Effective field theory description of light dilaton

In the absence of direct evidence for new physics at the electroweak scale, interest in light scalar and pseudo-scalar bosons has grown significantly in recent years. Such states emerge naturally in various extensions of the Standard Model (SM) as pseudo-Nambu-Goldstone bosons (pNGBs) resulting from the spontaneous symmetry breaking. Their existence leads to diverse, testable phenomenological signatures, motivating extensive theoretical and experimental efforts.

Axions and generic axion-like particles (ALPs) are the most prominent candidates for light pseudo-scalars beyond the SM. They arise from the spontaneous breaking of approximate global $ U(1) $ symmetries. The QCD axion was originally proposed within the Peccei-Quinn mechanism to resolve the strong CP problem in quantum chromodynamics (QCD) [1, 2]. However, QCD axions with masses between 1 MeV and tens of GeV are strongly disfavored [3, 4] due to the strict relation between their mass and decay constant. Consequently, attention has shifted towards generic ALPs, which are free from this constraint. Their low-energy effective field theories (EFTs) are well developed [5−7], and the corresponding renormalization group equations have been thoroughly studied [8, 9], providing a solid foundation for phenomenological analyses across a broad range of energy scales [10−12]. Particular focus has been placed on the ultralight regime [13, 14], with masses $ \lesssim $ eV, where ALPs are viable dark matter (DM) candidates.

Complementing the search for pseudo-scalar ALPs, scalar dilatons represent another compelling class of light bosons. As pNGBs associated with the spontaneous breaking of approximate scale (or conformal) invariance [15−19], dilatons are ubiquitous in beyond the Standard Model (BSM) constructions. Notably, lattice simulations of confining gauge theories near the conformal window imply the existence of a light scalar field [20−24], which is interpreted as a potential dilaton, and has been discussed in Refs. [25−29]. The resulting EFT for the dilaton offers a robust framework for realistic composite Higgs models [30, 31] and can accommodate forbidden dark matter candidates [32]. While the dilaton's static properties and couplings to SM fields have been established [33−36], the renormalization group (RG) evolution of these interactions remains comparatively understudied. Consequently, despite a wide array of experimental searches in atomic, molecular, and gravitational systems [37−47], a continuous EFT description that consistently connects the ultraviolet (UV) symmetry-breaking scale to low-energy infrared (IR) phenomena is still lacking.

In this work, we address this gap by establishing a consistent particle-physics ¹ framework for the dilaton, originating from an UV conformal sector. While the scale of symmetry breaking lies significantly above the electroweak scale, the dilaton manifests as a light degree of freedom in the IR. This hierarchy necessitates the construction of a sequence of EFTs to rigorously bridge the disparate energy scales. We begin by treating the SM as the effective theory of the broken phase of a quantum scale-invariant UV completion, and subsequently extend the formalism to include higher-dimensional operators. We then build a tower of dilaton EFTs matched at successive energy scales. This hierarchical structure facilitates a model-independent description of dilaton interactions across different regimes, as illustrated in Fig. 1.

Figure 1.  (color online) Schematic hierarchy of energy scales associated with the dilaton EFTs constructed in this work. The figure also shows the dilaton couplings to gauge bosons, quarks, and nucleons, which serve as inputs for the phenomenological analyses. These interactions can be probed via LHC searches, flavor-changing neutral currents (FCNCs), supernova cooling, and—assuming the dilaton accounts for the observed DM abundance—precision measurements of time-varying fundamental constants.

Analogous to searches for ALPs, the detection strategy for the dilaton depends on its mass. A key distinction arises from the dilaton's scalar (CP-even) nature, which contrasts with the pseudo-scalar (CP-odd) couplings characteristic of ALPs. In this study, we focus on two representative mass regimes to derive complementary bounds on the dilaton parameter space. For masses near the MeV scale, the dilaton behaves as a particle-like state with interactions governed by the trace anomaly. In this regime, we probe the dilaton through signatures at the Large Hadron Collider (LHC), rare meson decays, and cosmological observations. Conversely, in the sub-eV regime, we treat the dilaton as a coherent wave-like field that induces oscillations in fundamental constants, detectable via precision measurements in atomic systems. While intermediate masses offer rich phenomenology, such as constraints from stellar cooling and violations of the Weak Equivalence Principle (WEP), we reserve these topics for future investigations.

This paper is organized as follows: In Section II, we develop the theoretical framework for the dilaton, comparing two distinct construction approaches—the conformal compensator method and manifestly scale-invariant regularization—and verifying its universal coupling to the trace anomaly. In Section III, we derive its leading interactions with SM particles and extend the framework to higher-dimensional operators. The effective Lagrangian for the low-energy effective field theory (LEFT) extended by the dilaton is presented in Section III.D and subsequently matched onto a chiral perturbation theory (χPT) Lagrangian that includes the dilaton in Section IV. In Section V, we derive complementary phenomenological constraints on the dilaton parameter space. Finally, we draw conclusions in Section VI.

In this section, we present the theoretical framework to construct an EFT for a light dilaton. When the conformal group $ S O(2,4) $ spontaneously breaks down to the Poincaré group, although there are five broken generators associated with dilatations and special conformal transformations, only one physical Goldstone boson, the dilaton χ, is required to restore conformal invariance in the effective Lagrangian [50]. At leading order in the dilaton field, its interactions with the visible-sector fields are usually given by a single term [33]

where $ \chi,f_\chi $ refer to the physical dilaton field and the dilaton vacuum expectation value (VEV) $ f_\chi $ respectively and $ T^\mu_{\ \mu} $ is the trace of the improved energy-momentum tensor. Consequently, the linear dilaton couplings are controlled by the new parameter, the dilaton VEV. This universal structure can be derived in two physically equivalent ways: the conformal compensator method and the scale-invariant regularization using a dynamical subtraction scale $ \mu(x) $. Moreover, the renormalization-group evolution of all couplings in the dilaton effective theory coincides, at the one-loop level, with the running of the conventional theory without the dilaton once the renormalization scale is identified with the dilaton VEV in the broken phase. The following subsections present these results in detail.

In this section, we present the theoretical framework to construct an EFT for a light dilaton. When the conformal group $ S O(2,4) $ spontaneously breaks down to the Poincaré group, although there are five broken generators associated with dilatations and special conformal transformations, only one physical Goldstone boson, the dilaton χ, is required to restore conformal invariance in the effective Lagrangian [50]. At leading order in the dilaton field, its interactions with the visible-sector fields are usually given by a single term [33]

where $ \chi,f_\chi $ refer to the physical dilaton field and the dilaton vacuum expectation value (VEV) $ f_\chi $ respectively and $ T^\mu_{\ \mu} $ is the trace of the improved energy-momentum tensor. Consequently, the linear dilaton couplings are controlled by the new parameter, the dilaton VEV. This universal structure can be derived in two physically equivalent ways: the conformal compensator method and the scale-invariant regularization using a dynamical subtraction scale $ \mu(x) $. Moreover, the renormalization-group evolution of all couplings in the dilaton effective theory coincides, at the one-loop level, with the running of the conventional theory without the dilaton once the renormalization scale is identified with the dilaton VEV in the broken phase. The following subsections present these results in detail.

In this section, we present the theoretical framework to construct an EFT for a light dilaton. When the conformal group $ S O(2,4) $ spontaneously breaks down to the Poincaré group, although there are five broken generators associated with dilatations and special conformal transformations, only one physical Goldstone boson, the dilaton χ, is required to restore conformal invariance in the effective Lagrangian [50]. At leading order in the dilaton field, its interactions with the visible-sector fields are usually given by a single term [33]

where $ \chi,f_\chi $ refer to the physical dilaton field and the dilaton vacuum expectation value (VEV) $ f_\chi $ respectively and $ T^\mu_{\ \mu} $ is the trace of the improved energy-momentum tensor. Consequently, the linear dilaton couplings are controlled by the new parameter, the dilaton VEV. This universal structure can be derived in two physically equivalent ways: the conformal compensator method and the scale-invariant regularization using a dynamical subtraction scale $ \mu(x) $. Moreover, the renormalization-group evolution of all couplings in the dilaton effective theory coincides, at the one-loop level, with the running of the conventional theory without the dilaton once the renormalization scale is identified with the dilaton VEV in the broken phase. The following subsections present these results in detail.

In this section, we review the structure of the energy-momentum tensor and its trace in a general field theory. Consider a theory described by a Lagrangian $ {\cal{L}}(x)= {\cal{L}}(\{\phi(x),\partial_\mu\phi(x)\}) $. Under scale transformation $x^\mu\rightarrow {\rm e}^\lambda x^\mu$, a field ϕ of scaling dimension $ d_\phi $ transforms as $\phi(x)\rightarrow {\rm e}^{d_\phi\lambda}\phi({\rm e}^\lambda x)$. The associated Noether current in d dimensions is

where $ T^{\mu\nu} $ is the symmetric, conserved energy-momentum tensor [51]. Its trace then equals the divergence of the scale current,

This relation implies that exact scale invariance requires a vanishing trace $ T^\mu{}_\mu=0 $. As an illustration, consider the QCD Lagrangian in $ d=4-2\epsilon $ dimensions,

where the field strength tensor $ G^a_{\mu\nu} $ and covariant derivative $ D_\mu $ are defined without explicit dependence on the gauge coupling g. Applying Eq. (3), the corresponding trace of the energy-momentum tensor is

Renormalization of the composite operator $ G^2\equiv G^a_{\mu\nu} G^{a\mu\nu} $ in the modified minimal subtraction ($ \overline{{\rm{MS}}} $) scheme gives the identity [52]

where $ \beta(g) $ is the QCD beta-function and $ \gamma_m $ is the quark mass anomalous dimension. Inserting this relation and taking the $ \epsilon\rightarrow0 $ limit, one recovers the well-known QCD trace anomaly

The above expression reveals that scale invariance is broken both classically by dimensional parameters (quark masses) and by the running of couplings at the quantum level. From this heuristic perspective of understanding the effects of scale transformations on the theory, a practical form of $ T^\mu{}_\mu $ without the dimensional regulator $ \epsilon $ can be derived by writing the Lagrangian in terms of the anomalous couplings and operators

where the coupling g depends on the renormalization scale μ and the operator $ {\cal{O}}_i $ has scaling dimension $ d_i $. The scale transformation $x\rightarrow {\rm e}^{\lambda}x$ is equivalent to shifting the renormalization scale as $\mu\rightarrow {\rm e}^{\lambda} \mu$ while the operators transform as ${\cal{O}}_i(x)\rightarrow {\rm e}^{\lambda d_i}{\cal{O}}_i({\rm e}^\lambda x)$. The variation of the Lagrangian under this combined transformation directly yields [33]

where $\beta(g_i)=\mu \dfrac{\partial g_i}{\partial\mu}$. This form manifestly separates explicit breaking by dimensional parameters from quantum anomalous breaking via running couplings.

In this section, we review the structure of the energy-momentum tensor and its trace in a general field theory. Consider a theory described by a Lagrangian $ {\cal{L}}(x)= {\cal{L}}(\{\phi(x),\partial_\mu\phi(x)\}) $. Under scale transformation $x^\mu\rightarrow {\rm e}^\lambda x^\mu$, a field ϕ of scaling dimension $ d_\phi $ transforms as $\phi(x)\rightarrow {\rm e}^{d_\phi\lambda}\phi({\rm e}^\lambda x)$. The associated Noether current in d dimensions is

where $ T^{\mu\nu} $ is the symmetric, conserved energy-momentum tensor [51]. Its trace then equals the divergence of the scale current,

This relation implies that exact scale invariance requires a vanishing trace $ T^\mu{}_\mu=0 $. As an illustration, consider the QCD Lagrangian in $ d=4-2\epsilon $ dimensions,

where the field strength tensor $ G^a_{\mu\nu} $ and covariant derivative $ D_\mu $ are defined without explicit dependence on the gauge coupling g. Applying Eq. (3), the corresponding trace of the energy-momentum tensor is

Renormalization of the composite operator $ G^2\equiv G^a_{\mu\nu} G^{a\mu\nu} $ in the modified minimal subtraction ($ \overline{{\rm{MS}}} $) scheme gives the identity [52]

where $ \beta(g) $ is the QCD beta-function and $ \gamma_m $ is the quark mass anomalous dimension. Inserting this relation and taking the $ \epsilon\rightarrow0 $ limit, one recovers the well-known QCD trace anomaly

The above expression reveals that scale invariance is broken both classically by dimensional parameters (quark masses) and by the running of couplings at the quantum level. From this heuristic perspective of understanding the effects of scale transformations on the theory, a practical form of $ T^\mu{}_\mu $ without the dimensional regulator $ \epsilon $ can be derived by writing the Lagrangian in terms of the anomalous couplings and operators

where the coupling g depends on the renormalization scale μ and the operator $ {\cal{O}}_i $ has scaling dimension $ d_i $. The scale transformation $x\rightarrow {\rm e}^{\lambda}x$ is equivalent to shifting the renormalization scale as $\mu\rightarrow {\rm e}^{\lambda} \mu$ while the operators transform as ${\cal{O}}_i(x)\rightarrow {\rm e}^{\lambda d_i}{\cal{O}}_i({\rm e}^\lambda x)$. The variation of the Lagrangian under this combined transformation directly yields [33]

where $\beta(g_i)=\mu \dfrac{\partial g_i}{\partial\mu}$. This form manifestly separates explicit breaking by dimensional parameters from quantum anomalous breaking via running couplings.

