Twist-3 fragmentation contribution to polarized hyperon production in unpolarized hadronic collisions

TL;DR

This paper calculates the contribution of twist-3 fragmentation functions to the polarization of hyperons produced in unpolarized proton-proton collisions, addressing a long-standing challenge in describing hyperon polarization.

Contribution

It provides the first leading-order calculation of twist-3 fragmentation effects on hyperon polarization within collinear factorization, incorporating QCD relations among functions.

Findings

Derived the leading-order cross section for twist-3 fragmentation contributions.

Clarified the role of twist-3 functions in hyperon polarization.

Enhanced understanding of polarization mechanisms in hadronic collisions.

Abstract

It has been known for a long time that hyperons produced in hadronic collisions are polarized perpendicular to the production plane of the reaction. This effect cannot be described by using twist-2 collinear parton correlators only. Here we compute the contribution of twist-3 fragmentation functions to the production of transversely polarized hyperons in unpolarized proton-proton collisions in the framework of collinear factorization. By taking into account the relations among the relevant twist-3 fragmentation functions which follow from the QCD equation of motion and the Lorentz invariance property of the correlators, we present the leading-order cross section for this term.

Equations

$${\Delta }_{ij}(z)$$

$${\Delta }_{\partial ij}^{\alpha }(z)=\frac{1}{N}\sum _X\int \frac{d\lambda }{2\pi }{e}^{-i\frac{\lambda }{z}}⟨0|[\infty w,0]{\psi }_i(0)|h(P_h,S_{\perp })X⟩$$

$$\times ⟨h(P_h,S_{\perp })X|{\bar{\psi }}_j(\lambda w)[\lambda w,\infty w]|0⟩{\overleftarrow{\partial }}^{\alpha }$$

$$=-i{M}_h{ϵ}^{\alpha S_{\perp }w{P}_h}(to0.0pt/\text{hss}{P}_h{)}_{ij}\frac{D_{1T}^{\perp (1)}(z)}{z}$$

$$+i{M}_h{S}_{\perp }^{\alpha }({\gamma }_{5}to0.0pt/\text{hss}{P}_h{)}_{ij}\frac{G_{1T}^{\perp (1)}(z)}{z}+\cdots .$$

$$[0,\lambda \omega ]=\mathcal{P}\exp \left\{i\mathrm{g}{\int }_{\lambda }^{0}dt{\omega }_{\mu }{A}^{\mu }(t\omega )\right\},$$

$$D_{1T}^{\perp (1)}(z)=z^2\int d^2{\vec{p}}_{\perp }\frac{{{\vec{p}}_{\perp }}^2}{2M_h^2}{D}_{1T}^{\perp }(z,z^2{\vec{p}}_{\perp }^2),$$

$${\Delta }_{Fij}^{\alpha }(z,z_1)$$

$$\mathrm{Im}{\hat{D}}_{DT}(z,z_1)$$

$$\mathrm{Im}{\hat{G}}_{DT}(z,z_1)$$

$$\int\frac{dz_{1}}{z_{1}^{2}}\Bigl{(}{\rm Im}\,\widehat{D}_{DT}(z,z_{1})-{\rm Im}\,\widehat{G}_{DT}(z,z_{1})\Bigr{)}=\frac{D_{T}(z)}{z}\,.$$

$$\displaystyle\int_{z}^{\infty}\frac{dz_{1}}{z_{1}^{2}}\frac{1}{1/z-1/z_{1}}\Bigl{(}{\rm Im}\,\widehat{D}_{FT}$$

$$=\frac{D_T(z)}{z}+D_{1T}^{\perp (1)}(z).$$

$$-\frac{2}{z}{\int }_z^{\infty }\frac{d{z}_1}{z_1^2}\frac{\mathrm{Im}{\hat{D}}_{FT}(z,z_1)}{(1/z_1-1/z{)}^2}=\frac{D_T(z)}{z}+\frac{d\left(D_{1T}^{\perp (1)}(z)/z\right)}{d(1/z)}.$$

