$\Omega^{2}_{(u,v)}\coloneqq\\{(u+2hv,v)+k(h\eta,\eta)+l(0,\eta)\mid
k,l\in[-1,1]\\}.$ (34)

When $\tfrac{v_{\min}-v}{\eta}<-1$ and $1<\tfrac{v_{\max}-v}{\eta}$, then
$\Omega^{2}_{(u,v)}\subset R$ and we carry on with the induction to Step 2.
Otherwise, we go directly to Step 3 as $\Omega^{n+1}\cap R^{c}\neq\emptyset$.

##### Step 2

Lower–bound for $n\geq 3$, while $\Omega^{n}_{(u,v)}\cap R^{c}=\emptyset$
We start by defining the parallelograms $\Omega^{n}_{(u,v)}$ and showing some
properties of the vectors that generate them. Then, by induction that for
$n\geq 2$, we will show the following statement:

For
$0<\epsilon<\min\left(\tfrac{1}{4},\tfrac{\eta}{2}(v_{\max}-v_{\min})\right)$
and $\eta<(v_{\max}-v_{\min})/2$, $\tilde{P}^{n}$ has a density $f^{\star n}$
lower–bounded by $\left(\tfrac{\eta\epsilon}{2}\right)^{n-2}\mu_{0}^{n}$ on
$\Omega^{n}_{(u,v)}$. (35)

Our induction is valid while $\Omega^{n}_{u,v}\subset R$ and Step 3 shows how
to modify it when it ceases to be the case. Our arguments become progressively
more geometric, for what we find the illustration of the proof in Figure 9
helpful.

[width=1.0]mixing_proof_2

Figure 9: Illustration of $\Omega^{n}_{(u,v)}$ for $n=2,\,3$ and of the
segment $P_{\lambda}$. Our argument consists in showing that for any
$(z_{1},z_{2})\in\Omega^{n+1}_{(u,v)}$, the length of the intersection of
$P_{\lambda}$ with $\Omega^{n}_{(u,v)}$ is at least $\eta\epsilon/2$. While
the dark green region is $\Omega^{3}$, the lighter colour shows a larger
region where the lower–bound is valid.

To define $\Omega^{n}_{(u,v)}$, let $v_{2}\coloneqq
T\left(0,\eta\right)=\left(h\eta,\eta\right)$ and for $n\geq 3$,

$v_{n}=(1-\epsilon)\left(T\left(0,\eta\right)+T(v_{n-1})\right)\in\mathbb{R}^{2}.$
(36)

For $n\geq 3$, we define

$\Omega^{n}_{(u,v)}\coloneqq\left\\{T^{n}\left(u,v\right)+l\left(0,\eta\right)+kv_{n}\mid
l,k\in[-1,1]\right\\}.$ (37)

Notice that if we take $n=2$ in (37), we get $\Omega^{2}_{(u,v)}$ from as
defined in (34).

While one can obtain an explicit expression of $v_{n}$, it is of little
pratical interest: we only need to ensure that the horizontal component of
$v_{n}$ remains sufficiently large. This is detailed in the proof of Lemma
E.5.

Since we have shown the statement (35) for $n=2$, we proceed with the
induction step. For $\left(z_{1},z_{2}\right)\in\Omega^{n+1}_{(u,v)}\cap R$,
we calculate

$\displaystyle\tilde{P}^{n+1}((u,v),]-\infty,z_{1}]\times[v_{\min},z_{2}])$
$\displaystyle=\int_{R\cap\\{x+yh\leq
z_{1}\\}}\tilde{P}^{n}((u,v),dxdy)P(y,[v_{\min},z_{2}])$
$\displaystyle=\int_{R\cap\\{x+yh\leq
z_{1}\\}}f_{(u,v)}^{\star{n}}(x,y)P(y,[v_{\min},z_{2}])dxdy.$

We can rewrite $R\cap\\{x+yh\leq z_{1}\\}=\\{(x,y)\in\mathbb{R}\mid y\in
I,\,x\leq z_{1}-yh\\}$. Differentiating with respect to $z_{1}$, we obtain

$\frac{\partial\tilde{P}^{n+1}((u,v),]-\infty,z_{1}]\times[v_{\min},z_{2}])}{\partial
z_{1}}=\int_{y}f_{(u,v)}^{\star{n}}(z_{1}-yh,y)P(y,[v_{\min},z_{2}])dxdy,$

where $f_{v}$ is defined in (12). For $z_{2}\geq v_{\min}$, we get

$f_{(u,v)}^{\star
n+1}(z_{1},z_{2})=\frac{\partial^{2}\tilde{P}^{n+1}((u,v),]-\infty,z_{1}]\times[v_{\min},z_{2}])}{\partial
z_{1}\partial z_{2}}=\int_{I}f_{(u,v)}^{\star{n}}(z_{1}-yh,y)f_{y}(z_{2})dy.$
(38)

The expression in (38) is similar to that from Step 1, except that it is
integrated over $I$. We can lower–bound the integrand: $f_{(u,v)}^{\star{n}}$
is lower–bounded by $\left(\tfrac{\eta\epsilon}{2}\right)^{n-2}\mu_{0}^{n}$ on
$\Omega^{n}_{(u,v)}$ for $n\geq 2$ and $f_{y}$ by $\mu_{0}$ on
$[y-\eta,y+\eta]\cap I$. To lower-bound $f_{(u,v)}^{\star n+1}$, it remains to
lower–bound the length of the integration domain. For the calculations, we
take the following parametrisation of $[z_{2}-\eta,z_{2}+\eta]\cap I$,

$\left\\{\lambda\in[-1,1]\mid P_{\lambda}\coloneqq
P_{0}+\lambda\eta(-h,1)\in\Omega^{n}_{(u,v)}\right\\},\qquad\text{where
}P_{0}=T^{-1}(z_{1},z_{2}).$ (39)

######  Lemma E.5.

The length of the segment (39) is at least $\tfrac{\epsilon}{2}$.

For the sake of readablity, we differ the proof of Lemma E.5 to Section E.2.1.
Finally, going back to (38), we have the desired lower–bound

$f_{(u,v)}^{\star(n+1)}(z_{1},z_{2})\geq\left(\tfrac{\eta\epsilon}{2}\right)^{n-2}\mu_{0}^{n}\times\mu_{0}\times\left(\tfrac{\eta\epsilon}{2}\right)=\left(\tfrac{\eta\epsilon}{2}\right)^{(n+1)-2}\mu_{0}^{n+1},\qquad\text{
for }(z_{1},\,z_{2})\in\Omega^{n+1}_{u,v}.$

In addition, for
$\epsilon<1\left/\left(1+\tfrac{3(v_{\max}-v_{\min})}{2\eta}\right)\right.$,

${\left(0,1\right)}\cdot{(v_{n+1}-v_{n})}=(1-\epsilon)\eta-\epsilon{\left(0,1\right)}\cdot{v_{n}}>(1-\epsilon)\eta-\epsilon(v_{\max}-v_{\min})>\epsilon\tfrac{v_{\max}-v_{\min}}{2}.$

The height of $\Omega^{n}_{u,v}$ grows with $n$, by at least a constant,
positive term. Hence, it eventually reaches ${v_{\max}-v_{\min}}$, in which
case $\Omega^{n}\cap R^{c}\neq\emptyset$.

##### Step 3

First non–empty intersection with the boundary
Let $n_{0}\coloneqq\min\\{n\in\mathbb{N}\mid\mu(\Omega^{n}\cap R^{c})>0\\}$.
Without loss of generality, $\Omega^{n_{0}}$ extends beyond $v_{\min}$. We
will now construct a region $\tilde{\Omega}^{n_{0}+1}\subset R$ such that
$f_{(u,v)}^{\star(n_{0}+1)}\geq\left(\tfrac{\eta\epsilon}{2}\right)^{n_{0}-2}\mu_{0}^{n_{0}}$
on $\tilde{\Omega}^{n_{0}+1}$ and for which
$\tilde{\Omega}^{n_{0}+1}\cap(\mathbb{R}\times\\{v_{\min}\\})$ is
lower–bounded. Since we can choose $\eta$ arbitrarily small, we can treat the
lower and upper boundaries independently, so we focus on the construction of
$\tilde{\Omega}^{n_{0}+1}_{u,v}$ on the boundary
$\mathbb{R}\times\\{v_{\min}\\}$ first.

For $P\in\mathbb{R}\times\\{v_{\min}\\}$, we consider $P_{\lambda}$ as in
(39), under the constraint that the integration segment lies within $R$, that
is, $\\{\lambda\in[0,1]\mid P_{\lambda}\in\Omega^{n_{0}}\\}$. We denote the
length of this segment by $L(P)$

$L(P)\coloneqq\left|\left\\{\lambda\in[-1,1]\mid
P+\eta\lambda\left(-h,1\right)\in\Omega^{n_{0}}\cap R\right\\}\right|,$

and we let $A,B$ be the endpoints of
$\Omega^{n_{0}}\cap(\mathbb{R}\times\\{v_{\min}\\})$. We rely on the following
claim, whose proof is in Section E.2.2. Figure 10 illustrates the situation.

The set $\\{P\in\Omega^{n_{0}}\cap\mathbb{R}\times\\{v_{\min}\\}\mid
L(P)\geq\tfrac{\epsilon}{2}\\}$ is not empty. If we denote by
$D=A+\left(x_{D},0\right)$ and $E=A+\left(x_{E},0\right)$ its right- and left-
endpoints, then for some $c_{0}>0$ independent of $(u,v)$, we have
$x_{B}+c_{0}\leq x_{D}-x_{E}.$ (40)

[width=1.0]mixing_proof_boundary

Figure 10: Illustration of Step 3 and the proof of (40). The leftmost polygon
represents $\Omega^{n_{0}-1}$, at the iteration before the first non-trivial
intersection occurs. The two middle parallelograms illustrate the two cases,
$x_{E}\leq x_{B}$ and $x_{B}\leq x_{E}$ respectively, from the proof of (40).
On the right, the bottom part of the polygon $\tilde{\Omega}^{n_{0}+1}$ as
constructed in Step 3. The dashed lines represent the integration segments,
whose length is measured by $L$.

In particular, $L(P)\geq\tfrac{\epsilon}{2}$ implies that $f_{(u,v)}^{\star
n_{0}+1}(P)\geq\left(\tfrac{\eta\epsilon}{2}\right)^{n_{0}-1}\mu_{0}^{n_{0}+1}$.
By convexity of $\Omega^{n_{0}}\cap R$, we have $L(P)\geq\tfrac{\epsilon}{2}$
for $P$ on the segment $T(E)T(D)$, so we can include that segment in
$\tilde{\Omega}^{n_{0}+1}$. As $L(E)=L(P)$, where
$P=E+(1-\epsilon/2)k\eta\left(-h,1\right),$ for $k\in[0,1]$, we have the same
lower–bound on the density holds on $T(P)$. So, we can include a segment of
height $\eta(1-\epsilon/2)$ above $T(E)$ in $\tilde{\Omega}^{n_{0}+1}$.
Therefore, we can define $\tilde{\Omega}^{n_{0}+1}$ as the polygon with
vertices $T(E)+\left(0,\eta(1-\epsilon/2)\right)$, $T(E)$, $T(D)$,
$T^{n_{0}+1}\left(u,v\right)-\left(0,\eta\right)+v_{n_{0}+1}$ and
$T^{n_{0}+1}\left(u,v\right)+\left(0,\eta\right)+v_{n_{0}+1}$.

