Quark models and meson cloud in deep inelastic scattering

NUCLEAR, PHYSICS A ELSEVIER

Nuclear Physics A 628 (1998) 296-310

Quark models and meson cloud in deep inelastic scattering Andreas Mair, Marco Traini * Dipartimento di Fisica, Universit?t degli Studi di Trento, 1-38050 Povo (Trento), Italy Istituto Nazionale di Fisica Nucleate, G.C. Trento, Italy

Received 23 August 1996; revised 16 October 1997;accepted 4 November1997

Abstract We propose a quark model description of the nucleon which includes, in a unified picture, valence quarks and a core of q~ pairs and gluons at the hadronic (non-perturbative) scale. The resulting parton distributions are evolved to high Q2 by means of an appropriate next-to-leading order evolution and compared with the data. The procedure is carried out for several, rather different, quark model pictures to analyze the model dependence. We conclude that there is not a smooth transition between the degrees of freedom of low-energy nuclear physics (baryons and mesons) and a description in terms of quarks and gluons. © 1998 Elsevier Science B.V.

1. Introduction In the recent past a formalism has been developed to describe deep inelastic scattering (DIS) in terms of models of hadron structure [1]. This formalism has been quite successful in understanding the missing ingredients of the naive models developed to explain DIS data. In particular, it has been shown that gluons also play an important rfle at the low-energy scale, a conclusion supported by phenomenological parametrizations [2]. In the process of obtaining a deeper understanding of the relation between the low-energy descriptions of hadron structure and the asymptotic behavior of the theory, the fine details of the q~ sea came under scrutiny. Since the gluon is blind with respect to the flavor degrees of freedom, the nucleon (light) sea, regarded as a pure radiative product of the (leading order) QCD processes, * Corresponding author. E-mail: [email protected]. 0375-9474/98/$19.00 (~ 1998 Elsevier Science B.V. All rights reserved. Pll S0375-9474(97) 00629-5

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is expected to be SU(3)f symmetric 1 . Of course, the plausible hypothesis that, at the hadronic /z2 scale, the mass of the strange quark cannot be assumed degenerate with the light u and d masses, is a reason for the suppression of the s~ pairs, but SU(2)f remain a good symmetry. Experimental data [4-6], accumulated during the last few years, imply the asymmetry of the ~, d distributions and, therefore, the breaking of the SU(2)f symmetry. Let us summarize these findings. (i) The recent analysis of neutrino charm production by the CCFR Collaboration shows that the nucleon's strange quark content is suppressed with respect to its nonstrange sea by a factor 0.48 -t- 0.06 [4]. (ii) The measurement [5] of the Gottfried sum rule of the proton's and neutron's structure functions FP'n(x, Q2), 1

GSR=

1

1

- - ( F ~ ( x , Q 2) - F~(x, O2)) X 0

0

0

1

=' + 2 f

d x ( ~ - d) =0.235-4-0.026,

(1)

0

implies that f~ d x ( ~ - d) ~ and f~(x, Q2) ~ ( x , Q2) [7]. (iii) The NA51 experiment at CERN extracted, for Fc/d, the value 0.51 4-0.06 at x = 0.18, Q2 ~ 27 GeV 2, by measuring the ratio of cross sections for muon pair production using 450 GeV protons colliding with proton and deuteron targets [6]. The presence of a (non-perturbative) mesonic cloud around the nucleon gives a quite natural explanation [8] of the S U ( 2 ) f asymmetry of the proton's antiquark sea, and much theoretical work has been devoted to the study of such problems [9-11]. In particular, the convolution models, despite their unresolved problems [ 12], have been widely used [ 10,11]. The basic hypothesis of these models is that the physical nucleon wave function (in the infinite momentum frame) can be written as a superposition of Fock states, which include meson-baryon components with the same quantum numbers as the original proton. The partonic structure of these states is incorporated by means of convolution using a generalized Sullivan scheme [ 10,13]. Quite recently, Koepf et al. [ 14] addressed the question of whether the nucleon's antiquark sea can be attributed entirely to its virtual meson cloud and whether there exists a smooth transition between hadronic and quark-gluon degrees of freedom. To calculate the antiquark sea these authors take into account both 7r and K mesons in the Sullivan process [ 13 ] of deep inelastic lepton-nucleon scattering, therefore including 7rNN, KNY (with Y E {A, 27, 2;*}) and rrNA in the flavor SU(3) and SU(2) breaking components of the nucleon's antiquark sea and its strange quark content. Their formalism can be subject to some criticism regarding their definition of the meson light-cone distribution, which lacks some of the desirable properties of a generalized scheme, such as gauge I The present paper will discuss next-to-leading order evolution. At that order a quite small violation of S U ( 3 ) y symmetry occurs (cf. Ref. [3] ).

