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Spontaneous symmetry breaking in Lorentz violating background

Syed Masood

http://orcid.org/0000-0002-7246-706X

Department of Physics, International Islamic University (IIU), Islamabad 44000, Pakistan

E-mail Address: thamersyed@gmail.com

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Mushtaq B. Shah

Department of Physics, National Institute of Technology, Srinagar, Kashmir 190006, India

E-mail Address: Mbstheory@gmail.com

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, and

Prince A. Ganai

Department of Physics, National Institute of Technology, Srinagar, Kashmir 190006, India

E-mail Address: princeganai@nitsri.net

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https://doi.org/10.1142/S0219887818500214Cited by:5 (Source: Crossref)

Abstract

In this paper, we study spontaneous symmetry breaking of gauge theories in a Lorentz violating background. Here, Lorentz symmetry will be broken down to its subgroup using the formalism of very special relativity. The breaking of the Lorentz symmetry will modify the gauge theories, and this will in turn modify the Higgs mechanisms for such theories. We will explicitly demonstrate that the different Hilbert spaces in various gauges of this theory can be related to each other through the gaugeon formalism. We will also discuss the FFBRST transformation for this theory, and observe that gaugeon formalism can be obtained from same. Thus, by making the BRST parameter finite and field dependent, we can relate different Hilbert spaces in different gauges for a gauge theory with spontaneous symmetry breaking in VSR.

Keywords:

AMSC: 81T13

References

1. P. Horava, Phys. Rev. D 79 (2009) 084008, arXiv: 0901.3775 [hep-th]. Web of Science, Google Scholar
2. R. Gambini and J. Pullin, Phys. Rev. D 59 (1999) 124021, arXiv: gr-qc/9809038. Web of Science, Google Scholar
3. G. ’t Hooft, Class. Quantum Grav. 13 (1996) 1023, arXiv: gr-qc/9601014. Web of Science, Google Scholar
4. G. Amelino-Camelia, J. R. Ellis, N. E. Mavromatos, D. V. Nanopoulos and S. Sarkar, Nature 393 (1998) 763, arXiv: astro-ph/9712103. Web of Science, Google Scholar
5. N. Kajuri, Int. J. Mod. Phys. A 30(34) (2015) 1550204, arXiv: 1406.7400 [gr-qc]. Link, Web of Science, Google Scholar
6. O. Sargın and M. Faizal, Europhys. Lett. 113(3) (2016) 30007, arXiv: 1509.00843 [gr-qc]. Google Scholar
7. S. M. Carroll, J. A. Harvey, V. A. Kostelecky, C. D. Lane and T. Okamoto, Noncommutative field theory and Lorentz violation, Phys. Rev. Lett. 87 (2001) 141601, arXiv: hep-th/0105082. Web of Science, Google Scholar
8. S. Chakravarty, K. Dasgupta, O. J. Ganor and G. Rajesh, Pinned branes and new nonLorentz invariant theories, Nucl. Phys. B 587 (2000) 228, arXiv: hep-th/0002175. Web of Science, Google Scholar
9. A. G. Cohen and S. L. Glashow, Very special relativity, Phys. Rev. Lett. 97 (2006) 021601, arXiv: hep-ph/0601236. Web of Science, Google Scholar
10. J. Vohánka and M. Faizal, Chern–Simons theory in SIM(1) superspace, Eur. Phys. J. C 75(12) (2015) 592, arXiv: 1503.04761 [hep-th]. Web of Science, Google Scholar
11. J. Vohánka and M. Faizal, Supersymmetric Chern–Simons theory in presence of a boundary in the light-like direction, Nucl. Phys. B 904 (2016) 327, arXiv: 1505.08112 [hep-th]. Web of Science, Google Scholar
12. J. Vohánka and M. Faizal, Super-Yang–Mills theory in SIM(1) superspace, Phys. Rev. D 91(4) (2015) 045015, arXiv: 1409.6334 [hep-th]. Web of Science, Google Scholar
13. A. G. Cohen and D. Z. Freedman, SIM(2) and SUSY, J. High Energy Phys. 0707 (2007) 039, arXiv: hep-th/0605172. Web of Science, Google Scholar
14. S. Petras, R. von Unge and J. Vohanka, SIM(2) and supergraphs, J. High Energy Phys. 1107 (2011) 015, arXiv: 1102.3856 [hep-th]. Web of Science, Google Scholar
15. J. Vohanka, Gauge theory and SIM(2) superspace, Phys. Rev. D 85 (2012) 105009. Web of Science, Google Scholar
16. W. Muck, Very special relativity in curved space-times, Phys. Lett. B 670 (2008) 95, arXiv: 0806.0737 [hep-th]. Web of Science, Google Scholar
17. D. V. Ahluwalia and S. P. Horvath, Very special relativity as relativity of dark matter: The Elko connection, J. High Energy Phys. 1011 (2010) 078. Web of Science, Google Scholar
18. Z. Chang, M. H. Li, X. Li and S. Wang, Cosmological model with local symmetry of very special relativity and constraints on it from supernovae, Eur. Phys. J. C 73(6) (2013) 2459, arXiv: 1303.1593 [astro-ph.CO]. Web of Science, Google Scholar
19. J. Alfaro and V. O. Rivelles, Non Abelian fields in very special relativity, Phys. Rev. D 88 (2013) 085023, arXiv: 1305.1577 [hep-th]. Web of Science, Google Scholar
20. J. Alfaro, P. Gonzaleź and R. Avilá, Electroweak standard model with very special relativity, Phys. Rev. D 91 (2015) 105007, Addendum 91(12) (2015) 129904, arXiv:1504.04222 [hep-ph]. Google Scholar
21. C. Becchi, A. Rouet and R. Stora, Renormalization of gauge theories, Annals Phys. 98 (1976) 287. Web of Science, Google Scholar
22. I. V. Tyutin, Gauge invariance in field theory and statistical physics in operator formalism, arXiv:0812.0580 [hep-th]. Google Scholar
23. G. Costa, J. Julve, T. Marinucci and M. Tonin, Nonabelian gauge theories and triangle anomalies, Nuovo Cimento A 38 (1977) 373. Web of Science, Google Scholar
24. T. Kugo and I. Ojima, Manifestly covariant canonical formulation of Yang–Mills field theories: Physical state subsidiary conditions and physical S matrix unitarity, Phys. Lett. B 73 (1978) 459. Web of Science, Google Scholar
