\left[ \omega_{n}^2 C_{\gamma \gamma \gamma \gamma}-  2\omega_{\gamma}^2 C_{\gamma \gamma nn} \right]\left[ \omega_{2\gamma-n}^2 C_{\gamma \gamma \gamma \gamma}-  2\omega_{\gamma}^2 C_{\gamma ,\gamma ,2\gamma-n,2\gamma-n}  \right]- \left[ \omega_{\gamma}^2 C_{\gamma, 2\gamma-n, \gamma,n} \right]^2
\end{align*}

are non–zero for all $  n \in \{0,1,\dots, \gamma -1 \}$, that completes the proof.

Next, we establish the non–degeneracy condition for this model.

Let $\gamma \geq 0$ be any integer. Then, the non–degeneracy condition  (<ref>) holds true.

Let $\gamma \geq 0$ be any integer. Also, pick any integers $m \geq 2\gamma+1$ and $n \in \{0,1,\dots,\gamma-1\}$. Then, according to Lemma <ref>, we have

\begin{align*}
C_{ijkm}= \omega_{\min \{i,j,k,m\}}
\end{align*}

provided that either

\begin{align*}
% 2 minus
& \omega_{i} + \omega_{j} - \omega_{k} - \omega_{m} = 0, \\
& \omega_{i} - \omega_{j} + \omega_{k} - \omega_{m} = 0, \\
& \omega_{i} - \omega_{j} - \omega_{k} + \omega_{m} = 0.
\end{align*}

One can easily show that all the indices $(i,j,k,m)$ of the Fourier coefficients that appear in Lemma <ref> satisfy at least one of these conditions and hence we infer

\begin{align*}
C_{\gamma\gamma\gamma\gamma} &= \omega_{\gamma}  , \\
C_{\gamma\gamma nn} &=\omega_{n}   , \\
C_{\gamma\gamma mm} &=  \omega_{\gamma}    , \\
C_{\gamma,\gamma,2\gamma-n,2\gamma-n} &=   \omega_{\gamma}   , \\
C_{\gamma,2\gamma-n,\gamma,n} &=  \omega_{n}   .
\end{align*}

Putting all together, yields

\begin{align*}
& \omega_{m}^2 C_{\gamma \gamma \gamma \gamma}-  2\omega_{\gamma}^2 C_{\gamma \gamma mm} = \omega_{\gamma} \left(\omega_{m}^2 -2 \omega_{\gamma}^2 \right) \geq \omega_{\gamma} \left(\omega_{2\gamma+1}^2 -2 \omega_{\gamma}^2 \right) =2\omega_{\gamma}^3  \geq 2,  \\
& D_{\gamma n} = \omega_{n}\omega_{\gamma}^2  (n-3-4 \gamma ) (n-\gamma )^2  \neq 0 ,
\end{align*}

for all $m \geq 2\gamma+1$ and $n \in \{0,1,\dots,\gamma-1\}$, that competes the proof.

§.§ Conformal cubic wave equation out of spherical symmetry

Next, we consider the conformal cubic wave equation out of spherical symmetry and show that the non–degeneracy condition is a condition on the Fourier coefficients.

Let $\gamma$ and $\mu_1,\mu_2 $ be any integers and define $ \xi $ according to (<ref>)–(<ref>). Then, the non–degeneracy condition

\begin{align*}
\ker \left( d \mathcal{M}( \xi ) \right) = \{ 0 \}
\end{align*}

is equivalent to

\begin{align}\label{NonDegeneracyConditionCHModel2}
\begin{dcases}
\left(\somega_{m}^{(\mu_1,\mu_2)} \right)^2 \mathsf{C}_{\gamma \gamma \gamma \gamma}^{(\mu_1,\mu_2)}-  2 \left(\somega_{\gamma}^{(\mu_1,\mu_2)} \right)^2 \mathsf{C}_{\gamma \gamma mm}^{(\mu_1,\mu_2)}   \neq 0, \text{ for all }   m \geq 2\gamma+1, \\
\mathsf{D}_{\gamma n}^{(\mu_1,\mu_2)}  \neq 0 ,  \text{ for all }   n \in \{0,1,\dots, \gamma -1 \},
\end{dcases}
\end{align}

\begin{align*}
\mathsf{D}_{\gamma n}^{(\mu_1,\mu_2)}  & :=
\left[ \left( \somega_{n}^{(\mu_1,\mu_2)} \right)^2 \mathsf{C}_{\gamma \gamma \gamma \gamma}^{(\mu_1,\mu_2)} -  2 \left(\somega_{\gamma}^{(\mu_1,\mu_2)} \right)^2 \mathsf{C}_{\gamma \gamma nn}^{(\mu_1,\mu_2)} \right]\cdot \\
& \cdot \left[ \big(\somega_{2\gamma-n}^{(\mu_1,\mu_2)} \big)^2 \mathsf{C}_{\gamma \gamma \gamma \gamma}^{(\mu_1,\mu_2)}  -  2 \left( \somega_{\gamma}^{(\mu_1,\mu_2)} \right)^2 \mathsf{C}_{\gamma ,\gamma ,2\gamma-n,2\gamma-n}^{(\mu_1,\mu_2)}  \right]
- \left[ \left( \somega_{\gamma}^{(\mu_1,\mu_2)} \right)^2 \mathsf{C}_{\gamma, 2\gamma-n, \gamma,n}^{(\mu_1,\mu_2)} \right]^2.
\end{align*}

The proof is similar to the one of Lemma <ref>.

Next, we establish the non–degeneracy condition for this model.

Let $\gamma,\mu_1,\mu_2 \geq 0$ be any integers with $\gamma \in \{0,1,2,3,4,5\}$ and $\mu_1=\mu_2=:\mu$ where $\mu$ is either sufficiently small with $\mu \in \{0,1,2,3,4,5\}$ or sufficiently large. Then, the non–degeneracy condition (<ref>) holds true.

Let $\gamma,\mu_1,\mu_2 \geq 0$ be any integers with $\gamma \in \{0,1,2,3,4,5\}$ and $\mu_1=\mu_2=:\mu$ where $\mu$ is either sufficiently small with $\mu \in \{0,1,2,3,4,5\}$ or sufficiently large. Also, pick any integer $m \geq 2\gamma+1$ and, according to Lemmata <ref> and <ref>, we have that

\begin{align*}
\mathsf{C}_{\gamma \gamma \gamma\gamma}^{(\mu,\mu)}
& =  \frac{1}{2}      \sum_{\lambda=0}^{\gamma} \mathsf{M}_{ \gamma}^{(\mu)} (\lambda) \mathsf{M}_{ \gamma}^{(\mu)} (\lambda) \xi_{\lambda}(\mu), \\\mathsf{C}_{\gamma \gamma mm}^{(\mu,\mu)}
& =  \frac{1}{2}      \sum_{\lambda=0}^{\gamma} \mathsf{M}_{ \gamma}^{(\mu)} (\lambda) \mathsf{M}_{ m}^{(\mu)} (\lambda) \xi_{\lambda}(\mu),
\end{align*}

where the function

\begin{align*}
\mathsf{M}_{ m}^{(\mu)} (\lambda) &=\frac{1}{2 \pi ^{3/2}} \frac{(4 \lambda +4 \mu +1) \Gamma \left(\lambda +\frac{1}{2}\right) \Gamma \left(2 \mu +\frac{1}{2}\right) \Gamma \left(\lambda +\mu +\frac{1}{2}\right)}{\Gamma (\lambda +\mu +1) \Gamma (\lambda +2 \mu +1)} \cdot \\
& \cdot \frac{(2 \mu +2 m+1) \Gamma \left(m-\lambda +\frac{1}{2}\right) \Gamma (m+\lambda +2 \mu +1)}{\Gamma (m-\lambda +1) \Gamma \left(m+\lambda +2 \mu +\frac{3}{2}\right)}
\end{align*}

is decreasing with respect to $m$. In addition, recall that the eigenvalues are given by

\begin{align*}
\left(  \somega_{m}^{(\mu,\mu)} \right)  ^2:=(2m+1+2\mu)^2
\end{align*}

and they are clearly increasing with respect to $m \geq 0$. In other words, the function

\begin{align*}
\mathsf{P}_{m}^{(\mu)}(\lambda):=\frac{\mathsf{M}_{ m}^{(\mu)} (\lambda)}{\big( \somega_{m}^{(\mu,\mu)} \big)^2}
\end{align*}

is decreasing with respect to $m$ for all $m \geq \gamma$ as a product of two positive and decreasing functions. In the following, we show that

\begin{align*}
\big( \somega_{m}^{(\mu,\mu)} \big)^2 \mathsf{C}_{\gamma \gamma \gamma \gamma}^{(\mu,\mu)}-   2 \left( \somega_{\gamma}^{(\mu,\mu)} \right)^2 \mathsf{C}_{\gamma \gamma mm}^{(\mu,\mu)}
\end{align*}

stays away from zero for all $m \geq 2\gamma+1$ and $\gamma \in \{0,1,2,3,4,5\}$ provided that $\mu$ is either sufficiently small, say $\mu \in \{1,2,\dots,10\}$ or sufficiently large. To this end, we use the monotonicity of $\mathsf{P}_{m}^{(\mu)}(\lambda)$ with respect to $m$ to infer

\begin{align*}
&	\big( \somega_{m}^{(\mu,\mu)} \big)^2 \mathsf{C}_{\gamma \gamma \gamma \gamma}^{(\mu,\mu)}-   2 \left( \somega_{\gamma}^{(\mu,\mu)} \right)^2 \mathsf{C}_{\gamma \gamma mm}^{(\mu,\mu)} = \\
&	\big( \somega_{m}^{(\mu,\mu)} \big)^2\big( \somega_{\gamma}^{(\mu,\mu)} \big)^2 \left[\frac{\mathsf{C}_{\gamma \gamma \gamma \gamma}^{(\mu,\mu)}}{\big( \somega_{\gamma}^{(\mu,\mu)} \big)^2 } -2 \frac{\mathsf{C}_{\gamma \gamma mm }^{(\mu,\mu)}}{\big( \somega_{m}^{(\mu,\mu)} \big)^2 } \right]= \\
& \frac{1}{2} 	\big( \somega_{m}^{(\mu,\mu)} \big)^2\big( \somega_{\gamma}^{(\mu,\mu)} \big)^2 \left[     \sum_{\lambda=0}^{\gamma}\frac{ \mathsf{M}_{ \gamma}^{(\mu)} (\lambda)}{\big( \somega_{\gamma}^{(\mu,\mu)} \big)^2 } \mathsf{M}_{ \gamma}^{(\mu)} (\lambda) \xi_{\lambda}(\mu) -2 \sum_{\lambda=0}^{\gamma} \mathsf{M}_{ \gamma}^{(\mu)} (\lambda) \frac{ \mathsf{M}_{ m}^{(\mu)} (\lambda)}{\big( \somega_{m}^{(\mu,\mu)} \big)^2 }  \xi_{\lambda}(\mu)\right]= \\
& \frac{1}{2} 	\big( \somega_{m}^{(\mu,\mu)} \big)^2\big( \somega_{\gamma}^{(\mu,\mu)} \big)^2 \left[     \sum_{\lambda=0}^{\gamma}\mathsf{P}_{\gamma}^{(\mu)}(\lambda) \mathsf{M}_{ \gamma}^{(\mu)} (\lambda) \xi_{\lambda}(\mu) -2 \sum_{\lambda=0}^{\gamma} \mathsf{M}_{ \gamma}^{(\mu)} (\lambda) \mathsf{P}_{m}^{(\mu)}(\lambda) \xi_{\lambda}(\mu)\right]= \\
& \frac{1}{2} 	\big( \somega_{m}^{(\mu,\mu)} \big)^2\big( \somega_{\gamma}^{(\mu,\mu)} \big)^2  \sum_{\lambda=0}^{\gamma} \mathsf{M}_{ \gamma}^{(\mu)} (\lambda)\left[    \mathsf{P}_{\gamma}^{(\mu)}(\lambda)    -2    \mathsf{P}_{m}^{(\mu)}(\lambda)\right] \xi_{\lambda}(\mu) \geq  \\
& \frac{1}{2} 	\big( \somega_{m}^{(\mu,\mu)} \big)^2\big( \somega_{\gamma}^{(\mu,\mu)} \big)^2  \sum_{\lambda=0}^{\gamma} \mathsf{M}_{ \gamma}^{(\mu)} (\lambda)\left[    \mathsf{P}_{\gamma}^{(\mu)}(\lambda)    -2    \mathsf{P}_{2\gamma+1}^{(\mu)}(\lambda)\right] \xi_{\lambda}(\mu)  =
\big( \somega_{m}^{(\mu,\mu)} \big)^2 \mathsf{S}_{\gamma}^{(\mu)},
\end{align*}

where we set

\begin{align*}
\mathsf{S}_{\gamma}^{(\mu)}:=\frac{1}{2} \big( \somega_{\gamma}^{(\mu,\mu)} \big)^2  \sum_{\lambda=0}^{\gamma} \mathsf{M}_{ \gamma}^{(\mu)} (\lambda)\left[    \mathsf{P}_{\gamma}^{(\mu)}(\lambda)    -2    \mathsf{P}_{2\gamma+1}^{(\mu)}(\lambda)\right] \xi_{\lambda}(\mu).
\end{align*}

On the one hand,  for all $\gamma \in \{0,1,2,3,4,5 \}$ and $\mu \in \{0,1,2,3,4,5\}$ we compute $\mathsf{S}_{\gamma}^{(\mu)}$ and verify that all  $\mathsf{S}_{\gamma}^{(\mu)}$ are strictly positive. On the other hand, for all $\gamma \in \{0,1,\dots,10 \}$ and sufficiently large $\mu$, we firstly compute $\mathsf{S}_{\gamma}^{(\mu)}$ in terms of $\mu$ and then derive its asymptotic expansion as $\mu \longrightarrow \infty$. For $\gamma=0$, we find

\begin{align*}
\mathsf{S}_{0}^{(\mu)}=	\frac{4^{\mu } (2 \mu +1) (10 \mu +7) \Gamma \left(\mu +\frac{1}{2}\right)^2 \Gamma \left(\mu +\frac{5}{2}\right)}{\pi  (2 \mu +3)^2 \Gamma (\mu +1) \Gamma \left(2 \mu +\frac{5}{2}\right)}  ,
\end{align*}

that is strictly positive, for all $\mu \geq 0$, and for $\gamma \in \{1,2,3,4,5\}$, we expand

\begin{align*}
\mathsf{S}_{\gamma}^{(\mu)}=
\sigma_{\gamma} \mu ^{\frac{1}{2}}+ \mathcal{O} \left(\mu^{-\frac{1}{2}} \right),
\end{align*}

as $\mu \longrightarrow \infty$, for some strictly positive constants $\sigma_{\gamma}$. Figure <ref> illustrates the constants $\sigma_{\gamma}$ as $\gamma $ varies within $ \{1,2,\dots,10\}$.