$$p(p)+p(p^{\prime })\to {\Lambda }^{\uparrow }(P_h,S_{\perp })+X.$$

$$P_h^0\frac{d\sigma (P_h,S_{\perp })}{d^3{P}_h}$$

$${\Delta }_{ij}(k)$$

$${\Delta }_{A,ij}(k,k_1)=\frac{1}{N}\sum _X\int d^4\xi \int d^4\eta e^{-i{k}_1\cdot \xi }{e}^{-i(k-k_1)\cdot \eta }$$

$$\times ⟨0|{\psi }_i(0)|h(P_h,S_{\perp })X⟩⟨h(P_h,S_{\perp })X|{\bar{\psi }}_j(\xi )g{A}^{\alpha }(\eta )|0⟩,$$

$$P_h^0$$

$$P_h^0\frac{d\sigma (P_h,S_{\perp })}{d^3{P}_h}=\frac{2{\alpha }_s^2{M}_h}{s^2}{ϵ}^{P_{h}p{p}^{\prime }{S}_{\perp }}\int \frac{dx}{x}{f}_1(x)\int \frac{d{x}^{\prime }}{x^{\prime }}{f}_1(x^{\prime })$$

$$\times \int \frac{dz}{z^3}\delta (\hat{s}+\hat{t}+\hat{u})[\frac{D_T(z)}{z}{\hat{\sigma }}_1-\left\{\frac{d}{d(1/z)}\frac{D_{1T}^{\perp (1)}(z)}{z}\right\}{\hat{\sigma }}_2$$

$$-D_{1T}^{\perp (1)}(z){\hat{\sigma }}_3+{\int }_z^{\infty }\frac{d{z}_1}{z_1^2}\left(\frac{z_1}{1/z-1/z_1}\right)$$

$$\times \left(\mathrm{Im}{\hat{D}}_{FT}(z,z_1)+\mathrm{Im}{\hat{G}}_{FT}(z,z_1)\right){\hat{\sigma }}_4].$$

$${\hat{\sigma }}_1$$

$${\hat{\sigma }}_2$$

$${\hat{\sigma }}_3$$

$${\hat{\sigma }}_4$$

$${\hat{\sigma }}_1$$

$${\hat{\sigma }}_2$$

$${\hat{\sigma }}_3$$

$${\hat{\sigma }}_4$$

$${\hat{\sigma }}_1$$

$${\hat{\sigma }}_2$$

$${\hat{\sigma }}_3$$

$${\hat{\sigma }}_4$$

$${\hat{\sigma }}_1$$

$${\hat{\sigma }}_2$$

$${\hat{\sigma }}_3$$

$${\hat{\sigma }}_4$$

$${\hat{\sigma }}_1$$

Full text

††preprint: APS/123-QED

I Introduction

The first observation of transversely polarized hyperons in unpolarized hadronic collisions was already made in the 1970s. Specifically, when colliding protons with a beryllium target and detecting $\Lambda$ hyperons, it was found that the $\Lambda$’s show a transverse polarization asymmetry (often denoted as $A_{N}$), which is largest (up to 30%) for polarization perpendicular to the reaction plane and vanishes in the reaction plane Bunce:1976yb . This pioneering measurement was followed by a number of corresponding experiments which, in particular, also covered different kinematic ranges Heller:1978ty ; Erhan:1979xm ; Heller:1983ia ; Lundberg:1989hw ; Yuldashev:1990az ; Ramberg:1994tk ; Fanti:1998px ; Abt:2006da ; Aaij:2013oxa ; ATLAS:2014ona . Some of the earlier data are reviewed in Pondrom:1985aw ; Panagiotou:1989sv . We also refer to Metz:2016swz for a list of relevant papers. Generally $A_{N}$ vanishes for exact mid-rapidity of the hyperon in a process like $p\,p\to\Lambda^{\uparrow}X$, and it increases with increasing rapidity. Hyperon polarization was also studied in related reactions such as $\gamma\,p\to\Lambda^{\uparrow}X$ Aston:1981em ; Abe:1983jy , quasi-real photo-production of $\Lambda$’s in lepton scattering off nucleons and nuclei Airapetian:2007mx ; Airapetian:2014tyc , and in electron-positron collisions Ackerstaff:1997nh ; Abdesselam:2016nym .

