Zeitschrift
Partides
Dr PhysikC
Z. Phys. C - Particles and Fields 28, 321-333 (1985)
andF ds
@ Springer-Verlag1985
Q2 Dependence of the Proton and Neutron Structure Functions
from Neutrino and Antineutrino Scattering in Deuterium
D. Allasia v, C. Angelini 5, A. Baldini s, L. Bertanza 5, A. Bigi s, V. Bisi 7, F. Bobisut 4, T. Bolognese 6,
A. Borg 6, E. Calimani 4, p. Capiluppi 3, R. Casali s, S. Ciampolillo 4, R. Cirio 7a, j. Derkaoui 3 b,
M.L. Faccini-Turluer 6, V. Flaminio s, A.G. Frodesen 2, D. Gamba 7, G. Giacomelli 3, H. Huzita 4,
B. Jongejans 1, I. Lippi 6c M. Loreti 4, C. Louedec 6, G. Mandrioli a, A. Margiotta 3,
A. Marzari-Chiesa 7, A. Nappi s, R. Pazzi s, L. Riccati 7, A. Romero 7, A.M. Rossi 3, A. Sconza 4,
P. Serra-Lugaresi 3, A, Tenner I, G.W. van Apeldoorn i, P. van Dam 1, N. van Eijndhoven 1, D. Vignaud 6,
C. Visser 1a, R. Wigmans 1a
1 N I K H E F - H , NL-1098 SJ Amsterdam, The Netherlands
z Institute of Physics, University, N-5014 Bergen, Norway
3 Dipartimento di Fisica dell'Universit/tand INFN, 1-40126Bologna, Italy
4 Dipartimento di Fisica dell'Universit/tand INFN, 1-35100 Padova, Italy
s Dipartimento di Fisica dell'Universit/~and INFN, 1-56100 Pisa, Italy
6 D6partement de Physique des Particules E16mentaires,CEN Saclay, F-91191 Bures-sur-Yvette,France
7 Istituto di Fisica dell'Universitfiand INFN, 1-10100Torino, Italy
Received 8 March 1985
Abstract. 12,100 vD and 10,500 VD charged current
interactions in deuterium measured in the BEBC
bubble chamber were used to obtain the complete
set of structure functions of proton and neutron. The
x and Q2 dependence of the structure functions of
up and down valence quarks and antiquarks are
presented and discussed. The Adler and Gross-Llewellyn Smith sum rules have been tested at different
Q2 values. A QCD analysis of the four non singlet
structure functions xF~N, xuv, xd~ and F~"-F~ v has
been performed yielding values of ALO between 100
a n d 300 MeV.
1. Introduction
The Q2 dependence of the nucleon structure functions has been studied in deep inelastic scattering
experiments and compared to the predictions of
QCD in the last years. Those results were obtained
using mainly neutrino I-1], electron and muon [2]
scattering on isoscalar targets or on Hydrogen [3].
" Now at CERN, Geneva, Switzerland
u Now at Facult6 de Sciences, Universit6 Mohammed I, Oujda,
Maroc
Now at Dipartimento di Fisica dell'Universit/t, Padova, Italy
d Now at Ministry of Education and Sciences, The Hague, The
Netherlands
In this report we present the first results concerning the study of the x and Q2 dependence of the
proton and neutron structure functions as well as
those of the up and down quarks and antiquarks
obtained from vD and ~D interactions.
The (anti)neutrino-nucleon charged current (CC)
cross section can be written as:
d2a v(~ G2ME[(
Mxy
dxdy- ~
1-Y-2E-+
y2 \
2(R~)
" F2(x, Q2)'~] ( Y - ~ -) xF3(x, Q2)]
(1)
where R=(F2-2xF~)/2xF 1 is the parameter that
measures the Callan-Gross relation [4] violation
and will be taken equal to zero for the whole range
of x and Q2 unless explicitely mentioned. Assuming
isospin symmetry, s = ~ - a n d neglecting the contribution of charm and heavier flavours, one obtains
four effective functions P s that are expressed very
simply in terms of quark distributions:
ff~P(x, QZ)~-2x[d + ~ + ~s](x, Q2),
-FV"tx2,,QZ)~-2x[u+d+~s]( x, Q2),
~P~(x, Q~) ~- 2~ [d -~] (x, Q?),
xff~"(x, Q2)~_2x[u-d](x, Q2).
(2)
322
D. Allasia et al.: QZ Dependence of the Proton and Neutron Structure Functions
The formulae are approximate because the strange
quark contribution depends on the particular x and
Q2bin (see Appendix).
The combination of these four quantities allows
to separate the distributions of different flavours of
valence and sea quarks.
Results concerning the x-dependence of the
structure functions were already published [5]. Hereafter we describe briefly the experimental procedure (Sect. 2) and present the structure functions
at different x and Q2 values in Sect. 3. Two quark
patton model sum rules, that give the number of
quarks in the nucleon, were tested and are discussed
in Sect. 4.
Section 5 describes the analysis of the four non
singlet structure functions xF~ N, xu~, xd v and (F~"
- F ~ p) in the framework of QCD: systematic effects
that affect the determination of the parameter A are
discussed. Section 6 summarizes our conclusions.
2. Experimental Procedure
The experiment was performed exposing the BEBC
bubble chamber filled with deuterium to the C E R N SPS wide band neutrino and antineutrino beams.
The beams were produced by 400 GeV protons incident on a beryllium target (60 cm long for the
neutrino beam, l l 0 c m long for the antineutrino
beam).
The present data are based on the analysis of
75,000 pictures from neutrino runs and 272,000 from
antineutrino runs. A fiducial volume of 18.14m 3,
corresponding to 2.52 tons of deuterium, was used
for primary interactions in the bubble chamber.
The selection of CC events was based on the
two-plane External Muon Identifier (EMI). Only
events with muon momentum, p,, greater than 4
GeV/c were used in this analysis. With this cut the
geometrical acceptance of the EMI is about 97 % for
v interactions and 99 % for ~ interactions.
The whole film was scanned twice for all topologies, except for one-prong events without V~ The
overall scanning efficiency, after two scans, was 99 %;
for two prongs it was 96% and for one prong plus
V ~ it was 85%. All measured events were reconstructed using the standard chain of the C E R N H Y D R A programs; the overall passing rate was
98 % for antineutrino and 96 % for neutrino induced
events. Appropriate weights were applied for scanning efficiencies, passing rates and EMI geometrical
acceptance.
Radiative corrections were applied using the
method proposed by de R6jula et al. [6].
A partial and special scan was performed to
search for one prong events coming from gp--./z + X ~
Table 1
Number of events
<QZ>
(GeV/c) z
vp
vn
~p
~n
raw
corrected
5,571
6,568
7,317
3,189
4,908 _+106
10,497 _+141
9,013 ___203
4,781 _+ 91
9.4
11.2
5.1
4.2
reactions. The one prong contribution to the sample
was found to be (12-t-2)%, about half of which can
be attributed to the elastic process Vp~#+n. The x
and Q2 distributions of the quasi-elastic and deep
inelastic components were estimated using the 3
prong events from this experiment [7] and independently the one prong events with at least one neutral
particle from the Gargamelle-SPS experiment [8].
