Regularized Kalb-Ramond magnetic monopole with finite energy

Abstract

In a previous work we suggested a self-gravitating electromagnetic monopole solution in a string-inspired model involving global spontaneous breaking of a SO(3) internal symmetry and Kalb-Ramond (KR) axions, stemming from an antisymmetric tensor field in the massless string multiplet. These axions carry a charge, which, in our model, also plays the role of the magnetic charge. The resulting geometry is close to that of a Reissner-Nordstr\"om (RN) black hole with charge proportional to the KR-axion charge. We proposed the existence of a thin shell structure surrounding a (large inner) core as the dominant mass contribution to the energy functional. Although the resulting energy was finite, and proportional to the KR-axion charge, the size of the shell was not determined and left as a phenomenological parameter. In the current article, we propose a new way to calculate the size of the thin shell: string theory considerations suggest that the short-distance physics inside the inner core may be dominated by a positive cosmological constant term proportional to the scale of the spontaneous symmetry breaking of SO(3). The size of the shell is estimated by matching the RN metric of the shell to the de Sitter metric inside the core. The matching entails the Israel junction conditions for the metric and its first derivatives at the inner boundary of the shell, and determines the inner mass-shell radius. The axion charge plays an important role in guaranteeing the positivity of the ``mass coefficient'' of the gravitational potential term appearing in the metric; so, the KR electromagnetic monopole shows normal attractive gravitational effects. This is to be contrasted with the axion-less global monopole case where such a matching is known to yield a negative ``mass coefficient'' (and, hence, a repulsive gravitational effect). The total energy of our monopole within the shell is calculated. As a result of the violation of Birkhoff's theorem (due to the formal divergence of the energy functional in the absence of a large distance cutoff), the total energy does not have to equal the mass coefficient. However, for phenomenologically relevant sets of parameters, the ratio of the total energy and the mass coefficient in the shell is close to 1. The gravitational ``effective mass coefficient'' in the shell can be made equal to the total energy outside the core by a small decrease in the cosmological constant in the de Sitter region. This is achieved through a dilaton potential which is suitably negative inside the de-Sitter region, but vanishes outside that region.