Spin Physics at PHENIX ^†

Abstract

Situated at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory, the PHENIX experiment has for almost two decades been at the forefront of investigations into spin structure and dynamics in high-energy nuclear physics. Although decommissioned in 2016, the PHENIX collaboration has released a number of new results over the past several years that continue to inform the field. Recent longitudinal spin measurements uncover the role of gluon and sea quark polarization in the proton. Transverse spin measurements probe the transverse momentum-dependent (TMD) distributions and higher-twist multiparton correlators that are needed to fully explain partonic dynamics in the initial and final state. Additionally, the effects of heavy ions on spin have been studied by comparing transverse spin measurements between p+p and p+A collisions. These recent results and their wider implications are presented.

1. Introduction

As the lightest stable bound state in quantum chromodynamics (QCD), the proton serves as an excellent laboratory for investigating quarks and gluons—together known as partons—and the hadronic bound states that they form. By polarizing the proton, additional spin degrees of freedom become available as a tool to further test this interplay between the partons and hadrons. The research field of nuclear spin physics takes advantage of these tools to explore fundamental questions related to the structure and dynamics of QCD and its bound states. One of the central questions in the field is how the spin of the proton arises from its fundamental constituents. The proton spin can be decomposed into contributions from the quark polarization, the gluon polarization, and their respective orbital angular momentum [1],

$${\displaystyle \frac{1}{2}}={\displaystyle \frac{1}{2}}\sum ∆q+∆g+L_q+L_g,$$

(1)

where the quark and gluon polarization contributions can be written as the integrals of their respective helicity distributions over Bjorken-x, $∆q={\int }_0^1∆q\left(x\right)dx$ and $∆g={\int }_0^1∆g\left(x\right)dx$. The quark polarization sum runs over the valence and sea quarks, $\sum ∆q=\sum \left(∆u+∆\overline{u}+∆d+∆\overline{d}+∆s+∆\overline{s}\right)$. These helicity distributions describe the difference in longitudinally polarized proton densities with partons of positive versus negative helicity. Measurements in polarized deep-inelastic scattering (DIS) [2,3,4,5,6] and semi-inclusive deep-inelastic scattering (SIDIS) [7,8] have helped constrain the contribution from the quark helicity to ∼30% of the total proton spin. While polarized DIS can access $∆g$ at next-to-leading order (NLO) through photon–gluon fusion, polarized proton collisions at RHIC offer improved sensitivity to the gluon helicity through leading-order (LO) quark–gluon and gluon–gluon interactions. Measurements sensitive to $∆g$ from PHENIX [9,10,11] and its counterpart experiment STAR [12,13,14] led to the first evidence of significant nonzero gluon polarization at $x>0.05$ [15,16]. Recent measurements at RHIC, with both larger luminosity datasets and higher collision center-of-mass energies, are focused on pushing towards smaller uncertainties and exploring the small-x region of the gluon helicity.

Helicity distributions are just one of a set of distributions that generalize the standard unpolarized parton distribution functions (PDFs) to account for the polarizations of both the partons and the proton. These PDFs can in general depend on the partonic transverse momentum $k_T$ and are therefore called Transverse Momentum-Dependent (TMD) PDFs. In the case of the unpolarized PDFs, the helicity distributions, and the transversity distributions (the analogue of helicity distributions for transversely polarized protons), the $k_T$ dependence integrates out and the distributions describe correlations between the spin of the parton and the spin of the proton. The remaining TMDs explicitly depend on $k_T$ and encode correlations between the partonic momentum and the proton spin. A set of corresponding TMD fragmentation functions (FFs) exists in the final state, describing spin–momentum correlations between the struck parton and the hadron that it fragments into. The two most prominent examples of TMDs in spin physics are the Sivers TMD PDF [17,18], which correlates the transverse momentum of an initial-state parton with the transverse spin of the proton, and the Collins TMD FF [19], which correlates the transverse spin of a struck quark with preferential directions of the fragmenting hadrons. Both of these functions have been measured to be nonzero through azimuthal asymmetries observed in particle production from SIDIS [20,21,22,23,24,25,26] and electron-positron annihilation [27].

