Longitudinal-transverse separations of deep-inelastic structure functions at low Q2 for hydrogen and deuterium.
ABSTRACT We report on a study of the longitudinal to transverse cross section ratio, R=sigmaL/sigmaT, at low values of x and Q2, as determined from inclusive inelastic electron-hydrogen and electron-deuterium scattering data from Jefferson Laboratory Hall C spanning the four-momentum transfer range 0.06<Q2<2.8 GeV2. Even at the lowest values of Q2, R remains nearly constant and does not disappear with decreasing Q2, as might be expected. We find a nearly identical behavior for hydrogen and deuterium.
- Physical Review Letters 07/1988; 60(25):2591-2594. · 7.94 Impact Factor
arXiv:nucl-ex/0611023v1 13 Nov 2006
Longitudinal-Transverse Separations of Structure Functions at Low Q2for Hydrogen
V. Tvaskis,1,2M. E. Christy,3J. Arrington,4R. Asaturyan,5O. K. Baker,3H. P. Blok,1,2P. Bosted,6M. Boswell,7
A. Bruell,8A. Cochran,3L. Cole,3J. Crowder,9J. Dunne,10R. Ent,6H. C. Fenker,6B. W. Filippone,11
K. Garrow,6A. Gasparian,3J. Gomez,6H.E. Jackson,4C. E. Keppel,3,6E. Kinney,12Y. Liang,3,13
W. Lorenzon,14A. Lung,6D. J. Mack,6J. W. Martin,8K. McIlhany,8D. Meekins,6R. G. Milner,8
J. H. Mitchell,6H. Mkrtchyan,5B. Moreland,6V. Nazaryan,3I. Niculescu,15A. Opper,16R. B. Piercey,10
D.H. Potterveld,4B. Rose,6Y. Sato,3W. Seo,17G. Smith,6K. Spurlock,10G. van der Steenhoven,2
S. Stepanyan,5V. Tadevosian,5A. Uzzle,3W. F. Vulcan,6S. A. Wood,6B. Zihlmann,1and V. Ziskin8
1Vrije Universiteit, 1081 HV Amsterdam, The Netherlands
2National Instituut voor Kernfysica en Hoge-Energiefysica (NIKHEF), 1009 DB Amsterdam, The Netherlands
3Hampton University, Hampton, Virginia 23668
4Argonne National Laboratory, Argonne, Illinois 60439
5Yerevan Physics Institute, 375036, Yerevan, Armenia
6Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606
7Randolph-Macon Woman’s College, Lynchburg, Virginia 24503
8Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
9Juniata College, Huntingdon, Pennsylvania 16652
10Mississippi State University, Mississippi State, Mississippi 39762
11California Institute of Technology, Pasadena, California 91125
12University of Colorado, Boulder, Colorado 80309
13American University, Washington, D.C. 20016
14University of Michigan, Ann Arbor, Michigan 48109
15The George Washington University, Washington, D.C. 20052
16Ohio University, Athens, Ohio 45071
17Kyungpook National University, Taegu 702-701, Korea
(Dated: February 4, 2008)
We report on a study of the longitudinal to transverse cross section ratio, R = σL/σT, at low
values of x and Q2, as determined from inclusive inelastic electron-hydrogen and electron-deuterium
scattering data from Jefferson Lab Hall C spanning the four-momentum transfer range 0.06 < Q2<
2.8 GeV2. Even at the lowest values of Q2, R remains nearly constant and does not disappear with
decreasing Q2, as expected. We find a nearly identical behaviour for hydrogen and deuterium.
PACS numbers: 13.60.-r,12.38.Qk,13.90.+i,13.60.Hb
Since the early experiments at the Stanford Linear Ac-
celerator Center (SLAC), which discovered the substruc-
ture of the nucleon and led to the development of the
quark parton model , deep inelastic scattering (DIS)
has been a powerful tool in the investigation of the par-
tonic substructure of the nucleon. After decades of ex-
periments with electron and muon beams, the nucleon
structure function F2(x,Q2) is known with high preci-
sion over many orders of magnitude in x and Q2.
Here, Q2is the negative square of the four-momentum
transfer of the exchanged virtual photon in the scatter-
ing process. The Bjorken scaling variable x = Q2/2Mν,
with M the nucleon mass and ν the energy transfer, can
be interpreted as the fraction of the nucleon momentum
carried by the struck parton.
