Dilaton Effective Theory

• Dilaton Effective Theory is a low-energy framework where a scalar compensator nonlinearly realizes broken scale or conformal symmetry.
• It systematically incorporates anomaly matching, soft theorems, and controlled power counting to describe dynamics from lattice simulations to holographic models.
• The approach finds applications in electroweak symmetry breaking, dark sector phenomenology, and entanglement entropy by reconstructing dilaton potentials and observables.

Dilaton Effective Theory denotes a family of low-energy effective field theories in which a scalar compensator realizes spontaneously broken exact or approximate scale or conformal symmetry. In the anomaly-matching literature, the dilaton is the Weyl compensator for RG flows between CFT fixed points; in near-conformal confining gauge theories, it appears together with pNGBs from internal symmetry breaking; and in supersymmetric, holographic, Standard-Model, and dark-sector constructions it is embedded into more structured multiplets or sectors. Across these realizations, the common mechanism is that operators are dressed by powers of a compensator field, explicit breaking is encoded by anomaly terms or a dilaton potential, and observables depend on the balance between spontaneous and explicit breaking of dilatations (Chacko et al., 2012, Golterman et al., 2021, Appelquist et al., 2022).

1. Compensator structure, field content, and symmetry realization

A standard starting point is the conformal compensator

$\chi(x)=f\,e^{\sigma(x)/f},$

for which $\sigma$ shifts under dilatations and $\chi$ transforms with scaling dimension one. In near-conformal confining gauge theories, the light sector is enlarged from ordinary chiral EFT to include both the dilaton and the pNGB multiplet,

$\Sigma=\exp\!\left[\frac{2 i \pi}{f_\pi}\right],$

with $\Sigma\Sigma^\dagger=\mathbf 1$ and $\Sigma\to U_L\Sigma U_R^\dagger$ under $SU(N_f)_L\times SU(N_f)_R$. The leading lattice-oriented dEFT then takes the form

$${\cal L}_{\rm LO}=\frac12 \partial_\mu \chi \partial^\mu \chi+\frac{f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^2 {\rm Tr}\!\left[\partial_\mu\Sigma(\partial^\mu\Sigma)^\dagger\right]+\frac{m_\pi^2f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^y{\rm Tr}\!\left[\Sigma+\Sigma^\dagger\right]-V(\chi),$$

with $\langle\chi\rangle=f_d$, $m_\pi^2=2B_\pi m$, and $\sigma$0 the scaling dimension associated with the explicit chiral-breaking deformation (Appelquist et al., 2017, Appelquist et al., 2022).

Dilaton chiral perturbation theory (dChPT) uses an equivalent but distinct parametrization in terms of a dimensionless dilaton field $\sigma$1. Its LO Lagrangian is

$\sigma$2

with the power counting

$\sigma$3

Here $\sigma$4 measures the distance to the conformal sill in the Veneziano limit, and $\sigma$5 is the mass anomalous dimension at the sill (Golterman et al., 2021, Golterman et al., 2024).

This compensator logic also underlies more general EFT constructions. In the Standard Model with hidden scale invariance, all dimensionful parameters, including the cutoff, are promoted to suitable powers of $\sigma$6. In anomaly-based dilaton EFT, the same compensator organizes couplings to the trace of the EMT, $\sigma$7. In all cases, the dilaton is not merely a singlet scalar added to the spectrum; it is the field that nonlinearly realizes broken dilatations (Kobakhidze et al., 2017, Chacko et al., 2012).

2. Anomaly matching, Wess–Zumino structure, and soft constraints

In anomaly-matching problems, the dilaton EFT is organized around the fact that RG flows between fixed points break Weyl symmetry explicitly while allowing it to be restored by a spurion-like compensator. In four dimensions, the action is naturally decomposed as

$\sigma$8

where $\sigma$9 reproduces Weyl and gauge anomalies and $\chi$0 is built from Weyl- and gauge-invariant local operators. Bobev, Elvang, and Olson showed that for $\chi$1 supersymmetric flows the low-energy bosonic sector contains not only the dilaton $\chi$2 but also the Goldstone mode $\chi$3 of broken $\chi$4, with invariant building block $\chi$5. They further showed that this enlarged EFT does not obstruct the Komargodski–Schwimmer proof of the four-dimensional $\chi$6-theorem, and that in flat space the four-dilaton amplitude remains

$\chi$7

so the forward-limit positivity argument still gives $\chi$8 (Bobev et al., 2013).

