Polarized hyperons from pA scattering in the gluon saturation regime

We study the production of transversely polarized Λ hyperons in high-energy collisions of protons with large nuclei. The large gluon density of the target at saturation provides an intrinsic semi-hard scale which should naturally allow for a weakcoupling QCD description of the process in terms of a convolution of the quark distribution of the proton with the elementary quark–nucleus scattering cross section (resummed to all twists) and a fragmentation function. In this case of transversely polarized Λ production we employ a so-called polarizing fragmentation function, which is an odd function of the transverse momentum of the Λ relative to the fragmenting quark. Due to this kt -odd nature, the resulting Λ polarization is essentially proportional to the derivative of the quark–nucleus cross section with respect to transverse momentum, which peaks near the saturation momentum scale. Such processes might therefore provide generic signatures for high parton density effects and for the approach to the “black-body” (unitarity) limit of hadronic scattering.  2003 Elsevier Science B.V. All rights reserved. It has been known for over 25 years that Λ’s produced in collisions of unpolarized hadrons exhibit polarization perpendicular to the production plane. As of yet, such data are not available for very high energies where one expects that hadronic cross sections are close to their geometrical values (the “black body limit”). However, the BNL-RHIC collider will soon collide protons and deuterons on gold nuclei at energies of ∼ 200 GeV in the nucleon–nucleon center of mass frame; later on, much higher energies will be accessible at the CERN-LHC. In this Letter, we demonstrate that the polarization of Λ hyperons produced in the forward region in high-energy collisions of protons and heavy nuclei may generically be a sensitive probe of high-density effects and gluon saturation in the target. The wave function of a hadron (or nucleus) boosted to large rapidity exhibits a large number of gluons at small x , which is the fraction of the light-cone momentum carried by the gluon. The density of gluons is expected to saturate when it becomes, parametrically, of the order of the inverse QCD coupling constant αs [1]. The parton density at saturation is denoted by Qs , the so-called saturation momentum. This provides an intrinsic momentum scale [2] E-mail address: [email protected] (D. Boer). 0370-2693/03/$ – see front matter  2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0370-2693(03)00081-9 34 D. Boer, A. Dumitru / Physics Letters B 556 (2003) 33–40 Fig. 1. Kinematics of the pA→ΛX process. The direction of positive Λ polarization is indicated for each quadrant in the Λ production plane. which grows with atomic number and with rapidity because more gluons can be radiated in the initial state when phase space is big. For sufficiently high energies and/or large nuclei, the saturation momentum Qs can become much larger than ΛQCD, such that weak coupling methods are applicable. Forward Λ production in pA collisions is dominated by high-x quarks from the proton traversing the high gluon density region of the heavy nucleus. The quarks typically experience interactions with momentum transfers of the order of the saturation momentum. Thus, for large gluon densities in the target, such that the saturation momentum is in the perturbative regime,Qs 1 GeV, the coherence of the projectile is lost, and the scattered quarks (having an average transverse momentum proportional to Qs ) fragment independently [3]. While nonperturbative constituentquark and diquark scattering and hadronization models [4] have been employed to understand hyperon polarization in collisions of protons with dilute targets, we expect that in the high-energy limit the presence of the intrinsic semihard scale Qs should naturally allow for a weak-coupling QCD description of the process. One can thus calculate the cross section for qA scattering in this kinematical domain within pQCD [5], and the deflected, outgoing quark will subsequently fragment into hadrons, which is described by a fragmentation function. In order to explain the transverse Λ polarization in unpolarized hadron collisions within such a factorized pQCD description, it has been suggested that unpolarized quarks can fragment into transversely polarized hadrons, for instance Λ hyperons. The associated probability [6,7] is described by a so-called polarizing fragmentation function, sometimes also called Sivers (effect) fragmentation function. Its main properties are that it is an odd function of the transverse momentum relative to the quark, kt , and that the Λ polarization is orthogonal to kt , because of parity invariance. The polarizing fragmentation function is defined as [7]1 (1) ∆Dh↑/q(z, kt )≡ D̂h↑/q(z, kt)− D̂h↓/q(z, kt )= D̂h↑/q(z, kt)− D̂h↑/q(z,− kt ), and denotes the difference between the densities D̂h↑/q(z, kt ) and D̂h↓/q(z, kt ) of spin-1/2 hadrons h, with longitudinal momentum fraction z, transverse momentum kt and transverse polarization ↑ or ↓, in a jet originating from the fragmentation of an unpolarized parton q . Clearly, this kt -odd function vanishes when integrated over transverse momentum and also when the transverse momentum and the transverse spin are parallel. In order to set the sign convention for the Λ polarization we define (2) ∆Dh↑/q(z, kt )≡∆Dh↑/q ( z, | kt | ) Ph · ( q × kt ) | q × kt | , where q is the momentum of the unpolarized quark that fragments and Ph is the direction of the polarization vector of the hadron h (the ↑ direction). Fig. 1 shows the kinematics of the process under consideration