KAHIBARO
Table of Contents
Why this energy matters
In collider physics, the most useful measure of how much energy is available to create new particles is the center of mass energy. It is the total energy of the system measured in a special frame, the center of mass frame, where the total momentum is zero.
This quantity is important because particle creation depends on the energy that is actually available in the collision itself, not just on how energetic one particle looks in the laboratory. A machine may accelerate particles to very high speeds, but if much of the energy is tied up in the motion of the whole system, that energy cannot all be used to produce new mass.
The center of mass frame
Imagine two particles moving toward each other. If their momenta are equal in size and opposite in direction, the total momentum is zero. In that frame, called the center of mass frame, the collision is as direct as possible. The full energy of both particles contributes to the interaction.
If the particles have energies $E_1$ and $E_2$, then in the center of mass frame the total energy is simply
$$
E_{\text{cm}} = E_1 + E_2
$$
because the total momentum there is zero.
For relativistic particles, the more general and very important relation is
$$
E_{\text{cm}}^2 = \left(E_1 + E_2\right)^2 - \left(\mathbf{p}_1 + \mathbf{p}_2\right)^2 c^2
$$
where $\mathbf{p}_1$ and $\mathbf{p}_2$ are the momenta of the two particles.
The center of mass energy is the total relativistic energy measured in the frame where the total momentum is zero.
A key invariant formula is
$$
E_{\text{cm}}^2 = \left(E_1 + E_2\right)^2 - \left(\mathbf{p}_1 + \mathbf{p}_2\right)^2 c^2
$$
This value is the same in every reference frame, even though $E_1$, $E_2$, and the momenta separately may change from one frame to another.
Equal beam collider
The simplest case is a collider with two identical particles moving directly toward each other with equal energies. Their momenta cancel, so the laboratory frame is already the center of mass frame.
If each beam particle has energy $E$, then
$$
E_{\text{cm}} = 2E
$$
This is one of the big advantages of colliders. Nearly all of the beam energy is available for the collision.
For example, if two protons each have energy $7 \text{ TeV}$ and collide head on, then
$$
E_{\text{cm}} = 14 \text{ TeV}
$$
Fixed target compared with collider
To see why colliders are so powerful, compare them with a fixed target experiment. In a fixed target setup, one particle moves and the other is initially at rest. A large part of the incoming energy goes into the motion of the combined system after collision, so less energy is available for creating new particles.
For a beam particle with energy $E$ and momentum $p$, striking a target particle of rest mass $m$ at rest, the center of mass energy is
$$
E_{\text{cm}}^2 = \left(E + mc^2\right)^2 - p^2 c^2
$$
Using the relativistic relation
$$
E^2 = p^2 c^2 + m^2 c^4
$$
for identical beam and target particles, this becomes
$$
E_{\text{cm}}^2 = 2m^2 c^4 + 2Emc^2
$$
If the beam energy is much larger than the rest energy, then approximately
$$
E_{\text{cm}} \approx \sqrt{2Emc^2}
$$
This grows only as the square root of the beam energy, not directly with it.
For a fixed target experiment at high beam energy,
$$
E_{\text{cm}} \approx \sqrt{2Emc^2}
$$
For a head on collider with equal beams,
$$
E_{\text{cm}} = 2E
$$
This is why colliders are far more efficient than fixed target machines for reaching high collision energies.
A numerical comparison
Suppose a proton beam has energy $1000 \text{ GeV}$ and strikes a proton at rest. The proton rest energy is about $0.938 \text{ GeV}$. Then
$$
E_{\text{cm}} \approx \sqrt{2 \times 1000 \times 0.938}\ \text{GeV}
$$
$$
E_{\text{cm}} \approx \sqrt{1876}\ \text{GeV} \approx 43.3\ \text{GeV}
$$
So a $1000 \text{ GeV}$ fixed target beam gives only about $43 \text{ GeV}$ of center of mass energy.
Now compare that with two counter moving proton beams, each of energy $1000 \text{ GeV}$. Then
$$
E_{\text{cm}} = 2000\ \text{GeV}
$$
The difference is enormous.
| Setup | Beam energy description | Center of mass energy |
|---|---|---|
| Fixed target proton on proton | $1000\ \text{GeV}$ on stationary target | $\approx 43.3\ \text{GeV}$ |
| Collider proton on proton | two beams, each $1000\ \text{GeV}$ | $2000\ \text{GeV}$ |
Threshold for producing new particles
One major use of center of mass energy is deciding whether a reaction can happen at all. To produce a new particle, the collision must provide at least enough energy to account for the rest energy of the final particles.
If the final state contains particles with masses $m_1, m_2, \dots$, then the minimum possible center of mass energy is at least
$$
E_{\text{cm}} \ge \left(m_1 + m_2 + \cdots \right)c^2
$$
This is called the threshold condition.
For example, to produce a particle of mass $M$, the collision must have enough center of mass energy so that
$$
E_{\text{cm}} \ge Mc^2
$$
In practice, often more energy is needed because final particles may also carry kinetic energy.
A reaction is only possible if the center of mass energy is at least as large as the total rest energy of the produced particles:
$$
E_{\text{cm}} \ge \sum_i m_i c^2
$$
This threshold idea is central in high energy physics.
Invariant mass connection
The center of mass energy is closely related to invariant mass. For a system of particles, the invariant mass $M$ satisfies
$$
M c^2 = E_{\text{cm}}
$$
or equivalently
$$
M^2 c^4 = E_{\text{tot}}^2 - p_{\text{tot}}^2 c^2
$$
where $E_{\text{tot}}$ and $p_{\text{tot}}$ are the total energy and total momentum in any frame.
This means that the center of mass energy is not just a convenient quantity, it is a fundamental property of the particle system.
Visual picture
A fixed target collision has a moving center of mass, while a symmetric collider can have the center of mass nearly at rest in the laboratory frame.
Summary relation table
| Situation | Formula for $E_{\text{cm}}$ |
|---|---|
| General two particle system | $E_{\text{cm}}^2 = \left(E_1 + E_2\right)^2 - \left(\mathbf{p}_1 + \mathbf{p}_2\right)^2 c^2$ |
| Equal head on beams | $E_{\text{cm}} = 2E$ |
| Fixed target, identical particles | $E_{\text{cm}}^2 = 2m^2 c^4 + 2Emc^2$ |
| High energy fixed target approximation | $E_{\text{cm}} \approx \sqrt{2Emc^2}$ |
Final idea
Center of mass energy tells us the true energy budget of a collision. It explains why colliders are the preferred machines for discovering heavy particles and exploring new physics. Two beams colliding head on make much better use of beam energy than a beam striking a stationary target, which is why modern high energy physics relies so heavily on collider experiments.