GOOFy -- a systematic approach

1. Problem Statement

The paper addresses the search for new, systematic symmetries in quantum field theories (QFT) that lead to stable relations between model parameters under Renormalization Group Equation (RGE) running. Previous work (Ref. [1]) identified a specific symmetry in the Two-Higgs-Doublet Model (2HDM), termed "GOOFy" (or $r_0$), which enforced specific relations among scalar potential parameters (e.g., $m_{11}^2 + m_{22}^2 = 0$, $\lambda_1 = \lambda_2$). These relations were found to be fixed points of the RGE, remaining stable up to all loop orders in perturbation theory.

However, the original formulation was limited to the scalar sector of the 2HDM and relied on a somewhat "non-orthodox" transformation that mixed real and imaginary fields. The open questions were:

1. Can this symmetry be formalized and generalized to arbitrary scalar and fermionic field theories?
2. How does this symmetry constrain Yukawa couplings and fermionic mass terms?
3. Is the Standard Model (SM) compatible with this symmetry?
4. Do the resulting constraints in extended models (like the 2HDM) remain RGE-stable when fermionic sectors are included?

2. Methodology

The authors developed a systematic framework for constructing "GOOFy" transformations based on the following core principles:

• Coordinate Rescaling: The fundamental definition of a GOOFy transformation includes an imaginary rescaling of spacetime coordinates: $x^\mu \to i x^\mu$.
• Non-Consistent Field Transformations: Unlike standard symmetries where the Hermitian conjugate of a transformed field equals the transformation of the conjugate field ($( \Phi^\dagger )' = ( \Phi' )^\dagger$), GOOFy transformations allow for independent transformations of fields and their conjugates.
  • For scalars: $\Phi \to X_\phi \Phi^*$ and $\Phi^\dagger \to -\Phi^T X_\phi^\dagger$.
  • For fermions: $\Psi \to -X_\psi \gamma^0 C \Psi^*$ and $\bar{\Psi} \to -\Psi^T C^{-1} i X_\psi^\dagger$.
• Kinetic Term Invariance: The transformation matrices ($X_\phi, X_\psi$) are determined by requiring the kinetic terms ($\mathcal{L}_{kin}$) to remain invariant under the combined coordinate and field transformations. This fixes the relationship between the transformation of a field and its conjugate (e.g., requiring $\bar{X} = -X$ for scalars).
• Constraint Derivation: By demanding the invariance of mass terms (scalar and fermionic) and Yukawa interactions under these specific transformations, the authors derive algebraic constraints on the model parameters (mass matrices, coupling constants).
• RGE Stability Verification: The derived constraints are tested for stability under RGE running using the PyR@TE3 software package. The authors verify if the beta functions of the constrained parameters vanish (or satisfy the fixed-point condition) up to two-loop and, in specific cases, three-loop orders.

3. Key Contributions

A. Formalization of GOOFy Symmetry

The paper generalizes the GOOFy concept from a specific 2HDM observation to a universal strategy for constructing field theories. It defines a class of transformations that are "non-consistent" (independent transformation of $\Phi$ and $\Phi^\dagger$) but consistent with kinetic term invariance via coordinate rescaling.

B. Constraints on Mass Terms

• Scalars: The symmetry forbids the standard $\Phi^\dagger M^2 \Phi$ mass term for a single complex scalar ($N_\phi=1$) because it requires $\text{Tr}(M^2)=0$. For $N_\phi \ge 2$, it allows mass matrices with specific structures (e.g., traceless or off-diagonal forms).
• Fermions:
  • Dirac Mass: The analysis proves that Dirac mass terms are strictly forbidden under GOOFy symmetry for any number of fermion families ($N_\psi$). The transformation properties lead to the condition $M_D = 0$.
  • Majorana Mass: Majorana mass terms are allowed, provided the mass matrix satisfies specific symmetry conditions related to the transformation matrix $X_\psi$.