We have obtained a convex pentagon $\tilde{\Omega}^{n_{0}+1}$ on which
$\tilde{P}^{n_{0}+1}((u,v),\cdot)$ is lower–bounded by a measure with density
lower–bounded by
$\left(\tfrac{\eta\epsilon}{2}\right)^{n_{0}-1}\mu_{0}^{n_{0}+1}$. Because $T$
preserves lengths on horizontal cross-sections, (40) implies that the length
of $T(D)T(E)$ is equal to that of $ED$, which is longer by $c_{0}=\eta h/4$
than the intersection at $n_{0}$.

##### Step 4

Induction for $n>n_{0}+1$
Assume that
$\tilde{\Omega}^{n_{0}+1}\cap(\mathbb{R}\times\\{v_{\max}\\})=\emptyset$. As a
consequence of calculations for Step 2, $\tilde{\Omega}^{n}_{u,v}$ is growing
upwards. Indeed, the calculations rely on Assumption (12) and the fact that
$v_{n}$ has a horizontal component whose length we control. Therefore, they
adapt to $\tilde{\Omega}^{n}_{u,v}$, with $v_{n}$ being the vector from $T(D)$
to ${T^{n_{0}+1}\left(u,v\right)-\left(0,\eta\right)+v_{n_{0}+1}}$.

In addition, (40) still holds. Indeed, redefine $A,\,B,\,D$ and $E$, except
with $n_{0}$ replaced by $n_{0}+1$ in the expression of $L$. We notice that
$A,\,B$ coincide with $T(E)$ and $T(D)$ from the previous iteration. Because
$AB$ is now of length at least $c_{0}=h\eta/4$, the proof is easier as we fall
in the first case. We define $\tilde{\Omega}^{n_{0}+2}$ as in Step 3.

We can now iterate this procedure, obtaining a lower–bound of $f_{u,v}^{\star
n}$ by a uniform constant, on a convex and polygonal domain
$\tilde{\Omega}^{n}$. Crucially, both the height of $\tilde{\Omega}^{n}$ and
the length of its intersection with $\mathbb{R}\times\\{v_{\min}\\}$ grow, by
uniformly lower–bounded amounts.

##### Step 5

Intersection with both boundaries
For some $n_{1}\in\mathbb{N}$, the intersection
$\tilde{\Omega}^{n_{1}}_{u,v}\cap(\mathbb{R}\cap\\{v_{\max}\\})$ is not
trivial. By a procedure analogue to that in Step 3, we can define
$\tilde{\Omega}^{n_{1}+1}$, which non–trivially intersects both boundaries.
Using the procedure from Step 4, it is clear that the intersection will not
only remain non–trivial with $n$, but also increase.

##### Step 6

Cross-sections with length at least $1$
By definition, $\tilde{\Omega}^{n}$ is delimited by a convex, polygonal
domain. The length of any horizontal cross-section of $\tilde{\Omega^{n}}$ is
lower–bounded by the minimum of the lengths of the intersections with the
lower and upper boundaries111To see this, consider the parallelogram on the 4
vertices of $\tilde{\Omega}^{n}$ which belong to the boundary. That
parallelogram is included in $\tilde{\Omega^{n}}$ by convexity, so the lengths
of the horizontal sections between the length of both bases.. Recall that by
Step 4, these two are increasing, and this by at least $h\eta/4$ at each
iteration. Hence, for some $n=n_{u,v}$, all horizontal sections of
$\tilde{\Omega}^{n_{u,v}}_{u,v}$ are of length at least 1.

By construction of $\tilde{\Omega}^{n}_{(u,v)}$, we have obtained a region
such that for any $n\geq n_{u,v}$,,

  1. 1.

$\tilde{P}^{n}((u,v),\cdot)$ is lower–bounded by
$\left(\frac{\eta\epsilon}{2}\right)^{n-2}\mu_{0}^{n}\mu$ on
$\tilde{\Omega}^{n}_{u,v}$, ($\mu$ being the Lebesgue measure)

  2. 2.

$\\{\tilde{\Omega}^{n}_{(u,v)}+(k,0)\\}_{k\in\mathbb{Z}}$ is a cover of $R$.

##### Step 7

Uniform lower–bound
We now show that we can choose a uniform $N\in\mathbb{N}$, such that
$n_{u,v}\leq N$ for all $(u,v)\in R$. Fix $(u,v)\in R$ and let
$\bar{\Omega}^{2}_{u,v}$ be defined as in (33), except with $\tfrac{\eta}{2}$
instead of $\eta$. We can then perform Step 1 to Step 6, so we obtain a domain
$\bar{\Omega}^{\bar{n}_{u,v}}_{u,v}$ with cross-sections of length at least 1,
for some $\bar{n}_{u,v}\geq n_{u,v}$.

Notice that the shrinked parallelogram at $n=2$ is contained in parallelograms
for different initial conditions. Specifically, we have
$\bar{\Omega}^{2}_{u,v}\subset\Omega^{2}_{x,y}$ for $(x,y)\in C_{u,v}$, where
$C_{u,v}=T^{-2}(\bar{\Omega}^{2}_{u,v})$. In particular,
$n_{x,y}\leq\bar{n}_{u,v}$, for all $(x,y)\in C_{u,v}$. Since
$(\overset{\circ}{C_{u,v}})_{(u,v)\in[0,1]\times I}$ is an open cover of
$[0,1]\times I$, by compacity, we can find a finite cover
$\\{C_{u_{k},v_{k}}\\}_{k=1}^{K}$. Clearly, $N=\max_{1\leq k\leq
K}\bar{n}_{u_{k},v_{k}}<\infty$ gives a uniform bound on
$(n_{u,v})_{(u,v)\in[0,1]\times I}$. The bound is also valid on $R\times I$,
because the whole construction is invariant with respect to horizontal
translations.

Finally, for $(u,v)\in R$, we have that $\tilde{P}^{N}((u,v),\cdot)$ is
lower–bounded by $\left(\frac{\eta\epsilon}{2}\right)^{N-2}\mu_{0}^{N}\mu$, on
$\Omega_{u,v}$ and $\\{\Omega_{(u,v)}+(k,0)\\}_{k\in\mathbb{Z}}$ is a cover of
$R$, where $\Omega_{u,v}\coloneqq\tilde{\Omega}^{N}_{u,v}$.

##### Step 8

Conclusion
We can now go back to $(\mathrm{frac}\gamma_{n},V_{n})$. By lower–bounding
$\tilde{P}^{N}$ with a uniform measure, we can use the same arguments as in
Section E.1 to conclude. For $A\in\mathcal{B}([0,1]\times I)$, we have

$\displaystyle\mathrm{frac}_{\star}\tilde{P}^{N}((u,v),A)$
$\displaystyle=\tilde{P}^{N}((u,v),\mathrm{frac}^{-1}(A))$
$\displaystyle\geq\tilde{P}^{N}((u,v),\mathrm{frac}^{-1}(A)\cap\Omega_{(u,v)})$
$\displaystyle\geq C\mu(\mathrm{frac}^{-1}(A)\cap\Omega_{(u,v)})$
$\displaystyle(\text{minorating on $\Omega_{(u,v)}$})$
$\displaystyle=C\mu(\cup_{k\in Z}A+(k,0)\cap\Omega_{(u,v)})$
$\displaystyle(\\{A+(k,0)\\}_{k}\text{ disjoint})$ $\displaystyle=C\sum_{k\in
Z}\mu(A+(k,0)\cap\Omega_{(u,v)})$ $\displaystyle=C\sum_{k\in
Z}\mu(A\cap(\Omega_{(u,v)}-(k,0)))$ $\displaystyle(\mu\text{ translation--
invariant})$ $\displaystyle\geq C\mu(\cup_{k\in
Z}A\cap(\Omega_{(u,v)}-(k,0)))$ $\displaystyle=C\mu(A)$
$\displaystyle(\\{\Omega_{(u,v)}+(k,0)\\}_{k\in\mathbb{Z}}\text{ is a cover of
$R$}),$

where
$C=C_{\eta,\epsilon,\mu_{0},N}\coloneqq\left(\frac{\eta\epsilon}{2}\right)^{N-2}\mu_{0}^{N}$.
The lower–bound is uniform in $(u,v)$ and also shows that the measure is non-
trivial. We conclude the proof of Proposition E.1 by applying Theorem E.2.

####  E.2.1 Proof of Lemma E.5

We recall that for some $l,k\in[-1,1]$,

$\left(z_{1},z_{2}\right)=T^{n+1}\left(u,v\right)+l\left(0,\eta\right)+kv_{n},$

where $v_{n}$ is given in (36), so

$P_{\lambda}\coloneqq
T^{-1}\left(z_{1},z_{2}+\lambda\eta\right)=T^{n}\left(u,v\right)+\eta(l+\lambda)\left(-h,1\right)+k(1-\epsilon)\left(\left(0,\eta\right)+v_{n}\right).$
(41)

For a parallelogram $\Omega$ generated by vectors $x,\,y$ and centered around
the origin, we have

$P\in\Omega\iff\left\\{\begin{array}[]{ccccc}{x}^{T}{y^{\perp}}&\leq&{P}^{T}{y^{\perp}}&\leq&{-x}^{T}{y^{\perp}}\\\
{-y}^{T}{x^{\perp}}&\leq&{P}^{T}{x^{\perp}}&\leq&{y}^{T}{x^{\perp}},\\\
\end{array}\right.$ (42)

where $(x_{1},x_{2})^{\perp}=(x_{2},-x_{1})$. Combining (41) with (42), we
have that $P_{\lambda}\in\Omega^{n}_{(u,v)}$ if and only if

$\displaystyle\left\\{\begin{array}[]{c}\eta{\left(0,1\right)}\cdot{v_{n}^{\perp}}\leq\eta(l+\lambda){\left(-h,1\right)}\cdot{v_{n}^{\perp}}+k(1-\epsilon)\left(\eta{\left(0,1\right)}\cdot{v_{n}^{\perp}}+{v_{n}}\cdot{v_{n}^{\perp}}\right)\leq-\eta{\left(0,1\right)}\cdot{v_{n}^{\perp}}\\\
-\eta{v_{n}}\cdot{\left(0,1\right)^{\perp}}\leq\eta^{2}(l+\lambda){\left(-h,1\right)}\cdot{\left(0,1\right)^{\perp}}+k(1-\epsilon)\left(\eta^{2}{\left(0,1\right)}\cdot{\left(0,1\right)^{\perp}}+\eta{v_{n}}\cdot{\left(0,1\right)^{\perp}}\right)\leq\eta{v_{n}}\cdot{\left(0,1\right)^{\perp}}\\\
\end{array}\right.$ $\displaystyle\iff$
$\displaystyle\left\\{\begin{array}[]{ccccc}{\left(1,0\right)}\cdot{v_{n}}(-1+k(1-\epsilon))&\leq&-(l+\lambda){\left(1,h\right)}\cdot{v_{n}}&\leq&{\left(1,0\right)}\cdot{v_{n}}(1+k(1-\epsilon))\\\
{v_{n}}\cdot{\left(1,0\right)}(-1-k(1-\epsilon))&\leq&(-h\eta)(l+\lambda)&\leq&{v_{n}}\cdot{\left(1,0\right)}(1-k(1-\epsilon)).\\\
\end{array}\right.$

