Yang–Mills–Proca Theory Overview

• Yang–Mills–Proca theory is a non-Abelian framework that introduces a Proca-type mass term to Yang–Mills dynamics, breaking traditional local gauge invariance.
• The theory employs constant solution techniques and algebraic formulations, using structures like Dirac and Pauli matrices to illustrate massive spin-1 field behavior.
• Recent extensions include gauge-invariant multi-field formulations, supersymmetric tensor-dual models, and Lorentz-violating embeddings, broadening its theoretical and phenomenological scope.

Searching arXiv for the cited papers to ground the article in current preprints and related work. arxiv_search(query="(Marchuk, 18 Jun 2026)", max_results=5, sort_by="relevance") Searching arXiv for the main paper and closely related formulations. {"query":"(Marchuk, 18 Jun 2026)", "max_results": 5, "sort_by": "relevance"} Yang–Mills–Proca theory denotes the non-Abelian massive vector-field generalization obtained by adjoining a Proca-type mass term to Yang–Mills dynamics, and, more broadly, a family of formulations for massive Lie-algebra-valued spin-1 fields. In its ordinary one-connection form, the field equations are

$$\partial_\mu A_\nu-\partial_\nu A_\mu-[A_\mu,A_\nu]=F_{\mu\nu},\qquad \partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]+m^2A^\nu=0,$$

or, in the notation of a related associative-algebra formulation,

$$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$

The defining structural feature is that the mass term makes the theory non-gauge-invariant in the usual local Yang–Mills sense, because $A^\nu$ transforms inhomogeneously under local gauge transformations (Marchuk, 18 Jun 2026, Marchuk et al., 2016). Recent work has extended this basic structure in several directions, including gauge-invariant multi-field enlargements, supersymmetric tensor-dual formulations, Lorentz-violating BRST-controlled embeddings, and gravitating generalized-Proca systems (Marchuk, 18 Jun 2026, Sezgin et al., 2012, Santos et al., 2014, Martinez et al., 2022).

1. Canonical equations and loss of local gauge symmetry

The standard Yang–Mills equations used in the recent gauge-invariant reformulation literature are

$$\partial_\mu A_\nu-\partial_\nu A_\mu-[A_\mu,A_\nu]=F_{\mu\nu},\qquad \partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]=J^\nu,$$

with non-Abelian gauge transformation

$$A_\mu\to U^{-1}A_\mu U-U^{-1}\partial_\mu U,\qquad F_{\mu\nu}\to U^{-1}F_{\mu\nu}U,\qquad J^\mu\to U^{-1}J^\mu U.$$

Ordinary Yang–Mills–Proca is obtained by replacing the current with a mass term, so that the second equation becomes

$$\partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$

The mass term $m^2A^\nu$ is not gauge covariant because $A^\nu$ transforms with the inhomogeneous piece $-U^{-1}\partial^\nu U$, and this is the precise reason ordinary Yang–Mills–Proca is not gauge invariant (Marchuk, 18 Jun 2026).

A closely related formulation on pseudo-Euclidean space $\mathbb R^{p,q}$ introduces a coupling $$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$0 and writes the same theory as

$$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$1

In that setting the system is explicitly described as not gauge invariant, while retaining invariance under global transformations with constant $$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$2,

$$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$3

The same analysis yields a generalized Lorentz condition $$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$4, directly paralleling the divergence constraint of the Abelian Proca equation (Marchuk et al., 2016).

The Abelian precursor clarifies the mechanism. For the Proca system

$$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$5

taking the divergence gives $$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$6, and hence

$$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$7

The non-Abelian theory is the direct Yang–Mills analogue of this massive deformation, but without an accompanying gauge-restoration mechanism in its ordinary one-field form (Marchuk, 18 Jun 2026).

2. Algebraic structure and constant solutions

A mathematically distinctive treatment of Yang–Mills–Proca theory studies constant solutions, meaning $$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$8 is independent of spacetime position. In that case the second-order Yang–Mills–Proca equation reduces to the purely algebraic condition

$$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$9

If

$A^\nu$0

this becomes a cubic system for the coefficients $A^\nu$1. Commuting constant fields satisfy the equation with $A^\nu$2, and therefore give constant solutions of the massless Yang–Mills case (Marchuk et al., 2016).

