fulltext

NNLO QCD analysis of CCFR data on xF

structure function

and Gross–Llewellyn-Smith sum rule with higher twist

and nuclear corrections

A. V. SIDOROV(*) and M. V. TOKAREV(**)

Joint Institute for Nuclear Research - 141980 Dubna, Moscow Region, Russia

(ricevuto il 20 Ottobre 1997; approvato il 15 Dicembre 1997)

Summary. — A detailed NNLO QCD analysis of new CCFR data on the xF3

structure function including the target mass, higher twist and nuclear corrections was performed and parametrizations of the perturbative and power terms of the structure function were constructed. The results of QCD analysis of the structure function were used to study the Q2_{-dependence of the Gross–Llewellyn-Smith sum}

rule. The aS/p-expansion of SGLS(Q2) was studied and parameters of the expansion

were found to be s14 2.74 6 0.01, s24 22.22 6 0.23, s34 27.86 6 1.74 which are in

good agreement with the perturbative QCD predictions for the Gross–Llewellyn-Smith sum rule in the next-to-leading and next-to-next-to-leading order.

PACS 12.38.Qk – Experimental tests. PACS 12.38.Bx – Perturbative calculations. PACS 13.15 – Neutrino interactions.

1. – Introduction

The progress of perturbative Quantum ChromoDynamics (QCD) in the description of the high-energy physics of strong interactions is considerable [1]. Recent experimental data on the structure function (SF) of neutrino deep-inelastic scattering obtained at the Fermilab Tevatron [2] provide a good possibility to precisely verify the QCD predictions for scaling violation with experiment data. The QCD predictions for SF evolution of deep-inelastic scattering (DIS) are calculated now up to the next-to-next-to-leading order (NNLO) of a perturbative theory [3-6]. The method of comparison of 3-loop QCD predictions with SF experimental data has been developed in [7-9] based on the Jacobi polynomial structure function expansion [10]. It is well known that beyond the perturbative QCD there are other effects (higher twist effects,

(*) E-mail: sidorovHthsun1.jinr.dubna.su (**) E-mail: tokarevHsunhe.jinr.dubna.su

nuclear corrections, target mass corrections, etc.) that should be included in the joint QCD analysis of SF.

In this paper, we present the results of NNLO QCD analysis of the data on the xF3

structure function obtained by the CCFR Collaboration taking into account the target mass, higher twist and nuclear corrections. The Q2_{-evolution of SF is studied and}

parametrizations of perturbative (leading twist) and power terms of the structure function are constructed. The results of our NNLO QCD analysis of the structure function are in good qualitative agreement with NNLO theoretical predictions for the

Q2_{-evolution of the Gross–Llewellyn-Smith sum rule [11]: S}theor

GLS (Q2) 43Q [12aS/p 2

3.25 Q (aS/p)2] [12].

In sect. 2, the method of NNLO QCD analysis of SF based on the SF Jacobi polynomial expansion including the target mass, higher twist and nuclear corrections is described. The results of NNLO QCD analysis of SF are presented in sect. 3. In sect. 4, the aS/p-expansion of the Gross–Llewellyn-Smith sum rule is considered and

expansion parameters are found.

2. – Method of QCD analysis

2.1. Jacobi polynomial expansion method. – We use, for the QCD analysis, the Jacobi polynomial expansion method proposed in [10]. It was developed in [13] and applied for the 3-loop order of perturbative QCD (pQCD) to fit F2[7] and xF3

data [8, 9, 14, 15].

Following the method [13], we can write the structure function xF3in the form

xF₃pQCD(x , Q2 ) 4xa ( 1 2x)b

!

!

n (x) is a set of Jacobi polynomials and cjn(a , b) are coefficients of the series of

Ua , b n (x) in powers of x : Una , b(x) 4

!

The Q2_{-evolution of the moments M}pQCD 3 (N , Q

2_{) is given by the well-known}

perturbative QCD [16, 17] formula

.

