Impact picture for the analyzing power a N in very forward p p elastic scattering The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Bourrely, Claude, Jacques Soffer, and Tai Tsun Wu. 2007. “Impact Picture for the Analyzing powerANin Very Forwardppelastic Scattering.” Physical Review D 76 (5). https://doi.org/10.1103/ physrevd.76.053002. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:41555830 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA CPT-P27-2007 Impact picture for the analyzing power AN in very forward pp elastic scattering Claude Bourrely∗ Centre de Physique Th´eorique†, CNRS Luminy case 907, F-13288 Marseille Cedex 09, France Jacques Soffer‡ Department of Physics, Temple University, Philadelphia, PA 19122-6082, USA Tai Tsun Wu§ Harvard University, Cambridge, MA 02138, USA and Theoretical Physics Division, CERN, 1211 Geneva 23, Switzerland In the framework of the impact picture we compute the analyzing power AN for pp elastic scattering at high energy and in the very forward direction. We consider the full set of Coulomb amplitudes and show that the interference between the hadronic non-flip amplitude and the single-flip Coulomb amplitude is sufficient to obtain a good agreement with the present experimental data. This leads us to conclude that the single-flip hadronic amplitude is small in this low momentum transfer region and it strongly suggests that this process can be used as an absolute polarimeter at the BNL-RHIC pp collider. PACS numbers: 13.40.-f, 13.85.Dz, 13.88.+e I. INTRODUCTION energy √s = 500 GeV, offers a unique oppor- tunity to measure single- and double-spin ob- servables, and thus to provide the determination The measurement of spin observables in of the spin-dependent amplitudes, which remain hadronic exclusive processes is the only way to unknown so far. obtain the full knowledge on the corresponding For instance, for an elastic collision of trans- set of scattering amplitudes, and in particular, versely polarized protons, the differential cross their relative size and phase difference. Taking section as a function of the momentum transfer the specific case of proton-proton elastic scat- t and the azimuthal angle φ, reads tering, a reconstruction of the five amplitudes has been worked out in the low-energy domain d2σ dσ [1]. This situation is very different at high en- 2π = [1 + (PB + PY )AN cos φ (1) dtdφ dt ergy; due to the lack of data, in the range plab ≃ + P P (A cos2 φ + A sin2 φ)] , 100-300 GeV, besides the non-flip hadronic am- B Y NN SS h plitude φ1 , only the hadronic helicity-flip ampli- h where PB and PY are the beam polarizations, tude φ5 is known and to a rather poor level of AN the analyzing power and ANN , ASS are dou- accuracy. The advent of the BNL-RHIC pp col- ble spin asymmetries (see Ref. [2] for definitions). lider, where the two proton beams can be polar- In this expression, the values of the beam po- ized, longitudinally and transversely, up to an larizations have to be known accurately in or- der to reduce the errors on the spin asymme- ∗Electronic address: [email protected] tries. So new measurements are indeed required † Unit´e Mixte de Recherche 6207 du CNRS et des Univer- to achieve an amplitude analysis of pp elastic sit´es Aix-Marseille I, Aix-Marseille II et de l’Universit´e du Sud Toulon-Var - Laboratoire affili´e `a la FRUMAM. scattering at high energy, and the success of the ‡Electronic address: jsoff[email protected] vast BNL-RHIC spin programme [3] also relies §Electronic address: [email protected] heavily on the precise determination of the beam 2 polarizations. One possibility for an absolute po- energy dependence is given by the crossing sym- larimeter1 is provided by the measurement of the metric expression [10, 11] analyzing power AN , in the very forward t re- | | c c gion, where significant Coulomb nuclear interfer- s u S0(s) = c′ + c′ , (4) ence (CNI) occurs [5–7]. (ln s) (ln u) In the calculation of the analyzing power an where u is the third Mandelstam variable. The important question arises: is the interference choice one makes for F (b2) is crucial and, as ex- fully dominated by the hadronic non-flip ampli- plained in Ref. [8], we take the Bessel transform tude with the one-photon exchange helicity-flip of amplitude or must one also take into account the 2 contribution of the hadronic helicity-flip