Abstract
Here we take the point of view of calculating perturbatively in an EFT the discontinuity of a PWA along the LHC, that we denoted above as \(\varDelta (p^2)\). Once \(\varDelta (p^2)\) is approximated in this way one can then solve the IE that follows from the N / D method in order to calculate D(s) along the LHC, and then the full T(s).
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- 1.
By this we mean that
$$\begin{aligned} \int _0^1dy \int _0^1 dx\, K(x, y)^2<\infty ~,\\ \int _0^1 dy\, f(y)^2 <\infty ~.\nonumber \end{aligned}$$(11.17) - 2.
The only exception might be if the factor \(\frac{\lambda m}{4\pi ^2}(-L)^{\gamma + \frac{1}{2}}\) multiplying the integral in Eq. (11.15) coincided with an eigenvalue of the kernel K(y, x). Nonetheless, since \(\lambda \) is continuous and the eigenvalues of a kernel are discrete we could employ a smooth continuation to find the solution in the a priori unlikely case of such coincidence.
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© 2019 The Author(s), under exclusive licence to Springer Nature Switzerland AG
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Oller, J.A. (2019). The N / D Method with Perturbative \(\varDelta (p^2)\). In: A Brief Introduction to Dispersion Relations. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-13582-9_11
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DOI: https://doi.org/10.1007/978-3-030-13582-9_11
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