arXiv:1103.5704v2 [hep-ex] 2 May 2011
Preprint typeset in JHEP style - HYPER VERSION
May 3, 2011
Inclusive Measurements of Inelastic Electron and
Positron Scattering from Unpolarized Hydrogen and
The HERMES Collaboration
Abstract: Results of inclusive measurements of inelastic electron and positron scattering
from unpolarized protons and deuterons at the HERMES experiment are presented. The
structure functions Fp
2are determined using a parameterization of existing data for
the longitudinal-to-transverse virtual-photon absorption cross-section ratio. The HERMES
results provide data in the ranges 0.006 ≤ x ≤ 0.9 and 0.1 GeV2≤ Q2≤ 20 GeV2, covering
the transition region between the perturbative and the non-perturbative regimes of QCD
in a so-far largely unexplored kinematic region. They are in agreement with existing world
data in the region of overlap. The measured cross sections are used, in combination with
data from other experiments, to perform fits to the photon-nucleon cross section using the
functional form of the ALLM model. The deuteron-to-proton cross-section ratio is also
Keywords: Lepton-Nucleon Scattering.
The HERMES Collaboration
A. Airapetian12,15, N. Akopov26, Z. Akopov5, E.C. Aschenauer6,a,
W. Augustyniak25, R. Avakian26, A. Avetissian26, E. Avetisyan5,
S. Belostotski18, N. Bianchi10, H.P. Blok17,24, A. Borissov5, J. Bowles13,
V. Bryzgalov19, J. Burns13, M. Capiluppi9, G.P. Capitani10, E. Cisbani21,
G. Ciullo9, M. Contalbrigo9, P.F. Dalpiaz9, W. Deconinck5, R. De Leo2,
L. De Nardo11,5, E. De Sanctis10, M. Diefenthaler14,8, P. Di Nezza10,
M. D¨ uren12, M. Ehrenfried12, G. Elbakian26, F. Ellinghaus4, R. Fabbri6,
A. Fantoni10, L. Felawka22, S. Frullani21, D. Gabbert6, G. Gapienko19,
V. Gapienko19, F. Garibaldi21, G. Gavrilov5,18,22, V. Gharibyan26,
F. Giordano5,9, S. Gliske15, M. Golembiovskaya6, C. Hadjidakis10,
M. Hartig5,b, D. Hasch10, G. Hill13, A. Hillenbrand6, M. Hoek13, Y. Holler5,
I. Hristova6, Y. Imazu23, A. Ivanilov19, H.E. Jackson1, H.S. Jo11,
S. Joosten14,11, R. Kaiser13,c, G. Karyan26, T. Keri13,12, E. Kinney4,
A. Kisselev18, V. Korotkov19, V. Kozlov16, P. Kravchenko8,18,
V.G. Krivokhijine7, L. Lagamba2, R. Lamb14, L. Lapik´ as17, I. Lehmann13,
P. Lenisa9, L.A. Linden-Levy14, A. L´ opez Ruiz11, W. Lorenzon15, X.-G. Lu6,
X.-R. Lu23, B.-Q. Ma3, D. Mahon13, N.C.R. Makins14, S.I. Manaenkov18,
L. Manfr´ e21, Y. Mao3, B. Marianski25, A. Martinez de la Ossa8,4,
H. Marukyan26, C.A. Miller22, Y. Miyachi23, A. Movsisyan26, V. Muccifora10
M. Murray13, A. Mussgiller5,8, E. Nappi2, Y. Naryshkin18, A. Nass8,
M. Negodaev6, W.-D. Nowak6, L.L. Pappalardo9, R. Perez-Benito12,
N. Pickert8, M. Raithel8, P.E. Reimer1, A.R. Reolon10, C. Riedl6, K. Rith8,
G. Rosner13, A. Rostomyan5, J. Rubin14, D. Ryckbosch11, Y. Salomatin19,
F. Sanftl23,20, A. Sch¨ afer20, G. Schnell6,11,d, K.P. Sch¨ uler5, B. Seitz13,
T.-A. Shibata23, V. Shutov7, M. Stancari9, M. Statera9, E. Steffens8,
J.J.M. Steijger17, H. Stenzel12, J. Stewart6, F. Stinzing8, S. Taroian26,
A. Trzcinski25, M. Tytgat11, A. Vandenbroucke11, Y. Van Haarlem11,
C. Van Hulse11, D. Veretennikov18, V. Vikhrov18, I. Vilardi2, C. Vogel8,
S. Wang3, S. Yaschenko6,8, H. Ye3, Z. Ye5, S. Yen22, W. Yu12, D. Zeiler8,
B. Zihlmann5, P. Zupranski25.
1Physics Division, Argonne National Laboratory, Argonne, Illinois 60439-4843, USA
2Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70124 Bari, Italy
3School of Physics, Peking University, Beijing 100871, China
4Nuclear Physics Laboratory, University of Colorado, Boulder, Colorado 80309-0390,
aNow at: Brookhaven National Laboratory, Upton, New York 11772-5000, USA
bNow at: Institut f¨ ur Kernphysik, Universit¨ at Frankfurt a.M., 60438 Frankfurt a.M., Germany
cPresent address: International Atomic Energy Agency, A-1400 Vienna, Austria
dNow at: Department of Theoretical Physics, University of the Basque Country UPV/EHU, 48080
Bilbao, Spain and IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain
– 1 –
5DESY, 22603 Hamburg, Germany
6DESY, 15738 Zeuthen, Germany
7Joint Institute for Nuclear Research, 141980 Dubna, Russia
8Physikalisches Institut, Universit¨ at Erlangen-N¨ urnberg, 91058 Erlangen, Germany
9Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara and Dipartimento di Fisica,
Universit` a di Ferrara, 44100 Ferrara, Italy
10Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati, 00044 Frascati,
11Department of Subatomic and Radiation Physics, University of Gent, 9000 Gent,
12Physikalisches Institut, Universit¨ at Gießen, 35392 Gießen, Germany
13Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ,
14Department of Physics, University of Illinois, Urbana, Illinois 61801-3080, USA
15Randall Laboratory of Physics, University of Michigan, Ann Arbor, Michigan
16Lebedev Physical Institute, 117924 Moscow, Russia
17National Institute for Subatomic Physics (Nikhef), 1009 DB Amsterdam, The
18Petersburg Nuclear Physics Institute, Gatchina, Leningrad region 188300, Russia
19Institute for High Energy Physics, Protvino, Moscow region 142281, Russia
20Institut f¨ ur Theoretische Physik, Universit¨ at Regensburg, 93040 Regensburg, Germany
21Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Gruppo Collegato Sanit` a and
Istituto Superiore di Sanit` a, 00161 Roma, Italy
22TRIUMF, Vancouver, British Columbia V6T 2A3, Canada
23Department of Physics, Tokyo Institute of Technology, Tokyo 152, Japan
24Department of Physics, VU University, 1081 HV Amsterdam, The Netherlands
25Andrzej Soltan Institute for Nuclear Studies, 00-689 Warsaw, Poland
26Yerevan Physics Institute, 375036 Yerevan, Armenia
– 2 –
Over the past decades, lepton-nucleon scattering has played a major role in the development
of our present understanding of nucleon structure. For a review on the subject see for
example . In lowest order perturbation theory, scattering of charged leptons l off nucleons
N proceeds via the exchange of a neutral boson (γ∗, Z0). At the HERMES lepton-nucleon
centre-of-mass energy of√s = 7.2 GeV, contributions from Z0-exchange to the cross section
can be neglected. Therefore, only the electromagnetic interaction in the approximation
of one-photon exchange is considered here. In this approximation, the differential cross
section of unpolarized inclusive charged-lepton-nucleon scattering, l + N → l′+ X (where
X denotes the undetected final state), is parameterized by two structure functions F1(x,Q2)
and F2(x,Q2). Here x = Q2/2Mν is the Bjorken variable, with −Q2being the square of
the four-momentum transferred by the virtual photon and ν its energy in the target rest
frame. The variable x is a measure for the inelasticity of the process with 0 ≤ x ≤ 1, and
x = 1 for elastic scattering.
