Single Spin Asymmetry $A_N$ in Polarized Proton-Proton Elastic Scattering at $\sqrt{s}=200$ GeV

We report a high precision measurement of the transverse single spin asymmetry $A_N$ at the center of mass energy $\sqrt{s}=200$ GeV in elastic proton-proton scattering by the STAR experiment at RHIC. The $A_N$ was measured in the four-momentum transfer squared $t$ range $0.003 \leqslant |t| \leqslant 0.035$ $\GeVcSq$, the region of a significant interference between the electromagnetic and hadronic scattering amplitudes. The measured values of $A_N$ and its $t$-dependence are consistent with a vanishing hadronic spin-flip amplitude, thus providing strong constraints on the ratio of the single spin-flip to the non-flip amplitudes. Since the hadronic amplitude is dominated by the Pomeron amplitude at this $\sqrt{s}$, we conclude that this measurement addresses the question about the presence of a hadronic spin flip due to the Pomeron exchange in polarized proton-proton elastic scattering.

We report a high precision measurement of the transverse single spin asymmetry AN at the center of mass energy √ s = 200 GeV in elastic proton-proton scattering by the STAR experiment at RHIC. The AN was measured in the four-momentum transfer squared t range 0.003 |t| 0.035 (GeV/c) 2 , the region of a significant interference between the electromagnetic and hadronic scattering amplitudes. The measured values of AN and its t-dependence are consistent with a vanishing hadronic spin-flip amplitude, thus providing strong constraints on the ratio of the single spin-flip to the non-flip amplitudes. Since the hadronic amplitude is dominated by the Pomeron amplitude at this √ s, we conclude that this measurement addresses the question about the presence of a hadronic spin flip due to the Pomeron exchange in polarized proton-proton elastic scattering.

I. INTRODUCTION
High energy diffractive hadronic scattering at small values of four-momentum transfer squared t, is dominated by an exchange of the Pomeron trajectory, a color-singlet object with the quantum numbers of the vacuum [1,2]. The calculation of cross-sections for small-t scattering requires a non-perturbative approach in QCD and its theoretical treatment is still being developed. The experimental data therefore provide significant constraints for theoretical approaches and models [3,4]. The coupling of the Pomeron to the nucleon spin is of special interest since it is predicted to be sensitive to the internal dynamics of the nucleon [3,4]. Studies of the spin dependence of protonproton (pp) scattering at small momentum transfers and at the highest energies presently available at RHIC offer an opportunity to reveal important information on the nature of the Pomeron.
There are several theoretical approaches which predict non-zero spin-dependent Pomeron amplitudes for elastic scattering. Examples include an approach in which the Pomeron-proton coupling is modeled via two-pion exchange [5], an impact picture model assuming that the spin-flip contribution is sensitive to the impact parameter distribution of matter in a polarized proton [6], and a model which treats the Pomeron helicity coupling analogously to that of the isoscalar anomalous magnetic moment of the nucleon [7]. Still another approach assumes a diquark enhanced picture of the proton [8], in which a non-zero spin-flip amplitude may arise if the proton wave function is dominated by an asymmetric configuration, such as a quark-diquark.
Here we present a high precision measurement of the transverse single spin asymmetry A N in elastic scattering of polarized protons at √ s = 200 GeV in the t-range 0.003 |t| 0.035 (GeV/c) 2 by the STAR experiment [9] at RHIC. The single spin asymmetry A N is defined as the left-right cross-section asymmetry with respect to the transversely polarized proton beam. In this range of t, A N originates predominantly from the interference between electromagnetic (Coulomb) spin-flip and hadronic (nuclear) non-flip amplitudes [3]. However, it was realized that A N in the Coulomb-nuclear interference (CNI) region is also a sensitive probe of the hadronic spin-flip amplitude [8], which will be discussed in more detail in Section II.
