# Baumslag-Solitar groups and residual nilpotence

C.E. Kofinas, V. Metaftsis and A.I. Papistas  Department of Mathematics,
University of the Aegean, Karlovassi, 832 00 Samos, Greece<EMAIL_ADDRESS>Department of Mathematics, University of the Aegean, Karlovassi, 832 00 Samos,
Greece<EMAIL_ADDRESS>Department of Mathematics, Aristotle University of
Thessaloniki, 541 24 Thessaloniki, Greece<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a Baumslag-Solitar group. We calculate the intersection
$\gamma_{w}(G)$ of all terms of the lower central series of $G$. Using this,
we are able to show that $[\gamma_{w}(G),G]=\gamma_{w}(G)$; thus answering a
question of Bardakov and Neschadim [1]. Finally, for any $c\in\mathbb{N}$,
with $c\geq 2$, we show that the quotient group
$\gamma_{c}(G)/\gamma_{c+1}(G)$ of the lower central series of $G$ is finite.

##  1\. Introduction

Baumslag-Solitar groups are groups that admit a presentation of the form

$\mathrm{BS}(m,n)=\langle a,t\mid t^{-1}a^{m}t=a^{n}\rangle$

where $m,n$ are non-zero integers. They were introduced in [2] as examples of
two-generator one-relator groups with proper quotients isomorphic to the group
itself (that is, the groups don’t satisfy the Hopf property). Since then,
Baumslag-Solitar groups and their properties have been extensively studied by
various authors and they have been the testbed for various conjectures and
theories.

Our work is concerned with the residual nilpotence of these groups. A survey
about the residual properties of these groups is given in [11]. In [1],
Bardakov and Neschadim studied the lower central series of Baumslag-Solitar
groups and computed the intersection of all terms of the lower central series
for some special cases of the non-residually nilpotent Baumslag-Solitar
groups. In the present paper, we explicitly compute the intersection of all
terms of the lower central series for any Baumslag-Solitar group which is not
residually nilpotent.

Throughout this paper, a Baumslag-Solitar group is denoted by
$\mathrm{BS}(m,n)$. Since $\mathrm{BS}(m,n)$, $\mathrm{BS}(n,m)$ and
$\mathrm{BS}(-m,-n)$ are pairwise isomorphic, we may assume, without loss of
generality, that the integers $m$ and $n$ in the presentation of
$\mathrm{BS}(m,n)$ satisfy the condition $0<m\leq|n|$.

Our main result is the following.

######  Theorem 1.

Let $G=\mathrm{BS}(m,n)$, with $0<m\leq|n|$, ${\rm gcd}(m,n)=d\geq 1$,
$m=dm_{1}$ and $n=dn_{1}$.

  1. (1)

If $n_{1}\not\equiv m_{1}\pmod{p}$ for every prime number $p$, then
$\gamma_{w}(G)$ is the normal closure in $G$ of the set
$\\{a^{d},[t^{-k}a^{\mu}t^{k},a^{\nu}]\mid
k\in\mathbb{Z},\mu,\nu\in\mathbb{N},{\rm gcd}(\mu,\nu)=1~{}{\rm
and}~{}\mu\nu=d\\}$.

  2. (2)

If there is a prime number $p$ such that $n_{1}\equiv m_{1}\pmod{p}$, then
$\gamma_{w}(G)$ is the normal closure in $G$ of the set
$\\{[t^{-k}a^{\mu}t^{k},a^{\nu}]\mid k\in\mathbb{Z},\mu,\nu\in\mathbb{N},{\rm
gcd}(\mu,\nu)=1~{}{\rm and}~{}\mu\nu=d\\}$.

Next, we are concerned with the following question of [1].

Is true that if $G$ is a Baumslag-Solitar group, then
$\gamma_{\omega}(G)=[\gamma_{\omega}(G),G]$?

In fact, we are able to answer the above question affirmatively.

######  Theorem 2.

Let $G={\rm BS}(m,n)$, with $0<m\leq|n|$. Then
$[\gamma_{w}(G),G]=\gamma_{w}(G)$.

In section $5$, we show that the quotient groups
$\gamma_{i}(G)/\gamma_{i+1}(G)$, with $i\geq 2$, of the lower central series
of $G$ are finite.

##  2\. Auxiliary results

Let $G$ be a group. For elements $a,b$ of $G$, we write $[a,b]$ for the
commutator of $a$ and $b$, that is $[a,b]=a^{-1}b^{-1}ab$. We denote $\langle
g_{1},\ldots,g_{c}\rangle$ the subgroup of $G$ generated by the elements
$g_{1},\ldots,g_{c}$. For subgroups $A$ and $B$ of $G$, we write
$[A,B]=\langle[a,b],a\in A,b\in B\rangle$. For a positive integer $c$, let
$\gamma_{c}(G)$ be the $c$-th term of the lower central series of $G$. We
point out that $\gamma_{2}(G)=[G,G]=G^{\prime}$; that is, the derived group of
$G$. We write $\gamma_{\omega}(G)$ for the intersection of all terms of the
lower central series of $G$, that is, $\gamma_{\omega}(G)=\bigcap_{c\geq
1}\gamma_{c}(G)$. We say $G$ is a residually $\mathcal{P}$ group if for every
element $1\neq g\in G$ there is a normal subgroup $N_{g}$ of $G$ not
containing $g$ such that $G/N_{g}$ has the property $\mathcal{P}$. In case
${\mathcal{P}}$ is nilpotency, we say that the group is residually nilpotent.
Equivalently, we say that $G$ is a residually nilpotent group, if
$\gamma_{\omega}(G)=\\{1\\}$. For the rest of the paper, $N_{w}$ denotes the
intersection of all finite index (normal) subgroups of $G$ and $(Np)_{w}$
denotes the intersection of all finite index normal subgroups of $G$ with
index some power of a prime number $p$.

The following proposition summarizes some residual properties concerning
Baumslag-Solitar groups.

######  Proposition 1.

Let $G$ be the Baumslag-Solitar group with presentation

$G=\mathrm{BS}(m,n)=\langle t,a\mid t^{-1}a^{m}t=a^{n}\rangle,$

with $0<m\leq|n|$. Then,

  1. (1)

The group $G$ is residually finite if and only if $m=1$ or $|n|=m$.

  2. (2)

The group $G$ is residually nilpotent if and only if $m=1$ and $n\neq 2$ or
$|n|=m=p^{r}$, $r>0$ for some prime number $p$.

  3. (3)

The group $G$ is residually $p$ for some prime number $p$ if and only if $m=1$
and $n\equiv 1\pmod{p}$ or $n=m$ and $m=p^{r}$ or $n=-m$, $p=2$ and $m=2^{r}$,
$r\geq 1$.

######  Remark 1.

  1. (1)

The residual finiteness of the Baumslag-Solitar groups originally studied in
[2] and completed in [8]. Recently, Moldavanskii in [9] calculated $N_{w}$ for
Baumslag-Solitar groups.

  2. (2)

In [12], Raptis and Varsos gave necessary conditions for the residual
nilpotence of HNN-extensions with base group a finitely generated abelian
group. Theorem 1 (1) follows from [12]: it is a special case of [12, Corollary
2.7].

  3. (3)

Theorem 1 (3) follows from the study of Kim and McCarron in certain one
relator groups (see [5, Main Theorem]). Also, Moldavanskii in [10] (see also
[11]) calculated $(N_{p})_{w}$ for $\mathrm{BS}(m,n)$.

######  Lemma 1.

Let $G$ be a group and $N$ be a normal subgroup of $G$. If $x,y\in G$ such
that $[x,y]\in N$, then

  1. (1)

$[x^{\kappa},y],[x,y^{\kappa}]\in N$ for all $\kappa\in\mathbb{N}$.
Furthermore, $[x^{\kappa},y]\equiv[x,y^{\kappa}]\equiv[x,y]^{\kappa}~{}({\rm
mod}~{}[N,G])$ for all $\kappa\in\mathbb{N}$.

  2. (2)

If $[x,y^{m}]\in[N,G]$ for some $m\in\mathbb{N}$, then $[x,y]^{m}\in[N,G]$.

  3. (3)

If $[x^{m},y]\in[N,G]$ for some $m\in\mathbb{N}$, then $[x,y]^{m}\in[N,G]$.

Proof.

  1. (1)

This is straightforward.

  2. (2)

Let $[x,y^{m}]\in[N,G]$ for some $m\in\mathbb{N}$. By Lemma 1 (1) (for
$\kappa=m$), we have $[x,y^{m}]=[x,y]^{m}w$, with $w\in[N,G]$, and so,
$[x,y]^{m}\in[N,G]$.

  3. (3)

Let $[x^{m},y]\in[N,G]$. By Lemma 1 (1) (for $\kappa=m$), we have
$[x^{m},y]=[x,y]^{m}w_{1}$, with $w_{1}\in[N,G]$, and so, $[x,y]^{m}\in[N,G]$.
$\Box$

The following result gives us a relation among residually finite, residually
nilpotent and residually $p$ for some prime number $p$.

######  Lemma 2.

Let $G$ be a finitely generated group. Then
$N_{w}\leq\gamma_{w}(G)\leq(Np)_{w}$. Moreover, $\bigcap\limits_{p\ {\rm
prime}}(Np)_{w}\leq\gamma_{w}(G)$.

Proof. Since a finite $p$-group is nilpotent, we have every residually
$p$-group is also residually nilpotent. Hence, $G/(Np)_{w}$ is residually
nilpotent and so, $\gamma_{w}(G)\leq(Np)_{w}$. We know that $G/\gamma_{w}(G)$
is residually nilpotent and hence residually finite. Since $G/N_{w}$ is
residually finite, we obtain $N_{w}\leq\gamma_{w}(G)$. Set
$B=\bigcap\limits_{p\ {\rm prime}}(Np)_{w}$. To get a contradiction, we assume
that $g\in B$ and $g\notin\gamma_{\omega}(G)$. Since
$\widetilde{G}=G/\gamma_{w}(G)$ is residually nilpotent, there exists an
epimorphism $\phi$ from $\widetilde{G}$ onto a finitely generated nilpotent
group $H$ with $\phi(g\gamma_{\omega}(G))\neq 1$. Since $H$ is polycyclic, we
have $H$ is residually finite. Thus, there exists a finite nilpotent group
$\hat{H}$ and an epimorphism $\hat{\phi}:\widetilde{G}\rightarrow\hat{H}$ with
$\hat{\phi}(g\gamma_{\omega}(G))\neq 1$. Since $\hat{H}$ is the direct product
of its Sylow $p$-subgroups, there exist a prime number $p$ and a Sylow
$p$-subgroup $S_{p}$ of $\hat{H}$ such that
$\hat{\phi}(g\gamma_{\omega}(G))\in S_{p}\setminus\\{1\\}$. Since $S_{p}$ is a
finite $p$-group, we have $g\notin(Np)_{w}$, and so $g\not\in B$, a
contradiction. $\Box$

######  Lemma 3.

Let

$G=\langle t,x_{1},\ldots,x_{n}\mid
x_{i}^{p_{i}^{r_{i}}}=1,i=1,\ldots,n,[x_{i},x_{j}]=1\rangle\cong\mathbb{Z}*\left(\mathbb{Z}_{p_{1}^{r_{1}}}\times\dots\times\mathbb{Z}_{p_{n}^{r_{n}}}\right)$

where $n\geq 2$, $p_{1},\ldots,p_{n}$ are distinct prime numbers. Then
$\gamma_{w}(G)$ is the normal closure in $G$ of the set
$\\{[t^{-k}x_{i}t^{k},x_{j}]:i,j\in\\{1,\ldots,n\\};i\neq
j;k\in\mathbb{Z}\\}$.

Proof. The elements $x_{i}$ have orders $p_{i}^{r_{i}}$ and so, the orders of
$x_{i}$ and $x_{j}$ are coprime for every $i\neq j$. As in the proof of Lemma
2, we write $\widetilde{G}=G/\gamma_{\omega}(G)$. Let
$\widetilde{g}=g\gamma_{\omega}(G)$ and $g\notin\gamma_{\omega}(G)$. Since
$\widetilde{G}$ is residually nilpotent, there exists an epimorphism $\phi$
from $\widetilde{G}$ onto a finitely generated nilpotent group $H$ with
$\phi(\widetilde{g})\neq 1$. Since $H$ is polycyclic, we have $H$ is
residually finite. Thus, there exists a finite nilpotent group $\hat{H}$ and
an epimorphism $\hat{\phi}:\widetilde{G}\rightarrow\hat{H}$ with
$\hat{\phi}(\widetilde{g})\neq 1$. Since $\hat{H}$ is the direct product of
its Sylow $p$-subgroups, there exist a prime number $p$ and a Sylow
$p$-subgroup $S_{p}$ of $\hat{H}$ such that $\hat{\phi}(\widetilde{g})\in
S_{p}\setminus\\{1\\}$. Hence, the elements $[t^{-k}x_{i}t^{k},x_{j}]$, with
$i\neq j$, are trivial under any homomorphism onto $S_{p}$. Therefore,
$[t^{-k}x_{i}t^{k},x_{j}]\in\gamma_{\omega}(G)$ for all $k\in\mathbb{Z}$ and
$i,j\in\\{1,\ldots,n\\}$, with $i\neq j$.

For the converse, we will show that $G/N$ is residually nilpotent, where $N$
is the normal closure in $G$ of the elements $[t^{-k}x_{i}t^{k},x_{j}]$ for
all $i,j\in\\{1,\ldots,n\\}$ with $i\neq j$ and $k\in\mathbb{Z}$. Let $g\in
G/N$ with $g\neq 1$. If the exponent sum of $t$ in $g$ is non-zero then we can
take the homomorphism $\phi:G/N\rightarrow\mathbb{Z}$ with $x_{i}\mapsto 0$
and $t\mapsto 1$. Then $\phi(g)\neq 1$ and since $\mathbb{Z}$ is residually
nilpotent the result follows. On the other hand, assume that the exponent sum
of $t$ in $g$ is zero. Notice that the relations $[t^{-k}x_{i}t^{k},x_{j}]$
are equivalent to $[t^{-s}x_{i}t^{s},t^{l}x_{j}t^{-l}]$ for every
$s,l\in\mathbb{Z}$. Using these relations, $g$ can be written as

$g=t^{-s_{1}}x_{1}^{w_{1}}t^{s_{1}}\ldots
t^{-s_{m_{1}}}x_{1}^{w_{m_{1}}}t^{s_{m_{1}}}\cdot\ldots\cdot
t^{-q_{1}}x_{k}^{z_{1}}t^{q_{1}}\ldots
t^{-q_{m_{k}}}x_{k}^{z_{m_{k}}}t^{q_{m_{k}}}$

where the words

$t^{-s_{1}}x_{1}^{w_{1}}t^{s_{1}}\ldots
t^{-s_{m_{1}}}x_{1}^{w_{m_{1}}}t^{s_{m_{1}}},\ldots,t^{-q_{1}}x_{k}^{z_{1}}t^{q_{1}}\ldots
t^{-q_{m_{k}}}x_{k}^{z_{m_{k}}}t^{q_{m_{k}}}$

are reduced. Since $g\neq 1$ at least one of the
$w_{1},\ldots,w_{m_{1}},\ldots,z_{1},\ldots,z_{m_{k}}\neq 0$. Then we take the
homomorphism $\phi:G/N\rightarrow\mathbb{Z}*\mathbb{Z}_{p_{i}^{r_{i}}}$ with
$\phi(\langle t\rangle)=\mathbb{Z}$, $\phi(\langle
x_{i}\rangle)=\mathbb{Z}_{p_{i}^{r_{i}}}$ and $\phi(x_{j})=0$ for all $j\neq
i$. Since $\mathbb{Z}*\mathbb{Z}_{p_{i}^{r_{i}}}$ are residually nilpotent
(see [4, Theorem 4.1]), the result follows. $\Box$

######  Lemma 4.

For a positive integer $m$, with $m\geq 2$, let $G=\langle t,a\mid
a^{m}=1\rangle$. Then $\gamma_{\omega}(G)$ is the normal closure in $G$ of the
set $\\{[t^{-k}a^{\mu}t^{k},a^{\nu}]:k\in\mathbb{Z};\mu,\nu\in\mathbb{N};{\rm
gcd}(\mu,\nu)=1;\mu\nu=m\\}$.

Proof. Let $m=p_{1}^{r_{1}}\ldots p_{n}^{r_{n}}$, $n\geq 2$ be the prime
number decomposition of $m$. Since
$G\cong\mathbb{Z}*(\mathbb{Z}_{p_{1}^{r_{1}}}\times\ldots\times\mathbb{Z}_{p_{n}^{r_{n}}})$,
$G$ admits a presentation as in Lemma 3 and the isomorphism between the two
presentations implies that each $x_{i}$ maps to
$a^{{p_{1}^{r_{1}}}\ldots{p_{i-1}^{r_{i-1}}}{p_{i+1}^{r_{i+1}}}\ldots{p_{n}^{r_{n}}}}=u_{i}$.

