Nuclear dependence of the transverse-single-spin asymmetry for forward neutron production in polarized p + A collisions at √ s NN = 200 GeV

C. Aidala, Y. Akiba, 51, ∗ M. Alfred, V. Andrieux, K. Aoki, N. Apadula, H. Asano, 50 C. Ayuso, B. Azmoun, V. Babintsev, A. Bagoly, N.S. Bandara, K.N. Barish, S. Bathe, 51 A. Bazilevsky, M. Beaumier, R. Belmont, A. Berdnikov, Y. Berdnikov, D.S. Blau, M. Boer, J.S. Bok, M.L. Brooks, J. Bryslawskyj, 8 V. Bumazhnov, C. Butler, S. Campbell, V. Canoa Roman, R. Cervantes, C.Y. Chi, M. Chiu, I.J. Choi, J.B. Choi, † Z. Citron, M. Connors, 51 N. Cronin, M. Csanád, T. Csörgő, T.W. Danley, M.S. Daugherity, G. David, K. DeBlasio, K. Dehmelt, A. Denisov, A. Deshpande, 56 E.J. Desmond, A. Dion, D. Dixit, J.H. Do, A. Drees, K.A. Drees, M. Dumancic, J.M. Durham, A. Durum, T. Elder, 20 A. Enokizono, 52 H. En’yo, S. Esumi, B. Fadem, W. Fan, N. Feege, D.E. Fields, M. Finger, M. Finger, Jr., S.L. Fokin, J.E. Frantz, A. Franz, A.D. Frawley, Y. Fukuda, C. Gal, P. Gallus, P. Garg, 56 H. Ge, F. Giordano, Y. Goto, 51 N. Grau, S.V. Greene, M. Grosse Perdekamp, T. Gunji, H. Guragain, T. Hachiya, 51 J.S. Haggerty, K.I. Hahn, H. Hamagaki, H.F. Hamilton, S.Y. Han, J. Hanks, S. Hasegawa, T.O.S. Haseler, X. He, T.K. Hemmick, J.C. Hill, K. Hill, R.S. Hollis, K. Homma, B. Hong, T. Hoshino, N. Hotvedt, J. Huang, S. Huang, K. Imai, J. Imrek, M. Inaba, A. Iordanova, D. Isenhower, Y. Ito, D. Ivanishchev, B.V. Jacak, M. Jezghani, Z. Ji, X. Jiang, B.M. Johnson, 20 V. Jorjadze, D. Jouan, D.S. Jumper, J.H. Kang, D. Kapukchyan, S. Karthas, D. Kawall, A.V. Kazantsev, V. Khachatryan, A. Khanzadeev, C. Kim, 31 D.J. Kim, E.-J. Kim, M. Kim, M.H. Kim, D. Kincses, E. Kistenev, J. Klatsky, P. Kline, T. Koblesky, D. Kotov, 53 S. Kudo, K. Kurita, Y. Kwon, J.G. Lajoie, E.O. Lallow, A. Lebedev, S. Lee, M.J. Leitch, Y.H. Leung, N.A. Lewis, X. Li, S.H. Lim, 63 L. D. Liu, M.X. Liu, V-R Loggins, V.-R. Loggins, S. Lokos, K. Lovasz, D. Lynch, T. Majoros, Y.I. Makdisi, M. Makek, M. Malaev, V.I. Manko, E. Mannel, H. Masuda, M. McCumber, P.L. McGaughey, D. McGlinchey, C. McKinney, M. Mendoza, W.J. Metzger, A.C. Mignerey, D.E. Mihalik, A. Milov, D.K. Mishra, J.T. Mitchell, G. Mitsuka, S. Miyasaka, 58 S. Mizuno, 59 P. Montuenga, T. Moon, D.P. Morrison, S.I.M. Morrow, T. Murakami, 50 J. Murata, 52 K. Nagai, K. Nagashima, T. Nagashima, J.L. Nagle, M.I. Nagy, I. Nakagawa, 51 H. Nakagomi, 59 K. Nakano, 58 C. Nattrass, T. Niida, R. Nouicer, 51 T. Novák, 62 N. Novitzky, R. Novotny, A.S. Nyanin, E. O’Brien, C.A. Ogilvie, J.D. Orjuela Koop, J.D. Osborn, A. Oskarsson, G.J. Ottino, K. Ozawa, 59 V. Pantuev, V. Papavassiliou, J.S. Park, S. Park, 54, 56 S.F. Pate, M. Patel, W. Peng, D.V. Perepelitsa, 12 G.D.N. Perera, D.Yu. Peressounko, C.E. PerezLara, J. Perry, R. Petti, M. Phipps, 24 C. Pinkenburg, R.P. Pisani, A. Pun, M.L. Purschke, P.V. Radzevich, K.F. Read, 57 D. Reynolds, V. Riabov, 49 Y. Riabov, 53 D. Richford, T. Rinn, S.D. Rolnick, M. Rosati, Z. Rowan, J. Runchey, A.S. Safonov, T. Sakaguchi, H. Sako, V. Samsonov, 49 M. Sarsour, K. Sato, S. Sato, B. Schaefer, B.K. Schmoll, K. Sedgwick, R. Seidl, 51 A. Sen, 57 R. Seto, A. Sexton, D. Sharma, I. Shein, T.