$D[\gamma^{k-1}]$ are inherited by the resulting expression in terms of a
poset $D[\gamma^{k}]$.

We begin by characterizing the extremal time integrals in Eq. (5.21). In fact,
we need only repeat the steps in Eqs. (5.4) to (5.9) that we applied to the
original integral form for $F_{D}(\\{E_{\alpha}\\},{\cal L}_{G})$ in Eq.
(5.2). We can do so because these steps depend only on the poset structure of
the diagram $D[\gamma^{k-1}]$.

To organize the $k$th set of integrals, given the poset $D[\gamma^{k-1}]$, we
identify the set of its embedded extrema $M^{k}$ (See App. A). Next, we pick a
maximal antichain (a set of extrema that are not related pairwise), which we
label $A^{k}$. Generically, it contains both embedded minima and embedded
maxima. We denote the number of minima in $A^{k}$ by $r_{k}$ and the number of
maxima by $s_{k}$. Analogous to Eq. (5.4), we can write

$\displaystyle A^{k}\ $ $\displaystyle=$ $\displaystyle\ \\{x\in M^{k}|x,y\in
A^{k}\implies x\sim y\\}$ (5.27) $\displaystyle=$
$\displaystyle\left\\{b_{k,1},\dots,b_{k,r_{k}}\,;\,d_{k,1},\dots,d_{k,s_{k}}\right\\}\
\equiv\ \left\\{e_{k,i}\right\\}\,,$

where, as before, in Eq. (5.27) the $b_{k,i}$ represent the minima in $A^{k}$
and the $d_{k,i}$ represent the maxima in $A^{k}$. We use the symbol $e_{k,i}$
to represent elements of the combined set.

Next, we identify a companion set $C^{k}$, which is the set of next-to-
extremal vertices. Analogous to Eqs. (5.5) and (5.6) we define

$\displaystyle C^{k}$ $\displaystyle=$ $\displaystyle\left\\{x\in
V[\gamma^{k-1}]\bigg{|}\;\exists v\in A^{k}:(v\prec_{\gamma^{k-1}}x)\,{\rm
or}\,(x\prec_{\gamma^{k-1}}v)\right\\}$ (5.28) $\displaystyle=$
$\displaystyle\bigcup\limits_{i=1}^{r_{k}}C_{b_{k,i}}\bigcup\limits_{j=1}^{s_{k}}C_{d_{k,j}},$

where we have defined the covering (or covered) set for each element in
$A^{k}$ by

$\displaystyle C^{k}_{b_{k,i}}$ $\displaystyle=$ $\displaystyle\left\\{x\in
V[\gamma^{k-1}]\bigg{|}\;b_{k,i}\prec_{\gamma^{k-1}}x\right\\},$
$\displaystyle C^{k}_{d_{k,i}}$ $\displaystyle=$ $\displaystyle\left\\{x\in
V[\gamma^{k-1}]\bigg{|}\;x\prec_{\gamma^{k-1}}d_{k,i}\right\\}.$ (5.29)

As before, we identify every consistent set of next-to extremal elements. We
label the next-to-extremal element of $e_{k,i}$ by $x^{(e_{k,i})}_{a}$. As
above, the index $a$ labels different choices in the next-to-extremal element
$x$, given the choice in next-to-extremal elements for all $e_{l,m},\;l\leq
k-1,m\leq r_{l}+s_{l}$, as well as $e_{k,m},\;m\leq i-1$ (See App. B for more
details). We can now define the object $\gamma^{k}$ representing one of the
consistent choices for next-to-extremal vertices, up to the stage $k$,

$\displaystyle\gamma^{k}\ =\
\left\\{\gamma^{k-1}\bigcup_{i=1}^{r_{k}+s_{k}}\;x_{a}^{(e_{k,i})}\right\\}\,.$
(5.30)

We now observe that a choice in $\gamma^{k}$ naturally induces a poset
structure on the set of vertices $V\left[\gamma^{k}\right]$, which is defined
by analogy to Eq. (5.9) as

$V\left[\gamma^{k}\right]=V\left[\gamma^{k-1}\right]\bigg{\backslash}\bigcup_{i=1}^{r_{k}+s_{k}}e_{k,i}.$
(5.31)

The corresponding binary relationship for the set of vertices $V[\gamma^{k}]$
is uniquely defined by requiring that the chosen covering vertex of a given
minimum, $b_{k,i}$, say $x^{(b_{k,i})}_{j}$, is itself a minimum in the set
$C_{b_{k,i}}$ and the chosen covered vertex of a given maximum, $d_{k,i}$, say
$x^{(d_{k,i})}_{j}$ is itself a maximum in the set $C_{d_{k,i}}$. We may
therefore conclude that the new binary relation, constructed by analogy to Eq.
(5.8), and labeled $\geq_{\gamma^{k}}$, is given by

$\displaystyle\hskip 28.45274pt{\rm for}\ a,\,b\in V[\gamma^{k}],\
\mathrm{and}\ a\geq_{\gamma^{k-1}}b,\ {\rm then}\ a\geq_{\gamma^{k}}b$
$\displaystyle\hskip 28.45274pt\forall
y\in{C_{b_{k,i}}}:y\geq_{\gamma^{k}}x_{j}^{(b_{k,i})}>_{\gamma^{k}}b_{k,i}\,,$
$\displaystyle\hskip 28.45274pt\forall
y\in{C_{d_{k,i}}}:d_{k,i}>_{\gamma^{k}}x_{j}^{(d_{k,i})}\geq_{\gamma^{k}}y\,.$
(5.32)

Together with the set of vertices, $V^{k}$, the binary relation $\geq_{k}$
defines a new poset $D[\gamma^{k}]$.

With the new poset structure in place, we can carry out the time integrals
corresponding to the extrema $e_{k,i}$. To do so, we rewrite Eq. (5.21),
separating out the integrals over extremal elements in $A^{k}$, in analogy
with Eq. (5.12). As before such a decomposition will enable us to carry out
the extremal integrals, here in

$\displaystyle
2\pi\delta\left(\sum_{\alpha=1}^{n}E_{\alpha}\right)\,F_{D}\left(\\{E_{\alpha}\\},{\cal
L}_{G}\right)$ $\displaystyle=$ $\displaystyle\
\sum_{\gamma^{k}}\;\prod_{l=1}^{k-1}\prod_{i=1}^{r_{l}}\frac{i}{\Delta_{b_{l,i}}[\gamma^{l}]}\;\prod_{j=1}^{s_{l}}\frac{i}{\Delta_{d_{l,j}}[\gamma^{l}]}$
$\displaystyle\times\prod_{\alpha=1}^{|V[\gamma^{k}]|}\int_{X_{D}[\gamma^{k}]}dt_{\alpha}e^{-i(E[\gamma^{k-1}]_{\alpha}+\eta[\gamma^{k-1}]^{(\alpha)}_{j}(\omega_{j}-i\epsilon))t_{\alpha}}$
$\displaystyle\times\
\prod_{i=1}^{r_{k}}\int_{-\infty}^{t[x^{(b_{k,i})}_{a}]}dt_{k,i}\,e^{-i\left(E_{i}[\gamma^{k-1}]+\sum_{j}\eta[\gamma^{k-1}]^{(b_{k,i})}_{j}(\omega_{j}-i\epsilon)\right)t_{k,i}}$
$\displaystyle\times\
\prod_{j=1}^{s_{k}}\int^{\infty}_{t[x^{(d_{k,i})}_{a}]}dt_{k,i}\,e^{-i\left(E_{i}[\gamma^{k-1}]+\sum_{j}\eta[\gamma^{k-1}]^{(d_{k,i})}_{j}(\omega_{j}-i\epsilon)\right)t_{k,i}}\,.$

We can repeat the steps from Eqs. (5.13) to (5.16) to perform extremal time
integrals and define the denominators that arise at this stage of the
integration procedure. The results are of exactly the same form as for the
case $k=2$, and reproduce the inductive forms of momentum dependence in
denominators and phases in Eqs. (5.22) to (LABEL:eq:E-eta-gamma-k),

$\displaystyle
2\pi\delta\left(\sum_{\alpha=1}^{n}E_{\alpha}\right)\,F_{D}\left(\\{E_{\alpha}\\},{\cal
L}_{G}\right)$ $\displaystyle=$ $\displaystyle\
\sum_{\gamma^{k}}\;\prod_{l=1}^{k}\prod_{i=1}^{r_{l}}\frac{i}{\Delta_{b_{l,i}}[\gamma^{l}]}\;\prod_{j=1}^{s_{l}}\frac{i}{\Delta_{d_{l,j}}[\gamma^{l}]}$
(5.34) $\displaystyle\times\
\prod_{\alpha=1}^{|V[\gamma^{k}]|}\int_{X_{D}[\gamma^{k}]}dt_{\alpha}e^{-i(E[\gamma^{k}]_{\alpha}+\eta[\gamma^{k}]^{(\alpha)}_{j}(\omega_{j}-i\epsilon))t_{\alpha}}\,.$

Comparing the expression in Eq. (5.34) with the form of the integral after
$k-1$ time integrals in Eq. (5.21), and the four conditions that follow that
expression, we can check that we have completed an inductive construction of
Eq. (5.34) with the quantities
$\Delta_{b_{k,i}}[\gamma^{k}],\Delta_{d_{k,i}}[\gamma^{k}]$ and $E_{\alpha}\
[\gamma^{k}],\eta[\gamma^{k}]^{(\alpha)}_{j}$ defined recursively through Eqs.
(5.22), (5.23) and (LABEL:eq:E-eta-gamma-k) respectively. We also see that the
four conditions that follow Eq. (5.21) remain true after the $k$th integral.

(1) Poset structure. By construction $D[\gamma^{k}]$ is a new poset, which
again represents a diagram with all vertices local in time, with a non-empty
set of embedded extremal vertices $M^{k+1}$.

(2) Functional dependence. The time integrals in Eq. (LABEL:eq:integral-form-
poset-gamma-k) leading to Eq. (5.34) give precisely the results for
denominators and phases specified by Eqs. (5.22) - (LABEL:eq:E-eta-gamma-k),
with $l=k$.

(3) Denominators, cuts, and merged sets. The new denominators
$\Delta_{b_{k,i}}$ and $\Delta_{d_{k,i}}$ arise from the time integrals of
embedded extremal vertices in the chosen set $A^{k}$ of $D[\gamma^{k-1}]$.
Their explicit expressions in Eq. (5.22) show that they have the standard
forms of energy deficits. We can verify that they correspond to cuts of the
original diagram, $D$ that separate $D$ into two connected components as
follows. Consider first the minimal vertices, $b_{k,i}$ of $D[\gamma^{k-1}]$.
Each emits lines from the set of vertices in the merged set,
$\lambda[b_{k,i}]$, all of whose elements are unordered with respect to the
vertices in any other merged set $\lambda[b_{k,j}]$, $j\neq i$. As a result,
all vertices in $\lambda[b_{k,i}]$ are connected to the remainder of the
original diagram $D$ only through lines emitted by vertex $b_{k,i}$, which may
connect to any non-minimal vertices in $D[\gamma^{k-1}]$. Then, cutting all
lines emitted by $b_{k,i}$ separates all vertices in $\lambda[b_{k,i}]$ from
the remainder of the diagram $D$. But $\lambda[b_{k,i}]$ is connected by
construction, and so is $V[\gamma^{k-1}]\setminus b_{k,i}$ since $b_{k,i}$ is
by construction an embedded minimal vertex. Thus, cutting the set of lines
emerging from any $b_{k,i}$ cuts the diagram into two connected components.
Identical considerations apply to the embedded maximal vertices, $d_{k,i}$ of
$D[\gamma^{k-1}]$. Lines emerging from (for minimal) or absorbed into (for
maximal) extremal vertices of the next poset $D[\gamma^{k}]$, labeled
$e_{k+1,i}$, are emitted or absorbed by sets of merged vertices, which we
label $\lambda[e_{k+1,i}]$, of the original diagram, $D$. These sets are
defined. as in Eq. (5.25), by

$\displaystyle\lambda[e_{k+1,i}]\ =\
\left\\{e_{k+1,i}\,\prod_{e_{k^{\prime},j}}\,\cup\,\lambda[e_{k^{\prime},j}]\,\bigg{|}\,x^{(e_{k^{\prime},j})}_{a}=e_{k+1,i}\
,\,k^{\prime}\leq k-1\right\\}\,,$ (5.35)

where index $i$ varies over all choices of the set $\gamma^{k}$.