The (anti)neutrino energy was determined for
each event using the method described in [9], except
for low-Q 2 and low multiplicity events for which
another method was used 1-10]. The events with
incident neutrino energy smaller than 10 GeV were
not used.
To account for resolution smearing effects arising
from the method of energy determination, from Fermi motion and from measurement errors, a Monte
Carlo written by Grant [11] was adapted to our
experimental conditions and correction factors were
calculated for the numbers of events in the x and Q2
bins used in the analysis. For x>0.8, these corrections are large ( > 50 %) and data from this region
are not included in the analysis.
(-)
An event was classified as a v - n interaction if
it had either an even number of prongs, or an odd
number including a proton in the backward direction or with a momentum smaller than 150 MeV/c
(spectator proton). All the remaining odd-prong
(-)
events were classified as v - p
interactions. The
(-)
contamination of ( ~ ) - p interactions from the v - n
interactions and the loss of the latter ones due to
rescattering were corrected for using the procedure
described in [12] with a rescattering fraction f = 0 . 1 2
+0.03. The number of events as well as the mean
values of Q2 for the four samples are given in Table 1.
The experimental structure functions (S.F.)
ff2(x ' Qz) and xff'3(x, Q2) a r e obtained from the measured numbers of v interactions, n ~, and g interactions, n ~, in each x and QZ bin. The properly normalized number of events n v, n ~ can be expressed by
formula (1) multiplied by the v(~) flux q~(E) and
integrated over E and y. The procedure is explained
with some more details in the Appendix.
D. Allasia et al.:
Q2Dependence
of the Proton and Neutron Structure Functions
323
c a l c u l a t i o n w i t h different fluxes o b t a i n e d by statistically d i s t o r t i n g the o r i g i n a l e x p e r i m e n t a l h i s t o g r a m .
T h e e s t i m a t e d errors o b t a i n e d o n the i n d i v i d u a l F i
in each x a n d Q2 bin are m u c h s m a l l e r t h a n the
statistical ones.
T h e flux s h a p e w a s o b t a i n e d b y fitting the experi m e n t a l energy d i s t r i b u t i o n s of the events, corrected
for e n e r g y r e s o l u t i o n s m e a r i n g a n d the cut in p , .
T h e p o s s i b l e s y s t e m a t i c u n c e r t a i n t y i n h e r e n t in such
p r o c e d u r e w a s tested r e p e a t i n g several t i m e s the S.F.
Table 2. Isoscalar structure functions P~N(x, Q2) and xP~N(x, Q2). Systematic errors include the errors on the absolute
normalization, given in the text, and the uncertainty on the shapes of the energy spectra. Ac represents the difference
between the value of the S.F. at the quoted x and Q2 and the average value over the x - Q z bin. p~N is obtained
assuming R~-0.0; the difference A R between the values of p~N obtained with R-=0.1 and those obtained with R m0
is also reported. The x bins used are such that the x in column 2 are the values at the center; the Q2 bin limits are:
1 - 2 - 4 - 8 - 16 - 40 - 100 (GeV/c) 2
Q2 (GeV/c) 2 x
"vN
F2
stat
sys
Ac
AR
xF~ N
stat
sys
Ac
1.5
0.030
0.080
O. 125
0.175
0 9250
0.350
0.450
O. 550
0.700
1.064
1.064
1.032
1.092
O. 907
0.736
0,470
O. 342
0.169
0.047
0.053
0.051
0.060
O. 045
0.050
0.044
O. 040
0.018
0.008
0.010
0.012
0.014
0.012
0.010
0.005
O. 002
0.002
-0.033
-0.013
-0.007
-0.003
O. 004
0.007
0.008
O. 006
0.011
0.027
0.013
0.007
0.004
O. 002
0.001
0.000
O. 000
0.000
0.380
0.611
O. 645
0.717
0.432
0.975
0.076
0.117
O. 151
0.241
0. 260
0.405
0.013
0.021
0.039
0.050
0 9060
0.082
0.016
-0.009
-0.007
-0.004
-0. 002
0.003
3.0
0.030
0.080
0.125
0.175
0.250
0.350
0.450
0.550
0.700
1.106
1.153
1.154
1.114
0.911
0.760
0.453
0.331
0.130
0.062
0.055
0.049
0.050
0.036
0.037
0.030
0.028
0.012
0.028
0.004
0.009
0.013
0.013
0.010
0.005
0.003
0.002
-0.035
-0.015
-0.008
-0.002
0.007
0.011
0.011
0.008
0.O13
0.038
0.027
0.018
0.012
0.006
0.003
0.001
0.000
0.000
0.398
0.508
0.611
0.704
0.874
0.365
0.084
0.089
0.095
O.117
0.105
0.146
0.038
0.007
0.018
0.030
0.040
0,040
0.043
-0.004
-0.004
-0.002
0.001
0.003
6.0
0.030
0.080
0.125
0.175
0.250
0.350
0.450
0.550
0.700
1.188
1.091
1.144
1.066
0.938
0.706
0.531
0.351
0.115
0.099
0.064
0.052
0.049
0.033
0.030
0.028
0.023
0.009
0.052
0.020
0.009
0.006
0.004
0.007
0.006
0.004
0.001
-0.038
-0.013
-0.008
-0.002
0.007
0.011
0.013
0.009
O.O14
0.054
0.037
0.032
0.024
0.015
0.008
0.004
0.002
0.OOO
O.296
0.542
0.688
0.755
0.831
0.624
0.573
0.258
0.099
0.120
0.089
0.080
0.082
0.065
0.068
0.b76
0.073
0.038
0.063
0.028
0.008
0.010
0.019
0.022
0.020
0.015
0.006
0.048
-0,001
-0.002
0.000
0.004
0.008
0.011
0.010
0.012
11.0
0.030
0.080
0.125
0.175
0.250
0.350
0.450
0.550
0.700
1.241
1.098
0.979
0.938
0,859
0.688
0.481
0.296
0.101
O.216
0.098
0.063
0.054
0.034
0.030
0.025
0.020
0.008
0.123
O.O63
O.O28
0.O13
0.004
0.005
0,004
0.003
0.002
-0.041
-0.011
-0.006
-0.002
-0.001
0.003
0,004
O.OOi
0.009
0.081
0.054
0.038
O.O31
0.024
0.015
0.009
0.004
0.001
0.255
0.632
0.720
0.669
0.679
0.639
0.467
0.300
0.099
0.237
0.116
0.082
0.076
0.051
0.049
0.045
0.040
0.019
0.076
0.053
0.036
0.017
0.005
0.008
0.008
0.005
0.003
0.056
0.004
0.000
0.000
-0.001
0.002
0.003
0.006
0.009
24.0
0.080
0.125
0.175
0.250
0.350
0.450
0.550
0.700
1.505
1.083
0.784
0.860
0.567
0.384
0.265
0.088
0.267
0.104
0.068
0.041
0.028
0.022
0.017
0.007
0.117
0.065
0.034
0.026
0.010
0.001
0.001
O.OOO
-0.035
-0.015
-0.004
-0.007
0.000
0.002
0.000
0.007
0.098
0.063
0.039
0.034
0.019
0.012
0.007
0.002
0.660
0.485
0.565
0.699
0.442
0.372
0.269
0.090
0.286
0.117
0.079
0.052
0.038
0.031
0.026
0.011
0.120
0.073
0.046
0.033
0.013
0.004
0.003
0.001
0.013
0.002
0.000
-0.002
0.000
0.001
0.005
0.007
55.0
0.250
0.350
0.450
0.550
0.700
0.697
0.593
0.329
0.189
0.059
0.082
0.051
0.032
0.021
0.008
0.050
0.039
0.025
0.015
0.001
-0.023
-0.008
-0.002
-0.002
0.003
0.043
0.032
0.016
0.008
0.002
0.535
0.575
0.305
0.180
0.059
0.089
0.058
0.038
0.026
0.010
0.055
0.044
0.015
0.010
0.001
-0.008
-0.006
-0.002
0.002
0.003
324
D. Allasia et al.: Q2 Dependence of the Proton and Neutron Structure Functions
Absolute, normalization was imposed using the
average values of the total v and 9- cross sections,
assumed to depend linearly on the energy:
a~/E = (0.636 _ 0.012) 910- 38 cm2/GeV,
a~/E-- (0.306 +0.007)- 10- 3s cmZ/GeV [13].