ampli- ˜ 2 a + t h F (t) = f[G(t)] 2 . (5) tude φ5 , mentioned above? Several arguments a t h − concerning the magnitude and phase of φ5 in the small t-region, have been discussed in great Here G(t) stands for the proton electromagnetic detail in Ref. [6] and it was concluded that the form factor, parametrized as measurement of AN in the CNI region was badly 1 G(t) = . (6) needed to get the answer. The purpose of this (1 t/m2)(1 t/m2) paper is to study this problem in the framework − 1 − 2 of the impact picture developed almost three The slowly varying function occurring in Eq. (5) decades ago [8], which has led to a very success- reflects the approximate proportionality between ful phenomenology, repeatedly verified by high- the charge density and the hadronic matter dis- energy experiments, including near the forward tribution inside a proton [12]. So the Pomeron direction. 2 part of the amplitude depends on only six pa- ′ rameters c, c , m1, m2, f, and a. The asymptotic energy regime of hadronic interactions are con- II. THE IMPACT-PICTURE APPROACH trolled by c and c′, which will be kept, for all elastic reactions, at the values obtained in 1984 In the impact picture, the spin-independent [13], namely h h hadronic amplitude φ1 = φ3 for pp and pp¯ elastic scattering reads as [8] c = 0.167 and c′ = 0.748 . (7) h is −iq·b −Ω0(s,b) The remaining four parameters are related, more φ , (s, t) = e (1 e )db , (2) 1 3 2π ! − specifically to the reaction pp (pp¯ ) and they have been fitted in [14] by the use of a large set of where q is the momentum transfer (t = q2) − elastic data. and Ω0(s, b) is the opaqueness at impact param- We now turn to the Regge background. A eter b and at a given energy s. We take generic Regge exchange amplitude has an ex- pression of the form 2 Ω0(s, b) = S0(s)F (b ) + R0(s, b) . (3) bit −iπαi(t) s αi(t) R˜i(s, t) = Cie 1 e [ ] , (8) Here the first term is associated with the " ± # s0 Pomeron exchange, which generates the diffrac- bit where Cie is the Regge residue, refers to tive component of the scattering and the second ± an even- or odd-signature exchange, αi(t) = term is the Regge background. The Pomeron ′ α0i + αit, is a standard linear Regge trajectory 2 and s0 = 1 GeV . If R˜0(s, t) = R˜i(s, t) is 1 i Proton-Helium elastic scattering has been also consid- the sum over all the allowed Regge$trajectories, ered as a possible high-energy polarimeter [4]. 2 b pp the Regge background R0(s, ) in Eq. (3) is the An accurate measurement for the real part of the ˜ forward scattering amplitude is a real challenge for the Bessel transform of R0(s, t). In pp (pp¯ ) elas- LHC [9]. tic scattering, the allowed Regge exchanges are 3 A2, ρ, ω, so the Regge background involves sev- eral additional parameters, which are given in Ref. [14]. In earlier work, spin-dependent hadronic am- plitudes were implemented [8, 15, 16], using the notion of rotating matter inside the pro- ton, which allowed us to describe the polar- izations and spin correlation parameters, but for the present purpose hadronic spin-dependent amplitudes will be ignored. In order to de- scribe the very small t-region we are interested in, one adds to the hadronic amplitude consid- ered above, the full set of Coulomb amplitudes C φi (s, t), whose expressions are given in Ref. [17] and the Coulomb phase in Ref. [18]. The two observables of interest are the un- polarized cross section dσ/dt and the analyzing power AN , whose expressions in terms of the hadronic and Coulomb amplitudes are respec- tively dσ(s, t) π = φh(s, t) + φC (s, t) 2 (9) dt s2 | i i | i=1%,··· ,5 and h ∗ C 4Im((φ1 (s, t)) φ5 (s, t)) AN (s, t)= . (10) φh(s, t)+φC (s, t) 2 i=1,··· ,5| i i | $ The numerator of this last expression is not fully h h general because we have assumed that φ1 = φ3 h and φ2,4,5 = 0. FIG. 1: The differential cross section versus the mo- mentum transfer t for different energies. Data from III. NUMERICAL RESULTS Refs. [19–24]. The analyzing power AN has been measured at high energy for √s = 13.7, 19.4, 200 GeV, Note that the momentum transfer runs over four but before turning to the calculation of this decades and the cross section over eleven orders quantity, it is necessary to look at the predic- of magnitude, which is a good illustration of the tions for the differential cross section, at the validity of the impact picture.