11institutetext: Yale University, New Haven CT 06511, USA
11email<EMAIL_ADDRESS>
# ALMA: Automata Learner using Modulo 2 Multiplicity Automata

Nevin George

###### Abstract

We present ALMA (Automata Learner using modulo 2 Multiplicity Automata), a
Java-based tool that can learn any automaton accepting regular languages of
finite or infinite words with an implementable membership query function.
Users can either pass as input their own membership query function, or use the
predefined membership query functions for modulo 2 multiplicity automata and
non-deterministic Büchi automata. While learning, ALMA can output the state of
the observation table after every equivalence query, and upon termination, it
can output the dimension, transition matrices, and final vector of the learned
modulo 2 multiplicity automaton. Users can test whether a word is accepted by
performing a membership query on the learned automaton.

ALMA follows the polynomial-time learning algorithm of Beimel et. al.
(Learning functions represented as multiplicity automata. J. ACM 47(3), 2000),
which uses membership and equivalence queries and represents hypotheses using
modulo 2 multiplicity automata. ALMA also implements a polynomial-time
learning algorithm for strongly unambiguous Büchi automata by Angluin et. al.
(Strongly unambiguous Büchi automata are polynomially predictable with
membership queries. CSL 2020), and a minimization algorithm for modulo 2
multiplicity automata by Sakarovitch (Elements of Automata Theory. 2009).

###### Keywords:

automata theory finite automata Büchi automata multiplicity automata learning

##  1 Introduction

Angluin’s exact learning model [1] has been studied extensively in the context
of learning theory, and it can be used to learn automata representing regular
languages of finite and infinite words. In the model, a learner interacts with
an oracle to learn a regular language using membership and equivalence
queries. In a membership query, the learner learns from the oracle whether a
word is in the language. In an equivalence query, the learner forms a
hypothesis on what the language is, and the oracle either confirms that the
hypothesis is correct or returns a counterexample, i.e, a word for which the
hypothesis and language differ.

As one application of the exact learning model, Beimel et. al. [4] detail a
polynomial-time algorithm to learn multiplicity automata using membership and
equivalence queries. We provide a high-level overview of the algorithm: the
algorithm begins with a trivial hypothesis of the language, represented as a
multiplicity automaton of dimension $1$ defined over some field $\cal K$. On
each iteration of a loop, the algorithm performs an equivalence query. If the
hypothesis is equivalent to the language, the algorithm terminates and outputs
the hypothesis. Otherwise, the algorithm receives a counterexample from the
oracle, which is used to improve the hypothesis. The algorithm terminates
after $d$ iterations, where $d$ is the dimension of the smallest possible
multiplicity automaton that can represent the language.

Strongly unambiguous Büchi automata (SUBAs) (defined in Section 2.1) are a
type of non-deterministic Büchi automaton (NBA) first introduced by Bosquet
and Löding [6]. SUBAs are useful for modeling reactive systems, as they are
fully expressive, i.e., they can represent any regular $\omega$-language, and
can often represent regular $\omega$-languages more succinctly than other NBA
representations [2]. Angluin et. al. [2] also showed that SUBAs are learnable
in polynomial time, further increasing their importance.

In this paper, we present ALMA, a Java-based tool that implements the
algorithm of Beimel et. al. and extends it to learn any arbitrary automaton
representing regular languages of finite or infinite words with an
implementable membership query function. Membership query functions have been
implemented for M2MAs and NBAs, and users can enter as input any arbitrary
membership query function that 1) takes as input any possible word that can be
formed from the given alphabet, and 2) outputs 0 or 1 to indicate whether the
word is in the language. To improve the run time of learning SUBAs, ALMA does
not use the general learning algorithm but rather the SUBA learning algorithm
of Angluin et. al. [2]. Also when learning M2MAs, ALMA first implements the
algorithm of Sakarovitch [10] to minimize the M2MA before running the learning
algorithm. This is because in the SUBA learning algorithm, converting the
input SUBA into an equivalent M2MA incurs a quadratic increase size.
Minimizing M2MAs before learning enables the algorithm to learn significantly
larger SUBAs, especially because empirically the M2MAs have been seen to often
minimize to M2MAs of much smaller dimension. The effect of the minimization
algorithm on a sample of input SUBAs can be seen in Table 2 in Section 5.

####  1.0.1 Usefulness/Novelty

In the verification community, M2MAs can be useful for representing regular
$\omega$-languages, as they are learnable in polynomial time and often
relatively succinct. They are also useful for tasks such as model checking,
since performing operations such as intersection, union, complementation,
emptiness, and equivalence on M2MAs is cheap [3]. Through the minimization and
learning algorithms, ALMA enables researchers in the verification community to
easily test and verify these properties of M2MAs, promoting further
exploration into these useful automata. As an example, ALMA was used in the
paper by Angluin et. al. [3] to explore the suitableness of representing
regular $\omega$-languages using M2MAs. The authors used ALMA to convert
SUBAs, NBAs, and DBAs into equivalent M2MAs, and they compared the
succinctness of these M2MAs with that of DFAs accepting the same language.
ALMA can similarly be used by other researchers to gain insights into M2MAs
and their benefits/drawbacks as compared to other representations.

ALMA is the first publicly available implementation of the novel SUBA learning
algorithm by Angluin et. al. [2]. Using ALMA, users can run the algorithm to
learn any regular $\omega$-language. Since learned automata are represented as
M2MAs, users can use the many desirable properties of M2MAs to gain insights
into the initial SUBA and regular $\omega$-language. In addition, since
membership query functions are often relatively easy to implement and ALMA
already provides a membership query function for the general NBA case, the
scope of what ALMA can learn is large, promoting its usefulness in a wide
variety of settings.

Many tools such as ROLL ($\omega$-Regular Language Learning Library) [9] and
libalf [5] already exist that can learn automata representing regular
languages of finite and infinite words. However, ALMA is the first tool that
uses M2MAs to represent hypotheses and the learned automaton in the learning
algorithm. ALMA’s usefulness lies not with necessarily being the fastest tool
available to learn regular languages, but with exploring the practical
benefits of M2MAs and the features of the algorithms by Beimel et. al.,
Angluin et. al., and Sakarovitch.

##  2 Useful Definitions

###  2.1 Finite and Büchi Automata

Let $\Sigma$ be a finite alphabet. Then $\Sigma^{*}$ and $\Sigma^{\omega}$ are
the sets of all finite and infinite words, respectively, that can be formed
using elements from $\Sigma$. A finite language is a subset of $\Sigma^{*}$,
and an $\omega$-language is a subset of $\Sigma^{\omega}$. If $w$ is a word in
$\Sigma^{*}$ or $\Sigma^{\omega}$, let $|w|$ be the length of $w$ and $w[i]$
be the $i$’th character of $w$.

A finite-state automaton $A$ is represented as a tuple
($\Sigma,Q,I,\Delta,F$), where $\Sigma$ is the alphabet, $Q$ is the finite set
of states, $I\subseteq Q$ is the set of initial states, $\Delta\subseteq
Q\times\Sigma\times Q$ is the set of transitions, and $F\subseteq Q$ is the
set of final states. The automaton $A$ is deterministic if every pair
$(q,\sigma)\in Q\times\Sigma$ appears as the first two elements in at most one
triple in $\Delta$.

A run on $A$ for a word $w$ is a series of states $q_{0},q_{1},\ldots\in Q$
such that $\forall i$ satisfying $1\leq i\leq|w|$,
$(q_{i-1},w[i],q_{i})\in\Delta$. A run for a finite word is final if it ends
in a final state. For infinite words, a run is final if it passes infinitely
often through a final state. A run is accepting if it is final and begins at
an initial state. The automaton $A$ accepts the finite/infinite word $w$ if
there exists an accepting run for $w$.

Non-deterministic finite automata (NFAs) and non-deterministic Büchi automata
(NBAs) are automata accepting finite and infinite words, respectively.
Deterministic finite automata (DFAs) are deterministic NFAs, and deterministic
Büchi automata (DBAs) are deterministic NBAs. Unambiguous finite automata
(UFAs) and unambiguous Büchi automata (UBAs) are NFAs and NBAs, respectively,
for which every word has at most one accepting run. A UBA is a strongly
unambiguous Büchi automaton (SUBA) if every word has at most one final run.

###  2.2 Modulo 2 Multiplicity Automata

Assume a field $\cal K$ and some dimension $d$. A multiplicity automaton $A$
is represented as a tuple
($\Sigma,v_{I},\\{\mu_{\sigma}\\}_{\sigma\in\Sigma},v_{F}$), where $\Sigma$ is
the alphabet, $v_{I}$ is the initial vector, each $\mu_{\sigma}$ is a
transition matrix, and $v_{F}$ is the final vector. $v_{I}$ and $v_{F}$ have
dimension $d\times 1$, and each $\mu_{\sigma}$ has dimension $d\times d$. The
set of states is all row vectors $v\in\\{0,1\\}^{d}$, and the initial state is
$v_{I}^{\top}$. For a given word $w=\sigma_{1}\sigma_{2}\ldots\sigma_{n}$, let
$\mu(w)=\mu_{\sigma_{1}}\mu_{\sigma_{2}}\ldots\mu_{\sigma_{n}}$. The set of
reachable states is all vectors of the form $v_{I}^{\top}\mu(w)$. Associated
with the automaton $A$ is a function $f_{A}:\Sigma^{*}\to\mathcal{K}$, where
$\forall w\in\Sigma^{*}$,

$f_{A}(w)=v_{I}^{\top}\mu(w)v_{F}.$

A modulo 2 multiplicity automaton (M2MA) is a multiplicity automaton where
$\mathcal{K}=$ GF(2) and all calculations are done modulo 2. An M2MA $A$
accepts a word $w\in\Sigma^{*}$ if and only if $f_{A}(w)=1$.

As an example, consider the following M2MA $M$ adapted from Angluin et. al.
[2].

