# Hard Diffraction in S  HERPA

Frank<EMAIL_ADDRESS>Institute for Particle Physics
Phenomenology, Durham University, Durham DH1 3LE, UK  Peter
<EMAIL_ADDRESS>Institute for Particle Physics
Phenomenology, Durham University, Durham DH1 3LE, UK

###### Abstract

We present the first complete simulation framework for the simulation of jet
production in diffractive events at next-to leading order in QCD, matched to
the parton shower. We validate the implementation in the S  HERPA event
generator with data from the H1 and Z  EUS experiments for diffractive DIS and
diffractive photoproduction. For the latter, we review different models aiming
to explain the observed factorisation breaking and we argue that at NLO the
direct component must also be suppressed. We provide predictions for
diffractive jet production both in DIS and in photoproduction events for the
upcoming E  IC .

##  1 Introduction

Diffractive events constitute a significant part of the cross section at
electron-hadron colliders such as H  ERA in the past and the upcoming E  IC
and at hadron-hadron colliders such as the currently operating L  HC .
Especially the production of jets in diffractive events also offers potential
additional insights into the dynamics of the strong interaction in its
interplay between the hard perturbative scales of jet production and the soft
non-perturbative scales of hadronic structures [1]. In this publication we
introduce the first hadron-level simulation of diffractive jet-production at
next-to leading order accuracy in the perturbative expansion of QCD at
electron-hadron colliders. At such experiments, the interaction of incident
electrons and protons is mediated by a virtual photon; in dependence of its
virtual mass $q^{2}=-Q^{2}$ the interaction will either be classified as
electro-production (at large $Q^{2}$, in the following somewhat misnamed as
deep-inelastic scattering, DIS), or as photo-production (at small $Q^{2}$). In
the latter the photon may fluctuate into a system with hadronic structure,
like, for example, a vector meson, and, correspondingly it must be described
by a photon parton distribution function (PDF). We will therefore discuss the
simulation of diffractive events in both event classes and we will show
results for both.

In general, the description of hard processes involving incident hadrons
relies on the factorisation of the cross section calculation into PDFs which
encode the transition of the hadrons into the partons, their constituents
which are usually identified with the quanta of the strong interaction, quarks
and gluons, and into the parton-level cross sections. This factorization is
necessary to allow systematic calculations of cross sections from first
principles in quantum field theory, and their systematic improvement through
perturbative expansions in the limit of small couplings, which act as
expansion parameter. This factorization picture has been proven to hold true
for DIS [2], and has been a cornerstone of QCD theory ever since. The proof
was extended for diffractive DIS [3], underpinning calculations for di-jet
production cross sections at next-to (NLO) [4] and next-to–next-to leading
order (NNLO) in QCD [5]. Diffractive PDFs were extracted at NLO accuracy
mainly from H  ERA data in [6]; it is customary to factorise them into the
flux of a pomeron (or reggeon) and the PDF of this intermediate ”particle”
[7].

However successful for the description of diffractive DIS, though, it appears
as if the factorization assumption does not extrapolate to the case of
diffractive photo-production events. There are a number of underlying physics
effects which may render factorization invalid, including, for example, (i)
additional scattering between the hadronic structure of the photon and the
proton rump, broadly speaking part of the unitarisation of the hard
diffractive cross section [8, 9]; (ii) the impact of hadronization effects
which may impact on the emergence and survival of rapidity gaps necessary for
the definition of diffractive events [10]. The apparent breakdown of
factorisation in diffractive photo-production also led to approaches where the
different components of the cross section, usually the resolved component
where the photon assumes a hadronic structure, are rescaled to fit the data
[11, 12]. A similar rescaling approach was also taken by the authors of [13]
who analysed differences between H1 and Z  EUS data as well. Generally
speaking, the resolved components were found to need a suppression by a factor
of about 3, however, it remains unclear whether a rescaling of the resolved
component only correctly reflects the process of factorization breaking.

Our publication is organised as follows: In Sec. 2 we will briefly review the
theory ingredients for the calculation, in particular the expressions for the
pomeron and reggeon fluxes and resulting cross sections, followed by a
presentation of our simulation framework and the definition of fiducial phase
spaces and experimental observables in Sec. 3. We validate our diffractive DIS
simulation in Sec. 4, before turning to a more in-depth discussion of photo-
production and the factorization breaking there in Sec. 5. Predictions for the
E  IC for dijet production in both diffractive DIS and diffractive
photoproduction in Sec. 6 provide an outlook to the future. We summarise our
findings in Sec. 7.

##  2 Theory framework for jet production in diffraction

Diffractive events are characterised by the presence of an intact beam proton,
or a typically low-mass excitation thereof in the final state. In [3],
factorisation has been shown to hold for these type of processes with the
introduction of so-called Diffractive PDFs (DPDF),
$f_{i}^{D}(x,\mu_{F},x_{I\\!\\!P},t)$. In this factorised approach, the cross-
section for the process $ep\to eXY$, where $Y$ denotes the elastically
scattered proton or its excitation, can be calculated through

${\rm d}\sigma_{ep\to
eXY}(x_{I\\!\\!P},t)=\sum_{i}f_{i}^{D}(x,\mu_{F},x_{I\\!\\!P},t)\otimes\hat{\sigma}_{ei\to
eX}(x,\mu_{F})\ .$ (2.1)

Due to the colour-disconnection of the final state systems $X$ and $Y$, the
events typically show a large rapidity gap (LRG). The tagging of such events
can be experimentally achieved either through dedicated detection of the
system $Y$ or by vetoing activity in certain rapidity regions, hence selecting
events with LRGs.

