolute
masses of neutrinos. All these features make $0\nu\beta\beta$ an exciting
process for probing BSM physics. It thus becomes essential to disentangle the
competing underlying mechanisms inducing $0\nu\beta\beta$ in order to extract
these new neutrino physics parameters arising from BSM physics Faessler et al.
(2011). The inverse half-life formula for $0\nu\beta\beta$ has the following
general structure:

$\displaystyle\Big{[}T_{1/2}^{0\nu}\Big{]}^{-1}=\absolutevalue{\sum_{i}\Big{(}\text{PPP}\Big{)}_{i}\times\Big{(}\text{PSF}\Big{)}_{i}^{\frac{1}{2}}\times\Big{(}\text{NME}\Big{)}_{i}}^{2}.$
(2)

Here, PPP are the particle physics parameters arising from BSM physics, the
phase-space factors (PSF) take into account the kinematical factors of the two
outgoing electrons, and NME are the nuclear matrix elements for the nuclear
transition between the initial and final nuclei. The summation $i$ is over all
possible amplitudes that could induce the $0\nu\beta\beta$ process. Because of
the modulus squared, interference between different terms in Eq. (2) also
contribute to the total decay rate of the process. In Ref. Ahmed et al. (2017)
we studied the interference between the standard mass mechanism and heavy RH
neutrino exchange mechanism. Our analysis in Ahmed et al. (2017) showed
dependence of the relative interference factor on the Q value of
$0\nu\beta\beta$ ($Q_{\beta\beta}$). A contribution no larger than 12$\%$ was
found for all the nuclei considered. Here we extend our study of interference
to other relevant pairs of mechanisms, inducing $0\nu\beta\beta$ in the LRSM
for six nuclei of current experimental interest.

The paper is organized as follows: Section II gives a brief outline of the
LRSM followed by the general formalism for $0\nu\beta\beta$ in LRSM in Sec.
III. In Sec. IV we present the analysis of the interference terms with the
numerical results.

##  II Brief Review of the Left-Right Symmetric Model

In the LRSM the SM gauge group $\mathcal{G}_{SM}\equiv SU(3)_{C}\otimes
SU(2)_{L}\otimes U(1)_{Y}$ is extended to $SU(3)_{C}\otimes\mathcal{G}_{LR}$
with $\mathcal{G}_{LR}\equiv SU(2)_{L}\otimes SU(2)_{R}\otimes U(1)_{B-L}$
Mohapatra and Senjanovic (1981); Pati and Salam (1974); Mohapatra and Pati
(1975). Restoring parity above the electroweak scale, the extended group
$SU(2)_{R}$ allows us to form the RH fermions as doublets. We have the
following fermion particle content in LRSM with the corresponding
representation under $SU(3)_{C}\otimes\mathcal{G}_{LR}$ Barry and Rodejohann
(2013); Bhupal Dev et al. (2015); Borah et al. (2018),

$\displaystyle
SU(2)_{L}\,\,\text{Leptons}:\,L_{Lj}=\left(\begin{array}[]{c}{\nu_{Lj}}\\\
{e_{Lj}}\end{array}\right)\in(\mathbf{1},\mathbf{2},\mathbf{1},-1),$ (5)
$\displaystyle
SU(2)_{R}\,\,\text{Leptons}:\,L_{Rj}=\left(\begin{array}[]{c}{\nu_{Rj}}\\\
{e_{Rj}}\end{array}\right)\in(\mathbf{1},\mathbf{1},\mathbf{2},-1),$ (8)
$\displaystyle
SU(2)_{L}\,\,\text{Quarks}:\,Q_{Lj}=\left(\begin{array}[]{c}{u_{Lj}}\\\
{d_{Lj}^{\prime}}\end{array}\right)\in\left(\mathbf{3},\mathbf{2},\mathbf{1},\tfrac{1}{3}\right),$
(11) $\displaystyle
SU(2)_{R}\,\,\text{Quarks}:\,Q_{Rj}=\left(\begin{array}[]{c}{u_{Rj}}\\\
{d_{Rj}^{\prime}}\end{array}\right)\in\left(\mathbf{3},\mathbf{1},\mathbf{2},\tfrac{1}{3}\right),$
(14)

where the generations are defined as:
$\nu_{j=1,2,3}\equiv\\{\nu_{e},\nu_{\mu},\nu_{\tau}\\}$,
$e_{j=1,2,3}\equiv\\{e,\mu,\tau\\}$, $u_{j=1,2,3}\equiv\\{u,c,t\\}$,
$d^{\prime}_{j=1,2,3}\equiv\\{d^{\prime},s^{\prime},b^{\prime}\\}$. The
subscripts $L$ and $R$ are associated with the chiral projection operators
$P_{L}=\tfrac{1}{2}(1-\gamma^{5})$ and $P_{R}=\tfrac{1}{2}(1+\gamma^{5})$,
respectively. The first three entries of the quadruplet of numbers denote the
dimension of the representation under each of the gauge groups $SU(3)_{C}$,
$SU(2)_{L}$, $SU(2)_{R}$, respectively Burgess and Moore (2006). The fourth
entry denotes the quantum number associated with the group $U(1)_{B-L}$: the
difference between the baryon and lepton number, $B-L=2(Q-T_{3L}-T_{3R})$,
with $Q$ being the electromagnetic charge and $T_{3L}$ and $T_{3R}$ being the
third component of the isospin corresponding to $SU(2)_{L}$ and $SU(2)_{L}$,
respectively. For example, $(\mathbf{3},\mathbf{1},\mathbf{2},\frac{1}{3})$
for $SU(2)_{R}$ quarks denote a triplet under $SU(3)_{C}$, a singlet under
$SU(2)_{L}$, a doublet under $SU(2)_{R}$, and has a charge $\frac{1}{3}$ under
$U(1)_{B-L}$ Borah et al. (2018). The seven massless gauge bosons along with
their respective couplings for the $\mathcal{G}_{LR}^{EW}$ sector are,

$\displaystyle
SU(2)_{L}:g_{L},\,\,\\{W_{L\mu}^{1},W_{L\mu}^{2},W_{L\mu}^{3}\\},$ (15)
$\displaystyle
SU(2)_{R}:g_{R},\,\,\\{W_{R\mu}^{1},W_{R\mu}^{2},W_{R\mu}^{3}\\},$ (16)
$\displaystyle U(1)_{B-L}:g^{\prime},\,\,B_{\mu}.$ (17)

The interaction Lagrangian before spontaneous symmetry breaking (SSB) between
fermions and gauge bosons for the $\mathcal{G}_{LR}$ sector is constructed in
parallel to the SM electroweak Lagrangian (a summation of repeated indices,
$j,a=1,2,3$ and $\mu=0,1,2,3$, is implied),

$\displaystyle\mathcal{L}^{\text{\tiny{EW}}}_{\text{\tiny{LR}}}$
$\displaystyle=g_{L}\left[\overline{L}_{Lj}\gamma^{\mu}\frac{\sigma_{a}}{2}L_{Lj}+\overline{Q}_{Lj}\gamma^{\mu}\frac{\sigma_{a}}{2}Q_{Lj}\right]W_{L\mu}^{a}$
$\displaystyle+g_{R}\left[\overline{L}_{Rj}\gamma^{\mu}\frac{\sigma_{a}}{2}L_{Rj}+\overline{Q}_{Rj}\gamma^{\mu}\frac{\sigma_{a}}{2}Q_{Rj}\right]W^{a}_{R\mu}$
$\displaystyle+g^{\prime}\left[\overline{L}_{Lj}\gamma^{\mu}\tfrac{B-L}{2}L_{Lj}+\overline{Q}_{Lj}\gamma^{\mu}\tfrac{B-L}{2}Q_{Lj}\right.$
$\displaystyle\left.+\overline{L}_{Rj}\gamma^{\mu}\tfrac{B-L}{2}L_{Rj}+\overline{Q}_{Rj}\gamma^{\mu}\tfrac{B-L}{2}Q_{Rj}\right]B_{\mu}.$
(18)

The charge-current part of
$\mathcal{L}^{\text{\tiny{EW}}}_{\text{\tiny{LR}}}$, which is relevant for
$0\nu\beta\beta$, takes the following form (confining ourselves to only the
first generation),

$\displaystyle\mathcal{L}_{\text{\tiny{LR}}}^{\text{\tiny{CC}}}\supseteq\frac{g_{L}}{\sqrt{2}}$
$\displaystyle\Big{[}\Big{(}\overline{\nu_{eL}}\gamma^{\mu}e_{L}+\overline{u_{L}}\gamma^{\mu}d^{\prime}_{L}\Big{)}W_{\mu
L}^{+}$
$\displaystyle+\Big{(}\overline{e_{L}}\gamma^{\mu}\nu_{eL}+\overline{d^{\prime}_{L}}\gamma^{\mu}u_{L}\Big{)}W_{\mu
L}^{-}\Big{]}$
$\displaystyle+\frac{g_{R}}{\sqrt{2}}\Big{[}\Big{(}\overline{\nu_{eR}}\gamma^{\mu}e_{R}+\overline{u_{R}}\gamma^{\mu}d^{\prime}_{R}\Big{)}W_{\mu
R}^{+}$
$\displaystyle+\Big{(}\overline{e_{R}}\gamma^{\mu}\nu_{eR}+\overline{d^{\prime}_{R}}\gamma^{\mu}u_{R}\Big{)}W_{\mu
R}^{-}\Big{]},$ (19)

where the charged vector bosons are defined in terms of the
$W_{L(R)\mu}^{a=1,2,3}$ fields as,

