##  Appendix A Dedekind eta function its multiplier system and theta function

In this appendix we collect some relevant facts about the Dedekind eta
function $\eta(\tau)$ and the theta function $\theta(\tau)$, used in section
2. Under the modular transformation, both will pick a square root of
$c\tau+d$, and the branch of the square root needs to be specified. In the
main text we have already defined the argument of $z\in\mathbb{C}$ as
$\text{Arg}\,z\in[-\pi,\pi)$. Thus the square root for $z\in\mathbb{C}$ is

$\begin{gathered}z^{\frac{1}{2}}=\sqrt{|z|}e^{\frac{i}{2}\text{Arg}\,z}\,,\end{gathered}$
(153)

and this convention will be used throughout the discussion.

The Dedekind eta function can be written in the form of infinite products,

$\begin{gathered}\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n})\,.\end{gathered}$
(154)

Since two $\text{SL}(2,\mathbb{Z})$-matrices generate the whole group, its
modular properties can be captured by

$\begin{gathered}\eta(T\tau)=\eta(\tau+1)=e^{\frac{\pi
i}{12}}\eta(\tau)\,,\quad\eta(S\tau)=\eta\left(-\frac{1}{\tau}\right)=\sqrt{-i\tau}\eta(\tau)\,,\end{gathered}$
(155)

where

$\begin{gathered}T=\left(\begin{matrix}1&1\\\ 0&1\end{matrix}\right)\,,\quad
S=\left(\begin{matrix}0&-1\\\ 1&0\end{matrix}\right)\,.\end{gathered}$ (156)

More generally, the modular properties of $\eta(\tau)$ under
$\text{SL}(2,\mathbb{Z})$ can be written as Kohler2011

$\begin{gathered}\eta(M\tau)=\chi_{\eta}(M)(c\tau+d)^{1/2}\eta(\tau)\,\end{gathered}$
(157)

with a nontrivial multiplier system $\chi_{\eta}(M)$. Let $c$ and $d$ be
integers such that $\gcd(c,d)=1$, $d$ is odd and $c\neq 0$. Let
$\text{sgn}(x)=\frac{x}{|x|}$ be the sign of a real number $x\neq 0$. Then

$\begin{gathered}\left(\frac{c}{d}\right)^{*}=\left(\frac{c}{|d|}\right)\,,\quad\text{and}\quad\left(\frac{c}{d}\right)_{*}=\left(\frac{c}{|d|}\right)\cdot(-1)^{\frac{1}{4}\left(\text{sgn}(c)-1\right)(\text{sgn}(d)-1)}\,,\end{gathered}$
(158)

where $\left(\frac{c}{d}\right)$ is the Legendre symbol and we set

$\begin{gathered}\left(\frac{0}{1}\right)^{*}=\left(\frac{0}{-1}\right)^{*}=1\,,\quad\left(\frac{0}{1}\right)_{*}=1\,,\quad\left(\frac{0}{-1}\right)_{*}=-1\,.\end{gathered}$
(159)

For arbitrary element in $M=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\text{SL}(2;\mathbb{Z})$, the multiplier system
of the Dedekind eta function is given by

$\displaystyle\chi_{\eta}(M)=\left(\frac{d}{c}\right)^{*}q\left(\frac{1}{24}\left[(a+d)c-bd(c^{2}-1)-3c\right]\right)$
if $c$ is odd, (160)
$\displaystyle\chi_{\eta}(M)=\left(\frac{c}{d}\right)_{*}q\left(\frac{1}{24}\left[(a+d)c-bd(c^{2}-1)+3d-3-3cd\right]\right)$
if $c$ is even,

where $q(z)=e^{2\pi iz}$. It should be noted that $\chi_{\eta}(M)$ cannot form
the homomorphism from $\text{SL}(2;\mathbb{Z})$ to $\text{U}(1)$. Since the
$S=\left(\begin{smallmatrix}0&-1\\\ 1&0\end{smallmatrix}\right)$
transformation satisfy $S^{2}=-\mathbb{1}$, we have

$\begin{gathered}\eta(\tau)=\eta((-\mathbb{1})\tau)=\chi_{\eta}(-\mathbb{1})(-1)^{1/2}\eta(\tau)=\chi_{\eta}(-\mathbb{1})(-i)\eta(\tau)\\\
\Rightarrow\chi_{\eta}(-\mathbb{1})=i\neq\chi_{\eta}(S)^{2}=-i\,,\end{gathered}$
(161)

In the main text, the congruence subgroup $\Gamma_{0}(4)$ of
$\text{SL}(2;\mathbb{Z})$ was introduced:

$\begin{gathered}\Gamma_{0}(N)=\left\\{\left.\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in\text{SL}(2,\mathbb{Z})\right|c\equiv 0\mod
N\right\\}\end{gathered}$ (162)

for any positive integer $N$. Within the congruence subgroup $\Gamma_{0}(4)$
the weight $1/2$ modular form is well-defined Shimrua1973 ; JPSerre1977 . For
element $M\in\Gamma_{0}(4)$, the transformation of the square of the theta
function, given as

$\begin{gathered}\theta(\tau)=\sum_{n\in\mathbb{Z}}q^{n^{2}}=1+2q^{2}+2q^{4}+\ldots\,,\end{gathered}$
(163)

takes the form

$\begin{gathered}\theta^{2}(M\tau)=\left(\frac{-1}{d}\right)(c\tau+d)\theta^{2}(\tau)\,,\end{gathered}$
(164)

where $\left(\frac{-1}{d}\right)$ denotes the Legendre symbol,
$\left(\frac{-1}{d}\right)=(-1)^{\frac{d-1}{2}}$.

##  Appendix B Orthogonal modular forms

Some necessary properties of orthogonal modular forms were reviewed in
subsection 3.3. In order to make the paper more self-contained, more
background material is collected in this Appendix. Definitions and theorems
are given without proofs. Our presentation follows closely
bruinier2002borcherds , which can be consulted for detailed explanations.

Throughout this section, as in the main text, we denote by $L$ an even lattice
of signature $(2,l)$ and assume $l\geq 3$.

###  B.1 The Weil representation

We denote the complex upper-half plane
$\mathbb{H}=\\{\tau\in\mathbb{C};\,\imaginary\tau>0\\}$. $\tau$ is the
standard variable on $\mathbb{H}$ and we use $x$ and $y$ for its real and
imaginary parts respectively ($\tau=x+iy$). For $z\in\mathbb{C}$ we define
$e(z)=e^{2\pi iz}$ and denote by $\sqrt{z}=z^{1/2}$ the principal branch of
the square root. For arbitrary $b\in\mathbb{C}$, we define
$z^{b}=e^{b\text{Ln}\,z}$ where $\text{Ln}\,z$ denotes the principal branch of
the logarithm. We denote by $\text{Mp}(2;\mathbb{R})$ the metapletic group,
i.e. the double cover of group $\text{SL}(2;\mathbb{R})$, realized by the two
choices of holomorphic square roots of $\tau\rightarrow c\tau+d$ for arbitrary
element $M\in\text{SL}(2;\mathbb{R})$,

$\begin{gathered}M=\left(\begin{matrix}a&b\\\ c&d\end{matrix}\right),\quad
a,b,c,d\in\mathbb{R},\quad\det M=ad-bc=1\,.\end{gathered}$ (165)

Any element in $\text{Mp}(2;\mathbb{R})$ can be written as
$\left(M,\phi(\tau)\right)$ where $M\in\text{SL}(2,\mathbb{R})$ and
$\phi(\tau)^{2}=c\tau+d$. The multiplication in the group
$\text{Mp}(2;\mathbb{R})$ is defined as

$\begin{gathered}\left(M_{1},\phi_{1}(\tau)\right)\left(M_{2},\phi_{2}(\tau)\right)=\left(M_{1}M_{2},\phi_{1}(M_{2}\tau)\phi_{2}(\tau)\right)\,,\end{gathered}$
(166)

where $M\tau=(a\tau+b)/(c\tau+d)$ denotes the usual action of
$\text{SL}(2;\mathbb{R})$. By fixing the choice $\phi(\tau)=\sqrt{c\tau+d}$,
we actually define a locally isomorphic embedding
$\text{SL}(2;\mathbb{R})\xhookrightarrow{}\text{Mp}(2;\mathbb{R})$

$\begin{gathered}M\mapsto\widetilde{M}=\left(M,\sqrt{c\tau+d}\right).\end{gathered}$
(167)

$\text{Mp}(2;\mathbb{Z})$ is generated by two elements $T,S$

$\begin{gathered}T=\left(\left(\begin{matrix}1&1\\\
0&1\end{matrix}\right),1\right)\,,\quad S=\left(\left(\begin{matrix}0&-1\\\
1&0\end{matrix}\right),\sqrt{\tau}\right)\,.\end{gathered}$ (168)

One has the relation $S^{2}=(ST)^{3}=Z$, where

$\begin{gathered}Z=\left(\left(\begin{matrix}-1&0\\\
0&-1\end{matrix}\right),i\right)\end{gathered}$ (169)

is the standard generator of the center of $\text{Mp}(2;\mathbb{Z})$. For
convenience we define $\Gamma_{1}=\text{SL}(2;\mathbb{Z})$,

$\begin{gathered}\Gamma_{\infty}=\left\\{\left(\begin{matrix}1&n\\\
0&1\end{matrix}\right);\,n\in\mathbb{Z}\right\\}\leq\Gamma_{1}\,,\\\
\widetilde{\Gamma}_{\infty}=\langle
T\rangle=\left\\{\left(\left(\begin{matrix}1&n\\\
0&1\end{matrix}\right),1\right);\,n\in\mathbb{Z}\right\\}\,,\end{gathered}$
(170)

where $\langle T\rangle$ denotes the group generated by $T$.

Suppose $L$ is an even lattice equipped with a symmetric $\mathbb{Z}$-valued
bilinear form $\left(z_{1},z_{2}\right)$ for $z_{1},z_{2}\in L$ and the
associated quadratic form $q(z)=(z,z)/2$ is integer for arbitrary $z\in L$. We
denote by $L^{\prime}$ the dual lattice. The quotient $L^{\prime}/L$ is a
finite Abelian group, the so-called discriminant group. Since the quadratic
form can be extended to the dual lattice, we can define the quadratic form on
$L^{\prime}/L$, which takes values in $\mathbb{Q}/\mathbb{Z}$. There is a
unitary representation $\varrho$ of $\text{Mp}(2;\mathbb{Z})$ on the algebra
$\mathbb{C}[L^{\prime}/L]$. If we denote the standard basis of
$\mathbb{C}[L^{\prime}/L]$ by $\left\\{\mathfrak{e}_{\gamma}|\gamma\in
L^{\prime}/L\right\\}$, then $\varrho$ can be defined by the action of the
generators $S,T\in\text{Mp}(2;\mathbb{Z})$ as follows

$\displaystyle\varrho(T)\mathfrak{e}_{\gamma}=e(q(\gamma))\,,$ (171)
$\displaystyle\varrho(S)\mathfrak{e}_{\gamma}=\frac{\sqrt{i}^{b^{-}-b^{+}}}{\sqrt{|L^{\prime}/L|}}\sum_{\delta\in
L^{\prime}/L}e(-(\gamma,\delta))\mathfrak{e}_{\delta}\,.$

This is the so-called Weil representation. Based on the relation $S^{2}=Z$, we
have

$\displaystyle\varrho(Z)\mathfrak{e}_{\gamma}$
$\displaystyle=\frac{i^{b^{-}-b^{+}}}{|L^{\prime}/L|}\sum_{\delta,\lambda\in
L^{\prime}/L}e(-(\gamma,\delta))e(-(\delta,\lambda))\mathfrak{e}_{\lambda}$
(172) $\displaystyle=i^{b^{-}-b^{+}}\mathfrak{e}_{-\gamma}\,.$

We denote by $\langle\cdot,\cdot\rangle$ the standard product of
$\mathbb{C}[L^{\prime}/L]$, i.e.