Such event topologies are usually attributed to the exchange of Pomeron (and
subleadingly, a Reggeon); motivated by this physics model the factorisation
picture has been extended [7], to separate the DPDF further into products of
flux factors and PDFs:

$f^{D}_{i}\left(x,\mu_{F},x_{I\\!\\!P},t\right)=f_{{I\\!\\!P}/P}\left(x_{I\\!\\!P},t\right)f_{i/{I\\!\\!P}}\left(x,\mu_{F}\right)+n_{I\\!\\!R}f_{{I\\!\\!R}/P}\left(x_{I\\!\\!P},t\right)f_{i/{I\\!\\!R}}\left(x,\mu_{F}\right)$
(2.2)

Inspired by Regge theory, the flux factors assume the form

$f_{{I\\!\\!P},{I\\!\\!R}/P}\left(x_{I\\!\\!P},t\right)=A_{{I\\!\\!P},{I\\!\\!R}}\frac{e^{B_{{I\\!\\!P},{I\\!\\!R}}t}}{x_{{I\\!\\!P},{I\\!\\!R}}^{2\alpha_{{I\\!\\!P},{I\\!\\!R}}(t)-1}}$
(2.3)

with the parameters describing the trajectories
$\alpha_{{I\\!\\!P},{I\\!\\!R}}(t)=\alpha_{{I\\!\\!P},{I\\!\\!R}}(0)+\alpha_{{I\\!\\!P},{I\\!\\!R}}^{\prime}\,t$
fitted to data. Integrating over $t$ in the interval $[t_{\rm min},\,t_{\rm
cut}]$ yields

$f_{{I\\!\\!P},{I\\!\\!R}/P}\left(x_{I\\!\\!P}\right)=\frac{A_{{I\\!\\!P},{I\\!\\!R}}x^{1.\,-2.\alpha_{{I\\!\\!P},{I\\!\\!R}}(0)}}{B_{{I\\!\\!P},{I\\!\\!R}}-2.\alpha_{{I\\!\\!P},{I\\!\\!R}}^{\prime}\log(x)}\left(e^{B_{{I\\!\\!P},{I\\!\\!R}}t_{\mathrm{min}}}x^{-2.\alpha_{{I\\!\\!P},{I\\!\\!R}}^{\prime}t_{\mathrm{min}}}-e^{B_{{I\\!\\!P},{I\\!\\!R}}t_{\mathrm{cut}}}x^{-2.\alpha_{{I\\!\\!P},{I\\!\\!R}}^{\prime}t_{\mathrm{cut}}}\right)\,.$
(2.4)

The integration boundaries are given by
$t_{\mathrm{min}}=-\frac{m_{p}^{2}x^{2}}{1-x}$, with $m_{p}$ the proton mass,
and $t_{\mathrm{cut}}$ the experimental boundary of the momentum transfer of
the proton. While the PDF for the Pomeron
$f_{i/{I\\!\\!P}}\left(x,\mu_{F}\right)$ has to be fitted to data, the Reggeon
PDF is usually approximated by the pion PDF,
$f_{i/{I\\!\\!R}}\left(x,\mu_{F}\right)\approx
f_{i/\pi}\left(x,\mu_{F}\right)$ [14].

Depending on the virtuality of the exchanged photon, events can be further
differentiated into the deep-inelastic scattering (DIS) or the photoproduction
regime. In the latter case, the factorisation includes a photon flux and,
potentially, a photon PDF. The overall phase space setup and the
implementation of the photoproduction events follows [15].

This results in the following factorisation formula for the cross-section for
Diffractive DIS

$\sigma^{\mathrm{(DDIS)}}\left(ep\to
eXY\right)=\int_{0}^{x_{{I\\!\\!P}{},\mathrm{max}}}{\rm
d}x_{I\\!\\!P}\int_{t_{\mathrm{cut}}}^{t_{\mathrm{min}}}{\rm
d}t\int_{0}^{1}{\rm
d}x_{i}f^{D}_{i}\left(x_{i},\mu_{F},x_{I\\!\\!P},t\right)\,\hat{\sigma}\left(ei\to
eXY\right)$ (2.5)

and for Diffractive Photoproduction

$\displaystyle\sigma^{\mathrm{(DPHO)}}\left(ep\to eXY\right)=$
$\displaystyle\int_{0}^{1}{\rm
d}xf^{(e)}_{\gamma}\left(x\right)\int_{0}^{x_{{I\\!\\!P}{},\mathrm{max}}}{\rm
d}x_{I\\!\\!P}\int_{t_{\mathrm{cut}}}^{t_{\mathrm{min}}}{\rm d}t$
$\displaystyle\int_{0}^{1}{\rm
d}x_{j}f^{(\gamma)}_{j}\left(x_{j},\mu_{F}\right){\rm
d}x_{i}f^{D}_{i}\left(x_{i},\mu_{F},x_{I\\!\\!P},t\right)\
\hat{\sigma}\left(ji\to XY\right)$ (2.6)

where in the latter case $f^{(\gamma)}_{j}$ is replaced by a Delta
distribution for direct photoproduction.

