- Email: [email protected]
Spin-dependent quark and gluon distributions
Ph}.sit~ Lettet'~ B 319 ( 1993"1 .85-.9(I Norlh-Ilolland
PHYSICS LETTERS B
Spin-dependent quark and gluon distributions * D. de F l o r t a n . L.N. Epele, H. F a n c h i o t t i , C.A. G a r c i a C a n a l t a n d R. Sassot Laboratorto de Fistca Tedrwa. Departamento de P'istca. ('mverstdad Naztonal de La Plata. C C'. 67. 1900 La Plata. 4r~entma Received 29 June 1993; revtvM manuscnpt rccet,,'ed 28 September 1993 Editor:. M. Dine
Spin dependent quark and $1uon dlstributaons constructed m the frame~ork of thc spin ddut~on model and m agreement w~th recent SMC data on the deuteron polarised asymmetries are presented. This model, when included the anomalous $1uon comrtbuhon produces quark distributions that g~,,e aver) good de~-'nptlon of all available experimental data verifying Bjorken and Ellis-Jaffe sum rules.
I. Introduction Recently the Spin Muon Collaboration (SMC) [ I ] at ('ERN has reported on the first measurement of the deuteron polansed asymmetr) in deep inelastic scattering of polarised muons off polarised deuterons. The kinematical range covered is 0.006 < x < 0.6. I GeV: < Q: < 30GeV 2. From this measurement, the first moment of the spin-dependent structure funchon gta has been found to be I
r( = [ 0 = 0.023 -- 0.020(slat.) ± 0.015(syst.).
(I)
two standard deviations smaller than the prediction of the Ellis-Jaffc sum rules [ 2 ] I] a'tm,,-j,ff~ = 0.085 -t: 0.005
(2)
confirming the violatton of these sum rules as it was suggested by EMC data on the spin-dependent structure function for h.~drogen targets [3J ~PIFMC = 0 . 1 2 6 + 0 . 0 1 0 ( s l a t . ) ~ 0 . 0 1 5 1 s ~ s t . ) . P
It-ah,-J,fr¢ = 0.189 -r_ 0.005. Parlmlb supported b~ CONICET-Argentina. Fellow of the Fundactrn Anlorchas. Argentina.
I'-.l~evtcrScience Pubhshc~ B.V SSI)I 1)37(1-26'93[9 ~,)F I .LI5.x
(3) ~4)
Notwithstanding the flagrant violation of these parton model sum rules. SMC data for the deuteron combined with that of EMC for thc proton gives for the difference between the first moments of the proton and the neutron structure functions /i p - G"I'~MC+FMC = 0.20 ± 0.05(star.) + O.04(syst.) (5) in excellent agreement ~ith the Bjorken sum rule prediction [4J. As the average Q: values of the two experiments are d~fferent, some corrections should be done in order to combine the data [5 ]. However. in the case of EMC and SMC data, these corrections are smaller than the experimental errors and therefore neglected. Another group, EI42 at SLAC [6], has also tested recently th~s sum rule. Their estimate is based on an independent measurement of the neutron as}mm e t o from helium targets but in a veD low Q' range. It has been shown [ 5 ] that when the pertinent scaling violating contributions to the structure functions are taken into account, their data is also consistent with the the Bjorken sum rule. Taking into account that the Ellis-Jaffe prediction for the the difference between proton and neutron structure functions is perfectly consistent wath that of the Bjorken sum rule, the obliged question is: what is missing m the Ellis-Jaffe prediction for the sum that almost cancel in the difference? -t _,~,>
Volume 319. number 1.2.3
PHYSICS I.FrTER.',,;B
The answer comes from the proposal, based on the axial anomaly [7]. that has been made to explain the startling conclusions of the EMC experiment. The claim ~s that the matrix elements of the flavour singlet axial current involves not onl.~ the net quark spin, as m the na,ve quark model, but also a gluomc contrtbution, namely
,q~(.):) = ~ ~ e ~ A q ' ( x )
2 2n ' V ' ' A g ( x ) "
(6)
t
Aq, and Ag are the polarised quark and gluon densities respectively, the first term is the naive parton model contribution and the second is the anomalous one. Isospin symmetry at the hadron level implies that the anomalous contribution above should be the same for the proton and for the neutron structure functions. Then m order to conciliate EMC data and the Ellis-Jaffe sum rule for the proton. F~p ~ - J , n , - A ~ w ~ ~ F~r EMt •
(7)
one needs AF~v ~
= 0.063.