The constants $\sigma_{\gamma}$ as $\gamma $ varies within $ \{1,2,\dots,10\}$. They are all strictly positive.

Consequently, in both cases, we can ensure that $\mathsf{S}_{\gamma}^{(\mu)} >0$ and hence we conclude that

\begin{align*}
&	\big( \somega_{m}^{(\mu,\mu)} \big)^2 \mathsf{C}_{\gamma \gamma \gamma \gamma}^{(\mu,\mu)}-   2 \left( \somega_{\gamma}^{(\mu,\mu)} \right)^2 \mathsf{C}_{\gamma \gamma mm}^{(\mu,\mu)}  \geq  	\big( \somega_{m}^{(\mu,\mu)} \big)^2 \mathsf{S}_{\gamma}^{(\mu)} \geq \big( \somega_{2\gamma+1}^{(\mu,\mu)} \big)^2 \mathsf{S}_{\gamma}^{(\mu)}>0,
\end{align*}

for all $m \geq 2\gamma+1$. Finally, it remains to show that the determinants

\begin{align*}
\mathsf{D}_{\gamma n}^{(\mu,\mu)}  & :=
\left[ \left( \somega_{n}^{(\mu,\mu)} \right)^2 \mathsf{C}_{\gamma \gamma \gamma \gamma}^{(\mu,\mu)} -  2 \left(\somega_{\gamma}^{(\mu,\mu)} \right)^2 \mathsf{C}_{\gamma \gamma nn}^{(\mu,\mu)} \right]\cdot \\
& \cdot \left[ \big(\somega_{2\gamma-n}^{(\mu,\mu)} \big)^2 \mathsf{C}_{\gamma \gamma \gamma \gamma}^{(\mu,\mu)}  -  2 \left( \somega_{\gamma}^{(\mu,\mu)} \right)^2 \mathsf{C}_{\gamma ,\gamma ,2\gamma-n,2\gamma-n}^{(\mu,\mu)}  \right]
- \left[ \left( \somega_{\gamma}^{(\mu,\mu)} \right)^2 \mathsf{C}_{\gamma, 2\gamma-n, \gamma,n}^{(\mu,\mu)} \right]^2.
\end{align*}

are all non–zero for $ n \in \{0,1,\dots, \gamma -1 \}$. To this end, for all $\gamma \in \{1,2,3,4,5\}$ and $ n \in \{0,1,\dots,   \gamma -1 \}$, we firstly compute each of the Fourier coefficients in the determinants above in terms of $\mu$ and then either compute the determinants when $\mu$ is sufficiently small $\mu \in \{0,1,2,3,4,5\}$ or compute their limit as $\mu \longrightarrow \infty$. For example, for $\gamma=1$, we have $n=0$ and compute

\begin{align*}
\mathsf{D} _{10}^{(\mu,\mu)} &= -\frac{3\ 16^{\mu -1}}{\pi ^2}(\mu +1) (2 \mu +3)^4 (2 \mu +5) (4 \mu +7)\cdot \\
& \cdot \left(20 \mu ^4+328 \mu ^3+1029 \mu ^2+1155 \mu +435\right) \frac{\Gamma \left(\mu +\frac{1}{2}\right)^2 \Gamma \left(\mu +\frac{3}{2}\right)^3 \Gamma \left(\mu +\frac{3}{2}\right)}{\Gamma (\mu +2)^2 \Gamma \left(2 \mu +\frac{9}{2}\right)^2}
\end{align*}

and it is clearly strictly negative for all integers $\mu \geq 0$. For all $\gamma \in \{1,2,\dots,10\}$ and $n \in \{0,1,\dots,\gamma-1\}$, we find either $\mathsf{D} _{\gamma n}^{(\mu,\mu)} < 0$ when $\mu$ is sufficiently small and $\mathsf{D} _{\gamma n}^{(\mu,\mu)} \longrightarrow \pm \infty$ when $\mu \longrightarrow \infty$, that completes the proof.

§.§ Yang–Mills equation in spherical symmetry

Finally, we consider the Yang–Mills equation in spherical symmetry and show that the non–degeneracy condition is a condition on the Fourier coefficients.

Let $\gamma \in \{0,1,2,3,4,5\}$ and define $ \xi $ according to (<ref>)–(<ref>). Then, the non–degeneracy condition

\begin{align*}
\ker \left( d  \textswab{M}_{-}( \xi ) \right) =\{ 0 \}
\end{align*}

is equivalent to

\begin{align}\label{NonDegeneracyConditionCHModel3}
\begin{dcases}
\mathfrak{u}_{\gamma \gamma}   + \mathfrak{v}_{\gamma \gamma}  \neq 0,\\
\mathfrak{u}_{\gamma m}  \neq 0, \text{ for all }   m \geq 2\gamma+1  , \\
\mathfrak{D}_{\gamma n}    \neq 0 ,  \text{ for all }   n \in \{0,1,\dots, \gamma -1 \} ,
\end{dcases}
\end{align}

\begin{align*}
\mathfrak{D}_{\gamma n}  & : =
\mathfrak{u}_{\gamma n} \mathfrak{u}_{\gamma ,2\gamma-n} - \mathfrak{v}_{\gamma n} \mathfrak{v}_{\gamma ,2\gamma-n}
\end{align*}

\begin{align*}
\mathfrak{u}_{\gamma m}  &:=
\left(\frac{\ssomega_{m}}{  \mathfrak{K}_{\gamma}} \right)^{2}+\frac{3}{4} \mathfrak{C}_{\gamma \gamma mm}  - \frac{9}{2}   \sum_{ \substack{ \nu = 0  }}^{m+\gamma}    \frac{\left(\overline{\mathfrak{C}}_{\gamma \nu m} \right)^2 }{\ssomega_{\nu}^2-(\ssomega_{m}+\ssomega_{\gamma})^2 }-  \frac{9}{4}  \sum_{\substack{ \nu = 0  } }^{2\gamma}
\frac{\overline{\mathfrak{C}}_{m \nu m}\overline{\mathfrak{C}}_{\gamma\gamma \nu }}{\ssomega_{\nu}^2 }  \\
& -\frac{9}{2}  \sum_{ \substack{ \nu = 0 \\ \nu \neq \pm(m-\gamma)-2  }}^{m+\gamma}   \frac{\left(\overline{\mathfrak{C}}_{m\gamma \nu } \right)^2}{\ssomega_{\nu}^2-(\ssomega_{m}-\ssomega_{\gamma})^2  }   , \\
\mathfrak{v}_{\gamma m}  &:=   \frac{3}{8} \mathfrak{C}_{\gamma ,2\gamma-m,\gamma, m} -\frac{9}{4}   \sum_{\substack{ \nu = 0  } }^{2\gamma}
\frac{\overline{\mathfrak{C}}_{2\gamma-m, \nu, m} \overline{\mathfrak{C}}_{\gamma\gamma \nu }}{\ssomega_{\nu}^2-(2\ssomega_{\gamma})^2  }  -\frac{9}{2} \sum_{ \substack{ \nu = 0 \\ \nu \neq \pm(m-\gamma)-2  }}^{m+\gamma}   \frac{\overline{\mathfrak{C}}_{\gamma \nu m} \overline{\mathfrak{C}}_{2\gamma-m,\gamma ,\nu }}{\ssomega_{\nu}^2-(\ssomega_{2\gamma-m}-\ssomega_{\gamma})^2  } .
\end{align*}

Let $\gamma \in \{0,1,2,3,4,5\}$ and define $ \xi $ according to (<ref>)–(<ref>). Furthermore, assume that $\ker ( d  \textswab{M}_{-}( \xi ) ) =\{ 0 \}$ and pick any $h=\{h^j: j \geq 0\}\in  l_{s+3}^{2}  $ such that $d  \textswab{M}_{-}( \xi )[h]=0$. Also, fix any integer $m \geq 0 $. Then, according to Lemma <ref>, we have that the system

\begin{align}
&\mathfrak{u}_{\gamma m}  h^m
+\mathfrak{v}_{\gamma m}  h^{2\gamma-m}    =0, \Hquad \text{ for } 0\leq m \leq \gamma-1,\label{Eq11NDCModel3YM} \\
&\left( \mathfrak{u}_{\gamma \gamma}   + \mathfrak{v}_{\gamma \gamma}  \right)h^{\gamma}
=0, \Hquad \text{ for } m =\gamma,\label{Eq22NDCModel3YM}  \\
&\mathfrak{u}_{\gamma m}  h^m	 +
\mathfrak{v}_{\gamma m} 	h^{2\gamma-m}    =0,  \Hquad \text{ for } \gamma+1\leq m \leq 2\gamma,\label{Eq33NDCModel3YM}  \\
&	\mathfrak{u}_{\gamma m} h^m
=0, \Hquad \text{ for }   m \geq  2\gamma+1 , \label{Eq44NDCModel3YM}
\end{align}

has $h=\{h^i: i\geq 0\}=0$ as the unique solution, where

\begin{align*}
\mathfrak{u}_{\gamma m}  &:=
\left(\frac{\ssomega_{m}}{  \mathfrak{K}_{\gamma}} \right)^{2}+\frac{3}{4} \mathfrak{C}_{\gamma \gamma mm} -  \mathfrak{a}_{\gamma m}
\mathfrak{v}_{\gamma m}  &:=
\frac{3}{8} \mathfrak{C}_{\gamma ,2\gamma-m,\gamma, m} -\mathfrak{b}_{\gamma m}
\end{align*}

Furthermore, $ \mathfrak{a}_{\gamma m} $ and $\mathfrak{b}_{\gamma m} $ are given by Lemma <ref> and putting all together yields

\begin{align*}
\mathfrak{u}_{\gamma m}  &:=
\left(\frac{\ssomega_{m}}{  \mathfrak{K}_{\gamma}} \right)^{2}+\frac{3}{4} \mathfrak{C}_{\gamma \gamma mm}  - \frac{9}{2}   \sum_{ \substack{ \nu = 0  }}^{m+\gamma}    \frac{\left(\overline{\mathfrak{C}}_{\gamma \nu m} \right)^2 }{\ssomega_{\nu}^2-(\ssomega_{m}+\ssomega_{\gamma})^2 \ssomega^2}-  \frac{9}{4}  \sum_{\substack{ \nu = 0  } }^{2\gamma}
\frac{\overline{\mathfrak{C}}_{m \nu m}\overline{\mathfrak{C}}_{\gamma\gamma \nu }}{\ssomega_{\nu}^2 }  \\
& -\frac{9}{2}  \sum_{ \substack{ \nu = 0 \\ \nu \neq \pm(m-\gamma)-2  }}^{m+\gamma}   \frac{\left(\overline{\mathfrak{C}}_{m\gamma \nu } \right)^2}{\ssomega_{\nu}^2-(\ssomega_{m}-\ssomega_{\gamma})^2  }   , \\
\mathfrak{v}_{\gamma m}  &:=   \frac{3}{8} \mathfrak{C}_{\gamma ,2\gamma-m,\gamma, m} -\frac{9}{4}   \sum_{\substack{ \nu = 0  } }^{2\gamma}
\frac{\overline{\mathfrak{C}}_{2\gamma-m, \nu, m} \overline{\mathfrak{C}}_{\gamma\gamma \nu }}{\ssomega_{\nu}^2-(2\ssomega_{\gamma})^2  }  -\frac{9}{2} \sum_{ \substack{ \nu = 0 \\ \nu \neq \pm(m-\gamma)-2  }}^{m+\gamma}   \frac{\overline{\mathfrak{C}}_{\gamma \nu m} \overline{\mathfrak{C}}_{2\gamma-m,\gamma ,\nu }}{\ssomega_{\nu}^2-(\ssomega_{2\gamma-m}-\ssomega_{\gamma})^2  } .
\end{align*}

Now, (<ref>) and (<ref>) yield

\begin{align*}
\mathfrak{u}_{\gamma \gamma}   + \mathfrak{v}_{\gamma \gamma}  \neq 0
\end{align*}

\begin{align*}
\mathfrak{u}_{\gamma m}  \neq 0
\end{align*}

for all $m\geq 2\gamma+1$. Next, we rearrange (<ref>) and (<ref>) by setting $m=n$ and $m=2\gamma-n$ respectively to obtain

\begin{align*}
&\mathfrak{u}_{\gamma n}  h^n
+\mathfrak{v}_{\gamma n}  h^{2\gamma-n}   =0,   \\
&\mathfrak{u}_{\gamma ,2\gamma-n}  h^{2\gamma-n}	 +
\mathfrak{v}_{\gamma ,2\gamma-n} 	h^{n}      =0,
\end{align*}

for all $n\in \{0,1,\dots,\gamma-1\}$, that can be written in the matrix form

\begin{align*}
\begin{bmatrix}
\mathfrak{u}_{\gamma n} & \mathfrak{v}_{\gamma n}  \\
\mathfrak{v}_{\gamma ,2\gamma-n}    &   \mathfrak{u}_{\gamma ,2\gamma-n}
\end{bmatrix}
\begin{bmatrix}
h^n   \\
\end{bmatrix}=
\begin{bmatrix}
0  \\
\end{bmatrix},
\end{align*}

for all $  n \in \{0,1,\dots, \gamma -1 \}$. Observe that these are in total $\gamma$ $(2 \times 2)-$linear systems where the unknowns are $h^{m}$ for $m \in  \{0,1,\cdots,2\gamma\}\setminus \{\gamma\} $. Finally, these systems have only the trivial solution $h^{m}=0$ for all $m \in \{0,1,\cdots,2\gamma\}\setminus \{\gamma\} $ if and only if the determinants

\begin{align*}
\mathfrak{D}_{\gamma n}  &  =
\mathfrak{u}_{\gamma n} \mathfrak{u}_{\gamma ,2\gamma-n} - \mathfrak{v}_{\gamma n} \mathfrak{v}_{\gamma ,2\gamma-n}
\end{align*}

are non–zero for all $  n \in \{0,1,\dots, \gamma -1 \}$, that completes the proof.

