- Email: [email protected]
Instanton effects on extracting the nucleon parton distributions
Available online at www.sciencedirect.com
Nuclear Physics B (Proc. Suppl.) 219–220 (2011) 174–177 www.elsevier.com/locate/npbps
Instanton effects on extracting the nucleon parton distributions A. Mirjalilia,c , M. Dehghania , M. M. Yazdanpanahb,c a Physics
Department, Yazd University, P.O.B 89195-741, Yazd, Iran Department, Kerman Shahid-Bahonar University, Kerman, Iran c School of Particles and Accelerators, IPM (Institute for Research in Fundamental Sciences) P.O. B 19395-5531, Tehran, Iran b Physics
Abstract In considering the nucleon parton distributions at low energy values, the non-perturbative effects will have an important role. It is usual to assume for the related vacuum a structure and attribute it a back ground field. It is very like to the situation which we have for Bohm-Aharnov effect when an electron moved in a region where there is not any magnetic field (vacuum). In this case we can attribute to the related vacuum a back ground field which is in fact the gauge field of electrodynamics force. The fluctuations which exist with respect to the concerned vacuum in QCD can appear as instantons and will effect on then nucleon parton distributions especially at low Bjorken- x values. A comparison between the proton structures when we consider the instanton effect with respect to the usual one, indicates the non-perturbative effect on extracting these functions. Considering this effect will yield a better result for the F2p structure function. Keywords: Instanton effect, parton distribution, non-perturbative region, nucleon structure function, chirality violation 1. Introduction It was known that t’Hooft interaction can be treated as the mechanism for baryon-number violation [1]. This interaction is induced by vacuum fluctuations of gauge fields-instantons. The renewal of interest in this problem happens basically due to the fact that at high energy the multiple production of gauge bosons from instanton leads to anomalous growth of the baryon-number violation cross-section [2]. In QCD, the instantons violate the chirality and therefore the same mechanism, evidently leads to anomalous quarks chirality violation at high energy. The most interesting fact is that in contrast of problems of the experimental searches of baryon-number violation, in strong interaction we already observe multiple anomalous spin effects, which can be connected only with a large quarks chirality violation at high energy [3]. The perturbative QCD predicts the disappearance of chirality violation at high energy and therefore the instanton interaction have been proposed as the funda0920-5632/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2011.10.092
mental mechanism for the spin effects at high energy [4]. The experimental evident lead to the conclusion that the behavior of the sea quarks at the small x differs strongly from the perturbative QCD predictions. The main goal of this article is the estimation of the contribution to the quark distribution functions of the nucleon induced by instantons. Here, we will present the arguments that these surprising results can be explained by manifestation of a complicated structure of QCD vacuum, namely by taking into account the instanton-induced interaction between quarks. 2. t’Hooft Lagrangian It is well known that behavior of quark-quark cross section at high energy determines the behavior of the nucleon structure functions at low x [5]. Thus the anomalous behavior of the quark-nucleon crosssections should provide the anomalous behavior of the structure functions at x → 0. It is connected with the
A. Mirjalili et al. / Nuclear Physics B (Proc. Suppl.) 219–220 (2011) 174–177
fact that at very high energy the small size instanton approximation is incorrect and then one needs to take into account the overlapping of instantons [6]. It was supposed that QCD instantons induce the following form for growth of cross section σqq ∼ sαI −1 where αI > 1 and the value for αI will be found from the fit of the experimental data on the structure functions. By using results of [7] one can connect the contribution of the instantons to the quark distribution functions with the value of the quark-nucleon cross section through the instanton ∞ k2 max 2 ImT qN 1 ¯ (S , k ) q(x) = 3 dS dk2 2 2 2 2π S 0 (k − mq ) k2 min × −x+
m2q − k2 S−
MN2
−
k2
.
