Generating Einstein gravity, cosmological constant and Higgs mass from restricted Weyl invariance

Recently, it has been pointed out that dimensionless actions in four dimensional curved spacetime possess a symmetry which goes beyond scale invariance but is smaller than full Weyl invariance. This symmetry was dubbed {\it restricted Weyl invariance}. We show that starting with a restricted Weyl invariant action that includes a Higgs sector with no explicit mass, one can generate the Einstein-Hilbert action with cosmological constant and a Higgs mass. The model also contains an extra massless scalar field which couples to the Higgs field (and gravity). If the coupling of this extra scalar field to the Higgs field is negligibly small, this fixes the coefficient of the nonminimal coupling $R \Phi^2$ between the Higgs field and gravity. Besides the Higgs sector, all the other fields of the standard model can be incorporated into the original restricted Weyl invariant action.Comment: 7 pages, no figure

Enlarging the symmetry of pure R2R^2 gravity, BRST invariance and its spontaneous breaking

Pure $R^2$ gravity was considered originally to possess only global scale symmetry. It was later shown to have the larger restricted Weyl symmetry where it is invariant under the Weyl transformation $g_{\mu\nu} \to \Omega^2(x)\, g_{\mu\nu}$ when the conformal factor $\Omega(x)$ obeys the harmonic condition $\Box \Omega(x)=0$. Restricted Weyl symmetry has an analog in gauge theory. Under a gauge transformation $A_{\mu}\to A_{\mu} + \frac{1}{e}\partial_{\mu} f(x)$, the gauge-fixing term $(\partial_{\mu}A^{\mu})^2$ has a residual gauge symmetry when $\Box f=0$. In this paper, we consider scenarios where the symmetry of pure $R^2$ gravity can be enlarged even further. In one scenario, we add a massless scalar field to the pure $R^2$ gravity action and show that the action becomes on-shell Weyl invariant when the equations of motion are obeyed. We then enlarge the symmetry to a BRST symmetry where no on-shell or restricted Weyl condition is required. The BRST transformations here are not associated with gauge transformations (such as diffeomorphisms) but with Weyl (local scale) transformations where the conformal factor consists of a product of Grassmann variables. BRST invariance in this context is a generalization of Weyl invariance that is valid in the presence of the Weyl-breaking $R^2$ term. In contrast to the BRST invariance of gauge theories like QCD, it is not preserved after quantization since renormalization introduces a scale (leading to the well-known Weyl (conformal) anomaly). We show that the spontaneous breaking of the BRST symmetry yields an Einstein action; this still has a symmetry which is also anomalous. This is in accord with previous work that shows that there is conformal anomaly matching between the unbroken and broken phases when conformal symmetry is spontaneously broken.Comment: 13 pages, no figure