$Z^{\epsilon}:=\Phi((\log\log(1/\epsilon))^{-1}Q_{(\epsilon\log\log(1/\epsilon))^{1/2}}A\;,\;(\log\log(1/\epsilon))^{-1/2}R_{\epsilon}W),$

where $Q_{\epsilon}f(t)=\epsilon^{-1}f(\epsilon t)$,
$R_{\epsilon}f(t)=\epsilon^{-1/2}f(\epsilon t)$, and $\Phi(f,g)=f\circ g$ is
the composition map as above. We remark that if we view $A=\psi(B_{1},B_{2})$
as a chaos of order 2, then we have a commutation relation
$Q_{\epsilon}\psi=\psi\tilde{R}_{\epsilon}$ where
$\tilde{R}_{\epsilon}(f_{1},f_{2})(t)=(\epsilon^{-1/2}f_{1}(\epsilon
t),\epsilon^{-1/2}f_{2}(\epsilon t))$ for $(f_{1},f_{2})\in C[0,1]\times
C[0,1]$. Note in the above expression that the exponent of $-1$ on the double
logarithm in the first coordinate is consistent with the fact that $A$ is a
chaos of order 2.

Hence, mimicking the previous example but using the full generality of the
above theorem, we see that $Z^{\epsilon}$ has the set of limit points given by
the closure of functions of the form $f\circ g$ where $f,g\in C[-1,1]$ are
smooth and $f$ is of the form
$\frac{1}{2}\int_{0}^{\bullet}h_{2}h_{1}^{\prime}-h_{1}h_{2}^{\prime}$ where
$\int_{-1}^{1}g^{\prime}(t)^{2}+h_{1}^{\prime}(t)^{2}+h_{2}^{\prime}(t)^{2}dt\leq
1$, and $g(0)=h_{i}(0)=0$ for $i=1,2$.

##  Appendix A Ergodic properties of measure-preserving linear operators

In this appendix we review the fact that the ergodic properties of measure-
preserving linear operators on an abstract Wiener space are necessarily
determined by their action on the Cameron-Martin space.

######  Definition A.1.

Let $(X,H,\mu)$ be an abstract Wiener space. We say that a Borel-measurable
map $S:X\to X$ is a measure-preserving a.e. linear map if there exists a
Borel-measurable linear subspace $E\subset X$ with $\mu(E)=1$ such that
$S:E\to X$ is linear and $\mu(S^{-1}(F))=\mu(F)$ for all Borel subsets
$F\subset X.$

######  Lemma A.2.

Let $(X,H,\mu)$ be an abstract Wiener space. The set of measure-preserving
a.e. linear maps (modulo a.e. equivalence) is in bijection with the set of
bounded linear maps from $H\to H$ satisfying $SS^{*}=I$. Moreover the
bijection is given by simply restricting $S$ to $H$.

###### Proof.

We need to show that any a.e. defined measure-preserving linear transformation
from $X\to X$ necessarily maps $H\to H$ boundedly and satisfies $SS^{*}=I$ on
$H$. To prove this, let $S:E\to X$ be a measure preserving Borel measurable
linear map, where $E\subset X$ is a Borel measurable linear subspace
satisfying $\mu(E)=1$. It is known [Hai09, Section 3.4] that $H$ equals the
intersection of all Borel-measurable linear subspaces of $X$ of measure $1$.
If $F$ is any Borel-measurable linear subspace of measure $1$, then so is
$E\cap S^{-1}(F)$, and thus $x\in H$ implies $x\in S^{-1}(F)$ so that $Sx\in
F$. Since $F$ is arbitrary, we have shown that $Sx\in H$ for all $x\in H$.
Thus $S$ is a globally defined Borel measurable linear map from $H\to H$, from
which it follows that $S$ is automatically bounded [Sch66]. In order for $S$
to be measure-reserving, it must clearly satisfy $SS^{*}=I$ (e.g. by computing
the covariance structure of the pushforward measure $S_{*}\mu$).

Conversely, given any bounded linear map $S:H\to H$ satisfying $SS^{*}=I$,
there exists a $\mu$-a.e.-defined Borel-measurable linear extension $\hat{S}$
on $X$ which is unique up to a.e. equivalence [Hai09, Sections 3.5 and 3.6].
Moreover the condition $SS^{*}=I$ guarantees that this extension is measure-
preserving, as this is precisely the condition under which the covariance
operator of the underlying Gaussian measure is unchanged.∎

Below and in the main body of the paper we always write $\hat{S}=S$ without
specifying that it actually denotes the unique extension. Next we have a lemma
about the structure of such measure-preserving transformations, namely that
they can be orthogonally decomposed into a unitary part and a part converging
strongly to zero.

######  Lemma A.3.

Let $H$ be a Hilbert space and consider any linear operator $S:H\to H$
satisfying $SS^{*}=I$. Then we can orthogonally decompose $H=A\oplus B$ where
$A$ and $B$ are both invariant under $S$ and $S^{*}$. Moreover $S:A\to A$ is
unitary and $\lim_{n\to\infty}\|S_{n}x\|_{H}=0$ for all $x\in B.$ Explicitly
one can write $A=\bigcap_{n\in\mathbb{N}}\text{Im}(S_{n}^{*})$ and
$B=\overline{\bigcup_{n\in\mathbb{N}}\ker(S_{n})}$, where the bar denotes the
closure in $H$.

###### Proof.

Since $SS^{*}=I$, we have $S_{n}S_{n}^{*}=I$. It is clear that $S$ and $S^{*}$
both leave $\bigcap_{n\in\mathbb{N}}\text{Im}(S_{n}^{*})$ invariant and
$S^{*}S$ is the identity there. For each $n$, one has
$H=\text{Im}(S_{n}^{*})\oplus\ker(S_{n})$ via the decomposition
$x=S_{n}^{*}S_{n}x+(x-S_{n}^{*}S_{n}x)$. Hence, $S_{n}^{*}S_{n}$ is merely the
projection map from $H$ onto $\text{Im}(S_{n}^{*})$ and indeed the two given
subspaces $A,B$ are orthogonal. Moreover it is clear that $\|S_{n}x\|\to 0$ on
the closure of $\bigcup_{n\in\mathbb{N}}\ker(S_{n})$, since $\|S_{n}x\|$ is
eventually zero for all $x$ in the dense subspace $\bigcup\ker(S_{n})$. ∎

Below, we will refer to the subspace $A$ as the unitary part. Given Lemma A.2,
it is natural then to ask what conditions on $S$, when viewed as a map from
$H\to H$, ensure that the measure-preserving extension $S$ is ergodic, weakly
mixing and strongly mixing. The next proposition addresses this.

######  Proposition A.4.

Let $(X,H,\mu)$ be an abstract Wiener space. Consider any linear operator
$S:H\to H$ satisfying $SS^{*}=I$. Then we have the following equivalences.

  1. (1)

$\bigcap_{n\geq 1}\sigma(S_{n})$ is a 0-1 $\sigma$-algebra (i.e., consists
only of sets of $\mu$-measure 0 or 1) if and only if $\|S_{n}x\|_{H}\to 0$ as
$n\to\infty$ for all $x\in H.$

  2. (2)

$S$ is strongly mixing if and only if $\langle S_{n}x,y\rangle_{H}\to 0$ as
$n\to\infty$ for all $x,y\in H$.

  3. (3)

$S$ is ergodic if and only if any of the following five equivalent conditions
hold:

    1. (a)

$\frac{1}{n}\sum_{j=1}^{n}\langle S_{j}x,y\rangle_{H}^{k}\to 0$ as
$n\to\infty$ for all $x,y\in H$ and $k\in\mathbb{N}.$

    2. (b)

$\frac{1}{n}\sum_{j=1}^{n}|\langle S_{j}x,y\rangle_{H}|\to 0$ as $n\to\infty$
for all $x,y\in H.$

    3. (c)

$S$ is weakly mixing.

    4. (d)

$S$ admits no invariant subspace of dimension two on which it acts by a
rotation matrix.

    5. (e)

The spectral measure $\mu_{x}$ of $S$ is atomless for every $x$ in the unitary
part of $S$.

These statements are classical, but we give a proof of the proposition for
completeness. For example, Item (3e) is a reformulation of the well-known
Maruyama theorem on ergodicity of shifts of Gaussian fields [Mar49]. The
equivalence of ergodicity and weak mixing is a special case of e.g. [RZ97].
Regarding (2) and (3), Ustunel and Zakai have proved stronger results even for
nonlinear maps of the Wiener space, see e.g. [UZ95, UZ00, UZ01].

In the proof we will use the standard fact that for each $h\in H$ there exists
an a.e. defined linear extension of the map from $H\to\mathbb{R}$ given by
$v\mapsto\langle v,h\rangle_{H}$. By an abuse of notation we denote this
linear extension as $\langle\cdot,h\rangle$ as well, and the map from $H\to
L^{2}(X,\mu)$ given by $h\mapsto\langle
h,{\raisebox{-2.15277pt}{\scalebox{1.8}{$\cdot$}}}\rangle$ is a linear
isometry. In particular, the law of each $\langle
h,{\raisebox{-2.15277pt}{\scalebox{1.8}{$\cdot$}}}\rangle$ is a Gaussian of
variance $\|h\|_{H}^{2}$ with respect to $\mu$, see [Hai09, Section 3.4].

###### Proof of Item (1).

Assume first that $\|S_{n}x\|_{H}\to 0$ for all $x\in H.$ Then one can
decompose $H$ into an orthogonal direct sum: $H=\bigoplus_{n\geq 0}H_{n}$
where $H_{n}:=\ker(S_{n+1})\cap\text{Im}(S_{n}^{*}),$ via the formula

$x=\sum_{k=0}^{\infty}S_{k}^{*}S_{k}x-S_{k+1}^{*}S_{k+1}x.$

The series converges to $x$ in $H$ because the $N^{th}$ partial sum equals
$x-S_{N+1}^{*}S_{N+1}x$ and we know that
$\|S_{N+1}^{*}S_{N+1}x\|_{H}\leq\|S_{N+1}x\|_{H}\to 0$. Moreover one easily
checks that
$S_{k}^{*}S_{k}x-S_{k+1}^{*}S_{k+1}x\in\ker(S_{k+1})\cap\text{Im}(S_{k}^{*})$,
and that $\ker(S_{i})\cap\text{Im}(S_{i-1}^{*})$ is orthogonal to
$\ker(S_{j})\cap\text{Im}(S_{j-1}^{*})$ for $i<j$.

Let $\xi$ be a $B$-valued random variable with law $\mu$ and define $\xi_{n}$
to be the projection of $\xi$ onto $H_{n}$. Note that the $\xi_{n}$ are
independent $X$-valued random variables (but not necessarily i.i.d.). Note
also that $S_{n}\xi$ is measurable with respect to $\\{\xi_{j}\\}_{j\geq n}$.
Consequently $\bigcap_{n\geq 1}\sigma(S_{n}\xi)\subseteq\bigcap_{n\geq
1}\sigma(\\{\xi_{j}:j\geq n\\}),$ which by Kolmogorov’s 0-1 law is a 0-1 sigma
algebra.

Conversely, suppose that $\|S_{n}x\|\not\to 0$ as $n\to\infty$ for some $x\in
H.$ Then by Lemma A.3, the closed subspace $A:=\bigcap_{n\geq
0}\text{Im}(S_{n}^{*})$ is nonzero. Letting $\xi$ denote a random variable in
$X$ with law $\mu$, let $\xi_{A}$ denote the projection onto $A$ applied to
$\xi$. Since $S|_{A}$ is unitary (again by Lemma A.3) it is clear that
$\bigcap_{n}\sigma(S_{n})$ contains at least $\sigma(\xi_{A})$, which is
nontrivial since $A\neq\\{0\\}$ so that at least one nonzero Gaussian random
variable is measurable with respect to it. ∎

###### Proof of Item (2).

Suppose that $\langle S_{n}x,y\rangle_{H}\to 0$ for all $x,y\in H$. We wish to
show that

$\lim_{n\to\infty}\int_{X}f(S_{n}x)g(x)\mu(dx)=0$

for all bounded measurable functions $f,g:X\to\mathbb{R}$ such that
$\int_{X}fd\mu=\int_{X}gd\mu=0$. By an application of Cauchy-Schwarz and the
measure-preserving property of $S_{n}$, it suffices to prove this in a dense
subspace of $L^{2}(X,\mu)$.

By using the Wick forumla (aka Isserlis’ theorem), one can easily show that
the claim is at least true whenever $f$ and $g$ are both of the form $x\mapsto
p(\langle x,e_{1}\rangle,...,\langle x,e_{k}\rangle)$ for some
$k\in\mathbb{N}$, some polynomial $p:\mathbb{R}^{k}\to\mathbb{R}$ and some
orthonormal set of vectors $e_{1},...,e_{k}$ in $H$. Then by a density
argument and the fact that the $\langle x,e_{i}\rangle$ has Gaussian tails,
one can extend this from $k$-variable polynomials $p$ to all continuous
functions $f,g:\mathbb{R}^{k}\to\mathbb{R}$ with at-worst polynomial growth at
infinity. One can then extend to all bounded Borel-measurable functions
$f,g:\mathbb{R}^{k}\to\mathbb{R}$ by density of continuous functions in
$L^{2}(\mathbb{R}^{k},\gamma_{k})$ where $\gamma_{k}$ is the standard Gaussian
measure on $\mathbb{R}^{k}$.

Thus, to finish the argument, it suffices to show that the set of all
functions of the form $x\mapsto f(\langle x,e_{1}\rangle,...,\langle
x,e_{k}\rangle)$, where $f:\mathbb{R}^{k}\to\mathbb{R}$ is bounded and
measurable, is dense in $L^{2}(X,\mu)$. To show this, choose an orthonormal
basis $\\{e_{j}\\}$ for $H$ and Let $\mathcal{F}_{n}$ denote the sigma algebra
generated by $\langle x,e_{1}\rangle,...,\langle x,e_{n}\rangle$. Let $f\in
L^{\infty}(X,\mu)$ and let $f_{n}:=\mathbb{E}[f|\mathcal{F}_{n}].$ Then
$f_{n}$ is bounded and measurable and of the form $x\mapsto h(\langle
x,e_{1}\rangle,...,\langle x,e_{n}\rangle)$. Furthermore by martingale
convergence $\|f_{n}-f\|_{L^{2}}\to 0$, completing the proof.

The converse direction is striaghtforward: if $S$ is strongly mixing, then
apply the mixing definition to $f(x):=\langle x,a\rangle$ and $g(x):=\langle
x,b\rangle$ to conclude that $\langle S_{n}a,b\rangle\to 0$. ∎

###### Proof of Item (3).

We are going to show that ergodicity implies (a) which implies (b) which
implies (c). Clearly (c) implies ergodicity. Then we show that (d) is
equivalent to (e) for the unitary part of the operator from the decomposition
in Lemma A.3. Then we will show that for unitary operators, (e) holds if and
only if (b) holds.

So assume that $S$ is ergodic as a map from $X\to X.$ This is equivalent to
the statement that
$\int_{X}\big{(}\frac{1}{n}\sum_{j=1}^{n}f(S_{j}x)\big{)}g(x)\mu(dx)\to 0$ for
all $f,g\in L^{2}(X,\mu)$ such that $\int_{X}fd\mu=\int_{X}gd\mu=0$. Now fix
$k\in\mathbb{N}$ and $a,b\in H$. Letting $H_{k}$ denote the $k^{th}$ Hermite
polynomial, we set $f(x):=\frac{1}{\sqrt{k!}}H_{k}(\langle x,a\rangle)$ and
$g(x):=\frac{1}{\sqrt{k!}}H_{k}(\langle x,b\rangle).$ Then
$\int_{X}f(S_{j}x)g(x)d\mu=\langle a,S_{j}b\rangle_{H}^{k}$ (see e.g.
[Nua06]), so by ergodicity we obtain (a).

