Numerical study of the twist-3 asymmetry ALTin single-inclusive electron-nucleon and proton-proton collisions Brandon Bauer Daniel Pitonyak and Cody Shay

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Numerical study of the twist-3 asymmetry ALT in single-inclusive

electron-nucleon and proton-proton collisions

Brandon Bauer, Daniel Pitonyak, and Cody Shay

Department of Physics, Lebanon Valley College, Annville, PA 17003, USA

We provide the ﬁrst rigorous numerical analysis of the longitudinal-transverse double-spin asym-

metry ALT in electron-nucleon and proton-proton collisions for the case where only a single pion,

jet, or photon is detected in the ﬁnal state. Given recent extractions of certain, previously unknown,

non-perturbative functions, we are able to compute contributions from all terms relevant for ALT

and make realistic predictions for the observable at Jeﬀerson Lab (JLab) 12 GeV, COMPASS, the

future Electron-Ion Collider, and the Relativistic Heavy Ion Collider. We also compare our results

to a JLab 6 GeV measurement, which are the only data available for this type of reaction. The

twist-3 nature of ALT makes it a potentially fruitful avenue to probe quark-gluon-quark correlations

in hadrons as well as provide insights into dynamical quark mass generation in QCD.

I. INTRODUCTION

One of the earliest puzzles in spin physics research was the observation in the 1970s of large asymmetries in single-

inclusive reactions where one hadron is transversely polarized [1, 2] – so-called single transverse-spin asymmetries

(SSAs) AN. This eventually was recognized as a signature of multi-parton correlations in hadrons [3–7] and has

been a source of intense theoretical [3–23], phenomenological [7, 10, 16, 24–33], and experimental [34–49] activity for

decades. The collinear twist-3 formalism that underpins this work allows one to explore a rich set of non-perturbative

functions, of which SSAs are sensitive to a certain subset. Namely, the naïve time-reversal odd (T-odd) nature of SSAs

gives access to pole contributions from initial state multi-parton distribution functions (PDFs) (where typically one of

the partons’ momentum fractions vanishes [5–8, 10, 13, 16]1); or to the imaginary part of (non-pole) ﬁnal-state multi-

parton fragmentation functions (FFs) [14, 15].2For example, ANin p↑p→π X at forward rapidity is mainly sensitive

to the Qiu-Sterman PDF FF T (x, x)(where the two quarks carry the same momentum fraction x), as well as H⊥(1)

1(z)

(which is the ﬁrst-moment of the Collins function) and ˜

H(z), with zthe momentum fraction carried by the produced

hadron. The latter two functions are certain integrals over z1(from zto ∞) of the FF ˆ

F U (z, z1)[17], where =

indicates the imaginary part. There are a plethora of SSA measurements, not only in p↑p→h X but also semi-inclusive

deep-inelastic scattering (SIDIS) e N↑→e h X [51–58], electron-positron annihilation e+e−→h1h2X[59–63], and

Drell-Yan p↑p→ {W±, Z, or `+`−}X[64, 65]. Due to this data, as well as the connection between collinear twist-3

and transverse momentum dependent (TMD) functions [66–70], FF T (x, x),H⊥(1)

1(z), and ˜

H(z), along with the twist-2

transversity PDF h1(x), have all been extracted in various phenomenological analyses (see, e.g., [29, 32, 33, 71–74]).

A complimentary observable to study multi-parton correlations in hadrons is the longitudinal-transverse double-

spin asymmetry ALT in collisions like ~e N↑→π X and p↑~p →π X. These are T-even reactions that are sensitive to

the non-pole pieces of certain multi-parton PDFs (e.g., FF T (x, x1)with x6=x1) and the real part <of certain multi-

parton FFs (e.g., ˆ

F U (z, z1)). From the theoretical side, ALT has been well studied in electron-nucleon [17, 75, 76]

and proton-proton [77–81] collisions for various single-inclusive ﬁnal states (e.g., hadron, jet, or photon), with some

limited numerical work performed for the electron-nucleon case [75, 76], but none for proton-proton. The main

hindrance to more rigorous predictions has been the lack of input for important non-perturbative functions in ALT ,

which forces one to resort to approximations or the outright neglect of certain terms [75, 76]. For example, one of

the main PDFs that enters ALT is g(1)

