On the analytic computation of massless propagators in dimensional regularization

We comment on the algorithm to compute periods using hyperlogarithms, applied to massless Feynman integrals in the parametric representation. Explicitly, we give results for all three-loop propagators with arbitrary insertions including order ε⁴${\epsilon }^4$ and show examples at four and more loops.     

Non-cocyclic anomaly in the Gauss law commutator

The commutator of the Gauss law operator in chiral gauge theories with background gauge fields is calculated algebraically: the anomalous piece is not a cocycle.     

Quantum relaxation and the fundamental commutator

A simple consistency calculation based on the operator Langevin equation for a quantum oscillator coupled to an infinite thermal string yields the required exact result [x(t), p(t)]=iℏfor all t>0 given this value at t=0, contrary to recent suggestions.     

Gauss law commutator in the chirally gauged Wess-Zumino-Witten model

We evaluate the commutator of the Gauss law constraints starting from the chirally gauged Wess-Zumino-Witten action. The calculations are done at tree level, i.e. by evaluating corresponding Poisson brackets. The results are compared with commutators obtained by others directly from the gauged fermionic theory, and with Faddeev's results based on cohomology.     

Uncontrollable multichannel commutator in the output cascade of the SNOP-3 generator

A multichannel uncontrollable submersible commutator is described. The commutator is tested in the output cascade of the SNOP-3 generator in the discharge of a signal-forming line with a wave resistance of 0.65 Ω to a transmission line with the same wave resistance and an inductive load of 30 nH. The inductance and ohmic resistance of the commutator are determined (15–20 nH and 0.2 Ω, respectively). With the inductive load of 30 nH, a rate of current increase of 3·10¹³ A/sec is obtained. A special commutator design with a screened grounded plane is tested; this design permits reduction in the prepulse voltage at the load to 0.25% of the voltage at the signal-forming line. Institute of High-Power Electronics, Siberian Branch, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 67–72, December, 1995.     

On the equal-time commutator in the Lee model including the unstableV particle

A number of statements about properties relevant to the real physical theories, illustrated by the Lee model in a previous paper, have been completed and generalised to the case with the unstableV particle. It has been shown that for a general class of convergent theories the equal-time commutator of “charge” densities preserves the canonical value.     

Dimensional regularization in the Wess-Zumino-Witten model

A method is described for the dimensional regularization and renormalization of a two-dimensional non-linear sigma model with torsion, the Wess-Zumino-Witten model. The technique solves the problems generated by the presence of the antisymmetric tensor e^uv. The two-loop calculation agrees with the non-perturbative result.     

Ambiguities in the infrared regularization of QCD

The next-to-leading corrections of leading-log asymptotic freedom are determined in the known infrared (IR) regularization schemes. On- and off-shell calculation leads to different answers; the n-dimensional regularization of IR and mass singularities agrees with the on-shell results. The non-log corrections in Drell-Yan processes are important for Q² ? 10³ GeV².     

Hadron masses and the sigma commutator in light of chiral perturbation theory

We analyze carefully the impact of non-analytic chiral corrections to the mass spectrum of the pseudoscalar meson octet J^P = 0^? and the baryon octet $\text{J}{}^{\mathrm{<mtext>P</mtext>}}\text{=}\text{1}\text{2}{}^{\mathrm{+}}$. We find that the quark mass ratios must lie in the range $\text{21 ≤}\text{m}{}_{\mathrm{<mtext>s</mtext>}}\text{m}\text{?}\text{≤ 32}$ and $\text{1.6 ≤}\text{m}{}_{\mathrm{<mtext>d</mtext>}}\text{m}{}_{\mathrm{<mtext>u</mtext>}}\text{≤ 2.2}$. We also calculate the analogous corrections to the pion-nucleon sigma commutator σ_πN. It turns out that the value σ_πN = 60 MeV is not compatible with the structure of the meson and baryon spectrum, unless the nucleon mass is smaller than 600 MeV in the chiral limit m_u = m_d = m_s = 0.     

