New Recursion Relations for Tree Amplitudes of Gluons

New Recursion Relations for Tree Amplitudes of Gluons

December 2004 | Ruth Britto, Freddy Cachazo and Bo Feng
**Summary:** This paper presents new recursion relations for tree-level gluon amplitudes in gauge theory. The relations allow any tree amplitude to be expressed as a sum over terms constructed from products of two amplitudes with fewer particles and a Feynman propagator. These two amplitudes are physical, with all particles on-shell and momentum conservation preserved. The relations are shown to naturally produce the most compact forms of known amplitudes, including those for up to seven gluons and a new eight-gluon amplitude with alternating helicities. The recursion relations are derived from the infrared (IR) behavior of one-loop N=4 supersymmetric amplitudes, where IR equations are used to derive new recursion relations for tree-level amplitudes. The new relations are tested against known results and are shown to reproduce compact forms of amplitudes, including those for six and seven gluons. The paper also discusses the application of these relations to compute the eight-gluon amplitude with alternating helicities and highlights the surprising fact that tree-level amplitudes can be reduced to products of three-gluon amplitudes and propagators. The paper also explores the implications of these recursion relations for future research, including potential connections to string theory and supersymmetric amplitudes. It outlines a possible proof of the recursion relations based on the IR behavior of one-loop amplitudes and the decomposition of box integrals into products of tree amplitudes. The relations are shown to naturally lead to a trivalent-vertex representation of tree-level amplitudes, which may have connections to string theory and MHV diagrams. The paper concludes with a discussion of closed sets of amplitudes that are invariant under the recursion relations and the possibility of solving them explicitly.**Summary:** This paper presents new recursion relations for tree-level gluon amplitudes in gauge theory. The relations allow any tree amplitude to be expressed as a sum over terms constructed from products of two amplitudes with fewer particles and a Feynman propagator. These two amplitudes are physical, with all particles on-shell and momentum conservation preserved. The relations are shown to naturally produce the most compact forms of known amplitudes, including those for up to seven gluons and a new eight-gluon amplitude with alternating helicities. The recursion relations are derived from the infrared (IR) behavior of one-loop N=4 supersymmetric amplitudes, where IR equations are used to derive new recursion relations for tree-level amplitudes. The new relations are tested against known results and are shown to reproduce compact forms of amplitudes, including those for six and seven gluons. The paper also discusses the application of these relations to compute the eight-gluon amplitude with alternating helicities and highlights the surprising fact that tree-level amplitudes can be reduced to products of three-gluon amplitudes and propagators. The paper also explores the implications of these recursion relations for future research, including potential connections to string theory and supersymmetric amplitudes. It outlines a possible proof of the recursion relations based on the IR behavior of one-loop amplitudes and the decomposition of box integrals into products of tree amplitudes. The relations are shown to naturally lead to a trivalent-vertex representation of tree-level amplitudes, which may have connections to string theory and MHV diagrams. The paper concludes with a discussion of closed sets of amplitudes that are invariant under the recursion relations and the possibility of solving them explicitly.
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