C. Yukawa Coupling Constraints

The authors derived general conditions for Yukawa matrices ($\Gamma$) to be invariant. These conditions link different Yukawa matrices (e.g., $\Gamma_1$ and $\Gamma_2$ in 2HDM) via complex conjugation and unitary transformations.

• The paper explicitly solves these constraints for various combinations of scalar ($N_\phi$) and fermion ($N_\psi$) counts.
• It identifies specific forms of Yukawa matrices (e.g., involving complex phases and specific zero patterns) that satisfy the symmetry.

D. Application to the Standard Model (SM)

The authors tested the SM against GOOFy invariance:

• Failure: The SM is not viable under GOOFy symmetry.
  • The single Higgs doublet cannot support a non-zero mass term (violating the $\text{Tr}(M^2)=0$ requirement for $N_\phi=1$).
  • Even if the scalar sector were ignored, the resulting Yukawa constraints either force the Yukawa matrices to vanish or result in a model with no CP violation and no complex phases, rendering it phenomenologically unacceptable.

E. Application to the Two-Higgs-Doublet Model (2HDM)

The 2HDM ($N_\phi=2$) emerges as a viable candidate for a GOOFy-invariant theory:

• Scalar Sector: New relations between the 2HDM potential parameters are derived. Crucially, the paper identifies a new set of relations distinct from the original $r_0$ symmetry:
  $$m_{11}^2 + m_{22}^2 = 0, \quad m_{12}^2 = 0, \quad \lambda_1 = \lambda_2, \quad \lambda_6 = \lambda_7$$
  (This corresponds to $\Delta\theta = 0$).
• Fermionic Sector: The authors solved for the Yukawa matrices ($\Gamma_{1,2}$) compatible with these scalar constraints. They identified specific solutions (labeled A, D-1, D-2, L, M) that allow for non-zero masses for all quarks, avoiding the massless quark problem found in other solutions.

4. Results

1. RGE Stability: The most significant result is the verification that the derived parameter relations are RGE stable.
   • The constraints on Yukawa couplings and scalar masses are stable up to two-loop order for all tested models.
   • For the specific 2HDM potential relations (specifically the set with $\lambda_6 = \lambda_7$), the stability extends up to three-loop orders (when Yukawa couplings are neglected in the beta function calculation, consistent with previous findings for the original GOOFy symmetry).
2. Fixed Points: These relations constitute fixed points of the RGE flow. If the theory is initialized with these parameter relations at a high scale, they are preserved at lower energy scales.
3. Phenomenological Viability:
   • The SM is ruled out.
   • The 2HDM is viable. The new relations ($\lambda_6 = \lambda_7$) imply a softly broken CP symmetry, distinct from previously studied CP-conserving or CP-violating 2HDMs.
   • The existence of solutions where all quarks have non-zero masses suggests the model can be phenomenologically realistic, though Flavor Changing Neutral Currents (FCNC) constraints require further study.

5. Significance

• Theoretical Framework: The paper provides a rigorous, systematic method for generating new symmetries in QFT that go beyond standard unitary or discrete symmetries by utilizing non-consistent field transformations and imaginary coordinate scaling.
• Predictive Power: By identifying fixed points in the RGE flow, the theory reduces the number of free parameters in BSM models. For the 2HDM, it predicts specific relations between quartic couplings and mass parameters that must hold if the theory possesses this symmetry.
• RGE Stability: The demonstration of stability up to three loops is a strong theoretical argument for the naturalness of these relations, suggesting they are not accidental but fundamental features of the theory's structure.
• New Physics Directions: The work opens a new avenue for constructing BSM models (specifically 2HDMs) that are protected by a novel symmetry, potentially offering solutions to hierarchy problems or flavor puzzles while maintaining consistency with high-energy running.

In conclusion, the paper successfully generalizes the GOOFy symmetry, proves its incompatibility with the Standard Model, and establishes the Two-Higgs-Doublet Model as a robust, RGE-stable framework for physics beyond the Standard Model, characterized by a unique set of parameter relations.