Feynman Diagram Kinematics Calculator
Compute the momentum and energies of two decay products at a Feynman diagram vertex, given the parent and daughter particle masses.
🕸️ What is the Feynman Diagram Kinematics Calculator?
This Feynman diagram kinematics calculator computes the momentum and energies of two particles produced at a 1-to-2 Feynman diagram vertex, such as a parent particle decaying into two daughter particles at rest.
At a decay vertex M → m₁ + m₂, energy-momentum conservation in the parent's rest frame uniquely fixes the daughter momentum p* (shared equally and oppositely by both daughters) and each daughter's energy E₁*, E₂*. These formulas are the starting point for computing invariant masses, decay angular distributions, and detector acceptance in real particle physics analyses.
A common point of confusion is assuming the daughter energies split evenly between the two particles. They only split evenly when the daughter masses are equal, if one daughter is heavier, it carries away more of the available energy while the momentum magnitude stays exactly the same for both, since momentum conservation (not energy conservation alone) fixes p*.
This calculator is useful for particle physics students working through decay kinematics problems, and for anyone checking whether a proposed decay is kinematically allowed before computing its rate or branching ratio.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Higgs boson to b-quark pair
Example 2 - J/ψ meson to muon pair
Example 3 - Charged kaon to pion pair (unequal masses)
❓ Frequently Asked Questions
🔗 Related Calculators
What is Feynman diagram vertex kinematics?
Vertex kinematics describes the energies and momenta of the particles meeting at a point (vertex) in a Feynman diagram. For a 1-to-2 decay vertex, where a parent particle of mass M splits into two daughters of mass m₁ and m₂, energy-momentum conservation fixes the daughter momentum and energies uniquely in the parent's rest frame.
What is the formula for two-body decay momentum?
p* = √{[M²−(m₁+m₂)²][M²−(m₁−m₂)²]} / (2M), where M is the parent mass and m₁, m₂ are the daughter masses. Both daughters share this same momentum magnitude p*, moving in opposite directions in the parent's rest frame.
What is the formula for the daughter particle energies?
E₁* = (M²+m₁²−m₂²) / (2M) and E₂* = (M²+m₂²−m₁²) / (2M). These always satisfy E₁*+E₂*=M, confirming total energy is conserved at the vertex.
When is a two-body decay kinematically forbidden?
A decay M → m₁ + m₂ is kinematically forbidden whenever M is less than m₁+m₂, because there would not be enough rest energy in the parent to create both daughter particles' rest masses, let alone give them any momentum.
What happens when the two daughter masses are equal?
When m₁=m₂, the formula simplifies: p* = √(M²−4m₁²)/2 and both daughters get exactly half the parent's mass-energy, E₁*=E₂*=M/2. This is the case for symmetric decays like J/ψ→μ⁺μ⁻.
What is a real example of two-body decay kinematics?
The J/ψ meson (mass 3.0969 GeV) decaying to a muon pair (each 0.105658 GeV) produces daughter muons with momentum p*≈1.5448 GeV/c and energy E*≈1.5485 GeV each, since the muon mass is tiny compared to half the J/ψ mass.
How does this relate to Mandelstam variables?
Two-body decay kinematics is a special case (1-to-2, at rest) of the more general 2-to-2 scattering kinematics described by the Mandelstam variables s, t, and u. Setting one incoming particle's momentum to zero and using only two outgoing particles recovers the two-body decay formulas.
Why is the daughter momentum the same magnitude for both particles?
Momentum conservation requires the total momentum before and after the decay to be equal. Since the parent is at rest (zero total momentum), the two daughters must have exactly equal and opposite momentum vectors, so their magnitudes are identical.
Does this calculator work for massless daughter particles?
Yes, setting m₁=0 (or m₂=0) is valid, for example a photon in a radiative decay. The formula simplifies smoothly, giving p*=(M²−m₂²)/(2M) when m₁=0.
What units should I use for the masses?
Any consistent mass/energy unit works (typically GeV in particle physics, using natural units where c=1), as long as the parent mass and both daughter masses use the same unit.