Weyl–Dirac–Born–Infeld Action: Unified Gauge Theory

• WDBI action is a determinant-based, Weyl-invariant construction that unifies gravity with the Standard Model in arbitrary spacetime dimensions.
• Its leading expansion reproduces Weyl-invariant SM plus quadratic gravity, naturally using the Weyl-covariant curvature as a regulator.
• After Stueckelberg breaking, the framework yields Einstein–Hilbert gravity and SM couplings, eliminating Weyl anomalies below the Planck scale.

The Weyl–Dirac–Born–Infeld action (WDBI) is a determinant-based gauge-theoretic construction that extends the Dirac–Born–Infeld action to gravity and the Standard Model (SM) in Weyl conformal geometry. In its 2025 formulation, it is defined in arbitrary spacetime dimension $d$ as an exactly SM- and Weyl-gauge-invariant action built from Weyl-covariant curvature, gauge-field, Higgs, and fermion operators of mass dimension $2$. Its leading expansion reproduces Weyl-gauge-invariant SM plus Weyl quadratic gravity, while its broken Stueckelberg phase yields SM plus Einstein–Hilbert gravity below the Planck scale. A central claim of the construction is that, because the exact action is Weyl invariant in arbitrary $d$, it requires no external ultraviolet regulator and instead uses Weyl-covariant scalar curvature as the geometric regulator in $d=4-2\epsilon$ (Ghilencea, 14 Aug 2025).

1. Determinant construction in arbitrary dimension

The defining WDBI action is

$$S_{\bf d}=\int d^dx\,\Big[-\det A_{\mu\nu}\Big]^\frac{1}{2},$$

with $A_{\mu\nu}$ assembled from all operators of mass dimension $2$ that are simultaneously invariant under the SM gauge group and the Weyl gauge group in $d$ dimensions: $$\begin{aligned} A_{\mu\nu}&= a_0 \,\hat R\, g_{\mu\nu}+ a_1 \,\hat R_{\mu\nu}+ a_2 \,\hat F_{\mu\nu} + a_3\,F_{\mu\nu}^{(1)} + a_4^{(i)} \,F^{(i)}_{\alpha\beta} F^{(i)\,\alpha\beta}\,g_{\mu\nu}\,\hat R^{-1} \ &\quad + a_5\,\vert \hat\nabla_\alpha H\vert^2\, \hat R^{1-d/2} \,g_{\mu\nu} +a_6 \,\vert H\vert^2 \hat R^{2-d/2} g_{\mu\nu} +a_7 \vert H\vert^4\,\hat R^{3-d}\,g_{\mu\nu} \ &\quad + a_8\, \big( i\, \overline\psi \gamma^a\,e_a^\alpha\hat\nabla_\alpha\psi+\textrm{h.c.} \,\big) \, \hat R^{1-d/2}\, g_{\mu\nu} \ &\quad + a_9\,\big( \overline \psi_L\,Y_\psi\, H\psi_R+\overline \psi_L Y^\prime_\psi\tilde H\,\psi^\prime_R+\textrm{h.c.}\big) \hat R^{2-3\,d/4}\,g_{\mu\nu} \ &\quad + a_{10}\, \hat F_{\alpha\beta} \hat F^{\alpha\beta} \hat R^{-1} g_{\mu\nu} +a_{11}\, \hat F_{\alpha\beta} F^{(1)\,\alpha\beta} \hat R^{-1} g_{\mu\nu}. \end{aligned}$$

Here $g_{\mu\nu}$ is the metric, $2$0 the SM Higgs doublet, $2$1 the SM fermions, $2$2 the $2$3 field strength, $2$4 with $2$5 the SM gauge field strengths, $2$6 the Weyl field strength of the Weyl gauge boson $2$7, and $2$8, $2$9, $d$0 the Weyl-covariant curvature tensors and scalar. All coefficients $d$1 are dimensionless, and each term entering $d$2 is SM- and Weyl-gauge invariant of mass dimension $d$3 (Ghilencea, 14 Aug 2025).

This determinant structure generalizes earlier gravity-only Weyl-DBI constructions by incorporating the full SM operator content under the same determinant. In the terminology of the 2025 paper, this is a unified gauge theory of the SM and Weyl group, with the latter understood as the gauge theory of dilatations and Poincaré symmetry.

