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The value of QCD coupling constant
and power corrections
in the structure function F2 measurements
V.G. Krivokhijine and A.V. Kotikov
Joint Institute for Nuclear Research, 141980 Dubna, Russia
Abstract
We reanalyze deep inelastic scattering data of BCDMS Collaboration by includ-
ing proper cuts of ranges with large systematic errors. We perform also the ts of
high statistic deep inelastic scattering data of BCDMS, SLAC, NM and BFP Collab-
orations taking the data separately and in combined way and nd good agreement
between these analyses. We extract the values of both the QCD coupling constant
s (M 2
Z ) up to NLO level and of the power corrections to the structure function
F 2 . The ts of the combined data for the nonsinglet part of the structure function
F 2 predict the coupling constant value s (M 2
Z ) = 0:1174  0:0007 (stat) 0:0019
(syst) 0:0010 (normalization). The ts of the combined data for both: the nons-
inglet part of F 2 and the singlet one, lead to the values s (M 2
Z ) = 0:1177  0:0007
(stat) 0:0021 (syst) 0:0009 (normalization). Both above values are in very good
agreement with each other.
1 Introduction
The deep inelastic scattering (DIS) leptons on hadrons is the basical process to study
the values of the parton distribution functions (PDF) which are universal (after choosing
of factorization and renormalization schemes) and can be used in other processes. The
accuracy of the present data for deep inelastic structure functions (SF) reached the level
at which the Q 2 -dependence of logarithmic QCD-motivated terms and power-like ones
may be studied separately (for a review, see the recent papers [1] and references therein).
In the present letter we sketch the results of our analysis [2] at the next-to-leading
order (NLO) of perturbative QCD for the most known DIS SF F 2 (x; Q 2 ) 1 taking into
account experimental data [4]-[7] of SLAC, NM, BCDMS and BFP Collaborations. We
stress the power-like e ects, so-called twist-4 (i.e.  1=Q 2 ) contributions. To our purposes
we represent the SF F 2 (x; Q 2 ) as the contribution of the leading twist part F pQCD
2 (x; Q 2 )
described by perturbative QCD, when the target mass corrections are taken into account
(and coincides with F tw2
2 (x; Q 2 ) when the target mass corrections are withdrawn), and
the nonperturbative part (\dynamical" twist-four terms):
F 2 (x; Q 2 )  F full
2 (x; Q 2 ) = F pQCD
2 (x; Q 2 )

1 +
~ h 4 (x)
Q 2

; (1)
where ~
h 4 (x) is magnitude of twist-four terms.
Contrary to standard ts (see, for example, [8]- [10]) when the direct numerical calcu-
lations based on Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation [11] are
1 Here Q 2 = q 2 and x = Q 2 =(2pq) are standard DIS variables, where q and p are photon and hadron
momentums, respectively.
1

used to evaluate structure functions, we use the exact solution of DGLAP equation for
the Mellin moments M tw2
n (Q 2 ) of SF F tw2
2 (x; Q 2 ):
M k
n (Q 2 ) =
Z 1
0
x n 2 F k
2 (x; Q 2 ) dx (k = full; pQCD; tw2; :::) (2)
and the subsequent reproduction of F k
2 (x; Q 2 ) at every needed Q 2 -value with help of
the Jacobi Polynomial expansion method [12, 13] (see similar analyses at the NLO level
[13, 14] and at the next-next-to-leading order (NNLO) level and above [15].
In this letter we do not present exact formulae of Q 2 -dependence of SF F 2 which are
given in [2]. We note only that the moments M tw2
n (Q 2 ) at some Q 2
0 is theoretical input of
our analysis and the twist-four term ~ h 4 (x) is considered as a set of free parameters (one
constant ~ h 4 (x i ) per x i -bin): ~ h free
4 (x) =
P I
i=1
~ h 4 (x i ), where I is the number of bins.
2 Fits of F 2 : procedure
Having the QCD expressions for the Mellin moments M k
n we can reconstruct the SF F k
2 (x)
as
F k;Nmax
2 (x; Q 2 ) = x a (1 x) b
Nmax X
n=0
 a;b
n (x)
n
X
j=0
c (n)
j ( ; )M k
j+2 Q 2

