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Impact of CLAS and COMPASS data on Polarized Parton Densities and Higher Twist

a r X i v :h e p -p h /0612360v 2 10 A p r 2007

Elliot Leader ?

Imperial College,Prince Consort Road,London SW72BW,England.

Aleksander V.Sidorov ?

Bogoliubov Theoretical Laboratory Joint Institute for Nuclear Research 141980Dubna,Russia.

Dimiter B.Stamenov ?

Institute for Nuclear Research and Nuclear Energy

Bulgarian Academy of Sciences Blvd.Tsarigradsko Chaussee 72,So?a 1784,Bulgaria

(Dated:March 28,2007)

We have re-analyzed the world data on inclusive polarized DIS including the very precise CLAS proton and deuteron data,as well as the latest COMPASS data on the asymmetry A d 1,and have studied the impact of these data on polarized parton densities and higher twist e?ects.We demon-strate that the low Q 2CLAS data improve essentially our knowledge of higher twist corrections to the spin structure function g 1,while the large Q 2COMPASS data in?uence mainly the strange quark density.In our new analysis we ?nd that a negative polarized gluon density,or one that changes sign as a function of x ,cannot be ruled out on the basis of the present DIS data.

PACS numbers:13.60.Hb,12.38.-t,14.20.Dh

I.INTRODUCTION

The European Muon Collaboration (EMC)experiment [1]at CERN found that a surprisingly small fraction of the proton spin is carried by the spin of the quarks.This observation was a big challenge to our understanding of the partonic spin structure of the nucleon,i.e.,how the nucleon spin is built up out from the intrinsic spin and or-bital angular momentum of its constituents,quarks and gluons.Since that time substantial e?orts,both exper-imental and theoretical,have been made to answer this question.Our present knowledge about the spin struc-ture of the nucleon comes mainly from polarized inclusive and semi-inclusive DIS experiments at SLAC,CERN,DESY and JLab,polarized proton-proton collisions at RHIC and polarized photoproduction experiments.One of the important and best studied aspects of this knowl-edge is the determination of the longitudinal polarized parton densities in QCD and their ?rst moments [2,3],which correspond to the spins carried by the quarks and gluons in the nucleon.

One of the features of polarized DIS is that a lot of the present data are in the preasymptotic region (Q 2～1?5GeV 2,4GeV 2

account

where g 1(x,Q 2)pQCD is the well known (logarithmic in Q 2)NLO pQCD contribution g 1(x,Q 2)pQCD =

2π

δC q )+αs (Q 2

)

N f

(3)

and h TMC (x,Q 2)are the calculable kinematic target

mass

corrections

[7],which e?ectively belong to the LT term.In Eq.(3),?q (x,Q 2),?ˉq (x,Q 2)and ?G (x,Q 2)are quark,anti-quark and gluon polarized densities in the proton,which evolve in Q 2according to the spin-dependent NLO DGLAP equations.δC (x )q,G are the NLO spin-dependent Wilson coe?cient functions and the symbol ?denotes the usual convolution in Bjorken x space.N f is the number of active ?avors (N f =3in our analysis).In addition to the LT contribution,the dynamical higher twist e?ects

g 1(x,Q 2)HT =h (x,Q 2)/Q 2+O (Λ4/Q 4),

(4)

must be taken into account at low Q 2.The latter are non-perturbative e?ects and cannot be calculated in a model independent way.That is why we prefer to extract them directly from the experimental data.The method used to extract simultaneously the polarized parton densities and higher twist corrections to g 1is described in [4].Ac-cording to this method,the g 1/F 1and A 1(≈g 1/F 1)data have been ?tted using the experimental data for the un-polarized structure function F 1(x,Q 2)

g 1(x,Q 2)

F 1(x,Q 2)exp

(5)

As usual,F 1is replaced by its expression in terms of

the usually extracted from unpolarized DIS experiments F 2and R and phenomenological parametrizations of the experimental data for F 2(x,Q 2)[8]and the ratio R (x,Q 2)of the longitudinal to transverse γN cross-sections [9]are used.Note that such a procedure is equivalent to a ?t to (g 1)exp ,but it is more precise than the ?t to the g 1data themselves actually presented by the experimental groups because here the g 1data are extracted in the same way for all of the data sets.Note also,that in our analysis the logarithmic Q 2dependence of h (x,Q 2)in Eq.(5),which is not known in QCD,is https://www.wendangku.net/doc/468531365.html,pared to the principal 1/Q 2dependence it is expected to be small and the accuracy of the present data does not allow its determination.Therefore,the extracted from the data values of h (x )correspond to the mean Q 2for each x -bean (see Table II and the discussion below).