Now assume (a) holds. Let $a,b\in H$ with $\|a\|=\|b\|=1$. Fix $\epsilon>0$
and let $p:[-1,1]\to\mathbb{R}$ be a polynomial such that
$\sup_{x\in[-1,1]}\big{|}p(x)-|x|\big{|}<\epsilon.$ Since $|\langle
a,S_{j}b\rangle|\leq 1$ for all $j\in\mathbb{N}$, it follows that
$\frac{1}{n}\sum_{j=1}^{n}\big{|}p(\langle a,S_{j}b\rangle)-|\langle
a,S_{j}b\rangle|\big{|}<\epsilon.$ Moreover, since (a) holds we know that
$\frac{1}{n}\sum_{j=1}^{n}p(\langle a,S_{j}b\rangle)\to 0$. Consequently we
find that $\limsup_{n}\frac{1}{n}\sum_{j=1}^{n}|\langle
a,S_{j}b\rangle_{H}|\leq\epsilon.$ Since $\epsilon$ is arbitrary, it follows
that (b) holds.

Now assume that (b) holds. To show (c), we want that
$\frac{1}{n}\sum_{j=1}^{n}\big{|}\int_{X}f(S_{j}x)g(x)\mu(dx)\big{|}\to 0$ for
all $f,g\in L^{2}(X,\mu)$ that have mean zero. First note that, by essentially
the same series of density arguments given in the proof of Item (2), it
suffices to prove this whenever $f,g$ are of the form $x\mapsto p(\langle
x,e_{1}\rangle,...,\langle x,e_{k}\rangle)$ for some $k\in\mathbb{N}$, some
polynomial $p:\mathbb{R}^{k}\to\mathbb{R}$ and some orthonormal set of vectors
$e_{1},...,e_{k}$ in $H$. In turn, by Wick formula it suffices to show that
$\frac{1}{n}\sum_{j=1}^{n}|\langle a,S_{j}b\rangle_{H}|^{k}\to 0$ for all
$a,b\in H$ and all $k\in\mathbb{N}$. Note that we have $|\langle
a,S_{j}b\rangle|\leq\|a\|\|b\|,$ so $|\langle
a,S_{j}b\rangle_{H}|^{k}\leq\|a\|^{k-1}\|b\|^{k-1}|\langle
a,S_{j}b\rangle_{H}|$. Summing over $j$ and applying (b), we conclude
$\frac{1}{n}\sum_{j=1}^{n}|\langle a,S_{j}b\rangle_{H}|^{k}\to 0$.

It is straightforward to show that (using the spectral theorem of the unitary
operator) atoms of $\mu_{x}$ correspond precisely to complex eigenvalues of
$S$, i.e., two-dimensional subspaces on which $S$ acts by rotation. Thus (d)
implies (e) and vice versa (by focusing only on the Finally we explain why (e)
is equivalent to (b). The spectral measure $\mu_{x}$ of $S$ is supported on
$\mathbb{T}:=\\{z\in\mathbb{C}:|z|=1\\}$ and defined via its Fourier
transform: $\hat{\mu}_{x}(k):=\langle S_{k}x,x\rangle$ where $k\in\mathbb{Z}$
and $S_{k}:=S_{-k}^{*}$ if $k<0$. Thus, to show that (b) and (e) are
equivalent, we just need to show that a finite measure $\mu$ on $\mathbb{T}$
is atomless if and only if $\frac{1}{n}\sum_{k=-n}^{n}|\hat{\mu}(k)|\to 0$ as
$n\to\infty$. This is a direct consequence of Wiener’s Lemma. ∎

##  Appendix B Gaussian measures on spaces of distributions

The results of this appendix are included only for completeness and
convenience. Here we show under general conditions how to identify the
Cameron-Martin space of a given Gaussian measure on some space of
distributions, and how to identify when a semigroup of operators on some
Banach space of distributions is strongly continuous or satisfies the mixing
condition, as needed in Theorem 1.3. We let $U\subset\mathbb{R}^{d}$ be any
open set (possibly unbounded), and we let $\bar{U}$ denote its closure.

######  Definition B.1.

We shall consider a vector space $\mathcal{F}(U)$ of smooth bounded functions
$f:\bar{U}\to\mathbb{R}$ (not necessarily vanishing on $\partial U$) such that
all of the seminorms

$\|f\|_{\beta,n}:=\sup_{x\in\bar{U}}(1+|x|^{2n})|\partial^{\beta}f(x)|$

are finite, as we range over $n\in\mathbb{N}$ and multi-indices $\beta.$ Here
$|x|$ denotes Euclidean norm of the vector $x\in\mathbb{R}^{d}.$ We will
always assume the following:

  * •

$\mathcal{F}(U)$ contains all smooth functions with compact support contained
in $U$.

  * •

$\mathcal{F}(U)$ is complete with respect to the above family of seminorms.

Such a space $\mathcal{F}(U)$ has the structure of a Frechet space if we endow
it with these seminorms, hence it has a continuous dual which will be denoted
as $\mathcal{F}^{\prime}(U)$, endowed with its weak* topology.

Note that one has a natural embedding
$\mathcal{F}(U)\hookrightarrow\mathcal{F}^{\prime}(U)$ given by identifying a
smooth function $f$ with the element in the dual given by sending
$g\mapsto\int_{\bar{U}}fg$. Such an embedding is injective by the first bullet
point above. Below we will not distinguish $\mathcal{F}(U)$ from the image of
this embedding, instead simply viewing all spaces as subsets of
$\mathcal{F}^{\prime}(U)$.

A typical example of $\mathcal{F}(U)$ will be of the following form: assume
$U\subset\mathbb{R}^{d}$ is bounded with smooth boundary, then partition the
boundary of $U$ into some finite number of subsets, and let $\mathcal{F}(U)$
be the set of smooth functions whose derivatives satisfy some linear relation
on each element of the partition (e.g. $U=(0,1)$ and $\mathcal{F}(U)$ consists
of those smooth functions on $[0,1]$ whose derivatives up to order $k$ vanish
at the left endpoint, where $k\in\mathbb{N}$). Unless $U=\mathbb{R}^{d}$, the
spaces $\mathcal{F}(U)$ and $\mathcal{F}^{\prime}(U)$ will typically not be
closed under the partial differentiation operators in the same way that
$\mathcal{S}(\mathbb{R}^{d})$ and $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ are.

Note that the continuous dual $\mathcal{F}^{\prime}(U)$ as a topological space
(despite not being second-countable) has a Borel $\sigma$-algebra which is
generated by a countably infinite set of continuous linear functionals
$\mathcal{F}^{\prime}(U)\to\mathbb{R}$. Indeed continuous linear functionals
on $\mathcal{F}^{\prime}(U)$ are precisely elements of $\mathcal{F}(U)$. The
latter is second-countable and metrizable, hence admits a countable dense
subset. Since the topology of $\mathcal{F}^{\prime}(U)$ is (by definition)
generated by all continuous linear functionals on it, this countable set of
continuous linear functionals is sufficient to generate the Borel
$\sigma$-algebra on $\mathcal{F}^{\prime}(U)$.

######  Definition B.2.

We say that a Banach space $X$ is embedded between $\mathcal{F}(U)$ and
$\mathcal{F}^{\prime}(U)$ if $X$ is a Borel-measurable linear subspace of
$\mathcal{F}^{\prime}(U)$ which contains $\mathcal{F}(U),$ and such that the
norm of $X$ (defined to be infinity outside of $X$) is a Borel-measurable
function on $\mathcal{F}^{\prime}(U)$.

The norm topology on a Banach space is the one induced by the metric
$d(x,y)=\|x-y\|$. The weak topology on a Banach space $X$ is the one generated
by the linear functionals $f\in X^{*}$, and the weak* topology on $X^{*}$ is
the one generated by the linear functionals on $X^{*}$ given by
$\hat{x}:f\mapsto f(x)$ as $x$ ranges throughout $X.$

######  Lemma B.3.

A linear subspace of a Banach space is weak-dense if and only if it is norm-
dense.

The proof is immediate by Hahn-Banach. Our next lemma is a result which
roughly says that if $X$ is a Banach space of distributions that is embedded
between $\mathcal{F}(U)$ and $\mathcal{F}^{\prime}(U)$ in such a way that $X$
densely contains $\mathcal{F}(U)$, then its dual space $X^{*}$ is also
embedded between $\mathcal{F}(U)$ and $\mathcal{F}^{\prime}(U)$, up to an
isometric isomorphism. The dual space may no longer densely contain
$\mathcal{F}(U)$, but in the weak* topology it still does. We will use
$(\cdot,\cdot)$ to denote the natural pairing between $\mathcal{F}(U)$ and
$\mathcal{F}^{\prime}(U)$ with $(\phi,\psi)=\int_{\bar{U}}\phi\psi$ if
$\phi,\psi\in\mathcal{F}(U)$, i.e., the pairing in $L^{2}(\bar{U})$.

######  Lemma B.4.

Suppose $X$ is a Banach space that is embedded between $\mathcal{F}(U)$ and
$\mathcal{F}^{\prime}(U)$. Suppose that the image of the first embedding is
norm-dense in $X$. Then there exists a Banach space $X^{du}$ which is also
embedded between $\mathcal{F}(U)$ and $\mathcal{F}^{\prime}(U)$, and which
admits a ”canonical” bilinear pairing $B:X\times X^{du}\to\mathbb{R}$ in the
sense that following properties hold:

  1. (1)

$|B(x,f)|\leq\|x\|_{X}\|f\|_{X^{du}}$.

  2. (2)

The map from $X^{du}\to X^{*}$ given by $f\mapsto B(\cdot,f)$ is an
isomorphism and a linear isometry.

  3. (3)

$B(\phi,f)=(f,\phi)$ and $B(x,\phi)=(x,\phi)$ for all $x\in X$, all $f\in
X^{du},$ and all $\phi\in\mathcal{F}(U).$

The image of the first embedding $\mathcal{F}(U)\hookrightarrow X^{du}$ may
not be norm-dense in $X^{du}$, but it is always dense with respect to the
weak* topology on $X^{du}$.

Before proving the lemma, let us give examples when $U=\mathbb{R}^{d}$ so
$\mathcal{F}(U)=\mathcal{S}(\mathbb{R}^{d})$. If $X=L^{p}(\mathbb{R}^{d})$
with $1\leq p<\infty$, then $X^{du}=L^{q}(\mathbb{R}^{d})$ with
$\frac{1}{p}+\frac{1}{q}=1.$ If $X=C_{0}(\mathbb{R}^{d})$ then $X^{du}$
consists of finite signed Borel measures on $\mathbb{R}^{d}$ equipped with
total variation norm. Note in this case that the weak-closure of
$\mathcal{S}(\mathbb{R}^{d})$ in $X^{du}$ is $L^{1}(\mathbb{R}^{d})$, which is
a proper closed subspace of $X^{du}$, but still dense with respect to weak
(i.e. Prohorov) convergence of measures. Further examples are given by Sobolev
spaces $X=H^{s}(\mathbb{R}^{d})$ and $X^{du}=H^{-s}(\mathbb{R}^{d})$ with
$s\in\mathbb{R}$. A rich class of examples is given by more general Sobolev
spaces and Besov spaces including those defined with weight functions.

###### Proof.

Note that we have a map $\mathcal{G}$ from $X^{*}\to\mathcal{F}^{\prime}(U)$
given by restriction to the Schwartz class, i.e.,
$f\stackrel{{\scriptstyle\mathcal{G}}}{{\mapsto}}f|_{\mathcal{F}(U)}.$ The map
clearly defines a continuous linear operator from
$X^{*}\to\mathcal{F}^{\prime}(U)$. We define $X^{du}$ to be the image of
$X^{*}$ under $\mathcal{G}$. We also note that $\mathcal{G}$ is injective
since $\mathcal{F}(U)$ is dense in $X$ by assumption. We thus define the norm
on $X^{du}$ by $\|\mathcal{G}f\|_{X^{du}}=\|f\|_{X^{*}}.$ Then clearly
$X^{du}$ is a Banach space that is isometric to $X^{*}$ (via $\mathcal{G}$)
and the inclusion $X^{du}\hookrightarrow\mathcal{F}^{\prime}(U)$ is clearly
continuous.

We now claim that $\mathcal{F}(U)$ is contained in $X^{du}$. To prove this we
need to check that if $\phi\in\mathcal{F}(U)$ then the map $a_{\phi}$ from
$X\to\mathbb{R}$ given by $x\mapsto(x,\phi)$ is continuous. This is clear
because if $x_{n}\to x$ in $X$, then $x_{n}\to x$ in $\mathcal{F}^{\prime}(U)$
so that $(x_{n},\phi)\to(x,\phi).$ Next, we note that
$\phi=\mathcal{G}a_{\phi}\in X^{du}$, proving the claim. By the closed graph
theorem, it follows that the inclusion map $\mathcal{F}(U)\hookrightarrow
X^{du}$ is automatically continuous.

Now we construct the bilinear map $B$. For this, we simply define
$B(x,\mathcal{G}f):=f(x)$ whenever $x\in X$ and $f\in X^{*}$. Clearly
$|B(x,\mathcal{G}f)|\leq\|f\|_{X^{*}}\|x\|_{X}=\|\mathcal{G}f\|_{X^{du}}\|x\|_{X},$
so that $B$ is bounded. Note that if $\phi\in\mathcal{S}(\mathbb{R}^{d})$ then
$B(\phi,\mathcal{G}f)=f(\phi)=(f|_{\mathcal{S}(\mathbb{R}^{d})},\phi)=(\mathcal{G}f,\phi)$,
as desired. Also $B(x,\phi)=B(x,\mathcal{G}a_{\phi})=a_{\phi}(x)=(x,\phi),$
completing the proof. The map $f\mapsto B(\cdot,f)$ is an isometry from
$X^{du}\to X^{*}$ because it is inverse to the isometry $\mathcal{G}.$

Finally we need to show that $\mathcal{F}(U)$ is weak* dense in $X^{du}$. This
follows immediately from the fact that $\mathcal{F}(U)$ is a total set in
$X^{du}$, i.e., it separates points of $X$. This is clear because it separates
points of the larger space $\mathcal{S}^{\prime}$. It is known that totality
is equivalent to weak* density of the finite linear span of any given subset
of $X^{*}$, see [CA13, Corollary 5.108]. ∎

######  Remark B.5.

The assumptions in Lemma B.4 automatically imply separability of $X$ in its
norm topology, because $\mathcal{F}(U)$ is necessarily second-countable and
completely metrizable. However, we also remark that a linear subspace of
$X^{*}$ can certainly be weak* dense without being norm dense, e.g.,
$C_{0}(\mathbb{R})$ is dense in the weak* topology but not the norm topology
of $L^{\infty}(\mathbb{R})=L^{1}(\mathbb{R})^{*}.$ Consequently under the
assumptions of Lemma B.4, $X^{du}$ may not be separable in its norm topology
(e.g. take $U=\mathbb{R}$ and take $X$ to be $L^{1}(\mathbb{R})$ or
$C_{0}(\mathbb{R})$).

######  Remark B.6.

In the above proof, we stated the formula $B(x,\mathcal{G}f)=f(x)$. This
implies that for $f\in X^{du}$, the dual norm is given by the conjugate
formula

$\|f\|_{X^{du}}=\sup\\{(f,\phi):\phi\in\mathcal{F}(U),\|\phi\|_{X}\leq 1\\},$
(B.1)

since $(f,\phi)=B(f,\phi)$ if $\phi\in\mathcal{F}(U)$. Conversely, if
$f\in\mathcal{F}^{\prime}(U)$ such that the right side is finite, then $f\in
X^{du}$ with norm given by that supremum, since $(f,\cdot)\in X^{*}$ so that
$\mathcal{G}((f,\cdot))=f$.

A corollary Lemma B.4 is that if we want to check that a given Hilbert space
$H$ is the Cameron-Martin space of a Gaussian measure $\mu$ on some separable
Banach space $X$ of distributions as above, then it suffices to check the
action of the covariance function only on smooth test functions, as opposed to
the entire dual space of $X$.

