Deriving Graph-based BCJ Relation From String Theory

The calculation of scattering amplitudes has always been an important issue in many fields such as quantum field theory.When faced with a more complicated scattering process,using traditional methods to calculate the scattering amplitude may encounter some difficulties.In recent decades,domestic and foreign colleagues have made a series of important progress in the research of new structures and calculation methods of scattering amplitudes.These methods can solve some of the difficulties encountered in traditional methods.Discovering new scattering amplitude relations is an important part of it.It can not only effectively simplify the calculation of scattering amplitudes,but also reveal the hidden structure of scattering amplitudes.The Yang-Mills theory tree level color-ordered scattering amplitudes satisfy two important relations,which are the Bern-Carrasco-Johansson(BCJ)relation and the Kleiss-Kuijf(KK)relation.These two relations which together with the inherent cyclic symmetry of the gauge field can effectively simplify the calculation of scattering amplitudes.And these two relations can be proved by the method of string theory.The scattering amplitudes of Einstein-Yang-Mills theory satisfy the so-called recursive expansion relation.Using this relation,the Einstein-Yang-Mills tree-level scattering amplitudes can be converted into Einstein-Yang-Mills tree-level scattering amplitudes with fewer gravitons.And finally be transformed into a combination of pure Yang-Mills field scattering amplitudes.The obtained combination coefficients can be expressed in the form of graphs.On the basis of this development,further research on the gauge-invariant nature of field theory has led to a new relation called the graph-based BCJ relation.We first briefly review some of the important results in the field of scattering amplitudes.We further introduce some useful scattering amplitude relations in string theory and field theory.The string theory approach to the BCJ relation and the KK relation of the tree-level color-ordered gluon amplitudes will also be introduced in detail.Based on the above discussions,this thesis points out that the graph-based BCJ relation can also be proved by properties of string theory scattering amplitudes.This thesis first takes the two-point,three-point,four-point,and five-point cases as examples.By considering the combination of specific open string amplitudes in string theory,using the known relation between the string theory tree-level color-ordered scattering amplitudes,and taking the imaginary part of the obtained expression,a new string theory scattering amplitude relation is derived.This relation can be regarded as the graph-based BCJ relation of open string tree-level scattering amplitudes.At the same time,when we take the real part of the expression,we also get an additional relation.This relation can be regarded as the graph-based KK relation of the open string tree-level scattering amplitudes.In the field theory limit,the graph-based BCJ relation of the string treelevel scattering amplitudes can be transformed into the graph-based BCJ relation in Yang-Mills field theory.Furthermore,when the field theory limit is imposed on the graph-based KK relation of string tree-level scattering amplitudes,we get the graphbased KK relation in Yang-Mills field theory.This thesis also provides a recursive method to derive the general case of graph-based BCJ relation and KK relation.