the required result. ∎

We shall now give an explicit form of the polar parts $\phi_{m}^{P}$ and
$\psi_{n}^{P}$ of the mock Jacobi forms $\phi_{m}^{F}$ and $\psi_{n}^{F}$
respectively. To do this we need the coefficient of the Laurent expansion of
$\phi_{m},\psi_{n}$ at $z=0$. This can be obtained from (5.9). The coefficient
of the Laurent expansion of the partition function at $z=0$ is given by

$\frac{1}{\widetilde{\Phi}_{k}(\tau,z,\sigma)}\simeq-\frac{1}{4\pi^{2}}\tilde{f}^{(k)}(\tau)\tilde{g}^{(k)}(\sigma)\frac{1}{z^{2}}+O(z^{-4}),$
(5.40)

where $\tilde{f}^{(k)},\tilde{g}^{(k)}$ are the inverses of the modular forms
given by (5.10):

$\tilde{f}^{(k)}(\tau):=\frac{1}{f^{(k)}(N\tau)},\quad\tilde{g}^{(k)}(\sigma):=\frac{1}{g^{(k)}(\sigma/N)}.$
(5.41)

Consequently, the residue of $\phi_{m}$ at $z=0$ is given by (see (4.43))

$E_{0\leavevmode\nobreak\
m}^{\phi}(\sigma)=\hat{f}^{(k)}(m)\tilde{g}^{(k)}(\sigma),\qquad
D_{0\leavevmode\nobreak\ m}^{\phi}(\sigma)=0\,,$ (5.42)

where $\hat{f}^{(k)}(m)$ is the coefficient of $q^{m}$ in
$\tilde{f}^{(k)}(\tau)$. Similarly the residue of $\psi_{n}$ is given by

$E_{0\leavevmode\nobreak\
n/N}^{\psi}(\sigma)=\hat{g}^{(k)}(n)\tilde{f}^{(k)}(\tau),\qquad
D_{0\leavevmode\nobreak\ n/N}^{\psi}(\sigma)=0\,,$ (5.43)

where $\hat{g}^{(k)}(n)$ is the coefficient of $p^{n/N}$ in
$\tilde{g}^{(k)}(\sigma)$. Thus using (3.48) and (4.44), the polar part can be
written as

$\begin{split}&\phi_{m}^{P}(\sigma,z)={\hat{f}^{(k)}(m)}{\tilde{g}^{(k)}(\sigma)}\mathcal{B}^{0}_{m}(\sigma,z),\\\
&\psi_{n}^{P}(\tau,z)={\hat{g}^{(k)}(n)}{\tilde{f}^{(k)}(\tau)}\widetilde{\mathcal{B}}^{0}_{\frac{n}{N}}(\tau,z),\end{split}$
(5.44)

where the Appell-Lerch sums $\mathcal{B}^{0}_{m}$ and
$\widetilde{\mathcal{B}}^{0}_{\frac{n}{N}}$ can be computed from (3.47) and
(4.23) respectively. This gives

$\begin{split}&\phi_{m}^{P}(\sigma,z)={\hat{f}^{(k)}(m)}{\tilde{g}^{(k)}(\sigma)}\sum_{s\in\mathbb{Z}}\frac{p^{ms^{2}+s}\zeta^{2ms+1}}{(1-p^{s}\zeta)^{2}},\\\
&\psi_{n}^{P}(\tau,z)={\hat{g}^{(k)}(n)}{\tilde{f}^{(k)}(\tau)}\sum_{s\in\mathbb{Z}}\frac{q^{N(ns^{2}+s)}\zeta^{2ns+1}}{(1-q^{Ns}\zeta)^{2}}.\end{split}$
(5.45)

We now prove our main theorem. Let us introduce the following set of triplets
of charge. Let $A$ denote the set of integrs $(n,\ell,m)$ satisfying

$m,n>0,\quad 4\frac{n}{N}m-\ell^{2}>0,$ (5.46)

and

$\hbox{Min}_{m_{1}\in N\mathbb{Z},\,s\in\mathbb{Z},\,s(s+1)=0\,{\rm
mod}\,m_{1}}\left[\left({m_{1}\ell+2ms+m\over m}\right)^{2}+{m_{1}^{2}\over
m^{2}}\left(4{n\over N}m-\ell^{2}\right)\right]\geq 1\,.$ (5.47)

Let us also introduce the set $B$ of integers $(n,\ell,m)$ satisfying

$m,n>0,\quad 4\frac{n}{N}m-\ell^{2}>0,$ (5.48)

and

$\hbox{Min}_{n_{1},s\in\mathbb{Z},\,s(s+1)=0\,{\rm
mod}\,n_{1}}\left[\left({n_{1}N\ell-2ns-n\over
n}\right)^{2}+{n_{1}^{2}N^{2}\over n^{2}}\left(4{n\over
N}m-\ell^{2}\right)\right]\geq 1\,.$ (5.49)

We then have the following theorem.

######  Theorem 5.4.

  1. (a)

For $(n,\ell,m)\in A$, the single-centered black hole degeneracy with
$NQ^{2}/2=m,P^{2}/2=n$ and $Q\cdot P=\ell$ are Fourier coefficients666Up to a
constant multiple, see (5.57) below. of the mock Jacobi form $\phi_{m}^{F}$.

  2. (b)

For $(n,\ell,m)\in B$, the single-centered black hole degeneracy with
$NQ^{2}/2=m,P^{2}/2=n$ and $Q\cdot P=\ell$ are Fourier coefficients of the
mock Jacobi form $\psi_{n}^{F}$.

###### Proof.

Let us start with charges in the set $A$. We now want to use the first
Fourier-Jacobi expansion in (5.20) to perform the contour integral for single-
centered degeneracy. This requires us to deform the contour $\mathcal{C}$ to
$\mathcal{C}^{\prime}$ which is defined identical to $\mathcal{C}$ except that
Im$(\tau)\to\infty$. To do this we need to show that there are no poles of the
partition function between the region bounded by the contour $\mathcal{C}$ and
the deformed contour. This is proved in Appendix B.1. We can hence write

$d^{*}\left(\frac{n}{N},\ell,m\right)=\frac{(-1)^{\ell+1}}{N}\int_{\mathcal{C}^{\prime}}d\sigma
dzd\tau e^{-2\pi i\left(m\tau+\frac{n}{N}\sigma+\ell
z\right)}\sum_{m^{\prime}=-\widetilde{\alpha}}^{\infty}\phi_{m^{\prime}}(\sigma,z)q^{m^{\prime}}.$
(5.50)

Performing the $\tau$ integral, we get

$\begin{split}d^{*}\left(\frac{n}{N},\ell,m\right)=\frac{(-1)^{\ell+1}}{N}\int_{\mathcal{C}(n,\ell)}d\sigma
dze^{-2\pi i\left(\frac{n}{N}\sigma+\ell
z\right)}\phi_{m}(\sigma,z)\end{split}$ (5.51)

where $\mathcal{C}(n,\ell)$ is the projection of $\mathcal{C}$ onto
$\sigma$-$z$ plane.

Now we specify the $z$ integral. Note that the attractor contour (5.14)
implies that

$\text{Im}(z)=-\frac{\ell}{2m}\text{Im}(\sigma).$ (5.52)

Moreover we can shift the contour in purely horizontal direction since we do
not cross any pole in doing so. So we shift the contour such that on the
contour

$-\frac{\ell}{2m}\text{Re}(\sigma)\leq\text{Re}(z)\leq-\frac{\ell}{2m}\text{Re}(\sigma)+1.$
(5.53)

Thus we have

$\begin{split}d^{*}\left(\frac{n}{N},\ell,m\right)&=\frac{(-1)^{\ell+1}}{N}\int
d\sigma e^{-2\pi
i\frac{n}{N}\sigma}\int_{-\frac{\ell}{2m}\sigma}^{-\frac{\ell}{2m}\sigma+1}dze^{-2\pi
i\ell z}\phi_{m}\left(\sigma,z\right)\\\ &=\frac{(-1)^{\ell+1}}{N}\int d\sigma
e^{-2\pi i\frac{n}{N}\sigma}e^{2\pi
i\ell^{2}\sigma/4m}h_{\ell}^{(-\frac{\ell}{2m}\sigma)}(\sigma),\end{split}$
(5.54)

where

$h_{\ell}^{(-\frac{\ell}{2m}\sigma)}(\sigma):=e^{-2\pi
i\ell^{2}\sigma/4m}\int_{-\frac{\ell}{2m}\sigma}^{-\frac{\ell}{2m}\sigma+1}dze^{-2\pi
i\ell z}\phi_{m}(\sigma,z)\,.$ (5.55)

It follows from the discussion at the end of Appendix B.2 that
$h_{\ell}^{(-\frac{\ell}{2m}\sigma)}(\sigma)$ does not have a pole in the
upper half $\sigma$ plane. Hence the integration over $\sigma$ can be taken to
be over any horizontal line of length $N$.

Now, from (4.21), (4.22), and the fact that
$h_{\ell}^{(-\frac{\ell}{2m}\sigma)}(\sigma)$ only depends on $\ell\bmod 2m$
(see the discussion below (4.5)), we get

$\displaystyle\phi_{m}^{F}(\sigma,z)$ $\displaystyle=$
$\displaystyle\sum_{\ell=0}^{2m-1}h_{\ell}^{(-\frac{\ell}{2m}\sigma)}(\sigma)\vartheta_{m,\ell}(\sigma,z)$
(5.56) $\displaystyle=$
$\displaystyle\sum_{\ell=0}^{2m-1}\sum_{r\in\mathbb{Z}}e^{2\pi
i(\ell+2mr)^{2}\sigma/4m}e^{2\pi
i(\ell+2mr)z}h_{\ell}^{(-\frac{\ell}{2m}\sigma)}(\sigma)$ $\displaystyle=$
$\displaystyle\sum_{\ell\in\mathbb{Z}}e^{2\pi i\ell^{2}\sigma/4m}e^{2\pi i\ell
z}h_{\ell}^{(-\frac{\ell}{2m}\sigma)}(\sigma).$

Using this we can express (5.54) as

$\begin{split}d^{*}\left(\frac{n}{N},\ell,m\right)&=\frac{(-1)^{\ell+1}}{N}\int
d\sigma\int dz\,e^{-2\pi i\frac{n}{N}\sigma}\,e^{-2\pi i\ell
z}\phi_{m}^{F}(\sigma,z)\,.\end{split}$ (5.57)

In this form, the $\sigma$ and $z$ integrals run over any horizontal lines of
length $N$ and 1 respectively. This proves (a).

To prove (b), we shall use the fact that for charges in set $B$, the
integration contour $\mathcal{C}$ can be deformed to
$\mathcal{C}^{\prime\prime}$, where $\mathcal{C}^{\prime\prime}$ is defined
identical to $\mathcal{C}$ except that Im$(\sigma)\to\infty$. The validity of
such a deformation without picking up any residues has been proved in Appendix
B.1. Then following similar calculation as for (a) we get

$d^{*}(n/N,\ell,m)=\frac{(-1)^{\ell+1}}{N}\int_{\mathcal{C}(m,\ell)}d\tau
e^{-2\pi im\tau}e^{2\pi i\ell^{2}N\tau/4n}\tilde{h}_{\ell}^{(-\frac{\ell
N}{2n}\tau)}(\tau),$ (5.58)

where

$\tilde{h}_{\ell}^{(-\frac{\ell N}{2n}\tau)}(\tau):=e^{-2\pi
i\tau\frac{\ell^{2}N}{4n}}\int_{-\frac{\ell N}{2n}\tau}^{-\frac{\ell
N}{2n}\tau+1}\psi_{n}(\tau,z)\mathbf{e}(-\ell z)dz.$ (5.59)

Now, from (4.21), (4.22), we get

$\begin{split}\psi_{n}^{F}(\tau,z)&=\sum_{\ell=0}^{2n-1}\widetilde{h}_{\ell}^{(-\frac{\ell
N}{2n}\tau)}(\tau)\widetilde{\vartheta}_{\frac{n}{N},\ell}(\tau,z)\\\
&=\sum_{\ell=0}^{2n-1}\sum_{r\in\mathbb{Z}}e^{2\pi
i(\ell+2nr)^{2}N\tau/4n}e^{2\pi
i(\ell+2nr)z}\widetilde{h}_{\ell}^{(-\frac{\ell N}{2n}\tau)}(\tau)\\\
&=\sum_{\ell\in\mathbb{Z}}e^{2\pi i\ell^{2}N\tau/4n}e^{2\pi i\ell
z}\widetilde{h}_{\ell}^{(-\frac{\ell N}{2n}\tau)}(\tau).\end{split}$ (5.60)

Using this we can express (5.58) as

$\begin{split}d^{*}\left(\frac{n}{N},\ell,m\right)&=\frac{(-1)^{\ell+1}}{N}\int
d\tau\int dz\,e^{-2\pi im\tau}\,e^{-2\pi i\ell
z}\psi_{n}^{F}(\tau,z)\,,\end{split}$ (5.61)

where the $\tau$ and $z$ integrals run over any horizontal lines of length 1.
This proves (b). ∎

######  Remark 5.5.

Single-centered degeneracies for charges belonging to the set $A$ can be
obtained by performing contour integrals with integrand $\phi_{m}(\sigma,z)$
or $\phi_{m}^{F}(\sigma,z)$. The former is given by the expression (5.54)
where the limits on the $z$ integral depends explicitly on $\ell$. On the
other hand the latter is given by (5.57) where the limits of integration are
independent of $\ell$. Analogous statement holds for single-centered
degeneracies for charges belonging to the set $B$.