Let $\mathcal{M}=\\{[t^{-k}u_{i}t^{k},u_{j}];i,j\in\\{1,\ldots,n\\};i\neq
j;k\in\mathbb{Z}\\}$ be the generating set of $\gamma_{w}(G)$ described in
Lemma 3 and
$\mathcal{K}=\\{[t^{-k}a^{\mu}t^{k},a^{\nu}]:k\in\mathbb{Z};\mu,\nu\in\mathbb{N};{\rm
gcd}(\mu,\nu)=1;\mu\nu=m\\}$. Write $K$ for the normal closure in $G$ of the
set $\mathcal{K}$. Let also $b_{i}$ be non-zero integers such that $b_{i}\cdot
p_{1}^{r_{1}}\ldots p_{i-1}^{r_{i-1}}p_{i+1}^{r_{i+1}}\ldots
p_{n}^{r_{n}}\equiv 1\pmod{p_{i}^{r_{i}}}.$ Then $\mathcal{M}$ is in the
normal closure of the elements

$[t^{-k}u_{i}^{b_{i}}t^{k},u_{j}^{b_{j}}]=[t^{-k}a^{b_{i}\cdot
p_{1}^{r_{1}}\ldots p_{i-1}^{r_{i-1}}p_{i+1}^{r_{i+1}}\ldots
p_{n}^{r_{n}}}t^{k},a^{b_{j}\cdot p_{1}^{r_{1}}\ldots
p_{j-1}^{r_{j-1}}p_{j+1}^{r_{j+1}}\ldots p_{n}^{r_{n}}}]$

which can be written as

$[t^{-k}(a^{p_{1}^{r_{1}}\ldots p_{i-1}^{r_{i-1}}p_{j}^{r_{j}}})^{b_{i}\cdot
s_{1}}t^{k},(a^{p_{i}^{r_{i}}\ldots p_{j-1}^{r_{j-1}}p_{j+1}^{r_{j+1}}\ldots
p_{n}^{r_{n}}})^{b_{j}\cdot s_{2}}]$

for appropriate $s_{1},s_{2}$. But then, in $G/K$,

$[t^{-k}(a^{p_{1}^{r_{1}}\ldots
p_{i-1}^{r_{i-1}}p_{j}^{r_{j}}})t^{k},(a^{p_{i}^{r_{i}}\ldots
p_{j-1}^{r_{j-1}}p_{j+1}^{r_{j+1}}\ldots p_{n}^{r_{n}}})]=1$

for all $k\in\mathbb{Z}$. Therefore, by repeatedly using the commutator
identities $[xy,z]=[x,z]^{y}[y,z]$ and $[x,yz]=[x,z][x,y]^{z}$, we get

$[t^{-k}(a^{p_{1}^{r_{1}}\ldots p_{i-1}^{r_{i-1}}p_{j}^{r_{j}}})^{b_{i}\cdot
s_{1}}t^{k},(a^{p_{i}^{r_{i}}\ldots p_{j-1}^{r_{j-1}}p_{j+1}^{r_{j+1}}\ldots
p_{n}^{r_{n}}})^{b_{j}\cdot s_{2}}]=1$

for all $k\in\mathbb{Z}$ and hence, an element of $\mathcal{M}$. Consequently,
$\gamma_{\omega}(G)\subseteq K$.

For the converse, we change the presentation of $G$ to the presentation
$G=\langle t,x_{1},\ldots,x_{n}\mid
x_{i}^{p_{i}^{r_{i}}}=1,i=1,\ldots,n,[x_{i},x_{j}]=1\rangle$, where
$p_{1},\ldots,p_{n}$ are distinct prime numbers. In this presentation, the
elements of $\mathcal{K}$ are

$[t^{-k}(x_{1}\ldots x_{n})^{\mu}t^{k},(x_{1}\ldots
x_{n})^{\nu}]=[t^{-k}(x_{i_{1}}\ldots x_{i_{l}})^{\mu}t^{k},(x_{j_{1}}\ldots
x_{j_{n-l}})^{\nu}],$

with
$\\{x_{i_{1}},\ldots,x_{i_{l}}\\}\cup\\{x_{j_{1}},\ldots,x_{j_{n-l}}\\}=\\{x_{1},\ldots,x_{n}\\}$
and
$\\{x_{i_{1}},\ldots,x_{i_{l}}\\}\cap\\{x_{j_{1}},\ldots,x_{j_{n-l}}\\}=\emptyset.$
Now one can easily show, by repeatedly using the commutator identities
$[xy,z]=[x,z]^{y}[y,z]$ and $[x,yz]=[x,z][x,y]^{z}$ that the elements of $K$
belong to $\gamma_{\omega}(G)$. Therefore, $K\subseteq\gamma_{\omega}(G)$ and
so, $\gamma_{\omega}(G)=K$. $\Box$

###  2.1. Known results on Baumslag–Solitar groups

Moldavanskii in [9] has shown the following.

######  Proposition 2 ([9, Theorem 1]).

Let $G=\mathrm{BS}(m,n)$, with $0<m\leq|n|$ and $d={\rm gcd}(m,n)$. Then
$N_{w}$ coincides with the normal closure in $G$ of the set
$\\{[t^{k}a^{d}t^{-k},a]:k\in\mathbb{Z}\\}$.

By Proposition 2 and Lemma 2, we get the following result, which we will use
it repeatedly in the following.

######  Corollary 1.

Let $G=\mathrm{BS}(m,n)$, with $0<m\leq|n|$ and $d={\rm gcd}(m,n)$. Then for
all $k,x,y\in\mathbb{Z}$,
$[(t^{-k}a^{d}t^{k})^{x},a^{y}]\in\gamma_{\omega}(G)$ and
$[(t^{-k}at^{k})^{y},(a^{d})^{x}]\in\gamma_{\omega}(G)$.

Proof. Since $[t^{-k}a^{y}t^{k},a^{dx}]=[a^{y},(a^{dx})^{t^{-k}}]^{t^{k}}$ and
$\gamma_{\omega}(G)$ is normal in $G$, it suffices to prove that
$[t^{-k}a^{dx}t^{k},a^{y}]\in\gamma_{\omega}(G)$ for all $k,x,y\in\mathbb{Z}$.
By Proposition 2 and Lemma 2, we get
$[t^{-k}a^{d}t^{k},a]\in\gamma_{\omega}(G)$ for all $k\in\mathbb{Z}$. By using
a double induction argument on $x$ and $y$, we obtain the desired result.
$\Box$

Moreover Moldavanskii in [10] (see also [11]) has shown the following.

######  Proposition 3.

Let $G=\mathrm{BS}(m,n)$, $p$ be a prime number and let $m=p^{r}m_{1}$ and
$n=p^{s}n_{1}$ where $r,s\geq 0$ and $m_{1},n_{1}$ are not divided by $p$. Let
also $d={\rm gcd}(m_{1},n_{1})$, $m_{1}=du$ and $n_{1}=dv$. Then

  1. (1)

if $r\neq s$ or if $m_{1}\not\equiv n_{1}\pmod{p}$, then $(Np)_{w}$ coincides
with the normal closure in $G$ of $a^{p^{\xi}}$, where $\xi=\min\\{r,s\\}$

  2. (2)

if $r=s$ and $m_{1}\equiv n_{1}\pmod{p}$, then $(Np)_{w}$ coincides with the
normal closure in $G$ of the set
$\\{t^{-1}a^{p^{r}u}ta^{-p^{r}v},[t^{k}a^{p^{r}}t^{-k},a]:k\in{\mathbb{Z}}\\}$.

##  3\. Calculation of $\gamma_{w}(\mathrm{BS}(m,n))$

######  Proposition 4.

Let $G=\mathrm{BS}(m,n)$, with $0<m\leq|n|$ and ${\rm gcd}(m,n)=1$. Then

  1. (1)

If there is a prime $p$ such that $n\equiv m\pmod{p}$, then $\gamma_{w}(G)$ is
the normal closure in $G$ of the set $\\{[t^{-k}at^{k},a]:k\in\mathbb{Z}\\}$.

  2. (2)

If $n\not\equiv m\pmod{p}$ for any prime integer $p$, then $\gamma_{w}(G)$ is
the normal closure in $G$ of $a$.

Proof.

  1. (1)

Assume that there is a prime number $p$ such that $n\equiv m\pmod{p}$. Since
${\rm gcd}(m,n)=1$, we have $p$ divides neither $m$ nor $n$ and so, $m$ and
$n$ satisfy the conditions of Proposition 3 (2). Therefore, we have $(Np)_{w}$
is the normal closure in $G$ of $[t^{-k}at^{k},a]$. On the other hand, by
Proposition 2, we have $N_{w}$ is the normal closure in $G$ of the set
$\\{[t^{-k}at^{k},a]:k\in\mathbb{Z}\\}$. Hence, the description of
$\gamma_{w}(G)$ is an immediate consequence of Lemma 2.

  2. (2)

Assume that $m\not\equiv n\pmod{p}$ for every prime number $p$. Then by
Proposition 3 (1), we have $a\in\bigcap\limits_{p\ {\rm prime}}(Np)_{w}$ and
so, by Lemma 2, $a\in\gamma_{\omega}(G)$. $\Box$

######  Remark 2.

Notice that in the above Proposition, when $m=1$ and $n\neq 2$, then the
commutators $[t^{-k}at^{k},a]$ are trivial and hence, $\gamma_{w}(G)=\\{1\\}$.

######  Lemma 5.

Let $G=\mathrm{BS}(m,n)$, with $0<m\leq|n|$, let ${\rm gcd}(m,n)=d$ and let
$\mu,\nu$ be positive integers such that $1\leq\mu,\nu\leq d$, ${\rm
gcd}(\mu,\nu)=1$ and $\mu\nu=d$. Then
$[t^{-k}a^{\mu}t^{k},a^{\nu}]\in\gamma_{\omega}(G)$ for all $k\in\mathbb{Z}$.

Proof. By Corollary 1, we have $[t^{-k}a^{d}t^{k},a]\in\gamma_{\omega}(G)$.
Hence, in the case $d$ is a power of a prime number, that is, $\mu=1$ or
$\nu=1$, the required result follows. Thus, in what follows, we may assume
that $\mu,\nu>1$. Fix some $k\in\mathbb{Z}$ and let us denote
$u=t^{-k}a^{\mu}t^{k}$. Assume that $[u,a^{\nu}]\in\gamma_{i}(G)$ for some
$i\geq 2$. Since
$[u^{\nu},a^{\nu}]=[t^{-k}a^{\mu\nu}t^{k},a^{\nu}]=[t^{-k}a^{d}t^{k},a^{\nu}]$,
it follows from Corollary 1 that $[u^{\nu},a^{\nu}]\in\gamma_{w}(G)$ and
hence, $[u^{\nu},a^{\nu}]\in\gamma_{j}(G)$ for all $j\in{\mathbb{N}}$. In
particular, we have $[u^{\nu},a^{\nu}]\in\gamma_{i+1}(G)$. Since
$[u,a^{\nu}]\in\gamma_{i}(G)$ and
$[u^{\nu},a^{\nu}]\in\gamma_{i+1}(G)=[\gamma_{i}(G),G]$, we get, by Lemma 1
(3) (for $N=\gamma_{i}(G)$), $[u,a^{\nu}]^{\nu}\in\gamma_{i+1}(G)$. Similarly,
since $[u,a^{\mu\nu}]=[u,a^{d}]=[a^{\mu},t^{k}a^{d}t^{-k}]^{t^{k}}$ and
$\gamma_{\omega}(G)$ is normal in $G$, it follows from Corollary 1 that
$[u,a^{\mu\nu}]\in\gamma_{\omega}(G)$. In particular, we have
$[u,a^{\mu\nu}]\in\gamma_{i+1}(G)$. As before, by Lemma 1 (2), we get
$[u,a^{\nu}]^{\mu}\in\gamma_{i+1}(G)$. Thus,
$[u,a^{\nu}]^{\nu},[u,a^{\nu}]^{\mu}\in\gamma_{i+1}(G)$. Since ${\rm
gcd}(\mu,\nu)=1$, we have $[u,a^{\nu}]\in\gamma_{i+1}(G)$. We carry on this
process, and we obtain the required result. $\Box$

We are now able to give the proof of our main theorem.

######  Theorem 1.

Let $G=\mathrm{BS}(m,n)$, with $0<m\leq|n|$, ${\rm gcd}(m,n)=d\geq 2$,
$m=dm_{1}$ and $n=dn_{1}$. Let also $\mu,\nu$ be positive integers with ${\rm
gcd}(\mu,\nu)=1$ and $\mu\nu=d$. Then

  1. (1)

If $n_{1}\not\equiv m_{1}\pmod{p}$ for every prime number $p$, then
$\gamma_{w}(G)$ is the normal closure in $G$ of the set
$\\{a^{d},[t^{-k}a^{\mu}t^{k},a^{\nu}]:k\in\mathbb{Z},\mu,\nu\in\mathbb{N},{\rm
gcd}(\mu,\nu)=1~{}{\rm and}~{}\mu\nu=d\\}$.

  2. (2)

If there is a prime number $p$ such that $n_{1}\equiv m_{1}\pmod{p}$, then
$\gamma_{w}(G)$ is the normal closure in $G$ of the set
$\\{[t^{-k}a^{\mu}t^{k},a^{\nu}]:k\in\mathbb{Z},\mu,\nu\in\mathbb{N},{\rm
gcd}(\mu,\nu)=1~{}{\rm and}~{}\mu\nu=d\\}$.

Proof.

  1. (1)

Let $T$ be the normal closure in $G$ of the set
$\\{a^{d},[t^{-k}a^{\mu}t^{k},a^{\nu}]:k\in\mathbb{Z},\mu,\nu\in\mathbb{N},{\rm
gcd}(\mu,\nu)=1~{}{\rm and}~{}\mu\nu=d\\}$. We first show that
$T\subseteq\gamma_{\omega}(G)$. By Lemma 5, it is enough to show that
$a^{d}\in\gamma_{\omega}(G)$. Let $d=p^{r_{1}}_{1}\cdots
p^{r_{\kappa}}_{\kappa}$ be the prime number decomposition of $d$. For
$i\in\\{1,\ldots,\kappa\\}$, we write $m=p^{r_{i}}_{i}m^{\prime}_{1i}$ and
$n=p^{r_{i}}_{i}n^{\prime}_{1i}$, where
$m^{\prime}_{1i}=\frac{d}{p^{r_{i}}_{i}}m_{1}$ and
$n^{\prime}_{1i}=\frac{d}{p^{r_{i}}_{i}}n_{1}$. Now,
$n^{\prime}_{1i}-m^{\prime}_{1i}=\frac{d}{p^{r_{i}}_{i}}(n_{1}-m_{1})$. Since
$p_{i}\nmid(n_{1}-m_{1})$, we obtain
$p_{i}\nmid(n^{\prime}_{1i}-m^{\prime}_{1i})$. By Proposition 3 (1), we have
$a^{p^{r_{i}}_{i}}\in(Np_{i})_{\omega}$ for every $i\in\\{1,\ldots,\kappa\\}$
and so, $a^{d}\in(Np_{i})_{\omega}$ for any $i\in\\{1,\ldots,\kappa\\}$. By
Proposition 3 (1), for every $q\notin\\{p_{1},\ldots,p_{\kappa}\\}$, we have
$a\in(Nq)_{\omega}$, which again implies that $a^{d}\in(Nq)_{\omega}$.
Therefore, $a^{d}\in\bigcap\limits_{p\ {\rm prime}}(Np)_{w}$. By Lemma 2, we
have $a^{d}\in\gamma_{\omega}(G)$ and so, $T\subseteq\gamma_{\omega}(G)$. On
the other hand,
$G/T\cong(\mathbb{Z}*\mathbb{Z}_{d})/\gamma_{w}(\mathbb{Z}*\mathbb{Z}_{d})$
which is residually nilpotent.

  2. (2)

Assume that there is some prime $p$ such that $n_{1}\equiv m_{1}\pmod{p}$ and
let $M$ be the normal closure in $G$ of the set
$\\{[t^{-k}a^{\mu}t^{k},a^{\nu}]:k\in\mathbb{Z},\mu,\nu\in\mathbb{N},{\rm
gcd}(\mu,\nu)=1~{}{\rm and}~{}\mu\nu=d\\}$. By Lemma 5, we have
$M\subseteq\gamma_{w}(G)$. We claim that $G/M$ is residually nilpotent. Notice
that $G/M$ has a presentation of the form

$G/M=\langle a,t\mid t^{-1}a^{m}t=a^{n},[t^{k}a^{\mu}t^{-k},a^{\nu}]=1\
k\in\mathbb{Z},\rangle$

with $\mu,\nu\in\mathbb{N},{\rm gcd}(\mu,\nu)=1~{}{\rm and}~{}\mu\nu=d\\}$ Let
$g$ be an element in $G/M$. Then $g$ is a word in $a,t$.

Assume first that the exponent sum of $t$ in $g$ is non-zero. Then there is a
map $\phi:G/M\rightarrow\mathbb{Z}$ such that $a\mapsto 0$ and $t\mapsto 1$.
It can easily be seen that $\phi$ is a homomorphism and that $\phi(g)\neq 1$.
Since $\mathbb{Z}$ is nilpotent and the result follows.

