# Pair production

Pair production refers to the creation of an elementary particle and its antiparticle, usually when a photon (or another neutral boson) interacts with a nucleus or another boson. For example an electron and its antiparticle, the positron, may be created. This is allowed, provided there is enough energy available to create the pair – at least the total rest mass energy of the two particles – and that the situation allows both energy and momentum to be conserved. Other pairs produced could be a muon and anti-muon or a tau and anti-tau. However, all other conserved quantum numbers (angular momentum, electric charge, lepton number) of the produced particles must sum to zero – thus the created particles shall have opposite values of each other. For instance, if one particle has electric charge of +1 the other must have electric charge of −1, or if one particle has strangeness of +1 then another one must have strangeness of −1. The probability of pair production in photon-matter interactions increases with increasing photon energy and also increases with atomic number approximately as Z2.

## Examples

γ + γ  → e + e+

In nuclear physics, this occurs when a high-energy photon interacts with a nucleus. The energy of this photon can be converted into mass through Einstein’s equation, E=mc2; where E is energy, m is mass and c is the speed of light. The photon must have enough energy to create the mass of an electron plus a positron. The rest mass of an electron is 9.11 × 10−31 kg (0.511 MeV), the same as a positron. Without a nucleus to absorb momentum, a photon decaying into electron-positron pair (or other pairs for that matter) can never conserve energy and momentum simultaneously.[1]

## Photon–nucleus interaction

There are different processes how an electron-positron pair can be produced. In air (e.g. in lightning discharges) the most important one is the scattering of photons at the nuclei of atoms or molecules. Quantum mechanically, the process of pair production can be described by the quadruply differential cross section:[2]

\begin{align} d^4\sigma &= \frac{Z^2\alpha_{fine}^3c^2}{(2\pi)^2\hbar}|\mathbf{p}_+||\mathbf{p}_-| \frac{dE_+}{\omega^3}\frac{d\Omega_+ d\Omega_- d\Phi}{|\mathbf{q}|^4}\times \\ &\times\left[- \frac{\mathbf{p}_-^2\sin^2\Theta_-}{(E_--c|\mathbf{p}_-|\cos\Theta_-)^2}\left (4E_+^2-c^2\mathbf{q}^2\right)\right.\\ &-\frac{\mathbf{p}_+^2\sin^2\Theta_+}{(E_+-c|\mathbf{p}_+|\cos\Theta_+)^2}\left (4E_-^2-c^2\mathbf{q}^2\right) \\ &+2\hbar^2\omega^2\frac{\mathbf{p}_+^2\sin^2\Theta_++\mathbf{p}_-^2\sin^2\Theta_-}{(E_+-c|\mathbf{p}_+|\cos\Theta_+)(E_--c|\mathbf{p}_-|\cos\Theta_-)} \\ &+2\left.\frac{|\mathbf{p}_+||\mathbf{p}_-|\sin\Theta_+\sin\Theta_-\cos\Phi}{(E_+-c|\mathbf{p}_+|\cos\Theta_+)(E_--c|\mathbf{p}_-|\cos\Theta_-)}\left(2E_+^2+2E_-^2-c^2\mathbf{q}^2\right)\right]. \\ \end{align}

with

\begin{align} d\Omega_+&=\sin\Theta_+\ d\Theta_+,\\ d\Omega_-&=\sin\Theta_-\ d\Theta_-. \end{align}

This expression can be derived by using a quantum mechanical symmetry between pair production and Bremsstrahlung.
$Z$ is the atomic number, $\alpha_{fine}\approx 1/137$ the fine structure constant, $\hbar$ the reduced Planck's constant and $c$ the speed of light. The kinetic energies $E_{kin,+/-}$ of the positron and electron relate to their total energies $E_{+,-}$ and momenta $\mathbf{p}_{+,-}$ via

