Mandelstam Variables Calculator

Formula used

Mandelstam variables are Lorentz-invariant combinations of four-momenta for a 2→2 process. Using metric signature (+,−,−,−):

For on-shell external particles with masses m1..m4, the identity holds:

In the center-of-mass frame, when you provide √s and the scattering angle θ (between incoming 1 and outgoing 3), the calculator uses:

• E1 = (s + m1² − m2²) / (2√s), E2 = (s + m2² − m1²) / (2√s)
• E3 = (s + m3² − m4²) / (2√s), E4 = (s + m4² − m3²) / (2√s)
• |p_in| = √λ(s,m1²,m2²) / (2√s), |p_out| = √λ(s,m3²,m4²) / (2√s)
• t = m1² + m3² − 2(E1E3 − |p_in||p_out| cosθ)
• u = m1² + m4² − 2(E1E4 + |p_in||p_out| cosθ)

Here λ(a,b,c)=a²+b²+c²−2ab−2ac−2bc is the Källén function.

How to use this calculator

1. Select a calculation method. Choose the CM option if you know √s and θ.
2. Enter the four particle masses m1..m4 with a consistent mass unit.
3. Pick the energy unit for √s and the output invariants.
4. Provide the required known invariants or the scattering angle, then press Calculate.
5. Review the identity check. A tiny Δ indicates good numerical consistency.
6. Use the download buttons to export results as CSV or PDF.

Example data table

Sample 2→2 case with identical masses. Values are shown in GeV and GeV².

In this example, s+t+u matches m1²+m2²+m3²+m4² within rounding.

Article

1. Mandelstam variables in scattering analysis

Two body scattering appears in nuclear, particle, and astroparticle studies. Mandelstam variables are Lorentz-invariant combinations of four-momenta that describe the same interaction in any inertial frame. They summarize kinematics in three scalars that pair naturally with theoretical amplitudes and measured angular distributions.

2. Definitions and sign conventions

For a 2→2 process with external four-momenta p1,p2 → p3,p4 and metric (+,−,−,−), the invariants are defined as s=(p1+p2)², t=(p1−p3)², and u=(p1−p4)². Different sign conventions exist, so always keep your metric choice consistent when comparing formulas.

3. Physical meaning of s

The variable s is the total invariant mass squared of the initial state. In the center-of-mass frame, s equals (total energy)², so √s is the collision energy available to create final-state masses and kinetic energy. Threshold behavior is controlled by √s relative to (m3+m4).

4. Physical meaning of t and momentum transfer

The variable t measures momentum transfer between particle 1 and particle 3. In many scattering problems, small deflection angles correspond to small |t|, and differential cross sections are often plotted as dσ/dt. With the (+,−,−,−) metric, t is commonly negative for physical scattering at nonzero angle.

5. u channel and crossing symmetry

The variable u represents the alternative exchange channel. Many amplitudes can be related by exchanging labels of final particles, and analytic continuation between s, t, and u supports crossing symmetry arguments. This is one reason theorists prefer writing results directly in terms of s, t, and u.

6. Identity check and mass consistency

For on-shell external particles, the invariants satisfy s+t+u=m1²+m2²+m3²+m4². This calculator reports the identity check and Δ difference so you can quickly spot input mistakes, unit mismatches, or kinematic inconsistencies when solving for a missing invariant from the other two.

7. Center-of-mass inputs and the Kallen function

When you provide √s and a scattering angle, the calculator reconstructs CM energies and momentum magnitudes using the Kallen function λ(a,b,c). These steps enforce thresholds and compute t and u through energy-momentum relations, which is useful for unequal masses and realistic final states.

8. Practical workflow and exporting results

A practical workflow is to select the method matching your known quantities, keep mass and energy units consistent, and verify that √s exceeds both initial and final thresholds. Use the identity check to validate results, then export CSV for analysis scripts or PDF for lab notes and reports.

FAQs

1) What are Mandelstam variables used for?

They provide Lorentz-invariant kinematic inputs for 2→2 scattering, so amplitudes, cross sections, and experimental distributions can be compared without choosing a specific reference frame.

2) Why can t and u be negative?

With the (+,−,−,−) metric, spacelike momentum transfer typically gives negative squared four-momentum. Many physical scattering angles lead to negative t and u values.

3) What does the identity check mean?

For on-shell external particles, s+t+u must equal m1²+m2²+m3²+m4². The calculator shows Δ so you can confirm numerical consistency and catch unit mistakes.

4) When should I use the CM method?

Use it when you know the collision energy √s and the CM scattering angle. The calculator then computes s, t, and u while respecting thresholds and unequal masses.

5) Can I compute a missing variable from two known ones?

Yes. Choose the s–t, s–u, or t–u method, enter your two known invariants, and the tool computes the third using the on-shell identity.

6) Which units should I use?

Use a consistent energy unit for √s and outputs, and a consistent mass unit for m1..m4. The calculator converts internally and reports s, t, u in the selected unit squared.

7) Why is there a threshold warning?

If √s is below the required kinematic threshold, the final state cannot be produced with the given masses. The calculator warns you to prevent unphysical results.