Precision Small Scattering Angle Measurements of Elastic Proton-Proton Single and Double Spin Analyzing Powers at the RHIC Hydrogen Jet Polarimeter
Precision Small Scattering Angle Measurements of Elastic Proton-Proton Single and Double Spin...
Poblaguev, A. A.;Zelenski, A.;Aschenauer, E.;Atoian, G.;Eyser, K. O.;Huang, H.;Makdisi, Y.;Schmidke, W. B.;Alekseev, I.;Svirida, D.;Buttimore, N. H.;
2019-10-16 00:00:00
PHYSICAL REVIEW LETTERS 123, 162001 (2019) Precision Small Scattering Angle Measurements of Elastic Proton-Proton Single and Double Spin Analyzing Powers at the RHIC Hydrogen Jet Polarimeter A. A. Poblaguev , A. Zelenski, E. Aschenauer, G. Atoian, K. O. Eyser, H. Huang, Y. Makdisi, and W. B. Schmidke Brookhaven National Laboratory, Upton, New York 11973, USA I. Alekseev and D. Svirida Alikhanov Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia N. H. Buttimore School of Mathematics, Trinity College, Dublin 2, Ireland (Received 29 May 2019; revised manuscript received 26 August 2019; published 16 October 2019) The Polarized Atomic Hydrogen Gas Jet Target polarimeter is employed by the Relativistic Heavy Ion Collider (RHIC) to measure the absolute polarization of each colliding proton beam. Polarimeter detectors and data acquisition were upgraded in 2015 to increase solid angle, energy range, and energy resolution. These upgrades and advanced systematic error analysis along with improved beam intensity and polarization in RHIC runs 2015 (E ¼ 100 GeV) and 2017 (255 GeV) allowed us to greatly reduce beam the statistical and systematic uncertainties for elastic spin asymmetries, A ðtÞ and A ðtÞ, in the Coulomb- N NN nuclear interference momentum transfer range 0.0013 < −t< 0.018 GeV . For the first time hadronic single spin-flip r and double spin-flip r amplitude parameters were reliably isolated at these energies and 5 2 momentum transfers. Measurements at two beam energies enable a separation of Pomeron and Regge pole contributions to r ðsÞ and r ðsÞ, indicating that the spin component may persist at high energies. 5 2 DOI: 10.1103/PhysRevLett.123.162001 Introduction.—Study of the spin-averaged elastic pp data [8–11] was insufficient to identify a Pomeron con- hadronic amplitude at high energies has a more than 50 year tribution, if any, to the pp spin-dependent amplitudes. history [1] and is continuing at the Large Hadron Collider. In this Letter, we report new measurements of the single An essential contribution to this study relates to forward spin A ðtÞ and double spin A ðtÞ analyzing powers in the N NN scattering for which the optical theorem and Coulomb- small angle elastic collision of RHIC’s polarized proton beams with Polarized Atomic Hydrogen Gas Jet Target nuclear interference (CNI) provide an opportunity to pffiffiffi separate the real and imaginary parts of an amplitude. (HJET) [12] at s ¼ 13.76 and 21.92 GeV. The precision Regge theory, based on the analyticity of a scattering has improved significantly by comparison with previous amplitude, is a recognized method of understanding the HJET publications [9,10] and this has allowed us to not only energy dependence of amplitudes [2]. isolate hadronic spin-flip amplitudes but also to incorporate An explanation of the unexpected discovery in the spin dependence in a Regge pole analysis. It appears that seventies of a growing pp cross section at high energies forward elastic pp scattering has nonvanishing single and [3] was found [4] in the Pomeron concept, which is now double spin-flip hadronic amplitudes at high energy where associated with the exchange of nonperturbative QCD the Pomeron dominates. The results of the analysis facilitate gluons [5]. Currently, the Pomeron and Regge pole picture extrapolation of the measured A ðtÞ to a wide range of of unpolarized elastic pp scattering is commonly consid- energies, essential for CNI polarimetry. Additional mea- pffiffiffi surements at the RHIC injection energy (E ¼ 24 GeV) ered as well established in the s ¼ 5 GeV–13 TeV c:m: beam might yield an improved Reggeon fit and the possibility [13] energy range [1], though some new results, e.g., from the of experimentally resolving the Odderon issue [7]. TOTEM experiment [6], call for a revision [7]. However, The HJET provides an absolute proton beam polarization the accuracy of existing polarized high energy experimental measurement averaged across a beam. Typically, hP i ∼ beam 55 2.0 0.3 Þ% [14] for an 8-h RHIC store. The stat syst achieved accuracy satisfies the requirements of hadron Published by the American Physical Society under the terms of polarimetry for planned and future accelerators such as the the Creative Commons Attribution 4.0 International license. Electron Ion Collider (EIC) [15]. This work is based on Further distribution of this work must maintain attribution to the technique of high energy beam polarization measure- the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP . ment developed at RHIC. The methodology can be 0031-9007=19=123(16)=162001(6) 162001-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 123, 162001 (2019) recommended for EIC including a possible extension of it respectively. To determine analyzing powers A ðtÞ and using other polarized nuclei such as He. A ðtÞ, the single spin (jet and beam) and double spin NN HJET polarimeter at RHIC.