p(\widetilde{t}+u\hslash,\bm{s}_{3})\,d\bm{s}_{3}du\right]^{\frac{1}{2}}$
$\displaystyle\stackrel{{\scriptstyle\text{(ii)}}}{{\leq}}C_{1}|v|\left\\{\int_{\mathcal{S}}\left(x-\widetilde{t}\right)^{2}K_{T}^{2}\left(\frac{x-\widetilde{t}}{h}\right)K_{S}^{2}\left(\frac{\bm{z}-\bm{s}_{3}}{b}\right)p_{S}(\bm{s}_{3})\,d\bm{s}_{3}\right\\}^{\frac{1}{2}}\left[\int_{\mathcal{S}(\widetilde{t})}\frac{1}{p(\widetilde{t},\bm{s}_{3})^{2}}\,d\bm{s}_{3}\right]^{\frac{1}{2}}$
$\displaystyle\stackrel{{\scriptstyle\text{(iii)}}}{{\leq}}C_{2}|v|\left\\{\int_{\mathcal{S}}\left(x-\widetilde{t}\right)^{2}K_{T}^{2}\left(\frac{x-\widetilde{t}}{h}\right)K_{S}^{2}\left(\frac{\bm{z}-\bm{s}_{3}}{b}\right)p_{S}(\bm{s}_{3})\,d\bm{s}_{3}\right\\}^{\frac{1}{2}},$

where $C_{1},C_{2}>0$ are some absolute constants under our Assumption A4. In
addition, the inequality (i) follows from Cauchy-Schwarz inequality, the
inequality (ii) utilizes the facts that $\bar{K}_{T},p(t,\bm{s})$ is bounded
and $p_{T}(\widetilde{t})$ is uniformly bounded away from 0 in
$\mathcal{T}^{\prime}$, and (iii) leverages the fact that the support
$\mathcal{S}(t)$ of the conditional density $p(\bm{s}|t)$ is compact and
$p(t,\bm{s})$ is uniformly bounded away from 0 within $\mathcal{S}(t)$ for any
$t\in\mathcal{T}^{\prime}$. Now, since the supports of $K_{T},K_{S}$ are
compact (or $K_{T},K_{S}$ are square integrable) under Assumptions A6, we know
that
$\int_{\mathcal{S}}\left(x-\widetilde{t}\right)^{2}K_{T}^{2}\left(\frac{x-\widetilde{t}}{h}\right)K_{S}^{2}\left(\frac{\bm{z}-\bm{s}_{3}}{b}\right)p_{S}(\bm{s}_{3})\,d\bm{s}_{3}$
is again bounded, which implies that

$\displaystyle\left|\mathbb{E}_{\bm{U}_{i_{3}}}\left[\frac{\bm{e}_{2}^{T}\bm{M}_{q}^{-1}\bm{\Psi}_{\widetilde{t},\bm{S}_{i_{3}}}(v,x,\bm{z})}{p(\widetilde{t},\bm{S}_{i_{3}})\cdot
p_{T}(\widetilde{t})}\cdot\frac{1}{\hslash}\bar{K}_{T}\left(\frac{\widetilde{t}-T_{i_{3}}}{\hslash}\right)\right]\right|$
$\displaystyle\leq
C_{2}|v|\left\\{\int_{\mathcal{S}}\left[\bm{e}_{2}^{T}\bm{M}_{q}^{-1}\bm{\psi}_{\widetilde{t},\bm{s}_{3}}(x,\bm{z})\right]^{2}p_{S}(\bm{s}_{3})\,d\bm{s}_{3}\right\\}^{\frac{1}{2}}$
$\displaystyle\leq C_{2}^{\prime}|v|$

for some absolute constant $C_{2}^{\prime}>0$. Finally, plugging this result
into our original display (44) yields that

$\displaystyle\left|f_{t_{1}}(v,x,\bm{z})-f_{t_{2}}(v,x,\bm{z})\right|$
$\displaystyle\leq C_{3}\cdot|t_{1}-t_{2}|\cdot|v|$

for some absolute constant $C_{3}>0$. Since
$\mathbb{E}_{Q}|Y|^{2}=\mathbb{E}_{Q}\left[\left(\mu(T,\bm{S})+\epsilon\right)^{2}\right]\leq
2\mathbb{E}_{Q}\left[\mu(T,\bm{S})^{2}\right]+2\sigma^{2}<\infty$ for any
probability measure $Q$ under Assumption A3 and the diameter of
$\mathcal{T}^{\prime}$ is a finite constant, we conclude from Example 19.7 in
van der Vaart (1998) that

$\sup_{Q}N_{[]}\left(\widetilde{\mathcal{F}},L_{2}(Q),\epsilon\right)\leq\frac{C_{3}^{\prime}}{\epsilon}$

for some constant $C_{3}^{\prime}>0$. Additionally, since any $2\epsilon$
bracket $[f_{t_{1}},f_{t_{2}}]$ of $\mathcal{F}$ is contained in a ball of
radius $\epsilon$ centered at $\frac{f_{t_{1}}+f_{t_{2}}}{2}$, we know that

$\sup_{Q}N\left(\widetilde{\mathcal{F}},L_{2}(Q),\epsilon\right)\leq\sup_{Q}N_{[]}\left(\widetilde{\mathcal{F}},L_{2}(Q),2\epsilon\right)\leq\frac{C_{4}}{\epsilon}$

for some constant $C_{4}>0$. Thus, $\widetilde{\mathcal{F}}$ is a bounded VC-
type class of functions. Finally, the above calculations tell us that for any
$f_{t}\in\widetilde{\mathcal{F}}$,

$\displaystyle\left|f_{t}(v,x,\bm{z})\right|$
$\displaystyle=\left|\mathbb{E}_{\bm{U}_{i_{2}}}\left[\int_{T_{i_{2}}}^{t}\mathbb{E}_{\bm{U}_{i_{3}}}\left[\frac{\bm{e}_{2}^{T}\bm{M}_{q}^{-1}\bm{\Psi}_{\widetilde{t},\bm{S}_{i_{3}}}(v,x,\bm{z})}{p(\widetilde{t},\bm{S}_{i_{3}})\cdot
p_{T}(\widetilde{t})}\cdot\frac{1}{\hslash}\bar{K}_{T}\left(\frac{\widetilde{t}-T_{i_{3}}}{\hslash}\right)\right]d\widetilde{t}\right]\right|$
$\displaystyle\leq
C_{3}\cdot\mathbb{E}_{\bm{U}_{i_{2}}}\left[|t-T_{i_{2}}|\cdot|v|\right]$
$\displaystyle\leq C_{5}\cdot|v|,$

where $C_{5}>0$ is some absolute constant. Thus, an envelope function of
$\widetilde{\mathcal{F}}$ can be given by $(v,x,\bm{z})\mapsto
F_{1}(v,x,\bm{z})=C_{5}\cdot|v|$. The similar argument applies to the case
where $q>2$ in the local polynomial regression for estimating
$\frac{\partial}{\partial t}\mu(t,\bm{s})$, because $\mathcal{K}_{q,d}$ in
Assumption A6(b) is a bounded class of functions. The result thus follows for
$\widetilde{\mathcal{F}}$.

As for $\widetilde{\mathcal{F}}_{\theta}$, we note that when $q=2$,

$\displaystyle g_{t}(v,x,\bm{z})$
$\displaystyle=\sqrt{h^{3}b^{d}}\cdot\bar{\varphi}_{t}(v,x,\bm{z})$
$\displaystyle=\mathbb{E}_{(T_{i_{3}},\bm{S}_{i_{3}})}\left[\frac{h\cdot\bm{e}_{2}^{T}\bm{M}_{q}^{-1}\bm{\Psi}_{t,\bm{S}_{i_{3}}}\left(Y,T,\bm{S}\right)}{p(t,\bm{S}_{i_{3}})\cdot
p_{T}(t)}\cdot\frac{1}{\hslash}\bar{K}_{T}\left(\frac{t-T_{i_{3}}}{\hslash}\right)\right]$
$\displaystyle=\mathbb{E}_{(T_{i_{3}},\bm{S}_{i_{3}})}\left[\frac{C_{K}v\left(x-t\right)K_{T}\left(\frac{x-t}{h}\right)K_{S}\left(\frac{\bm{z}-\bm{S}_{i_{3}}}{b}\right)}{p(t,\bm{S}_{i_{3}})\cdot
p_{T}(t)}\cdot\frac{1}{\hslash}\bar{K}_{T}\left(\frac{t-T_{i_{3}}}{\hslash}\right)\right]$

for each $g_{t}\in\widetilde{\mathcal{F}}_{\theta}$. Furthermore, by
Assumptions A3 and A6, we know that the function inside the above expectation
is Lipschitz with respect to $t\in\mathcal{T}^{\prime}$, because
$\frac{1}{p(t,\bm{S}_{i_{3}})\cdot p_{T}(t)}$ is differentiable with a bounded
partial derivative with respect to $t$ and the kernel functions are Lipschitz.
Hence, $\left|g_{t_{1}}(v,x,\bm{z})-g_{t_{2}}(v,x,\bm{z})\right|\leq C_{6}|v|$
for some constant $C_{6}>0$. Again, the analogous argument applies to the case
where $q>2$ in the local polynomial regression for estimating
$\frac{\partial}{\partial t}\mu(t,\bm{s})$. The result follows for
$\widetilde{\mathcal{F}}_{\theta}$. ∎

######  Lemma 10 (Corollary 2.2 in Chernozhukov et al. 2014).

Let $\mathcal{G}$ be a collection of functions that is pointwise measurable
and of VC-type (see Assumption A6) with an envelop function $\widetilde{G}$
and constants $A\geq e$ and $\upsilon\geq 1$. Suppose also that for some
$\widetilde{A}\geq\widetilde{\sigma}>0$ and $q^{\prime}\in[4,\infty]$, we have
$\sup_{\widetilde{g}\in\mathcal{G}}\mathbb{E}\left[\widetilde{g}(\bm{U})^{k}\right]\leq\widetilde{\sigma}^{2}\cdot\widetilde{A}^{k-2}$
for $k=2,3,4$ and
$\left|\left|G\right|\right|_{P,q^{\prime}}\leq\widetilde{A}$. Let
$\mathbb{B}$ be a centered Gaussian process defined on $\mathcal{G}$ with
covariance function

$\mathrm{Cov}\left(\mathbb{B}(\widetilde{g}_{1}),\mathbb{B}(\widetilde{g}_{2})\right)=\mathbb{E}\left[\widetilde{g}_{1}(\bm{U})\cdot\widetilde{g}_{2}(\bm{U})\right],$

where $\widetilde{g}_{1},\widetilde{g}_{2}\in\mathcal{G}$ and
$\bm{U}=(Y,T,\bm{S})$. Then, for every $\widetilde{\gamma}\in(0,1)$ and
sufficiently large, there exists a random variable
$\widetilde{B}\stackrel{{\scriptstyle
d}}{{=}}\sup_{\widetilde{g}\in\mathcal{G}}\mathbb{B}(\widetilde{g})$ such that

$\mbox{$\mathrm{P}$}\left(\left|\sup_{\widetilde{g}\in\mathcal{G}}\left|\mathbb{G}_{n}(\widetilde{g})\right|-\widetilde{B}\right|>\frac{C_{1}\cdot\widetilde{A}^{\frac{1}{3}}\widetilde{\sigma}^{\frac{2}{3}}\log^{\frac{2}{3}}n}{\widetilde{\gamma}^{\frac{1}{3}}n^{\frac{1}{6}}}\right)\leq
C_{2}\cdot\widetilde{\gamma},$

where $C_{1},C_{2}>0$ are two constants that only depend on $q^{\prime}$.
Here, $\widetilde{B}_{1}\stackrel{{\scriptstyle d}}{{=}}$ $\widetilde{B}_{2}$
for two random variables $\widetilde{B}_{1},\widetilde{B}_{2}$ means that they
have the same distribution.

######  Lemma 11 (Lemma 2.3 in Chernozhukov et al. 2014).

Under the same setup for $\mathcal{G}$ as in Lemma 10, we assume that there
exist constants $\underline{\sigma},\bar{\sigma}>0$ such that
$\underline{\sigma}^{2}\leq\mathbb{E}\left[\widetilde{g}^{2}(\bm{U})\right]\leq\bar{\sigma}^{2}$
for all $\widetilde{g}\in\mathcal{G}$. Moreover, suppose that there exist
constants $r_{1},r_{2}>0$ such that

$\mbox{$\mathrm{P}$}\left(\left|\sup_{\widetilde{g}\in\mathcal{G}}|\mathbb{G}_{n}(f)|-\sup_{\widetilde{g}\in\mathcal{G}}|\mathbb{B}(f)|\right|>r_{1}\right)\leq
r_{2}.$

Then,

$\displaystyle\sup_{u\geq
0}\left|\mbox{$\mathrm{P}$}\left(\sup_{\widetilde{g}\in\mathcal{G}}|\mathbb{G}_{n}(f)|\leq
u\right)-\mbox{$\mathrm{P}$}\left(\sup_{\widetilde{g}\in\mathcal{G}}|\mathbb{B}(f)|\geq
u\right)\right|$ $\displaystyle\leq
C_{\sigma}r_{1}\left\\{\mathbb{E}\left[\sup_{\widetilde{g}\in\mathcal{G}}|\mathbb{B}(f)|\right]+\sqrt{\max\left\\{1,\,\log\left(\frac{\underline{\sigma}}{r_{1}}\right)\right\\}}\right\\}+r_{2},$

where $C_{\sigma}>0$ is a constant depending only on $\underline{\sigma}$ and
$\bar{\sigma}$.

######  Lemma 12.

Let $d\geq 1$. If $h\asymp b\asymp n^{-\frac{1}{\gamma}}$ and $\hslash\asymp
n^{-\frac{1}{\varpi}}$ for some $\gamma\geq\varpi>0$ such that
$\frac{nh^{d+5}}{\log n}\to c_{1}$ and $\frac{n\hslash^{5}}{\log n}\to c_{2}$
for some finite number $c_{1},c_{2}\geq 0$ and $\frac{n\hslash^{2}}{\log
n},\frac{\hslash}{h^{d+3}\log n},\hslash
n^{\frac{1}{4}},\frac{\hslash^{2}}{h^{d+3}}\to\infty$ as $n\to\infty$, then

$\max\left\\{\sqrt{nh^{d+7}},\sqrt{\frac{\log
n}{n\hslash^{2}}},\sqrt{\frac{h^{d+3}\log
n}{\hslash}},\sqrt{\frac{h^{d+3}}{\hslash^{2}}}\right\\}\lesssim\frac{\log^{\frac{2}{3}}n}{\left(nh^{d+3}\right)^{\frac{1}{6}}}$
(45)

when $n$ is sufficiently large.