(8)
The Ellis-Jaffe prediction for the moments does not include an eventual contribution from the strange quarks which would lead to an extra term m eq. ( 71. However there are strong arguments that constraint th~s contribuuon to be much smaller than that expected from gluons [81. This will be reflected in our anal.vs~s. Consequently one can predict the sum of proton and neutron moments to be Fip + Fi)'l I ~n-),ne - 2~r~t¢~°~ "" 0.061 .
23 l)~.'x"ember199 ~,
The existence of a gluonic contribution affects greatly the extraction of polarised quark densities, ~,hich in principle should be done using spindependent structure functions as input. Unfortunately, the magnitude of Ag has not been measured independently in other processes and one has to rely on the analysis of EMC and SMC data. In order to compute the cross section asymmetries m processes capable of yielding good estimates of the polansed gluon distribution, Sridhar and Leader [9] have recently presented three sets of quark and gluon distributions. They are consistent with recent unpolansed quark and gluon distributions, and also fit the previous data on g~t Ix ) from EMC. In their analysis. they pay special attention to the constraint on the polarised dtstributions coming from the exact form and magmtude of the unpolarised ones. In the present work we try an alternative method to produce polarised distributions from the unpolarised ones. The proposal is to apply spin dilution functions [ 10 ] which are adjusted in order to fit EMC and SMC data. We also take into account the anomalous $1uon contnbution, with different polarised gluon distributtons, and compare its effects in the fits of the structure functton data, in the constraints given by the hyperon ,a-decay data and in the saturation of the polarised sum rules. In this wa.~. we can construct a set of polansed quark and gluon distributions, compatible with the unpolarised distributions. This set fits EMC, SMC and E142 data on the proton, deuteron and neutron polansed asymmetries respectively, satisfying the constraints given by the hyperonic semileptonic decay constants, and verifying the Bjorken and Ellis Jaffe sum rules.
(9)
which is in agreement with the expertmental value
2. Spin dilution fnnctions
Ftp + ~"I~Mc = 0.049-]: 0 . 0 4 4 ( s t a t . ) : 0.032(syst.).
The idea of relating spin.dependent quark distributions to the corresponding spin-independent distributions was originally implemented in a model for the spin-dependent nucleon structure functions by Carlitz and Kaur [ 10]. The starting point in their model are the S U ( 6 ) wavefunctions for the nucleons, which are extended in order to have a dependence on the Bjorken variable x. This dependence can easily be related to that of the spin-independent distributions
(I0) The anomalous contr,bmion obviously cancels in the dtfference between the proton and neutron moments. This explains the apparent paradox. Consequently, the SMC data is also consistent with the suggestion of an anomalous gluonic contribution to the spindependent structure functions. 2~e')
Volume 319. number 1.2.3
PHYSICS LETI ElLSB
and the SU(6) wavefunction is relaxed, to take into account the quark spin-flip due to gluon emission, by means of a spin diluhon function. The shape of the spin dilutmn function can be fixed in order to satis~' the constraints on the polarised distributions for x 0 and x ~ 1 and, in this wa,,, the model provides a prediction for the spin dependent valence quark distributmns. Since the original attempt of Carlitz and Kaur some variations of the idea have been developed in order to incorporate flavour dependence to the spin dilution function [11 l, polarisation of the sea quarks [8] and the use of EMC and SLAC data on the spin-dependent proton structure function to fix the shape of the dilution funetmns. In the present work we follow these suggestions and we also incorporate the anomalous gluonic contribution to the spin-dependent structure functions. Th~s last contributmn depends strongly on the net gluon polarisation ( A g ( x ) in eq. (6)) which. m the spirit of the dilution model, is related to the unpolarised gluon distribution g ( x ) b) Ag (x. O" ) = .f~ (x, Q-~ )g (.v, Q2 ).
(11 )
where .ft (x. Q") is the spin dilution function for the gluon. As wc will work at a fixed Q: value of I0 GeV "~. from now on we can drop the Q: dependence of the dilution functions and the quark d~stributmns. The Q2 dependence of the dilution function takes into account the difference in the evolution of g~ and F:. Although the analysis of the intrinsic gluon distribution in a hadron is essentially non-perturbatwe, several theoretical constraints limit its form. It has been shown [ 12 ] that in the low x domain, the quarks in the hadron radiate gluons coherentl'~. Then computing the emission of gluons from the quark lines by taking into account the interference between amplitudes, the spin dilution function for the gluon vanish linearly w~th x: lira .fg (x) ,~, x.
( 12 )
t'~0
In the large x limit, the coupling of quarks to gluons tends to match the sign of the quark helicity to the gluon helicity impl~,ing
lim I -./~(.~)
,-,I + .f~(x)
(I- x)".