(1)
In Eq. (1), k2 is defined by k2 = x
S k2 + MN2 + ⊥ , x−1 1−x
(2)
where k is the momentum of the soft quark which was created by the virtual photon and S = (P + k)2 in which P is the momentum of initial quark. Therefore we can write S + MN2 ) , x−1 S Q2 + MN2 ) + . = x( x−1 1−x
2 kmax = x( 2 kmin
(3)
The imaginary part of the scattering amplitude ImT qN is connected with the total cross section of the quark nucleon interaction through the relation σtot =
1 √ ImT qN (S , k2 ) , 2pc.m S
(4)
where pc.m = (−k2 + (S −M4S+k ) )( 2 ) is the threemomentum in the quark-nucleon center of mass system. At low energies the cross section induced by instantons is determined by the specific t’Hooft interaction between quarks, which for the number of flavors N f = 3 and for massless quarks has the following form 4 (x) = dρ n(ρ)( π2 ρ3 )2 (5) L(2) ef f 3 3 3 × u¯ R uL d¯R dL 1 + (1 − σuμν σdμν )λau λad 32 4 + R ←→ L . 2
2 2
1
175
Here ρ is the size of instanton and n(ρ) is its density, consists in the specific chirality and flavor properties of the interaction. Namely, it changes chirality by the value and it acts only between different flavors of the quarks. Using the insertion of the additional instantonantiinstanton pairs, it was shown that the contribution to the structure functions violates both the Ellis-Jaffe sum rule, related to the value of chirality carried by proton quarks, and the Gottfried sum rule connected with flavor properties of the quark sea. The value of the instanton contribution to the violation of the Ellis-Jaffe sum rule and Gottfried sum rules is determined by the value of the quark-nucleon instanton induced cross section. It should be stressed, that interaction (5) is the point like interaction with a dimensional coupling constant. Therefore it should lead, similarly to the case of the Fermi weak interaction, to an anomalous growth of the cross section of the quark quark interaction with increasing energy. At low energies, we can estimate this growth by using the instanton liquid model of the QCD vacuum [8]. 3. Instanton-induced interaction For the quark-quark scattering due to instantons, the chirality and flavor properties of the sea quarks are determined by the valence quark lines incoming t’Hooft vertex [4]. So, to obtain the relation between the various chirality and flavor sea quark distribution functions it is enough to consider the contribution to the quark sea. Let us consider the quark sea created by interaction (5) from the S U(6)W proton symmetrical wave function 5 1 1 2 u↑+ u↓+ d↑+ d↓. (6) 3 3 3 3 We will use the properties of the vertex which lead to the opposite chirality and different flavor as compare with the chirality and flavor of the initial valence quark [4] p ↑=
2 NI dv (x)xαv fu (x) , 3 xα I αv 1 NI dv (x)x 2¯u−I (x) = fu (x), 3 xα I αv 1 NI uv (x)x fu (x) , 2d¯+I (x) = 6 xα I αv 5 NI uv (x)x 2d¯−I (x) = fu (x), 6 xα I 2 s¯+I (x) = 2(¯u+I (x) + d¯+I (x)),
2¯u+I (x) =
2 s¯−I (x) = 2(¯u−I (x) + d¯−I (x)),
(7)
where NI is any constant, uv (x), dv (x) are the valence distribution functions,which we choose in the usual form
A. Mirjalili et al. / Nuclear Physics B (Proc. Suppl.) 219–220 (2011) 174–177
176
uv (x) =Nu x−αv (1 − x)bu , dv (x) =Nd x−αv (1 − x)bd .
(8)
The function fu (x) provides the correct unitarity limit at x → 0 . This function can be chosen in the following form ⎧ ⎪ ⎪ if x > x0 ; ⎨1, (9) fu (x) = ⎪ ⎪exp[−(x /x − 1)], if x < x . ⎩ 0
q(x, Q2 ) =
0
Instantons also lead to the valence quark chirality flipping due to the annihilation processes. Its contribution to the valence quark distributions functions is [4] 5 NI uv (x)xαv 6 xα I αv 1 u N I v (x)x I (x) = uv− α I 6 x 2 NI dv (x)xαv I dv+ (x) = 3 xα I αv 1 N d I v (x)x I dv− (x) = 3 xα I I uv+ (x) =
fu (x) , fu (x),
(14)
where M ≈ mη = 0.96 GeV is the characteristic hadron scale which is connected to the scale of the violation of the U A (1) symmetry due to instantons.
