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Discrete Laplacian structure and kernel reduction of the gap equation in odd-dimensional large-N vector models at imaginary chemical potential

Year: 2026
Type: article
Abstract: We identify a mathematical structure in the gap equations of the large- N Gross–Neveu model at imaginary chemical potential in odd spacetime dimensions. Using this model as the primary example, the central observation is an elementary dimensional ladder: a single centred first difference with respect to $$\sigma ^2$$ σ 2 relates the thermal part of the gap equation in dimension d to that in dimension $$d-2$$ d - 2 , for every odd d . Iterating this elementary step generates the higher-order finite-difference (discrete Laplacian) relations; for the distinguished sequence $$d=4k+3$$ d = 4 k + 3 , whose kernel exponent is even, the closed stencil hierarchy connects d to $$2d-3$$ 2 d - 3 through 2 k iterates. We show that the gap equations can be written as the sum of two parts: one defined by higher-order finite-difference patterns and a cutoff-dependent part given by a truncated asymptotic expansion of a hypergeometric function. The same discrete finite-difference hierarchy is shown to arise in the thermal sector of large- N bosonic models, indicating that the underlying structure is not restricted to fermionic theories. We further argue that this picture suggests a relation between thermal field theories in these odd d dimensions and exactly solvable one-dimensional quantum problems. We find that the thermal mass at specific imaginary chemical potential values corresponds to points where internal-energy and entropy contributions balance, which is formally analogous to OPE inversion formula techniques where thermal mass values arise from sets of transcendental equations.
DOI: 10.1140/epjc/s10052-026-16430-4
Language: English
Open Access status: diamond
OpenAlex ID: W7223142571
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Cites: 31
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Related to: 10