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Gluon Virtuality and Heavy Sea Quark Contributions to the Spin-Dependent g


SLAC-PUB-8025 December 1998

Gluon Virtuality and Heavy Sea Quark Contributions to the Spin-Dependent g1 Structure Function

arXiv:hep-ph/9901244v1 7 Jan 1999

Steven D. Bass

a, ?

, Stanley J. Brodsky

b, ?

and Ivan Schmidt

c, ?

a

Max Planck Institut f¨ ur Kernphysik,

Postfach 103980, D-69029 Heidelberg, Germany
b

Stanford Linear Accelerator Center,

Stanford University, Stanford, California 94309, U.S.A.
c

Departmento de F? ?sica, Universidad T? ecnica Federico Santa Mar? ?a, Casilla 110-V, Valpara? ?so, Chile

Abstract We analyze the quark mass dependence of photon gluon fusion in polarized deep inelastic scattering for both the intrinsic and extrinsic gluon distributions of the nucleon. We calculate the e?ective number of ?avors for each of the heavy and light quark photon gluon fusion contributions to the ?rst moment of the spin-dependent structure function g1 (x).

?

Steven.Bass@mpi-hd.mpg.de; present address: Physik Department, Technische Universit¨ at M¨ unchen, D-85747 Garching, Germany. sjbth@slac.stanford.edu; work supported by the Department of Energy under contract number DE–AC03–76SF00515. ischmidt@?s.utfsm.cl; work supported by Fondecyt (Chile) under grant 1960536 and by a C? atedra Presidencial (Chile).

?

?

1

Introduction

One of the most interesting aspects of deep inelastic lepton-proton scattering is the
p contribution to the g1 spin-dependent structure function from photon-gluon fusion

subprocesses γ ? (q )g (p) → q q ?. Naively, one would expect zero contributions from light mass q q ? pairs to the ?rst moment
1 0 p dx g1 (x, Q2 ) since the q and q ? have opposite

helicities. In fact, this is not the case if the quark mass mq is small compared to a scale set by the spacelike gluon virtuality p2 . This is the origin of the so-called targets assuming three light ?avors. Here ?g is the helicity carried by gluons in the hadron target, ?g (Q) =
1 0
s anomalous correction ?3 α ?g [1]-[4] to the Ellis-Ja?e sum rule [5] for isospin-zero 2π

the language of the operator product expansion, the photon-gluon subprocess contrigraph [6, 7] contribution to the hadronic matrix element of the local axial current. distinct contributions to the ?rst moment of g1 in polarized deep inelastic scattering. For ?xed gluon virtuality P 2 = ?p2 the photon-gluon fusion process induces two

dx[g↑ (x, Q) ? g↓ (x, Q)], at the factorization scale Q. In

butions to the ?rst moment of g1 (x, Q) correspond to the anomalous VVA triangle

Let mq denote the mass of the struck sea quark. When Q2 is much greater than both
2 m2 q and P the box graph contribution to the ?rst moment of g1 for a gluon target is

[8]:
1 0

dxg1

γ?g

αs The ?rst, mass-independent term (? 2 ) in Eq. (1) comes from the region of phase π

2m2 αs ? q =? 1+ 2 2π P

?

1
2 1 + 4m2 q /P

ln ?

?

2 1 + 4m2 q /P ? 1 2 1 + 4m2 q /P + 1

?? ??

.

(1)

relative to the photon-gluon direction. It measures a contact photon-gluon interaction and is associated [3] with the axial anomaly [6, 7]. The second mass-dependent term comes from the region of phase space where the struck quark carries transverse
2 2 2 momentum kT ? P 2 , m2 q . This mass dependent term vanishes in the limit P ? mq
s and tends to + α when P 2 ? m2 q . The “soft” mass dependent term in Eq. (1) is 2π

2 space where the struck quark carries large transverse momentum squared kT ? Q2

associated with the quark parton distribution of the gluon ?q (gluon) ; it can safely be neglected for the light (up and down) quarks. On the other hand, the magnitude of the gluon virtuality is important for gauging the contribution of the massive sea quarks. If the sea quark mass is heavy compared
2 to the gluon virtuality 4m2 = ?p2 , the photon-gluon fusion contribution q ? P

