Abstract
We study an evaporating black hole in the boundary conformal field theory (BCFT) model under the fully time-dependent AdS-Vaidya spacetime geometry. We introduce the time-dependent finite bath termed the effective Hawking radiation region. This is described by a nontrivial BCFT solution that acts as a time-dependent brane which we call the moving end-of-the-radiation (METR) brane that leads to a new type of Hubeny-Rangamani-Takayanagi surface. We further examine the island formulation in this particular time-dependent spacetime. The Page curve is calculated by using Holographic Entanglement Entropy (HEE) in the context of double holography.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
S.W. Hawking, Black Holes and Thermodynamics, Phys. Rev. D 13 (1976) 191 [INSPIRE].
D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [INSPIRE].
D.N. Page, Time Dependence of Hawking Radiation Entropy, JCAP 09 (2013) 028 [arXiv:1301.4995] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
K. Papadodimas and S. Raju, An Infalling Observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
D. Harlow and P. Hayden, Quantum Computation vs. Firewalls, JHEP 06 (2013) 085 [arXiv:1301.4504] [INSPIRE].
A. Almheiri et al., An Apologia for Firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
S. Lloyd and J. Preskill, Unitarity of black hole evaporation in final-state projection models, JHEP 08 (2014) 126 [arXiv:1308.4209] [INSPIRE].
K. Papadodimas and S. Raju, Black Hole Interior in the Holographic Correspondence and the Information Paradox, Phys. Rev. Lett. 112 (2014) 051301 [arXiv:1310.6334] [INSPIRE].
N. Bao et al., Rescuing Complementarity With Little Drama, JHEP 12 (2016) 026 [arXiv:1607.05141] [INSPIRE].
N. Bao et al., Branches of the Black Hole Wave Function Need Not Contain Firewalls, Phys. Rev. D 97 (2018) 126014 [arXiv:1712.04955] [INSPIRE].
C. Akers, N. Engelhardt and D. Harlow, Simple holographic models of black hole evaporation, JHEP 08 (2020) 032 [arXiv:1910.00972] [INSPIRE].
R. Bousso and M. Tomašević, Unitarity From a Smooth Horizon?, Phys. Rev. D 102 (2020) 106019 [arXiv:1911.06305] [INSPIRE].
J.-M. Bae, D.J. Lee, D.-H. Yeom and H. Zoe, Before the Page Time: Maximum Entanglements or the Return of the Monster?, Symmetry 14 (2022) 1649 [arXiv:2002.03543] [INSPIRE].
H. Geng, Y. Nomura and H.-Y. Sun, Information paradox and its resolution in de Sitter holography, Phys. Rev. D 103 (2021) 126004 [arXiv:2103.07477] [INSPIRE].
X.-L. Qi, Entanglement island, miracle operators and the firewall, JHEP 01 (2022) 085 [arXiv:2105.06579] [INSPIRE].
L. Aalsma et al., Shocks and information exchange in de Sitter space, JHEP 10 (2021) 104 [arXiv:2105.12737] [INSPIRE].
C. Akers and G. Penington, Quantum minimal surfaces from quantum error correction, SciPost Phys. 12 (2022) 157 [arXiv:2109.14618] [INSPIRE].
I. Aref’eva and I. Volovich, A note on islands in Schwarzschild black holes, Teor. Mat. Fiz. 214 (2023) 500 [arXiv:2110.04233] [INSPIRE].
K. Okuyama and K. Sakai, Page curve from dynamical branes in JT gravity, JHEP 02 (2022) 087 [arXiv:2111.09551] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
N. Engelhardt and S. Fischetti, Surface Theory: the Classical, the Quantum, and the Holographic, Class. Quant. Grav. 36 (2019) 205002 [arXiv:1904.08423] [INSPIRE].
G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 (2020) 002 [arXiv:1905.08255] [INSPIRE].
A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063 [arXiv:1905.08762] [INSPIRE].
A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, JHEP 03 (2020) 149 [arXiv:1908.10996] [INSPIRE].
H.Z. Chen et al., Information Flow in Black Hole Evaporation, JHEP 03 (2020) 152 [arXiv:1911.03402] [INSPIRE].
T.J. Hollowood and S.P. Kumar, Islands and Page Curves for Evaporating Black Holes in JT Gravity, JHEP 08 (2020) 094 [arXiv:2004.14944] [INSPIRE].