$M=(\\{a,b\\},\begin{pmatrix}1&0&0\end{pmatrix}^{\top},\\{\mu_{a},\mu_{b}\\},\begin{pmatrix}1&1&0\end{pmatrix}^{\top}\\}$

where

$\mu_{a}=\begin{pmatrix}0&0&1\\\ 1&0&0\\\ 1&1&1\end{pmatrix}\text{and
}\mu_{b}=\begin{pmatrix}0&1&0\\\ 1&0&1\\\ 1&1&0\end{pmatrix}.$

$M$ is equivalent to the DFA in Fig. 1. We explain Fig. 1: the computation
begins at the initial state $\begin{pmatrix}1&0&0\end{pmatrix}$. On reading an
$a/b$ from the initial state, the DFA visits the states

$\begin{pmatrix}1&0&0\end{pmatrix}\mu_{a}=\begin{pmatrix}1&0&0\end{pmatrix}\begin{pmatrix}0&0&1\\\
1&0&0\\\ 1&1&1\end{pmatrix}=\begin{pmatrix}0&0&1\end{pmatrix}$
$\begin{pmatrix}1&0&0\end{pmatrix}\mu_{b}=\begin{pmatrix}1&0&0\end{pmatrix}\begin{pmatrix}0&1&0\\\
1&0&1\\\ 1&1&0\end{pmatrix}=\begin{pmatrix}0&1&0\end{pmatrix}.$

Other states are visited similarly by multiplying the current state vector by
$\mu_{a}$ or $\mu_{b}$. $\begin{pmatrix}1&0&0\end{pmatrix}$,
$\begin{pmatrix}0&1&0\end{pmatrix}$, and $\begin{pmatrix}1&0&1\end{pmatrix}$
are accepting states since

$\begin{pmatrix}1&0&0\end{pmatrix}\begin{pmatrix}1&1&0\end{pmatrix}^{\top}=\begin{pmatrix}0&1&0\end{pmatrix}\begin{pmatrix}1&1&0\end{pmatrix}^{\top}=\begin{pmatrix}1&0&1\end{pmatrix}\begin{pmatrix}1&1&0\end{pmatrix}^{\top}=1,$

where $v_{F}=\begin{pmatrix}1&1&0\end{pmatrix}^{\top}$.

$100$start$010$$101$$001$$111$$110$$b$$b$$b$$a$$a$$a$$b$$b$$a$$a$$a$$b$ Figure
1: DFA for the M2MA $M$

M2MAs have many important properties described in Section 1 that make them
useful in verification communities, e.g., learnable in polynomial time and
cheap intersection, union, complementation, emptiness, and equivalence. For
more information on M2MAs, we recommend reading the paper by Angluin et. al.
[3], which studies these properties of M2MAs extensively and performs a
detailed analysis of M2MAs’ ability to succinctly represent regular finite and
$\omega$-languages.

###  2.3 $L_{\$}$ Language

Since Büchi automata accept infinite words and M2MAs accept finite words,
Büchi automata and M2MAs cannot accept words from the same language $L$.
However, Büchi [7] showed that two regular $\omega$-languages are equivalent
if and only if they agree on a set of ultimately periodic words, i.e., words
of the form $u(v)^{\omega}$ where $u$ and $v$ are finite words and $v$ is non-
empty. We then consider the language of finite words $L_{\$}=\\{u\$v\mid
u(v)^{\omega}\in L\\}$, which Calbrix et. al. [8] showed is regular. If a
Büchi automaton accepts an infinite language $L$, a finite automaton is said
to also represent $L$ if it accepts $L_{\$}$. The $L_{\$}$ language is used by
Angluin et. al. [2] in their SUBA learning algorithm in order to obtain a
finite automaton equivalent to the initial SUBA, and the learned M2MA accepts
words from $L_{\$}$.

##  3 Usage

###  3.1 Access and How to Run

ALMA is an open-source library freely available at the following GitHub
repository: https://github.com/nevingeorge/Learning-Automata. The repository
contains the executables, source files, and sample input files for ALMA, and
the README document contains detailed information on how to use the
executables and the required format for the input files.

ALMA is a Java-based tool designed to be used on the command line. To run for
example the executable M2MA.jar, a user should enter the command java -jar
M2MA.jar within a Terminal/Command Prompt window. This will start the program,
and the program will then print instructions on how to input the desired input
file and flags.

###  3.2 Executables

ALMA consists of five main executables: 1) M2MA.jar, 2) SUBA.jar, 3)
minimize.jar, 4) NBA.jar, and 5) arbitrary.jar. We provide a basic overview of
each of the jar files and their use cases. For each executable, the output is
always the dimension, final vector, and transition matrices of the
learned/minimized M2MA. Also after the algorithms terminate, each executable
allows users to check whether a word is accepted by the outputted M2MA.
Infinite words are represented using the $L_{\$}$ language explained in
Section 2.3.

####  3.2.1 M2MA.jar

is used to minimize and learn M2MAs. The alphabet, dimension, final vector,
and transition matrices of the input M2MA must be specified (the initial
vector is always the vector with all zeros except for a 1 in the first row).

####  3.2.2 SUBA.jar

is used to learn SUBAs using the algorithm of Angluin et. al. [2]. The
alphabet, number of states, final states, and transitions of the input SUBA
must be specified (the only initial state is the first state).

####  3.2.3 minimize.jar

is used to minimize M2MAs using the algorithm of Sakarovitch [10]. The input
is the same as that for M2MA.jar. minimize.jar can also be used to output the
minimized M2MA equivalent to the input SUBA in the SUBA learning algorithm
(i.e., it runs every step of the SUBA learning algorithm except for learning
the final M2MA). In this case, the input is the same as that for SUBA.jar.

####  3.2.4 NBA.jar

is used to learn NBAs. The alphabet, number of states, final states, and
transitions of the input NBA must be specified (the only initial state is the
first state). NBA.jar uses approximate equivalence queries (described in
Section 4.5.3), which requires as input the number of tests to run, maximum
length of a test, and maximum limit on the number of equivalence queries.

####  3.2.5 arbitrary.jar

is used to learn arbitrary automata representing regular languages of finite
and infinite words with an implementable membership query function. The
membership query function must be defined in MQ.java. The only constraints on
the function is that it must accept as input any possible word formed from
letters in the alphabet, and it must output 0 or 1 depending on whether the
word is contained in the language. arbitrary.jar also uses approximate
equivalence queries, which requires the same input parameters as described for
NBA.jar.

###  3.3 Flags

Users can enter the following optional flags to add or remove information from
the output of the algorithm.

-v:

The algorithm outputs the state of the observation table (a matrix consisting
of the words whose membership in the language is known) after every
equivalence query in the learning algorithm.

-m:

The algorithm outputs detailed information on the progress of the minimization
algorithm. It gives status updates at different points in the algorithm, such
as when it finishes creating the state/co-state spaces and the observation
table. It also outputs the initial M2MA to be minimized and the final
minimized observation table.

-d:

When running minimize.jar on a SUBA, the algorithm outputs only the dimension
of the minimized M2MA instead of the dimension, final vector, and transition
matrices of the M2MA.

-a:

After minimizing the M2MA, the algorithm outputs the number of states of the
minimal deterministic finite automaton (DFA) that represents the same language
as the M2MA.

##  4 Implementation Details

###  4.1 Architecture

Figure 2: M2MA.jar Architecture Figure 3: SUBA.jar Architecture Figure 4:
minimize.jar Architecture Figure 5: NBA.jar and arbitrary.jar Architecture

Figures 2-5 provide a high-level overview of the architecture for each of the
five executables. The learning algorithm of Beimel et. al. [4], described in
more detail in Section 4.3, is implemented in every executable except for
minimize.jar. The M2MA minimization algorithm of Sakarovitch [10] (described
in Section 4.2) is implemented in M2MA.jar, SUBA.jar, and minimize.jar. All
five executables first take in as input from the command line the name of the
input file and optional flags. The input is passed to a parser, which then
reads the input and sends the parameters of the automaton to be
learned/minimized to the algorithms. The executables also all run final checks
(described in Section 4.6) on the outputted M2MA. Once the checks complete,
users can test whether a word is accepted by the automaton.

###  4.2 M2MA Minimization Algorithm

The M2MA minimization algorithm, implemented in the minimize function of
M2MA.java, is described in abstract terms in the work by Sakarovitch [10] and
in more concrete detail in the appendix of [3]. The implemented minimization
algorithm follows the pseudo-code in the appendix of [3] closely.

The algorithm requires a heavy use of linear algebra, much of which is done
using the Apache Commons Math package. While testing the executables, however,
many of the unminimized M2MAs were seen to be relatively sparse with few 1’s
in the transition matrices. To take advantage of the sparseness, ALMA
implements custom linear algebra functions that use a sparse representation of
matrices. Matrix rows are represented using arrays containing the locations of
the 1’s in the row. For example, the row
$\begin{bmatrix}0&1&0&0&0&1\end{bmatrix}$ is represented as
$\begin{bmatrix}2&6\end{bmatrix}$, indicating that the row has a 1 in columns
2 and 6. Matrix multiplication, dot products, and tests for linear
independence are implemented using the sparse matrix representation. Since the
minimization function deals with M2MAs, the custom linear algebra functions
perform all calculations modulo 2, which further improves the run time.

###  4.3 General Learning Algorithm

The learning algorithm of Beimel et. al. [4] is the core algorithm
underpinning ALMA, and it is implemented in the learn function of M2MA.java.
We provide a high-level overview of the algorithm in Section 1, and more
details on the algorithm are found in [4]. Like the minimization algorithm,
the linear algebra is implemented using a combination of the Apache Commons
Math package and custom functions using the sparse representation for
matrices.

The default membership query function the learning algorithm uses is that for
M2MAs. As described in Section 2.2, an M2MA
$A=(\Sigma,v_{I},\\{\mu_{\sigma}\\}_{\sigma\in\Sigma},v_{F})$ accepts a word
$w$ if $v_{I}^{\top}\mu(w)v_{F}=1$. Equivalence queries for M2MAs are easy,
since as a by-product of running the minimization algorithm we get a complete
observation table for the automaton. The equivalence query function checks
whether the current hypothesis agrees with the M2MA being learned on every
word in the observation table, as well as all possible one-letter extensions
of the words. If they agree on every word and one-letter extension, the
algorithm terminates; otherwise, the algorithm returns a word for which they
disagree on as a counterexample.

###  4.4 SUBA Learning Algorithm

The SUBA learning algorithm of Angluin et. al. [2] works as follows: the input
SUBA is first converted into an equivalent UFA using a simple construction by
Bosquet and Löding [6]. If the SUBA has $n$ states, the constructed UFA has
size $2n^{2}+n$. Next, the UFA is converted to an equivalent M2MA of the same
size. Lastly, the M2MA is learned using the algorithm of Beimel et. al. [4].
The algorithms for the SUBA to UFA and UFA to M2MA conversions are found in
SUBA.java, and the outputted M2MA is sent to M2MA.java to be minimized and
learned.

