Energy dependence of CNI analyzing power for proton-carbon scattering

We use a simple Regge model to determine the energy dependence of the analyzing power for pC scattering in the CNI region. We take the model of Cudell et al which determines the Regge couplings and intercepts for the I = 0, non-flip Regge exchanges (Pomeron, f1 and ω) and extend it to the spin-flip amplitudes by allowing each of these exchanges to have independent spin-flip factors τP , τf and τω. Using this we show that by making measurments at two separate energies, with polarization known at one energy, one can fix the ratios of the analyzing power at any energy. By making an additional assumption that is reasonable, but not necessarily true, namely τω = τf , we show that one can predict the energy dependence of the analyzing power using the existing E950 data. We present the corresponding predictions for beam energies of 100 GeV and 250 GeV protons on a fixed carbon target based on a fit to the Spin 2000 data. Finally, we discuss the relation of these results to the pp CNI analyzing power. ∗This manuscript has been authored under contract number DE-AC02-76CH00016 with the U.S. Department of Energy. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. 1 We begin with the parametrization of pp elastic scattering given by Cudell et al [1], though one might do the same thing using other parametrizations such as that of Block et al [2]. Since it is known that the elastic, non-flip scattering is overwhelmingly I = 0 exchange, even at 24 GeV/c [3], we will assume the Regge couplings that they determine are for the I = 0 families and so directly applicable to pC scattering. The form they assume for the forward amplitude then has the form g0(s, 0) = gP (s) + gf(s) + gω(s) (1) with gP (s) = −Xs(cot π 2 (1 + ǫ)− i), (2) gf(s) = −Y s(cot π 2 (1− η)− i), (3) gω(s) = −Y s ′ (tan π 2 (1− η′) + i) (4) normalized that Im(g0(s)) = σtot(s). The values of the parameters given by them are ǫ = 0.0933, η = 0.357, η′ = 0.560 (5) X = 18.79, Y = 63.0, Y ′ = 36.2. (6) Our model is that the spin-flip pp I = 0 exchange amplitude g5(s, t) is given by g5(s, t) = τ(s) √ −t m g0(s, t) (7) = √ −t m {τP gP (s) + τf gf(s) + τω gω(s)}. (8) where τ(s) depends on energy but not on t over the CNI range. It is in general neither real nor constant in s and is given by 1 τ(s) = {τP gP (s) + τf gf(s) + τω gω(s)}/g0(s, 0) (9) where the τi’s are energy-independent, real constants. The phases of the amplitudes come only from the energy dependence as given in Eq.(1). This is the key assumption from Regge theory which we need: as a result the real and imaginary parts of τ(s) are given at each energy in terms of the three real constants τP , τf and τω. In a recent paper [4] it was shown under rather general assumptions that the spin-flip factor for proton-nucleus scattering τpA(s) is equal to the I = 0 part of the proton-proton spin-flip factor τ(s). From here on we will use this result to study the energy dependence of the pC analyzing power, and will return to the question of pp analyzing power at the end of the note. To determine the three real parameters τP , τf and τω we need three equations. At each energy, the fit to the small t behaviour determines two quantities, P (s)(κ/2−Re[τ(s)]) and P (s)Im[τ(s)]. Thus if we know the polarization at one energy, s0, then we have two of the needed equations: Re[τ(s)] = τP Re[gP (s)/g0(s)] + τf Re[gf (s)/g0(s)] + τω Re[gω(s)/g0(s)], (10) Im[τ(s)] = τP Im[gP (s)/g0(s)] + τf Im[gf(s)/g0(s)] + τω Im[gω(s)/g0(s)], (11) evaluated at s = s0. If we measure the asymmetry but not the polarization at some other energy s then all we can obtain is the shape of the curve characterized by the energydependent parameter S(s) S(s) = Im[τ(s)] κ/2− Re[τ(s)] . (12) Given a measured value S(s1), s1 6= s0 we can use κ 2 S(s) = τP Im[gP (s)/g0(s)] + τf Im[gf(s)/g0(s)] + τω Im[gω(s)/g0(s)] (13) + S(s){τP Re[gP (s)/g0(s)] + τf Re[gf(s)/g0(s)] + τω Re[gω(s)/g0(s)]}, (14) evaluated at s = s1 to provide a third independent equation which can be used with Eqs.(10) and (11) to determine τP , τf and τω. One should note that Re[gP (s)/g0(s)] +Re[gf (s)/g0(s)] +Re[gω(s)/g0(s)] = 1, (15) 2 Im[gP (s)/g0(s)] + Im[gf (s)/g0(s)] + Im[gω(s)/g0(s)] = 0. (16) Then one can solve Eq.(10) for τP in terms of the differences ∆f = τf −τP and ∆ω = τω−τP . The two remaining equations can then be solved for ∆f and ∆ω. It is clear that this method is not limited to the specific model chosen here and one could carry through the exercise even if more terms are needed in the Regge fit by using measurements at additional energies. The spin-flip factors for the different Regge poles are interesting quantities to know within the context of any given model. 2 Here we would like to be a little more adventuresome and see if we can determine the energy dependence from existing data by making an additional, plausible assumption; namely, it is easy to see that if τf = τω that the previously described process can be carried through using measurements at only one energy. This assumption is not completely arbitrary; it follows from the assumption of exchange degeneracy for Regge couplings and trajectories and has been much used in the past [5]. It is, however, on shaky foundations and not always successful phenomenologically [6]. We use it here without further apologies because we need it, and, anyhow, we will soon know if it it true or not. We hope at the least that the results to be given below will be of some use in giving some realistic possiblilities. Obviously, they should not be used for polarimetry without some confirmation. We will proceed in the following way: we will first determine the real and imaginary parts of τpC(42) using the data from E950 reported at Spin 2000 [7], with its error ellipse. This will then be converted into values for τP and τR = τf = τω , with errors. Subsequently, this will be used to calculate the analyzing power at the higher RHIC energies with lab momentum pL = 100GeV/c and pL = 250GeV/c for proton on fixed carbon target. We use a modification of the method given in the paper of Buttimore et al [8] to extract the value of τpC from the data. Starting from the formula of [4]