Tree-level contribution to \bar{B} -> X_d gamma using fragmentation functions

We evaluate the most important tree-level contributions connected with the b-> u \bar{u} d gamma transition to the inclusive radiative decay \bar{B}-> X_d gamma using fragmentation functions. In this framework the singularities arising from collinear photon emission from the light quarks (u, \bar{u} and d) can be absorbed into the (bare) quark-to-photon fragmentation function. We use as input the fragmentation function extracted by the ALEPH group from the two-jet cross section measured at LEP, where one of the jets is required to contain a photon. To get the quark-to-photon fragmentation function at the fragmentation scale \mu_F \sim m_b, we use the evolution equation, which we solve numerically. We then calculate the (integrated) photon energy spectrum for b-> u \bar{u} d gamma related to the operators P^u_{1,2}. For comparison, we also give the corresponding results when using nonzero (constituent) masses for the light quarks.Comment: 13 pages, 4 figure

Inclusive Decay Rate for B→Xd+γB \to X_d + \gamma in Next-to-Leading Logarithmic Order and CP Asymmetry in the Standard Model

We compute the decay rate for the CKM-suppressed electromagnetic penguin decay $B \to X_d + \gamma$ (and its charge conjugate) in NLO QCD, including leading power corrections in $1/m_b^2$ and $1/m_c^2$ in the standard model. The average branching ratio of the decay $B \to X_d\gamma$ and its charge conjugate is estimated to be in the range $6.0 \times 10^{-6} \leq \leq 2.6 \times 10^{-5}$, obtained by varying the CKM-Wolfenstein parameters $\rho$ and $\eta$ in the range $-0.1 \leq \rho \leq 0.4$ and $0.2 \leq \eta \leq 0.46$ and taking into account other parametric dependence. In the stated range of the CKM parameters, we find the ratio $R(d\gamma/s\gamma) = <BR(B \to X_d\gamma)>/$ to lie in the range between 0.017 and 0.074. Theoretical uncertainties in this ratio are found to be small. Hence, this ratio is well suited to provide independent constraints on the CKM parameters. The CP-asymmetry in the $B \to X_d \gamma$ decay rates is found to be in the range $(7 - 35)$. Both the decay rates and CP asymmetry are measurable in forthcoming experiments at $B$ factories and possibly at HERA-B.Comment: 17 pages including 7 postscript figures; uses epsfig; The changes w.r.t the previous version are: A comment about the Bremsstrahlung corrections is added as well as a note on the feasibility of the measurement $B -> X_d gamma

Towards next-to-next-to-leading-log accuracy for the width difference in the Bs−BˉsB_s-\bar{B}_s system: fermionic contributions to order (mc/mb)0(m_c/m_b)^0 and (mc/mb)1(m_c/m_b)^1

We calculate a class of three-loop Feynman diagrams which contribute to the next-to-next-to-leading logarithmic approximation for the width difference $\Delta\Gamma_{s}$ in the $B_s-\bar{B}_s$ system. The considered diagrams contain a closed fermion loop in a gluon propagator and constitute the order $\alpha_s^2 N_f$, where $N_f$ is the number of light quarks. Our results entail a considerable correction in that order, if $\Delta\Gamma_{s}$ is expressed in terms of the pole mass of the bottom quark. If the $\overline{MS}$ scheme is used instead, the correction is much smaller. As a result, we find a decrease of the scheme dependence. Our result also indicates that the usually quoted value of the NLO renormalization scale dependence underestimates the perturbative error.Comment: We corrected a typographical mistake in Eq. (4.18), made larger axis labels in Fig.2. Version accepted by JHE

Towards the NNLL precision in Bˉ→Xsγ\bar B \to X_s \gamma

The present NLL prediction for the decay rate of the rare inclusive process $\bar B \to X_s \gamma$ has a large uncertainty due to the charm mass renormalization scheme ambiguity. We estimate that this uncertainty will be reduced by a factor of 2 at the NNLL level. This is a strong motivation for the on-going NNLL calculation, which will thus significantly increase the sensitivity of the observable $\bar B \to X_s \gamma$ to possible new degrees of freedom beyond the SM. We also give a brief status report of the NNLL calculation.Comment: 5 pages, 2 figures, contribution to the proceedings of EPS-HEP 200

Reduction of Charm Quark Mass Scheme Dependence in Bˉ→Xsγ\bar B \to X_s \gamma at the NNLL Level

The uncertainty of the theoretical prediction of the $\bar B \to X_s \gamma$ branching ratio at NLL level is dominated by the charm mass renormalization scheme ambiguity. In this paper we calculate those NNLL terms which are related to the renormalization of $m_c$, in order to get an estimate of the corresponding uncertainty at the NNLL level. We find that these terms significantly reduce (by typically a factor of two) the error on ${BR}(\bar B \to X_s \gamma)$ induced by the definition of $m_c$. Taking into account the experimental accuracy of around 10% and the future prospects of the $B$ factories, we conclude that a NNLL calculation would increase the sensitivity of the observable $\bar B \to X_s \gamma$ to possible new degrees of freedom beyond the SM significantly.Comment: 13 pages including 3 figure

NLL QCD contribution of the electromagnetic dipole operator to B -> X_s gamma gamma with a massive strange quark

We calculate the O(alpha_s) corrections to the double differential decay width dGamma_{77}/(ds_1 ds_2) for the process B -> X_s gamma gamma originating from diagrams involving the electromagnetic dipole operator O_7. The kinematical variables s_1 and s_2 are defined as s_i=(p_b - q_i)^2/m_b^2, where p_b, q_1, q_2 are the momenta of b-quark and two photons. We introduce a nonzero mass m_s for the strange quark to regulate configurations where the gluon or one of the photons become collinear with the strange quark and retain terms which are logarithmic in m_s, while discarding terms which go to zero in the limit m_s -> 0. When combining virtual- and bremsstrahlung corrections, the infrared and collinear singularities induced by soft and/or collinear gluons drop out. By our cuts the photons do not become soft, but one of them can become collinear with the strange quark. This implies that in the final result a single logarithms of m_s survives. In principle the configurations with collinear photon emission could be treated using fragmentation functions. In a related work we found that similar results can be obtained when simply interpreting m_s appearing in the final result as a constituent mass. We do so in the present paper and vary m_s between 400 MeV and 600 MeV in the numerics. This work extends a previous paper of us, where only the leading power terms w.r.t. the (normalized) hadronic mass s_3=(p_b-q_1-q_2)^2/m_b^2 were taken into account in the underlying triple differential decay width dGamma_{77}/(ds_1 ds_2 ds_3).Comment: arXiv admin note: substantial text overlap with arXiv:1110.125