††footnotetext:  File: main.tex, printed: 2024-8-28, 4.03

# Second Hankel determinant of Logarithmic coefficients for Starlike and
Convex functions associated with lune

Sanju Mandal  Sanju Mandal, Department of Mathematics, Jadavpur University,
Kolkata-700032, West Bengal,India<EMAIL_ADDRESS>and
Molla Basir Ahamed  Molla Basir Ahamed, Department of Mathematics, Jadavpur
University, Kolkata-700032, West Bengal,India.
<EMAIL_ADDRESS>
###### Abstract.

The Hankel determinant $H_{2,1}(F_{f}/2)$ is defined as:

$\displaystyle H_{2,1}(F_{f}/2):=\begin{vmatrix}\gamma_{1}&\gamma_{2}\\\
\gamma_{2}&\gamma_{3}\end{vmatrix},$

where $\gamma_{1},\gamma_{2},$ and $\gamma_{3}$ are the first, second and
third logarithmic coefficients of functions belonging to the class
$\mathcal{S}$ of normalized univalent functions. In this article, we establish
sharp inequalities $|H_{2,1}(F_{f}/2)|\leq 1/16$ and $|H_{2,1}(F_{f}/2)|\leq
23/3264$ for the logarithmic coefficients of starlike and convex functions
associated with lune.

###### Key words and phrases:

Univalent functions, Starlike functions, Convex functions, Hankel Determinant,
Logarithmic coefficients, Schwarz functions

###### AMS Subject Classification: Mathematics Subject Classification:

Primary 30A10, 30H05, 30C35, Secondary 30C45

##  1\. Introduction

Suppose $\mathcal{H}$ be the class of functions $f$ which are holomorphic in
the open unit disk $\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$ of the form

(1.1) $\displaystyle
f(z)=\sum_{n=1}^{\infty}a_{n}z^{n},\;\mbox{for}\;z\in\mathbb{D}.$

Then $\mathcal{H}$ is a locally convex topological vector space endowed with
the topology of uniform convergence over compact subsets of $\mathbb{D}$. Let
$\mathcal{A}$ denote the class of functions $f\in\mathcal{H}$ such that
$f(0)=0$ and $f^{\prime}(0)=1$. That is, the functions $f$ of the form

(1.2) $\displaystyle
f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n},\;\mbox{for}\;z\in\mathbb{D}.$

Let $\mathcal{S}$ denote the subclass of all functions in $\mathcal{A}$ which
are univalent. For a general theory of univalent functions, we refer the
classical books [8, 12].

Let

(1.3) $\displaystyle
F_{f}(z):=\log\dfrac{f(z)}{z}=2\sum_{n=1}^{\infty}\gamma_{n}(f)z^{n},\;\;z\in\mathbb{D},\;\;\log
1:=0,$

be a logarithmic function associated with $f\in\mathcal{S}$. The numbers
$\gamma_{n}:=\gamma_{n}(f)$ are called the logarithmic coefficients of $f$.
Although the logarithmic coefficients $\gamma_{n}$ play a critical role in the
theory of univalent functions, it appears that there are only a limited number
of exact upper bounds established for them. As is well known, the logarithmic
coefficients play a crucial role in Milin’s conjecture ([19], see also[8,
p.155]). We note that for the class $\mathcal{S}$ sharp estimates are known
only for $\gamma_{1}$ and $\gamma_{2}$, namely

$\displaystyle|\gamma_{1}|\leq
1,\;\;|\gamma_{2}|\leq\dfrac{1}{2}+\dfrac{1}{e}=0.635\ldots$

Estimating the modulus of logarithmic coefficients for $f\in\mathcal{S}$ and
various sub-classes has been considered recently by several authors. We refer
to the articles [1, 2, 6, 11, 29] and references therein.

######  Definition 1.1.

Let $f$ and $g$ be two analytic functions. Then $f$ is subordinated by $g$ and
written as $f(z)\prec g(z)$, if there exists a self map $w$ such that $w(0)=0$
such that $f(z)=g(w(z))$. Moreover, if $g$ is univalent and $f(0)=g(0)$, then
$f(\mathbb{D})\subseteq g(\mathbb{D})$.

Raina and Sokol [24] introduced the class $\mathcal{S}^{*}_{\mbox{\leftmoon}}$
given by

$\displaystyle\mathcal{S}^{*}_{\mbox{\leftmoon}}:=\left\\{f\in\mathcal{S}:\;\vline\left(\frac{zf^{\prime}(z)}{f(z)}\right)^{2}-1\vline\leq
2\vline\frac{zf^{\prime}(z)}{f(z)}\vline,\;\;z\in\mathbb{D}\right\\}.$

Geometrically, a function $f\in\mathcal{S}^{*}_{\mbox{\leftmoon}}$ is that,
for any $z\in\mathbb{D}$, the ratio $\frac{zf^{\prime}(z)}{f(z)}$ contains the
region which is bounded by the lune. It is given by the relation
$\left\\{w\in\mathbb{C}:|w^{2}-1|\leq 2|w|\right\\}$. By using the definition
of the subordination, the class $\mathcal{S}^{*}_{\mbox{\leftmoon}}$ is
defined as

$\displaystyle\mathcal{S}^{*}_{\mbox{\leftmoon}}:=\left\\{f\in\mathcal{S}:\frac{zf^{\prime}(z)}{f(z)}\prec
z+\sqrt{1+z^{2}}=q(z),\;\;z\in\mathbb{D}\right\\},$

where branch of the square root is chosen to be $q(0)=1$. The class
$\mathcal{C}^{*}_{\mbox{\leftmoon}}$ of convex function $q$ is defined as

$\displaystyle\mathcal{C}^{*}_{\mbox{\leftmoon}}:=\left\\{f\in\mathcal{S}:1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\prec
q(z),\;\;z\in\mathbb{D}\right\\},$

The class $\mathcal{S}^{*}_{\mbox{\leftmoon}}$ has been the subject of
extensive investigation by several authors. The coefficient estimates of the
class $\mathcal{S}^{*}_{\mbox{\leftmoon}}$ were investigated by Raina and
Sokol [22, 23], whereas Gandhi and Ravichandran [10] examined the radius
issues associated with the same class. Certain differential subordinations
related to the class $\mathcal{S}^{*}_{\mbox{\leftmoon}}$ were studied by
Sharma et al. [27]. Raina et al. [21] give integral representation and
sufficient conditions for the functions in the class
$\mathcal{S}^{*}_{\mbox{\leftmoon}}$. A recent contribution by Cho et al. [7]
proposed a conjecture regarding the coefficients of this particular class.

In geometric function theory, a lot of emphasis have been given to evaluate
the bounds of Hankel determinants, whose elements are the coefficients of
analytic functions $f$ characterize in $\mathbb{D}$ of the form (1.2). Hankel
matrices (and determinants) play a key role in several branches of mathematics
and have various applications [31].

This study is dedicated to providing the sharp bound for the second Hankel
determinant, whose entries are the logarithmic coefficients. We commence by
presenting the definitions of Hankel determinants in the case where
$f\in\mathcal{A}$.

The Hankel determinant $H_{q,n}(f)$ of Taylor’s coefficients of functions
$f\in\mathcal{A}$ represented by (1.1) is defined for $q,n\in\mathbb{N}$ as
follows:

$\displaystyle H_{q,n}(f):=\begin{vmatrix}a_{n}&a_{n+1}&\cdots&a_{n+q-1}\\\
a_{n+1}&a_{n+2}&\cdots&a_{n+q}\\\ \vdots&\vdots&\vdots&\vdots\\\
a_{n+q-1}&a_{n+q}&\cdots&a_{n+2(q-1)}\end{vmatrix}.$

Kowalczyk and Lecko [13] recently proposed a Hankel determinant whose elements
are the logarithmic coefficients of $f\in\mathcal{S}$, realizing the extensive
use of these coefficients. This determinant is expressed as follows:

$\displaystyle
H_{q,n}(F_{f}/2)=\begin{vmatrix}\gamma_{n}&\gamma_{n+1}&\cdots&\gamma_{n+q-1}\\\
\gamma_{n+1}&\gamma_{n+2}&\cdots&\gamma_{n+q}\\\
\vdots&\vdots&\vdots&\vdots\\\
\gamma_{n+q-1}&\gamma_{n+q}&\cdots&\gamma_{n+2(q-1)}\end{vmatrix}.$

The study of Hankel determinants for starlike, convex, or many other functions
has been done extensively (see [20, 15, 25, 28, 13]), their sharp bounds have
been established. Recently, the Hankel determinants with logarithmic
coefficients have been examined for certain subclasses of starlike, convex,
univalent, strongly starlike and strongly convex functions (see [3, 13, 14]
and references therein). However, a little is known about sharp bounds of
Hankel determinants of logarithmic coefficients and need to explore them for
many classes of functions.

Differentiating (1.3) and using (1.2), a simple computation shows that

$\displaystyle\begin{cases}\gamma_{1}=\dfrac{1}{2}a_{2},\vspace{1.5mm}\\\
\gamma_{2}=\dfrac{1}{2}\left(a_{3}-\dfrac{1}{2}a^{2}_{2}\right),\vspace{1.5mm}\\\
\gamma_{3}=\dfrac{1}{2}\left(a_{4}-a_{2}a_{3}+\dfrac{1}{3}a^{3}_{2}\right),\vspace{1.5mm}\\\
\gamma_{4}=\dfrac{1}{2}\left(a_{5}-a_{2}a_{4}+a^{2}_{2}a_{3}-\dfrac{1}{2}a^{2}_{3}-\dfrac{1}{4}a^{4}_{2}\right),\vspace{1.5mm}\\\
\gamma_{5}=\dfrac{1}{2}\left(a_{6}-a_{2}a_{5}-a_{3}a_{4}+a_{2}a^{2}_{3}+a^{2}_{2}a_{4}-a^{3}_{2}a_{3}+\dfrac{1}{5}a^{5}_{2}\right).\end{cases}$

Due to the great importance of logarithmic coefficients in the recent years,
it is appropriate and interesting to compute the Hankel determinant whose
entries are logarithmic coefficients. In particular, the second Hankel
determinant of $F_{f}/2$ is defined as

(1.4) $\displaystyle
H_{2,1}(F_{f}/2)=\gamma_{1}\gamma_{3}-\gamma^{2}_{2}=\dfrac{1}{48}\left(a^{4}_{2}-12a^{2}_{3}+12a_{2}a_{4}\right).$

In this paper, we aim to explore by examining the sharp bound of the Hankel
determinant $H_{2,1}(F_{f}/2)$ for two class of functions, namely, starlike
and convex functions associated with lune.

It is known that for the Koebe function $f(z)=z/(1-z)^{2}$, the logarithmic
coefficients are $\gamma_{n}=1/n$, for each positive integer $n$. Since the
Koebe function appears as an extremal function in many problems of geometric
theory of analytic functions, one could expect that $\gamma_{n}=1/n$ holds for
functions in $\mathcal{S}$. But this is not true in general, even in order of
magnitude. The problem of computing the bound of the logarithmic coefficients
are studied recently by several authors in different contexts, for instance
see [1, 2, 6, 29, 20].

As usual, instead of the whole class $\mathcal{S}$, one can take into account
their subclasses for which the problem of finding sharp estimates of Hankel
determinant of logarithmic coefficients can be studied. The problem of
computing the sharp bounds of $H_{2,1}(F_{f}/2)$ was considered in [13] for
starlike and convex functions. It is now appropriate to remark that
$H_{2,1}(F_{f}/2)$ is invariant under rotation since for
$f_{\theta}(z):=e^{-i\theta}f(e^{i\theta}z)$, $\theta\in\mathbb{R}$ when
$f\in\mathcal{S}$ we have

$\displaystyle
H_{2,1}(F_{f_{\theta}}/2)=\frac{e^{4i\theta}}{48}\left(a^{4}_{2}-12a^{2}_{3}+12a_{2}a_{4}\right)=e^{4i\theta}H_{2,1}(F_{f}/2).$

##  2\. Preliminary results

The Carath$\acute{e}$odory class $\mathcal{P}$ and its coefficients bounds
plays a significant roles in establishing the bounds of Hankel determinants.
The class $\mathcal{P}$ of analytic functions $h$ defined for $z\in\mathbb{D}$
is given by

(2.1) $\displaystyle p(z)=1+\sum_{n=1}^{\infty}c_{n}z^{n}$

with positive real part in $\mathbb{D}$. A member of $\mathcal{P}$ is called a
Carath$\acute{e}$odory function. It is known that $c_{n}\leq 2$, $n\geq 1$ for
a function $p\in\mathcal{P}$ (see [8]).

In this section, we present key lemmas which will be used to prove the main
results of this paper. Parametric representations of the coefficients are
often useful in finding the bound of Hankel determinants, in this regard,
Libera and Zlotkiewicz [17, 18] derived the following parameterizations of
possible values of $c_{2}$ and $c_{3}$.

######  Lemma 2.1.

[17, 18] If $p\in\mathcal{P}$ is of the form (2.1) with $c_{1}\geq 0$, then

(2.2) $\displaystyle c_{1}=2\tau_{1},$ (2.3) $\displaystyle
c_{2}=2\tau^{2}_{1}+2(1-\tau^{2}_{1})\tau_{2}$

and

(2.4) $\displaystyle
c_{3}=2\tau^{3}_{1}+4(1-\tau^{2}_{1})\tau_{1}\tau_{2}-2(1-\tau^{2}_{1})\tau_{1}\tau^{2}_{2}+2(1-\tau^{2}_{1})(1-|\tau_{2}|^{2})\tau_{3}$

for some $\tau_{1}\in[0,1]$ and
$\tau_{2},\tau_{3}\in\overline{\mathbb{D}}:=\\{z\in\mathbb{C}:|z|\leq 1\\}$.