M. Rozali et al., Information radiation in BCFT models of black holes, JHEP 05 (2020) 004 [arXiv:1910.12836] [INSPIRE].
A. Almheiri, R. Mahajan and J.E. Santos, Entanglement islands in higher dimensions, SciPost Phys. 9 (2020) 001 [arXiv:1911.09666] [INSPIRE].
K. Hashimoto, N. Iizuka and Y. Matsuo, Islands in Schwarzschild black holes, JHEP 06 (2020) 085 [arXiv:2004.05863] [INSPIRE].
M. Alishahiha, A. Faraji Astaneh and A. Naseh, Island in the presence of higher derivative terms, JHEP 02 (2021) 035 [arXiv:2005.08715] [INSPIRE].
H. Geng and A. Karch, Massive islands, JHEP 09 (2020) 121 [arXiv:2006.02438] [INSPIRE].
H.Z. Chen et al., Quantum Extremal Islands Made Easy, Part I: Entanglement on the Brane, JHEP 10 (2020) 166 [arXiv:2006.04851] [INSPIRE].
V. Chandrasekaran, M. Miyaji and P. Rath, Including contributions from entanglement islands to the reflected entropy, Phys. Rev. D 102 (2020) 086009 [arXiv:2006.10754] [INSPIRE].
D. Bak, C. Kim, S.-H. Yi and J. Yoon, Unitarity of entanglement and islands in two-sided Janus black holes, JHEP 01 (2021) 155 [arXiv:2006.11717] [INSPIRE].
R. Bousso and E. Wildenhain, Gravity/ensemble duality, Phys. Rev. D 102 (2020) 066005 [arXiv:2006.16289] [INSPIRE].
C. Krishnan, Critical Islands, JHEP 01 (2021) 179 [arXiv:2007.06551] [INSPIRE].
R. Emparan, A.M. Frassino and B. Way, Quantum BTZ black hole, JHEP 11 (2020) 137 [arXiv:2007.15999] [INSPIRE].
H.Z. Chen et al., Quantum Extremal Islands Made Easy, Part II: Black Holes on the Brane, JHEP 12 (2020) 025 [arXiv:2010.00018] [INSPIRE].
Y. Ling, Y. Liu and Z.-Y. Xian, Island in Charged Black Holes, JHEP 03 (2021) 251 [arXiv:2010.00037] [INSPIRE].
A. Bhattacharya et al., Topological shadows and complexity of islands in multiboundary wormholes, JHEP 02 (2021) 152 [arXiv:2010.04134] [INSPIRE].
J. Hernandez, R.C. Myers and S.-M. Ruan, Quantum extremal islands made easy. Part III. Complexity on the brane, JHEP 02 (2021) 173 [arXiv:2010.16398] [INSPIRE].
J. Kumar Basak et al., Islands for entanglement negativity, SciPost Phys. 12 (2022) 003 [arXiv:2012.03983] [INSPIRE].
E. Caceres, A. Kundu, A.K. Patra and S. Shashi, Warped information and entanglement islands in AdS/WCFT, JHEP 07 (2021) 004 [arXiv:2012.05425] [INSPIRE].
F. Deng, J. Chu and Y. Zhou, Defect extremal surface as the holographic counterpart of Island formula, JHEP 03 (2021) 008 [arXiv:2012.07612] [INSPIRE].
A. Karlsson, Concerns about the replica wormhole derivation of the island conjecture, arXiv:2101.05879 [INSPIRE].
R.-X. Miao, Codimension-n holography for cones, Phys. Rev. D 104 (2021) 086031 [arXiv:2101.10031] [INSPIRE].
C. Bachas and V. Papadopoulos, Phases of Holographic Interfaces, JHEP 04 (2021) 262 [arXiv:2101.12529] [INSPIRE].
A. May and D. Wakeham, Quantum tasks require islands on the brane, Class. Quant. Grav. 38 (2021) 144001 [arXiv:2102.01810] [INSPIRE].
K. Kawabata, T. Nishioka, Y. Okuyama and K. Watanabe, Probing Hawking radiation through capacity of entanglement, JHEP 05 (2021) 062 [arXiv:2102.02425] [INSPIRE].
A. Bhattacharya, A. Bhattacharyya, P. Nandy and A.K. Patra, Islands and complexity of eternal black hole and radiation subsystems for a doubly holographic model, JHEP 05 (2021) 135 [arXiv:2103.15852] [INSPIRE].
