Evaluation of a ln tan integral arising in quantum field theory

Coffey, Mark W.

doi:10.1063/1.2981311

Cited by 11 publications

(18 citation statements)

References 15 publications

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“…In the last step we applied the integral representation of Lemma 2(a) of Ref. [7]. By collecting terms, we then have Proposition C1 and by Eq.…”

Section: Concluding Brief Discussionmentioning

confidence: 99%

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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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“…In the last step we applied the integral representation of Lemma 2(a) of Ref. [7]. By collecting terms, we then have Proposition C1 and by Eq.…”

Section: Concluding Brief Discussionmentioning

confidence: 99%

“…The conjectural relations of Broadhurst [4] together with our previous study [7] of a certain ln tan integral of quantum field theory, provide significant portions of our motivation for the present investigation. We are also interested to advance the state of the art in certain areas of integration and special function theory.…”

Section: Introductionmentioning

confidence: 89%

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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“…In the end, these expressions and (9) together give the sought result given by (4). Observe that recently Coffey [19] evaluated differently this integral in terms of Cl 2 (θ). To obtain (5) It is obvious then, that, by making use of this expansion and the above definitions of L d (s) and Cl 2 (θ), (7) and (8), we would get the required formula (5).…”

Section: Formal Proof Of (2)mentioning

confidence: 91%

A dilogarithmic integral arising in quantum field theory

Cvijović¹

2009

Journal of Mathematical Physics

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Recently, an interesting dilogarithmic integral arising in quantum field theory has been closed-form evaluated in terms of the Clausen function $\text{Cl}_2(\theta)$ by Coffey [J. Math. Phys.} 49 (2008), 093508]. It represents the volume of an ideal tetrahedron in hyperbolic space and is involved in two intriguing equivalent conjectures of Borwein and Broadhurst. It is shown here, by simple and direct arguments, that this integral can be expressed by the triplet of the Clausen function values which are involved in one of the two above-mentioned conjectures.Comment: 6 page

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“…In contrast to (49), the expression in [9] for I 7 involves θ + ≡ tan −1 ( √ 7/3). With the various analytic evaluations now known for I 7 or L −7 (2), we have enlarged the set of possible relations amongst Cl 2 values.…”

Section: Bloch-wigner Dilogarithm [17]mentioning

confidence: 99%

“…In regard to the combination on the right side of (3), we comment on an observation given previously [9]. We have that ± sin(2π/7), ± sin(4π/7), and ± sin(6π/7)…”

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confidence: 99%

Alternative Evaluation of a ln tan Integral Arising in Quantum Field Theory

Coffey

2010

Math Phys Anal Geom

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A certain dilogarithmic integral I 7 turns up in a number of contexts including Feynman diagram calculations, volumes of tetrahedra in hyperbolic geometry, knot theory, and conjectured relations in analytic number theory. We provide an alternative explicit evaluation of a parameterized family of integrals containing this particular case. By invoking the Bloch-Wigner form of the dilogarithm function, we produce an equivalent result, giving a third evaluation of I 7 . We also alternatively formulate some conjectures which we pose in terms of values of the specific Clausen function Cl 2 .

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Evaluation of a ln tan integral arising in quantum field theory

Cited by 11 publications

References 15 publications

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

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

A dilogarithmic integral arising in quantum field theory

Alternative Evaluation of a ln tan Integral Arising in Quantum Field Theory

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