Experimental access to TMDs relies on the availability of two scales, scale Q to separate out the partonic-level hard scattering, and soft scale $k_T\ll Q$ that is sensitive to the non-perturbative partonic transverse momentum. In measurements with only one accessible scale (the hard scale Q), such as high $p_T$ production of inclusive particles from hadronic collisions, it is more natural to describe the PDFs and FFs in their collinear limit. In this framework, one encounters correlation terms at $n=3$ in the $1/Q^{n-2}$ factorization expansion of the cross-section (where n is the twist). These twist-3 correlation functions describe quantum interference between the scattering amplitudes of a single active parton and a composite parton state with an extra gluon. They can describe multiparton quark–gluon [28,29] or trigluon [30,31] correlations. When convoluted together with the standard PDFs and FFs, the twist-3 correlators can generate azimuthal asymmetries in particle production from $p^{\uparrow }+p$ collisions that are known as transverse single spin asymmetries,

$$\begin{array}{cc}\hfill A_N\propto & \sum _{a,b,c}{\varphi }_{a/A}^{\left(3\right)}(x_1,x_2,{\overrightarrow{s}}_{\perp })\otimes {\varphi }_{b/B}\left(x^{\prime }\right)\otimes \widehat{\sigma }\otimes D_{c\to C}\left(z\right)\hfill \\ \hfill +& \sum _{a,b,c}\delta q_{a/A}(x,{\overrightarrow{s}}_{\perp })\otimes {\varphi }_{b/B}^{\left(3\right)}(x_1^{\prime },x_2^{\prime })\otimes {\widehat{\sigma }}^{\prime }\otimes D_{c\to C}\left(z\right)\hfill \\ \hfill +& \sum _{a,b,c}\delta q_{a/A}(x,{\overrightarrow{s}}_{\perp })\otimes {\varphi }_{b/B}\left(x^{\prime }\right)\otimes {\widehat{\sigma }}^{\prime \prime }\otimes D_{c\to C}^{\left(3\right)}(z_1,z_2),\hfill \end{array}$$

(2)

where Superscript (3) terms represent the twist-3 correlators and all other terms are at the leading twist-2. From Equation (2), the asymmetry can originate from an initial-state “Sivers-like” twist-3 correlator, ${\varphi }^{\left(3\right)}$, coupled with a leading-twist fragmentation function, D, or a final-state “Collins-like” correlator, $D^{\left(3\right)}$, coupled with a leading-twist transversity distribution, $\delta q$. The twist-3 functions can be related to the $k_T$ moments of the TMD PDFs and FFs. Observables at PHENIX are chosen for their sensitivity to specific twist-3 correlators and, by extension, specific TMDs. As an example, the direct photon $A_N$ provides a considerably powerful constraint on the twist-3 quark–gluon and trigluon correlators because the clean final state minimizes any twist-3 contribution from fragmentation.

2. Experiment

The Relativistic Heavy Ion Collider (RHIC) provides the spin physics community with the unique opportunity to study spin observables from high-energy polarized proton collisions. The collection of polarized physics runs at RHIC relevant for PHENIX data taking are outlined in

Table 1. PHENIX polarized physics data taking runs.

Year	System	$\sqrt{\mathit{s}}\left(\mathbf{GeV}\right)$	Polarization	Recorded
			Direction	Luminosity [${\mathrm{pb}}^{-\mathbf{1}}$]
2006	p + p		transverse	0.02
		62.4	longitudinal	0.08
		200	transverse	2.7
			longitudinal	7.5
2008	p + p	200	transverse	5.2
2009	p + p	200	longitudinal	16
		500		14
2011	p + p	500	longitudinal	18
2012	p + p	200	transverse	9.7
		510	longitudinal	32
2013	p + p	510	longitudinal	155
	p + p	200	transverse	60
2015	p + Au			1.27
	p + Al			3.97

.

