Geodesic
BRST–BV approach for interacting higher-spin fields
Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 1, pp. 98-126 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We develop the BRST–BV approach to the construction of the general off-shell Lorentz covariant cubic, quartic, and $e$-tic interaction vertices for irreducible higher-spin fields on $d$-dimensional Minkowski space. We consider two different cases for interacting integer higher-spin fields with both massless and massive fields. The deformation procedure to find a minimal BRST–BV action for interacting higher-spin fields, defined with help of a generalized Hilbert space, is based on the preservation of the master equation in each power of the coupling constant $g$ starting from the Lagrangian formulation for a free gauge theory. For illustration, we consider the construction of local cubic vertices for $k$ irreducible massless fields of integer helicities, and $k-1$ massless fields and one massive field of spins $s_1, \dots, s_{k-1}, s_k$. For a triple of two massless scalars and a tensor field of integer spin, the BRST–BV action with cubic interaction is explicitly found. In contrast to the previous results on cubic vertices, following our results for the BRST approach to massless fields, we use a single BRST–BV action instead of the classical action with reducible gauge transformations. The procedure is based on the complete BRST operator that includes the trace constraints used in defining the irreducible representation with a definite integer spin.
Keywords: higher-spin field theory, gauge theories, BRST operator, field–antifield formalism, totally symmetric higher-spin fields, cubic interaction vertices. 
@article{TMF_2023_217_1_a6,
     author = {A. A. Reshetnyak},
     title = {BRST{\textendash}BV approach for interacting higher-spin fields},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {98--126},
     year = {2023},
     volume = {217},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_217_1_a6/}
}
TY  - JOUR
AU  - A. A. Reshetnyak
TI  - BRST–BV approach for interacting higher-spin fields
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2023
SP  - 98
EP  - 126
VL  - 217
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2023_217_1_a6/
LA  - ru
ID  - TMF_2023_217_1_a6
ER  - 
%0 Journal Article
%A A. A. Reshetnyak
%T BRST–BV approach for interacting higher-spin fields
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2023
%P 98-126
%V 217
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2023_217_1_a6/
%G ru
%F TMF_2023_217_1_a6
A. A. Reshetnyak. BRST–BV approach for interacting higher-spin fields. Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 1, pp. 98-126. http://geodesic.mathdoc.fr/item/TMF_2023_217_1_a6/

[1] M. A. Vasiliev, “Higher spin gauge theories in any dimension”, C. R. Phys., 5:9–10 (2004), 1101–1109, arXiv: hep-th/0409260 | DOI | MR

[2] X. Bekaert, S. Cnockaert, C. Iazeolla, M. A. Vasiliev, “Nonlinear higher spin theories in various dimensions”, Higher Spin Gauge Theories, Proceedings of 1st Solvay Workshop (Brussels, Belgium, 12–14 May, 2004), eds. R. Argurio, G. Barnich, G. Bonelli, M. Grigoriev, International Solvay Institutes for Physics and Chemistry, Brussels, 2006, 132–197, arXiv: hep-th/0503128

[3] A. Fotopoulos, M. Tsulaia, “Gauge-invariant Lagrangians for free and interacting higher spin fields: a review of the BRST formulation”, Internat. J. Modern Phys. A, 24:1 (2008), 1–60, arXiv: 0805.1346 | DOI | MR | Zbl

[4] X. Bekaert, N. Boulanger, P. Sundell, “How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples”, Rev. Modern Phys., 84:3 (2012), 987–1009, arXiv: 1007.0435 | DOI

[5] M. A. Vasiliev, “Higher-spin theory and space-time metamorphoses”, Modifications of Einstein's Theory of Gravity at Large Distances, Lecture Notes in Physics, 892, ed. E. Papantonopoulos, Springer, Cham, 2015, 227–264, arXiv: 1404.1948 | DOI | MR

[6] X. Bekaert, N. Boulanger, A. Campaneoli, M. Chodaroli, D. Francia, M. Grigoriev, E. Sezgin, E. Skvortsov, Snowmass white paper: Higher spin gravity and higher spin symmetry, arXiv: 2205.01567

