Self-consistent dynamical foundation of nonlinear Goldstone realizations: the reducing transformation, vacuum parameters, and Goldstone content for Lagrangians without explicit Goldstone fields (Salam–Strathdee, 1969)

Salam and Strathdee's 1969 paper 'Nonlinear Realizations. I. The Role of Goldstone Bosons' introduced the reducing-matrix (coset-representative) construction of nonlinear realizations of a spontaneously broken symmetry $G\to H$ and explicitly acknowledged three unresolved issues: no general method defines the reducing transformation $L_\phi$ when the Lagrangian contains no explicit Goldstone-boson fields with the requisite quantum numbers; the self-consistent computation of vacuum parameters such as $\langle\chi\rangle$ (or of a self-generated vector mass) has no general algorithm, convergence proof, or uniqueness statement; and whether such a self-consistency procedure introduces additional Goldstone particles is stated to be an open question. Later literature closed the kinematic half of this bundle: the Coleman–Wess–Zumino / Callan–Coleman–Wess–Zumino (CCWZ) construction makes the Goldstone effective theory depend only on the breaking pattern (the Goldstones may be composite), and the hidden-local-symmetry program shows that extra Goldstone-like fields in fully chiral-invariant constructions are would-be (unphysical) modes absorbed by gauge bosons. The surviving open core is the dynamical half: for a microscopic strongly coupled theory without elementary Goldstone fields, no general convergent self-consistent procedure is known that determines the breaking pattern, order parameters, and resulting composite-Goldstone effective theory while establishing the physical massless spectrum, and no general no-go theorem exists either. The problem asks for such a foundation, or for a precise limitation result.

Background

In relativistic quantum field theory, a continuous internal symmetry group $G$ can act on fields linearly (ordinary matrix representations) or nonlinearly. The distinction becomes essential when a symmetry is spontaneously broken: the Lagrangian and equations of motion are invariant under $G$, but the vacuum is invariant only under a subgroup $H$. Goldstone's theorem then asserts, for a Lorentz-invariant vacuum and positive-norm states, that the spectrum must contain one massless spin-zero particle — a Goldstone boson — for each broken generator, i.e. $\dim(G/H)$ of them, where $G/H$ is the coset space (the quotient of the group by its unbroken subgroup, with one dimension per broken generator); the pions of two-flavor QCD (quantum chromodynamics, the strong-interaction gauge theory of quarks) in the chiral (massless-quark) limit are the canonical composite example, and the symmetry-breaking order parameter is the nonvanishing vacuum expectation value $\langle\chi\rangle$ of a suitable field (analogous to the pion decay constant $f_\pi$).

Salam and Strathdee (Phys. Rev. 184, 1750 (1969)) gave a compact construction, modeled on Wigner's description of a general four-momentum as a Lorentz boost acting on a standard momentum, of the nonlinear realizations of $G$ that become linear on $H$. The central object is the reducing matrix (coset representative) $L_\phi$: a group-valued matrix whose independent entries are the Goldstone fields $\phi$ parametrizing the coset space $G/H$; matter fields are projected through it and then transform linearly under $H$ while $G$ acts nonlinearly. In their Section 3, starting from a linearly $G$-covariant Lagrangian containing a spin-zero multiplet $M$ whose components include the Goldstone fields (their 'preferred fields'), $L_\phi$ is defined by solving $\langle (D(L_\phi^{-1})M)_0\rangle = 0$, i.e. by transforming to variables in which the Goldstone components have zero vacuum value. The contemporaneous papers of Coleman, Wess and Zumino (I) and Callan, Coleman, Wess and Zumino (II) — together the 'CCWZ construction' — classified all such realizations for compact connected semisimple $G$ and built the most general invariant phenomenological (low-energy effective) Lagrangian from covariant derivatives, with no reference to any underlying microscopic Lagrangian; this is today the standard tool for constructing Goldstone effective field theories.

Salam and Strathdee themselves emphasized the dynamical gap in this kinematic machinery, in three passages. First (footnote to their Eq. (3.4)): 'We do not have a general method which defines the transformation $L_\phi$ in cases where the Lagrangian contains explicitly no Goldstone boson fields with the requisite quantum numbers. For the above method to work, it is necessary to introduce extra fields with the requisite quantum numbers, perhaps by some self-consistent method which we have not investigated.' Second, the numerical order parameters the construction needs — e.g. the vacuum expectation value $\langle\chi\rangle$ of the preferred chiral four-vector, whose nonvanishing encodes the vacuum asymmetry while the massive fluctuation $\chi'$ may be discarded — are to be found by a self-consistent (tadpole) procedure: substitute $\chi=\langle\chi\rangle+\chi'$ and demand that quantum corrections give $\langle\chi'\rangle=0$. That method goes back to Goldstone, Salam and Weinberg (Phys. Rev. 127, 965 (1962)) and was later formalized through the effective potential $V(\phi)$, whose stationary points give the exact vacuum expectation values (Coleman and Weinberg, Phys. Rev. D 7, 1888 (1973)); but no general algorithm, convergence proof, or uniqueness statement for such procedures exists. Third, at the end of their Section 3, in the context of a fully $G$-invariant Lagrangian in which a vector-meson ($\rho$) mass is generated self-consistently (bare mass zero, self-mass equal to the physical mass), they state: 'Whether this self-consistency procedure will introduce other Goldstone particles into the theory is an open question.'