(4) Partial nesting. Finally, the sets $\lambda[e_{k+1,i}]$, given by Eq.
(5.35), associated with posets $D[\gamma^{k}]$, have the same properties as
the $\lambda[e_{k,i}]$ associated with the posets $D[\gamma^{k-1}]$, as
described in Eq. (5.26). The nesting features of Eq. (5.26) are inherited by
the sets defined by Eq. (5.35) precisely because the $\lambda[e_{k+1,i}]$ are
disjoint unions of smaller sets that satisfy (5.26). A consequence is that
elements from different $\lambda[b_{k+1,i}]$s are unordered (form an
antichain), and similarly for elements in different $\lambda[d_{k+1,i}]$. In
addition, the $k+1$st layer of merged sets satisfy the same relation as the
$k$th, Eq. (5.26),

$\displaystyle
V[\gamma^{k}]\prod_{i}\,\cup\,\lambda[b_{k+1,i}]\,\prod_{l}\,\cup\,\lambda[d_{k+1,l}]\
$ $\displaystyle=$ $\displaystyle\ V\,,$ (5.36)

because the difference between the set $V[\gamma^{k}]$ and the corresponding
quantity $V[\gamma^{k-1}]$ in Eq. (5.26) is precisely the chosen set of
embedded extremal vertices of $D[\gamma^{k-1}]$, $A^{k-1}$, which are
identified with the new vertices in the sets $\lambda[e_{k+1,i}]$. Thus, any
vertices that are in $V[\gamma^{k-1}]$ but not in $V[\gamma^{k}]$ are absorbed
into the union of the $\lambda[e_{k+1,i}]$, along with all of the
$\lambda[e_{k,i}]$.

In summary, all of the recursive features of the integrals of extrema have
been confirmed. In making the steps from the $(k-1)$st to $k$th sets of time
integrals, we used only the partial ordering and the existence of at least one
embedded extremal vertex at each step. Starting with any finite-order diagram,
however, it is clear that eventually, the process terminates at $k=\kappa$,
when the poset $D[\gamma^{\kappa}]$ consists of a single vertex,
$e_{\kappa+1}$. Such a vertex will connect to no internal lines (none remain)
but will connect to all external energies, $E_{\alpha}$. Its time integral
yields the momentum conserving delta function in Eq. (5).

The general form for an arbitrary poset $D$ is thus the energy conserving
delta function in Eq. (5.2), times the amplitude function,

$\displaystyle F_{D}\left(\\{E_{\alpha}\\},{\cal L}_{G}\right)$
$\displaystyle=$ $\displaystyle\
\sum_{\gamma^{\kappa}}\;\prod_{l=1}^{\kappa}\prod_{i=1}^{r_{l}}\frac{i}{E_{b_{l,i}}[\gamma^{l-1}]-\sum_{n}\left|\eta[\gamma^{l-1}]^{(b_{k,i})}_{n}\right|\omega_{j}+i\epsilon\
}$ (5.37)
$\displaystyle\times\prod_{j=1}^{s_{l}}\frac{i}{\left(-E_{d_{l,i}}[\gamma^{l-1}]\right)-\sum_{n}\eta[\gamma^{l-1}]^{(d_{l,i})}_{n}\omega_{j}+i\epsilon},$

where we have used the explicit forms of denominators in Eqs. (5.22) and
(5.23). Here all denominators correspond to cuts that divide the graph into
exactly two connected components, and the sum over $\gamma^{\kappa}$
represents all complete, recursive choices of next-to-extremal sets, starting
with those of the original embedded extremal vertices of $D$. Each denominator
is of the form of an energy deficit, either with respect to a sum of energies
flowing out of or into the diagram. In contrast to the normal TOPT form,
however, singularities associated with any diagram are all physical, in that
they divide the diagram into two connected subdiagrams.

In practical cases, most external energies are zero to begin with
(corresponding to collections of internal vertices). Such denominators are
negative semi-definite even for massless theories. An intriguing feature of
Eq. (5.37) is that denominators for which $E_{i}[\gamma^{k}]=0$ are universal,
in the sense that they are independent of the underlying process. Such
denominators describe a series of states that emerge from the vacuum, and
which couple diagrammatically to the process described by amplitude $F$ at
composite vertices of the sort illustrated in Figs. 4 and 8 above. We shall
not pursue this concept of universality here, and leave it as a subject for
future work.

As noted above, an equivalent expression for a general amplitude has been
derived in Ref. [18] by a somewhat different graphical analysis. In the next
section, we apply the method developed above to study weighted cross sections
in leptonic annihilation.

We close this section by remarking briefly on the application of these methods
to light-cone ordered perturbation theory (LCOPT) [28, 29, 30]. In LCOPT,
vacuum orderings are absent altogether. The method here may be applied,
however, whenever there are more than two external momenta, which leads, as in
time ordering, to pseudo-physical cuts. The result is just of the form of Eq.
(5.37), with the $E_{i}$ and $\omega_{j}$ replaced by external and on-shell
minus momenta, respectively (in $x^{+}$ ordering). In this way, pseudo-
physical cuts are again eliminated.

##  6 Vacuum polarization graphs, leptonic annihilation, and weighted cross
sections

In this section, we will adapt the general result for poset contributions to
amplitudes, Eq. (5.37), to lepton annihilation processes, and their weighted
cross sections. The essential feature of this analysis is that it eliminates
the pseudo-physical denominators altogether. A specific diagram in the form of
Eq. (5.37), however, does not have a manifest ordering between potential
physical cuts. To provide such an order, we will adapt our procedure to the
process at hand, in this case, leptonic annihilation.

Given a vacuum polarization graph $G$, we single out two vertices $i$ and $o$,
where the current carries positive and negative energy into the diagram,
respectively. As in the vacuum polarization diagrams of Sec. 2, the external
energies of all vertices $\alpha$ vanish, with the exception of $i$ and $o$,

$\displaystyle E_{\alpha}\ =\ Q\delta_{\alpha
i}-Q^{\prime}\delta_{\alpha,o}\,,$ (6.1)

where energy conservation will require $Q=Q^{\prime}$. Next, we define the
first set of embedded extrema $\hat{M}^{1}$, using the extremal set
$\hat{S}=\left(\text{Min}(D)\cup\text{Max}(D)\right)\setminus\\{i,o\\}$. We
then continue with our process of integrating out the largest antichain in
$\hat{M}^{1}$. The main idea is to remove the two extrema $i,o$ at each stage
in the procedure until all other extrema, whose time integrations result in
vacuum denominators (and soft pinches) only. This separates out, at the level
of the integrand, long time processes that only carry soft singularities in
the IR from short time processes that have the interpretation of taking place
at the “hard scale”. As in Sec. 5, we define a set of next-to-extremal
vertices $\gamma^{1}$, and carry out the first set of integrals. In general,
$\hat{M}^{1}$ contains $r_{1}$ embedded minima and $s_{1}$ embedded maxima,
and we find

$\displaystyle 2\pi\delta\left(Q-Q^{\prime}\right)\,\pi_{D}\left(Q,{\cal
L}_{G}\right)$ $\displaystyle=$
$\displaystyle\sum_{\gamma^{1}}\;\prod_{i=1}^{r_{1}}\frac{i}{\tilde{\Delta}_{b_{1,i}}[\gamma^{1}]}\;\prod_{i=1}^{s_{1}}\frac{i}{\tilde{\Delta}_{d_{1,i}}[\gamma^{1}]}$
(6.2) $\displaystyle\times\
\prod_{\alpha=1}^{|V[\gamma^{1}]|}\int_{X_{D}[\gamma^{1}]}dt_{\alpha}e^{-i(E_{\alpha}+\eta^{(\alpha)}_{j}[\gamma^{1}](\omega_{j}-i\epsilon))t_{\alpha}},$

where we have used notation identical to that introduced in Sec. 5. The
difference from Eq. (LABEL:eq:integral-min-max-gamma-1), however is that
$E_{\alpha}[\gamma^{1}]=E_{\alpha}$, given in Eq. (6.1), by construction. This
follows from our choice to integrate out a set of vertices that do not carry
external energy. As a consequence, the denominators $\tilde{\Delta}_{b_{1,i}}$
and $\tilde{\Delta}_{d_{1,i}}$ are of the “vacuum” form,

$\displaystyle\tilde{\Delta}_{b_{1,i}}[\gamma^{1}]\ =\
-\sum_{j}\left|\eta^{(b_{1,i})}_{j}\right|\omega_{j}\,\,.$ (6.3)

for minima and

$\displaystyle\tilde{\Delta}_{d_{1,i}}[\gamma^{1}]$ $\displaystyle=$
$\displaystyle\ -\sum_{j}\eta^{(d_{1,i})}_{j}\omega_{j}\,,$ (6.4)

for maxima, where once again we observe that the difference from Eqs. (5.14)
and (5.16) is simply the absence of external energy. These denominators
therefore do not vanish anywhere except when all lines in the state carry zero
momenta (which is a genuine pinch surface in massless theories). In Eqs. (6.3)
and (6.4) we have explicitly dropped the $i\epsilon$’s to emphasize the
reality of these state denominators.

We now continue the process that we have described in Sec. 5, but continue to
leave out vertices $i,o$ at each stage, until these are the only extrema
remaining. We emphasize that at each stage we only generate denominators of
the kind we encountered in Eqs. (6.3), (6.4). The process terminates after
$\kappa_{1}$ steps, when the set of embedded extrema is empty,
$\hat{M}^{\kappa_{1}+1}=\emptyset$. This does not mean that all vertices have
been exhausted since we have explicitly excluded $i$ and $o$ from the set of
extrema at each stage and non-embedded extremal vertices may also remain. We
thus can write for the poset integrand $\pi_{D}$,

$\displaystyle 2\pi\delta\left(Q-Q^{\prime}\right)\,\pi_{D}\left(Q,{\cal
L}_{G}\right)$ $\displaystyle=$ $\displaystyle\
\sum_{\gamma^{\kappa_{1}}}\;\prod_{l=1}^{\kappa_{1}}\prod_{i=1}^{r_{l}}\frac{i}{-\sum_{k}\left|\eta\;[\gamma^{l}]^{(b_{l,i})}_{k}\right|\omega_{k}}\;\prod_{j=1}^{s_{l}}\frac{i}{-\sum_{k}\eta\;[\gamma^{l}]^{(d_{l,j})}_{k}\omega_{k}}$
(6.5) $\displaystyle\times\
\prod_{\alpha=1}^{|V[\gamma^{\kappa_{1}}]|}\int_{X_{D}[\gamma^{\kappa_{1}}]}dt_{\alpha}e^{-i(E_{\alpha}+\eta[\gamma^{\kappa_{1}}]^{(\alpha)}_{j}(\omega_{j}-i\epsilon))t_{\alpha}}\,.$

Having separated the vacuum denominators we would like to order the remaining
integrand of the vacuum polarization poset diagram. The poset
$X_{D}[\gamma^{\kappa_{1}}]$ has no tadpole subdiagrams, lacking external
momenta and connected to the remaining diagram by a single vertex only. This
is because, as shown in App. A, any nontrivial tadpole has an embedded
extremal vertex. But the times of all embedded extrema except $i$ and $o$ have
been integrated over in arriving at Eq. (6.5).