25. H. Hata and T. Kugo, Subsidiary conditions and physical S matrix unitarity in covariant canonical formulation of supergravity, Nucl. Phys. B 158 (1979) 357. Web of Science, Google Scholar
26. M. Faizal and M. Khan, A superspace formulation of the BV action for higher derivative theories, Eur. Phys. J. C 71 (2011) 1603, arXiv: 1103.0221 [hep-th]. Web of Science, Google Scholar
27. L. Baulieu and J. Thierry-Mieg, The principle of BRS symmetry: An alternative approach to Yang–Mills theories, Nucl. Phys. B 197 (1982) 477. Web of Science, Google Scholar
28. T. Azevedo and R. L. Jusinskas, Background constraints in the infinite tension limit of the heterotic string, J. High Energy Phys. 1608 (2016) 133, arXiv: 1607.06805 [hep-th]. Web of Science, Google Scholar
29. I. Oda, Covariant matrix model of superparticle in the pure spinor formalism, Mod. Phys. Lett. A 18 (2003) 1023. Link, Web of Science, Google Scholar
30. M. Faizal, Aspects of ABJ theory, J. High Energy Phys. 1301 (2013) 156, arXiv: 1301.4305 [hep-th]. Web of Science, Google Scholar
31. M. Faizal, Harmonic superspace Gaugeon formalism for the ABJM theory, Mod. Phys. Lett. A 27 (2012) 1250147, arXiv: 1207.2120 [hep-th]. Link, Web of Science, Google Scholar
32. M. Faizal, $M$ $M$ -Theory in the Gaugeon formalism, Commun. Theor. Phys. 57 (2012) 637, arXiv: 1201.1220 [hep-th]. Web of Science, Google Scholar
33. M. Faizal, M-Theory on deformed superspace, Phys. Rev. D 84 (2011) 106011, arXiv: 1111.0213 [hep-th]. Web of Science, Google Scholar
34. M. Faizal, Chern–Simons-matter theory, Int. J. Mod. Phys. A 28 (2013) 1350012, arXiv: 1301.5664 [math-ph]. Link, Web of Science, Google Scholar
35. M. Faizal, S. Upadhyay and B. P. Mandal, IR finite graviton propagators in de Sitter spacetime, Eur. Phys. J. C 76(4) (2016) 189, arXiv: 1604.00390 [hep-th]. Web of Science, Google Scholar
36. L. N. Chang and C. P. Soo, BRST cohomology and invariants of 4-D gravity in Ashtekar variables, Phys. Rev. D 46 (1992) 4257–4262. Web of Science, Google Scholar
37. S. Kachru, Extra states and symmetries in $D < 2$ $D < 2$ closed string theory, Nucl. Phys. B 390 (1993) 173, arXiv: hep-th/9204053. Web of Science, Google Scholar
38. H. Y. Lee, A. Nakamichi and T. Ueno, Topological two form gravity in four-dimensions, Phys. Rev. D 47 (1993) 1563, arXiv: hep-th/9205066. Web of Science, Google Scholar
39. N. Ohta and H. Suzuki, Interactions of discrete states with nonzero ghost number in $c = 12 - D$ $c = 1 2 - D$ gravity, Mod. Phys. Lett. A 7 (1992) 2723, arXiv: hep-th/9205101. Link, Web of Science, Google Scholar
40. M. Faizal, Spontaneous breaking of Lorentz symmetry by ghost condensation in perturbative quantum gravity, J. Phys. A 44 (2011) 402001, arXiv: 1108.2853 [gr-qc]. Google Scholar
41. M. Faizal, BRST and anti-BRST symmetries in perturbative quantum gravity, Found. Phys. 41 (2011) 270, arXiv: 1010.1143 [gr-qc]. Web of Science, Google Scholar
42. M. Faizal, Noncommutative quantum gravity, Mod. Phys. Lett. A 28 (2013) 1350034, arXiv: 1302.5156 [gr-qc]. Link, Web of Science, Google Scholar
43. M. Faizal, Noether‘s charge in the super-group field cosmology, Grav. Cosmol. 20(2) (2014) 132, arXiv: 1303.5478 [gr-qc]. Web of Science, Google Scholar
44. M. Faizal, Absence of black holes information paradox in group field cosmology, Int. J. Geom. Meth. Mod. Phys. 11 (2014) 1450010, arXiv: 1301.0224 [gr-qc]. Link, Web of Science, Google Scholar
45. M. Faizal, Super-group field cosmology, Class. Quantum Grav. 29 (2012) 215009, arXiv: 1209.2346 [gr-qc]. Web of Science, Google Scholar
46. J. Troost, Massless particles on supergroups and $A d S_{3} x S^{3}$ $A d S_{3} x S^{3}$ supergravity, J. High Energy Phys. 1107 (2011) 042, arXiv: 1102.0153 [hep-th]. Web of Science, Google Scholar
