Strong Massive-Massless Continuation in QFT

• Strong Massive-Massless Continuation is a framework in quantum field theory and gravity that establishes an analytic and gauge-invariant connection between massive and massless regimes.
• It employs methods such as nonlocal field redefinitions and algebraic deformations to reveal new couplings and non-decoupling fields in models like massive gravity and higher-spin systems.
• The approach unifies disparate phenomena—from DVZ discontinuities to kinetic reformulations—providing universal insights that guide the development of consistent quantum models.

Strong massive-massless continuation refers, across a broad spectrum of quantum field theory and gravitational models, to structural features and technical mechanisms that allow for a continuous, regular, or even exact analytic connection between massive and massless regimes. Contrary to the naive expectation that the $m\to0$ limit of a massive theory should generally reduce to its corresponding massless version, strong massive-massless continuation captures the circumstances where (i) new couplings, degrees of freedom, or corrections survive and modify the limit, or (ii) exact mathematical correspondences (via analytic continuation, algebraic deformation or field redefinitions) allow massive and massless quantities to be interrelated in a robust way.

1. Gauge-Invariant Continuations and Non-Decoupling Fields in Gravity

In massive gravity, especially in quantum gauge-theoretic (spin-2) formulations, the standard expectation is that "additional" degrees of freedom present in the massive case decouple as $m\to0$, leaving only the Einstein–Hilbert action of general relativity. However, when enforcing full gauge invariance via cohomological methods (for example, nilpotent gauge charges and descent equations), it is possible to classify all admissible couplings between gravitons and matter, including nonstandard couplings via a vector graviton $v_\mu$ field. The persistence of such fields in the $m\to0$ limit leads to modifications of the Einstein–Hilbert theory, as additional quartic couplings between scalar matter and $v_\mu$ survive. Explicitly, the modified Lagrangian is schematically

$$\mathcal{L} = -\frac{2}{\kappa^2} \sqrt{-g}\,R + \sqrt{-g}\big[\mathcal{L}_{\text{matter}} + \mathcal{L}_{\text{new quartic couplings}}\big],$$

where the new matter couplings alter both the Einstein equations and the matter field equations. Invariance under the underlying gauge symmetry (not general coordinate invariance in this construction, which is formulated on a Minkowski background) strictly restricts allowable couplings and demands the scaling of certain couplings with $m$ so that the massless limit yields a nontrivial, modified theory. This process—where a consistent, gauge-invariant deformation from massive to massless gravity produces new interactions and non-decoupling fields—is a prototypical realization of strong massive-massless continuation (0912.1112).

2. Non-Existence and Structural Obstructions in Nonlinear Gravity Limits

Not all models permit a strong or unambiguous massive-massless continuation. In $f$–$g$ massive gravity—especially when seeking to realize nonlinear partially massless (PM) gauge symmetries by tuning the graviton mass in a de Sitter background—there exists a "fifth constraint" that, while necessary to avoid a sixth ghost-like degree of freedom, simultaneously obstructs the desired nonlinear gauge invariance. At the linearized level, PM symmetry can remove the helicity-0 excitation; nonlinearly, the fifth constraint fails to become a Bianchi identity, and the decoupling of the unwanted mode does not occur. Moreover, this constraint structure is also the root of acausal propagation in these theories. Thus, in such nonlinear scenarios the "continuation" from massive to (partially) massless is disrupted by algebraic and analytic obstructions: the naive $m\to0$ theory is neither physically consistent nor smooth (Deser et al., 2013).

3. Algebraic and Kinetic Realizations: Higher Spin and Kinetic Theory

In higher-spin systems, strong connections between massive and massless sectors exist through algebraic and geometric constructions:

• For higher-spin particles, models with extended worldline supersymmetry (SO(N)) allow for a dimensional reduction introducing mass, where the resulting field equations (of Fierz-Pauli type) for the massive sector smoothly limit to multiplets of massless Fronsdal-Labastida fields as $m\to0$. The constraints enforce the correct matching of physical degrees of freedom, and compensator fields organize the transition between massive and massless descriptions (Bastianelli et al., 2014).
• In kinetic theory for fermions, the full set of spin degrees of freedom for massive particles (including transverse spin components) is naturally encoded in the dipole-moment tensor. By a decomposition with respect to a reference vector $u^\mu$, "magnetic" and "electric" parts are isolated; the transverse magnetic dipole term, which is nonvanishing for $m\neq0$, vanishes as $m\to0$, so that the kinetic equations describing four degrees of freedom reduce smoothly to the two chiral degrees of freedom of massless fermions (Sheng et al., 2020).

4. Analytic Continuation, Nonlocal Field Redefinitions, and Distributional Methods

Strong massive-massless continuation also encompasses analytic and algebraic techniques that relate massive and massless theories:

• Nonlocal field redefinitions—e.g., infinite series in derivatives or coordinates—can map massive scalar, Dirac, or oscillator systems to their massless counterparts, demonstrating an invertible transformation between the equations of motion or actions. The essential mechanism involves constructing operators (e.g., using Pochhammer symbols and dilatation generators) such that the mass parameter is absorbed or removed at the action level. This transformation is invertible and functionally exact for free theories where physical degrees of freedom match (Najafizadeh, 2022).
• The analytic continuation of Mellin moments, Appell functions, and other multi-variable special functions enables direct transition between massive and massless operator matrix elements, conformal blocks, or celestial amplitudes. In Mellin-transform-based approaches, the transition $t\to \pm 1/x$ unifies massive and massless OMEs in $x$-space via analytic functions, smoothing the threshold between the two regimes (Behring et al., 2023). Similarly, in celestial conformal field theory, analytic continuation and projection (e.g., via monodromy techniques) allow for a decomposition of massless scalar correlators into conformal blocks even when the underlying hypergeometric structures are nontrivial (Fan, 2023). In the massless limit of celestial amplitudes, poles in Gamma functions associated with the scaling dimension compensate vanishing phase factors, enforcing a well-defined matching only for special ("conformally soft" or "generalized conformal primary") operator dimensions (Fan, 8 Apr 2024).

