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Parton evolution model and its explanation of the EMC effect and the nuclear Drell-Yan process
PHYSICS REPORTS EL.SEVIER
Physics Reports 242 (1994) 505—517
____________________
model and its explanation of the EMC and the nuclear Drell—Yan process
Parton evolution
effect
G.L. Lj~,,c,* J.P. Shen”, J.J. Yang~’,H.Q. Shend Center of Theoretical Physics, CCAST (World Laboratory), Be(/ing 100080, China b Institute of High Energy Physics, Academia Sinica, Be(/ing 100039, China C Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky, KY 40506, USA d Department of Physics, Nanjing Normal University, Nanjing 210024, China a
Abstract The evolution of partons in the nuclear environment is investigated. It is shown that the evolution of valence and sea (including charm) quarks and gluons is determined by different interaction vertex factors and their effects on their own momentum distributions are also different. By using this parton evolution model, together with the nuclear shadowing effect due to the spatial overlap of partons from neighboring nucleons in the nucleus, the experimental data of the EMC effect and the nuclear Drell—Yan process can be well explained. Therefore, the change of the momentum distribution of partons in the bound nucleon caused by the evolution ofpartons in the nuclear environment is possibly the origin ofthe 2-rescaling model are EMC effect. The similarities and differences between the parton evolution model and the Q discussed.
1. Introduction Since the experimental discovery of the EMC effect [1], physicists have proposed many models [2—14]to explain it. In our previous work [13, 14], we analyzed the contribution of the energy— momentum distribution of nucleons in nuclei to the EMC effect by using the Hartree—Fock single-particle energies and wave functions calculated with a density-dependent Skyrme interaction, and pointed out that, with these correct single-particle energies and light-cone normalized wave functions, one cannot explain the EMC effect well by considering the Fermi motion correction including the nuclear binding effect only. We also found that, based on the consideration of the Fermi motion correction and the nuclear binding effect, one can qualitatively explain the EMC effect by introducing the Q2 or x-rescaling mechanism. One obtains better agreement with the experimental data by using the latter mechanism than the former in the region of 0.2 < x < 0.5. * Corresponding
author.
0370-1573/94/$7.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0370-1573(94)00046-6
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However, in these cases, the momentum of the nucleus is no longer conserved, i.e., the momentum of the nucleus is no longer equal to the sum of momenta of valence and sea (including charm) quarks and gluons in the nucleus. In order to compensate for the loss of nuclear momentum, we proposed a double x-rescaling model [14]; i.e., we employed different x-rescaling parameters for the momentum distributions of valence and sea (including charm) quarks in the nucleon structure function. This calculation showed that, by using the double x-rescaling model and considering the nuclear shadowing effect caused by the spatial overlap of partons from neighboring nucleons in the nucleus in the small x region, one can keep the nuclear momentum conservation and, at the same time, well explain the experimental data of the EMC effect in the whole x region. In this paper, we investigate the dynamical mechanism of the double x-rescaling model. In our opinion, the EMC effect can be caused by the evolution of partons in the nuclear environment. By taking into account the relation between the evolution of partons and their confinement scale in the nucleon, we evaluate the ratio of the average effective radius of the nucleon in the nuclear medium to the radius of the free nucleon according to the ratio of the effective mass of the nucleon bound in nucleus 209Bi to the mass of the free nucleon, m~/mN~ 0.75, given by Brown and Rho [15] in the investigation of the enhancement of the spin—isospin interaction between nucleons in the nuclear medium. The result is FN(~)/rN ~ 1.3. This implies that the average effective radius rN(A) of the nucleon bound in the nucleus with mass number A is greater than the free nucleon radius rN. Comparing with the free nucleon case, the momentum distribution of partons in the bound nucleon, according to Heisenberg’s uncertainty relation, moves to the small x region, i.e., the structure function of the bound nucleon is apparently different from that of the free nucleon. By studying the influence of the evolution of partons in the nuclear environment on the structure function of the bound nucleon, we find that the evolution of valence and sea (including charm) quarks and gluons are determined by different interaction vertex factors and their effects on their own momentum distributions are also different. This is just the dynamical mechanism of the double x-rescaling model. By using this parton evolution model (PEM), together with the nuclear shadowing effect due to the spatial overlap of partons from neighboring nucleons in the nucleus, we calculate the ratios of the average nuclear structure functions of nuclei 12C, 56Fe, 64Cu, and ‘165n to the deuteron structure function. The calculated results are in good agreement with the experimental data of the EMC effect. Bickerstaffet al. [16] pointed out that different models with very similar deep-inelastic scattering lead to very different lepton-pair production ratios of the Drell—Yan processes on a bound nucleon to a free nucleon. Thus, one can reasonably conclude that the models could be distinguished by an accurately performed Drell—Yan experiment. Recently, a precise measurement of lepton-pair production in 800 GeV proton—nucleus collisions was completed by the E772 collaboration at Fermilab [17]. By using the PEM, we calculate the ratios of the Drell—Yan cross sections of nuclei C, Ca, Fe, and W to 2H. The results from our model agree better with the experimental data of the nuclear Drell—Yan process than those [17] from the pion-excess and quark-cluster models. By considering the fact that the Q2-rescaling model [9—11]also relates to the swelling of the quark confinement scale in the bound nucleon, we discuss similarities and differences between the PEM and the Q2-rescaling model. The organization of this paper is as follows: the PEM is briefly described in Section 2. The explanation of the EMC effect and the nuclear Drell—Yan process by using the PEM are presented in Section 3. The similarities and differences between the PEM and the Q2-rescaling model are discussed in Section 4. Summary is given in Section 5.
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2. The parton evolution model The EMC effect reveals that the structure function of the nucleon bound in the nucleus is obviously different from that of the free nucleon. In our opinion, this difference is due to the evolution of partons in the nuclear environment as briefly discussed in the introduction. Let us now discuss the effect of the evolution of partons in the nuclear environment on the parton distribution functions of the bound nucleon. According to the QCD theory, we take the Lagrangian density of the quark—gluon interaction as ~ —a~B~~)] (1) = —ig[~y~B~t2q ,
where q, t~and B~ are field operators of the quark, antiquark and gluon, respectively, g the quark—gluon coupling constant,f 8p~the SU(3) structure constants, t~= ,~/2,and 2~(c~ = 1,2, 8) the Gell—Mann matrices. To describe the evolution among partons, we define the evolution matrix from parton a to partons b and c as follows: ...