298

A. Mair, M. Traini/Nuclear Physics A 628 (1998) 296-310

invariance and charge conservation [ 15]. However, due to the limited aim of their paper, namely to calculate solely the antiquark sea, the criticism has no practical consequence. Their conclusions can be summarized as follows. (i) Once the various cut-offs are adjusted to the large-x tails of the antiquark distributions at Q2 = 1 GeV 2, the agreement remains satisfactory for smaller x values and for increasing momentum transfer in the case of the S U ( 3 ) f xq8 = x ( ~ + d - 2~) and S U ( 2 ) f xq3 = x ( d - ~) non-singlet components, but the xg distribution is quite poorly described and the deviations grow rapidly with increasing Q2. (ii) The entire sea quark distribution in the nucleon cannot be attributed to its mesonic cloud, and other degrees of freedom, notably gluon splitting into qgl pairs, are relevant even at moderate x values and momentum transfer. Are there additional mechanisms beyond the meson cloud which contribute to the formation of the antiquark sea? Answering this question is the main motivation of this paper. We take, for this purpose, a complementary approach with respect to previous investigations (e.g., Ref. [ 11 ] ), namely we look at the problem from the point of view of the quark model picture. Our line of research, motivated by our previous experience, follows the path opened up some time ago by Kuti and Weisskopf (KW) [ 16]. In the KW scheme the nucleon is considered as a three-quark structure accompanied by a core of virtual quark-antiquark pairs and gluons. The valence quarks carry the internal quantum numbers of the nucleon, whereas the core has vacuum quantum numbers. In formulating a complete description which includes more than one mechanism, care must be taken to avoid double counting. Therefore, we specify in detail the mechanisms contributing to the three partonic structures. While KW proceeds to a statistical description of the core, we will adhere to a valence description for the partons at the hadronic scale, as in our previous work [ 1 ]. The new ingredient, the primordial sea, will be constructed by taking a symmetric combination of quarks and antiquarks, according to the KW philosophy. We use as input the antiquark distribution of Koepf et al. [ 14] evolved to the hadronic scale, as demanded by consistency in our scheme [ 17] 2. Within this unified description of the parton distributions of the nucleon incorporating valence quarks, gluons and the sea at the hadronic scale, we approach the experimental data at high Q2 by performing a perturbative QCD evolution, thereby answering the previously posed questions. The paper is organized as follows. In Section 2 the unpolarized parton distributions are evaluated, at the hadronic scale, including the meson cloud qgl pairs and gluon components. The evolution procedure is discussed in Section 3 where a symmetric formulation (in the initial and final scale) is given to evoid ill-behaviour at low scale. The predictions obtained using a variety of constituent quark models are discussed in Section 4 and compared with the experimental data.

2The model for the sea arises, within conventionalquark physics, from a 3,°0 pair creation model [ 18] assuming degeneracy of the quark masses (mu = md = ms). The result of Koepf et al. is taken as a good parametrization at a certain ~ale, not as anythingfundamental.

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2. The quarks, gluons and pair cloud at the hadronic scale Our general framework is based on the idea that the high-energy parton distributions when evolved (in QCD) to a low-energy scale, indicate that a valence picture of hadron structure arises [ 19]. This valence picture is represented, theoretically, by the quark models (QM) which are very successful in explaining the low-energy properties of hadrons.

2.1. Valence quarks and partons A simple description which connects the parton distributions and the momentum density of the constituents has been developed in the recent past [ 1]. Within that approach the valence quark distributions can be written as q v ( x ' / x 2 ) - (1 -1x ) 2 f d3knq(k)6( 1 -x- x

k~)

o~

-

dlk[lklnq(Ik[),

2~M (i--_ x)2 f

(2)

k,,(x) where

M km(x)

x

= "~

(1 - x )

/m,,',2(1-x)

.