W. Kim and M. Nam, Entanglement entropy of asymptotically flat non-extremal and extremal black holes with an island, Eur. Phys. J. C 81 (2021) 869 [arXiv:2103.16163] [INSPIRE].
L. Aalsma and W. Sybesma, The Price of Curiosity: Information Recovery in de Sitter Space, JHEP 05 (2021) 291 [arXiv:2104.00006] [INSPIRE].
D. Neuenfeld, The Dictionary for Double Holography and Graviton Masses in d Dimensions, arXiv:2104.02801 [INSPIRE].
H. Geng, S. Lüst, R.K. Mishra and D. Wakeham, Holographic BCFTs and Communicating Black Holes, jhep 08 (2021) 003 [arXiv:2104.07039] [INSPIRE].
X. Wang, R. Li and J. Wang, Page curves for a family of exactly solvable evaporating black holes, Phys. Rev. D 103 (2021) 126026 [arXiv:2104.00224] [INSPIRE].
V. Balasubramanian, A. Kar and T. Ugajin, Entanglement between two gravitating universes, Class. Quant. Grav. 39 (2022) 174001 [arXiv:2104.13383] [INSPIRE].
C.F. Uhlemann, Islands and Page curves in 4d from Type IIB, JHEP 08 (2021) 104 [arXiv:2105.00008] [INSPIRE].
D. Neuenfeld, Homology conditions for RT surfaces in double holography, Class. Quant. Grav. 39 (2022) 075009 [arXiv:2105.01130] [INSPIRE].
K. Kawabata, T. Nishioka, Y. Okuyama and K. Watanabe, Replica wormholes and capacity of entanglement, JHEP 10 (2021) 227 [arXiv:2105.08396] [INSPIRE].
J. Chu, F. Deng and Y. Zhou, Page curve from defect extremal surface and island in higher dimensions, JHEP 10 (2021) 149 [arXiv:2105.09106] [INSPIRE].
Y. Lu and J. Lin, Islands in Kaluza-Klein black holes, Eur. Phys. J. C 82 (2022) 132 [arXiv:2106.07845] [INSPIRE].
J. Kruthoff, R. Mahajan and C. Murdia, Free fermion entanglement with a semitransparent interface: the effect of graybody factors on entanglement islands, SciPost Phys. 11 (2021) 063 [arXiv:2106.10287] [INSPIRE].
I. Akal et al., Holographic moving mirrors, Class. Quant. Grav. 38 (2021) 224001 [arXiv:2106.11179] [INSPIRE].
J. Kumar Basak et al., Page curve for entanglement negativity through geometric evaporation, SciPost Phys. 12 (2022) 004 [arXiv:2106.12593] [INSPIRE].
E. Caceres, A. Kundu, A.K. Patra and S. Shashi, Page curves and bath deformations, SciPost Phys. Core 5 (2022) 033 [arXiv:2107.00022] [INSPIRE].
H. Omiya and Z. Wei, Causal structures and nonlocality in double holography, JHEP 07 (2022) 128 [arXiv:2107.01219] [INSPIRE].
H. Geng et al., Inconsistency of islands in theories with long-range gravity, JHEP 01 (2022) 182 [arXiv:2107.03390] [INSPIRE].
C.-J. Chou, H.B. Lao and Y. Yang, Page curve of effective Hawking radiation, Phys. Rev. D 106 (2022) 066008 [arXiv:2111.14551] [INSPIRE].
F. Deng, Y.-S. An and Y. Zhou, JT gravity from partial reduction and defect extremal surface, JHEP 02 (2023) 219 [arXiv:2206.09609] [INSPIRE].
A. Bhattacharjee and M. Saha, JT gravity from holographic reduction of 3D asymptotically flat spacetime, JHEP 01 (2023) 138 [arXiv:2211.13415] [INSPIRE].
A. Bhattacharya, A. Bhattacharyya and A.K. Patra, Holographic complexity of Jackiw-Teitelboim gravity from Karch-Randall braneworld, JHEP 07 (2023) 060 [arXiv:2304.09909] [INSPIRE].
M. Afrasiar et al., Islands and dynamics at the interface, JHEP 11 (2023) 192 [arXiv:2306.12476] [INSPIRE].
V. Balasubramanian et al., Geometric secret sharing in a model of Hawking radiation, JHEP 01 (2021) 177 [arXiv:2003.05448] [INSPIRE].
A. Bhattacharya, Multipartite purification, multiboundary wormholes, and islands in AdS3/CFT2, Phys. Rev. D 102 (2020) 046013 [arXiv:2003.11870] [INSPIRE].