In the deep-inelastic scattering (DIS) regime,
typical hadronic scale, usually set to be the mass M of the nucleon, and the invariant mass
W of the photon-nucleon system is much larger than the masses of nucleon resonances. In
the Quark-Parton Model (QPM), the DIS process is viewed as the incoherent superposition
of elastic lepton scattering from quasi-free point-like quarks of any flavor q. The variable
x can then be interpreted as the fraction of the longitudinal nucleon momentum carried
by the struck quark in a frame where the nucleon moves with infinite momentum in the
direction opposite to that of the virtual photon. In this picture, quark distribution functions
fq(x,Q2) describe the number density of quarks of flavor q in a fast-moving nucleon at a
given value of (x,Q2) and experimental values of F2(x,Q2) have been used to constrain
these. At low values of Q2, where this picture of incoherent quasi-free scattering does not
apply, phenomenological models have been developed (see e.g. Refs.  and ) to describe
the measured structure functions.
?Q2and ν are much larger than the
A wealth of unpolarized inclusive charged-lepton DIS data is available from the collider
experiments H1 [4, 5, 6, 7, 8] and ZEUS [9, 10, 11, 12, 13, 14] at HERA with lepton-
nucleon centre-of-mass energies√s up to 320 GeV, the muon experiments BCDMS ,
EMC , NMC  and E665  (√s∼= 12 − 31 GeV), experiments with electrons at
SLAC  (√s ≤ 7 GeV) and at JLAB [20, 21, 22, 23]) (√s ≤ 3.25 GeV). The HERMES
experiment  at HERA collected a large data set for positron and electron scattering
on a variety of nuclear targets, including the proton and deuteron data presented here. In
particular, the HERMES data cover the transition region between the perturbative and
non-perturbative regimes of QCD in a kinematic region so far largely unexplored by other
experiments. In this work, these data are presented together with fits to the world data for
the photon-nucleon cross section using the Regge-motivated approach of the ALLM [3, 25]
model. The paper is organized as follows: the formalism leading to the extraction of
the structure function F2is briefly reviewed in Sect. 2; Sect. 3 deals with the HERMES
experimental arrangement and the data analysis is described in Sect. 4. The systematic
– 3 –
uncertainties in the resulting cross sections and cross-section ratios are discussed in Sect. 5.
Section 6 offers a discussion of the results and Sect. 7 provides a summary.
Lepton mass (taken to be negligible)
Mass of target nucleon
k = (E,?k), k′= (E′,?k′)
4–momenta of the initial and final state leptons
θ, φ Polar and azimuthal angle of the scattered lepton
4–momentum of the initial target nucleon
q = k − k′
4–momentum of the virtual photon
ν =P · q
2P · q=
y =P · q
P · k
W2= (P + q)2= M2+ 2Mν − Q2
≈ 4EE′sin2 θ
Negative squared 4–momentum transfer
= E − E′
Energy of the virtual photon in the target rest
Bjorken scaling variable
Fractional energy of the virtual photon
Squared invariant mass of the photon–nucleon
Table 1: Kinematic variables used in the description of lepton scattering.
In the approximation of one-photon exchange, the inclusive differential cross section
for scattering unpolarized charged leptons on unpolarized nucleons can be conveniently
parameterized in terms of the structure functions F1(x,Q2) and F2(x,Q2):
y2· F1(x,Q2) +
?1 − y
where αemis the fine-structure constant and all other variables are described in Tab. 1.
The quantities x and Q2are fully determined by the kinematic conditions of the incident
and scattered leptons and the target nucleon. This cross section can also be written in
terms of longitudinal (L) and transverse (T) virtual-photon contributions
dx dQ2= Γ[σT(x,Q2) + ǫ σL(x,Q2)] ,(2.2)
where σLand σT are the absorption cross sections for longitudinal and transverse virtual
photons, Γ is the flux of transverse virtual photons and the virtual-photon polarization
– 4 –
parameter ǫ is the ratio of virtual-photon fluxes for longitudinal and transverse polariza-
tions . The structure functions F1(x,Q2) and F2(x,Q2) can then be expressed in terms
of the two virtual-photon absorption cross sections σL(x,Q2) and σT(x,Q2):
MK · σT(x,Q2) ,
·?σL(x,Q2) + σT(x,Q2)?
where K = ν(1 − x) in the Hand convention [27, 28].
