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In Search of the Quark Spins in the Nucleon A Next--to--Next--to-- Leading Order QCD Analys

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hep–ph/9512440DFUPG–95–GEN–01December 1995In Search of the Quark Spins in the Nucleon:A Next–to–Next–to–Leading Order QCD Analysis of the Ellis–Ja?e Sum Rule ?Paolo M.Gensini Dip.di Fisica dell’Universit`a di Perugia,Perugia,Italy,and Sezione di Perugia dell’I.N.F.N.,Perugia,Italy To be submitted to Zeitschrift f¨u r Physik C:Particles and Fields

In Search of the Quark Spins in the Nucleon:

A Next–to–Next–to–Leading Order QCD Analysis

of the Ellis–Ja?e Sum Rule

Paolo M.Gensini

Dip.di Fisica dell’Universit`a di Perugia,Perugia,Italy,and

Sezione di Perugia dell’I.N.F.N.,Perugia,Italy

ABSTRACT

The data from the last seven experiments performed on polarized deep–inelastic scattering on proton and neutron(or deuteron)targets have been analyzed in search of

a precise determination of the spin fraction carried by the quarks in the nucleon.We

?nd that this fraction can be of the size expected from na¨?ve quark model arguments,

provided the gluon axial anomaly is explicitly included and the isosinglet axial charge

normalization is?xed at a suitably low momentum scale,such that a)the running,

strong coupling constant is about unit,and b)the orbital angular momentum inside

the nucleon vanishes.

We also?nd that,despite the appeal of this solution of the“nucleon spin crisis”,a solution where the axial anomaly is absent and its e?ects are traded for an

appreciable strange quark polarization can not however be excluded—because of

the limited accuracy of the data—unless this latter and/or the gluon polarization

in the nucleon are explicitly measured.

1.Introduction.

Just after the1988publication by the European Muon Collaboration(EMC) at CERN of their preliminary1results on the asymmetry in muon polarized deep–inelastic scattering(PDIS)on a polarized hydrogen(actually spin–frozen ammonia) target,the particle physics community was somewhat shocked to learn that the total sum of the quark spins in the proton seemed to be very close to zero,rather than somewhere between1/2and3/4,as generally expected from na¨?ve quark–model arguments.

As for any unexpected discovery in our?eld,to begin perhaps with that of the muon,the following years have seen both the planning and running of new experiments,and the deepening of the theoretical studies(not completely free of much heated controversies)trying to clarify this mystery,often know under the name of“nucleon spin crisis”.

The aim of this paper,relying heavily on the results thus accumulated,is

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to use i)all PDIS data taken on both proton and neutron(or deuteron)targets, ii)current phenomenological ideas on the behaviour of the parton distributions at both large and small values of the Bjorken variable x,and iii)the results from perturbative QCD(PQCD)at next–to–next–to–leading order(NNLO),to produce an internally consistent estimate of the quark spin content of the nucleon.

It will turn out that one can not,at the present level of experimental accuracy, unambiguously separate the di?erent?avour components from PDIS data alone, nor directly verify the PQCD predictions for the isoscalar sum rule:namely,one can not decide on a purely experimental basis on the nature of its unitary–singlet component,given the(basically two)di?erent possibilities o?ered in principle by the stage at which one decides to send the quark masses(used as regulators in the calculations of the splitting functions)to zero.

Despite this persistent lack of experimental proof of all PQCD arguments,it can however be shown that,for the“most natural”(in the parton model framework) of these two possibilities,one can interpret the data as being consistent with a negligible intrinsic strange component of the quarks’spins at a suitably low mass scale,such thatαs?1,consistent with the na¨?ve quark model expectation.

There are two primary reasons why this result contradicts the initial?ndings from the EMC data1:the?rst,of experimental nature,is that it is now clear that the g p1values given by the EMC were too low because of their choice of the F p2data employed to normalize them;the second,of theoretical nature,is that PQCD corrections were included only at leading order,while it is only at next–to–leading order that peculiar features of the polarized deep–inelastic scattering(and particularly for its unitary–singlet piece)become to emerge,as it will be clear in next section.

The rest of this paper will be divided in three parts:a presentation of the PQCD NNLO corrections to the?rst–moment isovector and isoscalar sum rules,a careful enumeration of the constraints that can be imposed on the various polarized distribution functions of the quarks,in terms of which one can describe the polarized

structure functions g p,n

1,and then a short discussion on the results obtained?tting

to the data simple parametrizations satisfying these constraints.This discussion will be centred on the spin composition of the nucleon,and in particular on the need(or absence of any need)for an intrinsic strange component?s in it.