(13)
23 I)¢ccmbcr 1993
An interpolation betwecn these two limits provides I
.fg(x) = I + ag(I - x ) 2 / x
"
(14)
where a~ is a parameter to be adjusted. For dilution functions of the valence quark distributions we adopt the x dependence given by [ 101, 1 f,/(.x.S3) = 1 + a,/(l - x ) ' - / x " ¢ '
(15)
where S.a indicates the total spin projection of the spectator pair of quarks and ,~q gives the behaviour of the ratio between the quark and gluon densities as x --, 0, q, (.s'_.__~))-,. x"¢. g(x )
( 16 )
For the sea quarks we take I . ~ ( x ) ~ I + a~(I - . r ) 4 / x ~'-"
(17)
The strange quark contribution has also been included for completeness albeit we expect it to be suppressed. In th~s wa~. the spin-dependent structure functions take the form
2 x ~ I x ) = ~xu,.(.x')f~(x:0) - ~.rd,.(x)[4.f~(.~.;01 + 4 f u ( x ; I ) + 2 f , t(.x':l)]
- .f,~(x;0)
(~:) '~' N f . r A g ( x ) + ~ x ' f f ( x ) f ~ ( x )
+ g.rd(x)f~(x)+
gx~(x).f,-(x).
(18)
2x g ~ ( x ) = ~ x u, (x).fd ( x: 0 ) . ~ x d ~ ( . t ) [ - 4 f , ( x ; 0 ) + 8 f , ( x ; l) + f a ( x ; O ) + l~(.x':l)l •
(52) ,,, N / x a g ( x ) + .
T~
+ ?~.x-~(.,-).t-~(.s-)+ ~x~(.,.).h(x).
~.~3(x)f~(.~) "
(19)
Notice that these equations hold only at the Q2 value where the parameters were fixed. The functional form for the dilution functions together with the verT recent C T Q P D parametrisation [ 13 ] for the spin-independent patton distributions allow to fix the values for the parameters a¢ and a~ which give the best fit to EMC and SMC polarised asymmetries. For this aim we proceed as follows: first 287
Volume 319. n u m b e r 1.2.3
P H Y S I C S I . E ' I I ERS B
"3 lM.'ccmber 1993
,q "'4
'
4
\
c.
I I • 4
, ,,
1
1,1
?
I1'
I
we fix the value of the parameter a s in order to compatlbfli;,e the Ellis-Jaffe prediction for the sum of the proton and neutron moments ~P + [ ~ . corrected b~, the anomalous contribution - 2 A / ~ ¢'x~, with the SMC value. In this way we obtain a polarised gluon distribution similar to the onc of ref. [9] but that satisfies the above mentioned constraints on the asymptotic bchaviour of polariscd gluons. The result for the spin dilution function is presented m fig. I. As can be seen m the figure, the value for a s forced by the convergence of the Ellis-Jaffe sum rule and the gluomc as~ mptotic constraints imply a strong correlation between the gluon and valence quark polarisatlon factors. In fig. 2 the resulting polarised gluon distribution is shown together with the one suggested in ref. [9]. Our polarised gluon distribution implies a stronger net polarisatlon for low values of x. With the gluon parameter a¥ fixed, we determine the other parameters needed to fit EMC and SMC data. In order to avoid uncertainties related with higher twist contributions to the polarised structure functions, we have performed the analysis using polarised asymmetries, which have a negligible Q2 dependence. Figs. 3-5 compare the asymmetries obtained with our choice of parameters with those obtamcd experimentally. The dashed lines in both figures correspond to a spin dilution model with no anomalous gluon contribution [8]. Although this last model gives a good description of the proton asymm e t e , it fails to reproduce the shape of the neutron and deuteron asymmetries. 288
,¢,
I'
Fig. I. The spin dilution function fx(x) for a¢ = 0.027.
i
\
I
.'
o
\, •
~ , . ,
..
11
Ft& 2. The spin-dependent $1uon distribution Ag(x) compared to the the one proposed in ref. [9] (dashes).
s
•
v Ul;
IVf I
I
:('
~t
L.
