![](https://csdnimg.cn/release/download_crawler_static/12298219/bg4.jpg)
F
Dð3Þ
2;L
are dimensionless. In this analysis, we neglect Z
0
exchange, though it should be included in future studies.
The reduced cross sections σ
D
r
depend on center-of-mass
energy via y ¼
Q
2
ξβs
. The Y
þ
factors ensure that in the region
where y is not too close to unity
σ
D
r
≃ F
D
2
ð6Þ
to good approximation.
Both σ
Dð3Þ
r
and σ
Dð4Þ
r
have been measured at the HERA
collider [1,2,25–31] and used to obtain QCD-inspired
parametrizations.
The standard perturbative QCD approach to diffractive
cross sections is based on collinear factorization [5–7].It
was demonstrated that, similarly to the inclusive DIS cross
section, the diffractive cross section can be written, up to
terms of order Oð1=Q
2
Þ, in a factorized form
dσ
ep→eXY
ðβ; ξ;Q
2
;tÞ
¼
X
i
Z
1
β
dzdˆσ
ei
β
z
;Q
2
f
D
i
ðz; ξ;Q
2
;tÞ; ð7Þ
where the sum is performed over all parton flavors (gluon,
d-quark, u-quark, etc.). The hard scattering partonic cross
section dˆσ
ei
can be computed perturbatively in QCD and is
the same as in the inclusive deep inelastic scattering case.
The long distance part f
D
i
corresponds to the diffractive
parton distribution functions, which can be interpreted as
conditional probabilities for partons in the proton, provided
the proton is scattered into the final state system Y with
specified 4-momentum p
0
. They are evolved using the
DGLAP evolution equations [32–35] similarly to the
inclusive case. The analogous formula for the t-integrated
structure functions reads
F
Dð3Þ
2=L
ðβ; ξ;Q
2
Þ¼
X
i
Z
1
β
dz
z
C
2=L;i
β
z
f
Dð3Þ
i
ðz;ξ;Q
2
Þ; ð8Þ
where the coefficient functions C
2=L;i
are the same as in
inclusive DIS.
III. SIMULATIONS FOR THE
ELECTRON-PROTON DIS
A. Diffractive PDF parametrizations and HERA data
Fits to the diffractive structure functions were performed
by H1 [25] and ZEUS [29]. They both parametrize the
diffractive PDFs in a two-component model, which is a
sum of two exchange contributions, P and R:
f
Dð4Þ
i
ðz; ξ;Q
2
;tÞ¼f
p
P
ðξ;tÞf
P
i
ðz; Q
2
Þþf
p
R
ðξ;tÞf
R
i
ðz; Q
2
Þ:
ð9Þ
For both of these terms, proton vertex factorization is
assumed, meaning that the diffractive exchange can be
interpreted as colorless objects called a “Pomeron” or a
“Reggeon” with parton distributions f
P;R
i
ðβ;Q
2
Þ. The flux
factors f
p
P;R
ðξ;tÞ represent the probability that a Pomeron/
Reggeon with given values ξ, t couples to the proton. They
are parametrized using the form motivated by Regge
theory,
f
p
P;R
ðξ;tÞ¼A
P;R
e
B
P;R
t
ξ
2α
P;R
ðtÞ−1
; ð10Þ
with a linear trajectory α
P;R
ðtÞ¼α
P;R
ð0Þþα
0
P;R
t. The
diffractive PDFs relevant to the t-integrated cross sections
read
f
Dð3Þ
i
ðz; ξ;Q
2
Þ¼ϕ
p
P
ðξÞf
P
i
ðz; Q
2
Þþϕ
p
R
ðξÞf
R
i
ðz; Q
2
Þ;
ð11Þ
with
ϕ
p
P;R
ðξÞ¼
Z
dtf
p
P;R
ðξ;tÞ: ð12Þ
Note that the notions of Pomeron and Reggeon used here to
model hard diffraction in DIS are, in principle, different
from those describing the soft hadron-hadron interactions;
in particular, the parameters of the fluxes may be different.
The diffractive parton distributions of the Pomeron at the
initial scale μ
2
0
¼ 1.8 GeV
2
are parametrized as
zf
P
i
ðz; μ
2
0
Þ¼A
i
z
B
i
ð1 − zÞ
C
i
; ð13Þ
where i is a gluon or a light quark. In the diffractive
parametrizations, all the light quarks (antiquarks) are
assumed to be equal. For the treatment of heavy flavors,
a variable flavor number scheme (VFNS), in which the
charm and bottom-quark DPDFs are generated radiatively
via DGLAP evolution and no intrinsic heavy-quark dis-
tributions are assumed, is adopted. The structure functions
are calculated in a general-mass variable flavor number
scheme (GM-VFNS) [36,37], which ensures a smooth
transition of F
2;L
across the flavor thresholds by including
Oðm
2
h
=Q
2
Þ corrections. The parton distributions for the
Reggeon component are taken from a parametrization
which was obtained from fits to the pion structure function
[38,39].
In Eq. (9), the normalization factors of fluxes, A
P;R
and
of DPDFs, A
i
enter in the product. To resolve the
ambiguity, we fix
1
A
P
and use f
R
i
ðz; Q
2
Þ normalized to
1
Here, as in the HERA fits, A
P
is fixed by normalizing
ϕ
p
P
ð0.003Þ¼1.
ARMESTO, NEWMAN, SŁOMIŃSKI, and STAŚTO PHYS. REV. D 100, 074022 (2019)
074022-4