A causal approximation scheme for the commutator of the axial vector-currents

An approximation scheme which fulfils the causality condition is used to compute the matrix elements of the commutator of two axial-vector currents between one proton state. The commutator sum rule is supposed to be saturated by taking into account the nucleon, theN ^*(3/2, 3/2) resonance, and the partially disconnected pion-proton intermediate states. For the axial current form factors pion pole formulas are assumed. It is shown that only the subtraction constants are relevant for the equal time limit of the commutator and so the equivalence with similar earlier methods is proven.     

Low-energy kaon nucleon scattering and the magnitude of the sigma commutator

A direct expression between low-energy parameters and scattering lengths is presented for kaon nucleon scattering. As an application, the σ-term is calculated and agreement with various dispersive approaches is obtained.     

Point-splitting analysis of commutator anomalies in non-abelian chiral gauge theories

A gauge covariant point-splitting regularisation is employed to calculate different anomalous commutators in four dimensional chiral gauge theories. For an external gauge field the fixed time anomalous commutator of the gauge group generators is seen to violate the Jacobi identity. The cohomological prediction can be confirmed provided the electric fields do not commute. Other commutators like the current-current and current-electric field are consistent with the Bjorken-Johnson-Low (BJL) derivation.     

Infrared singularities in the dimensional regularization scheme

Infrared singularities arising in some renormalized amplitudes of quantum electrodynamics are analyzed using the dimensional regularization method. We define infrared and ultraviolet convergent regions in the ν complex plane (ν is the number of dimensions of space time). It turns out that these regions do not overlap for quantum electrodynamics. Nevertheless, it is shown that there exists a unique analytic continuation from the infrared convergent region which allows us to interpret the infrared divergence in the renormalized electron self-energy amplitude as an isolated singularity at ν = 4. This statement seems to be true at all orders of perturbation theory. We also prove that the double limit μ → 0, ν → 4 (μ is the auxiliary photon mass) does not exist in quantum electrodynamics and we conjecture that this lack of uniformity provides theoretical support for the ansatz of Marciano and Sirlin.     

Conformal regularization of the Kepler Problem

The manifoldM of null rays through the origin of ^2,n+1 is diffeomorphic toS ¹×S ⁿ , and it is a homogeneous space of SO(2,n+1). This group therefore acts onT*M, which we show to be the generating manifold of the extended phase space of the regularized Kepler Problem. A local canonical chart inT*M is found such that the restriction to the subbundle of the null nonvanishing covectors is given byp ₀+H(q,p)=0, whereH(q,p) is the Hamiltonian of the Kepler Problem. By means of this construction, we get some results that clarify and complete the previous approaches to the problem.     

On the regularization of the Kepler problem

An intimate relationship between Moser's regularization [1] and the KS-regularization [3] of the 3-dimensional Kepler problem is established. Explicit formulae linking Moser's and the KS-transformation are obtained in the case of negative as well as in the case of positive energies. As a side result it is shown that the KS-transformation owes its existence to the local isomorphism of SO(2,4) and SU(2,2).     

On the inversion of diffusion NMR data: Tikhonov regularization and optimal choice of the regularization parameter

The analysis of diffusion NMR data in terms of distributions of diffusion coefficients is hampered by the ill-posed nature of the required inverse Laplace transformation. Na?ve approaches such as multiexponential fitting or standard least-squares algorithms are numerically unstable and often fail. This paper updates the CONTIN approach of the application of Tikhonov regularization to stabilise this numerical inversion problem and demonstrates two methods for automatically choosing the optimal value of the regularization parameter. These approaches are computationally efficient and easy to implement using standard matrix algebra techniques. Example analyses are presenting using both synthetic data and experimental results of diffusion NMR studies on the azo-dye sunset yellow and some polymer molecular weight reference standards.     

Dimensional regularization in massless QED

Using dimensional regularization, the cross section for scattering of massless electrons in an external potential is examined to order α³. The masses of the photon and the electron are put equal to zero from the beginning, and the continuous number of dimensions, n, serves as a regularizing parameter for all ultraviolet divergences as well as for those associated with the two kinds of massless particles. The result is a double- and single-pole structure around n = 4 for the separate terms in the cross section. These poles are found to cancel provided the hard bremsstrahlung is also taken into account.