A sketch of the generic singularity structure of g(τ ) The singularities are generically branch points, and the cuts are oriented away from the region τ > 0, where the correlator is analytic. The black dashed contour is the original one in eq. (3.31), while the blue dashed one is shifted to the left in order to leverage the exponential suppression at large Λ. Finally, the red contour is K in eq. (3.34).

Contexts in source publication

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... the other hand, further singularities may appear on the imaginary τ axis. A generic example of the analytic structure in the τ plane is depicted in figure 7. Let us now shift the integration contour in (3.31) to the left, as shown in figure 7. It is clear from eq. (3.31) that the part of the contour which runs in the left half plane is exponentially suppressed at large Λ. ...

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... the other hand, further singularities may appear on the imaginary τ axis. A generic example of the analytic structure in the τ plane is depicted in figure 7. Let us now shift the integration contour in (3.31) to the left, as shown in figure 7. It is clear from eq. (3.31) that the part of the contour which runs in the left half plane is exponentially suppressed at large Λ. ...

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... us now come back to the additional singularities on the Im τ axis in figure 7. Contrary to the singularity at τ = 0, the contribution of each of them is not enhanced by the 1/τ pole in (3.31). ...

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... last comment is in order. Even if the leftover singularities in figure 7 are softer than the one at τ = 0, one could still worry that there is an infinite sum over all of them at τ = it n for real t n . However, this sum is suppressed by the presence of the rapidly oscillating phases e iΛtn . ...

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... for finite Λ the eigenvalues do pick up an imaginary part when the radius is larger than a certain value ¯ R * . For the vacuum energy, this can be seen in figure 17. ...

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... the δ-function in eq. (C.11) has support at most in two points, at fixed θ 1 , see fig. 37. We can then evaluate the integral in dθ 2 by changing variables, being careful about the sign of the Jacobian ∂ξ(τ, θ 1 , θ 2 )/∂θ 2 . Referring again to fig. 37, the geodesic distance of the marked point at angle θ 2 from the point at angle θ 1 is decreasing while the former enters the solid circle, and increasing while it exits it. ...

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... the δ-function in eq. (C.11) has support at most in two points, at fixed θ 1 , see fig. 37. We can then evaluate the integral in dθ 2 by changing variables, being careful about the sign of the Jacobian ∂ξ(τ, θ 1 , θ 2 )/∂θ 2 . Referring again to fig. 37, the geodesic distance of the marked point at angle θ 2 from the point at angle θ 1 is decreasing while the former enters the solid circle, and increasing while it exits it. This fixes the sign of the Jacobian. We obtain the following expression for the ...

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... used the reflection symmetry θ → π − θ to restrict the integration region to θ 1 < π/2. At the minimal angle θ min the solid circle of fig. 37 becomes tangent to the semi-circle of radius 1. As promised, eq. (C.13) expresses the kernel in terms of the incomplete elliptic integral of the third kind: The full matrix element at fixed τ may now be obtained by integrating the kernel as in eq. (C.10). Alternatively, one can recover the small τ limits in eqs. (C.8,C.9) as limits of ...

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... conclude this subsection, let us describe how the introduction of a non vanishing cutoff changes the kernel (C.11). Referring to fig. 37, now the integration region of the potential only extends on the wedge defined by θ ∈ [, π −]. Depending on the values of ξ, τ and , one of three cases happens: none, one or both of the intersections marked in red lie inside this wedge. Hence, the kernel is a piecewise continuous function. We shall focus on the limit , τ → 0, while ...

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... the only non-analyticity along the path is the branch point at t = πn, n ∈ Z. We conclude that the analytic structure of g χ (τ ) is of the form presented in figure 7, with power law monodromies along the imaginary axis, all with the same exponent. ...