######  Lemma B.7.

Let $H$ be a Hilbert space and let $X$ be a Banach space such that both spaces
are embedded between $\mathcal{F}(U)$ and $\mathcal{F}^{\prime}(U).$ Suppose
that $\mathcal{F}(U)$ is norm-dense in both $H$ and $X$. Let $\mu$ be a
Gaussian measure on $X$. If

$\int_{X}(x,\phi)(x,\psi)\mu(dx)=\langle\phi,\psi\rangle_{H}$ (B.2)

for all $\phi,\psi\in\mathcal{F}(U),$ then $\mu$ has Cameron-Martin space
$H^{du},$ which is necessarily contained in $X$.

###### Proof.

The Cameron-Martin norm may be defined for $h\in X$ by

$\|h\|_{CM}=\sup\\{f(h):f\in X^{*},\int_{X}f^{2}d\mu\leq 1\\}.$ (B.3)

By the separability of $X$ and weak* density of $\mathcal{F}(U)$ in $X^{du}$,
for any $f\in X^{*}$, there exist $f_{n}\in\mathcal{F}(U)$ such that $f_{n}$
converges to $f$ pointwise. Since for a sequence of Gaussian random variables,
pointwise convergence implies the convergence in variance, hence we have
$\lim_{n\to\infty}\int_{X}f_{n}^{2}d\mu=\int_{X}f^{2}d\mu.$ Using this
together with (B.3), we have

$\displaystyle\|h\|_{CM}$
$\displaystyle=\sup\\{(h,\phi):\phi\in\mathcal{F}(U),\int_{X}(x,\phi)^{2}\mu(dx)\leq
1\\}$
$\displaystyle=\sup\\{(h,\phi):\phi\in\mathcal{F}(U),\|\phi\|_{H}^{2}\leq
1\\}.$

The second equality is due to (B.2). By the weak density of $\mathcal{S}$ in
$H$, the right hand side above equals the operator norm of the linear
functional on $H$ given by $(\cdot,h)$, which by (B.1) equals the conjugate
norm $\|h\|_{H^{du}}$ (to be understood as $+\infty$ if $h\notin H^{du}$).
Hence, for $h\in X\cap H^{du}$, we have $\|h\|_{CM}=\|h\|_{H^{du}}$.

So far this argument shows that the Cameron-Martin space of $\mu$ equals
$X\cap H^{du}$ (with the norm of $H^{du}),$ which is therefore closed in
$H^{du}$. To finish the proof we need to show that $H^{du}$ does not contain
any vectors outside of $X$. Assume for contradiction that such a vector does
exist; then $X\cap H^{du}$ would have a nonzero orthogonal complement with
respect to the inner product of $H^{du}$. Take some nonzero bounded linear
functional $u:H^{du}\to\mathbb{R}$ which vanishes on the closed subspace
$X\cap H^{du}$. Since $H$ is reflexive, every linear functional on $H^{du}$ is
represented as $B(f,\cdot)$ for some $f\in H$ where $B$ is the bilinear form
constructed in the previous lemma. Thus write $u=B(f,\cdot)$ for some $f\in
H$. Since $X\cap H^{du}$ contains $\mathcal{F}(U)$ we see that
$(f,\phi)=B(f,\phi)=0$ for all $\phi\in\mathcal{F}(U)$. This means that $f=0$
so that $u=0,$ a contradiction. ∎

######  Lemma B.8.

Suppose that $Q$ is a bounded linear operator from $L^{2}(\bar{U})\to
L^{2}(\bar{U})$ such that $(Q\phi,\phi)>0$ for all $\phi\in\mathcal{F}(U).$
Let $X$ be a Banach space that is embedded between $\mathcal{F}(U)$ and
$\mathcal{F}^{\prime}(U)$. Assume that

  * •

$\mu$ is a Gaussian measure on $X$ such that
$\int_{X}(x,\phi)(x,\psi)\mu(dx)=(Q\phi,\psi)$ for all
$\phi,\psi\in\mathcal{F}(U)$.

  * •

The completion of $L^{2}(\bar{U})$ with respect to $(Qf,f)^{1/2}$ embeds
continuously into $\mathcal{F}^{\prime}(U)$.

Then $Q$ has a unique positive square root $Q^{1/2}$, and the image of
$L^{2}(\bar{U})$ under $Q^{1/2}$ contains $\mathcal{F}(U)$, and Cameron-Martin
space of $\mu$ is the completion of $\mathcal{F}(U)$ under the norm
$\|Q^{-1/2}\phi\|_{L^{2}(\bar{U})}.$

We remark that the second bullet point is a nontrivial condition. For example
if $U=\mathbb{R}^{d}$ and $Q$ denotes convolution with heat kernel at time 1,
then the image of $Q^{1/2}$ does not contain
$\mathcal{F}(U)=\mathcal{S}(\mathbb{R}^{d}).$ As always $(\cdot,\cdot)$ here
denotes the pairing in $L^{2}(\bar{U}).$

###### Proof.

First let us verify that $Q$ is closable and positive-definite on
$L^{2}(\bar{U})$. The assumption that that
$\int_{X}(x,\phi)(x,\psi)\mu(dx)=(Q\phi,\psi)$ guarantees that $Q$ is self-
adjoint. It is positive-definite since we assumed that $(Q\phi,\phi)>0$ for
all $\phi$ in the dense subspace $\mathcal{F}(U)$ of $L^{2}(\bar{U})$. By
functional calculus (or just spectral theorem) we can define $Q^{\alpha}$ for
any $\alpha\in\mathbb{R}$, and this is also a closable and densely defined
operator, such that the domain of $Q^{\alpha}$ is contained in the domain of
$Q^{\beta}$ if $\alpha<\beta<0$. Let $\mathcal{I}$ denote the domain of
$Q^{-1}$.

Let $H$ be the Hilbert space obtained by completing $\mathcal{F}(U)$ with
respect to the norm $(Q\phi,\phi)^{1/2}.$ By assumption $H$ embeds into the
continuous dual space $\mathcal{F}^{\prime}(U)$. Then for $\phi\in\mathcal{I}$
one has

$\displaystyle\|Q^{-1/2}\phi\|_{L^{2}(\bar{U})}=\sqrt{(Q^{-1}\phi,\phi)}$
$\displaystyle=\sup\\{(\phi,\psi):\psi\in\mathcal{F}(U),(Q\psi,\psi)\leq 1\\}$
$\displaystyle=\sup\\{(\phi,\psi):\psi\in\mathcal{F}(U),\|\psi\|_{H}\leq
1\\}=\|\phi\|_{H^{du}},$

where the last equality follows from (B.1). Consequently we may conclude that
$H^{du}$ is simply the completion of $\mathcal{I}$ equipped with the norm
given by $\|Q^{-1/2}\phi\|_{L^{2}(\bar{U})},$ and by Lemma B.4 the latter
Hilbert space is in turn necessarily embedded between $\mathcal{F}(U)$ and
$\mathcal{F}^{\prime}(U)$ in the sense of Definition B.2. Lemma B.4 also
implies that $\mathcal{F}(U)$ is dense in the weak* topology, but as $H$ is a
Hilbert space, the weak and weak* topology coincide. As $\mathcal{F}(U)$ is a
linear subspace, Lemma B.3 implies it is norm dense. ∎

Next we identify conditions for a semigroup $S_{t}$ of operators on a Banach
space $X$ embedded in $\mathcal{F}^{\prime}(U)$ to be strongly continuous. The
following lemma is straightforward given Remark B.5.

######  Lemma B.9.

Let $X$ be a Banach space that is embedded between $\mathcal{F}(U)$ and
$\mathcal{F}^{\prime}(U)$, such that $\mathcal{F}(U)$ is weakly dense in $X$.
Let $S_{t}:X\to X$ be a semigroup of bounded operators for $t\geq 0$ such that

  1. (1)

each $S_{t}$ maps $\mathcal{F}(U)$ to itself.

  2. (2)

$S_{t}\phi\stackrel{{\scriptstyle\mathcal{F}(U)}}{{\to}}\phi$ as $t\to 0$ for
all $\phi\in\mathcal{F}(U)$.

  3. (3)

$\sup_{t\leq 1}\|S_{t}\|_{X\to X}<\infty.$

Then $(S_{t})$ is a strongly continuous semigroup on $X$.

In the main body of the paper, we often use the above lemma in multiplicative
form, where the assumptions translate to
$R_{\epsilon}\phi\stackrel{{\scriptstyle\mathcal{F}(U)}}{{\to}}\phi$ as
$\epsilon\uparrow 1$ and $\sup_{\epsilon\in[e^{-1},1]}\|R_{\epsilon}\|_{X\to
X}<\infty$. Finally the following lemma gives an easy way to check the mixing
condition which is required in Theorem 1.3.

######  Lemma B.10.

Let $H$ be a Hilbert space that is embedded between $\mathcal{F}(U)$ and
$\mathcal{F}^{\prime}(U)$, such that $\mathcal{F}(U)$ is weakly dense in $H$.
Let $\mathcal{F}^{o}(U)$ be a dense subset of $\mathcal{F}(U)$. Let $S:H\to H$
be a bounded operator such that $SS^{*}=I$ and $\langle
S^{n}\phi,\psi\rangle_{H}\to 0$ for all $\phi,\psi\in\mathcal{F}^{o}(U)$. Then
$\langle S^{n}\phi,\psi\rangle_{H}\to 0$ for all $\phi,\psi\in H.$

Since $\mathcal{F}^{o}(U)$ is necessarily norm-dense in $H$ (Remark B.5), the
proof is clear by a density argument and the fact that $\|S^{n}\|_{H\to H}\leq
1$ for all $n$. The lemma makes the mixing condition easy to check, since in
examples of interest we often have that $\mathcal{F}^{o}(U)$ consists of
compactly supported functions, and often $\langle S^{n}\phi,\psi\rangle_{H}=0$
for large enough $n$ whenever $\phi,\psi$ are compactly supported.

## References

  * [AG93] Miguel A. Arcones and Evarist Giné.  On decoupling, series expansions, and tail behavior of chaos processes.  J. Theoret. Probab., 6(1):101–122, 1993.
  * [Alb90] J.M.P. Albin.  On the general law of iterated logarithm with application to selfsimilar processes and to gaussian processes in n and hilbert space.  Stoch. Proc. Appl., 1990.
  * [Ale89] Kenneth S. Alexander.  Characterization of the cluster set of the LIL sequence in Banach space.  Ann. Probab., 17(2):737–759, 1989.
  * [Arc95] Miguel A. Arcones.  On the law of the iterated logarithm for Gaussian processes.  J. Theoret. Probab., 8(4):877–903, 1995.
  * [BBAK92] P. Baldi, G. Ben Arous, and G. Kerkyacharian.  Large deviations and the Strassen theorem in Hölder norm.  Stochastic Process. Appl., 42(1):171–180, 1992.
  * [Bog98] Vladimir I. Bogachev.  Gaussian measures.  Mathematical Surveys and Monographs, 1998.
  * [Bur93] Krzysztof Burdzy.  Some path properties of iterated Brownian motion.  In Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992), volume 33 of Progr. Probab., pages 67–87. Birkhäuser Boston, Boston, MA, 1993.
  * [CA13] D Charalambos and Border Aliprantis.  Infinite Dimensional Analysis: A Hitchhiker’s Guide.  Springer-Verlag Berlin and Heidelberg GmbH & Company KG, 2013.
  * [CCFR95] Endre Csáki, Miklós Csörgo, Antónia Földes, and Pál Révész.  Global Strassen-type theorems for iterated Brownian motions.  Stochastic Process. Appl., 59(2):321–341, 1995.
  * [Cha68] S. D. Chatterji.  Martingale convergence and the Radon-Nikodym theorem in Banach spaces.  Math. Scand., 22:21–41, 1968.
  * [Cor12] Ivan Corwin.  The Kardar-Parisi-Zhang equation and universality class.  Random Matrices Theory Appl., 1(1):1130001, 76, 2012.
  * [CS20] Ivan Corwin and Hao Shen.  Some recent progress in singular stochastic partial differential equations.  Bull. Amer. Math. Soc. (N.S.), 57(3):409–454, 2020.
  * [CW15] Ajay Chandra and Hendrik Weber.  Stochastic PDEs, Regularity Structures, and Interacting Particle Systems.  arXiv e-prints, page arXiv:1508.03616, August 2015.
  * [DG21] Sayan Das and Promit Ghosal.  Law of Iterated Logarithms and Fractal Properties of the KPZ Equation.  arXiv e-prints, page arXiv:2101.00730, January 2021.
  * [DU99] L. Decreusefond and A. S. Üstünel.  Stochastic analysis of the fractional Brownian motion.  Potential Anal., 10(2):177–214, 1999.
  * [Erd54] Arthur Erdelyi.  Tables of Integral Transforms (Vol. 1).  McGraw Hill, 1954.
  * [Fat21] Parisa Fatheddin.  The law of the iterated logarithm for a class of SPDEs.  Stoch. Anal. Appl., 39(1):113–135, 2021.
  * [FH20] Peter K. Friz and Martin Hairer.  A course on rough paths.  Universitext. Springer, Cham, [2020] ©2020.  With an introduction to regularity structures, Second edition of [ 3289027].
  * [GH21] Massimiliano Gubinelli and Martina Hofmanová.  A pde construction of the euclidean $\phi$ 3 4 quantum field theory.  Communications in Mathematical Physics, 384(1):1–75, 2021.
  * [GK91] Victor Goodman and James Kuelbs.  Rates of clustering for some Gaussian self-similar processes.  Probab. Theory Related Fields, 88(1):47–75, 1991.
  * [GK93] V. Goodman and J. Kuelbs.  Gaussian chaos and functional laws of the iterated logarithm for Itô-Wiener integrals.  Ann. Inst. H. Poincaré Probab. Statist., 29(4):485–512, 1993\.
  * [GKZ81] V. Goodman, J. Kuelbs, and J. Zinn.  Some results on the LIL in Banach space with applications to weighted empirical processes.  Ann. Probab., 9(5):713–752, 1981.
  * [GMW23] Paolo Grazieschi, Konstantin Matetski, and Hendrik Weber.  Martingale-driven integrals and singular spdes, 2023.
  * [Gro67] Leonard Gross.  Abstract Wiener spaces.  In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), volume II: Contributions to Probability Theory, Part 1, pages 31–42. Univ. California Press, Berkeley, Calif., 1967.
  * [Hai09] Martin Hairer.  An introduction to stochastic pdes.  arXiv preprint arXiv:0907.4178, 2009.
  * [Hai13] Martin Hairer.  Solving the KPZ equation.  Ann. of Math. (2), 178(2):559–664, 2013.
  * [Hai16] M. Hairer.  Advanced stochastic analysis.  Unpublished lecture notes, 2016.
  * [HL18] Martin Hairer and Cyril Labbé.  Multiplicative stochastic heat equations on the whole space.  J. Eur. Math. Soc. (JEMS), 20(4):1005–1054, 2018.
  * [HP15] Martin Hairer and Étienne Pardoux.  A Wong-Zakai theorem for stochastic PDEs.  J. Math. Soc. Japan, 67(4):1551–1604, 2015.
  * [HW15] Martin Hairer and Hendrik Weber.  Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions.  Ann. Fac. Sci. Toulouse Math. (6), 24(1):55–92, 2015.
  * [Kam00] David Kammeler.  A First Course in Fourier Analysis.  Prentice Hall, 2000.
  * [KL73] J. Kuelbs and R. Lepage.  The law of the iterated logarithm for Brownian motion in a Banach space.  Trans. Amer. Math. Soc., 185:253–265, 1973.
  * [KMT76] J. Komlós, P. Major, and G. Tusnády.  An approximation of partial sums of independent RV’s, and the sample DF. II.  Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 34(1):33–58, 1976\.
  * [Kue77] J. Kuelbs.  Kolmogorov’s law of the iterated logarithm for Banach space valued random variables.  Illinois J. Math., 21(4):784–800, 1977.
  * [Lai74] Tze Leung Lai.  Reproducing kernel Hilbert spaces and the law of the iterated logarithm for Gaussian processes.  Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 29:7–19, 1974\.
  * [Led96] Michel Ledoux.  Isoperimetry and gaussian analysis.  In Lectures on probability theory and statistics, pages 165–294. Springer, 1996.
  * [LT11] Michel Ledoux and Michel Talagrand.  Probability in Banach spaces.  Classics in Mathematics. Springer-Verlag, Berlin, 2011.  Isoperimetry and processes, Reprint of the 1991 edition.
  * [LT21] Yier Lin and Li-Cheng Tsai.  Short time large deviations of the KPZ equation.  Communications in Mathematical Physics, 386(1):359–393, mar 2021\.
  * [LWC04] Zhengyan Lin, Wensheng Wang, and Yong-Kab Choi.  Strassen-type laws of iterated logarithm for a fractional Brownian sheet.  Stochastic Anal. Appl., 22(1):193–210, 2004.
  * [Man76] Gian-Carlo Mangano.  On strassen-type laws of the iterated logarithm for gaussian elements in abstract spaces.  Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 36(3):227–239, 1976.
  * [Mar49] Gisiro Maruyama.  The harmonic analysis of stationary stochastic processes.  Mem. Fac. Sci. Kyūsyū Univ. A, 4:45–106, 1949.
  * [MO86] T. Mori and H. Oodaira.  The law of the iterated logarithm for self-similar processes represented by multiple wiener integrals.  Prob. Th. Rel. Fields., 1986.
  * [MQR21] Konstantin Matetski, Jeremy Quastel, and Daniel Remenik.  The KPZ fixed point.  Acta Math., 227(1):115–203, 2021.
  * [MW17] Jean-Christophe Mourrat and Hendrik Weber.  Global well-posedness of the dynamic $\Phi^{4}$ model in the plane.  Ann. Probab., 45(4):2398–2476, 2017.
  * [Neu98] Daniel Neuenschwander.  Law of the iterated logarithm for Lévy’s area process composed with Brownian motion.  Statist. Probab. Lett., 40(4):371–377, 1998.
  * [Nis67] Makiko Nisio.  On the extreme values of Gaussian processes.  Osaka Math. J., 4:313–326, 1967.
  * [Nua06] David Nualart.  The Malliavin calculus and related topics.  Probability and its Applications (New York). Springer-Verlag, Berlin, second edition, 2006.
  * [OM05] M. Ait Ouahra and M. Mellouk.  Strassen’s law of the iterated logarithm for stochastic Volterra equations and applications.  Stochastics, 77(2):191–203, 2005.
  * [Ood72] Hiroshi Oodaira.  On Strassen’s version of the law of the iterated logarithm for Gaussian processes.  Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 21:289–299, 1972\.
  * [Par75] W. J. Park.  The law of the iterated logarithm for Brownian sheets.  J. Appl. Probability, 12(4):840–844, 1975.
  * [Pic67] James Pickands, III.  Maxima of stationary Gaussian processes.  Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 7:190–223, 1967\.
  * [PR19] Nicolas Perkowski and Tommaso Cornelis Rosati.  The KPZ equation on the real line.  Electron. J. Probab., 24:Paper No. 117, 56, 2019.
  * [QT22] Jeremy Quastel and Li-Cheng Tsai.  Hydrodynamic large deviations of tasep, 2022.
  * [R7́9] P. Révész.  A generalization of Strassen’s functional law of iterated logarithm.  Z. Wahrsch. Verw. Gebiete, 50(3):257–264, 1979.
  * [RY99] Daniel Revuz and Marc Yor.  Continuous martingales and Brownian motion.  Number 293 in Grundlehren der mathematischen Wissenschaften. Springer, Berlin [u.a.], 3. ed edition, 1999.
  * [RZ97] Jan Rosiński and Tomasz Żak.  The equivalence of ergodicity and weak mixing for infinitely divisible processes.  J. Theoret. Probab., 10(1):73–86, 1997.
  * [Sch66] Laurent Schwartz.  Sur le théorème du graphe fermé.  C. R. Acad. Sci. Paris Sér. A-B, 1966.
  * [Str64] V. Strassen.  An invariance principle for the law of the iterated logarithm.  Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 3:211–226 (1964), 1964.
  * [Tak89] K. Takashima.  Sample path properties of ergodic self-similar processes.  Osaka J. Math., 1989.
  * [Tem23] Markus Tempelmayr.  Characterizing models in regularity structures: a quasilinear case, 2023\.
  * [UZ95] Ali Süleyman Üstünel and Moshe Zakai.  Random rotations of the Wiener path.  Probab. Theory Related Fields, 103(3):409–429, 1995.
  * [UZ00] Ali Süleyman Üstünel and Moshe Zakai.  Ergodicité des rotations sur l’espace de Wiener.  C. R. Acad. Sci. Paris Sér. I Math., 330(8):725–728, 2000.
  * [UZ01] Ali Süleyman Üstünel and Moshe Zakai.  Some ergodic theorems for random rotations on wiener space.  arXiv preprint arXiv 0110162, 2001.
  * [Wal86] John B. Walsh.  An introduction to stochastic partial differential equations.  In École d’été de probabilités de Saint-Flour, XIV—1984, volume 1180 of Lecture Notes in Math., pages 265–439. Springer, Berlin, 1986.
  * [Wea18] Nik Weaver.  Lipschitz algebras.  World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018.  Second edition of [ MR1832645].