Located at the 8 o’clock interaction point on the RHIC ring, the PHENIX detector, shown in Figure 1, was built for the high-resolution measurement of hadrons, photons, electrons, and muons. Two central arms, each with $∆\varphi =\pi /2$ azimuthal and $\left|\eta \right|<0.35$ pseudorapidity coverage, were outfitted with high granularity electromagnetic calorimeters, drift and pad chambers for track reconstruction, a Ring-Imaging Cherenkov Detector for e/$\pi$ discrimination, and a silicon vertex detector for primary and secondary vertex reconstruction. Muon arms with full azimuthal coverage at intermediate pseudorapidity, $1.2 < \left|\eta \right| < 2.4$, measured charged particle momentum and provided muon and charged hadron identification. At forward rapidity, the Muon Piston Calorimeters were full azimuth electromagnetic calorimeters from $3.1 < \left|\eta \right| < 3.8$, and the beam–beam counter was a global event characterization detector responsible for luminosity counting and providing a minimum bias trigger. Upstream of the interaction region in the RHIC tunnel, the zero-degree calorimeters provided hadronic calorimetry intended for far-forward neutron detection at $\left|\eta \right| > 6.8$. A thorough description of the PHENIX detector and its capabilities can be found in the detector overview [32].

3. Results

3.1. Longitudinal Spin

At RHIC energies, hard scattering occurs mostly through quark–gluon and gluon–gluon interactions. As such, PHENIX observables from longitudinally polarized proton collisions provide unparalleled access to the gluon helicity distribution through measurements of their longitudinal double spin asymmetries,

$$A_{LL}={\displaystyle \frac{{\sigma }^{++}-{\sigma }^{+-}}{{\sigma }^{++}+{\sigma }^{+-}}}\propto {\displaystyle \frac{∆\sigma }{\sigma }}.$$

(3)

Here, ${\sigma }^{++}\left({\sigma }^{+-}\right)$ denotes the cross-section from collisions of protons with like (opposite) helicities and $∆\sigma \left(\sigma \right)$ is the helicity (unpolarized) cross-section. The helicity cross-section is the convolution of the partonic helicity distributions with the partonic cross-section,

$$∆\sigma =\sum ∆f_{a/A}\otimes ∆f_{b/B}\otimes ∆{\sigma }_{ab},$$

(4)

for partons $a,b$ and hadrons $A,B$, where we neglected any fragmentation effects. In most cases at RHIC, at least one of the parton helicity distributions is $∆g\left(x\right)$.

3.1.1. Direct Photon $A_{LL}$

The cleanest measurement of the gluon helicity is the longitudinal double spin asymmetry of direct photons. The production of direct photons comes predominantly from quark–gluon Compton scattering, leading to minimal contributions from fragmentation in the final state. Additionally, the $A_{LL}$ from direct photons has a linear dependence in $∆g\left(x\right)$ and is therefore sensitive to its sign.

This is particularly relevant considering the recent study [33] that found consistent positive and negative solutions for the gluon helicity after relaxing a positivity constraint. A response paper [34] argues that the relaxing of such a constraint leads to unphysical predictions. The direct photon $A_{LL}$ can serve as an experimental arbiter to this question. In Run 13, PHENIX measured the direct photon $A_{LL}$ at $\sqrt{s}=510$ GeV, the first measurement of its kind [35]. Figure 2 shows the direct photon $A_{LL}$, which favors the positive solution of $∆g$ at 2.8$\sigma$.

3.1.2. Pion $A_{LL}$

While the first evidence of nonzero gluon polarization was discovered at high x, the low-x behavior of the gluon helicity is still largely unconstrained due to a lack of experimental data. In order to probe the $x<0.05$ region of $∆g\left(x\right)$, PHENIX measured the ${\pi }^{\pm }$$A_{LL}$ at the increased center of mass energy of $\sqrt{s}=510$ GeV [38] to go along with a previous measurement from $\sqrt{s}=200$ GeV [39]. This improved the downward reach in $x_T$ by about a factor of two (Figure 3).

