Nuclear dependence of the transverse single-spin asymmetry in the production of charged hadrons at forward rapidity in polarized $p+p$, $p+$Al, and $p+$Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV

TL;DR

This study investigates how the transverse single-spin asymmetry (TSSA) in charged hadron production varies across different nuclear targets in polarized proton collisions, revealing a nuclear dependence that informs understanding of TSSAs in small systems.

Contribution

It provides the first detailed measurement of nuclear dependence of TSSAs in charged hadron production at forward rapidity in polarized proton-nucleus collisions.

Findings

Positive asymmetry in p+p collisions

Reduced asymmetry in p+Al and p+Au collisions

Nuclear dependence observed in TSSA measurements

Abstract

We report on the nuclear dependence of transverse single-spin asymmetries (TSSAs) in the production of positively-charged hadrons in polarized $p^{\uparrow}+p$, $p^{\uparrow}+$Al and $p^{\uparrow}+$Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV. The measurements have been performed at forward rapidity ($1.4<\eta<2.4$) over the range of $1.8<p_{T}<7.0$ GeV$/c$ and $0.1<x_{F}<0.2$. We observed a positive asymmetry $A_{N}$ for positively-charged hadrons in \polpp collisions, and a significantly reduced asymmetry in $p^{\uparrow}$+$A$ collisions. These results reveal a nuclear dependence of charged hadron $A_N$ in a regime where perturbative techniques are relevant. These results provide new opportunities to use \polpA collisions as a tool to investigate the rich phenomena behind TSSAs in hadronic collisions and to use TSSA as a new handle in studying small-system collisions.

Tables

Table 1. Table 1: A N subscript 𝐴 𝑁 A_{N} and sources of systematic uncertainty for positively-charged hadrons for 1.4 < η < 2.4 1.4 𝜂 2.4 1.4<\eta<2.4 , 0.1 < x F < 0.2 0.1 subscript 𝑥 𝐹 0.2 0.1<x_{F}<0.2 , and 1.8 < p T < 7.0 1.8 subscript 𝑝 𝑇 7.0 1.8<p_{T}<7.0 GeV / c absent 𝑐 /c in p ↑ superscript 𝑝 ↑ p^{\uparrow} + p 𝑝 p , p ↑ superscript 𝑝 ↑ p^{\uparrow} +Al , and p ↑ superscript 𝑝 ↑ p^{\uparrow} +Au collisions at s N ​ N = 200 ​ GeV subscript 𝑠 𝑁 𝑁 200 GeV \sqrt{s_{{}_{NN}}}=200~{}{\rm GeV} .

	$p^{\uparrow }$+$p$	$p^{\uparrow }$+Al	$p^{\uparrow }$+Au
$A_N$	$3.14\times {10}^{-2}$	$1.42\times {10}^{-2}$	$0.12\times {10}^{-2}$
$\delta A_N^{\mathrm{stat}}$	$0.37\times {10}^{-2}$	$0.72\times {10}^{-2}$	$0.55\times {10}^{-2}$
$\delta A_N^{\mathrm{syst}}$	${}_{-0.18}{}^{+0.05}\times {10}^{-2}$	${}_{-0.02}{}^{+0.02}\times {10}^{-2}$	${}_{-0.06}{}^{+0.06}\times {10}^{-2}$
$\delta A_N^{\mathrm{method}}$	${}_{-0.05}{}^{+0.05}\times {10}^{-2}$	${}_{-0.02}{}^{+0.02}\times {10}^{-2}$	${}_{-0.06}{}^{+0.06}\times {10}^{-2}$
$\delta A_N^{\mathrm{smear}}$	${}_{-0.17}{}^{+0.00}\times {10}^{-2}$	${}_{-0.00}{}^{+0.01}\times {10}^{-2}$	${}_{-0.00}{}^{+0.01}\times {10}^{-2}$

