Research ArticlesNo Access

A study of 3-form gauge theories in the Lorentz violating background

Mushtaq Bashir Shah

http://orcid.org/0000-0001-5286-4183

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

E-mail Address: mbstheory72@gmail.com

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and

Prince Ahmad 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/S0219887818501062Cited by:1 (Source: Crossref)

Abstract

We study the Lorentz symmetry breaking of the 3-form gauge theory down to its sub-group. A 3-form gauge theory is studied in such a Lorentz violating background and these symmetry violation effects will affect the aspects of such a gauge theory. Also, we study the gaugeon formalism and FFBRST of 3-form theory in such a background. It is seen that the generating functional gets modified. With this, we obtain a connection between covariant and noncovariant gauges of such a gauge theory. Furthermore, we study the Batalin–Vilkovisky (BV) formulation of such a gauge theory in such a Lorentz violating background.

Keywords:

AMSC: 81T13

References

1. G. ’t Hooft, Quantization of point particles in $(2 + 1)$ -dimensional gravity and spacetime discreteness, Class. Quantum Grav. 13 (1996) 1023. Crossref, Web of Science, Google Scholar
2. G. Amelino-Camelia, J. R. Ellis, N. E. Mavromatos, D. V. Nanopoulos and S. Sarkar, Tests of quantum gravity from observations of gamma-ray bursts, Nature 393 (1998) 763. Crossref, Web of Science, Google Scholar
3. R. Gambini and J. Pullin, Nonstandard optics from quantum spacetime, Phys. Rev. D 59 (1999) 124021. Crossref, Web of Science, Google Scholar
4. 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. Crossref, Web of Science, Google Scholar
5. P. Horava, Quantum gravity at a Lifshitz point, Phys. Rev. D 79 (2009) 084008. Crossref, Web of Science, Google Scholar
6. S. Chakravarty, K. Dasgupta, O. J. Ganor and G. Rajesh, Pinned branes and new nonLorentz invariant theories, Nuclear Phys. B 587 (2000) 228. Crossref, Web of Science, Google Scholar
7. V. A. Kostelecky and S. Samuel, Gravitational phenomenology in higher dimensional theories and strings, Phys. Rev. D 40 (1989) 1886. Crossref, Web of Science, Google Scholar
8. A. G. Cohen and S. L. Glashow, Very special relativity, Phys. Rev. Lett. 97 (2006) 021601. Crossref, Web of Science, Google Scholar
9. M. M. Sheikh-Jabbari and A. Tureanu, Realization of Cohen–Glashow very special relativity on noncommutative space-time, Phys. Rev. Lett. 101 (2008) 261601. Crossref, Web of Science, Google Scholar
10. A. G. Cohen and D. Z. Freedman, SIM(2) and SUSY, J. High Energy Phys. 0707 (2007) 039. Crossref, Web of Science, Google Scholar
11. S. Petras, R. von Unge and J. Vohanka, SIM(2) and supergraphs, J. High Energy Phys. 1107 (2011) 015. Crossref, Web of Science, Google Scholar
12. J. Vohanka, Gauge theory and SIM(2) superspace, Phys. Rev. D 85 (2012) 105009. Crossref, Web of Science, Google Scholar
13. J. Vohnka and M. Faizal, Chern–Simons theory in SIM(1) superspace, Eur. Phys. J. C 75(12) (2015) 592. Crossref, Web of Science, Google Scholar
14. J. Vohnka and M. Faizal, Supersymmetric Chern–Simons theory in presence of a boundary in the light-like direction, Nuclear Phys. B 904 (2016) 327. Crossref, Web of Science, Google Scholar