~
=
,
(2)
where T represents the time-ordered product. Substituting Eq. (1) into Eq. (2), we obtain 1/
=
I’~ \3~3I
(3) i~) where Pa,Pb,Pc, Ea,(27t) Eb, and E~are the momenta and energies of partons a, b, and c, respectively. fr~—CbCis the interaction vertex factor among partons a, b, and c in the momentum space. Taking the 4-momenta of partons a and b as ~PaPbPc 912 1gva_~b~ \/2Ea2Eb2Ec Vit, (Ea (I — Eb — E~
Pa(P,P,0) Pb
=
(zp +
(4)
p~/2zp),zp,p
1) (5) in the infinite momentum frame, one can express the evolution probability from parton a to parton b as dFa~~(Z)= where color.
~
z(1
—
—
z)~dp~,
(6)
represents the average over initial states and the sum over final states for both spin and To be more specific, ~‘
2
~I Vq~gI -
El
=
2c
2 z) 1+(1—z)2 1+2
2(R) z(1
~_.g~l2 2C
~l ~~i2
2
=
2(R) (1
=
(7) (8)
z)z
—
Z2~
2p~, 1(1~)Z)
(9)
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~lggl24C
(10)
~
(11)
2(G)(11)[1Z+1+(1z)z]p~,
C2(G) =
3,
=
~fly
(12)
where q, ~j,and g stand for the quark, antiquark, and gluon, respectively. N~is the color number of quarks.
When a nucleon enters into the nucleus with mass number A, its effective radius increases from rN to rN(A), consequently, the restriction on the change of the nucleon transverse momentum satisfies l/rN(A) p1 i/rN. Therefore, we can only consider the variation of the nucleon structure function caused by the evolution from parton a to parton b with the transverse momentum in the above range. Then, we can obtain the probability of the above evolution by integrating Eq. (6), Fa_Cb(Z)
=
2(
2 ~ 7t)
J
~l Va~.bcl z(i
‘1/r1
—
z)
P±
1/rN(A)
2
dp1
(13)
.
Considering the conservation of valence quark number and nuclear momentum in the evolution process, we obtain the parton evolution probability functions as follows:
Fq.q(y,z)~C2(R)ln1[(~+Z) 2,A)
—
+~5(1 _z)]
1]ö(l
—
z)
(14)
,
+ [W~(y,Q
c
Fq~g(z)=
2 ln~-~,
(15)
2(R) 1 + (1— z)
Fg_.q(z) =~~[z2 + (1— z)2]ln~-~~~1
(16)
,
Fg~+g(y,z)=
~ C
2(G)ln~ {(1 —z)~ + 2,A)
—
1]~5(1— z)
Z —
+ z(1
—
Dg~5(1 z) —
z) + ~[ii ,
—
~‘~]~(1
—
z)}
(17)
+ [Wg(y,Q
with dz
Jo =
F(z) = ~ dzF~ F(1) (1—z)÷ Jo 1—z
g2/4~t.
,
(18)
(19)
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N~is the flavor number of quarks. Dg in Eq. (17), introduced to have the nuclear momentum conserved, can be expressed as 1] GN(x, Q2)}dx S~”{[l4’~(x,Q2,A) 1][SN(x,Q2) + CN(x,Q2)] + [Wg(x,Q2,A) Dg S~xGN(x,Q2)dx —
—
(20) JJ~(g)(X,Q2, A) in Eqs. (14) and (17), similar to that in Ref. [14], is the nuclear shadowing factor due to the spatial overlap of partons from neighboring nucleons in the nucleus in the small x region, i.e.,
Jf~(g)(X, Q2)
1, A~(x) X > Xfl lKq~(g) OXXfl,
=
(21)
,
YA~X)
with the spatial overlap factor 1
=
(22)
—~xmNrN(A).
The x~in Eq. (21) is the critical value of the Bjorken variable x, where nuclear shadowing starts to occur. It is determined by setting 0.
=
(23)
By assuming that the valence quarks cannot be deconfined and released from the bound nucleon, we obtain ~4’~~(x,Q2,A)ml
(24)
.
Thus the parton evolution probability function for valence quarks becomes Fqv~qv(z)~C 2(R)[(~~)
+~c5(1_z)]ln~~1.
(25)
We can now give the parton distribution functions of the bound nucleon by using the parton evolution probability functions mentioned above. If we let VA(N), ~ CA(N), and GA(N) denote the distribution functions of valence, sea, and charm quarks and gluons in the bound nucleon in the nucleus with mass number A (in the free nucleon), respectively, then
2)
=
VA(x,Q
SA(x,Q2)
=
J J
VN(x,Q2) + SN(x,Q2) +
+
f
Fqv~qv(x/y)vN(y,Q2)d
(26)
Fq~q~(y,x/y)S~dy
6 Fgq (x/y) GN(y,Q2)
~
(27)
)
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cA(x, Q2)
=
cN(x, Q)2 +
+
GA(x,Q2)
=
/ Physics Reports 242 (1994) 505—517
Fq~q,(y, x/y) C(y,
GN(y,Q2)
J
2Fg.q(x/y)
J
GN(x,Q2) +
dy,
Q2),
(28)
Fq~g(X/y)![VN(y,Q2)
+ SN(y, Q2) + CN(y, Q2)]
where VN(x,
Q2
dy +
Fg~g(y,x/y) GN(y, ~
dy,
(29)
SN(x, Q2), CN(x, Q2) and Gt~~(x, Q2) are taken from set 2 of Ref. [18].
3. Explanation of the EMC effect and the nuclear Drell—Yan process Let us explain the EMC effect and the nuclear Drell—Yan process by using the PEM. Define the ratio of the average nuclear structure function to the deuteron structure function as [13] R~4~(x, Q2) F~(x,Q2)
F~(x,Q2)/F~(x,Q2),
=
2)—~(N Z)[F~(x, Q2) —F~(x,Q2)]},
{F
=
(30) (31)
—
2~(x,Q
where F2A is the nuclear structure function, F~and F~are the free neutron and proton structure functions, respectively. The second term compensates for the neutron excess, so that F~’(x,Q2) represents the average structure function of a hypothetical nucleus with equal numbers of protons and neutrons. By considering the Fermi motion of nucleons in the nucleus, F 2A can be written as [13] 2) F2A(x,Q
=
~j~ç3
(32)
l~(p)l2zF~(x/z,Q2),
where z = (Po + p3)/mN, Po = mN + mN and ~ are the rest mass and separation energy of a nucleon in the single-particle state )~,respectively, and ~‘A( p) is the single-particle wave function of the nucleon in the momentum space, which satisfies the light-cone normalization: ~,
J~
2luI’2(P)l~ 1.