Assume now that the exponent sum of $t$ in $g$ is zero. Then $g$ can be
written in the form

$None$
$g=g_{0}(t^{\varepsilon_{1}}g_{1}t^{-\varepsilon_{1}})(t^{\varepsilon_{1}+\varepsilon_{2}}g_{2}t^{-(\varepsilon_{1}+\varepsilon_{2})})\cdots(t^{\varepsilon_{1}+\cdots+\varepsilon_{\kappa-1}}g_{\kappa-1}t^{-(\varepsilon_{1}+\cdots+\varepsilon_{\kappa-1}})g_{\kappa},$

with $g_{1}\in Y$ if $\varepsilon_{1}=-1$ or $g_{1}\in Z$ if
$\varepsilon_{1}=+1$, $g_{2}\in Y$ if $\varepsilon_{2}=-1$ or $g_{2}\in Z$ if
$\varepsilon_{2}=+1$ etc. Thus, each $g_{j}$, with
$j\in\\{1,\ldots,\kappa\\}$, has the form $a^{\rho}$ where
$\rho\leq\min\\{m-1,n-1\\}$. Write each $a^{\rho}=a^{d\lambda}a^{r}$, with
$r\in\\{0,\ldots,d-1\\}$,
$t^{-k}a^{\rho}t^{k}=(t^{-k}(a^{d})^{\lambda}t^{k})(t^{-k}a^{r}t^{k})$,
$t^{-k}a^{\zeta}t^{k}=a^{\zeta}[a^{\zeta},t^{k}]$ and replace them in the
expression $(3.1)$. Then by using the identity
$\alpha\beta=\beta\alpha[\alpha,\beta]$ as many times as it is needed and the
identities

$None$ $[ab,c]=[a,c]~{}[[a,c],b]~{}[b,c],$ $None$
$[a,bc]=[a,c]~{}[a,b]~{}[[a,b],c],$

$g$ has an expression of the form

$g=a^{\lambda}~{}[(a^{d})^{\lambda_{1}},t^{k_{1}}]\ldots[(a^{d})^{\lambda_{s}},t^{k_{s}}]\cdot
w$

where $\lambda_{1},\ldots,\lambda_{s}\in\mathbb{N}$ and $w$ is a product of
group commutators of the form $[h_{1},\ldots,h_{r}]$, with $r\geq 2$ and
$h_{1},\ldots,h_{r}\in\\{a,\ldots,a^{d-1}\\}\cup\\{t^{k}:k\in\mathbb{Z}\setminus\\{0\\}\\}$.
Note that $d\lambda_{1},\ldots,d\lambda_{s}<m_{1},n_{1}$. Next, we separate
two cases.

    1. (a)

Let $w=1$. For the next few lines, let $G_{1}={\rm
BS}(m_{1},n_{1})=\langle\bar{t},\bar{a}:(\bar{t})^{-1}(\bar{a})^{m_{1}}\bar{t}=(\bar{a})^{n_{1}}\rangle$.
Since ${\rm gcd}(m_{1},n_{1})=1$ and $n_{1}\equiv m_{1}\pmod{p}$ for a prime
integer $p$, we have, by Proposition 4 (1) and equation $(3.2)$,
$\gamma_{\omega}(G_{1})$ is the normal closure in $G_{1}$ of the set
$\\{[\bar{t}^{-k}\bar{a}\bar{t}^{k},\bar{a}]:k\in\mathbb{Z}\\}$. Then there is
a natural homomorphism $\psi:G/M\rightarrow G_{1}/\gamma_{w}(G_{1})$ with
$a\mapsto\bar{a}$ and $t\mapsto\bar{t}$ such that $\psi(g)\neq 1$. Since
$G_{1}/\gamma_{w}(G_{1})$ is residually nilpotent, the result follows.

    2. (b)

Let $w\neq 1$. Then it suffices to map $G/M$ to $(G/M)/\langle
a^{d}\rangle\cong(\mathbb{Z}*\mathbb{Z}_{d})/\gamma_{w}(\mathbb{Z}*\mathbb{Z}_{d})$.
The image of $g$ is $w\neq 1$ which is reduced. The result follows from that
fact that $(G/H)/\langle
a^{d}\rangle\cong(\mathbb{Z}*\mathbb{Z}_{d})/\gamma_{w}(\mathbb{Z}*\mathbb{Z}_{d})$
is residually nilpotent. $\Box$

##  4\. The group $[\gamma_{w}(G),G]$

######  Lemma 6.

Let $G={\rm BS}(m,n)$, with $0<m\leq|n|$, and let $d={\rm gcd}(m,n)$. Then

  1. (1)

$[t^{-k}a^{d}t^{k},a^{d}]\in[\gamma_{\omega}(G),G]$ for all $k\in\mathbb{Z}$.

  2. (2)

$[t^{-k}a^{d}t^{k},a]^{d}\in[\gamma_{\omega}(G),G]$ for all $k\in\mathbb{Z}$.

  3. (3)

$[t^{-k}at^{k},a^{d}]^{d}\in[\gamma_{\omega}(G),G]$ for all $k\in\mathbb{Z}$.

Proof. Throughout the proof, we write $u_{k}=t^{-k}a^{d}t^{k}$, with
$k\in\mathbb{Z}$. By Corollary 1, $[u_{k},a^{d}]\in\gamma_{\omega}(G)$ for all
$k\in\mathbb{Z}$. Furthermore, we write $m=dm_{1}$ and $n=dn_{1}$, where
$\gcd(m_{1},n_{1})=1$.

  1. (1)

Since $[u_{k},a^{d}]=([u_{-k},a^{d}]^{-1})^{t^{k}}$ and
$[\gamma_{\omega}(G),G]$ is normal in $G$, it suffices to show that
$[u_{k},a^{d}]\in[\gamma_{\omega}(G),G]$ for all $k\in\mathbb{N}$. We use
induction on $k$. Assume at first that $k=1$. Since, as aforementioned,
$[u_{1},a^{d}]\in\gamma_{\omega}(G)$ and since
$[u_{1}^{m_{1}},a^{d}]=[t^{-1}a^{m}t,a^{d}]=[a^{n},a^{d}]=1$, it follows from
Lemma 1 that $[u_{1},a^{d}]^{m_{1}}\in[\gamma_{\omega}(G),G]$. Since
$[u_{1},a^{dn_{1}}]=[a^{d},ta^{n}t^{-1}]^{t}=[a^{d},a^{m}]^{t}=1$, by Lemma 1,
we get $[u_{1},a^{d}]^{n_{1}}\in[\gamma_{\omega}(G),G]$. But
$\gcd(m_{1},n_{1})=1$ and so, the result follows for $k=1$. Assume that
$[u_{k},a^{d}]\in[\gamma_{\omega}(G),G]$ for some $k\in\mathbb{N}$. By
Corollary 1, we get
$[u_{k}^{n_{1}},a^{d}]=[t^{-k}a^{n}t^{k},a^{d}]\in\gamma_{\omega}(G)$ and
$[u_{k},a^{dm_{1}}]=[u_{k},a^{m}]=[t^{-k}a^{d}t^{k},a^{m}]\in\gamma_{\omega}(G)$.
Since

$[u_{k+1}^{m_{1}},a^{d}]=[t^{-(k+1)}a^{m}t^{k+1},a^{d}]=[t^{-k}(t^{-1}a^{m}t)t^{k},a^{d}]=[u_{k}^{n_{1}},a^{d}]\in\gamma_{w}(G),$

it follows from Lemma 1 that $[u_{k+1},a^{d}]^{m_{1}}\in[\gamma_{w}(G),G]$.
Since

$[u_{k+1},a^{dn_{1}}]^{t^{-1}}=[t^{-(k+1)}a^{d}t^{k+1},a^{n}]^{t^{-1}}=[u_{k},ta^{n}t^{-1}]=[u_{k},a^{m}]=[u_{k},a^{dm_{1}}]\in\gamma_{w}(G)$

and $\gamma_{\omega}(G)$ is normal in $G$, we have
$[u_{k+1},a^{dn_{1}}]\in\gamma_{w}(G)$. Hence, by Lemma 1, we get
$[u_{k+1},a^{d}]^{n_{1}}\in[\gamma_{w}(G),G]$. But $\gcd(m_{1},n_{1})=1$ and
so, the result follows.

  2. (2)

By Corollary 1, $[u_{k},a]\in\gamma_{\omega}(G)$ and, by Lemma 6 (1),
$[u_{k},a^{d}]\in[\gamma_{\omega}(G),G]$ for all $k\in\mathbb{Z}$, the result
follows from Lemma 1.

  3. (3)

By Corollary 1, $[t^{-k}at^{k},a^{d}]\in\gamma_{\omega}(G)$. Since
$[t^{-k}at^{k},a^{d}]=([t^{k}a^{d}t^{-k},a]^{-1})^{t^{k}}$, the result follows
from Lemma 6 (2). $\Box$

######  Proposition 5.

Let $G={\rm BS}(m,n)$, with $0<m\leq|n|$,and let $d=\gcd(m,n)$ be a power of a
prime integer $p$. Then
$[t^{-k}a^{d}t^{k},a],[t^{-k}at^{k},a^{d}]\in[\gamma_{\omega}(G),G]$ for all
$k\in\mathbb{N}$.

Proof. Let $d=p^{\mu}$, with $\mu\geq 1$. Thus, we may write $m=p^{r}m_{1}$
and $n=p^{s}n_{1}$, where $\mu=\min\\{r,s\\}$ and
$\gcd(p,m_{1})=\gcd(p,n_{1})=1$. Throughout the proof, we write
$u_{k}=t^{-k}a^{d}t^{k}$ and $v_{k}=t^{-k}at^{k}$. By Corollary 1, we have
$[u_{k},a],[v_{k},a^{d}]\in\gamma_{\omega}(G)$ for all $k\in\mathbb{N}$. We
separate several cases.

  1. (1)

Let $r=s$. In this case, we have $d=p^{r}$. At first, we show that
$[u_{k},a]\in[\gamma_{\omega}(G),G]$ for all $k\in\mathbb{N}$. We use
induction on $k$. Let $k=1$. Since
$[u_{1}^{m_{1}},a]=[t^{-1}a^{m}t,a]=[a^{n},a]=1\in[\gamma_{\omega}(G),G]$ and
$[u_{1},a]\in\gamma_{\omega}(G)$, we have from Lemma 1 that
$[u_{1},a]^{m_{1}}\in[\gamma_{\omega}(G),G]$. Furthermore, by Lemma 6 (2),
$[u_{1},a]^{p^{r}}\in[\gamma_{\omega}(G),G]$. But, $\gcd(m_{1},p^{r})=1$ and
so, we have $[u_{1},a]\in[\gamma_{\omega}(G),G]$. Thus, our claim is valid for
$k=1$. Assume that $[u_{k},a]\in[\gamma_{\omega}(G),G]$ for some
$k\in\mathbb{N}$. Using $(3.2)$ as many times as it is needed and since
$[u_{k},a]\in[\gamma_{\omega}(G),G]$, we get
$[u_{k}^{n_{1}},a]\in[\gamma_{\omega}(G),G]$. Since
$[u_{k+1}^{m_{1}},a]=[t^{-(k+1)}a^{m}t^{k+1},a]=[t^{-k}a^{n}t^{k},a]=[u_{k}^{n_{1}},a]\in[\gamma_{\omega}(G),G]$
and $[u_{k+1},a]\in\gamma_{\omega}(G)$, it follows from Lemma 1 that
$[u_{k+1},a]^{m_{1}}\in[\gamma_{\omega}(G),G]$. Furthermore, by Lemma 6 (2),
$[u_{k+1},a]^{p^{r}}\in[\gamma_{\omega}(G),G]$. But, $\gcd(m_{1},p^{r})=1$ and
so, we have $[u_{k+1},a]\in[\gamma_{\omega}(G),G]$. Therefore,
$[u_{k},a]\in[\gamma_{\omega}(G),G]$ for all $k\in\mathbb{N}$. By $(3.2)$, we
get $[u_{k},a]=[a^{p^{r}},t^{k},a]\in[\gamma_{\omega}(G),G]$ for all
$k\in\mathbb{N}$.

Next, we show that $[v_{k},a^{p^{r}}]\in[\gamma_{\omega}(G),G]$ for all $k>0$.
As before, we use induction on $k$. Let $k=1$. Since
$[v_{1},a^{p^{r}n_{1}}]=[v_{1},a^{n}]=[a,ta^{n}t^{-1}]^{t}=[a,a^{m}]^{t}=1\in[\gamma_{\omega}(G),G]$
and $[v_{1},a^{p^{r}}]\in\gamma_{\omega}(G)$, it follows from Lemma 1 that
$[v_{1},a^{p^{r}}]^{n_{1}}\in[\gamma_{\omega}(G),G]$. Furthermore, by Lemma 6
(2), $[v_{1},a^{p^{r}}]^{p^{r}}\in[\gamma_{\omega}(G),G]$. But,
$\gcd(p^{r},n_{1})=1$ and so, we have
$[v_{1},a^{p^{r}}]\in[\gamma_{\omega}(G),G]$. Thus, our claim is true for
$k=1$. Assume that $[v_{k},a^{p^{r}}]\in[\gamma_{\omega}(G),G]$ for some
$k\in\mathbb{N}$. Using $(3.3)$ as many times as it is needed and since
$[v_{k},a]\in[\gamma_{\omega}(G),G]$, we get
$[v_{k},(a^{p^{r}})^{m_{1}}]=[v_{k},a^{m}]\in[\gamma_{\omega}(G),G]$. Since
$[t^{-(k+1)}at^{k+1},a^{n}]=[t^{-k}at^{k},a^{m}]^{t}=[v_{k},a^{m}]^{t}$ and
$[\gamma_{\omega}(G),G]$ is normal in $G$, we get
$[v_{k+1},a^{n}]=[v_{k+1},(a^{p^{r}})^{n_{1}}]\in[\gamma_{\omega}(G),G]$.
Since $[v_{k+1},a^{p^{r}}]\in\gamma_{\omega}(G)$, we have from Lemma 1 that
$[v_{k+1},a^{p^{r}}]^{n_{1}}\in[\gamma_{\omega}(G),G]$. Furthermore, by Lemma
6 (2), $[v_{k+1},a^{p^{r}}]^{p^{r}}\in[\gamma_{\omega}(G),G]$. But,
$\gcd(n_{1},p^{r})=1$ and so, we have
$[v_{k+1},a^{p^{r}}]\in[\gamma_{\omega}(G),G]$. Therefore,
$[v_{k},a^{p^{r}}]\in[\gamma_{\omega}(G),G]$ for all $k\in\mathbb{N}$. By
$(3.2)$, we get
$[v_{k},a^{p^{r}}]=[a,t^{k},a^{p^{r}}]\in[\gamma_{\omega}(G),G]$ for all
$k\in\mathbb{N}$.

  2. (2)

Let $r<s$. In this case, we have $d=p^{r}$. By similar arguments as in case
(1), we get $[u_{k},a]\in[\gamma_{\omega}(G),G]$ for all $k\in\mathbb{N}$.
Thus, it remains to show that $[v_{k},a^{p^{r}}]\in[\gamma_{\omega}(G),G]$ for
all $k\in\mathbb{N}$. By Corollary 1 (for $y=x=1$ and $d=p^{r}$),
$[v_{k},a^{p^{r}}]\in\gamma_{\omega}(G)$ for all $k\in\mathbb{N}$. By Lemma 6
(3) (for $d=p^{r}$), $[v_{k},a^{p^{r}}]^{p^{r}}\in[\gamma_{\omega}(G),G]$ for
all $k\in\mathbb{N}$. We separate two cases.

    1. (a)

Let $2r\leq s$ and fix a positive integer $k\geq 1$. By Lemma 1 (1) (for
$N=\gamma_{\omega}(G)$, $x=v_{k+1}$, $y=a^{p^{r}}$,
$\kappa=p^{r}p^{s-2r}n_{1}$), we have

$None$
$[v_{k+1},(a^{p^{r}})^{p^{r}p^{s-2r}n_{1}}]\equiv[v_{k+1},a^{p^{r}}]^{p^{r}p^{s-2r}n_{1}}~{}({\rm
mod}~{}[\gamma_{\omega}(G),G]).$

Since $[v_{k+1},a^{p^{r}}]^{p^{r}}\in[\gamma_{\omega}(G),G]$, we get
$[v_{k+1},a^{p^{r}}]^{p^{r}p^{s-2r}n_{1}}\in[\gamma_{\omega}(G),G]$ and so, by
$(4.1)$, we have $[v_{k+1},a^{p^{s}n_{1}}]\in[\gamma_{\omega}(G),G]$. But
$[v_{k+1},a^{p^{s}n_{1}}]=[v_{k},ta^{p^{s}n_{1}}t^{-1}]^{t}=[v_{k},a^{p^{r}m_{1}}]^{t}$.
Since $[\gamma_{\omega}(G),G]$ is normal in $G$, we get

$None$ $[v_{k},a^{p^{r}m_{1}}]\in[\gamma_{\omega}(G),G].$

Since $[v_{k},a^{p^{r}}]\in\gamma_{\omega}(G)$, it follows from Lemma 1 (1)
(for $N=\gamma_{\omega}(G)$, $x=v_{k}$, $y=a^{p^{r}}$, $\kappa=m_{1}$) that

$[v_{k},a^{p^{r}m_{1}}]\equiv[v_{k},a^{p^{r}}]^{m_{1}}~{}({\rm
mod}~{}[\gamma_{\omega}(G),G]).$

By $(4.2)$, we obtain $[v_{k},a^{p^{r}}]^{m_{1}}\in[\gamma_{\omega}(G),G]$.
Since $[v_{k},a^{p^{r}}]^{p^{r}}\in[\gamma_{\omega}(G),G]$ and ${\rm
gcd}(m_{1},p^{r})=1$, we have
$[v_{k},a^{p^{r}}]=[t^{-k}at^{k},a^{d}]\in[\gamma_{\omega}(G),G]$.

