- Email: [email protected]
Single-spin azimuthal asymmetries in the “Reduced twist-3 approximation”
15 June 2000
Physics Letters B 483 Ž2000. 69–73
Single-spin azimuthal asymmetries in the ‘‘Reduced twist-3 approximation’’ E. De Sanctis a , W.-D. Nowak b, K.A. Oganessyan a,b,c a
INFN-Laboratori Nazionali di Frascati, Õia Enrico Fermi 40, I-00044 Frascati, Italy b DESY Zeuthen, Platanenallee 6, D-15738 Zeuthen, Germany c YereÕan Physics Institute, Alikhanian Br. St. 2, 375036 YereÕan, Armenia Received 28 February 2000; accepted 11 May 2000 Editor: R. Gatto
Abstract We consider the single-spin azimuthal asymmetries recently measured at the HERMES experiment for charged pions produced in semi-inclusive deep inelastic scattering of leptons off longitudinally polarized protons. Guided by the experimental results and assuming a vanishing twist-2 transÕerse quark spin distribution in the longitudinally polarized nucleon, denoted as ‘‘reduced twist-3 approximation’’, a self-consistent description of the observed single-spin asymmetries is obtained. In addition, predictions are given for the z dependence of the single target-spin asymmetry. q 2000 Elsevier Science B.V. All rights reserved.
Semi-inclusive deep inelastic scattering ŽSIDIS. of leptons off a polarized nucleon target is a rich source of information on the spin structure of the nucleon and on parton fragmentation. In particular, measurements of azimuthal asymmetries in SIDIS allow the further investigation of the quark and gluon structure of the polarized nucleon. The HERMES collaboration has recently reported on the measurement of single target-spin asymmetries in the distribution of the azimuthal angle f relative to the lepton scattering plane, in semi-inclusive charged pion production on a longitudinally polarized hydrogen target w1x. The sin f moment of this distribution was found to be significant for pq-production. For py it was found to be consistent with zero within present experimental uncertainties, as it was the case
E-mail address: [email protected] ŽK.A. Oganessyan..
for the sin2 f moments of both pq and py. Singlespin asymmetries vanish in models in which hadrons consist of non-interacting collinear partons Žquarks and gluons., i.e. they are forbidden in the simplest version of the parton model and perturbative QCD. Non-vanishing and non-identical intrinsic transverse momentum distributions for oppositely polarized partons play an important role in most explanations of such non-zero single-spin asymmetries; they are interpreted as the effects of ‘‘naive time-reversal-odd’’ ŽT-odd. fragmentation functions w2–6x, arising from non-perturbative hadronic final-state interactions. In Refs. w7,8x these asymmetries were evaluated and it was shown that a good agreement with the HERMES data can be achieved by using only twist-2 distribution and fragmentation functions. In this letter the single target-spin sin f h and sin2 f h azimuthal asymmetries are investigated in the light of the recent HERMES results w1x. It will be
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 5 9 6 - 7
E. De Sanctis et al.r Physics Letters B 483 (2000) 69–73
70
shown that these results may be interpreted towards a vanishing twist-2 quark transverse spin distribution in the longitudinally polarized nucleon1. Under this assumption, which will be called hereafter ‘‘reduced twist-3 approximation’’, the sub-leading order in 1rQ single target-spin sin f h asymmetry reduces to the twist-2 level and is interpreted as the effect of the convolution of the transversity distribution and the T-odd fragmentation function. In this situation, also measurements with a longitudinally polarized target at HERMES may be used to extract the transversity distribution in a way similar to that proposed in Ref. w11x for a transversely polarized target, once enough statistics will be collected. The sin f h and sin2 f h moments of experimentally observable single target-spin asymmetries in the SIDIS cross-section can be related to the parton distribution and fragmentation functions involved in the parton level description of the underlying process w3,4x. Their anticipated dependence on p T Ž k T ., the intrinsic transverse momentum of the initial Žfinal. parton, reflects into the distribution of PhT , the transverse momentum of the semi-inclusively measured hadron. The moments are defined as appropriately weighted integrals over this observable, of the cross section asymmetry:
²
< PhT < Mh
sin f h : < PhT <
2
hT
'
Mh 2
sin f h Ž d sqy d sy . ,
Ž 1.
q y hT Ž d s q d s .