(10)
p p p ¯ +,− u¯ (x)+,− = d(x) = 2 s¯(x)+,− = N s (1 − x)5 /xαP (0)−1 , (11)
where by taking into account the suppression of the strange sea, one usually connects with the belated start of the evolution due to the large mass of the strange quark [9]. The amount of α p (0) = 1.08 is representing the intercept of the soft pomeron [10]. From parton distributions in above, one can obtain the unpolarized distribution functions q(x) ¯ = q¯ + (x) + q¯ − (x) and the polarized Δq(x) ¯ = q¯ + (x) − q¯ − (x) distribution functions. So, the nucleon structure functions F2μN (x) have the following form F2μp (x) =e2u (urv (x) + 2¯u p (x)) + e2d (dvr (x) + 2d¯p (x))
+ 2e2s s¯ p (x) + ( e2q )A(x), q
F2μn (x) =e2d (urv (x) + 2¯u p (x)) + e2u (dvr (x) + 2d¯p (x))
+ 2e2s s¯ p (x) + ( e2q )A(x), (12) q
where NI (uv (x) + dv (x))xαv +1 fu (x), xα I
Q2 q(x) , + M2
From parton distributions in above, one can obtain the following sum rules for the unpolarized distributions:
fu (x) , fu (x),
Q2
4. Sum rules
The quark sea induced by the perturbative gluons does not differ in chirality or flavor, and therefore it can be taken in the form using quark-counting rules
A(x) =
and qrv (x) is renormalized valence quark distribution function . At low Q2 some investigations should be paid. We know that in this region the conventional valence quark model for the quark distribution functions works well and therefore we will assume that the instanton contribution has the following dependence on Q2 [12]
(13)
1 0 1
0
dx(urv (x) + uvI (x)) =2, dx(dvr (x) + dvI (x)) =1.
(15)
Fits of the NMC and H1 experimental data for the unpolarized structure function of the proton F2p (x) and the neutron-proton ratio F2μn (x)/F2μp (x) is used in the calculations. We obtain a large excess of αI in comparison with 1, which provides the growth of F2N (x) at low x. We conclude, that the nonpertubative sea gives the large contribution to quark sea distribution functions in the interval 0.0001 < x < 0.1 and the growth of the structure functions in this region is determined by the anomalous dependence on the energy of the instanton-induced interaction between quarks.
5. Results The flavor dependence of the instanton sea leads to large violation of the S U(2)-flavour symmetry quark sea, which, of course, provides a strong violation of the Gottfried sum rule for the integrated difference of the structure functions 1 dx μp (F (x) − F2μn (x)), IG (t) = x 2 t 1 (16) IG (0) = . 3
A. Mirjalili et al. / Nuclear Physics B (Proc. Suppl.) 219–220 (2011) 174–177
The model predicts a very large violation of the Gottfried sum rule: IGI (0) = 0.247 which can be compared with the NMC result [13] IGN MC
60 With the instanton effect In the standard case
= 0.257 ± 0.017.
