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The proton spin at high energies
Volume 227, number 3,4
PHYSICS LETTERS B
31 August 1989
THE PROTON SPIN AT HIGH ENERGIES B. LAMPE
CERN. CH-1211Geneva23. Switzerland Received 9 May 1989
It is conceivable that a large part of the proton spin is carried by gluons. We show how the gluon component can be determined by a measurement of the polarized structure functions at very high energies.
Recently, the European Muon Collaboration has measured the polarized proton structure function g~ using a polarized muon beam on a polarized target. They found [ 1 ] I
~dxg~ (x, Q2) = 0 . 1 1 4 _+0.012+0.026
( 1)
0
at Q 2 = 10.7 GeV 2. In the naive parton model the first m o m e n t ( g l ) ofg~ is given by [2] 1
(g,)=
f
dxg~(x)=½ ~
Q~(Aq)
(2)
q=u
0
where
Qq is
the electric charge o f the quark q.
Aq=q+--q_ is the difference of the densities of quarks with helicities + ½ and - ½ in a proton with helicity + ½ and ( A q ) is the first m o m e n t of Aq. This naive parton result can in principle get corrections from several sources. There may be a sea quark contribution [2], a gluon contribution [ 3 - 5 ] and a contribution from the orbital angular m o m e n t u m [ 6 ] to the spin of the proton. The effect o f the sea quarks can be included by extending the sum in eq. (2) to nr excited flavours. Possible effects of the orbital angular m o m e n t u m have been studied in the framework of generalized Skyrme models [ 7 ]. In this letter we will concentrate on the gluon contribution and on the question of how it can be determined by a measurement at high energies. The gluon helicity content Ag in the polarized pro0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
ton is given by the difference of the gluon densities with helicities + 1 and - 1. It can be shown that to leading order the product o%(Ag) is scale-independent [ 8 ], which means that it does not decrease with increasing energy like a typical O (c~s) contribution. This has led the authors of ref. [ 3 ] to conjecture that o%(Ag) is large ( o f order one), rather than of order c~s. In this case, a consistent lowest-order calculation should contain the contributions from the diagram in fig. 1, which are of order o~s.Ag, whereas contributions of order a~Aq, o~Ag, etc., can be safely neglected. One can calculate the contribution from fig. 1 to the first m o m e n t of gl [3-5,9]. The net outcome is that the first moments ( A q ) should be replaced by the differences (Aq-- (O~s/2n)Ag). It is conceivable that the bulk of the proton spin is carried by the valence quarks and the gluon. However, it is clear that a measurement of ( g l ) does not give sufficient information to check this possibility or to determine ( A g ) separately ~. The main aim of this letter is to ~' One may get some information by sweepingover a heavy quark threshold [4], or from jet production in polarized deep inelastic scattering [ 5 ].
+ crossed q(p')
g(p) Fig. 1. 469
Volume 227, number 3,4
v(pd
PHYSICS LETTERS B
~-
=
e(p.z)
lowing decomposition into three structure functions gJ, g3 and g4:
pv t W*lq) p(P)
~
~p)
;
31 August 1989
u(p')
H , , , = - i e ( / ~ v P q ) gpqJ + ( - g , , . +
q"q"'~ q2 ] g3
Fig. 2. + (P,, -- Pq show that by going to HERA energies and including weak effects, (Ag> can be uniquely determined. Before I discuss ep scattering, let me say a few words about neutrino proton scattering, because there the situation is very simple. In fig. 2, I have drawn the relevant lowest order process and defined my notation for the momenta. The process can be described in terms of an effective interaction between a lefthanded lepton and a left-handed quark current ~.~= G_~j,,j~, ,/2
(3) '
Z, =~7~,( 1 - 7s) u , L , = u y ~ , ( 1 - T s ) d .