Next, we establish the non–degeneracy condition for this model.

Let $\gamma \in \{0,1,2,3,4,5\}$. Then, the non–degeneracy condition (<ref>) holds true.

Let $\gamma \in \{0,1,2,3,4,5\}$, $ \alpha = 1/ \sqrt{6}$ and pick any frequency $\ssomega \in \mathcal{W}_{\alpha}$ with $\ssomega<1$. Also, pick an integer $m \geq 2\gamma+1$ and define

\begin{align*}
\mathfrak{u}_{\gamma m}  &:=
\left(\frac{\ssomega_{m}}{  \mathfrak{K}_{\gamma}} \right)^{2}+\frac{3}{4} \mathfrak{C}_{\gamma \gamma mm}  - \frac{9}{2}   \sum_{ \substack{ \nu = 0  }}^{m+\gamma}    \frac{\left(\overline{\mathfrak{C}}_{\gamma \nu m} \right)^2 }{\ssomega_{\nu}^2-(\ssomega_{m}+\ssomega_{\gamma})^2  }-  \frac{9}{4}  \sum_{\substack{ \nu = 0  } }^{2\gamma}
\frac{\overline{\mathfrak{C}}_{m \nu m}\overline{\mathfrak{C}}_{\gamma\gamma \nu }}{\ssomega_{\nu}^2 }  \\
& -\frac{9}{2}  \sum_{ \substack{ \nu = 0 \\ \nu \neq \pm(m-\gamma)-2  }}^{m+\gamma}   \frac{\left(\overline{\mathfrak{C}}_{m\gamma \nu } \right)^2}{\ssomega_{\nu}^2-(\ssomega_{m}-\ssomega_{\gamma})^2  }   , \\
\mathfrak{v}_{\gamma m}  &:=   \frac{3}{8} \mathfrak{C}_{\gamma ,2\gamma-m,\gamma, m} -\frac{9}{4}   \sum_{\substack{ \nu = 0  } }^{2\gamma}
\frac{\overline{\mathfrak{C}}_{2\gamma-m, \nu, m} \overline{\mathfrak{C}}_{\gamma\gamma \nu }}{\ssomega_{\nu}^2-(2\ssomega_{\gamma})^2  }  -\frac{9}{2} \sum_{ \substack{ \nu = 0 \\ \nu \neq \pm(m-\gamma)-2  }}^{m+\gamma}   \frac{\overline{\mathfrak{C}}_{\gamma \nu m} \overline{\mathfrak{C}}_{2\gamma-m,\gamma ,\nu }}{\ssomega_{\nu}^2-(\ssomega_{2\gamma-m}-\ssomega_{\gamma})^2  } .
\end{align*}

Firstly, we show that

\begin{align*}
\mathfrak{u}_{\gamma \gamma}   + \mathfrak{v}_{\gamma \gamma}  \neq 0.
\end{align*}

In this case, all the sums above are finite and in particular $\nu$ varies within $\{0,1,\dots,2\gamma\}$. Hence, we compute $\mathfrak{u}_{\gamma \gamma}   + \mathfrak{v}_{\gamma \gamma}$ for all $\gamma \in \{0,1,2,3,4,5\}$ and verify that its has a strictly positive lower bound. Figure <ref>

The constants $\mathfrak{u}_{\gamma \gamma}   + \mathfrak{v}_{\gamma \gamma}$ as $\gamma$ varies within $\{0,1,\dots,10 \}$. They decrease and stay away from zero.

illustrates the constants $\mathfrak{u}_{\gamma \gamma}   + \mathfrak{v}_{\gamma \gamma}$ as $\gamma$ varies within $\{0,1,2,3,4,5\}$. Secondly, we show that

\begin{align*}
\mathfrak{u}_{\gamma m}  \neq 0,
\end{align*}

for $ m \geq 2\gamma+1$. To this end, we note that $\overline{\mathfrak{C}}_{ijm}=0$ for all integers $i,j,m \geq 0$ with either $m>i+j$ or $|i-j|>m$ (Lemma <ref>). Specifically, for any integer $m\geq 2\gamma+1$, we focus on

\begin{align*}
\sum_{ \substack{ \nu = 0  }}^{m+\gamma}    \frac{\left(\overline{\mathfrak{C}}_{\gamma \nu m} \right)^2 }{\ssomega_{\nu}^2-(\ssomega_{m}+\ssomega_{\gamma})^2   } , \Hquad
\sum_{ \substack{ \nu = 0 \\ \nu \neq \pm(m-\gamma)-2  }}^{m+\gamma}   \frac{\left(\overline{\mathfrak{C}}_{m\gamma \nu } \right)^2}{\ssomega_{\nu}^2-(\ssomega_{m}-\ssomega_{\gamma})^2   }
\end{align*}

and note that we must have

\begin{align*}
m-\gamma	\leq \nu \leq m+\gamma
\end{align*}

* $ m-\gamma=|m-\gamma|>\nu   \Longrightarrow \overline{\mathfrak{C}}_{\gamma   \nu m}=0$,

* $ \nu>m+\gamma   \Longrightarrow \overline{\mathfrak{C}}_{\gamma   \nu m}=0$.

In addition, for all such $\nu$, the conditions $\nu \neq  \pm(m-\gamma)-2 $ are satisfied since

* $ \nu \geq m-\gamma \Longrightarrow (m-\gamma)-2 <m-\gamma \leq \nu  \Longrightarrow \nu \neq (m-\gamma)-2$,

* $m\geq 2\gamma+1 \Longrightarrow -(m-\gamma)-2 <0 \Longrightarrow \nu \neq  -(m-\gamma)-2 $.

Consequently, for all $m\geq 2\gamma+1$, we have

\begin{align*}
\mathfrak{u}_{\gamma m}  &:=
\left(\frac{\ssomega_{m}}{  \mathfrak{K}_{\gamma}} \right)^{2}+\frac{3}{4} \mathfrak{C}_{\gamma \gamma mm} - \frac{9}{2}   \sum_{ \substack{ \nu = m-\gamma  }}^{m+\gamma}    \frac{\left(\overline{\mathfrak{C}}_{\gamma \nu m} \right)^2 }{\ssomega_{\nu}^2-(\ssomega_{m}+\ssomega_{\gamma})^2   }-  \frac{9}{4}  \sum_{\substack{ \nu = 0  } }^{2\gamma}
\frac{\overline{\mathfrak{C}}_{m \nu m}\overline{\mathfrak{C}}_{\gamma\gamma \nu }}{\ssomega_{\nu}^2 }  \\
& -\frac{9}{2}  \sum_{ \substack{ \nu = m-\gamma  }}^{m+\gamma}   \frac{\left(\overline{\mathfrak{C}}_{m\gamma \nu } \right)^2}{\ssomega_{\nu}^2-(\ssomega_{m}-\ssomega_{\gamma})^2   }
\end{align*}

and by setting $\nu= \sigma+m-\gamma  $ and $\nu=\sigma$ respectively we can rewrite the latter as follows

\begin{align*}
\mathfrak{u}_{\gamma m}  &:=
\left(\frac{\ssomega_{m}}{  \mathfrak{K}_{\gamma}} \right)^{2}+\frac{3}{4} \mathfrak{C}_{\gamma \gamma mm} - \frac{9}{2}   \sum_{ \substack{
\sigma=0  }}^{2\gamma}    \frac{\left(\overline{\mathfrak{C}}_{\gamma, \sigma+m-\gamma  , m} \right)^2 }{\ssomega_{ \sigma+m-\gamma  }^2-(\ssomega_{m}+\ssomega_{\gamma})^2   }-  \frac{9}{4}  \sum_{\substack{ \sigma = 0  } }^{2\gamma}
\frac{\overline{\mathfrak{C}}_{m \sigma m}\overline{\mathfrak{C}}_{\gamma\gamma \sigma }}{\ssomega_{\sigma}^2 }  \\
& -\frac{9}{2}  \sum_{ \substack{ \sigma = 0  }}^{2\gamma}   \frac{\left(\overline{\mathfrak{C}}_{m,\gamma, \sigma+m-\gamma   } \right)^2}{\ssomega_{ \sigma+m-\gamma  }^2-(\ssomega_{m}-\ssomega_{\gamma})^2   }.
\end{align*}

Now, recall from Lemma <ref> that

\begin{align*}
\mathfrak{K}_{\gamma}^{-2}
& =\frac{3 \mathfrak{C}_{\gamma\gamma\gamma\gamma}-8\mathfrak{q}_{\gamma} }{-8\ssomega_{\gamma}^2}   = - \frac{ 3 }{8\ssomega_{\gamma}^2}   \mathfrak{C}_{\gamma\gamma\gamma\gamma} + \frac{ 1 }{\ssomega_{\gamma}^2}  \mathfrak{q}_{\gamma}   \\
& = - \frac{ 3 }{8\ssomega_{\gamma}^2}   \mathfrak{C}_{\gamma\gamma\gamma\gamma} +\frac{9}{4 \ssomega_{\gamma}^2}  \sum_{  \nu = 0}^{2\gamma}  \left(  \overline{\mathfrak{C}}_{\gamma \gamma \nu} \right)^2
\left(\frac{2}{\ssomega_{\nu}^2 }  +  \frac{1}{\ssomega_{\nu}^2-(2\ssomega_{\gamma})^2    } \right)
\end{align*}

that yields

\begin{align*}
\left(\frac{\ssomega_{m}}{  \mathfrak{K}_{\gamma}} \right)^{2} =
- \frac{ 3\ssomega_{m}^2 }{8\ssomega_{\gamma}^2}   \mathfrak{C}_{\gamma\gamma\gamma\gamma} +\frac{9\ssomega_{m}^2}{4 \ssomega_{\gamma}^2}  \sum_{  \nu = 0}^{2\gamma}  \left(  \overline{\mathfrak{C}}_{\gamma \gamma \nu} \right)^2
\left(\frac{2}{\ssomega_{\nu}^2 }  +  \frac{1}{\ssomega_{\nu}^2-(2\ssomega_{\gamma})^2    } \right) .
\end{align*}

Putting all together, we obtain

\begin{align*}
\mathfrak{u}_{\gamma m}  &=
- \frac{ 3\ssomega_{m}^2 }{8\ssomega_{\gamma}^2}   \mathfrak{C}_{\gamma\gamma\gamma\gamma} +\frac{9\ssomega_{m}^2}{4 \ssomega_{\gamma}^2}  \sum_{  \sigma = 0}^{2\gamma}  \left(  \overline{\mathfrak{C}}_{\gamma \gamma \sigma} \right)^2
\left(\frac{2}{\ssomega_{\sigma}^2 }  +  \frac{1}{\ssomega_{\sigma}^2-(2\ssomega_{\gamma})^2    } \right) \\
& +\frac{3}{4} \mathfrak{C}_{\gamma \gamma mm} - \frac{9}{2}   \sum_{ \substack{
\sigma=0  }}^{2\gamma}    \frac{\left(\overline{\mathfrak{C}}_{\gamma, \sigma+m-\gamma  , m} \right)^2 }{\ssomega_{ \sigma+m-\gamma  }^2-(\ssomega_{m}+\ssomega_{\gamma})^2   }-  \frac{9}{4}  \sum_{\substack{ \sigma = 0  } }^{2\gamma}
\frac{\overline{\mathfrak{C}}_{m \sigma m}\overline{\mathfrak{C}}_{\gamma\gamma \sigma }}{\ssomega_{\sigma}^2 }  \\
& -\frac{9}{2}  \sum_{ \substack{ \sigma = 0  }}^{2\gamma}   \frac{\left(\overline{\mathfrak{C}}_{m,\gamma, \sigma+m-\gamma   } \right)^2}{\ssomega_{ \sigma+m-\gamma  }^2-(\ssomega_{m}-\ssomega_{\gamma})^2   }.
\end{align*}

In addition, we also note that $\overline{\mathfrak{C}}_{ijm}=0$ for all integers $i,j,m \geq 0$ with $i+j-m \notin 2 \mathbb{N}\cup \{0\}$ (Lemma <ref>). Specifically, we must have

\begin{align*}
\sigma \in 2 \mathbb{N}\cup \{0\}
\end{align*}

* $ \sigma =\sigma+\gamma-\gamma \notin  2 \mathbb{N}\cup \{0\}  \Longrightarrow \overline{\mathfrak{C}}_{\gamma \gamma \sigma}=0$,

* $ \sigma =\gamma+\sigma+m-\gamma-m  \notin  2 \mathbb{N}\cup \{0\}  \Longrightarrow \overline{\mathfrak{C}}_{\gamma,\sigma+m-\gamma,m}=\overline{\mathfrak{C}}_{m,\gamma,\sigma+m-\gamma}=0$.

