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While Pad\'e approximation is a general method for improving convergence ofseries expansions, Gell-Mann--Low renormalization group normally relies on thepresence of special symmetries. We show that in the single-variable case, thelatter becomes an integral Hermite-Pad\'e approximation, needing no specialsymmetries. It is especially useful for interpolating between expansions forsmall values of a variable and a scaling law of known exponent for largevalues. As an example, we extract the scaling-law prefactor for the one-bodydensity matrix of the Lieb-Liniger gas. Using a new result for the 4th-orderterm in the short-distance expansion, we find a remarkable agreement with knownab initio numerical results.

A Hermite-Pad\'e perspective on Gell-Mann--Low renormalization group: an application to the correlation function of Lieb-Liniger gas