EFT asymptotics: the growth of operator degeneracy

Melia, Tom; Pal, Sridip

doi:10.21468/SciPostPhys.10.5.104

SciPost Physics

EFT asymptotics: the growth of operator degeneracy

Tom Melia, Sridip Pal

SciPost Phys. 10, 104 (2021) · published 14 May 2021

doi: 10.21468/SciPostPhys.10.5.104
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Abstract

We establish formulae for the asymptotic growth (with respect to the scaling dimension) of the number of operators in effective field theory, or equivalently the number of $S$-matrix elements, in arbitrary spacetime dimensions and with generic field content. This we achieve by generalising a theorem due to Meinardus and applying it to Hilbert series -- partition functions for the degeneracy of (subsets of) operators. Although our formulae are asymptotic, numerical experiments reveal remarkable agreement with exact results at very low orders in the EFT expansion, including for complicated phenomenological theories such as the standard model EFT. Our methods also reveal phase transition-like behaviour in Hilbert series. We discuss prospects for tightening the bounds and providing rigorous errors to the growth of operator degeneracy, and of extending the analytic study and utility of Hilbert series to EFT.

TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.10.5.104
TI  - EFT asymptotics: the growth of operator degeneracy
PY  - 2021/05/14
UR  - https://www.scipost.org/SciPostPhys.10.5.104
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 10
IS  - 5
SP  - 104
A1  - Melia, Tom
AU  - Pal, Sridip
AB  - We establish formulae for the asymptotic growth (with respect to the scaling dimension) of the number of operators in effective field theory, or equivalently the number of $S$-matrix elements, in arbitrary spacetime dimensions and with generic field content. This we achieve by generalising a theorem due to Meinardus and applying it to Hilbert series -- partition functions for the degeneracy of (subsets of) operators. Although our formulae are asymptotic, numerical experiments reveal remarkable agreement with exact results at very low orders in the EFT expansion, including for complicated phenomenological theories such as the standard model EFT. Our methods also reveal phase transition-like behaviour in Hilbert series. We discuss prospects for tightening the bounds and providing rigorous errors to the growth of operator degeneracy, and of extending the analytic study and utility of Hilbert series to EFT.
ER  -

@Article{10.21468/SciPostPhys.10.5.104,
	title={{EFT asymptotics: the growth of operator degeneracy}},
	author={Tom Melia and Sridip Pal},
	journal={SciPost Phys.},
	volume={10},
	pages={104},
	year={2021},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.10.5.104},
	url={https://scipost.org/10.21468/SciPostPhys.10.5.104},
}

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¹ ² Tom Melia,
³ Sridip Pal

Funders for the research work leading to this publication