###### Proof of Lemma 12.

We consider controlling the four quantities on the left hand side of (45) by
the right hand side separately.

$\bullet$ Quantity I: Note that
$\sqrt{nh^{d+7}}\lesssim\frac{\log^{\frac{2}{3}}n}{\left(nh^{d+3}\right)^{\frac{1}{6}}}$
is equivalent to $nh^{d+6}\lesssim\log n$. Under our conditions, we know that
the slowest rate of convergence for $h$ is $O\left(n^{-\frac{1}{d+5}}\right)$
up to some logarithmic factors of $n$. Thus,
$nh^{d+6}=O\left(n^{-\frac{1}{d+5}}\right)$ will be dominated by $O(\log n)$.

$\bullet$ Quantity II: Note that $\sqrt{\frac{\log
n}{n\hslash^{2}}}\lesssim\frac{\log^{\frac{2}{3}}n}{\left(nh^{d+3}\right)^{\frac{1}{6}}}$
is equivalent to $\frac{h^{d+3}}{n^{2}\hslash^{6}}\lesssim\log n$. Again,
under our conditions, the slowest rate of convergence for $h$ is
$O\left(n^{-\frac{1}{d+5}}\right)$ up to some logarithmic factors of $n$.
Furthermore, given that $\hslash n^{\frac{1}{4}}\to\infty$, the fastest rate
of convergence for $\hslash$ should be smaller than the order
$O\left(n^{-\frac{1}{3}}\right)$. Thus, under these rates,
$\frac{h^{d+3}}{n^{2}\hslash^{6}}$ is dominated by
$O\left(n^{-\frac{d+3}{d+5}}\right)$ and thus by $O\left(\log n\right)$.

$\bullet$ Quantity III: Note that $\sqrt{\frac{h^{d+3}\log
n}{\hslash}}\lesssim\frac{\log^{\frac{2}{3}}n}{\left(nh^{d+3}\right)^{\frac{1}{6}}}$
is equivalent to $\frac{nh^{4(d+6)}}{\hslash^{3}}\lesssim\log n$. Again, under
our conditions, the slowest rates of convergence for $h$ is
$O\left(n^{-\frac{1}{d+5}}\right)$, and the fastest rate of convergence for
$\hslash$ is smaller than the order $O\left(n^{-\frac{1}{3}}\right)$ up to
some logarithmic factors of $n$. Under these rates,
$\frac{nh^{4(d+6)}}{\hslash^{3}}=O\left(n^{-\frac{2d+2}{d+5}}\right)$ will be
dominated by $O\left(\log n\right)$.

$\bullet$ Quantity IV: Note that
$\sqrt{\frac{h^{d+3}}{\hslash^{2}}}\lesssim\frac{\log^{\frac{2}{3}}n}{\left(nh^{d+3}\right)^{\frac{1}{6}}}$
is equivalent to $n^{\frac{1}{2}}h^{2d+6}\hslash^{-3}\lesssim\log^{2}n$.
Again, under our conditions, the slowest rate of convergence for $h$ is
$O\left(n^{-\frac{1}{d+5}}\right)$ up to some logarithmic factors of $n$.
Furthermore, given that $\hslash n^{\frac{1}{4}}\to\infty$, the fastest rate
of convergence for $\hslash$ should be smaller than the order
$O\left(n^{-\frac{1}{4}}\right)$. Thus,
$n^{\frac{1}{2}}h^{2d+6}\hslash^{-3}=O\left(n^{\frac{1-3d}{4(d+5)}}\right)$,
which is dominated by $O\left(\log^{2}n\right)$ when $d\geq 1$.

In summary, the result follows from combining the above four cases. ∎

######  Lemma 13.

Let $\mathcal{T}^{\prime}\subset\mathcal{T}$ be a compact set so that
$p_{T}(t)$ is uniformly bounded away from 0 within $\mathcal{T}^{\prime}$.
Suppose that Assumptions A3, A4, A5, and A6 hold. Assume that
$\mathrm{Var}(\epsilon)=\sigma^{2}>0$ under the general confounding model (2).
Then, when $h,b,\hslash$ are sufficiently small as $n\to\infty$, there exist
constants $\underline{\sigma},\bar{\sigma}>0$ such that
$\underline{\sigma}^{2}\leq\mathbb{E}\left[\varphi_{t}(Y,T,\bm{S})^{2}\right]\leq\bar{\sigma}^{2}$
for any $t\in\mathcal{T}^{\prime}$, where $\varphi_{t}$ is defined in (17).
Here, all the constants $\underline{\sigma},\bar{\sigma}>0$ are independent of
$h,b,\hslash$ and $n$. Furthermore, the same upper and lower bounds apply to
$\mathrm{Var}\left[\varphi_{t}(Y,T,\bm{S})\right]$. Finally, the same result
holds true when we replace $\varphi_{t}$ with $\bar{\varphi}_{t}$ for any
$t\in\mathcal{T}^{\prime}$.

###### Proof of Lemma 13.

We first compute the upper bound on
$\mathbb{E}\left[\varphi_{t}(Y,T,\bm{S})^{2}\right]$. Recall from (17) that

$\displaystyle\mathbb{E}\left[\varphi_{t}(Y,T,\bm{S})^{2}\right]$
$\displaystyle=\mathbb{E}\left\\{\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\mathbb{E}_{(T_{i_{3}},\bm{S}_{i_{3}})}\left[\frac{\bm{e}_{2}^{T}\bm{M}_{q}^{-1}\bm{\Psi}_{\widetilde{t},\bm{S}_{i_{3}}}(Y,T,\bm{S})}{\sqrt{hb^{d}}\cdot
p(\widetilde{t},\bm{S}_{i_{3}})\cdot
p_{T}(\widetilde{t})}\cdot\frac{1}{\hslash}\bar{K}_{T}\left(\frac{\widetilde{t}-T_{i_{3}}}{\hslash}\right)\right]d\widetilde{t}\right)\right]^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\frac{1}{hb^{d}}\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathcal{T}\times\mathcal{S}}\frac{\bm{e}_{2}^{T}\bm{M}_{q}^{-1}\bm{\psi}_{\widetilde{t},\bm{s}_{3}}(T,\bm{S})\cdot
Y}{p(\widetilde{t},\bm{s}_{3})\cdot
p_{T}(\widetilde{t})}\cdot\frac{1}{\hslash}\bar{K}_{T}\left(\frac{\widetilde{t}-t_{3}}{\hslash}\right)p(t_{3},\bm{s}_{3})\,dt_{3}d\bm{s}_{3}d\widetilde{t}\right)\right]^{2}\right\\}.$

By the definition of $\bm{M}_{q}$ in (14) when $q=2$, we know that
$\bm{e}_{2}^{T}\bm{M}_{q}^{-1}=\left(0,C_{K},0,\underbrace{0,...,0}_{d}\right)$
for some constant $C_{K}\neq 0$ that only depends on the kernels
$K_{T},K_{S}$. Then, we proceed the above display as:

$\displaystyle\mathbb{E}\left[\varphi_{t}(Y,T,\bm{S})^{2}\right]$
$\displaystyle=\frac{C_{K}^{2}}{hb^{d}}\cdot\mathbb{E}\left\\{\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\frac{Y\left(\frac{T-\widetilde{t}}{h}\right)K_{T}\left(\frac{T-\widetilde{t}}{h}\right)K_{S}\left(\frac{\bm{S}-\bm{s}_{3}}{b}\right)}{p(\widetilde{t},\bm{s}_{3})\cdot
p_{T}(\widetilde{t})}\cdot\bar{K}_{T}\left(u\right)\cdot
p(\widetilde{t}+u\hslash,\bm{s}_{3})\,dud\bm{s}_{3}d\widetilde{t}\right)\right]^{2}\right\\}$
$\displaystyle\leq\frac{C_{K}^{2}}{hb^{d}}\cdot\mathbb{E}\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\frac{Y^{2}\left(\frac{T-\widetilde{t}}{h}\right)^{2}K_{T}^{2}\left(\frac{T-\widetilde{t}}{h}\right)K_{S}^{2}\left(\frac{\bm{S}-\bm{s}_{3}}{b}\right)}{p(\widetilde{t},\bm{s}_{3})^{2}\cdot
p_{T}(\widetilde{t})^{2}}\cdot\bar{K}_{T}^{2}\left(u\right)\cdot
p(\widetilde{t}+u\hslash,\bm{s}_{3})^{2}\,dud\bm{s}_{3}d\widetilde{t}\right)\right]$
$\displaystyle\leq\frac{2C_{K}^{2}}{hb^{d}}\cdot\int_{\mathcal{T}\times\mathcal{S}}\int_{\mathcal{T}}\int_{t_{2}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\frac{\left[\mu(t_{1},\bm{s}_{1})^{2}+\sigma^{2}\right]\left(\frac{t_{1}-\widetilde{t}}{h}\right)^{2}K_{T}^{2}\left(\frac{t_{1}-\widetilde{t}}{h}\right)K_{S}^{2}\left(\frac{\bm{s}_{1}-\bm{s}_{3}}{b}\right)}{p(\widetilde{t},\bm{s}_{3})^{2}\cdot
p_{T}(\widetilde{t})^{2}}$
$\displaystyle\quad\quad\times\bar{K}_{T}^{2}\left(u\right)\cdot
p(\widetilde{t}+u\hslash,\bm{s}_{3})^{2}\,dud\bm{s}_{3}d\widetilde{t}\cdot
p_{T}(t_{2})\,dt_{2}\cdot p(t_{1},\bm{s}_{1})\,dt_{1}d\bm{s}_{1}$
$\displaystyle=2C_{K}^{2}\int_{\mathcal{T}}\int_{t_{2}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\int_{\mathbb{R}\times\mathbb{R}^{d}}\frac{\left[\mu(\widetilde{t}+xh,\bm{s}_{3}+b\bm{z})^{2}+\sigma^{2}\right]x^{2}K_{T}^{2}\left(x\right)K_{S}^{2}\left(\bm{z}\right)}{p(\widetilde{t},\bm{s}_{3})^{2}\cdot
p_{T}(\widetilde{t})^{2}}$
$\displaystyle\quad\quad\times\bar{K}_{T}^{2}\left(u\right)\cdot
p(\widetilde{t}+u\hslash,\bm{s}_{3})^{2}p(\widetilde{t}+xh,\bm{s}_{3}+b\bm{z})\,dxd\bm{z}\,dud\bm{s}_{3}d\widetilde{t}\cdot
p_{T}(t_{2})\,dt_{2}$ $\displaystyle<\infty,$

because all the terms inside the integrand are bounded under Assumptions A3,
A4, and A6. Hence, $\mathbb{E}\left[\varphi_{t}(Y,T,\bm{S})^{2}\right]$ can be
upper bounded by a constant $\bar{\sigma}^{2}>0$ that is independent of the
bandwidths $h,b,\hslash$.

For the lower bound on $\mathbb{E}\left[\varphi_{t}(Y,T,\bm{S})^{2}\right]$,
we compute that

$\displaystyle\mathbb{E}\left[\varphi_{t}(Y,T,\bm{S})^{2}\right]$
$\displaystyle=\frac{C_{K}^{2}}{hb^{d}}\cdot\mathbb{E}\left\\{\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\frac{\left[\mu(T,\bm{S})+\epsilon\right]\left(\frac{T-\widetilde{t}}{h}\right)K_{T}\left(\frac{T-\widetilde{t}}{h}\right)K_{S}\left(\frac{\bm{S}-\bm{s}_{3}}{b}\right)}{p(\widetilde{t},\bm{s}_{3})\cdot
p_{T}(\widetilde{t})}\cdot\bar{K}_{T}\left(u\right)\cdot
p(\widetilde{t}+u\hslash,\bm{s}_{3})\,dud\bm{s}_{3}d\widetilde{t}\right)\right]^{2}\right\\}$
$\displaystyle=\frac{C_{K}^{2}}{hb^{d}}\cdot\mathbb{E}\left\\{\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\frac{\mu(T,\bm{S})\left(\frac{T-\widetilde{t}}{h}\right)K_{T}\left(\frac{T-\widetilde{t}}{h}\right)K_{S}\left(\frac{\bm{S}-\bm{s}_{3}}{b}\right)}{p(\widetilde{t},\bm{s}_{3})\cdot
p_{T}(\widetilde{t})}\cdot\bar{K}_{T}\left(u\right)\cdot
p(\widetilde{t}+u\hslash,\bm{s}_{3})\,dud\bm{s}_{3}d\widetilde{t}\right)\right]^{2}\right\\}$
$\displaystyle\quad+\frac{C_{K}^{2}\sigma^{2}}{hb^{d}}\cdot\mathbb{E}\left\\{\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\frac{\left(\frac{T-\widetilde{t}}{h}\right)K_{T}\left(\frac{T-\widetilde{t}}{h}\right)K_{S}\left(\frac{\bm{S}-\bm{s}_{3}}{b}\right)}{p(\widetilde{t},\bm{s}_{3})\cdot
p_{T}(\widetilde{t})}\cdot\bar{K}_{T}\left(u\right)\cdot
p(\widetilde{t}+u\hslash,\bm{s}_{3})\,dud\bm{s}_{3}d\widetilde{t}\right)\right]^{2}\right\\}$
$\displaystyle=\frac{C_{K}^{2}}{hb^{d}}\int_{\mathcal{T}\times\mathcal{S}}\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\frac{\mu(t_{1},\bm{s}_{1})\left(\frac{t_{1}-\widetilde{t}}{h}\right)K_{T}\left(\frac{t_{1}-\widetilde{t}}{h}\right)K_{S}\left(\frac{\bm{s}_{1}-\bm{s}_{3}}{b}\right)}{p(\widetilde{t},\bm{s}_{3})\cdot
p_{T}(\widetilde{t})}\bar{K}_{T}\left(u\right)p(\widetilde{t}+u\hslash,\bm{s}_{3})dud\bm{s}_{3}d\widetilde{t}\right)\right]^{2}p(t_{1},\bm{s}_{1})dt_{1}d\bm{s}_{1}$
$\displaystyle\quad+\frac{C_{K}^{2}\sigma^{2}}{hb^{d}}\int_{\mathcal{T}\times\mathcal{S}}\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\frac{\left(\frac{t_{1}-\widetilde{t}}{h}\right)K_{T}\left(\frac{t_{1}-\widetilde{t}}{h}\right)K_{S}\left(\frac{\bm{s}_{1}-\bm{s}_{3}}{b}\right)}{p(\widetilde{t},\bm{s}_{3})\cdot
p_{T}(\widetilde{t})}\cdot\bar{K}_{T}\left(u\right)\cdot
p(\widetilde{t}+u\hslash,\bm{s}_{3})\,dud\bm{s}_{3}d\widetilde{t}\right)\right]^{2}p(t_{1},\bm{s}_{1})dt_{1}d\bm{s}_{1}$
$\displaystyle=\frac{C_{K}^{2}}{hb^{d}}\int_{\mathcal{T}\times\mathcal{S}}\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\frac{\mu(t_{1},\bm{s}_{1})\left(\frac{t_{1}-\widetilde{t}}{h}\right)K_{T}\left(\frac{t_{1}-\widetilde{t}}{h}\right)K_{S}\left(\frac{\bm{s}_{1}-\bm{s}_{3}}{b}\right)}{p(\widetilde{t},\bm{s}_{3})\cdot
p_{T}(\widetilde{t})}\bar{K}_{T}\left(u\right)p(\widetilde{t}+u\hslash,\bm{s}_{3})\sqrt{p(t_{1},\bm{s}_{1})}\,dud\bm{s}_{3}d\widetilde{t}\right)\right]^{2}dt_{1}d\bm{s}_{1}$
$\displaystyle\quad+\frac{C_{K}^{2}\sigma^{2}}{hb^{d}}\int_{\mathcal{T}\times\mathcal{S}}\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\frac{\left(\frac{t_{1}-\widetilde{t}}{h}\right)K_{T}\left(\frac{t_{1}-\widetilde{t}}{h}\right)K_{S}\left(\frac{\bm{s}_{1}-\bm{s}_{3}}{b}\right)}{p(\widetilde{t},\bm{s}_{3})\cdot
p_{T}(\widetilde{t})}\bar{K}_{T}\left(u\right)p(\widetilde{t}+u\hslash,\bm{s}_{3})\sqrt{p(t_{1},\bm{s}_{1})}\,dud\bm{s}_{3}d\widetilde{t}\right)\right]^{2}dt_{1}d\bm{s}_{1}$
$\displaystyle\stackrel{{\scriptstyle\text{(i)}}}{{\geq}}C_{K}^{2}\sigma^{2}\int_{\mathbb{R}\times\mathbb{R}^{d}}\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\frac{xK_{T}\left(x\right)\cdot
K_{S}\left(\bm{z}\right)}{p(\widetilde{t},\bm{s}_{3})\cdot
p_{T}(\widetilde{t})}\cdot\bar{K}_{T}\left(u\right)p(\widetilde{t}+u\hslash,\bm{s}_{3})\sqrt{p(\widetilde{t}+hx,\bm{s}_{3}+b\bm{z})}\,dud\bm{s}_{3}d\widetilde{t}\right)\right]^{2}dxd\bm{z},$