The same anomaly-centered perspective appears holographically. In holographic RG flows, the bulk Goldstone mode $\chi$9 associated with broken radial translations plays the AdS analogue of the Goldstone mode in the EFT of Inflation, and its UV boundary value is the field-theory spurion/dilaton $\Sigma=\exp\!\left[\frac{2 i \pi}{f_\pi}\right],$0. In two dimensions this construction reproduces the full Wess–Zumino action

$\Sigma=\exp\!\left[\frac{2 i \pi}{f_\pi}\right],$1

while in higher even dimensions, in a slow-flow limit, it reproduces the universal $\Sigma=\exp\!\left[\frac{2 i \pi}{f_\pi}\right],$2-anomaly term proportional to $\Sigma=\exp\!\left[\frac{2 i \pi}{f_\pi}\right],$3 (Kaplan et al., 2014).

Soft-theorem methods impose a separate set of model-independent constraints. A recent double-soft dilaton theorem takes into account the full dilation commutator

$\Sigma=\exp\!\left[\frac{2 i \pi}{f_\pi}\right],$4

including the spacetime-dependent term that is invisible in ordinary single-soft limits evaluated at $\Sigma=\exp\!\left[\frac{2 i \pi}{f_\pi}\right],$5. The resulting matrix element is

$\Sigma=\exp\!\left[\frac{2 i \pi}{f_\pi}\right],$6

If a single operator is responsible for generating the pseudo-dilaton mass, compatibility of the single-soft and double-soft relations forces

$\Sigma=\exp\!\left[\frac{2 i \pi}{f_\pi}\right],$7

This sharpens the admissible structure of explicit-breaking terms in pseudo-dilaton EFTs (Zwicky, 22 Aug 2025).

3. Near-conformal confining dynamics and the dChPT power counting

The most developed non-anomaly application of dilaton EFT is the low-energy description of confining gauge theories just below the conformal window. Here the light scalar is interpreted as a dilatonic meson, while the pions arise from spontaneous chiral symmetry breaking. dChPT is designed precisely for this setting, and its defining feature is that proximity to the conformal sill furnishes an additional small parameter, so that the EFT can expand systematically in both $\Sigma=\exp\!\left[\frac{2 i \pi}{f_\pi}\right],$8 and $\Sigma=\exp\!\left[\frac{2 i \pi}{f_\pi}\right],$9 (Golterman et al., 2021).

A distinctive consequence is the existence of a large-mass regime. In the notation of the theory,

$\Sigma\Sigma^\dagger=\mathbf 1$0

can hold while the loop-expansion parameter remains small because

$\Sigma\Sigma^\dagger=\mathbf 1$1

In this regime the classical dilaton vev satisfies

$\Sigma\Sigma^\dagger=\mathbf 1$2

and the theory exhibits approximate hyperscaling,

$\Sigma\Sigma^\dagger=\mathbf 1$3

even though the massless theory is assumed to remain confining and chirally broken. This explains why lattice data can look mass-deformed conformal over accessible fermion masses without requiring infrared conformality (Golterman et al., 2018, Golterman et al., 2021).

The consistency of this construction depends on power counting. Golterman and Shamir examined two alternative proposals: a $\Sigma\Sigma^\dagger=\mathbf 1$4 counting and a $\Sigma\Sigma^\dagger=\mathbf 1$5 counting. They found that the $\Sigma\Sigma^\dagger=\mathbf 1$6 counting fails because the scale-invariant quartic dilaton coupling enters the actual loop expansion parameter unsuppressed, so higher-order graphs are not parametrically smaller than the terms kept at leading order. By contrast, the $\Sigma\Sigma^\dagger=\mathbf 1$7 counting coincides with dChPT. A central corollary is that the widely used $\Sigma\Sigma^\dagger=\mathbf 1$8-potential coincides with the tree-level potential of the invalid $\Sigma\Sigma^\dagger=\mathbf 1$9 counting; it can function as a controlled EFT potential only if one simultaneously assumes $\Sigma\to U_L\Sigma U_R^\dagger$0 is small, in which case one has effectively returned to dChPT (Golterman et al., 2024).

