Imaginary scaling invariance of the one-loop effective potential

This paper investigates the one-loop effective potential of the r0 (or "GOOFy") symmetric two-Higgs Doublet Model and a minimal symmetric model, concluding that the symmetry remains valid at the one-loop level provided the UV cutoff squared transforms non-trivially under r0 and the minimal model includes two real fields.

The Gist

Imagine the universe as a giant, complex machine built from invisible building blocks called "fields." Physicists use mathematical recipes, called potentials, to describe how these blocks interact and what rules they follow. Usually, these recipes have strict symmetries—like a snowflake that looks the same if you rotate it. If you change the recipe slightly, the symmetry breaks, and the machine behaves differently.

This paper introduces a very strange, new kind of symmetry called "GOOFy" (or r0). The name comes from the first letters of the authors' last names, but the concept is anything but silly. It's a "bizarre" symmetry that physicists had never seen before.

Here is the breakdown of what the paper claims, using simple analogies:

1. The "Magic Mirror" Transformation

In normal physics, if you want to flip a sign (like turning a positive number into a negative one), you just multiply by -1. But in this specific model, the authors found a way to flip the sign of a fundamental quantity called $r_0$ (which represents the "size" or "energy" of the fields) without breaking the laws of physics.

To do this, they have to perform a "magic trick" on two things at once:

• The Fields: They turn real, physical fields into "imaginary" ones (a mathematical concept where numbers involve the square root of -1).
• The Space and Time: They also turn the coordinates of space and time into imaginary numbers.

The Analogy: Imagine you are looking at a reflection in a mirror. Usually, a mirror flips left and right. But in this "GOOFy" world, the mirror doesn't just flip left and right; it turns the entire room into a ghostly, translucent version of itself, and you (the observer) also turn into a ghost. Surprisingly, even though everything looks "ghostly" and imaginary, the rules of the game (the physics) remain exactly the same.

2. Why This Matters: The "Fixed Points"

The authors discovered that if you apply this strange "ghostly" transformation, certain relationships between the numbers in the recipe (the parameters) become locked in place.

The Analogy: Think of a recipe for a cake. Usually, you can change the amount of sugar or flour, and the cake will still bake, just tasting different. But with this new symmetry, it's as if the universe has a rule that says: "If you have 2 cups of flour, you must have exactly 1 cup of sugar, no matter what."

These locked relationships are special because they are stable. Even if you look at the recipe through a microscope (quantum loops) or zoom out to a telescope (high energy), these relationships don't break. They are "renormalization group stable," meaning they survive all the messy calculations physicists usually have to do to make sense of the quantum world.

3. The One-Loop Test: Does the Ghost Hold Up?

The main goal of this paper was to check if this symmetry works not just at the "tree level" (the basic, simple version of the theory) but also at the "one-loop level" (a more complex version that includes quantum fluctuations, like tiny ripples in a pond).

The authors tested this using two models:

1. The 2HDM (Two-Higgs-Doublet Model): A complex extension of the Standard Model of particle physics.
2. The Toy Model (2RSM): A simplified version with just two real fields, used to prove the math works in a smaller sandbox.

The Result: They found that the symmetry does hold up. However, there is a catch. For the math to work out perfectly, the "cutoff" (a limit the physicists use to stop their calculations from blowing up to infinity) must also turn into a negative number when the fields turn imaginary.

The Analogy: Imagine you are balancing a scale. You put a heavy weight on one side (the fields). To keep it balanced, you have to move the fulcrum (the cutoff) to a strange, negative position. If you don't move the fulcrum, the scale tips. But if you move the fulcrum exactly as the "GOOFy" rules demand, the scale stays perfectly balanced, even in the quantum world.

4. The Conclusion

The paper concludes that this "GOOFy" symmetry is real and robust.

• It creates new, stable rules for how particles interact.
• It forces certain masses and forces to be equal or related in specific ways.
• It requires a very unusual transformation where space, time, and matter all turn "imaginary" together.

The authors argue that even though this transformation looks weird and "bizarre" compared to standard symmetries, it produces real, physical consequences (like mass degeneracies, where different particles end up having the same mass). Therefore, they insist it deserves to be called a symmetry.

In a nutshell: The paper says, "We found a weird, ghostly way to flip the universe inside out. Surprisingly, if you do it right, the laws of physics stay exactly the same, and it locks certain numbers together forever. We proved this works even when we add the messy details of quantum mechanics."

Technical Summary

Technical Summary: Imaginary Scaling Invariance of the One-Loop Effective Potential

Problem Statement
Recent investigations into the Two-Higgs-Doublet Model (2HDM) revealed a hitherto unknown class of relations between scalar potential parameters that remain stable under Renormalization Group Equation (RGE) evolution to all orders of perturbation theory. These relations, identified as fixed points of the beta functions, could not be explained by the six known symmetries of the 2HDM scalar potential. Consequently, a new class of transformations, dubbed "$r_0$" or "GOOFy" symmetries, was hypothesized. These transformations involve mapping real bosonic fields and spacetime coordinates into purely imaginary ones. While the tree-level Lagrangian and potential were shown to be invariant under these transformations, it remained an open question whether this invariance persists at the quantum level, specifically within the one-loop effective potential. The central problem addressed in this work is to verify the invariance of the one-loop effective potential under $r_0$ transformations and to determine the necessary conditions for this symmetry to hold.