In the region of large Q2and ν, the results of DIS
measurements are typically interpreted in terms of par-
tons (quarks and gluons). In this case, the theoretical
framework is provided by perturbative Quantum Chromo
Dynamics (pQCD), which includes logarithmic scaling
violations.This description starts to fail when non-
perturbative effects such as higher-twist interactions be-
tween the struck quark in the scattering process and
other quarks or gluons in the nucleon become impor-
tant. The sensitivity for higher twist effects increases
with decreasing Q2, since they are proportional to pow-
ers of 1/Q2.
There is great interest in the behaviour of the nucleon
structure functions in the low Q2region where the tran-
sition from perturbative to non-perturbative QCD takes
place. However, little is known about this behaviour,
since at large invariant mass W of the hadronic system
there are few data points in this region, except for the
(transverse) cross section σT at exactly Q2= 0, which is
accessible through real photon absorption experiments.
The more plentiful data at low W are typically inter-
preted in terms of nucleon resonance excitations.
The differential cross section for inclusive electron scat-
tering, after dividing by the virtual photon flux factor
(Γ), can be written as
dΩdE′= σT+ εσL, (1)
where ε is the virtual photon polarization and σL(σT) is
the longitudinal (transverse) virtual-photon absorption
cross section, which depends on x and Q2. Current con-
servation determines the behaviour of the structure func-
tions for Q2→ 0, leading to
= O(Q2), (2)
where FLand F1are the longitudinal and transverse nu-
cleon structure functions. The value of Q2at which this
behavior becomes manifest is however neither predictable
nor yet observed.
While there is a wealth of data for F2 = (2xF1+
FL)/(1 + Q2/ν2), relatively few data exist for FL, or
equivalently R. Data on R(x,Q2) on hydrogen and deu-
terium are available in a limited x and Q2(Q2> 1 GeV2)
range, with a typical uncertainty of 0.1-0.2 [3, 4], compa-
rable to the size of R itself. For scattering from point-like
spin-1/2 particles, R should vanish at large values of Q2
because of helicity conservation. At low values of Q2,
however, R is not small, and typical values are about
Precision data on R are necessary for several reasons.
Most importantly, determinations of the structure func-
tion F2from cross section measurements, and the parton
distributions derived therefrom, need numerical values
for R. If the former are not based on longitudinally and
transversely separated measurements, the uncertainties
in F2introduced by assumptions for R can be as large as
20%. Furthermore, especially in the low Q2region, data
are needed to study higher twist effects and to search for
the onset of the current conservation behaviour of the
structure functions at low Q2described above.
The determination of R is typically accomplished via
a Rosenbluth-type separation technique, which requires
high precision measurements of the cross section at fixed
values of x and Q2, but at different values of ε. This
technique requires the use of at least two beam energies
and correspondingly different scattered-electron angles,
θ. Only in some experiments have such measurements
actually been performed [3, 4, 5, 6, 7, 8, 9, 10].
In the framework of perturbative QCD, there is no
requirement that R be the same for protons and neutrons.
Previous results have shown RD- RH, the difference in
R from hydrogen and deuterium targets, to be consistent
with zero [3, 9, 10]. However, these measurements were
carried out mainly at higher Q2, where R itself is small
and any difference, therefore, difficult to ascertain.
In this paper we present results from a study of R
for both hydrogen and deuterium at low values of Q2.
The experiment (E99-118) was carried out at the Thomas
Jefferson National Accelerator Facility (Jlab). Data were
obtained at 0.007< x < 0.55 and 0.06 < Q2< 2.8 GeV2,
by utilizing 2.301, 3.419 and 5.648 GeV electron beams at
a current of I = 25 µA. The minimum scattered electron
energy was E′≈ 0.4 GeV and the range of the invariant
mass squared of the hadronic system W2was between 3.5
and 10 GeV2. Electrons scattered from 4 cm long liquid
hydrogen (H) and deuterium (D) targets were detected
in the High-Momentum Spectrometer (HMS) in Hall C
at various angles between 10◦and 60◦.
The inclusive double differential cross section for each
energy and angle bin within the spectrometer acceptance
was determined from
where ∆Ω (∆E′) is the bin width in solid angle (scattered
energy), L is the total integrated luminosity, and Ycorris
the measured electron yield after correcting for detector
inefficiencies, background events, and radiative effects.
To account for backgrounds from π0production and
its subsequent decay into two photons followed by pair
production of electron-positron pairs, positron data were
also taken by reversing the polarity of the spectrometer.
