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Fermionic NNLL corrections to b -> s γ

Kay Bieri, Christoph Greub, Matthias Steinhauser

TL;DR

This paper advances the NNLL program for B -> X_s γ by computing fermionic α_s^2 n_f corrections to the matrix elements of O1, O2, O7, and O8, including both virtual and real contributions. Using an effective Hamiltonian and regulator masses, the authors obtain renormalized, finite results for the matrix elements, with explicit logarithmic and z-dependent structures. They find significant cancellations between current-current and dipole operator contributions, leading to substantial but μ-sensitive shifts in the branching ratio when a photon-energy cut is applied. The work demonstrates that matrix-element corrections can dominate the NNLL effects and suggests that naive non-abelianization may provide a good estimate of the full α_s^2 corrections, marking an important initial step toward a complete NNLL prediction for BR(B -> X_s γ).

Abstract

In this paper we take the first step towards a complete next-to-next-to-leading logarithmic (NNLL) calculation of the inclusive decay rate for $B \to X_sγ$. We consider the virtual corrections of order $\alphas^2 n_f$ to the matrix elements of the operators ${O}_1$, ${O}_2$ and ${O}_8$ and evaluate the real and virtual contributions to ${O}_7$. These corrections are expected to be numerically important. We observe a strong cancelation between the contributions from the current-current operators and $O_7$ and obtain, after applying naive non-abelianization, a reduction of the branching ratio of 3.9% (for $μ=3.0$ GeV) and an increase of 3.4% (for $μ=9.6$ GeV).

Fermionic NNLL corrections to b -> s γ

TL;DR

This paper advances the NNLL program for B -> X_s γ by computing fermionic α_s^2 n_f corrections to the matrix elements of O1, O2, O7, and O8, including both virtual and real contributions. Using an effective Hamiltonian and regulator masses, the authors obtain renormalized, finite results for the matrix elements, with explicit logarithmic and z-dependent structures. They find significant cancellations between current-current and dipole operator contributions, leading to substantial but μ-sensitive shifts in the branching ratio when a photon-energy cut is applied. The work demonstrates that matrix-element corrections can dominate the NNLL effects and suggests that naive non-abelianization may provide a good estimate of the full α_s^2 corrections, marking an important initial step toward a complete NNLL prediction for BR(B -> X_s γ).

Abstract

In this paper we take the first step towards a complete next-to-next-to-leading logarithmic (NNLL) calculation of the inclusive decay rate for . We consider the virtual corrections of order to the matrix elements of the operators , and and evaluate the real and virtual contributions to . These corrections are expected to be numerically important. We observe a strong cancelation between the contributions from the current-current operators and and obtain, after applying naive non-abelianization, a reduction of the branching ratio of 3.9% (for GeV) and an increase of 3.4% (for GeV).

Paper Structure

This paper contains 14 sections, 89 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Virtual corrections to $\bm{b \to s \gamma}$ associated with $\bm{O_1}$ and $\bm{O_2}$
  3. Regularized three-loop corrections to $\bm{\langle s \gamma|O_2|b \rangle$
  4. Counterterm contributions to $\bm{\langle s \gamma|O_2|b \rangle}$
  5. Renormalized result for $\bm{\langle s \gamma|O_2|b \rangle}$
  6. Real and virtual corrections to $\bm{\langle s \gamma|O_7|b \rangle}$
  7. Virtual corrections to $\bm{\langle s \gamma|O_8|b \rangle}$
  8. Numerical impact of the $\bm{{\cal O}(\alpha_s^2 n_f)}$ corrections
  9. Conclusions
  10. Building Blocks
  11. Regularized three-loop results for $\bm{\langle s \gamma|O_2|b \rangle}$
  12. Correction functions needed for the NLL result
  13. $\bm{{\cal O}(\alpha_{\text{s}}^2 n_f)}$ contributions to various $\bm{Z}$ factors
  14. Implementing the photon energy cut-off in the $\bm{{\cal O}(\alpha_{\text{s}} n_f)}$ terms

Figures (10)

  • Figure 1: Diagrams 1a--c and 2a--c associated with the operator $O_2$. The photon is represented by a wavy line and is emitted from a down-type quark in all the diagrams. The virtual gluons are represented by curly lines. The sum of the first three graphs is denoted with $M_{2,{\rm bare}}^{(2)}(1)$, whereas the sum of the second three diagrams is called $M_{2,{\rm bare}}^{(2)}(2)$.
  • Figure 2: Diagrams 3a--b and 4a--b associated with the operator $O_2$. The photon is represented by a wavy line and is emitted from an up-type quark in all the diagrams. The virtual gluons are represented by curly lines. The sum of the first two graphs is denoted with $M_{2,{\rm bare}}^{(2)}(3)$, whereas the sum of the second two diagrams is called $M_{2,{\rm bare}}^{(2)}(4)$.
  • Figure 3: Counterterm diagrams to $O_2$ involving the operator $O_4$. The crosses denote the possible places for photon emission. Note that the diagrams where the photon is emitted from the fermion-loop are zero due to Furry's theorem.
  • Figure 4: Virtual (a), gluon-bremsstrahlung (b) and quark-pair radiation (c) graphs to the operator $O_7$. In (b) and (c), the diagrams where the gluon is emitted from the $s$-quark are not shown.
  • Figure 5: Graphs associated with virtual corrections to the operator $O_8$. The crosses denote the possible places where the photon can be emitted.
  • ...and 5 more figures