where (i) applies the changes of variables
$x=\frac{t_{1}-\widetilde{t}}{h},\bm{z}=\frac{\bm{s}_{1}-\bm{s}_{3}}{b}$. Now,
since the variance $\sigma^{2}=\mathbb{E}(\epsilon)>0$, we have that for any
$t\in\mathcal{T}^{\prime}$,

$\displaystyle\begin{split}&\mathbb{E}\left[\varphi_{t}(Y,T,\bm{S})^{2}\right]\\\
&\geq
C_{K}^{2}\sigma^{2}\int_{\mathbb{R}\times\mathbb{R}^{d}}x^{2}K_{T}^{2}\left(x\right)K_{S}^{2}\left(\bm{z}\right)\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\frac{\bar{K}_{T}\left(u\right)p(\widetilde{t}+u\hslash,\bm{s}_{3})\sqrt{p(\widetilde{t}+hx,\bm{s}_{3}+b\bm{z})}}{p(\widetilde{t},\bm{s}_{3})\cdot
p_{T}(\widetilde{t})}\,dud\bm{s}_{3}d\widetilde{t}\right)\right]^{2}dxd\bm{z}\\\
&=\widetilde{C}_{K}^{2}\sigma^{2}\int_{\mathbb{R}^{d}}K_{S}^{2}\left(\bm{z}\right)\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathbb{R}\times\mathcal{S}}\frac{\bar{K}_{T}\left(u\right)p(\widetilde{t}+u\hslash,\bm{s}_{3})\sqrt{p(\widetilde{t},\bm{s}_{3})}}{p(\widetilde{t},\bm{s}_{3})\cdot
p_{T}(\widetilde{t})}\,dud\bm{s}_{3}d\widetilde{t}\right)\right]^{2}d\bm{z}+O\left(\max\\{h,b\\}\right)\\\
&=\widetilde{C}_{K}^{2}\sigma^{2}\left[\mathbb{E}_{T_{i_{2}}}\left(\int_{T_{i_{2}}}^{t}\int_{\mathcal{S}(\widetilde{t})}\frac{\sqrt{p(\widetilde{t},\bm{s}_{3})}}{p_{T}(\widetilde{t})}\,d\bm{s}_{3}d\widetilde{t}\right)\right]^{2}+O\left(\max\\{h,b\\}+\hslash^{2}\right)\end{split}$
(46)

where $\widetilde{C}_{K}^{2}>0$ is again some constant that only depends on
the kernels $K_{T},K_{S},\bar{K}_{T}$. By Assumption A4, $p(t,\bm{s})$ is
upper bounded and also uniformly bounded away from 0 within it support
$\mathcal{E}$. Thus, there exists a constant $C^{\prime}>0$ such that

$\int_{\mathcal{S}(\widetilde{t})}\frac{\sqrt{p(\widetilde{t},\bm{s}_{3})}}{p_{T}(\widetilde{t})}\,d\bm{s}_{3}\geq
C^{\prime}$ (47)

for some $\widetilde{t}\in\mathcal{T}^{\prime}$. Let $A_{T}$ be an subset of
$\mathcal{T}$ on which (47) holds. By the non-degeneracy of $p_{T}$ under
Assumption A4, we can proceed (46) as:

$\displaystyle\mathbb{E}\left[\varphi_{t}(Y,T,\bm{S})^{2}\right]$
$\displaystyle\geq\widetilde{C}_{K}^{2}\sigma^{2}C^{\prime
2}\left\\{\mathbb{E}_{T_{i_{2}}}\left[(t-T_{i_{2}})\cdot\mathbbm{1}_{A_{T}}\right]\right\\}^{2}+O\left(\max\\{h,b\\}\right)+O(\hslash^{2})>\underline{\sigma}^{2}$

for some constant $\underline{\sigma}^{2}>0$.

The similar argument applies to the case where $q>2$ in the local polynomial
regression for estimating $\frac{\partial}{\partial t}\mu(t,\bm{s})$, because
$\mathcal{K}_{q,d}$ in Assumption A6(b) is a bounded class of functions.
Furthermore, $\mathbb{E}\left[\varphi_{t}(Y,T,\bm{S})\right]\to 0$ as $h,b\to
0$ with $n\to\infty$. The result thus follows for the influence function
$\varphi_{t}$ of $\widehat{m}_{\theta}(t)$.

As for the influence function $\bar{\varphi}_{t}$ of
$\widehat{\theta}_{C}(t)$, we use the similar arguments to obtain its upper
bound as:

$\displaystyle\mathbb{E}\left[\bar{\varphi}_{t}(Y,T,\bm{S})^{2}\right]$
$\displaystyle=\mathbb{E}\left\\{\left[\mathbb{E}_{(T_{i_{3}},\bm{S}_{i_{3}})}\left(\frac{\bm{e}_{2}^{T}\bm{M}_{q}^{-1}\bm{\Psi}_{t,\bm{S}_{i_{3}}}(Y,T,\bm{S})}{\sqrt{hb^{d}}\cdot
p(t,\bm{S}_{i_{3}})\cdot
p_{T}(t)}\cdot\frac{1}{\hslash}\bar{K}_{T}\left(\frac{t-T_{i_{3}}}{\hslash}\right)\right)\right]^{2}\right\\}$
$\displaystyle=\frac{1}{hb^{d}}\cdot\mathbb{E}\left\\{\left[\int_{\mathcal{T}\times\mathcal{S}}\frac{\bm{e}_{2}^{T}\bm{M}_{q}^{-1}\bm{\psi}_{t,\bm{s}_{3}}(T,\bm{S})\cdot
Y}{p(t,\bm{s}_{3})\cdot
p_{T}(t)}\cdot\frac{1}{\hslash}\bar{K}_{T}\left(\frac{t-t_{3}}{\hslash}\right)p(t_{3},\bm{s}_{3})\,dt_{3}d\bm{s}_{3}\right]^{2}\right\\}$
$\displaystyle=\frac{C_{K}^{2}}{hb^{d}}\cdot\mathbb{E}\left\\{\left[\int_{\mathbb{R}\times\mathcal{S}}\frac{Y\left(\frac{T-t}{h}\right)K_{T}\left(\frac{T-t}{h}\right)K_{S}\left(\frac{\bm{S}-\bm{s}_{3}}{b}\right)}{p(t,\bm{s}_{3})\cdot
p_{T}(t)}\cdot\bar{K}_{T}(u)\cdot
p(t+u\hslash,\bm{s}_{3})\,dud\bm{s}_{3}\right]^{2}\right\\}$
$\displaystyle\leq\frac{C_{K}^{2}}{hb^{d}}\cdot\mathbb{E}\left[\int_{\mathbb{R}\times\mathcal{S}}\frac{Y^{2}\left(\frac{T-t}{h}\right)^{2}K_{T}^{2}\left(\frac{T-t}{h}\right)K_{S}^{2}\left(\frac{\bm{S}-\bm{s}_{3}}{b}\right)}{p(t,\bm{s}_{3})^{2}\cdot
p_{T}(t)^{2}}\cdot\bar{K}_{T}^{2}(u)\cdot
p(t+u\hslash,\bm{s}_{3})^{2}\,dud\bm{s}_{3}\right]$
$\displaystyle\leq\frac{2C_{K}^{2}}{hb^{d}}\int_{\mathcal{T}\times\mathcal{S}}\int_{\mathbb{R}\times\mathcal{S}}\frac{\left[\mu(t_{1},\bm{s}_{1})^{2}+\sigma^{2}\right]\left(\frac{t_{1}-t}{h}\right)^{2}K_{T}^{2}\left(\frac{t_{1}-t}{h}\right)K_{S}^{2}\left(\frac{\bm{s}_{1}-\bm{s}_{3}}{b}\right)}{p(t,\bm{s}_{3})^{2}\cdot
p_{T}(t)^{2}}\cdot\bar{K}_{T}^{2}(u)$ $\displaystyle\quad\times
p(t+u\hslash,\bm{s}_{3})^{2}\cdot
p(t_{1},\bm{s}_{1})\,dud\bm{s}_{3}dt_{1}d\bm{s}_{1}$
$\displaystyle=2C_{K}^{2}\int_{\mathbb{R}\times\mathcal{S}}\int_{\mathbb{R}\times\mathbb{R}^{d}}\frac{\left[\mu(t+hx,\bm{s}_{3}+b\bm{z})^{2}+\sigma^{2}\right]x^{2}K_{T}^{2}\left(x\right)K_{S}^{2}\left(\bm{z}\right)}{p(t,\bm{s}_{3})^{2}\cdot
p_{T}(t)^{2}}\cdot\bar{K}_{T}^{2}(u)$ $\displaystyle\quad\times
p(t+u\hslash,\bm{s}_{3})^{2}\cdot
p(t+hx,\bm{s}_{3}+b\bm{z})\,dxd\bm{z}dud\bm{s}_{3}$ $\displaystyle<\infty$

when $q=2$ under Assumptions A3, A4, and A6. Hence,
$\mathbb{E}\left[\bar{\varphi}_{t}(Y,T,\bm{S})^{2}\right]$ can be upper
bounded by a constant $\bar{\sigma}^{2}>0$ that is independent of the
bandwidths $h,b,\hslash$.

For the lower bound on
$\mathbb{E}\left[\bar{\varphi}_{t}(Y,T,\bm{S})^{2}\right]$, we again compute
that