Therefore, by setting $\sigma =2\tau$, we arrive at

\begin{align*}
\mathfrak{u}_{\gamma m}  &=
- \frac{ 3\ssomega_{m}^2 }{8\ssomega_{\gamma}^2}   \mathfrak{C}_{\gamma\gamma\gamma\gamma} +\frac{9\ssomega_{m}^2}{2 \ssomega_{\gamma}^2}  \sum_{  \tau = 0}^{\gamma}
\frac{ \left(  \overline{\mathfrak{C}}_{\gamma ,\gamma , 2\tau} \right)^2}{\ssomega_{2\tau}^2 }  + \frac{9\ssomega_{m}^2}{4 \ssomega_{\gamma}^2}  \sum_{  \tau = 0}^{\gamma}  \frac{ \left(  \overline{\mathfrak{C}}_{\gamma ,\gamma , 2\tau} \right)^2}{\ssomega_{2\tau}^2-(2\ssomega_{\gamma})^2    }  \\
& +\frac{3}{4} \mathfrak{C}_{\gamma \gamma mm} - \frac{9}{2}   \sum_{ \substack{
\tau=0  }}^{\gamma}    \frac{\left(\overline{\mathfrak{C}}_{\gamma, 2\tau+m-\gamma  , m} \right)^2 }{\ssomega_{ 2\tau+m-\gamma  }^2-(\ssomega_{m}+\ssomega_{\gamma})^2   }-  \frac{9}{4}  \sum_{\substack{ \tau = 0  } }^{\gamma}
\frac{\overline{\mathfrak{C}}_{m, 2\tau, m}\overline{\mathfrak{C}}_{\gamma,\gamma,2\tau }}{\ssomega_{2\tau}^2 }  \\
& -\frac{9}{2}  \sum_{ \substack{ \tau = 0  }}^{\gamma}   \frac{\left(\overline{\mathfrak{C}}_{m,\gamma, 2\tau+m-\gamma   } \right)^2}{\ssomega_{ 2\tau +m-\gamma  }^2-(\ssomega_{m}-\ssomega_{\gamma})^2   }.
\end{align*}

Now, all the Fourier coefficients above are non–zero and, according to Lemma <ref>, we have

\begin{align*}
\overline{\mathfrak{C}}_{\gamma,\gamma,2\tau} &=
2 \sqrt{\frac{2}{\pi }} \frac{(\tau +1)^2 (\gamma -\tau +1) (\gamma +\tau +3)}{(\gamma +1) (\gamma +3) \sqrt{4 \tau  (\tau +2)+3}}, \\
\overline{\mathfrak{C}}_{m ,2\tau , m} & =2 \sqrt{\frac{2}{\pi }} \frac{ (\tau +1)^2 (m-\tau +1) (m+\tau +3)}{(m+1) (m+3) \sqrt{4 \tau  (\tau +2)+3}}, \\
\overline{\mathfrak{C}}_{\gamma, 2\tau +m-\gamma  , m}&=2 \sqrt{\frac{2}{\pi }}  \frac{(\tau +1) (\gamma -\tau +1) (m+\tau +3) (-\gamma +m+\tau +1)}{\sqrt{(\gamma +1) (\gamma +3) (m+1) (m+3) (-\gamma +m+2 \tau +1) (-\gamma +m+2 \tau +3)}}.
\end{align*}

The latter allows us to compute

\begin{align*}
\sum_{  \tau  = 0}^{\gamma}  \frac{\left(  \overline{\mathfrak{C}}_{\gamma ,\gamma, 2\tau  } \right)^2}{\ssomega_{2\tau  }^2 } &=\frac{(\gamma +2) (2 \gamma +3) (2 \gamma +5)}{15 \pi  (\gamma +1) (\gamma +3)}, \\
\sum_{\substack{ \tau  = 0  } }^{\gamma}
\frac{\overline{\mathfrak{C}}_{m ,2\tau , m}\overline{\mathfrak{C}}_{\gamma,\gamma, 2\tau }}{\ssomega_{2\tau }^2 }	&=  \frac{(\gamma +2) (-\gamma  (\gamma +4)+5 m (m+4)+15)}{15 \pi  (m+1) (m+3)},
\end{align*}

for all integers $\gamma \geq 0$. Also, recall that $\mathfrak{C}_{\gamma\gamma mm}$ and $\mathfrak{C}_{\gamma\gamma\gamma\gamma}$ are given by closed formulas (Remark <ref>). Consequently, we rescale $\mathfrak{u}_{\gamma m} $,

\begin{align*}
\frac{\mathfrak{u}_{\gamma m} }{\ssomega_{m}^2} &=
- \frac{ 3  }{8\ssomega_{\gamma}^2}   \mathfrak{C}_{\gamma\gamma\gamma\gamma} +\frac{9 }{2 \ssomega_{\gamma}^2}  \sum_{  \tau = 0}^{\gamma}
\frac{ \left(  \overline{\mathfrak{C}}_{\gamma ,\gamma , 2\tau} \right)^2}{\ssomega_{2\tau}^2 }  + \frac{9 }{4 \ssomega_{\gamma}^2}  \sum_{  \tau = 0}^{\gamma}  \frac{ \left(  \overline{\mathfrak{C}}_{\gamma ,\gamma , 2\tau} \right)^2}{\ssomega_{2\tau}^2-(2\ssomega_{\gamma})^2    }  \\
& +\frac{3}{4\ssomega_{m}^2} \mathfrak{C}_{\gamma \gamma mm} - \frac{9}{2\ssomega_{m}^2}   \sum_{ \substack{
\tau=0  }}^{\gamma}    \frac{\left(\overline{\mathfrak{C}}_{\gamma, 2\tau+m-\gamma  , m} \right)^2 }{\ssomega_{ 2\tau+m-\gamma  }^2-(\ssomega_{m}+\ssomega_{\gamma})^2   }-  \frac{9}{4\ssomega_{m}^2}  \sum_{\substack{ \tau = 0  } }^{\gamma}
\frac{\overline{\mathfrak{C}}_{m, 2\tau, m}\overline{\mathfrak{C}}_{\gamma,\gamma,2\tau }}{\ssomega_{2\tau}^2 }  \\
& -\frac{9}{2\ssomega_{m}^2}  \sum_{ \substack{ \tau = 0  }}^{\gamma}   \frac{\left(\overline{\mathfrak{C}}_{m,\gamma, 2\tau+m-\gamma   } \right)^2}{\ssomega_{ 2\tau +m-\gamma  }^2-(\ssomega_{m}-\ssomega_{\gamma})^2   }
\end{align*}

and obtain that

\begin{align}\label{SplitingofMathfrakU}
\frac{\mathfrak{u}_{\gamma m} }{\ssomega_{m}^2} = \mathfrak{I}_{\gamma m}  + \mathfrak{E}_{\gamma m}  ,
\end{align}

where $\mathfrak{I}_{\gamma m} $ stands for the part that can be explicitly computed,

\begin{align*}
\mathfrak{I}_{\gamma m}  &:= - \frac{ 3  }{8\ssomega_{\gamma}^2}   \mathfrak{C}_{\gamma\gamma\gamma\gamma} +\frac{9 }{2 \ssomega_{\gamma}^2}  \sum_{  \tau = 0}^{\gamma}
\frac{ \left(  \overline{\mathfrak{C}}_{\gamma ,\gamma , 2\tau} \right)^2}{\ssomega_{2\tau}^2 }  + \frac{9 }{4 \ssomega_{\gamma}^2}  \sum_{  \tau = 0}^{\gamma}  \frac{ \left(  \overline{\mathfrak{C}}_{\gamma ,\gamma , 2\tau} \right)^2}{\ssomega_{2\tau}^2-(2\ssomega_{\gamma})^2    }  \\
& +\frac{3}{4\ssomega_{m}^2} \mathfrak{C}_{\gamma \gamma mm} -  \frac{9}{4\ssomega_{m}^2}  \sum_{\substack{ \tau = 0  } }^{\gamma}
\frac{\overline{\mathfrak{C}}_{m, 2\tau, m}\overline{\mathfrak{C}}_{\gamma,\gamma,2\tau }}{\ssomega_{2\tau}^2 },
\end{align*}

and $\mathfrak{E}_{\gamma m}$ stands for the part that cannot be explicitly computed,

\begin{align*}
\mathfrak{E}_{\gamma m}   & :=    -  \frac{9}{2\ssomega_{m}^2}   \sum_{ \substack{
\tau=0  }}^{\gamma}    \frac{\left(\overline{\mathfrak{C}}_{\gamma, 2\tau+m-\gamma  , m} \right)^2 }{\ssomega_{ 2\tau+m-\gamma  }^2-(\ssomega_{m}+\ssomega_{\gamma})^2  }
-\frac{9}{2\ssomega_{m}^2}  \sum_{ \substack{ \tau = 0  }}^{\gamma}   \frac{\left(\overline{\mathfrak{C}}_{m,\gamma, 2\tau+m-\gamma   } \right)^2}{\ssomega_{ 2\tau +m-\gamma  }^2-(\ssomega_{m}-\ssomega_{\gamma})^2  }.
\end{align*}

Now, using the elementary inequalities

\begin{align*}
|\ssomega_{ 2\tau+m-\gamma  }^2-(\ssomega_{m}+\ssomega_{\gamma})^2|&=|4 (\gamma -\tau +1) (m+\tau +3)| \geq 4(m+3) , \\
|\ssomega_{ 2\tau +m-\gamma  }^2-(\ssomega_{m}-\ssomega_{\gamma})^2|&=|4 (\tau +1) (-\gamma +m+\tau +1)| \geq 2(m+3),
\end{align*}

for all $0 \leq \tau \leq \gamma$ and $\gamma \geq 0$, we estimate

\begin{align*}
& \left|	\mathfrak{E}_{\gamma m}   \right| \leq   \frac{9}{2\ssomega_{m}^2}   \sum_{ \substack{
\tau=0  }}^{\gamma}    \frac{\left( \overline{ \mathfrak{C}}_{\gamma, 2\tau+m-\gamma  , m} \right)^2 }{\left| \ssomega_{ 2\tau+m-\gamma  }^2-(\ssomega_{m}+\ssomega_{\gamma})^2     \right|}  +\frac{9}{2\ssomega_{m}^2}  \sum_{ \substack{ \tau = 0  }}^{\gamma}   \frac{\left(\overline{\mathfrak{C}}_{m,\gamma, 2\tau+m-\gamma   } \right)^2}{\left| \ssomega_{ 2\tau +m-\gamma  }^2-(\ssomega_{m}-\ssomega_{\gamma})^2     \right|} \\
& \leq \frac{9  }{2\ssomega_{m}^2} \frac{1}{4(m+3)}  \sum_{ \substack{
\tau=0  }}^{\gamma}  \left( \overline{ \mathfrak{C}}_{\gamma, 2\tau+m-\gamma  , m} \right)^2 + \frac{9  }{2\ssomega_{m}^2} \frac{1}{2(m+3)}\sum_{ \substack{
\tau=0  }}^{\gamma}  \left( \overline{ \mathfrak{C}}_{\gamma, 2\tau+m-\gamma  , m} \right)^2 \\
& = \frac{9  }{2\ssomega_{m}^2} \left(\frac{1}{4(m+3)}+\frac{1}{2(m+3)}\right) \sum_{ \substack{
\tau=0  }}^{\gamma}  \left( \overline{ \mathfrak{C}}_{\gamma, 2\tau+m-\gamma  , m} \right)^2  \\
& =\frac{9 (\gamma +2)}{70 \pi  (m+1) (m+2)^2 (m+3)^2}\Big[-3 \gamma ^4-24 \gamma ^3-40 \gamma ^2+32 \gamma +7 \gamma ^2 m^2+28 \gamma  m^2+35 m^2 \\
&+28 \gamma ^2 m+112 \gamma  m+140 m+105 \Big] :=  \mathfrak{P}_{\gamma m},
\end{align*}

where we used the closed formula for $\overline{ \mathfrak{C}}_{\gamma, 2\tau+m-\gamma  , m} $ from above. Hence, for all $m \geq 2\gamma+1$, we obtain

\begin{align}\label{ProveRHSpositive}
\frac{\mathfrak{u}_{\gamma m} }{\ssomega_{m}^2}   = \mathfrak{I}_{\gamma m}  +  \mathfrak{E}_{\gamma}  \geq  \mathfrak{I}_{\gamma m} -  \mathfrak{P}_{\gamma m}   :=\mathfrak{O}_{\gamma m}.
\end{align}