T. Hartman, E. Shaghoulian and A. Strominger, Islands in Asymptotically Flat 2D Gravity, JHEP 07 (2020) 022 [arXiv:2004.13857] [INSPIRE].
T.J. Hollowood, S. Prem Kumar and A. Legramandi, Hawking radiation correlations of evaporating black holes in JT gravity, J. Phys. A 53 (2020) 475401 [arXiv:2007.04877] [INSPIRE].
A. Karlsson, Replica wormhole and island incompatibility with monogamy of entanglement, arXiv:2007.10523 [INSPIRE].
X. Wang, R. Li and J. Wang, Islands and Page curves of Reissner-Nordström black holes, JHEP 04 (2021) 103 [arXiv:2101.06867] [INSPIRE].
S. Fallows and S.F. Ross, Islands and mixed states in closed universes, JHEP 07 (2021) 022 [arXiv:2103.14364] [INSPIRE].
T.J. Hollowood, S.P. Kumar, A. Legramandi and N. Talwar, Islands in the stream of Hawking radiation, JHEP 11 (2021) 067 [arXiv:2104.00052] [INSPIRE].
D. Marolf and H. Maxfield, The page curve and baby universes, Int. J. Mod. Phys. D 30 (2021) 2142027 [arXiv:2105.12211] [INSPIRE].
M. Cadoni and A.P. Sanna, Unitarity and Page Curve for Evaporation of 2D AdS Black Holes, arXiv:2106.14738 [https://doi.org/10.3390/e24010101] [INSPIRE].
M.-H. Yu and X.-H. Ge, Islands and Page curves in charged dilaton black holes, Eur. Phys. J. C 82 (2022) 14 [arXiv:2107.03031] [INSPIRE].
D.-Q. Sun, Hawking-Page phase transition, Page curve and islands in black holes, arXiv:2107.05218 [INSPIRE].
B. Ahn et al., Islands in charged linear dilaton black holes, Phys. Rev. D 105 (2022) 046012 [arXiv:2107.07444] [INSPIRE].
N.H. Cao, Entanglement entropy and Page curve of black holes with island in massive gravity, Eur. Phys. J. C 82 (2022) 381 [arXiv:2108.10144] [INSPIRE].
A. Bhattacharya, A. Bhattacharyya, P. Nandy and A.K. Patra, Partial islands and subregion complexity in geometric secret-sharing model, JHEP 12 (2021) 091 [arXiv:2109.07842] [INSPIRE].
S. He, Y. Sun, L. Zhao and Y.-X. Zhang, The universality of islands outside the horizon, JHEP 05 (2022) 047 [arXiv:2110.07598] [INSPIRE].
A. Bhattacharya, A. Bhattacharyya, P. Nandy and A.K. Patra, Bath deformations, islands, and holographic complexity, Phys. Rev. D 105 (2022) 066019 [arXiv:2112.06967] [INSPIRE].
D. Basu, H. Parihar, V. Raj and G. Sengupta, Defect extremal surfaces for entanglement negativity, Phys. Rev. D 108 (2023) 106005 [arXiv:2205.07905] [INSPIRE].
D. Basu, J. Lin, Y. Lu and Q. Wen, Ownerless island and partial entanglement entropy in island phases, SciPost Phys. 15 (2023) 227 [arXiv:2305.04259] [INSPIRE].
G.R. Dvali, G. Gabadadze and M. Porrati, 4-D gravity on a brane in 5-D Minkowski space, Phys. Lett. B 485 (2000) 208 [hep-th/0005016] [INSPIRE].
D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B 455 (1995) 522 [cond-mat/9505127] [INSPIRE].
J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
W. Kim and M. Nam, Temperatures of AdS2 black holes and holography revisited, arXiv:2304.04427 [INSPIRE].
Y. Matsuo, Islands and stretched horizon, JHEP 07 (2021) 051 [arXiv:2011.08814] [INSPIRE].
A. Saha, S. Gangopadhyay and J.P. Saha, Mutual information, islands in black holes and the Page curve, Eur. Phys. J. C 82 (2022) 476 [arXiv:2109.02996] [INSPIRE].
M.-H. Yu, C.-Y. Lu, X.-H. Ge and S.-J. Sin, Island, Page curve, and superradiance of rotating BTZ black holes, Phys. Rev. D 105 (2022) 066009 [arXiv:2112.14361] [INSPIRE].
