Goldstone boson

In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu in the context of the BCS superconductivity mechanism,^[1] and subsequently elucidated by Jeffrey Goldstone,^[2] and systematically generalized in the context of quantum field theory.^[3]

These spinless bosons correspond to the spontaneously broken internal symmetry generators, and are characterized by the quantum numbers of these. They transform nonlinearly (shift) under the action of these generators, and can thus be excited out of the asymmetric vacuum by these generators. Thus, they can be thought of as the excitations of the field in the broken symmetry directions in group space—and are massless if the spontaneously broken symmetry is not also broken explicitly.

If, instead, the symmetry is not exact, i.e. if it is explicitly broken as well as spontaneously broken, then the Nambu–Goldstone bosons are not massless, though they typically remain relatively light; they are then called pseudo-Goldstone bosons or pseudo-Nambu–Goldstone bosons (abbreviated PNGBs).

Goldstone's theorem

Goldstone's theorem examines a generic continuous symmetry which is spontaneously broken; i.e., its currents are conserved, but the ground state is not invariant under the action of the corresponding charges. Then, necessarily, new massless (or light, if the symmetry is not exact) scalar particles appear in the spectrum of possible excitations. There is one scalar particle—called a Nambu–Goldstone boson—for each generator of the symmetry that is broken, i.e., that does not preserve the ground state. The Nambu–Goldstone mode is a long-wavelength fluctuation of the corresponding order parameter.

By virtue of their special properties in coupling to the vacuum of the respective symmetry-broken theory, vanishing momentum ("soft") Goldstone bosons involved in field-theoretic amplitudes make such amplitudes vanish ("Adler zeros").

In theories with gauge symmetry, the Goldstone bosons are "eaten" by the gauge bosons. The latter become massive and their new, longitudinal polarization is provided by the Goldstone boson.

Examples

Natural

• In fluids, the phonon is longitudinal and it is the Goldstone boson of the spontaneously broken Galilean symmetry. In solids, the situation is more complicated; the Goldstone bosons are the longitudinal and transverse phonons and they happen to be the Goldstone bosons of spontaneously broken Galilean, translational, and rotational symmetry with no simple one-to-one correspondence between the Goldstone modes and the broken symmetries.
• In magnets, the original rotational symmetry (present in the absence of an external magnetic field) is spontaneously broken such that the magnetization points into a specific direction. The Goldstone bosons then are the magnons, i.e., spin waves in which the local magnetization direction oscillates.
• The pions are the pseudo-Goldstone bosons that result from the spontaneous breakdown of the chiral-flavor symmetries of QCD effected by quark condensation due to the strong interaction. These symmetries are further explicitly broken by the masses of the quarks, so that the pions are not massless, but their mass is significantly smaller than typical hadron masses.
• The longitudinal polarization components of the W and Z bosons correspond to the Goldstone bosons of the spontaneously broken part of the electroweak symmetry SU(2)⊗U(1), which, however, are not observable. Because this symmetry is gauged, the three would-be Goldstone bosons are "eaten" by the three gauge bosons corresponding to the three broken generators; this gives these three gauge bosons a mass, and the associated necessary third polarization degree of freedom. This is described in the Standard Model through the Higgs mechanism. An analogous phenomenon occurs in superconductivity, which served as the original source of inspiration for Nambu, namely, the photon develops a dynamical mass (expressed as magnetic flux exclusion from a superconductor), cf. the Ginzburg–Landau theory.

Theory

Consider a complex scalar field φ, with the constraint that φ^*φ = v², a constant. One way to impose a constraint of this sort is by including a potential interaction term in its Lagrangian density,

$\lambda (\phi ^{*}\phi -v^{2})^{2}~,$

and taking the limit as λ → ∞ (this is called the "Abelian nonlinear σ-model". It corresponds to the Goldstone sombrero potential where the tip and the sides shoot to infinity, preserving the location of the minimum at its base).

The constraint, and the action, below, are invariant under a U(1) phase transformation, δφ=iεφ. The field can be redefined to give a real scalar field (i.e., a spin-zero particle) θ without any constraint by

$\phi =ve^{{i\theta }}\,$

where θ is the Nambu–Goldstone boson (actually vθ is), and the U(1) symmetry transformation effects a shift on θ, namely

$\delta \theta =\epsilon ~,$

but does not preserve the ground state |0〉, (i.e. the above infinitesimal transformation does not annihilate it—the hallmark of invariance), as evident in the charge of the current below.

Thus, the vacuum is degenerate and noninvariant under the action of the spontaneously broken symmetry.

