Goldstone theorem

The Goldstone theorem is a theorem of theoretical physics , which in solid state physics and quantum field theory is applied. It says that mass- less particles appear in theories with spontaneously broken symmetry :

• If the broken symmetry group is a group above the usual number range, then these particles obey the Bose-Einstein statistics and are therefore referred to as Goldstone or Nambu-Goldstone bosons .
• If an approximate symmetry is also explicitly broken, then particles with low mass appear, which can be interpreted as pseudo Goldstone bosons .
• If the symmetry is based on a super algebra ( supersymmetry ), particles also appear which satisfy the Fermi-Dirac statistics ; such particles are called Goldstone fermions .

The Goldstone bosons were discovered by Yoichiro Nambu as part of investigations into superconductivity . Jeffrey Goldstone worked the theory further and expanded it to the field of quantum field theory.

Solid state physics

One application of Goldstone's theorem in solid-state physics concerns ferromagnetism : In ferromagnetic materials the laws that describe them are invariant under rotations in space. Above the Curie temperature , the magnetization is equal to zero - that is, it is also invariant under spatial rotations. Below the Curie temperature, however, the magnetization has a constant value different from zero and points in a certain direction, the preferred direction; the invariance (symmetry) under spatial rotations is broken. In this case, the Goldstone bosons are magnons : quasiparticles that represent a magnetic spin wave.

Particle physics

Spontaneous symmetry breaking in particle physics is equivalent to the fact that the Lagrangian of theory is invariant under the operation of a symmetry group , but the vacuum state is not. For every broken generator of the symmetry group, an additional particle arises in the theory. The mass of the Goldstone boson is protected by symmetry; no mass term is generated by quantum corrections . All Goldstone bosons of the Standard Model have spin  0 and parity  −1, so are pseudoscalar bosons .

If a global symmetry is broken, the Goldstone bosons appear as physically observable particles in the particle zoo . This is the case with the approximate symmetry breaking in quantum chromodynamics , in which the pions represent the quasi-Goldstone bosons.

If a local gauge symmetry is broken, the Goldstone bosons do not appear as observable particles. With the unitary calibration , a calibration can always be chosen in which the Goldstone bosons decouple, i.e. H. are inert and not subject to any interaction. Due to the Higgs mechanism , spontaneous symmetry breaking gives the gauge bosons their mass, and there are as many massive gauge bosons as Goldstone bosons. Therefore one speaks of the Goldstone bosons corresponding to the gauge bosons. In the context of the Goldstone boson equivalence theorem , the Goldstone bosons correspond to the longitudinal modes of the massive gauge bosons; In the technical jargon this is so called that the gauge bosons “eat up” the Goldstone bosons.

In the context of perturbation theory and its visualization with the aid of Feynman diagrams , the Goldstone bosons appear as virtual particles that propagate . The Feynman rules assign a mass to these virtual Goldstone bosons depending on the calibration:

• in unitary calibration the Goldstone bosons have an infinitely large mass;
• the calibration that leaves the Goldstone bosons massless is called the Landau calibration ;
• In the Feynman calibration , in which the propagator of the massive gauge bosons is structurally identical to that of the massless gauge bosons, the Goldstone bosons have the same mass as the corresponding gauge bosons.

The Goldstone fermions of the supersymmetric theories are called Goldstinos . In the case of global symmetry, this is an ordinary particle; in the case of local symmetry, it gives the gravitino its mass, analogous to the Higgs mechanism . The boson super partners of the Goldstinos are called Sgoldstinos .

Example: Chiral symmetry breaking in QCD

An example of Goldstone bosons in quantum chromodynamics  (QCD) are the pions : the masses of the two light u and d quarks are almost 0 compared to the mass scale of the strong interaction , so that the strong interaction has an approximately global symmetry (a chiral symmetry which transforms left and right handed fields independently), d. i.e., it is invariant under the transformation ${\displaystyle SU(2)_{L}\times SU(2)_{R}}$

${\displaystyle {\begin{pmatrix}u_{L}\\d_{L}\end{pmatrix}}\mapsto U{\begin{pmatrix}u_{L}\\d_{L}\end{pmatrix}},}$

${\displaystyle {\begin{pmatrix}u_{R}\\d_{R}\end{pmatrix}}\mapsto V{\begin{pmatrix}u_{R}\\d_{R}\end{pmatrix}},}$

wherein and independent - matrices are. ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle SU(2)}$

The QCD vacuum breaks this symmetry spontaneously, only one is observed in the particle spectrum , which rotates left- and right-handed components simultaneously (i.e. the matrices and in the above transformation must be identical. This remaining symmetry is known as isospin symmetry). The pions play the role of the Goldstone bosons. ${\displaystyle SU(2)}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle SU(2)_{L+R}}$

However, since u and d quarks are not exactly massless (only then can left and right-handed quark fields be transformed independently of each other), the symmetry is not only spontaneous, but also explicitly broken, so that the pions are not exactly massless either. ${\displaystyle SU(2)_{L}\times SU(2)_{R}}$

However, their mass is very small compared to the mass of a proton or neutron ( ) and, in particular, considerably smaller than would be expected if one counts constituents: a pion is composed of a quark-antiquark pair as a meson , while a proton or neutron as Baryons each consist of three quarks; naively calculated, a pion should have about the mass of a proton or neutron. ${\displaystyle m_{\pi ^{+}}/m_{p^{+}}=0{,}15}$ ${\displaystyle 2/3}$

This effect also occurs in a weakened form with the kaons , i.e. with mesons with a strange quark , which is also one of the light quarks : They are only about half as heavy as the Λ-baryon or the Σ-baryons .

See also

• Goldstone Boson Equivalence Theorem

literature

• J. Goldstone: In: Il Nuovo Cimento. 19/1961, p. 154
• J. Goldstone, A. Salam , S. Weinberg : Broken Symmetries . In: Physical Review , 127/3/1962, pp. 965-970, ISSN  0031-899X

Web links

• Goldstone Boson. (No longer available online.) In: Astronomy Knowledge Data Base (U Ottawa). Archived from the original on March 19, 2017 ; accessed on January 18, 2018 . 
• A Goldstone Boson Primer . (English) arxiv : hep-ph / 9812468
• Breaking symmetry and gold stone theorem. Seminar lecture

Individual evidence

1. ^ Y Nambu: Quasiparticles and Gauge Invariance in the Theory of Superconductivity . In: Physical Review . 117, 1960, pp. 648-663. doi : 10.1103 / PhysRev.117.648 .
2. ^ J Goldstone: Field Theories with Superconductor Solutions . In: Nuovo Cimento . 19, 1961, pp. 154-164. doi : 10.1007 / BF02812722 .
3. ^ J Goldstone, Abdus Salam, Steven Weinberg: Broken Symmetries . In: Physical Review . 127, 1962, pp. 965-970. doi : 10.1103 / PhysRev.127.965 .
4. ↑ P. Abreu et al .: Search for the goldstino at √ s from 189 to 202 . In: CERN-EP / 2000-110 . August 16, 2000, p. 12 ( archives-ouvertes.fr [PDF] Accepted by Phys.Lett.B).
5. ↑ Daniel M. Kaplan: GeVEvidence for the Sgoldstino in the decay Σ ⁺ → pμ ⁺ μ ^- from the HyperCP experiment . In: SUSY 2005, University of Durham . July 18, 2005, p. 25 ( ppd.fnal.gov [PDF]).