As in our previous analyses,for the input NLO polar-ized parton densities at Q 20=1GeV

we have adopted a simple parametrization

x ?u v (x,Q 20)=

ηu A u x a u xu v (x,Q 20),x ?d v (x,Q 20)=ηd A d x a d xd v (x,Q 20),x ?s (x,Q 20)=ηs A s x a s xs (x,Q 20),x ?G (x,Q 20)

ηg A g x a g xG (x,Q 20),

(6)

where on the RHS of (6)we have used the MRST99(cen-tral gluon)[10]parametrizations for the NLO(

MS (n f

=4)=300MeV,which

corresponds to αs (M 2

z )

=0.1175,as obtained by the MRST NLO QCD analysis [11]of the world unpolarized data.This is in excellent agreement with the current

world average αs (M 2

z )=0.1176±0.002[12].

III.RESULTS OF ANALYSIS

In this section we will discuss how inclusion of the

CLAS proton and deuteron g 1/F 1data [5]and the new COMPASS data on A d 1[6]in?uence our previous results [3]on polarized PD and higher twist obtained from the NLO QCD ?t to the world data [1,13,14],before the CLAS and the latest COMPASS data were available.

Impact of CLAS data

The CLAS EG 1/p ,d data (633experimental points)we have used in our analysis are high-precision data in the following kinematic region:{x ～0.1?0.6,Q 2～1?5GeV 2,W >2GeV }.As the CLAS data are mainly low Q 2data where the role of HT becomes im-portant,they should help to ?x better the higher twist e?ects.Indeed,due to the CLAS data,the determina-tion of HT corrections to the proton and neutron spin structure functions,h p (x )and h n (x ),is signi?cantly im-proved in the CLAS x region,compared to the values of HT obtained from our LSS’05analysis [3]in which a

0.00.20.40.60.8

(x )[G e V

]

FIG.1:E?ect of CLAS data on the higher twist values.

NLO(

MS)LSS’05PPD.As

expected,the central values of the polarized PD are prac-tically not a?ected by the CLAS data (see Table I).This is a consequence of the fact that at low Q 2the devia-tion from logarithmic in Q 2pQCD behaviour of g 1is accounted for by the higher twist term (4)in g 1.Indeed,if one calculates the χ2-probability for the combined world+CLAS data set using the LSS’05polarized PD and corresponding HT values,the result for χ2is 938.9for 823experimental points,which signi?cantly decreases to 718.0after the ?t.As seen from Table I,the best ?t to the combined data is achieved mainly through the changes in the HT values.This supports the theoretical framework in which the leading twist QCD contribution

is supplemented by higher twist terms of O (Λ2QCD /Q 2

).One can see also from Table I,that the accuracy of the determination of polarized PD is essentially improved.This improvement (illustrated in Fig.2)is a consequence of the much better determination of higher twist contri-butions to the spin structure function g 1,as discussed above.Due to the good accuracy of the CLAS data,one can split the measured x region of the world+CLAS data set into 7bins instead of 5,as used up to now,and there-

TABLE I:The parameters of the NLO(

190-16823-16χ2154.5718.0χ2/DF 0.8880.890

x i

h p (x i )[GeV 2]

x i

h n (x i )[GeV 2]

4 0.00.20.40.60.8 1.0

0.00.20.40.60.8 1.0

x 0.00.20.40.60.8 1.0 0.00.20.40.60.8 1.0

FIG.2:Impact of CLAS data on the uncertainties for NLO(

MS)LSS’06

strange quark and gluon densities corresponding to?ts of the

data using5and7x-bins for higher twist.

covering the low x region:0.004

the behaviour of the spin structure function g d1should be

more sensitive to the sign of the gluon polarization.Note

also,that due to the larger statistics the latest COM-

PASS data give more precise and detailed information

about A d1and g d1in the above experimental region(see

0.010.1

-0.0

-0.00.00.00.00.0(a)

0.010.1

(b)

FIG.5:Comparison of our NLO(

MS)

LSS’06polarized parton densities.

that reason this x region is not shown in Fig.5(a).The best ?t to the new g 1data is illustrated in Fig.5(b).The e?ect of the new data on the polarized parton densities and the higher twist corrections is illustrated in Fig.6and Fig.7,respectively.While (?u +?ˉu )

and (?d +?ˉd )parton densities do not change in the

experimental region (for that reason they are not shown in Fig.6),the magnitudes of both the polarized gluon and strange quark sea densities and their ?rst moments slightly decrease (see Fig.6and Table II).As a con-sequence,?Σ(Q 2=1GeV 2)increases from (0.165±0.044)to (0.207±0.040)for ?G >0and (0.243±0.065)for ?G <0(see below the discussion about ?G <0).As the COMPASS data are mainly at large Q 2,the im-pact of the new data on the values of higher twist correc-tions is negligible,and as expected,they do not improve the uncertainties of HT.The new central values practi-cally coincide with the old ones (see Fig.7(a)).The only exception are the central values of HT at small x for both the proton and the neutron targets which are slightly lower than the old ones.Note that this is the only region where the COMPASS DIS events are at small Q 2:1-4GeV 2.As a result,in the small x region two opposite ten-dencies occur.In order to make g d