The corresponding Lagrangian density is given by

${{\mathcal L}}=-{\frac {1}{2}}(\partial ^{\mu }\phi ^{*})\partial _{\mu }\phi +m^{2}\phi ^{*}\phi =-{\frac {1}{2}}(-ive^{{-i\theta }}\partial ^{\mu }\theta )(ive^{{i\theta }}\partial _{\mu }\theta )+m^{2}v^{2},$

and thus

$=-{\frac {v^{2}}{2}}(\partial ^{\mu }\theta )(\partial _{\mu }\theta )+m^{2}v^{2}~.$

Note that the constant term m²v² in the Lagrangian density has no physical significance, and the other term in it is simply the kinetic term for a massless scalar.

The symmetry-induced conserved U(1) current is

$J_{\mu }=-v^{2}\partial _{\mu }\theta ~.$

The charge, Q, resulting from this current shifts θ and the ground state to a new, degenerate, ground state. Thus, a vacuum with 〈θ〉 = 0 will shift to a different vacuum with 〈θ〉 = −ε. The current connects the original vacuum with the Nambu–Goldstone boson state, 〈0|J₀(0)|θ〉≠ 0.

In general, in a theory with several scalar fields, φ_j, the Nambu–Goldstone mode φ_g is massless, and parameterises the curve of possible (degenerate) vacuum states. Its hallmark under the broken symmetry transformation is nonvanishing vacuum expectation 〈δφ_g〉, an order parameter, for vanishing 〈φ_g〉 = 0, at some ground state |0〉 chosen at the minimum of the potential, 〈∂V/∂φ_i〉 = 0. Symmetry dictates that all variations of the potential with respect to the fields in all symmetry directions vanish. The vacuum value of the first order variation in any direction vanishes as just seen; while the vacuum value of the second order variation must also vanish, as follows. Vanishing vacuum values of field symmetry transformation increments add no new information.

By contrast, however, nonvanishing vacuum expectations of transformation increments, 〈δφ_g〉, specify the relevant (Goldstone) null eigenvectors of the mass matrix,

$\left\langle {\partial ^{2}V \over \partial \phi _{i}\partial \phi _{j}}\right\rangle \langle \delta \phi _{j}\rangle =0~,$

and hence the corresponding zero-mass eigenvalues.

Goldstone's argument

The principle behind Goldstone's argument is that the ground state is not unique. Normally, by current conservation, the charge operator for any symmetry current is time-independent,

${d \over dt}Q={d \over dt}\int _{x}J^{0}(x)=0~.$

Acting with the charge operator on the vacuum either annihilates the vacuum, if that is symmetric; else, if not, as is the case in spontaneous symmetry breaking, it produces a zero-frequency state out of it, through its shift transformation feature illustrated above. (Actually, here, the charge itself is ill-defined, cf. the Fabri-Picasso argument below. But its better behaved commutators with fields, that is, the transformation shifts, are still time-invariant, d〈δφ_g〉/dt=0 , thus generating a δ(k⁰) in its Fourier transform.^[4])

Thus, if the vacuum is not invariant under the symmetry, action of the charge operator produces a state which is different from the vacuum chosen, but which has zero frequency. This is a long-wavelength oscillation of a field which is nearly stationary: there are physical states with zero frequency, k⁰, so that the theory cannot have a mass gap.

This argument is further clarified by taking the limit carefully. If an approximate charge operator acting in a huge but finite region A is applied to the vacuum,

${d \over dt}Q_{A}={d \over dt}\int _{x}e^{{-x^{2} \over 2A^{2}}}J^{0}(x)=-\int _{x}e^{{-x^{2} \over 2A^{2}}}\nabla \cdot J=\int _{x}\nabla (e^{{-x^{2} \over 2A^{2}}})\cdot J~,$

a state with approximately vanishing time derivative is produced,

$\|{d \over dt}Q_{A}|0\rangle \|\approx {1 \over A}\|Q_{A}|0\rangle \|.$

Assuming a nonvanishing mass gap m₀, the frequency of any state like the above, which is orthogonal to the vacuum, is at least m₀,

$\|{d \over dt}|\theta \rangle \|=\|H|\theta \rangle \|\geq m_{0}\|\;|\theta \rangle \|~.$

Letting A become large leads to a contradiction. Consequently m₀ = 0.

Exception: This argument fails, however, when the symmetry is gauged, because then the symmetry generator is only performing a gauge transformation. A gauge transformed state is the same exact state, so that acting with a symmetry generator does not get one out of the vacuum. See Higgs mechanism.

The Fabri-Picasso theorem: Q does not properly exist in the Hilbert space, unless Q|0〉 = 0.
The argument^[5] requires both the vacuum and the charge Q to be translationally invariant, P|0〉 = 0, [P,Q]= 0.