As ${\left(1,h\right)}\cdot{v_{n}}>0$ and denoting

$a_{n}\coloneqq\frac{1}{h\eta}{\left(1,0\right)}\cdot{v_{n}},\qquad
b_{n}\coloneqq
1-\frac{{\left(0,h\right)}\cdot{v_{n}}}{{\left(1,h\right)}\cdot{v_{n}}},$

we have

$P_{\lambda}\in\Omega^{n}\iff\left\\{\begin{array}[]{ccccc}b_{n}(-1-k(1-\epsilon))-l&\leq&\lambda&\leq&b_{n}(1-k(1-\epsilon))-l\\\
a_{n}(-1+k(1-\epsilon))-l&\leq&\lambda&\leq&a_{n}(1+k(1-\epsilon))-l.\\\
\end{array}\right.$

Finally, taking into account that $\lambda\in[-1,1]$, we obtain that

$\lambda\in[\max(-1,b_{n}(-1-k(1-\epsilon))-l,a_{n}(-1+k(1-\epsilon))-l),\min(1,b_{n}(1-k(1-\epsilon))-l,a_{n}(1+k(1-\epsilon))-l)],$

which is of length

$\displaystyle\min($ $\displaystyle
1,b_{n}(1-k(1-\epsilon))-l,a_{n}(1+k(1-\epsilon))-l)+$ $\displaystyle+\min($
$\displaystyle 1,b_{n}(1+k(1-\epsilon))+l,a_{n}(1-k(1-\epsilon))+l)=$
$\displaystyle=\min($ $\displaystyle
2\min(1,a_{n},b_{n}),(a_{n}+b_{n})(1-k(1-\epsilon)),1+b_{n}(1+k(1-\epsilon))+l,$
$\displaystyle
1+a_{n}(1-k(1-\epsilon))+l,1+b_{n}(1-k(1-\epsilon))-l,1+a_{n}(1+k(1-\epsilon)-l)$

We claim that for $n\geq 2$,

$a_{n}\geq 1,\qquad b_{n}\geq\frac{1}{2}.$ (43)

Combining (43) with $l,k\in[-1,1]$, $0<\epsilon\leq\tfrac{1}{2}$, we conclude
that the length of (39) is at least $\tfrac{\epsilon}{2}$.

It remains to show (43). For $a_{n}$, we proceed by induction. Using
$v_{2}=T\left(0,\eta\right)$ for $n=2$ and
$v_{3}=(1-\epsilon)\eta\left(3h,2\right)$, we verify that $a_{2},\,a_{3}\geq
1$. Notice that
${\left(T\left(x,y\right)\right)}\cdot{\left(1,0\right)}={\left(\left(x,y\right)+\left(yh,0\right)\right)}\cdot{\left(1,0\right)}={\left(x,y\right)}\cdot{\left(1,h\right)}$.
Then,

${v_{n+1}}\cdot{\left(1,0\right)}=(1-\epsilon){\left(T\left(0,\eta\right)+T(v_{n})\right)}\cdot{\left(1,0\right)}=(1-\epsilon)\left[h\eta+{v_{n}}\cdot{\left(1,h\right)}\right].$

Using the induction hypothesis, ${v_{n}}\cdot{\left(1,0\right)}\geq h\eta$
combined with ${v_{n}}\cdot{\left(0,h\right)}\geq 0$,

${v_{n+1}}\cdot{\left(1,0\right)}\geq 2h\eta(1-\epsilon)\geq h\eta,$

since $\epsilon\leq\tfrac{1}{2}$.

For $b_{n}$, we can calculate directly
$b_{2}=\tfrac{h\eta}{h\eta+h\eta}=\tfrac{1}{2}$. For $n\geq 3$, we can express
$v_{n}$ using (36), so that

$\displaystyle\frac{{\left(1,0\right)}\cdot{v_{n}}}{{\left(1,h\right)}\cdot{v_{n}}}=$
$\displaystyle\frac{{\left(1,0\right)}\cdot{\left(T\left(0,\eta\right)+T(v_{n-1})\right)}}{{\left(1,h\right)}\cdot{\left(T\left(0,\eta\right)+T(v_{n-1})\right)}}$
$\displaystyle=$
$\displaystyle\frac{h\eta+{\left(1,0\right)}\cdot{T(v_{n-1})}}{2h\eta+{\left(1,h\right)}\cdot{T(v_{n-1})}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{1}{2}\frac{{\left(\left(1,0\right)-\left(0,h\right)\right)}\cdot{T(v_{n-1})}}{2h\eta+{\left(1,h\right)}\cdot{T(v_{n-1})}}$
$\displaystyle\geq$ $\displaystyle\frac{1}{2},$

where the last inequality follows from

${\left(\left(1,0\right)-\left(0,h\right)\right)}\cdot{T(v_{n-1})}={\left(1,0\right)}\cdot{v_{n-1}}+{\left(0,h\right)}\cdot{v_{n-1}}-{\left(0,h\right)}\cdot{v_{n-1}}\geq
0.$

∎

####  E.2.2 Proof of (40)

Notice first that $L(A)=0$ and $L(P)=0$ for any $P\in R\times\\{v_{\min}\\}$
to the left of $A$, so that $0\leq x_{B},x_{D},x_{E}$. Second, consider
$P=A+\left(x_{B}+(1-\epsilon/2)\eta h\tfrac{1}{1-b},0\right)$. As
$L(P)=\tfrac{\epsilon}{2}$, we have that $\\{P\mid
L(P)\geq\tfrac{\epsilon}{2}\\}\neq\emptyset$, so $D$ and $E$ exist. In
addition, we know that $x_{D}\geq x_{B}+(1-\epsilon/2)\eta h\tfrac{1}{1-b}$.
In particular, $b\leq 1$ implies that $x_{B}<x_{D}$.

Since $A\in\Omega^{n_{0}}_{u,v}$, we can write
$A=T^{n_{0}}\left(u,v\right)+l_{A}\eta\left(0,\eta\right)-v_{n_{0}}$. By
definition, $n_{0}$ is the first time such that $\Omega^{n}_{0}\cap R^{c}$ has
non-trivial measure, so, using the relation between $\Omega^{n_{0}-1}$ and
$\Omega^{n_{0}}$, we can conclude that $l_{A}\leq 0\leq 1-\epsilon/2$.

We distinguish two cases, depending which one of $\eta\epsilon h/2$ or $x_{B}$
is greater. First, if $\eta h\epsilon/2\leq x_{B}$, then $x_{E}=\eta
h\epsilon/2$. Indeed, the triangle formed by $A$, $E$ and
$A+\left(0,\eta\epsilon/2\right)$ is in $\Omega^{n_{0}}$, so
$L\left(A+\left(\eta h\epsilon/2,0\right)\right)\geq\epsilon/2$. Therefore,

$\displaystyle x_{D}-x_{E}$ $\displaystyle\geq
x_{B}+(1-\tfrac{\epsilon}{2})\eta h\tfrac{1}{1-b}-\eta h\tfrac{\epsilon}{2}$
$\displaystyle\geq x_{B}+\eta h(2(1-\tfrac{\epsilon}{2})-\tfrac{\epsilon}{2})$
$\displaystyle\geq x_{B}+\tfrac{5}{4}\eta h,$

where in the last two inequalities, we have use that $\tfrac{1}{2}\leq b$ and
$\epsilon\leq\tfrac{1}{2}$.

Next, if $x_{B}<\eta h\epsilon/2$, then $\eta h\epsilon/2\leq x_{E}$. So, for
$x\leq\eta h$,

$L\left(A+\left(x,0\right)\right)=\frac{1}{\eta
h}\left(x-\tfrac{{\left(0,h\right)}\cdot{v_{n}}}{{\left(1,h\right)}\cdot{v_{n}}}\
(x-x_{B})_{+}\right)=\frac{1}{\eta h}(xb-x_{B}(1-b)).$ (44)

Notice that $L\left(A+\left(\eta h,0\right)\right)\geq\frac{\epsilon}{2}$ for
$\epsilon\leq\frac{2}{5}$ small enough,

$\displaystyle L\left(A+\left(\eta h,0\right)\right)-\frac{\epsilon}{2}$
$\displaystyle=\frac{1}{\eta h}(\eta hb-x_{B}(1-b))-\frac{\epsilon}{2}$
$\displaystyle=b-(1-b)\frac{x_{B}}{\eta h}-\frac{\epsilon}{2}$
$\displaystyle\geq b-(1-b)\frac{\epsilon}{2}-\frac{\epsilon}{2}$
$\displaystyle=b\left(1+\frac{\epsilon}{2}\right)-\frac{3}{2}\epsilon$
$\displaystyle\geq\frac{1}{2}\left(1-\frac{5}{2}\epsilon\right),$

so $x_{E}\leq\eta h$. Using (44), we find that $x_{E}=\frac{1}{b}(\eta
h\epsilon/2+x_{B}(1-b))$. Finally,

$\displaystyle x_{D}-x_{E}-x_{B}$
$\displaystyle=x_{B}\left(1-\tfrac{1-b}{b}\right)+\eta
h\tfrac{1-\epsilon/2}{1-b}-\tfrac{1}{b}\eta h\epsilon/2-\eta
h\tfrac{\epsilon}{2}$ $\displaystyle=x_{B}(2-\tfrac{1}{b})+\eta
h\left(\tfrac{1}{1-b}+\tfrac{\epsilon}{2}(\tfrac{1}{b}-\tfrac{1}{1-b})\right)-\eta
h\tfrac{\epsilon}{2}$ $\displaystyle=x_{B}(2-\tfrac{1}{b})+\tfrac{\eta
h}{1-b}\left(1-\tfrac{\epsilon}{b}(b-\tfrac{1}{2}))\right)-\eta
h\tfrac{\epsilon}{2}.$

Since $\tfrac{1}{2}\leq b\leq 1$, we have $\tfrac{b-1/2}{b}\leq 1$ and
$\tfrac{1}{1-b}\geq 2$, so that

$x_{D}-x_{E}-x_{B}\geq 2(1-\epsilon)\eta h-\eta h\epsilon/2\geq\eta
h(1-\tfrac{3}{2}\epsilon).$

Combining the two cases with $\epsilon<\tfrac{1}{2}$, we conclude that

$x_{D}-x_{E}\geq x_{B}+\eta
h\min\left(\left(1-\tfrac{\epsilon}{2}\right),\tfrac{5}{4}\right)\geq
x_{B}+\tfrac{1}{4}\eta h.$

∎

##  Appendix F Mixing-preserving operations: mixing coefficients of
$(X_{n})_{n\in\mathbb{N}}$

######  Proposition F.1.

Let $X_{n}$ be as in(13). For any $k\in\mathbb{N}$,

$\beta_{X}(k+M-1)\leq\beta_{S}(k)\leq\beta_{\mathrm{frac}(\gamma)}(k)+\beta_{W}(k).$

The proposition is a consequence of Lemmata F.2 and F.3, combined with the
fact that $\phi$ is continuous, so
$\beta_{\phi(\gamma)}(k)\leq\beta_{\mathrm{frac}(\gamma)}(k)$. The proofs of
the lemmata essentially consist in manipulating the definitions.

