Renormalization Group Procedure For Effective Particles Research Articles

Baryon Masses Estimate in Heavy Flavor QCD

We apply the renormalization group procedure for effective particles (RGPEP) to the QCD eigenvalue problem for only heavy quarks. We derive the effective Hamiltonian that acts on the Fock space by solving the RGPEP equation up to second order in powers of the coupling constant. The eigenstates that contain three quarks and two or more gluons are eliminated by inserting a gluon-mass term in the component with one gluon and the eigenvalue problem for baryons is formulated. We estimate masses for bbb and ccc states and find that the results match the estimates obtained in lattice QCD and in quark models.

Asymptotic freedom using a gluon mass as a regulator

Front-Form Hamiltonian dynamics provides a framework in which QCD’s vacuum is simple and states are boost invariant. However, canonical expressions are divergent and must be regulated in order to establish well-defined eigenvalue problems. The Renormalization Group Procedure for Effective Particles (RGPEP) provides a systematic way of finding counterterms and obtaining regulated Hamiltonians. Among its achievements is the description of asymptotic freedom, with a running coupling constant defined as the coefficient in front of the three gluon-vertex operators in the regulated Hamiltonian. However, the obtained results need a deeper understanding, since the coupling exhibits a finite dependence on the regularization functions, at least at the third-order term in the perturbative expansion. Here we present a similar derivation using a different regularization scheme based on massive gluons. The procedure can be extended to incorporate contributions from virtual fermions.

Elementary example of exact effective-Hamiltonian computation

We present an exact computation of effective Hamiltonians for an elementary model obtained from the Yukawa theory by going to the limit of bare fermions being infinitely heavy and bare bosons being at rest with respect to the fermions that emit or absorb them. The coupling constant can be arbitrarily large. The Hamiltonians are computed by solving the differential equation of the renormalization group procedure for effective particles (RGPEP). Physical fermions, defined in the model as eigenstates of the effective Hamiltonians, are obtained in the form of an effective fermion dressed with a coherent state of effective bosons. The model computation illustrates the method that can be used in perturbative computations of effective Hamiltonians for realistic theories. It shows the mechanism by which the perturbative expansion and Tamm-Dancoff approximation increase in accuracy along the RGPEP evolution.

Computation of effective front form Hamiltonians for massive Abelian gauge theory

Renormalization group procedure for effective particles (RGPEP) is applied in terms of a second-order perturbative computation to an Abelian gauge theory, as an example of application worth studying on the way toward derivation of a dynamical connection between the spectroscopy of bound states and their parton-model picture in the front form of Hamiltonian dynamics. In addition to the ultraviolet transverse divergences that are handled using the RGPEP in previously known ways, the small-x divergences are handled by introducing a mass parameter and a third polarization state for gauge bosons using a mechanism analogous to spontaneous breaking of global gauge symmetry, in a special limit that simplifies the theory to Soper's front form of massive QED. The resulting orders of magnitude of scales involved in the dynamics of effective constituents or partons in the simplified theory are identified for the fermion and boson mass counter terms, effective masses and self-interactions, as well as for the Coulomb-like effective interactions in bound states of fermions. Computations in orders higher than second are mentioned but not described in this article.

Approximate Hamiltonian for baryons in heavy-flavor QCD

Aiming at relativistic description of gluons in hadrons, the renormalization group procedure for effective particles (RGPEP) is applied to baryons in QCD of heavy quarks. The baryon eigenvalue problem is posed using the Fock-space Hamiltonian operator obtained by solving the RGPEP equations up to second order in powers of the coupling constant. The eigenstate components that contain three quarks and two or more gluons are heuristically removed at the price of inserting a gluon-mass term in the component with one gluon. The resulting problem is reduced to the equivalent one for the component of three quarks and no gluons. Each of the three quark–quark interaction terms thus obtained consists of a spin-dependent Coulomb term and a spin-independent harmonic oscillator term. Quark masses are chosen to fit the lightest spin-one quarkonia masses most accurately. The resulting estimates for bbb and ccc states match estimates obtained in lattice QCD and in quark models. Masses of ccb and bbc states are also estimated. The corresponding wave functions are invariant with respect to boosts. In the ccb states, charm quarks tend to form diquarks. The accuracy of our approximate Hamiltonian can be estimated through comparison by including components with two gluons within the same method.