50
It is obvious that the instanton- induced interaction will change the behavior of F2N (x) at low x values. The plot in below confirms this reality. 6. Conclusion The perturbative QCD predicts the disappearance of chirality violation at high energy and therefore the instanton interaction have been proposed as the fundamental mechanism for the spin effects at high energy. Consequently an un-symmetrized distributions for sea quark densities will be resulted and the violation of the GSR will be justified. Due to instanton effect the result for F2N (x) structure function at low x values will differ with respect to the usual one as can be seen in Fig. 1. The instanton effect on structure function can be extended to high order approximation while the renormalized coupling constant depends on the ρ as the size of instanton. This quantity can be found by fitting the structure function on the available experimental data. 7. Acknowledgment Authors acknowledge the institute for research in fundamental sciences (IPM) for their hospitality whilst this research was performed. A. M. is indebted Yazd university to support him financially to present this research result. He is also grateful to A. E. Dorokhov to give his constructive comments. References [1] ’t Hooft Phys. Rev. D14 (1976) 3432. [2] A. Ringwald Nucl. Phys. B330 (1990) 1; O. Espinosa Nucl. Phys. B343 (1990) 310. [3] G. Bunce et al. Part. World 3 (1992) 1 . [4] A. E. Dorokhov, N. I. Kochelev Phys. Lett. B259 (1991) 335; Phys. Lett. B304 (1993) 167; Int. J. of Mod. Phys. A8 (1993) 603. [5] P. V. Landshoff, J. C. Polkinghorne, R.D. Short Nucl. Phys. B28 (1971) 225. [6] I. I. Balitsky, M. G. Ryskin Phys. Lett. B296 (1992) 185. The contribution of the small size instantons to the coefficient function in front of parton distributions for DIS was calculated in paper I. I. Balitsky, V. M. Braun Phys. Rev. D47 (1993) 1879. [7] S. A. Kulagin, G. Piller, W. Weise, Phys.Rev.C50 (1994) 1154. [8] E. V. Shuryak. Rev.Mod.Phys. 65 (1993)1 and references therein.
177
40
F2
30
20
10
0 0.0001
0.001
0.01
0.1
1
x
Figure 1: Proton structure function in the standard case (solid line) and its comparison when the instanton effects has been considered (dashed line).
[9] [10] [11] [12]
V. Barone at el. Phys. Lett. B268 (1991)279. A. Donnachie, P. V. Landshoff Nucl. Phys. B231 (1984) 189. R. D. Carlitz, J. Kaur Phys. Rev. Lett. 38 (1977) 673. N. I. Kochelev, Procceding of 2nd Meeting on Possible Measurements of Singly Polarized pp and pn Collisions at HERA (1995) Aug.31-Sep.1 [13] NMC, P. Amaudruz et al. Phys. Rev. Lett. 66 (1991) 2712; Preprint CERN-PPE/93-1993; H1 Coll. Preprint DESY 93-113.
Nuclear Physics B (Proc. Suppl.) 219–220 (2011) 174–177 www.elsevier.com/locate/npbps
Instanton effects on extracting the nucleon parton distributions A. Mirjalilia,c , M. Dehghania , M. M. Yazdanpanahb,c a Physics
Department, Yazd University, P.O.B 89195-741, Yazd, Iran Department, Kerman Shahid-Bahonar University, Kerman, Iran c School of Particles and Accelerators, IPM (Institute for Research in Fundamental Sciences) P.O. B 19395-5531, Tehran, Iran b Physics
Abstract In considering the nucleon parton distributions at low energy values, the non-perturbative effects will have an important role. It is usual to assume for the related vacuum a structure and attribute it a back ground field. It is very like to the situation which we have for Bohm-Aharnov effect when an electron moved in a region where there is not any magnetic field (vacuum). In this case we can attribute to the related vacuum a back ground field which is in fact the gauge field of electrodynamics force. The fluctuations which exist with respect to the concerned vacuum in QCD can appear as instantons and will effect on then nucleon parton distributions especially at low Bjorken- x values. A comparison between the proton structures when we consider the instanton effect with respect to the usual one, indicates the non-perturbative effect on extracting these functions. Considering this effect will yield a better result for the F2p structure function. Keywords: Instanton effect, parton distribution, non-perturbative