###  4.5 Learning NBA and Arbitrary Automata

In M2MA.jar, minimize.jar, and SUBA.jar, the learning algorithm runs
membership and equivalence queries on M2MAs. The learning algorithm for
NBA.jar, however, uses a membership query function designed specifically for
NBAs which we describe in this section. arbitrary.jar passes to the learning
algorithm a membership query function specified by the user from MQ.java. Also
since the equivalence query function for M2MAs doesn’t work in the general
setting, NBA.jar and arbitrary.jar use approximate equivalence queries that
rely on testing a random sample of words from the language.

####  4.5.1 NBA Membership Query Function

The NBA membership query function is described in Algorithm 1. As explained in
Section 2.1, an NBA accepts a word if there exists a path for the word that
begins at an initial state and passes infinitely often through a final state.
The input parameters $u,v$ represent a word $w=u\$v$ in the $L_{\$}$ language.

Algorithm 1 NBA Membership Query Function

1:function main($u$, $v$)

2: $S_{u}\leftarrow$ states reachable from the initial state on reading $u$

3: $S_{uv}\leftarrow S_{u}\cup$ reachable($S_{u}$, $v$)

4: for all $s\in S_{uv}$ do

5: $S_{uv}^{\prime}\leftarrow$ reachable($\\{s\\}$, $v$)

6: if $s\in S_{uv}^{\prime}$ $\And$ passed a final state to reach $s$ then

7: return 1

8: end if

9: end for

10: return 0

11:end function

12:

13:function reachable($S$, $v$)

14: $S_{v}\leftarrow$ states reachable from a state in $S$ on reading $v$

15: $S_{v}^{\prime}\leftarrow\varnothing$

16: repeat

17: $S_{v}\leftarrow S_{v}\cup S_{v}^{\prime}$

18: $S_{v}^{\prime}\leftarrow$ states reachable from a state in $S_{v}$ on
reading $v$

19: until $S_{v}^{\prime}\subseteq S_{v}$

20: return $S_{v}$

21:end function

In Algorithm 1, the reachable function finds all the states reachable from a
set of initial states $S$ on reading some positive number of $v$’s. On every
iteration of the loop defined in line 16, the function finds the states
reachable on reading another $v$, and the function terminates after an
iteration of the loop where no new states are found.

In the main function, in lines 2-3 Algorithm 1 stores in $S_{uv}$ all the
states reachable from the initial state on reading a $u$ and a non-negative
number of $v$’s. Then in lines 4-10, the algorithm determines whether there is
an accepting loop on any of the states in $S_{uv}$ (i.e., a path from a state
in $S_{uv}$ to itself that passes through a final state).

####  4.5.2 Run Time Analysis

Let $n$ be the number of states, $m$ be the number of transitions, and $l$ be
the length of $u\$v$. Line 2 runs in $O(nml)$ \- we read each of the $O(l)$
characters in $u$ one at a time, and with each character we calculate the new
states that can be reached using the $O(m)$ transitions from the $O(n)$ states
reached so far. reachable runs in $O(n^{2}ml)$ \- the loop runs at most $n$
times since there are at most $n$ states to add to $S$, and line 18 runs in
$O(nml)$. The loop in line 4 terminates after $O(n)$ iterations, and since
reachable is called in line 5, the loop runs in $O(n^{3}ml)$. Therefore,
Algorithm 1 runs in $O(n^{3}ml)$.

####  4.5.3 Approximate Equivalence Queries

The approximate equivalence query function is defined in arbitrary.java. Along
with the hypothesis, it requires two parameters $n$ and $l$ as input. The
function generates $n$ random words of length at most $l$, and it tests
whether the hypothesis and automaton being learned agrees on these words.
Increasing $n$ improves the accuracy of the function at the expense of the run
time. In the input file, users also give a limit $m$ on the number of
equivalence queries that can be run. Since there are no guarantees on the size
of the learned M2MA, the parameter $m$ prevents the algorithm from running for
an arbitrarily long length of time.

###  4.6 Checks

The code performs checks at various points of the algorithm. If a check fails,
the code throws an exception and terminates the program. Example checks
include checking the validity of the input (e.g., correct number of transition
matrices, matrices only contain 0’s and 1’s, etc.), whether a matrix is
invertible before running a linear equation solver, and if the dimension of
the minimized M2MA equals that of the final learned M2MA.

After learning an M2MA, the code checks whether the outputted M2MA agrees with
the input automaton on the membership of a random sample of words. By default,
the code generates 1000 random words of length at most 25, but these constants
can be modified. To perform this check, the code uses the previously described
membership query functions for M2MAs, NBAs, and arbitrary automata, as well as
a membership query function for SUBAs described in a paper by Bosquet and
Löding [6]. Users can also manually confirm whether the outputted M2MA accepts
words from the language in the “Ready for Operations” phase of the executables
described in Section 4.1.

##  5 Experimental Evaluation

The GitHub repository contains many input files that can be used to test each
of the executables. There exist input files for many different sizes of M2MAs,
SUBAs, and NBAs, as well as input files for different edge cases (e.g., M2MAs
of dimension 1).

Table 1: M2MA.jar Run Time M2MA Dimension | Average Run Time
---|---
10 | 0.33s
20 | 2.44s
30 | 9.76s
40 | 28.57s
50 | 82.57s
60 | 177.00s
70 | 355.09s
80 | 657.82s
90 | 1084.85s
100 | 1819.80s
Table 2: SUBA.jar Run Time SUBA Language | SUBA Size |  | Unminimized
---
M2MA Dim
| Learned
---
M2MA Dim
Run Time
$a\Sigma^{*}(\Sigma^{*}b\Sigma^{*})^{\omega}$ | 2 | 10 | 5 | 0.20s
$\Sigma^{*}a\Sigma^{5}ab^{\omega}$ | 8 | 136 | 10 | 0.30s
| $((a+b)^{*}(a(a+b)a(a+b)c$
---
$+b(a+b)b(a+b)d))^{\omega}$
9 | 171 | 92 | 50.86s
$(a^{*}a^{4}b)^{\omega}$ | 5 | 55 | 32 | 0.57s
$(a^{*}a^{5}b)^{\omega}$ | 6 | 78 | 44 | 2.04s
$(a^{*}a^{6}b)^{\omega}$ | 7 | 105 | 58 | 6.72s
$a^{\omega}$ | 1 | 3 | 3 | 0.05s
$(ab^{5})^{\omega}$ | 6 | 78 | 43 | 0.45s
$(ab^{10})^{\omega}$ | 11 | 253 | 133 | 25.11s
$(ab^{15})^{\omega}$ | 16 | 528 | 273 | 411.48s
$(ab^{20})^{\omega}$ | 21 | 903 | 463 | 3064.32s
Table 3: NBA.jar Run Time NBA Size |  | Number of
---
tests/EQ
| Learned M2MA
---
Dimension
Run Time
2 | 10000 | 3 | 1.02s
4 | 10000 | 7 | 0.25s
6 | 10000 | 24 | 1.80s
8 | 100000 | 27 | 14.00s
10 | 100000 | 83 | 983.39s

Tables 1-3 give a sense of the run times for M2MA.jar, SUBA.jar, and NBA.jar.
The experiments were run on a standard laptop with 8 GB RAM, 8 cores, and a
CPU frequency of 3200 MHz.

Table 1 details the run time of M2MA.jar. Fifty random M2MAs with the alphabet
$\\{a,b,c\\}$ were generated for each of the dimensions $10,20,\ldots,50$, and
the average run times were calculated. For the dimensions $60,70,\ldots,100$,
ten random M2MAs were generated. These results can be reproduced using the
executable M2MA_experiments.jar in the Experimental Evaluation folder of the
GitHub repository. Instructions on how to use the jar file are printed to
standard output once the executable is run.

Table 2 details the run time of SUBA.jar. The SUBA input files used to
generate the results are found in the Experimental Evaluation folder of the
GitHub repository.

Table 3 details the run time of NBA.jar. The NBAs in the table were randomly
generated, with one NBA generated per size. Since NBA.jar uses approximate
equivalence queries, the table contains a column for the number of tests to
run for every equivalence query. The maximum length of a test for every NBA in
the table is 25. The NBA input files used to generate the results are found in
the Experimental Evaluation folder of the GitHub repository.

####  5.0.1 Practical Capabilities

The run times for M2MA.jar increase at a roughly cubic rate. For smaller
M2MAs, the program runs quickly, but for M2MAs of dimension larger than 100,
the program can take hours to terminate. For the purpose of using ALMA to
explore the properties of M2MAs, it is unlikely that one will need to work
with M2MAs of dimension much larger than 100, so the program’s run time should
not pose a significant constraint.

One may think that experimenting with the SUBA learning algorithm, which
incurs a $2n^{2}+n$ blow up in size going from the initial SUBA to the
converted M2MA, will be impractical when dealing with SUBAs of size even
greater than 10. However, as can be seen in Table 2, the unminimized M2MA in
the SUBA learning algorithm often minimizes to a much smaller M2MA, which is
why the tool implements the minimization algorithm of Sakarovitch [10]. For
example, the SUBA of size 21 representing the language $(ab^{20})^{\omega}$
converts into a large unminimized M2MA of dimension 903. However, it minimizes
to an M2MA of dimension 463, and SUBA.jar runs in less than an hour for this
SUBA.

The run times for NBA.jar depend heavily on the number of tests performed
every equivalence query. To get decent approximations on the learned M2MA
dimension for NBAs of size larger than 10, the number of tests per equivalence
query should be at least on the order of 10 to the 5th or 6th power. One can
try to change the maximum length of the tests to get better results. For
randomly generated NBAs of size much larger than 10, the trade-off between the
run time and accuracy becomes much more apparent, and the number of tests per
equivalence query may have to be decreased for the run time to be practical.