$\displaystyle\mathbb{E}\left[\bar{\varphi}_{t}(Y,T,\bm{S})^{2}\right]$
$\displaystyle=\frac{C_{K}^{2}}{hb^{d}}\cdot\mathbb{E}\left\\{\left[\int_{\mathbb{R}\times\mathcal{S}}\frac{\left[\mu(T,\bm{S})+\epsilon\right]\left(\frac{T-t}{h}\right)K_{T}\left(\frac{T-t}{h}\right)K_{S}\left(\frac{\bm{S}-\bm{s}_{3}}{b}\right)}{p(t,\bm{s}_{3})\cdot
p_{T}(t)}\cdot\bar{K}_{T}(u)\cdot
p(t+u\hslash,\bm{s}_{3})\,dud\bm{s}_{3}\right]^{2}\right\\}$
$\displaystyle\geq\frac{C_{K}^{2}\sigma^{2}}{hb^{d}}\int_{\mathcal{T}\times\mathcal{S}}\left[\int_{\mathbb{R}\times\mathcal{S}}\frac{\left(\frac{t_{1}-t}{h}\right)K_{T}\left(\frac{t_{1}-t}{h}\right)K_{S}\left(\frac{\bm{s}_{1}-\bm{s}_{3}}{b}\right)}{p(t,\bm{s}_{3})\cdot
p_{T}(t)}\cdot\bar{K}_{T}(u)\cdot
p(t+u\hslash,\bm{s}_{3})\,dud\bm{s}_{3}\right]^{2}p(t_{1},\bm{s}_{1})\,dt_{1}d\bm{s}_{1}$
$\displaystyle=\frac{C_{K}^{2}\sigma^{2}}{hb^{d}}\int_{\mathcal{T}\times\mathcal{S}}\left[\int_{\mathbb{R}\times\mathcal{S}}\frac{\left(\frac{t_{1}-t}{h}\right)K_{T}\left(\frac{t_{1}-t}{h}\right)K_{S}\left(\frac{\bm{s}_{1}-\bm{s}_{3}}{b}\right)}{p(t,\bm{s}_{3})\cdot
p_{T}(t)}\cdot\bar{K}_{T}(u)\cdot
p(t+u\hslash,\bm{s}_{3})\sqrt{p(t_{1},\bm{s}_{1})}\,dud\bm{s}_{3}\right]^{2}dt_{1}d\bm{s}_{1}$
$\displaystyle\geq
C_{K}^{2}\sigma^{2}\int_{\mathbb{R}\times\mathbb{R}^{d}}\left[\int_{\mathbb{R}\times\mathcal{S}}\frac{xK_{T}\left(x\right)K_{S}\left(\bm{z}\right)}{p(t,\bm{s}_{3})\cdot
p_{T}(t)}\cdot\bar{K}_{T}(u)\cdot
p(t+u\hslash,\bm{s}_{3})\sqrt{p(t+hx,\bm{s}_{3}+b\bm{z})}\,dud\bm{s}_{3}\right]^{2}dxd\bm{z}$
$\displaystyle=\widetilde{C}_{K}^{2}\sigma^{2}\int_{\mathbb{R}\times\mathbb{R}^{d}}x^{2}K_{T}^{2}(x)K_{S}^{2}(\bm{z})\left[\int_{\mathbb{R}\times\mathcal{S}}\frac{\bar{K}_{T}(u)\cdot
p(t,\bm{s}_{3})\sqrt{p(t,\bm{s}_{3})}}{p(t,\bm{s}_{3})\cdot
p_{T}(t)}\,dud\bm{s}_{3}\right]^{2}dxd\bm{z}+O\left(\max\\{h,b,\hslash^{2}\\}\right)$
$\displaystyle=\widetilde{C}_{K}^{2}\sigma^{2}\int_{\mathbb{R}\times\mathbb{R}^{d}}x^{2}K_{T}^{2}(x)K_{S}^{2}(\bm{z})\left[\int_{\mathcal{S}(t)}\frac{\sqrt{p(t,\bm{s}_{3})}}{p_{T}(t)}\,d\bm{s}_{3}\right]^{2}dxd\bm{z}+O\left(\max\\{h,b,\hslash^{2}\\}\right)$
$\displaystyle\geq\underline{\sigma}^{2}$

for some $\underline{\sigma}^{2}>0$ by (47) when $h,b,\hslash$ are
sufficiently small as $n\to\infty$. The result thus follows for the influence
function $\bar{\varphi}_{t}$ of $\widehat{\theta}_{C}(t)$. ∎

######  Theorem 6 (Gaussian approximation).

Let $q\geq 2$ in the local polynomial regression for estimating
$\frac{\partial}{\partial t}\mu(t,\bm{s})$ and
$\mathcal{T}^{\prime}\subset\mathcal{T}$ be a compact set so that $p_{T}(t)$
is uniformly bounded away from 0 within $\mathcal{T}^{\prime}$. Suppose that
Assumptions A1, A2, A3, A4, A5, and A6 hold. If $h\asymp b\asymp
n^{-\frac{1}{\gamma}}$ and $\hslash\asymp n^{-\frac{1}{\varpi}}$ for some
$\gamma\geq\varpi>0$ such that $\frac{nh^{d+5}}{\log n}\to c_{1}$ and
$\frac{n\hslash^{5}}{\log n}\to c_{2}$ for some finite number $c_{1},c_{2}\geq
0$ and $\frac{n\hslash^{2}}{\log n},\frac{\hslash}{h^{d+3}\log n},\hslash
n^{\frac{1}{4}},\frac{\hslash^{2}}{h^{d+3}}\to\infty$ as $n\to\infty$, then
there exist Gaussian processes $\mathbb{B},\bar{\mathbb{B}}$ such that

$\displaystyle\sup_{u\geq
0}\left|\mbox{$\mathrm{P}$}\left(\sqrt{nh^{3}b^{d}}\cdot\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}(t)-m(t)\right|\leq
u\right)-\mbox{$\mathrm{P}$}\left(\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|\leq
u\right)\right|=O\left(\left(\frac{\log^{5}n}{nh^{d+3}}\right)^{\frac{1}{8}}\right),$
$\displaystyle\sup_{u\geq
0}\left|\mbox{$\mathrm{P}$}\left(\sqrt{nh^{3}b^{d}}\cdot\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{\theta}_{C}(t)-\theta_{C}(t)\right|\leq
u\right)-\mbox{$\mathrm{P}$}\left(\sup_{g\in\mathcal{F}_{\theta}}|\bar{\mathbb{B}}(g)|\leq
u\right)\right|=O\left(\left(\frac{\log^{5}n}{nh^{d+3}}\right)^{\frac{1}{8}}\right),$

where $\mathcal{F},\mathcal{F}_{\theta}$ are defined in (18).

###### Proof of Theorem 6.

We only prove the Gaussian approximation for $\widehat{m}_{\theta}(t)$, since
the result for its derivative estimator $\widehat{\theta}_{C}(t)$ follows from
an identical argument. At a high level, given the asymptotic linearity of
$\widehat{m}_{\theta}(t)$ in Lemma 5, we will use Lemma 10 to establish the
coupling between $\sup_{f\in\mathcal{F}}|\mathbb{G}_{n}(f)|$ and
$\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|$ for the Gaussian process $\mathbb{B}$
defined in the theorem statement and then utilize Lemma 11 to translate the
coupling to a bound on the Kolmogorov distance between
$\sqrt{nh^{3}b^{d}}\cdot\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}(t)-m(t)\right|$
and $\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|$.

By Lemma 9, we know that the class of scaled influence functions

$\widetilde{\mathcal{F}}=\left\\{(v,x,\bm{z})\mapsto\sqrt{h^{3}b^{d}}\cdot\varphi_{t}(v,x,\bm{z}):t\in\mathcal{T}^{\prime}\right\\}=\left\\{\sqrt{h^{3}b^{d}}\cdot
f:f\in\mathcal{F}\right\\}$

is a VC-type class with an envelope function $(v,x,\bm{z})\mapsto
F_{1}(v,x,\bm{z})=C_{5}\cdot|v|$ for some constant $C_{5}>0$ that is
independent of $n,h,b,\hslash$. In addition, recalling the definition of
$\varphi_{t}$ in (17) together with our Assumption A6 on moments of the kernel
functions $K_{T},K_{S}$ and Lemma 13, we obtain that

$\displaystyle\sup_{f\in\widetilde{\mathcal{F}}}\mathbb{E}\left[f(\bm{U})^{2}\right]\leq
C_{6}^{2}\cdot h^{3}b^{d}:=\widetilde{\sigma}^{2}<\infty$
$\displaystyle\;\text{ and
}\;\left[\mathbb{E}\left(C_{5}|Y|\right)^{4}\right]^{\frac{1}{4}}\leq
C_{5}^{\prime}\left(\mathbb{E}\left|\mu(T,\bm{S})\right|^{4}+\mathbb{E}|\epsilon|^{4}\right)^{\frac{1}{4}}:=\widetilde{A}<\infty,$

where $C_{5}^{\prime},C_{6}>0$ are some constants that are independent of
$n,h,b,\hslash$. By Lemma 10, we know that for any
$\widetilde{\gamma}\in(0,1)$,

$\mbox{$\mathrm{P}$}\left(\left|\sup_{f\in\widetilde{\mathcal{F}}}|\mathbb{G}_{n}(f)|-\sup_{f\in\widetilde{\mathcal{F}}}|\mathbb{B}(f)|\right|>\frac{C_{1}\cdot\widetilde{A}^{\frac{1}{3}}\left(h^{3}b^{d}\right)^{\frac{1}{3}}\log^{\frac{2}{3}}n}{\widetilde{\gamma}^{\frac{1}{3}}n^{\frac{1}{6}}}\right)\leq\frac{C_{2}\cdot\widetilde{\gamma}}{2}.$

Dividing $\sqrt{h^{3}b^{d}}$ on both sides of the inequality inside
$\mathrm{P}$ gives us that

$\mbox{$\mathrm{P}$}\left(\left|\sup_{t\in\mathcal{T}^{\prime}}|\mathbb{G}_{n}(\varphi_{t})|-\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|\right|>\frac{C_{1}\cdot\widetilde{A}^{\frac{1}{3}}\log^{\frac{2}{3}}n}{\widetilde{\gamma}^{\frac{1}{3}}\left(nh^{3}b^{d}\right)^{\frac{1}{6}}}\right)\leq\frac{C_{2}\cdot\widetilde{\gamma}}{2}.$

On the other hand, we know from Lemma 5 that for any
$\widetilde{\gamma}\in(0,1)$,

$\mbox{$\mathrm{P}$}\left(\frac{\left|\sqrt{nh^{3}b^{d}}\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}(t)-m(t)\right|-\sup_{t\in\mathcal{T}^{\prime}}|\mathbb{G}_{n}\varphi_{t}|\right|}{\max\left\\{\sqrt{nh^{d+7}},\sqrt{\frac{\log
n}{n\hslash^{2}}},\sqrt{\frac{h^{d+3}\log
n}{\hslash}},\sqrt{\frac{h^{d+3}}{\hslash^{2}}}\right\\}}>\frac{C_{3}}{\widetilde{\gamma}^{\frac{1}{3}}}\right)\leq\frac{C_{2}\cdot\widetilde{\gamma}}{2},$

where $C_{3}>0$ is some large constant. Combining the above two inequalities
yields that

$\displaystyle\mbox{$\mathrm{P}$}\Bigg{(}\left|\sqrt{nh^{3}b^{d}}\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}(t)-m(t)\right|-\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|\right|>$
$\displaystyle\quad\frac{C_{3}}{\widetilde{\gamma}^{\frac{1}{3}}}\cdot\max\left\\{\sqrt{nh^{d+7}},\sqrt{\frac{\log
n}{n\hslash^{2}}},\sqrt{\frac{h^{d+3}\log
n}{\hslash}},\sqrt{\frac{h^{d+3}}{\hslash^{2}}}\right\\}+\frac{C_{1}^{\prime}\cdot\log^{\frac{2}{3}}n}{\widetilde{\gamma}^{\frac{1}{3}}\left(nh^{3}b^{d}\right)^{\frac{1}{6}}}\Bigg{)}\leq
C_{2}\cdot\widetilde{\gamma},$

where $C_{1}^{\prime}=C_{1}\cdot\widetilde{A}^{\frac{1}{3}}>0$ is again a
constant. Now, if $h\asymp b\asymp n^{-\frac{1}{\gamma}}$ and $\hslash\asymp
n^{-\frac{1}{\varpi}}$ for some $\gamma\geq\varpi>0$ such that
$\frac{nh^{d+5}}{\log n}\to c_{1}$ and $\frac{n\hslash^{5}}{\log n}\to c_{2}$
for some finite number $c_{1},c_{2}\geq 0$ and $\frac{n\hslash^{2}}{\log
n},\frac{\hslash}{h^{d+3}\log n},\hslash
n^{\frac{1}{4}},\frac{\hslash^{2}}{h^{d+3}}\to\infty$ as $n\to\infty$, then we
know from Lemma 12 that

$\max\left\\{\sqrt{nh^{d+7}},\sqrt{\frac{\log
n}{n\hslash^{2}}},\sqrt{\frac{h^{d+3}\log
n}{\hslash}},\sqrt{\frac{h^{d+3}}{\hslash^{2}}}\right\\}\leq\frac{C_{1}^{\prime}\cdot\log^{\frac{2}{3}}n}{\left(nh^{d+3}\right)^{\frac{1}{6}}}$

when $n$ is sufficiently large. Hence, we conclude that when $n$ is
sufficiently large,

$\mbox{$\mathrm{P}$}\Bigg{(}\left|\sqrt{nh^{3}b^{d}}\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}(t)-m(t)\right|-\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|\right|>\frac{C_{4}\cdot\log^{\frac{2}{3}}n}{\widetilde{\gamma}^{\frac{1}{3}}\left(nh^{d+3}\right)^{\frac{1}{6}}}\Bigg{)}\leq
C_{2}\cdot\widetilde{\gamma},$

for some large constants $C_{2},C_{4}>0$. To upper bound the Kolmogorov
distance between
$\sqrt{nh^{3}b^{d}}\cdot\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}(t)-m(t)\right|$
and $\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|$, we leverage Lemmas 11 and 13 to
obtain that

$\displaystyle\sup_{u\geq
0}\left|\mbox{$\mathrm{P}$}\left(\sqrt{nh^{3}b^{d}}\cdot\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}(t)-m(t)\right|\leq
u\right)-\mbox{$\mathrm{P}$}\left(\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|\leq
u\right)\right|$
$\displaystyle\leq\frac{C_{5}\cdot\log^{\frac{5}{6}}n}{\widetilde{\gamma}^{\frac{1}{3}}\left(nh^{d+3}\right)^{\frac{1}{6}}}+C_{2}\cdot\widetilde{\gamma},$

where $C_{5}>0$ is some constant that depends only on
$\bar{\sigma}\geq\underline{\sigma}>0$ in Lemma 13. Here, we also utilize the
fact that $\log\left(\frac{1}{r_{1}}\right)=\log n$ when
$r_{1}=\frac{C_{4}\cdot\log^{\frac{2}{3}}n}{\widetilde{\gamma}^{\frac{1}{3}}\left(nh^{d+3}\right)^{\frac{1}{6}}}$
and use the Dudley’s entropy inequality for Gaussian processes (Corollary
2.2.8 in van der Vaart and Wellner 1996) to argue that
$\mathbb{E}\left[\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|\right]=\mathbb{E}\left[\sup_{t\in\mathcal{T}}|\mathbb{B}(\varphi_{t})|\right]=O\left(\sqrt{\log
n}\right)$. We take
$\widetilde{\gamma}=O\left(\left(\frac{\log^{5}n}{nh^{d+3}}\right)^{\frac{1}{8}}\right)$
to optimize the right hand side of the above inequality and deduce that

$\sup_{u\geq
0}\left|\mbox{$\mathrm{P}$}\left(\sqrt{nh^{3}b^{d}}\cdot\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}(t)-m(t)\right|\leq
u\right)-\mbox{$\mathrm{P}$}\left(\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|\leq
u\right)\right|=O\left(\left(\frac{\log^{5}n}{nh^{d+3}}\right)^{\frac{1}{8}}\right).$

