Collins Analyzing Power and Azimuthal
Asymmetries
A. V. Efremova 1, K. Goekeb , and P. Schweitzerc 2
b
a Joint Institute for Nuclear Research, Dubna, 141980 Russia
Institute for Theoretical Physics II, Ruhr University Bochum, Germany
c Dipartimento di Fisica Nucleare e Teorica, Università di Pavia, Italy
Abstract. Spin azimuthal asymmetries in pion electro-production in deep inelastic scattering off
longitudinally polarized protons, measured by HERMES, are well reproduced theoretically with
no adjustable parameters. Predictions for azimuthal asymmetries for a longitudinally polarized
deuteron target are given. Using this the z-dependence of the Collins fragmentation function is
extracted for the first time. The first information on ex is extracted from CLAS A LU asymmetry.
INTRODUCTION
Recently azimuthal asymmetries have been observed in pion electro-production in semi
inclusive deep-inelastic scattering off longitudinally (with respect to the beam) [1, 2]
and transversely polarized protons [3]. These asymmetries contain information on the Todd “Collins” fragmentation function H1a z and on the transversity distribution h a1 x
[4]3 . H1a z describes the left-right asymmetry in fragmentation of transversely polarized quarks into a hadron [5, 6, 7] (the "Collins asymmetry"), and h a1 x describes the
distribution of transversely polarized quarks in nucleon [4]. Both H 1a z and ha1 x are
twist-2, chirally odd, and not known experimentally. Only some years ago experimental
indications to H1 in e e -annihilation have appeared [8], while the HERMES and SMC
data [1, 2, 3] provide first experimental indications to h a1 x.
Here we explain the observed azimuthal asymmetries [1, 2] and predict pion and kaon
asymmetries from a deuteron target for HERMES by using information on H 1 from
DELPHI [8] and the predictions for the transversity distribution h a1 x from the chiral
quark-soliton model (χ QSM) [9]. Our analysis is free of any adjustable parameters.
Moreover, we use the model prediction for ha1 x to extract H1 z from the z-dependence
of HERMES data. For more details and complete references see Ref.[10, 11, 12]. Finally,
using the new information on H1 z, we extract the twist-3 distribution e a x from very
recent CLAS data [13].
1
Supported by grants RFBR-00-02-16696, INTAS-00/587 and Heisenberg-Landau Programm.
Supported by the contract HPRN-CT-2000-00130 of the European Commission.
3 We use the notation of Ref.[5, 6] with H z normalized to P instead of M .
1
h
h
2
CP675, Spin 2002: 15th Int'l. Spin Physics Symposium and Workshop on Polarized Electron
Sources and Polarimeters, edited by Y. I. Makdisi, A. U. Luccio, and W. W. MacKay
© 2003 American Institute of Physics 0-7354-0136-5/03/$20.00
459
TRANSVERSITY AND COLLINS PFF
The χ QSM is a quantum field-theoretical relativistic model with explicit quark and
antiquark degrees of freedom. This allows an unambiguous identification of quark and
antiquark distributions in the nucleon, which satisfy all general QCD requirements
due to the field-theoretical nature of the model [14]. The results of the parameter-free
calculations for unpolarized and helicity distributions agree within 10 20% with
parameterizations, suggesting a similar reliability of the model prediction for h a1 x [9].
H1 is responsible in e e annihilation for a specific azimuthal asymmetry of a hadron
in a jet around the axis in direction of the second hadron in the opposite jet [5]. This
asymmetry was probed using the DELPHI data collection [8]. For the leading particles
in each jet of two-jet events, averaged over quark flavors, the most reliable value of the
analyzing power is given by 63 20%. However, the larger “optimistic” value is not
excluded
H 1 (1)
D1 125 14%
with unestimated but presumably large systematic errors.
THE AZIMUTHAL ASYMMETRY
In [1, 2] the cross section for lp l π X was measured in dependence of the azimuthal
angle φ , i.e. the angle between lepton scattering plane and the plane defined by momentum of virtual photon q and momentum Ph of produced pion. The twist-2 and twist-3
azimuthal asymmetries read [6]4
sin 2φ
AUL
x ∝
1a xH aπ ∑ e2ah1L
∑ e2a f1a xDa1π 1
(2)
sin φ
AUL
1 x ∝
M
e2a xhaL xH1aπ ∑ e2a f1a xDa1π ∑
Q a
a
(3)
sin φ
AUL
2 x ∝
a
sin θγ ∑ e2aha1 xH1aπ ∑ e2a f1axDa1π a
a
sin φ
with sin θγ 2x 1 yM Q and AUL
h
a
(4)
Asin φ Asin φ . In Eqs.(2-4) the pure twist-3
UL1
UL2
L terms are neglected. The results of Ref.[16] justify to use this WW-type approxima1
1
tion in which xhL 2h
2x2 x dξ h1 ξ ξ 2 .