For high-energy collisions and sufficiently large transverse momentum $P_{T}$ of the hyperon, $A_{N}$ can be computed in perturbative quantum chromodynamics (QCD). However, it has been known for a long time that by only using collinear leading-twist (twist-2) parton correlators one cannot describe this type of transverse single-spin asymmetry (SSA) Kane:1978nd . As $A_{N}$ is a genuine twist-3 observable, one rather needs the full machinery of collinear higher-twist factorization Efremov:1981sh ; Ellis:1982wd ; Ellis:1982cd . Already in the early 1980s this approach was used in connection with transverse SSAs Efremov:1981sh ; Efremov:1984ip . Later works further elaborated on these twist-3 calculations, where a main focus was on the transverse target SSA for processes like $p^{\uparrow}p\to\pi X$ — see for instance Qiu:1991pp ; Qiu:1991wg ; Qiu:1998ia ; Kanazawa:2000kp ; Eguchi:2006qz ; Eguchi:2006mc ; Kouvaris:2006zy ; Koike:2009ge ; Metz:2012ct ; Beppu:2013uda . An overview of these calculations can be found in Pitonyak:2016hqh .

In collinear factorization, the transverse SSAs receive, a priori, twist-3 contributions from two-parton and three-parton correlation functions which are associated with either the initial-state or final-state hadrons. These correlators are parameterized in terms of twist-3 parton distribution functions (PDFs) and fragmentation functions (FFs), respectively. While the complete leading-order (LO) twist-3 cross section for $p^{\uparrow}p\to\pi X$ can be found in the literature Kouvaris:2006zy ; Koike:2009ge ; Metz:2012ct ; Beppu:2013uda , only part of the twist-3 cross section for $p\,p\to\Lambda^{\uparrow}X$ is available Kanazawa:2000cx ; Zhou:2008fb ; Koike:2015zya . The present work is a major step towards completing the calculation of all possible terms.

The numerator of the transverse SSA for $p\,p\to\Lambda^{\uparrow}X$ has two types of contributions. The first one, which contains a twist-3 PDF for one of the unpolarized protons combined with the unpolarized twist-2 PDF for the other proton and the spin-dependent twist-2 “transversity” FF, was derived in Refs. Kanazawa:2000cx ; Zhou:2008fb ; Koike:2015zya . Here we focus on the second contribution, which involves twist-3 FFs and the twist-2 unpolarized PDFs of the protons. (A short version of the present work was presented in Yabe:talk ; Yabe:proceedings .) Specifically, we compute all the LO terms that are related to quark-gluon-quark (qgq) fragmentation correlators, while terms given by quark-antiquark-gluon ($q\bar{q}g$) correlators and pure gluon (gg and ggg) correlators will be considered elsewhere.

Not only is our study important for obtaining a complete analytical result, but it may also be critical for the phenomenology of this observable. In this context we emphasize that recent work strongly suggests numerical dominance of the collinear twist-3 fragmentation contribution for $A_{N}$ in $p^{\uparrow}p\to\pi X$ Kanazawa:2014dca ; Gamberg:2017gle .

The remainder of the paper is organized as follows: In Sec. II, we list the definitions of the twist-3 FFs that are relevant for the present study. In that section we also present relations among the FFs which are based on the QCD equation of motion and Lorentz invariance Kanazawa:2015ajw . These relations are crucial for, in particular, obtaining a frame-independent result for $A_{N}$. In Sec. III, we discuss the calculation for the twist-3 fragmentation contribution to the cross section for $p\,p\to\Lambda^{\uparrow}X$, while Sec. IV is devoted to a brief summary.