In this section, we review the structure of the energy-momentum tensor and its trace in a general field theory. Consider a theory described by a Lagrangian $ {\cal{L}}(x)= {\cal{L}}(\{\phi(x),\partial_\mu\phi(x)\}) $. Under scale transformation $x^\mu\rightarrow {\rm e}^\lambda x^\mu$, a field ϕ of scaling dimension $ d_\phi $ transforms as $\phi(x)\rightarrow {\rm e}^{d_\phi\lambda}\phi({\rm e}^\lambda x)$. The associated Noether current in d dimensions is

where $ T^{\mu\nu} $ is the symmetric, conserved energy-momentum tensor [51]. Its trace then equals the divergence of the scale current,

This relation implies that exact scale invariance requires a vanishing trace $ T^\mu{}_\mu=0 $. As an illustration, consider the QCD Lagrangian in $ d=4-2\epsilon $ dimensions,

where the field strength tensor $ G^a_{\mu\nu} $ and covariant derivative $ D_\mu $ are defined without explicit dependence on the gauge coupling g. Applying Eq. (3), the corresponding trace of the energy-momentum tensor is

Renormalization of the composite operator $ G^2\equiv G^a_{\mu\nu} G^{a\mu\nu} $ in the modified minimal subtraction ($ \overline{{\rm{MS}}} $) scheme gives the identity [52]

where $ \beta(g) $ is the QCD beta-function and $ \gamma_m $ is the quark mass anomalous dimension. Inserting this relation and taking the $ \epsilon\rightarrow0 $ limit, one recovers the well-known QCD trace anomaly

The above expression reveals that scale invariance is broken both classically by dimensional parameters (quark masses) and by the running of couplings at the quantum level. From this heuristic perspective of understanding the effects of scale transformations on the theory, a practical form of $ T^\mu{}_\mu $ without the dimensional regulator $ \epsilon $ can be derived by writing the Lagrangian in terms of the anomalous couplings and operators

where the coupling g depends on the renormalization scale μ and the operator $ {\cal{O}}_i $ has scaling dimension $ d_i $. The scale transformation $x\rightarrow {\rm e}^{\lambda}x$ is equivalent to shifting the renormalization scale as $\mu\rightarrow {\rm e}^{\lambda} \mu$ while the operators transform as ${\cal{O}}_i(x)\rightarrow {\rm e}^{\lambda d_i}{\cal{O}}_i({\rm e}^\lambda x)$. The variation of the Lagrangian under this combined transformation directly yields [33]

where $\beta(g_i)=\mu \dfrac{\partial g_i}{\partial\mu}$. This form manifestly separates explicit breaking by dimensional parameters from quantum anomalous breaking via running couplings.

The dilaton is introduced through the conformal compensator field [33]

where $ f_\chi=\langle\Phi\rangle $ is the order parameter for spontaneous conformal symmetry breaking, determined by the underlying conformal sector. Under scale transformations, χ transforms non-linearly as $\chi(x)\rightarrow \chi({\rm e}^\lambda x)+f_\chi\lambda$, while the compensator transforms linearly as

To make the Lagrangian in Eq. (8) manifestly invariant, the dimensional couplings are treated as spurions by assigning them a fictitious scaling dimension $ 4-d_i $ and the anomalous breaking via running couplings is incorporated via renormalization scale's dependence on the dilaton field:

Expanding this scale-invariant Lagrangian to linear order in the dilaton field, we obtain

where the first term is the original Lagrangian $ {\cal{L}} $, and the second term precisely gives the universal dilaton coupling to the trace of the energy-momentum tensor, $ -\dfrac{\chi}{f_\chi}T^{\mu}_{\ \mu} $, with $ T^{\mu}_{\ \mu} $ given in Eq. (9).

This construction ensures that low-energy dilaton interactions are dictated solely by the pattern of spontaneous scale (conformal) symmetry breaking, independent of the microscopic origin of the breaking sector.

The dilaton is introduced through the conformal compensator field [33]

where $ f_\chi=\langle\Phi\rangle $ is the order parameter for spontaneous conformal symmetry breaking, determined by the underlying conformal sector. Under scale transformations, χ transforms non-linearly as $\chi(x)\rightarrow \chi({\rm e}^\lambda x)+f_\chi\lambda$, while the compensator transforms linearly as

To make the Lagrangian in Eq. (8) manifestly invariant, the dimensional couplings are treated as spurions by assigning them a fictitious scaling dimension $ 4-d_i $ and the anomalous breaking via running couplings is incorporated via renormalization scale's dependence on the dilaton field:

Expanding this scale-invariant Lagrangian to linear order in the dilaton field, we obtain

where the first term is the original Lagrangian $ {\cal{L}} $, and the second term precisely gives the universal dilaton coupling to the trace of the energy-momentum tensor, $ -\dfrac{\chi}{f_\chi}T^{\mu}_{\ \mu} $, with $ T^{\mu}_{\ \mu} $ given in Eq. (9).

This construction ensures that low-energy dilaton interactions are dictated solely by the pattern of spontaneous scale (conformal) symmetry breaking, independent of the microscopic origin of the breaking sector.

The dilaton is introduced through the conformal compensator field [33]

where $ f_\chi=\langle\Phi\rangle $ is the order parameter for spontaneous conformal symmetry breaking, determined by the underlying conformal sector. Under scale transformations, χ transforms non-linearly as $\chi(x)\rightarrow \chi({\rm e}^\lambda x)+f_\chi\lambda$, while the compensator transforms linearly as

To make the Lagrangian in Eq. (8) manifestly invariant, the dimensional couplings are treated as spurions by assigning them a fictitious scaling dimension $ 4-d_i $ and the anomalous breaking via running couplings is incorporated via renormalization scale's dependence on the dilaton field:

Expanding this scale-invariant Lagrangian to linear order in the dilaton field, we obtain

where the first term is the original Lagrangian $ {\cal{L}} $, and the second term precisely gives the universal dilaton coupling to the trace of the energy-momentum tensor, $ -\dfrac{\chi}{f_\chi}T^{\mu}_{\ \mu} $, with $ T^{\mu}_{\ \mu} $ given in Eq. (9).

This construction ensures that low-energy dilaton interactions are dictated solely by the pattern of spontaneous scale (conformal) symmetry breaking, independent of the microscopic origin of the breaking sector.

While the conformal compensator method offers a powerful and symmetry-driven framework for constructing the low-energy effective theory of the dilaton, it does not automatically provide a regularization and renormalization scheme that preserves scale invariance at the quantum level. A complementary approach, particularly well-suited for perturbative quantum computations in such effective theories, is the adoption of a manifestly scale-invariant regularization [53, 54]. In this section, we illustrate this method using QCD as a toy model.

Assuming a UV-conformal sector beyond QCD, our starting point is the scale-invariant extension of the QCD Lagrangian in $ d=4-2\epsilon $ dimensions, where the dilaton field Φ is introduced via analytical continuation:

where y quantifies the tree-level Yukawa interaction between Φ and quarks. As in standard dimensional regularization, the couplings are rescaled by a renormalization scale μ to remain dimensionless. However, to preserve manifest scale invariance, we promote a constant scale μ to a field-dependent function

where z is an arbitrary dimensionless parameter. The exponent in Eq. (15) ensures that $ \mu(\Phi) $ has mass dimension $ 1 $.

After spontaneous scale symmetry breaking, the dilaton field Φ can be expanded around its VEV $ f_\chi=\langle\Phi\rangle $, which represents the scale of new physics,

where the fluctuation field χ denotes the physical dilaton particle. The scale function $ \mu(\Phi) $ now generates the usual renormalization scale $ \mu_0\equiv \mu(f_\chi) $. For small fluctuations $ \chi\ll f_\chi $, the effective Lagrangian reduces to

where higher-order terms in $ \chi,\epsilon $ are neglected. The above Lagrangian corresponds to a renormalized theory at the scale $ \mu_0 $ and has two distinct components. The first three terms reproduce the standard QCD Lagrangian in Eq. (4) with a fermion mass $ m=yf_\chi $ emerging dynamically. The dilaton interaction term is precisely proportional to the trace of the energy-momentum tensor $ T^\mu{}_\mu $ given in Eq. (5). This identification simplifies the Lagrangian in the limit $ \epsilon\rightarrow0 $

This structure is consistent with the expectations from the conformal compensator method.

In this way, the standard QCD theory is embedded into a quantum scale-invariant framework, naturally generating linear dilaton couplings to the trace of the energy-momentum tensor. This approach preserves scale invariance manifestly during intermediate calculations, making it ideal for perturbative studies of dilaton effective theories.

While the conformal compensator method offers a powerful and symmetry-driven framework for constructing the low-energy effective theory of the dilaton, it does not automatically provide a regularization and renormalization scheme that preserves scale invariance at the quantum level. A complementary approach, particularly well-suited for perturbative quantum computations in such effective theories, is the adoption of a manifestly scale-invariant regularization [53, 54]. In this section, we illustrate this method using QCD as a toy model.

Assuming a UV-conformal sector beyond QCD, our starting point is the scale-invariant extension of the QCD Lagrangian in $ d=4-2\epsilon $ dimensions, where the dilaton field Φ is introduced via analytical continuation:

where y quantifies the tree-level Yukawa interaction between Φ and quarks. As in standard dimensional regularization, the couplings are rescaled by a renormalization scale μ to remain dimensionless. However, to preserve manifest scale invariance, we promote a constant scale μ to a field-dependent function

where z is an arbitrary dimensionless parameter. The exponent in Eq. (15) ensures that $ \mu(\Phi) $ has mass dimension $ 1 $.

After spontaneous scale symmetry breaking, the dilaton field Φ can be expanded around its VEV $ f_\chi=\langle\Phi\rangle $, which represents the scale of new physics,

where the fluctuation field χ denotes the physical dilaton particle. The scale function $ \mu(\Phi) $ now generates the usual renormalization scale $ \mu_0\equiv \mu(f_\chi) $. For small fluctuations $ \chi\ll f_\chi $, the effective Lagrangian reduces to

where higher-order terms in $ \chi,\epsilon $ are neglected. The above Lagrangian corresponds to a renormalized theory at the scale $ \mu_0 $ and has two distinct components. The first three terms reproduce the standard QCD Lagrangian in Eq. (4) with a fermion mass $ m=yf_\chi $ emerging dynamically. The dilaton interaction term is precisely proportional to the trace of the energy-momentum tensor $ T^\mu{}_\mu $ given in Eq. (5). This identification simplifies the Lagrangian in the limit $ \epsilon\rightarrow0 $

This structure is consistent with the expectations from the conformal compensator method.

In this way, the standard QCD theory is embedded into a quantum scale-invariant framework, naturally generating linear dilaton couplings to the trace of the energy-momentum tensor. This approach preserves scale invariance manifestly during intermediate calculations, making it ideal for perturbative studies of dilaton effective theories.

While the conformal compensator method offers a powerful and symmetry-driven framework for constructing the low-energy effective theory of the dilaton, it does not automatically provide a regularization and renormalization scheme that preserves scale invariance at the quantum level. A complementary approach, particularly well-suited for perturbative quantum computations in such effective theories, is the adoption of a manifestly scale-invariant regularization [53, 54]. In this section, we illustrate this method using QCD as a toy model.