It is useful to list some special properties of the set $A$ and $B$ proved in
appendix B.

  1. 1.

The sets $A$ and $B$ are invariant under the elliptic transformations777This
is just the S-duality transformation (5.17) for the matrix
$\begin{pmatrix}1&\lambda_{1}\\\ 0&1\end{pmatrix}\in\Gamma_{1}(N)$.

$(n/N,\ell,m)\to(n/N+m\lambda_{1}^{2}+\lambda_{1}\ell,\ell+2m\lambda_{1},m),\qquad\lambda_{1}\in\mathbb{Z}\,$
(5.62)

and888This is just the S-duality transformation (5.17) for the matrix
$\begin{pmatrix}1&0\\\ \lambda_{2}N&1\end{pmatrix}\in\Gamma_{1}(N)$.

$(n/N,\ell,m)\to(n/N,\ell+2n\lambda_{2},m+Nn\lambda_{2}^{2}+N\lambda_{2}\ell),\qquad\lambda_{2}\in\mathbb{Z}\,$
(5.63)

respectively.

  2. 2.

For given $(n,\ell,m)$ it only requires a finite amount of work to determine
if it is in set $A$. The second term in (5.47) is bounded by 1 only for a
finite number of choices of $m_{1}$, so we can restrict to this set for
testing the inequality given in (5.47). For a given value of $m_{1}$ in this
set, only a finite number of choices of $s$ can violate this inequality. Once
we have exhausted the list of $m_{1}$ and $s$ that could possibly violate the
inequality and found no candidate, we can declare that the corresponding
$(n,\ell,m)$ belongs to set $A$. A similar remark holds for set $B$.

  3. 3.

For given $m$, it requires only a finite amount of work to classify all the
orbits under (5.62) that belong to the set $A$. Since (5.62) can bring $\ell$
to the range $0\leq\ell<2m$, we can choose $\ell$ to be in this range. For
each of these values of $\ell$, only finite number of allowed values of $n$
can keep the second term in (5.47) below one for $m_{1}\neq 0$. For each of
these choices of $n$ and $m_{1}$, we then examine the list of $s$ for which
the inequality can be violated. This is again a finite operation. For
$m_{1}=0$ the inequality is satisfied for all $n$ and $s$ and no further work
is needed. A similar result holds for the set $B$.

  4. 4.

It is also shown in Appendix B.3 and B.4 that the set of charges

$\begin{split}&A_{1}:=\left\\{(n,\ell,m):\leavevmode\nobreak\
m,n>0,\leavevmode\nobreak\
4\frac{n}{N}m-\ell^{2}\geq\frac{m^{2}}{N^{2}}\right\\},\\\
&A_{2}:=\left\\{(n,\ell,m):\leavevmode\nobreak\ m,n>0,\leavevmode\nobreak\
\frac{n}{N}\geq m,\leavevmode\nobreak\ 0\leq|\ell|\leq
2m\right\\},\end{split}$ (5.64)

are subsets of $A$ and

$\begin{split}&B_{1}:=\left\\{(n,\ell,m):\leavevmode\nobreak\
m,n>0,\leavevmode\nobreak\
4\frac{n}{N}m-\ell^{2}\geq\frac{n^{2}}{N^{2}}\right\\},\\\
&B_{2}:=\left\\{(n,\ell,m):\leavevmode\nobreak\ m,n>0,\leavevmode\nobreak\
m\geq\frac{n}{N},\leavevmode\nobreak\ 0\leq|\ell|\leq 2n\right\\},\end{split}$
(5.65)

are subsets of $B$. Note that $A_{1}$ is invariant under the elliptic
transformation (5.62) and $B_{1}$ is invariant under the elliptic
transformation (5.63). But $A_{2},B_{2}$ are not invariant under the
transformations (5.62) and (5.63) respectively. It follows that the images of
$A_{2}$ under (5.62) and $B_{2}$ under (5.63) are also subsets of $A$ and $B$
respectively.

###  5.3 Comparison with Dabholkar-Murthy-Zagier

We have found the set of charges for which the degeneracies are given by the
coefficients of the mock Jacobi forms. In particular, for $N=1$ we must
recover the result of [9, Section 11.1]. From our analysis, it turns out that
the italicized statement below eq. (11.17) of [9] in its most general form is
not correct. The general claim in [9, Section 11.1] is that the degeneracy for
every charge $(n,\ell,m)$ with $n>m$ is given by the corresponding coefficient
of 999Note that the mock Jacobi form $\psi_{m}^{F}$ of [9] is denoted as
$\phi_{m}^{F}$ in this paper. Also the roles of $Q$ and $P$ are interchanged
compared to [9] since we have $Q^{2}/2=m,P^{2}/2=n$. $\phi_{m}^{F}$. We will
give some counterexamples to demonstrate that this is not true. In particular
we will find charges $(n,\ell,m)$ with $n>m$ which do not belong to the set
$A$.

Consider for example the charge $(n,\ell,m)=(10,14,5)$. It can easily be
checked that $(n,\ell,m)=(10,14,5)$ does not belong to the set $A$, since the
choice $m_{1}=1,s=-2$ violates (5.47). This can also be seen more explicitly
as follows. For the choice of charges given above, the attractor contour is at

$\displaystyle\text{Im}(\tau)=\frac{20}{\varepsilon},\quad\text{Im}(\sigma)=\frac{10}{\varepsilon},\quad\text{Im}(z)=-\frac{14}{\varepsilon}.$
(5.66)

To express the degeneracy for this charge as the coefficient of
$\phi_{5}^{F}$, we need to deform the contour from
$\text{Im}(\tau)=\frac{20}{\varepsilon}$ to $\text{Im}(\tau)\to i\infty$. But
using (5.7) we see that there is a pole at $\text{Im}(\tau)=22/\varepsilon$
corresponding to $n_{2}=0,m_{1}=1,n_{1}=-2,j=-3$. Thus this deformation is not
allowed. More precisely, wall crossing formula gives a nonzero jump in the
degeneracy if we cross this pole. Let us explicitly demonstrate this. If the
single-centered degeneracy for charges $(n,\ell,m)$ had been given by the
Fourier expansion coefficients of $\phi^{F}_{m}$, then the degeneracy
$d^{*}(10,14,5)$ would have been given by the coefficient of
$p^{10}\zeta^{14}$ of $\phi_{5}^{F}(\sigma,z)$ where $p=e^{2\pi
i\sigma},\zeta=e^{2\pi iz}$ as before. By elliptic invariance (5.62), this is
same as the coefficient of $p\zeta^{4}$. By explicit computation on
Mathematica, we found the coefficient to be $2023536$. Thus we find 101010The
first sign of trouble comes from the negative sign, since it violates the
positivity conjecture of [25].

$d^{*}_{\text{DMZ}}(10,14,5)=-2023536.$ (5.67)

On the other hand, following the prescription in [25], $d^{*}(10,14,5)$ can be
computed by first bringing the charge in the range given in Eq. (3.7) of [25]
and then computing the corresponding coefficient of the partition function. By
the S-duality transformation (which leaves the single-centered degeneracies
invariant), the charge can be brought to

$\displaystyle(10,14,5)\to(1,0,1).$ (5.68)

From [25, Table 1], we find

$\displaystyle d^{*}(10,14,5)=d^{*}(1,0,1)=50064.$ (5.69)

Let us now explicitly calculate the jump in the degeneracy across the wall at
$\text{Im}(\tau)=\frac{22}{\varepsilon}$. The wall crossing formula for the
decay
$(\vec{Q},\vec{P})\to(\vec{Q}_{1},\vec{P}_{1})+(\vec{Q}_{2},\vec{P}_{2})$
across a wall of marginal stability is given by [22, Eq. (5.5.17)]

$(-1)^{\vec{Q}_{1}\cdot\vec{P}_{2}-\vec{Q}_{2}\cdot\vec{P}_{1}+1}\left|\vec{Q}_{1}\cdot\vec{P}_{2}-\vec{Q}_{2}\cdot\vec{P}_{1}\right|d_{\text{half}}(\vec{Q}_{1},\vec{P}_{1})d_{\text{half}}(\vec{Q}_{2},\vec{P}_{2}),$
(5.70)

where $d_{\text{half}}(\vec{Q}_{i},\vec{P}_{i})$ denotes the number of bosonic
minus fermionic half BPS supermultiplets carrying charges
$(\vec{Q}_{i},\vec{P}_{i})$. The pole at 111111This corresponds to a pole of
the partition function with $n_{2}=0$ in (5.7).

$n_{1}\sigma+jz-m_{1}\tau+m_{2}=0,$ (5.71)

corresponds to the wall of marginal stability associated with the decay (see
for example Eq. (2.21), Eq. (2.22) of [19]) 121212The convention used in [19]
is related to the current convention by an exchange of $\rho$ and $\sigma$.:

$(\vec{Q},\vec{P})\to(\alpha\vec{Q}+\beta\vec{P},\gamma\vec{Q}+\delta\vec{P})+((1-\alpha)\vec{Q}-\beta\vec{P},-\gamma\vec{Q}+(1-\delta)\vec{P})\,,$
(5.72)

where

$\gamma=n_{1},\qquad\beta=m_{1},\qquad\delta={1-j\over
2},\qquad\alpha={j+1\over 2}\,.$ (5.73)

For $m_{1}=1,n_{1}=-2,j=-3$ the decay is

$(\vec{Q},\vec{P})\to(\vec{P}-\vec{Q},2(\vec{P}-\vec{Q}))+(2\vec{Q}-\vec{P},2\vec{Q}-\vec{P}).$
(5.74)

By S-duality, the charges $(\vec{P}-\vec{Q},2(\vec{P}-\vec{Q}))$ and
$(2\vec{Q}-\vec{P},2\vec{Q}-\vec{P}))$ can be transformed to purely magnetic
charges $(0,\vec{P}-\vec{Q})$ and $(0,2\vec{Q}-\vec{P})$ respectively. The
degeneracies $d_{\text{half}}(\vec{Q}_{i},\vec{P}_{i})$ can thus be computed
using the coefficients of the reciprocal $\Delta^{-1}$ of the Ramanujan’s
discriminant function. To compute these degeneracies for our triplet of
charges $(n,\ell,m)=(10,14,5)$, we compute:

$\begin{split}&\frac{(\vec{P}-\vec{Q})^{2}}{2}=\frac{Q^{2}+P^{2}-2Q\cdot
P}{2}=n+m-\ell=1,\\\
&\frac{(2\vec{Q}-\vec{P})^{2}}{2}=\frac{4Q^{2}+P^{2}-4Q\cdot
P}{2}=4m+n-2\ell=2.\end{split}$ (5.75)

Thus $d_{\text{half}}(\vec{Q}_{1},\vec{P}_{1})$ and
$d_{\text{half}}(\vec{Q}_{2},\vec{P}_{2})$ are the coefficients of $q^{2}$ and
$q^{3}$ respectively of $\Delta(q)^{-1}$. Explicitly, we have

$\displaystyle d_{\text{half}}(\vec{Q}_{1},\vec{P}_{1})=324,\quad
d_{\text{half}}(\vec{Q}_{2},\vec{P}_{2})=3200.$ (5.76)

Moreover, we have

$\displaystyle(-1)^{\vec{Q}_{1}\cdot\vec{P}_{2}-\vec{Q}_{2}\cdot\vec{P}_{1}+1}\left|\vec{Q}_{1}\cdot\vec{P}_{2}-\vec{Q}_{2}\cdot\vec{P}_{1}\right|=-2.$
(5.77)

Thus the jump is given by

$\displaystyle d_{\text{jump}}(10,14,5)=-2\cdot 324\cdot 3200=-2073600.$
(5.78)

From this, we see that

$\displaystyle
d^{*}_{\text{DMZ}}(10,14,5)-d_{\text{jump}}(10,14,5)=-2023536+2073600=50064$
(5.79)

which matches with the degeneracy (5.69).

Another such example is the charge $(n,\ell,m)=(13,19,7)$ where we have
nonzero jump in the degeneracy.

It is easy to check that neither (10,14,5) nor (13,19,7) belong to the set
$B$. Thus, they provide examples of charges that cannot be directly related to
the Fourier coefficients of mock Jacobi forms.

Finally, we note that while the coefficients of mock modular forms do not
capture single centered black hole degeneracies for these charges, they do
capture the index in a different chamber of the moduli space. The precise
description of this chamber can be read out from the prescription given in [6]
relating the location of the integration contour to the asymptotic values of
the closed string moduli. As we change the asymptotic moduli from this value
to the attractor values where only single centered black holes exist, we
encounter a codimensional one wall of marginal stability across which the
index jumps[21, 8].

###  5.4 Mock modularity in $\mathbb{Z}_{M}\times\mathbb{Z}_{N}$ orbifolds CHL
models

Finally, we comment on the validity of the above prescription to get single-
centered degeneracy for $\mathbb{Z}_{M}\times\mathbb{Z}_{N}$ orbifolds of Type
IIB superstring theory compactified on $\mathcal{M}\times
S^{1}\times\widetilde{S}^{1}$ where $\mathcal{M}=K3$ or $T^{4}$ and
$\mathbb{Z}_{M}$ acts on $S^{1}$ by shifts of $1/M$ units and $\mathbb{Z}_{N}$
acts on $\widetilde{S}^{1}$ by shifts of $1/N$ units, besides respective
actions on $K3$. We can determine the S-duality group by demanding that they
preserve the vectors $(1/M,0)$ and $(0,1/N)$ on the fundamental parallelogram
defining the torus $T^{2}$ in the compactification space $\mathcal{M}\times
T^{2}$. It is easy to see that under the action

$\begin{pmatrix}x\\\ y\end{pmatrix}\mapsto\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}x\\\ y\end{pmatrix},\quad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z})$ (5.80)

the subgroup of $\mathrm{SL}(2,\mathbb{Z})$ which leaves $(1/M,0)$ and
$(0,1/N)$ invariant is given by

$\Gamma(M,N):=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z}):\begin{array}[]{c}a\equiv
1(\bmod\leavevmode\nobreak\ M),\leavevmode\nobreak\ \leavevmode\nobreak\
b\equiv 0(\bmod\leavevmode\nobreak\ M)\\\ c\equiv 0(\bmod\leavevmode\nobreak\
N),\leavevmode\nobreak\ \leavevmode\nobreak\ d\equiv
1(\bmod\leavevmode\nobreak\ N)\end{array}\right\\}.$ (5.81)

It can be seen that $\Gamma(M,N)$ is a congruence subgroup of level $MN$.