Hd P < PhT < 2 MMh
sin2 f h : < PhT < 2
2
Hd P
hT
'
²
< PhT < Mh
sin f h : Ž x , y, z . 1
I1 LŽ x , y, z . q I1T Ž x , y, z . ,
s I0 Ž x , y, z . ²
< PhT < 2 MMh
MMh 2
sin2 f h Ž d sqy d sy . .
Ž 2.
q y hT Ž d s q d s .
Hd P
Here qŽy. denote the antiparallel Žparallel. longitu-
8 s I0 Ž x , y, z .
SLŽ 1 y y . h1HLŽ1. Ž x . z 2 H1H Ž1. Ž z . ,
Ž 4. where I0 Ž x , y, z . s Ž 1 q Ž 1 q y . I1 LŽ x , y, z . s 4SL
M Q
2
. f 1Ž x . D1Ž z . ,
Ž 2 y y . '1 y y
= xh LŽ x . zH1H Ž1. Ž z . yh1HLŽ1. Ž x . H˜ Ž z . ,
Ž 5. Ž 6.
With k 1 Ž k 2 . being the 4-momentum of the incoming Žoutgoing. charged lepton, Q 2 s yq 2 where q s k 1 y k 2 is the 4-momentum of the virtual photon. P Ž Ph . is the momentum of the target Žfinal hadron., x s Q 2r2Ž Pq ., y s Ž Pq .rŽ Pk 1 ., z s Ž PPh .rŽ Pq ., k 1T the incoming lepton transverse momentum with respect to the virtual photon momentum direction, and f h is the azimuthal angle between PhT and k 1T around the virtual photon direction. Note that the azimuthal angle of the transverse Žwith respect to the virtual photon. component of the target polarization, fS , is equal to 0 Žp . for the target polarized parallel Žanti-parallel. to the beam w12x. The components of the longitudinal and transverse target polarization in the virtual photon frame are denoted by SL and ST x , respectively.
1
After this work has been completed we became aware of Refs. w9,10x where this possibility has also been considered.
Ž 3.
sin2 f h : Ž x , y, z .
I1T Ž x , y, z . s 2 ST x Ž 1 y y . h1 Ž x . zH1H Ž1. Ž z . .
Hd P
²
dinal polarization of the target and M Ž Mh . is the mass of the target Žfinal hadron.. For both polarized and unpolarized leptons these asymmetries are given by w3,4,12x 2
2
We omit the current quark mass dependent terms.
E. De Sanctis et al.r Physics Letters B 483 (2000) 69–73
Twist-2 distribution and fragmentation functions have a subscript ‘1’: f 1Ž x . and D 1Ž z . are the usual unpolarized distribution and fragmentation functions, while h1HLŽ1. Ž x . and h1Ž x . describe the quark transverse spin distribution in longitudinally and transversely polarized nucleons, respectively. The twist-3 distribution function in the longitudinally polarized nucleon is denoted by h LŽ x . w14x. The spin dependent fragmentation function H1H Ž1. Ž z ., describing transversely polarized quark fragmentation ŽCollins effect w2x., can be interpreted as the production probability of an unpolarized hadron from a transversely polarized quark w15x. The fragmentation function H˜ Ž z . is the interaction-dependent part of the twist-3 fragmentation function: H Ž z . s y2 zH1H Ž1. Ž z . q H˜ Ž z .. The functions with superscript Ž1. denote p T2and k T2-moments, respectively: h1HLŽ1. Ž x . ' d 2 p T
H
p T2
ž / 2M2
H1H Ž1. Ž z . ' z 2 d 2 k T
H
h1HL Ž x , p T2 . ,
k T2
ž / 2 Mh2
H1H Ž z , z 2 k T2 . .
Ž 7. Ž 8.
The function h LŽ x . can be split into a twist-2 part, h1HLŽ1. Ž x ., and an interaction-dependent part, h˜ LŽ x . w14,16x: h LŽ x . s y2
h1HLŽ1. Ž x . x
q h˜ LŽ x . .
Ž 9.
As it was shown in Refs. w4,16x this relation can be rewritten as h L Ž x . s h1 Ž x . y
d dx
h1HLŽ1. Ž x . .
Ž 10.
