The Gross-Llewellyn Smith sum rule up to ${cal O}(alpha_s^4)$-order QCD corrections

In the paper, we analyze the properties of Gross-Llewellyn Smith (GLS) sum rule by using the $mathcal{O}(alpha_s^4)$-order QCD corrections with the help of principle of maximum conformality (PMC). By using the PMC single-scale approach, we obtain an accurate renormalization scale-and-scheme independent fixed-order pQCD contribution for GLS sum rule, e.g. $S^{rm GLS}(Q_0^2=3{rm GeV}^2)|_{rm PMC}=2.559^{+0.023}_{-0.024}$, where the error is squared average of those from $Deltaalpha_s(M_Z)$, the predicted $mathcal{O}(alpha_s^5)$-order terms predicted by using the Pad{e} approximation approach. After applying the PMC, a more convergent pQCD series has been obtained, and the contributions from the unknown higher-order terms are highly suppressed. In combination with the nonperturbative high-twist contribution, our final prediction of GLS sum rule agrees well with the experimental data given by the CCFR collaboration.