The partition function $\Phi^{(M,N)}(\tau,z,\sigma)$ and the prescription for
the calculation of single-centered degeneracy appeared in [24]. The partition
function is again a Siegel modular form for some subgroup $G^{\prime}$ of
$\mathrm{Sp}(2,\mathbb{Z})$ that contains the S-duality group. Expanding the
partition function in $p=e^{2\pi i\sigma}$ first, we obtain a family of
meromorphic Jacobi forms of rational index $n/M$ for
$\Gamma(M,N)\ltimes(M\mathbb{Z}\times\mathbb{Z})$. The location of poles of
these meromorphic Jacobi forms are again at $M\mathbb{Z}\sigma+\mathbb{Z}$.
One can then apply the general formalism of Section 3 and Section 4 to write
down the polar and finite parts of these meromorphic Jacobi forms. The finite
part is again a mock Jacobi form. Similarly, one can expand in $q=e^{2\pi
i\tau}$ first to obtain mock Jacobi forms for the group
$S\Gamma(M,N)S^{-1}\ltimes(N\mathbb{Z}\times\mathbb{Z})=\Gamma(N,M)\ltimes(N\mathbb{Z}\times\mathbb{Z})$
with index $m/N$, where

$S=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z}).$
(5.82)

Acknowledments. We would like to thank Nabamita Banerjee and Suvankar Dutta
for discussions and collaboration at the initial stages of this work. We would
also like to thank Atish Dabholkar and Sameer Murthy for useful discussions.
R.K.S. would like to thank IISER Bhopal and ICTS Bangalore for hospitality
where part of this work was done. R.K.S. would also like to thank Madhav Sinha
and Spencer Stubbs for help with Mathematica code. The work of R.K.S. is
supported by the US Department of Energy under grant DE-SC0010008. The work of
A.B. was supported by grant MTR/2019/000582 from the SERB, Government of
India. A.B. would like to thank ICTS Bangalore for hospitality where part of
this work was done. The work of A.S. was supported by ICTS-Infosys Madhava
Chair Professorship, the J. C. Bose fellowship of the Department of Science
and Technology, India and the Department of Atomic Energy, Government of
India, under project no. RTI4001.

##  Appendix A Transformation properties of $\widetilde{\Phi}_{k}$

In this appendix, we prove some transformation properties of the reciprocal of
the partition function given by $\widetilde{\Phi}_{k}$. The Siegel modular
form $\widetilde{\Phi}_{k}$ is related to another Siegel modular form
$\Phi_{k}$ as 131313We have ignored a constant in the relation which will not
be important for the discussion below. follows [22, Eq. (C.21)]

$\widetilde{\Phi}_{k}(\tau,z,\sigma)=(N\tau)^{-k}\Phi_{k}\left(\frac{\tau\sigma-z^{2}}{N\tau},\frac{\tau\sigma-z^{2}+z}{N\tau},\frac{\tau\sigma-(z-1)^{2}}{N\tau}\right).$
(A.1)

$\Phi_{k}$ is a Siegel modular form of weight $k$ with respect to a subgroup
$G$ of $\mathrm{Sp}(2,\mathbb{Z})$ generated by

$\displaystyle g_{1}(a,b,c,d)$
$\displaystyle\equiv\left(\begin{array}[]{cccc}a&0&b&0\\\ 0&1&0&0\\\
c&0&d&0\\\ 0&0&0&1\end{array}\right),\leavevmode\nobreak\ \leavevmode\nobreak\
ad-bc=1,\quad c=0\bmod N,\quad a,d=1\bmod N,$ (A.2) $\displaystyle g_{2}$
$\displaystyle\equiv\left(\begin{array}[]{cccc}0&1&0&0\\\ -1&0&0&0\\\
0&0&0&1\\\ 0&0&-1&0\end{array}\right),$ $\displaystyle g_{3}(\lambda,\mu)$
$\displaystyle\equiv\left(\begin{array}[]{cccc}1&0&0&\mu\\\ \lambda&1&\mu&0\\\
0&0&1&-\lambda\\\ 0&0&0&1\end{array}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \lambda,\mu\in\mathbb{Z}.$

It follows from (A.1), (A.2) that these Siegel modular forms satisfy the
following periodicity properties [25]:

$\begin{split}\Phi_{k}(\tau+1,z,\sigma)=\Phi_{k}(\tau,z+1,\sigma)=\Phi_{k}(\tau,z,\sigma+1)\\\
\widetilde{\Phi}_{k}(\tau+1,z,\sigma)=\widetilde{\Phi}(\tau,z+1,\sigma)=\widetilde{\Phi}(\tau,z,\sigma+N).\end{split}$
(A.3)

One can use the relation (A.1) to find the subgroup $\widetilde{G}$ from $G$.
We will content ourselves by proving certain transformation properties of
$\widetilde{\Phi}_{k}$ used in Section 5. We begin by noting that
$\widetilde{G}$ has a subgroup consisting of the elements [12, Section 5]

$g_{4}(a,b,cN,d):=\begin{pmatrix}a&b&0&0\\\ cN&d&0&0\\\ 0&0&d&-cN\\\
0&0&-b&a\end{pmatrix},\quad ad-bcN=1,\quad a,d\equiv
1(\bmod\leavevmode\nobreak\ N).$ (A.4)

This means that

$\widetilde{\Phi}_{k}\underset{k}{|}g_{4}(a,b,cN,d)=\widetilde{\Phi}_{k}.$
(A.5)

This will be useful for proving the proposition below.

######  Proposition A.1.

We have that

$\begin{split}&\widetilde{\Phi}_{k}\underset{k}{|}g_{1}(a,b,cN,d)=\widetilde{\Phi}_{k}\\\
&\widetilde{\Phi}_{k}\underset{k}{|}g^{1}(a,bN,c,d)=\widetilde{\Phi}_{k},\quad
ad-bcN=1,\quad a,d\equiv 1(\bmod\leavevmode\nobreak\ N),\end{split}$ (A.6)

and

$\begin{split}&\widetilde{\Phi}_{k}\underset{k}{|}g_{3}(\lambda
N,\mu)=\widetilde{\Phi}_{k}\\\
&\widetilde{\Phi}_{k}\underset{k}{|}g^{3}(\lambda,\mu)=\widetilde{\Phi}_{k},\quad\lambda,\mu\in\mathbb{Z},\end{split}$
(A.7)

where

$\begin{split}g^{1}(a,b,c,d):=\begin{pmatrix}1&0&0&0\\\ 0&a&0&b\\\ 0&0&1&0\\\
0&c&0&d\end{pmatrix},\quad
g^{3}(\lambda,\mu):=\begin{pmatrix}1&\lambda&0&\mu\\\ 0&1&\mu&0\\\ 0&0&1&0\\\
0&0&-\lambda&1\end{pmatrix}.\end{split}$ (A.8)

###### Proof.

Since $\Phi_{k}$ transforms as a Siegel modular form with respect to $G$, we
can write

$\begin{split}&\Phi_{k}\underset{k}{|}g_{1}(a,b,cN,d)=\Phi_{k},\quad ad-
bcN=1,\quad a,d\equiv 1(\bmod\leavevmode\nobreak\ N);\\\
&\Phi_{k}\underset{k}{|}g_{2}=\Phi_{k};\\\
&\Phi_{k}\underset{k}{|}g_{3}(\lambda,\mu)=\Phi_{k},\quad\lambda,\mu\in\mathbb{Z}.\end{split}$
(A.9)

We begin by simplifying (A.1). Using
$\Phi_{k}\underset{k}{|}g_{3}(-1,0)=\Phi_{k}$ we get

$\Phi_{k}(\tau,z-\tau,\sigma+\tau-2z)=\Phi_{k}(\tau,z,\sigma).$ (A.10)

Using this along with (A.1), we get

$\widetilde{\Phi}_{k}(\tau,z,\sigma)=(N\tau)^{-k}\Phi_{k}\left(\frac{\sigma}{N}-\frac{z^{2}}{N\tau},\frac{z}{N\tau},-\frac{1}{N\tau}\right).$
(A.11)

Now put

$W_{N}:=\frac{1}{\sqrt{N}}\begin{pmatrix}0&N&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\
0&0&N&0\end{pmatrix},\quad\mu_{N}:=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&0&0&-\frac{1}{N}\\\ 0&0&1&0\\\ 0&N&0&0\end{array}\right).$ (A.12)

Note that

$\begin{split}(g_{2}g_{1}(0,-1,1,0)W_{N}g_{2})\circ\Omega=\left(\begin{array}[]{cr}\frac{\sigma}{N}-\frac{z^{2}}{\tau
N}&\frac{z}{\tau N}\\\ &\\\ \frac{z}{\tau N}&-\frac{1}{\tau
N}\end{array}\right)\end{split}$ (A.13)

Hence using (A.11) we have that

$\widetilde{\Phi}_{k}(\Omega)=\left(\Phi_{k}\underset{k}{|}g_{2}g_{1}(0,-1,1,0)W_{N}g_{2}\right)(\Omega).$
(A.14)

Since $g_{2}\in G$, we can write this as

$\widetilde{\Phi}_{k}(\Omega)=\left(\left(\Phi_{k}\underset{k}{|}g_{2}\right)\underset{k}{|}g_{1}(0,-1,1,0)W_{N}g_{2}\right)(\Omega)=\left(\Phi_{k}\underset{k}{|}g_{1}(0,-1,1,0)W_{N}g_{2}\right)(\Omega),$
(A.15)

where we used (A.9) in the last equality. Moreover, we have that

$g_{1}(0,-1,1,0)W_{N}=W_{N}\mu_{N}.$ (A.16)

This implies that

$\widetilde{\Phi}_{k}(\Omega)=\left(\Phi_{k}\underset{k}{|}W_{N}\mu_{N}g_{2}\right)(\Omega).$
(A.17)

It is easy to see that

$\begin{split}g_{2}g^{1}(a,b,c,d)=g_{1}(a,b,c,d)g_{2},\\\
\mu_{N}g_{1}(a,b,c,d)=g_{1}(a,b,c,d)\mu_{N},\\\
W_{N}g_{1}(a,bN,c,d)=g_{2}^{-1}g_{1}(a,b,cN,d)g_{2}W_{N};\end{split}$ (A.18)

and

$\begin{split}g_{2}g^{3}(\lambda,\mu)=g_{3}(-\lambda,-\mu)g_{2},\\\
W_{N}g_{3}(-\lambda,-\mu)=g^{3}(-N\lambda,-\mu)W_{N},\\\
g_{1}(0,-1,1,0)g^{3}(0,-\mu)=g_{3}(\mu,0)g_{1}(0,-1,1,0).\end{split}$ (A.19)

Using (A.17), (A.18), and (A.9), we get

$\widetilde{\Phi}_{k}\underset{k}{|}g^{1}(a,bN,c,d)=\Phi_{k}\underset{k}{|}W_{N}\mu_{N}g_{2}g^{1}(a,bN,c,d)=\widetilde{\Phi}_{k},\quad
ad-bcN=1,\quad a,d\equiv 1(\bmod\leavevmode\nobreak\ N).$ (A.20)

Also using (A.15), (A.19) and (A.9), we get,

$\widetilde{\Phi}_{k}\underset{k}{|}g^{3}(0,\mu)=\Phi_{k}\underset{k}{|}g_{1}(0,-1,1,0)W_{N}g_{2}g^{3}(0,\mu)=\widetilde{\Phi}_{k}.$
(A.21)

Moreover we have

$g^{3}(\lambda,\mu)=g_{1}(1,-\lambda\mu,0,1)g_{4}(1,\lambda,0,1)g^{3}(0,\mu).$
(A.22)

$g_{1}(1,-\lambda\mu,0,1)$ leaves $\widetilde{\Phi}_{k}$ invariant as a
consequence of the $\tau\to\tau+1$ symmetry described in the second line of
(A.3). The other elements on the right hand side of (A.22) have already been
shown to leave $\widetilde{\Phi}_{k}$ invariant. Therefore, we have

$\widetilde{\Phi}_{k}\underset{k}{|}g^{3}(\lambda,\mu)=\widetilde{\Phi}_{k}.$
(A.23)

We can also check that

$\begin{split}g_{2}g_{1}(a,b,c,d)=g^{1}(a,b,c,d)g_{2},\\\
\mu_{N}g^{1}(a,b,c,d)=g^{1}(d,-c/N^{2},-bN^{2},a)\mu_{N},\\\
W_{N}g^{1}(d,-c/N^{2},-bN^{2},a)=g_{1}(d,-c/N,-bN,a)W_{N};\\\ \end{split}$
(A.24)

and

$\begin{split}g_{2}g_{3}(\lambda,\mu)=g^{3}(-\lambda,-\mu)g_{2},\\\
W_{N}g^{3}(-N\lambda,-\mu)=g_{3}(-\lambda,-\mu)W_{N},\\\
g_{1}(0,-1,1,0)g_{3}(\lambda,\mu)=g_{3}(-\mu,\lambda)g_{1}(0,-1,1,0).\end{split}$
(A.25)

These can be used to prove the remaining parts of the statement following the
same procedure. ∎

##  Appendix B Deformation of the contour $\mathcal{C}$ to
$\mathcal{C}^{\prime}$ and $\mathcal{C}^{\prime\prime}$ without crossing a
pole

In this appendix, we determine under what condition the attractor contour
$\mathcal{C}$ can be deformed to the contour $\mathcal{C}^{\prime}$ at
$\tau=i\infty$ with fixed $\text{Im}(\sigma)$ and $\text{Im}(z)$ and to the
contour $\mathcal{C}^{\prime\prime}$ at $\sigma=i\infty$ with fixed
$\text{Im}(\tau)$ and $\text{Im}(z)$ without crossing a pole. We shall focus
on charges $(\vec{Q},\vec{P})$ satisfying $Q^{2}P^{2}-(Q\cdot P)^{2}>0$, since
single-centered black holes exist only for such charges. For
$P^{2}=2n,Q^{2}=\frac{2m}{N},Q\cdot P=\ell$, this translates to

$\displaystyle\frac{4mn}{N}-\ell^{2}>0.$ (B.1)

###  B.1
$\mathcal{C}\longrightarrow\mathcal{C}^{\prime},\mathcal{C}^{\prime\prime}$

Let us first consider the deformation from
$\mathcal{C}\longrightarrow\mathcal{C}^{\prime}$.

