Optimized regulator for the quantized anharmonic oscillator

Abstract

The energy gap between the first excited state and the ground state is calculated for the quantized anharmonic oscillator in the framework of the functional renormalization group method. The compactly supported smooth regulator is used which includes various types of regulators as limiting cases. It was found that the value of the energy gap depends on the regulator parameters. We argue that the optimization based on the disappearance of the false, broken symmetric phase of the model leads to the Litim's regulator. The least sensitivity on the regulator parameters leads however to an IR regulator being somewhat different of the Litim's one, but it can be described as a perturbatively improved, or generalized Litim's regulator and provides analytic evolution equations, too.Comment: 8 pages, 4 figure

Vacuum energy and renormalization of the field-independent term

“…[24] and additional discussions in refs. [25,26]], the application of a suitable subtraction reduces the quadratic (k 2 ) divergence of the RG evolution in the UV limit and produces correct results for the free energy in the IR limit. However, it is very important to observe that the non-physical behaviour, i.e., the k 2 divergence of the UV limit is the consequence of the absence of the Gaussian fixed point in the RG flow once the constant term is included.…”

“…The nonperturbative RG is a possible method, but there are also others, including lattice and Monte Carlo methods. In general, the nonperturbative RG method [17][18][19][20][21] has been used successfully in many areas of physics from statistical mechanics to high energy physics [22][23][24][25][26][27][28][29][30] and in cosmology [31][32][33][34][35][36][37][38] with particular attention to the cosmological constant problem [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57]. It is obvious that the vacuum energy density induced by the quantum fluctuations would JCAP03(2022)062 add to the cosmological constant (see refs.…”

Quantum–classical transition in the Caldeira–Leggett model

“…The numerical solution of the Schrödinger equation for the anharmonic oscillator shows up a convex effective potential, independently of the strength of the anharmonic coupling. In the case of the RG method we should use higher orders of the gradient expansion in order to get acceptable results, nevertheless there always remains a region of the parameter space where the effective potential is concave, independently of the renormalization scheme [17,18].…”

“…Generally the O(N ) model has a Wilson-Fisher (WF) fixed point which separates two phases, the symmetric and the spontaneously broken (or simply broken) ones. In the case of d = 1 the model has a single phase, since there is no spontaneous symmetry breaking due to the quantum tunneling effect [14][15][16][17][18][19]. However, the RG treatment runs into difficulty when the initial double-well potential meets a weak anharmonic coupling [15].…”