AN in proton-proton collisions and the role of twist-3 fragmentation

We review and give an update on the current status of what causes transverse single-spin asymmetries (TSSAs) in semi-inclusive processes where a single hadron is detected in the final state, especially those involving proton-proton (pp) collisions. In particular, we provide a new analysis within collinear factorization of TSSAs in high transverse momentum charged and neutral pion production in pp collisions at the Relativistic Heavy Ion Collider (RHIC). This study incorporates the so-called twist-3 fragmentation term and shows that one can describe RHIC data through this mechanism. Moreover, by fixing other non-perturbative inputs through extractions of transverse momentum dependent functions in e+e− → h1h2X and semi-inclusive deep-inelastic scattering (SIDIS), we provide for the first time a consistency between certain spin/azimuthal asymmetries in all three reactions (i.e., pp, e+e−, and SIDIS).


Introduction
Transverse single-spin asymmetries (TSSAs) have received much attention from both the experimental and theoretical side starting in the mid-1970s. Initially, large effects were seen in transversely polarized Λ production in proton-beryllium collisions [1]. These results were thought to contradict perturbative Quantum Chromodynamics (pQCD) because for such asymmetries (denoted by A N ) one should have A N ∼ α s m q /P h⊥ , with α s = g 2 /4π (g is the strong coupling constant), m q the mass of the quark, and P h⊥ the transverse momentum of the outgoing hadron/particle [2]. Later it was shown how twist-3 quarkgluon-quark correlations in the nucleon could cause significant asymmetries [3], with benchmark calculations first performed in [4,5] using collinear factorization. Experimental measurements of TSSAs in single-inclusive hadron production in proton-(anti)proton collisions also continued to show a sizable A N [6][7][8][9][10]. This led to much (still ongoing) theoretical work on these and similar observables -see, e.g., [4,5,[11][12][13][14][15][16][17]. In Sec. 2 we summarize the theoretical formalism used to describe TSSAs, namely collinear twist-3 factorization. We also review attempts to explain this effect and why it has remained a puzzle for close to 40 years. We next show in Sec. 3 how the fragmentation mechanism could play a crucial role in TSSAs for single-inclusive pion production from protona e-mail: dpitonyak@quark.phy.bnl.gov b e-mail: koichi.kanazawa@temple.edu c e-mail: koike@nt.sc.niigata-u.ac.jp d e-mail: metza@temple.edu proton (pp) collisions. This is the main content of the manuscript and is based on the work in [18]. In particular, we demonstrate that one can obtain a very good fit of all high transverse momentum RHIC data in p ↑ p → πX by including this fragmentation term while also fixing certain non-perturbative inputs from transverse momentum dependent (TMD) functions extracted in e + e − → h 1 h 2 X and semi-inclusive deep-inelastic scattering (SIDIS). Thus we provide for the first time in pQCD a simultaneous description of certain spin/azimuthal asymmetries in all three reactions (i.e., pp, e + e − , and SIDIS). In Sec. 4 we summarize our results and provide an outlook on the much fruitful work that lies ahead in order to fully understand TSSAs.

