# Inclusive Measurements of Inelastic Electron and Positron Scattering from Unpolarized Hydrogen and Deuterium Targets

**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 $F_2^p$ and $F_2^d$ are 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 $0.006\leq x\leq 0.9$ and 0.1 GeV$^2\leq Q^2\leq$ 20 GeV$^2$,

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 determined.

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**ABSTRACT:**We use all the available new precise data for deep inelastic and related hard scattering processes to perform NLO global parton analyses. These new data allow an improved determination of partons and, in particular, the inclusion of the recent measurements of the structure functions at HERA and of the inclusive jets at the Tevatron help to determine the gluon distribution and alpha_S better than ever before. We find a somewhat smaller gluon at low x than previous determinations and that alpha_S(M_Z^2) = 0.119 +/- 0.002 (expt.) +/- 0.003 (theory).European Physical Journal C 10/2001; 23. · 5.25 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Measurements of the totel cross section for the processes e + p yields ; e' + n + pi /sup +/ and e + p yields e' + p + pi Â° are reported for a ; wide range of center-of-mass energies and momentum transfers extending above the ; first pionnucleon resonance and to momentum transfers of 20 Fâ»Â². Only the ; final electron is observed in this experiment. Results are analyzed in terms of ; nucleon form factors using experimental pion-nucleon phase shifts and the theory ; of Fubini, Nambu, and Wataghin. In general, the data seem consistent with ; current picture of nucleon structure, except for a preference for a negative ; rather than positive neutron-electric form factor, G/sub En/. It is demonstrated ; from the electron angular distribution for constant momentum transfer and ; constant center-of-mass energy that pion electroproduction does in fact occur ; primarily through transverse currents. The general form of the separation into ; transverse and scalar photons for inelastic or elastic electron scattering is ; discussed. In addition, an approximate formula for the background process of ; wide-angle bremsstrahlung is quoted which appears to be accurate to 1--2% over a ; very wide range of electron and photon energies when compared to a numerical ; computation by a digital computer. (auth);Physical Review - PHYS REV X. 01/1963; 129(4):1834-1846. - SourceAvailable from: ArXiv
##### Article: New Global Fit to the Total Photon-Proton Cross-Section sigma L+T and to the Structure Function F2

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**ABSTRACT:**A fit to world data on the photon-proton cross section sigma L+T and the unpolarised structure function F2 is presented. The 23-parameter ALLM model based on Reggeon and Pomeron exchange is used. Cross section data were reconstructed to avoid inconsistencies with respect to R of the published F2 data base. Parameter uncertainties and correlations are obtained.09/2007;

Page 1

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

Deuterium Targets

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

determined.

2and Fd

Keywords: Lepton-Nucleon Scattering.

Page 2

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,

USA

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 –

Page 3

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,

Italy

11Department of Subatomic and Radiation Physics, University of Gent, 9000 Gent,

Belgium

12Physikalisches Institut, Universit¨ at Gießen, 35392 Gießen, Germany

13Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ,

United Kingdom

14Department of Physics, University of Illinois, Urbana, Illinois 61801-3080, USA

15Randall Laboratory of Physics, University of Michigan, Ann Arbor, Michigan

48109-1040, USA

16Lebedev Physical Institute, 117924 Moscow, Russia

17National Institute for Subatomic Physics (Nikhef), 1009 DB Amsterdam, The

Netherlands

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 –

Page 4

1. Introduction

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 [1]. 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. [2] and [3]) 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 [15],

EMC [16], NMC [17] and E665 [18] (√s∼= 12 − 31 GeV), experiments with electrons at

SLAC [19] (√s ≤ 7 GeV) and at JLAB [20, 21, 22, 23]) (√s ≤ 3.25 GeV). The HERMES

experiment [24] 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 –

Page 5

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.

2. Formalism

ml

Lepton mass (taken to be negligible)

M

Mass of target nucleon

k = (E,?k), k′= (E′,?k′)

4–momenta of the initial and final state leptons

θ, φPolar and azimuthal angle of the scattered lepton

P

lab

= (M,0)

4–momentum of the initial target nucleon

q = k − k′

4–momentum of the virtual photon

Q2= −q2lab

ν =P · q

M

Q2

2P · q=

y =P · q

P · k

W2= (P + q)2= M2+ 2Mν − Q2

≈ 4EE′sin2 θ

2

Negative squared 4–momentum transfer

lab

= E − E′

Energy of the virtual photon in the target rest

frame

x =

Q2

2Mν

ν

E

Bjorken scaling variable

lab

=

Fractional energy of the virtual photon

Squared invariant mass of the photon–nucleon

system

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):

d2σ

dx dQ2=4πα2

em

Q4

?

y2· F1(x,Q2) +

?1 − y

x

−My

2E

?

