Calculations of Higher Order Quantum Chromodynamics Corrections

Abstract

Our article is devoted to the computation of two three-loop diagrams that contribute to the decay $b \to s\gamma$ at $\alpha _{s}^{2}$ order. We use differential equations for master integrals (MI) to compute these diagrams for an arbitrary c-quark mass. The program CANONICA is used to obtain differential equations on a canonical basis. Using them, it is possible to solve the differential equations and obtain expressions for MI-s in terms of GPL functions. We hope that the same method can be used for other three-loop diagrams which contribute to the decay $b \to s\gamma$ at order $\alpha _{s}^{2}$.

APPENDIX

Below is the sum of the two diagrams in Fig. 1 as a series from $\varepsilon$ to ${{\varepsilon }^{{ - 1}}}$. The expression for ${{\varepsilon }^{0}}$ is too long to be reproduced here. If necessary, it can be provided by the authors.

$$\begin{gathered} {{S}_{m}} = \frac{{10}}{{3{{\varepsilon }^{2}}}} + \frac{1}{\varepsilon }( - \frac{{23G( - 1,x)}}{{{{x}^{3}}}} + \frac{{23G(1,x)}}{{{{x}^{3}}}} + \frac{{13G( - 1,x)}}{x} - 10G( - 1,x) + 20G(0,x) \hfill \\ \quad \quad \; - \frac{{13G(1,x)}}{x} - 10G(1,x) + \frac{{39}}{{{{x}^{2}}}} + 10i\pi + \frac{8}{3} + 20\log (2) \hfill \\ \quad \quad \; + \frac{{({{x}^{2}} - 1)(13{{x}^{2}} - 1)}}{{2{{x}^{4}}}}( - G( - 1, - 1,x) + G( - 1,1,x) + G(1, - 1,x) - G(1,1,x)) \hfill \\ \quad \quad \; + \frac{{2\left( {5{{x}^{4}} - 6{{x}^{2}} + 1} \right)}}{{{{x}^{4}}}}( - G( - 1, - 1, - 1,x) + G( - 1, - 1,1,x) + G( - 1,1, - 1,x) \hfill \\ \quad \quad \; - G( - 1,1,1,x) + 2G(0, - 1, - 1,x) - 2{\text{ }}G(0, - 1,1,x) - 2G(0,1, - 1,x) \hfill \\ \quad \quad \; + 2G(0,1,1,x) - G(1, - 1, - 1,x) + G(1, - 1,1,x) + G(1,1, - 1,x) - G(1,1,1,x)) \hfill \\ \quad \quad \; + \frac{{2\left( {3{{x}^{4}} - 4{{x}^{2}} + 1} \right)}}{{{{x}^{4}}}}(G( - 1, - 1, - 1, - 1,x) - G( - 1, - 1, - 1,1,x) - G( - 1, - 1,1, - 1,x) \hfill \\ \quad \quad \; + G( - 1, - 1,1,1,x) - 2{\text{ }}G( - 1,0, - 1, - 1,x) + 2{\text{ }}G( - 1,0, - 1,1,x) + 2{\text{ }}G( - 1,0,1, - 1,x) \hfill \\ \quad \quad \; - 2G( - 1,0,1,1,x) + G( - 1,1, - 1, - 1,x) - G( - 1,1, - 1,1,x) - G( - 1,1,1, - 1,x) \hfill \\ \quad \quad \; + G( - 1,1,1,1,x) - 2{\text{ }}G(0, - 1, - 1, - 1,x) + 2G(0, - 1, - 1,1,x) + 2{\text{ }}G(0, - 1,1, - 1,x) \hfill \\ \quad \quad \; - 2{\text{ }}G(0, - 1,1,1,x) + 4{\text{ }}G(0,0, - 1, - 1,x) - 4{\text{ }}G(0,0, - 1,1,x) - 4{\text{ }}G(0,0,1, - 1,x) \hfill \\ \quad \quad \; + 4G(0,0,1,1,x) - 2G(0,1, - 1, - 1,x) + 2G(0,1, - 1,1,x) + 2G(0,1,1, - 1,x) \hfill \\ \quad \quad \; - 2{\text{ }}G(0,1,1,1,x) + G(1, - 1, - 1, - 1,x) - G(1, - 1, - 1,1,x) - G(1, - 1,1, - 1,x) \hfill \\ \quad \quad \; + G(1, - 1,1,1,x) - 2{\text{ }}G(1,0, - 1, - 1,x) + 2G(1,0, - 1,1,x) + 2{\text{ }}G(1,0,1, - 1,x) \hfill \\ \quad \quad \; - 2G(1,0,1,1,x) + G(1,1, - 1, - 1,x) - G(1,1, - 1,1,x) - G(1,1,1, - 1,x) + G(1,1,1,1,x))). \hfill \\ \end{gathered} $$