# Photoproduction of $J/\psi$ in p-p ultra-peripheral collisions at LHC energy

Zhi-Lei Ma<EMAIL_ADDRESS>Zhun Lu  Jia-Qing Zhu<EMAIL_ADDRESS>Li
Zhang<EMAIL_ADDRESS>
###### Abstract

$J/\psi$ production in p-p ultra-peripheral collisions through the elastic and
inelastic photoproduction processes, where the virtual photons emitted from
the projectile interact with the target, are studied. The comparisions between
the exact treatment results and the ones of equivalent photon approximation
are expressed as $Q^{2}$ (virtuality of photon), $z$ and $p_{T}$
distributions, and the total cross sections are also estimated. The method
developed by Martin and Ryskin is employed to avoid double counting when the
different production mechanism are considered simultaneously. The numerical
results indicate that, the equivalent photon approximation can be only applied
to the coherent or elastic electromagnetic process, the improper choice of
$Q^{2}_{\mathrm{max}}$ and $y_{\mathrm{max}}$ will cause obvious errors. And
the exact treatment is needed to deal accurately with the $J/\psi$
photoproduction.

††journal: Nuclear Physics B

[1]organization=Department of Physics, Yunnan University,city=Kunming,
postcode=650091, country=China

[2]organization=School of Physics, Southeast University,city=Nanjing,
postcode=211189, country=China

[3]organization=Department of Astronomy, Key Laboratory of Astroparticle
Physics of Yunnan Province, Yunnan University,city=Kunming, postcode=650091,
country=China

##  1 Introduction

It is well known that the photoproduction processes in ultra-peripheral
collisions (UPCs) can be theoretically studied by using the equivalent photon
approximation (EPA), which can be traced back to early works by Fermi [1],
Weizsäcker and Williams [2], and Landau and Lifshitz [3, 4]. The central idea
of EPA is that the electromagnetic field of a fast moving charged particle can
be interpreted as an equivalent flux of photon distributed with some density
$n(\omega)$ on a frequency spectrum [5, 6, 7]. Thus, the cross section is
approximated by the convolution of the photon flux with the relevant real
photoproduction cross section. Since its convenience and simplicity, EPA has
been widely applied to many topics, for instance, heavy quarkonium
photoproduction, two-photon particle production, the determination of the
nuclear parton distributions, and small-$x$ physics. [10, 9, 8, 16, 20, 13,
14, 15, 17, 18, 19, 11, 12]. EPA is sensitive to the value of kinematical
variables, and can only be employed in the very restricted kinematical regions
and in some special processes [13]. In most of physically interesting cases,
the EPA accuracy is usually denoted by a dynamical cut off
$\Lambda_{\gamma}^{2}$ of the photon virtuality $Q^{2}$. At
$Q^{2}<\Lambda_{\gamma}^{2}$, virtual photo-absorption cross sections differ
slightly from their values on the mass shell and quickly decrease at
$Q^{2}>\Lambda_{\gamma}^{2}$. Thus, EPA is a reasonable approximation
comparing with the exact treatment, and is used precisely for the description
of the cross sections only at $Q^{2}<\Lambda_{\gamma}^{2}$ [17, 18, 19].
However, the applicability range of EPA and of its accuracy are not always
considered in most works where the improper bounds of involved variables are
used, such as $Q^{2}_{\mathrm{max}}$ is usually set as $\hat{s}/4$ ($\hat{s}$
is the squared centre-of-mass (CM) frame energy of the photo-absorption
processes) or even infinity, $y_{\mathrm{max}}=1$ are also adopted directly,
and a number of imprecise statements and some widely used equivalent photon
spectra are obtained beyond the EPA validity range [17, 18, 19, 21, 23, 16,
20, 22, 24, 25, 26]. Furthermore, when the different types of photoproduction
processes are considered, the serious double counting exist which will cause
the results include the large fictitious contributions.

In this work, we consider the photoproduction of $J/\psi$ in p-p UPCs at LHC
energy, which has received many researches in EPA. We present the comparisons
between EPA and the exact treatment which reduces to the EPA in the limit
$Q^{2}\rightarrow 0$ and can be considered as the generalization of
Leptoproduction [27, 28, 29]. Actually, as always with photons, the situation
is quite complex. We have to deal with the several kinds of interactions
simultanesously, and the method developed by Martin and Ryskin [30] is adopted
to avoid double counting. There are two types of photon emission mechanisms
should be distinguished [31, 9]: coherent and incoherent emissions. In the
first type, virtual photon are radiated coherently by the whole proton which
remains intact after the photon emitted. In the second type, virtual photon
are radiated incoherently by the charged constituents inside the proton, and
the proton will dissociate or excite after the photon emitted. Otherwise, two
kinds of photoproduction mechanisms need to be considered: elastic and
inelastic reactions. In the first case, a certain particle is produced while
the target remains in the ground state (or is only internally excited). On the
other hand, in the inelastic interactions the particle produced is accompanied
by one or more particles from the breakup of the target.

This paper is organized as follow. Section 2 presents the formalism of exact
treatment for the $J/\psi$ photoproduction in p-p UPCs. Based on the method of
Martin and Ryskin, the coherent and incoherent contributions, and the elastic
and inelastic interactions are considered simultaneously. EPA is also
introduced by taking $Q^{2}\rightarrow 0$ in the exact treatment. Section 3
illustrates the numerical results, the distributions of $Q^{2}$, $z$ and
$p_{T}$, and the total cross sections are presented. Finally, the summary and
conclusions are given in Section 4. To avoid confusion, the terminology
”coherent” or ”incoherent” is always used to describe the photon emission
types in this paper, which is different in many literatures where the
”coherent” and ”incoherent” usually refer to the case that the target nucleus
remains intact or break up after scattering with photon.

##  2 Formalism

Heavy quarkonium production involves both perturbative and nonperturbative
aspects of quantum chromodynamics (QCD), which can be calculated by using the
nonrelativistic QCD (NRQCD) factorization formalism [32]. This formalism
implies a separation of short-distance coefficients, which can be calculated
perturbatively as expansions in the strong-coupling constant $\alpha_{s}$,
from long-distance matrix elements (LDMEs), which must be extracted from
experiment. The LDMEs are process independent, and can be classified in terms
of their scaling in $\nu$, the relative velocity of the heavy quarks in the
bound state. A crucial feature for this formalism is that it takes into
account the $Q\bar{Q}$ Fock space, which is spanned by the states
$n=^{2S+1}L_{J}^{(c)}$ with definite spin $S$, orbital angular momentum $L$,
total angular momentum $J$, and color multiplicity $c$. In particular, this
formalism predicts the existence of color-octet processes by the
nonperturbative emission of soft gluons in nature.

In this section, we employ the NRQCD factorization formalism to describe the
inelastic $J/\psi$ photoproduction. And for elastic interactions, we adopt a
general method of high energy scattering developed by Dosch and Ferreira [33,
34, 35, 36], which based on the functional integral approach [37, 38] to QCD
and on the WKB method. This approach is capable of incorporating both the
perturbative and nonperturbative aspects. The functional integrals are
calculated nonperturbatively, in an extended stochastic vacuum model [39, 40],
that has been successfully applied in many fields, from hadron spectroscopy to
high energy scattering.

Figure 1: The $J/\psi$ photonproduction in ultra-peripheral p-p collision. The
virtual photon emitted from the projectile $\alpha$ interacts with target
$\beta$, $\alpha$ can be the proton or it’s charged parton for coherent and
incoherent photon emissions mechanisms, respectively. And $\beta$ can also be
the proton or it’s parton (quark or gluon) for elastic and inelastic
interactions, respectively. $\alpha^{\prime}$ and $\beta^{\prime}$ are the
scattered $\alpha$ and $\beta$, respectively.

###  2.1 Exact treatment

As a generalization of Leptoproduction framework, the exact treatment for the
$J/\psi$ photoproduction in p-p UPCs is based on the expansion of the proton
or quark tensor (multiplied by $Q^{-2}$) by using the transverse and
longitudinal polarization operators, and thus the virtual photon radiated from
the projectile is off mass shell and no longer transversely polarized. The
formalism is analogous with Ref. [29].

The general form of cross section for $J/\psi$ photoproduction in p-p UPCs,
which is described in Fig. 1, can be written as

$\displaystyle d\sigma(\alpha+\beta\rightarrow\alpha+J/\psi+\beta)\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\frac{|\mathcal{M}_{\alpha\beta}|^{2}}{4p_{\mathrm{CM}}\sqrt{s_{\alpha\beta}}}d\textrm{PS}_{3}(p_{\alpha}+p_{\beta};p_{\alpha}^{\prime},J/\psi,p_{\beta}^{\prime})$
$\displaystyle=$
$\displaystyle\\!\\!\frac{|\mathcal{M}_{\alpha\beta}|^{2}}{4p_{\mathrm{CM}}\sqrt{s_{\alpha\beta}}}\frac{d^{3}p_{\alpha}^{\prime}}{(2\pi)^{3}2E_{\alpha}^{\prime}}d\textrm{PS}_{2}(q+p_{\beta};J/\psi,p_{\beta}^{\prime}),$

where
$d\mathrm{PS}_{3}(p_{\alpha}+p_{\beta};p_{\alpha}^{\prime},J/\psi,p_{\beta}^{\prime})$
is the Lorentz-invariant phase-space measure.
$p_{\mathrm{CM}}=\sqrt{(s_{\alpha\beta}-m_{\alpha}^{2}-m_{\beta}^{2})^{2}-4m_{\alpha}^{2}m_{\beta}^{2}}/2\sqrt{s_{\alpha\beta}}$
and $s_{\alpha\beta}=(p_{\alpha}+p_{\beta})^{2}$ are the momentum and energy
square of $\alpha\beta$ CM frame, respectively. $E_{\alpha}^{\prime}$ is the
energy of the scattered projectile $\alpha$.

