The spin structure of the nucleon

Nuclear Physics B (Proc. Suppl.) 74 (1999) 142-148

The Spin Structure

PROCEEDINGS SUPPLEMENTS

of the Nucleon

J.-M. Le GOP, on behalf of the SMC “CEA Saclay, 91191 Gif-stir-Yvette,

Prance, E-mail: [email protected]

We present the final results of the spin asymmetries A1 and the spin structure functions g1 of the proton and the deuteron from the SMC. In the region I < 0.02 we use a new method for the determination of Al. The usual method employs inclusive scattering events and includes a large radiative background at low z, while the new method minimises the radiative background by selecting events with at least one hadron as well as a muon in the Snal state. Using these data and other presently available data we performed a next-to-leading order QCD analysis of the spin structure function 91. We present results for the first moments of the proton, deuteron and neutron structure functions, and determine singlet and non-singlet parton distributions. We also test the Bjorken sum rule and find agreement with the theoretical prediction at the level of 10%.

1. INTRODUCTION

from the spin of the gluons, Ag, and from the orbital momentum, L,,

A lot of our understanding of hadron structure comes through the Quark Model which relies on SU(3) flavour and SU(2) spin symmetries. Of course, we now believe that Quantum ChromoDynamics (QCD) is the proper theory to describe the strong interaction. However we are still not able to solve the equations of QCD in the general case. Models are then still very useful and it is important to understand the relations between QCD and the models, and in particular the Quark Model. In this framework, the spin of the nucleon appears as a very interesting issue. In the Quark Model the spin of the nucleon is obtained from the spin of the constituent quarks, whereas in QCD the question of what is carrying the nucleon spin is still an open question. The Quark Model, through its use of SU(2) spin symmetry, relies explicitely on the spin being carried by the quark. However, the results of the EMC experiment [l] were interpreted as indicating that the contribution of the spin of the quark to the spin of the nucleon is close to zero. If this is indeed the case, how is it that the Quark Model is so successful ?

The contribution from the spin of the quarks can be further decomposed according to the different flavours, AX = Au + Ad + As. In the simplest version of the Quark Model, AX = 1. However when relativistic corrections are included it is brought down to AE kc 0.75. On the other hand the combination as = Au + Ad 2As can be expressed in terms of the hyperons p decay constants F and D. If we believe that there is little strangeness in the nucleon and that the strange sea should not be polarised we may assume that As = 0. In this case AC = us = 3F - D, which experimentally is found to be 0.58. Before the publication of the polarised deep inelastic scattering (DIS) data of the EMC, AE was then expected to be large. It was known that gluons were contributing, for instance, to half of the nucleon momentum. However all the gluons that appear in the nucleon contribute to its momentum while they were expected to appear with either orientations of their spin and then not to contribute to the spin of the nucleon.

2. The Spin of the Nucleon

3. Polarised DIS and QCD

The spin of the nucleon can be decomposed in contributions from the spin of the quarks, AC,

Polarised DIS experiments measure the spin structure function gi . In the Quark Parton Model

0920-5632/99/$ - see front matter 0 1999 Elsevier Science B.V. PII SO920-5632(99)00151-6

1

1

z = zAC+Ag+L,.

All rights reserved.

(1)

J-M. Le Gaff/Nuclear Physics B (Proc. Suppl.) 74 (1999) 142-148

(QPM) g1 can be written in terms of the polarised distribution functions weighted by the square of the charge of the corresponding flavour, ’ [!A+)

g1(4=2

9

+ $A+)

,

+ ;A.,.,]

(2)

where AU(Z) = ZJ~- & + tif - til represents the number of quarks of flavour ‘1~,with a fraction 2 of the nucleon momentum and a spin parallel to that of the nucleon, minus that with the spin antiparallel. Then Au = [Au(z) is indeed the contribution of flavour u to the spin of the nucleon. In QCD however, the parton distribution functions depend both on x and Q2. The Q2 dependence is predicted by the DGLAP equations [2] which involve separately the singlet contribution AX = Au + Ad + As and non-singlet contributions like Au - Ad or Au + Ad - 2As: d