~-"M)

'

x

(3)

k+ = ko - kz is the light-cone quark momentum fraction, nq(k) the quark momentum density distribution predicted by the specific QM wave functions, and M and mq are the nucleon and constituent quark masses, respectively. One can check that f dx qv (x, tx2) = fd3knq(k) = Nq, i.e. the particle sum rule is preserved and the distributions (2) are defined within the correct support 0 ~ 4. We will assume that a gluon distribution G(x, IX~) contributes to the total momentum sum rule at the scale/x20 in such a way that f G(x,/z02) = .Afg= 2, the minimum number of gluons required to make a singlet. The valence gluons are assumed to be proportional to the total valence contribution [ 1,2]

N.

G(x'tz2) = T

[uv(x'/x°2) + dv(x'/x°2)] "

(4)

The last step in our formalism is provided by the meson cloud contribution to deep inelastic scattering, as developed, within the convolution approach, by Koepf et al. [ 14,? ] 1

xON(x,~) = ~7_o~qsf clyfM~(y)yOM (y,~g ) , M,B

(5)

x

where ceqB are the spin-flavor SU(6) Clebsch-Gordan coefficients, XglM(X, txg) is the meson's valence antiquark distribution fit [ 17], and fMn(Y) is the meson's light-cone

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distribution in the nucleon cloud as evaluated in Ref. [ 14]. The antiquark distribution, obtained in this way, is taken as a parametrization of the neutral pair-core [ 16] by incorporating an exactly equal amount of sea quarks for each flavor. All these components must be consistent with the momentum sum rule 1

dx x(uv(x, #2) + dv(x,

+ G(x, Ixg ) + 2(t~(x, #~) + d(x, /z2) + g(x,/z~) ) = 1,

o

(6) which acts as a renormalization of the valence quark content. Using Eq. (4) the previous expression reduces to

/ dxx(uv+dv)-3_~A/.g

dxx(~+d+g)

1-2

o

.

(7)

0

2.2. The hadronic scale Eq. (7) fixes the hadronic scale /z~. In our case (using, for the nth moment, the notation (f), = f dx f (x)x n-I ), one obtains at the hadronic scale/x02 (uv + dv)2 = 0.53,

2(/i + d + ~)2 = 0.11,

(G)2 = 0.36.

(8)

The actual values o f / x 2 are fixed [21] evolving back (at the appropriate perturbative order) unpolarized data fits of Ref. [22], until the valence distribution matches the required momentum (8) (uv + dv}2 = 0.53. The resulting NLO parameters are: ces(/X2)nqr NLO = 0.040,

as (/x°2)4rr LO = 0.058,

/X2INLO= 0.37 GeV 2,

#02ILO = 0.35 GeV 2,

ANLO= 248 MeV;

ALO = 232 MeV,

(9)

and the LO has been added for completeness. The A values are suggested by the analysis of Gliick et al. [23], Ces(/Z0Z)INLO is obtained by evolving back the valence distribution as previously discussed, and /z~ is found by solving numerically the NLO transcendental equation

In /z~

A2NLO--

4~r

ill, [ 4~r + ~202

+

fill

=0,

(lO)

which assumes the more familiar expression

as(a 2) 1 ( flo lnln(Q2/A2Lo) ~ 4~r - floln(QZ/a2Lo ) _1 -- fl~ In(Q2/A2Lo ) ]

(11)

only in the limit QZ >> AzLO" Gliick et al. in their analysis of the NLO parton distributions [23] made use of the simplified expression (11). We checked [21] that we

A. Mair, M. Traini/Nuclear Physics A 628 (1998) 296-310

301

0.12

0.1

0.08

x

0.06

0.02

o

,"

"

dl

o12

-

_

x

03

o.,

o.~

x

Fig, I. The distributions xft(x, iz2) (continuous curve), xd(x, ix~) (dot-dashed curve), and x~(x, ~ ) (dashed curve) as a function of x at the scale ,tt02 = 0.37 GeV 2. They are evaluated within the virtual meson cloud model of Koepf et al. [ 14,201 according to Eq. (5).