To obtain the $Q^{2}$ distribution, it is convenient to do the calculations in
the rest frame of $\alpha$, where
$|\textbf{q}|=|\textbf{p}_{\alpha^{\prime}}|=r$,
$Q^{2}=-q^{2}=(p_{\alpha}-p_{\alpha^{\prime}})^{2}=2m_{\alpha}(\sqrt{r^{2}+m_{\alpha}^{2}}-m_{\alpha})$,
$d^{3}p_{\alpha}^{\prime}=r^{2}drd\cos\theta d\varphi$, and $y=(q\cdot
p_{\beta})/(p_{\alpha}\cdot
p_{\beta})=(q_{0}-|\textbf{p}_{\beta}|r\cos\theta/E_{\beta})/m_{\alpha}$. By
using the Jacobian determinant,

$\displaystyle
dydQ^{2}=\left|\frac{D(Q^{2},y)}{D(\cos\theta,r)}\right|d\cos\theta
dr=\frac{2|\textbf{p}_{\beta}|r^{2}}{E_{\alpha^{\prime}}E_{\beta}}d\cos\theta
dr,$ (2)

where $E_{\beta}=(s_{\alpha\beta}-m_{\alpha}^{2}-m_{\beta}^{2})/2m_{\alpha}$,
the cross section of subprocess $\alpha+\beta\rightarrow\alpha+J/\psi+\beta$
can be expressed as

$\displaystyle\\!\\!\\!\\!d\sigma(\alpha+\beta\rightarrow\alpha+J/\psi+\beta)$
$\displaystyle=\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\frac{e_{\alpha}^{2}\alpha}{Q^{2}}\frac{y\rho^{\mu\nu}T_{\mu\nu}}{8\pi^{2}}\frac{d\mathrm{PS}_{2}(q+p_{\beta};J/\psi,p_{\beta}^{\prime})}{4\hat{p}_{\mathrm{CM}}\sqrt{\hat{s}}}f(s_{\alpha\beta},p_{\mathrm{CM}},\hat{s},\hat{p}_{\mathrm{CM}},m_{\alpha},m_{\beta})dydQ^{2}d\varphi,$

where $e_{\alpha}$ is the charge of projectile, $\alpha=1/137$ is the
electromagnetic coupling constant, $\hat{p}_{\mathrm{CM}}$ and
$\hat{s}=(q+p_{\beta})^{2}$ are the momentum and energy square of
$\gamma^{*}\beta$ CM frame, and

$\displaystyle
f(s_{\alpha\beta},p_{\mathrm{CM}},\hat{s},\hat{p}_{\mathrm{CM}},m_{\alpha},m_{\beta})=\frac{\hat{p}_{\mathrm{CM}}\sqrt{\hat{s}}}{yp_{\mathrm{CM}}\sqrt{s_{\alpha\beta}}}\frac{s_{\alpha\beta}-m_{\alpha}^{2}-m_{\beta}^{2}}{\sqrt{(s_{\alpha\beta}-m_{\alpha}^{2}+m_{\beta}^{2})^{2}-4m_{\alpha}^{2}m_{\beta}^{2}}},$
(4)

the decomposition
$|\mathcal{M}_{\alpha\beta}|^{2}=4\pi\alpha^{2}e_{\alpha}^{2}\rho^{\mu\nu}T_{\mu\nu}/Q^{2}$
is used, $T_{\mu\nu}$ is the amplitude of reaction
$\gamma^{*}+\beta\rightarrow J/\psi+\beta$, and $\rho^{\mu\nu}$ is the tensor
of projectile $\alpha$ (multiplied by $Q^{-2}$)

$\displaystyle\rho^{\mu\nu}=(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^{2}})F_{2}(Q^{2})-\frac{(2P_{\alpha}-q)^{\mu}(2P_{\alpha}-q)^{\nu}}{q^{2}}F_{1}(Q^{2}),$
(5)

$F_{1}(Q^{2})$ and $F_{2}(Q^{2})$ are the general expressions of form factors
for the projectile $\alpha$.

With the help of the linear combinations [13]

$\displaystyle Q^{\mu}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\sqrt{\frac{-q^{2}}{(q\cdot
p_{\beta})^{2}-q^{2}p_{\beta}^{2}}}(p_{\beta}-q\frac{q\cdot
p_{\beta}}{q^{2}})^{\mu},$ $\displaystyle R^{\mu\nu}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-g^{\mu\nu}+\frac{(q\cdot
p_{\beta})(q^{\mu}p_{\beta}^{\nu}+q^{\nu}p_{\beta}^{\mu})-q^{2}p_{\beta}^{\mu}p_{\beta}^{\nu}-p_{\beta}^{2}q^{\mu}q^{\nu}}{(q\cdot
p_{\beta})^{2}-q^{2}p_{\beta}^{2}},$

they satisfy the relations: $q_{\mu}Q^{\mu}=q_{\mu}R^{\mu\nu}=0$,
$Q^{\mu}Q_{\mu}=1$, $\rho^{\mu\nu}$ can be expended as

$\displaystyle\rho^{\mu\nu}=\rho^{00}Q^{\mu}Q^{\nu}+\rho^{++}R^{\mu\nu},$ (6)

where $\rho^{++}=R^{\mu\nu}\rho_{\mu\nu}/2$,
$\rho^{00}=Q^{\mu}Q^{\nu}\rho_{\mu\nu}$. It can be seen that $R^{\mu\nu}$ and
$Q^{\mu}Q^{\nu}$ are equivalent to the transverse and longitudinal
polarizations [29]: $R^{\mu\nu}=\varepsilon_{T}^{\mu\nu}$,
$Q^{\mu}Q^{\nu}=-\varepsilon_{L}^{\mu\nu}$, respectively. On the other hand,
we adopt Martin-Ryskin method by using the weighting factor $P_{\beta}$ to
distinguish the different part in the whole processes [30]. Thus, the
differential cross section of subprocess
$\alpha+\beta\rightarrow\alpha+J/\psi+\beta$ can accordingly be expressed as

$\displaystyle\\!\\!\\!\\!\frac{d\sigma}{dydQ^{2}}(\alpha+\beta\rightarrow\alpha+J/\psi+\beta)$
$\displaystyle=\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\frac{e_{\alpha}^{2}\alpha}{2\pi}\frac{1}{2}\left(\frac{y\rho^{++}}{Q^{2}}R^{\mu\nu}T_{\mu\nu}+\frac{y\rho^{00}}{Q^{2}}Q^{\mu}Q^{\nu}T_{\mu\nu}\right)P_{\beta}f(s_{\alpha\beta},p_{\mathrm{CM}},\hat{s},\hat{p}_{\mathrm{CM}},m_{\alpha},m_{\beta})$
$\displaystyle\\!\\!\\!\\!\times\frac{d\mathrm{PS}_{2}(q+p_{\beta};J/\psi,p_{\beta}^{\prime})}{4\hat{p}_{\mathrm{CM}}\sqrt{\hat{s}}}$
$\displaystyle=\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!d\hat{t}\frac{e_{\alpha}^{2}\alpha}{2\pi}\left[\frac{y\rho^{++}}{Q^{2}}\frac{d\sigma_{T}}{d\hat{t}}(\gamma^{*}+\beta\rightarrow
J/\psi+\beta)+\frac{y\rho^{00}}{Q^{2}}\frac{d\sigma_{L}}{2d\hat{t}}(\gamma^{*}+\beta\rightarrow
J/\psi+\beta)\right]$ (7) $\displaystyle\\!\\!\\!\\!\times
P_{\beta}f(s_{\alpha\beta},p_{\mathrm{CM}},\hat{s},\hat{p}_{\mathrm{CM}},m_{\alpha},m_{\beta}),$

where the relations

$\displaystyle d\sigma_{T}(\gamma^{*}+\beta\rightarrow J/\psi+\beta)\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2}R^{\mu\nu}T_{\mu\nu}\frac{d\mathrm{PS}_{2}(q+p_{\beta};J/\psi,p_{\beta}^{\prime})}{4\hat{p}_{\mathrm{CM}}\sqrt{\hat{s}}},$
$\displaystyle d\sigma_{L}(\gamma^{*}+\beta\rightarrow J/\psi+\beta)\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!Q^{\mu}Q^{\nu}T_{\mu\nu}\frac{d\mathrm{PS}_{2}(q+p_{\beta};J/\psi,p_{\beta}^{\prime})}{4\hat{p}_{\mathrm{CM}}\sqrt{\hat{s}}},$
(8)

are used, and

$\displaystyle\rho^{++}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!F_{2}(Q^{2})+\frac{1}{2}\left[\frac{(2-y)^{2}}{y^{2}+Q^{2}m_{p}^{2}/(p_{\alpha}\cdot
p_{\beta})^{2}}-\frac{4m_{\alpha}^{2}}{Q^{2}}-1\right]F_{1}(Q^{2}),$
$\displaystyle\rho^{00}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-F_{2}(Q^{2})+\frac{(2-y)^{2}}{y^{2}+Q^{2}m_{p}^{2}/(p_{\alpha}\cdot
p_{\beta})^{2}}F_{1}(Q^{2}).$ (9)

Here, $d\sigma_{T}/d\hat{t}$ and $d\sigma_{L}/d\hat{t}$ represent the
transverse and longitudinal cross sections of subprocess
$\gamma^{*}+\beta\rightarrow J/\psi+\beta$, respectively.

For the elastic reactions, the weighting factor in Eq. (2.1) is chosen as
$P_{\beta}=G_{\mathrm{E}}^{2}(Q^{2})$ for recognizing the elastic part from
the whole processes, where $G_{\mathrm{E}}$ is the elastic form factor of
proton and can be parameterized by the dipole form

$\displaystyle
G_{\mathrm{E}}(Q^{2})=\frac{1}{(1+Q^{2}/0.71~{}\textrm{GeV})^{2}}.$ (10)

And we adopt a general method developed by Dosch and Ferreira, in terms of the
overlaps of the photon and vector meson light cone wave functions, the
production amplitude of vector mesons can be written as

$\displaystyle T_{\gamma^{*}p\rightarrow Vp,\lambda}(\hat{t})$
$\displaystyle=\\!\\!\\!\\!$ $\displaystyle\\!\\!\\!\\!\int
d^{2}R_{1}dz_{1}\psi_{V\lambda}(z_{1},R_{1})^{*}\psi_{\gamma^{*}\lambda}(z_{1},R_{1},Q^{2})J(\mathbf{q},z,\mathbf{R}_{1})$
$\displaystyle\approx\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!(-2is)\frac{Ce^{-a|\hat{t}|}}{1+b|\hat{t}|}\mathrm{exp}\left[\frac{-12\eta|\hat{t}|}{Q^{2}+M_{V}^{2}}\right]\left(T_{h}^{\lambda}(Q^{2})\left(\frac{\hat{s}}{s_{0}}\right)^{\epsilon_{h}}+T_{s}^{\lambda}(Q^{2})\left(\frac{\hat{s}}{s_{0}}\right)^{\epsilon_{s}}\right),$

the details can be found in Refs. [33, 34, 35, 36]. Thus, the transverse and
longitudinal cross sections of subprocess $\gamma^{*}+p\rightarrow J/\psi+p$
can be cast into

$\displaystyle\frac{d\sigma_{T(L)}}{d\hat{t}}(\gamma^{*}+p\rightarrow
J/\psi+p)=\frac{1}{16\pi\hat{s}^{2}}|T_{\gamma^{*}p\rightarrow J/\psi
p}^{\pm(0)}|^{2},$ (12)

where the helicities $\lambda=\pm 1$ and $0$ denote the transverse and
longitudinal polarizations of the photon. Considering the different photon
emission mechanisms, the elastic $J/\psi$ photoproduction should be divided
into the elastic-coherent (el-coh.) and elastic-incoherent (el-incoh.)
processes. In the elastic-coherent process, virtual photon radiated coherently
by the whole incident proton interacts with the another incident proton, and
proton remains intact after scattering. The cross section of $J/\psi$ produced
by this channel is

$\displaystyle\frac{d\sigma^{\mathrm{el-coh.}}}{dQ^{2}}(p+p\rightarrow
p+J/\psi+p)=\int dy\frac{d\sigma}{dydQ^{2}}(p+p\rightarrow p+J/\psi+p).$

$d\sigma/dydQ^{2}(p+p\rightarrow p+J/\psi+p)$ can be obtained from Eqs. (2.1)
and (12), where both the projectile and target are the proton:
$m_{\alpha}=m_{\beta}=m_{p}$, and the general expressions of form factor
$F_{1}(Q^{2})$ and $F_{2}(Q^{2})$ turn into the elastic proton form factors
$H_{1}(Q^{2})$ and $H_{2}(Q^{2})$ accordingly. Then the proton tensor
multiplied by $Q^{-2}$ reads

$\displaystyle\rho^{++}_{\mathrm{el-
coh.}}=H_{2}(Q^{2})+\frac{1}{2}\left[\frac{(y-2)^{2}}{y^{2}+Q^{2}m_{p}^{2}/(p_{\alpha}\cdot
p_{\beta})^{2}}-\frac{4m_{p}^{2}}{Q^{2}}-1\right]H_{1}(Q^{2}),$
$\displaystyle\rho^{00}_{\mathrm{el-
coh.}}=-H_{2}(Q^{2})+\frac{(y-2)^{2}}{y^{2}+Q^{2}m_{p}^{2}/(p_{\alpha}\cdot
p_{\beta})^{2}}H_{1}(Q^{2}),$ (14)

where

$\displaystyle
H_{1}(Q^{2})=\frac{G_{\mathrm{E}}^{2}(Q^{2})+(Q^{2}/4m_{p}^{2})G_{\mathrm{M}}^{2}(Q^{2})}{1+Q^{2}/4m_{p}^{2}},$
$\displaystyle
H_{2}(Q^{2})=G_{\mathrm{M}}^{2}(Q^{2})=2.79^{2}G_{\mathrm{E}}^{2}(Q^{2}),$
(15)

In the Martin-Ryskin method [30], the coherent probability, or weighting
factors (WF), is given by the square of the form factor $G_{E}^{2}(Q^{2})$,
while the effect of magnetic form factor is neglected

$\displaystyle H_{1}(Q^{2})=H_{2}(Q^{2})=G_{E}^{2}(Q^{2}).$ (16)

This approximation has been widely applied for the derivation of the
equivalent photon fluxes of proton, such as the photon spectrum Eq.(34) that
we will discuss in section 2.3, which developed by Drees and Zeppenfeld [18].
However, in the present paper, we take into account the complete expressions
given in Eq. (2.1).

In the elastic-incoherent processes, virtual photon, emitted from quarks
inside incoming proton, interacts with the target proton which remains intact
after scattering. Accordingly, the cross section of $J/\psi$ produced by el-
incoh. is

$\displaystyle\\!\\!\\!\\!\frac{d\sigma^{\mathrm{el-
incoh.}}}{dQ^{2}}(p+p\rightarrow X_{A}+J/\psi+p)$ $\displaystyle=\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\sum_{a}\int
dydx_{a}f_{a/p}(x_{a},\mu^{2})\frac{d\sigma}{dydQ^{2}}(a+p\rightarrow
a+J/\psi+p),$ (17)

where $x_{a}=p_{a}/p_{A}$ is the parton’s momentum fraction,
$f_{a/p}(x_{a},\mu^{2})$ is the parton distribution function of proton A, and
the factorized scale is chosen as $\mu=\sqrt{m_{J/\psi}^{2}+Q^{2}}$, the cross
section $d\sigma/dydQ^{2}(a+p\rightarrow a+J/\psi+p)$ can also be obtained
from Eqs. (2.1) and (12) with $m_{\alpha}=m_{a}=0$, $m_{\beta}=m_{p}$. And
$F_{1}(Q^{2})=F_{2}(Q^{2})=L_{1}(Q^{2})=L_{2}(Q^{2})$, then the massless quark
tensor multiplied by $Q^{-2}$ is

$\displaystyle\rho^{++}_{\mathrm{el-
incoh.}}=L_{2}(Q^{2})-\frac{1}{2}\left[1-\frac{(y-2)^{2}}{y^{2}+Q^{2}m_{p}^{2}/(p_{\alpha}\cdot
p_{\beta})^{2}}\right]L_{1}(Q^{2}),$ $\displaystyle\rho^{00}_{\mathrm{el-
incoh.}}=-L_{2}(Q^{2})+\frac{(y-2)^{2}}{y^{2}+Q^{2}m_{p}^{2}/(p_{\alpha}\cdot
p_{\beta})^{2}}L_{1}(Q^{2}),$ (18)

for incoherent contribution, the ”remaining” probability has to be considered
for avoiding double counting, thus $L_{1}(Q^{2})$ and $L_{2}(Q^{2})$ take the
following forms

$\displaystyle L_{1}(Q^{2})=L_{2}(Q^{2})=1-G_{E}^{2}(Q^{2}).$ (19)