Other background contributions include electron scatter-
ing from the aluminum walls of the cryogenic target cells
and electroproduced negatively charged pions. Events
from the former were subtracted by performing substi-
tute empty target runs, while events from the latter were
identified and removed by use of both a gas Cherenkov
counter and an electromagnetic calorimeter. Additional
details regarding the analysis and the standard Hall C
apparatus employed in this experiment can be found in
Radiative effects including bremsstrahlung, vertex cor-
rections and loop diagrams are calculated using the ap-
proach by Bardin et al . Additional radiative effects
in the target and its exit windows were determined us-
ing the formalism of Mo and Tsai . The calculation
of such effects includes the emission of one hard photon.
There is, however, the possibility that the electron could
emit two hard photons. The calculation of this process is
unfortunately not fully established and the correspond-
ing effect was therefore treated in the present analysis as
a separate uncertainty.
For each energy bin, a weighted average cross section
over θ within the spectrometer acceptance was obtained
by using a model to correct for the angular variation
of the cross section from the central angle value.
minimize the dependence on the model used to compute
both this correction and the radiative effects, an iterative
procedure was employed.
The total systematic uncertainty in the differential
cross section was taken as the quadratic sum of all the
systematic uncertainties contributing to the cross section
measurement. In a Rosenbluth separation one needs to
distinguish between uncertainties that are correlated be-
tween measurements at different ε, such as uncertainties
in target thickness and integrated charge, and uncorre-
lated ones. Not including the contributions from radia-
tive corrections, the uncorrelated systematic uncertain-
ties on the cross section measurements in this experiment
amounted to 0.9%, while the total systematic uncertainty
The size, and consequently the uncertainty, of the ra-
diative effects strongly depends on the kinematics and is
largest at low values of E′where the measured cross sec-
tion is dominated by events from elastic or quasi-elastic
scattering with the emission of one or more photons in
the initial or final state. The estimate of these uncertain-
ties was determined by varying all relevant input cross
sections within their uncertainties, and amounted to as
much as 1.5% for hydrogen and 8.5% for deuterium in the
most extreme cases considered. It rapidly decreased for
higher values of E′, i.e. higher values of x and Q2. The
much larger uncertainty in the deuterium cross section
is due to the contribution from quasielastic scattering
which can only be modelled approximately.
The extractions of purely longitudinal and transverse
cross sections and structure functions were accomplished
via the Rosenbluth technique, where the reduced cross
section is fit linearly as a function of ε, as in Eq.(1). The
intercept of such a fit gives the transverse cross section σT
(and therefore the structure function F1(x,Q2)), while
the slope gives the longitudinal cross section σL, from
which the structure functions R(x,Q2) or FL(x,Q2) can
The results for R(x,Q2) in hydrogen are shown in Fig.
1 as a function of Q2for fixed values of x (red squares).
The inner error bars represent the combined statistical
and uncorrelated systematic uncertainties. The total er-
ror bars represent the statistical and systematic uncer-
tainties added in quadrature. The data are compared
to the results of previous measurements at Jlab (E94-
110) , SLAC , and by the EMC , NMC  and
BCDMS  collaborations at CERN.
The structure function F2 determined through the
Rosenbluth separation technique was found to agree to
better than 2% with a Regge-motivated parameteriza-
tion of all previously available deep-inelastic scattering
data , even at very low values of Q2. This fa-
cilitated the utilization of an alternative approach where
R was calculated using this parametrization (including
a 2% uncertainty) for the structure function F2(x,Q2)
1 + εR
1 + R
The results (shown by the blue circles in Fig. 1) cover
a larger kinematic range than the results from the Rosen-
bluth separation, since the measurement of this experi-
ment is combined via the model with results of previous
experiments at different energies. The inner error bars
represent the statistical uncertainty and the total error
bars represent the quadratic sum of the statistical un-
certainty and the uncertainty in the radiative corrections
x = 0.030
E99-118 (Ros. Sep.) E99-118 (Ros. Sep.)
E99-118 (Mod. Dep.)E99-118 (Mod. Dep.)
x = 0.050
x = 0.075x = 0.100
x = 0.175x = 0.250
x = 0.325
x = 0.400
FIG. 1: Comparison of the values of R(x,Q2) for hydrogen
from the present experiment (E99-118) to the results of other
experiments. The dashed curve represents the parametriza-
developed in  (see text for details.)
e99118(x,Q2) and the solid curve represents the model
due to the possible emission of two hard photons. The
shaded bands represent the uncertainties in the radiative
effects due to uncertainties in the input cross sections.