-A. Shibata, 58 K. Shigaki, M. Shimomura, 41 T. Shioya, P. Shukla, A. Sickles, C.L. Silva, D. Silvermyr, B.K. Singh, C.P. Singh, V. Singh, M. J. Skoby, M. Slunečka, K.L. Smith, M. Snowball, R.A. Soltz, W.E. Sondheim, S.P. Sorensen, I.V. Sourikova, P.W. Stankus, S.P. Stoll, T. Sugitate, A. Sukhanov, T. Sumita, J. Sun, S. Syed, J. Sziklai, A Takeda, K. Tanida, 51, 54 M.J. Tannenbaum, S. Tarafdar, 61 G. Tarnai, R. Tieulent, A. Timilsina, T. Todoroki, M. Tomášek, C.L. Towell, R.S. Towell, I. Tserruya, Y. Ueda, B. Ujvari, H.W. van Hecke, S. Vazquez-Carson, J. Velkovska, M. Virius, V. Vrba, 26 N. Vukman, X.R. Wang, 51 Z. Wang, Y. Watanabe, 51 Y.S. Watanabe, C.P. Wong, C.L. Woody, C. Xu, Q. Xu, L. Xue, S. Yalcin, Y.L. Yamaguchi, 56 H. Yamamoto, A. Yanovich, P. Yin, J.H. Yoo, I. Yoon, H. Yu, 48 I.E. Yushmanov, W.A. Zajc, A. Zelenski, S. Zharko, and L. Zou

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 þ p collisions at RHIC, and the existing theoretical framework that was successful in describing the single-spin asymmetry in p þ p collisions predicts only a moderate atomic-mass-number (A) dependence. In contrast, the asymmetries observed at RHIC in p þ A collisions showed a surprisingly strong A dependence in inclusive forward neutron production. The observed asymmetry in p þ Al collisions is much smaller, while the asymmetry in p þ Au collisions is a factor of 3 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. DOI: 10.1103/PhysRevLett.120.022001 Understanding forward particle production in highenergy 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 ultrahigh-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 [1][2][3]. 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, the 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 þ p collisions from ffiffi ffi s p ¼ 4. Another important approach in forward particle production is to study the nuclear dependence in p þ A collisions. In the perturbative region, theoretical approaches based on color-glass-condensate models predicted that hadronic A N should decrease with increasing A [20-24], while some approaches based on pQCD factorization predicted that A N would stay approximately the same for all nuclear targets [25]. On the other hand, almost no theoretical or 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 þ p collisions, production cross sections [26][27][28] were successfully explained in terms of one-pion exchange [29][30][31][32][33]. However, that model could not explain the sizable A N in very forward (near zero degree) neutron production, discovered at RHIC in p þ p collisions at ffiffi ffi s p ¼ 200 GeV [28]. To reproduce the experimental asymmetry, an interference between the spin-flip π exchange and a nonspin-flip a 1 -Reggeon exchange was necessary [33]. 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 [34].
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 ffiffiffiffiffiffiffi ffi s NN p ¼ 200 GeV recorded in 2015 with the PHENIX detector [35]. For p þ p collisions 18 RHIC stores were used and one store each for p þ Al and p þ Au measurements, with a typical store length of 8 h. The average beam polarization in p þ p, p þ Al, and p þ Au data samples was 0.515 AE 0.002, 0.59 AE 0.02, and 0.59 AE 0.04, respectively, with an additional global uncertainty of 3% from the polarization normalization [36,37].
