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The hard QCD contribution to nucleon structure functions at x→1
Physics Letters B 317 ( 1993) 617-621 Noah-Holland
PHYSICS LETTERS B
The hard Q C D contribution to nucleon structure functions at x--, 1 S.V. Esaibegyan 1 Department de Fisica de Particulas, Universalize de Santiago de Composite, E- 15 706 Santiago de Composite, Spain
and N.L. T e r - I s a a k y a n Yerevan Physics Institute, Alikhanyan Brother street 2, 375036 Yerevan 36, Armenia
Received 28 May 1993; revised manuscript received 20 August 1993 Editor: P.V. Landshoff
The hard QCD contribution to the nucleon structure functions at X~ 1 expressedin terms of the V - A projection of the nucleon wave function has been obtained in the frameworkof the quark-patton model in the infinite momentum frame. Comparisonwith experiment shows that for any reasonablenucleon wave function the data cannot be describedby hard QCD contributions.
It is well known that in hard QCD processes the contributions of small and large distances are separated. The hard diagrams of QCD perturbation theory determine the small distance contribution whereas the nonperturbative effects correspond to the large distance contribution, which is usually described by phenomenological or semiphenomenological models. Therefore the investigations of such processes and the clarification of the range of their correspondence to experiment are important not only for checking the QCD predictions but also for revealing the structure of nonperturbative interactions and their space-time scale. On the other hand there are some experimental and theoretical evidences that in some processes at sufficiently large momentum transfer such a separation is not yet achieved. So in elastic proton proton scattering at large angles the polarization effects cannot be described only by the hard QCD contribution [ 1 ]. For electromagnetic form factors the possibility of a hard description at present remains disputable [2-6 ]. Possibly that means that our concept of the space-time domain ofnonperturbative effects must be corrected and such effects could be essential at rather small distances, where we traditionally suppose the dominance of the hard QCD contributions. In this paper we actually find additional arguments in favor of such a point of view. We found that the nucleon structure function at x ~ 0.95 cannot be described by a hard QCD contribution with any reasonable nucleon wave function, in particular with the asymmetric wave function of ref. [ 6 ] which describes the asymptotics of nucleon form factors. It is well known that the hard contribution to the nucleon structure function F 2 ( x ) at x--, 1 corresponds to quark diagrams with two hard gluon exchanges and has the following form [ 7 ]: F2(x)~
(1-x)
3.
(1)
Here we have found exact formulae which express the value of F 2 ( x ) in terms of integrals of nucleon wave functions. It makes it possible not only to establish the power behaviour of ( 1 ), but also to find the numerical estimates for different wave functions and to compare our predictions with experiment. It is convenient to work Present address: YerevanPhysicsInstitute, Alikhanyan Brother street 2, 375036 Yerevan 36, Armenia. ElsevierScience Publishers B.V.
617
Volume 317, number4
PHYSICSLETTERSB
18 November 1993
in the old-fashioned perturbation theory in the infinite momentum frame with P~--,~ , P . = O, qo= - q , = - q 2 / 4P (Pu and q~, are the four-momenta of the nucleon and virtual photon, respectively) [8 ]. For the selection of terms leading at x--, 1, it is easy to consider explicit forms of the energy denominators and the quark-gluon vertices at x--, 1 in the Coulomb gauge or in the gauge with Go=0 (G u is the gluon field) [9]. In that ease only the diagrams of the type presented in fig. la survive. The diagrams of fig. lb are suppressed as ( 1 - x ) 2 at x--, 1 [91. For quark momenta in the IMF we introduce the standard parameterization (see fig. la) k~=giP+k~_L , k~±P=O,
pi=xiP+pi± , pi±P=O.