These world averages, and the underlying hypothesis, are consistent with a cross section measurement obtained in this experiment [10].
The S.F.'s were computed in finite x and Q2
intervals and represent therefore values averaged
over the bins. To get the values of the S.F. at fixed x
and Q2, a simple model was used to parametrize the
scaling violation of the valence and sea quarks 1-14]
and to calculate the correction to be applied to the
various structure functions in each bin.
I IIIIEI I
0.3f
In this section we present the structure functions
obtained according to the procedure explained
above and in the Appendix.
I
l,+
++ {~
X=O.03
~ §
0.0
~
o.o;{
+{ §162t'{
•
+{ ++ +{ {+
X=0.125
+{ § § §
0.2 I
0.0
0.11
0.0
i
i
i
I
0.2
I
I
0.4
I
0.6
0.8
i
i
1.0
x
i
0.2
i
-
9
Vp
o
V Iq
o o
-0.2
-
0.4
0.0
b
0.2
0.4
0.6
0.8
X=0.25
0,1
0.0 ................ §
......
x:o.35
_-..............~.{...,,{....,,+.....,{
......
x:o.,5
: .............
x:o.,o
0.05
0,00
0.02
........
I I [llltll
I I IIIII1[
100
Q2(GeV//c
)2
Fig. 2. x and Q2 dependence of the separate antiquark structure
functions. The errors are purely statistical. Some points with very
large errors, reported in Table 5, are not drawn
-0.4
0.0
X=0.175
+t *+'* '+ tt
I 1111111]
10
i
o x~io{x.Q~/
0.1
000
9
~)O2 Lhis exp.
o
"v Fe CDHSEref.lg]
0.2
{ t ~ l 9 .t~-,LIFeEMC [ref.2c]
I I TIIIH I
~P(x ,(32)
9 •
0.05
0.00
3. Results on the Structure Functions
I J II1111 /
1.0
X
Fig. 1. a The logarithmic derivative d(InF~N)/d(inQ 2) versus x.
Also shown are data from C D H S and E M C experiments. The
line is the Q C D prediction for A = 0 . 2 GeV. b The same derivative
for P~P a n d / ~ " separately
The extraction of the ff/s requires the knowledge
of the parameter R. There are measurements of R in
e,/~ [15] and v experiments [16], performed at different energies and momentum transfers, that provide somewhat different results. Recent analyses of
high statistics v experiments [17] indicate a small
violation of the Gallan-Gross relation at small x and
a possible x dependence, in agreement with theoretical expectations. In view of the present experimental
uncertainties, in the present analysis R - 0 was assumed. The systematic shifts on the values of the
S.F. obtained assuming R - 0 . 1 0 , independent of x
and Q2, will be explicitely given in the Tables.
Table 2 presents the S.F. I~N(x,Q 2) and
2F~r~(x, Q2) for an isoscalar target, i.e. using vD and
gD interactions. Present data from experiments with
larger statistics Ill, g, i] show a common pattern of
scaling violation, although they disagree in absolute
normalization in certain kinematical regions. Our
data exhibit also the same qualitative Q2 dependence: this can be seen in Fig. la, where the logarithmic derivative of ff~N is plotted as a function of
x. The Q2 dependence of ln/72 has been fitted at
fixed x by using a linear dependence on In Q2 in the
Q2 range of our experiment starting from Q2
D. Allasia et al.: Q2 Dependence of the Proton and Neutron Structure Functions
325
Tab|e3. Proton structure functions P~p(x, QZ) and xP~p(x,Q2). The systematic errors contain, in addition to the
isoscalar case, a contribution from the uncertainty on the rescattering fraction
Q2(GeV/c)2 x
1.5
3.0
6.0
11.0
24.0
55.0
F~ p
stat
sys
Ac
AR
xF~ p
stat
sys
Ac
0.030
0.080
0.125
0.995
0.818
0.826
0.079
0.090
0.081
0.016
0.014
0.018
-0.039
-0.008
-0.002
0.025
0.010
0.005
0.344
0.291
0.676
0.127
0.]99
0.243
0.025
0.029
0.054
0.019
-0.002
-0.004
0.175
0,250
0.350
0.450
0.550
0.700
0.784
0.711
0,458
O.191
O.212
O.109
0.092
0.067
0.061
0.057
0.052
0.023
0,021
0.015
0.017
O,O11
O.O11
O.013
0.000
0.007
0.005
0.003
0.002
O.OOO
0.003
0.001
0.000
O.OOO
O.OOO
O.0OO
1.200
0.450
0.311
0.373
0.392
0.508
0.089
0.093
0.149
-0,004
-0.001
0.001
0.030
0.080
0.125
0.909
1.076
0.877
0.102
0.093
0.076
0.027
0.009
0.009
-0.041
-0.012
-0.004
0.032
0.025
0.014
0.022
0.708
0.539
0.440
0.078
0.050
0.046
0.012
0.010
0.009
0.000
0.006
0.006
0.008
0.004
0.002
0.141
0.153
0.149
0.185
0.150
0.187
0.037
0.175
0.250
0.350
0.186
0.162
0.354
0.169
0.525
0.222
0.014
0.020
0.028
0.029
0.048
0.000
-0.001
0.000
0.002
0.002
0.450
0.550
0.700
0.239
0.179
0.067
0.040
0.038
0.012
0.006
0.006
0.006
0.004
0.002
0.000
0.001
0.000
0.000
0.030
0.080
0.125
0.175
0.250
0.350
0.450
0.550
0.700
0.881
0.910
0.822
0.761
0.662
0.498
0.310
0.]50
O.O51
0.144
0.099
0.089
0.072
0.050
0.044
0.035
0.024
0.010
0.055
0.019
0.009
0.008
0.008
0.008
0.005
0.004
0.003
-0.045
-0.011
-0.006
-0.002
0.004
0.006
0.005
0.001
O.OO0
0.040
0.031
0.022
0.017
0.010
0.005
