In a 1993 model calculation, Głazek and Wilson showed that the overlapping transverse ultraviolet divergences of a light-front Hamiltonian (a Yukawa-type model — fermions coupled by scalar-meson exchange — truncated to two-fermion states) can be cancelled to all orders in Hamiltonian perturbation theory by a counterterm local in the transverse direction that contains an arbitrary function of the fermions' longitudinal momenta, with the renormalized two-fermion interaction obeying the renormalization-group flow equation dVλ/dλ=−VλKλVλdV_\lambda/d\lambda = -V_\lambda K_\lambda V_\lambda. They explicitly left open the extension of this program to full light-front QCD, and the authors' program — the source paper together with its immediate same-author continuations (Wilson et al. 1994; Perry 1994) — itemized what is missing: removal of the longitudinal (small-p+p^+) cutoff dependence, including the enhanced logarithms and the omitted self-interaction/seagull terms; determination of the finite counterterm functions from physical conditions such as rotational invariance about the transverse axes; generalization of the counterterm structure and flow to Hamiltonians acting in Fock spaces with variable numbers of constituents; and nonperturbative control of the flow, including whether the theory is asymptotically free in the transverse-flow sense (the effective coupling of the λ\lambda-dependent Hamiltonian vanishing at short transverse distances) — the behavior the program expects a correctly renormalized realistic gauge-theory Hamiltonian flow to exhibit, and whose determination it treats as part of the open agenda. The subsequent literature (similarity renormalization group, also called the renormalization group procedure for effective particles, RGPEP; coupling coherence; perturbative equivalence proofs; sector-dependent renormalization; basis light-front quantization; front-form asymptotic freedom) resolves pieces only in models, in low perturbative order, or with phenomenologically fitted counterterms: no complete, regulator-independent, symmetry-preserving renormalized light-front QCD Hamiltonian exists. The problem remains open.

Background

In Dirac's 1949 classification of forms of relativistic dynamics, the front form (light-front quantization) uses the light-like coordinate x+=t+zx^+=t+z as time, with conjugate momentum p−p^-; the Hamiltonian is the generator P−P^- of translations in x+x^+, and the invariant mass squared is M2=P+P−−P⊥2M^2=P^+P^- - P_\perp^2. The attraction of this formulation for hadron physics is that the vacuum is trivial for states of nonzero p+p^+ (so bound states admit a clean Fock-space expansion — a sum over states with fixed particle content and momenta), boosts along the front direction are kinematical, and light-front wave functions directly encode the parton distributions (the momentum distributions of quarks and gluons inside hadrons) measured in high-energy scattering. The price is that the canonical light-front Hamiltonian of an interacting field theory is ill-defined: local interactions generate ultraviolet divergences whose light-front perturbation theory is not simply the covariant one. In particular, integrals over the transverse momenta p⊥p_\perp of constituents produce logarithmic divergences nested inside one another — analogous to the overlapping divergences of covariant perturbation theory, in which divergent subgraphs cannot be separated into individually renormalizable pieces — and a cutoff on transverse momenta (here λ\lambda, the scale of the relative transverse momenta at which interactions are chopped off) makes the theory cutoff dependent. A counterterm is an operator added to the Hamiltonian whose divergent part cancels these cutoff dependences. Głazek and Wilson (1993) constructed a deliberately simplified model: a light-front Hamiltonian acting on states of two fermions, with one-boson exchange, fermion self-interactions dropped, and the eigenvalue in the exchange term replaced by a constant. They proved that a single Hamiltonian counterterm, local in the transverse coordinates, removes the overlapping transverse divergences to all orders in Hamiltonian perturbation theory. Two features of their result drive the open problem. First, the counterterm is fixed by the divergences only up to a finite, arbitrary function of the fermions' longitudinal momenta p+p^+; this function is a new, infinite set of free parameters. Second, the renormalized two-fermion interaction VλV_\lambda obeys a nonlinear renormalization-group equation, dVλ/dλ=−VλKλVλdV_\lambda/d\lambda = -V_\lambda K_\lambda V_\lambda with a known kernel KλK_\lambda, so the cutoff dependence is traded for a flow problem in λ\lambda whose solutions might develop singularities at finite λ\lambda. The model also ignored the longitudinal direction entirely: a separate cutoff ε\varepsilon on small p+p^+ (below which longitudinal momenta are chopped) remains in the theory, and when boson longitudinal momenta can become arbitrarily small, longitudinal integrals produce logarithms enhanced at small ε\varepsilon that the transverse-local counterterm does not touch. Finally, in the front form, rotations about axes perpendicular to the light front (the xx and yy axes) are dynamical — they depend on the interaction — so any cutoff or particle-number truncation generically violates rotational invariance of the spectrum, and Głazek and Wilson suggested that the arbitrary longitudinal function in their counterterm could be chosen to partly remove exactly this violation. The same authors and collaborators developed the machinery intended to scale this construction up: the similarity renormalization group (also called the renormalization group procedure for effective particles, RGPEP), which builds a band-diagonal effective Hamiltonian HλH_\lambda in Fock space free of small energy denominators (Głazek and Wilson 1993b, 1994), and related works introduced coupling coherence (Perry 1994) — the requirement that the number of couplings that run explicitly with the cutoff stays limited — and sector-dependent renormalization (Karmanov, Mathiot, and Smirnov 2008), in which masses and couplings are assigned per Fock sector so that a truncated theory can still be renormalized. The renormalized light-front QCD Hamiltonian is widely regarded in the light-front community as the central unsolved construction problem standing between QCD and nonperturbative, Minkowski-space hadron calculations (see the reviews by Brodsky, Pauli, and Pinsky 1998, and by Głazek 2026).