##  6 Conclusion

ALMA has limitations - for example, the run time of M2MA.jar becomes
impractical for random M2MAs of size much larger than 100, and ALMA doesn’t
implement an exact equivalence query function for learning NBAs and arbitrary
automata. However, ALMA is highly efficient for most standard use cases, and
it can be used to promote further research into M2MAs and their properties.
For example, in the paper by Angluin et. al. [3], ALMA is used to find the
dimension of the minimum M2MA that can represent a regular $\omega$-language.
Angluin et. al. compare this dimension with the size of other finite automata
that represent the same language to analyze the succinctness of the M2MA
representation. Along with serving as a useful tool for investigating M2MAs,
ALMA confirms the theoretical results of Beimel et. al. [4] and Angluin et. al
[2, 3], and is the first publicly available tool that can be used to explore
these learning algorithms.

####  6.0.1 Acknowledgements

I would like to thank Dana Angluin, Timos Antonopoulos, and Dana Fisman for
their help with this paper and all the feedback and advice they gave. Dana
Angluin especially helped me significantly during the entire process of
creating ALMA, and I would like to thank her greatly for her support and
mentorship.

## References

  * [1] Angluin, D.: Queries and concept learning. Machine Learning 2(4), 319–342 (1987)
  * [2] Angluin, D., Antonopoulos, T., Fisman, D.: Strongly unambiguous Büchi automata are polynomially predictable with membership queries. In: 28th EACSL Annual Conference on Computer Science Logic, CSL. pp. 8:1–8:17 (2020)
  * [3] Angluin, D., Antonopoulos, T., Fisman, D., George, N.: Representing regular languages of infinite words using mod 2 multiplicity automata. In: Foundations of Software Science and Computation Structures - 25th International Conference, FOSSACS. p. 1–20 (2022)
  * [4] Beimel, A., Bergadano, F., Bshouty, N.H., Kushilevitz, E., Varricchio, S.: Learning functions represented as multiplicity automata. J. ACM 47(3), 506–530 (May 2000)
  * [5] Bollig, B., Katoen, J.P., Kern, C., Leucker, M., Neider, D., Piegdon, D.R.: libalf: The automata learning framework. In: Computer Aided Verification - 22nd International Conference, CAV. pp. 360–364 (2010)
  * [6] Bousquet, N., Löding, C.: Equivalence and inclusion problem for strongly unambiguous Büchi automata. In: Language and Automata Theory and Applications, 4th International Conference, LATA. Proceedings. pp. 118–129 (2010). https://doi.org/10.1007/978-3-642-13089-2_10
  * [7] Büchi, J.: On a decision method in restricted second order arithmetic. In: International Congress on Logic, Methodology and Philosophy. pp. 1–11. Stanford Univ. Press (1962)
  * [8] Calbrix, H., Nivat, M., Podelski, A.: Ultimately periodic words of rational w-languages. In: Proceedings of the 9th International Conference on Mathematical Foundations of Programming Semantics. pp. 554–566. Springer-Verlag (1994)
  * [9] Li, Y., Sun, X., Turrini, A., Chen, Y.F., Xu, J.: Roll 1.0: $\omega$-regular language learning library. In: Tools and Algorithms for the Construction and Analysis of Systems - 25th International Conference, TACAS, Part I. pp. 365–371 (2019)
  * [10] Sakarovitch, J.: Elements of Automata Theory. Cambridge University Press, USA (2009)


# Manifestations of non-zero Majorana CP violating phases in oscillations of
supernova neutrinos

Artem Popov<EMAIL_ADDRESS>Department of Theoretical Physics, Moscow
State University, 119991 Moscow, Russia  Alexander Studenikin
<EMAIL_ADDRESS>Department of Theoretical Physics, Moscow State
University, 119991 Moscow, Russia  Joint Institute for Nuclear Research,
141980 Dubna, Russia

###### Abstract

We investigate effects of non-zero Dirac and Majorana CP violating phases on
neutrino-antineutrino oscillations in a magnetic field of astrophysical
environments. It is shown that in the presence of strong magnetic fields and
dense matter, non-zero CP phases can induce new resonances in the oscillations
channels $\nu_{e}\leftrightarrow\bar{\nu}_{e}$,
$\nu_{e}\leftrightarrow\bar{\nu}_{\mu}$ and
$\nu_{e}\leftrightarrow\bar{\nu}_{\tau}$. We also consider all other possible
oscillation channels with $\nu_{\mu}$ and $\nu_{\tau}$ in the initial state.
The resonances can potentially lead to significant phenomena in neutrino
oscillations accessible for observation in experiments. In particular, we show
that neutrino-antineutrino oscillations combined with Majorana-type CP
violation can affect the $\bar{\nu}_{e}$/$\nu_{e}$ ratio for neutrinos coming
from the supernovae explosion. This effect is more prominent for the normal
neutrino mass ordering. The detection of supernovae neutrino fluxes in the
future experiments, such as JUNO, DUNE and Hyper-Kamiokande, can give an
insight into the nature of CP violation and, consequently, provides a tool for
distinguishing the Dirac or Majorana nature of neutrinos.

##  I Introduction

CP symmetry implies that the equations of motion of a system remain invariant
under the CP-transformation, that is a combination of charge conjugation (C)
and parity inversion (P). In 1964, with the discovery of the neutral kaon
decay [1], it was confirmed that CP is not an underlying symmetry of the
electroweak interactions theory, thus opening a vast field of research in CP
violation. Currently, CP violation is a topic of intense studies in particle
physics that also has important implications in cosmology. In 1967, Sakharov
proved that the existence of CP violation is a necessary condition for
generation of the baryon asymmetry through baryogenesis in the early Universe
[2]. A review of possible baryogenesis scenarios can be found in [3].

Today we have solid understanding of CP violation in the quark sector, that
appears due to the complex phase in the Cabibbo–Kobayashi–Maskawa matrix
parametrisation. Its magnitude is expressed by the Jarlskog invariant
$\mathcal{J}_{CKM}=(3.18\pm 0.15)\times 10^{-5}$ [4], which seems to be
excessively small to engender baryogenesis at the electroweak phase transition
scale [3]. However, in addition to experimentally confirmed CP violation in
the quark sector, CP violation in the lepton (neutrino) sector hypothetically
exists (see [5] for a review). Leptonic CP violation is extremely difficult to
observe due to weakness of neutrino interactions. In 2019, a first
breakthrough happened when T2K collaboration reported constraints on the Dirac
CP violating phase in neutrino oscillations [6]. Hopefully, future gigantic
neutrino experiments, such as JUNO, DUNE and Hyper-Kamiokande, will have a
good chance significantly improve this results. Note that leptonic CP
violation plays an important role in baryogenesis through leptogenesis
scenarios [7].

The CP violation pattern in the neutrino sector depends on whether neutrino is
a Dirac or Majorana particle. The Pontecorvo–Maki–Nakagawa–Sakata neutrino
mixing matrix in the most common parameterization has the following form

$U=\begin{pmatrix}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{pmatrix}\begin{pmatrix}c_{13}&0&s_{13}e^{-i\delta}\\\
0&1&0\\\
-s_{13}e^{i\delta}&0&c_{13}\end{pmatrix}\begin{pmatrix}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}e^{i\alpha_{1}}&0&0\\\
0&e^{i\alpha_{2}}&0\\\ 0&0&1\end{pmatrix},$ (1)

where $\delta$ is the Dirac CP violating phase, the additional phases
$\alpha_{1}$ and $\alpha_{2}$ are the so-called Majorana CP violating phases,
which can be non-zero only for the case of Majorana neutrinos, and
$c_{ik}=\cos\theta_{ik}$ and $s_{ik}=\sin\theta_{ik}$. As it was shown in [8,
9], it is impossible to observe the Majorana CP phases in the neutrino flavour
oscillations (see also [10, 11] for a recent discussion). Nevertheless, in
[12] the authors stated that in principle it is possible to measure Majorana
CP phases in neutrino-antineutrino oscillations due to tiny effects of non-
zero neutrino masses. However, the probability of this oscillation process for
Majorana neutrinos is extremely low to be observed in the near future. Recall
that presently the experiments on neutrinoless double beta-decay are
considered to be the most promising potential way of measuring the Majorana CP
violating phases.

Thus, to measure the Majorana CP violating phases is a challenging task for
future physics. In this paper, we study neutrino-antineutrino oscillations
engendered by the interaction with a magnetic field in astrophysical
environments and search for possible manifestations of the Majorana CP
violating phases in neutrino fluxes from supernovae. We show that the effects
of the non-zero Majorana CP violating phases indeed can be observed in the
neutrino oscillations. Therefore, the detection of supernovae neutrino fluxes
can give an insight into the nature of CP violation and, consequently, provide
a tool for distinguishing the Dirac or Majorana nature of neutrinos.

The paper is organized as follows. In Sec. II we present a brief introduction
into the theory of Majorana neutrinos interactions and mixing. In Sec. III we
develop the formalism for description of neutrino oscillations in external
fields. Sec. IV presents our numerical results on effects of the Majorana CP
violating phases in neutrino-antineutrino oscillations in astrophysical media.
Finally, Sec. V summarises our results.

##  II Interactions of Majorana neutrinos

In this section we briefly introduce the theory of Majorana neutrino and its
interactions. An arbitrary spinor can be represented as the sum of two
independent chiral components

$\Psi_{D}=\Psi_{L}+\Psi_{R}$ (2)

and has four degrees of freedom (corresponding to particle and antiparticle
with two helicities). The Majorana theory of fermions implies that the left
and right components of a field are no longer independent and satisfy the
relation: $\Psi_{R}=\Psi_{L}^{c}$. Thus, a Majorana spinor has the following
form

$\Psi_{M}=\Psi_{L}+\Psi_{L}^{c}$ (3)

and the neutrality relation $\Psi_{M}^{c}=\Psi_{M}$ holds. A Majorana field
has only two degrees of freedom.

The Majorana mass term is introduced as

$m_{i}\overline{\nu_{i}}\nu_{i}=m_{i}\overline{(\nu^{L}_{i})^{c}}\nu^{L}_{i}+m_{i}\overline{\nu^{L}_{i}}(\nu^{L}_{i})^{c},$
(4)

where $i=1,2,3$. The Majorana-type neutrino mass term can be generated by the
seesaw mechanism, which naturally appears in the low-energy limit of certain
beyond Standard Model theories (see [13] for a review). An interesting feature
of the Majorana mass term is that it violates the total lepton number by two
units, which makes possible lepton number violating processes, such as the
neutrinoless double beta-decay (0$\nu\beta\beta$). At the moment, experiments
on 0$\nu\beta\beta$ are considered to be the most prospective way to probe the
nature of the neutrino mass. Furthermore, these experiments are potentially
sensitive to the Majorana CP violating phases difference
$\alpha_{1}-\alpha_{2}$ [5].

