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##  Appendix A A Note on the Definition of Insdel LDCs

Recall the definition of Insdel LDCs from Definition 1.

See 1

In our definition of Insdel LDCs we assume that the decoder $\mathsf{Dec}$ is
directly given $m^{\prime}$, the length of the corrupted codeword $y$.
Arguably it may be more reasonable to require $\mathsf{Dec}$ to recover
$x_{i}$ without a priori knowledge of the length $m^{\prime}$. This question
does not arise in the definition of Hamming LDCs as the length of the
corrupted codeword is fixed. If we do not give the Insdel decoder the length
$m^{\prime}$ then $\mathsf{Dec}$ may query for an out of range index
$j>m^{\prime}$ and we would need to define how such queries are handled e.g.,
if $\mathsf{Dec}$ queries for $y[j]$ for $j>m^{\prime}$ we might return $\bot$
to indicate that the query is out of range. In this case the decoder could
always recover $m^{\prime}$ after $O(\log m)$ queries by using binary search
to find the maximum $j$ such that $y[j]\neq\bot$.

We stress that giving the local decoder access to $m^{\prime}$ can only help
the decoder. If the information is not helpful the decoder can always chose to
ignore the extra information $m^{\prime}$. Since our focus is on proving lower
bounds we chose to give $\mathsf{Dec}$ access to the length $m^{\prime}$ which
only makes the lower bounds stronger.

Another modification of Definition 1 might allow for the Insdel codewords
$C(x)$ to have variable length i.e., $C\mathrel{\mathop{\mathchar
58\relax}}\Sigma^{n}\rightarrow\Sigma^{\leq m}$. Now if we require that this
insdel distance between $C(x)$ and the corrupted codeword $y$ is at most
$2\delta|C(x)|$ and if we additionally give $\mathsf{Dec}$ access to the
length $m^{\prime}=|y|$ of of the corrupted codeword then the encoding
algorithm can “cheat” and use codeword length to encode $x$. For example, when
$\Sigma=\\{0,1\\}$ and $\delta<1/6$ we could define a
$(q=0,\delta,\epsilon=\frac{1}{2})$-insdel LDC $C\mathrel{\mathop{\mathchar
58\relax}}\\{0,1\\}^{n}\rightarrow\\{0,1\\}^{\leq m}$ with $m=2^{2^{n}}$ as
follows: define a bijective mapping $\mathtt{Int}\mathrel{\mathop{\mathchar
58\relax}}\\{0,1\\}^{n}\rightarrow\\{0,\ldots,2^{n}-1\\}$ in the natural way
and then set $C(x)=1^{2^{\mathtt{Int}(x)+1}}$ i.e., $1$ repeated
$2^{{\mathtt{Int}(x)+1}}$ times. If $\delta<1/6$ then $m^{\prime}=|y|$ must
lie in the range

$\frac{2}{3}2^{{\mathtt{Int}(x)+1}}<m^{\prime}<\frac{4}{3}2^{{\mathtt{Int}(x)+1}}=\frac{2}{3}2^{\mathtt{Int}(x)+2}$

since we require that the insdel distance between $C(x)$ and $y$ is at most
$2\delta|C(x)|<2^{\mathtt{Int}(x)+1}/3$. This allows the local insdel decoder
to recover the entire message $x$ (and any particular bit $x_{i}$) from
$m^{\prime}$ without any queries to the corrupted codeword $y$. In particular,
the decoder could find the unique integer $k$ such that
$\frac{2}{3}2^{2^{k+1}}<m^{\prime}<\frac{2}{3}2^{k+2}$ and then recover
$x=\mathtt{Int}^{-1}(k)$.

Arguably the above construction “cheats” by encoding the message $x$ in the
length of the codeword (unary) and allowing the decoder to directly learn the
length of the (corrupted) codeword. If we do not allow the decoder to directly
learn the length $m^{\prime}$ of the corrupted message then there are no known
constructions of $(q=2,\delta,\epsilon)$-insdel LDCs for any constants
$\delta,\epsilon>0$ — for any information rate $n/m$.

##  Appendix B Definition of Private-Key (Insdel) LDCs

In the private-key setting the encoder $C(x;R)$ and decoder
$\mathsf{Dec}(y,m^{\prime},i;R)$ are given access to shared (secret) set of
random coins $R$ which is not given to the channel $\mathcal{A}$. Fixing $x$
and $R$ we say that a corrupted codeword $y$ $\epsilon$-fools the decoder if
there exists an index $i\leq n$ such that
$\Pr[\mathsf{Dec}(y,m^{\prime},i;R)=x_{i}]<\frac{1}{2}+\epsilon$. In the
classical setting (no shared randomness) we require that no corrupted codeword
$y$ with $\mathsf{ED}\left(C(x;R),y\right)$ has the property that it
$\epsilon$-fools the decoder. In the private-key setting we relax this
requirement and allow that $\epsilon$-fooling codewords $y$ exist so long as
the probability the channel $\mathcal{A}$ outputs such a codeword is
negligible. Formally, let $\mathtt{FoolED}(\mathcal{A},x,R)=1$ (resp.
$\mathtt{FoolHamm}(\mathcal{A},x,R)=1$) denote the event that the corrupted
codeword $y=\mathcal{A}(x,C(x;R))$ output by the channel $\epsilon$-fools the
decoder and $\mathsf{ED}\left(C(x;R),y\right)\leq 2\delta m$ (resp.
$\mathtt{Hamm}(C(x;R),y)\leq\delta m$). Crucially, the random coins $R$ are
not known to the channel otherwise $\mathcal{A}$ could do a brute-force search
to find such an $\epsilon$-fooling string.

We say that the pair $(C,\mathsf{Dec})$ is a $(q,\delta,\epsilon)$-private-key
LDC for Insdel (resp. Hamming) errors if there is a negligible function
$\mu(\cdot)$ such that for all channels $\mathcal{A}$ and messages
$x\in\Sigma^{n}$ we have $\Pr_{R}[\mathtt{FoolED}(\mathcal{A},x,R)]\leq\mu(n)$
(resp. $\Pr_{R}[\mathtt{FoolHamm}(\mathcal{A},x,R)]\leq\mu(n)$).

##### Prior Private-Key Constructions

[OPS07] gives a $(q,\delta,\epsilon)$-private-key Hamming LDC with query
complexity $q=\log^{2}n$ and $\epsilon>\frac{1}{2}-\mu(n)$ for a negligible
function $\mu(n)$ i.e., except with negligible probability $\mu(n)$ the
channel outputs a codeword $y$ such that $\forall i\leq n$ we have
$\Pr[\mathsf{Dec}(y,m^{\prime},i;R)=x_{i}]\geq 1-\mu(n)$. [BB21] gives a
$(q,\delta,\epsilon)$-private-key Insdel LDC with query complexity
$q=\log^{c}n$ and $\epsilon>\frac{1}{2}-\mu(n)$ for a negligible function
$\mu(n)$. The result is obtained by applying the Hamming to Insdel compiler of
[BBG+20] to [OPS07].

#####  $q=1$ query private-key Hamming LDC

We can also extend ideas from [OPS07] to obtain a
$(q=1,\delta,\epsilon)$-private-key Hamming LDC for suitable constants
$\delta+\epsilon<\frac{1}{2}$. The length of the codeword is just
$m=n\log^{2}n$. In specific we define
$C(x;R=(\pi,\mathtt{OTP}))=\pi\left(x^{t}\oplus\mathtt{OTP}\right)$. Here,
$\mathtt{OTP}\in\\{0,1\\}^{tn}$ is a uniformly random $tn$-bit string used as
a one-time-pad and $x^{t}\in\\{0,1\\}^{tn}$ denotes the string $x$
concatenated to itself $t$ times i.e., $x^{1}\doteq x$ and $x^{i+1}\doteq
x^{i}\circ x$. Note that for any $x$ the string $\mathtt{OTP}\oplus x^{t}$ is
distributed uniformly at random. The random permutation
$\pi\mathrel{\mathop{\mathchar 58\relax}}[tn]\rightarrow[tn]$ randomly
shuffles the bits of $z=x^{t}\oplus\mathtt{OTP}$ e.g., if
$z=z[1]\circ\ldots\circ z[tn]$ then $\pi(z)=z[\pi(1)]\circ\ldots\circ
z[\pi(tn)]$.

Fixing $\pi$ and an index $i\leq n$ we can define
$S_{\pi,i}\doteq\\{\pi(i),\ldots,\pi(i+tn)\\}$ to be the set of indices of the
codeword $C(x;\pi,\mathtt{OPT})$ which correspond to $x[i]$ i.e., such that
$C(x;\pi,\mathtt{OPT})[j]\oplus\mathtt{OTP}[j]=x[i]$. The decoder
$\mathsf{Dec}(y,m^{\prime},i;R=(\pi,\mathtt{OTP}))$ will randomly pick $j\in
S_{\pi,i}$ and return $y[j]\oplus\mathtt{OTP}[j]$ as our guess for $x[i]$. Now
a corrupted codeword $y$ $\epsilon$-fools the decoder if and only if for some
$i\leq n$ at least $\left(\frac{1}{2}-\epsilon\right)$-fraction of the bits in
$S_{\pi,i}$ were flipped i.e., $\left|\left\\{j\in
S_{\pi,i}\mathrel{\mathop{\mathchar
58\relax}}y[j]=C(x;\pi,\mathtt{OPT})[j]\right\\}\right|\leq
t\left(\frac{1}{2}+\epsilon\right)$.

Because the channel $\mathcal{A}$ does not have $\pi$ or $\mathtt{OTP}$ and
can flip at most $\delta m$ bits in the codeword the expected number of bit
flips in $S_{\pi,i}$ is just $\delta t<\left(\frac{1}{2}-\epsilon\right)t$.
Applying concentration bounds and union bounding we have

$\Pr[\exists i.~{}\left|\left\\{j\in S_{\pi,i}\mathrel{\mathop{\mathchar
58\relax}}y[j]=C(x;\pi,\mathtt{OPT})[j]\right\\}\right|\leq
t\left(\frac{1}{2}+\epsilon\right)]\leq\mu(n)$

for a negligible function $\mu(n)$ whenever we set $t=\log^{2}n$. Here, the
randomness is taken over the selection of $\pi$ and $\mathtt{OTP}$.

##  Appendix C A Note on Hadamard Codes and Type 1 Errors

In this section, we consider the following error pattern corresponding to
error type 1 mentioned in the technique overview i.e., pick arbitrary
$e\leq\delta m/2$ and delete the first $e$ bits from the codeword and append
$e$ arbitrary bits at the end of the codeword. In particular, if $y$ is a
codeword with length $m$ then the error pattern $D_{e}$ deletes the first $e$
bits of the codeword $y$ and then insert $e$ arbitrary bits to the end. We
denote the the corrupted codeword we obtained after this error pattern by
$D_{e}(y)$.