1consistent with zero

for x <0.03,the HT contribution h d

=(h p +h n )0.925/2,

which is positive,decreases slightly,while (g d

)LT ,which is negative,grows slightly due to the smaller negative contribution of ?s (x,Q 2)and the smaller contribution of

TABLE II:E?ect of the new COMPASS data on polarized PD and HT.The parameters of the NLO(

?G>0?G>0?G<0ηu0.926?0.926?0.926?

a u0.264±0.0270.273±0.0280.273±0.028

ηd-0.341??0.341??0.341?

a d0.172±0.1180.202±0.1180.160±0.108

ηs-0.070±0.006-0.063±0.005-0.057±0.010

a s0.674±0.0530.715±0.0520.746±0.088

ηg0.173±0.1840.129±0.166-0.200±0.414

a g 2.969±1.437 3.265±1.6680.698±0.806

0.028 2.00.034±0.0400.010±0.0390.017±0.041 0.075 2.4-0.001±0.030-0.016±0.030-0.019±0.037 0.150 1.7-0.046±0.010-0.050±0.009-0.056±0.018 0.250 1.8-0.055±0.011-0.059±0.010-0.067±0.013 0.350 2.4-0.050±0.013-0.054±0.012-0.060±0.013 0.450 3.2-0.012±0.015-0.016±0.015-0.020±0.015 0.625 4.10.018±0.0150.016±0.0150.014±0.015 0.028 1.80.187±0.0640.165±0.0640.180±0.065 0.075 2.40.190±0.0440.173±0.0440.174±0.049 0.150 1.40.104±0.0410.107±0.0390.092±0.040 0.250 1.50.018±0.0300.019±0.0300.006±0.029 0.350 2.20.028±0.0260.031±0.0250.019±0.023 0.450 3.00.010±0.0200.013±0.0200.005±0.019 0.625 3.90.014±0.0130.016±0.0130.012±0.012

?G(x,Q2),convoluted with its Wilson coe?cient func-tionδC G(x),which is negative in this x range(see Eq.

(3)).

We have also checked the stability of our results with respect to a change inαs(M2z),which in our analysis coincides with its current world average,as mentioned in Section II.Whenαs(M2z)is varied by one standard deviation±0.002,the change of the values of the free pa-rameters is within their errors.In particular,the change ofηs andηg,the?rst moments of the polarized quark sea and gluon densities,is smaller than10%of their standard deviations.

C.The sign of the gluon polarization

We have also studied the possibility of a negative po-larized gluon density.Starting with a negative value for ηg=?G(Q20)(the?rst moment for the input gluon po-larized density?G(x,Q20)),we have found a minimum inχ2corresponding to a negative solution forηg,and to negative?G(Q2)and x?G(x,Q2).The values ofχ2 corresponding to the?ts with?G>0and?G<0are practically the same(see Table II)and the data cannot distinguish between these two solutions for?G(see Fig.5(a)).Note that in our previous analyses we also found solutions with negative?G,but they were not presented because the correspondingχ2were signi?cantly larger than those corresponding to the solutions with positive ?G.

In Fig.8the negative polarized gluon density is com-pared with the positive one.As seen,the shape of the negative gluon density di?ers from that of positive one, but in both cases the magnitude of x?G is small.Con-sequently the parton densities obtained in the?ts with ?G>0and?G<0are almost identical.For the strange quarks this is illustrated in Fig.8.Thus the the-oretical curves(g d1)LT for the two types of gluon polariza-tion are practically identical,even in the region x<0.01 (see Fig.5(b)).

Furthermore,as seen in Fig.7(b),the extracted HT values corresponding to the?ts with?G>0and?G< 0,are e?ectively identical.Thus also the total theoretical expression(g d1)tot is essentially the same for?G>0and ?G<0,even at very small x<0.01.