47. M. Faizal, The BV formalization of Chern–Simons theory on deformed superspace, Commun. Theor. Phys. 58 (2012) 704, arXiv: 1207.6564 [hep-th]. Web of Science, Google Scholar
48. R. Banerjee and S. Deguchi, A superspace formulation of Yang–Mills theory on sphere, J. Math. Phys. 51 (2010) 052301, arXiv: 0905.3050 [hep-th]. Web of Science, Google Scholar
49. K. Ulker, $N = 2$ $N = 2$ SYM action as a BRST exact term, topological Yang–Mills and instantons, Phys. Rev. D 68 (2003) 085005, arXiv: hep-th/0304154. Web of Science, Google Scholar
50. M. B. Shah, M. Faizal, P. A. Ganai, Z. Zaz, A. Bhat and S. Masood, Boundary effects in super-Yang–Mills theory, Eur. Phys. J. C 77(5) (2017) 309. Web of Science, Google Scholar
51. A. C. Lehum, J. R. Nascimento, A. Y. Petrov and A. J. da Silva, Supergauge theories in aether superspace, Phys. Rev. D 88 (2013) 045022, arXiv: 1305.1812 [hep-th]. Web of Science, Google Scholar
52. P. Weinreb and M. Faizal, Generalized Faddeev–Popov method for a deformed supersymmetric Yang–Mills theory, Phys. Lett. B 748 (2015) 102, arXiv: 1506.07618 [hep-th]. Web of Science, Google Scholar
53. S. Upadhyay and P. K. Panigrahi, Quantum gauge freedom in very special relativity, Nucl. Phys. B 915 (2017) 168, arXiv: 1608.03947 [hep-th]. Web of Science, Google Scholar
54. S. Upadhyay, M. B. Shah and P. A. Ganai, Lorentz violating $p$ $p$ -form gauge theories in superspace, Eur. Phys. J. C 77(3) (2017) 157, arXiv: 1702.05755 [hep-th]. Web of Science, Google Scholar
55. K. I. Yokoyama, Canonical Yang–Mills field theory with invariant gauge-families: Introduction of gauge parameters as a group vector, Prog. Theor. Phys. 59 (1978) 1699. Web of Science, Google Scholar
56. K. I. Yokoyama, Renormalization scheme of a canonical Yang–Mills field theory with invariant gauge families, Prog. Theor. Phys. 60 (1978) 1167; K. I. Yokoyama, Gauge independence as a consequence of gauge covariance, Phys. Lett. B 79 (1978) 79. Google Scholar
57. K. I. Yokoyama and R. Kubo, Gauge-structure of canonical quantum electrodynamics, Prog. Theor. Phys. 52 (1974) 290. Google Scholar
58. K. I. Yokoyama, M. Takeda and M. Monda, On finite local gauge transformations in a Yang–Mills field theory with invariant gauge families, Prog. Theor. Phys. 60 (1978) 927 Erratum: [Prog. Theor. Phys. 60 (1978) 1609] Erratum: [Prog. Theor. Phys. 61 (1979) 697]. Google Scholar
59. K. I. Yokoyama, M. Takeda and M. Monda, Gauge covariance in nonabelian gauge theories, Prog. Theor. Phys. 64 (1980) 1412. Web of Science, Google Scholar
60. K. I. Yokoyama, Canonical quantum electrodynamics with invariant one-parameter gauge families, Prog. Theor. Phys. 51 (1974) 1956. Google Scholar
61. M. Hayakawa and K. I. Yokoyama, Gauge, renormalization and the goto-imamura-schwinger term in quantum electrodynamics, Prog. Theor. Phys. 44 (1970) 533. Google Scholar
62. T. Kugo and I. Ojima, Local covariant operator formalism of nonabelian gauge theories and quark confinement problem, Prog. Theor. Phys. Suppl. 66 (1979) 1. Web of Science, Google Scholar
63. T. Kugo and I. Ojima, Subsidiary conditions and physical S matrix unitarity in indefinite metric quantum gravitational theory, Nucl. Phys. B 144 (1978) 234. Web of Science, Google Scholar
64. K. I. Izawa, Simple BRS gauge fixing, Prog. Theor. Phys. 88 (1992) 759. Google Scholar
65. M. Koseki, M. Sato and R. Endo, Gaugeon formalism with BRST symmetry, Prog. Theor. Phys. 90 (1993) 1111. Web of Science, Google Scholar
66. M. Koseki, M. Sato and R. Endo, BRST symmetric gaugeon formalism for Yang–Mills fields, arXiv:hep-th/9510102. Google Scholar
67. Y. Nakawaki, Quantum gauge invariant formalism of the Schwinger model: Gaugeon theory of the Schwinger model, Prog. Theor. Phys. 98 (1997) 5. Google Scholar
68. R. Endo and M. Koseki, Gaugeon formalism for spin - $3 / 2$ $3 / 2$ Rarita-Schwinger gauge field, Prog. Theor. Phys. 103 (2000) 685, arXiv: hep-th/9912014. Google Scholar