This section shows that a variant of the Hadamard code allows the local
decoder to recover from type 1 errors using just $q=2$ queries.

We prove the following theorem.

######  Theorem 6.

There exist explicit $(2,\delta,1/2-\delta-2^{-t})$ insdel LDCs
$C\colon\mathinner{\left\\{0,1\right\\}}^{n}\rightarrow\mathinner{\left\\{0,1\right\\}}^{m}$
with $m=2^{nt}$, which corrects error Type 1.

Let $\textsf{Had}_{n}$ be the Hadamard encoding for message length $n$. Given
a string $\mathbf{x}=x_{0}x_{2}\dots
x_{n-1}\in\mathinner{\left\\{0,1\right\\}}^{n}$, we view the codeword
$\textsf{Had}_{n}(\mathbf{x})$ as a function
$f_{\mathbf{x}}\colon[2^{n}-1]\rightarrow\mathinner{\left\\{0,1\right\\}}^{n}$,
where

$\displaystyle\forall a\in[2^{n}-1]\textup{ with binary representation
}a=\sum_{j=0}^{n-1}a_{j}\cdot 2^{j},\quad
f_{\mathbf{x}}(a)=\bigoplus_{j=0}^{n-1}a_{j}x_{j}.$

Encoder Let $t\in\mathbb{N}$ be a parameter. For a message
$\mathbf{x}=x_{0}x_{1}\cdots x_{n-1}\in\mathinner{\left\\{0,1\right\\}}^{n}$,
let
$\mathbf{x}^{(t)}\coloneqq(x_{0}0^{t-1})(x_{1}0^{t-1})\dots(x_{n-1}0^{t-1})$,
i.e. the string $\mathbf{x}$ with $t-1$ zeros following each bit. The encoder
function is given by

$\displaystyle\mathsf{Enc}(\mathbf{x})=\textsf{Had}_{tn}(\mathbf{x}^{(t)}).$

Therefore the codeword length is $m=2^{tn}$. Note that $x_{i}$ has index $ti$
in $\mathbf{x}^{(t)}$, we can recover $x_{i}=f_{\mathbf{x}^{(t)}}(2^{ti})$.

Decoder Consider the following decoder $\mathsf{Dec}$. Let
$f\colon[2^{tn}-1]\rightarrow\mathinner{\left\\{0,1\right\\}}$ be the received
string. To decode $x_{i}$, $\mathsf{Dec}$ picks a random $a\in[2^{tn}-1]$ such
that $a+2^{ti}\leq 2^{tn}-1$ and outputs $f(a)\oplus f(a+2^{ti})$.

Analysis For integers $a,b\in\mathbb{N}$ with binary representations $a_{j}$
and $b_{j}$ we write $a\leq_{2}b$ if $a_{j}\leq b_{j}$ for all $j$. We say
$a\leq 2^{tn}-2^{ti}-1$ is _bad_ for $i$ if $2^{ti}(2^{t}-1)\leq_{2}a$, i.e.
$a_{j}=1$ for all $ti\leq j\leq t(i+1)-1$. Otherwise we say $a$ is _good_ for
$i$.

The success rate of the decoder is implied by the following two claims.

######  Claim 12.

If $a$ is good for $i$, then for any
$\mathbf{x}\in\mathinner{\left\\{0,1\right\\}}^{n}$,
$f_{\mathbf{x}^{(t)}}(a)\oplus f_{\mathbf{x}^{(t)}}(a+2^{ti})=x_{i}$.

###### Proof.

Let $a_{j}$ and $a_{j}^{\prime}$ be the binary representations of $a$ and
$a+2^{ti}$, respectively. Note that $a_{j}=a_{j}^{\prime}$ for $j<ti$, and
$a_{ti}\neq a_{ti}^{\prime}$. If $a$ is good for $i$, we must also have
$a_{j}=a_{j}^{\prime}$ for $j\geq t(i+1)$, since otherwise there exists
$j_{0}\geq t(i+1)$ such that $a_{j_{0}}^{\prime}=1$ and $a_{j_{0}}=0$, and
then

$\displaystyle
2^{ti}=(a+2^{ti})-a=\sum_{j=ti}^{j_{0}}(a_{j}^{\prime}-a_{j})2^{j}>2^{j_{0}}-\sum_{j=ti}^{j_{0}-1}2^{j}=2^{ti},$

which is a contradiction. The inequality is strict because $a_{j}=0$ for at
least one $ti\leq j\leq j_{0}-1$.

Finally, note that $x^{(t)}_{j}=0$ for $ti<j<t(i+1)$. We thus have

$\displaystyle f_{\mathbf{x}^{(t)}}(a)\oplus
f_{\mathbf{x}^{(t)}}(a+2^{ti})=\bigoplus_{j=ti}^{t(i+1)-1}a_{j}x^{(t)}_{j}\oplus\bigoplus_{j=ti}^{t(i+1)-1}a_{j}^{\prime}x^{(t)}_{j}=a_{ti}x^{(t)}_{ti}\oplus
a_{ti}^{\prime}x^{(t)}_{ti}=\left(a_{ti}\oplus
a_{ti}^{\prime}\right)x_{i}=x_{i}.$

∎

######  Claim 13.

For any $0\leq i\leq n-1$, $\Pr_{a\in[2^{tn}-2^{ti}-1]}\left[a\textup{ is good
for }i\right]\geq 1-2^{-t}$.

###### Proof.

Note that for $a\geq 2^{t(i+1)}$, $2^{ti}(2^{t}-1)\leq_{2}a$ if and only if
$2^{ti}(2^{t}-1)\leq_{2}a-2^{t(i+1)}$. Therefore we only need to show
$\Pr_{a\in[2^{t(i+1)}-1]}\left[a\textup{ is good for }i\right]\geq 1-2^{-t}$.
Clearly $a$ is good for $i$ when $a<2^{ti}(2^{t}-1)$, and hence

$\displaystyle\Pr_{a\in[2^{tn}-2^{ti}-1]}\left[a\textup{ is good for
}i\right]\geq\frac{2^{ti}(2^{t}-1)}{2^{t(i+1)}}=\frac{2^{t}-1}{2^{t}}=1-2^{-t}.$

∎

If the distances among the indices are preserved, each deletion at the
beginning of the codeword reduces the number of good indices by at most 1. So
after $\delta m$ deletions we are left with at least

$\displaystyle(1-2^{-t})(2^{tn}-2^{ti})-\delta\cdot
2^{tn}=(1-\delta)2^{tn}-2^{t(n-1)}-2^{ti}\geq(1-\delta-2^{-t+1})2^{tn}$

good $a$’s for $i$. Therefore it is a $(2,\delta,1/2-\delta-2^{-t+1})$ LDC for
such errors.

##  Appendix D Proof of Theorem 4

The following theorem says that from any linear (resp. affine) insdel LCC, we
can obtain a linear (resp. affine) insdel LDC that has the same parameter. The
theorem is essentially Lemma 2.3 from [Yek12].

######  Theorem 7.

Let $\mathbb{F}$ be a finite field. Suppose $C\subseteq\mathbb{F}^{m}$ is a
linear (resp. affine) $(q,\delta,\varepsilon)$-insdel LCC, then there exists a
linear (resp. affine) $(q,\delta,\varepsilon)$-insdel LDC $C^{\prime}$
encoding messages of length $\mathsf{dim}(C)$ to codewords of length $m$. The
same holds for the insdel LCCs and insdel LDCs in the private-key setting.

###### Proof of Theorem 7.

For any linear code $C\subseteq\mathbb{F}^{m}$, it encodes a message
$\mathbb{x}\in\mathbb{F}^{n}$ to a codeword $\mathbb{y}\in\mathbb{F}^{m}$
through encoding function $\mathbb{y}=\mathbb{x}\cdot G$ with generating
matrix $G\in\mathbb{F}^{n\times m}$. Let $I\subseteq[m]$ be a set of
$\mathsf{dim}(C)$ information coordinates of $C$ (i.e. a set of coordinates
whose value uniquely determines an element in $C$). For $\mathbb{y}\in C$, let
$\mathbb{y}|_{I}\in\mathbb{F}^{n}$ be the restriction of $\mathbb{y}$ to the
coordinates in $I$. We can find another generator matrix $G^{\prime}$ such
that for any message $\mathbb{x}\in\mathbb{F}^{n}$,
$\mathbb{y}=\mathbb{x}\cdot G^{\prime}\in C$ and $\mathbb{y}|_{I}=\mathbb{x}$.
It is easy to verify that the locally correctability of $C$ implies the
locally decodability of $C^{\prime}$.

If $C$ is an affine code, the encoding function becomes
$\mathbb{y}=\mathbb{x}\cdot G+\mathbb{b}$ for some
$\mathbb{b}\in\mathbb{F}^{m}$. Again, let $I\subseteq[m]$ be a set of
$\mathsf{dim}(C)$ information coordinates of $C$. Similarly, we can pick
generator matrix $G^{\prime}$ such that for any message
$\mathbb{x}\in\mathbb{F}^{n}$, $\mathbb{y}=\mathbb{x}\cdot
G^{\prime}+\mathbb{b}\in C$ and $\mathbb{y}|_{I}-\mathbb{b}|_{I}=\mathbb{x}$.
The locally correctability of $C$ implies the locally decodability of
$C^{\prime}$.

We note the above argument holds for insdel LCCs and insdel LDCs in the
private-key setting.

∎

The proof of Theorem 4 uses the same reduction introduced by [BGT16]. We first
introduce two lemmas. The following lemma says that insdel LCCs must have
large Hamming distance.

######  Lemma 5.

If $C\in\Sigma^{m}$ is a $(q,\delta,\varepsilon)$-LCC, then for any two
codewords $c,c^{\prime}\in C$, the Hamming distance between $c$ and
$c^{\prime}$ is larger than $2\delta m$.

###### Proof.

Assume there are two codewords $c,c^{\prime}\in C$ such that $c\neq
c^{\prime}$ and the Hamming distance between $c$ and $c^{\prime}$ is at most
$2\delta m$. Then we can conclude that $\mathsf{ED}(c,c^{\prime})\leq 4\delta
m$. This is because we can transform $c$ into $c^{\prime}$ with less then
$2\delta m$ symbol substitutions. Meanwhile, each symbol substitution can be
replaced by an insertion and a deletion. Thus it takes at most $4\delta m$
insertions/deletions to transform $c$ into $c^{\prime}$.