—The HJET [12] acts like a asymmetries fixed target that measures absolute polarization of 24– a ¼A jP j;a ¼A jP j;a ¼A jP P j; ð2Þ N j N N b NN NN j b 255 GeV proton beams at RHIC. It consists of three main N components: an atomic beam source, a Breit-Rabi polarimeter were derived [13] from the selected elastic event counts to measure atomic hydrogen polarization, and a recoil ð↑↓Þðþ−Þ N discriminated by the right or left (RL) detector RL spectrometer to determine the beam and vertically polarized location and the beam (↑↓) and jet (þ−) spin directions. atomic hydrogen target (the jet) spin correlated asymmetries For CNI elastic pp scattering at high energies, the of the detected recoil protons. Polarizations of both RHIC theoretical basis for an experimental parametrization of beams (alternating spin up or down bunches), so-called blue the analyzing powers was introduced in Refs. [18,19] and and yellow, are measured concurrently and continuously. updated [20] for the RHIC spin program [21]. The The jet density profile in the horizontal direction is well analyzing powers can be written in terms of the anomalous approximated by a Gaussian distribution (σ ≈ 2.6 mm), jet magnetic moment of a proton ϰ ¼ 1.793, the unpolarized 12 2 with 1.2 × 10 atoms=cm in the center. Since the rf- pp scattering parameters ρðsÞ (forward real-to-imaginary transition efficiency exceeds 99.9%, the polarization, amplitude ratio), σ ðsÞ (total cross section), BðsÞ (the tot P ≈ 0.96, is defined by the strength (1.2 kG) of the holding jet nuclear slope), and hadronic single, r ¼ R þ iI , and 5 5 5 field magnet [10]. The atomic hydrogen spin direction is double, r ¼ R þ iI , spin-flip amplitude parameters: 2 2 2 reversed every 5–10 min. The recoil spectrometer is sketched in Fig. 1. For elastic pp pffiffiffiffiffi A ðtÞ scattering, the spectrometer geometry allows us to detect −t recoil protons with kinetic energy up to T ≈ 10–11 MeV, 0 0 0 0 2 ½ϰ ð1 − ρ δ Þ − 2ðI − δ R Þt =t − 2ðR − ρ I Þ C 5 C 5 c 5 5 i.e., to −t ¼ 2m T ∼ 0.02 GeV . To reconstruct the kinetic p R ¼ ; 2 2 ðt =tÞ − 2ðρ ˜ þ δ Þt =t þ 1 þ ρ ˜ energy of punch through protons (T > 7.8 MeV), signal c C c waveform shape analysis was carried out. ð3Þ A detailed description of the HJET data analysis is given in Ref. [14]. A crucial part of the analysis relates to an A ðtÞ NN accurate determination of the background event rate in every 0 0 0 0 0 2 −2ðR þδ I Þt =tþ2ðI þρ R Þ−ðρ ϰ −4R Þϰ t =2m Si detector as a function of the measured energy and the 2 C 2 c 2 2 5 c p ¼ : 2 2 spins of the jet and beam. Hence, to a subpercent level, spin ðt =tÞ −2ðρ ˜ þδ Þt =tþ1þρ ˜ c C c effects were properly treated in the background subtraction. ð4Þ Spin correlated asymmetries.—To measure the proton beam polarization, we studied the spin-correlated differ- 0 0 0 In Ref. [20], terms ϰ , ρ , ρ ˜, and t in Eqs. (3)–(4) ential cross section [16,17], appeared as ϰ, ρ, ρ, and t , respectively. For the HJET d σ measurements, −t ¼ 8πα=σ ≈ 0.0018 GeV and the c tot ∝ ½1 þ A ðtÞ sin φðP þ P Þþ A ðtÞsin φP P ; N j b NN j b dtdφ Coulomb phase is δ ¼ −α ln j0.8905ðB þ 8=Λ Þtj ∼ 0.02 [20]. ð1Þ Recently, it has been pointed out [22] that Eqs. (3)–(4) dependence on azimuthal angle φ. At HJET, sin φ ¼1 were derived in Ref. [20] with some simplifications. For the depending on right or left position of the Si detector relative increased precision of the HJET measurements, corrections to the beam. P are the jet and beam polarizations, j;b to A ðtÞ and A ðtÞ should be applied. Some of them have N NN been outlined in Ref. [23], in particular, (i) the difference between pp electromagnetic and hadronic form factors and (ii) an additional term ∼m =s in the single spin-flip electromagnetic amplitude. These corrections can be rep- resented by the following substitutions: 0 2 2 t ¼ t × ½1 þðr =3 − B=2 − ϰ=2m Þt; ð5Þ c c p p 0 2 2 2 2 2 ρ ¼ ρ þðr =3 − 4=Λ − ϰ=2m − ϰ =4m Þt ≈ ρ; ð6Þ p p p c ρ ˜ ¼ ρ − ð4=Λ − B=2Þt ; ð7Þ FIG. 1. A schematic view of the HJET recoil spectrometer consisting of eight silicon detectors with 12 vertically oriented strips 0 2 2 ϰ ¼ðϰ − 2m =sÞ=ð1 − μ t=4m Þ; ð8Þ p p p (readout channels) each. The distance between beams is ∼2 mm. 162001-2 PHYSICAL REVIEW LETTERS 123, 162001 (2019) Run15 (100 GeV) 2 2 where Λ ¼ 0.71 GeV , and r ¼ 0.875 fm (CODATA [24]) 0.8 1.2 is a proton charge radius. 0.7 In most measurements of ρ, the pp electromagnetic form 0.6 1.0 em factor F ðtÞ was approximated in data analysis by 0.5 2 −4 Blue beam Yellow beam Blue beam Yellow beam ð1 − t=Λ Þ derived from the electric form factor in dipole 2 2 2 2 0.8 χ = 76.2 / 86 χ = 74.3 / 86 χ = 88.1 / 86 χ = 72.1 / 86 ∼ ∼ 0.4 ∼ ∼ form [25]. Therefore, the value of ρ − ρ ≈ 0.002 should be α = 0.9643(52) α = 0.9548(51) α = 0.5470(43) α = 0.5768(42) 5 5 5 5 β = 0.0169(14) β = 0.0207(14) β = 0.0198(20) β = 0.0189(18) 5 5 5 5 interpreted as a systematic correction to be applied to the 2468 10 2468 10 value of ρ obtained from these experiments. This correction T = −t / 2m [MeV] T = −t / 2m [MeV] R p R Run17 (255 GeV) might be essential for the Regge pole fit of the unpolarized 1.2 0.7 data; however, it is completely negligible for this work. em The absorptive corrections to F ðtÞ, due to the initial 0.6 1.0 and final state hadronic interactions between the colliding 0.5 protons [22], are currently unavailable [26] and, conse- Blue beam Yellow beam Blue beam Yellow beam 0.8 2 2 2 2 χ = 57.4 / 74 χ = 96.7 / 74 χ = 87.6 / 86 χ = 84.3 / 86 quently, are not included in the fits to the analyzing powers. 0.4 ∼ ∼ ∼ ∼ α = 0.9382(46) α = 0.9315(45) α = 0.5334(29) α = 0.5426(28) 5 5 5 5 em em β = 0.0074(10) β = 0.0091(10) β = 0.0096(13) β = 0.0104(13) However, if they effectively modify F → F × ½1 þ 5 5 5 5 2468 10 2468 10 aðsÞt=t then the result of the fit using Eq. (3) should be T = −t / 2m [MeV] T = −t / 2m [MeV] p p R R corrected [23] by FIG. 2. Measured normalized asymmetries in RHIC Run 15 Δ R ¼ a ϰ=2; Δ I ¼ −a δ ϰ=2 ≈ 0; ð9Þ a 5 sf a 5 nf C (100 GeV) and Run 17 (255 GeV). The fit energy range is 1.9 < jet where “sf” and “nf” denote the spin-flip and non-flip T < 9.9 MeV for the 255 GeV a ðT Þ and 0.7 <T < R R R absorptive corrections, respectively. 9.9 MeV for the other graphs. The fit parameter α ˜ is defined pffiffiffi as α ˜ ¼hPiα . Analyzing power measurements at s ¼ 13.76 GeV 5 5 pffiffiffi and s ¼ 21.92 GeV.—Here we analyze HJET data acquired in two RHIC proton-proton runs: Run 15 Eq. (10). In the fits with ρ being a free parameter (100 GeV) [27] and Run 17 (255 GeV) [28]. About we obtained ρ ¼ −0.050 0.025 (100 GeV) and ρ ¼ 2 × 10 elastic pp events were selected at HJET in each −0.028 0.018 (255 GeV), values which agree with run. In the data analysis, the values of σ ðsÞ and ρðsÞ were tot unpolarized pp data to about 1 standard deviation. So, taken from the pp and pp ¯ data fit [29]. The slopes BðsÞ this test does not indicate any statistically significant were derived from Ref. [30]. The run specific conditions of discrepancy with the theoretical expectation (10). the measurements can be briefly summarized as Run 15: pffiffiffi To determine the hadronic spin-flip amplitude ratio r , s ¼ 13.76 GeV, ρ ¼ −0.079, σ ¼ 38.39 mb, B ¼ tot j;b pffiffiffi −2 eff we fit all four measured asymmetries a ðtÞ¼ P A ðt; r Þ j;b N 5 11.2 GeV , P ¼ 0.954; Run 17: s ¼ 21.92 GeV, N jet with unknown blue and yellow beam polarizations as free −2 eff ρ ¼ −0.009, σ ¼ 39.19 mb, B ¼ 11.6 GeV , P ¼ tot jet parameters. Nonzero values of r ¼ R þ iI were found, 5 5 5 eff 0.953; where P is the effective jet polarization after jet −3 systematic corrections. 