Let $A$ be the set of triples $(n,\ell,m)\in\mathbb{Z}^{3}$ satisfying
$m,n>0,\left(4\frac{n}{N}m-\ell^{2}\right)>0$ and

$\hbox{Min}_{m_{1}\in N\mathbb{Z},\,s\in\mathbb{Z},\,s(s+1)=0\,{\rm
mod}\,m_{1}}\left[\left({m_{1}\ell+2ms+m\over m}\right)^{2}+{m_{1}^{2}\over
m^{2}}\left(4{n\over N}m-\ell^{2}\right)\right]\geq 1\,.$ (B.2)

We have the following result.

######  Proposition B.1.

If $(n,\ell,m)\in A$, then the partition function $\widetilde{\Phi}_{k}^{-1}$
does not have poles in the region bounded by the contours $\mathcal{C}$ and
$\mathcal{C}^{\prime}$.

###### Proof.

We have seen in (5.7) that the general structure of poles of the partition
function is given by

$\displaystyle n_{2}\left(\sigma\tau-z^{2}\right)+jz+n_{1}\sigma-
m_{1}\tau+m_{2}=0$ (B.3)

where

$m_{1}\in N\mathbb{Z},\quad n_{1}\in\mathbb{Z},\quad j\in 2\mathbb{Z}+1,\quad
m_{2},n_{2}\in\mathbb{Z},\quad
m_{1}n_{1}+m_{2}n_{2}+\frac{j^{2}}{4}=\frac{1}{4}.$ (B.4)

We also saw in (5.14) that on the contour $\mathcal{C}$ we have

$\text{Im}(\tau)={2n\over
N\varepsilon},\qquad\text{Im}(\sigma)={2m\over\varepsilon},\qquad\text{Im}(z)=-{\ell\over\varepsilon}\,,$
(B.5)

and the real parts of the variables are bounded. We first claim that if
$(\tau,z,\sigma)$ is a pole of the partition function, then we must have

$n_{1}\sigma-m_{1}\tau+jz+m_{2}=0,$ (B.6)

where

$m_{1}\in N\mathbb{Z},\quad n_{1},m_{2}\in\mathbb{Z},\quad j\in
2\mathbb{Z}+1,\quad m_{1}n_{1}+\frac{j^{2}}{4}=\frac{1}{4}.$ (B.7)

To see this, note that for sufficiently small $\varepsilon$, the dominant
contribution to the LHS of (B.3) comes from the term

$-n_{2}\left(\text{Im}(\sigma)\text{Im}(\tau)-\text{Im}(z)^{2}\right).$ (B.8)

Now since $\text{Im}(\sigma)\text{Im}(\tau)-\text{Im}(z)^{2}>0$ in
$\mathbb{H}_{2}$, the coefficient of $n_{2}$ in (B.3) does not vanish. For
sufficiently small $\varepsilon$ this term dominates in the region between
$\mathcal{C}$ and $\mathcal{C}^{\prime}$, and (B.3) is not satisfied if
$n_{2}\neq 0$. This means that if we want to look for poles in the region
bounded by $\mathcal{C}$ and $\mathcal{C}^{\prime}$ we must take $n_{2}=0$. So
the only relevant poles are given by (B.6).

Taking imaginary part of (B.6) gives

$\begin{split}&n_{1}\operatorname{Im}(\sigma)-m_{1}\operatorname{Im}(\tau)+j\operatorname{Im}(z)=0.\end{split}$
(B.9)

Since the imaginary parts of $\sigma$ and $z$ are fixed at their values on the
attractor contour given in (B.5), we get

$\displaystyle\operatorname{Im}(\tau)=\frac{2mn_{1}-j\ell}{m_{1}\varepsilon}=\frac{2m}{\varepsilon}\left(\frac{n_{1}-j\frac{\ell}{2m}}{m_{1}}\right).$
(B.10)

Comparing (B.10) with (B.5) we see that, in order that there are no poles
between $\mathcal{C}$ and $\mathcal{C}^{\prime}$, we must have

$\frac{n_{1}-j\frac{\ell}{2m}}{m_{1}}\leq{n\over Nm}\,,$ (B.11)

for all allowed values of $j$, $n_{1}$ and $m_{1}$, so that all the relevant
poles occur for $\operatorname{Im}(\tau)\leq 2n/(N\varepsilon)$. Special care
is needed to address the situation when the inequality in (B.11) is replaced
by an equality. In this case there is a pole on the attractor contour, which
signals that the attractor point is on a wall of marginal stability. However
the jump in the index across this wall vanishes, reflecting that
$\widetilde{\Phi}_{k}^{-1}$ has vanishing residue at the pole. This will be
demonstrated in Section B.2.

Using (B.7) and $n_{2}=0$, we get

$\frac{n_{1}}{m_{1}}=\frac{1-j^{2}}{4m_{1}^{2}}$ (B.12)

Substituting this in (B.11), the inequality becomes

$\frac{1-j^{2}}{4m_{1}^{2}}-\frac{j\frac{\ell}{2m}}{m_{1}}-{n\over Nm}\leq 0.$
(B.13)

By multiplying this by $4m_{1}^{2}$ we can express this as

$\left(j+\ell\,{m_{1}\over m}\right)^{2}+{m_{1}^{2}\over m^{2}}\left(4{n\over
N}m-\ell^{2}\right)\geq 1\,.$ (B.14)

This condition must hold for all $m_{1}\in N\mathbb{Z}$, $j\in 2\mathbb{Z}+1$,
$m_{1}|(j^{2}-1)/4$. Writing $j=2s+1$ with $s\in\mathbb{Z}$ we can express
this as

$\hbox{Min}_{m_{1}\in N\mathbb{Z},\,s\in\mathbb{Z},\,s(s+1)=0\,{\rm
mod}\,m_{1}}\left[\left({m_{1}\ell+2ms+m\over m}\right)^{2}+{m_{1}^{2}\over
m^{2}}\left(4{n\over N}m-\ell^{2}\right)\right]\geq 1\,.$ (B.15)

This is invariant under the transformation

$(n/N,\ell,m)\to(n/N+m\lambda_{1}^{2}+\lambda_{1}\ell,\ell+2m\lambda_{1},m),\qquad\lambda_{1}\in\mathbb{Z}\,,$
(B.16)

since the term inside the square bracket as well as the conditions on
$m_{1},s$ are invariant under this transformation as long as we transform $s$
to $s-m_{1}\lambda_{1}$. This proves the desired result. ∎

Next we shall consider the deformation from $\mathcal{C}$ to
$\mathcal{C}^{\prime\prime}$.

Let $B$ be the set of triples $(n,\ell,m)\in\mathbb{Z}^{3}$ satisfying
$m,n>0,\left(4\frac{n}{N}m-\ell^{2}\right)>0$ and

$\hbox{Min}_{n_{1},s\in\mathbb{Z},\,s(s+1)=0\,{\rm
mod}\,n_{1}}\left[\left({n_{1}N\ell-2ns-n\over
n}\right)^{2}+{n_{1}^{2}N^{2}\over n^{2}}\left(4{n\over
N}m-\ell^{2}\right)\right]\geq 1\,.$ (B.17)

Then we have the following result.

######  Proposition B.2.

If $(n,\ell,m)\in B$, then the partition function $\widetilde{\Phi}_{k}^{-1}$
does not have poles in the region bounded by the contours $\mathcal{C}$ and
$\mathcal{C}^{\prime\prime}$.

###### Proof.

The analysis is similar except that we deform the imaginary part of $\sigma$
to $i\infty$ keeping the imaginary parts of $\tau$ and $z$ fixed at the values
given in (B.5). This leads to the analog of (B.11)

$\frac{m_{1}+j\frac{N\ell}{2n}}{n_{1}}\leq{Nm\over n}\,,$ (B.18)

and the analog of (B.14),

$\left(j-\ell\,{n_{1}N\over n}\right)^{2}+{n_{1}^{2}N^{2}\over
n^{2}}\left(4{n\over N}m-\ell^{2}\right)\geq 1\,.$ (B.19)

Demanding that this holds for all $j$ and $n_{1}$, we get

$\hbox{Min}_{n_{1},s\in\mathbb{Z},\,s(s+1)=0\,{\rm
mod}\,n_{1}}\left[\left({n_{1}N\ell-2ns-n\over
n}\right)^{2}+{n_{1}^{2}N^{2}\over n^{2}}\left(4{n\over
N}m-\ell^{2}\right)\right]\geq 1\,.$ (B.20)

This is invariant under the transformation

$(n/N,\ell,m)\to(n/N,\ell+2n\lambda_{2},m+Nn\lambda_{2}^{2}+N\lambda_{2}\ell),\qquad\lambda_{2}\in\mathbb{Z}\,,$
(B.21)

since the term inside the square bracket remains invariant under this
transformation accompanied by a shift $s\to s+Nn_{1}\lambda_{2}$. This proves
the desired result. ∎

###  B.2 Poles on the attractor contour

We shall now show that when there is a pole on the attractor contour, the
residue of $\widetilde{\Phi}_{k}^{-1}$ at the pole vanishes and we can deform
the contour through such a pole. For this we first note that the if the pole
(B.6) lies on the attractor contour (B.5), then we have,

$2mn_{1}-2{n\over N}m_{1}-j\ell=0\,.$ (B.22)

We now use (5.72) and (5.73) to conclude that the pole at (B.6) corresponds to
the wall of marginal stability associated with the decay:

$(Q,P)\to(\alpha Q+\beta P,\gamma Q+\delta P)+((1-\alpha)Q-\beta P,-\gamma
Q+(1-\delta)P)\,,$ (B.23)

where

$\gamma=n_{1},\qquad\beta=m_{1},\qquad\delta={1-j\over
2},\qquad\alpha={j+1\over 2}\,.$ (B.24)

On the other hand on a marginal stability wall corresponding to the decay
$(Q,P)\to(Q_{1},P_{1})+(Q_{2},P_{2})$ the jump across the wall is proportional
to

$(Q_{1}\cdot P_{2}-Q_{2}\cdot P_{1})\,,$ (B.25)

which for (B.23) evaluates to

$\beta P^{2}-\gamma Q^{2}+(\alpha-\delta)Q\cdot P=-2mn_{1}+2m_{1}{n\over
N}+j\ell=0\,,$ (B.26)

where in the last step we used (B.22). This shows that when a pole of
$\widetilde{\Phi}_{k}^{-1}$ lies on the attractor contour, the jump across the
corresponding wall of marginal stability vanishes, and hence the residue of
$\widetilde{\Phi}_{k}^{-1}$ at the pole also vanishes. This proves the desired
result.

As a corollary of this, we can also prove the following useful result. When
(B.15) is satisfied and we can deform the $\tau$ contour to $i\infty$ keeping
the $\sigma$ and $z$ contours fixed, there remains the possibility that there
will be poles on the contour in the $(\sigma,z)$ plane. To explore this
possibility, we note that to satisfy (B.10) for $\tau=i\infty$ we have to set
$m_{1}=0$. (B.4) now sets $j=1$ so that the poles are at:

$n_{1}\operatorname{Im}(\sigma)+\operatorname{Im}(z)=0\,.$ (B.27)

On the other hand the integration contour for $z$ and $\sigma$ are at

$\operatorname{Im}(\sigma)={2m\over\varepsilon},\qquad\operatorname{Im}(z)=-{\ell\over\varepsilon}=-{\ell\over
2m}\operatorname{Im}(\sigma)\\\ ,.$ (B.28)

Now since we have set $m_{1}=0$, it follows that if any of the poles (B.27)
lie on (B.28), they will also lie on the attractor contour (B.10). Therefore
by our earlier result the residue at the pole vanishes and they are harmless.
We also see from (B.27), (B.28) that as long as we keep the ratio
$\operatorname{Im}(z)/\operatorname{Im}(\sigma)$ fixed at $-\ell/(2m)$, we can
deform the imaginary part of $\sigma$ without crossing any pole.