By construction, the remaining diagram has at most two embedded extremal
vertices, $i$ and $o$. Any additional extremal vertices must be non-embedded.
Each non-embedded vertex, whether extremal or not, connects two internally
connected subdiagrams, one of which must include $i$ and the other $o$. This
is because there are no tadpole subdiagrams.

We have three possibilities,

  1. 1.

$i\leq_{\gamma^{\kappa_{1}}}o$,

  2. 2.

$o\leq_{\gamma^{\kappa_{1}}}i$,

  3. 3.

$o\sim_{\gamma^{\kappa_{1}}}i$.

These three situations have been represented diagrammatically in Fig. 9.

(a)  (b)  (c)  (d)

Figure 9:  LABEL:sub@ileqo A poset ordered graph after $\kappa_{1}$ steps
representing the situation when $i<o$. Here, $G$ is any graph containing no
extrema.
LABEL:sub@oleqi A poset ordered graph after $\kappa_{1}$ steps representing
the situation when $o<i$. Here, $G^{\prime}$ is any graph containing no
extrema.
LABEL:sub@osimi A poset ordered graph with $i\sim o$. Here $i$ is a minimum
and $o$ is a maximum, but $i\nleq o$ and $G_{1},G_{2}$ and $G_{3}$ are
subgraphs containing no embedded extrema other than possibly $i$ or $o$. This
represents a situation with two non-embedded extrema. Generically, $i,o$ are
either minima or maxima and there may be an arbitrary number of intervening
non-embedded extrema.
LABEL:sub@osimiex An example of a poset ordered graph with $i\sim o$ in QED at
three loops. Integrating the times of vertices $4,1$ yields a poset of the
type in Fig. 9c.

Let us analyze cases (1), (2) first. Since we have integrated out all embedded
extrema excluding $i,o$, one of $i,o$ is an embedded extremum. Suppose
$i\leq_{\gamma^{\kappa_{1}}}o$, and $i$ were not a minimum. This would imply
there exists a minimal vertex $x<_{\gamma^{\kappa_{1}}}i$ which is not
embedded. By definition the cut succeeding $x$, $\Delta_{x}$ would split the
graph into two or more disconnected subdiagrams, say $V_{1}\dots V_{k}$,
$k\geq 2$. Because $x$ is a minimum, vertices in different $V_{i}$ are all
mutually unordered. Then $i,o$ would be in the same disconnected component,
say $V_{1}$, since $i\leq_{\gamma^{\kappa_{1}}}o$. This would therefore imply
that all the other components $V_{2}\dots V_{k}$ carry no external momenta,
that is, that there are tadpole subdiagrams, which we have shown cannot be
present in $D[\gamma^{\kappa_{1}}]$. The same reasoning shows that $i$ is
itself an embedded minimum and that there are no other minima in the poset
$D[\gamma^{\kappa_{1}}]$. A similar argument reveals that $o$ is the unique
maximum in $D[\gamma^{\kappa_{1}}]$ for case (1).

One can use similar reasoning to argue that if $o\leq_{\gamma^{\kappa_{1}}}i$,
$i$ is the unique maximum and $o$ is the unique minimum in
$D[\gamma^{\kappa_{1}}]$ . In cases $(1),(2)$ we can either impose any total
order (time order) or continue along the lines of the arguments made in Sec.
5. The remaining integrals in Eq. (6.5) can therefore be carried out either by
constructing new posets or through the imposition of a sum of time orders.
Every time-ordered graph in cases (1) and (2) has a unique minimum and a
unique maximum. Carrying out the procedure outlined in Sec. 5 is exactly
equivalent to generating all possible time orders.

Turning to case (3), we treat the possibility that
$o\sim_{\gamma^{\kappa_{1}}}i$. It is easy to see using the arguments made for
the first two cases that both $i,o$ are embedded extrema of the graph, using
the absence of tadpole subgraphs. Suppose $o$ was an embedded extremum and $i$
was not an extremal element. This means there is an
$x\in(\text{Min}D[\gamma^{\kappa_{1}}])$ and
$y\in\text{Max}(D[\gamma^{\kappa_{1}}])$ such that
$x\leq_{\gamma^{\kappa_{1}}}i\leq_{\gamma^{\kappa_{1}}}y$. By assumption, both
$x,y$ are not embedded, and therefore, in the absence of tadpole subdiagrams,
$x$ divides the graph into precisely two additional components, $V_{1}$ and
$V_{2}$, with $i$ and $y$ in the same component, say $V_{1}$, and $o\in
V_{2}$. However, this means $y$ cannot divide the graph into two components
without a tadpole subgraph. Thus, $y$ must have been an embedded extremum,
contrary to our assumption. Thus, $i$ must be an embedded extremum.

In summary, we learn that in all three cases, $i,o$ are both unique embedded
extrema. In cases (1) and (2), vertices $i$ and $o$ are in a single, connected
diagram, in which they are the only extremal vertices. That is, there are no
non-embedded extrema. In these cases, any other vertex $y$ satisfies

$\displaystyle i<_{\gamma^{\kappa_{1}}}$ $\displaystyle y$
$\displaystyle<_{\gamma^{\kappa_{1}}}o\,,{\rm case}\ (1)\,,$ $\displaystyle
o<_{\gamma^{\kappa_{1}}}$ $\displaystyle y$
$\displaystyle<_{\gamma^{\kappa_{1}}}i\,,{\rm case}\ (2)\,.$ (6.6)

Suppose we integrate $t_{i}$ first, (up or down) to time
$t_{x_{i^{\prime}}^{(\kappa_{1}+1)}}$. In the resulting poset,
$D[\gamma^{\kappa_{1}+1}]$, any remaining vertices $y$ continue to satisfy the
same inequalities. That is,

$\displaystyle x_{i^{\prime}}^{(\kappa_{1}+1)}<_{\gamma^{\kappa_{1}+1}}$
$\displaystyle y$ $\displaystyle<_{\gamma^{\kappa_{1}+1}}o\,,\;\;\,{\rm case}\
(1)\,,$ $\displaystyle o\ <_{\gamma^{\kappa_{1}+1}}$ $\displaystyle y$
$\displaystyle<_{\gamma^{\kappa_{1}+1}}x_{i^{\prime}}^{(\kappa_{1}+1)}\,,{\rm
case}\ (2)\,.$ (6.7)

Therefore, vertex $x_{i^{\prime}}^{(\kappa_{1}+1)}$ is now an embedded
extremum. The argument that $x_{i^{\prime}}^{(\kappa_{1}+1)}$ is also an
embedded extremum for case (3) in which $i$ and $o$ are in different
subdiagrams, follows the same pattern, using the fact that $i$ and $o$ are
unique embedded extrema.

In order to now be able to order the resulting expressions diagrammatically,
we will only integrate the embedded extremum $i$ (and not $o$). Following the
procedure outlined in Sec. 5, we integrate over a single-element
$M^{\kappa+1}=\\{i\\}$. After integrating the time $t_{i}$, we will find a new
extremal $i^{\prime}$ with external energy $Q$ flowing into it. Because there
are no additional embedded extremal vertices other than vertices $i$ and $o$,
this will also be an embedded extremal vertex, although it may be a minimum or
a maximum. If we again choose $\kappa$ to represent the total number of
sequences in the process, the procedure terminates after $\kappa-\kappa_{1}+1$
steps (including the final integral integral that gives the energy delta
function). At each integration step $\kappa_{1}+1\leq l\leq\kappa$, there is a
single time integral only, corresponding to the sequence that begins with
vertex $i$ at step $\kappa_{1}+1$. This results in the formula,

$\displaystyle\pi_{D}\left(Q,{\cal L}_{G}\right)$ $\displaystyle=$
$\displaystyle\
\sum_{\gamma^{\kappa_{1}}}\;\prod_{l=1}^{\kappa_{1}}\prod_{i=1}^{r_{l}}\frac{i}{-\sum_{n}\left|\eta\;[\gamma^{l}]^{(b_{l,i})}_{n}\right|\omega_{n}}\;\prod_{j=1}^{s_{l}}\frac{i}{-\sum_{n}\eta\;[\gamma^{l}]^{(d_{l,j})}_{n}\omega_{n}}$
(6.8) $\displaystyle\times\
\sum_{\gamma^{\kappa}|\gamma^{\kappa_{1}}}\;\prod_{l=\kappa_{1}+1}^{\kappa}\frac{i}{\tilde{\Delta}_{e_{l}}[\gamma^{l}]+i\epsilon}\;,$

where the sum over $\gamma^{\kappa}$ is restricted to sets of next-to-extremal
vertices consistent with the choice $\gamma^{\kappa_{1}}$ (see App. B). Here
$e_{l}$ is defined to be the vertex that satisfies $E_{e_{l}}[\gamma^{l}]=Q>0$
in the notation of Eq. (LABEL:eq:E-eta-gamma-k), and the denominator is
defined by

$\tilde{\Delta}_{e_{l}}[\gamma^{l}]=\lambda_{e_{l}}Q\ -\
\sum_{k}\left|\eta[\gamma^{l}]^{(e_{l})}_{k}\right|\omega_{k},$ (6.9)

where $\lambda_{e_{l}}=+1$ when vertex $e_{l}$ is a minimum, and $-1$ when it
is a maximum. In the latter case, the state denominators represented by
$\tilde{\Delta}_{e_{l}}$ are negative definite. We note that as $l$ increases,
the denominators change sign when vertex $e_{l}$ reaches a previously non-
embedded extremal vertex (in case (3) above), which then becomes embedded. We
also notice that the denominators $\tilde{\Delta}$ in Eq. (6.9) are defined
slightly differently from the definition of $\Delta$ in Eqs. (5.14) and (5.16)
in order to explicitly display the $i\epsilon$.

Having obtained the expression for the forward scattering locally in phase
space, we immediately write down the contribution of a poset $D$ to the
$e^{+}e^{-}$ total cross section’s integrand, in terms of a sum over ordered
final states, $C$,

$\displaystyle\text{Im}\,\left[\pi_{D}(Q,{\cal L}_{G})\right]$
$\displaystyle=$ $\displaystyle\
\sum_{\gamma^{\kappa_{1}}}\;\prod_{l=1}^{\kappa_{1}}\prod_{i=1}^{r_{l}}\frac{i}{-\sum_{n}\left|\eta\;[\gamma^{l}]^{(b_{l,i})}_{n}\right|\omega_{n}}\;\prod_{j=1}^{s_{m}}\frac{i}{-\sum_{n}\eta\;[\gamma^{l}]^{(d_{l,j})}_{n}\omega_{n}}$
(6.10) $\displaystyle\times\
\sum_{\gamma^{\kappa}|\gamma^{\kappa_{1}}}\sum_{C=\kappa_{1}+1}^{\kappa}\;\prod_{l=C+1}^{\kappa}\frac{i}{\tilde{\Delta}_{e_{l}}[\gamma^{l}]-i\epsilon}2\pi\delta(\tilde{\Delta}_{C})\;\prod_{l=\kappa_{1}+1}^{C-1}\frac{i}{\tilde{\Delta}_{e_{l}}[\gamma^{l}]+i\epsilon}\;.$

This result is equivalent to the TOPT expression in Eq.
(LABEL:eq:forwardscattering-cuts), but has been reorganized through partial
ordering to eliminate all pseudo-physical cuts.

