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Concept# Auxiliary field

Summary

In physics, and especially quantum field theory, an auxiliary field is one whose equations of motion admit a single solution. Therefore, the Lagrangian describing such a field A contains an algebraic quadratic term and an arbitrary linear term, while it contains no kinetic terms (derivatives of the field):
:\mathcal{L}_\text{aux} = \frac{1}{2}(A, A) + (f(\varphi), A).
The equation of motion for A is
:A(\varphi) = -f(\varphi),
and the Lagrangian becomes
:\mathcal{L}_\text{aux} = -\frac{1}{2}(f(\varphi), f(\varphi)).
Auxiliary fields generally do not propagate, and hence the content of any theory can remain unchanged in many circumstances by adding such fields by hand.
If we have an initial Lagrangian \mathcal{L}_0 describing a field \varphi, then the Lagrangian describing both fields is
:\mathcal{L} = \mathcal{L}*0(\varphi) + \mathcal{L}*\text{aux} = \mathcal{L}_0(\varphi) - \frac{1}{

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The main topics discussed in this thesis are supersymmetric low-energy effective theories and metastability conditions in generic non-renormalizable models with global and local supersymmetry. In the first part we discuss the conditions under which the low-energy expansion in space-time derivatives preserves supersymmetry implying that heavy multiplets can be more efficiently integrated out directly at the superfield level. These conditions translate into the requirements that also fermions and auxiliary fields should be small compared to the heavy mass scale. They apply not only to the matter sector, but also to the gravitational one if present, and imply in that case that the gravitino mass should be small. We finally give a simple prescription to integrate out heavy chiral and vector superfields consisting respectively in imposing stationarity of the superpotential and of the Kähler potential; the procedure holds in the same form both for global and local supersymmetry. In the second part we study general criteria for the existence of metastable vacua which break global supersymmetry in models with local gauge symmetries. In particular we present a strategy to define an absolute upper bound on the mass of the lightest scalar field which depends on the geometrical properties of the Kähler target manifold. This bound can be saturated by properly tuning the superpotential and its positivity therefore represents a necessary and sufficient condition for the existence of metastable vacua. It is derived by looking at the subspace of all those directions in field space for which an arbitrary supersymmetric mass term is not allowed and scalar masses are controlled by supersymmetry-breaking splitting effects. This subspace includes not only the direction of supersymmetry breaking, but also the directions of gauge symmetry breaking and the lightest scalar is in general a linear combination of fields spanning all these directions. Our purpose is to show that the largest value for the lightest mass is in general achieved when the lightest scalar is a combination of the Goldstone and the Goldstino partners. We conclude by computing the effects induced by the integration of heavy multiplets on the light masses. In particular we focus on the sGoldstino partners and we show that heavy chiral multiplets induce a negative level-repulsion effect that tends to compromise vacuum stability, whereas heavy vector multiplets in general induce a positive-definite contribution. Our results find application in the context of string-inspired supergravity models, where metastability conditions can be used to discriminate among different compactification scenarios and supersymmetric effective theories can be used to face the problem of moduli stabilization.

We establish a classical analog of the Nambu-Goldstone theorem for spontaneous breaking of spacetime symmetries. It provides a counting rule for independent Nambu-Goldstone fields and states which of them are gapped. We demonstrate that only those symmetry group generators give rise to independent Nambu-Goldstone fields that act nontrivially on a vacuum at the origin of coordinates. Other generators give rise to auxiliary fields that must be excluded from a theory by the means of inverse Higgs constraints. The physical meaning of the inverse Higgs phenomenon and an application of our results to theories of massive gravity are discussed.

Leonardo Brizi, Claudio Scrucca

We reconsider the general question of how to characterize most efficiently the low-energy effective theory obtained by integrating out heavy modes in globally and locally supersymmetric theories. We consider theories with chiral and vector multiplets and identify the conditions under which all approximately supersymmetric low-energy effective theory can exist. These conditions translate into the requirements that all the derivatives, fermions and auxiliary fields should be small in units of the heavy mass scale. They apply not only to the matter sector, but also to the gravitational one if present, and imply in that case that the gravitino mass should he small. We then show how to determine the unique exactly supersymmetric theory that approximates this effective theory at the lowest order in the counting of derivatives, fermions and auxiliary fields, by working both at the superfield level and with component fields. As a result we give a simple prescription tor integrating out heavy superfields in an algebraic and manifestly supersymmetric way, which turns out to hold in the same form both for globally and locally supersymmetric theories. meaning that the process of integrating out heavy modes commutes with the process of switching on gravity. More precisely, for heavy chiral and vector multiplets one has to impose respectively stationarity of the superpotential and the Kahler potential. (C) 2009 Elsevier B.V. All rights reserved.

2009