(2)
The contributions of the diagrams of fig. la (as well as other diagrams which differ from the previous by quark line permutations) have the following form:
S
t
2
¢.
d]"-- dx2dx3 d2p2 d2p3 (27f) 64x IX2X3
( 3)
Here s and s~ stand for the projections of the spins of the nucleon and quarks on the third axis; Q1 is the charge of the first quark, which interacts with a virtual photon, the factor 3 arises due to the fact that the interaction with only one of the quarks is considered. The wave functions ~,(2t) are defined through the nucleon-quark transition vertex V~,(;h, k~± ) as follows: s ~ (27t) 6 1 1 ~ d 2 k 2d2k3± x Vs t~ 1 q~,l( l) = sA ~,ki±) 2 p ( E _ e l _ e 2 _ e 3 ) ,
(4)
where E = (p2 _I.M 2) 1/2, ¢i__ (k 2 + m 2 )1/2, M and m; denote the nucleon and ith quark masses. The separation of integrals in (3) by transverse momenta actually means the separation of large and small distances and is connected with the fact that the "Born term" B,~,~ in leading approximation does not depend on k~±: 2 2 256(4~a,) 4 x2x3 Js~ J ~ J ~ J ~ , - ~ Bs,~,(21,2~;xi, P~±)= 81i1_21)2( 1 - 2 1,) 2m2± 2 m3± 2 ( m2x 2 xa + m]± x2) 2
X ( l - A 1 ) ( 1 - 2 ] ) + (22-23) ( X ~ - 2 ~ ) -
2 2 t p 4x2x3m2±m3±(A:-2a)(A2-23)~ 1 (--m]+-----------~ ± x, ~--~x~)-5 ,] ~2~3~a~ .
(5)
Here m 2± = m 2 + p 2±. Since the integrations over p~± in (3) are divergent at the lower limits, we kept in (5) the
P2
f
k3
t
/
/
Ps
~"
) /
I
I
Fig. I. diagramsof old-fasluonedperturbationtheorythat determinethe hard contributionto nucleonstructurefunctions. 618
Volume 317, number4
PHYSICSLETTERSB
18 November 1993
quark masses, considering them as effective parameters, determined by nonperturbative interactions. The running QCD coupling txs= tx,(4m 2 / ( 1 - x ) ) is taken at typical virtualities of the hard quark and gluon lines (see eq. (11)). It is convenient to expand the wave function ~,(2,.) in the vector V(4i), axialA (2~), and tensor T(4~) projections [ 5 ]: q~/2~( 4~) = ~4 [2 Tu?uTdJ, - ( V-A )u?u~,d? - ( V+ A )u,Lutd? + 2 ~ 3 + I ~ 3 ]s~s,
(6)
which are determined as the matrix elements of the corresponding nonlocal operators (see for example ref. [5 ] ). These values are connected by the following relations [6 ]: V(41,42,43)=V(42,4t,43) , A(41,42,43)m--A(42,41,43),
2T(21,42, 4'!,3)=~N(4t, 23,22) + ~,v(42,23, 2t) ,
(7)
where ~gN=V(41,42,43)-A(41,42,43) .
(8)
We did not write the color part of the wave functions (4) and (6) and included the colour factor ~ in the Born term ( 5 ). Using eqs. (7), ( 8 ) the structure function (3) may be finally presented in the form or, ~'(I_x)3{2Q~[A(A+B)+½C(C+D)]+(Q2+Q~t)(B2+ID2)} F°tx) = -128( ~ ,,,~--~:
(9)
where
f d~2d~3~gN(4 I,42, 43)
A=~4
4243(1__41 )
'
C = ~ j f d~2d~3(42--43) ~ _ - - ~ ) 2 ~v(2,,22,43),
D--~4 fj
d'/l'2d"~'3(41-42) BN(41,42,43)4,42(1_43) 2 f,
,
(10)
We present (9) for the proton, for the neutron the replacements Q , * , Qa must be inserted. Here rh2± = rn2+/~., where/~± is the lower limit for p± integrations. We do not know the value of m, but it seems possible to find the lower bound on r~ ~ considering the kinematic region where our approximations are valid and the process is really hard. Let us consider the energy denominator corresponding to the dash-dotted line in fig. la
2p(E--el--E2--~3)fM2- E m2"L ~ M 2 - 4rn2 Xi
l--x"
x~l
(11)
For the validity of the results obtained it is necessary to require rh 2 : ~ M 2 ( 1 - x )
(atx~l).