15. J. Vohnka and M. Faizal, Super-Yang-Mills theory in SIM(1) superspace, Phys. Rev. D 91(4) (2015) 045015. Crossref, Web of Science, Google Scholar
16. A. P. Kouretsis, M. Stathakopoulos and P. C. Stavrinos, The general very special relativity in Finsler cosmology, Phys. Rev. D 79 (2009) 104011. Crossref, Web of Science, Google Scholar
17. W. Muck, Very special relativity in curved space-times, Phys. Lett. B 670 (2008) 95. Crossref, Web of Science, Google Scholar
18. 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. Crossref, Web of Science, Google Scholar
19. 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. Crossref, Web of Science, Google Scholar
20. J. Alfaro and V. O. Rivelles, Non-Abelian fields in very special relativity, Phys. Rev. D 88 (2013) 085023. Crossref, Web of Science, Google Scholar
21. J. Alfaro, P. Gonzlez and R. Vila, Electroweak standard model with very special relativity, Phys. Rev. D 91 (2015) 105007; Addendum: [Phys. Rev. D 91(12) (2015) 129904]. Google Scholar
22. C. Becchi, A. Rouet and R. Stora, Renormalization of gauge theories, Ann. Phys. 98 (1976) 287. Crossref, Web of Science, Google Scholar
23. I. V. Tyutin, gauge invariance in field theory and statistical physics in operator formalism, 2008 (Submitted). Google Scholar
24. G. Costa, J. Julve, T. Marinucci and M. Tonin, Nonabelian gauge theories and triangle anomalies, Nuovo Cim. A 38 (1977) 373. Crossref, Google Scholar
25. 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. Crossref, Web of Science, Google Scholar
26. H. Hata and T. Kugo, Subsidiary conditions and physical S-matrix unitarity in covariant canonical formulation of supergravity, Nuclear Phys. B 158 (1979) 357. Crossref, Web of Science, Google Scholar
27. M. Faizal and M. Khan, A superspace formulation of the BV action for higher derivative theories, Eur. Phys. J. C 71 (2011) 1603. Crossref, Web of Science, Google Scholar
28. L. Baulieu and J. Thierry-Mieg, The principle of BRS symmetry: An alternative Approach to Yang–Mills theories, Nuclear Phys. B 197 (1982) 477. Crossref, Web of Science, Google Scholar
29. T. Azevedo and R. L. Jusinskas, Background constraints in the infinite tension limit of the heterotic string, J. High Energy Phys. 1608 (2016) 133. Crossref, Web of Science, Google Scholar
30. C. R. Mafra and O. Schlotterer, One-loop superstring six-point amplitudes and anomalies in pure spinor superspace, J. High Energy Phys. 1604 (2016) 148. Google Scholar
31. R. L. Jusinskas, On the field-antifield (a)symmetry of the pure spinor superstring, J. High Energy Phys. 1512 (2015) 136. Google Scholar
32. M. Faizal, Chern–Simons-matter theory, Int. J. Modern Phys. A 28 (2013) 1350012. Link, Web of Science, Google Scholar
33. M. Faizal and D. J. Smith, Supersymmetric Chern–Simons theory in presence of a boundary, Phys. Rev. D 85 (2012) 105007. Crossref, Web of Science, Google Scholar
34. M. Faizal, S. Upadhyay and B. P. Mandal, IR finite graviton propagators in de Sitter spacetime, Eur. Phys. J. C 76(4) (2016) 189. Crossref, Web of Science, Google Scholar
35. L. N. Chang and C. P. Soo, BRST cohomology and invariants of 4-D gravity in Ashtekar variables, Phys. Rev. D 46 (1992) 4257. Crossref, Web of Science, Google Scholar
36. S. Kachru, Extra states and symmetries in $D < 2$ closed string theory, Nuclear Phys. B 390 (1993) 173. Crossref, Web of Science, Google Scholar
