Hadronic branching fraction measurements are fast approaching
1%
precision levels, at which proper treatment of
Final State (electromagnetic) Radiation (FSR) plays a significant role. We present a simultaneous determination
of averages for the D^{0} branching fractions to the final states K^{−}π^{+},
π^{+}π^{−} and K^{+} K^{−} with a uniform treatment FSR in these
modes. In particular, the measurements have all been corrected, if necessary, to correspond to h^{+}
h^{−} (nγ), with a uniform modeling of FSR.

Many (but not all) of the measurements in use apply the PHOTOS package to estimate the FSR contributions. However, PHOTOS itself has evolved over the past 20 years, and an experiment's configuration of PHOTOS within their Monte Carlo suite affects the reliability of the PHOTOS output, particularly whether interference effects are included when the final state includes multiple charged particles. In some cases, interference in FSR has been turned off because of problems in J/ψ decays with a low value of the PHOTOS parameter XPHCUT.

A crude history of the PHOTOS 2.XX releases used in the current measurements, along with the status of interference and correction of the measurements is

- 11/1993 - v2.00: accounts for interference in particle-antiparticle final states
- 03/1999 - v2.02: extends interference to two body, all charged final states (eg., K
^{−}π^{+}) - 10/2005 - v2.15: extends interference to > 2 body final states

The following photon energy and hadronic invariant mass spectra are shown for the different PHOTOS versions and
configurations in use in many of the current measurements. For 2.00, you can see the interference term
present in π^{+}π^{−}, but not in K^{−}π^{+}. For the two body
mass plots shown, a typical experimental smearing of 10 MeV has been applied to illustrate the size of the FSR tail
relative to experimental resolution. The arrow at 30 MeV in the photon energy spectrum indicates where the spectrum
becomes important relative to experimental resolution (i.e., very quickly!).

The FSR photons energy spectrum extends to very sizeable energies, causing a very long low side mass tail on the
hadronic invariant mass. Without accounting for this effect in experimental efficiency determinations, the
K^{−}π^{+} branching fraction would be biased low in the range of 2-3% for typical selection
criteria. Absolute π^{+}π^{−} and K^{+}K^{−} rates would be affected
at levels of approximately 4-6% and 1-2%, respectively. The most precise determinations of the branching fractions for
these modes are via ratios to K^{−}π^{+}, for which there is some cancellation of the effect.

To obtain new average branching fractions for these three modes, we have performed a simultaneous average of the
branching fraction measurements for *B*(D^{0} → K^{−}π^{+}) and the
branching ratio measurements *B*(D^{0} → π^{+}π^{−}) /
*B*(D^{0} → K^{−}π^{+}) and *B*(D^{0} →
K^{+}K^{−}) / *B*(D^{0} → K^{−}π^{+}).
All measurements have been corrected to correspond to the use of PHOTOS 2.15 with interfence effects included.
Typically, experiments have either reported the change in branching fraction with and without incorporation of FSR,
or have made explicit requirements on kinematic variables like the K−π invariant mass. In these cases, any
needed correction is straightforward to evaluate. The correction is very difficult to evaluate for two measurements
(from E791 and CLEO II) in which signal yields were obtained from fits to kinematic variables, but no information
is provided regarding the effects of FSR. These measurements are older measurements with uncertainties
significantly larger than recent measurements, so have been excluded from the averages presented here.
FOCUS measurements of the *B*(D^{0} → π^{+}π^{−}) /
*B*(D^{0} → K^{−}π^{+}) and *B*(D^{0} →
K^{+}K^{−}) / *B*(D^{0} → K^{−}π^{+}) branching ratios
do contribute significantly to current world averages, so using information provided by the authors, we have
implemented a toy Monte Carlo procedure to evaluate the necessary corrections.

In the averaging procedure, we have assumed that the dominant uncertainty in the FSR corrections come from the
fact that the mesons are treated like structureless particles, so no contribution from structure-dependent terms
in the decay process (*eg.* radiation off individual quarks). Internal studies done by various experiments
have indicated that in Kπ decay, the PHOTOS corrections agree with data at the 20-30% level. We therefore
attribute a 25% uncertainty to the FSR prediction from potential structure-dependent contributions. For the
other two modes, the only difference in structure is the final state valence quark content. While radiative
corrections typically come in with a 1/M dependence, one would expect the additional contribution from the
structure terms to come in on time scales shorter than the hadronization time scale. In this case, you might
expect Λ_{QCD} to be the relevant scale, rather than the quark masses, and therefore that the
amplitude is the same for the three modes. In treating the correlations among the measurements this is what
we assume. We also assume that PHOTOS and structure amplitudes are relatively real with constructive
interference. The uncertainties largely cancel in the branching fraction ratios, and on the final
average branching fractions, the FSR uncertainty on Kπ dominates. Note that because of the relative
sizes of FSR in the different modes, the ππ/Kπ branching ratio uncertainty is positively correlated
with the Kπ branching fraction uncertanty, while the KK/Kπ branching ratio uncertainty is negatively correlated.

