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arXiv:hep-ph/0009071

The Similarity Renormalization Group

S. Szpigel, Robert J. Perry

Published Sep 6, 2000• 8 citations
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Abstract

Quantum field theories require a cutoff to regulate divergences that result from local interactions, and yet physical results can not depend on the value of this cutoff. The renormalization group employs a transformation that changes the cutoff to isolate hamiltonians that produce cutoff-independent eigenvalues. The similarity renormalization group is based on similarity transformations that regulate off-diagonal matrix elements, forcing the hamiltonian towards a band-diagonal form as the cutoff is lowered. This avoids pathologies that plagued tradition transformations acting on hamiltonians, making it possible to produce a well-behaved perturbative approximation of renormalized hamiltonians in asymptotically free theories. We employ a simple two-dimensional delta function example to illustrate this new renormalization technique.

DOI
10.48550/arxiv.hep-ph/0009071

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Quantum field theories require a cutoff to regulate divergences that result from local interactions, and yet physical results can not depend on the value of this cutoff. The renormalization group employs a transformation that changes the cutoff to isolate hamiltonians that produce cutoff-independent eigenvalues. The similarity renormalization group is based on similarity transformations that regulate off-diagonal matrix elements, forcing the hamiltonian towards a band-diagonal form as the cutoff is lowered. This avoids pathologies that plagued tradition transformations acting on hamiltonians, making it possible to produce a well-behaved perturbative approximation of renormalized hamiltonians in asymptotically free theories. We employ a simple two-dimensional delta function example to illustrate this new renormalization technique.
The Schrodinger equation with a two-dimensional delta-function potential is a simple example of an asymptotically free theory that undergoes dimensional transmutation. Renormalization requires the introduction of a mass scale, which can be lowered perturbatively until an infrared cutoff produced by non-perturbative effects such as bound state formation is encountered. We outline the effective field theory and similarity renormalization group techniques for producing renormalized cutoff hamiltonians, and illustrate the control of logarithmic and inverse-power-law errors both techniques provide.
The renormalization of the effective field theories (EFTs) in many-body systems is the most pressing and challenging problem in modern nuclear ab initio calculation. For general non-relativistic EFTs, we prove that the renormalization group (RG) invariance can be achieved if and only if all single-particle momenta are regulated with a universal cutoff Λ. For a numerical demonstration, we construct a series of N^{2}LO chiral forces with Λvarying from 250 MeV to 400 MeV. With all low energy constants fixed in two- and three-nucleon systems, we reproduce the experimental binding energies of ^{4}He and ^{16}O nearly independently of Λ. In contrast, all recent nuclear EFT constructions regulate the relative momenta for Galilean invariance, thus inherently break the RG invariance. This explains the unpleasantly strong cutoff-dependences observed in recent ab initio calculations. Our method can also be used to build RG-invariant EFTs with non-perturbative interactions.
Abstract Chapter 5 first recalls the importance of the concept of scale decoupling in physics. It then emphasizes that quantum field theory and the theory of critical phenomena have provided two examples where this concepts fails. To deal with such a situation, a new tool has been invented: the renormalization group. In the framework of effective quantum field theory, a perturbative renormalization group has been formulated. Its implementation has led to the discovery of fixed points as zeros of beta functions, and calculations of critical exponents of a class of macroscopic phase transitions in the form of Wilson–Fisher epsilon or fixed dimension expansions. These expansions being divergent, they could summed by methods based on the Borel transformation and the determination of the large order behaviour of perturbation theory.
The Similarity Renormalization Group (SRG), introduced in its modern Hamiltonian form by G■azek & Wilson and by Wegner, and pedagogically clarified by Szpigel & Perry (2000) through the two-dimensional delta-function laboratory, has become one of the dominant tools of ab-initio nuclear theory. Continuous unitary transformations that suppress off-diagonal matrix elements with respect to an energy-difference cutoff systematically soften nuclear interactions, accelerate the convergence of many-body methods, and generate a controlled hierarchy of induced many-body forces. This independent study traces the trajectory from the original pedagogical prototype to the present state of the art. It examines the SRG evolution of two-, three- and four-body forces, the in-medium similarity renormalization group (IMSRG), applications to nuclear structure and reactions, and the adaptation of the error-scaling diagnostics pioneered by Szpigel and Perry to the quantification of residual cutoff dependence. Emphasis is placed on the systematic inclusion of higher-body forces and on the necessity of rigorous uncertainty quantification if SRG-based predictions are to meet modern standards of scientific reliability. The study concludes with an assessment of open frontiers in continuum observables, electroweak currents, and the interface with lattice QCD and chiral effective field theory. Also available on Academia.edu: https://www.academia.edu/attachments/133749625/download_file?s=portfolio