Sivers effect in semiinclusive deeply inelastic scattering

The Sivers function is extracted from HERMES data on single spin asymmetries in semi-inclusive deeply inelastic scattering. Our analysis use a simple Gaussian model for the distribution of transverse parton momenta, together with the flavor dependence given by the leading $1/N_c$ approximation and a neglect of the Sivers antiquark distribution. We find that within the errors of the data these approximations are sufficient.

Figure 1

Kinematics of the SIDIS process $lp\to l^{\prime }hX$ and the definitions of the azimuthal angles in the lab frame.

Figure 2

The average transverse momentum $⟨P_{h\perp }(z)⟩$ of pions produced in SIDIS as measured by HERMES from a deuterium target [6] vs $z$. The dashed curve is the $⟨P_{h\perp }(z)⟩$ in the Gaussian model (10) with the parameters as fixed here, see Eq. (11). The dotted curve is the $⟨P_{h\perp }(z)⟩$ Gaussian model with the parameters as obtained from a study of data on the Cahn effect [33].

Figure 3

(a) The best fit and the, respectively, 1- and 2-$\sigma$ range for the parameters $A$ and $b$ in the ansatz (20) for the Sivers function. (b) The dependence of the parameter $A$ on the Gaussian width $p_{\mathrm{Siv}}^2$ characterizing the transverse momentum distribution in the Sivers function in the Gaussian model (5). The parameter $b$ is practically $p_{\mathrm{Siv}}^2$-independent.

Figure 4

Left: The $u$-quark Sivers function $x{f}_{1T}^{\perp (1)u}(x)$ as a function of $x$, as extracted from the HERMES data [9]. Shown are the best fit, and its 1- and 2-$\sigma$ regions. Right: Here it is shown that the absolute value of the extracted Sivers function does not exceed half of the positivity bound in the Gaussian model in Eq. (16).

Figure 5

The azimuthal SSA $A_{UT}^{\sin ({\varphi }_h-{\varphi }_S)}$ as function of $x$ for charged pions as obtained from the fit (23) in comparison to the final HERMES data [9].

Figure 6

$A_{UT}^{\sin ({\varphi }_h-{\varphi }_S)}$ as a function of $z$. The data are from the HERMES experiment [9]. The curves show the 1-$\sigma$ variation of our large-$N_c$ constrained fit (23) for the Sivers function, and the impact of varying the Gaussian width $p_{\mathrm{Siv}}^2$ of the Sivers function in the range (24). The $z$-dependent data were not used for the fit, i.e., the comparison serves as a cross check for our results.

Figure 7

$A_{UT}^{\sin ({\varphi }_h-{\varphi }_S)}$ as a function of $x$. The data are from the HERMES experiment [9]. The curves show the effect of modifying the result of large-$N_c$ constrained fit (23) by using the Sivers antiquark distribution functions given in Eq. (25). The figures show that the data [9] have little sensitivity to the Sivers $\overline{q}$-distributions, so that the neglect of the antiquark distributions in the fit was justified.

Figure 8

(a) and (b) $A_{UT}^{\sin ({\varphi }_h-{\varphi }_S)}$ as a function of $x$. The data are from the HERMES experiment [9]. The curves show our best fit (23) for the Sivers function, and the effect of $1/N_c$-corrections as modeled in Eq. (27). The figures show that the statistical uncertainty of the data [9] is of comparable magnitude as the $1/N_c$-corrections. (c) and (d) The same as in Figs. 8a, 8b but for the $z$-dependence of the Sivers SSA.

Figure 9

The Sivers SSA at COMPASS. The theoretical curves indicate the order of magnitude of the effect according to the rough model for $1/N_c$-corrections in Eq. (27). Thus, taking the large-$N_c$ counting rule (18) into account we conclude that the HERMES [10] and COMPASS [10] data are compatible with each other.

Figure 10

The 1-$\sigma$ region of our best fit for $f_{1T}^{\perp (1)u}(x)$ in comparison to a variation of $\pm 30\%$ of our best fit (23), which roughly corresponds to the generic size of $1/N_c$ corrections.

Figure 11

The fits from Ref. 32 of $f_{1T}^{\perp (1)a}(x)$ to the $P_{h\perp }$-weighted preliminary HERMES data [8] lie within the 1-$\sigma$ region of the fit obtained here to the final (non-$P_{h\perp }$-weighted) HERMES data [9].

Figure 12

The Sivers SSA for neutral pions at HERMES as function of $x$ and $z$, respectively, as predicted on the basis of the fit result (23). The error band arises from the 1-$\sigma$ uncertainty of the fit (23).