1T(x), which is the ﬁrst-moment of the worm-gear TMD g1T, and it has only

been extracted recently [82, 83].3Previous numerical computations utilizing g(1)

1T(x)relied on a Wandzura-Wilczek

approximation [17, 84–86] that neglects quark-gluon-quark correlators to approximate g(1)

1T(x)in terms of an integral

of the helicity PDF g1(x):g(1)

1T(x) = xR1

xdy g1(y)/y. In addition, the twist-3 fragmentation piece to ALT is sensitive

1The poles are due to propagators in the hard scattering going on shell. While usually this causes a momentum fraction in the multi-

parton PDF to vanish (“soft poles”), there are certain processes that also lead to “hard poles” [9, 11, 50], where all parton momentum

fractions remain nonzero.

2We will still refer to initial-state twist-3 functions as parton distribution functions (PDFs) and ﬁnal-state twist-3 functions as fragmen-

tation functions (FFs), even though they do not have a strict probability interpretation.

3We mention that the authors of Ref. [83] did not directly extract the twist-3 function g(1)

1T(x)needed in our analysis.

arXiv:2210.14334v2 [hep-ph] 4 Jan 2023

to a coupling of the chiral-odd twist-3 FF E(z)with h1(x)[80]. No extractions exist of E(z), but recent knowledge

obtained about the closely related FF ˜

H(z)[33] allows us for the ﬁrst time to develop a realistic input for E(z)(in

past numerical work, this function had been simply set to zero [76]). The potential for future measurements of ALT ,

particularly in electron-nucleon collisions, to provide more direct information about E(z)are intriguing due to the

connection of this FF to dynamical quark mass generation in QCD [87–89].

From the experimental side, measurements of ALT in single-inclusive processes like those introduced above are

unfortunately lacking. The only data available are from Jeﬀerson Lab 6 GeV (JLab6) on ALT in ~e n↑→π X [90].

Therefore, in this paper we give rigorous numerical predictions for ALT in a variety of reactions and kinematic

conﬁgurations in order to motivate future measurements. Namely, we will present results for ~e N↑→π X for JLab 12

GeV (JLab12) with N=n, COMPASS with N=p, and the future Electron-Ion Collider (EIC) with N=p(along

with ~e p↑→jet X), as well as for the Relativistic Heavy Ion Collider (RHIC) for p↑~p → {π, jet, or γ}X. Even with

the new information about g(1)

1T(x)and ˜

H(z)previously mentioned, we still must employ approximations for or neglect

certain twist-3 PDFs or FFs due to lack of input for them. Thus, one stands to gain further insight into multi-parton

correlations through measurements of ALT . Especially with only a few years of running left at RHIC, the world’s

only polarized proton-proton collider, one may forever lose the chance to measure ALT in p↑~p → {π, jet, or γ}X.

The paper is organized as follows: in Sec. II we review the analytical formulas for ALT that have been derived in

the literature for the processes of interest along with the twist-3 PDFs and FFs that enter them. We also discuss the

inputs and approximations used for these various non-perturbative functions as well as our strategy for computing the

average values and uncertainties of our predictions. We examine the main selected results for ALT in electron-nucleon

and proton-proton collisions, and their implications for future measurements, in Sec. III. The plots themselves can

be can be found in Appendix A (for electron-nucleon) and Appendix B (for proton-proton). In Sec. IV we close with

our conclusions and outlook.

II. THEORETICAL AND COMPUTATIONAL BACKGROUND

In this section we review the analytical formulas for ALT needed for our computational work along with the relevant

non-perturbative functions and certain relations between them. The asymmetry itself is generically deﬁned as

ALT ≡

4n[dσLT (+,↑)−dσLT (−,↑)] −[dσLT (+,↓)−dσLT (−,↓)]o

dσunp

,(1)

where dσLT (λ, ~

ST)(dσunp) is the longitudinal-transverse spin-dependent (unpolarized) cross section, with +(−)

indicating a particle with positive (negative) helicity λ, and ↑(↓) denoting a particle with transverse spin ~

STalong

the designated positive (negative) transverse axis (e.g., ±y). Moving forward, the numerator of Eq. (1) will be denoted

by dσLT (without any arguments). We break this section down into the electron-nucleon and proton-proton cases.