Finally, for each $\gamma \in \{0,1,2,3,4,5\}$, we use the closed formulas for $\mathfrak{C}_{\gamma\gamma mm}$ and $\mathfrak{C}_{\gamma\gamma\gamma\gamma}$ we derived in Remark <ref> to firstly explicitly compute $\mathfrak{I}_{\gamma m} $ in terms of $m$ and then explicitly compute $\mathfrak{O}_{\gamma m}$ in terms of $m$. Once the closed formula is derived, we then show that $\mathfrak{O}_{\gamma m}>0$ for all $m \geq 2\gamma+1$. For example, for $\gamma=0$, we find

\begin{align*}
\mathfrak{I}_{0 m} =	\frac{m (m+4) (5 m (m+4)+29)+66}{12 \pi  (m+1) (m+2)^2 (m+3)}
\end{align*}

and hence

\begin{align*}
\mathfrak{O}_{0 m}=\frac{5 m^4+40 m^3+109 m^2+8 m-42}{12 \pi  (m+1) (m+2)^2 (m+3)},
\end{align*}

that is greater than $10^{-3}$ provided that $m \geq 1$. Similarly, for $\gamma=1$, we compute

\begin{align*}
\mathfrak{I}_{1 m} =\frac{\left(m^2+2\right) (m (m+8)+18)}{4 \pi  (m+1) (m+2)^2 (m+3)}
\end{align*}

and hence

\begin{align*}
\mathfrak{O}_{1m}=\frac{m \left(m^4+11 m^3+44 m^2-32 m-348\right)}{4 \pi  (m+1) (m+2)^2 (m+3)^2},
\end{align*}

that is greater than $10^{-3}$ provided that $m \geq 3$.
For all the other cases with $\gamma \in \{2,3,4,5\}$, we find

\begin{align*}
\mathfrak{O}_{2m} &= \frac{109 m^5+1199 m^4+4523 m^3-30347 m^2-132936 m+107244}{600 \pi  (m+1) (m+2)^2 (m+3)^2} ,\\
\mathfrak{O}_{3m} &= \frac{43 m^5+473 m^4+1646 m^3-33554 m^2-129372 m+238248}{300 \pi  (m+1) (m+2)^2 (m+3)^2} ,\\
\mathfrak{O}_{4m} &= \frac{83 m^5+913 m^4+2851 m^3-139159 m^2-515982 m+1611198}{700 \pi  (m+1) (m+2)^2 (m+3)^2} ,\\
\mathfrak{O}_{5m} &=  \frac{17 m^5+187 m^4+505 m^3-53329 m^2-194760 m+905292}{168 \pi  (m+1) (m+2)^2 (m+3)^2}
\end{align*}

and the claim follows similarly[For $\gamma \in \{2,3\}$, the estimates $\mathfrak{O}_{\gamma m} \geq 10^{-3}$ hold true for all integers $m \geq 2\gamma+1$, whereas, for $\gamma \in \{4,5\}$, we have that $\mathfrak{O}_{\gamma m} \geq 10^{-3}$ provided that $m \geq 2\gamma+3$ instead of $m \geq 2\gamma+1$. In this case, we use the definition of the Fourier coefficients to explicitly compute $	 \mathfrak{u}_{\gamma m}  \ssomega_{m}^{-2} $ for $m\in \{2\gamma+1,2\gamma+2\}$ and verify that it is still strictly positive.]. Finally, it remains to show that the determinants

\begin{align*}
\mathfrak{D}_{\gamma n} & : =
\mathfrak{u}_{\gamma n} \mathfrak{u}_{\gamma ,2\gamma-n} - \mathfrak{v}_{\gamma n} \mathfrak{v}_{\gamma ,2\gamma-n}
\end{align*}

are all non–zero for $ n \in \{0,1,\dots, \gamma -1 \}$ that follows by a direct computation using the definition of the Fourier coefficients. Specifically, we  compute $\mathfrak{D}_{\gamma n}$ for all $\gamma \in \{0,1,2,3,4,5\}$  and $ n \in \{0,1,\dots, \gamma -1 \}$ and verify that they are all strictly negative, that completes the proof. Figure <ref>

The determinants $\mathfrak{D}_{\gamma n}  $ for $\gamma =5$ (blue), $\gamma =6$ (orange) and $\gamma =7$ (green) as $n$ varies within $ \{0,1,\dots,\gamma-1\}$. They are all in fact non–zero.

illustrates the determinants $\mathfrak{D}_{\gamma n}  $ for $\gamma \in \{5,6,7\}$ as $n$ varies within $ \{0,1,\dots,\gamma-1\}$.

[1]
Michael T. Anderson.
On the uniqueness and global dynamics of AdS spacetimes.
Classical Quantum Gravity, 23(23):6935–6953, 2006.

[2]
D. Bambusi and S. Paleari.
Families of periodic solutions of resonant PDEs.
J. Nonlinear Sci., 11(1):69–87, 2001.

[3]
J. Ben Achour, E. Huguet, J. Queva, and J. Renaud.
Explicit vector spherical harmonics on the 3-sphere.
J. Math. Phys., 57(2):023504, 7, 2016.

[4]
Massimiliano Berti and Philippe Bolle.
Periodic solutions of nonlinear wave equations with general
Comm. Math. Phys., 243(2):315–328, 2003.

[5]
Massimiliano Berti and Philippe Bolle.
Multiplicity of periodic solutions of nonlinear wave equations.
Nonlinear Anal., 56(7):1011–1046, 2004.

[6]
Massimiliano Berti and Philippe Bolle.
Cantor families of periodic solutions for completely resonant
nonlinear wave equations.
Duke Math. J., 134(2):359–419, 2006.

[7]
Piotr Bizoń, Ben Craps, Oleg Evnin, Dominika Hunik, Vincent Luyten, and
Maciej Maliborski.
Conformal flow on $\rm S^3$ and weak field integrability in $\rm
Comm. Math. Phys., 353(3):1179–1199, 2017.

[8]
Piotr Bizoń, Dominika Hunik-Kostyra, and Dmitry Pelinovsky.
Stationary states of the cubic conformal flow on $\Bbb S^3$.
Discrete Contin. Dyn. Syst., 40(1):1–32, 2020.

[9]
Piotr Bizoń, Dominika Hunik-Kostyra, and Dmitry E. Pelinovsky.
Ground state of the conformal flow on $\Bbb S^3$.
Comm. Pure Appl. Math., 72(6):1123–1151, 2019.

[10]
Piotr Bizoń and Andrzej Rostworowski.
On weakly turbulent instability of anti-de Sitter space.
Phys. Rev. Lett., 107:031102, 2011.

[11]
Piotr and Bizoń.
Talk "Instability of AdS and simple semilinear wave equations on
compact manifolds", workshop Dynamics in Geometric Dispersive Equations and
the Effects of Trapping, Scattering and Weak Turbulence, BIRS, Banff, May

[12]
Thierry Cazenave, Jalal Shatah, and A. Shadi Tahvildar-Zadeh.
Harmonic maps of the hyperbolic space and development of
singularities in wave maps and Yang-Mills fields.
Ann. Inst. H. Poincaré Phys. Théor., 68(3):315–349,

[13]
Kung Ching Chang and Chong Wei Hong.
Periodic solutions for the semilinear spherical wave equation.
Acta Math. Sinica (N.S.), 1(1):87–97, 1985.

[14]
Athanasios Chatzikaleas.
On time periodic solutions to the conformal cubic wave equation on
the Einstein cylinder.
J. Math. Phys., 61(11):111505, 32, 2020.

[15]
Matthew W. Choptuik, Jorge E. Santos, and Benson Way.
Charting islands of stability with multioscillators in anti–de
sitter space.
Phys. Rev. Lett., 121:021103, Jul 2018.

[16]
Y. Choquet-Bruhat, S. M. Paneitz, and I. E. Segal.
The Yang-Mills equations on the universal cosmos.
J. Funct. Anal., 53(2):112–150, 1983.

[17]
Yvonne Choquet-Bruhat.
Global solutions of Yang-Mills equations on anti-de Sitter
Classical Quantum Gravity, 6(12):1781–1789, 1989.

[18]
Demetrios Christodoulou and Sergiu Klainerman.
The global nonlinear stability of the Minkowski space,
volume 41 of Princeton Mathematical Series.
Princeton University Press, Princeton, NJ, 1993.

[19]
Piotr T. Chruściel and Jalal Shatah.
Global existence of solutions of the Yang-Mills equations on
globally hyperbolic four-dimensional Lorentzian manifolds.
Asian J. Math., 1(3):530–548, 1997.

[20]
Howard S. Cohl.
Report from the open problems session at OPSFA13.
SIGMA Symmetry Integrability Geom. Methods Appl., 12:Paper No.
091, 12, 2016.
Held in Gaithersburg, MD, June 4, 2015.

[21]
Mihalis Dafermos and Gustav Holzegel.
Dynamic instability of solitons in 4+1 dimensional gravity with
negative cosmological constant.
(unpublished), 2006.

[22]
Oleg Evnin.
Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS
arXiv e-prints, page arXiv:2104.09797, April 2021.

[23]
Gyula Fodor, Péter Forgács, and Philippe Grandclément.
Self-gravitating scalar breathers with a negative cosmological
Phys. Rev. D, 92:025036, Jul 2015.

[24]
Helmut Friedrich.
On the existence of $n$-geodesically complete or future complete
solutions of Einstein's field equations with smooth asymptotic structure.
Comm. Math. Phys., 107(4):587–609, 1986.

[25]
Gustav Holzegel, Jonathan Luk, Jacques Smulevici, and Claude Warnick.
Asymptotic properties of linear field equations in anti–de Sitter
Comm. Math. Phys., 374(2):1125–1178, 2020.

[26]
Wolfram Research, Inc.
Mathematica, Version 12.2.
Champaign, IL, 2020.

[27]
Maciej Maliborski and Andrzej Rostworowski.
Time-Periodic Solutions in an Einstein
AdS–Massless-Scalar-Field System.
Phys. Rev. Lett., 111:051102, 2013.

[28]
Georgios Moschidis.
A proof of the instability of Ads for the Einstein–massless
Vlasov system.
arXiv:1812.04268, 2018.

[29]
Georgios Moschidis.
A proof of the instability of AdS for the Einstein-null dust
system with an inner mirror.
Anal. PDE, 13(6):1671–1754, 2020.

[30]
Georgios Moschidis.
The instability of anti-de sitter spacetime for the einstein-scalar
field system.
Princeton University, Seminar, Nov. 2021.

[31]
Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark,
NIST handbook of mathematical functions.
U.S. Department of Commerce, National Institute of Standards and
Technology, Washington, DC; Cambridge University Press, Cambridge, 2010.
With 1 CD-ROM (Windows, Macintosh and UNIX).

[32]
Simone Paleari, Dario Bambusi, and Sergio Cacciatori.
Normal form and exponential stability for some nonlinear string
Z. Angew. Math. Phys., 52(6):1033–1052, 2001.

[33]
Paul H. Rabinowitz.
Free vibrations for a semilinear wave equation.
Comm. Pure Appl. Math., 31(1):31–68, 1978.

[34]
Paul H. Rabinowitz.
Periodic solutions of Hamiltonian systems.
Comm. Pure Appl. Math., 31(2):157–184, 1978.

[35]
Mizan Rahman.
A nonnegative representation of the linearization coefficients of the
product of Jacobi polynomials.
Canadian J. Math., 33(4):915–928, 1981.

[36]
J. Sánchez-Ruiz.
Linearization and connection formulae involving squares of
Gegenbauer polynomials.
Appl. Math. Lett., 14(3):261–267, 2001.

[37]
Jalal Shatah.
Normal forms and quadratic nonlinear Klein-Gordon equations.
Comm. Pure Appl. Math., 38(5):685–696, 1985.

[38]
Gábor Szegő.
Orthogonal polynomials.
American Mathematical Society, Providence, R.I., fourth edition,
American Mathematical Society, Colloquium Publications, Vol. XXIII.

[39]
Zhengfang Zhou.
The existence of periodic solutions of nonlinear
wave equations on S('N).
ProQuest LLC, Ann Arbor, MI, 1986.
Thesis (Ph.D.)–New York University.


# A Method for a Pseudo-Local Measurement of the Galactic Magnetic Field

Steven R. Spangler Department of Physics and Astronomy, University of Iowa

###### Abstract

Much of the information about the magnetic field in the Milky Way and other
galaxies comes from measurements which are path integrals, such as Faraday
rotation and the polarization of synchrotron radiation of cosmic ray
electrons. The measurement made at the radio telescope results from the
contributions of volume elements along a long line of sight. The inferred
magnetic field is therefore some sort of average over a long line segment. A
magnetic field measurement at a given spatial location is of much more
physical significance. In this paper, we point out that HII regions
fortuitously offer such a “point” measurement, albeit of one component of the
magnetic field, and averaged over the sightline through the HII region.
However, the line of sight (LOS) through an HII region is much smaller (e.g.
30 - 50 pc) than one through the entire Galactic disk, and thus constitutes a
“pseudo-local” measurement. We use published HII region Faraday rotation
measurements to provide a new constraint on the magnitude of
magnetohydrodynamic (MHD) turbulence in the Galaxy, as well as to raise
intriguing speculations about the modification of the Galactic field during
the star formation process.

interstellar medium — interstellar magnetic fields — interstellar plasma

##  1 1\. Introduction

It has been known for decades that the interstellar medium is threaded by a
magnetic field. This is not surprising, since almost all phases of the
interstellar medium (ISM) have sufficient level of ionization to satisfy the
conditions for a plasma parameter $\Lambda_{e}\gg 1$ (number of particles per
DeBye sphere large, Nicholson, 1983), and thus constitute plasmas. Currents
flowing in these plasmas generate a magnetic field. Measurements that provide
information on the magnitude and functional form of the Galactic field include
Faraday rotation of radio sources, both in the Galaxy and beyond, Zeeman
splitting of magnetically-sensitive transitions of the hydrogen atom and the
hydroxyl, methanol and water molecules, polarization of the Galactic
nonthermal synchrotron radiation, and polarization of visible and infrared
light due to the alignment of interstellar grains in the Galactic magnetic
field. Recent reviews on the Galactic field include Ferrière (2011) and Han
(2017).