For $\tau_{1}\in\mathbb{T}:=\\{z\in\mathbb{C}:|z|=1\\}$, there is a unique
function $p\in\mathcal{P}$ with $c_{1}$ as in (2.2), namely

$\displaystyle p(z)=\frac{1+\tau_{1}z}{1-\tau_{1}z},\;\;z\in\mathbb{D}.$

For $\tau_{1}\in\mathbb{D}$ and $\tau_{2}\in\mathbb{T}$, there is a unique
function $p\in\mathcal{P}$ with $c_{1}$ and $c_{2}$ as in (2.2) and (2.3),
namely

$\displaystyle
p(z)=\frac{1+(\overline{\tau_{1}}\tau_{2}+\tau_{1})z+\tau_{2}z^{2}}{1+(\overline{\tau_{1}}\tau_{2}-\tau_{1})z-\tau_{2}z^{2}},\;\;z\in\mathbb{D}.$

For $\tau_{1},\tau_{2}\in\mathbb{D}$ and $\tau_{3}\in\mathbb{T}$, there is a
unique function $p\in\mathcal{P}$ with $c_{1},c_{2}$ and $c_{3}$ as in
(2.2)–(2.3), namely

$\displaystyle
p(z)=\frac{1+(\overline{\tau_{2}}\tau_{3}+\overline{\tau_{1}}\tau_{2}+\tau_{1})z+(\overline{\tau_{1}}\tau_{3}+\tau_{1}\overline{\tau_{2}}\tau_{3}+\tau_{2})z^{2}+\tau_{3}z^{3}}{1+(\overline{\tau_{2}}\tau_{3}+\overline{\tau_{1}}\tau_{2}-\tau_{1})z+(\overline{\tau_{1}}\tau_{3}-\tau_{1}\overline{\tau_{2}}\tau_{3}-\tau_{2})z^{2}-\tau_{3}z^{3}},\;\;z\in\mathbb{D}.$

######  Lemma 2.2.

[5] Let $A,B,C$ be real numbers and

$\displaystyle
Y(A,B,C):=\max\\{|A+Bz+Cz^{2}|+1-|z|^{2}:z\in\overline{\mathbb{D}}\\}.$

(i) If $AC\geq 0$, then

$\displaystyle
Y(A,B,C)=\begin{cases}|A|+|B|+|C|,\;\;\;\;\;\;\;\;\;\;\;\;\;|B|\geq
2(1-|C|),\vspace{2mm}\\\
1+|A|+\dfrac{B^{2}}{4(1-|C|)},\;\;\;\;\;|B|<2(1-|C|).\end{cases}$

(ii) If $AC<0$, then

$\displaystyle
Y(A,B,C)=\begin{cases}1-|A|+\dfrac{B^{2}}{4(1-|C|)},\;\;\;\;-4AC(C^{-2}-1)\leq
B^{2}\land|B|<2(1-|C|),\vspace{2mm}\\\
1+|A|+\dfrac{B^{2}}{4(1+|C|)},\;\;\;\;B^{2}<\min\\{4(1+|C|)^{2},-4AC(C^{-2}-1)\\},\vspace{2mm}\\\
R(A,B,C),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise,\end{cases}$

where

$\displaystyle
R(A,B,C):=\begin{cases}|A|+|B|-|C|,\;\;\;\;\;\;\;\;\;\;\;\;\;|C|(|B|+4|A|)\leq|AB|,\vspace{2mm}\\\
-|A|+|B|+|C|,\;\;\;\;\;\;\;\;\;\;\;|AB|\leq|C|(|B|-4|A|),\vspace{2mm}\\\
(|C|+|A|)\sqrt{1-\dfrac{B^{2}}{4AC}},\;\;\;otherwise.\end{cases}$

For a better clarity in our presentation, we divide this into two section
consisting of different families of functions from the class $\mathcal{A}$ and
prove our main results for starlike functions and convex functions associated
with lune.

##  3\. The second Hankel determinant of logarithmic coefficients of functions
in the class $\mathcal{S}^{*}_{\mbox{\leftmoon}}$

We obtain the following result finding the sharp bound of $|H_{2,1}(F_{f}/2)|$
for functions in the class $\mathcal{S}^{*}_{\mbox{\leftmoon}}$.

######  Theorem 3.1.

Let $f\in\mathcal{S}^{*}_{\mbox{\leftmoon}}$. Then

(3.1) $\displaystyle|H_{2,1}(F_{f}/2)|\leq\frac{1}{16}.$

The inequality is sharp for the function
$g\in\mathcal{S}^{*}_{\mbox{\leftmoon}}$ given by

$\displaystyle
g(z)=z\exp\left(\int_{0}^{z}\frac{x^{2}+\sqrt{1+x^{4}}-1}{x}dx\right)=z+\frac{z^{3}}{2}+\frac{z^{5}}{4}+\cdots.$

###### Proof.

Let $f\in\mathcal{S}^{*}_{\mbox{\leftmoon}}$. Then in view Definition 1.1, it
follows that

(3.2) $\displaystyle\frac{zf^{\prime}(z)}{f(z)}=w(z)+\sqrt{1+w^{2}(z)},$

where $w$ is a Schwarz function with $w(0)=0$ and $|w(z)|\leq 1$ in
$\mathbb{D}$. Let $h\in\mathcal{P}$. Then we can write

(3.3) $\displaystyle w(z)=\frac{h(z)-1}{h(z)+1}.$

From (3.2) and (3.3), a simple computation shows that

(3.4) $\displaystyle\begin{cases}a_{2}=\dfrac{1}{2}c_{1},\vspace{2mm}\\\
a_{3}=\dfrac{1}{16}c^{2}_{1}+\dfrac{1}{4}c_{2},\vspace{2mm}\\\
a_{4}=\dfrac{1}{24}c_{1}c_{2}+\dfrac{1}{6}c_{3}-\dfrac{1}{96}c^{3}_{1}.\end{cases}$

Since the class $\mathcal{P}$ and $H_{2,1}(F_{f}/2)$ is invariant under
rotation, we may assume that $c_{1}\in[0,2]$(see [4]; see also [12, Vol. I, p.
80, Theorem 3]), that is, in view of (2.2), that $\tau_{1}\in[0,1]$. Using
(1.4) and (3.4), it is easy to see that

(3.5) $\displaystyle H_{2,1}(F_{f}/2)$
$\displaystyle=\frac{1}{48}\left(a^{4}_{2}-12a^{2}_{3}+12a_{2}a_{4}\right)$
$\displaystyle=\frac{1}{3072}\left(-3c^{4}_{1}-8c^{2}_{1}c_{2}-48c^{2}_{2}+64c_{1}c_{3}\right).$

By the Lemma 2.1 and (3.5), a straightforward computation shows that

(3.6) $\displaystyle H_{2,1}(F_{f}/2)$
$\displaystyle=\frac{1}{192}\left(-3\tau^{4}_{1}+4(1-\tau^{2}_{1})\tau^{2}_{1}\tau_{2}-4(1-\tau^{2}_{1})(3+\tau^{2}_{1})\tau^{2}_{2}\right.$
$\displaystyle\left.\quad+16\tau_{1}\tau_{3}(1-\tau^{2}_{1})(1-|\tau^{2}_{2}|)\right).$

Below, we discuss the following possible cases on $\tau_{1}$:

Case 1. Suppose that $\tau_{1}=1$. Then from (3.6), we easily obtain

$\displaystyle|H_{2,1}(F_{f}/2)|=\frac{1}{64}.$

Case 2. Let $\tau_{1}=0$. Then from (3.6), we see that

$\displaystyle|H_{2,1}(F_{f}/2)|=\frac{1}{16}|\tau_{2}|^{2}\leq\frac{1}{16}.$

Case 3. Suppose that $\tau_{1}\in(0,1)$. Applying the triangle inequality in
(3.6) and by using the fact that $|\tau_{3}|\leq 1$, we obtain

$\displaystyle|H_{2,1}(F_{f}/2)|$
$\displaystyle=\frac{1}{192}\left(|-3\tau^{4}_{1}+4(1-\tau^{2}_{1})\tau^{2}_{1}\tau_{2}-4(1-\tau^{2}_{1})(3+\tau^{2}_{1})\tau^{2}_{2}|\right.$
$\displaystyle\left.\quad+16\tau_{1}(1-\tau^{2}_{1})(1-|\tau^{2}_{2}|)\right)$
(3.7)
$\displaystyle=\frac{1}{12}\tau_{1}(1-\tau^{2}_{1})\left(|A+B\tau_{2}+C\tau^{2}_{2}|+1-|\tau_{2}|^{2}\right),$

where

$\displaystyle
A:=\frac{-3\tau^{3}_{1}}{16(1-\tau^{2}_{1})},\;\;B:=\frac{\tau_{1}}{4}\;\;\mbox{and}\;\;C:=\frac{-(3+\tau^{2}_{1})}{4\tau_{1}}.$

Observe that $AC>0$, so we can apply case (i) of Lemma 2.2. Now we check all
the conditions of case (i).

3(a) We note the inequality

$\displaystyle|B|-2(1-|C|)$
$\displaystyle=\frac{\tau_{1}}{4}-2\left(1-\frac{(3+\tau^{2}_{1})}{4\tau_{1}}\right)$
$\displaystyle=\frac{3\tau^{2}_{1}-8\tau_{1}+6}{4\tau_{1}}>0$

which is true for all $\tau_{1}\in(0,1)$. Thus it follows from Lemma 2.2 and
the inequality (3) that

$\displaystyle|H_{2,1}(F_{f}/2)|$
$\displaystyle\leq\frac{1}{12}\tau_{1}(1-\tau^{2}_{1})\left(|A|+|B|+|C|\right)$
$\displaystyle=\frac{1}{192}\left(12-4\tau^{2}_{1}-5\tau^{4}_{1}\right)$
$\displaystyle\leq\frac{1}{16}.$

3(b) Next, it is easy to check that

$\displaystyle|B|-2(1-|C|)$
$\displaystyle=\frac{\tau_{1}}{4}-2\left(1-\frac{(3+\tau^{2}_{1})}{4\tau_{1}}\right)$
$\displaystyle=\frac{3\tau^{2}_{1}-8\tau_{1}+6}{4\tau_{1}}<0$

which is not true for all $\tau_{1}\in(0,1)$.

Summarizing cases $1$, $2$, and $3$, the inequality (3.1) is established.

To complete the proof, it is sufficient to show that the bound is sharp. In
order to show that we consider the function
$g\in\mathcal{S}^{*}_{\mbox{\leftmoon}}$ as follows

$\displaystyle
g(z)=z\exp\left(\int_{0}^{z}\frac{x^{2}+\sqrt{1+x^{4}}-1}{x}dx\right)=z+\frac{z^{3}}{2}+\frac{z^{5}}{4}+\cdots,$

with $a_{2}=a_{4}=0$ and $a_{3}=1/2$. By a simple computation, it can be
easily shown that $|H_{2,1}(F_{g}/2)|=1/16$. This completes the proof. ∎

##  4\. The second Hankel determinant of logarithmic coefficients of functions
in the class $\mathcal{C}_{\mbox{\leftmoon}}$

We obtain the following sharp bound of $|H_{2,1}(F_{f}/2)|$ for functions in
the class $\mathcal{C}_{\mbox{\leftmoon}}$.

######  Theorem 4.1.

Let $f\in\mathcal{C}_{\mbox{\leftmoon}}$. Then

(4.1) $\displaystyle|H_{2,1}(F_{f}/2)|\leq\frac{23}{3264}$

The inequality is sharp for the function $h\in\mathcal{C}_{\mbox{\leftmoon}}$
is given by

$\displaystyle
h(z)=\int_{0}^{z}\frac{h_{0}(x)}{x}dx=z+\frac{\sqrt{69}}{12\sqrt{17}}z^{3}+\frac{1}{20}\left(\frac{69}{136}+\frac{\sqrt{69}}{4\sqrt{17}}\right)z^{5}+\cdots,$

where $h_{0}(z)$ is given by (4.8).

###### Proof.

Let $f\in\mathcal{C}_{\mbox{\leftmoon}}$. Then by the Definition 1.1, we see
that

(4.2) $\displaystyle
1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}=w(z)+\sqrt{1+w^{2}(z)},$

where $w$ is a Schwarz function with $w(0)=0$ and $|w(z)|\leq 1$ in
$\mathbb{D}$. Let $h\in\mathcal{P}$. Then we can write

(4.3) $\displaystyle w(z)=\frac{h(z)-1}{h(z)+1}.$

In view of (4.2) and (4.3), a simple computation shows that

(4.4) $\displaystyle\begin{cases}a_{2}=\dfrac{1}{4}c_{1},\vspace{2mm}\\\
a_{3}=\dfrac{1}{48}c^{2}_{1}+\dfrac{1}{12}c_{2},\vspace{2mm}\\\
a_{4}=\dfrac{1}{96}c_{1}c_{2}+\dfrac{1}{24}c_{3}-\dfrac{1}{384}c^{3}_{1}.\end{cases}$

Since the class $\mathcal{P}$ and $H_{2,1}(F_{f}/2)$ is invariant under
rotation, we may assume that $c_{1}\in[0,2]$(see [4]; see also [12, Volume I,
page 80, Theorem 3]), that is, in view of (2.2), that $\tau_{1}\in[0,1]$. From
(1.4) and (4.4), an easy computation leads to

(4.5) $\displaystyle H_{2,1}(F_{f}/2)$
$\displaystyle=\dfrac{1}{48}\left(a^{4}_{2}-12a^{2}_{3}+12a_{2}a_{4}\right)$
$\displaystyle=\dfrac{1}{36864}\left(-7c^{4}_{1}-8c^{2}_{1}c_{2}-64c^{2}_{2}+96c_{1}c_{3}\right).$

In view of Lemma 2.1 and (4.5), we obtain

(4.6) $\displaystyle H_{2,1}(F_{f}/2)$
$\displaystyle=\frac{1}{2304}\left(-3\tau^{4}_{1}+12(1-\tau^{2}_{1})\tau^{2}_{1}\tau_{2}-8(1-\tau^{2}_{1})(2+\tau^{2}_{1})\tau^{2}_{2}\right.$
$\displaystyle\left.\quad+24\tau_{1}\tau_{3}(1-\tau^{2}_{1})(1-|\tau^{2}_{2}|)\right).$

Now, we may have the following cases on $\tau_{1}$:

Case 1. Suppose that $\tau_{1}=1$. Then from (4.6), we obtain

$\displaystyle|H_{2,1}(F_{f}/2)|=\frac{1}{768}.$

Case 2. Let $\tau_{1}=0$. Then from (3.6), we have the following estimate

$\displaystyle|H_{2,1}(F_{f}/2)|=\frac{1}{144}|\tau_{2}|^{2}\leq\frac{1}{144}.$

Case 3. Suppose that $\tau_{1}\in(0,1)$. Applying the triangle inequality in
(4.6) and by using the fact that $|\tau_{3}|\leq 1$, we obtain

$\displaystyle H_{2,1}(F_{f}/2)$
$\displaystyle\leq\frac{1}{2304}\left(|-3\tau^{4}_{1}+12(1-\tau^{2}_{1})\tau^{2}_{1}\tau_{2}-8(1-\tau^{2}_{1})(2+\tau^{2}_{1})\tau^{2}_{2}|\right.$
$\displaystyle\left.\quad+24\tau_{1}(1-\tau^{2}_{1})(1-|\tau^{2}_{2}|)\right)$
(4.7)
$\displaystyle=\frac{1}{96}\tau_{1}(1-\tau^{2}_{1})\left(|A+B\tau_{2}+C\tau^{2}_{2}|+1-|\tau_{2}|^{2}\right),$

where

$\displaystyle
A:=\frac{-\tau^{3}_{1}}{8(1-\tau^{2}_{1})},\;\;B:=\frac{\tau_{1}}{2}\;\;\mbox{and}\;\;C:=\frac{-(2+\tau^{2}_{1})}{3\tau_{1}}.$

Observe that $AC>0$, so we can apply case (i) of Lemma 2.2. Now we check all
the conditions of case (i).

