Coulomb phase corrections to the transverse analyzing power $A_N(t)$ in high energy forward proton-proton scattering

Study of polarized proton-proton elastic scattering in the Coulomb-nuclear interference region allows one to measure the forward hadronic single spin-flip amplitude including its phase. However, in a precision experimental data analysis, a phase shift correction $\delta_C$ due to the long distance Coulomb interaction should be taken into account. For unpolarized scattering, $\delta_C$ is commonly considered as well established. Here, we evaluate the Coulomb phase shifts for the forward elastic proton-proton single spin-flip electromagnetic and hadronic amplitudes. Only a small discrepancy between the spin-flip and non-flip phases was found which can be neglected in the high energy forward elastic $\mathit{pp}$ studies involving transverse spin. Nonetheless, the effective alteration of the hadronic spin-flip amplitude by the long distance electromagnetic corrections can be essential for interpretation of the experimental results.


I. INTRODUCTION
Since electromagnetic amplitude can substantially contribute to the elastic forward proton-proton (pp) scattering at high energies, experimental study of the Coulombnuclear interference (CNI) in the pp scattering allows one to reveal the hadronic amplitude structure. For the unpolarized scattering, the CNI pp amplitude can be approximated as [1] φ CNI pp (s, t) = Im φ(s, 0) (i+ρ)e Bt/2 + t c t e iδ C + Bt/2 . (1) Here, φ(s, t) stands for the hadronic amplitude and the electromagnetic component is identified by t c /t term, where t c = −8πα/σ tot (s), α is the fine structure constant and σ tot is the total pp cross section. Generally, φ CNI pp and the parameters used are functions of total energy squared s and momentum transfer squared t.
In this paper, numerical estimates will be done for a 100 GeV proton beam (typical for the Relativistic Heavy Ion Collider and the future Electron Ion Collider) scattering off a fixed proton target. Therefore, ρ = −0.079 [2], σ tot = 39.2 mb [2], t c = −1.86 × 10 −3 GeV 2 , and B = 11.2 GeV −2 [3]. The electromagnetic form factor is expressed, via rms charge radius of a proton, r E = 0.841 fm [4]. For the unpolarized scattering, a theoretical understanding of the Coulomb phase shift δ C (t) was developed in many works, particularly in [5]. Following Ref. [6] and neglecting terms ∼ t 2 ln t,  (3) is commonly used in experimental data analysis for many years. The next to leading order corrections (4) can be disregarded in this paper. It was shown in Ref. [7] that the Coulomb phase should be independent of the helicity structure of the experimentally measured scattering amplitudes though subsequent determination of the pure hadronic amplitudes may involve order-α corrections resulting from the spin of the particles involved.
Recently, long-distance electromagnetic corrections, including the absorption, to the spin-flip amplitudes were derived [8] in the eikonal model. However, the results were presented as Fourier integrals (which were calculated numerically) and, thus, cannot be implemented, in a simple way, to an experimental data analysis. Also, it was noted [8] that the corrections to the spin-flip Coulomb phase "are so large, that hardly can be treated as a phase shift", which may be understood as a suggestion to obsolete a commonly used expression for the analyzing power A N (t) given in Eq. (7) and, consequently, to reanalyze all previous measurements of A N (t).
Here, we found compact algebraic approximations for the long-distance electromagnetic corrections, separately, to the spin-flip Coulomb phase and to the hadronic spinflip amplitude parameter r 5 [Eq. (6)]. It was shown that the difference between the spin-flip and nonflip Coulomb phases is small. Although, the effective correction to Re r 5 was found to be essential for the experimental accuracy already achieved [9], it can be applied directly to the value of r 5 obtained in experimental data fit. The corresponding changes in the Regge fit [9] of the measured values of r 5 (s) will be discussed. flip (sf) helicity amplitudes [7,10,11] Here, hadronic φ and electromagnetic φ parts of an amplitude are discriminated by tilde symbol. For t → 0, there is a simple relation between sf and nf amplitudes [11]: where m p is a proton mass, κ p = µ p − 1 = 1.793 is anomalous magnetic moment of a proton, and complex r 5 = R 5 +iI 5 , |r 5 | ∼ 0.02 [9], parameterize hadronic spinflip amplitude [11]. Omitting some small corrections [12], the analyzing power can be approximated [11] as where δ C is given in Eq. (3) while δ em C and δ h C are spin-flip phase shifts in the φ sf φ * nf and φ sf φ * nf interference terms, respectively.
All recent experimental studies [9,[13][14][15] of the forward elastic proton-proton A N (t) had been done using Eq. (7) and assuming δ em Spin-flip phases, δ em C and δ h C , can be evaluated in a simple way using expressions derived in Ref. [6] to study δ C (t). To provide a framework for the calculations, some results of Ref. [6] are briefly overviewed in Sec. III A.
A. Theoretical approach used [6] to calculate Coulomb corrections to the nonflip amplitudes Considering multiple photon exchange in the elastic pp scattering and neglecting the higher order corrections O(α 3 ), the net long-range Coulomb (C) amplitude (see Fig.1) can be presented as [6] where q T ≈ √ −t is transverse momentum and the eikonal phase is a Fourier transform of the Coulomb part of the amplitude calculated in Born approximation [6] f C (q T ) = −2α Here,f C is defined as sum of two nonflip helicity amplitudes ++|++ and +−|+− [11]. A small photon mass λ was included to Eq. (11) to keep the integrals finite. The multiphoton exchange results in an acquired Assuming Φ C (q T ) 1, one finds Similarly, to calculate the Coulomb corrections to the hadronic amplitude, one can use the following relations, Equations (14) and (19) were analytically integrated in Ref. [6]. Both, Φ C (q T ) and Φ NC (q T ), contain the divergent term ln q 2 /λ 2 which, however, cancels in final expression (3) for the Coulomb phase difference (20)

