Quarkonium polarization in low- pThadro-production from past data to future opportunities Pietro Faccioli1 Ilse Kr atschmer2and Carlos Louren co3

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Quarkonium polarization in low-pThadro-production:

from past data to future opportunities

Pietro Faccioli1), Ilse Kr¨atschmer2) and Carlos Louren¸co3)

Abstract

Several ﬁxed-target experiments reported J/ψand Υ polarization measurements, as

functions of Feynman x(xF) and transverse momentum (pT), in three diﬀerent po-

larization frames, using diﬀerent combinations of beam particles, target nuclei and

collision energies. The data form such a diverse and heterogeneous picture that, at

ﬁrst sight, no clear trends can be observed. A more detailed look, however, allows

us to discern qualitative physical patterns that inspire and support a simple inter-

pretation: the directly-produced quarkonia result from either gluon-gluon fusion or

from quark-antiquark annihilation, with the former mesons being fully longitudinally

polarized and the latter being fully transversely polarized. This hypothesis provides

a reasonable quantitative description of the J/ψand Υ(1S) polarizations measured in

the xF.0.5 kinematical domain. We provide predictions that can be experimentally

tested, using proton and/or pion beams, and show that improved J/ψand ψ(2S) po-

larization measurements in pion-nucleus collisions can provide signiﬁcant constraints

on the poorly known parton distribution functions of the pion.

1) LIP, Lisbon, Portugal, Pietro.Faccioli@cern.ch

2) ISTA, Klosterneuburg, Austria, Ilse.Kraetschmer@gmx.at

3) CERN, Geneva, Switzerland, Carlos.Lourenco@cern.ch

arXiv:2210.09845v1 [hep-ph] 18 Oct 2022

1 Introduction

Charmonium and bottomonium production provides an ideal case study for the understand-

ing of hadron formation in quantum chromodynamics (QCD) [1]. Its theoretical description

is based on the generally agreed assumption that the charm and beauty quarks (the heav-

iest ones capable of forming bound states) are heavy enough to allow the factorization of

short- and long-distance eﬀects. Within the non-relativistic QCD (NRQCD) framework [2],

in particular, perturbative QCD computations provide the production cross sections of the

QQ pre-resonance (the “short-distance coeﬃcients”, SDCs), while the non-perturbative

evolution of the QQ state to the observed meson (the hadronization step) is described by

phenomenological parameters (the “long-distance matrix elements”, LDMEs), determined

from ﬁts to experimental data. Other theoretical approaches have been considered, such as

the colour-singlet model (CSM) [3,4] and the colour-evaporation model (CEM) [5,6]. These

theoretical models diﬀer in the choice and classiﬁcation of the allowed pre-resonance con-

ﬁgurations. The NRQCD approach foresees the contribution of all possible spin, S, orbital

angular momentum, L, total angular momentum, J, and colour (c= 1,8) conﬁgurations,

QQ(2S+1L[c]

J), organized in an expansion in powers of the relative QQ velocity, v < 1, so

that only a small number of leading and sub-leading terms remain quantitatively important.

Instead, the CSM considers that the ﬁnal-state hadron can only result from a colour-neutral

(singlet) pre-resonance having the same quantum numbers and the CEM is built upon the

assumption that one universal hadronization factor per quarkonium state (independent of

the S, L, J conﬁguration) multiplies the short-distance QQ production cross section.

The fundamental question that all models address is: how are the observable kinematic

properties of the produced quarkonium meson related to the quantum state of the unob-

servable QQ pre-resonance? The answers are diﬀerent because, among other factors, the

several contributing short-distance processes are scaled by diﬀerent long-distance weights.

The observable polarization of the quarkonium state provides particularly signiﬁcant in-

formation regarding the hadronization model, given that it directly reﬂects the mixture

of S, L, J conﬁgurations (and polarizations) of the contributing pre-resonance states. The

polarizations of ﬁve vector quarkonia (J/ψ,ψ(2S), Υ(1S), Υ(2S) and Υ(3S)) have recently

been measured at relatively high transverse momentum, pT, both at the Tevatron [7] and

at the LHC [8–12]. These measurements, showing no signiﬁcant signs of polarization, have

been addressed in many studies, including analyses based on the NRQCD [13–20] and

CEM [21] approaches.