2.The First Moments of g p,n

1

and PQCD at NNLO.

The sum rules for the?rst moments I p,n

0(Q2)= 10dxg p,n1(x,Q2)are often

discussed separating the non–singlet,(p?n)combination from the isoscalar one, (p+n),which contains both a non–singlet and a singlet part.Actually,despite the technical di?culties involved in dealing with the latter,both sum rules have equal footing in PQCD and should not be considered as separate entities,apart

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from historical,or practical,considerations.

Their only di?erence lies indeed in the fact that the axial coupling involved in the ?rst is extremely well measured from neutron β–decay,while the couplings involved in the second are outside direct measurement,and can be arrived at only through more or less founded theoretical arguments.It is therefore expedient to separate them,since the ?rst can either be considered a good test of higher–order PQCD,or,alternatively (and this will be the attitude taken in the present paper),a useful normalization for the non–singlet part of the moments I p,n 0(Q 2).

The ?rst–moment sum rule for the di?erence between proton and neutron po-larized structure functions,known as the Bjorken sum rule (BjSR),reads 2(written in its full QCD garb)

I p ?n 0(Q 2)= 1

0dx [g p 1(x,Q 2)?g n 1(x,Q 2)]=1

π·[1+c (8)1(N f )·

αs π)2+...],(2)where the numerical values of c (8)1,2are listed,for N f from 3to 5,in Table I,and their

complete expressions can be found in the original paper by Larin and Vermaseren 11.Due to the large values of the constants c (8)1,2for N f =3,4,C 8(αs )is evidently decreasing with αs much more steeply than expected on the basis of the leading order estimate C 8=1?αs /πat low values of Q 2,such as those of SLAC experiments E1424(=2.0GeV 2)and E14312,13(=3.0GeV 2),and of the preliminary deuteron data from the Spin Muon Collaboration 14(SMC)at CERN (=4.6GeV 2).

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An initial comment is in order here:actually,the expansion inαs for C8(αs)is known to one order beyond15,16what has been written in Eq.(2),but its unitary–singlet partner,C1(αs),has been estimated16only to O(α3s),as most of the quanti-ties computed in PQCD to extract the couplingαs:for its consistency,any PQCD analysis must be performed to the same,?xed order in the coupling,using the β–function at that order to express the running of the couplingαs(Q2).

Accordingly,here only NNLO expansions will be used,together with the ex-pansion forαs(Q2)(and the estimate of its scaleΛ

(5)=200±50MeV,and therefore to the valueΛ

MS

(3)from the slightly older compilation of Ref.17.

MS

Table I

Coe?cients of higher QCD corrections to BjSR

A question better addressed at this point is the actual value of N f,the number of active?avours,to be used at each Q2in the PQCD expansions of the coe?cients C1,8(αs)and of the anomalous dimension of the unitary–siglet axial charge.For time–like Q2,there is no ambiguity,since for heavy quarks the?avour thresholds at Q2=4m2i can be?xed by setting the mass m i of the i–th?avour quark ap-proximately equal to that of its lowest–mass pseudoscalar meson.In deep–inelastic scattering,however,a?avour is active only when appreciably contributing to the moment sum rules,i.e.when produced a)in a really inclusive manner(read:not only in low–multiplicity events),and b)over an appreciable range in Bjorken’s vari-able x(say up to x?1/3).If one sets the beginning of the scaling region at Q2?2GeV2(as indicated by the“classic”SLAC–MIT experiments),the previous re-quirements ask for a Q2>~17GeV2for charm to be an active?avour in DIS,and so mandates N f=3for the PQCD analysis of all available PDIS data.

The choice of N f=3(rather than4)has no great e?ect on C8(αs),as one can read from Table I,but for the Q2–evolution of the unitary–singlet piece the

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coe?cient of the ?rst–order term in the expansion in powers of αs of its integrated anomalous dimension almost cancels,for N f =4,the ?rst–order one in C 1(αs ).It is therefore important,for the extraction of the quarks’spin fraction,

Σ=N f i ?q i =?u +?d +?s for N f =3,(3)

to use the value of N f appropriate to the range of values of Q 2where the actual data have been taken.This,unfortunately,has not always been done consistently by some authors 19.

