Perry and Wilson (1993) proposed coupling-coherence conditions under which a Hamiltonian quantum field theory can be renormalized perturbatively even though symmetry-violating regulators generate infinitely many relevant and marginal operators: only a finite number of running couplings and masses are independent variables, and all other (dependent) operator coefficients are fixed functions of them that vanish when the independent couplings vanish, so that counterterms restoring Lorentz covariance, gauge invariance, and other hidden symmetries are determined automatically. The source left open four issues: existence and uniqueness of solutions of the coupled nonlinear coefficient equations; whether perturbatively suppressed dependent operators remain negligible when the Hamiltonian is used nonperturbatively at low energy; behavior at strong coupling and for particle-number-changing interactions; and the construction of the full light-front QCD realization. The problem remains open.

Background

In the renormalization group (RG) approach to quantum field theory one introduces a cutoff that limits which degrees of freedom are kept in the Hamiltonian and studies how the parameters of the theory must change as the cutoff is lowered. Operators are classified by this behavior: relevant operators (whose couplings, such as masses, grow as the cutoff is lowered), marginal operators (whose couplings change only logarithmically, such as gauge couplings or quartic scalar couplings), and irrelevant operators (whose couplings vanish). In the usual covariant, Euclidean-space formulations of renormalizable theories only finitely many relevant and marginal operators appear, so finitely many measurements fix the theory.

Light-front (or front-form) quantization, introduced by Dirac, formulates field theory on the null plane x+=t+z=constx^+ = t + z = \mathrm{const} instead of equal-time slices. The Fock space (the space built by acting with free-particle creation operators on the vacuum) then has two decisive advantages for bound-state physics: boosts are kinematic (center-of-mass motion factors out), and the vacuum mixes only with modes of exactly zero longitudinal momentum. Hadrons (the strongly interacting bound states, such as the proton and neutron) can therefore be attacked as eigenvalue problems of a Hamiltonian, Hψ=M2ψH \psi = M^2 \psi, without first solving the quantum chromodynamics (QCD) vacuum. The price is that every practical regulator on light-front momenta (limits on longitudinal momentum fractions and transverse momenta) violates continuous symmetries of the underlying theory: Lorentz covariance, gauge invariance, and cluster decomposition (the requirement that distant subsystems decouple). The renormalized Hamiltonian must then be allowed to contain every operator consistent with the principles that are not violated, and there are infinitely many relevant and marginal operators, including coefficient functions that are arbitrary functions of longitudinal momenta. Naively this destroys predictivity, since infinitely many parameters would have to be measured.

Perry and Wilson proposed a way out, which became known as coupling coherence: seek solutions of the Hamiltonian RG equations in which only a finite number of variables (the canonical running couplings and masses) depend on the cutoff independently, while every other, dependent, coupling or mass depends on the cutoff only through its dependence on these running variables and vanishes when they vanish. This generalizes the earlier reduction-of-couplings method of Oehme and Zimmermann, in which all couplings of a renormalizable theory are constrained to be functions of a single running coupling. In one-loop examples the 1993 paper showed that these conditions fix the symmetry-violating counterterms automatically: O(N)O(N) symmetry (invariance under rotations of the NN real scalar fields) emerges in scalar theories, fermions are driven massless when the regulator breaks chiral invariance, gauge bosons become massless when gauge invariance is broken, and ϕ→−ϕ\phi \to -\phi symmetry can be recovered in the broken-symmetry phase. The successor formalism is the similarity renormalization group, developed into the renormalization group procedure for effective particles (RGPEP) by Głazek and collaborators: a continuous unitary transformation that rescales the size of the interacting quanta (effective particles) and suppresses changes in free invariant mass beyond a cutoff, with the counterterms fixed by computing matrix elements of the effective Hamiltonian order by order in the coupling. In this framework asymptotic freedom (the weakening of the strong coupling at short distances) has been reproduced directly in the effective Hamiltonian for gluons, and complete renormalized Hamiltonians have recently been obtained to second order in the coupling for Yukawa theory (fermions coupled to a scalar boson) and for light-front QCD. The unresolved question is whether the whole construction is sound and complete: whether the coupling-coherence equations always have unique, symmetry-restoring solutions, whether the resulting Hamiltonian is mathematically well defined after the cutoff is removed, whether perturbatively suppressed dependent operators stay negligible in nonperturbative applications, and whether the procedure can actually be carried through for QCD, where confinement (the absence of isolated quarks and gluons in the physical spectrum), chiral symmetry breaking (the dynamical generation of hadron-scale masses despite an approximately left-right symmetric bare theory), and zero longitudinal momentum modes (zero modes) lie outside perturbation theory.

Problem Statement

The source question is whether the coupling-coherence conditions proposed by Perry and Wilson for perturbative renormalization of Hamiltonian field theories with infinitely many relevant and marginal operators can be established as a sound and complete renormalization principle, with light-front QCD as the decisive application. The single common objective is a mathematically well-defined, symmetry-restoring, nonperturbatively validated renormalized effective Hamiltonian for light-front QCD obtained through the coupling-coherence (similarity renormalization group / RGPEP) construction; the following results jointly serve that objective and are not independent of one another: (1) Well-posedness and uniqueness: prove, for the whole class of theories and regulators of interest (the light-front Hamiltonian theories with symmetry-violating regulators described in the Background, namely scalar, Yukawa, QED, and QCD with invariant-mass or longitudinal/transverse-momentum cutoffs), or disprove by an explicit counterexample in any single member of that class, that the coupled nonlinear coefficient equations of the Hamiltonian renormalization group with the coupling-coherence constraints (finitely many independent running variables; all other dependent coefficients functions of them that vanish when they vanish) admit a solution, that it is unique, that it restores the symmetries broken by the regulator, and that after cutoff removal the resulting Hamiltonian is a well-defined self-adjoint operator (an operator equal to its adjoint, so that it has a real spectrum and generates unitary time evolution), or a quadratic form together with the self-adjoint extension fixed by requiring that the cutoff-removal limit preserve the restored symmetries and unitary time evolution. This replaces the present case-by-case verification at each perturbative order. (2) Nonperturbative reliability: establish whether the dependent relevant and marginal operators that are perturbatively suppressed at the scale where the Hamiltonian is fixed remain negligible when the Hamiltonian is solved nonperturbatively (diagonalized) for low-energy observables, or exhibit where they do not, including the role of zero longitudinal-momentum modes, chiral symmetry breaking, and confinement, none of which perturbation theory generates by itself. (3) Light-front QCD realization: construct the renormalized effective light-front QCD Hamiltonian beyond the currently achieved second order in the coupling (the fourth-order computation is the stated next milestone of the program), and verify that it reproduces standard covariant perturbative QCD at the corresponding order while yielding convergent nonperturbative predictions for hadron observables, including the strongly coupled, particle-number-changing interactions associated with confinement. The three components are jointly posed, and establishing only some of them does not resolve the source question; a demonstration, in a theory and regulator of interest, that the coupling-coherence equations have no solution, admit non-unique solutions, fail to restore the broken symmetries, fail to yield a well-defined self-adjoint operator (or quadratic form with such a selected self-adjoint extension) after cutoff removal, or leave the perturbatively suppressed dependent operators non-negligible at low energy would refute the principle.

References

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