Nuclear dependence of the transverse-single-spin asymmetry for forward neutron production in polarized $p$$+$$A$ collisions at $\sqrt{s_{_{NN}}}=200$ GeV

TL;DR

This paper reports on the first measurements of transverse-single-spin asymmetries in forward neutron production during polarized proton collisions with heavy nuclei at RHIC, revealing a surprisingly strong atomic-mass-number dependence.

Contribution

It presents the first experimental investigation of $A$ dependence of single-spin asymmetries in $p+A$ collisions at RHIC, challenging existing theoretical predictions.

Findings

Asymmetry in $p$+Al is small.

Asymmetry in $p$+Au is three times larger and opposite in sign.

Strong $A$ dependence observed in forward neutron asymmetries.

Abstract

During 2015 the Relativistic Heavy Ion Collider (RHIC) provided collisions of transversely polarized protons with Au and Al nuclei for the first time, enabling the exploration of transverse-single-spin asymmetries with heavy nuclei. Large single-spin asymmetries in very forward neutron production have been previously observed in transversely polarized $p$$+$$p$ collisions at RHIC, and the existing theoretical framework that was successful in describing the single-spin asymmetry in $p$$+$$p$ collisions predicts only a moderate atomic-mass-number ($A$) dependence. In contrast, the asymmetries observed at RHIC in $p$$+$$A$ collisions showed a surprisingly strong $A$ dependence in inclusive forward neutron production. The observed asymmetry in $p$$+$Al collisions is much smaller, while the asymmetry in $p$$+$Au collisions is a factor of three larger in absolute value and of opposite sign. The interplay of different neutron production mechanisms is discussed as a possible explanation of the observed $A$ dependence.

Tables

Table 1. Table 1: A N subscript 𝐴 𝑁 A_{N} for forward neutron production in p 𝑝 p + + p 𝑝 p , p 𝑝 p + + Al , and p 𝑝 p + + Au collisions, for ZDC inclusive, ZDC ⊗ tensor-product \otimes BBC-tag, and ZDC ⊗ tensor-product \otimes BBC-veto samples.

			$p$$+$$p$				$p$$+$Al				$p$$+$Au
		Inclusive	BBC Tag	BBC Veto		Inclusive	BBC Tag	BBC Veto		Inclusive	BBC Tag	BBC Veto
$A_N$		-0.054	-0.064	-0.031		-0.013	-0.057	0.073		0.157	-0.015	0.234
Statistical error		$\pm$0.001	$\pm$0.002	$\pm$0.004		$\pm$0.002	$\pm$0.003	$\pm$0.003		$\pm$0.002	$\pm$0.005	$\pm$0.002
Systematic error:
Background		$\pm$0.007	$\pm$0.009	$\pm$0.017		-0.001	-0.010	+0.004		+0.015	-0.003	+0.012
Smearing		$\pm$0.002	$\pm$0.003	$\pm$0.001			$\pm$0.002	$\pm$0.003		$\pm$0.007		$\pm$0.010
Beam pos.		$\pm$0.009	$\pm$0.006	$\pm$0.010		$\pm$0.004	$\pm$0.004	$\pm$0.006		$\pm$0.002	$\pm$0.004	$\pm$0.008
Polarization							$\pm$0.002	$\pm$0.003		$\pm$0.011	$\pm$0.001	$\pm$0.017
Calibration		$\pm$0.003	$\pm$0.001	$\pm$0.007		$\pm$0.001	$\pm$0.004	$\pm$0.004		$\pm$0.004	$\pm$0.009	$\pm$0.006
Total systematic		$\pm$0.012	$\pm$0.011	$\pm$0.021		${}_{-0.004}{}^{}{}_{}{}^{+0.004}$	${}_{-0.012}{}^{}{}_{}{}^{+0.007}$	${}_{-0.008}{}^{}{}_{}{}^{+0.009}$		${}_{-0.014}{}^{}{}_{}{}^{+0.020}$	${}_{-0.010}{}^{}{}_{}{}^{+0.009}$	${}_{-0.022}{}^{}{}_{}{}^{+0.025}$

