# Diffractive longitudinal structure function at the Electron Ion Collider

Néstor Armesto  Instituto Galego de Física de Altas Enerxías IGFAE,
Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia-
Spain Paul R. Newman  School of Physics and Astronomy, University of
Birmingham, UK  Wojciech Słomiński  Institute of Theoretical Physics,
Jagiellonian University, Kraków, Poland  Anna M. Staśto  Department of
Physics, Penn State University, University Park, PA 16802, USA

###### Abstract

Possibilities for the measurement of the longitudinal structure function in
diffraction $F_{\mathrm{L}}^{\mathrm{D}}$ at the future US Electron Ion
Collider are investigated. The sensitivity to $F_{\mathrm{L}}^{\mathrm{D}}$
arises from the variation of the reduced diffractive cross section with
centre-of-mass energy. Simulations are performed with various sets of beam
energy combinations and for different assumptions on the precision of the
diffractive cross section measurements. Scenarios compatible with current EIC
performance expectations lead to an unprecedented precision on
$F_{\mathrm{L}}^{\mathrm{D}}$ at the 5-10% level in the best measured regions.
While scenarios with data at a larger number of centre-of-mass energies allow
the extraction of $F_{\mathrm{L}}^{\mathrm{D}}$ in the widest kinematic domain
and with the smallest uncertainties, even the more conservative assumptions
lead to precise measurements. The ratio $R^{\mathrm{D}}$ of photoabsorption
cross sections for longitudinally to transversely polarised photons can also
be obtained with high precision using a separate extraction method.

##  1 Introduction

Diffraction in deep inelastic scattering (DIS) was studied extensively at the
Hadron-Elektron-Ringanlage (HERA) collider at DESY. The measurements showed
that it gives a large contribution, of about $\sim 10$%, to the total cross
section [1, 2], see the review [3] and refs. therein. Diffractive events are
characterised by the measurement of either a proton (in the case of coherent
diffraction) or a state with the proton quantum numbers (incoherent
diffraction). Experimentally, diffractive events are defined either by the
identification of a proton in dedicated far-forward detectors housed in Roman
pot insertions to the beampipe (see for example [4, 5, 6, 7]), or by a lack of
hadronic activity in a sizeable kinematic region adjacent to the outgoing
proton beam, i.e. the presence of a large rapidity gap (LRG) (see for example
[7, 8, 9]). Based on the experimental results from HERA, it was possible to
analyse the partonic structure of the $t$-channel colourless exchange in such
events. A successful description of the diffractive structure functions was
achieved at high $Q^{2}$ based on the collinear factorization and Dokshitzer-
Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution of the corresponding
diffractive parton densities (DPDF) [4, 10]. The latter quantities parametrize
the partonic content of the colourless exchange in the diffractive events.

Diffraction has been a central subject in investigations of strong
interactions for many decades [11, 12]. As a pure quantum phenomenon, some
properties derive from basic requirements like unitarity. On the other hand,
the microscopic dynamics by which a composite object, a hadron or nucleus, is
able to undergo a high energy collision and remain colourless with its
constituents bound, is closely related to the confinement mechanism [13].
Besides, diffraction is very sensitive to the high energy behaviour of Quantum
Chromodynamics (QCD), specifically to the low-$x$ distribution of partons and
its energy evolution [14]. Therefore, it is a promising observable for
observing deviations from linear evolution like higher twist effects or parton
saturation. Diffractive $ep$ scattering is also related to nuclear shadowing
on deuterons [15] and tests the validity of perturbative factorisation [16,
17, 18] – known to be violated in diffractive dijet photoproduction [19].
Furthermore, due to the simplicity of the final state, diffractive events may
offer new opportunities for the detection of rare phenomena, see [20] and
refs. therein.

Among the various diffractive observables that can be measured in DIS, a very
interesting one, yet experimentally challenging, is the diffractive
longitudinal structure function $F_{\mathrm{L}}^{\mathrm{D}}$. Given by the
coupling of virtual photons with longitudinal polarisation to the hadron that
undergoes the diffractive interaction, it is – as in the case of inclusive
events – a more sensitive probe of the gluon content of the target and of the
QCD evolution than the diffractive structure function
$F_{\mathrm{2}}^{\mathrm{D}}$, which is only sensitive to the gluon content
via evolution. $F_{\mathrm{L}}^{\mathrm{D}}$ also probes contributions from
higher twists, similarly to the inclusive case, see [21]. Thus, its
measurement gives the opportunity of constraining the gluon contribution to
diffraction in DIS and the dynamics beyond linear evolution driving this kind
of interaction.

$F_{\mathrm{L}}^{\mathrm{D}}$ is a very poorly known quantity, with the only
existing experimental study done by the H1 Collaboration [22] at HERA. On top
of the intrinsic difficulty of disentangling diffractive from inclusive
events, measuring the longitudinal structure function requires variation of
the centre-of-mass energy of the lepton-hadron collisions. At HERA the former
was determined by the LRG method. For the latter, four energies of the proton
beam were employed, additionally selecting those events with high inelasticity
of the electron (see below) where the contribution of the longitudinal
structure function to the total reduced cross section is largest. This region
is difficult experimentally, since it is associated with low electron energies
and with the hadronic final state being produced in the same backward
pseudorapidity region as the scattered electron.

Planned DIS colliders like the Electron Ion Collider (EIC) [23, 24], the Large
Hadron-electron Collider (LHeC) [25, 26] or the Future-Circular-Collider in
its electron-hadron option (FCC-eh) [27, 28] will benefit from larger
integrated luminosities exceeding those at HERA by factors ${\cal O}(1000)$
and new detector techniques providing enhanced possibilities for separating
diffractive from non-diffractive events.

In previous works we analysed the potential of the measurements of the
diffractive reduced cross section at the LHeC and FCC-he and at the EIC [29,
30] as well as the possibility for constraining the diffractive parton
distribution functions. In this work we focus on the possibilities for the
determination of $F_{\mathrm{L}}^{\mathrm{D}}$ in coherent diffraction on
protons at the EIC, where simulations of the forward detectors, including
their effect on particle reconstruction, are available [24].

