# Group action and $L^{2}$-norm estimates of geometric problems

Thang Pham  Hanoi University of Science, Vietnam National University
<EMAIL_ADDRESS>
###### Abstract.

In 2017, by using the group theoretic approach, Bennett, Hart, Iosevich,
Pakianathan, and Rudnev obtained a number of results on the distribution of
simplices and sum-product type problems. The main purpose of this paper is to
give a series of new applications of their powerful framework, namely, we
focus on the product and quotient of distance sets, the $L^{2}$-norm of the
direction set, and the $L^{2}$-norm of scales in difference sets.

###### 2010 Mathematics Subject Classification:

52C10, 42B05, 11T23

##  1\. Introduction

Let $\mathbb{F}_{q}$ be a finite field of order $q$, where $q$ is a prime
power. Let $O(d)$ be the orthogonal group of $d$ by $d$ matrices with entries
in $\mathbb{F}_{q}$.

Given an integer $k\geq 1$, we say that two $k$-simplices in
$\mathbb{F}_{q}^{d}$ with vertices $(x_{1},\ldots,x_{k+1})$ and
$(y_{1},\ldots,y_{k+1})$ are in the same congruence class if
$||x_{i}-x_{j}||=||y_{i}-y_{j}||$ for all $1\leq i\neq j\leq k+1$. Here
$||x_{i}-x_{j}||=(x_{i1}-x_{j1})^{2}+\cdots+(x_{id}-x_{jd})^{2}$. This is
equivalent to say that there exist $\theta\in O(d)$ and
$z\in\mathbb{F}_{q}^{d}$ such that $y_{i}=\theta x_{i}+z$ for all $1\leq i\leq
k+1$. Given a set $E\subset\mathbb{F}_{q}^{d}$, we denote the set of
congruence classes of $k$-simplcies determined by $E$ by $T_{k}^{d}(E)$.

The question of finding the smallest threshold $\alpha$ such that
$|T_{k}^{d}(E)|\gg q^{\binom{k+1}{2}}$ whenever $|E|\gg q^{\alpha}$ has a rich
history, for instance, see [9, 24, 10, 3, 21]. The best current result is due
to Bennett, Hart, Iosevich, Pakianathan, and Rudnev in [2]. More precisely, by
using the group theoretic approach, they proved that for all $1\leq k\leq d$,
if $|E|\gg q^{d-\frac{d-1}{k+1}}$, then $|T_{k}^{d}(E)|\gg
q^{\binom{k+1}{2}}$. In two dimensions, they are able to obtain better
exponents, namely, $8/5$ for $k=2$ and $4/3$ for $k=1$. We also note that when
$k=1$, this problem is known as the Erdős-Falconer distance problem in the
literature, we refer the interested reader to [20, 15] for the recent
progress.

It is worth noting that the group action techniques are not only useful in
discrete setting, but it is also very powerful in the continuous setting. One
example, we have to mention here, is an $L^{2}$ identity due to Liu [17],
which plays an important role in the recent breakthrough on the pinned
Falconer distance problem [5, 6]. Similar results of this approach can also be
found in [7, 8, 18].

The main purpose of this paper is to provide a series of new applications of
this group theoretic approach, namely, on the product and quotient of distance
sets, the $L^{2}$-norm of the direction set, and the $L^{2}$-norm of scales in
difference sets.

Throughout this paper, we denote the set of square elements in
$\mathbb{F}_{q}$ by $(\mathbb{F}_{q})^{2}$, and by $X\gg Y$ we mean there
exists an absolutely positive constant $C$ such that $X\geq CY$.

###  1.1. Product and Quotient of distance sets

For $E\subset\mathbb{F}_{q}^{2}$, the distance set $\Delta(E)$ is the set of
all distances determined by pairs of points in $E$, namely,

$\Delta(E):=\left\\{||x-y||\colon x,y\in E\right\\}.$

We define the product, quotient, and sum of $\Delta(E)$ as follows:

$\frac{\Delta(E)}{\Delta(E)}:=\left\\{\frac{a}{b}\colon
a,b\in\Delta(E)\right\\},~{}\Delta(E)\cdot\Delta(E):=\left\\{a\cdot b\colon
a,b\in\Delta(E)\right\\},$

and

$\Delta(E)+\Delta(E):=\left\\{a+b\colon a,b\in\Delta(E)\right\\}.$

Our first result is on the size of $\Delta(E)\cdot\Delta(E)$.

######  Theorem 1.1.

For $E\subset\mathbb{F}_{q}^{2}$ with $|E|\gg q^{8/7}$, we have

$|\Delta(E)\cdot\Delta(E)|\gg q.$

For comparison, one can use graph theoretic techniques or discrete Fourier
analysis to prove the same exponent for the sum set $\Delta(E)+\Delta(E)$, for
instance, see [22]. It is necessary to say that the study of the product set
would be much harder than the case of the sum set, the main reason comes from
the observation that $\Delta(E)+\Delta(E)=\Delta(E\times E)$, so the sum set
can be reduced to the original distance problem for Cartesian product sets,
which possesses some nicely additive structures. As a consequence, when $q$ is
a prime, it has been shown in [4] that the exponent $8/7$ can be improved to
$11/10$. For the sum of two different distance sets, we refer the reader to
[14] for more details. We also want to emphasize here that it is difficult to
extend the methods from [4, 14, 22] to study the case of product set
$\Delta(E)\cdot\Delta(E)$. As in the Erdős-Falconer distance problem, the
conjectured exponent for product of distance sets should be $1$.

For quotient of distance sets, by using discrete Fourier analysis, Iosevich,
Koh, and Parshall [11] proved that if $|E|\geq 9q$, then

(1.1) $\frac{\Delta(E)}{\Delta(E)}=\mathbb{F}_{q}.$

They also constructed examples to show that the condition $|E|\gg q$ is
optimal. There are two perspectives we want to mention here: their proof is
very sophisticated, and it only tells us that the quotient contains the whole
field, so given an element $r\in\mathbb{F}_{q}$, it is not clear how many
tuples $(a,b,c,d)\in E^{4}$ such that $||a-b||/||c-d||=r$. Using the group
theoretic method, we prove the following.

######  Theorem 1.2.

For $E\subset\mathbb{F}_{q}^{2}$ with $q\equiv 3\mod 4$. Assume that $|E|\gg
q$, then for each non-zero square $r$ in $\mathbb{F}_{q}$, the number of
quadruples $(a,b,c,d)\in E^{4}$ such that $||a-b||/||c-d||=r$ is at least
$\gg|E|^{4}q^{-1}$. In particular,

$\left(\mathbb{F}_{q}\right)^{2}\subseteq\frac{\Delta(E)}{\Delta(E)}.$

We remark that the condition $q\equiv 3\mod 4$ is needed in the proof of this
theorem, which helps to avoid pairs of zero distances.

###  1.2. Distribution of directions

Given $E\subset\mathbb{F}_{q}^{2}$, the $L^{2}$-norm of the direction set
bounds the number of quadruples $(u,v,x,y)\in E^{4}$ such that

(1.2) $(u-v)=\lambda\cdot(x-y),~{}x\neq y,~{}u\neq v,$

for some $\lambda\neq 0$. We denote the number of such quadruples by
$L^{2}(D_{E})$. In the next theorem, we give an upper bound for this quantity.

######  Theorem 1.3.

For $E\subset\mathbb{F}_{q}^{2}$, we have

$\left|L^{2}(D_{E})-\frac{|E|^{4}}{q}\right|\ll q^{2}|E|^{2}.$

If we are interested in the number of directions spanned by a set
$E\subset\mathbb{F}_{q}^{2}$, then Theorem 1.3 says that we have at least $\gg
q$ directions as long as $|E|\gg q^{3/2}$. However, it is known in the
literature that the condition $|E|\gg q$ would be sufficient, see [12] for
example. When the size of $E$ is very small, say $|E|\ll q$, we refer the
reader to [23] for related results.

###  1.3. Scales in difference sets

For $E\subset\mathbb{F}_{q}^{2}$, an element $\lambda\in\mathbb{F}_{q}$ is
called a scale in the difference set $E-E$ if there exist
$u_{1},v_{1},u_{2},v_{2}\in E$ such that

$(u_{1}-v_{1})=\lambda(u_{2}-v_{2}).$

We denote the set of all scales in $E-E$ by $\frac{E-E}{E-E}$. In this paper,
we are interested in bounding the $L^{2}$-norm, i.e. the number of tuples
$(u_{1},v_{1},u_{2},v_{2},u_{3},v_{3},u_{4},v_{4})\in E^{8}$ such that

$(u_{1}-v_{1})=\lambda(u_{2}-v_{2}),~{}(u_{3}-v_{3})=\lambda(u_{4}-v_{4}),$

for some $\lambda\neq 0$. We denote the number of such tuples by
$L^{2}(S_{E})$.

With the same approach, we have an upper bound for the $L^{2}$-norm of scales.

######  Theorem 1.4.

For $E\subset\mathbb{F}_{q}^{2}$, we have

$\left|L^{2}(S_{E})-\frac{|E|^{8}}{q^{3}}\right|\ll|E|^{6}+q^{2}|E|^{5}.$

For the set of scales, it is not hard to show that if $|E|\gg q^{3/2}$, then
$E-E/E-E$ covers the whole field. Roughly speaking, under the condition
$|E|\gg q^{3/2}$, one can find a line with at least $q^{1/2}$ points from $E$,
then without loss of generality, we assume that line is defined by $y=0$. So
the problem is reduced to one dimensional version, namely, $A-A/A-A$ for some
$A\subset\mathbb{F}_{q}$. It is well-known that $A-A/A-A=\mathbb{F}_{q}$ when
$|A|\gg q^{1/2}$, see [1, Lemma 1] for example. We leave this to the reader
for checking in detail. The exponent $3/2$ is also optimal, for example, take
$q=p^{2}$ and $E=\mathbb{F}_{p}\times\mathbb{F}_{q}$, then
$\left|\frac{E-E}{E-E}\right|=p=o(q)$.

It follows from Theorems 1.3 and 1.4 that $L^{2}(D_{E})=(1+o(1))|E|^{4}/q$ and
$L^{2}(S_{E})=(1+o(1))|E|^{8}/q^{3}$ whenever $|E|\gg q^{3/2}$ and $|E|\gg
q^{5/3}$, respectively. We do not believe these two theorems are sharp, and
offer the following conjectures.

######  Conjecture 1.5.

Let $E$ be a subset of $\mathbb{F}_{q}^{2}$. If $|E|\gg q^{1+\epsilon}$ for
any $\epsilon>0$, then

$L^{2}(D_{E})=(1+o(1))\frac{|E|^{4}}{q}.$

######  Conjecture 1.6.

Let $E$ be a subset of $\mathbb{F}_{q}^{2}$. If $|E|\gg
q^{\frac{3}{2}+\epsilon}$ for any $\epsilon>0$, then

$L^{2}(S_{E})=(1+o(1))\frac{|E|^{8}}{q^{3}}.$

The final remark is that all results in this paper can be extended to higher
dimensions in the same way, but the proofs will become much complicated, for
instance, in the proofs of Theorems 1.1 and 1.2, one has to count the number
of pairs of zero distances, which can be done by using a number of results
from Restriction theory [13]. Therefore, to keep this paper simple, we only
present results in two dimensions.

##  2\. Notations from discrete Fourier analysis

In this section, we recall some notations and results from discrete Fourier
analysis which will be needed for our coming proofs in next sections.

Given a complex function $f\colon\mathbb{F}_{q}^{d}\to\mathbb{C}$, the Fourier
transform of $f$ is defined by

$\widehat{f}(\xi)=\frac{1}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}f(x)\chi(-x\cdot\xi),$

where $\chi$ is a fixed non-trivial additive character of $\mathbb{F}_{q}$.

Using the orthogonality property of $\chi$, i.e.

$\sum_{x\in\mathbb{F}_{q}^{d}}\chi(x\cdot\xi)=\begin{cases}0&\mbox{if}~{}\xi\neq
0,\\\ q^{d}&\mbox{if}~{}\xi=0,\end{cases}$

one can prove that

$f(x)=\sum_{\xi\in\mathbb{F}_{q}^{d}}\widehat{f}(\xi)\chi(\xi\cdot x).$

In this setting, the Plancherel formula reads as

$\sum_{\xi\in\mathbb{F}_{q}^{d}}|\widehat{f}(\xi)|^{2}=q^{-d}\sum_{x\in\mathbb{F}_{q}^{d}}|f(x)|^{2}.$

When $f$ is the indicator of a given set $E\subset\mathbb{F}_{q}^{d}$, one has

$\sum_{\xi\in\mathbb{F}_{q}^{d}}|\widehat{E}(\xi)|^{2}=q^{-d}|E|.$

The following lemma is known as the finite field analog of the spherical
average in the classical Falconer distance problem [19, Chapter 3].

######  Lemma 2.1.

$\max_{t\in\mathbb{F}_{q}\setminus\\{0\\}}\sum_{||\eta||=t}|\widehat{E}(\eta)|^{2}\ll\frac{|E|^{3/2}}{q^{3}}.$

A proof of this lemma can be found in [3, Lemma 4.4].

