RGPEP: Effective Particle Renormalization

• RGPEP is a Hamiltonian-based renormalization group technique that introduces effective particles with a well-defined size via continuous unitary transformations.
• It produces scale-dependent, band-diagonal Hamiltonians with exponential form factors that suppress large invariant-mass transitions, enabling nonperturbative analyses.
• Applications include modeling relativistic bound states, heavy flavor QCD spectra, and parton distributions by linking low-energy confining potentials with high-energy asymptotic freedom.

The Renormalization Group Procedure for Effective Particles (RGPEP) is a nonperturbative, Hamiltonian-based renormalization group technique developed for quantum field theory, with particular application to the front-form (light-front) dynamics. Unlike Wilsonian approaches that integrate out high-energy modes, RGPEP introduces a continuous unitary transformation of the canonical Hamiltonian, producing a family of effective Hamiltonians acting in the Fock space of "dressed" particles of finite size. The effective interactions are band-diagonal in invariant mass, characterized by smooth vertex form factors that suppress transitions involving large energy transfer, thereby yielding a well-defined, renormalized (possibly nonlocal) Hamiltonian suitable for nonperturbative studies of relativistic bound states, spectrum, and parton structure.

1. Conceptual Foundation of the RGPEP

RGPEP is formulated as an operator evolution that transforms the canonical Hamiltonian $\mathcal{H}_0$ (expressed in terms of bare field operators $a_0$) into a resolution-dependent effective Hamiltonian $\mathcal{H}_s$ (in terms of effective operators $a_s$): $a_s = U_s a_0 U_s^\dagger\,,$ where $U_s$ is a unitary operator parameterized by a scale $s$ (with units of length). The effective Hamiltonians $\mathcal{H}_s(a_s)$ all describe the same physics (unitary equivalent) but have vertices carrying exponential form factors

$f_s = \exp\left[ - s^4 (\Delta M^2)^2 \right]$

which suppress transitions that change the invariant mass by more than $1/s$ (Glazek, 2011). The parameter $s$ thus controls the effective "size" of the particles—bare (pointlike) for $s \to 0$, fully dressed (maximal nonlocality) as $s \to \infty$. This "dressing" reorganizes the theory's degrees of freedom and systematically suppresses large off-diagonal invariant-mass transitions.

The central RGPEP evolution (flow) equation for the effective Hamiltonian is

$$\frac{d\mathcal{H}_s}{ds^4} = [G_s, \mathcal{H}_s]\,,$$

where the generator $G_s$ is constructed to preserve kinematical symmetries (especially boost invariance in front-form dynamics), often chosen as

$$G_s = [\mathcal{H}_\text{free}, \mathcal{H}_s^+ ]\,,$$

with $\mathcal{H}_\text{free}$ the free Hamiltonian and $\mathcal{H}_s^+$ the modified interaction vertex (weighted by total $P^+$ as needed for front-form boost invariance) (Glazek, 2011, Glazek, 2011, Glazek et al., 2016). This structure guarantees that the resulting $\mathcal{H}_s$ is "narrow"—band-diagonal in free invariant mass differences.

2. Technical Structure and Perturbative Expansion

While the RGPEP evolution can be implemented nonperturbatively for certain bilinear models (Glazek, 2012, Glazek, 2013, Glazek, 2013), practical application to interacting quantum field theories often relies on a perturbative expansion in the coupling $g$: $$\mathcal{H}_s = \mathcal{H}_\text{free} + g \mathcal{H}_{s,1} + g^2 \mathcal{H}_{s,2} + g^3 \mathcal{H}_{s,3} + \cdots$$ Each term in this expansion acquires scale-dependent form factors in its vertices, and the flow equations generate increasingly complex operator structures at higher orders (Glazek, 2012). Explicit recursive formulas for $\mathcal{H}_{s,n}$ up to fourth order are specified in a basis labeled by Fock-sector indices and quantum numbers. For example, the coefficient for a two-body term at second order is given by

$$\mathcal{H}_{s,ab}^{(2)} = \int_0^s d\tau\, A_{s, a x b} \, \mathcal{H}_{0,1,a x} \mathcal{H}_{0,1,x b},$$

with $A_{s,a x b}$ determined by the free invariant mass differences, and similar nested integrals for higher orders (Glazek, 2012). Counterterms are inserted order by order to remove ultraviolet divergences, resulting in a renormalized, finite effective Hamiltonian at each $s$ (Serafin et al., 5 Aug 2025).