(12)
At large values of m this condition is automatically fulfilled, otherwise it must be provided taking the proper value of/~±. The restrictions from other energy denominators proved to be of the same order. The QCD evolution of the structure function, connected with gluon emission that we take into account, using the results ofref. [ 10] is a follows: 1
Q2
F~(u)V(x/u,~) du,
F~(x)-x
o~,(Q2)
(13)
U
where F~ (u) is the structure function without gluon emissions,
V(z, ~) is the possibility of finding a quark with 619
Volume 317, number 4
PHYSICS LETTERSB
18 November 1993
a fraction of momentum z of the original quark; Qo denotes the normalization point of the evolution equation. This quantity is expected to be of the order of typical quark virtualities in the nucleon. The precise value of Qo for given A Qc° we estimated from experiment requiring eq. (13) to describe well the logarithmic Q2 dependence at x--, 1 (for Fo ( u ) ~ ( 1 - u) 3). For such a choice of Qo V(z, ~) is practically independent ofA °c°. The creation of quark-antiquark pairs at x--, 1 may be neglected. The effects of the QCD evolution of the nucleon wave function are supposed to be included into the phenomenological functions for which numerical estimates will be given. An analysis of the terms omitted in our approximation shows that our results are valid at x~> 0.9. Therefore for comparison with experiment it is necessary to extract from the data the lowest twist contribution to the structure function and to extrapolate it into this region. For this purpose we parameterized the SLAC [ 11 ], SLAC-MIT [ 12 ] and EMC [ 13 ] data taking into account logarithmic and power terms in Q2, excluded the latter from the parameterization and extrapolated the lowest twist contribution into the region of large x. It is interesting that the power terms turned out to be very large and made up for example 30% at x = 0.75 and Q2 = 30 GeV 2. The results of such an extrapolation are presented in table 1. We tried different functions for the parameterization. The corresponding results differ from each other within 30%. So, the separation of the logarithmic and power terms and the extrapolation can be considered to be good enough. In our estimates we take r~ 2 >/½M( 1 - x ) . This is a rather weak condition and therefore the real estimates on F2 (x) could be considerably lower. We take the world average values of QCD coupling, as ( M z ) = 0.1134 + 0.0035 [ 14] ( M z is the Z°-boson mass). To translate this value of ot (/~) into the range #2= 4m 2 / ( 1 - x ) we used the next to leading order formulae in the modified minimal subtraction (MS) scheme [ 15 ]. The corresponding values of ors(/~) are presented in table 1 (A ~-~ = 0.316 + 0.054 ). For the asymptotic wave function [ 16 ] ~ o = 120foxl x2x3 ,
(14)
where fo = ( 5 + 0.3 ) × 10- 3 GeV 2 [ 6 ], the theoretical predictions are negligible compared with experiment and we do not present these in table 1. The situation resembles very much the case of the asymptotics of nucleon form factors which cannot be described by the hard contribution with asymptotic wave function (14) [ 17 ]. It seems possible to increase considerably the theoretical predictions taking the asymmetric wave function with a sharp peak at 2 ~~ 1 (see eq. (10) ). A function of such type has been proposed in ref. [ 6 ] from QCD dispersion sum rules: ~=~(23.814x~
(15)
+ 12.978x~ +6.174x] +5.88x3 - 7 . 0 9 8 ) .
This function describes the nucleon form factors at large 0 2 , but it is difficult to make it conform with the description of the nucleon low energy parameters in the quark model [ 5 ]. In the case of the nucleon structure function the wave function ( 15 ) leads to the increase of the theoretical estimates by a factor of 25 only. The numerical results for such a function are presented in table 1. One can see that the theoretical predictions remain much smaller than for the experiment. If we even increase the value of as (/z) by four standard deviations, it will be impossible to describe the experiment at least for x ~<0.9 5. The use of functions that are more asymmetric than (15) seems unreasonable, since they lead to a contradicTable 1 Theoretical predictions for proton structure functions (for wave function (15)) and its extrapolated experimental values at Q2= 30 GeV2.