A. ALT in Electron-Nucleon Collisions

We consider the reaction ~e N↑→ {πor jet}X, where the produced ﬁnal-state particle has a transverse momentum

PT, which sets the hard scale for the process. We deﬁne the +z-axis to be the direction of N↑’s momentum in the

electron-nucleon center-of-mass (c.m.) frame. In addition to PT, the asymmetry also depends on the c.m. energy √S

and rapidity η(which can also be written in terms of xF= 2PTsinh(η)/√S). The coordinate system is such that at

ﬁxed-target experiments like JLab and COMPASS, the ﬁnal-state particle is produced in the backward region (i.e.,

negative rapidity). The two other Mandelstam variables at the hadronic level are T=−√SpP2

T+x2

FS/4 + xFS/2

and U=−√SpP2

T+x2

FS/4−xFS/2. We can then write ALT for the case of pion production as [17, 76],

A~eN↑

→πX

LT =Z1

zmin

z3−4PT

S+T/z 1

aM

ˆuDπ/a

1(z)Ga/N(x, ˆs, ˆ

t, ˆu) + Mπ

zˆ

tha/N

1(x)Eπ/a(z)−ˆs

t

zmin

S+T/z

afa/N

1(x)Dπ/a

1(z)ˆs2+ ˆu2

t2,(2)

where

G(x, ˆs, ˆ

t, ˆu) = g(1)

1T(x)−xdg(1)

1T(x)

dx !ˆs(ˆs−ˆu)

2ˆ

t2+x gT(x)−ˆsˆu

t2+x g1

(x)ˆu(ˆs−ˆu)

2ˆ

t2,(3)

with x=−(U/z)/(S+T /z),zmin =−(T+U)/S, and the partonic Mandelstam variables ˆs=xS, ˆ

t=xT/z, ˆu=U/z.

The sum Pais over all light quark and antiquark ﬂavors (a=qor ¯q), eais the quark or antiquark charge (in units

of the positron charge e), and M(Mπ) is the nucleon (pion) mass.

The non-perturbative functions in Eqs. (2), (3) include the (twist-2) unpolarized PDF f1(x)and FF D1(z), helicity

PDF g1(x), and transversity PDF h1(x), along with the kinematical twist-3 PDF g(1)

1T(x)(ﬁrst-moment of the worm-

gear TMD g1T), intrinsic twist-3 PDF gT(x), and (chiral-odd) intrinsic twist-3 FF E(z). We see that Eq. (2) can be

separated into two terms: one involving twist-3 PDFs (what we will call the “distribution term”) and one involving a

twist-3 FF (what we will call the “fragmentation term”). We note that the case of jet production [75] can be readily

obtained from Eq. (2) by replacing D1(z)with δ(1 −z)and setting the fragmentation term to zero.

Some readers may be familiar with the more widely studied/measured ALT asymmetry in inclusive DIS ~e N↑→

e X [91–98], where the scattered electron is detected in the ﬁnal state instead of a pion. In that process, the entire

result depends only on gT(x), which is connected to the color Lorentz force on a struck quark in DIS [99]. Already

Eq. (2) makes apparent the rich structure of multi-parton correlators one is sensitive to in ALT for ~e N↑→π X that

cannot be accessed in inclusive DIS. This presents both a challenge, in that one has several unknown twist-3 functions,

but also an opportunity to probe diﬀerent aspects of multi-parton correlations in hadrons.

