## Theoretical Aspects of Azimuthal and Transverse Spin Asymmetries

THEORETICAL ASPECTS OF AZIMUTHAL AND TRANSVERSE SPIN ASYMMETRIESa

arXiv:hep-ph/0106171v2 17 Dec 2001

P.J. MULDERS1 , A.A. HENNEMAN1 , and D. BOER2

1 Division of Physics and Astronomy, Vrije Universiteit De Boelelaan 1081, NL-1081 HV Amsterdam, the Netherlands 2 RIKEN-BNL Research Center Brookhaven National Laboratory, Upton, NY 11973, U.S.A.

We use Lorentz invariance and the QCD equations of motion to study the evolution of functions that appear at leading (zeroth) order in a 1/Q expansion in azimuthal asymmetries. This includes the evolution equation of the Collins fragmentation function. The moments of these functions are matrix elements of known twist two and twist three operators. We present the evolution in the large Nc limit, restricted to the non-singlet case for the chiral-even functions.

In this contribution I want to present one possible way to investigate the QCD evolution of azimuthal asymmetries 1 . These asymmetries appear in hard scattering processes with at least two relevant hadrons and constitute a rich phenomenology, suitable for studying quark and gluon correlations in hadrons. By relevant hadrons we mean hadrons used as target or detected in the ?nal state. A well-known azimuthal asymmetry appears in the semi-inclusive deep inelastic polarized leptoproduction of pions (ep↑ → e′ πX) generated by the socalled Collins e?ect 2 . This asymmetry is one of the possibilities to gain access to the so-called transversity or transverse spin distribution function 3,4 , which is the third distribution function needed for the complete characterization of the (collinear) spin state of a proton as probed in hard scattering processes. In contrast to the transversity function, the evolution of the Collins fragmentation function had not been investigated sofar. Knowledge of this evolution is indispensable for relating measurements at di?erent energies. For azimuthal asymmetries 5 in processes like semi-inclusive leptoproduction, often appearing coupled to the spin of the partons and/or hadrons, it is important to take transverse momentum of partons into account, ?rst studied by Ralston and Soper 3 for the Drell-Yan process at tree level. Also the Collins function involves transverse momenta. Furthermore, it is a socalled T-odd function allowed because time-reversal symmetry does not pose constraints for fragmentation functions. Its evolution will be one of the new results presented here, although we limit ourselves to the large Nc limit, in which case

a Talk

at the 9th International Workshop on Deep Inelastic Scattering and QCD (DIS2001), Bologna, Italy, Apr. 27- May 1, 2001

1

the evolution for T-odd pT -dependent functions is autonomous. Factorization crucially depends on the presence of a large energy scale in the process, such as the space-like momentum transfer squared q 2 = ?Q2 in leptoproduction. In this paper we will be concerned with functions that appear in processes which have, apart from such a hard scale, an additional soft momentum scale, related to the transverse momentum of the partons. In one-hadron inclusive leptoproduction this scale appears because one deals with three momenta: the large momentum transfer q, the target momentum P and the momentum of the produced hadron Ph . The noncollinearity at the quark level appears via qT = q + xB P ? Ph /zh , where xB = Q2 /2P · q and zh = P · Ph /P · q are the usual semiinclusive scaling variables, at large Q2 identi?ed with lightcone momentum fractions. The hadron momenta P and Ph de?ne in essence two lightlike directions n+ and n? , respectively. The soft 2 scale is Q2 = ?qT . T To study the scale dependence of the various distribution and fragmentation functions appearing in these (polarized) processes we construct speci?c moments in both pT and x, employ Lorentz invariance and use the QCD equations of motion. The moments in x for leading (collinear) distribution functions (appearing for instance in inclusive leptoproduction) are related to matrix elements of twist two operators. On the other hand, for the transverse moments entering the azimuthal asymmetry expressions of interest, one ?nds relations to matrix elements of twist two and twist three operators, for which the evolution, however, is known. In the large Nc limit this evolution becomes particularly simple. In hard processes the e?ects of hadrons can be studied via quark and gluon correlators. In inclusive deep inelastic scattering (DIS), these are lightcone correlators depending on x ≡ p+ /P + of the type Φij (x) ≡ dξ ? i p·ξ e P, S| ψ j (0) U(0, ξ) ψi (ξ)|P, S 2π .