The manuscript is organised as follows. In Sec. 2 we present the general
expressions and kinematics that will be used in our analysis and discuss the
experimental aspects of proton tagging at the EIC. In Sec. 3 we discuss the
generation of simulated EIC data (‘pseudodata’) and the method of extraction
of $F_{\mathrm{L}}^{\mathrm{D}}$ as well as the choices of beam energies.
Results are presented in Sec. 4, first for the reduced cross section
$\sigma_{\mathrm{red}}^{\mathrm{D(3)}}$ and then for
$F_{\mathrm{L}}^{\mathrm{D(3)}}$. We then proceed to discuss the influence of
the systematic error assumed in the pseudodata and the assumptions on the beam
configurations. Results for
$R^{\mathrm{D(3)}}=F_{\mathrm{L}}^{\mathrm{D(3)}}/F_{\mathrm{T}}^{\mathrm{D(3)}}$
are also presented. We end with conclusions in Sec. 5.

##  2 Definitions and kinematics

###  2.1 Diffractive variables and definitions

In this work we focus on neutral current diffractive deep inelastic scattering
(DDIS) in the one photon exchange approximation, neglecting radiative
corrections whose contribution can be corrected. For an electron or positron
with four momentum $l$ and a proton with four-momentum $P$, the diagram is
shown in Fig. 1. A characteristic feature of the diffractive process, as
illustrated in Fig. 1, is the presence of the rapidity gap between the final
proton (or its dissociated state) $Y$ and the system $X$. It is mediated by
the colourless object, indicated by $P/R$, to which we refer generally as
‘diffractive exchange’.

Figure 1: Diagram showing the neutral current diffractive DIS process and the
relevant kinematic variables in the one photon exchange approximation.

In DDIS several variables can be defined in terms of the four-momenta
indicated in Fig. 1 and the usual Mandelstam variables:

$\displaystyle Q^{2}$ $\displaystyle=$ $\displaystyle-q^{2}\ ,$ $\displaystyle
y$ $\displaystyle=$ $\displaystyle\frac{P\cdot q}{P\cdot\ell}\ ,$
$\displaystyle x$ $\displaystyle=$ $\displaystyle\frac{Q^{2}}{2P\cdot
q}=\frac{Q^{2}}{ys}\ ,$ $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{Q^{2}}{2\,(P-P^{\prime})\cdot q}\ ,$ $\displaystyle\xi$
$\displaystyle=$ $\displaystyle\frac{x}{\beta}\ ,$ $\displaystyle t$
$\displaystyle=$ $\displaystyle(P^{\prime}-P)^{2}\ .$ (1)

Besides the standard DIS variables $s,Q^{2},y,x$, in DDIS some additional
variables appear: $t$ is the squared four-momentum transfer at the proton
vertex, $\xi$ (alternatively denoted by $x_{I\\!\\!P}$) can be interpreted as
the momentum fraction of the ‘diffractive exchange’ with respect to the beam
hadron, and $\beta$ is the momentum fraction of the parton (probed by the
virtual photon) with respect to the diffractive exchange. In Fig. 2 we show
the kinematic coverage in $x$ and $Q^{2}$ of the EIC for three selected
energies compared to that of HERA. Since HERA was operating at higher centre-
of-mass energy than the EIC, it could reach lower values of $x$. The EIC can
operate at several energy combinations, which will result in a wide coverage
of $x$ also towards moderate and large $x$, and which is essential for
$F_{\mathrm{L}}^{\mathrm{D}}$ measurement. In Fig. 2 only three beam energy
combinations are shown, a subset of a wider range of combinations possible at
the EIC, see the discussion below.

Figure 2: Kinematic $x-Q^{2}$ plane showing different choices of beam energies
at the EIC and the region covered by HERA experiments. Note that
$\eta_{e}>-3.5$ corresponds to an angular acceptance of 176.5 degrees for the
electron.

Only four variables, usually chosen to be $\beta,\xi,Q^{2},t$, are needed to
characterise the reduced cross section, related to the measured cross section
by

$\frac{d^{4}\sigma^{\mathrm{D}}}{d\xi d\beta dQ^{2}dt}=\frac{2\pi\alpha_{\rm
em}^{2}}{\beta Q^{4}}\,Y_{+}\,\sigma_{\mathrm{red}}^{\mathrm{D}(4)}\,,$ (2)

where $Y_{+}=1+(1-y)^{2}$. It is also customary to perform an integration over
$t$, defining

$\frac{d^{3}\sigma^{\mathrm{D}}}{d\xi d\beta dQ^{2}}=\frac{2\pi\alpha_{\rm
em}^{2}}{\beta Q^{4}}\,Y_{+}\,\sigma_{\mathrm{red}}^{\mathrm{D}(3)}\,.$ (3)

In the one photon exchange approximation, the reduced cross sections can be
expressed in terms of two diffractive structure functions $F_{2}^{\mathrm{D}}$
and $F_{\mathrm{L}}^{\mathrm{D}}$:

$\sigma_{\mathrm{red}}^{\mathrm{D}(4)}=F_{2}^{\mathrm{D}(4)}(\beta,\xi,Q^{2},t)-\frac{y^{2}}{Y_{+}}F_{\mathrm{L}}^{\mathrm{D}(4)}(\beta,\xi,Q^{2},t)\;,$
(4)
$\sigma_{\mathrm{red}}^{\mathrm{D}(3)}=F_{2}^{\mathrm{D}(3)}(\beta,\xi,Q^{2})-\frac{y^{2}}{Y_{+}}F_{\mathrm{L}}^{\mathrm{D}(3)}(\beta,\xi,Q^{2})\;,$
(5)

where $F_{\mathrm{2,L}}^{\mathrm{D}(4)}$ have dimension $\mathrm{GeV}^{-2}$
and $F_{\mathrm{2,L}}^{\mathrm{D}(3)}$ are dimensionless.

