Recent Measurements of Longitudinal and
Transverse Unpolarized Structure Functions,
and Their Impact on Spin Asymmetry
Measurements
1
C. E. Keppel
Hampton University / Jefferson Lab
Abstract. New measurements are presented of the separated longitudinal and transverse proton structure functions in the nucleon resonance region (1 < W 2 < 4 GeV2 ), spanning the fourmomentum transfer range 0.2 < Q 2 < 4.0 (GeV/c)2 . The results are from Jefferson Lab experiment
E94-110, which measured unpolarized inclusive electron-proton cross sections in Hall C for the purpose of performing Rosenbluth-type separations. Results of the analysis of the data are presented,
as well as a discussion of their non-trivial relevance to spin asymmetry measurements.
MOTIVATION
The determination of the nucleon spin structure functions, g 1 and g2 , from electron spin
asymmetry measurements requires knowledge of the longitudinal/transverse (L/T) separated unpolarized structure functions. At lower values of Q 2 , the region of the nucleon
resonances covers larger fractions of the Bjorken x range. Determining the small Q2
behavior of the structure functions at large x, therefore, requires precision L/T measurements in the resonance region. High precision measurements of Rx Q 2 σL σT , the
ratio of longitudinal to transverse cross sections, have been available for over a decade
in the deep inelastic scattering (DIS) regime. However, in the region of the resonances
there is very little data on Rx Q2 and, therefore, the longitudinal and transverse structure functions, FL x Q2 and F1 x Q2 . The small number of measurements that exist
vary in the range (01 R 04) and have typical errors of 100% or more. The world’s
R data prior to the recent E94-110 experiment at Jefferson Lab are plotted in figure 1,
for all Q2 9 GeV c2 . This data set can provide neither knowledge of the resonance
structure nor information on the Q2 dependence of Rx Q2 in this regime.
UNPOLARIZED STRUCTURE FUNCTIONS AND SPIN
ASYMMETRIES
Both the virtual photon spin asymmetries, A 1 and A2 , and the spin structure functions
can be determined from measurements of the parallel and perpendicular electron spin
1
Work supported in part by a grant from the National Science Foundation.
CP675, Spin 2002: 15th Int'l. Spin Physics Symposium and Workshop on Polarized Electron
Sources and Polarimeters, edited by Y. I. Makdisi, A. U. Luccio, and W. W. MacKay
© 2003 American Institute of Physics 0-7354-0136-5/03/$20.00
294
FIGURE 1. World’s data on RW 2 Q2 in the resonance region for all Q 2 9 GeV c2 prior to E94110.
asymmetries, A and A . For the transverse asymmetry,
A1 C
A dA D (1)
The factor C is kinematic only, while the photon depolarization factor,
1 ε E E 1 εR
¼
D
(2)
is a function of the unpolarized structure function ratio R. ε is defined in the following
section. For ε R 1, the fractional uncertainty in A 1 coming from an uncertainty in R
of δ R is
∆A1 δ R
ε
δ R εδ R
(3)
A1
1 εR
Using either the average of the previous world’s resonance region data, R = 0.06, or an
extrapolation of SLAC DIS fits, R 021, both of which are used in the literature, results
in a difference of δ R 015. At ε 05 this leads to an uncertainty in A 1 of 9%.
The spin structure function g1 can be extracted via
g1 F 1 ε R
F1 A1 γ A2 ∝ F1 1 ε R ∝ 2
2
1 γ 1R
(4)
Therefore, at ε 1, only knowledge of F2 is needed for the extraction of g1 . However, F2
can not be measured independently from R except at ε = 1, and all previous extractions
295
of F2 in the resonance region have required an assumption for the value of R. The percent
difference in F2 extracted from cross section measurements assuming a value for R of
either 0.2 or 0.0 (both choices, again, are used in the literature) varies from 2% to
16% at typical JLab resonance region kinematics. [10]
EXPERIMENT
In the one photon exchange approximation, the cross section for unpolarized inclusive
electron-proton scattering can be expressed in terms of the helicity coupling between the
photon and proton as
dσ
dΩdE
¼
Γ
σT x Q2 εσL x Q2 (5)
where σT and σL are the photo-absorption cross sections for pure transversely and
longitudinally polarized photons, respectively, Γ is the flux of transverse virtual photons
and ε is the photon relative longitudinal polarization. In terms of the structure functions
F1 x Q2 and FL x Q2 , the double differential cross section can be written as
dσ
dΩdE
¼
Γ
4π 2 α
2xF1 x Q2 ε FL x Q2 2
2
xW M p (6)
Direct correspondence between equations 5 and 6 shows that F1 x Q2 is purely transverse, while the combination
FL x Q2 1 4M 2p x2
F2 x Q2 2xF1 x Q2 Q2
(7)
is purely longitudinal.