    2. (b)

Let $2r>s$ and fix a positive integer $k\geq 1$. Since
$[v_{k},a^{p^{r}}]\in\gamma_{\omega}(G)$, it follows from Lemma 1 (1) (for
$x=v_{k}$, $y=a^{p^{r}}$, $\kappa=m_{1}$) that
$[v_{k},a^{p^{r}m_{1}}]\in\gamma_{\omega}(G)$ and

$[v_{k},a^{p^{r}m_{1}}]\equiv[v_{k},a^{p^{r}}]^{m_{1}}~{}({\rm
mod}~{}[\gamma_{\omega}(G),G]).$

Since $[\gamma_{\omega}(G),G]$ is normal in $G$,

$None$
$[v_{k},a^{p^{r}m_{1}}]^{p^{2r-s}}\equiv[v_{k},a^{p^{r}}]^{p^{2r-s}m_{1}}~{}({\rm
mod}~{}[\gamma_{\omega}(G),G]).$

Since
$[v_{k+1},a^{p^{s}n_{1}}]=[v_{k},ta^{p^{s}n_{1}}t^{-1}]^{t}=[v_{k},a^{p^{r}m_{1}}]^{t}$
and $[v_{k},a^{p^{r}m_{1}}]\in\gamma_{\omega}(G)$, we have

$[v_{k+1},a^{p^{s}n_{1}}]\equiv[v_{k},a^{p^{r}m_{1}}]~{}({\rm
mod}~{}[\gamma_{\omega}(G),G]).$

By $(4.3)$, we get

$None$
$[v_{k+1},a^{p^{s}n_{1}}]^{p^{2r-s}}\equiv[v_{k},a^{p^{r}}]^{p^{2r-s}m_{1}}~{}({\rm
mod}~{}[\gamma_{\omega}(G),G]).$

Since $[v_{k+1},a^{p^{r}}]\in\gamma_{\omega}(G)$ and $r<s$, it follows from
Lemma 1 (1) (for $x=v_{k+1}$, $y=a^{p^{r}}$, $\kappa=p^{s-r}n_{1}$) that

$None$
$[v_{k+1},a^{p^{s}n_{1}}]\equiv[v_{k+1},a^{p^{r}}]^{p^{s-r}n_{1}}~{}({\rm
mod}~{}[\gamma_{\omega}(G),G]).$

By $(4.4)$ and $(4.5)$, we have

$None$
$[v_{k},a^{p^{r}}]^{p^{2r-s}m_{1}}\equiv[v_{k+1},a^{p^{r}}]^{p^{r}}~{}({\rm
mod}~{}[\gamma_{\omega}(G),G]).$

Since $[v_{k+1},a^{p^{r}}]^{p^{r}}\in[\gamma_{\omega}(G),G]$, we obtain from
$(4.6)$ that $[v_{k},a^{p^{r}}]^{p^{2r-s}m_{1}}\in[\gamma_{\omega}(G),G]$.
Since ${\rm gcd}(p^{r},p^{2r-s}m_{1})=p^{2r-s}$, we get
$[v_{k},a^{p^{r}}]^{p^{2r-s}}\in[\gamma_{\omega}(G),G]$. If $3r\leq 2s$, then,
by applying similar arguments as in case $2r\leq s$, we have
$[v_{k},a^{p^{r}}]\in[\gamma_{\omega}(G),G]$. If $3r>2s$, then, by applying
similar arguments as in case $2r>s$, we get
$[v_{k},a^{p^{r}}]^{p^{3r-2s}}\in[\gamma_{\omega}(G),G]$. Since
$2r-s>3r-2s>\cdots$ and since there is $y$ such that $(y+1)r\leq ys$ (for
$\frac{r}{s-r}\in\mathbb{N}$, let $y=\frac{r}{s-r}$ and, for
$\frac{r}{s-r}\notin\mathbb{N}$, let $y$ be the integral part of
$\frac{r}{s-r}$), by continuing this process, we obtain
$[v_{k},a^{p^{r}}]\in[\gamma_{\omega}(G),G]$.

  3. (3)

Let $s<r$. By applying similar arguments as in case $(2)$, we obtain the
required result.

By cases (1), (2) and (3), we get the desired result. $\Box$

######  Proposition 6.

Let $G={\rm BS}(m,n)$, with $0<m\leq|n|$ and let $N_{\omega}$ be the
intersection of all finite index subgroups of $G$. Then
$N_{\omega}\leq[\gamma_{\omega}(G),G]$.

Proof. Let $d=\gcd(m,n)$. Since, by Lemma 2,
$N_{\omega}\leq\gamma_{\omega}(G)$ and, by Proposition 2, $N_{\omega}$
coincides with the normal closure in $G$ of the set
$\\{[t^{-k}a^{d}t^{k},a]:k\in\mathbb{Z}\\}$, it suffices to show that
$[t^{-k}a^{d}t^{k},a]\in[\gamma_{\omega}(G),G]$ for all $k\in\mathbb{Z}$.
Furthermore, since
$[t^{-k}a^{x}t^{k},a^{y}]=([t^{k}a^{y}t^{-k},a^{x}]^{-1})^{t^{k}}$ and
$[\gamma_{\omega}(G),G]$ is normal in $G$, it is enough to show that
$[t^{-k}a^{d}t^{k},a],[t^{-k}at^{k},a^{d}]\in[\gamma_{\omega}(G),G]$ for all
$k\in\mathbb{N}$. For $d=1$, the required result follows from Lemma 6 (1) and
so, from now on, we may assume that $d>1$. Let $d=p_{1}^{\mu_{1}}\ldots
p_{\lambda}^{\mu_{\lambda}}$ be the prime factor decomposition of $d$. To
prove the result, we use induction on $\lambda$. For $\lambda=1$, the required
result follows from Proposition 5. Assume that the result is true for some
$\lambda\geq 1$ and let $d=p_{1}^{\mu_{1}}\ldots
p_{\lambda+1}^{\mu_{\lambda+1}}$. Thus, we may write $m=p_{1}^{r_{1}}\ldots
p_{\lambda+1}^{r_{\lambda+1}}\mu$ and $n=p_{1}^{s_{1}}\ldots
p_{\lambda+1}^{s_{\lambda+1}}\nu$, where $\mu_{i}=\min\\{r_{i},s_{i}\\}$ and
$\gcd(p_{i},\mu)=\gcd(p_{i},\nu)=1$, with $i=1,\ldots,\lambda+1$. For
$j\in\\{1,\ldots,\lambda+1\\}$, let $u=a^{p_{j}^{\mu_{j}}}$ and let $K_{j}$ be
the subgroup of $G$ generated by the set $\\{u,t\\}$. For convenience, we
write $m_{j}=m/({p_{j}^{\mu_{j}}})$, $n_{j}=n/({p_{j}^{\mu_{j}}})$ and
$d_{j}=d/({p_{j}^{\mu_{j}}})$. We point out that $K_{j}={\rm BS}(m_{j},n_{j})$
(see [7, Lemma 7.10]). Since $\gcd(m_{j},n_{j})=p_{1}^{\mu_{1}}\ldots
p_{j-1}^{\mu_{j-1}}p_{j+1}^{\mu_{j+1}}\ldots p_{\lambda+1}^{\mu_{\lambda+1}}$,
by our inductive argument, we get
$[t^{-k}u^{d_{j}}t^{k},u]\in[\gamma_{\omega}(K_{j}),K_{j}]\subseteq[\gamma_{\omega}(G),G]$,
that is,
$[t^{-k}a^{p_{j}^{\mu_{j}}d_{j}}t^{k},a^{p_{j}^{\mu_{j}}}]\in[\gamma_{\omega}(G),G]$
and so, $[t^{-k}a^{d}t^{k},a^{p_{j}^{\mu_{j}}}]\in[\gamma_{\omega}(G),G]$.
Since $[t^{-k}a^{d}t^{k},a]\in\gamma_{\omega}(G)$, it follows that
$[t^{-k}a^{d}t^{k},a]^{p_{j}^{\mu_{j}}}\in[\gamma_{\omega}(G),G]$. Therefore,
for $j_{1},j_{2}\in\\{1,\ldots,\lambda+1\\}$, with $j_{1}\neq j_{2}$, we have
$[t^{-k}a^{d}t^{k},a]^{p_{j_{1}}^{\mu_{j_{1}}}},[t^{-k}a^{d}t^{k},a]^{p_{j_{2}}^{\mu_{j_{2}}}}\in[\gamma_{\omega}(G),G]$.
Since ${\rm gcd}(p_{j_{1}}^{\mu_{j_{1}}},p_{j_{2}}^{\mu_{j_{2}}})=1$, we get
$[t^{-k}a^{d}t^{k},a]\in[\gamma_{\omega}(G),G]$ and the result follows. $\Box$

######  Corollary 2.

Let $G=\mathrm{BS}(m,n)$, with $0<m\leq|n|$ and ${\rm gcd}(m,n)=1$. Then
$[\gamma_{w}(G),G]=\gamma_{w}(G)$.

Proof. Let us assume first that there is some prime number $p$ such that
$m\equiv n\pmod{p}$. Then by Proposition 4 (1), $\gamma_{w}(G)=N_{w}$ and so,
by Proposition 6, $\gamma_{w}(G)=[\gamma_{w}(G),G]$. On the other hand, if
$m\not\equiv n\pmod{p}$ for every prime integer $p$, then $m-n=\pm 1$ and so,
the relation $t^{-1}a^{m}t=a^{n}$ becomes $t^{-1}a^{n\pm 1}t=a^{n}$ or
equivalently, $t^{-1}a^{n\pm 1}ta^{-(n\pm 1)}=a^{\mp 1}$ or $[t,a^{-(n\pm
1)}]=a^{\mp 1}$. By Proposition 4 (2), we have $a\in\gamma_{w}(G)$. Since
$[t,a^{-(n\pm 1)}]=a^{\mp 1}$, we obtain $a\in[\gamma_{w}(G),G]$. By
Proposition 4 (2), $\gamma_{\omega}(G)\subseteq[\gamma_{\omega}(G),G]$ and the
result follows. $\Box$

######  Proposition 7.

Let $G={\rm BS}(m,n)$, with $0<m\leq|n|$, let $d=\gcd(m,n)$ and let $\mu$,
$\nu$ be positive integers, with $1\leq\mu,\nu\leq d$, such that
$\gcd(\mu,\nu)=1$ and $d=\mu\nu$. Then
$[t^{-k}a^{\mu}t^{k},a^{\nu}]\in\gamma_{\omega}(G)$ for all $k\in\mathbb{Z}$.
In particular, $[t^{-k}a^{\mu}t^{k},a^{\nu}]\in[\gamma_{\omega}(G),G]$ for all
$k\in\mathbb{Z}$.

Proof. The first part of the proposition was shown in Lemma 5. We shall show
that $[t^{-k}a^{\mu}t^{k},a^{\nu}]\in[\gamma_{\omega}(G),G]$ for all
$k\in\mathbb{Z}$. Fix some $k\in\mathbb{Z}$. By Proposition 2,
$[t^{-k}a^{d}t^{k},a]\in N_{\omega}$ and so, by Proposition 6, we have
$[t^{-k}a^{d}t^{k},a]=[t^{-k}a^{\mu\nu}t^{k},a]\in[\gamma_{\omega}(G),G]$.
Using $(3.3)$ as many times as it is needed and since $[\gamma_{\omega}(G),G]$
is a normal subgroup of $G$, we obtain
$[t^{-k}a^{\mu\nu}t^{k},a^{\nu}]\in[\gamma_{\omega}(G),G]$. By Lemma 5,
$[t^{-k}a^{\mu}t^{k},a^{\nu}]\in\gamma_{\omega}(G)$ and so, by Lemma 1,
$[t^{-k}a^{\mu}t^{k},a^{\nu}]^{\nu}\in[\gamma_{\omega}(G),G]$. By Proposition
2, Proposition 6 and since $N_{\omega}$ is a normal subgroup of $G$, we get
$[t^{-k}at^{k},a^{d}]\in[\gamma_{\omega}(G),G]$. Using $(3.2)$ as many times
as it is needed and since $[\gamma_{\omega}(G),G]$ is a normal subgroup of
$G$, we obtain
$[(t^{-k}at^{k})^{\mu},a^{\mu\nu}]=[t^{-k}a^{\mu}t^{k},a^{\mu\nu}]\in[\gamma_{\omega}(G),G]$.
By Lemma 5, $[t^{-k}a^{\mu}t^{k},a^{\nu}]\in\gamma_{\omega}(G)$ and so, by
Lemma 1, we have
$[t^{-k}a^{\mu}t^{k},a^{\nu}]^{\mu}\in[\gamma_{\omega}(G),G]$. But
$\gcd(\mu,\nu)=1$, therefore we get the result. $\Box$

So now we may give an answer to the question of Bardakov and Neschadim.

######  Theorem 2.

Let $G={\rm BS}(m,n)$, with $0<m\leq|n|$. Then
$[\gamma_{w}(G),G]=\gamma_{w}(G)$.

Proof. Let $d={\rm gcd}(m,n)$, with $m=dm_{1}$, $n=dn_{1}$ and ${\rm
gcd}(m_{1},n_{1})=1$. Let $n=m$. By Theorem 1 (1), Proposition 7 and
$[\gamma_{\omega}(G),G]\leq\gamma_{\omega}(G)$, we have
$[\gamma_{\omega}(G),G]=\gamma_{\omega}(G)$. If there is a prime number $p$
such that $m_{1}\equiv n_{1}\pmod{p}$, then, by Theorem 1 (2), Proposition 7
and $[\gamma_{\omega}(G),G]\leq\gamma_{\omega}(G)$, we have the required
result. Finally, let $m_{1}\not\equiv n_{1}\pmod{p}$ for every prime number
$p$. By Proposition 7 and Theorem 1 (2), it is sufficient to show that
$a^{d}\in[\gamma_{w}(G),G]$. Since $m_{1}\not\equiv n_{1}\pmod{p}$ for every
prime number $p$, we have $m_{1}-n_{1}=\pm 1$. Hence, the relation
$t^{-1}a^{m}t=a^{n}$ implies that $t^{-1}a^{d(n_{1}\pm 1)}t=a^{dn_{1}}$ or
equivalently $t^{-1}a^{d(n_{1}\pm 1)}t=a^{d(n_{1}\pm 1)}a^{\mp d}$ or
$[a^{d(n_{1}\pm 1)},t]=a^{\mp d}$. By Theorem 1 (3), $a^{d}\in\gamma_{w}(G)$
and so, we obtain the required result. $\Box$

##  5\. Quotient groups

Throughout this section, for any group $G$, we write ${\rm
gr}_{c}(G)=\gamma_{c}(G)/\gamma_{c+1}(G)$, with $c\in\mathbb{N}$. Let ${\rm
gr}(G)$ denote the (restricted) direct sum of the abelian groups ${\rm
gr}_{c}(G)$. It is well known that ${\rm gr}(G)$ has the structure of a Lie
algebra (over $\mathbb{Z}$) by defining a Lie multiplication
$[a\gamma_{r+1}(G),b\gamma_{s+1}(G)]$ $=[a,b]\gamma_{r+s+1}(G)$, where
$a\gamma_{r+1}(G)$ and $b\gamma_{s+1}(G)$ are the images of the elements
$a\in\gamma_{r}(G)$ and $b\in\gamma_{s}(G)$ in the quotient groups ${\rm
gr}_{r}(G)$ and ${\rm gr}_{s}(G)$, respectively, and $[a,b]\gamma_{r+s+1}(G)$
is the image of the group commutator $[a,b]$ in the quotient group ${\rm
gr}_{r+s}(G)$. Multiplication is then extended to ${\rm gr}(G)$ by linearity.

Let $F$ be a free group of finite rank $n\geq 2$, with a free generating set
$\\{x_{1},\ldots,x_{n}\\}$. It is well known that ${\rm gr}(F)$ is a free Lie
algebra of rank $n$; freely generated by the set $\\{y_{1},\ldots,y_{n}\\}$,
where $y_{i}=x_{i}\gamma_{2}(F)$ and $i\in\\{1,\ldots,n\\}$. Let $N$ be a
normal subgroup of $F$. For $c\in\mathbb{N}$, let $N_{c}=N\cap\gamma_{c}(F)$.
Note that $N_{1}=N$. Furthermore, for all $c\in\mathbb{N}$, we write ${\rm
I}_{c}(N)=N_{c}\gamma_{c+1}(F)/\gamma_{c+1}(F)$. Form the (restricted) direct
sum ${\mathcal{L}}(N)$ of the abelian groups ${\rm I}_{c}(N)$. Since $N$ is
normal in $F$, ${\mathcal{L}}(N)$ is an ideal of ${\rm gr}(F)$ (see [6]).

The following result is probably known, but we give a proof for completeness.

######  Lemma 7.

With the previous notations, ${\rm gr}_{c}(F/N)\cong{\rm gr}_{c}(F)/{\rm
I}_{c}(N)$ for all $c\in\mathbb{N}$.

Proof. Throughout this proof, we write $G=F/N$. For all $c\in\mathbb{N}$,
$\gamma_{c}(G)=\gamma_{c}(F)N/N$. We have the following natural isomorphisms
as abelian groups

$\begin{array}[]{rll}{\rm
gr}_{c}(G)&\cong&\frac{\gamma_{c}(G)N}{\gamma_{c+1}(G)N}=\frac{\gamma_{c}(F)(\gamma_{c+1}(F)N)}{\gamma_{c+1}(F)N}\\\
&\cong&\frac{\gamma_{c}(F)}{\gamma_{c}(F)\cap(\gamma_{c+1}(F)N)}.\end{array}$

Since $\gamma_{c+1}(F)\subseteq\gamma_{c}(F)$, by the modular law, we have
$\gamma_{c}(F)\cap(\gamma_{c+1}(F)N)=\gamma_{c+1}(F)(\gamma_{c}(F)\cap N)$.
Therefore, for all $c\in\mathbb{N}$,

${\rm gr}_{c}(G)\cong\frac{\gamma_{c}(F)}{\gamma_{c+1}(F)(\gamma_{c}(F)\cap
N)}\cong\frac{{\rm
gr}_{c}(F)}{\frac{\gamma_{c+1}(F)N_{c}}{\gamma_{c+1}(F)}}=\frac{{\rm
gr}_{c}(F)}{{\rm I}_{c}(N)}$

as abelian groups in a natural way. $\Box$

For the rest of this section, let $F$ be a free group of rank $2$, with a free
generating set $\\{x,y\\}$.

