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Register a new userA Fast Way to Compute Functional Determinants of Radially Symmetric Partial Differential Operators in General Dimensions
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Recently the partial wave cutoff method was developed as a new calculational scheme for a functional determinant of quantum field theory in radial backgrounds. For the contribution given by an infinite sum of large partial waves, we derive explicitly radial WKB series in the angular momentum cutoff for $d=2,3,4$ and 5 ($d$ is the spacetime dimension), which has uniform validity irrespectively of any specific values assumed for other parameters. Utilizing this series, precision evaluation of the renormalized functional determinant is possible with a relatively small number of low partial wave contributions determined separately. We illustrate the power of this scheme in numerically exact evaluation of the prefactor (expressed as a functional determinant) in the case of the false vacuum decay of 4D scalar field theory.
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Let s 1 ,. .. , s k be the elementary symmetric functions of the complex variables x 1 ,. .. , x k. We say that F $in$ C[s 1 ,. .. , s k ] is a trace function if their exists f $in$ C[z] such that F (s 1 ,. .. , s k ] = k j=1 f (x j) for all s $in$ C k. We give an explicit finite family of second order differential operators in the Weyl algebra W 2 := C[s 1 ,. .. , s k ] $partial$ $partial$s 1 ,. .. , $partial$ $partial$s k which generates the left ideal in W 2 of partial differential operators killing all trace functions. The proof uses a theorem for symmetric differential operators analogous to the usual symmetric functions theorem and the corresponding map for symbols. As a corollary, we obtain for each integer k a holonomic system which is a quotient of W 2 by an explicit left ideal whose local solutions are linear combinations of the branches of the multivalued root of the universal equation of degree k: z k + k h=1 (--1) h .s h .z k--h = 0.
A class of noncanonical effective potentials is introduced allowing stable, radially symmetric, solutions to first order Bogomolnyi equations for a real scalar field in a fixed spacetime background. This class of effective potentials generalizes those found previously by Bazeia, Menezes, and Menezes [Phys.Rev.Lett. 91 (2003) 241601] for radially symmetric defects in a flat spacetime. Use is made of the on-shell method introduced by Atmaja and Ramadhan [Phys.Rev.D 90 (2014) 10, 105009] of reducing the second order equation of motion to a first order one, along with a constraint equation. This method and class of potentials admits radially symmetric, stable solutions for four dimensional static, radially symmetric spacetimes. Stability against radial fluctuations is established with a modified version of Derricks theorem, along with demonstrating that the radial stress vanishes. Several examples of scalar field configurations are given.
We study four-point correlation functions of half-BPS operators of arbitrary weight for all dimensions d=3,4,5,6 where superconformal theories exist. Using harmonic superspace techniques, we derive the superconformal Ward identities for these correlators and present them in a universal form. We then solve these identities, employing Jack polynomial expansions. We show that the general solution is parameterized by a set of arbitrary two-variable functions, with the exception of the case d=4, where in addition functions of a single variable appear. We also discuss the operator product expansion using recent results on conformal partial wave amplitudes in arbitrary dimension.
We prove that $p$-determinants of a certain class of differential operators can be lifted to power series over $mathbb{Q}$. We compute these power series in terms of monodromy of the corresponding differential operators.
We present H1jet, a fast code that computes the total cross section and differential distribution in the transverse momentum of a colour singlet. In its current version, the program implements only leading-order $2to 1$ and $2to 2$ processes, but could be extended to higher orders. We discuss the processes implemented in H1jet, give detailed instructions on how to implement new processes, and perform comparisons to existing codes. This tool, mainly designed for theorists, can be fruitfully used to assess deviations of selected new physics models from the Standard Model behaviour, as well as to quickly obtain distributions of relevance for Standard Model phenomenology.
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