As alluded to above, there are diﬀerent categories of twist-3 correlators: kinematical, intrinsic, and also dy-

namical [17]. The kinematical twist-3 functions are generically ﬁrst-moments of twist-2 TMDs (f(1)(x)≡

Rd2~

kT~

T/(2M2)f(x,~

T)); intrinsic use a twist-3 Dirac projection in a quark-quark correlator; and dynamical are

quark-gluon-quark or tri-gluon correlators. These twist-3 PDFs or FFs are not independent of each other and can

be related through QCD equation-of-motion relations (EOMRs) and Lorentz invariance relations (LIRs). We refer

the reader to Ref. [17] (and references therein) for an extensive overview of collinear twist-3 functions, including

their correlator deﬁnitions, derivations of EOMRs and LIRs, and how to express kinematical and intrinsic twist-3

functions in terms of the dynamical ones. For the PDFs relevant to our study (see Eq. (3)), we note the following

relations [17, 85, 100–103]:

gq/N

T(x) = gq/N

1(x) + dg(1)q/N

1T(x)

dx −2PZ1

−1

dy Gq/N

F T (x, y)

(x−y)2,(4)

g(1)q/N

1T(x) = xgq/N

T(x)−mq

Mhq/N

1(x) + PZ1

−1

dx1

Fq/N

F T (x, x1)−Gq/N

F T (x, x1)

x−x1

,(5)

gq/N

T(x) = Z(x)

dy gq/N

1(y)

y+mq

M hq/N

1(x)

x+Zx

(x)

dy hq/N

1(y)

y2!

+Z(x)

dx1

1PZ1

−1

dx21−x1δ(x1−x)

x1−x2

Fq/N

F T (x1, x2)−3x1−x2−x1(x1−x2)δ(x1−x)

(x1−x2)2Gq/N

F T (x1, x2),

(6)

g(1)q/N

1T(x) = xZ(x)

dy gq/N

1(y)

y+mq

MxZx

(x)

dy hq/N

1(y)

+xZ(x)

dx1

1PZ1

−1

dx2"Fq/N

F T (x1, x2)

x1−x2−(3x1−x2)Gq/N

F T (x1, x2)

(x1−x2)2#,(7)

where Pdenotes the principal value prescription, (x)≡2θ(x)−1,mqis the quark mass, and FF T (x, x1),GF T (x, x1)

are dynamical twist-3 PDFs (with FF T (x, x1)giving the Qiu-Sterman function when x=x1). The twist-2, kinematical

twist-3, and intrinsic twist-3 PDFs all have support −1≤x≤1, where gq/N

1(−x) = g¯q/N

1(x),gq/N

T(−x) = g¯q/N

T(x),

g(1)q/N

1T(−x) = −g(1)¯q/N

1T(x), and hq/N

1(−x) = −h¯q/N

1(x). The dynamical twist-3 PDFs have support |x| ≤ 1,|x1| ≤ 1,

and |x−x1| ≤ 1, with Fq/N

F T (−x1,−x) = F¯q/N

F T (x, x1)and Gq/N

F T (−x1,−x) = −G¯q/N

F T (x, x1)[17]. The ﬁrst expression (4)

is a LIR and (5) is an EOMR, while (6), (7) are the result of solving Eqs. (4), (5) for the respective functions [17]

so that they only involve dynamical twist-3 correlators (with possibly a twist-2 term, as above with R(x)

xdy g1(y)/y).

Neglecting the quark mass terms and dynamical twist-3 PDFs in Eqs. (6), (7) leads to the well-known Wandzura-

Wilczek (WW) approximations [84, 85, 100–106]

ga/N

T(x)WW

≈Z1

dy ga/N

1(y)

y, g(1)a/N

1T(x)WW

≈xZ1

dy ga/N

1(y)

y,(8)

where a=qor ¯q. Until recently, the WW approximation was the only input available for g(1)

1T(x). Now with the

extraction of g(1)

1T(x)in Ref. [82], we do not necessarily have to resort to the WW approximation. The expression in

Eq. (7) makes clear there is more structure embedded in g(1)

1T(x)than what is accounted for in the WW approximation.

Likewise, using the extracted g(1)

1T(x)from Ref. [82] in Eq. (4) in principle inserts information about multi-parton

correlators into the expression for gT(x), which the WW approximation does not encode. Even so, we do not have

complete information on gT(x)because GF T (x, x1)is not known. In Ref. [107], gu−d

T(x)was extracted for the ﬁrst

time in lattice QCD using the so-called quasi-distribution approach [108]. An interesting prospect is one in principle

could obtain information on GF T (x, x1)through a ﬂavor-separated computation of gT(x)on the lattice (taking g1(x)

and g(1)

1T(x)as known functions).