Equations

$$\log \mathcal{L}=\sum _i\log (1+P\cdot A_N\sin ({\varphi }_{\mathrm{pol}}-{\varphi }_h^i))+\text{constant},$$

$${\sigma }^2(A_N)=(-\frac{{\partial }^2\mathcal{L}}{(\partial A_N{)}^2}{)}^{-1}.$$

$$\begin{gathered} A_N(\varphi )=\frac{{\sigma }^{\uparrow }(\varphi )-{\sigma }^{\downarrow }(\varphi )}{{\sigma }^{\uparrow }(\varphi )+{\sigma }^{\downarrow }(\varphi )} \\ =\frac{1}{P}\cdot \frac{N^{\uparrow }(\varphi )-R\cdot N^{\downarrow }(\varphi )}{N^{\uparrow }(\varphi )+R\cdot N^{\downarrow }(\varphi )}, \end{gathered}$$

$$A_N^{\mathrm{reco},m}=\sum _i{f}^{i\to m}\cdot A_N^{\mathrm{truth},i},$$

Full text

Understanding the transverse-single-spin asymmetries (TSSAs), that describe the azimuthal-angular dependence of particle production relative to the transverse-spin direction of the polarized proton in the reaction $p^{\uparrow}+p\rightarrow h+X$, has been a long-standing puzzle. The first observations in pion production at large Feynman-$x$ ($x_{F}$) Klem et al. (1976) showed measured TSSAs that were considerably larger than early theoretical predictions (in collinear leading twist approach) Kane et al. (1978). Surprisingly large measured TSSAs continued to persist in hadronic collisions at high energies up to $\sqrt{s}=500~{}{\rm GeV}$ Antille et al. (1980); Adams et al. (1991a, b); Allgower et al. (2002); Adams et al. (2004); Lee and Videbaek (2007); Abelev et al. (2008); Arsene et al. (2008); Adamczyk et al. (2012); Adare et al. (2014a, b); Mondal (2014). To explain these large TSSAs, two approaches were proposed within perturbative quantum chromodynamics (pQCD). One approach is called transverse-momentum-dependent factorization, in which TSSAs are generated by correlations between the nucleon’s transverse spin direction and the transverse momentum of a parton in the polarized nucleon (Sivers effect Sivers (1990, 1991)), and from the fragmentation of a transversely polarized parton into a final-state hadron (Collins effect Collins (1993)). Another approach, directly applicable to single-hadron production (with $p_{T}\gg\Lambda_{\rm QCD}$) presented in this paper is the twist-3, collinear-factorization framework Efremov and Teryaev (1982). The full description of TSSAs in \mbox{p^{\uparrow}$+$p}\rightarrow h+X in the twist-3 collinear factorization includes twist-3 functions from the polarized proton, the unpolarized proton, and the parton fragmentation into final-state hadrons. The twist-3 functions describe quark-gluon-quark correlations and trigluon correlations in the polarized proton and have been studied in detail Qiu and Sterman (1998); Kouvaris et al. (2006); Koike and Tanaka (2007); Koike and Tomita (2009); Kanazawa and Koike (2010); Kang et al. (2011); Kanazawa and Koike (2011); Kang and Prokudin (2012); Beppu et al. (2014). Recently, calculations of the twist-3 contribution from parton fragmentation have been carried out and have shown this to be an important mechanism for understanding TSSA measurements Metz and Pitonyak (2013); Kanazawa et al. (2014); Gamberg et al. (2017).

The Relativistic Heavy Ion Collider (RHIC) is a unique facility that can accelerate polarized protons and collide them with other (polarized) protons or nuclei Alekseev et al. (2003). The extension of TSSA measurements to $p^{\uparrow}$+$A$ collisions not only gives us a crucial tool for understanding the nature of TSSAs, but also provides a new handle for studying $p$+$A$ collisions and the parton dynamics inside nuclei, where many emergent effects remain to be understood. These include the so-called “Cronin” effect, an enhancement in the inclusive hadron $p_{T}$ spectrum with respect to $p$+$p$ collisions at moderate $p_{T}$ of approximately $2<p_{T}<6$ GeV$/c$ that is proposed to be due to multiple scattering effects in the nuclear medium and modified hadronization mechanism Cronin et al. (1975); Adler et al. (2003); Aad et al. (2016); Adare et al. (2013). Another exciting observation is that the collective behavior across large pseudorapidity ranges in high multiplicity $p$+$A$ collisions that hints formation the quark-gluon plasma  Dusling et al. (2016); Nagle and Zajc (2018); Aidala et al. (2019a). Furthermore, when hadron production is measured in the proton-going direction, the properties of nuclear gluons in the small-$x$ region can be probed, where $x$ is the fraction of the proton’s longitudinal momentum carried by the parton. The dynamics of gluons in the small-$x$ regime, where the gluon density is predicted to increase drastically, can be described by the color-glass condensate (CGC) formalism Gelis et al. (2010) at the saturation scale $Q_{s}$, where $Q^{2}_{sA}\propto A^{1/3}$ for the target nucleus Schäfer and Zhou (2014); Zhou (2015). In recent years, substantial attention has been given to an interplay between small-$x$ physics and spin physics by studying TSSAs in transversely-polarized proton and ion collisions ($p^{\uparrow}$+$A$) and gluon saturation effects in a nucleus are taken into account for various calculations of TSSAs in $p^{\uparrow}$+$A$ collisions Boer et al. (2006); Kang and Yuan (2011); Kovchegov and Sievert (2012); Kang and Xiao (2013); Zhou (2014); Altinoluk et al. ; Schäfer and Zhou (2014); Zhou (2015); Boer et al. (2016); Hatta et al. (2016, 2017); Benić and Hatta (2019). An $A$-dependence of TSSAs can arise from the $A$-dependence of $Q_{s}$ when the probe is at or below $Q_{s}$, while TSSAs are expected to be $A$-independent at higher scales Boer et al. (2006); Kang and Yuan (2011); Hatta et al. (2016, 2017); Benić and Hatta (2019). Therefore, experimental data on hadron TSSAs measured in $p$+$A$ collisions with varying $A$-size will provide valuable information testing these models and bring new insights in understanding the dynamics of the $p$+$A$ collisions.