/

´

y

z

Here aS(Q2) is the next-to-next-to-leading order strong-interaction constant, g( 0 ), N

are the nonsinglet leading order anomalous dimensions, and the factor HN(Q02, Q2)

contains next- and next-to-next-to-leading order QCD corrections to the coefficient functions and anomalous dimensions (1) and is constructed in accordance with [8, 9] based on theoretical results of [3-6].

The unknown coefficients M3(N , Q02) in (3) could be parametrized as Mellin

moments of some function:

M₃pQCD(N , Q2 0) 4



Here coefficients aishould be found by the QCD fit of experimental data.

The target mass corrections (TMC) are included into our fits through the Nachtmann moments [19] of the SFs:

MpQCD n (Q2) 4



(

)

k

nuclx2/Q2and Mnuclis the mass of a nucleon. We are

taking into account the order O(M2

nucl/Q2) corrections: Mn(Q2) 4MnpQCD(Q2) 1 n(n 11) n 12 Mnucl2 Q2 M pQCD n 12 (Q 2 ) , (6)

where Mnare the Mellin moments of measured xF3SF.

2.2. Higher twist correction. – We take into account the higher twist (HT) contribution following the method of [20] and [14, 15, 9]. To extract the HT contribution, the SF is parametrized as follows:

xF3(x , Q2) 4xF pQCD 3 (x , Q 2 ) 1 h(x) Q2 , (7)

where the Q2_{dependence of the first term in the r.h.s. is determined by perturbative}

QCD.

The function h(x) as well as the parameters a1-a4and scale parameter L should be

determined by fitting the experimental data.

2.3. Nuclear correction. – We have used the covariant approach in the light-cone variables [21, 22] to estimate the ratio of structure functions

RFD/N4 FD 3 (x , Q2) FN 3 (x , Q2) (8)

and to perform the joint NNLO QCD analysis of the data [2] on the structure function

F3(for the NLO analysis, see ref. [23]).

We would like to remind that the ratio RA/N

F describes the influence of nuclear

medium on the structure of a free nucleon in the process. We use the approximation

RFD/N4 RFFe/N. It gives an estimation from below of the effect of nucleon Fermi motion

in a heavy nucleus.

The covariant approach in the light-cone variables is based on the relativistic deuteron wave function (RDWF) with one nucleon on mass shell. The RDWF depending on one variable, the virtuality of nucleon k2_{(x , k}

»), can be expressed via the

neutrino-deuteron scattering FD

3 in the approach can be written as follows:

FD 3 (a , Q2) 4



The nucleon SF is defined as FN

3 4 (F3nN1 F3n N) /2, a 42 q2/2(pq). The function

D(x , k») describes the left (right)-helicity distribution for an active nucleon

(antinucleon) that carries away the fraction of deuteron momentum x 4k1 1/p1 and

transverse momentum k». It is expressed via the RDWF ca(k1) as follows:

D(x , k») PSp]c

a

(k1) Q (m 1k×)Qcb(k1) Q q× QsmnQ g5Q r(S)abQ emngdqgpd( ,

(10)

where r(S)_ab is the polarization density matrix for an unpolarized deuteron. Note that in the approach used the distribution function D(x , k») includes not only usual S- and

D-wave components of the deuteron but also a P-wave component. The latter describes

the contribution of N N-pair production. The contribution of this mechanism is small over a momentum range (x E1).

The nuclear effect in a deuteron for the n 1DKm2

1 X process has been estimated in [23]. It has been found that the ratio RD/N

F is practically independent of the

parametrization of parton distributions [24-26] and the nucleon SF [27] over a wide kinematic range of x 41023

2 0.7, Q24 1–500 (GeV/c )2. The curve has an oscillatory feature and cross-over point x0: RFD/N(x0) 41, x0C 0.03.

