[Paper Review] Higher Derivative Corrections to Lower Order RG Flow Equations

S. P. de Alwis|arXiv (Cornell University)|Sep 12, 2018

advanced mathematical theories3 references3 citations

TL;DR

This paper demonstrates that higher derivative corrections to the renormalization group (RG) flow of the cosmological constant in quantum gravity arise not from $R^N$ or $R_{\mu\nu}^N$ terms with $N>2$, but from operators like $R \Box^n R$ for $n = 0,1,2,\ldots$, which have been previously overlooked. It further argues that perturbative ghosts from curvature-squared terms are spurious, removable via cutoff-dependent field redefinitions, and thus do not invalidate the asymptotic safety program.

ABSTRACT

We show that the RG flow equation for the cosmological constant (CC) receives contributions (in addition to those coming from the CC the Einstein-Hilbert term and $R^{2}$ and $R_{μν}^{2}$ terms) only from terms with just two powers of curvature, but having also powers of the covariant derivative, in the Wilsonian effective action. In pure gravity our argument implies that just considering $f(R)$ theories will miss this effect which arises from terms such as $"R"\square^{n}"R",\,n=0,1,2,\ldots$. We expect similar contributions for the flow equation of the Einstein-Hilbert term as well. Finally we argue that the perturbative ghosts coming from curvature squared terms in the action are in fact spurious since they are at the cutoff scale and can be removed by (cutoff dependent) field redefinitions.

Motivation & Objective

• To identify previously neglected higher derivative corrections to the RG flow of the cosmological constant in pure gravity.
• To address the apparent inconsistency of perturbative ghosts in higher-derivative gravity theories within the asymptotic safety program.
• To clarify the role of curvature-squared terms and their ghost states in effective field theory with a UV cutoff.
• To argue that these ghosts are artifacts of the cutoff and can be removed via field redefinitions, preserving unitarity.
• To emphasize that $f(R)$-type truncations miss key contributions from $R\Box^nR$-type operators.

Proposed method

• Uses the Wilsonian effective action with a momentum cutoff $\Lambda$ to express quantum gravity as an infinite series of local operators.
• Applies a specific form of the RG equation derived in [11], analogous to Polchinski’s equation, to analyze the running of couplings.
• Identifies that only operators with two curvatures and arbitrary powers of the covariant derivative, such as $R\Box^nR$, contribute to the cosmological constant's RG flow.
• Demonstrates that curvature-squared terms ($R^2$, $R_{\mu\nu}^2$) introduce spurious ghosts at the cutoff scale $\Lambda$, which are removable via cutoff-dependent field redefinitions.
• Argues that such field redefinitions preserve the S-matrix and eliminate unphysical states, consistent with string theory and asymptotic safety.
• Relies on the Deser-Redlich argument to justify the removal of ghosts, assuming the existence of a UV fixed point.

Experimental results

Research questions

• RQ1 Which higher derivative operators contribute to the RG flow of the cosmological constant beyond $R^2$ and $R_{\mu\nu}^2$ terms?
• RQ2 Why do $f(R)$-type theories miss critical corrections to the cosmological constant's RG flow?
• RQ3 Are the perturbative ghosts associated with curvature-squared terms in the effective action physical or artifacts of the cutoff?
• RQ4 Can the apparent ghosts from $R^2$ terms be removed without altering physical scattering amplitudes?
• RQ5 What is the role of $R\Box^nR$ operators in the Wilsonian effective action for quantum gravity?

Key findings

• The RG flow of the cosmological constant receives contributions from $R\Box^nR$ operators for $n = 0,1,2,\ldots$, which are not captured by $f(R)$-type truncations.
• Higher derivative terms like $R^N$ or $R_{\mu\nu}^N$ for $N>2$ do not contribute to the cosmological constant's RG equation.
• Perturbative ghosts from $R^2$ terms are spurious and arise only at the UV cutoff scale $\Lambda$, making them unphysical in low-energy calculations.
• These ghosts can be removed via a cutoff-dependent field redefinition that eliminates all $R^2$-type terms from the action.
• The procedure of ghost elimination via field redefinition is valid only under the assumption of a UV fixed point, which is central to the asymptotic safety program.
• The S-matrix remains invariant under such field redefinitions, ensuring physical consistency despite the removal of higher-derivative quadratic terms.