Since $\mathsf{ED}(c,c^{\prime})\leq 4\delta m$, there must exists a
$y\in\Sigma^{m^{\prime}}$ such that $\mathsf{ED}(c,y)\leq 2\delta m$ and
$\mathsf{ED}(c^{\prime},y)\leq 2\delta m$. Let $i$ be one of the positions
that $c$ and $c^{\prime}$ differs. By the definition of insdel LCCs, there is
a probabilistic algorithm $\mathsf{Dec}$ such that
$\Pr\left[\mathsf{Dec}(y,m^{\prime},i)=c_{i}\right]\geq\frac{1}{2}+\varepsilon$
and
$\Pr\left[\mathsf{Dec}(y,m^{\prime},i)=c^{\prime}_{i}\right]\geq\frac{1}{2}+\varepsilon$.
This is impossible because $c_{i}\neq c^{\prime}_{i}$. Thus, the Hamming
distance between $c$ and $c^{\prime}$ must be larger than $2\delta m$.

∎

The reduction needs the following notion of VC-dimension.

######  Definition 6 (VC-dimension).

Let $A\subseteq\\{0,1\\}^{n}$, then the VC-dimension of $A$, denoted by
$\mathsf{vc}(A)$ is the cardinality of the largest set $I\subseteq[n]$ which
is shattered by $A$. That is, the restriction of $A$ to $I$,
$A|_{I}=\\{0,1\\}^{|I|}$.

The following lemma due to Dudley [Dud78] is the key to the reduction.

######  Lemma 6 (Theorem 14.12 in [LT13]).

Let $A\subseteq\\{0,1\\}^{n}$ such that for every distinct $x,y\in A$,
$\|x-y\|_{\ell_{2}}\geq\varepsilon\sqrt{n}$. Then,

$\mathsf{vc}(A)=\Omega\left(\frac{\log(|A|)}{\log(2/\varepsilon)}\right).$

###### Proof of Theorem 4.

Assume $C\mathrel{\mathop{\mathchar
58\relax}}\\{0,1\\}^{k}\rightarrow\Sigma^{m}$ is a
$(q,\delta,\varepsilon)$-insdel LCC. Without loss of generality, we can assume
$\Sigma=\\{0,1\\}^{s}$. Consider an error correcting code
$C^{0}\mathrel{\mathop{\mathchar
58\relax}}\\{0,1\\}^{s}\rightarrow\\{0,1\\}^{t}$ with constant distance
$\delta_{0}$, i.e. any two codewords in $C^{0}$ has Hamming distance
$\delta_{0}t$. Let $C^{1}\mathrel{\mathop{\mathchar
58\relax}}\\{0,1\\}^{k}\rightarrow\\{0,1\\}^{mt}$ be the concatenation of code
$C$ and $C^{0}$. That is, every codeword in $C^{1}$ is obtained by first
encoding the message with code $C$ and then encoding each symbol in $\Sigma$
with code $C^{0}$. By Lemma 5, any two codewords in $C^{1}$ must have Hamming
distance at least $2\delta\delta_{0}mt$. Thus, let
$\varepsilon=\sqrt{2\delta\delta_{0}}$, for any $c,c^{\prime}\in C^{1}$, we
have

$\|c-c^{\prime}\|_{\ell_{2}}\geq\varepsilon\sqrt{mt}.$

By Lemma 6, the VC-dimension of $C^{1}$ ( $\mathsf{vc}(C^{1})$) is

$\Omega\left(\frac{\log(|C^{1}|)}{\log(2/\varepsilon)}\right)=\Omega\left(\frac{\log(|C|)}{\log(2/\varepsilon)}\right)=\Omega\left(\frac{k}{\log(1/\delta)}\right)$

Let $k^{\prime}=\mathsf{vc}(C^{1})$. By the definition of VC-dimension, there
exists a set $I\subseteq[mt]$ of size $k^{\prime}$ that shatters $C^{1}$, i.e.
$C^{1}|_{I}=\\{0,1\\}^{|I|}$.

We can build a $(q,\delta,\varepsilon)$-insdel LDC
$C^{\prime}\mathrel{\mathop{\mathchar
58\relax}}\\{0,1\\}^{k^{\prime}}\rightarrow\Sigma^{m}$ as follows. For any
message $x\in\\{0,1\\}^{k^{\prime}}$, let $C^{\prime}(x)=z\in C$ such that
$C^{0}(z)|_{I}=x$. Here, by $C^{0}(z)$, we mean encoding each symbol of $z$
with code $C^{0}$. By the property of set $I$, we know such a codeword $z\in
C$ must exist. If there are more than one $z\in C$ satisfying
$C^{0}(z)|_{I}=x$, we pick one of them arbitrarily. Assuming
$I=\\{t_{1},t_{2},\dots,t_{k^{\prime}}\\}$, for any message
$x\in\\{0,1\\}^{k^{\prime}}$ and any $i\in[k^{\prime}]$, we have
$x_{i}=\left(C^{0}(C^{\prime}(x))\right)_{t_{i}}$.

We now show that $C^{\prime}$ is indeed a $(q,\delta,\varepsilon)$-insdel LDC.
To decode a message bit $x_{i}$, we only need to look at the block of
$\left(\\{0,1\\}^{t}\right)^{m}$ that contains $t_{i}$, say it is the $j$-th
block. Assume we are given a $y\in\Sigma^{*}$ with
$\mathsf{ED}\left(y,C^{\prime}(x)\right)\leq 2\delta m$, and notice that
$C^{\prime}(x)$ is also a codeword of $C$. The decoding algorithm for $C$ can
recover $C^{\prime}(x)_{j}$ with probability of at least $1/2+\varepsilon$,
using at most $q$ queries to $y$. Applying $C^{0}$ to $C^{\prime}(x)_{j}$ will
give us $x_{i}$. Thus, $x_{i}$ can be decoded with probability at least
$1/2+\varepsilon$ using at most $q$ queries to $y$.

∎


# Minimal Left-Right Symmetric Model with $A_{4}$ modular symmetry

Ankita Kakoti<EMAIL_ADDRESS>Bichitra Bijay Boruah
<EMAIL_ADDRESS>Mrinal Kumar Das<EMAIL_ADDRESS>Department of
Physics, Tezpur University, Tezpur 784028, India

###### Abstract

In this paper, we have realized the left-right symmetric model with modular
symmetry. We have used $\Gamma$(3) modular group which is isomorphic to non-
abelian discrete symmetry group $A_{4}$. The advantage of using modular
symmetry is the non-requirement for the use of extra particles called
’flavons’. In this model, the Yukawa couplings are expressed in terms of
modular forms $(Y_{1},Y_{2},Y_{3})$. In this work, we have studied minimal
Left-Right Symmetric Model for both type-I and type-II dominances. Here, we
have calculated the values for the Yukawa couplings and then plotted it
against the sum of the neutrino masses. The results obtained are well within
the experimental limits for the desired values of sum of neutrino masses. We
have also briefly analyzed the effects of the implications of modular symmetry
on neutrinoless double beta decay with the new physics contributions within
Left-Right Symmetric Model.

###### pacs:

12.60.-i,14.60.Pq,14.60.St

##  I Introduction

Despite the huge and continued success of the Standard Model (SM) of particle
physics, it leaves some of the puzzles unanswered like the existence of
neutrino masses, baryon asymmetry of the universe, existence of dark matter
etc. The discovery of neutrino oscillation by Sudbury neutrino observatory and
Super-Kamiokande experiments was a milestone discovery in the area of neutrino
physics. The experiments like MINOS [1], T2K [2], Daya-Bay [3], Double-Chooz
[4], RENO [5] etc. provided evidence on the neutrinos being massive which is
one of the most compelling revelation that we need to go beyond Standard
Model. However inspite of the huge achievements in determining the neutrino
oscillation parameters in solar, atmospheric , reactor and accelerator
neutrino experiments, many questions related to neutrino still remain
unsolved. Among these lies the question regarding the absolute mass scale of
neutrinos, exact nature of the particle (Dirac or Majorana), hierarchical
pattern of the mass spectrum (Normal or Inverted) and leptonic CP violation.
The absolute mass scale of the neutrinos is not yet known. However experiments
like Planck has given an upper bound on the sum of the light neutrino masses
to be $\Sigma|m_{\nu_{i}}|<0.23eV$ in 2012 [6] and recently the bound has been
constarined to $\Sigma|m_{\nu_{i}}|<0.11eV$ [7]. The most successful data
pertaining to neutrino oscillation parameters is found in the 3$\sigma$ global
fit data [8] as shown in table (1) .

Parameters |  Normal Ordering |  Inverted Ordering
---|---|---
$\Delta$ $m_{21}^{2}$ $(10^{-5}eV^{2})$ |  $6.82$ $\rightarrow$ $8.04$ |  $6.82$ $\rightarrow$ $8.04$
$\Delta$ $m_{3l}^{2}$ $(10^{-5}eV^{2})$ |  $2.435$ $\rightarrow$ $2.598$ |  $-2.581$ $\rightarrow$ $-2.414$
$sin^{2}$ $\theta_{12}$ |  $0.264$ $\rightarrow$ $0.343$ |  $0.269$ $\rightarrow$ $0.343$
$sin^{2}$ $\theta_{23}$ |  $0.415$ $\rightarrow$ $0.616$ |  $0.419$ $\rightarrow$ $0.617$
$sin^{2}$ $\theta_{13}$ |  $0.02032$ $\rightarrow$ $0.02410$ |  $0.02052$ $\rightarrow$ $0.02428$
Table 1: Global fit 3$\sigma$ values for neutrino oscillation parameters.