2. Weyl geometry, charges, and covariant differentiation

The geometric setting is the Weyl gauge covariant metric formulation of Weyl geometry. Weyl gauge transformations are

$d$4

with $d$5. Scalars and fermions transform with Weyl weights

$d$6

so for the Higgs doublet $d$7, $d$8. In $d$9, $d=4-2\epsilon$0 and $d=4-2\epsilon$1.

The non-metricity condition is

$d=4-2\epsilon$2

with symmetric affine connection

$d=4-2\epsilon$3

The physical Weyl-covariant derivative acting on a tensor $d=4-2\epsilon$4 of total Weyl charge $d=4-2\epsilon$5 is

$d=4-2\epsilon$6

This formulation is metric, $d=4-2\epsilon$7, but non-affine.

For matter fields, the Weyl-covariant derivatives are

$d=4-2\epsilon$8

A specific simplification occurs for fermions: $d=4-2\epsilon$9 so fermions do not couple to $$S_{\bf d}=\int d^dx\,\Big[-\det A_{\mu\nu}\Big]^\frac{1}{2},$$0 at tree level.

The Weyl-covariant curvatures are

$$S_{\bf d}=\int d^dx\,\Big[-\det A_{\mu\nu}\Big]^\frac{1}{2},$$1

$$S_{\bf d}=\int d^dx\,\Big[-\det A_{\mu\nu}\Big]^\frac{1}{2},$$2

and $$S_{\bf d}=\int d^dx\,\Big[-\det A_{\mu\nu}\Big]^\frac{1}{2},$$3. Their transformation laws imply that $$S_{\bf d}=\int d^dx\,\Big[-\det A_{\mu\nu}\Big]^\frac{1}{2},$$4 has Weyl weight $$S_{\bf d}=\int d^dx\,\Big[-\det A_{\mu\nu}\Big]^\frac{1}{2},$$5, while $$S_{\bf d}=\int d^dx\,\Big[-\det A_{\mu\nu}\Big]^\frac{1}{2},$$6, $$S_{\bf d}=\int d^dx\,\Big[-\det A_{\mu\nu}\Big]^\frac{1}{2},$$7, and $$S_{\bf d}=\int d^dx\,\Big[-\det A_{\mu\nu}\Big]^\frac{1}{2},$$8 are invariant in the sense required by the determinant construction (Ghilencea, 14 Aug 2025).

3. Series expansion and recovery of Weyl-invariant SM plus quadratic gravity

A controlled expansion is obtained by defining

$$S_{\bf d}=\int d^dx\,\Big[-\det A_{\mu\nu}\Big]^\frac{1}{2},$$9

and using $A_{\mu\nu}$0. The action becomes

$A_{\mu\nu}$1

After curvature reorganization, the canonical leading-order form is

$A_{\mu\nu}$2

The couplings satisfy

$A_{\mu\nu}$3

$A_{\mu\nu}$4

Thus the leading order reproduces Weyl-gauge-invariant SM plus Weyl quadratic gravity in arbitrary $A_{\mu\nu}$5, with dimensionless couplings $A_{\mu\nu}$6. In particular, there is no $A_{\mu\nu}$7–hypercharge kinetic mixing at leading order because $A_{\mu\nu}$8 by choice of coefficients. The expansion is controlled by

$A_{\mu\nu}$9

so subleading corrections are suppressed by powers of $2$0 (Ghilencea, 14 Aug 2025).

In $2$1, the explicit leading action is

$2$2

4. Geometric regularisation and absence of Weyl anomaly

A distinctive claim of the WDBI framework is that exact Weyl invariance in arbitrary $2$3 removes the need for an external ultraviolet regulator. The paper states that the action is mathematically well-defined in $2$4 dimensions and that, in $2$5, the scalar curvature itself provides the regularising factor. In the canonical leading-order form, operators acquire the precise curvature powers required for dimensional continuation: $2$6

Accordingly, the gravitational and gauge operators come with $2$7, the Higgs quartic with $2$8, and Yukawa terms with $2$9. The paper emphasizes that to regularize the $d$0 theory one simply replaces $d$1, with no dimensional-regularization scale $d$2 and no compensator field added. Because every operator entering $d$3 is exactly Weyl invariant in arbitrary $d$4, $d$5 is invariant, $d$6 is invariant, and therefore $d$7 is exactly Weyl invariant in any $d$8. On that basis, the construction claims the absence of a Weyl anomaly (Ghilencea, 14 Aug 2025).