For E94-110, the separation of the measured differential cross section into longitudinal and transverse strengths was accomplished via the Rosenbluth technique [1]. Measurements were made over a range in ε at fixed x, Q2 and d σ Γ was fit linearly in ε .
The intercept of the fit gave σT (and therefore F1 x Q2 ), while the slope gave the structure function ratio Rx Q2 σL σT FL x Q2 2xF1 x Q2 . These separations were
performed at all Q2 , W 2 , where enough range in ε existed to allow a good fits. A single
Rosenbluth fit and the extracted value of R is shown in figure 2. In total, over 180 Rosenbluth fits were performed and the separated F1 x Q2 and FL x Q2 structure functions
were fit globally as a function of x and Q2 . This data allows, for the first time, both the
resonance structure and the Q2 dependence of the separated structure functions to be
studied.
RESULTS AND CONCLUSIONS
The Rosenbluth extracted values (triangles) of 2xF1 and FL are plotted in figure 3 as a
function of x for various Q2 bins and include 80% of the data. Also plotted are the
results of a two-dimensional fitting of the data in Q 2 and W 2 (solid line). A fit to DIS data
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80000
70000
60000
50000
40000
30000
20000
10000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FIGURE 2. A example Rosenbluth separation, with the W 2 Q2 and extracted value for the RW 2 Q2 on the plot.
from SLAC [9] (dashed line) is seen to extend smoothly to the current data. Combining
the DIS and resonance regimes will allow for the lower Q2 moments of the separated
structure functions to be extracted for the first time. Resonant structure can be seen for
the first time in RW 2 Q2 , which is plotted versus W 2 in figure 4.
The data represent the first high precision measurement of the resonance region L/T
separated unpolarized structure functions for the proton. The large kinematic range measured allows for a determination of the Q2 dependence of individual resonance regions.
The new data will allow for the systematic uncertainties in asymmetry measurements
associated with uncertainties in the unpolarized structure functions to be reduced significantly.
REFERENCES
1. M. N. Rosenbluth, Phys. Rev. 79, 615 (1956)
2. F.W. Brasse et al , Nucl. Phys. B110, 413 (1976)
3. L. W. Whitlow et al , Phys. Lett B250, 193 (1990)
4. L.H. Tao, Ph.D. Thesis, The American University (1994)
5. C.E. Keppel, Ph.D. Thesis, The American University (1994)
6. I. Niculescu, R. Ent, C.E. Keppel, Phys. Rev. Lett. 85, 1186 (2000)
7. C.E. Carlson and N.C. Mukhopadhyay, Phys. Rev. D41, 2343 (1990)
8. C.E. Carlson and N.C. Mukhopadhyay, Phys. Rev. D47, 1737 (1993)
9. K. Abe et al , Phys. Lett. B452, 194 (1999)
10. I. Niculescu. Ph.D. Thesis, Hampton University (1999)
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FIGURE 3. Extracted 2xF1 (Left), and FL (Right), as a function of Bjorken x for various ranges in
Q2 . The Rosenbluth separated data (blue triangles) are plotted with the full uncertainties (statistical +
systematic). Also plotted are the results of a two-dimensional fit in Q 2 and W 2 to the data (solid curve).
The position of the ∆P33 1232 resonance is indicated by the red arrow.
FIGURE 4. R as a function of W 2 for various ranges in Q 2 . The Rosenbluth separated data (triangles)
are plotted with the full uncertainties (statistical + systematic).
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