The RGPEP form factors ensure that as $s$ increases, the effective Hamiltonian becomes more band-diagonal and nonlocal, facilitating the decoupling of high-invariant-mass transitions and providing natural IR/UV regulators.

3. Physical Interpretation and Applications

RGPEP provides a scale-dependent "interpolating" basis between the canonical, UV-divergent description of quantum fields and the effective theories appropriate at any given scale. In the front-form dynamics central to RGPEP, vacuum structure is trivial (no zero-mode or pair creation due to $p^+>0$ for all excitations), allowing a nonperturbative construction of bound states and spectral properties without the complications of instant-form vacuum condensates (Glazek, 2012, Glazek, 2013, Glazek, 2013).

Hadronic bound states: At scales $s \gtrsim 1/\Lambda_\text{QCD}$ (i.e., $\lambda \sim \Lambda_\text{QCD}$), RGPEP yields a constituent quark picture with a nonperturbative oscillator-like confining potential arising from the gluon condensate inside hadrons, not the vacuum. The operator structure of RGPEP naturally produces harmonic oscillator forms for mesons and baryons, with frequencies matching AdS/QCD soft-wall predictions when the gluon condensate expectation value is related to the AdS scale (Glazek, 2011, Glazek, 2011).

Heavy flavor QCD: In the context of heavy quarkonia and baryons, RGPEP with an effective gluon mass ansatz produces an effective inter-quark Hamiltonian that is confining (harmonic oscillator) at long distances and reduces to a Coulomb form (with Breit–Fermi corrections) at short distances. Successful predictions of mass spectra for doubly and triply heavy baryons have been established with excellent agreement with lattice QCD and phenomenological quark models (Gómez-Rocha et al., 2017, Gómez-Rocha et al., 2023).

Spectroscopy and parton phenomenology: The Fock-space wave functions of bound states in the RGPEP basis encode both the spectroscopy (discrete mass eigenvalues) and the momentum-space parton distributions, thus inherently linking the low-energy and high-energy views of hadron structure (Glazek, 2020). In the Abelian gauge sector, RGPEP-generated effective Hamiltonians recover the correct nonrelativistic limits (e.g., for lepton–proton bound states) with scale-dependent corrections to the extraction of physical observables (such as the proton radius in hydrogen versus muonic hydrogen) (Glazek, 2014).

4. Running Coupling, Asymptotic Freedom, and Universality

The RGPEP construction systematically produces the running of effective couplings as a function of the resolution scale $\lambda = 1/s$. Perturbative calculations in QCD yield the canonical logarithmic running characteristic of asymptotic freedom: $$g_\lambda = g_0 - \frac{g_0^3}{48 \pi^2} N_c \left[ 11 + h(x_0)\right] \ln \frac{\lambda}{\lambda_0}$$ where $h(x_0)$ is a finite function capturing scheme dependence in small-$x$ regularization in light-front quantization (Gomez-Rocha et al., 2015, Gómez-Rocha, 2016). For suitable regularizations (e.g., $h(x_0)=0$), the standard beta function appears: $$\lambda\frac{dg_\lambda}{d\lambda} = -\frac{11 N_c}{48\pi^2} g_\lambda^3\,.$$ This demonstrates that the RGPEP procedure is fully consistent with the fundamental renormalization group properties of non-Abelian gauge theories and is universal with respect to technical details (such as choice of generator) (Gomez-Rocha et al., 2015).