; (3)
where  a;b
n are the Jacobi polynomials 2 and a; b are tted parameters.
First of all, we choose the cut Q 2  1 GeV 2 in all our studies. For Q 2 < 1 GeV 2 , the
applicability of twist expansion is very questionable. Secondly, we choose quite large values
of the normalization point Q 2
0 : our perturbative formulae should be applicable at the value
of Q 2
0 . Moreover, the higher order corrections  k
s (Q 2
0 ) and  ( s (Q 2 ) s (Q 2
0 )) k (k  2)
should be less important at higher Q 2
0 values.
We use MINUIT program [19] for minimization of  2 (F 2 ) = j(F exp
2 F teor
2 )=F exp
2 j 2 .
We consider free normalizations of data for di erent experiments. For the reference, we
use the most stable deuterium BCDMS data at the value of energy E 0 = 200 GeV (E 0
is the initial energy lepton beam). Using other types of data as reference gives negligible
changes in our results. The usage of xed normalization for all data leads to ts with a
bit worser  2 .
3 Results of ts
Hereafter we choose Q 2
0 = 90 GeV 2 (Q 2
0 = 20 GeV 2 ) for the nonsinglet (combine nonsinglet
and singlet) evolution, that is in good agreement with above conditions. We use also
N max = 8.
3.1 BCDMS 12 C +H 2 +D 2 data
We start our analysis with the most precise experimental data [6] obtained by BCDMS
muon scattering experiment at the high Q 2 values. The full set of data is 762 (607) points
(for the bounded x range: x  0:25).
It is well known that the original analyses given by BCDMS Collaboration itself (see
also Ref. [9]) lead to quite small values s (M 2
Z ) = 0:113. Although in some recent papers
2 We note here that there is similar method [16], based on Bernstein polynomials. The method has
been used in the analyses at the NLO level in [17] and at the NNLO level in [18].
2

# # # # # # # #
####
####
####
####
a s
(Q 0
2 )
N Y cut
Figure 1: The study of systematics at dif-
ferent Y cut values in the ts based on non-
singlet evolution. The QCD analysis of
BCDMS 12 C; H 2 ; D 2 data (nonsinglet case)
is given at x cut = 0:25 and Q 2
0 = 90 GeV 2 .
The inner (outer) error-bars show statisti-
cal (systematic) errors.
# # # # # # #
####
####
####
####
####
a S (Q 0
2
)
N Y cut
Figure 2: The study of systematics at dif-
ferent Y cut values in the ts based on com-
bine singlet and nonsinglet evolution. All
other notes are as in Fig. 1 with two ex-
ceptions: no a x cut and Q 2
0 = 20 GeV 2 .
Moreover, the points N Y cut = 1; 2; 3; 4; 5
correspond the values N = 1; 2; 4; 5; 6 in
the Table 1.
(see, for example, [8, 20]) more higher values of the coupling constant s (M 2
Z ) have been
observed, we think that an additional reanalysis of BCDMS data should be very useful.
Based on study [21] we proposed in [2] that the reason for small values of s (M 2
Z )
coming from BCDMS data was the existence of the subset of the data having large sys-
tematic errors. We studied this subject by introducing several so-called Y -cuts 3 (see [2]).
Excluding this set of data with large systematic errors leads to essentially larger values of
s (M 2
Z ) and very slow dependence of the values on the concrete choice of the Y -cut (see
below).
We use the following x-dependent Y -cuts:
y  0:14 when 0:3 < x  0:4; y  0:16 when 0:4 < x  0:5
y  Y cut3 when 0:5 < x  0:6; y  Y cut4 when 0:6 < x  0:7
y  Y cut5 when 0:7 < x  0:8 (4)
and several N sets for the cuts at 0:5 < x  0:8:
N 0 1 2 3 4 5 6
Y cut3 0 0.14 0.16 0.16 0.18 0.22 0.23
Y cut4 0 0.16 0.18 0.20 0.20 0.23 0.24
Y cut5 0 0.20 0.20 0.22 0.22 0.24 0.25
Table 1: The values of Y cut3 , Y cut4 and Y cut5 .
The systematic errors for BCDMS data were given [6] as multiplicative factors to be
applied to F 2 (x; Q 2 ): f r ; f b ; f s ; f d and f h are the uncertainties due to spectrometer res-
olution, beam momentum, calibration, spectrometer magnetic eld calibration, detector
3 Hereafter we use the kinematical variable Y = (E 0 E)=E 0 , where E is scattering energies of lepton.
3