dQ2) = TP,,

-AqNS dlnQ2

NS

@AqNS,

143

where the higher order QCD corrections are not detailed. This relation is very similar to that of the &PM (Eq. 2) but it involves the up instead of the Aq. Naively one would indeed identify aq with Aq, and the flavour singlet matrix element a0 = a, + ad + a, with AE. Experimentally a0 is found to be small and a, to be negative which was often advertised as the “spin crisis”: the contribution of the spin of the quark to the spin of the nucleon seems to be small and that of the strange sea to be negative ! However in QCD the parton distributions are only defined in a given factorisation scheme (the way one splits between the soft and the hard part of the interaction), while the matrix elements and g1 are measurable quantities which are therefore scheme independent. The relation between a0 and AX then depends on the chosen scheme. In the so called MS scheme a0 = AC but in the Adler Bardeen (AB) scheme [3] a0 receives a gluonic contribution through the axial anomaly of &CD. a0 = ACAB - nf zAsAi+

(3) where the Pij are the splitting functions, nj the number of active flavours and @ denotes the convolution product. The structure function g1 can then be written as a convolution of the parton distribution functions with the so-called coefficient functions,

+$-

‘8 bvs]

(4

The formalism of the operator product expansion (OPE) allows to relate the first moment I?1 (Q2) = Ji g1 (x, Q2)dx to the axial matrix elements a, which are obtained from the mean value of the axial current over the nucleon wave function as sfiaq =< NJ@ypysqlN > :

I?; = f

(ia”

+

iad+ia8)

(1 +a,.

. .)

(5)

(6)

The AB scheme is actually designed in such a way that ACAB is independent of Q2. One can then consider ACAB as the intrinsic contribution of the spin of the quark, the one that can be compared to the &PM expectation. If Ag is large enough the experimental results (UO small and a, < 0) could then be reconciliated with the &PM expectations (AX large and As zero). Using the OPE results one can also derive sum rules. In the framework of SU(2) isospin symmetry the isovector combination I’: - I’;l can only involve the isovector combination of matrix elements u3 = a, - Ud. Neutron beta decay provides a3 = gA/gv which gives the Bjorken sum rule [4], r;

- r;

1 gA = --cN& 6sv

where CNS is the non-singlet should be stressed that this is diction of &CD. If we now use SU(3) fl avour aa = a, + ad - 2a, = 3F -

QCD correction. It a fundamental presymmetry D which

we have together

I-M. Le Gof/Nuclear Physics B (Pmt. Suppl.) 74 (1999) 142-148

144

Absorber

Targets Beam Definition

Spectrometer

F’OD

F’OE

muon ID

F’OA Hl’

-30

-20

-10

0

10

H3’

20

Figure 1. A schematic view of the experimental setup of the SMC with a3 = gA/gv and the assumption a, = 0 provides three equations out of which the value of each aq can be obtained. Introducing these values in the OPE relations gives the Ellis and Jaffe sum rules [5] for I’l; and I’? separately. However they are not as fundamental as the Bjorken sum rule and the fact that they are actually violated only means that the assumption a, = 0 should be released. Instead, the experimental values of as and as can be used together with Pi to obtain an experimental result for the a*, i.e. a0 small and a* < 0. 4. THE SMC EXPERIMENT The SMC experiment was designed to check the result of the EMC experiment on I?: and to measure I’: in order to test for the first time the Bjorken sum rule. The 190 GeV secondary muon beam from the CERN SPS had an intensity of 4.5 lo7 ,u+ per 2.4 s spill. The momentum of the muons was measured in a magnetic spectrometer upstream of the experimental hall. Fig. 1 presents a schematic view of the experimental setup [6]. The incident muon track was defined using scintillator hodoscopes and a proportional chamber. The polarised target was made of two cells polarised in opposite directions by dynamic nuclear polarisa-

tion and whose polarisations were reversed every 5 hours by rotation of the magnetic field. To measure the trajectory of the scattered muon more than 150 planes of proportional chambers, drift chambers and streamer tubes were installed before, after and inside a dipole magnet. The muon was identified by its track behind a 2 m thick iron absorber. In previous publications the determination of Al from SMC data was done using an inclusive event selection, requiring only a scattered muon. In addition to deep inelastic scattering events, the resulting sample includes scattering events which are elastic on free target nucleons, or elastic or quasi-elastic on target nuclei and which are accompanied by the radiation of a hard photon. These radiative events do not carry any informs tion on the spin structure of the nucleon and only degrade the statistical accuracy of the measurement. They dilute the spin effects in the cross section for the inclusive sample, similarly to the non-polarisable nuclei in the target, accounted for by the dilution factor f. The effective dilution factor f’,

(8) accounts for both diluting sources.