In 1974,Ellis and Ja?e 20,faced with the problem of how to use a sum rule akin to Eq.(1)with only hydrogen data,used parton–model ideas,?avour SU(3)symmetry and the Okubo–Zweig (OZI)rule to derive a sum rule for the ?rst moment of g p 1alone ;as stated above,the PQCD–corrected version of such a sum rule (when freed of OZI–rule restrictions)is a part of PQCD as fundamental as the BjSR.For the isoscalar combination of PDIS structure functions this sum rule becomes the (PQCD corrected)Ellis–Ja?e sum rule (EJSR)

I p +n 0(Q 2)=

10

dx [g p 1(x,Q 2)+g n 1(x,Q 2)]==19

·C 1(αs )·g 0(Q 2)+(h.t.)I t =0,

(4)not to be confused with the original EJSR 20,which was derived for I p 0(Q 2)only ,

and without any PQCD corrections to the parton–model plus OZI–rule predictions.

Additional complications with respect to the BjSR,Eq.(1),arise from the facts a)that the isoscalar axial charges of the nucleon are not directly measurable,and g 8is indeed derived via ?avour–symmetry arguments (apart from symmetry–breaking e?ects),and b)that the unitary–singlet one g 0(Q 2)couples not only to the quarks,but also to the gluons via the axial anomaly and possesses therefore anomalous dimensions 21,so that its evolution with Q 2is not exausted by the PQCD coe?cient C 1(αs )and must be explicitly computed.

Since this coupling is scheme dependent,this point has been the focus of a very heated theoretical debate 22.To cut a long history short,one can summarise it by saying that,in conventional parton language where the masses of the partons m i are neglected with respect to the momentum scale Q ,and wishing to identify Σwith the spin fraction carried by the quarks actually present in the target proton,one has to put g 0(Q 2)=Σ?N f αs dt g 0(t )=?N f αs

which relates to the anomalous dimension of the axial anomaly via

?N fαsα

s

(6′)

where the variousγij({i,j}={g,q})represent the coe?cients giving dΣ/dt and d?G/dt in terms ofΣand?G,whose matrix is diagonalized(in any scheme where quark mass regulators are sent to zero)building the combination in Eq.(5)on one side,which evolves anomalously as in Eq.(6),and leavingΣon the other,free of anomalous dimensions since the constraint in Eq.(6′)makes the determinant of the2×2matrix of the(?nal)coe?cients(whose eigenvalues are the anomalous dimensions of the two operators mixed by the evolution)vanish identically.This has been veri?ed step by step at next–to–leading order22:that it should hold also at higher orders is inferred from the fact that conserved(and partially conserved) charges should be free of anomalous dimensions on one side,and on the other there is nothing,apart from the mixing with the unitary singlet of the pure gluonic world,to distinguish an SU(3)–from a U(3)–symmetric fermionic world,so that it is“natural”to expect all purely fermionic operators to be free of anomalous dimensions.

After integrating inαs from a normalization scaleμ2to Q2,one obtains the integrated anomalous dimension at NNLO

log g0(Q2)

33?2N f

αs(Q2)?αs(μ2) 24

+

N f

8(153?19N f)

)

αs(Q2)+αs(μ2)

g0(Q2)=g0(μ2)·exp[?γ(αs(Q2))??γ(αs(μ2))],where?γ(αs)is the integrated anoma-lous dimension,but has caused unnecessary problems whenever exp?γ(αs)has been

expanded in powers of the coupling,since then one had either to go one step further in the expansion of the anomalous dimension to?nd the“missing”power15,16,or

to truncate“prematurely”the series in the coe?cient C1(αs).Both procedures are inconsistent from a PQCD point of view,since they tend to mix one order with the

next,while the running couplingαs must be de?ned order by order:it is clear that in this way one will end up usingαs at a given order in at least one perturbative

expansion calculated at a di?erent order(not to be confused with the power of

αs appearing in the?nal expression,which could depend,as is the case here,on additional mathematical manipulations).