######  Lemma F.2.

For two random variables $U:(\Omega_{U},\sigma^{U})\rightarrow\mathbb{R}$,
$V:(\Omega_{V},\sigma^{V})\rightarrow\mathbb{R}$ with
$(\beta_{U}(k))_{k\in\mathbb{N}},\,(\beta_{V}(k))_{k\in\mathbb{N}}\in\ell^{1}$
summable, we have ${\beta_{U+V}(k)\leq\beta_{U}(k)+\beta_{V}(k)}$. If $U$ and
$V$ are defined on the same probability space, but are independent, the same
holds true.

###### Proof.

Define $Z\coloneqq U+V$. Then, $Z$ is $(\Omega_{Z},\sigma^{Z})$-measurable,
where $\Omega_{Z}=\Omega_{U}\times\Omega_{V}$ and
$\sigma^{Z}=\sigma^{U}\otimes\sigma^{V}$. As $\sigma^{Z}$ is generated by
products of elements from $\sigma^{U}$ and $\sigma^{V}$, we only need to
consider (countable) partitions $\mathcal{A}_{U},\mathcal{B_{U}}$ and
$\mathcal{A}_{V},\mathcal{B}_{V}$ of
$\sigma^{U}_{-\infty,0},\sigma^{U}_{k,\infty}$ and
$\sigma^{V}_{-\infty,0},\sigma^{V}_{k,\infty}$ respectively. For any
$A_{U}\in\mathcal{A}_{U},A_{V}\in\mathcal{A}_{V}$ and
$B_{U}\in\mathcal{B}_{U},B_{V}\in\mathcal{B}_{V}$, by definition of the
product probability measure,

$\displaystyle P((A_{U}{\times}A_{V})\cap(B_{U}\times
B_{V})){-}P(A_{U}{\times}A_{V})P(B_{U}{\times}B_{V})$
$\displaystyle=(P_{U}(A_{U}\cap
B_{U}){-}P_{U}(A_{U})P_{U}(B_{U}))P_{V}(A_{V}\cap B_{V})$
$\displaystyle{+}P_{U}(A_{U})P_{U}(B_{U})(P_{V}(A_{V}\cap
B_{V}){-}P_{V}(A_{V})P_{V}(B_{V})).$

Since $\beta_{U}$ is is summable, $\sum_{A_{U},B_{U}}|P_{U}(A_{U}\cap
B_{U})-P_{U}(A_{U})P_{U}(B_{U})|<\infty$ (idem for $V$), so we can regroup
terms and

$\begin{array}[]{rlc}{\sum\limits_{\begin{subarray}{c}A_{U}\in\mathcal{A}_{U},A_{V}\in\mathcal{A}_{V},\\\
B_{U}\in\mathcal{B}_{U},B_{V}\in\mathcal{B}_{V}\end{subarray}}}P((A_{U}{\times}A_{V})\cap(B_{U}{\times}B_{V}))-P(A_{U}{\times}A_{V})P(B_{U}{\times}B_{V}){=}&\lx@intercol\sum\limits_{\begin{subarray}{c}A_{U}\in\mathcal{A}_{U},\\\
B_{U}\in\mathcal{B}_{U}\end{subarray}}(P_{U}(A_{U}\cap
B_{U}){-}P_{U}(A_{U})P_{U}(B_{U}))\hfil\lx@intercol\\\
&{\times}\sum\limits_{\begin{subarray}{c}A_{V}\in\mathcal{A}_{V},\\\
B_{V}\in\mathcal{B}_{V}\end{subarray}}P_{V}(A_{V}\cap B_{V})&(=1)\\\
&{+}\sum_{\begin{subarray}{c}A_{U}\in\mathcal{A}_{U},\\\
B_{U}\in\mathcal{B}_{U}\end{subarray}}P_{U}(A_{U})P_{U}(B_{U})&(=1)\\\
&\lx@intercol\times\sum\limits_{\begin{subarray}{c}A_{V}\in\mathcal{A}_{V},\\\
B_{V}\in\mathcal{B}_{V}\end{subarray}}(P_{V}(A_{V}\cap
B_{V}){-}P_{V}(A_{V})P_{V}(B_{V}))\hfil\lx@intercol\\\
\leq&\beta_{U}(k)+\beta_{V}(k).&\end{array}$

We conclude by taking the sup over partitions of $\Omega_{Z}.$ ∎

######  Lemma F.3.

Consider $S=(S_{i})_{i\in\mathbb{N}}$ with coefficients $\beta_{S}(k)$ and
define $X_{n}=(S_{n},\ldots,S_{n+M-1})$. Then,

${\beta_{X}(k+M-1)\leq\beta_{S}(k).}$

###### Proof.

First, note that the $\sigma$-algebra generated by a vector coincides with the
$\sigma$-algebra generated by its components

$\displaystyle\sigma(X_{n_{1}},\ldots X_{n_{2}})$
$\displaystyle=\sigma((S_{n_{1}},\ldots,S_{n_{1}+M-1}),\ldots,(S_{n_{2}},\ldots,S_{n_{2}+M-1}))$
$\displaystyle=\sigma(S_{n_{1}},\ldots,S_{n_{2}+M-1})$
$\displaystyle=\sigma^{S}_{n_{1},n_{2}+M-1}.$

Then, any partition $\mathcal{A}\subset\sigma^{X}_{n_{1},n_{2}}$ is also in
$\sigma^{S}_{n_{1},n_{2}+M-1}$. Since $\beta_{X}$ is defined as a sup over
such partitions, $\beta_{X}(k+M-1)\leq\beta_{S}(k)$. For $k\leq M$, we can
take $\mathcal{A}=\mathcal{B}\subset\sigma(S_{k})$. Since $S_{k}$ is a
continuous random variable, $\beta_{X}(k)=1$. ∎

##  Appendix G Gaussian approximation for dependent data

######  Theorem G.1 ([Kosorok, 2008, Theorem 11.22]).

Let $(X_{n})_{n\in\mathbb{N}}\subset\mathbb{R}^{d}$ be a stationary sequence
and consider a functional family $\mathcal{F}=(F_{t})_{t\in\mathbb{T}}$ with
finite bracketing entropy. Suppose there exists $r\in]2,\infty[$, such that

$\sum_{k=1}^{\infty}k^{\tfrac{2}{r-2}}\beta_{X}(k)<\infty,$ (45)

Then, $\sqrt{N}(\hat{F}_{t}-F_{t})$ converges to a tight, zero–mean Gaussian
$G_{d}$ process with covariance (15).

######  Theorem G.2 ([Bühlmann, 1995, Theorem1]).

Let $(X_{n})_{n\in\mathbb{N}}\subset\mathbb{R}^{d}$ be a stationary sequence
and consider a functional family $\mathcal{F}=(F_{t})_{t\in\mathbb{T}}$ with
finite bracketing entropy. Suppose that
$\beta_{X}(k)\xrightarrow[k\to\infty]{}0$ decrease exponentially and that
$\mathcal{F}$ satisfies (19,21). Let the bootstrap sample be generated with
the Moving Block Bootstrap, where the block size $L(n)$ satisfying
$L(n)\rightarrow\infty$ and $L(n)=\mathcal{O}(n^{1/2-\epsilon})$ for some
$0<\epsilon<\tfrac{1}{2}$. Then,

$\sqrt{N}(\hat{F}_{N}^{*}-\mathbb{E}^{*}[\hat{F}_{N}^{*}])\rightarrow^{*}G_{d}\qquad\text{
in probability,}$

where $G_{d}$ is the zero-mean Gaussian Process with the covariance (15).

##  Appendix H Proofs of Propositions 4.4 and 4.5

###### Proof of Proposition 4.4.

We first note that when $A_{h}\leq\epsilon$, then
$\mathrm{pers}_{p,\epsilon}^{p}(h)=0$. For the non-trivial case, we follow the
proof of Theorem 4.13 in [Perez, 2022]. An upper-bound of the covering number
of the image of $h$, at radius $\tau>0$ is $T(2\Lambda/\tau)^{1/\alpha}+1$, so
that

$\mathrm{pers}_{p,\epsilon}^{p}(h)\leq
p\int_{\epsilon}^{A(f)}\left(T\left(\frac{2\Lambda}{\tau}\right)^{1/\alpha}+1\right)(\tau-\epsilon)^{p-1}d\tau=(A_{h}-\epsilon)^{p}+pT(2\Lambda)^{1/\alpha}\int_{\epsilon}^{A(f)}\frac{(\tau-\epsilon)^{p-1}}{\tau^{1/\alpha}}d\tau$

We recall that since $\frac{A_{h}}{\tau}\geq 1$ and $\frac{1}{\alpha}\leq
p-1$, $(\frac{A_{h}}{\tau})^{1/\alpha}\leq(\frac{A_{h}}{\tau})^{p-1}$, so

$\frac{(\tau-\epsilon)^{p-1}}{\tau^{1/\alpha}}=\frac{1}{A_{h}^{1/\alpha}}\left(\frac{A_{h}}{\tau}\right)^{1/\alpha}(\tau-\epsilon)^{p-1}\leq
A_{h}^{p-1-1/\alpha}\left(1-\frac{\epsilon}{\tau}\right)^{p-1}.$

Finally, by recognizing that $1-\epsilon/\tau\leq 1-\epsilon/A_{h},$ we obtain

$\displaystyle\mathrm{pers}_{p,\epsilon}^{p}(h)$
$\displaystyle\leq(A_{h}-\epsilon)^{p}+pT(2\Lambda)^{1/\alpha}A_{h}^{p-1-1/\alpha}(1-\epsilon/A_{h})^{p-1}(A_{h}-\epsilon)$
$\displaystyle\leq(A_{h}-\epsilon)(1-\epsilon/A_{h})^{p-1}[A_{h}^{p-1}+pT(2\Lambda)^{1/\alpha}A_{h}^{p-1-1/\alpha}]$
$\displaystyle\leq(A_{h}-\epsilon)^{p}\left(1+pT\left(\tfrac{2\Lambda}{A_{h}}\right)^{1/\alpha}\right)$
$\displaystyle\leq(A_{h}-\epsilon)^{p}\left(1+pT\left(\tfrac{2\Lambda}{\epsilon}\right)^{1/\alpha}\right),$

where we have used that $\epsilon^{1/\alpha}\leq A_{h}^{1/\alpha}$ ∎

By Hölder continuity, $A_{h}\leq T^{\alpha}\Lambda$, so the ratio
$\tfrac{T\Lambda^{1/\alpha}}{A_{h}^{1/\alpha}}\geq 1$ denotes how small the
amplitude of $h$ is relative to what it could be, under the Hölder assumption.
Interestingly, that term increases as $A_{h}$ gets smaller, but the whole
bound is indeed increasing in $A_{h}$, which is of the order of
$A_{h}^{p}+A_{h}^{2-1/\alpha}$.