photon-absorption cross-section ratio R = σL/σT can be expressed in terms of F1and F2:
2xF1(x,Q2)− 1 .(2.5)
A determination of the structure functions F1(x,Q2) and F2(x,Q2) requires in principle
cross-section measurements made at the same x and Q2but at two or more different values
of y (see Eq. (2.1)), i.e., with different beam energies. The HERMES data used for this
analysis were taken at a single beam energy. In such a situation, it is common practice to
re-parameterize the cross section, Eq. (2.1), as a function of F2and R using Eq. (2.5):
1 − y −Q2
2[1 + R(x,Q2)]
The structure function F2(x,Q2) can then be extracted from a single cross-section mea-
surement at a given (x,Q2), by using a parameterization for R(x,Q2) obtained from the
available world data. This approach has been used in the analysis presented in this paper.
3. The Experiment
The HERA facility at DESY comprised a proton and a lepton storage ring. HERMES was
a fixed-target experiment using only the lepton beam, which consisted of either electrons
or positrons at an energy of 27.6 GeV, while the proton beam passed through the non-
instrumented horizontal mid-plane of the HERMES spectrometer. An open-ended storage
cell that could be fed with either polarized or unpolarized gas was installed internally to
the lepton ring.
The HERMES spectrometer, which consisted of two identical halves above and below
the electron beam, was a forward spectrometer  with multiple tracking stages before and
after a 1.5 Tm dipole magnet. It had a geometrical acceptance of ±170 mrad horizontally
and ±(40 − 140) mrad vertically for particles originating from the center of the target
cell, resulting in polar scattering angles θ ranging from about 40 to 220 mrad. Particle
identification (PID) capabilities were provided by combining the responses of a lead-glass
calorimeter, a pre-shower hodoscope (H2), a transition-radiation detector (TRD), and a
thresholdˇCerenkov detector that was upgraded to a dual-radiator ring-imagingˇCerenkov
detector (RICH) [29, 30] in the year 1998. The lead-glass calorimeter and the pre-shower
hodoscope were included in the trigger together with two other hodoscopes (H0 and H1).
– 5 –
In this experiment target gases of hydrogen and deuterium were used. Part of the data
were taken with polarized hydrogen and deuterium, with the target spin being reversed in 1-
3 min time intervals so that the target was effectively unpolarized. In the case of hydrogen,
using areal densities of the order of 1014nucleons cm−2and lepton currents of typically
about 30 mA, luminosities of the order of 2·1031cm−2s−1were achieved for the polarized
running, and about 10 times higher values for unpolarized running. The luminosity was
measured by scattering the lepton beam off the atomic electrons of the target gas, i.e.,
Møller scattering e−e−→ e−e−for an electron beam and Bhabha scattering e+e−→ e+e−
together with the annihilation process e+e−→ γγ for a positron beam. The cross sections
for these processes are precisely known in Quantum Electrodynamics, including radiative
corrections. The scattered particles were detected in coincidence by two identical small
calorimeters  located symmetrically with respect to the beam pipe. The coincidence
rate of the pairs of leptons (and photons) provided a relative monitor of the luminosity.
An absolute calibration of the luminosity measurement was provided by correlating the
coincidence rate with the yields of the Møller, Bhabha and annihilation processes.
4. Data Analysis
An event is accepted if it contains a track identified as a lepton by the PID system (see
Sect. 4.3), and satisfies the selection criteria described in Sect. 4.1. The number of mea-
sured events Nmeasin each (x,Q2) bin is corrected by subtracting the charge-symmetric
background from secondary processes Ncs and by dividing the resulting number by the
corresponding trigger and lepton-identification efficiencies Etrigger and Elep, while taking
into account the hadron contamination Chad:
Nevents= (Nmeas− Ncs) ·
·1 − Chad
These corrections are described in Sects. 4.2 to 4.4.
The experimental cross section is then obtained as the ratio of the number of events
Neventsin each (x,Q2) bin of widths ∆x and ∆Q2, and the integrated luminosity L (see
dx dQ2(x,Q2) =Nevents(x,Q2)
An unfolding procedure for disentangling instrumental and radiative effects from the
measured cross section is then applied in order to obtain the Born cross sections σp,d
σp,d(see Sect. 4.6). The structure functions Fp
cross sections through Eq. (2.6) using the parameterization R = R1998 . Two more
corrections related to detector geometry and alignment are discussed in Sects. 4.7 and 4.8.
