Infrared problem

The infrared problem is an apparent problem in quantum field theories with massless particles.

The contributions of massless particles such as photons or gluons with very low energy lead to divergent components of the scattering amplitudes in quantum field theories. The cause of the problem is that, due to their vanishing mass, the particles can assume any low energy, or - equivalent to this - that the electromagnetic interaction is long-range.

The name of the problem comes from the fact that low energy photons have a proportionally low frequency . Electromagnetic waves of low frequency, i.e. long wavelengths , are known as infrared radiation .

Two different effects contribute to the infrared problem: On the one hand, the radiation or absorption of such low-energy particles leads to singularities, and on the other hand, these also appear as virtual particles with any small energy in quantum corrections. In quantum chromodynamics , the case also occurs that the gluons show self- interaction, i.e. massless particles themselves can emit massless particles. In all cases, the singularities can be avoided by introducing a small mass of the photon or gluon to regularize the theory ( Pauli-Villars regularization ) so that the smallest possible energy of the particle corresponds to this mass.

If these different contributions are added up, it turns out that the infrared problem is only an apparent problem; all divergent contributions cancel each other out exactly. In quantum electrodynamics this is known as the Bloch-Nordsieck theorem , in the general case, which includes quantum chromodynamics and quantum electrodynamics, the Kinoshita-Lee-Nauenberg theorem .

In axiomatic quantum field theory , the infrared problem is a problem that has been investigated until today (2008), for which there is still no generally accepted solution in the axiomatic framework.

example

In the annihilation of an electron-positron pair and the subsequent generation of a muon-antimuon pair , the renormalized scattering cross-section reads through virtual corrections ${\displaystyle e^{+}e^{-}\to \mu ^{+}\mu ^{-}}$

${\displaystyle \sigma _{V}={\frac {\alpha }{2\pi }}\sigma _{0}\left[-\ln ^{2}{\frac {m_{\gamma }^{2}}{Q^{2}}}-3\ln {\frac {m_{\gamma }^{2}}{Q^{2}}}-{\frac {7}{2}}+{\frac {\pi ^{2}}{3}}\right]}$

and that by the emission of an additional photon

${\displaystyle \sigma _{R}={\frac {\alpha }{2\pi }}\sigma _{0}\left[\ln ^{2}{\frac {m_{\gamma }^{2}}{Q^{2}}}+3\ln {\frac {m_{\gamma }^{2}}{Q^{2}}}+5-{\frac {\pi ^{2}}{3}}\right]}$,

where the fine structure constant , the scattering cross section are in leading order and the center of gravity energy. The parameter is the regularization parameter introduced as the photon mass. Both of these terms are individually divergent, but in their sum the contributions exactly cancel each other out. ${\displaystyle \alpha }$ ${\displaystyle \sigma _{0}}$ ${\displaystyle Q^{2}}$ ${\displaystyle m_{\gamma }}$

${\displaystyle \sigma _{V}+\sigma _{R}={\frac {3\alpha }{4\pi }}\sigma _{0}}$

Individual evidence

1. ^ Felix Bloch and Arold Nordsieck: Note on the Radiation Field of the Electron . In: Physical Review . tape 52 , no. 2 , 1937, pp. 54–59 , doi : 10.1103 / PhysRev.52.54 (English).
2. ↑ Toichiro Kinoshita: Mass Singularities of Feynman Amplitudes . In: Journal of Mathematical Physics . tape 3 , no. 4 , 1962, pp. 650 - 677 , doi : 10.1063 / 1.1724268 (English).
3. ↑ Tsung-Dao Lee and Michael Nauenberg: Degenerate Systems and Mass Singularities . In: Physical Review D . tape 133 , 6B, 1964, pp. B1549 - B1562 , doi : 10.1103 / PhysRev.133.B1549 (English).

literature

• Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 , pp. 355-380 (English).