Thus, the obtained results give evidence that the function RD/N

F is defined by the

structure of the relativistic deuteron wave function and can be used to extract the nucleon SF F3Nfrom the experimentally known deuteron one:

F3N(x , Q2) 4 [RFD/N(x) ]21Q F3D(x , Q2) .

(11)

The performed analysis of the nuclear correction for the nucleon SF also allows one to consider the influence of the nuclear effect on the GLS sum rule [11]:

SGLS4



We have used the result on the RFD/Nratio to study the x- and Q2-dependences of the

GLS integral SGLS(x , Q2) 4



It has been shown in [23] that the nuclear effect of Fermi motion is very important for the NLO QCD analysis of the structure function F3 and verification of the

Gross–Llewellyn-Smith sum rule over a wide region of Q2

4 3–500 ( GeV/c )2. Therefore in this paper, we include the nuclear effect in the NNLO QCD analysis of both the structure function and the Gross–Llewellyn-Smith sum rule.

3. – NNLO QCD analysis of structure function xFN 3

In this section, we perform the QCD analysis of the xFN

3 experimental data [2]

Fig. 1. – Dependence of the coefficients a1-a4on Q2. The black circles are experimental points, the

lines are results of the fit.

as a first approximation that RFe/N

F 4 RFD/NfR: xF3N(x , Q2) 4a1(Q2) xa2(Q 2₎ ( 1 2x)a3(Q2)

(

)

The constants h(xi) (one per x-bin) parametrize the HT x-dependence. The points xiare

chosen in accordance with [2]: 0.0075–0.75.

The values of the parameters ai(Q02) (i 41, R , 4), scale parameter L and

constants h(xi) have been determined by fitting the whole set of xF3 data [2] (116

experimental points) for different values of Q2

0 in the kinematic region: 2 ( GeV/c )2G

Q2_{G 200 (GeV/c )}2 (2). Only statistical errors are taken into account. The x2parameter is found to be about 105 for 116 experimental points.

Figure 1a)-d) shows the parameters of SF ai at the points Q24 1.5–200 (GeV/c)2.

The LMS parameter was found in the interval ( 210 –250) MeV with statistical error

about 620 MeV.

Figure 2 shows the x-dependence of the HT contribution h(xi).

Fig. 2. – The function h(x) vs. log (x) describing a higher twist correction. The black circles are experimental points, the line is the result of the fit.

We use the result of our NNLO QCD analysis presented in figs. 1 and 2 to obtain the parametrisation of xF3in form (14).

The Q2-dependence of the coefficients aiis parametrized as follows:

ai(Q2) 4

!

2

cjQ zj, z 4log (Q2)

(15)

and is shown in fig. 1 by solid lines. The coefficients ciare presented in table I.

Figure 1 shows the dependence of coefficients ai on Q2. The black circles are

experimental points, the lines are results of the fit.

The parameter a2 slowly depends on Q2 and corresponds to the theoretical

estimation atheor

2 C 0.68 [28].

Our results for a3(Q2) could be compared to the well-known Buras-Gaemers

parametrization [29]: a34 h11 h2Q ln (Q2_/L2₎ ln (Q02/L2) . (16)

At the point Q024 1.8 GeV2it gives [16] h14 2.6 and h24 0.8. Our result is h14 3.48 6

0.03 and h24 1.18 6 0.09. It is not far from the pQCD theoretical estimation [30] based

TABLEI. – Coefficients cifor the parametrization of the function xF3(x , Q2).

a1 a2 a3 a4

c0 3.1430 6.6704 3 1021 3.2753 2.92880

c1 1.6794 4.0720 3 1023 8.2294 3 1021 21.65755

TABLEII. – Coefficients difor the parametrization of the function h(x). d0 d1 d2 d3 0.2329 0.8060 0.7308 0.1842 on the relation d d ln (Q2₎ a3(Q 2 ) 4 4 3 p aS(Q 2 ) 1O

(

)

and gives htheor2 4 0.64.

