A particle-physics preprint reports that a mathematical problem in a model of quarks and diquarks is associated with the model’s flow itself, rather than being simply numerical noise. In the calculation, regions of negative diffusion appeared alongside strong oscillations in the derivative of the model’s effective potential. A higher-derivative term called hyperdiffusion removed the oscillatory pattern in the reported comparison, after which the researchers calculated a phase diagram for the model.
The finding is narrower than a claim about full quantum chromodynamics, or QCD. The work uses a Quark-Diquark Model within the Local Potential Approximation, and the authors acknowledge that calculations beyond this approximation might remove the effect. Whether negative diffusion is a genuine feature of the broader model therefore remains unsettled.
The instability appears at lower scales
The study examines a functional-renormalization-group flow, tracking how the model’s equations behave as the renormalization-group scale changes. The key quantity is a diffusion coefficient in a conservative advection-diffusion equation for the derivative of the effective potential. In the formulation used here, the flow contains an advection flux, a diffusion-like term and a source term, so the coefficient’s sign is central to the numerical behavior.
At the ultraviolet starting scale of 1 GeV, all the derivative values examined were in a positive-diffusion region. As the flow moved down to 0.39 GeV and then 0.075 GeV, some values entered regions of negative diffusion. The same flow showed strong oscillations in the effective-potential derivative and its derivatives, leading the authors to interpret the effect as more than an isolated numerical artifact.
This was a deterministic model calculation, not an experiment or a study involving participants. The researchers evaluated the diffusion coefficient analytically at fixed renormalization-group scale, temperature and chemical potential, then solved the flow numerically for the QDM parameter sets listed in the study. The central test was whether the negative regions coincided with the oscillations and whether the flow could be regularized well enough to calculate phase boundaries.
A deliberately limited model
The QDM contains quark and diquark degrees of freedom but no scalar or pseudoscalar mesons. It is not intended to describe chiral symmetry breaking; instead, the setup is used to study diquark condensation. The model has 2 flavors and 3 colors in 3+1 dimensions and retains only the fully antisymmetric scalar diquark pairing channel.
For the numerical calculation, the derivative field was placed on a uniform grid with 2 MeV spacing over the interval from zero to the maximum pairing field. The implementation used HLLE fluxes, fifth-order weighted Essentially Non-Oscillatory interpolation, a fourth-order central-difference stencil and LSODA integration from scipy. These choices describe how the equations were solved; they are not measurements of a physical system.
When negative diffusion appeared, the authors added hyperdiffusion, a higher-derivative regularizer, to the derivative flow. That extra term can affect the calculation, so the zero-regularization result was approached rather than read directly from a single run. The researchers calculated observables at different values of the regularization parameter and applied a least-squares fit as the parameter was taken toward zero.
The stabilized calculation maps two modeled phases
In the reported regularized comparison, the oscillatory behavior seen in the unregularized flow was absent. The stabilized calculation was then used to construct phase diagrams for the three listed QDM parameter sets. The diagrams contained first- and second-order transition lines between a normal-conducting phase, marked by a zero pairing field, and a color-superconducting phase, marked by a nonzero pairing field. A critical point separated the two types of transition line.
The transition lines also served as a check on the renormalization-group setup. For parameter set 3, the authors compared results obtained with an ultraviolet cutoff of 5 GeV with results using 1 GeV and judged the agreement to be reasonably consistent with renormalization-group expectations. The assessment was qualitative, and no quantitative uncertainty for the comparison was reported.
The strength of the modeled quark-diquark coupling was associated with the size of the negative-diffusion region. The reported minimum diffusion values were −0.0044 GeV, −0.014 GeV and −0.021 GeV. The stronger-coupling cases reached farther into negative diffusion at higher temperatures, and their unregularized flows showed stronger oscillations than parameter set 1.
A numerical procedure, not a settled physical effect
The authors interpret negative diffusion as a feature of the QDM within the studied Local Potential Approximation rather than as a numerical artifact. They also conclude that hyperdiffusion supports low-temperature, high-chemical-potential phase-diagram calculations. Those conclusions apply to the tested formulation: beyond-Local-Potential-Approximation truncations were not examined.
The calculation has technical limits as well. It uses a finite field grid and an infrared cutoff of 0.075 GeV. At high chemical potential and low temperature, solver termination can occur at earlier renormalization-group times because of convexity restoration. Hyperdiffusion introduces systematic error and numerical stiffness, while the zero-regularization result is an extrapolation rather than an exact evaluation.
Only the listed parameter sets and this hyperdiffusion scheme were studied. Other regularization methods, additional parameter choices and additional observables remain untested. The supplied analysis reports no statistical uncertainty intervals for the phase boundaries or critical point, and the work does not validate the phase diagram against full-QCD calculations or empirical data.
The work therefore does not establish the phase structure of full QCD, show that negative diffusion persists beyond the Local Potential Approximation or demonstrate that color-superconducting or inhomogeneous phases occur in full QCD. Its practical message is more limited: within this QDM calculation, hyperdiffusion offers one way to obtain a non-oscillatory flow and examine the resulting phase diagram.
Further tests would need to compare hyperdiffusion with other regularization schemes and examine more parameter choices and observables. The analysis also leaves open how the result would transfer to the QMDM, full QCD or calculations of inhomogeneous phases.
The paper’s status
The work is a preprint identified as arXiv:2608.20196v1 and dated 20 August 2026. No journal or peer-review publication is listed in the supplied metadata. The work received partial support from the Deutsche Forschungsgemeinschaft through CRC-TR 211, project number 315477589-TRR 211, and one author received support from the Stiftung Giersch.
Paper data and sources
Original title: Negative diffusion in the Functional Renormalization Group flow for the Quark-Diquark Model
Authors: Johannes Poeplau, Ashutosh Dash, Dirk H. Rischke
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text