##  3 Validation: simulation and data

###  3.1 Event generation

Events are generated with a pre-release of S  HERPA [16, 17] v3.0.0; the code
will be made available in a future release. The matrix element part of the
simulation used A  MEGIC [18, 19] and C  OMIX [20] for tree-level matrix
elements, O  PEN L  OOPS [21, 22, 23, 24] for loop matrix elements, and
Catani-Seymour dipole subtraction [25, 26] automated in [19] for the treatment
of infrared divergences. The matching to the parton shower [27] is achieved
through the M  C @N  LO formalism [28] in its implementation presented in
[29]. For the PDFs we used built-in interfaces to the SAS1M set [30, 31] for
the photon and the H1 2007 Fit B set [6] for the pomeron. For the reggeon and
the proton, we used the sets GRVPI0 [14] and PDF4LHC21_40_pdfas [32],
respectively, interfaced through L  HAPDF [33]. The parameters of pomeron flux
were taken from the fit in [6]. We calculated in the 3-flavour scheme in
accordance with the PDF and additionally allowed for massive $c$-quarks at NLO
and massive $b$-quarks at LO in the final state. We consistently used the
current default value for $\alpha_{S}=0.118$ with three-loop running, which is
also in-line with the H1 2006 PDF. The factorisation scale and the
renormalisation scale were set to $\mu_{F}=\mu_{R}=H_{T}/2$ for
photoproduction events and to
$\mu_{F}^{2}=\mu_{R}^{2}=\frac{1}{4}(Q^{2}+H_{T,\mathrm{hadr}}^{2})$ for DIS
events, with $H_{T,\mathrm{hadr}}$ as the scalar sum over the transverse
momenta of all hadronic particles, and were varied by factors of 2 in a
7-point scale variation to estimate the uncertainties. The partonic final
states were hadronized with the cluster fragmentation model of A  HADIC [34],
fitted to LEP data.

Diffractive events are simulated with an assumed intact beam proton. Low-mass
excitations have been seen to account for an additional 20% in cross-section
flat in phase space and can therefore be taken into account with an overall
scaling of 0.83 [35]. Diffractive photoproduction is composed of two
components, the direct and the resolved photon contributions, with the latter
simulated through a photon PDF. However appealing this picture of combining
two PDFs is – one for the photon, one for the pomeron or reggeon – the assumed
factorisation underpinning it is expected to break down [3] as a consequence
of additional soft interactions between the photon and the proton beam. To
account for the suppression of these events, we generalized the multiple-
interactions modelling in S  HERPA to allow for this kind of interactions.
Naturally, this argument can only be applied to the resolved component and
therefore it has been conjectured that factorisation might still hold for the
direct component [9]. We will study this ansatz in Sec. 5.3.

###  3.2 Experimental observables and datasets

Our implementation of diffractive events in DIS and photoproduction was
validated with data from the H1 [36, 4] and Z  EUS [37] collaborations.

#### H1, JHEP05 (2015) 056

The data from the H1 collaboration in [36] detected the out-going proton in
the Very Forward Proton Spectrometer (VFPS) and measured di-jet production in
both the DIS, $4\,\mathrm{GeV}^{2}<Q^{2}<80\,\mathrm{GeV}^{2}$, and
photoproduction, $Q^{2}<2\,\mathrm{GeV}^{2}$, regime. To describe the
kinematics of the diffractive exchange, the variable $x_{I\\!\\!P}$ was
defined as

$x_{I\\!\\!P}=1-\frac{E_{p}^{\prime}}{E_{p}}$ (3.1)

with $E_{p}^{(\prime)}$ the energy of the incoming (outgoing) proton; the
acceptance of the VFPS yielded a range of $0.010<x_{I\\!\\!P}<0.024$. The
partonic momentum fractions with respect to the diffractive exchange and, in
the case of photoproduction, with respect to the electron and the photon have
been defined as

$z_{I\\!\\!P}=\frac{Q^{2}+M_{12}^{2}}{Q^{2}+M_{X}^{2}}$ (3.2)

for diffractive DIS events and

$\displaystyle z_{I\\!\\!P}$
$\displaystyle=\frac{\sum_{j=1,2}\left(E+p_{z}\right)_{j}}{\sum_{i\in
X}\left(E+p_{z}\right)_{i}}\;,\;\;\;y=\frac{\sum_{i\in
X}\left(E-p_{z}\right)_{i}}{2E_{e}}\;,\;\mbox{\rm
and}\;\;x_{\gamma}=\frac{\sum_{j=1,2}\left(E-p_{z}\right)_{j}}{\sum_{i\in
X}\left(E-p_{z}\right)_{i}}$ (3.3)

for diffractive photoproduction events, where $M_{12}$ denotes the di-jet
invariant mass, $M_{X}$ the invariant mass of $X$ defined below; the index $j$
runs over the leading jets and the index $i$ over the final states in $X$. The
momentum transfer $t$ from the proton was required to be
$\left|t\right|<0.6\,\mathrm{GeV}^{2}$. The leading and subleading jets were
clustered with the $k_{T}$-algorithm with $R=1$ in the photon-proton rest
frame and were required to have transverse energies of
$E_{T}^{*}>5.5\,\mathrm{GeV}$ and $4.0\,\mathrm{GeV}$222By the superscript ’∗’
we denote quantities measured in the photon-proton rest frame., respectively,
and lie within the pseudorapidity range of $-1<\eta<2.5$ in the laboratory
frame. The inelasticity $y$ was required to be within $0.2<y<0.7$. The
invariant mass of the system $X$ was calculated as

