Abstract
We point out that the one-particle-irreducible vacuum correlation functions of a QFT are the structure constants of an L∞-algebra, whose Jacobi identities hold whenever there are no local gauge anomalies. The LSZ prescription for S-matrix elements is identified as an instance of the “minimal model theorem” of L∞-algebras. This generalises the algebraic structure of closed string field theory to arbitrary QFTs with a mass gap and leads to recursion relations for amplitudes (albeit ones only immediately useful at tree-level, where they reduce to Berends-Giele-style relations as shown in [1]).
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References
T. Macrelli, C. Sämann and M. Wolf, Scattering Amplitude Recursion Relations in BV Quantisable Theories, arXiv:1903.05713 [INSPIRE].
H. Lehmann, K. Symanzik and W. Zimmermann, On the formulation of quantized field theories, Nuovo Cim.1 (1955) 205 [INSPIRE].
C. Itzykson and J.B. Zuber, Quantum Field Theory, International Series In Pure and Applied Physics, McGraw-Hill, New York (1980) [INSPIRE].
J. Zinn-Justin, Renormalization of Gauge Theories, Lect. Notes Phys.37 (1975) 1 [INSPIRE].
I.A. Batalin and G.A. Vilkovisky, Quantization of Gauge Theories with Linearly Dependent Generators, Phys. Rev.D 28 (1983) 2567 [Erratum ibid.D 30 (1984) 508] [INSPIRE].
I.A. Batalin and G.A. Vilkovisky, Existence Theorem for Gauge Algebra, J. Math. Phys.26 (1985) 172 [INSPIRE].
I.A. Batalin and G.A. Vilkovisky, Closure of the Gauge Algebra, Generalized Lie Equations and Feynman Rules, Nucl. Phys.B 234 (1984) 106 [INSPIRE].
I.A. Batalin and G.A. Vilkovisky, Gauge Algebra and Quantization, Phys. Lett.102B (1981) 27 [INSPIRE].
I.A. Batalin and G.A. Vilkovisky, Relativistic S Matrix of Dynamical Systems with Boson and Fermion Constraints, Phys. Lett.69B (1977) 309 [INSPIRE].
B. Zwiebach, Closed string field theory: Quantum action and the B-V master equation, Nucl. Phys.B 390 (1993) 33 [hep-th/9206084] [INSPIRE].
T. Lada and J. Stasheff, Introduction to SH Lie algebras for physicists, Int. J. Theor. Phys.32 (1993) 1087 [hep-th/9209099] [INSPIRE].
R. D’Auria and P. Fré, Geometric Supergravity in d = 11 and Its Hidden Supergroup, Nucl. Phys.B 201 (1982) 101 [Erratum ibid.B 206 (1982) 496] [INSPIRE].
R. D’Auria, P. Fré and T. Regge, Graded Lie Algebra Cohomology and Supergravity, Riv. Nuovo Cim.3N12 (1980) 1 [INSPIRE].
L. Castellani, P. Fré, F. Giani, K. Pilch and P. van Nieuwenhuizen, Gauging of d = 11 Supergravity?, Annals Phys.146 (1983) 35 [INSPIRE].
H. Sati, U. Schreiber and J. Stasheff, L ∞algebra connections and applications to String- and Chern-Simons n-transport, in Quantum Field Theory, B. Fauser, J. Tolksdorf and E. Zeidler eds., Birkhauser (2009) [arXiv:0801.3480] [INSPIRE].
F.A. Berends, G.J.H. Burgers and H. van Dam, On the Theoretical Problems in Constructing Interactions Involving Higher Spin Massless Particles, Nucl. Phys.B 260 (1985) 295 [INSPIRE].
M. Schlessinger and J. Stasheff, Deformation theory and rational homotopy type, arXiv:1211.1647.
A. Sen, Equations of Motion in Nonpolynomial Closed String Field Theory and Conformal Invariance of Two-dimensional Field Theories, Phys. Lett.B 241 (1990) 350 [INSPIRE].