100 GeV∶ R ¼ð−16.4 0.8 1.5 Þ × 10 ; ð11Þ 5 stat syst For visual control of consistency between the measured −3 j;b I ¼ð−5.3 2.9 4.7 Þ × 10 ; ð12Þ 5 stat syst single spin asymmetries a and theoretical expectations, it is convenient to use the normalized asymmetry −3 255 GeV∶ R ¼ð−7.9 0.5 0.8 Þ × 10 ; ð13Þ 5 stat syst a ðT Þ¼ a ðtÞ=A ðt; r ¼ 0Þ¼ Pα ð1 þ β t=t Þ; ð10Þ n R N N 5 5 5 c −3 I ¼ð19.4 2.5 2.5 Þ × 10 : ð14Þ 5 stat syst which is well approximated by a linear function of t with stat The correlation parameters between R and I are ρ ¼ 5 5 parameters α ðr Þ ≈ 1–2I =ϰ and β ðr Þ ≈−2R =ϰ. The 5 5 5 5 5 5 syst stat −0.884, ρ ¼ −0.868 (100 GeV) and ρ ¼ −0.882, measured β must be the same for jet and beam asymme- 5 syst tries. The maximum of A ðt; r ¼ 0) is about 0.045 at ρ ¼ 0.075 (255 GeV). The specified systematic errors N 5 T ¼ −t=2m ∼ 1.7 MeV (see Fig. 6). do not include the effects of uncertainties in the external R p j;b parameters (ρ, σ , B, and r ). For both beam energies, the Shown in Fig. 2, the experimental dependencies a ðT Þ tot p corresponding corrections to r can be approximated with are linear functions of T in good agreement with expect- jet sufficient accuracy by ations. For the 255 GeV a ðT Þ, the outlier points at T < 1.9 MeV (presumably due to interference of the −1 ΔR ¼ −0.11 × Δρ − ð0.0019 mb Þ × Δσ 5 tot magnetic field and inelastic background effects) were 2 −1 eliminated from the data analysis. þð0.0010 GeV Þ × ΔB − ð0.024 fm Þ × Δr ; An incorrect value of ρ used in the calculation of ð15Þ A ðt; r ¼ 0Þ may result in a false nonlinearity of N 5 162001-3 jet jet a T a T n n R R beam beam a T a T n R n R PHYSICAL REVIEW LETTERS 123, 162001 (2019) 13.76 21.92 13.76 21.92 100 GeV 255 GeV P (Pomeron) P (Pomeron) 10 10 R (a , f ) 2 2 1 1 −R (a , f ) 2 2 0 −1 −1 −R (ρ, ω) 2 2 10 −R (ρ, ω) 10 χ = 49.4 / 41 χ = 41.8 / 44 2 2 02 46 8 10 02 46 8 10 10 10 10 10 s [GeV] s [GeV] T = −t / 2m [MeV] T = −t / 2m [MeV] R p R p FIG. 4. The Reggeon contributions RðsÞ¼ R ðsÞþ iI ðsÞ to R R FIG. 3. Double spin asymmetry a measured at HJET. The fit NN Eq. (22) defined by the AU-Lγ ¼ 2ðTÞ fit of Ref. [29]. used values of P and P from the single spin analysis. j b −1 Here though, we use functions RðsÞ as shown in Fig. 4 ΔI ¼ 0.86 × Δρ − ð0.0085 mb Þ × Δσ 5 tot where [29] the Pomeron is represented by a Froissaron − ð0.0011 GeV Þ × ΔB: ð16Þ parametrization Assessing the values of the external parameters is beyond 2 2 2 PðsÞ ∝ πf ln s=4m þ ið1 þ f ln s=4m Þ; ð24Þ F p F p the scope of this work. For the double spin asymmetry a (Fig. 3), the jet spin þ with f ¼ 0.0090 and the R intercepts are α ¼ 0.65 NN F R correlated systematic uncertainties cancel in the ratio and α − ¼ 0.45. In the HJET measurements, jImr j (i.e., both jImr j and a ðT Þ=a ðT Þ. This statement was verified by 5;2 5 NN R R jImr j) grew with energy indicating that there is a notice- comparing the ratio for data with and without background able Pomeron contribution to both single and double subtraction. Therefore, for the experimental determination spin-flip amplitudes. Moreover, an increasing jr j suggests of the double spin analyzing power A ðtÞ it is convenient NN that the Pomeron component dominates in r already at to use the following relation: HJET energies. A ðt; r Þ a ðtÞ N 5 NN Because of a limited number of the experimental spin- A ðtÞ¼ × : ð17Þ NN hP i a ðtÞ flip entries and following Ref. [32], we expanded r ðsÞ 5;2 using the above nonflip functions RðsÞ scaled by real For r and hP i taken from the single spin fit, the 5 b (because of analyticity in s) spin-flip factors f experimental uncertainty in Eq. (17) is strongly dominated 5;2 by the statistical uncertainties of a ðT Þ: NN R σ ðsÞ × r ðsÞ¼ f RðsÞ: ð25Þ tot 5;2 5;2 −3 R¼P;R 100 GeV∶ R ¼ð−3.65 0.28 Þ × 10 ; ð18Þ 2 stat In a combined fit of the 100 and 255 GeV HJET data, −3 I ¼ð−0.10 0.12 Þ × 10 ; ð19Þ 2 stat we find −3 255 GeV∶ R ¼ð−2.15 0.20 Þ × 10 ; ð20Þ f ¼ 0.045 0.002 0.003 ; ð26Þ 2 stat stat syst −3 I ¼ð−0.35 0.07 Þ × 10 : ð21Þ 2 stat f ¼ −0.032 0.007 0.014 ; ð27Þ stat syst stat f ¼ 0.622 0.023 0.024 : ð28Þ The correlation parameters are ρ ¼ 0.860 (100 GeV) and stat syst stat ρ ¼ 0.808 (255 GeV). Obviously, nonzero values of jr j Similarly, for the double spin-flip amplitude expansion are well established for both beam energies. we obtain Energy dependence of r ðsÞ and r ðsÞ.—For unpolarized 5 2 f ¼ −0.0020 0.0002 ; ð29Þ protons, elastic pp (pp ¯ ) scattering can be described at low stat −t with a Pomeron P and the subleading C ¼1 Regge þ − f ¼ 0.0162 0.0007 ; ð30Þ stat poles for I ¼ 0,1, encoded by R for (f , a ) and R for 2 2 (ω, ρ) [31]. In this approach, the unpolarized pp amplitude f ¼ 0.0297 0.0041 : ð31Þ stat may be presented as a sum of Reggeon contributions X At high energies where the contributions R decay, the σ ðsÞ × ½ρðsÞþ i¼ RðsÞ: ð22Þ tot model (25) used gives the following spin-flip parameters: R¼P;R r ðsÞ¼ f × ½ρðsÞþ i: ð32Þ 5;2 5;2 A basic simple pole approximation assumes In terms of the Pomeron anomalous magnetic moment −iπα 2 α −1 R R RðsÞ ∝ ð1 þ ζ e Þðs=4m Þ ð23Þ R p introduced in Ref. [33], the fit yields M ¼ 2f ¼ with signature factors ζ ¼1, ζ ¼þ1 and “standard” 0.09 0.01. The provisional value of r ∼ 0.03 [20] P P intercepts α ¼ 0.5 and α ¼ 1.1. derived from πp data [34] at 6–14 GeV=c can, using R P 162001-4 a T × 10 NN R a T × 10 NN R R (s) [mb] I (s) [mb] R PHYSICAL REVIEW LETTERS 123, 162001 (2019) 1. HJET s=13.76 GeV 3. HJET (Fig. 4 ) → s=200 GeV 0.05 4 2. HJET s=21.92 GeV 4. HJET (Eq.(23)) → s=200 GeV A (t) A (t) 0.004 N NN 5. STAR s=200 GeV s=13.76 GeV 0.0 5 4 s=21.92 GeV 0.04 5.5 6.0 6.5 7.0 −0.5 40 3 Re r × 10 0.002 4 s=13.76 GeV −1.0 20 2 0.03 s=21.92 GeV −1.5 1 3 0.000 −20 −10 010 −4 −3 −2 −1 0 0.005 0.010 0.015 0.005 0.010 0.015 3 3 Re r × 10 Re r × 10 2 2 5 2 −t GeV −t GeV FIG. 5. Δχ ¼ 1 correlation (stat þ syst) contours for r and r . 