Studies extending back over several decades indicate that the Galactic field
consists, in part, of a Galaxy-wide, large scale field that can be described,
at least empirically, by analytic functions of galactocentric coordinates, and
a superposed, spatially-random component that is plausibly interpreted as
magnetohydrodynamic (MHD) turbulence, presumably similar to that which exists
in the solar wind. An example of a study that prescribes the form of the large
scale, “deterministic” component of the field is van Eck et al. (2011).
Reviews which consider the state of knowledge of both the large scale and
turbulent components are given in Ferrière (2011) and Han (2017). A very
recent review which summarizes what we know and can speculate about the
turbulent component is Ferrière (2020).

It is probably safe to say that all observational investigations have
concluded that the turbulent component is comparable in magnitude to the large
scale component. The exact value of the magnitude of the turbulent component
depends on what is assumed for the outer scale of the turbulence. Values for
the ratio $\frac{\delta b}{B_{0}}$ where $\delta b$ is the rms value of the
turbulent component and $B_{0}$ is the magnitude of the large scale component,
range from a few tens of percent to a number in excess of unity. For example,
Ferrière (2011) and Ferrière (2020) review our knowledge of the Galactic
field, and pay attention to the relative magnitudes of the regular and
turbulent fields, citing results of $\frac{\delta b}{B_{0}}\geq 3$ from two
independent investigations. Interestingly, Beck (2016), in a review of what is
known about magnetic fields in external galaxies similar to the Milky Way,
argues that the turbulent component is substantially larger than the large
scale, deterministic component. In Section 6 of his review, Beck examines the
state of knowledge of the Galactic field in the context of results for
external galaxies. He concludes that the amplitude of isotropic magnetic
fluctuations is $5\mu G$, that of anisotropic fluctuations $2\mu G$, and that
of the regular component (ordered over Galactic scales) is $2\mu G$. It should
be emphasized that these estimates are dependent on assumptions on the outer
scale of the turbulence, i.e. the scale which distinguishes a Galactic-scale
“deterministic” component from a turbulent component. Nonetheless, these
reviews make clear that there is substantial observational evidence for a
turbulent component to the Galactic magnetic field that is comparable in
magnitude, or even larger than an ordered, large scale field. In this paper,
we will consider the dimensionless turbulent amplitude $\frac{\delta
b}{B_{0}}$ to be a free parameter, subject to the observational constraints
cited above. Knowledge of the magnitude and spectrum of the Galactic MHD
turbulence is important to Galactic astrophysics, and astrophysics in general,
since this turbulence contributes to processes such as heating of the ISM
through dissipation of turbulence, confinement and propagation of the cosmic
rays, turbulent transport of heavy elements, and modification or regulation of
the star formation process.

One of the difficulties in determining the relative roles of a large scale and
a turbulent component of the Galactic field is the fact that some of the main
diagnostic techniques such as Faraday rotation, polarization of Galactic
synchrotron radiation, and dust polarization consist of path integrals over
long lines of sight through the Galaxy. What this means is that we get some
sort of average of the Galactic field over a long line of sight (LOS) through
the Galaxy, rather than the value at a set of points along that line of sight.
Zeeman splitting of OH and methanol maser transitions are not subject to this
limitation, since the region of emission is very localized (Crutcher and
Kemball, 2019). However, as discussed in Crutcher and Kemball (2019) regions
of maser emission have very high densities compared to the general ISM, and
are products of the star formation process. We will return to this point in
Section 4.

In this paper, we discuss a type of observation which utilizes the Faraday
rotation technique to obtain an estimate of the Galactic field (strictly
speaking one component of the field) in a localized region, rather than an
entire line of sight through the Galaxy. The localized region is comprised of
the volume of an HII region. In this sense, the technique provides a “local”
measurement. In what follows, we first summarize the basis of the technique of
Faraday rotation, then discuss recent observational results which allow it to
be used in a “local” measurement of the Galactic field.

Faraday rotation is a fundamental plasma process which consists of rotation in
the plane of polarization of radio waves due to the presence of a magnetized
plasma (Nicholson, 1983). It is used as a diagnostic technique in laboratory
plasmas as well as in radio astronomy. In the present context, it is applied
to radio waves emitted by extragalactic radio sources and which propagate
through the plasma of the interstellar medium (ISM). The change in the
position angle relative to what would be measured in the absence of the
plasma, $\Delta\chi$, is given by

$\Delta\chi=\left[\left(\frac{e^{3}}{2\pi
m_{e}^{2}c^{4}}\right)\int_{L}n\vec{B}\cdot\vec{ds}\right]\lambda^{2}$ (1)

In Equation (1) $e,m_{e},\mbox{ and }c$ are the customary fundamental physical
constants, $n$ is the electron density, $\vec{B}$ is the vector magnetic
field, both functions of position along the line of sight, and $\vec{ds}$ is
an increment in the line of sight from the source to the radio telescope.
Faraday rotation is a path integral measurement rather than a point
measurement, so there are contributions to plasma throughout the ISM along the
line of sight. The quantity in square brackets in Equation (1) is termed the
rotation measure (RM), and is usually quoted in SI units of radians/m2. For
most lines of sight through the ISM, the magnetic field measured is not only
an average over a long line through the ISM, it is also an average weighted by
the plasma density. Information on many lines of sight around the sky is used
to obtain estimates of the large scale structure of the Galactic field, as
well as the contribution of the turbulent component.

The observational impetus for this paper is a set of studies of the HII
regions Rosette Nebula and W4 by A. Costa and coworkers (Savage et al., 2013;
Costa et al., 2016; Costa and Spangler, 2018). Costa and co-workers carried
out Faraday rotation measurements with the Very Large Array (VLA) of
background radio sources viewed through these two HII regions. The main
results of these investigations were: (1) rotation measures (RM) for sources
viewed through the HII regions were typically much larger than for nearby
sources whose lines of sight probed the Galaxy, but not the HII regions, (2)
although there were large variations in the RM from source-to-source when the
LOS passed through the HII region, there was a well-defined average RM
(magnitude and sign) for the HII region, particularly the Rosette, (3) the
sign of the HII region-induced RM (which dominated the total RM) was
consistent with the sign of the large scale Galactic field in that part of the
sky. The RM due to the Rosette Nebula (Galactic longitude = 206.5 degrees) was
dominantly positive, while that for W4 (Galactic longitude = 135 degrees) was
dominantly negative.

The important point is that the magnetic field in the “RM anomaly” associated
with the HII region is confined to the volume of the HII region rather than
being an average over a line of sight of many kiloparsecs in the Galactic
plane. The radii of both HII regions are approximately 20-25 parsecs, so the
magnetic field is “local” in this sense. As discussed in the text earlier in
this section, in view of the apparent large amplitude, turbulent component of
the Galactic field, a reasonable a-priori expectation would be that the
magnetic field at a point in the Galactic plane, and averaged over a volume of
radius 20 - 25 parsecs, would have poor correlation with the ordered, large
scale Galactic field. The observations of the Rosette Nebula and W4 do not
bear out this expectation. In this paper, we explore the consequences of this
result for our inference about the Galactic magnetic field.

##  2 2\. Additional Observational Results on Faraday Rotation through HII
Regions

Two measurements is, of course, very poor statistics, even by the base
standards of astronomy. There is a need for additional measurements of the
sort presented by Costa and collaborators. Such measurements were provided by
the observations of Harvey-Smith et al. (2011), who carried out similar
observations, for similar goals, of somewhat larger HII regions like the
$\lambda$ Orionis HII region. In collecting HII regions from the sample of
Harvey-Smith et al. (2011), we restricted attention to objects in the outer
two quadrants of the Galactic plane, i.e. those with $90^{\circ}\leq l\leq
270^{\circ}$. Harvey-Smith et al. (2011) reported the the average value of the
line-of-sight component of the magnetic field $<B_{\parallel}>$ for each HII
region, and this quantity was also available from the analyses of Savage et
al. (2013); Costa et al. (2016); Costa and Spangler (2018).

In the studies of Costa and coworkers, the mean LOS component of the field,
$<B_{\parallel}>$, was not explicitly measured; analyses of the data
concentrated on fitting plasma shell models to the sets of RM values for each
HII region. In addition to other parameters, these models were dependent on
the magnitude and orientation of the magnetic field outside the HII region
(see Section 5.1 of Costa et al. (2016)). A limiting type of model was one in
which there was no amplification of the external magnetic field in the HII
region (referred to by Costa as the “Harvey-Smith model”). In this case, the
model fits give exactly the same quantity as the parameter $<B_{\parallel}>$
in Harvey-Smith et al. (2011). We took values from Table 4 of Costa et al.
(2016) and Table 6 of Costa and Spangler (2018) to obtain values of
$<B_{\parallel}>$, which should be directly comparable to this quantity in
Harvey-Smith et al. (2011). Once again, $<B_{\parallel}>$ has a sign and a
magnitude. The values of $<B_{\parallel}>$ from Harvey-Smith et al. (2011),
Costa et al. (2016), and Costa and Spangler (2018) are plotted in Figure 1,
and given in Table 1.

Figure 1: Measurements of $<B_{\parallel}>$ from Faraday rotation measurements through 6 HII regions in the 2nd and 3rd Galactic quadrants. Solid (blue) symbols indicate $<B_{\parallel}>>0$ (Field pointing towards observer), and open (red) symbols indicate $<B_{\parallel}><0$ (field pointing away from observer). The size of the plotted symbol is proportional to $\sqrt{|<B_{\parallel}>|}$. The locations of the 6 sources used are plotted in Galactic coordinates. In the simplest case of a purely azimuthal field, the projection of the large scale Galactic field on the LOS would change sign at $l=180^{\circ}$. In all cases, the polarity of $<B_{\parallel}>$ is the same as that for the large scale Galactic field in that quadrant. Table 1: HII Region Average Magnetic Field Components HII region | l | b | $<B_{\parallel}>\mu G$ | Ref.
---|---|---|---|---
Rosette | 206.5 | -2.1 | +2.73 | Costa et al. (2016)
W4 | 135 | +1.0 | -2.29 | Costa and Spangler (2018)
Sh2-264 | 195 | -12 | +2.2 | Harvey-Smith et al. (2011)
Sivan 3 | 144.5 | +14 | -2.5 | Harvey-Smith et al. (2011)
Sh2-171 | 118.0 | +6 | -2.3 | Harvey-Smith et al. (2011)
Sh2-220 | 161.3 | -12.5 | -6.3 | Harvey-Smith et al. (2011)

The clear result from Figure 1 is that for all 6 HII regions, the “local”
magnetic field defined by the average over the volume of the HII region, has
the same polarity as the Galactic-scale, mean magnetic field. In none of these
6 cases are fluctuations on scales of a few tens of parsecs capable of making
the “local” polarity opposite to the average over several kiloparsecs.

##  3 3\. Statistics of a Local, Line of Sight Measurement of $\vec{B}$ in a
Turbulent Medium

In this section, we discuss the statistics of the LOS component of the
magnetic field at a given point, when the magnetic field consists of the
vector sum of a large scale, ordered magnetic field $\vec{B_{0}}$, and a
superposed, spatially random turbulent field $\vec{\delta b}$. For ease of
following the discussion, the Appendix gives a glossary of mathematical
symbols and variables used in the following.

###  3.1 Geometry of the Line of Sight and the Galactic Magnetic Field

The geometry of the situation is shown in Figure 2. The z axis is defined by
the path from the radio source through the HII region and to the observer,
with the positive direction towards the observer. The x axis is in the plane
of the sky, and points toward decreasing galactic longitude. We also simplify
the situation by adopting the good approximation that the x-z plane coincides
with the Galactic plane. Finally, the y axis is perpendicular to the Galactic
plane and completes a right-handed coordinate system.

Figure 2: Cartoon illustrating geometry of observer, polarized extragalactic
radio source, and HII region. A Galactic magnetic field is illustrated,
consisting of a large scale, ordered component $\vec{B}_{0}$ inclined at an
angle $\Theta$ with respect to the line of sight, and a superposed turbulent
component $\vec{\delta b}$. The plane of the paper is the Galactic plane, the
z axis is defined by a ray from the radio source to the observer. The
coordinates x and y complete a right-handed coordinate system. In this
illustration, only one component of the turbulent field is shown, $\delta
b_{p}$, which is the component perpendicular to the large scale field and in
the Galactic plane. To lend realism, the trace shown represents 6 hours of
turbulent solar wind magnetic fluctuations measured by the WIND spacecraft on
September 25, 2020.

The observational results of Harvey-Smith et al. (2011); Savage et al. (2013);
Costa et al. (2016); Costa and Spangler (2018) show that there is a well-
defined mean field throughout the HII regions, with a magnitude equal to or
greater than the general Galactic field. Given the known presence of plasma
turbulence in the interstellar medium (ISM), we interpret this average HII
region field to be the vector sum of the large scale galactic field
$\vec{B_{0}}$ and a turbulent field $\vec{\delta b}$. As indicated in Figure
2, the angle between the line-of-sight (LOS) and the large scale field is
$\Theta$. We also assume, for this analysis, that the large-scale field is in
the Galactic plane.

Given the configuration illustrated in Figure 2, we define a second coordinate
system which is convenient for describing the turbulence. The $\parallel$
direction is in the direction of the mean field, the $p$ direction is
perpendicular to the mean field, and in the Galactic plane, and the $\perp$
direction completes the right-hand coordinate system and is perpendicular to
the Galactic plane. These three directions are defined by the unit vectors
$\hat{e}_{\parallel}$, $\hat{e}_{p}$, and $\hat{e}_{\perp}$.