$\begin{gathered}\left\langle\sum_{\gamma\in
L^{\prime}/L}\lambda_{\gamma}\mathfrak{e}_{\gamma},\sum_{\gamma\in
L^{\prime}/L}\mu_{\gamma}\mathfrak{e}_{\gamma}\right\rangle=\sum_{\gamma\in
L^{\prime}/L}\lambda_{\gamma}\bar{\mu}_{\gamma}\,.\end{gathered}$ (173)

For $\gamma,\delta\in L^{\prime}/L$, we can define the representation matrix
element
$\varrho_{\gamma\delta}(M,\phi)=\left\langle\varrho(M,\phi)\mathfrak{e}_{\delta},\mathfrak{e}_{\gamma}\right\rangle$.

###  B.2 Vector-valued modular forms

######  Definition B.1

(Petersson slash operator) Let $\kappa\in\frac{1}{2}\mathbb{Z}$ and $f$ be a
$\mathbb{C}[L^{\prime}/L]$-valued function on $\mathbb{H}$. For
$(M,\phi)\in\text{Mp}(2;\mathbb{Z})$ we define the Petersson slash operator
$|_{\kappa}(M,\phi)$ by

$\begin{gathered}\left(f|_{\kappa}(M,\phi)\right)(\tau)=\phi(\tau)^{-2\kappa}\varrho(M,\phi)^{-1}f(M\tau)\,.\end{gathered}$
(174)

We denote by $\varrho^{*}$ the dual representation of $\varrho$. If we think
of $\varrho(M,\phi)$ as a matrix with entries in $\mathbb{C}$, then
$\varrho^{*}(M,\phi)$ is simply the complex conjugate of $\varrho(M,\phi)$.
The "dual operation" of $\text{Mp}(2;\mathbb{Z})$ on functions
$f:\mathbb{H}\rightarrow\mathbb{C}[L^{\prime}/L]$ is given by

$\begin{gathered}\left(f|^{*}_{\kappa}(M,\phi)\right)(\tau)=\phi(\tau)^{-2\kappa}\varrho^{*}(M,\phi)^{-1}f(M\tau)\,.\end{gathered}$
(175)

If we assume that the function
$f:\mathbb{H}\rightarrow\mathbb{C}[L^{\prime}/L]$ is a holomorphic function
which is invariant under the $|_{\kappa}^{*}$ operation of
$T\in\text{Mp}(2;\mathbb{Z})$. Since $f$ can be expanded by the basis
$\mathfrak{e}_{\gamma}$ of $L^{\prime}/L$, we have
$f=\sum_{\gamma}f_{\gamma}\mathfrak{e}\gamma$. The invariance is satisfied if
and only if

$\begin{gathered}f_{\gamma}(\tau)=f_{\gamma}|_{\kappa}^{*}T(\tau)=e^{*}(q(\gamma))^{-1}f_{\gamma}(\tau+1)\\\
\Leftrightarrow\quad
e(q(\gamma)\tau)f_{\gamma}(\tau)=e\left(q(\gamma)(\tau+1)\right)f_{\gamma}(\tau+1)\,,\end{gathered}$
(176)

which means the invariance of $f$ under $T$ implies that the function
$e(q(\gamma)\tau)f_{\gamma}(\tau)$ is periodic with period $1$. We can
directly Fourier expand $f$ by

$\begin{gathered}f(\tau)e(q(\gamma)\tau)=\sum_{\gamma\in
L^{\prime}/L}\sum_{n\in\mathbb{Z}}c(\gamma,n)e(n\tau)\mathfrak{e}_{\gamma}\,.\end{gathered}$
(177)

To have a compact expression, we define
$\mathfrak{e}_{\gamma}(n\tau)=e(n\tau)\mathfrak{e}_{\gamma}$ and write

$\begin{gathered}f(\tau)=\sum_{\gamma\in
L^{\prime}/L}\sum_{n\in\mathbb{Z}-q(\gamma)}c(\gamma,n)\mathfrak{e}_{\gamma}(n\tau)\,,\end{gathered}$
(178)

with Fourier coefficients

$\begin{gathered}c(\gamma,n)=\int_{0}^{1}\langle
f(\tau),\mathfrak{e}_{\gamma}(n\bar{\tau})\rangle dx\,.\end{gathered}$ (179)

######  Definition B.2

(holomorphic modular form of dual Weil representation) Let
$\kappa\in\frac{1}{2}\mathbb{Z}$. A function
$f:\mathbb{H}\rightarrow\mathbb{C}[L]$ is called a modular form of weight
$\kappa$ with respect to $\varrho^{*}$ and $\text{Mp}(2;\mathbb{Z})$ if

  1. i)

$f|_{\kappa}^{*}(M,\phi)=f$ for all $(M,\phi)\in\text{Mp}(2;\mathbb{Z})$,

  2. ii)

$f$ is holomorphic on $\mathbb{H}$,

  3. iii)

$f$ is holomorphic at the cusp $\infty$. If $c(\gamma,0)\equiv 0$, $f$ is
called a cusp form.

The condition $(iii)$ requires $f$ has a Fourier expansion of the form

$\begin{gathered}f(\tau)=\sum_{\gamma\in
L^{\prime}/L}\sum_{\begin{subarray}{c}n\in\mathbb{Z}-q(\gamma)\\\ n\geq
0\end{subarray}}c(\gamma,n)\mathfrak{e}_{\gamma}(n\tau)\,.\end{gathered}$
(180)

The $\mathbb{C}$-vector space of modular forms of weight $\kappa$ with respect
to $\varrho^{*}$ and $\text{Mp}(2;\mathbb{Z})$ is denoted by $M_{\kappa,L}$
and the subspace of cusp forms is denoted by $S_{\kappa,L}$ . Similar to the
usual complex valued modular form of $\text{SL}(2;\mathbb{Z})$, the linear
space $M_{\kappa,L}$ is finite dimensional.

###  B.3 Nearly holomorphic modular forms

######  Definition B.3

(nearly holomorphic modular form) A function
$f:\mathbb{H}\rightarrow\mathbb{C}[L]$ is called a nearly holomorphic modular
form of weight $k$ (with respect to $\varrho$ and $\text{Mp}(2;\mathbb{Z})$),
if

  1. i)

$f|_{k}(M,\phi)=f$ for all $(M,\phi)\in\text{Mp}(2;\mathbb{Z})$,

  2. ii)

$f$ is holomorphic on $\mathbb{H}$,

  3. iii)

$f$ has a pole in $\infty$, i.e. $f$ has a Fourier expansion of the form

$\begin{gathered}f(\tau)=\sum_{\gamma\in
L^{\prime}/L}\sum_{\begin{subarray}{c}n\in\mathbb{Z}+q(\gamma)\\\
n\gg-\infty\end{subarray}}c(\gamma,n)\mathfrak{e}_{\gamma}(n\tau)\,.\end{gathered}$
(181)

The space of these nearly holomorphic modular forms is denoted by
$M_{k,L}^{!}$. The summation $n\gg-\infty$ indicates that there exists a
finite negative number $n_{0}$ such that all $n\geq n_{0}$. This condition
implies that the pole at the cusp ($\infty$) has finite order. The Fourier
polynomial

$\begin{gathered}\sum_{\gamma\in
L^{\prime}/L}\sum_{\begin{subarray}{c}n\in\mathbb{Z}+q(\gamma)\\\
n<0\end{subarray}}c(\gamma,n)\mathfrak{e}_{\gamma}(n\tau)\end{gathered}$ (182)

is called the principal part of $f$.

As shown in bruinier2002borcherds , the space of nearly holomorphic modular
form is generated by the Poincaré series, thus is finite dimensional. The
principal part should satisfy the Theorem B.3.

###  B.4 Modular forms on generalized upper-half plane

The orthogonal modular forms and Borcherds product were introduced in the main
text. Recall the definition of $j(M,Z)$ in (LABEL:eq:sec4_action_M4)). More
generally we can rewrite it as

$\begin{gathered}j(M,Z)=\left(MZ_{L},z\right)\,,\end{gathered}$ (183)

where $z=\left(1,0,\ldots,0\right)^{T}$ is the $l+2$ vector and
$Z_{L}=(-q_{0}(Z),Z,1)^{T}$. Suppose $L$ is an even lattice of signature
$(2,l)$ and $V=L\otimes\mathbb{R}$. The function $j(M,Z)$ on
$\text{O}^{+}(V)\times\mathbb{H}_{l}$ is an automorphy factor for
$\text{O}^{+}(V)$, i.e. it satisfies the cocycle relation

$\begin{gathered}j(M_{1}M_{2},Z)=j(M_{1},M_{2}\langle
Z\rangle)j(M_{2},Z)\,.\end{gathered}$ (184)

For an arbitrary $a\in\mathbb{C}$, we have already specified
$\text{Arg}\,a\in[-\pi,\pi)$, which is the principal value of argument of $a$.
We denote by Ln the logarithm of the principal branch, which is defined as
$\text{Ln}\,a=\ln|a|+i\text{Arg}\,a$. For an arbitrary $a,b\in\mathbb{C}$, we
define $a^{b}=e^{b\,\text{Ln}\,a}$. Let $r\in\mathbb{Q}$, if
$M\in\text{O}^{+}(V)$ and $Z\in\mathbb{H}_{l}$, then
$j(M,Z)^{r}=e^{r\text{Ln}\,j(M,Z)}$. There exists a map $w_{r}$ from
$\text{O}^{+}(V)\times\text{O}^{+}(V)$ to the set of roots of unity (of order
bounded by the denominator of $r$) such that

$\begin{gathered}j(M_{1}M_{2},Z)^{r}=w_{r}(M_{1},M_{2})j(M_{1},M_{2}\langle
Z\rangle)^{r}j(M_{2},Z)^{r}\,.\end{gathered}$ (185)

######  Definition B.4

(multiplier system) Let $\Gamma\leq\text{O}^{+}(V)$ be a subgroup and
$r\in\mathbb{Q}$ as above. By a multiplier system of weight $r$ we mean a map

$\begin{gathered}\chi:\Gamma\longrightarrow
S^{1}=\left\\{t\in\mathbb{C}|\,|t|=1\right\\}\end{gathered}$ (186)

satisfying

$\begin{gathered}\chi(M_{1}M_{2})=w_{r}(M_{1},M_{2})\chi(M_{1})\chi(M_{2})\,,\quad
M_{1},M_{2}\in\Gamma\,.\end{gathered}$ (187)

If $r\in\mathbb{Z}$, then $\chi$ is actually a character of $\Gamma$, then
$\chi(M)j(M,Z)^{r}$ is a cocycle of $\Gamma$.