A previous measurement of A N in a similar t-range and the same √ s, but with limited statistics, has been reported by the pp2pp collaboration [10]. Other measurements of A N performed at small t were obtained at significantly lower energies. They include high precision results from the RHIC polarimeters obtained at √ s = 6.8 − 21.7 GeV for elastic proton-proton [11][12][13] and proton-carbon [14] scattering, as well as earlier results from the BNL AGS for pC scattering [15] at √ s = 6.4 GeV and from FNAL E704 for pp scattering [16] at √ s = 19.4 GeV. The combined analysis of all results, which covers a wide energy range and different targets, will help to disentangle contributions of various exchange mechanisms relevant for elastic scattering in the forward region [17]. In particular, such an analysis will allow us to extract information on the spin dependence of the diffractive mechanism which dominates at high energies.

II. HADRONIC SPIN-FLIP AMPLITUDE IN ELASTIC COLLISIONS
Elastic scattering of two protons is described by five independent helicity amplitudes: two helicity conserving (φ 1 and φ 3 ), two double helicity-flip (φ 2 and φ 4 ), and one single helicity-flip amplitude (φ 5 ) -see [3] for definitions. At very high √ s, such as available at RHIC, and very small |t| < 0.05 (GeV/c) 2 , the proton mass m can be neglected with respect to √ s and t can be neglected with respect to m, which simplifies kinematical factors in the following formulas. The elastic spin-averaged cross-section is given by: while the single spin-flip amplitude φ 5 gives rise to the single spin asymmetry, A N , through interference with the remaining amplitudes: Each of the amplitudes consists of Coulomb and hadronic contributions: , with the electromagnetic one-photon exchange amplitudes φ em i described by QED using the measured anomalous magnetic moment of the proton. [18]. The optical theorem relates the hadronic amplitudes to the total cross-section: which provides an important constraint on the parameterization of these dominant helicity conserving hadronic amplitudes.
The contribution of the two double spin-flip hadronic amplitudes φ had 2 and φ had 4 to the asymmetry A N is small, as indicated by both experimental results [19,20] and theoretical predictions [21]. Thus, the main contribution to A N is given by: where m is the proton mass. Thus r 5 is the measure of the ratio of the hadronic single spin-flip amplitude (φ 5 ) to hadronic single non-flip amplitudes (φ 1 and φ 3 ). Using this parametrization the following representation of A N can be derived [3]: where t c = −8πα/σ total , κ is the anomalous magnetic moment of the proton, ρ = Re φ + /Im φ + is the ratio of the real to imaginary parts of the non-flip elastic amplitude, and δ is the relative phase between the Coulomb and hadronic amplitudes [3]: where B is the slope of the forward peak in elastic scattering, α = 1/137 is the fine structure constant, γ = 0.5772 is Euler's constant, and Λ 2 = 0.71 (GeV/c) 2 .

III. DETECTION OF ELASTIC PROTON-PROTON COLLISIONS AT RHIC
The protons, which scatter elastically at small angles ( 2 mrad), follow the optics of the RHIC magnets and are detected by a system of detectors placed close to the beam inside movable vessels known as "Roman Pots" (RPs) [22]. The Roman Pot stations are located on either side of the STAR interaction point (IP) at 55.5 m and 58.5 m with horizontal and vertical insertions of the detectors, respectively. The coordinate system of the experiment is described in Fig. 1. There are eight Roman Pots, four on each side of the IP. Four approach the beam horizontally WHI, WHO (EHI, EHO) and four approach the beam vertically WVU, WVD (EVU, EVD) as shown in Fig. 1. The location of the RPs was optimized so that, combined with proper accelerator magnet settings, it provides so-called "parallel-to-point focusing", i.e. the (x, y) position of the scattered protons at the RPs depends almost exclusively on their scattering angles and is nearly insensitive to the transverse position of the interaction point. As shown in Fig. 1, there are five major magnets between the RPs and the collision point, two dipole magnets DX and D0, which bend beams into collision, and the focusing triplet of quadrupoles Q1-Q3. The dipole magnets scatter out particles with momentum which is not close to the beam momentum. The detector package inside each RP consists of four 0.4 mm thick silicon micro-strip detector planes with a strip pitch of about 100 µm, two of them measuring the horizontal (x) and two the vertical (y) position of a scattered proton. The sensitive area of the detectors is 79 × 48 mm 2 . Scintillation counters covering this area are used to form a trigger for elastic events. More details on the experiment and the technique can be found in Refs. [22,23].