2are finally derived from the Born
4.1 Event selection
The kinematic range of the events selected for this analysis is shown in Fig. 1, together
with the requirements imposed on the kinematic and geometrical variables.
– 6 –
Q2 [ GeV2]
Figure 1: Binning in (x,Q2) used in the analysis and kinematic acceptance of events. The kine-
matic region covered is limited by the geometrical acceptance in θ and constraints on y and W2.
The symbols mark the locations of the average values of (x,Q2) of each bin. The symbols A to F
denote bins with increasing Q2at given x.
The tracks are required to be fully contained within
the fiducial geometric acceptance of the HERMES spec-
trometer. The constraint W2> 5 GeV2excludes the
region of nucleon resonances and acts as a selection
for y > 0.1.The constraint y ≤ 0.85 discards the
low-momentum region, where radiative effects increase
and where the trigger efficiency has not yet reached its
plateau. The requirements imposed on y select the
momentum range 4.1GeV< p < 24.8GeV for the de-
tected particle. The resulting (x,Q2) region, 0.006 ≤
x ≤ 0.9 and 0.1 GeV2≤ Q2≤ 20 GeV2, is subdivided
into 19 bins in x and each x bin into up to six bins in
Table 2 shows the numbers of events for each year
of measurements used in this analysis, before the application of any of the corrections
discussed in the following sections.
Year events, in million
Nmeas used in this analysis, sepa-
rated into the years of data taking.
The numbers correspond to the total
luminosities of about 450pb−1on the
proton and about 460pb−1on the
Number of raw events
– 7 –
The trigger used for the recording of inelastic scattering events required signals from the
hodoscopes H0, H1, and H2 and a sufficiently large energy deposition in the calorimeter.
The efficiencies of the trigger detectors are extracted individually from special calibration
triggers and combined to obtain the total trigger efficiency. It is assumed that inefficiencies
in the electronic trigger logics are negligible. Trigger efficiencies are sensitive to, among
others, misalignment effects, detector-voltage setting, and radiation damage, the latter
especially in the H0 hodoscope. This last effect is responsible for the reduced efficiencies
seen at small scattering angles and for the differences between the top and bottom detector.
Such differences are shown in Fig. 2 for data taken in the year 2000, for the kinematic
binning used in the analysis.
Figure 2: Trigger efficiencies Etrigger for data taken in the year 2000 shown separately for the
top and bottom spectrometer halves. The error bars represent only statistical uncertainties. The
symbols refer to the Q2bins shown in Fig. 1.
4.3 Particle Identification
The scattered lepton (positron or electron) is identified by a combination of the responses
of the transition-radiation detector TRD, the pre-shower hodoscope H2, and the lead-
glass calorimeter. Each of these elements used alone gives a high rejection of hadrons. A
ˇCerenkov detector provides additional hadron identification. (A thresholdˇCerenkov was
used for pion identification in 1996-97, and a ring-imagingˇCerenkov detector was used
– 8 –
thereafter to identify pions, kaons, and protons.) The detector response of an individual
PID element is determined by placing very restrictive constraints on the response of the
remaining elements, thereby generating a clean sample of a given kind of particles with
which the unit under study is calibrated.
In combination, the array of detectors provides an average lepton identification with
an efficiency Elepof about 98% and an average hadron contamination Chadbelow 1% over
the full kinematic range of the HERMES acceptance. At low values of x, lepton efficiencies
as low as 94% and hadron contaminations as high as 3% are reached.
The efficiency for lepton identification and the fractional hadron contamination as a
function of x for the various Q2bins are presented in Fig. 3 for representative data taken
in the year 2000. The figure shows that for smaller values of x (x < 0.1) a lower lepton
identification efficiency appears correlated to a larger hadron contamination.
lepton id. efficiency[%]
Figure 3: Lepton identification efficiency Elep and hadron contamination Chadin the year 2000.
The symbols refer to the Q2binning shown in Fig. 1.