The weighted single target-spin asymmetries defined above are related to the ones measured by HERMES w1x through the following relations: fh Asin UL f
2 Mh ² PhT :
2fh Asin f UL
²
2 MMh 2 : ² PhT
< PhT < Mh ²
sin f h : ,
2 < < PhT
MMh
sin2 f h : ,
71
When combining the HERMES experimental results of a significant target-spin sin f h asymmetry for pq and of a vanishing sin2 f h asymmetry with the preliminary evidence from Z 0 2-jet decay on a non-zero T-odd transversely polarized quark fragmentation function w17x, it follows immediately from Eq. Ž4. that h1HLŽ1. Ž x ., the twist-2 transverse quark spin distribution in a longitudinally polarized nucleon, should vanish. Consequently, from Eqs. Ž9., Ž10. follows that
™
h LŽ x . s h˜ LŽ x . s h1 Ž x . .
Ž 13 .
In this situation the single target-spin sin f h asymmetry given by Eq. Ž3. reduces to the twist-2 level Ž‘‘reduced twist-3 approximation’’.. The fact that h1HLŽ1. Ž x . Žsee Eq. Ž7.. vanishes may be interpreted as follows: the distribution function h1HLŽ x, p T2 ., which is non-zero itself, vanishes at any x when it is averaged over the intrinsic transverse momentum of the initial parton, p T . As a matter of fact, in a longitudinally polarized nucleon partons polarized transversely at large p T may indeed have a polarization opposite to that at smaller p T , at any x. It is important to mention that the ‘‘reduced twist3 approximation’’ does not require H˜ Ž z . s 0, which otherwise would lead to the inconsistency that H1H Ž1. Ž z . would be required to vanish w4,18x. For the numerical calculations the non-relativistic approximation h1Ž x . s g 1Ž x . is taken as lower limit 3, and h1Ž x . s Ž f 1Ž x . q g 1Ž x ..r2 as an upper limit w23x. Here g 1Ž x . is the customary Žchiral even. polarized distribution function. For the sake of simplicity, Q 2-independent parameterizations were chosen for the distribution functions f 1Ž x . and g 1Ž x . w24x. To calculate the T-odd fragmentation function H1H Ž1. Ž z ., the Collins parameterization w2x for the
Ž 11 . Ž 12 .
where the subscripts U and L indicate unpolarized beam and longitudinally polarized target, respectively.
3 For non-relativistic quarks, h1Ž x . s g 1Ž x .. Several models suggest that h1Ž x . has the same order of magnitude as g 1Ž x . w13,19,20x. The evolution properties of h1 and g 1 , however, are very different w21,22x. At the Q 2 values of the HERMES measurement the assumption h1 s g 1 is fulfilled at large, i.e. valence-like, x values, while large differences occur at lower x w11x.
E. De Sanctis et al.r Physics Letters B 483 (2000) 69–73
72
fh Fig. 1. The single target-spin asymmetry Asin for pq producUL tion as a function of Bjorken x, evaluated using MC s 0.28 GeV in Eq. Ž14.. The solid line corresponds to h1 s g 1 , the dashed one to h1 s Ž f 1 q g 1 .r2. Data are from Ref. w1x.
analyzing power of transversely polarized quark fragmentation was adopted: < k T < H1H Ž z ,k T2 .
A C Ž z ,k T . '
Mh
D 1 z ,k T2
Ž
.
s
MC < k T < MC2 q k T2
.
Ž 14 .
For the distribution of the final parton’s intrinsic transverse momentum, k T , in the unpolarized fragmentation function D 1Ž z,k T2 . a Gaussian parameterization was used w25x with ² z 2 k T2 : s b 2 Žin the numerical calculations b s 0.36 GeV was taken w26x.. q For Dp1 Ž z . the parameterization from Ref. w27x was adopted. In Eq. Ž14. MC is a typical hadronic mass whose value may range from mp to M p . Using MC s 2 mp for the analyzing power of Eq. Ž14. results in 1
Hz s0.1dzH
H 1
Ž z.