$M_{X}^{2}=\left(\sum_{i\in X}E_{i}\right)^{2}-\left(\sum_{i\in
X}\vec{p}_{i}\right)^{2}\ .$ (3.4)

#### H1, EPJC51 (2007) 549

A similar measurement was undertaken in [4], where the mass of the system $Y$
was restricted to $M_{Y}^{2}<1.6\,\mathrm{GeV}^{2}$. Similarly to before, di-
jets were measured in the DIS,
$4\,\mathrm{GeV}^{2}<Q^{2}<80\,\mathrm{GeV}^{2}$, and photoproduction,
$Q^{2}<0.01\,\mathrm{GeV}^{2}$, regimes and jets were clustered as before,
demanding $E_{T}^{*}>5(4)\,\mathrm{GeV}$ for the (sub)leading jet within
pseudorapditiy $-1<\eta^{\mathrm{lab}}<2$. The photon-proton c.m.s.-energy $W$
was restricted to $165\,\mathrm{GeV}<W<242\,\mathrm{GeV}$ and the proton
momentum transfer to $\left|t\right|<1\,\mathrm{GeV}^{2}$. The kinematical
variables were defined as

$\displaystyle x_{I\\!\\!P}=\frac{Q^{2}+M_{X}^{2}}{Q^{2}+W^{2}}\;,\;\mbox{\rm
and}\;\;z_{I\\!\\!P}=\frac{Q^{2}+M_{12}^{2}}{Q^{2}+M_{X}^{2}}\ ,$ (3.5)

with $M_{12}$ the dijet mass, for diffractive DIS events and

$\displaystyle x_{I\\!\\!P}=\frac{\sum_{i\in
X}\left(E+p_{z}\right)_{i}}{2E_{p}}\;,\;\;\;z_{I\\!\\!P}=\frac{\sum_{j=1,2}\left(E+p_{z}\right)_{j}}{2x_{I\\!\\!P}E_{p}}\;,\;\;\;y=1-\frac{E_{e}^{\prime}}{E_{e}}\;,\;\mbox{\rm
and}\;\;x_{\gamma}=\frac{\sum_{j=1,2}\left(E-p_{z}\right)_{j}}{2yE_{e}}$

for diffractive photoproduction events, where $E_{e}^{(\prime)}$ denotes the
incoming (outgoing) lepton’s energy; $M_{X}$ was defined as in the previous
analysis, c.f. Eq. 3.4.

#### ZEUS, EPJC55 (2008) 177

The Z  EUS collaboration measured di-jet production in diffractive
photoproduction [37] with photon virtualities of $Q^{2}<1\,\mathrm{GeV}^{2}$
and an inelasticity of $0.20<y<0.85$. The out-going proton was not detected,
instead diffractive events were selected by requiring LRGs. To correct for
proton-dissociative events, a fraction of $16\%$ had been subtracted from the
total cross-section. The following variables were defined:

$\displaystyle
x_{I\\!\\!P}=\frac{\sum_{h}\left(E+p_{z}\right)_{h}}{2E_{p}}\;,\;\;\;z_{I\\!\\!P}=\frac{\sum_{j}E_{T}^{(j)}\mathrm{e}^{\eta^{(j)}}}{2x_{I\\!\\!P}E_{p}}\;,\;\;\;y$
$\displaystyle=\frac{\sum_{h}\left(E-p_{z}\right)_{h}}{2E_{e}}\;,\;\;\;x_{\gamma}=\frac{\sum_{j}E_{T}^{(j)}\mathrm{e}^{-\eta^{(j)}}}{2yE_{e}}\;,$
(3.7)

and

$\displaystyle
M_{X}^{2}=\sum_{h}\left(E-p_{z}\right)_{h}\left(E+p_{z}\right)_{h}\;,$ (3.8)

where the index $h$ runs over reconstructed Energy Flow Objects in the main
detectors and it was required $x_{I\\!\\!P}<0.025$. Jets were clustered with
the $k_{T}$ algorithm using $R=1$ in the laboratory frame, with cuts of
$E_{T}>7.5(6.5)\,\mathrm{GeV}$ for the (sub)leading jet and within
pseudorapidity $-1.5<\eta<1.5$.

##  4 Diffractive DIS

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Figure 1: Differential DDIS cross-sections with respect to the photon-proton
centre-of-mass energy $W$, the photon virtuality $Q^{2}$, and the leading jet
transverse momentum $E_{T}^{*\rm{jet1}}$ (upper row, left to right, data from
[4]), to momentum ratios $x_{I\\!\\!P}$, $z_{I\\!\\!P}$ and inelasticity $y$
(second row, left to right, data from [36]), to the average pseudorapidity
$\langle\eta\rangle$ and the pseudorapidity difference $\Delta\eta$ of the
jets as well as the transverse momentum of the leading jet $E_{T}^{\rm{jet1}}$
(third row, left to right, data from [36]).