The dependence of the reduced cross sections
$\sigma_{\mathrm{red}}^{\mathrm{D}(4,3)}$ on the centre-of-mass energy comes
via the inelasticity $y=\frac{Q^{2}}{\xi\beta s}$. Due to the $Y_{+}$ factor,
$\sigma_{\mathrm{red}}^{\mathrm{D}(4,3)}\simeq F_{2}^{\mathrm{D}(4,3)}$ when
$y$ is not too close to unity.

Both reduced cross sections $\sigma_{\mathrm{red}}^{\mathrm{D}(3)}$ and
$\sigma_{\mathrm{red}}^{\mathrm{D}(4)}$ have been measured at HERA [1, 2, 31,
4, 32, 33, 10, 5, 34]. These data have been used for perturbative QCD analyses
based on collinear factorization [16, 17, 18], where the diffractive cross
section reads

$d\sigma^{ep\rightarrow
eXY}(\beta,\xi,Q^{2},t)\;=\;\sum_{i}\int_{\beta}^{1}dz\
d\hat{\sigma}^{ei}\left(\frac{\beta}{z},Q^{2}\right)\,f_{i}^{\rm
D}(z,\xi,Q^{2},t)\;,$ (6)

up to terms of order ${\cal O}(1/Q^{2})$. Here, the sum is performed over all
parton species (gluon and all quark flavours). The hard scattering partonic
cross section $d\hat{\sigma}^{ei}$ can be computed perturbatively in QCD and
is the same as in the inclusive deep inelastic scattering case. The long
distance part $f_{i}^{\rm D}$ corresponds to the DPDFs, which can be
interpreted as conditional probabilities for partons in the proton, provided
the proton is scattered into the final state system $Y$ with four-momentum
$P^{\prime}$. They are non-perturbative objects to be extracted from data, but
their evolution through the DGLAP evolution equations [35, 36, 37, 38] can be
computed perturbatively, similarly to the inclusive case. The analogous
formula for the $t$-integrated structure functions reads

$F_{2/\mathrm{L}}^{\mathrm{D}(3)}(\beta,\xi,Q^{2})=\sum_{i}\int_{\beta}^{1}\frac{dz}{z}\,C_{2/\mathrm{L},i}\Big{(}\frac{\beta}{z}\Big{)}\,f_{i}^{\mathrm{D}(3)}(z,\xi,Q^{2})\;,$
(7)

where the coefficient functions $C_{2/\mathrm{L},i}$ are the same as in
inclusive DIS and the DPDFs $f_{i}^{\mathrm{D}(3)}(z,\xi,Q^{2})$ have been
determined from comparisons to HERA data [1, 2, 31, 4, 32, 33, 10, 5, 34].

###  2.2 Experimental Considerations

As can be inferred from Eq. 5, sensitivity to $F_{\mathrm{L}}^{\mathrm{D}}$ is
strongest as $y\rightarrow 1$. Experimentally, this is a region in which
backgrounds are hard to control, since it corresponds to the lowest scattered
electron energies and also to cases where hadronic final state particles are
produced in the same (backward) pseudorapidity region as the scattered
electron. Extractions of the inclusive and diffractive longitudinal structure
functions therefore place strong challenges on the performance of
electromagnetic calorimetry, tracking and particle identification in the
backward region of the detector. H1 achieved measurements down to electron
energies of around 3 GeV. At the EIC, where charged pion rejection factors
relative to electrons of the order of $10^{-4}$ are targeted, the aim is to go
substantially lower, even into the sub-GeV range. Here, we apply an upper $y$
cut of 0.96, which is typical of current EIC studies. The targeted $\eta$
range of the EIC experiments, with calorimeter and tracking coverage to at
least as far backwards as $\eta=-3.5$, provides full coverage for scattered
electrons with $Q^{2}>1$ GeV2.

In the H1 measurement [22], the use of the LRG method of selecting diffractive
events led to a normalisation uncertainty of 7%. This uncertainty can in
principle be eliminated through the use of beamline proton tagging based on
instrumentation housed in Roman pot insertions to the beampipe. In contrast to
the situation at previous colliders, beamline instrumentation has been a
fundamental consideration from the outset at the EIC. It is essential to the
success of the diffractive programme, as illustrated in Fig. 3, where the
rapidity ranges covered by the undecayed final state system $X$ and proton are
shown as a function of $\xi$ for four combinations of electron and proton beam
energies. The bands correspond to ranges $\beta\in[0.1,0.9]$ and $p_{\perp}$
of the final state proton below $4\,\mathrm{GeV}$. The decay of the $X$ system
into a multi-particle hadronic system extends its extent forwards in
pseudorapidity in a manner that scales logarithmically with $\xi$, reducing
the size the rapidity gap. For comparison, gaps smaller than about 3
pseudorapidity units could not be used reliably at HERA due to the poorly
modelled contributions from gaps produced from hadronisation fluctuations in
non-diffractive processes. Whilst fairly large rapidity gaps exist at the
lowest $\xi$ values and highest EIC centre-of-mass energies, it is clear that
throughout most of the EIC phase space and for most of the expected beam
configurations, LRG methods will yield poor performance.

Figure 3: Ranges in the rapidity of the scattered proton and the undecayed
dissociative system $X$ as a function of $\xi$ for three different beam energy
combinations at the EIC and for HERA. The bands correspond to all cases where
the proton transverse momentum is lower than 4 GeV, $0.005<y<0.96$ and
$0.1<\beta<0.9$.

The most recent studies of the physics and detector requirements at the EIC
envisage multiple beamline proton spectrometers, allowing a full determination
of the outgoing proton kinematics with good measurements of both $\xi$ and
$t$. In Fig. 4 we show the kinematic coverage for the forward proton
considered in [24]. In contrast to the LRG method, the multiple planned
detector stations with a combined angular acceptance $0.5-20$ mrad lead to a
wide potential measurement range in $\xi$ and $t$ for all beam energies.
Measurements up to $\xi$ values as large as 0.5 may be possible, well beyond
the range in which diffractive ($t$-channel exchange) processes are expected
to be the dominant mechanism for leading proton production. We therefore
assume here that events will be selected based on the scattered proton and
that the accessible phase space in $\xi$ is not strongly limited
experimentally.

