Effective action from the functional renormalization group
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Regular Article - Theoretical Physics
Effective action from the functional renormalization group Nobuyoshi Ohta1,a , Lesław Rachwał2,b 1 2
Department of Physics, Kindai University, Higashi-Osaka, Osaka 577-8502, Japan Department of Nuclear Physics and Physical Engineering, Czech Technical University, Bˇrehova 7, Prague 11519, Czech Republic
Received: 1 March 2020 / Accepted: 4 August 2020 © The Author(s) 2020
Abstract We study the quantum gravitational system coupled to a charged scalar, Dirac fermions, and electromagnetic fields. We use the “exact” or “functional” renormalization group equation to derive the effective action 0 by integrating the flow equation from the ultraviolet scale down to k = 0. The resulting effective action consists of local terms and nonlocal terms with unique coefficients.
1 Introduction It is one of the most urgent problems in theoretical physics to understand quantum property of the gravitational system interacting with various matter fields. The Einstein gravity is non-renormalizable in perturbation theory. If one believes that gravity must be described by some quantum theory at some level, then we are led to expect that the theory will contain terms quadratic in curvatures which are necessarily generated by quantum effects, with coefficients of order unity. In fact, we expect to find in the action all possible diffeomorphism-invariant terms constructed with the metric and its derivatives. It had been shown long ago that if one includes such higher derivative terms, then the theory is perturbatively renormalizable [1]. The price one must pay is that perturbative unitarity is lost. The superstring theory is supposed to circumvent the problem, but it is still difficult to understand quantum geometric aspects of the spacetime physics in superstrings. Another approach to get insight into the quantum effects of gravity is to use the functional renormalization group (FRG). It enables us to study the RG flow of infinitely many couplings as functions of a cutoff k. It has been used to study the ultraviolet (UV) behavior of gravity and establish the existence of a nontrivial fixed point (FP) which may be used to define a continuum limit [2–4]. To formulate this, one defines the a e-mail:
[email protected] (corresponding author)
b e-mail:
[email protected]
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effective average action (EAA) k by performing the path integral over field modes with the momentum scales equal or bigger than a scale given by a cutoff k [5,6], and following the usual procedure of defining effective action. This EAA itself is not the usual effective action and it is still divergent in general in the UV limit. The k-dependence of the EAA is described by the FRG equation (FRGE) −1 δ 2 k 1 + Rk ∂t R k , (1.1) ∂t k = STr 2 δφδφ where t = ln k, and Rk is a cutoff kernel which goes to zero when its argument z is greater than the cutoff scale k 2 . Because the FRGE only sees the variation of the EAA, it is free from the UV divergence and well defined in contrast to EAA. Technically
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