In order to make the \(\xi R\) sizable contribution at the high-energy collider plausible, we will explore the theoretical motivation to drop *R* term, namely \(\mathcal {L}_g=0\) due to renormalizable criteria.

Renormalizability and associated infinity are usually thought as annoying; however, it can be treated as the tool, even a principle to construct a meaningful theory. In order to illustrate the key difficulty to renormalize gravity, we utilize a toy model with only one real scalar field \(\phi\). \(\kappa\) is taken as 1 in Eq. (9). The Lagrangian of matter of Eq. (10) is replaced by

\(\begin{aligned} \mathcal {L}_m= \sqrt{g}\left\{ -\frac{1}{2} g^{\mu \nu } \partial _\mu \phi \partial _\nu \phi + \frac{1}{2} \phi M^2 \phi \right\} . \end{aligned}\)
(13)

Treating *g* as the external source, the counter-terms at one-loop level can be extracted from Ref. [11]

\(\begin{aligned} \Delta {\mathcal {L}}= & {} \frac{\sqrt{g}}{\epsilon } \left\{ \frac{1}{4} \left( M^2 -\frac{1}{6} R \right) ^2 \right. \nonumber \\{} & {} \left. +\frac{1}{120} \left( R_{\mu \nu } R^{\mu \nu } -\frac{1}{3} R^2 \right) \right\} \end{aligned}\)
(14)

where \(\epsilon =8\pi ^2 (D-4)\) and *D* is the dimension of the space-time.

Reference [11] has argued that the unrenormalizable term in Eq. (14), namely the \(R M^2\) term, can be eliminated by adding the specific term \(\frac{1}{12}R \phi ^2\) to the original Lagrangian of Eq. (13). However, the unrenormalizable terms \(R^2\) and \(R_{\mu \nu } R^{\mu \nu }\) remain. The situation becomes even worse after including contributions from the gravitons in the loops. It seems impossible to generally eliminate all unrenormalizable terms by modifying the original Lagrangian. This is the key argument that gravity is an unrenormalizable theory. As shown in this simple excise, there exists fundamental difficulty to renormalize the gravity in this way. Some fundamental aspect of the gravity has to be changed.

As the basic requirement of a renormalizable theory, the new form counter-terms beyond the origin Lagrangian are not allowed. It seems there is only possible by treating the metric as the coupling parameter instead of the dynamical field. Under this assumption, the metric is not the dynamical field, namely the kinetic term *R* will be dropped. Provided that the metric acts only as the parameter, the form of counter-terms in Eq. (14) is the same with original Lagrangian in Eq. (13). All the previous unrenormalizable terms \(R M^2\), \(R^2\), and \(R_{\mu \nu } R^{\mu \nu }\) are the functions of the \(g_{\mu \nu }\), which is the building block of the original Lagrangian. From this point of view, the model must be renormalizable as it should be. The renormalizability of toy model of Eq. (13) is guaranteed by the properties of the dynamical quantum field \(\phi\). The metric only becomes the dynamical field after the electro-weak symmetry breaking, as shown in last section.

In principle, a realistic model should include all theoretical allowed terms. In Eqs. (9) and (10), the kinetic term *R* and the higher power of *R* terms are not allowed, since these terms break either the theory renormalizability or vacuum stability. In this sense, the renormalizabilty is treated as the principle to construct a physical theory.

For the general case, the quantum behavior can be written as the path integral of dynamical field \(\phi\)

\(\begin{aligned} Z=\int D\phi \exp \left\{ i S\right\} . \end{aligned}\)
(15)

Note that the metric field *g* is not a priori assumed as dynamic field. The action *S* can be divided as metric and other (matter) parts

\(\begin{aligned} S= S(g)+ S(g,\phi )+ S(\phi ). \end{aligned}\)
(16)

As such, \(\exp \{i S(g)\}\) is independent on the quantum field (\(\phi\)) and can be dropped and the path integral can be simplified as

\(\begin{aligned} Z=\int D\phi \exp \left\{ i S(g, \phi ) +i S(\phi ) \right\} . \end{aligned}\)
(17)

The metric, as the dynamical field after electro-weak symmetry breaking, manifests itself only classically. Equation (10) is only the specific realization of the general case.