I-i"
I
. . . . . . . . 1,,
g
'iI
I
...I :f
,:
Ft$, 3. The spm-dependent proton asymmetr~ given by the model compared to EMC and earlier SL~IC data [ 14 ]. FIg. 6 shows valence quark polarised densities obtained from the model developed in this paper together with the ones obtained in ref. [8]. As expected, the effect of gluon polarisation allows valence quark to carT')' more net spin. In fact, the quantity L" = Au + Ad + As takes the value L" = 0.63 whereas for reference [8] X = 0.22. The sea quark polansed densities (fig. 7) show a behaviour similar to that of gluons, as could be expected. At variance with other analysis, the three denslties have heliclties positively correlated with that of the proton. The differences in shape between flavours can be explained arguing that least massive quarks
Vtflumc 319. number 1.2.3
PHYSICS LETTERS B
23 December 1993
~4
I
. . . . . . . .
/
i ( , .5
.. ,,,,.
-,
•
"
,,,
(i.,
~
t
I, (i,'
I
~,',[
v,
\
~
Ix
/t
,],
I
,~ ,i~
fl
~"1 c, n
I
,
"'I
F~
()
1
~",
~ .\d.
I 6
I
6
=
,
,"
Ftg. 4. The spin-dependent deuteron asymmetry given by the model compared to SMC data.
.
6
6
. . . .
-"
I
1( I
a
I ! "
Fig. 6. The spin-dependent valence quark distributions with and wtthout {dashes) net gluon polansatton m the model.
• ;:;I r ,'~ i. '
i-~'; "7"
"
!
.14
i
/1 I
..r[
t
•
\
\.
•
"~,:"
I I
",;
I11 '
1 ~"t :
I
Ftg. 5. The spin-.dependent neutron asymmetry gtven b~, the model cornpart-d to E-142 data.
F~g. 7. The same as fig. 6 for the sea quark &stributions.
are most hip,hi.,, polariscd m the sea, and this has been taken into account m the dilution functions. The unpolansed sea quark distributions in the C T Q P D parametrisation are not isospln invariant: this produces also a difference m the polarised densities. Table I shov,'s the set of parameters chosen here together with the set of ref. [8]. Notice that the differences between the distributions come not only from the different parameters but also from the unpolarised distributions used in each case. In table 2 we present the values expected for sum rules involving polarlsed structure functions together with those obtained integrating our spin..dcpendent
Table I
Parameter
Yh,s model
Ref. [8]
a,,,~.o = ~ au.._., aa~.o = a~
0 22 0.07 0.6
a,¢~.]
0. I
0.216 0.02.$ 0.55 0.062
at
0.027
-
distributions. Ellis--Jaffe sum rules have been corrected with the anomalous gluon contribution. We 28q
Volume 31V. number 1.2.3
23 December 1993
PHYSICS LETTERS B
Table 2
Experimental data
Sum rules pred~cttons
Spm-ddution model with
FrY - F:' /Tat + /"1"
Fit' F/D F + D
SMC + EMC 0.20 - 0.05 ± 0.04 SMC 0.049 + 0.044 ± 0.032 EMC 0.126 ~ 0.01 = 0.015 0.631 ± 0 0 2 4 1.232±0015
also show the same quantities for a set o f distributions in which the gluon c o n t r i b u t i o n has been fixed | o zero. in s u m m a r y , we have presenled a set o f polarised quark and gluon d~stributions o b t a i n e d in the framework o f a spin dilution m o d e l that includes the a n o m a lous gluon c o n t r i b u t i o n to polarised observables. T h e set gives a very good description o f recent polarised data, satisfies Bjorken and Ellis Jaffe sum rules and the constraints given by h.vperonic decays.
References 111SM Collab.. B Ade~a el al. Phys. Lelt. B 302 (1993) 533. [2]J. Elhs and R.L. Jaffe. Phys. Rev. D 9 (1974) 1444: D 10 (1974) 1669(E)
2()0
BSR 0.191 = 0.002 EJ 0.061 = 0.008 EJ 0.124 = 0.007
gluons
0.192 0.052 0.122 0.634 1.25
without gluons 0.199 0.063 0.131 0.422 1.30
[3] EM Collab.. J. Ashman et al., Ph!,s. Left. B 206 ( 19881 364. 14] J.D. Bjorken. Phys. Rev. 148 (19661 1467. [51 J. Ellis and M. Karliner. CERN-TH-b898/93 (1993). [61 EI42 Collab., P.L. Anthony el al.. SLAC.PUB-6101
(1993). [7] G. Altarelh and G.G. Ross, Ph.~. Left. B 212 (1988) 883. 18] R.M. Woloshyn, Nucl. Phys. A 496 (1989) 749. {9] K. Sridhar and E. Leader. Phys. Leu. B 295 (1992) 283. [ 10] R. Carhtz and J Kaur. Phys. Rev. Left. 37 (1976) 673. [ I I ] A . Schafer. Phys. Left. B 208 (1988) 175. [12]$3. BrodskT and I.A. Schm~dt, Ph)s. Left. B 234. (1990) 144. [13] CTEQ Collab., J. BOltS el al., preprinl FERMILABPUB-92-371 (1993). 1141M.J. Alguard et al., Phys. Rev. Left. 37 (1976) 1261; 41 (1978) 70. G. Baum et al., Phys. Rev. Left. 45 (1980) 2000;, 51 11983) 1135.