region, nucleon structure function, chirality violation 1. Introduction It was known that t’Hooft interaction can be treated as the mechanism for baryon-number violation [1]. This interaction is induced by vacuum fluctuations of gauge fields-instantons. The renewal of interest in this problem happens basically due to the fact that at high energy the multiple production of gauge bosons from instanton leads to anomalous growth of the baryon-number violation cross-section [2]. In QCD, the instantons violate the chirality and therefore the same mechanism, evidently leads to anomalous quarks chirality violation at high energy. The most interesting fact is that in contrast of problems of the experimental searches of baryon-number violation, in strong interaction we already observe multiple anomalous spin effects, which can be connected only with a large quarks chirality violation at high energy [3]. The perturbative QCD predicts the disappearance of chirality violation at high energy and therefore the instanton interaction have been proposed as the funda0920-5632/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2011.10.092
mental mechanism for the spin effects at high energy [4]. The experimental evident lead to the conclusion that the behavior of the sea quarks at the small x differs strongly from the perturbative QCD predictions. The main goal of this article is the estimation of the contribution to the quark distribution functions of the nucleon induced by instantons. Here, we will present the arguments that these surprising results can be explained by manifestation of a complicated structure of QCD vacuum, namely by taking into account the instanton-induced interaction between quarks. 2. t’Hooft Lagrangian It is well known that behavior of quark-quark cross section at high energy determines the behavior of the nucleon structure functions at low x [5]. Thus the anomalous behavior of the quark-nucleon crosssections should provide the anomalous behavior of the structure functions at x → 0. It is connected with the
A. Mirjalili et al. / Nuclear Physics B (Proc. Suppl.) 219–220 (2011) 174–177
fact that at very high energy the small size instanton approximation is incorrect and then one needs to take into account the overlapping of instantons [6]. It was supposed that QCD instantons induce the following form for growth of cross section σqq ∼ sαI −1 where αI > 1 and the value for αI will be found from the fit of the experimental data on the structure functions. By using results of [7] one can connect the contribution of the instantons to the quark distribution functions with the value of the quark-nucleon cross section through the instanton ∞ k2 max 2 ImT qN 1 ¯ (S , k ) q(x) = 3 dS dk2 2 2 2 2π S 0 (k − mq ) k2 min × −x+
m2q − k2 S−
MN2
−
k2
.
(1)
In Eq. (1), k2 is defined by k2 = x
S k2 + MN2 + ⊥ , x−1 1−x
(2)
where k is the momentum of the soft quark which was created by the virtual photon and S = (P + k)2 in which P is the momentum of initial quark. Therefore we can write S + MN2 ) , x−1 S Q2 + MN2 ) + . = x( x−1 1−x
2 kmax = x( 2 kmin
(3)
The imaginary part of the scattering amplitude ImT qN is connected with the total cross section of the quark nucleon interaction through the relation σtot =
1 √ ImT qN (S , k2 ) , 2pc.m S
(4)
where pc.m = (−k2 + (S −M4S+k ) )( 2 ) is the threemomentum in the quark-nucleon center of mass system. At low energies the cross section induced by instantons is determined by the specific t’Hooft interaction between quarks, which for the number of flavors N f = 3 and for massless quarks has the following form 4 (x) = dρ n(ρ)( π2 ρ3 )2 (5) L(2) ef f 3 3 3 × u¯ R uL d¯R dL 1 + (1 − σuμν σdμν )λau λad 32 4 + R ←→ L . 2
2 2
1
175