$E_{+,-}=E_{kin,+/-}+m_e c^2=\sqrt{m_e^2 c^4+\mathbf{p}_{+,-}^2 c^2}.$

Conservation of energy yields

$\hbar\omega=E_{+}+E_{-}.$

The momentum $\mathbf{q}$ of the virtual photon between incident photon and nucleus is:

\begin{align} -\mathbf{q}^2&=-|\mathbf{p}_+|^2-|\mathbf{p}_-|^2-\left(\frac{\hbar}{c}\omega\right)^2+2|\mathbf{p}_+|\frac{\hbar}{c} \omega\cos\Theta_+ +2|\mathbf{p}_-|\frac{\hbar}{c} \omega\cos\Theta_- \\ &-2|\mathbf{p}_+||\mathbf{p}_-|(\cos\Theta_+\cos\Theta_-+\sin\Theta_+\sin\Theta_-\cos\Phi), \end{align}

where the directions are given via:

\begin{align} \Theta_+&=\sphericalangle(\mathbf{p}_+,\mathbf{k}),\\ \Theta_-&=\sphericalangle(\mathbf{p}_-,\mathbf{k}),\\ \Phi&=\text{Angle between the planes } (\mathbf{p}_+,\mathbf{k}) \text{ and } (\mathbf{p}_-,\mathbf{k}), \end{align}

where $\mathbf{k}$ is the momentum of the incident photon.

In order to analyse the relation between the photon energy $E_+$ and the emission angle $\Theta_+$ between photon and positron, Köhn and Ebert integrated [3] the quadruply differential cross section over $\Theta_-$ and $\Phi$. The double differential cross section is:

\begin{align} \frac{d^2\sigma (E_+,\omega,\Theta_+)}{dE_+d\Omega_+} = \sum\limits_{j=1}^{6} I_j \end{align}

with

\begin{align} I_1&=\frac{2\pi A}{\sqrt{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+}} \\ &\times \ln\left(\frac{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+-\sqrt{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2 \Theta_+}(\Delta^{(p)}_1+\Delta^{(p)}_2)+\Delta^{(p)}_1\Delta^{(p)}_2}{-(\Delta^{(p)}_2) ^2-4p_+^2p_-^2\sin^2\Theta_+ -\sqrt{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2 \Theta_+}(\Delta^{(p)}_1-\Delta^{(p)}_2)+\Delta^{(p)}_1\Delta^{(p)}_2 }\right) \\ &\times\left[-1-\frac{c\Delta^{(p)}_2}{p_-(E_+-cp_+\cos\Theta_+)}+\frac{p_+^2c^2\sin^2\Theta_+} {(E_+-cp_+\cos\Theta_+)^2}-\frac{2\hbar^2\omega^2p_-\Delta^{(p)}_2}{c(E_+-cp_+\cos \Theta_+)((\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+)}\right], \\ I_2&=\frac{2\pi Ac}{p_-(E_+-cp_+\cos\Theta_+)}\ln\left( \frac{E_-+p_-c}{E_--p_-c}\right), \\ I_3&=\frac{2\pi A}{\sqrt{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+ }} \\ &\times\ln\Bigg(\Big((E_-+p_-c)(4p_+^2p_-^2\sin^2\Theta_+(E_--p_-c)+(\Delta^{(p)}_1+\Delta^{(p)}_2) ((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c) \\ &-\sqrt{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+}))\Big)\Big((E_--p_-c) (4p_+^2p_-^2\sin^2\Theta_+(-E_--p_-c) \\ &+(\Delta^{(p)}_1-\Delta^{(p)}_2) ((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)-\sqrt{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+}))\Big)^{-1}\Bigg) \\ &\times\left[\frac{c(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)}{p_-(E_+-cp_+\cos\Theta_+)}\right.\\ &+\Big[((\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+)(E_-^3+E_-p_-c)+p_-c(2 ((\Delta^{(p)}_1)^2-4p_+^2p_-^2\sin^2\Theta_+)E_-p_-c \\ &+\Delta^{(p)}_1\Delta^{(p)}_2(3E_-^2+p_-^2c^2))\Big]\Big[(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+\Big]^{-1} \\ &+\Big[-8p_+^2p_-^2m^2c^4\sin^2\Theta_+(E_+^2+E_-^2)-2\hbar^2\omega^2p_+^2\sin^2\Theta_+p_-c(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c) \\ &+2\hbar^2\omega^2p_- m^2c^3(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)\Big] \Big[(E_+-cp_+\cos\Theta_+)((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)\Big]^{-1} \\ &+\left.\frac{4E_+^2p_-^2(2(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2-4m^2c^4p_+^2p_-^2\sin^2\Theta_+)(\Delta^{(p)}_1E_-+\Delta^{(p)}_2p_-c)}{((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)^2}\right], \\ I_4&=\frac{4\pi Ap_-c(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)}{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+}+\frac{16\pi E_+^2p_-^2 A(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2}{((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)^2}, \\ I_5&=\frac{4\pi A}{(-(\Delta^{(p)}_2)^2+(\Delta^{(p)}_1)^2-4p_+^2p_-^2\sin^2\Theta_+) ((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)} \\ &\times\left[\frac{\hbar^2\omega^2p_-^2}{E_+cp_+\cos\Theta_+} \Big[E_-[2(\Delta^{(p)}_2)^2((\Delta^{(p)}_2)^2-(\Delta^{(p)}_1)^2)+8p_+^2p_-^2\sin^2\Theta_+((\Delta^{(p)}_2)^2+(\Delta^{(p)}_1)^2)] \right.\\ &+p_-c[2\Delta^{(p)}_1\Delta^{(p)}_2((\Delta^{(p)}_2)^2-(\Delta^{(p)}_1)^2)+16\Delta^{(p)}_1\Delta^{(p)}_2p_+^2p_-^2\sin^2\Theta_+]\Big]\Big[(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+\Big]^{-1}\\ &+ \frac{2\hbar^2\omega^2 p_{+}^2 \sin^2\Theta_+(2\Delta^{(p)}_1\Delta^{(p)}_2 p_-c+2(\Delta^{(p)}_2)^2E_-+8p_+^2p_-^2\sin^2\Theta_+ E_-)}{E_+-cp_+\cos\Theta_+}\\ &-\Big[2E_+^2p_-^2\{2((\Delta^{(p)}_2)^2-(\Delta^{(p)}_1)^2)(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2 +8p_+^2p_-^2\sin^2\Theta_+[((\Delta^{(p)}_1)^2+(\Delta^{(p)}_2)^2)(E_-^2+p_-^2c^2)\\ &+4\Delta^{(p)}_1\Delta^{(p)}_2E_-p_-c]\}\Big]\Big[(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+\Big]^{-1}\\ &-\left.\frac{8p_+^2p_-^2\sin^2\Theta_+(E_+^2+E_-^2)(\Delta^{(p)}_2p_-c +\Delta^{(p)}_1 E_-)}{E_+-cp_+\cos\Theta_+}\right], \\ I_6&=-\frac{16\pi E_-^2p_+^2\sin^2\Theta_+ A}{(E_+-cp_+\cos\Theta_+)^2 (-(\Delta^{(p)}_2)^2+(\Delta^{(p)}_1)^2-4p_+^2p_-^2\sin^2\Theta_+)} \end{align}