PHYSICS LETTERS B
Spin-dependent quark and gluon distributions * D. de F l o r t a n . L.N. Epele, H. F a n c h i o t t i , C.A. G a r c i a C a n a l t a n d R. Sassot Laboratorto de Fistca Tedrwa. Departamento de P'istca. ('mverstdad Naztonal de La Plata. C C'. 67. 1900 La Plata. 4r~entma Received 29 June 1993; revtvM manuscnpt rccet,,'ed 28 September 1993 Editor:. M. Dine
Spin dependent quark and $1uon dlstributaons constructed m the frame~ork of thc spin ddut~on model and m agreement w~th recent SMC data on the deuteron polarised asymmetries are presented. This model, when included the anomalous $1uon comrtbuhon produces quark distributions that g~,,e aver) good de~-'nptlon of all available experimental data verifying Bjorken and Ellis-Jaffe sum rules.
I. Introduction Recently the Spin Muon Collaboration (SMC) [ I ] at ('ERN has reported on the first measurement of the deuteron polansed asymmetr) in deep inelastic scattering of polarised muons off polarised deuterons. The kinematical range covered is 0.006 < x < 0.6. I GeV: < Q: < 30GeV 2. From this measurement, the first moment of the spin-dependent structure funchon gta has been found to be I
r( = [ 0 = 0.023 -- 0.020(slat.) ± 0.015(syst.).
(I)
two standard deviations smaller than the prediction of the Ellis-Jaffc sum rules [ 2 ] I] a'tm,,-j,ff~ = 0.085 -t: 0.005
(2)
confirming the violatton of these sum rules as it was suggested by EMC data on the spin-dependent structure function for h.~drogen targets [3J ~PIFMC = 0 . 1 2 6 + 0 . 0 1 0 ( s l a t . ) ~ 0 . 0 1 5 1 s ~ s t . ) . P
It-ah,-J,fr¢ = 0.189 -r_ 0.005. Parlmlb supported b~ CONICET-Argentina. Fellow of the Fundactrn Anlorchas. Argentina.
I'-.l~evtcrScience Pubhshc~ B.V SSI)I 1)37(1-26'93[9 ~,)F I .LI5.x
(3) ~4)
Notwithstanding the flagrant violation of these parton model sum rules. SMC data for the deuteron combined with that of EMC for thc proton gives for the difference between the first moments of the proton and the neutron structure functions /i p - G"I'~MC+FMC = 0.20 ± 0.05(star.) + O.04(syst.) (5) in excellent agreement ~ith the Bjorken sum rule prediction [4J. As the average Q: values of the two experiments are d~fferent, some corrections should be done in order to combine the data [5 ]. However. in the case of EMC and SMC data, these corrections are smaller than the experimental errors and therefore neglected. Another group, EI42 at SLAC [6], has also tested recently th~s sum rule. Their estimate is based on an independent measurement of the neutron as}mm e t o from helium targets but in a veD low Q' range. It has been shown [ 5 ] that when the pertinent scaling violating contributions to the structure functions are taken into account, their data is also consistent with the the Bjorken sum rule. Taking into account that the Ellis-Jaffe prediction for the the difference between proton and neutron structure functions is perfectly consistent wath that of the Bjorken sum rule, the obliged question is: what is missing m the Ellis-Jaffe prediction for the sum that almost cancel in the difference? -t _,~,>
Volume 319. number 1.2.3
PHYSICS I.FrTER.',,;B
The answer comes from the proposal, based on the axial anomaly [7]. that has been made to explain the startling conclusions of the EMC experiment. The claim ~s that the matrix elements of the flavour singlet axial current involves not onl.~ the net quark spin, as m the na,ve quark model, but also a gluomc contrtbution, namely
,q~(.):) = ~ ~ e ~ A q ' ( x )
2 2n ' V ' ' A g ( x ) "
(6)
t
Aq, and Ag are the polarised quark and gluon densities respectively, the first term is the naive parton model contribution and the second is the anomalous one. Isospin symmetry at the hadron level implies that the anomalous contribution above should be the same for the proton and for the neutron structure functions. Then m order to conciliate EMC data and the Ellis-Jaffe sum rule for the proton. F~p ~ - J , n , - A ~ w ~ ~ F~r EMt •
(7)
one needs AF~v ~
= 0.063.