###  3.2 Model for the Magnetic Field in the Vicinity of an HII Region

Given this latter coordinate system, we express the magnetic field (mean field
plus turbulence) by

$\vec{B}=(B_{0}+\delta b_{\parallel})\hat{e}_{\parallel}+\delta
b_{p}\hat{e}_{p}+\delta b_{\perp}\hat{e}_{\perp}$ (2)

In analogy with the Alfvènic turbulence in the solar wind, we expect $<(\delta
b_{p})^{2}>=<(\delta b_{\perp})^{2}>$ and $<(\delta b_{p})^{2}>\gg<(\delta
b_{\parallel})^{2}>$.

The model we adopt is that $\delta b_{\parallel},\delta b_{p},\mbox{ and
}\delta b_{\perp}$ are random, 0-mean quantities with an amplitude dominated
by fluctuations on the outer scale l, $l\geq 2R$, where $R$ is the radius of
the HII region. In the cases of the Rosette Nebula (Costa et al., 2016) and W4
(Costa and Spangler, 2018), the values of $2R$ are 38 and 50 parsecs,
respectively. These values are comparable to, or slightly less than the outer
scale of the 2D component of interstellar turbulence claimed by Minter and
Spangler (1996).

We adopt a very simple model for the average observed RM to a set of
extragalactic radio sources viewed through an HII region,

$RM_{obs}=RM_{bck}+RM_{neb}$ (3)

where $RM_{bck}$ is the background RM due to the plasma in the Galaxy which
would be observed in the absence of the nebula, and $RM_{neb}$ is the average
RM for lines of sight through the HII region. $RM_{neb}$ is given by the
simple expression

$RM_{neb}=2CnRAB_{z}$ (4)

where $C$ is the set of fundamental constants in curved brackets in Equation
(1), $n$ is the mean electron density in the ionized portion of the HII region
(determined by independent measurements such as intensity of thermal radio
emission in Savage et al. (2013); Costa et al. (2016); Costa and Spangler
(2018)), $R$ again is the radius of the HII region, and $B_{z}$ is the z
component of the interstellar field (mean field plus turbulence) at the
location of the HII region. The variable $A\geq 1$ is a factor describing
whether the magnetic field within the HII region is amplified over the value
in the surrounding ISM. Harvey-Smith et al. (2011) assumed that it was not
($A=1$), while Costa et al. (2016); Costa and Spangler (2018) considered
physically-based models in which $A\leq 4$. The case $A=1$ was considered as
well.

$B_{z}$ is obtained from Equation (2) as

$B_{z}=\vec{B}\cdot\hat{e}_{z}$ (5)

Substituting Equation (2) into this expression and evaluating the dot products
of unit vectors gives

$B_{z}=(B_{0}+\delta b_{\parallel})\cos\Theta-\delta b_{p}\sin\Theta$ (6)

The $\delta b_{\perp}$ component of the turbulent field makes no contribution
because, for lines of sight in or close to the Galactic plane, it is
perpendicular to the LOS. The negative sign in the second term arises because
positive $\delta b_{p}$ is pointing away from the observer (Figure 2).

Equation (6) shows the important, though obvious fact that the polarity of
$B_{z}$ (i.e. the same as $\vec{B_{0}}$ or opposite) depends on both the
amplitude of the fluctuations relative to the mean field as well as the angle
$\Theta$ between the large scale field and the LOS.

This expression can be rewritten in terms of dimensionless turbulence
amplitudes (turbulent “modulation indices”) as

$\displaystyle B_{z}=B_{0}(\cos\Theta+x\cos\Theta-y\sin\Theta)$ (7)
$\displaystyle x\equiv\frac{\delta b_{\parallel}}{B_{0}}$ (8) $\displaystyle
y\equiv\frac{\delta b_{p}}{B_{0}}$ (9)

where $x,y$ are dimensionless, 0 mean random variables with assumed properties
$\sqrt{<x^{2}>},\sqrt{<y^{2}>}\leq 1$. With all of this, the observed RM is
then

$\displaystyle
RM_{obs}=RM_{bck}+2CnRAB_{0}(\cos\Theta+x\cos\Theta-y\sin\Theta)$ (10)
$\displaystyle RM_{obs}=RM_{bck}+RM_{n}(\cos\Theta+x\cos\Theta-y\sin\Theta)$
(11)

with an obvious definition of the net nebular $RM_{n}$. It is worth
emphasizing that $RM_{n}$ is a property of the nebula itself, independent of
the ISM or the geometry of the line of sight.

The question considered in this paper then is whether the second term has the
same sign as the first, and if not, if it is large enough to cause the
observed RM to have the opposite sign as $RM_{bck}$. Equation (11) shows that
this depends on two factors.

  1. 1.

The first is the relative magnitudes of $RM_{bck}$ and $RM_{n}$. Savage et al.
(2013); Costa et al. (2016); Costa and Spangler (2018) found that for both the
Rosette Nebula and W4, the largest absolute magnitudes of the RMs through the
nebula were of the order of 1000-1500 rad/m2, whereas the background Galactic
RMs in those parts of the sky had absolute magnitudes of the order of 100 -
150 rad/m2. This indicates that the magnitude of $RM_{n}$ could be as high as
10 times that of $RM_{bck}$.

  2. 2.

The statistics of the quantity

$\xi=\cos\Theta+x\cos\Theta-y\sin\Theta$ (12)

are obviously crucial. For the observed RM to be of the opposite polarity to
$RM_{bck}$, it is necessary that $\xi<0$, and in fact $\xi<-\Delta$ where
$\Delta$ is defined as the ratio of background to nebular RM.

For the remainder of this section, we turn to a discussion of the statistics
of $\xi$.

###  3.3 Statistics of the Random Variable $\xi$

We begin by making a further simplification and ignoring the x term in $\xi$.
The justification for this is the assumption stated above that for Alfvènic
turbulence, the fluctuations in the parallel component (to the mean field) are
small compared to the transverse components. In addition, both the mean field
and the parallel component contributions to the stochastic variable $\xi$ are
proportional to $\cos\Theta$. We therefore proceed to consider the statistics
of the variable

$\xi=\cos\Theta-y\sin\Theta$ (13)

Only $y$ is a random variable, so we model it as a Gaussian-distributed,
random variable with zero mean, with a probability distribution function

$p(y)=\frac{1}{\sqrt{2\pi}\sigma}\exp(-y^{2}/2\sigma^{2})$ (14)

The quantity $\sigma$ is the dimensionless amplitude of the fluctuations of
the p component of the turbulence, and is a crucial parameter in our analysis.

Given this expression we can generate the probability distribution function
(pdf) of the stochastic variable $\xi$

$p(\xi)=\frac{1}{\sqrt{2\pi}\sigma}\frac{1}{\sin\Theta}\exp\left(-\frac{(\cos\Theta-\xi)^{2}}{2\sigma^{2}\sin^{2}\Theta}\right)$
(15)

For economy of notation, let $a\equiv\cos\Theta$, $b\equiv\sin\Theta$, with
the obvious identity $b=\sqrt{1-a^{2}}$, so that

$\displaystyle p(\xi)=\frac{1}{\sqrt{2\pi}\sigma
b}\exp\left(-\frac{(\xi-a)^{2}}{2\sigma^{2}b^{2}}\right)$ (16) $\displaystyle
p(\xi)=\frac{1}{\sqrt{2\pi}d}\exp\left(-\frac{(\xi-a)^{2}}{2d^{2}}\right)$
(17)

with $d\equiv\sigma b$. The distribution function of $\xi$ is also Gaussian,
with a nonzero mean value $<\xi>=a=\cos\Theta$.

A sample plot of $p(\xi)$ is shown in Figure 2. This figure shows that for
sufficiently large-amplitude turbulence ($\sigma=0.70$ in this case), and
sufficiently large angles of the mean field with respect to the line of sight
($\Theta=70^{\circ}$) the probability of the local field being reversed with
respect to the mean field is substantial.

Figure 3: The probability distribution function of the variable $\xi$ for the
case in which the dimensionless amplitude $\sigma$ of the magnetic
fluctuations in the p component of the magnetic field is $\sigma=0.70$ and the
angle $\Theta$ between the line of sight and the mean magnetic field is 70
degrees. In this case, the probability of $\xi$ being negative is substantial.

###  3.4 Expression for $P_{-}$, the Probability of a Polarity Reversal

Given the expression in Equation (16) or (17) for the probability distribution
function of the parameter $\xi$, we have the quantity of interest, the total
probability that the line-of-sight component of the magnetic field at a given
location in the ISM will be reversed from that of the mean field, We note this
quantity by $P_{-}$, and the obvious expression for it is

$P_{-}=\int_{-\infty}^{-\Delta}d\xi p(\xi)$ (18)

By inspection, it can be seen that this integral is the same as

$P_{-}=\int_{-\infty}^{-(\Delta+a)}dyp_{G}(y,d)=\int_{\Delta+a}^{\infty}dyp_{G}(y,d)$
(19)

where $p_{G}(y,d)$ is a zero mean, normalized Gaussian pdf with standard
deviation $d$.

If we make a final change of variables, $y\rightarrow
t\equiv\frac{y}{\sqrt{2}d}$, then Equation (19) becomes

$\displaystyle P_{-}=\frac{1}{\sqrt{\pi}}\int_{X}^{\infty}dte^{-t^{2}}$ (20)
$\displaystyle
X\equiv\frac{\Delta+a}{\sqrt{2}d}=\frac{(\Delta+\cos\Theta)}{\sqrt{2}\sigma\sin\Theta}$
(21)

Equation (20) is very close to the standard form of the Error Function
(Beckmann, 1967)

$erf(X)\equiv\frac{2}{\sqrt{\pi}}\int_{0}^{X}dte^{-t^{2}}$ (22)

Comparison of Equations (20) and (22) allows the final, compact formula for
the total probability of polarity reversal,

$P_{-}=\frac{1}{2}\left[1-erf(X)\right]=\frac{1}{2}erfc(X)$ (23)

where $erfc(X)$ is the complement of the error function.

Equation (23) is plotted in Figure 4.

Figure 4: The probability that the line of sight component of the magnetic
field at the location of an HII region is opposite to that of the global,
Galactic field. The probability $P_{-}$ is a function of the parameter
$X=\frac{(\Delta+\cos\Theta)}{\sqrt{2}\sigma\sin\Theta}$

.

As the argument $X\rightarrow 0$, the probability of reversal
$P_{-}\rightarrow 0.5$. This makes complete sense, since small argument $X$
corresponds to strong turbulence and/or a mean field nearly perpendicular to
the line of sight. In this case, the local magnetic field is dominated by the
turbulent component $\delta b_{p}$, which has a 50 percent change of being
positive and and a 50 percent chance of being negative.

The case of $X\rightarrow\infty$ corresponds to weak turbulence, and/or a mean
field increasingly directed towards the observer. In this case, the
probability of a reversal in the line-of-sight component of the magnetic field
approaches zero.

The probability of a reversal in the line of sight component is determined by
the parameter $X$, and it is of interest to calculate this quantity for some
observationally-relevant cases.

###  3.5 Calculations Relevant to Observations of HII Regions

In this section, we briefly apply these formulas to the observations presented
in Section 2. A total of 6 HII regions distributed over a range of Galactic
longitudes precludes a convincing and sophisticated statistical analysis.
However, the results presented in Figure 1 suggest that theoretical values of
$P_{-}\geq 0.20$ would begin to have an uncomfortable confrontation with the
observations. On the other hand, if characteristics of the HII regions,
amplitude of ISM turbulence and direction of the line of sight indicate a
value of $P_{-}\sim 0.01$, our calculations are completely consistent with the
observations discussed in Section 2. The formula given in Eq. (21) shows that
the parameter $X$ depends on $\Theta$, $\sigma$, and $\Delta$. In what
follows, we adopt a value of $\Delta=0.1$, which is in accord with the
observations of the Rosette Nebula and W4 discussed above.

Savage et al. (2013); Costa et al. (2016) in their study of the Rosette
nebula, found that reasonable fits of simple magnetized shell models to the
set of RM measurements gave a value of $\Theta\simeq 70^{\circ}$ if the
interstellar magnetic field is amplified in the shell by shock compression,
and $\Theta\simeq 50^{\circ}$ if there were no change in the magnitude of the
magnetic field in the shell, with respect to the general ISM (see Table 5 of
Costa et al., 2016). Savage et al. (2013) noted that an independent estimate,
based on models of the Galactic magnetic field, would give $\Theta\simeq
60^{\circ}$ at the location of the Rosette. Costa and Spangler (2018) used a
value of $\Theta=55^{\circ}$ for W4, if the Galactic magnetic field is
approximated as azimuthal. Thus values of $50^{\circ}\leq\Theta\leq
70^{\circ}$ are relevant to the results shown in Figure 1.

The amplitude of the turbulent magnetic field on spatial scales of tens of
parsecs is not well known, but results from the literature discussed in
Section 1 indicate that it is of order the mean, large scale field. Thus our
parameter $\sigma$ if probably of the order of unity, or a large fraction
thereof. A value of $\sigma=0.70\simeq\frac{1}{\sqrt{2}}$ would have the
magnitude of fluctuating field (quadratic sum of $\delta b_{p}$ and $\delta
b_{\perp}$ components) equal to the magnitude of the large scale field.

A value of $\sigma=0.70$ and $\Theta=70^{\circ}$ gives $X=0.48$; reference to
Figure 4 shows that the probability of reversal $P_{-}$ is about 24 %. In such
a case it is mildly surprising that all 6 HII regions discussed in Section 2
produce “Faraday rotation anomalies” with the same polarity as the large scale
field. On the other hand, a slightly smaller value of $\sigma=0.50$ and
$\Theta=50^{\circ}$, also consistent with the Costa et al. (2016) results for
the Rosette Nebula and W4, would have $X=1.37$. In this case, Figure 4 shows
that the probability of a polarity reversal for a single source is less than 5
%, and the results of Figure 1 are unremarkable.