S.R. Coleman and E.J. Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking, Phys. Rev.D 7 (1973) 1888 [INSPIRE].
E. Witten and B. Zwiebach, Algebraic structures and differential geometry in 2-D string theory, Nucl. Phys.B 377 (1992) 55 [hep-th/9201056] [INSPIRE].
E.P. Verlinde, The Master equation of 2-D string theory, Nucl. Phys.B 381 (1992) 141 [hep-th/9202021] [INSPIRE].
H. Kajiura, Noncommutative homotopy algebras associated with open strings, Rev. Math. Phys.19 (2007) 1 [math/0306332] [INSPIRE].
K. Münster and I. Sachs, Homotopy Classification of Bosonic String Field Theory, Commun. Math. Phys.330 (2014) 1227 [arXiv:1208.5626] [INSPIRE].
S. Konopka, The S-matrix of superstring field theory, JHEP11 (2015) 187 [arXiv:1507.08250] [INSPIRE].
A. Nützi and M. Reiterer, Scattering amplitudes in YM and GR as minimal model brackets and their recursive characterization, arXiv:1812.06454 [INSPIRE].
A. Jevicki and C.-k. Lee, The S Matrix Generating Functional and Effective Action, Phys. Rev.D 37 (1988) 1485 [INSPIRE].
O. Hohm and B. Zwiebach, L ∞Algebras and Field Theory, Fortsch. Phys.65 (2017) 1700014 [arXiv:1701.08824] [INSPIRE].
B. Jurčo, L. Raspollini, C. Sämann and M. Wolf, L ∞-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism, arXiv:1809.09899 [INSPIRE].
P. Ritter and C. Sämann, L ∞-Algebra Models and Higher Chern-Simons Theories, Rev. Math. Phys.28 (2016) 1650021 [arXiv:1511.08201] [INSPIRE].
D. Fiorenza, C.L. Rogers and U. Schreiber, A Higher Chern-Weil derivation of AKSZ σ-models, Int. J. Geom. Meth. Mod. Phys.10 (2013) 1250078 [arXiv:1108.4378] [INSPIRE].
A. Kotov and T. Strobl, Characteristic classes associated to Q-bundles, Int. J. Geom. Meth. Mod. Phys.12 (2014) 1550006 [arXiv:0711.4106] [INSPIRE].
O. Hohm, V. Kupriyanov, D. Lüst and M. Traube, Constructions of L ∞algebras and their field theory realizations, Adv. Math. Phys.2018 (2018) 9282905 [arXiv:1709.10004] [INSPIRE].
R. Blumenhagen, M. Fuchs and M. Traube, \( \mathcal{W} \)algebras are L ∞algebras, JHEP07 (2017) 060 [arXiv:1705.00736] [INSPIRE].
M. Cederwall and J. Palmkvist, L ∞algebras for extended geometry from Borcherds superalgebras, Commun. Math. Phys.369 (2019) 721 [arXiv:1804.04377] [INSPIRE].
Y. Cagnacci, T. Codina and D. Marques, L ∞algebras and Tensor Hierarchies in Exceptional Field Theory and Gauged Supergravity, JHEP01 (2019) 117 [arXiv:1807.06028] [INSPIRE].
S. Lavau, H. Samtleben and T. Strobl, Hidden Q-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions, J. Geom. Phys.86 (2014) 497 [arXiv:1403.7114] [INSPIRE].
S. Lavau, Tensor hierarchies and Leibniz algebras, J. Geom. Phys.144 (2019) 147 [arXiv:1708.07068] [INSPIRE].
A.S. Arvanitakis, Brane Wess-Zumino terms from AKSZ and exceptional generalised geometry as an L ∞-algebroid, arXiv:1804.07303 [INSPIRE].
M. Alexandrov, A. Schwarz, O. Zaboronsky and M. Kontsevich, The Geometry of the master equation and topological quantum field theory, Int. J. Mod. Phys.A 12 (1997) 1405 [hep-th/9502010] [INSPIRE].