We have used the definition,

$\Delta m_{3l}^{2}=\Delta m_{31}^{2};\Delta m_{31}^{2}>0;NO$ (1)

$\Delta m_{3l}^{2}=\Delta m_{32}^{2};\Delta m_{32}^{2}<0;IO$ (2)

The simplest way to look for neutrino masses is by the seesaw mechanism. The
mechanism may be of type I [9], [10],type II [11], [12],type III [13] and
Inverse Seesaw [14]. These are extensions of the SM where we incorporate extra
particles like right-handed fermions,scalar fermion triplets, gauge singlet
neutral fermions etc. The BSM physics also sheds light upon the phenomena like
baryon asymmetry of the universe (BAU) [15], Lepton Number Violation (LNV)
[16], Lepton Flavor violation (LFV) [17], existence of dark matter [18], [19]
etc. A BSM framework which has been successful in explaining the first three
of the phenomenologies is the Left- Right Symmetric Model (LRSM) [20, 21, 22,
23, 24], an extension of the SM corresponding to the addition of $SU(2)_{R}$
group into the theory. The gauge group of LRSM is $SU(3)_{C}\otimes
SU(2)_{R}\otimes SU(2)_{L}\otimes U(1)_{B-L}$. The type I and type II seesaw
masses appear naturally in the model. The right-handed neutrinos are an
essential part of the model, which acquires Majorana mass when $SU(2)_{R}$
symmetry is broken. LRSM provides a natural framework to understand the
spontaneous breaking of parity and origin of small neutrino masses by seesaw
mechanism [25].
Another concerning aspect is the ambiguity regarding nature of neutrinos which
has not been yet predicted by the SM of particle physics, that whether
neutrinos are Dirac or Majorana fermions. This problem is directly connected
to the issue of lepton number conservation. One of the process of fundamental
importance which arises in almost any extension of the SM is Neutrinoless
Double Beta Decay(NDBD) [26], [27] which when verified can assure that
neutrinos are Majorana fermions. NDBD is a slow, radiative process that
transforms a nuclide of atomic number Z into its isobar with atomic number Z+2
[28],

$N(A,Z)\rightarrow N(A,Z+2)+e^{-}+e^{-}$ (3)

The main aim in the search of NDBD (0$\nu\beta\beta$) is the measurement of
effective Majorana neutrino mass, which is a combination of the neutrino mass
eigenstates and neutrino mixing matrix terms [28]. However, no experimental
evidence regarding the decay has been in picture till date. In addition to the
determination of the effective masses, the half-life of the decay [29]
combined with sufficient knowledge of the nuclear matrix elements (NME), we
can set a constraint on the neutrino masses. The experiments like KamLAND-Zen
[30] and GERDA [31] which uses Xenon-136 and Germanium-76 respectively have
improved the lower bound on the half-life of the decay process. However,
KamLAND-Zen imposes the best lower limit on the half life as
$T_{1/2}^{0\nu}>1.07\times 10^{26}$ yr at 90 % CL and the corresponding upper
limit of the effective Majorana mass in the range (0.061-0.165)eV. There are
several contributions in LRSM that appear due to additional RH current
interactions, giving rise to sizeable LFV rates for TeV scale RH neutrino that
occur at rates accessible in current experiments. It has been found that the
most significant constraints has been provided by the decays, $\mu\rightarrow
3e$ and $\mu\rightarrow\gamma e$. In the Standard Model, these LFV decays are
suppressed by the tiny neutrino masses. No experiment has so far observed any
flavor violating processes including charged leptons. However, many
experiments are currently going on to set strong limits on the most relevant
LFV observables that will constrain the parameter space of many new models.
The best bounds on the branching ratio for LFV decays of the form
$\mu\rightarrow\gamma e$ comes from MEG experiment and it is set at
$BR(\mu\rightarrow\gamma e)<4.2\times 10^{-13}$. In case of the decay
$\mu\rightarrow 3e$, the bound is set by the SINDRUM experiment at
$BR(\mu\rightarrow 3e)<1.0\times 10^{-12}$.

As mentioned LRSM is an important theory that incorporates the above mentioned
phenomenologies, i.e., the phenomenologies related to neutrinos. There are
many works where the authors make use of discrete symmetry groups like $A_{4}$
[32],$S_{4}$ [33],$Z_{2}$ etc. [34] to analyze the problem of flavor structure
of fermions and to study various related phenomenologies. In our work, we have
used $A_{4}$ modular symmetry to study neutrino masses and mixings and hence
study Neutrinoless Double Beta Decay within the model. The advantage of using
modular symmetry over discrete flavor symmetries is that the study of the
model using symmetries can be done without the introduction of extra particles
called ’flavons’. Hence the model is minimal.

However, in this work we have not done a very detailed analysis of the above
mentioned phenomenologies, but only realized the left-right symmetric model
with the help of $A_{4}$ modular symmetry and studied the variations of new
physics contributions of neutrinoless double beta decay within LRSM with the
range of values for Yukawa couplings, which in our model is expressed as
modular forms. In section (II), we have given a detailed explanation of the
left-right symmetric model, the associated Lagrangian and the mass terms. We
begin section (III) by introducing

modular symmetry and then in section (IV), we incorporate modular symmetry
into LRSM and determine the associated mass matrices. In section (V), we
present a very brief discussion of neutrinoless double beta decay and its
associated contributions and their relations with the modular forms. In
section (VI), the numerical analysis and results of this work has been
discussed and the last section reads the conclusion for the present work.

##  II Minimal Left-Right Symmetric Model

The Left-Right Symmetric Model (LRSM) was first introduced around 1974 by Pati
and Salam. Rabindra N. Mohapatra and Goran Senjanovic were also some pioneers
of this very elegant theory. LRSM is an extension of the Standard Model of
particle physics, the gauge group being $SU(3)_{C}\otimes SU(2)_{R}\otimes
SU(2)_{L}\otimes U(1)_{B-L}$, which has been studied by several groups since
1970’s [25], [21, 22, 23, 24]. The usual type-I and type-II seesaw neutrino
masses arises naturally in the model. The seesaw scale is identified by the
breaking of $SU(2)_{R}$. Some other problems are also addressed in LRSM like
parity, CP violation of weak interaction, massive neutrinos, hierarchy
problems, etc. LRSM removes the disparity between the left and right-handed
fields by considering the RH fields to be doublet under the additional
$SU(2)_{R}$ keeping the right sector couplings same as the left-one by left-
right symmetry. In this model, the electric charge is given by
$Q=T_{3L}+T_{3R}+\frac{B-L}{2}$, where $T_{3L}$ and $T_{3R}$ are the
generators of $SU(2)_{L}$ and $SU(2)_{R}$ respectively. $B-L$ refers to baryon
number minus lepton number. The particle content of the model along with their
respective charge assignments are given in table(III). The matrix
representation for the scalar sector is given by,

$\phi=\begin{pmatrix}\phi_{1}^{0}&\phi_{1}^{+}\\\
\phi_{2}^{-}&\phi_{2}^{0}\end{pmatrix}$ (4)
$\Delta_{L,R}=\begin{pmatrix}\frac{\delta_{L,R}^{+}}{\sqrt{2}}&\delta_{L,R}^{++}\\\
\delta_{L,R}^{0}&-\frac{\delta_{L,R}^{+}}{\sqrt{2}}\end{pmatrix}$ (5)

In order for the fermions to attain mass, a Yukawa Lagrangian is necessary
which couples to the bidoublet $\phi$. The Yukawa Lagrangian incorporating the
bidoublet is given by,

$\mathcal{L_{D}}=\overline{l_{iL}}(Y_{ij}^{l}\phi+\widetilde{Y_{ij}^{l}}\widetilde{\phi})l_{jR}+\overline{Q_{iL}}(Y_{ij}^{q}\phi+\widetilde{Y_{ij}^{q}}\widetilde{\phi})Q_{jR}+h.c$
(6)

where, $l_{L}$ and $l_{R}$ are the left-handed and right-handed lepton fields,
$Q_{L}$ and $Q_{R}$ are the left-handed and right-handed quark fields. $Y^{l}$
being the Yukawa coupling corresponding to leptons and $Y^{q}$ being the
Yukawa coupling for the quarks. The Yukawa Lagrangian incorporating the scalar
triplets which play a role in providing Majorana mass to the neutrinos is
given by,

$\mathcal{L_{M}}=f_{L,ij}{\Psi_{L,i}}^{T}Ci\sigma_{2}\Delta_{L}\Psi_{L,j}+f_{R,ij}{\Psi_{R,i}}^{T}Ci\sigma_{2}\Delta_{R}\Psi_{R,j}+h.c$
(7)

$f_{L}$ and $f_{R}$ are the Majorana Yukawa couplings and are equal subjected
to discrete left-right symmetry. The scalar potential in LRSM is a combination
of interaction terms consisting the potential and after spontaneous symmetry
breaking the scalars attain VEVs given by,

$<\Delta_{L,R}>=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0\\\
v_{L,R}&0\end{pmatrix}$ (8) $<\phi>=\begin{pmatrix}k&0\\\
0&e^{i\theta}k^{\prime}\end{pmatrix}$ (9)

The magnitudes of the VEVs follows the relation, $|v_{L}|^{2}<|k^{2}+k^{\prime
2}|<|v_{R}|^{2}$. The breaking pattern of the LRSM gauge group takes place in
two steps. The LRSM gauge group is first broken down to the Standard Model
gauge group by the vev of the scalar triplet $\Delta_{R}$, and then the
Standard Model gauge group is broken down to the electromagnetic gauge group
i.e., $U(1)_{em}$ by the vev of the bidoublet and a tiny vev of the scalar
triplet $\Delta_{L}$.
The Dirac mass terms for the leptons come from the Yukawa Lagrangian, which
for the charged leptons and the neutrinos are given by,

$M_{l}=\frac{1}{\sqrt{2}}(k^{\prime}Y_{l}+k\tilde{Y_{l}})$ (10)
$M_{D}=\frac{1}{\sqrt{2}}(kY_{l}+k^{\prime}\tilde{Y_{l}})$ (11)

The light neutrino mass after spontaneous symmetry breaking (SSB), generated
within a type (I+II) seesaw can be written as,

$M_{\nu}={M_{\nu}}^{I}+{M_{\nu}}^{II},$ (12)

$M_{\nu}=M_{D}{M_{RR}}^{-1}{M_{D}}^{T}+M_{LL}$ (13)

where,

$M_{LL}=\sqrt{2}v_{L}f_{L}$ (14)

and,

$M_{RR}=\sqrt{2}v_{R}f_{R}$ (15)

The first and second terms in corresponds to type-I seesaw and type-II seesaw
masses respectively. It is an interesting fact that in the context of LRSM
both type-I and type-II terms can be equally dominant or either of the two
terms can be dominant, but under certain conditions [35, 36]. It has been
demonstrated in the Appendix A. In the context of LRSM however, both the
type-I and type-II mass terms can be expressed in terms of the heavy right-
handed Majorana mass matrix, so equation (13) will follow,

$M_{\nu}=M_{D}M_{RR}^{-1}M_{D}^{T}+\gamma\Biggl{(}\frac{M_{W}}{v_{R}}\Biggl{)}^{2}M_{RR}$
(16)

where, $\gamma$ is a dimensionless parameter which is a function of various
couplings, appearing in the VEV of the triplet Higgs $\Delta_{L}$, i.e.,
$v_{L}=\gamma(\frac{v^{2}}{v_{R}})$ and here, $v=\sqrt{k^{2}+k^{\prime 2}}$,
and

$\gamma=\frac{\beta_{1}kk^{\prime}+\beta_{2}k^{2}+\beta_{3}k^{\prime
2}}{(2\rho_{1}-\rho_{3})(k^{2}+k^{\prime 2})}$ (17)

In our model, the dimensionless parameter $\gamma$ has been fine tuned to
$\gamma\approx 10^{-6}$ and $v_{R}$ is of the order of $TeV$.