###### Proof of Proposition 4.5.

Let $f,g\in C([0,T])$ such that $\|f-g\|_{\infty}<\epsilon/4$. Let
$\Gamma:D(f)\rightarrow D(g)$ be a matching. Recall that
$|w_{\epsilon}(b,d)-w_{\epsilon}(\eta_{b},\eta_{d})|\leq|b-\eta_{b}|+|d-\eta_{d}|\leq
2\|(b,d)-(\eta_{b},\eta_{d})\|_{\infty}$. In addition, if $d-b<\epsilon/2$,
then both $w_{\epsilon}(b,d)=0=w_{\epsilon}(\Gamma(b,d))$. Using the bound on
the difference of $p$-powers as in the proof of Proposition 4.3,

$\displaystyle\left|\sum_{(b,d)\in
D(f)}w_{\epsilon}(b,d)^{p}-\sum_{(b^{\prime},d^{\prime})\in
D(g)}w_{\epsilon}(b^{\prime},d^{\prime})^{p}\right|$ $\displaystyle\leq
p\sum_{(b,d)\in
D(f)}|w_{\epsilon}(b,d)-w_{\epsilon}(\Gamma(b,d))|\max\\{w_{\epsilon}(b,d)^{p-1},w_{\epsilon}(\Gamma(b,d))^{p-1}\\}$
$\displaystyle\leq 2p\|f-g\|_{\infty}\sum_{\begin{subarray}{c}(b,d)\in D(f)\\\
d-b\geq\epsilon/2\end{subarray}}\max\\{w_{\epsilon}(b,d)^{p-1},w_{\epsilon}(\Gamma(b,d))^{p-1}\\}$
$\displaystyle\leq p\left(\sum_{\begin{subarray}{c}(b,d)\in D(f)\\\
d-b\geq\epsilon/2\end{subarray}}(w_{\epsilon}(b,d)+2\epsilon/4)^{p-1}\right)\|f-g\|_{\infty}.$

Since $f$ is continuous on a compact domain, it is uniformly continuous, so
the right-hand side is finite and depends only on $f$.

For the Lipschitz character, we follow the proof of [Perez, 2022, Lemma 3.20].
For $f,g\in C^{\alpha}_{\Lambda}([0,T])$,

$\displaystyle\left|\sum_{(b,d)\in
D(f)}w_{\epsilon}(b,d)^{p}-\sum_{(b^{\prime},d^{\prime})\in
D(g)}w_{\epsilon}(b^{\prime},d^{\prime})^{p}\right|$ $\displaystyle\leq
p\sum_{(b,d)\in
D(f)}|w_{\epsilon}(b,d)-w_{\epsilon}(\Gamma(b,d))|\max\\{w_{\epsilon}(b,d)^{p-1},w_{\epsilon}(\Gamma(b,d))^{p-1}\\}$
$\displaystyle\leq 2p\|f-g\|_{\infty}\left(\sum_{(b,d)\in
D(f)}w_{\epsilon}(b,d)^{p-1}+\sum_{(b^{\prime},d^{\prime})\in
D(g)}w_{\epsilon}(b^{\prime},d^{\prime})^{p-1}\right)$
$\displaystyle=2p(\mathrm{pers}_{p-1,\epsilon}^{p-1}(D(f))+\mathrm{pers}_{p-1,\epsilon}^{p-1}(D(g))\|f-g\|_{\infty}.$

By Lemma 4.4,
$\mathrm{pers}_{p-1,\epsilon}^{p-1}(D(f))\leq\frac{2^{1/\alpha}}{1-1/(p-1)\alpha}\Lambda^{p-1}T^{(p-1)\alpha-1}$,
so that

$|\mathrm{pers}_{p,\epsilon}^{p}(D(f))-\mathrm{pers}_{p,\epsilon}^{p}(D(g))|\leq\frac{2^{2+1/\alpha}}{1-1/(p-1)\alpha}\Lambda^{p-1}T^{(p-1)\alpha-1}\|f-g\|_{\infty}.$

∎

##  Appendix I Lipschitz constant for $k^{pi}$ and $k^{pi,t}$

First, $(x,y)\mapsto\exp(-(x^{2}+y^{2}))$ is $\tfrac{2\sqrt{2}}{e}-$Lipschitz
with respect to the Euclidean norm, so $\tfrac{4}{e}-$Lipschitz for the
Minkowski norm. Let us now consider
$k^{pi,t}(b,d)(x,y)=\tfrac{1}{2\pi\sigma^{2}}\left(2-\tfrac{\|(b,d)-(x,y)\|_{\infty}}{\sigma}\right)_{+}^{r}\exp\left(-\tfrac{(b-x)^{2}+(d-y)^{2}}{2\sigma^{2}}\right)$.
Then, for ${r>1}$,

$\displaystyle\left|\left(2-\tfrac{\|(b,d)-(x,y)\|_{\infty}}{\sigma}\right)_{+}^{r}-\left(2-\tfrac{\|(b^{\prime},d^{\prime})-(x,y)\|_{\infty}}{\sigma}\right)_{+}^{r}\right|=\left|\int_{0}^{1}\tfrac{d}{dt}\left(2-\tfrac{\|(b,d)+(b^{\prime}-b,d^{\prime}-d)t-(x,y)\|_{\infty}}{\sigma}\right)_{+}^{r}dt\right|$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}\left|r\left(2-\tfrac{|b+(b^{\prime}-b)t-x|}{\sigma}\right)_{+}^{r-1}(-1)^{b-x>b^{\prime}-bt}\tfrac{(b^{\prime}-b)}{\sigma}1_{|b+(b^{\prime}-b)t-x|\geq|d+(d-d^{\prime})t-y|}\right.$
$\displaystyle+\left.r\left(2-\tfrac{|d+(d^{\prime}-d)t-y|}{\sigma}\right)_{+}^{r-1}(-1)^{d-y>d^{\prime}-dt}\tfrac{(d^{\prime}-d)}{\sigma}1_{|b+(b^{\prime}-b)t-x|\leq|d+(d-d^{\prime})t-y|}\right|dt.$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}\tfrac{r}{\sigma}\left(\left(2-\tfrac{|b+(b^{\prime}-b)t-x|}{\sigma}\right)_{+}^{r-1}|b-b^{\prime}|+r\left(2-\tfrac{|d+(d^{\prime}-d)t-y|}{\sigma}\right)_{+}^{r-1}|d-d^{\prime}|\right)dt$
$\displaystyle\leq$
$\displaystyle\tfrac{r}{\sigma}\left((2-\tfrac{\min(|b-x|,|b^{\prime}-x|)}{\sigma})_{+}^{r-1}|b-b^{\prime}|+r(2-\tfrac{\min(|d-y|,|d^{\prime}-y|)}{\sigma})_{+}^{r-1}|d-d^{\prime}|\right)$
$\displaystyle\leq$
$\displaystyle\tfrac{2r}{\sigma}(2-\tfrac{\min(\|(b,d)-(x,y)\|_{\infty},\|(b^{\prime},d^{\prime})-(x,y)\|_{\infty})}{\sigma})_{+}^{r-1}\|(b,d)-(b^{\prime},d^{\prime})\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\tfrac{2^{r}r}{\sigma}\|(b,d)-(b^{\prime},d^{\prime})\|_{\infty}.$

Then, we obtain

$\displaystyle|k^{pi,t}(b,d)(x,y)-k^{pi,t}(b^{\prime},d^{\prime})(x,y)|\leq$
$\displaystyle\tfrac{1}{2\pi\sigma^{2}}\left|\left(2-\tfrac{\|(b,d)-(x,y)\|_{\infty}}{\sigma}\right)_{+}^{r}-\left(2-\tfrac{\|(b^{\prime},d^{\prime})-(x,y)\|_{\infty}}{\sigma}\right)_{+}^{r}\right|\exp\left(-\tfrac{(b-x)^{2}+(d-y)^{2}}{2\sigma^{2}}\right)$
$\displaystyle+\tfrac{1}{2\pi\sigma^{2}}\left(2-\tfrac{\|(b^{\prime},d^{\prime})-(x,y)\|_{\infty}}{\sigma}\right)_{+}^{r}\left|\exp\left(-\tfrac{(b-x)^{2}+(d-y)^{2}}{2\sigma^{2}}\right)-\exp\left(-\tfrac{(b^{\prime}-x)^{2}+(d^{\prime}-y)^{2}}{2\sigma^{2}}\right)\right|$
$\displaystyle\leq$
$\displaystyle\tfrac{1}{2\pi\sigma^{2}}\tfrac{2^{r}r}{\sigma}\|(b,d)-(b^{\prime},d^{\prime})\|_{\infty}+\tfrac{1}{2\pi\sigma^{2}}2^{r}\tfrac{4}{e}\left\|\left(\tfrac{b-x}{\sigma},\tfrac{d-y}{\sigma}\right)-\left(\tfrac{b^{\prime}-x}{\sigma},\tfrac{d^{\prime}-y}{\sigma}\right)\right\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\tfrac{2^{r-1}}{\pi\sigma^{3}}\left(r+2\right)\|(b,d)-(b^{\prime}d^{\prime})\|_{\infty}.$

##  Appendix J Moments of the Hölder constant of a stochastic process

Let $(W_{t})_{t\in[0,T]}$ be a stochastic process. A path $t\mapsto
W_{t}(\omega)$ is said to be $\alpha$-Hölder if
$|W_{t}(\omega)-W_{s}(\omega)|\leq\Lambda_{W(\omega)}|s-t|^{\alpha}$, for any
$s,t\in[0,T]$. Many processes, for example Gaussian processes, do not admit a
uniform constant. Based on [Azäis and Wschebor, 2009, Hu and Le, 2013,
Shevchenko, 2017], we will now give a condition under which
$\Lambda_{W,\omega}$ is a random variable and we will calculate its moments.