Collinear twist-3 formalism
We consider a process of the type A(P, S ⊥ ) + B(P ) → C(P h ) + X, where the 4-momenta and polarizations of the incoming protons A, B and outgoing hadron C are specified. In collinear twist-3 QCD factorization one has where f a/A(t) denotes the twist-t distribution function for parton a in proton A, and similarly for the other distribution function f b/B(t) and fragmentation function D C/c(t) for hadron C in parton c. fractions. In Eq. (1) a sum over partonic channels and parton flavors in each channel is understood.
For quite some time it was assumed that the first term in (1) dominates A N in p ↑ p → hX for the production of light hadrons (see, e.g., Refs. [5,12,14]), where the qgq SGP matrix element, called the Qiu-Sterman (QS) function T F [4,5], is generally considered the most important part. The QS function can be related to the TMD Sivers function f ⊥ 1T [22,23]. One has [24] T q where q is the quark flavor, and M is the nucleon mass. Because of the relation in (2), one should be able to extract f ⊥ 1T from the Sivers TSSA A Siv SIDIS in SIDIS, evaluate the r.h.s. of (2) and obtain the QS function on the l.h.s. that has been extracted directly from A N in pp collisions. It was quite a shock, then, when such an attempt failed, i.e., one could not simultaneously explain both A N and A Siv SIDIS [25]. Rather, what was found in Ref. [25] was that the two extractions for T F differ in sign. This "sign-mismatch" puzzle could not be resolved by more flexible parameterizations of f ⊥ 1T [26]. Also tri-gluon correlations are unlikely to fix this issue [17], while SFPs may play some role [13].
At this point one may start to question the assumption that the QS function is the dominant source of A N in p ↑ p → πX. In fact, data on the TSSA in inclusive DIS [27,28] seems to support this point of view, i.e., that the first term in (1) is not the main cause of A N . This is seen clearly through the analysis in [15], where one obtains the wrong sign for the neutron target TSSA when using T F extracted directly from A N data on pp collisions. Therefore, in the next section we analyze the impact of the third term in (1), i.e., the collinear twist-3 fragmentation part, to see if one can describe charged and neutral pion RHIC A N data through this mechanism.
However, before we proceed to the phenomenology, let us first recall the important details of the analytical result for this fragmentation term. Its contribution to the cross section in (1) reads [21] Here i denotes the channel, where P hz is the longitudinal momentum of the hadron, as well as pseudo-rapidity η = − ln tan(θ/2), where θ is the scattering angle. The variables x F , η are further related by There are several non-perturbative functions that enter into Eq. (3). They are the transversity distribution h 1 , the unpolarized parton density f 1 , and the three (twist-3) fragmentation functions (FFs)Ĥ, H, andĤ FU , with the last one being the imaginary part of a 3-parton correlator. In Ref. [21] one can also find the definition of those functions and the results for the hard scattering coefficients S i for each channel i. (An alternative notation of the relevant FFs can also be found in Ref. [29], where twist-3 fragmentation effects in SIDIS were computed.) Similar to the relation between T F and the Sivers function f ⊥ 1T in Eq. (2), the functionĤ can be written in terms of the TMD Collins function H ⊥ 1 [30] according to [21,31] Exploiting the universality of the Collins function [32], one can simultaneously extract (see [33] and references therein) H ⊥ 1 and h 1 from data [34,35] on the Collins TSSA A Col SIDIS in SIDIS [36] and data [37,38] on the cos(2φ) modulation A cos(2φ) . Such information for H ⊥ 1 and h 1 , as well as that for the f ⊥ 1T [40,41], will be useful when describing A N . The FFsĤ, H, andĤ FU are not independent, but rather satisfy [21] implying that in the collinear twist-3 framework one has two independent FFs. It is important to realize that this is different from the so-called TMD approach for A N , where only H ⊥ 1 enters the fragmentation piece [42]. We note that the Sivers effect in the TMD formalism has also been applied to A N in p ↑ p → hX [41]. However, given that for single-inclusive processes there is only one large scale, using TMD factorization (which requires two different scales) in such reactions can only be considered a phenomenological model. In that sense, the collinear twist-3 formalism is the more rigorous theoretical framework.

