Calculator Inputs
Example Data Table
| Case | √s GeV | Mass GeV | Angle | Charge | Color Factor | Use |
|---|---|---|---|---|---|---|
| Muon pair | 10 | 0.1057 | 60 | -1 | 1 | Lepton study |
| Electron pair | 5 | 0.000511 | 45 | -1 | 1 | Light mass check |
| Quark pair model | 20 | 1.27 | 90 | 0.6667 | 3 | Color comparison |
Formula Used
This calculator uses a simplified tree level electromagnetic pair annihilation model.
s = √s²
β = √(1 - 4m² / s)
A = 1 + cos²θ + (1 - β²)sin²θ
|M|² = Nc × (QiQf)² × e⁴ × A × K × P
dσ / dΩ = Nc × (QiQf)² × α² × β × A × K × P / 4s
σ = Nc × (QiQf)² × 4πα²β × (3 - β²) × K × P / 6s
Here Nc is the color factor. K is a correction multiplier. P is the polarization scale.
How To Use This Calculator
- Enter the center of mass energy in GeV.
- Enter the final particle mass in GeV.
- Add the scattering angle in degrees.
- Keep inverse alpha near 137 unless another value is needed.
- Choose charge factors and color factor.
- Use optional correction and polarization multipliers.
- Press the calculate button.
- Download the result as CSV or PDF.
Pair Annihilation Amplitude Guide
What This Calculator Explains
Pair annihilation is a clean way to study particle interactions. An electron and a positron can meet, vanish, and create another particle pair. The invariant amplitude describes the core probability pattern behind that event. It is called invariant because the result is written with quantities that keep their meaning across valid reference frames.
Why The Amplitude Matters
The amplitude is not just one number. It connects charge, beam energy, scattering angle, final particle mass, and optional correction factors. When its squared value is larger, the modeled interaction is stronger at that selected angle. This page focuses on a tree level electromagnetic channel. It is useful for assignments, quick checks, reports, and early research notes. It is not a replacement for a full detector simulation.
Main Physical Inputs
The center of mass energy sets the Mandelstam variable s. The final particle mass controls the velocity factor beta. The angle changes the shape through cosine and sine terms. The charge values scale the result because electromagnetic coupling depends on particle charge. The color factor helps compare leptons and quarks. A correction multiplier can represent a simple higher order or efficiency adjustment.
Reading The Results
The squared invariant amplitude is dimensionless in this simplified model. The differential cross section is shown per steradian. The total cross section is integrated over angle. Event yield is estimated from the selected integrated luminosity. The Mandelstam t and u values are included for reference. They help connect the angular form with standard scattering formulas.
Careful Use
Use consistent GeV units for energy and mass. Keep the final mass below half the center of mass energy. Otherwise, pair production is below threshold. Use alpha inverse near 137 for low energy estimates. Use another value only when a running coupling is intended. Select color factor one for leptons. Select three for quark color counting. Review assumptions before publishing any number.
Practical Benefit
The calculator gives a transparent workflow. You enter inputs, submit the form, inspect the result block, then export a table. The CSV file supports spreadsheets. The PDF file supports classroom notes. The example table gives safe starting values. Clear formulas make each step easier to audit. It helps reduce routine manual checking work.
FAQs
What is pair annihilation?
Pair annihilation occurs when a particle and antiparticle interact and transform into other allowed final particles. This calculator models a simple electromagnetic channel.
What does invariant amplitude mean?
It is the central scattering quantity used to describe interaction strength. Its squared value connects directly with cross section estimates.
Which units should I use?
Use GeV for energy and mass. The cross section output is converted to picobarns for easier comparison.
What is beta in the result?
Beta is the final particle velocity factor in the center of mass frame. It approaches one for very light final particles.
Why is there a threshold warning?
Pair creation needs enough energy to create both final particles. The energy must exceed twice the final particle mass.
What color factor should I choose?
Use one for leptons. Use three for a simple quark color count when modeling quark pair production.
What does the correction multiplier do?
It scales the result for simple adjustment studies. It can represent a rough correction, efficiency, or comparison factor.
Is this a full collider simulation?
No. It is a transparent educational calculator. Detailed studies need full matrix elements, detector effects, and radiative corrections.