Loop Amplitudes and Quantum Homotopy Algebras
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Springer
Received: February 11, 2020 Accepted: June 2, 2020 Published: July 1, 2020
Branislav Jurˇ co,a Tommaso Macrelli,b Christian S¨ amannc and Martin Wolfb a
Charles University Prague, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovsk´e 83, Prague 186 75, Czech Republic b Department of Mathematics, University of Surrey, Guildford GU2 7XH, U.K. c Maxwell Institute for Mathematical Sciences, Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, U.K.
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We derive a recursion relation for loop-level scattering amplitudes of Lagrangian field theories that generalises the tree-level Berends-Giele recursion relation in Yang-Mills theory. The origin of this recursion relation is the homological perturbation lemma, which allows us to compute scattering amplitudes from minimal models of quantum homotopy algebras in a recursive way. As an application of our techniques, we give an alternative proof of the relation between non-planar and planar colour-stripped scattering amplitudes. Keywords: Scattering Amplitudes, BRST Quantization, Gauge Symmetry ArXiv ePrint: 1912.06695
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)003
JHEP07(2020)003
Loop Amplitudes and Quantum Homotopy Algebras
Contents 1 Introduction
1
2 Scalar field theory 2.1 Homotopy algebra 2.2 Tree-level scattering amplitudes 2.3 Loop-level scattering amplitudes
2 2 3 4 6 6 8 10
4 Conclusions
12
1
Introduction
Batalin-Vilkovisky (BV) quantisation [1, 2] does not only help in gauge fixing and quantising complicated quantum field theories, but it also provides an important link between classical and quantum field theories and homotopy algebras even for theories without gauge symmetries. At the classical level, the BV formalism associates to every Lagrangian field theory an L∞ -algebra which captures both the kinematics and the dynamics of the field theory [3–7]. The action of the classical field theory translates to the homotopy Maurer-Cartan action of its L∞ -algebra, having the same set of fields, symmetries, equations of motions and Noether currents. Physically equivalent classical field theories have quasi-isomorphic L∞ algebras, which is the appropriate notion of equivalence from a mathematical perspective. The tree-level scattering amplitudes of a quantum field theory are encoded in the minimal models (i.e. smallest quasi-isomorphic forms) of its L∞ -algebra. Recently, it was shown that the quasi-isomorphism between both induces recursion relations for these amplitudes [8] (see also [9] for related discussions of the S-matrix in the L∞ -language, [10] for the tree-level perturbiner expansion, and [11, 12] for an L∞ -interpretation of tree-level onshell recursion relations). In the context of Yang-Mills (YM) theory, this recursion relation is known as the Berends-Giele recursion relation [13]. In this article, we generalise the results of [8] to loop-level
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