620
x
r~± (GeV)
oq(4r~2 / ( 1- x ) )
F2(x )
F~"P(x)
0.9 0.95
0.18 0.12
0.34_+0.04 0.34+0.04
(6.6+3.0)×10 -6 (2.6_+ 1.2)× 10-6
(4.6+0.5)X 10-4 (3.6+0.6) × 10-5
Volume 317, number 4
PHYSICS LETTERS B
18 November 1993
tion with experiment, not only for the low energetic p a r a m e t e r s o f the nucleon, but also for the asymptotics o f the form factors. So, for the experimentally available values o f x ( a n d at least up to x~<0.95 i f the observed experimental d e p e n d e n c e r e m a i n s valid in this range) the e x p e r i m e n t a l d a t a cannot be described by h a r d Q C D contributions. One o f us (S.E.) is grateful to the theory group o f D e p a r t a m e n t o de Fisica de Particulas, U n i v e r s i d a d e de Santiago de Compostela, where the last part o f this work was carried out, for its k i n d hospitality.
References [ 1] A. Yokasawa, Phys. Rep. 64 (1980) 47. [ 2 ] A.S. Bagdasaryan, S.V, Esaibegyan and N.L. Ter-Isaakyan, Yad. Fiz. 38 (1983) 402. [3] V.A. Nesterenko and A.V. Radyushkin, Phys. Lett. B 115 (1982) 410; B 128 (1983) 439. [ 4 ] B.L. Ioffe and A.N. Smilga, Nucl. Phys. B 216 ( 1982 ) 373. [5 ] I.G. Aznauryan, A.S. Bagdasaryan, S.V. Esaibegyan and N.L. Ter-Isaakyan, Soy. J. Nucl. Phys. 55 (1992) 1099. [6] V.L. Chernyak, A.A. Ogloblin and I.R. Zhitnitsky, Yad. Fiz. 48 (1988) 841. [ 7 ] G.R. Farrar and D.R. Jackson, Phys. Rev. Lett. 35 ( 1975 ) 1416. [ 8 ] I.G. Aznauryan, A.S. Bagdasaryan and N.L. Ter-Isaakyan, Yad. Fiz. 36 (1982) 1278; Phys. Lett. B 112 ( 1982 ) 393. [ 9 ] S.G. Grigoryan, S.V. Esaibegyan and N.L. Ter-Isaakyan, Yad. Fiz. 27 (1978) 1312. [10 Yu.L. Dokshitser, D.I. Dyakonov and S.I. Troyan, Phys. Rep. 58 (1980) 269. [11 A. Bodek et al., Phys. Rev. D 20 (1979) 1471. [12 M.D. Mestuyer el al., Phys. Rev. D 27 ( 1983 ) 285. [13 I.I. Aubert et al., Nucl. Phys. B 259 ( 1965 ) 189. [14 Particle Data Group, K. Hikasa el al., Review of particle properties, Phys. Rev. D 45, part II (1992). [15 W.A. Bardeen A.J. Buras, D.W. Duke and T. Muta, Phys. Rev. D 18 (1978) 3998. [16 S.J. Brodsky and G.L. Lcpage, Phys. Rev. D 22 (1980) 2157. [17 I.G. Aznauryan, S.V. Esaibegyan and N.L. Ter-Isaakyan, Phys. Lett. B 90 (1980) 151; B 92 (1980) 371 (E).
621
PHYSICS LETTERS B
The hard Q C D contribution to nucleon structure functions at x--, 1 S.V. Esaibegyan 1 Department de Fisica de Particulas, Universalize de Santiago de Composite, E- 15 706 Santiago de Composite, Spain
and N.L. T e r - I s a a k y a n Yerevan Physics Institute, Alikhanyan Brother street 2, 375036 Yerevan 36, Armenia
Received 28 May 1993; revised manuscript received 20 August 1993 Editor: P.V. Landshoff
The hard QCD contribution to the nucleon structure functions at X~ 1 expressedin terms of the V - A projection of the nucleon wave function has been obtained in the frameworkof the quark-patton model in the infinite momentum frame. Comparisonwith experiment shows that for any reasonablenucleon wave function the data cannot be describedby hard QCD contributions.