######  Proposition J.1 ([Azäis and Wschebor, 2009, Proposition 1.11]).

Suppose $W$ satisfies (9) with $K_{r_{2},r_{1}}$ and let
$\alpha\in]0,\tfrac{r_{1}}{r_{2}}[$. Then, there exists a version
$(V_{t})_{t\in[0,1]}$ of $W$ and a random variable $\Lambda_{V,\alpha}>0$,
such that, for all $s,t\in[0,1]$,

$P(|V_{t}-V_{s}|\leq\Lambda_{V,\alpha}|t-s|^{\alpha})=1\quad\text{and}\quad
P(W(t)=V(t))=1.$

######  Theorem J.2 ([Shevchenko, 2017]).

Let $r_{2}\in\mathbb{N}$ be such that $K_{r_{2},\alpha r_{2}}<\infty$ and
$1-\alpha>\tfrac{1}{r_{2}}$, $r_{2}\geq 2$,

$\mathbb{E}[\Lambda_{W}]\leq 16\
\tfrac{\alpha+1}{\alpha}TK_{r_{2},r_{2}\alpha+1}^{1/r_{2}}.$

In addition,

$\mathbb{E}[\Lambda_{W}^{k}]\leq\begin{cases}\left(2^{3+2/r_{2}}\
\tfrac{\alpha+2/r_{2}}{\alpha}\right)^{k}K_{r_{2},r_{2}\alpha+1}^{k/r_{2}},&\text{for
}0<k\leq r_{2},\\\ \left(2^{3+2/r_{2}}\
\tfrac{\alpha+2/r_{2}}{\alpha}\right)^{k}K_{k,k(\alpha+2/r_{2})-1},&\text{for
}k>r_{2}.\end{cases}$

######  Lemma J.3 (Garsia–Rodemich–Rumsey Inequality [Hu and Le, 2013, Lemma
1.1]).

Let $G:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ be a non–decreasing function
with $\lim_{x\rightarrow\infty}G(x)=\infty$ and $\delta:[0,T]\rightarrow[0,T]$
continuous and non–decreasing with $\delta(0)=0$. Let $G^{-1}$ and
$\delta^{-1}$ be lower–inverses. Let $f:[0,T]\rightarrow\mathbb{R}$ be a
continuous functions such that

$\int_{0}^{T}\int_{0}^{T}G\left(\frac{|f(x)-f(y)|}{\delta(x-y)}\right)dxdy\leq
B<\infty.$

Then, for any $s,t\in[0,T]$,

$|f(s)-f(t)|\leq 8\int_{0}^{|s-t|}G^{-1}(4B/u^{2})d\delta(u).$

###### Proof of Theorem J.2.

Consider a path $W(\omega)$ of the stochastic process and set
$B(\omega)\coloneqq\int_{0}^{T}\int_{0}^{T}G\left(\frac{|W_{t}(\omega)W_{s}(\omega)|}{\delta(t-s)}\right)dtds$,
where $G(u)=u^{r_{2}}$ and $\delta(u)=u^{\alpha+2/r_{2}}$. Then,
$G^{-1}(u)=u^{1/r_{2}}$ and
$\tfrac{d}{du}\delta=(\alpha+2/r_{2})u^{\alpha+2/r_{2}-1}$. Applying Lemma
J.3,

$\displaystyle|W_{t}(\omega)-W_{s}(\omega)|$ $\displaystyle\leq
8\int_{0}^{|s-t|}G^{-1}(4B(\omega)/u^{2})d\delta(u)$ $\displaystyle\leq
8\int_{0}^{|t-s|}\left(\frac{4B(\omega)}{u^{2}}\right)^{1/r_{2}}(\alpha+2/p)u^{\alpha+2/r_{2}-1}du$
$\displaystyle\leq
8(4B(\omega))^{1/r_{2}}(\alpha+2/r_{2})\int_{0}^{|t-s|}u^{\alpha-1}du$
$\displaystyle=8(4B(\omega))^{1/r_{2}}\
\tfrac{\alpha+2/r_{2}}{\alpha}|t-s|^{\alpha}.$

As this is valid for any $s,\,t\in[0,T]$, $\Lambda_{W}(\omega)\leq
8(4B(\omega))^{1/r_{2}}\ \tfrac{\alpha+2/r_{2}}{\alpha}$. By Jensens’
inequality,

$\mathbb{E}[\Lambda_{W}]\leq 2^{3+2/r_{2}}\
\tfrac{\alpha+2/r_{2}}{\alpha}\mathbb{E}[B(\omega)^{1/r_{2}}]\leq
2^{3+2/r_{2}}\ \tfrac{\alpha+2/r_{2}}{\alpha}\mathbb{E}[B(\omega)]^{1/r_{2}}.$
(46)

By linearity of expectation,

$\displaystyle\mathbb{E}\left[\int_{0}^{T}\int_{0}^{T}G\left(\frac{|W_{t}(\omega)W_{s}(\omega)|}{\delta(t-s)}\right)dtds\right]$
$\displaystyle=\int_{0}^{T}\int_{0}^{T}\frac{\mathbb{E}[|W_{t}(\omega)W_{s}(\omega)|^{r_{2}}]}{\delta(t-s)^{r_{2}}}dtds$
$\displaystyle=\int_{0}^{T}\int_{0}^{T}\frac{\mathbb{E}[|W_{t}(\omega)W_{s}(\omega)|^{r_{2}}]}{|t-s|^{p\alpha+2}}dtds$
$\displaystyle\leq\int_{0}^{T}\int_{0}^{T}K_{p,p\alpha+1}dtds$
$\displaystyle=T^{2}K_{r_{2},r_{2}\alpha+1}.$

Finally, $\mathbb{E}[\Lambda_{W}]\leq 2^{3+2/r_{2}}\
\tfrac{\alpha+2/r_{2}}{\alpha}T^{2/r_{2}}K_{r_{2},r_{2}\alpha+1}^{1/r_{2}}$,
as long as $r_{2}\alpha+1\leq r_{2}$ and we can simplify the constants if
$r_{2}>2$. Consider now the higher moments. If $k\leq r_{2}$, we can still
apply Jensens’ inequality in (46):

$\mathbb{E}[\Lambda_{W}^{k}]\leq\left(2^{3+2/r_{2}}\
\tfrac{\alpha+2/r_{2}}{\alpha}\right)^{r_{2}}\mathbb{E}[B(\omega)^{k/r_{2}}]\leq\left(2^{3+2/r_{2}}\
\tfrac{\alpha+2/r_{2}}{\alpha}\right)^{k}\mathbb{E}[B(\omega)]^{k/r_{2}}\leq\left(2^{3+2/r_{2}}\
\tfrac{\alpha+2/r_{2}}{\alpha}\right)^{k}K_{r_{2},r_{2}\alpha+1}^{k/r_{2}}.$

However, if $k\geq r_{2}$,

$\displaystyle\mathbb{E}\left[\left(\int_{0}^{T}\int_{0}^{T}G\left(\frac{|W_{t}(\omega)W_{s}(\omega)|}{\delta(t-s)}\right)dtds\right)^{k/r_{2}}\right]$
$\displaystyle=\int_{0}^{T}\int_{0}^{T}\frac{\mathbb{E}[|W_{t}(\omega)W_{s}(\omega)|^{k}]}{\delta(t-s)^{k}}dtds$
$\displaystyle=\int_{0}^{T}\int_{0}^{T}\frac{\mathbb{E}[|W_{t}(\omega)W_{s}(\omega)|^{k}]}{|t-s|^{k\alpha+2k/r_{2}}}dtds$
$\displaystyle\leq\int_{0}^{T}\int_{0}^{T}K_{k,k(\alpha+2/r_{2})-1}dtds$
$\displaystyle=T^{2}K_{k,k(\alpha+2/r_{2})-1}.$

∎


∎

11institutetext: J. Arrington 22institutetext: Lawrence Berkeley National
Laboratory, 22email<EMAIL_ADDRESS>33institutetext: M. Yurov
44institutetext: Los Alamos National Laboratory

# A measurement of two-photon exchange in Super-Rosenbluth separations with
positron beams

John R Arrington  Mikhail Yurov

(Received: date / Accepted: date)

###### Abstract

The proton electric and magnetic form factors, $G_{E}$ and $G_{M}$, are
intrinsically connected to the spatial distribution of charge and
magnetization in the proton. For decades, Rosenbluth separation measurement,
based on measurements of the angular dependence of elastic electron-proton
scattering, were used to separate $G_{E}$ and $G_{M}$. More recently,
polarized electron scattering measurements were used to improve the precision
of the separation but showed significant disagreement with Rosenbluth
measurements. The discrepancy was confirmed using a new ‘Super-Rosenbluth’
technique to improve the precision of the Rosenbluth extraction.

This large and unexpected discrepancy raised significant questions about our
understanding of the proton, and became a high-priority issue for the field.
Since then, a string of theoretical and experimental studies attempting to
understand the discrepancy have led to the conclusion that two-photon exchange
(TPE) corrections are most likely responsible for the discrepancy. TPE can be
measured directly by comparing positron-proton (e+p) and electron-proton (e-p)
scattering, but these measurements are extremely challenging. To date, TPE
contributions have been directly observed only at lower $Q^{2}$ values, below
the region where the Rosenbluth and polarization measurements strongly
disagree.

In this work, we show that there are significant benefits to combining the
Super-Rosenbluth technique with positron beam measurements. In addition to
providing improved precision and a greater kinematic reach, the use of the
Super-Rosenbluth technique provides a comparison of e+ and e- scattering that
is insensitive to some of the key systematic uncertainties associated with
direct comparisons of e+p to e-p scattering.

###### Keywords:

Two Photon Exchange Proton Form Factors Elastic Scattering

###### pacs:

13.60.Fz 13.40.Gp 13.40.Ks

##  1 Nucleon form factors measurements

The proton electromagnetic form factors provide unique insight into the
proton’s spatial structure, encoding the radial distribution of its charge and
magnetization Kelly:2002if . In the one-photon exchange approximation, the
proton form factors can be related to the reduced cross section for electron-
proton (e-p) or positron-proton (e+p) elastic scattering Perdrisat:2006hj ;
Arrington:2006zm ,

$\sigma_{R}(Q^{2},\varepsilon)=\tau G_{M}^{2}(Q^{2})+\varepsilon
G_{E}^{2}(Q^{2}),$ (1)

where $G_{E}(Q^{2})$ and $G_{M}(Q^{2})$ are the charge and magnetic form
factors of the proton, $Q^{2}$ is the four-momentum transfer squared,
$\tau=Q^{2}/4M_{p}^{2}$, $M_{p}$ is the mass of the proton, and
$\varepsilon=1/\left[1\,+\,2(1+\tau)\tan^{2}(\theta_{e}/2)\right]$ is the
virtual photon polarization parameter, and $\theta_{e}$ is the electron
scattering angle. The form factors $G_{E}$ and $G_{M}$ are normalized at
$Q^{2}=0$ to the proton’s charge (1) and magnetic moment ($\mu_{p})$ in units
of elementary charge and nuclear magnetons, respectively.

At fixed $Q^{2}$, $\sigma_{R}$ depends linearly on $\varepsilon$. By keeping
$Q^{2}$ fixed but making measurements at multiple $\varepsilon$ values (by
varying the beam energy and electron scattering angle) one can map out the
$\varepsilon$ dependence of the reduced cross section. A linear fit in
$\varepsilon$ allows for the extraction of $G_{M}(Q^{2})$ from the value of
the fit at $\varepsilon=0$ and $G_{E}(Q^{2})$ from the slope. This method of
extracting the form factors is known as a Rosenbluth separation
Rosenbluth:1950yq  or Longitudinal-Transverse (LT) separation.

Measurements of the elastic e-p cross section from the 1960s through the 1990s
showed that the form factors approximately followed the dipole form,

$G_{E}(Q^{2})\approx G_{M}(Q^{2})/\mu_{p}^{2}\approx
G_{D}(Q^{2})=1/(1+Q^{2}/0.71)^{2},$ (2)

with $Q^{2}$ in units of GeV2. Therefore, even for $\varepsilon=1$, the
contribution of the charge form factor is suppressed relative to the magnetic
form factor by a factor of $1/(\tau\mu_{p}^{2})\approx 0.5/Q^{2}$. At
$Q^{2}\approx 1$ GeV2, $G_{E}$ contributes at most 30% to the cross section,
and its contribution decreases as $1/Q^{2}$ at larger $Q^{2}$ values. Because
of this, direct Rosenbluth separations are severely limited in precision above
$Q^{2}=3$-4 GeV2 Arrington:2003df . In addition, the extraction of $G_{E}$ is
very sensitive to any experimental corrections that vary with $\varepsilon$,
which was a particular problem for analyses that combined different
experiments to obtain data covering a range of $\varepsilon$.