37. H. Terao, Quantum analysis of Jackiw and Teitelboim’s model for $(1 + 1)$ -D gravity and topological gauge theory, Nuclear Phys. B 395 (1993) 623. Crossref, 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. Crossref, Web of Science, Google Scholar
39. M. Faizal, Noncommutative quantum gravity, Modern Phys. Lett. A 28 (2013) 1350034. Link, Web of Science, Google Scholar
40. J. Troost, Massless particles on supergroups and $A d S_{3} x S^{3}$ supergravity, J. High Energy Phys. 1107 (2011) 042. Crossref, Web of Science, Google Scholar
41. M. Faizal, The BV formalization of Chern–Simons theory on deformed superspace, Commun. Theor. Phys. 58 (2012) 704. Crossref, Web of Science, Google Scholar
42. R. Banerjee and S. Deguchi, A superspace formulation of Yang–Mills theory on sphere, J. Math. Phys. 51 (2010) 052301. Crossref, Web of Science, Google Scholar
43. K. Ulker, $N = 2$ SYM action as a BRST exact term, topological Yang–Mills and instantons, Phys. Rev. D 68 (2003) 085005. Crossref, Web of Science, Google Scholar
44. 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. Crossref, Web of Science, Google Scholar
45. 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. Crossref, Web of Science, Google Scholar
46. P. Weinreb and M. Faizal, Generalized Faddeev-Popov method for a deformed supersymmetric Yang–Mills theory, Phys. Lett. B 748 (2015) 102. Crossref, Web of Science, Google Scholar
47. S. Upadhyay and P. K. Panigrahi, Quantum gauge freedom in very special relativity, Nuclear Phys. B 915 (2017) 168. Crossref, Web of Science, Google Scholar
48. M. B. Shah and P. A. Ganai, Quantum gauge freedom in the Lorentz violating background, Int. J. Geom. Methods Mod. Phys. 15(01) (2017) 1850009. Link, Web of Science, Google Scholar
49. S. Masood, M. B. Shah and P. A. Ganai, Spontaneous symmetry breaking in Lorentz violating background, Int. J. Geom. Methods Mod. Phys. 15(02) (2017) 1850021. Link, Web of Science, Google Scholar
50. S. Upadhyay, M. B. Shah and P. A. Ganai, Lorentz violating $p$ -form gauge theories in superspace, Eur. Phys. J. C 77(3) (2017) 157. Crossref, Web of Science, Google Scholar
51. G. Leibbrandt, Introduction to noncovariant gauges, Rev. Modern Phys. 59 (1987) 1067. Crossref, Google Scholar
52. G. Leibbrandt, Noncovariant Gauges: Quantization of Yang–Mills and Chern–Simons Theory in Axial-Type Gauges (World Scientific, 1994). Link, Google Scholar
53. V. N. Gribov, Nuclear Phys. B 139 (1978) 1. Crossref, Web of Science, Google Scholar
54. A. P. Szczepaniak and E. S. Swanson, Coulomb gauge QCD, confinement, and the constituent representation, Phys. Rev. D 65 (2002) 025012. Crossref, Web of Science, Google Scholar
55. R. R. Metsaev, C. B. Thorn and A. A. Tseytlin, Light cone superstring in AdS space-time, Nuclear Phys. B 596 (2001) 151. Crossref, Web of Science, Google Scholar
56. L. Brink, O. Lindgren and B. E. W. Nilsson, The ultraviolet finiteness of the $N = 4$ Yang–Mills theory, Phys. Lett. B 123B (1983) 323. Crossref, Web of Science, Google Scholar
57. A. Gustavsson, M5 brane on $R^{1, 2} \times S^{3}$ , J. High Energy Phys. 1201 (2012) 057. Crossref, Web of Science, Google Scholar
58. H. C. Kim, S. Kim, E. Koh, K. Lee and S. Lee, On instantons as Kaluza-Klein modes of M5-branes, J. High Energy Phys. 1112 (2011) 031. Crossref, Web of Science, Google Scholar