The result follows. ∎

###  B.7 Proof of Theorem 7

######  Theorem 7 (Bootstrap consistency).

Let $q\geq 2$ in the local polynomial regression for estimating
$\frac{\partial}{\partial t}\mu(t,\bm{s})$,
$\mathcal{T}^{\prime}\subset\mathcal{T}$ be a compact set so that $p_{T}(t)$
is uniformly bounded away from 0 within $\mathcal{T}^{\prime}$, and
$\mathbb{U}_{n}=\left\\{(Y_{i},T_{i},\bm{S}_{i})\right\\}_{i=1}^{n}$ be the
observed data. Suppose that Assumptions A1, A2, A3, A4, A5, and A6 hold. If
$h\asymp b\asymp n^{-\frac{1}{\gamma}}$ and $\hslash\asymp
n^{-\frac{1}{\varpi}}$ for some $\gamma,\varpi>0$ such that
$\frac{nh^{d+5}}{\log n}\to c_{1}$ and $\frac{n\hslash^{5}}{\log n}\to c_{2}$
for some finite number $c_{1},c_{2}\geq 0$ and $\frac{n\hslash^{2}}{\log
n},\frac{\hslash}{h^{d+3}\log n},\hslash
n^{\frac{1}{4}},\frac{\hslash^{2}}{h^{d+3}}\to\infty$ as $n\to\infty$, then

$\displaystyle\sup_{u\geq
0}\left|\mbox{$\mathrm{P}$}\left(\sqrt{nh^{3}b^{d}}\cdot\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}^{*}(t)-\widehat{m}_{\theta}(t)\right|\leq
u\Big{|}\mathbb{U}_{n}\right)-\mbox{$\mathrm{P}$}\left(\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|\leq
u\right)\right|=O_{P}\left(\left(\frac{\log^{5}n}{nh^{d+3}}\right)^{\frac{1}{8}}\right)$
$\displaystyle\sup_{u\geq
0}\left|\mbox{$\mathrm{P}$}\left(\sqrt{nh^{3}b^{d}}\cdot\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{\theta}_{C}^{*}(t)-\widehat{\theta}_{C}(t)\right|\leq
u\Big{|}\mathbb{U}_{n}\right)-\mbox{$\mathrm{P}$}\left(\sup_{g\in\mathcal{F}_{\theta}}|\bar{\mathbb{B}}(g)|\leq
u\right)\right|=O_{P}\left(\left(\frac{\log^{5}n}{nh^{d+3}}\right)^{\frac{1}{8}}\right),$

where $\widehat{m}_{\theta}^{*}(t)$ and $\widehat{\theta}_{C}^{*}(t)$ are the
integral estimator (8) and localized derivative estimator (12) based on a
bootstrap sample
$\mathbb{U}_{n}^{*}=\left\\{\left(Y_{i}^{*},T_{i}^{*},\bm{S}_{i}^{*}\right)\right\\}_{i=1}^{n}$
respectively, and $\mathbb{B},\bar{\mathbb{B}}$ are the same Gaussian
processes as in Theorem 6.

###### Proof of Theorem 7.

We only prove the bootstrap consistency for $\widehat{m}_{\theta}(t)$, since
the result for its derivative estimator $\widehat{\theta}_{C}(t)$ follows from
an identical argument. Our proof here is similar to the proof of Theorem 7 in
Chen et al. (2015) and the proof of Theorem 4 in Chen et al. (2017). The key
difference is that the functional space remains unchanged as $\mathcal{F}$ in
our scenario here for both the original Gaussian approximation (Theorem 6) and
the bootstrapped version, because the index set $\mathcal{T}^{\prime}$ of
$\mathcal{F}$ is fixed.

Let $\mathbb{U}_{n}=\left\\{(Y_{i},T_{i},\bm{S}_{i})\right\\}_{i=1}^{n}$ be
the observed data and
$\mathbb{U}_{n}^{*}=\left\\{(Y_{i}^{*},T_{i}^{*},\bm{S}_{i}^{*})\right\\}_{i=1}^{n}$
be the bootstrap sample. We also denote
$\mathbb{G}_{n}^{*}(\mathbb{U}_{n})=\sqrt{n}\left(\mathbb{P}_{n}^{*}-\mathbb{P}_{n}\right)$,
where $\mathbb{P}_{n}^{*}$ is the empirical measure defined by the bootstrap
sample $\mathbb{U}_{n}^{*}$. Assume that
$\mathbb{U}_{n}\subset\mathcal{Y}\times\mathcal{T}\times\mathcal{S}$ is fixed
for a moment. Then, we can apply our arguments in Lemma 5 by replacing the
probability measure $\mathrm{P}$ by $\mathbb{P}_{n}$ and obtain that

$\displaystyle\left|\sqrt{nh^{3}b^{d}}\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}^{*}(t)-\widehat{m}_{\theta}(t)\right|-\sup_{t\in\mathcal{T}^{\prime}}|\mathbb{G}_{n}^{*}(\mathbb{U}_{n})\varphi_{t}|\right|$
$\displaystyle=O_{P}\left(\sqrt{nh^{d+7}}+\sqrt{\frac{\log
n}{n\hslash^{2}}}+\sqrt{\frac{h^{d+3}\log
n}{\hslash}}+\sqrt{\frac{h^{d+3}}{\hslash^{2}}}\right).$

Following the same argument in Theorem 6, we obtain that

$\sup_{u\geq
0}\left|\mbox{$\mathrm{P}$}\left(\sqrt{nh^{3}b^{d}}\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}^{*}(t)-\widehat{m}_{\theta}(t)\right|\leq
u\Big{|}\mathbb{U}_{n}\right)-\mbox{$\mathrm{P}$}\left(\sup_{f\in\mathcal{F}}|\mathbb{B}_{n}(f)|\leq
u\Big{|}\mathbb{U}_{n}\right)\right|=O_{P}\left(\left(\frac{\log^{5}n}{nh^{d+3}}\right)^{\frac{1}{8}}\right),$
(48)

where $\mathbb{B}_{n}$ is a Gaussian process on $\mathcal{F}$ such that for
any $f_{1},f_{2}\in\mathcal{F}$, it has

$\mathbb{E}\left[\mathbb{B}_{n}(f_{1})|\mathbb{U}_{n}\right]=0\quad\text{ and
}\quad\mathrm{Cov}\left[\mathbb{B}_{n}(f_{1}),\mathbb{B}_{n}(f_{2})\big{|}\mathbb{U}_{n}\right]=\frac{1}{n}\sum_{i=1}^{n}f_{1}(Y_{i},T_{i},\bm{S}_{i})\cdot
f_{2}(Y_{i},T_{i},\bm{S}_{i}).$

Notice that the difference between $\sup_{f\in\mathcal{F}}|\mathbb{B}_{n}(f)|$
and $\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|$ is small, because these two
Gaussian processes differ in their covariance and

$\displaystyle\mathrm{Cov}\left[\mathbb{B}_{n}(f_{1}),\mathbb{B}_{n}(f_{2})\big{|}\mathbb{U}_{n}\right]$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}f_{1}(Y_{i},T_{i},\bm{S}_{i})\cdot
f_{2}(Y_{i},T_{i},\bm{S}_{i})$
$\displaystyle\to\mathrm{Cov}(\mathbb{B}(f_{1}),\mathbb{B}(f_{2}))=\mathbb{E}\left[f_{1}(Y,T,\bm{S})\cdot
f_{2}(Y,T,\bm{S})\right]$

as $n\to\infty$. More precisely, by Corollary 9 in Giessing (2023), we know
that

$\displaystyle\begin{split}&\sup_{u\geq
0}\left|\mbox{$\mathrm{P}$}\left(\sup_{f\in\mathcal{F}}|\mathbb{B}_{n}(f)|\leq
u\Big{|}\mathbb{U}_{n}\right)-\mbox{$\mathrm{P}$}\left(\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|\leq
u\right)\right|\\\ &\leq
C_{6}\left[\frac{\sup_{f_{1},f_{2}\in\mathcal{F}}\left|\mathrm{Cov}\left[\mathbb{B}_{n}(f_{1}),\mathbb{B}_{n}(f_{2})\big{|}\mathbb{U}_{n}\right]-\mathrm{Cov}\left[\mathbb{B}(f_{1}),\mathbb{B}(f_{2})\right]\right|}{\max\left\\{\mathrm{Var}\left(\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|\right),\mathrm{Var}\left(\sup_{f\in\mathcal{F}}|\mathbb{B}_{n}(f)|\right)\right\\}}\right]^{\frac{1}{3}}\\\
&\leq
C_{6}^{\prime}\left[\sup_{f_{1},f_{2}\in\mathcal{F}}\left|\frac{1}{n}\sum_{i=1}^{n}f_{1}(Y_{i},T_{i},\bm{S}_{i})\cdot
f_{2}(Y_{i},T_{i},\bm{S}_{i})-\mathbb{E}\left[f_{1}(Y,T,\bm{S})\cdot
f_{2}(Y,T,\bm{S})\right]\right|\right]^{\frac{1}{3}}\\\
&=C_{6}^{\prime}\left[\sup_{f\in\mathcal{F}^{2}}\left|\frac{1}{n}\sum_{i=1}^{n}f(Y_{i},T_{i},\bm{S}_{i})-\mathbb{E}\left[f(Y,T,\bm{S})\right]\right|\right]^{\frac{1}{3}}\end{split}$
(49)

where the last inequality follows from Lemma 13 and $C_{6},C_{6}^{\prime}>0$
are two absolute constants. Here, $\mathcal{F}^{2}\equiv\left\\{f_{1}\cdot
f_{2}:f_{1},f_{2}\in\mathcal{F}\right\\}$. Now, by symmetrization (Lemma 2.3.1
in van der Vaart and Wellner 1996) and maximal inequality (Corollary 2.2.8 in
van der Vaart and Wellner 1996), we also know that

$\displaystyle\sup_{f\in\mathcal{F}^{2}}\left|\frac{1}{n}\sum_{i=1}^{n}f(Y_{i},T_{i},\bm{S}_{i})-\mathbb{E}\left[f(Y,T,\bm{S})\right]\right|$
$\displaystyle\leq
2\mathbb{E}\left[\sup_{f\in\mathcal{F}^{2}}\left|\frac{1}{n}\sum_{i=1}^{n}\chi_{i}f(Y_{i},T_{i},\bm{S}_{i})\right|\right]$
$\displaystyle\leq\frac{C_{7}}{\sqrt{n}}\int_{0}^{\infty}\sup_{Q}\sqrt{\log
N\left(\mathcal{F}^{2},L_{2}(Q),\epsilon\right)}\,d\epsilon$
$\displaystyle\leq\frac{C_{7}^{\prime}}{\sqrt{n}}\int_{0}^{\infty}\sup_{Q}\sqrt{\log
N\left(\mathcal{F},L_{2}(Q),\epsilon\right)}\,d\epsilon$
$\displaystyle=\frac{C_{7}^{\prime}}{\sqrt{n}}\int_{0}^{\sqrt{h^{3}b^{d}}\cdot
M_{\mathcal{F}}}\sup_{Q}\sqrt{\log
N\left(\widetilde{\mathcal{F}},L_{2}(Q),\sqrt{h^{3}b^{d}}\cdot\epsilon\right)}\,d\epsilon$
$\displaystyle=\frac{C_{7}^{\prime}}{\sqrt{nh^{3}b^{d}}}\int_{0}^{M_{\mathcal{F}}}\sup_{Q}\sqrt{\log
N\left(\widetilde{\mathcal{F}},L_{2}(Q),u\right)}\,du$
$\displaystyle\leq\frac{C_{8}}{\sqrt{nh^{3}b^{d}}},$

where $\chi_{1},...,\chi_{n}$ are Rademacher random variables,
$\widetilde{\mathcal{F}}=\sqrt{h^{3}b^{d}}\cdot\mathcal{F}=\left\\{\sqrt{h^{3}b^{d}}f:f\in\mathcal{F}\right\\}$
by Lemma 9, and $C_{7},C_{7}^{\prime},C_{8}>0$ are absolute constants. In
addition,
$M_{\mathcal{F}}=\sup_{f\in\mathcal{F}}\mathbb{E}\left[f(Y_{1},T_{1},\bm{S}_{1})^{2}\right]<\infty$
by Lemma 13 and the last inequality follows from the VC-type property of
$\widetilde{\mathcal{F}}$ by Lemma 9. Combining the above result with (49), we
obtain that

$\sup_{u\geq
0}\left|\mbox{$\mathrm{P}$}\left(\sup_{f\in\mathcal{F}}|\mathbb{B}_{n}(f)|\leq
u\Big{|}\mathbb{U}_{n}\right)-\mbox{$\mathrm{P}$}\left(\sup_{f\in\mathcal{F}}|\mathbb{B}(f)|\leq
u\right)\right|=O_{P}\left(\frac{1}{\left(nh^{3}b^{d}\right)^{\frac{1}{6}}}\right).$

Together with (48), the result thus follows. ∎

###  B.8 Proof of Corollary 8

######  Corollary 8 (Uniform confidence band).

Under the setup of Theorem 7, we have that

$\displaystyle\mbox{$\mathrm{P}$}\left(\theta(t)\in\left[\widehat{\theta}_{C}(t)-\bar{\xi}_{1-\alpha}^{*},\widehat{\theta}_{C}(t)+\bar{\xi}_{1-\alpha}^{*}\right]\text{
for all
}t\in\mathcal{T}^{\prime}\right)=1-\alpha+O\left(\left(\frac{\log^{5}n}{nh^{d+3}}\right)^{\frac{1}{8}}\right),$
$\displaystyle\mbox{$\mathrm{P}$}\left(m(t)\in\left[\widehat{m}_{\theta}(t)-\xi_{1-\alpha}^{*},\widehat{m}_{\theta}(t)+\xi_{1-\alpha}^{*}\right]\text{
for all
}t\in\mathcal{T}^{\prime}\right)=1-\alpha+O\left(\left(\frac{\log^{5}n}{nh^{d+3}}\right)^{\frac{1}{8}}\right).$

###### Proof of Corollary 8.

By Theorem 6 and Theorem 7, we have the Berry-Esseen bounds related to
bootstrap estimates for the distributions of
$\sqrt{nh^{3}b^{d}}\cdot\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}(t)-m(t)\right|$
and
$\sqrt{nh^{3}b^{d}}\cdot\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{\theta}_{C}(t)-\theta_{C}(t)\right|$
as:

$\displaystyle\sup_{u\geq
0}\left|\mbox{$\mathrm{P}$}\left(\sqrt{nh^{3}b^{d}}\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}^{*}(t)-\widehat{m}_{\theta}(t)\right|\leq
u\Big{|}\mathbb{U}_{n}\right)-\mbox{$\mathrm{P}$}\left(\sqrt{nh^{3}b^{d}}\cdot\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{m}_{\theta}(t)-m(t)\right|\leq
u\right)\right|$
$\displaystyle\quad=O_{P}\left(\left(\frac{\log^{5}n}{nh^{d+3}}\right)^{\frac{1}{8}}\right),$

and

$\displaystyle\sup_{u\geq
0}\left|\mbox{$\mathrm{P}$}\left(\sqrt{nh^{3}b^{d}}\cdot\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{\theta}_{C}^{*}(t)-\widehat{\theta}_{C}(t)\right|\leq
u\right)-\mbox{$\mathrm{P}$}\left(\sqrt{nh^{3}b^{d}}\cdot\sup_{t\in\mathcal{T}^{\prime}}\left|\widehat{\theta}_{C}(t)-\theta_{C}(t)\right|\leq
u\right)\right|$
$\displaystyle=O_{P}\left(\left(\frac{\log^{5}n}{nh^{d+3}}\right)^{\frac{1}{8}}\right).$

The results thus follow. ∎


# Investigations on the charmless decays of $X(3872)$ in intermediate meson
loops model

Yan Wang  College of Physics and Engineering, Qufu Normal University, Qufu
273165, China  Qi Wu  School of Physics and Center of High Energy Physics,
Peking University, Beijing 100871, China  Gang Li<EMAIL_ADDRESS>College of
Physics and Engineering, Qufu Normal University, Qufu 273165, China  Wen-Hua
Qin<EMAIL_ADDRESS>College of Physics and Engineering, Qufu Normal
University, Qufu 273165, China  Xiao-Hai Liu<EMAIL_ADDRESS>Department of Physics, School of Science, Tianjin University, Tianjin 300350,
China  Chun-Sheng An<EMAIL_ADDRESS>School of Physical Science and
Technology, Southwest University, Chongqing 400715, China  Ju-Jun Xie
<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 101408, China  School of
Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001,
China

###### Abstract

The charmless decay processes of $X(3872)$ provide us a good platform to study
the nature and the decay mechanism of $X(3872)$. Based on a molecular nature
of $X(3872)$ as a $\bar{D}D^{*}$ bound state, we have investigated the
charmless decays $X(3872)\to VV$ and $VP$ via intermediate
$D^{*}{\bar{D}}+c.c.$ meson loops, where $V$ and $P$ stand for light vector
and pseudoscalar mesons, respectively. We discuss three cases, i.e., pure
neutral components ($\theta=0$), isospin singlet ($\theta=\pi/4$) and neutral
components dominant ($\theta=\pi/6$), where $\theta$ is a phase angle
describing the proportion of neutral and charged constituents. The proportion
of neutral and charged constituent have an influence on the decay widths of
$X(3872)\to VV$ and $VP$. With the coupling constant of $X(3872)$ to the
$\bar{D}D^{*}$ channel obtained under the molecule ansatz of $X(3872)$
resonance, the predicted decay widths of $X(3872)\rightarrow VV$ are about
tens of keVs, while the decay width can reach a few hundreds of keVs for
$X(3872)\to VP$. The dependence of these ratios between different decay modes
of $X(3872)\to VV$ and $X(3872)\to VP$ to the mixing angle $\theta$ is also
investigated. It is expected that the theoretical calculations here can be
tested by future experiments.

##  I Introduction

In 2003, the $X(3872)$ state was first observed by the Belle Collaboration in
the $J/\psi\pi^{+}\pi^{-}$ invariant mass spectrum of the $B\to KX(3872)\to
K\pi^{+}\pi^{-}J/\psi$ decay Choi:2003ue . Then, it was confirmed in the
$J/\psi\pi^{+}\pi^{-}$ channel from $p\bar{p}$ collisions by CDF and D0
Collaborations Acosta:2003zx ; Abazov:2004kp , and $e^{+}e^{-}$ collisions by
BABAR Collaboration Aubert:2004ns ; Aubert:2008gu . Its quantum numbers was
determined to be $I^{G}(J^{PC})=0^{+}(1^{++})$ by LHCb Collaboration
Aaij:2013zoa . There are two salient feature of $X(3872)$, one is that it has
very narrow width $(\Gamma_{X}<1.2~{}\mathrm{MeV})$, the other one is that its
mass is extremely closing to the mass threshold of $D^{0}\bar{D}^{\ast 0}$
channel.

The interpretation of the nature of $X(3872)$ is still an open question. Since
its quantum numbers are $J^{PC}=1^{++}$ and its mass is very close to the
$D^{0}\bar{D}^{\ast 0}$ threshold, one naturel explanation is that it is a
$D\bar{D}^{\ast}$ hadronic molecule as discussed in Refs. Close:2003sg ;
Pakvasa:2003ea ; Swanson:2004pp ; Swanson:2003tb ; Tornqvist:2004qy ;
Voloshin:2003nt ; Wong:2003xk ; AlFiky:2005jd ; Braaten:2006sy ;
Fleming:2007rp ; Ding:2009vj ; Dong:2009yp ; Lee:2009hy ; Lee:2011rka ;
Liu:2008tn ; Zhang:2009vs ; Gamermann:2009uq ; Mehen:2011ds ; Nieves:2011vw ;
Nieves:2012tt ; Li:2012cs ; Sun:2012sy ; Guo:2013sya ; Guo:2014hqa ;
He:2014nya ; Zhao:2014gqa ; Guo:2014taa ; Braaten:2003he . In general, a
hadronic molecule can couple to other components which have the same quantum
numbers. For instance, the possibility of a charmonium $c\bar{c}$ excited
state admixture was investigated in Refs. Dong:2008gb ; Suzuki:2005ha . It was
also pointed out that the $D^{\pm}D^{\ast\mp}$ and $D^{+}_{s}D^{\ast-}_{s}$
components are necessary to explain the branching ratio of $X(3872)$ to
$J/\psi\rho$ and $J/\psi\omega$ Gamermann:2007fi ; Gamermann:2009fv ;
Aceti:2012cb . On the other hand, the $X(3872)$ is also considered as a
tetraquark state Maiani:2004vq ; Maiani:2005pe ; Maiani:2007vr ;
Terasaki:2007uv . However, searching for the charged partners of $X(3872)$
shows negative results Aubert:2004zr . Besides, the $X(3872)$ was also viewed
as a conventional charmonium state Barnes:2003vb ; Eichten:2004uh ;
Barnes:2005pb .

In Ref. Liu:2006df , the isospin violating decay process of
$X(3872)\rightarrow J/\psi\rho$ was estimated using final state interactions
(FSI) by consider intermediate $D\bar{D}^{\ast}$ meson loop, where it was
found that the contribution from FSI is tiny. The radiative decays
$X(3872)\rightarrow\gamma\psi/\psi^{\prime}$ were investigated in Refs.
Dong:2009uf ; Dong:2009yp ; Guo:2014taa , and the results support the
molecular picture of $X(3872)$. While in Refs. Dubynskiy:2007tj ;
Fleming:2008yn ; Mehen:2015efa , the pionic transition from $X(3872)$ to
$\chi_{cJ}$ was studied. In Ref. Dubynskiy:2007tj  it was concluded that these
decay rates exhibit significantly different patterns depending on a pure
charmonium or a multi-quark structure of $X(3872)$.

All these above theoretical studies of $X(3872)$ focus on its charmful decay
modes. To better understand the nature of $X(3872)$, the study of its other
decay modes is needed. For example, the charmless decays can also provide us a
good platform to further study the nature of $X(3872)$. In this work, under
the molecule ansatz of the $X(3872)$, which is a bound state of
$\bar{D}D^{*}$, we will investigate the charmless decays of $X(3872)\to VV$
and $VP$ ($V$ and $P$ stand for the vector meson and pseudoscalar meson) via
intermediate charmed meson loops in an effective Lagrangian approach.

This article is organized as follows. In Sec. II, based on a molecular nature
of $X(3872)$ as a $\bar{D}D^{*}$ bound state, we present the related decay
amplitudes obtained with the effective Lagrangians constructed in the heavy
quark limit and chiral symmetry. In Sec. III, we show our numerical results
and discussions, and last section is devoted to a short summary.

##  II Theoretical framework

###  II.1 Coupling constant and decay diagrams

For a state slightly below an S-wave two-hadron threshold, the effective
coupling constant of this state to the two-body channel, $g_{\rm eff}$, is
related to the probability of finding the two-hadron component in the physical
wave function of the bound state, $c_{i}^{2}$, and the binding energy
$\epsilon=m_{1}+m_{2}-M$ Weinberg:1965zz ; Baru:2003qq ; Guo:2013zbw

$\displaystyle g_{{\rm eff}}^{2}=16\pi
c_{i}^{2}(m_{1}+m_{2})^{2}\sqrt{\frac{2\epsilon}{\mu}},$ (1)

where $\mu=m_{1}m_{2}/(m_{1}+m_{2})$ is the reduced mass of $m_{1}$ and
$m_{2}$.

Assuming that the $X(3872)$ is a $S$-wave molecular state with quantum numbers
$J^{PC}=1^{++}$ given by the superposition of $D^{0}{\bar{D}}^{*0}$ and
$D^{\pm}D^{\ast\mp}$ hadronic configurations as

$\displaystyle|X(3872)\rangle$ $\displaystyle=$
$\displaystyle\frac{\cos\theta}{\sqrt{2}}|D^{*0}{\bar{D}}^{0}+D^{0}{\bar{D}}^{*0}\rangle$
(2)
$\displaystyle+\frac{\sin\theta}{\sqrt{2}}|D^{*+}D^{-}+D^{-}D^{*+}\rangle,$

where $\theta$ is a phase angle describing the proportion of neutral and
charged constituents. For example, $\theta=0$ stands for $X(3872)$ as a pure
${\bar{D}}^{*0}D^{0}/{\bar{D}}^{0}D^{*0}$, while $\theta=\pi/4$ and
$\theta=-\pi/4$ correspond to the isospin singlet and isospin triplet states,
respectively. Then, one can parameterize the coupling of $X(3872)$ to the
charmed mesons in terms of the following Lagrangian:

$\displaystyle{\cal L}_{X(3872)}$ $\displaystyle=$
$\displaystyle\frac{g_{n}}{\sqrt{2}}X_{\mu}^{\dagger}(D^{*0\mu}{\bar{D}}^{0}+D^{0}{\bar{D}}^{*0\mu})$
(3)
$\displaystyle+\frac{g_{c}}{\sqrt{2}}X_{\mu}^{\dagger}(D^{*+\mu}D^{-}+D^{+}D^{*-\mu}),$

where $g_{n}$ and $g_{c}$ are the coupling constants of $X(3872)$ with its
neutral and charged components, respectively.

Using the masses of the $X(3872)$ and the charmed mesons as in Refs.
Zyla:2020zbs ; Guo:2013zbw , we obtain the mass difference between the
$X(3872)$ and the ${\bar{D}}^{*0}D^{0}/{\bar{D}}^{0}D^{*0}$ (neutral) and
$D^{*-}D^{+}/D^{*+}D^{-}$ threshold to be $0.16$ MeV and $8.21$ MeV,
respectively. Assuming that $X(3872)$ is a pure $D^{0}{\bar{D}}^{*0}$ or
$D^{\pm}D^{*\mp}$ molecule, we obtain

$\displaystyle|g_{{\rm eff}}^{n}|$ $\displaystyle=$ $\displaystyle 3.70~{}{\rm
GeV},~{}~{}~{}{\rm with}~{}c^{2}_{D^{0}{\bar{D}}^{*0}}=1,$ (4)
$\displaystyle|g_{{\rm eff}}^{c}|$ $\displaystyle=$ $\displaystyle 9.91~{}{\rm
GeV},~{}~{}~{}{\rm with}~{}c^{2}_{D^{\pm}D^{*\mp}}=1.$ (5)

As a result, the coupling constants appearing in Eq. (3) are as follows
111These coupling constants are assumed to be real.,

$\displaystyle g_{n}=|g_{{\rm eff}}^{n}|\cos\theta,~{}~{}~{}g_{c}=|g_{{\rm
eff}}^{c}|\sin\theta\,.$ (6)

Figure 1: Diagrams contributing to the charmless decay $X(3872)\to VV$ with
$D{\bar{D}}^{*}+c.c.$ as intermediate states.

With the above $\bar{D}D^{*}$ molecular picture for $X(3872)$, these
$X(3872)\to VV$ and $VP$ decays can proceed via $X(3872)\to\bar{D}D^{*}\to VV$
or $VP$ through triangle loop diagrams, which are shown in Figs. 1 and 2,
respectively. In this mechanism, $X(3872)$ goes into $\bar{D}D^{*}$ at a first
step, then $\bar{D}$ and $D^{*}$ are converged to $VV$ or $VP$ in the final
state by exchanging a charmed meson. Note that, in Figs. 1 and 2, we have
considered only the leading contributions as discussed in Refs. Lipkin:1986bi
; Lipkin:1988tg ; Lipkin:1986av .

Figure 2: Diagrams contributing to the charmless decay $X(3872)\to VP$ with
$D{\bar{D}}^{*}+c.c.$ as intermediate states.