The preliminary alignment was done by surveying the detector packages during their assembly and after installation inside the Roman Pots with respect to the beam line of the accelerator. The displacement of the RPs during data taking was measured by linear variable differential transformers (LVDTs). The final alignment was done using elastic events in the overlapping regions of horizontal and vertical RPs, which allowed a relative position measurement of the RPs on each side of the IP with a precision better than 0.1 mm. Collinearity of the elastic events and Monte-Carlo simulations of the acceptance boundaries due to limiting apertures in the quadrupole magnets were used to further constrain the geometry and to estimate systematic errors.
The data were taken during four dedicated RHIC stores between June 30 and July 4, 2009 with special beam optics of β * = 22 m in order to minimize the angular divergence at the IP [24]. The average luminosity over the four stores during which the data were collected was L ≈ 2·10 29 cm −2 s −1 . The closest approach of the first strip to the center of the beam was about 10 mm or about 12 σ of the transverse beam size. A total of 33 million elastic triggers were recorded.

IV. DATA SELECTION AND RECONSTRUCTION OF ELASTIC SCATTERING EVENTS
The selection of elastic events in this experiment is based on the collinearity of the scattered proton tracks. A single track was required on each side of the IP. Noisy and dead strips were rejected, with a total of five out of ≈ 14,000 in the active detector area. Track reconstruction started with the search for hits in the silicon detectors. First, adjacent strips with collected charge values above 5 σ from their pedestal averages were found and combined into clusters. A threshold depending on the cluster width was applied to the total charge of the cluster, thus improving the signal-tonoise ratio for clusters of 3 to 5 strips, while wider clusters were rejected. The cluster position was determined as a charge weighted average of strip coordinates. For each RP a search was performed for matching clusters in the pairs of planes measuring the same coordinate. Two clusters in such planes were considered matched if the distance between them was smaller than 200 µm, approximately the width of two strips. A matching pair with the smallest matching distance was chosen and its cluster coordinates were averaged. If only one cluster in the pair of planes was found, we just use its coordinate for the analysis. If more than one cluster or no match was found, no output from this RP was selected. An (x, y) pair found in an RP was considered a track. About 1/3 of all reconstructed tracks were found in the region of overlapping acceptance between the horizontal and the vertical RPs; for those tracks the average of the kinematic variables was used. To minimize the background contribution from beam halo particles, products of beam-gas interactions, and detector noise, fiducial areas were selected to cut edges of the silicon detectors near the beam and boundaries of the magnet apertures.
Planar angles θ RP x , θ RP y and coordinates x RP , y RP of protons at a given RP relate to the angles θ x , θ y and coordinates x, y at the IP by the transport matrix M: For the case of parallel-to-point focusing, and in the absence of x-y mixing, the transport matrix is simplified and the so-called "effective" length, L eff , terms dominate. The L eff values are in the range of 22-26 m for this experiment. The angles of the scattered protons at the IP can then be reconstructed independently for the East (E) and West (W) arms with respect to the IP: Because non-dominant terms in the transport matrix are small and result in a negligible correction of about 4 µrad to the reconstruction of the scattering angles, we used a 2×2 matrix (L eff x , a 14 ; a 32 , L eff y ), which was obtained by neglecting those small terms of the transport matrix.
Once the planar angles at IP were reconstructed, a collinearity requirement was imposed using χ 2 defined as: where δθ x,y = [θ W x,y − θ E x,y ] and the mean values δθ x,y and widths σ θx,y are taken from the fits to data performed for each data sample. An example is shown in Fig. 2. The small non-zero mean values (≈ 10 µrad) are consistent with the uncertainties of angle determinations discussed in the next section. Fig. 2 shows a typical distribution of δθ y vs. δθ x and its projections, fitted with a Gaussian and a linear background. Based on these fits, the non-collinear background contribution is estimated to be 0.3-0.5%. The requirement of χ 2 < 9 left about 21 million events for the asymmetry calculations.