Another,equally unnecessary,but luckily only semantic problem has been

created by the persons who?rst misnamed the scaleμ2in the same equation a renormalization scale,while it is clear enough that one has to do with just a normal-ization scale,which has absolutely nothing to share with the actual renormalization procedure15.The scaleμ2is thus free both to appear explicitly in the physical

expressions,and to be chosen to follow the author’s(or authors’)theoretical preju-

dices;certain precautions must however be followed:for instance,working at a?xed number of?avours N f=3,it would be rather unwise to set it to in?nity,where N f

will be six at the best of our knowledge,so that the g0(∞)N f=3so obtained would hardly be connected to the physical limit of g0(Q2)for Q2→∞.Note also that this g0(∞)N f=3(whatever its meaning)can not be identi?ed withΣas de?ned by Eq.(3):if one believed the singlet axial coupling g0(Q2)in the EJSR,Eq.(4),

to be given byΣalone,from the diagonalization of the unitary–singlet operators one had also to put?γ=0accordingly.In line of principle,these two de?nitions for g0(Q2)could be distinguished on the basis of the di?erent evolutions with Q2

they predict for the EJSR integrals I p+n

0(Q2):making the two to coincide at a scale

Q20,the di?erence for the EJSR between the case of a running g0(Q2)and that of g0=Σat a scale Q2would correspond to

?I p+n

0(Q2)=

2

π·[1+c

(1)

1

(N f)·

αs

π

)2+...].(9)

There are hovewer additional complications,for this coe?cient receives contribu-tions from graphs,not included in the other,whose number grows with the order of the calculation.Thus Kataev16has produced only an estimate of the constant c(1)2

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for N f=3(note that one of the conditions under which this was possible breaks down for N f=4).

To close this section,the numerical values of the constants for the perturbative expansions in orders of(αs/π)of C8(αs),C1(αs),and?γ(αs)= kγk·(αs/π)k,are listed in Table II.Two points must be noted before going to the next section:the constants in the singlet part are systematically lower than in the corresponding, non–singlet one,giving a slower evolution with Q2,and the trends of the inte-grated anomalous dimension exp?γ(αs)and of the coe?cient C1(αs)run in opposite directions,tending to some extent to compensate each other.

Also,the?rst line in Table II makes clear enough the essential di?erence be-tween leading–and higher–order PQCD treatments:besides introducing an anoma-lous dimension for g0(Q2),the latter break strongly the accidental,lowest–order degeneracy of the coe?cient functions C1,8(αs).

Table II

Constants in the perturbative expansions at NNLO

3.Measurements and parametrizations for g1(x).

What is actually measured by experiments in PDIS are not the structure functions g1themselves,but rather the polarization asymmetries A1,related to the polarized cross sectionsσ↑↑,σ↑↓by

D·A1=(σ↑↑?σ↑↓)/(σ↑↑+σ↑↓),(10)

where D is the target polarization fraction,and g2is neglected:g1is then related to A1by

A1·F2(x,Q2)

g1(x,Q2)=

intercept being close to zero.However,since one does not expect the sea distri-

butions to couple dominantly to an isovector,pseudoscalar trajectory,but rather to an isoscalar one such as the eta,one should rather have for these a behaviour

x?αη(0),withαη(0)??1/4,which,together with the expected negative sign for the sea contribution,produces a spike in the isoscalar part of g1as x→0,of the

type g1~α?β·x1/4(withα,β>0),perhaps just appearing24at very low values of the Bjorken variable x in the proton data taken by the Spin Muon Collaboration at CERN at=10.0GeV2.Of course,such a spiky behaviour does not show in the integrand of the BjSR,which can therefore be extrapolated smoothly to x=0 according to the conventional practice in this matter.

This point could have a non–negligible in?uence on the evaluation of the EJSR,

Eq.(4),raising the low–x contribution to its left–hand side well above the conven-tional estimates.A special comment is in order here on the EMC published data5 for g p1:rather than using them,one should use instead only their values for A p1with an adequate set of values for F p2and R p.Indeed,the EMC calculated g p1using a)the ratio R p predicted by PQCD,systematically smaller than experiment since ?nite–mass corrections are dominant at low values of Q2and x(though the e?ect of this choice is not too important at=10.7GeV2),and b)their values for the unpolarized structure function F p2,systematically lower than those by the BCDMS collaboration25(and than the recent New Muon Collaboration(NMC)data26as well)by as much as13%at the lowest values of x.Even using their measured values of A1together with a phenomenological parametrization for F p2(x,Q2)(and R p(x,Q2))to produce g p1(x,Q2)at a reference,?xed value of Q2(and assuming A1 to vary little6,15,27with Q2,an assumption which a recent analysis from the E143 Collaboration at SLAC seem to corroborate28)is not completely free of the above, last source of error:indeed only the latest,post–NMC parametrizations29have dropped the unpolarized EMC data altogether,while all previous analyses ended up averaging over the two,con?icting sets of data for small values of x.We have therefore re–normalized the published EMC data5for g p1with the known ratio of BCDMS to EMC data,and this will be the data set referred to as EMC p rev in the rest of the paper.

