Goldstone modes and the Higgs mechanism

A Goldstone mode is a massless excitation that appears when a continuous symmetry of the laws is broken by the ground state. Moving along the family of equally good ground states costs no energy, so a slow, long-wavelength wave of that motion costs arbitrarily little, and the particle it describes has no mass. If a small explicit term tilts the family, the mode acquires a small mass and is called a pseudo-Goldstone mode. When the broken symmetry is a gauge symmetry, the Higgs mechanism applies instead: the gauge field absorbs the would-be Goldstone modes as extra polarizations and becomes massive. The program uses all three ideas. In its coset toy model the polarization angle is a Goldstone direction, which an added potential turns into a pseudo-Goldstone mode, and its momentum graviton X gets its mass from a gravitational version of the Higgs mechanism.

Status: Established — [T] — textbook field theory, explained here from general knowledge; the program’s applications are the Hessian count of the coset toy model [V] (verify/toy_coset.py; the toy-model scripts) and the Debt-1 derivation of X’s mass, tagged [D + T] in #11 §7.1 and “[D — structural]” in its calc note.

Goldstone’s theorem#

When a continuous global symmetry G is spontaneously broken to a subgroup G₀, the vacuum manifold G/G₀ has one flat direction for each broken generator (spontaneous symmetry breaking). Goldstone’s theorem says that in a Lorentz-invariant field theory each of these directions carries a massless spin-0 particle [T].

The reason is clearest in terms of waves. A uniform shift of the whole system along the vacuum manifold costs no energy, because it only exchanges one vacuum for another. A shift that varies slowly in space, with wavelength λ, costs only gradient energy, which falls toward zero as λ grows. The frequency of such a wave therefore goes to zero with its wavenumber k, and with Lorentz invariance the frequency is c|k|, the dispersion of a massless particle [T]. Spin waves in a magnet and sound waves in a crystal are the condensed-matter counterparts; without Lorentz invariance their dispersion, and sometimes their number, can differ [T].

Counting with the Hessian#

Classically the count is a statement about the potential V at its minimum Φ₀ [T]. The matrix of second derivatives there, the Hessian, is the mass-squared matrix of small oscillations:

M²ᵢⱼ = ∂²V/∂Φᵢ∂Φⱼ   at Φ = Φ₀

massive modes        = rank of M²
flat directions      = number of fields − rank of M²
Goldstone directions = dim G − dim G₀  (each one flat)

Every broken generator moves the vacuum along the vacuum manifold, so each gives a null direction of the Hessian [T]. The converse fails. A null direction that does not come from a broken symmetry is accidental, and nothing guarantees that it survives higher-order terms or quantum corrections [T]. The individual eigenvalues also depend on how the fields are parametrized; at a critical point the rank and the number of negative eigenvalues do not (Sylvester’s law of inertia) [T]. The toy-model record shows both facts. For the same saddle of model I with its first, unbounded potential (the naive potential, now dead), the round-4 referee reports the negative eigenvalue −3.3723 (review/round4/referee/audit-notes.md, Step 2(a)) and the adjudication −6.47, “in my coordinates” (review/round4-adjudication.md, F2); the values differ, and both are negative.

The count in model I#

The program’s coset toy model (model I) is a field theory of a phase-space metric H and a real structure K on a 1+1-dimensional base, with the Euclidean Lagrangian (toy-model note §2)

L = (f_H²/8) Tr(H⁻¹∂H H⁻¹∂H) + (f_K²/4) Tr(H⁻¹ ∂Kᵀ H ∂K) + (λ/8) Tr(H⁻¹ Cᵀ H C),   C = KI + IK

where I ≡ ω⁻¹H. “The potential vanishes exactly when K is compatible with H” (toy-model note §2), and at H = 1 the gate checks that it is a sum of squares (verify/toy_coset.py). The compatible pairs, the vacua, form “the upper half-plane times a circle of K’s, a 3-dimensional space” (worklog, ‘Chunk 1 results’). The fields span four directions, two for H and two for K, whose target is “two-dimensional de Sitter space” (toy-model note §2; de Sitter space). So at a vacuum the potential is flat along three directions and curves upward in one. The gate’s spectrum section is headed “4 fields, 3-dim vacuum manifold”, and its check reads:

PASS  potential Hessian has rank 1: three flat directions (Goldstones), one massive  [[0, 4, 0, 0]]

