Ⅱ-5. Hadron Physics (Theory)

RIKEN Accel. Prog. Rep. 47 (2014)

Twist-3 fragmentation and transverse single-spin asymmetries†
D. Pitonyak∗1Transverse single-spin asymmetries (TSSAs) in inclusive hadron production (denoted by AN ) have been
the subject of intense study since the late 1970s. These
are defined asAN =⊥ ) − dσ(−S⊥ )dσ(S,2 dσunp(1)⊥ )) is the cross section with⊥ ) (dσ(−Swhere dσ(Stransverse spin S⊥ oriented “up” (“down”) and dσunp
is the unpolarized cross section. Experiments havemeasured large effects for these observables (with themost recent results from proton-proton collisions atRHIC1–3) ), which contradict the prediction of the naı̈vecollinear parton model4) . However, a framework using twist-3 multi-parton correlators can potentially describe these large TSSAs5–7) .
The assumption for many years was that the socalled soft-gluon pole (SGP) piece dominates over
the other contributions7,8) . This part involves thenon-perturbative twist-3 Qiu-Sterman (QS) function
TF (x, x)6,7) , which was extracted several years ago8) .
However, a later analysis revealed that this extractionof TF (x, x) does not satisfy the model-independent relation with the Sivers function extracted from semiinclusive deep-inelastic scattering (SIDIS) off a transversely polarized proton: the two different extractions
disagree in sign9) . This “sign mismatch” crisis has led
to a reexamination of whether the QS function is themost significant part of TSSAs in inclusive hadron production — see, e.g., the recent discussion10) . The focus
has now shifted to whether a contribution involving
twist-3 fragmentation functions can resolve the “signmismatch” and provide the dominant effect.
The complete analytic result for the twist-3 fragmentation term in the single-spin dependent cross section for p↑ p → hX was given for the first time by thepresent author and A. Metz11) :⊥ )Ph0 dσ(S2α2 Mhα β⊥,αβ S⊥=− sPh⊥3

Sd Ph   1 dz  1 dx 11×3
z xmin x x x S + T /zi a,b,c zmindĤ c (z) i1ab cSĤĤ (z) − z×h1 (x) f1 (x )dz−x t̂ − xû
1i+ H c (z) SHz†

∗1Condensed in part from the article in Phys. Lett. B, Vol. 723,
365 (2013), whose work was funded by the NSF under Grant
No. PHY-1205942
RIKEN Nishina Center, Brookhaven National Lab+ 2z2



11 i
dz1
SPV 1ĤFc,
U (z, z1 )12z1
ξ ĤF Uz − z1



. (2)

See the paper11) for more details. In particular, Appendix A of the aforementioned reference contains thehard scattering coefficients S i in (2).
The piece in (2) also involves two independent nonperturbative functions: Ĥ(z) and ĤFU (z, z1 ). (Thefunction H(z) can be written in terms of the othertwo.) In principle one has information on Ĥ(z)through its relation to the Collins function in SIDIS.
One must then parameterize the unknown function
ĤFU (z, z1 ) and see if a fit to the data1–3) is possible.
We propose the following form for this (3-parton) fragmentation correlator that is consistent with its support
properties:
ĤFU (z, z1 ) = N z α (z/z1 )β (1 − z)δ (1 − z/z1 )γ× D1 (z) D1 (z/z1 ) ,(3)where D1 is the unpolarized fragmentation function.
We are in the process of carrying out a numerical studyof AN in p↑ p → πX using (3). This will be an important step towards solving an almost 40 year problemof what causes large TSSAs in inclusive hadron production from proton-proton collisions.
References
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