##  3\. Proof of Theorem 1.1

It is well-known that for $u,v,x,y\in\mathbb{F}_{q}^{d}$, if
$||u-v||=||x-y||$, then there exist $\theta\in O(d)$ and
$z\in\mathbb{F}_{q}^{d}$ such that

$\theta u+z=x,~{}\theta v+z=y.$

We now show that a similar statement holds for the case of product of distance
sets.

######  Lemma 3.1.

Set

$G_{1}=\bigg{\\{}diag(r_{1}\theta_{1},r_{2}\theta_{2})\in M_{2d\times
2d}\colon r_{1}\cdot r_{2}=1,\theta_{1},\theta_{2}\in O(d)\bigg{\\}},$

and $G=G_{1}\times\mathbb{F}_{q}^{2d}$. For
$x_{1},x_{2},x_{3},x_{4},y_{1},y_{2},y_{3},y_{4}\in\mathbb{F}_{q}^{d}$, if

$||x_{1}-y_{1}||\cdot||x_{2}-y_{2}||=||x_{3}-y_{3}||\cdot||x_{4}-y_{4}||\neq
0,~{}\frac{||x_{4}-y_{4}||}{||x_{2}-y_{2}||}\in(\mathbb{F}_{q})^{2},$

then there exists $g\in G$ such that

$g(x_{1},x_{2})^{T}=(x_{3},x_{4})^{T},~{}g(y_{1},y_{2})^{T}=(y_{3},y_{4})^{T}.$

###### Proof.

If

$||x_{1}-y_{1}||\cdot||x_{2}-y_{2}||=||x_{3}-y_{3}||\cdot||x_{4}-y_{4}||\neq
0,$

then

$\frac{||x_{1}-y_{1}||}{||x_{3}-y_{3}||}=\frac{||x_{4}-y_{4}||}{||x_{2}-y_{2}||}.$

Set $r=\frac{||x_{4}-y_{4}||}{||x_{2}-y_{2}||}$. From our assumptions, we know
that $r$ is a square, say, say, $r=t^{2}$ for some $t\neq 0$. This infers that

$||(1/t)x_{1}-(1/t)y_{1}||=||x_{3}-y_{3}||.$

Hence, we can find $\theta_{1}\in O(d)$ and $z_{1}\in\mathbb{F}_{q}^{d}$ such
that

$(1/t)\theta_{1}x_{1}+z_{1}=x_{3},~{}(1/t)\theta_{1}y_{1}+z_{1}=y_{3}.$

Similarly, from the fact that

$||x_{4}-y_{4}||=||tx_{2}-ty_{2}||,$

we can find $\theta_{2}\in O(d)$ and $z_{2}\in\mathbb{F}_{q}^{d}$ such that

$t\theta_{2}x_{2}+z_{2}=x_{4},~{}t\theta_{2}y_{2}+z_{2}=y_{4}.$

In other words, $g=(1/t\theta_{1},t\theta_{2})\times(z_{1},z_{2})\in G$ is the
element we want to find. ∎

###### Proof of Theorem 1.1.

Let

$A=\\{(a,b)\in E\times
E\colon||a-b||\in(\mathbb{F}_{q})^{2}\setminus\\{0\\}\\},$

and

$B=E\times E\setminus\left(A\cup\\{(a,b)\in E\times
E\colon||a-b||=0\\}\right).$

Let $N_{0}(E)$ be the number of pairs $(a,b)\in E\times E$ such that
$||a-b||=0$. We know from [16, Proposition 2.4] that

$N_{0}(E)\ll\frac{|E|^{2}}{q}+q|E|\ll|E|^{2},$

when $|E|\gg q$. Thus, either $|A|$ or $|B|$ is bounded from below by
$|E|^{2}/2$. Without loss of generality, we assume that $|B|\gg|E|^{2}/2$. For
any $\lambda\in(\mathbb{F}_{q})^{2}\setminus\\{0\\}$, define

$\nu(\lambda):=\\#\\{(x_{1},y_{1},x_{2},y_{2})\in
B^{2}\colon||x_{1}-y_{1}||\cdot||x_{2}-y_{2}||=\lambda\\}.$

We have

$\sum_{\lambda\in\left(\Delta(E)\cdot\Delta(E)\right)\cap(\mathbb{F}_{q})^{2}\setminus\\{0\\}}\nu(\lambda)\gg|E|^{4}.$

By the Cauchy-Schwarz inequality, one has

$\sum_{\lambda\in\Delta(E)\cap(\mathbb{F}_{q})^{2}\setminus\\{0\\}}\nu(\lambda)\ll|\Delta(E)|^{1/2}\cdot\left(\sum_{\lambda\in\mathbb{F}_{q}}\nu(\lambda)^{2}\right)^{1/2}.$

In the next step, we are going to bound
$\sum_{\lambda\in\mathbb{F}_{q}}\nu(\lambda)^{2}$ from above. For
$\theta=diag(r_{1}\theta_{1},r_{2}\theta_{2})\in G_{1}$ and
$z=(z_{1},z_{2})\in\mathbb{F}_{q}^{2}\times\mathbb{F}_{q}^{2}$, we define
$\mu_{\theta}(z)$ to be the number of tuples $(x_{1},x_{2},x_{3},x_{4})\in
E^{4}$ such that

$\theta(x_{1},x_{2})^{T}+(z_{1},z_{2})^{T}=(x_{3},x_{4})^{T}.~{}$

Then we observe that

$\sum_{\lambda}\nu(\lambda)^{2}\leq\sum_{\theta\in
G_{1},z\in\mathbb{F}_{q}^{4}}\mu_{\theta}(z)^{2}.$

It suffices to prove that

$\sum_{\theta\in
G_{1},z\in\mathbb{F}_{q}^{4}}\mu_{\theta}(z)^{2}\ll\frac{|E|^{8}}{q},$

whenever $|E|\gg q^{8/7}$.

Set $F=E\times E$. One has

$\displaystyle\mu_{\theta}(z)$
$\displaystyle=\sum_{x_{1},x_{2}\in\mathbb{F}_{q}^{2}}F(x_{1},x_{2})F\left(\theta(x_{1},x_{2})^{T}+z\right)$
$\displaystyle=\sum_{x_{1},x_{2}\in\mathbb{F}_{q}^{2}}F(x_{1},x_{2})\sum_{m\in\mathbb{F}_{q}^{4}}\widehat{F}(m)\chi\left(m\cdot(\theta(x_{1},x_{2})^{T}+z)\right)$
$\displaystyle=q^{4}\sum_{m\in\mathbb{F}_{q}^{4}}\widehat{F}(m)\widehat{F}(-\theta^{T}m)\chi(m\cdot
z).$

Hence,

$\widehat{\mu_{\theta}}(\xi)=q^{4}\widehat{F}(-\xi)\widehat{F}(\theta^{T}\xi).$

Therefore, for a fixed $\theta\in G_{1}$,

$\sum_{\xi\in\mathbb{F}_{q}^{4}}|\widehat{\mu_{\theta}}(\xi)|^{2}=\frac{|F|^{4}}{q^{8}}+\sum_{\xi\neq
0}|\widehat{\mu_{\theta}}(\xi)|^{2}.$

By writing
$\xi=(\xi_{1},\xi_{2})\in\mathbb{F}_{q}^{2}\times\mathbb{F}_{q}^{2}$ and
recall that $F=E\times E$, we have

$\displaystyle\sum_{\theta\in
G_{1},~{}\xi\in\mathbb{F}_{q}^{4}\setminus\\{0\\}}|\widehat{\mu_{\theta}}(\xi)|^{2}=q^{8}\sum_{\theta,\xi\neq
0}|\widehat{F}(-\xi)|^{2}|\widehat{F}(\theta^{T}\xi)|^{2}$
$\displaystyle=q^{8}\sum_{\theta}\sum_{\xi_{1}=0,\xi_{2}\neq
0}+q^{8}\sum_{\theta}\sum_{\xi\neq
0,\xi_{2}=0}+q^{8}\sum_{\theta}\sum_{\xi_{1}\neq 0,\xi_{2}\neq 0}=:I+II+III.$

For $I$, by Plancherel, one has

$\displaystyle I$ $\displaystyle=q^{8}\sum_{\theta_{1},\theta_{2}\in
O(2)}~{}\sum_{r_{1},r_{2}:~{}r_{1}\cdot r_{2}=1}~{}\sum_{\xi_{1}=0,\xi_{2}\neq
0}|\widehat{E}(-\xi_{1})|^{2}|\widehat{E}(-\xi_{2})|^{2}|\widehat{E}(r_{1}\theta_{1}^{T}\xi_{1})|^{2}|\widehat{E}(r_{2}\theta_{2}^{T}\xi_{2})|^{2}$
$\displaystyle=q|E|^{4}\sum_{\theta_{2}}\sum_{r_{2}\neq 0}\sum_{\xi_{2}\neq
0}|\widehat{E}(-\xi_{2})|^{2}|\widehat{E}(r_{2}\theta_{2}^{T}\xi_{2})|^{2}$
$\displaystyle=q|E|^{4}\sum_{\xi_{2}\neq
0}|\widehat{E}(-\xi_{2})|^{2}\sum_{r\neq
0}\sum_{||\eta||=||\xi_{2}||}|\widehat{E}(r\eta)|^{2}$ $\displaystyle\ll
q^{-3}|E|^{6}.$

Similarly, $II\ll q^{-3}|E|^{6}.$ For $III$, we proceed as follows:

$\displaystyle III$
$\displaystyle=q^{8}\sum_{\theta}\sum_{r_{1},r_{2}:r_{1}\cdot
r_{2}=1}\sum_{\xi_{1}\neq 0,\xi_{2}\neq
0}|\widehat{E}(-\xi_{1})|^{2}|\widehat{E}(-\xi_{2})|^{2}|\widehat{E}(r_{1}\theta_{1}^{T}\xi_{1})|^{2}|\widehat{E}(r_{2}\theta_{2}^{T}\xi_{2})|^{2}$
$\displaystyle=q^{8}\sum_{\theta_{2}}\sum_{\xi_{2}\neq 0}\sum_{r_{2}\neq
0}|\widehat{E}(\xi_{2})|^{2}|\widehat{E}(r_{2}\theta_{2}^{T}\xi_{2})|^{2}\sum_{\xi_{1}\neq
0}|\widehat{E}(\xi_{1})|^{2}\sum_{\eta:||\eta||=\frac{||\xi_{1}||}{r_{2}^{2}}}|\widehat{E}(\eta)|^{2}.$

This implies

$III\ll
q^{8}\left(\sum_{\xi_{1}}|\widehat{E}(\xi_{1})|^{2}\right)^{3}\cdot\left(\max_{t\in\mathbb{F}_{q}\setminus\\{0\\}}\sum_{||\eta||=t}|\widehat{E}(\eta)|^{2}\right).$

Using Lemma 2.1, we obtain

$III\ll q^{-1}|E|^{9/2}.$

In other words,

$\sum_{\theta,\xi}|\widehat{\mu}_{\theta}(\xi)|^{2}\ll\frac{|E|^{8}}{q^{5}}+\frac{|E|^{6}}{q^{3}}+\frac{|E|^{9/2}}{q}.$

This infers

$\sum_{\theta\in
G_{1},z\in\mathbb{F}_{q}^{4}}\mu_{\theta}(z)^{2}\ll\frac{|E|^{8}}{q}+q|E|^{6}+q^{3}|E|^{9/2}\ll\frac{|E|^{8}}{q},$

whenever $|E|\gg q^{8/7}$. ∎

##  4\. Proof of Theorem 1.2

We note that $r=0$ is trivially in $\Delta(E)/\Delta(E).$ For $\theta\in O(2)$
and $z\in\mathbb{F}_{q}^{2}$, assume $r$ is a nonzero square, we define

$\eta_{\theta}(z):=\\#\\{(u,v)\in E\times E\colon u-\sqrt{r}\theta v=z\\}.$

We first observe that

$\sum_{\theta\in
O(2),z\in\mathbb{F}_{q}^{2}}\eta_{\theta}(z)=|E|^{2}\cdot|O(2)|.$

So by the Hölder inequality, one has

$\sum_{\theta,z}\eta_{\theta}(z)^{2}\geq\frac{|E|^{4}|O(2)|}{q^{2}}.$

On the other hand, the sum $\sum_{\theta,z}\eta_{\theta}(z)^{2}$ counts the
number of tuples $(a,b,c,d)\in E^{4}$ such that $a-\sqrt{r}\theta c=z$ and
$b-\sqrt{r}\theta d=z$, so $||a-b||=r||c-d||$. For such a tuple, we have
$r\in\Delta(E)/\Delta(E)$, unless $||a-b||=||c-d||=0$.

For each tuple $(a,b,c,d)\in E^{4}$ such that $||a-b||=r||c-d||$, we have two
possibilities.

If $a-b\neq 0$, then this tuple is counted $\sim|O(1)|\sim 1$ times in the sum
$\sum_{\theta,z}\eta_{\theta}(z)^{2}$, since the size of the stabilizer of
each non-zero element is $\sim|O(1)|$.