$\displaystyle W_{L(R)\mu}^{\pm}=\frac{1}{\sqrt{2}}\Big{(}W_{L(R)\mu}^{1}\mp
iW_{L(R)\mu}^{2}\Big{)}.$ (20)

The scalar sector consists of two Higgs triplets and a bi-doublet Grimus
(1993),

$\displaystyle\Delta_{L(R)}=\left[\begin{array}[]{cc}{\frac{1}{\sqrt{2}}\Delta^{+}_{L(R)}}&{\Delta_{L(R)}^{++}}\\\
{\Delta_{L(R)}^{0}}&{\frac{-1}{\sqrt{2}}\Delta_{L(R)}^{+}}\end{array}\right]\,,\,\Phi=\left[\begin{array}[]{cc}{\phi_{1}^{0}}&{\phi_{2}^{+}}\\\
{\phi_{1}^{-}}&{\phi_{2}^{0}}\end{array}\right],$ (25)

with $\Delta_{L}\in(1,\mathbf{3},1,2)$, $\Delta_{R}\in(1,1,\mathbf{3},2)$ and
$\Phi\in(1,\mathbf{2},\mathbf{2},0)$. The gauge symmetry $\mathcal{G}_{LR}$ is
broken in two stages by the scalar sector of the theory. Above the SM
electroweak scale the SSB: $SU(2)_{L}\otimes SU(2)_{R}\otimes
U(1)_{B-L}\rightarrow SU(2)_{L}\otimes U(1)_{Y}$ takes place through the
vaccum expectation value (VEV) of the two Higgs triplets,

$\displaystyle\left\langle\Delta_{L}\right\rangle=\left(\begin{array}[]{cc}{0}&{0}\\\
{\frac{1}{\sqrt{2}}v_{L}e^{i\theta_{L}}}&{0}\end{array}\right)\,,\,\left\langle\Delta_{R}\right\rangle=\left(\begin{array}[]{cc}{0}&{0}\\\
{\frac{1}{\sqrt{2}}v_{R}}&{0}\end{array}\right),$ (30)

This breaks the parity and also allows Majorana mass terms for neutrinos. In
the second stage, the SM electroweak SSB: $SU(2)_{L}\otimes
U(1)_{Y}\rightarrow U(1)_{EM}$ takes place through the VEV of the bi-doublet
Higgs,

$\langle\Phi\rangle=\left(\begin{array}[]{cc}{\frac{1}{\sqrt{2}}\kappa_{1}}&{0}\\\
{0}&{\frac{1}{\sqrt{2}}\kappa_{2}e^{i\alpha}}\end{array}\right).$ (31)

Here we have written the Lagrangian in the flavor basis. After SSB
$\mathcal{L}_{\text{\tiny{LR}}}^{\text{\tiny{EW}}}$ acquires mass terms for
the fermions and gauge bosons. For the neutrino sector, type I + II seesaw
scenario is assumed, giving rise to small masses for light neutrinos due to
the presence of heavy Majorana neutrinos Barry and Rodejohann (2013). The
mass-matrix for neutrinos ($\nu_{e,\mu,\tau}$), d type quarks
($d^{\prime}_{j}$) and the charged vector bosons ($W_{L(R)}^{\pm}$) are not
diagonal in the flavor basis. We thus reexpress the flavor-basis fields in
terms of fields in the mass-basis diagonalizing the mass matrices, for
$d^{\prime}$ quarks:

$\displaystyle d_{L}^{\prime}=V_{ud}d_{L}+V_{us}s_{L}+V_{ub}b_{L},$ (32)
$\displaystyle
d_{R}^{\prime}=V^{\prime}_{ud}d_{R}+V^{\prime}_{us}s_{R}+V^{\prime}_{ub}b_{R},$
(33)

for electron-neutrinos:

$\displaystyle\nu_{eL}=\sum_{i=1,2,3}^{\text{light}}U_{ei}\nu_{Li}+\sum_{i=1,2,3}^{\text{heavy}}S_{ei}(N_{Ri})^{c},$
(34)
$\displaystyle\nu_{eR}=\sum_{i=1,2,3}^{\text{light}}T^{*}_{ei}(\nu_{Li})^{c}+\sum_{i=1,2,3}^{\text{heavy}}V^{*}_{ei}N_{Ri},$
(35)

and for $W$ bosons:

$\displaystyle\left(\begin{array}[]{c}{W_{L}^{\pm}}\\\
{W_{R}^{\pm}}\end{array}\right)=\left(\begin{array}[]{cc}{\cos\xi}&{\sin\xi
e^{i\alpha}}\\\ {-\sin\xi
e^{-i\alpha}}&{\cos\xi}\end{array}\right)\left(\begin{array}[]{c}{W_{1}^{\pm}}\\\
{W_{2}^{\pm}}\end{array}\right).$ (42)

Here Eq. (32) is the first row of the Cabibbo-Kobayashi-Maskawa (CKM) matrix
for LH quark mixing with Eq. (33) being the first row of an equivalent CKM
matrix for RH quark mixing Senjanovic and Tello (2016). The matrix elements
$V_{ud}$ and $V^{\prime}_{ud}$ can be approximated as
$V_{ud}\simeq\cos{\theta_{c}}$ and
$V^{\prime}_{ud}\simeq\cos{\theta_{c}^{\prime}}$ in terms of the Cabibbo angle
$\theta_{c}$ for LH $d$ quarks and analogous $\theta_{c}^{\prime}$ for RH $d$
quarks Doi et al. (1985). We have considered the
$(3\text{-light}+3\text{-heavy})$ scenario for Majorana neutrino mixing
wherein the mass basis of the light-neutrinos are $\nu_{i}$ with masses
$m_{i}$, and the heavy neutrinos are $N_{i}$ with masses $M_{i}$. The $S,T,V$
mixing matrices are generalization of the PMNS matrix $U$ for the LH-light
neutrino mixing. The charged $W$ bosons, $W^{\pm}_{L(R)}$ are linear
combination of physical bosons $W^{\pm}_{1(2)}$ with definite masses
$m_{W_{1}}$ and $m_{W_{2}}$, respectively. We can further assume a discrete LR
symmetry where the Lagrangian is invariant under the exchange
$L\leftrightarrow R$. This assumption requires that the two gauge couplings be
equal, $g=g_{L}=g_{R}$. The case of $g_{L}\neq g_{R}$ leads to different
expressions for the effective couplings $G_{F}$, $\lambda$, $\eta$ (see
below), but the form of the $0\nu\beta\beta$ amplitudes are the same as for
$g_{L}=g_{R}$ (see Ref. Hirsch and Klapdor-Kleingrothaus (1996) for details).
Thus, under these assumptions we can write the charged-current Lagrangian for
the first fermion generation in the mass basis as,

$\displaystyle\mathcal{L}^{\text{\tiny{CC}}}_{\text{\tiny{LR}}}\supseteq\frac{g}{\sqrt{2}}\sum_{i=1}^{3}$
$\displaystyle\Bigg{[}\bigg{[}\Big{(}U^{*}_{ei}\overline{\nu_{Li}}+S^{*}_{ei}\overline{(N_{Ri})^{c}}\Big{)}\gamma^{\mu}e_{L}+\cos{\theta_{c}}\overline{u_{L}}\gamma^{\mu}d_{L}\bigg{]}\Big{(}\cos{\xi}W_{1\mu}^{+}+\sin{\xi}e^{i\alpha}W_{2\mu}^{+}\Big{)}$
$\displaystyle+\bigg{[}\overline{e_{L}}\gamma^{\mu}\Big{(}U_{ei}\nu_{Li}+S_{ei}(N_{Ri})^{c}\Big{)}+\cos{\theta_{c}}\overline{d_{L}}\gamma^{\mu}u_{L}\bigg{]}\Big{(}\cos{\xi}W_{1\mu}^{-}+\sin{\xi}e^{i\alpha}W_{2\mu}^{-}\Big{)}$
$\displaystyle+\bigg{[}\Big{(}T_{ei}\overline{(\nu_{Li})^{c}}+V_{ei}\overline{N_{Ri}}\Big{)}\gamma^{\mu}e_{R}+\cos{\theta^{\prime}_{c}}\overline{u_{R}}\gamma^{\mu}d_{R}\bigg{]}\Big{(}-\sin{\xi}e^{-i\alpha}W_{1\mu}^{+}+\cos{\xi}W_{2\mu}^{+}\Big{)}$
$\displaystyle+\bigg{[}\overline{e_{R}}\gamma^{\mu}\Big{(}T^{*}_{ei}(\nu_{Li})^{c}+V^{*}_{ei}N_{Ri}\Big{)}+\cos{\theta^{\prime}_{c}}\overline{d_{R}}\gamma^{\mu}u_{R}\bigg{]}\Big{(}-\sin{\xi}e^{-i\alpha}W_{1\mu}^{-}+\cos{\xi}W_{2\mu}^{-}\Big{)}\Bigg{]}.$
(43)

##  III Formalism for $0\nu\beta\beta$ in the Left-Right Symmetric Model

###  III.1 $\beta$-decay in left-right symmetric model

(a) $W_{L}$ mediation for purely LH fields.