4. Lattice phenomenology and reconstruction of the dilaton potential

Dilaton EFT became practically important through lattice fits. A key observation is that pNGB data alone already constrain the scalar sector tightly. In the lattice-oriented framework,

$\Sigma\to U_L\Sigma U_R^\dagger$1

is independent of the detailed form of the scalar potential and therefore allows a direct extraction of the scaling exponent $\Sigma\to U_L\Sigma U_R^\dagger$2. The same framework also relates $\Sigma\to U_L\Sigma U_R^\dagger$3 to $\Sigma\to U_L\Sigma U_R^\dagger$4, so plots of $\Sigma\to U_L\Sigma U_R^\dagger$5 against $\Sigma\to U_L\Sigma U_R^\dagger$6 probe the shape of the dilaton potential directly (Appelquist et al., 2017).

Updated analyses of the $\Sigma\to U_L\Sigma U_R^\dagger$7 theory with $\Sigma\to U_L\Sigma U_R^\dagger$8 fundamental fermions and the $\Sigma\to U_L\Sigma U_R^\dagger$9 theory with two sextet fermions found $SU(N_f)_L\times SU(N_f)_R$0 close to $SU(N_f)_L\times SU(N_f)_R$1, large-field behavior consistent with $SU(N_f)_L\times SU(N_f)_R$2, and $SU(N_f)_L\times SU(N_f)_R$3 near $SU(N_f)_L\times SU(N_f)_R$4. For the eight-flavor theory, representative fits gave

$SU(N_f)_L\times SU(N_f)_R$5

while the sextet analysis gave

$SU(N_f)_L\times SU(N_f)_R$6

These fits were interpreted as evidence for universal near-conformal behavior rather than ordinary QCD-like chiral dynamics (Appelquist et al., 2017).

A complementary six-parameter fit to LSD $SU(N_f)_L\times SU(N_f)_R$7, $SU(N_f)_L\times SU(N_f)_R$8 data used the potential

$SU(N_f)_L\times SU(N_f)_R$9

and obtained the central values

$${\cal L}_{\rm LO}=\frac12 \partial_\mu \chi \partial^\mu \chi+\frac{f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^2 {\rm Tr}\!\left[\partial_\mu\Sigma(\partial^\mu\Sigma)^\dagger\right]+\frac{m_\pi^2f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^y{\rm Tr}\!\left[\Sigma+\Sigma^\dagger\right]-V(\chi),$$0

with

$${\cal L}_{\rm LO}=\frac12 \partial_\mu \chi \partial^\mu \chi+\frac{f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^2 {\rm Tr}\!\left[\partial_\mu\Sigma(\partial^\mu\Sigma)^\dagger\right]+\frac{m_\pi^2f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^y{\rm Tr}\!\left[\Sigma+\Sigma^\dagger\right]-V(\chi),$$1

An important conclusion of that analysis was that current data do not require the explicit scale-breaking deformation in the chiral limit to be marginal or nearly marginal; values of $${\cal L}_{\rm LO}=\frac12 \partial_\mu \chi \partial^\mu \chi+\frac{f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^2 {\rm Tr}\!\left[\partial_\mu\Sigma(\partial^\mu\Sigma)^\dagger\right]+\frac{m_\pi^2f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^y{\rm Tr}\!\left[\Sigma+\Sigma^\dagger\right]-V(\chi),$$2 well below $${\cal L}_{\rm LO}=\frac12 \partial_\mu \chi \partial^\mu \chi+\frac{f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^2 {\rm Tr}\!\left[\partial_\mu\Sigma(\partial^\mu\Sigma)^\dagger\right]+\frac{m_\pi^2f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^y{\rm Tr}\!\left[\Sigma+\Sigma^\dagger\right]-V(\chi),$$3 are allowed, with the conservative range

$${\cal L}_{\rm LO}=\frac12 \partial_\mu \chi \partial^\mu \chi+\frac{f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^2 {\rm Tr}\!\left[\partial_\mu\Sigma(\partial^\mu\Sigma)^\dagger\right]+\frac{m_\pi^2f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^y{\rm Tr}\!\left[\Sigma+\Sigma^\dagger\right]-V(\chi),$$4

That is, present lattice evidence permits clearly non-marginal deformations (Appelquist et al., 2019).