LC

(1)

where the subscript ‘LC’ indicates ξ + = ξT = 0 and U(0, ξ) is a gauge link with the path running along the minus direction. The parametrization relevant for DIS at leading (zeroth) order in a 1/Q expansion is Φtwist?2 (x) = 1 f1 (x) n+ + SL g1 (x) γ5 n+ + h1 (x) γ5 S T n+ , 2 (2)

where longitudinal spin SL refers to the component along the same lightlike direction as de?ned by the hadron. Specifying also the ?avor one also encounters q q the notations q(x) = f1 (x), ?q(x) = g1 (x) and δq(x) = ?T q(x) = hq (x). The 1 evolution equations for these functions are known to next-to-leading order and 2

for the singlet f1 and g1 there is mixing with the unpolarized and polarized gluon distribution functions g(x) and ?g(x), respectively. For DIS up to order 1/Q one needs also the M/P + parts in the parameterization of Φ(x), Φtwist?3 (x) = + M 2P + M 2P + [ n+ , n? ] 2 [n , n ] . ?i SL eL (x)γ5 ? fT (x) ?ρσ γρ ST σ + i h(x) + ? (3) T 2 e(x) + gT (x) γ5 S T + SL hL (x) γ5

We have not imposed time-reversal invariance in order to study also the Todd functions, which are particularly important in the study of fragmentation. The functions e, gT and hL are T-even, the functions eL , fT and h are Todd. The leading order evolution of e, gT and hL is known 6 and for the non-singlet case this also provides the evolution of the T-odd functions eL , fT and h respectively, for which the operators involved di?er only from those of the T-even functions by a γ5 matrix. The twist assignment is more evident by connecting these functions to the Fourier transforms of matrix elements of α the form P, S|ψ j (0) U(0, η) iDT (η) U(η, ξ) ψi (ξ)|P, S via the QCD equations of motion. For a semi-inclusive hard scattering process in which two hadrons are identi?ed (in either initial or ?nal state) the treatment of transverse momentum is important. Instead of lightcone correlations one needs lightfront correlations (where only ξ + = 0). The parametrization of the x and pT dependent correlator becomes3,7,8 Φ(x, pT ) = 1 2

⊥ f1 (x, p2 ) n+ + f1T (x, p2 ) T T σ ??νρσ γ ? nν pρ ST + T M

? ? g1s (x, pT ) n+ γ5 ? h1T (x, p2 ) iσ?ν γ5 ST nν T +

? h⊥ (x, pT ) 1s

σ?ν p? nν iσ?ν γ5 p? nν T + T + . + h⊥ (x, p2 ) 1 T M M

(4)

(p · S ) We used the shorthand notation g1s (x, pT ) ≡ SL g1L (x, p2 )+ TM T g1T (x, p2 ), T T and similarly for h⊥ . The parameterization contains two T-odd functions, the 1s ⊥ Sivers function f1T 9 and the function h⊥ , the distribution function analogue 1 ⊥ of the Collins fragmentation function H1 . The whole treatment of the fragmentation functions is analogous with dependence on the quark momentum ? fraction z = Ph /k ? and kT . We use capital letters for the fragmentation functions. At measured qT one deals with a convolution of two transverse momentum dependent functions, where the transverse momenta of the partons

3

from di?erent hadrons combine to qT 3,7,10 . A decoupling is achieved by studyα ing cross sections weighted with the momentum qT , leaving only the directional (azimuthal) dependence. The functions that appear in that case are contained pα T in Φα (x) ≡ d2 pT M Φ(x, pT ) which projects out the functions in Φ(x, pT ) ? where pT appears linearly, Φα (x) ? = 1 2

α ?g1T (x) ST n+ γ5 ? SL h1L (x) (1) ⊥(1)

[γ α , n+ ]γ5 2 (x) [γ α , n+ ] , 2 (5)

ρ ? f1T (x) ?α ?νρ γ ? nν ST ? i h1 ?

⊥(1)

⊥(1)

p2 n T and transverse moments are de?ned as f (n) (x) = d2 pT 2M 2 f (x, pT ). At this point one can invoke Lorentz invariance as a possibility to rewrite some functions. All functions in Φ(x) and Φα (x) involve nonlocal matrix el? ements of two quark ?elds. Before constraining the matrix elements to the light-cone or lightfront only a limited number of amplitudes can be written down. This leads to the following Lorentz-invariance relations gT = g1 + fT = ? d (1) g , dx 1T hL = h1 ? h=? d ⊥(1) , h dx 1L (6) (7)

d ⊥(1) , f dx 1T

d ⊥(1) h . dx 1

From these relations, it is clear that the p2 /2M 2 moments of the pT -dependent T functions, appearing in Φα (x), involve both twist-2 and twist-3 operators. ? Another useful set of functions is obtained as the di?erence between the correlator ΦD (x) which via equations of motion is connected to Φtwist?3 and Φ? . This di?erence corresponds in A+ = 0 gauge to correlators ΦA , involving P, S|ψ j (0) U(0, η)Aα (η) U(η, ξ) ψi (ξ)|P, S . The di?erence de?nes interactionT dependent (tilde) functions, m (1) ⊥(1) ? (8) h1 (x) ? g1T (x) + i x fT (x) + f1T (x) ≡ x gT (x) + ix fT (x), ? M m ⊥(1) ? x hL (x) ? g1 (x) + 2 h1L (x) ? ix eL (x) ≡ x hL (x) ? ix eL (x), ? (9) M m ⊥(1) ? x e(x) ? f1 (x) + i x h(x) + 2 h1 (x) ≡ x e(x) + ix h(x). ? (10) M x gT (x) ? Using the equations of motion relations in Eqs. (8) - (10) and the relations based on Lorentz invariance in Eqs. (6) - (7), it is straightforward to relate 4

the various twist-3 functions and the p2 /2M 2 (transverse) moments of pT T dependent functions. The results e.g. for the h-functions are (omitting quark mass terms) are