1L
We assume isospin symmetry and favoured fragmentation for D a1 and H1a , i.e. Dπ1 0
Du1π Dd1 π 2Dū1π etc. and Dū1π Du1π 0 etc.
Note a sign-misprint in Eq.(115) of [6] for the sin φ -term Eq.(3). It was corrected in Eq.(2) of [15]. The
conventions in Eqs.(2–4) agree with [1, 2]: Target polarization opposite to beam is positive, and z axis is
parallel to q (in [6] it is anti-parallel).
4
460
W(φ)
0
W(φ)
AUL (x, π , p)
+
W(φ)
AUL (x, π , p)
0.05
0.05
0.05
sinφ
sinφ
0
sin2φ
0
0
sin2φ
sinφ
sin2φ
x
0
-
AUL (x, π , p)
x
0.1 0.2 0.3 0.4
0
0.1 0.2 0.3 0.4
x
0
0.1 0.2 0.3 0.4
φ FIGURE 1. Azimuthal asymmetries AW
weighted by W φ sin φ , sin 2φ for pions as function of
UL
sin
φ
sin2φ ).
x. Rhombus (squares) denote data for A UL (AUL
HERMES ASYMMETRIES
When using Eq.(1) to explain HERMES data, we assume a weak scale dependence of
the analyzing power. We take ha1 x from the χ QSM [9] and f 1a x from Ref.[17], both
LO-evolved to the average scale Q2av 4 GeV2 .
sin φ x, Asin 2φ x [1, 2] are compared with the results
In Fig.1 HERMES data for AUL
UL
of our analysis. We conclude that the azimuthal asymmetries obtained with h a1 x from
the χ QSM [9] combined with the “optimistic” DELPHI result Eq.(1) for the analyzing
power are consistent with data.
We exploit the z-dependence of HERMES data for π 0 , π azimuthal asymmetries to
extract H1 zD1 z. For that we use the χ QSM prediction for ha1 x, which introduces
a model dependence of order 10 20%. The result is shown in Fig.2a. The data can
be described by a linear fit H1 z 033 006zD1z. The average H1 D1 138 28% is in good agreement with DELPHI result Eq.(1) 5 . The errors are the
statistical errors of the HERMES data. Numerically it is close to those calculated from
chiral perturbation theory [18].
The approach can be applied to predict azimuthal asymmetries in pion and kaon
production off a longitudinally polarized deuterium target, which are under current study
π
π
at HERMES. The additional assumption used is that H 1K DK
1 H1 D1 . The
predictions are shown in Fig.2b. The "data points" estimate the expected error bars.
Asymmetries for K̄ 0 and K are close to zero in our approach.
Interestingly all sin φ asymmetries change sign at x 05 (unfortunately the HERMES
cut is x 04). This is due to the negative sign in Eq.(4) and the harder behaviour of h 1 x
with respect to hL x. This prediction however is sensitive to the favoured fragmentation
approximation.
We learn that transversity could be measured also with a longitudinally polarized
target, e.g. at COMPASS, simultaneously with ∆G.
5
H 10 5%, however, seem less reliable.
SMC data [3] yield an opposite sign, D1 1
461
W(φ)
H(z)/D(z)
W(φ)
A UL,D(x,π)
+ 0
π π
π0
ππ
0.03
0.3
0.02
0.2
0.01
A UL,D(x,K)
+
0
0.2
0.4
0.6
0
-0.01
sin2φ
sin2φ
x
z
0.8
sinφ
0.01
-0.01
0
K
0
K
0.02
sinφ
0
0.1
+
0.03
0
x
0.1 0.2 0.3 0.4
0
0.1 0.2 0.3 0.4
(a)
(b)
FIGURE 2. (a). H1 D1 vs. z, as extracted from HERMES data for π and π 0 production. (b).
Predictions for A sin φ , Asin2φ from a deuteron target for HERMES. Asymmetries for K̄ 0 , K are close
UL
to zero in our approach.