######  Definition B.5

(modular form on generalized upper-half plane) Let $\Gamma\leq\Gamma(L)$ be a
subgroup of finite index and $\chi$ a multiplier system for $\Gamma$ of weight
$r\in\mathbb{Q}$. A meromorphic function $F$ on $\mathbb{H}_{l}$ is called a
meromorphic modular from of weight $r$ and multiplier system $\chi$ with
respect to $\Gamma$, if

$\begin{gathered}\Psi(M\langle
Z\rangle)=\chi(M)j(M,Z)^{r}\Psi(Z)\end{gathered}$ (188)

for all $M\in\Gamma$. If $\Psi$ is even holomorphic on $\mathbb{H}_{l}$ then
it is called a holomorphic modular form.

The Borcherds product can lift a nearly holomorphic modular form
$f(\tau)=\sum_{\gamma\in
L^{\prime}/L}f_{\gamma}\mathfrak{e}_{\gamma}:\mathbb{H}\rightarrow\mathbb{C}[L^{\prime}/L]$
(see Definition B.3) of weight $1-l/2$ with Fourier expansion

$\begin{gathered}f(\tau)=\sum_{\gamma\in
L^{\prime}/L}\sum_{n\in\mathbb{Z}+q(\gamma)}c(\gamma,n)\mathfrak{e}_{\gamma}(n\tau)\,,\end{gathered}$
(189)

to the meromorphic function $\Psi(Z):\mathbb{H}\rightarrow\mathbb{C}$ of
weight $c(0,0)/2$. The precise theorem is stated as follows.

######  Theorem B.1

(Theorem 13.3 (1) in Borcherds1998  or Theorem 3.22 (i) in
bruinier2002borcherds ) Let $L$ be an even lattice of signature $(2,l)$ with
$l\geq 3$, and $z\in L$ a primitive isotropic vector. Let $z^{\prime}\in
L^{\prime}$ and $K=L\cap z^{\perp}\cap{z^{\prime}}^{\perp}$. Moreover, assume
that $K$ also contains an isotropic vector. Let $f$ be a nearly holomorphic
modular form of weight $k=1-l/2$ whose Fourier coefficients $c(\gamma,n)$ are
integral for $n<0$. Then

$\begin{gathered}\Psi(Z)=\prod_{\beta\in
L^{\prime}/L}\prod_{\begin{subarray}{c}m\in\mathbb{Z}+q(\beta)\\\
m<0\end{subarray}}\Psi_{\beta,m}(Z)^{c(\beta,m)/2}\end{gathered}$ (190)

is a meromorphic function on $\mathbb{H}_{l}$ of (rational) weight $c(0,0)/2$
for the modular group $\Gamma(L)$ with some multiplier systems $\chi$ of
finite order. If $c(0,0)\in 2\mathbb{Z}$, then $\chi$ is the character of
group $\Gamma(L)$.

For functions $\Psi_{\beta,m}(Z)$ see Definition 3.14 in bruinier2002borcherds
.

We can turn to the zeros and poles of $\Psi(Z)$. A nowhere-vanishing
holomorphic modular forms $\Psi(Z)$ obtained through Borcherds product cannot
exist since there is no input nonzero holomorphic modular form $f(\tau)$ of
negative weight $1-l/2$. Before determining the positions of poles and zeroes,
it is necessary to explain the concept of rational quadratic divisors (Heegner
divisors).

Let $z\in L$ be a primitive norm $0$ vector, $z^{\prime}\in L^{\prime}$ with
$(z,z^{\prime})=1$. Let $N$ be unique positive integer with such that
$(z,L)=N\mathbb{Z}$. Then we have $z/N\in L^{\prime}$. Denote by $K$ the
lattice

$\begin{gathered}K=L\cap z^{\perp}\cap z^{\prime\perp}\,.\end{gathered}$ (191)

$K$ has signature $(b^{+}-1,b^{-}-1)=(1,l-1)$. For an arbitrary vector $n\in
V=L\otimes\mathbb{R}$, $n_{K}$ denotes the orthogonal projection $n$ to
$K\otimes\mathbb{R}$ and

$\begin{gathered}n_{K}=n-(n,z)z^{\prime}+(n,z)(z^{\prime},z^{\prime})z-(n,z^{\prime})z\,.\end{gathered}$
(192)

If $n\in L^{\prime}$, then $n_{K}$ lies in the dual lattice $K^{\prime}$ of
$K$. Let $\zeta\in L$ be a lattice vector with $(\zeta,z)=N$. Let $n\in L$,
then the vector

$\begin{gathered}\tilde{n}=n-(n,z/N)\zeta-(n,z^{\prime})z+(n,z/N)(\zeta,z^{\prime})z\end{gathered}$
(193)

lies in $L$ and easy to verify that $\tilde{n}\perp z$ and $\tilde{n}\perp
z^{\prime}$. Hence $\tilde{n}\in K$ and each element $n\in L$ can be uniquely
decomposed in this way, or equivalently,
$L=K\oplus\mathbb{Z}\zeta\oplus\mathbb{Z}z$. Now let $\lambda\in L^{\prime}$
be a vector of negative norm, i.e. $q(\lambda)<0$. Then the orthogonal
complement $\lambda^{\perp}\subset L\otimes\mathbb{R}$ is a rational quadratic
space of type $(2,l-1)$. With these settings we can define the rational
quadratic divisors

######  Definition B.6

(rational quadratic divisor or Heegner divisor) Let $\lambda\in L^{\prime}$ be
a vector of negative norm $m$, we set

$\begin{gathered}H_{\lambda}=\left\\{[Z_{L}]\in\mathcal{K}^{+}|\,(Z_{L},\lambda)=0\right\\}\,.\end{gathered}$
(194)

Moreover, due to the decomposition $Z_{L}=(-q(Z)-q(z^{\prime}))z+Z+z^{\prime}$
(recall the equation (40)) and $\lambda=bz+\lambda_{K}+az^{\prime}$, expanding
the inner product $(Z_{L},\lambda)$ yields

$\begin{gathered}H_{\lambda}\cong\left\\{Z\in\mathbb{H}_{l}|\,aq(Z)-(Z,\lambda_{K})-aq(z^{\prime})-b=0\right\\}\,\end{gathered}$
(195)

in coordinates on $\mathbb{H}_{l}$. This set defines a prime divisor on
$\mathbb{H}_{l}$. Suppose $\beta\in L^{\prime}/L$ and $m$ is a negative
rational number; the sum

$\begin{gathered}H(\beta,m)=\sum_{\begin{subarray}{c}\lambda\in\beta+L\\\
q(\lambda)=m\end{subarray}}H_{\lambda}\end{gathered}$ (196)

is called the rational quadratic divisor (or Heegner divisor) of discriminant
$(\beta,m)$, which is a $\Gamma(L)$-invariant divisor on $\mathbb{H}_{l}$.
When $\beta=0$, we usually denote $H(m)=\frac{1}{2}H(0,m)$.

This definition is suitable for lattices of signature $(2,l)$ with arbitrary
Gram matrix. If we specify the Gram matrix of $L=\Pi_{1,1}\oplus L_{0}$ as
defined in the equation (42) and the vector $z,z^{\prime}$, equivalently we
have

$\begin{gathered}H_{\lambda}=\left\\{Z\in\mathbb{H}_{l}|\,aq_{0}(Z)-(Z,\lambda_{K})_{0}-b=0\right\\}\,,\end{gathered}$
(197)

where the subscript emphasizes that the inner product is associated with the
quadratic form $S_{0}$. With this definition we can describe the position of
the zeros and poles by the following theorem.

######  Theorem B.2

(Theorem 13.3 (2) in Borcherds1998  or Theorem 3.22 (ii) in
bruinier2002borcherds ) The zeros and poles of $\Psi(Z)$ lies on the divisor
of $\Psi(Z)$ on $\mathbb{H}_{l}$, which is the linear combinations of Heegner
divisors determined by the principal part of the nearly holomorphic modular
form $f$

$\begin{gathered}\left(\Psi\right)=\frac{1}{2}\sum_{\beta\in
L^{\prime}/L}\sum_{\begin{subarray}{c}m\in\mathbb{Z}+q(\beta)\\\
m<0\end{subarray}}c(\beta,m)H(\beta,m)\,.\end{gathered}$ (198)

The multiplicities of $H(\beta,m)$ are $2$, if $2\beta=0$ in $L^{\prime}/L$,
and $1$, if $2\beta\neq 0$ in $L^{\prime}/L$.

As we saw from the above theorems, the properties of the Borcherds product
$\Psi(Z)$ are completely captured by the nearly holomorphic modular form
$f(\tau)$, in particular by the principal part of $f(\tau)$:

$\begin{gathered}\sum_{\gamma\in
L^{\prime}/L}\sum_{\begin{subarray}{c}n\in\mathbb{Z}+q(\gamma)\\\
n<0\end{subarray}}c(\gamma,n)\mathfrak{e}_{\gamma}(n\tau)\,.\end{gathered}$
(199)

Pairing the form $f(\tau)$ with a vector valued cusp form of weight $1+l/2$
for the dual Weil representation (see Definition B.2) gives a meromorphic
elliptic modular form of weight $2$ for $\text{SL}(2;\mathbb{Z})$, hence its
constant term must vanish by the residue theorem (no nonzero
$\text{SL}(2;\mathbb{Z})$ modular form of weight $2$) and this gives the
conditions on the principal part on $f$, stated as the following theorem. By
setting $\kappa=1+l/2$ and denoting the space of the vector valued modular
cusp form of weight $\kappa$ with respect to lattice $L$ as $S_{\kappa,L}$, we
have

######  Theorem B.3

(Theorem 1.17 in bruinier2002borcherds ) There exists a nearly holomorphic
modular form $f\in M_{k,L}^{!}$ with prescribed principal part

$\begin{gathered}\sum_{\beta\in
L^{\prime}/L}\sum_{\begin{subarray}{c}m\in\mathbb{Z}+q_{1}(\beta)\\\
m<0\end{subarray}}c(\beta,m)\mathfrak{e}_{\beta}(m\tau)\end{gathered}$ (200)

($c(\beta,m)\in\mathbb{C}$ with $c(\beta,m)=c(-\beta,m)$), if and only if the
functional

$\begin{gathered}\sum_{\beta\in
L^{\prime}/L}\sum_{\begin{subarray}{c}m\in\mathbb{Z}+q_{1}(\beta)\\\
m<0\end{subarray}}c(\beta,m)a_{\beta,-m},\end{gathered}$ (201)

equals zero in $S_{\kappa,L}^{*}$. For $\gamma\in D(L)$ and
$n\in\mathbb{Z}-q(\gamma)$ with $n>0$,
$a_{\gamma,n}:S_{\kappa,L}\rightarrow\mathbb{C}$ denote the functional in the
dual space $S_{\kappa,L}^{*}$ of $S_{\kappa,L}$ which maps a cusp form $f$ to
its $(\gamma,n)$-th Fourier coefficient $a_{\gamma,n}(f)$.

