Scattering Amplitude

The scattering amplitude $ \mathcal{M} $ is the central quantity that quantifies the likelihood that two particles, upon colliding, undergo a specific interaction. The squared modulus of this amplitude yields the physical probability of the process, from which the experimentally observable cross section is obtained.

In quantum physics, when two particles collide, several different reaction outcomes may occur.

For example, a beam of particles impinges on a target. Some particles continue essentially undeflected, while others are scattered into different directions.

Each scattering direction is associated with a certain probability. These probabilities are not calculated directly. Instead, one computes probability amplitudes.

An amplitude is a complex number whose squared modulus gives the probability of a specific outcome.

Note. In quantum physics, all processes are formulated in terms of probability amplitudes. When two or more distinct processes lead to the same final state, their amplitudes add coherently, not their probabilities. Only after summing the amplitudes does one take the squared modulus to obtain the physical probability.

In scattering processes, that is, collisions between particles, the scattering amplitude $ \mathcal{M} $ encodes all the dynamical information relevant to the interaction.

Once $ \mathcal{M} $ is known, any observable quantity can be computed. In particular, one can determine the cross section, which measures how likely the particles are to interact through a given channel.

The cross section $ \sigma $ is proportional to the squared modulus of the scattering amplitude.

\[ \sigma \propto |\mathcal{M}|^2 \]

Accordingly, a larger amplitude corresponds to a higher interaction probability, and a higher probability implies a larger cross section.

A Practical Example

Consider pion-proton scattering, involving a pion $ \pi $ and a proton $ p $.

$$ \pi + p \to \pi + p $$

Two isospin channels are allowed: one with total isospin $ I = \tfrac{3}{2} $ and one with $ I = \tfrac{1}{2} $.

Why are there two channels? In this case, the pion carries isospin $ I_1 = 1 $, while the proton carries isospin $ I_2 = \tfrac{1}{2} $. When isospins are combined, the total isospin $ I $ can assume all values between $ |I_1 - I_2| $ and $ I_1 + I_2 $, in integer or half-integer steps. Here: \[ |1 - \tfrac{1}{2}| = \tfrac{1}{2} \] \[ 1 + \tfrac{1}{2} = \tfrac{3}{2} \] Thus, the only allowed values of the total isospin are: \[ I = \tfrac{1}{2}, \ \tfrac{3}{2} \] This result is conventionally written as: \[1 \otimes \tfrac{1}{2} = \tfrac{3}{2} \oplus \tfrac{1}{2} \] indicating that pion-proton scattering can occur only in one of these two total isospin states.

Each channel is associated with its own scattering amplitude $ \mathcal{M} $:

$ \mathcal{M}_{3/2} $ for the channel with isospin $ \tfrac{3}{2} $
$ \mathcal{M}_{1/2} $ for the channel with isospin $ \tfrac{1}{2} $

Depending on the initial state, the scattering process may proceed through a single isospin channel or through a linear superposition of several channels.

In pion-proton scattering, the initial state can occur in three distinct configurations, determined by the electric charge of the pion:

$ \pi^+ + p $
$ \pi^0 + p $
$ \pi^- + p $

Each configuration corresponds to a different value of the third component of isospin and, as a consequence, to a different decomposition into the total isospin channels $ I = \tfrac{3}{2} $ and $ I = \tfrac{1}{2} $.

This follows from the fact that the pion forms an isospin triplet. It has isospin $ I = 1 $ and three possible values of $ I_3 $, which correspond directly to its three charge states:

$ \pi^+ $ has $ I_3 = +1 $
$ \pi^0 $ has $ I_3 = 0 $
$ \pi^- $ has $ I_3 = -1 $

The proton has isospin $ I = \tfrac{1}{2} $ and $ I_3 = +\tfrac{1}{2} $.

Combining the pion and the proton yields three initial states with different values of the total isospin projection:

$ I_3^{\text{tot}} = 1 + \frac12 = +\tfrac{3}{2} $ for $ \pi^+ + p $
$ I_3^{\text{tot}} = 0 + \frac12 = +\tfrac{1}{2} $ for $ \pi^0 + p $
$ I_3^{\text{tot}} = -1 + \frac12 = -\tfrac{1}{2} $ for $ \pi^- + p $

From the standpoint of isospin symmetry, these are physically distinct states, each characterized by a different decomposition into the total isospin channels $ I = \tfrac{3}{2} $ and $ I = \tfrac{1}{2} $.

Among them, only $ \pi^+ + p $ is a pure isospin state with $ I = \tfrac{3}{2} $. The remaining two configurations are mixed states, meaning they are linear superpositions of multiple isospin channels.

What distinguishes a pure from a mixed initial state?

If the initial state is pure, that is, it has a definite total isospin, only a single scattering amplitude contributes.

For example, when the initial state is $ \pi^+ + p $, the scattering amplitude is simply $ \mathcal{M}_{3/2} $.

\[ \mathcal{M} = \mathcal{M}_{3/2} \]

By contrast, if the initial state is mixed, as in the case of $ \pi^- + p $, multiple amplitudes contribute. These amplitudes add coherently and may interfere with one another.

\[ \mathcal{M} = c_{3/2} \mathcal{M}_{3/2} + c_{1/2} \mathcal{M}_{1/2} \]

Note. In certain situations, even when the initial state is mixed, the dynamics may effectively select a single isospin channel, leading to a simplified description. This scenario requires a more detailed dynamical analysis and is not discussed here.

In all cases, the squared modulus of the scattering amplitude $ \mathcal{M} $ determines the cross section $ \sigma $, which represents the probability that the reaction takes place.

\[ \sigma \propto |\mathcal{M}|^2 \]

For a more detailed discussion of this example, see my notes on pion-nucleon scattering.

And so on.