As just mentioned two paragraphs above,to build the?rst moments the actual

measurements of g p,n

1have to be extrapolated in x to x=0and to x=1,to cover

parts of the integration range not coverable by the experiments on the asymmetry

A p,n 1.The second extrapolation does not pose any problem,since A1tends to

unit(and R to zero)in the limit x→1(this is a well–known feature30of the nucleon wavefunction:at high values of x the proton—neutron—structure is

dominated by the u+—d+—quark distribution:some parametrizations31violate this constraint gaining thus additional but unphysical freedom in dealing with the

small sea components),and Eq.(11)is thus reducing simply to g1~F2/(2x), whose behaviour as x→1is largely determined by well known“counting rules”of

conventional parton models.

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In this paper,instead of extrapolating separately in the two extreme ranges of x,the choice has been made to?t the data to simple functional forms incorporating at least three elements:a)the counting rules,b)reasonable Regge(or other,QCD–motivated)behaviours for x→0(separately for valence and sea contributions), and c)constraints on the integrals of the parton distribution functions coming from a connection between the constituent–quark picture and the quark–parton model, originally introduced by Altarelli,Cabibbo,Maiani and Petronzio32,and recently recovered in this context by Fritzsch33.

In this picture,the structure functions g p,n

1are decomposed in terms of the

helicity distribution fuctionsδq′i for each active quark?avour(q i=u,d,s),where the prime indicates that the contribution from the axial anomaly,formally of order αs,has been included in each?avour’s sea distribution,δq′i=δq i+k qg?δG(the symbol?will stand from here on for a convolution integral in Bjorken variables), with k qg the appropriate gluon–to–quark splitting function.It has been sometimes said in the literature that the anomaly contributes to g1only at very small x values, so that it should be hardly seen in the x–regions covered by experiments:this is true(in the scheme where quark masses m i are sent to zero)only if the polarized gluon distribution is assumed to peak at x=0,as e.g.in the intrinsic gluon distribution proposed by Brodsky and Schmidt34.Unfortunately,their distribution forδG(x)has the wrong Regge behaviour for x~0,contrary to their statement, since it requires dominance ofδG by the pion trajectory,with interceptαπ(0)?0, while it should be dominated instead by a pseudoscalar–glueball trajectory,with an expected interceptαG(0)<~αη′(0)??1.When this constraint is imposed on the Brodsky–Schmidt formul?,ceteris paribus,most of the anomaly contribution falls in the x interval covered by the experiments(e.g.80%of it in the case of the EMC x–range),making it virtually indistinguishable from the other,intrinsic sea distributions,while the integral?G remains of the same magnitude as in Ref. 34,being solidly tied to the momentum fraction?1/2carried by the gluons at the momentum scaleSLAC?1~2GeV2at which these intrinsic components are de?ned.

Separating further valence(u v,d v)from sea(u′s,d′s,s′)components one can write,with self–explanatory notations,

g p1(x,Q2)=

2

18·[δd v(x,Q2)+δd′s(x,Q2)+δs′(x,Q2)],(12) and

g n1(x,Q2)=

2

18·[δu v(x,Q2)+δu′s(x,Q2)+δs′(x,Q2)].(13)

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The valence and sea distributions(with the latter primed to distinguish them from

the intrinsic ones,free of the gluonic contribution,which relate to the quarks spin contents?q i entering e.g.in the angular momentum sum rule),will in general be

expressed by the general functional formδq′v,s(x,Q2)= j x?αj(0)·(1?x)n v,s·P j(x,Q2),where the sum runs over the Regge singularities assumed to dominate the

distribution at x~0,and the P j(x,Q2)will be polynomials in x with Q2–dependent coe?cients35,36.