##  III Modular Symmetry

Modular symmetry has gained much importance in aspects of model building [37],
[38]. This is because it can minimize the extra particle called ’flavons’
while analyzing a model with respect to a particular symmetry group. An
element $q$ of the modular group acts on a complex variable $\tau$ which
belongs to the upper-half of the complex plane given as [38] [39]

$q\tau=\frac{a\tau+b}{c\tau+d}$ (18)

where $a,b,c,d$ are integers and $ad-bc=1$, Im$\tau$$>$0.

The modular group is isomorphic to the projective special linear group
PSL(2,Z) = SL(2,Z)/$Z_{2}$ where, SL(2,Z) is the special linear group of
integer $2\times 2$ matrices having determinant unity and $Z_{2}=({I,-I})$ is
the centre, $I$ being the identity element. The modular group can be
represented in terms of two generators $S$ and $T$ which satisfies
$S^{2}=(ST)^{3}=I$. $S$ and $T$ satisfies the following matrix
representations:

$S=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ (19) $T=\begin{pmatrix}1&1\\\
0&1\end{pmatrix}$ (20)

corresponding to the transformations,

$S:\tau\rightarrow-\frac{1}{\tau};T:\tau\rightarrow\tau+1$ (21)

Finite modular groups (N $\leq$ 5) are isomorphic to non-abelian discrete
groups, for example, $\Gamma(3)\approx A_{4}$, $\Gamma(2)\approx S_{3}$,
$\Gamma(4)\approx S_{4}$. While using modular symmetry, the Yukawa couplings
can be expressed in terms of modular forms, and the number of modular forms
present depends upon the level and weight of the modular form. For a modular
form of level N and weight 2k, the table below shows the number of modular
forms associated within and the non-abelian discrete symmetry group to which
it is isomorphic [39].

N | No. of modular forms | $\Gamma(N)$
---|---|---
2 | k + 1 | $S_{3}$
3 | 2k + 1 | $A_{4}$
4 | 4k + 1 | $S_{4}$
5 | 10k + 1 | $A_{5}$
6 | 12k |
7 | 28k - 2 |
Table 2: No. of modular forms corresponding to modular weight 2k.

In our work, we will be using modular form of level 3, that is, $\Gamma(3)$
which is isomorphic to $A_{4}$ discrete symmetry group. The weight of the
modular form is taken to be 2, and hence it will have three modular forms
$(Y_{1},Y_{2},Y_{3})$ which can be expressed as expansions of q given by,

$Y_{1}=1+12q+36q^{2}+12q^{3}+84q^{4}+72q^{5}+36q^{6}+96q^{7}+180q^{8}+12q^{9}+216q^{10}$
(22)
$Y_{2}=-6q^{1/3}(1+7q+8q^{2}+18q^{3}+14q^{4}+31q^{5}+20q^{6}+36q^{7}+31q^{8}+56q^{9})$
(23)
$Y_{3}=-18q^{2/3}(1+2q+5q^{2}+4q^{3}+8q^{4}+6q^{5}+14q^{6}+8q^{7}+14q^{8}+10q^{9})$
(24)

where, $q=\exp(2\pi i\tau)$.

##  IV Minimal LRSM with $A_{4}$ modular symmetry

In particle physics, symmetries have always played a very crucial role. The
realization of LRSM with the help of discrete flavor symmetries have been done
in earlier works [40], [41]. In our work we have incorporated $A_{4}$ modular
symmetry into LRSM. The advantage of using modular symmetry rather than flavor
symmetry is the minimal use of extra particles (flavons) and hence the model
is minimal. The model contains usual particle content of LRSM [42]. The lepton
doublets transform as triplets under $A_{4}$ and the bidoublet and scalar
triplets transform as 1 under $A_{4}$ [43]. As we have considered modular
symmetry, we assign modular weights to the particles, keeping in mind that
matter multiplets corresponding to the model can have negative modular
weights, but the modular forms cannot be assigned negative weights. The
assignment of these weights are done in such a way that in the Lagrangian the
sum of the modular weights in each term is zero. Modular weights corresponding
to each particle is shown in table (III). The Yukawa Lagrangian for the
leptonic and quark sector in LRSM is given by equation (6),(7) and with a
reference to that we can write the Yukawa Lagrangian of our $A_{4}$ modular
symmetric LRSM, for the fermionic sector, by introducing Yukawa coupling in
the form of modular forms $Y$ is given as,

$\mathcal{L_{Y}}=\overline{l_{L}}\phi{l_{R}}Y+\overline{l_{L}}\tilde{\phi}{l_{R}}Y+\overline{Q_{L}}\phi{Q_{R}}Y+\overline{Q_{L}}\tilde{\phi}{Q_{R}}Y+{{l_{R}}^{T}}Ci{\tau_{2}}{\Delta_{R}}{l_{R}}{Y}+{{l_{L}}^{T}}Ci{\tau_{2}}{\Delta_{L}}{l_{L}}{Y}$
(25)

Gauge group | $Q_{L}$ | $Q_{R}$ | $l_{L}$ | $l_{R}$ | $\phi$ | $\Delta_{L}$ | $\Delta_{R}$
---|---|---|---|---|---|---|---
$SU(3)_{C}$ | 3 | 3 | 1 | 1 | 1 | 1 | 1
$SU(2)_{L}$ | 2 | 1 | 2 | 1 | 2 | 3 | 1
$SU(2)_{R}$ | 1 | 2 | 1 | 2 | 2 | 1 | 3
$U(1)_{B-L}$ | 1/3 | 1/3 | -1 | -1 | 0 | 2 | 2
$A_{4}$ | 3 | 3 | 3 | 3 | 1 | 1 | 1
$k_{I}$ | 0 | -2 | 0 | -2 | 0 | -2 | 2
Table 3: Charge assignments for the particle content of the model.

The Yukawa couplings $Y=(Y_{1},Y_{2},Y_{3})$ are expressed as modular forms of
level 3.

| Y (modular forms)
---|---
$A_{4}$ | $3$
$k_{I}$ | $2$
Table 4: Charge assignment and modular weight for the corresponding modular
Yukawa form for the model.

In our work, we are concerned with the mass of the neutrinos and as such,
using $A_{4}$ modular symmetry and using the multiplication rules for $A_{4}$
group, we construct the Dirac and Majorana mass matrices as given below. The
Dirac mass matrix is given by,

$M_{D}=v\begin{pmatrix}2Y_{1}&-Y_{3}&-Y_{2}\\\ -Y_{2}&-Y_{1}&2Y_{3}\\\
-Y_{3}&2Y_{2}&-Y_{1}\end{pmatrix}$ (26)

where, $v$ is considered to be the VEV for the Higgs bidoublet.

The right-handed Majorana mass matrix is given by,

$M_{R}=v_{R}\begin{pmatrix}2Y_{1}&-Y_{3}&-Y_{2}\\\ -Y_{3}&2Y_{2}&-Y_{1}\\\
-Y_{2}&-Y_{1}&2Y_{3}\end{pmatrix}$ (27)

where, $v_{R}$ is the VEV for the scalar triplet $\Delta_{R}$. As it is seen
that the Majorana mass matrix for our model is found to be symmetric in nature
as it should be. Under these assumptions for modular symmetric LRSM and the
basis that we have considered, our charged lepton mass matrix is also found to
be diagonal.

The type-I seesaw mass is then given by,

$M_{\nu_{I}}=M_{D}.{M_{R}}^{-1}.{M_{D}}^{T}$ (28)

and, the type-II seesaw mass is given by,

$M_{\nu_{II}}=M_{LL}$ (29)

As mentioned above, in LRSM type-II seesaw mass can also be expressed in terms
of the right-handed mass $M_{R}$ as,

$M_{\nu_{II}}=\gamma\Biggl{(}\frac{M_{W}}{v_{R}}\Biggl{)}^{2}M_{R}$ (30)

###  IV.1 Type-I dominanace

In LRSM, the type-I seesaw mass dominates when the vev of the left-handed
triplet is taken to be negligibly small and hence the type-II term is absent.
In such a case the lightest neutrino mass can be given in terms of the type-I
seesaw mass term given by,

$M_{\nu}=M_{D}{M_{R}}^{-1}{M_{D}}^{T}$ (31)

and the heavy right-handed Majorana mass term can be given as,

$M_{R}=f_{R}v_{R}$ (32)

where, $f_{R}$ is the right-handed Majorana Yukawa coupling.
In the approximation that $k^{\prime}<<k$, and if we consider that our Yukawa
coupling $Y^{l}$ corresponding to the neutrino masses is $y_{D}$ and the
coupling $\widetilde{Y^{l}}$ for the charged fermion masses is denoted by
$y_{L}$, so considering $y_{D}k>>y_{L}k^{\prime}$ we can write the type-I mass
term as [44],

$M_{\nu}=\frac{k^{2}}{v_{R}}y_{D}f_{R}^{-1}y_{D}^{T}$ (33)

If we consider that $U_{R}$ is a unitary matrix that diagonalizes $M_{R}$, so
since the VEV $v_{R}$ is a constant the same matrix will also diagonalize the
coupling matrix $f_{R}$. Taking $f_{R}=f_{L}=f$, so

$f=U_{R}f^{dia}U_{R}^{T}$ (34)

If we take inverse on both sides and taking into account the property of a
unitary matrix $(U_{R}^{-1}=U_{R}^{T})$, we get,

$f^{-1}=U_{R}^{T}(f^{dia})^{-1}U_{R}$ (35)

Therefore, we get

$M_{\nu}=\frac{k^{2}}{v_{R}}y_{D}U_{R}^{T}(f^{dia})^{-1}U_{R}y_{D}^{T}$ (36)

Multiplying both sides of the equation with $U_{R}^{T}$ from the right and
with $U_{R}$ from left, we finally arrive at the following equation,

$U_{R}M_{\nu}U_{R}^{T}=(M_{\nu})^{dia}$ (37)

where we have used $U_{R}y_{D}U_{R}^{T}=y_{D}$. So, the unitary matrix
diagonalizing the matrix $M_{R}$ also diagonalizes the light neutrino mass
matrix. So in this case it can be determined that if $m_{i}$ denotes the light
neutrino mass and $M_{i}$ denotes the heavy neutrino mass, then they are
related as

$m_{i}\propto\frac{1}{M_{i}}$ (38)

For our model, the Yukawa couplings are modular forms expressed as expansions
of $q$, and the mass matrices are expressed in terms of the modular forms
$(Y_{1},Y_{2},Y_{3})$. So, the light neutrino mass matrix, $M_{\nu}$ for
type-I dominance is given by the equation (31). As already stated in equations
(26) and (27), the Dirac and Majorana mass matrices are determined by the
application of multiplication rules for the $A_{4}$ group. So, for type-I
dominance, our light neutrino mass matrix will be given by,

$M_{\nu}=\frac{v^{2}}{v_{R}}\begin{pmatrix}2Y_{1}&-Y_{2}&-Y_{3}\\\
-Y_{2}&2Y_{3}&-Y_{1}\\\ -Y_{3}&-Y_{1}&2Y_{2}\end{pmatrix}$ (39)

As mentioned previously, the value for $v_{R}$ is of the order of $TeV$ and
that for $v$ is in $GeV$. We have computed the values of the sum of the
neutrino masses for type-I dominance and checked the correctness of our model
by plotting it against the Yukawa couplings and the result was found to match
the experimental bounds.