(4)
At the high energies in which we are interested we can neglect all fermion masses. Therefore the only mass in the problem is the W mass mw. The coupling in eq. (3) is the Q2-dependent "Fermi coupling" G e2 1 , f 2 - 8 sin20w Q 2 + m w "
(5)
The cross-section 2rcG 2 d3p2 da= ~ L,,. W " " (2x)3p o
(6)
factorizes into a lepton tensor L,,. ={p, P2 },,u - i e (/zup, P2),
(7)
{Pl P2 }i,.: = PI I, P2. + P2I, Pl . - gt,. Pl P2
(8)
~-q,)
(10)
Formally this looks identical to the decomposition of the unpolarized hadron tensor into the well-known structure functions Fb F: and F3. For simple photon exchange where one has no Y5coupling, the structure functions g3 and g4 are zero. However, for the W exchange and the neutrino scattering of fig. 2 one gets gl = A d ,
(lla)
g3=--Ad.
(llb)
It is clear that in antineutrino scattering one would measure Au instead of Ad. One can get g4 from g3 by a Callan-Gross-like relation [ 11 ] g4 = 2 X g 3 ,
(12)
which means that, instead of the two independent tensors -g~,.+ (qj,q.)/q2 and [ P u - (Pq/q2)qu] X [p_(pq/q2)q.], only their linear combination {Pp' }~,. appears. We will see that the gluon contributions respect this relation. This is in contrast to the unpolarized case, where the gluons are well known to leads to deviations from the Callan-Gross relation [12]. According to the Altarelli-Ross presumption, eq. ( 11 ) has to be modified by adding the gluon contributions drawn in fig. 3, where also the particle momenta and indices are defined. The quark trace which enter the calculation is a sum of two terms, one with and one without a Ys:
and a hadron tensor W~,., which in general depends
on eight structurefunctions [ 10]. Here we are interested only in that part of the hadron tensor which de-
V(l~)
~
pends on the polarization vector s of the proton, and for simplicity we will assume a longitudinally polarized proton beam, s=2pP, with average helicity 2p. The hadron tensor is then of the form W,,, = W,,, (2p = 0 ) + 2 p H , , , .
(9)
The polarization part Hu~ of this tensor has the
fol-
470
• pv ~ ~'(ql ;
g(p) ~
eft)z) k
p'
~a Fig. 3.
÷ crossed
Volume 227, number 3,4
PHYSICS LETTERS B
31 August 1989
APqg=½(2z-1) F<5) = t r # ' ( ~ 5 ) \ ~
x,,( g \y~ 2 ~
2pk Y~'+YP
7,+,,~
Y~
7=) .
(13)
To get the contribution proportional to Ag one has to calculate the difference of the cross-sections a(Wg+ ) - a ( W g _ ) for a gluon with helicity + I and - i . Therefore one has to contract the traces (I 3) with
( g,e*) + - ( ~,,~*)_ ~ie(papq) /pq ,
(14)
where e + are the polarization vectors of the gluon for helicity + 1. The parton-level hadron tensor is then O~s n I .
h,,, ~ ~ TlE (papq) [F(#V, pa) +/'5 (/tv, pc) ]. (15) I have included a factor of nJ2 for the number of generations which may run in the quark loop of fig. 3. The product of the e-tensor and/'5 is symmetric in # and v. Therefore it can give contributions only to g3 and g4. Similarly, the product of the e-tensor and F gives the contribution to g~, which is known from the photon exchange calculations [3-5]. If one adds the two contributions, one gets
h,,~ - nl a, A P~ 2 2~r " x{_iE(ltvpq)[~+~l+p2Z2
1
]
~-)Pq --~
X
_ _ +p2
is the polarized splitting function [ 8 ]. To get the contributions to the first moments of the structure functions g~, g3 and g4, one has to integrate this expression with the measure f6 dzf~ dy. We take the calculation for g~ from ref. [4], and what we get is that the ( A d ) in the expression for the first moment ofgl is effectively replaced by (Ad-3Ag(as/ 2~r) ). The infra-red singularities cancel, essentially due to the antisymmetry of APqg under the exchange z,--, 1 - z . On the other hand, for g3 and g4, not only the infra-red singularities, but the whole contribution drops out due to the antisymmetry of the integrand under the exchange y,--,1-y. Therefore eqs. (1 lb) and (12) remain unchanged and the conclusion is that a measurement of ( g l ) and (g3) one can determine ( A g ) . A reasonable procedure is to use g3 or g4 to define the quark densities and to calculate ( A g ) from ( g l ) . The problem is that for small energies the crosssection (6) is of the order (QZ/m2)2, which is too small for the neutrino energies feasible in the near future. The situation is better for ep scattering where, as we shall see, the relevant terms are of the order Q2/m2. So let me say a few words about ep scattering. For longitudinally polarized electrons (with average helicity 2e), one can formally maintain the cross-section formula (6) and the general decomposition of the polarized hadron tensor (10) with the proviso that the structure functions will depend on the electron charges and degrees of polarization (but not on the electron momenta). One gets G
+2z(p,-Pqq,,)(p. -
--
(17)
e2
x/~ - Q2,
Pqq,,'~l q2 ]] +...