Figure 4: Final proton tagging. $x_{\mathrm{L}},t$ range of the proton tagged
by the EIC detector for three proton energies, 275 GeV, 100 GeV and 41 GeV.
The brown strip marks a small ($\sim$ 1 mrad) region not covered by the
current detector design.

The standard Rosenbluth method of extracting longitudinal structure functions
involves fits to data at the same $\xi$, $\beta$ and $Q^{2}$ values, but
different centre-of-mass energies. Systematic uncertainties that are not
correlated between different beam energies therefore tend to propagate into
larger uncertainties on $F_{\mathrm{L}}^{\mathrm{D}}$ than those that are
positively correlated between beam energies. Since statistical uncertainties
are completely uncorrelated between different beam energies,
$F_{\mathrm{L}}^{\mathrm{D}}$ measurements are also particularly sensitive to
the available sample sizes.

Due to the relatively small integrated luminosities in the reduced proton beam
energy runs, the HERA measurement of $F_{\mathrm{L}}^{\mathrm{D(3)}}$ [22] was
limited by statistical uncertainties throughout most of the phase space. Since
the integrated luminosity expected at the EIC is around three orders of
magnitude larger than that at HERA, the sample sizes will be much larger
(integrated luminosities of 10 fb-1 per beam energy are assumed here) and
statistical uncertainties are expected to be unimportant.

A detailed systematic uncertainty analysis was carried out in the HERA
measurement, with the conclusion that no single source dominated, but also
giving some baseline from which to extrapolate to the likely precision
achievable on the cross sections at a future collider such as the EIC. The
best precision achieved in diffractive reduced cross section measurements at
HERA was at the 4% level, with uncorrelated sources contributing as little as
2%, arising primarily from track-cluster linking and vertex finding
efficiencies. Recent and ongoing studies of proposed EIC instrumentation
solutions [24] already indicate that uncertainties of this kind will be
dramatically reduced at the EIC. We therefore consider scenarios in which the
uncertainties that are uncorrelated between beam energies are either 1% or 2%.
With sources related to the LRG method eliminated, correlated systematic
uncertainties are also expected to be reduced significantly. The alignment and
calibration procedures required in Roman pot methods inherently lead to
correlated systematics, but using methods developed at HERA [39, 40, 41],
coupled with the substantial further evolution of proton-tagging techniques at
the LHC [42, 43, 44, 45] and future EIC-specific work, we estimate that these
are controllable to the sub-2% level, and will thus have a negligible effect
on the $F_{\mathrm{L}}^{\mathrm{D}}$ extraction compared with the uncorrelated
sources.

##  3 Method

###  3.1 Pseudodata generation

We shall first describe the pseudodata generation for our simulations. The
momentum transfer $t$ is integrated over in this analysis. Let us rewrite Eq.
(5) as

$\sigma_{\mathrm{red}}^{\mathrm{D}(3)}=F_{2}^{\mathrm{D}(3)}(\beta,\xi,Q^{2})-Y_{\mathrm{L}}\,F_{\mathrm{L}}^{\mathrm{D}(3)}(\beta,\xi,Q^{2})\;,$
(8)

where

$Y_{\mathrm{L}}=\frac{y^{2}}{Y_{+}}=\frac{y^{2}}{1+(1-y)^{2}}\;.$ (9)

As mentioned previously, the extraction of the longitudinal diffractive
structure function relies on the possibility of disentangling it from
$F_{2}^{D}$, as is evident in the formula above for the reduced cross section.
This is possible if, for fixed $(\beta,Q^{2},\xi)$, one can vary
$Y_{\mathrm{L}}$, and hence $y$, in a sufficiently wide range. Given that
$y=Q^{2}/(s\beta\xi)$ it is therefore necessary to perform measurements of the
reduced cross section using different centre-of-mass energies. The EIC is
uniquely positioned to perform such a measurement, thanks to its design, which
allows for a wide range of different beam energies.

We have considered several beam energies for both the electrons and the
protons, within the range expected for the EIC:

$\displaystyle E_{e}$ $\displaystyle=5,10,18\,\mathrm{GeV}\,,$ $\displaystyle
E_{p}$ $\displaystyle=41,100,120,165,180,275\,\mathrm{GeV}\,.$ (10)

These beam energies combine to give 17 distinct centre-of-mass energies (there
is a degeneracy in this choice since two combinations $10\times 180$ and
$18\times 100$ lead to the same centre-of-mass energy, $85\,\mathrm{GeV}$).
The centre-of-mass energies corresponding to all combinations are given in
Table 1. In order to test the sensitivity of $F_{\mathrm{L}}^{\mathrm{D(3)}}$
to the available beam energies, we consider three different subsets in the
analysis :

  * S-17)

17 values — all combinations from Table 1 except for $10\times 180$.

  * S-9)

9 values — marked bold in Table 1,

  * S-5)

5 values — marked bold against a green background in Table 1.

Set S-17 contains the widest range of possibilities. S-5 is the set of
combinations that has often been assumed in EIC studies to date [24].
Additionally, we consider an intermediate set S-9, which restricts the list to
three proton and three electron beam energies, whilst maintaining the same
overall kinematic range as S-17.

|  | $E_{p}\,[\mathrm{GeV}]$
---|---|---
|  | 41 | 100 | 120 | 165 | 180 | 275
$E_{e}\,[\mathrm{GeV}]\;$ |  5 | 29 | 45 | 49 | 57 | 60 | 74
10 | 40 | 63 | 69 | 81 | 85 | 105
18 | 54 | 85 | 93 | 109 | 114 | 141
Table 1: Centre-of-mass energies (in GeV) for various combinations of beam
energies.

The pseudodata for the reduced diffractive cross section at the EIC were
generated using Eqs. (5) and (7). The diffractive parton distribution used for
the evaluation of the cross section is the ZEUS-SJ set [46]. This fit uses
inclusive diffractive data together with diffractive DIS dijet data, which are
added to improve the constraints on the diffractive gluon distribution.