[7] D. Ponomarev, “Basic intoroduction to higher-spin theories”, Internat. J. Theor. Phys., 62 (2023), 146, 141 pp., arXiv: 2206.15385 | DOI

[8] R. Manvelyan, K. Mkrtchyan, W. Rühl, “General trilinear interaction for arbitrary even higher spin gauge fields”, Nucl. Phys. B, 836:3 (2010), 204–221, arXiv: 1003.2877 | DOI | MR

[9] R. Manvelyan, K. Mkrtchyan, W. Rḧl, “A generating function for the cubic interactions of higher spin fields”, Phys. Lett. B, 696:4 (2011), 410–415, arXiv: 1009.1054 | DOI | MR

[10] E. Joung, M. Taronna, “Cubic interactions of massless higher spins in (A)dS: metric-like approach”, Nucl. Phys. B, 861:1 (2012), 145–174, arXiv: 1110.5918 | DOI | MR

[11] M. Vasiliev, “Cubic vertices for symmetric higher-spin gauge fields in $(A)dS_d$”, Nucl. Phys. B, 862:2 (2012), 341–408, arXiv: 1108.5921 | DOI | MR

[12] A. Fotopoulos, M. Tsulaia, “Current exchanges for reducible higher spin multiplets and gauge fixing”, JHEP, 10 (2009), 050, 25 pp., arXiv: 0907.4061 | DOI

[13] R. R. Metsaev, “BRST-BV approach to cubic interaction vertices for massive and massless higher-spin fields”, Phys. Lett. B, 720:1–3 (2013), 237–243, arXiv: 1205.3131 | DOI | MR

[14] M. V. Khabarov, Yu. M. Zinoviev, “Cubic interaction vertices for massless higher spin supermultiplets in $d=4$”, JHEP, 02 (2021), 167, 17 pp., arXiv: 2012.00482 | DOI | MR

[15] I. L. Buchbinder, V. A. Krykhtin, M. Tsulaia, D. Weissman, “Cubic vertices for $\mathcal{N} = 1$ supersymmetric massless higher spin fields in various dimensions”, Nucl. Phys. B, 967 (2021), 115427, 25 pp., arXiv: 2103.08231 | DOI | MR

[16] R. R. Metsaev, “Interacting massive and massless arbitrary spin fields in 4d flat space”, Nucl. Phys. B, 984 (2022), 115978, 25 pp., arXiv: 2206.13268 | DOI

[17] I. L. Buchbinder, A. A. Reshetnyak, “General cubic interacting vertex for massless integer higher spin fields”, Phys. Lett. B, 820 (2021), 136470, 8 pp., arXiv: 2105.12030 | MR

[18] A. A. Reshetnyak, “K strukture kubichnoi vershiny vzaimodeistviya bezmassovykh polei vysshikh tselykh spinov”, Pisma v EChAYa, 19:6(245) (2022), 499–508, arXiv: 2205.00488

[19] I. L. Buchbinder, A. A. Reshetnyak, Covariant cubic interacting vertices for massless and massive integer higher spin fields, arXiv: 2212.07097

[20] I. L. Buchbinder, V. A. Krykhtin, T. V. Snegirev, “Cubic interactions of d4 irreducible massless higher spin fields within BRST approach”, Eur. Phys. J. C, 82:11 (2022), 1007, 7 pp., arXiv: 2208.04409 | DOI

[21] E. Skvortsov, T. Tran, M. Tsulaia, “A stringy theory in three dimensions and massive higher spins”, Phys. Rev. D, 102:12 (2020), 126010, 6 pp., arXiv: 2006.05809 | DOI | MR

[22] M. Taronna, “Higher-spin interactions: four-point functions and beyond”, JHEP, 04 (2012), 029, 75 pp., arXiv: 1107.5843 | DOI | MR

[23] P. Dempster, M. Tsulaia, “On the structure of quartic vertex for massless higher spin fields on Minkowski background”, Nucl. Phys. B, 865:2 (2012), 353–375, arXiv: 1203.5597 | DOI