Obviously this imposes non-trivial condition on the principal part of the
nearly holomorphic modular form $f$.

###  B.5 Character of the lattice

If the weight of the modular form is integer, which is the case of interest,
the multiplier system is actually the character of the modular group
$\Gamma(L)$, or the character of the lattice $L$. This forms a homomorphism
from the modular group to $\text{U}(1)$. From the well-known Pontryagin
duality, the abelianisation $G^{ab}=G/[G,G]$ of the group $G$ is isomorphic to
the character group $\text{Hom}(G,\mathbb{C}^{\times})$. Thus to obtain the
character we need to consider the abelianisation of the modular group
$\Gamma(L)$. As discussed in section 5, the anomaly cancellation imposes the
triviality of the character for the admissible lattices. To the best of our
knowledge, the sufficient and necessary conditions for a lattice of signature
$(2,l)$ to have trivial characters are not known. A sufficient condition is
known:

######  Theorem B.4

(Theorem 1.7 in GRITSENKO2009JA ) Let $L$ be an even integral lattice
containing at least two hyperbolic planes ($\Pi_{1,1}$), such that
$\text{rank}_{3}(L)\geq 5$262626For any prime $p$ the $p$-rank of L, denoted
by $\text{rank}_{p}(L)$, is the maximal rank of the sublattices $M$ such that
$\det(M)$ is coprime to $p$. and $\text{rank}_{2}(L)\geq 6$, then the
$\Gamma(L)^{ab}\cong\mathbb{Z}/2\mathbb{Z}$ and $\text{S}\Gamma(L)^{ab}$ is
trivial, where $\text{S}\Gamma(L)$ is the modular group intersect with the
special orthogonal group of lattice $L$, i.e.
$\text{S}\Gamma(L):=\Gamma(L)\cap\text{SO}(L)$.

An immediate corollary is that if $L=\Pi_{1,1}\oplus\Pi_{1,1}\oplus\hat{L}$
and $\hat{L}$ contains a sublattice isomorphic to $A_{2}$, it satisfies the
so-called Kneser conditions Kneser1984 ; GRITSENKO2009JA  and the character
for group $S\Gamma(L)$ is trivial. Notably, if the lattice
$L=\Pi_{1,1}\oplus\Pi_{1,1}\oplus\hat{L}$ is an even unimodular lattice of
rank at least $6$, we have the same conclusion that the
$\Gamma(L)^{ab}\cong\mathbb{Z}/2\mathbb{Z}$ and $\text{S}\Gamma(L)^{ab}$ is
trivial.

##  Appendix C Alternative $l=2$ counterterm from Hilbert modular forms

In this appendix, we will provide a brief overview of the alternative
construction for the case $l=2$ mentioned at the end of section 6, which
follows a similar path to the procedure for $l\geq 3$ cases. For this we need
to introduce Hilbert modular forms, following closely to these good references
Bruinier2003 ; Mayer2007 ; bruinier20081 .

Let $\mathbb{K}=\mathbb{Q}(\sqrt{p})$, $p\in\mathbb{N}$, $p>1$ squarefree, be
a real quadratic number field with the ring of integers and discriminant

$\begin{gathered}\mathcal{O}_{\mathbb{K}}=\mathbb{Z}+\mathbb{Z}\omega_{\mathbb{K}},\,\omega_{\mathbb{K}}=\begin{cases}(1+\sqrt{p})/2\,,\\\
\sqrt{p}\,,\end{cases}\quad d_{\mathbb{K}}=\begin{cases}p,&p\equiv
1\,\text{mod}\,4\,,\\\ 4p,&\text{else}\,.\end{cases}\end{gathered}$ (202)

The non-trivial automorphism $\mathbb{K}\rightarrow\mathbb{K}$ is given by

$\begin{gathered}\alpha=\alpha_{0}+\alpha_{1}\sqrt{p}\longmapsto\alpha^{*}=\alpha_{0}-\alpha_{1}\sqrt{p}\,,\quad\alpha_{0},\alpha_{1}\in\mathbb{Q}.\end{gathered}$
(203)

The Hilbert modular group is given by
$\Gamma_{\mathbb{K}}=\text{SL}(2;\mathcal{O}_{\mathbb{K}})$. With respect to
this group, we can define the Hilbert modular form

######  Definition C.1

(Hilbert modular form) Let
$\mu:\text{SL}(2;\mathcal{O}_{\mathbb{K}})\rightarrow\mathbb{C}$ be a map of
finite order (multiplier system). A Hilbert (Blumenthal) modular form for
$\mathbb{K}$ of weight $r=(r_{1},r_{2})\in\mathbb{Q}^{2}$ with multiplier
system $\mu$ is a holomorphic function $f:\mathbb{H}^{2}\rightarrow\mathbb{C}$
with the properties

  1. i)

$f(M\tau)=\mu(M)(c\tau+d)^{r_{1}}(c^{*}\tau_{2}+d^{*})^{r_{2}}f(\tau)$ for all
$\tau\in\mathbb{H}^{2},M\in\text{SL}(2;\mathcal{O}_{\mathbb{K}})$, where

$\begin{gathered}M\tau:=\left(\frac{a\tau_{1}+b}{c\tau_{1}+d},\frac{a^{*}\tau_{2}+b^{*}}{c^{*}\tau_{2}+d^{*}}\right)\,,\quad\tau=(\tau_{1},\tau_{2})\,,\quad
M=\left(\begin{matrix}a&b\\\ c&d\end{matrix}\right)\,.\end{gathered}$ (204)

  2. ii)

$f$ is regular at cusps of $\text{SL}(2;\mathcal{O}_{\mathbb{K}})$.

If $f$ vanishes at all cusps, we call $f$ a cusp form. If $f$ has homogeneous
weight $r=(k,k)\in\mathbb{Q}^{2}$ we will also say that $f$ has weight
$k\in\mathbb{Q}$.

If we want to use such Hilbert modular forms to cancel the anomaly, several
things need to be clarified. First, the action of Hilbert modular group
$\Gamma_{\mathbb{K}}$ on the generalized upper-half plane $\mathbb{H}_{2}$ is
obtained through the homomorphism $\Omega$ in (147). For arbitrary
$M\in\text{SL}(2;\mathbb{K})$ (or $\Gamma_{\mathbb{K}}$),

$\displaystyle\Omega(M,M^{*})\langle Z\rangle$
$\displaystyle=\left(\frac{\alpha Z_{1}+\beta}{\gamma
Z_{1}+\delta}\,,\frac{\alpha^{*}Z_{2}+\beta^{*}}{\gamma^{*}Z_{2}+\delta^{*}}\right)^{T},\quad
M=\left(\begin{matrix}\alpha&\beta\\\ \gamma&\delta\end{matrix}\right)\,,$
(205) $\displaystyle j(M,Z)$ $\displaystyle=(\gamma
Z_{1}+\delta)(\gamma^{*}Z_{2}+\delta^{*})\,.$

In other words, the symmetry group now is the Hilbert modular group
$\Gamma_{\mathbb{K}}$, different with the previous
$\text{SL}(2,\mathbb{Z})\times\text{SL}(2,\mathbb{Z})$. Theorem 2 of Hauffe-
Waschbusch2022  proves the group $\Gamma_{\mathbb{K}}$ is actually isomorphic
to the discriminant kernel of the orthogonal group, so the symmetry further
shrinks to $\Gamma_{\mathbb{K}}$.

Furthermore, non-trivial lattice structure emerges. The modular group
$\Gamma_{\mathbb{K}}$ is the spin group $\text{Spin}(L)$ of lattice $L$
(section 2.7 of Chapter 2 in bruinier20081 ), where $L$ can be written as
$L=\mathbb{Z}\oplus\mathbb{Z}\oplus\mathcal{O}_{\mathbb{K}}$ with quadratic
form $q((a,\nu,b))=ab-\nu\nu^{*}$ for $a,b\in\mathbb{Z}$ and
$\nu\in\mathcal{O}_{\mathbb{K}}$. For example, suppose $p\equiv 1\mod 4$, and
we write the lattice explicitly in terms of matrices

$\begin{gathered}L=\left\\{\left.\left(\begin{matrix}a&\nu\\\
\nu^{*}&b\end{matrix}\right)\right|\,a,b\in\mathbb{Z},\,\nu\in\mathcal{O}_{\mathbb{K}}\right\\}\,,\quad
q(X)=\det(X)\,\,\text{for}\,\,X\in L\,.\end{gathered}$ (206)

The basis of this lattice is easily written in terms of the matrices

$\begin{gathered}e_{1}=\left(\begin{matrix}1&0\\\ 0&0\end{matrix}\right),\quad
e_{2}=\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right),\quad
e_{3}=\left(\begin{matrix}0&\frac{1+\sqrt{p}}{2}\\\
\frac{1-\sqrt{p}}{2}&0\end{matrix}\right),\quad
e_{4}=\left(\begin{matrix}0&0\\\ 0&1\end{matrix}\right)\,.\end{gathered}$
(207)

One can easily obtain the Gram matrix in terms of these basis, i.e.
$S_{ij}=(e_{i},e_{j})=q(e_{i}+e_{j})-q(e_{i})-q(e_{j})$, then (we use the
symbol $S$ as defined in the previous sections)

$\begin{gathered}S=\left(\begin{matrix}&&&1\\\ &-2&-1&\\\ &-1&\frac{p-1}{2}\\\
1&&&\end{matrix}\right)\,.\end{gathered}$ (208)

Though $S$ contains a $\Pi_{1,1}$ as usual, the rest presents a non-trivial
lattice structure, not realized by known string-theoretic constructions.

The next step is to construct the suitable Hilbert modular forms of non-
trivial weight. The conditions in section 5 still need to be satisfied,
namely:

  * •

The character $\mu$ of the group $\Gamma_{\mathbb{K}}$ (of the lattice $L$)
must be trivial.

  * •

The zeros and poles of the Hilbert modular form correspond to the symmetry
enhancement point of the theory.