We report here on the observation of a nuclear dependence of TSSAs of positively-charged-hadron production at forward rapidity ($0.1<x_{F}<0.2$ and $1.4<\eta<2.4$, probing $0.004\lesssim x\lesssim 0.1$ in the nuclei) in collisions between transversely polarized protons and unpolarized protons or nuclei, $p^{\uparrow}$+$p$, $p^{\uparrow}$+Al, $p^{\uparrow}$+Au at $\sqrt{s_{{}_{NN}}}=200{\rm\ GeV}$ measured with the PHENIX detector. The positively charged hadron is preferred in the nuclear-dependence measurement because the significance of TSSAs for negatively charged hadrons will be reduced by the partial cancellation of the asymmetry due to opposite signs of TSSAs for $\pi^{-}$ and $K^{-}$ in $p^{\uparrow}$+$p$ collisions Arsene et al. (2008); Lee and Videbaek (2007). In this measurement, we follow the convention to quantify TSSAs as $A_{N}$, where $A_{N}$ is the modulation of the azimuthal angle of the hadron ($\phi_{h}$) relative to the azimuthal angle of the transverse spin of the proton ($\phi_{\rm pol}$), i.e., hadron-production cross section $\sigma\propto 1+A_{N}\sin(\phi_{\rm pol}-\phi_{h})$.

The data from transversely polarized $p^{\uparrow}$+$p$, $p^{\uparrow}$+Al, and $p^{\uparrow}$+Au collisions at \sqrt{s_{{}_{NN}}}=200\GeV were collected with the PHENIX detector during the RHIC 2015 running period. Proton beams were polarized vertically with respect to the beam direction with an average polarization of 58% (clockwise-beam) or 57% (counterclockwise-beam) for $p^{\uparrow}$+$p$, 58% for $p^{\uparrow}$+Al, and 61% for $p^{\uparrow}$+Au collisions, with a relative uncertainty of 3% due to uncertainty in the polarization normalization. The beams are bunched. To minimize systematic effects due to time dependence of machine and detector performance, the spin configuration of the colliding bunches is alternated every 106 ns.

The PHENIX detector comprises two central arms at midrapidity and two muon arms at forward rapidity Adcox et al. (2003); Akikawa et al. (2003); only reconstructed tracks from the muon arms are used for this analysis. The two muon spectrometers cover $1.2<\eta<2.4$ (polarized $p$-going direction) and $-2.2<\eta<-1.2$ ($A$-going direction) in pseudorapidity with full azimuthal angle coverage. Each muon arm has 7.5 nuclear interaction lengths ($\lambda_{I}$) of hadron absorber followed by a muon tracker (MuTr), which is a set of three stations of cathode strip chambers for momentum measurements of charged particles. The MuTr determines the momentum of a charged particle in a radial magnetic field of $\int{B\cdot dl}=0.72~{}{\rm T\cdot m}$ with a momentum resolution of $\delta p/p\approx 0.05$ for hadrons in the kinematic range of this analysis. A Muon Identifier (MuID), located behind the MuTr, comprises five layers of stainless-steel absorbers ($\sim 5\lambda_{I}$ total) and Iarocci tube planes. The MuID helps to identify muons and hadrons based on the penetration depth of the tracks at $p_{z}\gtrsim 3.5$ GeV$/c$ Adare et al. (2012).