If $a=b$, then $c=d$. Moreover, the number of tuples $(a,b,c,d)\in E^{4}$ with
$a=b$ and $c=d$ in the sum $\sum_{\theta,z}\eta_{\theta}(z)^{2}$ is equal to
$\sum_{\theta,z}\eta_{\theta}(z)=|E|^{2}|O(2)|$.

From these observations, the number of tuples $(a,b,c,d)\in E^{4}$ such that
$||a-b||=r||c-d||\neq 0$ is at least

$\frac{1}{|O(1)|}\left(\sum_{\theta,z}\eta_{\theta}(z)^{2}-|E|^{2}|O(2)|-|O(1)|N_{0}\right),$

where $N_{0}$ is the number of tuples with $||a-b||=r||c-d||=0$, $a\neq b$,
and $c\neq d$.

Since $q\equiv 3\mod 4$, it is clear that $N_{0}=0$. Thus,

$\frac{1}{|O(1)|}\left(\sum_{\theta,z}\eta_{\theta}(z)^{2}-|E|^{2}|O(2)|-|O(1)|N_{0}\right)\gg\frac{|E|^{4}}{q},$

whenever $|E|\gg q$.

##  5\. Proof of Theorem 1.3

Define the group $G=\mathbb{F}_{q}\times\mathbb{F}_{q}^{2}$ with the operator

$(\lambda,z_{1})\ast(\beta,z_{2})=(\lambda\cdot\beta,z_{1}+z_{2}).$

For a fixed $\lambda$, we define the counting function

$\gamma_{\lambda}(z):=\\#\\{(a,b)\in E^{2}\colon\lambda a+z=b\\}.$

By a direct computation, the tuple $(u,v,x,y)$ satisfies (1.2) if and only if
there exist $\lambda\neq 0$ and $z\in\mathbb{F}_{q}^{2}$ such that $\lambda
u+z=x$ and $\lambda v+z=y$. Thus,

$L^{2}(S_{E})=\sum_{\lambda,z}\gamma_{\lambda}(z)^{2}.$

For a fixed $\lambda$, we have

$\displaystyle\sum_{z\in\mathbb{F}_{q}^{2}}\gamma_{\lambda}(z)^{2}$
$\displaystyle=q^{2}\sum_{\xi\in\mathbb{F}_{q}^{2}}|\widehat{\gamma_{\lambda}(\xi)}|^{2}=\frac{|\sum_{z}\gamma_{\lambda}(z)|^{2}}{q^{2}}+q^{2}\sum_{\xi\neq
0}|\widehat{\gamma_{\lambda}(\xi)}|^{2}$
$\displaystyle=\frac{|E|^{4}}{q^{2}}+q^{2}\sum_{\xi\neq
0}|\widehat{\gamma_{\lambda}(\xi)}|^{2}.$

On the other hand,

$\displaystyle\gamma_{\lambda}(z)$ $\displaystyle=\sum_{u}E(u)E(\lambda
u+z)=\sum_{u}E(u)\sum_{\xi}\widehat{E}(\xi)\chi(\xi\cdot(\lambda u+z))$
$\displaystyle=\sum_{\xi}\widehat{E}(\xi)\chi(\xi\cdot
z)\sum_{u}E(u)\chi(\lambda\xi\cdot u)$
$\displaystyle=q^{2}\sum_{\xi}\widehat{E}(\xi)\widehat{E}(-\lambda\xi)\chi(\xi\cdot
z).$

Hence,

$\widehat{\gamma_{\lambda}(\xi)}=q^{2}\widehat{E}(-\xi)\widehat{E}(\lambda\xi).$

So,

$\displaystyle\sum_{\xi\neq
0}|\widehat{\gamma_{\lambda}(\xi)}|^{2}=q^{4}\sum_{\xi\neq
0}|\widehat{E}(-\xi)|^{2}|\widehat{E}(\lambda\xi)|^{2}.$

Hence,

$\sum_{\lambda\neq 0}\sum_{\xi\neq
0}|\widehat{\gamma_{\lambda}(\xi)}|^{2}=q^{4}\sum_{\xi\neq
0}|\widehat{E}(\xi)|^{2}\sum_{m\in<\xi>\setminus\\{0\\}}|\widehat{E}(m)|^{2},$

where $<\xi>$ is the line generated by $\xi$.

To proceed further, we set $L:=<\xi>$ and $T_{L}(E):=\sum_{m\in
L}|\widehat{E}(m)|^{2}$. It follows that

$\displaystyle T_{L}(E)$ $\displaystyle=q^{-4}\sum_{m\in L}\sum_{x,y\in
E}\chi(m\cdot(x-y))=q^{-4}\sum_{x,y\in
E}\sum_{t\in\mathbb{F}_{q}}\chi(t(\xi\cdot(x-y)))=q^{-4}(I+II),$

where

$I=\sum_{x,y\in
E,~{}\xi\cdot(x-y)=0}\sum_{t\in\mathbb{F}_{q}}\chi(t(\xi\cdot(x-y)))=q\sum_{x,y\in
E,~{}\xi\cdot(x-y)=0}1,$

and

$II=\sum_{x,y\in E,~{}\xi\cdot(x-y)\neq
0}\sum_{t\in\mathbb{F}_{q}}\chi(t(\xi\cdot(x-y)))=0.$

We note that the equation $\xi\cdot(x-y)=0$ with $x,y\in E$ means that $x-y$
lies on the line with the normal vector $\xi$ and passing through the origin.
Using the fact that each line in $\mathbb{F}_{q}^{2}$ contains exactly $q$
points, we have $I\leq q^{2}|E|$. Thus, using Plancherel formula, we obtain

$\sum_{\lambda\neq 0}\sum_{\xi\neq
0}|\widehat{\gamma_{\lambda}(\xi)}|^{2}\leq|E|^{2}.$

In other words,

$\left|\sum_{\lambda,z}\gamma_{\lambda}(z)^{2}-\frac{|E|^{4}}{q}\right|\ll
q^{2}|E|^{2}.$

This completes the proof of the theorem.

##  6\. Proof of Theorem 1.4

Using the notations as in the previous section, we observe that

$L^{2}(S_{E})=\sum_{\lambda\neq
0}\left(\sum_{z}\gamma_{\lambda}(z)^{2}\right)^{2}.$

As in the previous section, we know that

$\sum_{z\in\mathbb{F}_{q}^{2}}\gamma_{\lambda}(z)^{2}=\frac{|E|^{4}}{q^{2}}+q^{2}\sum_{\xi\neq
0}|\widehat{\gamma_{\lambda}}(\xi)|^{2},$

and

$\sum_{\lambda}\sum_{\xi\neq
0}|\widehat{\gamma_{\lambda}}(\xi)|^{2}\ll|E|^{2}.$

Thus,

$\left|\sum_{\lambda\neq
0}\left(\sum_{z}\gamma_{\lambda}(z)^{2}\right)^{2}-\frac{|E|^{8}}{q^{3}}\right|\ll|E|^{6}+q^{4}\sum_{\lambda}\left(\sum_{\xi\neq
0}|\widehat{\gamma_{\lambda}}(\xi)|^{2}\right)^{2}.$

As we computed in the previous section, one has

$\displaystyle\left(\sum_{\xi\neq
0}|\widehat{\gamma_{\lambda}}(\xi)|^{2}\right)^{2}=q^{8}\left(\sum_{\xi\neq
0}|\widehat{E}(\xi)|^{2}|\widehat{E}(-\lambda\xi)|^{2}\right)^{2}.$

Therefore,

$\displaystyle\sum_{\lambda}\left(\sum_{\xi\neq
0}|\widehat{E}(\xi)|^{2}|\widehat{E}(-\lambda\xi)|^{2}\right)^{2}=\sum_{\xi_{1}\neq
0,\xi_{2}\neq
0}|\widehat{E}(\xi_{1})|^{2}|\widehat{E}(\xi_{2})|^{2}\sum_{\lambda}|\widehat{E}(\lambda\xi_{1})|^{2}|\widehat{E}(\lambda\xi_{1})|^{2}$
$\displaystyle=\sum_{\xi_{1}\neq 0,\xi_{2}\neq
0}|\widehat{E}(\xi_{1})|^{2}|\widehat{E}(\xi_{2})|^{2}\cdot
L(\xi_{1},\xi_{2}),$

where

$L(\xi_{1},\xi_{2})=\sum_{\lambda}|\widehat{E}(\lambda\xi_{1})|^{2}|\widehat{E}(\lambda\xi_{2})|^{2}.$

To proceed further, we need the following lemma.

######  Lemma 6.1.

For $\xi_{1},\xi_{2}\neq 0$, one has

$L(\xi_{1},\xi_{2})=\sum_{\lambda}|\widehat{E}(\lambda\xi_{1})|^{2}|\widehat{E}(\lambda\xi_{2})|^{2}\ll\frac{|E|^{3}}{q^{6}}.$

###### Proof.

By the Cauchy-Schwarz inequality, we have

$L(\xi_{1},\xi_{2})\leq\left(\sum_{\lambda}|\widehat{E}(\lambda\xi_{1})|^{4}\right)^{1/2}\cdot\left(\sum_{\lambda}|\widehat{E}(\lambda\xi_{2})|^{4}\right)^{1/2}.$

Moreover, for $m\neq 0$,

$\displaystyle\sum_{\lambda}|\widehat{E}(\lambda
m)|^{4}=\frac{1}{q^{8}}\sum_{x,y,z,w\in
E}\sum_{\lambda}\chi(m\lambda(x+y-z-w))$
$\displaystyle=\frac{1}{q^{7}}\sum_{x,y,z,w\in E}1_{m\cdot(x+y-z-w)=0}$
$\displaystyle\ll\frac{|E|^{3}}{q^{6}},$

where the last inequality follows from the number of tuples $(x,y,z,w)\in
E^{4}$ such that $x+y-z-w$ lies on the line with the normal vector $m$ and
passing through the origin. This completes the proof. ∎

To continue, we apply the Plancherel formula,

$\displaystyle\sum_{\lambda}\left(\sum_{\xi\neq
0}|\widehat{E}(\xi)|^{2}|\widehat{E}(-\lambda\xi)|^{2}\right)^{2}=\sum_{\xi_{1}\neq
0,\xi_{2}\neq 0}|\widehat{E}(\xi_{1})|^{2}|\widehat{E}(\xi_{2})|^{2}\cdot
L(\xi_{1},\xi_{2})$ $\displaystyle\ll\frac{|E|^{5}}{q^{10}}.$

In other words, we obtain

$\left|\sum_{\lambda}\left(\sum_{z}\gamma_{\lambda}(z)^{2}\right)^{2}-\frac{|E|^{8}}{q^{3}}\right|\ll|E|^{6}+q^{2}|E|^{5}.$

##  7\. Acknowledgement

The author would like to thank to Vietnam Institute for Advanced Study in
Mathematics (VIASM) for the hospitality and for the excellent working
condition.