could also use the A values suggested by those authors in our approach at the cost of a re-evaluation of the static point #02. The NLO values for ~02 obtained making use of the approximation (11) would be larger: as an example, #20INLO = 0.37 GeV 2 in Eq. (9) would become /Z~INLO ~ 0.42 GeV 2 (an interesting discussion on the effects of the approximation (11) can be found in Refs. [21,24] ). In Fig. 1 the sea distributions at the hadronic scale ~02 = 0.37 GeV 2 are shown as a function of x. They are obtained within the model of Koepf et al. without modifying their parametrization for the MNB vertices. In particular, we keep their choice of the cut-offs A~tNB which are softer than those used in most meson-exchange models of the NN interaction. At the hadronic scale we assume that the only non-vanishing antiquark distributions are x~(x, I~), xd(x, t,2) and x~(x, lz~) of Fig. 1. As a consequence the total non-perturbative sea at the hadronic scale results in xS(x, tz~) = 2x(~(x,tz02) + d(x, #~) + g(x, I~)), where we have incorporated an exactly equal amount of sea quarks for each flavor. They represent the input distributions for the perturbative NLO evolution together with the corresponding valence (2) and gluon components (4).

2.3. Quark models In the following we discuss the results obtained using different models for the valence quark contributions, namely the Isgur-Karl (IK) model [25], which has been largely used in the past to study the low-energy properties of hadrons and also deep inelastic polarized and unpolarized scattering [ 1 ], a relativistic Coulomb-like potential model used by Franklin and Ierano [26] in the context of deep inelastic scattering, and an algebraic model proposed recently [27], the wave functions of which give a rather good description of the electromagnetic elastic and transition form factors [28]. All

A. Mair, M. Traini/NuclearPhysicsA 628 (1998) 296-310

302

these models were originally adjusted without including mesonic and gluonic degrees of freedom. Adding, in our framework, such new ingredients implies a simple rescaling of the size parameters of the models according to the momentum sum rule (6) as discussed in Ref. [ 1 ], namely: (i) IK model: the harmonic oscillator basis constant changes from a 2 = 1.23 fm -e to a 2 = 1.09 fm-2; all the other parameters remain unchanged, in particular the amplitude of the various SU(6) components of the wave function. The momentum distribution is from Ref. [29]. (ii) Algebraic model: in this case 8aq3

A/'u

nq(k) = ,rr2 (1 + k2a2q)4'

(12)

and the parameters au = 0.258 fm and ad = 0.285 fm [30] become a~ = 0.372au and a~ = 0.372ad SO that the ratio au/ad is unchanged. This ratio is related to the SU(2)f breaking properties of the model and remains unmodified analogously to the IK model. (iii) Relativistic Coulomb-like potential model: the momentum distributions reads

b2q 2aq -tnq(k) = N'q

(13)

and the parameters b, = ba = 0.172 GeV are used instead of bu = ba = 0.4 GeV [26]. In Fig. 2 we show the NLO 3 parton distributions at the hadronic scale/.L02 = 0.37 GeV 2 for the three models considered in the present investigation. The dependence of these parton distributions on the model wave function is rather strong. The IK model wave functions are basically of harmonic oscillator type and therefore do not contain high momentum components. The resulting valence distributions are rather narrow. Their xuv and xdv distributions differ because the model Hamiltonian contains non-symmetric SU(6) terms due to the color-magnetic hyperfine interaction associated with the one-gluon exchange (OGE). The algebraic model wave functions contain higher momentum components and an SU(6) breaking mechanism which is rather different from that of the OGE in the IK model, and which leads to different behavior for the valence quark distributions xuv and xdv. The Coulomb potential model contains quite high momentum components and is SU(6) symmetric (b, = ba), therefore the gluon distribution xG is numerically equal to the xuv distribution (cf. Eq. (4)). The distributions of Figs. 1 and 2 are assumed as the non-perturbative input distributions at the hadronic scale for the NLO evolution performed within the DIS factorization scheme. In the calculation of Koepf et al. [ 14] agreement with the experimental data 3The most natural factorization scheme for the calculation of structure functions from quark models is the so-called DIS scheme (cf. Refs. [ 1,23,24] and Section 3).

A. Mair, M. Traini/Nuclear Physics A 628 (1998) 296-310

303

1.4

1.2

1

0.8

x"

0.6

0.4

0,2

o16

o17

ola

o19

0.5 X

0.6

0.7

0.8

019

0.5 X

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5 x

0.1

0.2

0.3

0.4

0.1

0.2

0.3

0.4

1.2

1

~o :=.