A similar argument can be used to show that when (B.17) is satisfied and we
deform the $\sigma$ integration contour to $i\infty$, there is no pole with
non-vanishing residue in the $(\tau,z)$ plane as long as the ratio
$\operatorname{Im}(z)/\operatorname{Im}(\tau)$ fixed at $-N\ell/(2n)$.

###  B.3 Special elements of the set $A$

We shall now construct certain special elements in the set $A$. For this we
need to find $(n,\ell,m)$ which satisfy the inequality (B.14) for every
allowed value of $j,m_{1}$. The following set of charges are easily seen to be
in the set $A$:

$\begin{split}&A_{1}:=\left\\{(n,\ell,m):\leavevmode\nobreak\
m,n>0,\leavevmode\nobreak\
4\frac{n}{N}m-\ell^{2}\geq\frac{m^{2}}{N^{2}}\right\\}.\end{split}$ (B.29)

Indeed since $m_{1}\in N\mathbb{Z}$, for charges in $A_{1}$ the second term of
(B.14) is bounded below by 1 and hence satisfies the inequality unless
$m_{1}=0$. However, for $m_{1}=0$, the first term is bounded from below by 1
since $j$ is odd. Note that $A_{1}$ is invariant under elliptic transformation
(B.16).

We now construct a slightly nontrivial subset of $A$. We prove that the set

$A_{2}:=\left\\{(n,\ell,m):\leavevmode\nobreak\ m,n>0,\leavevmode\nobreak\
\frac{n}{N}\geq m,\leavevmode\nobreak\ 0\leq|\ell|\leq 2m\right\\}$ (B.30)

is contained in141414The set $A_{2}$ contains charges with $\ell\equiv 0\bmod
2m$ which corresponds to a pole lying on the attractor contour. This is
harmless as discussed in section B.2 above. $A$. Since $\frac{n}{N}\geq m$, we
have

$\frac{1-j^{2}}{4m_{1}^{2}}-\frac{j\frac{\ell}{2m}}{m_{1}}-{n\over
Nm}\leq\frac{1-j^{2}}{4m_{1}^{2}}-\frac{j\frac{\ell}{2m}}{m_{1}}-1.$ (B.31)

Thus to prove (B.13), it suffices to show that

$\displaystyle\frac{1-j^{2}}{4m_{1}^{2}}-\frac{j\frac{\ell}{2m}}{m_{1}}-1\leq
0.$ (B.32)

Put $a=\frac{\ell}{2m}$. Then $-1\leq a\leq 1$ for charges in $A_{2}$. The
inequality can be written as

$j^{2}+4ajm_{1}+4m_{1}^{2}-1\geq 0.$ (B.33)

Since $|a|\leq 1$, we have

$\begin{split}j^{2}+4ajm_{1}+4m_{1}^{2}-1&\geq j^{2}-4|jm_{1}|+4m_{1}^{2}-1\\\
&=(|j|-2|m_{1}|)^{2}-1\geq 0,\end{split}$ (B.34)

since $|j|-2|m_{1}|$ is an odd integer. This proves the desired result.

It follows from (B.16) that any other $(n,\ell,m)$ obtained from the set
$A_{2}$ by the transformation (B.16) will also belong to the set $A$.

###  B.4 Special elements of the set $B$

We shall now construct certain special elements in the set $B$ which consists
of integers $(n,\ell,m)$ satisfying the inequality (B.19) for every allowed
value of $j,n_{1}$. The following set of charges are easily seen to be in the
set $B$:

$\begin{split}&B_{1}:=\left\\{(n,\ell,m):\leavevmode\nobreak\
m,n>0,\leavevmode\nobreak\
4\frac{n}{N}m-\ell^{2}\geq\frac{n^{2}}{N^{2}}\right\\}.\end{split}$ (B.35)

Indeed for charges in $B_{1}$, the second term of (B.19) is bounded below by 1
and hence satisfies the inequality unless $n_{1}=0$. On the other hand, for
$n_{1}=0$, the first term is bounded from below by 1. Note that $B_{1}$ is
invariant under elliptic transformation (B.21).

We now prove that the set

$B_{2}:=\left\\{(n,\ell,m):\leavevmode\nobreak\ m,n>0,\leavevmode\nobreak\
m\geq\frac{n}{N},\leavevmode\nobreak\ 0\leq|\ell|\leq 2n\right\\}$ (B.36)

is contained in $B$. Since $m\geq\frac{n}{N}$, we have

$\frac{m_{1}+j\frac{N\ell}{2n}}{n_{1}}-{Nm\over
n}\leq\frac{m_{1}+j\frac{N\ell}{2n}}{n_{1}N^{2}}-1.$ (B.37)

Thus to prove (B.18), it suffices to prove that

$\frac{m_{1}+j\frac{N\ell}{2n}}{n_{1}N^{2}}-1\leq 0$ (B.38)

for all allowed values of $n_{1},m_{1},j$ satisfying (B.7). Using (B.7) we get

$\frac{m_{1}}{n_{1}}=\frac{1-j^{2}}{4n_{1}^{2}}$ (B.39)

Substituting this in (B.37), the inequality becomes

$\frac{1-j^{2}}{4n_{1}^{2}N^{2}}-\frac{j\frac{\ell}{2n}}{n_{1}N}-1\leq 0.$
(B.40)

Put $a=\frac{\ell}{2n}$. Then $-1\leq a\leq 1$ for charges in $B_{2}$. The
inequality can be written as

$j^{2}+4ajNn_{1}+4n_{1}^{2}N^{2}-1\geq 0.$ (B.41)

To prove this, we note that

$\begin{split}j^{2}+4ajNn_{1}+4n_{1}^{2}N^{2}-1&\geq
j^{2}-4|jNn_{1}|+4n_{1}^{2}N^{2}-1\\\ &=(|j|-2|n_{1}N|)^{2}-1\geq
0.\end{split}$ (B.42)

This proves the desired relation.

As before, any other $(n,\ell,m)$ related to charges in the set $B_{2}$ by the
transformation (B.21) will also be part of set $B$.

## References

  * [1] Hiroki Aoki  “On Jacobi forms of real weights and indices”  In _Osaka Journal of Mathematics_ 54.3 Osaka UniversityOsaka Metropolitan University, Departments of Mathematics, 2017, pp. 569–585
  * [2] Ajit Bhand and Ranveer Kumar Singh  “Zagier’s weight $3/2$ mock modular form”, 2022  arXiv:2012.00539 [math.NT]
  * [3] K. Bringmann, A. Folsom, K. Ono and L. Rolen  “Harmonic Maass Forms and Mock Modular Forms: Theory and Applications”, Colloquium Publications  American Mathematical Society, 2017  URL: https://books.google.co.in/books?id=dTdDDwAAQBAJ
  * [4] Aradhita Chattopadhyaya and Justin R. David  “Properties of dyons in $\mathcal{N}=4$ theories at small charges”  In _Journal of High Energy Physics_ 2019.5 Springer ScienceBusiness Media LLC, 2019  DOI: 10.1007/jhep05(2019)005
  * [5] Shyamoli Chaudhuri, George Hockney and Joseph D. Lykken  “Maximally supersymmetric string theories in D $<$ 10”  In _Phys. Rev. Lett._ 75, 1995, pp. 2264–2267  DOI: 10.1103/PhysRevLett.75.2264
  * [6] Miranda C.. Cheng and Erik Verlinde  “Dying Dyons Don’t Count”  In _JHEP_ 09, 2007, pp. 070  DOI: 10.1088/1126-6708/2007/09/070
  * [7] H. Cohen and F. Strömberg  “Modular Forms: A Classical Approach”, Graduate studies in mathematics  American Mathematical Society, 2017  URL: https://books.google.co.in/books?id=1MmctQEACAAJ
  * [8] Atish Dabholkar and Davide Gaiotto  “Spectrum of CHL dyons from genus-two partition function”  In _JHEP_ 12, 2007, pp. 087  DOI: 10.1088/1126-6708/2007/12/087
  * [9] Atish Dabholkar, Sameer Murthy and Don Zagier  “Quantum Black Holes, Wall Crossing, and Mock Modular Forms”, 2012  arXiv:1208.4074 [hep-th]
  * [10] Soumya Das and B. Ramakrishnan  “Jacobi forms and differential operators”  In _Journal of Number Theory_ 149, 2015, pp. 351–367  DOI: https://doi.org/10.1016/j.jnt.2014.10.006
  * [11] Justin R. David, Dileep P. Jatkar and Ashoke Sen  “Product representation of Dyon partition function in CHL models”  In _JHEP_ 06, 2006, pp. 064  DOI: 10.1088/1126-6708/2006/06/064
  * [12] Justin R. David, Dileep P. Jatkar and Ashoke Sen  “Dyon spectrum in generic N=4 supersymmetric Z(N) orbifolds”  In _JHEP_ 01, 2007, pp. 016  DOI: 10.1088/1126-6708/2007/01/016
  * [13] Justin R. David and Ashoke Sen  “CHL Dyons and Statistical Entropy Function from D1-D5 System”  In _JHEP_ 11, 2006, pp. 072  DOI: 10.1088/1126-6708/2006/11/072
  * [14] M. Eichler and D. Zagier  “The Theory of Jacobi Forms”, Progress in Mathematics  Birkhäuser Boston, 2013  URL: https://books.google.co.in/books?id=p_PTBwAAQBAJ
  * [15] Amanda Folsom  “Perspectives on mock modular forms”  In _Journal of Number Theory_ 176, 2017, pp. 500–540  DOI: https://doi.org/10.1016/j.jnt.2017.02.001
  * [16] V. Gritsenko  “Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms”, 1999  arXiv:math/9906190
  * [17] Dileep P. Jatkar and Ashoke Sen  “Dyon spectrum in CHL models”  In _JHEP_ 04, 2006, pp. 018  DOI: 10.1088/1126-6708/2006/04/018
  * [18] N.I. Koblitz  “Introduction to Elliptic Curves and Modular Forms”, Graduate Texts in Mathematics  Springer New York, 2012  URL: https://books.google.com/books?id=0hTSBwAAQBAJ
  * [19] Ipsita Mandal and Ashoke Sen  “Black Hole Microstate Counting and its Macroscopic Counterpart”  In _Class. Quant. Grav._ 27, 2010, pp. 214003  DOI: 10.1088/0264-9381/27/21/214003
  * [20] M.R. Murty, M. Dewar and H. Graves  “Problems in the Theory of Modular Forms”, HBA Lecture Notes in Mathematics  Springer Nature Singapore, 2016  URL: https://books.google.com/books?id=v8eWDQAAQBAJ
  * [21] Ashoke Sen  “Walls of Marginal Stability and Dyon Spectrum in N=4 Supersymmetric String Theories”  In _JHEP_ 05, 2007, pp. 039  DOI: 10.1088/1126-6708/2007/05/039
  * [22] Ashoke Sen  “Black Hole Entropy Function, Attractors and Precision Counting of Microstates”  In _Gen. Rel. Grav._ 40, 2008, pp. 2249–2431  DOI: 10.1007/s10714-008-0626-4
  * [23] Ashoke Sen  “A Twist in the Dyon Partition Function”  In _JHEP_ 05, 2010, pp. 028  DOI: 10.1007/JHEP05(2010)028
  * [24] Ashoke Sen  “Discrete Information from CHL Black Holes”  In _JHEP_ 11, 2010, pp. 138  DOI: 10.1007/JHEP11(2010)138
  * [25] Ashoke Sen  “How Do Black Holes Predict the Sign of the Fourier Coefficients of Siegel Modular Forms?”  In _Gen. Rel. Grav._ 43, 2011, pp. 2171–2183  DOI: 10.1007/s10714-011-1175-9
  * [26] Carl Ludwig Siegel  “Symplectic Geometry”  In _American Journal of Mathematics_ 65.1 Johns Hopkins University Press, 1943, pp. 1–86  URL: http://www.jstor.org/stable/2371774
  * [27] Don Zagier  “Nombres de classes et formes modulaires de poids 3/2.”, 1975  URL: https://api.semanticscholar.org/CorpusID:117393824
  * [28] Sander Zwegers  “Mock Theta Functions”, 2008  arXiv:0807.4834 [math.NT]


# The Alpha-Proton Differential Flow in the Alfvénic Young Solar Wind: From
Sub-Alfvénic to Super-Alfvénic

Hao Ran State Key Laboratory of Space Weather, National Space Science Center,
Chinese Academy of Sciences, Beijing 100190, People’s Republic of China;
<EMAIL_ADDRESS>University of Chinese Academy of Sciences, Beijing 100049,
People’s Republic of China  Ying D. Liu State Key Laboratory of Space Weather,
National Space Science Center, Chinese Academy of Sciences, Beijing 100190,
People’s Republic of China<EMAIL_ADDRESS>University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China  Chong Chen  School of
Microelectronics and Physics, Hunan University of Technology and Business,
Changsha 410205, People’s Republic of China  Parisa Mostafavi Johns Hopkins
Applied Physics Laboratory, Laurel, MD 20723, USA

###### Abstract

Data obtained from Parker Solar Probe (PSP) since 2021 April have shown the
first in situ observation of the solar corona, where the solar wind is formed
and accelerated. Here we investigate the alpha-proton differential flow and
its characteristics across the critical Alfvén surface (CAS) using data from
PSP during encounters 8-10 and 12-13. We first show the positive correlation
between the alpha-proton differential velocity and the bulk solar wind speed
at PSP encounter distances. Then we explore how the characteristics of the
differential flow vary across the CAS and how they are affected by Alfvénic
fluctuations including switchbacks. We find that the differential velocity
below the CAS is generally smaller than that above the CAS, and the local
Alfvén speed well limits the differential speed both above and below the CAS.
The deviations from the alignment between the differential velocity and the
local magnetic field vector are accompanied by large-amplitude Alfvénic
fluctuations and decreases in the differential speed. Moreover, we observe
that $V_{\alpha p}$ increases from $M_{A}<1$ to $M_{A}\simeq 2$ and then
starts to decrease, which suggests that alphas may remain preferentially
accelerated well above the CAS. Our results also reveal that in the sub-
Alfvénic solar wind both protons and alphas show a strong correlation between
their velocity fluctuations and magnetic field fluctuations, with a weaker
correlation for alphas. In contrast, in the super-Alfvénic regime the
correlation remains high for protons, but is reduced for alphas.