· F2(x,Q2)

?

, (2.1)

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

d2σ

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 –

Page 6

parameter ǫ is the ratio of virtual-photon fluxes for longitudinal and transverse polariza-

tions [26]. 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):

F1(x,Q2) =

1

4π2αem

1

4π2αem

MK · σT(x,Q2) ,

νK

Q2

4M2x2

(2.3)

F2(x,Q2) =

1 +

·?σL(x,Q2) + σT(x,Q2)?

, (2.4)

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:

The longitudinal-to-transverse

R(x,Q2) =σL

σT

=

?

1 +4M2x2

Q2

?

F2(x,Q2)

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):

d2σ

dx dQ2=4πα2

em

Q4

F2(x,Q2)

x

?

1 − y −Q2

4E2+

y2+ Q2/E2

2[1 + R(x,Q2)]

?

.(2.6)

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 [24] 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 –

Page 7

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 [31] 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

Etrigger

·1 − Chad

Elep

.(4.1)

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

Sect. 4.5):

d2σExp

dx dQ2(x,Q2) =Nevents(x,Q2)

∆x ∆Q2

·1

L.

(4.2)

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 [32]. Two more

corrections related to detector geometry and alignment are discussed in Sects. 4.7 and 4.8.

Born≡

2and Fd

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 –

Page 8

1

10

10

-2

10

-1

1

x

Q2 [ GeV2]

y<0.85

Θ<0.22 rad

W2>5.0 GeV2

y>0.1

Θ>0.04 rad

Q2=1 GeV2

A

B

C

D

E

F

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

Q2.

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.

Yearevents, in million

proton

2.3

4.6

9.5

16.4

deuteron

1996

1997

2000

total

2.8

3.1

12.6

18.5

Table 2:

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

deuteron.

Number of raw events

– 7 –

Page 9

4.2 Trigger

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.

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

10

-2

10

-1

trigger efficiency

Top Detector

x

trigger efficiency

Bottom Detector

A

B

C

D

E

F

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 –

Page 10

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.

92

94

96

98

100

102

lepton id. efficiency[%]

x

hadron contamination[%]

A

B

C

D

E

F

0

0.5

1

1.5

2

2.5

3

3.5

10

-2

10

-1

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 –

Page 11

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.

4.5 Luminosity

The integrated luminosity L per nucleon is calculated as flollows:

L =

?

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)

cs[%]

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

Z.

(4.3)

x

A

B

C

D

E

F

0

2

4

6

8

10

12

14

10

-2

10

-1

Figure 4:

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 –

Page 12

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[25] and the parameterization R1990[33] for R, while

that for the deuteron is derived from the same parameterizations in conjunction with the

fit [34] to Fd

2data from NMC, SLAC and BCDMS. Radiative effects are simulated with

RADGEN [35]. 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. [36] and [37].

When using the more recent parameterizations for the proton from Ref. [38], the results

are essentially the same.

The probabilistic information about event migration can be summarized in a smearing

matrix [39],

2/Fp

S(i,j) =

∂σExp(i)

∂σBorn(j)=

n(i,j)

nBorn(j). (4.4)

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

acceptance.

The inverted squared submatrix S′(i,j) = S(i,j > 0) relates the measured distribu-

tions to the distributions at Born level:

σBorn(j) =

?

i

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 –

Page 13

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

uncertainty.

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 –

Page 14

5. Systematic Uncertainties

5.1 Inclusive inelastic scattering cross sections

Particle identification.

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.

Nevertheless,

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).

Misalignment.

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

events.

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

radiative corrections.

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 –

Page 15

sets result in an overall normalization uncertainty of 7.6% for the data taken on a hydrogen

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

data sets.

The efficiencies Ee.m.for proton and deuteron are different [41], 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

2and Fd

2

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 [18], BCDMS [15], NMC [17], SLAC [19], 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.

2and Fd

2are listed in Tabs. 5

– 14 –

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