QCD-Conformal Dilaton Dynamics

• QCD-conformal dilaton is a scalar mode associated with spontaneous or anomaly-mediated conformal symmetry breaking in QCD-like gauge theories, unifying EFT and holographic perspectives.
• It emerges near the conformal window in walking theories and is characterized as a pseudo-Nambu–Goldstone boson with a mass controlled by explicit symmetry breaking.
• Effective actions and anomaly poles in perturbation theory provide measurable insights into its role in hadronic form factors and gravitational interactions.

The QCD-conformal dilaton is the effective or composite scalar mode associated with the spontaneous, approximate, or anomaly-mediated breaking of scale/conformal symmetry in QCD-like gauge theories. In the strict Goldstone limit it is the dilaton, the Goldstone boson of spontaneously broken conformal symmetry; in more realistic settings it is typically a pseudo-Nambu–Goldstone boson whose mass is controlled by explicit breaking from the trace anomaly, relevant deformations, or other slowly running couplings. The concept is most natural in walking theories, gauge theories near the conformal window, and holographic models near a conformal edge, whereas ordinary QCD is not conformal in the UV or IR in the strict sense because the running coupling generates a scale and the trace anomaly is nonzero (Gretsch et al., 2013, Rose et al., 2012, Rojas et al., 2023).

1. Symmetry content and basic definition

The standard field-theoretic definition starts from spontaneous conformal-symmetry breaking. In that setting the dilaton is the corresponding Goldstone boson, and a convenient parametrization is

$X = v e^{\tau},$

with $v$ the dilaton vacuum expectation value and $\tau$ the fluctuation field. In the compensator formulation, one may pass to dressed variables

$$\hat \Phi(x)=e^{-\tau}\Phi(x),\qquad \hat g_{\mu\nu}(x)=e^{2\tau(x)}g_{\mu\nu}(x),\qquad \hat X=v,$$

which makes explicit that the dilaton nonlinearly realizes the broken Weyl/conformal symmetry (Gretsch et al., 2013).

This framework is directly relevant to QCD-like gauge dynamics near a conformal fixed point. If such a theory is close to conformality and the breaking is spontaneous, the low-energy spectrum contains a light dilaton in close analogy with pions as Goldstone bosons of chiral symmetry breaking. The essential distinction is between spontaneous breaking, which leaves the dilaton massless in the exact limit, and explicit breaking, especially from the trace anomaly, which gives the dilaton a mass (Gretsch et al., 2013).

A more specialized proposal is the conformal dilaton phase, introduced as a third infrared phase distinct from both an ordinary QCD-like confining phase and an unbroken conformal phase. In that proposal, the theory sits at an IR fixed point, conformal symmetry is spontaneously broken, the dilaton remains massless in the chiral limit, and massive hadrons can still exist because the dilaton pole guarantees the conformal Ward identities in the infrared (Debbio et al., 2021). This suggests that the phrase “QCD-conformal dilaton” encompasses both conventional near-conformal pseudo-dilaton scenarios and more stringent spontaneously broken conformal phases.

2. Trace anomaly, dilatation current, and anomaly poles

The operator backbone of dilaton physics is the dilatation current

$$J_D^\mu(x)=x_\nu T^{\mu\nu}(x), \qquad \partial_\mu J_D^\mu = T^\mu{}_\mu,$$

so dilaton couplings are organized by the trace of the energy-momentum tensor. A standard effective interaction is

$$\mathcal L_{\text{int}} = -\frac{1}{\Lambda_\rho}\,\rho\, T^\mu{}_\mu,$$

or equivalently $\rho\,\partial_\mu J_D^\mu/\Lambda_\rho$ (Coriano et al., 2012, Rose et al., 2012).