For the inelastic reactions, the weighting factor is
$P_{\beta}=1-G_{E}^{2}(Q^{2})$. There are two types of photon contributions
that should be considered: direct-photon contribution and resolved-photon
contribution [9]. For the direct-photon process, the high-energy photon,
emitted from the projectile interacts with the partons of target proton
directly. For the resolved-photon process, the high-energy photon can
fluctuate into a color singlet state with multiple $q\bar{q}$ pairs and
gluons. Due to this fluctuation, the photon interacts with the partons in B
like a hadron, and the subprocesses are almost the purely strong interaction
processes. Together with the two different photon emissions mechanism
mentioned earlier, we have four topologies: inelastic-coherent direct (inel-
coh.dir.), inelastic-coherent resolved (inel-coh.res.), inelastic-incoherent
direct (inel-incoh.dir.) and inelastic-incoherent resolved (inel-incoh.res.)
processes. These abbreviations will appear in many places of remained content
and we do not explain its meaning again. According to NRQCD factorization
formalism, the cross section of $J/\psi$ produced by inel-coh.dir. is

$\displaystyle\\!\\!\\!\\!\frac{d\sigma^{\mathrm{inel-
coh.dir.}}}{dQ^{2}}(p+p\rightarrow p+J/\psi+X)$ $\displaystyle=\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\sum_{b}\int
dydx_{b}f_{b/p}(x_{b},\mu^{2})\sum_{n}\langle\mathcal{O}^{J/\psi}[n]\rangle\frac{d\sigma}{dydQ^{2}}(p+b\rightarrow
p+c\bar{c}[n]+b),$

where $\langle\mathcal{O}^{J/\psi}[n]\rangle$ is the long-distance matrix
elements of NRQCD. For inelastic reactions, $m_{\beta}=m_{b}=0$,
$x_{b}=p_{b}/p_{B}$ is the parton’s momentum fraction,
$f_{b/p}(x_{b},\mu^{2})$ is the parton distribution function of proton B.
Based on Eq. (2.1), the cross section of subprocess
$\alpha+b\rightarrow\alpha+c\bar{c}[n]+b$ should be rewritten as

$\displaystyle\\!\\!\\!\\!\\!\\!\frac{d\sigma}{dydQ^{2}}(\alpha+b\rightarrow\alpha+c\bar{c}[n]+b)$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!\frac{e_{\alpha}^{2}\alpha}{2\pi}\left[\frac{y\rho^{++}}{Q^{2}}\sigma_{T}(\gamma^{*}+b\rightarrow
c\bar{c}[n]+b)+\frac{y\rho^{00}}{2Q^{2}}\sigma_{L}(\gamma^{*}+b\rightarrow
c\bar{c}[n]+b)\right]P_{\beta}$ $\displaystyle\\!\\!\\!\\!\\!\\!\times
f(s,p_{\mathrm{CM}},\hat{s},\hat{p}_{\mathrm{CM}},m_{\alpha},m_{b})$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!d\hat{t}\frac{e_{\alpha}^{2}\alpha}{2\pi}F_{b}[n]\left[\frac{y\rho^{++}}{Q^{2}}T_{b}[n]-y\rho^{00}L_{b}[n]\right]P_{\beta}f(s,p_{\mathrm{CM}},\hat{s},\hat{p}_{\mathrm{CM}},m_{\alpha},m_{b}),$
(21)

$F_{b}[n]$, $T_{b}[n]$ and $L_{b}[n]$ can be found in Refs. [29] and the
proton tensor multiplied by $Q^{-2}$ is

$\displaystyle\rho^{++}_{\mathrm{inel-
coh.}}=H_{2}(Q^{2})+\frac{1}{2}\left[\frac{(y-2)^{2}}{y^{2}}-\frac{4m_{p}^{2}}{Q^{2}}-1\right]H_{1}(Q^{2}),$
$\displaystyle\rho^{00}_{\mathrm{inel-
coh.}}=-H_{2}(Q^{2})+\frac{(y-2)^{2}}{y^{2}}H_{1}(Q^{2}).$ (22)

In the same way, the cross section of $J/\psi$ produced by inel-incoh.dir.
reads

$\displaystyle\\!\\!\\!\\!\frac{d\sigma^{\mathrm{inel-
incoh.dir}}}{dQ^{2}}(p+p\rightarrow X_{A}+J/\psi+X)$
$\displaystyle=\\!\\!\\!\\!$ $\displaystyle\\!\\!\\!\\!\sum_{a,b}\int
dydx_{a}dx_{b}f_{a/p}(x_{a},\mu^{2})f_{b/p}(x_{b},\mu^{2})\sum_{n}\langle\mathcal{O}^{J/\psi}[n]\rangle$
$\displaystyle\times\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\frac{d\sigma}{dydQ^{2}}(a+b\rightarrow
a+c\bar{c}[n]+b),$ (23)

the differential cross section $d\sigma/dydQ^{2}(a+b\rightarrow
a+c\bar{c}[n]+b)$ is the same as Eq. (2.1) but with $m_{\alpha}=0$, the
massless quark tensor multiplied by $Q^{-2}$ is

$\displaystyle\rho^{++}_{\mathrm{inel-
incoh.}}=\left[1-G_{\mathrm{E}}^{2}(Q^{2})\right]\frac{1+(1-y)^{2}}{y^{2}},$
$\displaystyle\rho^{00}_{\mathrm{inel-
incoh.}}=\left[1-G_{\mathrm{E}}^{2}(Q^{2})\right]\frac{4(1-y)}{y^{2}}.$ (24)

In the inelastic-coherent resolved process, the parton $a^{\prime}$ of hadron-
like photon which emitted from proton $A$, interacts with the parton $b$ of
another incident proton $B$ via the interactions of quark-antiquark
annihilation, quark-gluon Compton scattering and gluon-gluon fusion. Therefore
the cross section of $J/\psi$ produced by inel-coh.res. has the form of

$\displaystyle\\!\\!\\!\\!\frac{d\sigma^{\mathrm{inel-
coh.res.}}}{dQ^{2}}(p+p\rightarrow p+J/\psi+X)$ $\displaystyle=\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\sum_{b}\sum_{a^{\prime}}\int
dydx_{b}dz_{a^{\prime}}d\hat{t}f_{b/p}(x_{b},\mu^{2})f_{a^{\prime}/\gamma}(z_{a^{\prime}},\mu^{2})\sum_{n}\langle\mathcal{O}^{J/\psi}[n]\rangle$
$\displaystyle\times\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\frac{e_{\alpha}^{2}\alpha}{2\pi}\frac{y\rho^{++}_{\mathrm{inel-
coh.}}}{Q^{2}}\frac{d\sigma}{d\hat{t}}(a^{\prime}+b\rightarrow
c\bar{c}[n]+b),$ (25)

where $z_{a}^{\prime}=p_{a^{\prime}}/q$ denotes the parton’s momentum fraction
of the resolved photon, $f_{\gamma}(z_{a^{\prime}},\mu^{2})$ is the parton
distribution function of the resolved photon [41], the complete list for the
cross section of partonic processes $a^{\prime}+b\rightarrow c\bar{c}[n]+b$
can be found in Refs. [42, 43]. The strong coupling constant is taken as the
one-loop form [44]

$\displaystyle\alpha_{S}=\frac{12\pi}{(33-2n_{f})\ln(\mu^{2}/\Lambda^{2})},$
(26)

with $n_{f}=3$ and $\Lambda=0.2~{}\mathrm{GeV}$. In the inelastic-incoherent
resolved process, the parton from hadron-like photon emitted by charged parton
of incident proton can interacts with the parton of target proton. The cross
section of $J/\psi$ produced by inel-incoh.res. has the form of

$\displaystyle\\!\\!\\!\\!\frac{d\sigma^{\mathrm{inel-
incoh.res.}}}{dQ^{2}}(p+p\rightarrow X_{A}+J/\psi+X)$
$\displaystyle=\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\sum_{a,b}\sum_{a^{\prime}}\int
dydx_{a}dx_{b}dz_{a^{\prime}}d\hat{t}f_{a/p}(x_{a},\mu^{2})f_{b/p}(x_{b},\mu^{2})f_{a^{\prime}/\gamma}(z_{a^{\prime}},\mu^{2})\sum_{n}\langle\mathcal{O}^{J/\psi}[n]\rangle$
$\displaystyle\times\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!e_{a}^{2}\frac{\alpha}{2\pi}\frac{y\rho^{++}_{\mathrm{inel-
incoh.}}}{Q^{2}}\frac{d\sigma}{d\hat{t}}(a^{\prime}+b\rightarrow
c\bar{c}[n]+b).$ (27)

###  2.2 The $p_{T}$ and $z$ distributions of $J/\psi$ production

It is straightforward to obtain the distributions of $p_{T}$ and $z$ by
accordingly reordering and redefining the integration variables. For deriving
the $p_{T}$ distribution, the Mandelstam variables of partonic process
$\gamma^{*}+\beta\rightarrow J/\psi+\beta$ should be written in the following
forms

$\displaystyle\hat{s}$ $\displaystyle=2\cosh
y_{r}m_{T}\sqrt{\cosh^{2}y_{r}m_{T}^{2}+m_{\beta}^{2}-M_{J/\psi}^{2}}$ (28)
$\displaystyle+2\cosh^{2}y_{r}m_{T}^{2}+m_{\beta}^{2}-M_{J/\psi}^{2},$
$\displaystyle\hat{t}$ $\displaystyle=M_{J/\psi}^{2}-Q^{2}-2m_{T}(E_{q}\cosh
y_{r}-\hat{p}_{\mathrm{CM}}\sinh y_{r}),$ $\displaystyle\hat{u}$
$\displaystyle=M_{J/\psi}^{2}+m_{\beta}^{2}-2m_{T}(E_{\beta}\cosh
y_{r}+\hat{p}_{\mathrm{CM}}\sinh y_{r}),$

where $y_{r}=(1/2)\ln(E+p_{z})/(E-p_{z})$ is the rapidity,
$m_{T}=\sqrt{M_{J/\psi}^{2}+p_{T}^{2}}$ is the $J/\psi$ transverse mass, the
expressions of $E_{q}$, $E_{\beta}$ and $\hat{p}_{\mathrm{CM}}$ can be found
in A. For elastic-coherent process, the variables $y$ and $\hat{t}$ should be
transformed into the following form by using the Jacobian determinant

$\displaystyle d\hat{t}dy\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\mathcal{J}_{\mathrm{el-coh.}}dy_{r}dp_{T}$ (29)
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\left|\frac{D(y,\hat{t})}{D(p_{T},y_{r})}\right|dy_{r}dp_{T}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{2\hat{p}_{\mathrm{CM}}\sqrt{\hat{s}}(\sqrt{\cosh^{2}y_{r}m_{T}^{2}+m_{\beta}^{2}-M_{J/\psi}^{2}}+\cosh
y_{r}m_{T})}{(s_{\alpha\beta}-m_{\alpha}^{2}-m_{\beta}^{2})\sqrt{\cosh^{2}y_{r}m_{T}^{2}+m_{\beta}^{2}-M_{J/\psi}^{2}}}dy_{r}dp_{T},$

the expressions of the Jacobian determinant for the rest processes are listed
as follow

$\displaystyle\mathrm{el-incoh.}\\!\\!\\!\\!$ $\displaystyle:$
$\displaystyle\\!\\!\\!\\!~{}~{}d\hat{t}dx_{a}=\frac{\mathcal{J}_{\mathrm{el-
coh.}}}{y}dy_{r}dp_{T},$ $\displaystyle\mathrm{inel-coh.dir.}\\!\\!\\!\\!$
$\displaystyle:$
$\displaystyle\\!\\!\\!\\!~{}~{}d\hat{t}dx_{b}=\frac{\mathcal{J}_{\mathrm{el-
coh.}}}{y}dy_{r}dp_{T},$ $\displaystyle\mathrm{inel-incoh.dir.}\\!\\!\\!\\!$
$\displaystyle:$
$\displaystyle\\!\\!\\!\\!~{}~{}d\hat{t}dx_{b}=\frac{\mathcal{J}_{\mathrm{el-
coh.}}}{yx_{a}}dy_{r}dp_{T},$ $\displaystyle\mathrm{inel-
coh.res.}\\!\\!\\!\\!$ $\displaystyle:$
$\displaystyle\\!\\!\\!\\!~{}~{}d\hat{t}dz_{a^{\prime}}=\frac{\mathcal{J}_{\mathrm{el-
coh.}}}{yx_{b}}dy_{r}dp_{T},$ $\displaystyle\mathrm{inel-
incoh.res.}\\!\\!\\!\\!$ $\displaystyle:$
$\displaystyle\\!\\!\\!\\!~{}~{}d\hat{t}dz_{a^{\prime}}=\frac{\mathcal{J}_{\mathrm{el-
coh.}}}{yx_{a}x_{b}}dy_{r}dp_{T},$ (30)

For the $z$ distribution, we can do the similar transformations, and the
Mandelstam variables in Eq. (28) should be rewritten as:
$\hat{s}=M_{J/\psi}^{2}/z+p_{T}^{2}/z(1-z)$,
$\hat{t}=-(1-z)(\hat{s}+Q^{2}-m_{\beta}^{2})$ and
$\hat{u}=M_{J/\psi}^{2}+m_{\beta}^{2}-z(\hat{s}+Q^{2}-m_{\beta}^{2})$.

###  2.3 The equivalent photon spectrum

The idea of EPA was first developed by Fermi [1], who replaced the
electromagnetic fields from a fast-moving charged particle with an equivalent
flux of photon. The number of photons with energy $\omega$, $n(\omega)$, is
given by the Fourier transform of the time-dependent electromagnetic field.
Therefore, the electromagnetic interaction between the charged particle and
the nucleus is reduced to the interaction between those photons and the
nucleus. This idea has been extended to include the interaction of
relativistic charged particles by Weizsäcker and Williams, and the method is
often known as the Weizsäcker-Williams method [2]. An essential advantage of
EPA consists in the fact that, when using it, it is sufficient to obtain the
photo-absorption cross section on the mass shell only. Details of its off
mass-shell behavior are not essential. Thus, the EPA approach, as a useful
technique, has been widely applied to obtain various cross sections for
charged particles production in relativistic heavy ion collisions [13].

Although tremendous successes have been achieved, the accuracy of EPA and its
applicability range are often neglected, and a number of imprecise statements
pertaining to the essence and the advantages of EPA were given [16, 20, 21,
24, 25, 22, 23]. Some limitations of the kinematic variables are often used
directly [29, 46, 17, 45, 18, 47], e.g., the choice of
$Q^{2}_{\mathrm{max}}\sim\hat{s}$ or $\infty$ is widely used instead of the
significant dynamical cut off $\Lambda^{2}_{\gamma}$ which represents the
precision of EPA [13]; $y_{\mathrm{max}}=1$ is used in the calculations,
actually it is far less than one in coherent or elastic processes; And the
integration of some widely used spectra are performed over the entire
kinematically allowed region, which leads to erroneous expressions. However,
the exact treatment developed in the previous sections can reduce to the EPA
when $Q^{2}\rightarrow 0$. Detailed discussion on this issue can be also found
in Ref. [13]. This provides us a powerful and overall approach for comparing
our results with the EPA ones, to study the features of EPA in the entire
kinematical ranges and give the appropriate kinematical relations.

Taking $Q^{2}\rightarrow 0$, the transverse and longitudinal polarization
tensors can reduce to

$\displaystyle\lim_{Q^{2}\rightarrow
0}\varepsilon^{\mu\nu}_{L}=-\frac{q^{\mu}q^{\nu}}{q^{2}},$
$\displaystyle\lim_{Q^{2}\rightarrow
0}\varepsilon^{\mu\nu}_{T}=-g^{\mu\nu}+\frac{(q^{\mu}p_{b}^{\nu}+q^{\nu}p_{b}^{\mu})}{q\cdot
p_{b}}.$ (31)

Since $q^{\mu}T_{\mu\nu}=0$, Eq. (2.1) becomes:

$\displaystyle\\!\\!\\!\\!\lim_{Q^{2}\rightarrow
0}\frac{d\sigma}{dydQ^{2}d\hat{t}}(\alpha+\beta\rightarrow\alpha+J/\psi+\beta)$
$\displaystyle=\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\left(e_{\alpha}^{2}\frac{\alpha}{2\pi}\frac{y\rho^{++}}{Q^{2}}\right)\frac{d\sigma_{T}}{d\hat{t}}(\gamma^{*}+\beta\rightarrow
J/\psi+\beta)\bigg{|}_{Q^{2}=0}$ $\displaystyle=\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\frac{df_{\gamma}(y)}{dQ^{2}}\frac{d\sigma_{T}}{d\hat{t}}(\gamma^{*}+\beta\rightarrow
J/\psi+\beta)\bigg{|}_{Q^{2}=0},$ (32)

where the general form of the equivalent photon flux function $f_{\gamma}(y)$
reads

$\displaystyle f_{\gamma}(y)\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\int
dQ^{2}\frac{e_{\alpha}^{2}\alpha}{2\pi}\frac{y\rho^{++}}{Q^{2}}$ (33)
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{e_{\alpha}^{2}\alpha}{2\pi}\int\frac{dQ^{2}}{Q^{2}}y\left[F_{2}(Q^{2})+\left(\frac{2(1-y)}{y^{2}}-\frac{2m_{\alpha}^{2}}{Q^{2}}\right)F_{1}(Q^{2})\right],$

we can see that the contribution of $d\sigma_{L}/d\hat{t}$ and the terms
proportional to $Q^{2}$ are neglected in the limit $Q^{2}\rightarrow 0$.