On the fragmentation side we have [17]

Eh/q(z) = −2z Z∞

dz1

H<,h/q

F U (z, z1)

z−1

z1−mq

2Mh

Dh/q

1(z)!,(9)

where ˆ

HF U (z, z1)is a quark-gluon-quark (dynamical twist-3) FF, and Mhis the hadron mass. The support properties

are 0≤z≤1and z < z1<∞[17]. We mention again that dynamical twist-3 FFs are complex valued because of the

lack of a time-reversal constraint in the fragmentation sector and have both real <and imaginary =parts. Recently,

the FF ˜

H(z)has been extracted [33], and it is connected to the imaginary part of the same underlying correlator

HF U (z, z1)as E(z)depends on [17]:

Hh/q(z)=2zZ∞

dz1

H=,h/q

F U (z, z1)

z−1

.(10)

We will use ˜

H(z)to build up plausible scenarios for E(z)in our numerical work.

B. ALT in Proton-Proton Collisions

We now consider the reaction p↑~p → {π, jet, or γ}X. We deﬁne the +z-axis to be the direction of p↑’s momentum

in the proton-proton c.m. frame. There are three pieces to this observable for the case of pion production, depending

on whether the twist-3 eﬀects occur in p↑,~p, or π(for jet and γ, one only has the ﬁrst two terms). We write ALT for

this case as

Ap↑~p→πX

LT =dσTdist

LT +dσLdist

LT +dσfrag

dσunp

,(11)

where in the numerator we have indicated whether the term contains twist-3 eﬀects from p↑(transversely polarized

distribution – “Tdist”) [78], from ~p (longitudinally polarized distribution – “Ldist”) [81], or from π(fragmentation –

“frag”) [80]. The expression for the unpolarized cross section reads

dσunp =α2

SZ1

zmin

dz Z1

xmin

x0z2(xS +U/z)X

a,b,c

fa/p

1(x)fb/p

1(x0)Dπ/c

1(z)Hi

U(ˆs, ˆ

t, ˆu),(12)

where zmin =−(T+U)/S,xmin =−(U/z)/(S+T/z),x0=−(xT /z)/(xS +U/z), and the summations are over

all channels iand parton ﬂavors a, b, c. The hard factors Hi

U(ˆs, ˆ

t, ˆu)depend on the partonic Mandelstam variables

ˆs=xx0S, ˆ

t=xT/z, ˆu=x0U/z, and they can be found in Ref. [10].

We next turn to the longitudinal-transverse polarized cross sections. For dσTdist

LT we have [78]

dσTdist

LT =−2α2

sMPT

SZ1

zmin

dz Z1

xmin

x0z3(xS +U/z)X

a,b,c

ˆmiGa/p↑

i(x, ˆs, ˆ

t, ˆu)gb/~p

1(x0)Dπ/c

1(z),(13)

where

Gi(x, ˆs, ˆ

t, ˆu) = g(1)

1T(x)−xdg(1)

1T(x)

dx !Hi

˜g(ˆs, ˆ

t, ˆu) + xgT(x)Hi

1,GDT (ˆs, ˆ

t, ˆu) + x

2(g1(x)−gT(x)) Hi

3,GDT (ˆs, ˆ

t, ˆu)

+g(1)

1T(x) + PZ1

−1

dx1

x(FF T (x, x1) + GF T (x, x1))

x−x1Hi

2,GDT (ˆs, ˆ

t, ˆu).(14)

Some comments are in order about the expressions (13), (14). First, the variable ˆmiin Eq. (13) is either ˆs,ˆ

t, or ˆu

depending on the channel i, with the speciﬁc values found in Table 1 of Ref. [78].4Second, the original expression

in Ref. [78] (see Eq. (17) of that paper) is written in terms of the functions ˜g(x)and FDT (x, x1), GDT (x, x1). The

former is just a diﬀerent notation for g(1)