2

to

1 0

dx g1 (x, Q2 ) vanishes to leading order in αs (Q2 ). This result follows from a

general theorem based on the Drell-Hearn-Gerasimov sum rule [9] which states that the logarithmic integral over the photoabsorption cross section
∞ νπ

dν ?σγa→bc (ν ) = 0(α3) ; ν

(2)

it vanishes at order α2 for any 2 → 2 Standard Model process [10, 11]. Here ?σ is present application, the gluon (for p2 = 0) takes the role of the on-shell photon γ

the cross section di?erence for parallel versus anti-parallel incident helicities. In the and the particle a can be taken as a real or virtual photon. As the photon virtuality

Q2 becomes large, the DHG integral evolves to the ?rst moment of the helicitydependent structure function g1 (x, Q2 ). Thus the fusion γ ? g → q q ? Born contribution to
1 0 2 2 dx g1 (x, Q2 ) vanishes for small gluon virtuality P 2 ? 4m2 q , P ? Q . Notice

that the Born photon-gluon fusion contribution to the Ellis-Ja?e moment is zero even for very light quarks as long as the gluon virtuality can be neglected.

The above application of the DHG theorem holds for any photon virtuality q 2 = ?Q2 , and is thus more general than leading twist [12]. In fact, the leading-order absorption cross sections vanishes even if Q2 < 4m2 q , as long as the gluon virtuality can be neglected. The result also holds for the weak as well as electromagnetic current probes [11, 13]. It is clearly important to ascertain the actual numerical contribution of heavy ? to the ?rst moment of g1 ; i.e., what is the e?ective number of sea quarks ss ?, cc ?, b? b, tt quark contributions to the Ellis-Ja?e moment? From the above discussion, the speci?c contribution of a given sea quark pair q q ? depends not only on Q2 , but more critically on the ratio of scales p2 /4m2 q . In a full QCD calculation of photon-gluon fusion contributions to the ?rst moment of g1 one needs to integrate over the distributions of extrinsic and intrinsic gluon virtualities in the target nucleon. For small gluon cancels with the “soft” mass dependent contribution. For deeply-virtual gluons the mass-independent anomaly contribution dominates over the mass-dependent term which tends to zero. Therefore, we shall investigate the e?ect of retaining the ?nite quark masses and performing a more exact analysis, in which we integrate over P 2 . Our aim is to understand the role of heavy quarks (e.g. strange and charm) in the 3
γ virtualities (P 2 ? m2 q ) the “hard” anomaly contribution to the ?rst moment of g1
?g

fusion contribution to the dν/ν moment of the di?erence of helicity-dependent photo-

photon-gluon fusion process. As we shall show in the next section, the exact form of the spectrum N (p2 ) of gluon virtuality in the target nucleon depends in detail on the physics of the nucleon wavefunction. “Extrinsic” gluon contributions, which arise from gluon bremmstrahlung qV → qV g of a valence quark, have a relatively hard virtuality dNext (p2 )/dp2 ? αs (p2 )/p2 , above a minimum virtuality p2 min . The mean virtuality of the extrinsic gluons depends on the upper limit of integration, which in turn depends on the kinematic phase space. On the other hand, intrinsic gluons, which are associated with the physics of the nucleon wavefunction (for example, gluons emitted by one valence quark and absorbed by another quark), have a relatively soft spectrum. We will characterize the shape of the intrinsic gluon virtuality by the convergent form dNint (p2 )/dp2 ? dNext (p2 )/dp2/[1 + p2 /M2 ], where M is a typical hadronic mass scale. We shall use such model forms for the extrinsic and intrinsic gluon distribug1 (x, Q2 ). In addition to the photon-gluon fusion contributions, additional contributions to the ?rst moment of g1 (x, Q2 ) arise from intrinsic heavy sea quarks associated with higher Fock states in the target hadron. For example, meson-baryon ?uctuations such [14]. In the case of charm, the small probability 0(1%) of intrinsic charm present in the proton implies a small intrinsic charm contribution to
1 0

tions to predict speci?c contributions of the heavy sea quarks to the ?rst moment of

as p → K Λ imply a negative intrinsic strange quark contribution to

1 0

dx g1 (x, Q2 )

dx g1 (x, Q2 ).