The Majorana condition $\nu^{c}_{i}=\nu_{i}$ puts significant constraints on
the structure of the flavour neutrino fields. Unlike in the case of Dirac
neutrinos, when the mixing matrix for the right component of the field can be
introduced arbitrarily since right-handed neutrinos are sterile, for Majorana
neutrinos the following relations hold

$\displaystyle\nu_{\alpha}^{L}$ $\displaystyle=$
$\displaystyle\sum_{i}U_{\alpha i}\nu_{i}^{L},$ (5)
$\displaystyle\nu_{\alpha}^{R}$ $\displaystyle=$
$\displaystyle(\nu_{\alpha}^{L})^{c}=\sum_{i}U_{\alpha
i}^{*}(\nu_{i}^{L})^{c},$ (6)

and

$\nu_{\alpha}=\nu_{\alpha}^{L}+\nu_{\alpha}^{R}=\sum_{i}U_{\alpha
i}\nu_{i}^{L}+\sum_{i}U_{\alpha i}^{*}(\nu_{i}^{L})^{c},$ (7)

where $\alpha=e,\mu,\tau$. Obviously, the flavour fields satisfy the Majorana
condition $\nu^{c}_{\alpha}=\nu_{\alpha}$.

Now consider the Majorana neutrino interactions. It is known that massive
neutrinos must possess a non-zero anomalous magnetic moment, and therefore
interact with a magnetic field (see [14] for a review on the electromagnetic
properties of neutrinos). The Majorana neutrino interaction with a magnetic
field has the following form

$\mathcal{L}_{mag}=-\sum_{i,k}\mu_{ik}\left[\overline{(\nu_{i}^{L})^{c}}\bm{\Sigma}\bm{B}\nu_{k}^{L}+\overline{\nu_{i}^{L}}\bm{\Sigma}\bm{B}(\nu_{k}^{L})^{c}\right]=-\sum_{\alpha,\beta}\mu_{\alpha\beta}\left[\overline{(\nu_{\alpha}^{L})^{c}}\bm{\Sigma}\bm{B}\nu_{\beta}^{L}+\overline{\nu_{\alpha}^{L}}\bm{\Sigma}\bm{B}(\nu_{\beta}^{L})^{c}\right].$
(8)

It is clear from the form of the Lagrangian that the interaction with a
magnetic field can induce neutrino-antineutrino oscillations
$\nu_{\alpha}\leftrightarrow\bar{\nu}_{\beta}$. The Majorana condition imposes
certain constraints on neutrino magnetic moments. Since a Majorana neutrino is
a truly neutral particle, it cannot posses diagonal electric and magnetic
dipole form factors. However, non-diagonal entries are possible, in particular
the transition magnetic moments. The magnetic moments matrix of a Majorana
neutrino $\mu_{ik}$ is antisymmetric and Hermitian, and then has only non-
diagonal entries which are purely imaginary quantities:
$\mu_{ik}=i|\mu_{ik}|=-\mu_{ki}$ for $i\neq k$. Numerical values of the
neutrino magnetic moments are discussed in Sec. IV. For a thorough review of
neutrino electromagnetic properties and spin oscillations see [14] and
references therein.

Interactions of neutrinos with matter are described by the Lagrangian

$\mathcal{L}_{mat}=-\sum_{\alpha}V^{(f)}_{\alpha}\overline{\nu_{\alpha}}\gamma_{0}(1+\gamma_{5})\nu_{\alpha},$
(9)

where

$V^{(f)}=\operatorname{diag}\left(\frac{G_{F}n_{e}}{\sqrt{2}}-\frac{G_{F}n_{n}}{2\sqrt{2}},-\frac{G_{F}n_{n}}{2\sqrt{2}},-\frac{G_{F}n_{n}}{2\sqrt{2}}\right)$
(10)

is the Wolfenstein potential. Here we consider a normal electrically neutral
matter composed of electrons, protons and neutrons. Since for a Majorana field
the relation $\bar{\Psi}\gamma_{\mu}\Psi=0$ holds, one can replace
$(1+\gamma_{5})$ with $\gamma_{5}$. Then, the matter interaction Lagrangian
takes the following forms for Majorana and Dirac neutrino respectively:

$\mathcal{L}_{mat}^{M}=-\sum_{\alpha}V^{(f)}_{\alpha}\left[\overline{\nu^{L}_{\alpha}}\gamma_{0}\nu_{\alpha}^{L}-\overline{(\nu^{L}_{\alpha})^{c}}\gamma_{0}(\nu_{\alpha}^{L})^{c}\right],$
(11)
$\mathcal{L}_{mat}^{D}=-\sum_{\alpha}V^{(f)}_{\alpha}\overline{\nu^{L}_{\alpha}}\gamma_{0}\nu_{\alpha}^{L}.$
(12)

It is well known [8, 9] that it is not possible to distinguish between Dirac
and Majorana neutrinos in studies of the neutrino flavour oscillations. Our
studies below are initiated by an expectation that the neutrino-antineutrino
oscillations induced by a magnetic field could provide an appropriate setup to
probe the nature of the neutrino mass term. In particular, we show that under
certain realistic astrophysical conditions the Majorana CP violating phases
affect neutrino oscillations pattern.

The effects of non-zero Majorana CP violating phases in neutrino-antineutrino
oscillations have been studied before in [12]. The authors considered
neutrino-antineutrino oscillations in vacuum induced by the Majorana mass term
$m_{\alpha\beta}\overline{\nu^{c}}_{\alpha}\nu_{\beta}$. Despite the fact that
the probabilities of such oscillations are strongly suppressed by the factor
of order $m^{2}/E^{2}$, they still can be possibly used to determine the
magnitudes of both Dirac and Majorana CP violating phases in future
terrestrial experiments, provided that neutrino is a Majorana fermion. In [15,
16] effects of non-zero Dirac CP violating phase were studied without
accounting for the interaction with a magnetic field. It was shown that the
magnitude of the Dirac CP phase does not affect supernovae neutrino fluxes
unless muon and tau (anti)neutrino fluxes differ at the neutrinosphere.

In turn, here below we show that supernovae neutrino fluxes can carry
significant information about CP violation, given that neutrino magnetic
moments are large enough to enable substantial $\nu\leftrightarrow\bar{\nu}$
oscillations in the supernova envelope. In what follows we develop a
consistent approach to the description of neutrino spin (or neutrino-
antineutrino in the Majorana case) oscillations in astrophysical environments
in the three neutrino framework.

##  III Formalism

In this section we extend the formalism developed in [17] to account for the
transition magnetic moments and neutrino interactions with matter. We derive
the following system of Dirac equations for the massive neutrino states

$\displaystyle(i\gamma^{\mu}\partial_{\mu}-m_{i}-V^{(m)}_{ii}\gamma^{0}\gamma_{5})\nu_{i}(x)-\sum_{k\neq
i}\big{(}\mu_{ik}\bm{\Sigma}\bm{B}+V^{(m)}_{ik}\gamma^{0}\gamma_{5}\big{)}\nu_{k}(x)=0,$
(13)

where $V^{(m)}=U^{{\dagger}}V^{(f)}U$ is the matter potential in the mass
basis, $i,k=\\{1,2,3\\}$. In the presence of electron matter component $n_{e}$
and/or interaction of the transition magnetic moments with a magnetic field
these three equations are coupled. As a result, the neutrino mass states under
these conditions are non-stationarity. Eq. (13) can be rewritten in the
Hamiltonian form:

$i\frac{\partial}{\partial t}\nu(x)=\begin{pmatrix}H_{11}&H_{12}&H_{13}\\\
H_{21}&H_{22}&H_{23}\\\ H_{31}&H_{32}&H_{33}\end{pmatrix}\nu(x)=H\nu(x),$ (14)

where $\nu=(\nu_{e},\nu_{\mu},\nu_{\tau})^{T}$ and

$H_{ik}=\delta_{ik}\gamma_{0}\bm{\gamma}\bm{p}+m_{i}\delta_{ik}\gamma_{0}+\mu_{ik}\gamma_{0}\bm{\Sigma}\bm{B}+V_{ik}^{(m)}\gamma_{5}.$
(15)

Our goal is to calculate the probabilities of oscillations between different
neutrino and antineutrino states. The probabilities are expressed as

$\displaystyle
P(\nu_{\alpha}^{s}\rightarrow\nu_{\beta}^{s^{\prime}})=\big{|}\braket{\nu_{\beta}^{s^{\prime}}(0)}{\nu_{\alpha}^{s}(x)}\big{|}^{2}=\Big{|}\sum_{i,k}\left(U^{s^{\prime}}_{\beta
k}\right)^{*}U_{\alpha
i}^{s}\braket{\nu_{k}^{s^{\prime}}(0)}{\nu_{i}^{s}(x)}\Big{|}^{2},$ (16)

where $U^{s}$ are the mixing matrices for left ($s=L$) and right ($s=R$)
neutrinos. Note that in the Majorana case right neutrino is an antineutrino:
$\nu_{i}^{R}=(\nu_{i}^{L})^{c}$ and then $U^{L}=U$, $U^{R}=U^{*}$.