69. H. Miura and R. Endo, BRST symmetric gaugeon formalism for the Higgs model, Prog. Theor. Phys. 117 (2007) 4. Google Scholar
70. S. Upadhyay, Field-dependent quantum gauge transformation, Europhys. Lett. 105 (2014) 21001. Google Scholar
71. S. Upadhyay, Gaugeon formalism for perturbative quantum gravity, Eur. Phys. J. C 74 (2014) 2737. Web of Science, Google Scholar
72. S. Upadhyay, The conformal gauge to the derivative gauge for worldsheet gravity, Phys. Lett. B 740 (2015) 341. Web of Science, Google Scholar
73. M. Faizal, S. Upadhyay and B. P. Mandal, IR finite graviton propagators in de Sitter spacetime, Eur. Phys. J. C 76(4) (2016) 189. Web of Science, Google Scholar
74. M. Faizal, S. Upadhyay and B. P. Mandal, Anti-FFBRST transformations for the BLG theory in presence of a boundary, Int. J. Mod. Phys. A 30(7) (2015) 1550032, arXiv: 1501.01616 [hep-th]. Link, Web of Science, Google Scholar
75. M. Faizal, S. Upadhyay and B. P. Mandal, Finite field-dependent BRST symmetry for ABJM theory in $N = 1$ $N = 1$ superspace, Phys. Lett. B 738 (2014) 201, arXiv: 1410.0671 [hep-th]. Web of Science, Google Scholar
76. M. Faizal, B. P. Mandal and S. Upadhyay, Finite BRST transformations for the Bagger–Lambert–Gustavasson theory, Phys. Lett. B 721 (2013) 159, arXiv: 1212.5653 [hep-th]. Web of Science, Google Scholar
77. S. Upadhyay, M. Faizal and P. A. Ganai, Interpolating between different gauges in the ABJM theory, Int. J. Mod. Phys. A 30(27) (2015) 1550185. Link, Web of Science, Google Scholar
78. S. Upadhyay and P. A. Ganai, Finite field-dependent symmetry in thirring model, Prog. Theor. Exp. Phys. 6 (2016) 063B04 https://doi.org/10.1093/ptep/ptw084 Google Scholar
79. P. W. Higgs, Spontaneous symmetry breakdown without massless bosons, Phys. Rev. 145 (1966) 1156. Web of Science, Google Scholar
80. T. W. B. Kibble, Symmetry breaking in nonAbelian gauge theories, Phys. Rev. 155 (1967) 1554. Web of Science, Google Scholar
81. K. Fujikawa, B. W. Lee and A. I. Sanda, Generalized renormalizable gauge formulation of spontaneously broken gauge theories, Phys. Rev. D 6 (1972) 2923. Web of Science, Google Scholar
82. S. D. Joglekar and B. P. Mandal, Finite field dependent BRS transformations, Phys. Rev. D 51 (1995) 1919. Web of Science, Google Scholar
83. J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231], arXiv:hep-th/9711200. Google Scholar
84. A. Gustavsson, Selfdual strings and loop space Nahm equations, J. High Energy Phys. 0804 (2008) 083, arXiv: 0802.3456 [hep-th]. Web of Science, Google Scholar
85. J. Bagger and N. Lambert, Comments on multiple M2-branes, J. High Energy Phys. 0802 (2008) 105, arXiv: 0712.3738 [hep-th]. Web of Science, Google Scholar
86. J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D 77 (2008) 065008, arXiv: 0711.0955 [hep-th]. Web of Science, Google Scholar
87. M. Faizal, Deformation of the ABJM theory, Europhys. Lett. 98 (2012) 31003, arXiv: 1204.1191 [hep-th]. Google Scholar
88. O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, $N = 6$ $N = 6$ superconformal Chern–Simons-matter theories, M2-branes and their gravity duals, J. High Energy Phys. 0810 (2008) 091, arXiv: 0806.1218 [hep-th]. Web of Science, Google Scholar
89. O. Aharony, O. Bergman and D. L. Jafferis, Fractional M2-branes, J. High Energy Phys. 0811 (2008) 043, arXiv: 0807.4924 [hep-th]. Web of Science, Google Scholar
90. M. Faizal, Non-anticommutative ABJ theory, Nucl. Phys. B 869 (2013) 598, arXiv: 1301.0223 [hep-th]. Web of Science, Google Scholar
91. J. M. Queiruga, A. C. Lehum and M. Faizal, Kählerian effective potentials for Chern–Simons-matter theories, Nucl. Phys. B 902 (2016) 58, arXiv: 1511.03586 [hep-th]. Web of Science, Google Scholar
92. E. Antonyan and A. A. Tseytlin, On 3d $N = 8$ $N = 8$ Lorentzian BLG theory as a scaling limit of 3d superconformal $N = 6$ $N = 6$ ABJM theory, Phys. Rev. D 79 (2009) 046002, arXiv: 0811.1540 [hep-th]. Web of Science, Google Scholar