(b) $W_{R}$-$W_{L}$ mediation for RH-LH mixed fields.

(c) $W_{L}$-$W_{R}$ mediation for LH-RH mixed fields.

(d) $W_{R}$ mediation for purely RH fields.

Figure 1: $\beta$-decay diagrams in LRSM at the $W$-boson and effective Fermi-
like four-fermion level.

Starting from the charge-current Lagrangian of Eq. (II) for the LRSM, after
applying second-order perturbation in the gauge coupling $g$, we get four
different types of $\beta$-decay diagrams due to the presence of RH currents
(see Fig. 1). We can then integrate out the heavy degrees of freedom for the
charged bosons ($m_{W_{L}},m_{W_{R}}\geq 80$ GeV) to get point-like Fermi
vertices. Fig. 1a shows the usual $\beta$-decay via $W_{L}^{-}$ exchange with
$G_{\beta}=G_{F}\cos{\theta_{c}}$ being the effective point-like coupling
between LH-quarks and LH-lepton currents, and $G_{F}$ is the Fermi constant.
Figures. 1b, 1d, 1c describe the presence of RH quarks and/or lepton currents.
In Fig. 1b the RH-quark and LH-lepton currents are coupled by $W_{R}$-$W_{L}$
mixing, mediated by the effective coupling $G_{\beta}\kappa$. Fig. 1c shows
the diagram of $W_{L}$-$W_{R}$ exchange between LH-quarks and RH-lepton
currents with effective coupling $G_{\beta}\eta$. Lastly, Fig. 1d shows the RH
counterpart for the usual $\beta$-decay of Fig. 1a with $W_{R}^{-}$ exchange,
and $G_{\beta}\lambda$ is the effective coupling between RH currents for
quarks and lepton. The exact expressions for the effective couplings, $G_{F}$,
$\lambda$, $\eta$, in terms of the LRSM parameters are given in Eqs. (7)-(9)
of Ref. Hirsch and Klapdor-Kleingrothaus (1996). For small $W_{L}$-$W_{R}$
mixing ($\xi\ll 1$) we get,

$\displaystyle
G_{F}\simeq\sqrt{2}g^{2}/8m^{2}_{W_{L}}\quad,\quad\eta=\kappa\simeq\tan{\xi},$
(44)
$\displaystyle\lambda\simeq(m_{W_{1}}/m_{W_{2}})^{2}\simeq(m_{W_{L}}/m_{W_{R}})^{2}.$
(45)

Thus at the level of effective couplings we can write an effective low-energy
($V\pm A$) Fermi-like current-current Lagrangian for $\beta$ decayHirsch and
Klapdor-Kleingrothaus (1996); Doi et al. (1985) considering the RH-currents.
Taking $\cos{\theta^{\prime}_{c}}/\cos{\theta_{c}}=1$ one gets,

$\displaystyle\mathcal{L}_{\text{\tiny{LR}}}^{\beta}=\frac{G_{\beta}}{\sqrt{2}}\Big{[}j^{\mu}_{L}J_{L\mu}^{\dagger}+\kappa
j^{\mu}_{L}J_{R\mu}^{\dagger}+\eta j^{\mu}_{R}J_{L\mu}^{\dagger}$
$\displaystyle+\lambda j^{\mu}_{R}J_{R\mu}^{\dagger}\Big{]}+\text{H.c.},$ (46)

where $j^{\mu}_{\alpha}=\overline{e_{\alpha}}\gamma^{\mu}\nu_{e\alpha}$ and
$J^{\dagger}_{\alpha,\mu}=\overline{u_{\alpha}}\gamma_{\mu}d_{\alpha}$ are
leptonic and hadronic currents respectively with $\alpha=L,R$. The four terms
in Eq. (46), in that order, correspond to the four diagrams of Fig. 1,
respectively. H.c. denotes the Hermitian conjugate terms, which do not
contribute to $0\nu\beta\beta$.

Notice that the neutrino fields are written in the flavor basis. The light and
heavy neutrino mixing parameters in Eq. (35), are part of the leptonic
currents and not of the effective BSM parameters $\eta$, $\lambda$. The LNV
parameters of neutrino mixing are realized at the amplitude level in our
analysis. 111See Sec. III.4 for the effective-field theory approach to
$0\nu\beta\beta$ where the LNV parameters are interpreted at the effective
coupling level but give us the same formula for the half-life.

###  III.2 Amplitudes and diagrams for $0\nu\beta\beta$ from
$\mathcal{L}_{\text{\tiny{LR}}}^{\beta}$.

At the effective Lagrangian level of Eq. (46) $0\nu\beta\beta$ amplitude
arises at second-order ($G^{2}_{\beta}$) of perturbation. The time-ordered
product of $\mathcal{L}_{\text{\tiny{LR}}}^{\beta}$ has ten distinct terms,

$\displaystyle\mathcal{T}\Big{(}\mathcal{L}_{\text{\tiny{LR}}}^{\beta}(x)\mathcal{L}_{\text{\tiny{LR}}}^{\beta}(y)\Big{)}=\frac{G^{2}_{\beta}}{2}\mathcal{T}\Big{(}[j_{L}J^{\dagger}_{L}]_{x}[j_{L}J^{\dagger}_{L}]_{y}+2\kappa[j_{L}J^{\dagger}_{L}]_{x}[j_{L}J^{\dagger}_{R}]_{y}+\kappa^{2}[j_{L}J^{\dagger}_{R}]_{x}[j_{L}J^{\dagger}_{R}]_{y}$
$\displaystyle+\lambda^{2}[j_{R}J^{\dagger}_{R}]_{x}[j_{R}J^{\dagger}_{R}]_{y}+2\lambda\eta[j_{R}J^{\dagger}_{R}]_{x}[j_{R}J^{\dagger}_{L}]_{y}+\eta^{2}[j_{R}J^{\dagger}_{L}]_{x}[j_{R}J^{\dagger}_{L}]_{y}$
$\displaystyle+\lambda[j_{L}J^{\dagger}_{L}]_{x}[j_{R}J^{\dagger}_{R}]_{y}+\eta[j_{L}J^{\dagger}_{L}]_{x}[j_{R}J^{\dagger}_{L}]_{y}+\kappa\lambda[j_{L}J^{\dagger}_{R}]_{x}[j_{R}J^{\dagger}_{R}]_{y}+\kappa\eta[j_{L}J^{\dagger}_{R}]_{x}[j_{R}J^{\dagger}_{L}]_{y}\Big{)}.$
(47)

From the above time-ordered product we see three types of combinations of
leptonic currents: $j_{L}j_{L}$, $j_{R}j_{R}$, and $j_{L}j_{R}$. After
applying Wick’s theorem to the time-ordered product, the neutrino fields in
the leptonic currents get contracted, giving rise to the virtual neutrino
propagator of $0\nu\beta\beta$. The flavor neutrinos are linear combinations
of mass eigenstates as in Eq. (35). Thus the virtual neutrino propagators
would be of two types: light or heavy massive Majorana neutrinos Borah et al.
(2018) for each of the three leptonic current combinations. Expressed in terms
of the usual Dirac propagator we get for the neutrino propagators Srednicki
(2007),

$\displaystyle\mathcal{T}\Big{(}j_{L}(x)j_{L}(y)\Big{)}\propto\contraction{}{\nu}{{}_{eL}(x)}{\nu}\nu_{eL}(x)\nu^{T}_{eL}(y$
$\displaystyle=\sum_{i}P_{L}\Big{[}U^{2}_{ei}\,\,S^{D}_{m_{i}}(x-y)+S^{2}_{ei}\,\,S^{D}_{M_{i}}(x-y)\Big{]}P_{L}C,$
(48)
$\displaystyle\mathcal{T}\Big{(}j_{R}(x)j_{R}(y)\Big{)}\propto\contraction{}{\nu}{{}_{eR}(x)}{\nu}\nu_{eR}(x)\nu^{T}_{eR}(y)$
$\displaystyle=\sum_{i}P_{R}\Big{[}T^{*2}_{ei}S^{D}_{m_{i}}(x-y)+V^{*2}_{ei}S^{D}_{M_{i}}(x-y)\Big{]}P_{R}C,$
(49)
$\displaystyle\mathcal{T}\Big{(}j_{L}(x)j_{R}(y)\Big{)}\propto\contraction{}{\nu}{{}_{eL}(x)}{\nu}\nu_{eL}(x)\nu^{T}_{eR}(y)$
$\displaystyle=\sum_{i}P_{L}\Big{[}U_{ei}T^{*}_{ei}S^{D}_{m_{i}}(x-y)+S_{ei}V^{*}_{ei}S^{D}_{M_{i}}(x-y)\Big{]}P_{R}C,$
(50)

where $C$ is the charge-conjugation matrix and the Dirac propagator
$S^{D}_{m^{\prime}_{i}}$ is defined as ($m_{i}^{\prime}=m_{i},M_{i}$),

$\displaystyle
S^{D}_{m^{\prime}_{i}}(x-y)=i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{e^{-iq\cdot(x-y)}}{q^{2}-m^{\prime
2}_{i}}(\not{q}+m_{i}^{\prime}).$ (51)

Because of the presence of the chiral projection operators $P_{L(R)}$ we will
have two categories of contributions to the amplitude,