The main explicit solution family is Clifford-type. If the algebra contains elements $A^\nu$3 satisfying

$A^\nu$4

then they satisfy the constant Yang–Mills–Proca equation with

$A^\nu$5

If $A^\nu$6 of the $A^\nu$7 components are set to zero, leaving $A^\nu$8, the same construction yields

$A^\nu$9

The scaling property is also explicit: if $$\partial_\mu A_\nu-\partial_\nu A_\mu-[A_\mu,A_\nu]=F_{\mu\nu},\qquad \partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]=J^\nu,$$0 solves the constant equation with parameter $$\partial_\mu A_\nu-\partial_\nu A_\mu-[A_\mu,A_\nu]=F_{\mu\nu},\qquad \partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]=J^\nu,$$1, then $$\partial_\mu A_\nu-\partial_\nu A_\mu-[A_\mu,A_\nu]=F_{\mu\nu},\qquad \partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]=J^\nu,$$2 solves it with $$\partial_\mu A_\nu-\partial_\nu A_\mu-[A_\mu,A_\nu]=F_{\mu\nu},\qquad \partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]=J^\nu,$$3 (Marchuk et al., 2016).

Concrete realizations are given in terms of Dirac and Pauli matrices. In $$\partial_\mu A_\nu-\partial_\nu A_\mu-[A_\mu,A_\nu]=F_{\mu\nu},\qquad \partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]=J^\nu,$$4, a tetrad $$\partial_\mu A_\nu-\partial_\nu A_\mu-[A_\mu,A_\nu]=F_{\mu\nu},\qquad \partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]=J^\nu,$$5 and Dirac matrices $$\partial_\mu A_\nu-\partial_\nu A_\mu-[A_\mu,A_\nu]=F_{\mu\nu},\qquad \partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]=J^\nu,$$6 produce

$$\partial_\mu A_\nu-\partial_\nu A_\mu-[A_\mu,A_\nu]=F_{\mu\nu},\qquad \partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]=J^\nu,$$7

while in $$\partial_\mu A_\nu-\partial_\nu A_\mu-[A_\mu,A_\nu]=F_{\mu\nu},\qquad \partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]=J^\nu,$$8, an orthonormal frame and Pauli matrices yield

$$\partial_\mu A_\nu-\partial_\nu A_\mu-[A_\mu,A_\nu]=F_{\mu\nu},\qquad \partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]=J^\nu,$$9

The same paper also embeds Grassmann algebra into Clifford algebra and observes that Grassmann-valued $$A_\mu\to U^{-1}A_\mu U-U^{-1}\partial_\mu U,\qquad F_{\mu\nu}\to U^{-1}F_{\mu\nu}U,\qquad J^\mu\to U^{-1}J^\mu U.$$0 satisfy the constant equation for $$A_\mu\to U^{-1}A_\mu U-U^{-1}\partial_\mu U,\qquad F_{\mu\nu}\to U^{-1}F_{\mu\nu}U,\qquad J^\mu\to U^{-1}J^\mu U.$$1, because nilpotency and anticommutation make the double commutator vanish (Marchuk et al., 2016).

Beyond exact constant backgrounds, the same framework develops a formal perturbation theory for Yang–Mills equations near a constant solution,

$$A_\mu\to U^{-1}A_\mu U-U^{-1}\partial_\mu U,\qquad F_{\mu\nu}\to U^{-1}F_{\mu\nu}U,\qquad J^\mu\to U^{-1}J^\mu U.$$2

with a linearized equation for the first correction $$A_\mu\to U^{-1}A_\mu U-U^{-1}\partial_\mu U,\qquad F_{\mu\nu}\to U^{-1}F_{\mu\nu}U,\qquad J^\mu\to U^{-1}J^\mu U.$$3. This part concerns Yang–Mills with a prescribed current near a constant background rather than the full massive theory itself, but it shows how Yang–Mills–Proca constant backgrounds can function as algebraic starting points for local expansions (Marchuk et al., 2016).