Assuming a UV-conformal sector beyond QCD, our starting point is the scale-invariant extension of the QCD Lagrangian in $ d=4-2\epsilon $ dimensions, where the dilaton field Φ is introduced via analytical continuation:

where y quantifies the tree-level Yukawa interaction between Φ and quarks. As in standard dimensional regularization, the couplings are rescaled by a renormalization scale μ to remain dimensionless. However, to preserve manifest scale invariance, we promote a constant scale μ to a field-dependent function

where z is an arbitrary dimensionless parameter. The exponent in Eq. (15) ensures that $ \mu(\Phi) $ has mass dimension $ 1 $.

After spontaneous scale symmetry breaking, the dilaton field Φ can be expanded around its VEV $ f_\chi=\langle\Phi\rangle $, which represents the scale of new physics,

where the fluctuation field χ denotes the physical dilaton particle. The scale function $ \mu(\Phi) $ now generates the usual renormalization scale $ \mu_0\equiv \mu(f_\chi) $. For small fluctuations $ \chi\ll f_\chi $, the effective Lagrangian reduces to

where higher-order terms in $ \chi,\epsilon $ are neglected. The above Lagrangian corresponds to a renormalized theory at the scale $ \mu_0 $ and has two distinct components. The first three terms reproduce the standard QCD Lagrangian in Eq. (4) with a fermion mass $ m=yf_\chi $ emerging dynamically. The dilaton interaction term is precisely proportional to the trace of the energy-momentum tensor $ T^\mu{}_\mu $ given in Eq. (5). This identification simplifies the Lagrangian in the limit $ \epsilon\rightarrow0 $

This structure is consistent with the expectations from the conformal compensator method.

In this way, the standard QCD theory is embedded into a quantum scale-invariant framework, naturally generating linear dilaton couplings to the trace of the energy-momentum tensor. This approach preserves scale invariance manifestly during intermediate calculations, making it ideal for perturbative studies of dilaton effective theories.

In this section, we demonstrate that a quantum theory with spontaneously broken scale invariance, realized via the dilaton, can reproduce the running couplings of conventional theories with explicit dimensional parameters. Perturbative calculations are well-defined only in the broken phase, where the dilaton acquires a VEV. Building upon the effective scale-invariant extension of QCD involving a dilaton obtained in the previous section, we analyze the scaling properties of renormalized Green's functions in the broken phase, closely following the approach of Ref. [53].

To illustrate the mechanism explicitly, we first recall the standard treatment in ordinary QCD, where the running of coupling g and fermion mass m with the renormalization scale μ is governed by the beta functions $ \beta_g,\beta_m $. These are encoded in the scaling properties of the renormalized connected n-point Green's function. The Ward identity corresponding to the scale transformations in momentum space is given by

where $ \phi=(G_\mu,\psi,\bar{\psi}) $ labels the gluon, quark and anti-quark fields, $ n_\phi=(n_G,n_\psi,n_{\bar{\psi}}) $ denote the numbers of external legs of each type, $c_\phi=(3,\dfrac{5}{2},\dfrac{5}{2})$, and $ p_k^\phi $ is the momentum for the k-th leg for the field ϕ. This identity ensures the correct engineering dimension $ 4-\sum c_\phi n_\phi $ of the Green's function, accounting for the implicit μ dependence introduced by renormalization. The μ dependence is further constrained by the Callan-Symanzik equation, which expresses the independence of physical observables from the arbitrary renormalization scale μ:

with the beta functions and anomalous dimensions defined as

where $ Z_\phi $ are the wave function renormalization constants. Combining Eq. (19) and Eq. (20) leads to the differential equation describing the scale dependence of the Green's functions:

This equation implies that the couplings run with momentum or the characteristic energy of the process:

In the subsequent analysis, we will show how an analogous derivation in the scale-invariant theory with a dilaton leads to identical running behavior in the broken phase.

In the scale-invariant extension of QCD constructed in the previous section, several important modifications arise in the treatment of renormalized n-point Green's function. The field content now includes the dilaton, so the momentum space version of the Ward identity becomes

where $ \phi=(G_\mu,\psi,\bar{\psi},\chi) $ with $c_\phi=(3,\dfrac{5}{2},\dfrac{5}{2},3)$, and we have explicitly included the derivative with respect to the VEV $ f_\chi $. A second important change is that the renormalization scale is now dependent on the VEV through the dimensionless parameter z, defining $ \mu_0=zf_\chi $. Physical observables must be independent of the arbitrary choice of z, leading to a Callan-Symanzik-like equation with respect to z:

where the hatted beta functions and anomalous dimensions are now defined as

Here, the dimensional fermion mass m of standard QCD is replaced by the dimensionless Yukawa coupling y, and the z dependence of the VEV $ f_\chi $ is given by the field renormalization of the dilaton (consistent with the counter-terms being identical to those in the symmetric phase under appropriate renormalization schemes [55]). Finally, in order to eliminate the explicit dependence on $ f_\chi $, we use the quantum action principle, which yields the additional scaling relation

The right-hand side corresponds to diagrams with a single insertion of the dilaton. We now consider the decoupling limit $ y\rightarrow0 $ while keeping the physical fermion mass $ m=y f_\chi $ finite. In this limit, the dilaton becomes non-propagating, the anomalous dimension $ \gamma_\chi $ vanishes at the relevant order, and the insertion term on the right-hand side of Eq. (27) can be neglected. Combining Eqs. (24)− (27) then produces a scaling equation of identical functional form to Eq. (22):

with the operator $ m\dfrac{\partial}{\partial m} $ being replaced by $ y\dfrac{\partial}{\partial y} $.

The beta functions $ \beta_g,\beta_m $ in standard QCD and their counterparts $ \hat{\beta}_g,\hat{\beta}_y $ in the scale-invariant theory are both determined by the $ 1/\epsilon $ poles in dimensional regularization that renormalize their respective couplings. The effective Lagrangian of our scale-invariant formulation in Eq. (17) differs from the standard QCD Lagrangian in Eq. (4) by an evanescent operator of $ {\cal{O}}(\epsilon) $ and by dilaton-mediated interactions proportional to the small Yukawa coupling y. In the decoupling limit, these additional terms do not contribute to the one-loop divergences. Consequently, the one-loop beta functions calculated in the scale-invariant framework coincide exactly with those of standard QCD. Differences emerge only starting at the two-loop level, where evanescent interactions can contribute to the renormalization group flow through finite $ 1/\epsilon $ poles. However, this difference is controlled by dilaton interactions that are suppressed either by $ 1/f_\chi $ or by the small coupling $ y = m_f/f_\chi $. Consequently, such higher-order effects on the dilaton couplings driven by the β-functions are suppressed by inverse powers of $ f_\chi $ and thus negligible.

We conclude that the theory with spontaneously broken scale invariance reproduces the standard one-loop running of couplings in the broken phase, with the conventional renormalization scale dependence $ \ln\mu $ being replaced by $ \ln (z f_\chi) $. This equivalence not only demonstrates the quantum consistency of the formalism, but also highlights its utility as a powerful top-down approach for constructing dilaton EFTs where scale symmetry is broken spontaneously.

In this section, we demonstrate that a quantum theory with spontaneously broken scale invariance, realized via the dilaton, can reproduce the running couplings of conventional theories with explicit dimensional parameters. Perturbative calculations are well-defined only in the broken phase, where the dilaton acquires a VEV. Building upon the effective scale-invariant extension of QCD involving a dilaton obtained in the previous section, we analyze the scaling properties of renormalized Green's functions in the broken phase, closely following the approach of Ref. [53].

To illustrate the mechanism explicitly, we first recall the standard treatment in ordinary QCD, where the running of coupling g and fermion mass m with the renormalization scale μ is governed by the beta functions $ \beta_g,\beta_m $. These are encoded in the scaling properties of the renormalized connected n-point Green's function. The Ward identity corresponding to the scale transformations in momentum space is given by

where $ \phi=(G_\mu,\psi,\bar{\psi}) $ labels the gluon, quark and anti-quark fields, $ n_\phi=(n_G,n_\psi,n_{\bar{\psi}}) $ denote the numbers of external legs of each type, $c_\phi=(3,\dfrac{5}{2},\dfrac{5}{2})$, and $ p_k^\phi $ is the momentum for the k-th leg for the field ϕ. This identity ensures the correct engineering dimension $ 4-\sum c_\phi n_\phi $ of the Green's function, accounting for the implicit μ dependence introduced by renormalization. The μ dependence is further constrained by the Callan-Symanzik equation, which expresses the independence of physical observables from the arbitrary renormalization scale μ:

with the beta functions and anomalous dimensions defined as

where $ Z_\phi $ are the wave function renormalization constants. Combining Eq. (19) and Eq. (20) leads to the differential equation describing the scale dependence of the Green's functions:

This equation implies that the couplings run with momentum or the characteristic energy of the process:

In the subsequent analysis, we will show how an analogous derivation in the scale-invariant theory with a dilaton leads to identical running behavior in the broken phase.

In the scale-invariant extension of QCD constructed in the previous section, several important modifications arise in the treatment of renormalized n-point Green's function. The field content now includes the dilaton, so the momentum space version of the Ward identity becomes

where $ \phi=(G_\mu,\psi,\bar{\psi},\chi) $ with $c_\phi=(3,\dfrac{5}{2},\dfrac{5}{2},3)$, and we have explicitly included the derivative with respect to the VEV $ f_\chi $. A second important change is that the renormalization scale is now dependent on the VEV through the dimensionless parameter z, defining $ \mu_0=zf_\chi $. Physical observables must be independent of the arbitrary choice of z, leading to a Callan-Symanzik-like equation with respect to z:

where the hatted beta functions and anomalous dimensions are now defined as

Here, the dimensional fermion mass m of standard QCD is replaced by the dimensionless Yukawa coupling y, and the z dependence of the VEV $ f_\chi $ is given by the field renormalization of the dilaton (consistent with the counter-terms being identical to those in the symmetric phase under appropriate renormalization schemes [55]). Finally, in order to eliminate the explicit dependence on $ f_\chi $, we use the quantum action principle, which yields the additional scaling relation

The right-hand side corresponds to diagrams with a single insertion of the dilaton. We now consider the decoupling limit $ y\rightarrow0 $ while keeping the physical fermion mass $ m=y f_\chi $ finite. In this limit, the dilaton becomes non-propagating, the anomalous dimension $ \gamma_\chi $ vanishes at the relevant order, and the insertion term on the right-hand side of Eq. (27) can be neglected. Combining Eqs. (24)− (27) then produces a scaling equation of identical functional form to Eq. (22):

with the operator $ m\dfrac{\partial}{\partial m} $ being replaced by $ y\dfrac{\partial}{\partial y} $.

The beta functions $ \beta_g,\beta_m $ in standard QCD and their counterparts $ \hat{\beta}_g,\hat{\beta}_y $ in the scale-invariant theory are both determined by the $ 1/\epsilon $ poles in dimensional regularization that renormalize their respective couplings. The effective Lagrangian of our scale-invariant formulation in Eq. (17) differs from the standard QCD Lagrangian in Eq. (4) by an evanescent operator of $ {\cal{O}}(\epsilon) $ and by dilaton-mediated interactions proportional to the small Yukawa coupling y. In the decoupling limit, these additional terms do not contribute to the one-loop divergences. Consequently, the one-loop beta functions calculated in the scale-invariant framework coincide exactly with those of standard QCD. Differences emerge only starting at the two-loop level, where evanescent interactions can contribute to the renormalization group flow through finite $ 1/\epsilon $ poles. However, this difference is controlled by dilaton interactions that are suppressed either by $ 1/f_\chi $ or by the small coupling $ y = m_f/f_\chi $. Consequently, such higher-order effects on the dilaton couplings driven by the β-functions are suppressed by inverse powers of $ f_\chi $ and thus negligible.

We conclude that the theory with spontaneously broken scale invariance reproduces the standard one-loop running of couplings in the broken phase, with the conventional renormalization scale dependence $ \ln\mu $ being replaced by $ \ln (z f_\chi) $. This equivalence not only demonstrates the quantum consistency of the formalism, but also highlights its utility as a powerful top-down approach for constructing dilaton EFTs where scale symmetry is broken spontaneously.