$\displaystyle\text{i)}\quad
P_{L(R)}\frac{\not{q}+m^{\prime}_{i}}{q^{2}-m_{i}^{\prime
2}}P_{L(R)}\propto\frac{m^{\prime}_{i}}{q^{2}-m_{i}^{\prime 2}},$ (52)
$\displaystyle\text{ii)}\quad
P_{L(R)}\frac{\not{q}+m^{\prime}_{i}}{q^{2}-m_{i}^{\prime
2}}P_{R(L)}\propto\frac{\not{q}}{q^{2}-m_{i}^{\prime 2}}.$ (53)

Thus we have (i) mass-dependent amplitudes where the two electrons have the
same chirality, and (ii) momentum-dependent amplitudes when the two electrons
have opposite chiralities Pas et al. (1999). The typical scale of momentum
transfer for the vitual neutrino is $\absolutevalue{q}\simeq 100$ MeV. Here we
assume $m_{i}\ll\absolutevalue{q}$ and $M_{i}\gg\absolutevalue{q}$ for the
light and heavy Majorana neutrinos, respectively. Depending on the mass of the
intermediate Majorana neutrinos, we have two categories of approximations for
both the mass and momentum dependent amplitudes,

i) mass-dependent propagators:
$\displaystyle\frac{m^{\prime}_{i}}{q^{2}-m_{i}^{\prime
2}}\simeq\left\\{\begin{array}[]{ll}{\frac{m_{i}}{q^{2}},}&{m_{i}^{2}\ll
q^{2}\qquad\text{light-}\nu_{i}}\\\ {-\frac{1}{M_{i}},}&{M_{i}^{2}\gg
q^{2}\qquad\text{heavy-}N_{i}}\end{array}\right.,$ (56) ii) momentum-dependent
propagators: $\displaystyle\frac{\not{q}}{q^{2}-m_{i}^{\prime
2}}\simeq\left\\{\begin{array}[]{cc}{\frac{1}{|q|},}&{m_{i}^{2}\ll
q^{2}\qquad\text{light-}\nu_{i}}\\\ {-\frac{|q|}{M_{i}^{2}},}&{M_{i}^{2}\gg
q^{2}\qquad\text{heavy-}N_{i}}\end{array}\right..$ (59)

For the keV scale ($M_{i}<\absolutevalue{q}$) neutrino case see Refs. Barry et
al. (2014); Borah et al. (2018). We now discuss the (i) mass-dependent and
(ii) momentum-dependent cases separately.

(a) Light neutrino exchange for purely LH currents. Diagram $\propto\eta_{m}$
arising from $j_{L}J_{L}^{\dagger}j_{L}J_{L}^{\dagger}$ term. (b) Heavy
neutrino exchange for purely RH currents. Diagram $\propto\eta_{N}$ arising
from $j_{R}J_{R}^{\dagger}j_{R}J_{R}^{\dagger}$ term.

Figure 2: Relevant diagrams for $0\nu\beta\beta$ in LRSM for both electron of
same chirality.

#### Mass-dependent mechanisms: Outgoing electrons having same chirality

The first six terms on the right-hand side of Eq. (III.2) are mass-dependent
terms where both the electrons are either LH or RH. We can ignore most of the
second-order terms because of the smallness of the BSM parameters ($\kappa$,
$\lambda$, $\eta$ $\ll$ 1) for both light and heavy neutrino exchange.
Moreover, the first-order term in $\kappa$ is further suppressed because of
its dependence on neutrino mass. Since the mixing matrix $S$ is small and
given the heavy mass $M_{i}$ being in the denominator, the heavy neutrino
exchange case for purely LH currents can be ignored. Thus, the first term
$[j_{L}J^{\dagger}_{L}]_{x}[j_{L}J^{\dagger}_{L}]_{y}$ gives rise to the
regular mass mechanism of Fig. 2a of light neutrino exchange for purely LH
hadronic and leptonic currents. The amplitude for the ‘mass-mechanism’ is
then,

$\displaystyle\mathcal{A}_{L}^{\nu}\propto
G_{\beta}^{2}\sum_{i}\frac{U_{ei}^{2}m_{i}}{q^{2}},$ (60)

where the dimensionless LNV complex parameter
$\eta_{m}=\absolutevalue{\eta_{m}}\exp(i\phi_{m})$ for the ‘mass-mechanism’
along with the phase are defined in terms of the BSM parameters of LRSM as
follows:

$\displaystyle\absolutevalue{\eta_{m}}$
$\displaystyle=\frac{1}{m_{e}}\absolutevalue{\langle
m_{\beta\beta}\rangle}=\frac{1}{m_{e}}\absolutevalue{\sum_{i}U^{2}_{ei}m_{i}},$
(61) $\displaystyle\phi_{m}$
$\displaystyle=\text{Arg}\Big{[}\sum_{i}U^{2}_{ei}m_{i}\Big{]}.$ (62)

The only second-order term considered in Eq. (III.2) is the $\lambda^{2}$ term
for the heavy neutrino exchange because the mixing matrix $V$ is assumed to be
large. Thus, from the term
$\lambda^{2}[j_{R}J^{\dagger}_{R}]_{x}[j_{R}J^{\dagger}_{R}]_{y}$ we get the
diagram of Fig. 2b. Then, the amplitude for the heavy neutrino exchange for
the purely RH currents is,

$\displaystyle\mathcal{A}_{R}^{N}\propto
G_{\beta}^{2}\lambda^{2}\sum_{i}\frac{V_{ei}^{*2}}{M_{i}},$ (63)

where the dimensionless LNV parameter
$\eta_{N}=\absolutevalue{\eta_{N}}\exp(i\phi_{N})$ for the heavy neutrino
exchange ($N_{i}$) is,

$\displaystyle\absolutevalue{\eta_{N}}$
$\displaystyle=m_{p}\lambda^{2}\absolutevalue{\sum_{i}\frac{V^{*2}_{ei}}{M_{i}}}=m_{p}\Big{(}\frac{m_{W_{L}}}{m_{W_{R}}}\Big{)}^{4}\absolutevalue{\sum_{i}\frac{V^{*2}_{ei}}{M_{i}}},$
(64) $\displaystyle\phi_{N}$
$\displaystyle=\text{Arg}\Big{[}\sum_{i}\frac{V^{*2}_{ei}}{M_{i}}\Big{]}.$
(65)

#### Momentum-dependent mechanisms: Outgoing electrons having opposite
chiralities

The last four terms in Eq. (III.2) are momentum-dependent terms. The first-
order terms $\lambda$ and $\eta$ can give competing contributions to
$0\nu\beta\beta$ compared with the regular mass-mechanism of Fig. 2a for light
neutrino exchange. Thus, the term
$\lambda[j_{L}J^{\dagger}_{L}]_{x}[j_{R}J^{\dagger}_{R}]_{y}$ gives rise to
the diagram of Fig. 3a, the so-called $\lambda$ mechanism, due to the
combination of LH and RH currents. The amplitude of Fig. 3a for the
$\lambda$-mechanism is then,

(a) $\lambda$-diagram due to both LH and RH currents. Diagram
$\propto\eta_{\lambda}$ arising from
$j_{L}J_{L}^{\dagger}j_{R}J_{R}^{\dagger}$ term. (b) $\eta$-diagram due to
gauge boson mixing. Diagram $\propto\eta_{\eta}$ arising from
$j_{L}J_{L}^{\dagger}j_{R}J_{L}^{\dagger}$ term.

Figure 3: Relevant diagrams for $0\nu\beta\beta$ in LRSM for both electrons of
opposite chirality.

$\mathcal{A}_{\lambda}^{\nu}\propto
G_{\beta}^{2}\lambda\sum_{i}U_{ei}T_{ei}^{*}\frac{1}{q},$ (66)

where the corresponding dimensionless LNV PPP,
$\eta_{\lambda}=\absolutevalue{\eta_{\lambda}}\exp(i\phi_{\lambda})$,

$\displaystyle\absolutevalue{\eta_{\lambda}}$
$\displaystyle=\lambda\absolutevalue{\sum_{i}U_{ei}T^{*}_{ei}}=\Big{(}\frac{m_{W_{L}}}{m_{W_{R}}}\Big{)}^{2}\absolutevalue{\sum_{i}U_{ei}T^{*}_{ei}},$
(67) $\displaystyle\phi_{\lambda}$
$\displaystyle=\text{Arg}\Big{[}\sum_{i}U_{ei}T^{*}_{ei}\Big{]}.$ (68)

The other first-order term
$\eta[j_{L}J^{\dagger}_{L}]_{x}[j_{R}J^{\dagger}_{L}]_{y}$ in Eq. (III.2)
gives rise to the diagram of Fig. 3b, the so-called $\eta$ mechanism due to
$W_{L}-W_{R}$ mixing. The amplitude for Fig. 3b for the $\eta$ mechanism is
then

$\mathcal{A}_{\eta}^{\nu}\propto
G_{\beta}^{2}\eta\sum_{i}U_{ei}T_{ei}^{*}\frac{1}{q},$ (69)

where the corresponding dimensionless LNV PPP,
$\eta_{\eta}=\absolutevalue{\eta_{\eta}}\exp(i\phi_{\eta})$, is

$\displaystyle\absolutevalue{\eta_{\eta}}$
$\displaystyle=\eta\absolutevalue{\sum_{i}U_{ei}T^{*}_{ei}}=\tan{\xi}\absolutevalue{\sum_{i}U_{ei}T^{*}_{ei}},$
(70) $\displaystyle\phi_{\eta}$
$\displaystyle=\text{Arg}\Big{[}\sum_{i}U_{ei}T^{*}_{ei}\Big{]}.$ (71)

Terms due to heavy neutrino exchange are suppressed, being proportional to
$S_{ei}V_{ei}^{*}q/M_{i}^{2}$ Barry and Rodejohann (2013).