ineфciencies and energy normalization, respectively. For this study each experimental
point of the undistorted set was multiplied by a factor characterizing a given type of un-
certainties and a new (distorted) data set was tted again in agreement with our procedure
considered in the previous section. The factors (f r ; f b ; f s ; f d ; f h ) were taken from papers [6]
(see CERN preprint versions in [6]). The s values for the distorted and undistorted sets
of data are given in the Figs. 1 and 2 (for the cases of nonsinglet and complete evolutions,
respectively) together with the total systematic error estimated in quadratures.
From the Figs. 1 and 2 we can see that the s values are obtained for N = 1  6 of
Y cut3 , Y cut4 and Y cut5 are very stable and statistically consistent. The case N = 6 of the
Table 1 reduces the systematic error in s by factor 1:8 and increases the value of s ,
while increasing the statistical error on the 30%.
After the cuts have been implemented (we use the set N = 6 of the Table 1), we have
590 (452) points (for the bounded x range: x  0:25). Fitting them in agreement with
the same procedure considered in the previous Section, we obtain the following results:
from ts, based on nonsinglet evolution (i.e. when x  0:25):
s (M 2
Z ) = 0:1153  0:0013 (stat)  0:0022 (syst)  0:0012 (norm);
from ts, based on combined singlet and nonsinglet evolution:
s (M 2
Z ) = 0:1175  0:0014 (stat)  0:0020 (syst)  0:0011 (norm); (5)
where hereafter the symbol \norm" marks the error of normalization of experimental data.
The results are agree each other within considered errors. In Ref. [2] we have also
analyzed the combine SLAC, NM and BFP data and found good agreement with (5).
So, we have a possibility to t together all the data. It is the subject of the following
subsection.
3.2 SLAC, BCDMS, NM and BFP data
After these Y -cuts have been incorporated (with N = 6) for BCDMS data, the full set of
combine data is 1309 (797) points (for the bounded x range: x  0:25).
To verify the range of applicability of perturbative QCD, we analyze rstly the data
without a contribution of twist-four terms, i.e. when F 2 = F pQCD
2 . We do several ts
using the cut Q 2  Q 2
cut and increase the value Q 2
cut step by step. We observe good
agreement of the ts with the data when Q 2
cut  10  15 GeV 2 (see the Figs. 3 and 4).
Later we add the twist-four corrections and t the data with the standard cut Q 2  1
GeV 2 . We have nd very good agreement with the data. Moreover the predictions for
s (M 2
Z ) in both above procedures are very similar (see the Figs. 3 and 4). The results of
the ts are compiled in Summary (see Eqs. (6)-(9)).
The values of twist-four terms are presented in the Fig. 5. To obtain the values we
used the approximate equality of twist-four terms for H 2 and D 2 targets (see [9, 2] and
references therein).
If we used at initial conditions at low x: F tw2 (x; Q 2
0 )  Const, we see a strong
rise of twist-four terms at lower x-bins, that illustrates a quite strong di erence between
experimental data (the recent H1 and ZEUS data [22, 23] demonstrate a rise of SF F 2 at
low values of x and Q 2 ) and these initial conditions.
4

0 2 4 6 8 10 12
1.0
1.5
2.0
2.5
3.0
(GeV 2 )
(GeV 2 )
Q 2
cut
cc 22
(F 2 )/DOF
0 2 4 6 8 10 12
0.112
0.113
0.114
0.115
0.116
0.117
0.118
0.119
0.120
aa s (M Z
2 )
Q 2
cut
Figure 3: The values of s (M 2
Z ) and  2
at di erent Q 2 -values of data cuts in the
ts based on nonsinglet evolution. The
black (white) points show the analyses of
data without (with) twist-four contribu-
tions. Only statistical errors are shown.
0 2 4 6 8 10 12 14 16
1.1
1.2
1.3
1.4
1.5
1.6
(GeV 2 )
(GeV 2 )
Q 2
cut
cc 22 (F 2 )/DOF
0 2 4 6 8 10 12 14 16
0.116
0.117
0.118
0.119
0.120
0.121
0.122
0.123
0.124
0.125
aa s (M Z
2 )
Q 2
cut
Figure 4: The values of s (M 2
Z ) and  2 at
di erent Q 2 -values of data cutes in the ts
based on combine singlet and nonsinglet
evolution. All other notes are as in Fig.
3.
If we continue our ts taking the asymptotics of initial condition at low x: F tw2 (x; Q 2
0 ) 
x ! and take free nonzero ! value, we have ! = 0:18 at Q 2
0 = 20 GeV 2 4 , that is in good
agreement with the recent HERA data and with theoretical studies [25, 24].
As it is possible to see in the Fig. 5, the e ect of strong rise of twist-four magnitude at
small x values observed in previous ts is completely absent here. So, the rise is replaced
by the small x rise of twist-two gluon and sea quark distributions. This replacement
seems due to a small number of experimental points at low x range and narrow range
of Q 2 values there. The quite strong cancellation of twist-four corrections at low x is in
good agreement with the recent studies [25, 26].
4 We would like to note that the ts contain strong correlations between the values of !, the coupling
constant and twist-four terms. These correlations come because of very limited numbers of experimental
data used here lie at the low x region. Indeed, only the NMC experimental data contribute there. Then,
the result ! = 0:18 can be considered seriously only when H1 and ZEUS data [22, 23] have been taken
into account. We hope to incorporate the HERA data [22, 23] in our future investigations.
5