The charm contribution to the nucleon helicity-dependent structure functions and sum rules will be addressed by several new experiments. The COMPASS [15] and HERMES [16] experiments will measure charm production [17]-[25] in polarized deep inelastic scattering. Experiments have also been proposed at SLAC [26]. The aim of these experiments is to learn about the gluon polarization in a nucleon through the photon-gluon fusion process.

2

Polarized gluons and g1

In order to analyze the sensitivity of the anomaly to the sea quark mass in the photon-gluon fusion subprocesses, let us start by expressing the contributing gluon distributions in terms of the corresponding bound-state wavefunctions. In general 4

the Q2 dependence of the parton distributions comes from the integral of the boundstate wavefunction over the virtuality of the corresponding parton up to the scale Q2 . Schematically, for the polarized gluon distribution, we have △G(x, Q ) = which means that d2 ?Ng/p ? 2 2 2 2 2 ? G ( x, P ) = | Ψ ( P , x ) | ? | Ψ ( P , x ) | ≡ . g ↑/p↑ g ↓/p↑ ?P 2 dP 2 dx (4)
2 Q2

dP 2[|Ψg↑/p↑ (P 2 , x)|2 ? |Ψg↓/p↑ (P 2 , x)|2 ],

(3)

Here Ψg↑/p↑ and Ψg↓/p↑ are the gluon wavefunctions for positive and negative helicities relative to the proton helicity as functions of the gluon virtuality P 2 = ?p2 and the fraction x of the plus component of the target nucleon’s momentum. In perturbative QCD the total photon-gluon fusion contribution to g1 for a nucleon target is given by g1 (x, Q2 ) = where g1
(Gq ) (G)

1 2

e2 q g1
q

(Gq )

(x, Q2 ) ,

(5)

is the contribution where the struck quark carries ?avor q g1
(Gq )

(x, Q2 ) =

Q2
2 Pmin

dP 2

? (?G(x, P 2 )) ? Aq (x, Q2 , P 2) . 2 ?P

(6)

structure function g1 of a “gluon target” with virtuality P 2 , where the struck quark
2 carries ?avor q . The infra-red cut-o? Pmin is the minimum gluon virtuality at which

Here ? denotes the convolution over x and Aq denotes the contribution to the spin

we can apply perturbative QCD—that is, where the current-quark and gluon degrees of freedom in perturbative QCD give way to dynamical chiral symmetry breaking and con?nement. The GRV [27] and Bag model [28] analyses of deep inelastic structure functions involve taking a QCD-inspired model-input for the leading twist parton
2 distributions at some low scale ?2 0 , evolving the distributions to deep inelastic Q and

comparing with data. The optimal GRV and Bag model ?ts to deep inelastic data
2 are found with ?2 0 ? 0.2 - 0.3 GeV . Motivated by this phenomenological observation, 2 we shall set Pmin = 0.3GeV2 .

photon-gluon fusion. We can de?ne the “hard” part of Aq by imposing a cuto? on 5

In the Born approximation Aq is calculated from the box graph contribution to

2 the transverse momentum squared of the struck quark kT > λ2 [8]:

αs Aq |hard (x, Q , P , λ ) = ? 2π
2 2 2

1? 1? ln
2

2 4(m2 q +λ ) W2 4x2 P 2 Q2

(2x ? 1)(1 ?
4x2 P 2 Q2 4x2 P 2 Q2

2xP 2 ) Q2

(7)

1?

1 1?
2 4(m2 q +λ ) W2

1+ 1?

1? 1?

1? 1?