3(a) We note the inequality

$\displaystyle|B|-2(1-|C|)$
$\displaystyle=\frac{\tau_{1}}{2}-2\left(1-\frac{(2+\tau^{2}_{1})}{3\tau_{1}}\right)$
$\displaystyle=\frac{7\tau^{2}_{1}-12\tau_{1}+8}{6\tau_{1}}>0,$

which is true for all $\tau_{1}\in(0,1)$. It follows from Lemma 2.2 and the
inequality (4) that

$\displaystyle|H_{2,1}(F_{f}/2)|$
$\displaystyle\leq\frac{1}{96}\tau_{1}(1-\tau^{2}_{1})\left(|A|+|B|+|C|\right)$
$\displaystyle=\frac{1}{2304}\left(16+4\tau^{2}_{1}-17\tau^{4}_{1}\right)$
$\displaystyle\leq\frac{23}{3264}.$

3(b) For the second inequality, we see that

$\displaystyle|B|-2(1-|C|)$
$\displaystyle=\frac{\tau_{1}}{2}-2\left(1-\frac{(2+\tau^{2}_{1})}{3\tau_{1}}\right)$
$\displaystyle=\frac{7\tau^{2}_{1}-12\tau_{1}+8}{6\tau_{1}}<0,$

which is not true for all $\tau_{1}\in(0,1)$. Summarizing, all the cases
discussed above, we obtain the inequality of the result.

In order to show the bound is sharp, we consider the function
$h\in\mathcal{C}_{\mbox{\leftmoon}},$ for

(4.8) $\displaystyle h_{0}(z)$
$\displaystyle=z\exp\left(\sqrt{\frac{69}{68}}\int_{0}^{z}\frac{x^{2}+\sqrt{1+x^{4}}-1}{x}dx\right)$
$\displaystyle=z+\frac{\sqrt{69}}{4\sqrt{17}}z^{3}+\frac{1}{4}\left(\frac{69}{136}+\frac{\sqrt{69}}{4\sqrt{17}}\right)z^{5}+\cdots,.$

Let

$\displaystyle
h(z)=\int_{0}^{z}\frac{h_{0}(x)}{x}dx=z+\frac{\sqrt{69}}{12\sqrt{17}}z^{3}+\frac{1}{20}\left(\frac{69}{136}+\frac{\sqrt{69}}{4\sqrt{17}}\right)z^{5}+\cdots,$

with $a_{2}=a_{4}=0$ and $a_{3}=\sqrt{69}/{12\sqrt{17}}$. A simple computation
shows that $|H_{2,1}(F_{h}/2)|=23/32640$, which demonstrates that the bound is
sharp. This completes the proof. ∎

Compliance of Ethical Standards:

Conflict of interest. The authors declare that there is no conflict of
interest regarding the publication of this paper.

Data availability statement. Data sharing is not applicable to this article as
no datasets were generated or analyzed during the current study.

## References

  * [1] M. F. Ali and V. Allu, On logarithmic coefficients of some close-to-convex functions, Proc. Amer. Math. Soc. 146 (2018), 1131–1142.
  * [2] M.F. Ali, V. Allu and D. K. Thomas, On the third logarithmic coefficients of close- to- convex functions, Curr. Res. Math. Comput. Sci. II, Publisher UWM, Olsztyn, (2018) 271–278.
  * [3] V. Allu, V. Arora, and A. Shaji, On the Second Hankel Determinant of Logarithmic Coefficients for Certain Univalent Functions, Mediterr. J. Math. 20 (2023), 81.
  * [4] C. Carathéodory, Über den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann. 64 (1907), 95–115.
  * [5] N. E. Cho, Y. C. Kim and T. Sugawa, A general approach to the Fekete-Szego problem, J. Math. Soc. Japan. 59 (2007), 707-727.
  * [6] N. E. Cho, B. Kowalczyk, O. S. Kwon, A. Lecko and Y. J. Sim, On the third logarithmic coefficient in some subclasses of close-to-convex functions, Rev. R. Acad. Cienc. Exactas Fís. Nat.(Esp.) 114, Art: 52, (2020), 1–14.
  * [7] N. E. Cho, V. Kumar, O.S. Kwon and Y.J. Sim, Coefficient bounds for certain subclasses of starlike functions, J. Inequal. Appl. 2019 (2019) 276.
  * [8] P. T. Duren, Univalent Functions. Springer-Verlag, New York Inc (1983).
  * [9] I. Efraimidis, A generalization of Livingston’s coefficient inequalities for functions with positive real part, J. Math. Anal. Appl. 435 (2016), 369–379.
  * [10] S. Gandhi and V. Ravichandran, Starlike functions associated with a lune, Asian-Eur. J. Math. 10 (2017) 1750064.
  * [11] D. Girela, Logarithmic coefficients of univalent functions, Ann. Acad. Sci. Fenn. 25 (2000), 337–350.
  * [12] A. W. Goodman, Univalent Functions (Mariner, Tampa, FL, 1983).
  * [13] B. Kowalczyk and A. Lecko, Second Hankel determinant of logarithmic coefficients of convex and starlike functions, Bull. Aust. Math. Soc. 105 (2022), no. 3, 458–467.
  * [14] B. Kowalczyk and A. Lecko, Second Hankel determinant of logarithmic coefficients of convex and starlike functions of order alpha, Bull. Malays. Math. Sci. Soc. 45 (2022), no. 2, 727–740.
  * [15] B. Kowalczyk and A. Lecko, The second Hankel determinant of the logarithmic coefficients of strongly starlike and strongly convex functions, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 91 (2023).
  * [16] B. Kowalczyk, A. Lecko and Y. J. Sim, The sharp bound for the Hankel determinant of the third kind for convex functions, Bull. Aust. Math. Soc. 97 (2018), 435–445.
  * [17] R. J. Libera and E. J. Zlotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), 225-230.
  * [18] R. J. Libera and E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivatives in $\mathcal{P}$, Proc. Amer. Math. Soc. 87 (1983), 251-257.
  * [19] I. M. Milin, Univalent Functions and Orthonormal Systems (Nauka, Moscow, 1971) (in Russian); English translation, Translations of Mathematical Monographs, 49 (American Mathematical Society, Providence, RI, 1977).
  * [20] S.Ponnusamy and T. Sugawa, Sharp inequalities for logarithmic coefficients and their applications, Bull. Sci. Math. 166 (2021), 23.
  * [21] R. K. Raina, P. Sharma and J. Sokół, Certain classes of analytic functions related to the crescent-shaped regions, J. Contemp. Math. Anal. 53 (2018) 355–362.
  * [22] R. K. Raina and J. Sokół, On coefficient for certain class of starlike functions, Hacet. J. Math. Stat. 44(6) (2015) 1427–1433.
  * [23] R. K. Raina and J. Sokół, Some coefficient properties relating to a certain class of starlike functions, Miskolc Math. Notes 44(6) (2015) 1427–1433.
  * [24] R. K. Raina and J. Sokół, Some properties related to a certain class of starlike functions, C. R. Math. Acad. Sci. Paris 353(11) (2015) 973–978.
  * [25] M. Raza, A. Riaz and D. K. Thomas, The third Hanekl determinant for inverse coefficients of convex functions, Bull. Aust. Math. Soc. (2023), 1-7.
  * [26] K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan 11 (1959), 72–75.
  * [27] P. Sharma, R. K. Raina and J. Sokół, Certain Ma–Minda type classes of analytic functions associated with the crescent-shaped region, Anal. Math. Phys. 9 (2019) 1–17.
  * [28] Y. J. Sim, A. Lecko, and D. K. Thomas, The second Hankel determinant for strongly convex and Ozaki close to convex functions, Ann. Mat. Pura Appl. 200(6) (2021) 2515-2533.
  * [29] D. K. Thomas, On logarithmic coefficients of close to convex functions, Proc. Amer. Math. Soc. 144 (2016), 1681–1687.
  * [30] O. Toeplitz, Zur Transformation der Scharen bilinearer Formen von unendlichvielen Ver¨anderlichen. Mathematischphysikalis- che, Klasse, Nachr. der Kgl. Gessellschaft derWissenschaften zu G¨ottingen, (1907), 110–115.
  * [31] K. Ye and L. H. Lim, Every matrix is a product of Toeplitz matrices, Found. Comput. Math. 16 (2016), no. 3, 577–598.
  * [32] P. Zaprawa, Initial logarithmic coefficients for functions starlike with respect to symmetric points, Bol. Soc. Mat. Mex. 27 (2021) 62.


# Signature of Collapsars as Sources for High-energy Neutrinos and $r$-process
Nuclei

Gang Guo<EMAIL_ADDRESS>School of Mathematics and Physics, China
University of Geosciences, Wuhan 430074, China  Yong-Zhong Qian
<EMAIL_ADDRESS>School of Physics and Astronomy, University of Minnesota,
Minneapolis, MN 55455, USA  Meng-Ru Wu<EMAIL_ADDRESS>Institute of
Physics, Academia Sinica, Taipei, 11529, Taiwan  Institute of Astronomy and
Astrophysics, Academia Sinica, Taipei, 10617, Taiwan

###### Abstract

If collapsars are sources for both high-energy (HE) neutrinos and $r$-process
nuclei, the profuse low-energy antineutrinos from $\beta$-decay of the newly-
synthesized nuclei can annihilate the HE neutrinos. Considering HE neutrinos
produced at internal shocks induced by intermittent mildly-magnetized jets, we
show that such annihilation suppresses the overall HE neutrino spectrum at
$\gtrsim 300$ TeV and produces a corresponding flavor composition of
$(F_{\nu_{e}+\bar{\nu}_{e}}:F_{\nu_{\mu}+\bar{\nu}_{\mu}}:F_{\nu_{\tau}+\bar{\nu}_{\tau}})_{\star}\approx(1:10:1)$
at source. We find that the emergent HE neutrino flux can well fit the diffuse
flux observed at IceCube if contributions from all similar sources are taken
into account. Our results highlight the unique role of HE neutrinos in
supporting collapsars as sources for $r$-process nuclei, and can be tested by
detection of HE neutrinos from individual sources and accurate measurement of
the diffuse HE neutrino flux spectrum and flavor composition.

##  I Introduction

Collapsars produced by the collapse of massive stars into black holes have
long been considered a leading candidate for powering Type Ibc supernovae and
long gamma-ray bursts (GRBs) of both the bright and low-luminosity (LLGRBs)
varieties Woosley (1993); MacFadyen and Woosley (1999); Woosley and Bloom
(2006); Campana  _et al._ (2006); Hjorth and Bloom (2012); Kumar and Zhang
(2014a). Shocks associated with the propagation of the collapsar jet through
the stellar envelope and/or the black-hole accretion disk wind were also
proposed as candidate sites for producing $\sim$ TeV–PeV high energy (HE)
neutrinos Mészáros and Waxman (2001); Razzaque  _et al._ (2003, 2004); Ando
and Beacom (2005); Razzaque  _et al._ (2005); Horiuchi and Ando (2008); Enberg
_et al._ (2009); Bartos  _et al._ (2012); Murase and Ioka (2013); Fraija
(2014); Xiao and Dai (2014); Bhattacharya  _et al._ (2015); Varela  _et al._
(2015); Tamborra and Ando (2016); Senno  _et al._ (2016, 2018); Denton and
Tamborra (2018a, b); He  _et al._ (2018); Chang  _et al._ (2022); Guarini  _et
al._ (2023). These neutrinos may contribute significantly to the diffuse flux
detected by IceCube, whose astrophysical origin remains unknown despite recent
reports of potential association of several events with a blazar Aartsen  _et
al._ (2018a, b), three tidal disruption events Stein  _et al._ (2021); Reusch
_et al._ (2022); van Velzen  _et al._ (2021), and an active galaxy Abbasi  _et
al._ (2022a).