Problem Statement

The problem left open by Głazek and Wilson (1993) is to extend their demonstrated transverse renormalization of a model light-front Hamiltonian to the full light-front Hamiltonian of quantum chromodynamics. Their result — cancellation, to all orders in Hamiltonian perturbation theory, of the overlapping transverse ultraviolet divergences of a two-fermion model by a transverse-local counterterm, with the renormalized interaction VλV_\lambda flowing according to dVλ/dλ=−VλKλVλdV_\lambda/d\lambda=-V_\lambda K_\lambda V_\lambda — must be augmented by four linked, jointly necessary developments that the authors itemized as open in the source paper and in the immediate continuations of the same program (Wilson et al., Phys. Rev. D 49, 6720 (1994); Perry, Ann. Phys. (N.Y.) 232, 116 (1994)): (1) removal of the dependence on the longitudinal cutoff ε\varepsilon on small longitudinal momenta p+p^+, including the logarithmically enhanced longitudinal integrals that arise when the boson longitudinal momentum can become arbitrarily small, and including the fermion self-interaction (seagull) terms omitted in the model; (2) determination of the finite functions of longitudinal momenta that the transverse-local counterterm contains but the renormalization procedure itself leaves arbitrary, as distinct from fitting the functions to experimental data in lieu of a first-principles determination; (3) generalization of the counterterm structure and of the renormalization-group flow to Hamiltonians acting in Fock spaces with variable numbers of constituents, so that the symmetry violations introduced by constituent-number truncations can be controlled rather than assumed away; and (4) nonperturbative control of the renormalization-group flow in the transverse cutoff λ\lambda — excluding or taming uncontrolled singularities of the renormalized interactions at finite λ\lambda, and, for realistic gauge theories, determining the asymptotic behavior of the flow — in particular whether the theory is asymptotically free in the sense of this Hamiltonian flow. The common objective is a single, well-defined object: an effective light-front QCD Hamiltonian on the Fock space of quarks and gluons (untruncated, or with truncations whose renormalization and errors are systematically controlled) that is finite, independent of the transverse and longitudinal regulators when they are removed, respects the Poincaré symmetries not kinematically guaranteed in the front form (above all rotational invariance), and whose λ\lambda-flow is under control, with its asymptotic behavior determined (in particular whether it is asymptotically free). The target object is the full four-dimensional light-front QCD Hamiltonian itself, with its finite counterterm functions determined from first-principles conditions such as coupling coherence or restoration of the Poincaré symmetries, rather than a model-theory analogue (Yukawa, QED, pure glue), a low-order or lower-dimensional truncation, or a counterterm fixed by phenomenological fitting to data.

References

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