It is well known that in hard QCD processes the contributions of small and large distances are separated. The hard diagrams of QCD perturbation theory determine the small distance contribution whereas the nonperturbative effects correspond to the large distance contribution, which is usually described by phenomenological or semiphenomenological models. Therefore the investigations of such processes and the clarification of the range of their correspondence to experiment are important not only for checking the QCD predictions but also for revealing the structure of nonperturbative interactions and their space-time scale. On the other hand there are some experimental and theoretical evidences that in some processes at sufficiently large momentum transfer such a separation is not yet achieved. So in elastic proton proton scattering at large angles the polarization effects cannot be described only by the hard QCD contribution [ 1 ]. For electromagnetic form factors the possibility of a hard description at present remains disputable [2-6 ]. Possibly that means that our concept of the space-time domain ofnonperturbative effects must be corrected and such effects could be essential at rather small distances, where we traditionally suppose the dominance of the hard QCD contributions. In this paper we actually find additional arguments in favor of such a point of view. We found that the nucleon structure function at x ~ 0.95 cannot be described by a hard QCD contribution with any reasonable nucleon wave function, in particular with the asymmetric wave function of ref. [ 6 ] which describes the asymptotics of nucleon form factors. It is well known that the hard contribution to the nucleon structure function F 2 ( x ) at x--, 1 corresponds to quark diagrams with two hard gluon exchanges and has the following form [ 7 ]: F2(x)~
(1-x)
3.
(1)
Here we have found exact formulae which express the value of F 2 ( x ) in terms of integrals of nucleon wave functions. It makes it possible not only to establish the power behaviour of ( 1 ), but also to find the numerical estimates for different wave functions and to compare our predictions with experiment. It is convenient to work Present address: YerevanPhysicsInstitute, Alikhanyan Brother street 2, 375036 Yerevan 36, Armenia. ElsevierScience Publishers B.V.
617
Volume 317, number4
PHYSICSLETTERSB
18 November 1993
in the old-fashioned perturbation theory in the infinite momentum frame with P~--,~ , P . = O, qo= - q , = - q 2 / 4P (Pu and q~, are the four-momenta of the nucleon and virtual photon, respectively) [8 ]. For the selection of terms leading at x--, 1, it is easy to consider explicit forms of the energy denominators and the quark-gluon vertices at x--, 1 in the Coulomb gauge or in the gauge with Go=0 (G u is the gluon field) [9]. In that ease only the diagrams of the type presented in fig. la survive. The diagrams of fig. lb are suppressed as ( 1 - x ) 2 at x--, 1 [91. For quark momenta in the IMF we introduce the standard parameterization (see fig. la) k~=giP+k~_L , k~±P=O,
pi=xiP+pi± , pi±P=O.
(2)
The contributions of the diagrams of fig. la (as well as other diagrams which differ from the previous by quark line permutations) have the following form:
S
t
2
¢.
d]"-- dx2dx3 d2p2 d2p3 (27f) 64x IX2X3
( 3)
Here s and s~ stand for the projections of the spins of the nucleon and quarks on the third axis; Q1 is the charge of the first quark, which interacts with a virtual photon, the factor 3 arises due to the fact that the interaction with only one of the quarks is considered. The wave functions ~,(2t) are defined through the nucleon-quark transition vertex V~,(;h, k~± ) as follows: s ~ (27t) 6 1 1 ~ d 2 k 2d2k3± x Vs t~ 1 q~,l( l) = sA ~,ki±) 2 p ( E _ e l _ e 2 _ e 3 ) ,
(4)
where E = (p2 _I.M 2) 1/2, ¢i__ (k 2 + m 2 )1/2, M and m; denote the nucleon and ith quark masses. The separation of integrals in (3) by transverse momenta actually means the separation of large and small distances and is connected with the fact that the "Born term" B,~,~ in leading approximation does not depend on k~±: 2 2 256(4~a,) 4 x2x3 Js~ J ~ J ~ J ~ , - ~ Bs,~,(21,2~;xi, P~±)= 81i1_21)2( 1 - 2 1,) 2m2± 2 m3± 2 ( m2x 2 xa + m]± x2) 2
X ( l - A 1 ) ( 1 - 2 ] ) + (22-23) ( X ~ - 2 ~ ) -
2 2 t p 4x2x3m2±m3±(A:-2a)(A2-23)~ 1 (--m]+-----------~ ± x, ~--~x~)-5 ,] ~2~3~a~ .