###  II.2 The interaction Lagrangians and decay amplitudes

The Lagrangians relevant to the light vector and pseudoscalar mesons can be
constructed based on the heavy quark limit and chiral symmetry,

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle-ig_{{\cal
D}^{\ast}{\cal D}{\mathcal{P}}}\left({\cal
D}^{i}\partial^{\mu}{\mathcal{P}}_{ij}{\cal D}_{\mu}^{\ast j\dagger}-{\cal
D}_{\mu}^{\ast i}\partial^{\mu}{\mathcal{P}}_{ij}{\cal D}^{j{\dagger}}\right)$
(7) $\displaystyle+\frac{1}{2}g_{{\cal
D}^{\ast}D^{\ast}{\mathcal{P}}}\varepsilon_{\mu\nu\alpha\beta}{\cal
D}_{i}^{\ast\mu}\partial^{\nu}{\mathcal{P}}^{ij}{\overset{\leftrightarrow}{\partial}}{\\!{}^{\alpha}}{\cal
D}_{j}^{\ast\beta{\dagger}}$ $\displaystyle-
ig_{\mathcal{D}\mathcal{D}\mathcal{V}}\mathcal{D}_{i}^{\dagger}{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}}{\\!{}_{\mu}}\mathcal{D}^{j}(\mathcal{V}^{\mu})^{i}_{j}$
$\displaystyle-2f_{\mathcal{D}^{*}\mathcal{D}\mathcal{V}}\epsilon_{\mu\nu\alpha\beta}(\partial^{\mu}\mathcal{V}^{\nu})^{i}_{j}(\mathcal{D}_{i}^{\dagger}{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}}{\\!{}^{\alpha}}\mathcal{D}^{*\beta
j}-\mathcal{D}_{i}^{*\beta\dagger}{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}}{\\!{}^{\alpha}}{\cal
D}^{j})$
$\displaystyle+ig_{\mathcal{D}^{*}\mathcal{D}^{*}\mathcal{V}}\mathcal{D}^{*\nu\dagger}_{i}{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}}{\\!{}_{\mu}}\mathcal{D}^{*j}_{\nu}(\mathcal{V}^{\mu})^{i}_{j}$
$\displaystyle+4if_{\mathcal{D}^{*}\mathcal{D}^{*}\mathcal{V}}\mathcal{D}^{*\dagger}_{i\mu}(\partial^{\mu}\mathcal{V}^{\nu}-\partial^{\nu}\mathcal{V}^{\mu})^{i}_{j}\mathcal{D}^{*j}_{\nu}+{\rm
H.c.},$

with the convention $\varepsilon_{0123}=1$, where $\mathcal{P}$ and
${\mathcal{V}}_{\mu}$ are $3\times 3$ matrices for the octet pseudoscalar and
nonet vector mesons, respectively,

$\displaystyle\mathcal{P}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&K^{0}\\\
K^{-}&{\bar{K}}^{0}&-\sqrt{\frac{2}{3}}\eta\\\ \end{array}\right),$ (11)
$\displaystyle\mathcal{V}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{*0}\\\
K^{*-}&{\bar{K}}^{*0}&\phi\\\ \end{array}\right)\,.$ (15)

In the heavy quark and chiral limits, the couplings of the charmed meson to
the light vector mesons have the relationship Casalbuoni:1996pg ; Cheng:2004ru
,

$\displaystyle g_{{\cal D}{\cal D}V}$ $\displaystyle=$ $\displaystyle g_{{\cal
D}^{*}{\cal D}^{*}V}=\frac{\beta g_{V}}{\sqrt{2}},$ (16) $\displaystyle
f_{{\cal D}^{*}{\cal D}V}$ $\displaystyle=$ $\displaystyle\frac{f_{{\cal
D}^{*}{\cal D}^{*}V}}{m_{{\cal D}^{*}}}=\frac{\lambda g_{V}}{\sqrt{2}}\,,$
(17) $\displaystyle g_{\mathcal{D}^{*}\mathcal{D}\mathcal{P}}$
$\displaystyle=$
$\displaystyle\frac{2g}{f_{\pi}}\sqrt{m_{\mathcal{D}}m_{\mathcal{D}^{*}}},$
(18) $\displaystyle g_{\mathcal{D}^{*}\mathcal{D}^{*}\mathcal{P}}$
$\displaystyle=$ $\displaystyle\frac{g_{{\cal
D}^{*}{\mathcal{D}}{\mathcal{P}}}}{\sqrt{m_{\mathcal{D}}m_{\mathcal{D}^{*}}}}\,.$
(19)

In this work, we take parameters $\beta=0.9$, $\lambda=0.56\,{\rm GeV}^{-1}$,
$g=0.59$, and $g_{V}={m_{\rho}/f_{\pi}}$ with $f_{\pi}=132$ MeV, as used in
previous works Casalbuoni:1996pg ; Isola:2003fh .

Then one can easily write the explicit transition amplitudes for
$X(3872)(p_{1})\to[D^{(*)}(q_{1}){\bar{D}}^{(*)}(q_{3})]D^{(*)}(q_{2})\to
V_{1}(p_{2})V_{2}(p_{3})$ shown in Fig. 1 as follows:

$\displaystyle\mathcal{M}_{a}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q_{2}}{(2\pi)^{4}}[g_{\rm
eff}\epsilon_{1\alpha}][g_{DDV}(q_{1}-q_{2})_{\mu}\epsilon_{2}^{*\mu}]$ (20)
$\displaystyle\times[2f_{D^{*}DV}\epsilon_{\kappa\lambda\rho\sigma}ip_{3}^{\kappa}\epsilon_{3}^{*\lambda}(q_{2}+q_{3})^{\rho}]\frac{1}{q_{1}^{2}-m_{1}^{2}}$
$\displaystyle\times\frac{1}{q_{2}^{2}-m_{2}^{2}}\frac{(g^{\alpha\sigma}-{q_{3}^{\alpha}q_{3}^{\sigma}}/{m_{3}^{2}})}{q_{3}^{2}-m_{3}^{2}}{\cal
F}(q^{2}),$ $\displaystyle\mathcal{M}_{b}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q_{2}}{(2\pi)^{4}}[g_{\rm
eff}\epsilon_{1\alpha}][2f_{D^{*}DV}\epsilon_{\mu\nu\xi\phi}p_{2}^{\mu}\epsilon_{2}^{*\nu}(q_{1}-q_{2})^{\xi}]$
(21)
$\displaystyle\times[g_{D^{*}D^{*}V}(q_{3}+q_{2})^{\rho}g_{\lambda\sigma}\epsilon^{*}_{3\rho}+4f_{D^{*}D^{*}V}(p_{3\lambda}g_{\sigma}^{\rho}$
$\displaystyle-
p_{3\sigma}g_{\lambda}^{\rho})\epsilon^{*}_{3\rho}]\frac{i}{q_{1}^{2}-m_{1}^{2}}\frac{(g^{\phi\sigma}-{q_{2}^{\phi}q_{2}^{\sigma}}/{m_{2}^{2}})}{q_{2}^{2}-m_{2}^{2}}$
$\displaystyle\times\frac{(g^{\alpha\lambda}-{q_{3}^{\alpha}q_{3}^{\lambda}}/{m_{3}^{2}})}{q_{3}^{2}-m_{3}^{2}}{\cal
F}(q^{2}),$ $\displaystyle\mathcal{M}_{c}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q_{2}}{(2\pi)^{4}}[g_{\rm
eff}\epsilon_{1\alpha}][-2f_{D^{*}DV}\epsilon_{\mu\nu\xi\phi}p_{2}^{\mu}\epsilon_{2}^{*\nu}(q_{1}-q_{2})^{\xi}]$
(22)
$\displaystyle\times[(g_{DDV}(q_{3}+q_{2})_{\kappa}\epsilon_{3}^{*\kappa}]\frac{(g^{\alpha\phi}-{q_{1}^{\alpha}q_{1}^{\phi}}/{m_{1}^{2}})}{q_{1}^{2}-m_{1}^{2}}$
$\displaystyle\times\frac{i}{q_{2}^{2}-m_{2}^{2}}\frac{1}{q_{3}^{2}-m_{3}^{2}}{\cal
F}(q^{2}),$ $\displaystyle\mathcal{M}_{d}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q_{2}}{(2\pi)^{4}}[g_{\rm
eff}\epsilon_{1\alpha}][g_{D^{*}D^{*}V}(q_{1}-q_{2})^{\xi}g_{\nu\phi}\epsilon_{2\xi}^{*}$
(23)
$\displaystyle-4f_{D^{*}D^{*}V}(p_{2\nu}g_{\phi}^{\xi}-p_{2\phi}g_{\nu}^{\xi})\epsilon_{2\xi}^{*}]$
$\displaystyle\times[-2f_{D^{*}DV}\epsilon_{\kappa\lambda\rho\sigma}p^{\kappa}_{3}\epsilon_{3}^{*\lambda}(q_{2}+q_{3})^{\rho}]$
$\displaystyle\times\frac{(g^{\alpha\phi}-{q_{1}^{\alpha}q_{1}^{\phi}}/{m_{1}^{2}})}{q_{1}^{2}-m_{1}^{2}}\frac{(g^{\nu\sigma}-{q_{2}^{\nu}q_{2}^{\sigma}}/{m_{2}^{2}})}{q_{2}^{2}-m_{2}^{2}}$
$\displaystyle\times\frac{i}{q_{3}^{2}-m_{3}^{2}}{\cal F}(q^{2})\,,$

where $p_{1}$ ($\varepsilon_{1}$), $p_{2}$ ($\varepsilon_{2}$) and $p_{3}$
($\varepsilon_{3}$) are the four-momenta (polarization vector) of the initial
state $X(3872)$, final state $V_{1}$ and $V_{2}$, respectively. $q_{1}$,
$q_{2}$ and $q_{3}$ are the four-momenta of the up, right and down charmed
mesons in the triangle loop, respectively.

The explicit transition amplitudes for
$X(3872)(p_{1})\to[D^{(*)}(q_{1}){\bar{D}}^{(*)}(q_{3})]D^{(*)}(q_{2})\to
V(p_{2})P(p_{3})$ shown in Fig. 2 are as follows:

$\displaystyle\mathcal{M}_{a}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q_{2}}{(2\pi)^{4}}[g_{\rm
eff}\epsilon_{1\alpha}][-g_{DDV}(q_{1}-q_{2})_{\mu}\epsilon_{2}^{*\mu}]$ (24)
$\displaystyle\times[g_{D^{*}DP}p_{3}^{\kappa}]\frac{1}{q_{1}^{2}-m_{1}^{2}}\frac{i}{q_{2}^{2}-m_{2}^{2}}$
$\displaystyle\times\frac{(g_{\kappa}^{\alpha}-{q_{3}^{\alpha}q_{3\kappa}}/{m_{3}^{2}})}{q_{3}^{2}-m_{3}^{2}}{\cal
F}(q^{2}),$ $\displaystyle\mathcal{M}_{b}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q_{2}}{(2\pi)^{4}}[g_{\rm
eff}\epsilon_{1\alpha}][2f_{D^{*}DV}\epsilon_{\mu\nu\xi\phi}p_{2}^{\mu}\epsilon_{2}^{*\nu}(q_{1}-q_{2})^{\xi}]$
(25)
$\displaystyle\times[\frac{1}{2}g_{D^{*}D^{*}P}\epsilon_{\kappa\lambda\rho\sigma}p_{3}^{\lambda}(q_{3}+q_{2})^{\rho}]\frac{-i}{q_{1}^{2}-m_{1}^{2}}$
$\displaystyle\times\frac{(g^{\phi\kappa}-{q_{2}^{\phi}q_{2}^{\kappa}}/{m_{2}^{2}})}{q_{2}^{2}-m_{2}^{2}}\frac{(g^{\alpha\sigma}-{q_{3}^{\alpha}q_{3}^{\sigma}}/{m_{3}^{2}})}{q_{3}^{2}-m_{3}^{2}}{\cal
F}(q^{2}),$ $\displaystyle\mathcal{M}_{c}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q_{2}}{(2\pi)^{4}}[g_{\rm
eff}\epsilon_{1\alpha}][g_{D^{*}D^{*}V}(q_{1}-q_{2})^{\xi}g_{\nu\phi}\epsilon_{2\xi}^{*}$
(26)
$\displaystyle-4f_{D^{*}D^{*}V}(p_{2\nu}g_{\phi}^{\xi}-p_{2\phi}g_{\nu}^{\xi})\epsilon_{2\xi}^{*}][g_{D^{*}DP}p_{3}^{\kappa}]$
$\displaystyle\times\frac{(g^{\alpha\phi}-{q_{1}^{\alpha}q_{1}^{\phi}}/{m_{1}^{2}})}{q_{1}^{2}-m_{1}^{2}}\frac{(g_{\kappa}^{\nu}-{q_{2}^{\nu}q_{2\kappa}}/{m_{2}^{2}})}{q_{2}^{2}-m_{2}^{2}}$
$\displaystyle\times\frac{i}{q_{3}^{2}-m_{3}^{2}}{\cal F}(q^{2}),$

where $\mathcal{F}(q^{2})$ is the form factor introduced to depict the off-
shell effects of the exchanged mesons as well as the structure effects of the
involved mesons. The form factor $\mathcal{F}(q^{2})$ is parameterized as

$\displaystyle\mathcal{F}\left(q^{2}\right)=\left(\frac{m^{2}-\Lambda^{2}}{q^{2}-\Lambda^{2}}\right)^{n},$
(27)

normalized to unity at $q^{2}=m^{2}$ Cheng:2004ru , where $m$ and $q$ are mass
and momenta of the exchanged mesons. The cutoff $\Lambda$ can be further
reparameterized as $\Lambda=m_{D^{(\ast)}}+\alpha\Lambda_{\rm QCD}$ with
$\Lambda_{\rm QCD}=0.22\ {\rm GeV}$. The model parameter $\alpha$ is usually
expected to be of order of unity Cheng:2004ru ; Tornqvist:1993vu ;
Tornqvist:1993ng ; Locher:1993cc ; Li:1996yn , but its concrete value cannot
be estimated by the first principle. In practice, the value of $\alpha$ is
usually determined by comparing theoretical estimates with the corresponding
experimental measurements. However, no charmless decay mode of $X(3872)$ is
known so far. For the rescattering processes studied in this work, it is found
that the monopole form ($n=1$) or dipole form ($n=2$) for $\mathcal{F}(q^{2})$
is utilized, the numerical results are much sensitive to the values of
parameter $\alpha$, and we have to use a very small value, otherwise, these
partial decay widths will be very large, even more than the total width of
$X(3872)$. In order to avoid too large dependence of the parameter $\alpha$,
we take $n=3$ in the numerical calculations.

##  III Numerical Results and discussions

Figure 3: The $\alpha$-dependence of decay widths (in unit of keV) of
$X(3872)\rightarrow VV$ and $X(3872)\rightarrow VP$ with $\theta=0$.

Figure 4: The $\alpha$-dependence of decay widths (in unit of keV) of
$X(3872)\rightarrow VV$ and $X(3872)\rightarrow VP$ with $\theta=\pi/4$

Figure 5:  The $M_{X(3872)}$-dependence of the decay widths (in unit of keV)
of $X(3872)\rightarrow VV$ with $\alpha=1.0$.

Figure 6: The same as Fig 5 but for $X(3872)\rightarrow VP$.

In this section, we will mainly discuss three cases where $\theta$ is $0$,
$\pi/6$ and $\pi/4$. When $\theta=0$, it indicates that $X(3872)$ is a pure
bound state with only neutral component. When $\theta=\pi/4$, the proportions
of the neutral and charged components are the same. There are both neutral and
charged components at $\theta=\pi/6$, but the proportion of the neutral
component is dominant.