Phenomenological fit of pion data
We analyze A N data for p ↑ p → πX in the forward region of the polarized proton, which has been studied by EPJ Web of Conferences 02013-p.2 the STAR [8], BRAHMS [9], and PHENIX [10] collaborations at RHIC. The data at √ S = 200 GeV typically has P h⊥ > 1 GeV, and we therefore focus on those measurements in order to safely apply pQCD. Throughout we use the GRV98 unpolarized parton distributions [43] and the DSS unpolarized FFs [44]. The GRV98 functions were also used in Refs. [33,40,41] for extracting the Sivers function and the transversity, which we take as input in our calculation, so we adhere with this choice as a matter of consistency. The qgq SGP contribution to (1) is computed by fixing T F through Eq. (2) with two different inputs for the Sivers function -SV1: f ⊥ 1T from Ref. [40], obtained from SIDIS data on A Siv SIDIS [45,46]; and SV2: f ⊥ 1T from Ref. [41], "constructed" such that, in the TMD approach, the contribution of the Sivers effect to A N is maximized while maintaining a good fit of A Siv SIDIS . The input SV1 has a flavor-independent large-x behavior, while SV2 in that region has a flavor dependence and also falls off slower. To compute the fragmentation contribution we take h 1 and H ⊥ 1 (which fixesĤ through (4)) from [33]. For favored fragmentation into π + we make forĤ FU the ansatẑ with the parameters N fav , α fav , α fav , β fav , β fav and the unpolarized FF D. This parameterization follows the standard procedure of modifying the small and large "x" behavior of twist-2 unpolarized functions when trying to fit an unknown function. Note that z and z/z 1 are chosen as our "variables" because their allowed range is [0, 1] [47] and that our ansatz satisfies the constraintĤ FU (z, z) = 0 [47,48]. With the use of DSS FFs [44], the factor I fav reads I fav ≡ I u+ū − Iū where I i (i = u +ū,ū) is defined as and B[a, b] the Euler β-function. The parameters N i , α i , β i , γ i , and δ i come from D FFs at the initial scale and are given in Table III of [44]. Note that D π + /u in Ref. [44] differs from D π + /d . J fav in (6) is similarly defined as J fav ≡ J u+ū − Jū, where J i (i = u +ū,ū) follows from I i through α fav → (α fav + 4), β fav → (β fav + 1). The factor 1/(2I fav J fav ) in (6) is convenient and implies 1 0 dz z H π + /u (3) (z) = N fav at the initial scale, where H (3) represents the entire second term on the r.h.s. of (5). For the disfavored FFsĤ π + /(d,ū),

FU
we make an ansatz in full analogy to (6), introducing the additional parameters N dis , α dis , α dis , β dis , β dis . (I dis and J dis are calculated using D π + /d = D π + /ū from [44].) The π − FFs are then fixed through charge conjugation, and the π 0 FFs are given by the average of the FFs for π + and π − . The FFs H π/q are computed by means of (5). All parton correlation functions are evaluated at the scale P h⊥ with leading order evolution of the collinear functions. x F θ = 4°π + π - Figure 1. Fit results for A π 0 N (data from [8]) and A π ± N (data from [9]) for the SV1 input. The dashed line (dotted line in the case of π − ) meansĤ FU switched off.  Figure 2. Results for the FFs H π + /q andH π + /q FU (defined in the text) for the SV1 input. Also shown is H π + /q without the contribution fromĤ FU (dashed line).
Using the MINUIT package we fit the fragmentation contribution to data for A π 0 N [8] and A π ± N [9]. In order to limit the number of free parameters, we only keep 7 of them free inĤ π + /q,

