# Nuclear shell model study of neutrinoless double beta decay under Left-Right
symmetric model

Dong-Liang Fanga,b, B. Alex Brownc and Fedor Šimkovicd,e aInstitute of Modern
Physics, Chinese Academy of sciences, Lanzhou, 730000, China  bUniversity of
Chinese Academy of Sciences, Beijing, 100049,China  cDepartment of Physics and
Astronomy, FRIB Laboratory, Michigan State University, East Lansing, MI
42284-1321, USA  dFaculty of Mathematics, Physics and Informatics, Comenius
University in Bratislava, 842 48 Bratislava, Slovakia  eInstitute of
Experimental and Applied Physics, Czech Technical University in Prague, 110 00
Prague, Czech Republic

###### Abstract

We use the large scale nuclear shell model to calculate the nuclear matrix
elements for the neutrino mediated neutrinoless double beta decay within the
Left-Right symmetric model for four nuclei: 76Ge, 82Se, 130Te and 136Xe. We
perform a systematic analysis on the general magnitude of different terms for
related mechanisms. For the $\eta$ mechanism, we find that the weak magnetism
$R$ term dominates the decay rate while the $p$-wave effect is suppressed.
While for the $\lambda$ mechanism, the $\omega$ and the $q$ terms are with
equal importance. For the latter $q$ term, important contributions from weak-
magnetism MM part are observed. Finally, we give the constraints on the new
physics parameters $m_{\beta\beta}$, $\lambda$ and $\eta$ from current
experiments.

##  I Introduction

The neutrinoless double beta ($0\nu\beta\beta$) decay is one of the rarest and
most mysterious processes in nature. With the construction of the deep-
underground laboratories and advancements of many up-to-ton-scale experiments
Zhang  _et al._ (2023); Cao  _et al._ (2023); Ni  _et al._ (2019); Adhikari
_et al._ (2022), we are approaching the so-called inverted mass hierarchy
region Agostini  _et al._ (2023a). In the next few decades, we may see the
signal of this lepton-number violating process and answer the questions of
whether the neutrino is a Majorana particle and subsequently why the universe
is dominated by matter instead of antimatter.

The discovery of neutrinoless double beta decay will pave our way to physics
beyond the standard model (SM). While this discovery will prove lepton number
violation, we still need to understand the details of this process and what is
actually happening, whether it is mediated by the simple light neutrino with
SM weak currents or with more complicated weak currents. These are closely
related to new physics models. One of the promising models of this kind is the
so-called Left-Right symmetric model Mohapatra and Senjanovic (1981). For this
model, the see-saw mechanism can be naturally incorporated Mohapatra  _et al._
(2007), addressing the issue of the smallness of neutrino mass. Such studies
have already attracted much attention from the community, and the implications
for future observations of $0\nu\beta\beta$-decay have been intensively
discussed Tello  _et al._ (2011); Chakrabortty  _et al._ (2012); Awasthi  _et
al._ (2013); Deppisch  _et al._ (2015); Pritimita  _et al._ (2016); Fukuyama
and Sato (2023).

Another issue is that if this model turns out to be the naturally chosen one,
how could we confirm this? Various approaches have been proposed. For example,
one could compare the decay half-lives of different isotopes by taking
advantage of the different NME dependencies on various mechanisms for
different nuclei Simkovic  _et al._ (2010); Faessler  _et al._ (2011a, b);
Meroni  _et al._ (2013); Gráf  _et al._ (2022); Agostini  _et al._ (2023b).
One could measure the angular correlation of the two emitted electrons Gráf
_et al._ (2022). One could also resort to comparing the different decay modes
of the same isotope Simkovic and Faessler (2002); Fang and Faessler (2023).

While most discussions focus on the new physics side, it is important to note
that calculations from the nuclear side, which are crucial for our
understanding of this process, are still lacking. Compared to the extensive
nuclear many-body calculations for $0\nu\beta\beta$-decay with a standard
neutrino mass mechanism Yao  _et al._ (2022); Agostini  _et al._ (2023a), the
nuclear structural calculation for the LR symmetric model related to the
$\lambda$ and $\eta$ mechanisms are limited and incomplete. Earlier
publications have utilized methods such as large scale shell models (LSSM)
Caurier  _et al._ (1996), quasi-particle random phase approximation (QRPA)
Muto  _et al._ (1989); Pantis  _et al._ (1996); Suhonen and Civitarese (1998)
and projected Hatree-Fock Boglyubov (PHFB) Tomoda (1991). However, these
studies have omitted the important leading order (LO) component, such as the
Pseudo-Scalar component of the nuclear current. Subsequent calculations
Šimkovic  _et al._ (2017); Sarkar  _et al._ (2020); Iwata and Sarkar (2021)
still lack a thorough investigation of all the relevant terms.

Especifically for the LSSM calculations, we have abundant calculations of
various nuclei for a standard neutrino mass mechanism, some with traditional
Hamiltonians obtained by reproducing the nuclear properties (for a complete
list, see e.g. Horoi (2022); Coraggio  _et al._ (2020a)), while others are the
so-called ab initio calculations Belley  _et al._ (2021, 2023) using the
renormalized interactions derived from bare nucleon-nucleon interactions
adjusted to produce nucleon scattering phase shifts. However, there are
discrepancies among these calculations especially between the conventional and
ab initio ones. This is not the topic of current study. For the mechanisms
related to LR symmetric models, such calculations are limited Caurier  _et
al._ (1996); Sarkar  _et al._ (2020); Iwata and Sarkar (2021); Horoi and
Neacsu (2018). Systematic studies of $0\nu\beta\beta$ with the LR symmetric
model including comprehensive comparisons across various models and detailed
error analysis, are still lacking.

In this work, we carry out LSSM calculations with traditional Hamiltonians to
explore the long-range mechanism mediated by neutrinos in the LR symmetric
model, incorporating key components of nuclear currents. We give a detailed
analysis of the relative magnitude of each term associated with each mechanism
and discuss how this difference originates from the nuclear side.

This article is arranged as follows, first the mechanism and related
expression for the decay width, and then we give the NME results and some
analysis of relative importance of each term, finally we give a brief
discussion on how current results could be used to constrain the new physics
parameters.

##  II $0\nu\beta\beta$ under LR symmetric model

Starting from the basic LR symmetric model $SU(2)_{L}\otimes SU(2)_{R}\otimes
U(1)_{B-L}$, one has extra gauge bosons $W_{R}$, $Z_{R}$, the triplet and the
bi-doublet Higgs bosons $\phi$ and $H_{R}$, as well as right handed neutrinos
$\nu_{R}$. In such a model, besides the electro-weak breaking energy scale
$v_{L}$, there is another symmetry breaking energy scale $v_{R}$ which
determines the general mass of right-handed gauge bosons. The Yukawa coupling
between the neutrinos and Higgs bosons gives rise to the neutrino mass, and
the see-saw mechanism can be naturally realized Mohapatra and Senjanovic
(1981).

In such a model, the weak and the mass eigenstates of the gauge bosons are
different, this then leads to the mixing between the left- and right-handed
gauge bosons:

$\displaystyle\left(\begin{array}[]{c}W_{L}\\\
W_{R}\end{array}\right)=\left(\begin{array}[]{cc}\cos\zeta&\sin\zeta\\\
-\sin\zeta&\cos\zeta\end{array}\right)\left(\begin{array}[]{c}W_{1}\\\
W_{2}\end{array}\right)$ (7)

$W_{L}$ and $W_{R}$ are the gauge eigenstates while $W_{1}$ and $W_{2}$ are
the mass eigenstates respectively. The masses for $W_{1}$ and $W_{2}$ Bosons
are $M_{1}$ and $M_{2}$ respectively. And $\zeta$ is the mixing angle between
the mass and weak eigenstates of weak bosons.

The interaction of the lepton and nucleon currents with $W$ Bosons can be
written as Doi  _et al._ (1985); Tomoda (1991):

$\displaystyle\mathcal{L}_{CC}=\frac{g}{2\sqrt{2}}[(j_{L}^{\mu}+J_{L}^{\mu})W^{-}_{L\mu}+(j_{R}^{\mu}+J_{R}^{\mu})W^{-}_{R\mu}]+h.c.$
(8)

The lepton currents have the form Tomoda (1991):

$\displaystyle j_{L(R)}^{\mu}=\bar{e}_{L(R)}\gamma^{\mu}\nu_{eL(R)}$ (9)

The Yukawa coupling of the neutrinos with the triplet Higgs Boson leads to the
mixing between the left- and the right-handed neutrinos, but the Pontecorvo-
Maki-Nakagawa-Sakata (PMNS) matrix Maki  _et al._ (1962); Pontecorvo (1967) is
still with unknown origin:

$\displaystyle\nu_{eL}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{3}(U_{ej}\nu_{j}+S_{ei}N_{j}^{C})$
$\displaystyle\nu_{eR}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{3}(T^{*}_{ej}\nu_{j}^{C}+V_{ej}^{*}N_{j})$ (10)

Here $\nu_{j}$ and $N_{j}$ are the mass eigenstates of the light and the heavy
neutrinos respectively. And U, S, T and V constitute a generalized 6$\times$6
PMNS matrix Xing (2012).

The nucleon currents can be obtained by the Lorentz invariance with the
impulse approximation Doi  _et al._ (1985); Simkovic  _et al._ (1999):

$\displaystyle J_{L(R)}^{\mu}(\vec{x})$ $\displaystyle=$
$\displaystyle\bar{p}(\vec{x})(g_{V}\gamma^{\mu}-ig_{W}\sigma^{\mu\nu}q_{\nu}$
$\displaystyle\mp g_{A}\gamma^{\mu}\gamma_{5}\pm
g_{P}\gamma_{5}q^{\mu})n(\vec{x})$

The form factors $g_{V}$, $g_{A}$, $g_{P}$ and $g_{W}$ have the form given in
Tomoda (1991), and when reduced to the non-relativistic form, we have weak
magnetism coupling constant $g_{M}(q^{2})=g_{V}(q^{2})+g_{W}(q^{2})$.

By integrating out the heavy gauge boson, we obtain the current-current
interactions for these weak interactions as:

$\displaystyle H_{\rm int}$ $\displaystyle=$
$\displaystyle\frac{G\cos\theta_{C}}{\sqrt{2}}(j_{L\mu}J_{L}^{\mu\dagger}+\kappa
j_{L\mu}J_{R}^{\mu\dagger}+\eta j_{R\mu}J_{L}^{\mu\dagger}$ (11)
$\displaystyle+\lambda j_{R\mu}J_{R}^{\mu\dagger})+h.c.$

Here the parameters $\lambda$, $\eta$ and $\kappa$ are combinations of
parameters of the LR-symmetric model:

$\displaystyle\frac{G}{\sqrt{2}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{8M_{1}^{2}}(\cos^{2}\zeta+(M_{1}/M_{2})^{2}\sin^{2}\zeta)\approx\frac{g^{2}}{8M_{1}^{2}}$
$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\frac{(M_{1}/M_{2})^{2}+\tan^{2}\zeta}{1+(M_{1}/M_{2})^{2}\tan^{2}\zeta}\approx(M_{1}/M_{2})^{2}+\tan^{2}\zeta$
$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\kappa=-\frac{(1-(M_{1}/M_{2})^{2})\tan\zeta}{1+(M_{1}/M_{2})^{2}\tan^{2}\zeta}\approx-\tan\zeta$
(12)

If $\lambda$ has the same order as $\eta$, then the $\tan^{2}\zeta$ term in
$\lambda$ can be safely neglected, on the other hand if the first and second
term in the r.h.s. of $\lambda$ have a similar size
($M_{W1}/M_{W2}\sim\tan\zeta$), then the contribution from $\lambda$ to
$0\nu\beta\beta$-decay can be safely neglected.

To better understand how the nuclear part and the lepton part contribute to
the reaction matrix $R$ (the definition see Appendix C of Doi  _et al._
(1985)), we rewrite the decay width in Tomoda (1991); Štefánik  _et al._
(2015) in the form:

$\displaystyle[T_{1/2}^{0\nu}]^{-1}$ $\displaystyle=$
$\displaystyle\mu_{\beta\beta}^{2}\mathcal{C}_{mm}+\mu_{\beta\beta}\langle\lambda\rangle\cos\psi_{1}\mathcal{C}_{m\lambda}$
(13) $\displaystyle+$
$\displaystyle\mu_{\beta\beta}\langle\eta\rangle\cos\psi_{2}\mathcal{C}_{m\eta}+\langle\lambda\rangle^{2}\mathcal{C}_{\lambda\lambda}$
$\displaystyle+$
$\displaystyle\langle\eta\rangle^{2}\mathcal{C}_{\eta\eta}+\langle\lambda\rangle\langle\eta\rangle\cos(\psi_{1}-\psi_{2})\mathcal{C}_{\lambda\eta}$

The new physics parameters directly related to the decay are
$\mu_{\beta\beta}=m_{\beta\beta}/m_{e}$
($m_{\beta\beta}=|\sum_{j}U^{2}_{ej}m_{j}|$ is the effective neutrino mass
defined in the literature Simkovic  _et al._ (1999)),
$\langle\lambda\rangle=|\lambda\sum_{j}U_{ej}V_{ej}|$ and
$\langle\eta\rangle=|\eta\sum_{j}U_{ej}V_{ej}|$, the two phase angles have the
form
$\psi_{1}=arg[(\sum_{j}m_{j}U_{ej}^{2})(\sum_{j}U_{ej}T_{ej}^{*}(g_{V}/g_{V}^{\prime}))^{*}]$
and $\psi_{2}=arg[(\sum_{j}m_{j}U_{ej}^{2})(\sum_{j}U_{ej}T_{ej}^{*})^{*}]$.
By assuming $g_{V}=g_{V}^{\prime}$, we have $\psi_{1}\approx\psi_{2}$.

