Dilaton Model: Scale and Anomaly Dynamics

• Dilaton models are theoretical frameworks where a scalar field tracks scale or conformal symmetry breaking, acting as a Goldstone or pseudo-Goldstone boson.
• They encompass diverse realizations such as graviscalars, radions, Higgs-portal scalars, and cosmological dilatons, each with distinct dynamics and phenomenology.
• These models employ trace couplings and anomaly pole structures to connect the scalar sector to the energy–momentum tensor, influencing collider signals and dark sector physics.

A dilaton model is a theoretical construction in which a scalar degree of freedom is associated with scale or conformal symmetry. In the narrow sense, the dilaton is the Goldstone or pseudo-Nambu–Goldstone boson of spontaneous or approximate scale-symmetry breaking; in broader usage, the term also covers graviscalar or radion constructions, anomaly-pole effective descriptions tied to the dilatation current, Higgs-portal scalars that control dimensionful parameters, and cosmological scalars embedded in modified gravity or hidden sectors (Cresswell-Hogg et al., 31 Jan 2025, Rose et al., 2012, Coriano et al., 2012, Kobakhidze et al., 2017, Brax et al., 2023). This terminological breadth is not accidental: the common organizing datum is coupling to the trace sector of the theory, but the microscopic ontology, symmetry structure, and phenomenology differ substantially across realizations.

1. Taxonomy and defining structures

Across the literature, “dilaton model” does not denote a single canonical Lagrangian. It denotes a class of models whose unifying feature is that a scalar mode tracks the breaking of scale invariance, either directly as an order parameter fluctuation or indirectly through the trace of the energy-momentum tensor. A recurrent distinction is between a true Goldstone dilaton at exact spontaneous breaking, a pseudo-dilaton in the presence of small explicit breaking, and an effective anomaly-induced scalar mode inferred from correlator structure rather than from a manifest scalar bound state (Cresswell-Hogg et al., 31 Jan 2025, Rose et al., 2012, Coriano et al., 2012).

Class	Defining structure	Representative sources
Microscopic Goldstone or pseudo-dilaton	Spontaneous or approximate conformal breaking in a controllable QFT or EFT	(Cresswell-Hogg et al., 31 Jan 2025, Grinstein et al., 2011, Rojas et al., 2023)
Trace-coupled fundamental dilaton or graviscalar	Interaction organized by $\frac{1}{\Lambda_\rho}\rho\,T^\mu_{\ \mu}$	(Rose et al., 2012, Coriano et al., 2012, Barger et al., 2011)
Radion or extra-dimensional linear-dilaton scalar	Scalar from warped geometry, often mixed with bulk dilaton fluctuations	(Cox et al., 2012)
Higgs-portal or hidden-scale scalar	Scalar controls the Higgs-sector mass scale or hidden conformal scale	(Kobakhidze et al., 2017, Brax et al., 2023)
Dark-sector or cosmological dilaton	Dilaton EFT or modified-gravity scalar used for relic density, dark energy, or phase transitions	(Appelquist et al., 2024, Khatua et al., 2011, Phong et al., 2019)

A standard misconception is that every dilaton automatically couples to the anomaly in the same way. The detailed analyses distinguish at least three cases: a graviscalar dilaton whose trace coupling is built in, a classically scale-invariant fundamental scalar that couples at tree level to explicit breaking terms but not automatically to the anomaly, and an effective non-gravitational dilaton whose anomaly coupling is encoded through pole structure in $J_DVV$ or $TVV$ correlators (Coriano et al., 2012).

2. Microscopic quantum-field-theoretic realizations

A particularly clean first-principles realization is the asymptotically free three-dimensional Gross–Neveu–Yukawa model with $N$ two-component Dirac fermions $\psi_a$ and one real scalar $\phi$,

$S=\int d^3x \left[\bar\psi_a \slashed{\partial} \psi_a + \frac12 (\partial \phi)^2 + H\, \phi\, \bar\psi_a \psi_a + \frac1{3!}\bar\lambda_3 \phi^3 \right].$

In the large-$N$ limit, the cubic scalar coupling becomes exactly marginal in the IR, and the low-energy theory contains a finite line of strongly interacting fixed points parameterized by $\lambda_3=\bar\lambda_3/H^3$. After the rescaling $\varphi=H\phi$, the fixed-point effective potential becomes

$J_DVV$0

with stability bound $J_DVV$1. The endpoint $J_DVV$2 is a continuous quantum critical endpoint with a half-sidedly flat potential and a moduli space of vacua $J_DVV$3, so spontaneous scale breaking occurs precisely there; the scalar two-point function then develops a $J_DVV$4 pole, identifying a true massless dilaton in the microscopic spectrum (Cresswell-Hogg et al., 31 Jan 2025).