Now we focus on the calculation of the amplitudes
$\braket{\nu_{k}^{s^{\prime}}(0)}{\nu_{i}^{s}(x)}$ using Eq. (13). Consider
the neutrino mass states with a definite helicity

$\displaystyle\ket{\psi^{L}_{1}(0)}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\ket{L}\\\ 0\\\ 0\end{pmatrix},\ \ \
\ket{\psi^{R}_{1}(0)}=\begin{pmatrix}\ket{R}\\\ 0\\\ 0\end{pmatrix},$
$\displaystyle\ket{\psi^{L}_{2}(0)}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0\\\ \ket{L}\\\ 0\end{pmatrix},\ \ \
\ket{\psi^{R}_{2}(0)}=\begin{pmatrix}0\\\ \ket{R}\\\ 0\end{pmatrix},$ (17)
$\displaystyle\ket{\psi^{L}_{3}(0)}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0\\\ 0\\\ \ket{L}\end{pmatrix},\ \ \
\ket{\psi^{R}_{3}(0)}=\begin{pmatrix}0\\\ 0\\\ \ket{R}\end{pmatrix},$

where $\ket{L}$ and $\ket{R}$ are the eigenvectors of the helicity operator
$\bm{\Sigma}\bm{p}/p$, the eigenvalues are $-1$ and $+1$, respectively. We
consider astrophysical neutrinos with energies of order 10 MeV [18] and mass
to be 1.1 eV, that is equal to the upper-bound reported by the KATRIN
collaboration [19]. Thus, ultrarelativistic assumption is justified, and we
can write out

$\ket{L}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ -1\\\ 0\\\ 1\end{pmatrix},\ \ \
\ket{R}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ 0\\\ 1\\\ 0\end{pmatrix}.$ (18)

The formal solution of the evolution equation with the initial states (17) is

$\ket{\psi^{L}_{i}(x)}=e^{-iHx}\ket{\psi^{L}_{i}(0)}.$ (19)

Indices $i$ and $L$ refer only to the initial conditions (17), and, since the
massive neutrino state with a definite helicity is generally not a stationary
quantum state, for $x>0$ the state $\ket{\psi_{i}^{L}(x)}$ is, strictly
speaking, no longer a massive neutrino state with a certain polarization. This
state rather accounts for possible transitions between neutrino mass states
and helicity states due to interactions with matter and the magnetic field.
The amplitudes of the transitions in Eq. (16) are

$\braket{\nu^{s^{\prime}}_{k}(0)}{\nu_{i}^{s}(x)}=\braket{\psi_{k}^{s^{\prime}}(0)}{\psi_{i}^{s}(x)}.$
(20)

The easiest way to compute the amplitudes of interest (20) is to use the
eigendecomposition of the Hamiltonian

$H=\sum_{n}E_{n}\ket{n}\bra{n},\ \ \ \ H\ket{n}=E_{n}\ket{n}$ (21)

and the following relation for the matrix exponential

$e^{-iHx}=\sum_{n}e^{-iE_{n}x}\ket{n}\bra{n}=\sum_{n}e^{-iE_{n}x}P_{n},$ (22)

where

$P_{n}=\ket{n}\bra{n}.$ (23)

The amplitudes of the transitions between massive neutrino states with a
definite helicity can be represented in the form of the plane wave
decomposition

$\displaystyle\braket{\nu_{k}^{s^{\prime}}(0)}{\nu_{i}^{s}(x)}=\sum_{n}\braket{\psi_{k}^{s^{\prime}}(0)}{P_{n}}{\psi_{i}^{s}(0)}e^{-iE_{n}x}=\sum_{n}C_{nki}^{ss^{\prime}}e^{-iE_{n}x},$
(24)

where the coefficients

$C_{nki}^{ss^{\prime}}=\braket{\psi_{k}^{s^{\prime}}(0)}{P_{n}}{\psi_{i}^{s}(0)}$
(25)

are used.

The probabilities of neutrino oscillations are

$P(\nu_{\alpha}^{s}\rightarrow\nu_{\beta}^{s^{\prime}})=\Big{|}\sum_{n}\sum_{i,k}\left(U_{\beta
k}^{s^{\prime}}\right)^{*}U_{\alpha
i}^{s}C_{nki}^{ss^{\prime}}e^{-iE_{n}x}\Big{|}^{2}.$ (26)

They can be expressed in the explicit form

$P(\nu_{\alpha}^{s}\rightarrow\nu_{\beta}^{s^{\prime}};x)=\delta_{\alpha\beta}\delta_{ss^{\prime}}-4\sum_{n>m}\operatorname{Re}(A_{\alpha\beta
nm}^{ss^{\prime}})\sin^{2}\left(\frac{\pi
x}{L^{osc}_{nm}}\right)+2\sum_{n>m}\operatorname{Im}(A_{\alpha\beta
nm}^{ss^{\prime}})\sin\left(\frac{2\pi x}{L^{osc}_{nm}}\right),$ (27)

where

$A_{\alpha\beta nm}^{ss^{\prime}}=\sum_{i,j,k,l}\left(U_{\beta
k}^{s^{\prime}}\right)^{*}U_{\alpha i}^{s}U_{\beta
l}^{s^{\prime}}\left(U_{\alpha
j}^{s}\right)^{*}\big{(}C_{nki}^{ss^{\prime}}\big{)}^{*}C_{mlj}^{ss^{\prime}}$
(28)

and $L^{osc}_{nm}=2\pi/(E_{n}-E_{m})$.

This formula generalizes the well-known expression for the probabilities of
vacuum neutrino oscillations. A similar expression was obtained in [20] for
the case of Dirac neutrinos with only diagonal magnetic moments. The last term
of (27) is T violating and, provided that CPT symmetry is conserved, is also
CP violating. CPT conservation, however, is not true for the case of the
neutrino interaction with particle-antiparticle asymmetric media, such as a
supernova environment, because of the matter-induced (the extrinsic) CPT
violation [21]. Thus, for a realistic astrophysical environment we can not
assume that the last term of (27) encapsulates the CP violating effects.

For the further considerations it is useful to introduce two additional
quantities. Firstly, using (26) one can calculate the amplitudes of
oscillations:

$P(\nu_{\alpha}^{s}\rightarrow\nu_{\beta}^{s^{\prime}})_{max}=\Big{(}\sum_{n}|\mathcal{I}_{n\alpha\beta}^{ss^{\prime}}|\Big{)}^{2},$
(29)

where

$\mathcal{I}_{n\alpha\beta}^{ss^{\prime}}=\sum_{i,k}(U_{\beta
k}^{s^{\prime}})^{*}U^{s}_{\alpha i}C^{ss^{\prime}}_{nki}.$ (30)

Secondly, from (27) we derive the distance-averaged probabilities:

$\langle
P(\nu_{\alpha}^{s}\rightarrow\nu_{\beta}^{s^{\prime}})\rangle=\delta_{\alpha\beta}\delta_{ss^{\prime}}-2\sum_{n>m}\operatorname{Re}(A_{\alpha\beta
nm}^{ss^{\prime}}).$ (31)

In what follows we apply the developed formalism for the Majorana neutrino
oscillations to study neutrino fluxes in astrophysical media, peculiar, for
instance, for supernovae. Note that the developed approach with a
straightforward modification can be also applied to the case of Dirac
neutrinos.

##  IV Astrophysical applications

Here below we present the numerical results on neutrino-antineutrino
oscillations in supernovae neutrino fluxes. First we analyse possible
resonances in the neutrino-antineutrino oscillations channels. Then we
investigate the effects of these resonances on the flavour composition of the
neutrino fluxes for a certain supernova model.

The magnitudes of the oscillations parameters in the PMNS matrix (1) are given
in Table 1 (see [4, 22]).

Table 1: Neutrino oscillation parameters according to [22]. Parameter | $\sin^{2}\theta_{12}$ | $\sin^{2}\theta_{23}$ | $\sin^{2}\theta_{13}$ | $\Delta m^{2}_{12}/\text{eV}^{2}$ |  $\left|\Delta m^{2}_{13}\right|$/eV2
---|---|---|---|---|---
Value | 0.310 | 0.558 | 0.022 | 7.39$\times 10^{-5}$ | 2.52$\times 10^{-3}$

The magnetic moments matrix in the case of Majorana neutrinos is antisymmetric
and consists of the purely imaginary entries (see [14] for a detailed
discussion). In the studies below we use the following representation:

$\mu_{ij}=\begin{pmatrix}0&i|\mu_{12}|&i|\mu_{13}|\\\
-i|\mu_{12}|&0&i|\mu_{23}|\\\ -i|\mu_{13}|&-i|\mu_{23}|&0\end{pmatrix}.$ (32)

The best terrestrial experiment upper bounds on the neutrino magnetic moments,
obtained by the GEMMA reactor neutrino experiment [23] and Borexino
collaboration [24] by measuring the solar neutrino fluxes, are on the level
$\mu_{\nu}<2.8\div 2.9\times 10^{-11}\mu_{B}$. An order of magnitude more
stringent upper bound is provided by the observed properties of the globular
cluster stars [25, 26, 27]. For our further analyses we fix the values of the
transition magnetic moments in the neutrino mass basis accordingly:
$|\mu_{12}|=|\mu_{13}|=|\mu_{23}|=10^{-12}\mu_{B}$. The particular features of
the neutrino oscillations described below are generally appropriate for the
case of an arbitrary choice of non-zero transition magnetic moments.

###  IV.1 Resonances in neutrino-antineutrino oscillations

Consider the amplitudes (29) of neutrino-antineutrino oscillations. Since the
neutrino magnetic moments are, generally speaking, small, we are principally
interested in oscillations under the extreme external conditions peculiar to
astrophysical objects, supernovae in particular. A supernova inner region is
characterized by the baryon number density $n_{B}$ that is of order $10^{30}$
cm-3 and even higher, with $n_{B}=n_{p}+n_{n}$, where $n_{p}$ and $n_{n}$ are
the proton and neutron number densities, respectively. For a neutral media, as
one of a supernova, the proton density $n_{p}$ is equal to the electron
density $n_{e}$. Magnetic fields during a core-collapse can reach magnitudes
up to $10^{15}-10^{16}$ G right after $\approx 9$ ms after bounce (see [28]).
In our analysis below we use a more conservative value for the magnetic field
$B\sim 10^{13}$ G and also chose the baryon number density $n_{B}=10^{32}$
cm-3.

We are particularly interested in the neutrino-antineutrino oscillations
engendered by a magnetic field because the MSW resonances in the neutrino
flavour oscillations are not possible under extreme densities in the inner
supernova regions. However, the flavour oscillations dominate in the outer
regions, where the densities are lower and the magnetic field is excessively
weak to engender the spin-flip transitions. The MSW effect contribution is
considered in Subsec. B.

Fig. 1 shows the amplitudes (29) of the neutrino-antineutrino oscillations as
functions of the electron fraction $Y_{e}=n_{e}/n_{B}$ for the CP violating
phases given by $\delta=0$, $\alpha_{1}=0$, $\alpha_{2}=0$. The resonant curve
in Fig. 1 reproduces the well-known resonant behaviour of the spin-flavor
($RSF$) conversion studied in [29, 30], with the resonance in the
$\nu_{e}\to\bar{\nu}_{\mu}$ channel for $Y_{e}\approx 0.5$, described by [29,
30]

$P_{max}=\mu^{2}B^{2}/((\mu B)^{2}+\Delta H^{2}),\ \ \ \Delta
H=\sqrt{2}G_{F}n_{B}(1-2Y_{e})-\Delta m^{2}\cos 2\theta/2p.$ (33)

However, the CP conserving values $\delta=0$ and $\delta=\pi$ are disfavored
at the 95$\%$ confidence level by the T2K collaboration [6], and the data
shows a preference for near maximal CP violation $\delta=\pi/2$. Therefore, it
is worth to proceed with consideration of the CP violating effects in neutrino
spin oscillations.

