What is a cross section, really?

Question

Upon looking at different resources, there is a common definition of a cross section (in the context of QFT) to be the probability that some scattering process occurs. For example, here is a definition given in Wikipedia

“The cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation (e.g. a particle beam, sound wave, light, or an X-ray) intersects a localized phenomenon (e.g. a particle or density fluctuation”.

But if that is the case, then how does it differ conceptually from the square of the matrix element in QFT? It’s the square of the matrix element that physically corresponds to the probability that some scattering process occurs. The Fermi’s golden rule relates the cross section to the square of the matrix element, so how can they both be probabilities?

Answer 1

Cross sections are quoted in area (usually barn with some prefix), which is like a measure of how small of a target you have to hit. The matrix element gives a probability of a scattering event, that depends on some variables, usually incidence angles, speeds etc. The matrix element is proportional to the differential cross section. This is the infinitesimal cross sectional area per infinitesimal output solid angle, $\frac{d\sigma}{d\Omega}$. Once you integrate that over all solid angles, you get your cross section (an area)

Answer 2

The (differential) cross section for some scattering process is directly proportional to the (square of the) matrix element for that process. Conceptually they are both proportional to the probability that the process occurs. But they have different units and have slightly different meanings. Neither is a pure probability by itself, but both are closely related to the probabilities of a transition.

Let's be explicit about our terms.$% \newcommand{\bra}[1]{\left< #1 \right|} \newcommand{\ket}[1]{\left| #1 \right>} \newcommand{\braket}[1]{\left< #1 \right>} $ The "matrix element" connecting two states $\ket a$ and $\ket b$ is generally the integral $\mathcal H_{ab} = \braket{a\middle|\hat H\middle| b}$, where $\hat H$ is the Hamiltonian operator. If the states are properly normalized, so that $\braket{a\middle|b} = \delta_{ab}$, then the matrix element has the same units has the Hamiltonian operator: an energy.

To turn a matrix element into a decay probability, you find the decay width, $\Gamma_{ab} \propto \left|\mathcal H_{ab} \right|^2 \rho(b)$, which also has units of energy. (Here $\rho$ is the density of states around the final state, which has units of states (dimensionless) per unit energy.) The dimensionless probability that a state has decayed through a particular pathway during some time interval $\Delta t$ is $e^{-\Delta t \,\Gamma / \hbar}$. This is quite a bit of massaging, and requires some specification about your experimental setup.

To turn a matrix element into a cross section, you have to do even more massaging, because the initial and final momenta and energies and the spin degeneracy and the phase space all come into the calculation. Here's a nice writeup that keeps all of the dimensionful constants, rather than the handwaving practice of setting $\hbar = c = \pi = 2\pi = 1$ which is common in order-of-magnitude estimates. Note that the linked result includes an unphysical "normalization volume," because the plane-wave initial states can't actually be normalized.

An operational definition of a cross section is to imagine a scattering experiment from a thin fixed target, with thickness $\ell$ and number density $n$. In the approximation where you can ignore multiple scattering (thus a "thin" target), you find that the beam is exponentially attenuated: the ratio of the incident beam intensity to the unscattered beam intensity goes like $e^{-n\ell s}$, where the $s$ stands for "something." Dimensional analysis tells you that $s$ must have units of area. For the case of hard-sphere scattering, $s$ actually does turn out to be the cross-sectional area of the spheres, so we retain that name and promote the $s$ to its Greek-alphabet counterpart to make it seem like we're educated.

Probabilities are dimensionless. Neither the cross-section nor the matrix element is dimensionless, so neither is a probability.