(8)
The Ellis-Jaffe prediction for the moments does not include an eventual contribution from the strange quarks which would lead to an extra term m eq. ( 71. However there are strong arguments that constraint th~s contribuuon to be much smaller than that expected from gluons [81. This will be reflected in our anal.vs~s. Consequently one can predict the sum of proton and neutron moments to be Fip + Fi)'l I ~n-),ne - 2~r~t¢~°~ "" 0.061 .
23 l)~.'x"ember199 ~,
The existence of a gluonic contribution affects greatly the extraction of polarised quark densities, ~,hich in principle should be done using spindependent structure functions as input. Unfortunately, the magnitude of Ag has not been measured independently in other processes and one has to rely on the analysis of EMC and SMC data. In order to compute the cross section asymmetries m processes capable of yielding good estimates of the polansed gluon distribution, Sridhar and Leader [9] have recently presented three sets of quark and gluon distributions. They are consistent with recent unpolansed quark and gluon distributions, and also fit the previous data on g~t Ix ) from EMC. In their analysis. they pay special attention to the constraint on the polarised dtstributions coming from the exact form and magmtude of the unpolarised ones. In the present work we try an alternative method to produce polarised distributions from the unpolarised ones. The proposal is to apply spin dilution functions [ 10 ] which are adjusted in order to fit EMC and SMC data. We also take into account the anomalous $1uon contnbution, with different polarised gluon distributtons, and compare its effects in the fits of the structure functton data, in the constraints given by the hyperon ,a-decay data and in the saturation of the polarised sum rules. In this wa.~. we can construct a set of polansed quark and gluon distributions, compatible with the unpolarised distributions. This set fits EMC, SMC and E142 data on the proton, deuteron and neutron polansed asymmetries respectively, satisfying the constraints given by the hyperonic semileptonic decay constants, and verifying the Bjorken and Ellis Jaffe sum rules.
(9)
which is in agreement with the expertmental value
2. Spin dilution fnnctions
Ftp + ~"I~Mc = 0.049-]: 0 . 0 4 4 ( s t a t . ) : 0.032(syst.).
The idea of relating spin.dependent quark distributions to the corresponding spin-independent distributions was originally implemented in a model for the spin-dependent nucleon structure functions by Carlitz and Kaur [ 10]. The starting point in their model are the S U ( 6 ) wavefunctions for the nucleons, which are extended in order to have a dependence on the Bjorken variable x. This dependence can easily be related to that of the spin-independent distributions
(I0) The anomalous contr,bmion obviously cancels in the dtfference between the proton and neutron moments. This explains the apparent paradox. Consequently, the SMC data is also consistent with the suggestion of an anomalous gluonic contribution to the spindependent structure functions. 2~e')
Volume 319. number 1.2.3
PHYSICS LETI ElLSB
and the SU(6) wavefunction is relaxed, to take into account the quark spin-flip due to gluon emission, by means of a spin diluhon function. The shape of the spin dilutmn function can be fixed in order to satis~' the constraints on the polarised distributions for x 0 and x ~ 1 and, in this wa,,, the model provides a prediction for the spin dependent valence quark distributmns. Since the original attempt of Carlitz and Kaur some variations of the idea have been developed in order to incorporate flavour dependence to the spin dilution function [11 l, polarisation of the sea quarks [8] and the use of EMC and SLAC data on the spin-dependent proton structure function to fix the shape of the dilution funetmns. In the present work we follow these suggestions and we also incorporate the anomalous gluonic contribution to the spin-dependent structure functions. Th~s last contributmn depends strongly on the net gluon polarisation ( A g ( x ) in eq. (6)) which. m the spirit of the dilution model, is related to the unpolarised gluon distribution g ( x ) b) Ag (x. O" ) = .f~ (x, Q-~ )g (.v, Q2 ).
(11 )
where .ft (x. Q") is the spin dilution function for the gluon. As wc will work at a fixed Q: value of I0 GeV "~. from now on we can drop the Q: dependence of the dilution functions and the quark d~stributmns. The Q2 dependence of the dilution function takes into account the difference in the evolution of g~ and F:. Although the analysis of the intrinsic gluon distribution in a hadron is essentially non-perturbatwe, several theoretical constraints limit its form. It has been shown [ 12 ] that in the low x domain, the quarks in the hadron radiate gluons coherentl'~. Then computing the emission of gluons from the quark lines by taking into account the interference between amplitudes, the spin dilution function for the gluon vanish linearly w~th x: lira .fg (x) ,~, x.