The details of the ZEUS-SJ parametrization closely follow those of [8] and can
be found in [46]. Below we summarize a few important features. The diffractive
parton densities are parametrized using a two-component form:

$f_{i}^{\mathrm{D}(4)}(z,\xi,Q^{2},t)=f^{p}_{{I\\!\\!P}}(\xi,t)\,f_{i}^{{I\\!\\!P}}(z,Q^{2})+f^{p}_{{I\\!\\!R}}(\xi,t)\,f_{i}^{{I\\!\\!R}}(z,Q^{2})\;.$
(11)

The first term in Eq. (11) is interpreted as the exchange of a ‘Pomeron’ and
the second is a ‘Reggeon’ component. They dominate in different $\xi$ regions:
the ‘Pomeron’ is dominant for $\xi\leq 0.01$. The ‘Reggeon’ starts to be
important for $\xi>0.01$ and becomes dominant for $x>0.1$. For both terms,
proton vertex factorization is assumed, which means that the diffractive
parton density factorizes into a parton distribution in a diffractive exchange
$f_{i}^{{I\\!\\!P},{I\\!\\!R}}$ and a flux factor
$f^{p}_{{I\\!\\!P},{I\\!\\!R}}$. The parton distribution in the ‘Pomeron’ and
‘Reggeon’ $f_{i}^{{I\\!\\!P},{I\\!\\!R}}(\beta,Q^{2})$ only depend on the
longitudinal momentum fraction $\beta$ of the parton with respect to the
Pomeron/Reggeon and the photon virtuality $Q^{2}$. The flux factors
$f^{p}_{{I\\!\\!P},{I\\!\\!R}}(\xi,t)$, on the other hand, only depend on
$\xi$, which is related to the size of the rapidity gap, and the momentum
transfer at the proton vertex $t$. They represent the probability that a
Pomeron/Reggeon with given values of $\xi,t$ couples to the proton. The flux
factors are parametrized using a form motivated by Regge theory:

$f^{p}_{{I\\!\\!P},{I\\!\\!R}}(\xi,t)=A_{{I\\!\\!P},{I\\!\\!R}}\frac{e^{B_{{I\\!\\!P},{I\\!\\!R}}t}}{\xi^{2\alpha_{{I\\!\\!P},{I\\!\\!R}}(t)-1}}\,,$
(12)

with a linear trajectory
${\alpha_{{I\\!\\!P},{I\\!\\!R}}(t)=\alpha_{{I\\!\\!P},{I\\!\\!R}}(0)+\alpha_{{I\\!\\!P},{I\\!\\!R}}^{\prime}\,t}$.

The diffractive parton distributions are evolved using the NLO DGLAP
equations. For the case of the Pomeron at the initial scale
$\mu_{0}^{2}=1.8\,\mathrm{GeV}^{2}$ they are parametrized as

$zf_{i}^{I\\!\\!P}(z,\mu_{0}^{2})=A_{i}z^{B_{i}}(1-z)^{C_{i}}\;,$ (13)

where $i$ is a gluon or a light quark. In the diffractive parametrizations all
the light quarks (anti-quarks) are assumed to be equal. For the treatment of
heavy flavours, a variable flavour number scheme (VFNS) is adopted, where the
charm and bottom quark DPDFs are generated radiatively via DGLAP evolution.
There is no intrinsic heavy quark distribution present. The structure
functions are calculated in a General-Mass Variable Flavour Number scheme (GM-
VFNS) [47, 48] which ensures a smooth transition of $F_{\mathrm{2,L}}$ across
the flavour thresholds by including $\mathcal{O}(m_{h}^{2}/Q^{2})$
corrections.

The model that we use to compute the diffractive cross section is state of the
art. Still, it contains substantial uncertainties in the Reggeon contribution,
which was poorly constrained by HERA data. Whilst these uncertainties affect
the predicted cross sections and structure functions, they do not to first
order impact our assessment of the feasibility of measuring the longitudinal
diffractive structure function, which is our primary objective here.

The parton distributions for the Reggeon component are taken from a
parametrization which was obtained from fits to the pion structure function
[49, 50]. HERA data required the addition of the Reggeon contribution, but
could not constrain it. The high $\xi$ region where it dominates is accessible
in the EIC kinematics, and the possibilities for disentangling the Reggeon
contribution (or any contribution other than the Pomeron) were discussed in
[24]. This is an aspect demanding a dedicated study that we leave for the
future.

The pseudodata were generated as the extrapolation of the fit to HERA [46],
amended with a random Gaussian smearing with standard deviation corresponding
to the relative error $\delta$. The total error was assumed to be composed of
systematic and statistical components and computed as

$\delta=\sqrt{\delta^{2}_{\rm sys}+\delta^{2}_{\rm stat}}\,.$ (14)

The statistical error was evaluated assuming an integrated luminosity
$10\,{\rm fb}^{-1}$, see [24]. For the binning adopted in this study, the
statistical uncertainties have a very small effect on the extraction of the
longitudinal structure function. As discussed in Sec. 2.2, correlated
systematic uncertainties on the reduced cross section are also expected to be
relatively unimportant in the $F_{\mathrm{L}}^{\mathrm{D}}$ extraction and are
thus neglected here. For the uncorrelated systematic error we have considered
two scenarios, with $2\%$ and $1\%$ ascribed to each data point.

The cuts imposed for the data selection are:

  * •

$Q^{2}\geq 3\;\mathrm{GeV}^{2}$ : both the H1 [32] and the ZEUS [46] analyses
of the inclusive diffractive data observed that the quality of the DGLAP-based
fit deteriorates in the low $Q^{2}$ region, possibly because it is sensitive
to higher twist contributions. This cut is imposed to limit the sensitivity to
such effects. The EIC kinematics and expected scattered electron coverage lead
to full acceptance for all $x$ in the chosen $Q^{2}$ region at all beam
energies, see Fig. 2.

  * •

$y$ between 0.005 and 0.96, which is the expected coverage of a typical
measurement whilst maintaining well-controlled systematics.