II Twist-3 fragmentation functions and their relations

We first recall the definitions of the twist-3 FFs for a transversely polarized spin-$\frac{1}{2}$ hadron. One can identify, a priori, three different types of such functions, which in Kanazawa:2015ajw were referred to as intrinsic, kinematical, and dynamical FFs. We start with the intrinsic twist-3 FFs for $\Lambda^{\uparrow}$. They are defined through a quark-quark (qq) fragmentation correlator according to Mulders:1995dh ; Bacchetta:2006tn ; Kanazawa:2015ajw

[TABLE]

where $\psi_{i}$, $\psi_{j}$ are the quark fields carrying the spinor indices $i$, $j$, and a color average is implied in Eq. (1), with $N=3$ the number of quark colors. The hadron ($\Lambda$) is characterized by its four-momentum $P_{h}$ and (transverse) spin vector $S_{\perp}$, while $M_{h}$ is its mass. The four-vector $w^{\mu}$ is light-like ($w^{2}=0$) and satisfies $P_{h}\cdot w=1$. For simplicity, Wilson lines in the operator (1) are suppressed. Here and below we use the shorthand notation $\epsilon^{\alpha S_{\perp}wP_{h}}\equiv\epsilon^{\alpha\beta\gamma\delta}S_{\perp\beta}w_{\gamma}P_{h\delta}$, where our convention for the Levi-Civita tensor is $\epsilon^{0123}=+1$. The r.h.s. of Eq. (1) contains the twist-2 transversity FF $H_{1}$, which describes the probability for a transversely polarized quark to fragment into a transversely polarized hadron, and the intrinsic twist-3 FFs $D_{T}$ and $G_{T}$. These (dimensionless) FFs depend on the fraction $z$ of the quark momentum which is carried by the hadron. Hermiticity implies that they are real-valued. From the functions available in (1), it is actually only the naïve time-reversal-odd (T-odd) function $D_{T}$ which contributes to the piece of the transverse SSA we calculate here.

The kinematical twist-3 FFs parameterize the derivative of the qq correlator Mulders:1995dh ; Bacchetta:2006tn ; Kanazawa:2015ajw ,

[TABLE]

The derivative on the r.h.s. of (2) also acts on the Wilson line, which generally is defined through

[TABLE]

where $\mathcal{P}$ indicates path-ordering and $g$ is the strong coupling. The FFs $D_{1T}^{\perp(1)}$ and $G_{1T}^{\perp(1)}$ are also real-valued. This (T-odd) function is a particular moment of a transverse-momentum-dependent (TMD) FF Mulders:1995dh ; Bacchetta:2006tn ,

[TABLE]

with $D_{1T}^{\perp}$ describing the fragmentation of an unpolarized quark into a transversely polarized spin-$\frac{1}{2}$ hadron. Using the so-called generalized parton model, which exclusively works with TMD PDFs and FFs, in Ref. Anselmino:2000vs this function was fitted to $A_{N}$ data for $p\,p\to\Lambda^{\uparrow}X$. The result of the fit was then used to estimate transverse SSAs in semi-inclusive deep-inelastic scattering Anselmino:2001js , including neutrino-nucleon scattering $\nu N\to\ell^{\pm}\Lambda^{\uparrow}X$ for which some data are available Astier:2000ax . We note that, like with the intrinsic functions, only T-odd kinematical correlators can contribute to our calculation. Namely, only $D_{1T}^{\perp(1)}$ can enter from (2).

Let us finally discuss the dynamical twist-3 FFs. They parameterize the so-called $F$-type qgq correlator (see, for instance, Refs. Meissner:thesis ; Kanazawa:2015jxa ; Kanazawa:2015ajw ),

[TABLE]