The ground-breaking multi-messenger observations of GW170817 linked binary
neutron star mergers (BNSMs) to both short GRBs and the production of heavy
elements beyond iron via the rapid neutron-capture process ($r$-process)
Abbott  _et al._ (2017a, b); Kasen  _et al._ (2017); Drout  _et al._ (2017);
Cowperthwaite  _et al._ (2017); Villar  _et al._ (2017); Shibata  _et al._
(2017); Metzger (2020); Watson  _et al._ (2019). The astrophysical
environments of BNSMs and collapsars are rather similar in that both host
accretion disks and the associated jets, thereby producing GRBs and
potentially HE neutrinos. Interestingly, Ref. Siegel  _et al._ (2019) showed
that the physical conditions of disk outflows in collapsars resemble those of
outflows in BNSMs, and proposed that collapsars are a likely or even the
dominant site for producing $r$-process nuclei. This proposal remains hotly
debated and requires further theoretical and observational efforts to clarify
Siegel (2019); Macias and Ramirez-Ruiz (2019); Bartos and Marka (2019); Miller
_et al._ (2020); Fujibayashi  _et al._ (2020); Brauer  _et al._ (2021); Fraser
and Schönrich (2022); Barnes and Metzger (2022); Just  _et al._ (2022);
Fujibayashi  _et al._ (2022); Anand  _et al._ (2023).

In this work, we investigate for the first time a novel connection between HE
neutrinos and $r$-process nucleosynthesis in collapsars. Unstable neutron-rich
heavy nuclei are produced by rapid neutron capture during the $r$-process.
Their $\beta$ decay produces $\bar{\nu}_{e}$ with energy $E_{L}\sim 4$ MeV on
a timescale of $\sim 1$ s _everywhere_ inside the outflow that expands with a
typical velocity $v_{\rm ej}\sim(0.05$–$0.3)c$ (see Fig. 1). Following flavor
oscillations, these low energy (LE) antineutrinos can annihilate HE neutrinos
of the corresponding flavor produced by shocks at radius $R$. Efficient
annihilation via the $Z$-resonance requires $s=2E_{H}E_{L}(1-\cos\theta)\sim
m_{Z}^{2}$, where $\theta\sim\mathcal{O}(1)$ is the intersection angle and
$m_{Z}$ is the $Z$ mass. Therefore, the HE neutrino flux at
$E_{H}\gtrsim\mathcal{O}(100$–1000) TeV is expected to be affected. As we will
show, such annihilation can leave clear imprints on the energy spectrum and
flavor composition of the emergent HE neutrinos.

Figure 1: Sketch of production of HE neutrinos (with energy $E_{H}$) and their annihilation with LE antineutrinos (with energy $E_{L}$) from $\beta$-decay of $r$-process nuclei in a collapsar. Table 1: Proper values of magnetic field $B_{d(u)}$, proton number density $n_{p,d(u)}$, and photon temperature $T_{\gamma,d(u)}$ in the shocked (downstream) and unshocked (upstream) jets. | $B_{d(u),8}$ | $n_{p,d(u),19}$ | $T_{\gamma,d(u),{\rm keV}}$
---|---|---|---
downstream | $8.2(\epsilon_{B,d}L_{{\rm iso},52})^{1/2}/(R_{{\rm is},10}\Gamma_{2})$ | $1.8L_{\rm iso,52}(1-\epsilon_{e,d}-\epsilon_{B,d})R_{\rm is,10}^{-2}\Gamma_{2}^{-2}$ | $3.7(\epsilon_{e,d}L_{\rm iso,52})^{1/4}R_{\rm is,10}^{-1/2}\Gamma_{2}^{-1/2}$
upstream | $B_{d,8}/\xi$ | $n_{p,d,19}/\xi$ | $T_{\gamma,d,{\rm keV}}(\epsilon_{e,u}/\epsilon_{e,d})^{1/4}(\Gamma/\Gamma_{r})^{1/2}$

##  II Antineutrinos from $\beta$-decay

The $\beta$-decay of the newly-synthesized $r$-process nuclei not only
produces a profuse $\bar{\nu}_{e}$ flux, but also provides a dominant source
of energy to heat the associated ejecta Metzger  _et al._ (2010). We estimate
the $\bar{\nu}_{e}$ emission using the power generated by $\beta$-decay. In
our benchmark study, we simply assume that the ejecta expands with a constant
velocity $v_{\rm ej}$, forming a steady spherical “wind” with a constant mass
outflow rate $\dot{M}$ (see Appendix A). As the ejecta expands, the power
generated by $\beta$-decay per unit mass stays almost constant for a period of
$\sim T_{r}$ when the $r$-process produces neutron-rich nuclei far from
stability. After neutron capture ceases, that power approximately follows a
power-law decline. We take $\dot{\epsilon}_{\bar{\nu}_{e},0}\eta(t)$ as the
part of the power per unit mass that is carried away by $\bar{\nu}_{e}$, where
$\dot{\epsilon}_{\bar{\nu}_{e},0}$ sets the magnitude and $\eta(t)$
characterizes the time evolution with $\eta(0)\approx 1$. From numerical
calculations, $\eta(t)$ can be fitted as Rosswog  _et al._ (2014)

$\displaystyle\eta(t)\approx\left[\frac{1}{2}-\frac{1}{\pi}\arctan\left(\frac{t-T_{r}}{0.11~{}\rm
s}\right)\right]^{1.3}.$ (1)

Ignoring nuclear shell effects and Coulomb correction, we simply take the
$\bar{\nu}_{e}$ spectrum to be $\propto E^{2}(Q-E)^{2}$ with $Q$ being the
$Q$-value. The energy-differential emission rate of $\bar{\nu}_{e}$ per unit
mass (i.e., emissivity) at radius $r=v_{\rm ej}t$ can be estimated as

$\displaystyle j_{\bar{\nu}_{e}}(E,r)$
$\displaystyle=\frac{\dot{\epsilon}_{\bar{\nu}_{e},0}\eta(r/v_{\rm
ej})}{\langle E_{L}\rangle}\left[\frac{15(2\langle
E_{L}\rangle-E)^{2}E^{2}}{16\langle E_{L}\rangle^{5}}\right],$ (2)

where the term in the brackets is the approximate normalized $\bar{\nu}_{e}$
spectrum with an average energy of $\langle E_{L}\rangle=Q/2=4$ MeV. Note that
both $\dot{\epsilon}_{\bar{\nu}_{e},0}$ and $T_{r}$ depend on the electron
fraction $Y_{e}$ of the ejecta Rosswog  _et al._ (2014). Assuming a typical
value of $Y_{e}=0.2$ for collapsar outflows and guided by the power used in
Ref. Wu  _et al._ (2019), we take $\dot{\epsilon}_{\bar{\nu}_{e},0}=5\times
10^{18}$ erg/g/s and $T_{r}=0.4$ s.

The $\bar{\nu}_{e}$ emission is approximately isotropic. Referring to Fig. 1,
we estimate the $\bar{\nu}_{e}$ intensity (in units of cm-2 s-1 MeV-1 sr-1) at
radius $R$ and angle $\theta$ by integrating the emissivity along the path
length $l$:

$\displaystyle I_{\bar{\nu}_{e}}(E,R,\theta)$
$\displaystyle=\int\frac{\rho(r)j_{\bar{\nu}_{e}}(E,r)}{4\pi}dl$
$\displaystyle=\frac{R\dot{M}}{16\pi^{2}v_{\rm
ej}}\int\frac{j_{\bar{\nu}_{e}}(E,r)\sin\theta}{r^{2}\sin^{2}(\theta+\theta^{\prime})}d\theta^{\prime},$
(3)

where $\rho(r)=\dot{M}/(4\pi r^{2}v_{\rm ej})$ is the mass density of the
ejecta at radius $r$, and $r$ is a function of $R$, $\theta$, and
$\theta^{\prime}$. Unless noted otherwise, we take
$\dot{M}=0.02\,M_{\odot}/{\rm s}$ Hayakawa and Maeda (2018); Fujibayashi  _et
al._ (2022) and $v_{\rm ej}=0.05c$ MacFadyen and Woosley (1999); Miller  _et
al._ (2020). The corresponding total number density of LE $\bar{\nu}_{e}$ at
$R\sim 10^{10}$ cm is $\sim 10^{23}~{}{\rm cm}^{-3}$.

Following production, the LE $\bar{\nu}_{e}$ undergo flavor oscillations prior
to interacting with the HE neutrinos. The oscillation scenario depends on the
matter densities at the $\bar{\nu}_{e}$ emission site and the interaction
site. We consider three sets of probabilities $P_{\bar{\nu}_{\alpha}}$ for the
initial $\bar{\nu}_{e}$ to be a $\bar{\nu}_{\alpha}$ at the interaction site:
$(P_{\bar{\nu}_{e}},P_{\bar{\nu}_{\mu}},P_{\bar{\nu}_{\tau}})\approx(0.55,0.18,0.27)$
(pure vacuum oscillations), $(0.675,0.095,0.23)$ (NH), and
$(0.022,0.545,0.433)$ (IH). The latter two sets assume adiabatic flavor
evolution following emission at high densities for normal (NH) and inverted
(IH) neutrino mass hierarchy, respectively (see Appendix B).

Figure 2: HE neutrino spectra including annihilation with LE antineutrinos
(solid curves) or not (dashed curves). Vacuum oscillations of LE antineutrinos
are assumed. The benchmark case with $L_{\rm iso}=10^{53}$ erg/s is shown in
the left ($\nu_{e}+\nu_{\mu}+\nu_{\tau}$ and
$\bar{\nu}_{e}+\bar{\nu}_{\mu}+\bar{\nu}_{\tau}$) and middle
($\nu_{\alpha}+\bar{\nu}_{\alpha}$ with $\alpha=e,\mu,\tau$) panels. Cases of
$L_{\rm iso}=10^{50}$–$10^{53}$ erg/s are shown in the right panel (all
flavors). The inset shows the ratios of all-flavor spectra without
annihilation to those with annihilation.

##  III High-energy neutrino production

Relativistic jets in collapsars induce shocks that can accelerate electrons
and protons. The accelerated electrons can produce $\gamma$-rays or X-rays
through synchrotron radiation or inverse Compton scattering, thereby making
bright GRBs, while the HE protons can collide with photons or stellar matter
to make $\pi^{\pm}$ and $K^{\pm}$, thereby producing HE neutrinos through
meson decays. Shocks emerging at different sites during jet propagation lead
to different scenarios for HE neutrino production Waxman and Bahcall (1997,
2000); Li  _et al._ (2002); Guetta  _et al._ (2004); Murase and Nagataki
(2006); Murase  _et al._ (2006); Murase (2007); Hummer  _et al._ (2012); Zhang
and Kumar (2013); Liu and Wang (2013); Bustamante  _et al._ (2015); Nir  _et
al._ (2016); Biehl  _et al._ (2018); Guo  _et al._ (2020); Kimura (2022). To
demonstrate the effects of LE antineutrinos from $\beta$-decay, we focus on HE
neutrinos produced during jet propagation deep inside the progenitors of
collapsars. Specifically, we consider proton acceleration at internal shocks
before jet collimation Mészáros and Waxman (2001); Murase and Ioka (2013).

Ref. Murase and Ioka (2013) emphasized that shocks inside stars are likely
mediated by radiation and in contrast to collisionless shocks Levinson and
Bromberg (2008); Katz  _et al._ (2010), may not support efficient particle
acceleration and HE neutrino production. It was shown later, however, that if
the jets are mildly magnetized, a strong collisionless subshock may occur
within the radiation-mediated shock (RMS) Beloborodov (2017); Lundman and
Beloborodov (2019) to allow particle acceleration at, e.g., mildly
relativistic internal shocks Crumley  _et al._ (2019). Indeed, using
relativistic magnetohydrodynamic simulations of magnetized jets from BNSMs,
Ref. Gottlieb and Globus (2021) found that subshocks can emerge at small radii
and produce HE neutrinos at internal shocks, collimation shocks, and shock
breakout.

Interestingly, a mild magnetization is also essential to avoid heavy baryon
loading of the jets from mixing with the cocoon material Gottlieb  _et al._
(2020, 2021); Gottlieb and Globus (2021). With negligible jet-cocoon mixing,
the jet Lorentz factor could grow almost linearly with radius due to adiabatic
expansion (see, e.g., Piran (2004); Meszaros (2006); Kumar and Zhang (2014b))
and reach $\gtrsim 100$ deep inside the star Mizuta and Aloy (2009); Gottlieb
_et al._ (2021). Internal shocks caused by collisions of slow and rapid jets
can occur at $R_{\rm is}\approx\Gamma_{s}^{2}c\delta t=3\times
10^{9}\Gamma^{2}_{s,1}\delta t_{-3}$ cm, where $\Gamma_{s}$ is the typical
Lorentz factor of the slow jets and $\delta t$ is the variability timescale.
Here and below, $A_{x}=A/10^{x}$, where $A$, if dimensional, is in cgs units.
We assume that the typical Lorentz factor $\Gamma_{r}$ of the unshocked rapid
jets is twice the Lorentz factor $\Gamma$ of the shocked jet. The relative
Lorentz factor between the two is $\Gamma_{\rm
rel}\approx\Gamma_{r}/(2\Gamma)+\Gamma/(2\Gamma_{r})=1.25$, consistent with
the mildly relativistic shock.