Consider the correlation function of the charge with itself,

${\displaystyle \langle 0|QQ|0\rangle =\int d^{3}x\langle 0|j_{0}(x)Q|0\rangle =\int d^{3}x\langle 0|e^{iPx}j_{0}(0)e^{-iPx}Q|0\rangle }$

${\displaystyle =\int d^{3}x\langle 0|e^{iPx}j_{0}(0)e^{-iPx}Qe^{iPx}e^{-iPx}|0\rangle =\int d^{3}x\langle 0|j_{0}(0)Q|0\rangle ~,}$

so the integrand in the r.h.s. does not depend on the position.

Thus, its value is proportional to the total space volume, ‖Q|0〉‖² = ∞; so that Q does not properly exist in the Hilbert space, unless the symmetry is unbroken, Q|0〉 = 0.

Infraparticles

There is an arguable loophole in the theorem. If one reads the theorem carefully, it only states that there exist non-vacuum states with arbitrarily small energies. Take for example a chiral N = 1 super QCD model with a nonzero squark VEV which is conformal in the IR. The chiral symmetry is a global symmetry which is (partially) spontaneously broken. Some of the "Goldstone bosons" associated with this spontaneous symmetry breaking are charged under the unbroken gauge group and hence, these composite bosons have a continuous mass spectrum with arbitrarily small masses but yet there is no Goldstone boson with exactly zero mass. In other words, the Goldstone bosons are infraparticles.

Nonrelativistic theories

A version of Goldstone's theorem also applies to nonrelativistic theories (and also relativistic theories with spontaneously broken spacetime symmetries, such as Lorentz symmetry or conformal symmetry, rotational, or translational invariance).

It essentially states that, for each spontaneously broken symmetry, there corresponds some quasiparticle with no energy gap—the nonrelativistic version of the mass gap. (Note that the energy here is really H−μN−α→⋅P→ and not H.) However, two different spontaneously broken generators may now give rise to the same Nambu–Goldstone boson. For example, in a superfluid, both the U(1) particle number symmetry and Galilean symmetry are spontaneously broken. However, the phonon is the Goldstone boson for both.

In general, the phonon is effectively the Nambu–Goldstone boson for spontaneously broken Galilean/Lorentz symmetry. However, in contrast to the case of internal symmetry breaking, when spacetime symmetries are broken, the order parameter need not be a scalar field, but may be a tensor field, and the corresponding independent massless modes may now be fewer than the number of spontaneously broken generators, because the Goldstone modes may now be linearly dependent among themselves: e.g., the Goldstone modes for some generators might be expressed as gradients of Goldstone modes for other broken generators.

Nambu–Goldstone fermions

Spontaneously broken global fermionic symmetries, which occur in some supersymmetric models, lead to Nambu–Goldstone fermions, or goldstinos.^[6][7] These have spin ½, instead of 0, and carry all quantum numbers of the respective supersymmetry generators broken spontaneously.

Spontaneous supersymmetry breaking smashes up ("reduces") supermultiplet structures into the characteristic nonlinear realizations of broken supersymmetry, so that goldstinos are superpartners of all particles in the theory, of any spin, and the only superpartners, at that. That is, to say, two non-goldstino particles are connected to only goldstinos through supersymmetry transformations, and not to each other, even if they were so connected before the breaking of supersymmetry. As a result, the masses and spin multiplicities of such particles are then arbitrary.

See also

• Pseudo-Goldstone boson
• Majoron
• Higgs mechanism
• Mermin–Wagner theorem
• Vacuum expectation value
• Noether's theorem

References

1. Nambu, Y (1960). "Quasiparticles and Gauge Invariance in the Theory of Superconductivity". Physical Review. 117: 648–663. Bibcode:1960PhRv..117..648N. doi:10.1103/PhysRev.117.648.
2. Goldstone, J (1961). "Field Theories with Superconductor Solutions". Nuovo Cimento. 19: 154–164. doi:10.1007/BF02812722.
3. Goldstone, J; Salam, Abdus; Weinberg, Steven (1962). "Broken Symmetries". Physical Review. 127: 965–970. Bibcode:1962PhRv..127..965G. doi:10.1103/PhysRev.127.965.
4. Scholarpedia proof
5. Fabri, E and Picasso, L E (1966), "Quantum Field Theory and Approximate Symmetries", Phys. Rev. Lett. 16 (1966) 408 doi:10.1103/PhysRevLett.16.408.2
6. Volkov, D.V.; Akulov, V (1973). "Is the neutrino a goldstone particle?". Physics Letters. B46: 109–110. Bibcode:1973PhLB...46..109V. doi:10.1016/0370-2693(73)90490-5.
7. Salam, A; et al. (1974). "On Goldstone Fermion". Physics Letters. B49: 465–467. Bibcode:1974PhLB...49..465S. doi:10.1016/0370-2693(74)90637-6.