0.002
0.001
O.OOO
0.040
0.176
0.065
0.007
0.465
0.442
0.438
0.618
0.437
0.138
0,136
0.122
0.097
0.102
0.026
0.016
0.014
0.016
0.019
0.002
0.000
0.000
0.002
0.004
0.369
0.150
0.049
0.097
0.078
0.041
0.014
0.013
0.013
0.005
0.005
0.000
0.080
0.125
0.753
0.643
0.144
0.103
0.036
0.022
-0.009
-0.004
0.038
0.025
0.175
0.250
0.350
0.450
0.711
0.559
0.389
0.218
0,089
0.049
0.042
0.030
0.013
0.004
0.004
0.002
-0.002
-0.001
0.001
0.001
0.024
0.015
0.009
0.004
0.550
0.700
0.138
0.045
0.022
0.008
0.002
0.002
-0.001
0.000
0.002
0.000
0.368
0.328
0.547
0.352
0.382
0.233
O.131
0.032
0.172
0.135
0.124
0.076
0.071
0.055
0.045
O.O19
0.042
0.027
0.018
0.007
0.007
0.006
0.006
0.005
0.004
0.001
0.000
0.000
0.001
0.001
0.003
O.OOO
0.125
0.175
0.250
0.350
0.450
0.550
0.700
1.109
0.629
0.491
0.311
0.204
O.116
0.033
0.165
0.108
0.058
0.040
0.027
0.020
0.007
0.077
0.039
0.017
0.006
0.002
0.002
O.OO1
-0.019
-0.005
-0.004
-0.001
0.000
-O.OO1
O.OOO
0.063
0.031
0.020
0.011
0.006
0.003
0.O00
0.646
0.186
0.085
0.004
0.586
0.281
0.217
0.128
0.075
0.055
0.044
0.021
0.009
0.000
-0.001
0.000
O.195
0.108
0.040
0.O31
0.003
0.002
0.000
0.002
0.027
O.O12
0.002
0.0OO
0.250
01364
0.119
0.081
-0.014
0.023
0.350
0.450
0.550
0.305
0.184
0.102
0.066
0.040
0.026
0.058
0.016
0.000
-0.005
-0.001
-0.002
0.017
0.009
0.004
0.155
0.265
0.164
0.088
0.130
0.075
0,048
0.033
0.090
0.065
0.020
0.015
-0.002
-0.003
-0.001
0.001
= l(GeV/c) 2. Our data are compared with the results of C D H S [ l g ] and E M C [2c] and a g o o d
agreement is seen.
Tables 3 and 4 present the structure functions for
proton and neutron targets, ff~', P~P, xff~", xF~ p The
systematic errors quoted include the contribution
due to the uncertainty on the absolute normaliza-
tion, on the energy spectra shapes (see Sect. 2) and
on the rescattering fraction: the latter is of course
not present in the vD data.
The logarithmic derivative d(lnF2)/d(lnQ 2) for
both P~P and F~ n as a function of x is shown in
Fig. lb. Within the errors there is no difference in
shape between neutron and proton.
326
D. Allasia et al.: Qz Dependence of the Proton and Neutron Structure Functions
Table 4. Neutron structure functions F~"(x, Qz) and xP~"(x, Q2)
Q2 (GeV/c) 2 x
~n
stat
sys
Ac
AR
xF~ n
stat
sys
Ac
1.5
0.030
0.080
0.125
0.175
0.250
0.350
0.450
0.550
0.700
1.142
1.310
1.241
1.408
1.104
1.015
0.770
0.463
0.225
0.080
0.102
0.093
0.108
0.079
0.093
0.097
0.082
0.036
0.016
0.016
0.020
0.025
0.019
0.022
0.013
0.013
0.013
-0.028
-O.019
-0.011
-0.007
0.002
0.009
0.014
0.010
0.014
0.029
0.016
0.008
0.005
0.002
0.001
0.000
0.000
0.000
0.426
0.929
0.619
0,248
0.426
1.657
0.129
0.226
0.277
0.438
0.458
0.763
0.025
0.033
0.062
0.091
0.108
0.188
0.014
-0.017
-0.008
-0.002
-0.003
0.004
3.0
0.030
0.080
0.125
0.175
0.250
0.350
0.450
0.550
0.700
1.303
1.236
1.432
1.520
1,283
1.079
0.667
0.482
0.191
0.117
0.098
0.089
0.102
0.069
0.070
0.061
0.058
0.022
0.040
0.009
0.015
0.021
0.019
0.016
0.008
0,008
0.007
-0.026
-0.017
-0.011
-0.004
0.008
0.016
0.017
0.015
0.019
0.045
0.029
0.023
0.017
0.009
0.004
0.001
0.001
0.000
0.613
0.857
0.871
1.242
1.227
0.512
0.161
0.160
0.175
0.240
0.205
0,279
0.054
0.015
0.029
0.049
0.058
0.046
0.061
-0.010
-0.007
-0.004
0.000
0.005
6 .0
0.030
0.080
0.125
0.175
0.250
0,350
0,450
0.550
0.700
1.483
1.272
1.467
1,370
1.216
0.917
0.748
0.552
0.177
0.183
0.113
0.104
0,090
0.061
0.055
0,052
0.046
0.018
0.076
0.026
0,015
0.011
0.011
0,012
0.009
0.006
0.003
-0.024
-0.016
-0.011
-0.002
0.008
0.016
0.021
0.017
0.022
0.068
0.043
0.041
0.030
0.020
0.010
0.006
0.003
0,001
0.556
0.619
0.935
1.074
1.049
0.810
0.770
0.369
0.146
0.222
0.157
0.159
0.152
0.119
0.127
0.145
0.149
0.075
0.092
0.036
0.013
0.017
0.025
0.030
0.028
0.025
0.016
0.087
-0.003
-0.003
0.000
0.005
0.011
0.017
0.015
0.018
11.0
0.030
0.080
0.125
0.175
0.250
0.350
0.450
0.550
0.700
1.913
1.446
1.304
1.169
1.160
0.987
0.746
0.455
0.156
0.449
0.199
0.120
0.104
0.064
0.058
0.051
0.038
0.016
0.303
0.082
0.038
0.019
0.008
0.007
0.007
0.007
0,004
-0.021
-0.013
-0.007
-0.002
-0.001
0.005
0.007
0.003
0.013
0.124
0.070
0.050
0.038
0.032
0.022
0.013
0.007
0.002
0.900
1.093
0.797
1.005
0.894
0.705
0.472
0.162
0.236
0.157
0.146
0.097
0.097 9
0.094
0.078
0.039
0.078
0.054
0.026
0.009
0.012
0.013
0.012
0.006
0.004
-0.001
0.000
-0.001
0.003
0.005
0.009
0.014
0.080
0.125
0.175
0.250
0.350
0.450
0.550
0.700
2.400
1.104
0.942
1.227
0.882
0.564
0.415
0.143
0.570
0.154
0.117
0.079
0.056
0.042
0.035
0.015
0.209
0.064
0.046
0.036
0.016
0.003
0.003
0.001
-0.045
-0.013
-0.004
-0.008
0.003
0.003
0.001
0,011
0.159
0.066