## References

  * [1] A. Balog, Another sum-product estimate in finite fields, Proceedings of the Steklov Institute of Mathematics, 280(2) (2013): 23–29.
  * [2] M. Bennett, D. Hart, A. Iosevich, J. Pakianathan, and M. Rudnev, Group actions and geometric combinatorics in $\mathbb{F}_{q}^{d}$, Forum Mathematicum, 29(1) (2017): 91-110.
  * [3] J. Chapman, M. Burak Erdogan, D. Hart, A. Iosevich, and D. Koh, Pinned distance sets, $k$-simplices, Wolff ’s exponent in finite fields and sum-product estimates, Mathematische Zeitschrift, 271 (1-2) (2012): 63-–93.
  * [4] D. Cheong, D. Koh, and T. Pham, An asymmetric bound for sum of distance sets, Proceedings of the Steklov Institute of Mathematics, 314(1) (2021): 279–289.
  * [5] X. Du, A. Iosevich, Y. Ou, H. Wang, and R. Zhang, An improved result for Falconer’s distance set problem in even dimensions, Mathematische Annalen, 380 (2021): 1215–-1231.
  * [6] L. Guth, A. Iosevich, Y. Ou, and H. Wang, On Falconer’s distance set problem in the plane, Inventiones mathematicae, 219(3) (202): 779–830.
  * [7] A. Greenleaf, A. Iosevich, B. Liu, and E. Palsson, A group-theoretic viewpoint on Erdős–Falconer problems and the Mattila integral, Revista Matemática Iberoamericana, 31(3) (2015): 799–810.
  * [8] A. Greenleaf, A. Iosevich, and S. Mkrtchyan, Existence of similar point configurations in thin subsets of ${\mathbb{R}}^{d}$, Mathematische Zeitschrift, 297(1) (2021): 855–865.
  * [9] D. Hart and A. Iosevich, Ubiquity of simplices in subsets of vector spaces over finite fields, Analysis Mathematica, 34 (2007).
  * [10] D. Hart, A. Iosvich, D. Koh, S. Senger, and Ignacio Uriarte-Tuero, Distance graphs in vector spaces over finite fields, Recent advances in harmonic analysis and applications, pp. 139–160. Springer, New York, NY, 2012.
  * [11] A. Iosevich, D. Koh, and H. Parshall, On the quotient set of the distance set, Moscow Journal of Combinatorics and Number Theory, 8(2) (2019): 103–115.
  * [12] A. Iosevich, H. Morgan, and J. Pakianathan, On directions determined by subsets of vector spaces over finite fields, Integers, 11:A39, 2011.
  * [13] A. Iosevich, D. Koh, S. Lee, T. Pham, and C-Y. Shen, On restriction estimates for the zero radius sphere over finite fields, Canadian Journal of Mathematics, 73(3)(2021): 769–786.
  * [14] D. Koh, T. Pham, C-Y. Shen, and L. A. Vinh, A sharp exponent on sum of distance sets over finite fields, Mathematische Zeitschrift, 297(3) (2021): 1749–1765.
  * [15] D. Koh, T. Pham, and L. A. Vinh, Extension theorems and a connection to the Erdős-Falconer distance problem over finite fields, Journal of Functional Analysis, 281(8) (2021): 109137.
  * [16] D. Koh and H. Sun, Distance sets of two subsets of vector spaces over finite fields, Proceedings of the American Mathematical Society, 143(4) (2015): 1679–1692.
  * [17] B. Liu, An $L^{2}$-identity and pinned distance problem, Geometric and Functional Analysis, 29(1) (2019): 283–294.
  * [18] B. Liu, Group actions, the Mattila integral and applications, Proceedings of the American Mathematical Society, 147(6) (2019): 2503–2516.
  * [19] P. Mattila, Fourier analysis and Hausdorff dimension, Vol. 150, Cambridge University Press, 2015.
  * [20] B. Murphy, G. Petridis, T. Pham, M. Rudnev, and S. Stevens, On the pinned distances problem in positive characteristic, Journal of the London Mathematical Society, 105(1) (2022): 469–499.
  * [21] H. Parshall, Simplices over finite fields, Proceedings of the American Mathematical Society, 145(6) (2017): 2323–2334.
  * [22] T. Pham, Erdős distinct distances problem and extensions over finite spaces, Phd thesis, EPFL, Lausane Switzerland, 2017.
  * [23] T. Szőnyi, On the number of directions determined by a set of points in an affine Galois plane, Journal of Combinatorial Theory Series A, 74(1) (1996): 141–146.
  * [24] L. A. Vinh, On kaleidoscopic pseudo-randomness of finite Euclidean graphs, Discussiones Mathematicae: Graph Theory, 32(2) (2012).


2021

[1,2]Ruben López-Coto

[3]Emma de Oña Wilhelmi

1]Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Padova, I-35131,
Italy

2]Instituto de Astrofísica de Andalucía, CSIC, Granada, 18080, Spain

3]DESY, Platanenallee 6, Zeuthen, D-15738, Germany

4]Max-Planck-Institut für Kernphysik, P.O. Box 103980, Heidelberg, D-69029,
Germany

5]Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, Dublin, Ireland

6]INAF Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, Firenze,
I-50125, Italy

7]Università degli Studi di Firenze, Via Sansone 1, Sesto Fiorentino, I-50019,
Italy

# Gamma-ray halos around pulsars as the key to understanding cosmic ray
transport in the Galaxy

<EMAIL_ADDRESS><EMAIL_ADDRESS>Felix Aharonian  Elena Amato
Jim Hinton  [  [  [  [  [  [  [

###### Abstract

Pulsars are factories of relativistic electrons and positrons that propagate
away from the pulsar, permeating later our Galaxy. The acceleration and
propagation of these particles are a matter of intense debate. In the last few
years, we had the opportunity to directly observing the injection of these
particles into the interstellar medium through the discovery of gamma-ray
halos around pulsars. This new type of gamma-ray source is produced by
electrons and positrons diffusing out of the pulsar wind nebula and scattering
ambient photon fields to produce gamma rays. This new field of study comes
with a number of observations and constraints at different wavelengths and a
variety of theoretical models explaining the characteristics of these halos.
We examine the characteristics of the propagation of cosmic rays inferred from
the observations of halos and their local and global implications on particle
transport in the Galaxy. We also discuss the prospects for observations of
these sources with facilities such as LHAASO, or CTA or SWGO in the near
future.

##  1 Introduction

Pulsars, rapidly rotating neutron stars left behind in supernova explosions,
have played a critical role in the development of many areas of astrophysics.
In non-thermal astrophysics, the nebula around the Crab pulsar was the first
known TeV source (Whipple_Crab, ) and is by far the best-studied site of
astrophysical particle acceleration outside the solar system
(2008ARA&A..46..127H, ; 2014RPPh…77f6901B, ). The relativistic wind from
pulsars is halted at a termination shock and beyond this a synchrotron-
radiation-emitting pulsar wind nebula (PWN) forms, apparently dominated by
relativistic electron/positron pairs and magnetic fields (see (GaenslerSlane,
) for a review). The processes of energy conversion from the highly magnetised
flow in the vicinity of the pulsar magnetosphere (at $\sim 10^{6}$m) to the
particle-dominated one at the parsec-scale nebula are a matter of continuing
debate and intense theoretical study (see e.g. 2013MNRAS.431L..48P  and
references therein).

Ideally, in a steady, spherical flow, the gamma-radiation from the confined
region should reflect the dominant convective character of propagation
(relativistic MHD flow) and the morphology of the nebula - typically
elongated/asymmetric relative to the pulsar’s position, whereas beyond it a
diffusive propagation should dominate. Note however that this paradigm breaks
when looking at the emission of the highest energy particles, which,
especially in evolved objects, have large Larmor radii, that can be comparable
to the system size (more complex MHD simulations and particle transport models
are required when looking into PWNe in detail (2003astro.ph..8483R, ;
2017ASSL..446..215D, ; 2016MNRAS.460.4135P, ).

Figure 1 illustrates the main evolutionary stages of PWNe. During Stage 1 the
PWN is inside the SNR and has not been touched by the reverse shock yet. Stage
2 starts when the PWN is crushed by the reverse shock but the pulsar is still
within the SNR: due to the declining magnetic field and the different cooling
time of the emitting electrons, X-ray dim, gamma-ray bright ’relic’ bubbles
can be observed at this stage. Stage 3 starts when the pulsar leaves its
parent SNR and high energy particles escape in the ISM: Geminga and PSR
B0656+14 (HAWC_Science, ) are thought to be in this stage. The fringes between
Stage 2 and 3 are often unclear, and there has been considerable discussion of
when the term _halo_ can legitimately be applied. Here we adopt the term to
refer to a population of particles essentially free from their parent PWN, or
at least outside of the region in which the nebula is energetically dominant.

Understanding the physics of these halos has implications of fundamental
importance for a number of open problems in High Energy Astrophysics. Firstly,
halos can be used to probe the evolution of particle acceleration and escape
in pulsars and PWNe, which are themselves a unique laboratory for relativistic
astrophysics. Secondly, the observation of cooling electrons and positrons
traveling freely through the ISM from a well-defined source provides a unique
probe of the propagation of relativistic particles. This transport is
regulated by the diffusion coefficient, which for the ISM has been estimated
to be D(10 GeV)$\sim 8\times 10^{28}$cm2s-1 (1990acr..book…..B, ;
2011ApJ…729..106T, ), through comparison between spallation data and particle
transport models. However, this value refers to the average diffusion
coefficient in the ISM, and local variations should be expected. Observations
of halos provide information about these variations around powerful electron
and positron accelerators. Moreover, these measurements have strong
implications on the global electron spectrum.

In this review, we explore the existing experimental constraints on pulsar
halos and discuss the theoretical expectations for particle escape and
propagation, finally considering the prospects in this area given the powerful
new instrumentation on the horizon.

Figure 1: Evolutionary stages in the life of a PWN, adapted from Giacinti et
al., 2020 Halo_fraction , illustrating the early confinement of particles and
the later escape of (at least) the higher energy relativistic particles to
form a halo visible in TeV gamma rays.

##  2 Current experimental results

The observations of our Galaxy in the TeV energy range with moderate to large
field of view sensitive telescopes have revealed a large number of extended
regions of gamma rays. In particular, the Galactic plane survey performed by
the H.E.S.S. array of Cherenkov telescopes (2005Sci…307.1938A, ;
2006ApJ…636..777A, ; 2018A&A…612A…1H, ) was the first milestone in the study
of such multi-parsec structures, unveiling a large population of PWNe, which
dominates the TeV emission of the Galactic plane (2018A&A…612A…2H, ). The
presence of TeV structures much larger than the X-ray ones was already
predicted by 1997MNRAS.291..162A . The large diversity of TeV PWNe discovered
by H.E.S.S. and others IACTs like VERITAS and MAGIC (see e.g (2014A&A…567L…8A,
; 2019ApJ…878..126C, ), promoted the classification of PWN evolution in
several evolutionary stages, according to the physical properties of the
region from which TeV emission originates during the lifetime of a pulsar.
This first classification included a free expansion stage, followed by a
second phase in which the PWN is interacting with the turbulent plasma left
behind by the SN explosion (Stage 1 in Fig. 1); examples of this stage are
PWNe like the Crab Nebula or 3C 58. The next stage, also called relic-stage is
one in which a large TeV nebula expands beyond the SNR, altering the
surrounding ISM (Stage 2 in Fig. 1). For objects in this stage, TeV
observations with Cherenkov telescopes, combining good energy and angular
resolution, allowed disentangling between emission properties in different
sub-regions, proving efficient particle cooling and diffusive propagation
within the nebula (2020A&A…644A.112H, ; 2019A&A…627A.100H, ;
2012A&A…548A..46H, ; 2020A&A…640A..76P, ).

Figure 2: Map of the sky region around the HAWC detection of Geminga. VHE
gamma-ray inset: Test-statistics (TS) map from the HAWC observations
(Abeysekara et al., 2017 (HAWC_Science, )). Optical inset: DSS2 obtained using
Aladin sky atlas (Bonnarel et al., 2000 (2000A&AS..143…33B, )). X-ray
NASA/CXC/PSU (Posselt et al. 2017 (Posselt_2017, )); Infrared Map: NASA/JPL-
Caltech

The advent of large field of view instruments like Milagro (2007ApJ…664L..91A,
) and currently HAWC and LHAASO in the TeV range (2020ApJ…905…76A, ;
2021Natur.594…33C, ), and the Fermi-LAT satellite in the GeV range
(2020ApJS..247…33A, ), opened a new window to complete the evolutionary
picture of PWNe, thanks to improved sensitivity to very large gamma-ray
structures (see Fig. 2). The discovery of a large $\sim$5∘ TeV emission around
the $\gtrsim 100$ kyr old pulsars Geminga and PSR B0656+14 triggered a new
understanding of the electrons injected within the pulsar environment. The
dimension of the halo ($\sim$25 pc for 100 TeV electrons) has been interpreted
as due to slow escape (HAWC_Science, ) of electrons and positrons accelerated
at the pulsar wind termination shock. The most natural explanation for the
propagation is that charged particles diffuse in the turbulent magnetic field
of the region. Under this assumption, HAWC_Science  constrained the diffusion
coefficient in the region surrounding Geminga and PSR B0656+14 to $D$(100
TeV)$\sim 5\times 10^{27}$ cm${}^{2}s^{-1}$, a value much lower than the
average in the ISM, which poses a problem on the origin of this increased
level of turbulence with respect to that inferred, from local CR spectra, as
the average in the Galaxy. Recently, an alternative interpretation to the slow
diffusion scenario has been proposed, in which a combination of ballistic plus
diffusive propagation at the same rate as the average in the ISM is used to
explain the observed size and TeV emission features (Recchia2021, ). The
region seen in TeV is characterized by an energy density below that of the
ISM, which evidences an outflow of escaping electrons in a region that is not
modified by the pulsar itself. The proximity of these pulsars (at $\sim$250
pc), combined with the large field of view makes possible the detection of the
otherwise very low TeV gamma-ray surface brightness ( $\sim 10^{-12}$TeV cm-2s
deg-2). These electrons diffuse away and fill up a large region or halo,
providing a unique clean scenario to study diffusive propagation in the
Galaxy. The transition between relic- and halo-stage is blurred and had
motivated different classification criteria (Halo_fraction, ; Linden17, ). An
example of this transitional stage is the very extended source HESS J1825–137
(21 kyrs) with an unusually large extension of $\gtrsim 100$ pc
(2019A&A…621A.116H, ; 2018ApJ…860…59K, ), exceeding the scales anticipated by
the standard hydro-dynamical paradigm of PWN formation. The energy-dependence
of the morphology strongly suggests advection-dominated transport within the
PWN, but the fringes of the emission extend to very large distances and may
indicate the presence of unconfined TeV-emitting particles (2019A&A…621A.116H,
). Similarly, the 11 kyr-old Vela X shows a compact TeV emission, coincident
with a bright X-ray nebula and an extended region with similar spectrum. The
extended region also coincides with a radio halo (1958AuJPh..11..550R, ),
which might be a sign of particles escaping (2006A&A…448L..43A, ;
2019A&A…627A.100H, ; 2018A&A…617A..78T, ; 2013ApJ…774..110G, ). However, the
majority of the TeV PWNe are consistent with a relic scenario. These naturally
stem from the observational bias towards $\sim$0.2∘ sources at $\sim$5 kpc
distance, for which Cherenkov telescope sensitivity is optimal, in terms of
size and flux: at shorter distances, the halo becomes too large to be fully
contained in the field of view (HESS_Geminga, ; 2015arXiv150904224F, ;
2016A&A…591A.138A, ), whereas, for distant sources, the flux might be too
faint to be detectable. The good angular resolution permits however a deeper
investigation of the morphology in different energy ranges, hinting at
transition regions between different propagation regimes. Although there are
only two firmly identified sources in which emission is produced by already
escaped electrons, there have been reports of other halo-candidates
(2017ATel10941….1R, ; 2018ATel12013….1B, ; 2019ICRC…36..797S, ; HAWC56TeV, ;
2021arXiv210609396L, ) and proposals that this could be the dominant mechanism
in several known sources (Hooper_HESS17, ; 2020PhRvD.101j3035D, ). Very
recently, the LHAASO collaboration has reported the detection of Geminga
(LHAASO_Geminga, ) and is currently studying it and PSR B0656+14 in detail
(Geminga_Monogem_LHAASO, ).