Turning first to the Diffractive DIS measurements, we compare the S  HERPA
results, obtained at M  C @N  LO accuracy with dijet production data from two
publications by the H1 collaboration [4, 36].

In the upper two rows of Fig. 1 we focus on more inclusive observables that
describe the overall kinematics of the events. They include very general
observables such as the centre-of-mass energy of the photon-proton system $W$,
the photon virtuality $Q^{2}$ or the transverse momentum of the leading jet,
$E_{T}^{*{\rm jet1}}$, all taken from [4], and extend to observables that
expose more of the underlying parton-level dynamics such as the momentum
fractions $x_{I\\!\\!P}$ and $z_{I\\!\\!P}$ and the inelasticity $y$, with
data from [36]. We consistently observe excellent agreement of the simulation
at M  C @N  LO

accuracy with data, with $K$-factors ranging from about 1.5 up to 2-3
depending on the observable and the associated phase space region. With the
exception of the $W$ and $y$ distributions, the $K$-factors are relatively
flat and do not overly change the shape of the distributions. This confirms
other findings which established good agreement of data and fixed-order
calculations at NLO [5] and the corresponding $K$-factors. The increasing size
of $K$-factors is readily understood and associated with the additional phase
space made available due to the asymmetric cuts and additional parton-level
channels with the largest enhancement of the cross-section seen in the forward
region, as expected.

##  5 Diffractive photoproduction

###  5.1 Diffractive photoproduction at M  C @N  LO accuracy

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Figure 2: Differential diffractive photoproduction cross-sections with respect
to momentum ratios $x_{I\\!\\!P}$, $z_{I\\!\\!P}$, $y$ and $x_{\gamma}$ (top
to bottom), obtained by H1 (left column, data from [36]) and Z  EUS (right
column, data from [37]).

Turning now to the description of diffractive photoproduction of dijets, we
observe that the convincing agreement of simulation and data does not hold
true anymore, supporting statements about the possible breakdown of
factorization in such processes [3]. To highlight this, let us first take a
look at the momentum ratios $x_{I\\!\\!P}$, $z_{I\\!\\!P}$, $y$ and
$x_{\gamma}$. They have been analysed by both the H1 and the Z  EUS
collaboration in [36] and [37], respectively, and we display the results in
the left and right column of Fig. 2. As already seen in previous NLO
calculations [11], the calculation of the cross-section in our M  C @N  LO
samples severely overestimates the data in diffractive photoproduction, by a
factor of up to 2-3, while the LO predictions are in somewhat better agreement
overall. In addition, we observe a sharp increase, amounting to a visible
shape distortion, of the M  C @N  LO simulation with respect to the
experiment, particularly at large values of
$x_{\gamma}\stackrel{{\scriptstyle>}}{{\sim}}0.75$, a regime usually
associated with ”direct” photoproduction, i.e. the photon acting as a point-
like particle. By far and large, however, the M  C @N  LO simulation describes
data reasonably well in the ”resolved” photoproduction regime of small
$x_{\gamma}\stackrel{{\scriptstyle<}}{{\sim}}0.5$.

We note that differences in the shape and normalisation of between the two
different sets of data - and therefore between the two simulations - stem from
the different phase spaces populated by the H1 and Z  EUS analyses, mainly
related to the difference in the range of $Q^{2}$ and the definition of the
diffractively scattered proton.

Our findings so far of NLO predictions overshooting data by large factors, are
in agreement also with an analysis of final state observables, such as the
leading jet transverse energy, $E_{T}^{\rm jet1}$, the diffractive invariant
mass $M_{X}$, and the average pseudorapidity of jets $\langle\eta\rangle$ and
the pseudorapidity difference on the two leading jets $|\Delta\eta|$, which we
show in Fig. 3. Again the M  C @N  LO predictions have a $K$-factor of about 2
with respect to their leading-order counterparts, and they exceed data (taken
from H1 [36]), again, by a factor of about 2.

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Figure 3: Differential dijet Diffractive Photoproduction cross-sections with
respect to leading jet transverse energy $E_{T}^{\text{jet1}}$, invariant mass
$M_{X}$, difference in dijet-pseudorapidity $|\Delta\eta|$ and average dijet-
pseudorapditiy $<\eta>$. Results of the S  HERPA simulation with M  C @N  LO
accuracy are compared with results at LO and with data from H1 [36].

We observe that the shapes of the distributions do not agree between data and
theory, especially in the pseudorapditiy distributions and the momentum
fractions.

To compensate for the massive difference in measured and calculated cross
sections, H1 applied a global factor of about 1/2 to the calculation [4, 38,
36], improving its agreement with the data. This rescaling does not improve
the theory agreement with the Z  EUS data, where the calculation even slightly
undershoots the data for low $x_{\gamma}$; therefore a global scaling can not
adequately be used to describe the factorisation breaking. We will turn to
this problem in more detail in subsubsection 5.3 by further examining the
interplay seen in this observable.

###  5.2 Reggeon contribution

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Figure 4: Distributions of leading jet transverse energy $E_{T}^{\text{jet1}}$
(top row) and $x_{\gamma}$ (bottom row) with and without Reggeon contribution,
compared to data by H1 from [4] (left) and [36] (middle) and from Z  EUS [37]
(right) in their respective definitions of fiducial phase space.