In the following calculation, c~and i~’~(p) are taken from Ref. [14]. By using the 4~(x, Q2) parton can be distribution as functions given in Section 2, the bound nucleon structure function F~ represented F~’””~(x, Q2) =
x V~(x,
Q2)
±~ xV~’(x, Q2) +
~x[SA(x, Q2) + 2C-~(x,Q2)]
,
(34)
G.L. Li et al. / Physics Reports 242 (1994) 505—517
where + ( —) stands for the proton (neutron) and V~(x,Q2) = u~(x,Q2) + d~(x,Q2), V~(x,Q2)
(35)
u~”(x,Q2) — d~”(x,Q2),
=
511
(36)
where u~and d~’are the distribution functions of valence u and d quarks in the bound proton, respectively. The functions V~( 3),SinA, Eq. and(30). CA are Eqs.account (26)—(28). 41’~(x,Q2) By given takingbyinto the nuclear surface effect, We now calculate R’ similar to Ref. [19], we take rN(A)
=
rN[l + 0.43 ln(2
—
A113)]
(37)
,
where the constant 0.43 is determined by rN(~,)~ 1.3 rN. The values of the parameter Kq~(g)~fl Eq. (21) are taken as 0.25, 0.30, 0.35, and 0.55 for nuclei C, Fe, Cu, and Sn, respectively. The comparison between the ratios R~4~(x,Q2) for nucleus 56Fe calculated by the PEM and the experimental data of the EMC effect [20] is plotted in Fig. 1. Similarly, the results for nuclei ‘2C, 64Cu, and ‘16Sn are presented in Fig. 2. In the calculations, we take Q2 = 20 Gev2/c2. From these figures, one finds that in the PEM, by considering both nuclear shadowing and Fermi-motion corrections, the experimental data of the EMC effect can be well explained in the whole x region. The double x-rescaling model [14], which has no definite physical mechanism, can also well explain the EMC effect. In our opinion, the different evolution modes of valence and sea (including charm) quarks can cause different x-rescaling for the momentum distributions of valence and sea (including charm) quarks in the nucleon structure function. By assuming xvA(x, Q2) XSA(X,
Q2)
=
=
5~xVN(o~x,Q2),
(38)
(39)
o~xSN(~,x,Q2),
we find out that x-rescaling parameters 5~and ~5,,which slightly depend on x, can be estimated in the PEM. They are ~i5~>1,
ö,<1.
(40,41)
This is qualitatively consistent with the result from the double x-rescaling model in Ref. [14]. Therefore, the evolution of partons in the nuclear environment is just the dynamical mechanism of the double x-rescaling model. Let us now discuss the nuclear Drell—Yan process in terms of the PEM. The nuclear Drell—Yan process is a collision between the beam hadron (h) and the target hadron (A) in the nucleus. According to the Drell—Yan model [23], the differential cross section for the nuclear Drell—Yan process: h+A—~t4j.C+X
(42)
is given by d2a~’4(x 1 x2) dx1 dx2
1 /4~2\ =
2
~ ~-~-) ~ e~[q~(xj)c~(x2)
+ ~(x1)q~(x2)],
(43)
where M denotes the total mass of the lepton pair (in its rest frame). The e, is the fractional quark charge (in units of e) and the factor ~ comes from the fact that only a pair of q~which have opposite
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(a) C/D
1.~___
(b) Cu/D
1,2 ______________ 1.1
_________
(c) Sn/D
~,
1,0
O.~4O!6O~8
I.0
Fig. 1. Comparison between the ratio R’~”(x,Q2) for nucleus 56Fe calculated by the PEM and the experimental data of the EMC effect [20]. (LISLAC 1983 Fe/D, .BCDMS 1986 Fe/D, 0 EMC 1986 Cu/D.) Fig. 2. Comparison between the ratios RA~(x, Q2) for nuclei ‘2C, 64Cu and ~6Sn calculated by the. PEM and the experimental data of the EMC effect. The solid dots and the open circles are taken from SLAC [21] and EMC [22], respectively.
colors annihilate into a virtual photon. The functions q~4~(x)and ~~x) are the quark and antiquark distribution functions in the beam hadron (h) (the target hadron (A)), respectively; x 1(x2) is the fraction of the longitudinal momentum carried by the quark or antiquark in the beam hadron (the target hadron). If we let ~
,
(44)
then 2a~(xj,x d
2) dx1dx2
=
11 4~2 \ 3\3x1x2M 2 ji HhA(xl,x2)
—1
.
(45)
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By considering the fact that the quark and antiquark distribution functions in the hadron are Q2-dependent, Hh_A(xl, x2) should be replaced by HhA(xl, x2, Q2). To be more specific, Hh_~(xj,x 2) can be expressed as 2, Q H~A(xj,x 2) = ~-~x 2) + d~(x 2)]x 4(x 2) 2, Q 1[4u~(xj, Q 1,Q 2S’ 2,Q +
x
2)
2)[(N + 4Z)x
2u~(x2,Q
1SN(x1, Q
2) + 2Ax 2)], 2d~(x2,Q 2S’~(x2,Q 31~x 2) + 4d~(x 2)]x 2) 1[u~(x1,Q 1,Q 2S”(x2,Q + (4N + Z)x
HflA(xl, x
2) 2, Q
=
+
x
2)[(N + 4Z)x 1SN(x~,Q
(46)
2) 2u~(x2,Q
+ (4N + Z)x
2) + 2Ax 2)], (47) 2d~(x2,Q 2SA(x2, Q where p(n) labels the beam proton (neutron), N and Z are, respectively, the neutron and proton number of the target nucleus with mass number A. The functions ut’, d~,and S’~are given by Eqs. (26), (27), (35), and (36). Although various models with very different physics assumptions are all able to explain the EMC effect, they give different predictions for the nuclear Drell—Yan process. Thus, one can use the information from the nuclear Drell—Yan process to test the models of the EMC effect. Recently, the E772 collaboration [17] precisely measured the atomic-mass dependence of dimuon production induced by 800 GeV protons in the kinematic region (0.025 < x 2 < 0.3) where the result is sensitive to the antiquark distribution in the target nucleus. They gave the nuclear Drell—Yan ratios in the mass regions 4 ~ M 9 GeV and M 11 GeV, where there is no contribution from the decay of quarkonium resonances. In order to compare with the experimental data mentioned above, we define the nuclear Drell—Yan ratio as 2~”~4(xi,x AID Sdxi(d2 a~(x 2)/dxi dx2) T (x2) = $dxi (d 1, x2)/dx1 dx2)’ (48) where the integral range for x1 is determined according to Ref. [17]. By using the PEM, we calculate the nuclear Drell—Yan ratios for nuclei C, Ca, Fe and W. The values of the parameter Kq~(g)in Eq. (21) are taken as 0.25, 0.28, 0.30, and 0.35 for nuclei C, Ca, Fe and W, respectively. In Fig. 3, we present the resultant nuclear Drell—Yan ratios together with the experimental data from the E772 collaboration and the predictions from the other EMC models [17]. The solid curves are our results, the dashed, dotted,2-rescaling and dash-dotted are, respectively, theagree resultsbetter from with the pion models.curves Apparently, our results the excess, quark data cluster, andnuclear Q experimental of the Drell—Yan ratios than those from the pion excess and quark cluster models. This shows that the PEM can reasonably describe the sea quark distribution in the bound nucleon. In Fig. 3(c), on comparing the PEM’s result with that from the the Q2-rescaling model, we see that the influence of the nuclear shadowing on the nuclear Drell—Yan ratios is evident. We also consider the Fermi-motion correction in the calculation of T~~(x 2) and find that it is negligible in the region of x < 0.3.