##  1 Introduction

The two main ion components of the solar wind, protons and alpha particles,
are different in terms of their velocity and temperature (e.g., Marsch et al.,
1982; Steinberg et al., 1996; Neugebauer et al., 1996; Kasper et al., 2007,
2012; Mostafavi et al., 2022). Despite being heavier, alpha particles tend to
travel faster than protons, which generates a differential streaming between
them. This phenomenon was first reported by Robbins et al. (1970), and remains
an unsolved puzzle. Several mechanisms have been proposed to explain the cause
of the differential streaming. For example, Isenberg & Vasquez (2009) proposed
a cyclotron-resonant Fermi process that is capable of providing the
preferential acceleration of minor ions, Verscharen et al. (2013) discussed
the role of the parallel Alfvénic drift instabilities in the formation and
evolution of the differential streaming, and Chandran (2010) claimed that
stochastic heating caused by low-frequency Alfvén-wave turbulence also
contributes to the preferential heating of minor ions. According to Hu &
Habbal (1999), the preferential acceleration and heating of the alpha
particles close to the Sun are both due to the resonant cyclotron interaction.

Before the Parker Solar Probe (PSP; Fox et al., 2016) era, the differential
flow was studied with data from 0.3 to 5.5 au. Some common features of the
differential velocity emerged in this range of distances. For example, using
data from Helios 1 and 2, Marsch et al. (1982) reported a clear tendency of
the differential speed increasing with the solar wind speed. The same
conclusion was drawn by Steinberg et al. (1996) and Ďurovcová et al. (2017)
with Wind observations. In addition, previous observations suggested that the
differential speed is of the order of, but generally lower than, the local
Alfvén speed (e.g., Marsch et al., 1982; Neugebauer et al., 1996; Berger et
al., 2011; Ďurovcová et al., 2017). One possible interpretation was given by
Gary et al. (2000) that a sufficiently fast differential flow can lead to
alpha-proton instabilities that could drive wave-particle scattering. This
scattering reduces the differential speed and thus limits it to be lower than
the local Alfvén speed. Another interesting feature is the alignment between
the differential velocity and the local magnetic field vector (e.g., Marsch et
al., 1982; Yamauchi et al., 2004; Matteini et al., 2015). Yamauchi et al.
(2004), for example, revealed a high correlation between the differential
velocity and the magnetic field vector both inside and outside pressure
balance structures with Ulysses observations.

The PSP mission, launched in 2018, has enabled us to investigate the
differential flow at unprecedented proximity to the Sun (McManus et al., 2022;
Mostafavi et al., 2022). Mostafavi et al. (2022) confirmed the positive
correlation between the differential velocity and the bulk solar wind speed,
the dependence of the differential flow on heliocentric distance, and the
limitation of the differential speed by the local Alfvén speed inside 0.3 AU,
using PSP data from encounters 3-7. However, sub-Alfvénic solar wind
measurements are not incorporated into their study, and the impacts of
Alfvénic fluctuations on the differential flow are not discussed. Ofman et al.
(2023) studied the nonlinear evolution of ion kinetic instabilities and the
transfer of energy between ions and waves in the sub-Alfvénic regime, which
suggests that the locally produced ion instabilities contribute to the
anisotropic heating of alphas. One of the striking observations of PSP is the
common presence of Alfvénic switchbacks in the young solar wind, which are
characterized by a deflection in the magnetic field direction and an increase
in the radial velocity (Bale et al., 2019; Kasper et al., 2019). McManus et
al. (2022) examined the 3D nature of the velocity fluctuations of alphas and
protons in switchbacks using PSP observations from encounters 3 and 4. They
revealed that the velocity of alphas may increase, remain constant, or
decrease in switchbacks depending on whether the differential speed is
smaller, equal to, or larger than the local wave speed. Here our study
includes the behaviors of alphas and protons inside switchbacks at closer
distances to the Sun. Another important achievement of PSP is the crossing of
the critical Alfvén surface (CAS) for the first time in 2021 April (Kasper et
al., 2021), which happened at a surprisingly large distance ($\sim$ 19
$R_{S}$). Liu et al. (2023) suggested that the sub-Alfvénic solar wind
detected by PSP is a special type of wind emanating from boundaries inside a
coronal hole, termed a low Mach-number boundary layer (LMBL). An LMBL is
associated with an enhanced Alfvén radius, which explains the crossing of the
CAS at $\sim$ 19 $R_{S}$.

In this paper, we focus on the characteristics of the alpha-proton
differential flow across the CAS and how they are affected by Alfvénic
fluctuations including switchbacks. Given the magnetic nature of switchbacks
and the unprecedented measurements of the sub-Alfvénic solar wind, it is of
great interest to investigate the above mentioned properties of the
differential flow at PSP encounter distances, especially in the new domain
below the CAS, and to explore the effects of Alfvénic fluctuations on the
differential flow. Specifically, this work aims to address the following
questions: (i) How does the alpha-proton differential flow vary with the solar
wind speed in the most recent orbits of PSP? (ii) How do the characteristics
of the differential flow vary when the solar wind changes from sub-Alfvénic to
super-Alfvénic (i.e., across the CAS)? (iii) How do Alfvénic fluctuations
including switchbacks affect the differential flow? This work is organized as
follows. In Section 2, we present our analysis and results. We summarize our
conclusions in Section 3.

##  2 Observations and Results

Data used in this study are from the FIELDS (Bale et al., 2016) instrument
suite and the SWEAP (Kasper et al., 2016) package aboard PSP. The level 2
product from FIELDS provides us the RTN magnetic field data. The ion data are
from the level 3 moment products from SPAN-I, which is an electrostatic
analyzer on the ram side of PSP (Livi et al., 2022). We consider an interval
of observations to be reliable if the cores of the ions (protons and alphas)
are within the field of view and energy range of SPAN-I. Further, transient
events such as coronal mass ejections (CMEs) or heliospheric current sheet
(HCS) crossings in the selected intervals are excluded. The details of the
data selection and curation in this study are given in the Appendix. Densities
measured by SPAN-I are derived from partial moments of the particle velocity
distributions, which may not be accurate as the full ion distribution is not
captured by SPAN-I. Therefore, we adopt the electron density from quasi-
thermal noise (QTN) spectroscopy to represent the plasma density (Moncuquet et
al., 2020). The local Alfvén speed is thus computed as
$V_{A}=B/\sqrt{\mu_{0}n_{e}m_{p}}$, where $B$ is the local magnetic field
strength, $\mu_{0}$ is the permeability of vacuum, $n_{e}$ is the electron
density, and $m_{p}$ is the proton mass. Use of electron density as a
representation of the plasma density may introduce an overestimate of the
local Alfvén speed by 2% to 5% (Mostafavi et al., 2022). We use PSP data from
encounters 8-10 and encounters 12-13, during which sub-Alfvénic solar wind
measurements are reported (Jiao et al., 2024). We exclude encounter 11 due to
the absence of the QTN density for this encounter. The data are linearly
interpolated to 2 s resolution. All vector data are in the RTN coordinate
system.

The magnitude of the differential velocity is calculated as:

$|\bm{V_{\alpha
p}}|=\mathrm{sign}(|\bm{V_{\alpha}}|-|\bm{V_{p}}|)(|\bm{V_{\alpha}}-\bm{V_{p}}|).$
(1)

This derivation takes the direction of both the protons and alphas into
consideration (e.g., Steinberg et al., 1996; Berger et al., 2011; Ďurovcová et
al., 2017; Mostafavi et al., 2022). The method for calculating the deflection
angle (i.e., $\theta$) of the magnetic field is adopted from Liu et al.
(2023). The calculation is capable of indicating the deflection direction of
the magnetic field. The angle $\theta$ varies from $-\pi$ to $\pi$, with a
negative one corresponding to a clockwise deflection and a positive one
corresponding to a counter-clockwise deflection when viewed from the north. A
value of $-\pi$ or $\pi$ implies a completely reversed magnetic field. We
include all possible deflection angles instead of only those larger than
$90^{\circ}$ (e.g., Dudok de Wit et al., 2020; Liu et al., 2023). Liu et al.
(2023) showed that with this derivation of the deflection angle the
enhancement in the proton radial velocity can be written as $\delta
V_{pR}/V_{A}=1-\cos\theta$, which well explains the one-sided nature of the
velocity spikes observed within switchbacks (e.g., Gosling et al., 2009;
Horbury et al., 2018; Kasper et al., 2019). Readers are directed to Liu et al.
(2023) for a detailed description of the derivation of the deflection angle
and its relationship with the velocity enhancement. Also, following Liu et al.
(2023), we derive the variation of the radial velocity of the particles
$\delta V_{R}$ by subtracting a low-pass filtered radial velocity $V_{Rf}$
from $V_{R}$. The cuttoff frequency of the filter is $2\times 10^{-4}$ Hz,
which is below the break frequency (about $2\times 10^{-3}$ Hz) between the
energy-containing range and inertial range in the power spectrum of solar wind
fluctuations (e.g., Kasper et al., 2021; Zank et al., 2022).

###  2.1 Overview of Measurements at Encounters 9 and 10

We first present an overview of the measurements at encounters 9 and 10, which
show a slow solar wind case and a fast solar wind case, respectively. Figure 1
displays the in situ measurements near the perihelion of encounter 9 made from
19:40 UT on 2021 August 5 to the beginning of 2021 August 10, during which PSP
distances range from about 42 $R_{S}$ to 16 $R_{S}$. We select this time
range, because it covers a relatively wide range of $V_{\alpha p}$ and
contains sub-Alfvénic intervals. The radial velocity of protons in this
interval starts from around 300 km/s, then rises up to about 380 km/s at
around 05:00 UT on August 6, and stays at this level until around 12:00 UT on
August 7. After this plateau, the radial velocity of protons drops to around
220 km/s and varies in the range from 170 km/s to 300 km/s. As can be seen in
Figure 1(c), in the faster solar wind the radial speed of alphas is higher
than that of protons, while in the slower solar wind the two radial speeds are
close. In the sub-Alfvénic intervals the solar wind is very slow but presents
a relatively obvious $V_{\alpha p}$. These slow but fast wind like flows are
consistent with the theory of Liu et al. (2023), who term such sub-Alfvénic
intervals LMBLs. An LMBL, characterized by a drop in the Alfvén Mach number
resulting from its decreased plasma density and relatively low velocity, may
originate from the peripheral region of a coronal hole. Figure 1(d) shows the
relationship between $V_{\alpha p}$ and $V_{A}$. The local Alfvén speed acts
as an upper limit for $V_{\alpha p}$. For the faster wind from 06:00 UT on
2021 August 6 to 00:00 UT on 2021 August 7, $V_{\alpha p}$ fluctuates near
$V_{A}$. The average speed of this interval is 358 km/s. Figure 1(f) displays
the cosine value of the angle between the magnetic field vector and the
differential velocity. The two vectors are generally well aligned (i.e., close
to -1 or 1) except when $V_{\alpha p}$ is close to 0 and changes sign
frequently. Such frequent changes in the sign of $V_{\alpha p}$ cause the
cosine value to fluctuate between -1 and 1, which makes the determination of
the alignment difficult. Therefore, we mark the spikes of the curve in Figure
1(f) every 50 data points with red crosses when $V_{\alpha p}$ is overall
positive. The red crosses then represent the times when the alignment
deviates. We also mark those points in Figure 1(d) and (e) with crosses too.
The locations of the crosses indicate that the spikes in Figure 1 (f) are
associated with drops in $V_{\alpha p}$ and large deflection angles. This
implies that the deviations from the alignment are related to switchbacks and
the velocity fluctuations within switchbacks. Figure 1(i) shows that the
velocity variations of protons are largely one-sided as in Liu et al. (2023).
The velocity variations of alphas (Figure 1(h)) are similar to those of
protons, but appear more two-sided. This may suggest a weaker modulation of
alphas by Alfvénic fluctuations. McManus et al. (2022) showed cases where the
velocity of alphas may increase, remain constant, or decrease in switchbacks
depending on the relationship between the differential speed and the wave
speed. This explains the more two-sided velocity variations of alphas. We will
discuss this result in more details below.

Figure 2 presents the measurements at encounter 10 from 08:10 UT on 2021
November 19 to the end of 2021 November 21. The reason for selecting this time
range is similar as for encounter 9, but here we focus on a fast wind
interval. The radial speed of protons first starts from around 450 km/s, and
rises to over 500 km/s at the beginning of 2021 November 20. Our magnetic
mapping results (not shown here) suggest that PSP was connected to the
interior of a large coronal hole during this interval. The magnetic mapping of
the same encounter can also be found in Badman et al. (2023). Then it drops to
around 250 km/s at the end of 2021 November 20, and soon rises up again to
over 400 km/s. This dip of the radial speed of protons corresponds to an
interval when PSP was surfing around the CAS (See Figure 2(g)). The radial
speed of protons finally drops to a very low level after 14:00 UT on 2021
November 21. As at encounter 9, the radial velocity of alphas is higher than
that of protons in the faster solar wind. The local Alfvén speed is an upper
limit for $V_{\alpha p}$ in most cases (98.2 %). The alignment between the
magnetic field vector and the differential velocity is in general well-kept
when $V_{\alpha p}$ does not change sign frequently. The locations of the
crosses indicate that the deviations from the alignment are accompanied by
large deflection angles and drops in $V_{\alpha p}$. The velocity variations
of alphas (Figure 2(h)) again appear more two-sided than those of protons. It
is important to mention that, although PSP was traveling in the vicinity of
the CAS, we have not identified a steady sub-Alfvénic interval during this
time period.