To further elucidate the components of the cross-section, we show in Fig. 4
the leading jet transverse energy, comparing the calculation with and without
the Reggeon contribution, in three different definitions of phase space.
Depending on it we find sizeable positive contribution for the full range of
the observable.

###  5.3 Modelling factorisation breaking in diffractive photoproduction

The overshoot of the cross-section hints at the breakdown of the factorisation
and different models have been brought forward to explain the discrepancies,
which we will review in the following. The variable $x_{\gamma}$ has been used
by the experiments as a discriminator between the direct and resolved photon
components and our calculation confirms that it indeed works well to discern
the different contribution. Generally, a small value of $x_{\gamma}$ will
correspond to a dominant contribution by the resolved photon and its PDF,
while values close to unity will mostly stem from direct contributions, i.e.
the photon acting as a point-like particle. In this section we use this
observable the handle it provides on the dynamics of factorisation breaking
and its interplay with the direct and/or resolved components.

In [10] it was argued that the factorization breaking is a consequence of
hadronisation, bin migration and NLO effects, which we exemplify in the
distribution of $x_{\gamma}$ in Fig. 5. While hadronisation certainly plays a
big role in the bin migration between the two largest $x_{\gamma}$ bins and
the total cross-section is decreased, the overall overestimation of the total
cross-section is still present after taking these effects into account. In
fact, hadronisation is essential for the reconstruction of $x_{\gamma}$ in the
Z  EUS analysis, as a significant number of events in the parton-level
simulation end up in the the region $x_{\gamma}>1$. It is also somewhat
amusing to note that hadronisation effects tend to reduce the overshoot of the
simulation in the highest $x_{\gamma}$ bins for the H1 analyses, while they
tend to actually create it for the Z  EUS analysis.

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Figure 5: Differential distribution of $x_{\gamma}$ in diffractive
photoproduction. Results of the S  HERPA simulation with M  C @N  LO accuracy
are compared with results at LO and with data by H1 from [4] (left) and [36]
(middle) and Z  EUS [37] (right).

As mentioned previously, it has furthermore been argued that the decrease of
the cross-section is due to soft interactions between the resolved photon and
the proton [3]. We implemented a simplistic generalized multiple-parton
interactions modelling in S  HERPA to veto events, which have an additional
scattering between the photon and the proton and would hereby destroy the
rapidity gap. In Fig. 6 we show the effect of this rejection. Naturally this
mechanism only applies to the resolved component and affects only those
regions of the phase space where the resolved component dominates. We also
recall that the resolved component can be further decomposed into the point-
and hadron-like component, where the difference is that the backwards
evolution would collapse the former to a photon and the latter to a meson-like
state. Furthermore, it has been pointed out in [10] that this so-called
anomalous component, i.e. splittings of $\gamma\to q\bar{q}$, would not
exhibit further interactions with the proton. This would lead to a smaller
suppression in the resolved component, depending on the size of these
splittings in the radiation off these quarks111 These splittings are currently
not included in the simulation in S  HERPA and, as indicated, we leave the
study of this effect to future work.. We therefore expect that the generalized
MPI modelling would only apply to meson-like states, whereas the point-like
contribution would see a suppression mechanism similar to the direct
component. A comprehensive study of the suppression mechanism will have to
disentangle these two components; while the hadron-like states will undergo
the MPI-based breaking of the factorisation, the point-like state will have to
interpolate between the direct and the hadron-like regimes. The implementation
of this modelling and the details of the suppression of the point-like
resolved component are left for future work. However, while our naive model
depends on some assumptions of the impact parameter and other unconstrained
parameters and some additional simplifications, which certainly deserve
further investigation, it does not appear as if these effects alone can
accommodate the observed large discrepancy of simulation and data.

|  |
---|---|---
Figure 6: Differential diffractive photoproduction cross-sections with respect
to momentum ratio $x_{\gamma}$ as measured by H1 in [4] (left) and [36]
(middle) and $x_{\gamma}^{\mathrm{obs}}$ as measured by Z  EUS [37] (right),
compared to results of the S  HERPA simulation at M  C @N  LO accuracy with
and without a veto on $\gamma p$ interactions.

The authors of [13] conducted a study which found that, even though there is a
slight dependence, the different phase space cuts are not the cause of the
discrepancy in the suppression between the H1 and Z  EUS measurements.

| H1, EPJC51 (2007) 549 [4] | H1, JHEP05 (2015) 056 [36] | Z  EUS , EPJC55 (2008) 177 [37]
---|---|---|---
$R_{\mathrm{res}}$ | $0.4\pm 0.1$ | $0.6\pm 0.3$ | $1.3\pm 0.1$
$R_{\mathrm{dir}}$ | $0.4\pm 0.1$ | $0.3\pm 0.2$ | $0.5\pm 0.1$

Table 1:  Scaling factors for the direct, $R_{\mathrm{dir}}$, and resolved,
$R_{\mathrm{res}}$, component in diffractive photoproduction for the
respective experimental data.

|  |
---|---|---
Figure 7: Differential diffractive photoproduction cross-sections with respect
to momentum ratio $x_{\gamma}$ as measured by H1 in [4] (left) and [36]
(middle) and $x_{\gamma}^{\mathrm{obs}}$ as measured by Z  EUS [37] (right),
compared to results of the S  HERPA simulation at M  C @N  LO accuracy with
the direct and resolved component scaled separately.