The note states the result: “The potential’s Hessian has rank 1 at the vacuum. That leaves three Goldstone directions: the two non-compact half-plane directions, and the compact angle φ, which parametrizes the Lagrangian Grassmannian Λ(1).” (toy-model note §2). The Lagrangian Grassmannian Λ(d) is the space of possible position subspaces, a circle when d = 1. The worklog marks the result “Spectrum [V]” and identifies the compact direction: “The compact Goldstone is the rotation of K at fixed H: the Lagrangian Grassmannian Λ(1), which is the checkpoint’s Maslov circle (the τ field of §10).” (‘Chunk 1 results’). The round-4 referee, in other coordinates, found “rank 1, eigenvalues {3:1, 0:3} -> 3 flat directions” (audit notes, Step 2(a)): a different nonzero eigenvalue, the same rank.

Here every flat direction is a Goldstone direction. The stabilizer of a vacuum in Sp(2, ℝ) is the discrete group {±1} (toy-model note §2), so the number of broken directions equals the dimension of the vacuum manifold. Their characters differ:

  • The compact angle φ is the program’s polarization angle. At H = 1 its kinetic term is “(f_K²/4) Tr(dKᵀ dK) = (f_K²/2)(dφ)²”, “an XY stiffness” (verify/toy_s4.py, docstring). In 1+1 dimensions this compact mode cannot order (the Mermin–Wagner–Coleman theorem).
  • The two half-plane directions belong to the H-sector, which round 4 identified as “a hyperbolic SO(1,2) model” that can break even in two dimensions (review/round4-adjudication.md, “Mermin–Wagner vs [11]”; coset and hyperbolic sigma models).

Details are on Model I vacua, the order parameter and the Goldstone spectrum.

Pseudo-Goldstone modes#

A small explicit breaking of the symmetry tilts the vacuum manifold [T]. The would-be Goldstone mode then moves in a shallow potential and acquires a small mass, which vanishes as the explicit breaking is switched off. The pions are the standard example: the chiral symmetry of the strong interaction is broken spontaneously, and the small quark masses break it explicitly, so the pions are light rather than massless [T]. A Goldstone mode couples through derivatives of its field, and a pseudo-Goldstone mode keeps that property apart from terms proportional to the explicit breaking [T]. The case of a field that is an angle, with a periodic potential, is axion-like fields and pseudo-Goldstone bosons.

The program meets the idea twice:

  • The polarization angle with a ρ-preserving potential. To make the polarization definite, model I adds a potential that keeps ρ and reduces the continuous rotation of K to a discrete group. In its bounded form, at H = 1, “V_b = −(μ/2)(1 + cos 2φ)” (toy-model note §4), with its minima at the two vacua ±K_ref. The worklog draws the conclusion: the toy supports “broken reciprocity” in the form “a pseudo-Goldstone polarization angle with a ρ-symmetric potential”, not “only ω rigid, nothing else” (‘Chunk 3 results’). See the bounded potential V_b and Open: the origin of the ρ-preserving potential.
  • The dark-energy field τ. The lab notebook: “τ is a (pseudo-)Goldstone: derivatively coupled, no tree-level Yukawa to SM matter, safe from torsion-balance bounds despite being ultralight. [D]” (#1 §4; no coupling, so no Yukawa force), where SM is the Standard Model. See fifth-force safety of τ.

A zero mode that no symmetry protects#

With V_b and H dynamical, the reference vacuum is “a stable minimum: Hessian eigenvalues 0, 0.764, 2, 5.236” (toy-model note §4; verify/toy_coset.py). The zero is the accidental kind described above: “The zero mode is the classical flat line H = diag(y, 1/y), K = K_ref. It is not symmetry-protected, because V_b breaks the squeeze away from K = ±K_ref. Boosts are penalized.” (toy-model note §4). The round-4 adjudication worded it differently, as “the residual diagonal squeeze, which is an exact symmetry” (F2). The note v2 outranks the adjudication, and the wiki follows the note (the bounded potential V_b; conflicts between sources; see flag 12 in review/wiki-build-flags.md).

