# Knots and 4-manifolds

Brendan Owens  School of Mathematics and Statistics
University of Glasgow
Glasgow, G12 8SQ, United Kingdom<EMAIL_ADDRESS>
###### Abstract.

These notes are based on the lectures given by the author during Winter Braids
IX in Reims in March 2019. We discuss slice knots and why they are
interesting, as well as some ways to decide if a given knot is or is not
slice. We describe various methods for drawing diagrams of double branched
covers of knots in the 3-sphere and surfaces in the 4-ball, and how these can
be useful to decide if an alternating knot is slice. We include a description
of the computer search for slice alternating knots due to the author and Frank
Swenton.

These notes are a written accompaniment to the series of three lectures given
by the author during the Winter Braids IX school and conference held in Reims,
France in March 2019. The goal of the lectures was to give an inviting
introduction to 4-dimensional aspects of classical knot theory, with a focus
on slice knots and some ways to find or obstruct them. We prioritise intuition
over precision, and urge the interested reader to consult other sources for
further reading and detail. We aim to provide such sources as we go along.

There are many good introductory books on knot theory. Three that are
particularly useful for the material in these notes are those by Lickorish,
Livingston, and Rolfsen [38, 41, 55]. An excellent introduction to smooth
4-dimensional topology is the book of Gompf and Stipsicz [19].

##  1\. Lecture 1

### Slice knots

A knot is a smooth embedding of $S^{1}$ in $S^{3}$, considered up to smooth
isotopy, and sometimes also up to reversal. Some examples are shown below,
drawn in the usual way as projections with crossing information at the double
points. If our knots are oriented we indicate this with an arrow, as shown.

The above diagrams are _alternating_ : as you traverse the knot, you alternate
between over- and under-crossings. A knot is called alternating if it admits
an alternating diagram.

A 2-knot is an embedding of $S^{2}$ in $S^{4}$, again up to isotopy. This can
be either smooth or topologically locally flat, the latter meaning that at
each point of the 2-knot, one can find continuous local coordinates in which
the 2-knot maps to a coordinate plane [14]. A knot $K$ is _slice_ if $K$ is
the intersection of a 2-knot with the equatorial $S^{3}$ in $S^{4}$, as
indicated in the cartoon below. We say $K$ is smoothly or topologically slice,
according to whether the 2-knot is smooth or just locally flat.

Equivalently $K$ is the boundary of an embedded $D^{2}$ in $D^{4}$. Perhaps
surprisingly, this simple concept has fundamental significance in
4-dimensional topology. We will give some examples, and then briefly describe
two applications of slice knots.

### Slice and ribbon surfaces in the 4-ball

It turns out that surfaces in the 4-ball can be represented and studied using
knot and link diagrams. We begin with a trivial example: the unknot bounds an
embedded disk in $S^{3}$. Thinking of $S^{3}$ as the boundary of the 4-ball,
this also gives a disk in $D^{4}$; we modify it to obtain a properly embedded
disk, with boundary mapping to boundary and interior to interior, by pushing
points in the interior of the disk into the interior of the 4-ball. Two
nontrivial slice knots, the stevedore knot and the square knot, are shown
below.

Consider the top left diagram. One can view this as a narrow strip or band
connecting two disks in $S^{3}$. The band passes through each of the disks
once, making what is called a ribbon singularity: this is a self-intersection
of a surface along an interval, such that the preimage of the interval
consists of one internal interval in the domain surface and one properly
embedded interval, with its endpoints on the boundary. Thus we see an immersed
disk in $S^{3}$ bounded by the stevedore knot. As with the trivial disk
bounded by the unknot, we can push the interior of this disk into the interior
of the 4-ball. Doing this carefully enables us to resolve the ribbon
intersections and obtain an embedded disk. We simply make sure that the part
of the disk containing the internal interval gets pushed further into the
4-ball than the part of the disk containing the properly embedded interval.
Thus the pictures above demonstrate that the stevedore and square knots are
slice. A surface in $S^{3}$ is called _ribbon-immersed_ if it is embedded
except for some ribbon singularities as in these examples. We see that in
general the boundary of a ribbon-immersed disk is a slice knot.

It is helpful to view these disks from the point of view of Morse theory (we
will give a brief introduction to Morse theory and handlebodies in Lecture 2).
Given any smooth surface $F$ in the 4-ball, we may isotope it so that the
radial distance function restricts to give a Morse function on $F$. Then we
can draw the intersection of $F$ with 3-spheres of fixed distance from the
origin; the result is a sequence of link diagrams in $S^{3}$ called a “movie”.
The movie version of the same slice disk for the stevedore is shown below:

What are the steps in this sequence? From $r=0$ to $r=1/2$, we have passed two
minima of $r|_{F}$, resulting in a two component unlink. In general a minimum
gives rise to an extra unknot component in the next frame of the movie.
Correspondingly, if we pass a maximum, then an unknot, which is contained in a
3-ball disjoint from the rest of the link, disappears. What happens between
$r=1/2$ and $r=1$? On the level of diagrams, we perform a _band move_ : that
is to say, we find an embedded rectangle in $S^{3}$ which intersects the link
shown in the $r=1/2$ picture on two opposite sides, and then we erase those
two sides of the rectangle and replace them by the other two sides. In terms
of Morse theory, we pass a saddle point. To see this it may help to look at
the following picture which might occur at $r=0.9$: an isotopy has occurred
between $r=1/2$ and $r=0.9$, and we can now see that the $r=1$ frame results
from this by passing a saddle point.

An embedded surface $F$ in $B^{4}$ is called _ribbon_ if the radial distance
function restricts to be Morse with no maxima. From the preceding example, we
see that this gives rise to a ribbon-immersed surface in $S^{3}$, and that by
pushing the interior into the 4-ball to resolve the singularities we may
recover $F$.

As usual one can turn such handlebody decompositions upside down. In this
context this simply means we run the movie backwards. Now maxima result in
adding split disjoint unknots to our movie frame, and minima correspond to
removing them. (The split union of two links in $S^{3}$ is the link which
contains the given links in disjoint 3-balls.) Saddles still correspond to
band moves: the operation of replacing two sides of an embedded rectangle with
the other two sides is its own inverse. Also note that one may push minima
deep into the 4-ball and maxima close to the surface, so that the
corresponding movie events are sorted with minima followed by saddles followed
by maxima, or the reverse. Thus every smooth compact surface in $B^{4}$
bounded by a given link $L$ is given by a movie, which in turn is given by a
finite set of band moves applied to a split union of $L$ with an unlink, which
results in another unlink. Here the first unlink corresponds to maxima and the
second to minima.

We say a knot is _ribbon_ if it bounds a ribbon disk in $B^{4}$, or
equivalently if it bounds a ribbon-immersed disk in $S^{3}$. Since the Euler
characteristic of a disk is one, being ribbon is equivalent to the existence
of $k$ band moves applied to the knot which convert it to the
$(k+1)$-component unlink. More generally, a knot $K$ is (smoothly) slice if
and only if there exists a set of $k$ band moves applied to the split union of
$K$ and an $l$-component unlink, converting it to the $(k-l+1)$-component
unlink. A longstanding question of Fox asks if every slice knot is ribbon
[11].

### Exotic ${\mathbb{R}}^{4}$ from Khovanov homology

From the work of many mathematicians, including Radó, Moise, and Stallings, we
know that in any dimension but 4, there is a unique smooth structure on
${\mathbb{R}}^{n}$ [46, 52, 59]. It is an amazing fact that ${\mathbb{R}}^{4}$
admits more than one smooth structure. In fact it follows from combining deep
work of Donaldson, Freedman, Gompf, and Taubes [60] that in fact
${\mathbb{R}}^{4}$ admits uncountably many smooth structures! This requires
gauge theory and in particular a great deal of analysis. We will give a brief
sketch here of a gauge-theory-free proof due to Rasmussen, following a
suggestion of Gompf, in which the existence of an exotic smooth structure on
${\mathbb{R}}^{4}$ follows from the existence of a knot which is topologically
slice but _not_ smoothly slice. Examples are the $(-3,5,7)$ pretzel knot,
shown below, and the untwisted Whitehead double of the trefoil [19, Figure
6.13]. In both cases topological sliceness follows from a deep theorem of
Freedman [13], which tells us that knots with trivial Alexander polynomial are
topologically slice, while smooth nonsliceness follows from a computation of
Rasmussen’s $s$-invariant, defined using Khovanov homology [53].

Given a knot $K$ in $S^{3}$ we obtain a 4-manifold with boundary called the
_knot trace_ $X_{K}$; this is obtained by gluing a 2-handle to the 4-ball
along $K$. To be slightly more precise, we have

$X_{K}=D^{4}\cup(D^{2}\times D^{2}),$

where we quotient by a gluing map defined on $\partial D^{2}\times D^{2}$
which takes $\partial D^{2}\times 0$ to $K$, and $\partial D^{2}\times 1$ to a
0-framing, or if you prefer to a push-off or longitude of $K$ which is
nullhomologous in $S^{3}\setminus K$, or equivalently which lies on a Seifert
surface for $K$. Cartoon below:

Then we have the following lemma, which holds in either the smooth or
topological category:

######  Lemma 1.1 (Kirby-Melvin).

$K$ is slice $\iff X_{K}\hookrightarrow S^{4}\iff
X_{K}\hookrightarrow{\mathbb{R}}^{4}$.

A literal sketch of the proof is as follows:

The cartoon on the left gives the forward direction, and the converse comes
from the sketch on the right. The smaller $D^{4}$ inside $X_{K}$ is standard
(it is a ball neighbourhood of the centre of $D^{4}\subset X_{K}$) so its
complement in $S^{4}$ is a ball in which we see a slice disk. For details see
Kirby-Melvin or Miller-Piccirillo [32, 44].

