Implicit vs explicit renormalization and effective interactions

Abstract

Effective interactions can be obtained from a renormalization group analysis in two complementary ways. One can either explicitly integrate out higher energy modes or impose given conditions at low energies for a cut-off theory. While the first method is numerically involved, the second one can be solved almost analytically. In both cases we compare the outcoming effective interactions for the two nucleon system as functions of the cut-off scale and find a strikingly wide energy region where both approaches overlap, corresponding to relevant scales in light nuclei Λ 200MeV. This amounts to a great simplification in the determination of the effective interaction parameters.

Fixed points of the SRG evolution and the on-shell limit of the nuclear force

“…However, in the infrared region we will suggest that many features are fairly general and model independent. Here we focus on the Wilson and the Wegner SRG generators, since the evolution with a block-diagonal generator [30] in the infrared region has already been studied in previous works [31,32], where the explicit renormalization of a simple NN force and the implicit renormalization of a pionless effective field theory (EFT) at next-to-leading order were shown to be equivalent over a wide range of the renormalization scale. The SRG equations are mostly solved numerically on a finite momentum grid with sufficiently many points as to approach the continuum and therefore the grid is viewed as an auxiliary means.…”

Uncertainty quantification of effective nuclear interactions

“…Of course, one possible interpretation of the Wigner symmetry as a long distance symmetry is Wilsonian in nature as suggested in Ref. 49 and checked within the Similarity Renormalization group approach [70][71][72][73] at relatively small energy scales.…”

Low Scale Saturation of Effective NN Interactions and Their Symmetries

“…Finally, it would be very interesting to check this behavior of the counterterms by the explicit renormalization method based on the Block-Diagonal Similarity Renormalization Group, as was done in a toy model in Ref. [36,37], as the problem becomes extremely stiff and computationally expensive for Λ 1 fm −1 . There, it was found that the simple analytical formulas overlap in the physically interesting Λ ∼ 1 fm −1 region and allow for sidestepping the stiff equations for lower values.…”

“…We will call the coefficients in the expansion counterterms, although, properly speaking, the name is justified when the potential v(p , p) is used to solve the problem in a restricted Hilbert space p, p ≤ Λ, which means, in particular, fitting scattering data up to CM momentum p ≤ Λ [36] and providing a prescription to ensure hermiticity of the interaction. If a is the range of the interaction, only under these conditions is a truly universal behavior of the counterterms guaranteed for Λ ∼ 1/a as will be shown below.…”

“…[36][37][38], which has been positively tested for a simple toy model with just S-waves with the Block Diagonal (BD) formulation of the Similarity Renormalization Group (SRG) [39], which is an upgraded version of the V lowk -approach (see also [40] for a review). In previous work [36][37][38], it has explicitly been shown that the scale dependence deduced for the parameters of Equation (2) is not expected to change when higher orders in the expansion are included in a wide range of cut-offs Λ. This is our motivation to pursue the present analysis.…”