0.047
0.049
0.028
0.017
0.011
0.003
0.373
0.562
1.1t5
0.667
0.549
0.430
0.153
0.174
0.140
0.101
0.076
0.060
0.053
0.025
0.073
0.055
0.046
0.020
0.006
0.005
0.003
0.001
0.000
-0.004
0,000
0.003
0.008
0.012
0.250
0.350
0.450
0.550
0.700
1.025
0.881
0.474
0.276
0.107
0.172
0.097
0.059
0.041
0.017
0.081
0.058
0.016
0.010
0.003
-0.030
-0.011
-0.003
-0.003
0.005
0.063
0.047
0.023
0.012
0.004
0.901
0.883
0.446
0.270
0.105
0.187
0.111
0.071
0.051
0.022
0.090
0.065
0.020
0.010
0.003
-0.014
-0.008
-0,002
0.002
0.005
24 .O
55.0
From the complete set of the ffi's one can extract
the m o m e n t u m distributions of the valence quarks
and the antiquarks of different flavours:
x E I P = 1/4(ff~2 p - x f f ~ p ) = x ~ + e " x s
xY/" = 1/4(ff~" - xff~ n) = x d + e "x s
9 x u ~ = 1/4(ff;" + xF~") -- 1/4(ff~ p - - x f f ; p)
xdv = 1 / 4 ( P ; .
- 1/4 ( ; F - xPj")
(3)
(see the Appendix for a discussion on the amount of
strange quarks). The results concerning the antiquarks are reported in Table 5 and are shown in
Fig. 2 where the errors are purely statistical.
The behaviour of x ~ and x d is remarkably similar in all the Q 2 range and do not exibit a sizeable
Q2 dependence. The fractional m o m e n t u m carried
by both ~ and d is practically zero for x > 0.4: this is
in agreement with other v experiments [ l g ] . These
D. A/lasia et al.: Q2 Dependence of the Proton and Neutron Structure Functions
327
Table 5. Antiquark structure functions: xglP=xg+sxg, Xgl"=Xd+exs; see the text and the Appendix for a discussion on the amount
of strange quark
Q2 (GeV/c)
2
x
xq p
stat
sys
Ac
AR
xq n
stat
sys
Ac
AR
1.5
0.030
0.080
0.125
0.175
0.163
0.132
0,038
-0.104
0.037
0.054
0.064
0.096
0.004
0,004
0,009
0.020
-0.014
-0.001
0.000
o.001
-0.017
-0.016
-0.018
-0,017
0.179
0.095
0.155
0.290
0,038
0.062
0,073
0.112
0.004
0.005
0.010
0,021
-0.011
0.000
-0.001
-0.00t
-0.019
-0.026
--0.026
-0,031
3.0
0.030
0,080
0.125
0.175
0.250
0.181
0.228
0.131
0,135
0.003
0.044
0.045
0.042
0.050
0,040
0.007
0.002
0.003
0.004
0.008
-0.016
-0.003
-0.001
0.000
0.001
-0.013
-0.019
-0.017
-0.015
-0.011
0.172
0.095
0.140
0.070
0.014
0.050
0.047
0.049
0.065
0.054
0.010
0.003
0.004
0.008
0.010
-0.022
-0.002
-0.001
0.000
0.002
-0.019
-0.022
-0.027
-0.031
-0.027
0.350
0.055
0.048
0.008
0.001
-0.010
0.142
0.072
0.014
0.003
-0.024
6.0
0.030
0.080
0.125
0.175
0.250
0.350
0.450
0.550
0.700
0.210
0.111
0.095
0.081
0.011
0.015
-0.015
0.000
0.001
0.057
0,042
0.041
0.035
0.027
0.028
0.026
0.020
0.011
0.013
0.005
0.002
0.002
0.001
0.002
0.002
0.002
0.001
-0,013
-0,003
-0.002
-0.001
0.001
0.001
0.000
-0.001
0.000
-0.011
-0.014
-0.014
"-0.013
-0.013
-0.010
-0.007
-0.003
-0.001
0.232
0.163
0.133
0.074
0.042
0.027
-0,005
0.046
0.008
0.072
0.048
0.047
0,044
0.033
0.034
0,038
0.039
0.019
0,019
0,008
0.004
0,002
0.003
0.004
0.005
0.004
0.003
-0.026
-0.003
-0.002
0.000
0.001
0.001
0.001
0.001
0.001
-0.018
-0.019
-0.024
-0.024
-0.023
-0.019
-0.016
-0,012
-0.004
11.0
0.080
0.125
0.175
0,250
0.350
O,450
0.550
0.700
0.096
0.079
O.041
0.052
0.002
-0.004
0.002
0.003
0.056
0.042
0,038
0.023
0.021
0.016
0,013
0,005
0.007
0.005
0.002
0,002
0.002
O.OO1
O.OO1
0,001
-0.003
-0.001
-0.001
0.000
0.000
O.OOO
-O.OO1
0.000
-0.008
-0.009
-0.011
-0.009
-0.007
-O.OO4
-0.003
-0.001
0.137
0.053
0.093
0.039
0.023
O.O10
-0.004
-0,002
0.076
0.049
0.045
0.029
0.028
0.027
0,022
0.010
0.011
0.007
0.006
0.002
0.002
0.002
0,002
0.001
-0,004
-0.002
-0,001
0.000
0,000
u.ool
-0.002
0,000
-0.017
-0.018
-0.018
-0.019
-0.017
-0,O14
-0.009
-0.003
24.0
0.125
0.175
0.250
0.350
0.450
0.550
0.700
0.116
0.011
0.052
0.024
0.002
0.002
0.001
0.062
0.042
0.024
0,017
0.012
0.009
0.004
0.016
0.003
0.003
0.002
0.001
0,001
0.000
-0.006
-0.001
-0.001
0,000
0.000
-0.001
0.000
-0.011
-0.007
-0.007
-0.005
-0.003
-0.002
-0.001
0.183
0.095
0.028
0,039
0.004
-0,004
-0.002
0.058
0.045
0.032
0.023
0.018
0.016
0.007
0.013
0.008
0.006
0.003
0.002
0,002
0.001
-0.004
-0.001
-0.001
0.000
0.000
-0.002
0.000
-0.010
-0,011
-0.017
-0.014
-0.009
-0,007
-0.003
55.0
0.250
0.350
0.450
0,550
0.700
0.052
0.010
0.005
0.004
O.044
0.025
0.015
0.010
0.002
0.002
0.001
0.001
-0.003
-0,001
0.000
-0.001
-0.003
-0.003
-0.002
-0.001
0.031
-0.001
0.007
0.001
0.001
0.063
0,037
0.023
0.016
0.007
0.006
0.004
0.002
0.001
0.001
-0.004
-0.001
0.000
-0.001
0.000
-0.009
-0.009
-0.006
-0.004
-0.002
results, apart from their intrinsic interest, will allow
in the analysis of the non singlet structure functions
the use at large x of the more precisely measured
-#~v/2, ff~"/2 and ,~N instead of xdo, xu~ and xP;"N
respectively.