The lattice program has also extended beyond the p-regime. In the sextet theory, a dilaton EFT analysis of the $${\cal L}_{\rm LO}=\frac12 \partial_\mu \chi \partial^\mu \chi+\frac{f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^2 {\rm Tr}\!\left[\partial_\mu\Sigma(\partial^\mu\Sigma)^\dagger\right]+\frac{m_\pi^2f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^y{\rm Tr}\!\left[\Sigma+\Sigma^\dagger\right]-V(\chi),$$5-regime used RMT fits to the lowest Dirac eigenvalues and found consistency with the quartic-like potential

$${\cal L}_{\rm LO}=\frac12 \partial_\mu \chi \partial^\mu \chi+\frac{f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^2 {\rm Tr}\!\left[\partial_\mu\Sigma(\partial^\mu\Sigma)^\dagger\right]+\frac{m_\pi^2f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^y{\rm Tr}\!\left[\Sigma+\Sigma^\dagger\right]-V(\chi),$$6

while the logarithmic potential

$${\cal L}_{\rm LO}=\frac12 \partial_\mu \chi \partial^\mu \chi+\frac{f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^2 {\rm Tr}\!\left[\partial_\mu\Sigma(\partial^\mu\Sigma)^\dagger\right]+\frac{m_\pi^2f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^y{\rm Tr}\!\left[\Sigma+\Sigma^\dagger\right]-V(\chi),$$7

predicted too small a condensate. This provided a regime-crossing test of the EFT beyond conventional spectroscopy (Fodor et al., 2020).

More recently, dEFT has been used “across the conformal edge” as a diagnostic rather than purely descriptive framework. Fitting the potential

$${\cal L}_{\rm LO}=\frac12 \partial_\mu \chi \partial^\mu \chi+\frac{f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^2 {\rm Tr}\!\left[\partial_\mu\Sigma(\partial^\mu\Sigma)^\dagger\right]+\frac{m_\pi^2f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^y{\rm Tr}\!\left[\Sigma+\Sigma^\dagger\right]-V(\chi),$$8

to lattice data, one analysis found that $${\cal L}_{\rm LO}=\frac12 \partial_\mu \chi \partial^\mu \chi+\frac{f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^2 {\rm Tr}\!\left[\partial_\mu\Sigma(\partial^\mu\Sigma)^\dagger\right]+\frac{m_\pi^2f_\pi^2}{4}\left(\frac{\chi}{f_d}\right)^y{\rm Tr}\!\left[\Sigma+\Sigma^\dagger\right]-V(\chi),$$9 with $\langle\chi\rangle=f_d$0 favors a symmetry-breaking minimum at nonzero $\langle\chi\rangle=f_d$1, hence confinement, whereas $\langle\chi\rangle=f_d$2 with one adjoint Dirac fermion favors an origin-centered minimum, hence infrared conformality. The reported central values were

$\langle\chi\rangle=f_d$3

for the eight-flavor confining case, and

$\langle\chi\rangle=f_d$4

for the adjoint IR-conformal case (Appelquist et al., 18 Dec 2025).

System	Representative EFT output	Source
$\langle\chi\rangle=f_d$5, $\langle\chi\rangle=f_d$6 fundamental	$\langle\chi\rangle=f_d$7, $\langle\chi\rangle=f_d$8, $\langle\chi\rangle=f_d$9	(Appelquist et al., 2017)
$m_\pi^2=2B_\pi m$0, $m_\pi^2=2B_\pi m$1 fundamental	$m_\pi^2=2B_\pi m$2, $m_\pi^2=2B_\pi m$3, $m_\pi^2=2B_\pi m$4, $m_\pi^2=2B_\pi m$5	(Appelquist et al., 2019)
$m_\pi^2=2B_\pi m$6, $m_\pi^2=2B_\pi m$7 adjoint	fit favors IR conformality with $m_\pi^2=2B_\pi m$8, $m_\pi^2=2B_\pi m$9	(Appelquist et al., 18 Dec 2025)

5. Naturalness, radiative control, and unitarity limits

The lightness of a dilaton is not generic in strongly coupled theories. A controlled EFT analysis of light dilatons in approximately conformal ultraviolet completions of electroweak symmetry breaking showed that a parametrically light dilaton is natural only when the operator that explicitly breaks conformal symmetry is close to marginal at the symmetry-breaking scale. In the scenarios of direct interest for electroweak symmetry breaking this condition is not generally expected, so a light dilaton is typically associated with mild tuning. At the same time, once a light dilaton exists, corrections to its couplings induced by conformal-symmetry-violating effects are suppressed by

$\sigma$00

and are usually subleading, except for couplings to marginal operators where symmetry-violating effects can sometimes dominate (Chacko et al., 2012).