[24] P. M. Lavrov, “On interactions of massless spin 3 and scalar fields”, Eur. Phys. J. C, 82 (2022), 1059, 7 pp., arXiv: 2208.05700 | DOI

[25] R. R. Metsaev, “Cubic interaction vertices for massive and massless higher spin fields”, Nucl. Phys. B, 759:1–2 (2006), 147–201, arXiv: hep-th/0512342 | DOI | MR

[26] I. A. Batalin, G. A. Vilkovisky, “Gauge algebra and quantization”, Phys. Lett. B, 102:1 (1981), 27–31 | DOI | MR

[27] I. A. Batalin, G. A. Vilkovisky, “Quantization of gauge theories with linearly dependent generators”, Phys. Rev. D, 28:10 (1983), 2567–2582 ; Erratum, 30:2 (1984), 508–508 | DOI | MR | DOI

[28] I. A. Batalin, G. A. Vilkovisky, “Existence theorem for gauge algebra”, J. Math. Phys., 26:1 (1985), 172–184 | DOI | MR

[29] G. Barnich, M. Grigoriev, A. Semikhatov, I. Tipunin, “Parent field theory and unfolding in BRST first-quantized terms”, Commun. Math. Phys., 260:1 (2005), 147–181, arXiv: hep-th/0406192 | DOI | MR

[30] K. Alkalaev, M. Grigoriev, I. Tipunin, “Massless Poincaré modules and gauge invariant equations”, Nucl. Phys. B, 823:3 (2009), 509–545, arXiv: 0811.3999 | DOI | MR

[31] A. Reshetnyak, “Constrained BRST-BFV and BRST-BV Lagrangians for half-integer HS fields on $R^{1,d-1}$”, Phys. Part. Nucl., 49:5 (2018), 952–957, arXiv: 1803.05173 | DOI

[32] C. Burdik, V. K. Pandey, A. Reshetnyak, “BRST-BFV and BRST-BV descriptions for bosonic fields with continuous spin on $\mathbb{R}^{1,d-1}$”, Internat. J. Modern Phys. A, 35:26 (2020), 2050154, 59 pp., arXiv: 1906.02585 | DOI | MR

[33] Č. Burdík, A. A. Reshetnyak, “BRST-BV quantum actions for constrained totally-symmetric integer HS fields”, Nucl. Phys. B, 965 (2021), 115357, 20 pp., arXiv: 2010.15741 | DOI | MR

[34] A. K. H. Bengtsson, “A unified action for higher spin gauge bosons from covariant string theory”, Phys. Lett. B, 182:3–4 (1986), 321–325 | DOI

[35] A. A. Reshetnyak, “Constrained BRST-BFV Lagrangian formulations for higher spin fields in Minkowski spaces”, JHEP, 09 (2018), 104, 63 pp., arXiv: 1803.04678 | MR

[36] C. Fronsdal, “Massless fields with integer spin”, Phys. Rev. D, 18:10 (1978), 3624–3629 | DOI

[37] I. L. Buchbinder, A. Pashnev, M. Tsulaia, “Lagrangian formulation of the massless higher integer spin fields in the AdS background”, Phys. Lett. B, 523:3–4 (2001), 338–346, arXiv: hep-th/0109067 | DOI | MR

[38] I. L. Buchbinder, V. A. Krykhtin, A. Pashnev, “BRST approach to Lagrangian construction for fermionic massless higher spin fields”, Nucl. Phys. B, 711:1–2 (2005), 367–391, arXiv: hep-th/0410215 | DOI | MR

[39] I. L. Buchbinder, V. A. Krykhtin, “Gauge invariant Lagrangian construction for massive bosonic higher spin fields in D dimentions”, Nucl. Phys. B, 727:3 (2005), 537–563, arXiv: hep-th/0505092 | DOI | MR

[40] I. L. Buchbinder, A. Fotopoulos, A. C. Petkou, M. Tsulaia, “Constructing the cubic interaction vertex of higher spin gauge fields”, Phys. Rev. D, 74:10 (2006), 105018, 16 pp., arXiv: hep-th/0609082 | DOI | MR