(5)
Here m 2± = m 2 + p 2±. Since the integrations over p~± in (3) are divergent at the lower limits, we kept in (5) the
P2
f
k3
t
/
/
Ps
~"
) /
I
I
Fig. I. diagramsof old-fasluonedperturbationtheorythat determinethe hard contributionto nucleonstructurefunctions. 618
Volume 317, number4
PHYSICSLETTERSB
18 November 1993
quark masses, considering them as effective parameters, determined by nonperturbative interactions. The running QCD coupling txs= tx,(4m 2 / ( 1 - x ) ) is taken at typical virtualities of the hard quark and gluon lines (see eq. (11)). It is convenient to expand the wave function ~,(2,.) in the vector V(4i), axialA (2~), and tensor T(4~) projections [ 5 ]: q~/2~( 4~) = ~4 [2 Tu?uTdJ, - ( V-A )u?u~,d? - ( V+ A )u,Lutd? + 2 ~ 3 + I ~ 3 ]s~s,
(6)
which are determined as the matrix elements of the corresponding nonlocal operators (see for example ref. [5 ] ). These values are connected by the following relations [6 ]: V(41,42,43)=V(42,4t,43) , A(41,42,43)m--A(42,41,43),
2T(21,42, 4'!,3)=~N(4t, 23,22) + ~,v(42,23, 2t) ,
(7)
where ~gN=V(41,42,43)-A(41,42,43) .
(8)
We did not write the color part of the wave functions (4) and (6) and included the colour factor ~ in the Born term ( 5 ). Using eqs. (7), ( 8 ) the structure function (3) may be finally presented in the form or, ~'(I_x)3{2Q~[A(A+B)+½C(C+D)]+(Q2+Q~t)(B2+ID2)} F°tx) = -128( ~ ,,,~--~:
(9)
where
f d~2d~3~gN(4 I,42, 43)
A=~4
4243(1__41 )
'
C = ~ j f d~2d~3(42--43) ~ _ - - ~ ) 2 ~v(2,,22,43),
D--~4 fj
d'/l'2d"~'3(41-42) BN(41,42,43)4,42(1_43) 2 f,
,
(10)
We present (9) for the proton, for the neutron the replacements Q , * , Qa must be inserted. Here rh2± = rn2+/~., where/~± is the lower limit for p± integrations. We do not know the value of m, but it seems possible to find the lower bound on r~ ~ considering the kinematic region where our approximations are valid and the process is really hard. Let us consider the energy denominator corresponding to the dash-dotted line in fig. la
2p(E--el--E2--~3)fM2- E m2"L ~ M 2 - 4rn2 Xi
l--x"
x~l
(11)
For the validity of the results obtained it is necessary to require rh 2 : ~ M 2 ( 1 - x )
(atx~l).