59. C. S. Chu and D. J. Smith, Towards the quantum geometry of the M5-brane in a constant C-field from multiple membranes, J. High Energy Phys. 0904 (2009) 097. Crossref, Web of Science, Google Scholar
60. J. Bagger and N. Lambert, Comments on multiple M2-branes, J. High Energy Phys. 0802 (2008) 105. Crossref, Web of Science, Google Scholar
61. J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D 77 (2008) 065008. Crossref, Web of Science, Google Scholar
62. A. Gustavsson, Selfdual strings and loop space Nahm equations, J. High Energy Phys. 0804 (2008) 083. Crossref, Web of Science, Google Scholar
63. M. Faizal, Gauge and supersymmetric invariance of a boundary Bagger-Lambert-Gustavsson theory, J. High Energy Phys. 1204 (2012) 017. Crossref, Web of Science, Google Scholar
64. M. Faizal, B. P. Mandal and S. Upadhyay, Finite BRST transformations for the Bagger-Lambert-Gustavasson theory, Phys. Lett. B 721 (2013) 159. Crossref, Web of Science, Google Scholar
65. P.-M. Ho, Chin. J. Phys. 48 (2010) 1. Web of Science, Google Scholar
66. M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Vols. 1 and 2 (Cambridge University Press, Cambridge, 1987). Google Scholar
67. J. Polchinski, String Theory, Vols. 1 and 2 (Cambridge University Press, Cambridge, 1998). Crossref, Google Scholar
68. D. Lust and S. Theisen, Lectures in String Theory (Springer-Verlag, New York, 1989). Google Scholar
69. S. Upadhyay and B. P. Mandal, BV formulation of higher form gauge theories in a superspace, Eur. Phys. J. C 72 (2012) 2059. Crossref, Web of Science, Google Scholar
70. S. D. Joglekar and B. P. Mandal, Phys. Rev. D 51 (1995) 1919. Crossref, Web of Science, Google Scholar
71. R. Banerjee and B. P. Mandal, Phys. Lett. B 27 (2000) 488. Google Scholar
72. S. D. Joglekar and B. P. Mandal, Int. J. Modern Phys. A 17 (2002) 1279. Link, Web of Science, Google Scholar
73. S. Upadhyay and B. P. Mandal, Relating Gribov–Zwanziger theory to effective Yang–Mills theory, Eur. Phys. Lett. 93(3) (2011) 31001. Crossref, Web of Science, Google Scholar
74. S. Upadhyay, S. K. Rai and B. P. Mandal, Off-shell nilpotent finite BRST/Anti-BRST transformations, J. Math. Phys. 52 (2011) 022301. Crossref, Web of Science, Google Scholar
75. S. Upadhyay and B. P. Mandal, Eur. Phys. Lett. 93 (2011) 31001; Modern Phys. Lett. A 40 (2010) 3347. Google Scholar
76. B. P. Mandal, S. K. Rai and S. Upadhyay, Finite nilpotent symmetry in Batalin–Vilkovisky formalism, Eur. Phys. Lett. 92(2) (2010) 21001. Crossref, Web of Science, Google Scholar
77. S. Upadhyay and B. P. Mandal, Field dependent nilpotent symmetry for gauge theories, Eur. Phys. J. C 72 (2012) 2065. Crossref, Web of Science, Google Scholar
78. S. Upadhyay and B. P. Mandal, Finite BRST transformation and constrained systems, Ann. Phys. 327 (2012) 2885. Crossref, Web of Science, Google Scholar
79. S. Upadhyay and B. P. Mandal, Relating Gribov–Zwanziger theory and Yang–Mills theory in Batalin–Vilkovisky formalism, AIP Conf. Proc. 1444 (2011) 213. Google Scholar
80. M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, USA, 1992). Crossref, Google Scholar
81. S. Weinberg, The Quantum Theory of Fields, Vol-II: Modern Applications (Cambridge University Press, UK, 1996). Crossref, Google Scholar
82. 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]. Google Scholar