1

hL (x) = 2x

x

dy

1

h1 (y) ? + hL (x) ? 2x y2 dy

1

dy

x

? hL (y) , y2

(11) (12) (13) (14)

1 ? hL (y) h1 (y) + dy , 2 y y2 x x 1 ? h(y) ? h(x) = h(x) ? 2x dy 2 , y x

h1L (x) =? x2

⊥(1)

h1

⊥(1)

(x)

1

x2

=

x

dy

? h(y) . y2

Actually, we need not consider the T-odd functions separately. They can be simply considered as imaginary parts of other functions, when we allow complex functions. In particular one can expand the correlation functions into matrices in Dirac space 11 to show that the relevant combinations are ⊥ (g1T ?i f1T ) which we can treat together as one complex function g1T . Similarly we can use complex functions (h⊥ + i h⊥ ) → h⊥ , (gT + i fT ) → gT , (hL + i h) 1 1L 1L → hL , (e + i eL) → e. The functions f1 , g1 and h1 remain real, they don’t have T-odd partners. As mentioned the evolution of the twist-2 functions and the tilde functions in known. The twist-2 functions have an autonomous evolution of the form αs (τ ) d f (x, τ ) = dτ 2π

1 x

dy [f ] P y

x y

f (y, τ ),

(15)

where τ = ln Q2 and P [f ] are the splitting functions. In the large Nc limit, also the tilde functions have an autonomous evolution. Using the relations given above, we then ?nd the evolution of the transverse moments, d (1) αs (τ ) g (x, τ ) = Nc dτ 1T 4π

1

dy

x

x2 + xy 1 δ(y ? x) + 2 2 y (y ? x)+ +

g1T (y, τ )

(1)

x2 g1 (y, τ ) , (16) y2 h1L (y, τ )

⊥(1)

d ⊥(1) αs (τ ) h (x, τ ) = Nc dτ 1L 4π

1

dy

x

3x2 ? xy 1 δ(y ? x) + 2 2 y (y ? x)+ 5

x ? h1 (y, τ ) . y

(17)

Next we note that apart from a γ5 matrix the operator structures of the T-odd ⊥(1) ⊥(1) (1) ⊥(1) functions f1T and h1 are in fact the same as those of g1T and h1L (or as mentioned before, they can be considered as the imaginary part of these functions 11 ). This implies that for the non-singlet functions, one immediately obtains the (autonomous) evolution of these T-odd functions. In particular we obtain for the Collins fragmentation function (at large Nc ), αs d ⊥(1) zH1 (z, τ ) = Nc dτ 4π

1

dy

z

3y ? z 1 δ(y ? z) + 2 y(y ? z)+

yH1

⊥(1)

(y, τ ),

(18) which should prove useful for the comparison of data on Collins function asymmetries from di?erent experiments, performed at di?erent energies. Summarizing, we have obtained evolution equations of the pT -dependent functions that appear in asymmetries and that are not suppressed by explicit powers of the hard momentum. But as functions of transverse momentum they are not of de?nite twist 1. 2. 3. 4. 5. A. Henneman, D. Boer and P.J. Mulders, hep-ph/0104271. J.C. Collins, Nucl. Phys. B 396 (1993) 161. J.P. Ralston and D.E. Soper, Nucl. Phys. B 152 (1979) 109. R.L. Ja?e and X. Ji, Nucl. Phys. B 375 (1992) 527. A.V. Efremov, O.G. Smirnova and L.G. Tkachev, Nucl. Phys. B (Proc. Suppl.) 79 (1999) 554 and Nucl. Phys. B (Proc. Suppl.) 74 (1999) 49; A. Bravar (for the SMC Collaboration), Nucl. Phys. B (Proc. Suppl.) 79 (1999) 520; A. Airapetian et al., HERMES Collaboration, Phys. Rev. Lett. 84 (2000) 4047. see e.g. A.V. Belitsky, Lectures given at the XXXI PNPI Winter School on Nuclear and Particle Physics, St. Petersburg, Repino, February, 1997, hep-ph/9703432. P.J. Mulders and R.D. Tangerman, Nucl. Phys. B 461 (1996) 197 and Nucl. Phys. B 484 (1997) 538 (E). D. Boer and P.J. Mulders, Phys. Rev. D 57 (1998) 5780. D. Sivers, Phys. Rev. D 41, 83 (1990); Phys. Rev. D 43 (1991) 261; M. Anselmino, M. Boglione and F. Murgia, Phys. Lett. B 362 (1995) 164. D. Boer, Phys. Rev. D 62 (2000) 094029 and hep-ph/0102071. A. Bacchetta, M. Boglione, A. Henneman and P.J. Mulders, Phys. Rev. Lett. 85 (2000) 712. 6

6.

7. 8. 9. 10. 11.

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