In Fig. 3, we plot the $\alpha$-dependence of the partial decay widths of
$X(3872)\rightarrow VV$ and $X(3872)\rightarrow VP$ with $\theta=0$,
respectively. In the range of $\alpha=0.6\sim 1.2$, the predicted partial
decay widths of $X(3872)\rightarrow VV$ are about a few KeV, while the partial
decay widths can reach a few tens of KeV for $X(3872)\to VP$. Since the
$X(3872)\to K^{\ast 0}\bar{K}^{\ast 0}$ transition proceeds via
$[D^{+}D^{*-}]D^{(\ast)}_{s}$ intermediate mesons, while the $X(3872)\to
K^{\ast+}K^{\ast-}$ transition proceeds via
$[D^{0}\bar{D}^{*0}]D^{(\ast)}_{s}$ intermediate mesons. So in the case of
$\theta=0$, there is no neutral $K^{\ast 0}\bar{K}^{\ast 0}$ channel as shown
in Fig. 3 (a). The same reason for $X(3872)\rightarrow K^{\ast 0}\bar{K}^{0}$
in Fig. 3 (b). From Fig. 3 (a), one can see that the partial decay width of
$X(3872)\to\rho\rho$ is larger than those of $X(3872)\to K^{\ast+}K^{\ast-}$
and $\omega\omega$ decay modes. This is because both the charged
$\rho^{+}\rho^{-}$ and neutron $\rho^{0}\rho^{0}$ channels contribute to the
$\rho\rho$ channel. While for the $X(3872)\to\rho^{0}\rho^{0}$ decay, its
partial decay width is almost equal to the decay of $X(3872)\to\omega\omega$.
In addition, for the $X(3872)\to K^{\ast+}K^{\ast-}$ decays, there are only
contributions from the exchanging of charged charm mesons. In the case of
$\theta=0$, only neutral charmed meson loops contribute to the isospin-
violating channel $X(3872)\to\rho^{0}\omega$. As a result, the obtained decay
widths are almost the same as that of the channel $X(3872)\to\omega\omega$.

In Fig. 4, we plot the $\alpha$-dependence of the partial decay widths of
$X(3872)\rightarrow VV$ and $X(3872)\rightarrow VP$ with $\theta=\pi/4$. In
the range of $\alpha=0.6\sim 1.2$, the predicted partial decay widths of
$X(3872)\rightarrow VV$ are about a few tens of KeV, while the partial decay
widths can reach several hundred KeV for $X(3872)\to VP$. The behavior is
similar to that of Fig. 3. Since the case of $\theta=\pi/4$ corresponds to
equal neutral and charged components in $X(3872)$, so the channels $X(3872)\to
K^{\ast+}K^{\ast-}$ and $X(3872)\to K^{\ast 0}\bar{K}^{\ast 0}$ have non-zero
decay widths. The $X(3872)\to K^{\ast 0}\bar{K}^{\ast 0}$ transition proceed
via $[D^{+}D^{*-}]D^{(\ast)}_{s}$ intermediate mesons, while the $X(3872)\to
K^{\ast+}K^{\ast-}$ transition proceed via $[D^{0}\bar{D}^{*0}]D^{(\ast)}_{s}$
intermediate mesons. The mass of $X(3872)$ is much closer the mass threshold
of $D^{0}\bar{D}^{*0}$ than $D^{+}D^{*-}$, so the threshold effects of
$X(3872)\to K^{\ast+}K^{\ast-}$ will be larger than that of $X(3872)\to
K^{\ast 0}\bar{K}^{\ast 0}$. However, the couplings constant values obtained
from Eq. (6) have the relation $g_{n}\textless g_{c}$. Thus with the same
value of $\alpha$, the obtained partial decay width of $X(3872)\rightarrow
K^{\ast 0}\bar{K}^{\ast 0}$ is about several times larger than that of
$X(3872)\rightarrow K^{\ast+}K^{\ast-}$. However, for the $X(3872)\to\rho\rho$
decay, there are contributions from the exchanging both charged charm mesons
and neutral charm mesons, and these two contributions give the instructive
interference of the decay amplitudes. A similar situation occurs in
$X(3872)\rightarrow VP$ as shown in Fig. 3 (b). A similar situation occurs in
$X(3872)\rightarrow VP$ as shown in Fig. 4 (b). In the case of $\theta=\pi/4$,
the charged and neutral charmed meson loops should cancel out exactly in the
isospin symmetry limit for the isospin-violating channel
$X(3872)\to\rho^{0}\omega$. In other words, the mass difference between the
$u$ and $d$ quark will lead to $m_{D^{(*)\pm}}\neq m_{D^{(*)0}}$ due to the
isospin symmetry breaking. As a result, the charged and neutral charmed meson
loops cannot completely cancel out, and the residue part will contribute to
the isospin-violating amplitudes. The partial widths of the isospin-violating
channel $X(3872)\to\omega\rho^{0}$ as shown in Fig. 4 (a) are suppressed.

Using the center value of the total decay width of $X(3872)$ that was reported
recently by the LHCb Collaboration Aaij:2020qga ; Aaij:2020xjx , we obtain the
branching ratios for $X(3872)\rightarrow VV$ and $VP$ in the cases of
$\theta=0$, $\pi/6$ and $\pi/4$ , respectively. We take the range of $\alpha$
as $0.6\sim 1.2$, then the numerical results are shown in the Table. 1. Our
theoretical numerical results show that with the increase of $\theta$, the
partial decay widths of $K^{\ast+}K^{\ast-}$ and $K^{\ast+}K^{-}+c.c.$
channels decrease. Because there is only neutral charmed mesons loops in
$X(3872)\rightarrow K^{\ast+}K^{\ast-}$ and $X(3872)\rightarrow
K^{\ast+}K^{-}$. And also the $X(3872)$ coupling constant to the neutral
channel $g_{n}$ is proportional to $\cos\theta$.

Table 1: The branching ratios for $X(3872)\rightarrow VV$ and $X(3872)\rightarrow VP$ with different $\theta$ values. The $\alpha$ range is taken to be $0.6\sim 1.2$ here. Final states | $\theta=0$ | $\theta=\pi/6$ | $\theta=\pi/4$
---|---|---|---
$\rho\rho$ | $(0.15-7.86)\times 10^{-3}$ | $(0.06-3.20)\times 10^{-2}$ | $(0.83-4.29)\times 10^{-2}$
$K^{\ast+}K^{\ast-}$ | $(0.08-4.11)\times 10^{-3}$ | $(0.06-3.08)\times 10^{-3}$ | $(0.04-2.05)\times 10^{-3}$
$K^{\ast 0}\bar{K}^{\ast 0}$ | $--$ | $(0.11-5.36)\times 10^{-3}$ | $(0.02-1.07)\times 10^{-2}$
$\omega\omega$ | $(0.03-1.55)\times 10^{-3}$ | $(0.12-6.28)\times 10^{-3}$ | $(0.16-8.41)\times 10^{-3}$
$\rho^{0}\omega$ | $(0.03-1.56)\times 10^{-3}$ | $(0.02-1.25)\times 10^{-4}$ | $(0.03-1.31)\times 10^{-3}$
$\rho^{\pm}\pi^{\mp}$ | $(0.09-4.40)\times 10^{-2}$ | $(0.004-1.87)\times 10^{-1}$ | $(0.05-2.53)\times 10^{-1}$
$K^{\ast+}K^{-}+c.c.$ | $(0.08-3.99)\times 10^{-2}$ | $(0.06-2.99)\times 10^{-2}$ | $(0.04-1.99)\times 10^{-2}$
$K^{\ast 0}\bar{K}^{0}+c.c.$ | $--$ | $(0.11-5.66)\times 10^{-2}$ | $(0.02-1.13)\times 10^{-1}$

In Fig. 5, we present the partial decay widths of the $X(3872)\rightarrow VV$
in terms of the mass of $X(3872)$, where we have fixed the value of $\alpha$
as $1.0$. The coupling constant of $X(3872)$ in Eq. (1) and the threshold
effects can simultaneously influence the the mass of $X(3872)$ dependence of
the decay widths. Generally speaking, with increasing the mass difference
between $X(3872)$ and $D^{*0}{\bar{D}}^{0}$ mesons, i.e., increasing the
binding energy, the coupling strength of $X(3872)$ increases, and the
threshold effects decrease. Both the coupling strength of $X(3872)$ and the
threshold effects vary quickly in the small binding energy region and slowly
in the large binding energy region. As a result, the behavior of the partial
widths is relatively sensitive at small binding energy, while it becomes
smooth at large binding energy. The single-cusp structure locates at the the
thresholds of the $D^{*0}{\bar{D}}^{0}$ mesons for most of the decay channels
except for $K^{*0}\bar{K}^{*0}$ channel. This is because the $X(3872)\to
K^{\ast 0}\bar{K}^{\ast 0}$ transition proceed via
$[D^{+}D^{*-}]D^{(\ast)}_{s}$ intermediate mesons. A similar behavior of
partial widths occur in $X(3872)\rightarrow VP$ as shown in the Fig. 6.

It would be interesting to further clarify the uncertainties arising from the
introduction of form factors by studying the $\alpha$ dependence of the ratios
between different partial decay widths. For the decays $X(3872)\to VV$, we
define the following ratios to the partial decay widths of
$X(3872)\to\omega\omega$

$\displaystyle R_{1}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(X(3872)\to\omega\rho^{0})}{\Gamma(X(3872)\to\omega\omega)}\,,$
$\displaystyle R_{2}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(X(3872)\to\rho\rho)}{\Gamma(X(3872)\to\omega\omega)}\,,$
$\displaystyle R_{3}$ $\displaystyle=$ $\displaystyle\frac{\Gamma(X(3872)\to
K^{\ast+}K^{\ast-})}{\Gamma(X(3872)\to\omega\omega)}\,,$ $\displaystyle R_{4}$
$\displaystyle=$ $\displaystyle\frac{\Gamma(X(3872)\to K^{\ast 0}\bar{K}^{\ast
0})}{\Gamma(X(3872)\to\omega\omega)}\,.$ (28)

For the decays of $X(3872)\to VP$, the following ratios are defined:

$\displaystyle r_{1}$ $\displaystyle=$ $\displaystyle\frac{\Gamma(X(3872)\to
K^{\ast+}K^{-}+c.c.)}{\Gamma(X(3872)\to\rho\pi)}\,,$ $\displaystyle r_{2}$
$\displaystyle=$ $\displaystyle\frac{\Gamma(X(3872)\to K^{\ast
0}\bar{K}^{0}+c.c.)}{\Gamma(X(3872)\to\rho\pi)}\,.$ (29)

The ratios $R_{1}$ in terms of $\alpha$ are plotted in Fig. 7. The results of
Fig. 7 show that the ratios are completely insensitive to this dependence.
This stabilities of the ratios in terms of $\alpha$ indicate a reasonably
controlled cutoff for each channel by the form factor to some extent. On the
other hand, one can see that, in Fig. 7, there are extremely strong dependence
of the ratio on the isospin mixing angle, $\theta$, which is of more
fundamental significance than the parameter $\alpha$. This stability stimulate
us to study the mixing angle $\theta$ dependence.

Figure 7: The $\alpha$-dependence of the ratio $R_{1}$ defined in Eq. (28).

Figure 8: (a). The ratio $R_{i}$ $(i=1,2,3,4)$ defined in Eq. (28) as a
function of the mixing angle $\theta$ with $\alpha=1.0$. (b). The ratio
$r_{i}$ $(i=1,2)$ defined in Eq. (29) as a function of the mixing angle
$\theta$ with $\alpha=1.0$.

Next, we turn to the dependence of these ratios defined in Eqs. (28) and (29)
to the mixing angle $\theta$ with a fixed $\alpha$. In Fig. 8, we present the
theoretical results of the ratio $R_{i}$ $(i=1,2,3,4)$ defined in Eq. (28) and
$r_{i}$ $(i=1,2)$ defined in Eq. (29) as a function of the mixing angle
$\theta$ with a fixed value $\alpha=1.0$. It is interesting to note that the
results of the ratio
$R_{2}=\frac{\Gamma(X(3872)\to\rho\rho)}{\Gamma(X(3872)\to\omega\omega)}$ are
not dependent on the value of $\theta$. These ratios shown in Fig. 8 may be
tested by the future experimental measurements and can be used to determine
the value of the mixing angle.

##  IV Summary

Based on a molecular nature of $X(3872)$, we have investigated the charmless
decays of $X(3872)\to VV$ and $VP$. For $X(3872)$, we considered three cases,
i.e., pure neutral components ($\theta=0$), isospin singlet ($\theta=\pi/4$)
and neutral components dominant ($\theta=\pi/6$), where $\theta$ is a phase
angle describing the proportion of neutral and charged constituents. We
explore the rescattering mechanism within the effective Lagrangian based on
the heavy quark symmetry and chiral symmetry. We can see that although the
decay widths increase with the increase of $\alpha$ when we fix the phase
angle $\theta$, our theoretical results show that the cutoff parameter
$\alpha$ dependence of the partial widths is not drastically sensitive, which
indicates the dominant mechanism driven by the intermediate meson loops with a
fairly good control of the ultraviolet contributions. When $X(3872)$ is a pure
neutral bound state, the predicted partial decay widths of $X(3872)\rightarrow
VV$ are about a few keV, while the partial decay widths can reach a few tens
of keV for $X(3872)\to VP$. When there are both neutral and charged components
in $X(3872)$, the predicted decay widths of $X(3872)\rightarrow VV$ are about
tens of keV. while the decay widths can reach a few hundreds of keV for
$X(3872)\rightarrow VP$.

Moreover, the dependence of these ratios between different charmless decay
modes of $X(3872)$ to the charged and neutral mixing angle for the $X(3872)$
in the molecular picture is also investigated, which may be tested by future
experiments and can be used to determine the value of the mixing angle.

## Acknowledgements

We thank the anonymous referee for very constructive comments on the
manuscript. This work is supported by the National Natural Science Foundation
of China, under Grants Nos. 12075133, 11835015, 11975165, 12075288, 11735003,
11675131, and 11961141012. It is also partly supported by Taishan Scholar
Project of Shandong Province (Grant No. tsqn202103062), the Higher Educational
Youth Innovation Science and Technology Program Shandong Province (Grant No.
2020KJJ004), and the Youth Innovation Promotion Association CAS.

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