Finally we take a knot $K$ as above which is topologically but not smoothly
slice. Thus $X_{K}$ embeds topologically in ${\mathbb{R}}^{4}$. Freedman-Quinn
tell us that ${\mathbb{R}}^{4}\setminus X_{K}$ admits a smooth structure, and
this can be extended over all of ${\mathbb{R}}^{4}$, with $X_{K}$ as a smooth
submanifold [12]. Since Kirby-Melvin tell us that $X_{K}$ does not embed
smoothly in standard ${\mathbb{R}}^{4}$, this smooth structure must be exotic.

As a final note about the above, we must mention the proof by Piccirillo that
the Conway knot is not slice, which made a similar use of Lemma 1.1 and
Rasmussen’s invariant [26, 51]. She exhibited a knot $K$ with nonzero
Rasmussen invariant which is therefore not slice, and showed that the knot
trace of $K$ is diffeomorphic to that of the Conway knot.

### The concordance group of knots

The set of oriented knots in $S^{3}$ is an abelian monoid with the operation
given by connected sum. This can be realised diagrammatically as shown below,
or one can excise two standard pairs $(D^{3},D^{1})$ from each of
$(S^{3},K_{1})$ and $(S^{3},K_{2})$ and glue the complements together matching
orientations.

Sadly there are no inverses for this operation at the level of knots and
isotopy: knot genus is additive by [57] (or see [38, Theorem 2.4]), so the
only way to get an unknot by connected sum is if each summand is the unknot.

An isotopy of knots $K\simeq K^{\prime}$ is equivalent to an embedding

$S^{1}\times I\hookrightarrow S^{3}\times I$

which gives $K$ and $K^{\prime}$ on the boundary, and which is the identity on
the interval summand. We can remove this restriction and allow any smooth (or
locally flat) proper embedding of the annulus into $S^{3}\times I$ which
cobounds $K$ and $K^{\prime}$. Such an embedded annulus is called a
_concordance_ between $K$ and $K^{\prime}$. It is quite straightforward to see
that this gives an equivalence relation on the set of oriented knots, which we
will write $K\sim K^{\prime}$.

It is also straightforward to see that a knot $K$ is slice if and only if $K$
is concordant to the unknot. More generally, $K\sim K^{\prime}$ if and only if
$K\\#-K^{\prime}$ is slice, where $-K$ denotes the mirror reverse of the
oriented knot $K$. Recall that the mirror of a knot in $S^{3}$ is its image
under reflection in a plane; thus one obtains a diagram for $-K$ from one for
$K$ by changing all crossings and reversing the orientation. If $K\sim
K^{\prime}$, then drilling out a regular neighbourhood of the image in
$S^{3}\times I$ of an arc $\\{1\\}\times I$ from the concordance $S^{1}\times
I$ results in a slice disk in $D^{4}$ bounded by $K\\#-K^{\prime}$.
Conversely, if $K\\#-K^{\prime}$ is slice then we can add a $(3,1)$ handle
pair to $(D^{4},K\\#-K^{\prime})$ to obtain a concordance. Breaking that down:
a slice disk for $K\\#-K^{\prime}$ gives rise to an annulus in $D^{4}$ bounded
by the disjoint union of $K$ and $-K^{\prime}$ by applying a band move to undo
the connected sum. We then attach a 3-handle to the 4-ball along a 2-sphere
separating $K$ and $-K^{\prime}$ to obtain the required embedded annulus in
$S^{3}\times I$.

One then has that $\mathcal{C}=\\{\mbox{oriented knots}\\}/\\!\sim$ is a group
with operation given by connected sum, and with the inverse of $K$ represented
by $-K$. There are two versions of this, depending on whether the concordances
are smoothly or locally flatly embedded. For further reading on knot
concordance, please see for example [27, 41, 42].

One can also consider concordance of links, with an equivalence relation given
by embeddings of disjoint unions of annuli in $S^{3}\times I$. This does not
give rise to a group in the same way for links of more than one component. For
various approaches to defining concordance groups of links, see [8, 24, 28].

Basic problem: how to decide if a given knot $K$ is or is not slice?

Showing $K$ is slice:

  * •

exhibit a slice or ribbon disk as described above

  * •

exhibit a handle decomposition of the 4-ball involving $0$-, $1$-, and
$2$-handles. Then the boundaries of the cocores of the 2-handles are slice
knots in $S^{3}$, and one can apply Kirby calculus to obtain diagrams of them
in the standard picture of $S^{3}$. This method has been successfully used by
Gompf-Scharlemann-Thompson and also Abe-Tange [1, 18].

To show $K$ is nonslice, there are a plethora of obstructions, some of which
obstruct topological sliceness and some just smooth sliceness. These include

  * •

determinant (square for slice knots)

  * •

signature and Levine-Tristram signatures

  * •

Fox-Milnor condition on the Alexander polynomial

  * •

Herald-Kirk-Livingston condition on twisted Alexander polynomials

  * •

Heegaard Floer concordance invariants

  * •

Rasmussen’s invariant and other invariants coming from variants and
refinements of Khovanov homology.

Some sources for the above include [25, 27, 36, 37, 39, 42, 53].

Some notable progress in recent years includes Lisca’s classification of slice
two-bridge knots, together with his verification that all such are ribbon
[40]. The obstruction used by Lisca is an application of Donaldson’s
diagonalisation theorem, and double branched covers of the 4-ball; we will
look at this in more detail in Lecture 3. Subsequently, Lecuona applied
Lisca’s method to prove that slice implies ribbon for a family of alternating
Montesinos knots [35].

Sample problems:

  * •

Classify slice knots in some chosen family of knots

  * •

Classify ribbon knots in some chosen family of knots

  * •

Prove slice implies ribbon for some chosen family of knots

  * •

Classify slice disks (or ribbon disks, or ribbon disks with up to a certain
number of critical points) for some chosen knot.

The following is the most basic example of the last question above.

######  Question 1.2.

Is there only one isotopy class of ribbon disk for the unknot?

A positive answer for ribbon disks with at most three critical points follows
from theorems of Gabai and Scharlemann [15, 56]. These show that the only way
to apply a band move to the unknot and obtain the $2$-component unlink is the
obvious planar diagram, which gives rise to the standard disk pushed in from
$S^{3}$.

For other work on classification of slice disks, see [6, 30, 43].

Our focus for the rest of these lectures will be on alternating knots, and
whether we can determine if they are slice. The powerful obstruction from
Donaldson’s diagonalisation theorem and double branched covers used by Lisca
extends to this case, as we will describe. We will see that in addition to
giving us an obstruction, Donaldson’s theorem actually helps us to find ribbon
disks.

##  2\. Lecture 2

The main goal of this lecture is to look at double branched covers of knots
and links in the 3-sphere and surfaces in the 4-ball. We begin with a very
brief introduction to Morse theory and Kirby calculus, which enables us to
draw pictures of smooth 3- and 4-dimensional manifolds. For more details we
refer the reader to [2, 19, 45].

Let $M^{n}$ be a smooth compact manifold, possibly with boundary. We may
choose a Morse function

$f:M\to{\mathbb{R}}.$

This has the property that at each critical point $p$ of $f$, there exist
local coordinates in which

$f=-x_{1}^{2}-\dots-x_{k}^{2}+x_{k+1}^{2}+\dots+x_{n}^{2}.$

We say that the index $i(p)$ of $p$ is $k$. This leads to a _handlebody
decomposition_ of $M$ as follows. We consider sublevel sets
$M_{b}=f^{-1}(-\infty,b]$. These are the same up to diffeomorphism for nearby
regular values of $b$, and change when the level $b$ passes a critical point
of index $k$ by attachment of a $k$-handle, which is $D^{k}\times D^{n-k}$
attached to $M_{b}$ along $\partial D^{k}\times D^{n-k}$. This is illustrated
for the most standard example (the height function on a torus) below.

There are 4 critical points, of index 0, 1, 1, and 2. The first sublevel set
is the empty set, to which we attach a 0-handle $D^{2}$ when we pass the
minimum of the height function $f$. Then each time we pass an index one
critical point we attach a band $D^{1}\times D^{1}$ along two opposite faces
of the rectangle. Finally we pass a maximum and attach a 2-handle, which in
this dimension is just a 2-disk attached along its boundary circle.

A huge benefit of this description for low-dimensional topologists is that it
enables us to draw $(n-1)$-dimensional pictures of $n$-dimensional manifolds.
Let’s see this for the above example. There is a single minimum point, so
there is a single $0$-handle. The boundary of the $0$-handle is a copy of
$S^{1}$, and this is where we draw our diagram of the torus. We just need to
specify how the two 1-handles are attached; we will then have built a surface
with a single boundary component, and there is then just one manifold that we
can obtain by attaching a 2-handle, so we don’t need to specify any more
information except that there is a 2-handle to be attached. This results in
the following picture of the torus.

In a similar way we can obtain 3-dimensional diagrams for 4-manifolds, which
is of great benefit for those of us who have difficulty seeing in 4
dimensions! These are commonly referred to as Kirby diagrams. Suppose that $M$
is a compact connected oriented 4-manifold. A suitable Morse function gives
rise to a handle decomposition

$M^{4}=0\mbox{-handle}\cup\\{1\mbox{-handles}\\}\cup\\{2\mbox{-handles}\\}\cup\\{3\mbox{-handles}\\}(\cup
4\mbox{-handle}?),$

where there is a 4-handle attached if the manifold is closed. We draw the
attaching regions in $\partial(0\mbox{-handle})=S^{3}$. The attaching region
for a 1-handle is $\partial D^{1}\times D^{3}$, which is a pair of disjoint
3-balls. The attaching region for a 2-handle is $\partial D^{2}\times D^{2}$
which is a solid torus. We specify this by giving a knot in $S^{3}$ and a
framing coefficient which is an integer; the $0$-framing of a knot in $S^{3}$
is equal to a push-off on its Seifert surface, or in other words a push-off of
the knot which is nullhomologous in the knot complement. More generally the
$m$-framing is a push-off of the knot which has linking number $m$ with the
knot. Here are some examples.