5. Limitations, Discontinuities, and the Role of Nonlinearities

There are cases where the massive-to-massless transition exhibits significant subtleties:

• The van Dam–Veltman–Zakharov (DVZ) discontinuity arises when, despite taking the graviton mass $m\to0$, the amplitude for gravitational interactions between conserved sources does not agree with the strictly massless graviton theory due to residual longitudinal or scalar components. This discontinuity is understood within frameworks employing string-localized fields, wherein the separation and smoothing of helicities is explicit—yet a scalar remnant remains unless the potential is redefined by a "fattening" procedure, re-coupling the decoupled massless components (Mund et al., 2017).
• In massive Yang-Mills theory, the apparent discontinuity (divergence of amplitudes or propagators as $m\to0$ in perturbation theory) is tamed via the phenomenon of strong coupling: nonlinear field redefinitions and the emergence of a "Vainshtein scale" ($L_{\rm str}\sim g/m$) result in the decoupling of the longitudinal mode, restoring the massless Yang-Mills theory up to small corrections (Hell, 2021).
• In the context of harmonic oscillators and related quantum mechanical or field-theoretic path integrals, the $m\to 0$ limit produces singular ("runaway") behavior where dominant configurations (zero-modes) are non-classical and are connected to stochastic or Fokker–Planck-like dynamics. Such models exemplify cases where the naive massive-massless transition is singular and must be understood through the lens of distributions, zero-modes, and generalized functional integrals (Modanese, 2016).

6. Phenomenological and Universality Implications

A strong massive-massless continuation is often associated with the preservation or emergence of universality:

• In the tunneling approach to Hawking radiation in cosmological and black hole backgrounds, for example, the emission probability and Hawking temperature associated with vector bosons (obtained via semiclassical solutions to the Proca equation and WKB approximation) coincide precisely with the scalar and Dirac cases. The resulting temperature $T_H = (1/2\pi r_A)$ is strictly independent of whether vector bosons are massive or massless, underscoring the universality of horizon thermodynamics and quantum effects (Jusufi et al., 2017).
• In gravitational lensing, the proper identification of so-called Wheelerian mass—effective mass arising from field energy rather than Newtonian sources—enables otherwise massless wormhole spacetimes to mimic black hole observables in the strong-field regime, provided certain charges or parameters are tuned. This further illustrates that effective, observable quantities may interpolate nontrivially between massive and massless constructions (Izmailov et al., 2019).

7. Cubic Vertices and Higher-Spin Gauge Symmetry

Strong massive-massless continuation is also central to the construction of interacting higher-spin field theories. In the BRST approach, using a complete (including trace) BRST operator, cubic vertices for irreducible higher-spin fields can be constructed such that, upon deformation of the mass parameters, the same algebraic structure covers both massive and massless interactions for all possible external states. The imposition of irreducibility and trace constraints in the cubic vertex is necessary for consistent Lagrangian dynamics; otherwise, auxiliary or unphysical modes contaminate the massless limit. The explicit construction and projection mechanisms in Fock space enable a smooth "continuation" between regimes (Buchbinder et al., 2022).

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Summary Table: Key Manifestations of Strong Massive-Massless Continuation

Mechanism / Regime	Structural Behavior	References
Gauge-invariant limit in massive gravity	Extra vector field $v_\mu$ survives	(0912.1112)
Nonlinear PM gravity in bimetric models	Obstruction due to fifth constraint	(Deser et al., 2013)
Higher-spin field constructions	Dimensional reduction, compensators	(Bastianelli et al., 2014)
Kinetic theory with spin	Transverse spin decouples smoothly	(Sheng et al., 2020)
Nonlocal redefinitions and analytic continuation	Exact mass-massless mapping for free field	(Najafizadeh, 2022, Behring et al., 2023)
String-localized fields, "fattening"	Smooth but nontrivial limit; DVZ effect	(Mund et al., 2017)
Massless limit in path integral	Runaway zero-modes dominate	(Modanese, 2016)
Tunneling across horizons in cosmology/black holes	Universal Hawking temperature	(Jusufi et al., 2017)
Wheelerian mass in lensing by wormholes	Effective mimicry via field charge	(Izmailov et al., 2019)
BRST-constructed higher-spin cubic vertices	Mass deformation, projectors assure smooth	(Buchbinder et al., 2022)
Celestial amplitudes, conformal primaries	Gamma function poles enable matching	(Fan, 8 Apr 2024)

A strong massive-massless continuation thus encompasses (i) the mathematical control of the transition between mass regimes via symmetry or analytic techniques, (ii) the persistence, disappearance, or emergence of degrees of freedom (both physical and unphysical), and (iii) universality and observable consequences in both field theory and gravity. Realizing such a continuation generally requires careful treatment of gauge invariance, anomalies, nonlocal structures, and the full algebraic and analytic structure of the theory across all scales.