(18)
(
( g l ) = ½ q=l ~ bq A q - ~-~Ag °~s
)
,
19)
(16) Q2
where q=y-p2z2/Q2 and z = 1 -y-p2z2/Q2. Here y = ½( 1 - c o s 0) is a measure of the scattering angle 0 in the W - g centre-of-mass system and z = Q2/ 2pq is the parton level x. To regulate the infra-red singularities at y ~ 0 and y ~ 1 the gluon has been put offshell, p 2 ¢ 0 . The ellipses indicate irrelevant p2 contributions:
bq=Q2q)'e- 2Qqvq ( ae + ve'~'e) Q2 + m2z +(aZ+vZ)[2aeve+2e(a2e+V2)] (
02
~2
\ Q2 + m27.] '
(20)
Q2 q~ (g3) =
Q2Wm z
1 aqCq ( A q )
,
(21)
471
Volume 227, number 3,4
PHYSICS LETTERS B
1
fi,
o 0.06 O.Ot,
0.02 XQ=O
0 no giuon (b) -0.02 " t -
, "I ¢ ke=-"0h"6
A'
-0.04.
0
-0.04
-nOB
-0.06
-0.02
' 0.02
) 0.0~ I 0.06
Fig. 4. The first moments of the structure functions gl and g3 as a function of the electron degree of polarization 2e at x/Q 5 =60 GeV. ( a ) A u = 0 . 9 7 , Ad=-O.28, As=Ac=Ab=0, (as/2n)Ag= 0.23. (b) Aq=Aq(a)-0.23, Ag=0. Note that g~ is the same for both cases.
Cq= -Qq(ve +a~2c)
Q2
+vq(v~. +G2 +22~v~G) _Q 2 _+ m z2 ' g4=2×g3 •
(22) (23)
The weak charges vc, ao, vq and aq are defined with the normalization ae= (4 sin 0w cos 0w) - l [ 13 ]. We see that the effect of the Z in g~ is only to change the factor in front of the difference (Aq--Ag(OZs/21r) ). For (g3) and (g4) we again find no gluon contribution at all. To get a qualitative feeling of what the effect of the gluon may be, I have calculated the first moments of g~ and g3 for two cases: (a) with a large gluon component and (b) with a small gluon component and a large contribution from the sea quarks. It is clear from fig. 4 that that there is a large difference between the two cases for large enough electron polarization. One should note, however, that a large antiquark component would complicate the analysis. Finally, I would like to make some comments on the regularization scheme dependence of the result. It is in fact well known from unpolarized deep inelastic scattering that different regularization schemes lead to different QCD corrections to the structure functions. Only the difference between these corrections are scheme independent [ 14 ]. Physical results 472
31 August 1989
must be independent of the regularization scheme, and indeed only those difference are physically meaningful. One is free to redefine the quark densities in such a way that the very simple relation between the structure function 1:2 and these densities which holds in lowest order is maintained also in higher order. Now we come back to the polarized case. The null result which I have derived for g3 and g4 is in fact independent of the regularization scheme. On the other hand, the correction to gj is scheme dependent [ 5 ]. However, among all possible results the result of the off-shell regularization is preferred, because it is in agreement with what is dictated by the operator product expansion and the anomaly. We have seen that this result is finite. Therefore it is possible to attribute it solely to the gluon. Let me now come to the conclusions. We have seen how by a measurement of the polarized structure functions at high energies the gluon component of the proton spin can be determined. Since HERA is intended to work at some point with polarized jet targets, it is possible that this can be achieved in the near future. We have collected here only the leading contributions to the structure functions. It would be interesting to know also the terms of the form a2Ag and asAq and the scale corrections (anomalous dimensions) beyond the leading order [ 15 ]. I would like to thank G. Altarelli and W. Hollik for helpful discussions.