In this section, we demonstrate that a quantum theory with spontaneously broken scale invariance, realized via the dilaton, can reproduce the running couplings of conventional theories with explicit dimensional parameters. Perturbative calculations are well-defined only in the broken phase, where the dilaton acquires a VEV. Building upon the effective scale-invariant extension of QCD involving a dilaton obtained in the previous section, we analyze the scaling properties of renormalized Green's functions in the broken phase, closely following the approach of Ref. [53].

To illustrate the mechanism explicitly, we first recall the standard treatment in ordinary QCD, where the running of coupling g and fermion mass m with the renormalization scale μ is governed by the beta functions $ \beta_g,\beta_m $. These are encoded in the scaling properties of the renormalized connected n-point Green's function. The Ward identity corresponding to the scale transformations in momentum space is given by

where $ \phi=(G_\mu,\psi,\bar{\psi}) $ labels the gluon, quark and anti-quark fields, $ n_\phi=(n_G,n_\psi,n_{\bar{\psi}}) $ denote the numbers of external legs of each type, $c_\phi=(3,\dfrac{5}{2},\dfrac{5}{2})$, and $ p_k^\phi $ is the momentum for the k-th leg for the field ϕ. This identity ensures the correct engineering dimension $ 4-\sum c_\phi n_\phi $ of the Green's function, accounting for the implicit μ dependence introduced by renormalization. The μ dependence is further constrained by the Callan-Symanzik equation, which expresses the independence of physical observables from the arbitrary renormalization scale μ:

with the beta functions and anomalous dimensions defined as

where $ Z_\phi $ are the wave function renormalization constants. Combining Eq. (19) and Eq. (20) leads to the differential equation describing the scale dependence of the Green's functions:

This equation implies that the couplings run with momentum or the characteristic energy of the process:

In the subsequent analysis, we will show how an analogous derivation in the scale-invariant theory with a dilaton leads to identical running behavior in the broken phase.

In the scale-invariant extension of QCD constructed in the previous section, several important modifications arise in the treatment of renormalized n-point Green's function. The field content now includes the dilaton, so the momentum space version of the Ward identity becomes

where $ \phi=(G_\mu,\psi,\bar{\psi},\chi) $ with $c_\phi=(3,\dfrac{5}{2},\dfrac{5}{2},3)$, and we have explicitly included the derivative with respect to the VEV $ f_\chi $. A second important change is that the renormalization scale is now dependent on the VEV through the dimensionless parameter z, defining $ \mu_0=zf_\chi $. Physical observables must be independent of the arbitrary choice of z, leading to a Callan-Symanzik-like equation with respect to z:

where the hatted beta functions and anomalous dimensions are now defined as

Here, the dimensional fermion mass m of standard QCD is replaced by the dimensionless Yukawa coupling y, and the z dependence of the VEV $ f_\chi $ is given by the field renormalization of the dilaton (consistent with the counter-terms being identical to those in the symmetric phase under appropriate renormalization schemes [55]). Finally, in order to eliminate the explicit dependence on $ f_\chi $, we use the quantum action principle, which yields the additional scaling relation

The right-hand side corresponds to diagrams with a single insertion of the dilaton. We now consider the decoupling limit $ y\rightarrow0 $ while keeping the physical fermion mass $ m=y f_\chi $ finite. In this limit, the dilaton becomes non-propagating, the anomalous dimension $ \gamma_\chi $ vanishes at the relevant order, and the insertion term on the right-hand side of Eq. (27) can be neglected. Combining Eqs. (24)− (27) then produces a scaling equation of identical functional form to Eq. (22):

with the operator $ m\dfrac{\partial}{\partial m} $ being replaced by $ y\dfrac{\partial}{\partial y} $.

The beta functions $ \beta_g,\beta_m $ in standard QCD and their counterparts $ \hat{\beta}_g,\hat{\beta}_y $ in the scale-invariant theory are both determined by the $ 1/\epsilon $ poles in dimensional regularization that renormalize their respective couplings. The effective Lagrangian of our scale-invariant formulation in Eq. (17) differs from the standard QCD Lagrangian in Eq. (4) by an evanescent operator of $ {\cal{O}}(\epsilon) $ and by dilaton-mediated interactions proportional to the small Yukawa coupling y. In the decoupling limit, these additional terms do not contribute to the one-loop divergences. Consequently, the one-loop beta functions calculated in the scale-invariant framework coincide exactly with those of standard QCD. Differences emerge only starting at the two-loop level, where evanescent interactions can contribute to the renormalization group flow through finite $ 1/\epsilon $ poles. However, this difference is controlled by dilaton interactions that are suppressed either by $ 1/f_\chi $ or by the small coupling $ y = m_f/f_\chi $. Consequently, such higher-order effects on the dilaton couplings driven by the β-functions are suppressed by inverse powers of $ f_\chi $ and thus negligible.

We conclude that the theory with spontaneously broken scale invariance reproduces the standard one-loop running of couplings in the broken phase, with the conventional renormalization scale dependence $ \ln\mu $ being replaced by $ \ln (z f_\chi) $. This equivalence not only demonstrates the quantum consistency of the formalism, but also highlights its utility as a powerful top-down approach for constructing dilaton EFTs where scale symmetry is broken spontaneously.

Having established the framework for the dilaton effective theory, we now apply it to the SM to derive the low-energy dilaton-mediated interactions. The spontaneous breaking of scale symmetry occurs at the new physics (NP) scale Λ, determined by the dilaton VEV $ f_\chi $. All new degrees of freedom responsible for this breaking are assumed to be significantly heavier than Λ and can be safely integrated out.

Since Λ is taken to lie well above the electroweak scale $ v\simeq 246 $ GeV, the theory just below Λ consists of the SM fields plus the dilaton, described by an effective Lagrangian that respects both $S U(3)_c\times S U(2)_L\times U(1)_Y$ and approximate scale invariance. After scale symmetry breaking, the interactions among the SM fields themselves are encoded within the SMEFT framework, while the universal dilaton couplings to the SM trace anomaly provide the leading dilaton-mediated interactions.

To analyze dilaton phenomenology at colliders and in low-energy measurements, we derive the effective Lagrangian at different energy scales. This is achieved by renormalization group evolving the full $S U(3)_c\times S U(2)_L\times U(1)_Y$-invariant Lagrangian (including the dilaton) down to the electroweak scale and subsequent matching onto an $S U(3)_c\times U(1)_{\rm{em}}$-invariant EFT after electroweak symmetry breaking. In the following, we adopt the manifestly scale-invariant regularization framework, which allows a systematic treatment of renormalization group effects in the dilaton couplings.

Having established the framework for the dilaton effective theory, we now apply it to the SM to derive the low-energy dilaton-mediated interactions. The spontaneous breaking of scale symmetry occurs at the new physics (NP) scale Λ, determined by the dilaton VEV $ f_\chi $. All new degrees of freedom responsible for this breaking are assumed to be significantly heavier than Λ and can be safely integrated out.

Since Λ is taken to lie well above the electroweak scale $ v\simeq 246 $ GeV, the theory just below Λ consists of the SM fields plus the dilaton, described by an effective Lagrangian that respects both $S U(3)_c\times S U(2)_L\times U(1)_Y$ and approximate scale invariance. After scale symmetry breaking, the interactions among the SM fields themselves are encoded within the SMEFT framework, while the universal dilaton couplings to the SM trace anomaly provide the leading dilaton-mediated interactions.

To analyze dilaton phenomenology at colliders and in low-energy measurements, we derive the effective Lagrangian at different energy scales. This is achieved by renormalization group evolving the full $S U(3)_c\times S U(2)_L\times U(1)_Y$-invariant Lagrangian (including the dilaton) down to the electroweak scale and subsequent matching onto an $S U(3)_c\times U(1)_{\rm{em}}$-invariant EFT after electroweak symmetry breaking. In the following, we adopt the manifestly scale-invariant regularization framework, which allows a systematic treatment of renormalization group effects in the dilaton couplings.

Having established the framework for the dilaton effective theory, we now apply it to the SM to derive the low-energy dilaton-mediated interactions. The spontaneous breaking of scale symmetry occurs at the new physics (NP) scale Λ, determined by the dilaton VEV $ f_\chi $. All new degrees of freedom responsible for this breaking are assumed to be significantly heavier than Λ and can be safely integrated out.

Since Λ is taken to lie well above the electroweak scale $ v\simeq 246 $ GeV, the theory just below Λ consists of the SM fields plus the dilaton, described by an effective Lagrangian that respects both $S U(3)_c\times S U(2)_L\times U(1)_Y$ and approximate scale invariance. After scale symmetry breaking, the interactions among the SM fields themselves are encoded within the SMEFT framework, while the universal dilaton couplings to the SM trace anomaly provide the leading dilaton-mediated interactions.

To analyze dilaton phenomenology at colliders and in low-energy measurements, we derive the effective Lagrangian at different energy scales. This is achieved by renormalization group evolving the full $S U(3)_c\times S U(2)_L\times U(1)_Y$-invariant Lagrangian (including the dilaton) down to the electroweak scale and subsequent matching onto an $S U(3)_c\times U(1)_{\rm{em}}$-invariant EFT after electroweak symmetry breaking. In the following, we adopt the manifestly scale-invariant regularization framework, which allows a systematic treatment of renormalization group effects in the dilaton couplings.

The SM Lagrangian is classically scale-invariant, with the exception of the negative quadratic term in the Higgs potential which explicitly breaks the symmetry. To restore scale invariance, the dilaton field Φ is introduced:

where the most general, scale-invariant potential involving the Higgs doublet and the scalar field is given by [56]

The terms proportional to $ \lambda_\phi $ and $ \lambda_\chi $ describe the self-interactions of the Higgs and the dilaton respectively, while the term proportional to $ \lambda_m $ acts as the portal interaction responsible for generating the Higgs mass upon spontaneous breaking of scale symmetry. To obtain a charge-neutral VEV, we consider the potential involving the neutral components

Spontaneous symmetry breaking requires $ V_0 $ to develop a non-trivial vacuum with nonzero expectation values $ v\equiv\langle\phi\rangle $ and $ f_\chi\equiv\langle\Phi\rangle $, where $ v\simeq246 $ GeV is fixed by electroweak phenomenology. The minimization conditions

lead to the relations

These conditions guarantee the presence of a classically flat direction in the field space. Crucially, manifestly scale-invariant regularization schemes preserve this flatness at the quantum level, preventing the standard Coleman-Weinberg mechanism [57] from triggering scale symmetry breaking. Instead, spontaneous symmetry breaking may occur through alternative means such as inertial breaking [58], where the vacuum is selected along the flat direction by external factors such as cosmological initial conditions or non-minimal couplings to gravity. Finally, vacuum stability requires $ \lambda_\phi>0 $ and $ \lambda_\chi>0 $, while the phenomenological hierarchy $ v\ll f_\chi $ implies the coupling relation $ \lambda_\chi\ll-3\lambda_m\ll\lambda_\phi $.

The SM Lagrangian is classically scale-invariant, with the exception of the negative quadratic term in the Higgs potential which explicitly breaks the symmetry. To restore scale invariance, the dilaton field Φ is introduced:

where the most general, scale-invariant potential involving the Higgs doublet and the scalar field is given by [56]

The terms proportional to $ \lambda_\phi $ and $ \lambda_\chi $ describe the self-interactions of the Higgs and the dilaton respectively, while the term proportional to $ \lambda_m $ acts as the portal interaction responsible for generating the Higgs mass upon spontaneous breaking of scale symmetry. To obtain a charge-neutral VEV, we consider the potential involving the neutral components

Spontaneous symmetry breaking requires $ V_0 $ to develop a non-trivial vacuum with nonzero expectation values $ v\equiv\langle\phi\rangle $ and $ f_\chi\equiv\langle\Phi\rangle $, where $ v\simeq246 $ GeV is fixed by electroweak phenomenology. The minimization conditions

lead to the relations

These conditions guarantee the presence of a classically flat direction in the field space. Crucially, manifestly scale-invariant regularization schemes preserve this flatness at the quantum level, preventing the standard Coleman-Weinberg mechanism [57] from triggering scale symmetry breaking. Instead, spontaneous symmetry breaking may occur through alternative means such as inertial breaking [58], where the vacuum is selected along the flat direction by external factors such as cosmological initial conditions or non-minimal couplings to gravity. Finally, vacuum stability requires $ \lambda_\phi>0 $ and $ \lambda_\chi>0 $, while the phenomenological hierarchy $ v\ll f_\chi $ implies the coupling relation $ \lambda_\chi\ll-3\lambda_m\ll\lambda_\phi $.