2 4(m2 q +λ ) W2 2 4(m2 q +λ ) W2 2

4x P 2xP 2 2 xP 2 2mq (1 ? Q2 ) ? P x(2x ? 1)(1 ? Q2 ) +(x ? 1 + 2 ) xP 2 4x2 P 2 2 2 Q (m2 q + λ )(1 ? Q2 ) ? P x(x ? 1 + Q2 )

1?

4x2 P 2 Q2
2

.

x Here mq is the fermion mass, x is the Bjorken variable and W 2 = Q2 ( 1? ) ? P 2 is x

the center of mass energy for the photon-gluon collision. The running coupling, αs ,

in Eq. (7) is evaluated at the scale P 2 . Following Parisi and Petronzio [29] we shall use a modi?ed running αs (P 2 )—see Eq. (15) below—which freezes in the infrared, to describe Aq when P 2 becomes small. Keeping contributions where the struck quark carries transverse momentum squared
(Gq )

2 kT ≥ λ2 , the photon-gluon fusion contribution to the ?rst moment of g1

is obtained (8)

from (6):

Γq (Q2 , λ2 ) = Here

Q2
2 Pmin

dP 2 Iq (P 2 , λ2 )
xmax 0

d?Ng/p 2 (P ) . dP 2

Iq (P 2 , λ2 ) = and

dx Aq (x, Q2 , P 2, λ2 )

(9)

The cuto?s xmax

zmax (P 2 ) d?Ng/p 2 d2 ?Ng/p dz ( P ) = (z, P 2 ) . (10) dP 2 dP 2dz 0 and zmax in Eqs. (9,10) come from the kinematics. For the box

from the phase space factor

2 graph term Aq , the cuto? xmax = Q2 /(Q2 + P 2 + 4(m2 q + λ )) in Eq. (9) is obtained

derived from the explicit form of the polarized gluon distribution—see below. In the rest of this Section we discuss the contribution of q q ? pairs with small transverse momentum (when we relax the λ2 cut-o?), the size of higher-twist contributions to the ?rst moment of g1 the ?rst moment. When we integrate over the full range of possible impact parameters we need to
2 include small values of kT in Eqs. (8–10). This necessarily involves extrapolating (Gq )

1?

2 4(m2 q +λ ) W2

in Eq. (8). The cuto? zmax in Eq. (10) is

, and the jet signature of the di?erent contributions to

6

the calculation into the domain of non-perturbative QCD. Shore and Veneziano [30] have considered the analogous process of the spin structure function of the polarized photon for a virtual photon target. They argue that the target photon virtuality where
1 0 γ N depends on the realization of dxg1 grows from zero (at P 2 = 0) to ? α π c 1 0 γ dxg1 (x, Q2 ) grow rapidly from zero when P 2 ? 4m2 q . In full QCD,

chiral symmetry breaking in QCD. In perturbative QCD the individual quark ?avor contributions to

spontaneous chiral symmetry breaking means that the scale of the transition virtuality is set by the constituent quark mass rather than by the current quark mass — that is, we expect
1 0 γ dxg1 (x, Q2 ) to grow rapidly from zero when P 2 ? m2 ρ . Motivated by this

result, one might expect the gluon-virtuality where Iq grows rapidly to depend on
2 any possible diquark structure of the gluon at low kT . Spontaneous chiral symmetry

breaking is a considerably more dramatic e?ect in the light quark masses than it is in the heavy quark masses. Thus we can expect that perturbative QCD will provide a reasonable model-independent estimate of the heavy-quark Aq when λ2 becomes small. When Q2 → ∞ the expression for Aq simpli?es to the leading twist (=2) contriAq (x, Q2 , P 2, λ2 ) = αs Q2 1?x (2x ? 1) ln 2 + ln ?1 2π λ x +(2x ? 1) ln λ2 2 x(1 ? x)P 2 + (m2 q +λ ) (11)

bution:

2 2m2 q ? P x(2x ? 1) +(1 ? x) 2 x(1 ? x)P 2 + m2 q +λ

which has the ?rst moment 2m2 αs q Iq (P 2 , λ2 ) = ? ?1 + 2 2π P
?