The amplitudes of the neutrino-antineutrino oscillations for the case of non-
zero Dirac CP violating phase are shown in Fig. 2. The resonance in the
channel $\nu_{e}\to\bar{\nu}_{\mu}$ is persistent for all values of $\delta$.
There is also a new resonance in the channel $\nu_{e}\to\bar{\nu}_{e}$ that
appears at $Y_{e}\approx 0.35$. The location of the resonance does not depend
either on the magnetic field strength $B$ or the baryon density $n_{B}$ and
the neutrino energy $p$. This resonance occurs even for values of the Dirac CP
violating phase which are only slightly different from the CP conserving
values, i.e. $\delta=0$ or $\pi$. Thus we can expect significant
$\nu_{e}\rightarrow\bar{\nu}_{e}$ conversions at a certain point of a
supernova if neutrinos are Majorana particles, Dirac CP violating phase
$\delta$ is non-zero and the interaction with the stellar magnetic field is
strong enough ($B\sim 10^{12}\div 10^{13}\ G$).

Although the effects of the Dirac CP violating phase may be important in
astrophysics, the phase itself is likely to be measured in terrestrial
experiments (such as JUNO, DUNE and Hyper-Kamiokande) in the near future.

We focus below on manifestations of the Majorana CP violating phases. The
amplitudes (29) of neutrino-antineutrino oscillations for the cases $\delta=0$
and $\delta=\pi/2$ correspondingly as functions of the electron fraction for
the particular values of Majorana CP violating phases $\alpha_{1}$ and
$\alpha_{2}$ are shown in Fig. 3 and Fig. 4 . A particular feature is that now
the resonant peak in the $\nu_{e}\leftrightarrow\bar{\nu}_{\tau}$ oscillations
can appear at $Y_{e}\approx 0.5$ for certain magnitudes of the Majorana phases
in addition to the resonances in the $\nu_{e}\leftrightarrow\bar{\nu}_{e}$ and
$\nu_{e}\leftrightarrow\bar{\nu}_{\mu}$ conversions. Fig. 5 and Fig. 6 present
the amplitudes of $\nu\leftrightarrow\bar{\nu}$ oscillations as functions of
both $\alpha_{1}$ and $\alpha_{2}$.

From (29) we have also found that the amplitudes of
$\nu_{e}\leftrightarrow\bar{\nu}_{\mu}$ and
$\nu_{e}\leftrightarrow\bar{\nu}_{\tau}$ conversions do not depend on
$\delta$. However, the appearance of a non-zero $\delta$ breaks the central
symmetry of $P(\nu_{e}\rightarrow\bar{\nu}_{e})_{max}$ shown in Fig. 5.

The resonant values of the electron fraction $Y_{e}=0.35$ and $Y_{e}=0.5$
found above are robust and do not depend on the magnetic field strength,
neutrino energy or baryon number density. However, the width of the resonant
curves $P_{max}(Y_{e})$ varies with the field-density ratio $\mu
B/G_{F}n_{B}$: the resonances become wider as it increases. Additionally, the
$\nu_{e}\leftrightarrow\bar{\nu}_{e}$ resonance peak is inherently narrower
than two others.

It is important to estimate the scale $L_{res}$ of the supernova region where
$\nu\leftrightarrow\bar{\nu}$ oscillations exhibit the resonance behavior and
compare it to the corresponding oscillations length $L_{osc}$. In the case
$L_{osc}\ll L_{res}$ there is no room for the resonance behavior of the
$\nu\leftrightarrow\bar{\nu}$ oscillations in the considered astrophysical
environment. The scale of the resonant region is given by

$L_{res}\approx\left(\frac{dY_{e}}{dr}\right)^{-1}\Delta Y_{e},$ (34)

where $\Delta Y_{e}$ is the width of the resonant curve $P_{max}(Y_{e})$. The
value $dY_{e}/dr$ typically is of order $10^{-8}$ cm-1 [18] and it is natural
to modify (34) accordingly

$L_{res}\approx\left(\frac{dY_{e}/dr}{10^{-8}\operatorname{cm}^{-1}}\right)^{-1}\Delta
Y_{e}\times 10^{3}\operatorname{km}.$ (35)

Our estimations show that $\Delta Y_{e}\approx 0.3$ for the
$\nu_{e}\leftrightarrow\bar{\nu}_{e}$ oscillations, and $\Delta Y_{e}\approx
1$ for the $\nu_{e}\leftrightarrow\bar{\nu}_{\mu}$ and
$\nu_{e}\leftrightarrow\bar{\nu}_{\tau}$ oscillations for the set of
parameters described above (see Figs. 1-4). The corresponding resonant regions
scales $L_{res}$ are found to be 30 km and 100 km, respectively. At the same
time, the oscillation length $L_{osc}$ is below 1 km. Thus, the resonance
oscillations $\nu\leftrightarrow\bar{\nu}$ are expected to proceed not far
from the neutrinosphere.

Finally, we conclude that depending on the particular values of the Dirac and
Majorana CP violating phases three resonances may occur:
$\nu_{e}\leftrightarrow\bar{\nu}_{e}$ resonance (at $Y_{e}=0.35$),
$\nu_{e}\leftrightarrow\bar{\nu}_{\mu}$ resonance and
$\nu_{e}\leftrightarrow\bar{\nu}_{\tau}$ resonance (both at $Y_{e}=0.5$). Note
that the amplitudes of oscillations described above do not significantly vary
with the neutrino energy, provided that it is greater than 0.1 MeV. For
neutrinos with energies far below this threshold the oscillations patterns
become drastically different, but this is not the case of interest for the
neutrinos from supernovae. The effects of different mass hierarchies have also
been found to be subtle. We qualitatively show below how the appearance of the
new resonances affects observable neutrino fluxes.

###  IV.2 Effects of CP violation on supernova neutrino fluxes

Consider the influence of the $\nu\leftrightarrow\bar{\nu}$ oscillations on
the neutrino fluxes emitted at the later stages of a supernova evolution. We
compute neutrino fluxes as follows

$\Phi_{\nu_{\alpha}}=\sum_{\beta}\Phi_{\nu_{\beta}}^{0}P(\nu_{\beta}\rightarrow\nu_{\alpha}),$
(36)

where $\alpha,\beta=e,\bar{e},\mu,\bar{\mu},\tau,\bar{\tau}$ and
$\Phi^{0}_{\nu_{\beta}}$ are the neutrino fluxes at the neutrinosphere. As it
follows from (35), the resonant regions scales are much larger then the
oscillation length: $L_{res}\gg L_{osc}$. Thus, the probabilities
$P(\nu_{\beta}\rightarrow\nu_{\alpha})$ can be replaced with the averaged
probabilities (31). Besides, both the $\nu\to\bar{\nu}$ and MSW resonant
regions are smaller than the distance between them. Thus, in the evaluation of
the neutrino fluxes one can first calculate the neutrino fluxes within the
inner supernova region accounting for the $\nu\leftrightarrow\bar{\nu}$
oscillations, and then proceed with accounting for the MSW oscillations.

For our simulation we use the following values of the initial neutrino fluxes:
$\Phi_{\nu_{e}}^{0}=\Phi_{\bar{\nu}_{e}}^{0}=4.1\times 10^{51}$ erg/sec,
$\Phi_{\nu_{x}}^{0}=7.9\times 10^{51}$ erg/sec,
$\Phi_{\nu_{\mu},\nu_{\tau},\overline{\nu}_{\mu},\overline{\nu}_{\tau}}^{0}=\Phi_{\nu_{x}}^{0}$
[31]. Supernovae models predict that $Y_{e}\approx 0.4$ during the later
stages of a supernova explosion [32]. This value lies between the resonant
values $Y_{e}=0.35$ and $Y_{e}=0.5$. Since the resonance at $Y_{e}=0.35$ is
substantially narrower than the resonance at $Y_{e}=0.5$, we expect that the
latter contribution is more significant. Accounting for the probability
conservation relation $\sum_{\alpha}P(\nu_{\alpha}\to\nu_{\beta})=1$, one can
simplify (36) as follows

$\Phi_{\nu_{\alpha}}=\Phi^{0}_{\nu_{x}}+(\Phi^{0}_{\nu_{e}}-\Phi^{0}_{\nu_{x}})\left[P(\nu_{e}\rightarrow\nu_{\alpha})+P(\overline{\nu}_{e}\rightarrow\nu_{\alpha})\right].$
(37)

Since it is impossible to detect the muon and tau neutrino and antineutrino
fluxes separately for the energy range of a supernova neutrino emission (see
in [18]), one have to construct observables using only $\Phi_{\nu_{e}}$,
$\Phi_{\bar{\nu}_{e}}$ and $\Phi_{\nu_{x}}$. From our analysis it follows that
the most pronounced effect of the Majorana CP violation phases can appear in
the ratio of $\bar{\nu_{e}}$ and $\nu_{e}$ fluxes that can be written in the
following form

$\frac{\Phi_{\bar{\nu}_{e}}}{\Phi_{\nu_{e}}}=\frac{1+(\Phi^{0}_{\nu_{e}}/\Phi^{0}_{\nu_{x}}-1)\langle
P(\bar{\nu}_{e}\to\bar{\nu}_{e})\rangle+(\Phi^{0}_{\nu_{e}}/\Phi^{0}_{\nu_{x}}-1)\langle
P(\nu_{e}\to\bar{\nu}_{e})\rangle}{1+(\Phi^{0}_{\nu_{e}}/\Phi^{0}_{\nu_{x}}-1)\langle
P(\nu_{e}\to\nu_{e})\rangle+(\Phi^{0}_{\nu_{e}}/\Phi^{0}_{\nu_{x}}-1)\langle
P(\bar{\nu}_{e}\to\nu_{e})\rangle}.$ (38)

Note that a deviation from unity of the ratio
$\Phi_{\bar{\nu}_{e}}/\Phi_{\nu_{e}}$
($\Phi_{\bar{\nu}_{e}}/\Phi_{\nu_{e}}\neq 1$) indicates for CP violating
effects. Two types of the CP violating effects can take place in neutrino
oscillations: the extrinsic (matter-induced) and intrinsic [21]. We are
interested only in the intrinsic CP violation, i.e. the effects of non-zero CP
violating phases. Our numerical results presented below show that for neutrino
energies above 0.1 MeV the extrinsic contribution to (38) is negligible.