The valence quarks m o m e n t u m distributions xu~
and xd~ are shown in Fig. 3. Many experiments performed with e,/~ and v beams measured the ratio r
---xdJxu~ averaging over different Q2 range. In
Fig. 4 we present r as a function of x and Q2: the
data are consistent with no Q2 dependence in the
range explored and are well represented by the values obtained averaging over all Q2's in this experiment and already published [5].
4. Sum Rules
Using the isoscalar data a test of the Gross-Llewellyn Smith sum rule [18] as a function of Q2
[%+ ~](Q2)_
i xFff(x, Q2) dx
0
(4)
X
was performed, og~ and ~ are the number of valence quarks.
For each Qe bin above QZ=l(GeV/c)2, the
quantity
s ~ = Z,(xP~N(x, Q2)/x), ~ x,
(5)
328
D. Allasia et al.: Q2 Dependenceof the Proton and Neutron Structure Functions
i
i
i i i TIll I
i
03
i I I IIil[
i
]
i
i
i i~ll
I
i
r
i
llli[
I
0.4
x d v (x ,Q2)
XUv(X'QZ)
x= 0.03
0.2
0.6
-4
,
,
{-
X=O08
0.2
+ + ,
01
0.3
0.1
IT
t
X=OA25
'
X= 0.25
01
~'
o.1
~
~
{
Jt
0.2
X=0.125
:It
X= 0.175
Q3
X= 045
I
X=0.35
;
,I, - -
0.6
I
l-
X= 045
~
X=0.55
I
0,1
~
T
0.3 I
X=0.55
o
0.1
~
0.0
--"~r"-"
I
I
I
I IIIII
1
~
9
I
X:070
I
I
I IIIll
10
O~( GeV/c )2
a
100
t
o.~I
~
0.05
I
I
I
•
I I ttll
1
t
I
10
I
t I Etll
100
Q2(GeV/c)2
b
Fig. 3a and b. x and Q2 dependence of the separate xu,, and xdo structure functions. The curves represent the results of the Q C D fits
obtained with W 2 > 3 GeV 2, using F~"/2 and F~P/2for x >0.4, and given in the first column of Table 6
was calculated in the interval 0 . 0 6 > x > 0 . 8 0 , using
iF~N instead of xff~ ~' for x>0.4. The contribution
coming from the regions x < 0 . 0 6 and x > 0 . 8 0 and
the effect of using the discrete sum (5) instead of the
integral (4) were evaluated by means of a simple
model of scaling violations for quark distributions
[14] that is able to reproduce reasonably well our
structure functions by means of t h e following parametrizations:
3Q2
(0~,2"1
+4Q2~
~..i~2 ]
xu~(x, Q2) =2x~(1 --X)2"I+Q2/B
xdv(x, 2
3e=
x ~ 1-x)~176
O )= (
[
0'75 +4Q2 ~
(6)
~' o . ~ !
where B is the Euler beta function. The parameter c~
is assumed to be equal for up and down quarks and
was set t o 0.55, a value that provides the best fit to
the combined set of our data on xdv and xu,.
The correction factors to (5) obtained following
this method increase slowly with Q2 and represent
from 45 % to 55 % of the total integral. Because of
the diverging 1/x weight, the evaluation of integral
(4) for x <0.06 is very sensitive to the exponent a in
the formulae (6). An estimate of the systematic uncertainty on the correction factors was obtained repeating the calculation with c~=0.50 and c~=0.60,
which are more than three standard deviations from
the preferred value. The effect on the sum rule is a
decrease of 7-8 % with c~=0.50 and an increase of 910~o with c~=0.60. The results for 1 < Q 2 < 4 0
(GeV/c) 2 are given in Fig. 5a and provide a positive
test of the Q P M model, which predicts d#v+ ~v = 3 constant. Second order Q C D corrections to the sum
rule [19] require a value 3(1-a~(Q2)/n) for the integral (4). The dotted line shows the prediction for A
=0.2 GeV - a value compatible with the results to
be presented in Sect. 5. This prediction is in good
agreement with the data. Data on the Gross-Llewellyn Smith sum rule at fixed Q2's were obtained by
the BEBC-Gargamelle Collaboration [20]: they are
consistent with our results.
Using the same procedure, a test of the Adler
sum rule [21] as a function of Q2
[% "~]
({2 =) =
i FVn~x
o
2x
dx
(7)
D. Allasia et al.: Q2 Dependence of the Proton and Neutron Structure Functions
329
I
a8 I
{ ' i ......
...........
04
Ir :[dv/Uvl
........
(x'Oz)
-+- - -
I
4.0
X=O05
3.0
04 --{ ..... §..... §.... ~. . . . . -{---
i //~ PM
,,
f
08 I
I
F3 (x,Q~)d x
-{
I
i
',__-f--+
\o i
x:o.ls
2.0
co
0.0~-
I
I
5
10
i-
=o.2 GeV'
50
100
Q8
04
................
O0
§
{
§ .....
+- . . . . .
f___
x:o.35
I
I
,1
Vn
i t F2 - -~-F2
'o
1.5
0.6
......
+ .....
§.....
{___
1
x : 0.45
a2
1.0
t
i
06
vp
dx
2X
+
I
i
I
__F__+ ..... +..... +___
02
0.5
04
__t___ J_ .... ~_ .......
•
I
I
I
5
10
50
100
O0
I
I
I
I
IIIll
I
10
I
Q2(GeV/c)2
t
~ Jlsrl
100
Fig. 4. The ratio r=xdv/xu~ versus x and QZ. The dashed lines
are the values averaged over all Q2 in the relative x bins
was also performed; in this case at Q2=24 (GeV/c) 2
only the data in the range 0 . 1 0 < x < 0 . 8 0 were used
and the correction factor was applied accordingly.
The Q P M predicts ~ v - ~ v - - 1 ; this should hold at
all orders of QCD. The results are shown in Fig. 5b
for 1 < Q 2 < 4 0 (GeV/c) z and are in agreement with
the theoretical expectations.