where $F^{\alpha w}\equiv F^{\alpha\beta}w_{\beta}$ represents a component of the gluon field strength tensor. The (two-argument) FFs $\widehat{D}_{FT}(z,z_{1})$ and $\widehat{G}_{FT}(z,z_{1})$ have support for $1>z>0$ and $z_{1}>z$ Gamberg:2008yt ; Meissner:2008yf ; Gamberg:2010uw ; Kanazawa:2013uia . These support properties imply, in particular, that the functions themselves Meissner:2008yf and their $z_{1}$-partial derivatives Kanazawa:2015ajw vanish for a vanishing gluon momentum. Therefore, for fragmentation one has no so-called gluonic poles Qiu:1991pp ; Qiu:1991wg , which play a very important role for contributions to SSAs caused by twist-3 effects coming from initial-state hadrons. The vanishing of gluonic poles in fragmentation is also intimately connected with the universality of TMD FFs (see, for instance, Refs. Metz:2002iz ; Collins:2004nx ; Gamberg:2008yt ; Meissner:2008yf ; Gamberg:2010uw ). In general, the dynamical twist-3 FFs are complex Meissner:thesis ; Yuan:2009dw ; Kang:2010zzb ; Metz:2012ct ; Kanazawa:2013uia , and we use their complex conjugate in Eq. (5). (In this paper we follow the convention of Kanazawa:2015ajw for $\widehat{D}_{FT}$ and $\widehat{G}_{FT}$.) One can also define D-type twist-3 FFs by replacing $gF^{\alpha w}(\mu w)$ in (5) with the covariant derivative $D^{\alpha}(\mu w)$. These functions, however, do not represent new independent objects, but they can rather be related to the F-type functions. For the imaginary parts of the FFs, which matter in the present study, one has the relations Eguchi:2006qz ; Eguchi:2006mc ; Meissner:thesis ; Yuan:2009dw ; Kang:2010zzb ; Metz:2012ct ; Kanazawa:2013uia

[TABLE]

where here $P$ indicates the principal-value prescription. For the calculation in Sec. III we will use the F-type FFs.

Although, a priori, all three types of FFs as defined in (1), (2), (5) appear in the derivation of the twist-3 cross section for $p\,p\to\Lambda^{\uparrow}X$, these functions are not independent. There exist relations based on the QCD equation of motion (e.o.m.) and so-called Lorentz invariance relations (LIRs). A comprehensive derivation of these relations as well as a list of references can be found in Kanazawa:2015ajw . The e.o.m. relation which is relevant for the present study takes its simplest form for the D-type functions,

[TABLE]

Using Eqs. (6), (7) one finds the e.o.m. relation involving the F-type functions,

[TABLE]

Making use of Lorentz invariance of the parton correlation functions one can further derive the LIR

[TABLE]

These relations generally simplify the form of twist-3 cross sections and, in particular, guarantee their frame-independence Kanazawa:2014tda ; Kanazawa:2015jxa ; Kanazawa:2015ajw ; Gamberg:2017gle .

III Twist-3 cross section for $pp\to\Lambda^{\uparrow}X$

We now sketch the derivation and present the results of the twist-3 spin-dependent cross section for

[TABLE]

Applying the Feynman gauge formalism developed in Kanazawa:2013uia , one can obtain the corresponding cross section as

[TABLE]

where the summation over all channels and parton types is implicit. In Eq. (12), $s=(p+p^{\prime})^{2}$ is the square of the center of mass energy and $\Omega^{\alpha}_{\beta}=g^{\alpha}_{\beta}-P_{h}^{\alpha}w_{\beta}$. The unpolarized twist-2 PDF is denoted by $f_{1}$, and the correlators $\Delta(z)$, $\Delta^{\beta}_{\partial}(z)$ and $\Delta^{\beta}_{F}(z,z_{1})$ are defined in (1), (2) and (5). The symbol ${\rm Tr}$ indicates the trace over color and spinor indices. In deriving Eq. (12), we introduced the partonic hard scattering parts (before collinear expansion), $S(k)$ and $S_{L\alpha}\!\!\left(k_{1},k\right)$, corresponding, respectively, to the fragmentation matrix elements

[TABLE]

and

[TABLE]

as shown in Figs. 1 (a) and (b). In Ref. Kanazawa:2013uia , it has been proven that, after the collinear expansion, $S$ and $S_{L\alpha}$ eventually constitute the partonic hard cross section for the gauge-invariant correlation functions $\Delta(z)$, $\Delta^{\beta}_{\partial}(z)$ and $\Delta^{\beta}_{F}(z,z_{1})$, as shown in (12). Note $S_{L\alpha}\!\!\left(\frac{P_{h}}{z_{1}},\frac{P_{h}}{z}\right)$ is the hard part for the diagram in which the coherent gluon line from $\Delta^{\beta}_{F}(z,z_{1})$ is located to the left of the cut, and the effect of the mirror diagram is taken into account by the principal value prescription and the factor 2 for the third term in (12). Substituting (1), (2) and (5) into Eq. (12), one can cast the cross section in the following form:

[TABLE]

where $A^{(1)}(w)\equiv{p^{\prime}\cdot P_{h}\over p\cdot p^{\prime}}\epsilon^{P_{h}pwS_{\perp}}$, $A^{(2)}(w)\equiv{p\cdot P_{h}\over p\cdot p^{\prime}}\epsilon^{P_{h}wp^{\prime}S_{\perp}}$, and each partonic cross section $\hat{\sigma}^{(i)}_{Y}$ ($i=1,\,2$, $Y=T,\,D,\,ND,\,\dots$) is a function of the partonic Mandelstam variables defined as $\hat{s}=(xp+x^{\prime}p^{\prime})^{2}$, $\hat{t}=(xp-P_{h}/z)^{2}$, $\hat{u}=(x^{\prime}p^{\prime}-P_{h}/z)^{2}$. The lowest-order Feynman diagrams for the partonic hard parts $S$ and $S_{L\alpha}$ in each channel are shown in Figs. 2–5. Several comments are in order here: (i) Unlike in the case of the twist-3 PDFs, for twist-3 FFs the nonpole term of the hard scattering coefficients contributes to $A_{N}$. (In this context, see also the discussion in the paragraph after Eq. (5).) In particular, the imaginary part of the complex functions $\widehat{D}_{FT}$ and $\widehat{G}_{FT}$ contributes to the spin-dependent cross section, reflecting the naïve T-odd nature of $A_{N}$.

(ii) One finds that the $z_{1}$-dependence of the hard parts $S_{L\alpha}$ in Eq. (12) has a relatively simple structure that does not “mix” with the partonic Mandelstam variables. Therefore, the contributions from ${\rm Im}\,\widehat{D}_{FT}$ and ${\rm Im}\,\widehat{G}_{FT}$ can be brought into the form shown in (15), where the hard partonic cross sections $\hat{\sigma}^{(i)}_{Y}$ are independent of $z_{1}$ and only depend on $\hat{s}$, $\hat{t}$, $\hat{u}$.

(iii) Calculation of the diagrams provides the relations $\hat{\sigma}^{(i)}_{DF1}=-\hat{\sigma}_{GF1}^{(i)}$, $\hat{\sigma}_{DSFP}^{(i)}=\hat{\sigma}_{GSFP}^{(i)}$, $\hat{\sigma}_{GF2}^{(i)}=0$ ($i=1,\,2$) for all channels. Using these relations in combination with the e.o.m. relation (9) and the LIR (10) one can rewrite the cross section in (15) in a very compact manner — see Eq. (16) below — by introducing the combinations $\hat{\sigma}_{1}^{(i)}\equiv\hat{\sigma}_{T}^{(i)}+\hat{\sigma}_{DF1}^{(i)}+\hat{\sigma}_{DF2}^{(i)}$, $\hat{\sigma}_{2}^{(i)}\equiv\hat{\sigma}_{D}^{(i)}-\hat{\sigma}_{DF2}^{(i)}$, $\hat{\sigma}_{3}^{(i)}\equiv\hat{\sigma}_{ND}^{(i)}-\hat{\sigma}_{DF1}^{(i)}$ and $\hat{\sigma}_{4}^{(i)}\equiv\hat{\sigma}_{DSFP}^{(i)}$.