The coefficients $\mathcal{C}$’s have the forms Štefánik  _et al._ (2015):

$\displaystyle\mathcal{C}_{mm}$ $\displaystyle=$
$\displaystyle\mathcal{G}_{01}|M^{0\nu}_{m}|^{2}$
$\displaystyle\mathcal{C}_{m\lambda}$ $\displaystyle=$
$\displaystyle-\mathcal{G}_{03}M^{0\nu}_{m}M^{0\nu}_{\omega-}+\mathcal{G}_{04}M^{0\nu}_{m}M^{0\nu}_{q+}$
$\displaystyle\mathcal{C}_{m\eta}$ $\displaystyle=$
$\displaystyle\mathcal{G}_{03}M^{0\nu}_{m}M^{0\nu}_{\omega+}-\mathcal{G}_{04}M^{0\nu}_{m}M^{0\nu}_{q-}-\mathcal{G}_{05}M^{0\nu}_{m}M^{0\nu}_{P}$
$\displaystyle+$ $\displaystyle\mathcal{G}_{06}M^{0\nu}_{m}M^{0\nu}_{R}$
$\displaystyle\mathcal{C}_{\lambda\lambda}$ $\displaystyle=$
$\displaystyle\mathcal{G}_{02}|M^{0\nu}_{\omega-}|^{2}+\mathcal{G}_{011}|M^{0\nu}_{q+}|^{2}$
$\displaystyle\mathcal{C}_{\eta\eta}$ $\displaystyle=$
$\displaystyle\mathcal{G}_{02}|M^{0\nu}_{\omega+}|^{2}+\mathcal{G}_{011}|M^{0\nu}_{q-}|^{2}+\mathcal{G}_{08}|M^{0\nu}_{P}|^{2}$
$\displaystyle+$
$\displaystyle\mathcal{G}_{09}|M^{0\nu}_{R}|^{2}-\mathcal{G}_{07}M^{0\nu}_{P}M^{0\nu}_{R}$
$\displaystyle\mathcal{C}_{\lambda\eta}$ $\displaystyle=$
$\displaystyle-2\mathcal{G}_{02}M^{0\nu}_{\omega-}M^{0\nu}_{\omega+}-\mathcal{G}_{010}(M^{0\nu}_{q+}M^{0\nu}_{\omega+}+M^{0\nu}_{q-}M^{0\nu}_{\omega-})$
(14) $\displaystyle-$ $\displaystyle
2\mathcal{G}_{011}M^{0\nu}_{q+}M^{0\nu}_{q-}$

Here we use a different form of phase space factors from that of Štefánik  _et
al._ (2015) to better separate the effects of the $s$ and $p$-wave electrons.
Their correspondence to the phase space factors of Štefánik  _et al._ (2015)
is as following:

$\displaystyle\mathcal{G}_{01,02,03}$ $\displaystyle=$ $\displaystyle
G_{01,02,03}$ $\displaystyle\mathcal{G}_{04}$ $\displaystyle=$
$\displaystyle(3m_{e}R)G_{04}$ $\displaystyle\mathcal{G}_{05,06}$
$\displaystyle=$ $\displaystyle(m_{e}R)G_{05,06}$
$\displaystyle\mathcal{G}_{07,08,09}$ $\displaystyle=$
$\displaystyle(m_{e}R)^{2}G_{07,08,09}$ $\displaystyle\mathcal{G}_{10}$
$\displaystyle=$ $\displaystyle 3(m_{e}R)^{2}\tilde{G}_{010}$
$\displaystyle\mathcal{G}_{11}$ $\displaystyle=$
$\displaystyle(3m_{e}R)^{2}\tilde{G}_{011}$

Meanwhile, the NMEs can be written as sums from different individual NMEs:

$\displaystyle M^{0\nu}_{m}$ $\displaystyle=$ $\displaystyle-
M_{F}+M_{GT}+M_{T}$ $\displaystyle M^{0\nu}_{\omega\pm}$ $\displaystyle=$
$\displaystyle M_{\omega GT\pm}+M_{\omega T\pm}\pm M_{\omega F}$
$\displaystyle M^{0\nu}_{q\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{3m_{e}R}(M_{qGT\pm}-6M_{qT\pm}\pm 3M_{qF})$
$\displaystyle M^{0\nu}_{R}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{e}R}(M_{RGT}+M_{RT})$ $\displaystyle M^{0\nu}_{P}$
$\displaystyle=$ $\displaystyle\frac{1}{m_{e}R}M_{P}$ (15)

We notice that the reaction matrix has the form $R\sim
Mf_{e}(\epsilon_{1},\epsilon_{2})$ Doi  _et al._ (1985), where M contains the
information of the neutrino potential as well as nuclear transition amplitude,
while $f_{e}$ is a functional of electron wave functions. Then, for the
diagonal terms such as $\mathcal{C}_{mm}$, it is reasonable that
$R\sim\mathcal{R}\equiv M\sqrt{\mathcal{G}}$. This newly defined term
$\mathcal{R}$ can therefore be used for the comparison of general magnitude of
components from various mechanisms, as we shall show in subsequent sections.

The individual NMEs $M_{I}$ can be written in a general form Štefánik  _et
al._ (2015) :

$\displaystyle M_{I}$ $\displaystyle=$ $\displaystyle\langle
f||h_{I}(r,r_{+})\mathcal{O}_{I}||i\rangle$ (16) $\displaystyle=$
$\displaystyle\sum_{p_{1}p_{2}n_{1}n_{2}J}\langle
f||[c_{p1}^{\dagger}c_{p2}^{\dagger}]_{J}[\tilde{c}_{n2}\tilde{c}_{n1}]_{J}||i\rangle$
$\displaystyle\times$ $\displaystyle\langle
p_{1}p_{2}J||h_{I}(r,r_{+})\mathcal{O}_{I}||n_{1}n_{2}J\rangle$

In such a treatment, the input from nuclear many-body approaches are the so-
called two-body transition densities(TBTD) $\langle
f||[c_{p1}^{\dagger}c_{p2}^{\dagger}]_{J}[\tilde{c}_{n2}\tilde{c}_{n1}]_{J}||i\rangle$.
For our calculations, the TBTDs are obtained from the NuShellX code Brown and
Rae (2014).

The neutrino expotential $h_{I}(r,r+)$ can be written for the $m$, $q$, $R$
and $P$ terms in a general form as:

$\displaystyle h_{Ik}(r,r_{+})=f_{src}^{2}(r)\sum_{k}\frac{2R}{\pi}\int
f_{Ik}(q,r,r_{+})\frac{qdq}{q+\tilde{A}_{m}}$ (17)

Here $k$ refers to contributions from different components of the weak
current. A closure energy $\tilde{A}_{m}$ is introduced to average the
excitation energies from the intermediate states. And
$\vec{r}=\vec{r}_{m}-\vec{r}_{n}$ and
$\vec{r}_{+}=(\vec{r}_{m}+\vec{r}_{n})/2$ are the relative distance and the
center of mass coordinate of the two decaying nucleons respectively. The short
range correlations (src’s) between the two nucleons is taken into account by
the radial function $f_{src}(r)$. In our calculation, we adopt the CD-Bonn src
Simkovic  _et al._ (2008).

The radial functions $f_{I}$ in above $h_{I}$ have the form [with the usual
convention Štefánik  _et al._ (2015), where a factor $g_{A}(0)$ is taken out]
for the mass term:

$\displaystyle f_{F}$ $\displaystyle=$ $\displaystyle
j_{0}(qr)g_{V}^{2}(q^{2})$ $\displaystyle f_{GT}$ $\displaystyle=$
$\displaystyle f_{GT,AA}+f_{GT,AP}+f_{GT,PP}+f_{GT,MM}$ $\displaystyle=$
$\displaystyle
j_{0}(qr)[g_{A}^{2}(q^{2})-\frac{2}{3}g_{A}(q^{2})g_{P}(q^{2})\frac{q^{2}}{2m_{N}}$
$\displaystyle+$
$\displaystyle\frac{1}{3}g_{P}^{2}(q^{2})\frac{q^{4}}{4m_{N}^{2}}+\frac{2}{3}g_{M}^{2}(q^{2})\frac{q^{2}}{4m_{N}^{2}}]$
$\displaystyle f_{T}$ $\displaystyle=$ $\displaystyle
f_{T,AP}+f_{T,PP}+f_{T,MM}$ $\displaystyle=$ $\displaystyle
j_{2}(qr)[\frac{2}{3}g_{A}(q^{2})g_{P}(q^{2})\frac{q^{2}}{2m_{N}}-\frac{1}{3}g_{P}^{2}(q^{2})\frac{q^{4}}{4m_{N}^{2}}$
$\displaystyle+$
$\displaystyle\frac{1}{3}g_{M}^{2}(q^{2})\frac{q^{2}}{4m_{N}^{2}}]$

for the $q$ term:

$\displaystyle f_{qF}$ $\displaystyle=$ $\displaystyle
j_{1}(qr)qrg_{V}^{2}(q^{2})$ $\displaystyle f_{qGT\pm}$ $\displaystyle=$
$\displaystyle f_{qGT,AA}+f_{qGT,AP}+f_{qGT,PP}\mp f_{qGT,MM}$
$\displaystyle=$ $\displaystyle
j_{1}(qr)qr(g_{A}^{2}(q^{2})+2g_{A}(q^{2})g_{P}(q^{2})\frac{q^{2}}{2m_{N}}$
$\displaystyle-$ $\displaystyle g_{P}^{2}(q^{2})\frac{q^{4}}{4m_{N}^{2}}\mp
g_{M}^{2}(q^{2})\frac{q^{2}}{2m_{N}^{2}})$ $\displaystyle f_{qT\pm}$
$\displaystyle=$ $\displaystyle f_{qT,AA}+f_{qT,AP}+f_{qT,PP}\mp f_{qT,MM}$
$\displaystyle=$ $\displaystyle
j_{1}(qr)qr\frac{1}{3}g_{A}^{2}(q^{2})-j_{1}(qr)qr\frac{1}{3}g_{A}(q^{2})g_{P}(q^{2})\frac{q^{2}}{2m_{N}}$
$\displaystyle+$
$\displaystyle\frac{1}{10}(\frac{2}{3}j_{1}(qr)+j_{3}(qr))qrg_{P}^{2}(q^{2})\frac{q^{4}}{4m_{N}^{2}}$
$\displaystyle\pm$
$\displaystyle\frac{1}{30}(j_{1}(qr)+3j_{3}(qr))qrg_{M}^{2}(q^{2})\frac{q^{2}}{2m_{N}^{2}}$

for the $R$ term:

$\displaystyle f_{RGT}$ $\displaystyle=$ $\displaystyle-
j_{0}(qr)qR\frac{q}{3m_{N}}g_{M}(q^{2})g_{A}(q^{2})$ $\displaystyle f_{RT}$
$\displaystyle=$ $\displaystyle-
j_{2}(qr)qR\frac{q}{6m_{N}}g_{M}(q^{2})g_{A}(q^{2})$

and finally for the $P$ term:

$\displaystyle f_{P}$ $\displaystyle=$ $\displaystyle
j_{1}(qr)qr_{+}g_{V}(q^{2})g_{A}(q^{2})$ (18)

While the radial functions for the $\omega$ term are a bit different:

$\displaystyle h_{I}(r)=f_{src}^{2}(r)\frac{2R}{\pi}\sum_{k}\int
f_{Ik}(q,r)\frac{q^{2}dq}{[q+\tilde{E}-(E_{i}+E_{f})/2]^{2}}$ (19)

With

$\displaystyle f_{\omega F}$ $\displaystyle=$ $\displaystyle
j_{0}(qr)g_{V}^{2}(q^{2})$ $\displaystyle f_{GT\pm}$ $\displaystyle=$
$\displaystyle f_{GT,AA}+f_{GT,AP}+f_{GT,PP}\pm f_{GT,MM}$ $\displaystyle=$
$\displaystyle
j_{0}(qr)[g_{A}^{2}(q^{2})-\frac{2}{3}g_{A}(q^{2})g_{P}(q^{2})\frac{q^{2}}{2m_{N}}$
$\displaystyle+$
$\displaystyle\frac{1}{3}g_{P}^{2}(q^{2})\frac{q^{4}}{4m_{N}^{2}}\pm\frac{2}{3}g_{M}^{2}(q^{2})\frac{q^{2}}{4m_{N}^{2}}]$
$\displaystyle f_{T\pm}$ $\displaystyle=$ $\displaystyle f_{T,AP}+f_{T,PP}\pm
f_{T,MM}$ $\displaystyle=$ $\displaystyle
j_{2}(qr)[\frac{2}{3}g_{A}(q^{2})g_{P}(q^{2})\frac{q^{2}}{2m_{N}}-\frac{1}{3}g_{P}^{2}(q^{2})\frac{q^{4}}{4m_{N}^{2}}$
$\displaystyle\pm$
$\displaystyle\frac{1}{3}g_{M}^{2}(q^{2})\frac{q^{2}}{4m_{N}^{2}}]$

The angular part of the matrix element $O_{I}$ has the forms:

$\displaystyle\mathcal{O}_{iF}$ $\displaystyle=$ $\displaystyle 1$
$\displaystyle\mathcal{O}_{iGT}$ $\displaystyle=$
$\displaystyle\vec{\sigma}_{m}\cdot\vec{\sigma}_{n}$
$\displaystyle\mathcal{O}_{iT}$ $\displaystyle=$ $\displaystyle
3(\vec{\sigma}_{m}\cdot\hat{r})(\vec{\sigma}_{n}\cdot\hat{r})-\vec{\sigma}_{m}\cdot\vec{\sigma}_{n}$
$\displaystyle\mathcal{O}_{P}$ $\displaystyle=$ $\displaystyle
i(\vec{\sigma}_{m}-\vec{\sigma}_{n})\cdot(\hat{r}\times\hat{r}_{+})$ (20)

Here $i$ refers to different terms such as the $\omega$ term.