3. Gauge-invariant generalizations

A recent reformulation replaces the standard Stueckelberg scalar compensation by an additional vector field. In the Abelian case, one introduces $$A_\mu\to U^{-1}A_\mu U-U^{-1}\partial_\mu U,\qquad F_{\mu\nu}\to U^{-1}F_{\mu\nu}U,\qquad J^\mu\to U^{-1}J^\mu U.$$4 with its own Maxwell field strength $$A_\mu\to U^{-1}A_\mu U-U^{-1}\partial_\mu U,\qquad F_{\mu\nu}\to U^{-1}F_{\mu\nu}U,\qquad J^\mu\to U^{-1}J^\mu U.$$5,

$$A_\mu\to U^{-1}A_\mu U-U^{-1}\partial_\mu U,\qquad F_{\mu\nu}\to U^{-1}F_{\mu\nu}U,\qquad J^\mu\to U^{-1}J^\mu U.$$6

and couples it to the original Proca potential through

$$A_\mu\to U^{-1}A_\mu U-U^{-1}\partial_\mu U,\qquad F_{\mu\nu}\to U^{-1}F_{\mu\nu}U,\qquad J^\mu\to U^{-1}J^\mu U.$$7

The mass term thereby appears through the gauge-invariant difference $$A_\mu\to U^{-1}A_\mu U-U^{-1}\partial_\mu U,\qquad F_{\mu\nu}\to U^{-1}F_{\mu\nu}U,\qquad J^\mu\to U^{-1}J^\mu U.$$8. The gauge symmetry is

$$A_\mu\to U^{-1}A_\mu U-U^{-1}\partial_\mu U,\qquad F_{\mu\nu}\to U^{-1}F_{\mu\nu}U,\qquad J^\mu\to U^{-1}J^\mu U.$$9

with $$\partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$0 and $$\partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$1 invariant. The construction contains all Proca solutions as the special sector $$\partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$2, but it also admits extra solutions such as $$\partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$3 with both fields massless. The system is therefore presented as a gauge-invariant generalization containing Proca, not as a demonstrated strict equivalence. No Lagrangian or variational derivation is given (Marchuk, 18 Jun 2026).

The same idea is lifted to the non-Abelian case by introducing $$\partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$4 connection-like fields $$\partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$5, corresponding curvatures $$\partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$6, and currents $$\partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$7, assembled into columns $$\partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$8, $$\partial_\mu F^{\mu\nu}-[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$9, and $m^2A^\nu$0. The generalized equations are

$m^2A^\nu$1

$m^2A^\nu$2

where the mass-coupling matrix $m^2A^\nu$3 satisfies

$m^2A^\nu$4

All copies transform under a single gauge function $m^2A^\nu$5 as

$m^2A^\nu$6

so every component carries the same inhomogeneous term. The row-sum-zero condition ensures that $m^2A^\nu$7 transforms homogeneously,

$m^2A^\nu$8

and the mass-like term is therefore gauge covariant (Marchuk, 18 Jun 2026).

For $m^2A^\nu$9, the explicit example

$A^\nu$0

produces equations in which the mass couplings depend on differences such as $A^\nu$1. The derived relations

$A^\nu$2

$A^\nu$3

are covariant-divergence-type constraints on the difference field. As written, the theory is a coupled multi-connection gauge-invariant system rather than a demonstrated one-copy reformulation of ordinary Yang–Mills–Proca (Marchuk, 18 Jun 2026).

This gauge-restoration strategy is explicitly contrasted with Stueckelberg theory. What is similar is the use of a compensating field and a mass term depending on a gauge-invariant combination. What is different is that the compensator is a vector field, not a scalar, and that in the non-Abelian case the compensating structure is distributed over several connection-like fields (Marchuk, 18 Jun 2026).