In our scenario, a strong collisionless subshock forms within the internal RMS
to facilitate particle acceleration. This scenario requires $\sigma_{u}\equiv
B_{u}^{2}/(4\pi\rho_{u}c^{2})\gtrsim 0.01$ Beloborodov (2017); Lundman and
Beloborodov (2019), where $B_{u}$ and $\rho_{u}$ are the proper magnetic field
and mass density, respectively, of the unshocked upstream region. Due to shock
compression, the magnetic field is amplified to $B_{d}=\xi B_{u}$ in the
shocked downstream region with the compression ratio $\xi\sim 5$ for
$\Gamma_{\rm rel}\approx 1.25$ Beloborodov (2017). Because HE neutrino flux
would be strongly suppressed by synchrotron cooling of $\pi^{\pm}$ and
$K^{\pm}$ in the enhanced magnetic field $B_{d}$, we focus on HE neutrino
production via $pp$ and $p\gamma$ reactions in the upstream region (i.e., the
unshocked jet), which was barely discussed in previous literature. For
radiation-dominated jets at launching, the initial jet temperature is $\sim
1.3L^{1/4}_{\rm iso,52}$ MeV at $r=10^{7}$ cm Kumar and Zhang (2014a). The
jets expand adiabatically and the temperature decreases as $r^{-1}$. At
$r=10^{9}$–$10^{10}$ cm, the jet temperature remains a few keV, corresponding
to a thermal energy fraction $\epsilon_{e,u}\gtrsim 0.1$. Proper values of
magnetic field, proton number density, and photon temperature in the shocked
and unshocked jets are given in Table 1.

Protons can be accelerated to high energies by crossing the shock fronts
repeatedly. During this process, protons have significant probabilities to
escape from both the upstream and downstream regions, resulting in a power-law
spectrum Bell (1978). Recent simulations of mildly relativistic shocks showed
that the accelerated protons in the upstream region are much more abundant
than those in the downstream region (see Figs. 7 and 8 of Crumley  _et al._
(2019)). To obtain a conservative estimate of the maximal proton energy
$E_{p,{\rm max}}$, we compare the timescales of shock acceleration and
different cooling processes for protons in the shocked jet (see Appendix C).
For all the parameter sets adopted in this work, the $E_{p,{\rm max}}$
computed in the rest frame of the shocked jet is always sufficient to make PeV
neutrinos ($E_{\nu}\sim 0.05\Gamma E_{p}$ from $pp$ and $p\gamma$ reactions).

We take the (unnormalized) spectrum of the accelerated protons to be
$\phi_{p}(E_{p})=E_{p}^{-2}\exp(-E_{p}/E_{p,\rm max})$, and use PYTHIA 8.3
Bierlich  _et al._ (2022) and SOPHIA Mucke  _et al._ (2000) for $pp$ and
$p\gamma$ reactions, respectively, to obtain the $\pi^{\pm}$ and $K^{\pm}$
yields at different centre-of-mass energies. Because the shocked jet is only
mildly relativistic relative to the rapid jet ($\Gamma_{\rm rel}=1.25$), we
ignore the Lorentz transformation for $E_{p}$ and $\phi_{p}(E_{p})$ in our
calculations. We include cooling of $\pi^{\pm}$ and $K^{\pm}$ due to
synchrotron radiation, inverse Compton scattering, $e^{\pm}$ pair production
on photons ($\pi^{\pm}/K^{\pm}+\gamma\to\pi^{\pm}/K^{\pm}+e^{+}+e^{-}$),
hadronic scattering, and adiabatic expansion (see Appendix C).

In our benchmark study, we take $R_{\rm is}=3\times 10^{9}$ cm, $L_{\rm
iso}=10^{53}$ erg/s, $\Gamma_{r}=2\Gamma\approx 90L^{0.18}_{\rm iso,49}\approx
472$ Yi  _et al._ (2017), $\sigma_{d}=\xi\sigma_{u}\approx
2\epsilon_{B,d}=0.25$, $\xi=5$, $\epsilon_{e,d}=0.5$, and $\epsilon_{e,u}=0.3$
(see Table 1). The corresponding $E_{p,{\rm max}}$ is $\approx 10^{6}$ GeV,
and we take $E_{p,{\rm min}}=100$ GeV without affecting the results. We find
that $e^{\pm}$ production on photons and synchrotron radiation dominate the
cooling of $\pi^{\pm}$ and $K^{\pm}$ (see Appendix C). The benchmark HE
neutrino spectra are shown in Fig. 2. The competition between $pp$ and
$p\gamma$ reactions depends on the effective threshold for meson production
and the number density of targets. The $pp$ reaction dominates at low proton
energies and produces almost equal neutrino and antineutrino fluxes at
$E\lesssim 30$ TeV (left panel) due to equal production of $\pi^{+}\,(K^{+})$
and $\pi^{-}\,(K^{-})$. Once the proton energy exceeds the effective threshold
for the $p\gamma$ reaction, this reaction dominates due to the much larger
number density of thermal photons and gives rise to higher fluxes of neutrinos
than antineutrinos (left panel) due to dominant production of
$\pi^{+}\,(K^{+})$ over $\pi^{-}\,(K^{-})$ and stronger synchrotron cooling of
$\mu^{\pm}$. The latter effect also suppresses production of $\nu_{e}$ and
$\bar{\nu}_{e}$ from $\mu^{\pm}$ decay compared to that of $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ from $\pi^{\pm}\,(K^{\pm})$ decay (middle panel).

To cover a wide range of GRB luminosities, we compare the all-flavor neutrino
spectra in the right panel of Fig. 2 using $L_{\rm iso}=10^{50}$–$10^{53}$
erg/s with other parameters unchanged. The variations with increasing $L_{\rm
iso}$ are caused by the boost of neutrino energy due to a larger Lorentz
factor and by more efficient cooling of charged mesons at relatively low
energies due to a larger photon density or $B_{u}$. For $L_{\rm iso}=10^{53}$
erg/s, cooling by pair production and synchrotron radiation suppresses
neutrino production from $\pi^{\pm}$ decay so much that $K^{\pm}$ decay
produces half of the neutrinos at 100 TeV.

##  IV Annihilation of High-Energy neutrinos with Low-Energy antineutrinos

Simulations show that the nonrelativistic radioactive outflows occur within a
few seconds of jet launching and last for a time comparable to the jet
duration Just  _et al._ (2022); Fujibayashi  _et al._ (2022). Therefore, most
of the HE neutrinos produced at $R_{\rm is}\sim 10^{9}$–$10^{10}$ cm can meet
the LE antineutrinos from the radioactive outflows. The resulting annihilation
takes $\lesssim$ 1 s, which is significantly shorter than the outflow
duration. We use a one-dimensional Monte Carlo code to simulate the
propagation of HE neutrinos in the LE antineutrino background and the
accompanying processes, which include $\nu_{\alpha}\bar{\nu}_{\alpha}$
annihilation and decays of the produced $\pi^{\pm}$ and $K^{\pm}$. Note that
we also include the annihilation of the regenerated HE $\nu_{\alpha}$ from the
earlier annihilation processes. The radial step $\Delta R$ to the next
interaction point for an HE $\nu_{\alpha}$ with energy $E_{H}$ at radius $R$
is determined by $\tau_{\alpha}(E_{H},R,\Delta R)=-\ln\lambda$, where
$\tau_{\alpha}$ is the optical depth due to $\nu_{\alpha}\bar{\nu}_{\alpha}$
annihilation and $\lambda$ is a random number between 0 and 1. The relevant
cross section $\sigma_{\nu_{\alpha}\bar{\nu}_{\alpha}}(s)$ peaks at
$s=m_{Z}^{2}$, and we use PYTHIA 8.3 Bierlich  _et al._ (2022) to track the
final products for all $s$ values.

Assuming vacuum oscillations of LE antineutrinos prior to annihilation, we
show the spectra of HE neutrinos following annihilation in Fig. 2. The
$Z$-resonance starts to occur at $E\sim 300$ TeV. Beyond this energy, the HE
neutrino flux drops but the HE antineutrino flux slightly increases due to
production from $Z$ decay (left panel). HE neutrinos at these energies are
mostly $\nu_{\mu}$, which are annihilated by the LE $\bar{\nu}_{\mu}$ (middle
panel). Strikingly, the $\nu_{e}$ and $\bar{\nu}_{e}$ regenerated from such
annihilation are even more abundant than those produced without annihilation.
Consequently, the flux of $\nu_{e}+\bar{\nu}_{e}$ is very similar to that of
the regenerated $\nu_{\tau}+\bar{\nu}_{\tau}$ at $E\gtrsim 300$ TeV following
annihilation (middle panel). The above effects produce a unique flavor
composition of
$(F_{\nu_{e}+\bar{\nu}_{e}}:F_{\nu_{\mu}+\bar{\nu}_{\mu}}:F_{\nu_{\tau}+\bar{\nu}_{\tau}})_{\star}\approx(1:10:1)$
for $E\gtrsim 300$ TeV at source, which corresponds to a normalized
composition of $\approx(0.21:0.42:0.37)$ at Earth following vacuum
oscillations. The corresponding all-flavor spectrum with the suppression due
to annihilation is shown in the right panel of Fig. 2 for different values of
$L_{\rm iso}$. With an optical depth of 3–8 for $\nu_{\mu}\bar{\nu}_{\mu}$
annihilation for $E_{H}\sim 0.4$–1 PeV, the suppression factor can be as high
as 3–3.5 (see the inset in the right panel of Fig. 2).

##  V Diffuse neutrino flux

Although prompt HE neutrinos associated with bright GRBs are tightly
constrained by IceCube Abbasi  _et al._ (2012); Aartsen  _et al._ (2015, 2016,
2017); Abbasi  _et al._ (2022b), subleading contributions of precursor
neutrinos from jet propagation inside stellar matter still remain possible.
The most recent analyses showed that precursor neutrinos preceding the prompt
$\gamma$-rays by tens of seconds are limited to $\lesssim 10$% of the diffuse
flux at $E=100$ TeV (see Fig. 7 of Abbasi  _et al._ (2022b)). On the other
hand, LLGRBs, whose rate is $\sim 10$ times higher than that of classical GRBs
Soderberg  _et al._ (2006); Liang  _et al._ (2007); Nakar (2015), could be
caused by a shock breakout driven by a relativistic jet propagating into an
extended envelope Campana  _et al._ (2006). They may have a common collapsar
origin and host jets with similar luminosities inside stars to those for
bright GRBs Nakar (2015); Senno  _et al._ (2016). Note that, in the shock
breakout scenario, the observed $\gamma$-ray luminosities of LLGRBs
($L_{\gamma,\rm llgrb}^{\rm obs}\sim 10^{46}$–$10^{48}$ erg/s) are much lower
than those of bright GRBs because the emission for LLGRBs spans a longer
duration ($\Delta T_{\gamma,\rm llgrb}\sim 10^{3}$–$10^{4}$ s) as determined
by the large shock breakout radius and is over a wider opening angle
($\theta_{\rm llgrb}\sim 1$ rad). Considering the duration correction, the
beam factor correction, and a baryon loading factor $f_{p}\sim 10$, the true
luminosity of LLGRB jets initially launched inside stars could be $L_{\rm
iso}\sim L_{\gamma,\rm llgrb}^{\rm obs}f_{p}\frac{\Delta T_{\gamma,\rm
llgrb}}{\Delta T_{\rm eng}}\frac{\theta^{2}_{\rm llgrb}}{\theta^{2}_{\rm
j}}\sim 10^{51}$–$10^{53}$ erg/s, where $\Delta T_{\rm eng}\sim 10$–100 s is
the central engine duration and $\theta_{\rm j}\sim 0.1$ rad is the initial
jet opening angle at launch. As HE neutrinos are produced deep inside stars in
our work, LLGRBs and bright GRBs are similar sources for HE neutrinos, except
that they have different occurrence rates. Further, choked collapsar jets
could be even more common than bright GRBs and LLGRBs if the central engine is
not active for too long Sobacchi  _et al._ (2017). To incorporate
contributions from all similar sources, we introduce the parameter $f_{\star}$
as the ratio of the total rate of collapsars with mildly-magnetized jets to
that of bright GRBs.

The unnormalized HE neutrino spectrum from a single source is shown in Fig. 2.
To obtain the diffuse neutrino flux from all similar sources over cosmic
history, we multiply the unnormalized spectrum by the following factor Murase
and Ioka (2013)

$\displaystyle\frac{c}{4\pi H_{0}}\frac{f_{z}f_{\star}\dot{n}_{\rm GRB}f_{\rm
cr}E_{\rm iso}}{\ln(E_{p,\rm max}/E_{p,\rm min})}\frac{1}{{\rm GeV}\ {\rm
cm}^{-2}\ {\rm s}^{-1}\ {\rm sr}^{-1}}$ $\displaystyle\sim 7\times
10^{-9}\left(\frac{f_{z}f_{\star}\dot{n}_{\rm GRB}}{\rm
Gpc^{-3}~{}yr^{-1}}\right)f_{\rm cr}E_{\rm iso,53},$ (4)

where $f_{z}\sim 3$ accounts for evolution of source population and neutrino
energy with redshift $z$ Waxman and Bahcall (1999), $\dot{n}_{\rm GRB}\sim 1$
Gpc-3 yr-1 is the rate of bright GRBs per unit volume, $E_{\rm iso}$ is the
total isotropic energy of jets, and $f_{\rm cr}$ is the fraction of jet energy
converted into HE protons. Taking $f_{\star}=10$, $f_{\rm cr}E_{\rm
iso,53}\sim 1$, and assuming that contributions from sources at $z\sim 1$
dominate, we compare the all-flavor $E^{2}F_{\nu}(E)$ from our model to the
IceCube data Aartsen  _et al._ (2020) in Fig. 3. It can be seen that LLGRBs
and choked jets with collapsar origin may explain the observed TeV–PeV flux at
IceCube. More interestingly, the spectrum with suppression at $E\gtrsim
100$–200 TeV due to annihilation (and redshift) leads to better agreement with
data than that without annihilation. Results using $\dot{M}=0.01$ and
$0.05~{}M_{\odot}$/s are also shown for comparison. Note that spherical winds
and vacuum oscillations of LE antineutrinos are assumed for our benchmark
study. The annihilation effect could be similar or even enhanced for
asymmetric winds while it will be reduced for the NH oscillation scenario with
a smaller $P_{\bar{\nu}_{\mu}}$ (see Appendices A and B). We find that
annihilation is significant under a range of assumptions (see Fig. 3). Our
model, however, cannot account for the PeV events detected at IceCube despite
the production of a small bump above $\sim 1$ PeV associated with a minimum in
the timescale for cooling of $K^{\pm}$ by pair production (see Fig. 6 in
Appendix C). Another source is required for these events Chen  _et al._
(2015); Palladino  _et al._ (2017); Chianese  _et al._ (2017); Denton and
Tamborra (2018b); Riabtsev and Troitsky (2023).