With the equations above, we give a complete expression for the NME with weak-
current including all LO terms and a sub-leading order (NLO) weak-magnetism
term Cirigliano  _et al._ (2017).

NME |  76Ge$\rightarrow^{76}$Se |  82Se$\rightarrow^{82}$Kr |  130Te$\rightarrow^{130}$Xe |  136Xe$\rightarrow^{136}$Ba
---|---|---|---|---
|  |  |  jun45 |  jj44b |  jun45 |  jj44b |  jj55a | GCN50:82 |  jj55a | GCN50:82
$M_{m}$ | F |  | -0.665 | -0.601 | -0.624 | -0.523 | -0.668 | -0.701 | -0.574 | -0.567
GT | AA | 3.584 | 3.278 | 3.360 | 2.860 | 3.147 | 3.180 | 2.648 | 2.549
AP | -1.090 | -0.960 | -1.021 | -0.834 | -0.979 | -1.034 | -0.820 | -0.829
PP | 0.344 | 0.300 | 0.321 | 0.261 | 0.313 | 0.335 | 0.260 | 0.268
MM | 0.247 | 0.215 | 0.229 | 0.188 | 0.227 | 0.244 | 0.188 | 0.194
total | 3.085 | 2.833 | 2.889 | 2.474 | 2.708 | 2.724 | 2.277 | 2.183
T | AP | -0.013 | -0.004 | -0.014 | -0.012 | 0.008 | 0.015 | 0.002 | 0.014
PP | 0.002 | -0.001 | 0.003 | 0.003 | -0.006 | -0.007 | -0.003 | -0.006
MM | -0.001 | -0.000 | -0.001 | -0.002 | 0.003 | 0.003 | 0.001 | 0.002
total | -0.012 | -0.004 | -0.013 | -0.010 | 0.004 | 0.010 | -0.000 | 0.010
$M_{\omega\pm}$ | F |  | -0.637 | -0.575 | -0.597 | -0.500 | -0.637 | -0.669 | -0.545 | -0.540
GT | AA | 3.276 | 2.980 | 3.073 | 2.596 | 2.883 | 2.931 | 2.427 | 2.351
AP | -1.044 | -0.919 | -0.978 | -0.798 | -0.939 | -0.993 | -0.786 | -0.795
PP | 0.333 | 0.290 | 0.310 | 0.252 | 0.303 | 0.324 | 0.252 | 0.259
MM | 0.239 | 0.208 | 0.221 | 0.181 | 0.220 | 0.236 | 0.182 | 0.188
GT+total | 2.803 | 2.558 | 2.626 | 2.231 | 2.466 | 2.498 | 2.075 | 2.002
GT-total | 2.325 | 2.172 | 2.184 | 1.789 | 2.026 | 2.026 | 2.711 | 2.626
T | AP | -0.012 | -0.003 | -0.013 | -0.011 | 0.009 | 0.015 | 0.003 | 0.014
PP | 0.002 | -0.001 | 0.003 | 0.003 | -0.006 | -0.007 | -0.003 | -0.006
MM | -0.001 | -0.000 | -0.001 | -0.002 | 0.003 | 0.003 | 0.001 | 0.002
T+total | -0.011 | -0.004 | -0.012 | -0.010 | 0.005 | 0.010 | 0.000 | 0.010
T-total | -0.0013 | -0.004 | -0.014 | -0.014 | -0.001 | 0.004 | -0.001 | 0.006
$M_{q\pm}$ | F |  | -0.379 | -0.351 | -0.359 | -0.304 | -0.408 | -0.417 | -0.358 | -0.342
GT | AA | 3.210 | 2.981 | 3.016 | 2.605 | 2.781 | 2.751 | 2.348 | 2.209
AP | 4.842 | 4.317 | 4.571 | 3.741 | 4.267 | 4.425 | 3.607 | 3.563
PP | -1.943 | -1.706 | -1.829 | -1.479 | -1.731 | -1.827 | -1.454 | -1.468
MM | -1.874 | -1.636 | -1.745 | -1.426 | -1.708 | -1.825 | -1.419 | -1.456
GT+total | 7.983 | 7.228 | 7.502 | 6.293 | 7.026 | 7.173 | 5.920 | 5.760
GT-total | 4.235 | 3.956 | 4.012 | 3.441 | 3.610 | 3.523 | 3.082 | 2.848
T | AA | -0.056 | -0.033 | -0.055 | -0.042 | -0.031 | -0.009 | -0.031 | 0.002
AP | 0.004 | -0.001 | 0.006 | 0.008 | -0.018 | -0.018 | -0.007 | -0.015
PP | 0.000 | 0.001 | -0.001 | -0.003 | 0.007 | 0.005 | 0.002 | 0.003
MM | 0.000 | -0.000 | -0.000 | -0.001 | 0.001 | 0.001 | 0.000 | 0.001
T+total | -0.051 | -0.034 | -0.050 | -0.035 | -0.043 | -0.023 | -0.036 | -0.012
T-total | -0.051 | -0.034 | -0.050 | -0.037 | -0.041 | -0.021 | -0.036 | -0.009
$M_{R}$ | GT |  | 4.256 | 3.713 | 4.037 | 3.314 | 4.686 | 5.048 | 3.948 | 4.080
T |  | 0.014 | 0.004 | 0.018 | 0.028 | -0.056 | -0.056 | -0.014 | -0.042
$M_{P}$ |  |  | -0.431 | -0.279 | -0.428 | -0.152 | -0.498 | -0.425 | -0.289 | -0.255
Table 1: $0\nu\beta\beta$-decay NMEs for 76Ge, 82Se, 130Te and 136Xe from LSSM
calculations. Here, for each term we list all the components from different
parts as defined in eq.(16). For each nuclei, we present results from two
different Hamiltonians as indicated in text.

##  III Results and Discussion

###  III.1 The NMEs for LR symmetric model

For the calculations of nuclear matrix elements, we use the large-scale shell
model (LSSM) method. For 76Ge and 82Se, we use the $jj44$ model space that
include four orbitals (0$f_{5/2}$, 1$p_{3/2}$, 1$p_{1/2}$ and 0$g_{9/2}$) for
both protons and neutrons. We adopt two Hamiltonians, $jj44b$ Lisetskiy  _et
al._ (2004) and $jun45$ Honma  _et al._ (2009), to understand the possible
theoretical uncertainties arising from nuclear interactions within the $jj44$
model space. For 130Te and 136Xe, we use the $jj55$ model space that contains
five orbitals (0$g_{7/2}$, 1$d_{5/2}$, 1$d_{3/2}$, 2$s_{1/2}$ and
0$h_{11/2}$), and the two Hamiltonians used for this model space are $jj55pn$
Brown  _et al._ (2005) and GCN50:82 Menendez  _et al._ (2009) respectively.

The corresponding NMEs are presented in Table.1. Here, to make comparisons
with previous calculations, we divide the NMEs by corresponding functions of
$g_{A}(0)$ or $g_{V}(0)$. For the neutrino mass mechanism, there are already
LSSM calculations for these four nuclei (e.g. Sen’kov and Horoi (2016);
Coraggio  _et al._ (2020b); Horoi and Brown (2013); Caurier  _et al._ (2012);
Menéndez (2018)). Our results agree well with previous calculations and we
also find that the choice of the closure energy may be the origin of
uncertainties, in some cases, leading to a deviation of more than 15%. In this
work, we adopt a closure energy of $7$ MeV. In agreement with results from
other calculations, the NME is dominated by the GT part, and the Tensor part
is negligible. Meanwhile, the Fermi part is about 1/4 to 1/5 to the GT part,
smaller than a naive estimation of $1/3$ using Fierz rearrangement Prezeau
_et al._ (2003) for the short range mechanism case. As found in the
literatures Brown  _et al._ (2015), the uncertainties for NMEs with the
neutrino-mass mechanism from different Hamiltonians are about 10%. In our
calculation, among the four nuclei, the biggest deviation related the choice
of Hamiltonian comes from 82Se, where the two Hamiltonians lead to an
uncertainty of about 20% for the Fermi part and about 15% for the GT part.
Meanwhile, the ratios of corresponding components inside the GT part from the
two Hamiltonians are basically the same. For other nuclei, such uncertainties
are limited to less than 10% for all components.

For other mechanisms, the LSSM calculations are relatively limited, dating
back to $90^{\prime}s$ Caurier  _et al._ (1996) and recently some calculations
concerning $\lambda$ mechanism Sarkar  _et al._ (2020); Iwata and Sarkar
(2021). However, systematic studies with nuclear currents including even the
weak magnetism term are not yet available. So in this work, we give a thorough
analysis of all the relevant long-range terms related to the LR symmetric
model.

In the LR symmetric model, different mechanisms exhibit distinct current-
current interaction structures. For the nuclear part, both the mass and the
$\eta$ mechanisms have left-handed (V-A) nuclear currents at both vertices.
However, for the $\lambda$ mechanism, one has the left-handed (V-A) nuclear
current at one vertex and right-handed (V+A) nuclear current at another. Given
that the weak-magnetism current is induced by the vector current, this implies
that for the $\lambda$ mechanism, the MM components has a different sign
relative to the other GT components, in contrast to the mass or the $\eta$
mechanism. Therefore, as expressed in eq.(15), the GT and tensor parts have MM
components differed by the sign for the two mechanisms.

For the $\omega_{+}$ term, which has the same nuclear current structure as the
mass term, its difference with the mass term comes from the energy denominator
[eq.(19)]. If we set the closure energy to be zero, then these two terms
become identical. With the chosen closure energy, the $\omega$ term is
slightly smaller than the mass term. For the F part, we have about 5%
reduction for all cases, irrespective of nuclei or Hamiltonian. For the GT
part, the reduction is less than 10% and is mostly from the AA component,
either because it is large in magnitude and also due to the fact that its
radial integration is dominated by the low $q$ parts as we will show in
subsequent discussions. The MM component is about 10% in magnitude to the AA
component and then the difference between $\omega_{-}$ and $\omega_{+}$ terms
is about 20% due to the different signs of the MM component.

The Fermi part is about 1/5 of the GT part for the mass or the $\omega$ terms.
Its relative magnitude of the $q$ terms is much smaller, just about 1/10 or
less for the two $fp$ nuclei and 1/9 or less for the two heavier ones. Their
absolute magnitude is also suppressed, its absolute value is only about 50% to
60% of counterparts in the mass and the $\omega$ terms.

A similar situation occurs for the AA component of the GT part. Compared to
its counterpart in mass term, it is reduced by more than 10%. Meanwhile, other
components in the $q$ term are significantly enhanced, especially the AP
component, which now becomes the dominant contribution to the GT part. Unlike
the mass term, the four components accounted for here contribute with similar
magnitudes. The major reason is that these components now have similar angular
coupling coefficients, in contrast to the mass term. Due to the largeness of
the MM component, previous calculations without the inclusion of this weak-
magnetism term give an under or over-estimation for $q$ terms Sarkar  _et al._
(2020); Iwata and Sarkar (2021). This weak-magnetism component contributes
more than half of the AA component, and less than half of the dominant AP
component. This implies that we have to treat it as an equal contribution as
other LO terms and include it in future calculations. For $q_{+}$ term, the PP
and the MM components almost cancels each other. Then, $M_{qGT+}$ is 2-3 times
larger than $M_{GT}$ in the mass term. However, for $q_{-}$ term, these two
terms nearly cancel contributions from the AP part, allowing for a similar
magnitude of $M_{qGT-}$ to $M_{GT}$ in the mass term.

On the other hand, the tensor part, as in the mass or $\omega$ terms, can be
safely neglected. This is common in LSSM calculations while QRPA suggests that
the magnitude of tensor part is about 10%. The reason may stem from the
missing orbitals in LSSM compared to QRPA and this still needs further
investigation.

For the GT part of the so-called relativistic ($R$-) term Tomoda (1991),
despite the fact that the weak-magnetism current is NLO, its magnitude is of
the LO size irrespective of the nucleus or the Hamiltonian. For the two
lighter nuclei, $M_{RGT}$ is basically the same as $M_{qGT-}$ and for the two
heavier nuclei, it is even larger than the latter. This suggests that although
it is counted as NLO because of the factor $q/M_{N}$, it is actually giving LO
contributions mainly due to a large $\mu_{p}-\mu_{n}$ Cirigliano  _et al._
(2017). We will give a more detailed analysis in subsequent sections. For the
$R$ term, as other terms, we can safely neglect the tensor part in LSSM
calculations.

The last term to be discussed is the $P$ term. Compared to other terms, it is
somehow heavily suppressed due to its special operator structure. In previous
calculation Caurier  _et al._ (1996), the change of NME signs for different
nuclei is observed, but for our calculation, we find for different nuclei, it
has always the same negative sign for NME values. While $jun45$ and $jj44b$
gives quite different predictions, the results from $jj55a$ and GCN50:82 are
closer, the difference is about 10%. Meanwhile, our results differs largely
from that of LSSM calculation in Caurier  _et al._ (2012) by more than a
factor of two, but close to QRPA calculation in Muto  _et al._ (1989). These
discrepancies need further investigations.