The SM Lagrangian is classically scale-invariant, with the exception of the negative quadratic term in the Higgs potential which explicitly breaks the symmetry. To restore scale invariance, the dilaton field Φ is introduced:

where the most general, scale-invariant potential involving the Higgs doublet and the scalar field is given by [56]

The terms proportional to $ \lambda_\phi $ and $ \lambda_\chi $ describe the self-interactions of the Higgs and the dilaton respectively, while the term proportional to $ \lambda_m $ acts as the portal interaction responsible for generating the Higgs mass upon spontaneous breaking of scale symmetry. To obtain a charge-neutral VEV, we consider the potential involving the neutral components

Spontaneous symmetry breaking requires $ V_0 $ to develop a non-trivial vacuum with nonzero expectation values $ v\equiv\langle\phi\rangle $ and $ f_\chi\equiv\langle\Phi\rangle $, where $ v\simeq246 $ GeV is fixed by electroweak phenomenology. The minimization conditions

lead to the relations

These conditions guarantee the presence of a classically flat direction in the field space. Crucially, manifestly scale-invariant regularization schemes preserve this flatness at the quantum level, preventing the standard Coleman-Weinberg mechanism [57] from triggering scale symmetry breaking. Instead, spontaneous symmetry breaking may occur through alternative means such as inertial breaking [58], where the vacuum is selected along the flat direction by external factors such as cosmological initial conditions or non-minimal couplings to gravity. Finally, vacuum stability requires $ \lambda_\phi>0 $ and $ \lambda_\chi>0 $, while the phenomenological hierarchy $ v\ll f_\chi $ implies the coupling relation $ \lambda_\chi\ll-3\lambda_m\ll\lambda_\phi $.

Following the scale-invariant regularization framework from the previous section, we analytically continue to $ d=4-2\epsilon $ dimensions and expand the dilaton field Φ around its VEV after spontaneous scale symmetry breaking. The resulting effective Lagrangian is given by

where $ {\cal{L}}_{\rm{SM}} $ corresponds to the full SM Lagrangian, with the Higgs quadratic term generated proportionally to the VEV $ f_\chi $ and $ {\cal{L}}_{\chi,{\rm{SM}}} $ describes the interaction between the dilaton χ and the SM fields. The dilaton couplings are determined by the trace anomaly and are thus proportional to the beta functions, which coincide with those of the conventional SM at one-loop level, as required by our formalism.

Using the known renormalization-group evolution of the couplings, we evolve the effective Lagrangian down to the weak scale $ \mu_w $, where electroweak symmetry is spontaneously broken. In the broken phase, it is appropriate to express the dilaton interactions in the mass-eigenstate basis of the physical fields.

For the gauge boson sector, this requires rotating to the physical $ W^\pm,Z $ and photon fields F. For the fermions, it involves unitary transformations that diagonalize the Yukawa matrices, e.g., $ U_u^{\dagger} Y_u R_{u} = Y_u^{{\rm{diag}}} $. In the scalar sector, h and χ mix to form mass eigenstates $ \tilde{h} $ and $ \tilde{\chi} $,

where the mixing angle α diagonalizes the scalar mass-squared matrix and satisfies $ \tan^2\alpha=v^2/f_\chi^2 $. For notational convenience, we henceforth drop the tildes and redefine h and χ to denote their respective physical mass eigenstates. We identify h as the SM Higgs boson observed at the LHC with a mass $ m_h=125 $ GeV, and χ as the massless pseudo-Goldstone dilaton. Therefore, in the physical mass-eigenstate basis, the dilaton χ couples to the SM fields both via the trace anomaly and through the scalar mixing with the Higgs. The complete dilaton-mediated interactions incorporating both effects are given by

where $ m_W=g_2v/2 $, $ m_Z=\sqrt{g_2^2+g_1^2}v/2 $ and $ m_\psi=Y_\psi^{\rm{diag}}v/\sqrt{2} $ are the conventional SM masses. Crucially, the dilaton-Higgs mixing is responsible for the terms explicitly proportional to the standard mass scales: the $ m_W^2 $ and $ m_Z^2 $ terms for the vector bosons, and the leading unit contribution ($ 1\cdot m_\psi $) to the fermion couplings. The remaining terms, specifically the fermion contribution proportional to the anomalous dimension $ \gamma_{m_\psi} $ and the field strength tensor interactions, originate from the trace anomaly. The coefficients $ c_{ij} $ are determined by the beta functions and the weak mixing angle $ \theta_w $,

with $ s_w=\sin\theta_w $ and $ c_w=\cos\theta_w $. The effective Lagrangian $ {\cal{L}}_\chi $ at the weak scale, with all parameters defined at $ \mu\gtrsim \mu_w $, provides the foundation for our subsequent phenomenological analysis.

The above discussion assumes exact scale invariance. Through the scale-invariant regularization scheme, the dilaton remains exactly massless even at the quantum level after spontaneous scale symmetry breaking [56, 59, 60]. When explicit scale-symmetry breaking effects are present, the dilaton acquires a mass accordingly. A general analysis can be carried out by introducing a nearly marginal operator of scaling dimension $ d_\mathcal{O} $ to the initial scale-invariant theory [36]

The renormalization group evolution of the coupling $ \lambda_{\cal{O}} $ determines the scaling of this operator. After spontaneous scale symmetry breaking, the resulting dilaton potential leads to a mass $ m_\chi\propto\sqrt{4-d_\mathcal{O} } f_\chi $. Corrections to the effective Lagrangian from a non-zero dilaton mass are of order $ {\cal{O}}(m_\chi^2/f_\chi^2) $ and are neglected in the following analysis.

Following the scale-invariant regularization framework from the previous section, we analytically continue to $ d=4-2\epsilon $ dimensions and expand the dilaton field Φ around its VEV after spontaneous scale symmetry breaking. The resulting effective Lagrangian is given by

where $ {\cal{L}}_{\rm{SM}} $ corresponds to the full SM Lagrangian, with the Higgs quadratic term generated proportionally to the VEV $ f_\chi $ and $ {\cal{L}}_{\chi,{\rm{SM}}} $ describes the interaction between the dilaton χ and the SM fields. The dilaton couplings are determined by the trace anomaly and are thus proportional to the beta functions, which coincide with those of the conventional SM at one-loop level, as required by our formalism.

Using the known renormalization-group evolution of the couplings, we evolve the effective Lagrangian down to the weak scale $ \mu_w $, where electroweak symmetry is spontaneously broken. In the broken phase, it is appropriate to express the dilaton interactions in the mass-eigenstate basis of the physical fields.

For the gauge boson sector, this requires rotating to the physical $ W^\pm,Z $ and photon fields F. For the fermions, it involves unitary transformations that diagonalize the Yukawa matrices, e.g., $ U_u^{\dagger} Y_u R_{u} = Y_u^{{\rm{diag}}} $. In the scalar sector, h and χ mix to form mass eigenstates $ \tilde{h} $ and $ \tilde{\chi} $,

where the mixing angle α diagonalizes the scalar mass-squared matrix and satisfies $ \tan^2\alpha=v^2/f_\chi^2 $. For notational convenience, we henceforth drop the tildes and redefine h and χ to denote their respective physical mass eigenstates. We identify h as the SM Higgs boson observed at the LHC with a mass $ m_h=125 $ GeV, and χ as the massless pseudo-Goldstone dilaton. Therefore, in the physical mass-eigenstate basis, the dilaton χ couples to the SM fields both via the trace anomaly and through the scalar mixing with the Higgs. The complete dilaton-mediated interactions incorporating both effects are given by

where $ m_W=g_2v/2 $, $ m_Z=\sqrt{g_2^2+g_1^2}v/2 $ and $ m_\psi=Y_\psi^{\rm{diag}}v/\sqrt{2} $ are the conventional SM masses. Crucially, the dilaton-Higgs mixing is responsible for the terms explicitly proportional to the standard mass scales: the $ m_W^2 $ and $ m_Z^2 $ terms for the vector bosons, and the leading unit contribution ($ 1\cdot m_\psi $) to the fermion couplings. The remaining terms, specifically the fermion contribution proportional to the anomalous dimension $ \gamma_{m_\psi} $ and the field strength tensor interactions, originate from the trace anomaly. The coefficients $ c_{ij} $ are determined by the beta functions and the weak mixing angle $ \theta_w $,

with $ s_w=\sin\theta_w $ and $ c_w=\cos\theta_w $. The effective Lagrangian $ {\cal{L}}_\chi $ at the weak scale, with all parameters defined at $ \mu\gtrsim \mu_w $, provides the foundation for our subsequent phenomenological analysis.

The above discussion assumes exact scale invariance. Through the scale-invariant regularization scheme, the dilaton remains exactly massless even at the quantum level after spontaneous scale symmetry breaking [56, 59, 60]. When explicit scale-symmetry breaking effects are present, the dilaton acquires a mass accordingly. A general analysis can be carried out by introducing a nearly marginal operator of scaling dimension $ d_\mathcal{O} $ to the initial scale-invariant theory [36]

The renormalization group evolution of the coupling $ \lambda_{\cal{O}} $ determines the scaling of this operator. After spontaneous scale symmetry breaking, the resulting dilaton potential leads to a mass $ m_\chi\propto\sqrt{4-d_\mathcal{O} } f_\chi $. Corrections to the effective Lagrangian from a non-zero dilaton mass are of order $ {\cal{O}}(m_\chi^2/f_\chi^2) $ and are neglected in the following analysis.

Following the scale-invariant regularization framework from the previous section, we analytically continue to $ d=4-2\epsilon $ dimensions and expand the dilaton field Φ around its VEV after spontaneous scale symmetry breaking. The resulting effective Lagrangian is given by

where $ {\cal{L}}_{\rm{SM}} $ corresponds to the full SM Lagrangian, with the Higgs quadratic term generated proportionally to the VEV $ f_\chi $ and $ {\cal{L}}_{\chi,{\rm{SM}}} $ describes the interaction between the dilaton χ and the SM fields. The dilaton couplings are determined by the trace anomaly and are thus proportional to the beta functions, which coincide with those of the conventional SM at one-loop level, as required by our formalism.

Using the known renormalization-group evolution of the couplings, we evolve the effective Lagrangian down to the weak scale $ \mu_w $, where electroweak symmetry is spontaneously broken. In the broken phase, it is appropriate to express the dilaton interactions in the mass-eigenstate basis of the physical fields.

For the gauge boson sector, this requires rotating to the physical $ W^\pm,Z $ and photon fields F. For the fermions, it involves unitary transformations that diagonalize the Yukawa matrices, e.g., $ U_u^{\dagger} Y_u R_{u} = Y_u^{{\rm{diag}}} $. In the scalar sector, h and χ mix to form mass eigenstates $ \tilde{h} $ and $ \tilde{\chi} $,

where the mixing angle α diagonalizes the scalar mass-squared matrix and satisfies $ \tan^2\alpha=v^2/f_\chi^2 $. For notational convenience, we henceforth drop the tildes and redefine h and χ to denote their respective physical mass eigenstates. We identify h as the SM Higgs boson observed at the LHC with a mass $ m_h=125 $ GeV, and χ as the massless pseudo-Goldstone dilaton. Therefore, in the physical mass-eigenstate basis, the dilaton χ couples to the SM fields both via the trace anomaly and through the scalar mixing with the Higgs. The complete dilaton-mediated interactions incorporating both effects are given by

where $ m_W=g_2v/2 $, $ m_Z=\sqrt{g_2^2+g_1^2}v/2 $ and $ m_\psi=Y_\psi^{\rm{diag}}v/\sqrt{2} $ are the conventional SM masses. Crucially, the dilaton-Higgs mixing is responsible for the terms explicitly proportional to the standard mass scales: the $ m_W^2 $ and $ m_Z^2 $ terms for the vector bosons, and the leading unit contribution ($ 1\cdot m_\psi $) to the fermion couplings. The remaining terms, specifically the fermion contribution proportional to the anomalous dimension $ \gamma_{m_\psi} $ and the field strength tensor interactions, originate from the trace anomaly. The coefficients $ c_{ij} $ are determined by the beta functions and the weak mixing angle $ \theta_w $,

with $ s_w=\sin\theta_w $ and $ c_w=\cos\theta_w $. The effective Lagrangian $ {\cal{L}}_\chi $ at the weak scale, with all parameters defined at $ \mu\gtrsim \mu_w $, provides the foundation for our subsequent phenomenological analysis.