0.8

0,6

0.4

0.2

O'

1.2

1

~ o..¢

~

0.8

o.e

0.4

0.2

Fig. 2. The valence distributions xuv(x,t, 2) (continuous curves) and xdv(x,l.t 2) (dot-dashed curve) as a function of the Bjorken variable x at the hadronic scale ,~2o = 0.37 GeV 2 for the lsgur-Karl model (upper panel), the algebraic model (middle panel), and the Coulomb potential model (lower panel). For comparison the valence-like gluon distributions of Eq. (4) are also shown (dashed curves), xG(x, I~ ) = xuv (x, I-~) in the lower panel (see text).

A. Mair, M. Traini/Nuclear Physics A 628 (1998) 296-310

304

on the sea distribution seems to decrease as Q2 increases. The questions posed, which our formalism has to answer, will appear in a more nitid fashion if a long evolution is considered. Therefore, we have chosen to show results at a rather high value of the momentum transfer, namely Q2 = 10 GeV 2.

3. NLO evolution procedure In this section we briefly illustrate the evolution procedure we have been using. Although alternative factorization schemes have been investigated [ 1,21,23,24], we remain within the MS renormalization and DIS factorization scheme as already mentioned. In this case the moments of the F2 proton (neutron) structure functions take the simple form

(FP'(n)(Q2)>n= Z e2q
[+(-)h(xq3(x, Q2))n + ~(xqs(x, Q2)>n + ln],

=2

where + ( - )

refers to proton and neutron, respectively; X

component and contributions.

q3 = u + fi -- ( d + d ) ,

q8 = u + fi +

(14) =

~-]~q(q+ gl)

is a singlet

d + d - 2(s + g) are non-singlet

3.1. Non-singlet distributions The NLO evolution of the unpolarized distributions is performed following the solution of the renormalization group equation. Since, in our case, the starting point for the evolution (tz~) is rather low, the form of the equations must guarantee complete symmetry for the evolution f r o m / x 2 to Q2 >>/z02 and back avoiding additional approximations associated with Taylor expansions but not with the genuine perturbative QCD expansion [21 ]. In particular, for the non-singlet (NS) sector we have

(qNs(Q2))n l,n 1 + [ ('YNS/2/30) 1,n

0n

2

0n

2

- (Y~s/3,/2/3 o) ]

[ a ( Q Z ) / 4 " n "]

1 + [ (TNS/2/30) -- (y~,/Sfl]/2/30)] [ce(a02)/47r] (15) where ,~ r y(0,1),n s are the anomalous dimensions at LO and NLO in the DIS scheme 4 , and /30 and /31 are the expansion coefficients (up to NLO) of the function /3(Q2): • In _ 1.n--,~ r) r,(I),NS 4 The yL~ are redefined in the DIS scheme in such a way that Eq. (14) holds, 1.e. YNS ~ 7'NS -r zpOt,n . The Wilson coefficients C(nl)'q and Cn(l)'g, in the ~ for example, in Refs. [2,24].

renormalization and factorization scheme, can be found,

A. Mair, M. Traini/Nuclear Physics A 628 (1998) 296-310

305

/30 = 11 - 2/3Nf, /31 = 102 38/3Nf for Nf active flavors. Eq. (15) reduces to the -

more familiar form (e.g., Refs. [2,23,24] ) (qNS(Q2)),, = [ (~e(Q2)"~ ~'~/s/Zt~°( ( Y ~ / ~

Y~~/31"~ (ce(Q2)4-~ a(/z~) ) ) ] (qNs(/z~))n 2/32 J (16)

after performing a Taylor expansion for both ce(pz)/47r << 1 and a(Q2)/47r << 1.

3.2. Singlet distributions The singlet sector is more involved. The singlet and gluon moments mix and the corresponding evolution equations become (X(Q2))n "~ = ~,l(Q2,n)An~,l-l(tz2,n) ( I'(Iz2))n )
(17)

where ce(Q z) [(~+R"~+ +pnR"W_) 21~(Q2, n) = 1 + - 41r [ -2flo

PnR"W +

-

+ 2/30 + a " - ,t~

+

P~_R"P+

--

2/30 + U++- a o

]

,

A"= (a(Q2) )a'~/2/3o (a(Q2) ~ a"_/2/~0 \ a--~o2) j ~+ + \ a--~02) j W_.