Figure 3: All-flavor diffuse neutrino flux spectra from our collapsar model
with and without annihilation compared to IceCube data Aartsen  _et al._
(2020). Contributions from jets with $L_{\rm iso,53}=1$ at redshift $z=1$ are
assumed to dominate. Results for different values of
$\dot{m}\equiv\dot{M}/(M_{\odot}\rm/s)$, asymmetric winds confined to
$\theta^{\prime}=[0^{\circ},45^{\circ}]$ and $[30^{\circ},45^{\circ}]$, and
the NH oscillation scenario for LE antineutrinos are also shown.

##  VI Summary and discussion

Considering collapsars as sources for both HE neutrinos and $r$-process
nuclei, we have investigated a novel effect on HE neutrinos caused by their
annihilation with LE antineutrinos from the $\beta$-decay of $r$-process
nuclei. Assuming typical jet parameters, we have also discussed production of
HE neutrinos at internal subshocks induced by mildly-magnetized jets deep
inside collapsars. Bright GRBs, LLGRBs, and choked GRBs may have common origin
in collapsars, and contributions to the diffuse HE neutrino flux are expected
to be dominated by the latter two. Our collapsar model suggests that these
sources can well account for the observed flux at IceCube, especially when
annihilation with LE antineutrinos from $\beta$-decay of $r$-process nuclei is
included. Specifically, an excess at 10–100 TeV and potentially, a deficit at
0.1–1 PeV can be explained consistently for the IceCube events without
overproducing the diffuse $\gamma$-ray background (see e.g., Murase  _et al._
(2016, 2020); Capanema  _et al._ (2020); Fang  _et al._ (2022)).

A critical test of our HE neutrino production model is detection of precursor
HE neutrinos (with a lead time of $\sim 10$–$10^{3}$ s) from nearby LLGRBs or
bright GRBs. For an LLGRB at 100 Mpc with canonical parameters, IceCube-Gen2
can detect $\sim 10$ cascade events within 10–100 s if the jets are mildly
magnetized. This short emission would also favor the speculation that LLGRBs
host very similar jets to those in bright GRBs. Our model predicts a flavor
composition at Earth close to that in the so-called muon-damped scenario at 10
TeV to 1 PeV, which can be distinguished from those in the standard pion and
neutron decay scenarios Ackermann (2017); Aartsen  _et al._ (2021); Song  _et
al._ (2021) by precise measurements. The expected spectrum decays as
$E^{-\gamma}$ at energies below $\sim 100$ TeV with $\gamma\sim 0.5$–1,
followed by a steepening above $100$–200 TeV due to the annihilation effect.
Anticipating 10 years of data at IceCube-Gen2, we find that a spectral
steepening with $\Delta\gamma>1$ for our scenario can be observed at 95% CL
(see Appendix D). In contrast, no clear break is expected without
annihilation. The above neutrino signatures together with the $r$-process
imprints in collapsar lightcurves can provide important support of collapsars
as sources for both HE neutrinos and $r$-process nuclei.

We have made several simplifications in this initial study. Future
improvements include detailed modeling of LE antineutrino emission, spatial
and temporal variations of $r$-process production in collapsars, and self-
consistent collapsar simulations for HE neutrino production. Finally, the same
annihilation effect should occur in BNSMs that host short GRBs and the
$r$-process, and can in principle be tested with a future nearby event.

###### Acknowledgements.

We thank Brian Metzger and Daniel Siegel for useful discussions. We are also
grateful to the anonymous referees for their constructive comments. This work
was supported in part by the National Natural Science Foundation of China
(12205258) and the Natural Science Foundation of Shandong Province, China
[ZR2022JQ04 (G.G.)], the US Department of Energy [DE-FG02-87ER40328 (Y.Z.Q.)],
and the National Science and Technology Council (No. 110-2112-M-001-050,
111-2628-M-001-003-MY4), the Academia Sinica (No. AS-CDA-109-M11), and the
Physics Division of the National Center for Theoretical Sciences, Taiwan
(M.R.W.).

##  Appendix A Geometry of disk winds

We have assumed in the main text that the disk winds are spherically
symmetric. These winds could be non-spherical and be channelled into a certain
range of angles off the jet axis. For example, Ref. MacFadyen and Woosley
(1999) found that $\theta^{\prime}$ (see Fig. 1 of the main text) could range
from $\sim 10^{\circ}$ to $\sim 40^{\circ}$, while Ref. Siegel  _et al._
(2019) assumed $30^{\circ}\leq\theta^{\prime}\leq 45^{\circ}$. A collimated
wind was also considered in Ref. Hayakawa and Maeda (2018) with
$\theta^{\prime}\leq 30^{\circ}$. Approximating asymmetric winds as
“isotropic” winds confined to $\theta^{\prime}=[\theta^{\prime}_{\rm
min},\theta^{\prime}_{\rm max}]$, we have shown the results in Fig. 3 of the
main text. For winds confined within $[0^{\circ},45^{\circ}]$, the LE neutrino
intensity is enhanced by a factor of $3$ compared to the spherical case; while
for $\theta^{\prime}=[30^{\circ},45^{\circ}]$, the intensity and the
annihilation effect are close to the spherical case.

##  Appendix B Oscillation scenarios for LE antineutrinos

Oscillations of LE antineutrinos prior to interaction with HE neutrinos depend
on the local densities at the $\bar{\nu}_{e}$ emission site and the
interaction site. We consider annihilation inside the jets where the typical
density is far below 1 g/cm3. The wind density at the $\bar{\nu}_{e}$ emission
site is $\rho\sim\dot{M}/(4\pi r^{2}v_{\rm ej})\sim 10^{3}$ g/cm3 at $r\sim
10^{9}$ cm assuming $\dot{M}\sim 0.01M_{\odot}$/s and $v_{\rm ej}\sim 0.05c$.
On the other hand, the density at the emission site could be dominated by
stellar matter and can reach $\sim 10^{6}$ g/cm3 at $10^{9}$ cm, although
stellar matter may be blown away by the jets and/or disk winds. The relevant
density could also be low if $\bar{\nu}_{e}$ are emitted at larger radii.
Given these complexities and uncertainties, we consider the extreme cases
where $\bar{\nu}_{e}$ are emitted at very high and very low densities,
respectively.

The $\bar{\nu}_{e}$, if emitted at densities of $\gg 10^{4}$ g/cm3, almost
coincides with the effective mass eigenstate in matter $\bar{\nu}_{1m}$ or
$\bar{\nu}_{3m}$ for normal (NH) or inverted (IH) neutrino mass hierarchy,
respectively. For adiabatic evolution during propagation, the emitted
$\bar{\nu}_{e}$ becomes the vacuum mass eigenstate $\bar{\nu}_{1}$ (NH) or
$\bar{\nu}_{3}$ (IH) at very low densities for the interaction site. The
probability $P_{\bar{\nu}_{\alpha}}$ to be a $\bar{\nu}_{\alpha}$ at the
interaction site is
$(P_{\bar{\nu}_{e}},P_{\bar{\nu}_{\mu}},P_{\bar{\nu}_{\tau}})\approx(0.675,0.095,0.23)$
(NH) or (0.022, 0.545, 0.433) (IH) with the mixing parameters from Ref.
Workman  _et al._ (2022). If $\bar{\nu}_{e}$ are emitted at very low
densities,
$(P_{\bar{\nu}_{e}},P_{\bar{\nu}_{\mu}},P_{\bar{\nu}_{\tau}})\approx(0.55,0.18,0.27)$
as expected from pure vacuum oscillations. As HE neutrinos are mainly
$\nu_{\mu}$ in our model, the flux of LE $\bar{\nu}_{\mu}$ is the most
relevant for annihilation. The NH and IH scenarios are the most pessimistic
and optimistic, respectively. For $\bar{\nu}_{e}$ emitted at intermediate
densities, $P_{\bar{\nu}_{\mu}}$ ranges from 0.095 to 0.545. Compared to pure
vacuum oscillations, the NH scenario reduces the LE $\bar{\nu}_{\mu}$ flux by
a factor of $\sim 2$. As shown in Fig. 3 of the main text, the NH scenario
with $\dot{m}=0.02$ gives rise to a similar annihilation effect to that for
pure vacuum oscillations with $\dot{m}=0.01$.

##  Appendix C Timescales for acceleration and cooling

Figure 4 shows the timescales for proton acceleration and cooling in the rest
frame of the shocked jet. The acceleration timescale is approximated by
$E_{p}/(eB_{d}c)$. The cooling timescale due to $p\gamma$ reactions is given
by

$\displaystyle t^{-1}_{p\gamma}(E_{p})=$ $\displaystyle c\int d\epsilon
d\Big{(}{\cos\theta_{p\gamma}\over 2}\Big{)}$
$\displaystyle\times\kappa(\tilde{\epsilon})\sigma_{p\gamma}(\tilde{\epsilon}){dn(\epsilon)\over
d\epsilon}(1-\cos\theta_{p\gamma}),$ (5)

where $\theta_{p\gamma}$ is the intersection angle, $dn(\epsilon)/d\epsilon$
is the differential density of thermal photons, $\tilde{\epsilon}$ is the
photon energy in the rest frame of the proton, and $\kappa$ and
$\sigma_{p\gamma}$ are the inelasticity and cross section, for which we take
the expressions from Guo  _et al._ (2020). The cooling timescale due to the
Bethe-Heitler (BH) process can be obtained similarly and we use the
inelasticity and cross section from Chodorowski  _et al._ (1992). For the
cooling timescales due to $pp$ scattering, synchrotron radiation, inverse
Compton (IC) scattering, and adiabatic expansion, we use the expressions given
in Ref. Razzaque  _et al._ (2005). For the parameters chosen, the maximal
proton energy achieved in the comoving frame of the shocked jet is about
$10^{6}$ GeV.

Figure 4: Timescales for shock acceleration of protons and their cooling by
various processes in the rest frame of the shocked jet. We have taken $L_{\rm
iso}=10^{53}$ erg/s and the benchmark values for the other parameters.

To avoid strong synchrotron cooling of mesons, we consider $\pi^{\pm}$ and
$K^{\pm}$ production via $pp$ and $p\gamma$ reactions in the unshocked rapid
jet. Figure 5 shows the timescales for $\pi^{\pm}$ cooling by various
processes. Synchrotron radiation, $e^{\pm}$ pair production on photons,
$\pi^{\pm}p$ collision, and adiabatic cooling are calculated similarly to the
case of protons. For $\pi^{\pm}\gamma$ process, we use the cross sections for
IC scattering and multi-meson production (dominated by
$\gamma+\pi^{\pm}\to\pi^{0}+\pi^{\pm}$ at low energy) from Kaiser and
Friedrich (2008). Pair production is studied similarly to that for protons
using the cross section and inelasticity from Chodorowski  _et al._ (1992).
Note that the expressions in Chodorowski  _et al._ (1992) apply equally to
proton and charged mesons, as they were derived for pair production in a
Coulomb field Maximon (1968). As the observed neutrino energy $E_{\rm
ob}\sim\Gamma_{r}E_{\pi}/4$, both pair production and synchrotron radiation
are relevant for determining the HE neutrino fluxes at $0.1\lesssim E\lesssim
1$ PeV. Cooling timescales for $K^{\pm}$ have been estimated similarly to
those for $\pi^{\pm}$ (see Fig. 6) except that $K^{\pm}\gamma$ multi-meson
production and IC are ignored compared to pair production and synchrotron
radiation.

Figure 5: Timescales for $\pi^{\pm}$ decay and their cooling by various
processes in the rest frame of the unshocked rapid jet. We have taken $L_{\rm
iso}=10^{53}$ erg/s and the benchmark values for the other parameters. Figure
6: Timescales for $K^{\pm}$ decay and their cooling by various processes in
the rest frame of the unshocked rapid jet. We have taken $L_{\rm iso}=10^{53}$
erg/s and the benchmark values for the other parameters.

##  Appendix D Detecting spectral steepening with IceCube-Gen2

With a much better energy resolution, the cascade events at IceCube-Gen2 can
be used to explore the potential of observing a spectral break in the HE
neutrino flux Abbasi  _et al._ (2021); Kochocki (2023). Lacking the detailed
detector response of IceCube-Gen2, we carry out a relatively crude study. The
only information we rely on is the cascade effective area of IceCube-Gen2 from
Aartsen  _et al._ (2021), which is about 1 order of magnitude larger than that
of IceCube. Note that for the cascade effective area given, the astrophysical
HE neutrinos are typically assumed to be distributed isotropically in sky and
equally among all flavors. Processes producing cascade events include neutral-
current (NC) reactions of all flavors, charged-current (CC) reactions of
$\nu_{e}+\bar{\nu}_{e}$, and part (with a probability of $\sim 83$%) of the CC
reactions of $\nu_{\tau}+\bar{\nu}_{\tau}$. The effective areas for the CC and
NC reactions of HE neutrinos at a given zenith angle $\theta_{z}$ can be
obtained with consideration of Earth absorption. The deposited energy $E_{d}$
in the detector depends on the neutrino energy $E$ and the interaction type,
and we follow the expressions [Eqs. (B16-B23)] given in Ref. Palomares-Ruiz
_et al._ (2015) to calculate $E_{d}$. When generating the $E_{d}$
distribution, we also consider an energy resolution of 10%. The expected
spectrum of $E_{d}$ can then be obtained for any given astrophysical neutrino
flux.

