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NNLO QCD corrections to the B -> X_s gamma matrix elements using interpolation in m_c

Mikolaj Misiak, Matthias Steinhauser

Abstract

One of the most troublesome contributions to the NNLO QCD corrections to B -> X_s gamma originates from three-loop matrix elements of four-quark operators. A part of this contribution that is proportional to the QCD beta-function coefficient beta_0 was found in 2003 as an expansion in m_c/m_b. In the present paper, we evaluate the asymptotic behaviour of the complete contribution for m_c >> m_b/2. The asymptotic form of the beta_0-part matches the small-m_c expansion very well at the threshold m_c = m_b/2. For the remaining part, we perform an interpolation down to the measured value of m_c, assuming that the beta_0-part is a good approximation at m_c=0. Combining our results with other contributions to the NNLO QCD corrections, we find BR(B -> X_s gamma) = (3.15 +_ 0.23) x 10^-4 for E_gamma > 1.6 GeV in the B-meson rest frame. The indicated error has been obtained by adding in quadrature the following uncertainties: non-perturbative (5%), parametric (3%), higher-order perturbative (3%), and the interpolation ambiguity (3%).

NNLO QCD corrections to the B -> X_s gamma matrix elements using interpolation in m_c

Abstract

One of the most troublesome contributions to the NNLO QCD corrections to B -> X_s gamma originates from three-loop matrix elements of four-quark operators. A part of this contribution that is proportional to the QCD beta-function coefficient beta_0 was found in 2003 as an expansion in m_c/m_b. In the present paper, we evaluate the asymptotic behaviour of the complete contribution for m_c >> m_b/2. The asymptotic form of the beta_0-part matches the small-m_c expansion very well at the threshold m_c = m_b/2. For the remaining part, we perform an interpolation down to the measured value of m_c, assuming that the beta_0-part is a good approximation at m_c=0. Combining our results with other contributions to the NNLO QCD corrections, we find BR(B -> X_s gamma) = (3.15 +_ 0.23) x 10^-4 for E_gamma > 1.6 GeV in the B-meson rest frame. The indicated error has been obtained by adding in quadrature the following uncertainties: non-perturbative (5%), parametric (3%), higher-order perturbative (3%), and the interpolation ambiguity (3%).

Paper Structure

This paper contains 8 sections, 52 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The effective theory
  3. The corrections $P_2^{(1)}$ and $P_3^{(2)}$
  4. The $\beta_0$-part of $P_2^{(2)}$
  5. The full correction $P_2^{(2)}$ in the limit $m_c \gg m_b/2$
  6. Interpolation in $m_c$
  7. Estimating the uncertainties
  8. Conclusions

Figures (3)

  • Figure 1: ${\rm Re}(a+b)$ (left plot) and ${\rm Re}\,r_2^{(2)}$ (right plot) as functions of $m_c/m_b = \sqrt{z}$. See the text for explanation of the curves.
  • Figure 2: $P_2^{(2)\rm rem}$, $P_2^{(2)\beta_0}$, $P_1^{(2)}$ and $P_3^{(2)}$ as functions of $m_c/m_b = \sqrt{z}$ for $\mu_0=2 M_W$, $\mu_b = m_b^{1S}/2$ and $\mu_c = m_c(m_c)$. The other input parameters are set to their central values from Appendix A. See the text below Eq. (\ref{['p32z']}).
  • Figure 3: Renormalization scale dependence of ${\cal B}(\bar{B} \to X_s \gamma)$ in units $10^{-4}$ at the LO (dotted lines), NLO (dashed lines) and NNLO (solid lines). The upper-left, upper-right and lower plots describe the dependence on $\mu_c$, $\mu_b$ and $\mu_0$ [GeV], respectively.