The same construction makes the pseudo-dilaton regime explicit. Approaching the endpoint by the correlated double-scaling limit

$J_DVV$5

keeps the condensate finite while removing explicit deformations. At the endpoint, the dilaton decay constant is

$J_DVV$6

while small scalar-mass or fermion-mass perturbations generate the same linearized pseudo-dilaton mass formula and satisfy the soft-dilaton relation

$J_DVV$7

The same analysis also shows that exact masslessness is not generic: it requires the endpoint and the removal of explicit breakings $J_DVV$8 (Cresswell-Hogg et al., 31 Jan 2025).

A distinct perturbative four-dimensional realization exhibits the same near-conformal logic. In the $J_DVV$9 gauge theory with vectorlike fermions and two real singlet scalars of “A Very Light Dilaton,” RG trajectories approach a perturbative IR fixed point but can undergo spontaneous symmetry breaking before the flow reaches it. The dilaton mass is controlled by the distance to the critical surface,

$TVV$0

so $TVV$1 can be made parametrically small without taking the interactions to zero. This model is explicitly organized around proximity to criticality rather than decoupling (Grinstein et al., 2011).

3. Trace couplings, anomaly structure, and the dilatation current

The standard low-energy organizing principle for many phenomenological dilaton models is the trace coupling

$TVV$2

Because the dilatation current satisfies

$TVV$3

this interaction may equally be viewed as a coupling to $TVV$4. In Standard-Model-oriented analyses, the trace contains both explicit mass terms and anomaly terms proportional to gauge kinetic operators, so the same coupling generates interactions with fermion masses, weak-boson masses, and $TVV$5-type operators (Rose et al., 2012, Coriano et al., 2012).

A key point is that anomaly-induced dilaton couplings can be extracted directly from three-point functions. In QCD, the amputated on-shell dilaton–two-gluon vertex is written as

$TVV$6

with

$TVV$7

In the massless limit, the $TVV$8 term isolates the anomaly pole. This is the basis for the anomaly-pole interpretation of an effective dilaton mode: the correlator behaves as if a scalar Nambu–Goldstone degree of freedom were being exchanged, even though the analysis does not by itself prove a physical bound state (Rose et al., 2012).

That caveat is essential. The more detailed correlator analysis shows that the pole must be infrared coupled to have direct physical significance. For massless internal fermions and on-shell photons, the $TVV$9 and $N$0 anomaly poles are infrared coupled; for off-shell photons, the residue vanishes; for gluons, perturbative on-shell pole structure exists, but confinement makes the interpretation subtler. This is why anomaly enhancement is straightforward for $N$1 channels and more delicate for $N$2 in effective, non-gravitational dilaton models (Coriano et al., 2012).

Another technical distinction concerns renormalization. In the graviscalar analysis of electroweak dilaton vertices, one-loop renormalization is consistent only if the Higgs is conformally coupled, $N$3, so that the improved energy-momentum tensor is used. Without the improvement term, the dilaton–Higgs mixing sector remains UV divergent. This requirement is model-specific, but it sharply illustrates that the coupling $N$4 is not sufficient by itself to define a consistent electroweak dilaton framework (Coriano et al., 2012).

4. Composite, holographic, and extra-dimensional constructions

In holographic near-conformal models, the dilaton is often the unique parametrically light scalar while the rest of the composite spectrum remains at the infrared dynamical scale. In the AdS$N$5 model “Holographic light dilaton at the conformal edge,” the bulk scalar dual to $N$6 is taken slightly below the Breitenlohner–Freedman bound,

$N$7

and the lightest scalar mode satisfies

$N$8

while higher scalar and tensor excitations remain $N$9. Under an explicit condition relating IR and UV parameters, the low-energy two-point function is saturated by the dilaton pole and obeys the PCDC relation

$\psi_a$0

The result is explicitly conditional: the light dilaton appears only in a subclass of nearly conformal models with appropriate boundary conditions and small backreaction (Rojas et al., 2023).

Composite-Higgs dilaton EFTs formulate the relation between the Higgs compositeness scale $\psi_a$1 and the conformal-breaking scale $\psi_a$2. In the large-$\psi_a$3 counting used for composite Higgs plus dilaton phenomenology,

$\psi_a$4

so the scale that controls electroweak tuning need not coincide with the scale that suppresses dilaton couplings. This distinction feeds directly into collider reach: the same analysis finds that LHC sensitivity to the dilaton “reaches ~3 TeV in dilaton mass for generic parameter choices,” and that dilaton searches are complementary to Higgs compositeness probes (Bruggisser et al., 2022).

Extra-dimensional linear-dilaton models furnish a different realization. In the five-dimensional linear dilaton model, the bulk dilaton simultaneously supports the warped background and stabilizes the interval by a Goldberger–Wise-type mechanism. The radion is a coupled metric–dilaton scalar mode with mass of order the curvature scale, and because the brane action contains a direct dilaton factor, the radion couples not only to $\psi_a$5 but also directly to the Standard Model Lagrangian. A notable consequence is a larger branching fraction to photons than in Randall–Sundrum radion phenomenology (Cox et al., 2012).