1T(x). The latter are “D-type” dynamical twist-3 PDFs that use the covariant

derivative, whereas we have chosen to write the result in terms of “F-type” functions FF T (x, x1), GF T (x, x1)that use

the ﬁeld strength tensor. They are related via [9]

FDT (x, x1) = P1

x−x1

FF T (x, x1),(15)

GDT (x, x1) = P1

x−x1

GF T (x, x1) + δ(x−x1)g(1)

1T(x).(16)

Lastly, we continued to “optimize” Eq. (14) from the original version in Ref. [78] so that it is written in terms of a

maximal set of functions for which there is input for from the literature. An observation made in Ref. [78] was that

the hard factors Hi

FDT ,Hi

GDT found in Appendix A5of that paper can be broken down into three types of terms,

namely, Hi=Hi

1+Hi

2/(1 −ξ) + Hi

3/ξ, where ξ= (x−x1)/x, with Hi

1,FDT =Hi

1,GDT ,Hi

2,FDT =−Hi

2,GDT , and

3,FDT = 0. This insight allows one to use the LIR (4) and EOMR (5) to obtain the ﬁnal form in Eq. (14), where

now the only non-perturbative functions we lack input for are FF T (x, x1), GF T (x, x1), and we will then ignore those

terms in our numerical work.

We now give the formulas for the remaining two terms in the numerator of Eq. (11). For dσLdist

LT we have [81]

dσLdist

LT =−2α2

sMPT

SZ1

zmin

dz Z1

xmin

z3(xS +U/z)X

a,b,c

ha/p↑

1(x)Hb/~p(x0,ˆs, ˆ

t, ˆu)Dπ/c

1(z),(17)

where

H(x0,ˆs, ˆ

t, ˆu) = h1(x0)Hi

1L(ˆs, ˆ

t, ˆu) + hL(x0)Hi

2L(ˆs, ˆ

t, ˆu) + dh⊥(1)

1L(x0)

dx0Hi

3L(ˆs, ˆ

t, ˆu).(18)

The hard factors Hi

{1,2,3}Lcorrespond to ˆσ{1,2,3}in Eqs. (16)–(21) of Ref. [81]. The function hL(x)is an intrinsic

twist-3 function while h⊥(1)

1L(x)is kinematical twist-3 (ﬁrst-moment of the other worm-gear TMD function h⊥

1L). Unlike

g(1)

1T(x), there are no phenomenological extractions of h⊥(1)

1L(x). Therefore, in our numerical work we must use WW

approximations that connect hL(x)and h⊥(1)

1L(x)to the twist-2 transversity PDF h1(x)[17, 100, 101, 103]:

ha/N

L(x)WW

≈2xZ1

dy ha/N

1(y)

y2, h⊥(1)a/N

1L(x)WW

≈x2Z1

dy ha/N

1(y)

y2,(19)

where a=qor ¯q. Finally, for dσfrag

LT we have [80]

dσfrag

LT =2α2

sMPT

SZ1

zmin

dz Z1

xmin

x0z4(xS +U/z)X

a,b,c

ha/p↑

1(x)gb/~p

1(x0)Eπ/c(z)Hi

f(ˆs, ˆ

t, ˆu),(20)

where the hard factors Hi

fcorrespond to ˆσiin Eq. (15) of Ref. [80], and E(z)is the same dynamical twist-3 FF

introduced in the electron-nucleon case (2) (see also Eq. (9)).

4We note a typo in the last row for the ˆ

tcolumn of Table 1 in Ref. [78], where the channel should read q¯q→¯q0q0.

5The hard factors Hi

˜gcan also be found in Appendix A of Ref. [78].

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Numericalstudyofthetwist-3asymmetryALTinsingle-inclusiveelectron-nucleonandproton-protoncollisionsBrandonBauer,DanielPitonyak,andCodyShayDepartmentofPhysics,LebanonValleyCollege,Annville,PA17003,USAWeprovidetherstrigorousnumericalanalysisofthelongitudinal-transversedouble-spinasym-metryALTinelectro...

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Numerical study of the twist-3 asymmetry ALTin single-inclusive electron-nucleon and proton-proton collisions Brandon Bauer Daniel Pitonyak and Cody Shay

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