In perturbation theory the conformal anomaly has a characteristic pole structure. The correlator $\langle J_D^\mu V V' \rangle$, or equivalently the trace sector of the $T V V$ amplitude, contains an effective massless pole interpreted as the exchange of the dilaton, in close analogy with the pion pole in anomalous axial-current correlators. In the QCD example studied for the on-shell dilaton–gluon–gluon vertex, the form factor contains a term proportional to

$\frac{1}{s}(11N_C-2n_f),$

which is identified as the anomaly pole; the paper states that this pole “completely accounts for the trace anomaly” and is inherited by the QCD dilatation current (Rose et al., 2012).

The UV/IR interpretation of this structure was sharpened by the formulation of anomaly actions for dilaton interactions. In that perspective, the UV description is a nonlocal $v$0 anomaly pole in 1PI correlators such as $v$1, while the IR description is a local Wess–Zumino action for a composite dilaton. The same anomalous symmetry underlies both pictures, with the pole representing the perturbative signature and the Wess–Zumino action the nonlinear effective realization (Rose et al., 2014).

Recent QCD analyses of the non-Abelian $v$2 correlator at order $v$3 further isolated a spin-0 anomaly form factor,

$v$4

and established a mass-independent dispersive sum rule,

$v$5

In the conformal/massless limit the spectral density collapses to a $v$6 term, so the trace sector behaves as an effective scalar exchange. These works explicitly interpret the spin-0 channel as dilaton-like, while also emphasizing that this need not correspond to a new elementary scalar inserted into the QCD Lagrangian (Corianò et al., 2 Apr 2025, Corianò et al., 25 Mar 2026).

3. Quantum consistency, regulators, and mechanisms for lightness

A central technical problem is whether a dilaton can remain massless, or parametrically light, after renormalization. One explicit solution is to promote all mass scales in dimensional regularization, including the renormalization scale, to powers of the dilaton. In the prototype construction, the regulator is chosen so that the regularized action is Weyl invariant in $v$7 dimensions, with the renormalization scale effectively replaced by the dilaton vev,

$v$8

If the theory is free of gravitational (Diff) anomaly, the counterterms can be chosen conformally invariant order by order, and the dilaton remains massless to all orders in perturbation theory (Gretsch et al., 2013).

A closely related flat-space program replaces the usual external renormalization scale by a dynamical dilaton field $v$9, so that exact quantum conformal symmetry is preserved in flat space and all physical scales arise only after $\tau$0. In that construction even the QCD transmutation scale is generated from the dilaton background,

$\tau$1

The same work argues that once gravity is dynamical, local Weyl symmetry is anomalous and only the global scale subgroup remains anomaly free (Shaposhnikov et al., 2022).

Beyond exact constructions, several controlled mechanisms generate a light pseudo-dilaton. In $\tau$2 SQCD near the lower edge of the conformal window, weakly gauging a flavor subgroup provides an almost marginal deformation. In the magnetic description the singlet meson $\tau$3 reaches the unitarity bound,

$\tau$4

and, after stabilization by soft SUSY-breaking terms, the leading dilaton mass is proportional to the almost marginal coupling,

$\tau$5

That construction is presented as a purely four-dimensional realization of the Contino–Pomarol–Rattazzi mechanism in a QCD-like setting (Cleary et al., 2015).

The opposite conclusion appears in walking technicolor analyses near the conformal edge. There the nonperturbative trace anomaly remains finite, $\tau$6, and the PCDC relation implies

$\tau$7

The result is that a true massless techni-dilaton is obtained only in a decoupling limit $\tau$8; otherwise the dilaton mass is generically of order the dynamical mass scale $\tau$9, not parametrically zero (Hashimoto et al., 2010). This remains one of the clearest formulations of the tension between approximate scale invariance and a genuinely isolated light scalar.