######  Proposition 8.

For a non-zero integer $\kappa$, let
${\mathcal{R}}_{\kappa}=\\{x\gamma_{2}(F),$ $\kappa(y\gamma_{2}(F))\\}$ and
let ${\mathcal{L}}_{\mathcal{R}_{\kappa}}$ be the Lie subalgebra of ${\rm
gr}(F)$ generated by the set $\mathcal{R}_{\kappa}$. Then
${\mathcal{L}}_{\mathcal{R}_{\kappa}}$ is free on ${\mathcal{R}}_{\kappa}$
and, for any $c\in\mathbb{N}$, ${\mathcal{L}}_{\mathcal{R}_{\kappa}}\cap{\rm
gr}_{c}(F)$ has finite index in ${\rm gr}_{c}(F)$. Moreover, for any
$c\in\mathbb{N}$, ${\mathcal{L}}_{\mathcal{R}_{\kappa}}\cap{\rm gr}_{c}(F)$ is
spanned by all Lie commutators of degree $c$ containing $\kappa\overline{y}$.

Proof. Let ${\rm gr}_{1,R}(F)$ be the subgroup of ${\rm gr}_{1}(F)$ generated
by the set ${\mathcal{R}}_{\kappa}=\\{x\gamma_{2}(F),$
$\kappa(y\gamma_{2}(F))\\}$. It is a free abelian group of rank $2$ and has a
finite index $|\kappa|$ in ${\rm gr}_{1}(F)$. For $c\in\mathbb{N}$, let
${\mathcal{L}}_{c,\mathcal{R}_{\kappa}}={\mathcal{L}}_{\mathcal{R}_{\kappa}}\cap{\rm
gr}_{c}(F)$. Then ${\mathcal{L}}_{\mathcal{R}_{\kappa}}=\bigoplus_{c\geq
1}{\mathcal{L}}_{c,\mathcal{R}_{\kappa}}$ and each
${\mathcal{L}}_{c,\mathcal{R}_{\kappa}}$ is a free abelian subgroup of ${\rm
gr}_{c}(F)$. We observe that

$None$ ${\mathbb{Q}}\otimes_{\mathbb{Z}}{\rm
gr}(F)={\mathbb{Q}}\otimes_{\mathbb{Z}}{\mathcal{L}}_{\mathcal{R}_{\kappa}}.$

Since ${\rm gr}(F)$ is a free Lie algebra, we get
${\mathbb{Q}}\otimes_{\mathbb{Z}}{\rm gr}(F)$ is a free Lie algebra over
$\mathbb{Q}$ on any $\mathbb{Q}$-basis of $\mathbb{Q}\otimes_{\mathbb{Z}}{\rm
gr}_{1}(F)$. By $(5.1)$ and since ${\rm gr}(F)=\bigoplus_{c\geq 1}{\rm
gr}_{c}(F)$ and ${\mathcal{L}}_{\mathcal{R}_{\kappa}}=\bigoplus_{c\geq
1}{\mathcal{L}}_{c,\mathcal{R}_{\kappa}}$, we have, for all $c\in\mathbb{N}$,

$None$ ${\mathbb{Q}}\otimes_{\mathbb{Z}}{\rm
gr}_{c}(F)={\mathbb{Q}}\otimes_{\mathbb{Z}}{\mathcal{L}}_{c,\mathcal{R}_{\kappa}}.$

Hence, for all $c\in\mathbb{N}$, ${\mathcal{L}}_{c,\mathcal{R}_{\kappa}}$ has
finite index in ${\rm gr}_{c}(F)$.

Write $\overline{x}=x\gamma_{2}(F)$ and $\overline{y}=y\gamma_{2}(F)$. Since
${\rm gr}(F)$ is a free Lie algebra on $\\{\overline{x},\overline{y}\\}$ and
${\mathcal{L}}_{\mathcal{R}_{\kappa}}$ is generated by the set
$\\{\overline{x},\kappa\overline{y}\\}$, the natural Lie homomorphism $\psi$
from ${\rm gr}(F)$ into ${\mathcal{L}}_{\mathcal{R}_{\kappa}}$, with
$\psi(\overline{x})=\overline{x}$ and $\psi(\overline{y})=\kappa\overline{y}$,
is onto. It is easily verified that, for all $c\in\mathbb{N}$, $\psi$ induces
a $\mathbb{Z}$-linear mapping $\psi_{c}$ from ${\rm gr}_{c}(F)$ onto
${\mathcal{L}}_{c,{\mathcal{R}}_{\kappa}}$. Namely,
$\psi_{c}(u(\overline{x},\overline{y}))=u(\overline{x},\kappa\overline{y})$
for all $u(\overline{x},\overline{y})\in{\rm gr}_{c}(F)$. Since
${\mathcal{L}}_{c,{\mathcal{R}}_{\kappa}}$ has finite index in ${\rm
gr}_{c}(F)$ and ${\rm gr}_{c}(F)$ is a free abelian group of finite rank, we
obtain $\psi_{c}$ is an isomorphism of abelian groups.

Let $\overline{w}\in{\rm Ker}\psi$. Since ${\rm gr}(F)$ is graded, without
loss of generality, we may assume that $\overline{w}=w\gamma_{c+1}(F)\in{\rm
gr}_{c}(F)$ for some $w\in\gamma_{c}(F)$ and $c\in\mathbb{N}$. To get a
contradiction, we assume that
$w=w(x,y)\in\gamma_{c}(F)\setminus\gamma_{c+1}(F)$. Then
$\psi(\overline{w})=\psi(w(\overline{x},\overline{y})=\psi_{c}(w(\overline{x},\overline{y}))=w(\overline{x},\kappa\overline{y})$.
By bilinearity of the Lie bracket, we get
$\psi_{c}(w(\overline{x},\overline{y}))=m(\kappa)w(\overline{x},\overline{y})$,
where $m(\kappa)\in\mathbb{Z}\setminus\\{0\\}$ depends on $\kappa$. Since
$\overline{w}\in{\rm Ker}\psi$, we have
$\psi(\overline{w})=m(\kappa)\overline{w}=0$ in ${\rm gr}_{c}(F)$. That is,
$w(x,y)^{m(\kappa)}\in\gamma_{c+1}(F)$, which is a contradiction, since ${\rm
gr}_{c}(F)$ is a free abelian group (of finite rank). Therefore, $\psi$ is an
isomorphism of Lie algebras (over) $\mathbb{Z}$. Hence,
${\mathcal{L}}_{\mathcal{R}_{\kappa}}$ is a free Lie algebra with a free
generating set $\mathcal{R}_{\kappa}$. By the elimination theorem (see, for
example, [3, Chapter 2, Section 2.9, Proposition 10]),
${\mathcal{L}}_{\mathcal{R}_{\kappa}}=\langle\overline{x}\rangle\oplus
L(\\{\kappa\overline{y}\\}\wr\\{\overline{x}\\})$, where the free Lie algebra
$L(\\{\kappa\overline{y}\\}\wr\\{\overline{x}\\})$ is the ideal in
${\mathcal{L}}_{\mathcal{R}_{\kappa}}$ generated by $\kappa\overline{y}$.
Clearly,

$None$
$L(\\{\kappa\overline{y}\\}\wr\\{\overline{x}\\})=\langle\kappa\overline{y}\rangle\oplus\left(\bigoplus_{c\geq
2}{\mathcal{L}}_{c,\mathcal{R}_{\kappa}}\right).$

and so, we obtain ${\mathcal{L}}_{c,\mathcal{R}_{\kappa}}$ is spanned by all
Lie commutators of degree $c$ containing $\kappa\overline{y}$. $\Box$

######  Remark 3.

The fact that $\psi_{c}$ is an isomorphism in the above proof can also be
shown as follows. Since ${\rm gr}_{c}(F)$ is a free abelian group of finite
rank, we have ${\mathbb{Q}}\otimes_{\mathbb{Z}}{\rm gr}_{c}(F)$ is a finite-
dimensional vector space over $\mathbb{Q}$, and any $\mathbb{Z}$-basis of
${\rm gr}_{c}(F)$ may be regarded as a $\mathbb{Q}$-basis of
${\mathbb{Q}}\otimes_{\mathbb{Z}}{\rm gr}_{c}(F)$. Thus, $\psi_{c}$ may be
extended to a $\mathbb{Q}$-linear mapping $\overline{\psi}_{c}$ from
${\mathbb{Q}}\otimes_{\mathbb{Z}}{\rm gr}_{c}(F)$ onto
${\mathbb{Q}}\otimes_{\mathbb{Z}}{\mathcal{L}}_{c,\mathcal{R}_{\kappa}}$. By
$(5.2)$, we obtain $\overline{\psi}_{c}$ is an isomorphism of vector spaces
and so, $\psi_{c}$ is an isomorphism of abelian groups.

Let $N$ be the normal closure in $F$ of the set $\\{r=y^{n-m}[x,y^{m}]\\}$,
with $m\in\mathbb{N}$ and $n\in\mathbb{Z}$. Write $\delta=n-m$, $t=xN$ and
$a=yN$. Clearly, the quotient group $G=F/N$ has a presentation $G=F/N=\langle
t,a\mid a^{\delta}[t,a^{m}]=1\rangle$ of a Baumslag-Solitar group.

######  Proposition 9.

With the previous notations, let $G=\mathrm{BS}(m,n)=\langle t,a\mid
a^{\delta}[t,a^{m}]=1\rangle$. Then for any $c\in\mathbb{N}$, with $c\geq 2$,
${\rm gr}_{c}(G)$ is a finite abelian group.

Proof. Assume that $\delta\neq 0$ and let
${\mathcal{R}}_{\delta}=\\{x\gamma_{2}(F),\delta(y\gamma_{2}(F))\\}$. By
Proposition 8, ${\mathcal{L}}_{\mathcal{R}_{\delta}}$ is a free Lie algebra on
the set ${\mathcal{R}}_{\delta}$. By $(5.3)$ (for $\kappa=\delta\neq 0$),
$I_{1}(N)=\langle\delta\overline{y}\rangle$ and since ${\mathcal{L}}(N)$ is an
ideal in ${\rm gr}(F)$, we have
$L(\\{\delta\overline{y}\\}\wr\\{\overline{x}\\})\subseteq{\mathcal{L}}(N)$.
Hence, for all $c\geq 2$, ${\mathcal{L}}_{c,{\mathcal{R}}_{\delta}}\subseteq
I_{c}(N)$. By Proposition 8, for any $c\geq 2$,
${\mathcal{L}}_{c,{\mathcal{R}}_{\delta}}$ has finite index in ${\rm
gr}_{c}(F)$ and so, $I_{c}(N)$ has finite index in ${\rm gr}_{c}(F)$. By Lemma
7, we obtain, for any $c\geq 2$, ${\rm gr}_{c}(G)$ is a finite abelian group.

Assume now that $\delta=0$. In this case, let
${\mathcal{R}}_{m}=\\{x\gamma_{2}(F),m(y\gamma_{2}(F))\\}$ and
$I_{1}(N)=\\{0\\}$. We use similar arguments as before. By Proposition 8,
${\mathcal{L}}_{\mathcal{R}_{m}}$ is a free Lie algebra on the set
${\mathcal{R}}_{\delta}$. By $(5.3)$ (for $\kappa=m$),
$I_{2}(N)=\langle[\overline{x},m\overline{y}]\rangle$ and since
${\mathcal{L}}(N)$ is an ideal in ${\rm gr}(F)$, we have $\bigoplus_{c\geq
2}\mathcal{L}_{c,{\mathcal{R}}_{m}}\subseteq{\mathcal{L}}(N)$. Hence, for all
$c\geq 2$, ${\mathcal{L}}_{c,{\mathcal{R}}_{m}}\subseteq I_{c}(N)$. By
Proposition 8, for any $c\geq 2$, ${\mathcal{L}}_{c,{\mathcal{R}}_{m}}$ has
finite index in ${\rm gr}_{c}(F)$ and so, $I_{c}(N)$ has finite index in ${\rm
gr}_{c}(F)$. By Lemma 7, we obtain, for any $c\geq 2$, ${\rm gr}_{c}(G)$ is a
finite abelian group. $\Box$

## References

  * [1] V.G. Bardakov and M.V. Neshchadim, On the lower central series of Baumslag-Solitar groups, Algebra and Logic 59 (2020), 281–294.
  * [2] G. Baumslag and D. Solitar, Some two-generator one-relator non-Hopfian groups, Bull.Amer. Math. Soc. 68 (1962), 199–201.
  * [3] N. Bourbaki, Lie groups and Lie algebras Part I, Hermann, Paris, 1987 (Chapters 1-3).
  * [4] K.W. Gruenberg, residual properties of infinite soluble groups, Proc. LMS 7 (3) (1957), 29–62.
  * [5] G. S. Kim and J. McCarron, On Amalgamated Free Products of Residually p-Finite Groups, J. Algebra 162 (1) (1993), 1–11.
  * [6] M. Lazard, Sur les groups nilpotents et les anneaux de Lie, Ann. Sci. Ecole Norm. Sup., 71 (3) (1954), 101–190.
  * [7] G. Levitt, Quotients and subgroups of Baumslag-Solitar groups, J. Group theory 18 (2015), 1–43.
  * [8] S. Meskin, (1972). Nonresidually finite one-relator groups. Trans. Am. Math. Soc. 164 (1972), 105–114.
  * [9] D.I. Moldavanskii, The intersection of the subgroups of finite index in Baumslag–Solitar groups, Mathematical Notes, 87 No. 1 (2010), 88–95.
  * [10] D.I. Moldavanskii, The intersection of the subgroups of finite $p$-index in Baumslag–Solitar groups, Vestnik Ivanovo State Univ. Ser. Natural, Social Sci., 2 (2010) 106–111. (Russian)
  * [11] D.I. Moldavanskii, On the residual properties of Baumslag–Solitar groups, Communications in Algebra, 46 (2018), no. 9, 3766–3778.
  * [12] E. Raptis and D. Varsos, Residual properties of HNN-extensions with base group an abelian group, J. Pure and Applied Algebra 59 (1989), 285–290.


# Detecting Boosted Dark Photons with Gaseous Detectors

Michael L. Graesser  Theoretical Division, Los Alamos National Laboratory, Los
Alamos, NM 87545, USA  R. Andrew Gustafson  Center for Neutrino Physics,
Department of Physics, Virginia Tech University, Blacksburg, Virginia 24601,
USA  Kate Hildebrandt  School of Physics and Astronomy, University of
Minnesota, Minneapolis, MN 55455, USA  Varun Mathur  Center for Neutrino
Physics, Department of Physics, Virginia Tech University, Blacksburg, Virginia
24601, USA  Ian M. Shoemaker  Center for Neutrino Physics, Department of
Physics, Virginia Tech University, Blacksburg, Virginia 24601, USA

###### Abstract

We search for indirect signals of $\mathscr{O}$(keV) dark matter annihilating
or decaying into $\mathscr{O}$(eV) dark photons. These dark photons will be
highly boosted and have decay lengths larger than the Milky Way, and can be
absorbed by neutrino or dark matter experiments at a rate dependent on the
photon-dark photon kinetic mixing parameter and the optical properties of the
experiment. We show that current experiments can not probe new parameter
space, but future large-scale gaseous detectors with low backgrounds (i.e.
CYGNUS, NEXT, PANDAX-III) may be sensitive to this signal when the
annihilation cross section is especially large.

††preprint: LA-UR-24-20773

##  I Introduction

Dark photons are a renormalizable extension of the Standard Model (SM), with a
Lagrangian

$\mathscr{L}\supset-\frac{1}{4}F^{\prime}_{\mu\nu}F^{\prime\mu\nu}+\frac{1}{2}\epsilon
F^{\prime}_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m_{A^{\prime}}^{2}A^{\prime\mu}A^{\prime}_{\mu}~{}.$
(1)

Here $A^{\prime}$ and $F^{\prime}$ are the dark field and dark field strength
respectively, $F$ is the Standard Model (SM) photon field strength,
$m_{A^{\prime}}$ is the mass of the dark photon, and $\epsilon$ is the kinetic
mixing (first explored in Holdom (1986), see Caputo  _et al._ (2021) for a
recent review). This kinetic mixing gives the dark photon a coupling to SM
electromagnetic currents, leading to observable interactions. The dark photon
mass is assumed throughout to be of a Stückelberg-type.

In this paper we focus on models in which cold dark matter (CDM) is partially
comprised of fermionic $\chi$ particles that are charged particles of the dark
sector with Lagrangian

$\mathscr{L}\supset\bar{\chi}(i\partial_{\mu}\gamma^{\mu}-m_{\chi})\chi-
g_{D}A^{\prime}_{\mu}\bar{\chi}\gamma^{\mu}\chi,$ (2)

where $m_{\chi}$ is the mass of the fermion, and $g_{D}$ is the dark electric
charge of $\chi$.

We consider a scenario where both $\chi$ and its anti-particle $\bar{\chi}$
are present in the Milky Way. If $m_{\chi}\gg m_{A}$, then the annihilation of
a $\chi$ and $\bar{\chi}$ will produce ultra-relativistic dark photons,
allowing them to be seen in terrestrial, low-background experiments (i.e.
neutrino experiments). Similar analyses have been done using other
decay/annihilation products of dark matter Huang and Zhao (2014); Agashe  _et
al._ (2014); Cherry  _et al._ (2015) or with other astrophysical sources
Agashe  _et al._ (2021).