As the atmospheric muon background will be strongly suppressed by veto
Ackermann (2017); Aartsen  _et al._ (2020, 2021); Kochocki (2023), we only
consider the background from atmospheric neutrinos, using the fluxes from the
MCEq tool Fedynitch  _et al._ (2015). The expected $E_{d}$ spectrum for this
background is calculated similarly to that for astrophysical neutrinos. The
veto can also be used to suppress the atmospheric neutrino background
Ackermann (2017); Aartsen  _et al._ (2021); Kochocki (2023); Aartsen  _et al._
(2020), and we take an overall suppression factor of $1/3$ for
$\cos\theta_{z}\geq 0.2$ (see e.g., Fig. 4.10 of Niederhausen (2018)).

Figure 7: Sensitivity for measuring the break energy $E_{b}$ and the spectral
index change $\delta\gamma$ at 95% CL with 10 years of IceCube-Gen2 data. The
$\times$ symbol represents the best-fit point with
$\chi^{2}(E_{b},\delta\gamma)=\chi^{2}_{\rm min}$.

Based on Fig. 3 of the main text, we simply model our astrophysical HE
neutrino flux with a broken power-law at $10~{}{\rm TeV}\leq E\leq 500$ TeV:

$E^{2}F_{\nu}=A\times\begin{cases}({E/100~{}\rm TeV})^{-\gamma},&E<E_{b},\\\
({E_{b}/100~{}\rm
TeV})^{-\gamma}({E/E_{b}})^{-\gamma-\delta\gamma},&E>E_{b}.\end{cases}$ (6)

We generate the so-called Asimov dataset assuming $A_{0}=2\times 10^{-8}~{}\rm
GeV~{}cm^{-2}~{}s^{-2}~{}sr^{-1}$, $E_{b0}=150$ TeV, $\gamma_{0}=0.5$, and
$\delta\gamma_{0}=2.5$. We take 30 equal-size bins for $E_{d}$ between 10 TeV
and 500 TeV on logarithmic scale, and 10 equal-size bins for
$\cos\theta_{z}=[-1,1]$. The $\chi^{2}$ function is defined as

$\displaystyle\chi^{2}(E_{b},\delta\gamma)=\min_{\\{A,\gamma,\alpha\\}}\Big{[}-2\sum_{i,j}(n_{ij}\ln{\mu_{ij}}-\mu_{ij})\Big{]},$
(7)

where $n_{ij}$ is the event number of the Asimov dataset in the $i$th
$E_{d}$-bin and the $j$th $\cos\theta$-bin, and
$\mu_{ij}(A,E_{b},\gamma,\delta\gamma,\alpha)$ is the expected event number
with $\alpha$ being the relative uncertainty of the atmospheric neutrino
background. Note that
$n_{ij}=\mu_{ij}(A_{0},E_{b0},\gamma_{0},\delta\gamma_{0},0)$. As suggested by
Eq. (7), the parameters $A$, $\gamma$, and $\alpha$ are allowed to vary freely
to minimize the $\chi^{2}$ function for a given combination of $E_{b}$ and
$\delta\gamma$.

The $E_{b}$-$\delta\gamma$ contour in Fig. 7 shows the sensitivity for
measuring the spectral break with 10 years of IceCube-Gen2 data. This contour
corresponds to $\chi^{2}(E_{b},\delta\gamma)-\chi^{2}_{\rm min}\approx 5.99$
(95% CL). We see that a steepening of the spectrum with $\delta\gamma>1$ at
$100~{}{\rm TeV}\lesssim E_{b}\lesssim 240$ TeV can be measured.