1
2 2 1 + 4(m2 q + λ )/P

ln ?

?

2 2 1 + 4(m2 q + λ )/P ? 1 2 2 1 + 4(m2 q + λ )/P + 1

??

?? .

(12)

For ?nite quark masses, the cuto? xmax protects Aq from reaching the ln(1 ? x) di?erent values of P 2 and λ2 . The “Q2 = ∞” values are obtained by keeping only cut-o? itself acts as a major source of higher-twist. When we relax the cut-o? by setting λ2 to zero we ?nd a large ? 63% higher twist suppression of Ic at Q2 = 10 7

singularity in Eq. (8). To quantify this e?ect, in Table 1 we list the values of Iq for the leading twist contribution, Eq. (13). For the “hard” cut-o? λ2 = 1 GeV2 the

(P ) Table 1: Heavy quark e?ects in Iq (in units of ? αs2 ) π

2

P2

λ2

Q2 10.0 ∞

light 0.77 0.98 1.00 0.99 1.00 1.00

strange 0.74 0.94 0.96 0.61 0.63 0.63

charm 0.16 0.30 0.32 0.013 0.033 0.035

0.5 1.0 0.5 1.0 0.5 0.0 0.5 0.0

0.5 1.0 100.0 10.0 ∞

0.5 0.0 100.0

GeV2 . Ste?ens and Thomas [24] have observed that Ic is, to good approximation, independent of P 2 for values of P 2 between zero and 1 GeV2 . The rise in Ic with increasing λ2 corresponds to removing a greater amount of the mass dependent term in Eq. (1) which cancels against the mass-independent (anomaly) term which is
2 associated with kT ? Q2 in the limit Q2 → ∞. When we increase the cut-o? λ2 on

the transverse momentum squared of the struck quark we increase the infrared cut-o?

on the invariant mass of the qq pairs produced by the photon-gluon fusion. Quark mass dependent contributions to Iq go to zero when we increase the invariant mass suppressed below unity with the λ2 = 1 GeV2 cut-o? at Q2 = 10 GeV2 . If we 0.96 (λ2 = 0.1 GeV2 ) when we use our modi?ed αs (—see Eq. (15) below) together with the current light-quark mass through-out.
γ g Eq. (1), the “hard” anomaly contribution to Iq (the ?rst moment of g1 ) cancels with
?

Mqq much greater than 4m2 q . Note that the light-quark Il in Table 1 is signi?cantly

decrease the cut-o? this Il grows to 0.87 (λ2 = 0.5 GeV2 ), 0.91 (λ2 = 0.3 GeV2 ) and

2 Consider the large Q2 limit (Q2 ? 4m2 q ). When we set λ = 0 in Eq. (12) to obtain

the “soft” mass dependent contribution for small gluon virtualities (P 2 ? m2 q ). For deeply virtual gluons the mass-independent anomaly contribution dominates over the mass-dependent term which tends to zero in the limit P 2 ? 4m2 q . It is interesting to observe that in a semi-inclusive experiment one can in principle identify events which 8

correspond speci?cally to the contributions to the ?rst moment of g1 (x). These are events with three jets recoiling, taking up the large momentum qT transferred by the plus the quark that emitted the gluon, with mass M and transverse momentum pT . only the q q ? pair recoils produce no contribution to the ?rst moment of g1 (x). This corresponds to events with small transverse momentum of the quark that emitted moment of g1 (x) for heavy quark production implies that there must be a polarization asymmetry zero, which in principle can be measured experimentally [31].
2 2 2 the gluon p2 T ? mq , or with gluon virtuality ?p ? mq . The vanishing of the ?rst

lepton. The ?nal state consists of the q q ? pair with perpendicular momentum qT ? pT

These events have gluon virtuality ?p2 ≥ m2 q . As shown in [31], the events where