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1.3
.2
(a) C/D
-
(b) Ca/D
I
—I
(c) Fe/D
.2
/
I
I
(d) W/D
07
03
Fig. 3. The nuclear Drell—Yan ratios T’~(x
2)for nuclei: C, Ca, Fe, W are plotted versus x2. The solid curves are the results calculated by the PEM. The dashed, dotted, dash—dotted are, respectively, thecollaboration results form the pion 2-rescaling models. The and experimental datacurves are taken from the E772 [17]. excess, quark cluster, and Q
4. Similarities and differences between the parton evolution model and the Q2-rescaling model Since the Q2-rescaling model also relates to the swelling of the quark confinement scale in the bound nucleon, we should discuss the relation between the PEM and the Q2-rescaling model. To do this, let us take nth order moment of the valence quark distribution function VA(X,
Q2): M,~’(Q2)= (1 + F~)M~(Q2),
(49)
M~~(Q2) =
(50)
where
F~=
x~t VA~(X, Q2)dx,
Z’~’ Fqv~qv(z)dz=
B~5= C 2(R)
1~ ~ ~‘
~
If we assume 2)= M~(~(Q2)Q2), M~(Q
~-
B~ln(i~(A)/r~),
+
ó(1 — z)] dz.
(51)
(52)
(53)
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/ Physics Reports 242 (1994) 505—517
= 1 +F M~(~(Q2)Q2)
515
(54)
M~(Q2) According to the QCD theory, we obtain ______________
[1 + ;(Q2)b
—
2)
0t’ 1B~/(2itbo)
[1~(Q2)/
M~(Q
0t]
‘
(56) 2/p2) ln(Q ln(~(Q2)Q2/p2)
(57)
,
t
= =
(58)
,
where p is the mass dimension parameter. For sufficiently large Q2, Eq. (55) can approximately be expressed by M~(~(Q2)Q2) ~ +
B~ln~(Q2).
Thus, by taking the reference value of; as ~c(Q2 = /
—2 rN(A)
\;(~2)/;(Q2)
}
(59) we obtain, from Eqs. (54) and (59), (60)
\ rN /
This is just the expression of the Q2-rescaling parameter in the Q2-rescaling model. Therefore, we can conclude that the Q2-rescaling model is an approximation of the PEM, which is valid for sufficiently large Q2 and considering only the evolution mode of valence quarks for all types of quarks. The differences between the PEM and the Q2-rescaling model are as follows: (1) In the PEM, the evolution of valence and sea (including charm) quarks and gluons are determined by different interaction vertex factors and their effects on their own momentum distributions are also different. However, the Q2-rescaling model, ignoring the different evolution modes for different types of quarks, describes the influence of the nuclear environment on the distribution functions of valence and sea (including charm) quarks by using the same Q2-rescaling parameter determined by the approximate behavior of the evolution of valence quarks at sufficiently large Q2. (2) In the Q2-rescaling model, rN(A)/rN is a free parameter, while in the PEM, FN(A)/rN is determined by the empirical formula Eq. (37) with FN(~)/rN~ 1.3, which is consistent with the ratio of the effective mass of the nucleon bound in nucleus 209Bi to the mass of the free nucleon, m~/mN~ 0.75, given by Brown and Rho [15]. There is no free parameter such as the Q2-rescaling parameter in the PEM. (3) As pointed out in Ref. [14], in the Q2-rescaling model, the nuclear momentum is not conserved, i.e., the nuclear momentum is not equal to the sum of momenta of valence and sea (including charm) quarks and gluons in the nucleus. While in the PEM, as a restraint condition, the
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nuclear momentum is automatically conserved. Therefore, the PEM is more consistent than the Q2-rescaling model. In addition, the PEM has a definite physical mechanism. 5. Summary We investigate the influence of the evolution of partons in the nuclear environment on the structure function of the bound nucleon and find that the evolution of valence and sea (including charm) quarks and gluons are determined by different interaction vertex factors and their effects on their own momentum distributions are also different. It brings the physical nature of the double x-rescaling model [14] to light, i.e., the evolution of partons in the nuclear environment is the dynamical mechanism of the double x-rescaling model. By using the PEM and considering the nuclear shadowing and Fermi-motion corrections, the experimental data of the EMC effect and the nuclear Drell—Yan process can be well explained and, at the same time, the nuclear momentum is automatically conserved. Therefore, the change of the momentum distribution of partons in the bound nucleon caused by the evolution of partons in the nuclear environment is possibly the origin of the EMC effect.
Acknowledgments The authors would like to thank Professor A.D. MacKellar for critical reading of the manuscript tand professor K.F. Liu for helpful discussions. This work was supported by the National Science Foundation under grant No. 8820088, the Center for Computational Sciences, University of Kentucky, and the National Natural Science Foundation of China.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
J.J. Aubert et al., EMC, Phys. Lett. 123 B (1983) 275. C.H. Llewellyn Smith, Phys. Lett. 128 B (1983) 107. E. Ericson and A.W. Thomas, Phys. Lett. 128 B (1983) 112. R.L. Jaffe, Phys. Rev. Lett. 50 (1983) 228. C.E. Carlson and T.J. Havens, Phys. Rev. Lett. 51(1983) 261. M. Ericson, Prog. Particle NucI. Phys. 11(1984) 277. A.W. Thomas, Prog. Particle Nucl Phys. 11(1984) 325. EL. Berger, F. Coester and RB. Wiringa, Phys. Rev. D 29 (1984) 398. FE. Close, R.G. Roberts and G.G. Ross, Phys. Lett. 129 B (1983) 346. R.L. Jaffe, FE. Close, R.G. Roberts and G.G. Ross, Phys. Lett. 134 B (1984) 449. F.E. Close, R.L. Jaffe, R.G. Roberts and G.G. Ross, Phys. Rev. D 31(1985)1004. S.V. Akulinichev, G.M. Vagradov and S.A. Kulagin, JETP Lett. 42 (1985) 127; S.V. Akulinichev, S.A. Kulagin and G.M. Vagradov, Phys. Lett. 158 B (1985) 485; S.V. Akulinichev, S. Shlomo, S.A. Kulagin and G.M. Vagradov, Phys. Rev. Lett. 55 (1985) 2239. [13] G.L. Li, K.F. Liu and G.E. Brown, Phys. Lett. 213 B (1988) 531. [14] G.L. Li, Z.J. Cao and CS. Zhong, Nucl. Phys. A 509 (1990) 757. [15] G.E. Brown and M. Rho, Phys. Lett. 237 B (1990) 3.