4. Supersymmetric, geometric, and Lorentz-violating formulations

A supersymmetric realization constructs a tensor–Yang–Mills system in which a tensor multiplet and an additional vector multiplet both transform in a nontrivial representation $A^\nu$4 of a semi-simple Yang–Mills group $A^\nu$5. After Stückelberg gauge fixing, the extra vector is eaten by the tensor field, yielding an off-shell massive tensor multiplet. Dualization then replaces the massive tensor by an on-shell massive vector multiplet $A^\nu$6 coupled to an off-shell Yang–Mills multiplet. The resulting bosonic Proca–Yang–Mills sector contains

$A^\nu$7

with

$A^\nu$8

The construction is explicitly contrasted with the Higgs mechanism: the massive vector arises from tensor dualization rather than spontaneous symmetry breaking, and after auxiliary elimination the action contains an infinite nonpolynomial tower in the scalar $A^\nu$9. The same work states that tree-level amplitudes for longitudinal massive vectors still grow with energy, so high-energy unitarity is not restored in the Yang–Mills–Higgs manner (Sezgin et al., 2012).

A geometrically different direction starts from a MacDowell–Mansouri gauge theory of gravity with gauge group $-U^{-1}\partial^\nu U$0 or $-U^{-1}\partial^\nu U$1. The broken part of the connection is modified according to

$-U^{-1}\partial^\nu U$2

After the explicit breaking $-U^{-1}\partial^\nu U$3, the resulting four-dimensional action becomes a fixed combination of generalized Proca and beyond-generalized-Proca interactions coupled to gravity. In flat space the theory is written as

$-U^{-1}\partial^\nu U$4

This is Yang–Mills-type in origin because the fundamental object is a gauge connection with curvature, but it is not ordinary internal Yang–Mills with a Proca mass added by hand; the vector is part of the broken connection sector and behaves as a Proca-like field because no $-U^{-1}\partial^\nu U$5 gauge symmetry survives for $-U^{-1}\partial^\nu U$6 (Chagoya et al., 2022).

A third extension arises in Lorentz-violating pure Yang–Mills theory. There the ordinary physical action is embedded into a BRST-invariant enlarged theory with external sources, and algebraic renormalization requires a local composite operator sector that is quadratic in the gauge field. When the external sources are set to their physical values, the physical action acquires both an isotropic Proca-like term

$-U^{-1}\partial^\nu U$7

and a Lorentz-violating tensorial quadratic term

$-U^{-1}\partial^\nu U$8

The same analysis states that the theory is renormalizable to all orders in Landau gauge, that the standard Yang–Mills renormalization remains unchanged, and that the analogous Abelian construction does not generate a Proca-like photon mass. The result is therefore not textbook Yang–Mills–Proca but a renormalizable Lorentz-violating Yang–Mills theory whose physical limit contains Proca-like mass parameters (Santos et al., 2014).

5. Gravitating and solitonic realizations

A direct curved-spacetime $-U^{-1}\partial^\nu U$9 realization appears in the Einstein–Dirac system coupled to Yang–Mills or Yang–Mills–Proca magnetic fields. The vector-field sector is

$\mathbb R^{p,q}$0

with

$\mathbb R^{p,q}$1

The Proca mass term is described as a genuine mass term for the non-Abelian vector field and as explicitly breaking local $\mathbb R^{p,q}$2 gauge invariance. Under a static spherically symmetric magnetic monopole ansatz, the field equations reduce to a boundary-value problem for coupled ordinary differential equations. The Proca mass enters through

$\mathbb R^{p,q}$3

appearing both in the energy density and in the gauge equation. For the Proca case the asymptotic gauge-field profile is

$\mathbb R^{p,q}$4

so the field is finite-ranged, and the paper states that there are no kinklike solutions in the Proca case. For the parameter ranges studied, the $\mathbb R^{p,q}$5 magnetic field contributes only weakly to the total energy density and mass, so the solutions are interpreted as magnetized Dirac stars dominated by the spinor sector (Dzhunushaliev et al., 2019).

