Heavy-to-light chromomagnetic matrix element

Exclusive chromomagnetism in heavy-to-light FCNCs

“…For the LC-OPE dispersion representation one introduces an effective continuum threshold s 0 [11, 1] 9 , which corresponds to the duality approximation mentioned above. The crucial point in connection with the anomalous threshold, which results in branch cuts extending into the complex plane, is that its real part is above the continuum threshold, m 2 b + m 2 B /2 > s 0 , and therefore it is entirely included in Γ C and will not contribute to the final sum rule 10 . Therefore the path Γ minus the path Γ C corresponds to the path 8 In principle there might be further isolated states, with B(0 − ) quantum numbers, between m 2 B and (m B + 2m π ) 2 .…”

“…Realistically one can expect s 0 to be somewhere between say (m B + 2m π ) 2 and (m B + m ρ ) 2 . Whether or not this affects the final result depends on the convergence of the LC-OPE and Borel parameters and has to be analysed and is discussed in section 4 10 It is also suppressed by the Borel transformation, both due to the large real part of s and the Γ P that encircles the real line segment from m 2 b to s 0 . The final sum rule can be written as…”

“…functions in use, can be found in appendices G and H respectively. A shorter write-up of some of the main points of this paper can be found in [10].…”