Corrections to the Elastic Proton-Proton Analyzing Power Parametrization at High Energies

The HJET Polarized Atomic Hydrogen Gas Jet Target polarimeter (HJET) polarimeter was designed to measure the absolute polarization of the proton beams at the Relativistic Heavy Ion Collider. In these measurements, the small scattering angle elastic $pp$ single $A_N(t)$ and double $A_{NN}(t)$ spin analyzing powers can be precisely determined. The experimental accuracy achieved at HJET requires corrections to the $A_N(t)$ parametrization, conventionally used for such studies. In this paper we evaluate the corrections to the analyzing powers due to (i) the differences between the electromagnetic and hadronic form factors and (ii) the $m_p^2/s$ terms in the elastic spin-flip $pp$ electromagnetic amplitude. The corresponding alterations of the evaluated hadronic spin-flip amplitudes are about the same as the experimental uncertainties of the HJET measurements. The proposed corrections may have implications for the elastic $pp$ forward real-to-imaginary amplitude ratios $\rho$ determined in unpolarized $pp$ experiments.


I. INTRODUCTION
The Polarized Atomic Hydrogen Gas Jet Target [1] polarimeter (HJET) is employed to measure the absolute polarization of proton beams at the Relativistic Heavy Ion Collider (RHIC). For that, the vertically polarized proton beam is elastically scattered at small angles ( Fig. 1) on the vertically polarized target (the jet) with well-determined polarization |P j | = 0.957 ± 0.001 and the beam and jet spin correlated asymmetries of the recoil protons [Eq. (20)] are studied.
The major upgrade of HJET in 2015, along with the development of new methods in data analysis, allowed us to reduce the systematic uncertainties of the beam polarization measurements to a σ syst P /P 0.5% [2] level. Such a small systematic uncertainty of measurements, combined with large statistics of approximately 2 × 10 9 elastic pp events per RHIC run accumulated in 2015 (E lab = 100 GeV) and 2017 (E lab = 255 GeV), allowed us to precisely measure the single A N and double A NN spin analyzing powers [3] in the Coulomb-nuclear interference (CNI) region.
Generally, A N (s, t) and A NN (s, t) are functions of the invariant variables s, center-of-mass energy squared, and t, 4-momentum transfer squared. An important part of the experimental study of the analyzing power is isolation of the hadronic spin-flip amplitudes. The theoretical basis for such studies was developed in Refs. [4,5]. An update [6] for the RHIC spin program provided a parametrization of A N (s, t) which was used in all previous experimental evaluations [7][8][9] of the hadronic spin-flip amplitudes in high energy near-forward elastic pp scattering.
Recently it was pointed out [10] that analyzing power * poblaguev@bnl.gov A N (t) given in Ref. [6] was derived with some simplifications, which might be essential for the experimental accuracy achieved at HJET: (i) it was implicitly assumed that the electromagnetic form factor is equal to the hadronic form factor exp(Bt/2), and (ii) the elastic pp electric form factor, G pp E , was approximated, G pp E =G 2 E (t), by an electric form factor G E (t) determined in electron-proton scattering experiments. The absorptive corrections, due to the initial and final state inelastic hadronic interactions between the colliding protons [10], were not considered in Ref. [6].
After this paper was accepted for publication, a theoretical evaluation of the absorptive corrections in elastic pp scattering was given in Ref. [10]. These new results were not taken into account below.
Here, we analyze the effect of the possible corrections to the A N (t) parametrization on the results of the recent HJET measurements. The evaluated alteration of the measured hadronic spin-flip amplitudes suggests including the corrections in the data analysis. Also it was found that the discussed corrections may be important for experimental determination of the real-to-imaginary ratio ρ of the pp forward elastic scattering amplitude.

II. PARAMETRIZATION OF THE CNI ANALYZING POWERS AT HIGH ENERGIES
Elastic p ↑ p ↑ scattering is described by five helicity amplitudes [6] φ 1 (s, t) = + + |M | + + ,  electromagnetic components of the elastic pp amplitude should be explicitly indicated, The Coulomb phase is approximately independent of helicity [5,11] where γ = 0.5772 is Euler's constant and Λ 2 = 0.71 GeV 2 . For numerical estimates in Eq. (3) and below, we assume the HJET measurement values of s and t. The differential cross section slope B(s) depends on energy as B 0 + B 1 ln s [12] and is about 11.5 GeV −2 . To the lowest order in α, the fine structure constant, the electromagnetic amplitudes were calculated in Ref. [5]. For very low t, the hadronic amplitude is dominated by the term. According to the optical theorem, where m p is proton mass and σ tot (s) is the total pp cross section. Therefore, φ h + (s, t) can be presented as where and exp (Bt/2) is the nuclear form factor. Similarly, hadronic single and double spin-flip amplitudes may be parametrized by the dimensionless factors and respectively. Using the expression for the elastic pp cross section the single spin analyzing power can be presented [6] as where In Eq. (13), κ = µ p − 1 = 1.793 is the proton's anomalous magnetic moment. The A N (t) dependence on r 5 appears in a linear function of t while the dependence on r 2 is negligible. Similarly [6],

III. ANALYZING POWER MEASUREMENTS AT HJET
For elastic scattering of vertically polarized beam and target protons, the recoil proton azimuthal angle ϕ distribution is given [3] by Here, ϕ used is defined in accordance with Fig. 1, and P j and P b are jet and beam polarizations, respectively. For HJET detectors, sin ϕ = ±1 and, thus, three spin correlated asymmetries A N P j , A N P b , and A NN P j P b can be experimentally determined in the momentum transfer range 0.001 −t 0.020 GeV 2 . Consequently, one can derive the beam polarization P b (the main purpose of HJET) as well as analyzing powers A N (t) and A NN (t).
The preliminary analysis of the HJET data acquired in RHIC runs 2015 and 2017 has been done using the analyzing power formulas of Ref. [6]. The values of σ tot (s) and ρ(s) were taken from Ref. [13] fit. The slope B(s) was derived from Ref. [14]. Only for numerical estimates below, these preliminary results could be summarized as For r 2 , systematic errors are small in these measurements.