Equations

$${ϵ}_N(\varphi )=P{A}_N^{\mathrm{fit}}\sin (\varphi -{\varphi }_0),$$

$$A_{N}^{\rm S}=\frac{\mbox{$A_{N}$}^{\rm fit}-r_{\rm eff}A_{N}^{\rm B}}{1-r_{\rm eff}},$$

Full text

Understanding forward particle production in high energy hadron collisions is of great importance, because most of the energy goes in the forward direction, and therefore informs our understanding of overall particle production. This has particular importance in studies of ultra-high energy cosmic rays, where extraction of the cosmic ray distributions from air shower measurements depends on models of forward particle production in the interaction with nuclei in the air d’Enterria et al. (2011); Kampert and Unger (2012); Adriani et al. (2016). Mechanisms for forward particle production are not well understood, as perturbative quantum chromodynamics (pQCD) is not applicable at small momentum transfers and diffractive production mechanisms are not well modeled. To better understand production mechanisms, measurement of the single spin asymmetry $A_{N}$, describing the azimuthal asymmetry of particle production relative to the spin direction of the transversely polarized beam or target provides crucial tests and deeper insight beyond just cross-section measurements. The spin degree of freedom has served as a strong discriminator between theoretical models. For example, the origin of the large asymmetries discovered in forward meson production in p$$+$$p collisions from $\sqrt{s}$ =4.9–19.4 GeV Klem et al. (1976); Dragoset et al. (1978); Antille et al. (1980); Saroff et al. (1990); Allgower et al. (2002); Adams et al. (1991a, b, 1998) and later confirmed at $\sqrt{s}$ =62.4–500 GeV at the Relativistic Heavy Ion Collider (RHIC) Arsene et al. (2008); Abelev et al. (2008a); Adamczyk et al. (2012); Adare et al. (2014a); Heppelmann et al. (2013); Adare et al. (2014b) has been under intensive discussion for three decades and still remains an open question Aschenauer et al. . Despite substantial theoretical attempts to reproduce data in the pQCD regime using the conventional $2\rightarrow 2$ parton scattering processes, the latest multiplicity dependent $A_{N}$ measurements from RHIC Mondal (2014) indicate that a significant contribution to the asymmetry may be of a diffractive nature.

Another important approach in forward particle production is to study the nuclear dependence in p$$+A collisions. In the perturbative region, theoretical approaches based on color-glass-condensate models predicted that hadronic $A_{N}$ should decrease with increasing $A$ Boer et al. (2006); Boer and Dumitru (2003); Boer et al. (2009); Kang and Yuan (2011); Kovchegov and Sievert (2012), while some approaches based on pQCD factorization predicted that $A_{N}$ would stay approximately the same for all nuclear targets Qiu (2012). On the other hand, almost no theoretical/experimental studies are available in the nonperturbative region or diffractive scattering with polarized probes on nuclei, and interesting phenomena may be hidden in this unexplored region.

In the case of forward neutron production in p$$+$$p collisions, production cross sections Engler et al. (1975); Flauger and Monnig (1976); Fukao et al. (2007) were successfully explained in terms of one-pion exchange Capella et al. (1975); Kopeliovich et al. (1996); Nikolaev et al. (1999); Kaidalov et al. (2006); Kopeliovich et al. (2011). However, that model could not explain the sizable $A_{N}$ in very forward (near zero degree) neutron production, discovered at RHIC in p$$+$$p collisions at $\sqrt{s}=200$ GeV Fukao et al. (2007). To reproduce the experimental asymmetry, an interference between the spin-flip $\pi$ exchange and a non spin-flip $a_{1}$-Reggeon exchange was necessary Kopeliovich et al. (2011). Kopeliovich, Potashnikova, and Schmidt considered nuclear absorption effects as a source for a possible $A$ dependence of $A_{N}$, and found only a small effect Kopeliovich et al. (2017).