We recall that to get the total cross section as defined in Eq. (2.3), we
multiply the expression in Eq. (6.10) by the poset numerator factor, sum over
posets and finally over graphs, giving the integrand

$\displaystyle\sigma(Q,{\cal L}_{G})$ $\displaystyle=$ $\displaystyle\
\sum_{G}\sum_{D}\mathbb{N}_{D}\,\;\prod_{i=1}^{N_{G}}\frac{1}{2\omega_{i}}\;\sum_{\gamma^{\kappa_{1}}}\;\prod_{l=1}^{\kappa_{1}}\prod_{i=1}^{r_{l}}\frac{i}{-\sum_{n}\left|\eta\;[\gamma^{l}]^{(b_{l,i})}_{n}\right|\omega_{k}}$
(6.11)
$\displaystyle\times\prod_{j=1}^{s_{l}}\frac{i}{-\sum_{n}\eta\;[\gamma^{l}]^{(d_{l,j})}_{n}\omega_{n}}$
$\displaystyle\times\
\sum_{\gamma^{\kappa}|\gamma^{\kappa_{1}}}\sum_{C=\kappa_{1}+1}^{\kappa}\;\prod_{l=C+1}^{\kappa}\frac{i}{\tilde{\Delta}_{e_{l}}[\gamma^{l}]-i\epsilon}\
2\pi\delta(\tilde{\Delta}_{C})\;\prod_{l=\kappa_{1}+1}^{C-1}\frac{i}{\tilde{\Delta}_{e_{l}}[\gamma^{l}]+i\epsilon}\,.$

This is the analog of the TOPT expression for the integrand of the cross
section, again without pseudo-physical cuts.

It is now straightforward to obtain a new expression for the integrand of the
weighted cross section defined in Eq. (2.9). The relation (6.11) is fully
local in loop momenta ${\cal L}_{G}$, and hence specifies all contributions to
the squared amplitude from each final state, $C$. We thus have

$\displaystyle\Sigma[f,Q]$ $\displaystyle=$ $\displaystyle\ \sum_{G}\int
d{\cal
L}_{G}\;\prod_{i=1}^{N_{G}}\frac{1}{2\omega_{i}}\;\sum_{D}\mathbb{N}_{D}\sum_{\gamma^{\kappa_{1}}}\;\prod_{l=1}^{\kappa_{1}}\prod_{i=1}^{r_{l}}\frac{i}{-\sum_{n}\left|\eta\;[\gamma^{l}]^{(b_{l,i})}_{n}\right|\omega_{n}}\;$
(6.12)
$\displaystyle\times\prod_{j=1}^{s_{l}}\frac{i}{-\sum_{n}\eta\;[\gamma^{l}]^{(d_{l,j})}_{n}\omega_{n}}\
\sum_{\gamma^{\kappa}|\gamma^{\kappa_{1}}}\sum_{C=\kappa_{1}+1}^{\kappa}\;\prod_{l=C+1}^{\kappa}\frac{i}{\tilde{\Delta}_{e_{l}}[\gamma^{l}]-i\epsilon}\;2\pi\delta(\tilde{\Delta}_{C})$
$\displaystyle\times f_{C}(\vec{q}_{1}\dots\vec{q}_{k_{C}})\
\left(\frac{1+\lambda_{e_{C}}}{2}\right)\;\prod_{l=\kappa_{1}+1}^{C-1}\frac{i}{\tilde{\Delta}_{e_{l}}[\gamma^{l}]+i\epsilon}\,.$

In Eq. (6.12) we have inserted a factor of
$\left(\frac{1+\lambda_{e_{C}}}{2}\right)$ to set the weight of cuts with
negative external energy, and hence $\lambda_{e_{C}}=-1$, to zero. We can also
keep them, but they will always give zero because the argument of the energy
conservation delta function is negative definite for such states. In the sum
over posets $D$, only those posets with $i<_{\gamma^{\kappa_{1}}}o$ or
$i\sim_{\gamma^{\kappa_{1}}}o$ can have $\lambda_{eC}=1$.

Using Eq. (6.12) and the $\delta$ function identity in Eq. (3.1) we can
collect terms with the same denominator structure to write the analog of Eq.
(3.4)

$\displaystyle\Sigma[f,Q]$ $\displaystyle=$ $\displaystyle\ \sum_{G}\int
d{\cal
L}_{G}\;\prod_{i=1}^{N_{G}}\frac{1}{2\omega_{i}}\;\sum_{D}\mathbb{N}_{D}\sum_{\gamma^{\kappa_{1}}}\;\prod_{l=1}^{\kappa_{1}}\prod_{i=1}^{r_{l}}\frac{i}{-\sum_{n}\left|\eta\;[\gamma^{l}]^{(b_{l,i})}_{n}\right|\omega_{n}}\;$
$\displaystyle\times\prod_{j=1}^{s_{l}}\frac{i}{-\sum_{n}\eta\;[\gamma^{l}]^{(d_{l,j})}_{n}\omega_{n}}\
\sum_{\gamma^{\kappa}|\gamma^{\kappa_{1}}}\Bigg{(}\sum_{C=\kappa_{1}+1}^{\kappa-1}\prod_{j=C+1}^{\kappa-1}\frac{i}{\tilde{\Delta}_{e_{j}}[\gamma^{j}]-i\epsilon}$
$\displaystyle\times\left(f_{C}(\vec{q}_{1}\dots\vec{q}_{k_{C}})\left(\frac{1+\lambda_{e_{C}}}{2}\right)-f_{C+1}(\vec{q}_{1}\dots\vec{q}_{k_{C+1}})\
\left(\frac{1+\lambda_{e_{C+1}}}{2}\right)\right)\;\prod_{i=\kappa_{1}+1}^{C}\;\frac{i}{\tilde{\Delta}_{e_{i}}[\gamma^{i}]+i\epsilon}$
$\displaystyle-\prod_{j=\kappa_{1}+1}^{\kappa}\frac{i}{\tilde{\Delta}_{e_{j}}[\gamma^{j}]-i\epsilon}f_{1}(\vec{q}_{1}\dots\vec{q}_{k_{1}})\left(\frac{1+\lambda_{e_{1}}}{2}\right)+\prod_{i=\kappa_{1}+1}^{\kappa}\frac{i}{\tilde{\Delta}_{e_{i}}[\gamma^{i}]+i\epsilon}f_{\kappa}(\vec{q}_{1}\dots\vec{q}_{k_{\kappa}})\left(\frac{1+\lambda_{e_{\kappa}}}{2}\right)\Bigg{)}\,.$

This represents our final result for leptonic annihilation cross sections. It
carries no unphysical singularities and is power counting finite everywhere in
the region of integration. It is a sum of terms corresponding to all cuts of a
vacuum polarization diagram with, in general, elementary vertices and
composite vertices that include vacuum denominators. Examples can be found in
Figs. 8a and 8c of Sec. 4. As in the original form of Eq. (3.21), the final
two terms in this expression only require contour deformations to manifest
finiteness.

##  7 Summary

In this work, we have used TOPT to re-express infrared safe cross sections in
electroweak annihilation in a form that is manifestly power-counnting finite
in four dimensions. We observed that contributions to cross sections, in which
the amplitude or complex conjugate are disconnected, vanish after a sum over
appropriate time orders. Generalizing to arbitrary amplitudes, we reorganized
TOPT expression using their poset structure, to derive results similar to Ref.
[18]. We then applied this formalism to electroweak annihilation, to derive a
locally finite diagrammatic expression in which all pseudo-physical cuts are
eliminated.

We anticipate that it will be possible to use an expression like Eq. (6) as
the starting point for the numerical evaluation of weighted cross sections in
leptonic annihilation, complementing the loop-tree duality treatments of Refs.
[7]-[9]. This, of course, will require a systematic method to use contour
deformation or related methods to avoid or cancel non-pinched singularities
[31]. Beyond leptonic annihilation, the methods described here may complement
those of Refs. [5, 6] to control infrared singularities locally in hadronic
cross sections.

### Acknowledgements

We thank Charalampos Anastasiou for many helpful discussions on these and
related topics. We thank Zeno Capatti and Hoffie Hannesdottir, Valentin
Hirschi, and Dario Kermanschah for very useful conversations. This work was
supported in part by National Science Foundation Awards PHY-1915093 and
PHY-2210533.

##  Appendix A Embedded Extrema

We would like to show that every connected, finite-order poset diagram has at
least one embedded extremal vertex.

First, we recall that embedded vertex, $v$ in poset diagram $D$ is one whose
elimination from $D$ leaves the remaining diagram, $D\setminus v$, connected.
If, on the contrary, $u$ is a “non-embedded” vertex, we must have $D\setminus
u=\cup_{i=1}^{p_{u}}D^{[u]}_{i}$, with $p_{u}$ the number of subdiagrams of
$D$ that share vertex $u$ and are otherwise disconnected. This is the case
whether $u$ is an extremum or not.

We begin our argument by picking an arbitrary extremum, $e_{0}$, which for
definiteness we may choose to be a minimum. Say $e_{0}$ is not embedded. Then,
starting at $e_{0}$, we choose arbitrarily a path $x_{1},x_{2}\dots$, that
follows the relation $x_{j+1}>x_{i}$. We refer to this as an “increasing”
path. Because $D$ is finite-order, this path must terminate at some maximum
vertex, $e_{1}$. Either $e_{1}$ is embedded, or not. If not, we know that
$D\setminus e_{1}=\cup_{i=1}^{p_{e_{1}}}D^{[e_{1}]}_{i}$. We repeat the
process for vertex $e_{1}$ in subdiagram, $D^{[e_{1}]}_{i}\subset D$, chosen
arbitrarily, where $e_{0}\not\in D^{[e_{1}]}_{i}$. If $e_{1}$ is not a maximum
in $D^{[e_{1}]}_{i}$, we can again follow an increasing path in the
$D^{[e_{1}]}$ to a second maximum. If $e_{1}$ happens to be a maximum in
$D^{[e_{1}]}_{i}$, we can follow a decreasing path until we reach a minimum.
In either case, the extremum, $e_{2}$ at which we arrive may or may not be
embedded. If it is not embedded we repeat the process by choosing any of the
subdiagrams $D^{[e_{2}]}_{j}\subset D_{i}^{[e_{1}]}$, $e_{1}\not\in
D^{[e_{2}]}_{j}$. Because the complete poset $D$ is of finite order,

$\displaystyle 0\,<\,|D^{[e_{2}]}_{j}|\,<\,|D^{[e_{1}]}_{i}|\,,$ (A.1)

and this sequence of steps must terminate at an extremal vertex, $e_{n}$ in
some subdiagram $D_{k}^{[e_{n-1}]}$, whose removal does not disconnect
$D_{k}^{[e_{n-1}]}$. This is because Eq. (A.1) represents the first two terms
in a sequence of strictly reducing positive integers. Such a sequence must
terminate at $0$. The extremal vertex at which the sequence terminates is
embedded.555Notice that extremum $e_{n-1}$ is considered part of
$D_{k}^{[e_{n-1}]}$.

We conclude by observing that the arguments above apply to diagrams that are
of the tadpole topology: no external momenta and connected to the remainder of
the diagram by only a single vertex, which may or may not be extremal. Thus,
as observed in Sec. 6, every tadpole has an embedded extremal vertex, by
definition without an external momentum. In the intermediate expression of Eq.
(6.5), all such time integrals have been carried out, and in the process all
tadpoles subdiagrams have been eliminated in poset $D[\gamma^{\kappa_{1}}]$.

