On tadpoles and vacuum redefinitions in String Theory

Abstract

“Issues on tadpoles and vacuum redefinitions in String Theory” M. Nicolosi This Thesis is devoted to the problem of NS-NS tadpoles, bosonic one-point functions going into the vacuum that typically emerge in String Theory after supersymmetry breaking. These theories contain bosonic fields in two sectors, commonly denoted with NS-NS and R-R. While R-R tadpoles typically signal an inconsistency, like the presence of quantum anomalies in the case of a compact internal space, and thus in general must be cancelled, NS-NS tadpoles are associated to redefinitions of the background, as first stressed by Fischler and Susskind in the eighties. In particular, in Type I String Theory NS-NS tadpoles emerge already at the disk level and, from a space-time viewpoint, correspond to configurations of D-branes and orientifold planes with a non-vanishing tension giving rise to a net gravitational attraction that curves the background space-time. Up to now one is able to perform efficient string computations only in a flat Minkowski background, a case that is allowed and protected by supersymmetry. Hence, the (closed) infrared divergences emerging after supersymmetry breaking in string amplitudes, due to the propagation of NS-NS massless states that are absorbed by tadpoles at vanishing momentum, are just the signal that the flat Minkowski background is no more a vacuum of the theory. In this context our proposal is to keep quantizing the string around the Minkowski background, recovering the proper results after suitable tadpole resummations that cancel the infrared divergences. This procedure is still very difficult to carry out in String Theory, because the higher-order tadpole corrections correspond to Riemann surfaces of increasing genus, and efficient calculations can be only carried out up to genus one (one-loop amplitudes). Moreover, in most models that realize supersymmetry breaking, tadpoles arise already at the disk level, and thus, even in a perturbative region of small string coupling, the first tadpole corrections can be large. Hence, it is interesting to search for models with “small” tadpoles. Examples of this kind seem are provided by models with suitable internal fluxes, for which reliable perturbative results can be recovered just considering the first tadpole corrections. Another line that one can pursue is to search for quantities that are protected against the infrared divergences. An example of this kind is provided by the one-loop string corrections to gauge couplings, commonly known as threshold corrections, for supersymmetry breaking models with parallel branes, a case that we have widely discussed in this Thesis. The Thesis is organized in the following way. There is a general Introduction to String Theory, where we summarize the main ideas of the Theory, trying to underline its successes and its open problems. Then in the first Chapter we recall the basic properties of string spectra and discuss some simple examples of toroidal and orbifold compactifications. The second Chapter is devoted to reviewing a number of different mechanisms to break supersymmetry. In the third Chapter we begin to analyze our resummation program in a number of field theory toy models, trying to recover the right results, at least at the classical level, starting from a “wrong vacuum”. The cases of cubic and quartic potentials are simple and interesting, and display some general features concerning tadpole resummations and convergence domains around inflection points of the potential, where the tadpole expansion breaks down. Our analysis shows that, starting from an arbitrary initial value of the field, classical tadpole resummations typically drive the quantities we are computing towards an extremum of the potential, not necessary a minimum. In addition, for the case of a quartic potential we find some very special “non-renormalization” points for which all higher order tadpole corrections cancel. We then analyze our procedure for a sting-inspired toy model with tadpoles localized on lower dimensional D-branes, performing explicitly the resummations. We also consider the introduction of gravity, that should give further complications related to the graviton mass terms, but seems to not affect substantially our program, and indeed tadpole resummations prove still to work in this case. Finally, in Chapter four we begin to face the tadpole problem in String Theory itself. In the first Section, we describe an example where the vacuum redefinition can be understood not only at the level of the low energy effective field theory, but also at the full string theory level. In particular, we show that the vacuum of a Type II orientifold with a compact dimension and local tadpoles is a Type 0 orientifold without compact dimensions. These results are contained in a paper to appear in Nuclear Physics B. Finally, in the last Section we begin the analysis of one-loop threshold corrections in a number of models with supersymmetry breaking with parallel branes and no closed tachyons propagating in the bulk. The result is that the one-loop threshold corrections in all these cases are always (closed) infrared finite, in spite of the presence of NS-NS tadpoles. These computations will be included in a paper that is currently in preparation.