In this Letter, we report the first measurements of $A_{N}$ for very forward neutron production in collisions between polarized protons and nuclei (Al and Au) at $\sqrt{s_{{}_{NN}}}=200$ GeV recorded in 2015 with the PHENIX detector Adcox et al. (2003). For p$$+$$p collisions 18 RHIC stores were used and 1 store each for p$$+Al and p$$+Au measurements, with a typical store length of 8 hours. The average beam polarization in p$$+$$p, p$$+Al, and p$$+Au data samples was $0.515\pm 0.002$, $0.59\pm 0.02$ and $0.59\pm 0.04$, respectively, with additional global uncertainty of 3% from the polarization normalization sch ; W. Schmidke (private communication).

The experimental setup using a zero-degree calorimeter (ZDC) Adler et al. (2001) and a position-sensitive shower-maximum detector (SMD) is similar to the one used for p$$+$$p data Adare et al. (2013). The ZDC comprises three modules located in series at $\pm$18 m away from the collision point. The ZDC has an acceptance in the transverse plane of 10 $\times$ 10 cm2, with a total of 5.1 nuclear interaction lengths (or 149 radiation lengths), and an energy resolution of $\sim$25%–20% for 50–100 GeV neutrons. The SMD comprises $x$-$y$ (horizontal-vertical) scintillator strip hodoscopes inserted between the first and second ZDC modules (approximately at the position of the maximum hadronic shower), and provides a position resolution of $\sim 1$ cm for 50–100 GeV neutrons. These detectors are located downstream of the RHIC DX beam splitting magnet, so that near beam-momentum charged particles from collisions are expected to be swept into the beam lines and out of the ZDC acceptance (see Fig. 1).

To accommodate asymmetric p$$+A collisions of beams with different rigidity, the DX magnets were moved horizontally Liu et al. (2016). In this special setup for the present measurement, the proton beam was angled off axis by ${\sim}2$ mrad relative to the nominal beam direction at the collision point, with a crossing angle with the Au (Al) beam of 2.0 mrad (1.1 mrad). Correspondingly, the ZDC was moved by 3.6 cm (2 mrad) to keep zero-degree neutrons at the ZDC center (see Fig. 1).

The data was collected with triggers employing the ZDC and beam-beam counters (BBCs) Allen et al. (2003). Only the north ZDC detector, facing the incoming polarized proton beam was used in this analysis. Two BBC counters are located at $\pm 144$ cm from the nominal collision point along the beam pipe and are designed to detect charged particles in the pseudorapidity range of $\pm$(3.0–3.9) with full azimuthal coverage. The ZDC inclusive trigger required the energy deposited in the ZDC to be greater than 15 GeV. The ZDC$\otimes$BBC-tag trigger in addition required at least one hit in each of the BBCs, and ZDC$\otimes$BBC-veto trigger required no hits in both BBCs. The latter two sets represent mutually exclusive but not complete subsets of the ZDC inclusive triggered data.

As described in detail in Ref. Adare et al. (2013), event selection and neutron identification cuts include: (1) a total ZDC energy cut of 40–120 GeV; (2) at least two SMD strips fired (above threshold) in both $x$ and $y$ directions, and a nonzero (above threshold) energy in the second ZDC module (to reject photons); and (3) an acceptance cut of 0.5 $<r<$ 4.0 cm for the reconstructed radial distance $r$ from the determined beam center (to reduce the impact of the position resolution and edge effects in the asymmetry measurements).

The raw asymmetry ($\epsilon_{N}(\phi)$) is calculated using the square-root formula Adare et al. (2013) for each azimuthal angle ($\phi$) bin. The polarization normalized $A_{N}^{\rm fit}$ is then extracted from the fit to a sine function

[TABLE]

where $P$ is the proton beam polarization and $\phi_{0}$ is the polarization direction in the transverse plane.

