Ⅱ-5. Hadron Physics (Theory)

RIKEN Accel. Prog. Rep. 47 (2014)

Polarized fragmentation functions and electron-positron annihilation†
D. Pitonyak∗1
Electron-positron annihilation into (possibly polarized) hadrons, where one or more hadrons are identified, gives one access to fragmentation functions (FFs),
which embody the process of a parton forming a
hadron, and they contain important information about
the strong interaction in the non-perturbative regime.
(These are analogous to parton distribution functions(PDFs) that look at partons inside of hadrons.) Suchexperiments have been run by both the Belle Collaboration at KEK in Japan and the BABAR Collaboration at SLAC in the US. In order to have a complete
framework to analyze FFs in electron-positron annihilation, one must write the cross section in a generalform involving structure functions, which can then becalculated (for small transverse momentum of the exchanged boson) in terms of twist-2 transverse momentum dependent (TMD) FFs. For the future International Linear Collider (ILC), it is also beneficial to havecomplete results for polarized leptons including electroweak effects. Given its similarity to Drell-Yan, one
should also be able to transcribe such results to thatreaction, which would be useful for double-polarized
Drell-Yan experiments at RHIC.
Here we extend earlier works1–5) in order to addressthese issues. The cross section for the reaction e+ e− →ha hb X is given by422αem
Pa0 Pb0 dσ=(Lµν W µν )γγ
q2d3 Pa d3 Pb

MZ4 G2F(Lµν W µν )ZZ
64π 2 q 2
√αem 2MZ2 GF+((Lµν W µν )γZ + h.c.) , (1)8πq 2+

where one has leptonic tensors Lµν and hadronic tensors W µν for γ-exchange, Z-exchange, and their interference. As the present author, M. Schlegel, andA. Metz discuss in detail6) , one can write down thefirst term in (1) in terms of 72 structure functions using electromagnetic gauge invariance, hermiticity, andparity (see Eq. (3.21) of 6) ). The second and thirdterms are more involved due to the fact that one nolonger has the parity constraint. Nevertheless, one cancalculate these terms (as well as the first) within theparton model at twist-2. The result leads to 128 structure functions (see Eqs. (4.34), (4.35) and Appendix Aof 6) ). Where relevant, we have checked our resultswith those in1–4) . Thus, we have for the first time acomplete framework for the study of TMD FFs withine+ e− → ha hb X including electroweak terms and the†∗1

Condensed from the article arXiv:1310.6240 [hep-ph] submitted to Physical Review D
RIKEN Nishina Center, Brookhaven National Lab

polarization of all particles.
We also note that if we make the replacements(4Pa0 Pb0 dσ/d3 Pa d3 Pb )e+ e− −→ (4l0 l 0 dσ/d3l d3l  )DY ,za (zb ) → xa (xb ), and Nc → 1/Nc , the structure functions associated with unpolarized leptons for the pure
electromagnetic case are the same as those given inDrell-Yan5) with the TMD PDFs replaced by their
TMD FF analogues. The only additional change one
must remember is that ha (hb ) in the e+ e− case has alarge minus- (plus-) component of momentum, whereasfor Drell-Yan one normally uses the reverse convention.
Along the same lines, one can easily transcribe the results in Appendix A of 6) to obtain the relevant expressions for Drell-Yan when one allows the q q̄ pair toannihilate into a Z-boson. Thus, we have for the firsttime full results for double-polarized Drell-Yan that
include electroweak effects, which would be needed if
such experiments were conducted at RHIC.
We would finally like to highlight a structure function that appears in 6) for first time: q−
2φ,ewq
w3 H1⊥ H̄1⊥ ,(2)=F5 (s) Cew
GcosUUq

where F5q− (s) is a prefactor that is nonzero if Zq[· · ·] is the convolution of aexchange is included, Cew
weight w3 and the Collins function H1⊥ . One could access (2) at a future ILC and extract the Collins function. This result could then be checked against theCollins function that has been obtained recently fromBelle and BABAR data7) . Given that the ILC would
have around two orders of magnitude higher center-ofmass energy than Belle and BABAR, such an analysis
would also be an important test of the TMD evolution formalism and its application to phenomenology,which has been of recent interest.
References
1) D. Boer, R. Jakob and P. J. Mulders, Nucl. Phys. B
504, 345 (1997) [hep-ph/9702281]; Phys. Lett. B 424,
143 (1998) [hep-ph/9711488].
2) D. Boer, R. Jakob and P. J. Mulders, Phys. Lett. B 424,
143 (1998) [hep-ph/9711488].
3) D. Boer, Phys. Rev. D 60, 014012 (1999) [hepph/9902255].
4) D. Boer, Nucl. Phys. B 806, 23 (2009) [arXiv:0804.2408[hep-ph]].
5) S. Arnold, A. Metz and M. Schlegel, Phys. Rev. D 79,034005 (2009) [arXiv:0809.2262 [hep-ph]].
6) D. Pitonyak, M. Schlegel and A. Metz, arXiv:1310.6240[hep-ph].
7) M. Anselmino, M. Boglione, U. D’Alesio, S. Melis,F. Murgia and A. Prokudin, Phys. Rev. D 87, 094019(2013) [arXiv:1303.3822 [hep-ph]].

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