Apart from the diagrams considered in Figs. dia-same and 3, there could be
additional contributions due to exchange of $SU(2)_{R}$ and $SU(2)_{L}$ Higgs
triplets in LRSM, see Fig. 3 of Ref. Barry and Rodejohann (2013). These
diagrams are suppressed Barry and Rodejohann (2013); Bhupal Dev et al. (2015);
Tello et al. (2011) and hence we will not consider them in the subsequent
analysis.

###  III.3 Half-Life for $0\nu\beta\beta$

Considering the total amplitude for $0\nu\beta\beta$ for the four diagrams of
Fig. 2 and 3,

$\displaystyle\mathcal{A}^{0\nu}=\mathcal{A}_{L}^{\nu}+\mathcal{A}_{R}^{N}+\mathcal{A}_{\lambda}^{\nu}+\mathcal{A}_{\eta}^{\nu}$
(72)

we arrive at the following inverse half-life formula for $0\nu\beta\beta$,

$\displaystyle[T^{0\nu}_{1/2}]^{-1}=g^{4}_{A}$
$\displaystyle\Big{[}C_{m}\absolutevalue{\eta_{m}}^{2}+C_{N}\absolutevalue{\eta_{N}}^{2}+C_{\lambda}\absolutevalue{\eta_{\lambda}}^{2}+C_{\eta}\absolutevalue{\eta_{\eta}}^{2}$
$\displaystyle+\sum_{i\neq
j}^{\\{m,N,\lambda,\eta\\}}C_{ij}\absolutevalue{\eta_{i}}\absolutevalue{\eta_{j}}\cos{(\phi_{i}-\phi_{j})}\Big{]},$
(73)

where we have factorized $g_{A}^{4}=(1.27)^{4}$ to be consistent with our
definitions of the PSFs Stoica and Mirea (2013); Neacsu and Horoi (2016a), see
below. The first four terms are contributions of the individual mechanisms.
The rest of the terms are due to the interference between pairs of mechanisms,
we have six such combinations. The differences in phases for the LNV
parameters $\eta_{i}$s (Eqs. (61), (64), (67), (70)) may produce interference
effects. The $C_{i}$ and $C_{ij}$ are products of relevant NME and PSF for
individual and interference terms, respectively Doi et al. (1985); Suhonen and
Civitarese (1998):

$\displaystyle
C_{m}=G_{01}\Big{[}M_{GT}-\Big{(}\frac{g_{V}}{g_{A}}\Big{)}^{2}M_{F}+M_{T}\Big{]}^{2},$
(74) $\displaystyle
C_{N}=G_{01}\Big{[}M_{GTN}-\Big{(}\frac{g_{V}}{g_{A}}\Big{)}^{2}M_{FN}+M_{TN}\Big{]}^{2},$
(75) $\displaystyle
C_{\lambda}=G_{02}\mathcal{M}^{2}_{2-}-\frac{2}{9}G_{03}\mathcal{M}_{1+}\mathcal{M}_{2-}+\frac{1}{9}G_{04}\mathcal{M}_{1+}^{2}$
(76) $\displaystyle
C_{\eta}=G_{02}\mathcal{M}^{2}_{2+}-\frac{2}{9}G_{03}\mathcal{M}_{1-}\mathcal{M}_{2+}+\frac{1}{9}G_{04}\mathcal{M}^{2}_{1-}$
$\displaystyle\qquad\qquad-G_{07}M_{P}M_{R}+G_{08}M_{P}^{2}+G_{09}M_{R}^{2},$
(77) $\displaystyle
C_{mN}=-2G^{\prime}_{01}\Big{[}M_{GT}-\Big{(}\frac{g_{V}}{g_{A}}\Big{)}^{2}M_{F}+M_{T}\Big{]}$
$\displaystyle\qquad\qquad\qquad\times\Big{[}M_{GTN}-\Big{(}\frac{g_{V}}{g_{A}}\Big{)}^{2}M_{FN}+M_{TN}\Big{]},$
(78) $\displaystyle
C_{m\lambda}=-\Big{[}M_{GT}-\Big{(}\frac{g_{V}}{g_{A}}\Big{)}^{2}M_{F}+M_{T}\Big{]}$
$\displaystyle\qquad\qquad\times\Big{[}G_{03}\mathcal{M}_{2-}-G_{04}\mathcal{M}_{1+}\Big{]},$
(79) $\displaystyle
C_{N\lambda}=-\Big{[}M_{GTN}-\Big{(}\frac{g_{V}}{g_{A}}\Big{)}^{2}M_{FN}+M_{TN}\Big{]}$
$\displaystyle\qquad\qquad\times\Big{[}G_{03}\mathcal{M}_{2-}-G_{04}\mathcal{M}_{1+}\Big{]},$
(80) $\displaystyle
C_{m\eta}=\Big{[}M_{GT}-\Big{(}\frac{g_{V}}{g_{A}}\Big{)}^{2}M_{F}+M_{T}\Big{]}$
$\displaystyle\,\,\,\,\,\,\,\times\Big{[}G_{03}\mathcal{M}_{2+}-G_{04}\mathcal{M}_{1-}-G_{05}M_{P}+G_{06}M_{R}\Big{]},$
(81) $\displaystyle
C_{N\eta}=\Big{[}M_{GTN}-\Big{(}\frac{g_{V}}{g_{A}}\Big{)}^{2}M_{FN}+M_{TN}\Big{]}$
$\displaystyle\,\,\,\,\,\,\,\times\Big{[}G_{03}\mathcal{M}_{2+}-G_{04}\mathcal{M}_{1-}-G_{05}M_{P}+G_{06}M_{R}\Big{]},$
(82) $\displaystyle
C_{\lambda\eta}=-2G_{02}\mathcal{M}_{2-}\mathcal{M}_{2+}+\frac{2}{9}G_{03}\Big{[}\mathcal{M}_{1+}\mathcal{M}_{2+}$
$\displaystyle\qquad\qquad\qquad+\mathcal{M}_{2-}\mathcal{M}_{1-}\Big{]}-\frac{2}{9}G_{04}\mathcal{M}_{1+}\mathcal{M}_{1-},$
(83)

where the following definitions are used,

$\displaystyle\mathcal{M}_{1\pm}=M_{GTq}\pm
3\Big{(}\frac{g_{V}}{g_{A}}\Big{)}^{2}M_{Fq}-6M_{Tq},$ (84)
$\displaystyle\mathcal{M}_{2\pm}=M_{GT\omega}\pm\Big{(}\frac{g_{V}}{g_{A}}\Big{)}^{2}M_{F\omega}-\frac{1}{9}\mathcal{M}_{1\mp}.$
(85)

Note that the term $\frac{1}{9}\mathcal{M}_{1\mp}$ in Eq. (85) above is the
correct expression (see footnote on p.146 of Ref. Doi and Kotani (1993)); it
was incorrectly written as $\frac{1}{9}\mathcal{M}_{1\pm}$ in Eq. (3.5.16) of
Ref. Doi et al. (1985). Detailed expressions for the thirteen NME $\\{M_{F}$,
$M_{GT}$, $M_{T}$, $M_{F\omega}$, $M_{Fq}$, $M_{GT\omega}$, $M_{GTq}$,
$M_{Tq}$, $M_{P}$, $M_{R}$, $M_{FN}$, $M_{GTN}\\}$, $M_{TN}\\}$ are given in
the appendix of Ref. Horoi and Neacsu (2018). The expressions for the nine PSF
integrals $\\{G_{01}-G_{09}\\}$ are Neacsu and Horoi (2016a)

$\displaystyle
G_{0k}=\frac{g^{0\nu}}{r_{A}^{2}}\int_{1}^{T+1}\\!\\!\\!\\!\\!\\!\\!\\!b_{0k}F_{0}(Z_{s},\epsilon_{1})F_{0}(Z_{s},\epsilon_{2})p_{1}p_{2}\epsilon_{1}\epsilon_{2}\text{d}\epsilon_{1},$
(86)

with

$\displaystyle
g^{0\nu}=\frac{(G_{F}\cos{\theta_{c}})^{4}m_{e}^{9}}{(2\pi)^{5}\ln
2}=2.8\times 10^{-22}\quad\text{yr}^{-1},$ (87)

where the expressions for the nine kinematical factors $b_{0k}$ ($k=1\sim 9$)
and definitions of other terms are given in Appendix A of Ref. Neacsu and
Horoi (2016a). The PSF $G^{\prime}_{01}$ in $C_{mN}$ (Eq. (78)) for the
interference between regular ‘mass-mechanism’ (Fig. 2a) and heavy-neutrino
exchange for purely RH currents (Fig. 2b) has the same expression as $G_{01}$
of Eq. (86) without the factors $\epsilon_{1}\epsilon_{2}$ Ahmed et al.
(2017). Because of our definitions of the PSFs and NMEs, the products $C_{i}$s
and $C_{ij}$s are reported in the units of $\text{y}^{-1}$.