This scenario allows us to explore the signal from low-mass dark matter (DM)
that couple to dark photons. Until recent years, this low-mass DM was
relatively unconstrained by direct detection experiments. The difficulty low-
mass DM presents is that the recoil energy deposited is proportional to the DM
mass, typically falling below the detector threshold for masses less than a
few GeV. While low-threshold detector technologies have made advances in
recent years, new strategies and materials have great promise to lead the
field in constraining low-mass DM Essig  _et al._ (2012a, b); Graham  _et al._
(2012); An  _et al._ (2015); Essig  _et al._ (2016); Hochberg  _et al._
(2016a); Derenzo  _et al._ (2017); Bloch  _et al._ (2017); Hochberg  _et al._
(2017, 2016b); Kouvaris and Pradler (2017); Essig  _et al._ (2017); Budnik
_et al._ (2018); Bunting  _et al._ (2017); Knapen  _et al._ (2018); Hochberg
_et al._ (2018); Hertel  _et al._ (2019); Dolan  _et al._ (2018); Bringmann
and Pospelov (2019); Emken  _et al._ (2019); Essig  _et al._ (2019); Ema  _et
al._ (2019); Bell  _et al._ (2020); Trickle  _et al._ (2020a, b); Griffin  _et
al._ (2020); Baxter  _et al._ (2020); Kurinsky  _et al._ (2019); Catena  _et
al._ (2020); Griffin  _et al._ (2021); Flambaum  _et al._ (2020); Bell  _et
al._ (2021).

The layout of this paper is as follows: In Sec. II, we will discuss the
distribution of $\chi$ within the Milky Way in light of annihilations and the
corresponding dark photon flux. In Sec. III we describe the interaction of
dark photons with matter, in particular how the optical properties of
experiments can enhance or suppress dark photon absorption. In Sec. IV we show
results from existing and projected experiments. Sec. V covers existing
constraints on this model, while Sec. VI discusses a similar signal arising
from decaying dark matter.

##  II $\chi$ Distribution and Dark Photon Flux

In this work, we consider the distributions of $\chi$ and $\bar{\chi}$ to be
identical, and will define $\rho_{\chi}$ to be the combined mass density
distribution of $\chi$ and $\bar{\chi}$ (i.e. the density distribution of just
$\chi$ is $\rho_{\chi}/2$). In this paper, we consider large $\chi$
annihilation cross sections, so this necessarily makes our distribution time-
dependent. Annihilations deplete the energy distribution of $\chi$ according
to Kaplinghat  _et al._ (2000); Chan (2016)

$\frac{\partial\rho_{\chi}}{\partial t}=-\frac{\langle\sigma
v\rangle}{m_{\chi}}\rho^{2}_{\chi}$ (3)

for $s-$wave annihilation. We take $\rho_{\chi}$ to be proportional to the NFW
profile Navarro  _et al._ (1996) at every location in space, with a
proportionality constant which depends on time (see Appendix A for more
comments on this assumption),

$\rho_{\chi}(r,t)=f_{\chi}(t)\rho_{\mathrm{NFW}}(r)$ (4)

$\rho_{\mathrm{NFW}}(r)=\rho_{s}\frac{r_{s}}{r}\bigg{(}1+\frac{r}{r_{s}}\bigg{)}^{-2}$
(5)

where $\rho_{s}=$ 0.184 GeV $\mathrm{cm}^{-3}$ and $r_{s}=$ 24.42 kpc Cirelli
_et al._ (2011). If we let $f_{\chi}(t=0)=f_{i}$, then the time evolution of
this proportionality constant is

$f_{\chi}(t)=\bigg{(}\frac{1}{f_{i}}+\frac{\langle\sigma v\rangle
t}{m_{\chi}}\frac{\int\rho_{\mathrm{NFW}}^{2}(\mathbf{x})dV}{\int\rho_{\mathrm{NFW}}(\mathbf{x})dV}\bigg{)}^{-1},$
(6)

Where $\langle\sigma v\rangle$ is the present-day thermally averaged cross
section and the volume integral is taken over a sphere 60 kpc in radius. We
can use this to find the present-day ($t=t_{0}$) density of $\chi$, and
compute the mono-energetic flux of dark photons using a line-of-sight integral
Cirelli  _et al._ (2011)

$\Phi_{A^{\prime}}=\frac{1}{4\pi}\frac{\langle\sigma
v\rangle}{2m^{2}_{\chi}}\int
d\Omega\int_{\mathrm{LOS}}dx\rho^{2}_{\chi}(x,\theta,t_{0}),$ (7)

where $x$ is the distance from the Sun, and $\theta$ is the angle between our
line-of-sight and a line pointed towards the galactic center. We can relate
this to $r$ by $r=\sqrt{x^{2}+r_{\odot}^{2}-2xr_{\odot}\cos(\theta)}$, with
$r_{\odot}$ being the galactic radius of the sun. We can decompose this flux
using Eq. 4, and then find the annihilation cross section which provides the
largest flux, namely

$\langle\sigma
v\rangle_{\max}=\frac{m_{\chi}}{f_{i}t_{0}}\frac{\int\rho_{\mathrm{NFW}}(\mathbf{x})dV}{\int\rho^{2}_{\mathrm{NFW}}(\mathbf{x})dV}.$
(8)

This result is seen clearly in Fig. 1. For much of the paper, we will assume
this optimum annihilation cross section in order to make strong statements
about which experiments are unable to probe this signal. However, if we were
to take other values of $\langle\sigma v\rangle$, the flux roughly scales as

$\log(\Phi_{A^{\prime}})\sim-\bigg{|}\log\bigg{(}\frac{\langle\sigma
v\rangle}{\langle\sigma v\rangle_{\max}}\bigg{)}\bigg{|}.$ (9)

In this work we remain agnostic about the origin of $\chi$ in the early
Universe. While thermal relics are already excluded for most dark matter
coupled to dark photons de la Vega  _et al._ (2023) and for warm dark matter
below 5 keV Dekker  _et al._ (2022), a small window remains. We note that the
scenario considered here lies outside of this region, as the values of
$\langle\sigma v\rangle_{\max}$ are simply too large.

Figure 1: Flux of dark photons at Earth vs. $\langle\sigma v\rangle$ obtained
by computing Eq. 7 for different $\chi$ masses, all with $f_{i}$ = 0.1. We can
clearly see the optimum $\langle\sigma v\rangle$ for each mass.

In principle, the dark photon flux at our detector would also be shaped by
dark photon decays en route to Earth and attenuation when passing through
Earth. However, for the parameters considered in this paper, the decay length
is much larger than galactic scales and the cross section is not large enough
for significant attenuation to occur.

##  III Dark Photon Interactions with Matter

The absorption rate of dark photons is affected by the optical properties of
the detector. The in-medium photon propagator leads to an effective mixing
parameter of

$\epsilon_{\mathrm{eff}}=\frac{m^{2}_{A^{\prime}}}{m^{2}_{A^{\prime}}-\Pi_{T/L}}\times\epsilon,$
(10)

where $\Pi_{T}$ ($\Pi_{L}$) is the transverse (longitudinal) polarization
tensor of the medium An  _et al._ (2013a, b); Weldon (1982). In our work, we
are interested in transversely polarized dark photons, as in the annihilation
of $\chi$ particles at the Galactic Center these are the dominant annihilation
products in the limit $E_{\chi}\simeq m_{\chi}\gg m_{A^{\prime}}$. This result
in demonstrated in Appendix D. For an isotropic and non-magnetic (the relative
permeability is 1) material, we can relate the polarization tensor to the
(complex) index of refraction $n_{\mathrm{ref}}$ via Weldon (1982) (see also
Appendix A of Hochberg  _et al._ (2016c) or Coskuner  _et al._ (2021))

$\Pi_{T}(\omega)=-\omega^{2}(1-n_{\mathrm{ref}}^{2}).$ (11)

The index of refraction for a single element can be related to atomic
scattering factors $f_{1}(\omega)$ and $f_{2}(\omega)$ (available from the
Lawrence Berkeley Lab database Center for X-Ray Optics ), and calculated via

$n_{\mathrm{ref}}(\omega)=1-\frac{r_{0}}{2\pi}\bigg{(}\frac{hc}{\omega}\bigg{)}^{2}n_{A}(f_{1}(\omega)+if_{2}(\omega)),$
(12)

where $n_{A}$ is the number density of atoms, and $r_{0}=2.82\times 10^{-15}$
m is the classical radius of the electron. For molecular detectors, the
scattering factors for each atom are added111For helium gas, the index of
refraction using Eq. 12 is different than the index of refraction data given
by Center for X-Ray Optics  for energies above $\sim$ 5 keV. In this
discrepancy, we use the explicit index of refraction data from Center for
X-Ray Optics .. Fig. 2 gives an example of the index of refraction over the
range of energies for which we are interested. While in general the electric
permittivity
$\varepsilon(\omega,\vec{k})/\varepsilon_{0}=n^{2}_{\mathrm{ref}}(\omega,\vec{k})$
(not to be confused with the kinetic mixing parameter $\epsilon$) depends on
the dark photon energy $\omega$ and three-momentum $\vec{k}$, for the dark
photon energies and momentum transfer considered here the dipole approximation
is a good approximation, and consequently the dependence of the electric
permittivity on $\vec{k}$ is suppressed. A further discussion on this point
can be found in Appendix B.

Figure 2: The real and imaginary part of $1-n_{\mathrm{ref}}$ for a xenon gas
at $\rho_{Xe}=5\,\mathrm{kg}\,\mathrm{m}^{-3}$.

The absorption rate for a single dark photon within our material is
$\Gamma=-|\epsilon_{\mathrm{eff}}|^{2}\mathrm{Im}(\Pi_{T}(\omega))/\omega$ An
_et al._ (2013c). Our rate of dark photon events within the detector is
therefore

$R_{A^{\prime}}=|\epsilon_{\mathrm{eff}}|^{2}V_{\mathrm{det}}\Phi_{A^{\prime}}\frac{-\mathrm{Im}(\Pi_{T}(m_{\chi}))}{m_{\chi}v_{A^{\prime}}},$
(13)

where $V_{\mathrm{det}}$ is the volume of our detector and we have used the
fact that $\omega=m_{\chi}$ for our flux of dark photons. As $m_{\chi}\gg
m_{A^{\prime}}$, $v_{A^{\prime}}\approx c$.

We can see from Eq. 10 that for $m^{2}_{A^{\prime}}\gg\Pi_{T}$, we have
$\epsilon_{\mathrm{eff}}=\epsilon$, while for $m^{2}_{A^{\prime}}\ll\Pi_{T}$,
the effective kinetic mixing is suppressed by
$\mathscr{O}(m^{2}_{A^{\prime}}/\Pi_{T})$. Avoiding this suppression leads us
to look at low density detectors where $n_{\mathrm{ref}}$ is close to unity.
We also note that there is an enhancement when
$m_{A^{\prime}}^{2}=\mathrm{Re}(\Pi_{T})$.

(a) Liquid xenon constraints/sensitivities  (b) Liquid argon
constraints/sensitivities  (c) Solid germanium constraints/sensitivities  (d)
Gaseous xenon constraints/sensitivities ($\rho_{Xe}=5\,\mathrm{kg\,m}^{-3}$)

(e) Helium constraints/sensitivities ($\rho_{He}=0.2\,\mathrm{kg\,m}^{-3}$)
(f) Sulfur hexafluoride constraints/sensitivities
($\rho_{SF_{6}}=6.2\,\mathrm{kg\,m}^{-3}$)

Figure 3: Exclusion plots for various materials and assumed sensitivities for
$m_{\chi}=1$ keV, $f_{i}$ = 0.1, and the annihilation cross section taken to
be $\langle\sigma v\rangle_{\max}$. Included are existing dark photon bounds
from Caputo  _et al._ (2021) and $\chi$ bounds from Davidson  _et al._ (2000).

##  IV Results

Our model is dependent upon 5 parameters

$(m_{A^{\prime}},\epsilon,m_{\chi},g_{D},f_{i}).$ (14)

Oftentimes, we will mention $\langle\sigma v\rangle$ instead of $g_{D}$. We
can relate the two up to an order 1 factor via Feng  _et al._ (2010)

$\langle\sigma v\rangle\sim\frac{\pi\alpha_{D}^{2}}{m_{\chi}^{2}},$ (15)

where $\alpha_{D}=g_{D}^{2}/4\pi$.

In Fig. 3, we fix the value of $f_{i}=0.1$ and $m_{\chi}=1$ keV, and let
$\langle\sigma v\rangle=\langle\sigma v\rangle_{\max}$ be given by Eq. 8 with
our chosen $m_{\chi}$ value. We vary $m_{A^{\prime}}$ from 0.01 to 100 eV, and
find the corresponding value of $\epsilon$ which gives the a priori event
rate.

If we were to consider to consider arbitrary $\langle\sigma v\rangle$, then
because $\epsilon\sim\Phi_{A^{\prime}}^{-1/2}$, our sensitivity would roughly
scale as

$\log(\epsilon^{\prime})\sim\frac{1}{2}\bigg{|}\log\bigg{(}\frac{\langle\sigma
v\rangle}{\langle\sigma v\rangle_{\max}}\bigg{)}\bigg{|}.$ (16)

Alternatively, in Fig. 4, we instead fix the values of $f_{i}=0.1$ and
$m_{A^{\prime}}$. We vary $m_{\chi}$ from 0.3 to 30 keV, and at every value,
we set $\langle\sigma v\rangle$ to be the corresponding value of
$\langle\sigma v\rangle_{\max}$. As before, the value of $\epsilon$ which
provides the desired event rate is found and graphed.

(a) Liquid xenon constraints/sensitivities ($m_{A^{\prime}}$ = 25 eV)  (b)
Gaseous xenon constraints/sensitivities ($\rho_{Xe}=5\,\mathrm{kg\,m}^{-3}$ ;
$m_{A^{\prime}}$ = 1 eV)  (c) Helium constraints/sensitivities
($\rho_{He}=0.2\,\mathrm{kg\,m}^{-3}$ ; $m_{A^{\prime}}=0.3$ eV)  (d) Sulfur
hexafluoride constraints/sensitivities
($\rho_{SF_{6}}=6.2\,\mathrm{kg\,m}^{-3}$ ; $m_{A^{\prime}}$ = 2 eV)

Figure 4: Exclusion plots for various materials and assumed sensitivities for
$f_{i}$ = 0.1, with the annihilation cross section taken to be $\langle\sigma
v\rangle_{\max}$. Compared to Fig. 3, here the $\chi$ mass is varied with
values for $m_{A^{\prime}}$ taken to be near the resonance (which is
material–dependent as seen in Fig. 3). The horizontal line shows the existing
dark photon bound from Caputo  _et al._ (2021) for our chosen
$m_{A^{\prime}}$. Also included are bounds from $\chi$ scattering in the early
universe, obtained from Buen-Abad  _et al._ (2022). We shade the helium
results beyond 10 keV because our dipole assumption for absorption breaks down
beyond this point (see Appendix B for more details). Calculating the true
sensitivity bounds in this region is beyond the scope of this paper, but using
our method, the sensitivity is already subdominant to solar emitted
$A^{\prime}$ by $m_{\chi}=$ 10 keV. The cooling and CMB bounds should still be
valid in this shaded region. Figure 5: Exclusion and sensitivity plots for
gaseous xenon detectors for a variety of gaseous xenon densities and with
$m_{\chi}=1$ keV, $f_{i}$ = 0.1, and assuming a sensitivity of 2 events
$\mathrm{ton}^{-1}\mathrm{yr}^{-1}$. Included are bounds from Caputo  _et al._
(2021) and Davidson  _et al._ (2000).

###  IV.1 Liquid Xenon/Argon

The first detector materials we consider are liquid xenon and argon, which are
used in low background ton-scale experiments such as dark matter direct
detection Aprile  _et al._ (2022, 2018); Agnes  _et al._ (2023a, b) and
neutrinoless double-beta decay (for the case of argon Abgrall  _et al._ (2021)
and xenon Albert  _et al._ (2018)). For these noble liquids, the polarization
tensor $\Pi_{T}\sim\mathscr{O}(100\mathrm{eV}^{2})$, which we can see from the
kinks in $\epsilon-m_{A^{\prime}}$ plots in Figs. 3a & 3b. Because of the
relatively large (compared to gasses) in-medium effect, these liquid detectors
are unable to probe any new parameter space.

###  IV.2 Solid Germanium

Similar to liquid xenon and argon, germanium detectors have been used in dark
matter direct detection Arnaud  _et al._ (2020) and neutrinoless double-beta
decay experiments Edzards (2020). While germanium crystals are not isotropic
and so do not satisfy our assumption of isotropy, here we only consider
incident dark photons with $\cal{O}($keV$)$ energies and higher. For
absorption the momentum-transfer is also of this scale, and thus we expect
neglecting the band structure and the anisotropy of the crystals to be a good
approximation. Again, the relatively high densities of these detectors make
them unable to probe new parameter space, as can be seen in Fig. 3c.