Turning back to the distributions of $x_{\gamma}$, we point out that, while
the data from Z  EUS and H1 do not agree with each other on the overall size
of the necessary suppression, they both show an overshoot for large
$x_{\gamma}$ values. This hints at factorisation breaking happening in the
direct component too. In fact, the distinction between direct and resolved
components of photoproduction can not be maintained at NLO as the real
correction to ”direct” photoproduction, $\gamma j\to jjj$, can not be
distinguished from contributions of photon splitting combined with a two-jet
matrix element, $\left(\gamma\to
q\bar{q}\right)_{\mathrm{PDF}}\otimes\left(qj\to jj\right)_{\mathrm{ME}}$,
where $j$ denotes any parton. This led us to revisit the logic outlined in
[4], to further elucidate the impact of factorisation breaking on the
different photonic components. We fitted two prefactors, one each for direct
and resolved components in the simulation, to the data to quantify the effect
of the suppression in the two components separately, with results shown in
Tab. 1. While the scaling of the resolved component does vary among the
different measurements, the direct component seems to support a somewhat
universal suppression by a factor of 0.5. The data covered different cuts on
the photon virtuality, hence the suppression seems to be independent of the
kinematics at the electron-photon vertex.

In Fig. 7 we exhibit the results of the fit; we also did not observe any
significant discrepancies between simulation and data in other distributions
as a results of the rescaling.

##  6 Predictions for the E  IC

###  6.1 Diffractive DIS

For the analysis of diffractive DIS events we implemented a routine for R
IVET [39], loosely based on the measurement in [36], with the following phase
space: The photon virtuality was restricted to
$4\,\mathrm{GeV}^{2}<Q^{2}<110\,\mathrm{GeV}^{2}$ and we clustered jets in the
lab frame using the $k_{T}$ algorithm with $R=1$ within pseudorapditiy
$\left|\eta\right|<4$, demanding at least two jets with transverse energy
$E_{T}$ of at least 5.5 and 4 GeV, respectively. We assumed proton-tagging and
reconstructed the $x_{I\\!\\!P}$ as

$x_{I\\!\\!P}=1-\frac{E_{p}^{\prime}}{E_{p}}$ (6.1)

which had to satisfy $x_{I\\!\\!P}<0.1$, to allow for more phase space for jet
production compensating for the lower beam energies. The momentum transfer was
restricted to $\left|t\right|<0.6\,\mathrm{GeV}^{2}$ and we defined

$z_{I\\!\\!P}=\frac{Q^{2}+M_{12}^{2}}{Q^{2}+M_{X}^{2}}\ .$ (6.2)

|  |
---|---|---
|  |
|  |
Figure 8: Predictions at LO and M  C @N  LO accuracy of S  HERPA for a range
of observables relevant for diffractive DIS at the E  IC , namely: the
momentum ratios $x_{I\\!\\!P}$, $z_{I\\!\\!P}$ and inelasticity $y$ (upper
row, left to right), leading jet transverse momentum $E_{T}^{\rm jet1}$,
average jet rapidity $\langle\eta\rangle$ and diffractive mass $M_{X}$ (middle
row, left to right), transverse thrust $T_{\perp}$, thrust minor $T_{\perp,m}$
and transverse sphericity $S_{\perp}$ (bottom row, left to right).

In the upper two rows of Fig. 8 we exhibit the distributions in the momentum
fractions $x_{I\\!\\!P}$ and $z_{I\\!\\!P}$ and the inelasticity $y$ and the
distributions in the leading jet transverse energy $E_{T}^{\rm jet1}$, the
average jet pseudorapidity $\langle\eta\rangle$ and the diffractive mass
$M_{X}$. We not, again, significant $K$-factors between LO and M  C @N  LO
predictions ranging up to values of about 5 in the forward regime, testaments
to the lower energy scales of the processes we study here. In addition to
these observables we also display in the lower row of the same figure event
shape distributions, in particular transverse thrust $T_{\perp}$, transverse
thrust minor $T_{\perp m}$ and transverse sphericity $S_{\perp}$. These
observables are defined by

$T_{\perp}=\mathrm{max}_{\vec{n}_{\mathrm{T}}}\frac{\sum_{i}\left|\vec{p}_{\mathrm{T},i}\cdot\vec{n}_{\mathrm{T}}\right|}{\sum_{i}\vec{p}_{\mathrm{T},i}}\;,\;\;\;T_{\perp,m}=\mathrm{max}_{\vec{n}_{\mathrm{T}}}\frac{\sum_{i}\left|\vec{p}_{\mathrm{T},i}\times\vec{n}_{\mathrm{T}}\right|}{\sum_{i}\vec{p}_{\mathrm{T},i}}\;,\;\;\;S_{\perp}=\frac{2\lambda_{2}}{\lambda_{1}+\lambda_{2}}\;,$
(6.3)

and $n_{\mathrm{T}}$ is the transverse-thrust axis that maximises the
$T_{\perp}$ and $\lambda_{1,2}$ are the eigenvalues of the transverse
linearised sphericity tensor $\mathbf{S_{\alpha\beta}}$