The polar scattering angle θ and azimuthal angle ϕ (measured counterclockwise from the positive x-axis) for an event were then calculated as an average of those obtained from East and West arms, and the four-momentum transfer squared, t, was assigned to the event using t = −2p 2 (1 − cosθ) ≈ −p 2 θ 2 with p = 100.2 GeV/c.  Table I (a) -(e). The asymmetry ε ′ (ϕ) for the whole measured t-range (f). The red curves represent the best fit to Eq. (12) (a) -(e) and Eq. (13) (f).

V. SINGLE SPIN ASYMMETRIES
The azimuthal angle dependence of the cross-section for the elastic collision of vertically polarized protons is given [25] by: where higher order terms are ignored, dσ/dt is the spin-averaged cross-section, P B and P Y are the beam polarizations for the two colliding beams (called Blue and Yellow). The double spin asymmetry A N N is defined as the cross-section asymmetry for scattering of protons with spin orientations parallel and antiparallel with respect to the unit vectorn, normal to the scattering plane. The asymmetry A SS is defined analogously for both beams fully polarized along the unit vectorŝ in the scattering plane and normal to the beam. For each of the four RHIC stores, the event sample satisfying the requirements for elastic scattering was divided into five t-bins. Within each t-bin, the ϕ distributions were subdivided into bins of 10 • . The raw asymmetry, ε N (ϕ), was calculated using geometric means [26], the so-called "square root formula" for each pair of ϕ and π − ϕ bins in the range −π/2 < ϕ < π/2: where the "↑" and "↓" indicate the spin direction of the transversely polarized colliding proton beam bunches, N is the number of events detected in the respective spin and respective ϕ states and ν(ϕ) = P B P Y (A N N cos 2 (ϕ) + A SS sin 2 (ϕ)).
In the square root formula (12), the relative luminosities of different spin direction combinations cancel out. In addition, the detector acceptance and efficiency also cancel out, provided they do not depend on the bunch polarization. Results of Ref. [19] and preliminary results of this experiment [20] show that both A N N and A SS are very small ≈ 0.005 (and compatible with zero), constraining ν(ϕ) to ≈ 0.002, which can be safely neglected.
For each RHIC store, the obtained raw asymmetries were divided by the sum of polarizations of both beams for this particular store, and then averaged over the stores. The resulting asymmetries for each t bin are shown in Fig. 3(a-e) as a function of ϕ. The solid lines represent the best fits to Eq. (12).
Along with the raw asymmetry, ε N , which is proportional to the sum of the beam polarizations (P B + P Y ), other asymmetries can be obtained using different combinations of bunch spin directions. For instance, the asymmetry proportional to the beam polarization difference (P B − P Y ) is defined as follows: Provided that the beam polarizations (P B and P Y ) have the same values, which is approximately valid in this experiment, one would expect ε ′ = 0. The derived values of ε ′ may be used to estimate false asymmetries, which remain after applying the "square root" method. The distribution of the asymmetry ǫ ′ , obtained for the whole t-range, together with its fit, is shown in Fig. 3(f). During data taking, 64 bunches (16↑↑, 16↓↓, 16↑↓, 16↓↑) of the 90 proton beam bunches collided with usable spin patterns, and were used for ε N and ε ′ calculations.
The major systematic uncertainties of the experiment are due to the error of the beam polarization measurement, the reconstruction of t and a small background contribution as shown in Fig. 2. The two main contributions to the uncertainty in the t reconstruction are due to the uncertainties of the L eff values and the position of the beam center at the RP location. The former is mostly due to the uncertainty on values of the magnetic field strength in the Q1-Q3 focusing quadrupoles, which is mainly due to uncertainties in the magnet current and field measurements. The correction to the strength was derived using the correlation between the angle and position in the RPs for the tracks in the regions where the detector acceptance overlaps. An overall correction to the strength of the focusing quadrupoles of 0.5% was applied. The residual systematic error of the field calculation was estimated to be ≈ 0.5%, leading to ≈ 1% uncertainty in L eff and ≈ 1.4% uncertainty in t [27].