(iv) In order to test the frame-independence of our result, we have computed the cross section in two different frames: $w_{1}^{\mu}=\frac{p^{\prime\mu}}{p^{\prime}\cdot P_{h}}$ and $w_{2}^{\mu}=\frac{p^{\mu}}{p\cdot P_{h}}$. In the first frame, $A^{(1)}(w_{1})$ is nonzero while $A^{(2)}(w_{1})$ vanishes, so the result depends only on $A^{(1)}(w_{1})$ and $\hat{\sigma}_{1,2,3,4}^{(1)}(w_{1})$. In the second frame, $A^{(2)}(w_{2})$ is nonzero while $A^{(1)}(w_{2})$ vanishes, so the result depends only on $A^{(2)}(w_{2})$ and $\hat{\sigma}_{1,2,3,4}^{(2)}(w_{2})$. Since $A^{(1)}(w_{1})=A^{(2)}(w_{2})$, and we also found $\hat{\sigma}_{1,2,3,4}^{(1)}(w_{1})=\hat{\sigma}_{1,2,3,4}^{(2)}(w_{2})\equiv\hat{\sigma}_{1,2,3,4}$ for all channels, the result is the same in both frames.

Our final expression for the frame-independent twist-3 cross section reads

[TABLE]

This represents the complete result of the cross section caused by twist-3 effects of the qq and qgq fragmentation correlators depicted in Fig. 1. The partonic cross sections for each channel are given as follows:

(1) $qq^{\prime}\to qq^{\prime}\,$:

[TABLE]

(2) $q^{\prime}q\to qq^{\prime}\,$:

[TABLE]

(3) $qq\to qq\,$:

[TABLE]

(4) $q\bar{q}\to q^{\prime}\bar{q}^{\,\prime}\,$:

[TABLE]

(5) $\bar{q}q\to q^{\prime}\bar{q}^{\,\prime}\,$:

[TABLE]

(6) $q\bar{q}^{\,\prime}\to q\bar{q}^{\,\prime}\,$:

[TABLE]

(7) $\bar{q}^{\,\prime}q\to q\bar{q}^{\,\prime}\,$:

[TABLE]

(8) $q\bar{q}\to q\bar{q}\,$:

[TABLE]

(9) $\bar{q}q\to q\bar{q}\,$:

[TABLE]

(10) $qg\to qg\,$:

[TABLE]

(11) $gq\to qg\,$:

[TABLE]

(12) $gg\to q\bar{q}$ :

[TABLE]

Note that for a LO calculation of the fragmentation term, more channels contribute to $A_{N}$ for $p\,p\to\Lambda^{\uparrow}X$ than for $p^{\uparrow}p\to\pi X$ Metz:2012ct due to the chiral-odd nature of the parton correlators involved in the latter case.

IV Summary

We have calculated, to leading order in perturbation theory, the twist-3 fragmentation contribution to the transverse SSA $A_{N}$ for hyperon production in unpolarized proton-proton collisions. Specifically, we have taken into account all contributions arising from qq and qgq fragmentation correlators. We have verified that the result of the cross section is frame-independent when taking into account relations between the twist-3 FFs which are based on the QCD equation of motion and the Lorentz invariance of the parton correlators.

In order to complete the calculation of this spin-dependent twist-3 fragmentation effect, one also needs to include $q\bar{q}g$ correlators (see Fig. 6) as well as gg and ggg correlators. We mention that the $q\bar{q}g$ graphs need the pure gluon ones in order to have a gauge invariant subset of diagrams. (In the case of $A_{N}$ for $p^{\uparrow}p\to\pi X$ the contribution from the former vanishes after summing over all graphs, while the latter do not contribute at all Metz:2012ct .) Let us finally mention that so far the properties of tri-gluon FFs have only been studied to some extent Meissner:thesis . In particular, the LIRs in this case are not yet known. We plan to address these topics elsewhere.

Acknowledgements.

This work has been supported by the Grant-in-Aid for Scientific Research from the Japanese Society of Promotion of Science under Contract No. 26287040 (Y.K.), the National Science Foundation under Contract No. PHY-1516088 (A.M.), the U.S. Department of Energy, Office of Science, Office of Nuclear Physics within the framework of the TMD Topical Collaboration (D.P.), and in part by the U.S. Department of Energy, Office of Science under contract No. DE-AC52-06NA25396 and the LANL LDRD Program (S.Y.).