Abstract
We compute the lowest operator dimension ∆(J; D) at large global charge J in the O(2) Wilson-Fisher model in D = 4 − ϵ dimensions, to leading order in both 1/J and ϵ. While the effective field theory approach of [1] could only determine ∆(J; 3) as a series expansion in 1/J up to an undetermined constant in front of each term, this time we try to determine the coefficient in front of J3/2 in the ϵ-expansion. The final result for ∆(J; D) in the (resummed) ϵ-expansion, valid when J ≫ 1/ϵ ≫ 1, turns out to be
where next-to-leading order onwards were not computed here due to technical cumbersomeness, despite there are no fundamental difficulties. We also compare the result at ϵ = 1,
to the actual data from the Monte-Carlo simulation in three dimensions [2], and the discrepancy of the coefficient 0.293 from the numerics turned out to be 13%. Additionally, we also find a crossover of ∆(J; D) from ∆(J) ∝ \( {J}^{\frac{D}{D-1}} \) to ∆(J) ∝ J, at around J ∼ 1/ϵ, as one decreases J while fixing ϵ (or vice versa), reflecting the fact that there are no interacting fixed-point at ϵ = 0. Based on this behaviour, we propose an interesting double-scaling limit which fixes λ ≡ Jϵ, suitable for probing the region of the crossover. I will give ∆(J; D) to next-to-leading order in perturbation theory, either in 1/λ or in λ, valid when λ ≫ 1 and λ ≪ 1, respectively.
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Watanabe, M. Accessing large global charge via the ϵ-expansion. J. High Energ. Phys. 2021, 264 (2021). https://doi.org/10.1007/JHEP04(2021)264
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DOI: https://doi.org/10.1007/JHEP04(2021)264