Formula Used
This calculator uses natural units, so c = 1 and hbar = 1. Use the same energy scale for every mass, energy, and momentum input.
Mandelstam s: s = 2mᵢ² + 2(E1E2 - p1p2 cos θᵢ)
Center of mass energy: √s
Final beta factor: β = √(1 - 4mᶠ² / s)
Angular term: A = 1 + cos²θ + (1 - β²)sin²θ
Coupling: e² = 4πα
Spin averaged amplitude model: |M|² = e⁴(QᵢQᶠ)²A
Differential cross section density: dσ/dΩ = α²(QᵢQᶠ)²βA / 4s
The model is a transparent tree level learning form. Advanced work may need full spinor traces, propagator factors, weak exchange, loop corrections, and channel specific constants.
How to Use This Calculator
Enter the incoming particle rest mass and final particle rest mass. Keep all values in one energy unit scale. Add both incoming energies and momentum magnitudes. Use 180 degrees for a head on collision. Use the final scattering angle for the outgoing pair. Press the calculate button. Review s, beta, threshold status, and the invariant amplitude squared.
Use the CSV option for spreadsheets. Use the PDF option for a compact record. Check the consistency rows if the result looks unusual. A large difference between E² - p² and m² often means mixed units or unsuitable input values.
Example Data Table
| Example |
mᵢ |
mᶠ |
E1 |
E2 |
p1 |
p2 |
θᵢ |
θ |
Use Case |
| Muon pair study |
0.51099895 |
105.6583755 |
150 |
150 |
149.999 |
149.999 |
180 |
60 |
Electron positron annihilation |
| Higher beam check |
0.51099895 |
105.6583755 |
500 |
500 |
499.999 |
499.999 |
180 |
90 |
Angular comparison |
| Threshold test |
0.51099895 |
105.6583755 |
106 |
106 |
105.999 |
105.999 |
180 |
45 |
Near threshold behavior |
Advanced pair annihilation study
Pair annihilation is a core idea in quantum field practice. A particle meets its antiparticle. Their four momenta combine. The process can create another pair, photons, or other allowed states. This calculator gives a compact study model for an s channel exchange. It is built for classroom checks, quick notes, and early Feynman diagram work.
Natural unit workflow
The tool uses natural units. That means c and hbar are treated as one. Energies, momenta, and masses should share the same scale. The incoming angle controls the invariant mass of the collision. A head on setup uses one hundred eighty degrees. A same direction setup uses zero degrees. The final scattering angle controls the angular factor in the amplitude model.
Reading the output
The result begins with the Mandelstam s value. Its square root is the center of mass energy. The calculator also estimates the final beta factor. This shows whether the final pair is allowed above threshold. If the energy is below threshold, the beta value is clamped to zero. The output also lists t and u estimates for a center of mass view.
Amplitude meaning
The invariant amplitude shown here is spin averaged. It follows a common tree level QED style form. The charge factor lets you scale the interaction strength. The fine structure constant sets the electromagnetic coupling. The angular term becomes simple when the produced particles are light. It changes near threshold because beta becomes small.
Best use
Use the result as a guide, not a full event generator. Real calculations may need spinors, polarization sums, loop terms, widths, and detector cuts. Some reactions also need weak exchange, color factors, or identical particle rules. This page is best for transparent learning. It shows each main term before the final number. CSV and PDF exports help save examples for assignments, lab notes, and comparison tables.
Good input practice
Start with known rest masses. Then enter beam energies from your problem statement. Use momenta that match the same unit scale. Check the energy consistency line after every run. Large mismatch values usually mean mixed units. Try one clean example first. Then change one variable at a time. This makes trends easy to see. It also prevents hidden setup errors during revision before sharing final results.
FAQs
What does invariant amplitude mean?
It is the Lorentz invariant matrix element for a reaction. This page reports a spin averaged squared model for a simple pair annihilation channel.
Which units should I use?
Use one consistent energy scale. MeV is common. If mass is in MeV, then energies and momenta should also use MeV based natural units.
Why is the incoming angle important?
It changes the total invariant energy. A head on collision gives higher s than particles moving in the same direction at equal energies.
What does beta show?
Beta is the final state speed factor in natural units. It becomes zero below threshold and grows as center of mass energy increases.
Why is my process below threshold?
The center of mass energy is lower than twice the final particle mass. Raise beam energy or choose a lighter final particle pair.
Is this a complete Feynman calculation?
No. It is a study model. Full work can require spinor traces, propagators, widths, loop terms, and reaction specific corrections.
What does the charge factor do?
It scales the interaction strength. Use one for a basic unit charge example. Other values can approximate charge product effects.
Can I export the result?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a simple printable calculation summary.