5. Q C D Analysis
The evolution of the structure functions to leading
order in perturbative Q C D is given by the AltarelliParisi equations [22].
The evolution of the flavour singlet and the antiquark structure functions are coupled also to the
gluon distribution, which is poorly known and cannot be accurately determined from this experiment.
The S.F. reported in this paper give the opportunity
of analysing four non-singlet combinations: xff~ N,
xu~, xdv, F~"-E~ v. The first one, although not independent from the others, has more statistical power
and is not affected by the uncertainty on the rescattering fraction. As mentioned before, to improve
Fig. 5. a Test of the Gross-Llewellyn Smith sum rule versus
Qz: ~xf~ dx.
x
The dashed line is the second order QCD pre-
diction for A = 0 . 2
Fz FB
J~ x
dx.
vn _
GeV. b Test
of the Adler
sum rule:
vp
In both figures the estimated systematic uncertain-
ties (dashed lines) discussed in the text are shown separately
the accuracy of the analysis, instead of xF3, the
relevant ffz'S were used for x>0.4. For such high
values of x the theory predicts a very small value for
R: the use of the ff2 structure functions obtained
assuming R - 0 is therefore legitimate.
In the flavour non-singlet case the evolution
equation has the following form:
dFNs(X, Q ) )
d(lnQ 2)
_
C(S(Q 2)
2n
i Pqq(X/y)FNs(Y,Q2) d y
x
y
(8)
where Pqq(X/y) is one of the splitting functions given
by the theory and the strong coupling constant in
leading order (LO) is:
C~s(Q2) = 12 n/[(33
-2Nf)
in (Q2/A2)],
where A is a free parameter of the theory and NI
represents the number of quark flavours, which we
fix to three.
Different procedures were suggested in the last
few years to determine A by means of the evolution
equation (8). To obtain ALo we used a program
330
D. Allasia et al.: Q2 Dependence of the Proton and Neutron Structure Functions
Table 6. Results of leading order QCD fits, The first and second columns give the results obtained
using Furmanski-Petronzio's and Odorico's programs retaining the data points with W2>3 GeV 2.
The last column gives the results from Abbott and Barnett's program including target mass
corrections and a higher twist term. The cut Q2> 1 (GeV/c) a is always applied. The errors are
purely statistical
Structure
function
xff~N
W z > 3 GeV 2
All W 2
Furmanski-Petronzio
Odorico
ALO
x2/DOF
ALO
z2/DOF
ALO
)~2/DOF
180+ 60
65/38
220+ 70
57/41
250+ 90
80
62/44
53/41
-
F2v. - F 2~p
2
Abbott-Barnett
-
-
55• 120
55
54/37
68 + 120
- 68
52/38
170•
xu v
110
175• 70
53/38
120
170• 70
72/40
240•
xd~
170•
25/37
260•
29/39
140•
150
written by Odorico 1-23] and one written by Abbott
and Barnett [24], that require a parametrization of
Fys at a fixed Q2= Qg, that was chosen to be
140
56/42
26/41
latter is parametrized as:
h'x
(10)
Fys(X, Q2o)= Axe(1 -x)~(1 + yx)
at Q2 = 1 (GeV/c) 2.
(9)
~,/3 and 7 are free parameters, whereas A is given by
the QPM sum rules that fix the number of uv and d~
quarks. Alternatively a program written by Furmanski and Petronzio [25] was used: in that case
FNs at fixed Q~ does not require an explicit phenomenological parametrization like (9), since it is expressed by means of a series of Laguerre polynomials.
Non perturbative QCD effects are expected to be
most important for low values of the hadronic invariant mass W. Only data with W2>3 GeV 2 were
retained to perform fits using Odorico's and
Furmanski-Petronzio's programs.
The two programs yielded consistent results: in
the case of xdv and xF~ N the difference in ALO is
about 1/2 of the error. The four values of ALo, with
their statistical errors, are given in Table 6. There is
reasonable agreement among them, leading to the
conclusion that the leading order QCD fits are able
to represent the data. Figures 3 and 6 show the xu~,
x d v and F ~ " - F ~ p structure functions with Wa>3
GeV2; the curves represent the results of the QCD
fits with the values of ALo reported in Table 6.
An analysis using all W 2 values (always with
Q 2 > l (GeV/c) 2) was performed using Abbott and
Barnett's program, where target mass corrections
and contribution of higher twist terms were taken
into account. Retaining only the twist-4 term the
The parameter h was let free to vary between - 1
and + 1. The values of Aeo obtained with this method are also shown in Table 6. They are consistent
with the results of the two other methods previously
discussed and the fits have improved only slightly. It
should be remarked that there is a strong correlation between ALo and h: if h is negative, ALo
increases and viceversa. In fact the favoured solutions reported in Table 6 require h small and negative for the fits to the first three structure functions
and h close to zero, but positive for x d v. Given the
present statistics and the number of parameters involved, it is impossible to unambiguously separate
the logarithmic and the power-like Qz dependence of
the structure functions.
Before being tempted to reach any quantitative
conclusion from Table 6, one should discuss possible
systematic effects on the determination of ALo.
Concerning the experimental uncertainties, we
can make the following comments: i) a change of
20~o in the energy smearing corrections does not
change the results; ii) if one uses the values of the
S.F.'s averaged over the x and Q2 bins instead of the
ones at the bin centers, Aeo increases by about 20
MeV; iii) a shift of the rescattering fraction by the
amount of the uncertainty we assign to it, changes
ALo determined by the different flavour structure
functions by 30-40 MeV.
The analysis procedure itself gives rise to systematic uncertainties: i) if, using the data at all W 2, one
D. Allasia et al.: Q2 Dependence of the Proton and Neutron Structure Functions
o.a
0.1
F2vn-F2vP
~
2
+
~
i
I
I
+
o.~ ;
+
§
x=o.2s
{--.--
x=oas
'
X=0.45
0.1
02 1
0.1
' I t '
-
~
v
~[
X=0.55
0.1
X=0.70
0.03~1
I
I I I flfllf
10
100
O2(GeV/cl2
Fig. 6. x and Q2dependence of (F~"-F~P)/2. The curves are the
I
I
f ftll[
331
functions F 2 and x F 3 of neutron and proton separately, thereby obtaining the individual xuv, xdv, x g
and x d distributions. F r o m these measurements it
emerges that
a) x~ and x d have a remarkably similar Q2 behaviour and are both negligible at all Q>s for
x>0.4;
b) the ratio r = x d J x u v , which drops from about
0.5 at x-~0 to much smaller values at large x, is
consistent with being constant with Q2, for every
value of x;
c) the Q P M sum rules, which measure the number of quarks, have been verified at various Q2 for
5 (xF~N/x) d x and 5 ( F ~ " - F ~ ) / 2 x 9dx;
d) Q C D analyses were performed using four non
singlet S.F. in terms of the parameter ALo. Although
systematic uncertainties from the data themselves
and from the methods of fitting the data prevent us
from providing a precise value of ALo , one can
reach the conclusion that all the data are well described by perturbative Q C D and compatible with
values of ALo ranging from 100 to 300 MeV.