The later literature transformed these questions. The CCWZ/coset framework makes the phenomenological construction universal: given the pattern $G\to H$, the Goldstone effective theory is fixed by symmetry alone, whether or not the microscopic theory contains elementary scalars — the Goldstones may be composite bound states. The question of extra Goldstone-like fields was clarified by the hidden local symmetry (HLS) reformulation of the nonlinear sigma model, the generic effective theory of the Goldstone fields alone (Bando, Kugo, Uehara, Yamawaki, Yanagida, Phys. Rev. Lett. 54, 1215 (1985)): a fully chiral-invariant vector-mass term — exactly the construction of Salam–Strathdee's last footnote — can be written by gauging an auxiliary local symmetry $H_{\rm local}$, and the additional Goldstone-like fields this introduces are would-be Nambu–Goldstone bosons absorbed into the HLS gauge bosons via the Higgs mechanism, i.e. unphysical gauge artifacts rather than additional particles. Relatedly, McArthur (JHEP 11 (2010) 140) showed for broken spacetime symmetries that nonlinear realizations can contain unphysical Goldstone bosons — removable by the inverse Higgs constraint (a symmetry-based rule that eliminates a Goldstone field when the commutator of two broken generators closes on an unbroken one, making that field redundant) or by the equations of motion — and that these are gauge degrees of freedom of an enlarged isotropy (stabilizer) subgroup. What has no general solution is the dynamical half: deriving, from a microscopic strongly coupled Lagrangian that contains no elementary Goldstone fields, the actual breaking pattern, the order parameters, and the resulting composite-Goldstone effective theory, with controlled convergence and uniqueness, together with the physical massless spectrum.

Problem Statement

The problem, in the source-faithful formulation of Salam and Strathdee (1969), is to give the nonlinear-realization method a dynamical, self-consistent foundation applicable to microscopic Lagrangians that contain no explicit Goldstone-boson fields. Concretely: consider a Lorentz-invariant quantum field theory whose Lagrangian is invariant under a continuous internal group $G$ but contains no elementary spin-zero fields carrying the quantum numbers of the Goldstone bosons of a presumed spontaneous breaking $G\to H$ (QCD with massless quarks is the archetype, the pions being composite). Provide a general prescription that:

(1) constructs the Goldstone sector — introducing the (in general composite) Goldstone fields and defining the reducing transformation / nonlinear realization, equivalently the low-energy effective theory with its symmetry-determined structure — from the microscopic Lagrangian alone, without inserting elementary Goldstone fields by hand;

(2) determines the numerical vacuum parameters the construction requires — order parameters such as $\langle\chi\rangle$ (a vacuum expectation value encoding the symmetry breaking) and dynamically generated masses, e.g. a self-consistent vector-boson mass with zero bare value — by a procedure whose convergence is controlled and for which a uniqueness or vacuum-selection criterion is established; and

(3) settles whether that self-consistent procedure, applied to the microscopic Lagrangian, can introduce additional physical Goldstone particles beyond those corresponding to actually broken generators of the emergent vacuum, or whether all extra Goldstone-like modes are removable gauge or parametrization artifacts.

The common objective uniting the three deliverables is a bottom-up derivation of the spontaneously broken phase — its symmetry realization, its order parameters, and its massless spectrum — from the microscopic Lagrangian alone, replacing the by-hand phenomenological input of the coset construction.

References

1. A. Salam and J. Strathdee, 'Nonlinear Realizations. I. The Role of Goldstone Bosons', Physical Review 184, 1750–1759 (1969). DOI 10.1103/PhysRev.184.1750, https://doi.org/10.1103/PhysRev.184.1750
2. S. Coleman, J. Wess, and B. Zumino, 'Structure of Phenomenological Lagrangians. I', Physical Review 177, 2239–2247 (1969). DOI 10.1103/PhysRev.177.2239, https://doi.org/10.1103/PhysRev.177.2239
3. C. G. Callan, Jr., S. Coleman, J. Wess, and B. Zumino, 'Structure of Phenomenological Lagrangians. II', Physical Review 177, 2247–2250 (1969). DOI 10.1103/PhysRev.177.2247, https://doi.org/10.1103/PhysRev.177.2247
4. J. Goldstone, A. Salam, and S. Weinberg, 'Broken Symmetries', Physical Review 127, 965–970 (1962). DOI 10.1103/PhysRev.127.965, https://doi.org/10.1103/PhysRev.127.965
5. S. Coleman and E. Weinberg, 'Radiative Corrections as the Origin of Spontaneous Symmetry Breaking', Physical Review D 7, 1888–1910 (1973). DOI 10.1103/PhysRevD.7.1888, https://doi.org/10.1103/PhysRevD.7.1888
6. M. Bando, T. Kugo, S. Uehara, K. Yamawaki, and T. Yanagida, 'Is the ρ Meson a Dynamical Gauge Boson of Hidden Local Symmetry?', Physical Review Letters 54, 1215–1218 (1985). DOI 10.1103/PhysRevLett.54.1215, https://doi.org/10.1103/PhysRevLett.54.1215
7. I. N. McArthur, 'Nonlinear realizations of symmetries and unphysical Goldstone bosons', Journal of High Energy Physics 2010, 140 (2010). DOI 10.1007/JHEP11(2010)140, arXiv:1009.3696, https://arxiv.org/abs/1009.3696
8. D. Naegels, 'An introduction to Goldstone boson physics and to the coset construction', lecture notes (2021). arXiv:2110.14504, https://arxiv.org/abs/2110.14504
9. K. Yamawaki, 'Proving Rho Meson Is a Dynamical Gauge Boson of Hidden Local Symmetry', Symmetry 15, 2209 (2023). DOI 10.3390/sym15122209, arXiv:2310.09487, https://arxiv.org/abs/2310.09487