4. Effective actions, soft theorems, and self-interactions

Dilaton EFTs are highly constrained once the anomaly is treated systematically. In the anomaly-action approach, Weyl gauging of the renormalized effective action yields a Wess–Zumino functional for the dilaton. In four dimensions the resulting flat-space action contains terms such as $$\hat \Phi(x)=e^{-\tau}\Phi(x),\qquad \hat g_{\mu\nu}(x)=e^{2\tau(x)}g_{\mu\nu}(x),\qquad \hat X=v,$$0, $$\hat \Phi(x)=e^{-\tau}\Phi(x),\qquad \hat g_{\mu\nu}(x)=e^{2\tau(x)}g_{\mu\nu}(x),\qquad \hat X=v,$$1, and $$\hat \Phi(x)=e^{-\tau}\Phi(x),\qquad \hat g_{\mu\nu}(x)=e^{2\tau(x)}g_{\mu\nu}(x),\qquad \hat X=v,$$2, and the dilaton $$\hat \Phi(x)=e^{-\tau}\Phi(x),\qquad \hat g_{\mu\nu}(x)=e^{2\tau(x)}g_{\mu\nu}(x),\qquad \hat X=v,$$3-point vertices satisfy

$$\hat \Phi(x)=e^{-\tau}\Phi(x),\qquad \hat g_{\mu\nu}(x)=e^{2\tau(x)}g_{\mu\nu}(x),\qquad \hat X=v,$$4

Accordingly, the entire hierarchy of dilaton self-interactions in $$\hat \Phi(x)=e^{-\tau}\Phi(x),\qquad \hat g_{\mu\nu}(x)=e^{2\tau(x)}g_{\mu\nu}(x),\qquad \hat X=v,$$5 is fixed by the first four terms, rather than by an arbitrary infinite tower of EFT couplings (Rose et al., 2014).

A complementary derivation uses anomalous Ward identities. In the nearly conformal limit, where explicit breaking is reduced to the dilatation anomaly, successive functional derivatives of the anomaly equation generate a hierarchy for $$\hat \Phi(x)=e^{-\tau}\Phi(x),\qquad \hat g_{\mu\nu}(x)=e^{2\tau(x)}g_{\mu\nu}(x),\qquad \hat X=v,$$6, $$\hat \Phi(x)=e^{-\tau}\Phi(x),\qquad \hat g_{\mu\nu}(x)=e^{2\tau(x)}g_{\mu\nu}(x),\qquad \hat X=v,$$7, and $$\hat \Phi(x)=e^{-\tau}\Phi(x),\qquad \hat g_{\mu\nu}(x)=e^{2\tau(x)}g_{\mu\nu}(x),\qquad \hat X=v,$$8, which in turn determine the quadratic, cubic, and quartic dilaton interactions. In this formulation the cubic and quartic vertices are “essentially fixed by the anomaly,” and the construction extends in principle to arbitrarily high order (Coriano et al., 2012).

A newer development is the model-independent double-soft dilaton theorem,

$$\hat \Phi(x)=e^{-\tau}\Phi(x),\qquad \hat g_{\mu\nu}(x)=e^{2\tau(x)}g_{\mu\nu}(x),\qquad \hat X=v,$$9

derived by retaining the full spacetime dependence of the dilatation commutator

$$J_D^\mu(x)=x_\nu T^{\mu\nu}(x), \qquad \partial_\mu J_D^\mu = T^\mu{}_\mu,$$0

In the single-operator setup for pseudo-dilaton mass generation, compatibility of the one-soft and two-soft relations requires

$$J_D^\mu(x)=x_\nu T^{\mu\nu}(x), \qquad \partial_\mu J_D^\mu = T^\mu{}_\mu,$$1

For QCD-like theories in the chiral limit, the quark bilinear is argued to satisfy precisely

$$J_D^\mu(x)=x_\nu T^{\mu\nu}(x), \qquad \partial_\mu J_D^\mu = T^\mu{}_\mu,$$2

which makes the dilaton interpretation internally consistent within that specific framework (Zwicky, 22 Aug 2025).

The same logic constrains hadronic gravitational form factors in a conformal dilaton phase. For a scalar hadron,

$$J_D^\mu(x)=x_\nu T^{\mu\nu}(x), \qquad \partial_\mu J_D^\mu = T^\mu{}_\mu,$$3

while under explicit quark-mass deformation the dilaton mass hyperscales as

$$J_D^\mu(x)=x_\nu T^{\mu\nu}(x), \qquad \partial_\mu J_D^\mu = T^\mu{}_\mu,$$4

These relations were proposed as lattice diagnostics for separating a conformal dilaton phase from both ordinary QCD-like confinement and an unbroken conformal window (Debbio et al., 2021).