4.4 Charge-symmetric background
The observed event sample is contaminated by background coming mostly from charge-
symmetric processes, such as meson Dalitz decays (e. g. π0→ e+e−γ) or photon conversions
into e+e−pairs. Since these positrons and electrons originate from secondary processes,
they typically have lower momenta and are thus concentrated at high y. A correction for
charge-symmetric background events Ncsis applied in each kinematic bin by counting with
– 9 –
negative weight leptons with a charge opposite to that of the beam particle. It is assumed
that acceptance and inefficiencies are the same for background electrons and positrons, even
though their spatial distributions after the magnet are quite different. The x dependence
of CS, the ratio of charge-symmetric events to the total number of events in each kinematic
bin, is shown for the six Q2bins in Fig. 4. The charge-symmetric background is negligible
at large particle momenta, but reaches up to 12% at low particle momenta of about 6 GeV.
The integrated luminosity L per nucleon is calculated as flollows:
Ldt = (RLR− 2∆t · RL· RR) · clive· CLumi· ∆b ·A
Here, RLand RRare the count rates in the left and right luminosity detector, respectively,
RLRis the coincidence rate measured within a time window of ∆t = 40 ns, cliveis the trigger
livetime factor, CLumiis the year-dependent luminosity factor, ∆b is the time interval in
which the luminosity rates were obtained, and A/Z is the ratio of the numbers of nucleons
(A) and electrons (Z) in the target gas atoms.
The term 2∆t·RL·RRin Eq. (4.3)
cording to the statistical expectation and
is of the order of 0.1-0.5%. The physics
trigger livetime contribution cliveis de-
fined as the fraction of the physics events
that are accepted by the data acquisi-
tion system out of all events generating
a physics trigger. This quantity is typi-
cally above 90%.
The data aquisition system of the
luminosity detector worked independently
of the physics triggers. It is assumed
that the inefficiency of the luminosity
event trigger was negligible. The lumi-
nosity factor CLumiaccounts for the ge-
ometric acceptance of the luminosity de-
tector, the beam position and the abso-
lute Møller and Bhabha cross sections.
Its year dependence derives from the age-
ing of the luminosity detector and dif-
ferent running conditions, i.e., changes in beam charge and beam optics. The dependence
of the coincidence rate from beam position and slope was measured in order to disentangle
the dependence of the measured coincidence rate from beam orbits and geometrical ac-
ceptance of the luminosity detector. The uncertainty in the measurement of the absolute
luminosity is dominated by the uncertainty on the acceptance of the detector, which de-
pends sensitively on the impact coordinates of the particle. The uncertainty on the latter
ground, calculated from the ratio of the charge-
symmetric events to the total events in each bin, for
the 2000 deuterium data. The symbols refer to the
Q2binning shown in Fig. 1.
Percentage of charge-symmetric back-
corrects for accidental coincidences ac-
– 10 –
is about 2.5 mm, which propagates into an uncertainty of about 7% on the integrated cross
section and therefore on the luminosity.
4.6 Instrumental smearing and radiative effects
Instrumental smearing is due to intrinsic detector resolution and multiple scattering in the
various detector elements of the particles emerging from the DIS process and identified as
the scattered lepton. Radiative effects include vertex corrections to the QED hard scat-
tering amplitude and radiation of one or more real photons by the incoming or outgoing
lepton. Radiative effects and instrumental smearing both modify the Born kinematic con-
ditions resulting in altered reconstructed kinematic variables. Migration probabilities for
the relevant kinematic variables are determined from a Monte Carlo simulation and used
to correct the measured distributions.
The Born cross section for inelastic scattering on the proton is simulated according to
the ALLM97 parameterization of Fp
2 and the parameterization R1990 for R, while
that for the deuteron is derived from the same parameterizations in conjunction with the
fit  to Fd
2data from NMC, SLAC and BCDMS. Radiative effects are simulated with
RADGEN . The electric and magnetic form factors of the proton and neutron, from
which the elastic cross sections are derived, are taken from the fits in Refs.  and .
When using the more recent parameterizations for the proton from Ref. , the results
are essentially the same.
