The island formula, derived from replica wormhole saddles in the gravitational path integral, successfully reproduces the unitary Page curve for black hole radiation entropy in the large- <italic>N</italic> (large central charge) limit. However, the resulting entropy profile contains an unphysical sharp kink at the Page time <italic>t</italic> <inf>Page</inf>&#8212;a mathematical artefact of the <italic>min</italic> operation that dominates in that limit. Any finite quantum system must exhibit a smooth, analytic transition. In this paper we work within Jackiw&#8211;Teitelboim (JT) gravity coupled to a holographic CFT bath and derive systematically the leading finite- <italic>N</italic> correction to the radiation entropy, <display><formula id='ueq0001'><math altimg='si9.svg'><mrow><msub><mi>S</mi><mtext>rad</mtext></msub><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>N</mi><mo>)</mo></mrow><mo linebreak='goodbreak'>=</mo><msub><mi>S</mi><mtext>Hawk</mtext></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo linebreak='goodbreak'>+</mo><mi>&#948;</mi><mi>S</mi><mrow><mo>(</mo><mi>N</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo><mspace width='2.em' /><mi>&#948;</mi><mi>S</mi><mo>&#8764;</mo><mi mathvariant='script'>O</mi><mspace width='-0.16em' /><mrow><mo stretchy='true'>(</mo><msup><mrow><mi mathvariant='normal'>e</mi></mrow><mrow><mo>&#8722;</mo><msub><mi>S</mi><mtext>BH</mtext></msub></mrow></msup><mo stretchy='true'>)</mo></mrow><mo>,</mo></mrow></math></formula></display>originating from non-perturbative interference between the Hawking and replica wormhole saddles. Using a soft-mode resummation of the gravitational path integral we show that these corrections replace the sharp kink with a log-sum-exp interpolation controlled by 1/ <italic>N</italic>. We further study an interference term arising when both saddles contribute at comparable weight, providing the first analytic description of a smooth cross-over in the island formula. Our predictions are confirmed by a random-unitary-circuit simulation of an <italic>n</italic>-qubit Page transition and by direct numerical evaluation of the partition function in a matrix-model truncation of JT gravity. The results imply that information begins to leak gradually from the black hole interior long before <italic>t</italic> <inf>Page</inf>, consistent with unitarity but without invoking a firewall.

Bangladesh

Snigdha Saha

Non-perturbative transitions in the island formula: A finite- <italic>N</italic> resolution of the page curve kink

It is widely known that the Saha equation is not suitable for describing plasmas out of thermodynamic equilibrium. The primordial hydrogen recombination plasma is an example of this. In this work, we propose a theoretical modification to the standard Saha curve motivated by Tsallis statistics. In particular, we explore the possibility that a time-dependent <italic>q</italic>-parameter may serve as an effective proxy for the evolving thermodynamic conditions during recombination, especially considering that hydrogen recombination occurs from excited states. Within this framework, the <italic>q</italic>-parameter could be interpreted as encoding departures from equilibrium and could play the role of effective time-dependent temperature. This indicates that the time evolution of the <italic>q</italic>-parameter could provide a phenomenological mechanism for incorporating non-equilibrium effects into the recombination history. Our findings suggest that the Tsallis parameterization provides an alternative path to fit the distribution of free electrons by using an effective temperature. The implications of this approach might go beyond its immediate applications, as the Saha equation is widely used in various scientific fields like astrophysics, cosmology, plasma physics, and condensed matter physics.

Brazil

L.L. Sales

F.C. Carvalho

H.T.C.M. Souza

A possible correction of the Saha curve for non-equilibrium states

The formation and decay of metastable bound states can significantly decrease the thermal-relic dark matter density, particularly for dark matter masses around and above the TeV scale. Incorporating bound-state effects in the dark matter thermal decoupling requires in principle a set of coupled Boltzmann equations for the bound and unbound species. However, decaying bound states attain and remain in a quasi-steady state. Here we prove in generality that this reduces the coupled system into a single Boltzmann equation of the standard form, with an effective cross-section that describes the interplay among bound-state formation, ionisation, transitions and decays. We derive a closed-form expression for the effective cross-section for an arbitrary number of bound states, and show that bound-to-bound transitions can only increase it. Excited bound levels may thus decrease the dark matter density more significantly than otherwise estimated. Our results generalise the Saha ionisation equilibrium to metastable bound states, potentially with applications beyond the dark matter thermal decoupling.

Japan

Tobias Binder

Netherlands

Anastasiia Filimonova

France

Kalliopi Petraki

Graham White

Saha equilibrium for metastable bound states and dark matter freeze-out

The production of light (anti-)(hyper-)nuclei in heavy-ion collisions at the LHC is considered in the framework of the Saha equation, making use of the analogy between the evolution of the early universe after the Big Bang and that of “Little Bangs” created in the lab. Assuming that disintegration and regeneration reactions involving light nuclei proceed in relative chemical equilibrium after the chemical freeze-out of hadrons, their abundances are determined through the famous cosmological Saha equation of primordial nucleosynthesis and show no exponential dependence on the temperature typical for the thermal model. A quantitative analysis, performed using the hadron resonance gas model in partial chemical equilibrium, shows agreement with experimental data of the ALICE collaboration on d, $^{3}$He, <math><mmultiscripts><mrow><mtext>H</mtext></mrow><mprescripts></mprescripts><mrow><mi>Λ</mi></mrow><mrow><mn>3</mn></mrow></mmultiscripts></math>, and $^{4}$He yields for a very broad range of temperatures at <math><mi>T</mi><mo>≲</mo><mn>155</mn></math> MeV. The presented picture is supported by the observed suppression of resonance yields in central Pb–Pb collisions at the LHC.