FU
. Since the large-x behavior of h 1 is mostly unconstrained by current SIDIS data, we also allow the β-parameters β T u = β T d of the transversity to vary within the error range given in [33]. All integrations are done using the Gauss-Legendre method with 250 steps.
For the SV1 input the result of our 8-parameter fit is shown in Tab. 1. For the SV2 input the values of the fit parameters are similar, with an equally successful fit (χ 2 /d.o.f. = 1.10).
The very good description of the RHIC A N data is explicitly evident in Fig. 1. We emphasize this is a non-trivial outcome if one keeps in mind the constraint in (5) and the need to simultaneously fit data for A π 0 N and A π ± N . Results for the FFs H π + /q andH π + /q FU ≡ ∞ z dz 1 z 2  Figure 3. Individual contributions to A π 0 N (data from [8]) for SV1 and SV2 inputs.  Figure 4. Individual contributions to A π ± N from favored and disfavored fragmentation (data from [9]) for SV1 input.
are displayed in Fig. 2. Similar to the Collins function H ⊥ 1 , in either case the favored and disfavored FFs have opposite signs. Such reversed signs are actually "preferred" by the Schäfer-Teryaev (ST) sum rule h S h 1 0 dz z M hĤ h/q (z) = 0 [49]. Note that the ST sum rule, in combination with (5), implies a constraint on a certain linear combination of H h/q and (an integral of)Ĥ h/q, FU . In view of that, one benefits from favored and disfavored FFs having opposite signs like in Fig. 2. Also depicted in Fig. 2 is H π + /q whenĤ π + /q, FU is switched off. One seeŝ H π + /q, FU causes a reasonable increase from this scenario. As shown in Fig. 1, when the 3-parton FF is turned off, one has difficulty describing the data for A N . According to Fig. 3, theĤ term (including its derivative) contributes only very little to A N . Also the (qgq) SGP pole term is small, except for the SV2 input at large x F , where its contribution is opposite to the data. Note that with a Sivers function similar to SV2, there would definitely be serious issues with trying to match the A N data without the 3parton FF. Clearly A N is governed by the H-term. (Recall from (5) that this function involves bothĤ π/q andĤ π/q, FU .) This result can mainly be traced back to the hard scattering coefficients: e.g., for the dominant qg → qg channel one has S H ∝ 1/t 3 , but SĤ ∝ 1/t 2 [21] in the forward region wheret is small. Note also SĤ FU ∼ 1/t 3 for that channel, but it is suppressed by a color factor of 1/(N 2 c − 1). Next, Fig. 4 shows the breakdown of A π ± N into favored and disfavored fragmentation contributions. One can see that x F = 0.50 Figure 5. A N as function of P h⊥ for SV1 input at √ S = 500 GeV (data from [50]).
is dominated by favored (disfavored) fragmentation. Finally, Fig. 5 shows the P h⊥ -dependence of A N for √ S = 500 GeV. Preliminary data from STAR, extending to almost P h⊥ = 10 GeV, shows that A N is rather flat [50]. Oftentimes it is stated that the collinear twist-3 calculation cannot reproduce this flat P h⊥ dependence of A N due to the naïve expectation that A N ∼ 1/P h⊥ for a subleading twist effect. However, as was first argued in [5] and later shown in [14], this does not have to be the case. Our calculation indeed does lead to a flat P h⊥ dependence, and also the magnitude of A N is in line with the data. Note that the data of Ref. [50] were not included in our fit and that only statistical errors are shown in Fig. 5 [50].

Summary and outlook
For many years it was unclear what mechanism causes large TSSAs in hadron production from proton-proton collisions. Collinear twist-3 QCD factorization can be considered the most natural and rigorous approach to describe this observable, yet the sign-mismatch issue [25] threatened the validity of this formalism. Here we have shown for the first time that the fragmentation contribution in twist-3 factorization actually can describe high-energy RHIC data for A π N very well. By using a Sivers function fully consistent with SIDIS, we have demonstrated that this mechanism could also resolve the sign-mismatch crisis. We used the TMD Sivers, Collins, and transversity functions, which were extracted through spin/azimuthal asymmetries in SIDIS and e + e − → h 1 h 2 X, to fix certain non-perturbative inputs in our calculation. Together with the collinear 3-parton FF, these functions allowed for a very good fit of p ↑ p → πX data. Thus we have shown that at present a simultaneous description of all three observables is possible (i.e., pp, e + e − , and SIDIS). We leave an analysis of A N for kaons and etas and incorporation of SFPs for future work.
Ultimately in order to truly determine what mechanism underlies TSSAs, one must obtain information from other reactions in order to independently determine the relevant collinear twist-3 functions and/or verify that previously extracted functions are consistent with other measurements. In this context, one already has data on A N in p ↑ p → jet X available from the A N DY Collaboration [51]. Experiments to determine A N for Drell-Yan and direct photon production would also be beneficial. Even measurements of the Sivers and Collins asymmetries at large P h⊥ would be helpful and could be performed at Jefferson Lab (JLab) 12, COMPASS, or a future Electron-Ion Collider. In addition, data on TSSAs for single-inclusive hadron production from lepton-nucleon collisions is currently available from JLab [52] and HERMES [53]. This reaction was also recently analyzed in [54] using the collinear twist-3 approach and in [55] within the TMD framework. The main question then becomes if one can find a formalism that can consistently describe TSSAs in all of these processes. Much work is left to be done on both the theoretical and experimental side in order to answer this.