The first condition can be assured by an appropriate choice of the value of
$p$. However, identifying the rational quadratic divisor in this case is not
straightforward. Necessary details can be found in references bruinier20081 ;
Bruinier2003 ; Mayer2007 . It can be verified (see section 1.3 in Mayer2007 )
that the generalized upper-half plane $\mathbb{H}_{2}$ (isomorphic to
$\mathcal{K}^{+}$ defined in (35)) is exactly of the form of

$\begin{gathered}\mathbb{H}_{2}=\left\\{\left.\delta\left(\begin{matrix}Z_{1}Z_{2}&Z_{1}\\\
Z_{2}&1\end{matrix}\right)\right|\,\imaginary(Z_{1})>0,\,\,\imaginary(Z_{2})>0,\,\,\delta\in\mathbb{C}^{*}\right\\}\,.\end{gathered}$
(209)

Since we are working within the projective space, usually the factor $\delta$
is ignored. It is also obvious that we can use $(Z_{1},Z_{2})$ to label the
element in $\mathbb{H}_{2}$. In the previous sections we used rational
quadratic divisors to describe the position of zeroes and poles of the
Borcherds product. The corresponding object here is called Hirzebruch-Zagier
divisor Hirzebruch1976 .

######  Definition C.2

(Hirzebruch-Zagier divisor) For $(a,h,b)\in L$ and
$Z\in\mathbb{H}\times\mathbb{H}$, we have the innerproduct

$\begin{gathered}\left(\left(\begin{matrix}Z_{1}Z_{2}&Z_{1}\\\
Z_{2}&1\end{matrix}\right),\left(\begin{matrix}a&h\\\
h^{*}&b\end{matrix}\right)\right)=bZ_{1}Z_{2}-h^{*}Z_{1}-hZ_{2}+a\,.\end{gathered}$
(210)

The zero locus of the right hand side defines an analytic divisor on
$\mathbb{H}\times\mathbb{H}$. For a positive number $m$, in the space
$\mathbb{H}\times\mathbb{H}$ we define the set

$\displaystyle T(m)=\bigcup_{\begin{subarray}{c}(a,b,h)\in L^{\prime}/\\{\pm
1\\}\\\
q(a,b,h)=ab-h^{*}h=-m/p\end{subarray}}\left\\{(Z_{1},Z_{2})\in\mathbb{H}^{2}|\,aZ_{1}Z_{2}+hZ_{1}+h^{*}Z_{2}+b=0\right\\}\,.$
(211)

$T(m)$ is called Hirzebruch-Zagier divisor of discriminant $m$.

Before we extend the Borcherds product (Theorem B.1) to this case, we set up
some basic notations. In section 2 the congruence subgroup $\Gamma_{0}(p)$ is
defined. Corresponding modular forms may be defined as well.

######  Definition C.3

(Modular forms for congruence subgroups) Let $\mu$ be an abelian character
$\Gamma_{0}(p)\rightarrow\mathbb{C}^{*}$ and $k\in\mathbb{N}_{0}$ a non
negative integer. A holomorphic map $f:\mathbb{H}\rightarrow\mathbb{C}$ with
the transformation law

$\begin{gathered}f(M\tau)=\mu(M)(c\tau+d)^{k}f(\tau)\quad\text{for
all}\,\,M\in\Gamma_{0}(p)\,,\end{gathered}$ (212)

for which $f(\infty):=\lim_{\imaginary(z)\rightarrow\infty}f(z)$ and
$f(0):=\lim_{z\rightarrow 0}z^{k}f(z)$ exist in $\mathbb{C}\cup\\{\infty\\}$
(it has finite order at the infinity) is called nearly holomorphic modular
form for $\Gamma_{0}(p)$ of weight $k$ with character $\mu$. If $f(\infty)$
and $f(0)$ are complex numbers, then $f$ is called a holomorphic modular from
for $\Gamma_{0}(p)$ of weight $k$ with character $\mu$. If $f(\infty)=f(0)=0$,
then $f$ is called a cusp form.

We define the spaces

$\displaystyle A_{k}(p,\mu)\quad\text{nearly holomorphic modular forms for
$\Gamma_{0}(p)$ of weight $k$ with character $\mu$}\,,$ $\displaystyle
M_{k}(p,\mu)\quad\text{holomorphic modular forms for $\Gamma_{0}(p)$ of weight
$k$ with character $\mu$}\,,$ $\displaystyle S_{k}(p,\mu)=\left\\{f\in
M_{k}(p,\mu)|\,f\,\,\text{cusp form}\right\\}\,,$ $\displaystyle
A_{k}^{\pm}(p,\chi_{p})=\left\\{\left.f(z)=\sum_{n\in\mathbb{Z}}a(n)e^{2\pi
inz}\in A_{k}(p,\chi_{p})\right|\,a(n)=0\,\,\text{for}\,\,\chi_{p}(n)=\mp
1\right\\}\,,$ $\displaystyle
S_{k}^{\pm}(p,\chi_{p})=A^{\pm}_{k}(p,\chi_{p})\cap S_{k}(p,\chi_{p})\,.$

One can show that $A_{k}(p,\chi)p)=A_{k}^{+}(p,\chi_{p})\oplus
A_{k}^{-}(p,\chi_{p})$. If $f=\sum_{n\in\mathbb{Z}}a(n)q^{n}$ is a modular
form in $A_{k}^{\epsilon}(p,\chi_{p})$, then we call $\sum_{n<0}a(n)q^{n}$ the
principal part of $f$ (at $\tau\rightarrow\infty$). For all integers $n$ we
define

$\begin{gathered}s(n)=1+\sum_{j=0}^{p-1}\frac{e^{2\pi
inj/p}}{p}=2-\left(\frac{n}{p}\right)^{2}=\begin{cases}2,&\text{if}\,\,n\equiv
0\mod p\\\ 1,&\text{if}\,\,n\not\equiv 0\mod p\\\
\end{cases}\,.\end{gathered}$ (213)

Similar to the Theorem B.3, the principal part of $f$ has non trivial
restrictions (Theorem 6 in Bruinier2003 ). There exists a nearly holomorphic
modular form $f\in A_{k}^{+}(p,\chi_{p})$ with prescribed principal part
$\sum_{n<0}a(n)q^{n}$ (where $a(0)=0$ if $\chi_{p}(n)=-1$), if and only if

$\begin{gathered}\sum_{n<0}s(n)a(n)b(-n)=0\end{gathered}$ (214)

for every cusp form $g=\sum_{m>0}b(m)q^{m}$ in $S_{\kappa}^{+}(p,\chi_{p})$,
where $\kappa=2-k$. The case $k=0$ and $\kappa=2$ is of particular interest
for us. For prime number $p\equiv 1\mod 4$, the dimension of
$S_{2}(p,\chi_{p})$ is $2\left[\frac{p-5}{24}\right]$. Thus the space is empty
for $p=5$ case. Hence for $p=5$, for all $m\in\mathbb{N}$ there is a nearly
holomorphic modular form $f\in A_{0}^{+}(p,\chi_{p})$ with prescribed
principal part $\sum_{n<0}a(n)q^{n}$ if and only if $a(n)=0$ for all
$n\in\mathbb{N}$ with $\chi_{p}(n)=-1$. Such nearly holomorphic modular form
is unique Mayer2007 , and up to the normalization is the function $f$ given in
(151).

######  Theorem C.1

(Borcherds product for Hilbert modular forms, Theorem 9 in Bruinier2003 ) Let
$f=\sum_{n\in\mathbb{Z}}a(n)q^{n}\in A_{0}^{+}(p,\chi_{p})$ and assume that
$s(n)a(n)\in\mathbb{Z}$ for all $n<0$. Then there is a meromorphic function
$\Psi$ on $\mathbb{H}\times\mathbb{H}$ with the following properties:

  1. i)

$\Psi$ is a meromorphic modular form for $\Gamma_{\mathbb{K}}$ (the Hilbert
modular group defined in section 6) with some multiplier system of finite
order. The weight of $\Psi$ is equal to the constant coefficient $a(0)$ of
$f$.

  2. ii)

The divisor of $\Psi$ is determined by the principal part of $f$. It equals

$\begin{gathered}\sum_{n<0}s(n)a(n)T(-n)\,.\end{gathered}$ (215)

We can now verify the two required properties. The first is satisfied for
$p=5$, since the multiplier system of Hilbert modular form for
$\Gamma_{\mathbb{K}}$ is trivial (Corollary 5.2.1 in Mayer2007 ). The second
condition necessitates a confirmation that the positions of poles and zeros of
the Hilbert modular form correspond to the symmetry enhancements. The explicit
relationship between these points and the Hirzebruch-Zagier divisor is not yet
known. Naively, requiring that the symmetry enhancement appears at the
diagonal set $(Z_{1},Z_{2})=\\{(\tau,\tau)|\tau\in\mathbb{H}\\}$ similarly to
the choices in HARVEY1996315 , corresponds to the Hirzebruch-Zagier divisor
$T(1)$. There exists a weight $0$ modular form $f\in A_{0}^{+}(5,\chi_{5})$

$\begin{gathered}f(\tau)=q^{-1}+5+11q-54q^{4}+O(q^{5})\,,q=e^{2\pi
i\tau}\,,\end{gathered}$ (216)

that has only one term $(q^{-1})$ in the principal part. By Theorem C.1, we
arrive at a holomorphic Hilbert modular form of weight $5$ vanishing at
$T(1)$. However, as pointed out in the main text, unexpected symmetry
enhancements appear. For example, the point
$Z_{1}=Z_{2}=\frac{1}{2}+\frac{\sqrt{3}}{2}i$ may lead to $\text{SU}(3)$
symmetry, which does not appear in 8D (2,2) theories. Therefore, Hilbert
modular forms do not appear in the counterterms for the $\mathcal{N}=1$
theories with $l=2$.

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# Effect of magnetic field correlation length on the gamma-ray pulsar halo
morphology under anisotropic diffusion

Kun Fanga Hong-Bo Hua,b Xiao-Jun Bia,b En-Sheng Chena,b aKey Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China
bUniversity of Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

Anisotropic diffusion is one of the potential interpretations for the
morphology of the Geminga pulsar halo. It interprets the observed slow-
diffusion phenomenon through a geometric effect, assuming the mean magnetic
field direction around Geminga is closely aligned with the line of sight
toward it. However, this direction should not extend further than the
correlation length of the turbulent magnetic field $L_{c}$, which could be
$100$ pc or less. We first revisit the $L_{c}=\infty$ scenario and show that
the halo asymmetry predicted by this scenario is mainly contributed by the
electrons located beyond the “core” section around Geminga, which has a length
of $100$ pc. Then, considering the directional variation of the magnetic field
beyond the core section, we take one magnetic field configuration as an
example to investigate the possible halo morphology. The predicted morphology
has some different features compared to the $L_{c}=\infty$ scenario. The
current experiments may already be able to test these features. In addition,
we use a semi-analytical method to solve the anisotropic propagation equation,
which offers significant convenience compared to numerical approaches.

