Quark confinement in relativistic quantum field theory: the open questions of Parisi's 1975 soluble-model analysis

G. Parisi's 1975 paper 'Quark confinement in quantum field theory' showed, in soluble static (infinitely heavy, nonrelativistic) gauge models in $D\leqslant 3$ spacetime dimensions, that a long-range gauge interaction can confine quarks at the level of the propagator: the self-energy of an isolated quark diverges while neutral quark-antiquark pairs remain finite, and the quark propagator acquires an ever-accelerating phase, $\sim\exp(\mathrm{i}e^2 t^{4-D})$ for $2<D<3$ and $\sim\exp(\mathrm{i}e^2 t\ln t)$ at $D=3$, so that its momentum-space form is an entire function of $p^2$ with no single-particle pole, while the short-distance operator-product behaviour is unchanged. The paper then proposed a Ward-identity-preserving relativistic approximation for scalar electrodynamics, reducing the Schwinger-Dyson equation to a closed differential equation, deferred a detailed analysis of its corrections, and left open whether confinement of this type can be established in genuinely relativistic interacting gauge theories, whether the indefinite-metric Hilbert space it uses is essential, and whether the mechanism applies to real strong-interaction dynamics. Half a century later these questions remain open: exact confinement is established only in lower-dimensional models, confinement in $3+1$-dimensional quantum chromodynamics (QCD, the Standard-Model non-Abelian gauge theory of the strong interaction) is confirmed numerically but not derived from first principles, pole-free quark propagators survive mainly as phenomenological ansaetze, and the metric question reappears unresolved as the positivity-violation problem of confining gauge-field propagators.

Background

Quarks are the spin-1/2 constituents of hadrons (protons, neutrons, mesons) in the Standard Model, yet no isolated quark has ever been observed; this is the problem of quark confinement. A quantum field theory (QFT) predicts its particle spectrum through the poles of two-point Green functions (propagators): a stable particle of mass $m$ produces a pole of the propagator at $p^2=m^2$ with positive residue. Conversely, if the time-ordered two-point function oscillates with ever-increasing frequency at large time, its Fourier transform has no isolated pole and the corresponding excitation never appears as an asymptotic scattering state. QED already points at this behaviour: soft photons (the Bloch-Nordsieck infrared problem, the accumulation of infinitely many low-energy photons emitted in any charged scattering process) smear the electron's sharp mass pole into a continuum, producing an 'infraparticle' rather than a particle with a well-defined mass.

By 1975 there was no controlled relativistic QFT example of confinement. K. Johnson's paper 'Can Quarks Be Kept Inside Hadrons?' (1972) proposed a self-consistent dynamical mechanism, compatible with relativistic field theory, that removes the field's quanta from incoming and outgoing scattering states while producing an infinitely rising spectrum of bound states, all physical and of zero triality (zero net charge under the centre of the internal symmetry group, i.e. reachable from the vacuum); bag models confined quarks semi-classically inside a finite cavity; and massive-quark proposals postulated unconventional momentum-space Green functions. What was missing was a soluble, calculable field-theory laboratory showing that (i) the energy to create an isolated quark can be infinite while the quark field's Green functions remain nonzero and computable, (ii) the standard short-distance machinery - the Wilson operator-product expansion (OPE, the statement that products of fields at short distances expand into local operators with computable coefficients, together with its light-cone analogue, the basis of parton-model phenomenology) - is unaffected by confinement, and (iii) such constructions can be embedded in standard perturbative QFT machinery.

Parisi's 1975 Phys. Lett. B 60 paper supplied such a laboratory. In a static model - spin-0 quarks and antiquarks of charges $\pm e$ and mass $m\to\infty$ (no recoil), so that only the scalar potential $A^0$ couples, in $D$ spacetime dimensions with the free propagator $D(x^2)=\int \mathrm{d}^D K\, \mathrm{e}^{\mathrm{i}Kx}/K^2$ - the quark Green function is exactly

$$G(t)=\theta(t)\,\exp\!\left\{\mathrm{i}\left[E_0 t+\tfrac{e^2}{2}\int_0^t \mathrm{d}\tau \int_0^t \mathrm{d}\sigma\, D(0,\tau-\sigma)\right]\right\},$$

