The Effective Potential for the Conformal Factor in the Standard Model and Beyond

There is a general mechanism by which certain matter fields coupled to gravity can generate a nontrivial effective potential for the conformal factor of the metric. It is based on a nonstandard regularization method, with the cutoff being defined independently of the conformal factor. This mechanism produces a coupling of the matter fields to a dilaton, and a complicated interaction between matter, dilaton and metric. When it is applied to the standard model, it gives an effective potential which can be used to predict the top and Higgs masses. If the purely gravitational contribution to the potential is added, the mass of the dilaton is of the order of Planck’s mass and the large hierarchy between the Planck and Fermi scales appears to be due to the smallness of the Higgs-dilaton coupling. ∗ permanent address: SISSA, via Beirut 4, 34014 Trieste, Italy 1 It has been conjectured that the existence of a scalar dilaton might be related to the vanishing of the cosmological constant [1]. In particular, this has prompted studies of the coupling of a dilaton to the standard model [2]. Very recently, a new twist has been added to the subject: the existence of a dilaton and its classical coupling to the standard model have been derived from arguments of noncommutative geometry [3] and then, modulo some assumptions about the structure of radiative corrections, a very sharp prediction of the top quark mass was derived [4]. It is quite clear that the only role played by noncommutative geometry in this argument is to make a certain statement about the effective coupling of the dilaton to the standard model. Any theory leading to the same effective coupling will therefore also lead to the same prediction for the masses. In this note I will show that the dilaton can arise from the coupling of the standard model to gravity, if a nonstandard regularization method is adopted. Specifically, quantum fluctuations of the matter sector induce a nontrivial effective potential for the conformal factor of the metric, which then becomes an independent variable, the dilaton. The effective potential of [4] can be derived in this way. This idea has been studied before in the context of pure gravity [5]; very similar ideas have been discussed also in [6]. I will first illustrate the point in the case of a single real scalar field, since this example already has all the features of the more realistic models. Let us therefore start from the action S(φ, gμν) = − ∫ dx √ g [ 1 2 g∂μφ∂νφ+ 1 2 (m + ξR)φ + λ 4! φ ]