A geometrically distinct Weyl-gauged construction with local conformal $\psi_a$6 symmetry yields another nonstandard use of “dilaton model.” After Stückelberg breaking, the physical scalar is a nonlinear Higgs–dilaton mixture parameterized by $\psi_a$7 or $\psi_a$8. Two scenarios appear: the trigonometric scalar scenario (TSS) and the hyperbolic scalar scenario (HSS). In the TSS case with $\psi_a$9, current data still allow a dilaton-dominant $\phi$0 boson for

$\phi$1

and in that branch the quartic self-coupling is negative, $\phi$2. The same work identifies Higgs pair production at the HL-LHC as the decisive test of dilaton dominance (Wu et al., 14 Jan 2025).

5. Cosmology, hidden sectors, and very light dilatons

A hidden-scale-invariant reinterpretation of the Standard Model produces an ultralight dilaton by promoting all dimensionful parameters to functions of a field $\phi$3,

$\phi$4

After imposing $\phi$5 and $\phi$6, the dilaton mass is generated only through $\phi$7, so it is a two-loop effect: $\phi$8 The predicted mass range is $\phi$9 to $S=\int d^3x \left[\bar\psi_a \slashed{\partial} \psi_a + \frac12 (\partial \phi)^2 + H\, \phi\, \bar\psi_a \psi_a + \frac1{3!}\bar\lambda_3 \phi^3 \right].$0, depending on the hierarchy scale $S=\int d^3x \left[\bar\psi_a \slashed{\partial} \psi_a + \frac12 (\partial \phi)^2 + H\, \phi\, \bar\psi_a \psi_a + \frac1{3!}\bar\lambda_3 \phi^3 \right].$1, and the resulting light scalar is sufficiently feebly coupled that it can in principle play a dark-matter role (Kobakhidze et al., 2017).

A more specialized Higgs-portal construction uses a dilaton-like scalar to control the Higgs $S=\int d^3x \left[\bar\psi_a \slashed{\partial} \psi_a + \frac12 (\partial \phi)^2 + H\, \phi\, \bar\psi_a \psi_a + \frac1{3!}\bar\lambda_3 \phi^3 \right].$2 term. In that model,

$S=\int d^3x \left[\bar\psi_a \slashed{\partial} \psi_a + \frac12 (\partial \phi)^2 + H\, \phi\, \bar\psi_a \psi_a + \frac1{3!}\bar\lambda_3 \phi^3 \right].$3

and the stabilized background $S=\int d^3x \left[\bar\psi_a \slashed{\partial} \psi_a + \frac12 (\partial \phi)^2 + H\, \phi\, \bar\psi_a \psi_a + \frac1{3!}\bar\lambda_3 \phi^3 \right].$4 is chosen so that $S=\int d^3x \left[\bar\psi_a \slashed{\partial} \psi_a + \frac12 (\partial \phi)^2 + H\, \phi\, \bar\psi_a \psi_a + \frac1{3!}\bar\lambda_3 \phi^3 \right].$5. As a result, fluctuations $S=\int d^3x \left[\bar\psi_a \slashed{\partial} \psi_a + \frac12 (\partial \phi)^2 + H\, \phi\, \bar\psi_a \psi_a + \frac1{3!}\bar\lambda_3 \phi^3 \right].$6 couple only quadratically to Standard Model massive fields. The linear matter coupling vanishes, direct decays into Standard Model particles are absent, thermalization with the Standard Model bath fails in the sub-eV coherent-dark-matter regime, and the scalar can behave as misalignment dark matter while remaining compatible with local gravitational tests. The same analysis notes composition-dependent couplings to the oscillating condensate, possible equivalence-principle violation, and screening “akin to the symmetron mechanism” (Brax et al., 2023).

Dilaton effective field theory has also been repurposed directly as a dark-sector relic-density framework. In “Dilaton Forbidden Dark Matter,” the dark sector contains pNGBs $S=\int d^3x \left[\bar\psi_a \slashed{\partial} \psi_a + \frac12 (\partial \phi)^2 + H\, \phi\, \bar\psi_a \psi_a + \frac1{3!}\bar\lambda_3 \phi^3 \right].$7 and a singlet dilaton $S=\int d^3x \left[\bar\psi_a \slashed{\partial} \psi_a + \frac12 (\partial \phi)^2 + H\, \phi\, \bar\psi_a \psi_a + \frac1{3!}\bar\lambda_3 \phi^3 \right].$8,

$S=\int d^3x \left[\bar\psi_a \slashed{\partial} \psi_a + \frac12 (\partial \phi)^2 + H\, \phi\, \bar\psi_a \psi_a + \frac1{3!}\bar\lambda_3 \phi^3 \right].$9

The pNGBs are dark matter, and the dominant depletion process is the forbidden channel $N$0, with suppression

$N$1

For the lattice-motivated $N$2, $N$3 example, $N$4, benchmark choices include $N$5, $N$6, and $N$7, and the model reproduces $N$8 across viable MeV–GeV regions while requiring $N$9 to suppress the competing $\lambda_3=\bar\lambda_3/H^3$0 channel (Appelquist et al., 2024).