A result of Laudenbach and Poénaru [34] tells us that any self-diffeomorphism
of a connected sum of copies of $S^{1}\times S^{2}$ extends over the boundary
sum of copies of $S^{1}\times D^{3}$. This means that for a closed 4-manifold,
we do not need to specify how the 3- and 4-handles are attached. However, for
a 4-manifold with boundary, it is necessary to specify the embedded 2-spheres
along which we attach any 3-handles.

There is a useful alternative way to draw 1-handles in Kirby diagrams, namely
as dotted circles: one draws an unknot with a dot along it which is a meridian
of an arc in the boundary of the 0-handle connecting the two balls which are
the attaching region of the 1-handle. Attaching circles of 2-handles which go
over the 1-handle are instead drawn as passing through the dotted circle. The
following local picture gives a good idea of how to translate between these
two methods of drawing Kirby diagrams.

It is straightforward to calculate the homology groups and the intersection
pairing $Q_{M}$ on the second homology (Poincaré dual to the cup product
pairing) from a Kirby diagram of $M$. Kirby calculus is the application of
certain moves such as handleslides which are known to preserve $M$ and/or its
boundary 3-manifold. For more details on Kirby diagrams, see [2, 19].

There are two common and useful approaches for drawing pictures of
3-manifolds. One is to appeal to a theorem of Rohlin [54] which tells us that
every (closed oriented) 3-manifold is the boundary of a smooth 4-manifold, and
we can just draw a Kirby diagram of such a bounding 4-manifold. Another is to
take a 3-dimensional handle decomposition and draw our picture in the oriented
surface $\Sigma$ which is the boundary of the handlebody given by the union of
the $0$\- and $1$-handles, which is connected if there is a single $0$-handle.
We then draw two sets of circles on $\Sigma$: one set, commonly drawn in blue,
which shows where to attach the $2$-handles, and one set in red which if cut
along would undo the attachment of the $1$-handles. These are called Heegaard
diagrams. Here are some simple examples.

### Double branched covers

Let $L$ be a link of $l$ components in $S^{3}$. Denote a regular neighbourhood
of $L$ by $\nu(L)\cong L\times D^{2}$. Let $X$ be the link complement

$X=S^{3}\setminus\overset{\circ}{\nu(L)}.$

Then $H_{1}(X)\cong{\mathbb{Z}}^{l}$, generated by meridians
$\mu_{1},\dots,\mu_{l}$. Let $\widetilde{X}\rightarrow X$ be the double cover
of $X$ which is nontrivial on each meridian. The double cover
$\Sigma(S^{3},L)$ of $S^{3}$ branched along $L$ is then given as follows.

$\Sigma(S^{3},L)$$=$$\nu(L)$$\cup$$\widetilde{X}$$S^{3}$$=$$\nu(L)$$\cup$$X$$z^{2}$

In the same way we obtain the double branched cover $\Sigma(D^{4},F)$ of a
surface $F$, smoothly properly embedded in the 4-ball. The surface $F$ does
not have to be connected or orientable, and it may contain closed components.
We would like to learn to draw Kirby diagrams for these; this will be
described in more detail in ongoing work of the author with Sašo Strle [48],
and other accounts are given in [2, 3, 19]. We follow a method due to Akbulut-
Kirby, following a construction of Kauffman, also used by Gordon-Litherland
[3, 20, 31]. This applies to the special case of branch locus $F^{n-2}$ in
$D^{n}$ arising from an embedded submanifold in $\partial D^{n}$ by pushing
the interior of the submanifold into the interior of the $n$-ball.

We cut open the ball along the trace of the homotopy that pushed the interior
of the submanifold into the interior of $D^{n}$ to obtain the properly-
embedded $F$. This results in a copy of $D^{n}$ again, with a regular
neighbourhood $N$ of $F$, which is the total space of an $I$-bundle over $F$,
in its boundary. This cut-open $D^{n}$ has corners, as illustrated below in
the case $n=2$, with $F$ a single point, and $N$ drawn in blue.

The pair $(D^{n},F)$ may be recovered from this cut-open ball by gluing $N$ to
itself using the involution $\tau$ given by reflection on the fibres. We
obtain the double cover $\Sigma(D^{n},F)$ of $D^{n}$ branched along $F$ by
taking two copies of the cut-open $D^{2}$ and gluing them to each other via
the same involution $\tau$.

We also note for later that if one glues the two copies of the cut-open
$D^{n}$ by identifying copies of $N\times I$ instead of just $N$, one obtains
the same manifold $\Sigma(D^{n},F)$, up to diffeomorphism. Here is a sketch
for the $n=2$ example.

### The double branched cover of a link in $S^{3}$

As a 4-dimensional example of the above method, we will explain how to derive
a Kirby diagram and also a Heegaard diagram for the double branched cover of a
link in $S^{3}$. This Kirby diagram was first obtained by Ozsváth and Szabó,
and the closely related Heegaard diagram is due to Greene [22, 50]; their
proofs differed from that given here. We begin with a diagram of our link $L$;
let’s choose a diagram of the figure-8 knot for an example, shown in Figure 1
along with the resulting Kirby and Heegaard diagrams.

Figure 1. Kirby and Heegaard diagrams for the double branched cover of the
figure-8.

To draw these, we first choose a _chessboard colouring_ of the diagram: we
colour the regions of the diagram black or white, so that at each crossing the
colours meet like squares on a chessboard. Here is one of the two choices for
our example.

Let $F$ be the resulting black spanning surface embedded in $S^{3}$. We obtain
the Kirby diagram from the link diagram by copying the diagram with the
following substitutions at crossings (each crossing gets replaced by a clasp
between white regions).

This results in one 2-handle attaching circle for each white region of the
diagram. The framing coefficient of each is given by summing the local
contributions ($+1$ or $-1$) from each crossing. We then add one 3-handle; it
turns out that this may be cancelled with any of the 2-handles in the diagram
without changing the 4-manifold. We will see that this is in fact a Kirby
diagram for $\Sigma(D^{4},F)$, where by an abuse of notation we also use $F$
to refer to the properly embedded surface obtained by pushing the interior of
the black surface into the 4-ball.

The Heegaard diagram also makes use of the chessboard colouring and the black
surface $F$. For this we require $F$ to be connected. Let $N$ again be a
regular neighbourhood of $F$ which is the total space of an $I$-bundle over
$F$. The boundary of $N$ is our Heegaard surface $\Sigma$. The red curves are
exactly given by the boundaries of all but one of the white regions of the
chessboard-coloured diagram; the omitted region can be chosen arbitrarily. The
blue curves are obtained by taking a parallel copy of the red curves, and then
applying a Dehn twist for each crossing in the diagram to the one or two blue
curves passing through that part of $\Sigma$ (as in Figure 1).

######  Proposition 2.1.

The Kirby diagram described above represents the double cover
$\Sigma(D^{4},F)$ of the 4-ball branched along the pushed-in black surface.
The 3-handle may be cancelled with any of the 2-handles in the diagram. The
Heegaard diagram represents the double cover $\Sigma(S^{3},L)$ of $S^{3}$
branched along the link $L$.

###### Proof.

We note that the $n$-ball $D^{n}$ has a handle decomposition given by a single
$0$-handle and a cancelling pair consisting of an $(n-2)$-handle and an
$(n-1)$-handle. Equivalently we can build $D^{n}$ by starting with
$S^{n-2}\times D^{2}$ and adding a single $(n-1)$-handle, whose attaching
sphere is $S^{n-2}\times\\{\mbox{point}\\}$. Explicitly, we take a regular
neighbourhood of the equatorial $S^{n-2}$ — the intersection of the boundary
of $D^{n}$ with a hyperplane through the origin — and then attach a thickened-
up equatorial slice (a neighbourhood of a disk in the same hyperplane). In
particular we can obtain $D^{4}$ by starting with $S^{2}\times D^{2}$ and
attaching a 3-handle along $S^{2}\times\\{\mbox{point}\\}$.

So what does this have to do with double branched covers? Suppose we have a
copy of $D^{4}$ with a link $L$ in its boundary. We project $L$ to the
equatorial $S^{2}$ in the boundary sphere to obtain our diagram, and we
chessboard-colour it to obtain our black surface $F$. We take $N$ to be a
regular neighbourhood of $F$ in $S^{3}$ and $N\times I$ to be a regular
neighbourhood of $F$ in $D^{4}$. The intersection of $N\times I$ with the
2-sphere of the diagram contains the black regions, connected together at the
crossings; its complement in this 2-sphere consists of one disk for each white
region. By attaching one 2-handle for each white region of the diagram to
$N\times I$ we obtain a regular neighbourhood $S^{2}\times D^{2}$ of the
sphere containing the diagram, and then we attach a 3-handle to complete the
4-ball.

Now consider the description of $\Sigma(D^{4},F)$ given above, by gluing
together two copies of $D^{4}$ cut open along the homotopy pushing $F$ into
the interior. Each of these has a copy of $N$ in its boundary, and we recall
that $N$ is the total space of an $I$-bundle over $F$. We use the involution
$\tau$ coming from reflection on the fibres of this $I$-bundle to identify the
copies of $N\times I$ in each cut-open $D^{4}$; the quotient is
$\Sigma(D^{4},F)$. We want to describe a handle decomposition of this. We take
one of the cut-open 4-balls to be our 0-handle. This also contains $N\times I$
from the second copy, via the gluing map. We then add a 2-handle for each
white region and a 3-handle to complete the second copy of the cut-open
$D^{4}$.

To draw the Kirby diagram, we need to understand how these handles are
attached. To build the 4-ball starting with $N\times I$, the attaching circles
for the 2-handles lie in the plane of the diagram and are 0-framed. After
attaching all but one of the 2-handles to $N\times I$ the result is a 4-ball
$D^{2}\times D^{2}\subset S^{2}\times D^{2}$; then attaching the last 2-handle
changes this to $S^{2}\times D^{2}$, but this is cancelled by the attachment
of the 3-handle. To build the double branched cover, we need to see the image
of these attaching circles and framings under the gluing map $\tau$. The local
picture for this is shown in Figure 2.