This point is not merely formal. In the WDBI formulation, the regulator is not external to the geometry; it is carried by the Weyl-covariant curvature already present in the action. That feature is presented as the central distinction from ordinary Riemannian gauge theories, from conformal gravity regularised with an added dilaton, and from standard DBI constructions.

5. Stueckelberg breaking, emergence of Einstein–Hilbert gravity, and SM couplings

In four dimensions, the leading geometric sector is

$d$9

Introducing an auxiliary Stueckelberg/dilaton scalar $$\begin{aligned} A_{\mu\nu}&= a_0 \,\hat R\, g_{\mu\nu}+ a_1 \,\hat R_{\mu\nu}+ a_2 \,\hat F_{\mu\nu} + a_3\,F_{\mu\nu}^{(1)} + a_4^{(i)} \,F^{(i)}_{\alpha\beta} F^{(i)\,\alpha\beta}\,g_{\mu\nu}\,\hat R^{-1} \ &\quad + a_5\,\vert \hat\nabla_\alpha H\vert^2\, \hat R^{1-d/2} \,g_{\mu\nu} +a_6 \,\vert H\vert^2 \hat R^{2-d/2} g_{\mu\nu} +a_7 \vert H\vert^4\,\hat R^{3-d}\,g_{\mu\nu} \ &\quad + a_8\, \big( i\, \overline\psi \gamma^a\,e_a^\alpha\hat\nabla_\alpha\psi+\textrm{h.c.} \,\big) \, \hat R^{1-d/2}\, g_{\mu\nu} \ &\quad + a_9\,\big( \overline \psi_L\,Y_\psi\, H\psi_R+\overline \psi_L Y^\prime_\psi\tilde H\,\psi^\prime_R+\textrm{h.c.}\big) \hat R^{2-3\,d/4}\,g_{\mu\nu} \ &\quad + a_{10}\, \hat F_{\alpha\beta} \hat F^{\alpha\beta} \hat R^{-1} g_{\mu\nu} +a_{11}\, \hat F_{\alpha\beta} F^{(1)\,\alpha\beta} \hat R^{-1} g_{\mu\nu}. \end{aligned}$$0 to linearize $$\begin{aligned} A_{\mu\nu}&= a_0 \,\hat R\, g_{\mu\nu}+ a_1 \,\hat R_{\mu\nu}+ a_2 \,\hat F_{\mu\nu} + a_3\,F_{\mu\nu}^{(1)} + a_4^{(i)} \,F^{(i)}_{\alpha\beta} F^{(i)\,\alpha\beta}\,g_{\mu\nu}\,\hat R^{-1} \ &\quad + a_5\,\vert \hat\nabla_\alpha H\vert^2\, \hat R^{1-d/2} \,g_{\mu\nu} +a_6 \,\vert H\vert^2 \hat R^{2-d/2} g_{\mu\nu} +a_7 \vert H\vert^4\,\hat R^{3-d}\,g_{\mu\nu} \ &\quad + a_8\, \big( i\, \overline\psi \gamma^a\,e_a^\alpha\hat\nabla_\alpha\psi+\textrm{h.c.} \,\big) \, \hat R^{1-d/2}\, g_{\mu\nu} \ &\quad + a_9\,\big( \overline \psi_L\,Y_\psi\, H\psi_R+\overline \psi_L Y^\prime_\psi\tilde H\,\psi^\prime_R+\textrm{h.c.}\big) \hat R^{2-3\,d/4}\,g_{\mu\nu} \ &\quad + a_{10}\, \hat F_{\alpha\beta} \hat F^{\alpha\beta} \hat R^{-1} g_{\mu\nu} +a_{11}\, \hat F_{\alpha\beta} F^{(1)\,\alpha\beta} \hat R^{-1} g_{\mu\nu}. \end{aligned}$$1, one obtains the equivalent Weyl-invariant form