Figure 2 compares $\epsilon_{N}(\phi)/P$ results for ZDC inclusive samples from p$$+$$p, p$$+Al and p$$+Au collisions and shows the nuclear dependence of $A_{N}^{\rm fit}$, including a sign change from negative in p$$+$$p collisions to positive in p$$+Au collisions. The $A_{N}^{\rm fit}$ was measured separately in each PHENIX data taking segment, typically 60 min long, and then the weighted average was calculated. The obtained $A_{N}^{\rm fit}$ is then corrected for backgrounds and detector responses. The main background contribution comes from protons, generated by elastic, diffractive, and hard processes.

Protons from elastic and diffractive reactions travel close to the beam line and are swept by the DX magnet to the right (toward negative $x$ in Fig. 1). Only a small fraction of such protons scattered by large angles, larger than 4–5 mrad, fall in the ZDC acceptance. Because the cross section for these reactions falls sharply with scattering angle, these protons contribute mainly on the right side of the ZDC. This contribution was evaluated from the particle position distribution as measured by the SMD and found to be 9% and 32% in the inclusive ZDC and ZDC$\otimes$BBC-veto triggered samples respectively in p$$+$$p collisions, $<2\%$ in both samples in p$$+A collisions, and negligible in ZDC$\otimes$BBC-tag samples of both p$$+$$p and p$$+A collisions. The significant suppression of elastic and diffractive proton background relative to the neutron signal in p$$+A collisions can be understood as due to the stronger magnetic fields in the DX magnets. Correspondingly, the minimum scattering angle for the elastic and diffractive proton backgrounds to reach the ZDC acceptance increases from 3.8 mrad to 5 mrad, leading to a cross section reduction by an order of magnitude.

The contribution of charged hadron background from hard scattering processes, distributed nearly uniformly over the ZDC acceptance, was estimated using pythia6 Sjöstrand et al. (2001) with a geant3 Brun et al. detector simulation. However, from previous studies where a charge veto counter was installed in front of the ZDC to measure the charged hadron background, it was found that simulation underestimates the proton background by a factor of $\sim$2 Adare et al. (2013). Therefore the hard scattering background contribution from simulation was scaled by a factor of two with an uncertainty equal to the size of the increase. In p$$+$$p collisions this background fraction resulted in 6$\pm$3%, 3$\pm$1.5% and 12$\pm$6% in ZDC, ZDC$\otimes$BBC-veto and ZDC$\otimes$BBC-tag triggered samples, respectively. In p$$+A collisions due to increased neutron signal from electromagnetic (EM) processes (to be discussed later), the relative background contributions are expected to be smaller. Therefore the measured asymmetries in p$$+A collisions were not corrected for background, but one-sided systematic uncertainties (in the direction of asymmetry magnitude increase) equal to the upper $1\sigma$ limit of the background fractions taken from the p$$+$$p case, i.e. 9%, 4.5% and 18%, were conservatively assigned in ZDC, ZDC$\otimes$BBC-veto and ZDC$\otimes$BBC-tag triggered samples, respectively.

From the considerations above, only the p$$+$$p asymmetries were corrected for backgrounds according to

[TABLE]

where $A_{N}^{\rm S}$ and $A_{N}^{\rm B}$ stand for signal and background asymmetries, and $r_{\rm eff}$ is the “effective” background fraction in the reconstructed neutron sample. The parameter $r_{\rm eff}$ accounts for the dilution of the background effect in $A_{N}^{\rm fit}$ in the case when the background contributes preferably on one side of the detector (as from elastic or diffractive protons). This effect, which was studied in simulation, comes from a specific way the left and right sides of detector acceptance are combined in the square-root formula for asymmetry calculation. The background asymmetry $A_{N}^{\rm B}$ was evaluated from the comparison of asymmetries with and without the charge veto cut from the 2008 data when the charge veto counter was available, and then used in Eq. (2). The asymmetries $A_{N}^{\rm B}$ were found to be consistent with zero within statistical uncertainties for all triggers. After background correction, $A_{N}^{\rm S}$ results for p$$+$$p from 2008 and 2015 data were found to be consistent within statistical uncertainties. Asymmetries from 2015 data were used in the final results.