Fig. 7 (Left) shows the $\bar{\nu}_{e}/\nu_{e}$ ratio, calculated based on
(38), as a function of the Dirac CP phase $\delta$ for the case
$\alpha_{1}=0$, $\alpha_{2}=0$. The $\bar{\nu}_{e}/\nu_{e}$ ratio scarcely
reaches 12% at $\delta=\pi/2$ and $3\pi/2$. The relative insignificance of the
effect follows from the fact that $\nu_{e}\leftrightarrow\bar{\nu}_{e}$
oscillations amplitudes are suppressed at $Y_{e}=0.4$. Thus, one can neglect
the effect of the Dirac CP phase in the evaluation the $\bar{\nu}_{e}/\nu_{e}$
ratio.

Consider now the $\bar{\nu}_{e}/\nu_{e}$ ratio as a function of the Majorana
CP violating phases for $\delta=0$. The results are shown in Fig. 7 (Right).
Except the regions around $\alpha_{2}=\alpha_{1}\pm\pi$, the neutrino-
antineutrino oscillations induce significant asymmetry between $\nu_{e}$ and
$\bar{\nu}_{e}$ fluxes, that peaks at 50%. From (37) it also follows that the
muon and tau neutrino and antineutrino fluxes are still approximately equal to
each other after oscillations:
$\Phi_{\nu_{\mu}}=\Phi_{\nu_{\tau}}=\Phi_{{\bar{\nu}}_{\mu}}=\Phi_{{\bar{\nu}}_{\tau}}$.
Thus, neutrino-antineutrino oscillations in a magnetic field of the inner
supernova region can indeed induce significant asymmetry between the $\nu_{e}$
and $\bar{\nu}_{e}$ fluxes.

The next step is to compute the neutrino fluxes outside the supernova. For
this, one has to account for the Mikheev-Smirnov-Wolfenstein oscillations,
which take place in outer supernova regions with relatively low densities. The
adiabatic solution for the neutrino fluxes in the discussed case yields [18]:

$\begin{matrix}[l]\Phi^{out}_{\nu_{e}}=\Phi_{\nu_{x}}&\mbox{(NH)},\\\
\Phi^{out}_{\nu_{e}}=s_{12}^{2}\Phi_{\nu_{e}}+c_{12}^{2}\Phi_{\nu_{x}}&\mbox{(IH)},\\\
\Phi^{out}_{\bar{\nu}_{e}}=c_{12}^{2}\Phi_{\bar{\nu}_{e}}+s_{12}^{2}\Phi_{\nu_{x}}&\mbox{(NH)},\\\
\Phi^{out}_{\bar{\nu}_{e}}=\Phi_{\nu_{x}}&\mbox{(IH)},\end{matrix}$ (39)

where NH and IH refer to the normal and inverted mass hierarchy respectively,
and
$\Phi_{\nu_{\mu},\nu_{\tau},\overline{\nu}_{\mu},\overline{\nu}_{\tau}}=\Phi_{\nu_{x}}$.

The final results for $\bar{\nu}_{e}/\nu_{e}$ fluxes ratio in a supernova
emission are shown in Fig.8. It follows that the magnetic field induced
disproportion in the fluxes (see the above comments to Fig. 7) becomes smeared
after the MSW oscillations, specially for the case of the inverted mass
hierarchy: the maximal values for the ration are
$\Phi^{out}_{\bar{\nu}_{e}}/\Phi^{out}_{\nu_{e}}=0.23$ for NH and
$\Phi^{out}_{\bar{\nu}_{e}}/\Phi^{out}_{\nu_{e}}=0.23$ for IH. Table 2 shows
$\bar{\nu}_{e}/\nu_{e}$ ratio characteristic values (the “No magnetic field”
column stands for the case of $B=0$ and the “No CP” column stands for the case
of $\alpha_{1}=0$, $\alpha_{2}=0$). The minimal and maximal values of the
ratio $\Phi^{out}_{\bar{\nu}_{e}}/\Phi^{out}_{\nu_{e}}$ (depending on the
Majorana CP phases) are also shown in Table 2.

Table 2: $\bar{\nu}_{e}/\nu_{e}$ ratio characteristic values. $\Phi^{out}_{\bar{\nu}_{e}}/\Phi^{out}_{\nu_{e}}$ | No magnetic field | Min | No CP | Max
---|---|---|---|---
NH | 0.67 | 0.64 | 0.74 | 0.87
IH | 1.18 | 0.9 | 0.97 | 1.0

##  V Conclusion

In this paper we study the Majorana neutrinos oscillations in astrophysical
media with an emphasis on the CP violating effects. The semi-analytical
expressions for the neutrino oscillations probabilities are obtained. It is
shown that the appearance of the non-zero CP violating phases can give rise to
new resonances in the neutrino-antineutrino oscillations channels, namely:
$\nu_{e}\leftrightarrow\overline{\nu}_{e}$ resonance at $Y_{e}=0.35$ that
appears for both cases of the non-zero Dirac and Majorana phases; and
$\nu_{e}\leftrightarrow\overline{\nu}_{\mu},{\nu}_{\tau}$ resonances at
$Y_{e}=0.5$ that is possible only for the case of the non-zero Majorana CP
phases. These resonant values of $Y_{e}$ are persistent and do not depend on
the magnetic field strength $B$ or neutrino energy $p$.

The results obtained are applied to a particular physical situation: the
oscillations of neutrinos during the cooling stage of a supernova explosion.
It is shown that neutrino-antineutrino oscillations in the near-neutrinosphere
high-density region of a supernova can significantly modify the resulting
outcoming neutrino fluxes. Particularly, under certain non-zero values of the
Majorana CP phases, the $\bar{\nu}_{e}/\nu_{e}$ ratio reaches magnitudes up to
1.5 (as is shown in Fig. 7) within the inner supernova region. After the
consequent MSW oscillations, the effect becomes less pronounced, specially for
the case of the inverted neutrino mass hierarchy, but still is present. Our
results, however, only roughly estimate the $\bar{\nu}_{e}/\nu_{e}$ ratio. For
a precise calculation one have to account for several more factors,
particularly a realistic supernova density and the magnetic field profiles, as
well as the collective effects in neutrino oscillations. Latter can be
important, since nonlinear feedback due to self-interaction can enhance a
small effect. For now, we point out that the quantity
$\Phi_{\bar{\nu}_{e}}/\Phi_{\nu_{e}}$ can potentially be an important
observable for the supernova neutrino experiments.

One of the important new results is the conclusion that observations of the
ratio of supernovae fluxes $\Phi_{\bar{\nu}_{e}}/\Phi_{\nu_{e}}$ in the future
large volume neutrino detectors, such as JUNO and Hyper-Kamiokande, may
provide a tool for distinguishing between the Dirac and Majorana nature of
neutrinos.

Future neutrino experiments will hopefully not only provide us high-statistics
measurements of neutrino fluxes directly from a supernova explosion, but also
will be sensitive to the diffuse supernova neutrino background (DSNB). We
argue that DNSB flavour composition may depend on the CP violating phases.
Although to extract information about CP violation from DNSB is an utterly
complicated task, the diffuse neutrinos provide us a continuous source of
experimental data, while a supernova explosion is quite a rare event in the
Galaxy.

## Acknowledgements

The work is supported by the Russian Foundation for Basic Research under grant
No. 20-52-53022-GFEN-a. The work of A.P. is supported by the Foundation for
the Advancement of Theoretical Physics and Mathematics “BASIS” under grant No.
19-2-6-209-1.

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Figure 1: The amplitudes of neutrino-antineutrino oscillations as functions of
the electron fraction $Y_{e}$ for the case of CP conservation.

Figure 2: Left: The amplitudes of neutrino-antineutrino oscillations as
functions of the electron fraction $Y_{e}$ for the case of $\delta=\pi/2$,
$\alpha_{1}=0$ and $\alpha_{2}=0$. Right: The amplitudes of neutrino-
antineutrino oscillations as functions of the Dirac CP violating phase
$\delta$ for the case $Y_{e}=0.35$, $\alpha_{1}=0$ and $\alpha_{2}=0$.

Figure 3: The amplitudes of neutrino-antineutrino oscillations as functions of
the electron fraction $Y_{e}$ for the case of $\delta=0$. Left:
$\alpha_{1}=\alpha_{2}=\pi/2$. Right: $\alpha_{1}=\alpha_{2}=\pi$.

Figure 4: The amplitudes of neutrino-antineutrino oscillations as functions of
the electron fraction $Y_{e}$ for the case of $\delta=\pi/2$. Left:
$\alpha_{1}=\alpha_{2}=\pi/2$. Right: $\alpha_{1}=\alpha_{2}=\pi$.

Figure 5: The amplitude of $\nu_{e}\to\overline{\nu}_{e}$ oscillations for the
case $Y_{e}=0.35$ as a function of the Majorana CP violating phases
$\alpha_{1}$ and $\alpha_{2}$. Left: $\delta=0$. Right: $\delta=\pi/2$. Black
and yellow represent amplitudes equal to 0 and 1, respectively.

Figure 6: Left: The amplitude of $\nu_{e}\to\overline{\nu}_{\mu}$ oscillations
for the case $Y_{e}=0.5$ as a function of the Majorana CP violating phases
$\alpha_{1}$ and $\alpha_{2}$. Right: Same, but for the amplitude of
$\nu_{e}\to\overline{\nu}_{\tau}$ oscillations.

Figure 7: $\bar{\nu}_{e}/\nu_{e}$ disproportion engendered by neutrino-
antineutrino oscillations in a magnetic field. Left: as a function of the
Dirac CP violating phase $\delta$. Right: as a function of the Majorana phases
$\alpha_{1}$ and $\alpha_{2}$.

Figure 8: $\bar{\nu}_{e}/\nu_{e}$ ratio outside supernova as a function of the
Majorana CP violating phases $\alpha_{1}$ and $\alpha_{2}$ for NH and IH.