Lensing mechanism meets small- x physics: Single transverse spin asymmetry in p↑+p and p

Abstract

We calculate the single transverse spin asymmetry in polarized proton-proton (p ↑ þ p) and polarized proton-nucleus (p ↑ þ A) collisions (A N) generated by a partonic lensing mechanism. The polarized proton is considered in the quark-diquark model while its interaction with the unpolarized target is calculated using the small-x/saturation approach, which includes multiple rescatterings and small-x evolution. The phase required for the asymmetry is caused by a final-state gluon exchange between the quark and diquark, as is standard in the lensing mechanism of Brodsky, Hwang, and Schmidt [Phys. Lett. B 530, 99 (2002)]. Our calculation combines the lensing mechanism with small-x physics in the saturation framework. The expression we obtain for the asymmetry A N of the produced quarks has the following properties: (i) The asymmetry is generated by the dominant elastic scattering contribution and the 1=N 2 c suppressed inelastic contribution (with N c the number of quark colors). (ii) The asymmetry grows or oscillates with the produced quark's transverse momentum p T until the momentum reaches the saturation scale Q s , and then only falls off as 1=p T for larger momenta. (iii) The asymmetry decreases with increasing atomic number A of the target for p T below or near Q s , but is independent of A for p T significantly above Q s. We discuss how these properties may be qualitatively consistent with the data on A N published by the PHENIX Collaboration [

Quark sivers function at small x: spin-dependent odderon and the sub-eikonal evolution

“…One obtains the spin-dependent odderon amplitude O ζw by convoluting a square of the wave function (3.13) with the three-gluon exchange amplitude (3.14). Since the amplitude (3.14) is odd under the r ↔ 0 interchange, only the ∼ r part of the wave function squared contributes: this is exactly the part of the wave function dependent on the proton polarization X [43]. Putting the wave function squared and the amplitude (3.14) together we have…”

“…We want the coupling of the odderon to the transverse spin of the proton, so we need to insert the odderon exchange between the light-cone wave functions of the proton splitting into the quark-diquark pair, and the same wave function, but complex conjugate, describing the merger of the pair back into the proton. The light-cone wave function in the transverse spin basis for a proton at transverse position u and with transverse polarization X, with the quark carrying the fraction γ of the proton p + 1 momentum and the transverse polarization X (as labeled in figure 7), is [43]…”

Quark Sivers Function at Small-$x$: Leading contribution from the Spin-Dependent Odderon

“…We construct the odderon amplitude O ζw by calculating the diagram shown in Fig. 2, where the proton splits into a quark-diquark color dipole and exchanges three gluons in the symmetric color configuration d a bc = 2Tr[t a {t b , t c }] with the quark-antiquark dipole at and w. We need the light-cone wave function in the transverse spin basis for a proton at transverse position u and with transverse polarization X , with the quark carrying the fraction γ of the proton p + 1 momentum and the transverse polarization X [13]…”

Quark Sivers Function at Small $x$: Spin-Dependent Odderon and the Sub-Eikonal Evolution

“…The light-cone wave function in the transverse spin basis for a proton at transverse position u and with transverse polarization X, with the quark carrying the fraction γ of the proton p + 1 momentum and the transverse polarization X (as labeled in Fig. 7), is [43]…”

“…One obtains the spin-dependent odderon amplitude O ζw by convoluting a square of the wave function ( 55) with the three-gluon exchange amplitude (56). Since the amplitude ( 56) is odd under the r ↔ 0 interchange, only the ∼ r part of the wave function squared contributes: this is exactly the part of the wave function dependent on the proton polarization X [43]. Putting the wave function squared and the amplitude (56) together we have…”