The experimental setup using a zero-degree calorimeter (ZDC) [38] and a position-sensitive shower-maximum detector (SMD) is similar to the one used for p þ p data [39]. The ZDC comprises three modules located in series at AE18 m away from the collision point. The ZDC has an acceptance in the transverse plane of 10 × 10 cm 2 , with a total of 5.1 nuclear interaction lengths (or 149 radiation lengths), and an energy resolution of ∼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 ∼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 þ A collisions of beams with different rigidity, the DX magnets were moved horizontally [40]. In this special setup for the present measurement, the proton beam was angled off axis by ∼2 mrad relative to the nominal beam direction at the collision point, with a crossing angle with the Au (Al) beam of 2.0 (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 were collected with triggers employing the ZDC and beam-beam counters (BBCs) [41]. Only the north ZDC detector, facing the incoming polarized proton beam, was used in this analysis. Two BBCs are located at AE144 cm from the nominal collision point along the beam pipe and are designed to detect charged particles in the pseudorapidity range of AEð3.0-3.9Þ with full azimuthal coverage. The ZDC inclusive trigger required the energy deposited in the ZDC to be greater than 15 GeV. The ZDC ⊗ BBC-tag trigger in addition required at least one hit in each of the BBCs, and the ZDC ⊗ 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. [39], event selection and neutron identification cuts include (i) a total ZDC energy cut of 40-120 GeV, (ii) 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 (iii) 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 [ϵ N ðϕÞ] is calculated using the square-root formula [39] for each azimuthal angle (ϕ) bin. The polarization normalized A fit N is then extracted from the fit to a sine function where P is the proton beam polarization and ϕ 0 is the polarization direction in the transverse plane. Figure 2 compares ϵ N ðϕÞ=P results for ZDC inclusive samples from p þ p, p þ Al, and p þ Au collisions and shows the nuclear dependence of A fit N , including a sign change from negative in p þ p collisions to positive in p þ Au collisions. The A fit N was measured separately in each PHENIX data-taking segment, typically 60 min long, and then the weighted average was calculated. The obtained A fit N 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 the 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 ⊗ BBC-veto FIG. 1. ZDC location and beam orbits of a proton (blue) beam and a heavy-ion (yellow) beam in the special stores used for this analysis; the z axis shows the nominal beam direction, and the dashed line represents the zero-degree neutron trajectory. DX and D0 are the RHIC beam bending dipole magnets.
PHYSICAL REVIEW LETTERS 120, 022001 (2018) triggered samples, respectively, in p þ p collisions, < 2% in both samples in p þ A collisions, and negligible in ZDC ⊗ BBC-tag samples of both p þ p and p þ A collisions. The significant suppression of elastic and diffractive proton background relative to the neutron signal in p þ 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 to 5 mrad, leading to a cross section reduction by an order of magnitude.
The contribution of the charged hadron background from hard scattering processes, distributed nearly uniformly over the ZDC acceptance, was estimated using PYTHIA6 [42] with a GEANT3 [43] 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 the simulation underestimates the proton background by a factor of ∼2 [39]. Therefore, the hard scattering background contribution from the simulation was scaled by a factor of 2 with an uncertainty equal to the size of the increase. In p þ p collisions, this background fraction resulted in 6 AE 3%, 3 AE 1.5%, and 12 AE 6% in ZDC, ZDC ⊗ BBC-veto, and ZDC ⊗ BBC-tag triggered samples, respectively. In p þ A collisions, due to the 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 þ A collisions were not corrected for background, but one-sided systematic uncertainties (in the direction of the asymmetry magnitude increase) equal to the upper 1σ limit of the background fractions taken from the p þ p case, i.e., 9%, 4.5%, and 18%, were conservatively assigned in ZDC, ZDC ⊗ BBC-veto, and ZDC ⊗ BBC-tag triggered samples, respectively.