We adopt a uniform logarithmic binning with four bins per decade for each of
$\xi,\beta$ and $Q^{2}$ 111Resolution studies, performed using the Rapgap
Monte Carlo generator ([51], see also https://rapgap.hepforge.org/) with the
smearing routines available for the detector model in [24], show that this
binning is perfectly achievable.. This results in the numbers of
$\sigma_{\mathrm{red}}$ data points for each $(\xi,\beta,Q^{2})$ bin as shown
in Fig. 5. Taking into account that we require four points for the linear fit
(see Sec. 3.2) to extract $F_{\mathrm{L}}^{\mathrm{D(3)}}$, the analysis
therefore proceeds with a total of 364 $F_{\mathrm{L}}^{\mathrm{D(3)}}$ values
for set S-17, 285 $F_{\mathrm{L}}^{\mathrm{D(3)}}$ values for set S-9 and 160
$F_{\mathrm{L}}^{\mathrm{D(3)}}$ values for set S-5.

Figure 5: Count of different beam energy combinations from among set S-17 that
lead to measurable $\sigma_{\mathrm{red}}^{\mathrm{D}(3)}$ data points for
each $(\xi,\beta,Q^{2})$ bin. Only cases with a number of counts $\geq 4$ are
considered for the extraction of $F_{\mathrm{L}}^{\mathrm{D(3)}}$.

###  3.2 Extraction of the diffractive longitudinal structure function

The extraction of the diffractive longitudinal structure function
$F_{\mathrm{L}}^{\mathrm{D(3)}}$ is performed using the same method as in the
H1 analysis [22]. This method was adapted from the measurements of the
inclusive longitudinal structure function $F_{\mathrm{L}}$, see [52, 53, 54].
The reduced cross section is a linear function of $Y_{\mathrm{L}}$, see Eq.
(8). The structure function $F_{\mathrm{L}}^{\mathrm{D(3)}}$ can thus be found
by performing a linear fit, and extracting the slope of
$\sigma_{\mathrm{red}}^{\mathrm{D}(3)}$ as a function of $Y_{\mathrm{L}}$.
This is done for every set of values in $Q^{2}$, $\xi$ and $\beta$ for which
there are four or more available $\sigma_{\mathrm{red}}$ values (in the H1
analysis [22] only three points were required). In Figs. 6 and 7 examples of
fits in 4 bins of $(\xi,\beta,Q^{2})$ are shown, which are discussed in detail
in Subsec. 4.1.

Figure 6:  Examples of the
$\sigma_{\mathrm{red}}^{\mathrm{D}(3)}=F_{2}^{\mathrm{D(3)}}-Y_{\mathrm{L}}F_{\mathrm{L}}^{\mathrm{D(3)}}$
fit. Comparison between set S-17 (red line and points) and S-9 (blue line and
points).

Figure 7: Examples of the
$\sigma_{\mathrm{red}}^{\mathrm{D}(3)}=F_{2}^{\mathrm{D(3)}}-Y_{\mathrm{L}}F_{\mathrm{L}}^{\mathrm{D(3)}}$
fit. Comparison between set S-17 (red line and points) and S-5 (blue line and
points).

##  4 Results

In this Section we show the simulated results for
$F_{\mathrm{L}}^{\mathrm{D(3)}}$ and analyse the influence of choices of beam
energies, systematic errors and numbers of measurements. We also extract
results for $R=F_{\mathrm{L}}^{\mathrm{D(3)}}/F_{\mathrm{T}}^{\mathrm{D(3)}}$.
In obtaining uncertainties on $F_{\mathrm{L}}^{\mathrm{D(3)}}$ from fits to
Eq. 8, we take 68% confidence limits (CL). This practically corresponds to
$1\sigma$ errors for a number of degrees of freedom $\mathrm{NDF}\gtrsim 10$.
However, many $(\xi,Q^{2},\beta)$ bins where $F_{\mathrm{L}}^{\mathrm{D(3)}}$
is fitted contain even as few data points as 4, which results in 68% CL error
equal to $\simeq 1.3\,\sigma$.

###  4.1 Influence of systematic errors and choices of beam energies

In Figs. 6 and 7 examples of fits are shown in 4 selected bins of
$(\xi,\beta,Q^{2})$. In each figure two data sets are shown. The open red
circles correspond to set S-17, and the filled blue points are the subset that
is also present in S-9 (for Fig. 6) and S-5 (for Fig. 7). Uncorrelated
systematic uncertainties are considered at the level of $1\%$ on each data
point, with the influence of correlated sources taken to be negligible as
discussed previously. Separate fits are performed to each of the sets,
resulting in the two lines shown on each of the plots.

The S-17 set of beam energies contains the most points in $Y_{\mathrm{L}}$ and
therefore by construction gives the most precise results. Reducing the number
of beam energy combinations lowers the precision of the
$F_{\mathrm{L}}^{\mathrm{D(3)}}$ extraction. However, we observe that the fits
with S-9 and even S-5 do not deviate strongly from those of S-17 in most
cases. This is encouraging, since set S-5 is the current working hypothesis
for the EIC energy combinations. It is also evident that the strongest
variations with the choice of set arise in $(Q^{2},\xi,\beta)$ bins where
there is a limited range of $Y_{\mathrm{L}}$ available for the fit. For
example, the bin with $(\xi=0.032,\beta=0.56,Q^{2}=5.6\,\mathrm{GeV}^{2})$ has
a very small range in $Y_{\mathrm{L}}$, resulting in large variations between
the sets and correspondingly large uncertainties on the extracted
$F_{\mathrm{L}}^{\mathrm{D(3)}}$, despite the relatively large number of
$Y_{L}$ points. The relative accuracy of the $F_{\mathrm{L}}^{\mathrm{D(3)}}$
extraction also depends on its absolute size, which is reflected in the
steepness of the slope.

For each choice of beam energy combinations and systematic precision, ten
Monte Carlo simulations are carried out in order to get a feel for the
expected spread of the results after propagation through the Rosenbluth fits.
Fig. 8 shows the extracted values of $\xi$$F_{\mathrm{L}}^{\mathrm{D(3)}}$ as
a function of $\beta$ for selected values of $\xi$ and $Q^{2}$, with five
randomly chosen examples of the Monte Carlo replicas overlayed. In addition to
the three different beam energy sets with $\delta_{\mathrm{sys}}\ =1\%$
studied in Figs. 6 and 7, results are also shown for the S-17 set with
$\delta_{\mathrm{sys}}\ =2\%$. The results for the full set of
$(\beta,Q^{2},\xi)$ bins with S-17 and $\delta_{\mathrm{sys}}\ =1\%$ are shown
in Fig. 9.