###  IV.3 Gaseous Xenon

Ton-scale gaseous xenon detectors have been proposed in the context of
searching for double-beta decay such as the PANDAX-III Chen  _et al._ (2017)
and NEXT Gomez-Cadenas (2019) experiments. For gases, the size of $\Pi_{T}$
depends upon the target density. We can see in Figs. 3d & 4b that xenon at 5
kg $\mathrm{m}^{-3}$ (the density at standard temperature and pressure) can
probe new parameter space if the experiment is sensitive to rates
$\mathscr{O}$(10 $\mathrm{ton}^{-1}\mathrm{yr}^{-1}$) at keV scale energies.
This may be overly optimistic for double-beta decay experiments, which are
designed to be most sensitive near the $Q$-value of the isotope in question
(2.46 MeV for xenon-136). However, a future gasesous xenon detector with a low
energy threshold and strong background discrimination could look for this
signal.

We would also like to point out Fig. 5, in which we consider the hypothetical
sensitivities of gaseous xenon detectors with the same sensitivity but
different xenon densities. We see that lowering the density allows the
experiment to probe lower dark photon masses.

###  IV.4 Helium / Sulfur Hexafluoride

Gaseous helium and sulfur hexafluoride detectors interest us because of the
proposed CYGNUS experiment Vahsen  _et al._ (2020), a large volume dark matter
detector which contains a mixture of the two gasses. This experiment has been
considered to reach sizes of 1000 $\mathrm{m}^{3}$ and beyond, with very low
energy thresholds, making it an interesting detector to search for these
boosted dark photons. We would like to make a special note of helium gas in
Fig. 3e. Although helium has a relatively small cross-section for x-rays, that
same property leads to a strong resonance in Eq. 10. At a single density, the
helium detector would be sensitive to a small range of dark photon masses.
However, one could imagine a varying density (either in time or across
modules) that would allow for an improved search over a larger range of
masses.

##  V Existing constraints on $A^{\prime}$ and $\chi$

We will discuss how existing experiments/observations have constrained these
parameters.

###  V.1 Dark Photon Limits

One can consider an extension to the Standard Model given only by Eq. 1. This
model has a rich phenomenology and many constraints on $m_{A^{\prime}}$ and
$\epsilon$ as shown in Figure 1 of Caputo  _et al._ (2021) and regularly
updated in corresponding website. Note that we disregard the bounds which
arise when the dark photon is the dark matter candidate, as that is different
than the scenario considered here. In our region of interest
($m_{A^{\prime}}=\mathscr{O}(10^{-2}-10^{1}\mathrm{eV}))$ the most stringent
constraints consider the Sun as a source of dark photons, and constrain either
solar cooling An  _et al._ (2013d) or direct detection of these emitted
$A^{\prime}$ An  _et al._ (2020).

###  V.2 SM - $\chi$ Scattering

Our model allows for $\chi$ to scatter with electrically charged SM particles
by exchanging a virtual dark photon which mixes with the SM photon. We
consider $m_{\chi}\sim\mathscr{O}$(1-100 keV) with virialized velocity,
leading to kinetic energies of $\mathscr{O}(10^{-3}-10^{-1}$eV). These
energies are too small to probe with modern-day direct detection experiments.
However, in the early universe, scattering between $\chi$ and SM particles can
leave an observable imprint on the Cosmic Microwave Background (CMB)Buen-Abad
_et al._ (2022); Nguyen  _et al._ (2021). We can translate constraints on
cross section into constraints on $\epsilon$, $g_{D}$, $m_{\chi}$, and $f_{i}$
by assuming that $\chi$ is the only component of dark matter to interact with
SM particles.

We do not consider constraints in which $\chi$ interacts with long-range
electric or magnetic fields. Although there is mixing between the SM and dark
photons, the dark photon masses considered here are large enough that the
effects of long-range SM fields are exponentially suppressed.

###  V.3 $\chi$ Driven Stellar Cooling

Similar to dark photons, $\chi\bar{\chi}$ pairs can be produced in stars with
temperature $T\gtrsim m_{\chi}$. The strongest of these constraints come from
horizontal branch stars, which limit $q_{D}=g_{D}\epsilon/e<2\times 10^{-14}$
Davidson  _et al._ (2000).

###  V.4 $\chi$ \- $\chi$ Scattering

Gravitational measurements of galactic mergers and the ellipticity of galactic
dark matter halos place constraints on the self-interactions of dark matter
Lasenby (2020), which in our model can be translated into constraints on
$m_{\chi}$, $g_{D}$, $f_{i}$, and $m_{A^{\prime}}$. The simplest way to avoid
these constraints is to take $f_{i}\sim\mathscr{O}(0.1)$, so that any effects
of $\chi$ will be smaller than the uncertainty on the measurements.

##  VI Decaying Dark Matter

The analysis considered in this paper could similarly be used to constrain a
dark matter candidate $\phi$ which would decay into dark photons
($\phi\rightarrow A^{\prime}A^{\prime}$). If we assume that $\phi$ is the
entirety of dark matter, this model can be characterized by four parameters

$(m_{A^{\prime}},\epsilon,m_{\phi},\Gamma_{\phi}),$ (17)

where $\Gamma_{\phi}$ is the decay rate of $\phi$. This will produce a
galactic flux of dark photons which is given by

$\Phi_{A^{\prime},\mathrm{dec}}=\frac{1}{4\pi}\frac{2\Gamma_{\phi}}{m_{\phi}}\int
d\Omega\int_{\mathrm{LOS}}dx\rho_{\phi}(x,\theta).$ (18)

Taking a value for $\Gamma_{\phi}=0.03t_{0}^{-1}$, which is consistent with
Chen  _et al._ (2021), we obtain a flux of dark photons comparable with the
maximum allowed flux from $\chi\bar{\chi}$ annihilations. (In general, the
dark photon flux could also have a contribution from extragalactic $\phi$, but
for now, only the galactic contributions are compared). In Fig. 6 we compare
the two fluxes, along with the direct flux of dark matter at Earth, calculated
via $\Phi_{\chi}=\frac{\rho_{\chi}}{m_{\chi}}v_{\odot}$ where $v_{\odot}$ is
the velocity of the Sun around the Milky Way.

Figure 6: Comparing the galactic dark photon flux from annihilating dark
matter with annihilation cross section given by Eq. 8 and initial fraction
$f_{i}$ = 0.1, and decaying dark matter with $\Gamma_{\phi}$ = 0.03
$t_{0}^{-1}$. For annihilation and decay, “Energy” refers to the dark photon
energy, while for the direct $\chi$ flux, it refers to $m_{\chi}$.

##  VII Conclusions

In this paper, we have shown that for $\chi$ being a sub-component of dark
matter with large annihilation cross sections, there exists currently
unconstrained parameter space that can lead to visible signals in future
gaseous detectors. We do not provide exclusions or projections, as that would
require essential features of future experiments, such as energy thresholds
and estimates of backgrounds, that are beyond the scope of this work. However,
we do provide code to determine the rate of dark photon events at a particular
experiment (see the GitHub link in the abstract). We also wish to point out
some detector properties that would be most useful in looking for this signal

  * •

Low Densities and Large Volumes: Clearly, a larger target mass allows for more
dark photons to be absorbed, and we see in Fig. 5 that lower densities can be
sensitive to lower $m_{A^{\prime}}$. A variable density (with significant
exposure) could scan parameter space for an even larger range of
$m_{A^{\prime}}$.

  * •

Low Energy thresholds and Good Energy Resolution: The flux of $A^{\prime}$ is
largest for small $m_{\chi}$, but this flux would only be visible if the
energy threshold is below $m_{\chi}$. Moreover, the flux of dark photons would
be mono-energetic. Good energy resolution and reconstruction would have these
dark photon events appear as a peak, reducing the number of events necessary
to be significant.

  * •

Good Spatial Resolution: Typical x-ray backgrounds, which may come from
outside the detector or detector components, will mainly occur on the detector
edges and be unable to penetrate to the center of the fiducial volume. On the
other hand, the small interaction probability of dark photons would make them
equally likely to appear anywhere within the detector, so topological
discrimination could be used to reduce the background.

## Acknowledgements

The authors thank Kaustubh Agashe, Haipeng An, Steve Elliot, Yoni Kahn, Doojin
Kim, Ralph Massarczyk, and Sam Watkins for comments and discussions. The work
of MG is supported by the LDRD program at Los Alamos National Laboratory and
by the U.S. Department of Energy, Office of High Energy Physics, under
Contract No. DE-AC52-06NA25396. RAG, VM, and IMS are supported by the U.S.
Department of Energy Office of Science, Office of High Energy Physics, under
Award Number DE-SC0020262.

##  Appendix A Time Evolution of $\chi$ and Assumptions

We consider the parametrization of the $\chi$ density given by Eq. 4 (here
called the ”Diffusion Limit”). Using Eq. 3, we need to integrate over the
galaxy to find the time evolution of the scaling factor.

$\frac{df_{\chi}}{dt}\int\rho_{\mathrm{NFW}}(\textbf{x})dV=\frac{-\langle\sigma
v\rangle f_{\chi}^{2}(t)}{m_{\chi}}\int\rho^{2}_{\mathrm{NFW}}(\textbf{x})dV$
(19)

This differential equation is solved by Eq. 6.

An alternative approach to the time evolution of $\chi$ is what we call
Spatially-Independent Evolution (SIE). In this case, we consider the initial
distribution to be proportional to the NFW distribution
$\rho_{\chi,\mathrm{SIE}}(r,t=0)=f_{i,\mathrm{SIE}}\rho_{\mathrm{NFW}}(r)$,
and then each location in space evolves independent of other locations in
space, according to Eq. 3. This is solved by

$\rho_{\chi,\mathrm{SIE}}(r,t)=\frac{f_{i,\mathrm{SIE}}\times\rho_{\mathrm{NFW}}(r)m_{\chi}}{m_{\chi}+f_{i,\mathrm{SIE}}\times\rho_{\mathrm{NFW}}(r)\langle\sigma
v\rangle t}.$ (20)

Intuitively, SIE is the correct approach when annihilation happens very
quickly, whereas the Diffusion Limit is motivated when $\chi$ diffuses fast
enough to correct for any changes due to annihilations. To give a
quantification of this comparison, we will compare

$v_{\mathrm{NFW}}(r)f_{i}|\nabla\rho_{\mathrm{NFW}}(r)|\,\,\mathrm{vs}\,\,\frac{\langle\sigma
v\rangle f^{2}_{i}\rho^{2}_{\mathrm{NFW}}(r)}{m_{\chi}},$ (21)

where the left expression is the characteristic change in density due to
diffusion, and the right is due to annihilations. We define this
characteristic velocity as

$v_{\mathrm{NFW}}(r)=\sqrt{\frac{GM_{\mathrm{enc}}(r)}{r}},$ (22)

where $M_{\mathrm{enc}}$ is enclosed mass in a sphere centered at the galactic
center of radius $r$. Along with the dark matter density, we also need to
consider the baryonic density Berlin  _et al._ (2023); Hernquist (1990)

$\rho_{B}(r)=\frac{\rho_{B0}r^{4}_{0}}{r(r+r_{0})^{3}},$ (23)

where $\rho_{B0}=$ 26 GeV $\mathrm{cm}^{-3}$ and $r_{0}=2.7$ kpc. We now have
the information needed to compare the two expressions in Eq. 21, and we will
consider $\langle\sigma v\rangle=\langle\sigma v\rangle_{\max}$ as defined in
Eq. 8. As can be seen in Fig. 7, the diffusion occurs faster than
annihilation. For this reason, we assume the Diffusion Limit is a good
approximation, although a more detailed simulation would be needed to capture
all of the physics.

Figure 7: Comparing the two equations in Eq. 21 as a function of radius r from
the galactic center, with $f_{i}=0.1$ and $m_{\chi}=1$ keV.

##  Appendix B Electric Permittivity

In general the electric permittivity $\varepsilon$ is a function of both the
incident (dark) photon energy $\omega$ as well as its three momentum
$\vec{k}$. For most of the energies considered in this work, however, the
dipole approximation $e^{i\vec{k}\cdot\vec{x}}\simeq 1$ is valid and the
dependence of the electric permittivity on $\vec{k}$ can be neglected to
leading order Bethe and Salpeter (1957). The reasons are as follows.

Bound-state to bound-state transitions are necessarily for low incident
energies, since axiomatically, $\omega\simeq|\vec{k}|$ is bounded by the
atom’s ionization energy, and since very little energy is transferred to the
kinetic motion of the recoiling atom (because atoms are heavy). Thus
$\omega\simeq|\vec{k}|\simeq E_{nn^{\prime}}\sim(Z\alpha_{em})^{2}m_{e}$,
where $E_{nn^{\prime}}$ is the energy of the transition. With $x\simeq
a_{nn^{\prime}}$, i.e. the Bohr radius of the electron states in question, and
as $a_{nn^{\prime}}\simeq 1/(Z\alpha m_{e})$, one sees that $kx\ll 1$ provided
$Z\alpha\ll 1$ Bethe and Salpeter (1957). Moreover, bound-state to bound-state
electrons transitions are predominately by outer electrons, for which the
effective electric charge $Z\approx 1$. (This argument is also presented in An
_et al._ (2013b)).

For ionization, Bethe and Salpeter demonstrate that these processes are
dominated by inner electrons (K and L shells), and this is the key point
(Secs. 69-71,Bethe and Salpeter (1957)). Because these electrons see a higher
$Z$ charge, they therefore have a smaller Bohr radius and consequently $kx$
remains small. Namely, if $\omega\ll Z\alpha m_{e}$ (that is, the dark photon
wavelength is larger than the radius of that inner electron) then retardation
effects can be neglected and consequently $e^{ikx}=1$ to a good approximation
(Section 69, Eq. 69.9, Bethe and Salpeter (1957)).

For argon, $Z\alpha m_{e}\approx 65$ keV, for fluorine $Z\alpha m_{e}\approx
32$ keV, for germaniun $Z\alpha m_{e}\approx 116$ keV, for sulphur $Z\alpha
m_{e}\approx 60$ keV, and for xenon $Z\alpha m_{e}\approx 200$ keV. For these
elements, our analysis is accordingly restricted to $\omega\lesssim 30$ keV to
maintain the dipole approximation. For helium $Z\alpha m_{e}\approx 7$ keV,
however, such that the dipole approximation becomes invalid at energies
$\gtrsim$ 10 keV. In Fig. 4c, the region above $10$ keV is highlighted to
indicate the breakdown of the dipole approximation there.

##  Appendix C Comparison to Solar Bounds

The most stringent bounds in our region of interest come from the solar
production of dark photons An  _et al._ (2013d, 2020). We think it is
important to discuss the fundamental differences between solar production and
boosted production via $\chi$ annihilation at the Galactic Center.

First, in solar production, the flux of dark photons necessarily goes to zero
as the dark photon mass becomes small. This is because SM currents are the
source of the $A^{\prime}$, and the large solar in-medium effects suppresses
the effective mixing as seen in Eq. 10. Alternatively, $\chi$ annihilation is
scaled by $m_{\chi}$ and $g_{D}$, not $\epsilon$ or $m_{A^{\prime}}$ (assuming
$m_{A^{\prime}}\ll m_{\chi}$), so the flux of dark photons remains to
arbitrarily small dark photon masses.

Second, for solar production, dark photons are primarily in the longitudinal
polarization, which have an absorption rate linearly dependent on the density
An  _et al._ (2013c). This means that gaseous detectors would not offer an
improvement over liquid detectors for equal target masses. This is quite
different than the transversely dominated flux from $\chi$ annihilations, for
which we see a significant improvement at gaseous detectors.

##  Appendix D Non-relativistic Dark $\chi$ Annihilation and Dark Photon
Polarizations

This Appendix considers non-relativistic $\chi\overline{\chi}$ annihilation to
two dark photons. In the limit $E_{\chi}\simeq m_{\chi}\gg m_{A^{\prime}}$,
the final state is found to be dominated by transversely polarized dark
photons. Production of longitudinally polarized dark photons is highly
suppressed.