G.L. Li et a!. / Physics Reports 242 (1994) 505—517 [16] [17] [18] [19] [20] [21] [22] [23]
R.P. Bickerstaff, M.C. Birse and G.A. Miller, Phys. Rev. D 33 (1986) 3228. D.M. AIde et al., Phys. Rev. Lett. 64 (1990) 2479. D.W. Duke and J.F. Owens, Phys. Rev. D 30 (1984) 49. A.W. Hendry, D.B. Lichtenberg and E. Predazzi, Nuovo Cimento 92 A (1986) 427. EMC and BCDMS collab., CERN courier, 27 (2) (1987) 10. R.G. Arnord et al., Phys. Rev. Lett. 52 (1984) 727. J. Ashman et al., EMC, Phys. Lett. 202 B (1988) 603. S. Drell and T.M. Yan, Phys. Rev. Lett. 25 (1970) 316.
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Physics Reports 242 (1994) 505—517
____________________
model and its explanation of the EMC and the nuclear Drell—Yan process
Parton evolution
effect
G.L. Lj~,,c,* J.P. Shen”, J.J. Yang~’,H.Q. Shend Center of Theoretical Physics, CCAST (World Laboratory), Be(/ing 100080, China b Institute of High Energy Physics, Academia Sinica, Be(/ing 100039, China C Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky, KY 40506, USA d Department of Physics, Nanjing Normal University, Nanjing 210024, China a
Abstract The evolution of partons in the nuclear environment is investigated. It is shown that the evolution of valence and sea (including charm) quarks and gluons is determined by different interaction vertex factors and their effects on their own momentum distributions are also different. By using this parton evolution model, together with the nuclear shadowing effect due to the spatial overlap of partons from neighboring nucleons in the nucleus, the experimental data of the EMC effect and the nuclear Drell—Yan process can be well explained. Therefore, the change of the momentum distribution of partons in the bound nucleon caused by the evolution ofpartons in the nuclear environment is possibly the origin ofthe 2-rescaling model are EMC effect. The similarities and differences between the parton evolution model and the Q discussed.
1. Introduction Since the experimental discovery of the EMC effect [1], physicists have proposed many models [2—14]to explain it. In our previous work [13, 14], we analyzed the contribution of the energy— momentum distribution of nucleons in nuclei to the EMC effect by using the Hartree—Fock single-particle energies and wave functions calculated with a density-dependent Skyrme interaction, and pointed out that, with these correct single-particle energies and light-cone normalized wave functions, one cannot explain the EMC effect well by considering the Fermi motion correction including the nuclear binding effect only. We also found that, based on the consideration of the Fermi motion correction and the nuclear binding effect, one can qualitatively explain the EMC effect by introducing the Q2 or x-rescaling mechanism. One obtains better agreement with the experimental data by using the latter mechanism than the former in the region of 0.2 < x < 0.5. * Corresponding
author.
0370-1573/94/$7.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0370-1573(94)00046-6
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Reports 242 (1994) 505—517
However, in these cases, the momentum of the nucleus is no longer conserved, i.e., the momentum of the nucleus is no longer equal to the sum of momenta of valence and sea (including charm) quarks and gluons in the nucleus. In order to compensate for the loss of nuclear momentum, we proposed a double x-rescaling model [14]; i.e., we employed different x-rescaling parameters for the momentum distributions of valence and sea (including charm) quarks in the nucleon structure function. This calculation showed that, by using the double x-rescaling model and considering the nuclear shadowing effect caused by the spatial overlap of partons from neighboring nucleons in the nucleus in the small x region, one can keep the nuclear momentum conservation and, at the same time, well explain the experimental data of the EMC effect in the whole x region. In this paper, we investigate the dynamical mechanism of the double x-rescaling model. In our opinion, the EMC effect can be caused by the evolution of partons in the nuclear environment. By taking into account the relation between the evolution of partons and their confinement scale in the nucleon, we evaluate the ratio of the average effective radius of the nucleon in the nuclear medium to the radius of the free nucleon according to the ratio of the effective mass of the nucleon bound in nucleus 209Bi to the mass of the free nucleon, m~/mN~ 0.75, given by Brown and Rho [15] in the investigation of the enhancement of the spin—isospin interaction between nucleons in the nuclear medium. The result is FN(~)/rN ~ 1.3. This implies that the average effective radius rN(A) of the nucleon bound in the nucleus with mass number A is greater than the free nucleon radius rN. Comparing with the free nucleon case, the momentum distribution of partons in the bound nucleon, according to Heisenberg’s uncertainty relation, moves to the small x region, i.e., the structure function of the bound nucleon is apparently different from that of the free nucleon. By studying the influence of the evolution of partons in the nuclear environment on the structure function of the bound nucleon, we find that the evolution of valence and sea (including charm) quarks and gluons are determined by different interaction vertex factors and their effects on their own momentum distributions are also different. This is just the dynamical mechanism of the double x-rescaling model. By using this parton evolution model (PEM), together with the nuclear shadowing effect due to the spatial overlap of partons from neighboring nucleons in the nucleus, we calculate the ratios of the average nuclear structure functions of nuclei 12C, 56Fe, 64Cu, and ‘165n to the deuteron structure function. The calculated results are in good agreement with the experimental data of the EMC effect. Bickerstaffet al. [16] pointed out that different models with very similar deep-inelastic scattering lead to very different lepton-pair production ratios of the Drell—Yan processes on a bound nucleon to a free nucleon. Thus, one can reasonably conclude that the models could be distinguished by an accurately performed Drell—Yan experiment. Recently, a precise measurement of lepton-pair production in 800 GeV proton—nucleus collisions was completed by the E772 collaboration at Fermilab [17]. By using the PEM, we calculate the ratios of the Drell—Yan cross sections of nuclei C, Ca, Fe, and W to 2H. The results from our model agree better with the experimental data of the nuclear Drell—Yan process than those [17] from the pion-excess and quark-cluster models. By considering the fact that the Q2-rescaling model [9—11]also relates to the swelling of the quark confinement scale in the bound nucleon, we discuss similarities and differences between the PEM and the Q2-rescaling model. The organization of this paper is as follows: the PEM is briefly described in Section 2. The explanation of the EMC effect and the nuclear Drell—Yan process by using the PEM are presented in Section 3. The similarities and differences between the PEM and the Q2-rescaling model are discussed in Section 4. Summary is given in Section 5.