$$\begin{aligned} A_{\mu\nu}&= a_0 \,\hat R\, g_{\mu\nu}+ a_1 \,\hat R_{\mu\nu}+ a_2 \,\hat F_{\mu\nu} + a_3\,F_{\mu\nu}^{(1)} + a_4^{(i)} \,F^{(i)}_{\alpha\beta} F^{(i)\,\alpha\beta}\,g_{\mu\nu}\,\hat R^{-1} \ &\quad + a_5\,\vert \hat\nabla_\alpha H\vert^2\, \hat R^{1-d/2} \,g_{\mu\nu} +a_6 \,\vert H\vert^2 \hat R^{2-d/2} g_{\mu\nu} +a_7 \vert H\vert^4\,\hat R^{3-d}\,g_{\mu\nu} \ &\quad + a_8\, \big( i\, \overline\psi \gamma^a\,e_a^\alpha\hat\nabla_\alpha\psi+\textrm{h.c.} \,\big) \, \hat R^{1-d/2}\, g_{\mu\nu} \ &\quad + a_9\,\big( \overline \psi_L\,Y_\psi\, H\psi_R+\overline \psi_L Y^\prime_\psi\tilde H\,\psi^\prime_R+\textrm{h.c.}\big) \hat R^{2-3\,d/4}\,g_{\mu\nu} \ &\quad + a_{10}\, \hat F_{\alpha\beta} \hat F^{\alpha\beta} \hat R^{-1} g_{\mu\nu} +a_{11}\, \hat F_{\alpha\beta} F^{(1)\,\alpha\beta} \hat R^{-1} g_{\mu\nu}. \end{aligned}$$2

Gauge fixing with $$\begin{aligned} A_{\mu\nu}&= a_0 \,\hat R\, g_{\mu\nu}+ a_1 \,\hat R_{\mu\nu}+ a_2 \,\hat F_{\mu\nu} + a_3\,F_{\mu\nu}^{(1)} + a_4^{(i)} \,F^{(i)}_{\alpha\beta} F^{(i)\,\alpha\beta}\,g_{\mu\nu}\,\hat R^{-1} \ &\quad + a_5\,\vert \hat\nabla_\alpha H\vert^2\, \hat R^{1-d/2} \,g_{\mu\nu} +a_6 \,\vert H\vert^2 \hat R^{2-d/2} g_{\mu\nu} +a_7 \vert H\vert^4\,\hat R^{3-d}\,g_{\mu\nu} \ &\quad + a_8\, \big( i\, \overline\psi \gamma^a\,e_a^\alpha\hat\nabla_\alpha\psi+\textrm{h.c.} \,\big) \, \hat R^{1-d/2}\, g_{\mu\nu} \ &\quad + a_9\,\big( \overline \psi_L\,Y_\psi\, H\psi_R+\overline \psi_L Y^\prime_\psi\tilde H\,\psi^\prime_R+\textrm{h.c.}\big) \hat R^{2-3\,d/4}\,g_{\mu\nu} \ &\quad + a_{10}\, \hat F_{\alpha\beta} \hat F^{\alpha\beta} \hat R^{-1} g_{\mu\nu} +a_{11}\, \hat F_{\alpha\beta} F^{(1)\,\alpha\beta} \hat R^{-1} g_{\mu\nu}. \end{aligned}$$3 sets $$\begin{aligned} A_{\mu\nu}&= a_0 \,\hat R\, g_{\mu\nu}+ a_1 \,\hat R_{\mu\nu}+ a_2 \,\hat F_{\mu\nu} + a_3\,F_{\mu\nu}^{(1)} + a_4^{(i)} \,F^{(i)}_{\alpha\beta} F^{(i)\,\alpha\beta}\,g_{\mu\nu}\,\hat R^{-1} \ &\quad + a_5\,\vert \hat\nabla_\alpha H\vert^2\, \hat R^{1-d/2} \,g_{\mu\nu} +a_6 \,\vert H\vert^2 \hat R^{2-d/2} g_{\mu\nu} +a_7 \vert H\vert^4\,\hat R^{3-d}\,g_{\mu\nu} \ &\quad + a_8\, \big( i\, \overline\psi \gamma^a\,e_a^\alpha\hat\nabla_\alpha\psi+\textrm{h.c.} \,\big) \, \hat R^{1-d/2}\, g_{\mu\nu} \ &\quad + a_9\,\big( \overline \psi_L\,Y_\psi\, H\psi_R+\overline \psi_L Y^\prime_\psi\tilde H\,\psi^\prime_R+\textrm{h.c.}\big) \hat R^{2-3\,d/4}\,g_{\mu\nu} \ &\quad + a_{10}\, \hat F_{\alpha\beta} \hat F^{\alpha\beta} \hat R^{-1} g_{\mu\nu} +a_{11}\, \hat F_{\alpha\beta} F^{(1)\,\alpha\beta} \hat R^{-1} g_{\mu\nu}. \end{aligned}$$4 and yields the broken-phase action