5. Holographic and geometric realizations

Holography supplies several concrete realizations of QCD-conformal dilaton dynamics. One bottom-up model near the conformal edge introduces a bulk scalar $$J_D^\mu(x)=x_\nu T^{\mu\nu}(x), \qquad \partial_\mu J_D^\mu = T^\mu{}_\mu,$$5 dual to $$J_D^\mu(x)=x_\nu T^{\mu\nu}(x), \qquad \partial_\mu J_D^\mu = T^\mu{}_\mu,$$6 with mass slightly below the BF bound,

$$J_D^\mu(x)=x_\nu T^{\mu\nu}(x), \qquad \partial_\mu J_D^\mu = T^\mu{}_\mu,$$7

In the controlled regime of small backreaction and suitable boundary data, the lightest scalar satisfies

$$J_D^\mu(x)=x_\nu T^{\mu\nu}(x), \qquad \partial_\mu J_D^\mu = T^\mu{}_\mu,$$8

whereas the heavier scalar and tensor modes remain of order $$J_D^\mu(x)=x_\nu T^{\mu\nu}(x), \qquad \partial_\mu J_D^\mu = T^\mu{}_\mu,$$9. The same model shows that the light scalar saturates the low-energy dilatation Ward identity and obeys the PCDC relation, making it a parametrically light holographic dilaton (Rojas et al., 2023).

A related Einstein–dilaton holographic QCD model distinguishes spontaneous from explicit conformal breaking through the UV dimension $$\mathcal L_{\text{int}} = -\frac{1}{\Lambda_\rho}\,\rho\, T^\mu{}_\mu,$$0. When $$\mathcal L_{\text{int}} = -\frac{1}{\Lambda_\rho}\,\rho\, T^\mu{}_\mu,$$1, the renormalized trace vanishes, a massless scalar mode appears in the spectrum, and the scalar two-point function has a $$\mathcal L_{\text{int}} = -\frac{1}{\Lambda_\rho}\,\rho\, T^\mu{}_\mu,$$2 pole. When $$\mathcal L_{\text{int}} = -\frac{1}{\Lambda_\rho}\,\rho\, T^\mu{}_\mu,$$3, the bulk dilaton is massive, the AdS background is deformed, and the would-be Goldstone mode becomes a massive light scalar; for the model with $$\mathcal L_{\text{int}} = -\frac{1}{\Lambda_\rho}\,\rho\, T^\mu{}_\mu,$$4, the lightest scalar mass is well fit by

$$\mathcal L_{\text{int}} = -\frac{1}{\Lambda_\rho}\,\rho\, T^\mu{}_\mu,$$5

This provides an explicit holographic interpolation between spontaneous and explicit conformal-symmetry breaking (Mamani, 2019).

Improved holographic QCD and effective holographic QCD instead model QCD as a deformed CFT. In that formulation

$$\mathcal L_{\text{int}} = -\frac{1}{\Lambda_\rho}\,\rho\, T^\mu{}_\mu,$$6

and the deformed trace Ward identity

$$\mathcal L_{\text{int}} = -\frac{1}{\Lambda_\rho}\,\rho\, T^\mu{}_\mu,$$7

is matched to the QCD anomaly. One proposed dictionary is

$$\mathcal L_{\text{int}} = -\frac{1}{\Lambda_\rho}\,\rho\, T^\mu{}_\mu,$$8

This line of work interprets the bulk dilaton as the holographic carrier of QCD scale breaking, rather than as an isolated light hadron (Ballon-Bayona et al., 2018).

The holographic literature is not uniform. In V-QCD, studied in the Veneziano limit, all masses in the walking regime vanish with Miransky scaling as $$\mathcal L_{\text{int}} = -\frac{1}{\Lambda_\rho}\,\rho\, T^\mu{}_\mu,$$9, but their ratios approach nonzero constants. The conclusion is therefore explicit: there is no parametrically isolated dilaton in that spectrum (Arean et al., 2012). This negative result is an important counterweight to the positive constructions above.

