It is well-known that direct analytic continuation of the DGLAP evolution kernel(splitting functions)from space-like to time-like kinematics breaks down at three loops.We identify the origin of this breakdown as due t...It is well-known that direct analytic continuation of the DGLAP evolution kernel(splitting functions)from space-like to time-like kinematics breaks down at three loops.We identify the origin of this breakdown as due to splitting functions not being analytic functions of external momenta.However,splitting functions can be constructed from the squares of(generalized)splitting amplitudes.We establish the rules of analytic continuation for splitting amplitudes,and use them to determine the analytic continuation of certain holomorphic and anti-holomorphic part of splitting functions and transverse-momentum dependent distributions.In this way we derive the time-like splitting functions at three loops without ambiguity.We also propose a reciprocity relation for singlet splitting functions,and provide non-trivial evidence that it holds in QCD at least through three loops.展开更多
Hybrid density functional calculations was used to comprehensively study the electronic structure of S-,Snand Pb-monodoped and（Sn,S）-and（Pb,S）-codoped hexagonal WO_3（h-WO_3）in order to improve their visible ligh...Hybrid density functional calculations was used to comprehensively study the electronic structure of S-,Snand Pb-monodoped and（Sn,S）-and（Pb,S）-codoped hexagonal WO_3（h-WO_3）in order to improve their visible light photocatalytic activity.Results indicate that the（Sn,S）-and（Pb,S）-codoped h-WO_3 can realize a significant band gap reduction and prevent the formation of empty states in the valence band of h-WO_3,while Sn/Pb-monodoped h-WO_3 cannot,because in（Sn,S）-and（Pb,S）-codoping,the S-doping introduces the fully occupied S 3p states in the forbidden band gap of h-WO_3 and the acceptor metals（Sn and Pb）would assist the coupling of the introduced S with its nearest O.In particular,the（Sn,S）-codoped h-WO_3 has the narrowest band gap of 1.85 eV and highest reducing ability among the doped case.Moreover,the calculated optical absorption spectra show that（Sn,S）-codoping can improve the visible light absorption.In short,these results indicate that the（Sn,S）-codoped h-WO_3 is a promising material in solar-driven water splitting.展开更多
基金Supported in part by National Natural Science Foundation of China (11975200)the Zhejiang University Fundamental Research Funds for the Central Universities (2019QNA3005)。
文摘It is well-known that direct analytic continuation of the DGLAP evolution kernel(splitting functions)from space-like to time-like kinematics breaks down at three loops.We identify the origin of this breakdown as due to splitting functions not being analytic functions of external momenta.However,splitting functions can be constructed from the squares of(generalized)splitting amplitudes.We establish the rules of analytic continuation for splitting amplitudes,and use them to determine the analytic continuation of certain holomorphic and anti-holomorphic part of splitting functions and transverse-momentum dependent distributions.In this way we derive the time-like splitting functions at three loops without ambiguity.We also propose a reciprocity relation for singlet splitting functions,and provide non-trivial evidence that it holds in QCD at least through three loops.
基金supported by the National Natural Science Foundation of China (21476024, 21576008, 91334203 and 91634116)the National Key Technology Support Program (2014BAE12B01)+2 种基金Beijing Municipal Science and Technology Project (Z151100003315005)the Fundamental Research Funds for the Central Universities (PYCC1705)the “Chemical Grid Project” of BUCT
文摘Hybrid density functional calculations was used to comprehensively study the electronic structure of S-,Snand Pb-monodoped and（Sn,S）-and（Pb,S）-codoped hexagonal WO_3（h-WO_3）in order to improve their visible light photocatalytic activity.Results indicate that the（Sn,S）-and（Pb,S）-codoped h-WO_3 can realize a significant band gap reduction and prevent the formation of empty states in the valence band of h-WO_3,while Sn/Pb-monodoped h-WO_3 cannot,because in（Sn,S）-and（Pb,S）-codoping,the S-doping introduces the fully occupied S 3p states in the forbidden band gap of h-WO_3 and the acceptor metals（Sn and Pb）would assist the coupling of the introduced S with its nearest O.In particular,the（Sn,S）-codoped h-WO_3 has the narrowest band gap of 1.85 eV and highest reducing ability among the doped case.Moreover,the calculated optical absorption spectra show that（Sn,S）-codoping can improve the visible light absorption.In short,these results indicate that the（Sn,S）-codoped h-WO_3 is a promising material in solar-driven water splitting.