$$\begin{aligned} A_{\mu\nu}&= a_0 \,\hat R\, g_{\mu\nu}+ a_1 \,\hat R_{\mu\nu}+ a_2 \,\hat F_{\mu\nu} + a_3\,F_{\mu\nu}^{(1)} + a_4^{(i)} \,F^{(i)}_{\alpha\beta} F^{(i)\,\alpha\beta}\,g_{\mu\nu}\,\hat R^{-1} \ &\quad + a_5\,\vert \hat\nabla_\alpha H\vert^2\, \hat R^{1-d/2} \,g_{\mu\nu} +a_6 \,\vert H\vert^2 \hat R^{2-d/2} g_{\mu\nu} +a_7 \vert H\vert^4\,\hat R^{3-d}\,g_{\mu\nu} \ &\quad + a_8\, \big( i\, \overline\psi \gamma^a\,e_a^\alpha\hat\nabla_\alpha\psi+\textrm{h.c.} \,\big) \, \hat R^{1-d/2}\, g_{\mu\nu} \ &\quad + a_9\,\big( \overline \psi_L\,Y_\psi\, H\psi_R+\overline \psi_L Y^\prime_\psi\tilde H\,\psi^\prime_R+\textrm{h.c.}\big) \hat R^{2-3\,d/4}\,g_{\mu\nu} \ &\quad + a_{10}\, \hat F_{\alpha\beta} \hat F^{\alpha\beta} \hat R^{-1} g_{\mu\nu} +a_{11}\, \hat F_{\alpha\beta} F^{(1)\,\alpha\beta} \hat R^{-1} g_{\mu\nu}. \end{aligned}$$5

with

$$\begin{aligned} A_{\mu\nu}&= a_0 \,\hat R\, g_{\mu\nu}+ a_1 \,\hat R_{\mu\nu}+ a_2 \,\hat F_{\mu\nu} + a_3\,F_{\mu\nu}^{(1)} + a_4^{(i)} \,F^{(i)}_{\alpha\beta} F^{(i)\,\alpha\beta}\,g_{\mu\nu}\,\hat R^{-1} \ &\quad + a_5\,\vert \hat\nabla_\alpha H\vert^2\, \hat R^{1-d/2} \,g_{\mu\nu} +a_6 \,\vert H\vert^2 \hat R^{2-d/2} g_{\mu\nu} +a_7 \vert H\vert^4\,\hat R^{3-d}\,g_{\mu\nu} \ &\quad + a_8\, \big( i\, \overline\psi \gamma^a\,e_a^\alpha\hat\nabla_\alpha\psi+\textrm{h.c.} \,\big) \, \hat R^{1-d/2}\, g_{\mu\nu} \ &\quad + a_9\,\big( \overline \psi_L\,Y_\psi\, H\psi_R+\overline \psi_L Y^\prime_\psi\tilde H\,\psi^\prime_R+\textrm{h.c.}\big) \hat R^{2-3\,d/4}\,g_{\mu\nu} \ &\quad + a_{10}\, \hat F_{\alpha\beta} \hat F^{\alpha\beta} \hat R^{-1} g_{\mu\nu} +a_{11}\, \hat F_{\alpha\beta} F^{(1)\,\alpha\beta} \hat R^{-1} g_{\mu\nu}. \end{aligned}$$6