The above discussion assumes exact scale invariance. Through the scale-invariant regularization scheme, the dilaton remains exactly massless even at the quantum level after spontaneous scale symmetry breaking [56, 59, 60]. When explicit scale-symmetry breaking effects are present, the dilaton acquires a mass accordingly. A general analysis can be carried out by introducing a nearly marginal operator of scaling dimension $ d_\mathcal{O} $ to the initial scale-invariant theory [36]

The renormalization group evolution of the coupling $ \lambda_{\cal{O}} $ determines the scaling of this operator. After spontaneous scale symmetry breaking, the resulting dilaton potential leads to a mass $ m_\chi\propto\sqrt{4-d_\mathcal{O} } f_\chi $. Corrections to the effective Lagrangian from a non-zero dilaton mass are of order $ {\cal{O}}(m_\chi^2/f_\chi^2) $ and are neglected in the following analysis.

It is straightforward to extend the scale-invariant framework beyond the renormalizable terms of the SM to include higher-dimensional operators. A natural illustration can be provided by the dimension-$ 5 $ Weinberg operator $\dfrac{c^{(5)}}{\Lambda}L^T\tilde{H}^*C^{\dagger}\tilde{H}^{\dagger} L$ [61, 62], which generates Majorana neutrino masses.

To construct its scale-invariant counterpart, we consider the Type-I seesaw mechanism as a representative UV completion. For simplicity, we consider a single generation of leptons and the right-handed neutrino. In $ d=4-2\epsilon $ dimensions, the relevant scale-invariant mass terms are

where $ Y_\alpha $ are the Dirac-neutrino Yukawa couplings of the SM Higgs doublet H to the lepton doublet $ L_\alpha $, and y is the Majorana-neutrino Yukawa coupling of the dilaton field Φ to the right-handed neutrino N. After spontaneous scale symmetry breaking, $ \langle\Phi\rangle=f_\chi $, generating a heavy Majorana mass $ M_N=yf_\chi $. Integrating out N at energies $ \ll M_N $, substituting the classical equation of motion $ N\approx -(Y/y\Phi)C(\bar{L}\tilde{H})^T $ back into the Lagrangian directly produces

where the first term matches the standard Weinberg operator, identifying $ \Lambda\sim f_\chi $, and the second describes dilaton-neutrino interactions.

This result from the seesaw model is not a coincidence but reveals a powerful and general principle for constructing scale-invariant effective theories. In the absence of fundamental mass scales, all dimensional parameters arise dynamically from the dilaton VEV after spontaneous scale symmetry breaking. Therefore, any higher-dimensional operator in the SMEFT, conventionally written as $ {\cal{O}}^{(d)}/\Lambda^{d-4} $, acquires a scale-invariant form

where $ c_i $ are the respective coupling constants and every power of the suppression scale Λ in the denominator has been replaced by the same power of the dilaton field Φ.

It is straightforward to extend the scale-invariant framework beyond the renormalizable terms of the SM to include higher-dimensional operators. A natural illustration can be provided by the dimension-$ 5 $ Weinberg operator $\dfrac{c^{(5)}}{\Lambda}L^T\tilde{H}^*C^{\dagger}\tilde{H}^{\dagger} L$ [61, 62], which generates Majorana neutrino masses.

To construct its scale-invariant counterpart, we consider the Type-I seesaw mechanism as a representative UV completion. For simplicity, we consider a single generation of leptons and the right-handed neutrino. In $ d=4-2\epsilon $ dimensions, the relevant scale-invariant mass terms are

where $ Y_\alpha $ are the Dirac-neutrino Yukawa couplings of the SM Higgs doublet H to the lepton doublet $ L_\alpha $, and y is the Majorana-neutrino Yukawa coupling of the dilaton field Φ to the right-handed neutrino N. After spontaneous scale symmetry breaking, $ \langle\Phi\rangle=f_\chi $, generating a heavy Majorana mass $ M_N=yf_\chi $. Integrating out N at energies $ \ll M_N $, substituting the classical equation of motion $ N\approx -(Y/y\Phi)C(\bar{L}\tilde{H})^T $ back into the Lagrangian directly produces

where the first term matches the standard Weinberg operator, identifying $ \Lambda\sim f_\chi $, and the second describes dilaton-neutrino interactions.

This result from the seesaw model is not a coincidence but reveals a powerful and general principle for constructing scale-invariant effective theories. In the absence of fundamental mass scales, all dimensional parameters arise dynamically from the dilaton VEV after spontaneous scale symmetry breaking. Therefore, any higher-dimensional operator in the SMEFT, conventionally written as $ {\cal{O}}^{(d)}/\Lambda^{d-4} $, acquires a scale-invariant form

where $ c_i $ are the respective coupling constants and every power of the suppression scale Λ in the denominator has been replaced by the same power of the dilaton field Φ.

It is straightforward to extend the scale-invariant framework beyond the renormalizable terms of the SM to include higher-dimensional operators. A natural illustration can be provided by the dimension-$ 5 $ Weinberg operator $\dfrac{c^{(5)}}{\Lambda}L^T\tilde{H}^*C^{\dagger}\tilde{H}^{\dagger} L$ [61, 62], which generates Majorana neutrino masses.

To construct its scale-invariant counterpart, we consider the Type-I seesaw mechanism as a representative UV completion. For simplicity, we consider a single generation of leptons and the right-handed neutrino. In $ d=4-2\epsilon $ dimensions, the relevant scale-invariant mass terms are

where $ Y_\alpha $ are the Dirac-neutrino Yukawa couplings of the SM Higgs doublet H to the lepton doublet $ L_\alpha $, and y is the Majorana-neutrino Yukawa coupling of the dilaton field Φ to the right-handed neutrino N. After spontaneous scale symmetry breaking, $ \langle\Phi\rangle=f_\chi $, generating a heavy Majorana mass $ M_N=yf_\chi $. Integrating out N at energies $ \ll M_N $, substituting the classical equation of motion $ N\approx -(Y/y\Phi)C(\bar{L}\tilde{H})^T $ back into the Lagrangian directly produces

where the first term matches the standard Weinberg operator, identifying $ \Lambda\sim f_\chi $, and the second describes dilaton-neutrino interactions.

This result from the seesaw model is not a coincidence but reveals a powerful and general principle for constructing scale-invariant effective theories. In the absence of fundamental mass scales, all dimensional parameters arise dynamically from the dilaton VEV after spontaneous scale symmetry breaking. Therefore, any higher-dimensional operator in the SMEFT, conventionally written as $ {\cal{O}}^{(d)}/\Lambda^{d-4} $, acquires a scale-invariant form

where $ c_i $ are the respective coupling constants and every power of the suppression scale Λ in the denominator has been replaced by the same power of the dilaton field Φ.

For a dilaton significantly lighter than the weak scale, it can mediate novel signatures in low-energy processes at typical energies $ E\ll100 $GeV. In this regime, the heavy degrees of freedom, the top quark, the Higgs boson and the weak gauge bosons $ W^\pm $ and Z, are integrated out to obtain the corresponding low-energy effective theory. At tree-level matching, the Lagrangian for the dilaton is given by

where $ \beta_3 $ and $ \beta_e $ are the one-loop QCD and QED beta functions and $ \gamma_{m_\psi} $ are the fermion mass anomalous dimensions. All coefficients, including $ c_{\gamma\gamma} $, are evaluated at the matching scale.

Alternatively, these interactions can be parameterized by the LEFT [63−67] model-independently. In this section, we discuss the LEFT with dilaton and construct the interactions up to dimension 7.

The gauge group of this extended LEFT is the one of the QCD and the QED theories,

which guarantees color conservation and electric charge conservation. The building blocks are the two associated gauge bosons, the gluon G and the photon A, the leptons, the light quarks, and the dilaton. We present the building blocks and their representations in Table 1, where the baryon number symmetry group $ U(1)_B $ and the lepton number symmetry group $ U(1)_L $ are also given. In particular, we use the field strength tensors of the gauge bosons as the building blocks to maintain the gauge invariance

Similar to Eq. (41), the effective Lagrangian of the LEFT extended by the dilaton should take the form

where $ c_a^{(d)} $ are the dimensionless Wilson coefficients, and $ {\cal{O}}^{(d)}_a $ are the independent operators of d-dimension without the dilaton. According to the expansion form of the dilaton

the Lagrangian can be divided into two sectors

where $ {\cal{L}}_{{\rm{LEFT}},0} $ is the LEFT Lagrangian without the dilaton

while the Lagrangian $ {\cal{L}}_{{\rm{LEFT}},\chi\,} $ contains the operators with the dilaton, which can be expressed as an expansion in terms of the fluctuation field χ. Keeping the operators linear in the χ, the Lagrangian $ {\cal{L}}_{{\rm{LEFT}},\chi\,} $ takes the form

which implies for every specific dimension d, the fluctuation field χ interacts with the other fields in the LEFT in a universal way.

Accordingly, we can list the effective operators in $ {\cal{L}}_\chi $ linear in χ. Next, we present these operators up to dimension 7.

Dimension-4, 5, 6

Dimension-7

For a dilaton significantly lighter than the weak scale, it can mediate novel signatures in low-energy processes at typical energies $ E\ll100 $GeV. In this regime, the heavy degrees of freedom, the top quark, the Higgs boson and the weak gauge bosons $ W^\pm $ and Z, are integrated out to obtain the corresponding low-energy effective theory. At tree-level matching, the Lagrangian for the dilaton is given by

where $ \beta_3 $ and $ \beta_e $ are the one-loop QCD and QED beta functions and $ \gamma_{m_\psi} $ are the fermion mass anomalous dimensions. All coefficients, including $ c_{\gamma\gamma} $, are evaluated at the matching scale.

Alternatively, these interactions can be parameterized by the LEFT [63−67] model-independently. In this section, we discuss the LEFT with dilaton and construct the interactions up to dimension 7.

The gauge group of this extended LEFT is the one of the QCD and the QED theories,

which guarantees color conservation and electric charge conservation. The building blocks are the two associated gauge bosons, the gluon G and the photon A, the leptons, the light quarks, and the dilaton. We present the building blocks and their representations in Table 1, where the baryon number symmetry group $ U(1)_B $ and the lepton number symmetry group $ U(1)_L $ are also given. In particular, we use the field strength tensors of the gauge bosons as the building blocks to maintain the gauge invariance

Similar to Eq. (41), the effective Lagrangian of the LEFT extended by the dilaton should take the form

where $ c_a^{(d)} $ are the dimensionless Wilson coefficients, and $ {\cal{O}}^{(d)}_a $ are the independent operators of d-dimension without the dilaton. According to the expansion form of the dilaton

the Lagrangian can be divided into two sectors

where $ {\cal{L}}_{{\rm{LEFT}},0} $ is the LEFT Lagrangian without the dilaton

while the Lagrangian $ {\cal{L}}_{{\rm{LEFT}},\chi\,} $ contains the operators with the dilaton, which can be expressed as an expansion in terms of the fluctuation field χ. Keeping the operators linear in the χ, the Lagrangian $ {\cal{L}}_{{\rm{LEFT}},\chi\,} $ takes the form

which implies for every specific dimension d, the fluctuation field χ interacts with the other fields in the LEFT in a universal way.

Accordingly, we can list the effective operators in $ {\cal{L}}_\chi $ linear in χ. Next, we present these operators up to dimension 7.

Dimension-4, 5, 6

Dimension-7

For a dilaton significantly lighter than the weak scale, it can mediate novel signatures in low-energy processes at typical energies $ E\ll100 $GeV. In this regime, the heavy degrees of freedom, the top quark, the Higgs boson and the weak gauge bosons $ W^\pm $ and Z, are integrated out to obtain the corresponding low-energy effective theory. At tree-level matching, the Lagrangian for the dilaton is given by

where $ \beta_3 $ and $ \beta_e $ are the one-loop QCD and QED beta functions and $ \gamma_{m_\psi} $ are the fermion mass anomalous dimensions. All coefficients, including $ c_{\gamma\gamma} $, are evaluated at the matching scale.

Alternatively, these interactions can be parameterized by the LEFT [63−67] model-independently. In this section, we discuss the LEFT with dilaton and construct the interactions up to dimension 7.