and

\begin{align} A&=\frac{Z^2\alpha_{fine}^3c^2}{(2\pi)^2\hbar}\frac{|\mathbf{p}_+||\mathbf{p}_-|}{\omega^3},\\ \Delta^{(p)}_1&:=-|\mathbf{p}_+|^2-|\mathbf{p}_-|^2-\left(\frac{\hbar}{c}\omega\right) + 2\frac{\hbar}{c}\omega|\mathbf{p}_+|\cos\Theta_+,\\ \Delta^{(p)}_2&:=2\frac{\hbar}{c}\omega|\mathbf{p}_i|-2|\mathbf{p}_+||\mathbf{p}_-| \cos\Theta_+ + 2. \end{align}

This cross section can be applied in Monte Carlo simulations. An analysis of this expression shows that positrons are mainly emitted in the direction of the incident photon.

## Energy

Photon-nucleus pair production can only occur if the photons have an energy exceeding twice the rest energy (mec2) of an electron (1.022 MeV). These interactions were first observed in Patrick Blackett's counter-controlled cloud chamber, leading to the 1948 Nobel Prize in Physics. The same conservation laws apply for the generation of other higher energy particles such as the muon and tau.

In semiclassical general relativity, pair production was invoked to predict hypothetical Hawking radiation. According to quantum mechanics, particle pairs are constantly appearing and disappearing as a quantum foam. In a region of strong gravitational tidal forces, the two particles in a pair may sometimes be wrenched apart before they have a chance to mutually annihilate. When this happens in the region around a black hole, one particle may escape while its antiparticle partner is captured by the black hole.

Pair production is also the mechanism behind the hypothesized pair instability supernova type of stellar explosion, where pair production suddenly lowers the pressure inside a supergiant star, leading to a partial implosion, and then explosive thermonuclear burning. Supernova SN 2006gy is hypothesized to have been a pair production type supernova.

In 2008 the Titan laser aimed at a 1-millimeter-thick gold target was used to generate positron–electron pairs in large numbers.[4]