3.1.3. W Boson $A_L$

In addition to helping set constraints on the gluon polarization in the proton, PHENIX investigated the helicity of sea quarks through the measurement of the longitudinal single spin asymmetry of the W boson. The maximally parity-violating electroweak process $u_L{\overline{d}}_R\to W^{+}$, ${\overline{u}}_R{d}_L\to W^{-}$ fixes the handedness of the quarks and antiquarks such that reversing the polarization direction of the proton flips the corresponding helicities of the quarks. The longitudinal single spin asymmetry is thus proportional to a mixture of the helicities and unpolarized PDFs of the quarks and anti-quarks,

$$A_L^{W^{-}}=\frac{{\sigma }^{+}-{\sigma }^{-}}{{\sigma }^{+}+{\sigma }^{-}}\approx \frac{∆\overline{u}\left(x_1\right)d\left(x_2\right)-∆d\left(x_1\right)\overline{u}\left(x_2\right)}{\overline{u}\left(x_1\right)d\left(x_2\right)+d\left(x_1\right)\overline{u}\left(x_2\right)}$$

(5)

$W^{\pm }$ are detected through their leptonic decay modes such that the charge and kinematics of the decay lepton allow for the flavor separation of the sea quarks. This means that this measurement is sensitive to the polarized sea quark asymmetry $∆\overline{u}-∆\overline{d}$. Results of the W boson $A_L$ from PHENIX [40,41] and STAR [42] are shown in Figure 4, hinting at a positive polarized sea asymmetry. An additional high-precision W $A_L$ from STAR [43] was included in an update to the NNPDFpol1.1 global fit [16], which indicated for the first time the clear positive signature of the asymmetry.

3.2. Transverse Spin

In transversely polarized $p^{\uparrow }+p$ collisions, particle production is to first-order azimuthally dependent on a cosine modulation. The magnitude of this modulation is the same transverse single spin asymmetry $A_N$ of Equation (2), and it is measured experimentally by

$$A_{N}cos\varphi ={\displaystyle \frac{1}{P}}{\displaystyle \frac{d{\sigma }^{\uparrow }\left(\varphi \right)-d{\sigma }^{\downarrow }\left(\varphi \right)}{d{\sigma }^{\uparrow }\left(\varphi \right)+d{\sigma }^{\downarrow }\left(\varphi \right)}},$$

(6)

where P is the beam polarization and $d{\sigma }^{\uparrow ,\downarrow }\left(\varphi \right)$ are the cross-sections differential in the azimuth for polarized up and polarized down protons, respectively. The goal of the transverse spin physics program at PHENIX is to measure $A_N$ across an array of different final-state particles, which are each uniquely suited to provide constraints on some subset of the twist-3 correlators.

3.2.1. Direct Photon $A_N$

As stated in Section 1, direct photon observables mitigate any final-state color effects, leaving the direct photon $A_N$ directly sensitive to quark–gluon and trigluon correlators in the initial state only. A recent PHENIX measurement of the midrapidity direct photon $A_N$ at $\sqrt{s}=200$ GeV [44] is shown in Figure 5 with a comparison to theoretical predictions of the contributions to the asymmetry from the quark–gluon [45] and trigluon correlators [46]. The measurement is consistent with both the model predictions and zero asymmetry but with a 50-time statistical improvement from the previous best measurement at Fermilab E704 [47].

3.2.2. Open Heavy Flavor $A_N$

PHENIX also measured the midrapidity open heavy flavor $A_N$ at $\sqrt{s}=200$ GeV [48], shown in Figure 6. The open heavy flavor $A_N$ isolates the twist-3 initial-state trigluon correlators because (a) the process is dominated by gluon–gluon fusion and (b) gluons have no transversity within the spin 1/2 proton, hence eliminating any twist-3 effects from the final state. Not much is known of the trigluon functions, so a typical phenomenological strategy is to parameterize them in relation to the much better understood unpolarized gluon PDF [46,49]. Scanning through the parameter space and comparing the predictions to the experimental results enabled this measurement to place the first experimental constraints on these parameters: ${\lambda }_f,{\lambda }_d,K_G,K_G^{\prime }$.

3.2.3. Forward Charged Hadron and $\eta$ Meson $A_N$

At forward rapidities, PHENIX recently observed large transverse single spin asymmetries at high Feynman-x ($x_F$) in charged hadron [50] and preliminary $\eta$ meson results (Figure 7).

