Split dimensional regularization for the Coulomb gauge at two loops

Abstract We study two-to-two parton scattering amplitudes in the high-energy limit of perturbative QCD by iteratively solving the BFKL equation. This allows us to predict the imaginary part of the amplitude to leading-logarithmic order for arbitrary t-channel colour exchange. The corrections we compute correspond to ladder diagrams with any number of rungs formed between two Reggeized gluons. Our approach exploits a separation of the two-Reggeon wavefunction, performed directly in momentum space, between a soft region and a generic (hard) region. The former component of the wavefunction leads to infrared divergences in the amplitude and is therefore computed in dimensional regularization; the latter is computed directly in two transverse dimensions and is expressed in terms of single-valued harmonic polylogarithms of uniform weight. By combining the two we determine exactly both infrared-divergent and finite contributions to the two-to-two scattering amplitude order-by-order in perturbation theory. We study the result numerically to 13 loops and find that finite corrections to the amplitude have a finite radius of convergence which depends on the colour representation of the t-channel exchange.

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Vacuum Polarization in a Zero-Width Potential: Self-Adjoint Extension

Universe ◽

10.3390/universe7050127 ◽

2021 ◽

Vol 7 (5) ◽

pp. 127

Author(s):

Yuri V. Grats ◽

Pavel Spirin

Keyword(s):

Scalar Field ◽

Vacuum Polarization ◽

Dimensional Regularization ◽

Vacuum Expectation ◽

Vacuum Expectation Value ◽

Field Approximation ◽

Dimensional Euclidean Space ◽

Massless Scalar ◽

Massless Scalar Field ◽

Adjoint Extension

The effects of vacuum polarization associated with a massless scalar field near pointlike source with a zero-range potential in three spatial dimensions are analyzed. The “physical” approach consists in the usage of direct delta-potential as a model of pointlike interaction. We use the Perturbation theory in the Fourier space with dimensional regularization of the momentum integrals. In the weak-field approximation, we compute the effects of interest. The “mathematical” approach implies the self-adjoint extension technique. In the Quantum-Field-Theory framework we consider the massless scalar field in a 3-dimensional Euclidean space with an extracted point. With appropriate boundary conditions it is considered an adequate mathematical model for the description of a pointlike source. We compute the renormalized vacuum expectation value ⟨ϕ2(x)⟩ren of the field square and the renormalized vacuum averaged of the scalar-field’s energy-momentum tensor ⟨Tμν(x)⟩ren. For the physical interpretation of the extension parameter we compare these results with those of perturbative computations. In addition, we present some general formulae for vacuum polarization effects at large distances in the presence of an abstract weak potential with finite-sized compact support.

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The Non-Relativistic Many-Body Quantum-Mechanical Hamiltonian with Diamagnetic Current-Current Interaction

International Journal of Theoretical Physics ◽

10.1007/s10773-021-04842-9 ◽

2021 ◽

Author(s):

Ladislaus Alexander Bányai

Keyword(s):

Solid State ◽

Analogous Result ◽

Charged Particles ◽

Coulomb Gauge ◽

Quantum Mechanical ◽

Photon Propagator ◽

Coulomb Interactions ◽

Many Body ◽

Transverse Current ◽

Current Interaction

AbstractWe extend the standard solid-state quantum mechanical Hamiltonian containing only Coulomb interactions between the charged particles by inclusion of the (transverse) current-current diamagnetic interaction starting from the non-relativistic QED restricted to the states without photons and neglecting the retardation in the photon propagator. This derivation is supplemented with a derivation of an analogous result along the non-rigorous old classical Darwin-Landau-Lifshitz argumentation within the physical Coulomb gauge.

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