The observation of the multi-wavelength counterpart of pulsar halos poses
severe challenges. The corresponding size and expected flux are related to the
particles cooling regime: in the sub-100 GeV and radio energy band a large
emission region with a very hard spectral index is expected, whereas the
highest energy electrons are affected by strong cooling, resulting in compact
soft X-ray emission (see Fig. 2), as in the case of Geminga
(2003Sci…301.1345C, ; Posselt_2017, ). Geminga has been investigated in the
GeV regime by several authors using data from the Fermi-LAT telescope. The
results are however hampered by the expected number of photons in this energy
band: on one hand, the hard uncooled electron spectrum ($\sim$1.8) results in
low flux levels below a few tens of GeV; on the other hand, the overwhelming
gamma-ray background diffuse emission that dominates the Galactic plane makes
it difficult to disentangle the low surface brightness of these halos. Using
an energy-dependent, model-dependent template, (2019PhRvD.100l3015D, ) claimed
a detection of the Geminga halo above 10 GeV. A second analysis by
(2019ApJ…878..104X, ), using a smaller Region of Interest did not confirm such
an extended GeV emission, using a similar physics-motivated template approach
(two-zone diffusion spatial templates), although not including the proper
motion of the pulsar.Observations with instruments like CTA or SWGO in the
future should provide a clear picture of the evolution in size and spectrum
below hundreds of GeV.

In the search for halos, X-ray observations provide different diagnostic tools
at different scales. On one hand, the excellent angular resolution of
instruments like XMM-Newton and Chandra provides precise images of pulsars
propagating in the ISM, forming a bow-shock which evidences particle escape
through the observation of bright filaments (2010MNRAS.408.1216J, ;
2012ApJ…747…74H, ; 2014A&A…562A.122P, ). On the other, a few-degree soft X-ray
halo should emerge, corresponding to the synchrotron emission of electrons
powering the gamma-ray source. The detection of such diffuse emission requires
a deep observation program involving several pointings. In the case of
Geminga, (2019ApJ…875..149L, ) examined a 1∘ region around the pulsar and
obtained an upper limit on the magnetic field below $1\mu$G, based on X-ray
upper limits on the synchrotron emission by the electrons responsible for the
emission detected by HAWC. The derived magnetic field is below the mean one in
the ISM, pointing to a perturbed medium.

##  3 Implications in PWN theory

The experimental results described above have several implications, regarding
not only particle (electrons and positrons) acceleration efficiency, but also
their propagation. The maximum particle energy derived from the highest photon
energy measured in Geminga ($\approx 300$ TeV) has strong implication in the
acceleration mechanisms: it appears that particles are accelerated up to a
fraction close to 1 of the maximum potential drop available in the pulsar
magnetosphere $\Phi_{\rm PSR}=\sqrt{\dot{E}/c}$ (1969ApJ…157..869G, ), with
$c$ the speed of light and $\dot{E}$ the pulsar spin-down power. The bulk of
the particles making PWNe bright non-thermal sources are believed to be
accelerated at the pulsar wind termination shock (TS). The details of the
mechanisms are not clear. The three main proposals are: shock acceleration
(see 2015SSRv..191..519S  for a review), magnetic reconnection
(2011ApJ…741…39S, ) or resonant absorption of ion cyclotron waves
(1992ApJ…390..454H, ; 2006ApJ…653..325A, ). All of these mechanisms may in
principle reach the required energies (see e.g. (Amato19, ) for a recent
review), but not easily. In fact, the maximum achievable energy at the TS
(ignoring all dissipative effects) is determined by the condition that the
particle Larmor radius be smaller than the characteristic size of the
accelerator (”Hillas criterion”), namely $E_{\rm max,TS}=eR_{\rm TS}\ B_{\rm
TS}$, where $R_{\rm TS}$ is the TS radius and $B_{\rm TS}$ the local magnetic
field strength. Writing the magnetic pressure at the TS as a fraction
$\eta_{B}$ of th ram pressure of the wind $\dot{E}/(4\pi R_{\rm TS}^{2}c$),
one finds $B_{\rm TS}=(\eta_{B}^{1/2}/R_{\rm TS})\sqrt{\dot{E}/c}$. From which
we derive that the maximum achievable energy at the TS ($\eta_{B}$=1) is:
$E_{\rm max,TS}\approx e\Phi_{\rm PSR}$, i.e., in the absence of losses, the
maximum energy depends only on the potential drop in the pulsar magnetosphere.
Note that this constraint does not depend on the acceleration mechanism.
Writing the pulsar spin-down luminosity in units of $10^{36}$ erg/s, we have
$E_{\rm max}\approx 1.8\eta_{B}^{1/2}\ \dot{E}_{36}^{1/2}$ PeV (with
$\dot{E}_{36}$ in units of $10^{36}$ erg/s). In particular, for Geminga, with
$\dot{E}_{36}=0.03$, energies of a few hundreds TeV electrons correspond to
maximally efficient acceleration.

Once these particles have been accelerated, their escape in the ISM is again a
dive into poorly understood physics. The common view of PWNe is that these
sources can well be modeled within the framework of relativistic MHD (see e.g.
(Olmi16, ) for a review), where the propagation of particles inside them is
governed by advection. This picture is bound to fail at the highest energies
and indeed computation of the particle dynamics on top of the e.m. field
structure derived from MHD simulations shows that only in a narrow energy
range close to $E_{\rm max,TS}$ the fraction of particles that can escape from
a PWN becomes sizeable (Olmi19a, ). Very interestingly the escaping population
is charge-separated, with electrons and positrons escaping in about equal
amounts, but along different paths (Olmi19b, ). In principle, this would
create the conditions for the development, in the PWN vicinity, of a current
large enough to have interesting consequences. In fact, one of the proposed
explanations for the reduced diffusion coefficient constrained by the HAWC
observations is that this results from an enhanced turbulence level produced
by the particles escaping the PWN. The well-known resonant streaming
instability (1969ApJ…156..445K, ) does not seem to be effective enough
(Evoli18, ). The existence of a net current opens the door to the possibility
that the fast-growing non-resonant streaming instability (Bell04, ) can be at
work. This requires that the energy density in the current carrying particles
is larger than that in the local magnetic field. Such a condition, which is
possible in principle to satisfy (Olmi19b, ), appears at odds with estimates
of the energy density in very high energy electrons derived from modeling of
the Geminga halo emission (Halo_fraction, ). An alternative explanation for
the reduced diffusion coefficient is that it results from a local reduction of
the magnetic field coherence length, down to $pc$ values, a factor $\approx
10^{-2}$ of what is commonly adopted for MHD turbulence in the Galaxy
(2018MNRAS.479.4526L, ). In fact, within quasi-linear theory, the diffusion
coefficient can be written as

$D(E)\approx 2\times 10^{28}{\rm cm}^{2}{\rm
s}^{-1}\xi_{B,0.1}^{-1}\left(\frac{\lambda_{\rm pc}}{E_{\rm
TeV}}\right)^{\alpha-1}B_{\mu}^{\alpha-2}$ (1)

where $\xi_{B,0.1}=(\delta B/B)^{2}$ is the ratio between the power of the
turbulent and ordered magnetic field normalized to 0.1, $\lambda_{\rm pc}$ is
the outer scale of the turbulence in units of pc, $B_{\mu}$ is the large scale
magnetic field in units of $\mu G$ and $E_{\rm TeV}$ is the particle energy in
TeV; finally $\alpha$ is the turbulence spectral index with $\alpha=5/3(3/2)$
for a Kolmogorov (Kraichnan) phenomenology. It is clear then that a reduced
diffusivity might result from an increased turbulence level (larger $\xi_{B}$)
or a smaller coherence length $\lambda$ of the turbulence. Distinguishing
between these two scenarios would only be feasible by looking at particles of
higher energies, with Larmor radii comparable to the turbulence coherence
length. In that sense, observations with LHAASO should provide crucial
information in the understanding of the diffusive regime. Recently, a new
solution different from the diffusion-only regime for the propagation of
electrons has been proposed. Recchia2021  studied the halo morphology taking
into account that a significant fraction of the propagation of multi-TeV
electrons could take place in the ballistic or ballistic-to-diffusive regimes.
The ballistic propagation of these electrons in a turbulent magnetic field for
distances larger than their Larmor radius could make the gamma-ray morphology
of Geminga compatible with the standard diffusion coefficient derived from CR
measurements.

##  4 Halos and Galactic Cosmic Rays

The escaping particles diffuse further within the Galaxy, ultimately adding up
to the sea of Galactic CRs. These CRs can be described as a low-density plasma
whose propagation is governed by the diffusion-loss equation
(Strong&Moskalenko, ; Grenier&Strong, ). Local measurements of CR fluxes and
relative abundances provide insights into the distribution of sources
generating them. What pulsar halos directly probe is the transport of
electrons and positrons, but since CR propagation in the Galaxy is thought to
depend on particle rigidity alone, we expect the propagation of hadrons and
leptons to be the same for a given rigidity. Irrespective of the cause of
enhanced scattering in the halos, these regions can have important
implications for the galactic transport of both leptonic and hadronic CRs, and
even put crucial constraints on the origin of the locally measured CR fluxes.
In this sense, it is important to understand how common pulsar halos are
(Sudoh19, ; Halo_fraction, ).

###  4.1 Local effects

Electrons and positrons that escape from these sources can significantly
contribute to the local all-electron (e±) spectrum (Atoyan95, ). This provides
information about the propagation of these e±, while the e+ fraction (e+ flux
divided by e±flux), provides the ratio between the flux of secondary
positrons, produced in the collisions of CRs with the ISM, and that of the
dominant primary electrons. There is, however, an anomaly known as the
“positron excess” in the e+ fraction, produced by an excess of positrons above
the CR induced background (Adriani09, ; Ackermann12, ; Aguilar14, ) above
energies of a few GeV that has led to intense speculation on their origin.
This excess has been postulated to arise from PWNe (Aharonian95, ;
2009PhRvL.103e1101Y, ), microquasar jets (Gupta14, ) or dark matter
annihilation (Bergstrom08, ). According to the most commonly accepted
propagation theories within the ISM, the highest-energy positrons measured by
satellites ($\sim$1 TeV) must originate from a region within $\leq$ few kpc
from the Earth ($r_{\rm d}(E)=\sqrt{2D(E)t_{\rm cool}}(E)$, where $D$(1
TeV)$\sim 10^{29}-10^{30}$ cm-2 s-1, depending on the assumed energy
dependence of the diffusion coefficient, and $t_{\rm cool}$(1 TeV)$\sim 300$
kyr, also depending on the assumed energy losses), limiting the number of
possible sources behind this phenomenon (Evoli_Amato, ).

In Fig. 3 and 4, we can see different estimations from the literature for the
local all-electron and positron spectrum, some other recent works on local
electron/positron spectra estimations can be found in the literature
(2017PhRvD..95f3009L, ; 2019PhRvD..99d3005L, ; 2020PhRvL.125e1101E, ;
2021PhRvD.103h3010E, ; 2020PhRvD.102b3015M, ). The HAWC results on Geminga and
PSR B0656+14 (HAWC_Science, ) argued against a significant contribution to the
electron and positron spectrum at the Earth by these two pulsars, if assuming
a uniform (one-zone) diffusion coefficient from these two pulsars to the
Earth. There has later been extensive literature arguing otherwise
(2019ApJ…879…91J, ; 2020PhRvD.102b3015M, ; 2017PhRvD..96j3013H, ; Profumo2018,
), by invoking a two-zone diffusion model to describe the propagation of these
escaping electrons. In this scenario, the totality of the high-energy
electrons and positrons measured at the Earth can be explained using nearby
pulsars.