The beam-beam counters (BBCs) Allen et al. (2003), at $z=\pm 144~{}{\rm cm}$ from the nominal interaction point, comprise two arrays of 64 quartz Cherenkov detectors and cover the full azimuth and the pseudorapidity range $3.1<|\eta|<3.9$. The BBCs are used to determine the collision vertex $z$-position ($|z|<30~{}{\rm cm}$ cut was used in this analysis) as well as to provide a minimum-bias (MB) trigger with efficiencies of 55% for $p$+$p$, 72% for p$$+Al, and 84% for p$$+Au collisions. The $A$-going side of the BBC is also used to determine the event centrality based on the distribution of the charge sum Adare et al. (2014c). The recorded events are sampled by the MB trigger combined with muon triggers to enrich good muon and hadron tracks. The MuID provides a trigger for events containing one or more hadron or muon candidates. Momentum-sensitive triggering is provided by hit information from the MuTr to enrich tracks with \mbox{p_{T}}>3~{}{\rm GeV}/c Adachi et al. (2013).

This analysis uses only charged tracks that stop in the middle of the MuID planes (third or fourth plane out of five planes) due to a hadronic interaction with the absorber material. In the kinematic region of $0.1<x_{F}<0.2$, where the longitudinal momentum of particles is larger than $10~{}{\rm GeV}/c$, positively-charged hadron candidates mostly comprise $\pi^{+}$ and $K^{+}$.

The particle composition in the measured charged-hadron sample was estimated with a method developed in Ref. Adare et al. (2012, 2014d), based on identified charged-hadron spectra measured at midrapidity in $p$+$p$ and d$$+Au collisions at RHIC Adare et al. (2011); Agakishiev et al. (2012); Adare et al. (2013), and extrapolated to PHENIX muon arm rapidity region of $1.2<\eta<2.4$ for $p$+$p$, p$$+Al and p$$+Au collisions using pythia Sjöstrand et al. and hijing Gyulassy and Wang (1994) event generators. The $K^{+}/\pi^{+}$ ratio of $\sim 0.35$, as measured at RHIC at midrapidity at \mbox{p_{T}}\sim 2~{}{\rm GeV}/c (typical for our data) Adare et al. (2011); Agakishiev et al. (2012); Adare et al. (2013), was found approximately unchanged when extrapolated to forward rapidity in both $p$+$p$ and $p$+$A$ collisions. The $p/\pi^{+}$ ratio of $\sim 0.25$ ($\sim 0.35$) at midrapidity in $p$+$p$ (d$$+Au) collisions Adare et al. (2011); Agakishiev et al. (2012); Adare et al. (2013) was extrapolated to the value of $\sim 0.3$ ($\sim 0.5$) at the muon arm rapidity, with ratios in p$$+Al and p$$+Au collisions being in between values for $p$+$p$ and d$$+Au collisions. The initial charged hadron composition is significantly modified due to particle interaction in the detector material, which according to geant4-based Allison et al. (2016) detector simulation modifies the initial $K^{+}/\pi^{+}$ ($p/\pi^{+}$) ratio by a factor of 2.7 (0.4), which varies by ${\approx}5\%$ for different hadron interaction models Allison et al. (2016). As a result, the $\pi^{+}$/$K^{+}$/$p$ particle composition in our measured positively charged-hadron sample is evaluated to be 45%/47%/5% in $p$+$p$ collisions, with increased proton fraction to 7% (9%) in p$$+Al (p$$+Au) collisions.

The unbinned maximum-likelihood method for extracting $A_{N}$ was established in a previous study Aidala et al. (2017) which used the same detectors. Compared to binned approaches, this method is robust even for low-statistics data. The extended log-likelihood is defined to be

[TABLE]

where $\phi_{h}^{i}$ is the azimuthal angle of the $i$-th hadron with respect to the direction of the polarized proton beam, $\phi_{\rm pol}$ is the azimuthal angle for the beam polarization direction, which in the 2015 PHENIX run takes the values $+/-\frac{\pi}{2}$ for $\uparrow/\downarrow$ spin-signed beam bunches, respectively, and $P$ is the beam polarization. The asymmetry $A_{N}$ is determined by maximizing $\log\mathcal{L}$. For $p^{\uparrow}$+$p$ collisions, both beams are polarized, therefore the values of $A_{N}$ were measured separately for each beam, found to be consistent, and were averaged in the final result. For $p^{\uparrow}$+$A$ collisions, only the clockwise proton beam was polarized. The statistical uncertainty was calculated from the second derivative of the log-likelihood estimator,

