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Dimension-Six Matrix Elements from Sum Rules

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Abstract

As we have seen in Sect. 2.4.2, the SM contribution to meson mixing arises at the 1-loop level and is both CKM and GIM suppressed. This makes these observables highly sensitive to new physics contributions (an issue which we will explore further in Chap. 7), and so a precise knowledge of the theoretical predictions is very important.

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Notes

  1. 1.

    As discussed below the sum rule reproduces the VSA at LO. Therefore the factors \(A_{\tilde{Q}_i}\) appear at leading order in the expansion of the results in \(\epsilon \). However, the correlator is computed in d dimensions and corrections can appear. We find that this happens only for \(\tilde{Q}_1\) where the contraction of the two \(\gamma \) matrices inside the trace yields a d-dimensional factor.

  2. 2.

    The arbitrariness of the weight function is a mathematical statement which holds for the dispersion relation. The sum rule in Eq. 6.3.7 does however also assume quark-hadron duality and breaks down if pathological weight functions are used, e.g. rapidly oscillating ones. In the following we only use slowly varying weight functions with support on the complete integration domain.

  3. 3.

    This choice, while technically original, is a relatively straightforward modification of the previous usage of weight functions in sum rule calculations.

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Kirk, M.J. (2019). Dimension-Six Matrix Elements from Sum Rules. In: Charming New Physics in Beautiful Processes?. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-19197-9_6

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