We include systematic correlations in the CLEO II measurements, and for the ALEPH measurements we take the
systematic uncertainties in the D^{*} direction relative to the jet axis as correlated.

The input branching fractions, applied corrections, and references are summarized below. If a published result was already based on version 2.15 of PHOTOS with interference enabled, no correction is applied.

Experiment | BF (rescaled) [%] | correction [%] | PHOTOS version/Interf. | Reference |
---|---|---|---|---|

CLEO-c | 3.934 ± 0.021 ± 0.061(0.031) | -- | 2.15/Yes | PRD 89, 072002, 2014 [Spires] |

BaBar | 4.035 ± 0.037 ± 0.074(0.024) | 0.69% | 2.02/No | PRL 100, 051802, 2008 [Spires] |

CLEO II | 3.920 ± 0.154 ± 0.168(0.032) | 2.80% | none | PRL 80, 3193, 1998 [Spires] |

ALEPH | 3.930 ± 0.091 ± 0.125(0.032) | 0.79% | 2.0/No | Phys.Lett.B403:367-376,1997 [Spires] |

ARGUS | 3.490 ± 0.123 ± 0.288(0.024) | 2.33% | none | Phys Lett B340, 125, 1994 [Spires] |

CLEO II | 3.960 ± 0.080 ± 0.171(0.015) | 0.38% | 2.0/No | PRL 71, 3070, 1993 [Spires] |

ALEPH | 3.730 ± 0.351 ± 0.455(0.034) | 3.12% | none | Phys Lett, B266, 218, 1991 [Spires] |

Experiment | ππ/Kπ (rescaled) | correction [%] | PHOTOS version/Interf. | Reference |
---|---|---|---|---|

CLEO-c | 0.0370 ± 0.0006 ± 0.0009 (0.0002) | -- | 2.15/Yes | PRD 81, 052013, 2010 [Spires] |

CDF | 0.03594 ± 0.00054 ± 0.00043(0.00015) | -- | 2.15/Yes | PRL 94, 122001,2005 [Spires] |

FOCUS | 0.0364 ± 0.0012 ± 0.0006 (0.0002) | 3.10% | none | PL B555, 167, 2003 [Spires] |

Experiment | KK/Kπ (rescaled) | correction [%] | PHOTOS version/Interf. | Reference |
---|---|---|---|---|

CLEO-c | 0.1041 ± 0.0011 ± 0.0012(0.0003) | -- | 2.15/Yes | PRD 81, 052013, 2010 [Spires] |

CDF | 0.0992 ± 0.0011 ± 0.0012(0.0001) | -- | 2.15/Yes | PRL 94, 122001,2005 [Spires] |

FOCUS | 0.0982 ± 0.0014 ± 0.0014(0.0001) | -1.12% | none | PL B555, 167, 2003 [Spires] |

The fit for the three branching fractions has a final χ^{2} of 11.0 for 13 - 3 degrees of freedom.
The statistical and systematic covariance matrices (including the correlations noted above) have been summed
to obtain the full covariance matrix used in this fit.

The table below summarizes the branching fraction averages (in percent). The errors listed are statistical, systematic excluding FSR, and the FSR-related uncertainty. Note that the FSR uncertainty is comparable in magnitude to the statistical uncertainty on the average.

B(D^{0} → K^{−}π^{+}) | (3.962 ± 0.017 ± 0.038 ± 0.027)% |
---|---|

B(D^{0} → π^{+}π^{−}) | (0.144 ± 0.002 ± 0.002 ± 0.002)% |

B(D^{0} → K^{+}K^{−}) | (0.399 ± 0.003 ± 0.005 ± 0.002)% |

The correlation matrix for the fit, including statistical and systematic uncertainties and the correlations among inputs, is given by

B(D^{0} → K^{−}π^{+}) | 1.00 | 0.71 | 0.76 |
---|---|---|---|

B(D^{0} → π^{+}π^{−}) | 0.71 | 1.00 | 0.53 |

B(D^{0} → K^{+}K^{−}) | 0.76 | 0.53 | 1.00 |

The following figures show the comparison of the Kπ, KK, and ππ measurements to the average returned by the fit for each branching fraction. For each measurement and for the average branching fraction, the statistical uncertainties, the combined statistical and systematic uncertainties excluding FSR uncertainties, and the total uncertainties are separately indicated.

(The fit was also performed for the Kπ branching fraction alone, using only the Kπ measurements; the corresponding result of