[TABLE]

The $A_{N}$ calculated from the likelihood method is compared with the following azimuthal-fitting method based on the polarization formula Ohlsen and Keaton (1973):

[TABLE]

where $A_{N}(\phi)$ is the simple count-based transverse single-spin asymmetry in each of the 16 azimuthal $\phi$-bins, $\sigma^{\uparrow}$, $\sigma^{\downarrow}$ are cross sections for each polarization of spin up or down, $N^{\uparrow}$, $N^{\downarrow}$ are yields, and $R=L^{\uparrow}/L^{\downarrow}$ is the luminosity ratio (relative luminosity) between bunches with spin up and down, determined from the number of sampled MB triggers corresponding to different spin orientations. From this, $A_{N}$ is extracted from the fit of Eq. (3) with a function $A_{N}\cdot\sin(\phi_{\rm pol\uparrow}-\phi)$, where $\phi_{\rm pol\uparrow}=\pi/2$ is the azimuthal direction of the upward polarized bunches. Because every detector element is simultaneously used for the measurements with spin up and down, the possible systematic effects from acceptance nonuniformity and acceptance variation versus time are largely canceled. The relative variation between this method and the log likelihood method is included in the systematic uncertainty.

Figure 1 shows the reconstructed azimuthal modulation of positively-charged hadrons for $0.1<x_{F}<0.2$ and $1.8<p_{T}<7.0\rm{\ GeV/}c$ in $p^{\uparrow}$+$p$, $p^{\uparrow}$+Al, and $p^{\uparrow}$+Au collisions at $\sqrt{s_{{}_{NN}}}=200~{}{\rm GeV}$, as calculated using Eq. (3). The relatively larger statistical uncertainty in the bin at $\phi\sim 0.6$ rad is caused by a known detector inefficiency. The $\chi^{2}/NDF$ of the fits are 10.1/15 for $p^{\uparrow}$+$p$, 13.5/15 for $p^{\uparrow}$+Al, and 9.8/15 for $p^{\uparrow}$+Au. The $p^{\uparrow}$+$p$ results show a clear nonzero modulation, while the $p^{\uparrow}$+Al results show a weaker modulation. In $p^{\uparrow}$+Au collisions, the modulation is consistent with zero within the statistical uncertainty.

The finite momentum and azimuthal angle $\phi$ resolution in the MuTr and the interactions of particles with the materials prior to entering the MuTr lead to a kinematic smearing for the $A_{N}$ measurement. This smearing effect was studied and corrected with a full detector geant4 simulation. The effect due to the $\phi$ smearing was found to be negligible. The momentum smearing effect was evaluated by resolving a set of linear equations connecting $A_{N}$ for the true $x_{F}$ bins ($A_{N}^{\rm truth}$) and $A_{N}$ for the reconstructed $x_{F}$ bins ($A_{N}^{\rm reco}$):

[TABLE]

where $A_{N}^{{\rm reco},m}$ is $A_{N}$ for the $m$-th reconstructed $x_{F}$ bin from this measurement and $A_{N}^{{\rm truth},i}$ is that for the $i$-th true $x_{F}$ bin. $f^{i\rightarrow m}$ represents the fraction of charged particles whose true $x_{F}$ at the collision vertex belongs to the $i$-th true $x_{F}$ bin and is reconstructed as being in the $m$-th $x_{F}$ bin. $f^{i\rightarrow m}$ is obtained from the geant4 detector simulation. For calculating $A_{N}^{\rm truth}$ by solving Eq. (4), the $A_{N}^{\rm reco}$ is measured in a wider $x_{F}$ range $0.035<x_{F}<0.3$, by including two bins at lower $x_{F}$ and one bin at higher $x_{F}$. The resulting smearing-corrected $A_{N}^{\rm truth}$ of the positively-charged hadrons in bin $0.1<x_{F}<0.2$ are shown in Table 1. The difference between the obtained $A_{N}^{\rm truth}$ and the measured $A_{N}^{\rm reco}$ is small compared to the statistical uncertainty and is accounted for in the systematic uncertainty.