The recent measurement of the CR all-electron spectrum up to $\sim$20 TeV
(HESS_electrons, ), in addition to those by DAMPE (2017Natur.552…63D, ) and
CALET (Adriani17, ) indicates that these electrons and positrons must be
generated nearby the Earth because of their cooling due to their interaction
with interstellar magnetic and photon fields. Assuming fast diffusion, the
origin of these high energy electrons may be the aforementioned pulsars, or
sources such as SNRs (2019PhRvD..99j3022R, ; 2020JCAP…02..009F, ). The key to
discerning between these two source types is the positron flux, which seems to
point to a decrease at the highest energies, disfavoring pulsars as the origin
of the highest energy electrons (2019PhRvL.122d1102A, ). If on the other hand,
we consider a slower diffusion, even an undiscovered pulsar in the Local
Bubble, the explanation of the local high energy CR All-Electron Spectrum
might still be dominated by pulsars (2018PhRvL.121y1106L, ). Other observable
directly related to the local contribution of a particular source type is the
dipole anisotropy, which should in principle pinpoint the origin of the
primary accelerator. The current measurements are however still compatible
with the different scenarios proposed (Manconi17, ; 2018PhRvL.121y1106L, ).

It remains an open question whether known pulsars (Manconi18, ), SNRs
(2019PhRvD..99j3022R, ), unknown pulsars (2018PhRvL.121y1106L, ) or DM
(Bergstrom08, ) are still a viable explanation for the local e± flux.

Figure 3: Local all-electron spectrum measured by AMS-02 (Aguilar et al., 2019
(2019PhRvL.122j1101A, )), CALET (Adriani et al., 2017 (Adriani17, )), DAMPE
(Ambrosi et al., 2017 (2017Natur.552…63D, )), Fermi-LAT (Abdollahi et al.,
2017 (2017PhRvD..95h2007A, )) and H.E.S.S. (Kerszberg et al., 2017
(HESS_electrons, )), together with predictions from Reccia et al., 2019 and
Fornieri et al., 2020 (2019PhRvD..99j3022R, ; 2020JCAP…02..009F, ), and the
flux inferred by Abeysekara et al., 2017 HAWC_Science . Errorbars represent
1-sigma statistical errors in the data points. Figure 4: Local positron
spectrum measured by AMS-02 (Aguilar et al., 2019 (2019PhRvL.122d1102A, )) and
PAMELA (Adriani et al., 2013 (2013PhRvL.111h1102A, )), together with
predictions from Evoli et al., 2021 and Manconi et al., 2020
(2021PhRvD.103h3010E, ; 2020PhRvD.102b3015M, ), and the flux inferred by
Abeysekara et al., 2017 HAWC_Science . Errorbars represent 1-sigma statistical
errors in the data points.

###  4.2 Global effects

The level of magnetic field turbulence with respect to the average in the
Milky Way is expected to increase in regions dominated by an active CR
accelerator (Fermicocoon, ), but it is difficult to find an efficient
mechanism that could explain such an increase in the turbulence in the case of
halos around pulsars, as discussed in Section 3. Assuming that the existence
of pulsar halos implies a slow diffusion coefficient, there are two scenarios
in which the existence of pulsar halos can affect the global propagation of
CRs in our Galaxy. First of all, we assume that this slow diffusion is only a
signature of regions surrounding sources of this type. In this first scenario,
if these halos cover a significant fraction of the Galaxy, they could
significantly slow down Galactic CR propagation. It is however unlikely that
they can impact the overall residence time of hadronic CRs in the Galaxy,
which is dominated by propagation in the halo (2018MNRAS.479.4526L, ;
2017PhRvD..96j3013H, ). The second scenario would be the case that the
diffusion coefficient derived from the observation of pulsar halos is a more
accurate representation of the average in the Galaxy. This may have profound
implications, like a much larger accumulated grammage by CR nuclei (see e.g.
2016PhRvD..94h3003D  for a discussion of these effects around SNRs), which is
in any case mostly determined by propagation in the disk, difficult to
reconcile with the results obtained with CR propagation codes such as GALPROP
(GALPROP, ), DRAGON2 (DRAGON2, ), PICARD (PICARD, ) or USINE (USINE, ). This
constraint must be carefully taken into account by ”swiss-cheese” like models
of diffusion in the galactic disk (Profumo2018, ).

##  5 Conclusions and Prospects

The ”Pulsar - Pulsar Wind - Pulsar Wind Nebula” concept is a successful
paradigm explaining the link between two major galactic source populations:
pulsars, compact relativistic objects, and PWNe, diffuse nonthermal structures
filled by magnetic fields and relativistic electrons. The link is realized
through the ultrarelativistic (most likely, cold) electron-positron wind with
bulk motion Lorentz factor $\gamma\sim 10^{5}-10^{6}$. The typical magnetic
field of almost all PWNe is rather modest - about $10\mu$G or even less (the
Crab Nebula is an atypical PWN; its average field of 200-300 $\mu$G is a rare
exception - note also that magnetic fields higher than $10\mu$G have been
inferred from modeling efforts, see e.g. (2011MNRAS.410..381B, )) but the
general trend of low magnetic field holds. The favorable combination of the
low magnetic field and injection of ultrarelativistic electrons at the rate
comparable to the pulsar’s spin-down luminosity, allows electrons to travel to
tens of parsecs from their acceleration sites and form large scale gamma-ray
structures which can be detected by the current ground-based instruments. In
halo-type emission, we expect spherically symmetric morphology of gamma rays
with radial distribution of electrons close to $1/r$ taken into account the
continuous injection of electrons with a constant rate over $\geq 10^{4}$
years. However, at the highest energies, the radiative losses become an
essential factor, and we should see an energy cutoff toward the outskirts of
the halo. The model-independent information about the spatial and spectral
distribution of electrons provides a unique tool for the extraction of the
diffusion coefficient characterizing the propagation of CRs in the galactic
disk. The multi-hundred TeV electrons in the interstellar magnetic fields
produce X-ray synchrotron extended sources. The detection of these objects
with angular extensions $\geq 1$ degree is challenging but feasible, for
bright gamma-ray halos, by eRosita, or planned experiments like AMEGO and
AdEPT (2019BAAS…51c.183D, ). Thus, the combined X-ray and multi-TeV
observations could provide exact independent measurements of the interstellar
magnetic field throughout the galactic disk on $\leq 100$ pc scales. Several
authors (Linden17, ; Sudoh19, ; 2020PhRvD.101j3035D, ) have computed prospects
for the number of halos that could be detected by current and future
facilities, using different assumptions. This number ranges from a few, as
currently detected, up to hundreds of halos, under the most optimistic
assumptions. It is expected that the next-generation detectors, in particular
CTA and SWGO, together with the partly completed LHAASO, will dramatically
increase the number of identified pulsar halos. Likewise the continuous
increment of the already vast Fermi-LAT dataset will increase the sensitivity
to low-surface-brightness sources, not only unveiling more Geminga-like
objects but also characterizing the spectrum in the sub-100 GeV energy range.

It would not be an exaggeration to argue that the very task alone of
exploration of these standard candles, containing direct information about the
energy budget of pulsars in the relativistic electrons, as well as about the
CR diffusion coefficient and the magnetic field strength in the interstellar
medium, would justify these ambitious ground-based projects.

## Acknowledgements

This article is the result of fruitful discussions during the 1st Workshop on
Gamma-ray Halos Around Pulsars (see https://agenda.infn.it/e/GammaHalos). We
would first and foremost thank the Scientific Organizing Committee of the
Workshop: Alison Mitchell, Stefano Profumo, Diego Torres, Hao Zhou and Roberta
Zanin; the Local Organizing Committee: Maria Isabel Bernardos, Alessandro de
Angelis, Alessia Spolon and to all the participants: Soheila Abdollahi, Amal
Abdulrahman, Fabio Acero, Felix Aharonian, Andrea Albert, Elena Amato, Miguel
Araya, Thomas Armstrong, Vardan Baghmanyan, C Baheeja, Ali Baktash, Yiwei Bao,
Monica Barnard, Ulisses Barres de Almeida, Ivana Batkovic, Mabel Bernardos
Martin, Pasquale Blasi, Pere Blay, Mischa Breuhaus, Chad Brisbois, Daniel
Burgess, Senem Çabuk, Francesca Calore, Toms Capistran Rojas, Patrizia
Caraveo, Martina Cardillo, Alberto Carramiñana, Maria Carreon Gonzalez,
Sabrina Casanova, Franca Cassol, Safa Chagren, Pauline Chambery, Tej Chand,
Songzhan Chen, Ji-Gui Cheng, Sidika Merve Colak, Heide Costantini, Roland
Crocker, Alessandro De Angelis, Eduardo de la Fuente Acosta, Emma de Oña
Wilhelmi, Agnibha De Sarkar, Justine Devin, Mattia Di Mauro, Brenda Dingus,
Fiorenza Donato, Jordan Eagle, Christopher Eckner, Kathrin Egberts, Gabriel
Emery, Anant Eungwanichayapant, Carmelo Evoli, Youssef Eweis, Kwok Lung Fan,
Kun Fang, Luis Fariña, Youliang Feng, Michele Fiori, Henrike Fleischhack,
Ottavio Fornieri, Yves Gallant, Gwenael Giacinti, Magda Gonzalez, Eric
Gotthelf, Jaziel Goulart Coelho, David Green, Isabelle Grenier, Pietro
Grespan, Marie-Helene Grondin, Yingying Guo, Nayantara Gupta, Alexander Hahn,
Hend Hamed, Ian Herzog, Jim Hinton, Bohdan Hnatyk, Werner Hofmann, Binita
Hona, Dezhi Huang, Zhiqiu Huang, Armelle Jardin-Blicq, He Jiachun, Huang
Jiankun, Gudlaugur Johannesson, Vikas Joshi, Francois Kamal Youssef, Gottfried
Kanbach, Dmitry Khangulyan, Bruno Khelifi, Shota Kisaka, Tobias Kleiner,
Jurgen Knödlseder, Dmitriy Kostunin, Anu Kundu, Michael Kuss, Paul Chong Wa
Lai, Stefan Lalkovski, Federico Lavorenti, Marianne Lemoine-Goumard, Francesco
Leone, Manuel Linares, Tim Linden, Ruoyu Liu, Sheridan Lloyd, Rubén López-
Coto, Iryna Lypova, Kelly Malone, Silvia Manconi, Vincent Marandon, Alexandre
Marcowith, Jonatan Martin, Pierrick Martin, Francesco Massaro, Razmik
Mirzoyan, Alison Mitchell, Kaya Mori, Giovanni Morlino, Reshmi Mukherjee,
Kaori Nakashima, Lara Nava, Amid Nayerhoda, Michael Newbold, Melania Nynka,
Barbara Olmi, Elena Orlando, Ziwei Ou, Maura Pilia, Fabio Pintore, Illya
Plotnikov, Troy Porter, Raul Ribeiro Prado, Elisa Prandini, Giacomo Principe,
Stefano Profumo, Habeeb Rahman K. K, Brian Reville, Chiara Righi, Gonzalo
Rodríguez Fernández, Luca Romanato, Bronek Rudak, Samar Safi-Harb, Takayuki
Saito, Andres Sandoval, Andres Scherer, Pooja Sharma, Stefano Silvestri,
Atreyee Sinha, Hugh Spackman, Alessia Spolon, Giulia Stratta, Andy Strong,
Marcel Strzys, Takahiro Sudoh, Xiaona Sun, P.H. Thomas Tam, Shuta Tanaka,
Fabrizio Tavecchio, Regis Terrier, Luigi Tibaldo, Ge Tingting, Diego Torres,
Ramiro Torres Escobedo, Michelle Tsirou, Naomi Tsuji, Antonio Tutone, Allard
Jan van Marle, Juliane van Scherpenberg, Gaia Verna, Jacco Vink, Eda Vurgun,
Sarah Maria Wagner, Jieshuang Wang, Xiaojie Wang, Jie Xia, Gabrijela
Zaharijas, Silvia Zane, Roberta Zanin, Davit Zargaryan, Darko Zaric, Jianli
Zhang, Yi Zhang, Yi Zhang, Hao Zhou.

R.L.-C. acknowledges the financial support of the European Union Horizon 2020
research and innovation program under the Marie Skłodowska-Curie grant
agreement No. 754496 - FELLINI. R.L.-C. also acknowledges the financial
support from the State Agency for Research of the Spanish MCIU through the
‘Center of Excellence Severo Ochoa’ award to the Instituto de Astrofśica de
Andalucía (SEV-2017-0709). EA acknowledges support from ASI-INAF under grant
n.2017-14-H.0) and from INAF under grants “PRIN SKA-CTA”,“INAF Mainstream
2018”and “PRIN-INAF 2019”.

Author Contributions: R. López-Coto and E. de Oña Wilhelmi coordinated the
manuscript writing. All authors meet the journal’s authorship criteria and
have reviewed, discussed, and commented on the review content.

Competing Interests: The authors declare that they have no competing
interests.