Figure 2. The involution $\tau$ near a crossing.

We see that a crossing in the diagram results in a clasp between the 2-handle
attaching circles coming from the white regions on either side of the
crossing. The effect on a framing curve, which is just a pushoff on the
surface $\partial N$, is similar and results in the framing curve getting a
full twist around the attaching circle. These local contributions exactly
correspond to the description given above of the 2-handle attaching circles
and their framing coefficients.

Finally the boundary $\Sigma(S^{3},L)$ of $\Sigma(D^{4},F)$ is the union of
the boundaries of the two 4-balls that we glue together, minus the interior of
$N$. The complement of $N$ in $S^{3}$ is a 3-dimensional handlebody, and thus
the construction naturally yields a Heegaard decomposition of
$\Sigma(S^{3},L)$. The red curves of the Heegaard diagram bound disks in the
first copy of $S^{3}\setminus N$, and via Figure 2 we see that the blue curves
bound in the second copy. ∎

### More double branched covers of $D^{4}$

We have seen how to draw a simple Kirby diagram for the double cover of
$D^{4}$ branched along the black surface of a link diagram. One can also draw
Kirby diagrams of more general surfaces in $D^{4}$ and hence compute their
homological invariants including intersection form. For details on this see
[2, 3, 19, 48]. We give a brief description here for the case of a ribbon
surface. We take $F$ to be a ribbon-immersed surface in $S^{3}$ with a handle
decomposition chosen so that all ribbon singularities consist of 1-handles
passing through 0-handles. Draw this in ${\mathbb{R}}^{3}$ with the 0-handles
in the $xz$-plane, lying just below the $x$-axis. All attachments of 1-handles
to 0-handles should be on the $x$-axis and all ribbon singularities below the
$x$-axis in the interiors of the 0-handles. It helps to also “comb the
1-handles up” so that they mostly lie near the $xz$-plane and above the
$x$-axis. We then take the preimage of this diagram under the double branched
covering map

$\phi:{\mathbb{R}}^{3}\to{\mathbb{R}}^{3},\quad(x,\zeta=y+iz)\mapsto(x,\zeta^{2}).$

An example is shown in Figure 3 for the stevedore ribbon disk encountered in
Lecture 1. In practice one can draw the preimage of the diagram under $\phi$
by taking the diagram in the upper half plane, away from the $x$-axis, and
revolving it around the $x$-axis, then connecting the tangle diagrams in the
upper half plane and the lower half plane with a half twist of 4 strands for
each ribbon singularity, and with no twist on two strands at the end of each
1-handle (see Figure 3).

Figure 3. The double branched cover of the stevedore ribbon disk.

From this we draw a Kirby diagram as follows. Each 1-handle of $F$ gives rise
to an embedded annulus in $\phi^{-1}({\mathbb{R}}^{3})$; we take the core of
this annulus to be the attaching circle for a 2-handle, and the linking number
of the boundary of the annulus is the framing coefficient. We also draw
1-handles in dotted circle notation, one for all but one of the 0-handles of
$F$; these are drawn around the boundary of the preimage under $\phi$ of the
corresponding 0-handle of $F$. Again, see Figure 3 for an example. For details
and more examples see the sources cited above.

The main point here is that given a surface $F$ in $D^{4}$, there are known
methods for obtaining a Kirby diagram of $\Sigma(D^{4},F)$ and hence computing
the homology groups and the intersection pairing of this manifold. In [48] the
author and Strle give a definition of _disoriented homology_ groups associated
to such a surface $F$, together with a bilinear pairing on one of the groups.
We show that these are isomorphic to the homology groups of the double
branched cover, together with the intersection pairing. This enables these
invariants to be computed directly from a diagram of $F$, without needing to
first obtain a Kirby diagram.

For the remainder of these notes, if $F$ is a properly embedded surface in
$D^{4}$ or a surface embedded or ribbon-immersed in $S^{3}$ giving rise to a
pushed-in surface in $D^{4}$, then we will use the notation $\Sigma(D^{4},F)$
to refer to the resulting double branched cover. We will also use
$\Lambda_{F}$ to refer to the intersection lattice of this manifold, or in
other words the second homology group modulo torsion of $\Sigma(D^{4},F)$
together with its intersection pairing.

##  3\. Lecture 3

Our goal in this final lecture is to explain how Donaldson’s diagonalisation
theorem [9] gives rise to a very useful slice obstruction for alternating
knots, and also to an obstruction to any given band move being part of a
sequence that yields a ribbon disk. A computer search for ribbon alternating
knots based on this has been implemented in joint work of the author and Frank
Swenton [49] which has resulted in a database of over 200,000 previously
unknown slice knots.

### Goeritz, Gordon-Litherland, and Donaldson

We begin by describing the Goeritz lattice of a connected chessboard-coloured
diagram. Label the white regions $R_{0},\dots,R_{m}$ and then write down a
square matrix $\widehat{G}$ as follows. The rows and columns of $\widehat{G}$
are numbered from $0$ to $m$. The diagonal entries $\widehat{G}_{ii}$ are
given by a signed count of the crossings around region $R_{i}$, with signs
given as follows.

The off-diagonal terms $\widehat{G}_{ij}$ are given by a signed count of the
crossings between $R_{i}$ and $R_{j}$, with the opposite signs to those above.
The Goeritz matrix $G$ is obtained from $\widehat{G}$ by deleting the $0$th
row and column. Here is an example.

We get

$\widehat{G}=\begin{bmatrix}2&-1&-1\\\ -1&-2&3\\\ -1&3&-2\end{bmatrix},\mbox{
and }G=\begin{bmatrix}-2&3\\\ 3&-2\end{bmatrix}.$

This was introduced by Goeritz in 1933 [17]. He showed that the determinant
and more generally the _Minkowski units_ [47] of $G$ give rise to knot
invariants. The astute reader may notice that the matrices above represent the
intersection pairing which may be computed from the Kirby diagram for
$\Sigma(D^{4},F)$ in Figure 1.

The Goeritz lattice of a chessboard-coloured diagram is described as follows.
One takes the (free) abelian group $\Lambda$ with one generator given by each
white region, and one relation given by the sum of all of the generators. Note
this has basis given by all but one of the generators. The pairing

$\lambda:\Lambda\times\Lambda\to{\mathbb{Z}}$

is given by the matrix $\widehat{G}$. Comparing to the Kirby diagram from
Lecture 2, we recover Gordon-Litherland’s result that the Goeritz lattice is
the intersection lattice on the second homology group of the double branched
cover $\Sigma(D^{4},F)$ [20].

We also observe the following. The author learned this proof from Josh Greene.
For more topological proofs see [23, Prop. 4.1] or [49, Lemma 3.1].

######  Lemma 3.1.

The Goeritz lattice of a connected alternating diagram is definite. More
precisely, for one choice of chessboard-colouring the Goeritz lattice is
positive definite, and for the other it is negative definite.

###### Proof.

The Goeritz lattice of the crossingless diagram of the unknot is trivial and
hence both positive and negative definite. Assume now that our diagram has at
least one crossing. Being alternating is equivalent to all crossings having
the same sign when computing the Goeritz matrix. Assume that all of these
signs are $+1$, so that the diagonal entries of $\widehat{G}$ are all positive
and the off-diagonal entries are all nonpositive. The rows of $\widehat{G}$
sum to zero (for any diagram). Let $r_{i}$ denote the sum of the absolute
values of the off-diagonal entries of $G$ in the $i$th row. Then
$r_{i}=G_{ii}-|\widehat{G}_{0i}|$ is less than or equal to $G_{ii}$. The
eigenvalues of the symmetric matrix $G$ are real and by Gershgorin’s circle
theorem [16], they are contained in

$\bigcup[G_{ii}-r_{i},G_{ii}+r_{i}]$

and hence nonnegative. Since their product is the determinant of a link with
connected alternating diagram and hence nonzero [7], the eigenvalues are all
positive. The proof for the other colour is similar. ∎

We note that Josh Greene and Josh Howie independently proved in 2015 that the
property in Lemma 3.1 characterises alternating links: a link admits a
connected alternating diagram if and only if it bounds two embedded surfaces
in $S^{3}$ for which the double branched cover of $D^{4}$ has definite
intersection pairing (one positive and one negative) [23, 29].

We now describe a key slice obstruction for alternating knots. The double
cover of $S^{3}$ branched along the unknot is $S^{3}$, and the double cover of
$D^{4}$ branched along an unknotted properly embedded $D^{2}$ — or in other
words along the standard slice disk for the unknot — is again $D^{4}$. An
imprecise slogan that has some validity is that “rational homology does not
see knotting”. This manifests itself here in the fact that the double cover of
$S^{3}$ branched along any knot is a rational homology $S^{3}$ [38, Chapter
9], and the following fact due to Casson-Gordon [4]: the double cover
$\Sigma(D^{4},\Delta)$ branched along any properly embedded disk has the same
rational homology as the 4-ball.

We can combine this with Donaldson’s celebrated diagonalisation theorem [9],
which states that if $X$ is a closed smooth 4-manifold with a positive-
definite intersection form then in fact its intersection lattice
$\Lambda_{X}=(H_{2}(X;{\mathbb{Z}})/{\it Tors},Q_{X})$ is isomorphic to the
standard ${\mathbb{Z}}^{n}$ lattice; in other words, $\Lambda_{X}$ admits an
orthonormal basis. This gives us the following.