Table 1 also summarizes the systematic uncertainties for the $A_{N}$ measurements. The difference of $A_{N}$ extracted with two methods, Eqs. (1) and (3), is shown as $\delta A_{N}^{\rm method}$. The difference between the obtained $A_{N}^{\rm truth}$ and measured $A_{N}^{\rm reco}$ is assigned as a conservative systematic uncertainty due to the smearing effect, $\delta A_{N}^{\rm smear}$. The total systematic uncertainty $\delta A_{N}^{\rm syst}$ is calculated as a quadratic sum of these two uncertainties.

Figure 2 shows $A_{N}$ of positively-charged hadrons in $p^{\uparrow}$+$p$, $p^{\uparrow}$+Al, and $p^{\uparrow}$+Au collisions vs $A^{1/3}$ and the average number of nucleon-nucleon collisions $N_{\rm coll}^{\rm avg}$. The $N_{\rm coll}^{\rm avg}$ is calculated using the Glauber model Miller et al. (2007) for each centrality class in $p^{\uparrow}$+$A$ collisions Adare et al. (2014c). The figure caption and legends denote the ranges of parameters and give the determined values of the power parameters $\alpha$ and $\beta$. Panels (b) and (d) show the $\chi^{2}$ distributions with only statistical uncertainties included.

The recent efforts to calculate $A_{N}$ in $p^{\uparrow}$+$p$ and $p^{\uparrow}$+$A$ collisions, accounting for gluon saturation effects  Gamberg et al. (2017); Hatta et al. (2016, 2017); Benić and Hatta (2019) suggested that $A_{N}$ could be $A$-independent or $A^{-1/3}$-dependent for the different contributions to $A_{N}$ in the region where $p_{T}<Q_{s}$. However, $\langle p_{T}\rangle\sim 2.9~{}{\rm GeV}/c$ in our results is much larger than the saturation scale in the Au nucleus ($Q_{s}^{\rm Au}\sim 0.9$ GeV) for the kinematics of this measurement and would lead to no strong $A$ dependence of TSSAs under these models, as calculated in Ref. Benić and Hatta (2019). Nevertheless, the results in this paper strongly disfavor the $A$-independent scenario.

The $N_{\rm coll}^{\rm avg}$-dependence of $A_{N}$ also suggests the decrease of $A_{N}$ is related to the density of nuclear matter inside the target nucleus which the projectile proton traverses. This $N_{\rm coll}^{\rm avg}$-dependence of $A_{N}$ could be related to novel effects in $p$+$A$ collisions, such as multiple scattering of partons in the initial and/or final stages of the hard scattering, which is also indicated in the recent results of the nuclear modification of single hadron production and transverse momentum broadening in dihadron correlations in $p$+$A$ collisions Adare et al. (2013); Aidala et al. (2019b, ). Another possibility is interaction of the parton with hot QCD matter produced in $p$+$A$ collisions, as suggested by recent results in small systems Dusling et al. (2016); Nagle and Zajc (2018); Aidala et al. (2019a).

We note preliminary results from the STAR collaboration Heppelmann (2016) of measured $A_{N}$ for $\pi^{0}$ in $p^{\uparrow}$+$p$ and $p^{\uparrow}$+Au collisions in more forward kinematics at $2.6<\eta<4.0$, $0.2<x_{F}<0.7$, and $p_{T}>1.5$ GeV/$c$ that show small or no $A$-dependence. The dramatic difference in $A$-dependence of TSSAs in different particle species and kinematic range emphasizes the importance of further detailed studies of $A_{N}$ for different particle species over wide kinematics.

To summarize, we have reported $A_{N}$ of positively-charged hadrons for $1.4<\eta<2.4$, $0.1<x_{F}<0.2$, and $1.8<p_{T}<7.0$ GeV$/c$ in $p^{\uparrow}$+$p$, $p^{\uparrow}$+Al, and $p^{\uparrow}$+Au collisions at \sqrt{s_{{}_{NN}}}=200\GeV. For the first time, we observed an $A$-dependent $A_{N}$ in light hadron production in p$$+$$A collisions, with the asymmetry values dropping from $\sim$3% in $p$+$p$ collisions to a value consistent with zero in p$$+Au collisions. These results may provide new insights into the origin of $A_{N}$ and a unique tool to investigate the rich phenomena behind TSSAs in hadronic collisions and to use TSSAs as a new approach to studying the small-system collisions.