obtained by applying the Gribov-Levin-Migdal solvable-model result. For $1<D\leqslant 3$ the static self-potential $V(0)$ diverges while $V(x)-V(0)$ is finite: an isolated quark costs infinite energy, a neutral pair does not - confinement inside the soluble model. The large-time phase $\sim \mathrm{i} e^2 t^{4-D}$ (for $2<D<3$) or $\sim \mathrm{i} e^2 t\ln t$ (for $D=3$) makes the momentum propagator an 'integer' (entire, i.e. pole-free analytic) function of $p^2$. The construction is carried out in a Hilbert space with indefinite metric - quantizing the scalar potential produces negative-norm states, as in the Gupta-Bleuler covariant quantization of QED - and the short-distance (Wilson and light-cone expansion) behaviour remains that of free fields. To go beyond the static limit, the paper proposes, for scalar electrodynamics (a charged scalar field $\phi$ coupled to the electromagnetic field $A_\mu$), an approximation replacing the interacting three-point correlator by the ratio of free correlators times the interacting propagator,

$$\langle 0|T[J_\mu\phi^+\phi]|0\rangle_{\mathrm{I}} = \left(\langle 0|T[J_\mu \phi^+\phi]|0\rangle_{\mathrm{F}}/\langle 0|T[\phi^+\phi]|0\rangle_{\mathrm{F}}\right) \langle 0|T[\phi^+\phi]|0\rangle_{\mathrm{I}},$$

which preserves the Ward identities (relations among Green functions that follow from gauge invariance) at every step and converts the exact Schwinger-Dyson equation (the exact functional equation $(\square+m^2)G(x)=\mathrm{i}e\,\langle 0|T[A_\mu(x)\partial^\mu\phi^+(x)\phi(0)]|0\rangle$ for the propagator) into the closed differential equation $(\square+m^2)G(x)=e^2 V_\mu(x)\partial^\mu G(x)$ - where $V_\mu(x)$ is the long-range gauge potential generated by the exchanged soft gauge quanta, the relativistic analogue of the static self-potential $V$ of the soluble model - whose solution again shows the pole-free signature; the eikonal approximation (the high-energy, small-angle approximation resumming repeated soft exchanges) gives the same qualitative result. A detailed study of the corrections to this approximation, its rigorous status, and the realism of the models were explicitly left open by the paper, and it is these deferrals that constitute the open problem.

Problem Statement

The problem left open by Parisi's 1975 analysis is to turn the propagator-level picture of confinement demonstrated in static soluble models into an established result of relativistic quantum field theory. As one coherent objective - a standard-QFT account of quark confinement that is controlled, physically consistent, and realistic - the following linked questions remain open.

1. Controlled relativistic scheme: justify (or replace) the proposed relativistic approximation for scalar electrodynamics - the free-correlator ratio ansatz for the three-point function and the resulting closed differential equation $(\square+m^2)G(x)=e^2 V_\mu(x)\partial^\mu G(x)$, with $V_\mu$ the long-range gauge potential of soft gauge-boson exchange (the relativistic analogue of the static self-potential $V$ of the soluble model) - so that the absence of single-quark poles is a consequence of the dynamics rather than an assumption, and the status of the corrections to the approximation - the detailed analysis explicitly deferred by the paper - is established.

2. Rigorous relativistic generalization: determine whether confinement of this type can be established in genuinely relativistic, interacting gauge theories (ultimately of $3+1$-dimensional quantum chromodynamics (QCD), the non-Abelian gauge theory of the strong interaction) with the same level of control as the static soluble models - either exhibiting such a construction, or proving that no construction of this kind exists. The confinement sought is characterized by isolated coloured one-particle states absent from the physical spectrum, a criterion formulated (since the gauge-fixed propagator is gauge dependent) in terms of gauge-invariant observables or the physical-state condition, with the propagator-level signatures (divergent self-energy, no pole in the quark propagator, propagator an entire function of $p^2$) as its gauge-fixed indicators, while colour-neutral composites have finite energy, the quark Green functions remain nonzero and computable, and the short-distance operator-product behaviour is unchanged.

3. Hilbert-space structure: determine whether the indefinite metric (the negative-norm states produced by the scalar potential) is essential for constructions of this type - either exhibit a confinement construction of the same propagator-level kind realized in a positive-definite Hilbert space, or prove that confinement of this type necessarily entails indefinite metric or positivity violation.

4. Realism: determine whether the mechanism applies to the real strong-interaction dynamics (QCD) rather than only to low-dimensional or scalar models - either exhibiting the mechanism at work in QCD, or identifying the obstruction that prevents such an application.

These four questions are scientifically linked: item 1 is the analytic engine, item 2 its target, item 3 its consistency condition with unitarity and probability, and item 4 its physical endpoint. Item 2 concerns genuinely relativistic, interacting gauge theories, not the confinement already established in lower-dimensional models.

References

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