and indicates the direction of positive Λ polarization for each quadrant in the Λ production plane. It should be emphasized that such a nonzero probability difference ∆Dh↑/q(z, kt ) is allowed by both parity and time reversal invariance. Generally it is expected to occur due to final state interactions in the fragmentation process, 1 Another commonly used notation for the polarizing fragmentation function is D⊥ 1T , but with a slightly different definition [6]. D. Boer, A. Dumitru / Physics Letters B 556 (2003) 33–40 35 where the direction of the transverse momentum yields an oriented orbital angular momentum compensated by the transverse spin of the final observed hadron. This polarizing fragmentation function is the analogue of the so-called Sivers effect for parton distribution functions [8], which yields different probabilities of finding an unpolarized quark in a transversely polarized hadron, depending on the directions of the transverse spin of the hadron and the transverse momentum of the quark. The Sivers effect can lead to single spin asymmetries, for instance in p↑p→ πX, a process for which such (large) asymmetries have been observed in several experiments. Recently, such a single spin asymmetry in ep↑ → e′πX has been calculated in a one-gluon exchange model [9]. Shortly afterwards it was understood [10] as providing a model for the Sivers effect distribution function. A similar calculation has recently been performed by Metz [11] for the production of polarized spin-1/2 hadrons in unpolarized scattering, which can be viewed as providing a model for the polarizing fragmentation function. Here we will not employ such a model calculation, but rather use a parametrization for the polarizing fragmentation functions obtained from a fit to data [7]. However, these model calculations do demonstrate that nonzero Sivers effect functions can arise in principle. Due to the kt -odd nature of the polarizing fragmentation function it is accompanied by a different part of the partonic cross section (essentially the first derivative w.r.t. kt ) compared to the ordinary, unpolarized Λ fragmentation function, which is kt -even. The characteristics of the resulting Λ polarization will turn out to be rather different from presently available data for hadronic collisions at moderately high energies and with “dilute” targets. These data show a Λ polarization that increases approximately linearly as a function of the transverse momentum lt of the Λ, up to lt ∼ 1 GeV/c, after which it becomes flat, up to the highest measured lt values: lt ∼ 4 GeV/c. No indication of a decrease at these high lt values has been observed. Furthermore, the polarization increases with the longitudinal momentum fraction ξ and is to a large extent √ s independent. These features do not change with increasing A [12–14]. The only A dependence observed is a slight overall suppression of the Λ polarization for large A and higher energies. For Cu and Pb fixed targets, probed with a 400 GeV/c proton beam [13,14], the magnitude of the polarization is about 30% lower than for light nuclei. This effect is usually attributed to secondary Λ production through π−N interactions [14]. The slight suppression shows no evidence for a dependence on lt in the investigated range 0.9 < lt < 2.6 GeV/c, albeit with rather low statistical accuracy. It is clear that this data on heavy nuclei is not in the kinematic region where saturation is expected to play a dominant role and the main differences to the results presented below are that in the saturation regime the transverse Λ polarization will depend on the collision energy and no plateau region is expected. We shall now present our calculation of Λ polarization in the gluon saturation regime, following Ref. [7] regarding the treatment of the polarizing fragmentation functions. As mentioned above, in the calculation of the qA cross section one is dealing with small coupling if the target nucleus is very dense; however, the well-known leading-twist pQCD cannot be used when the density of gluons is large. Rather, scattering amplitudes have to be resummed to all orders in α2 s times the density. When the target is probed at a scale Qs , scattering cross sections approach the geometrical “black body” limit, while for momentum transfer far above Qs the target appears dilute and cross sections are approximately determined by the known leading-twist pQCD expressions. At high energies, and in the eikonal approximation, the transverse momentum distribution of quarks is essentially given by the correlation function of two Wilson lines V running along the light-cone at transverse separation rt (in the amplitude and its complex conjugate), (3) σ = ∫ dqt dq + (2π)2 δ ( q+ − p+)〈 1 Nc tr ∣∣∣ ∫ dzt e i qt · zt [V (zt )− 1] ∣∣∣2 〉 . Here, P+ is the large light-cone component of the momentum of the incident proton, and that of the incoming quark is p+ = xP+ (q+ for the outgoing quark). The correlator of Wilson lines has to be evaluated in the background field of the target nucleus. A relatively simple closed expression can be obtained [5] in the “Color Glass Condensate” model of the small-x gluon distribution of the dense target [2]. In that model, the small-x gluons are described as a classical non-Abelian Yang–Mills field arising from a stochastic source of color charge on the light-cone which 36 D. Boer, A. Dumitru / Physics Letters B 556 (2003) 33–40 is averaged over with a Gaussian distribution. The quark qt distribution is then given by [5] q+ dσ qA dq+ dqt d2b = q + P+ δ ( p+ − q+ P+ ) 1 (2π)2 C(qt ), (4) C(qt )= ∫ drt e i qt · rt { exp [ −2Qs ∫ dpt (2π)2 1 p4 t ( 1 − exp(i pt · rt ) )]− 2 exp−Q2s ∫ dpt (2π)2 1 p4 t ] + 1 }