Besides charged hadrons, the other background sources are photons and $K^{0}$ mesons. From pythia6 simulation their contribution after the analysis cuts was evaluated to be below 3% in all collision systems and triggers, and was neglected in the asymmetry results.

The measured asymmetries are affected by detector resolutions and other detector systematic effects (e.g. edge effects), as well as by the uncertainty in the shape of the neutron production cross section vs $p_{T}$ and $x_{F}$, the size of the asymmetry, and the assumption for the shape of $A_{N}(p_{T})$ within the $p_{T}$ range sampled in this analysis. These effects were studied in detail with a geant3 Monte Carlo simulation. The fully corrected transverse single spin asymmetry $A_{N}$ was calculated as $A_{N}=A_{N}^{S}/C_{\phi}$ where the correction factor $C_{\phi}$ was calculated in the simulation as the ratio of the measured asymmetry to the average input asymmetry over the neutron sample collected with experimental cuts used in the analysis. The biggest variation in $C_{\phi}$ comes from the position resolution uncertainty and the assumption for $A_{N}(p_{T})$. The position resolution in simulation vs data was confirmed from the comparison of shower shape and its fluctuations in SMD strips. The simulation was tuned to data by varying noise and thresholds in the SMD channels, as well as by introducing a cross talk effect, similar to Adare et al. (2013). An overall value of 3% was assigned to the $C_{\phi}$ uncertainty. For the shape of $A_{N}(p_{T})$, it was modeled as $A_{N}(p_{T})={\rm const}$ (as was assumed in Adare et al. (2013)) and $A_{N}(p_{T})\propto p_{T}$ (which is supported by theory in the $p_{T}$ range relevant here Kopeliovich et al. (2011)). The difference of 3% was included in the $C_{\phi}$ uncertainty. The final correction factor applied to the measured asymmetries is $C_{\phi}=0.855\pm 0.036$. Note, the $C_{\phi}$ value here is higher than the one in our previous publication Adare et al. (2013) mainly due to two reasons: first, more realistic $A_{N}(p_{T})\propto p_{T}$ assumption was used in this analysis, and, second, the optimized SMD thresholds reduced the smearing effect.

In addition to the beam polarization, background, and smearing correction ($C_{\phi}$) discussed above, the other sources of systematic uncertainties are the ZDC and SMD gain calibrations (including threshold variation) and location of the beam center on the ZDC plane. The latter is among the dominant uncertainties in this data, contributing 0.002–0.010 to the $A_{N}$ uncertainty. It was estimated by calculating the asymmetry for varying assumptions of the beam axis projection on the ZDC plane, $\pm 1$ cm in horizontal and $\pm 0.5$ cm in vertical directions from the ZDC center, which reflect the uncertainty in ZDC alignment relative to the beam axis.