The probabilistic information about event migration can be summarized in a smearing
Here, n(i,j) is the migration matrix representing the number of events originating from
kinematic bin j at Born level and measured in bin i. It is extracted from a Monte Carlo sim-
ulation with full track reconstruction that simulates the inelastic scattering cross section,
QED radiative effects and instrumental smearing. Material outside the detector acceptance
is excluded from this simulation for computational economy. The vector nBorn(j) contain-
ing the number of events at Born level is obtained from a second Monte Carlo calculation
that simulates only the (unradiated) inelastic cross section. An additional column j = 0
is defined for events that migrate into the acceptance from outside. The smearing matrix
S(i,j) has the property of being independent from the generated cross section within the
The inverted squared submatrix S′(i,j) = S(i,j > 0) relates the measured distribu-
tions to the distributions at Born level:
S′−1(j,i) × [σExp(i) − S(i,0)σBorn(0)]. (4.5)
The reconstruction of simulated tracks uses the same algorithm as for real data.
Tracking-related inefficiencies are taken into account in the unfolding procedure, assuming
– 11 –
that coincident particles outside the acceptance do not significantly affect the efficiency
and the simulation adequately models the physical processes in the tracking detectors.
The Monte Carlo generated data sample was a factor 10 larger than the experimental
data sample. The statistical uncertainties of the Monte Carlo data enter mainly via the
simulated experimental count rates in the migration matrices. A multi-sampling numer-
ical approach is used to propagate these statistical uncertainties through the unfolding
algorithm. The statistical uncertainties of the inelastic scattering Born cross section com-
ing from the experimental cross section and those originating from the finite statistical
precision of the Monte Carlo are summed in quadrature to produce the total statistical
4.7 Detection efficiency of specific radiative events
Radiative corrections include cases where the incoming electron radiates a high-energy
photon and then scatters elastically from the nucleon with negligible momentum transfer.
The efficiency to detect such events is reduced due to the following effect. The radiated
photon is emitted at small angles and has a large probability to hit the beam pipe, gen-
erating an electromagnetic shower that saturates the wire chambers. This makes the data
acquisition system skip the event as no tracking is possible. In order to compensate for this
omission, the detection efficiency for elastic and quasi-elastic radiative events is estimated
using a dedicated Monte Carlo simulation that includes a complete treatment of showers
in material outside the geometrical acceptance.
The resulting efficiencies Ee.m. are significantly less than 100% in the range 0.01 <
x < 0.1. They show a dip at x ≃ 0.02 where, in the case of the proton, they reach values
as low as 80% while in the case of the deuteron they are as low as 90% (60%) for elastic
(quasi-elastic) events. They are applied to the background term S(i,0)σBorn(0) in order
to not over-correct for radiative processes that are not observed in the spectrometer. More
details can be found in Refs. [39, 40].
4.8 Misalignment effects
Imperfect alignment of the two spectrometer halves and the beam with respect to their
ideal positions is studied in order to estimate the impact on the measured cross sections and
structure functions. Misalignment effects cannot be corrected for in the unfolding because
they are not of a stochastic nature. Rather, they are studied in a Monte Carlo simulation,
and the fractional change of the Born cross section in each kinematic bin is obtained from
the ratio of unfolded cross sections when using a MC with an aligned geometry and another
with a misaligned geometry. These fractional changes are used to rescale the experimental
Born cross sections on a bin-by-bin basis. Misalignment effects are most significant for
small scattering angles and high particle momenta, i.e., at small Q2in each bin of x. The
correction reaches values as high as 19% in the lowest Q2bin and decreases to about 3%
in the highest Q2bin.
– 12 –
5. Systematic Uncertainties
5.1 Inclusive inelastic scattering cross sections
Correlations between PID detectors as described in Sect.4.3 cannot be completely avoided.
They may bias the correction for particle identification. These effects are covered by the
assignment of a conservative PID uncertainty of the full size of the correction (see Eq. (4.1)).
Hadrons are predominantly produced at small momenta. Thus particle misidentification
occurs more likely at high y, i.e., towards higher Q2in each bin of x.
the uncertainty due to particle identification, δPID, is always smaller than 3%, because
contaminations somewhat compensate inefficiencies.
Instrumental smearing and radiative effects.