The Einstein–Hilbert term thus emerges with Planck mass $$\begin{aligned} A_{\mu\nu}&= a_0 \,\hat R\, g_{\mu\nu}+ a_1 \,\hat R_{\mu\nu}+ a_2 \,\hat F_{\mu\nu} + a_3\,F_{\mu\nu}^{(1)} + a_4^{(i)} \,F^{(i)}_{\alpha\beta} F^{(i)\,\alpha\beta}\,g_{\mu\nu}\,\hat R^{-1} \ &\quad + a_5\,\vert \hat\nabla_\alpha H\vert^2\, \hat R^{1-d/2} \,g_{\mu\nu} +a_6 \,\vert H\vert^2 \hat R^{2-d/2} g_{\mu\nu} +a_7 \vert H\vert^4\,\hat R^{3-d}\,g_{\mu\nu} \ &\quad + a_8\, \big( i\, \overline\psi \gamma^a\,e_a^\alpha\hat\nabla_\alpha\psi+\textrm{h.c.} \,\big) \, \hat R^{1-d/2}\, g_{\mu\nu} \ &\quad + a_9\,\big( \overline \psi_L\,Y_\psi\, H\psi_R+\overline \psi_L Y^\prime_\psi\tilde H\,\psi^\prime_R+\textrm{h.c.}\big) \hat R^{2-3\,d/4}\,g_{\mu\nu} \ &\quad + a_{10}\, \hat F_{\alpha\beta} \hat F^{\alpha\beta} \hat R^{-1} g_{\mu\nu} +a_{11}\, \hat F_{\alpha\beta} F^{(1)\,\alpha\beta} \hat R^{-1} g_{\mu\nu}. \end{aligned}$$7, the Weyl gauge boson acquires Stueckelberg mass

$$\begin{aligned} A_{\mu\nu}&= a_0 \,\hat R\, g_{\mu\nu}+ a_1 \,\hat R_{\mu\nu}+ a_2 \,\hat F_{\mu\nu} + a_3\,F_{\mu\nu}^{(1)} + a_4^{(i)} \,F^{(i)}_{\alpha\beta} F^{(i)\,\alpha\beta}\,g_{\mu\nu}\,\hat R^{-1} \ &\quad + a_5\,\vert \hat\nabla_\alpha H\vert^2\, \hat R^{1-d/2} \,g_{\mu\nu} +a_6 \,\vert H\vert^2 \hat R^{2-d/2} g_{\mu\nu} +a_7 \vert H\vert^4\,\hat R^{3-d}\,g_{\mu\nu} \ &\quad + a_8\, \big( i\, \overline\psi \gamma^a\,e_a^\alpha\hat\nabla_\alpha\psi+\textrm{h.c.} \,\big) \, \hat R^{1-d/2}\, g_{\mu\nu} \ &\quad + a_9\,\big( \overline \psi_L\,Y_\psi\, H\psi_R+\overline \psi_L Y^\prime_\psi\tilde H\,\psi^\prime_R+\textrm{h.c.}\big) \hat R^{2-3\,d/4}\,g_{\mu\nu} \ &\quad + a_{10}\, \hat F_{\alpha\beta} \hat F^{\alpha\beta} \hat R^{-1} g_{\mu\nu} +a_{11}\, \hat F_{\alpha\beta} F^{(1)\,\alpha\beta} \hat R^{-1} g_{\mu\nu}. \end{aligned}$$8