For the powers n v,s the parton–model counting rules36give n=2N s?1 (where N s is the number of“spectator”quarks),or n v=3and n s=7(for gluons the rule gives n g=5:their further suppression in the distributionsδq′s at high x–values comes from the splitting function):it is hovewer known35,36that the valence u–quark dominates at high x values over the d–quark in their unpolarized distribu-tions,and this fact is commonly“explained”as a consequence of the Pauli principle, barring two quarks of the same quantum numbers from being close to each other in phase space,and often expressed as a“penalty factor”(1?x)in the odd?avour distribution function.The same mechanism should operate in the polarized valence distributions as well,as in the sea distributions(both polarized and unpolarized), suppressing here u–quarks with respect to d–quarks,the“penalty”being now paid when an u–antiquark is produced at x~1;while there is undisputable evidence of this e?ect for unpolarized valence distributions in the ratio F n2/F p2,tending almost linearly to the value1/4as x→1,the same e?ect in the unpolarized sea ones could be responsible37for the defect of the so–called Gottfried sum rule(one can simply check that the?gures are indeed of the right order of magnitude,though a detailed model would require a complete re–?tting of all unpolarized parton distributions). For the PDIS data,this Pauli–principle“penalty factor”will be included only in the polarized valence distributions,for the e?ects of its presence in the sea ones would be so small vis–`a–vis the experimental errors that its inclusion would only lead to unnecessary mathematical complications in the?tting procedures.

With this additional factor omitted,and reducing the polynomials P j(x,Q2)to Q2–dependent factors,independent of x,the sea contributions to the PDIS structure functions will reduce to an isoscalar term,simply expressed as

(x,Q2)sea=P(Q2)·x1

g p,n

1

quark.Assuming SU(2)symmetry to hold for the partons inside the constituent quarks(Q=U,D only),and integrating over all Bjorken variables,one can put ?u U=?d D=?(q Q)v+?(q Q)s,?d U=?u D=?(q Q′)s and?s U=?s D=?s, and one?nds the relation for the spin content of the valence partons

?u v

?D

=?4(15)

from the constituent quark model results?U=4/3,?D=?1/3:note that,since these constituent quarks are structured objects,these values do not imply g A=5/3 (modulo small recoil corrections),as in na¨?ve treatments of the constituent quark model.The same hypothesis leads only to the generous bounds?1/4

With the constraint of Eq.(15)imposed on the polarized valence distributions joined with the neglet of the small isovector part in the polarized sea ones,the distributionsδu v andδd v can be normalized to the BjSR,independent of their functional forms,since Eq.(1)can be reduced to

I p?n 0(Q2)=

1

6·C8(αs)·g A+(h.t.)I t=1.(16)

For these forms,three parametrizations will be adopted,to check the systematic e?ects on the EJSR integrals of the behaviour assumed in the isoscalar valence distributions as x→0.

The?rst parametrization(which will be labeled FRP,for fully Reggeized parametrization)breaks down each of the polarized distributions for the valence quarks into an isoscalar and isovector part,the?rst dominated at x~0by the η–meson trajectory,and the second by the pion one,so that,after introducing the Pauli–principle“penalty factor”(1?x)inδd v and imposing the conditions in Eqs.

(13)and(14),with the x–polynomials P j(x,Q2)again reduced to Q2–dependent coe?cients,one gets

δu(1)v(x,Q2)=C(Q2)·[231

4

,4)?1·x141]·(1?x)3,(17)

δd(1)v(x,Q2)=C(Q2)·[231

4

,4)?1·x141]·(1?x)4,(17′)

where the normalization C(Q2)=C8(αs)·g A+6·(h.t.)I=1is?xed from Eq.(16)to automatically satisfy the BjSR,and the function B(α,β)is the well known Euler’s beta function.

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For the second set of model distribution functions(to be labeled SRP,or simpli?ed Regge–pole parametrization)one takes the limitαη(0)→απ(0)→0, still keeping the same constraints,so that Eqs.(17),(17′)become

δu(2)v(x,Q2)=

16

131log

1

1310 ·(1?x)3,(19)

δd(3)v(x,Q2)=C(Q2)· 66x?2817

MS (5)of Ref.17)and the

HTC of Ref.9as inputs,there is only one free parameter left for each set of data, i.e.the sea normalization P(Q2).Note that,to account for possible systematics either in the data or in the theoretical inputs(particularly the possible inadequacy both of the PQCD approximation and of the phenomenological parametrization used,as well as of the evaluation of the HTC),di?erent parameters will be used for di?erent data,even when these latter have been normalized at the same value of .

4.The isosinglet sum rule and the quark spins in the nucleon.