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2. The parton evolution model The EMC effect reveals that the structure function of the nucleon bound in the nucleus is obviously different from that of the free nucleon. In our opinion, this difference is due to the evolution of partons in the nuclear environment as briefly discussed in the introduction. Let us now discuss the effect of the evolution of partons in the nuclear environment on the parton distribution functions of the bound nucleon. According to the QCD theory, we take the Lagrangian density of the quark—gluon interaction as ~ —a~B~~)] (1) = —ig[~y~B~t2q ,
where q, t~and B~ are field operators of the quark, antiquark and gluon, respectively, g the quark—gluon coupling constant,f 8p~the SU(3) structure constants, t~= ,~/2,and 2~(c~ = 1,2, 8) the Gell—Mann matrices. To describe the evolution among partons, we define the evolution matrix from parton a to partons b and c as follows: ...
~
=
,
(2)
where T represents the time-ordered product. Substituting Eq. (1) into Eq. (2), we obtain 1/
I’~ \3~3I
(3) i~) where Pa,Pb,Pc, Ea,(27t) Eb, and E~are the momenta and energies of partons a, b, and c, respectively. fr~—CbCis the interaction vertex factor among partons a, b, and c in the momentum space. Taking the 4-momenta of partons a and b as ~PaPbPc 912 1gva_~b~ \/2Ea2Eb2Ec Vit, (Ea (I — Eb — E~
Pa(P,P,0) Pb
=
(zp +
(4)
p~/2zp),zp,p
1) (5) in the infinite momentum frame, one can express the evolution probability from parton a to parton b as dFa~~(Z)= where color.
~
z(1
—
—
z)~dp~,
(6)
represents the average over initial states and the sum over final states for both spin and To be more specific, ~‘
2
~I Vq~gI -
El
=
2c
2 z) 1+(1—z)2 1+2
2(R) z(1
~_.g~l2 2C
~l ~~i2
2
=
2(R) (1
=
(7) (8)
z)z
—
Z2~
2p~, 1(1~)Z)
(9)
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~lggl24C
(10)
~
(11)
2(G)(11)[1Z+1+(1z)z]p~,
C2(G) =
3,
=
~fly
(12)
where q, ~j,and g stand for the quark, antiquark, and gluon, respectively. N~is the color number of quarks.
When a nucleon enters into the nucleus with mass number A, its effective radius increases from rN to rN(A), consequently, the restriction on the change of the nucleon transverse momentum satisfies l/rN(A) p1 i/rN. Therefore, we can only consider the variation of the nucleon structure function caused by the evolution from parton a to parton b with the transverse momentum in the above range. Then, we can obtain the probability of the above evolution by integrating Eq. (6), Fa_Cb(Z)
=
2(
2 ~ 7t)
J
~l Va~.bcl z(i
‘1/r1
—
z)
P±
1/rN(A)
2
dp1
(13)
.
Considering the conservation of valence quark number and nuclear momentum in the evolution process, we obtain the parton evolution probability functions as follows:
Fq.q(y,z)~C2(R)ln1[(~+Z) 2,A)
—
+~5(1 _z)]
1]ö(l
—
z)
(14)
,
+ [W~(y,Q
c
Fq~g(z)=
2 ln~-~,
(15)
2(R) 1 + (1— z)
Fg_.q(z) =~~[z2 + (1— z)2]ln~-~~~1
(16)
,
Fg~+g(y,z)=
~ C
2(G)ln~ {(1 —z)~ + 2,A)
—
1]~5(1— z)
Z —
+ z(1
—
Dg~5(1 z) —
z) + ~[ii ,
—
~‘~]~(1
—
z)}
(17)
+ [Wg(y,Q
with dz
Jo =
F(z) = ~ dzF~ F(1) (1—z)÷ Jo 1—z
g2/4~t.
,
(18)
(19)
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N~is the flavor number of quarks. Dg in Eq. (17), introduced to have the nuclear momentum conserved, can be expressed as 1] GN(x, Q2)}dx S~”{[l4’~(x,Q2,A) 1][SN(x,Q2) + CN(x,Q2)] + [Wg(x,Q2,A) Dg S~xGN(x,Q2)dx —
—
(20) JJ~(g)(X,Q2, A) in Eqs. (14) and (17), similar to that in Ref. [14], is the nuclear shadowing factor due to the spatial overlap of partons from neighboring nucleons in the nucleus in the small x region, i.e.,
Jf~(g)(X, Q2)
1, A~(x) X > Xfl lKq~(g) OXXfl,
=
(21)
,
YA~X)
with the spatial overlap factor 1
=
(22)
—~xmNrN(A).
The x~in Eq. (21) is the critical value of the Bjorken variable x, where nuclear shadowing starts to occur. It is determined by setting 0.
=
(23)
By assuming that the valence quarks cannot be deconfined and released from the bound nucleon, we obtain ~4’~~(x,Q2,A)ml
(24)
.
Thus the parton evolution probability function for valence quarks becomes Fqv~qv(z)~C 2(R)[(~~)
+~c5(1_z)]ln~~1.
(25)
We can now give the parton distribution functions of the bound nucleon by using the parton evolution probability functions mentioned above. If we let VA(N), ~ CA(N), and GA(N) denote the distribution functions of valence, sea, and charm quarks and gluons in the bound nucleon in the nucleus with mass number A (in the free nucleon), respectively, then
2)
=
VA(x,Q
SA(x,Q2)
=
J J
VN(x,Q2) + SN(x,Q2) +
+
f
Fqv~qv(x/y)vN(y,Q2)d
(26)
Fq~q~(y,x/y)S~dy
6 Fgq (x/y) GN(y,Q2)
~
(27)
)
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cA(x, Q2)
=
cN(x, Q)2 +
+
GA(x,Q2)
=
/ Physics Reports 242 (1994) 505—517
Fq~q,(y, x/y) C(y,
GN(y,Q2)
J
2Fg.q(x/y)
J
GN(x,Q2) +
dy,
Q2),
(28)
Fq~g(X/y)![VN(y,Q2)
+ SN(y, Q2) + CN(y, Q2)]
where VN(x,
Q2
dy +
Fg~g(y,x/y) GN(y, ~
dy,
(29)
SN(x, Q2), CN(x, Q2) and Gt~~(x, Q2) are taken from set 2 of Ref. [18].