83. A. Gustavsson, Selfdual strings and loop space Nahm equations, J. High Energy Phys. 0804 (2008) 083. Crossref, Web of Science, Google Scholar
84. J. Bagger and N. Lambert, Comments on multiple M2-branes, J. High Energy Phys. 0802 (2008) 105. Crossref, Web of Science, Google Scholar
85. J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D 77 (2008) 065008. Crossref, Web of Science, Google Scholar
86. M. Faizal, Deformation of the ABJM theory, Europhys. Lett. 98 (2012) 31003. Crossref, Google Scholar
87. O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, $N = 6$ superconformal Chern–Simons-matter theories, M2-branes and their gravity duals, J. High Energy Phys. 0810 (2008) 091. Crossref, Web of Science, Google Scholar
88. O. Aharony, O. Bergman and D. L. Jafferis, Fractional M2-branes, J. High Energy Phys. 0811 (2008) 043. Crossref, Web of Science, Google Scholar
89. M. Faizal, Non-anticommutative ABJ theory, Nuclear Phys. B 869 (2013) 598. Crossref, Web of Science, Google Scholar
90. J. M. Queiruga, A. C. Lehum and M. Faizal, Khlerian effective potentials for Chern–Simons-matter theories, Nuclear Phys. B 902 (2016) 58. Crossref, Web of Science, Google Scholar
91. E. Antonyan and A. A. Tseytlin, On 3d $N = 8$ Lorentzian BLG theory as a scaling limit of 3d superconformal $N = 6$ ABJM theory, Phys. Rev. D 79 (2009) 046002. Crossref, Web of Science, Google Scholar
92. D. S. Berman, M. J. Perry, E. Sezgin and D. C. Thompson, Boundary conditions for interacting membranes, J. High Energy Phys. 1004 (2010) 025. Crossref, Web of Science, Google Scholar
93. F. Aprile and V. Niarchos, $𝒩 = 2$ supersymmetric field theories on 3-manifolds with A-type boundaries, J. High Energy Phys. 1607 (2016) 126. Crossref, Web of Science, Google Scholar
94. M. Faizal, Gauge and supersymmetric invariance of a boundary Bagger-Lambert-Gustavsson theory, J. High Energy Phys. 1204 (2012) 017. Crossref, Web of Science, Google Scholar
95. D. V. Belyaev and P. van Nieuwenhuizen, Simple $d = 4$ supergravity with a boundary, J. High Energy Phys. 0809 (2008) 069. Crossref, Web of Science, Google Scholar
96. M. Faizal, Y. Luo, D. J. Smith, M. C. Tan and Q. Zhao, Gauge and supersymmetry invariance of $𝒩 = 2$ boundary Chern–Simons theory, Nuclear Phys. B 914 (2017) 577. Crossref, Web of Science, Google Scholar
97. M. Faizal, Boundary effects in the BLG theory, Modern Phys. Lett. A 29(31) (2014) 1450154. Link, Web of Science, Google Scholar
98. D. S. Berman and D. C. Thompson, Membranes with a boundary, Nuclear Phys. B 820 (2009) 503. Crossref, Web of Science, Google Scholar
99. M. Faizal and D. J. Smith, Nonanticommutativity in the presence of a boundary, Phys. Rev. D 87(2) (2013) 025019. Crossref, Web of Science, Google Scholar
100. A. Bilal, Supersymmetric boundaries and junctions in four dimensions, J. High Energy Phys. 1111 (2011) 046. Crossref, Web of Science, Google Scholar
101. D. V. Belyaev and T. G. Pugh, The supermultiplet of boundary conditions in supergravity, J. High Energy Phys. 1010 (2010) 031. Crossref, Web of Science, Google Scholar
102. M. Faizal and A. Awad, Four dimensional supersymmetric theories in presence of a boundary, Phys. Lett. B 748 (2015) 414. Crossref, Web of Science, Google Scholar
103. D. V. Belyaev and P. van Nieuwenhuizen, Rigid supersymmetry with boundaries, J. High Energy Phys. 0804 (2008) 008. Crossref, Web of Science, Google Scholar