3

Extrinsic and intrinsic glue

Having established the theoretical framework, we now investigate photon-gluon fusion using simple models for the exclusive and inclusive gluon distributions of the nucleon. The “extrinsic” glue consists of gluons which are radiatively generated from individual valence quarks in the target whereas the “intrinsic” glue is associated with gluon exchange between valence quarks. For example, consider a gluon which is exchanged between two valence quarks in the proton. In constituent quark models these “gluon exchange currents” contribute to the proton-Delta mass di?erence [32]. They also renormalize the valence contributions to the nucleon’s axial charges, which are measured in β ?decays and in the ?rst moment of g1 (x, Q2 ). When cut, the exchanged when cut gives an extrinsic gluon. We now estimate the size of the extrinsic and intrinsic gluon contributions to the ?rst moment of g1 . We calculate the ratio ne? = Γh /Γl of the heavy to light quark contributions to photon gluon fusion for both the extrinsic and intrinsic glue. The extrinsic and intrinsic gluon distributions are dominated by gluons with small virtuality. The virtuality distribution
dNg/p dP 2

gluon gives an intrinsic gluon. A gluon which contributes to the quark self energy

for the extrinsic glue contains a logarith-

mic tale extending to the kinematic limits. Phenomenologically [33], the momentum distribution of the intrinsic glue is found to be weighted by the factor W (P 2 ) = Ng 9 1 P2 1+ M 2 (13)

GeV2 , the mass scale of the dipole ?t to the proton form factor [33]; Ng is a model dependent normalization constant. We start with a simple model for the gluon distributions taking into account their

relative to the extrinsic glue. The mass parameter M2 can be estimated as 0.71

correct P 2 distribution. Model dependent normalization uncertainties cancel in the ratios Γh /Γl . We treat the nucleon target as a three-quark system where the target quarks are treated as “elementary” with constituent quark mass M equal to 300 MeV. The polarized extrinsic gluon distribution is given by d2 ?Ng/p αs (z, P 2 ) = N CF 2 dP dz 2π
2 4 1 (1 ? (1 ? z )2 )p2 T +M z (1 ? z ) 2 2 2 z (p 2 T +M z )

(14)

P 2 in Eq. (8). The QCD factor CF = 4 ; N ? 0.6 [34, 35, 36] is the spin depolar3

where the (1 ? z ) factor is a Jacobian factor for the change of variables from p2 T to

ization factor found in relativistic quark models, which parametrizes the transfer of the proton’s angular momentum from intrinsic spin of the quarks to orbital angular momentum through relativistic e?ects and quark-pion coupling. Since the gluon

2 2 2 transverse momentum squared p2 T = P (1 ? z ) ? M z is non-negative, we obtain the √ zmax cuto? in Eq. (10): zmax (P 2 ) = (?1 + 1 + 4C )/2C , where C = M 2 /P 2 .

behavior predicted by color coherence [37]. By construction, it also exhibits the large x behavior associated with an elementary quark target. In a more sophisticated model one should also include gluon exchange between the valence quarks in addition to the present paper. gluon involved in the γ ? g → qq process. However, this is beyond the scope of the We work in the analytically extended [38] αV scheme [39]. This means that we use the running coupling αs (P 2) = 4π β0 ln(
P 2 +4m2 g ) Λ2 V

Our simple model, Eq. (14), for the gluon distributions exhibits the x → 0

,

(15)

2 in Eqs. (7,14). Here m2 g = 0.2GeV , ΛV = 0.16GeV and the number of ?avors

which contribute to β0 is taken as a continuous variable which depends on P 2 [38]: β0 = (11 ?
2 3 4 i=1

Ni ) where Ni ? 1 + 5 ρi
?1

, 10

(ρi = P 2 /m2 i) .

(16)

10

0

(a)

10

–1

10

–2

10
Neff for Extrinsic Glue

–3

(b)

10

–1

10

–2

10

–3

(c)

10

–1

10

–2

10
12–98 8460A3

–3

0

20

40 60 2 Q (GeV2)

80

100

Figure 1: The e?ective number of ?avors ne? for heavy sea quarks ss ?, cc ?, and b? b contributing to the ?rst moment of g1 (x, Q2 ), arising from γ ? -(extrinsic gluon) fusion, as a function of momentum transfers Q2 < 100GeV2 . In Fig. 1a, the cuto? on quark
2 transverse momentum kT > λ2 is set equal to zero. In Figs. 1b and 1c, λ2 =1 GeV2

and λ2 = 10 GeV2 , respectively.

