Heavy Flavor Averaging Group (HFLAV)

D0 hadronic branching fractions and Final State Radiation

(Updated 25 September 2021)

People working on this: Paras Naik

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 D0 branching fractions to the final states Kπ+, π+π and K+ K with a uniform treatment of 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 25 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

Also worth noting, an exponentiated multiple-photon mode was introduced in PHOTOS version 2.09, which allows PHOTOS to also simulate photons with low energies (i.e. allowing XPHCUT to approach zero); this mode can be turned on (default for PHOTOS 2.XX is off).

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 Kπ+. 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 Kπ+ 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. Several determinations of the branching fractions for these latter modes are via ratios to Kπ+, 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(D0 → Kπ+), B(D0 → π+π), and B(D0 → K+K), and also the branching ratio measurements B(D0 → π+π) / B(D0 → Kπ+) and B(D0 → K+K) / B(D0 → Kπ+). All measurements have been corrected to correspond to the use of PHOTOS 2.15 with interference effects included and exponentiated multiple-photon mode turned on. 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(D0 → π+π) / B(D0 → Kπ+) and B(D0 → K+K) / B(D0 → Kπ+) 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 ΛQCD 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 the statistical error. 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 also in the BES III 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 and exponentiated multiple-photon mode turned on, no correction is applied. nada

D0 → K- π+

Experiment BF (rescaled) [%] update shift [%] PHOTOS version/Interf. Reference
BES III 3.931 ± 0.006 ± 0.067(0.044) 1.25 2.03/Yes PRD 97, 072004, 2018 [INSPIRE]
CLEO-c 3.934 ± 0.021 ± 0.061(0.031) -- 2.15/Yes PRD 89, 072002, 2014 [INSPIRE]
BaBar 4.035 ± 0.037 ± 0.074(0.024) 0.69 2.02/No PRL 100, 051802, 2008 [INSPIRE]
CLEO II 3.917 ± 0.154 ± 0.167(0.027) 2.80 none PRL 80, 3193, 1998 [INSPIRE]
ALEPH 3.931 ± 0.091 ± 0.124(0.027) 0.79 2.0/No PLB 403, 367, 1997 [INSPIRE]
ARGUS 3.490 ± 0.123 ± 0.287(0.020) 2.33 none PLB 340, 125, 1994 [INSPIRE]
CLEO II 3.965 ± 0.080 ± 0.171(0.013) 0.38 2.0/No PRL 71, 3070, 1993 [INSPIRE]
ALEPH 3.733 ± 0.351 ± 0.455(0.028) 3.12 none PLB 266, 218, 1991 [INSPIRE]


D0 → π+ π-

Experiment BF (rescaled) [%] update shift [%] PHOTOS version/Interf. Reference
BES III 0.1529 ± 0.0018 ± 0.0032(0.0023) 1.39 2.03/Yes PRD 97, 072004, 2018 [INSPIRE]


D0 → π+π / D0 → K- π+

Experiment ππ/Kπ (rescaled) update shift [%] PHOTOS version/Interf. Reference
CLEO-c 0.0370   ± 0.0006   ± 0.0009  (0.0002) -- 2.15/Yes PRD 81, 052013, 2010 [INSPIRE]
CDF 0.03594 ± 0.00054 ± 0.00043(0.00015) -- 2.15/Yes PRL 94, 122001, 2005 [INSPIRE]
FOCUS 0.0364   ± 0.0012   ± 0.0006  (0.0002) 3.10 none PLB 555, 167, 2003 [INSPIRE]


D0 → K+ K-

Experiment BF (rescaled) [%] update shift [%] PHOTOS version/Interf. Reference
BES III 0.4271 ± 0.0021 ± 0.0069(0.0027) 0.89 2.03/Yes PRD 97, 072004, 2018 [INSPIRE]


D0 → K+K / D0 → K- π+

Experiment KK/Kπ (rescaled) update shift [%] PHOTOS version/Interf. Reference
CLEO-c 0.1041 ± 0.0011 ± 0.0012(0.0003) -- 2.15/Yes PRD 81, 052013, 2010 [INSPIRE]
CDF 0.0992 ± 0.0011 ± 0.0012(0.0001) -- 2.15/Yes PRL 94, 122001, 2005 [INSPIRE]
FOCUS 0.0982 ± 0.0014 ± 0.0014(0.0001) -1.12 none PLB 555, 167, 2003 [INSPIRE]

The fit for the three branching fractions has a final χ2 of 36.0 for 16 - 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 becoming large in magnitude to the statistical uncertainty on the average.

B(D0 → Kπ+) (3.999   ± 0.006   ± 0.031   ± 0.032)  %
B(D0 → π+π)(0.1490 ± 0.0012 ± 0.0015 ± 0.0019)%
B(D0 → K+K) (0.4113 ± 0.0017 ± 0.0041 ± 0.0025)%

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

B(D0 → Kπ+) 1.00 0.77 0.76
B(D0 → π+π)0.77 1.00 0.58
B(D0 → K+K) 0.76 0.58 1.00

The following figures show the comparison of the Kπ, ππ, and KK 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.

HFLAV eps pdf

HFLAV eps pdf             HFLAV eps pdf

(The fit was also performed for the Kπ branching fraction alone, using only the Kπ measurements; the corresponding result of B(D0 → Kπ+) = (3.949 ± 0.006 ± 0.032 ± 0.033)% has a final χ2 of 3.7 for 8 - 1 degrees of freedom.)

Lastly, we combine our HFLAV CPV-allowed RD average with our B(D0 → Kπ+) result, to provide a measurement of the branching fraction B(D0 → K+π).
B(D0 → K+π) (1.372 ± 0.017) × 10-4

Please note that the newest version of PHOTOS (PHOTOS++ v3.61) produces nearly identical results to PHOTOS v2.15, and thus a further update of results to PHOTOS++ v3.61 is unnecessary.
This page is maintained by P. Naik and was last updated

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