5 2 FIG. 6. Elastic pp analyzing powers A ðtÞ and A ðtÞ N NN Filled ellipses mean statistical error only. The HJET extrapola- measured in this work. The filled areas correspond to tions to 200 GeV are labeled 3 and 4. The STAR Collaboration σ .For A ðtÞ, the dashed lines refer to the expected statþsyst N result [11] for r was changed by us using Eqs. (5)–(8). analyzing powers if r ¼ 0. assumption (25), be related to f ≈ r in reasonable 5 ðsfÞ ðsfÞ obtained f ¼ 0.50.5 and α − ¼ 0.620.11. However, F R agreement with Eq. (26). ðsfÞ P ↑ ˜ f strongly depends on the α þ selection. The fit of the The value of f ¼ 0.10 0.01 [32] estimated from p C R Pomeron spin-flip intercept [using a simple pole for PðsÞ] data is noticeably larger than in Eq. (26). However, this is stable in a wide range of 0.3 < α þ < 0.8.Itgives estimate required a model dependent conversion from R proton-nucleus asymmetries to proton-proton r and, also, ðsfÞ ðsfÞ Δ ¼ α − 1 ¼ 0.117 0.031 ; ð33Þ P P statþsyst was strongly based on unpublished experimental results þ0.012 [35] with undetermined systematic uncertainties. which agrees with the unpolarized Δ ¼ 0.096 [36], −0.009 ðsfÞ The r ðsÞ and r ðsÞ dependencies on the beam energy 5 2 and α − ¼ 0.65 0.11. are illustrated in Fig. 5 where the extrapolations to pffiffiffi Summary.—In RHIC polarized proton runs 2015 s ¼ 200 GeV, based on the Froissaron parametrization (100 GeV) and 2017 (255 GeV), we have measured elastic (24), are labeled “3.” Consistency between the extrapola- pp analyzing powers in the CNI region 0.0013 < −t< tion of r and the STAR Collaboration measurement [11] 5 −4 0.018 GeV with accuracy jδA ðtÞj ∼ 2 × 10 [13] as N;NN was observed, though the STAR experimental uncertainties shown in Fig. 6. To graph A ðtÞ, we substituted the fitted are not inconsiderable. values of r from Eqs. (11)–(14) in Eq. (3), taking into It is interesting to note that the values of r and r , when 5 2 pffiffiffi account statistical and systematic uncertainties and projected from s 14–22 to 200 GeV, have smaller their covariances. In fact, this is equivalent to determining uncertainties than those of the measurements. This may A ðtÞ directly from the linear fit of the normalized be explained by decay of the R pole contributions at large asymmetries a ðT Þ. Thus, the result is not greatly affected n R s and by using functions RðsÞ that are too tightly con- by absorptive corrections, nor by possible variations in ρ, strained (which, for the selected model, is a good approxi- σ , B, and r . tot p mation in the energy range considered). However, many The accuracy achieved in the determination of A ðtÞ models [31] are used to parametrize σ ðsÞ and ρðsÞ which tot allows one to use a higher density unpolarized hydrogen may render RðsÞ more uncertain. jet target in a high precision absolute polarimeter, e.g., To estimate the dependence of a Reggeon analysis on a at a future EIC [15]. For a 30-fold increase in jet density, the particular model, we also fitted the HJET data using a sum expected statistical and systematic uncertainties of the of simple poles (23). These extrapolations of r and r to 5 2 stat pffiffiffi polarization measurement would be δ P ≲ 1%=h and s ¼ 200 GeV are labeled “4” in Fig. 5. Since, at HJET syst δ P=P ≲ 1%. energies, the double spin-flip amplitude is dominated by an The hadronic spin-flip amplitude ratios r and r were 5 2 R contribution, the r projection to 200 GeV is strongly reliably isolated at both energies. Applying the corrections affected by a variation of α þ. indicated in Eqs. (5)–(8) to the expression [20] for A ðtÞ The expansions (25) fit the measurements with statistically resulted in a change of the measured r by about the size of insignificant discrepancies χ ¼ 2.2 [Eqs. (26)–(28)]and the experimental uncertainty. The absorptive corrections χ ¼ 1.6 [Eqs. (29)–(31)]for ndf ¼ 1 showing consistency were not included in the data analysis, but, if they become between the experimental data and Eq. (25). available, a simple correction to Re r could be applied. To evaluate a possible difference between single spin-flip Measurements at two energies permitted a Regge pole (sf) and nonflip functions PðsÞ, we determined the ratio analysis of elastic pp scattering to be extended to the spin ðsfÞ ðsfÞ f ¼ f =f in a combined analysis including the dependent case. A Reggeon expansion of the spin-flip F F STAR Collaboration result. For a fixed α ¼ 0.65,we parameters r ðsÞ and r ðsÞ indicated that Pomeron single R 5 2 162001-5 Im r × 10 Im r × 10 Im r × 10 2 PHYSICAL REVIEW LETTERS 123, 162001 (2019) [15] A. Accardi et al., Eur. Phys. J. A 52, 268 (2016). and double spin-flip couplings were well determined and [16] J. Ashkin, E. Leader, M. L. Marshak, J. B. Roberts, J. Soffer, found to be significantly different from zero. However, the and G. H. Thomas, AIP Conf. Proc. 42, 142 (1978). absorptive corrections when available, might require a re- [17] E. Leader, in Spin in Particle Physics (Cambridge analysis of the expansion. University Press, Cambridge, England, 2001), p. 119. [18] B. Z. Kopeliovich and L. I. Lapidus, Yad. Fiz. 19, 218 We thank the Collider Accelerator Department and the (1974) [Sov. J. Nucl. Phys. 19, 114 (1974)], JINR-P2-72-34 RHIC/AGS Operation Groups. We also would like to thank [CERN-Trans-73-7]. A. Bazilevsky, B. Z. Kopeliovich, and M. Krelina for useful [19] N. H. Buttimore, E. Gotsman, and E. Leader, Phys. Rev. D discussions. B. Z. Kopeliovich read the manuscript and 18, 694 (1978); 35, 407 (1987). made valuable comments. This work was supported by [20] N. H. Buttimore, B. Z. Kopeliovich, E. Leader, J. Soffer, and Brookhaven Science Associates, LLC under Contract T. L. Trueman, Phys. Rev. D 59, 114010 (1999). No. DE-AC02-98CH 10886 with the U.S. Department of [21] G. Bunce, N. Saito, J. Soffer, and W. Vogelsang, Ann. Rev. Energy. Funding was also provided from the RIKEN BNL Nucl. Part. Sci. 50, 525 (2000); E. C. Aschenauer et al., Research Center. N. H. B. is grateful for partial support arXiv:1501.01220. from the University of Dublin. [22] M. Krelina and B. Z. Kopeliovich, Proc. Sci., SPIN2018 (2019) 033. [23] A. A. Poblaguev, arXiv:1910.02563. [24] P. J. Mohr, D. B. Newell, and B. N. Taylor, Rev. Mod. Phys. poblaguev@bnl.gov 88, 035009 (2016). [1] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, [25] L. H. Chan, K. W. Chen, J. R. Dunning, N. F. 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http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.pngPhysical Review LettersUnpaywallhttp://www.deepdyve.com/lp/unpaywall/precision-small-scattering-angle-measurements-of-elastic-proton-proton-4YVhD7T1GV