##  Appendix B An explicit algorithm to identify $\gamma^{k+1}$

In this Appendix, we would like to present one explicit algorithm to identify
all the possible consistent choices in $\gamma^{k+1}$, as defined for instance
in Eq. (5.7), given a poset
$D\left[\gamma^{k}\right]=\\{V\left[\gamma^{k}\right],\geq_{\gamma^{k}}\\}$
and a choice in maximal embedded anti-chain $A^{k+1}=\\{b_{k+1,1}\dots
b_{k+1,r_{k+1}},d_{k+1,1}\dots d_{k+1,s_{k+1}}\\}=\\{e_{k+1,j}\\}$. We
emphasize that the ordering of extremal elements in $A^{k+1}$ is arbitrary. As
in Sec. 5.1 we define next-to-extremal sets,

$\displaystyle C_{b_{k+1,j}}=\\{x\in
V\left[\gamma^{k}\right]\bigg{|}\;b_{k+1,j}\prec_{\gamma^{k}}x\\},$
$\displaystyle C_{d_{k+1,j}}=\\{x\in
V\left[\gamma^{k}\right]\bigg{|}\;x\prec_{\gamma^{k}}d_{k+1,j}\\},$
$\displaystyle
C^{k+1}=\bigcup\limits_{j=1}^{r_{k+1}}C_{b_{k+1,j}}\bigcup\limits_{l=1}^{s_{k+1}}C_{d_{k+1,l}}.$
(B.1)

We also define a set of next-to-extremal vertices,

$\displaystyle\gamma^{k}_{i}\ \equiv\
\bigcup\limits_{q\in[1,k]}\bigcup\limits_{a\in[1,\hat{r}_{q}(k,i)]}x_{{j_{q,a}}}^{(e_{q,a})}\,,$
(B.2)

where

$\displaystyle\hat{r}_{q}(k,i)$ $\displaystyle=$ $\displaystyle
r_{q}+s_{q}\text{ if }q\in[1,k-1]\,,$ $\displaystyle\hat{r}_{q}(k,i)$
$\displaystyle=$ $\displaystyle i\text{ if }q=k\,.$ (B.3)

In the notation of Eq. (B.3),
$D\left[\gamma^{k}_{0}\right]=D\left[\gamma^{k}\right]$, with $D[\gamma^{k}]$
defined in terms of the bilinear relation after the $k$th set of time
integrals, Eq. (5.32). We start with $D[\gamma^{k}]$ and construct the full
explicit sum over $D[\gamma^{k+1}]$, one extremal vertex at a time. We will
iteratively define the sum over $D\left[\gamma^{k+1}\right]$ in terms of a sum
over intermediate posets leading to
$D\left[\gamma^{k}_{r_{k+1}+s_{k+1}}\right]$.

Suppose we are given the poset $D\left[\gamma^{k}_{i}\right]$, which is, as
usual, a set of vertices $D\left[\gamma^{k}_{i}\right]$, and a binary relation
$\geq_{\gamma^{k}_{i}}$, defined by direct analogy to Eq. (5.32). To identify
the poset $D\left[\gamma^{k}_{i+1}\right]$, we would like the minimum
(maximum) $e_{k+1,i+1}$ to be covered by (to cover) a unique element
$x_{j}^{(e_{k+1,i+1})}$. To do this, we find the set

$\displaystyle\tilde{C}_{e_{k+1,i+1}}$ $\displaystyle=$ $\displaystyle\\{x\in
V\left[\gamma^{k}_{i+1}\right]\bigg{|}\;e_{k+1,i+1}\prec_{\gamma^{k}_{i}}x\\}\;\;\text{if
}\;\;i+1\leq r_{k+1}$ $\displaystyle\tilde{C}_{e_{k+1,i+1}}$ $\displaystyle=$
$\displaystyle\\{x\in
V\left[\gamma^{k}_{i+1}\right]\bigg{|}\;x\prec_{\gamma^{k}_{i}}e_{k+1,i+1}\\}\;\;\text{if
}\;\;i+1>r_{k+1}.$ (B.4) $\displaystyle=$
$\displaystyle\\{x^{(e_{k+1},i+1)}_{1},x^{(e_{k+1},i+1)}_{2}\dots
x^{(e_{k+1},i+1)}_{c_{k+1,i+1}}\\}.$

We notice that generically, $\tilde{C}_{e}$ is distinct from $C_{e}$ because
we use the binary $\geq_{\gamma^{k}}$ in Eq. (B.1) while we use the binary
$\geq_{\gamma^{k}_{i}}$ in Eq. (B.4). In fact, one generally finds
$\tilde{C}_{e}\subseteq C_{e}$ since we use the more restrictive binary
$\geq_{\gamma^{k}_{i}}$. We understand this to follow from the observation
that having identified unique covering (covered) vertices for the extrema
$e_{k+1,j}\;\;j\in[1,i]$, there are fewer consistent covering (covered)
vertices for $e_{k+1,i+1}$. We now define a new binary $\gamma^{k}_{i+1}$ that
satisfies

  * •

If $a\geq_{\gamma^{k}_{i}}b$, then $a\geq_{\gamma^{k}_{i+1}}b$

  * •

If $i+1\leq r_{k+1}$, we choose the earliest element
$x_{j_{k,i+1}}^{(e_{k,i+1})}\in\tilde{C}_{e_{k+1,i+1}}:y\geq_{\gamma^{k}_{i+1}}x_{j_{k,i+1}}^{(e_{k,i+1})},\forall
y\in\tilde{C}_{e_{k+1,i+1}}$,

  * •

If $i+1>r_{k+1}$, we choose the latest element
$x_{j_{k,i+1}}^{(e_{k,i+1})}\in\tilde{C}_{e_{k+1,i+1}}:y\leq_{\gamma^{k}_{i+1}}x_{j_{k,i+1}}^{(e_{k,i+1})},\forall
y\in\tilde{C}_{e_{k+1,i+1}}$.

With this definition of the binary relationship $\gamma^{k}_{i+1}$, we have
identified a unique element that provides a limit of integration for the
extremum element $e_{k,i+1}$. We may now define an intermediate poset
$D\left[\gamma^{k}_{i+1}\right]$,

$D\left[\gamma^{k}_{i+1}\right]=\left\\{V\left[\gamma^{k}\right]\setminus\left(\bigcup\limits_{j=1}^{i+1}e_{k,j}\right),\geq_{\gamma^{k}_{i+1}}\right\\}.$
(B.5)

Using this definition, we can start from $\gamma^{k}_{0}$ and recursively find
$\gamma^{k+1}=\gamma^{k}_{r_{k+1}+s_{k+1}}$ . It is clear from these
definitions, that every consistent set of next to extremal vertices is
generated. An inductive proof of this follows from assuming the result for
$\gamma^{k}_{i}$, and demonstrating that it follows for $\gamma^{k}_{i+1}$.

It is now possible to define a recursive notation that makes explicit the many
sums over sets $\gamma^{k}$ in the main text,

$\sum\limits_{\gamma^{k}}F\left(\left\\{\bigcup\limits_{m=1}^{k}\bigcup\limits_{l=1}^{r_{k}+s_{k}}x_{j_{m,l}}^{(e_{m,l})}\right\\}\right)=\sum\limits_{j_{k,r_{k}+s_{k}}=1}^{c_{k,r_{k}+s_{k}}}\dots\sum\limits_{j_{k,2}=1}^{c_{k,2}}\sum\limits_{j_{k,1}=1}^{c_{k,1}}\sum\limits_{\gamma^{k-1}}F\left(\left\\{\bigcup\limits_{m=1}^{k}\bigcup\limits_{l=1}^{r_{k}+s_{k}}x_{j_{m,l}}^{(e_{m,l})}\right\\}\right).\
$ (B.6)

In Eq. (B.6) $F$ is any function of the next-to-extremal vertices and
$c_{k,i}$ is the number of elements in the companion set $\tilde{C}_{k,i}$ as
in Eq. (B.4). Notice that the sum in Eq. (B.6) is a nested sum where the outer
summations depend on the inner summations. The summation over $\gamma^{k}$
depends also on our choice in anti-chain $A^{k}$.

## References

  * [1] G. Heinrich, “Collider Physics at the Precision Frontier,” Phys. Rept. 922, 1-69 (2021) doi:10.1016/j.physrep.2021.03.006 [arXiv:2009.00516 [hep-ph]].
  * [2] F. Caola, W. Chen, C. Duhr, X. Liu, B. Mistlberger, F. Petriello, G. Vita and S. Weinzierl, “The Path forward to N3LO,” [arXiv:2203.06730 [hep-ph]].
  * [3] R. Boughezal, Z. Ligeti, W. Altmannshofer, S. Das Bakshi, F. Caola, M. Chala, A. Diaz-Carmona, W. Chen, N. Darvishi and B. Henning, et al. “Theory Techniques for Precision Physics – Snowmass 2021 TF06 Topical Group Report,” [arXiv:2209.10639 [hep-ph]].
  * [4] W. J. Torres Bobadilla, G. F. R. Sborlini, P. Banerjee, S. Catani, A. L. Cherchiglia, L. Cieri, P. K. Dhani, F. Driencourt-Mangin, T. Engel and G. Ferrera, et al. “May the four be with you: Novel IR-subtraction methods to tackle NNLO calculations,” Eur. Phys. J. C 81, no.3, 250 (2021) doi:10.1140/epjc/s10052-021-08996-y [arXiv:2012.02567 [hep-ph]].
  * [5] C. Anastasiou, R. Haindl, G. Sterman, Z. Yang and M. Zeng, “Locally finite two-loop amplitudes for off-shell multi-photon production in electron-positron annihilation,” JHEP 04, 222 (2021) doi:10.1007/JHEP04(2021)222 [arXiv:2008.12293 [hep-ph]].
  * [6] C. Anastasiou and G. Sterman, ‘Locally finite two-loop QCD amplitudes from IR universality for electroweak production,” JHEP 05, 242 (2023) doi:10.1007/JHEP05(2023)242 [arXiv:2212.12162 [hep-ph]].
  * [7] Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni and B. Ruijl, “Numerical Loop-Tree Duality: contour deformation and subtraction”, JHEP 04 (2020) 096, [1912.09291].
  * [8] Z. Capatti, V. Hirschi, D. Kermanschah and B. Ruijl, “Loop-Tree Duality for Multiloop Numerical Integration”, Phys. Rev. Lett. 123 (2019) 151602, [1906.06138].
  * [9] Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni and B. Ruijl, “Manifestly Causal Loop-Tree Duality”, 2009.05509
  * [10] G. F. Sterman, “Mass Divergences in Annihilation Processes. 2. Cancellation of Divergences in Cut Vacuum Polarization Diagrams,” Phys. Rev. D 17, 2789 (1978) doi:10.1103/PhysRevD.17.2789
  * [11] G. F. Sterman, “Zero Mass Limit for a Class of Jet Related Cross-sections,” Phys. Rev. D 19, 3135 (1979) doi:10.1103/PhysRevD.19.3135
  * [12] G. T. Bodwin, “Factorization of the Drell-Yan Cross-Section in Perturbation Theory,” Phys. Rev. D 31, 2616 (1985) [erratum: Phys. Rev. D 34, 3932 (1986)] doi:10.1103/PhysRevD.34.3932
  * [13] J. C. Collins, D. E. Soper and G. F. Sterman, “Factorization for Short Distance Hadron - Hadron Scattering,” Nucl. Phys. B 261, 104-142 (1985) doi:10.1016/0550-3213(85)90565-6
  * [14] G. F. Sterman, “An Introduction to quantum field theory,” Cambridge University Press, 1993, ISBN 978-0-521-31132-8
  * [15] G. F. Sterman, “Partons, factorization and resummation, TASI 95,” [arXiv:hep-ph/9606312 [hep-ph]].
  * [16] S. Catani, T. Gleisberg, F. Krauss, G. Rodrigo and J. C. Winter, “From loops to trees by-passing Feynman’s theorem,” JHEP 09, 065 (2008) doi:10.1088/1126-6708/2008/09/065 [arXiv:0804.3170 [hep-ph]].
  * [17] M. Borinsky, Z. Capatti, E. Laenen and A. Salas-Bernárdez, JHEP 01, 172 (2023) doi:10.1007/JHEP01(2023)172 [arXiv:2210.05532 [hep-th]].
  * [18] Z. Capatti, “Exposing the threshold structure of loop integrals,” Phys. Rev. D 107, no.5, L051902 (2023) doi:10.1103/PhysRevD.107.L051902 [arXiv:2211.09653 [hep-th]].
  * [19] P. T. Komiske, E. M. Metodiev and J. Thaler, “The Hidden Geometry of Particle Collisions,” JHEP 07, 006 (2020) doi:10.1007/JHEP07(2020)006 [arXiv:2004.04159 [hep-ph]].
  * [20] S. Coleman and R. E. Norton, “Singularities in the physical region,” Nuovo Cim. 38, 438-442 (1965) doi:10.1007/BF02750472
  * [21] J. Collins, “A new and complete proof of the Landau condition for pinch singularities of Feynman graphs and other integrals,” [arXiv:2007.04085 [hep-ph]].
  * [22] G. F. Sterman, “Mass Divergences in Annihilation Processes. 1. Origin and Nature of Divergences in Cut Vacuum Polarization Diagrams,” Phys. Rev. D 17, 2773 (1978) doi:10.1103/PhysRevD.17.2773
  * [23] J. Collins, “Foundations of perturbative QCD,” Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 32, 1-624 (2011) Cambridge University Press, 2013, ISBN 978-1-00-940184-5 doi:10.1017/9781009401845
  * [24] G. F. Sterman and S. Weinberg, “Jets from Quantum Chromodynamics,” Phys. Rev. Lett. 39, 1436 (1977) doi:10.1103/PhysRevLett.39.1436
  * [25] P. M. Stevenson, “COMMENTS ON STERMAN-WEINBERG JET FORMULA,” Phys. Lett. B 78, 451-454 (1978) doi:10.1016/0370-2693(78)90483-5
  * [26] M. Dukes, E. Gardi, H. McAslan, D. J. Scott and C. D. White, “Webs and Posets,” JHEP 01, 024 (2014) doi:10.1007/JHEP01(2014)024 [arXiv:1310.3127 [hep-th]].
  * [27] O. Erdogan and G. Sterman, “Path description of coordinate-space amplitudes,” Phys. Rev. D 95, no.11, 116015 (2017) doi:10.1103/PhysRevD.95.116015 [arXiv:1705.04539 [hep-th]].
  * [28] S. J. Chang and S. K. Ma, “Feynman rules and quantum electrodynamics at infinite momentum,” Phys. Rev. 180, 1506-1513 (1969) doi:10.1103/PhysRev.180.1506
  * [29] J. B. Kogut and D. E. Soper, “Quantum Electrodynamics in the Infinite Momentum Frame,” Phys. Rev. D 1, 2901-2913 (1970) doi:10.1103/PhysRevD.1.2901
  * [30] S. J. Brodsky, H. C. Pauli and S. S. Pinsky, “Quantum chromodynamics and other field theories on the light cone,” Phys. Rept. 301, 299-486 (1998) doi:10.1016/S0370-1573(97)00089-6 [arXiv:hep-ph/9705477 [hep-ph]].
  * [31] D. Kermanschah, “Numerical integration of loop integrals through local cancellation of threshold singularities,” JHEP 01, 151 (2022) doi:10.1007/JHEP01(2022)151 [arXiv:2110.06869 [hep-ph]].


