Positivity properties of $O(a)$-improved Wilson fermion actions in interacting lattice QCD: spurious poles, sign-changing correlators, and their continuum-limit fate

A 1992/93 quenched lattice-QCD spectroscopy study by Lombardo, Parisi and Vladikas, performed with the Eguchi-Kawamoto/Hamber-Wu next-nearest-neighbor improved Wilson action, found a systematic reduction of finite-lattice-spacing effects but also stronger positivity violations than the standard Wilson action: the free quark propagator carries spurious complex and additional real poles, and at coarse coupling ($\beta=5.7$) the orbitally excited $a_1$ and $b_1$ meson correlators can change sign. The study called for a detailed investigation of the positivity properties of improved Wilson-type actions in the interacting theory, alongside four further follow-up needs: non-perturbative renormalization constants for the axial and vector currents, control of the heavy-mass chiral extrapolation, non-perturbative tuning of $O(a)$ improvement coefficients, and scaling tests at intermediate couplings. The renormalization, improvement-coefficient and scaling threads are now resolved in general form for the Sheikholeslami-Wohlert (clover) action by the ALPHA and UKQCD collaborations, the Rome-Southampton method and related programs of the 1990s, and the chiral-extrapolation thread has passed into standard chiral-perturbation-theory methodology. The positivity question - whether the free-field spurious-pole pattern persists in the interacting theory, with what channel and coupling dependence, and whether it decouples in the continuum limit - remains open in the interacting theory. The free theory is now understood in general terms (reflection positivity holds there if and only if all propagator poles are real, simple and carry non-negative residues; Handrack and Salmhofer, 2026), but no analogous statement exists for the interacting theory.

Background

Lattice QCD computes hadron properties by discretizing the QCD path integral on a hypercubic grid of spacing $a$ and extrapolating $a\to0$; the bare inverse gauge coupling is written $\beta=6/g_0^2$, with larger $\beta$ corresponding to finer lattices (in the quenched, valence-quark-only approximation used here, $\beta=5.7$ corresponds to $a\approx0.17$–$0.19$ fm and $\beta=6.0$ to $a\approx0.1$ fm). The standard Wilson fermion action removes the fermion-doubling problem of naive discretization (the appearance of extra, unwanted light fermion copies in the naive lattice derivative) through a nearest-neighbor term, at the price of leading discretization errors of order $a$ and explicit chiral-symmetry breaking. Symanzik's improvement program cancels these $O(a)$ errors by adding higher-dimension operators. Two improved Wilson-type discretizations emerged in the 1980s: the Sheikholeslami-Wohlert (clover) action, which adds a Pauli-type term $c_{SW}\,\bar\psi\sigma_{\mu\nu}F_{\mu\nu}\psi$ coupling the quark spin to the gauge field strength, and the Eguchi-Kawamoto/Hamber-Wu action, which adds a next-nearest-neighbor hopping term $S_{II}=\sum\bar\psi_n(C-D\gamma_\mu)U_{n,\mu}U_{n+\mu,\mu}\psi_{n+2\mu}+\mathrm{h.c.}$ with $C=-1/4k$ and $D=-1/8k$, where $k$ is the hopping parameter and $k_c$ (free-field value $1/6$) is its critical value at which the quark mass vanishes.

A structural property relevant to spectroscopy is reflection positivity (Osterwalder-Schrader positivity): Euclidean correlation functions of gauge-invariant operators then admit a spectral representation with non-negative weights, so effective-mass plateaus (the hadron-mass estimate extracted from correlator ratios at successive time separations) behave monotonically and variational bounds on hadron masses are valid. Extending an earlier proof by Osterwalder and Seiler for lattice gauge theories with fermions, Menotti and Pelissetto proved a general reflection-positivity statement for the standard Wilson action, valid for all gauge-invariant observables, and their analysis shows the proof depends on the nearest-neighbor structure of the Wilson action - a structure they argue is in a sense necessary for any such general statement. Improved actions therefore carry no comparable guarantee. In the free (non-interacting) theory a complete criterion is now known: for fermionic theories whose covariance is a real rational function of the Dirac operator, reflection positivity holds if and only if all propagator poles are real, simple and carry non-negative residues (Handrack and Salmhofer, 2026). This makes precise why the free Hamber-Wu propagator, with its complex pole pairs and its real poles of non-physical residue, violates reflection positivity at tree level; it says nothing about the interacting theory.