Precision Small Scattering Angle Measurements of Elastic Proton-Proton Single and Double Spin Analyzing Powers at the RHIC Hydrogen Jet Polarimeter
Precision Small Scattering Angle Measurements of Elastic Proton-Proton Single and Double Spin Analyzing Powers at the RHIC Hydrogen Jet Polarimeter
Abstract
PHYSICAL REVIEW LETTERS 123, 162001 (2019) Precision Small Scattering Angle Measurements of Elastic Proton-Proton Single and Double Spin Analyzing Powers at the RHIC Hydrogen Jet Polarimeter A. A. Poblaguev , A. Zelenski, E. Aschenauer, G. Atoian, K. O. Eyser, H. Huang, Y. Makdisi, and W. B. Schmidke Brookhaven National Laboratory, Upton, New York 11973, USA I. Alekseev and D. Svirida Alikhanov Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia N. H. Buttimore School of Mathematics, Trinity College, Dublin 2, Ireland (Received 29 May 2019; revised manuscript received 26 August 2019; published 16 October 2019) The Polarized Atomic Hydrogen Gas Jet Target polarimeter is employed by the Relativistic Heavy Ion Collider (RHIC) to measure the absolute polarization of each colliding proton beam. Polarimeter detectors and data acquisition were upgraded in 2015 to increase solid angle, energy range, and energy resolution. These upgrades and advanced systematic error analysis along with improved beam intensity and polarization in RHIC runs 2015 (E ¼ 100 GeV) and 2017 (255 GeV) allowed us to greatly reduce beam the statistical and systematic uncertainties for elastic spin asymmetries, A ðtÞ and A ðtÞ, in the Coulomb- N NN nuclear interference momentum transfer range 0.0013 < −t<
PHYSICAL REVIEW LETTERS 123, 162001 (2019) Precision Small Scattering Angle Measurements of Elastic Proton-Proton Single and Double Spin Analyzing Powers at the RHIC Hydrogen Jet Polarimeter A. A. Poblaguev , A. Zelenski, E. Aschenauer, G. Atoian, K. O. Eyser, H. Huang, Y. Makdisi, and W. B. Schmidke Brookhaven National Laboratory, Upton, New York 11973, USA I. Alekseev and D. Svirida Alikhanov Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia N. H. Buttimore School of Mathematics, Trinity College, Dublin 2, Ireland (Received 29 May 2019; revised manuscript received 26 August 2019; published 16 October 2019) The Polarized Atomic Hydrogen Gas Jet Target polarimeter is employed by the Relativistic Heavy Ion Collider (RHIC) to measure the absolute polarization of each colliding proton beam. Polarimeter detectors and data acquisition were upgraded in 2015 to increase solid angle, energy range, and energy resolution. These upgrades and advanced systematic error analysis along with improved beam intensity and polarization in RHIC runs 2015 (E ¼ 100 GeV) and 2017 (255 GeV) allowed us to greatly reduce beam the statistical and systematic uncertainties for elastic spin asymmetries, A ðtÞ and A ðtÞ, in the Coulomb- N NN nuclear interference momentum transfer range 0.0013 < −t< 0.018 GeV . For the first time hadronic single spin-flip r and double spin-flip r amplitude parameters were reliably isolated at these energies and 5 2 momentum transfers. Measurements at two beam energies enable a separation of Pomeron and Regge pole contributions to r ðsÞ and r ðsÞ, indicating that the spin component may persist at high energies. 5 2 DOI: 10.1103/PhysRevLett.123.162001 Introduction.—Study of the spin-averaged elastic pp data [8–11] was insufficient to identify a Pomeron con- hadronic amplitude at high energies has a more than 50 year tribution, if any, to the pp spin-dependent amplitudes. history [1] and is continuing at the Large Hadron Collider. In this Letter, we report new measurements of the single An essential contribution to this study relates to forward spin A ðtÞ and double spin A ðtÞ analyzing powers in the N NN scattering for which the optical theorem and Coulomb- small angle elastic collision of RHIC’s polarized proton beams with Polarized Atomic Hydrogen Gas Jet Target nuclear interference (CNI) provide an opportunity to pffiffiffi separate the real and imaginary parts of an amplitude. (HJET) [12] at s ¼ 13.76 and 21.92 GeV. The precision Regge theory, based on the analyticity of a scattering has improved significantly by comparison with previous amplitude, is a recognized method of understanding the HJET publications [9,10] and this has allowed us to not only energy dependence of amplitudes [2]. isolate hadronic spin-flip amplitudes but also to incorporate An explanation of the unexpected discovery in the spin dependence in a Regge pole analysis. It appears that seventies of a growing pp cross section at high energies forward elastic pp scattering has nonvanishing single and [3] was found [4] in the Pomeron concept, which is now double spin-flip hadronic amplitudes at high energy where associated with the exchange of nonperturbative QCD the Pomeron dominates. The results of the analysis facilitate gluons [5]. Currently, the Pomeron and Regge pole picture extrapolation of the measured A ðtÞ to a wide range of of unpolarized elastic pp scattering is commonly consid- energies, essential for CNI polarimetry. Additional mea- pffiffiffi surements at the RHIC injection energy (E ¼ 24 GeV) ered as well established in the s ¼ 5 GeV–13 TeV c:m: beam might yield an improved Reggeon fit and the possibility [13] energy range [1], though some new results, e.g., from the of experimentally resolving the Odderon issue [7]. TOTEM experiment [6], call for a revision [7]. However, The HJET provides an absolute proton beam polarization the accuracy of existing polarized high energy experimental measurement averaged across a beam. Typically, hP i ∼ beam 55 2.0 0.3 Þ% [14] for an 8-h RHIC store. The stat syst achieved accuracy satisfies the requirements of hadron Published by the American Physical Society under the terms of polarimetry for planned and future accelerators such as the the Creative Commons Attribution 4.0 International license. Electron Ion Collider (EIC) [15]. This work is based on Further distribution of this work must maintain attribution to the technique of high energy beam polarization measure- the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP . ment developed at RHIC. The methodology can be 0031-9007=19=123(16)=162001(6) 162001-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 123, 162001 (2019) recommended for EIC including a possible extension of it respectively. To determine analyzing powers A ðtÞ and using other polarized nuclei such as He. A ðtÞ, the single spin (jet and beam) and double spin NN HJET polarimeter at RHIC.