UL
e(x)
e(x)
5
4
3
2
u
f1 (x)
Soffer
bound
1
0
x
0
0.1
0.2
0.3
0.4
¯
FIGURE 3. The flavour combination exe u 14 ed x, with errorbars due to statistical error of CLAS
data, vs. x at Q2 15 GeV2 . For comparison f 1u x and the twist-3 Soffer bound are shown.
EXTRACTION OF EX FROM CLAS DATA
Very recently the sin φ asymmetry of π produced by scattering of polarized electrons
off unpolarised protons was reported by CLAS collaboration [13]. This asymmetry is
interesting since it allows to access the unknown twist-3 structure functions e a x which
are connected with nucleon σ -term:
1
0
dx ∑ ea x a
2σ
mu md
10
(5)
The asymmetry is given by [6]
φ
x ∝
Asin
LU
aπ 2 a
M ∑a ea e xH1
Q ∑a e2a f a xDaπ 1
1
(6)
Disregarding unfavored fragmentation and using the Collins analysing power extracted from HERMES in Sect., which yields for z-cuts of CLAS H1π Dπ1 462
020 004, we can extract eu x 14 ed x. The result is presented in Fig.3. For comparison the Soffer lower bound [19] from twist-3 density matrix positivity, e a x 2gaT x haL x,6 and the unpolarized distribution function f 1u x are plotted. The obtained points for ex are close to calculations as in bag model [21] as in χ QSM. One
can guess that the large number in the sum rule Eq.(5) might be due to, either a strong
rise of ex in the small x region, or a δ -function at x 0 [22].
¯
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Airapetian A. et. al., Phys. Rev. Lett. 84, 4047 (2000).
Airapetian A. et al., Phys. Rev. D64, 097101 (2001).
Bravar A., Nucl. Phys. (Proc. Suppl.) B79, 521 (1999).
Ralston J. and Soper D.E.,Nucl.Phys. B152, 109 (1979).
Jaffe R.L. and Ji X., Phys. Rev. Lett. 67, 552 (1991); Nucl. Phys. B375, 527 (1992). Cortes J.L., Pire
B. and Ralston J.P., Z. Phys. C55, 409 (1992).
Boer D., Jakob R. and Mulders P.J., Phys. Lett. B424, 143 (1998).
Mulders P. and Tangerman R., Nucl. Phys. B461, 197 (1996). [Erratum: ibid. B484, 538 (1996)].
Collins J., Nucl. Phys. B396, 161 (1993).
Artru X. and Collins J.C., Z. Phys. C69 277 (1996).
Efremov A.V., Smirnova O.G. and Tkatchev L.G., Nucl. Phys. (Proc. Suppl.) 74, 49 (1999) and 79,
554 (1999).
Pobylitsa P.V. and Polyakov M.V., Phys. Lett. B389, 350 (1996).
P. Schweitzer et al., Phys. Rev. D64, 034013 (2001).
Efremov A.V. et al., Phys. Lett. B478, 94 (2000).
Efremov A.V. et al., Phys. Lett. B522, 37 (2001) [Erratum-ibid. B544, 389 (2001)]
Efremov A.V. et al., Eur. Phys. J. C24, 407 (2002), hep-ph/0112166.
Avakian H. [CLAS-Coll.], Talk at "BARYONS-2002".
Diakonov D.I. et al., Nucl. Phys. B480, 341 (1996).
Boglione M. and Mulders P., Phys. Lett. B478, 114 (2000).
Dressler B. and Polyakov M.V., Phys. Rev. D61, 097501 (2000).
Glück M., Reya E. and Vogt A., Z. Phys. C 67, 433 (1995).
Bacchetta A., Kundu R., Metz A. and Mulders P.J., hep-ph/0206309.
Soffer J., Phys. Rev. Lett. 74, 1292 (1995).
Balla J., Polyakov M.V. and Weiss C., Nucl. Phys. B510, 327 (1998).
Signal A.I., Nucl. Phys. B497, 415 (1997), hep-ph/9610480.
Burkardt M. and Koike Y., Nucl. Phys. B632, 311 (2002).
Ê
For gaT x we use the Wandzura-Wilczek approximation g aT x x1 dξ ga1 ξ ξ and neglect consistently ga2 x which is strongly suppressed in the instanton vacuum [20]. For h aL x we use the analogous
approximation, as described in Sect.III.
6
463