0
1
Hz s0.1
w28x. By using z 0 s 0.01, the ratio reduces to 0.03; choosing a value of z 0 equal to 0.2 Ž0.3., the ratio increases to about 0.1 Ž0.12.. This behavior is mainly due to the fact that the fragmentation function D 1Ž z . diverges at small values of z. f hŽ . In Fig. 1, the asymmetry Asin x of Eq. Ž11. for UL q p production on a proton target is presented as a function of x-Bjorken and compared to HERMES data w1x, which correspond to 1 GeV 2 F Q 2 F 15 GeV 2 , 4 GeV F Ep F 13.5 GeV, 0.02 F x F 0.4, 0.2 F z F 0.7, and 0.2 F y F 0.8. The two theoretical curves are calculated by integrating over the same kinematic ranges taking ² PhT : s 0.365 GeV as input. The latter value is obtained in this kinematic region assuming a Gaussian parameterization of the distribution and fragmentation functions with ² p T2 : s Ž0.44. 2 GeV 2 w26x. From Fig. 1 it can be concluded that there is good agreement between the calculation in this letter and the HERMES data. Note that the ‘kinematic’ contrif hŽ . bution to Asin x , coming from the transverse UL component of the target polarization Žwith respect to the virtual photon direction. and given by I1T ŽEq. Ž6.., amounts to only 25%. fh The z dependence of the asymmetry Asin for UL q p production is shown in Fig. 2, where the two curves correspond to two limits for h1Ž x ., as introduced above. No data are available yet to constrain the calculations.
s 0.062,
Ž 15 .
dzD1 Ž z .
0
which is in good agreement with the experimental result 0.063 " 0.017 given for this ratio in Ref. w17x. Here H1H Ž z . is the unweighted polarized fragmentation function, defined as H1H Ž z . ' z 2 d 2 k T H1H Ž z , z 2 k T2 . .
H
Ž 16 .
It is worth mentioning that the ratio in Eq. Ž15. is rather sensitive to the lower limit of integration, z 0
fh Fig. 2. The single target-spin asymmetry Asin for pq producUL tion as a function of z evaluated using MC s 0.28 GeV. The solid line corresponds to h1 s g 1 , the dashed one to h1 s Ž f 1 q g 1 .r2.
E. De Sanctis et al.r Physics Letters B 483 (2000) 69–73
In conclusion, the recently observed single-spin azimuthal asymmetries in semi-inclusive deep inelastic lepton scattering off a longitudinally polarized proton target at HERMES are interpreted on the basis of the so-called ‘‘reduced twist-3 approximation’’, that is assuming a vanishing twist-2 transverse quark spin distribution in the longitudinally polarized nucleon. This leads to a self-consistent description of the observed single-spin asymmetries. In this approach the target-spin sin f h asymmetry is interpreted as the effect of the convolution of the transversity distribution, h1Ž x ., and a T-odd fragmentation function, H1H Ž1. Ž z ., and may allow to probe transverse spin observables in a longitudinally polarized nucleon. In addition, predictions are given for the z dependence of the single target-spin sin f h asymmetry, for which experimental data are not yet published.
Acknowledgements We would like to thank P. Mulders for many useful discussions, V. Korotkov for very useful comments and R. Kaiser for the careful reading of the manuscript. The work of K.A.O. was in part supported by INTAS contributions Žcontract numbers 93-1827 and 96-287. from the European Union.
References w1x HERMES Collaboration, A. Airapetian et al., hepexr9910062.
73