—The HJET [12] acts like a asymmetries fixed target that measures absolute polarization of 24– a ¼A jP j;a ¼A jP j;a ¼A jP P j; ð2Þ N j N N b NN NN j b 255 GeV proton beams at RHIC. It consists of three main N components: an atomic beam source, a Breit-Rabi polarimeter were derived [13] from the selected elastic event counts to measure atomic hydrogen polarization, and a recoil ð↑↓Þðþ−Þ N discriminated by the right or left (RL) detector RL spectrometer to determine the beam and vertically polarized location and the beam (↑↓) and jet (þ−) spin directions. atomic hydrogen target (the jet) spin correlated asymmetries For CNI elastic pp scattering at high energies, the of the detected recoil protons. Polarizations of both RHIC theoretical basis for an experimental parametrization of beams (alternating spin up or down bunches), so-called blue the analyzing powers was introduced in Refs. [18,19] and and yellow, are measured concurrently and continuously. updated [20] for the RHIC spin program [21]. The The jet density profile in the horizontal direction is well analyzing powers can be written in terms of the anomalous approximated by a Gaussian distribution (σ ≈ 2.6 mm), jet magnetic moment of a proton ϰ ¼ 1.793, the unpolarized 12 2 with 1.2 × 10 atoms=cm in the center. Since the rf- pp scattering parameters ρðsÞ (forward real-to-imaginary transition efficiency exceeds 99.9%, the polarization, amplitude ratio), σ ðsÞ (total cross section), BðsÞ (the tot P ≈ 0.96, is defined by the strength (1.2 kG) of the holding jet nuclear slope), and hadronic single, r ¼ R þ iI , and 5 5 5 field magnet [10]. The atomic hydrogen spin direction is double, r ¼ R þ iI , spin-flip amplitude parameters: 2 2 2 reversed every 5–10 min. The recoil spectrometer is sketched in Fig. 1. For elastic pp pffiffiffiffiffi A ðtÞ scattering, the spectrometer geometry allows us to detect −t recoil protons with kinetic energy up to T ≈ 10–11 MeV, 0 0 0 0 2 ½ϰ ð1 − ρ δ Þ − 2ðI − δ R Þt =t − 2ðR − ρ I Þ C 5 C 5 c 5 5 i.e., to −t ¼ 2m T ∼ 0.02 GeV . To reconstruct the kinetic p R ¼ ; 2 2 ðt =tÞ − 2ðρ ˜ þ δ Þt =t þ 1 þ ρ ˜ energy of punch through protons (T > 7.8 MeV), signal c C c waveform shape analysis was carried out. ð3Þ A detailed description of the HJET data analysis is given in Ref. [14]. A crucial part of the analysis relates to an A ðtÞ NN accurate determination of the background event rate in every 0 0 0 0 0 2 −2ðR þδ I Þt =tþ2ðI þρ R Þ−ðρ ϰ −4R Þϰ t =2m Si detector as a function of the measured energy and the 2 C 2 c 2 2 5 c p ¼ : 2 2 spins of the jet and beam. Hence, to a subpercent level, spin ðt =tÞ −2ðρ ˜ þδ Þt =tþ1þρ ˜ c C c effects were properly treated in the background subtraction. ð4Þ Spin correlated asymmetries.—To measure the proton beam polarization, we studied the spin-correlated differ- 0 0 0 In Ref. [20], terms ϰ , ρ , ρ ˜, and t in Eqs. (3)–(4) ential cross section [16,17], appeared as ϰ, ρ, ρ, and t , respectively. For the HJET d σ measurements, −t ¼ 8πα=σ ≈ 0.0018 GeV and the c tot ∝ ½1 þ A ðtÞ sin φðP þ P Þþ A ðtÞsin φP P ; N j b NN j b dtdφ Coulomb phase is δ ¼ −α ln j0.8905ðB þ 8=Λ Þtj ∼ 0.02 [20]. ð1Þ Recently, it has been pointed out [22] that Eqs. (3)–(4) dependence on azimuthal angle φ. At HJET, sin φ ¼1 were derived in Ref. [20] with some simplifications. For the depending on right or left position of the Si detector relative increased precision of the HJET measurements, corrections to the beam. P are the jet and beam polarizations, j;b to A ðtÞ and A ðtÞ should be applied. Some of them have N NN been outlined in Ref. [23], in particular, (i) the difference between pp electromagnetic and hadronic form factors and (ii) an additional term ∼m =s in the single spin-flip electromagnetic amplitude. These corrections can be rep- resented by the following substitutions: 0 2 2 t ¼ t × ½1 þðr =3 − B=2 − ϰ=2m Þt; ð5Þ c c p p 0 2 2 2 2 2 ρ ¼ ρ þðr =3 − 4=Λ − ϰ=2m − ϰ =4m Þt ≈ ρ; ð6Þ p p p c ρ ˜ ¼ ρ − ð4=Λ − B=2Þt ; ð7Þ FIG. 1. A schematic view of the HJET recoil spectrometer consisting of eight silicon detectors with 12 vertically oriented strips 0 2 2 ϰ ¼ðϰ − 2m =sÞ=ð1 − μ t=4m Þ; ð8Þ p p p (readout channels) each. The distance between beams is ∼2 mm. 162001-2 PHYSICAL REVIEW LETTERS 123, 162001 (2019) Run15 (100 GeV) 2 2 where Λ ¼ 0.71 GeV , and r ¼ 0.875 fm (CODATA [24]) 0.8 1.2 is a proton charge radius. 0.7 In most measurements of ρ, the pp electromagnetic form 0.6 1.0 em factor F ðtÞ was approximated in data analysis by 0.5 2 −4 Blue beam Yellow beam Blue beam Yellow beam ð1 − t=Λ Þ derived from the electric form factor in dipole 2 2 2 2 0.8 χ = 76.2 / 86 χ = 74.3 / 86 χ = 88.1 / 86 χ = 72.1 / 86 ∼ ∼ 0.4 ∼ ∼ form [25]. Therefore, the value of ρ − ρ ≈ 0.002 should be α = 0.9643(52) α = 0.9548(51) α = 0.5470(43) α = 0.5768(42) 5 5 5 5 β = 0.0169(14) β = 0.0207(14) β = 0.0198(20) β = 0.0189(18) 5 5 5 5 interpreted as a systematic correction to be applied to the 2468 10 2468 10 value of ρ obtained from these experiments. This correction T = −t / 2m [MeV] T = −t / 2m [MeV] R p R Run17 (255 GeV) might be essential for the Regge pole fit of the unpolarized 1.2 0.7 data; however, it is completely negligible for this work. em The absorptive corrections to F ðtÞ, due to the initial 0.6 1.0 and final state hadronic interactions between the colliding 0.5 protons [22], are currently unavailable [26] and, conse- Blue beam Yellow beam Blue beam Yellow beam 0.8 2 2 2 2 χ = 57.4 / 74 χ = 96.7 / 74 χ = 87.6 / 86 χ = 84.3 / 86 quently, are not included in the fits to the analyzing powers. 