The term describing the higher twist correction is written in the form

h(x) 4

!

i 40

3

diQ zi, z 4log (x)

(18)

and is shown in fig. 2. The values of coefficients di are presented in table II.

Thus, the NNLO QCD analysis of experimental data of the structure function xF3

has been performed and the parametrizations of the NNLO perturbative QCD, TMC, nuclear effect and higher twist corrections have been obtained.

4. – Experimental constraints on coefficients of aS-expansion of Gross–

Llewellyn-Smith sum rule

In this section, we would like to show the status of the NNLO QCD analysis of the experimental data [2].

The Q2_{-dependence of the parameters a}

i(Q2) allows us to study the behavior of

SGLS(Q2) in a wide kinematic region of the momentum transfer squared.

Figure 3 shows the dependence of SGLS(Q2) on aS/p in the 3-loop approximation.

The black and open circles correspond to the NNLO QCD analysis with and without

Fig. 3. – The aS/p-expansion of the Gross–Llewellyn-Smith sum rule SGLS(Q2). The black and

open circles are results of NNLO QCD analysis obtained with and without nuclear correction, respectively.

TABLE III. – The coefficients of the Gross-Llewellyn Smith integral SGLS(Q2) expansion in

aS/p.

NNLO+HT NNLO+HT+NC Theoretical [12]

s0 2.86 6 0.01 2.74 6 0.01 3.00

s1 24.93 6 0.32 22.22 6 0.23 23.00

s2 0.98 6 2.54 27.86 6 1.74 29.75

nuclear corrections, respectively. The statistical errors are about 60.4 and are not presented at the figure.

Figure 3 demonstrates the increase of SGLS(Q2) with decreasing aS/p. The result is

in qualitative agreement with the Q2_{-dependence of the sum rule found in [31] in the}

NLO QCD analysis without taking into account target mass corrections, higher twist and nuclear effect. A similar tendency in the NLO QCD approximation with target mass corrections and nuclear effect was found in [23]. For the estimation of the order

O(a4S) and power corrections, see [32]. Figure 3 shows a considerable sensitivity of

SGLS(Q2) to the nuclear correction.

The pQCD predictions for aS/p-expansion of the Gross–Llewellyn-Smith sum rule

up to (aS/p)2could be presented in the form

SGLStheor(Q2) 4c0Q

(

)

(19)

The coefficients c1and c2have been calculated in [12]: c14 2 1 , c24 2 3.25.

The aS-dependence of SGLS presented in fig. 3 could be parametrized by the

parabola:

Sexp

GLS(Q2) 4s01 s1Q (aS/p) 1s2Q (aS/p)2.

(20)

This expansion allows us to directly compare the results of the experimental data analysis with the NNLO QCD calculations. The values of s0, s1, s2 for the interval

aS/p E0.09 are presented in table III.

One can see from table III the high sensitivity of parameters si to the nuclear

correction. The obtained results for the coefficients si show that the consideration of

the nuclear correction gives good agreement with the theoretical calculation of the next-to-leading and next-to-next-leading order QCD corrections.

5. – Conclusion

A detailed NNLO QCD analysis of the structure function xF3 of new CCFR data

including the target mass, higher twist and nuclear corrections was performed. The parametrizations of perturbative and power terms of the structure function were constructed. The results of QCD analysis of the structure function were used to study the Q2_{-dependence of the Gross–Llewellyn-Smith sum rule. The a}

S/p-expansion of

SGLS(Q2) was examined and expansion parameters s1, s2, s3were found. We would like

to emphasize that the consideration of the nuclear correction allows us to achieve a good qualitative agreement between the results obtained from the experimental data by the NNLO QCD analysis and perturbative QCD predictions for the Gross–Llewellyn-Smith sum rule in next-to-leading and next-to-next-to-leading order.

* * *

This work was partially supported by Grants of the Russian Foundation for Fundamental Research under No. 95-02-05061 and No. 95-02-04314 and INTAS project No. 93-1180ext.

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