Renormalized quark–antiquark Hamiltonian induced by a gluon mass ansatz in heavy-flavor QCD

In response to the growing need for theoretical tools that can be used in QCD to describe and understand the dynamics of gluons in hadrons in the Minkowski space–time, the renormalization group procedure for effective particles (RGPEP) is shown in the simplest available context of heavy quarkonia to exhibit a welcome degree of universality in the first approximation it yields once one assumes that beyond perturbation theory gluons obtain effective mass. Namely, in the second-order terms, the Coulomb potential with Breit–Fermi spin couplings in the effective quark–antiquark component of a heavy quarkonium, is corrected in one-flavor QCD by a spin-independent harmonic oscillator term that does not depend on the assumed effective gluon mass or the choice of the RGPEP generator. The new generator we use here is much simpler than the ones used before and has the advantage of being suitable for studies of the effective gluon dynamics at higher orders than the second and beyond the perturbative expansion.

From Asymptotic Freedom Toward Heavy Quarkonia Within the Renormalization-Group Procedure for Effective Particles

The renormalization group procedure for effective particles (RGPEP), developed as a nonperturbative tool for constructing bound states in quantum field theories, is applied to QCD. The approach stems from the similarity renormalization group and introduces the concept of effective particles. It has been shown that the RGPEP passes the test of exhibiting asymptotic freedom. We present the running of the Hamiltonian coupling with the renormalization-group scale and summarize the basic elements needed in the formulation of the bound-state problem.

Asymptocic Freedom of Gluons in Hamiltonian Dynamics

We derive asymptotic freedom of gluons in terms of the renormalized $SU(3)$ Yang-Mills Hamiltonian in the Fock space. Namely, we use the renormalization group procedure for effective particles (RGPEP) to calculate the three-gluon interaction term in the front-form Yang-Mills Hamiltonian using a perturbative expansion in powers of $g$ up to third order. The resulting three-gluon vertex is a function of the scale parameter $s$ that has an interpretation of the size of effective gluons. The corresponding Hamiltonian running coupling constant exhibits asymptotic freedom, and the corresponding Hamiltonian $\beta$-function coincides with the one obtained in an earlier calculation using a different generator.

Asymptotic freedom in the front-form Hamiltonian for quantum chromodynamics of gluons

Asymptotic freedom of gluons in QCD is obtained in the leading terms of their renormalized Hamiltonian in the Fock space, instead of considering virtual Green's functions or scattering amplitudes. Namely, we calculate the three-gluon interaction term in the front-form Hamiltonian for effective gluons in the Minkowski space-time using the renormalization group procedure for effective particles (RGPEP), with a new generator. The resulting three-gluon vertex is a function of the scale parameter, $s$, that has an interpretation of the size of effective gluons. The corresponding Hamiltonian running coupling constant, $g_\lambda$, depending on the associated momentum scale $\lambda = 1/s$, is calculated in the series expansion in powers of $g_0 = g_{\lambda_0}$ up to the terms of third order, assuming some small value for $g_0$ at some large $\lambda_0$. The result exhibits the same finite sensitivity to small-$x$ regularization as the one obtained in an earlier RGPEP calculation, but the new calculation is simpler than the earlier one because of a simpler generator. This result establishes a degree of universality for pure-gauge QCD in the RGPEP.