(12)
At large values of m this condition is automatically fulfilled, otherwise it must be provided taking the proper value of/~±. The restrictions from other energy denominators proved to be of the same order. The QCD evolution of the structure function, connected with gluon emission that we take into account, using the results ofref. [ 10] is a follows: 1
Q2
F~(u)V(x/u,~) du,
F~(x)-x
o~,(Q2)
(13)
U
where F~ (u) is the structure function without gluon emissions,
V(z, ~) is the possibility of finding a quark with 619
Volume 317, number 4
PHYSICS LETTERSB
18 November 1993
a fraction of momentum z of the original quark; Qo denotes the normalization point of the evolution equation. This quantity is expected to be of the order of typical quark virtualities in the nucleon. The precise value of Qo for given A Qc° we estimated from experiment requiring eq. (13) to describe well the logarithmic Q2 dependence at x--, 1 (for Fo ( u ) ~ ( 1 - u) 3). For such a choice of Qo V(z, ~) is practically independent ofA °c°. The creation of quark-antiquark pairs at x--, 1 may be neglected. The effects of the QCD evolution of the nucleon wave function are supposed to be included into the phenomenological functions for which numerical estimates will be given. An analysis of the terms omitted in our approximation shows that our results are valid at x~> 0.9. Therefore for comparison with experiment it is necessary to extract from the data the lowest twist contribution to the structure function and to extrapolate it into this region. For this purpose we parameterized the SLAC [ 11 ], SLAC-MIT [ 12 ] and EMC [ 13 ] data taking into account logarithmic and power terms in Q2, excluded the latter from the parameterization and extrapolated the lowest twist contribution into the region of large x. It is interesting that the power terms turned out to be very large and made up for example 30% at x = 0.75 and Q2 = 30 GeV 2. The results of such an extrapolation are presented in table 1. We tried different functions for the parameterization. The corresponding results differ from each other within 30%. So, the separation of the logarithmic and power terms and the extrapolation can be considered to be good enough. In our estimates we take r~ 2 >/½M( 1 - x ) . This is a rather weak condition and therefore the real estimates on F2 (x) could be considerably lower. We take the world average values of QCD coupling, as ( M z ) = 0.1134 + 0.0035 [ 14] ( M z is the Z°-boson mass). To translate this value of ot (/~) into the range #2= 4m 2 / ( 1 - x ) we used the next to leading order formulae in the modified minimal subtraction (MS) scheme [ 15 ]. The corresponding values of ors(/~) are presented in table 1 (A ~-~ = 0.316 + 0.054 ). For the asymptotic wave function [ 16 ] ~ o = 120foxl x2x3 ,
(14)
where fo = ( 5 + 0.3 ) × 10- 3 GeV 2 [ 6 ], the theoretical predictions are negligible compared with experiment and we do not present these in table 1. The situation resembles very much the case of the asymptotics of nucleon form factors which cannot be described by the hard contribution with asymptotic wave function (14) [ 17 ]. It seems possible to increase considerably the theoretical predictions taking the asymmetric wave function with a sharp peak at 2 ~~ 1 (see eq. (10) ). A function of such type has been proposed in ref. [ 6 ] from QCD dispersion sum rules: ~=~(23.814x~
(15)
+ 12.978x~ +6.174x] +5.88x3 - 7 . 0 9 8 ) .
This function describes the nucleon form factors at large 0 2 , but it is difficult to make it conform with the description of the nucleon low energy parameters in the quark model [ 5 ]. In the case of the nucleon structure function the wave function ( 15 ) leads to the increase of the theoretical estimates by a factor of 25 only. The numerical results for such a function are presented in table 1. One can see that the theoretical predictions remain much smaller than for the experiment. If we even increase the value of as (/z) by four standard deviations, it will be impossible to describe the experiment at least for x ~<0.9 5. The use of functions that are more asymmetric than (15) seems unreasonable, since they lead to a contradicTable 1 Theoretical predictions for proton structure functions (for wave function (15)) and its extrapolated experimental values at Q2= 30 GeV2.
620
x
r~± (GeV)
oq(4r~2 / ( 1- x ) )
F2(x )
F~"P(x)
0.9 0.95
0.18 0.12
0.34_+0.04 0.34+0.04
(6.6+3.0)×10 -6 (2.6_+ 1.2)× 10-6
(4.6+0.5)X 10-4 (3.6+0.6) × 10-5
Volume 317, number 4
PHYSICS LETTERS B
18 November 1993
tion with experiment, not only for the low energetic p a r a m e t e r s o f the nucleon, but also for the asymptotics o f the form factors. So, for the experimentally available values o f x ( a n d at least up to x~<0.95 i f the observed experimental d e p e n d e n c e r e m a i n s valid in this range) the e x p e r i m e n t a l d a t a cannot be described by h a r d Q C D contributions. One o f us (S.E.) is grateful to the theory group o f D e p a r t a m e n t o de Fisica de Particulas, U n i v e r s i d a d e de Santiago de Compostela, where the last part o f this work was carried out, for its k i n d hospitality.
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