11institutetext:  Max-Planck-Institut für Sonnensystemforschung, Justus-von-
Liebig-Weg 3, 37077 Göttingen, Germany
11email<EMAIL_ADDRESS>22institutetext: Instituto Nacional de Técnica
Aeroespacial, Carretera de Ajalvir, km 4, E-28850 Torrejón de Ardoz, Spain
33institutetext: Univ. Paris-Sud, Institut d’Astrophysique Spatiale, UMR 8617,
CNRS, Bâtiment 121, 91405 Orsay Cedex, France 44institutetext: Instituto de
Astrofísica de Andalucía (IAA-CSIC), Apartado de Correos 3004, E-18080
Granada, Spain
44email<EMAIL_ADDRESS>55institutetext: Universitat de València, Catedrático José
Beltrán 2, E-46980 Paterna-Valencia, Spain 66institutetext: Institut für
Datentechnik und Kommunikationsnetze der TU Braunschweig, Hans-Sommer-Str. 66,
38106 Braunschweig, Germany 77institutetext: Instituto Universitario ”Ignacio
da Riva”, Universidad Politécnica de Madrid, IDR/UPM, Plaza Cardenal Cisneros
3, E-28040 Madrid, Spain 88institutetext: Leibniz-Institut für Sonnenphysik,
Schöneckstr. 6, D-79104 Freiburg, Germany 99institutetext: Institut für
Astrophysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077
Göttingen, Germany

# Wavefront error of PHI/HRT on Solar Orbiter at various heliocentric
distances

F. Kahil Corresponding author: F. Kahil.11<EMAIL_ADDRESS>A. Gandorfer 11 J.
Hirzberger 11 D. Calchetti 11 J. Sinjan 11 G. Valori 11 S.K. Solanki 11 M. Van
Noort 11 K. Albert 11 N. Albelo Jorge 11 A. Alvarez-Herrero 22 T. Appourchaux
33 L.R. Bellot Rubio  44 J. Blanco Rodríguez 55 A. Feller 11 B. Fiethe 66 D.
Germerott 11 L. Gizon  111010 L. Guerrero 11 P. Gutierrez-Marques 11 M.
Kolleck 11 A. Korpi-Lagg 11 H. Michalik 66 A. Moreno Vacas 44 D. Orozco Suárez
44 I. Pérez-Grande 88 E. Sanchis Kilders 55 J. Schou 11 U. Schühle 11 J. Staub
11 H. Strecker 44 J.C. del Toro Iniesta 44 R. Volkmer 99 J. Woch 11

###### Abstract

Aims.  We use wavefront sensing to characterise the image quality of the the
High Resolution Telescope (HRT) of the Polarimetric and Helioseismic Imager
(SO/PHI) data products during the second remote sensing window of the Solar
Orbiter’s (SO) nominal mission phase. The ultimate aims are to reconstruct the
HRT data by deconvolving with the HRT point spread function (PSF) as well as
to correct just for the effects of optical aberrations on the data.

Methods. We use a pair of focused-defocused images to compute the wavefront
error and derive the PSF of HRT by means of a phase diversity (PD) analysis.

Results. The wavefront error of HRT depends on the orbital distance of SO to
the Sun. At distances $>0.5$ au, the wavefront error is small, and stems
dominantly from the inherent optical properties of HRT. At distances $<0.5$ au
the thermo-optical effect of the Heat Rejection Entrance Window (HREW) becomes
noticeable. We develop an interpolation scheme for the wavefront error which
depends on the thermal variation of the HREW with the distance of SO to the
Sun. We also introduce a new level of image reconstruction, termed ”aberration
correction”, which aims to reduce the noise caused by image deconvolution,
while removing the aberrations caused by the HREW.

Conclusions. The computed PSF via phase diversity reduces significantly the
degradation caused by the HREW in the near-perihelion HRT data. In addition,
the aberration correction increases the noise by a factor of only $1.45$
compared to the factor of $3$ increase which results from the usual PD
reconstructions.

###### Key Words.:

Sun: photosphere, magnetic field, continuum – Techniques: Polarimetry,
Imagery, Phase diversity, wavefront error, PSF, MTF, Zernike, noise –
Instrumentation: SO, PHI, HRT

##  1 Introduction

Figure 1: The HRT PD image pair of 22 March 2022 at $0.334$ au. The focused
image is shown to the left, while the defocused image (by half a wavelength
with respect to the focused image) is shown to the right. We draw in yellow a
blow-up of the same region in both images.  Figure 2: Results of the PD
analysis. The wavefront error (upper panel, in units of wavelength) and
Modulation Transfer Function (MTF, lower panel) in four sub-regions, of
$750\times 750$ pixels each, of the entire FOV of HRT ($2048\times 2048$
pixels) .  Figure 3: The Zernike coefficients distribution as deduced from the
analysis of the PD datasets for each orbit. The dashed blue lines indicate the
RMS WFE values ($\pm\lambda/14$) which correspond to a diffraction limited
performance.

Solar Orbiter (SO, Müller et al., 2020) entered its low-orbital nominal
mission phase (NMP, Zouganelis et al., 2020) in late November 2021. During the
first orbit of the NMP most of the observations by the remote sensing
instruments were carried out in three remote sensing windows (RSWs) spanning
the period of 01 March 2022 until 06 April 2022. The closest approach of SO to
the Sun during these RSWs, of $0.32$ au, was reached on 26 March 2022. Among
the remote sensing instruments on-board, here we consider the Polarimetric and
Helioseismic Imager (SO/PHI, Solanki et al., 2020), which provides
measurements of the magnetic field, either of the full solar disc, or at
higher resolution of a small portion of the solar surface.

The latter observations are carried out with the High Resolution Telescope of
SO/PHI (HRT, see Gandorfer et al., 2018), which is a two-mirror system with a
decentered Ritchey-Chrétien configuration. The entrance aperture of the
telescope has a diameter of $140$ mm. With an effective focal length of $4125$
mm in the focal plane, the angular sampling corresponding to a working
wavelength of $\lambda=6173$ Å is $0.5^{\prime\\!\hskip 0.6pt\prime}$. This
angular sampling equals about $100$ km on the solar surface at the closest
perihelion of SO at $0.28$ au.

Changes in the very high image quality achieved by HRT are driven mainly by
the thermal environment. In particular, the Heat Rejection Entrance Window
(HREW, Solanki et al., 2020) acts as a passive thermal element in the heat-
shield assembly of the spacecraft, and exhibits a large temperature variation
along the highly elliptic orbit of Solar Orbiter. The HREW is designed such
that the temperature gradient across the glass plates of the window is
radially symmetric. Thus, the produced thermal lensing effect (the dependence
of the refractive index of the glass on the temperature) introduces only a
defocus term which can be compensated for by the HRT Refocus Mechanism (HRM,
see also Solanki et al., 2020). The amplitude of this gradient is estimated to
produce a defocus up to 4 $\lambda$ at perihelion (where the glass temperature
reaches about 200 ∘C in the center of the window with a 20 ∘C radial gradient
towards the outer edges).

These conditions, however, are not perfectly fulfilled in flight. Therefore,
it is expected that, at close solar proximity, the wavefront error (WFE) is
compromised by higher order residual optical aberrations, which the HRM is
incapable of removing. Then again, the HRM enables acquiring a pair of focused
and defocused images of the solar scene, which could be used, by means of
phase diversity analysis (PD, Paxman & Crippen, 1990; Löfdahl & Scharmer,
1994; Paxman et al., 1994), to determine the optical degradations of HRT due
to deformations of the HREW.

PD is a powerful technique which can be employed to capture the low-to-medium
order telescope aberrations (Gonsalves, 1983, 1985). These are usually
reflected in the total wavefront error at the exit pupil of a telescope, here
HRT, or in the point spread function (PSF) in the corresponding image plane
(Goodman, 1996). Internal optical aberrations such as coma or astigmatism are
inherent to the HRT, and originate from imperfections in the complex optical
system, mainly thermo-elastic despace errors in the two mirror system (Wilson,
1999). Residual defocus, spherical aberration or trefoil terms are not
expected to originate from the SO/PHI optics and are produced by thermal
gradients on the HREW.

During the Near-Earth Commissioning Phase (NECP, $0.8$ au) and in the second
remote sensing checkout window of the Cruise Phase (CP, $0.5$ au) the optical
aberrations of HRT were characterised. The results are published in Kahil et
al. (2022). They found a common wavefront error over the field of view (FOV)
of HRT and that the WFE is larger during CP when SO is closer to the Sun. From
March to April 2022, SO reached, for the first time, solar distances below
$0.5$ au. We aim in this work to evaluate the image quality of HRT data
products taken during the first two remote sensing windows of the NMP. In
Section 2 we present the PD data and our approach for fitting the WFE. In
Sections 3 and 4 we describe the methods we use to reconstruct the near-
perihelion data with the available PD measurements of HRT and show our
results. These are discussed along with conclusions in Section 5.

