Role of Cahn and Sivers effects in deep inelastic scattering

The role of intrinsic $\mathit{k}{}_{\perp }$ in inclusive and semi-inclusive Deep Inelastic Scattering processes ($\ell p\to \ell hX$) is studied with exact kinematics within QCD parton model at leading order; the dependence of the unpolarized cross section on the azimuthal angle between the leptonic and the hadron production planes (Cahn effect) is compared with data and used to estimate the average values of $k_{\perp }$ both in quark distribution and fragmentation functions. The resulting picture is applied to the description of the weighted single spin asymmetry $A_{UT}^{\sin ({\varphi }_{\pi }-{\varphi }_S)}$ recently measured by the HERMES collaboration at DESY; this allows to extract some simple models for the quark Sivers functions. These are compared with the Sivers functions which succeed in describing the data on transverse single spin asymmetries in $p^{\uparrow }p\to \pi X$ processes; the two sets of functions are not inconsistent. The extracted Sivers functions give predictions for the COMPASS measurement of $A_{UT}^{\sin ({\varphi }_{\pi }-{\varphi }_S)}$ in agreement with recent preliminary data, while their contribution to HERMES $A_{UL}^{\sin {\varphi }_{\pi }}$ is computed and found to be small. Predictions for $A_{UT}^{\sin ({\varphi }_K-{\varphi }_S)}$ for kaon production at HERMES are also given.

Figure 1

Kinematics of DIS processes.

Figure 2

Kinematics of the fragmentation process.

Figure 3

Three dimensional kinematics of the SIDIS process.

Figure 4

Fits to the $\cos {\varphi }_h$ dependence of the cross section: the dashed line is obtained with exact kinematics, while the solid bold line includes terms up to $\mathcal{O}(k_{\perp }/Q)$ only. The data are from Ref. 6.

Figure 5

The normalized cross section $d\sigma /d{P}_T^2$: the dashed line is obtained with exact kinematics, while the solid bold line includes terms up to $\mathcal{O}(k_{\perp }/Q)$ only. The shadowed region corresponds to varying the parameters $⟨k_{\perp }^2⟩$ and $⟨p_{\perp }^2⟩$, given in Eq. (46) of the text, by $20\%$. The data are from Ref. 31.

Figure 6

$⟨\cos {\varphi }_h⟩$, as given by Eq. (44), as a function of $P_T^{cut}$, left panel, and $x_F$, right panel: the dashed line is obtained with exact kinematics, while the solid bold line includes terms up to $\mathcal{O}(k_{\perp }/Q)$ only. The shadowed region corresponds to varying the parameters $⟨k_{\perp }^2⟩$ and $⟨p_{\perp }^2⟩$, given in Eq. (47) of the text, by $20\%$. $w_1(y)=\frac{(2-y)\sqrt{1-y}}{1+(1-y{)}^2}$. The data are from Ref. 24, left panel, and Ref. 6, right panel.

Figure 7

The product $h(k_{\perp })·g(k_{\perp })$, solid line, compared with the product $h^{\prime }(k_{\perp })·g(k_{\perp })$, dashed line, as a function of $k_{\perp }$.

Figure 8

HERMES data on $A_{UT}^{\sin ({\varphi }_{\pi }-{\varphi }_S)}$ [10] for scattering off a transversely polarized proton target and pion production. The curves are the results of our fits, with exact kinematics (dashed line) or keeping only terms up to $\mathcal{O}(k_{\perp }/Q)$ (solid bold line). The shadowed region corresponds to the theoretical uncertainty due to the parameter errors. Data on the $P_T$ dependence of $A_{UT}^{\sin ({\varphi }_{\pi }-{\varphi }_S)}$ are not available yet, and the third column of the figure gives our predictions.

Figure 9

COMPASS preliminary data [16] on $A_{UT}^{\sin ({\varphi }_h-{\varphi }_S)}$ for scattering off a transversely polarized deuteron target and the production of positively ($h^{+}$) and negatively ($h^{-}$) charged hadrons. The curves show our predictions, according to the values of the parameters for the Sivers functions given in Table  and obtained from fitting the HERMES data on $A_{UT}^{\sin ({\varphi }_{\pi }-{\varphi }_S)}$. Again, the dashed line refers to exact kinematics, Eq. (58), while the solid bold line is obtained by keeping only terms up to $\mathcal{O}(k_{\perp }/Q)$. The shadowed region shows the theoretical uncertainty due to the parameter errors.

Figure 10

Predictions for kaon asymmetries $A_{UT}^{\sin ({\varphi }_h-{\varphi }_S)}$ at HERMES kinematics for scattering off a transversely polarized proton target. The curves correspond to calculation with exact kinematics (dashed line) or keeping only terms up to $\mathcal{O}(k_{\perp }/Q)$ (solid bold line). The shadowed region corresponds to the theoretical uncertainty due to the parameter errors.

Figure 11

HERMES data on $A_{UL}^{\sin {\varphi }_{\pi }}$ [9] for scattering off a longitudinally polarized proton target and pion production. The curves show the contribution of our Sivers functions alone, with exact kinematics (Eq. (57), dashed line) or keeping only terms up to $\mathcal{O}(k_{\perp }/Q)$ (solid bold line). The shadowed region corresponds to the theoretical uncertainty due to the parameter errors.