w2x J. Collins, Nucl. Phys. B 396 Ž1993. 161. w3x A. Kotzinian, Nucl. Phys. B 441 Ž1995. 234. w4x P.J. Mulders, R.D. Tangerman, Nucl. Phys. B 461 Ž1996. 197. w5x X. Artru, J. Czyzevski, H. Yabuki, Z. Phys. C 73 Ž1997. 527. w6x R. Jaffe, X. Ji, J. Tang, Phys. Rev. Lett. 80 Ž1998. 1166. w7x A.M. Kotzinian, K.A. Oganessyan, A.R. Avakian, E. DeSanctis, hep-phr9908446; in: Proc. workshop Nucleon’99, Frascati, June 7–9, 1999, Nucl. Phys. A 666. w8x A.V. Efremov, M. Polyakov, K. Goeke, D. Urbano, hepphr0001119. w9x M. Boglione, P.J. Mulders, hep-phr0001196. w10x A.V. Efremov, hep-phr0001214. w11x V.A. Korotkov, W.-D. Nowak, K.A. Oganessyan, DESY 99-176, hep-phr0002268. w12x K.A. Oganessyan, A.R. Avakian, N. Bianchi, A.M. Kotzinian, hep-phr9808368; Proc. workshop Baryons’98, Bonn, September 22–26, 1998. w13x R. Jaffe, X. Ji, Nucl. Phys. B 375 Ž1992. 527. w14x R. Jaffe, X. Ji, Phys. Rev. Lett. 67 Ž1991. 552. w15x P.J. Mulders, Nucl. Phys. A 622 Ž1997. 239c. w16x R.D. Tangerman, P.J. Mulders, NIKHEF-94-P7, hepphr9408305. w17x A.V. Efremov, O.G. Smirnova, L.G. Tkachev, hepphr9812522; A.V. Efremov, in: Proc. of workshop DIS’99, Zeuthen, April 19–23, 1999. w18x A. Schaefer, O.V. Teryaev, hep-phr9908412. w19x P.V. Pobylitsa, M.V. Polyakov, Phys. Lett. B 389 Ž1996. 350. w20x V. Barone, T. Calarco, A. Drago, Phys. Lett. B 390 Ž1997. 287. w21x X. Artru, M. Mekhfi, Z. Phys. C 45 Ž1990. 669. w22x S. Scopetta, V. Vento, Phys. Lett. B 424 Ž1998. 25. w23x J. Soffer, Phys. Rev. Lett. 74 Ž1995. 1292. w24x S. Brodsky, M. Burkardt, I. Schmidt, Nucl. Phys. B 441 Ž1995. 197. w25x A.M. Kotzinian, P.J. Mulders, Phys. Lett. B 406 Ž1997. 373. w26x T. Sjostrand, Comp. Phys. Commun. 82 Ž1994. 74; CERNTH.7112r93; hep-phr9508391. w27x E. Reya, Phys. Rep. 69 Ž1981. 195. w28x E. Boglione, P.J. Mulders, Phys. Rev. D 60 Ž1999. 054007.
Physics Letters B 483 Ž2000. 69–73
Single-spin azimuthal asymmetries in the ‘‘Reduced twist-3 approximation’’ E. De Sanctis a , W.-D. Nowak b, K.A. Oganessyan a,b,c a
INFN-Laboratori Nazionali di Frascati, Õia Enrico Fermi 40, I-00044 Frascati, Italy b DESY Zeuthen, Platanenallee 6, D-15738 Zeuthen, Germany c YereÕan Physics Institute, Alikhanian Br. St. 2, 375036 YereÕan, Armenia Received 28 February 2000; accepted 11 May 2000 Editor: R. Gatto
Abstract We consider the single-spin azimuthal asymmetries recently measured at the HERMES experiment for charged pions produced in semi-inclusive deep inelastic scattering of leptons off longitudinally polarized protons. Guided by the experimental results and assuming a vanishing twist-2 transÕerse quark spin distribution in the longitudinally polarized nucleon, denoted as ‘‘reduced twist-3 approximation’’, a self-consistent description of the observed single-spin asymmetries is obtained. In addition, predictions are given for the z dependence of the single target-spin asymmetry. q 2000 Elsevier Science B.V. All rights reserved.
Semi-inclusive deep inelastic scattering ŽSIDIS. of leptons off a polarized nucleon target is a rich source of information on the spin structure of the nucleon and on parton fragmentation. In particular, measurements of azimuthal asymmetries in SIDIS allow the further investigation of the quark and gluon structure of the polarized nucleon. The HERMES collaboration has recently reported on the measurement of single target-spin asymmetries in the distribution of the azimuthal angle f relative to the lepton scattering plane, in semi-inclusive charged pion production on a longitudinally polarized hydrogen target w1x. The sin f moment of this distribution was found to be significant for pq-production. For py it was found to be consistent with zero within present experimental uncertainties, as it was the case
E-mail address: [email protected] ŽK.A. Oganessyan..