The cosmological use of dilatons extends beyond dark matter. In the Two-Time Physics model with a softly broken $\lambda_3=\bar\lambda_3/H^3$1 symmetry, the dilaton acts as the trigger of electroweak symmetry breaking and yields a first-order electroweak phase transition. The reported transition strength is $\lambda_3=\bar\lambda_3/H^3$2, and the associated dilaton mass window is $\lambda_3=\bar\lambda_3/H^3$3 (Phong et al., 2019). In a different terminological tradition, the Weyl-scaled induced-gravity “dilaton dark energy model” treats the dilaton as a quintessence-like scalar reconstructed inside $\lambda_3=\bar\lambda_3/H^3$4, $\lambda_3=\bar\lambda_3/H^3$5, and Hořava–Lifshitz gravities; that work reports that the reconstructed potential increases with the dilaton field in $\lambda_3=\bar\lambda_3/H^3$6 and $\lambda_3=\bar\lambda_3/H^3$7, while the Hořava–Lifshitz case is presented inconsistently, with both increasing and decreasing behavior stated in different parts of the paper (Khatua et al., 2011).

6. Collider phenomenology, minimal models, and interpretive disputes

At hadron colliders, the canonical trace-coupled dilaton is organized by

$\lambda_3=\bar\lambda_3/H^3$8

In that setup, couplings to Standard Model fermions and weak bosons are Higgs-like with the replacement $\lambda_3=\bar\lambda_3/H^3$9, while the $\varphi=H\phi$0 and $\varphi=H\phi$1 couplings receive direct trace-anomaly terms,

$\varphi=H\phi$2

This produces enhanced gluon-fusion production, a large low-mass $\varphi=H\phi$3 branching fraction, and a diphoton rate larger than the corresponding $\varphi=H\phi$4 and $\varphi=H\phi$5 rates across the mass range studied. In early LHC7 reinterpretations, “a scale of $\varphi=H\phi$6 less than $\varphi=H\phi$7 TeV is nearly excluded,” with $\varphi=H\phi$8 strongest below $\varphi=H\phi$9 GeV and $J_DVV$00 strongest above $J_DVV$01 GeV (Barger et al., 2011).

The decay $J_DVV$02 is a particularly sharp discriminator in such models. Because $J_DVV$03 contains Higgs-sector terms, this channel can open for $J_DVV$04 and lead to cascade final states $J_DVV$05, $J_DVV$06, and $J_DVV$07. By contrast, the minimal dilaton model proposed for the $J_DVV$08 GeV resonance is not a universal trace-coupled dilaton. It is a singlet scalar $J_DVV$09 mixed with a heavy Higgs and coupled to a vector-like top partner $J_DVV$10, with the light state’s couplings given by

$J_DVV$11

In that framework, the strict “dilaton like region”

$J_DVV$12

is disfavored by Higgs data, even though the broader singlet-plus-top-partner model remains viable (Abe et al., 2013).

A recurring interpretive dispute is therefore whether a given scalar should be called a dilaton at all. The literature itself draws the boundary differently in different settings. In the composite-Higgs context, the dilaton is explicitly a composite pseudo-Goldstone boson of approximate scale invariance (Bruggisser et al., 2022). In the anomaly-pole analyses, the effective scalar degree of freedom is inferred from $J_DVV$13 or $J_DVV$14 poles, but the authors explicitly do not claim a proof of a physical bound state (Rose et al., 2012, Coriano et al., 2012). In singlet or geometric models, “dilaton dominance” often means dominance in scalar mixing space rather than the presence of a universal trace-coupled Goldstone boson (Wu et al., 14 Jan 2025, Abe et al., 2013).

These distinctions matter phenomenologically. A plausible implication is that “dilaton model” is best understood as a model class organized by scale symmetry and trace couplings, not as a single EFT template. Some realizations predict an exactly massless scalar only at a critical endpoint; others predict a pseudo-dilaton whose mass is controlled by explicit breaking, proximity to a conformal edge, or radiative anomaly effects; still others use the name for radions, Higgs-portal scalars, or cosmological quintessence-like fields. The term is unified by symmetry language, but not by a unique microscopic definition (Cresswell-Hogg et al., 31 Jan 2025, Rojas et al., 2023, Cox et al., 2012, Brax et al., 2023).