These measurements agree with previous findings and have a broader kinematic reach to higher $x_F$. Forward production of hadrons is dominated by valence quark interactions, suggesting that a significant contribution to the asymmetries must originate from twist-3 quark–gluon correlators in either the initial state, final state, or both. Recent phenomenological studies [51,52] appear to indicate that, at least in the light meson sector, the initial-state twist-3 quark–gluon correlator by itself is insufficient to describe the size of the observed asymmetries and a consistent description between asymmetries in SIDIS and proton–proton collisions relies predominantly on twist-3 fragmentation.

3.2.4. Forward Charged Hadron $A_N$ in $p^{\uparrow }+A$

The forward charged hadron transverse single spin asymmetries were also measured in $p^{\uparrow }+Al$ and $p^{\uparrow }+Au$ [53]. When compared with $p^{\uparrow }+p$, PHENIX observed striking behavior suggesting a nuclear modification of $A_N$, which looks like $A^{-1/3}$, as shown in Figure 8 (left). A follow-up study measured $A_N$ differentially in $x_F$ with a much larger dataset and confirmed the suppression of the asymmetry as a function of mass number [50].

The proper theoretical description of this behavior is still elusive. Some calculations of $A_N$ in $p^{\uparrow }+A$ that account for gluon saturation effects find that such behavior is possible when in the saturation regime, but the PHENIX measurement is well above the saturation scale in the $Au$ nucleus, violating the condition needed for strong nuclear suppression [55]. Multiple scattering between the struck quark and nucleons has been suggested as a mechanism to suppress the azimuthal asymmetries in SIDIS [56]. This argument has yet to be extended theoretically to hadronic collisions, but it may warrant further investigation as the experimental observation of suppression of $A_N$ with respect to the average number of nucleon–nucleon collisions shown in Figure 8 (right) suggests dependence of $A_N$ on the density of the nuclear matter, a variable closely connected to the likelihood of multiple scatters.

3.2.5. Far-Forward Neutron $A_N$

At far-forward rapidities in PHENIX, the measurement of the nontrivial nuclear dependence of the neutron transverse single spin asymmetry [57] shown in Figure 9 has been the subject of considerable study. In this region, the mechanisms for producing particles are dominated by soft QCD processes, like one-pion exchange and diffractive scattering. While forward neutron yields have been well described using one-pion exchange models, the large negative neutron $A_N$ in $p^{\uparrow }+p$ collisions require a more subtle explanation of interference between the neutron production amplitudes from one-pion exchange and Reggeon exchange [58]. This mechanism, however, could not explain the large dependence and sign change of the asymmetry on the mass number in $p^{\uparrow }+A$ collisions.

The explanation for this initially unexpected behavior is that electromagnetic interactions from ultra-peripheral collisions (UPCs) generate large positive asymmetries that are highly dependent on the mass number [59]. This is reflected in Figure 9 where the ZDC⊗BBC-veto trigger requirement enhances the UPC contribution to $A_N$ while the ZDC⊗BBC-tag trigger requirement enhances the hadronic contribution. More recently, the very forward neutron asymmetries have been measured as a function of $p_T$ and $x_F$ with these separate trigger conditions [60]. Model calculations that include contributions from one-pion exchange and UPC qualitatively describe the behavior reasonably well, as seen in Figure 10 and Figure 11.

4. Conclusions

The PHENIX spin physics group has made a suite of recent measurements that explore the role of spin in QCD and nuclear physics. Measurements of longitudinal spin asymmetries offer evidence of nonzero gluon polarization and a positive polarized sea quark asymmetry. Measurements of transverse spin asymmetries in $p^{\uparrow }+p$ collisions constrain a wide variety of twist-3 multiparton correlators. In $p^{\uparrow }+A$ collisions, transverse spin asymmetries motivate new questions about the effects of nuclear matter in spin physics. As the PHENIX program winds down, the final spin analyses are underway. We will further explore the low-x behavior of the gluon helicity with forward measurements of longitudinal double spin asymmetries at $\sqrt{s}=510$ GeV.