104. D. Zwanziger, Nonperturbative modification of the Faddeev–Popov formula and banishment of the Naive vacuum, Nuclear Phys. B 209 (1982) 336. Crossref, Web of Science, Google Scholar
105. D. Zwanziger, Local and renormalizable action from the Gribov horizon, Nuclear Phys. B 323 (1989) 513. Crossref, Web of Science, Google Scholar
106. 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. Crossref, Web of Science, Google Scholar
107. D. Zwanziger, Action from the Gribov horizon, Nuclear Phys. B 321 (1989) 591. Crossref, Web of Science, Google Scholar
108. 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. Crossref, Web of Science, Google Scholar
109. R. Garattini, Distorting general relativity: Gravity’s rainbow and $f (R)$ theories at work, J. Cosmol. Astrophys. 1306 (2013) 017. Crossref, Web of Science, Google Scholar
110. R. Garattini, Self sustained traversable wormholes induced by Gravity’s rainbow and noncommutative geometry, EPJ Web Conf. 58 (2013) 01007. Crossref, Google Scholar
111. G. Amelino-Camelia, M. Arzano, G. Gubitosi and J. Magueijo, Rainbow gravity and scale-invariant fluctuations, Phys. Rev. D 88(4) (2013) 041303. Crossref, Web of Science, Google Scholar
112. J. Magueijo and L. Smolin, Gravity’s rainbow, Class. Quantum Grav. 21 (2004) 1725. Crossref, Web of Science, Google Scholar
113. S. H. Hendi, S. Panahiyan, B. Eslam Panah, M. Faizal and M. Momennia, Critical behavior of charged black holes in Gauss–Bonnet gravitys rainbow, Phys. Rev. D 94(2) (2016) 024028. Crossref, Web of Science, Google Scholar
114. S. H. Hendi, A. Dehghani and M. Faizal, Black hole thermodynamics in Lovelock gravity’s rainbow with (A)dS asymptote, Nuclear Phys. B 914 (2017) 117. Crossref, Web of Science, Google Scholar
115. A. S. Sefiedgar, From the entropic force to the Friedmann equation in rainbow gravity, Europhys. Lett. 117(6) (2017) 69001. Crossref, Google Scholar
116. S. Alsaleh, Thermodynamics of rotating Kaluza-Klein black holes in gravitys rainbow, Eur. Phys. J. Plus 132(4) (2017) 181. Crossref, Web of Science, Google Scholar
117. S. H. Hendi, B. Eslam Panah, S. Panahiyan and M. Momennia, $F (R)$ gravity’s rainbow and its Einstein counterpart, Adv. High Energy Phys. 2016 (2016) 9813582. Crossref, Web of Science, Google Scholar
118. S. Gangopadhyay and A. Dutta, Constraints on rainbow gravity functions from black hole thermodynamics, Europhys. Lett. 115(5) (2016) 50005. Crossref, Google Scholar
119. S. H. Hendi, M. Momennia, B. Eslam Panah and M. Faizal, Nonsingular universes In Gaussbonnet gravitys rainbow, Astrophys. J. 827(2) (2016) 153. Crossref, Web of Science, Google Scholar
120. A. F. Ali, M. Faizal, B. Majumder and R. Mistry, Gravitational collapse in gravity’s rainbow, Int. J. Geom. Methods Mod. Phys. 12(09) (2015) 1550085. Link, Web of Science, Google Scholar
121. A. F. Ali, M. Faizal and B. Majumder, Absence of an effective horizon for black holes in gravity’s rainbow, Europhys. Lett. 109(2) (2015) 20001. Crossref, Google Scholar
122. A. F. Ali, M. Faizal and M. M. Khalil, Remnants of black rings from gravity’s rainbow, J. High Energy Phys. 1412 (2014) 159. Crossref, Web of Science, Google Scholar
123. P. Galan and G. A. Mena Marugan, Entropy and temperature of black holes in a gravity’s rainbow, Phys. Rev. D 74 (2006) 044035. Crossref, Web of Science, Google Scholar