Figure 1: Variation of $|Y_{1}|$ with sum of neutrino masses.

Figure 2: Variation of $|Y_{2}|$ with sum of neutrino masses.

Figure 3: Variation of $|Y_{3}|$ with sum of neutrino masses.

###  IV.2 Type-II dominance

Type-II seesaw mass in LRSM dominates when the Dirac term connecting the
right-handed and left-handed parts is negligible as compared to that of the
type-II term [44]. In that case, our light neutrino mass $m_{\nu}$ will given
by the type-II seesaw mass term, i.e.,

$M_{\nu_{L}}=f_{L}v_{L}$ (40)

And the heavy mass matrix is given by,

$M_{R}=f_{R}v_{R}$ (41)

Again if we consider that $U_{L}$ and $U_{R}$ diagonalizes $M_{\nu_{L}}$ and
$M_{R}$ respectively, so for the reason mentioned above the same matrices will
also diagonalize $f_{L}$ and $f_{R}$ respectively and since in our model,
$f_{L}=f_{R}$, so we can consider $U_{L}=U_{R}$. In such a case, we arrive at
an important result that

$m_{i}\propto M_{i}$ (42)

Now using modular symmetry the light neutrino mass matrix for type-II
dominance in our model is given by,

$m_{\nu}=v_{L}\begin{pmatrix}2Y_{1}&-Y_{3}&-Y_{2}\\\ -Y_{3}&2Y_{2}&-Y_{1}\\\
-Y_{2}&-Y_{1}&2Y_{3}\end{pmatrix}$ (43)

where, $v_{L}$ is the vev for left-handed scalar triplet. The value of $v_{L}$
is taken to be of the order of $eV$. The sum of the neutrino masses is
computed for type-II dominance and plotting of the sum is done with the Yukawa
couplings which are found to be as shown under,

Figure 4: Variation of $|Y_{1}|$ with sum of neutrino masses.

Figure 5: Variation of $|Y_{2}|$ with sum of neutrino masses.

Figure 6: Variation of $|Y_{3}|$ with sum of neutrino masses.

##  V Neutrinoless Double Beta Decay $(0\nu\beta\beta)$ in minimal LRSM

Neutrinoless double beta decay is a lepton number violating process, which if
proven to exist will directly imply the Majorana nature of neutrinos.

$N(A,Z)\rightarrow N(A,Z+2)+e^{-}+e^{-}$ (44)

Many groups have however already done a lot of work on NDBD in the model ,
[21],[45, 46, 47, 48, 49, 50, 28]. In LRSM [51], there are several
contributions to NDBD in addition to the standard contribution via light
Majorana neutrino exchange owing to the presence of several heavy additional
scalar,vector and fermionic fields [52, 53, 54, 55]. Various contributions to
NDBD transition rate in LRSM are discussed as follows :

  * •

Standard Model contribution to NDBD where the intermediate particles are the
$W_{L}$ bosons and light neutrinos, the process in which the amplitude depends
upon the leptonic mixing matrix elements and light neutrino masses.

  * •

Heavy right-handed neutrino contribution in which the mediator particles are
the $W_{L}$ bosons and the amplitude depends upon the mixing between light and
heavy neutrinos as well as the mass of the heavy neutrino.

  * •

Light neutrino contribution to NDBD where the intermediate particles are
$W_{R}$ bosons and the amplitude depends upon the mixing between light and
heavy neutrinos as well as mass of the right-handed gauge boson $W_{R}$.

  * •

Heavy right-handed neutrino contribution where the mediator particles are the
$W_{R}$ bosons. The amplitude of this process is dependent on the elements of
the right handed leptonic mixing matrix and mass of the right-handed gauge
boson, $W_{R}$ as well as the mass of the heavy right handed Majorana
neutrino.

  * •

Light neutrino contribution from the Feynman diagram mediated by both $W_{L}$
and $W_{R}$, and the amplitude of the process depends upon the mixing between
light and heavy neutrinos,leptonic mixing matrix elements, light neutrino
masses and the mass of the gauge bosons, $W_{L}$ and $W_{R}$.

  * •

Heavy neutrino contribution from the Feynman diagram mediated by both $W_{L}$
and $W_{R}$, and the amplitude of the process depends upon the right handed
leptonic mixing matrix elements, mixing between the light and heavy neutrinos,
also the mass of the gauge bosons, $W_{L}$ and $W_{R}$ and the mass of the
heavy right handed neutrino.

  * •

Scalar triplet contribution ($\Delta_{L}$) in which the mediator particles are
$W_{L}$ bosons, and the amplitude for the process depends upon the masses of
the $W_{L}$ bosons, left-handed triplet Higgs, as well as their coupling to
leptons.

  * •

Right-handed scalar triplet contribution ($\Delta_{R}$) contribution to NDBD
in which the mediator particles are $W_{R}$ bosons, and the amplitude for the
process depends upon the masses of the $W_{R}$ bosons, right-handed triplet
Higgs, $\Delta_{R}$ as well as their coupling to leptons.

In our work, where we have incorporated $A_{4}$ modular symmetry to LRSM and
in our present work we have considered three of the above mentioned
contributions, one from the standard light neutrino contribution and the other
two new physics contribution mediated by $W_{R}^{-}$ and $\Delta_{R}$
respectively. For simple approximations, an assumption has been made in the
mass scales of heavy particles, where,

$M_{R}\approx M_{W_{R}}\approx M_{{\Delta_{L}}^{++}}\approx
M_{{\Delta_{R}}^{++}}\approx TeV$

. Under these assumptions, the amplitude for the light-heavy mixing
contribution which is proportional to $\frac{{m_{D}}^{2}}{M_{R}}$ remains very
small, since
$m_{\nu}\approx\frac{{m_{D}}^{2}}{M_{R}}\approx(0.01-0.1)eV,m_{D}\approx(10^{5}-10^{6})eV$
which implies $\frac{m_{D}}{M_{R}}$ $\approx$ $(10^{-7}-10^{-6})eV$. Thus in
our model, we ignore the contributions involving the light and heavy neutrino
mixings.

When NDBD is done in the framework of LRSM, the standard light neutrino
contribution is given by,

${m_{v}^{eff}}={U^{2}_{Li}}{m_{i}}$ (45)

where, $U_{Li}$ are the elements of the first row of the neutrino mixing
matrix $U_{PMNS}$, in which the elements are dependent on known mixing angles
$\theta_{13}$ , $\theta_{12}$ and the Majorana phases $\kappa$ and $\eta$. The
$U_{PMNS}$ matrix is given by,

$U_{PMNS}=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-c_{23}s_{12}-s_{23}s_{13}c_{12}e^{i\delta}&-c_{23}c_{12}-s_{23}s_{12}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{23}s_{12}-c_{23}s_{13}c_{12}e^{i\delta}&-s_{23}c_{12}-c_{23}s_{13}s_{12}e^{i\delta}&c_{23}c_{13}\end{pmatrix}P$
(46)

where, $P=diag(1,e^{i\kappa},e^{i\eta})$. So the effective mass can be
parametrized in terms of the elements of the diagonalizing matrix and the
eigenvalues as,

${m_{v}^{eff}}=m_{1}c_{12}^{2}c_{13}^{2}+m_{2}s_{12}^{2}c_{13}^{2}{e^{2i\kappa}}+m_{3}s_{13}^{2}{e^{2i\eta}}.$
(47)

##  VI Numerical Analysis and Results

In our present work, we have modified left-right symmetric model by
incorporating $A_{4}$ modular symmetry for both type-I and type-II dominances.
As we are using modular symmetry, the Yukawa couplings are expressed as
expansions of $q$ as shown in equations (22),(23) and (24). In our model, the
value of $q$ is found to be of the order of $10^{-1}$. The aboslute value of
the modulus should however be greater than 1.

$\tau=Re(\tau)+Im(\tau)$ (48)

| $Re(\tau)$ | $Im(\tau)$ | $|\tau|$
---|---|---|---
Range | [0.715,0.789] | [0.8,0.9] | [1.073,1.197]
Table 5:  Range of values corresponding to real and imaginary parts of the modulus $\tau$. Yukawa couplings | Normal Hierarchy | Inverted hierarchy
---|---|---
$Y_{1}(min)$ | $1.29155\times 10^{-9}$ |  $1.32276$ $\times$ $10^{-7}$
$Y_{1}(max)$ |  $8.22986$ $\times$ $10^{-7}$ |  $9.21382$ $\times$ $10^{-7}$
$Y_{2}(min)$ |  $3.02229$ $\times$ $10^{-9}$ |  $2.42826$ $\times$ $10^{-9}$
$Y_{2}(max)$ |  $1.32006$ $\times$ $10^{-6}$ |  $1.45952$ $\times$ $10^{-6}$
$Y_{3}(min)$ |  $3.39287$ $\times$ $10^{-9}$ |  $3.759$ $\times$ $10^{-9}$
$Y_{3}(max)$ |  $1.3165$ $\times$ $10^{-6}$ |  $1.50082$ $\times$ $10^{-6}$
Table 6:  Range of Yukawa couplings $(Y_{1},Y_{2},Y_{3})$ for both normal and
inverted hierarchy.