[41] I. L. Buchbinder, A. V. Galajinsky, V. A. Krykhtin, “Quartet unconstrained formulation for massless higher spin fields”, Nucl. Phys. B, 779:3 (2007), 155–177, arXiv: hep-th/0702161 | DOI | MR

[42] I. L. Buchbinder, V. A. Krykhtin, A. A. Reshetnyak, “BRST approach to Lagrangian construction for fermionic higher spin fields in AdS space”, Nucl. Phys. B, 787:3 (2007), 211–240, arXiv: hep-th/0703049 | DOI | MR

[43] I. L. Buchbinder, A. Reshetnyak, “General Lagrangian formulation for higher spin fields with arbitrary index symmetry. I. Bosonic fields”, Nucl. Phys. B, 862:1 (2012), 270–326, arXiv: 1110.5044 | DOI | MR

[44] B. S. Devitt, Dinamicheskaya teoriya grupp i polei, Nauka, M., 1987 | MR | MR | Zbl

[45] M. Grigoriev, P. H. Damgaard, “Superfield BRST charge and the master action”, Phys. Lett. B, 474:3–4 (2000), 323–330, arXiv: hep-th/9911092 | DOI

[46] D. M. Gitman, P. Yu. Moshin, A. A. Reshetnyak, “Local superfield Lagrangian BRST quantization”, J. Math. Phys., 46:7 (2005), 072302, 24 pp., arXiv: hep-th/0507160 | DOI | MR

[47] M. Alexandrov, A. Schwarz, O. Zaboronsky, M. Kontsevich, “The geometry of the master equation and topological quantum field theory”, Internat. J. Modern Phys. A, 12:7 (1997), 1405–1429, arXiv: hep-th/9502010 | DOI | MR

[48] L. P. S. Singh, C. R. Hagen, “Lagrangian formulation for arbitrary spin. 1. The boson case”, Phys. Rev. D, 9:4 (1974), 898–909 | DOI

[49] G. Barnich, M. Henneaux, “Consistent couplings between gauge fields and deformations of the master equation”, Phys. Lett. B, 311:1–4 (1993), 123–129, arXiv: hep-th/9304057 | DOI | MR

[50] M. Henneaux, “Consistent interactions between gauge fields: The cohomological approach”, Secondary Calculus and Cohomological Physics (August 24–31, 1997, Moscow, Russia), Contemporary Mathematics, 219, eds. M. Henneaux, J. Krasil'shchik, A. Vinogradov, AMS, Providence, RI, 1998, 93–109, arXiv: hep-th/9712226 | MR

[51] I. L. Buchbinder, P. M. Lavrov, “On a gauge-invariant deformation of a classical gauge-invariant theory”, JHEP, 06 (2021), 097, 17 pp., arXiv: 2104.11930 | DOI | MR

[52] D. Francia, A. Sagnotti, “On the geometry of higher-spin gauge fields”, Class. Quantum Grav., 20:12 (2003), S473–S485, arXiv: hep-th/0212185 | DOI | MR

[53] E. Skvortsov, T. Tran, M. Tsulaia, “Quantum chiral higher spin gravity”, Phys. Rev. Lett., 121:3 (2018), 031601, 5 pp., arXiv: 1805.00048 | DOI

[54] V. E. Didenko, O. A. Gelfond, A. V. Korybut, M. A. Vasiliev, “Limiting shifted homotopy in higher-spin theory”, JHEP, 12 (2019), 086, 49 pp., arXiv: 1909.04876 | DOI

[55] M. A. Vasiliev, “Projectively-compact spinor veritices and space-time spin-locality in higher-spin theory”, Phys. Lett. B, 834 (2022), 137401, 11 pp., arXiv: 2208.02004 | DOI | MR

[56] V. E. Didenko, A. V. Korybut, “On $z$-dominance, shift symmetry and spin locality in higher-spin theory”, JHEP, 05 (2023), 133, 32 pp., arXiv: 2212.05006 | DOI