The analyzed data correspond to the neutron sampled $p_{T}$ in the range smaller than $0.25$ GeV/$c$ peaked at about 0.1 GeV/$c$, which is defined mainly by detector acceptance and which is affected by detector resolutions. Due to the varying contribution of different processes to neutron production, the sampled $p_{T}$ distribution may vary in different collision systems and in different triggered data. Figure 3 shows the differences in the radial distributions, which is related to the neutron production cross section $d\sigma/dp_{T}$ by $p_{T}\propto r$ Adare et al. (2013). From a comparison with simulation assuming different slope parameter, $b$, in the parameterization $d\sigma/dp_{T}\sim e^{-b\cdot p_{T}}$, the data were found to be consistent with $b=4$ (GeV/$c$)-1 for all collision systems in ZDC$\otimes$BBC-tag triggered data, and $b=$4, 6 and 8 (GeV/$c$)-1 in p$$+$$p, p$$+Al and p$$+Au collisions, respectively, in ZDC$\otimes$BBC-veto triggered sample, with uncertainty $\sigma_{b}=1$ (GeV/$c$)-1 reflecting its sensitivity to SMD gain calibration and thresholds. These variations lead to a difference in the average $p_{T}$ sampled in different collision systems and triggers by as much as 10%. As can be also judged from Fig. 3, due to the small detector acceptance, the sampled $p_{T}$ distribution shows very modest dependence on the slope of the input $p_{T}$ distribution, particularly at low $p_{T}$ (or $r$), which is most responsible for the dilution of the measured asymmetry. As a consequence,  the variation of the correction factor $C_{\phi}$ due to different slope parameters $b$ discussed above was less than 1%.

Figure 4 and Table 1 summarize the results for $A_{N}$ in forward neutron production in p$$+$$p, p$$+Al and p$$+Au collisions, for ZDC inclusive, ZDC$\otimes$BBC-tag and ZDC$\otimes$BBC-veto samples. In addition to 3% scale uncertainty from polarization normalization, common to all points, the other part of the polarization uncertainty is correlated for different triggers in a particular collision system. The presented asymmetries in p$$+$$p collisions are consistent with our previous publication Adare et al. (2013), albeit with larger systematic uncertainties in this data due to larger background (unlike this measurement, charged veto counter was used in Adare et al. (2013) to suppress the background), and larger variations due to uncertainty of the beam position on the ZDC plane.

From Fig. 4, the $A$ dependence of $A_{N}$ for inclusive neutrons is strong. Compared to the $A_{N}$ of p$$+$$p collisions, the observed asymmetry in p$$+Al collisions is much smaller, while the asymmetry in p$$+Au collisions is a factor of three larger in absolute value and of opposite sign. This behavior is unexpected because the theoretical framework using $\pi$ and $a_{1}$-Reggeon interference can only predict a moderate nuclear dependence, and there is no known mechanism to flip the sign of $A_{N}$ within this framework Kopeliovich et al. (2017).

The asymmetries requiring BBC hits are remarkably different. Once BBC hits are required (ZDC$\otimes$BBC-tag), the drastic behavior of the inclusive $A_{N}$ vanishes and its sign stays negative, approaching $A_{N}=0$ at large $A$. In contrast, the strong $A$ dependence is amplified once no hits in the BBC are required (ZDC$\otimes$BBC-veto). While the BBCs cover a limited acceptance, the requirement (or veto) of hits in the BBC should place constraints on the activity near the detected neutron and thus the corresponding production mechanism.

One possibility to explain the present results is a contribution from EM interactions, which have been demonstrated to be important for reactions with small momentum transfer, e.g., in ultra-peripheral heavy ion collision at RHIC Abelev et al. (2008b); Afanasiev et al. (2009); Abelev et al. (2010); Agakishiev et al. (2012) and Large Hadron Collider Abelev et al. (2013); Abbas et al. (2013); Abelev et al. (2014); Adam et al. (2015), including forward neutron production in p+A collisions Mitsuka (2015), and polarization observables in fixed target experiments Alekseev et al. (2009); Carey et al. (1990). Although it was ignored in the interpretation for the p$$+$$p data Kopeliovich et al. (2017), EM interactions become increasingly important for large atomic number ($Z$) nuclei, as the EM field of the nucleus is a rich source of virtual photons, increasing as $Z^{2}$. Forward neutrons in the final state can be produced through nonresonant photo-$\pi^{+}$ production and neutron decay channel from photo-nucleon excitation processes, such as the $\Delta$ resonance Mitsuka (2017).

