Ⅱ-5.
Hadron Physics
Physics (Theory)
(Theory)-5. Hadron

RIKEN Accel.
Accel. Prog.
Prog. Rep.
Rep. 48
48 (2015)(2015)RIKEN

Twist-3 double-spin asymmetries in lepton-nucleon collisions†
D. Pitonyak,∗1 K. Kanazawa,∗2 A. Metz,∗2 and M. Schlegel∗3Hadrons, the strongly interacting particles that comprise almost all of the visible matter in the universe,
have been shown to possess a complex inner-structurethat goes beyond a simple quark picture. For example,
experimental results in the 1970s on transverse singlespin asymmetries (SSAs) revealed the crucial role thatquark-gluon-quark correlations could play in hadrons.
This is a consequence of the fact that such observables are twist-3 effects. Much work over the last 40years has been performed in the study of transverse
SSAs from both the experimental and theoretical sides.
In addition, one also has twist-3 double-spin asymmetries (DSAs), namely those where one particle islongitudinally polarized and the other is transverselypolarized. We will denote these by ALT . The classic process for which this effect has been analyzed is
ALT in inclusive deep-inelastic lepton-nucleon scattering (DIS). In that case the entire result can be written in terms of the collinear twist-3 function gT (x).
Furthermore, this asymmetry has been studied in theDrell-Yan process involving two incoming polarizedhadrons1–4) ; in inclusive lepton production from W boson decay in proton-proton scattering5) ; for jet production in lepton-nucleon collisions6) ; and for direct
photon production7) , jet/hadron production8) , and Dmeson production9) in nucleon-nucleon collisions.
Here we consider the reaction  N ↑ → h X, where
one can have twist-3 contributions from both the distribution (incoming nucleon) and the fragmentation (outgoing hadron) sides. The leading-order (LO) analyticalformulas for these terms are new results from this work,
but we refrain from showing them explicitly for brevity.
Based on this computation we will give numerical estimates for e N ↑ → π X, where N = p, n. We will
only look at the distribution piece, where we need LO
input for the non-perturbative functions D1 (z) (unpolarized fragmentation function), g̃(x) (“worm-gear”type function), gT (x), and g1 (x) (helicity distribution),where g̃(x) is the least known of these functions andhas gained quite some interest over the years.
Since we have little information on g̃(x), we lookat two scenarios: i) using the approximate relation⊥(1)⊥g̃(x) ≈ −f1T (x), where f1T
is the Sivers function;and ii) using a Wandzura-Wilczek (WW)-type approx1
imation g̃(x) ≈ x x dy
y g1 (y), which was also used elsewhere in the literature and holds relatively well incertain models. In both cases for gT (x) we use the1
WW approximation, gT (x) ≈ x dyy g1 (y). In Fig. 1†Condensed from the article arXiv:1411.6459 submitted to PLB∗1 RIKEN Nishina Center∗2 Department of Physics, Temple University∗3 Institute for Theoretical Physics, Tübingen University

S  7.25
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0.26S  3.45AALTLTWWΠΠ� WWΠΠ� WWWWΠΠ� Sivers
Sivers

0.2
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0.2
0.4
0.4

Fig. 1. ALT vs. xF for HERMES (top) and√ ALT vs. Ph⊥ for
JLab6 (bottom), where xF = 2Phz / S with Phz (Ph⊥ )the hadron’s longitudinal (transverse) momentum.

we show a sample of our results, namely for HERMES
and JLab6, where finalized data is expected soon fromboth groups.
We see from our plots that the “Sivers” input and“Wandzura-Wilczek” input can give quite different results due to the different behavior of g̃(x). Thus, evena qualitative comparison of our predictions with experiment could help distinguish between the Sivers andWW scenarios. Moreover, if the magnitude of the data
is in line with our results, one could have direct accessto the “worm-gear”-type function g̃(x), which has received some attention recently. If the magnitude is
not in agreement, this observable could give insight
into the importance of quark-gluon-quark correlationsin the nucleon and/or twist-3 fragmentation effects in
unpolarized hadrons. However, one always has to keep
in mind the potential large impact of next-to-leading
order terms. In general, we found the best chance tomeasure a nonzero asymmetry is at HERMES, JLab,and COMPASS, as the high center-of-mass energy ofan EIC leads to a very small effect. We expect thisconclusion to be rather robust.
References
1) R. L. Jaffe and X. D. Ji, Phys. Rev. Lett. 67, 552(1991); Nucl. Phys. B 375, 527 (1992).
2) R. D. Tangerman and P. J. Mulders, hep-ph/9408305.
3) Y. Koike et al., Phys. Lett. B 668, 286 (2008).
4) Z. Lu and I. Schmidt, Phys. Rev. D 84, 114004 (2011).
5) A. Metz and J. Zhou, Phys. Lett. B 700, 11 (2011).
6) Z.-B. Kang et al., Phys. Rev. D 84, 034046 (2011).
7) Z.-T. Liang et al., Phys. Lett. B 712, 235 (2012).
8) A. Metz et al., Phys. Rev. D 86, 114020 (2012).
9) Y. Hatta et al., Phys. Rev. D 88, 014037 (2013).

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