###  III.2 Enhancement of MM and PP components for the $q$ term

Figure 1: (Color online) Strength distribution of different components in $q$
space for $M_{mGT}$ and $M_{qGT+}$ for 76Ge. For $M_{qGT-}$, the MM part
changes the sign.

The weak magnetic current comes out to be an NLO contribution Cirigliano  _et
al._ (2017). This argument of power-counting is valid if the exchange momentum
is smaller or around $m_{\pi}$. For the NME, the MM component is a scalar
product of two weak magnetic currents, it is thus supposed to be suppressed.
For the light neutrino mass mechanism, this is the case as MM contributes only
several percent in Simkovic  _et al._ (1999) and for our calculation. However,
for the $q$ term, a different behavior is observed. Although the magnitude of
MM component is still smaller than AA or AP as well as PP, its relative ratio
to AA is now much larger than that of mass term. Such behavior has already
observed for the heavy neutrino mass mechanism Simkovic  _et al._ (1999) where
the higher exchange momentum dominates in the momentum ($q$) space
integration. As seen in above section, the relative ratios of the MM as well
as PP components to AA component have been shifted from about 10% to more than
50%.

To better understand the large contributions of the MM and PP components, in
Fig.1, we plot the GT NME’s strength distribution in $q$ space for both the
mass and $q$ terms, for the mass term such studies have already been done
Simkovic  _et al._ (2008); Engel  _et al._ (2014); Shimizu  _et al._ (2018).
We find that the results in Fig.1 agree with previous studies for mass
mechanism, while the behaviors look quite different for the MM parts of these
two terms in Fig.1. The maxima for different components appear at different
exchange momenta, this implies that for different mechanisms, the typical
exchange momentum is different. For the mass term, it is obvious that the AA
component is dominated by the low $q\sim 0\ll m_{\pi}$ region while AP and PP
components are dominated by the region $q\sim m_{\pi}$. The MM component in
this case are also dominated by contributions from the region $q\sim m_{\pi}$
but is suppressed by a factor of $q/M\sim m_{\pi}/M$. For the mass term, the
overall strength is proportional to the product of the form factor and the
spherical Bessel function $j_{0}(qr)$ which has a maximum at $q=0$ , this
explains the dominance of the AA component and the relative suppression of
other components, especially for the MM component.

For the $q$ terms, we find that the typical transfer momentum is around
$q\sim$400 MeV, this is due to the fact that the integrand is proportional to
the spherical Bessel function $j_{1}(qr)$ which has a node $qr\approx\pi/2$
instead of zero for rank 0 spherical Bessel function $j_{0}$. In Fig.1, we can
clearly see these nodes of the integrand functions

Therefore, as we have stated above, the naive power-counting may apply to the
NME calculations only if one is takes into account of the typical exchange
momentum $q$ for the process. For different terms and different components,
$q$ may be different, their effective power-counting may differ.

The discussion can be extended to the analysis of $M_{R}$, which is the
products of axial vector and weak magnetism currents. Compared to the GT
matrix elements from mass term, there is an additional factor $q^{2}$
multiplied to the Spherical Bessel function, this then enhances the
contribution from high $q>m_{\pi}$. When multiplied with an extra factor of
$g_{M}(0)\approx$4.7, we obtain $M_{RGT}$ similar or larger in size as
$M_{GT}$, although naively it is expected to be suppressed.

These analyses suggest that an explicit estimation of the magnitude of NMEs
requires our knowledge on the typical exchange momentum for the virtually
mediated neutrino.

###  III.3 Estimations of relative magnitudes of different terms

Table 2: A naive estimation of the magnitudes of different terms from $0\nu\beta\beta$-decay rates (see text). For each term, we have two sub rows indicating the results from the two different Hamiltonians: for 76Ge and 82Se, the first sub rows are results from jun45 and second sub rows results from jj44b; for 130Te and 136Xe, the first sub rows are results from jj55a and the second sub rows from GCN50:82. $r_{e}$, $r_{N}$ and $r_{R}$ are the ratios of square-rooted phase space factors, NMEs and reaction matrix elements of each term to the mass term respectively. |  | rough estimation |  76Ge |  82Se |  130Te |  136Xe
---|---|---|---|---|---|---
|  | lepton | nuclear | $\mathcal{R}$ | $G_{0\nu}$ | $M_{0\nu}$ |  | $G_{0\nu}$ | $M_{0\nu}$ |  | $G_{0\nu}$ | $M_{0\nu}$ |  | $G_{0\nu}$ | $M_{0\nu}$ |
$\mu_{\beta\beta}$ |  | $\mathcal{O}(1)$ | $\mathcal{O}(1)$ | $\mathcal{O}(1)$ | $0.24$ | 5.62 |  | $1.02$ | 5.26 |  | $1.43$ | 5.04 |  | $1.46$ | 4.25 |
|  | 5.16 |  |  | 4.50 |  |  | 5.11 |  |  | 4.10 |
|  |  |  |  | $r_{e}$ | $r_{N}$ | $r_{R}$ | $r_{e}$ | $r_{N}$ | $r_{R}$ | $r_{e}$ | $r_{N}$ | $r_{R}$ | $r_{e}$ | $r_{N}$ | $r_{R}$
$\langle\lambda\rangle$ | $M_{\omega}$ | $\mathcal{O}(\epsilon_{12}/m_{e})$ | $\mathcal{O}(1)$ | $\mathcal{O}(1)$ | 1.25 | 0.78 | 0.98 | 1.85 | 0.78 | 1.45 | 1.61 | 0.78 | 1.25 | 1.57 | 0.78 | 1.22
0.78 | 0.98 | 0.78 | 1.44 | 0.77 | 1.24 | 0.77 | 1.21
$M_{q}$ | $\mathcal{O}(\omega R)$ | $\mathcal{O}(q/m_{e})$ | $\mathcal{O}(1)$ | 0.010 | 55.1 | 0.53 | 0.012 | 54.0 | 0.65 | 0.013 | 44.2 | 0.59 | 0.013 | 43.4 | 0.58
53.6 | 0.51 | 52.6 | 0.63 | 43.7 | 0.58 | 42.5 | 0.57
$\langle\eta\rangle$ | $M_{\omega}$ | $\mathcal{O}(\epsilon_{12}/m_{e})$ | $\mathcal{O}(1)$ | $\mathcal{O}(1)$ | 1.25 | 0.69 | 0.86 | 1.85 | 0.69 | 1.27 | 1.61 | 0.66 | 1.07 | 1.57 | 0.66 | 1.04
0.69 | 0.86 | 0.69 | 1.27 | 0.66 | 1.06 | 0.66 | 1.04
$M_{q}$ | $\mathcal{O}(\omega R)$ | $\mathcal{O}(q/m_{e})$ | $\mathcal{O}(1)$ | 0.010 | 38.1 | 0.36 | 0.012 | 37.7 | 0.45 | 0.013 | 31.3 | 0.41 | 0.013 | 31.4 | 0.42
38.1 | 0.36 | 37.4 | 0.45 | 29.6 | 0.39 | 29.0 | 0.39
$M_{R}$ | $\mathcal{O}(1)$ | $\mathcal{O}(q^{2}/(M_{N}m_{e}))$ | $\mathcal{O}(\varepsilon^{-1})$ | 3.02 | 73.3 | 221.5 | 2.96 | 72.6 | 214.8 | 2.97 | 73.8 | 219.4 | 2.97 | 73.5 | 218.4
69.4 | 209.8 | 70.1 | 207.4 | 78.5 | 233.4 | 77.9 | 231.8
$M_{P}$ | $\mathcal{O}(\alpha Z)$ | $\mathcal{O}(q/m_{e})$ | $\mathcal{O}(\varepsilon^{-1})$ | 0.34 | 7.40 | 2.49 | 0.33 | 7.65 | 2.50 | 0.27 | 7.97 | 2.19 | 0.25 | 5.41 | 1.37
5.21 | 1.75 | 3.18 | 1.04 | 6.71 | 1.84 | 4.94 | 1.25

As has been demonstrated in Doi  _et al._ (1985); Tomoda (1991), different
terms may be of different relative magnitudes. Thus we adjust the expression
of the decay width to better explain the origin of these different magnitudes.
A rough estimation for the magnitude for the different terms is given in
Tomoda (1991), suggesting the dominance of $R$ terms and $P$ terms for $\eta$
mechanism, and they also suggest that $\lambda$ mechanism is dominated by
$\omega$ term. In this work, we make a more explicit study of the relative
magnitudes of various terms for different underlying mechanisms. We start by
using the standard mass mechanism as the baseline, since for this term at the
leading order, only one term contributes. Therefore, we define the ratios
$r_{eI}\equiv\sqrt{\mathcal{G}_{I}/\mathcal{G}_{01}}$ and $r_{NI}\equiv
M_{I}/M^{0\nu}_{m}$ to denote the relative importance of the lepton and
nuclear parts. The detailed values of these ratios are given in Table 2, with
the rough estimations given at first columns.

For the nuclear part, we use standard the mass mechanism NME as the baseline.
The $\omega$ term $M_{\omega}$ has $\mathcal{O}(1)$. The magnitude of the non-
helicity-suppressed NMEs $M^{0\nu}_{q}$ and $M^{0\nu}_{P}$ are proportional to
$q$, enhanced with a factor $q/m_{e}$ in our convention. Meanwhile, the
relativistic terms Tomoda (1991) $M^{0\nu}_{R}$ are suppressed by an extra
factor of $q/M_{N}$ from the weak-magnetism term compared to $M^{0\nu}_{q}$
and $M^{0\nu}_{P}$, here we neglect the possible contributions from the so-
called recoil term Doi  _et al._ (1985); Tomoda (1991) which is estimated to
be small in Tomoda (1991).

For the electron part, the $s$-wave appears at LO
[$j_{0}(kR)\sim\mathcal{O}(1)$], then the $p$-wave function which is connected
with $\vec{r}$ operator has a magnitude of $\mathcal{O}(\omega R)$, while the
one which is connected with the $\vec{r}_{+}$ operator should have the
magnitude $\alpha Z$, much larger than that for operators with $\vec{r}$, this
is the so-called $p$-wave effect Doi  _et al._ (1985).

To estimate these contributions, we introduce a parameter $\epsilon\approx
1/10$ like the $\epsilon_{\chi}$ from $\chi$EFT introduced in Cirigliano  _et
al._ (2017), then each of above contributions can be assigned an order. For
example, the $\mathcal{O}(q/M_{N})$ and $\mathcal{O}(\alpha Z)$ terms appears
at the order $\mathcal{O}(\varepsilon)$ while $\mathcal{O}(\omega R)$ and
$\mathcal{O}(m_{e}/q)$ appears at the order of $\mathcal{O}(\varepsilon^{2})$.

A detailed analysis based on our numerical results for NME and previous
numerical results for PSFs in Štefánik  _et al._ (2015) are presented in Table
2. Starting from the $\omega$ term, the electron part is basically the same as
the mass term, and so does the NME, agree with the assigned order
$\mathcal{O}(1)$ in Table. 2. Its contribution to the $\lambda$ mechanism is
generally larger than that to the $\eta$ mechanism due to opposite
contributions from the Fermi part. For the q terms, with assigned order
$\mathcal{O}(1)$, ratios of about 1/2 to standard mechanism for $\lambda$
mechanism and 1/2 to 1/3 for $\eta$ mechanism are observed, slightly smaller
than the naive estimation due to a suppression from the nuclear part. The
$p$-wave effect makes the electron part of the $P$ term more than one order of
magnitude larger than that of the $q$ term. Meanwhile, $P$ term’s NME is
heavily suppressed, order of magnitude smaller than expected. This is actually
observed by LSSM calculations earlier Caurier  _et al._ (1996). Therefore,
even with the so-called $p$-wave effect Doi  _et al._ (1985), the $P$ term
gives an $\mathcal{O}(1)$ contribution instead of the expected
$\mathcal{O}(\varepsilon^{-1})$ in Table 2. On the other hand, because of the
enhanced NMEs, the $R$ term gives an $\mathcal{O}(\varepsilon^{-2})$
contribution instead of the $\mathcal{O}(\varepsilon^{-1})$, one order of
magnitude larger than naive estimation. This makes the $R$ term dominate the
$\eta$ mechanism, while other terms two orders of magnitude smaller.

Therefore, without considerations of the new physics parameters, we need to
slightly modify the naive estimation of orders of contributions. From above
results, we find that $R$ has a much larger size than others, it is natural to
assume $R$ term to be at the leading order $\mathcal{O}(1)$, the $P$-term
which is supposed to be the same size as $R$ term is now suppressed by the
nuclear part and comes out to be of the similar size as $\omega$ term, they
all appear at the order $\mathcal{O}(\varepsilon^{2})$, so does the mass term.
Meanwhile, although the $q$ term is suppressed, it can still be account as
$\mathcal{O}(\varepsilon^{2})$.

In general, for LSSM calculations, we find that the $\eta$ mechanism is
dominated by the $R$ term, while corrections from other terms are at the
percent level (at the order of $\mathcal{O}(\varepsilon^{2})$). For the
$\lambda$ mechanism, the $\omega$ term at the order of
$\mathcal{O}(\varepsilon^{2})$ dominates but receives an additive correction
of about 50% from the $q$ term. Now, if we incorporate the new physics
parameters, we find that if these different mechanisms coexist, then one
requires that
$\langle\lambda\rangle\sim\mu_{\beta\beta}\sim(m_{e}/q)\langle\eta\rangle$.
This analysis helps to constrain the new physics model, and we will proceed to
this topic in the next section.

