Dirac delta regularization

I present a finite result for the Dirac delta "function."

A finite result for the set of associated prime ideals of formal local cohomology modules

Let [Formula: see text] be a Noetherian local ring, [Formula: see text] two ideals of [Formula: see text], and [Formula: see text] two finitely generated [Formula: see text]-modules. It is first shown that [Formula: see text] is a finite set. We also prove that except the maximal ideal [Formula: see text], the set [Formula: see text] is stable for large [Formula: see text], where we use [Formula: see text] to denote [Formula: see text]-module [Formula: see text] or [Formula: see text] and [Formula: see text] is the eventual value of [Formula: see text].

H → γγ, gauge invariance, and the hierarchy problem

The calculation of [Formula: see text] displays interesting behavior which depends on the regulator used in the integration over loop momenta. If calculated using a gauge-invariant regulator, such as dimensional regularization, the calculation yields a unique, finite, gauge-invariant result. If four-dimensional symmetric regulation is used without finite subtractions, additional pieces occur which spoil QED gauge invariance. In both cases, a finite result is obtained, but the particular finite result depends on the regulator utilized in the calculation. While gauge-invariant regulators such as dimensional regularization are normally used, four-dimensional symmetric integration is also physically motivated. Also, the gauge-invariance-violating terms that arise using four-dimensional symmetric integration are of the same form for the fermionic, scalar, and the SM [Formula: see text] loop calculated in renormalizable gauge. This presents an interesting possibility. Inspired by anomaly cancellation, we ask if it is possible that these gauge-invariance-violating terms may cancel in certain models when contributions from all diagrams are included. Here, we calculate the regulator-dependent contributions to [Formula: see text] arising from generic fermion and scalar loops, as well as the Standard Model [Formula: see text] loop contribution, which we evaluate in renormalizable gauge for general [Formula: see text]. We find that a cancellation between such terms is possible, and derive the cancellation condition. Additionally, we find that this cancellation condition ensures QED gauge invariance without finite subtractions for any regulator used, not just for four-dimensional symmetric integration. We additionally relate the regulator-dependent terms in [Formula: see text] to the behavior of quadratically-divergent Higgs tadpole diagrams under shifts of internal loop momentum. Thus, the cancellation condition for the gauge-invariance-violating terms in [Formula: see text] implies a relation between the quadratic divergences in Higgs tadpole diagrams; this has consequences for hypothesized solutions to the hierarchy problem. Lastly, we find that the MSSM obeys our cancellation condition.

Review on loop integrals which need regularization but yield finite results

In this pedagogical paper I will discuss one-loop integrals, where (i) different regions of the integration region lead to divergences and (ii) where these divergences cancel in the sum over all regions. These integrals cannot be calculated without regularization, in spite of the fact that they yield a finite result. A typical example where such integrals occur is the decay H →γγ.

FINITE GLUON FUSION AMPLITUDE IN THE GAUGE-HIGGS UNIFICATION

We show that the gluon fusion amplitude in the gauge-Higgs unification scenario is finite in any dimension regardless of its nonrenormalizability. This result is supported by the fact that the local operator describing the gluon fusion process is forbidden by the higher dimensional gauge invariance. We explicitly calculate the gluon fusion amplitude in an arbitrary dimensional gauge-Higgs unification model and indeed obtain the finite result.

BLACK HOLE ENTROPY FOR ANTI DE SITTER AND DE SITTER DILATONIC SPACETIME USING BRICK WALL METHOD

The entropy of a scalar field in the background of a dilatonic black hole both in anti de Sitter and de Sitter spacetime has been calculated using brick wall method. The form of divergent contributions to the entropy is in agreement with the previous calculations in the literature. The semiclassical approach used here is straightforward and produces finite result apart from an ambiguity in logarithmic terms.

Principal-value integrals – Revisited

The principal-value (PV) integral has proved a useful tool in many fields of physics. The PV is a specific method for obtaining a finite result for an improper integral. When the integration passes through a simple pole, one speaks of a "first-order" PV. In this paper, we examine first-order PV integrals and analyze several of their important properties. First, we discuss how the PV agrees with one's naïve expectation about these integrals. Next, we show that the basic definition of the first-order PV gives a generalized formula for the complex-variable PV expression. Finally, we show the correspondence between the finite-limit PV integral of x–1 along the real axis and the path integral of z–1 (where z = x + iy) in the complex plane.PACS Nos.: 02.90.+p, 05.90.+m

CONNECTION BETWEEN ζ AND CUTOFF REGULARIZATIONS OF CASIMIR ENERGIES

We study the connection between ζ- and cutoff-regularized Casimir energies for scalar fields. We show that, in general, both regularization schemes lead to divergent contributions, and to minimal finite parts which do not coincide. We determine the relationships among the various coefficients appearing in one approach and the other. We discuss the agreement with our predictions in the case of scalar fields in d-dimensional boxes under periodic boundary conditions. Finally, we apply our results to massless scalar fields in balls, an example where ambiguities remain under the physical prescriptions usually imposed to extract a finite result.

Friedel Sum Rule: Partition Function Approach

The Friedel sum rule has been derived using a grand canonical ensemble of electrons in the scattering states. The method, while illustrating the use of the statistical method of the grand canonical ensemble, clarifies the exact nature of the constraint which gives rise to this sum rule, viz. the charge neutrality of the entire crystal. An extra term has been found which cancels the contribution to the Friedel sum from the phase shifts at k = 0. This cancellation term is shown to exist in the conventional derivation if done more carefully, thus emphasizing the care needed when a finite result is sought by subtraction from two rather infinite terms. An expression for the excess charge density valid within the range of the scattering potential has been derived.