3. Explanation of the EMC effect and the nuclear Drell—Yan process Let us explain the EMC effect and the nuclear Drell—Yan process by using the PEM. Define the ratio of the average nuclear structure function to the deuteron structure function as [13] R~4~(x, Q2) F~(x,Q2)
F~(x,Q2)/F~(x,Q2),
=
2)—~(N Z)[F~(x, Q2) —F~(x,Q2)]},
{F
=
(30) (31)
—
2~(x,Q
where F2A is the nuclear structure function, F~and F~are the free neutron and proton structure functions, respectively. The second term compensates for the neutron excess, so that F~’(x,Q2) represents the average structure function of a hypothetical nucleus with equal numbers of protons and neutrons. By considering the Fermi motion of nucleons in the nucleus, F 2A can be written as [13] 2) F2A(x,Q
=
~j~ç3
(32)
l~(p)l2zF~(x/z,Q2),
where z = (Po + p3)/mN, Po = mN + mN and ~ are the rest mass and separation energy of a nucleon in the single-particle state )~,respectively, and ~‘A( p) is the single-particle wave function of the nucleon in the momentum space, which satisfies the light-cone normalization: ~,
J~
2luI’2(P)l~ 1.
In the following calculation, c~and i~’~(p) are taken from Ref. [14]. By using the 4~(x, Q2) parton can be distribution as functions given in Section 2, the bound nucleon structure function F~ represented F~’””~(x, Q2) =
x V~(x,
Q2)
±~ xV~’(x, Q2) +
~x[SA(x, Q2) + 2C-~(x,Q2)]
,
(34)
G.L. Li et al. / Physics Reports 242 (1994) 505—517
where + ( —) stands for the proton (neutron) and V~(x,Q2) = u~(x,Q2) + d~(x,Q2), V~(x,Q2)
(35)
u~”(x,Q2) — d~”(x,Q2),
=
511
(36)
where u~and d~’are the distribution functions of valence u and d quarks in the bound proton, respectively. The functions V~( 3),SinA, Eq. and(30). CA are Eqs.account (26)—(28). 41’~(x,Q2) By given takingbyinto the nuclear surface effect, We now calculate R’ similar to Ref. [19], we take rN(A)
=
rN[l + 0.43 ln(2
—
A113)]
(37)
,
where the constant 0.43 is determined by rN(~,)~ 1.3 rN. The values of the parameter Kq~(g)~fl Eq. (21) are taken as 0.25, 0.30, 0.35, and 0.55 for nuclei C, Fe, Cu, and Sn, respectively. The comparison between the ratios R~4~(x,Q2) for nucleus 56Fe calculated by the PEM and the experimental data of the EMC effect [20] is plotted in Fig. 1. Similarly, the results for nuclei ‘2C, 64Cu, and ‘16Sn are presented in Fig. 2. In the calculations, we take Q2 = 20 Gev2/c2. From these figures, one finds that in the PEM, by considering both nuclear shadowing and Fermi-motion corrections, the experimental data of the EMC effect can be well explained in the whole x region. The double x-rescaling model [14], which has no definite physical mechanism, can also well explain the EMC effect. In our opinion, the different evolution modes of valence and sea (including charm) quarks can cause different x-rescaling for the momentum distributions of valence and sea (including charm) quarks in the nucleon structure function. By assuming xvA(x, Q2) XSA(X,
Q2)
=
=
5~xVN(o~x,Q2),
(38)
(39)
o~xSN(~,x,Q2),
we find out that x-rescaling parameters 5~and ~5,,which slightly depend on x, can be estimated in the PEM. They are ~i5~>1,
ö,<1.
(40,41)
This is qualitatively consistent with the result from the double x-rescaling model in Ref. [14]. Therefore, the evolution of partons in the nuclear environment is just the dynamical mechanism of the double x-rescaling model. Let us now discuss the nuclear Drell—Yan process in terms of the PEM. The nuclear Drell—Yan process is a collision between the beam hadron (h) and the target hadron (A) in the nucleus. According to the Drell—Yan model [23], the differential cross section for the nuclear Drell—Yan process: h+A—~t4j.C+X
(42)
is given by d2a~’4(x 1 x2) dx1 dx2
1 /4~2\ =
2
~ ~-~-) ~ e~[q~(xj)c~(x2)
+ ~(x1)q~(x2)],
(43)
where M denotes the total mass of the lepton pair (in its rest frame). The e, is the fractional quark charge (in units of e) and the factor ~ comes from the fact that only a pair of q~which have opposite
512
G.L. Li et a!.
/ Physics Reports 242 (1994) 505—517 I
(a) C/D
1.~___
(b) Cu/D
1,2 ______________ 1.1
_________
(c) Sn/D
~,
1,0
O.~4O!6O~8
I.0
Fig. 1. Comparison between the ratio R’~”(x,Q2) for nucleus 56Fe calculated by the PEM and the experimental data of the EMC effect [20]. (LISLAC 1983 Fe/D, .BCDMS 1986 Fe/D, 0 EMC 1986 Cu/D.) Fig. 2. Comparison between the ratios RA~(x, Q2) for nuclei ‘2C, 64Cu and ~6Sn calculated by the. PEM and the experimental data of the EMC effect. The solid dots and the open circles are taken from SLAC [21] and EMC [22], respectively.
colors annihilate into a virtual photon. The functions q~4~(x)and ~~x) are the quark and antiquark distribution functions in the beam hadron (h) (the target hadron (A)), respectively; x 1(x2) is the fraction of the longitudinal momentum carried by the quark or antiquark in the beam hadron (the target hadron). If we let ~
,
(44)
then 2a~(xj,x d
2) dx1dx2
=
11 4~2 \ 3\3x1x2M 2 ji HhA(xl,x2)
—1
.
(45)
G.L. Li et al.