The gauge group of this extended LEFT is the one of the QCD and the QED theories,

which guarantees color conservation and electric charge conservation. The building blocks are the two associated gauge bosons, the gluon G and the photon A, the leptons, the light quarks, and the dilaton. We present the building blocks and their representations in Table 1, where the baryon number symmetry group $ U(1)_B $ and the lepton number symmetry group $ U(1)_L $ are also given. In particular, we use the field strength tensors of the gauge bosons as the building blocks to maintain the gauge invariance

Similar to Eq. (41), the effective Lagrangian of the LEFT extended by the dilaton should take the form

where $ c_a^{(d)} $ are the dimensionless Wilson coefficients, and $ {\cal{O}}^{(d)}_a $ are the independent operators of d-dimension without the dilaton. According to the expansion form of the dilaton

the Lagrangian can be divided into two sectors

where $ {\cal{L}}_{{\rm{LEFT}},0} $ is the LEFT Lagrangian without the dilaton

while the Lagrangian $ {\cal{L}}_{{\rm{LEFT}},\chi\,} $ contains the operators with the dilaton, which can be expressed as an expansion in terms of the fluctuation field χ. Keeping the operators linear in the χ, the Lagrangian $ {\cal{L}}_{{\rm{LEFT}},\chi\,} $ takes the form

which implies for every specific dimension d, the fluctuation field χ interacts with the other fields in the LEFT in a universal way.

Accordingly, we can list the effective operators in $ {\cal{L}}_\chi $ linear in χ. Next, we present these operators up to dimension 7.

Dimension-4, 5, 6

Dimension-7

Under the chiral scale $ \Lambda_{{\rm{QCD}}}\sim 1 $ GeV, the quarks and gluons are confined in the colorless bound states called hadrons, due to the quark condensate. Consequently, the approximate chiral symmetry $S U(3)_L\times S U(3)_R$ is spontaneously broken to the subgroup $S U(3)_V$. According to the Goldstone theorem [68, 69], the breaking degrees of freedom generate 8 Nambu-Goldstone bosons (NGBs), which compose the light pseudoscalar meson octets. The dilaton is not changed during the confinement of the quarks and gluons since it is free of the strong interaction. Thus, we expect it to participate in the interactions of hadrons in various low-energy processes, and we can obtain a chiral theory with dilatons [28, 29].

Adopting the CCWZ formalism [70−72] developed by Callan, Coleman, Wess, and Zumino, the meson octets can be collected in a unitary matrix

with

where $ \lambda^A $ are Gell-Mann matrices. This matrix is covariant as

where $(V_L\,,V_R)\in S U(3)_L\times S U(3)_R$, and $V\in S U(3)_V$. To construct the most general Lagrangian, we define the meson building blocks to be covariant under $S U(3)_V$ solely

which transforms under $S U(3)_V$ as

The covariant derivative $ D_\mu $ is defined as

where $ \Gamma_\mu $ is the chiral connection

The leading-order (LO) Lagrangian is composed of two $ u_\mu $:

which implies the meson momenta p, or the partial derivative of the mesons, is an expansion quantity of the Lagrangian. Thus, the building block $ u_\mu $ is of order p, and the LO Lagrangian is of order $ p^2 $.

Because the chiral symmetry is approximate, the mesons are not strictly massless. The mass terms are introduced by external sources

where

with M the quark mass matrix, and the subscript $ \pm $ implies their parity. The pure-meson Lagrangian has been constructed up to $ p^8 $ [73−82].

Furthermore, the baryon building blocks could be included by an octet,

which is covariant under $S U(3)_V$, $ B\rightarrow VBV^{\dagger} $. It is crucial that the baryon masses $ m_B $ are comparable to the scale $ \Lambda_{{\rm{QCD}}} $, and do not vanish under the chiral limit, thus, the power-counting scheme of the baryons is different from that of mesons. The momenta of the baryons are not small quantities because of the on-shell condition

thus, the derivatives of the baryons are powerless. One way to manage the powerless derivatives is the heavy baryon projection (HBP) formalism [83−86]. The HBP formalism adopts a time-like vector $ v^\mu $ to project out the large component of the momenta $ (v\cdot p)v^\mu $ and the small component $ v^\mu_T=v^\mu-(v\cdot p)v^\mu $. Consequently, the baryon B can also be projected into different components

where $ {\cal{B}}=P_v^+B $, and $ {\cal{H}}=P_v^-B $, with the projectors

Preserving the component $ {\cal{B}} $ and integrating out the component $ {\cal{H}} $, we obtain an expansion over a small quantity $ \dfrac{1}{v\cdot p} $, which is just the baryon mass $ m_B $ if we adopt the rest frame and set $ v=(1,0,0,0) $. Consequently, the Lagrangian in the HBP formalism can be regarded as a simultaneous expansion by both $ \Lambda_{{\rm{QCD}}} $ and $ m_B $. Thus, we define the power of an operator with baryons as the power of both the meson momenta p and the mass inverse $ 1/m_B $ in the HBP formalism. The LO meson-baryon Lagrangian is

where the subscript implies it is of power p. The meson-baryon operators up to $ p^5 $ have been constructed [84, 85, 87−93].

In addition to the building blocks above, the dilaton $ \Phi(x) $ is a scalar field transforming trivially under the $S U(3)_V$ symmetry. Since it is a Nambu-Goldstone boson, it can be expressed in the exponential form

similar to the mesons in Eq. (51). Utilizing these building blocks, the effective action involving dilatons should be scale- invariant, which means the effective operators $ {\cal{O}} $ should be scaled as

Because the dilaton transforms under the scale symmetry as

we obtain the LO scale-invariant Lagrangian [94, 95]

where y is related to the mass anomalous dimension $ \gamma_m $ by

The factor ${\rm e}^{\chi/f_\chi}$ is called the dimension compensator, whose powers are determined by the scale transformations of other fields.

Under the limit $ m=m_u=m_d $, the dilaton-meson and dilaton-nucleon interactions can be obtained via the expansion of Eq. (69)

Under the chiral scale $ \Lambda_{{\rm{QCD}}}\sim 1 $ GeV, the quarks and gluons are confined in the colorless bound states called hadrons, due to the quark condensate. Consequently, the approximate chiral symmetry $S U(3)_L\times S U(3)_R$ is spontaneously broken to the subgroup $S U(3)_V$. According to the Goldstone theorem [68, 69], the breaking degrees of freedom generate 8 Nambu-Goldstone bosons (NGBs), which compose the light pseudoscalar meson octets. The dilaton is not changed during the confinement of the quarks and gluons since it is free of the strong interaction. Thus, we expect it to participate in the interactions of hadrons in various low-energy processes, and we can obtain a chiral theory with dilatons [28, 29].

Adopting the CCWZ formalism [70−72] developed by Callan, Coleman, Wess, and Zumino, the meson octets can be collected in a unitary matrix

with

where $ \lambda^A $ are Gell-Mann matrices. This matrix is covariant as

where $(V_L\,,V_R)\in S U(3)_L\times S U(3)_R$, and $V\in S U(3)_V$. To construct the most general Lagrangian, we define the meson building blocks to be covariant under $S U(3)_V$ solely

which transforms under $S U(3)_V$ as

The covariant derivative $ D_\mu $ is defined as

where $ \Gamma_\mu $ is the chiral connection

The leading-order (LO) Lagrangian is composed of two $ u_\mu $:

which implies the meson momenta p, or the partial derivative of the mesons, is an expansion quantity of the Lagrangian. Thus, the building block $ u_\mu $ is of order p, and the LO Lagrangian is of order $ p^2 $.

Because the chiral symmetry is approximate, the mesons are not strictly massless. The mass terms are introduced by external sources

where

with M the quark mass matrix, and the subscript $ \pm $ implies their parity. The pure-meson Lagrangian has been constructed up to $ p^8 $ [73−82].

Furthermore, the baryon building blocks could be included by an octet,

which is covariant under $S U(3)_V$, $ B\rightarrow VBV^{\dagger} $. It is crucial that the baryon masses $ m_B $ are comparable to the scale $ \Lambda_{{\rm{QCD}}} $, and do not vanish under the chiral limit, thus, the power-counting scheme of the baryons is different from that of mesons. The momenta of the baryons are not small quantities because of the on-shell condition

thus, the derivatives of the baryons are powerless. One way to manage the powerless derivatives is the heavy baryon projection (HBP) formalism [83−86]. The HBP formalism adopts a time-like vector $ v^\mu $ to project out the large component of the momenta $ (v\cdot p)v^\mu $ and the small component $ v^\mu_T=v^\mu-(v\cdot p)v^\mu $. Consequently, the baryon B can also be projected into different components

where $ {\cal{B}}=P_v^+B $, and $ {\cal{H}}=P_v^-B $, with the projectors

Preserving the component $ {\cal{B}} $ and integrating out the component $ {\cal{H}} $, we obtain an expansion over a small quantity $ \dfrac{1}{v\cdot p} $, which is just the baryon mass $ m_B $ if we adopt the rest frame and set $ v=(1,0,0,0) $. Consequently, the Lagrangian in the HBP formalism can be regarded as a simultaneous expansion by both $ \Lambda_{{\rm{QCD}}} $ and $ m_B $. Thus, we define the power of an operator with baryons as the power of both the meson momenta p and the mass inverse $ 1/m_B $ in the HBP formalism. The LO meson-baryon Lagrangian is

where the subscript implies it is of power p. The meson-baryon operators up to $ p^5 $ have been constructed [84, 85, 87−93].

In addition to the building blocks above, the dilaton $ \Phi(x) $ is a scalar field transforming trivially under the $S U(3)_V$ symmetry. Since it is a Nambu-Goldstone boson, it can be expressed in the exponential form

similar to the mesons in Eq. (51). Utilizing these building blocks, the effective action involving dilatons should be scale- invariant, which means the effective operators $ {\cal{O}} $ should be scaled as

Because the dilaton transforms under the scale symmetry as

we obtain the LO scale-invariant Lagrangian [94, 95]

where y is related to the mass anomalous dimension $ \gamma_m $ by

The factor ${\rm e}^{\chi/f_\chi}$ is called the dimension compensator, whose powers are determined by the scale transformations of other fields.

Under the limit $ m=m_u=m_d $, the dilaton-meson and dilaton-nucleon interactions can be obtained via the expansion of Eq. (69)

Under the chiral scale $ \Lambda_{{\rm{QCD}}}\sim 1 $ GeV, the quarks and gluons are confined in the colorless bound states called hadrons, due to the quark condensate. Consequently, the approximate chiral symmetry $S U(3)_L\times S U(3)_R$ is spontaneously broken to the subgroup $S U(3)_V$. According to the Goldstone theorem [68, 69], the breaking degrees of freedom generate 8 Nambu-Goldstone bosons (NGBs), which compose the light pseudoscalar meson octets. The dilaton is not changed during the confinement of the quarks and gluons since it is free of the strong interaction. Thus, we expect it to participate in the interactions of hadrons in various low-energy processes, and we can obtain a chiral theory with dilatons [28, 29].

Adopting the CCWZ formalism [70−72] developed by Callan, Coleman, Wess, and Zumino, the meson octets can be collected in a unitary matrix

with

where $ \lambda^A $ are Gell-Mann matrices. This matrix is covariant as

where $(V_L\,,V_R)\in S U(3)_L\times S U(3)_R$, and $V\in S U(3)_V$. To construct the most general Lagrangian, we define the meson building blocks to be covariant under $S U(3)_V$ solely

which transforms under $S U(3)_V$ as

The covariant derivative $ D_\mu $ is defined as

where $ \Gamma_\mu $ is the chiral connection

The leading-order (LO) Lagrangian is composed of two $ u_\mu $:

which implies the meson momenta p, or the partial derivative of the mesons, is an expansion quantity of the Lagrangian. Thus, the building block $ u_\mu $ is of order p, and the LO Lagrangian is of order $ p^2 $.

Because the chiral symmetry is approximate, the mesons are not strictly massless. The mass terms are introduced by external sources

where

with M the quark mass matrix, and the subscript $ \pm $ implies their parity. The pure-meson Lagrangian has been constructed up to $ p^8 $ [73−82].

