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Weyl gauge-vector and complex dilaton scalar for conformal symmetry and its breaking

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Abstract

Instead of the scalar “dilaton” field that is usually adopted to construct conformally invariant Lagrangians for gravitation, we here propose a hybrid construction, involving both a complex dilaton scalar and a Weyl gauge-vector, in accord with Weyl’s original concept of a non-Riemannian conformal geometry with a transport law for length and time intervals, for which this gauge vector is required. Such a hybrid construction permits us to avoid the wrong sign of the dilaton kinetic term (the ghost problem) that afflicts the usual construction. The introduction of a Weyl gauge-vector and its interaction with the dilaton also has the collateral benefit of providing an explicit mechanism for spontaneous breaking of the conformal symmetry, whereby the dilaton and the Weyl gauge-vector acquire masses somewhat smaller than \(\textit{m}_\textit{P}\) by the Coleman–Weinberg mechanism. Conformal symmetry breaking is assumed to precede inflation, which occurs later by a separate GUT or electroweak symmetry breaking, as in inflationary models based on the Higgs boson.

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Notes

  1. The preservation of the dimension of the metric tensor requires a dimensional factor \(1/(\mathrm{mass})^{2}\) on the right side of this transformation equation, to compensate the dimension of \(\phi ^{2}\). This dimensional factor is of crucial importance in investigations of renormalizability. For the sake of simplicity, I here imitate Jackiw and Pi in omitting this factor.

  2. With the Lagrangian (4), additional conformally-invariant higher-order derivative terms, such as the term \(\alpha _{\textit{grav}} \sqrt{-g}(R_{\mu \nu } R^{\mu \nu }-R^{2}/3)\) often favored by theorists, merely add short-range Yukawa potentials to the usual macroscopic 1 / r Newtonian potential. If \(\alpha _{\textit{grav}} \) is of the order of magnitude of \({\sim }\)1, then the range of this Yukawa potential is about a Planck length and it produces no measurable macroscopic effects. However, the higher-order derivatives can lead to drastic modifications of the singularities found in general relativity.

  3. The masses are here expressed in terms of \(\Theta \), but they can be alternatively expressed in terms of \(\lambda \) because the condition for a minimum in the effective potential implies the relation \(\lambda =(33/128\pi ^{2})b^{4}\cosh ^{6}\Theta \) between the coupling constants [23].

  4. For conformal invariance, each of the Higgs scalar fields H associated with the breaking of GUT and electroweak symmetries requires the addition of an extra term \(-\sqrt{-g}{} \textit{HH}^{\dagger }R/6\) to the Lagrangian. This alters Eq. (7) and requires a small increase of \(\left\langle \chi \right\rangle \), which leads to small changes in the masses \(\textit{m}_{\textit{V}}\) and \(\textit{m}_{\textit{S}}\).

  5. Weyl did not include the adjustable coupling constant b in his law, and he used the symbol \(\ell \) for the length squared of a vector, whereas I prefer \(\ell ^{2}\).

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Acknowledgments

I am grateful to Hagen Kleinert, Philip Mannheim, and She-Sheng Xue for discussions of the difficulties in constructing standard gravity by symmetry breaking of a conformally-invariant theory, and I am grateful to two anonymous reviewers for bringing Refs. [18, 35] to my attention.

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  1. Department of Physics, University of Vermont, Burlington, VT, 05405-0125, USA

    Hans C. Ohanian

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Ohanian, H.C. Weyl gauge-vector and complex dilaton scalar for conformal symmetry and its breaking. Gen Relativ Gravit 48, 25 (2016). https://doi.org/10.1007/s10714-016-2023-8

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