References [ 1 ] European Muon Collab., J. Ashman et al., Phys. Lett. B 206 (1988) 364. [2] See e.g.F.E. Close and R.G. Roberts, Phys. Rev. Lett. 60 ( 1988 ) 1471; F.E. Close, Introduction to quarks and patrons (New York, 1979). [ 3 ] G. Altarelli and G.G. Ross, Phys. Lett. B 212 ( 1988 ) 391. [4] G. Altarelli and W.J. Stirling, CERN preprint TH.5249/88 (1988). [ 5 ] R.D. Carlitz, J.C. Collins and A.H. Mueller, Phys. Lett. B 214 (1988) 229. [6] P.G. Rathcliffe, Phys. Lett. B 192 (1987) 180. [7] G. Cldment and J. Stern, CERN preprint TH.5216/88 (1988); J. Ellis and M. Karliner, CERN preprint TH.5095/88 (1988).
Volume 227, number 3,4
PHYSICS LETTERS B
[8] G. Altarelli and G. Parisi, Nucl. Phys. B 126 (1977) 298. [9] P.G. Rathcliffe, Nucl. Phys. B 233 (1983) 45. [ 10] E. Derman, Phys. Rev. D 7 (1973) 2755. [ 11 ] C.G. Callan and D.J. Gross, Phys. Rev. Lett. 22 (1969) 156; W. BuchmiJller, B. Lampe and N. Vlachos, Phys. Lett. B 197 (1987) 379.
31 August 1989
[ 12 ] G. Altarelli and G. Martinelli, Phys. Lett. B 76 ( 1978 ) 89. [ 13 ] J.E. Kim, P. Langacker, M. Levine and H.H. Williams, Rev. Mod. Phys. 53 (1981) 211. [14] B. Humpert and W.L. van Neerven, Nucl. Phys. B 178 (1981)498;B 184 (1981) 225. [ 15 ] M.A. Ahmed and G.G. Ross, Nucl. Phys. B 111 (1976) 441.
473
PHYSICS LETTERS B
31 August 1989
THE PROTON SPIN AT HIGH ENERGIES B. LAMPE
CERN. CH-1211Geneva23. Switzerland Received 9 May 1989
It is conceivable that a large part of the proton spin is carried by gluons. We show how the gluon component can be determined by a measurement of the polarized structure functions at very high energies.
Recently, the European Muon Collaboration has measured the polarized proton structure function g~ using a polarized muon beam on a polarized target. They found [ 1 ] I
~dxg~ (x, Q2) = 0 . 1 1 4 _+0.012+0.026
( 1)
0
at Q 2 = 10.7 GeV 2. In the naive parton model the first m o m e n t ( g l ) ofg~ is given by [2] 1
(g,)=
f
dxg~(x)=½ ~
Q~(Aq)
(2)
q=u
0
where
Qq is
the electric charge o f the quark q.
Aq=q+--q_ is the difference of the densities of quarks with helicities + ½ and - ½ in a proton with helicity + ½ and ( A q ) is the first m o m e n t of Aq. This naive parton result can in principle get corrections from several sources. There may be a sea quark contribution [2], a gluon contribution [ 3 - 5 ] and a contribution from the orbital angular m o m e n t u m [ 6 ] to the spin of the proton. The effect o f the sea quarks can be included by extending the sum in eq. (2) to nr excited flavours. Possible effects of the orbital angular m o m e n t u m have been studied in the framework of generalized Skyrme models [ 7 ]. In this letter we will concentrate on the gluon contribution and on the question of how it can be determined by a measurement at high energies. The gluon helicity content Ag in the polarized pro0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
ton is given by the difference of the gluon densities with helicities + 1 and - 1. It can be shown that to leading order the product o%(Ag) is scale-independent [ 8 ], which means that it does not decrease with increasing energy like a typical O (c~s) contribution. This has led the authors of ref. [ 3 ] to conjecture that o%(Ag) is large ( o f order one), rather than of order c~s. In this case, a consistent lowest-order calculation should contain the contributions from the diagram in fig. 1, which are of order o~s.Ag, whereas contributions of order a~Aq, o~Ag, etc., can be safely neglected. One can calculate the contribution from fig. 1 to the first m o m e n t of gl [3-5,9]. The net outcome is that the first moments ( A q ) should be replaced by the differences (Aq-- (O~s/2n)Ag). It is conceivable that the bulk of the proton spin is carried by the valence quarks and the gluon. However, it is clear that a measurement of ( g l ) does not give sufficient information to check this possibility or to determine ( A g ) separately ~. The main aim of this letter is to ~' One may get some information by sweepingover a heavy quark threshold [4], or from jet production in polarized deep inelastic scattering [ 5 ].