11

Table 2: The e?ective number of heavy-?avors ne? = Γh /Γl .

Q2

λ2

ns

nc

nb

Extrinsic glue 10.0 10.0 10.0 100.0 100.0 100.0 Intrinsic glue 10.0 10.0 10.0 100.0 100.0 100.0 0.0 1.0 0.0 1.0 0.71 0.07 0.004 0.97 0.33 0.034 0.23 0.71 0.08 0.005 0.97 0.33 0.035 0.23 0.0 1.0 0.0 1.0 0.78 0.12 0.007 0.97 0.35 0.037 0.23 0.82 0.21 0.023 0.97 0.41 0.052 0.24

10.0 1.00 0.78

10.0 1.00 0.79

10.0 1.00 0.78

10.0 1.00 0.78

In Figs. 1a–1c we show the e?ective number of ?avors, ne? = Γh /Γl , for the heavy ?avor ss ?, cc ? and b? b production contributions to the ?rst moment of g1 (x, Q2 ) from γ ? -(extrinsic gluon) fusion up to Q2 = 100GeV2 . Figure 1a is obtained by setting λ2 equal to zero. (In this calculation we have used “modi?ed” αs , Eq. (15), together with the current light-quark mass through-out.) In Figs. 1b and 1c we set the cut-o? λ2 equal to 1 GeV2 and 10 GeV2 respectively. We repeat these calculations for γ ? (intrinsic gluon) fusion in Figs.2a–2c. The results in Figs. 1 and 2 are summarized in Table 2. The e?ective number of ?avors ne? = Γh /Γl increases for the heavy quarks when we increase the cuto? λ2 on the transverse momentum squared of the struck quark. 12

10

0

(a)

10

–1

10

–2

10 N eff for Intrinsic Glue

–3

(b)

10

–1

10

–2

10

–3

(c)

10

–1

10

–2

10
12–98 8460A4

–3

0

20

40 60 2 Q (GeV2)

80

100

Figure 2: The e?ective number of ?avors ne? for heavy sea quarks ss ?, cc ?, and b? b contributing to the ?rst moment of g1 (x, Q2 ), arising from γ ? -(intrinsic gluon) fusion.
2 In Fig. 2a, the cuto? on quark tranverse momentum kT > λ2 is set equal to zero. In

Figs. 2b and 2c, λ2 =1 GeV2 and λ2 = 10 GeV2 , respectively.

13

– (Extrinsic Glue integrand)

(a)

10–2

10–3

– (Intrinsic Glue integrand)

– 10 1

(b)

– 10 3

– 10 5

0
1–99 8460A5

20

40

60

80

100

P2

(GeV2)

Figure 3: The integrand in Eq. (8) as a function of P 2 for the light and charm quark contributions at Q2 = 100 GeV2 for extrinsic glue. Figure 3b shows the corresponding integrand for intrinsic glue. This corresponds to removing a greater amount of the mass dependent term in Eq.
2 (1) which cancels against the mass-independent (anomaly) term from kT ? Q2 in the

limit Q2 → ∞. By increasing λ2 we are increasing the cut on the invariant mass of the qq pairs produced by the photon-gluon fusion. Quark-mass dependent terms become less important when the invariant mass Mqq becomes much greater than 4m2 q.