In this paper we devote our attention to low-pTquarkonium hadro-production, a kine-

matical domain complementary to that explored at the LHC. We start by considering the

polarization measurements reported by several ﬁxed-target experiments, at CERN, DESY

and Fermilab, using proton or pion beams, in a broad energy range, colliding on targets

made of several materials. The question we address here is: can this multitude of low-pT

quarkonium polarization measurements be interpreted in a consistent physical picture? At

ﬁrst sight, we may think that it is very challenging to see coherent patterns emerging from

a collection of results obtained in such a diverse set of kinematical conditions, aﬀected by

several diﬃculties in the detection and analysis techniques, and reported using three dif-

ferent polarization frames. Nevertheless, a careful look at the experimental results allows

us to see that, while most data points ﬂuctuate around the unpolarized condition, there

are some tendencies towards strong polarizations in certain kinematical regions. These

qualitative patterns motivate us to consider a simple physical interpretation of low-pT

quarkonium production, as a superposition of two 2-to-1 processes: gluon-gluon fusion

and quark-antiquark annihilation, respectively leading to the production of fully longitu-

dinally polarized and fully transversely polarized mesons. Our study is exclusively focused

on the polarization data and deliberately follows a model-independent approach. Reports

on theoretical studies of low-pTquarkonium cross sections can be found, for example, in

Refs. [22–28].

The paper is structured as follows. Section 2 presents and reviews the experimental

measurements we have considered. Section 3 discusses possible qualitative indications

from the peculiar data patterns, which are then developed in Section 4 into a simple

model. Quantitative comparisons between the model and the experimental measurements

are shown in Sections 5 and 6, while predictions for future experiments are provided in

Section 7.

2 Experimental data

Figures 1 and 2 present, respectively for the J/ψ and Υ states, polarization measurements

made by ﬁxed-target experiments, listed in Table 1, using proton or pion beams and several

target materials. The considered observable, shown as functions of xFand pT, is the polar

anisotropy parameter λϑ[29]. Most of the measurements address J/ψ production [30–41],

with only one measurement of Υ production [42]. The ensemble of experiments covers an

overall kinematical domain deﬁned by −0.3.xF.1 and 0 < pT.5 GeV, with average

pTbetween 1.0 and 1.2 GeV and average pTsquared in the 1.5.hp2

Ti.2.2 GeV2range.

The polarizations were measured in three diﬀerent frames: Collins–Soper (CS) [43],

Gottfried–Jackson (GJ) [44] and centre-of-mass helicity (HX), where the polarization axis

zis deﬁned, respectively, as the relative direction of the colliding nucleons, the direction of

one of the two nucleons (generally the beam proton), and the direction of the quarkonium

itself with respect to the centre-of-mass of the system of the two nucleons.

Since each of the several experiments that measured the polarization of J/ψ mesons

used diﬀerent combinations of beam particles, target nuclei and collision energy, it is, a

priori, not surprising to see that the six panels of Fig. 1 display a rather scattered overall

picture. The collision energies span a broad range, from √s= 15.3 to 41.6 GeV, while the

target nuclei include eight elements between hydrogen and tungsten. The beam particles

include pions (both charges), protons and antiprotons, and even indium nuclei. And in

the case of secondary beams (e.g., the pion and antiproton cases), the beam composition

is contaminated by some fraction of other particles, which adds further complexity to the

picture. These complications can be illustrated with a few examples. E444 collected data

with a beam composed of several particles (π±,K±, p, ¯p) hitting a target system composed

of several materials (C, Cu, W), the combination π−-C being the most important. WA11

collected 40% of the data at 140 GeV beam momentum and 60% at 150 GeV. E537 collected

data with several beam-target conﬁgurations; the J/ψ sample is dominated by the π−-W

combination but there is also an important contribution (around 25% of the events) from

¯p-W collisions, while the data collected with Be and Cu targets, with both beams, is a

negligible contamination.

Besides the diversity of collision energies, beam particles and target nuclei, which surely

contributes to the visible spread of the data points, we also need to take into consideration

that polarization measurements are always very challenging and it is quite possible that

some of the reported systematic uncertainties are underestimated (in fact, some of the older

results were even published without mentioning systematic uncertainties). In particular,

most of the measurements were obtained from one-dimensional analyses, only considering

(GeV)

1 2 3 4 5

0 0.2 0.6 0.80.4-0.2

-1

-0.5

0.5

J/ψ

-1.5

-1

-0.5

0.5

-1.5

E866, p-Cu, x > 0.45

NA3, p-Pt

NA3, p-H

HERA-B, p-C, Ti, W

E444, π -C, Cu, W

−

E866, p-Cu, x < 0.45

HX frame

CS frame

GJ frame

-1

-0.5

0.5

WA92, π -Si, Cu, W

NA60, In-In

HERA-B, p-C, Ti, W

E444, π -C, Cu, W

−

E672/706, p,π -Be

WA11, π -Be

E537, p-W

−

HERA-B, p-C, Ti, W

E771, p-Si

E537, π -W

E615, π -W

−

Figure 1: The J/ψ polar anisotropy parameter λϑmeasured in the CS, GJ, and HX frames

(top to bottom), vs. xFand pT.