###  III.4 Current experimental constraints on LR-symmetric model

Current lower limits on half-lives of neutrinoless double beta decay collected
from various literatures are: $1.8\times 10^{26}$yr at 90% C.L. for 76Ge from
GERDA Agostini  _et al._ (2020), $4.6\times 10^{24}$yr at 90% C.L. for 82Se
from CUPID Azzolini  _et al._ (2022), $2.2\times 10^{25}$yr at 90% C.L. for
130Te from CUORE Adams  _et al._ (2022), and $2.3\times 10^{26}$yr at 90% C.L.
for 136Xe from KamLAND-Zen Abe  _et al._ (2023). The most stringent constraint
is from KamLAND-Zen, while GERDA also provide very tight constraints larger
than $10^{26}$yr. Assuming that the mass term dominates and set $g_{A}=1.27$,
we obtain the most stringent constraint on the effective neutrino mass for
$m_{\beta\beta}<66$ meV from 136Xe with the $jj55a$ Hamiltonian in our
calculation, these results are approaching the so-called inverted hierarchy
mass region. However, there are results with extremely small NMEs from other
many-body approaches Mustonen and Engel (2013); Fang  _et al._ (2018) for this
nucleus, which may push the effective neutrino mass limit to a higher value
region.

Current experiments still leave space for the coexistence of different
mechanisms if the neutrino mass indeed has an inverted hierarchy. Therefore,
in Fig.2, we choose typical effective neutrino masses and then provide the
constraints for $\lambda$ and $\eta$. With current LR symmetric model,
$\langle\lambda\rangle$ is defined as positive, and $\Psi_{1}=\Psi_{2}$ leaves
only one uncertain phase angle. We chose two typical effective neutrino masses
of $50$ meV and $3$ meV, which lie in the inverted and normal hierarchy
neutrino mass regions, respectively.

Figure 2: (Color online) Constraints on the LR symmetric model from current
experimental limitAgostini  _et al._ (2020); Azzolini  _et al._ (2022, 2022);
Abe  _et al._ (2023).Here the we take the double beta decay effective neutrino
mass $m_{\beta\beta}$ with two typical values 50meV(solid lines) and 3
meV(dashed lines) corresponding to IH and NH respectively. Here the shaded
area comes from the uncertainties of the phase angle $\psi$’s with the
assumption $\psi_{1}=\psi_{2}$. The different colors correspond to different
Hamiltonians one adopts: red for $jj44b$ and blue for $jun45$ for 76Ge and
82Se; red for $jj55a$ and blue for GCN50:82 for 130Te and 136Xe.

We first consider the inverted hierarchy case. The relatively large
$m_{\beta\beta}$ leaves a smaller room for the parameter space of
$\langle\lambda\rangle$ and $\langle\eta\rangle$. The most stringent
constraints on $\langle\lambda\rangle$ is from 136Xe, which requires that
$\langle\lambda\rangle$ be smaller than about $6-7\times 10^{-8}$ depending on
both the Hamiltonian and the phase angle. The uncertainties (shaded area)
caused by the phase angle are related to the interference terms (the second
and third terms in r.h.s. of eq.(13) ) and related to the magnitudes of
$\mathcal{C}_{m\lambda}$ and $\mathcal{C}_{m\eta}$, the latter determines the
width of the uncertainty band. Meanwhile the most stringent constraints on
$\langle\eta\rangle$ from our calculations come also from kamLAND-Zen, it
suggests that $|\langle\eta\rangle|<2-7\times 10^{-10}$, smaller than the
constraints on $\langle\lambda\rangle$. This can be explained by rough
analysis of magnitudes in previous sections, that the contributions for
$\langle\eta\rangle$ come out at LO dominated by $R$ term while that for
$\langle\lambda\rangle$ come out at next to sub leading order (N2LO). For the
other three nuclei, current constraints on $\langle\eta\rangle$ could be one
order of magnitude larger due to shorter half-life limits. Especially for
82Se, half-life limit needs to be improved in the near future from various
experiments proposed Cao  _et al._ (2023) to strengthen the constraints.

For a typical normal hierarchy case, we set $m_{\beta\beta}=3$meV then vary
$\langle\eta\rangle$ and $\langle\lambda\rangle$. In Fig.2, the most stringent
constraint is again from 136Xe, although now the constraints are a bit looser
than that of the inverted hierarchy case, the difference turns out to be
within a factor of two. Now the upper bounds of $\langle\lambda\rangle$ is
around $10^{-7}$, and the upper bound for $|\langle\eta\rangle|$ is a bit
larger than $6\times 10^{-10}$. The uncertainty caused by the phase angle is
now much smaller than at the inverted hierarchy case due to the smallness of
$\mu_{\beta\beta}$ providing $C_{m\eta}$ and $C_{m\lambda}$ don’t change.
Also, more stringent constraints on $\langle\eta\rangle$ are due to the fact
that the corresponding amplitude is at a lower order as mentioned above
because the NME for the $R$ term is enhanced making it a LO contribution.
Improved oscillation experiments on the neutrino mass and their hierarchy will
provide tighter constraints on these new physics parameters. With Fig.2, we
can see the uncertainty from NME arising from our choices of Hamiltonians on
the determination of new physics parameters.

##  IV Conclusion and Perspective

In this work, we calculate the NMEs of the neutrinoless double beta decay
process in the LR symmetric model for four nuclei (76Ge, 82Se, 130Te, and
136Xe) using the LSSM approach. We find that the weak magnetism component
plays a much more important role in the $R$ term of the $\eta$ mechanism and
the $q$ term of the $\lambda$ mechanism than expected. We compare the relative
magnitude of each term and discover that if the three new physics parameters
have a similar size, the relativistic term becomes the dominant one while the
$p$-wave effect is suppressed due to the smallness of the $P$ term’s NME.
Based on our nuclear structure calculations, combined with previous phase
space factor calculations, we provide the constraints on the three new physics
parameters under current experimental limits. Future calculations using more
many-body approaches are needed for calibrating the NME.

## Acknowledgement

This work is supported by National Key Research and Development Program of
China (2021YFA1601300).This work is also supported by Chinese Academy of
Sciences Project for Young Scientists in Basic Research (YSBR-099) and the
”from 0 to 1” innovation program ZDBS-LY-SLH014. B.A.B is supported by NSF
Grant PHY-2110365. F.Š. acknowledges support from the Slovak Research and
Development Agency under Contract No. APVV-22-0413, VEGA Grant Agency of the
Slovak Republic under Contract No. 1/0418/22 and by the Czech Science
Foundation (GACČR), project No. 24-10180S. We thank Prof. M. Horoi for the
helps on the TBTD calculation.

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# Group lattices over division rings
A tribute to the renowned Indian Mathematician K.S.S.Nambooripad

P.G. Romeo1 and Alanka Thomas2 1 Professor, Department of Mathematics, Cochin
University of Science and Technology, Kochi, Kerala, INDIA
2 Research Scholar, Department of Mathematics, Cochin University of Science
and Technology, Kochi, Kerala, INDIA.
$romeo_{-}parackal<EMAIL_ADDRESS>
###### Abstract.

Group action on an algebraic structure gives a representation of the group by
automorphisms on the structure. An action of a group on a lattice gives a
group lattice ($G$-lattice). K.S.S. Nambooripad introduced $G$-lattice in 1990
(cf.[6]). This paper revisits Nambooripad’s construction of $G$-lattice.

###### Key words and phrases:

Group, Lattice, Group lattice, Schreier extension

###### 2010 Mathematics Subject Classification:

06B05, 06C05, 18B40

Second author wishes to thank Council of Scientific and Industrial
Research(CSIR) INDIA, for providing financial support under JRF scheme through
CSIR-UGC NET

Prof.K.S.S. Nambooripad (1935-2020) is one of the most celebrated Indian
mathematicians who has made significant contributions to semigroup theory.
Nambooripad started his research at the University of Kerala, India, in 1965.
He submitted a brilliant doctoral dissertation on the structure theory of
regular semigroups to the University of Kerala in 1973 and was awarded the
doctoral degree in 1974. His thesis was published as a Memoir of the American
Mathematical society in 1979 (cf.[7]). Nambooripad’s contributions to
semigroup theory research were immense. He published outstanding research
papers in esteemed journals such as Semigroup Forum, Journal of algebra,
Proceedings of the Edinburgh Mathematical Society, Pacific Journal, and the
like. Professor Nambooripad was a champion of the free software movement and
was instrumental in popularizing Latex software in India. He also played a
vital role in establishing an internationally reputed school on semigroup
theory research in Kerala.

Nambooripad’s significant contributions were on the structure theory of
regular semigroups, and he described the structure of regular semigroups in
two ways. In the first method, he inspected the structure of a regular
semigroup using the notions of biordered sets and inductive groupoids
(cf.[7]). In the second way, his approach was categorical, where he described
the category of principal left and right ideals of regular semigroups, which
he called normal categories, and certain categorical relations called cross-
connections (cf.[8]). By using these normal categories and cross-connections,
he again described the structure of regular semigroups in a more abstract
(categorical) fashion.

Later, at the beginning of the ’90s, Nambooripad studied the action of groups
on lattices. He called such a structure a group lattice. The results regarding
this appeared in the proceedings of the Monash conference on semigroup theory
organized in honor of G.B. Preston in 1990 (cf.[6]).

Now let us recall that an action of a group on a set is a map from $G\times
X\rightarrow X$ which takes $(g,x)$ to $gx$ satisfies the following
properties,

  1. (1)

$ex=x$ for all $x\in X$

  2. (2)

$g(hx)=(gh)x$ for all $g,h\in G$ and $x\in X.$

Then $X$ is called a $G-$set. An action of $G$ on $X$ gives a homomorphism
that assigns a bijection on $X$ to each $g$ in $G$. In general, an action of a
group on an algebraic structure gives a representation of $G$ by automorphisms
of the respective algebraic structure. Nambooripad studied the action of a
group on a lattice, and he called the resulting structure a group lattice.
This paper is motivated by the mathematical work of Nambooripad (cf.[5]). It
is a revisit to the group lattice, introduced by Nambooripad, in which we
obtained some modifications on results and proofs. We dedicate this to the
memory of K.S.S. Nambooripad, who was the thesis adviser of the first author.

We have three sections in this paper. The first section gives the definition
and examples of $G-$lattice. The second section is devoted to discussing
$G$-lattices over arguesian geomodular lattices, the lattice of submodules of
a module $V$ over a division ring $K$ with $rank(V)\geq 3$. If $L\cong L(V)$,
$L$ is said to be coordinatized by $V$. If $G$ acts on an arguesian geomodular
lattice coordinatized by a module $V$ over $K$, it is called a $G-$lattice
over $K$. Nambooripad initiated the study of $G-$lattices over $K$. He
established a one-one correspondence between $G-$lattices over $K$ and
representations of $G$ over $K$ with rank at least $3$. For the third section,
one should have the notion of the Schreier extension. If $G$ and $N$ are two
groups, the Schreier extension of $N$ by $G$ is a group $H$, having $N$ as a
normal subgroup and $H/N\cong G$. Nambooripad used the concept of Schreier
extension (cf.[3]) of $K$ over $G$ while studying $G$-lattices over $K$ and
observed correspondence between Schreier extensions of $K$ by $G$ and
$G-$lattices over $K$. The third section studies Schreier extensions and their
correspondence with group lattices over a division ring.

##  1\. Group lattices

An action of a group on an algebraic structure is crucial as it gives a
representation of the group by automorphisms on the structure, in particular a
group acting on a lattice gives rise to a group lattice.

######  Definition 1.

Let $G$ be a group and $(L,\leq,\wedge,\vee)$ be a lattice. An action of $G$
on $L$ is defined as $gm\in L$ where $g\in G$ and $m\in L$ which satisfies the
following, for each $g,h\in G$ and $m,n\in L$

  1. (1)

$g(hm)=(gh)m$

  2. (2)

$em=m$ where $e$ is the identity in $G$

  3. (3)

$m\leq n$ if and only if $gm\leq gn$

  4. (4)

$g(m\wedge n)=gm\wedge gn$

  5. (5)

$g(m\vee n)=gm\vee gn$.

If a group $G$ acts on $L$, $L$ is called a $G-lattice$. A $group\,lattice$ is
a $G$-lattice for some group $G$. A $G-sublattice$ $L^{\prime}$ of a G-lattice
$L$ is a sublattice of $L$ which is itself a G-lattice under the same product
as that of $L$. For a $G-$sublattice $L^{\prime}$, $gm\in L^{\prime}$ for all
$g\in G$ and $m\in L^{\prime}$ and the properties $(1)$ to $(5)$ of
Definition:1 inherit from $L$.

######  Example 1.

Let $G$ be a group and $L(G)$ be the lattice of all subgroups of $G$. Then
$L(G)$ is a $G-lattice$ under conjugation, defined by
$gH=gHg^{-1}=\\{ghg^{-1}\mid h\in H\\}$ for all $g\in G$ and $H\leqslant G$.

The sublattice $LN(G)$ of all normal subgroups of $G$ is a $G$-sublattice of
$L(G)$. Since, the product $gH=gHg^{-1}=H\in LN(G)$ for all $H\in LN(G)$ and
$g\in G$.

######  Example 2.

If $X$ is a $G-set$, the power set $\mathscr{P}(X)$ is a $G-$lattice with
respect to the product $gA=\\{ga\mid a\in A\\}$ for all $g\in G$ and
$A\subseteq X$.

Before moving to more examples, recall some preliminary results required.