###  III.4 Effective-Field Theory approach to $0\nu\beta\beta$

Before proceeding to the analysis section we would like to point out that the
effective Lagrangian of Eq. (46) arises from an explicit LRSM charge-current
Lagrangian, Eq. (II). This is exactly the approach taken in the standard
literature, e.g. as in Ref. Doi et al. (1985), where RH neutrinos are assumed
to contribute besides the usual SM neutrinos. In the effective-field theory
(EFT) approach to $0\nu\beta\beta$ we encounter a dimension-six Lagrangian
Horoi and Neacsu (2018); Deppisch et al. (2012) that is similar in structure
to $\mathcal{L}_{\text{\tiny{LR}}}^{\beta}$ of Eq. (II),

$\displaystyle\mathcal{L}^{\text{\tiny{EFT}}}_{6}=$
$\displaystyle\frac{G_{\beta}}{\sqrt{2}}\Big{[}j^{\mu}_{V-A}J_{V-A,\mu}^{\dagger}+\text{\char
15\relax}_{V-A}^{V+A}j^{\mu}_{V+A}J_{V-A,\mu}^{\dagger}$
$\displaystyle+\text{\char
15\relax}_{V+A}^{V+A}j^{\mu}_{V+A}J_{V+A,\mu}^{\dagger}+\text{\char
15\relax}_{S-P}^{S+P}j_{S+P}J_{S-P}^{\dagger}$ $\displaystyle+\text{\char
15\relax}_{S+P}^{S+P}j_{S+P}J_{S+P}^{\dagger}+\text{\char
15\relax}_{T_{R}}^{T_{R}}j^{\mu\nu}_{T_{R}}J_{T_{R},\mu\nu}^{\dagger}\Big{]},$
(88)

which is the most general Lorentz-invariant Lagrangian responsible for
$0\nu\beta\beta$ in the second order of perturbation theory. The leptonic and
hadronic currents of the EFT Lagrangian are respectively
$j_{\beta}=\overline{e}\mathcal{O}_{\beta}\nu$ and
$J^{\dagger}_{\alpha}=\overline{u}\mathcal{O}_{\alpha}d$, with the
$\mathcal{O}_{\alpha,\beta}$ operators defined as,

$\displaystyle\mathcal{O}_{V\pm A}=$
$\displaystyle\gamma^{\mu}\left(1\pm\gamma_{5}\right),\quad\mathcal{O}_{S\pm
P}=\left(1\pm\gamma_{5}\right),$ $\displaystyle\mathcal{O}_{T_{R}}$
$\displaystyle=\frac{\mathrm{i}}{2}\left[\gamma_{\mu},\gamma_{v}\right]\left(1+\gamma_{5}\right).$
(89)

Note that the neutrino fields used in Eq. (III.4) are the SM LH-neutrinos in
the flavor basis. Heavy RH neutrinos in Eq. (35) are integrated out and any
related parameters are absorbed in the definition of the effective BSM
couplings $\text{\char 15\relax}_{\alpha}^{\beta}$s. EFT formalism allows us
to relate BSM physics parameters through the SM degrees of freedom. In the
case of LRSM we approximate the effective BSM couplings as,

$\displaystyle\text{\char
15\relax}_{V-A}^{V+A}=\eta_{\eta}\qquad,\qquad\text{\char
15\relax}_{V+A}^{V+A}=\eta_{\lambda}$ (90)

The scalar-pseudoscalar ($S\pm P$) and tensor ($T_{R}$) terms do not arise
from the LRSM charged-current Lagrangian, but from other BSM models. The term
related to the heavy-neutrino exchange in the presence of purely RH currents,
$\mathcal{A}_{R}^{N}$ (Fig. 2b), is not given by the
$\mathcal{L}^{\text{\tiny{EFT}}}_{6}$ since it is a short-range contribution
due to the exchange of heavy particles. $\mathcal{L}^{\text{\tiny{EFT}}}_{6}$
gives rise to long-range contributions to $0\nu\beta\beta$ due to the exchange
of light neutrinos, see Figs. (1(b)) and (1(c)) of Ref. Horoi and Neacsu
(2018). In the EFT approach to $0\nu\beta\beta$ the dimension-nine Lagrangian
is Horoi and Neacsu (2018)

$\displaystyle\mathcal{L}^{\text{\tiny{EFT}}}_{9}=$
$\displaystyle\frac{G^{2}_{\beta}}{2m_{P}}\Big{[}\epsilon_{1}JJj+\epsilon_{2}J^{\mu\nu}J_{\mu\nu}j+\epsilon_{3}^{LLz}J^{\mu}J_{\mu}j$
$\displaystyle+\epsilon_{3}^{RRz}J^{\mu}J_{\mu}j+\epsilon_{3}^{LRz}J^{\mu}J_{\mu}j+\epsilon_{3}^{RLz}J^{\mu}J_{\mu}j$
$\displaystyle+\epsilon_{4}J^{\mu}J_{\mu\nu}j^{\nu}+\epsilon_{5}J^{\mu}Jj_{\mu}\Big{]}.$
(91)

The expressions for the leptonic and hadronic currents are given in Ref. Horoi
and Neacsu (2018). The short-range contribution (see Fig. 1(d) of Ref. Horoi
and Neacsu (2018)) to $0\nu\beta\beta$, $\mathcal{A}_{R}^{N}$, arises from the
$J^{\mu}J_{\mu}j$ term of $\mathcal{L}^{\text{\tiny{EFT}}}_{9}$ in first-order
of perturbation where we approximate $\epsilon_{3}^{RRz}=\eta_{N}$. However,
the $0\nu\beta\beta$ half-life formula, Eq. (III.3), is the same in both
approaches. Thus, our analysis of the interference between different
mechanisms arising from $\mathcal{L}_{\text{\tiny{LR}}}^{\beta}$ can easily be
extended to a subset of terms of EFT approach to $0\nu\beta\beta$ Lagrangians
$\mathcal{L}^{\text{\tiny{EFT}}}_{6}$ and
$\mathcal{L}^{\text{\tiny{EFT}}}_{9}$. For a complete discussion of
$0\nu\beta\beta$ in the EFT approach see Refs. Cirigliano et al. (2018); del
Aguila et al. (2012); Deppisch et al. (2012). The contribution of the ($S\pm
P$) and $T_{R}$ terms of $\mathcal{L}^{\text{\tiny{EFT}}}_{6}$ to the total
decay rate of $0\nu\beta\beta$, along with the constraints on the effective
LNV couplings, have been studied with the assumption that the interference
terms are negligible Horoi and Neacsu (2018). As an extension of our current
work, we plan to explore in the future the contribution of all the possible
interference terms arising from $\mathcal{L}^{\text{\tiny{EFT}}}_{6}$. A
similar analysis can be also carried out for the interference terms arising
from $\mathcal{L}^{\text{\tiny{EFT}}}_{9}$; see, e.g., Eq. (5) of Horoi and
Neacsu (2018).

