arXiv subjects
Publications and source records attributed to D. J. Jeffrey.
We consider a sequence of polynomials appearing in expressions for the derivatives of the Lambert W function. The coefficients of each polynomial are shown to form a positive sequence that is log-concave and unimodal. This property implies that the positive real branch of the Lambert W function is a Bernstein function.
Based on a sequence of numerical computations, a conjecture is presented regarding the class of functions $H(x;a)=\exp(a)-(1+a/x)^x$, and the open problem of determining the values of $a$ for which the functions are completely monotonic with respect to $x$. The critical value of $a$ is determined here to sufficient accuracy to show that it is not a simple symbolic quantity.
We consider dominant 3-, 4-, and 5-loop contributions to $λ$, the quartic scalar coupling-constant's $β$-function in the Standard Model. We find that these terms accelerate the evolution of $λ$ to nonperturbative values, thereby lowering the unification bound for which scalar-couplings are still perturbative. We also find that these higher order contributions imply a substantial lowering of $λ$ itself before the anticipated onset of nonperturbative physics in the Higgs sector.