A dilogarithmic 3-dimensional ising tetrahedron

Broadhurst, David J.

doi:10.1007/s100520050470

Cited by 4 publications

(10 citation statements)

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“…The final integral needed for the two-loop self energy is 77 The first integral form in (C.69) was derived by Rajantie [32]. The second integral form and the relation to Clausen integrals was "experimentally" deduced (and verified to 1000 place precision) by Broadhurst [33]. This reduction to Clausen integrals has been proven analytically by Almkvist [34].…”

Section: C23 Even Worse Integralsmentioning

confidence: 91%

“…If one is only interested in thermodynamic quantities, or zero frequency correlators, then one may restrict the generalized chemical potentials to be time-independent. 33 Since the extended Hamiltonian of the system including the chemical potential terms is then time independent, the ensemble averages remain time-translationally invariant. Thus n a (r, τ ) β is independent of τ , and it may be replaced by the τ = 0 form n a (r) β .…”

Section: Quantum Theorymentioning

confidence: 99%

“…Alternatively, if one makes a compensating change of variables φ → φ + iλ in the functional integral (3.6), all dependence on λ disappears from the charged field Lagrangian L a , and the net effect is merely to shift the Gaussian measure for the electrostatic potential, 33 We will only need to use time-dependent chemical potentials when discussing the difference between zero-frequency and equal time correlators in section 7. 34 The final form shown for the second variation (3.10) involves an integral over imaginary time of the time-ordered correlation function…”

Section: Quantum Theorymentioning

confidence: 99%

“…These integrals are evaluated in Appendix C.2 (with help from Refs. [32,33]). Once again, the final contribution Γ i involving the single two-loop coupling g 0 ab removes the short-distance singularities in both the three-loop graphs Γ g and Γ h that diverge in three dimensions.…”

Section: Number Density Correlatorsmentioning

confidence: 99%

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Effective field theory for highly ionized plasmas

Brown

Yaffe

2001

Physics Reports

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We examine the equilibrium properties of hot, non-relativistic plasmas. The partition function and density correlation functions of a plasma with several species are expressed in terms of a functional integral over electrostatic potential distributions. This is a convenient formulation for performing a perturbative expansion. The theory is made well-defined at every stage by employing dimensional regularization which, among other virtues, automatically removes the unphysical (infinite) Coulomb self-energy contributions. The leading order, field-theoretic tree approximation automatically includes the effects of Debye screening. No further partial resummations are needed for this effect. Subleading, one-loop corrections are easily evaluated. The two-loop corrections, however, have ultraviolet divergences. These correspond to the short-distance, logarithmic divergence which is encountered in the spatial integral of the Boltzmann exponential when it is expanded to third order in the Coulomb potential. Such divergences do not appear in the underlying quantum theory -they are rendered finite by quantum fluctuations. We show how such divergences may be removed and the correct finite theory obtained by introducing additional local interactions in the manner of modern effective quantum field theories. We compute the two-loop induced coupling by exploiting a non-compact su(1, 1) symmetry of the hydrogen atom. This enables us to obtain explicit results for density-density correlation functions through two-loop order and thermodynamic quantities through three-loop order. The induced couplings are shown to obey renormalization group equations, and these equations are used to characterize all leading logarithmic contributions in the theory. A linear combination of pressure plus energy and number densities is shown to be described by a field-theoretic anomaly. The effective Lagrangian method that we employ yields a simple demonstration that, at long distance, correlation functions have an algebraic fall off (because of quantum effects) rather than the exponential damping of classical Debye screening. We use the effective theory to compute, easily and explicitly, this leading long-distance behavior of density correlation functions.The presentation is pedagogical and self-contained. The results for thermodynamic quantities at three-loop [or O(n 5/2 )] order, and for the leading long-distance forms of correlation functions, agree with previous results in the literature, but they are obtained in a novel and simple fashion using the effective field theory. In Preprint submitted to Elsevier Preprint 2 February 2008addition to the new construction of the effective field theory for plasma physics, we believe that the results we report for the explicit form of correlation functions at two-loop order, as well as the determination of higher-order leading-logarithmic contributions, are also original.

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Section: C23 Even Worse Integralsmentioning

confidence: 91%

Section: Quantum Theorymentioning

confidence: 99%

Section: Quantum Theorymentioning

confidence: 99%

Section: Number Density Correlatorsmentioning

confidence: 99%

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Effective field theory for highly ionized plasmas

Brown

Yaffe

2001

Physics Reports

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“…Even more closely related to our results below, (multiple) polylogarithms and their special values have occurred, among many others, in various ways in work of Broadhurst and Kreimer (e.g., [6]), occasionally even in connection with ladder relations (cf. [4,5]) as defined below. Recently, when calculating the two-loop hexagon Wilson loop in N = 4 supersymmetric Yang-Mills theory, Del Duca, Duhr and Smirnov [10] were led to a long expression in polylogarithms that has been subsequently enormously simplified by Goncharov et al [20] using Goncharov's notion of a symbol attached to a (multiple) polylogarithm (the first combinatorial description being given, under the name m -invariant, in [19], Section 4.4).…”

Section: Motivationmentioning

confidence: 99%

Functional equations and ladders for polylogarithms

Gangl

2013

Communications in Number Theory and Physics

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On a three-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions

Coffey

2008

Journal of Mathematical Physics

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Perturbative quantum field theory for the Ising model at the three-loop level yields a tetrahedral Feynman diagram C(a, b) with masses a and b and four other lines with unit mass. The completely symmetric tetrahedron C Tet ≡ C(1, 1) has been of interest from many points of view, with several representations and conjectures having been given in the literature. We prove a conjectured exponentially fast convergent sum for C(1, 1), as well as a previously empirical relation for C(1, 1) as a remarkable difference of Clausen function values. Our presentation includes Propositions extending the theory of the dilogarithm Li 2 and Clausen Cl 2 functions, as well as their relation to other special functions of mathematical physics. The results strengthen connections between Feynman diagram integrals, volumes in hyperbolic space, number theory, and special functions and numbers, specifically including dilogarithms, Clausen function values, and harmonic numbers.

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A dilogarithmic 3-dimensional ising tetrahedron

Cited by 4 publications

References 21 publications

Effective field theory for highly ionized plasmas

Effective field theory for highly ionized plasmas

Functional equations and ladders for polylogarithms

On a three-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions

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