Before turning to?tting the parametrizations to the data available(as of November1995)on the PDIS structure functions g1,it is better to consider the information we possess of the axial couplings appearing in the right–hand sides of the?rst–moment sum rules,Eqs.(1)and(4).All pieces of information available have been summarized in Table III,which deserves some https://www.sodocs.net/doc/2913864641.html,ually most

13

analyses39of these couplings stop at the information coming from asymmetries in the decay products momenta and/or polarizations:while this is enough in most of the cases,in some no such measurements are either available or possible,and disregard of the information coming from the rates can have drastic e?ects,even on the correctness of the conclusions inferred.Enough to say that dropping the information on the?S=0,Σ→Λcouplings leads to the conclusion that?avour SU(3)symmetry breaking e?ects are negligible39,relying in fact on a single point, being the coupling of theΞto bothΛandΣnot as well known as those of the latter two to the nucleon.Accordingly,Table III includes evidence from both asymmetries and rates,while the detailed analysis of these latter can be found elsewhere40.One further thing evident from Table III is that the two sources of information yield fully compatible values for the axial couplings,contrary to the statement of Ja?e and Manohar41,which originated from a completely outdated treatment of the decay rate data.

Table III

Data on octet baryon axial couplings

νe 1.2553±0.0018 1.2573±0.0028F+D

Σ?→Λe?3tanφF

Σ+→Λe+νe0.750±0.094D+√

νe?0.330±0.023?0.340±0.017F?D

Σ?→nμ?

νe0.729±0.0110.718±0.015F+1√

νμ0.756±0.139F+1√

νe0.265±0.0440.250±0.050F?1√

νμ0.76+0.47

?0.76F?1√

νe 1.216±0.147F+(1?4

3

tanφ)D

νe,yields the parameters F=0.4678and D=0.7876,or g8=3F?D=0.616±0.022and g A=F+D=1.2554±0.0020(note that this latter acts almost as a constraint due to the very high precision of neutron data on both asymmetries and rates);theχ2of the?t is not very high with respect to

14

the number of data points,being20.17(about2units better than withoutΣ0–Λmixing)versus15,but is concentrated almost exclusively in theΣ?→Λrate, whose datum on the axial coupling lies3σbelow the SU(3)–symmetric?t value. Coming this datum from a good–quality experiment44,this fact has to be taken as initial evidence for some?avour SU(3)breaking in these couplings,being the mixing required to explain this discrepancy more than?ve times the one calculated by Karl43,and thus giving a ratio g V/g A more than twice its experimental1σlimit. This must be remembered when using?avour symmetry to extract the nucleon spin composition.A further point raised30in this context is also the possible e?ect (particularly on the?S=1transitions)of the tensor components in the weak axial current:the good agreement between the two columns of data in Table III seems to indicate that the e?ect is not as large as indicated by some?ts45,since it would a?ect the axial couplings extracted from the asimmetries di?erently from those extracted from the rates.

Since it is the aim of the present paper to concentrate on the analysis of the spin composition of the nucleon,the PDIS structure functions will not be?t treating both C(Q2)and P(Q2)as free parameters,but rather taking the?rst from the right–hand side of the BjSR,Eq.(1),and?tting only the second to the data,experiment by experiment.Table IV will present the expectations for the right–hand side of the BjSR in NNLO PQCD,including the HTC of Ross and Roberts9,together with the integrals evaluated by two experimental groups,the E143Collaboration at SLAC13and the Spin Muon Collaboration at CERN46.In the same table,we shall also display the two expectations for the right–hand side of the EJSR obtained assuming validity of the OZI rule,i.e.Σ=g8,and either a) the na¨?ve parton model identi?cation g0=Σ(i.e.decoupling the contribution from the axial anomaly,as is the case for massive quarks),or b)the de?nition in Eq.(5) with the normalizations scaleμ2?xed so thatαs(μ2)=1(and=0):here g8is assumed equal to the above?avour SU(3)symmetry prediction.The choice a)amounts to using an isoscalar,PQCD corrected version of the original EJSR20, while b)serves to give an idea of the e?ect induced by a very reasonably sized axial anomaly contribution21,22.

From this table one can read two facts,already mentioned above:the running with Q2of the EJSR is slower than that of the BjSR,and therefore harder to see in the data unless higher precisions are reached,and the di?erence between the “anomalous”and“non–anomalous”versions of the EJSR,described by Eq.(8) (apart from the lower asymptotic value of the?rst,which can always be traded for a breaking of the OZI rule,i.e.a?s=0,in the second),is even smaller,and thus much harder to see.