Here ρ is the size of instanton and n(ρ) is its density, consists in the specific chirality and flavor properties of the interaction. Namely, it changes chirality by the value and it acts only between different flavors of the quarks. Using the insertion of the additional instantonantiinstanton pairs, it was shown that the contribution to the structure functions violates both the Ellis-Jaffe sum rule, related to the value of chirality carried by proton quarks, and the Gottfried sum rule connected with flavor properties of the quark sea. The value of the instanton contribution to the violation of the Ellis-Jaffe sum rule and Gottfried sum rules is determined by the value of the quark-nucleon instanton induced cross section. It should be stressed, that interaction (5) is the point like interaction with a dimensional coupling constant. Therefore it should lead, similarly to the case of the Fermi weak interaction, to an anomalous growth of the cross section of the quark quark interaction with increasing energy. At low energies, we can estimate this growth by using the instanton liquid model of the QCD vacuum [8]. 3. Instanton-induced interaction For the quark-quark scattering due to instantons, the chirality and flavor properties of the sea quarks are determined by the valence quark lines incoming t’Hooft vertex [4]. So, to obtain the relation between the various chirality and flavor sea quark distribution functions it is enough to consider the contribution to the quark sea. Let us consider the quark sea created by interaction (5) from the S U(6)W proton symmetrical wave function 5 1 1 2 u↑+ u↓+ d↑+ d↓. (6) 3 3 3 3 We will use the properties of the vertex which lead to the opposite chirality and different flavor as compare with the chirality and flavor of the initial valence quark [4] p ↑=
2 NI dv (x)xαv fu (x) , 3 xα I αv 1 NI dv (x)x 2¯u−I (x) = fu (x), 3 xα I αv 1 NI uv (x)x fu (x) , 2d¯+I (x) = 6 xα I αv 5 NI uv (x)x 2d¯−I (x) = fu (x), 6 xα I 2 s¯+I (x) = 2(¯u+I (x) + d¯+I (x)),
2¯u+I (x) =
2 s¯−I (x) = 2(¯u−I (x) + d¯−I (x)),
(7)
where NI is any constant, uv (x), dv (x) are the valence distribution functions,which we choose in the usual form
A. Mirjalili et al. / Nuclear Physics B (Proc. Suppl.) 219–220 (2011) 174–177
176
uv (x) =Nu x−αv (1 − x)bu , dv (x) =Nd x−αv (1 − x)bd .
(8)
The function fu (x) provides the correct unitarity limit at x → 0 . This function can be chosen in the following form ⎧ ⎪ ⎪ if x > x0 ; ⎨1, (9) fu (x) = ⎪ ⎪exp[−(x /x − 1)], if x < x . ⎩ 0
q(x, Q2 ) =
0
Instantons also lead to the valence quark chirality flipping due to the annihilation processes. Its contribution to the valence quark distributions functions is [4] 5 NI uv (x)xαv 6 xα I αv 1 u N I v (x)x I (x) = uv− α I 6 x 2 NI dv (x)xαv I dv+ (x) = 3 xα I αv 1 N d I v (x)x I dv− (x) = 3 xα I I uv+ (x) =
fu (x) , fu (x),
(14)
where M ≈ mη = 0.96 GeV is the characteristic hadron scale which is connected to the scale of the violation of the U A (1) symmetry due to instantons.
(10)
p p p ¯ +,− u¯ (x)+,− = d(x) = 2 s¯(x)+,− = N s (1 − x)5 /xαP (0)−1 , (11)
where by taking into account the suppression of the strange sea, one usually connects with the belated start of the evolution due to the large mass of the strange quark [9]. The amount of α p (0) = 1.08 is representing the intercept of the soft pomeron [10]. From parton distributions in above, one can obtain the unpolarized distribution functions q(x) ¯ = q¯ + (x) + q¯ − (x) and the polarized Δq(x) ¯ = q¯ + (x) − q¯ − (x) distribution functions. So, the nucleon structure functions F2μN (x) have the following form F2μp (x) =e2u (urv (x) + 2¯u p (x)) + e2d (dvr (x) + 2d¯p (x))
+ 2e2s s¯ p (x) + ( e2q )A(x), q
F2μn (x) =e2d (urv (x) + 2¯u p (x)) + e2u (dvr (x) + 2d¯p (x))
+ 2e2s s¯ p (x) + ( e2q )A(x), (12) q
where NI (uv (x) + dv (x))xαv +1 fu (x), xα I
Q2 q(x) , + M2
From parton distributions in above, one can obtain the following sum rules for the unpolarized distributions:
fu (x) , fu (x),
Q2
4. Sum rules
The quark sea induced by the perturbative gluons does not differ in chirality or flavor, and therefore it can be taken in the form using quark-counting rules
A(x) =
and qrv (x) is renormalized valence quark distribution function . At low Q2 some investigations should be paid. We know that in this region the conventional valence quark model for the quark distribution functions works well and therefore we will assume that the instanton contribution has the following dependence on Q2 [12]
(13)
1 0 1
0
dx(urv (x) + uvI (x)) =2, dx(dvr (x) + dvI (x)) =1.
