Renormalisation Group Invariants from Scaling and Non-overlapping Symmetries

Abstract: We show how the synergy of scaling and non-overlapping global symmetries can lead to Renormalisation Group Invariants (RGIs) among the parameters of potentials with multiple scalars. The instrumental role of scale-invariant field directions in the identification and construction of RGIs for bilinear field operators to all loops is demonstrated. We present a few illustrative examples to showcase our constructive spurion-field approach, which is applied to simple non-supersymmetric models as well as to scenarios reported recently in the literature that include the two-Higgs doublet model. The implications of our spurion-field formalism to address the gauge-hierarchy problem are discussed.

• The paper introduces a novel spurion-field approach to combine scaling and non-overlapping symmetries, yielding all-order renormalisation group invariants.
• Analyses in scalar models and the 2HDM framework demonstrate the protection of specific mass and coupling combinations against radiative corrections.
• The methodology offers a viable non-SUSY alternative for electroweak scale stabilization by ensuring RG independence in multi-scalar sectors.

Renormalisation Group Invariants from Scaling and Non-overlapping Symmetries

Overview and Motivation

This paper introduces a robust formalism based on the synergy between scaling and non-overlapping global symmetries for identifying and constructing Renormalisation Group Invariants (RGIs) in scalar theories, specifically focused on models with multiple scalars. The context is primarily the Two-Higgs Doublet Model (2HDM), though the framework is generically extensible to other multi-scalar potentials. The central methodological tool is the spurion-field approach, allowing for the parametric tracking of symmetry transformations even in the absence of exact global invariance. Theoretical implications relate directly to the gauge hierarchy problem, providing alternatives to supersymmetry (SUSY) for scale separation and stabilization at all orders.

Spurion Formalism and Symmetry Classification

The analysis begins with the introduction of the spurion technique, wherein explicit symmetry-breaking parameters are promoted to external fields (spurions) that transform to compensate for the non-invariant terms in the action. This construct allows the restoration of formal invariance under larger groups. Two symmetry classes are central:

• Non-overlapping Symmetries: These are global discrete or continuous groups with minimal or trivial overlap in their generators, e.g., $\mathbb{Z}_6$ and $\mathbb{Z}_4$ with overlap $\mathbb{Z}_2$.
• Scaling Symmetries: Classical scale (dilational) invariance emerges in the absence of mass terms. SI field directions, protected by such symmetries, are employed to find RGIs for dimensionful operators.

The synergy of these symmetries enables the identification of invariant combinations of parameters, with detailed spurion charge assignments tracking transformation properties.

Scalar Models: U(1) Examples and Invariant Identification

Initial illustrative models include complex scalar fields with global U(1) or extended discrete symmetries. For the U(1) model, the scalar potential with two fields is constructed, and the effect of turning off explicit symmetry-breaking parameters is discussed. The PQ-symmetric limit is analyzed, showing that $m^2_{12}$ and $\lambda_{6,7}$ remain radiatively zero, resulting from the synergy between the PQ and CP2 symmetries. The existence of an SI field direction $|\phi_1| = |\phi_2|$ enables the identification of the RGI condition $m_1^2 + m_2^2 = 0$, preserved at all orders.

In the general case with $m_{12}^2$ and $\lambda_{5,6,7}$ nonzero, RG invariance of $m_1^2 + m_2^2$ is protected by combining spurion charge analysis with symmetry arguments, even in the presence of complex or non-vanishing couplings, provided appropriate symmetry constraints are respected.

The notion is extended to scale-invariant rays such as $\mathbb{Z}_4$0 or $\mathbb{Z}_4$1, with RGIs identified for $\mathbb{Z}_4$2 and $\mathbb{Z}_4$3 in the decoupling limit, contingent on the absence of mixing terms. The explicit construction of spurion charges for all relevant operators, including gauge and Yukawa-induced contributions, clarifies which terms can and cannot contribute to RGI-violating corrections.

RGIs in the Two-Higgs Doublet Model

Extending the previous analysis, the 2HDM is scrutinized with general complex parameters for bilinear and quartic terms. The construction of the spurion charge framework allows the tracking of multiple global symmetries, notably U(1)$\mathbb{Z}_4$4, CP2, and CP3. The key results are as follows:

• CP2 Symmetric 2HDM: Along a special SI direction, the condition $\mathbb{Z}_4$5 can be maintained as an RGI to all orders, provided the Yukawa sector is appropriately structured. The combination is protected from radiative corrections by the combined action of scaling and non-overlapping symmetries. Explicit beta function decompositions verify that potentially dangerous terms are charged under symmetries and thus forbidden.
• CP3 Symmetric 2HDM: The joint action of U(1)$\mathbb{Z}_4$6 and CP2 is shown to be equivalent to CP3. RGIs similar to the CP2 case are found, but exact invariance of certain combinations can be violated at two-loop order if U(1)$\mathbb{Z}_4$7 is gauged, due to hypercharge-induced breaking of the necessary symmetry constraints.

The relationship between symmetry assignments and the structure of RGEs is made explicit using spurion charge tables, and selection rules are derived for all higher-order corrections. The approach generalizes directly to extended scalar sectors.

Implications for the Gauge Hierarchy Problem

A significant theoretical application is the stabilization of the electroweak scale in the 2HDM without invoking SUSY. By engineering the scalar potential and the Yukawa sector to align with appropriately chosen scaling and non-overlapping symmetries (specifically, softly broken CP2 plus PQ), the separation and RG independence of all mass scales is achieved. Explicit RGE structures for $\mathbb{Z}_4$8 and $\mathbb{Z}_4$9 are derived, showing that scale mixing is forbidden except through spurion-charged higher-dimensional operators. The possibility of little hierarchy emerges naturalistically from loop suppression in symmetry-violating spurion terms.

This mechanism allows the preservation of a stable low scale despite the presence of high mass thresholds — addressing the gauge hierarchy problem without the need for SUSY non-renormalization theorems. The general methodology is shown to apply to models with larger Higgs sectors, though with increasing complexity in symmetry identification and charge assignments.

Conclusions

The paper establishes a rigorous constructive methodology for identifying and proving the existence of RGIs in multi-scalar theories, with particular application to 2HDM. The approach leverages the interplay of scaling symmetries and non-overlapping global symmetries, formalized with a spurion analysis. Key claims include the all-order protection of specific mass and coupling combinations against RG evolution, contingent on symmetry structure. The formalism directly addresses the practical and theoretical problem of electroweak scale stabilization and provides a novel alternative pathway outside the scope of SUSY. Future research will extend the strategy to more complex Higgs sectors and further explore phenomenological applications and small CP2-violating deformations that preserve desirable RG properties.