for the sin2 f moments of both pq and py. Singlespin asymmetries vanish in models in which hadrons consist of non-interacting collinear partons Žquarks and gluons., i.e. they are forbidden in the simplest version of the parton model and perturbative QCD. Non-vanishing and non-identical intrinsic transverse momentum distributions for oppositely polarized partons play an important role in most explanations of such non-zero single-spin asymmetries; they are interpreted as the effects of ‘‘naive time-reversal-odd’’ ŽT-odd. fragmentation functions w2–6x, arising from non-perturbative hadronic final-state interactions. In Refs. w7,8x these asymmetries were evaluated and it was shown that a good agreement with the HERMES data can be achieved by using only twist-2 distribution and fragmentation functions. In this letter the single target-spin sin f h and sin2 f h azimuthal asymmetries are investigated in the light of the recent HERMES results w1x. It will be
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 5 9 6 - 7
E. De Sanctis et al.r Physics Letters B 483 (2000) 69–73
70
shown that these results may be interpreted towards a vanishing twist-2 quark transverse spin distribution in the longitudinally polarized nucleon1. Under this assumption, which will be called hereafter ‘‘reduced twist-3 approximation’’, the sub-leading order in 1rQ single target-spin sin f h asymmetry reduces to the twist-2 level and is interpreted as the effect of the convolution of the transversity distribution and the T-odd fragmentation function. In this situation, also measurements with a longitudinally polarized target at HERMES may be used to extract the transversity distribution in a way similar to that proposed in Ref. w11x for a transversely polarized target, once enough statistics will be collected. The sin f h and sin2 f h moments of experimentally observable single target-spin asymmetries in the SIDIS cross-section can be related to the parton distribution and fragmentation functions involved in the parton level description of the underlying process w3,4x. Their anticipated dependence on p T Ž k T ., the intrinsic transverse momentum of the initial Žfinal. parton, reflects into the distribution of PhT , the transverse momentum of the semi-inclusively measured hadron. The moments are defined as appropriately weighted integrals over this observable, of the cross section asymmetry:
²
< PhT < Mh
sin f h : < PhT <
2
hT
'
Mh 2
sin f h Ž d sqy d sy . ,
Ž 1.
q y hT Ž d s q d s .
Hd P < PhT < 2 MMh
sin2 f h : < PhT < 2
2
Hd P
hT
'
²
< PhT < Mh
sin f h : Ž x , y, z . 1
I1 LŽ x , y, z . q I1T Ž x , y, z . ,
s I0 Ž x , y, z . ²
< PhT < 2 MMh
MMh 2
sin2 f h Ž d sqy d sy . .
Ž 2.
q y hT Ž d s q d s .
Hd P
Here qŽy. denote the antiparallel Žparallel. longitu-
8 s I0 Ž x , y, z .
SLŽ 1 y y . h1HLŽ1. Ž x . z 2 H1H Ž1. Ž z . ,
Ž 4. where I0 Ž x , y, z . s Ž 1 q Ž 1 q y . I1 LŽ x , y, z . s 4SL
M Q
2
. f 1Ž x . D1Ž z . ,
Ž 2 y y . '1 y y
= xh LŽ x . zH1H Ž1. Ž z . yh1HLŽ1. Ž x . H˜ Ž z . ,
Ž 5. Ž 6.
With k 1 Ž k 2 . being the 4-momentum of the incoming Žoutgoing. charged lepton, Q 2 s yq 2 where q s k 1 y k 2 is the 4-momentum of the virtual photon. P Ž Ph . is the momentum of the target Žfinal hadron., x s Q 2r2Ž Pq ., y s Ž Pq .rŽ Pk 1 ., z s Ž PPh .rŽ Pq ., k 1T the incoming lepton transverse momentum with respect to the virtual photon momentum direction, and f h is the azimuthal angle between PhT and k 1T around the virtual photon direction. Note that the azimuthal angle of the transverse Žwith respect to the virtual photon. component of the target polarization, fS , is equal to 0 Žp . for the target polarized parallel Žanti-parallel. to the beam w12x. The components of the longitudinal and transverse target polarization in the virtual photon frame are denoted by SL and ST x , respectively.
1
After this work has been completed we became aware of Refs. w9,10x where this possibility has also been considered.
Ž 3.
sin2 f h : Ž x , y, z .
I1T Ž x , y, z . s 2 ST x Ž 1 y y . h1 Ž x . zH1H Ž1. Ž z . .
Hd P
²
dinal polarization of the target and M Ž Mh . is the mass of the target Žfinal hadron.. For both polarized and unpolarized leptons these asymmetries are given by w3,4,12x 2
2
We omit the current quark mass dependent terms.