In order to demonstrate the dominance of transverse polarizations, we subtract
off the transversely polarized contribution from the total rate and compare
the two. The process of $\chi(k_{1})\bar{\chi(k_{2})}\rightarrow
A^{\prime,\nu}(p1)A^{\prime,\mu}(p_{2})$ at tree-level consists of the matrix
element

$\begin{split}M=ig_{D}^{2}\bar{v(k_{2})}\bigg{(}\gamma^{\mu}\epsilon^{*}_{\mu}(p_{2})\frac{\not{p_{2}}-\not{k_{2}}+m_{\chi}}{t-m_{\chi}^{2}}\gamma^{\nu}\epsilon^{*}_{\nu}(p_{1})\\\
+\gamma^{\nu}\epsilon^{*}_{\nu}(p_{1})\frac{\not{p_{1}}-\not{k_{2}}+m_{\chi}}{u-m_{\chi}^{2}}\gamma^{\mu}\epsilon^{*}_{\mu}(p_{2})\bigg{)}u(k_{1}),\end{split}$
(24)

where $t$ and $u$ are the Mandelstam variables and $\epsilon_{\mu}(p)$ is a
polarization vector, not to be confused with the kinetic mixing parameter. To
compute the cross section of this process, we find $|M|^{2}$ and sum over the
outgoing polarizations. We know that for a massive vector boson, the sum over
polarizations (denoted by $\lambda$) returns

$\sum_{\lambda}\epsilon_{\lambda}^{*\mu}(p)\epsilon_{\lambda}^{\nu}(p)=\frac{p^{\mu}p^{\nu}}{p^{2}}-g^{\mu\nu}.$
(25)

We next define the longitudinal polarization to be

$\epsilon_{L}(p)=\frac{1}{\sqrt{\omega^{2}-|\overrightarrow{p}|^{2}}}\bigg{(}|\overrightarrow{p}|,\omega\frac{\overrightarrow{p}}{|\overrightarrow{p}|}\bigg{)}.$
(26)

It can be shown that

$\begin{split}\epsilon_{L}^{\mu}(p)\epsilon_{L}^{\nu}(p)=\delta^{\mu
0}\delta^{\nu 0}\bigg{(}\frac{p^{2}}{\omega^{2}-p^{2}}\bigg{)}-\delta^{\mu
0}\bigg{(}\frac{\omega p^{\nu}}{p^{2}}\bigg{)}\\\ -\delta^{\nu
0}\bigg{(}\frac{\omega
p^{\mu}}{p^{2}}\bigg{)}+\frac{p^{\mu}p^{\nu}}{\omega^{2}-p^{2}}+\frac{p^{\mu}p^{\nu}}{p^{2}}.\end{split}$
(27)

When we apply the on-shell condition, the first term will be
$\mathscr{O}(m_{A^{\prime}}^{2}/m_{\chi}^{2})$, which is small enough that we
will ignore it. To find final value of the polarization sum, it is useful to
define a new four-vector $\eta^{\mu}(\omega,p)$

$\eta^{\mu}(\omega,p)=\omega(1,0,0,0)-p^{\mu}.$ (28)

Thus we can define

$\begin{split}\sum_{\mathrm{transverse}}\epsilon_{\lambda}^{*\mu}(p)\epsilon_{\lambda}^{\nu}(p)=-g^{\mu\nu}-\frac{\eta^{2}p^{\mu}p^{\nu}}{(\eta\cdot
p)^{2}}\\\ +\frac{\eta^{\nu}p^{\mu}}{(\eta\cdot
p)}+\frac{\eta^{\mu}p^{\nu}}{(\eta\cdot
p)}+\mathscr{O}(m_{A^{\prime}}^{2}/m_{\chi}^{2}),\end{split}$ (29)

(this is easily implemented using FeynCalc Mertig  _et al._ (1991);
Shtabovenko  _et al._ (2016, 2020)). If we work in the low-velocity limit,
expanding in both $m_{A^{\prime}}$ and $K$ defined as $s=4(m_{\chi}+K)^{2}$,
we find,

$\frac{1}{4}\sum_{\mathrm{spin}}(\sum_{T+L}|M|^{2}-\sum_{T}|M|^{2})=\mathscr{O}\bigg{(}\frac{K^{2}}{m^{2}_{\chi}},\frac{m_{A^{\prime}}^{2}}{m^{2}_{\chi}}\bigg{)},$
(30)

where the term outside parentheses is to average over the incoming fermion
spins, and the latter sums are the polarization sums with and without the
longitudinal modes. Also, up to this level of precision

$\frac{1}{4}\sum_{\mathrm{spin}}\sum_{T}|M|^{2}=4g_{D}^{4}+16g_{D}^{4}\frac{K}{m_{\chi}}+\mathscr{O}\bigg{(}\frac{K^{2}}{m^{2}_{\chi}},\frac{m_{A^{\prime}}^{2}}{m^{2}_{\chi}}\bigg{)}.$
(31)

## References

  * Holdom (1986) B. Holdom, Phys. Lett. B 166, 196 (1986).
  * Caputo  _et al._ (2021) A. Caputo, A. J. Millar, C. A. J. O’Hare, and E. Vitagliano, Phys. Rev. D 104, 095029 (2021).
  * Huang and Zhao (2014) J. Huang and Y. Zhao, JHEP 02, 077 (2014), arXiv:1312.0011 [hep-ph] .
  * Agashe  _et al._ (2014) K. Agashe, Y. Cui, L. Necib, and J. Thaler, JCAP 10, 062 (2014), arXiv:1405.7370 [hep-ph] .
  * Cherry  _et al._ (2015) J. F. Cherry, M. T. Frandsen, and I. M. Shoemaker, Phys. Rev. Lett. 114, 231303 (2015), arXiv:1501.03166 [hep-ph] .
  * Agashe  _et al._ (2021) K. Agashe, S. J. Clark, B. Dutta, and Y. Tsai, Phys. Rev. D 103, 083006 (2021), arXiv:2007.04971 [astro-ph.CO] .
  * Essig  _et al._ (2012a) R. Essig, J. Mardon, and T. Volansky, Phys. Rev. D 85, 076007 (2012a), arXiv:1108.5383 [hep-ph] .
  * Essig  _et al._ (2012b) R. Essig, A. Manalaysay, J. Mardon, P. Sorensen, and T. Volansky, Phys. Rev. Lett. 109, 021301 (2012b), arXiv:1206.2644 [astro-ph.CO] .
  * Graham  _et al._ (2012) P. W. Graham, D. E. Kaplan, S. Rajendran, and M. T. Walters, Phys. Dark Univ. 1, 32 (2012), arXiv:1203.2531 [hep-ph] .
  * An  _et al._ (2015) H. An, M. Pospelov, J. Pradler, and A. Ritz, Phys. Lett. B 747, 331 (2015), arXiv:1412.8378 [hep-ph] .
  * Essig  _et al._ (2016) R. Essig, M. Fernandez-Serra, J. Mardon, A. Soto, T. Volansky, and T.-T. Yu, JHEP 05, 046 (2016), arXiv:1509.01598 [hep-ph] .
  * Hochberg  _et al._ (2016a) Y. Hochberg, Y. Zhao, and K. M. Zurek, Phys. Rev. Lett. 116, 011301 (2016a), arXiv:1504.07237 [hep-ph] .
  * Derenzo  _et al._ (2017) S. Derenzo, R. Essig, A. Massari, A. Soto, and T.-T. Yu, Phys. Rev. D 96, 016026 (2017), arXiv:1607.01009 [hep-ph] .
  * Bloch  _et al._ (2017) I. M. Bloch, R. Essig, K. Tobioka, T. Volansky, and T.-T. Yu, JHEP 06, 087 (2017), arXiv:1608.02123 [hep-ph] .
  * Hochberg  _et al._ (2017) Y. Hochberg, Y. Kahn, M. Lisanti, C. G. Tully, and K. M. Zurek, Phys. Lett. B 772, 239 (2017), arXiv:1606.08849 [hep-ph] .
  * Hochberg  _et al._ (2016b) Y. Hochberg, T. Lin, and K. M. Zurek, Phys. Rev. D 94, 015019 (2016b), arXiv:1604.06800 [hep-ph] .
  * Kouvaris and Pradler (2017) C. Kouvaris and J. Pradler, Phys. Rev. Lett. 118, 031803 (2017), arXiv:1607.01789 [hep-ph] .
  * Essig  _et al._ (2017) R. Essig, T. Volansky, and T.-T. Yu, Phys. Rev. D 96, 043017 (2017), arXiv:1703.00910 [hep-ph] .
  * Budnik  _et al._ (2018) R. Budnik, O. Chesnovsky, O. Slone, and T. Volansky, Phys. Lett. B 782, 242 (2018), arXiv:1705.03016 [hep-ph] .
  * Bunting  _et al._ (2017) P. C. Bunting, G. Gratta, T. Melia, and S. Rajendran, Phys. Rev. D 95, 095001 (2017), arXiv:1701.06566 [hep-ph] .
  * Knapen  _et al._ (2018) S. Knapen, T. Lin, M. Pyle, and K. M. Zurek, Phys. Lett. B 785, 386 (2018), arXiv:1712.06598 [hep-ph] .
  * Hochberg  _et al._ (2018) Y. Hochberg, Y. Kahn, M. Lisanti, K. M. Zurek, A. G. Grushin, R. Ilan, S. M. Griffin, Z.-F. Liu, S. F. Weber, and J. B. Neaton, Phys. Rev. D 97, 015004 (2018), arXiv:1708.08929 [hep-ph] .
  * Hertel  _et al._ (2019) S. A. Hertel, A. Biekert, J. Lin, V. Velan, and D. N. McKinsey, Phys. Rev. D 100, 092007 (2019), arXiv:1810.06283 [physics.ins-det] .
  * Dolan  _et al._ (2018) M. J. Dolan, F. Kahlhoefer, and C. McCabe, Phys. Rev. Lett. 121, 101801 (2018), arXiv:1711.09906 [hep-ph] .
  * Bringmann and Pospelov (2019) T. Bringmann and M. Pospelov, Phys. Rev. Lett. 122, 171801 (2019), arXiv:1810.10543 [hep-ph] .
  * Emken  _et al._ (2019) T. Emken, R. Essig, C. Kouvaris, and M. Sholapurkar, JCAP 09, 070 (2019), arXiv:1905.06348 [hep-ph] .
  * Essig  _et al._ (2019) R. Essig, J. Pérez-Ríos, H. Ramani, and O. Slone, Phys. Rev. Research. 1, 033105 (2019), arXiv:1907.07682 [hep-ph] .
  * Ema  _et al._ (2019) Y. Ema, F. Sala, and R. Sato, Phys. Rev. Lett. 122, 181802 (2019), arXiv:1811.00520 [hep-ph] .
  * Bell  _et al._ (2020) N. F. Bell, J. B. Dent, J. L. Newstead, S. Sabharwal, and T. J. Weiler, Phys. Rev. D 101, 015012 (2020), arXiv:1905.00046 [hep-ph] .
  * Trickle  _et al._ (2020a) T. Trickle, Z. Zhang, and K. M. Zurek, Phys. Rev. Lett. 124, 201801 (2020a), arXiv:1905.13744 [hep-ph] .
  * Trickle  _et al._ (2020b) T. Trickle, Z. Zhang, K. M. Zurek, K. Inzani, and S. Griffin, JHEP 03, 036 (2020b), arXiv:1910.08092 [hep-ph] .
  * Griffin  _et al._ (2020) S. M. Griffin, K. Inzani, T. Trickle, Z. Zhang, and K. M. Zurek, Phys. Rev. D 101, 055004 (2020), arXiv:1910.10716 [hep-ph] .
  * Baxter  _et al._ (2020) D. Baxter, Y. Kahn, and G. Krnjaic, Phys. Rev. D 101, 076014 (2020), arXiv:1908.00012 [hep-ph] .
  * Kurinsky  _et al._ (2019) N. A. Kurinsky, T. C. Yu, Y. Hochberg, and B. Cabrera, Phys. Rev. D 99, 123005 (2019), arXiv:1901.07569 [hep-ex] .
  * Catena  _et al._ (2020) R. Catena, T. Emken, N. A. Spaldin, and W. Tarantino, Phys. Rev. Res. 2, 033195 (2020), arXiv:1912.08204 [hep-ph] .
  * Griffin  _et al._ (2021) S. M. Griffin, Y. Hochberg, K. Inzani, N. Kurinsky, T. Lin, and T. Chin, Phys. Rev. D 103, 075002 (2021), arXiv:2008.08560 [hep-ph] .
  * Flambaum  _et al._ (2020) V. V. Flambaum, L. Su, L. Wu, and B. Zhu, (2020), arXiv:2012.09751 [hep-ph] .
  * Bell  _et al._ (2021) N. F. Bell, J. B. Dent, B. Dutta, S. Ghosh, J. Kumar, and J. L. Newstead, (2021), arXiv:2103.05890 [hep-ph] .
  * Kaplinghat  _et al._ (2000) M. Kaplinghat, L. Knox, and M. S. Turner, Phys. Rev. Lett. 85, 3335 (2000), arXiv:astro-ph/0005210 .
  * Chan (2016) M. H. Chan, Astropart. Phys. 80, 1 (2016), arXiv:1604.01494 [astro-ph.HE] .
  * Navarro  _et al._ (1996) J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J. 462, 563 (1996), arXiv:astro-ph/9508025 [astro-ph] .
  * Cirelli  _et al._ (2011) M. Cirelli, G. Corcella, A. Hektor, G. Hütsi, M. Kadastik, P. Panci, M. Raidal, F. Sala, and A. Strumia, Journal of Cosmology and Astroparticle Physics 2011, 051 (2011).
  * de la Vega  _et al._ (2023) L. M. G. de la Vega, R. Ferro-Hernandez, A. García-Viltres, E. Peinado, and E. Vázquez-Jáuregui, (2023), arXiv:2311.17987 [hep-ph] .
  * Dekker  _et al._ (2022) A. Dekker, S. Ando, C. A. Correa, and K. C. Y. Ng, Phys. Rev. D 106, 123026 (2022), arXiv:2111.13137 [astro-ph.CO] .
  * An  _et al._ (2013a) H. An, M. Pospelov, and J. Pradler, Phys. Lett. B 725, 190 (2013a), arXiv:1302.3884 [hep-ph] .
  * An  _et al._ (2013b) H. An, M. Pospelov, and J. Pradler, Phys. Rev. Lett. 111, 041302 (2013b), arXiv:1304.3461 [hep-ph] .
  * Weldon (1982) H. A. Weldon, Phys. Rev. D 26, 1394 (1982).
  * Hochberg  _et al._ (2016c) Y. Hochberg, M. Pyle, Y. Zhao, and K. M. Zurek, JHEP 08, 057 (2016c), arXiv:1512.04533 [hep-ph] .
  * Coskuner  _et al._ (2021) A. Coskuner, A. Mitridate, A. Olivares, and K. M. Zurek, Phys. Rev. D 103, 016006 (2021), arXiv:1909.09170 [hep-ph] .
  * (50) Center for X-Ray Optics, “X-ray interactions with matter,” https://henke.lbl.gov/optical_constants/.
  * An  _et al._ (2013c) H. An, M. Pospelov, and J. Pradler, Phys. Rev. Lett. 111, 041302 (2013c).
  * Davidson  _et al._ (2000) S. Davidson, S. Hannestad, and G. Raffelt, JHEP 05, 003 (2000), arXiv:hep-ph/0001179 .
  * Feng  _et al._ (2010) J. L. Feng, M. Kaplinghat, and H.-B. Yu, Phys. Rev. D 82, 083525 (2010).
  * Buen-Abad  _et al._ (2022) M. A. Buen-Abad, R. Essig, D. McKeen, and Y.-M. Zhong, Physics Reports 961, 1 (2022).
  * Aprile  _et al._ (2022) E. Aprile _et al._ ((XENON Collaboration)††, XENON), Phys. Rev. Lett. 129, 161805 (2022), arXiv:2207.11330 [hep-ex] .
  * Aprile  _et al._ (2018) E. Aprile _et al._ (XENON), Phys. Rev. Lett. 121, 111302 (2018), arXiv:1805.12562 [astro-ph.CO] .
  * Agnes  _et al._ (2023a) P. Agnes _et al._ (DarkSide), Phys. Rev. Lett. 130, 101001 (2023a), arXiv:2207.11967 [hep-ex] .
  * Agnes  _et al._ (2023b) P. Agnes _et al._ (DarkSide), Phys. Rev. Lett. 130, 101002 (2023b), arXiv:2207.11968 [hep-ex] .
  * Abgrall  _et al._ (2021) N. Abgrall _et al._ (LEGEND), (2021), arXiv:2107.11462 [physics.ins-det] .
  * Albert  _et al._ (2018) J. B. Albert _et al._ (nEXO), Phys. Rev. C 97, 065503 (2018), arXiv:1710.05075 [nucl-ex] .
  * Arnaud  _et al._ (2020) Q. Arnaud _et al._ (EDELWEISS), Phys. Rev. Lett. 125, 141301 (2020), arXiv:2003.01046 [astro-ph.GA] .
  * Edzards (2020) F. Edzards (LEGEND), J. Phys. Conf. Ser. 1690, 012180 (2020).
  * Chen  _et al._ (2017) X. Chen, C. Fu, J. Galan, K. Giboni, F. Giuliani, L. Gu, K. Han, X. Ji, H. Lin, J. Liu, _et al._ , Science China Physics, Mechanics & Astronomy 60, 1 (2017).
  * Gomez-Cadenas (2019) J. Gomez-Cadenas, arXiv preprint arXiv:1906.01743 (2019).
  * Vahsen  _et al._ (2020) S. E. Vahsen _et al._ , (2020), arXiv:2008.12587 [physics.ins-det] .
  * An  _et al._ (2013d) H. An, M. Pospelov, and J. Pradler, Physics Letters B 725, 190 (2013d).
  * An  _et al._ (2020) H. An, M. Pospelov, J. Pradler, and A. Ritz, Phys. Rev. D 102, 115022 (2020).
  * Nguyen  _et al._ (2021) D. V. Nguyen, D. Sarnaaik, K. K. Boddy, E. O. Nadler, and V. Gluscevic, Physical Review D 104, 103521 (2021).
  * Lasenby (2020) R. Lasenby, Journal of Cosmology and Astroparticle Physics 2020, 034 (2020).
  * Chen  _et al._ (2021) A. Chen, D. Huterer, S. Lee, A. Ferté, N. Weaverdyck, O. Alves, C. D. Leonard, N. MacCrann, M. Raveri, A. Porredon, _et al._ , Physical Review D 103, 123528 (2021).
  * Berlin  _et al._ (2023) A. Berlin, G. Krnjaic, and E. Pinetti, (2023), arXiv:2311.00032 [hep-ph] .
  * Hernquist (1990) L. Hernquist, Astrophys. J. 356, 359 (1990).
  * Bethe and Salpeter (1957) H. A. Bethe and E. E. Salpeter,  _Quantum Mechanics of One- and Two-Electron Atoms_ (1957).
  * Mertig  _et al._ (1991) R. Mertig, M. Bohm, and A. Denner, Comput. Phys. Commun. 64, 345 (1991).
  * Shtabovenko  _et al._ (2016) V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun. 207, 432 (2016), arXiv:1601.01167 [hep-ph] .
  * Shtabovenko  _et al._ (2020) V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun. 256, 107478 (2020), arXiv:2001.04407 [hep-ph] .