More specialized geometric models also exist. One QCD-inspired construction on $\rho\,\partial_\mu J_D^\mu/\Lambda_\rho$0 deforms the $\rho\,\partial_\mu J_D^\mu/\Lambda_\rho$1 metric by an exponential factor attributed to the dilaton,

$\rho\,\partial_\mu J_D^\mu/\Lambda_\rho$2

or equivalently by a cotangent potential $\rho\,\partial_\mu J_D^\mu/\Lambda_\rho$3, while preserving the $\rho\,\partial_\mu J_D^\mu/\Lambda_\rho$4 level degeneracy of high-lying unflavored mesons (Kirchbach et al., 2012). A different recent direction reconstructs the dilaton potential $\rho\,\partial_\mu J_D^\mu/\Lambda_\rho$5 in improved holographic QCD directly from lattice chiral-condensate data at $\rho\,\partial_\mu J_D^\mu/\Lambda_\rho$6 GeV and obtains a string-breaking distance $\rho\,\partial_\mu J_D^\mu/\Lambda_\rho$7 fm consistent with lattice expectations (Hashimoto et al., 2022).

6. Hadronic observables, large-$\rho\,\partial_\mu J_D^\mu/\Lambda_\rho$8 structure, and present status

Gravitational form factors and hard $\rho\,\partial_\mu J_D^\mu/\Lambda_\rho$9 kernels have become a central arena for QCD dilaton studies. For scalar hadrons, the trace form factor $\langle J_D^\mu V V' \rangle$0 provides an infrared test: if the infrared contains a massless dilaton, then

$\langle J_D^\mu V V' \rangle$1

so that $\langle J_D^\mu V V' \rangle$2, and the coupling to matter obeys the Goldberger–Treiman-like relation

$\langle J_D^\mu V V' \rangle$3

This was proposed as a practical probe of infrared conformality in QCD-like theories with particle content (Zwicky, 22 Aug 2025).

At large momentum transfer, the non-Abelian $\langle J_D^\mu V V' \rangle$4 three-point function enters factorized descriptions of pion and proton gravitational form factors. In that setting the trace sector carries an explicit $\langle J_D^\mu V V' \rangle$5 pole,

$\langle J_D^\mu V V' \rangle$6

which is interpreted as an anomaly-induced dilaton exchange in the $\langle J_D^\mu V V' \rangle$7-channel. The resulting CFT-informed, gauge-theory-adapted parameterization was proposed for future DVCS studies, especially at an Electron-Ion Collider (Corianò et al., 29 Apr 2025).

The modern status of the subject is therefore structurally mixed. Some constructions exhibit a parametrically light dilaton that saturates low-energy Ward identities (Rojas et al., 2023, Mamani, 2019); others find that walking dynamics scales down the entire spectrum without isolating a special scalar (Arean et al., 2012). In perturbative QCD, the anomaly pole and its sum rule support a dilaton-like interpretation of the spin-0 trace sector, but this interpretation is explicitly formulated as an emergent collective exchange rather than as evidence for a fundamental scalar degree of freedom (Corianò et al., 25 Mar 2026). For ordinary $\langle J_D^\mu V V' \rangle$8 QCD, one recent assessment describes a genuine massless dilaton in the chiral limit as unlikely, though not impossible, and regards the overall QCD case as uncertain pending further lattice and form-factor evidence (Zwicky, 22 Aug 2025).

In that sense, the QCD-conformal dilaton is best understood not as a single settled object but as a family of related constructions. Across EFT, anomaly matching, holography, and hadronic form factors, the recurring theme is that approximate conformality organizes a scalar trace sector with unusually constrained dynamics. Whether that sector is realized in any given QCD-like theory as a true Goldstone dilaton, a pseudo-dilaton, or only an anomaly-mediated interpolating mode remains a dynamical question rather than a purely kinematical one.