Two-Loop Matrix Element of the Current-Current Operator in the Decay B -> X_s gamma

TL;DR

This work computes the important two-loop matrix element $\langle s\gamma|Q_2|b\rangle$ for the inclusive decay $B\to X_s\gamma$ using the NDR scheme and an asymptotic-expansion approach in $z=\frac{m_c^2}{m_b^2}$, extending the expansion to $O(z^6)$. The calculation involves decomposing the contributing Feynman diagrams into multiple momentum regions (hard-hard, hard-soft, collinear-soft, soft-soft, etc.) and applying systematic expansions, with ultraviolet counterterms ensuring finiteness. The resulting expression for the matrix element takes the form $\langle s\gamma|Q_2|b\rangle = \langle s\gamma|Q_7|b\rangle_{\rm tree} \frac{\alpha_s}{4\pi}\left(\frac{416}{81}\ln\frac{m_b}{\mu} + r_2\right)$, where $r_2$ is given as a $z$-series up to $O(z^6)$ with both real and imaginary parts explicitly displayed. The new $z^4$–$z^6$ terms are numerically insignificant, and the results are consistent with the earlier GREUB, Hurth, and Wyler findings through $O(z^3)$, providing a robust confirmation and extending the precision of this critical ingredient for the NLO analysis of Br$(B\to X_s\gamma)$.

Abstract

We evaluate the important two-loop matrix element <s gamma| Q_2 | b> of the operator (cbar gamma^mu P_L b) (sbar gamma_mu P_L b) contributing to the inclusive radiative decay B -> X_s gamma. The calculation is performed in the NDR scheme, by means of asymptotic expansions method. The result is given as a series in z = m_c^2/m_b^2 up to O(z^6). We confirm the result of Greub, Hurth and Wyler obtained by a different method up to 0(z^3). Higher-order terms are found to be numerically insignificant.

Figures (6)

• Figure 1: Diagram $M_1$. Momentum assignments: $Q$, $q$, and $\gamma$ denote respectively four-momenta of the $b$ and $s$ quarks, and of the photon; $p$ and $k$ are loop momenta. Double lines denote a massive particle (the $b$ quark) and single lines are light particle propagators. The gluon is denoted by a dashed line.
• Figure 2: One-loop diagram (the $k$ propagator can have fractional power) corresponding to eq. (\ref{['eqk1']}).
• Figure 3: Subgraph corresponding to one of the two hard-soft contributions.
• Figure 4: Diagram $M_2$.
• Figure 5: Diagrams contributing to $M_3$, with photon emission from two $c$-quark lines.