+ crossed q(p')
g(p) Fig. 1. 469
Volume 227, number 3,4
v(pd
PHYSICS LETTERS B
~-
=
e(p.z)
lowing decomposition into three structure functions gJ, g3 and g4:
pv t W*lq) p(P)
~
~p)
;
31 August 1989
u(p')
H , , , = - i e ( / ~ v P q ) gpqJ + ( - g , , . +
q"q"'~ q2 ] g3
Fig. 2. + (P,, -- Pq show that by going to HERA energies and including weak effects, (Ag> can be uniquely determined. Before I discuss ep scattering, let me say a few words about neutrino proton scattering, because there the situation is very simple. In fig. 2, I have drawn the relevant lowest order process and defined my notation for the momenta. The process can be described in terms of an effective interaction between a lefthanded lepton and a left-handed quark current ~.~= G_~j,,j~, ,/2
(3) '
Z, =~7~,( 1 - 7s) u , L , = u y ~ , ( 1 - T s ) d .
(4)
At the high energies in which we are interested we can neglect all fermion masses. Therefore the only mass in the problem is the W mass mw. The coupling in eq. (3) is the Q2-dependent "Fermi coupling" G e2 1 , f 2 - 8 sin20w Q 2 + m w "
(5)
The cross-section 2rcG 2 d3p2 da= ~ L,,. W " " (2x)3p o
(6)
factorizes into a lepton tensor L,,. ={p, P2 },,u - i e (/zup, P2),
(7)
{Pl P2 }i,.: = PI I, P2. + P2I, Pl . - gt,. Pl P2
(8)
~-q,)
(10)
Formally this looks identical to the decomposition of the unpolarized hadron tensor into the well-known structure functions Fb F: and F3. For simple photon exchange where one has no Y5coupling, the structure functions g3 and g4 are zero. However, for the W exchange and the neutrino scattering of fig. 2 one gets gl = A d ,
(lla)
g3=--Ad.
(llb)
It is clear that in antineutrino scattering one would measure Au instead of Ad. One can get g4 from g3 by a Callan-Gross-like relation [ 11 ] g4 = 2 X g 3 ,
(12)
which means that, instead of the two independent tensors -g~,.+ (qj,q.)/q2 and [ P u - (Pq/q2)qu] X [p_(pq/q2)q.], only their linear combination {Pp' }~,. appears. We will see that the gluon contributions respect this relation. This is in contrast to the unpolarized case, where the gluons are well known to leads to deviations from the Callan-Gross relation [12]. According to the Altarelli-Ross presumption, eq. ( 11 ) has to be modified by adding the gluon contributions drawn in fig. 3, where also the particle momenta and indices are defined. The quark trace which enter the calculation is a sum of two terms, one with and one without a Ys:
and a hadron tensor W~,., which in general depends
on eight structurefunctions [ 10]. Here we are interested only in that part of the hadron tensor which de-
V(l~)
~
pends on the polarization vector s of the proton, and for simplicity we will assume a longitudinally polarized proton beam, s=2pP, with average helicity 2p. The hadron tensor is then of the form W,,, = W,,, (2p = 0 ) + 2 p H , , , .
(9)
The polarization part Hu~ of this tensor has the
fol-
470
• pv ~ ~'(ql ;
g(p) ~
eft)z) k
p'
~a Fig. 3.
÷ crossed
Volume 227, number 3,4
PHYSICS LETTERS B
31 August 1989
APqg=½(2z-1) F<5) = t r # ' ( ~ 5 ) \ ~
x,,( g \y~ 2 ~
2pk Y~'+YP
7,+,,~
Y~
7=) .