This discussion shows that future investigations on more HII regions could
provide more stringent limits on the large scale turbulent fluctuations in the
Galactic magnetic field. The formulas presented in this section provide a
mathematical vocabulary for discussing current as well as future observations.
For example, it is clear that the probability of a reversal becomes much
greater for $\Theta\sim\frac{\pi}{2}$. HII regions in the close vicinity of
the Galactic anticenter would provide particularly sensitive diagnostics for
the amplitude of large scale turbulence.

##  4 4\. Conclusions from Zeeman Effect Measurements of Star Formation Region
Masers

The analysis in Section 3 assumes that HII regions occur in the presence of an
unmodified Galactic magnetic field. In the simplified stellar bubble models
presented in Savage et al. (2013); Costa et al. (2016); Costa and Spangler
(2018) the expanding bubble modifies the Galactic field, but the RM
enhancement due to the bubble is proportional to the Galactic field (mean
field plus turbulent contribution at the location of the HII region) in the
absence of the HII region. In the model considered by Harvey-Smith et al.
(2011) the HII region consists of an ionization of the interstellar gas with
no modification of the Galactic field. Harvey-Smith et al. (2011) describe
this as the HII region “lighting up” the Galactic field that exists at the
location of the HII region.

These viewpoints may be physically naive. HII regions are produced by hot,
massive stars that are products of the star formation process. Star formation
regions, in turn, result from contraction of gas from the general ISM, its
conversion to molecular form, and an associated increase in gas density. It
therefore seems likely that the process of forming dense clumps in molecular
clouds from which stars form results in major modification of the Galactic
field, perhaps to the extent of totally randomizing the vector field from its
value in the absence of the star formation region. In this light, the
observational results presented in Section 2 are even more remarkable; the
preservation of the polarity of the large scale field occurs in spite of both
the presence of natural turbulence in the Galactic field, as well as the
possibly major variations to the field induced by the star formation process.

The HII region results presented in Section 2 are not the only “local”
measurements of the magnetic field in the ISM. Hydroxyl, methanol, and water
masers in star formation regions display the Zeeman effect, permitting
measurements of the magnitude and sign of the LOS component of the magnetic
field in the maser region (Crutcher and Kemball, 2019). These masers occur in
regions of very high density compared to the ISM far from star formation
regions, and thus probe gas that has been substantially modified in density,
and presumably magnetic field, by the star formation process (Crutcher and
Kemball, 2019).

In spite of this, observations of the polarity of the magnetic field in maser
regions show results similar to those presented in Section 2. Fish et al.
(2003) examined Zeeman effect measurements for more than 50 massive star
formation regions. Although they did not find a general Galactic-wide magnetic
field direction, they did find that “ …in the solar neighborhood the magnetic
field outside the solar circle is oriented clockwise as viewed from the north
Galactic pole …”. This is the same sense as shown in Figure 1 on the basis of
the HII region results. A subsequent study by Green et al. (2012) yielded
similar results. Green et al. (2012) reported measurements of the magnetic
field (sign and magnitude of the line-of-sight component of the magnetic
field) for 14 Zeeman pairs in 6 high mass star formation regions in the
Carina-Sagittarius arm (4th Galactic quadrant). Their results show a dominant
magnetic polarity in this spiral arm (See Figure 7 of Green et al. (2012) .
The significance of these observational results was summarized in Han (2017),
who noted that they implied “turbulence in molecular clouds and violent star-
forming regions seem not to alter the mean magnetic field direction, although
the field strength is much enhanced from a few $\mu$ G to a few mG ”.

##  5 5\. Speculations from the Study of Hydrodynamical Turbulence

The results for the HII regions presented above, and illustrated in Figure 1
indicate, at the very least, potentially significant limits on the magnitude
of turbulent magnetic field fluctuations in the Galactic field. The results on
star formation region OH masers from Fish et al. (2003) and Green et al.
(2012) indicate, and the HII regions results may indicate, a far stronger and
mysterious physical process: a memory of the large scale polarity of the
Galactic field as it is modified during the contraction and increase in
density associated with star formation.

The obvious question which then arises is the meaning of “the persistence of
memory” in the large scale, Galactic magnetic field. The problem arises
because of the assumed isotropic nature of turbulent fluctuations on scales
much smaller than the outer scale of the turbulence. Such fluctuations, deep
in the inertial subrange or close to the dissipation range, should have no
tendency to respect large scale asymmetries in the Galactic plasma. It is
worthwhile to look for insight from the study of hydrodynamical turbulence.
There are obvious physical differences between the turbulence in the diffuse
plasmas of the interstellar medium and the turbulence in gases and liquids in
laboratory and industrial settings. However, there are important similarities,
too, and both the quality of the diagnostics and the shear amount of effort
expended in the study of hydrodynamic turbulence exceeds that which is
possible for astrophysical turbulence.

A review of our understanding of “passive scalars” in hydrodynamic turbulence
is presented in Warhaft (2000). Passive scalars are quantities which are
convected by fluid turbulence, without affecting the flow field. Warhaft
(2000) considered as a prime example temperature fluctuations in hydrodynamic
turbulence. An astrophysical example of a passive scalar could be the plasma
density fluctuations responsible for interstellar radio scintillations.
Warhaft (2000) presents an illuminating discussion of the differences between
the engineering and physics views of turbulence, and offers an admonition that
is certainly relevant for Galactic astronomers as well: “ The physicists, with
their quest for universality, have concentrated on inertial and dissipation
scales, the hope being, if the Reynolds (Re) and Peclet (Pe) numbers are high
enough, that the small-scale behavior will be independent of the large scales.
We address both the large and small scales in this review and argue that this
division is artificial and dangerous”. Warhaft (2000) then goes to present
laboratory and theoretical evidence that small scale passive scalar
fluctuations (primarily temperature fluctuations) are anisotropic with an
anisotropy that reflects large scale asymmetry. In the cases discussed by
Warhaft (2000) this large scale asymmetry was most typically a transverse
temperature gradient imposed on a turbulent shear flow. Warhaft (2000) also
shows that the small scale passive scalar fluctuations possess a skewness and
kurtosis that exceeds that present in the fundamental turbulent velocity
field.

It is not clear to me if the clear and insightful presentation of Warhaft
(2000) holds a key to understanding the mysterious observation described in
this paper, or if any such comparison would be a hallucinogenic extrapolation
of laboratory results far outside their range of applicability. While
Warhaft’s review does describe a documented persistence of large scale
anisotropy on small turbulent scales, as appears to be indicated in our
Galactic ISM observations, it must be emphasized that the discussion of
Warhaft (2000) is concerned with passive scalars in the turbulence. In the
Galactic ISM context, that would be plasma density, temperature, ionization
fraction, and metallicity. The magnetic field, on the other hand, should share
status with fluid velocity as a “primary excitation” of the turbulence, if the
ISM turbulence is Alfvénic. Nonetheless, it would be worthwhile to consider
further if the extensive, laboratory based studies of turbulence from an
engineering perspective can illuminate the nature of plasma turbulence in the
interstellar medium.

##  6 6\. Summary and Conclusions

  1. 1.

In the case of the Galactic-anticenter-direction HII regions Rosette Nebula
and W4, the mean magnetic field in the HII region has the same polarity as the
general Galactic field in that part of space, despite the presence of strong
MHD turbulence which would be expected to randomize the polarity. An
additional 4 HII regions studied by Harvey-Smith et al (2011) share this
property.

  2. 2.

We derive formulas for the probability of the “point” magnetic field having
the same polarity as the general field, which depends on properties of the
turbulent magnetic fluctuations as well as the details of the line of sight.
For reasonable estimates, this probability could be non-negligible, being as
high as 25 % for plausible characteristics. As a result, the observation of 6
anticenter-direction HII regions, all having the same magnetic polarity of the
general field, becomes mildly curious.

  3. 3.

The results on HII regions support similar results on the polarity of star
formation OH masers presented by Fish et al. (2003) and Green et al. (2012).
In those studies, the polarity of the magnetic field in maser regions is
correlated among star formation regions that are spatially separated, and the
polarity is apparently determined by that of the large scale Galactic field.
The maser region results not only limit the properties of turbulent
fluctuations in the Galactic field, but also restrict the field randomization
that presumably occurs in the assembly of star formation cores during the
process of star formation.

  4. 4.

We point out that studies of turbulence in laboratory settings also show a
persistence of large scale asymmetries in small and inertial scale
fluctuations, which could be termed “the persistence of memory” in the
Galactic magnetic field. These results are discussed in Warhaft (2000), and
refer to passive scalars in hydrodynamic turbulence. Although it is unclear if
the results discussed in Warhaft (2000) are applicable to Galactic MHD
turbulence, it would appear to be a worthwhile path to investigate.

## References

  * Beck (2016) Beck, R. 2016, A&A Rev., 24, 4
  * Beckmann (1967) Beckmann, P. 1967, Probability in Communication Engineering, Harcourt, Brace & World, p36
  * Costa et al. (2016) Costa, A.H., Spangler, S.R., Sink, J.R. et al. 2016, ApJ, 821, 92
  * Costa and Spangler (2018) Costa, A.H. and Spangler, S.R. 2018, ApJ, 865, 65
  * Crutcher and Kemball (2019) Crutcher, R.M. and Kemball, A.J. 2019, Frontiers in Astronomy and Space Science, 6, 66
  * Ferrière (2011) Ferrière, K. 2011, Mem. Soc. Astron. Italiana, 82, 824
  * Ferrière (2020) Ferrière, K. 2020, Plasma Physics and Controlled Fusion, 62, 014014
  * Fish et al. (2003) Fish, V.L., Reid, M.J., Argon, A.L., and Menten, K.M. 2003, ApJ, 596, 328
  * Green et al. (2012) Green, J.A., McClure-Griffiths, N.M., Caswell, J.L. et al. 2012, MNRAS, 425, 2530
  * Han (2017) Han, J.L. 2017, ARA&A, 55, 111
  * Harvey-Smith et al. (2011) Harvey-Smith, L., Madsen, G.J., and Gaensler, B.M. 2011, ApJ, 736, 83
  * Minter and Spangler (1996) Minter, A.H. and Spangler, S.R. 1996, ApJ, 458, 194
  * Nicholson (1983) Nicholson, D.R.. 1983, Introduction to Plasma Theory, John Wiley & Sons, p3
  * Savage et al. (2013) Savage, A.H., Spangler, S.R., and Fischer, P.D. 2013, ApJ, 765, 42
  * van Eck et al. (2011) van Eck, C.L., Brown, J.C., Stil, J.M. et al. 2011, ApJ, 728, 97
  * Warhaft (2000) Warhaft, Z. 2000, Annual Review of Fluid Mechanics, 32, 203

##  Appendix A Glossary of Mathematical Variables

Variable | Definition
---|---
$\Delta\chi$ | change in polarization position angle due to Faraday rotation (Eq.1)
$\vec{B}$ | total magnetic field in ISM at location of HII region (Eq.2)
$\vec{B}_{0}$ | large scale, ordered component of Galactic magnetic field
$\vec{\delta b}$ | turbulent component of Galactic magnetic field
$\Theta$ | angle between LOS and large scale magnetic field $\vec{B_{0}}$ at location of HII region (Fig. 2)
$(\hat{e}_{\parallel},\hat{e}_{p},\hat{e}_{\perp})$ | unit vectors defining coordinate system aligned with large scale field $\vec{B_{0}}$
$(\delta b_{\parallel},\delta b_{p},\delta b_{\perp})$ | components of the turbulent magnetic field in a coordinate system aligned with large scale field $\vec{B_{0}}$
$RM_{obs}$ | observed RM for a background source observed through an HII region (Eq.3)
$RM_{bck}$ | RM due to ISM along LOS, but excluding HII region (Eq.3)
$RM_{neb}$ | RM due to plasma in HII region (Eq.3)
$C$ | set of atomic constants appearing in expression for Faraday rotation (Eq. 1)
$n$ | electron density in plasma causing Faraday rotation (Eq. 1)
$R$ | radius of HII region
$B_{z}$ | LOS component of Galactic magnetic field at location of HII region
$A$ | factor ($\geq 1$) determining whether Galactic field is amplified within HII region. Model dependent (Eq. 4)
$\hat{e}_{z}$ | unit vector in direction of LOS
$x$ | dimensionless amplitude of turbulent magnetic fluctuations in the direction of the large scale field (Eq. 8)
$y$ | dimensionless amplitude of turbulent magnetic fluctuations $\perp$ to large scale field (Eq. 9)
$\xi$ | derived variable dependent on $\Theta$, $x$, and $y$ (Eq. 12)
$\Delta$ | threshold value of $\xi$ such that sign of RM reversed for $\xi<-\Delta$
$\sigma$ | rms value of Gaussian-distributed $y$ (Eq. 14)
$a$ | secondary variable $a\equiv\cos\Theta$ (Eq. 16)
$b$ | secondary variable $b\equiv\sin\Theta$ (Eq. 16)
$d$ | secondary variable $d\equiv\sigma b$ (Eq. 17)
$p(\xi)$ | normalized, Gaussian pdf of $\xi$ (Eq. 15-17)
$p_{G}(\xi)$ | shifted, zero-mean version of $p(\xi)$ (Eq. 19)
$P_{-}$ | probability that turbulent fluctuations cause reversal of LOS magnetic polarity (Eq. 18)
$t$ | secondary variable $t\equiv\frac{y}{\sqrt{2}d}$ (Eq. 17)
$X$ | single parameter determining $P_{-}$ (Eq. 21)