The Higgs mechanism#

When the broken symmetry is a local, gauge symmetry, the would-be Goldstone fields are not physical [T]. A gauge transformation can remove them, so they never appear as massless particles. Their degrees of freedom reappear as the longitudinal polarization of the gauge field, which becomes massive, with a mass set by the gauge coupling and the vacuum value of the field that breaks the symmetry. A massless gauge field carries only transverse polarizations; the massive one carries one more, and that one is the absorbed Goldstone mode. The W and Z bosons of the weak interaction get their masses this way, and inside a superconductor the photon does, which is why magnetic fields are expelled from it beyond a penetration depth [T].

Stückelberg fields#

The absorbed mode can be kept in view as an explicit field, a Stückelberg field [T]. For a vector field the mass term is written with A_μ + ∂_μπ, which is unchanged under A_μ → A_μ + ∂_μα, π → π − α; the gauge π = 0, the unitary gauge, gives back the massive vector, and π is the Goldstone field the vector absorbs. For a massive spin-2 field h_μν the same device restores the linearized diffeomorphisms δh_μν = ∂_μξ_ν + ∂_νξ_μ through h_μν → h_μν + ∂_μπ_ν + ∂_νπ_μ, and the vector π_μ splits into a transverse vector and the gradient of a scalar, which carries the longitudinal (helicity-zero) mode [T]. The construction, the count of polarizations and the program’s use for the momentum graviton X are on Stückelberg fields.

Masses from a vacuum value#

In the Standard Model a fermion gets its mass from a Yukawa coupling to the Higgs field: once the Higgs field takes its vacuum value, the coupling becomes a mass term, proportional to the coupling constant times that value [T]. The toy-model note uses this as its analogy for matter locality: locality in the selected position variable “follows from the vacuum once matter couples this way, as fermion masses follow from a Higgs vacuum value once Yukawa couplings are given. Nothing here forces that coupling.” (toy-model note §5). See locality from an (H, K)-built coupling.

The gravitational Higgs mechanism of X#

The program’s momentum graviton X = Re Z, the real part of the modular field, gets its mass by this mechanism, with linearized diffeomorphisms in place of an internal gauge symmetry. #11 §7.1 gives the chain [D + T]:

  1. “The rigid Sp(2d,ℝ) contains Z → Z + B (B symmetric): X is shift-symmetric — axion-like, never auxiliary.” Constant shifts of X are exact symmetries, so “X has no potential and no non-derivative couplings from the geometric sector alone” (calc note #6 §1), and “The mass cannot be written by hand (shift symmetry) — but it doesn’t need to be.” (#6 §3; the shift symmetry).
  2. “Locally, dual diffeomorphisms act as δX_μν = ∂₍μλ_ν₎ — a second linearized-diff gauge field, because X is symmetric”. Massless, X would be a second graviton, which “is excluded by the multi-graviton no-go [8†]” (the multi-graviton no-go); without the gauge symmetry, it is excluded “by Lorentz signature” (#11 §7.1; the signature obstruction; dual diffeomorphisms; the forcing chain).
  3. “The polarization choice spontaneously breaks the dual diffs; X eats the four Goldstones and emerges massive, M_X = the polarization-breaking scale.”

The calc note gives the step in Higgs-mechanism terms, tagged “[D — structural]”: “The vacuum ⟨Z⟩ = i·g sits at X = 0 and is moved by the shifts: the polarization selection (η-emergence) spontaneously breaks the local dual diffeomorphisms. Gravitational Higgs mechanism: the four would-be Goldstones π_μ of the broken λ-gauge are eaten; X becomes a massive spin-2 with its five polarizations, mass M_X = the polarization-breaking scale.” (#6 §3). The roles map onto the textbook case: X is the gauge field, λ_μ the gauge parameter, the vacuum’s choice of polarization the symmetry-breaking vacuum value, and π_μ the absorbed Goldstone fields, the vector Stückelberg field of the spin-2 case. See the gravitational Higgs mechanism.

Two consequences in the record:

  • What is eaten. The later calc note words it differently: “the Higgsing of the dual diffeomorphisms eats the vector π_μ and the longitudinal scalar of the λ-gradient sector” (#8 §3). The sources do not reconcile the two wordings, and #11 §7.1 follows the first (conflicts between sources).
  • τ is not eaten. #6 already set the dark-energy field apart: “The Λ(d) moduli, τ included, are not the eaten modes — they carry the compact phase directions, which constant shifts do not generate; verifying τ survives un-eaten is flagged [O].” (#6 §3). #8 settled it to leading order: “τ is not eaten [D — leading order]: … τ is a J-modulus on the compact Λ(d), a different sector with different origin; it survives as the light field. (Residual τ–Stückelberg mixing through the potential: logged, expected small by compactness.)” (#8 §3). #11 §7.5: “τ un-eaten (compact J-sector, not the Stückelberg sector)”. The residual mixing is open problem 6, “τ–Stückelberg mixing matrix [O]” (#11 §14). See τ is not eaten and Open: τ–Stückelberg mixing.