###  2.2 Distributions and Correlations

After showing the typical examples for the slow and fast wind, here we look at
the distributions and correlations in the data. We also include data from
encounters 8, 12, and 13 to have enough statistics. Figure 3 contains the
histograms of $V_{\alpha p}/V_{A}$ and $\theta$ in the super-Alfvénic and sub-
Alfvénic solar wind. We divide the super-Alfvénic (1037029 measurements) and
sub-Alfvénic (76606 measurements) solar winds into four groups according to
their bulk speed at 100 km/s and 50 km/s intervals, respectively. The groups
are chosen such that each group has an acceptable proportion of the data. As
suggested by Figure 3(a), for the super-Alfvénic solar wind, most of
$V_{\alpha p}/V_{A}$ is between 0 and 1. Values lower than 0 (i.e.,
$V_{p}>V_{\alpha}$) usually occur in the slower solar winds ($V_{p}<320$
km/s). Values higher than 1 (i.e., $V_{\alpha p}>V_{A}$) are rare (only 2.2
%), and they usually occur in the faster solar winds ($V_{p}>320$ km/s).
McManus et al. (2022) also reported cases when $V_{\alpha p}>V_{A}$. They
suggested that alphas will move in antiphase with protons inside switchbacks
if $V_{\alpha p}$ is larger than the wave speed, which will result in a spike
in the proton velocity and a dip in the alpha velocity. The reason why
$V_{\alpha p}$ could exceed the local Alfvén speed, however, is unclear. The
peak in general moves to higher values as the solar wind speed increases.
These observations are consistent with previous studies (e.g., Marsch et al.,
1982; Mostafavi et al., 2022). Figure 3(b) shows the deflection angle
distributions of the super-Alfvénic solar wind. The deflection angle is
predominantly negative, which indicates a preferential clockwise rotation of
the magnetic field (also see Squire et al., 2022; Fargette et al., 2022).
According to Liu et al. (2023), this is due to the clockwise geometry of the
background Parker spiral field. The degree of this preference decreases as the
solar wind speed increases, as expected from the more radial Parker spiral
field in the faster solar wind. The peak also shifts to higher values as the
solar wind speed increases, which indicates that larger switchbacks tend to
appear in the faster solar winds. In Figure 3(c), we rarely observe data
points with $V_{\alpha p}/V_{A}>1$ in the sub-Alfvénic solar wind (less than
0.01 %), and data points with $V_{\alpha p}/V_{A}<0$ also exist in the slowest
solar winds ($V_{p}<175$ km/s). Akin to that of the super-Alfvénic solar wind,
the peak of the distributions moves to higher values as the solar wind speed
increases. For the deflection angle distributions of the sub-Alfvénic solar
wind shown in Figure 3 (d), the widths of all categories are roughly the same,
and the preference for clockwise rotation still exists. Equal widths agree
with the LMBL nature of the sub-Alfvénic intervals. As explained in Liu et al.
(2023), a low Alfvén Mach number suppresses the amplitudes of switchbacks,
regardless of the solar wind speed. The preference for clockwise rotation of
the magnetic field indicates that the influence brought by the clockwise
curvature is still valid below the CAS.

Figure 4 shows the scatter plots of $V_{\alpha p}$ versus $\theta$ and
$M_{A}$. We notice a rough decreasing trend of $V_{\alpha p}$ with increasing
$|\theta|$. This is consistent with Figure 1 and Figure 2 that large
switchbacks are accompanied by drops in $V_{\alpha p}$. The larger switchbacks
may disturb the alignment between the differential velocity and the magnetic
field vector in the inertial frame, and reduce the magnitude of the
differential velocity (Matteini et al., 2015; McManus et al., 2022). The sub-
Alfvénic solar wind is concentrated at small deflection angles as expected,
and the presence of large $V_{\alpha p}$ in the sub-Alfvénic wind may indicate
the existence of the fast wind like features in the slow sub-Alfvénic solar
wind. The right panel shows $V_{\alpha p}$ as a function of $M_{A}$. For a
better interpretation, we also plot the averages of $V_{\alpha p}$ for each
speed category. The curves show the dependence of the differential speed on
both $M_{A}$ and the solar wind speed. Firstly, as the solar wind becomes
faster at a fixed $M_{A}$, the differential speed generally increases
accordingly. On the other hand, the increase of $M_{A}$ can be considered as
the acceleration process of the solar wind from lower to higher altitudes. We
observe that the maximum value of $V_{\alpha p}$ increases until $M_{A}\simeq
2$, which may indicate that a strong preferential acceleration of alphas
occurs in this region. In the slower solar wind ($V_{p}<320$ km/s) the
preferential acceleration of alphas occurs below the CAS, and then decreases
above the CAS. In the faster solar wind ($V_{p}>320$ km/s) the preferential
acceleration region extends out above the CAS. Kasper & Klein (2019) suggested
that the preferential heating of alphas, which is highly related to the
preferential acceleration of alphas (Kasper et al., 2008; Bourouaine et al.,
2011), is predominantly active below the CAS. According to their discussions,
a fraction of the outward propagating waves in the expanding solar wind can be
reflected back toward the Sun, which, if below the CAS, can travel all the way
back to the Sun and interact with the outward propagating waves. Such
interactions can lead to wave-reflection-driven turbulence providing the
energy for the preferential heating of alphas. It is unclear why alphas are
still preferentially accelerated above the CAS in the fast solar wind. It may
be attributed to the combination of two factors. On the one hand, the
preferential acceleration of alphas is more intense and diverse in the fast
solar wind (Marsch et al., 1982). For example, Isenberg & Vasquez (2009)
proposed an ion cyclotron-resonant Fermi process that can perpendicularly heat
and accelerate minor ions in a high-speed stream originating from a coronal
hole. On the other hand, Coulomb collisions are practically negligible in the
fast solar wind due to its high-temperature and low-density conditions (Marsch
et al., 1982). Therefore, it takes longer for Coulomb collisions to thermalize
the plasma in the faster solar wind. This may also contribute to the
enhancement of $V_{\alpha p}$ above the CAS in the fast solar wind. After
$M_{A}\simeq 2$, the differential velocity decreases with increasing $M_{A}$.
The decreasing trend of $V_{\alpha p}$ with $M_{A}$ above this region might be
due to the instabilities that decelerate alpha particles as $V_{A}$ decreases
with distance (Gary et al., 2000), or due to Coulomb collisions that tend to
bring all species to equilibrium (Marsch et al., 1982). It is also important
to note that the amount of the sub-Alfvénic measurements is limited so far. We
expect more sub-Alfvénic observations for a more definite conclusion in the
future.

For better understanding the effects of Alfvénic fluctuations across the CAS
on the particles, we plot the behaviors of the two ion species in Figure 5.
Figure 5(a) displays the radial velocity enhancement of protons in units of
the local Alfvén speed versus the magnetic field deflection angle. Their
relationship is consistent with the equation $\delta
V_{R}/V_{A}=1-\mathrm{cos}\theta$ given by Liu et al. (2023). The red dashed
curve acts as an upper limit for the radial velocity enhancement because the
magnetic field is not exactly radial or anti-radial. Figure 5(a) confirms the
findings of Liu et al. (2023) but with larger statistics. In the sub-Alfvénic
solar wind, the trend is not very clear as the deflection angles are small.
The behaviors of alphas in the Alfvénic fluctuations, as shown in Figure 5(b),
are different from those of protons, and their velocity fluctuations do not
quite follow the red curve. This is consistent with the more two-sided
velocity variations of alphas shown in Figure 1 and Figure 2. Therefore, alpha
particles are less modulated by Alfvénic fluctuations than protons. Figure
5(c) and (d) present the relation between $B_{N}/B$ and $V_{N}/V_{A}$ for
protons and alphas, which can be used as another indicator of the effects of
Alfvénic fluctuations (Goldstein et al., 1995). Protons show a strong
correlation between $B_{N}/B$ and $V_{N}/V_{A}$ close to the expected Alfvén
wave relation (the red dashed line) for both below and above the CAS
(correlation coefficient CC = $0.93$ and $0.82$, respectively). A weaker
correlation is seen for alphas in the super-Alfvénic solar wind (CC=$0.61$),
but in the sub-Alfvénic solar wind the correlation appears strong (CC=$0.81$).
It is surprising that below the CAS alpha particles are also considerably
modulated by Alfvénic fluctuations. Therefore, alphas may be significantly
accelerated and heated by Alfvénic fluctuations below the CAS. The increased
correlation between $B_{N}/B$ and $V_{N}/V_{A}$ for alphas in the sub-Alfvénic
solar wind may be due to $V_{\alpha p}$ being closer to 0 in this region.
According to McManus et al. (2022), the behavior of alphas in Alfvénic
fluctuations is close to that of protons when $V_{\alpha p}$ is small. The CC
of alphas, however, is smaller than that of protons for both below and above
the CAS. Matteini et al. (2015) analyzed the motions of protons and alphas in
the Alfvén wave frame, and suggested that alphas are usually closer to the
center of the wave frame, which may result in the weaker modulation of alphas
by Alfvénic fluctuations. McManus et al. (2022) demonstrated three scenarios,
in which $V_{N}/V_{A}$ for alphas is correlated, uncorrelated, and anti-
correlated with $B_{N}/B$. The combination of these cases may lead to a
decrease in the CC for alphas.

##  3 Conclusions

We have investigated the characteristics of the alpha-proton differential flow
in the young solar wind across the CAS, and the effects of Alfvénic
fluctuations including switchbacks on the differential flow. We summarize our
findings, which address the three key questions that we have put forward in
Section 1, as follows:

  1. 1.

In general, alphas stream faster than protons (in 90.3 % of the cases), the
differential velocity between them is aligned with the local magnetic field
vector, and the local Alfvén speed provides an upper limit for the
differential velocity (in 97.9 % of the cases). These results are consistent
with previous studies. However, the situation changes with the solar wind
speed. In the slowest solar wind ($V_{p}<325$ km/s), we do observe occasions
when alphas stream slower than protons (in 14.0 % of the cases). As the solar
wind becomes faster, the differential velocity in units of the local Alfvén
speed also increases but remains smaller than 1.

  2. 2.

We find an overall enhancement of $V_{\alpha p}$ when $M_{A}\lesssim 2$, which
may indicate that a strong preferential acceleration of alphas occurs in this
region. The preferential acceleration of alphas extends to higher $M_{A}$ in
the faster solar wind. This may be attributed to the combination of two
factors: the more intense and diverse preferential acceleration mechanisms and
the less significant Coulomb collisions in the faster solar wind. After
$M_{A}\simeq 2$, $V_{\alpha p}$ decreases with increasing $M_{A}$. Below the
CAS, the differential velocity in units of the local Alfvén speed is generally
smaller than 1 (in 99.9 % of the cases), and the case of $V_{\alpha}<V_{p}$
exists only in the slowest solar wind ($V_{p}<175$ km/s). The situation is
similar above the CAS. The differential velocity is generally smaller than the
local Alfvén speed (in 97.8 % of the cases), and $V_{\alpha p}<V_{p}$ cases
are also seen only in the slowest solar wind ($V_{p}<325$ km/s).

  3. 3.

Alfvénic fluctuations including switchbacks could affect the characteristics
of the differential velocity. We observe deviations from the alignment between
the differential velocity and the local magnetic field vector, and they are
accompanied by large magnetic field deflection angles and drops in the
differential velocity. Moreover, protons and alphas are modulated by Alfvénic
fluctuations in different ways. The effects of Alfvénic fluctuations on
protons are stronger than on alphas. These results are consistent with the
findings of McManus et al. (2022). The radial velocity variations of alphas
are more two-sided than those of protons. In the sub-Alfvénic solar wind, the
correlation between $B_{N}/B$ and $V_{N}/V_{A}$ is strong for both protons and
alphas (CC = 0.93 and 0.81, respectively). It is surprising that the effects
of Alfvénic fluctuations on alphas appear strong in the sub-Alfvnéic solar
wind. In the super-Alfvénic solar wind, the correlation between $B_{N}/B$ and
$V_{N}/V_{A}$ is still strong for protons (CC = 0.82), but for alphas the
correlation is weakened (CC=0.61).

  4. 4.

Apart from the above conclusions that correspond to the three key questions in
Section 1, we also have some extra findings. We reveal that the deflection
angle increases with the solar wind speed in the super-Alfvénic solar wind,
while in the sub-Alfvénic solar wind the deflection angle is concentrated at
small values in all speed categories. We also find that the preference for
clockwise deflections is observed not only in the super-Alfvénic wind, but
also in the sub-Alfvénic wind. The degree of the preference decreases as the
solar wind speed increases, which may result from the more radial background
Parker spiral field in the faster solar wind.