5. Gauge Symmetry, Nonlocality, and Vacuum Problems

RGPEP is constructed to preserve gauge invariance and the kinematic symmetries of front-form dynamics. In strongly interacting, gauge-invariant theories, colored effective particles are always accompanied by appropriate gauge-field dressing. Notably, the approach replaces the problematic concept of a vacuum gluon (or quark) condensate with an effective, localized gluon condensate inside hadrons—which is critical for understanding confinement and the absence of vacuum instability in the light-front formulation (Glazek, 2011, Glazek, 2011).

The light-front vacuum remains trivial: pair creation from the vacuum is excluded by $p^+ > 0$ conservation, and all physical particles are built as excitations over the same vacuum. This avoids the vacuum divergence problems that plague instant-form quantization, particularly in models with mass mixing (no re-quantization or vacuum adjustment required) (Glazek, 2012, Glazek, 2013, Glazek, 2013).

The nonlocal character of the RGPEP Hamiltonian is essential: the form factors make the effective theory UV finite and introduce a natural scale for nonlocality, thus enabling the removal of explicit cutoff regularizers once appropriate counterterms are included. The resulting renormalized effective Hamiltonians are symmetric operators on Fock space, suitable for analytic, numerical, or quantum simulation algorithms (Serafin et al., 5 Aug 2025, Gustin et al., 20 Aug 2025).

6. Phenomenological and Practical Implications

RGPEP methods have been applied across a spectrum of relativistic quantum field theories:

• Bound-state calculations reduce to solving a Schrödinger-like equation with scale-dependent confining and Coulomb terms, allowing for ab initio predictions of spectra in QCD and other theories (Gómez-Rocha et al., 2017, Gómez-Rocha et al., 2023).
• Parton distribution functions (PDFs) are derived directly from the Fock-space expansion of eigenstates of the effective Hamiltonian, linking spectroscopy and high-energy scattering data (Glazek, 2020).
• Quantum simulation: RGPEP produces Hamiltonians that are band-diagonal and thus efficiently encodable for quantum computation. Techniques such as Ladder Operator Block Encoding (LOBE) achieve quantum resource estimates for the simulation of RGPEP-renormalized Hamiltonians comparable to those for canonical Hamiltonians, enabling practical digital quantum simulation studies (Gustin et al., 20 Aug 2025).
• Collider phenomenology: RGPEP-based effective degrees of freedom have been used to interpret jet and ridge phenomena in high-energy collisions, with mappings between low-energy effective (constituent) and high-energy partonic pictures via the $U_{s_2} U_{s_1}^\dagger$ transformation (Glazek et al., 2016).

7. Limitations, Extensions, and Ongoing Developments

RGPEP has broad applicability to both Abelian and non-Abelian field theories, as well as to systems with both bosonic and fermionic degrees of freedom. Its capacity for exact solutions is clearest in bilinear or mass-mixing models; for realistic theories (QCD, QED), it serves as a foundational structure for systematic perturbative or nonperturbative analysis, with higher-order corrections (in the coupling expansion) necessary for precision phenomenology (Serafin, 2017, Serafin et al., 5 Aug 2025).

The explicit form-factor structure of RGPEP facilitates clean separation of scales, but achieving nonperturbative solutions or resummations remains a central research challenge, particularly regarding the structure of the effective Hamiltonian at strong coupling or with explicit dynamical gluon (or gauge boson) masses (Gómez-Rocha et al., 2017, Gómez-Rocha et al., 2023).

In summary, RGPEP is a Hamiltonian renormalization group procedure that provides a rigorous, scale-dependent, symmetry-preserving basis for the construction of effective field theories in the front-form, producing well-defined, physically meaningful, nonlocal effective Hamiltonians suitable for the analysis of relativistic bound states, running couplings, and parton structures in diverse field-theoretic settings.