Relativistic Collisions and Threshold Energies

A collision conserves the total four-momentum of the incoming particles just as a decay does, but now two particles come in and (in general) different particles go out. The invariant mass of the initial state — its total four-momentum length — sets a hard ceiling on what the collision can produce, because it equals the total energy available in the center-of-momentum frame. This single invariant, written $s$, decides the threshold for creating a heavy particle, and its very different scaling for fixed-target versus colliding beams is the reason particle physics is done with colliders. The lesson closes with Compton scattering, the cleanest worked photon-electron collision.

Signature $(-,+,+,+)$; for the two-particle initial state $P^{\mu }=p_1^{\mu }+p_2^{\mu }$ and the invariant $s$ is defined below.

The invariant s as available energy

For a reaction with incoming four-momenta $p_1$ and $p_2$, the Mandelstam invariant $s$ is the squared invariant mass of the pair, written in energy-squared units as

Its square root is the total energy in the CM frame, $\sqrt{s}=E_{\text{CM}}$, because there ${\vec{p}}_1+{\vec{p}}_2=\vec{0}$ and $\sqrt{s}=E_1^{\ast }+E_2^{\ast }$. Since $s$ is built from the total four-momentum, it is conserved and frame-invariant: whatever the collision produces, the final state has the same $s$, so $\sqrt{s}$ is the total energy budget for making rest mass and kinetic energy in the CM frame.

An elastic collision keeps the same particles ($m_i$ unchanged, kinetic energy conserved in the CM frame); an inelastic collision converts kinetic energy into rest mass, producing heavier or additional particles. Relativity removes the classical distinction between kinetic and rest energy: the CM kinetic energy above threshold becomes the raw material for new mass.

Fixed target versus collider

Evaluate $s$ two ways. In a collider, two beams of equal energy $E$ meet head on with opposite momenta, so ${\vec{p}}_1+{\vec{p}}_2=\vec{0}$ and

The available energy is the full sum of beam energies. In a fixed-target setup, a beam particle of mass $m_1$ and lab energy $E_{\text{beam}}$ strikes a target of mass $m_2$ at rest. Then $P^{\mu }=(E_{\text{beam}}/c+m_{2}c,{\vec{p}}_{\text{beam}})$ and

At high beam energy the last term dominates, so

The available energy grows only as the square root of the beam energy: doubling $E_{\text{beam}}$ multiplies $\sqrt{s}$ by $\sqrt{2}$, not by $2$. A collider's $\sqrt{s}$ grows linearly with beam energy. Reaching $\sqrt{s}=14\text{TeV}$ head-on requires two $7\text{TeV}$ beams; matching it on a fixed target would need a beam of $E_{\text{beam}}=s/(2m_p{c}^2)\approx 10^{17}\text{eV}$, far beyond any accelerator. Most of a fixed-target beam's energy goes into moving the center of momentum forward, energy that cannot be tapped to make new particles.

Threshold energy for particle production

At threshold, the reaction has just enough energy to make the final-state particles, which are all produced at rest in the CM frame. The threshold condition is therefore

For a fixed target, insert $s=m_1^2{c}^4+m_2^2{c}^4+2m_2{c}^2{E}_{\text{beam}}$ and solve for the beam energy:

The threshold scales with the square of the produced mass — a heavy final state is expensive on a fixed target because the invariant mass must be built out of an inefficient $\sqrt{s}\sim \sqrt{E_{\text{beam}}}$.

Compton scattering

A photon of energy $E_{\gamma }=hc/\lambda$ strikes an electron at rest and scatters at angle $\theta$, the electron recoiling. Write the four-momenta before as $p_{\gamma }=(E_{\gamma }/c)(1,\hat{n})$ and $p_e=(m_{e}c,\vec{0})$, and after as $p_{\gamma }^{\prime }$ and $p_e^{\prime }$. Conservation gives $p_e^{\prime }=p_{\gamma }+p_e-p_{\gamma }^{\prime }$. Square both sides using the invariant $p_e^{\prime }\cdot p_e^{\prime }=-m_e^2{c}^2$ and the null condition $p_{\gamma }\cdot p_{\gamma }=p_{\gamma }^{\prime }\cdot p_{\gamma }^{\prime }=0$:

The electron terms give $p_e\cdot (p_{\gamma }-p_{\gamma }^{\prime })=-m_e(E_{\gamma }-E_{\gamma }^{\prime })$, and the photon cross term is $p_{\gamma }\cdot p_{\gamma }^{\prime }=-(E_{\gamma }{E}_{\gamma }^{\prime }/c^2)(1-\cos \theta )$. Substituting and cancelling,

Dividing by $E_{\gamma }{E}_{\gamma }^{\prime }{m}_e{c}^2$ and using $E_{\gamma }=hc/\lambda$ turns this into the wavelength shift

the Compton wavelength of the electron. The shift depends only on the scattering angle, not on the incident wavelength — a purely kinematic result of four-momentum conservation. Back-scattering ($\theta =180^{\circ }$) gives the maximum shift $2h/m_{e}c=4.85\text{pm}$; forward scattering ($\theta =0$) gives none. That the shift is measurable for X-rays but negligible for visible light is a matter of scale: $2.4\text{pm}$ against an X-ray's $\sim 100\text{pm}$ is a percent-level effect, against visible light's $\sim 500\text{nm}$ it is one part in $10^8$.

The recoil electron and the Compton edge

Solving the shift relation for the scattered photon energy gives

and energy conservation hands the balance to the electron as kinetic energy,

The electron takes the most energy when the photon back-scatters, $\theta =180^{\circ }$, giving the Compton edge

A monoenergetic photon beam scattering in a detector therefore deposits a continuous electron spectrum from zero up to $T_e^{\max }$, cut off sharply at the edge — never the full photon energy, because a real photon always survives with some energy. The gap between the edge and the full-energy photopeak is $E_{\gamma }-T_e^{\max }=E_{\gamma }/(1+2E_{\gamma }/m_e{c}^2)$, a signature used to calibrate gamma-ray spectrometers.

If instead a low-energy photon meets a highly relativistic electron, the same kinematics runs in reverse: the photon leaves the collision with far more energy than it entered, boosted by roughly ${\gamma }^2$. This inverse Compton scattering is how relativistic electrons in astrophysical sources upscatter starlight and microwave-background photons into X-rays and gamma rays, and it is the same four-momentum bookkeeping viewed from the electron's rest frame, where the process is ordinary Compton scattering.

Every result here reduced the kinematics to invariants — $s$, the squared four-momenta, the dot products $p_i\cdot p_j$ — that any frame computes to the same number. Systematizing those invariants for a general two-to-two process, and the single constraint that relates them, is the Mandelstam variables.