15

Table IV

Sum rule expectations for BjSR and na¨?ve EJSR In Table V there will be displayed the parameters P(Q2)obtained from the ?ts to the seven sets of PDIS data analyzed here:the re–normalized EMC set of data on proton1,5at=10.7GeV2,the E142neutron data4at= 2.0GeV2,the SMC preliminary deuteron data14at=4.6GeV2,the SMC proton data24at=10.0GeV2,the E143data on proton12and deuteron13 at=3.0GeV2,and the SMC new deuteron data40at=10.0GeV2. The very preliminary data of the SLAC–Yale E80and E130Collaborations47have not been included in the?ts,since they cover either very low values of Q2or a very limited range in x values,and would have been,besides,of very little statistical signi?cance;also,the recently appeared E143data28are not included,since they have become available only during the?nal redaction of the present paper.The factor1/2·(1?3

6·C(Q2),i.e.equal to the BjSR integral:from this and the previous table one can see that some negative sea is already expected

even at the level of a na¨?ve formulation of the(PQCD corrected)EJSR.The table displays also systematic variations in the parameter both with the model used for the valence distributions and with the nature of the target,variations which in the opinion of the author cast some doubt on the possibility of really testing PQCD in the isoscalar combination:since the two kind of variations are of the same order of magnitude,and comparable to the statistical errors from the?ts,their origin is di?cult to trace.

16

Table V

Values of P(Q2)from?ts to the data

(shown in parenthesis are theχ2–values per data point) These results for the sea parameter P(Q2)can now be turned into values for the EJSR integrals,since these can be written as

I p+n 0(Q2)=

1

4

,8)·P(Q2).(20)

Note however that the right–hand side of this equation corresponds to what is actually interpolated by the?ts only in the case of a deuterium target,when

I d0(Q2)=1

2ωD)·I p+n

(Q2):for a proton or neutron target the?ts measure,

respectively,

I p0(Q2)=1

4

,8)·P(Q2)(20′)

in the proton case,and

I n0(Q2)=B(

5

6·C(Q2)(21) in the?rst case,and

I p+n 0(Q2)=2·I n0(Q2)+

1

quantify48,but approximable with that of the most precise experimental determi-nation of the BjSR integral(i.e.with that of Ref.46).

Taking these aspects into account,the?ts yield for the EJSR the integrals listed in Table VI for the three valence parametrizations,which show a marked reduction with respect the na¨?ve expectations for this quantity,tabulated in the second column of Table IV.As already said,this is evidence for either presence of a sizeable reduction of g8and/or g0(due to a polarized strange–quark density or the breaking of?avour SU(3)symmetry49)or an appreciable contribution from the axial anomaly(or both).

From the previous table one can also see that the three parametrizations,

though behaving quite di?erently at x→0in the combination g p+n

1,do not produce

neither appreciably di?erentχ2’s(the BLP?t turns worse than the other two,wich statistically are on the same level,but only by a factor~2.5),nor large variations

(contrary to na¨?ve expectations)for I p+n

0(Q2):this is due to the normalization

of the valence components to the BjSR imposed by Eq.(15)–(16),valid for all parametrizations regardless of their behaviour as x→0and quite robust,since the only isovector contribution from the sea could come,under the hypotheses adopted here,from the Pauli principle“penalty factor”.The di?erent behaviours as x→0 of the valence parametrizations re?ect thus only on the quality of the?ts,tending to prefer distributions which stay?nite in this limit,since diverging ones become harder to accommodate to the behaviours of g1for neutron and deuteron targets at moderate and high values of the Bjorken variable x.

Table VI

Integrals of the EJSR from?ts to the data

Any interpretation of these data(beyond the existence of a sizeable,negative polarization in the sea)has to make clear the nature of the unitary–singlet piece in the EJSR.If one follows the na¨?ve parton model picture to the end,and identi?es g0withΣ,one must also have?γ=0from the vanishing of its anomalous dimension, so that the EJSR becomes now,since g8=Σ?3?s,

I p+n 0(Q2)=[

2

18·C8(αs)]·Σ?

1

which,puttingδI=?(h.t.)I

t =0

,F(Q2)=2

18·C8(αs)and R(Q2)=

C8(αs)/[6·F(Q2)],can be turned into a linear relation betweenΣand?s,

Σ=I p+n

(Q2)+δI

αs(Q2)·exp[?γ(αs(Q2))??γ(αs(μ2))]·

?G(μ2)+

+

1+3αs(μ2)18·C8(αs) ·Σ??16π·exp[?γ(αs)??γ(αs(μ2))]·C1(αs)(26)

The linear relation betweenΣand?s,Eq.(23)is still holding,but now F(Q2) is changed including the integrated anomalous dimension factor[1+3αs(μ2)/4π]·

19

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