## References

  * Woosley (1993) S. E. Woosley, Astrophys. J. 405, 273 (1993).
  * MacFadyen and Woosley (1999) A. I. MacFadyen and S. E. Woosley, Astrophys. J. 524, 262 (1999), arXiv:astro-ph/9810274 [astro-ph] .
  * Woosley and Bloom (2006) S. E. Woosley and J. S. Bloom, Ann. Rev. Astron. Astrophys. 44, 507 (2006), arXiv:astro-ph/0609142 .
  * Campana  _et al._ (2006) S. Campana, V. Mangano, A. J. Blustin, P. Brown, D. N. Burrows, G. Chincarini, J. R. Cummings, G. Cusumano, M. Della Valle, D. Malesani, P. Mészáros, J. A. Nousek, M. Page, T. Sakamoto, E. Waxman, B. Zhang, Z. G. Dai, N. Gehrels, S. Immler, F. E. Marshall, K. O. Mason, A. Moretti, P. T. O’Brien, J. P. Osborne, K. L. Page, P. Romano, P. W. A. Roming, G. Tagliaferri, L. R. Cominsky, P. Giommi, O. Godet, J. A. Kennea, H. Krimm, L. Angelini, S. D. Barthelmy, P. T. Boyd, D. M. Palmer, A. A. Wells, and N. E. White, Nature (London) 442, 1008 (2006), arXiv:astro-ph/0603279 [astro-ph] .
  * Hjorth and Bloom (2012) J. Hjorth and J. S. Bloom, in  _Chapter 9 in ”Gamma-Ray Bursts_ (2012) pp. 169–190.
  * Kumar and Zhang (2014a) P. Kumar and B. Zhang, Phys. Rept. 561, 1 (2014a), arXiv:1410.0679 [astro-ph.HE] .
  * Mészáros and Waxman (2001) P. Mészáros and E. Waxman, Physical Review Letters 87, 171102 (2001).
  * Razzaque  _et al._ (2003) S. Razzaque, P. Meszaros, and E. Waxman, Phys. Rev. D 68, 083001 (2003), arXiv:astro-ph/0303505 .
  * Razzaque  _et al._ (2004) S. Razzaque, P. Mészáros, and E. Waxman, Physical Review Letters 93, 181101 (2004).
  * Ando and Beacom (2005) S. Ando and J. F. Beacom, Phys. Rev. Lett. 95, 061103 (2005), arXiv:astro-ph/0502521 .
  * Razzaque  _et al._ (2005) S. Razzaque, P. Meszaros, and E. Waxman, Mod. Phys. Lett. A20, 2351 (2005), arXiv:astro-ph/0509729 [astro-ph] .
  * Horiuchi and Ando (2008) S. Horiuchi and S. Ando, Phys. Rev. D 77, 063007 (2008), arXiv:0711.2580 [astro-ph] .
  * Enberg  _et al._ (2009) R. Enberg, M. H. Reno, and I. Sarcevic, Phys. Rev. D 79, 053006 (2009), arXiv:0808.2807 [astro-ph] .
  * Bartos  _et al._ (2012) I. Bartos, B. Dasgupta, and S. Marka, Phys. Rev. D 86, 083007 (2012), arXiv:1206.0764 [astro-ph.HE] .
  * Murase and Ioka (2013) K. Murase and K. Ioka, Physical Review Letters 111, 121102 (2013).
  * Fraija (2014) N. Fraija, Mon. Not. Roy. Astron. Soc. 437, 2187 (2014), arXiv:1310.7061 [astro-ph.HE] .
  * Xiao and Dai (2014) D. Xiao and Z. G. Dai, Astrophysical Journal 790, 59 (2014).
  * Bhattacharya  _et al._ (2015) A. Bhattacharya, R. Enberg, M. H. Reno, and I. Sarcevic, JCAP 1506, 034 (2015), arXiv:1407.2985 [astro-ph.HE] .
  * Varela  _et al._ (2015) K. Varela, S. Sahu, A. F. O. Oliveros, and J. C. Sanabria, European Physical Journal C 75, 289 (2015), arXiv:1411.7992 .
  * Tamborra and Ando (2016) I. Tamborra and S. Ando, Phys. Rev. D93, 053010 (2016), arXiv:1512.01559 [astro-ph.HE] .
  * Senno  _et al._ (2016) N. Senno, K. Murase, and P. Meszaros, Phys. Rev. D93, 083003 (2016), arXiv:1512.08513 [astro-ph.HE] .
  * Senno  _et al._ (2018) N. Senno, K. Murase, and P. Mészáros, JCAP 01, 025 (2018), arXiv:1706.02175 [astro-ph.HE] .
  * Denton and Tamborra (2018a) P. B. Denton and I. Tamborra, Astrophys. J. 855, 37 (2018a), arXiv:1711.00470 [astro-ph.HE] .
  * Denton and Tamborra (2018b) P. B. Denton and I. Tamborra, JCAP 04, 058 (2018b), arXiv:1802.10098 [astro-ph.HE] .
  * He  _et al._ (2018) H.-N. He, A. Kusenko, S. Nagataki, Y.-Z. Fan, and D.-M. Wei, Astrophys. J. 856, 119 (2018), arXiv:1803.07478 [astro-ph.HE] .
  * Chang  _et al._ (2022) P.-W. Chang, B. Zhou, K. Murase, and M. Kamionkowski, (2022), arXiv:2210.03088 [astro-ph.HE] .
  * Guarini  _et al._ (2023) E. Guarini, I. Tamborra, and O. Gottlieb, Phys. Rev. D 107, 023001 (2023), arXiv:2210.03757 [astro-ph.HE] .
  * Aartsen  _et al._ (2018a) M. G. Aartsen _et al._ (IceCube, Fermi-LAT, MAGIC, AGILE, ASAS-SN, HAWC, H.E.S.S., INTEGRAL, Kanata, Kiso, Kapteyn, Liverpool Telescope, Subaru, Swift NuSTAR, VERITAS, VLA/17B-403), Science 361, eaat1378 (2018a), arXiv:1807.08816 [astro-ph.HE] .
  * Aartsen  _et al._ (2018b) M. G. Aartsen _et al._ (IceCube), Science 361, 147 (2018b), arXiv:1807.08794 [astro-ph.HE] .
  * Stein  _et al._ (2021) R. Stein _et al._ , Nature Astron. 5, 510 (2021), arXiv:2005.05340 [astro-ph.HE] .
  * Reusch  _et al._ (2022) S. Reusch _et al._ , Phys. Rev. Lett. 128, 221101 (2022), arXiv:2111.09390 [astro-ph.HE] .
  * van Velzen  _et al._ (2021) S. van Velzen _et al._ , (2021), arXiv:2111.09391 [astro-ph.HE] .
  * Abbasi  _et al._ (2022a) R. Abbasi _et al._ (IceCube), Science 378, 538 (2022a), arXiv:2211.09972 [astro-ph.HE] .
  * Abbott  _et al._ (2017a) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. Lett. 119, 161101 (2017a), arXiv:1710.05832 [gr-qc] .
  * Abbott  _et al._ (2017b) B. P. Abbott _et al._ (LIGO Scientific, Virgo, Fermi GBM, INTEGRAL, IceCube, AstroSat Cadmium Zinc Telluride Imager Team, IPN, Insight-Hxmt, ANTARES, Swift, AGILE Team, 1M2H Team, Dark Energy Camera GW-EM, DES, DLT40, GRAWITA, Fermi-LAT, ATCA, ASKAP, Las Cumbres Observatory Group, OzGrav, DWF (Deeper Wider Faster Program), AST3, CAASTRO, VINROUGE, MASTER, J-GEM, GROWTH, JAGWAR, CaltechNRAO, TTU-NRAO, NuSTAR, Pan-STARRS, MAXI Team, TZAC Consortium, KU, Nordic Optical Telescope, ePESSTO, GROND, Texas Tech University, SALT Group, TOROS, BOOTES, MWA, CALET, IKI-GW Follow-up, H.E.S.S., LOFAR, LWA, HAWC, Pierre Auger, ALMA, Euro VLBI Team, Pi of Sky, Chandra Team at McGill University, DFN, ATLAS Telescopes, High Time Resolution Universe Survey, RIMAS, RATIR, SKA South Africa/MeerKAT), Astrophys. J. Lett. 848, L12 (2017b), arXiv:1710.05833 [astro-ph.HE] .
  * Kasen  _et al._ (2017) D. Kasen, B. Metzger, J. Barnes, E. Quataert, and E. Ramirez-Ruiz, Nature 551, 80 (2017), arXiv:1710.05463 [astro-ph.HE] .
  * Drout  _et al._ (2017) M. R. Drout _et al._ , Science 358, 1570 (2017), arXiv:1710.05443 [astro-ph.HE] .
  * Cowperthwaite  _et al._ (2017) P. S. Cowperthwaite _et al._ , Astrophys. J. Lett. 848, L17 (2017), arXiv:1710.05840 [astro-ph.HE] .
  * Villar  _et al._ (2017) V. A. Villar _et al._ , Astrophys. J. Lett. 851, L21 (2017), arXiv:1710.11576 [astro-ph.HE] .
  * Shibata  _et al._ (2017) M. Shibata, S. Fujibayashi, K. Hotokezaka, K. Kiuchi, K. Kyutoku, Y. Sekiguchi, and M. Tanaka, Phys. Rev. D 96, 123012 (2017), arXiv:1710.07579 [astro-ph.HE] .
  * Metzger (2020) B. D. Metzger, Living Rev. Rel. 23, 1 (2020), arXiv:1910.01617 [astro-ph.HE] .
  * Watson  _et al._ (2019) D. Watson _et al._ , Nature 574, 497 (2019), arXiv:1910.10510 [astro-ph.HE] .
  * Siegel  _et al._ (2019) D. M. Siegel, J. Barnes, and B. D. Metzger, Nature 569, 241 (2019), arXiv:1810.00098 [astro-ph.HE] .
  * Siegel (2019) D. M. Siegel, Eur. Phys. J. A 55, 203 (2019), arXiv:1901.09044 [astro-ph.HE] .
  * Macias and Ramirez-Ruiz (2019) P. Macias and E. Ramirez-Ruiz, Astrophys. J. Lett. 877, L24 (2019), arXiv:1905.04315 [astro-ph.HE] .
  * Bartos and Marka (2019) I. Bartos and S. Marka, Astrophys. J. Lett. 881, L4 (2019), arXiv:1906.07210 [astro-ph.HE] .
  * Miller  _et al._ (2020) J. M. Miller, T. M. Sprouse, C. L. Fryer, B. R. Ryan, J. C. Dolence, M. R. Mumpower, and R. Surman, Astrophys. J. 902, 66 (2020), arXiv:1912.03378 [astro-ph.HE] .
  * Fujibayashi  _et al._ (2020) S. Fujibayashi, M. Shibata, S. Wanajo, K. Kiuchi, K. Kyutoku, and Y. Sekiguchi, Phys. Rev. D 102, 123014 (2020), arXiv:2009.03895 [astro-ph.HE] .
  * Brauer  _et al._ (2021) K. Brauer, A. P. Ji, M. R. Drout, and A. Frebel, Astrophys. J. 915, 81 (2021), arXiv:2010.15837 [astro-ph.HE] .
  * Fraser and Schönrich (2022) J. Fraser and R. Schönrich, Mon. Not. Roy. Astron. Soc. 509, 6008 (2022), arXiv:2111.11464 [astro-ph.GA] .
  * Barnes and Metzger (2022) J. Barnes and B. D. Metzger, Astrophys. J. Lett. 939, L29 (2022), arXiv:2205.10421 [astro-ph.HE] .
  * Just  _et al._ (2022) O. Just, M. A. Aloy, M. Obergaulinger, and S. Nagataki, Astrophys. J. Lett. 934, L30 (2022), arXiv:2205.14158 [astro-ph.HE] .
  * Fujibayashi  _et al._ (2022) S. Fujibayashi, Y. Sekiguchi, M. Shibata, and S. Wanajo, arXiv e-prints , arXiv:2212.03958 (2022), arXiv:2212.03958 [astro-ph.HE] .
  * Anand  _et al._ (2023) S. Anand _et al._ , (2023), arXiv:2302.09226 [astro-ph.HE] .
  * Metzger  _et al._ (2010) B. D. Metzger, G. Martínez-Pinedo, S. Darbha, E. Quataert, A. Arcones, D. Kasen, R. Thomas, P. Nugent, I. V. Panov, and N. T. Zinner, Monthly Notices of the Royal Astronomical Society 406, 2650 (2010), arXiv:1001.5029 [astro-ph.HE] .
  * Rosswog  _et al._ (2014) S. Rosswog, O. Korobkin, A. Arcones, F. K. Thielemann, and T. Piran, Mon. Not. Roy. Astron. Soc. 439, 744 (2014), arXiv:1307.2939 [astro-ph.HE] .
  * Wu  _et al._ (2019) M.-R. Wu, J. Barnes, G. Martinez-Pinedo, and B. D. Metzger, Phys. Rev. Lett. 122, 062701 (2019), arXiv:1808.10459 [astro-ph.HE] .
  * Hayakawa and Maeda (2018) T. Hayakawa and K. Maeda, Astrophys. J. 854, 43 (2018), arXiv:1801.09681 [astro-ph.HE] .
  * Waxman and Bahcall (1997) W. Waxman and J. Bahcall, Physical Review Letters 78, 2292 (1997).
  * Waxman and Bahcall (2000) E. Waxman and J. N. Bahcall, Astrophys. J. 541, 707 (2000), arXiv:hep-ph/9909286 .
  * Li  _et al._ (2002) Z. Li, Z. G. Dai, and T. Lu, Astronomy & Astrophysics 396, 303 (2002).
  * Guetta  _et al._ (2004) D. Guetta, D. Hooper, J. Alvarez-Muniz, F. Halzen, and E. Reuveni, Astropart. Phys. 20, 429 (2004), arXiv:astro-ph/0302524 .
  * Murase and Nagataki (2006) K. Murase and S. Nagataki, Phys. Rev. Lett. 97, 051101 (2006), arXiv:astro-ph/0604437 .
  * Murase  _et al._ (2006) K. Murase, K. Ioka, S. Nagataki, and T. Nakamura, Astrophys. J. Lett. 651, L5 (2006), arXiv:astro-ph/0607104 .
  * Murase (2007) K. Murase, Phys. Rev. D 76, 123001 (2007), arXiv:0707.1140 [astro-ph] .
  * Hummer  _et al._ (2012) S. Hummer, P. Baerwald, and W. Winter, Phys. Rev. Lett. 108, 231101 (2012), arXiv:1112.1076 [astro-ph.HE] .
  * Zhang and Kumar (2013) B. Zhang and P. Kumar, Phys. Rev. Lett. 110, 121101 (2013), arXiv:1210.0647 [astro-ph.HE] .
  * Liu and Wang (2013) R.-Y. Liu and X.-Y. Wang, Astrophys. J. 766, 73 (2013), arXiv:1212.1260 [astro-ph.HE] .
  * Bustamante  _et al._ (2015) M. Bustamante, P. Baerwald, K. Murase, and W. Winter, Nature Communications 6, 6783 (2015).
  * Nir  _et al._ (2016) G. Nir, D. Guetta, H. Landsman, and E. Behar, Astrophys. J. 817, 142 (2016), arXiv:1511.07010 [astro-ph.HE] .
  * Biehl  _et al._ (2018) D. Biehl, D. Boncioli, A. Fedynitch, and W. Winter, Astron. Astrophys. 611, A101 (2018), arXiv:1705.08909 [astro-ph.HE] .
  * Guo  _et al._ (2020) G. Guo, Y.-Z. Qian, and M.-R. Wu, Astrophys. J. 890, 83 (2020), arXiv:1911.07568 [astro-ph.HE] .
  * Kimura (2022) S. S. Kimura, (2022), arXiv:2202.06480 [astro-ph.HE] .
  * Levinson and Bromberg (2008) A. Levinson and O. Bromberg, Phys. Rev. Lett. 100, 131101 (2008), arXiv:0711.3281 [astro-ph] .
  * Katz  _et al._ (2010) B. Katz, R. Budnik, and E. Waxman, Astrophys. J. 716, 781 (2010), arXiv:0902.4708 [astro-ph.HE] .
  * Beloborodov (2017) A. M. Beloborodov, Astrophys. J. 838, 125 (2017), arXiv:1604.02794 [astro-ph.HE] .
  * Lundman and Beloborodov (2019) C. Lundman and A. Beloborodov, Astrophys. J. 879, 83 (2019), arXiv:1804.03053 [astro-ph.HE] .
  * Crumley  _et al._ (2019) P. Crumley, D. Caprioli, S. Markoff, and A. Spitkovsky, Mon. Not. Roy. Astron. Soc. 485, 5105 (2019), arXiv:1809.10809 [astro-ph.HE] .
  * Gottlieb and Globus (2021) O. Gottlieb and N. Globus, Astrophys. J. Lett. 915, L4 (2021), arXiv:2105.01076 [astro-ph.HE] .
  * Gottlieb  _et al._ (2020) O. Gottlieb, O. Bromberg, C. B. Singh, and E. Nakar, Mon. Not. Roy. Astron. Soc. 498, 3320 (2020), arXiv:2007.11590 [astro-ph.HE] .
  * Gottlieb  _et al._ (2021) O. Gottlieb, O. Bromberg, A. Levinson, and E. Nakar, Mon. Not. Roy. Astron. Soc. 504, 3947 (2021), arXiv:2102.00005 [astro-ph.HE] .
  * Piran (2004) T. Piran, Reviews of Modern Physics 76, 1143 (2004).
  * Meszaros (2006) P. Meszaros, Rept. Prog. Phys. 69, 2259 (2006), arXiv:astro-ph/0605208 [astro-ph] .
  * Kumar and Zhang (2014b) P. Kumar and B. Zhang, Phys. Rept. 561, 1 (2014b), arXiv:1410.0679 [astro-ph.HE] .
  * Mizuta and Aloy (2009) A. Mizuta and M. A. Aloy, Astrophys. J. 699, 1261 (2009), arXiv:0812.4813 [astro-ph] .
  * Bell (1978) A. R. Bell, Mon. Not. Roy. Astron. Soc. 182, 147 (1978).
  * Bierlich  _et al._ (2022) C. Bierlich _et al._ , (2022), arXiv:2203.11601 [hep-ph] .
  * Mucke  _et al._ (2000) A. Mucke, R. Engel, J. P. Rachen, R. J. Protheroe, and T. Stanev, Comput. Phys. Commun. 124, 290 (2000), arXiv:astro-ph/9903478 .
  * Yi  _et al._ (2017) S.-X. Yi, W.-H. Lei, B. Zhang, Z.-G. Dai, X.-F. Wu, and E.-W. Liang, JHEAp 13-14, 1 (2017), arXiv:1612.06586 [astro-ph.HE] .
  * Abbasi  _et al._ (2012) R. Abbasi _et al._ , Nature 484, 351 (2012).
  * Aartsen  _et al._ (2015) M. G. Aartsen _et al._ , Astrophysical Journal Letters 805, L5 (2015).
  * Aartsen  _et al._ (2016) M. G. Aartsen, , _et al._ , The Astrophysical Journal 824, 115 (2016), arXiv:1601.06484 .
  * Aartsen  _et al._ (2017) M. G. Aartsen _et al._ , The Astrophysical Journal 843, 112 (2017), arXiv:1702.06868 .
  * Abbasi  _et al._ (2022b) R. Abbasi _et al._ (IceCube, Fermi Gamma-ray Burst Monitor), Astrophys. J. 939, 116 (2022b), arXiv:2205.11410 [astro-ph.HE] .
  * Soderberg  _et al._ (2006) A. M. Soderberg _et al._ , Nature 442, 1014 (2006), arXiv:astro-ph/0604389 .
  * Liang  _et al._ (2007) E. Liang, B. Zhang, and Z. G. Dai, Astrophys. J. 662, 1111 (2007), arXiv:astro-ph/0605200 .
  * Nakar (2015) E. Nakar, Astrophys. J. 807, 172 (2015), arXiv:1503.00441 [astro-ph.HE] .
  * Sobacchi  _et al._ (2017) E. Sobacchi, J. Granot, O. Bromberg, and M. C. Sormani, Mon. Not. Roy. Astron. Soc. 472, 616 (2017), arXiv:1705.00281 [astro-ph.HE] .
  * Waxman and Bahcall (1999) E. Waxman and J. N. Bahcall, Phys. Rev. D 59, 023002 (1999), arXiv:hep-ph/9807282 .
  * Aartsen  _et al._ (2020) M. G. Aartsen _et al._ (IceCube), Phys. Rev. Lett. 125, 121104 (2020), arXiv:2001.09520 [astro-ph.HE] .
  * Chen  _et al._ (2015) C.-Y. Chen, P. S. Bhupal Dev, and A. Soni, Phys. Rev. D 92, 073001 (2015), arXiv:1411.5658 [hep-ph] .
  * Palladino  _et al._ (2017) A. Palladino, C. Mascaretti, and F. Vissani, Eur. Phys. J. C 77, 684 (2017), arXiv:1708.02094 [astro-ph.HE] .
  * Chianese  _et al._ (2017) M. Chianese, R. Mele, G. Miele, P. Migliozzi, and S. Morisi, Astrophys. J. 851, 36 (2017), arXiv:1707.05168 [hep-ph] .
  * Riabtsev and Troitsky (2023) K. Riabtsev and S. Troitsky, Phys. Lett. B 839, 137758 (2023), arXiv:2204.09339 [astro-ph.HE] .
  * Murase  _et al._ (2016) K. Murase, D. Guetta, and M. Ahlers, Phys. Rev. Lett. 116, 071101 (2016), arXiv:1509.00805 [astro-ph.HE] .
  * Murase  _et al._ (2020) K. Murase, S. S. Kimura, and P. Meszaros, Phys. Rev. Lett. 125, 011101 (2020), arXiv:1904.04226 [astro-ph.HE] .
  * Capanema  _et al._ (2020) A. Capanema, A. Esmaili, and K. Murase, Phys. Rev. D 101, 103012 (2020), arXiv:2002.07192 [hep-ph] .
  * Fang  _et al._ (2022) K. Fang, J. S. Gallagher, and F. Halzen, Astrophys. J. 933, 190 (2022), arXiv:2205.03740 [astro-ph.HE] .
  * Ackermann (2017) M. Ackermann, ed.,  _The IceCube Neutrino Observatory: Contributions to ICRC 2017 Part VI: IceCube-Gen2, the Next Generation Neutrino Observatory_ (2017) arXiv:1710.01207 [astro-ph.IM] .
  * Aartsen  _et al._ (2021) M. G. Aartsen _et al._ (IceCube-Gen2), J. Phys. G 48, 060501 (2021), arXiv:2008.04323 [astro-ph.HE] .
  * Song  _et al._ (2021) N. Song, S. W. Li, C. A. Argüelles, M. Bustamante, and A. C. Vincent, JCAP 04, 054 (2021), arXiv:2012.12893 [hep-ph] .
  * Workman  _et al._ (2022) R. L. Workman _et al._ (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Chodorowski  _et al._ (1992) M. J. Chodorowski, A. A. Zdziarski, and M. Sikora, Astrophys. J. 400, 181 (1992).
  * Kaiser and Friedrich (2008) N. Kaiser and J. M. Friedrich, Eur. Phys. J. A 36, 181 (2008), arXiv:0803.0995 [nucl-th] .
  * Maximon (1968) L. C. Maximon, J. Res. Natl. Bur. Stand. B 72, 79 (1968).
  * Abbasi  _et al._ (2021) R. Abbasi _et al._ (IceCube), PoS ICRC2021, 1129 (2021), arXiv:2107.10003 [astro-ph.HE] .
  * Kochocki (2023) A. Kochocki (IceCube-Gen2), PoS ECRS, 100 (2023).
  * Palomares-Ruiz  _et al._ (2015) S. Palomares-Ruiz, A. C. Vincent, and O. Mena, Phys. Rev. D 91, 103008 (2015), arXiv:1502.02649 [astro-ph.HE] .
  * Fedynitch  _et al._ (2015) A. Fedynitch, R. Engel, T. K. Gaisser, F. Riehn, and T. Stanev, EPJ Web Conf. 99, 08001 (2015), arXiv:1503.00544 [hep-ph] .
  * Niederhausen (2018) H. Niederhausen,  _Measurement of the High Energy Astrophysical Neutrino Flux Using Electron and Tau Neutrinos Observed in Four Years of IceCube Data_ , Ph.D. thesis, SUNY Stony Brook, New York (2018).