From table (V), it is seen that the absolute value of the modulus is greater
than unity, which is the expected result. The range of the Yukawa couplings
for our model is shown in the table above. It is seen from the table that the
minimum value of the Yukawa coupling $Y_{1}$ is in the scale of $10^{-10}$ for
normal hierarchy while for inverted hierarchy it is in the scale of $10^{-7}$.
However, for the maximum of $Y_{1}$ both the orderings are in the same scale.
For $Y_{2}$ and $Y_{3}$, the minimum and maximum values for both normal and
inverted hierarchies are within the same scale. We have plotted the effective
masses against the Yukawa couplings $(Y_{1},Y_{2},Y_{3})$ and it was found to
be well within the bound set by experiments.
As shown both for type-I and type-II dominances, we have plotted the absolute
values of the Yukawa couplings against the sum of the neutrino masses. The
range of the values for sum of neutrino masses for both the cases are given as
under,

$\sum m_{\nu}$ | Normal Hierarchy | Inverted hierarchy
---|---|---
$Type-I(min)$ | $0.000980556$ | $0.000437758$
$Type-I(max)$ | $0.177296$ | $0.186377$
$Type-II(min)$ | $0.000219304$ | $0.000035$
$Type-II(max)$ | $0.0200981$ | $0.0203081$
Table 7:  Range of values for sum of neutrino masses for type-I and type-II
dominances for both normal and inverted hierarchy.

###  VI.1 Standard Light Neutrino Contribution to $0\nu\beta\beta$

As mentioned above, in the standard light neutrino contribution to
$0\nu\beta\beta$, the intermediate particles are the $W_{L}$ bosons and light
neutrino. The effective mass for the contribution is given by equation (5.2).
Simplifying for the respective elements of $U_{Li}$ and $m_{i}$, the value of
the effective mass is obtained in terms of the modular forms
$(Y_{1},Y_{2},Y_{3})$ as,

$m_{\nu}^{eff}=m_{1}^{eff}+m_{2}^{eff}+m_{3}^{eff}$ (49)

where,

$m_{1}^{eff}=\frac{\nu(Y_{2}-Y_{3})^{2}(Y_{1}+Y_{2}+Y_{3})}{\nu_{R}(Y_{1}-Y_{3})^{2}}$
$m_{2}^{eff}=\frac{\nu^{2}(Y_{1}-Y_{2})^{2}(Y_{1}+Y_{2}+Y_{3}-\sqrt{3}\sqrt{3Y_{1}^{2}-2Y_{1}Y_{2}+3Y_{2}^{2}-2Y_{1}Y_{3}-2Y_{2}Y_{3}+3Y_{3}^{2}})}{2\nu_{R}(Y_{1}-Y_{3})^{2}}$
$m_{3}^{eff}=\frac{\nu^{2}(Y_{1}+Y_{2}+Y_{3}-\sqrt{3}\sqrt{3Y_{1}^{2}-2Y_{1}Y_{2}+3Y_{2}^{2}-2Y_{1}Y_{3}-2Y_{2}Y_{3}+3Y_{3}^{2}})}{2\nu_{R}}$

for type-I dominance, and the plots are shown as,

Figure 7: Variation of $|Y_{1}|$ with effective neutrino mass for standard
light neutrino contribution.

Figure 8: Variation of $|Y_{2}|$ with effective neutrino mass for standard
light neutrino contribution.

Figure 9: Variation of $|Y_{3}|$ with effective neutrino mass for standard
light neutrino contribution.

For type-II dominance, we have

$m_{1}^{eff}=-\frac{\nu_{L}(-Y_{2}+Y_{3})(Y_{1}+Y_{2}+Y_{3})}{Y_{1}-Y_{2}}$
$m_{2}^{eff}=-\frac{\nu_{L}(Y_{1}-Y_{3})(Y_{1}+Y_{2}+Y_{3}-\sqrt{3}\sqrt{3Y_{1}^{2}-2Y_{1}Y_{2}+3Y_{2}^{2}-2Y_{1}Y_{3}-2Y_{2}Y_{3}+3Y_{3}^{2}})}{2(Y_{1}-Y_{2})}$
$m_{3}^{eff}=\frac{\nu_{L}(Y_{1}-Y_{3})(Y_{1}+Y_{2}+Y_{3}+\sqrt{3}\sqrt{3Y_{1}^{2}-2Y_{1}Y_{2}+3Y_{2}^{2}-2Y_{1}Y_{3}-2Y_{2}Y_{3}+3Y_{3}^{2}})}{2}$

Figure 10: Variation of $|Y_{1}|$ with effective neutrino mass for standard
light neutrino contribution.

Figure 11: Variation of $|Y_{2}|$ with effective neutrino mass for standard
light neutrino contribution.

Figure 12: Variation of $|Y_{3}|$ with effective neutrino mass for standard
light neutrino contribution.

###  VI.2 Heavy Right-Handed Neutrino contribution to $0\nu\beta\beta$

In our work, we have considered contributions of heavy right-handed neutrino
and scalar Higgs triplet to NDBD. The effective mass for heavy right-handed
neutrino is given by,

$m_{R}^{eff}=p^{2}\Biggl{(}\frac{M_{W_{L}}^{4}}{M_{W_{R}}^{4}}\Biggl{)}\Biggl{(}\frac{U_{Rei}^{{*}^{2}}}{M_{i}}\Biggl{)}$
(50)

where, $p^{2}$ is the typical momentum exchange of the process. As it is known
that TeV scale LRSM plays a very important role in the process of neutrinoless
double beta decay (0$\nu\beta\beta$), we have considered the values as
$M_{W_{R}}=10TeV$ , $M_{W_{L}}=80GeV$ , $M_{\Delta_{R}}\approx 3TeV$ and after
calculation, the value for heavy right-handed neutrino is found to be in the
scale of $TeV$. The allowed value of p is in the range $(100-200)MeV$ and so
we consider, $p\approx 180MeV$. Thus, we get,

$p^{2}\Biggl{(}\frac{M_{W_{L}}^{4}}{M_{W_{R}}^{4}}\Biggl{)}=10^{10}eV^{2}$
(51)

where, $U_{Rei}$ refers to the first row elements of the diagonalizing matrix
of the heavy Majorana mass matrix and $M_{i}$ are its eigenvalues. The
effective mass corresponding to the heavy right-handed neutrino can be
expressed in terms of the modular forms as,

$m_{eff}^{R}=10^{10}(m_{eff}^{R_{1}}+m_{eff}^{R_{2}}+m_{eff}^{R_{3}})$ (52)

where,

$m_{eff}^{R_{1}}=\frac{2}{\nu_{R}(Y_{1}+Y_{2}+Y_{3}+\sqrt{3}\sqrt{3Y_{1}^{2}-2Y_{1}Y_{2}+3Y_{2}^{2}-2Y_{1}Y_{3}-2Y_{2}Y_{3}+3Y_{3}^{2}})}$
$m_{eff}^{R_{2}}=\frac{2(Y_{1}^{*}-Y_{3}^{*})^{2}}{\nu_{R}(Y_{1}+Y_{2}+Y_{3}-\sqrt{3}\sqrt{3Y_{1}^{2}-2Y_{1}Y_{2}+3Y_{2}^{2}-2Y_{1}Y_{3}-2Y_{2}Y_{3}+3Y_{3}^{2}})(Y_{1}^{*}-Y_{2}^{*})^{2}}$
$m_{eff}^{R_{3}}=\frac{(-Y_{2}^{*}+Y_{3}^{*})^{2}}{\nu_{R}(Y_{1}+Y_{2}+Y_{3})(Y_{1}^{*}-Y_{2}^{*})^{2}}$

The total effective mass is also calculated for the standard light and right-
handed heavy neutrino contribution, given by,

$|m_{\nu}^{{eff}^{total}}|=|m_{\nu}^{eff}+m_{eff}^{R}|$ (53)

which can be obtained in terms of the modular forms as a summation of the
above mentioned terms.

Figure 13: Variation of $|Y_{1}|$ with total effective neutrino mass.

Figure 14: Variation of $|Y_{2}|$ with total effective neutrino mass.

Figure 15: Variation of $|Y_{3}|$ with total effective neutrino mass.

The plots above are for type-I dominance.

Figure 16: Variation of $|Y_{1}|$ with total effective neutrino mass.

Figure 17: Variation of $|Y_{2}|$ with total effective neutrino mass.

Figure 18: Variation of $|Y_{3}|$ with total effective neutrino mass.

The figures above are the plots for type-II dominance.

###  VI.3 Scalar Triplet contribution to $0\nu\beta\beta$

The magnitude of $\Delta_{R}$ contribution is controlled by the factor
$\frac{M_{i}}{M_{\Delta_{R}}}$ [44]. However, scalar triplet contribution is
not included in the total contribution under the assumption
$\frac{M_{i}}{M_{\Delta_{R}}}<0.1$. But, some the mixing parameters in the
large part of the parameter space may result in a higher
$\frac{M_{i}}{M_{\Delta_{R}}}$ ratio and in such cases we will have to include
it in the total contribution. The impact of this contribution here is studied
in the limit, $M_{\Delta_{R}}\approxeq M_{heaviest}$.
The effective mass for scalar triplet contribution is given as,

$|m_{\Delta}^{eff}|=|p^{2}\frac{M_{W_{L}}^{4}}{M_{W_{R}}^{4}}\frac{U_{Rei}^{2}M_{i}}{M_{\Delta_{R}}^{2}}|$
(54)

The value of the mass for the right-handed scalar triplet is taken as,
$M_{\Delta_{R}}=3TeV$. So, the value of the coefficient results as,

$p^{2}\frac{M_{W_{L}}^{4}}{M_{W_{R}}^{4}}\frac{1}{M_{\Delta_{R}}^{2}}=\frac{10^{10}}{9\times
10^{24}}$ (55)

In terms of modular forms, the effective scalar mass can be expressed as,

$m_{eff}^{\Delta_{R}}=m_{eff_{1}}^{\Delta_{R}}+m_{eff_{2}}^{\Delta_{R}}+m_{eff_{3}}^{\Delta_{R}}$
(56)

where,

$m_{eff_{1}}^{\Delta_{R}}=\frac{\nu_{R}(Y_{2}+Y_{3})^{2}(Y_{1}+Y_{2}Y_{3})}{(Y_{1}-Y_{2})^{2}}$
$m_{eff_{2}}^{\Delta_{R}}=\frac{\nu_{R}(Y_{1}-Y_{3})^{2}(Y_{1}+Y_{2}+Y_{3}-\sqrt{3}\sqrt{3Y_{1}^{2}-2Y_{1}Y_{2}+3Y_{2}^{2}-2Y_{1}Y_{3}-2Y_{2}Y_{3}+3Y_{3}^{2}})}{2(Y_{1}-Y_{2})^{2}}$
$m_{eff_{3}}^{\Delta_{R}}=\frac{\nu_{R}(Y_{1}+Y_{2}+Y_{3}+\sqrt{3}\sqrt{3Y_{1}^{2}-2Y_{1}Y_{2}+3Y_{2}^{2}-2Y_{1}Y_{3}-2Y_{2}Y_{3}+3Y_{3}^{2}})}{2}$

The plots are shown as under.