These results are in contrast to those obtained in the COMPASS analysis[6]where there is a signi?cant di?er-ence between the theoretical curves corresponding to the cases?G>0and?G<0at very small x,i.e.in the region0.004

0.00.20.40.60.8

(a)

x h

(x )[G e V

]

0.00.20.40.60.8

(b)

(x )[G e V

]

FIG.7:E?ect of new COMPASS data on the higher twist values (a).Comparison between HT values corresponding to the ?ts

with ?G >0and ?G <0(b).

0.010.11

FIG.8:Strange quark sea densities x ?s (x )corresponding to the ?ts with ?G >0,?G <0and changing in sign x ?G .

question of HT contributions,which are not taken into account by COMPASS.In the above x region,Q 2is small (Q 2～1?3GeV 2)and we have found that the HT con-tribution to (g d

1)tot ,h d (x )/Q 2,is positive and large,up

to 40%of the magnitude of (g d

1)LT (see Fig.5(b)).Thus

what is ?tted by (g d

1)LT (COMPASS)is signi?cantly dif-ferent from what is ?tted by our (g d

1)LT (LSS)at small x ,

i.e.(g d 1)LT (COMPASS)=(g d

1)LT (LSS)+h d (x)/Q 2.As a result:i)The strange quark sea densities obtained in the two analyses are di?erent,especially in the case of ?G <0(see Fig.9and Fig.10).ii)The gluon den-sities obtained by COMPASS in both ?ts (?G >0and ?G <0)are more peaked than ours.

Finally,concerning the possible solution with negative ?G we would like to point out the much larger uncertain-ties in the determination of the strange quark sea and gluon densities,x ?s and x ?G ,and respectively,their ?rst moments (see Fig.11and Table II).As seen from Fig.11,the positive gluon density x ?G (x )lies in the er-ror band of the negative gluons except for x larger than 0.2.x ?s (x )corresponding to the positive ?G solution lies entirely in the error band of x ?s (?G <0).Bearing in mind the high precision of the CLAS and new COMPASS data over a large range in Q 2we have studied the possibility to obtain from the ?t to the world inclusive DIS data a gluon density which changes sign as a function of x .Such a density was discussed in [15]in order to describe the double longitudinal spin asym-metry A LL of inclusive π0production in polarized p+p collisions measured by the P HENIX [16]and STAR [17]Collaborations at RHIC.To that end we introduced a fac-tor (1+γx δ)in the input gluon density in (6)with two new free parameters,γand δ,to be determined from the ?t to the data.In Fig.8,the determined strange quark and gluon densities at Q 2=2.5GeV 2are compared with those corresponding to the positive and negative ?G so-lutions.As seen from Fig.8,the oscillating in sign gluon density lies between those of positive and negative ?G .The value of χ2per degree of freedom is 0.895,which coincides with the values obtained with purely positive or negative x ?G (x ).

0.010.11

FIG.9:Comparison between our strange quark sea and gluon densities

corresponding to?G>0and those obtained by COMPASS[6].

0.010.11

FIG.10:Comparison between our strange quark sea and gluon densities corresponding to?G<0and those obtained by COMPASS[6].

0.010.11

FIG.11:The uncertainties for the strange quark sea and gluon densities corresponding to a negative gluon polariza-tion.

Thus,we are forced to conclude that the accuracy and Q2range of the present DIS data is not good enough to discriminate between these three possibilities.At Q2=1GeV2,the shape of the oscillating in sign po-larized gluon density is consistent with that obtained by the AAC Collaboration from a combined analysis of DIS (CLAS and new COMPASS not included)andπ0asym-metry data[18].Note,however,that compared to the central value of the?rst moment?G AAC=?0.56±2.16 at Q2=1GeV2,presented in[18],the central value of our?G is positive,0.006,and much smaller in magni-tude.Under evolution in Q2neither?G(Q2)AAC,nor our?G(Q2)changes sign,and their magnitudes increase with increasing of Q2.As a result,the shape of the cor-responding gluon densities for Q2>Q20will follow dif-ferent tendencies:x?G(x,Q2)AAC becomes negative for larger x with increasing of Q2,while our gluon density for Q2>6GeV2is positive for any x in the experimental region(see Fig.12).

In Fig.13the ratio?G(x)/G(x)calculated for the di?erent?G(x)obtained in our analysis and using G(x)MRST′02taken from[19],is compared to the exist-ing direct measurements of?G/G[20].(Note that the MRST’02unpolarized parton densities were used also in the positivity constraints imposed on the polarized par-ton densities obtained in our analysis.)The theoretical curves are given for Q2=3GeV2.The most precise value for?G/G,the COMPASS one,is well consistent with any of the polarized gluon densities determined in our analysis.

0.010.11

FIG.12:Evolution in Q2of oscillating-in-sign gluon density.

FIG.13:Comparison between the experimental data and NLO(

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