(18)

(19)

R n, P~+ and A:k are given by: /31 ~ (o),n

R" =-7,~1)'" _ ~ r ,¢,(0) ,n ,~_ -

,

kn A_

'

a% = ~1,('Yqq(O)'n~--- ggg(O),n ^ 4- [ (~/(O),n

~,(k),n =

( ~ (k),n ~ (k),n l Yqq Yqg

,

__

rqq .

(0),n~2

~ ( O ) , n ~rgq , ( O ) , n l j 1/2), ) -4- 4¥qg

k = 0, 1,

(k),n ~/gg (k),n / Y gq

where A± are the eigenvalues of the matrix ~(o),, and ~ the projection operators. The 2 inverse matrix M----1 (/xo, n) has to be calculated numerically. Eq. (17) reduces to the more familiar form (e.g., Refs. [2,23,24] )

A. Mair, M. Traini/NuclearPhysics A 628 (1998) 296-310

306

'l't . . . . . . . . .

0.9

\~ "

0.8

\

'

~

/ / .J

~

0.6

_

'"..

0.5

0'40

0.05

0 1

0.15

02

0.25 X

03

0.35

04

0.45

0.5

Fig. 3. The ratio F~(x, Q2)IFP(x, Q2) as a function of x at Q2 = 10 G e V 2. The predictions of the IK (dashed curves), the algebraic (continuouscurve), and the Coulomb-likepotentialmodels (dot-dashedcurve) are comparedwith the CTEQ3Dexperimentalfit of Ref. [22]. (

<,~(Q2)> n -.~ = ( (o~(Q2),~ a'L/2.8o[p~

(G(Q2)),, J I. \

~ J

. 47r

x

2~o +

_

.4% - a"_

1 a(IZ2o) - a ( O 2) 2flo

t,,~--(-~~o2)) + (+<

> -)

p"__R~p~-

47r

//
'

again in the limiting case a(lz~)/47r << 1 and a(Q2)/4~r << 1.

4. N u m e r i c a l results a n d discussion

As a first illustrative example of the r61e of hadronic models in reproducing important effects in deep inelastic scattering, we show in Fig. 3 the ratio rn,p(x, Q2) = F~(x, Q 2 ) / F P ( x , Q 2) at Q2 = 10 GeV 2 for the three quark models investigated. If SU(6) is assumed to be a good symmetry, rn,p ~ 2/3 for large x values and, in fact, the SU(6) symmetric Coulomb potential model follows this behavior. On the other hand, rn,p 4 = 2//3 if SU(6) breaking effects are considered and experimentally one finds rn,p < 2//3 at large x. Both the IK and algebraic models embody SU(6) violations and their predictions for rn,p show the importance of modelling these breaking effects considering that, for x > 0.3, the ratio is dominated by the valence contributions and the effects of the evolution are very small. The results of the evolution for the singlet components are shown in Figs. 4 and 5. Despite the choice of rather different quark models, the x~(x, Q2) do not differ significantly and they are in reasonable agreement with the experimental CTEQ3D fit [22]

A. Mair, M. Traini/Nuclear Physics A 628 (1998) 296-310

307

0.14 0.12 0.1 A OX-0.08 0.06 0.04 0.02

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0,5

0.4

0.45

0.5

x

0.2

.

.

.

.

.

.

.

.

.

.

0.18

0.14[ 0"12t ..

x" 0.1

-. ~

0.08

o.o5

.. ~ ~ . " "" ~ %

0.04

0.0~ 0.05

0.1

0.15

0.2

0.25 x

0.3

0.35

0.1 0.09 0.08 0.07 O.OE ~ 0 0.05

J~ 0.04 0.03

0.02 0.01 O0

.

i

0.05

i

0.1

i

0.15

0.2

i

0.25 X

0.3

h

0.33

i

0.4

.......

i.... 0.45

0.5

Fig. 4. The sea distributions x~(x,Q 2) (upper panel), xd(x,Q 2) (middle panel), and x;(x,Q 2) (lower panel) as a function of x at Q2 : 10 GeV 2 as predicted by the IK (dashed curves), the algebraic (continuous curve), and the Coulomb-like potential models (dot-dashed curve). The dotted lines represent the CTEQ3D experimental tit of Ref. [22].