##  IV Analysis of interference terms

Table 1: Values of the product of NME and PSF, $C_{i}$ and $C_{ij}$, for various nuclei for the $0^{+}\to 0^{+}$ transition in units of $\text{y}^{-1}$. See Tables LABEL:PSF-values and LABEL:NME-values of the appendix.  |  48Ca |  76Ge |  82Se |  124Sn |  130Te |  136Xe
---|---|---|---|---|---|---
$C_{m}\cdot 10^{14}$ | $2.57$ | $3.00$ | $11.54$ | $4.14$ | $5.22$ | $4.39$
$C_{N}\cdot 10^{10}$ | $1.63$ | $0.87$ | $3.28$ | $1.84$ | $2.25$ | $1.86$
$C_{\lambda}\cdot 10^{13}$ | $1.22$ | $0.43$ | $3.52$ | $0.79$ | $1.24$ | $0.99$
$C_{\eta}\cdot 10^{09}$ | $1.45$ | $1.40$ | $5.11$ | $2.74$ | $3.67$ | $3.09$
$C_{mN}\cdot 10^{13}$ | $-1.82$ | $-4.11$ | $-9.35$ | $-6.10$ | $-6.64$ | $-5.75$
$C_{m\lambda}\cdot 10^{14}$ | $-0.90$ | $-1.13$ | $-5.68$ | $1.97$ | $-2.64$ | $-2.20$
$C_{m\eta}\cdot 10^{11}$ | $0.38$ | $0.64$ | $1.91$ | $-0.97$ | $1.19$ | $1.01$
$C_{N\lambda}\cdot 10^{12}$ | $-0.72$ | $-0.61$ | $-3.03$ | $1.31$ | $-1.74$ | $-1.43$
$C_{N\eta}\cdot 10^{10}$ | $3.05$ | $3.43$ | $10.19$ | $-6.45$ | $7.80$ | $6.58$
$C_{\lambda\eta}\cdot 10^{13}$ | $-1.51$ | $-0.60$ | $-5.05$ | $-1.06$ | $-1.65$ | $-1.31$
Table 2: Interference coefficients $\text{\char 15\relax}_{m\lambda}(\alpha)$ in $\%$ for specific $\alpha$ values. Nuclei | $\Large{\text{\char 15\relax}_{m\lambda}}(0.25)$ | $\Large{\text{\char 15\relax}_{m\lambda}}(0.5)$ | $\Large{\text{\char 15\relax}_{m\lambda}}(0.75)$ | $\Large{\text{\char 15\relax}_{m\lambda}}(1)$
---|---|---|---|---
48Ca | $6.42$ | $7.57$ | $7.95$ | $8.03$
76Ge | $12.68$ | $14.94$ | $15.69$ | $15.85$
82Se | $11.27$ | $13.28$ | $13.94$ | $14.08$
124Sn | $13.81$ | $16.28$ | $17.09$ | $17.27$
130Te | $13.16$ | $15.51$ | $16.28$ | $16.45$
136Xe | $13.33$ | $15.70$ | $16.49$ | $16.66$
Table 3: Interference coefficients $\text{\char 15\relax}_{m\eta}(\alpha)$ in $\%$ for specific $\alpha$ values. Nuclei | $\Large{\text{\char 15\relax}_{m\eta}}(0.25)$ | $\Large{\text{\char 15\relax}_{m\eta}}(0.5)$ | $\Large{\text{\char 15\relax}_{m\eta}}(0.75)$ | $\Large{\text{\char 15\relax}_{m\eta}}(1)$
---|---|---|---|---
48Ca | $25.11$ | $29.60$ | $31.07$ | $31.40$
76Ge | $39.27$ | $46.28$ | $48.59$ | $49.09$
82Se | $31.48$ | $37.10$ | $38.94$ | $39.35$
124Sn | $36.32$ | $42.80$ | $44.93$ | $45.40$
130Te | $34.29$ | $40.41$ | $42.42$ | $42.86$
136Xe | $34.75$ | $40.95$ | $42.99$ | $43.44$
Table 4: Interference coefficient $\text{\char 15\relax}_{\lambda\eta}(\alpha)$ in $\%$ for specific $\alpha$ values. Nuclei | $\Large{\text{\char 15\relax}_{\lambda\eta}}(0.25)$ | $\Large{\text{\char 15\relax}_{\lambda\eta}}(0.5)$ | $\Large{\text{\char 15\relax}_{\lambda\eta}}(0.75)$ | $\Large{\text{\char 15\relax}_{\lambda\eta}}(1)$
---|---|---|---|---
48Ca | $0.45$ | $0.54$ | $0.56$ | $0.57$
76Ge | $0.31$ | $0.37$ | $0.38$ | $0.39$
82Se | $0.48$ | $0.56$ | $0.59$ | $0.60$
124Sn | $0.29$ | $0.34$ | $0.35$ | $0.36$
130Te | $0.31$ | $0.36$ | $0.38$ | $0.39$
136Xe | $0.30$ | $0.35$ | $0.37$ | $0.38$
Figure 4: Coefficient of maximum interference $\text{\char
15\relax}_{m\lambda}(1)$ plotted against $Q_{\beta\beta}$ values.  Figure 5:
Coefficient of maximum interference $\text{\char 15\relax}_{m\eta}(1)$ plotted
against $Q_{\beta\beta}$ values.  Figure 6: Coefficient of maximum
interference $\text{\char 15\relax}_{\lambda\eta}(1)$ plotted against
$Q_{\beta\beta}$ values.

We now analyze the contribution of each of the interference terms in Eq.
(III.3) by comparison with the related pairs of squared amplitudes for each
individual mechanisms. The interference between light-LH and heavy-RH
neutrinos ($C_{mN}$ term in Eq. (III.3)) was analyzed in Ref. Ahmed et al.
(2017). Here we analyze the other five terms (three after symmetry, see
below). We write a generic approximate inverse half-life formula for a pair of
mechanisms in the following manner:

$\displaystyle[T^{0\nu}_{1/2}]^{-1}\simeq g^{4}_{A}$
$\displaystyle\Big{[}C_{i}\absolutevalue{\eta_{i}}^{2}+C_{j}\absolutevalue{\eta_{j}}^{2}$
$\displaystyle+C_{ij}\absolutevalue{\eta_{i}}\absolutevalue{\eta_{j}}\cos{(\phi_{i}-\phi_{j})}\Big{]},$
(124)

where $i,\,j=\\{m,\,N,\,\lambda,\,\eta\\}$ and $i\neq j$. We assume the
individual mechanism squared amplitude to be a factor $\alpha$ of each other
($0<\alpha\leq 1$),

$\displaystyle C_{j}\absolutevalue{\eta_{j}}^{2}=\alpha
C_{i}\absolutevalue{\eta_{i}}^{2}\Rightarrow\absolutevalue{\eta_{j}}=\sqrt{\alpha\frac{C_{i}}{C_{j}}}\absolutevalue{\eta_{i}}.$
(125)

Thus, our approximate generic half-life expression becomes,

$\displaystyle[T^{0\nu}_{1/2}]^{-1}\simeq
g^{4}_{A}(1+\alpha)C_{i}\absolutevalue{\eta_{i}}^{2}\Big{[}1+\text{\char
15\relax}_{ij}\cos{(\phi_{i}-\phi_{j})}\Big{]},$ (126)

where the interference coefficient

$\text{\char
15\relax}_{ij}(\alpha)=\frac{\sqrt{\alpha}}{1+\alpha}\frac{\absolutevalue{C_{ij}}}{\sqrt{\absolutevalue{C_{i}}\absolutevalue{C_{j}}}},$
(127)

would allow us to compare the contribution of the interference term with
respect to that of each individual mechanisms for maximum interference,
$|\cos{(\phi_{i}-\phi_{j})}|=1$. We numerically calculate the products of NME
and PSF, and the ten $C_{i}$ and $C_{ij}$ of Eq. (74)-(83), given in Table 1.

The NME for the six isotopes used in this study were calculated by shell-model
techniques Horoi and Stoica (2010); Horoi (2013) in three different model
spaces, using three different effective Hamiltonians Neacsu and Horoi (2015);
Horoi and Neacsu (2016a); Horoi and Neacsu (2018). Some of the NME are
sensitive to short-range correlations (SRC) effects entering the two-body
matrix elements. Here we used the CD-Bonn SRC parametrization Horoi (2013).
Using the AV18 SRC parametrization Horoi (2013), or/and the Strasbourg-Madrid
choice for the effective Hamiltonians Horoi and Neacsu (2018) does not
significantly change the results. The relevant NME and PSF used in this study
are given in the Appendix. Besides the values of Ref. Horoi and Neacsu (2018),
we have also considered the PSF of Ref. Stefanik et al. (2015) in conjungtion
to the various sets of NME. The results for the two sets of PSF do not have
any appreciable difference. As discussed in Ref. Horoi and Neacsu (2018),
competing contributions to the NME are always present but some are dominant,
such as those of the Gamow-Teller type operators, thus avoiding full
cancellations of the total NME. In addition, given that calculations in
different model spaces with different effective Hamiltonians lead to similar
results, we have confidence in the reliability of our conclusions.

Using Eq. (127) we then evaluate the interference coefficients, $\text{\char
15\relax}_{m\lambda}$, $\text{\char 15\relax}_{m\eta}$, $\text{\char
15\relax}_{N\lambda}$, $\text{\char 15\relax}_{N\eta}$, and $\text{\char
15\relax}_{\lambda\eta}$, for different nuclei and for some specific $\alpha$
values, in Tables 2-4. Note that the interference coefficients $\text{\char
15\relax}_{m\lambda}$ and $\text{\char 15\relax}_{N\lambda}$ are equal. Using
Eqs. (74), (76) and (79) we see that,

$\displaystyle\text{\char 15\relax}_{m\lambda}=\text{\char
15\relax}_{N\lambda}=\frac{\sqrt{\alpha}}{1+\alpha}\frac{\absolutevalue{G_{03}\mathcal{M}_{2-}-G_{04}\mathcal{M}_{1+}}}{\sqrt{G_{01}\absolutevalue{C_{\lambda}}}}.$
(128)

Similarly, using Eq. (75), (76) and (80) we get,

$\displaystyle\text{\char 15\relax}_{m\eta}=\text{\char 15\relax}_{N\eta}$
$\displaystyle=\frac{\sqrt{\alpha}}{1+\alpha}\frac{\absolutevalue{G_{03}\mathcal{M}_{2+}-G_{04}\mathcal{M}_{1-}-G_{05}M_{P}+G_{06}M_{R}}}{\sqrt{G_{01}\absolutevalue{C_{\eta}}}}.$
(129)

Using Eq. (74), (75) and (78), one sees from Eq. (127) that the interference
coefficient between the mass-mechanism and heavy neutrino exchange mechanism
($\text{\char 15\relax}_{mN}$) is $\propto 2G^{\prime}_{01}/G_{01}$, which was
considered in Ref. Ahmed et al. (2017). We observe that maximum interference
occurs for $\alpha=1$, i.e., when the pairs of individual mechanisms are equal
to each other. Moreover, $\text{\char 15\relax}_{ij}(\alpha)$ and $\text{\char
15\relax}_{ij}(1/\alpha)$ are the same, as one can verify from Eq. (127).