In the unfolding procedure an uncertainty can arise from uncertainties in the formalism to
calculate radiative effects and in the model used for the cross section outside the accep-
tance. The latter affect our results through the radiative tail. The uncertainty, δmodel, was
estimated by varying the input elastic and inelastic cross sections within their uncertain-
ties and found to be below 2%, except for a few bins, where it went up to 4.3% (3.1%)
at maximum for the proton (deuteron) case. This is negligible compared to the overall
normalization uncertainty of our data of about 7% (see below).
In each bin, half of the deviations of yields obtained in a Monte Carlo simulation with
estimated geometric misalignments from the yields obtained in a Monte Carlo simulation
with aligned (ideal) geometry serve as an estimate of the systematic uncertainty due to
misalignment. The uncertainty due to misalignment, δmis., reaches values of up to 5.4%.
However, the majority of the data points has an uncertainty due to misalignment that is
smaller than 3%.
Dependence on misalignment of the efficiency Ee.m.for elastic and quasi-elastic radiative
The efficiency Ee.m. and its dependence on misalignment were studied in Monte Carlo
simulations. The assignment of a corresponding systematic uncertainty is accomplished
by applying the values of Ee.m.extracted from a Monte Carlo simulation with aligned and
misaligned geometry to the high-multiplicity radiative events included in the background
term S(i,0)σBorn(0). The difference of the unfolded inelastic scattering Born cross sections
obtained for these efficiencies is assigned as a systematic uncertainty, δrad., due to these
Overall normalization uncertainty.
The normalization uncertainties of the absolute cross sections and the structure functions
are dominated by the uncertainty of the year-dependent luminosity constant CLumi in
Eq. (4.3). The uncertainties of the luminosity constants weighted with the sizes of the data
– 13 –
sets result in an overall normalization uncertainty of 7.6% for the data taken on a hydrogen Download full-text
target and 7.5% for the data taken on a deuterium target.
5.2 Inclusive inelastic scattering cross-section ratio σd/σp
The cross-section ratio can be determined with higher precision than the cross sections
themselves due to the cancellation of the misalignment uncertainty, the PID uncertainty
and, to a large extent, the overall normalization uncertainty. The remaining overall nor-
malization uncertainy of 1.4% is attributed to variations of the beam conditions between
The efficiencies Ee.m.for proton and deuteron are different , and therefore do not
cancel in the proton-to-deuteron cross-section ratio. The uncertainty δrad of the cross-
section ratio is obtained by propagating, for proton and deuteron cross sections, the un-
certainties of efficiencies for high-multiplicity radiative events due to misalignment. It is
found to be less than 2.5% in every kinematic bin.
6. Discussion of the Results
The kinematic conditions of the HERMES inclusive lepton-nucleon scattering cross sections
presented here overlap those of existing data over a large kinematic range. New information
is provided in the region with Q2? 1 GeV2and 15GeV2< W2< 45GeV2, corresponding
to 0.006 < x < 0.04.
6.1 Structure functions Fp
The differential cross sections d2σp,d/dxdQ2for inelastic scattering on the proton and
deuteron as well as the corresponding structure functions Fp
and 6 together with the statistical and systematic uncertainties. The statistical uncertain-
ties of the HERMES data range between 0.4% and 3.0%. Almost 80% of all data points
have a statistical uncertainty smaller than 1%. The overall normalization uncertainty of
7.6% (7.5%) for the inelastic scattering cross section on the proton (deuteron) and the
contribution from misalignment are the dominating systematic uncertainties.
The differential cross sections are shown in Figs. 5 and 6 as a function of Q2in bins of x.
The structure functions are shown in Figs. 7 and 8, together with the available world data
from fixed target (E665 , BCDMS , NMC , SLAC , JLAB [20, 21, 22, 23])
and collider experiments (H1 and ZEUS). The data are overlaid with new fits to world data,
including the data presented here, of inclusive proton (GD11-P) and deuteron (GD11-D)
cross sections. These functions are described in section 6.3.
In the region x ≥ 0.07 and Q2> 1 GeV2, HERMES data are in good agreement with
existing data from SLAC and NMC. The HERMES measurement provides also data in a
previously uncovered kinematic region between JLAB data on the one hand and NMC,
BCDMS, E665 and the collider experiments on the other. This can be clearly seen in
Figs. 7 and 8.
2are listed in Tabs. 5
– 14 –