$\mathbf{S_{\alpha\beta}}=\frac{1}{\sum_{i}\left|\vec{p}_{\mathrm{T},i}\right|}\sum_{i}\frac{1}{\left|\vec{p}_{\mathrm{T},i}\right|}\begin{pmatrix}p_{i,x}^{2}&p_{i,x}p_{i,y}\\\
p_{i,y}p_{i,x}&p_{i,y}^{2}\end{pmatrix}.$ (6.4)

While the events are broadly dominated by dijet kinematics, the event shape
distributions indicate a non-negligible contribution from three-jet events.
Clearly, the data taken at the E  IC will complement the H  ERA data in the
high-$x$ region [40] in updated fits to the DPDFs.

###  6.2 Diffractive photoproduction

Measuring diffractive photoproduction at the E  IC will shed new light on
factorisation breaking, and expand on some of the findings highlighted in Sec.
5.3. To obtain predictions for this process, we calculated an average over the
suppression factors for direct and resolved photon processes from Tab. 1:
$R_{\mathrm{res}}^{\mathrm{(EIC)}}=0.8\pm 0.2$ and
$R_{\mathrm{dir}}^{\mathrm{(EIC)}}=0.4\pm 0.1$ for the resolved and direct
component, respectively. Additionally to the scale uncertainty, we obtained
obtained envelopes for the uncertainties from the suppression factors. We used
the same cuts and settings as for the predictions for diffractive DIS, see the
previous subsection 6.1, restricting the virtuality to
$Q^{2}<4\,\mathrm{GeV}^{2}$ and defined different momentum fractions as

$x_{I\\!\\!P}=1-\frac{E_{p}^{\prime}}{E_{p}}\;,\;\;\;z_{I\\!\\!P}=\frac{\sum_{j}E_{T}^{(j)}\mathrm{e}^{\eta^{(j)}}}{2x_{I\\!\\!P}E_{p}}\;,\;\;\;y=1-\frac{E_{e}^{\prime}}{E_{e}}\;,\;\;\;x_{\gamma}=\frac{\sum_{j}E_{T}^{(j)}\mathrm{e}^{-\eta^{(j)}}}{2yE_{e}}$
(6.5)

In Fig. 9, we display the momentum fractions, inelasticity and jet observables
as well as event shapes in diffractive photoproduction. Uncertainties due to
the fitting to factorisation breaking and due to scale choices are of
comparable size, and the $K$ factors between the M  C @N  LO and LO accuracy
again reach values of about 5. The event shapes exhibit an anticipated effect,
namely that the limited phase space leads to lower multiplicities and an even
stronger dominance of dijet events compared to the DDIS events in the previous
chapter, manifest in the sharper peak at low values of $1-T_{\perp}$,
$T_{\perp m}$, and $S_{\perp}$.

|  |
---|---|---
|  |
|  |
Figure 9: Predictions at LO and M  C @N  LO accuracy of S  HERPA of momentum
ratio $x_{I\\!\\!P}$, $z_{I\\!\\!P}$ and $x_{\gamma}$ (upper row, left to
right), of inelasticity $y$, leading jet transverse momentum $E_{T}^{\rm
jet1}$ and average jet pseudorapidity $\langle\eta\rangle$ (middle row, left
to right), and of event shapes transverse thrust $T_{\perp}$, transverse
thrust minor $T_{\perp m}$ and transverse sphericity $S_{\perp}$ (bottom row,
left to right) in diffractive photoproduction at the E  IC .

##  7 Summary

Diffraction played an important role at H  ERA , making up 10% of the total
cross section. We simulated (hard) diffractive jet production at Next-to-
Leading-Order in S  HERPA for electron-proton collision in both the DIS and
photoproduction regimes, matching our calculation to the parton shower. This
results in the first fully differential hadron-level calculation of hard
diffraction at M  C @N  LO accuracy and provides an important accuracy
standard for future E  IC predictions.

Validating our simulation against data from the H1 and Z  EUS collaborations
we see excellent agreement for Diffractive DIS setups. In the Diffractive
Photoproduction regime, we observe significant discrepancies compared to the
data, confirming the findings in previous Fixed Order calculations. We review
and discuss different ansätze to explain the differences in view of our
hadron-level simulations and conclude that none of the proposed solutions
suffices to conclusively clarify the mechanism of factorisation breaking in
this regime. We argue that the factorisation breaking happens in fact in both
the direct and resolved component in diffractive photoproduction. A coherent
mechanism would need to take into account the suppression of the real
correction to the ”direct” component at NLO. We quantify the suppression in
this component by fitting the two components to the data.

Lastly, we presented predictions for Diffractive DIS and Diffractive
Photoproduction at the E  IC , where for the latter we estimated the
suppression due to factorisation breaking by means of the fits to the H1 and Z
EUS data. Data taking at the E  IC will provide more insights into the exact
mechanism of factorisation breaking and a thorough comparison to theory
predictions will determine the exact nature of the corresponding mechanism.
With this understanding it should be possible to apply this to hadron
colliders, hence allowing to study this phenomenon on the basis of the vast
data taken at the L  HC .

## 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 STFC
under grant agreement ST/P006744/1.

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