Calculation of size for bound-state constituents

Elements are given of a calculation that identifies the size of a proton in the Schroedinger equation for lepton-proton bound states, using the renormalization group procedure for effective particles (RGPEP) in quantum field theory, executed only up to the second order of expansion in powers of the coupling constant. Already in this crude approximation, the extraction of size of a proton from bound-state observables is found to depend on the lepton mass, so that the smaller the lepton mass the larger the proton size extracted from the same observable bound-state energy splitting. In comparison of Hydrogen and muon-proton bound-state dynamics, the crude calculation suggests that the difference between extracted proton sizes in these two cases can be a few percent. Such values would match the order of magnitude of currently discussed proton-size differences in leptonic atoms. Calculations using the RGPEP of higher order than second are required for a precise interpretation of the energy splittings in terms of the proton size in the Schroedinger equation. Such calculations should resolve the conceptual discrepancy between two conditions: that the renormalization group scale required for high accuracy calculations based on the Schroedinger equation is much smaller than the proton mass (on the order of a root of the product of reduced and average masses of constituents) and that the energy splittings due to the physical proton size can be interpreted ignoring corrections due to the effective nature of constituents in the Schr\"odinger equation.

Fermion Mass Mixing in Vacuum

Renormalization group procedure for effective particles (RGPEP) is applied to a theory of fermions that interact only through mass mixing terms in their Hamiltonian. Problems with virtual pair production in vacuum are avoided by using the front form of Hamiltonian dynamics. Masses and states of physical fermions emerge at the end of a calculation that is carried out exactly irrespective of the strength of the mass mixing terms. An a priori infinite set of renormalization group equations for all momentum modes of fermion quantum fields is reduced to just one equation for a two-by-two mass matrix. In distinction from scalars, fermions never become tachyons but appear chirally rotated when the mass mixing interaction term is sufficiently strong.

Hypothesis of Quark Binding by Condensation of Gluons in Hadrons

Hypothesis of quark binding through condensation of gluons inside hadrons is formulated in the context of a renormalization group procedure for effective particles (RGPEP) in the light-front (LF) Hamiltonian approach to QCD. At the momentum scales of relative motion of hadronic constituents that are comparable with Lambda_QCD, the hypothetical boost-invariant constituent dynamics is identified using gauge symmetry. The resulting picture of mesons and baryons closely resembles constituent quark models with harmonic oscillator potentials, shares some features of AdS/QCD, and can be systematically studied using RGPEP in QCD.

Harmonic oscillator force between heavy quarks

A renormalization-group procedure for effective particles is applied to quantum chromodynamics of one flavor of quarks with a large mass m in order to calculate light-front Hamiltonians for heavy quarkonia ${H}_{\ensuremath{\lambda}}$ using perturbative expansion in the coupling constant ${\ensuremath{\alpha}}_{\ensuremath{\lambda}}.$ $\ensuremath{\lambda}$ is the renormalization-group parameter with the interpretation of an inverse of the spatial size of the color charge distribution in the effective quarks and gluons. The eigenvalue equation for ${H}_{\ensuremath{\lambda}}$ couples quark-antiquark states with sectors of a larger number of constituents. The coupling to states with more than one effective gluon, and interactions in the quark-antiquark-gluon sector, are removed at the price of introducing an ansatz for the gluon mass ${\ensuremath{\mu}}^{2}.$ The simplified equation is used to evaluate a new Hamiltonian of order ${\ensuremath{\alpha}}_{\ensuremath{\lambda}}$ that acts only in the effective quark-antiquark sector and in the nonrelativistic limit turns out to contain the Coulomb term with Breit-Fermi corrections and a spin-independent harmonic oscillator term with a frequency $\ensuremath{\omega}=[(4/3)({\ensuremath{\alpha}}_{\ensuremath{\lambda}}/\ensuremath{\pi}){]}^{1/2}\ensuremath{\lambda}(\ensuremath{\lambda}{/m)}^{2}(\ensuremath{\pi}{/1152)}^{1/4}.$ The latter originates from the hole excavated in the overlapping quark self-interaction gluon clouds by the exchange of effective gluons between the quarks. The new term is largely independent of the details of ${\ensuremath{\mu}}^{2}$ and in principle can fit into the ballpark of phenomenology. The first approximation can be improved by including more terms in ${H}_{\ensuremath{\lambda}}$ and solving the eigenvalue equations numerically.