## References

  *   * (1) Weekes, T. C., et al.: Observation of TeV Gamma Rays from the Crab Nebula Using the Atmospheric Cerenkov Imaging Technique. ApJ 342, 379 (1989). https://doi.org/10.1086/167599
  * (2) Hester, J. J.: The Crab Nebula : an astrophysical chimera. ARA&A 46, 127–155 (2008). https://doi.org/10.1146/annurev.astro.45.051806.110608
  * (3) Bühler, R. et al.: The surprising Crab pulsar and its nebula: a review. Reports on Progress in Physics 77(6), 066901 (2014) arXiv:1309.7046 [astro-ph.HE]. https://doi.org/10.1088/0034-4885/77/6/066901
  * (4) Gaensler, B. M. et al.: The Evolution and Structure of Pulsar Wind Nebulae. ARA&A 44(1), 17–47 (2006) arXiv:astro-ph/0601081 [astro-ph]. https://doi.org/10.1146/annurev.astro.44.051905.092528
  * (5) Porth, O., et al.: Solution to the sigma problem of pulsar wind nebulae. MNRAS 431, 48–52 (2013) arXiv:1212.1382 [astro-ph.HE]. https://doi.org/10.1093/mnrasl/slt006
  * (6) Reynolds, S. P.: Magnetohydrodynamic Models for the Structure of Pulsar-Wind Nebulae. arXiv e-prints, 0308483 (2003) arXiv:astro-ph/0308483 [astro-ph]
  * (7) Del Zanna, L. et al.: Multidimensional Relativistic MHD Simulations of Pulsar Wind Nebulae: Dynamics and Emission 446, 215 (2017) arXiv:1703.10442 [astro-ph.HE]. https://doi.org/10.1007/978-3-319-63031-1_10
  * (8) Porth, O., et al.: Diffusion in pulsar wind nebulae: an investigation using magnetohydrodynamic and particle transport models. MNRAS 460(4), 4135–4149 (2016) arXiv:1604.03352 [astro-ph.HE]. https://doi.org/10.1093/mnras/stw1152
  * (9) Abeysekara, A. U., et al.: Extended gamma-ray sources around pulsars constrain the origin of the positron flux at Earth. Science 358(6365), 911–914 (2017) arXiv:1711.06223 [astro-ph.HE]. https://doi.org/10.1126/science.aan4880
  * (10) Berezinskii, V. S., et al.: Astrophysics of Cosmic Rays, (1990)
  * (11) Trotta, R., et al.: Constraints on Cosmic-ray Propagation Models from A Global Bayesian Analysis. ApJ 729(2), 106 (2011) arXiv:1011.0037 [astro-ph.HE]. https://doi.org/10.1088/0004-637X/729/2/106
  * (12) Giacinti, G., et al.: Halo fraction in TeV-bright pulsar wind nebulae. A&A 636, 113 (2020) arXiv:1907.12121 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/201936505
  * (13) Aharonian, F., et al.: A New Population of Very High Energy Gamma-Ray Sources in the Milky Way. Science 307(5717), 1938–1942 (2005) arXiv:astro-ph/0504380 [astro-ph]. https://doi.org/10.1126/science.1108643
  * (14) Aharonian, F., et al.: The H.E.S.S. Survey of the Inner Galaxy in Very High Energy Gamma Rays. ApJ 636(2), 777–797 (2006) arXiv:astro-ph/0510397 [astro-ph]. https://doi.org/10.1086/498013
  * (15) H. E. S. S. Collaboration, et al.: The H.E.S.S. Galactic plane survey. A&A 612, 1 (2018) arXiv:1804.02432 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/201732098
  * (16) H. E. S. S. Collaboration, et al.: The population of TeV pulsar wind nebulae in the H.E.S.S. Galactic Plane Survey. A&A 612, 2 (2018) arXiv:1702.08280 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/201629377
  * (17) Aharonian, F. A., et al.: Inverse Compton gamma radiation of faint synchrotron X-ray nebulae around pulsars. MNRAS 291(1), 162–176 (1997). https://doi.org/10.1093/mnras/291.1.162
  * (18) Aleksić, J., et al.: Discovery of TeV $\gamma$-ray emission from the pulsar wind nebula 3C 58 by MAGIC. A&A 567, 8 (2014) arXiv:1405.6074 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/201424261
  * (19) Coerver, A., et al.: Multiwavelength Investigation of Pulsar Wind Nebula DA 495 with HAWC, VERITAS, and NuSTAR. ApJ 878(2), 126 (2019) arXiv:1905.07327 [astro-ph.HE]. https://doi.org/10.3847/1538-4357/ab21d0
  * (20) H. E. S. S. Collaboration, et al.: An extreme particle accelerator in the Galactic plane: HESS J1826-130. A&A 644, 112 (2020) arXiv:2010.13101 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/202038851
  * (21) H. E. S. S. Collaboration, et al.: H.E.S.S. and Suzaku observations of the Vela X pulsar wind nebula. A&A 627, 100 (2019) arXiv:1905.07975 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/201935458
  * (22) H. E. S. S. Collaboration, et al.: Identification of HESS J1303-631 as a pulsar wind nebula through $\gamma$-ray, X-ray, and radio observations. A&A 548, 46 (2012) arXiv:1210.6513 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/201219814
  * (23) Principe, G., et al.: Energy dependent morphology of the pulsar wind nebula HESS J1825-137 with Fermi-LAT. A&A 640, 76 (2020) arXiv:2006.11177 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/202038375
  * (24) Bonnarel, F., et al.: The ALADIN interactive sky atlas. A reference tool for identification of astronomical sources. A&AS 143, 33–40 (2000). https://doi.org/%****␣main.bbl␣Line␣400␣****10.1051/aas:2000331
  * (25) Posselt, B., et al.: GEMINGA’s PUZZLING PULSAR WIND NEBULA. The Astrophysical Journal 835(1), 66 (2017). https://doi.org/10.3847/1538-4357/835/1/66
  * (26) Abdo, A. A., et al.: TeV Gamma-Ray Sources from a Survey of the Galactic Plane with Milagro. ApJ 664(2), 91–94 (2007) arXiv:0705.0707 [astro-ph]. https://doi.org/10.1086/520717
  * (27) Albert, A., et al.: 3HWC: The Third HAWC Catalog of Very-high-energy Gamma-Ray Sources. ApJ 905(1), 76 (2020) arXiv:2007.08582 [astro-ph.HE]. https://doi.org/10.3847/1538-4357/abc2d8
  * (28) Cao, Z., et al.: Ultrahigh-energy photons up to 1.4 petaelectronvolts from 12 $\gamma$-ray Galactic sources. Nature 594(7861), 33–36 (2021). https://doi.org/10.1038/s41586-021-03498-z
  * (29) Abdollahi, S., et al.: Fermi Large Area Telescope Fourth Source Catalog. ApJS 247(1), 33 (2020) arXiv:1902.10045 [astro-ph.HE]. https://doi.org/10.3847/1538-4365/ab6bcb
  * (30) Recchia, S., et al.: Do the Geminga, Monogem and PSR J0622+3749 $\gamma$ -ray halos imply slow diffusion around pulsars? Phys. Rev. D 104(12), 123017 (2021) arXiv:2106.02275 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.104.123017
  * (31) Linden, T., et al.: Using HAWC to discover invisible pulsars. Phys. Rev. D 96(10), 103016 (2017) arXiv:1703.09704 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.96.103016
  * (32) H. E. S. S. Collaboration, et al.: Particle transport within the pulsar wind nebula HESS J1825-137. A&A 621, 116 (2019) arXiv:1810.12676 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/201834335
  * (33) Khangulyan, D., et al.: On the Anomalously Large Extension of the Pulsar Wind Nebula HESS J1825-137. ApJ 860(1), 59 (2018) arXiv:1712.10161 [astro-ph.HE]. https://doi.org/10.3847/1538-4357/aac20f
  * (34) Rishbeth, H.: Radio Emission from the Vela-Puppis Region. Australian Journal of Physics 11, 550 (1958). https://doi.org/10.1071/PH580550
  * (35) Aharonian, F., et al.: First detection of a VHE gamma-ray spectral maximum from a cosmic source: HESS discovery of the Vela X nebula. A&A 448(2), 43–47 (2006) arXiv:astro-ph/0601575 [astro-ph]. https://doi.org/10.1051/0004-6361:200600014
  * (36) Tibaldo, L., et al.: Disentangling multiple high-energy emission components in the Vela X pulsar wind nebula with the Fermi Large Area Telescope. A&A 617, 78 (2018) arXiv:1806.11499 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/201833356
  * (37) Grondin, M. H., et al.: The Vela-X Pulsar Wind Nebula Revisited with Four Years of Fermi Large Area Telescope Observations. ApJ 774(2), 110 (2013) arXiv:1307.5480 [astro-ph.HE]. https://doi.org/10.1088/0004-637X/774/2/110
  * (38) Mitchell, A. (for the H. E. S. S. Collaboration): Search for extended gamma-ray emission around the Geminga pulsar with H.E.S.S. TeV Particle Astrophysics (2019). (2019)
  * (39) Flinders, A.: VERITAS Observations of the Geminga Supernova Remnant. arXiv e-prints, 1509–04224 (2015) arXiv:1509.04224 [astro-ph.HE]
  * (40) Ahnen, M. L., et al.: Search for VHE gamma-ray emission from Geminga pulsar and nebula with the MAGIC telescopes. A&A 591, 138 (2016) arXiv:1603.00730 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/201527722
  * (41) Riviere, C., et al.: HAWC detection of TeV emission near PSR B0540+23. The Astronomer’s Telegram 10941, 1 (2017)
  * (42) Brisbois, C., et al.: HAWC detection of TeV source HAWC J0635+070. The Astronomer’s Telegram 12013, 1 (2018)
  * (43) Smith, A.: A Systematic Search for TeV Halos associated with known pulsars 36, 797 (2019)
  * (44) Abeysekara, A. U., et al.: Multiple Galactic Sources with Emission Above 56 TeV Detected by HAWC. Phys. Rev. Lett. 124(2), 021102 (2020). https://doi.org/10.1103/PhysRevLett.124.021102
  * (45) Aharonian, F. et al.: Extended Very-High-Energy Gamma-Ray Emission Surrounding PSR J0622+3749 Observed by LHAASO-KM2A. Phys. Rev. Lett. 126(24), 241103 (2021) arXiv:2106.09396 [astro-ph.HE]. https://doi.org/10.1103/PhysRevLett.126.241103
  * (46) Hooper, D. et al.: Measuring the Local Diffusion Coefficient with H.E.S.S. Observations of Very High-Energy Electrons. Phys. Rev. D 98(8), 083009 (2018) arXiv:1711.07482 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.98.083009
  * (47) Di Mauro, M., et al.: Evidences of low-diffusion bubbles around Galactic pulsars. Phys. Rev. D 101(10), 103035 (2020) arXiv:1908.03216 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.101.103035
  * (48) Chen, S. (for the LHAASO Collaboration): LHAASO performance and first results on extended emission from known halos. 1st Workshop on Gamma-ray Halos Around Pulsars (2020). (2020)
  * (49) Guo, Y.: Observations of extended very-high-energy halos around Geminga and Monogem with the LHAAS0-KM2A 37, 964 (2021)
  * (50) Caraveo, P. A., et al.: Geminga’s Tails: A Pulsar Bow Shock Probing the Interstellar Medium. Science 301(5638), 1345–1348 (2003). https://doi.org/10.1126/science.1086973
  * (51) Di Mauro, M., et al.: Detection of a $\gamma$-ray halo around Geminga with the Fermi-LAT data and implications for the positron flux. Phys. Rev. D 100(12), 123015 (2019) arXiv:1903.05647 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.100.123015
  * (52) Xi, S.-Q., et al.: GeV Observations of the Extended Pulsar Wind Nebulae Constrain the Pulsar Interpretations of the Cosmic-Ray Positron Excess. ApJ 878(2), 104 (2019) arXiv:1810.10928 [astro-ph.HE]. https://doi.org/%****␣main.bbl␣Line␣800␣****10.3847/1538-4357/ab20c9
  * (53) Johnson, S. P. et al.: The pulsar B2224+65 and its jets: a two epoch X-ray analysis. MNRAS 408(2), 1216–1224 (2010) arXiv:1003.1724 [astro-ph.HE]. https://doi.org/10.1111/j.1365-2966.2010.17200.x
  * (54) Hui, C. Y., et al.: XMM-Newton Observation of PSR B2224+65 and Its Jet. ApJ 747(1), 74 (2012) arXiv:1112.5816 [astro-ph.HE]. https://doi.org/10.1088/0004-637X/747/1/74
  * (55) Pavan, L., et al.: The long helical jet of the Lighthouse nebula, IGR J11014-6103. A&A 562, 122 (2014) arXiv:1309.6792 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/201322588
  * (56) Liu, R.-Y., et al.: Constraining the Magnetic Field in the TeV Halo of Geminga with X-Ray Observations. ApJ 875(2), 149 (2019) arXiv:1904.11438 [astro-ph.HE]. https://doi.org/10.3847/1538-4357/ab125c
  * (57) Goldreich, P. et al.: Pulsar Electrodynamics. ApJ 157, 869 (1969). https://doi.org/10.1086/150119
  * (58) Sironi, L., et al.: Relativistic Shocks: Particle Acceleration and Magnetization. Space Sci. Rev. 191(1-4), 519–544 (2015) arXiv:1506.02034 [astro-ph.HE]. https://doi.org/10.1007/s11214-015-0181-8
  * (59) Sironi, L. et al.: Acceleration of Particles at the Termination Shock of a Relativistic Striped Wind. ApJ 741(1), 39 (2011) arXiv:1107.0977 [astro-ph.HE]. https://doi.org/10.1088/0004-637X/741/1/39
  * (60) Hoshino, M., et al.: Relativistic Magnetosonic Shock Waves in Synchrotron Sources: Shock Structure and Nonthermal Acceleration of Positrons. ApJ 390, 454 (1992). https://doi.org/10.1086/171296
  * (61) Amato, E. et al.: Heating and Nonthermal Particle Acceleration in Relativistic, Transverse Magnetosonic Shock Waves in Proton-Electron-Positron Plasmas. ApJ 653(1), 325–338 (2006) arXiv:astro-ph/0609034 [astro-ph]. https://doi.org/10.1086/508050
  * (62) Amato, E.: The theory of Pulsar Wind Nebulae: recent progress. arXiv e-prints, 2001–04442 (2020) arXiv:2001.04442 [astro-ph.HE]
  * (63) Olmi, B., et al.: Multi-D magnetohydrodynamic modelling of pulsar wind nebulae: recent progress and open questions. Journal of Plasma Physics 82(6), 635820601 (2016) arXiv:1610.07956 [astro-ph.HE]. https://doi.org/10.1017/S0022377816000957
  * (64) Olmi, B. et al.: Full-3D relativistic MHD simulations of bow shock pulsar wind nebulae: dynamics. MNRAS 484(4), 5755–5770 (2019) arXiv:1902.00442 [astro-ph.HE]. https://doi.org/10.1093/mnras/stz382
  * (65) Olmi, B. et al.: On the origin of jet-like features in bow shock pulsar wind nebulae. MNRAS 490(3), 3608–3615 (2019) arXiv:1910.01926 [astro-ph.HE]. https://doi.org/10.1093/mnras/stz2819
  * (66) Kulsrud, R. et al.: The Effect of Wave-Particle Interactions on the Propagation of Cosmic Rays. ApJ 156, 445 (1969). https://doi.org/10.1086/149981
  * (67) Evoli, C., et al.: Self-generated cosmic-ray confinement in TeV halos: Implications for TeV $\gamma$ -ray emission and the positron excess. Phys. Rev. D 98(6), 063017 (2018) arXiv:1807.09263 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.98.063017
  * (68) Bell, A. R.: Turbulent amplification of magnetic field and diffusive shock acceleration of cosmic rays. MNRAS 353(2), 550–558 (2004). https://doi.org/10.1111/j.1365-2966.2004.08097.x
  * (69) López-Coto, R. et al.: Constraining the properties of the magnetic turbulence in the Geminga region using HAWC $\gamma$-ray data. MNRAS 479(4), 4526–4534 (2018) arXiv:1712.04373 [astro-ph.HE]. https://doi.org/10.1093/mnras/sty1821
  * (70) Strong, A. W., et al.: Cosmic-ray propagation and interactions in the galaxy. Annual Review of Nuclear and Particle Science 57(1), 285–327 (2007) https://doi.org/10.1146/annurev.nucl.57.090506.123011. https://doi.org/10.1146/annurev.nucl.57.090506.123011
  * (71) Grenier, I. A., et al.: The Nine Lives of Cosmic Rays in Galaxies. ARA&A 53, 199–246 (2015). https://doi.org/10.1146/annurev-astro-082214-122457
  * (72) Sudoh, T., et al.: TeV halos are everywhere: Prospects for new discoveries. Phys. Rev. D 100(4), 043016 (2019) arXiv:1902.08203 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.100.043016
  * (73) Atoyan, A. M., et al.: Electrons and positrons in the galactic cosmic rays. Phys. Rev. D 52, 3265–3275 (1995). https://doi.org/10.1103/PhysRevD.52.3265
  * (74) Adriani, O., et al.: An anomalous positron abundance in cosmic rays with energies 1.5-100GeV. Nature 458, 607–609 (2009) arXiv:0810.4995. https://doi.org/10.1038/nature07942
  * (75) Ackermann, M., et al.: Measurement of Separate Cosmic-Ray Electron and Positron Spectra with the Fermi Large Area Telescope. Physical Review Letters 108(1), 011103 (2012) arXiv:1109.0521 [astro-ph.HE]. https://doi.org/10.1103/PhysRevLett.108.011103
  * (76) Aguilar, M., et al.: First Result from the Alpha Magnetic Spectrometer on the International Space Station: Precision Measurement of the Positron Fraction in Primary Cosmic Rays of 0.5-350 GeV. Physical Review Letters 110(14), 141102 (2013). https://doi.org/10.1103/PhysRevLett.110.141102
  * (77) Aharonian, F. A., et al.: High energy electrons and positrons in cosmic rays as an indicator of the existence of a nearby cosmic tevatron. A&A 294, 41–44 (1995)
  * (78) Yüksel, H., et al.: TeV Gamma Rays from Geminga and the Origin of the GeV Positron Excess. Phys. Rev. Lett. 103(5), 051101 (2009) arXiv:0810.2784 [astro-ph]. https://doi.org/10.1103/PhysRevLett.103.051101
  * (79) Gupta, N. et al.: p$\gamma$ interactions in Galactic jets as a plausible origin of the positron excess. MNRAS 441, 3122–3126 (2014) arXiv:1404.4188 [astro-ph.HE]. https://doi.org/10.1093/mnras/stu770
  * (80) Bergström, L., et al.: New positron spectral features from supersymmetric dark matter: A way to explain the PAMELA data? Phys. Rev. D 78(10), 103520 (2008) arXiv:0808.3725. https://doi.org/10.1103/PhysRevD.78.103520
  * (81) Evoli, C., et al.: Galactic factories of cosmic-ray electrons and positrons. Phys. Rev. D 103(8), 083010 (2021) arXiv:2010.11955 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.103.083010
  * (82) Lipari, P.: Interpretation of the cosmic ray positron and antiproton fluxes. Phys. Rev. D 95(6), 063009 (2017) arXiv:1608.02018 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.95.063009
  * (83) Lipari, P.: Spectral shapes of the fluxes of electrons and positrons and the average residence time of cosmic rays in the Galaxy. Phys. Rev. D 99(4), 043005 (2019) arXiv:1810.03195 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.99.043005
  * (84) Evoli, C., et al.: Signature of Energy Losses on the Cosmic Ray Electron Spectrum. Phys. Rev. Lett. 125(5), 051101 (2020) arXiv:2007.01302 [astro-ph.HE]. https://doi.org/10.1103/PhysRevLett.125.051101
  * (85) Evoli, C., et al.: Galactic factories of cosmic-ray electrons and positrons. Phys. Rev. D 103(8), 083010 (2021) arXiv:2010.11955 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.103.083010
  * (86) Manconi, S., et al.: Contribution of pulsars to cosmic-ray positrons in light of recent observation of inverse-Compton halos. Phys. Rev. D 102(2), 023015 (2020) arXiv:2001.09985 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.102.023015
  * (87) Jóhannesson, G., et al.: Cosmic-Ray Propagation in Light of the Recent Observation of Geminga. ApJ 879(2), 91 (2019) arXiv:1903.05509 [astro-ph.HE]. https://doi.org/%****␣main.bbl␣Line␣1325␣****10.3847/1538-4357/ab258e
  * (88) Hooper, D., et al.: HAWC observations strongly favor pulsar interpretations of the cosmic-ray positron excess. Phys. Rev. D 96(10), 103013 (2017) arXiv:1702.08436 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.96.103013
  * (89) Profumo, S., et al.: Lessons from HAWC pulsar wind nebulae observations: The diffusion constant is not a constant; pulsars remain the likeliest sources of the anomalous positron fraction; cosmic rays are trapped for long periods of time in pockets of inefficient diffusion. Phys. Rev. D 97(12), 123008 (2018) arXiv:1803.09731 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.97.123008
  * (90) Kerszberg, D. (for the H. E. S. S. Collaboration): The cosmic-ray electron spectrum measured with H.E.S.S. International Cosmic Ray Conference, [CRI215] (2017). (2017)
  * (91) DAMPE Collaboration, et al.: Direct detection of a break in the teraelectronvolt cosmic-ray spectrum of electrons and positrons. Nature 552(7683), 63–66 (2017) arXiv:1711.10981 [astro-ph.HE]. https://doi.org/10.1038/nature24475
  * (92) Adriani, O., et al.: Energy Spectrum of Cosmic-Ray Electron and Positron from 10 GeV to 3 TeV Observed with the Calorimetric Electron Telescope on the International Space Station. Physical Review Letters 119(18), 181101 (2017) arXiv:1712.01711 [astro-ph.HE]. https://doi.org/10.1103/PhysRevLett.119.181101
  * (93) Recchia, S., et al.: Local fading accelerator and the origin of TeV cosmic ray electrons. Phys. Rev. D 99(10), 103022 (2019) arXiv:1811.07551 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.99.103022
  * (94) Fornieri, O., et al.: Features in cosmic-ray lepton data unveil the properties of nearby cosmic accelerators. J. Cosmology Astropart. Phys 2020(2), 009 (2020) arXiv:1907.03696 [astro-ph.HE]. https://doi.org/10.1088/1475-7516/2020/02/009
  * (95) Aguilar, M., et al.: Towards Understanding the Origin of Cosmic-Ray Positrons. Phys. Rev. Lett. 122(4), 041102 (2019). https://doi.org/10.1103/PhysRevLett.122.041102
  * (96) López-Coto, R., et al.: Undiscovered Pulsar in the Local Bubble as an Explanation of the Local High Energy Cosmic Ray All-Electron Spectrum. Phys. Rev. Lett. 121(25), 251106 (2018) arXiv:1811.04123 [astro-ph.HE]. https://doi.org/10.1103/PhysRevLett.121.251106
  * (97) Manconi, S., et al.: Dipole anisotropy in cosmic electrons and positrons: inspection on local sources. J. Cosmology Astropart. Phys 1, 006 (2017) arXiv:1611.06237 [astro-ph.HE]. https://doi.org/10.1088/1475-7516/2017/01/006
  * (98) Manconi, S., et al.: Multi-messenger constraints to the local emission of cosmic-ray electrons. ArXiv e-prints (2018) arXiv:1803.01009 [astro-ph.HE]
  * (99) Aguilar, M., et al.: Towards Understanding the Origin of Cosmic-Ray Electrons.
prl 122(10), 101101 (2019). https://doi.org/10.1103/PhysRevLett.122.101101