A. Mair, M. Traini/Nuclear Physics A 628 (1998) 296-310

308 3

~

2.52

.

.

.

.

.

.

,

,

,

0.35

0.4

0.45

I

0.5i

LO ~ ~ ~' ~ i '

0

0.05

0.1

0,15

0.2

0.25 X

0.3

0.5

Fig. 5. The gluon distributions xG(x,Q 2) as a function o f x at Q2 = l0 GeV 2. Notation as in Fig. 4. The LO results for the Coulomb-like w.f. model have been added for comparison. 0.1

~,

0.09 0.08 0.07

h

0.06 ~ 0 0.05

"

I ~ 0.04

",\ "'"

"~"

0.03

'.. ,..

0.02 0.01 O0

0.()5

011

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x

Fig. 6. The strange sea distribution x~(x,Q 2) as a function of x at Q2 = 4 GeV 2. The NLO evolution predictions of the present work (continuous curve) are compared with the corresponding LO calculations (Coulomb-like w.f. model) (dot-dashed curve) and the results of Koepfet al. [14,201 within the convolution approach (dashed curve). The dotted line represents the CTEQ3D experimental fit of Ref. [22].

even at Q2 = 10 GeV 2. Good global agreement is obtained for the gluon distribution showing the importance of the NLO corrections 5. In particular, from the results shown in Fig. 5 one can conclude that the gluonic content of the nucleon at Q2 = 10 GeV 2 is consistent with the assumptions leading to Eq. (4). In Fig. 6 we show the strange component xg(x, Qa = 4 GeV2), comparing our results with the calculation of Ref. [ 14]. The distribution obtained by Koepf et al. is indeed 5 It is well known that the LO calculation produces gluon distributions which are too soft.

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tOO soft at x < 0.3. By adding the valence and gluon degrees of freedom the scenario changes radically: (i) the convolution picture is replaced by a complete QCD evolution and the different degrees of freedom can be consistently included; (ii) the identification of the mesonic cloud contribution with the nucleon sea is valid at the hadronic scale only; (iii) the resulting xgt(x, Q2) distributions are even larger than the experimental fit, in particular x g ( x , Q2), overcoming the large deficiency at the hadronic scale. This result is not an artefact of the NLO evolution as can be confirmed by looking into the LO evolution calculations shown in Fig. 6. The present calculation confirms that the gluon content of the nucleon at the hadronic scale plays a crucial r61e in the description of the behavior of the sea through perturbative evolution. The production of qO pairs due to gluon splitting generates a considerable amount of sea at high energies. Previous calculations of the same phenomenology [ 11 ] have not addressed the issue of the primordial gluon content of the nucleon, but have assumed that, at a certain scale, all the physics is determined by the meson-baryon Fock space through the Sullivan process. Despite the fact that the convolution models may be used to determine the primordial quark and sea contribution, we feel that primordial gluons are unavoidable: the data confirm that hypothesis and their origin may be attributed to the underlying structure of the constituent quarks [31,32]. Their existence implies, as shown by our calculation, a re-parametrization of the cut-off parameters for the meson-baryon theories. The trend of our results, overshooting of the sea distributions at large x, seems to indicate that they will approach those appearing in the more traditional domains of nuclear physics. In view of our calculation we conclude that there are mechanisms, not included in the convolution model description of the nucleon, which are relevant for explaining the data. The only mechanism discovered thus far that reproduces the gluon content at large momentum transfer is the existence of a primordial gluon cloud at the hadronic scale. This cloud, through perturbative evolution, influences in a notable manner the determination of the partonic structure of the nucleon, and at present cannot be substituted by hadronic mechanisms. Thus our calculation demonstrates that, with our present knowledge, there is not a smooth transition between the low-energy nuclear physics degrees of freedom (i.e., baryons and mesons) and a description in terms of quarks and gluons.

Acknowledgements We are indebted to Werner Koepf for supplying the fortran code generating the sea components within the model of Ref. [ 14], and to Vicente Vento for his encouragement and help during our calculations and for critical reading of the manuscript.

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