##  V Results and Discussion

From Tables 2-4 we observe an interference coefficient no larger than $\sim
18\%$ for the interference between the mass-mechanism and the $\lambda$
process ($\text{\char 15\relax}_{m\lambda}$). The same conclusions can be
drawn for the case for interference between RH-heavy neutrino exchange and the
$\lambda$ mechanism ($\text{\char 15\relax}_{N\lambda}=\text{\char
15\relax}_{m\lambda}$). The interference coefficient for $\lambda$ and $\eta$
mechanisms is negligible with a maximum of $0.59\%$ for 82Se. The interference
between the mass-mechanism and heavy neutrino exchange mechanism, $\text{\char
15\relax}_{mN}$, was considered in Ref. Ahmed et al. (2017) for $\alpha=1$,
see Eq. (25) and Table 1 of Ahmed et al. (2017). For the interference between
the mass mechanism and the $\eta$ mechanism, the maximum interference
coefficient ($\text{\char 15\relax}_{m\eta}(\alpha=1)$) ranges between $30\%$
to $50\%$ with a maximum of about $49\%$ for 76Ge. The interference
coefficient for RH-heavy neutrino exchange and the $\eta$ mechanism,
$\text{\char 15\relax}_{N\eta}$, has the same values.

We plot the coefficients for maximum interference, $\text{\char
15\relax}_{m\lambda}(1)$, $\text{\char 15\relax}_{m\eta}(1)$, and $\text{\char
15\relax}_{\lambda\eta}(1)$ as functions of $Q$ value ($Q_{\beta\beta}$) of
various nuclei in Fig. 4-6, respectively. We observe that $\text{\char
15\relax}_{m\lambda}(1)$ and $\text{\char 15\relax}_{m\eta}(1)$ decrease with
$Q_{\beta\beta}$. In our study of the interference between the mass mechanism
(Fig. 2a) and the heavy-neutrino exchange for purely RH currents (Fig. 2b) in
Ref. Ahmed et al. (2017) we found a similar dependence of $\text{\char
15\relax}_{mN}$ on $Q_{\beta\beta}$ (see Fig. 2 of Ahmed et al. (2017)). For
the $\text{\char 15\relax}_{\lambda\eta}(1)$ in Fig. 6 we do not observe any
particular dependence on $Q_{\beta\beta}$.

In summary, we studied the contributions of the interference effects to the
$0\nu\beta\beta$ decay rate for four competing mechanisms arising from LRSM:
(i) the regular mass mechanism for light-neutrino exchange of purely LH
currents ($\eta_{m}$), (ii) the heavy-neutrino exchange mechanism for purely
RH currents ($\eta_{N}$), (iii) the $\lambda$ mechanism ($\eta_{\lambda}$),
and (iv) the $\eta$ mechanism ($\eta_{\eta}$). We extended our analysis of
Ref. Ahmed et al. (2017) to interference effects between the ‘mass-mechanism’
($\eta_{m}$) and heavy-neutrino exchange mechanism ($\eta_{N}$) to the other
five contributions. Besides several BSM scenarios, the LRSM is being actively
investigated at the LHC Khachatryan et al. (2014). Several competing
mechanisms have been proposed to contribute to $0\nu\beta\beta$. It is
important to know if different mechanisms can be disentangled. To that goal,
analyzing the contribution of interference terms to the decay rate is
essential. By comparing the decay rate of several nuclei of experimental
interest one may be able to differentiate between two competing mechanisms,
provided that the contribution of interference term is negligible Horoi and
Neacsu (2018); Horoi and Neacsu (2016b). In the present study we have observed
that most of the two-mechanisms interference terms introduce a relatively
minor modification to the half-life, less than 20$\%$. However, the
interference between the neutrino exchange mechanisms (light and heavy) and
the $\eta$ mechanism are not small enough for the nuclei considered. In that
case, the angular distribution of the emitted electrons can be used to
distinguish between these two mechanisms, as has been discussed in Ref. Horoi
and Neacsu (2016b). One should emphasize that the interference coefficients we
found are not large enough to lead to a full cancellation of the decay rate
(see Eq. (IV)). Our conclusions are based on shell-model NME calculated with
different sets of effective Hamiltonians and short-range correlation
parametrizations, thus giving us confidence in their reliability.

Our analysis of the interference terms in $0\nu\beta\beta$ decay rate in the
context of LRSM can also be extended for the EFT approach to $0\nu\beta\beta$.
Specifically, as discussed in Sec. III.4, the interference of the amplitudes
arising from scalar-pseudoscalar ($S\pm P$) and tensor ($T_{R}$) terms in
$\mathcal{L}^{\text{\tiny{EFT}}}_{6}$ with the four amplitudes studied here.
This analysis will be reported separately.

###### Acknowledgements.

Support from the U.S. Department of Energy Grant No. DE-SC0015376 is
acknowledged. F.A. acknowledges the Science of Advanced Materials Program of
Central Michigan University for their support through a research
assistantship. F.A. would like to thank Andrei Neacsu for helpful discussions
and for providing the source files for the NME and PSF values.

*

## Appendix A

In this appendix we tabulate the values of the ten PSF, $\\{G_{01}-G_{09}$,
$G^{\prime}_{01}\\}$ and the thirteen NME, $\\{M_{F}$, $M_{GT}$, $M_{T}$,
$M_{F\omega}$, $M_{Fq}$, $M_{GT\omega}$, $M_{GTq}$, $M_{Tq}$, $M_{P}$,
$M_{R}$, $M_{FN}$, $M_{GTN}\\}$, $M_{TN}\\}$ taken from the literature.

Table 5: PSF in y-1 for $0^{+}\to 0^{+}$ transition. Values of ($G_{01}-G_{09}$) are taken from Ref. Horoi and Neacsu (2018) for all the isotopes except for 124Sn. Values of $G^{\prime}_{01}$ are taken from Ref. Ahmed et al. (2017). |  48Ca |  76Ge |  82Se |  124Sn Neacsu and Horoi (2016b) |  130Te |  136Xe
---|---|---|---|---|---|---
$G_{01}\cdot 10^{14}$ | $2.45$ | $0.23$ | $0.10$ | $0.89$ | $1.41$ | $1.45$
$G^{\prime}_{01}\cdot 10^{15}$ Ahmed et al. (2017) | $1.09$ | $0.29$ | $0.76$ | $0.98$ | $1.37$ | $1.46$
$G_{02}\cdot 10^{14}$ | $15.46$ | $0.35$ | $3.21$ | $1.68$ | $3.25$ | $3.15$
$G_{03}\cdot 10^{14}$ | $1.82$ | $0.12$ | $0.65$ | $0.50$ | $0.85$ | $0.85$
$G_{04}\cdot 10^{15}$ | $5.04$ | $0.42$ | $1.92$ | $1.56$ | $2.53$ | $2.58$
$G_{05}\cdot 10^{13}$ | $3.28$ | $0.60$ | $2.16$ | $2.70$ | $4.12$ | $4.36$
$G_{06}\cdot 10^{12}$ | $3.87$ | $0.50$ | $1.65$ | $1.47$ | $2.16$ | $2.21$
$G_{07}\cdot 10^{10}$ | $2.85$ | $0.28$ | $1.20$ | $1.11$ | $1.75$ | $1.80$
$G_{08}\cdot 10^{11}$ | $1.32$ | $0.17$ | $0.82$ | $1.04$ | $1.72$ | $1.83$
$G_{09}\cdot 10^{10}$ | $15.55$ | $1.12$ | $4.42$ | $2.95$ | $4.47$ | $4.44$
Table 6: Dimensionless NME for $0^{+}\to 0^{+}$ transition. Values taken from Ref. Horoi and Neacsu (2018); Horoi and Neacsu (2016a); Neacsu and Horoi (2016b). From Ref. Horoi and Neacsu (2018), we have considered the NME calculated with the CMU effective Hamiltonians and CD-Bonn SRC-parametrization. |  |  48Ca |  76Ge |  82Se |  124Sn Horoi and Neacsu (2016a); Neacsu and Horoi (2016b) |  130Te |  136Xe
---|---|---|---|---|---|---|---
| $M_{GT}$ | $-0.805$ | $-3.200$ | $-3.000$ | $-1.853$ | $-1.658$ | $1.501$
| $M_{F}$ | $0.233$ | $0.674$ | $0.632$ | $0.467$ | $0.438$ | $-0.400$
| $M_{T}$ | $-0.073$ | $-0.011$ | $-0.012$ | $-0.019$ | $0.006$ | $-0.007$
| $M_{GTN}$ | $-55.890$ | $-156.493$ | $-144.907$ | $-113.364$ | $-103.025$ | $92.565$
| $M_{FN}$ | $22.893$ | $62.649$ | $58.091$ | $43.295$ | $40.984$ | $-36.942$
| $M_{TN}$ | $-11.308$ | $-0.205$ | $-0.513$ | $-3.827$ | $2.022$ | $-2.178$
| $M_{GTq}$ | $-0.709$ | $-3.228$ | $-3.034$ | $1.793$ | $-1.587$ | $1.440$
| $M_{Fq}$ | $0.121$ | $0.383$ | $0.362$ | $-0.267$ | $0.249$ | $-0.230$
| $M_{Tq}$ | $0.173$ | $0.059$ | $0.058$ | $0.011$ | $0.013$ | $-0.012$
| $M_{GT\omega}$ | $-0.930$ | $-3.501$ | $-3.287$ | $2.053$ | $-1.855$ | $1.682$
| $M_{F\omega}$ | $0.232$ | $0.659$ | $0.618$ | $-0.456$ | $0.427$ | $-0.391$
| $M_{R}$ | $-1.001$ | $-3.243$ | $-3.088$ | $2.663$ | $-2.530$ | $2.312$
| $M_{P}$ | $-0.390$ | $2.435$ | $2.303$ | $-2.060$ | $1.707$ | $-1.600$

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