( 12 )
t'~0
In the large x limit, the coupling of quarks to gluons tends to match the sign of the quark helicity to the gluon helicity impl~,ing
lim I -./~(.~)
,-,I + .f~(x)
(I- x)".
(13)
23 I)¢ccmbcr 1993
An interpolation betwecn these two limits provides I
.fg(x) = I + ag(I - x ) 2 / x
"
(14)
where a~ is a parameter to be adjusted. For dilution functions of the valence quark distributions we adopt the x dependence given by [ 101, 1 f,/(.x.S3) = 1 + a,/(l - x ) ' - / x " ¢ '
(15)
where S.a indicates the total spin projection of the spectator pair of quarks and ,~q gives the behaviour of the ratio between the quark and gluon densities as x --, 0, q, (.s'_.__~))-,. x"¢. g(x )
( 16 )
For the sea quarks we take I . ~ ( x ) ~ I + a~(I - . r ) 4 / x ~'-"
(17)
The strange quark contribution has also been included for completeness albeit we expect it to be suppressed. In th~s wa~. the spin-dependent structure functions take the form
2 x ~ I x ) = ~xu,.(.x')f~(x:0) - ~.rd,.(x)[4.f~(.~.;01 + 4 f u ( x ; I ) + 2 f , t(.x':l)]
- .f,~(x;0)
(~:) '~' N f . r A g ( x ) + ~ x ' f f ( x ) f ~ ( x )
+ g.rd(x)f~(x)+
gx~(x).f,-(x).
(18)
2x g ~ ( x ) = ~ x u, (x).fd ( x: 0 ) . ~ x d ~ ( . t ) [ - 4 f , ( x ; 0 ) + 8 f , ( x ; l) + f a ( x ; O ) + l~(.x':l)l •
(52) ,,, N / x a g ( x ) + .
T~
+ ?~.x-~(.,-).t-~(.s-)+ ~x~(.,.).h(x).
~.~3(x)f~(.~) "
(19)
Notice that these equations hold only at the Q2 value where the parameters were fixed. The functional form for the dilution functions together with the verT recent C T Q P D parametrisation [ 13 ] for the spin-independent patton distributions allow to fix the values for the parameters a¢ and a~ which give the best fit to EMC and SMC polarised asymmetries. For this aim we proceed as follows: first 287
Volume 319. n u m b e r 1.2.3
P H Y S I C S I . E ' I I ERS B
"3 lM.'ccmber 1993
,q "'4
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4
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we fix the value of the parameter a s in order to compatlbfli;,e the Ellis-Jaffe prediction for the sum of the proton and neutron moments ~P + [ ~ . corrected b~, the anomalous contribution - 2 A / ~ ¢'x~, with the SMC value. In this way we obtain a polarised gluon distribution similar to the onc of ref. [9] but that satisfies the above mentioned constraints on the asymptotic bchaviour of polariscd gluons. The result for the spin dilution function is presented m fig. I. As can be seen m the figure, the value for a s forced by the convergence of the Ellis-Jaffe sum rule and the gluomc as~ mptotic constraints imply a strong correlation between the gluon and valence quark polarisatlon factors. In fig. 2 the resulting polarised gluon distribution is shown together with the one suggested in ref. [9]. Our polarised gluon distribution implies a stronger net polarisatlon for low values of x. With the gluon parameter a¥ fixed, we determine the other parameters needed to fit EMC and SMC data. In order to avoid uncertainties related with higher twist contributions to the polarised structure functions, we have performed the analysis using polarised asymmetries, which have a negligible Q2 dependence. Figs. 3-5 compare the asymmetries obtained with our choice of parameters with those obtamcd experimentally. The dashed lines in both figures correspond to a spin dilution model with no anomalous gluon contribution [8]. Although this last model gives a good description of the proton asymm e t e , it fails to reproduce the shape of the neutron and deuteron asymmetries. 288
,¢,
I'
Fig. I. The spin dilution function fx(x) for a¢ = 0.027.
i
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~ , . ,
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Ft& 2. The spin-dependent $1uon distribution Ag(x) compared to the the one proposed in ref. [9] (dashes).