From the considerations above, only the p þ p asymmetries were corrected for backgrounds according to where A S N and A B N stand for signal and background asymmetries, respectively, and r eff is the "effective" background fraction in the reconstructed neutron sample. The parameter r eff accounts for the dilution of the background effect in A fit N 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 the simulation, comes from a specific way the left and right sides of detector acceptance are combined in the squareroot formula for asymmetry calculation. The background asymmetry A B N 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 B N were found to be consistent with zero within statistical uncertainties for all triggers. After a background correction, A S N results for p þ 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 the 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 ðp T Þ 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 ¼ A S N =C ϕ , where the correction factor C ϕ 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 ϕ comes from the position resolution uncertainty and the assumption for A N ðp T Þ. The position resolution in the simulation vs data was confirmed from the comparison of the 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 Ref. [39]. An overall value of 3% was assigned to the C ϕ uncertainty. For the shape of A N ðp T Þ, it was modeled as A N ðp T Þ ¼ const (as was assumed in Ref. [39]) and A N ðp T Þ ∝ p T (which is supported by theory in the p T range relevant here [33]). The difference of 3% was included in the C ϕ uncertainty. The final correction factor applied to the measured asymmetries is C ϕ ¼ 0.855 AE 0.036. Note that the C ϕ value here is higher than the one in our previous publication [39] mainly due to two reasons: First, a more realistic A N ðp T Þ ∝ 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 ϕ ) 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, AE1 cm in the horizontal and AE0.5 cm in the vertical direction from the ZDC center, which reflect the uncertainty in ZDC alignment relative to the beam axis. PHYSICAL REVIEW LETTERS 120, 022001 (2018)

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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. Because of 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 dσ=dp T by p T ∝ r [39]. From a comparison with the simulation assuming different slope parameters b, in the parameterization dσ=dp T ∼ e −b·p T , the data were found to be consistent with b ¼ 4 ðGeV=cÞ −1 for all collision systems in ZDC ⊗ BBC-tag triggered data and b ¼ 4, 6, and 8 ðGeV=cÞ −1 in p þ p, p þ Al, and p þ Au collisions, respectively, in a ZDC ⊗ BBC-veto triggered sample, with uncertainty σ b ¼ 1 ðGeV=cÞ −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 a 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 ϕ due to different slope parameters b discussed above was less than 1%. Figure 4 and Table I summarize the results for A N in forward neutron production in p þ p, p þ Al, and p þ Au collisions, for ZDC inclusive, ZDC ⊗ BBC-tag, and ZDC ⊗ BBC-veto samples. In addition to the 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 þ p collisions are consistent with our previous publication [39], albeit with larger systematic uncertainties in these data due to a larger background (unlike this measurement, the charged veto counter was used in Ref. [39] to suppress the background) and larger variations due to the 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 þ p collisions, the observed asymmetry in p þ Al collisions is much smaller, while the asymmetry in p þ Au collisions is a factor of 3 larger in absolute value and of opposite sign. This behavior is unexpected, because the theoretical framework using π and a 1 -Reggeon interference can predict only a moderate nuclear dependence, and there is no known mechanism to flip the sign of A N within this framework [34].
The asymmetries requiring BBC hits are remarkably different. Once BBC hits are required (ZDC ⊗ BBC-tag), the drastic behavior of the inclusive A N vanishes and its sign stays negative, approaching A N ¼ 0 at large A. In contrast, the strong A dependence is amplified once no hits in the BBC are required (ZDC ⊗ 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 ultraperipheral heavy ion collision at RHIC [44][45][46][47] and Large Hadron Collider [48-51], including forward neutron production in p þ A collisions [52], and polarization observables in fixed target experiments [53,54]. Although it was ignored in the interpretation for the p þ p data [34], 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-π þ production and neutron decay channel from photonucleon excitation processes, such as the Δ resonance [55].
According to a Monte Carlo study [52], the neutron and its associated π þ 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 ⊗ BBC-tag events while enhanced in the ZDC ⊗ BBC-veto events. Here, it is noted that the importance of EM processes in p þ A collisions is also hinted at in the present data: The ratio between reconstructed neutrons in ZDC ⊗ BBC-veto and ZDC ⊗ BBC-tag samples increases from smaller than 0.5 in p þ p to ∼1 (∼5) in p þ Al (p þ Au) collisions. In addition, a faster drop of the neutron production cross section with p T in p þ A collisions in ZDC ⊗ BBC-veto triggered data discussed in Fig. 3(b) is consistent with the increasing role of EM processes that have a 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 ⊗ BBC-tag sample may be explained by the conventional pion and a 1 -Reggeon interference mechanism [34], that for the ZDC ⊗ BBCveto triggered sample could be explained by contributions from interference with EM amplitudes [55], which are expected to be enhanced in that data set. However, there could be other mechanisms, such as diffractive scattering, which is also expected to be enhanced by a ZDC ⊗ BBCveto 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 þ A collisions at ffiffiffiffiffiffiffi ffi s NN p ¼ 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 Křelina 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