######  Lemma 3.2.

Let $K$ be an alternating knot and let $F$ be the black surface of its
alternating diagram, with chessboard colouring chosen so that $\Lambda_{F}$ is
positive definite. Let $b$ be a band move applied to $K$. Then

  1. (1)

if $K$ is slice, then $\Lambda_{F}$ embeds as a finite index sublattice of a
standard ${\mathbb{Z}}^{m}$ lattice;

  2. (2)

if $b$ is part of a sequence of band moves giving a ribbon disk for $K$, then
there is a commutative diagram of lattice morphisms

$\Lambda_{F}$${\mathbb{Z}}^{m}$,$\Lambda_{F\cup b}$

where the horizontal arrow is the inclusion from (1), and $F\cup b$ is a
ribbon-immersed surface in $S^{3}$ obtained by attaching the band $b$ to $F$.

###### Proof sketch.

For part (1), we use the fact that the double branched cover $\Sigma(D^{4},F)$
of $F$ is positive definite and the double branched cover
$\Sigma(D^{4},\Delta)$ of a slice disk is a rational homology ball. Gluing
these along their common boundary gives a smooth closed manifold
$X=\Sigma(S^{4},F\cup\Delta)$; this is the double cover of the 4-sphere
branched along a properly embedded closed surface obtained from $F$ and
$\Delta$. The Mayer-Vietoris sequence shows that $\Lambda_{F}$ is a finite
index sublattice of the intersection lattice of $X$. It follows that the
latter is also positive definite, and thus diagonalisable by Donaldson. (This
is the slice obstruction used by Lisca in [40].)

Part (2) follows from the fact that the closed manifold
$X=\Sigma(S^{4},F\cup\Delta)$ is built from $\Sigma(D^{4},F)$ by adding a
2-handle for each 1-handle of $\Delta$ followed by a 3-handle for each minimum
of $\Delta$ and finally a 4-handle (see for example [19, 48]). This gives a
commutative diagram of manifold inclusions

$\Sigma(D^{4},F)$$X$,$\Sigma(D^{4},F\cup b)$

from which the commutative diagram of lattice morphisms follows. ∎

There are a couple of points to note about Lemma 3.2. The first is that the
mirror of a slice knot is slice, so one may change the crossings and the
colour and apply the lemma twice to each alternating knot and to each band
move. The second is that following Greene and Jabuka [21], one may strengthen
the lemma using Heegaard Floer homology. This puts extra conditions on the
lattice embedding in the lemma, namely that every equivalence class in
${\mathbb{Z}}^{m}/\Lambda_{F}$ is represented by an element of
$\\{0,1\\}^{m}$.

As an example let’s consider the stevedore knot $K$ together with the
following band move $b$.

We can calculate $\Lambda_{F}$ using the Goeritz matrix: we find this has rank
2 and the pairing is given by $G=\begin{bmatrix}5&-1\\\ -1&2\end{bmatrix}$,
using generators $v_{1}$ and $v_{2}$ corresponding to the white regions
$R_{1}$ and $R_{2}$. The first part of Lemma 3.2 tells us that this admits a
lattice embedding in ${\mathbb{Z}}^{2}$ with the dot product: sure enough if
we map $v_{1}$ to $(2,1)$ and $v_{2}$ to $(-1,1)$, this preserves the pairing.
Moreover, this embedding is unique up to symmetries. Now we apply the second
part of the lemma. The band $b$, shown in purple, would become part of the
black surface for the new diagram. This would split $R_{0}$ into two new
regions. We can choose the new generator $v_{3}$ to correspond to the
resulting small white region on the right side of the band. Then we can
calculate the Goeritz of the new diagram and we find that

$\displaystyle v_{3}\cdot v_{1}$ $\displaystyle=0$ $\displaystyle v_{3}\cdot
v_{2}$ $\displaystyle=-1.$

By part (2) of the lemma, if $b$ is part of a sequence giving a ribbon disk,
then there exists a vector $(x,y)\in{\mathbb{Z}}^{2}$ with

$\displaystyle(x,y)\cdot(2,1)$ $\displaystyle=0$
$\displaystyle(x,y)\cdot(-1,1)$ $\displaystyle=-1.$

Solving over the rationals we find $(x,y)=(1/3,-2/3)\notin{\mathbb{Z}}^{2}$.
We conclude that this band move is not part of any sequence giving a ribbon
disk for $K$.

The reader may compare this to the band move given below, and see that it
converts $K$ to the two-component unlink, and in fact is obtained by reversing
the movie shown for $K$ in Lecture 1. In this case the new white region to the
right of the band gets sent to $(0,-1)\in{\mathbb{Z}}^{2}$.

For more complicated band moves one cannot always compute $\Lambda_{F\cup b}$
using the Goeritz matrix. One may always either draw a Kirby diagram, or
calculate using disoriented homology [48]. This helpful tool does not seem to
be sufficient to give an answer to the following.

######  Question 3.3.

Is there a unique isotopy class of ribbon disk for the stevedore knot?

The search algorithm of [49] is based on the band obstruction of Lemma 3.2,
and on an optimistic notion that alternating knots like to stay alternating.
Recall that the nullity of a link in $S^{3}$ is the first Betti number of its
double branched cover. The nullity of a knot is zero, and that of an unlink is
one less than the number of components. In a movie of a ribbon disk starting
with a knot and ending with an unlink, each band move increases the nullity by
one. Connected alternating diagrams have nullity zero, and in fact the nullity
gives a lower bound on how many “nonalternating” crossings a diagram can have.
Our algorithm then applies the following steps to a given alternating knot:

  1. (1)

Check if the knot is obstructed by Lemma 3.2 (1) from being slice.

  2. (2)

Try all band moves which are unobstructed by Lemma 3.2 (2) and which keep the
resulting link as close to alternating as possible (i.e., the number of
nonalternating crossings and the nullity both increase by 1).

  3. (3)

For each such band move, simplify the resulting link diagram using Tsukamoto-
type moves.

  4. (4)

Repeat the previous two steps until all possibilities are exhausted or a
crossingless diagram is obtained. In the latter case, we declare the knot to
be _algorithmically ribbon_.

Some remarks are in order. We show in [49] that the bands to be considered in
step (2) take the following form,

so that there are always finitely many to consider. There are also finitely
many alternating diagrams of a given knot, and we have to deal with all of
them.

Also, what are Tsukamoto-type moves? These are a set of local modifications on
diagrams which preserve isotopy type, and also the property of being close to
alternating and unobstructed by a slight generalisation of Lemma 3.2 which
applies to such diagrams [49, Prop. 3.5]. They include Reidemeister I and II
moves and five others shown below, the first of which is the famous flype move
of P. G. Tait.

The third of these was called the _untongue move_ by Tsukamoto, who proved in
[61] that every almost-alternating111A diagram is almost alternating if it
contains exactly one nonalternating crossing. diagram of the two-component
unlink may be converted to the crossingless diagram by a sequence of flypes,
untongues, and Reidemeister I and II moves. It follows that if an alternating
knot admits a ribbon disk which can be presented by a single band move
converting it to an almost-alternating diagram of the unlink, then in fact
this disk will be detected by our algorithm. This together with experimental
evidence and optimism led us to state the following in [49].

######  Conjecture 3.4.

If an alternating knot admits a ribbon disk with three critical points, then
in fact it is algorithmically ribbon.

If true, this would give an effective classification of such alternating
knots: Tsukamoto’s theorem gives a recursive construction of all almost-
alternating diagrams of the two-component unlink, and each of these would
arise from four possible band moves from alternating diagrams as shown below.
This set of alternating diagrams would contain repetitions and would include
diagrams of links as well as knots.

The algorithmic search for alternating knots may be considered a work in
progress. We have applied it to all 250,912,342 prime alternating knots of up
to 20 crossings, using the prime alternating knot generator of Flint, Rankin,
and de Vries [10]. Of these, we find 438,375 are unobstructed from being slice
by Lemma 3.2 and 231,968 turn out to be algorithmically ribbon. We also know
of a further 628 examples of ribbon alternating knots of up to 20 crossings
which are _not_ algorithmically ribbon. The first of these was 12a631, shown
to be ribbon by Seeliger [58] using symmetric union diagrams (see also [5,
33]).

How can we improve the algorithm? One straightforward way is to implement more
obstructions, including Levine-Tristram signatures and the Fox-Milnor
condition (see for example [42]). Beyond this the hope is that we may
understand better how ribbon disks can arise and expand the algorithm to find
more of them. Thus we will expand the set of knots on which our algorithm is
able to determine the answer to the question “Slice: yes or no?” and hopefully
one day it will include all alternating knots.

For more details on the algorithm and to download Swenton’s software package
KLO in which it is implemented, we direct the reader to

www.klo-software.net/ribbondisks.

### Acknowledgements

I am very grateful to the organisers of Winter Braids IX for a wonderful
conference and for their patience in waiting for these notes. I also thank the
participants of the school for being an encouraging and helpful audience.
Thanks to Sašo Strle for helpful comments on an earlier draft, and to the
anonymous referee for many helpful suggestions to improve the exposition. I
thank Alex Zupan and Jeff Meier for pointing out that a claimed positive
answer to 1.2 in a previous version of these notes was unjustifed.

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# Resolved Photons in S  HERPA

Stefan Höche  Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Frank Krauss  Institute for Particle Physics Phenomenology, Durham University,
Durham DH1 3LE, UK  Peter Meinzinger  Institute for Particle Physics
Phenomenology, Durham University, Durham DH1 3LE, UK

###### Abstract

We present the first complete simulation framework, in the S  HERPA event
generator, for resolved photon interactions at next-to leading order accuracy.
It includes photon spectra obtained through the equivalent-photon
approximation, parton distribution functions to parametrize the hadronic
structure of quasi-real photons, the matching of the parton shower to next-to
leading order QCD calculations for resolved photon cross sections, and the
modelling of multiple-parton interactions. We validate our framework against a
wide range of photo-production data from L  EP and H  ERA experiments,
observing good overall agreement. We identify important future steps relevant
for high-quality simulations at the planned Electron-Ion Collider.