(13)
To get the contribution proportional to Ag one has to calculate the difference of the cross-sections a(Wg+ ) - a ( W g _ ) for a gluon with helicity + I and - i . Therefore one has to contract the traces (I 3) with
( g,e*) + - ( ~,,~*)_ ~ie(papq) /pq ,
(14)
where e + are the polarization vectors of the gluon for helicity + 1. The parton-level hadron tensor is then O~s n I .
h,,, ~ ~ TlE (papq) [F(#V, pa) +/'5 (/tv, pc) ]. (15) I have included a factor of nJ2 for the number of generations which may run in the quark loop of fig. 3. The product of the e-tensor and/'5 is symmetric in # and v. Therefore it can give contributions only to g3 and g4. Similarly, the product of the e-tensor and F gives the contribution to g~, which is known from the photon exchange calculations [3-5]. If one adds the two contributions, one gets
h,,~ - nl a, A P~ 2 2~r " x{_iE(ltvpq)[~+~l+p2Z2
1
]
~-)Pq --~
X
_ _ +p2
is the polarized splitting function [ 8 ]. To get the contributions to the first moments of the structure functions g~, g3 and g4, one has to integrate this expression with the measure f6 dzf~ dy. We take the calculation for g~ from ref. [4], and what we get is that the ( A d ) in the expression for the first moment ofgl is effectively replaced by (Ad-3Ag(as/ 2~r) ). The infra-red singularities cancel, essentially due to the antisymmetry of APqg under the exchange z,--, 1 - z . On the other hand, for g3 and g4, not only the infra-red singularities, but the whole contribution drops out due to the antisymmetry of the integrand under the exchange y,--,1-y. Therefore eqs. (1 lb) and (12) remain unchanged and the conclusion is that a measurement of ( g l ) and (g3) one can determine ( A g ) . A reasonable procedure is to use g3 or g4 to define the quark densities and to calculate ( A g ) from ( g l ) . The problem is that for small energies the crosssection (6) is of the order (QZ/m2)2, which is too small for the neutrino energies feasible in the near future. The situation is better for ep scattering where, as we shall see, the relevant terms are of the order Q2/m2. So let me say a few words about ep scattering. For longitudinally polarized electrons (with average helicity 2e), one can formally maintain the cross-section formula (6) and the general decomposition of the polarized hadron tensor (10) with the proviso that the structure functions will depend on the electron charges and degrees of polarization (but not on the electron momenta). One gets G
+2z(p,-Pqq,,)(p. -
--
(17)
e2
x/~ - Q2,
Pqq,,'~l q2 ]] +...
(18)
(
( g l ) = ½ q=l ~ bq A q - ~-~Ag °~s
)
,
19)
(16) Q2
where q=y-p2z2/Q2 and z = 1 -y-p2z2/Q2. Here y = ½( 1 - c o s 0) is a measure of the scattering angle 0 in the W - g centre-of-mass system and z = Q2/ 2pq is the parton level x. To regulate the infra-red singularities at y ~ 0 and y ~ 1 the gluon has been put offshell, p 2 ¢ 0 . The ellipses indicate irrelevant p2 contributions:
bq=Q2q)'e- 2Qqvq ( ae + ve'~'e) Q2 + m2z +(aZ+vZ)[2aeve+2e(a2e+V2)] (
02
~2
\ Q2 + m27.] '
(20)
Q2 q~ (g3) =
Q2Wm z
1 aqCq ( A q )
,
(21)
471
Volume 227, number 3,4
PHYSICS LETTERS B
1
fi,
o 0.06 O.Ot,
0.02 XQ=O
0 no giuon (b) -0.02 " t -
, "I ¢ ke=-"0h"6
A'
-0.04.
0
-0.04
-nOB
-0.06
-0.02
' 0.02
) 0.0~ I 0.06
Fig. 4. The first moments of the structure functions gl and g3 as a function of the electron degree of polarization 2e at x/Q 5 =60 GeV. ( a ) A u = 0 . 9 7 , Ad=-O.28, As=Ac=Ab=0, (as/2n)Ag= 0.23. (b) Aq=Aq(a)-0.23, Ag=0. Note that g~ is the same for both cases.