######  Definition 2.

Let $V$ be a module over a division ring $K$. A semilinear transformation $f$
on $V$ is a map $f:V\rightarrow V$ such that,

  1. (1)

$f(u+v)=f(u)+f(v)$ for all $u,v\in V$,

  2. (2)

there is an automorphism $\theta_{f}$ on $K$ such that $f(\alpha
v)=\theta_{f}(\alpha)f(v)$ for all $v\in V$ and $\alpha\in K$.

A bijective semilinear transformation is called a semilinear automorphism. The
set of all semilinear automorphisms on $V$ form a group under composition,
denoted by $SGL(V)$. A linear transformation $f$ on $V$ is a map
$f:V\rightarrow V$ such that,

  1. (1)

$f(u+v)=f(u)+f(v)$ for all $u,v\in V$,

  2. (2)

$f(\alpha v)=\alpha f(v)$ for all $v\in V$ and $\alpha\in K$.

Linear automorphisms are bijective linear transformations, and the set of all
linear automorpisms form a group under composition, denoted by $GL(V)$. Since,
every linear transformation is a semilinear transformation having $\theta_{f}$
as the identity automorphism on $K$, the group $GL(V)$ is a subgroup of
$SGL(V)$.

######  Definition 3.

A semilinear projective representation of a group $G$ is a map
$\rho:G\rightarrow SGL(V)$ such that for all $g,h\in G$, there exist
$\alpha(g,h)\in K^{*}$ such that $\rho(g)\rho(h)=\alpha(g,h)\rho(gh)$.

The map $\rho$ definied above is not a group homomorphism in general, but when
$\alpha(g,h)=1$ for all $g,h\in G$, $\rho$ is a group homomorphism and the
representation is called a semilinear representation of the group $G$.

A projective linear representation of $G$ is a semilinear projective
representation for which, each $\rho(g)$ is a linear automorphism. Here the
codomain of $\rho$ is restricted to the subgroup $GL(V)$. A linear
representation of a group $G$ is a map $\rho:G\rightarrow GL(V)$ such that
$\rho(g)\rho(h)=\rho(gh)$ for all $g,h\in G$. Clearly, a linear representation
is both projective and semilinear.

######  Definition 4.

Two semilinear projective representations $\rho$ and $\tilde{\rho}$ of a group
$G$ are equivalent, if there exists a map $\eta:G\rightarrow K^{*}$ such that
$\tilde{\rho}(g)=\eta(g)\rho(g)$.

For the remaining part of the paper, a representation of $G$ over $K$ means a
semilinear projective representation of $G$ on a module $V$ over $K$.

######  Example 3.

Let $V$ be module over a division ring $K$ and $\rho:G\rightarrow SGL(V)$ be a
representation of $G$ over $K$. Then, the lattice of submodules $L(V)$ of $V$,
is a $G-$lattice with respect to the product defined as,
$gW=\rho(g)W=\\{\rho(g)w\mid w\in W\\}$ for all $g\in G$ and $W\in L(V)$.

The $G$-lattice $L(V)$ defined above is called a semilinear projective
$G$-lattice. Similarly we have linear, projective and semilinear $G$-lattices
on $L(V)$, according to the choice of the representations.

It is well known that every group action provides a representation of the
group and vice versa, we can observe the same for group lattices also. For
that, consider a representation $\rho$ of $G$ by automorphisms on $L$, which
is a homomorphism from $G$ to $AutL$, the group of all lattice automorphisms
on $L$. Define the product $gm=\rho(g)m$ for all $g\in G$ and $m\in L$. Since
$\rho$ is a group homomorphism, the product satisfies properties $(1)$ and
$(2)$ in Definition: 1 and the rest follows from the fact that $\rho(g)$ is a
lattice automorphism for each $g\in G$. Hence $L$ is a $G-$lattice with
respect to this action.

For the converse part, consider a $G-$lattice $L$. For each $g\in G$ define
$\rho(g):L\rightarrow L$ such that $\rho(g)m=gm$. Properties $(3)$ to $(5)$ in
Definition: 1 implies that $\rho(g)$ is a lattice homomorphism.
$\rho(g)m=\rho(g)n$ implies $gm=gn$ and so
$m=(g^{-1}g)m=g^{-1}(gm)=g^{-1}(gn)=(g^{-1}g)n=n$. So $\rho(g)$ is injective.
$\rho(g)$ is surjective, since $g^{-1}m$ is the preimage of $m$ under
$\rho(g)$. Hence $\rho(g)$ is a lattice automorphism for each $g\in G$. From
property $(1)$ of Definition: 1, $\rho(gh)=\rho(g)\rho(h)$ and so the map
$\rho:G\rightarrow AutL$ which takes each $g\in G$ to $\rho(g)$ is a group
homomorphism. ie., $\rho$ is a representation of $G$ by automorphisms on $L$.
Hence there is a one-one correspondence between the $G-$lattices and the
representations of $G$ by lattice automorphisms. Thus we can summarize these
observations as the following theorem.

######  Theorem 1.

Let $G$ be a group and $L$ be a lattice. If $\rho:G\rightarrow AutL$ is a
representation of $G$ by automorphisms on $L$, for each $g\in G$ and $m\in L$,
the product $gm=\rho(g)m$ is an action of $G$ on $L$ so that $L$ is a
$G$-lattice. Conversely if $L$ is a $G$-lattice, the map $\rho:G\rightarrow
AutL$ such that $\rho(g)m=gm$ is a representaton of $G$ by automorphisms on
$L$.

##  2\. G-lattices over a division ring

Next we focus on group lattices over a division ring. Let us first familiarize
the terms and notations used in the sequel. Here a division ring is denoted by
$K$, and we use ${}^{\prime}V^{\prime}$ for a module over $K$. The set $L(V)$
of all submodules of $V$ form a lattice, and $L(V)$ is said to be
coordinatized by $V$. If $rank(V)\geq 3$, $L(V)$ is called an arguesian
geomodular lattice. If a group $G$ acts on an arguesian geomodular lattice, it
is called a $G$-lattice over $K$, more generally a group lattice over the
division ring $K$. A lattice morphism $f:L(V)\rightarrow L(V)$ is
coordinatizable if there exists a semilinear transformation
$f^{\prime}:V\rightarrow V$ such that $fW=f^{\prime}W=\\{f^{\prime}w\mid w\in
W\\}$. We assume, all lattice morphisms between arguesian geomodular lattices
considered here are coordinatizable.

The following theorem establishes a one-one correspondence between
$G$-lattices over $K$ and semilinear projective representations of $G$ over
$K$.

######  Theorem 2.

Let $G$ be a group and $K$ be a division ring. Every representation of $G$
over $K$ determines a $G$-lattice over $K$. Conversely, given a $G$-lattice
$L$ over $K$, there is a representation $\rho$ of $G$ over $K$ that determines
$L$.

###### Proof.

Let $\rho:G\rightarrow SGL(V)$ be a representation of $G$ over $K$. Then
$L(V)$ is a $G-$lattice with respect to the product $gW=\rho(g)W$ for all
$g\in G$ and $W\in L(V)$.
Conversely, $L$ be a $G$-lattice, there exist a module $V$ over $K$ having
$rank(V)\geq 3$ such that $L\cong L(V)$. By Theorem: 1 there exist a
representation $\rho:G\rightarrow AutL$ determines the $G-$lattice $L$. Also
there is a semilinear automorphism $\tilde{\rho}(g):V\rightarrow V$ that
coordinatizes $\rho(g):L\rightarrow L$, and $gW=\rho(g)W=\tilde{\rho}(g)W$ for
all $g\in G$ and $W\in L$. Let $\langle v\rangle$ be the submodule generated
by $v\in V$, then

$\tilde{\rho}(g)\tilde{\rho}(h)\langle v\rangle=\rho(g)\rho(h)\langle
v\rangle=\rho(gh)\langle v\rangle=\tilde{\rho}(gh)\langle v\rangle$

and so
$\langle\tilde{\rho}(g)\tilde{\rho}(h)v\rangle=\langle\tilde{\rho}(gh)v\rangle$.
Then there exist a $\alpha_{v}\in K^{*}$ such that
$\tilde{\rho}(g)\tilde{\rho}(h)v=\alpha_{v}\tilde{\rho}(gh)v$. Let $w\in V$ be
linearly independant with $v$. Since $\tilde{\rho}(gh)$ is an automorphism,
$\tilde{\rho}(gh)v$ and $\tilde{\rho}(gh)w$ are also linearly independant.
Consider

$\begin{array}[]{lcl}\mbox{$\alpha_{v+w}\tilde{\rho}(gh)v+\alpha_{v+w}\tilde{\rho}(gh)w$}&=&\mbox{$\alpha_{v+w}\tilde{\rho}(gh)(v+w)=\tilde{\rho}(g)\tilde{\rho}(h)(v+w)$}\\\
&=&\mbox{$\tilde{\rho}(g)\tilde{\rho}(h)v+\tilde{\rho}(g)\tilde{\rho}(h)w$}\\\
&=&\mbox{$\alpha_{v}\tilde{\rho}(gh)v+\alpha_{w}\tilde{\rho}(gh)w$}\end{array}$

then $\alpha_{v}=\alpha_{v+w}=\alpha_{w}.$ Similarly for any scalar $\beta$
and $w$ linearly independant with $\beta v$ we get $\alpha_{v}=\alpha_{\beta
v}=\alpha_{w}$. So the scalar $\alpha=\alpha_{v}$ is independant of the choice
of $v$. Hence $\tilde{\rho}(g)\tilde{\rho}(h)=\alpha\tilde{\rho}(gh)$ and
$\tilde{\rho}:G\rightarrow SGL(V)$ is a representation of $G$ over $K$. ∎

######  Theorem 3.

If $\rho$ and $\tilde{\rho}$ are two representations of $G$ over $K$, then
they determine the same $G-$lattice if and only if they are equivalent.

###### Proof.

If $\rho$ and $\tilde{\rho}$ are equivalent, there exist $\eta:G\rightarrow
K^{*}$ such that $\tilde{\rho}(g)=\eta(g)\rho(g)$. Then
$\tilde{\rho}(g)W=\eta(g)\rho(g)W=\rho(g)W$ for all $g\in G$ and $W\in L(V)$
and so $\rho$ and $\tilde{\rho}$ induce the same $G-$lattice.

Conversely if $\rho$ and $\tilde{\rho}$ induce the same $G-$lattice, then for
any $v\in V$, $\langle\rho(g)v\rangle=\rho(g)\langle
v\rangle=\tilde{\rho}(g)\langle v\rangle=\langle\tilde{\rho}(g)v\rangle$.
Similarly as in the proof of Theorem: 2, there exist $\alpha_{g}\in K^{*}$
independant of $v$ such that $\tilde{\rho}(g)v=\alpha_{g}\rho(g)v$ and so
$\tilde{\rho}(g)=\alpha_{g}\rho(g)$. $\eta:G\rightarrow K^{*}$ such that
$\eta(g)=\alpha_{g}$ satisfies the equation $\tilde{\rho}(g)=\eta(g)\rho(g)$
and so $\rho$ and $\tilde{\rho}$ are equivalent. ∎

##  3\. Schreier extension and associated Group lattices

Let $N$ and $G$ be groups. A Schreier extension of $N$ by $G$ is a group $H$
having $N$ as a normal subgroup and $H/N\cong G$ (cf.[3]). Let $H$ be a group
having $N$ as a normal subgroup and $H/N\cong G$. Then $H=\bigcup\limits_{g\in
G}N\overline{g}=\\{a\overline{g}\mid a\in N\text{ and }g\in G\\}$, where
$\\{\overline{g}\mid g\in G\\}$ is the set of all representatives from each
coset of $N$. For each $g\in G$ the map
$b\rightarrow\overline{g}b\overline{g}^{-1}$ is a group automorphism in $N$
and let it be denoted by $\chi(g)$. Also since $\overline{gh}$ is the
representative of the coset containing $\overline{g}\overline{h}$ there exist
$[g,h]\in N$ such that $\overline{g}\overline{h}=[g,h]\overline{gh}$. Then
$(a\overline{g})(b\overline{h})=a\overline{g}b\overline{g}^{-1}\overline{g}\overline{h}=a\chi(g)(b)[g,h]\overline{gh}$
and $H$ is a group with respect to this binary operation. Hence we have
functions,

  1. (1)

$\chi:G\rightarrow AutN$ defined by
$\chi(g)(a)=\overline{g}a\overline{g}^{-1}$

  2. (2)

$[_{-},_{-}]:G\times G\rightarrow N$ such that
$\overline{g}\overline{h}=[g,h]\overline{gh}$.

and we can see that $\chi$ and $[_{-},_{-}]$ has the following properties

  * $E1)$

$\chi(g)\chi(h)=[g,h]\chi(gh)[g,h]^{-1}$

  * $E2)$

$[g,h][gh,k]=\chi(g)([h,k])[g,hk]$

  * $E3)$

$[1,1]=1.$

In general, if $\chi:G\rightarrow AutN$ and $[_{-},_{-}]:G\times G\rightarrow
N$ are function satisfying above properties, $H=\\{a\overline{g}\mid a\in
N\text{ and }g\in G\\}$ is a group with respect to the binary operation
$(a\overline{g})(b\overline{h})=a\chi(g)(b)[g,h]\overline{gh}$. This is the
construction of Schreier extension of $N$ by $G$. The pair of functions
$(\chi,[_{-},_{-}])$ called a factor system for $(N,G)$ and the extension is
denoted by $H(\chi,[_{-},_{-}])$. Factor systems $(\chi,[_{-},_{-}])$ and
$(\chi^{\prime},[_{-},_{-}]^{\prime})$ are equivalent, if there exists a map
$\mu:G\rightarrow N$ such that

  1. $E4)$

$\chi^{\prime}(g)=\mu(g)^{-1}\chi(g)\mu(g)$

  2. $E5)$

$[g,h]\mu(gh)=\mu(g)\chi^{\prime}(g)(\mu(h))[g,h]^{\prime}$

  3. $E6)$

$\mu(1)=1$.