CS frame

(GeV)

1 2 3 4 5

0 0.2

-0.5

0.5

p-Cu 800 GeV

0.6 0.80.4-0.2

Υ(2S)+Υ(3S)

Υ(1S)

E866

Figure 2: The Υ(1S) and Υ(2S+3S) polar anisotropy parameter λϑmeasured by E866 in

the CS frame, vs. xFand pT.

Table 1: J/ψ and Υ polarization measurements in ﬁxed target experiments, characterized

by several beam energies (Elab) and angular coverages, denoted using xF, centre-of-mass

rapidity (ycms) or fractional momentum of the beam partons (x1).

Exp. [Ref.] Beam Target Elab √s∆xF∆pThpTi,hp2

(GeV) (GeV) (GeV) (GeV), (GeV2)

J/ψ

E537 [30] π−, ¯p W 125 15.3 0.0–0.7 0–2.5 hpTi= 1.04

WA11 [31] π−Be 146 16.6 0.0–0.4 0–2.4 hpTi= 1.0

NA60 [32] In In 158 17.2 ycms: 0–1 ≈0–4

E444 [33] π±C, Cu, W 225 20.6 x1: 0.2–1.0 0–2.5 hpTi= 1.2

E615 [34] π±W 252 21.8 0.25–1.0 0–5

NA3 [35] π−H, Pt 280 22.9 0.0–1.0 hp2

Ti= 1.52,1.85

WA92 [36] π−Si, Cu, W 350 25.6 ≈0.0–0.8 0–4

E672/706 [37] π−Be 515 31.1 0.1–0.8 0–3.5 hpTi= 1.17

E672/706 [38] p Be 530, 800 31.5, 38.8 0.0–0.6 hpTi= 1.15, 1.22

E771 [39] p Si 800 38.8 −0.05–0.25 0–3.5 hp2

Ti= 1.96

E866 [40] p Cu 800 38.8 ≈0.0–0.5 ≈0–4

HERA-B [41] p C, Ti, W 920 41.6 −0.34–0.14 0–5.4 hp2

Ti= 2.2

E866 [42] p Cu 800 38.8 0.0–0.6 0–4 hpTi= 1.3

the cos ϑobservable and neglecting acceptance correlations between the cos ϑand ϕvari-

ables of the dilepton angular distribution, a practice that can easily lead to signiﬁcantly

biased results, as discussed in Refs. [29,45]. This might explain why some of the data points

shown in Fig. 1 are outside of the physically allowed range (with λϑ<−1).

Despite the ﬁrst impression that the diversity of points form a rather scattered overall

picture, we can see that most of the J/ψ values ﬂuctuate around the λϑ= 0 limit (unpo-

larized production), with some trends towards strong polarizations in certain points of the

kinematic domain. Among all the J/ψ measurements, the one published by HERA-B [41]

stands out as the only one that considers all three polarization frames (CS, GJ and HX)

and that, furthermore, includes all three shape parameters of the angular distribution rel-

evant for parity-conserving decays, λϑ,λϕand λϑϕ [29]. It will, therefore, provide a very

useful beacon to guide our extraction of physically relevant trends (presented in the next

section) from the seemingly cryptic data collection depicted in Fig. 1.

The most salient feature that one can easily see as standing out of the global picture

is the polarization measurement reported by E866 for the (unresolved) Υ(2S) plus Υ(3S)

states [42]. While the values reported for the Υ(1S) mesons produced with xF<0.45

cluster around λϑ∼0.1 and are consistent with the J/ψ values provided by the same

experiment, for identical experimental conditions [40], those reported for the 2S+3S states

are surprisingly diﬀerent: λϑ∼+1. If we exclude the possibility of problems with the

experimental measurement, this observation reveals an astounding diﬀerence between the

polarizations observed for the excited states and for the ground state. Given that these

three S-wave states are expected to have identical polarizations when directly produced (or

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Quarkoniumpolarizationinlow-pThadro-production:frompastdatatofutureopportunitiesPietroFaccioli1),IlseKratschmer2)andCarlosLourenco3)AbstractSeveralxed-targetexperimentsreportedJ/andpolarizationmeasurements,asfunctionsofFeynmanx(xF)andtransversemomentum(pT),inthreedierentpo-larizationframes,usin...

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Quarkonium polarization in low- pThadro-production from past data to future opportunities Pietro Faccioli1 Ilse Kr atschmer2and Carlos Louren co3

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