There are not large differences between the results with the S-17 and S-9 beam
energy sets: S-9 leads to a small reduction in the access to small $\beta$
values and some increase of the uncertainties as measured by the spread
between different Monte Carlo sets. The differences are more pronounced in the
comparison between S-17 and S-5, due to the smaller number of bins in
$(\xi,\beta,Q^{2})$ accessible with S-5. In addition to larger uncertainties,
the kinematic range with S-5 is restricted to relatively larger $\beta$.
Therefore, although a larger number of energy combinations certainly yields
better results, an extraction of $F_{\mathrm{L}}^{\mathrm{D(3)}}$ is feasible
for the EIC-favoured set of energy combinations.

As expected, the influence of a factor 2 in $\delta_{\mathrm{sys}}$ translates
into a factor around 2 in the uncertainties in
$F_{\mathrm{L}}^{\mathrm{D(3)}}$. This reflects the fact that with the assumed
luminosity of 10 fb-1 per collision energy, systematic uncertainties are the
limiting factor throughout the kinematically accessible region.

$E_{\mathrm{CM}}\textrm{ set S-17},\delta_{\mathrm{sys}}=1\%,68\%\ {\rm CL}$ | $E_{\mathrm{CM}}\textrm{ set S-17},\delta_{\mathrm{sys}}=2\%,68\%\ {\rm CL}$
---|---
|
$E_{\mathrm{CM}}\textrm{ set S-9},\delta_{\mathrm{sys}}=1\%,68\%\ {\rm CL}$ | $E_{\mathrm{CM}}\textrm{ set S-5},\delta_{\mathrm{sys}}=1\%,68\%\ {\rm CL}$
|
Figure 8: Comparisons of the extracted $F_{\mathrm{L}}^{\mathrm{D(3)}}$ in
four selected $(\xi,Q^{2})$ bins, for five Monte Carlo samples (markers with
different colours, horizontally displaced from each other for clarity). The
first row compares the results for two different $\delta_{\mathrm{sys}}$
values, 1% on the left and 2% on the right, for the S-17 set of beam energies.
In the second row the results for smaller $E_{\mathrm{CM}}$ sets (S-9 and S-5)
are shown with $\delta_{\mathrm{sys}}=1\%$. The solid lines show the central
values of the model used for the generation of the pseudodata from which
$F_{\mathrm{L}}^{\mathrm{D(3)}}$ was extracted.

Figure 9: Extracted $F_{\mathrm{L}}^{\mathrm{D(3)}}$ for set S-17,
$\delta_{\mathrm{sys}}$=1%, 68% CL uncertainty bands, for 5 MC samples
(markers with different colours, horizontally displaced from each other for
clarity). The solid lines show the central values of the model used for the
generation of the pseudodata from which $F_{\mathrm{L}}^{\mathrm{D(3)}}$ was
extracted.

###  4.2 Estimated precision on $F_{\mathrm{L}}^{\mathrm{D(3)}}$

Figs. 8 and 9 illustrate the spread of results for
$F_{\mathrm{L}}^{\mathrm{D(3)}}$ extracted from 68% CL uncertainties on fits
to five different Monte Carlo pseudodata samples generated using the same
central values and uncertainties. The resulting distributions at each
$(Q^{2},\beta,\xi)$ point are complicated, since in addition to the
uncertainties propagated through the fits from the pseudodata, they also
reflect the available $Y_{\mathrm{L}}$ range and number of data points
available in the fits. In order to estimate the precision with which
$F_{\mathrm{L}}^{\mathrm{D(3)}}$ can be extracted at each $(Q^{2},\beta,\xi)$
point, we investigate the spread between the results obtained with the
different Monte Carlo samples, quantified using a direct arithmetic averaging
procedure, neglecting the uncertainties obtained from the fits. The mean and
variance are therefore taken to be

$\bar{v}=S_{1}/N,\quad(\Delta v)^{2}=\frac{S_{2}-S_{1}^{2}/N}{N-1}\ ,$ (15)

with $S_{n}=\sum_{i=1}^{N}v_{i}^{n}$ and $v_{i}$ the value of the extracted
$\xi$$F_{\mathrm{L}}^{\mathrm{D(3)}}$ in Monte Carlo sample $i$. The
uncertainty is then taken to be $\Delta v$.

Figure 10: Estimated uncertainties $\Delta v$ on
$F_{\mathrm{L}}^{\mathrm{D(3)}}$ extractions for beam energy sets S-17, S-9,
S-5, averaged over 10 MC samples for $\delta_{\mathrm{sys}}$ = 1%. The numbers
in each box give the relative accuracy of the $F_{\mathrm{L}}^{\mathrm{D(3)}}$
determination (in percent). Bins with errors exceeding 100% are indicated in
grey.

Fig. 10 shows the uncertainties $\Delta v$ in ($\beta,Q^{2},\xi$) bins for the
three sets of beam energies, obtained by averaging over 10 separate Monte
Carlo simulations. Even with 10 MC samples, there are some strong point-to-
point fluctuations in the estimated uncertainties. However, the general trends
become clear, in particular the regions in which reliable measurements can be
made. The best measured region for each $\xi$ value is at the lowest $\beta$
and $Q^{2}$, where the $Y_{\mathrm{L}}$ range is widest and statistical errors
are negligible. The uncertainties in a given bin do not depend strongly on the
energy sets, but the accessible kinematic range in which measurements can be
made decreases with decreasing numbers of energy combinations in the set.