Polarization measurements provided a way to overcome the limitations of the
Rosenbluth technique at larger $Q^{2}$ values Akhiezer:1957aa ; Arnold:1980zj
. Measurements utilizing polarized electron beams in combination with
polarized targets or recoil polarization measurements, referred to
collectively as PT measurements, are sensitive to the combination
$R_{p}=\mu_{p}G_{E}/G_{M}$, rather than the individual form factors. As such,
polarization measurements of $R_{p}$ combined with Rosenbluth separations,
which yield precise values for $G_{M}$ at large $Q^{2}$, were expected to
allow for reliable measurements of both form factors over a wide range of
$Q^{2}$.

Figure 1: Measurements of $R_{p}=\mu_{p}G_{E}/G_{M}$ for the proton. Red
triangles correspond to polarization measurements Jones:1999rz ; Gayou:2001qd
, Cyan crosses are Rosenbluth extractions from Ref. Arrington:2003qk , and
black circles are the Super-Rosenbluth measurements from Ref. Qattan:2004ht .
Figure adapted from Qattan:2004ht .

The first precision measurements of the form factors utilizing polarization
observables Jones:1999rz  showed a significant deviation from the Rosenbluth
observation of $R_{p}\approx 1$, with $R_{p}$ decreasing approximately
linearly with $Q^{2}$. This dramatic discrepancy led to reexaminations of
previous Rosenbluth extractions Arrington:2003df ; Arrington:2003qk  as well
as new high-$Q^{2}$ Rosenbluth extractions Qattan:2004ht ; Christy:2004rc ,
and new polarization measurements Punjabi:2005wq ; Puckett:2010ac ;
Puckett:2011xg ; Puckett:2017flj . These efforts demonstrated that there was a
clear inconsistency, illustrated in Figure 1, between the form factors
extracted using these two techniques up to $Q^{2}=6$ GeV2, and that the
discrepancy was above what could be explained by the experimental and
theoretical uncertainties quoted by the measurements Arrington:2003df ;
Arrington:2003qk ; Qattan:2004ht .

An important work in confirming and quantifying this discrepancy was the so-
called “Super-Rosenbluth” experiment Qattan:2004ht . The Super-Rosenbluth (SR)
separation is identical to conventional Rosenbluth measurements except for the
fact that the struck proton, rather than the scattered electron, is detected.
Proton detection provides a large number of benefits when interested in the
ratio $R_{p}$ Qattan:2004ht ; Qattan:2005zd , which depends only on the
relative $\varepsilon$ dependence of the reduced cross section. Figure 2
illustrates some of the advantages of the SR technique. At fixed $Q^{2}$, the
proton momentum is constant, so all momentum-dependent corrections cancel out
when examining the $\varepsilon$ dependence. The cross section for proton
detection has a dramatically smaller $\varepsilon$ dependence compared to
electron detection, making the measurements less sensitive to rate-dependent
effects and removing the need to increase beam current (and thus target
heating corrections) as the cross section decreases. The cross section is
generally much less sensitive to the knowledge of the beam energy and angle of
the detected particle. Finally, the cross section is insensitive to radiative
corrections where the scattered (undetected) electron radiates a photon,
reducing both the size and $\varepsilon$-dependence of the radiative
corrections Qattan:2005zd . Because of these advantages, the SR measurement
allowed for an extraction of $R_{p}$ with precision comperable to the
polarization measurements. These new, precise results were consistent with
conventional Rosenbluth extractions (Fig. 1) and were significantly less
sensitive to experimental systematics. This helped to rule out several
potential experimental corrections that may have caused the discrepancy,
provided a test of the standard radiative correction procedures, and gave a
better quantitative measurement of the discrepancy.

Figure 2: Comparison of electron and proton detection in elastic ep scattering
for a range of $Q^{2}$ values. The top left panel shows the detected particle
momentum vs $\varepsilon$, showing that proton detection at fixed $Q^{2}$ is
insensitive to momentum-dependent detector response. The top right panel shows
the cross section, which varies very little with $\varepsilon$ and is
significantly higher for proton detection in the low-$\varepsilon$ region. The
bottom left (right) panel show the cross section sensitivity to uncertainty in
beam energy (particle angle). The sensitivity to beam energy is lower for
proton detection in all kinematics, and the sensitivity to angle dramatically
reduced for large $\varepsilon$ measurements.

###  1.1 Two-photon exchange corrections

While experimental efforts were focusing on confirming the discrepancy, the
contribution of TPE diagrams were being examined as a potential explanation
for the discrepancy Guichon:2003qm ; Blunden:2003sp . TPE corrections are
expected to be small for both cross section and polarization measurements, but
it was demonstrated that a relatively small, $\varepsilon$-dependent TPE
correction could significantly impact the extraction of $G_{E}$ from
high-$Q^{2}$ Rosenbluth measurements. At large $Q^{2}$, the $\varepsilon$
dependence arising from $G_{E}$ is only 5–10% of the reduced cross section,
making even a percent-level correction significant if it modifies the
$\varepsilon$ dependence. In addition to changing the Rosenbluth slope, TPE
contributions can also cause the reduced cross section to deviate from the
linear behavior expected in the Born approximation.

As noted above, the Super-Rosenbluth experiment Qattan:2004ht  was
instrumental in confirming that the discrepancy was significant and could not
be easily explained by experimental uncertainties. The experiment gained
additional importance in the context of TPE as a likely source of the
discrepancy. The high-precision extraction of $R_{p}$ allowed for the best
quantification of the LT-PT discrepancy and thus the size of the linear TPE
contributions. In addition, it provided significantly improved tests for non-
linear contributions from TPE exchange Tvaskis:2005ex . The quantification of
the LT-PT discrepancy, combined with the limits on non-linear contributions,
allows for an extraction of the form factors Arrington:2007ux ;
Bernauer:2013tpr ; Ye:2017gyb  from combined analysis of polarization data and
cross section (with calculated and/or phenomenological TPE corrections). Under
the assumptions of these combined fits, the TPE corrections do not dominate
the uncertainties in the extracted form factors. However, this relies on the
assumption that TPE contributions fully explain the observed discrepancy, and
so far there is no direct observation of TPE for $Q^{2}\geq 2$ GeV2. Without
knowing that TPE fully resolve the discrepancy, we cannot be certain that
these extractions are reliable.

More direct tests were made by comparing positron-proton and electron-proton
scattering, where the TPE contributions change sign. A global analysis of
earlier measurements Arrington:2003ck  showed evidence for TPE contributions
with an $\varepsilon$ dependence consistent with the observed discrepancy, but
the data showing non-zero TPE contributions were limited to $Q^{2}$ values
below 1 GeV2. In addition, new experiments were proposed and carried out to
study TPE in e-p and e+p scattering Adikaram:2014ykv ; Rimal:2016toz ;
Rachek:2014fam ; Henderson:2016dea . These experiments confirmed the presence
of TPE contributions up to $Q^{2}\approx 1.5$ GeV2 and were in qualitative
agreement with TPE calculations Blunden:2005ew ; Zhou:2014xka , but did not
extend to the $Q^{2}$ region where a clear discrepancy was observed and lacked
sufficient precision to look for non-linear contributions in $\varepsilon$.

At the present time, while there are significant indications that TPE
corrections are responsible for the form factor discrepancy, there is no
direct confirmation. Direct extractions of TPE from e+p/e-p cross section
ratios indicate the presence of TPE corrections, but do not extend to $Q^{2}$
values where a large discrepancy is observed. Comparisons of LT and PT
measurements Arrington:2003df ; Qattan:2005zd , including a recent result with
improved radiative correction procedures Gramolin:2016hjt , show indications
of a discrepancy at the 2$\sigma$ level up to $Q^{2}\approx 6$ GeV2 (and only
one sigma at 8 GeV2) GMP12 , but cannot identify the source of the
discrepancy. Additional understanding is required to have reliable extractions
of the form factors and to validate calculations of these corrections for
other electron scattering observables Arrington:2003qk ; Arrington:2006hm ;
Blunden:2009dm ; Arrington:2011dn ; Arrington:2011kb ; Blunden:2012ty ;
Hall:2013loa ; Hall:2015loa ; Afanasev:2017gsk :

  * •

Confirmation of TPE as the source of the discrepancy above 1.5 GeV2.

  * •

Better constraints on the size of TPE from improved Rosenbluth separations for
$Q^{2}>2$-3 GeV2.

  * •

Improved constraints on non-linear contributions for all $Q^{2}$ values.

Additional data exist that can help address the second of these questions.
Data from high-$Q^{2}$ form factor measurements GMP12  can extend the $Q^{2}$
range of the LT-PT comparisons above 6 GeV2, while additional Super-Rosenbluth
measurements up to 6 GeV2 SR2  will improve the constraints on TPE in this
region as well as improve measurements of (or constraints on) non-linear
contributions. However, we still have no direct evidence that the source of
the discrepancy in the region of significant LT-PT discrepancy is entirely due
to TPE. Without a direct demonstration that TPE fully explains the
discrepancy, the assumptions currently being made to extract the form factors
could yield incorrect results. In addition, testing TPE calculations in
elastic e-p and e+p scattering at modest-to-high $Q^{2}$ values will give us
confidence that such calculations can be applied to other observables.

In the remainder of this paper, we lay out a proposal to combine the benefits
of the Super-Rosenbluth technique with the direct sensitivity of e+p/e-p cross
section comparisons. Several of the benefits provided by the SR measurement
address challenges in the direct e+p/e-p cross section comparisons, allowing
for a direct confirmation of the TPE hypothesis, as well as providing
significantly improved constraints on both the size and non-linear
contributions of TPE compared to electron Super-Rosenbluth measurements or
conventional direct comparisons of e+p and e-p scattering. The discussion here
expands on the ideas presented in Refs. yurov2017 ; Accardi:2020swt

##  2 Super-Rosenbluth measurements with positron beams

While e+p/e-p cross section comparisons provide the most direct measurement of
TPE, they are extremely challenging in practice. Both collider measurements
and fixed target experiments utilizing secondary positron beams provide modest
luminosity, making it challenging to reach larger $Q^{2}$ and low
$\varepsilon$ values where the cross section is low, but where TPE
contributions are expected to be largest. In some cases, there are different
corrections and systematic uncertainties associated with e+ and e- beams,
limiting the sensitivity even where sufficient statistics can be collected.
Differences between positron and electron running conditions can also limit
the precision of the measurements in cases where it is not possible to change
between e- and e+ beams quickly, which can lead to different run conditions
for the positron and electron data. Finally, measurements utilizing a fixed
beam energy do not allow for a direct extraction at fixed $Q^{2}$ at different
angles, and thus cannot directly measure the $\varepsilon$ dependence at fixed
$Q^{2}$.

