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The renormalization group procedure for effective particles (RGPEP) is a Hamiltonian approach to quantum field theory. It was developed as a nonperturbative tool for constructing bound states in quantum chromodynamics (QCD). The method stems from the similarity renormalization group procedure of Głazek and Wilson, uses light-front dynamics, and introduces the concept of effective particles in the Fock space.
Light-front QCD
Light-front QCD
The RGPEP is a Hamiltonian approach to quantum field theory and is formulated in the Dirac front form (or light front) of relativistic dynamics. The front form (FF) possesses several interesting features which are extremely useful in the calculation of hadron observables.
In the FF Hamiltonian there are no terms with only creation or only annihilation operators; therefore, the vacuum state |0> is an eigenstate of the Hamiltonian with eigenvalue 0. Furthermore, from the 10 generators of the Poincaré group, only 3 are dynamical (i.e. contain interactions). This is useful in situations in which one needs to consider bound states in different reference frames, as is the case, for example, in the calculation of electromagnetic form factors.
The use of light-front variables is essential in the description of the partonic structure of hadrons. Solving the FF Hamiltonian eigenvalue equation yields both hadron masses and light-front wave functions of the eigenstates. The light-front wave functions are necessary to calculate structure functions and other observables concerning parton distributions measured in deep inelastic scattering.
Renormalization group
Renormalization group
One of the central problems in describing bound states in QFT is the fact that one needs to deal with an infinite number of degrees of freedom. For example, a quarkonium state in the Fock space can be expressed by an infinite series with the structure
and there is no limit in the number of participating Fock components. The key idea of the RGPEP is that for describing observables characterized by a size scale s, one can derive an effective Hamiltonian Hs, which is written in a scale-dependent operator basis, such that the number of Fock components, relevant to the description, is sufficiently small for carrying out computations. The parameter s has the meaning of size of the effective particles. When infinitely many Fock components can be neglected, and only a few are significant, the bound state problem is drastically simplified and one can attempt to seek a numerical solution to the eigenvalue equation. The notion of effective particles is also used to explain the different behavior of interacting particles at different energy scales.
The notion of effective particles
The notion of effective particles
The initial Hamiltonian Ho , which is written in terms of (bare) creation and annihilation operators of point-like particles can be re-expressed in terms of particle operators of size s. Creation (annihilation) operators of effective particles labeled with s, acting in the Fock space, create (annihilate) effective particles of size s:
where Us is a symmetry transformation. It is also common to use the momentum-width parameter λ = 1/s. This parameter distinguishes different kinds of particles according to the rule that: effective particles of type λ can change their relative motion kinetic energy through a single effective interaction by no more than about λ.
It has been checked that the RGPEP passes the test of asymptotic freedom. Namely, it was shown that the strength of the running coupling constant depends on the size of the effective particles. Exhibiting asymptotic freedom is a precondition for any approach aiming at describing hadrons using QCD.
The figure shows the visualization of the three-quark configuration in a proton evolving with the RGPEP scale parameter s: (a) the picture at s much smaller than the scale s_c that corresponds to asymptotic freedom, (b) s somewhat smaller than s_c , and (c) s comparable with s_c, which corresponds to the constituent picture of hadrons. The consequence of the RGPEP is that the slow effective quarks must be large. (Picture adapted from APP42 (2011) 1933 ).
Heavy-flavor QCD
Heavy-flavor QCD
In order to start thinking about bound states in QCD, we focused on heavy quarkonia. Heavy quarkonium is the simplest bound state in QCD and it provides a fruitful ground for comparing QCD approaches with experimental data. A number of effective theories and phenomenological models are based on simplifications appearing due to the very large quark masses as compared with Λ_QCD. We use the hierarchy of scales illustrated below (Δ is an ultraviolet cutoff, and κ is the relative quark-antiquark momentum). This condition and the asymptotic freedom feature will allow us to construct an effective theory using RGPEP.
Exotic hadrons: quarkonium hybrids
Exotic hadrons: quarkonium hybrids
Another important feature of the RGPEP is that, since it uses a formulation in the Fock space which includes gluon degrees of freedom explicitly, the method provides an ideal framework for the description of hybrids.
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