##  1 Introduction

Photon–induced processes provide a rich testing ground for a wide range of
physics effects. This is, on one hand, because photons will couple to any
electromagnetically charged particle, resulting in a wide spectrum of
accessible final states. On the other hand, photons have the quantum numbers
of (neutral vector) mesons and their wave functions therefore have a sizeable
hadronic component which lets them interact strongly, with correspondingly
large cross sections. The production of low–multiplicity final states in
$\gamma\gamma$ collisions has been observed in many experiments [1, 2, 3, 4],
and yields interesting insights into the physics of hadrons and hadron
resonances described by effective theories of the strong interactions. At
increasing centre-of-mass energies of the colliding photons, new channels open
up, and the production of jets has been studied, for example, at L  EP [5, 6,
7, 8, 9, 10]. Similarly, the photo–production of various final states,
including jets, has been analysed by the H  ERA experiments [11, 12, 13, 14,
15, 16, 17].

The results obtained in these experiments have allowed to parameterise the
parton content of the hadronic component of quasi-real photons in the form of
parton distribution functions (PDFs), for example in the GRV [18], CJK [19,
20], SAL [21], and SaS [22, 23] sets111 To fix the (non–perturbative) inputs
for the PDFs at some infrared scale $Q_{0}^{2}\approx\mathcal{O}(1\,{\rm
GeV}^{2})$ different ansätze have been chosen. Most of them rely on the vector
meson dominance model which assumes that at these scales the hadronic
component of the photon wave function behaves like a vector meson, and use the
pion PDF with modifications inspired by the quark model. The more involved SaS
model for the photon structure function is based on the decomposition of the
photon wave function $|\gamma\rangle$ into three components: a ”bare” photon
component $|\gamma_{\rm bare}\rangle$ where the photon interacts indeed as a
point–particle, a non–perturbative component $|\gamma_{V}\rangle$ where the
photon fluctuates into various neutral vector mesons $V$ such as $\rho$,
$\omega$, $\phi$, $J/\psi$ etc., and a perturbative ”anomalous component”
$|\gamma_{q\bar{q}}\rangle$ in which the photon fluctuates into a $q\bar{q}$
pair. In all cases, the PDF at the low scale $Q_{0}^{2}$ is translated to PDFs
at larger scales through standard QCD evolution, by emitting secondary virtual
partons. . In fact, based on these PDFs, satisfying agreement between data and
calculations has been achieved, and a complete model for photon structure
functions and high–energy photon interactions [24] has been encoded in P
YTHIA [25]. There, the hard production of QCD final states at large scales,
i.e. jets, is simulated in the usual way by dressing the hard parton–level
matrix element with subsequent parton showers, the fragmentation of the
resulting partons into hadrons during hadronization, possibly including an
underlying event. Recently, the model was extended to also include the
perpendicular component of the photon momentum [26].

Pushing for higher accuracy, there have been a few predictions for inclusive
jet-production at fixed-order at H  ERA [27, 28, 29, 30, 31] and the E  IC
[32], while more attention has recently been paid to exclusive meson
production processes and photo-production at heavy-ion collisions [33, 34, 35,
36, 37, 38].

Anticipating the increased precision requirements for successfully operating a
possible future lepton collider such as F  CC-ee or the planned electron–ion
collider, E  IC , motivates to revisit the physics of photon–induced processes
and to arrive at fully-differential predictions at Next-To-Leading Order (NLO)
in QCD perturbation theory. We report here on the systematic inclusion of PDFs
for quasi–real photons into the S  HERPA event generation framework [39, 40],
the modelling of multiple-parton interactions, and the extenstion of the
calculation to Next-to-Leading Order matched to the parton shower. The paper
is organised as follows. In Sec. 2 we briefly discuss how S  HERPA combines
the equivalent photon approximation (EPA) and photon PDFs, with some emphasis
on the efficient integration over the resulting phase space, the matching to
the parton shower at NLO and the multiple-parton interactions (MPI) model. In
Sec. 3 we will show some first results of full Monte Carlo simulations,
comparing results obtained with S  HERPA to data from both the L  EP and H
ERA experiments at M  C @N  LO accuracy. We present comparisons to Leading
Order and study the effect of the MPI. We summarise our findings in Sec. 4.

##  2 Equivalent Photons and their PDFs

For the simulation of photo-production events in S  HERPA , we use its
existing EPA interface, improved the phase space handling for the initial
states for a more efficient integration, and added relevant photon PDFs to S
HERPA ’s internal PDF interface. The resulting code will be publically
available as part of the upcoming release of S  HERPA 3.0; in the meantime it
can be obtained from the authors upon request.

###  2.1 Phase Space Handling

In the following we detail the structures for efficient phase space sampling,
using the most involved example of doubly resolved photon–photon collisions at
lepton colliders, schematically depicted in Fig. 1.

$s_{12},y_{12}$$\rightarrow s^{\prime}_{12},y^{\prime}_{12}$$\rightarrow
s^{\prime\prime}_{12},y^{\prime\prime}_{12}$$p_{1}$$p^{\prime}_{1}$$p^{\prime\prime}_{1}$$p^{\prime\prime}_{2}$$p^{\prime}_{2}$$p_{2}$$q^{\prime\prime}_{1}$$q^{\prime\prime}_{N}$
Figure 1: Schematic sketch of the phase space mappings between the different
steps in the initial states, i.e. the Equivalent Photon Approximation (EPA)
and the Initial State Radiation (ISR), and the Matrix Element (ME). Each
coordinates pair of Mandelstam-$s^{\prime(\prime)}$ and rapidity
$y^{\prime(\prime)}$ is randomly sampled and the momenta are calculated as
functions of these.

The two incoming leptons have momenta $p_{1}$ and $p_{2}$, and a (beam)
c.m.-system characterised by the c.m.-energy squared $s_{12}$ and its rapidity
$y_{12}$ in the lab system. The momenta of the photons emitted by the leptons,
$p^{\prime}_{1}$ and $p^{\prime}_{2}$, create a (photon) c.m.-system
characterised by its c.m.-energy squared $s^{\prime}_{12}$ and rapidity
$y^{\prime}_{12}$ with respect to the beam system. The partonic structure of
the photons, as described by the PDFs, results in two partons with momenta
$p^{\prime\prime}_{1}$ and $p^{\prime\prime}_{2}$ to finally enter the hard
process which will result in final state particles with momenta
$q^{\prime\prime}_{i}$. The hard scattering is characterised by a c.m.-energy
of $s^{\prime\prime}_{12}$ and a rapidity $y^{\prime\prime}_{12}$ w.r.t. the
photon system. This structure requires two nested integrations for the two
successive ”initial states” (photons and partons): first an integration over
$s^{\prime}_{12}$ and $y^{\prime}_{12}$, with factors given by the EPA
spectra, and then an integration over $s^{\prime\prime}_{12}$ and
$y^{\prime\prime}_{12}$, with factors given by the PDFs, before adding the
integration over the final state phase space over the outgoing momenta
$q^{\prime\prime}_{i}$. Efficient integration over this complex phase space in
S  HERPA is facilitated through the multi-channel method [41] with
automatically generated integration channels that map out intrinsic structures
such as $s$-channel resonances etc.

After the successful generation of a phase space point, the corresponding
weight is calculated, given by the factors stemming from the EPA photon
spectra and the PDF weights.

###  2.2 Equivalent Photon Approximation

The equivalent photon approximation encoded in the Weizsäcker-Williams formula
[42, 43, 44] is based on the observation that quasi-virtual photons can be
approximated through real photons for small virtualities
$Q^{2}<\Lambda_{\text{cut}}^{2}$. As photo-production events are dominated by
the interaction of low-virtuality photons, the differential cross-section can
be substituted by $\text{d}\sigma_{eX}=\sigma_{\gamma X}(Q^{2}=0)\text{d}n$. S
HERPA uses an improved version of the spectrum, following [45], which
introduces the term proportional to $m_{e}^{2}$ to the spectrum, and the
photons are assumed to be collinear to the electron beam. The dependence of
the photon spectrum on the photon virtuality is integrated out. This results
in the following spectrum for electrons:

$\mathrm{d}n=\frac{\alpha_{\mathrm{em}}}{2\pi}\frac{\mathrm{d}x}{x}\left[\left(1+(1-x)^{2}\right)\log\left(\frac{Q^{2}_{\mathrm{max}}}{Q^{2}_{\mathrm{min}}}\right)-2m_{e}^{2}x^{2}\left(\frac{1}{Q^{2}_{\mathrm{min}}}-\frac{1}{Q^{2}_{\mathrm{max}}}\right)\right]$
(2.1)

Here, $x$ denotes the ratio of photon to electron energy,
$\frac{E_{\gamma}}{E_{e}}$, and $\alpha_{\mathrm{em}}$ is the electromagnetic
coupling constant. $Q^{2}_{\mathrm{max}}$ and $Q^{2}_{\mathrm{min}}$ denote
the maximal and minimal photon virtuality, where the latter can be calculated
from kinematic restrictions and is given by

$Q^{2}_{\mathrm{min}}=\frac{m_{e}^{2}x^{2}}{1-x}\,.$ (2.2)

The maximal virtuality is given by the experimental setup and the maximal
deflection angle of the electron, $\theta_{\mathrm{max}}$, below which the
hard process is still considered to be photon-induced. It is given by

$Q^{2}_{\mathrm{max}}=\mathrm{min}\left(Q^{2}_{\mathrm{min}}+E_{e}^{2}(1-x)\theta_{\mathrm{max}}^{2},Q^{2}_{\mathrm{max,fixed}}\right)\,.$
(2.3)

Defaults choices are $\theta_{\mathrm{max}}=0.3$ and the maximum virtuality to
$Q^{2}_{\mathrm{max,fixed}}=3\ \mathrm{GeV}^{2}$, but can be overwritten by
the user, cf. the S  HERPA manual [46].

###  2.3 Photon PDFs

To facilitate a comparison over different parameterisations, four PDF
libraries have been included in S  HERPA , see Table 1 for a summary.