  * (100) Abdollahi, S., et al.: Cosmic-ray electron-positron spectrum from 7 GeV to 2 TeV with the Fermi Large Area Telescope. Phys. Rev. D 95(8), 082007 (2017). https://doi.org/10.1103/PhysRevD.95.082007
  * (101) Adriani, O., et al.: Cosmic-Ray Positron Energy Spectrum Measured by PAMELA. Phys. Rev. Lett. 111(8), 081102 (2013) arXiv:1308.0133 [astro-ph.HE]. https://doi.org/10.1103/PhysRevLett.111.081102
  * (102) Ackermann, M., et al.: A Cocoon of Freshly Accelerated Cosmic Rays Detected by Fermi in the Cygnus Superbubble. Science 334(6059), 1103 (2011). https://doi.org/10.1126/science.1210311
  * (103) D’Angelo, M., et al.: Grammage of cosmic rays around Galactic supernova remnants. Phys. Rev. D 94(8), 083003 (2016) arXiv:1512.05000 [astro-ph.HE]. https://doi.org/10.1103/PhysRevD.94.083003
  * (104) Strong, A. W. et al.: Propagation of cosmic-ray nucleons in the galaxy. The Astrophysical Journal 509(1), 212 (1998)
  * (105) Evoli, C., et al.: Cosmic-ray propagation with DRAGON2: I. numerical solver and astrophysical ingredients. J. Cosmology Astropart. Phys 2017(2), 015 (2017) arXiv:1607.07886 [astro-ph.HE]. https://doi.org/10.1088/1475-7516/2017/02/015
  * (106) Kissmann, R.: PICARD: A novel code for the Galactic Cosmic Ray propagation problem. Astroparticle Physics 55, 37–50 (2014) arXiv:1401.4035 [astro-ph.HE]. https://doi.org/10.1016/j.astropartphys.2014.02.002
  * (107) Maurin, D., et al.: Cosmic Rays below Z=30 in a Diffusion Model: New Constraints on Propagation Parameters. ApJ 555(2), 585–596 (2001) arXiv:astro-ph/0101231 [astro-ph]. https://doi.org/10.1086/321496
  * (108) Bucciantini, N., et al.: Modelling spectral evolution of pulsar wind nebulae inside supernova remnants. MNRAS 410(1), 381–398 (2011) arXiv:1005.1831 [astro-ph.HE]. https://doi.org/10.1111/j.1365-2966.2010.17449.x
  * (109) Di Mauro, M., et al.: Prospects for the detection of synchrotron halos around middle-age pulsars. BAAS 51(3), 183 (2019) arXiv:1903.05699 [astro-ph.HE]