Especially in the model dependent extraction the uncer-
tainties are dominated by the unceratinties in the radia-
tive corrections which are correlated between data points.
The results from the second method agree very well with
those obtained from the Rosenbluth separation method.
Good agreement is also found with previous experiments
in the regions of x and Q2where the data overlap.
As mentioned above, at low values of Q2, current con-
servation requires R to be proportional to Q2. However,
in the data from the present experiment this behaviour is
not yet observed, and R remains nearly constant over the
measured range in Q2. Thus, the transition to R ∝ Q2
must occur below a Q2value of about 0.1 GeV2at low
x, or below 1 GeV2for x > 0.2. This result will have
a direct impact on structure function extractions at low
The dashed curve in Fig.
parametrization of R (RH
available data including those from this experiment. The
functional form of the parametrization has been chosen
to satisfy the condition that R vanishes as Q2goes to
zero. At Q2= 2 GeV2, it was connected to a previously
obtained parametrization from SLAC  that is based on
measurements at higher values of Q2. The solid curves
in the upper four panels of Fig. 1 show, within its range
of applicability (x ≤ 0.1), the model developed in ,
1 represents a new
e99118(x,Q2) ) based on all
RD - RH
FIG. 2: The difference RD− RHas a function of Q2from
the present experiment calculated via Rosenbluth separation.
The data from previous experiments are also shown for Q2<
which is based on the photon-gluon fusion mechanism
and extrapolated into the region of low Q2.
Both the Rosenbluth separation and the model depen-
dent extraction of R were also carried out for the deu-
terium data. While the precision of the results from the
Rosenbluth separation is comparable to that of the hy-
drogen data, the systematic uncertainty in the model de-
pendent extraction is much bigger for the deuterium data
due to a large uncertainty in the calculation of the quasi-
elastic radiation tail which is significant at low x and Q2.
Thus, the difference RD− RHwas calculated using
only the results from the Rosenbluth separation method,
compiled in Table I, and compared in Fig. 2 to previous
results from NMC  and SLAC . In this plot only
the most recent and precise data from SLAC  are
shown, while additional SLAC results  are included in
the statistical analysis of RD− RH, below. Previously,
the conclusion was drawn that there is no difference be-
tween RDand RH.However, most of the data from
NMC are at rather high Q2values, where R itself is small,
and the RD−RHvalues extracted from the SLAC mea-
surements  were averaged over all Q2, including high
Q2values, and were hence biased towards smaller differ-
ences. Including our results, the data are still consistent
with RDbeing identical to RH. However, at values of
Q2< 1.5 GeV2there is a hint both in the present data
and in the highest-precision data from SLAC  that
RDis smaller than RH. The global average (including
all data) yields RD− RH= -0.054 ± 0.029.
TABLE I: The values of RH and RD- RHcalculated via
the Rosenbluth separation. Note that the systematic error
in the difference accounts for the correlation between the un-
certainties in the hydrogen and deuterium data. Complete
data tables and the new parametrization of R(x,Q2) may be
requested via email (firstname.lastname@example.org, email@example.com).
Stat. Syst. RD- RHStat. Syst.
0.150 0.041 0.259 0.074 0.153
0.175 0.050 0.307 0.056 0.188
0.273 0.077 0.460 0.049 0.132
0.283 0.081 0.414 0.045 0.117
0.476 0.156 0.283 0.063 0.025
0.5080.091 0.406 0.038 0.168
1.0450.200 0.335 0.048 0.041
1.6700.320 0.211 0.038 0.021
The results presented here are measurements of the
longitudinal to transverse cross section ratio below Q2
of about 2 GeV2for hydrogen and deuterium targets.
These data appear in a region where R was expected
to disappear as Q2gets very small. However, a nearly
constant behaviour of RHand RDis observed down to
Q2of about 0.1 GeV2at low values of x. For Q2< 1.5
GeV2, the data hint at a small difference between RD
This work is supported in part by research grants
from U.S. Department of Energy, the U.S. National Sci-
ence Foundation, and the Stichting voor Fundamenteel
Oderzoek der Materie (FOM) of the Netherlands. The
Southeastern Universities Research Association operates
the Thomas Jefferson National Accelerator Facility un-
der the U.S. Department of Energy contract DEAC05-
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