s
•
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~t
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. . . . . . . . 1,,
g
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Ft$, 3. The spm-dependent proton asymmetr~ given by the model compared to EMC and earlier SL~IC data [ 14 ]. FIg. 6 shows valence quark polarised densities obtained from the model developed in this paper together with the ones obtained in ref. [8]. As expected, the effect of gluon polarisation allows valence quark to carT')' more net spin. In fact, the quantity L" = Au + Ad + As takes the value L" = 0.63 whereas for reference [8] X = 0.22. The sea quark polansed densities (fig. 7) show a behaviour similar to that of gluons, as could be expected. At variance with other analysis, the three denslties have heliclties positively correlated with that of the proton. The differences in shape between flavours can be explained arguing that least massive quarks
Vtflumc 319. number 1.2.3
PHYSICS LETTERS B
23 December 1993
~4
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Ftg. 4. The spin-dependent deuteron asymmetry given by the model compared to SMC data.
.
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Fig. 6. The spin-dependent valence quark distributions with and wtthout {dashes) net gluon polansatton m the model.
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Ftg. 5. The spin-.dependent neutron asymmetry gtven b~, the model cornpart-d to E-142 data.
F~g. 7. The same as fig. 6 for the sea quark &stributions.
are most hip,hi.,, polariscd m the sea, and this has been taken into account m the dilution functions. The unpolansed sea quark distributions in the C T Q P D parametrisation are not isospln invariant: this produces also a difference m the polarised densities. Table I shov,'s the set of parameters chosen here together with the set of ref. [8]. Notice that the differences between the distributions come not only from the different parameters but also from the unpolarised distributions used in each case. In table 2 we present the values expected for sum rules involving polarlsed structure functions together with those obtained integrating our spin..dcpendent
Table I
Parameter
Yh,s model
Ref. [8]
a,,,~.o = ~ au.._., aa~.o = a~
0 22 0.07 0.6
a,¢~.]
0. I
0.216 0.02.$ 0.55 0.062
at
0.027
-
distributions. Ellis--Jaffe sum rules have been corrected with the anomalous gluon contribution. We 28q
Volume 31V. number 1.2.3
23 December 1993
PHYSICS LETTERS B
Table 2
Experimental data
Sum rules pred~cttons
Spm-ddution model with
FrY - F:' /Tat + /"1"
Fit' F/D F + D
SMC + EMC 0.20 - 0.05 ± 0.04 SMC 0.049 + 0.044 ± 0.032 EMC 0.126 ~ 0.01 = 0.015 0.631 ± 0 0 2 4 1.232±0015
also show the same quantities for a set o f distributions in which the gluon c o n t r i b u t i o n has been fixed | o zero. in s u m m a r y , we have presenled a set o f polarised quark and gluon d~stributions o b t a i n e d in the framework o f a spin dilution m o d e l that includes the a n o m a lous gluon c o n t r i b u t i o n to polarised observables. T h e set gives a very good description o f recent polarised data, satisfies Bjorken and Ellis Jaffe sum rules and the constraints given by h.vperonic decays.
References 111SM Collab.. B Ade~a el al. Phys. Lelt. B 302 (1993) 533. [2]J. Elhs and R.L. Jaffe. Phys. Rev. D 9 (1974) 1444: D 10 (1974) 1669(E)
2()0
BSR 0.191 = 0.002 EJ 0.061 = 0.008 EJ 0.124 = 0.007
gluons
0.192 0.052 0.122 0.634 1.25
without gluons 0.199 0.063 0.131 0.422 1.30
[3] EM Collab.. J. Ashman et al., Ph!,s. Left. B 206 ( 19881 364. 14] J.D. Bjorken. Phys. Rev. 148 (19661 1467. [51 J. Ellis and M. Karliner. CERN-TH-b898/93 (1993). [61 EI42 Collab., P.L. Anthony el al.. SLAC.PUB-6101
(1993). [7] G. Altarelh and G.G. Ross, Ph.~. Left. B 212 (1988) 883. 18] R.M. Woloshyn, Nucl. Phys. A 496 (1989) 749. {9] K. Sridhar and E. Leader. Phys. Leu. B 295 (1992) 283. [ 10] R. Carhtz and J Kaur. Phys. Rev. Left. 37 (1976) 673. [ I I ] A . Schafer. Phys. Left. B 208 (1988) 175. [12]$3. BrodskT and I.A. Schm~dt, Ph)s. Left. B 234. (1990) 144. [13] CTEQ Collab., J. BOltS el al., preprinl FERMILABPUB-92-371 (1993). 1141M.J. Alguard et al., Phys. Rev. Left. 37 (1976) 1261; 41 (1978) 70. G. Baum et al., Phys. Rev. Left. 45 (1980) 2000;, 51 11983) 1135.