145

J-M. Le Gof/Nuclear Physics B (Proc. Suppl.) 74 (1999) 142-148

The new method was used for az < 0.02 while the standard inclusive method was used for larger z. The resulting asymmetries are presented in Fig. 3. They are compatible with data from SLAC/E143 experiment [9] but they extend down to lower value of z. The structure function gi is then obtained according to gi = AiF2/(2x(l + R)) , where parametrisations from unpolarised DIS experiments are used for Fz and R.

f 0.2

0.15

proton

0.1

?? SMC d>l.OGeV* 0.05

Figure 2. The dilution factor vs. x for inclusive and hadron tagged events

The total cross section utot and the one-photonexchange (Born) cross section crr7 are related by: qel + u$‘, where the = XaQ+a gt0t 2il + btail ctail terms are the cross sections from the radiative tails (elastic, quasi-elastic and inelastic reactions). The factor X, which does not depend on the polarisation, corrects for higher order contributions: virtual (vacuum and vertex corrections) and soft real photon radiation. For an effective measurement the dilution factor f’ should be large. A new method was then used to determine the asymmetry, it uses only events for which at least one hadron track has been reconstructed [7]. These hadrwn-tagged events do not include any contribution from a$, and &, since the recoil proton can not be observed in our spectrometer due to its small energy, and the total cross section for hadron-tagged events reduces to g:;:ge* = &7iy + ut’;;‘.

-0.2”“’

1O9

10'

(b)

deuteron

10-l

x

1

4 0a51

4

I

?? SMCd>l.OGeV*

(9)

In the calculation of the effective dilution factor f’ for hadron-tagged events, u::[~~* replaces atot in Eq. (8) and the effective dilution factor increases accordingly in particular at low z, as can be seen in Fig. 2.

Figure 3. The asymmetry AT and A$ vs. x. The ernw bars are statistical and the shaded band indicates the systematic uncertainty for SMC

.X-M. Le Gaff/Nuclear Physics B (Proc. Suppl.) 74 (1999) 142-148

146 5.

the

A was with and

QCD

analysis

next-to-leading order (NLO) QCD analysis performed [8] with SMC data [7] together data from EMC [I], SLAC El43 [9], El42 El54 [lo], and HERMES [ll].

IO -

The initial parton distributions are evolved to the Q2 of the data points using the DGLAP equations 3 and g1 is evaluated with Eq.(4). A x2 is computed from the difference between the calculated and the measured 91, using the statistical uncertainty. This x2 is minimised by changing the initial parton distribution parameters to get the best fit parton distributions at the initial Qi, Two programmes were used, programme 1 was provided by Ball, Forte and Ridolfi [3] while programme 2 was developed within SMC [8]. They give similar fits as illustrated in Fig 4 and a good description of the data. The x2 were 127 and 120 for 125 d.o.f. Programme 1 was used in the rest of the analysis.

10~’

?? SHC

0 A b 0 0

EMC El43 El42 El54 HERMES

_..

lo-*

10” :

lo-*

IO” :

-PRCQRAM 1 ---‘PROGRAM2

Figure 4. Data on gl from CERN experiments (left) SLAC and DESY experiments (right) at their measured Q2 together with the two &CD fits. The polarised parton scheme are parametrised 1 GeV2 as Af(x,

Q2)= N VIa?‘(1

distributions at an initial

- z)P’(l+

in the AB Q2 = Qf =

Of z),

(10)

where Af denotes AX, AqNs, or Ag, and N(o, /3, a) is a normalisation factor such that the parameter 11 is the integral of the corresponding parton distribution. The normalisation of the non-singlet quark densities f$z are fixed using the neutron and hyperon /I decay constants and assuming SU(3) flavour symmetry: Y$: = We use Ig,JgVj = F + D = (*)ie + +3. 1.2601 f 0.0025 and F/D = 0.575 f 0.016.

lo4

IO”

lo-* :

10-l :

Figure 5. Polarised parton distribution functions at Q: = 1 Gep. Their statistical uncertainty as obtained from the &CD fit is shown by a band with crossed hatch. The experimental systematic uncertainty is indicated by the vertically hatched band, and the theoretical uncertainty by the horizontally hatched band. Fig 5 presents the resulting parton distributions at the initial Q2. In order to estimate the uncertainty in the parton distribution functions due to the experimental systematic errors

147

J-M. Le Gaff/Nuclear Physics B (Proc. Suppl.) 74 (1999) 142-148

Table 1 r y’d’n at Qi = 5 Gep for the world set of data. ETTOTS are of statistical, systematical and theoretical origin. Proton Deuteron Neutron

investigated. The resulting contributions moments were found to be in the range tematic errors quoted.