Table 1: Summary of phase diversity data. First column: The acquisition dates of the PD image pair. Second column: The Heliocentric distance of SO. Third column: The optimal number of Zernike polynomials in the PD fitting routine. Fourth column: The total RMS wavefront error.  Date | Distance [au] | $Z$ | RMS WFE [$\lambda$]
---|---|---|---
20$-$04$-$2020 | 0.82 | 10 | 1/10
20$-$02$-$2021 | 0.523 | 10 | 1/7
22$-$03$-$2022 | 0.334 | 23 | 1/2.27
Figure 4: The feed-through containing the HREW during SO/PHI ground testing.
Two out of the three mounting interfaces to the heat-shield support panel (in
the lab setup replaced by the Aluminium plate) can be seen. Note that the
images of the HRT (and thus our wavefront plots) are rotated by $90^{\circ}$
with respect to the laboratory frame.  Figure 5: Reconstruction of the RSW2
data. The central region ($800\times 800$ pixels) of the focused image of the
PD pair of 22 March 2022 at $0.334$ au (left panel). The reconstructed region
with the PSF calculated by PD (right panel).  Figure 6: MURaM simulation of a
sunspot. The original synthesised continuum map (upper panel), rebinned to the
pixel size of HRT at $0.334$ au, which is equal to $121$ km. The rebinned and
degraded map (lower panel) with the PSF calculated from the PD analysis of the
PD image pair of 22 March 2022.

##  2 Phase Diversity analysis

We adopt the PD algorithm presented by Löfdahl & Scharmer (1994). The
procedure for fitting the wavefront error (and all corresponding references)
is described in Kahil et al. (2022). We use Noll’s expansion scheme of
orthonormal Zernike polynomials (Noll, 1976) to characterise the wavefront
error.

The PD pair is acquired during the second remote-sensing window (RSW2) on 22
March 2022 and at a distance of $0.334$ au. The image pair is taken in the
continuum of the SO/PHI spectral line. The artificial defocus introduced to
the focused images was chosen to be half of the SO/PHI wavelength
($\lambda/2$). The PD image pair is shown in Figure 1. Before fitting the WFE,
we align, using cross-correlation, the defocused image to sub-pixel accuracy
to the focused image. We therefore disregard the first three aberrations
(piston, tip, tilt) and start the WFE fitting from the fourth Zernike
polynomial (defocus, $Z4$).

We run the PD algorithm on four sub-regions of the FOV, each of an equal size
of $750\times 750$ pixels. The optimal number of the employed Zernike
polynomials which returns a valid WFE is $Z=23$ (from $Z4$ to $Z26$). Any
number larger than $Z=23$ results in an over-fitted WFE and an over-
reconstructed scene due to the noise amplification. The dependence of the WFE
fitting and restoration results on the employed number of Zernike polynomials
is discussed in Hirzberger et al. (2011). We show the results of the WFE
fitting in Figure 2. As expected, the spatial variation of the WFE across the
FOV is small and the images can be assumed to be isoplanatic, in agreement
with earlier results of Kahil et al. (2022).

To retrieve the set of Zernike coefficients to be used for characterising the
WFE of HRT during the RSW2 and for comparison with earlier results, we use the
averaged Zernike terms over the four sub-regions of Figure 2. The Zernike
coefficients are shown in the bar plot of Figure 3. Overplotted is the
distribution of the Zernike coefficients retrieved during earlier PD
measurements in NECP on 20 April 2021 ($0.82$ au) and during CP on 20 February
2021 ($0.52$ au). For these orbits, we employed only 10 Zernike polynomials.
This number was chosen because the total root mean square (RMS) of the WFE
saturates for $Z>10$. We summarize the results of the PD analysis and compare
them to earlier results in Table 1.

As expected, the Zernike terms that mostly increase in amplitude during the
RSW2 are the defocus (Z4), the first order trefoil (Z9, Z10) and spherical
aberration (Z11). This is a result of the deviation of the actual temperature
distribution across the HREW from a paraboloid shape. These deviations give
rise to spherical aberration, and to trefoil due to azimuthal inhomogeneities.

The contribution of the trefoil terms is apparent in the trigonal shape of the
wavefront error in Figure 2. This shape is related to the mount of the HREW in
the heat-shield of SO and attributed to a trigonal temperature inhomogeneity
on the HREW. The inhomogeneity is caused by heat conduction through the mount
points of the HREW within the feed-through of the spacecraft heat-shield
(which were designed to minimise thermal coupling), and of the hot feed-
through to the heat-shield support panel, which sees the strongest temperature
difference (see Figure 4).

Figure 5 shows a sub-region of the focused image of the PD pair acquired on 22
March 2022 at a solar distance of $0.334$ au. The trigonal pattern on top of
the solar scene can be observed (left panel). We construct the PSF of HRT from
the calculated best-fit Zernike polynomials and apply image restoration
through deconvolution with the Wiener filter. The restoration (right panel)
successfully removes the trigonal pattern caused by the HREW.

To test if the PSF deduced from the PD analysis of the pair taken during RSW2
represents the true aberrations emanating from the HREW of HRT, we degrade a
synthesised continuum image obtained using MURaM (Vögler, 2005) with the
calculated HRT PSF. The degradation is applied to the theoretical image, which
is then rebinned to the pixel size of HRT after adding Gaussian noise (upper
panel of Figure 6). The degraded image (lower panel of Figure 6) displays the
same trigonal pattern as seen in the degraded HRT data of the RSW2 (left panel
of Figure 5). This indicates that the wavefront fitting algorithm returns a
reasonable set of Zernike coefficients that describe the true aberrations
produced by the HREW of HRT.

Figure 7: The variation of the HRM best focus position in coarse and fine
focus (in units of $\lambda$) along the orbit of SO from 05 November 2021 to
22 March 2022. The curves correspond to the best-fit quadratic function to
both types of focus. The y-scaling is chosen such that the HRM best focus
position at the absolute maximum of the quadratic fit is equal to zero. Figure
8: The variation of the four Zernike coefficients (defocus, spherical
aberration, and the two components of 1st order trefoil) in units of $\lambda$
with distance of SO to the Sun (in au). The red curves are the quadratic fits
to the data points. The absolute value of the X-Trefoil aberration is plotted
in order to use the same scaling on the y-axis . Figure 9: Restoration with
the interpolated PSF. The continuum image of a sub-region of one dataset from
17 March 2022 at $0.379$ au (upper panel). The restored region with the PSF
calculated at $0.334$ au (middle panel). The restored region with the PSF
interpolated to a distance of $0.379$ au (lower panel). Figure 10: The
variation of the initial (lower data points) and reconstructed (upper data
points) contrast values in the SO/PHI continuum along the orbit of SO. The
dots correspond to science data while the crosses correspond to the PD
datasets. Each data point of the RSW2 data (green and magenta dots) is the
average of all daily observations on 17, 19, 20, 21 and 22 March 2022. The
dashed horizontal lines represent the averaged contrasts of all the
points/crosses of the corresponding colour.

##  3 PSF interpolation

The calculated PSF was estimated at a solar distance of $0.334$ au. Since the
aberrations introduced by the HREW are expected to increase with decreasing
solar distance of SO (see Figure 3), the HRT data obtained at different
orbital positions cannot be restored using the same calculated PSF. This will
result in an over-reconstruction for data recorded when SO was further from
the Sun than $0.334$ au. Therefore, we develop an interpolation scheme to
approximate the PSF for such distances where no PD image pairs were acquired.

We assume that the instrument does not change with orbital position and only
the temperature inhomogeneities on the HREW vary quadratically with solar
distance. This behaviour is apparent in the plot of the total WFE compensated
by the HRM along the orbit shown in Figure 7.

For constructing the interpolated PSF at a given distance below $0.5$ au
(where artifacts due to the HREW are significant), we use the 23 Zernike
coefficients deduced from the analysis of the RSW2 PD data at $0.334$ au
(orange bars in Figure 3) and interpolate with SO-Sun distance only the terms
which are mostly affected by the temperature of the HREW. These are: defocus
($Z4$), first order trefoil components ($Z9$, $Z10$) and spherical aberration
($Z11$). We use a quadratic function to model the variation of these
aberrations with distance. The choice of a quadratic function is motivated by
the following arguments: (1) The temperature across the HREW goes
quadratically with distance, and the WFE is temperature dependent; (2) the HRM
compensates for the defocus in HRT following a quadratic dependence on the
heliocentric distance (see Figure 7); (3) a quadratic function is the highest
order that could be fit unambiguously to three data points. The results of the
fits are shown in Figure 8. The interpolated values are plotted for the range
of 0.28 au (the closest perihelion of SO) to 1 au. We note that we are only
interested in reconstructing data from distances below 0.5 au, where the HREW
causes a significant reduction of the image quality. The HRT data taken at
distances larger than 0.5 au are corrected with the PSF calculated during the
cruise phase at 0.5 au (see Section 4.2) .

We show in Figure 9 a sub-region of a dataset taken on 17 March 2022 in the
SO/PHI continuum. The dataset was acquired at a solar distance of $0.379$ au.
The restoration is done with both, the PSF calculated at $0.334$ au and the
PSF interpolated to a distance of $0.379$ au. The aberrations due to the HREW
appear to be greatly reduced in both images, but without PSF interpolation,
images appear to be over-reconstructed. We quantify this effect by calculating
the normalized RMS contrast of the continuum intensity of quiet-Sun regions in
the RSW2 observations before restoration, after restoration with the PSF
determined at 0.334 au and after restoration with the interpolated PSF. The
contrast values are plotted in Figure 10. For comparison, we also show the
contrast values of reconstructed scenes from earlier orbits. These correspond
to the focused images of the PD datasets, and they are marked as crosses in
Figure 10.

The PD datasets are acquired by HRT shortly after refocusing with the HRM, so
that the obtained contrast of 9.3% to 9.7% in the corresponding reconstructed
solar scene are considered to be optimal. During the perihelion approach the
overall WFE is deteriorating rapidly, as depicted in Figure 7. Since re-
focusing with the HRM was not done between 16 March 2022 and 22 March 2022
(see Figure 7), we witness a significant decrease in contrast in the
corresponding initial images (lower green dots). Deconvolution with the
interpolated PSF (upper green dots) does increase the contrast significantly,
but the contrast values in these datapoints do not fully reach the optimum
value, which is only re-established once the system has been brought to
optimum focus again by the HRM (on 22 March 2022).

Using the nearest PSF estimated on 22 March 2022 for all data points (magenta
dots) shows clearly, that – with the natural exception of the March 22 point
itself – all data points suffer from over-reconstruction, since the applied
PSF corresponds to the worst WFE, while the data have been taken with a more
relaxed instrument.

Figure 11: Results of the aberration correction. A sub-region of an example
dataset taken on 17 March 2022 at 03:18:09 UT (first panel). The restored area
with the interpolated PSF (middle panel) and the aberration corrected area
(last panel). The maps are taken in the continuum of the SO/PHI spectral line
at $617.3$ nm.  Figure 12: Same as Figure 11, but for the LOS magnetograms
obtained from the inversions of the reconstructed Stokes maps.

##  4 Aberration correction

###  4.1 RSW2 data

The restoration of the full Stokes images with the interpolated PSF results in
amplified noise levels compared to non-restored data. This is an expected
drawback of deconvolution, which cannot be avoided (see discussion in Kahil et
al., 2022). Usual PD reconstructions are known to increase the noise in the
restored data by a factor of three with respect to the original degraded data
(see for example Martínez Pillet et al., 2011). Therefore, the polarisation
signals of small-scale magnetic structures in the restored data may lie within
the noise level, which is not convenient for studying such structures.

To reduce the noise level, while removing the aberrations caused by the HREW,
we convolve the restored images with the PSF of an ideal telescope of 140 mm
aperture, i.e. by an Airy function with its first zero at the diffraction
limit. This results in lower noise amplification at higher spatial frequencies
due to deconvolution. In this work, the term ”restored data” refers to the
reconstructed data with the corresponding interpolated PSF, and with a noise
level equal to roughly 3$\sigma$ ($\sigma$ being the noise level in the non-
restored data,which is estimated from the distribution of Stokes $V$ in the
continuum and of the LOS magnetic field ($\rm B_{\rm los}$) signals in quiet-
Sun areas, as described below), while ”aberration-corrected data” refers to
data corrected only for the aberrations caused by the HREW while keeping the
degradation caused by diffraction.

