The Fundaments of Unity: $\mathcal{O}(1)$ Couplings in Quantum Field Theories

Abstract

We critically examine the expectation that in a fundamental quantum field theory, dimensionless couplings in the Lagrangian density should all be of order unity. We propose a measure to quantify the adherence of a theory to this: the spread (the ratio of the largest to the smallest of the magnitudes) of such dimensionless couplings, obtaining various closed-form results. If we take independent and identically distributed (IID) couplings to parameterise our uncertainty on the values of the order-unity couplings, the spread can be much larger than one might naively expect. For a theory with 20 IID unit normal couplings, the probability that the spread is greater than 100 is 0.29, for example. Even when the IID couplings have exponentially suppressed tails, the distribution of the spread has fat power-law tails whose amplitude grows with the number of independent couplings.

Comments: 16 pages, 3 figures, v2 - minor revisions, v3 - version accepted for publication