0.4 ∼ ∼ ∼ ∼ α = 0.9382(46) α = 0.9315(45) α = 0.5334(29) α = 0.5426(28) 5 5 5 5 em em β = 0.0074(10) β = 0.0091(10) β = 0.0096(13) β = 0.0104(13) However, if they effectively modify F → F × ½1 þ 5 5 5 5 2468 10 2468 10 aðsÞt=t then the result of the fit using Eq. (3) should be T = −t / 2m [MeV] T = −t / 2m [MeV] p p R R corrected [23] by FIG. 2. Measured normalized asymmetries in RHIC Run 15 Δ R ¼ a ϰ=2; Δ I ¼ −a δ ϰ=2 ≈ 0; ð9Þ a 5 sf a 5 nf C (100 GeV) and Run 17 (255 GeV). The fit energy range is 1.9 < jet where “sf” and “nf” denote the spin-flip and non-flip T < 9.9 MeV for the 255 GeV a ðT Þ and 0.7 <T < R R R absorptive corrections, respectively. 9.9 MeV for the other graphs. The fit parameter α ˜ is defined pffiffiffi as α ˜ ¼hPiα . Analyzing power measurements at s ¼ 13.76 GeV 5 5 pffiffiffi and s ¼ 21.92 GeV.—Here we analyze HJET data acquired in two RHIC proton-proton runs: Run 15 Eq. (10). In the fits with ρ being a free parameter (100 GeV) [27] and Run 17 (255 GeV) [28]. About we obtained ρ ¼ −0.050 0.025 (100 GeV) and ρ ¼ 2 × 10 elastic pp events were selected at HJET in each −0.028 0.018 (255 GeV), values which agree with run. In the data analysis, the values of σ ðsÞ and ρðsÞ were tot unpolarized pp data to about 1 standard deviation. So, taken from the pp and pp ¯ data fit [29]. The slopes BðsÞ this test does not indicate any statistically significant were derived from Ref. [30]. The run specific conditions of discrepancy with the theoretical expectation (10). the measurements can be briefly summarized as Run 15: pffiffiffi To determine the hadronic spin-flip amplitude ratio r , s ¼ 13.76 GeV, ρ ¼ −0.079, σ ¼ 38.39 mb, B ¼ tot j;b pffiffiffi −2 eff we fit all four measured asymmetries a ðtÞ¼ P A ðt; r Þ j;b N 5 11.2 GeV , P ¼ 0.954; Run 17: s ¼ 21.92 GeV, N jet with unknown blue and yellow beam polarizations as free −2 eff ρ ¼ −0.009, σ ¼ 39.19 mb, B ¼ 11.6 GeV , P ¼ tot jet parameters. Nonzero values of r ¼ R þ iI were found, 5 5 5 eff 0.953; where P is the effective jet polarization after jet −3 systematic corrections. 100 GeV∶ R ¼ð−16.4 0.8 1.5 Þ × 10 ; ð11Þ 5 stat syst For visual control of consistency between the measured −3 j;b I ¼ð−5.3 2.9 4.7 Þ × 10 ; ð12Þ 5 stat syst single spin asymmetries a and theoretical expectations, it is convenient to use the normalized asymmetry −3 255 GeV∶ R ¼ð−7.9 0.5 0.8 Þ × 10 ; ð13Þ 5 stat syst a ðT Þ¼ a ðtÞ=A ðt; r ¼ 0Þ¼ Pα ð1 þ β t=t Þ; ð10Þ n R N N 5 5 5 c −3 I ¼ð19.4 2.5 2.5 Þ × 10 : ð14Þ 5 stat syst which is well approximated by a linear function of t with stat The correlation parameters between R and I are ρ ¼ 5 5 parameters α ðr Þ ≈ 1–2I =ϰ and β ðr Þ ≈−2R =ϰ. The 5 5 5 5 5 5 syst stat −0.884, ρ ¼ −0.868 (100 GeV) and ρ ¼ −0.882, measured β must be the same for jet and beam asymme- 5 syst tries. The maximum of A ðt; r ¼ 0) is about 0.045 at ρ ¼ 0.075 (255 GeV). The specified systematic errors N 5 T ¼ −t=2m ∼ 1.7 MeV (see Fig. 6). do not include the effects of uncertainties in the external R p j;b parameters (ρ, σ , B, and r ). For both beam energies, the Shown in Fig. 2, the experimental dependencies a ðT Þ tot p corresponding corrections to r can be approximated with are linear functions of T in good agreement with expect- jet sufficient accuracy by ations. For the 255 GeV a ðT Þ, the outlier points at T < 1.9 MeV (presumably due to interference of the −1 ΔR ¼ −0.11 × Δρ − ð0.0019 mb Þ × Δσ 5 tot magnetic field and inelastic background effects) were 2 −1 eliminated from the data analysis. þð0.0010 GeV Þ × ΔB − ð0.024 fm Þ × Δr ; An incorrect value of ρ used in the calculation of ð15Þ A ðt; r ¼ 0Þ may result in a false nonlinearity of N 5 162001-3 jet jet a T a T n n R R beam beam a T a T n R n R PHYSICAL REVIEW LETTERS 123, 162001 (2019) 13.76 21.92 13.76 21.92 100 GeV 255 GeV P (Pomeron) P (Pomeron) 10 10 R (a , f ) 2 2 1 1 −R (a , f ) 2 2 0 −1 −1 −R (ρ, ω) 2 2 10 −R (ρ, ω) 10 χ = 49.4 / 41 χ = 41.8 / 44 2 2 02 46 8 10 02 46 8 10 10 10 10 10 s [GeV] s [GeV] T = −t / 2m [MeV] T = −t / 2m [MeV] R p R p FIG. 4. The Reggeon contributions RðsÞ¼ R ðsÞþ iI ðsÞ to R R FIG. 3. Double spin asymmetry a measured at HJET. The fit NN Eq. (22) defined by the AU-Lγ ¼ 2ðTÞ fit of Ref. [29]. used values of P and P from the single spin analysis. j b −1 Here though, we use functions RðsÞ as shown in Fig. 4 ΔI ¼ 0.86 × Δρ − ð0.0085 mb Þ × Δσ 5 tot where [29] the Pomeron is represented by a Froissaron − ð0.0011 GeV Þ × ΔB: ð16Þ parametrization Assessing the values of the external parameters is beyond 2 2 2 PðsÞ ∝ πf ln s=4m þ ið1 þ f ln s=4m Þ; ð24Þ F p F p the scope of this work. For the double spin asymmetry a (Fig. 3), the jet spin þ with f ¼ 0.0090 and the R intercepts are α ¼ 0.65 NN F R correlated systematic uncertainties cancel in the ratio and α − ¼ 0.45. In the HJET measurements, jImr j (i.e., both jImr j and a ðT Þ=a ðT Þ. This statement was verified by 5;2 5 NN R R jImr j) grew with energy indicating that there is a notice- comparing the ratio for data with and without background able Pomeron contribution to both single and double subtraction. Therefore, for the experimental determination spin-flip amplitudes. Moreover, an increasing jr j suggests of the double spin analyzing power A ðtÞ it is convenient NN that the Pomeron component dominates in r already at to use the following relation: HJET energies. A ðt; r Þ a ðtÞ N 5 NN Because of a limited number of the experimental spin- A ðtÞ¼ × : ð17Þ NN hP i a ðtÞ flip entries and following Ref. [32], we expanded r ðsÞ 5;2 using the above nonflip functions RðsÞ scaled by real For r and hP i taken from the single spin fit, the 5 b (because of analyticity in s) spin-flip factors f experimental uncertainty in Eq. (17) is strongly dominated 5;2 by the statistical uncertainties of a ðT Þ: NN R σ ðsÞ × r ðsÞ¼ f RðsÞ: ð25Þ tot 5;2 5;2 −3 R¼P;R 100 GeV∶ R ¼ð−3.65 0.28 Þ × 10 ; ð18Þ 2 stat In a combined fit of the 100 and 255 GeV HJET data, −3 I ¼ð−0.10 0.12 Þ × 10 ; ð19Þ 2 stat we find −3 255 GeV∶ R ¼ð−2.15 0.20 Þ × 10 ; ð20Þ f ¼ 0.045 0.002 0.003 ; ð26Þ 2 stat stat syst −3 I ¼ð−0.35 0.07 Þ × 10 : ð21Þ 2 stat f ¼ −0.032 0.007 0.014 ; ð27Þ stat syst stat f ¼ 0.622 0.023 0.024 : ð28Þ The correlation parameters are ρ ¼ 0.860 (100 GeV) and stat syst stat ρ ¼ 0.808 (255 GeV). Obviously, nonzero values of jr j Similarly, for the double spin-flip amplitude expansion are well established for both beam energies. we obtain Energy dependence of r ðsÞ and r ðsÞ.