Name | # sets | Virtual? | NLO? | # flavours |  $x$-range |  $\mu_{F}^{2}$-range
---|---|---|---|---|---|---
GRV [18] | 2 | No | Yes | 5 | $[10^{-5},1]$ | $[0.25,10^{6}]$
SAL [21] | 1 | No | Yes | 6 | $[10^{-5},0.9999]$ | $[2,8\cdot 10^{4}]$
SaS [22, 23] | 4 | Yes | Yes | 6 | $[10^{-5},1]$ | $[0.25,10^{6}]$
CJK [19, 20] | 4 | No | Yes | 5 | $[10^{-5},1]$ | $[0.25,2\cdot 10^{5}]$

Table 1:  Photon PDF libraries included in S  HERPA and their properties.

Currently, all PDFs are evaluated at virtuality $Q^{2}=0$; the extension to
virtual photons, taking also into account longitudinal polarisations, will be
introduced in a later release. The extraction of a parton from the photon is
complemented by the corresponding treatment of the remaining partons. For this
a similar procedure is applied as in the remnant construction for a hadron,
however, with a few simplifications. As there are no valence quarks present in
the photon, the remnant is constructed as the anti-particle of the hard-
scattering quark. In the case of the gluons, a quark-antiquark pair is
constructed from one of the light quark generations. In the following
processing, the photon remnants is treated analogously to a hadron remnant,
i.e. its longitudinal momentum is distributed among the partons according to
the kinematics, and their transverse momenta, given by some non-perturbative
intrinsic $k_{T}$ distribution, compensate each other.

###  2.4 NLO and matching to the parton shower

The procedure for a $\mathrm{NLO}$ calculation in photo-production has been
presented in [47]. We will follow the same line of arguments for the matching
to the parton shower, summarising the procedure in the following paragraphs.
In contrast to the more familar case of hadronic PDFs, the evolution in the
photon PDFs has an additional ”source” term, related to the photon splitting
into a quark–anti-quark pair, which is proportional to $\alpha_{\mathrm{em}}$.
This means that the evolution of the parton distributions $f_{i}^{(\gamma)}$
for parton $i$ in the photon follow [47]

$\frac{\partial
f_{i}^{(\gamma)}}{\partial\mathrm{log}\mu^{2}}=\frac{\alpha_{\mathrm{em}}}{2\pi}P_{i\gamma}+\frac{\alpha_{S}}{2\pi}\sum_{j}P_{ij}\otimes
f_{j}^{(\gamma)}\,.$ (2.4)

The first term on the r.h.s. of the equation above gives rise to the
”anomalous component” mentioned before.

At Next-to-Leading-Order, there will be additional collinear divergences in
the real-corrections, stemming from the photon splitting. These divergences
appear only in the direct-photon component, but are cancelled against the
corresponding term in the photon PDF evolution.

To ensure an exact cancellation between these terms, the PDF has to use the
same factorisation scheme as the subtraction scheme, which in the case of S
HERPA is the $\overline{\mathrm{MS}}$ scheme. This reduces the number of
possible PDFs sets directly available for the NLO calculation in S  HERPA down
to SAS1M and SAS2M from the SaS library. We note that additional PDFs can be
made available easily by adding the respective factorization scheme dependent
correction terms [48]. Apart from this subtlety in the choice of PDFs both NLO
calculations and full simulations proceed in full analogy to the more familiar
case of, e.g., proton–proton collisions.

##  3 Validation of the framework

We will now turn to the validation of our implementation, comparing results at
M  C @N  LO precision with photo-production data from the O  PAL and Z  EUS
experiments. For each collider set-up and energy we generated samples of
$5\cdot 10^{6}$ events at M  C @N  LO accuracy. For the calculation of matrix
elements we used A  MEGIC [49] and C  OMIX [50] for the tree-level matrix
elements and subtraction terms [51] and O  PEN L  OOPS [52] for the one-loop
matrix elements. We added the cssparton shower [53] for the jet evolution,
matched with the M  C @N  LO method [54, 55] as implemented in S  HERPA [56].
Underlying event effects have been included through an implementation of the
Sjostrand-van Zijl model [57, 58] within S  HERPA 222For details and a future
tune to data we refer the reader to a forthcoming S  HERPA manual., and the
partons were hadronized with the cluster fragmentation of A  HADIC [59]. We
consistently used the current default value for $\alpha_{S}=0.118$ with three-
loop running. As described in 2.4, the event generation must currently use
PDFs based on the $\overline{\mathrm{MS}}$ scheme, so the resolved-photon
predictions were generated with, and averaged over, the SAS1M and SAS2M PDF
sets. The factorisation scale and the renormalisation scale were both set to
$\mu_{F}=\mu_{R}=H_{T}/2$ and we kept the 7-point variation for the scale
uncertainty estimate.

We used the R  IVET [60] framework with the existing analyses implementing [5,
6, 7, 11, 12]. For each experiment, different components of the cross-section
have to be summed over, for example,

$\sigma_{\text{tot}}=\sigma_{\gamma\gamma}+\sigma_{j\gamma}+\sigma_{\gamma
j}+\sigma_{jj}$ (3.1)

for L  EP and

$\sigma_{\text{tot}}=\sigma_{\gamma j}+\sigma_{jj}$ (3.2)

for H  ERA , where in both cases $j$ denotes a photon or proton resolved
through a PDF.

###  3.1 Comparison with L  EP data

The O  PAL analysis of photon-induced di-jet production at 198 GeV c.m.-energy
[5] offers the most differential observables, and we use it as the primary
reference for our validation of photo-production at L  EP 2\. To comply with
experimental cuts, the $k_{T}$ algorithm with a jet radius of $R=1.0$ is used
and cuts of $E_{T}>4.5\ (2.5)$ GeV for the leading (subleading) jet are
imposed.

$x_{\gamma}^{\pm}=\displaystyle{\frac{\sum\limits_{j=1,2}E^{(j)}\pm
p_{z}^{(j)}}{\sum\limits_{i\in{\rm hfs}}E^{(i)}\pm p_{z}^{(i)}}}\,,$ (3.3)

are used in this analysis to disentangle direct, singly, and doubly resolved
production. In their definition, the sum in the numerator is over the two
jets, and the sum in the denominator is over all hadronic final state
particles, thereby distilling the energy-fraction the jets have w.r.t. the
overall photon energies.

|
---|---
|
Figure 2: Differential dijet inclusive cross section with respect to
$\cos\theta^{*}$, defined in Eq. (3.5), comparing results of our S  HERPA M  C
@N  LO simulation at hadron-level with the LO simulation (top row) and with
simulations at parton-level (bottom row) and with data from O  PAL at an
$e^{-}e^{+}$ c.m.-energy of 198 GeV [5]. In the left and right panels the
requirement $x^{\rm obs}_{\gamma}<0.75$ and $x_{\gamma}^{\rm obs}>0.75$ are
applied and enhance resolved and direct contributions, respectively.

This is exemplified by, e.g., the distribution of events in

$\cos\Theta^{*}=\tanh\frac{\eta_{1}-\eta_{2}}{2}\,,$ (3.4)

an approximation of the angle between the two jets, and exhibited in Fig. 2.
Apart from satisfying agreement with data, a few things are worth noticing
here, which we will continue to observe also in the following: For
$x_{\gamma}^{\pm}>0.75$ the direct component dominates by about 1.5 orders of
magnitude, with only small scale uncertainties, indicated by the pink band.
Conversely, for $x_{\gamma}^{\pm}<0.75$ the doubly-resolved component
dominates with a significantly larger scale uncertainty, which, in this case,
also includes factorization scale uncertainties. Intuitively, the singly-
resolved component in both case ranges between the two other components. In
addition we observe that hadronization effects reduce the cross section in the
unresolved domanin, while the combination of hadronization and multiple parton
scattering increases it in the doubly-resolved regime. The visible effect in
the latter suggests that a careful retuning of the MPIs may further improve
agreement with data.

|  |
---|---|---
Figure 3: Distributions $x_{\gamma}^{\pm}$, collectively denoted as
$x_{\gamma}$ in different bins of average transverse jet energy:
$\bar{E}_{T}\in[5\,{\rm GeV},7\,{\rm GeV}]$ (left), $\bar{E}_{T}\in[7\,{\rm
GeV},11\,{\rm GeV}]$ (middle), $\bar{E}_{T}\in[11\,{\rm GeV},25\,{\rm GeV}]$
(right). Results of the S  HERPA simulation with M  C @N  LO accuracy are
compared with results at LO and with data from O  PAL at an $e^{-}e^{+}$
c.m.-energy of 198 GeV [5].

We report that distributions in $x_{\gamma}^{\rm obs}$ for three different
$\bar{E}_{T}$ experience a significant improvement in shape when going from
Leading to Next-to-Leading Order, cf. Fig. 3. However, in the transition
region between doubly resolved to unresolved events, we notice a clear
difference in shape: While for $x_{\gamma}^{\rm obs}<0.6-0.7$ the prediction
is relatively flat below the data, the underprediction at around
$x_{\gamma}^{\rm obs}\approx 0.8$ persists at NLO. Apart from possibly
insufficient photon PDFs – a point we will elucidate below – there are a
number of possible explanations: First of all, as before, a retuning of MPIs
may come to the rescue and fill up the gap.

|  |
---|---|---
Figure 4: Distributions of $|\Delta\eta|$ (left), $|\eta_{\rm cntr}|$
(middle), and $|\eta_{\rm fwd}|$ (right), comparing M  C @N  LO and LO.
Results of the S  HERPA simulation are compared with results from O  PAL at an
$e^{-}e^{+}$ c.m.-energy of 198 GeV [5].