Several of these limitations are reduced or eliminated when using the Super-
Rosenbluth technique. As shown in Fig. 2, the cross section for proton
detection is significantly higher than for electron detection at low
$\varepsilon$, where TPE are largest, offsetting the low luminosity and
extending the kinematic reach of the measurements. This also allows for
measurements to be taken at a fixed beam current, avoiding significant changes
in rate-dependent corrections or target heating effects that can be
significant when using high beam currents for measurements at low-cross
section kinematics. By focusing on the $\varepsilon$ dependence, the
extraction of $R_{p}=\mu_{p}G_{E}/G_{M}$ comes from comparison of positron
cross sections measured at different kinematics. Only after the extraction of
$R_{p}$ from the positron measurements do we compare it to electron results,
making the result significantly less sensitive to differences between electron
and positron beam characteristics, and eliminating the need for careful
comparisons of beam quality or frequent changes between electron and positron
beams to account for potential long-term drifts in the detector response.

Figure 3: Reduced cross section from E01-001 Qattan:2004ht  (magenta points)
along with linear fit. The dotted black line is the expected behavior based on
the $\varepsilon=1$ value from the cross section measurements combined with
the slope based on polarization transfer measurements of $\mu_{p}G_{E}/G_{M}$.
The red dashed line represents the expected positron reduced cross section
assuming that the LT-PT discrepancy is fully explained by TPE. The positron
Super-Rosenbluth measurement should yield uncertainties as good or better than
those of the E01-001 experiment. Figure adapted from Ref. yurov2017

To take advantage of these benefits, the Super-Rosenbluth measurement requires
measurements at multiple beam energies to perform Rosenbluth separations at
each $Q^{2}$ value measured. This allows for cancellation of many systematic
uncertainties that are exactly (or nearly) identical for different
$\varepsilon$ settings at fixed $Q^{2}$. Measurements with a larger number of
energies improve the sensitivity to the $\varepsilon$ dependence which is
especially beneficial when looking for non-linear contributions. Figure 3
shows the reduced cross section measurements from E01-001 Qattan:2004ht , the
slope expected based on polarization measurements (assuming no TPE corrections
to the polarization observables), and a projection for the expected positron
measurements. Electron measurements for $Q^{2}=4.1$ GeV2 suggest that the
contribution from $G_{E}(Q^{2})$ is extremely small and that the slope
extracted in the Rosenbluth separation mainly reflects TPE contributions,
yielding a negative slope (unphysical in the one-photon exchange
approximation) for positron measurements.

###  2.1 Proposed positron Super-Rosenbluth measurements

Because the main benefit of the Super-Rosenbluth technique relies on
cancellation between corrections at fixed $Q^{2}$ but different $\varepsilon$
values, rather than cancellation between corrections for positron and electron
beams, the experiment can be performed with only positrons and compared to
existing electron Super-Rosenbluth measurements. However, it is beneficial to
make positron and electron measurements using the same detectors, as the
resolution of the measured proton’s angle and momentum is important in
isolating elastic scattering and avoiding inelastic backgrounds Qattan:2004ht
. Thus, the approach taken here is to optimize a set of positron SR
measurements, and then to make the same measurements using electron beams. The
positron current will be limited by the source, while the electron beams can
be run at larger currents, such that the time is dominated by positron
measurements. For the following projections, we assume a 2 $\mu$A positron
beam current and use the existing SR measurements to make projections for
statistical and systematic uncertainties.

The initial SR measurements Qattan:2004ht  were performed in Hall A at
Jefferson Lab Alcorn:2004sb , with an average beam current of 60 $\mu$A
impinging on a 4 cm liquid hydrogen target, with an allocation of 10 days of
beamtime. Precise extractions of $\mu G_{E}/G_{M}$ were made at $Q^{2}$=2.64,
3.2, and 4.1 GeV2, with significantly smaller corrections and uncertainties
than any other Rosenbluth separations in this kinematic region. Accounting for
the reduction to 2 $\mu$A beam current for positrons and replacing the 4 cm
target with a 10 cm target gives a measurement with a factor of 12 reduction
in luminosity compared to the previous experiment. Because only one of the
High Resolution Spectrometers (HRSs) was used for these measurements in the
original experiment, we will make up a factor of two by using both
spectrometers. We can save another factor of two by reducing the statistics
since, even for the highest $Q^{2}$ setting, the statistical uncertainties
were below the systematic uncertainties of the measurement, usually by a
significant factor. In this scenario, we increase the run time by a factor of
three, yielding a 30 day measurement that would provide nearly identical final
uncertainties on the extracted value of $\mu G_{E}/G_{M}$ and slightly reduced
sensitivity to deviations from linearity in the reduced cross section due to
the slightly larger statistical uncertainties.

The Hall A measurement ran with five beam energies corresponding to two
different linac energy settings. The follow-up measurement E05-017 E05017  ran
in Hall C with 17 beam energies yurov_phd , allowing for a larger
$\varepsilon$ range and more $\varepsilon$ points at each $Q^{2}$, with two
dedicated linearity scans with 10 or more $\varepsilon$ points. The experiment
used similar energies and target as the Hall A experiment and covered $Q^{2}$
values from 0.4-5.8 GeV2 with 30 days of beamtime. A full version of this
measurement using positrons is not feasible: as with the original measurement
only the High-Momentum Spectrometer (HMS) can cover the necessary kinematic
range, and the high-$Q^{2}$ points are statistics limited, meaning a
significant reduction in statistics would significantly reduce the
sensitivity. As such, one would have to make up the full factor of 12 in
luminosity through increased run time. Thus, we base our projections on the
Hall A electron measurements presented above. Note that while the plan
presented here assumes data taking in Hall A with the two HRS spectrometers,
the experiment could also be performed in Hall C with the HMS spectrometer
with essentially the same figure of merit; Experiment E01-001 used the central
1.6 msr of the HRS, while E05-017 used 3.2 msr in the HMS.

Figure 4: Potential kinematics for the proposed measurement. The curves
indicate the elastic kinematics for beam energies corresponding to an energy
per pass of 2.2 GeV (solid line), 0.78 GeV (short-dash), and 0.6 GeV.
Horizontal lines represent $Q^{2}$ values that provide a good lever arm in
$\varepsilon$. Measurements up to $Q^{2}=4.5$ GeV2 are straightforward under
the assumptions given in the text, and higher beam currents or a longer target
would allow a precision measurements at $Q^{2}\approx 5.7$ GeV2. The red line
indicates the highest beam energy used in previous measurements E05017 ;
yurov_phd , and the red shaded region indicates the increased $\varepsilon$
coverage with higher energies. Above $Q^{2}\approx 3$ GeV2, the higher beam
energies will provide a significant increase in the $\varepsilon$ coverage and
a corresponding reduction in the uncertainty on $\mu_{p}G_{E}/G_{M}$.

Corresponding electron measurements could be taken with a factor of 10 or more
increase in beam current, meaning that the electron measurements could be
performed with minimal beam time for running, plus overhead for the required
beam energy changes. While one could compare the positron SR separations to
polarization measurements directly, as was done with the electron SR
(illustrated in Fig. 3), comparing electron and positron SR measurements
doubles the size of the TPE effects observed in extracting the Rosenbluth
slope or the deviations from linearity since the TPE contributions have the
opposite sign for positrons and electrons. It also makes the comparison
independent of TPE contributions to the polarization observables, although
these are believed to be very small Guichon:2003qm ; Afanasev:2005ex ;
Meziane:2010xc .

Note that the uncertainties in $R_{p}$ should match those from experiment
E01-001 Qattan:2004ht  assuming measurements at identical kinematics. However,
there is an additional gain that comes from the increased reach in
$\varepsilon$ possible with an 11 GeV beam (compared to the 4.7 GeV (5.15 GeV)
maximum beam energy from the Hall A (Hall C) measurement). This increases the
lever arm in $\varepsilon$ by a factor of 1.5 for $Q^{2}=4.1$ GeV2, reducing
the extracted uncertainty in $R_{p}$ by an identical factor, making the
positron measurement at least as precise as the completed electron
measurement, or allowing for comparable precision at higher $Q^{2}$ values.
Therefore, at the cost of additional overhead for beam energy changes, the
$Q^{2}$ range could be increased somewhat while yielding precision identical
to the previous measurement, and additional measurements could be added for
several $Q^{2}$ values below 3 GeV2, where the run times are minimal. Figure 4
shows an example of the kinematic coverage that would be possible using three
different values of the beam energy per pass. Note that these linac settings
also allow for a measurement at 5.7 GeV2 (not assumed in the scenario
presented above), given additional time or running with higher luminosity. It
would also allow for significantly improved checks on linearity, with more
points and a wider range in $\varepsilon$ for $Q^{2}$ values up to 2-3 GeV2.
Changing from 3 $Q^{2}$ points from 2.64-4.1 GeV2 to a total of 8 $Q^{2}$
values from 0.5-4.5 Gev2, with additional $\varepsilon$ points for
measurements below $Q^{2}=2.5$ GeV2, would only increase the measurements by
3-5 days.

Figure 5: The left figure shows $\mu_{p}G_{E}/G_{M}$ from the Bosted fit to
electron scattering data (top magenta curve), a parameterization of the
polarization transfer results (black curve), and a prediction for the results
of positron LT separations, assuming that TPE yields the difference. Note that
for $Q^{2}>2.7$ GeV2, the slope in the Rosenbluth separation for positrons
becomes negative, yielding an unphysical result for the form factor ratio. The
right figure shows the same curves, but for $(\mu_{p}G_{E}/G_{M})^{2}$. The
blue and black points represent uncertainties on existing SR and polarization
measurements, respectively (placed on the parameterizations), and the red and
magenta point indicate the projected uncertainties for the proposed
measurements.

In conclusion, we have presented a plan for a Super-Rosenbluth measurement
utilizing proposed positron beams at Jefferson Lab Accardi:2020swt . Based on
the results of the previous Super-Rosenbluth measurement, we show that a 2
$\mu$A positron beam at Jefferson Lab would allow for a series of positron SR
measurements over a wide range of $Q^{2}$ (0.5-4.5 GeV2) and $\varepsilon$,
covering the region where TPE are large and believed to explain the
discrepancy between polarization and Rosenbluth extractions of the form
factors. The measurement will provide improved precision compared to previous
SR measurements, and will include electron SR measurements to be made at the
same $Q^{2}$ values, to allow a direct comparison and to take advantage of the
enhanced range of $\varepsilon$ available with 11 GeV beams. Figure 5 shows
projections for the proposed measurements on positrons (and electrons),
compared to a subset of polarization measurements and the E01-001
Qattan:2004ht  Super-Rosenbluth results.

The existing electron Super-Rosenbluth measurement already provides the
world’s best precision on $\mu G_{E}/G_{M}$ from Rosenbluth experiments, as
well as the tightest constraints on the size and nonlinear contributions from
TPE through a comparison to polarization measurements. The TPE contribution
associated with a direct comparison of electron to positron measurements is
twice as large as in the comparison to polarization data, and the
uncertainties in the data set will be better, giving this approach
significantly more sensitivity to TPE. This measurement would provide the
first test of the hypothesis that TPE contributions explain the observed form
factor discrepancy at $Q^{2}>1.5$ GeV2, and would significantly increase our
quantitative extraction of TPE contributions as a function of $Q^{2}$ and
$\varepsilon$, validating form factor extractions and giving greater
confidence in TPE calculations that must be applied in other reactions where
direct, or even indirect, experimental constraints on TPE are not feasible.

###### Acknowledgements.

This work was supported by U.S. Department of Energy, Office of Science,
Office of Nuclear Physics, under contract number DE-AC02-05CH11231.

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