##  I Introduction

Pulsar halos are inverse Compton (IC) gamma-ray sources generated by high-
energy electrons and positrons111Electrons will denote both electrons and
positrons hereafter if not specified. escaping from pulsar wind nebulae (PWNe)
and diffusing in the interstellar medium (ISM) around the pulsars (see Ref.
Fang (2022); Liu (2022); López-Coto  _et al._ (2022) for reviews). The most
notable feature of pulsar halos is the extremely slow electron diffusion rate
inside the halo regions Abeysekara  _et al._ (2017); Aharonian  _et al._
(2021); Fang  _et al._ (2022); Albert  _et al._ (2023), which also enables
their visibility. Pulsar halos are ideal probes for studying cosmic ray
propagation in localized Galactic regions. They are believed to have critical
implications on the problems of the cosmic positron excess and the diffuse TeV
gamma-ray excess Hooper  _et al._ (2017); Fang  _et al._ (2018); Linden and
Buckman (2018), assuming we properly understand the slow-diffusion phenomenon.

As the diffusion coefficient is inversely proportional to the energy density
of the turbulent magnetic field Skilling (1971), the slow-diffusion phenomenon
may correspond to a significant turbulence injection. The escaping electron-
positron pairs themselves can excite turbulent waves through streaming
instability Evoli  _et al._ (2018); Mukhopadhyay and Linden (2022), while the
injection power may be too weak to significantly suppress the diffusion
coefficient considering the pulsar motion Fang  _et al._ (2019). The parent
supernova remnants (SNRs) of the pulsars are promising sources of the required
turbulence Fang  _et al._ (2019); Mukhopadhyay and Linden (2022), although
among the known pulsar halos, only the Monogem halo has an observable
associated SNR Plucinsky  _et al._ (1996); Knies  _et al._ (2018). The slow
diffusion inside the halo regions is inferred from the steep gamma-ray
profiles of the halos. It is also suggested that the steep profiles may be
reproduced with a typical diffusion rate in the Galaxy if a relativistic
correction to the diffusion equation is considered Recchia  _et al._ (2021).
However, the needed electron energy, in this case, is larger than what the
pulsars could provide, and the goodness of fit to the data is significantly
poorer than the slow-diffusion model Bao  _et al._ (2022).

Another interpretation that does not require a strong turbulent environment is
the anisotropic diffusion model Liu  _et al._ (2019); De La Torre Luque  _et
al._ (2022), where the diffusion coefficient perpendicular to the mean
magnetic field can be much smaller than parallel. If the mean magnetic field
around a pulsar coincides with the line of sight (LOS), perpendicular
diffusion could explain the steep profile of the halos, while parallel
diffusion remains typical for the Galaxy. For the Geminga halo, the canonical
pulsar halo, the large-scale Galactic field is not aligned with the LOS toward
it. However, the turbulent field direction could significantly deviate from
the large-scale field at the turbulence correlation length $L_{c}$ ($\lesssim
100$ pc), and alignment is still possible considering this fluctuation Liu
_et al._ (2019).

Along this line of thought, the mean-field direction beyond $\sim L_{c}$ from
the pulsar should no longer align with our LOS. In other words, the finiteness
of $L_{c}$ must be considered when calculating the pulsar halo morphology,
which was not specifically discussed in previous works Liu  _et al._ (2019);
De La Torre Luque  _et al._ (2022). In this work, we take the Geminga halo as
the object to study the impact of a finite $L_{c}$ on the pulsar halo
morphology under the anisotropic diffusion assumption. In Sec. II, we describe
our calculation and introduce a semi-analytical method for solving the
anisotropic diffusion equation, which is much more convenient than the
numerical method adopted by the previous works Liu  _et al._ (2019); De La
Torre Luque  _et al._ (2022). In Sec. III and IV, we revisit the model that
does not consider the finiteness of $L_{c}$ and show that the halo asymmetry
expected by this scenario is actually contributed by the electrons located
outside the typical scale of $L_{c}$ around the pulsar. As the direction of
the mean magnetic field far away from Geminga is unknown, we take one magnetic
field configuration as an example to discuss the possible halo morphology in
Sec. V. Finally, we conclude in Sec. VI.

##  II Semi-analytical solution of the anisotropic diffusion equation

After the accelerated electrons escape from the PWN, their propagation in the
interstellar medium (ISM) can be described by the diffusion-loss equation. We
solve the propagation equation to obtain the electron number density around
the pulsar and then do the LOS integration to obtain the electron surface
density, with which the gamma-ray emission can be derived from the standard IC
scattering calculation Blumenthal and Gould (1970).

The electron propagation equation under anisotropic diffusion can be expressed
by

$\displaystyle\frac{\partial N(E_{e},r,z,t)}{\partial t}=$
$\displaystyle\frac{D_{rr}(E_{e})}{r}\frac{\partial}{\partial
r}\left[r\frac{\partial N(E_{e},r,z,t)}{\partial
r}\right]+D_{zz}(E_{e})\frac{\partial^{2}N(E_{e},r,z,t)}{\partial z^{2}}$ (1)
$\displaystyle+\frac{\partial[b(E_{e})N(E_{e},r,z,t)]}{\partial
E_{e}}+Q(E_{e},r,z,t)\,,$

where $N$ is the electron number density, $E_{e}$ is the electron energy, $r$
and $z$ are the coordinates perpendicular and parallel to the mean magnetic
field with the pulsar position as the origin, and $t$ is the time coordinate
with the pulsar birth time as the origin. As suggested by Ref. Yan and
Lazarian (2008) and Xu and Yan (2013), the diffusion coefficient perpendicular
and parallel to the mean magnetic field have the relation of
$D_{rr}=D_{zz}M_{A}^{4}$, where $M_{A}$ is the Alfvénic Mach number. The
perpendicular diffusion coefficient is smaller than the parallel one in the
sub-Alfvénic regime. The energy-loss rate is denoted with
$b(E_{e})=b_{0}(E_{e})E_{e}^{2}$. We take a magnetic field strength of 3
$\mu$G for the synchrotron loss rate. We adopt the background photon fields
given in Ref. Abeysekara  _et al._ (2017) and the parametrization method given
in Ref. Fang  _et al._ (2021) to calculate the IC loss rate.

We divide the source function $Q$ into a spatial term $q_{x}$, a temporal term
$q_{t}$, and an energy term $q_{E}$. As the bow-shock PWN size is much smaller
than the gamma-ray halo, we can safely assume a point-like source as

$q_{x}(r,z)=\frac{1}{r}\delta(r)\delta(z)\,.$ (2)

The temporal term is assumed to follow the variation of the pulsar spin-down
luminosity as

$q_{t}(t)=\left\\{\begin{aligned} &[(1+t/t_{\rm sd})/(1+t_{p}/t_{\rm
sd})]^{-2}\,,\quad&t\geq 0\\\ &0\,,\quad&t<0\\\ \end{aligned}\right.\,,$ (3)

where $t_{p}$ is the pulsar age, and $t_{\rm sd}$ is the pulsar spin-down time
scale set to be $10$ kyr. The injection energy spectrum is assumed to be a
power law with an exponential cutoff as

$q_{E}(E_{e})=q_{e,0}\,E_{e}^{-p}\,{\rm
exp}\left[-\left(\frac{E_{e}}{E_{e,c}}\right)^{2}\right]\,,$ (4)

which is suggested by the relativistic shock acceleration theory of electrons
Dempsey and Duffy (2007).

We rescale the $r$ coordinate to $r^{\prime}=ar$, where $a=M_{A}^{-2}$. Then
Eq. (1) can be rewritten as

$\displaystyle\frac{\partial N^{\prime}(E_{e},r^{\prime},z,t)}{\partial t}=$
$\displaystyle\frac{D_{zz}(E_{e})}{r^{\prime}}\frac{\partial}{\partial
r^{\prime}}\left[r^{\prime}\frac{\partial
N^{\prime}(E_{e},r^{\prime},z,t)}{\partial
r^{\prime}}\right]+D_{zz}(E_{e})\frac{\partial^{2}N^{\prime}(E_{e},r^{\prime},z,t)}{\partial
z^{2}}$ (5)
$\displaystyle+\frac{\partial[b(E_{e})N^{\prime}(E_{e},r^{\prime},z,t)]}{\partial
E_{e}}+Q^{\prime}(E_{e},r^{\prime},z,t)\,,$

where $N^{\prime}(E_{e},r^{\prime},z,t)=N(E_{e},r^{\prime}/a,z,t)$, and
$Q^{\prime}(E_{e},r^{\prime},z,t)=q_{x}^{\prime}(r^{\prime},z)q_{E}(E_{e})q_{t}(t)$,
where

$q^{\prime}_{x}(r^{\prime},z)=q_{x}(r^{\prime}/a,z)=\frac{a^{2}}{r^{\prime}}\delta(r^{\prime})\delta(z)\,.$
(6)

Now Eq. (5) describes isotropic diffusion in cylindrical coordinates, and we
can straightforwardly write the solution given by the image method Delahaye
_et al._ (2010) as

$\displaystyle N^{\prime}(E_{e},r^{\prime},z,t)=$ $\displaystyle\int_{t_{\rm
ini}}^{t}dt_{0}\iint
r^{\prime}_{0}\,dr^{\prime}_{0}\,dz_{0}\,Q^{\prime}(E_{e}^{\star},r^{\prime}_{0},z_{0},t_{0})\,\frac{b(E_{e}^{\star})}{b(E_{e})}\,\frac{1}{(\pi\lambda^{2})^{3/2}}$
(7) $\displaystyle\times{\rm
exp}\left[-\frac{(r^{\prime}-r^{\prime}_{0})^{2}}{\lambda^{2}}\right]\,\sum_{n=-\infty}^{+\infty}{\rm
exp}\left\\{\frac{[z-(-1)^{n}z_{0}-2nz_{\rm max}]^{2}}{\lambda^{2}}\right\\}$
$\displaystyle=$ $\displaystyle\int_{t_{\rm
ini}}^{t}dt_{0}\,q_{t}(t_{0})\,q_{E}(E_{e}^{\star})\,\frac{b(E_{e}^{\star})}{b(E_{e})}\,\frac{a^{2}}{(\pi\lambda^{2})^{3/2}}\,{\rm
exp}\left(-\frac{r^{\prime
2}}{\lambda^{2}}\right)\,\sum_{n=-\infty}^{+\infty}{\rm
exp}\left[\frac{(z-2nz_{\rm max})^{2}}{\lambda^{2}}\right]\,,$

where

$E_{e}^{\star}\approx\frac{E_{e}}{[1-b_{0}E_{e}(t-t_{0})]}\,,\quad\lambda^{2}=4\int_{E_{e}}^{E_{e}^{\star}}\frac{D_{zz}(E^{\prime}_{e})}{b(E^{\prime}_{e})}\,dE^{\prime}_{e}\,,\quad
t_{\rm ini}={\rm max}\\{t-1/(b_{0}E_{e}),0\\}\,,$ (8)

and $z_{\rm max}$ is set to be $2$ kpc, which is significantly larger than
$\lambda$ within the energy range of interest. Finally, we obtain the solution
of Eq. (1), which is written as

$N(E_{e},r,z,t)=\int_{t_{\rm
ini}}^{t}dt_{0}\,q_{t}(t_{0})\,q_{E}(E_{e}^{\star})\,\frac{b(E_{e}^{\star})}{b(E_{e})}\,\frac{a^{2}}{(\pi\lambda^{2})^{3/2}}\,{\rm
exp}\left(-\frac{a^{2}r^{2}}{\lambda^{2}}\right)\,\sum_{n=-\infty}^{+\infty}{\rm
exp}\left[\frac{(z-2nz_{\rm max})^{2}}{\lambda^{2}}\right]\,.$ (9)

We denote the angle between the z axis and the direction to the pulsar with
$\Phi$. For the LOS direction with an angle of $\theta$ away from the pulsar,
the electron surface density can be calculated by

$S_{e}(\theta)=\int N(r(l,\theta,\Phi),z(l,\theta,\Phi))dl\,.$ (10)

One may refer to Ref. Liu  _et al._ (2019) for the transformation between
$(l,\theta,\Phi)$ and $(r,z)$.