Cq= -Qq(ve +a~2c)
Q2
+vq(v~. +G2 +22~v~G) _Q 2 _+ m z2 ' g4=2×g3 •
(22) (23)
The weak charges vc, ao, vq and aq are defined with the normalization ae= (4 sin 0w cos 0w) - l [ 13 ]. We see that the effect of the Z in g~ is only to change the factor in front of the difference (Aq--Ag(OZs/21r) ). For (g3) and (g4) we again find no gluon contribution at all. To get a qualitative feeling of what the effect of the gluon may be, I have calculated the first moments of g~ and g3 for two cases: (a) with a large gluon component and (b) with a small gluon component and a large contribution from the sea quarks. It is clear from fig. 4 that that there is a large difference between the two cases for large enough electron polarization. One should note, however, that a large antiquark component would complicate the analysis. Finally, I would like to make some comments on the regularization scheme dependence of the result. It is in fact well known from unpolarized deep inelastic scattering that different regularization schemes lead to different QCD corrections to the structure functions. Only the difference between these corrections are scheme independent [ 14 ]. Physical results 472
31 August 1989
must be independent of the regularization scheme, and indeed only those difference are physically meaningful. One is free to redefine the quark densities in such a way that the very simple relation between the structure function 1:2 and these densities which holds in lowest order is maintained also in higher order. Now we come back to the polarized case. The null result which I have derived for g3 and g4 is in fact independent of the regularization scheme. On the other hand, the correction to gj is scheme dependent [ 5 ]. However, among all possible results the result of the off-shell regularization is preferred, because it is in agreement with what is dictated by the operator product expansion and the anomaly. We have seen that this result is finite. Therefore it is possible to attribute it solely to the gluon. Let me now come to the conclusions. We have seen how by a measurement of the polarized structure functions at high energies the gluon component of the proton spin can be determined. Since HERA is intended to work at some point with polarized jet targets, it is possible that this can be achieved in the near future. We have collected here only the leading contributions to the structure functions. It would be interesting to know also the terms of the form a2Ag and asAq and the scale corrections (anomalous dimensions) beyond the leading order [ 15 ]. I would like to thank G. Altarelli and W. Hollik for helpful discussions.
References [ 1 ] European Muon Collab., J. Ashman et al., Phys. Lett. B 206 (1988) 364. [2] See e.g.F.E. Close and R.G. Roberts, Phys. Rev. Lett. 60 ( 1988 ) 1471; F.E. Close, Introduction to quarks and patrons (New York, 1979). [ 3 ] G. Altarelli and G.G. Ross, Phys. Lett. B 212 ( 1988 ) 391. [4] G. Altarelli and W.J. Stirling, CERN preprint TH.5249/88 (1988). [ 5 ] R.D. Carlitz, J.C. Collins and A.H. Mueller, Phys. Lett. B 214 (1988) 229. [6] P.G. Rathcliffe, Phys. Lett. B 192 (1987) 180. [7] G. Cldment and J. Stern, CERN preprint TH.5216/88 (1988); J. Ellis and M. Karliner, CERN preprint TH.5095/88 (1988).
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[8] G. Altarelli and G. Parisi, Nucl. Phys. B 126 (1977) 298. [9] P.G. Rathcliffe, Nucl. Phys. B 233 (1983) 45. [ 10] E. Derman, Phys. Rev. D 7 (1973) 2755. [ 11 ] C.G. Callan and D.J. Gross, Phys. Rev. Lett. 22 (1969) 156; W. BuchmiJller, B. Lampe and N. Vlachos, Phys. Lett. B 197 (1987) 379.
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[ 12 ] G. Altarelli and G. Martinelli, Phys. Lett. B 76 ( 1978 ) 89. [ 13 ] J.E. Kim, P. Langacker, M. Levine and H.H. Williams, Rev. Mod. Phys. 53 (1981) 211. [14] B. Humpert and W.L. van Neerven, Nucl. Phys. B 178 (1981)498;B 184 (1981) 225. [ 15 ] M.A. Ahmed and G.G. Ross, Nucl. Phys. B 111 (1976) 441.
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