Furthermore, the baryon building blocks could be included by an octet,

which is covariant under $S U(3)_V$, $ B\rightarrow VBV^{\dagger} $. It is crucial that the baryon masses $ m_B $ are comparable to the scale $ \Lambda_{{\rm{QCD}}} $, and do not vanish under the chiral limit, thus, the power-counting scheme of the baryons is different from that of mesons. The momenta of the baryons are not small quantities because of the on-shell condition

thus, the derivatives of the baryons are powerless. One way to manage the powerless derivatives is the heavy baryon projection (HBP) formalism [83−86]. The HBP formalism adopts a time-like vector $ v^\mu $ to project out the large component of the momenta $ (v\cdot p)v^\mu $ and the small component $ v^\mu_T=v^\mu-(v\cdot p)v^\mu $. Consequently, the baryon B can also be projected into different components

where $ {\cal{B}}=P_v^+B $, and $ {\cal{H}}=P_v^-B $, with the projectors

Preserving the component $ {\cal{B}} $ and integrating out the component $ {\cal{H}} $, we obtain an expansion over a small quantity $ \dfrac{1}{v\cdot p} $, which is just the baryon mass $ m_B $ if we adopt the rest frame and set $ v=(1,0,0,0) $. Consequently, the Lagrangian in the HBP formalism can be regarded as a simultaneous expansion by both $ \Lambda_{{\rm{QCD}}} $ and $ m_B $. Thus, we define the power of an operator with baryons as the power of both the meson momenta p and the mass inverse $ 1/m_B $ in the HBP formalism. The LO meson-baryon Lagrangian is

where the subscript implies it is of power p. The meson-baryon operators up to $ p^5 $ have been constructed [84, 85, 87−93].

In addition to the building blocks above, the dilaton $ \Phi(x) $ is a scalar field transforming trivially under the $S U(3)_V$ symmetry. Since it is a Nambu-Goldstone boson, it can be expressed in the exponential form

similar to the mesons in Eq. (51). Utilizing these building blocks, the effective action involving dilatons should be scale- invariant, which means the effective operators $ {\cal{O}} $ should be scaled as

Because the dilaton transforms under the scale symmetry as

we obtain the LO scale-invariant Lagrangian [94, 95]

where y is related to the mass anomalous dimension $ \gamma_m $ by

The factor ${\rm e}^{\chi/f_\chi}$ is called the dimension compensator, whose powers are determined by the scale transformations of other fields.

Under the limit $ m=m_u=m_d $, the dilaton-meson and dilaton-nucleon interactions can be obtained via the expansion of Eq. (69)

In this subsection, we present the next-to-leading (NLO) pure-meson operators, including the dilatons, which are of $ p^4 $ order. In particular, we use $ \langle \dots \rangle $ to represent the traces in the $S U(3)_V$ space.

$ {\cal{O}}(p^4) $

In this subsection, we present the next-to-leading (NLO) pure-meson operators, including the dilatons, which are of $ p^4 $ order. In particular, we use $ \langle \dots \rangle $ to represent the traces in the $S U(3)_V$ space.

$ {\cal{O}}(p^4) $

In this subsection, we present the next-to-leading (NLO) pure-meson operators, including the dilatons, which are of $ p^4 $ order. In particular, we use $ \langle \dots \rangle $ to represent the traces in the $S U(3)_V$ space.

$ {\cal{O}}(p^4) $

In this subsection, we present the next-to-leading (NLO) meson-baryon operators including the dilatons, which are of $ p^2 $ order.

$ {\cal{O}}(p^2) $

$ B^2 $:

$ B^2u^2 $:

$ B^2u\chi $:

$ B^2\chi^2 $:

In particular, the χPT operators presented here involve only the lightest mesons and baryons. The heavier mesons such as B can be introduced via the external sources.

These hadronic operators can be matched from the quark operators in the LEFT. Because of the confinement, such a matching is non-perturbative. The traditional matching method is to utilize various external sources implementing the chiral-symmetry-breaking effects. However, the external source method becomes involved for high-dimensional operators. An alternative method using spurions rather than the external sources has been developed, under which the matching is organized by the naive dimension analysis and can be extended to the high-dimensional matching systematically [96]. Besides, the non-perturbative effect can be taken into account via quantitative methods such as lattice computation. The systematic matching deserves more investigation in the future.

In this subsection, we present the next-to-leading (NLO) meson-baryon operators including the dilatons, which are of $ p^2 $ order.

$ {\cal{O}}(p^2) $

$ B^2 $:

$ B^2u^2 $:

$ B^2u\chi $:

$ B^2\chi^2 $:

In particular, the χPT operators presented here involve only the lightest mesons and baryons. The heavier mesons such as B can be introduced via the external sources.

These hadronic operators can be matched from the quark operators in the LEFT. Because of the confinement, such a matching is non-perturbative. The traditional matching method is to utilize various external sources implementing the chiral-symmetry-breaking effects. However, the external source method becomes involved for high-dimensional operators. An alternative method using spurions rather than the external sources has been developed, under which the matching is organized by the naive dimension analysis and can be extended to the high-dimensional matching systematically [96]. Besides, the non-perturbative effect can be taken into account via quantitative methods such as lattice computation. The systematic matching deserves more investigation in the future.

In this subsection, we present the next-to-leading (NLO) meson-baryon operators including the dilatons, which are of $ p^2 $ order.

$ {\cal{O}}(p^2) $

$ B^2 $:

$ B^2u^2 $:

$ B^2u\chi $:

$ B^2\chi^2 $:

In particular, the χPT operators presented here involve only the lightest mesons and baryons. The heavier mesons such as B can be introduced via the external sources.

These hadronic operators can be matched from the quark operators in the LEFT. Because of the confinement, such a matching is non-perturbative. The traditional matching method is to utilize various external sources implementing the chiral-symmetry-breaking effects. However, the external source method becomes involved for high-dimensional operators. An alternative method using spurions rather than the external sources has been developed, under which the matching is organized by the naive dimension analysis and can be extended to the high-dimensional matching systematically [96]. Besides, the non-perturbative effect can be taken into account via quantitative methods such as lattice computation. The systematic matching deserves more investigation in the future.

Building upon the EFT framework derived in the previous sections, we observe that the dilaton couples to a broad spectrum of SM fields across a wide range of energy scales, including gauge bosons, heavy quarks, mesons, and nucleons. This diverse interaction structure permits a comprehensive exploration by combining experimental results related to these particles. As mentioned before, while exact scale invariance yields a massless dilaton even at the quantum level, a physical mass $ m_\chi $ is generated via explicit symmetry breaking, typically through small deformations from nearly marginal operators. Throughout the parameter space considered in this section, the ratio $ m_\chi^2/f_\chi^2 $ remains sufficiently small, ensuring that the associated $ {\cal{O}}(m_\chi^2/f_\chi^2) $ corrections are negligible and the EFT description remains valid. In this work, we focus on two typical mass regimes where the dilaton behaves either as a particle-like state or as a coherent wave-like field.

For the particle-like regime, we concentrate on masses around the MeV scale. Motivated by the stringent constraints on ALPs from semi-invisible meson decays [12, 97, 98], we treat the dilaton as an invisible particle and investigate the specific mass window $ m_\chi\in[0.1, 200]\; $MeV. We restrict our analysis to this range for two reasons: for $ m_\chi>2\,m_\mu\simeq 200\; $MeV, the dilaton lifetime decreases significantly due to the opening of the decay channel $ \chi\to\mu^+\mu^- $, and the signal would be reconstructed from visible dimuon final states, which falls outside the scope of this study. For $ m_\chi<0.1\; $MeV, stellar cooling processes (e.g., in the Sun, red giants, white dwarfs) typically provide the dominant constraints [99, 100]. While a systematic analysis similar to those performed for ALPs [101] could be carried out for these mass regions, we focus here on the interplay between terrestrial experiments and supernova limits. Within the chosen mass window, we probe the dilaton via production at the LHC, rare meson decays and cosmological observations, using the dilaton EFTs derived earlier. The relevant Lagrangians governing these processes are given by

where the dilaton VEV satisfies $ f_\chi>1\; $TeV, since the scale symmetry breaking scale is much larger than v, and the coefficients $ c_{ij} $ are given explicitly in Eq. (37). At the LHC, we focus on the channel $ pp\to j\chi $, which has the largest production cross section (see Table 2). For rare meson decays, we study $ B^+\to K^+\chi $ and $ K^+\to\pi^+\chi $, and derive constraints from Belle II and NA62 measurements. Finally, we examine the astrophysical impact of dilaton production on the late-time neutrino signal from SN1987A. The resulting sensitivities for the conventional dilaton are presented in Fig. 4.

Figure 4.  (color online) Constraints on the dilaton VEV $ f_\chi $. The purple shaded region denotes the exclusion from HL-LHC. The region between the two blue dot-dashed lines corresponds to the 90% C.L. allowed region for the dilaton VEV when the excess observed at Belle II [114] is interpreted as a signal of new physics, while blue shaded region denotes the exclusion obtained under the assumption that the excess arises from statistical or systematic fluctuations. The green shaded region shows the exclusion from the NA62 collaboration [119]. The gray dashed contour indicates the approximate exclusion derived from SN1987A [126], where the input parameters used in the estimation are summarized in Table 5.

In contrast, for the wave-like ultralight regime, we focus on the mass range $ m_\chi<10^{-10}\; $eV. Above this threshold, constraints are typically dominated by fifth-force searches [40, 41] sensitive to long-range dilaton exchange, which manifests as violations of the WEP or deviations from the gravitational inverse-square law. In the ultralight regime considered here, the primary observable is the oscillation of the fine-structure constant driven by the dilaton-photon coupling. Constraints on the dilaton VEV, derived from precision measurements of these oscillations using atomic clocks and atom interferometers, are summarized in Fig. 5.

Figure 5.  (color online) Sensitivity projections for the dilaton VEV $ f_\chi $ as a function of dilaton mass $ m_\chi $. Red curves depict the SNR = 1 sensitivity envelopes of the Yb⁺/Al⁺ clock comparison experiment, for $ \tau_{\rm{int}}=10^4,10^8 $ s (deep red, light coral). Also depicted are sensitivities of different AION stages.

Building upon the EFT framework derived in the previous sections, we observe that the dilaton couples to a broad spectrum of SM fields across a wide range of energy scales, including gauge bosons, heavy quarks, mesons, and nucleons. This diverse interaction structure permits a comprehensive exploration by combining experimental results related to these particles. As mentioned before, while exact scale invariance yields a massless dilaton even at the quantum level, a physical mass $ m_\chi $ is generated via explicit symmetry breaking, typically through small deformations from nearly marginal operators. Throughout the parameter space considered in this section, the ratio $ m_\chi^2/f_\chi^2 $ remains sufficiently small, ensuring that the associated $ {\cal{O}}(m_\chi^2/f_\chi^2) $ corrections are negligible and the EFT description remains valid. In this work, we focus on two typical mass regimes where the dilaton behaves either as a particle-like state or as a coherent wave-like field.

For the particle-like regime, we concentrate on masses around the MeV scale. Motivated by the stringent constraints on ALPs from semi-invisible meson decays [12, 97, 98], we treat the dilaton as an invisible particle and investigate the specific mass window $ m_\chi\in[0.1, 200]\; $MeV. We restrict our analysis to this range for two reasons: for $ m_\chi>2\,m_\mu\simeq 200\; $MeV, the dilaton lifetime decreases significantly due to the opening of the decay channel $ \chi\to\mu^+\mu^- $, and the signal would be reconstructed from visible dimuon final states, which falls outside the scope of this study. For $ m_\chi<0.1\; $MeV, stellar cooling processes (e.g., in the Sun, red giants, white dwarfs) typically provide the dominant constraints [99, 100]. While a systematic analysis similar to those performed for ALPs [101] could be carried out for these mass regions, we focus here on the interplay between terrestrial experiments and supernova limits. Within the chosen mass window, we probe the dilaton via production at the LHC, rare meson decays and cosmological observations, using the dilaton EFTs derived earlier. The relevant Lagrangians governing these processes are given by

where the dilaton VEV satisfies $ f_\chi>1\; $TeV, since the scale symmetry breaking scale is much larger than v, and the coefficients $ c_{ij} $ are given explicitly in Eq. (37). At the LHC, we focus on the channel $ pp\to j\chi $, which has the largest production cross section (see Table 2). For rare meson decays, we study $ B^+\to K^+\chi $ and $ K^+\to\pi^+\chi $, and derive constraints from Belle II and NA62 measurements. Finally, we examine the astrophysical impact of dilaton production on the late-time neutrino signal from SN1987A. The resulting sensitivities for the conventional dilaton are presented in Fig. 4.