Cross sections
In quantum field theory we can predict matrix elements $\braket{\psi|\phi}$ which lead to probabilites of a state $\ket{\phi}$ evolving into another one $\bra{\psi}$. In the Heisenber picture the time evolution is absorbed into the operators.
In scattering experiments we consider intial and final states $\ket{i}$ and $\ket{f}$ for $t\to\pm\infty$ and assume that they are momentum eigenstates called asymptotic states. The probability of an initial state $\ket{i}$ evolving into a final state $\ket{f}$ can be described in terms of the Scattering matrix or S-matrix
Definition: _S-matrix
The scattering matrix is defined as $$S_{fi} = \braket{f|S|i},$$ where $S$ is the time evolution operator in the Heisenberg picture, which projects from $t=-\infty$ to $t=\infty$.
The S-matrix can be calculated in quantum field theory. To compare calculations with experiments we need a measureable quantaty
Definition: Cross section
The cross section $\sigma$ is a measure of the probability of incoming particles to be scattered to a certain final state. It can be written as $$\sigma = \frac{1}{T\cdot \Phi}N,$$ where $T$ is the timespan of the measurement, $\Phi$ is the flux of incoming particles and $N$ the measured number of scattered particles
Definition: Luminosity
The Luminosity $\mathcal{L}$ is a defined by the number of scattering events over time and cross section $$\mathcal{L} = \frac{1}{\sigma}\frac{dN}{dt}$$ This is a accelerator specific quantity Experiments usually report the integrated Luminosity (over time) for a measurement
To compare a theoretical calculation with the number of events measured by an experiment calculates one uses $$N = \int d\sigma \times \int \mathcal{L} dt,$$ so that we
- calculate the diff. cross section $d\sigma$
- Multiply by integraded Luminosity
The cross section can be related to the S-matrix. We seperate the S-matrix into a free part ($1$) and an interacting part ($\mathcal{T}$)
Definition: Transfer matrix
The Transfer matrix $\mathcal{T}$ is the interacting part of the S-matrix and defined by $$S = 1 + i\mathcal{T}$$ The $i$ is convention
We can factor out momentum conservatoin to get the Matrix element $\mathcal{M}$
Definition: matrix element
The matrix element $\braket{f|\mathcal{M}|i}$ is nontrivial part of the interaction, defined by $$\mathcal{T} = (2\pi)^4 \delta^4(\sum p^\mu_i - \sum p^\mu_f) \mathcal{M}.$$
Cross sections can directly be computed when we know the matrix element $|\mathcal{M}|^2 = |\braket{f|\mathcal{M}|i}|^2$
Cross section formula
For an $2 \to n$ process with inital and final states $\ket{i}$ and $\ket{f}$ we have $$d\sigma = \frac{(2\pi)^4 |\mathcal{M}|^2}{4\sqrt{(p_1 \cdot p_2)^2 - m^2_1m^2_2}}d\Phi,$$ with the lorentz invariant phase space $\Phi$
Definition: Lorentz invariant phase space
The lorentz invariant phase space $\Phi$ is $$\Phi = \delta^4(\sum p^\mu_i - \sum p^\mu_f)\prod \frac{d^3p_j}{(2\pi)^3\frac{1}{2E_j},$$ where the product runs over all final states $j$.
The derivation inserts the S-matrix-element $|\frac{f|S|i}|^2$ for the probability in the formula of the cross section
Derivation cross section formula
For a single particle the flux is its velocity over the Volume it inhabits $\Phi = \frac{|\vec{v}|}{V}$. The probability $dP$ for a process is $$dP = \frac{f|S|i}{\braket{f|f}\braket{i|i}}d\Phi.$$ Note that intial and final states are not normalized to one, but rather $\braket{p|p}=2E_pV$, with the volume $V$, which can be found from the normalization of momentum eigenstates Pluggin all this into the original formula $$d\sigma = \frac{1}{T}\frac{1}{\Phi}dP,$$ will cancel out the Volume and lead to the given expression.
Example: $2\to 2$ scattering
For a $2\to 2$ scattering process $$p_1 + p_2 \to p_3 + p_4$$ We use the center of mass frame $\vec{p_1} = -\vec{p_2}$. The phase space becomes $$d\Phi = \delta^4(p^\mu_1 + p^\mu_2 - p^\mu_3 - p^\mu_4)\frac{d^3p_3}{(2\pi)^3}\frac{1}{2E_3}\frac{d^3p_4}{(2\pi)^3}\frac{1}{2E_4}.$$ We can execute the $\delta^3$ for the momentum part to integrate over $\vec{p}4$. The resulting integral over $\vec{p}3$ can formally be solved by substituion resulting in $$\left(\frac{d\sigma}{d\Omega}\right)\text{CM} = \frac{1}{64\pi^2 E^2\text{CM}}\frac{p_f}{p_i} |\mathcal{M}|^2 \Theta(E_\text{CM}-m_3-m_4)$$