If $(\chi,[_{-},_{-}])$ and $(\chi^{\prime},[_{-},_{-}]^{\prime})$ are
equivalent, the map $a\overline{g}\rightarrow a\mu(g)\tilde{g}$ from
$H(\chi,[_{-},_{-}])$ to $H(\chi^{\prime},[_{-},_{-}]^{\prime})$ is an
isomorphism. Two factor systems are equivalent if and only if corresponding
extensions are isomorphic.

######  Definition 5.

A central extension of $N$ by $G$ is an extension $H(\chi,[_{-},_{-}])$ for
which, $[g,h]$ are in the center of $N$, for each $g,h$ in G.

Using $(E1)$, it is seen that the map $\chi:G\rightarrow AutN$ is a
homomorphism for a central extension. If $N$ is an abelian group, every
Schreier extension of $N$ is a central extension.

If $\chi(g)=I_{N}$ for all $g\in G$, the binary operation in $H$ reduces to
$(a\overline{g})(b\overline{h})=ab[g,h]\overline{gh}$ and $H$ is called a
projective extension of $N$ by $G$. If $[g,h]=1$ for all $g,h\in G$, then $H$
is the semidirect product of $N$ and $G$, this type of extensions are called
split extensions or semidirect products. It is easy to see that projective
extensions and semidirect products are central extensions. The direct product
of $N$ and $G$ is the Schreier extension of $N$ by $G$ which is both
projective and split.

Here we deal with Schreier extension of $K^{*}$ by $G$, where $K^{*}$ is the
group of nonzero elements of a division ring under multiplication. We shall
say that $H(\chi,[_{-},_{-}])$ is an extension of $K$ by $G$ and that
$(\chi,[_{-},_{-}])$ is a factor system for $(K,G)$, if it is a factor system
for $(K^{*},G)$ with the property that $\chi(g)\in AutK$ for each $g\in G$.
ie., every representation of $G$ over $K$ gives a Schreier extension of $K$ by
$G$. For a semilinear projective representation $\rho$, each $\rho(g)$ is a
semilinear transformation. So, there exist automorphism $\theta_{\rho(g)}$ on
$K$ such that $\rho(g)(\alpha v)=\theta_{\rho(g)}(\alpha)\rho(g)(v)$. For
brevity, let it be denoted by $\chi(g)$. Since the representation is
projective, for each $g,h\in G$ there exist $[g,h]\in K^{*}$ such that
$\rho(g)\rho(h)=[g,h]\rho(gh)$. We will prove in Proposition: 1 that this
$\chi$ and $[_{-},_{-}]$ together form a factor system for an extension of $K$
by $G$.

######  Proposition 1.

Let $\rho$ be a representation of $G$ over $K$. Then $(\chi,[_{-},_{-}])$ is a
factor system for $(K,G)$.

###### Proof.

Given $\rho$ is a representation of $G$ over $K$. For each $g\in G$ there
exist an automorphism $\chi(g)$ of $K$ such that $\rho(g)(\alpha
v)=\chi(g)(\alpha)\rho(g)(v)$ and $\rho(g)\rho(h)=[g,h]\rho(gh)$ for all
$g,h\in G$. $(\chi,[_{-},_{-}])$ is a factor system if it satisfies properties
$(E1)-(E3)$.

$\begin{array}[]{lcl}\mbox{$\chi(g)(\chi(h)(\alpha))\rho(g)\rho(h)(v)$}&=&\mbox{$\rho(g)\rho(h)(\alpha
v)=[g,h]\chi(gh)(\alpha)\rho(gh)(v)$}\\\
&=&\mbox{$[g,h]\chi(gh)(\alpha)[g,h]^{-1}[g,h]\rho(gh)(v)$}\\\
&=&\mbox{$[g,h]\chi(gh)(\alpha)[g,h]^{-1}\rho(g)\rho(h)(v)$}\end{array}$

Hence $\chi(g)\chi(h)=[g,h]\chi(gh)[g,h]^{-1}$ for all $g,h\in G$.

$\begin{array}[]{lcl}\mbox{$[g,h][gh,k]\rho(ghk)(v)$}&=&\mbox{$[g,h]\rho(gh)(\rho(k)(v)=\big{(}\rho(g)\rho(h)\big{)}\rho(k)(v)$}\\\
&=&\mbox{$\rho(g)\big{(}\rho(h)\rho(k)\big{)}(v)=\rho(g)\big{(}[h,k]\rho(hk)\big{)}(v)$}\\\
&=&\mbox{$\chi(g)([h,k])\rho(g)\rho(hk)(v)$}\\\
&=&\mbox{$\chi(g)([h,k])[g,hk]\rho(ghk)(v)$}.\end{array}$

Hence $[g,h][gh,k]\rho(ghk)(v)=[g,hk]\chi(g)[h,k]\rho(ghk)(v)$ for $g,h,k\in
G$ and $v\in V$ and so $[g,h][gh,k]=[g,hk]\chi(g)[h,k]$. It is clear that
$[1,1]=1$. Hence $(\chi,[_{-},_{-}])$ is a factor system for $(K,G)$. ∎

A representation $\rho$ of $G$ over $K$ is said to be associated with a factor
system $(\chi,[_{-},_{-}])$, if $\chi(g)=\theta_{\rho(g)}$ and
$\rho(g)\rho(h)=[g,h]\rho(gh)$ for each $g,h\in G$. A representation is
projective if and only if the Schreier extension associated with it is
projective. Similarly, a representation is semilinear if and only if the
Schreier extension associated with it is a semidirect product. So, direct
product $K\times G$ is the Schreier extension of $K$ by $G$ associated with a
linear representation of $G$ over $K$. Equivalent representations always give
equivalent factor systems. Proposition: 2 deals with this observation.

######  Proposition 2.

Let $(\chi,[_{-},_{-}])$ and $(\chi^{\prime},[_{-},_{-}]^{\prime})$ be factor
systems for $(K,G)$ and $\rho$ be the representation of $G$ over $K$
associated with $(\chi,[_{-},_{-}])$. Then $(\chi,[_{-},_{-}])$ is equivalent
to $(\chi^{\prime},[_{-},_{-}]^{\prime})$ if and only if there is a
representation of $\tilde{\rho}$ associated with
$(\chi^{\prime},[_{-},_{-}]^{\prime})$ which is equivalent to $\rho$.

###### Proof.

Let $\rho$ and $\tilde{\rho}$ be representation of $G$ over $K$ associated
with factor systems $(\chi,[_{-},_{-}])$ and
$(\chi^{\prime},[_{-},_{-}]^{\prime})$ for $(K,G)$ respectively. If $\rho$ is
equivalent to $\tilde{\rho}$, there exist a map $\mu:G\rightarrow K^{*}$ such
that $\rho(g)=\mu(g)\tilde{\rho}(g)$. Then $\mu$ is an equivalence of the
factor system $(\chi,[_{-},_{-}])$ with
$(\chi^{\prime},[_{-},_{-}]^{\prime})$; for, consider,

$\mu(g)\chi^{\prime}(g)(\alpha)\tilde{\rho}(g)(v)=\mu(g)\tilde{\rho}(g)(\alpha
v)=\rho(g)(\alpha v)=\chi(g)(\alpha)\mu(g)\tilde{\rho}(g)(v)$.

So $\mu(g)\chi^{\prime}(g)=\chi(g)\mu(g)$ for all $g\in G$ and this proves
$(E4)$. Now consider

$\begin{array}[]{lcl}\mbox{$[g,h]\mu(gh)\tilde{\rho}(gh)(v)$}&=&\mbox{$\rho(g)\rho(h)(v)=\mu(g)\tilde{\rho}(g)\mu(h)\tilde{\rho}(h)(v)$}\\\
&=&\mbox{$\mu(g)\chi^{\prime}(g)(\mu(h))[g,h]^{\prime}\tilde{\rho}(gh)(v)$}\end{array}$

Hence $[g,h]\mu(gh)=\mu(g)\chi^{\prime}(g)(\mu(h))[g,h]^{\prime}$ and this
proves $(E5)$. $\mu$ satisfies $(E4)-(E6)$ and so $\mu$ is an equivalence
between the factor systems $(\chi,[_{-},_{-}])$ and
$(\chi^{\prime},[_{-},_{-}]^{\prime})$.
Conversely let $(\chi,[_{-},_{-}])$ and $(\chi^{\prime},[_{-},_{-}]^{\prime})$
be equivalent and $\rho$ be a representation of $G$ over $K$ associated with
$(\chi,[_{-},_{-}])$. There exist an equivalence $\mu:G\rightarrow K^{*}$ from
$(\chi^{\prime},[_{-},_{-}]^{\prime})$ and $(\chi,[_{-},_{-}])$ satisfying
$(E4)-(E6)$. For each $g\in G$ define $\tilde{\rho}(g)=\mu(g)\rho(g)$, then
$\tilde{\rho}$ is a representation of $G$ over $K$. Since each $g\in G$,
$\tilde{\rho}(g)$ preserves addition as in the case of $\rho(g)$. Now using
$(E4)$ we get

$\tilde{\rho}(g)(\alpha
v)=\mu(g)\chi(g)(\alpha)\rho(g)(v)=\chi^{\prime}(g)(\alpha)\mu(g)\rho(g)(v)=\chi^{\prime}(g)(\alpha)\tilde{\rho}(g)(v)$

and so $\tilde{\rho}(g)$ is a semilinear transformation for each $g\in G$.
Using the bijectivity of $\rho(g)$, $\tilde{\rho}(g)$ is also bijective and so
$\tilde{\rho}(g)$ is a semilinear automorphism for each $g\in G$.

$\begin{array}[]{lcl}\mbox{$\tilde{\rho}(g)\tilde{\rho}(h)(v)$}&=&\mbox{$\mu(g)\rho(g)\mu(h)\rho(h)(v)=\mu(g)\chi(g)(\mu(h))[g,h]\rho(gh)(v)$}\\\
&=&\mbox{$[g,h]^{\prime}\mu(gh)\rho(gh)(v)=[g,h]^{\prime}\tilde{\rho}(gh)(v)$}\end{array}$

Hence $\tilde{\rho}(g)\tilde{\rho}(h)=[g,h]^{\prime}\tilde{\rho}(gh)$ and
$\tilde{\rho}$ is a representation of $G$ over $K$. It is clear that
$\tilde{\rho}$ is associated with the factor system
$(\chi^{\prime},[_{-},_{-}])$. Since $\mu:G\rightarrow K^{*}$ is a map such
that $\tilde{\rho}(g)=\mu(g)\rho(g)$ for all $g\in G$, $\rho$ and
$\tilde{\rho}$ are equivalent representations of $G$ over $K$. Hence the
proof. ∎

Theorem: 2 and Proposition: 1 together tells us that every $G$-lattice over
$K$ gives us a Schreier extension of $K$ by $G$. The converse is also true. We
need to have the following definition of a twisted group ring to prove that.

######  Definition 6.

Let $H=H(\chi,[_{-},_{-}])$ be an extension of $K$ by $G$. The twisted group
ring $K(G;H)$ is the free module generated by the set $\\{\bar{g}\mid g\in
G\\}$ over $K$.

For $u=\sum\limits_{g\in G}a_{g}\bar{g}$ and $v=\sum\limits_{h\in
G}b_{h}\bar{h}$ in $K(G;H)$ and $c\in K$ we have,

$u+v=\sum\limits_{g\in G}(a_{g}+b_{g})\bar{g}\qquad cu=\sum\limits_{g\in
G}ca_{g}\bar{g}$

$uv=\sum\limits_{k\in G}c_{k}\bar{k}$ where
$c_{k}=\sum\limits_{gh=k}a_{g}\chi(g)(b_{h})[g,h]$

The above product is obtained by distributively extending the product in $H$.
$K(G;H)$ is an associative ring and need not be an algebra in general. If
factor systems $(\chi,[_{-},_{-}])$ and $(\chi^{\prime},[_{-},_{-}]^{\prime})$
are equivalent, $H(\chi,[_{-},_{-}])$ and
$H(\chi^{\prime},[_{-},_{-}]^{\prime})$ are isomorphic and so $K(G;H)$ and
$K(G;H^{\prime})$ are also isomorphic. Hence $K(G;H)$ is uniquely determined
by an extension $H$.

######  Theorem 4.

Let $K$ be a division ring and $G$ a group. Every $G-$lattice $L$ over $K$
determines a unique extension of $K$ by $G$. Conversely, if $H$ is an
extension of $K$ by $G$, there is a $G-$lattice $L$ over $K$ such that $H$ is
determined by $L$.

###### Proof.

Corresponding to each $G-$lattice over $K$, we have a representation of $G$
over $K$ and vice versa. So the first part is clear. Let
$H=H(\chi,[_{-},_{-}])$ be an extension of $K$ by $G$. For each $g\in G$
define $\rho(g):K(G;H)\rightarrow K(G;H)$ such that for each $v\in V$,
$\rho(g)(v)=\overline{g}v$ where the product on the right is the product in
$K(G;H)$. For $v\in V$ and $\alpha\in K$, $\rho(g)(\alpha
v)=\overline{g}(\alpha
v)=\chi(g)(\alpha)\overline{g}v=\chi(g)(\alpha)\rho(g)(v).$ Hence $\rho(g)$ is
a semilinear automorphism having $\theta_{\rho(g)}=\chi(g)$. Also

$\rho(g)\rho(h)(v)=\overline{g}(\overline{h}(v))=\overline{g}\overline{h}(v)=[g,h]\overline{gh}(v)=[g,h]\rho(gh)(v)$

and so $\rho:G\rightarrow SGL(V)$ is a representation associated with the
factor system $(\chi,[_{-},_{-}])$. The lattice $L(K(G;H))$ is a $G-$lattice
with the action induced by $\overline{\rho}$. It is clear that the extension
determined by $L$ is the given $H$. ∎

The following example illustrate Theorem: 4 for the $C_{3}$-lattice,
$L(\mathbb{R}^{3})$.