Secondly, this drop around $x_{\gamma}\approx 0.7$ could be attributed to the
missing QED splitting kernel in the evolution of the parton shower. Including
this term would impact the backwards evolution of the photonic initial state
radiation leading to a photon being reconstructed as the initial state also in
the case of a resolved process. This again would lead to fewer radiation being
generated, therefore shifting the distribution of the resolved process towards
larger $x_{\gamma}$ values. The inclusion of this term in the evolution of the
initial state showering is left for future work. Finally, we should stress
that our singly resolved events are described by the $2\to 2$ scattering of
on-shell photons with partons from the resolved photons, an approximation
which is probably not entirely correct as virtual photons would lead to a DIS-
like scattering of the resolved photon, thereby inducing a somewhat different
kinematics and scale choices.

Fig. 4 shows distributions of jet pseudo-rapidities and their differences.
Again, the overall shape of the prediction is improved and the lowered NLO
cross-section is countered by the inclusion of Multiple-Parton Interactions
(MPIs).

###  3.2 Comparison with H  ERA data

For the further validation of our implementation in electron–proton collisions
we mainly rely on Z  EUS data [12] taken at H  ERA Run 2. The kinematic cuts
on the final states in the hard matrix element calculation were chosen to be a
minimal transverse momentum of $p_{T}>11(8)$ GeV for the (sub-)leading jets
using the $k_{T}$ clustering algorithm with radius $R=1.0$ to safely capture
the phase space cuts of $p_{T}^{(1)}>14$ GeV and $p_{T}^{(2)}>11$ GeV used in
the analysis of the Z  EUS Run-1 data [11], even after taking into account
MPIs. For the Z  EUS Run-2 data [12], we chose $p_{T}>13$ GeV to comply with
the experimental cut of $E_{T}>17$ GeV on the leading jet, with otherwise same
settings.

|
---|---
Figure 5: Differential dijet inclusive cross section with respect to
$\cos\theta^{*}$ for $x^{\rm obs}_{\gamma}<0.75$ (left) and $x_{\gamma}^{\rm
obs}>0.75$ (right), comparing results of our S  HERPA M  C @N  LO simulation
with Z  EUS Run-1 data [11].

We use the same scale setting algorithm as before, and we also evaluate the
theory uncertainties as combination of scale and PDF uncertainties like in the
case of electron–positron colliders in the section above. Defining, in analogy
to the case at lepton colliders above, Eqs. (3.3) and (3.4) , respectively,

$x_{\gamma}^{\rm
obs}=\frac{E_{T}^{(1)}e^{-\eta^{(1)}}+E_{T}^{(2)}e^{-\eta^{(2)}}}{2yE_{e}}\;\;\;\mbox{\rm
and}\;\;\;\cos\theta^{*}=\tanh\frac{\eta^{(1)}-\eta^{(2)}}{2}\,.$ (3.5)

Here (1,2) labels the leading and sub-leading jet, $E_{e}$ is the lepton
energy, and $y$ is the energy fraction of the photon w.r.t. the lepton. As
before we observe that the $x_{\gamma}$ is excellently suited to disentangle
unresolved and resolved photon interactions, cf. Fig. 5. Again we observe
satisfactory agreement with data, indicating that the summation over the
different components is correct. Interestingly, as suggested by the right
panel of the figure, it appears as if the direct component is not sufficient
to fully recover the experimental cross section. Possible explanations, as
before, are related to the missing ”anomalous” $\gamma\to q\bar{q}$ splitting
in the backwards evolution, or, possibly more relevant here, a failure of the
strictly on-shell approximation of the incident photons inherent to the
treatment through EPA.

|
---|---
Figure 6: Differential single-jet inclusive photo-production cross section with respect to the pseudo-rapidity of the leading jet, $\frac{\mathrm{d}\sigma}{\mathrm{d}\eta}$, comparing results of our S  HERPA simulation at parton (left) and hadron-level (right) with Z  EUS Run-2 data [12]. |
---|---
|
Figure 7: Single-jet inclusive transverse energy spectra for the leading jet
in different pseudo-rapidity bins of the $k_{T}$-jets: $0<\eta<1$ (top left),
$1<\eta<1.5$ (top right), $1.5<\eta<2$ (bottom left), and $2<\eta<2.5$ (bottom
right), comparing S  HERPA M  C @N  LO results with Z  EUS Run 2 data [12].

With this caveat in mind we will now turn into a more differential analysis of
QCD final states in photo-production at H  ERA . In Fig. 6 we compare the
parton- against the hadron-level results and as previously observed, the shape
improves significantly through the combined effect of hadronization and MPIs.
This is most visible in the phase space of $\eta<0$, rendering the
distribution of the simulation data flat compared to the experimental data.
While there remains a discrepancy between simulation and data of around
10-20%, it might be explained by three observations. First, this analysis cuts
requires only one jet in the final state which allows for contributions from
the DIS region to leak into this measurement, for example if the scattered
beam electron has not been correctly detected or identified. Secondly, the
precise value of the strong coupling in the fit of the photon PDFs is not
explicitly mentioned in the corresponding publications [22, 23]. An updated
photon PDF fit would be performed with the current world average of
$\alpha_{S}$ and might further reduce the discrepancy. Lastly, the modelling
of multiple-parton interactions for photon–proton interactions needs a fitting
to the data for both proton–proton and photon–proton data. As neither have
been tuned to data yet it can be suspected that the MPI will receive larger
contributions, thereby improving overall agreement of simulation and data.

The spectrum of the jet transverse energies $E_{T}$, displayed in Fig. 7
exhibits a slight shape in the distribution for forward jets and small
$E_{T}$. The same effect can be seen in Fig. 8, where the simulation describes
data well for central leading jets, but worsens as both go more forward. As
this part of the phase receives contributions from the MPIs, we again suspect
that the naive parameter choice underestimate the amount of additionally
generated radiation.

|  |
---|---|---
|  |
Figure 8: Differential dijet inclusive cross section with respect to the
transverse momentum of the leading jet, with $0<\eta^{(1)}<1$ (upper panel)
and $1<\eta^{(1)}<2.4$ (lower panel) and in different bins for $\eta^{(2)}$,
comparing results of our S  HERPA M  C @N  LO simulation at hadron-level incl.
MPI effects with the LO simulation and with data [11] taken by Z  EUS at H
ERA Run 1.

###  3.3 Photon PDF for precision phenomenology

|
---|---
Figure 9:  Distributions for $\cos\Theta^{*}$ at L  EP (left) and for
$\cos\theta^{*}$ at H  ERA (right), comparing S  HERPA ’s LO simulations with
data from O  PAL [5] and Z  EUS [11]. Scale uncertainties at LO are indicated
by the pink band, while PDF uncertainties are shown with the blue hatched
area.

The predictions of photo-production cross sections and distributions in
low-$x_{\gamma}$ space exhibit large variations depending on the used photon
PDF. In fact, these deviations can be as large or even larger in value than
the estimate of higher-order corrections through the scale variations,
especially for the simulation of photo-production at lepton colliders, see
Fig. 9. There we show the very inclusive $\cos\Theta^{*}$ and $\cos\theta^{*}$
distributions, obtained at leading order, at L  EP and H  ERA , respectively.,
indicating LO scale and PDF uncertainties separately. For the latter we
present the full range of results from all available leading order PDFs, and
we observe that the PDF uncertainties alone are of equal size as the LO scale
uncertainties. This underlines the need for a comprehensive retuning of the
photon PDFs with available data, and including higher-order calculations and
simulations.

It is also worth noting that a consistent simulation of photo-production at
hadron–lepton colliders necessitates a combined fit of the photon and proton
PDFs. Depending on the proton PDF and its value for $\alpha_{S}$ the inclusive
cross-section in a H  ERA Run 2 simulation gives deviations of about 20%, as
can be seen in Tab. 2. This underlines the necessity for a systematic refit of
photon PDFs to use in conjunction with modern proton PDFs. While no new data
has been taken since the retiring of the H  ERA collider, a consistent fit to
all the data with the updated values for $\alpha_{S}$ and including error
estimates should increase the confidence in precision phenomenology for photo-
production for the planned E  IC and other future colliders.

PDF | NNPDF23_lo_as_0130_qed | PDF4LHC21_40_pdfas
---|---|---
$\alpha_{S}$ | 0.13 | 0.118
Order | 1 | 3
$\sigma(\gamma j\to jj)$ / nb | $2.85\pm 0.02$ | $2.303\pm 0.016$
$\sigma(jj\to jj)$ / nb | $2.151\pm 0.002$ | $1.997\pm 0.002$

Table 2:  Inclusive cross-sections for one million events in a H  ERA Run 1
dijet photo-production setup with two different proton PDFs and the same PDF
for the photon.

##  4 Summary

In this paper we reported on developments of the S  HERPA event generator
towards the first hadron-level simulation of photo-production at NLO accuracy,
to be published in our upcoming S  HERPA 3.0 public release. We described how
the initial state phase space is treated to allow for a flexible yet efficient
integration of different spectra and distributions, based on the equivalent
photon approximation and the inclusion of a broad range of photon PDFs. We
validated our model by comparison to data from the L  EP and H  ERA
experiments and found satisfactory agreement between the data and the
simulation, noting that the non-perturbative aspects of the simulation, in
particular intrinsic $k_{T}$ and multiple-parton interactions, still require a
comprehensive tuning to data. Apart from possible improvement in this sector
of the simulation, our analysis of uncertainties underlined the necessity for
a refitting of the photon PDFs, especially in view of the anticipated
precision of the planned electron-ion collider. We believe that this step – a
first refitting of photon PDFs after 20 years – based on higher-order
calculations, modern simulation tools at least at NLO accuracy, and recent
proton PDFs will siginifcantly improve the quality of our theoretical
preprations for this new collider experiment.

## Acknowledgements

We are indebted to our colleagues in the S  HERPA collaboration, for numerous
discussions and technical support. F.K. gratefully acknowledges funding as
Royal Society Wolfson Research fellow. F.K. and P.M. are supported by the STFC
under grant agreement ST/P001246/1. S.H. is supported by Fermi Research
Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of
Energy, Office of Science, Office of High Energy Physics. P.M. would like to
thank the Fermilab Theory Division for hospitality while this work was
finalized.

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