0.121 f 0.003 f 0.005 f 0.017 0.021 f 0.004 f 0.003 IIZ0.016 -0.075 f 0.007 & 0.005 f 0.019 0.00,

the QCD fits were repeated with input values of A1 k gsyJt. We also estimated ‘theasymmetries oretical” uncertainties. The uncertainties due to cy,, the quark mass thresholds, and gA/gv were estimated as the experimental systematic errors. The sensitivity to the renormalisation and factorisation scales was estimated by setting pr = krQ2 and ,uf = k2Q2 and varying ICIand k2 in the range 0.5 5 kl, kz 5 2.0; this sensitivity is related to the neglect of higher order QCD corrections. The error due to the choice of a particular functional form for the initial parton distributions (Eq. 10) was estimated by adding a bfi term and by varying the initial scale Q,“. We use all available data in the kinematic region Q2 2 1 GeV2, 2 > 0.003 to evaluate l?r = Ji gr(z)dt at a fixed Q2. Starting from gr(z, Q2) at the measured x and Q2 we obtain gi at a fixed Q; as: sl(x 9;) = gdx Q2) + k$‘“(x Qi> gfit (t, Q2)]. We choose Qi = 5 GeV2 which is close to the average Q2 of the world data set used in the analysis. Data from different experiments are combined using a Monte Carlo procedure to take into account the correlations [6]. To estimate the contributions to the first moments from the unmeasured low z (z < 0.003) and high t (z > 0.8) regions, we integrate over gfit calculated at Q2 = 5 GeV2, as illustrated in Fig. 6 in the proton case. The results obtained for the first moments are presented in Tab. 1. The uncertainties in the low and high x integrals are obtained using the same procedure as for the estimation of the uncertainty in the QCD evolution. Had we taken the traditional approach of using Regge extrapolation in the low x region, we would get results consistent with those presented here but with significantly smaller uncertainties. Other functional behaviours of g1 at low 2 (x < 0.003) have been

wpoa

??

SMC

A El43

to the of sys-

0 EYC

-0.02

IO4

loJ

lo'*

10"

:

Figure 6. z& vs. x, the inset corresponds to the low x contribution. The parameter ns of the QCD fit is actually the gluon contribution to the spin of the nucleon, we find Ag = 0.99 ?A:$ (&a) ?E:$ (sys) ?A::: (th). The uncertainties are very large and points to the need of a direct measurement of Ag in processes where the gluon is involved in the leading order. The QCD analysis was actually also performed in the MS scheme where qa is ao, we find a0 = 0.19 f O.O5(stat) f O.O4(syst). On the other hand in the AB scheme we get ACAB = 0.38+0,$$ . ‘ -$$ +~~~. This is larger than ao but still significantly smaller than &PM expectations. So, it does not support the idea that a large enough Ag could explain the difference between the low experimental value of ae and the naive &PM expectation. The Bjorken sum rule cannot be tested directly from the estimations of I’1 of Tab. 1 because they were obtained with a QCD analysis which includes the constraint &,s - 17;s = %u. This constraint had to be released and the 6”t redone,

.Mf. Le Gaff/Nuclear Physics B (Proc. Suppl.) 74 (1999) 142-148

148

REFERENCES

giving at Q2 = 5 GeV2:

=

0.174 t”,:“,;‘, ,

(11)

which is in excellent agreement with the theoretical prediction FT -I?? = O.lSlf0.003 at the same

9;. 6. Conclusions The spin structure function gr is measured over the range 0.003 < z < 0.8 and enough data are now available to allow for a QCD analysis. The Bjorken sum rule is verified at the level of lo%, while the singlet axial matrix element is low, as = 0.19f0.05f0.04, which means that the Ellis Jaffe sum rules are violated. The QCD analysis provides a quark spin contribution to the nucleon spin in the AB scheme ACAB = 0.38_+~$I which is somewhat larger than a0 but still smaller than QPM expectations. There is little information on the polarised gluon parton distribution which points to the need for a direct measurement.

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