E. De Sanctis et al.r Physics Letters B 483 (2000) 69–73
Twist-2 distribution and fragmentation functions have a subscript ‘1’: f 1Ž x . and D 1Ž z . are the usual unpolarized distribution and fragmentation functions, while h1HLŽ1. Ž x . and h1Ž x . describe the quark transverse spin distribution in longitudinally and transversely polarized nucleons, respectively. The twist-3 distribution function in the longitudinally polarized nucleon is denoted by h LŽ x . w14x. The spin dependent fragmentation function H1H Ž1. Ž z ., describing transversely polarized quark fragmentation ŽCollins effect w2x., can be interpreted as the production probability of an unpolarized hadron from a transversely polarized quark w15x. The fragmentation function H˜ Ž z . is the interaction-dependent part of the twist-3 fragmentation function: H Ž z . s y2 zH1H Ž1. Ž z . q H˜ Ž z .. The functions with superscript Ž1. denote p T2and k T2-moments, respectively: h1HLŽ1. Ž x . ' d 2 p T
H
p T2
ž / 2M2
H1H Ž1. Ž z . ' z 2 d 2 k T
H
h1HL Ž x , p T2 . ,
k T2
ž / 2 Mh2
H1H Ž z , z 2 k T2 . .
Ž 7. Ž 8.
The function h LŽ x . can be split into a twist-2 part, h1HLŽ1. Ž x ., and an interaction-dependent part, h˜ LŽ x . w14,16x: h LŽ x . s y2
h1HLŽ1. Ž x . x
q h˜ LŽ x . .
Ž 9.
As it was shown in Refs. w4,16x this relation can be rewritten as h L Ž x . s h1 Ž x . y
d dx
h1HLŽ1. Ž x . .
Ž 10.
The weighted single target-spin asymmetries defined above are related to the ones measured by HERMES w1x through the following relations: fh Asin UL f
2 Mh ² PhT :
2fh Asin f UL
²
2 MMh 2 : ² PhT
< PhT < Mh ²
sin f h : ,
2 < < PhT
MMh
sin2 f h : ,
71
When combining the HERMES experimental results of a significant target-spin sin f h asymmetry for pq and of a vanishing sin2 f h asymmetry with the preliminary evidence from Z 0 2-jet decay on a non-zero T-odd transversely polarized quark fragmentation function w17x, it follows immediately from Eq. Ž4. that h1HLŽ1. Ž x ., the twist-2 transverse quark spin distribution in a longitudinally polarized nucleon, should vanish. Consequently, from Eqs. Ž9., Ž10. follows that
™
h LŽ x . s h˜ LŽ x . s h1 Ž x . .
Ž 13 .
In this situation the single target-spin sin f h asymmetry given by Eq. Ž3. reduces to the twist-2 level Ž‘‘reduced twist-3 approximation’’.. The fact that h1HLŽ1. Ž x . Žsee Eq. Ž7.. vanishes may be interpreted as follows: the distribution function h1HLŽ x, p T2 ., which is non-zero itself, vanishes at any x when it is averaged over the intrinsic transverse momentum of the initial parton, p T . As a matter of fact, in a longitudinally polarized nucleon partons polarized transversely at large p T may indeed have a polarization opposite to that at smaller p T , at any x. It is important to mention that the ‘‘reduced twist3 approximation’’ does not require H˜ Ž z . s 0, which otherwise would lead to the inconsistency that H1H Ž1. Ž z . would be required to vanish w4,18x. For the numerical calculations the non-relativistic approximation h1Ž x . s g 1Ž x . is taken as lower limit 3, and h1Ž x . s Ž f 1Ž x . q g 1Ž x ..r2 as an upper limit w23x. Here g 1Ž x . is the customary Žchiral even. polarized distribution function. For the sake of simplicity, Q 2-independent parameterizations were chosen for the distribution functions f 1Ž x . and g 1Ž x . w24x. To calculate the T-odd fragmentation function H1H Ž1. Ž z ., the Collins parameterization w2x for the
Ž 11 . Ž 12 .
where the subscripts U and L indicate unpolarized beam and longitudinally polarized target, respectively.
3 For non-relativistic quarks, h1Ž x . s g 1Ž x .. Several models suggest that h1Ž x . has the same order of magnitude as g 1Ž x . w13,19,20x. The evolution properties of h1 and g 1 , however, are very different w21,22x. At the Q 2 values of the HERMES measurement the assumption h1 s g 1 is fulfilled at large, i.e. valence-like, x values, while large differences occur at lower x w11x.
E. De Sanctis et al.r Physics Letters B 483 (2000) 69–73
72
fh Fig. 1. The single target-spin asymmetry Asin for pq producUL tion as a function of Bjorken x, evaluated using MC s 0.28 GeV in Eq. Ž14.. The solid line corresponds to h1 s g 1 , the dashed one to h1 s Ž f 1 q g 1 .r2. Data are from Ref. w1x.
analyzing power of transversely polarized quark fragmentation was adopted: < k T < H1H Ž z ,k T2 .