In Fig. 3a we show the integrand in Eq. (8) as a function of P 2 for the light

and charm quark contributions at Q2 = 100 GeV2 with the extrinsic glue. Figure 3b shows the corresponding integrand for the intrinsic glue. Figures 3a and 3b both involve λ2 = 0. Note that the heavy and light quark curves come closer together with increasing P 2 . This result corresponds to the fact that the mass-dependent term in Eq. (1) tends to zero in the limit P 2 ? 4m2 q. 14

4

Phenomenology and discussion
(0)

Gluon polarization o?ers a possible explanation for the small value of gA (the three?avor, singlet axial charge) extracted from polarized deep inelastic scattering [40, 41, 42, 43]: gA ? 0.2 ? 0.35.
(0) (0) (0)

(17)

Relativistic binding [36] and constituent-quark pion coupling [34, 35] models predict models gA is interpreted as the fraction of the nucleon’s helicity which is carried by its quark constituents. In QCD the axial anomaly [6, 7] induces various gluonic contributions to gA . One ?nds [1, 2, 3, 44] gA =
q (0) (0)

gA ? 0.6 — a factor of two larger than the measured gA . In these semi-classical
(0)

?q ? 3

αs ?g 2π

partons

+ C.

(18)

?q and ?g are the amount of spin carried by quark and gluon partons in the Here 1 2 local γ ? g interaction in Eq. (1) assuming three light ?avors. The soft mass-dependent
αs polarized proton. The ?3 2 ?gpartons term is associated with the mass-independent, π

contributions to photon-gluon fusion are included in ?qpartons . The last term, C ,

is associated with non-trivial gluon topology [44] and a possible δ (x) term in g1 . It is missed by polarized deep inelastic scattering experiments which measure the combination (gA ? C ).
(0)

How large are the photon-gluon fusion sea-quark contributions Γq to gA if we Since extrinsic glue is radiatively generated from single quark lines in the target,

(0)

allow for ?nite sea-quark masses and a spectrum of gluon virtuality? we believe that our model should provide a good order-of-magnitude estimate for the normalization of the extrinsic Γq . We ?nd Γext = ?0.0024 and (Γu + Γd + Γs )ext = c ?0.033 at Q2 = 100 GeV2 for the extrinsic glue. The magnitude of the intrinsic gluon contribution to gA depends on the normal(0)

ization of the gluon polarization. Taking the estimate ?gintrinsic = +0.5 at 1 GeV2
int [37], we obtain Γint = ?0.07 when Q2 = 100 GeV2 . c = ?0.0020 and (Γu + Γd + Γs )

It is interesting to compare these estimates of Γc with the results of heavy-quark

e?ective theory [21] and heavy-quark operator product expansion [45] calculations. These calculations express the total heavy-quark contribution to the ?rst moment of g1 in terms of the three-light-?avor singlet axial-charge gA . Using the most recent 15
(0)

value, Eq. (17), of gA , Manohar’s e?ective theory calculation of the heavy-charmquark axial-charge becomes gA
(charm)

(0)

(Q2 >> m2 c ) = ?0.0055 ± 0.0018 + O (1/mc ) .

(19)

Manohar gives an estimate ? 0.003 (magnitude) for the O (1/mc ) corrections.

is in good agreement with Eq. (19). However, with the same gluonic input, photongA extracted from polarized deep inelastic scattering and the quark model prediction.
(0)

int The sum of our extrinsic and intrinsic charm contributions (Γext c + Γc = ?0.0044)

gluon fusion can account for only about one-third of the di?erence between the value of Next-to-leading order QCD ?ts to the present world data for g1 are consistent with a value of ?g between zero and +2 at 1 GeV2 [46]. The value ?g = 2 would increase our estimate of the intrinsic gluon contribution by a factor of 4 and would bring theory into agreement with the empirical determinations of gA . We look forward to a more precise measurement of ?g from forthcoming experiments on open charm production. Finally, it is interesting to note that since the contributions due to heavy sea quarks come from highly virtual gluons, one expects minimal nuclear shadowing for
N their contribution to the ?rst moment of g1 . (0)

Acknowledgments
We thank R. J. Crewther, A. H. Mueller, F. M. Ste?ens, and A. W. Thomas for helpful conversations. This work was supported in part by the United States Department of Energy under contract number DE–AC03–76SF00515, by Fondecyt (Chile) under grant 1960536, and by a C? atedra Presidencial (Chile).

16

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