## Appendix: Data Selection and Curation

In order to have reliable velocity measurements, we need to select intervals
when the cores of the solar wind ions (alphas and protons) are primarily
within the field of view (FOV) and energy range of SPAN-I. For each encounter,
the total length of the reliable intervals is usually a few days.
Specifically, our selection is derived through the following steps. We select
intervals in which the peak of ion flux is in the energy range of SPAN-I. The
bulk of the 3D velocity distribution functions (VDFs) of the ions is also
confirmed to be within the FOV of SPAN-I during these intervals. Figure 6
shows the azimuthal fluxes of alphas and protons in the spacecraft frame. The
selected time intervals are indicated by the dashed lines and curly brackets
in Figure 6. Note that the CME at encounter 13 is already removed from the
data in this step. Readers are redirected to Liu et al. (2024) for the time
interval of the CME. Table 1 lists the detailed time ranges of the selected
intervals and the HCS crossings within them. The relatively long duration (up
to several hours) of the HCS crossings, as shown in Table 1, is a result of
their clustered occurrence. Our final dataset is obtained by subtracting the
times of the clustered HCS crossings from the selected intervals.

## Acknowledgments

This research was supported by NSFC under grant 42274201, the National Key R&D
Program of China (No.2021YFA0718600 and No.2022YFF0503800), the Strategic
Priority Research Program of the Chinese Academy of Sciences (No.XDB0560000),
and the Specialized Research Fund for State Key Laboratories of China. C.C.
was supported by the Research Foundation of Education Bureau of Hunan
Province, China (No.23B0593). P.M. acknowledges the support from NASA HGIO
grant 80NSSC23K0419. We are grateful to Dr. Huidong Hu and Dr. Rui Wang for
helpful discussions. We thank the NASA PSP mission and the SWEAP and FIELDS
teams for allowing uses of their data. The FIELDS RTN magnetic field data and
the QTN data are available at
https://research.ssl.berkeley.edu/data/psp/data/sci/fields/, and the SWEAP ion
data are available at http://sweap.cfa.harvard.edu/pub/data/sci/sweap/spi. We
acknowledge the work of the Sunpy team (The SunPy Community et al., 2020).

## References

  * Badman et al. (2023) Badman, S. T., Riley, P., Jones, S. I., et al. 2023, JGRA, 128, e2023JA031359, doi: https://doi.org/10.1029/2023JA031359
  * Bale et al. (2016) Bale, S. D., Goetz, K., Harvey, P. R., et al. 2016, SSRv, 204, 49, doi: 10.1007/s11214-016-0244-5
  * Bale et al. (2019) Bale, S. D., Badman, S. T., Bonnell, J. W., et al. 2019, Natur, 576, 237, doi: 10.1038/s41586-019-1818-7
  * Berger et al. (2011) Berger, L., Wimmer-Schweingruber, R. F., & Gloeckler, G. 2011, PhRvL, 106, 151103, doi: 10.1103/PhysRevLett.106.151103
  * Bourouaine et al. (2011) Bourouaine, S., Marsch, E., & Neubauer, F. M. 2011, ApJL, 728, L3, doi: 10.1088/2041-8205/728/1/L3
  * Chandran (2010) Chandran, B. D. G. 2010, ApJ, 720, 548
  * Dudok de Wit et al. (2020) Dudok de Wit, T., Krasnoselskikh, V. V., Bale, S. D., et al. 2020, ApJS, 246, 39, doi: 10.3847/1538-4365/ab5853
  * Ďurovcová et al. (2017) Ďurovcová, T., Šafránková, J., Němeček, Z., & Richardson, J. D. 2017, ApJ, 850, 164, doi: 10.3847/1538-4357/aa9618
  * Fargette et al. (2022) Fargette, N., Lavraud, B., Rouillard, A. P., et al. 2022, A & A, 663, A109, doi: 10.1051/0004-6361/202243537
  * Fox et al. (2016) Fox, N. J., Velli, M. C., Bale, S. D., et al. 2016, SSRv, 204, 7, doi: 10.1007/s11214-015-0211-6
  * Gary et al. (2000) Gary, S. P., Yin, L., Winske, D., & Reisenfeld, D. B. 2000, GeoRL, 27, 1355, doi: 10.1029/2000GL000019
  * Goldstein et al. (1995) Goldstein, B. E., Neugebauer, M., & Smith, E. J. 1995, GeoRL, 22, 3389, doi: 10.1029/95GL03182
  * Gosling et al. (2009) Gosling, J. T., McComas, D. J., Roberts, D. A., & Skoug, R. M. 2009, ApJ, 695, L213, doi: 10.1088/0004-637X/695/2/L213
  * Horbury et al. (2018) Horbury, T. S., Matteini, L., & Stansby, D. 2018, MNRAS, 478, 1980, doi: 10.1093/mnras/sty953
  * Hu & Habbal (1999) Hu, Y. Q., & Habbal, S. R. 1999, JGRA, 104, 17045, doi: https://doi.org/10.1029/1999JA900193
  * Isenberg & Vasquez (2009) Isenberg, P. A., & Vasquez, B. J. 2009, ApJ, 696, 591, doi: 10.1088/0004-637X/696/1/591
  * Jiao et al. (2024) Jiao, Y., Liu, Y. D., Ran, H., & Cheng, W. 2024, ApJ, 960, 42, doi: 10.3847/1538-4357/ad0dfe
  * Kasper et al. (2021) Kasper, J., Klein, K., Lichko, E., et al. 2021, PhRvL, 127, 255101
  * Kasper & Klein (2019) Kasper, J. C., & Klein, K. G. 2019, ApJL, 877, L35, doi: 10.3847/2041-8213/ab1de5
  * Kasper et al. (2008) Kasper, J. C., Lazarus, A. J., & Gary, S. P. 2008, PhRvL, 101, 261103, doi: 10.1103/PhysRevLett.101.261103
  * Kasper et al. (2012) Kasper, J. C., Stevens, M. L., Korreck, K. E., et al. 2012, ApJ, 745, 162, doi: 10.1088/0004-637X/745/2/162
  * Kasper et al. (2007) Kasper, J. C., Stevens, M. L., Lazarus, A. J., Steinberg, J. T., & Ogilvie, K. W. 2007, ApJ, 660, 901, doi: 10.1086/510842
  * Kasper et al. (2016) Kasper, J. C., Abiad, R., Austin, G., et al. 2016, SSRv, 204, 131, doi: 10.1007/s11214-015-0206-3
  * Kasper et al. (2019) Kasper, J. C., Bale, S. D., Belcher, J. W., et al. 2019, Natur, 576, 228, doi: 10.1038/s41586-019-1813-z
  * Liu et al. (2023) Liu, Y. D., Ran, H., Hu, H., & Bale, S. D. 2023, ApJ, 944, 116, doi: 10.3847/1538-4357/acb345
  * Liu et al. (2024) Liu, Y. D., Zhu, B., Ran, H., et al. 2024, arXiv:2401.06449.  https://arxiv.org/abs/2401.06449
  * Livi et al. (2022) Livi, R., Larson, D. E., Kasper, J. C., et al. 2022, ApJ, 938, 138, doi: 10.3847/1538-4357/ac93f5
  * Marsch et al. (1982) Marsch, E., Mühlhäuser, K.-H., Rosenbauer, H., Schwenn, R., & Neubauer, F. M. 1982, JGR, 87, 35, doi: 10.1029/JA087iA01p00035
  * Matteini et al. (2015) Matteini, L., Horbury, T. S., Pantellini, F., Velli, M., & Schwartz, S. J. 2015, ApJ, 802, 11, doi: 10.1088/0004-637X/802/1/11
  * McManus et al. (2022) McManus, M. D., Verniero, J. L., Bale, S. D., et al. 2022, ApJ, 933, 43, doi: 10.3847/1538-4357/ac6ba3
  * Moncuquet et al. (2020) Moncuquet, M., Meyer-Vernet, N., Issautier, K., et al. 2020, ApJS, 246, 44, doi: 10.3847/1538-4365/ab5a84
  * Mostafavi et al. (2022) Mostafavi, P., Allen, R. C., McManus, M. D., et al. 2022, ApJL, 926, L38, doi: 10.3847/2041-8213/ac51e1
  * Neugebauer et al. (1996) Neugebauer, M., Goldstein, B. E., Smith, E. J., & Feldman, W. C. 1996, JGR, 101, 17047, doi: 10.1029/96JA01406
  * Ofman et al. (2023) Ofman, L., Boardsen, S. A., Jian, L. K., et al. 2023, ApJ, 954, 109, doi: 10.3847/1538-4357/acea7e
  * Robbins et al. (1970) Robbins, D. E., Hundhausen, A. J., & Bame, S. J. 1970, JGR, 75, 1178, doi: 10.1029/JA075i007p01178
  * Squire et al. (2022) Squire, J., Johnston, Z., Mallet, A., & Meyrand, R. 2022, PhPl, 29, 112903, doi: 10.1063/5.0099924
  * Steinberg et al. (1996) Steinberg, J. T., Lazarus, A. J., Ogilvie, K. W., Lepping, R., & Byrnes, J. 1996, GeoRL, 23, 1183, doi: 10.1029/96GL00628
  * The SunPy Community et al. (2020) The SunPy Community, Barnes, W. T., Bobra, M. G., et al. 2020, ApJ, 890, 68
  * Verscharen et al. (2013) Verscharen, D., Bourouaine, S., Chandran, B. D. G., & Maruca, B. A. 2013, ApJ, 773, 8, doi: 10.1088/0004-637X/773/1/8
  * Yamauchi et al. (2004) Yamauchi, Y., Suess, S. T., Steinberg, J. T., & Sakurai, T. 2004, JGR, 109, doi: 10.1029/2003JA010274
  * Zank et al. (2022) Zank, G. P., Zhao, L.-L., Adhikari, L., et al. 2022, ApJL, 926, L16, doi: 10.3847/2041-8213/ac51da

Figure 1:  PSP measurements at encounter 9 showing a case of the slow solar
wind. The shaded areas indicate the sub-Alfvénic intervals. (a) Normalized (to
1 au values) magnetic field magnitude and radial component (red). (b)
Normalized electron density (from QTN). (c) Radial velocity of protons and
alpha particles (red). (d) Differential speed and local Alfvén speed (red).
(e) Magnetic field deflection angle. (f) Cosine of the angle between magnetic
field vector and differential velocity. (g) Alfvén Mach number. (h) Radial
velocity change of alpha particles in units of local Alfvén speed. (i) Radial
velocity change of protons in units of local Alfvén speed. The times of the
red crosses are derived by identifying the spikes in panel (f) every 50 data
points when the differential speed is overall larger than 0. Figure 2: PSP
measurements at encounter 10 showing a case of the fast solar wind. Similar to
Figure 1. Figure 3: Histograms of $V_{\alpha p}/V_{A}$ (left) and deflection
angle $\theta$ (right) in super-Alfvénic (top) and sub-Alfvénic (bottom) solar
wind. The super-Alfvénic and sub-Alfvénic solar winds are divided into 4
groups according to their bulk speed at 100 and 50 km/s intervals,
respectively. The percentages shown in panels (a) and (c) indicate the
proportion of each group to the total number of data points.  Figure 4:
Measurements of $V_{\alpha p}$ as functions of magnetic field deflection angle
(left) and radial Alfvén Mach number (right). The orange scatter in the left
panel refers to the sub-Alfvénic measurement. The curves shown in the right
panel indicate the averages of the alpha-proton differential flow as a
function of $M_{A}$ for different ranges of the solar wind speeds. Figure 5:
Effects of Alfvnéic fluctuations on protons (left) and alphas (right). Panels
(a) and (b) show the radial velocity variations of protons and alphas in units
of the local Alfvén speed as a function of magnetic field deflection angle.
The red dashed curves are derived from the equation $\delta
V_{R}/V_{A}=1-\mathrm{cos}\theta$ given by Liu et al. (2023). Panels (c) and
(d) refer to the correlations between the normal components of the velocity of
protons and alpha particles in units of the local Alfvén speed and the
magnetic field. The red dashed lines indicate the slope corresponding to the
local Alfvén speed (slope=1). The correlation coefficients (CCs) of the
scatters are also given. Again, the orange scatter refers to the sub-Alfvénic
measurement. Figure 6: The azimuthal fluxes of protons (upper) and alphas
(lower) in the spacecraft frame over time for each encounter. Dashed lines and
curly brackets indicate the intervals that are considered reliable.

Encounter | Selected time intervals | Clustered HCS crossings
---|---|---
Star time (UT) |  End time (UT) |  Start time (UT) |  End time (UT)
8 |  2021-04-25 14:10 |  2021-04-26 16:40 |  2021-04-29 00:30 |  2021-04-29 02:10
2021-04-28 15:00 |  2021-04-30 23:50 |  2021-04-29 08:00 |  2021-04-29 14:45
9 |  2021-08-05 19:40 |  2021-08-14 02:10 |  2021-08-10 00:10 |  2021-08-10 02:20
2021-08-10 09:00 |  2021-08-10 18:20
10 |  2021-11-19 08:10 |  2021-11-21 21:40 |  Not Found |
2021-11-22 11:40 |  2021-11-23 00:10 |
2021-11-24 01:10 |  2021-11-24 23:40 |
12 |  2022-05-28 19:40 |  2022-05-29 23:10 |  2022-06-02 11:30 |  2022-06-02 20:30
2022-05-30 03:10 |  2022-06-01 02:10
2022-06-01 07:40 |  2022-06-01 17:40
2022-06-02 08:40 |  2022-06-05 23:10
13 |  2022-09-03 06:10 |  2022-09-05 17:30 |  Not Found
2022-09-07 12:40 |  2022-09-08 13:10
Table 1: The selected reliable time intervals and clustered HCS crossings for
each encounter.