Lombardo, Parisi and Vladikas studied the quenched hadron spectrum with the Hamber-Wu action at $\beta=5.7$ and $\beta=6.0$ in the heavy-quark region. They found a systematic reduction of finite-$a$ effects (improved results at $\beta=5.7$ agree with standard Wilson results at $\beta=6.3$), but also stronger positivity violations than the standard Wilson action: the zero-momentum free quark propagator has four poles in the energy, two of them complex and two turning real for $k\gtrsim0.14$, producing ripples in the position-space propagator; in the simulation, the $L=1$ (orbitally excited) meson correlators $a_1$ and $b_1$ changed sign at $\beta=5.7$, with effective-mass wiggles that fade as $k\to k_c$ or $a\to0$. A free-field argument suggests the spurious poles are harmless - as $k\to k_c$ only the physical pole becomes massless - but this was never established in the interacting theory. The same study flagged four further follow-up needs: non-perturbative values of the renormalization constants $Z_A$ and $Z_V$ (the multiplicative factors relating the lattice axial and vector currents to their continuum counterparts, needed to normalize the pion and rho leptonic decay constants $f_\pi$ and $f_\rho$, which were taken here from one-loop perturbation theory); control of the delicate heavy-mass chiral extrapolation at $\beta=6.0$; non-perturbative tuning of the $O(a)$ improvement coefficient (the source paper recalls that the FNAL group, using the clover action, had observed strong sensitivity of spin splittings to the clover coefficient $c$); and scaling tests at intermediate couplings $\beta=6.3$–$6.5$ to confirm that the observed improvement is robust. Four of these five threads were later settled in general form; the positivity question is the surviving core.

Problem Statement

Determine the positivity properties of $O(a)$-improved Wilson-type fermion discretizations in the interacting lattice theory - the detailed investigation of the positivity properties of the improved actions for which the source study called - in a class of actions for which the general reflection-positivity guarantee that Menotti and Pelissetto proved for the unimproved Wilson action is no longer available. Concretely, taking the next-nearest-neighbor improved action of Eguchi-Kawamoto/Hamber-Wu used in the source study as the primary instance, the open question comprises the following three linked aspects, each of which a resolution must address - aspects (a) and (b) for that action, and aspect (c) for the full class of $O(a)$-improved Wilson-type actions:

(a) Persistence and structure in the interacting theory: does the free-field pattern - spurious complex-conjugate pole pairs and additional real poles with non-physical residues - survive once gauge interactions are included, and how do the induced positivity-violating contributions manifest in Euclidean hadron correlators? Since the poles of the interacting quark propagator are not themselves directly observable, persistence is to be understood through the reflection-positivity properties of gauge-invariant Euclidean correlators, in the sense of Menotti and Pelissetto; the free-field pole language characterizes the tree-level violation whose interacting fate is at issue. Quantify their channel dependence (which interpolating-field channels can produce sign-changing or oscillating correlators, the $a_1$ and $b_1$ orbitally-excited meson channels in the source study being the observed instances) and their dependence on the gauge coupling (lattice spacing) and on the quark mass.

(b) Continuum-limit fate: do the positivity-violating contributions decouple from hadron spectroscopy as $a\to0$, so that masses and matrix elements extracted at finite $a$ are unbiased in the continuum limit - as the free-field argument suggests - or do they leave a residual systematic? In particular, is the finite-$a$ contamination of an improved action smaller than, comparable to, or larger than that of the unimproved Wilson action (the source study observed stronger positivity violations for the improved action at $\beta=5.7$)?

(c) Structural statements: can any general reflection-positivity statement or quantitative bound on the violations be established for $O(a)$-improved Wilson-type actions generally (including the clover action and the next-nearest-neighbor Eguchi-Kawamoto/Hamber-Wu action), given that the Menotti-Pelissetto proof depends essentially on the nearest-neighbor structure of the Wilson action? The free theory is settled in general terms by the pole criterion of Handrack and Salmhofer, which applies only to non-interacting theories; the open content of this question concerns the interacting theory.

The common objective is a controlled account of reflection-positivity violation for improved Wilson-type fermion actions - its origin, its channel and coupling dependence, and its continuum-limit decoupling - that addresses all three aspects above, reaching a determinate conclusion on each: whether and how the free-field positivity-violating pattern persists in the interacting theory, whether the induced finite-lattice-spacing systematic that the improvement terms introduce into spectroscopy and correlator analyses decouples as $a\to0$ or survives as a quantified residual, and whether a general reflection-positivity statement or quantitative bound can be established for the class of $O(a)$-improved Wilson-type actions.

References

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