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KLT relations on $S_2$ factorize closed string amplitudes into product ofopen string tree amplitudes. The field theory limits of KLT factorizationrelations hold in minimal coupling theory of gauge field and gravity. In thispaper, we consider the field theory limits of relations on $D_2$. Though therelations on $D_2$ and KLT factorization relations hold on worldsheets withdifferent topologies, we find the field theory limits of $D_2$ relations alsohold in minimal coupling theory of gauge field and gravity. We use the $D_2$relations to give three- and four-point tree amplitudes where gluons areminimally coupled to gravitons. We also give a discussion on general treeamplitudes for minimal coupling of gauge field and gravity. In general, anytree amplitude with $M$ gravitons in addition to $N$ gluons can be given bypure-gluon tree amplitudes with $N+2M$ legs.

Disk relations for tree amplitudes in minimal coupling theory of gauge field and gravity