According to a Monte-Carlo study Mitsuka (2015), the neutron and its associated $\pi^{+}$ produced through this process are substantially boosted towards the proton beam direction, so that only a small fraction of pions would be detected by the BBC. Thus, a large fraction of EM processes are expected to be suppressed in the ZDC$\otimes$BBC-tag events while enhanced in the ZDC$\otimes$BBC-veto events. Here, it is noted that the importance of EM processes in p$$+A collisions is also hinted at in the present data: the ratio between reconstructed neutrons in ZDC$\otimes$BBC-veto and ZDC$\otimes$BBC-tag samples increases from smaller than $0.5$ in p$$+$$p to $\sim 1$ ($\sim 5$) in p$$+Al (p$$+Au) collisions. In addition, a faster drop of the neutron production cross section with $p_{T}$ in p$$+A collisions in ZDC$\otimes$BBC-veto triggered data discussed in Fig. 3b is consistent with increasing role of EM processes that have softer $p_{T}$ distribution than hadronic processes.

Similarly in the asymmetry measurements, contributions of different production mechanisms may be suppressed or enhanced by different event selection triggers. Hence, while the result for the ZDC$\otimes$BBC-tag sample may be explained by the conventional pion and $a_{1}$-Reggeon interference mechanism Kopeliovich et al. (2017), that for the ZDC$\otimes$BBC-veto triggered sample could be explained by contributions from interference with EM amplitudes Mitsuka (2017), which are expected to be enhanced in that dataset. However, there could be other mechanisms, such as diffractive scattering, which is also expected to be enhanced by a ZDC$\otimes$BBC-veto trigger. Therefore, further studies are needed to fully understand the present results.

In summary, we observe an unexpectedly strong $A$ dependence in $A_{N}$ of inclusive forward neutron production in polarized p$$+A collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV. Furthermore, a distinctly different behavior of $A_{N}$ was observed in two oppositely trigger-enhanced data sets. These surprising behaviors could be explained by a contribution of EM interactions, which may be sizable for heavy nuclei. Further studies of the production mechanism including EM contributions and diffractive scattering would have an impact not only to hadron physics, but also to cosmic-ray science, where measurements of high-energy cosmic rays depend on models of forward particle production in the interactions with nuclei in the air. Spin asymmetry measurements not only provide a unique discriminating power for the models of particle production, but also will contribute to our understanding of the origin of the transverse spin asymmetries in hadronic collisions.

We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory, especially the CA-D staff for providing beams with a special tune for these measurements, and the staff of the other PHENIX participating institutions for their vital contributions. We also thank Boris Kopeliovich and Michal K$\check{\rm r}$elina for providing us with theoretical calculations of the elastic proton cross sections and for useful discussions. We acknowledge support from the Office of Nuclear Physics in the Office of Science of the Department of Energy, the National Science Foundation, Abilene Christian University Research Council, Research Foundation of SUNY, and Dean of the College of Arts and Sciences, Vanderbilt University (U.S.A), Ministry of Education, Culture, Sports, Science, and Technology and the Japan Society for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento Científico e Tecnológico and Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil), Natural Science Foundation of China (People’s Republic of China), Croatian Science Foundation and Ministry of Science and Education (Croatia), Ministry of Education, Youth and Sports (Czech Republic), Centre National de la Recherche Scientifique, Commissariat à l’Énergie Atomique, and Institut National de Physique Nucléaire et de Physique des Particules (France), Bundesministerium für Bildung und Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), J. Bolyai Research Scholarship, EFOP, the New National Excellence Program (ÚNKP), NKFIH, and OTKA (Hungary), Department of Atomic Energy and Department of Science and Technology (India), Israel Science Foundation (Israel), Basic Science Research Program through NRF of the Ministry of Education (Korea), Physics Department, Lahore University of Management Sciences (Pakistan), Ministry of Education and Science, Russian Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and Wallenberg Foundation (Sweden), the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union, the Hungarian American Enterprise Scholarship Fund, the US-Hungarian Fulbright Foundation, and the US-Israel Binational Science Foundation.