Abstract
We discuss the symmetry factors of Feynman diagrams of scalar field theories with polynomial potential. After giving a concise general formula for them, we present an elementary and direct proof that when computing scattering amplitudes using the homological perturbation lemma, each contributing Feynman diagram is indeed included with the correct symmetry factor.
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I.A. Batalin and G.A. Vilkovisky, Gauge algebra and quantization, Phys. Lett. B 102 (1981) 27 [INSPIRE].
A.S. Schwarz, Geometry of Batalin-Vilkovisky quantization, Commun. Math. Phys. 155 (1993) 249 [hep-th/9205088] [INSPIRE].
B. Jurčo, L. Raspollini, C. Sämann and M. Wolf, L∞-algebras of classical field theories and the Batalin-Vilkovisky formalism, Fortsch. Phys. 67 (2019) 1900025 [arXiv:1809.09899] [INSPIRE].
A. Nützi and M. Reiterer, Scattering amplitudes in YM and GR as minimal model brackets and their recursive characterization, arXiv:1812.06454 [INSPIRE].
M. Reiterer, A homotopy BV algebra for Yang-Mills and color-kinematics, arXiv:1912.03110 [INSPIRE].
T. Macrelli, C. Sämann and M. Wolf, Scattering amplitude recursion relations in Batalin-Vilkovisky-quantizable theories, Phys. Rev. D 100 (2019) 045017 [arXiv:1903.05713] [INSPIRE].
A.S. Arvanitakis, The L∞-algebra of the S-matrix, JHEP 07 (2019) 115 [arXiv:1903.05643] [INSPIRE].
B. Jurčo, T. Macrelli, C. Sämann and M. Wolf, Loop amplitudes and quantum homotopy algebras, JHEP 07 (2020) 003 [arXiv:1912.06695] [INSPIRE].
L. Borsten, B. Jurčo, H. Kim, T. Macrelli, C. Sämann and M. Wolf, Double-copy from homotopy algebras, to appear.
H. Kajiura, Noncommutative homotopy algebras associated with open strings, Rev. Math. Phys. 19 (2007) 1 [math.QA/0306332] [INSPIRE].
V.K.A.M. Gugenheim and L.A. Lambe, Perturbation theory in differential homological algebra. I, Illinois J. Maths. 33 (1989) 566.
V.K.A.M. Gugenheim, L.A. Lambe and J.D. Stasheff, Perturbation theory in differential homological algebra. II, Illinois J. Math. 35 (1991) 357.
M. Crainic, On the perturbation lemma, and deformations, math.AT/0403266.
O. Gwilliam and T. Johnson-Freyd, How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism, arXiv:1202.1554 [INSPIRE].
T. Johnson-Freyd, Homological perturbation theory for nonperturbative integrals, Lett. Math. Phys. 105 (2015) 1605 [arXiv:1206.5319] [INSPIRE].
M. Doubek, B. Jurčo and J. Pulmann, Quantum L∞ algebras and the homological perturbation lemma, Commun. Math. Phys. 367 (2019) 215 [arXiv:1712.02696] [INSPIRE].
J. Pulmann, S-matrix and homological perturbation lemma, MSc Thesis, Charles University, Prague Czech Republic (2016).
B. Zwiebach, Closed string field theory: Quantum action and the B − V master equation, Nucl. Phys. B 390 (1993) 33 [hep-th/9206084] [INSPIRE].
M. Markl, Loop homotopy algebras in closed string field theory, Commun. Math. Phys. 221 (2001) 367 [hep-th/9711045] [INSPIRE].
C.D. Palmer and M.E. Carrington, A general expression for symmetry factors of Feynman diagrams, Can. J. Phys. 80 (2002) 847 [hep-th/0108088] [INSPIRE].
B. Jurčo, H. Kim, T. Macrelli, C. Sämann and M. Wolf, Perturbative quantum field theory and homotopy algebras, PoS CORFU2019 (2020) 199 [arXiv:2002.11168] [INSPIRE].
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Saemann, C., Sfinarolakis, E. Symmetry factors of Feynman diagrams and the homological perturbation lemma. J. High Energ. Phys. 2020, 88 (2020). https://doi.org/10.1007/JHEP12(2020)088
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DOI: https://doi.org/10.1007/JHEP12(2020)088