For the coherent case, there is a approximate analytic form which is developed
by Drees and Zeppenfeld (DZ) [18], this photon flux of proton is widely used
in the literatures [20, 21, 22, 23, 24, 25]. By taking
$Q^{2}_{\mathrm{max}}\rightarrow\infty$, setting
$F_{1}(Q^{2})=F_{2}(Q^{2})=G_{E}(Q^{2})$ (which equals to neglect the
contribution of magnetic form factor), and neglecting the $m_{\alpha}^{2}$
term in Eq. (33), they obtained

$\displaystyle
f_{\gamma}^{\mathrm{DZ}}(y)=\frac{\alpha}{2\pi}\frac{1+(1-y)^{2}}{y}\left(\ln
A-\frac{11}{6}+\frac{3}{A}-\frac{3}{2A^{2}}+\frac{1}{3A^{2}}\right),$ (34)

where $A=(1+0.71~{}\mathrm{GeV}/Q^{2}_{\mathrm{min}})$. For incoherent case,
the complete form of photon flux can be derived from Eq. (33) by setting
$F_{1}(Q^{2})=F_{2}(Q^{2})=1-G_{E}^{2}(Q^{2})$ and $m_{\alpha}=0$,

$\displaystyle f_{\gamma}^{\mathrm{incoh}}(y)\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\int
dQ^{2}\frac{\alpha}{2\pi}\frac{y\rho^{++}}{Q^{2}}$ (35) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!e_{\alpha}^{2}\frac{\alpha}{2\pi}\frac{1+(1-y)^{2}}{y}\int\frac{dQ^{2}}{Q^{2}}\left(1-G_{E}^{2}(Q^{2})\right).$

Actually, another approximate form of above equation is often used in the
practical calculations [20, 21, 22, 23, 24, 25, 26], which neglects the second
term of Eq. (35) and takes $Q_{\mathrm{min}}^{2}=1~{}\mathrm{GeV}^{2}$ and
$Q^{2}_{\mathrm{max}}=\hat{s}/4$

$\displaystyle
f_{\gamma}^{\mathrm{incoh}}(y)=e_{\alpha}^{2}\frac{\alpha}{2\pi}\frac{1+(1-y)^{2}}{y}\ln\frac{Q_{\mathrm{max}}^{2}}{Q_{\mathrm{min}}^{2}}.$
(36)

##  3 Numerical results

In this section, we provide the numerical results for the $J/\psi$
photoproductions in p-p UPCs at LHC energy. Several theoretical inputs and the
kinematic conditions need to be provided. The mass of the proton and $J/\psi$
are $m_{p}=0.938~{}\mathrm{GeV}$, $m_{J/\psi}=3.097~{}\mathrm{GeV}$ [49]. The
long-distance matrix elements of NRQCD used in this paper can be found in Ref.
[48], where

$\displaystyle\langle\mathcal{O}^{J/\psi}[^{3}S_{1}^{(1)}]\rangle=1.2\
\mathrm{GeV}^{3},$
$\displaystyle\langle\mathcal{O}^{J/\psi}[^{1}S_{0}^{(8)}]\rangle=1.8\times
10^{-2}\ \mathrm{GeV}^{3},$
$\displaystyle\langle\mathcal{O}^{J/\psi}[^{3}S_{1}^{(8)}]\rangle=1.3\times
10^{-3}\ \mathrm{GeV}^{3},$ (37)

and the multiplicity relations

$\displaystyle\langle\mathcal{O}^{J/\psi}[^{3}P_{0}^{(8)}]\rangle=m_{\mathrm{charm}}^{2}\langle\mathcal{O}^{J/\psi}[^{1}S_{0}^{(8)}]\rangle,$
$\displaystyle\langle\mathcal{O}^{J/\psi}[^{3}P_{J}^{(8)}]\rangle=(2J+1)\langle\mathcal{O}^{J/\psi}[^{3}P_{0}^{(8)}]\rangle,$
(38)

are considered. We choose the MMHT2014 NNLO set for the parton distribution
function of proton with $n_{f}=3$ [50, 51]. The complete kinematical relations
are given in A, one should note that the coherence condition [52] is
considered for coherent reactions, which means that the wavelength of the
photon is larger than the size of the nucleus, and the charged constituents
inside the nucleus should act coherently. This condition limits $Q^{2}$ and
$y$ to very low value ($Q^{2}\leq 1/R^{2}_{A}$, $R_{A}=A^{1/3}1.2\
\textrm{fm}$ is the size of the nucleus), $Q^{2}_{\mathrm{max}}=0.027\
\textrm{GeV}^{2}$ and $y_{\mathrm{max}}\sim 0.16$ for proton.

Figure 2: Comparisons of the exact results and the EPA ones for $J/\psi$
photoproduction in p-p UPCs. (a): For elastic reactions, the solid (black) and
dashed (red) lines represent the exact and EPA results of el-coh.,
respectively. The dotted (blue) and dashed-dotted (magenta) lines denote the
exact and EPA results of el-incoh., respectively. (b): Same as (a) but for
inelastic-direct processes. (c): Same as (a) but for inelastic-resolved
processes.

In Fig.2, the $Q^{2}$ distribution of $J/\psi$ photoproduction in p-p UPCs is
plotted. The curves of EPA and exact results share the same trend, and are
consistent with each other in small $Q^{2}$ region, since EPA can be obtained
from the exact treatment by setting $Q^{2}\rightarrow 0$ and only considering
the transverse photon contributions, this is one advantage of exact treatment.
For the case of coherent photon emission processes, we can see that the
contribution of el-coh. concentrates on the small $Q^{2}$ domain, and almost
have no difference between exact result and the EPA one [Fig.2 (a)].
Inversely, the contribution of inel-coh. [Fig.2 (b),(c)] can be neglected in
small $Q^{2}$ region compared to el-coh., and the difference between the exact
result and the EPA one appears when $Q^{2}>1~{}\mathrm{GeV}^{2}$. For
incoherent photon emission processes, its contributions are negligible by
comparing with coherent ones in small $Q^{2}$ region and become larger when
$Q^{2}>0.1~{}\mathrm{GeV}^{2}$, the difference between the exact results and
the EPA ones are obvious, especially for inel-incoh. where the difference is
prominent [Fig.2 (b),(c)]. Therefore, EPA is only applicable in the small
$Q^{2}$ domain, this feature permits one to use it for coherent or elastic
processes with proper choice of $Q^{2}_{\mathrm{max}}$, and reached the high
precision. However, EPA is not a effective approximation for incoherent photon
emission processes (especially for inel-incoh.), since its contribution
dominate the large $Q^{2}$ region where the EPA errors are prominent.

Table 1: Total cross sections of the $J/\psi$ photoproduction in the coherent
channel.

| el-coh. |  | inel-coh.(dir.+res.)
---|---|---|---
|  $\sigma~{}[\mathrm{nb}]$ |  $\Delta{\textsuperscript{a}}~{}[\mathrm{nb}]$ |  $\delta{\textsuperscript{b}}~{}[$%$]$ |  |  $\sigma~{}[\mathrm{nb}]$ |  $\Delta~{}[\mathrm{nb}]$ |  $\delta~{}[$%$]$
Exact |  13.3498 |  0.00 |  0.00 |  |  59.47 |  0.00 |  0.00
EPA CCc |  13.3506 |  0.0008 |  0.00 |  |  59.49 |  0.02 |  0.00
EPA ($Q^{2}_{\mathrm{max}}\backsim\hat{s}$) |  19.2987 |  5.9489 |  44.56 |  |  92.20 |  32.73 |  55.03
EPA $(y_{\mathrm{max}}=1)$ |  22.3444 |  8.9946 |  67.38 |  |  106.83 |  47.37 |  79.64
EPA No WF |  $\backslash$ |  $\backslash$ |  $\backslash$ |  |  1152.6764 |  1093.20 |  1838.16

  * a

Absolute error with respect to the exact result:
$\Delta=\sigma-\sigma_{\mathrm{Exact}}$.

  * b

Relative error with respect to the exact result:
$\delta=\Delta/\sigma_{\mathrm{Exact}}$.

  * c

The EPA result with the coherence condition (CC).

Table 2: Total cross sections of the $J/\psi$ photoproduction in the incoherent channel. | el-incoh. |  | inel-incoh.(dir.+res.)
---|---|---|---
|  $\sigma~{}[\mathrm{nb}]$ |  $\Delta~{}[\mathrm{nb}]$ |  $\delta~{}[$%$]$ |  |  $\sigma~{}[\mathrm{nb}]$ |  $\Delta~{}[\mathrm{nb}]$ |  $\delta~{}[$%$]$
Exact |  6.1573 |  0.00 |  0.00 |  |  422.75 |  0.00 |  0.00
EPA |  6.3722 |  0.2149 |  3.49 |  |  1939.50 |  1516.75 |  358.78
EPA No WF |  233.1243 |  226.9670 |  3686.14 |  |  9152.45 |  8729.70 |  2064.97
EPA $(Q^{2}_{\mathrm{min}}\backsim 1\mathrm{GeV^{2}})$ |  $\backslash$ |  $\backslash$ |  $\backslash$ |  |  3941.85 |  3519.10 |  832.43

To quantitatively estimate the EPA errors and discuss the influence from the
different choice of kinematic limitations, we also calculate the total cross
sections in Table 1 and 2. For coherent reactions in Table 1, we can see that
the EPA results with coherence condition (CC) nicely agree with exact ones,
since CC limits $Q^{2}_{\mathrm{max}}$ and $y_{\mathrm{max}}$ to very low
value which effectively cut the EPA errors. On the contrary, the usual choice
of $Q^{2}_{\mathrm{max}}\sim\hat{s}$ and $y_{\mathrm{max}}=1$ in most
literatures bring the large errors. For incoherent reactions in Table 2, the
EPA error is not obvious in el-incoh., since the elastic interactions dominate
the very small $Q^{2}$ region which effectively suppresses the EPA errors.
Unlikely el-incoh., the EPA error of inel-incoh. are prominent, which verifies
again that the essence of incoherent photon emission processes is in
contradiction with the feature of EPA. Besides, we can see that the coherent
and incoherent reactions are the main channel of elastic and inelastic
interactions, respectively. It should be emphasized that, if the Martin-Ryskin
method is not considered, the contribution of incoherent and inelastic
reactions will always much larger than coherent and elastic reactions in the
whole $Q^{2}$ region. This unphysical result is caused by the double counting,
and is most serious in inel-incoh.. Comparing with Martin-Ryskin method which
avoid this unphysical large value naturally, the results obtained by using the
artificial cutoff $Q^{2}_{\mathrm{min}}=1~{}\textrm{GeV}$, which is widely
applied in the literatures [20, 21, 22, 23, 24, 25, 26], still have large
errors ($\delta\sim 832\%$).

Figure 3: The $z$ distribution of $J/\psi$ photoproduction in p-p UPCs. (a):
The elastic reacionts. (b): The inelastic-direct processes, the solid (black),
dashed (red) and dotted (blue) lines represent the inel-coh.dir. results of
exact treatment, EPA and EPA without WF factors, respectively; those for inel-
incoh.dir. are depicted by the same type lines but are bolded and with
different colors. (c): Same as (b) but for inelastic-resolved processes.

In Fig.3, the $z$ distribution of $J/\psi$ photoproduction in p-p UPCs is
plotted. The EPA results nicely agree with exact ones in coherent reactions in
the whole $z$ regions. But in the incoherent reactions, the difference between
the exact results and EPA ones are significant, about an order of magnitude.
For elastic interactions [Fig.3(a)], the curves is negligible in the most
regions of $z$ and become important when $z>0.9$, especially near $z=1$, the
curves show a pronounced peak. For the case of inelastic interactions
[Fig.3(b),(c)], the contributions are important in the most of $z$ regions.
The curves of inelastic-direct processes is comparable with elastic reactions
near $z=0.9$, but rapidly deceased near $z=1$. These features are in agreement
with the traditional perspective that $z=0.9$ is the boundary to distinguish
the elastic and inelastic reactions. Finally, for the case of inelastic-
resolved processes, its contribution dominate the region $z<0.2$.

Figure 4: The $p_{T}$ distribution of $J/\psi$ photoproduction in p-p UPCs.
(a): The elastic reactions, the solid (black), dashed (red) and dotted (blue)
lines represent the el-coh. results of exact treatment, EPA and EPA with Eq.
(33), respectively; those for el-incoh. are depicted by the same type lines
but are bolded and with different colors. (b): Same as (a) but for inelastic-
direct processes. (c): Same as (a) but for inelastic-resolved processes.

In Fig.4, the $p_{T}$ distribution of $J/\psi$ photoproduction in p-p UPCs is
plotted, the equivalent photon fluxes Eqs.(34) and (36) are discussed. For the
case of coherent reactions, the exact results nicely agree with the EPA ones
in the whole $p_{T}$ region, since the coherence condition is adopted in the
calculations. However, the results based on the equivalent photon flux
function Eq.(34) are much larger than the results of exact treatment, since
$Q^{2}_{\mathrm{max}}=\infty$ is adopted in Eq.(34), which causes the large
EPA errors. For the case of incoherent reactions, the EPA results are greater
than the exact one in the whole $p_{T}$ region, but the difference between the
EPA result Eq. (36) and the exact one is much more evident (especially for the
inel-incoh.res. in Fig.4 (c)), since $Q^{2}_{\mathrm{max}}=\hat{s}/4$ is
employed in Eq.(36), which include the large fictitious contribution from the
large $Q^{2}$ domain. Otherwise, the option $y_{\mathrm{max}}=1$ is used
directly in these two fluxes, which is another important sources of errors.
Therefore, the choice of $Q^{2}_{\mathrm{max}}$ and $y_{\mathrm{max}}$ are
crucial to the accuracy of EPA, and the results based on Eq.(34) and (36) are
not accurate enough. For dealing with the incoherent processes, the exact
treatment has to be considered.

##  4 Summary and conclusions

In this paper, we calculate the $J/\psi$ photoproduction in p-p Ultra-
peripheral collisions at LHC energy. The elastic and inelastic interactions,
and coherent and incoherent photon emission processes are considered
simultaneously. The exact treatment, which can reduce to EPA in the limit
$Q^{2}\rightarrow 0$, is developed, where the proton or quark tensor is
expanded by using the transverse and longitudinal polarization operators, and
the square of electric form factor $G_{E}^{2}(Q^{2})$ is used as weighting
factor for avoiding double counting. Otherwise, the full kinematical relations
are also achieved. In order to study the feature of EPA in the entire
kinematical ranges, the comparisons between the exact results and the EPA ones
are expressed as the $Q^{2}$, $z$ and $p_{T}$ distributions, and total cross
sections are also estimated.