0.01 0.1 1
­0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
x
h 4
(x)
Figure 5: The values of the twist-four terms. The black and white points correspond
to the small-x asymptotics  x ! of sea quark and gluon distributions with ! = 0 and
! = 0:18, respectively. The statistical errors are displayed only.
4 Summary
We have demonstrated several steps of our study [2] of the Q 2 -evolution of DIS structure
function F 2 tting all modern experimental data at Bjorken variable x values: x  10 2 .
From the ts we have obtained the value of the normalization s (M 2
Z ) of QCD coupling
constant. First of all, we have reanalyzed the BCDMS data cutting the range with large
systematic errors. As it is possible to see in the Fig. 1, the value of s (M 2
Z ) rises strongly
when the cuts of systematics were incorporated. In another side, the value of s (M 2
Z )
does not dependent on the concrete type of the cut within modern statistical errors.
We have found that at Q 2  1015 GeV 2 the formulae of pure perturbative QCD (i.e.
twist-two approximation together with target mass corrections) are in good agreement
with all data. 5 The results for s (M 2
Z ) are very similar (see [2]) for the both types of
analyses: ones, based on nonsinglet evolution, and ones, based on combined singlet and
5 We note that at small x values, the perturbative QCD works well starting with Q 2 = 1:52 GeV 2 and
higher twist corrections are important only at very low Q 2 : Q 2  0:5 GeV 2 (see [25, 27] and references
therein). As it is was observed in [28, 24] (see also discussions in [25, 27, 29]) the good agreement between
perturbative QCD and experiment seems connect with large e ective argument of coupling constant at
low x range.
6

nonsinglet evolution. They have the following form:
 from ts, based on nonsinglet evolution:
s (M 2
Z ) = 0:1170  0:0009 (stat)  0:0019 (syst)  0:0010 (norm); (6)
 from ts, based on combined singlet and nonsinglet evolution:
s (M 2
Z ) = 0:1180  0:0013 (stat)  0:0021 (syst)  0:0009 (norm); (7)
When we have added twist-four corrections, we have very good agreement between
QCD (i.e. rst two coeфcients of Wilson expansion) and data starting already with
Q 2 = 1 GeV 2 , where the Wilson expansion should begin to be applicable. The results for
s (M 2
Z ) coincide for the both types of analyses: ones, based on nonsinglet evolution, and
ones, based on combined singlet and nonsinglet evolution. They have the following form:
 from ts, based on nonsinglet evolution:
s (M 2
Z ) = 0:1174  0:0007 (stat)  0:0019 (syst)  0:0010 (norm); (8)
 from ts, based on combined singlet and nonsinglet evolution:
s (M 2
Z ) = 0:1177  0:0007 (stat)  0:0021 (syst)  0:0009 (norm); (9)
Thus, there is very good agreement (see Eqs. (6), (7), (8) and (9)) between results
based on pure perturbative QCD at quite large Q 2 values (i.e. at Q 2  1015 GeV 2 ) and
the results based on rst two twist terms of Wilson expansion (at Q 2  1 GeV 2 , where
the Wilson expansion should be applicable).
We would like to note that we have good agreement also with the analysis [20] of
combined H1 and BCDMS data, which has been given by H1 Collaboration very recently.
Our results for s (M 2
Z ) are in good agreement also with the average value for coupling
constant, presented in the recent studies (see [8, 30, 18, 31] and references therein) and
in famous Altarelli and Bethke reviews [32].
At the end of our paper we would like to discuss the contributions of higher twist
corrections. In our study here we have reproduced well-known x-shape of the twist-four
corrections at the large and intermediate values of Bjorken variable x (see the Fig. 5 and
[9, 2]).
We would like to note about a small-x rise of the magnitude of twist-four corrections,
when we use at parton distributions at x ! 0. As we have discussed already in the Sec-
tion 3, there is a strong correlation between the small-x behavior of twist-four corrections
and the type of the corresponding asymptotics of the leading-twist parton distributions.
The possibility to have a singular type of the asymptotics leads (in our ts) to the ap-
pearance of the rise of sea quark and gluon distributions as  x 0:18 at low x values, that
is in full agreement with low x HERA data. At this case the rise of the magnitude of
twist-four corrections is completely canceled. This cancellation is in full agreement with
theoretical and phenomenological studies (see [25, 26, 27]).
Acknowledgments. Authors would like to express their sincerely thanks to the
Organizing Committee of the VIIIth International Workshop on advanced computing
and analysis techniques in physics research (ACAT 2002) for the kind invitation, the
nancial support at such remarkable Conferences, and for fruitful discussions. A.V.K.
was supported in part by Alexander von Humboldt fellowship and INTAS grant N366.
7

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9