IV. CORRECTIONS TO THE ANALYZING POWERS
To calculate corrections to Eq. (12), it is convenient to use the scaled amplitudes Since a possible dependence of ρ, r 2 , and r 5 on t may be neglected in the CNI region, the scaled hadronic amplitudes can be approximated by For the electromagnetic amplitudes, we should include the corrections of order of m 2 p /s which can be significant for E Lab = 100 GeV. Using the following expressions for the proton's electromagnetic form factors [6,15], and neglecting the t/s terms, one can derive from Ref. [5] ϕ em The following shorthand was used The proton's electric form factor G E (t) was approximated in (26) by the proton charge radius r E = r 2 E 1/2 . In Eq. (27), we did not distinguish between proton electric r E and magnetic r M radii.
The electromagnetic and hadronic form factors difference can be realized by the substitution t c → t c +bt where Since the electric form factor G E (t) in the dipole form [11,16] G D (t) = 1 − t/Λ 2 −2 , Λ 2 = 0.71 GeV 2 was commonly used in the elastic pp data analysis, it is convenient to explicitly isolate the corresponding term b D in (28) For the RHIC beam energies, The errors here correspond to the systematic uncertainties in the values of B(s) [14].
Approximating ϕ em 0 = (t c /t)e bt/tc , one finds a correction to the denominator (15) of the analyzing power expressions The term κ 2 t c /4m 2 p here is due to the spin-flip amplitude ϕ em 5 contribution to dσ/dt. The experimental determination of the real to imaginary ratio ρ at high energies is based on an analysis of the dσ/dt (t) ∝ f cs (t, ρ). The proton-proton electromagnetic form factor was approximated by G 2 D (t) in almost all experimental studies of ρ. Therefore, a biased value of ρ was measured in these experiments, ρ exp = ρ − b cs . The bias is small compared to the uncertainty of measurements in any of the experiments listed in PDG, but it may be substantial for the global fit [18]. Since the values of ρ from the global fit are used in the analyzing power measurements, we should replace in (38) as well as in the expressions for f 0 N , f 1 N , and f 1 NN above. Thus, the leading order corrections to the analyzing power A N (t) from Ref. [6] can be approximated in (12) as reflects the spin-flip contribution [see Eq. (27)] to the electromagnetic form factor. The specified error is dominated by the experimental uncertainties in the value of proton magnetic radius r M = 0.851±0.026 fm [19]. For A NN (t), the corrections are small compared to uncertainties of the measurement at HJET. Also, we can neglect the correction to the Coulomb phase δ C (r E , B).

V. NUMERICAL ESTIMATES OF THE CORRECTIONS
The effect of the substitution (41) can be parametrized by the effective correction b D ∆f N (t) to the linear func- The dependence of ∆f N (t) on ρ and b D can be neglected. The calculated value of this correction is shown in Fig. 2.
For the HJET values of b D given in Eqs. (32) and (33), the nonlinearity is not experimentally observable, and, thus, we can approximate or, equivalently, Obviously, the values of c 0 and c 1 depend on the t-range and experimental uncertainties. The HJET data analysis leads to c 0 ∼ −1.0 and c 1 ∼ 0.1. Combining (42), (43), and (47) we find the corrections to the measured hadronic form factors as follows: The errors here are due to uncertainties in values of B, r E , and r M . Each error is strongly correlated through Eqs. (50). The large corrections to I 5 are due to the term m 2 p /s in (48), which is 0.0047 for 100 GeV and 0.0018 for 255 GeV. Alterations of the measured r 5 are comparable with the experimental uncertainties (see Fig. 3) and, thus, should not be neglected.

VI. POSSIBLE EFFECT OF THE ABSORPTIVE CORRECTION
A dependence of the measured r 5 on the absorptive corrections could be readily estimated if the corre-sponding modification of the electromagnetic form factor F em (t) of an elastic pp amplitude can be approximated in the CNI region by a linear function of t, Generally, a(s) is spin dependent. It can be effected by the substitutions b nf → b nf +a nf , b κ → b κ +a sf −a nf , where a nf (s) and a sf (s) are absorptive corrections to nonflip and spin-flip amplitudes, respectively. The dominant absorptive corrections to r 5 and r 2 can be written as As it was underlined above, the correction, such as given in Eq. (51), does not modify f cs (t, ρ) but specifies the systematic errors in the experimental determinations of ρ.
In case of large corrections, say |a nf + b cs | 0.003, the results of all forward unpolarized proton-proton scattering measurements should be revised and, consequently, a new global fit of ρ(s) and σ tot (s) should be carried out.

VII. CONCLUSIONS
In this paper, the corrections to the analyzing powers given in Ref. [6] were studied. For the experimental results already published, Eqs. (48) and (49) allows one to evaluate with sufficient accuracy the corrections to the measured single spin-flip amplitude parameter r 5 .
The improved expressions for A N (t) and A NN (t) could be written in the same form as in Ref. [6] (if neglecting the absorptive correction terms ∆ a N = ∆ a NN = 0) may be introduced by the following substitutions: where a df (s) is the absorptive correction (51) to the double spin-flip electromagnetic form factor. The absorptive corrections may also affect the results of determination of ρ in unpolarized elastic pp scattering.