and a positive cosmological constant appears. Below $$\begin{aligned} A_{\mu\nu}&= a_0 \,\hat R\, g_{\mu\nu}+ a_1 \,\hat R_{\mu\nu}+ a_2 \,\hat F_{\mu\nu} + a_3\,F_{\mu\nu}^{(1)} + a_4^{(i)} \,F^{(i)}_{\alpha\beta} F^{(i)\,\alpha\beta}\,g_{\mu\nu}\,\hat R^{-1} \ &\quad + a_5\,\vert \hat\nabla_\alpha H\vert^2\, \hat R^{1-d/2} \,g_{\mu\nu} +a_6 \,\vert H\vert^2 \hat R^{2-d/2} g_{\mu\nu} +a_7 \vert H\vert^4\,\hat R^{3-d}\,g_{\mu\nu} \ &\quad + a_8\, \big( i\, \overline\psi \gamma^a\,e_a^\alpha\hat\nabla_\alpha\psi+\textrm{h.c.} \,\big) \, \hat R^{1-d/2}\, g_{\mu\nu} \ &\quad + a_9\,\big( \overline \psi_L\,Y_\psi\, H\psi_R+\overline \psi_L Y^\prime_\psi\tilde H\,\psi^\prime_R+\textrm{h.c.}\big) \hat R^{2-3\,d/4}\,g_{\mu\nu} \ &\quad + a_{10}\, \hat F_{\alpha\beta} \hat F^{\alpha\beta} \hat R^{-1} g_{\mu\nu} +a_{11}\, \hat F_{\alpha\beta} F^{(1)\,\alpha\beta} \hat R^{-1} g_{\mu\nu}. \end{aligned}$$9, $g_{\mu\nu}$0 decouples and the geometry becomes Riemannian.

The SM sector couples in a correspondingly Weyl-covariant manner. The Higgs Lagrangian is

$g_{\mu\nu}$1

reducing in $g_{\mu\nu}$2 to

$g_{\mu\nu}$3

The gauge sector is

$g_{\mu\nu}$4

and the Yukawa sector is

$g_{\mu\nu}$5

In $g_{\mu\nu}$6, the Yukawa interaction reduces to the standard form. After Stueckelberg breaking and electroweak symmetry breaking, masses arise as usual from the Higgs vacuum expectation value, while $g_{\mu\nu}$7 and $g_{\mu\nu}$8 arise from the $g_{\mu\nu}$9 vev. The paper further states that in the presence of the SM, the would-be Goldstone is a mixture of $2$00 and the neutral Higgs field; after symmetry breaking, one recovers SM plus Einstein–Hilbert gravity at low energies (Ghilencea, 14 Aug 2025).

Subleading operators generated by the determinant expansion include

$2$01

$2$02

which in $2$03 correspond to corrections suppressed by $2$04 or by $2$05.

6. Related formulations, earlier usages, and technical caveats

The 2025 gravity-plus-SM WDBI action extends a sequence of earlier Weyl-DBI constructions. A 2024 Weyl-DBI paper formulated the gravity-only determinant action in Weyl conformal geometry and argued that its leading expansion recovers Weyl quadratic gravity without introducing a UV regulator, while a 2026 review characterized that quadratic theory as the leading order of a more fundamental WDBI action. A further 2026 development described a non-local map from Weyl geometry to a Riemannian description based on a Weyl gauge invariant dressed metric

$2$06

for which

$2$07

and the WDBI action takes the same determinant form when written in terms of $2$08. In that dressed picture, however, ultraviolet non-commutativity appears through

$2$09

and the equation of motion of $2$10 does not commute with dressing. Earlier literature from 2010–2011 had also used Weyl-invariant DBI-Einstein constructions with a compensator scalar $2$11, a Weyl gauge field, and vector-inflation applications; those models employed Weyl-invariant combinations such as $2$12 and $2$13 (Ghilencea, 2024, Ghilencea, 8 Apr 2026, Ghilencea et al., 6 Jun 2026, Kan et al., 2010, Maki et al., 2011).

Several technical caveats recur across this literature. First, the WDBI construction is not merely ordinary Riemannian geometry in a different gauge: the dressed-metric description is explicitly non-local and path dependent in the symmetric phase. Second, the leading-order gravitational sector contains both $2$14 and $2$15, so after symmetry breaking one expects a massive spin-2 excitation of order $2$16; the data emphasize that possible issues associated with higher-derivative spin-2 modes are deferred to Planckian scales, and in related discussions suitable boundary conditions are invoked to remove a spin-2 ghost. Third, exact Weyl invariance and the absence of a Weyl anomaly are claimed for the exact action and for each order of its expansion, but in the broken phase, after $2$17 decouples and the theory reduces to Riemannian geometry, the usual Riemannian anomaly structure reappears. These points delimit the sense in which WDBI is presented as a Weyl-anomaly-free candidate for quantum gravity and as a unified determinant action for gravity and the Standard Model.