The distinction between a fundamental graviscalar dilaton and an effective pseudo-Nambu–Goldstone dilaton is technically important here. For a graviscalar arising from metric compactification, the coupling to the EMT trace is literal, and one-loop amplitudes exhibit anomaly-controlled enhancements in $\sigma$01 and $\sigma$02. In the electroweak sector, the perturbative analysis shows that renormalization is guaranteed only if the Higgs is conformally coupled, $\sigma$03, through the improvement term in the EMT. For an effective dilaton interpolated by the dilatation current $\sigma$04, the same anomaly logic appears through the identity $\sigma$05, but the physical realization of anomaly enhancement depends on whether the corresponding anomaly pole is infrared coupled in the channel under consideration (Coriano et al., 2012).

The low-energy expansion has a separate limitation from perturbative unitarity. A study of pure-dilaton $\sigma$06 scattering found that the tree-level amplitude is soft in the ultraviolet, approaching a constant of order $\sigma$07, but the one-loop amplitude grows as

$\sigma$08

Using the paper’s approximate amplitude-level criterion $\sigma$09, rather than a full partial-wave analysis, the EFT ceases to be self-consistent around

$\sigma$10

The same work argues that the natural next state to include is a spin-2 resonance with mass $\sigma$11, whose exchange can ameliorate the growth of the amplitude near the cutoff. The authors explicitly stress that this “unitarity bound” should be read as an NDA-like perturbative cutoff estimate, not as a rigorously derived partial-wave threshold (Hong et al., 2022).

Higher-order operator control has also been systematized. A review-and-extension of lattice dEFT developed explicit NLO operator bases for the pNGB sector, the dilaton-derivative sector, and the explicit scale-breaking sector, together with generalized NDA estimates for their coefficients. This elevated the framework from a purely LO fitting ansatz to a systematically improvable EFT suitable for loop calculations and precision applications (Appelquist et al., 2022).

6. Extensions, applications, and adjacent frameworks

Dilaton EFT has acquired a wide application range beyond near-conformal spectroscopy. In entanglement-entropy problems, a background dilaton couples directly to the integrated trace of the stress tensor, so the dilaton effective action computes the scale response of the replica free energy. Banerjee showed that in Gaussian theories this reproduces exactly the Green’s-function method, while in interacting theories it extends that method. For a two-dimensional flow to a nontrivial IR fixed point,

$\sigma$12

and in four dimensions, for the cylindrical geometry studied there,

$\sigma$13

The central lesson is that universal entanglement terms are anomaly-matching statements in dilaton EFT language (Banerjee, 2014).

In phenomenological SM constructions with hidden scale invariance, the EFT role of the dilaton is different but structurally related. One minimal framework promotes every dimensionful parameter in the Higgs sector and even the Wilsonian cutoff to powers of $\sigma$14, leading to the scalar potential

$\sigma$15

With the vacuum conditions

$\sigma$16

the hierarchy $\sigma$17 is technically natural, and the dilaton mass first appears at two-loop order,

$\sigma$18

For $\sigma$19 between $\sigma$20 GeV and $\sigma$21 GeV, the predicted dilaton mass range is $\sigma$22 to $\sigma$23 (Kobakhidze et al., 2017).

A separate application uses dEFT as a dark-sector theory. There the pNGBs are identified with dark matter and the dilaton with a slightly heavier scalar mediating forbidden annihilation,

$\sigma$24

The dark-sector EFT is

$\sigma$25

and for lattice-motivated choices such as $\sigma$26 and $\sigma$27, the observed relic density is obtained for pNGB masses from roughly tens of MeV up to order $\sigma$28 GeV, with the forbidden regime controlled by

$\sigma$29

and freezeout typically around $\sigma$30 (Appelquist et al., 2024).

Taken together, these developments show that Dilaton Effective Theory is not a single model but a technically constrained EFT language for broken scale symmetry across several domains. Its modern content includes anomaly-matching Wess–Zumino actions, systematic near-conformal pion-dilaton EFTs with nontrivial power counting, lattice-based potential reconstruction, controlled statements about naturalness and unitarity, and applications ranging from entanglement entropy and holographic RG flows to hidden-scale-invariant Higgs sectors and dark matter (Bobev et al., 2013, Golterman et al., 2024, Hong et al., 2022).