93. D. S. Berman, M. J. Perry, E. Sezgin and D. C. Thompson, Boundary conditions for interacting membranes, J. High Energy Phys. 1004 (2010) 025, arXiv: 0912.3504 [hep-th]. Web of Science, Google Scholar
94. F. Aprile and V. Niarchos, $𝒩 = 2$ supersymmetric field theories on 3-manifolds with A-type boundaries, J. High Energy Phys. 1607 (2016) 126, arXiv: 1604.01561 [hep-th]. Web of Science, Google Scholar
95. M. Faizal, Gauge and supersymmetric invariance of a boundary Bagger–Lambert–Gustavsson theory, J. High Energy Phys. 1204 (2012) 017, arXiv: 1204.0297 [hep-th]. Web of Science, Google Scholar
96. D. V. Belyaev and P. van Nieuwenhuizen, Simple $d = 4$ supergravity with a boundary, J. High Energy Phys. 0809 (2008) 069, arXiv: 0806.4723 [hep-th]. Web of Science, Google Scholar
97. M. Faizal, Y. Luo, D. J. Smith, M. C. Tan and Q. Zhao, Gauge and supersymmetry invariance of $𝒩 = 2$ boundary Chern–Simons theory, Nucl. Phys. B 914 (2017) 577, arXiv: 1601.05429 [hep-th]. Web of Science, Google Scholar
98. M. Faizal, Boundary effects in the BLG theory, Mod. Phys. Lett. A 29(31) (2014) 1450154, arXiv: 1303.5477 [hep-th]. Link, Web of Science, Google Scholar
99. D. S. Berman and D. C. Thompson, Membranes with a boundary, Nucl. Phys. B 820 (2009) 503, arXiv: 0904.0241 [hep-th]. Web of Science, Google Scholar
100. M. Faizal and D. J. Smith, Nonanticommutativity in the presence of a boundary, Phys. Rev. D 87(2) (2013) 025019, arXiv: 1211.3654 [hep-th]. Web of Science, Google Scholar
101. A. Bilal, Supersymmetric boundaries and junctions in four dimensions, J. High Energy Phys. 1111 (2011) 046, arXiv: 1103.2280 [hep-th]. Web of Science, Google Scholar
102. D. V. Belyaev and T. G. Pugh, The supermultiplet of boundary conditions in supergravity, J. High Energy Phys. 1010 (2010) 031, arXiv: 1008.1574 [hep-th]. Web of Science, Google Scholar
103. M. Faizal and A. Awad, Four dimensional supersymmetric theories in presence of a boundary, Phys. Lett. B 748 (2015) 414, arXiv: 1502.07717 [hep-th]. Web of Science, Google Scholar
104. D. V. Belyaev and P. van Nieuwenhuizen, Rigid supersymmetry with boundaries, J. High Energy Phys. 0804 (2008) 008, arXiv: 0801.2377 [hep-th]. Web of Science, Google Scholar
105. V. N. Gribov, Quantization of nonabelian gauge theories, Nucl. Phys. B 139 (1978) 1. Web of Science, Google Scholar
106. I. M. Singer, Some remarks on the Gribov ambiguity, Commun. Math. Phys. 60 (1978) 7. Web of Science, Google Scholar
107. K. Shigemoto, Field strength method and Gribov ambiguity in two-dimensional nonabelian gauge theory, Lett. Nuovo Cimento 24 (1979) 495. Web of Science, Google Scholar
108. T. P. Killingback, The Gribov ambiguity in gauge theories on the 4 torus, Phys. Lett. B 138 (1984) 87. Web of Science, Google Scholar
109. D. Zwanziger, Nonperturbative modification of the Faddeev–Popov formula and banishment of the naive vacuum, Nucl. Phys. B 209 (1982) 336. Web of Science, Google Scholar
110. D. Zwanziger, Local and renormalizable action from the Gribov horizon, Nucl. Phys. B 323 (1989) 513. Web of Science, Google Scholar
111. M. A. L. Capri, D. Dudal, A. D. Pereira, D. Fiorentini, M. S. Guimaraes, B. W. Mintz, L. F. Palhares and S. P. Sorella, Nonperturbative aspects of Euclidean Yang–Mills theories in linear covariant gauges: Nielsen identities and a BRST-invariant two-point correlation function, Phys. Rev. D 95(4) (2017) 045011. Web of Science, Google Scholar
112. D. Zwanziger, Action from the Gribov horizon, Nucl. Phys. B 321 (1989) 591. Web of Science, Google Scholar
113. M. S. Guimaraes, A. D. Pereira and S. P. Sorella, Remarks on the effects of the Gribov copies on the infrared behavior of higher dimensional Yang–Mills theory, Phys. Rev. D 94(11) (2016) 116011, arXiv: 1608.06979 [hep-th]. Web of Science, Google Scholar