A C Ž z ,k T . '
Mh
D 1 z ,k T2
Ž
.
s
MC < k T < MC2 q k T2
.
Ž 14 .
For the distribution of the final parton’s intrinsic transverse momentum, k T , in the unpolarized fragmentation function D 1Ž z,k T2 . a Gaussian parameterization was used w25x with ² z 2 k T2 : s b 2 Žin the numerical calculations b s 0.36 GeV was taken w26x.. q For Dp1 Ž z . the parameterization from Ref. w27x was adopted. In Eq. Ž14. MC is a typical hadronic mass whose value may range from mp to M p . Using MC s 2 mp for the analyzing power of Eq. Ž14. results in 1
Hz s0.1dzH
H 1
Ž z.
0
1
Hz s0.1
w28x. By using z 0 s 0.01, the ratio reduces to 0.03; choosing a value of z 0 equal to 0.2 Ž0.3., the ratio increases to about 0.1 Ž0.12.. This behavior is mainly due to the fact that the fragmentation function D 1Ž z . diverges at small values of z. f hŽ . In Fig. 1, the asymmetry Asin x of Eq. Ž11. for UL q p production on a proton target is presented as a function of x-Bjorken and compared to HERMES data w1x, which correspond to 1 GeV 2 F Q 2 F 15 GeV 2 , 4 GeV F Ep F 13.5 GeV, 0.02 F x F 0.4, 0.2 F z F 0.7, and 0.2 F y F 0.8. The two theoretical curves are calculated by integrating over the same kinematic ranges taking ² PhT : s 0.365 GeV as input. The latter value is obtained in this kinematic region assuming a Gaussian parameterization of the distribution and fragmentation functions with ² p T2 : s Ž0.44. 2 GeV 2 w26x. From Fig. 1 it can be concluded that there is good agreement between the calculation in this letter and the HERMES data. Note that the ‘kinematic’ contrif hŽ . bution to Asin x , coming from the transverse UL component of the target polarization Žwith respect to the virtual photon direction. and given by I1T ŽEq. Ž6.., amounts to only 25%. fh The z dependence of the asymmetry Asin for UL q p production is shown in Fig. 2, where the two curves correspond to two limits for h1Ž x ., as introduced above. No data are available yet to constrain the calculations.
s 0.062,
Ž 15 .
dzD1 Ž z .
0
which is in good agreement with the experimental result 0.063 " 0.017 given for this ratio in Ref. w17x. Here H1H Ž z . is the unweighted polarized fragmentation function, defined as H1H Ž z . ' z 2 d 2 k T H1H Ž z , z 2 k T2 . .
H
Ž 16 .
It is worth mentioning that the ratio in Eq. Ž15. is rather sensitive to the lower limit of integration, z 0
fh Fig. 2. The single target-spin asymmetry Asin for pq producUL tion as a function of z evaluated using MC s 0.28 GeV. The solid line corresponds to h1 s g 1 , the dashed one to h1 s Ž f 1 q g 1 .r2.
E. De Sanctis et al.r Physics Letters B 483 (2000) 69–73
In conclusion, the recently observed single-spin azimuthal asymmetries in semi-inclusive deep inelastic lepton scattering off a longitudinally polarized proton target at HERMES are interpreted on the basis of the so-called ‘‘reduced twist-3 approximation’’, that is assuming a vanishing twist-2 transverse quark spin distribution in the longitudinally polarized nucleon. This leads to a self-consistent description of the observed single-spin asymmetries. In this approach the target-spin sin f h asymmetry is interpreted as the effect of the convolution of the transversity distribution, h1Ž x ., and a T-odd fragmentation function, H1H Ž1. Ž z ., and may allow to probe transverse spin observables in a longitudinally polarized nucleon. In addition, predictions are given for the z dependence of the single target-spin sin f h asymmetry, for which experimental data are not yet published.
Acknowledgements We would like to thank P. Mulders for many useful discussions, V. Korotkov for very useful comments and R. Kaiser for the careful reading of the manuscript. The work of K.A.O. was in part supported by INTAS contributions Žcontract numbers 93-1827 and 96-287. from the European Union.
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