J.M.L. Fisch and M. Henneaux, Homological Perturbation Theory and the Algebraic Structure of the Antifield-Antibracket Formalism for Gauge Theories, Commun. Math. Phys.128 (1990) 627 [INSPIRE].
G. Barnich, R. Fulp, T. Lada and J. Stasheff, The sh Lie structure of Poisson brackets in field theory, Commun. Math. Phys.191 (1998) 585 [hep-th/9702176] [INSPIRE].
R. Fulp, T. Lada and J. Stasheff, sh-Lie algebras induced by gauge transformations, Commun. Math. Phys.231 (2002) 25 [INSPIRE].
M. Movshev and A.S. Schwarz, On maximally supersymmetric Yang-Mills theories, Nucl. Phys.B 681 (2004) 324 [hep-th/0311132] [INSPIRE].
M. Movshev and A.S. Schwarz, Algebraic structure of Yang-Mills theory, Prog. Math.244 (2006) 473 [hep-th/0404183] [INSPIRE].
A.M. Zeitlin, Homotopy Lie Superalgebra in Yang-Mills Theory, JHEP09 (2007) 068 [arXiv:0708.1773] [INSPIRE].
A.M. Zeitlin, Formal Maurer-Cartan Structures: From CFT to Classical Field Equations, JHEP12 (2007) 098 [arXiv:0708.0955] [INSPIRE].
A.M. Zeitlin, Batalin-Vilkovisky Yang-Mills theory as a homotopy Chern-Simons theory via string field theory, Int. J. Mod. Phys.A 24 (2009) 1309 [arXiv:0709.1411] [INSPIRE].
A.M. Zeitlin, String field theory-inspired algebraic structures in gauge theories, J. Math. Phys.50 (2009) 063501 [arXiv:0711.3843] [INSPIRE].
A.M. Zeitlin, Conformal Field Theory and Algebraic Structure of Gauge Theory, JHEP03 (2010) 056 [arXiv:0812.1840] [INSPIRE].
M. Roček and A.M. Zeitlin, Homotopy algebras of differential (super)forms in three and four dimensions, Lett. Math. Phys.108 (2018) 2669 [arXiv:1702.03565] [INSPIRE].
F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys.B 306 (1988) 759 [INSPIRE].
M. Kontsevich, Deformation quantization of Poisson manifolds. 1., Lett. Math. Phys.66 (2003) 157 [q-alg/9709040] [INSPIRE].
C. Braun and A. Lazarev, Unimodular homotopy algebras and Chern-Simons theory, J. Pure Appl. Algebra219 (2015) 5158 [arXiv:1309.3219] [INSPIRE].
T.V. Kadeishvili, On the homology theory of fibre spaces, Russ. Math. Surv.35 (1980) 231.
H. Kajiura, Homotopy algebra morphism and geometry of classical string field theory, Nucl. Phys.B 630 (2002) 361 [hep-th/0112228] [INSPIRE].
J.-L. Loday and B. Vallette, Algebraic operads, vol. 346, Springer Science & Business Media (2012).
J. Gomis, J. Paris and S. Samuel, Antibracket, antifields and gauge theory quantization, Phys. Rept.259 (1995) 1 [hep-th/9412228] [INSPIRE].
A.S. Arvanitakis, Chiral strings, topological branes and a generalised Weyl-invariance, Int. J. Mod. Phys.A 34 (2019) 1950031 [arXiv:1705.03516] [INSPIRE].
M. Henneaux and C. Teitelboim, Quantization of gauge systems, Princeton University Press (1992).
D.M. Jackson, A. Kempf and A.H. Morales, A robust generalization of the Legendre transform for QFT, J. Phys.A 50 (2017) 225201 [arXiv:1612.00462] [INSPIRE].
D. Anselmi, Removal of divergences with the Batalin-Vilkovisky formalism, Class. Quant. Grav.11 (1994) 2181 [INSPIRE].
G. Barnich, Classical and quantum aspects of the extended antifield formalism, Ann. U. Craiova Phys.10 (2000) 1 [hep-th/0011120] [INSPIRE].