W.-C. Gan, D.-H. Du and F.-W. Shu, Island and Page curve for one-sided asymptotically flat black hole, JHEP 07 (2022) 020 [arXiv:2203.06310] [INSPIRE].
C.-Z. Guo, W.-C. Gan and F.-W. Shu, Page curves and entanglement islands for the step-function Vaidya model of evaporating black holes, JHEP 05 (2023) 042 [arXiv:2302.02379] [INSPIRE].
R. Emparan et al., Holographic duals of evaporating black holes, JHEP 05 (2023) 182 [arXiv:2301.02587] [INSPIRE].
Y.C. Ong, B. McInnes and P. Chen, Cold black holes in the Harlow-Hayden approach to firewalls, Nucl. Phys. B 891 (2015) 627 [arXiv:1403.4886] [INSPIRE].
H.Z. Chen et al., Evaporating Black Holes Coupled to a Thermal Bath, JHEP 01 (2021) 065 [arXiv:2007.11658] [INSPIRE].
G. Grimaldi, J. Hernandez and R.C. Myers, Quantum extremal islands made easy. Part IV. Massive black holes on the brane, JHEP 03 (2022) 136 [arXiv:2202.00679] [INSPIRE].
B. Craps, J. Hernandez, M. Khramtsov and M. Knysh, Delicate windows into evaporating black holes, JHEP 02 (2023) 080 [arXiv:2209.15477] [INSPIRE].
S.B. Giddings and A. Nudelman, Gravitational collapse and its boundary description in AdS, JHEP 02 (2002) 003 [hep-th/0112099] [INSPIRE].
K. Maeda, M. Natsuume and T. Okamura, Extracting information behind the veil of horizon, Phys. Rev. D 74 (2006) 046010 [hep-th/0605224] [INSPIRE].
V.E. Hubeny, H. Liu and M. Rangamani, Bulk-cone singularities & signatures of horizon formation in AdS/CFT, JHEP 01 (2007) 009 [hep-th/0610041] [INSPIRE].
Y.C. Ong, The Persistence of the Large Volumes in Black Holes, Gen. Rel. Grav. 47 (2015) 88 [arXiv:1503.08245] [INSPIRE].
M. Visser, Gravitational vacuum polarization. 2: Energy conditions in the Boulware vacuum, Phys. Rev. D 54 (1996) 5116 [gr-qc/9604008] [INSPIRE].
N. Graham and K.D. Olum, Achronal averaged null energy condition, Phys. Rev. D 76 (2007) 064001 [arXiv:0705.3193] [INSPIRE].
A. Ishibashi, K. Maeda and E. Mefford, Achronal averaged null energy condition, weak cosmic censorship, and AdS/CFT duality, Phys. Rev. D 100 (2019) 066008 [arXiv:1903.11806] [INSPIRE].
N. Iizuka, A. Ishibashi and K. Maeda, Conformally invariant averaged null energy condition from AdS/CFT, JHEP 03 (2020) 161 [arXiv:1911.02654] [INSPIRE].
S. Gao and R.M. Wald, Theorems on gravitational time delay and related issues, Class. Quant. Grav. 17 (2000) 4999 [gr-qc/0007021] [INSPIRE].
A. Ishibashi and K. Maeda, The averaged null energy condition on holographic evaporating black holes, JHEP 03 (2022) 104 [arXiv:2111.05151] [INSPIRE].
T. Takayanagi, Holographic Dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602 [arXiv:1105.5165] [INSPIRE].
R.-X. Miao, C.-S. Chu and W.-Z. Guo, New proposal for a holographic boundary conformal field theory, Phys. Rev. D 96 (2017) 046005 [arXiv:1701.04275] [INSPIRE].
C.-S. Chu, R.-X. Miao and W.-Z. Guo, On New Proposal for Holographic BCFT, JHEP 04 (2017) 089 [arXiv:1701.07202] [INSPIRE].
W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical recipes 3rd edition: The art of scientific computing, Cambridge University Press (2007).
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
Acknowledgments
We thank Xiaoning Wu for the useful discussion. This work is supported by the Ministry of Science and Technology (MOST 109-2112-M-009-005) and National Center for Theoretical Sciences, Taiwan.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2306.16744
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Chou, CJ., Lao, H.B. & Yang, Y. Page curve of AdS-Vaidya model for evaporating black holes. J. High Energ. Phys. 2024, 342 (2024). https://doi.org/10.1007/JHEP05(2024)342
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2024)342