Abstract
The rôle of dilatation symmetry in explaining the smallness of the cosmological constant is investigated. Dilatation-invariant theories necessarily lead to a zero cosmological constant, but only if the vacuum is invariant. This is physically unacceptable, and dilatation invariance must be broken (at least) spontaneously, in which case we do not find any constraint on the cosmological constant. We show, too, that theories in which dilatation invariance is broken by anomalies due to renormalization do not lead to a vanishing cosmological constant without a fine-tuning of parameters. Similarly, theories with an (approximate) dilatation symmetry in the renormalizable sector require fine-tuning to eliminate the cosmological constant.
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