The restoration and aberration corrections are applied to the individual
Stokes images before the Milne-Eddington inversion of the radiative transfer
equation is performed. This step is being implemented in the on-ground HRT
data reduction pipeline presented in Sinjan et al. (2022). Example studies
which make use of the aberration-corrected HRT data have been carried out by
Calchetti et al. (2023) and Sinjan et al. (2023). We show in Figures 11 and 12
sub-regions of the continuum and $\rm B_{\rm los}$ maps of an example dataset.
The figures show the two levels of correction with the HRT PSF. The first
panels of each figure correspond to the non-restored data, while the panels in
the middle correspond to the restored version with noise levels of about
$3\sigma$. The last panels are the aberration-corrected data with lower noise
levels. We calculate the amount of noise reduction in the following
subsections.

We illustrate in Figure 13 the one dimensional ($1$D) power spectrum of a
quiet-Sun region taken on 17 March 2022 at 00:20:09 UT. The three curves
correspond to the same region in the original, restored and aberration-
corrected versions. The plot shows that the aberration correction lowers the
signal towards higher frequencies, which reduces the noise in the aberration
corrected data. It also lowers the signal at intermediate frequencies, which
results in lower image contrast: the RMS contrast of quiet-sun regions
decreases, on average, from 9% to 7% due to the aberration correction (the
initial contrast is equal to 3.4%).

To quantify the amount of noise reduction, we measure the distribution of
Stokes $V$ signals in the continuum in quiet-Sun areas. We also follow the
method in Liu et al. (2012) to compute the noise in the LOS magnetic field
maps. The results are shown in Figures 14 and 15 where the noise is measured
for all datasets of the RSW2 observations. Averaged over all days of the RSW2
observations, the noise in Stokes $V$ increases by a factor of only $1.45$ in
the aberration corrected data compared to the original data. This factor is
equal to $3$ upon full restoration of the data.

Figure 13: Power spectrum of the continuum intensity in a quiet-Sun region of
a dataset taken on 17 March 2022 for the two levels of data reconstruction
(data restoration and aberration correction).

###  4.2 RSW1 data

During the first remote sensing window of SO (RSW1), the HRT datasets were
acquired at heliocentric distances ranging from $0.547$ au to $0.489$ au.
Therefore, the degradation in these data by the HREW is smaller than for the
RSW2 data described earlier. However, for consistency with the released RSW2
data, we also apply the aberration correction. No PD measurements were
acquired during RSW1 so we use the PSF estimated during the cruise phase at a
similar distance from the Sun of $0.5$ au. This PSF had a smaller RMS WFE and
lower number of Zernike polynomials (see Table 1), so that the aberration
correction results in increased noise levels of only a factor of $1.2$ wrt
non-restored RSW1 data (see Figures 16 and 18). We show in Figures 17 and 19
an example of one HRT dataset taken on 03 March 2022 at a distance of $0.547$
au. We show the restored image and the aberration-corrected one.

##  5 Discussion and conclusions

In Sections 2 and 3 we have calculated, by means of phase diversity analysis,
the wavefront error of HRT when close to the Sun during the second remote
sensing window of the nominal phase mission. At such distances, the breaking
of the rotational symmetry of the temperature gradients across the HREW
introduces aberrations which cannot be corrected for by the HRT refocus
mechanism, but are well captured by PD calibration data.

By inspecting the variation of the Zernike terms describing the WFE with solar
distance, we could isolate the coefficients which are introduced by this
effect (defocus, trefoil and spherical aberration). The fact that we still see
a defocus term, even after refocusing by the HRM, can be explained by the
action of the autofocus system, which aims to optimize the RMS contrast in the
image by minimising the overall wavefront error. But, since the shape of the
wavefront is not parabolic, there is a trade-off between the rotationally
symmetric terms, mainly defocus and spherical aberration, such that after the
”refocusing” we see remaining contributions of both of these terms.

With the set of PD data available so far, we build an interpolation model for
the calculated Zernike coefficients in order to construct a PSF that depends
on the heliocentric distance. This PSF is employed to reconstruct the
corresponding HRT datasets close to perihelion ($<0.5$ au), when no PD
measurements are available. We showed that the deconvolution with an
interpolated PSF yields very good reconstruction of the original scene, as
long as the instrument has been brought to best focus by the HRM and the
residual WFE has been minimized. Refocusing by the HRM on a regular basis
especially during perihelion approach, where the temperature changes rapidly
is highly recommended in order to minimize the overall WFE prior to PSF
deconvolution. Under these preconditions, the smooth variation of the residual
WFE terms allows good reconstruction of the solar scene, without strong
amplification of noise, which would happen when the initial contrast gets too
low due to insufficient focusing.

In addition, we have proposed, in Section 4, a solution to avoid the noise
amplification which results from the reconstruction with the interpolated PSF.
The resulting data products are termed ”aberration corrected” with a noise
level that is on average $1.45$ larger than the noise in the non-reconstructed
RSW2 data. For RSW1 data where the degradation effect by the window is
smaller, the noise of the aberration corrected data increases by a factor of
$1.2$. This moderate increase of the noise level makes the aberration-
corrected data ideal for applying inversion methods, whereas studies based on
the intensity images might benefit from the full interpolated PSF restoration.

For future near-perihelion observations, HRT will acquire daily PD
measurements in order to study in more detail the variation of the aberrations
introduced by the HREW with solar distance, and with different pointings of
the spacecraft. In addition to these measurements, potential long-term
effects, which are not a direct function of distance alone cannot be ruled out
at the current time and they need to be further investigated in a future
study. Furthermore, acquiring multiple defocused datasets may increase the
accuracy of the wavefront error retrieval by the PD algorithm, as shown by
Bailén et al. (2023).

###### Acknowledgements.

Solar Orbiter is a space mission of international collaboration between ESA
and NASA, operated by ESA. We are grateful to the ESA SOC and MOC teams for
their support. The German contribution to SO/PHI is funded by the BMWi through
DLR and by MPG central funds. The Spanish contribution is funded by
FEDER/AEI/MCIU (RTI2018-096886-C5), a “Center of Excellence Severo Ochoa”
award to IAA-CSIC (SEV-2017-0709), and a Ramón y Cajal fellowship awarded to
DOS. The French contribution is funded by CNES.

## References

  * Bailén et al. (2023) Bailén, F., Orozco Suárez, D. V., Blanco Rodríguez, et al. 2023, A&A, in press
  * Calchetti et al. (2023) Calchetti, D., Stangalini, M., Jaafarzadeh, S., et al. 2023, A&A, in press
  * Gandorfer et al. (2018) Gandorfer, A., Grauf, B., Staub, J., et al. 2018, in Space Telescopes and Instrumentation 2018: Optical, Infrared, and Millimeter Wave, ed. M. Lystrup, H. A. MacEwen, G. G. Fazio, N. Batalha, N. Siegler, & E. C. Tong, Vol. 10698, International Society for Optics and Photonics (SPIE), 106984N
  * Gonsalves (1983) Gonsalves, R. A. 1983, in Wavefront Sensing, ed. N. Bareket & C. L. Koliopoulos, Vol. 0351, International Society for Optics and Photonics (SPIE), 56 – 65
  * Gonsalves (1985) Gonsalves, R. A. 1985, in Digital Image Processing, ed. A. G. Tescher, Vol. 0528, International Society for Optics and Photonics (SPIE), 202 – 215
  * Goodman (1996) Goodman, J. 1996, Introduction to Fourier Optics, Electrical Engineering Series (McGraw-Hill)
  * Hirzberger et al. (2011) Hirzberger, J., Feller, A., Riethmüller, T. L., Gandorfer, A., & Solanki, S. K. 2011, A&A, 529, A132
  * Kahil et al. (2022) Kahil, F., Gandorfer, A., Hirzberger, J., et al. 2022, in Space Telescopes and Instrumentation 2022: Optical, Infrared, and Millimeter Wave, ed. L. E. Coyle, S. Matsuura, & M. D. Perrin, Vol. 12180, International Society for Optics and Photonics (SPIE), 121803F
  * Liu et al. (2012) Liu, Y., Hoeksema, J. T., Scherrer, P. H., et al. 2012, Sol. Phys., 279, 295
  * Löfdahl & Scharmer (1994) Löfdahl, M. G. & Scharmer, G. B. 1994, A&AS, 107, 243
  * Martínez Pillet et al. (2011) Martínez Pillet, V., del Toro Iniesta, J. C., Álvarez-Herrero, A., et al. 2011, Sol. Phys., 268, 57
  * Müller et al. (2020) Müller, D., St. Cyr, O. C., Zouganelis, I., et al. 2020, A&A, 642, A1
  * Noll (1976) Noll, R. J. 1976, J. Opt. Soc. Am., 66, 207
  * Paxman & Crippen (1990) Paxman, R. G. & Crippen, S. L. 1990, in Digital Image Synthesis and Inverse Optics, ed. A. F. Gmitro, P. S. Idell, & I. J. LaHaie, Vol. 1351, International Society for Optics and Photonics (SPIE), 787 – 797
  * Paxman et al. (1994) Paxman, R. G., Thelen, B. J., & Seldin, J. H. 1994, Opt. Lett., 19, 1231
  * Sinjan et al. (2023) Sinjan, J., Calchetti, D., Hirzberger, J., et al. 2023, A&A, in press
  * Sinjan et al. (2022) Sinjan, J., Calchetti, D., Hirzberger, J., et al. 2022, in Software and Cyberinfrastructure for Astronomy VII, ed. J. Ibsen & G. Chiozzi, Vol. 12189, International Society for Optics and Photonics (SPIE), 121891J
  * Solanki et al. (2020) Solanki, S. K., del Toro Iniesta, J. C., Woch, J., et al. 2020, A&A, 642, A11
  * Vögler (2005) Vögler, A. 2005, Mem. Soc. Astron. Italiana, 76, 842
  * Wilson (1999) Wilson, R. N. 1999, in Reflecting Telescope Optics II: Manufacture, Testing, Alignment, Modern Techniques (Berlin, Heidelberg: Springer Berlin Heidelberg), 349–371
  * Zouganelis et al. (2020) Zouganelis, I., De Groof, A., Walsh, A. P., et al. 2020, A&A, 642, A3

Figure 14: Noise as measured in the Stokes $V$ continuum of the HRT spectral
line (in units of mean quiet-Sun intensity). The noise is measured in every
dataset of the RSW2 observations and for the three data levels: L2 degraded
data (green), restored with the interpolated PSF L2 data (orange), and
aberration-corrected L2 data (blue). The box in each plot displays the
computed mean of the measured noise over all daily observations.

Figure 15: Same as Figure 14 but for the noise measured in the LOS magnetic
field maps of the RSW2 datasets. The noise is measured from Gaussian fits to
the histograms of the $\rm B_{los}$ distribution as described in Liu et al.
(2012)

.

Figure 16: The noise levels in the original (green) and aberration-corrected
data (blue) of the RSW1 observations. The noise is measured in the continuum
of Stokes $V$ for each of the datasets. The restored data (with 3$\sigma$
noise level) are not fully available, so we do not show the corresponding
noise values. Figure 17: The restored and aberration-corrected RSW1 data. A
sub-region of an example dataset taken on 03 March 2022 at 09:40:09 UT in the
continuum (left panel). The restored region with the PSF estimated during the
cruise phase (middle panel) and the aberration-corrected region (right
panel).The SO-Sun distance was $0.547$ au  Figure 18:  The noise in the LOS
magnetograms of the aberration-corrected data of the RSW1 observations. Figure
19: Same as Figure 17 but for the LOS magnetogram.