A broader solitonic framework is provided by generalized $\mathbb R^{p,q}$6 Proca theory in Einstein gravity. Its matter Lagrangian contains the standard Yang–Mills–Proca sector

$\mathbb R^{p,q}$7

supplemented by derivative self-interactions and non-minimal curvature couplings controlled by parameters $\mathbb R^{p,q}$8 and $\mathbb R^{p,q}$9. The theory is explicitly invariant under diffeomorphisms and global internal $$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$00 transformations, not local $$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$01 gauge symmetry. With the purely magnetic t’Hooft–Polyakov/Bartnik–McKinnon-type ansatz

$$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$02

the field equations admit static, asymptotically flat, globally regular particle-like solutions. The regularity conditions imply

$$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$03

and solutions are classified by the number of nodes of $$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$04 (Martinez et al., 2022).

The generalized theory interpolates among several limits. If $$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$05 and all generalized couplings vanish, one recovers Einstein–Yang–Mills. If $$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$06 and all generalized couplings vanish, one obtains ordinary Einstein–$$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$07-Proca. The full generalized theory, however, exhibits phenomena absent in both of those limits, including regions of negative effective energy density, imaginary effective charge, and globally charged regular solutions. In the special sector with only $$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$08, constant-$$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$09 solutions yield a Reissner–Nordström-type metric,

$$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$10

with branches determined by

$$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$11

The paper emphasizes that the new global charge is induced by quartic non-derivative self-interactions rather than by the standard Proca mass term (Martinez et al., 2022).

6. Conceptual status and unresolved structure

Across the literature, Yang–Mills–Proca theory is not a single settled framework but a cluster of related constructions. Ordinary one-field Yang–Mills–Proca is the direct non-Abelian analogue of Proca and is explicitly non-gauge-invariant. The recent gauge-invariant enlargement with extra vector potentials restores gauge covariance by replacing a single inhomogeneously transforming field with several connection-like fields and mass couplings built from their differences. That enlarged theory contains ordinary Proca solutions as a subsector, but it also has additional solutions and is not shown to be dynamically equivalent either to ordinary Proca or to Stueckelberg theory. The same work does not present a Lagrangian, action, variational principle, canonical degree-of-freedom count, quantization procedure, BRST structure, renormalizability analysis, propagators, or phenomenological applications (Marchuk, 18 Jun 2026).

Other variants retain similar limitations in different forms. The supersymmetric tensor-dual construction yields an on-shell massive vector multiplet coupled to off-shell Yang–Mills, but does not reproduce the Higgs mechanism and does not remove the growth of longitudinal-vector scattering amplitudes at high energy (Sezgin et al., 2012). The Lorentz-violating BRST-controlled embedding proves all-order renormalizability of the enlarged theory, yet explicitly warns that the induced quadratic gauge-field terms should be interpreted as mass parameters rather than established physical masses, and does not settle unitarity, causality, or the pole structure of the massive modes (Santos et al., 2014). The broken-gauge vector–tensor gravity construction preserves second-order field equations and the intended generalized-Proca constraint structure, but leaves the full longitudinal-mode analysis for future work (Chagoya et al., 2022).

This suggests that the expression “Yang–Mills–Proca theory” has become an umbrella term for several non-equivalent strategies for treating massive non-Abelian vectors. In the narrow sense it denotes the direct equation

$$\partial_\mu A^\nu-\partial^\nu A_\mu-\rho[A_\mu,A^\nu]=F_\mu{}^\nu,\qquad \partial_\mu F^{\mu\nu}-\rho[A_\mu,F^{\mu\nu}]+m^2A^\nu=0.$$12

In broader current usage it also includes gauge-invariant multi-connection extensions, Higgsless supersymmetric tensor-dual models, Lorentz-violating renormalizable embeddings with induced Proca-like sectors, and generalized-Proca gravitating systems with only global internal symmetry. What unifies these constructions is the attempt to encode massive non-Abelian spin-1 dynamics; what differentiates them is the status of gauge symmetry, the number and type of compensating fields, and the extent to which equivalence to ordinary Yang–Mills–Proca is actually established (Marchuk, 18 Jun 2026, Martinez et al., 2022).