B. Calculation of the spin-flip Coulomb phase
To find the Coulomb corrected spin-flip amplitudes f sf C (q T ) and f sf NC (q T ), one can use the following eikonal phases [16] respectively. Here,f C /2 andf N /2 correspond to the nonflip amplitudes used in Eq. (6), and the azimuthal dependence of the scattering is defined by n, a unit vector orthogonal to the beam momentum and the proton spin. Considering the spin flip amplitudes for q T = nq T , one can readily determine the spin-flip phase Φ sf C (q T ) by including factor 2( q T q 1 )/q 2 T or 2( q T q 2 )/q 2 T to integral (14). Since q 1 q T + q 2 q T = q 2 T , we immediately find which leads to To calculate Φ sf NC (q T ), factor ( q 2 q T )/q 2 T = 1 − ( q 1 q T )/q 2 T should be applied in Eq. (19), which gives cos ϕ e u cos ϕ , (26) Expanding and using following definite integrals [17] π −π cos 2n+1 x dx = 0, one arrives to Thus, Evaluating δ em C and δ h C , we did not distinguish between nonflip B and spin-flip B sf hadronic slopes as well as between B [Eq. (2)] and where r M = 0.851±0.026 fm [18] is rms magnetic radius of a proton. For small t, i.e., omitting terms approaching zero if t → 0, one finds Since r M = r E within the current experimental accuracy of about 2%, we cannot distinguish between B and B sf . Also, there are arguments [8] to assume that B sf ≈ B.
Although in an extreme case, B sf = 2B (e.g., considered in Ref. [19] for proton-carbon scattering), δ h C −δ C can be increased by about a factor of 2, the effect will be invisible in the expression for analyzing power due to the strong suppression of term δ h C R 5 in (7) by a small value of |R 5 | 0.02.

C. The electromagnetic correction to r5
In Ref. [8], it was pointed out that the hadronic spinflip amplitude should also include the spin-flip photon exchange, i.e., one should replace The effective spin-flip amplitude r γ 5 can be related to the integral which is similar to that of calculated in (25)-(32). Thus, For the modified r γ 5 , Coulomb phase Φ sf NC (t) is the same as in Eq. (25).
Actually r γ 5 had being determined in all previous measurements of r 5 .
Thus, assuming spin-dependent measurements at high energies, we can agree with the approximation for the Coulomb phases suggested in Ref. [11]. Spin-flip photon exchange (37) results in an effective correction ∆ γ ≈ 0.003 to real part of the hadronic spin-flip parameter r 5 in Eq. (7). The correction found is about triple of the experimental accuracy for R 5 in the HJET measurements [9]. Since ∆ γ is independent of the experimental data analysis, any already published experimental value r γ 5 of the hadronic spin-flip amplitude, in particular given in [9,15], can be easily adjusted to a bare one, This might be especially important for a study of r 5 (s) dependence on energy, e.g., in the Regge fit [8,9].
To illustrate a possible effect of corrections (42), the Regge fit [9] of the HJET values of r 5 (for √ s = 13.76 and 21.92 GeV), was revisited. In Ref. [9], the spin-flip Reggeon, R + (s) and R − (s), and Pomeron (in a Froissaron parametrization), P (s), functions were approximated by the nonflip ones [2]. After applying corrections (42), the fit χ 2 = 2.2 (NDF=1) was improved to χ 2 = 0.9 and the central values A thorough refit of the r 5 measurements, including the STAR √ s = 200 GeV result [15], will be done elsewhere. It is interesting to note that an absorptive correction, to the electromagnetic spin-flip form factor, F sf pp (t) → F sf pp (t) × (1 + a sf t/t c ), results [12] in the same effective alteration of r 5 as shown in Eq. (42).
In Ref. [8], to calculate absorptive corrections, graphs in Fig. 1 were regrouped. In this approach, a spinflip photon contribution [Eq.
(37] to hadronic spin-flip amplitude f sf N was not considered. Nonetheless, the term χ sf C (b)γ nf N (b) appeared as an absorptive correction to electromagnetic spin-flip amplitude f sf C (q T ). The corresponding equation was integrated numerically in [8]. However, using result of calculation (38), one can readily find absorption parameter a sf to be in exact agreement with Eq. (47). Thus, we come to a conclusion [8] that Coulomb corrections to the hadronic spin-flip amplitude (42) and hadronic (absorption) correction to the spin-flip Coulomb amplitude (47) are two equivalent descriptions of the same final-state electromagnetic interaction of polarized protons.