######  Example 4.

Consider $\mathbb{R}^{3}$ and the the lattice $L(\mathbb{R}^{3})$ of its
subspaces. Let $C_{3}=\\{1,a,a^{2}\\}$ be the cyclic group of order $3$
generated by $a$. $L(\mathbb{R}^{3})$ is a $C_{3}$-lattice with respect to the
action defined as follows.

  1. (1)

$1W=W$ for all $W\in L(\mathbb{R}^{3}).$

  2. (2)

$aW$ is the subspace of $\mathbb{R}^{3}$ obtained by shifting the components
of each vector in $W$ one position to the right. For example,
$W=\\{(x,y,z)\in\mathbb{R}^{3}\mid x+y=z\\}$ is a subspace of $\mathbb{R}^{3}$
and $aW=\\{(x,y,z)\in\mathbb{R}^{3}\mid x+z=y\\}$.

  3. (3)

$a^{2}W$ is the subspace of $\mathbb{R}^{3}$ obtained by shifting the
components of each vector in $W$ two positions to the right. We have
$a^{2}W=\\{(x,y,z)\in\mathbb{R}^{3}\mid y+z=x\\}$ for the above subspace.

Let $v=(x,y,z)\in\mathbb{R}^{3}$ and $\langle v\rangle=\\{(kx,ky,kz)\mid
k\in\mathbb{R}\\}$ be the subspace spanned by $v$. Then $1\langle
v\rangle=\langle v\rangle=\langle(x,y,z)\rangle$,$a\langle
v\rangle=\\{(kz,kx,ky)\mid k\in\mathbb{R}\\}=\langle(z,x,y)\rangle$ and
$a^{2}\langle v\rangle=\\{(ky,kz,kx)\mid
k\in\mathbb{R}\\}=\langle(y,z,x)\rangle$. Define
$\rho(1)(x,y,z)=(x,y,x),\rho(a)(x,y,z)=(z,x,y)$ and $\rho(a^{2})(x,y,z)$
=$(y,z,x)$. It is easy to see that $\rho(g)$ is an automorphism in
$\mathbb{R}^{3}$ for each $g\in C_{3}$ and $\rho:C_{3}\rightarrow
Aut\mathbb{R}^{3}$ which takes each $g\in C_{3}$ to $\rho(g)$ defined above is
a group homomorphism. Hence $\rho$ is a linear representation of $C_{3}$ over
$\mathbb{R}$ and is the representation corresponding to the $C_{3}$-lattice
$L(\mathbb{R}^{3})$.
Now we discuss the Schreier extension associated with $\rho$. Since $\rho$ is
a linear representation, $\chi(g)=\theta_{\rho(g)}=I_{\mathbb{R}^{*}}$ and
$[g,h]=1$ for all $g,h\in C_{3}$. So Schreier extension of $\mathbb{R}^{*}$ by
$C_{3}$ is $H=\\{k\bar{1}\mid k\in\mathbb{R}^{*}\\}\cup\\{k\bar{a}\mid
k\in\mathbb{R}^{*}\\}\cup\\{k\bar{a^{2}}\mid k\in\mathbb{R}^{*}\\}$ and the
binary operation in $H$ is given by
$k\bar{g}k^{\prime}\bar{h}=k\chi(g)(k^{\prime})[g,h]\bar{gh}=kk^{\prime}\overline{gh}$.
Define a map $\phi:\mathbb{R}^{*}\times C_{3}\rightarrow H$ such that
$\phi(k,g)=k\bar{g}$. $\phi$ is a group homomorphism since
$\phi(k,g)\phi(k^{\prime},h)=k\bar{g}k^{\prime}\bar{h}=kk^{\prime}\overline{gh}=\phi(kk^{\prime},gh)$.
$\phi(k,g)=\phi(k^{\prime},h)$ means that $k\bar{g}=k^{\prime}\bar{h}$, so
$k=k^{\prime}$ and $g=h$ which implies $(k,g)=(k^{\prime},h)$ and hence $\phi$
is injective. $\phi$ is surjective since for $k\bar{g}\in H$ there is always
$(k,g)\in\mathbb{R}^{*}\times C_{3}$ such that $\phi(k,g)=k\bar{g}$. Hence $H$
is isomorphic to $\mathbb{R}^{*}\times C_{3}$.
The twisted group ring $\mathbb{R}(C_{3},\mathbb{R}^{*}\times
C_{3})=\\{x\bar{1}+y\bar{a}+z\bar{a^{2}}\\}$ is a vector space with respect to
the addition and scalar multiplication defined by

$\big{(}x_{1}\bar{1}+y_{1}\bar{a}+z_{1}\bar{a^{2}}\big{)}+\big{(}x_{2}\bar{1}+y_{2}\bar{a}+z_{2}\bar{a^{2}}\big{)}=(x_{1}+x_{2})\bar{1}+(y_{1}+y_{2})\bar{a}+(z_{1}+z_{2})\bar{a^{2}}$
$k\big{(}x\bar{1}+y\bar{a}+z\bar{a^{2}}\big{)}=kx\bar{1}+ky\bar{a}+kz\bar{a^{2}}$

Then $\theta:\mathbb{R}^{3}\rightarrow\mathbb{R}(C_{3},\mathbb{R}^{*}\times
C_{3})$ given by $\theta(x,y,z)=x\bar{1}+y\bar{a}+z\bar{a^{2}}$ is an
isomorphism. So $L(\mathbb{R}^{3})\cong
L(\mathbb{R}(C_{3},\mathbb{R}^{*}\times C_{3}))$. Note that
$L(\mathbb{R}(C_{3},\mathbb{R}^{*}\times C_{3}))$ is a $C_{3}$-lattice with
respect to the product defined as,

  1. (1)

$1W=W$

  2. (2)

$aW=\\{\bar{a}(x\bar{1}+y\bar{a}+z\bar{a^{2}})\mid
x\bar{1}+y\bar{a}+z\bar{a^{2}}\in W\\}=\\{z\bar{1}+x\bar{a}+y\bar{a^{2}}\mid
x\bar{1}+y\bar{a}+z\bar{a^{2}}\in W\\}$

  3. (3)

$a^{2}W=\\{\bar{a^{2}}(x\bar{1}+y\bar{a}+z\bar{a^{2}})\mid
x\bar{1}+y\bar{a}+z\bar{a^{2}}\in W\\}=\\{y\bar{1}+z\bar{a}+x\bar{a^{2}}\mid
x\bar{1}+y\bar{a}+z\bar{a^{2}}\in W\\}$

Hence $C_{3}$-lattices $L(\mathbb{R}(C_{3},\mathbb{R}^{*}\times C_{3}))$ and
$L(\mathbb{R}^{3})$ are isomorphic.

######  Remark 1.

$V$ need not be isomorphic with $K(G;H)$ in general. If we replace $C_{3}$ by
$S_{3}$ in the above example and define an action of $S_{3}$ on
$L(\mathbb{R}^{3})$ as above, then $\mathbb{R}(S_{3};\mathbb{R}^{*}\times
S_{3})$ is isomorphic with $\mathbb{R}^{6}$ but not with $\mathbb{R}^{3}$.

######  Theorem 5.

Let $G$ be a group and $K$ be a division ring. If $H$ is the extension
determined by the $G-$lattice $L$ over $K$,

  1. (1)

$H$ is a projective extension if and only if $L$ is a projective $G$-lattice

  2. (2)

$H$ is a semidirect product if and only if $L$ is a semilinear $G$-lattice

  3. (3)

$L$ is a linear $G$-lattice if and only if $H$ is a linear extension and so
$H\cong K^{*}\times G$.

###### Proof.

We can see that $L$ is a projective $G$-lattice if and only if corresponding
representation $\rho$ of $G$ over $K$ is projective, that is if and only if
the extension associated to $\rho$ is projective. Similarly, $L$ is a
semilinear $G$-lattice if and only if corresponding representation $\rho$ of
$G$ over $K$ is semilinear, that is if and only if the extension associated to
$\rho$ is a semidirect product. If $H$ is the direct product $K^{*}\times G$,
$H$ is a projective as well as a split extension of $K$ by $G$, and the
representation $\rho$ associated with $H$ is both projective and semilinear,
which is the linear representation of $G$ over $K$. So the $G$-lattice $L$
induced by $\rho$ is the linear $G$-lattice. ∎

The following theorem gives a necessary and sufficient condition so that the
twisted group ring is an algebra.

######  Theorem 6.

Let $L$ be a $G-$lattice over $K$ and $H$ be the extension determined by $L$.
Then, the twisted group ring $K(G;H)$ is an algebra over $K$ if and only if
$L$ is a projective $G-$lattice over the field $K$.

###### Proof.

Assume that $L$ is a projective $G-$lattice over the field $K$. To show that
$K(G;H)$ an algebra it remains to prove that $(au)v=u(av)=a(uv)$ for all $a\in
K$ and $u,v\in K(G;H)$. Since the product in $K(G;H)$ is obtained by
distributively extending the product in $H$, it is enough to check the above
property for the elements in the basis $\\{\overline{g}\mid g\in G\\}$ of
$K(G;H)$.
$\overline{g}(a\overline{h})=a(\overline{g}\overline{h})=(a\overline{g})\overline{h}$
since $H$ is a projective extension, this proves that $K(G;H)$ is a
$K-$algebra.
Conversely let $K(G;H)$ be an algebra. Then
$ba\overline{1}=(b\overline{1})(a\overline{1})=a(b\overline{1})=ab\overline{1}$
which implies that $ab=ba$ for all $a,b\in K^{*}$ and so $K$ is a field.
$\chi(g)(a)\overline{g}=(a\overline{1})\overline{g}=a\overline{g}$ and so
$\chi(g)(a)=a$ for all $a\in K^{*}$ and $g\in G$. Hence $\chi(g)=I_{K}$ for
each $g\in G$ and so $H$ is a projective extension of $K$ by $G$. ∎

Since $\rho:G\rightarrow SGL(V)$ is a semilinear projective representation,
$gv=\rho(g)(v)$ is not an action of $G$ on $V$ in general. However, it may
turn out to be an action by distributively extending the multiplication.

######  Proposition 3.

Let $G$ be a group, $K$ a division ring and $H$ be a Schreier extension of $K$
by $G$. If the representation $\rho:G\rightarrow SGL(V)$ of $G$ is associated
with the extension $H$, then $V$ is a $K(G;H)$ module. Conversely let $M$ be a
$K(G;H)$ module. Then $M$ is a module over $K$, for each $g\in G$ and $v\in
V$, $\rho(g)(v)=\overline{g}v$ defines a semilinear automorphism $\rho(g)$ on
$M$ and $\rho:G\rightarrow SGL(V)$ is a representation which determines the
extension $H$.

###### Proof.

Assume that $\rho:G\rightarrow SGL(V)$ is a representation of $G$ over $K$.

For $s=\sum\limits_{g\in G}a_{g}\overline{g}$ and $v\in V$ define
$sv=\sum\limits_{g\in G}a_{g}\rho(g)(v)$. Then $V$ is a $K(G;H)$-module with
respect to this product. Consider $s=\sum\limits_{g\in G}a_{g}\overline{g}$,
$t=\sum\limits_{g\in G}b_{g}\overline{g}$ in $K(G;H)$ and $u,v\in V$. Using
the semilinearity of $\rho(g)$ and the $K-$module structure on $V$ we get,

  1. (1)

$s(u+v)=\sum\limits_{g\in G}a_{g}\big{(}\rho(g)(u)+\rho(g)(v)\big{)}=su+sv$.

  2. (2)

$(s+t)v=\sum\limits_{g\in G}a_{g}\rho(g)(v)+b_{g}\rho(g)(v)=sv+tv$.

  3. (3)

$s(tv)=\sum\limits_{g\in G}a_{g}\rho(g)\bigg{(}\sum\limits_{h\in
G}b_{h}\rho(h)(v)\bigg{)}=\sum\limits_{g\in G}\sum\limits_{h\in
G}a_{g}\chi(g)(b_{h})\rho(g)\rho(h)(v)$
=$\sum\limits_{k\in
G}\bigg{(}\sum\limits_{gh=k}a_{g}\chi(g)(b_{h})[g,h]\bigg{)}\rho(k)(v)=(st)v$.

  4. (4)

$\bar{1}v=\rho(1)(v)=v$.

  5. (5)

$(bs)v=\sum\limits_{g\in G}ba_{g}\rho(g)(v)=b\sum\limits_{g\in
G}a_{g}\rho(g)(v)=b(sv)$.

Converse follows. ∎

Thus we summerize that given an extension $H$ of $G$ by $K$, every
$K(G;H)$-module $V$ induce a $G-$lattice on $L(V)$ and conversely, every
$G-$lattice induces an extension $H$ and a $K(G;H)$-module. Moreover, when $H$
and $H^{\prime}$ are isomorphic extensions, $K(G;H)$ and $K(G;H^{\prime})$ are
isomorphic, and they induce the same action on the module $V$ corresponding to
the $G-$lattice $L(V)$.

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