###  4.3 Results for the ratio of the longitudinal to transverse structure
functions

The ‘photoabsorption’ ratio in diffraction, $R^{\rm
D(3)}=F_{\mathrm{L}}^{\mathrm{D(3)}}/F_{\mathrm{T}}^{\mathrm{D(3)}}$, where
$F_{\mathrm{T}}^{\mathrm{D(3)}}=F_{2}^{\mathrm{D(3)}}-F_{\mathrm{L}}^{\mathrm{D(3)}}$,
is the ratio of the cross section for longitudinally polarised photons to that
for transversely polarised photons at the same $(\beta,Q^{2},\xi)$ values.
$R^{\rm D(3)}$ has a clear and intuitive physical meaning and can be compared
with similar quantities extracted from decay angular distributions in
exclusive processes such as vector meson production. It can be extracted from
a fit to the reduced cross section pseudodata as a function of
$Y_{\mathrm{L}}$ of the form

$\sigma_{\mathrm{red}}^{\mathrm{D}(3)}=\left[1+\left(1-Y_{\mathrm{L}}\right)\,R^{\rm
D(3)}\right]F_{\mathrm{T}}^{\mathrm{D(3)}}\;,$ (16)

with $R^{\rm D(3)}$ and $F_{\mathrm{T}}^{\mathrm{D(3)}}$ being the free fit
parameters. This alternative fit has different sensitivities to the
uncertainties in the measurements from those in the
$F_{\mathrm{L}}^{\mathrm{D(3)}}$ extraction.

Figure 11: Extracted $R^{\rm D(3)}$ data for beam energy set S-17 averaged
over 10 MC samples with $\delta_{\mathrm{sys}}$ = 1%. The solid lines show the
central values of the model used for the generation of the pseudodata.

Fig. 11 shows the results for the values of $R^{\rm D(3)}$ obtained from the
averaging method described in Sec. 4.2 using 10 Monte Carlo samples and the
largest set of beam energies, S-17 and $\delta_{\mathrm{sys}}$ = 1%. A precise
determination of $R^{\rm D(3)}$ over a large kinematic range will be possible
at the EIC.

##  5 Conclusions and Outlook

In this paper we have investigated the potential of the Electron Ion Collider
for the measurement of the longitudinal structure function in diffraction.
This is a challenging measurement that requires data with high statistics at
several different centre-of-mass energies, ideally well beyond those available
in the pioneering measurement by the H1 collaboration at HERA. We have
considered EIC scenarios with 17, 9 and 5 different values of $\sqrt{s}$, the
latter being the commonly assumed EIC scenario. Pseudodata for the reduced
diffractive cross section were generated using a model based on collinear
factorization with DGLAP evolution, assuming an integrated luminosity of
$10\,\rm fb^{-1}$ for each centre-of-mass energy. The uncorrelated systematic
errors are assumed to be either $1\%$ or $2\%$, which is challenging compared
with previous measurements, but consistent with the expected high level of
performance of the EIC detectors. The longitudinal structure function was
extracted using the standard Rosenbluth method of a linear fit to the reduced
cross section as a function of the $Y_{\mathrm{L}}$ variable, extracting
$F_{\mathrm{L}}^{\mathrm{D(3)}}$ from the slope and $F_{2}^{\mathrm{D}(3)}$
from the intercept. Fits were only performed in $(\beta,\xi,Q^{2})$ bins where
at least four centre-of-mass energies yielded data points at accessible $y$
values (in contrast to three in the H1 case). The scenarios with 17 and 9
centre-of-mass energies do not differ much in terms of the kinematic range in
which $F_{\mathrm{L}}^{\mathrm{D(3)}}$ can be extracted, whereas the scenario
with only 5 centre-of-mass energies results in a restricted kinematic range,
particularly at small values of $\beta$ and $\xi$. The precision on the
extracted value of $F_{\mathrm{L}}^{\mathrm{D(3)}}$ is strongly correlated
with the available range in $Y_{\mathrm{L}}$ in a given bin
$(\xi,\beta,Q^{2})$ and is not diminished substantially when going from 17 to
9 centre-of-mass energies. A larger difference is observed for the set with 5
centre-of-mass energies, mainly due to the smaller available range in $y$.
Nonetheless, for bins in which measurements are possible, the precision of the
extracted structure function $F_{\mathrm{L}}^{\mathrm{D(3)}}$ is comparable in
all scenarios. Given the very high target luminosity at the EIC, the exact
choices of running time at each beam energy will not be a strongly limiting
factor; the precision is likely to depend much more strongly on the size of
the systematic uncertainties that are not correlated between data points.

We have also performed a separate extraction of the ratio $R^{\rm
D(3)}=F_{\mathrm{L}}^{\mathrm{D(3)}}/(F_{2}^{\mathrm{D(3)}}-F_{\mathrm{L}}^{\mathrm{D(3)}})$
of the longitudinal to the transverse diffractive structure functions. A
precise extraction of this quantity is expected to be possible at the EIC.

As an outlook, we note that the currently foreseen forward instrumentation at
the EIC allows a precise determination of $-t$ in a wide range, see Sec. 2.2.
It will be very interesting to explore to what extent an extraction of
$F_{\mathrm{L}}^{\mathrm{D}(4)}(\beta,\xi,Q^{2},t)$ in a similar way to that
illustrated previously for $F_{\mathrm{L}}^{\mathrm{D(3)}}$, Eq. (4), will be
possible. This is a completely new study that we leave for the future.

## Acknowledgements

We thank Alex Jentsch, Hannes Jung, Kolja Kauder, and Mark Strikman for useful
discussions. NA acknowledges financial support by Xunta de Galicia (Centro
singular de investigación de Galicia accreditation 2019-2022); the ”María de
Maeztu” Units of Excellence program MDM2016-0692 and the Spanish Research
State Agency under project FPA2017-83814-P; European Union ERDF; the European
Research Council under project ERC-2018-ADG-835105 YoctoLHC; MSCA RISE 823947
”Heavy ion collisions: collectivity and precision in saturation physics”
(HIEIC); and European Union’s Horizon 2020 research and innovation programme
under grant agreement No. 824093. AMS is supported by the U.S. Department of
Energy grant No. DE-SC-0002145 and in part by National Science Centre in
Poland, grant 2019/33/B/ST2/02588.

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