If the finiteness of $L_{c}$ is considered, the $z$ axis should change
direction significantly at each $L_{c}$. We take an equivalent approach to
this problem in the LOS integration step, which will be described in detail in
Sec. V.

##  III Anisotropic diffusion model with infinite magnetic field correlation
length

Figure 1 shows a possible magnetic field configuration around Geminga for the
anisotropic-diffusion interpretation. The Galactic large-scale magnetic field
near the Galactic plane should follow the direction of the local spiral arm
(the Orion Spur), which deviates significantly from the LOS toward Geminga.
However, considering the fluctuation of the turbulent field at the scale of
$L_{c}$, there is a possibility that the mean field around Geminga coincides
with our LOS, although the field should change direction significantly outside
the scale of $L_{c}$ (the solid blue line in Fig. 1).

Figure 1: Schematic diagram showing a possible relationship between the
directions of the Galactic large-scale magnetic field, the turbulent magnetic
field around Geminga, and our LOS toward Geminga. The correlation length of
the turbulent field $L_{c}$ is assume to be $\sim 100$ pc, significantly
smaller than the distance between the observer and Geminga.

In the previous calculations of the anisotropic diffusion model Liu  _et al._
(2019); De La Torre Luque  _et al._ (2022), the magnetic field in the ISM
always remains in its direction near Geminga, as shown by the dashed blue line
in Fig. 1. It means that the finiteness of $L_{c}$ is not considered or that
$L_{c}$ is assumed to be much larger than indicated by observations Haverkorn
_et al._ (2008); Iacobelli  _et al._ (2013). In this section, we revisit this
scenario by comparing it with the HAWC measurement of the Geminga halo
Abeysekara  _et al._ (2017). The HAWC collaboration did not report significant
asymmetry in the Geminga halo, while the most distinct feature of anisotropic
diffusion is its expected asymmetry. The left panel of Fig. 2 shows an example
of the Geminga halo morphology with $\Phi=5^{\circ}$. The asymmetry increases
as $\Phi$ increases.

For a specific $\Phi$, we test if the asymmetry could be identified with the
data size used in the original paper of HAWC Abeysekara  _et al._ (2017). In
the initial step, we calculate the average gamma-ray profile in
$0^{\circ}<\zeta<360^{\circ}$ to fit the surface brightness profile given by
HAWC, where $\zeta$ is the azimuth marked in the left panel of Fig. 2. The
main parameters of the model are determined by this fitting procedure. Then in
the subsequent step, we calculate the integrated fluxes in three different
intervals of $0^{\circ}<\zeta<60^{\circ}$, $60^{\circ}<\zeta<120^{\circ}$, and
$120^{\circ}<\zeta<180^{\circ}$ and test if there is a significant difference
between them.

Figure 2: Left: Gamma-ray morphology of the Geminga halo expected by the
anisotropic diffusion model, assuming $L_{c}=\infty$, $\Phi=5^{\circ}$,
$M_{A}=0.2$, and $D_{rr,100}=3\times 10^{27}$ cm2 s-1. The units of the color
bar are $10^{-12}$ TeV cm-2 s-1 deg-2. The energy integration range is $8-40$
TeV. Right: Same as the left, but only consider the contribution of the
electrons located within the “core” section depicted in Fig. 5.

The diffusion coefficient take the form of $D_{i}=D_{i,100}(E/{\rm
100~{}TeV})^{\delta}$, where $i=zz$ or $rr$. The slope is assumed to be
$\delta=1/3$, as suggested by Kolmogorov’s theory. The parallel diffusion
coefficient $D_{zz}$ should be consistent with the cosmic-ray boron-to-carbon
ratio (B/C) measurements Aguilar  _et al._ (2016); Collaboration (2022);
Adriani  _et al._ (2022). The DAMPE experiment measures the B/C up to $\approx
5$ TeV/n and finds a spectral hardening at $\approx 100$ GeV/n. Assuming the
B/C spectrum above the hardening can extrapolate to 100 TeV, and the spectral
hardening is entirely attributed to the slope change of the diffusion
coefficient, we can get a lower limit of $D_{zz,100}$ of $\approx 7\times
10^{29}$ cm2 s-1 (see model B’ of Ref. Ma  _et al._ (2023)). We set
$D_{zz,100}$ as a free parameter in the fitting procedure. Another free
parameter is $M_{A}$, which together with $D_{zz,100}$ determines
$D_{rr,100}$.

For the injection spectrum, we set $p=1.0$ and $E_{e,c}=130$ TeV as suggested
by a fit to the HAWC gamma-ray spectrum Bao  _et al._ (2022). The conversion
efficiency from the pulsar spin-down energy to the injected electron energy is
set to be a free parameter and denoted with $\eta$, which mainly determines
the normalization of the injection spectrum. Since the gamma-ray profile
provided by HAWC is in a single energy bin of $8-40$ TeV, fixing the
parameters that mainly determine energy-dependent features ($p$, $E_{e,c}$,
and $\delta$) should hardly impact our results.

As an example, we show the fitting result for $\Phi=5^{\circ}$ in the left
panel of Fig. 3. The best-fit parameters are $D_{zz,100}=1.8\times 10^{30}$
cm2 s-1, $M_{A}=0.17$, and $\eta=8.2\%$. It can be seen that the full-azimuth
averaged profile can fit the HAWC data well (reduced $\chi^{2}$ is $0.89$).
However, the expected profiles in different subintervals are clearly
different.

Figure 3: Left: Fitting result to the HAWC measurement of the Geminga halo
profile Abeysekara  _et al._ (2017) assuming $\Phi=5^{\circ}$ and
$L_{c}=\infty$. Profiles in different azimuth intervals calculated with the
best-fit parameters are also shown. Right: Based on the subinterval profiles
in the left panel, but only consider the contribution of the electrons located
within the “core” section depicted in Fig. 5.

To estimate the significance of predicted flux differences between the
subintervals, we need to determine the statistical errors in the measurements.
We convert the HAWC measured flux into event counts and estimate the
background. The live time is set to be 507 days Abeysekara  _et al._ (2017),
and the observation time of the Geminga halo is about 6 hours on each transit.
The average photon energy of the measurement is 20 TeV Abeysekara  _et al._
(2017). According to of HAWC effective area of $22000$ m2 Zhou (2021), the
mean flux value can be converted to $n_{s}$, and the flux error roughly to
$\sqrt{n_{s}\Omega+n_{b}\Omega}/\Omega$, where $n_{s}$ and $n_{b}$ are the
number of excess events and background events per square degree, respectively,
and $\Omega$ is the solid angle of an angular interval. We assume that $n_{b}$
is uniform in all the angular intervals and obtain an average $n_{b}$ of $405$
deg-2.

Figure 4: Number of excess events in different regions of the Geminga halo
predicted by the anisotropic diffusion model, assuming an infinite extent for
the mean magnetic field around Geminga ($L_{c}=\infty$). The experiment
parameters adopted in the HAWC paper Abeysekara  _et al._ (2017) is used for
the estimation.

For each subinterval, the predicted excess integrated within $10^{\circ}$
around the pulsar can be expressed as $N_{s}\approx N_{\rm
model}\pm\sqrt{N_{\rm model}+N_{b}}$, where $N_{\rm model}$ is the integrated
model value, and $N_{b}$ is the integrated background. We compare the
predicted $N_{s}$ in different subintervals222As the model is symmetrical
relative to the horizontal axis (see Fig. 2), the region of
$0^{\circ}<\zeta<60^{\circ}$ actually represents
$-60^{\circ}<\zeta<60^{\circ}$. The same is true for other subintervals. in
Fig. 4. As shown, the difference between the subintervals of
$0^{\circ}<\zeta<60^{\circ}$ and $120^{\circ}<\zeta<180^{\circ}$ could be
identified with a significance of $3\sigma$ for $\Phi=2^{\circ}$. The
asymmetry is more significant for larger $\Phi$ as expected. For
$\Phi=1^{\circ}$, the excess difference is no longer significant between the
subintervals, which may not be detectable with the HAWC data. This restriction
given by this integrated-flux test could be more stringent than Ref. De La
Torre Luque  _et al._ (2022), where $\Phi$ is constrained to be smaller than
$\approx 2.5^{\circ}$.

##  IV Origin of the predicted halo asymmetry

Observations indicate that the correlation length of the turbulent magnetic
field in the Galactic ISM falls in $1-100$ pc Haverkorn  _et al._ (2008);
Iacobelli  _et al._ (2013). The correlation length is determined by the
turbulence injection scale, which is several times smaller than the actual
injection scale Harari  _et al._ (2002). As the main turbulence sources, SNRs
typically have scales of several dozen parsecs, so it is unlikely that $L_{c}$
near Geminga is significantly larger than $100$ pc, given that no large-scale
structure is found in that region. Furthermore, the mean magnetic field within
$20$ pc around the solar system, as measured by the Interstellar Boundary
Explorer (IBEX), forms a $\approx 60^{\circ}$ angle with the direction of
Geminga Funsten  _et al._ (2013). This local field direction could be
distorted by the Local Bubble Alves  _et al._ (2018). Thus, even if the mean
field near Geminga coincides with our LOS, it is unlikely that it extends all
the way to the vicinity of the solar system.

To interpret the halo morphology with the anisotropic diffusion model, the
mean magnetic field direction within the “core” section around Geminga, which
has a length of $L_{c}$ as depicted in Fig. 5, must align with the LOS
direction. We assume $L_{c}=100$ pc in the following calculations of the paper
unless specified. We evaluate the gamma-ray contribution of electrons located
within the core section, which is referred to as the core component. We
calculate the gamma-ray emission by only including the electrons located
within the range of $-L_{c}/2<z<L_{c}/2$ in the LOS integration step. In the
right panel of Fig. 2, we show an example of the gamma-ray morphology of the
core component. The parameters used here are the same as those in the left
panel of Fig. 2. It can be seen that the asymmetry is significantly reduced
when only the core component is retained. In the right panel of Fig. 3, we
show the gamma-ray profiles in three subintervals with the same parameters
used in the left panel but only include the core component in the calculation.
The difference between the profiles is significantly reduced compared with the
left, especially at large angles.