(15)
Fits of the NMC and H1 experimental data for the unpolarized structure function of the proton F2p (x) and the neutron-proton ratio F2μn (x)/F2μp (x) is used in the calculations. We obtain a large excess of αI in comparison with 1, which provides the growth of F2N (x) at low x. We conclude, that the nonpertubative sea gives the large contribution to quark sea distribution functions in the interval 0.0001 < x < 0.1 and the growth of the structure functions in this region is determined by the anomalous dependence on the energy of the instanton-induced interaction between quarks.
5. Results The flavor dependence of the instanton sea leads to large violation of the S U(2)-flavour symmetry quark sea, which, of course, provides a strong violation of the Gottfried sum rule for the integrated difference of the structure functions 1 dx μp (F (x) − F2μn (x)), IG (t) = x 2 t 1 (16) IG (0) = . 3
A. Mirjalili et al. / Nuclear Physics B (Proc. Suppl.) 219–220 (2011) 174–177
The model predicts a very large violation of the Gottfried sum rule: IGI (0) = 0.247 which can be compared with the NMC result [13] IGN MC
60 With the instanton effect In the standard case
= 0.257 ± 0.017.
50
It is obvious that the instanton- induced interaction will change the behavior of F2N (x) at low x values. The plot in below confirms this reality. 6. Conclusion The perturbative QCD predicts the disappearance of chirality violation at high energy and therefore the instanton interaction have been proposed as the fundamental mechanism for the spin effects at high energy. Consequently an un-symmetrized distributions for sea quark densities will be resulted and the violation of the GSR will be justified. Due to instanton effect the result for F2N (x) structure function at low x values will differ with respect to the usual one as can be seen in Fig. 1. The instanton effect on structure function can be extended to high order approximation while the renormalized coupling constant depends on the ρ as the size of instanton. This quantity can be found by fitting the structure function on the available experimental data. 7. Acknowledgment Authors acknowledge the institute for research in fundamental sciences (IPM) for their hospitality whilst this research was performed. A. M. is indebted Yazd university to support him financially to present this research result. He is also grateful to A. E. Dorokhov to give his constructive comments. References [1] ’t Hooft Phys. Rev. D14 (1976) 3432. [2] A. Ringwald Nucl. Phys. B330 (1990) 1; O. Espinosa Nucl. Phys. B343 (1990) 310. [3] G. Bunce et al. Part. World 3 (1992) 1 . [4] A. E. Dorokhov, N. I. Kochelev Phys. Lett. B259 (1991) 335; Phys. Lett. B304 (1993) 167; Int. J. of Mod. Phys. A8 (1993) 603. [5] P. V. Landshoff, J. C. Polkinghorne, R.D. Short Nucl. Phys. B28 (1971) 225. [6] I. I. Balitsky, M. G. Ryskin Phys. Lett. B296 (1992) 185. The contribution of the small size instantons to the coefficient function in front of parton distributions for DIS was calculated in paper I. I. Balitsky, V. M. Braun Phys. Rev. D47 (1993) 1879. [7] S. A. Kulagin, G. Piller, W. Weise, Phys.Rev.C50 (1994) 1154. [8] E. V. Shuryak. Rev.Mod.Phys. 65 (1993)1 and references therein.
177
40
F2
30
20
10
0 0.0001
0.001
0.01
0.1
1
x
Figure 1: Proton structure function in the standard case (solid line) and its comparison when the instanton effects has been considered (dashed line).
[9] [10] [11] [12]
V. Barone at el. Phys. Lett. B268 (1991)279. A. Donnachie, P. V. Landshoff Nucl. Phys. B231 (1984) 189. R. D. Carlitz, J. Kaur Phys. Rev. Lett. 38 (1977) 673. N. I. Kochelev, Procceding of 2nd Meeting on Possible Measurements of Singly Polarized pp and pn Collisions at HERA (1995) Aug.31-Sep.1 [13] NMC, P. Amaudruz et al. Phys. Rev. Lett. 66 (1991) 2712; Preprint CERN-PPE/93-1993; H1 Coll. Preprint DESY 93-113.