The spin-dependent structure function $g_1(x)$ of the proton from polarized deep-inelastic muon scattering

We present a new measurement of the virtual photon proton asymmetry $A_1^{\rm p}$ from deep inelastic scattering of polarized muons on polarized protons in the kinematic range $0.0008 < x < 0.7$ and $0.2 < Q^{2} < 100$ GeV$^{2}$. With this, the statistical uncertainty of our measurement has improved by a factor of $2$ compared to our previous measurements. The spin-dependent structure function $g_{1}^{\rm p}$ is determined for the data with $Q^{2} > 1$ GeV$^{2}$. A perturbative QCD evolution in next-to-leading order is used to determine $g_1^{\rm p}(x)$ at a constant $Q^2$. At $Q^{2} = 10$ GeV$^{2}$ we find, in the measured range, $\int_{0.003}^{0.7} g_{1}^{\rm p}(x){\rm d}x = 0.139 \pm 0.006~({\rm stat})\pm 0.008~({\rm syst)} \pm 0.006~({\rm evol})$. The value of the first moment $\Gamma_{1}^{\rm p} = \int_{0}^{1} g_{1}^{\rm p}(x){\rm d}x$ of $g_{1}^{\rm p}$ depends on the approach used to describe the behaviour of $g_{1}^{\rm p}$ at low $x$. We find that the Ellis-Jaffe sum rule is violated. With our published result for $\Gamma_{1}^{\rm d}$ we confirm the Bjorken sum rule with an accuracy of $\approx 15\%$ at the one standard deviation level.