Stokes's theorems for non-Abelian fields

Fishbane, Paul M.; Gasiorowicz, S.; Kaus, Peter

doi:10.1103/physrevd.24.2324

Cited by 62 publications

(74 citation statements)

References 9 publications

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“…Equation (5.1) does not depend on the choice of the position A, since cyclicity of the trace allows to change the reference point by substituting U Ay → U BA U Ay . The area ordering P area is defined by disecting the area enclosed by C in many (ultimately: infinitessimally small) areas in such an order that the parallel transporters C i along the infinitessimal areas combine to C. As a consequence of the non-Abelian Gauss theorem, (5.1) turns out to be independent of the orientation of the surface dσ µν (y) [ 36]. The total photoabsorption cross section (4.1) is closely related to the Stokes's theorem, as can be seen from the representation…”

Section: Photodissociation Via Non-abelian Stokes's Theoremmentioning

confidence: 99%

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Transverse dynamics of hard partons in nuclear media and the QCD dipole

Wiedemann

2000

Nuclear Physics B

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We derive the non-abelian generalization of the Furry approximation which describes the transverse dynamical evolution of a hard projectile parton inside a spatially extended colour target field. This provides a unified starting point for the target rest frame description of the nuclear dependence of a large class of observables. For the case of the virtual γ * → qq photoabsorption cross section, we investigate then in detail under which conditions the nuclear dependence encoded in the Furry wavefunctions can be parametrized by a qq QCD dipole cross section. The important condition is colour triviality, i.e., the property that for arbitrary N -fold rescattering contributions the only non-vanishing colour trace is Nc C N F . We give proofs for the colour triviality of the inelastic, diffractive and total photoabsorption cross section measured inclusively or with one jet resolved in the final state. Also, we list examples for which colour interference effects remain. Colour triviality allows us to write the γ * → qq contribution to the DIS nuclear structure function F2 for small Bjorken xBj in terms of a path integral which describes the transverse size evolution of the qq pair in the nuclear colour field. This expression reduces in an opacity expansion to the N = 1 result of Nikolaev and Zakharov, and in the eikonal approximation to the Glauber-type rescattering formulas first derived by Mueller. In the harmonic oscillator approximation of the path integral, we quantify deviations from the eikonal limit. Their onset is characterized by the scales L/l f and E tot ⊥ L which relate the longitudinal extension L of the nuclear target to the coherence length l f and the total transverse energy E tot ⊥ accumulated by the q-q-pair.

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Section: Photodissociation Via Non-abelian Stokes's Theoremmentioning

confidence: 99%

“…This relates the integral of the non-abelian vector potential A µ along a closed contour C to the area integral of the field strength tensor F µν , [ 34,35,36] Tr P exp…”

Section: Photodissociation Via Non-abelian Stokes's Theoremmentioning

confidence: 99%

Transverse dynamics of hard partons in nuclear media and the QCD dipole

Wiedemann

2000

Nuclear Physics B

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“…In ref. [4,18] one can find an alternative (generalizable to arbitrary dimensions and gauge groups) way to prove the equivalence between the Wilson and the standard formulation of the action through a non-Abelian formulation of the Stokes theorem.…”

Section: The Wilson Gauge Actionmentioning

confidence: 99%

“…Asymptotic stationary means that 18) where Π eq (q) is the unique equilibrium distribution ( Π eq (q) ∝ ξ 0 (q)). The consequence of this property is that with an algorithm that simulates an ergodic Markov process it is possible to sample configurations from a unique equilibrium distribution starting from an arbitrary distribution Π(q).…”

Section: Markov Processesmentioning

confidence: 99%

Critical slowing down and error analysis in lattice QCD simulations

2011

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, where a is the lattice spacing. In unquenched simulations with O(a) improved Wilson fermions we do not obtain a scaling law but find results compatible with the behavior that we find in the pure gauge theory. The discussion is supported by a large set of ensembles both in pure gauge and in the theory with two degenerate sea quarks. We have moreover investigated the effect of slow algorithmic modes in the error analysis of the expectation value of typical lattice QCD observables (hadronic matrix elements and masses). In the context of simulations affected by slow modes we propose and test a method to obtain reliable estimates of statistical errors. The method is supposed to help in the typical algorithmic setup of lattice QCD, namely when the total statistics collected is of O(10)τ exp . This is the typical case when simulating close to the continuum limit where the computational costs for producing two independent data points can be extremely large. We finally discuss the scale setting in N f = 2 simulations using the Kaon decay constant f K as physical input. The method is explained together with a thorough discussion of the error analysis employed. A description of the publicly available code used for the error analysis is included.ii ZusammenfassungIn dieser Arbeit untersuchen wir das Critical Slowing down der Gitter-QCD Simulationen. Wir führen eine Vorstudie in der quenched Approximation durch, in der wir feststellen, dass unsere Schätzung der exponentiellen Autokorrelation wie iii

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“…One can expandÛ(γ R×dτ ) in powers of loop area using the non-Abelian Stokes theorem [25,26], which shows that the first nonvanishing term in 1 − ReW k (γ R×dτ ) is proportional to the square of the loop area, therefore the action differential is of order dτ , and the integral in (24) is mathematically well-defined.…”

Section: Path Integral Solution and Two-dimensional Yang-mills Theorymentioning

confidence: 99%

Asymptotic behavior of Wilson loops from Schrödinger equation on the gauge group

Buividovich

Kuvshinov

2006

Physics Letters B

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Probability distribution of non-Abelian parallel transporters on the group manifold and the corresponding amplitude are investigated for quantum Yang-Mills fields. It is shown that when the Wilson area law and the Casimir scaling hold for the quantum gauge field, this amplitude can be obtained as the solution of the free Schrödinger equation on the gauge group. Solution of this equation is written in terms of the path integral and the corresponding action term is interpreted geometrically. We also note that the partition function of two-dimensional pure Yang-Mills theory living on the surface spanned on the loop solves the obtained equation.

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Stokes's theorems for non-Abelian fields

Cited by 62 publications

References 9 publications

Transverse dynamics of hard partons in nuclear media and the QCD dipole

Transverse dynamics of hard partons in nuclear media and the QCD dipole

Critical slowing down and error analysis in lattice QCD simulations

Asymptotic behavior of Wilson loops from Schrödinger equation on the gauge group

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