The position of the beam center is the reference point for the scattering angle calculations and effectively absorbs a large set of geometrical unknowns such as beam crossing angles and transverse beam positions at the IP, beam shifts from the beam pipe center at the RP location, as well as survey errors. To accommodate all these uncertainties, corrections to the survey were introduced based on the comparison of the simulated to the measured (x, y) distributions at the horizontal RPs on both sides of the IP. The simulation of the transport of elastically scattered protons through the RHIC magnets and the apertures was done and the detector acceptance was calculated. The acceptance boundaries from that simulation and the data were compared. No correction was found for the West side, while for the East side a correction of (∆x, ∆y) = (2.5, 1.5) mm was obtained. The uncertainty of that correction was estimated to be 400 µm. After applying that alignment correction, the collinearity, defined as the average angle difference δθ x,y (see Eq. (10)), was reduced from ≈ 55 µrad to ≈ 10 µrad. The remaining alignment uncertainty leads to a value of δt/t = 0.0020 [GeV/c]/ √ t and was added in quadrature to the uncertainty due to L eff . The number of background events in the data is less than 1% in all t-bins (e.g. see Fig. 2). Assuming the background is beam polarization independent, the asymmetry will be diluted by the same amount, δA N /A N < 0.01. This value results in a negligible contribution to the total error, when statistical and systematic errors are added in quadrature.
If the false asymmetry ε F were proportional to the beam polarization values, it would be indistinguishable from A N . On the contrary, if it does not depend on the polarization, it contributes equally to both ε N and ε ′ : and a direct estimate on the false asymmetry can be obtained: The values of the raw asymmetries, measured in the whole t-range, are ε N = 0.0276±0.0004 and ε ′ = −0.0007±0.0004. This gives a false asymmetry of ε F = −0.0004±0.0010. Thus the conclusion is that the false asymmetry is consistent with zero and very small compared to the measured raw asymmetry ε N . The results of the A N measurements in the five t-bins are summarized in Table I together with associated uncertainties and −t range boundaries. Two independent analyses of the data performed with slightly different selection criteria by two different groups gave consistent results. We have also done the cross checks to extract A N using the beam polarizations of the two beams. The resulting A N were found to be compatible with those in Table I

VI. RESULTS AND CONCLUSIONS
The measured values of A N are shown in Table I and presented in Fig. 4 together with parameterizations based on formula (5): the dashed line corresponds to no hadronic spin-flip contribution, i.e. r 5 = 0, while the solid line is the result of the fit using r 5 as a free parameter. Other parameter values used in the fit are: σ total = 51.79±0.12 mb, ρ = 0.1278 ± 0.0015 taken from fits to the world pp and pp data [29,30] and B = 16.3±1.8 (GeV/c) −2 from Ref. [23].
The value of r 5 resulting from the fit described above is shown in Fig. 5 together with 1σ confidence level contours.
In Table II, we show the central value of the fit and uncertainties on Re r 5 and Im r 5 due to the listed effects. In the first row of the table, the statistical error to the fit with the central value of the parameters is shown. The remaining rows show changes of Re r 5 and Im r 5 , when each parameter was varied one by one by ±1 σ during the fit procedure. Rows 2 and 3 show the effect due to the systematic uncertainty in L eff and alignment, row 4 due to the beam polarization (vertical scale uncertainty of A N ) and rows 5-7 systematic contributions due to the uncertainty of fit parameters. The dominant source of the systematic uncertainty is due to the beam polarization uncertainty. The total systematic uncertainty, including the effects related to rows 2-7 of Table II, is obtained by adding the error covariance matrices.
The final result on r 5 is shown in Fig. 5 together with both statistical and systematic uncertainties. The obtained values Re r 5 = 0.0017±0.0063 and Im r 5 = 0.007±0.057 are consistent with the hypothesis of no hadronic spin-flip contribution at the energy of this experiment.    Since the maximum A N in the CNI region can be evaluated as κ − 2Im r 5 in Eq. (5), theoretical calculations emphasize values of Im r 5 . Measurements of Im r 5 at different energies in the range 6.8 GeV √ s 200 GeV are shown in Fig. 6, together with predictions of theoretical models of the hadronic spin-flip amplitude as discussed above. All of the experimental results, including that reported here, are consistent with the assumption of no hadronic spin-flip contribution to the elastic proton-proton scattering. The high accuracy of the current measurement provides strong limits on the size of any hadronic spin-flip amplitude at this high energy, hence significantly constraining theoretical models which require hadronic spin-flip.