Persistence of the integer coefficient structure

Determine whether the integer-coefficient structure in the large-d expansion of the correction-to-scaling exponent \(\omega(\varepsilon)\) persists to higher orders, thereby establishing whether it reflects an underlying structure of the stochastic Navier–Stokes turbulence model rather than an accidental pattern.

Background

The paper computes the large-spatial-dimension expansion of the correction-to-scaling exponent through five loops and observes that, after a suitable rescaling, the known coefficients are integers: $1,5,28,170$. The authors note that these coefficients arise from many diagrams containing transcendental constants, making the integrality unexpected.

On the basis of this observation, the paper conjectures that the integrality reflects an underlying structure of the model and continues to higher orders. Identifying whether this conjecture is correct would support or undermine the proposed closed-form continuation of $\omega(\varepsilon)$, which is used to analyze the reality and stability of RG fixed points.