Mandelstam Variables Calculator
Calculate the Mandelstam variables s, t, and u for massless 2-to-2 particle scattering from the center-of-mass energy and angle.
📐 What is the Mandelstam Variables Calculator?
This Mandelstam variables calculator computes the three Lorentz-invariant kinematic variables s, t, and u for a massless 2-to-2 particle scattering process, given the center-of-mass energy √s and the scattering angle θ.
Mandelstam variables are the standard language for describing scattering kinematics in particle physics. s (the square of the total center-of-mass energy) sets the energy scale of a collision, resonances in s-channel processes appear as peaks when s equals the square of an intermediate particle's mass. t and u describe momentum transfer in forward and backward exchange (t-channel and u-channel) processes, and are the natural variables in which differential cross sections dσ/dt are written.
A useful sanity check is the identity s + t + u = Σm² (the sum of the squares of all four particle masses in the process). For massless particles, that sum is exactly zero, so a correct calculation always returns s + t + u = 0 up to floating-point rounding.
This calculator is useful for particle physics students working through scattering kinematics problems, and for anyone verifying a hand calculation of s, t, and u for a collider process.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Symmetric 90° scattering
Example 2 - LHC-scale collision at small angle
Example 3 - Full backscattering
❓ Frequently Asked Questions
🔗 Related Calculators
What are the Mandelstam variables?
The Mandelstam variables s, t, and u are three Lorentz-invariant quantities used to describe the kinematics of a 2-to-2 particle scattering or decay process. s is the square of the total center-of-mass energy, while t and u are squared momentum transfers between the incoming and outgoing particles.
What is the formula for s, t, and u in massless 2-to-2 scattering?
For massless particles, s=(√s)², t=−s/2×(1−cosθ), and u=−s/2×(1+cosθ), where √s is the center-of-mass energy and θ is the scattering angle in the center-of-mass frame.
Why does s + t + u always equal zero for massless particles?
In general, s+t+u equals the sum of the squares of all four particle masses involved in the scattering. For massless particles that sum is zero, so s+t+u=0 exactly, a useful identity for checking a calculation.
What do s, t, and u physically represent?
s is the square of the total energy available in the center-of-mass frame (relevant for resonance production). t is the squared momentum transfer in the forward-scattering direction (relevant for t-channel exchange processes). u is the squared momentum transfer in the backward direction (relevant for u-channel exchange processes).
What happens to t and u at forward and backward scattering angles?
At θ=0° (the outgoing particle continues forward, no deflection), t=0 and u=−s. At θ=180° (the outgoing particle is scattered straight back), t=−s and u=0. These are the two extreme kinematic limits.
Why are t and u negative?
By convention, t and u represent squared four-momentum transfers for physical (spacelike) exchanges in 2-to-2 scattering, which are negative or zero. Only s, the squared total energy, is positive for a physical process.
Does this calculator work for massive particles?
This calculator uses the massless approximation, which is highly accurate whenever √s is much larger than all four particle masses, as is typical at high-energy colliders. For scattering near threshold, where masses are comparable to √s, the full mass-dependent Mandelstam formulas are needed instead.
What is a real example use of Mandelstam variables?
At the LHC, proton-proton collisions at √s=13,000 GeV probe parton-level scattering processes described by s, t, and u. The differential cross section dσ/dt for many QCD and electroweak processes is written directly in terms of these variables.
How is the scattering angle θ defined?
θ is the polar scattering angle measured in the center-of-mass frame, between the direction of one incoming particle and the direction of one outgoing particle. It ranges from 0° (no deflection) to 180° (full backscattering).
Are s, t, and u independent of each other?
No, for a 2-to-2 process with fixed masses, only two of the three variables are independent, the third is fixed by the identity s+t+u=Σm². Once you know √s and the scattering angle, all three are determined.