Acknowledgments. We thank the CERN staff for running the SPS,
the neutrino beam, the EMI and the BEBC bubble chamber. We
are grateful to our programming, scanning and measuring teams
for their painstaking work. We are grateful to E. Longo and
P. Fritze for providing us with the programs, we used in the QCD
analysis.
results of the QCD fits obtained with W2>3 GeV and given in
the first column of Table 6
Appendix
does not allow for target mass corrections and higher twist terms, ALO descreases by about one standard
deviation for xF~ N, xuv and F ~ " - F ~ p and increases
by 200 MeV for xdv; ii) higher cuts in the hadronic
mass, although recommended to avoid non-perturbative effects, lead to much less reliable results due
to fluctuations in the data points and to the reduced
Q2 range; iii) allowing for a violation of the CallanGross relation and using R=0.1, ALO decreases by
about 20 MeV in the analysis of F~"-F~P; iv) assuming N / = 4 , ALo decreases by 30-50 MeV, depending on the structure function.
The conclusion of these analyses in the framework of perturbative Q C D is that up and down
quarks have compatible Q2 dependence and that the
non singlet structure functions measured in this experiment provide values of ALo between 100 and 300
MeV. The x and Q2 behaviour of the structure functions can also be explained by a combination of
perturbative Q C D and a twist-4 term.
The proton and neutron structure functions (S.F.)
measured by neutrino and antineutrino interactions
are defined in the Q P M in terms of quark and
antiquark fractional m o m e n t u m distributions*:
2xF~" = 2xF~ p = 2 x ( d + ~ + s)
2xF~" = 2xF~ p = 2x(u + d + s),
(A 1)
xF~ p = 2 x ( d - ~,+ s)
x f ~ " = 2x(u - d + s)
xF~ p = 2x(u - d - s )
(A2)
xF~"=2x(d-g-s)
and F 2 = 2 x F 1 (1 + R).
For an isoscalar target N, it follows:
2xF~ N = 2xF~ N = x (u + d + ~ + d + 2s)
xF~ N = x (u + d - ~ - d + 2 s)
(A 3)
xF~N=x(u+d-R-d-2s)
6. Conclusions
where we have neglected charm and heavier flavours
and assumed s - sp = s, = 2.
12,100 vD and 10,500 7D interactions were used to
measure the x and Q2 dependence of the structure
* The explicit x and Q2 dependence of the S.F.'s will be dropped
in the following
332
D. Allasia et al.: Q2 Dependenceof the Proton and Neutron Structure Functions
I
I
I
'
0,12
ment. The event energy distributions, proportional
to EO(E), were fitted with a twelve parameter polynomial: both the data and the fitted curve are
shown in Fig. 7. Absolute external normalization
was imposed as explained in Sect. 2.
The Ni integrals are actually calculated in given
x - Q 2 bins, say for xj<x<x~+i, QZ<Q2<Q2+1 , and
therefore the limits of integration depend on the
extreme values of the x - Q 2 bin and on the experimental cut p , > 4 GeV/c, using the equations y
= Q2/2MEx and Ymax< 1 __p,min/E.
The strange quarks give different contributions
to xF~ and xF~, as shown by formulae (A2) and
(A3). If one neglects the strange quarks, the following relations would hold:
0.10 ~
vp
9n -xF 3 ~ xF 3 =xf
I
I
f
l
I
t
I
t
I
0.12
>
m 0.10
0.08
~0.06
0.04
0.02
Ev(GeV)
0.14
~
,.-n
p
3,
-~ o.o8
and
z 0.0g
vN
~N
N
x F 3 = x F 3 - = x F 3.
0.04
0.02
i
0,00
'
0
'
~"
20
40
60
BO
100
120
140
160
"
'
180
'"
200
E~(GeV)
Fig. 7. Experimental v and 7 events energy spectra. The curves are
~2
obtained from fits to polynomials: dN/dE=~ia~x1-1, where x
= E/200, E in GeV
1
To measure these S.F.'s at various x and Q2
values one uses suitable linear combinations of neutrino, n*(x, Q2), and antineutrino, n~(x, Q2), number
of events from the vp, vn, ~p and 5n channels.
n * and n ~ can be expressed as:
n~(x, Q 2 )
=
2xF; N 1-4r F~N 2 4- x F ~
N3
(A4)
ng(x, Q2) = 2xF~ 1V, + V~ N2 -xF~ N3.
The kinematical integrals N~, N 2, N 3 are given by:
NI:S
G2M ~
r~ y2
~ Ec~(E)dE~ y d y
7~
E1
"E
E~
N2-=S
Yl
~ E4~(E)dE
N3=S
~ E~(E)dEIy
7~
1-y
1-
2E !
dy
Et
and likewise for N1, N 2 and N3.
S is the number of scattering centers, (~)(E) represents the (anti)neutrino flux measured in the experi-
vn
xf 3 =xF
9p
--
n
3 =xF 3
(A6)
Formulae (A4) would therefore simplify and can
be solved providing F 2 and xF 3 for n,p and N
targets for a given value of R. In the general case,
where the strange quark contribution is different
from zero, the experimental ffi's obtained from (A4)
are related to the ones defined by (A1), (A2) and
(A3), using the shorthand notation (A6), as follows:
~ V = F g +~7. xs
x P ~ = xF~ + ~ "xs
(A7)
and likewise for n and N targets, q and ~ are kinematical factors equal for all targets, that depend
on x and Q2: ~ is at most few percent, and therefore
the strange quark contribution to x F 3 is negligible;
t/ on the other hand varies between one and three.
We did not attempt to subtract the strange quark
contribution to F 2 using arbitrary estimates of
xs(x, Q2). The analysis performed on the Q2 dependence of the S.F.'s, discussed in Sect. 5, uses F 2
only at x>0.4, where the sea quark, and afortiori
the strange quark, contribution can safely be neglected.
It can be remarked that the difference (F~"
-F~V)/2=xuv-xdv is free from any xs contribution.
The antiquark momentum distributions in the
proton, x~/p, and in the neutron, x~/", given by
x~lP= 1/4(ff~p -- Xff~P)= x~ + e "XS
x~/"= 1 / 4 ( f f ~ " - x f f ~ " ) = x d + e ' x s
(AS)
are not pure ~ or, respectively, d distributions, but
contain a contribution of strange quark e. xs, with e
varying between 0.6 and 0.9, depending on x and Q2.
Integrated over Q2, s---3/4: this is the amount of
D. Allasia et al.: Q2 Dependence of the Proton and Neutron Structure Functions
strange quark we had in the g and d distributions
measured in our previous paper [5].
The amount of strange quark contribution in x u v
and xdv determined in this analysis is 1 / 2 ( ~ . x s ) , and
therefore negligible, as discussed above.
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