—For unpolarized 5 2 f ¼ −0.0020 0.0002 ; ð29Þ protons, elastic pp (pp ¯ ) scattering can be described at low stat −t with a Pomeron P and the subleading C ¼1 Regge þ − f ¼ 0.0162 0.0007 ; ð30Þ stat poles for I ¼ 0,1, encoded by R for (f , a ) and R for 2 2 (ω, ρ) [31]. In this approach, the unpolarized pp amplitude f ¼ 0.0297 0.0041 : ð31Þ stat may be presented as a sum of Reggeon contributions X At high energies where the contributions R decay, the σ ðsÞ × ½ρðsÞþ i¼ RðsÞ: ð22Þ tot model (25) used gives the following spin-flip parameters: R¼P;R r ðsÞ¼ f × ½ρðsÞþ i: ð32Þ 5;2 5;2 A basic simple pole approximation assumes In terms of the Pomeron anomalous magnetic moment −iπα 2 α −1 R R RðsÞ ∝ ð1 þ ζ e Þðs=4m Þ ð23Þ R p introduced in Ref. [33], the fit yields M ¼ 2f ¼ with signature factors ζ ¼1, ζ ¼þ1 and “standard” 0.09 0.01. The provisional value of r ∼ 0.03 [20] P P intercepts α ¼ 0.5 and α ¼ 1.1. derived from πp data [34] at 6–14 GeV=c can, using R P 162001-4 a T × 10 NN R a T × 10 NN R R (s) [mb] I (s) [mb] R PHYSICAL REVIEW LETTERS 123, 162001 (2019) 1. HJET s=13.76 GeV 3. HJET (Fig. 4 ) → s=200 GeV 0.05 4 2. HJET s=21.92 GeV 4. HJET (Eq.(23)) → s=200 GeV A (t) A (t) 0.004 N NN 5. STAR s=200 GeV s=13.76 GeV 0.0 5 4 s=21.92 GeV 0.04 5.5 6.0 6.5 7.0 −0.5 40 3 Re r × 10 0.002 4 s=13.76 GeV −1.0 20 2 0.03 s=21.92 GeV −1.5 1 3 0.000 −20 −10 010 −4 −3 −2 −1 0 0.005 0.010 0.015 0.005 0.010 0.015 3 3 Re r × 10 Re r × 10 2 2 5 2 −t GeV −t GeV FIG. 5. Δχ ¼ 1 correlation (stat þ syst) contours for r and r . 5 2 FIG. 6. Elastic pp analyzing powers A ðtÞ and A ðtÞ N NN Filled ellipses mean statistical error only. The HJET extrapola- measured in this work. The filled areas correspond to tions to 200 GeV are labeled 3 and 4. The STAR Collaboration σ .For A ðtÞ, the dashed lines refer to the expected statþsyst N result [11] for r was changed by us using Eqs. (5)–(8). analyzing powers if r ¼ 0. assumption (25), be related to f ≈ r in reasonable 5 ðsfÞ ðsfÞ obtained f ¼ 0.50.5 and α − ¼ 0.620.11. However, F R agreement with Eq. (26). ðsfÞ P ↑ ˜ f strongly depends on the α þ selection. The fit of the The value of f ¼ 0.10 0.01 [32] estimated from p C R Pomeron spin-flip intercept [using a simple pole for PðsÞ] data is noticeably larger than in Eq. (26). However, this is stable in a wide range of 0.3 < α þ < 0.8.Itgives estimate required a model dependent conversion from R proton-nucleus asymmetries to proton-proton r and, also, ðsfÞ ðsfÞ Δ ¼ α − 1 ¼ 0.117 0.031 ; ð33Þ P P statþsyst was strongly based on unpublished experimental results þ0.012 [35] with undetermined systematic uncertainties. which agrees with the unpolarized Δ ¼ 0.096 [36], −0.009 ðsfÞ The r ðsÞ and r ðsÞ dependencies on the beam energy 5 2 and α − ¼ 0.65 0.11. are illustrated in Fig. 5 where the extrapolations to pffiffiffi Summary.—In RHIC polarized proton runs 2015 s ¼ 200 GeV, based on the Froissaron parametrization (100 GeV) and 2017 (255 GeV), we have measured elastic (24), are labeled “3.” Consistency between the extrapola- pp analyzing powers in the CNI region 0.0013 < −t< tion of r and the STAR Collaboration measurement [11] 5 −4 0.018 GeV with accuracy jδA ðtÞj ∼ 2 × 10 [13] as N;NN was observed, though the STAR experimental uncertainties shown in Fig. 6. To graph A ðtÞ, we substituted the fitted are not inconsiderable. values of r from Eqs. (11)–(14) in Eq. (3), taking into It is interesting to note that the values of r and r , when 5 2 pffiffiffi account statistical and systematic uncertainties and projected from s 14–22 to 200 GeV, have smaller their covariances. In fact, this is equivalent to determining uncertainties than those of the measurements. This may A ðtÞ directly from the linear fit of the normalized be explained by decay of the R pole contributions at large asymmetries a ðT Þ. Thus, the result is not greatly affected n R s and by using functions RðsÞ that are too tightly con- by absorptive corrections, nor by possible variations in ρ, strained (which, for the selected model, is a good approxi- σ , B, and r . tot p mation in the energy range considered). However, many The accuracy achieved in the determination of A ðtÞ models [31] are used to parametrize σ ðsÞ and ρðsÞ which tot allows one to use a higher density unpolarized hydrogen may render RðsÞ more uncertain. jet target in a high precision absolute polarimeter, e.g., To estimate the dependence of a Reggeon analysis on a at a future EIC [15]. For a 30-fold increase in jet density, the particular model, we also fitted the HJET data using a sum expected statistical and systematic uncertainties of the of simple poles (23). These extrapolations of r and r to 5 2 stat pffiffiffi polarization measurement would be δ P ≲ 1%=h and s ¼ 200 GeV are labeled “4” in Fig. 5. Since, at HJET syst δ P=P ≲ 1%. energies, the double spin-flip amplitude is dominated by an The hadronic spin-flip amplitude ratios r and r were 5 2 R contribution, the r projection to 200 GeV is strongly reliably isolated at both energies. Applying the corrections affected by a variation of α þ. indicated in Eqs. (5)–(8) to the expression [20] for A ðtÞ The expansions (25) fit the measurements with statistically resulted in a change of the measured r by about the size of insignificant discrepancies χ ¼ 2.2 [Eqs. (26)–(28)]and the experimental uncertainty. The absorptive corrections χ ¼ 1.6 [Eqs. (29)–(31)]for ndf ¼ 1 showing consistency were not included in the data analysis, but, if they become between the experimental data and Eq. (25). available, a simple correction to Re r could be applied. To evaluate a possible difference between single spin-flip Measurements at two energies permitted a Regge pole (sf) and nonflip functions PðsÞ, we determined the ratio analysis of elastic pp scattering to be extended to the spin ðsfÞ ðsfÞ f ¼ f =f in a combined analysis including the dependent case. A Reggeon expansion of the spin-flip F F STAR Collaboration result. 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