A massive X also has a finite range. #11 §7.5 gives its couplings to ordinary matter as of gravitational strength and “Yukawa-cut at 1/M_X”, the range of a massive mediator (fifth-force safety of X).

Inverse-Higgs constraints#

When the broken symmetries act on spacetime, as translations and rotations do, an inverse-Higgs constraint can fix some would-be Goldstone fields as derivatives of others, so that there are fewer Goldstone modes than broken generators [T]. A flat membrane breaks the translation perpendicular to itself and the rotations that would tilt it, but its tilt is set by the slope of its displacement, and the displacement is its only independent mode. Calc note #5 expects such constraints to act on the moduli of the broken vacuum, “Λ(d) σ-modes (10: τ plus nine, of which inverse-Higgs constraints likely eliminate most [O])” (#5 §D), and lists the “inverse-Higgs accounting of the nine heavy moduli” as open (§F); no document works it out, and #11 §14 does not carry the item (Open: field content of the broken vacuum). The count, the commutator criterion, the membrane and the dilaton, and the program’s expectation are on inverse-Higgs constraints.

Standard references#

Added by the wiki from model memory (†); the program’s documents cite none of them for the general theory.

  • J. Goldstone (1961)†; J. Goldstone, A. Salam and S. Weinberg (1962)†: the massless modes of a broken continuous symmetry.
  • P. W. Anderson (1963)†: the photon mass in a superconductor. F. Englert and R. Brout (1964)†; P. W. Higgs (1964)†; G. S. Guralnik, C. R. Hagen and T. W. B. Kibble (1964)†: the mechanism in relativistic gauge theory.
  • E. C. G. Stueckelberg (1938)†: the field that bears his name.
  • N. Arkani-Hamed, H. Georgi and M. D. Schwartz (2003)†: Stückelberg fields for massive gravity.
  • E. A. Ivanov and V. I. Ogievetsky (1975)†: the inverse Higgs phenomenon. I. Low and A. V. Manohar (2002)†: Goldstone counting when spacetime symmetries break.

Used by#

Sources#

  • papers/broken-born-checkpoint-v2.md §7.1 — the chain: shift symmetry, dual diffeomorphisms, X eats the four Goldstones [D + T]; §7.5 — τ un-eaten; the Yukawa cut at 1/M_X; §14 item 6 — τ–Stückelberg mixing [O]
  • papers/reciprocity-toy-models.md §2 — model I’s Lagrangian, rank-1 Hessian and three Goldstone directions; §4 — V_b at H = 1, its Hessian with H dynamical and its unprotected zero mode; §5 — the Yukawa analogy
  • notes/debt1a-ii-X-classification.md §1 — X has no potential from geometry; §3 — the gravitational Higgs mechanism, mass M_X, τ not among the eaten modes [D — structural]
  • notes/debt1a-iv-beta-coefficients.md §3 — what is eaten; τ not eaten at leading order
  • notes/debt1a-geometric-core.md §D — the Λ(d) σ-modes and inverse-Higgs constraints [O]; §F — “inverse-Higgs accounting of the nine heavy moduli” as open
  • notes/broken-born-theory.md §4 — τ as a (pseudo-)Goldstone [D]
  • notes/toy-lagrangian.md ‘Chunk 1 results’ — “Spectrum [V]” and the compact Goldstone as Λ(1); ‘Chunk 3 results’ — “a pseudo-Goldstone polarization angle with a ρ-symmetric potential”
  • review/round4-adjudication.md ‘Rulings’ F2 — the zero mode called an exact symmetry; eigenvalues “in my coordinates”; “Mermin–Wagner vs [11]” — the hyperbolic H-sector
  • review/round4/referee/audit-notes.md Step 2(a) — the referee’s independent Hessian count
  • verify/toy_coset.py — the Hessian-rank check and V_b’s stability; verify/toy_s4.py — the docstring’s XY stiffness of φ
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