Figure 19: Variation of $|Y_{1}|$ with effective neutrino mass for scalar
triplet contribution.

Figure 20: Variation of $|Y_{2}|$ with effective neutrino mass for scalar
triplet contribution.

Figure 21: Variation of $|Y_{3}|$ with effective neutrino mass for scalar
triplet contribution.

##  VII Conclusion

The discovery of neutrino oscillations paved the gateway for physics beyond
the Standard Model. In our paper, we have realized LRSM with the help of
modular $A_{4}$ symmetry for both type-I and type-II dominance. Using modular
symmetry provides the advantage of using no extra particles called ’flavons’.
The Yukawa couplings are represented as modular forms expressed as expansions
of $q$. The values of the Yukawa couplings $(Y_{1},Y_{2},Y_{3})$ are
calculated using ’Mathematica’. The mass matrices are then determined using
the multiplication rules for $A_{4}$ group stated in the Appendix. The
Majorana mass matrix is found to be symmetric and under the considered basis,
the charged lepton mass matrix is also diagonal. We have expressed the light
neutrino and heavy right-handed neutrino mass matrix in terms of the modular
forms. We have also studied briefly the contributions of $0\nu\beta\beta$ in
LRSM. The effective masses corresponding to standard light neutrino
contribution, right-handed contribution and scalar triplet contributions are
determined in terms of $(Y_{1},Y_{2},Y_{3})$ and we have plotted the effective
mass corresponding to the considered contributions against the Yukawa
couplings. To summarize our work, some results are stated as under,

  * •

The absolute value of the modulus was found to be within the range $1.073$ to
$1.197$, which is greater than unity, that is the desired result.

  * •

The Yukawa couplings, expressed in terms of modular forms ranges from
$10^{-9}$ to $10^{-6}$.

  * •

The sum of the neutrino masses for type-I dominance ranges from the order of
$10^{-4}$ to $10^{-1}$ for both normal and inverted hierarchy.

  * •

The sum of the neutrino masses for type-II dominance ranges from the order of
$10^{-4}$ to $10^{-2}$ for both normal and inverted hierarchy.

The effective masses for the $0\nu\beta\beta$ contributions are calculated and
by determining their relations with the modular forms, we have plotted the
effective masses with the three Yukawa couplings and it has been found that
the values for the effective mass corresponding to each contribution is well
within the experimental bounds, which infact makes us clearly state that the
building of the model with modular symmetry is advantageous to that of flavor
symmetries. In this model, we have not used any extra particles and the
analysis has been done taking into consideration the calculated and computed
values for the model parameters and the results are found to be satisfactory,
so it can be stated that the Left-Right Symmetric Model can be constructed
with modular symmetry while satisfying the experimental bounds on the desired
parameters.

##  VIII Appendix A

Let us consider the Higgs potential of our model that has quadratic and
quartic coupling terms given by [36],

$V_{\phi,\Delta_{L},\Delta_{R}}=-\mu_{ij}^{2}Tr[\phi_{i}^{\dagger}\phi_{j}]+\lambda_{ijkl}Tr[\phi_{i}^{\dagger}\phi_{j}]Tr[\phi_{k}^{\dagger}\phi_{l}]+\lambda_{ijkl}^{{}^{\prime}}Tr[\phi_{i}^{\dagger}\phi_{j}\phi_{k}^{\dagger}\phi_{l}]-\mu_{ij}^{2}Tr[\Delta_{L}^{\dagger}\Delta_{L}+\Delta_{R}^{\dagger}\Delta_{R}]$
$\rho_{1}[(Tr[\Delta_{L}^{\dagger}\Delta_{L}])^{2}+(Tr[\Delta_{L}^{\dagger}\Delta_{L}])^{2}]+\rho_{2}(Tr[\Delta_{L}^{\dagger}\Delta_{L}\Delta_{L}^{\dagger}\Delta_{L}]+Tr[\Delta_{R}^{\dagger}\Delta_{R}\Delta_{R}^{\dagger}\Delta_{R}])+\rho_{3}Tr[\Delta_{L}^{\dagger}\Delta_{L}\Delta_{R}^{\dagger}\Delta_{R}]+$
$\alpha_{ij}Tr[\phi_{i}^{\dagger}\phi_{j}](Tr[\Delta_{L}^{\dagger}\Delta_{L}]+Tr[\Delta_{R}^{\dagger}\Delta_{R}])+\beta_{ij}(Tr[\Delta_{L}^{\dagger}\Delta_{L}\phi_{i}\phi_{j}^{\dagger}]+Tr[\Delta_{R}^{\dagger}\Delta_{R}\phi_{i}\phi_{j}^{\dagger}])+\gamma_{ij}(Tr[\Delta_{L}^{\dagger}\phi_{i}\Delta_{R}\phi_{j}^{\dagger}]+h.c)$
(57)

where, i,j,k,l runs from 1 to 2 with $\phi_{1}=\phi$ and
$\phi_{2}=\tilde{\phi}$. As mentioned above after SSB, the scalar sector
obtains VEV. So after the substitution of the respective VEVs and determining
the traces, so after simplification the potential can be written as,

$V=-\mu^{2}(v_{L}^{2}+v_{R}^{2})+\frac{\rho}{4}(v_{L}^{4}+v_{R}^{4})+\frac{\rho^{\prime}}{2}+\frac{\alpha}{2}(v_{L}^{2}+v_{R}^{2})k_{1}^{2}+\gamma
v_{L}v_{R}k^{2}$ (58)

where, we have used the approximation $k^{\prime}<<k$, and
$\rho^{\prime}=2\rho_{3}$. Our minimization conditions are, $\frac{\delta
V}{\delta v_{L}}=\frac{\delta V}{\delta v_{R}}=\frac{\delta V}{\delta
k}=\frac{\delta V}{\delta k^{\prime}}=0$

Therefore, we get,

$\frac{\delta V}{\delta v_{L}}=-2\mu^{2}v_{L}+\rho
v_{L}^{3}+\rho^{\prime}v_{L}k^{2}+\gamma v_{R}k^{2}$ (59)

Here, it is evident that the Majorana mass of the left-handed neutrino
$M_{LL}$ is dependent on the vev $v_{L}$ as already defined above. Again, we
have

$\frac{\delta V}{\delta v_{R}}=-2\mu^{2}v_{R}+\rho
v_{R}^{3}+\rho^{\prime}v_{R}k^{2}+\gamma v_{L}k^{2}$ (60)

So, the right handed Majorana mass $M_{RR}$ is dependent on the vev $v_{R}$.
Similarly, the calculations for the same can be carried out and it can be
found out the Dirac mass term $M_{D}$ can be expressed in terms of the vev for
the Higgs bidoublet as also defined previously.
Now, we are to determine a relation between the VEVs for the scalars and so
after using the minimization conditions and simplifying the equations, we come
to a relation given by,

$v_{L}v_{R}=\frac{\gamma}{\xi}k$ (61)

where, $\xi=\rho-\rho^{\prime}$.
The neutrino mass for LRSM is given as a summation of the type-I and type-II
term as already mentioned above. So, in the approximation that
$k^{\prime}<<k$, and if we consider that our Yukawa coupling $Y^{l}$
corresponding to the neutrino masses is $y_{D}$ and the coupling
$\widetilde{Y^{l}}$ for the charged fermion masses is denoted by $y_{L}$, so
considering $y_{D}k>>y_{l}k^{\prime}$ we can write,

$M_{\nu}=\frac{k^{2}}{v_{R}}y_{D}f_{R}^{-1}y_{D}^{T}+f_{L}v_{L}$ (62)

Since, for due to left-right symmetry, we can consider $f_{L}=f_{R}=f$, so the
above equation can be written as,

$M_{\nu}=\frac{k^{2}}{v_{R}}y_{D}f^{-1}y_{D}^{T}+fv_{L}$ (63)

So, from this equation we can come to a relation given by,

$M_{\nu}=(f\frac{\gamma}{\xi}+y_{D}f^{-1}y_{D}^{T})\frac{k^{2}}{v_{R}}$ (64)

Here, we can consider two situations, namely

  * •

If $f(\frac{\gamma}{\xi})<<y_{D}f^{-1}y_{D}^{T}$, the light neutrino mass is
given by the type-I term $M_{D}M_{RR}^{-1}M_{D}^{T}$. That is, here type-I is
dominant and the light neutrino mass is from the suppression of heavy
$\nu_{R}$.

  * •

If $f(\frac{\gamma}{\xi})>>y_{D}f^{-1}y_{D}^{T}$, the light neutrino mass is
given by the type-II term $fv_{L}$. That is, in this case type-II mass term is
dominant and the light neutrino mass is because of the tiny value of
$\nu_{L}$.

##  IX Appendix B

### Properties of $A_{4}$ group

$A_{4}$ is a non-abelian discrete symmetry group which represents even
permuatations of four objects. It has four irreducible representations, three
out of which are singlets $(1,1^{\prime},1^{\prime\prime})$ and one triplet
$3$ ($3_{A}$ represents the anti-symmetric part and $3_{S}$ the symmetric
part). Products of the singlets and triplets are given by,

$1\otimes 1=1$ $1^{\prime}\otimes 1^{\prime}=1^{\prime\prime}$
$1^{\prime}\otimes 1^{\prime\prime}=1$ $1^{\prime\prime}\otimes
1^{\prime\prime}=1^{\prime}$ $3\otimes 3=1\oplus 1^{\prime}\oplus
1^{\prime\prime}\oplus 3_{A}\oplus 3_{S}$

If we have two triplets under $A_{4}$ say, $(a_{1},a_{2},a_{3})$ and
$(b_{1},b_{2},b_{3})$ , then their multiplication rules are given by,

$1\approx a_{1}b_{1}+a_{2}b_{3}+a_{3}b_{2}$ $1^{\prime}\approx
a_{3}b_{3}+a_{1}b_{2}+a_{2}b_{1}$ $1^{\prime\prime}\approx
a_{2}b_{2}+a_{3}b_{1}+a_{1}b_{2}$
$3_{S}\approx\begin{pmatrix}2a_{1}b_{1}-a_{2}b_{3}-a_{3}b_{2}\\\
2a_{3}b_{3}-a_{1}b_{2}-a_{2}b_{1}\\\
2a_{2}b_{2}-a_{1}b_{3}-a_{3}b_{1}\end{pmatrix}$
$3_{A}\approx\begin{pmatrix}a_{2}b_{3}-a_{3}b_{2}\\\ a_{1}b_{2}-a_{2}b_{1}\\\
a_{3}b_{1}-a_{1}b_{3}\end{pmatrix}$

##  X References

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