M. Markl, Loop homotopy algebras in closed string field theory, Commun. Math. Phys.221 (2001) 367 [hep-th/9711045] [INSPIRE].
R. Fukuda, M. Komachiya and M. Ukita, On-shell Expansion of the Effective Action: S Matrix and the Ambiguity-Free Stability Criterion, Phys. Rev.D 38 (1988) 3747 [INSPIRE].
C.-j. Kim and V.P. Nair, Recursion rules for scattering amplitudes in nonAbelian gauge theories, Phys. Rev.D 55 (1997) 3851 [hep-th/9608156] [INSPIRE].
V.P. Nair, Quantum field theory: A modern perspective. Springer (2005).
W. Siegel, Fields, hep-th/9912205 [INSPIRE].
I.Y. Arefeva, L.D. Faddeev and A.A. Slavnov, Generating Functional for the s Matrix in Gauge Theories, Theor. Math. Phys.21 (1975) 1165 [INSPIRE].
G. Kallen, On the definition of the Renormalization Constants in Quantum Electrodynamics, Helv. Phys. Acta25 (1952) 417 [INSPIRE].
H. Lehmann, On the Properties of propagation functions and renormalization contants of quantized fields, Nuovo Cim.11 (1954) 342 [INSPIRE].
M. Reed and B. Simon, Methods of modern mathematical physics, vol. III: Scattering theory, Academic Press, New York, San Francisoco, London (1979).
E. Zeidler ed., Quantum field theory. I: Basics in mathematics and physics. A bridge between mathematicians and physicists, Springer (2006).
T. Adamo, S. Nakach and A.A. Tseytlin, Scattering of conformal higher spin fields, JHEP07 (2018) 016 [arXiv:1805.00394] [INSPIRE].
T. Adamo, E. Casali, L. Mason and S. Nekovar, Scattering on plane waves and the double copy, Class. Quant. Grav.35 (2018) 015004 [arXiv:1706.08925] [INSPIRE].
K.J. Costello, Renormalisation and the Batalin-Vilkovisky formalism, arXiv:0706.1533 [INSPIRE].
D. Anselmi, Master Functional And Proper Formalism For Quantum Gauge Field Theory, Eur. Phys. J.C 73 (2013) 2363 [arXiv:1205.3862] [INSPIRE].
K. Costello, Renormalization and effective field theory, No. 170, American Mathematical Soc. (2011).
C. Braun and J. Maunder, Minimal models of quantum homotopy Lie algebras via the BV-formalism, J. Math. Phys.59 (2018) 063512 [arXiv:1703.00082] [INSPIRE].
J.S.R. Chisholm, Change of variables in quantum field theories, Nucl. Phys.26 (1961) 469.
S. Kamefuchi, L. O’Raifeartaigh and A. Salam, Change of variables and equivalence theorems in quantum field theories, Nucl. Phys.28 (1961) 529.
R.E. Kallosh and I.V. Tyutin, The Equivalence theorem and gauge invariance in renormalizable theories, Yad. Fiz.17 (1973) 190 [INSPIRE].
M. Penkava and A.S. Schwarz, A ∞algebras and the cohomology of moduli spaces, hep-th/9408064 [INSPIRE].
R. Blumenhagen, I. Brunner, V. Kupriyanov and D. Lüst, Bootstrapping non-commutative gauge theories from L ∞algebras, JHEP05 (2018) 097 [arXiv:1803.00732] [INSPIRE].
R. Blumenhagen, M. Brinkmann, V. Kupriyanov and M. Traube, On the Uniqueness of L ∞bootstrap: Quasi-isomorphisms are Seiberg-Witten Maps, J. Math. Phys.59 (2018) 123505 [arXiv:1806.10314] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett.B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
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Arvanitakis, A.S. The L∞-algebra of the S-matrix. J. High Energ. Phys. 2019, 115 (2019). https://doi.org/10.1007/JHEP07(2019)115
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DOI: https://doi.org/10.1007/JHEP07(2019)115