Branching Ratio Calculator

The branching ratio for a decay channel i is the fraction of decays that proceed through that channel. Using decay widths:

BRᵢ = Γᵢ / Γ_total

If a lifetime τ is known, the total width can be obtained from:

Γ_total = ħ / τ

In counting experiments with detection efficiency εᵢ, the corrected yield is:

cᵢ = Nᵢ / εᵢ,   BRᵢ = cᵢ / Σ cⱼ

1. Select a method: widths/lifetime or counts/efficiencies.
2. Choose how many decay channels you want to compare.
3. Enter channel labels and values, plus optional uncertainties.
4. For widths: provide Γ_total or lifetime τ.
5. For counts: add εᵢ to correct detector acceptance.
6. Click Calculate to view results above the form.
7. Use CSV to export tables, or print to PDF.

1) Why branching ratios matter

Branching ratios summarize how an unstable state distributes its decays among competing channels. They convert detailed detector observables into a dimensionless probability model, enabling comparisons across experiments and energies. A well-normalized set of branching ratios also supports simulation tuning and background budgeting.

2) Widths, lifetimes, and physical scale

In many systems, the total decay width sets the overall timescale. Using the relation Γ_total=ħ/τ, a lifetime of 3.0×10⁻²⁵ s corresponds to a total width near 2.19×10³ MeV. Converting τ to Γ is useful when lifetimes are reported more precisely than widths.

3) Computing ratios from partial widths

When partial widths are available, each channel ratio is BRᵢ=Γᵢ/Γ_total. For example, with Γ_total=0.250 MeV and partial widths 0.083, 0.125, 0.042 MeV, the corresponding branching ratios are 0.332, 0.500, 0.168. The closure metric ΣΓᵢ/Γ_total highlights missing channels or inconsistent inputs.

4) Using event counts with efficiencies

Counting analyses often start from observed yields Nᵢ and efficiencies εᵢ. The corrected yield is cᵢ=Nᵢ/εᵢ, then BRᵢ=cᵢ/Σcⱼ. As an illustration: N=(12000, 8000, 2000) with ε=(0.60, 0.50, 0.40) gives corrected yields (20000, 16000, 5000) and ratios near (0.488, 0.390, 0.122).

5) Uncertainty propagation and reporting

This calculator propagates uncertainties assuming independent inputs. For widths, it applies standard ratio propagation using σ(BRᵢ)/BRᵢ ≈ √[(σΓᵢ/Γᵢ)² + (σΓ_total/Γ_total)²]. For counts, it uses a delta-method estimate that accounts for the shared denominator Σcⱼ.

6) Normalization checks and diagnostics

A reliable result set should satisfy 0 ≤ BRᵢ ≤ 1 and ΣBRᵢ ≈ 1. Deviations often indicate missing channels, overlapping selections, or efficiency mismodeling. If one channel dominates, small absolute errors in that channel can drive noticeable shifts across all other ratios.

7) Practical analysis workflow

Start by defining non-overlapping channels and stable selection rules. Enter preliminary widths or yields, then review closure and normalization. Next, refine efficiency estimates using control samples and re-run the calculator to quantify the impact on BRᵢ. Finally, export CSV for documentation and reproducible review.

8) Interpreting results for decisions

Branching ratios guide trigger allocation, detector upgrades, and model selection. For rare channels, a small BR may still be high-impact if it provides a clean signature. Compare BR values alongside uncertainties and expected event totals to decide where improved statistics or better efficiency calibration will deliver the largest gain.

1) What is a branching ratio?

A branching ratio is the fraction of decays that occur through a specific channel, expressed as BRᵢ=Γᵢ/Γtotal or as corrected-yield fractions from event counts.

2) Should my branching ratios sum to one?

Yes, if you include all channels contributing to the total. If you omit channels, the sum will be below one. If selections overlap or inputs are inconsistent, the sum may exceed one.

3) When should I use lifetime instead of total width?

Use lifetime when it is the reported quantity or when it is measured more precisely. The calculator converts τ to Γtotal using ħ/τ, then computes BR from your partial widths.

4) How does the counts method handle efficiencies?

It corrects each yield as cᵢ=Nᵢ/εᵢ, then forms BRᵢ=cᵢ/Σcⱼ. This reduces bias from unequal detector acceptance or selection efficiency across channels.

5) What uncertainty should I enter for counts?

If you have a fitted yield uncertainty, enter it directly. If you leave it blank, the calculator defaults to Poisson √N, which is common for simple counting statistics.

6) Why do uncertainties change across all channels together?

All BR values share the same denominator, so changing one channel affects the normalization of others. This coupling is especially strong when one channel is large or uncertain.

7) What does the closure check mean in width mode?

Closure is ΣΓᵢ/Γtotal. Values near one indicate the provided partial widths account for the total width. Values below one suggest missing channels, while values above one suggest inconsistent inputs.