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513
By considering the fact that the quark and antiquark distribution functions in the hadron are Q2-dependent, Hh_A(xl, x2) should be replaced by HhA(xl, x2, Q2). To be more specific, Hh_~(xj,x 2) can be expressed as 2, Q H~A(xj,x 2) = ~-~x 2) + d~(x 2)]x 4(x 2) 2, Q 1[4u~(xj, Q 1,Q 2S’ 2,Q +
x
2)
2)[(N + 4Z)x
2u~(x2,Q
1SN(x1, Q
2) + 2Ax 2)], 2d~(x2,Q 2S’~(x2,Q 31~x 2) + 4d~(x 2)]x 2) 1[u~(x1,Q 1,Q 2S”(x2,Q + (4N + Z)x
HflA(xl, x
2) 2, Q
=
+
x
2)[(N + 4Z)x 1SN(x~,Q
(46)
2) 2u~(x2,Q
+ (4N + Z)x
2) + 2Ax 2)], (47) 2d~(x2,Q 2SA(x2, Q where p(n) labels the beam proton (neutron), N and Z are, respectively, the neutron and proton number of the target nucleus with mass number A. The functions ut’, d~,and S’~are given by Eqs. (26), (27), (35), and (36). Although various models with very different physics assumptions are all able to explain the EMC effect, they give different predictions for the nuclear Drell—Yan process. Thus, one can use the information from the nuclear Drell—Yan process to test the models of the EMC effect. Recently, the E772 collaboration [17] precisely measured the atomic-mass dependence of dimuon production induced by 800 GeV protons in the kinematic region (0.025 < x 2 < 0.3) where the result is sensitive to the antiquark distribution in the target nucleus. They gave the nuclear Drell—Yan ratios in the mass regions 4 ~ M 9 GeV and M 11 GeV, where there is no contribution from the decay of quarkonium resonances. In order to compare with the experimental data mentioned above, we define the nuclear Drell—Yan ratio as 2~”~4(xi,x AID Sdxi(d2 a~(x 2)/dxi dx2) T (x2) = $dxi (d 1, x2)/dx1 dx2)’ (48) where the integral range for x1 is determined according to Ref. [17]. By using the PEM, we calculate the nuclear Drell—Yan ratios for nuclei C, Ca, Fe and W. The values of the parameter Kq~(g)in Eq. (21) are taken as 0.25, 0.28, 0.30, and 0.35 for nuclei C, Ca, Fe and W, respectively. In Fig. 3, we present the resultant nuclear Drell—Yan ratios together with the experimental data from the E772 collaboration and the predictions from the other EMC models [17]. The solid curves are our results, the dashed, dotted,2-rescaling and dash-dotted are, respectively, theagree resultsbetter from with the pion models.curves Apparently, our results the excess, quark data cluster, andnuclear Q experimental of the Drell—Yan ratios than those from the pion excess and quark cluster models. This shows that the PEM can reasonably describe the sea quark distribution in the bound nucleon. In Fig. 3(c), on comparing the PEM’s result with that from the the Q2-rescaling model, we see that the influence of the nuclear shadowing on the nuclear Drell—Yan ratios is evident. We also consider the Fermi-motion correction in the calculation of T~~(x 2) and find that it is negligible in the region of x < 0.3.
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1.3
.2
(a) C/D
-
(b) Ca/D
I
—I
(c) Fe/D
.2
/
I
I
(d) W/D
07
03
Fig. 3. The nuclear Drell—Yan ratios T’~(x
2)for nuclei: C, Ca, Fe, W are plotted versus x2. The solid curves are the results calculated by the PEM. The dashed, dotted, dash—dotted are, respectively, thecollaboration results form the pion 2-rescaling models. The and experimental datacurves are taken from the E772 [17]. excess, quark cluster, and Q
4. Similarities and differences between the parton evolution model and the Q2-rescaling model Since the Q2-rescaling model also relates to the swelling of the quark confinement scale in the bound nucleon, we should discuss the relation between the PEM and the Q2-rescaling model. To do this, let us take nth order moment of the valence quark distribution function VA(X,
Q2): M,~’(Q2)= (1 + F~)M~(Q2),
(49)
M~~(Q2) =
(50)
where
F~=
x~t VA~(X, Q2)dx,
Z’~’ Fqv~qv(z)dz=
B~5= C 2(R)
1~ ~ ~‘
~
If we assume 2)= M~(~(Q2)Q2), M~(Q
~-
B~ln(i~(A)/r~),
+
ó(1 — z)] dz.
(51)
(52)
(53)
G.L. Li et al.
then
/ Physics Reports 242 (1994) 505—517
= 1 +F M~(~(Q2)Q2)
515
(54)
M~(Q2) According to the QCD theory, we obtain ______________
[1 + ;(Q2)b
—
2)
0t’ 1B~/(2itbo)
[1~(Q2)/
M~(Q
0t]
‘
(56) 2/p2) ln(Q ln(~(Q2)Q2/p2)
(57)
,
t
= =
(58)
,
where p is the mass dimension parameter. For sufficiently large Q2, Eq. (55) can approximately be expressed by M~(~(Q2)Q2) ~ +
B~ln~(Q2).
Thus, by taking the reference value of; as ~c(Q2 = /
—2 rN(A)
\;(~2)/;(Q2)
}
(59) we obtain, from Eqs. (54) and (59), (60)
\ rN /
This is just the expression of the Q2-rescaling parameter in the Q2-rescaling model. Therefore, we can conclude that the Q2-rescaling model is an approximation of the PEM, which is valid for sufficiently large Q2 and considering only the evolution mode of valence quarks for all types of quarks. The differences between the PEM and the Q2-rescaling model are as follows: (1) In the PEM, the evolution of valence and sea (including charm) quarks and gluons are determined by different interaction vertex factors and their effects on their own momentum distributions are also different. However, the Q2-rescaling model, ignoring the different evolution modes for different types of quarks, describes the influence of the nuclear environment on the distribution functions of valence and sea (including charm) quarks by using the same Q2-rescaling parameter determined by the approximate behavior of the evolution of valence quarks at sufficiently large Q2. (2) In the Q2-rescaling model, rN(A)/rN is a free parameter, while in the PEM, FN(A)/rN is determined by the empirical formula Eq. (37) with FN(~)/rN~ 1.3, which is consistent with the ratio of the effective mass of the nucleon bound in nucleus 209Bi to the mass of the free nucleon, m~/mN~ 0.75, given by Brown and Rho [15]. There is no free parameter such as the Q2-rescaling parameter in the PEM. (3) As pointed out in Ref. [14], in the Q2-rescaling model, the nuclear momentum is not conserved, i.e., the nuclear momentum is not equal to the sum of momenta of valence and sea (including charm) quarks and gluons in the nucleus. While in the PEM, as a restraint condition, the
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nuclear momentum is automatically conserved. Therefore, the PEM is more consistent than the Q2-rescaling model. In addition, the PEM has a definite physical mechanism. 5. Summary We investigate the influence of the evolution of partons in the nuclear environment on the structure function of the bound nucleon and find that the evolution of valence and sea (including charm) quarks and gluons are determined by different interaction vertex factors and their effects on their own momentum distributions are also different. It brings the physical nature of the double x-rescaling model [14] to light, i.e., the evolution of partons in the nuclear environment is the dynamical mechanism of the double x-rescaling model. By using the PEM and considering the nuclear shadowing and Fermi-motion corrections, the experimental data of the EMC effect and the nuclear Drell—Yan process can be well explained and, at the same time, the nuclear momentum is automatically conserved. Therefore, the change of the momentum distribution of partons in the bound nucleon caused by the evolution of partons in the nuclear environment is possibly the origin of the EMC effect.
Acknowledgments The authors would like to thank Professor A.D. MacKellar for critical reading of the manuscript tand professor K.F. Liu for helpful discussions. This work was supported by the National Science Foundation under grant No. 8820088, the Center for Computational Sciences, University of Kentucky, and the National Natural Science Foundation of China.
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