TY - JOUR

T1 - Spontaneously broken symmetry restoration of quantum fields in the vicinity of neutral and electrically charged black holes

AU - Quinta, Gonçalo M.

AU - Flachi, Antonino

AU - Lemos, José P.S.

N1 - Funding Information:
Article funded by SCOAP3.
Funding Information:
We thank Peter Taylor and Cormac Breen for sharing the coefficients for a constant mass term in four dimensions that helped us deriving 〈φ2(x)〉ren and gnlS(r) above. GQ acknowledges the support of the Fundacão para a Ciência e Tecnologia (FCT Portugal) through Grant No. SFRH/BD/92583/2013. AF acknowledges the support of the Japanese Ministry of Education, Culture, Sports, Science Program for the Strategic Research Foundation at Private Universities ‘Topological Science’ Grant No. S1511006 and of the JSPS KAKENHI Grant No. 18K03626. JPSL acknowledges FCT for financial support through Project No. UID/FIS/00099/2013, Grant No. SFRH/BSAB/128455/2017, and Coordenac¸ão de Aperfeic¸oamento do Pessoal de Ńıvel Superior (CAPES), Brazil, for support within the Programa CSF-PVE, Grant No. 88887.068694/2014-00. The authors thankfully acknowledge the computer resources, technical expertise, and assistance provided by CENTRA/IST. Computations were performed at the cluster Baltasar-Sete-Sois supported by the H2020 ERC Consolidator Grant “Matter and strong field gravity: New frontiers in Einstein’s theory” grant agreement No. MaGRaTh-646597.

PY - 2019/4/1

Y1 - 2019/4/1

N2 - We consider the restoration of a spontaneously broken symmetry of an interacting quantum scalar field around neutral, i.e., Schwarzschild, and electrically charged, i.e., Reissner-Nordström, black holes in four dimensions. This is done through a semi-classical self-consistent procedure, by solving the system of non-linear coupled equations describing the dynamics of the background field and the vacuum polarization. The black hole at its own horizon generates an indefinitely high temperature which decreases to the Hawking temperature at infinity. Due to the high temperature in its vicinity, there forms a bubble around the black hole in which the scalar field can only assume a value equal to zero, a minimum of energy. Thus, in this region the symmetry of the energy and the field is preserved. At the bubble radius, there is a phase transition in the value of the scalar field due to a spontaneous symmetry breaking mechanism. Indeed, outside the bubble radius the temperature is low enough such that the scalar field settles with a nonzero value in a new energy minimum, indicating a breaking of the symmetry in this outer region. Conversely, there is symmetry restoration from the outer region to the inner bubble close to the horizon. Specific properties that emerge from different black hole electric charges are also noteworthy. It is found that colder black holes, i.e., more charged ones, have a smaller bubble length of restored symmetry. In the extremal case the bubble has zero length, i.e., there is no bubble. Additionally, for colder black holes, it becomes harder to excite the quantum field modes, so the vacuum polarization has smaller values. In the extremal case, the black hole temperature is zero and the vacuum polarization is never excited.

AB - We consider the restoration of a spontaneously broken symmetry of an interacting quantum scalar field around neutral, i.e., Schwarzschild, and electrically charged, i.e., Reissner-Nordström, black holes in four dimensions. This is done through a semi-classical self-consistent procedure, by solving the system of non-linear coupled equations describing the dynamics of the background field and the vacuum polarization. The black hole at its own horizon generates an indefinitely high temperature which decreases to the Hawking temperature at infinity. Due to the high temperature in its vicinity, there forms a bubble around the black hole in which the scalar field can only assume a value equal to zero, a minimum of energy. Thus, in this region the symmetry of the energy and the field is preserved. At the bubble radius, there is a phase transition in the value of the scalar field due to a spontaneous symmetry breaking mechanism. Indeed, outside the bubble radius the temperature is low enough such that the scalar field settles with a nonzero value in a new energy minimum, indicating a breaking of the symmetry in this outer region. Conversely, there is symmetry restoration from the outer region to the inner bubble close to the horizon. Specific properties that emerge from different black hole electric charges are also noteworthy. It is found that colder black holes, i.e., more charged ones, have a smaller bubble length of restored symmetry. In the extremal case the bubble has zero length, i.e., there is no bubble. Additionally, for colder black holes, it becomes harder to excite the quantum field modes, so the vacuum polarization has smaller values. In the extremal case, the black hole temperature is zero and the vacuum polarization is never excited.

KW - Black Holes

KW - Effective Field Theories

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U2 - 10.1007/JHEP04(2019)139

DO - 10.1007/JHEP04(2019)139

M3 - Article

AN - SCOPUS:85064940676

VL - 2019

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

SN - 1126-6708

IS - 4

M1 - 139

ER -