The numerical results indicate that the EPA is only applicable in a very
restricted domain (small $y$ and $Q^{2}$ domain), and is very sensitive to the
value of $y_{\mathrm{max}}$ and $Q^{2}_{\mathrm{max}}$, this feature permits
one to use it for coherent or elasitc processes with appropriate choice of
$Q^{2}_{\mathrm{max}}$ and $y_{\mathrm{max}}$, and reached the high precision.
Coherence condition is a good choice which effectively cut the errors from
large $y$ and $Q^{2}$ domain. The common options
$Q^{2}_{\mathrm{max}}\sim\hat{s}$ or $\infty$, and $y_{\mathrm{max}}=1$ will
cause the large errors. Besides, the validity condition of EPA is in
contradiction with the essence of incoherent photon emission processes
(especially for inel-incoh.), where the serious double counting problem also
exist. Therefore, EPA can not provide accurate enough results for $J/\psi$
photoproduction in p-p Ultra-peripheral collisions, and the exact treatment
needs to be adopted which has the widely applicability range and can naturally
avoid double counting.

## Acknowledgements

This work is supported in part by National Key R & D Program of China under
grant No. 2018YFA0404204, the National Natural Science Foundation of China
(Grant Nos. 11747086 and 11575043), and by the Young Backbone Teacher Training
Program of Yunnan University. Z. M. is supported by Yunnan Provincial New
Academic Researcher Award for Doctoral Candidates.

##  Appendix A Full kinematical relations

For the $Q^{2}$ distribution, the bounds of integration variables for elastic-
coherent process are given by

$\displaystyle\hat{t}_{\mathrm{min}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!M_{J/\psi}^{2}-Q^{2}-2(E_{q}E_{J/\psi}+p_{\mathrm{CM}}p_{\mathrm{CM}}^{\prime}),$
$\displaystyle\hat{t}_{\mathrm{max}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!M_{J/\psi}^{2}-Q^{2}-2(E_{q}E_{J/\psi}-p_{\mathrm{CM}}p_{\mathrm{CM}}^{\prime}),$
$\displaystyle y_{\mathrm{min}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{\hat{s}_{\mathrm{min}}+Q^{2}-m_{\beta}^{2}}{s-m_{\alpha}^{2}-m_{\beta}^{2}},$
$\displaystyle y_{\mathrm{max}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2m_{\alpha}^{2}}\left(\frac{\sqrt{Q^{2}(4m_{\alpha}^{2}+Q^{2})[(s-m_{\alpha}^{2}-m_{\beta}^{2})^{2}-4m_{\alpha}^{2}m_{\beta}^{2}]}}{s-m_{\alpha}^{2}-m_{\beta}^{2}}-Q^{2}\right),$

where $s_{\mathrm{min}}=(M_{T\mathrm{min}}+p_{T\mathrm{min}})^{2}$, $s$ is the
energy square of p-p CM frame, $y_{\mathrm{max}(\mathrm{min})}$ is same for
all kinds of reactions and not list in the following again, the energy and
momentum in $\gamma^{*}\beta$ CM frame are

$\displaystyle E_{q}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2\sqrt{\hat{s}}}(\hat{s}-Q^{2}-m_{\beta}^{2}),$
$\displaystyle E_{J/\psi}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2\sqrt{\hat{s}}}(\hat{s}+M_{J/\psi}^{2}-m_{\beta}^{2}),$
$\displaystyle\hat{p}_{\mathrm{CM}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2\sqrt{\hat{s}}}\sqrt{(\hat{s}+Q^{2}-m_{\beta}^{2})^{2}+4Q^{2}m_{\beta}^{2}},$
$\displaystyle\hat{p}_{\mathrm{CM}}^{\prime}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2\sqrt{\hat{s}}}\sqrt{(\hat{s}-M_{J/\psi}^{2}-m_{\beta}^{2})^{2}-4M_{J/\psi}^{2}m_{\beta}^{2}},$
(40)

For the elastic-incoherent case, the bounds of variables are the same as Eq.
(A), but for
$x_{a\mathrm{min}}=(\hat{s}_{\mathrm{min}}+Q^{2}-m_{\beta}^{2})/y(s_{\alpha\beta}-m_{\alpha}^{2}-m_{\beta}^{2})$,
$x_{a\mathrm{max}}=1$. The bounds of variables for inelastic interactions are
given in Table 3.

Table 3:  The bounds of variables for inelastic interactions. $\hat{s}_{\mathrm{min}}=\hat{s}_{\gamma\mathrm{min}}$, $p_{T}^{2}=\hat{t}(\hat{s}\hat{u}+Q^{2}M^{2})/(\hat{s}+Q^{2})^{2}$ and $\hat{t}_{\gamma}\hat{u}_{\gamma}/\hat{s}_{\gamma}$ for inelastic-direct and -resolved process, respectively.  variables |  coh.dir |  incoh.dir |  coh.res |  incoh.res
---|---|---|---|---
$z_{q\mathrm{min}}$ | $\frac{M_{J/\psi}^{2}+\hat{s}}{2\hat{s}}-\frac{\sqrt{(\hat{s}-M_{J/\psi}^{2})^{2}-4p_{T\mathrm{min}}^{2}\hat{s}}}{2\hat{s}}$
$z_{q\mathrm{max}}$ | $\frac{M_{J/\psi}^{2}+\hat{s}}{2\hat{s}}+\frac{\sqrt{(\hat{s}-M_{J/\psi}^{2})^{2}-4p_{T\mathrm{min}}^{2}\hat{s}}}{2\hat{s}}$
$\hat{t}_{\mathrm{min}}$ | $(z_{q\mathrm{min}}-1)(\hat{s}+Q^{2})$ | $(z_{q\mathrm{min}}^{\prime}-1)\hat{s}_{\gamma}$
$\hat{t}_{\mathrm{max}}$ | $(z_{q\mathrm{max}}-1)(\hat{s}+Q^{2})$ | $(z_{q\mathrm{max}}^{\prime}-1)\hat{s}_{\gamma}$
$z_{a^{\prime}\mathrm{min}}$ |  $\backslash$ |  $\backslash$ |  $\frac{\hat{s}_{\gamma\mathrm{min}}}{yx_{b}(s-2m_{p}^{2})}$ |  $\frac{\hat{s}_{\gamma\mathrm{min}}}{yx_{a}x_{b}(s-2m_{p}^{2})}$
$z_{a^{\prime}\mathrm{max}}$ |  $\backslash$ |  $\backslash$ |  1 |  1
$x_{b\mathrm{min}}$ |  $\frac{\hat{s}_{\mathrm{min}}+Q^{2}}{y(s-2m_{p}^{2})}$ |  $\frac{\hat{s}_{\mathrm{min}}+Q^{2}}{yx_{a}(s-2m_{p}^{2})}$ |  $\frac{\hat{s}_{\gamma\mathrm{min}}}{z_{a\mathrm{max}}y(s-2m_{p}^{2})}$ |  $\frac{\hat{s}_{\gamma\mathrm{min}}}{z_{a\mathrm{max}}yx_{a}(s-2m_{p}^{2})}$
$x_{b\mathrm{max}}$ | 1
$x_{a\mathrm{min}}$ |  $\backslash$ |  $\frac{\hat{s}_{\mathrm{min}}+Q^{2}}{y(s-2m_{p}^{2})}$ |  $\backslash$ |  $\frac{\hat{s}_{\gamma\mathrm{min}}}{z_{a\mathrm{max}}y(s-2m_{p}^{2})}$
$x_{a\mathrm{max}}$ |  $\backslash$ |  1 |  $\backslash$ |  1

For the $p_{T}$ distribution, the bounds of $y$, $x_{a}$ and $x_{b}$ are the
same as $Q^{2}$ distribution, but for
$z_{a^{\prime}\mathrm{max}}=1/(1+Q^{2}/4p_{T}^{2})$ [53, 54]. And instead of
$\hat{t}$, $Q^{2}$ and $y_{r}$ should be integrated out

$\displaystyle\\!\\!\\!\\!Q^{2}_{\mathrm{min}}|=\frac{x_{1}^{2}}{1-x_{1}}m_{\alpha}^{2},$
$\displaystyle\\!\\!\\!\\!Q^{2}_{\mathrm{max}}|_{\mathrm{coh}}=0.027\
\mathrm{GeV}^{2},~{}~{}Q^{2}_{\mathrm{max}}|_{\mathrm{incoh}}\\!\\!=(1-x_{1})s,$
$\displaystyle\\!\\!\\!\\!|y_{r\mathrm{max}}|=$
$\displaystyle\\!\\!\\!\\!\frac{1}{2}\ln\frac{\hat{s}_{\mathrm{max}}+M_{J/\psi}^{2}-m_{\beta}^{2}+\sqrt{(\hat{s}_{\mathrm{max}}-M_{J/\psi}^{2}-m_{\beta}^{2})^{2}-4(M_{J/\psi}^{2}m_{\beta}^{2}+p^{2}_{T}\hat{s}_{\mathrm{max}})}}{\hat{s}_{\mathrm{max}}+M_{J/\psi}^{2}-m_{\beta}^{2}-\sqrt{(\hat{s}_{\mathrm{max}}-M_{J/\psi}^{2}-m_{\beta}^{2})^{2}-4(M_{J/\psi}^{2}m_{\beta}^{2}+p^{2}_{T}\hat{s}_{\mathrm{max}})}},$

where $x_{1}=\hat{s}/s$.

For the $z$ distribution, the bounds of variables are the same as $Q^{2}$
distribution, but instead of $\hat{t}$, $Q^{2}$ should be integrated out and
it’s bounds are the same as Eq. (A).

## References

  * [1] E. Fermi, Z. Phys. 29, 315-327 (1924).
  * [2] C. F. von Weizsacker, Z. Phys. 88, 612-625 (1934).
  * [3] E. J. Williams, Phys. Rev. 45, 729 (1934).
  * [4] L.D. Landau and E.M. Lifshitz, Sov. Phys. 6, 244 (1934).
  * [5] G. Nordheim et al., Phys. Rev. 51, 1037 (1937).
  * [6] R. H. Dalitz and D. R. Yennie, Phys. Rev. 105, 1598 (1957).
  * [7] I. Ya. Pomeranchuk and I. M. Shmushkevich, Nucl. Phys. 23, 1295 (1961).
  * [8] R. B. Curtis, Phys. Rev. 104, 211 (1956).
  * [9] Z. L. Ma and J. Q. Zhu, Phys. Rev. D 97, no.5, 054030 (2018).
  * [10] Z. L. Ma, Z. Lu, J. Q. Zhu and L. Zhang, [arXiv:1910.04509 [hep-ph]].
  * [11] J. Nystrand, Nucl. Phys. A 752, 470-479 (2005).
  * [12] J. Nystrand, Nucl. Phys. A 787, 29-36 (2007).
  * [13] V. M. Budnev, I. F. Ginzburg, G. V. Meledin and V. G. Serbo, Phys. Rept. 15, 181-281 (1975).
  * [14] S. J. Brodsky, T. Kinoshita and H. Terazawa, Phys. Rev. D 4, 1532-1557 (1971).
  * [15] H. Terazawa, Rev. Mod. Phys. 45, 615-662 (1973).
  * [16] J. Q. Zhu, Z. L. Ma, C. Y. Shi and Y. D. Li, Nucl. Phys. B 900, 431-445 (2015).
  * [17] M. Drees, J. R. Ellis and D. Zeppenfeld, Phys. Lett. B 223, 454-460 (1989).
  * [18] M. Drees and D. Zeppenfeld, Phys. Rev. D 39, 2536 (1989).
  * [19] S. Frixione, M. L. Mangano, P. Nason and G. Ridolfi, Phys. Lett. B 319, 339-345 (1993).
  * [20] J. Q. Zhu, Z. L. Ma, C. Y. Shi and Y. D. Li, Phys. Rev. C 92, no.5, 054907 (2015).
  * [21] G. M. Yu and Y. D. Li, Phys. Rev. C 91, no.4, 044908 (2015).
  * [22] G. M. Yu, Y. C. Yu, Y. D. Li and J. S. Wang, Nucl. Phys. B 917, 234-240 (2017).
  * [23] G. M. Yu, Y. B. Cai, Y. D. Li and J. S. Wang, Phys. Rev. C 95, no.1, 014905 (2017).
  * [24] Y. P. Fu and Y. D. Li, Nucl. Phys. A 865, 76-82 (2011).
  * [25] Y. P. Fu and Y. D. Li, Chin. Phys. C 36, 721-726 (2012).
  * [26] Y. P. Fu and Y. D. Li, Phys. Rev. C 84, 044906 (2011).
  * [27] D. Graudenz, Phys. Rev. D 49, 3291-3319 (1994).
  * [28] S. Fleming and T. Mehen, Phys. Rev. D 57, 1846-1857 (1998).
  * [29] B. A. Kniehl and L. Zwirner, Nucl. Phys. B 621, 337-358 (2002).
  * [30] A. D. Martin and M. G. Ryskin, Eur. Phys. J. C 74, 3040 (2014).
  * [31] G. Baur, K. Hencken and D. Trautmann, J. Phys. G 24, 1657-1692 (1998).
  * [32] G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D 51, 1125-1171 (1995) [erratum: Phys. Rev. D 55, 5853 (1997)].
  * [33] H. G. Dosch and E. Ferreira, Phys. Lett. B 576, 83-89 (2003).
  * [34] A. Di Giacomo, H. G. Dosch, V. I. Shevchenko and Y. A. Simonov, Phys. Rept. 372, 319-368 (2002).
  * [35] H. G. Dosch, E. Ferreira and A. Kramer, Phys. Rev. D 50, 1992-2015 (1994).
  * [36] H. G. Dosch, T. Gousset, G. Kulzinger and H. J. Pirner, Phys. Rev. D 55, 2602-2615 (1997).
  * [37] C. Ewerz and O. Nachtmann, Annals Phys. 322, 1635-1669 (2007).
  * [38] O. Nachtmann, Annals Phys. 209, 436-478 (1991).
  * [39] H. G. Dosch, Phys. Lett. B 190, 177-181 (1987).
  * [40] H. G. Dosch and Y. A. Simonov, Phys. Lett. B 205, 339-344 (1988).
  * [41] M. Gluck, E. Reya and I. Schienbein, Phys. Rev. D 60, 054019 (1999) [erratum: Phys. Rev. D 62, 019902 (2000)].
  * [42] M. Klasen, B. A. Kniehl, L. N. Mihaila and M. Steinhauser, Phys. Rev. D 68, 034017 (2003).
  * [43] M. Klasen, B. A. Kniehl, L. N. Mihaila and M. Steinhauser, Phys. Rev. D 77, 117501 (2008)..
  * [44] Z. L. Ma, J. Q. Zhu, C. Y. Shi and Y. D. Li, Chin. Phys. Lett. 32, no.12, 121202 (2015).
  * [45] M. Drees, R. M. Godbole, M. Nowakowski and S. D. Rindani, Phys. Rev. D 50, 2335-2338 (1994).
  * [46] B. A. Kniehl, Phys. Lett. B 254, 267-273 (1991) doi:10.1016/0370-2693(91)90432-P
  * [47] A. Winther, K. Alder, Nucl. Phys. A 319, 518 (1979); C. A. Bertulani, G. Baur, Nucl. Phys. A 442, 739 (1985); C. A. Bertulani, G. Baur, Nucl. Phys. A 458, 725 (1986); E. Papageorgiu, Phys. Lett. B 250, 155 (1990).
  * [48] R. Sharma and I. Vitev, Phys. Rev. C 87, no.4, 044905 (2013).
  * [49] K. A. Olive et al. [Particle Data Group], Chin. Phys. C 38, 090001 (2014)
  * [50] L. A. Harland-Lang, A. D. Martin, P. Motylinski and R. S. Thorne, Eur. Phys. J. C 75, no.5, 204 (2015).
  * [51] L. A. Harland-Lang, A. D. Martin, P. Motylinski and R. S. Thorne, Eur. Phys. J. C 75, no.9, 435 (2015).
  * [52] G. Baur, K. Hencken, D. Trautmann, S. Sadovsky and Y. Kharlov, Phys. Rept. 364, 359-450 (2002).
  * [53] G. Rossi, Phys. Rev. D 29, 852 (1984).
  * [54] M. Gluck, E. Reya and M. Stratmann, Phys. Rev. D 51, 3220-3229 (1995).