Two-loop self-dual Euler-Heisenberg Lagrangians (I): Real part and helicity amplitudes

The fermion determinant in an instanton background for a quark field of arbitrary mass is studied using the Schwinger proper-time representation with WKB scattering phase shifts for the relevant partial-wave differential operators. Previously, results have been obtained only for the extreme small and large quark mass limits, not for intermediate interpolating mass values. We show that consistent renormalization and large-mass asymptotics requires up to third-order in the WKB approximation. This procedure leads to an almost analytic answer, requiring only modest numerical approximation, and yields excellent agreement with the well-known extreme small and large mass limits. We estimate that it differs from the exact answer by no more than 6% for generic mass values. In the philosophy of the derivative expansion the same amplitude is then studied using a Heisenberg-Euler-type effective action, and the leading order approximation gives a surprisingly accurate answer for all masses.

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“…(2.11) in [23]). Substituting the instanton form we obtain the derivative expansion (DE) approximation (here we set ρ = 1, as before)…”

Section: Field-theoretic Derivative Expansion Approachmentioning

confidence: 96%

Instanton determinant with arbitrary quark mass: WKB phase-shift method and derivative expansion

2004

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“…This function plays an important role in the quantum field theory computations in [15,16]. From the asymptotic (large x ) expansion of the digamma function [1] it follows that…”

Section: Generating Function Proof Of Miki's Identitymentioning

confidence: 99%

Bernoulli number identities from quantum field theory and topological string theory

Dunne

Schubert

2013

Communications in Number Theory and Physics

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We present a new method for the derivation of convolution identities for finite sums of products of Bernoulli numbers. Our approach is motivated by the role of these identities in quantum field theory and string theory. We first show that the Miki identity and the Faber-Pandharipande-Zagier (FPZ) identity are closely related, and give simple unified proofs which naturally yield a new Bernoulli number convolution identity. We then generalize each of these three identities into new families of convolution identities depending on a continuous parameter. We rederive a cubic generalization of Miki's identity due to Gessel and obtain a new similar identity generalizing the FPZ identity. The generalization of the method to the derivation of convolution identities of arbitrary order is outlined. We also describe an extension to identities which relate convolutions of Euler and Bernoulli numbers.

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“…It is also well-known that many remarkable simplifications occur for such helicity amplitudes [28,29]. Recently it has been found that analogous simplifications occur in the two-loop effective Lagrangian itself [18,19,20]. At one-loop, the on-shell renormalized effective Lagrangians for a constant self-dual background can be deduced from the results of Euler and Heisenberg [22] and Schwinger [23]:…”

Section: Strong-field Limits and Beta Functions A General Argumentmentioning

confidence: 99%

“…At two-loop, the on-shell renormalized effective Lagrangians for a constant self-dual background can be expressed in closed-form [18,19,20] in terms of the digamma function,…”

Section: Strong-field Limits and Beta Functions A General Argumentmentioning

confidence: 99%

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Zero modes, beta functions and IR/UV interplay in higher-loop QED

Dunne

Gies

Schubert

2002

J. High Energy Phys.

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We analyze the relation between the short-distance behavior of quantum field theory and the strong-field limit of the background field formalism, for QED effective Lagrangians in self-dual backgrounds, at both one and two loop. The self-duality of the background leads to zero modes in the case of spinor QED, and these zero modes must be taken into account before comparing the perturbative β function coefficients and the coefficients of the strong-field limit of the effective Lagrangian. At one-loop this is familiar from instanton physics, but we find that at two-loop the role of the zero modes, and the interplay between IR and UV effects in the renormalization, is quite different. Our analysis is motivated in part by the remarkable simplicity of the two-loop QED effective Lagrangians for a self-dual constant background, and we also present here a new independent derivation of these two-loop results.

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Two-loop self-dual Euler-Heisenberg Lagrangians (I): Real part and helicity amplitudes

Cited by 64 publications

References 71 publications

Instanton determinant with arbitrary quark mass: WKB phase-shift method and derivative expansion

Instanton determinant with arbitrary quark mass: WKB phase-shift method and derivative expansion

Bernoulli number identities from quantum field theory and topological string theory

Zero modes, beta functions and IR/UV interplay in higher-loop QED

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