We provide a qualitative explanation for the above results. For the
anisotropic diffusion model, the electron number density decreases slowly
along the $z$ direction and rapidly along the $r$ direction. Consequently, for
a position not very far from Geminga, the electron number density is mainly
determined by its $r$ coordinate. We plot two sets of points that are
symmetric with respect to Geminga in Fig. 1. The green set of points, located
within the core section, exhibits a similar $r$ coordinate and hence an
equivalent electron number density. This explains why the asymmetry of the
core component is not significant. The yellow set of points is outside the
core section. The $r$ coordinate of the point on the left is significantly
smaller than that on the right, so the electron number density of the point on
the left is significantly larger than that on the right. This means that the
electrons outside the core section mainly determine the asymmetry feature of
the halo.

Based on the analysis presented above, we highlight that a reasonable
prediction of the pulsar halo morphology features requires accounting for the
impact of the directional fluctuation of the mean magnetic field beyond the
core section around the pulsar.

Figure 5: Schematic diagram showing a possible magnetic field configuration in
the ISM around Geminga. Electrons propagating along the core part, which
roughly aligns with the LOS and has a length of $L_{c}=100$ pc, may interpret
the observed steep profile of the Geminga halo. Electrons leaving the core
part will be deflected to the directions of the wing parts, contributing
additional asymmetry to the halo morphology.

##  V Possible pulsar halo morphology considering the variation of magnetic
field direction

Given that the direction of the mean magnetic field outside the core section
cannot be restricted, we investigate the possible morphology of the Geminga
halo under the anisotropic diffusion model based on a simple magnetic field
configuration. As shown in Fig. 5, the direction of the mean magnetic field
experiences significant variations outside the core part, becoming
perpendicular to the LOS, with symmetric variations on both sides. This
implies that the propagation of electrons will undergo considerable deflection
after leaving the core section. The gamma-ray emission of the electrons
propagating in the direction of the yellow lines depicted in Fig. 5 are
referred to as the “wing” components. This scenario is referred to as the
$L_{c}=100$ pc model.

Figure 6: Same as Fig. 2, but assuming the magnetic field configuration
depicted in Fig. 5.

The calculation of the core component has been explained in Sec. IV. For the
wing components, we first assume that the $z$-axis direction remains unchanged
compared to the core part and get the electron number density distribution. To
achieve the effect of altering the $z$-axis direction, we transform the $z$
coordinate during the LOS integration step. For the left wing part in Fig. 5,
the origin of the new $z$ coordinate, $\tilde{z}$, is taken at the position of
the yellow dot in Fig. 5, and the $\tilde{z}$ axis is along the direction of
the wing. The angle between the $\tilde{z}$ axis and the LOS is
$\tilde{\Phi}=90^{\circ}$. The electron number density in the $\tilde{z}$
coordinate is denoted by $\tilde{N}$. Then the relationship between
$\tilde{N}$ and $N$ is given by

$\tilde{N}(\tilde{z})=N\left(z+\frac{L_{c}}{2}-\frac{L_{c}}{2}\sin\Phi\right)\,.$
(11)

During the LOS integration of this wing component, we consider only the
electron number density within the range of
$(L_{c}/2)\sin\Phi<\tilde{z}<L_{c}(1+1/2\sin\Phi)$. The wing component on the
right side can be obtained similarly.

In Fig. 6, We show the morphology of the Geminga halo predicted by the above
magnetic field configuration. The parameters used are the same as those in
Fig. 2. It can be seen that the differential fluxes of the wing components are
lower than that of the core component, so the predicted halo profile remains
steep around the pulsar and can be consistent with the observation. However,
due to the large angular extent of the wing components, the integrated fluxes
in different azimuth intervals may exhibit notable differences. In addition,
we can see that the flux of the wing component on the left is higher than that
on the right due to the closer distance of the former to our observation
point.

Figure 7: Left: Same as the left panel of Fig. 3, but assuming the magnetic
field configuration depicted in Fig. 6 and $L_{c}=100$ pc. Right: Same as Fig.
4, but assuming the magnetic field configuration depicted in Fig. 6 and
$L_{c}=100$ pc.

We repeat the calculations in Sec. III, first fitting the model with various
$\Phi$ to the HAWC data and then predicting the number of excess events in
different azimuth intervals to determine whether the asymmetry could be
detected by the integrated-flux test. Note that the direction of the wing
parts is always perpendicular to the LOS, that is, $\tilde{\Phi}=90^{\circ}$.

For comparison with the results presented in Fig. 3, we show the fitting
result of this new model in the left panel of Fig. 7, assuming
$\Phi=5^{\circ}$. The best-fit parameters are $D_{zz,100}=1.2\times 10^{30}$
cm2 s-1, $M_{A}=0.26$, and $\eta=19\%$. The difference between the predicted
profiles in different subintervals is considerably smaller than that of
$L_{c}=\infty$. As shown in the right panel of Fig. 7, only when $\Phi$
reaches $10^{\circ}$, does the difference between the predicted excesses in
different subintervals has a significance of $3\sigma$.

There are several remarkable differences in the expected halo morphology
between the $L_{c}=100$ pc and $L_{c}=\infty$ scenarios. First, for $\Phi$
values that are not very small, the $L_{c}=100$ pc model predicts a smaller
halo asymmetry, which is less likely to be detected. Current observations of
pulsar halos do not indicate significant asymmetry, indicating that the
possibility of anisotropic diffusion explaining the pulsar halo is relatively
higher after considering the finiteness of $L_{c}$.

Second, by comparing Fig. 4 and the right panel of Fig. 7, it is evident that
for the $L_{c}=\infty$ scenario, the asymmetry of the halo will vanish as
$\Phi$ approaches $0^{\circ}$, whereas for the case of $L_{c}=100$ pc, the
halo will still retain a certain degree of asymmetry even if $\Phi=0^{\circ}$,
due to the presence of the wing components. This implies that if the
anisotropic diffusion model is correct, the asymmetry of the halo will be
inevitably detected when the data size is larger. Assuming a live time of 500
days, we estimate the data size collected by the Water Cherenkov Detector
Array of the Large High Altitude Air Shower Observatory (LHAASO-WCDA), which
has an effective area of $\approx 78000$ m2 Ma  _et al._ (2022). If
$\Phi=0^{\circ}$, the expected number of excess events in the subinterval of
$60^{\circ}<\zeta<120^{\circ}$ is $\approx 1964\pm 382$, and that in the
subinterval of $120^{\circ}<\zeta<180^{\circ}$ is $\approx 3155\pm 382$.
Therefore, the difference between the two regions would be detected with a
significance of $3\sigma$ even for $\Phi=0^{\circ}$.

Third, the halo morphology expected by the $L_{c}=100$ pc scenario could be
more complex at large angles than $L_{c}=\infty$. Although we show only one
magnetic field configuration as an example and cannot go through all the
possibilities, it is easy to infer that due to the presence of the wing
components, the halo morphology will not exhibit a specific regularity as in
the $L_{c}=\infty$ case.

If $L_{c}$ is several times smaller than $100$ pc, the core component will be
dimmer and no longer dominate the halo morphology. The expected gamma-ray
morphology will then display a strong asymmetry as illustrated by the
simulations of Ref. López-Coto and Giacinti (2018), which may no longer
explain the observations. If $L_{c}$ is further decreased to the order of $1$
pc, the model will regress to the slow-diffusion scenario López-Coto and
Giacinti (2018) as the diffusion coefficient is positively correlated with
$L_{c}$ when the electron Larmor radius is much smaller than $L_{c}$ Aloisio
and Berezinsky (2004). There is no need to assume anisotropic diffusion in
this case.

##  VI Conclusion

In this study, we examine the anisotropic diffusion model as a potential
interpretation of the pulsar halo morphology. The main point of this work is
to illustrate the significance of accounting for the finiteness of $L_{c}$ in
comparison to ignoring it, where $L_{c}$ is the correlation length of the
turbulent magnetic field in the ISM. Our analysis focuses on the Geminga halo,
a canonical example of pulsar halos.

First, we discuss the model with $L_{c}=\infty$, which assumes that the mean
magnetic field around the pulsar is aligned closely with the LOS and extends
infinitely. In the initial step, we calculate the azimuth-averaged gamma-ray
profile for various $\Phi$ to fit the HAWC data. $\Phi$ denotes the angle
between the mean field and the LOS. In the subsequent step, we predict the
profiles in different azimuth intervals using the best-fit parameters in the
previous step and then evaluate the halo asymmetry by analyzing the difference
of the predicted integrated number of excess events within the subintervals,
assuming the data size presented in the HAWC paper Abeysekara  _et al._
(2017). The expected asymmetry of the model would be detected with a $3\sigma$
significance as long as $\Phi\gtrsim 2^{\circ}$.

Observations suggest that $L_{c}$ is on the order of $100$ pc or less.
Meanwhile, we find that the expected asymmetry in the $L_{c}=\infty$ scenario
mainly comes from the contribution of the electrons located beyond the “core”
section around the pulsar, which has a length of $100$ pc. Considering the
finiteness of $L_{c}$, the electron propagation beyond the core part should
have significantly deviated from the LOS. Therefore, we highlight that in
order to reasonably predict the pulsar halo morphology in the anisotropic
diffusion model, it is necessary to consider the variation of the field
direction beyond the core section.

Given that the field direction beyond the core part cannot be restricted, we
assume one simple magnetic field configuration and $L_{c}=100$ pc to
investigate the possible morphology of the Geminga halo under anisotropic
diffusion. We refer to the gamma-ray emission generated by the electrons
traveling beyond the core part as the “wing” components. The steep profile of
the halo could be explained by the core component, while the wing components
introduce different asymmetric features from the $L_{c}=\infty$ model. The
results show that the expected asymmetry in the $L_{c}=100$ pc case can be
smaller than the $L_{c}=\infty$ case. Thus, in the absence of significant halo
asymmetry found at present333The recently released H.E.S.S. result also does
not indicate significant asymmetry in the Geminga halo Aharonian  _et al._
(2023)., the possibility of interpreting observations using anisotropic
diffusion is enhanced. On the other hand, unlike the $L_{c}=\infty$ case, the
presence of the wing components introduces a certain degree of asymmetry even
for $\Phi=0^{\circ}$. If the anisotropic model is correct, the halo asymmetry
may already be detectable using the updated HAWC data Zhou (2021) or the
LHAASO-WCDA data.

In addition, we introduce a semi-analytical method to solve the anisotropic
propagation equation, which simplifies anisotropic diffusion to isotropic
diffusion through a coordinate transformation. This method is much more
convenient than numerical methods.

This work is supported by the National Natural Science Foundation of China
under the grants No. 12105292, No. U1738209, and No. U2031110.

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