Methodology
The authors employ two primary theoretical frameworks to investigate the one-loop properties of the $r_0$ symmetry:

1. The Two-Higgs-Doublet Model (2HDM): The authors utilize the bilinear formalism, representing the scalar sector using four gauge-invariant bilinears ($r_\mu$) and a Minkowski-like metric. They analyze the transformation properties of the scalar mass-squared matrix ($M_S^2$) and the one-loop effective potential under the $r_0$ transformation, which includes the mapping $r_0 \to -r_0$ and the imaginary scaling of coordinates ($x^\mu \to i x^\mu$).
2. A Minimal Toy Model (2RSM): To provide explicit verification, the authors introduce a minimal model consisting of two real scalar fields ($\phi_1, \phi_2$). They construct a potential invariant under a specific $r_0$-like transformation, diagonalize the mass matrix explicitly, and calculate the one-loop effective potential using cutoff regularization.

Key methodological steps include:

• Deriving the transformation rules for the 4-momentum ($p_\mu \to -i p_\mu$) and the UV cutoff squared ($\Lambda_{UV}^2 \to -\Lambda_{UV}^2$) as consequences of the imaginary scaling of spacetime coordinates.
• Analyzing the eigenvalues of the mass-squared matrix $M_S^2$ to determine their behavior under $r_0$.
• Calculating the trace of powers of $M_S^2$ to establish the parity of terms in the effective potential expansion.
• Performing renormalization of the toy model to demonstrate how counterterms cancel divergences while preserving the symmetry.

Key Contributions and Results

1. Invariance of the One-Loop Effective Potential: The paper demonstrates that the one-loop effective potential is invariant under $r_0$ transformations, provided that the UV cutoff squared ($\Lambda_{UV}^2$) transforms nontrivially (specifically, $\Lambda_{UV}^2 \to -\Lambda_{UV}^2$).

   • In the 2HDM, the authors show that the eigenvalues of $M_S^2$ transform in pairs such that the set of eigenvalues $\{\lambda_i(r_0)\}$ maps to $\{-\lambda_i(r_0)\}$. Consequently, traces of odd powers of $M_S^2$ are odd functions of $r_0$, while traces of even powers are even.
   • Since the momentum squared $p_E^2$ also transforms as $p_E^2 \to -p_E^2$ (due to $p_\mu \to -i p_\mu$), the terms in the effective potential expansion involving $(M_S^2/p_E^2)^n$ maintain invariance when combined with the transformation of the cutoff.

2. Explicit Verification in the Toy Model: For the minimal two-real-scalar model, the authors explicitly diagonalize the mass matrix. They find that the eigenvalues $M_{1,2}^2$ transform as $M_1^2 \to -M_2^2$ and $M_2^2 \to -M_1^2$.

   • The calculation confirms that the quadratically divergent term (proportional to $\Lambda_{UV}^2 \sum M_i^2$) and the logarithmically divergent terms are invariant under the combined transformation of fields, coordinates, and the cutoff.
   • The renormalized effective potential is shown to be $r_0$ invariant, with the symmetry realized in the counterterms required to cancel divergences.

3. Renormalization Group Stability: The study confirms that the relations defining the $r_0$ symmetry (e.g., $m_{11}^2 + m_{22}^2 = 0$, $\lambda_1 = \lambda_2$, $\lambda_6 = -\lambda_7$ in the 2HDM) correspond to fixed points of the beta functions. The toy model analysis explicitly shows that the beta function for the sum of squared masses vanishes when the symmetry is imposed, consistent with previous two-loop results.

Significance and Claims
The authors argue that the term "symmetry" is appropriate for these transformations, despite their unconventional nature involving imaginary scalings of real fields and coordinates. The significance of this work lies in:

• Validation of the Symmetry: Proving that the $r_0$ symmetry is not merely a tree-level artifact but holds for the one-loop effective potential, thereby establishing it as a genuine quantum symmetry of the theory.
• Mechanism of Invariance: Identifying that the invariance relies crucially on the nontrivial transformation of the regularization scale (cutoff or renormalization scale). Specifically, the imaginary scaling of spacetime coordinates implies an imaginary scaling of 4-momenta, which necessitates $\Lambda_{UV}^2 \to -\Lambda_{UV}^2$ (or $\mu^2 \to -\mu^2$ in dimensional regularization) to maintain invariance.
• Phenomenological Implications: The existence of these symmetries implies specific relations between parameters that are stable under RGE evolution to all orders. In the context of the 2HDM, this leads to mass degeneracies in the scalar sector and constraints on the decoupling of heavy degrees of freedom (masses of extra scalars must be below $\sim 700$ GeV).
• Novelty: The work highlights that these symmetries represent a new class of relations in bosonic field theories, distinct from standard gauge or global symmetries, which may offer new perspectives on the hierarchy problem and the structure of the Standard Model.

The paper concludes that while the transformations appear "bizarre" due to the complexification of real fields and coordinates, the resulting parameter relations are robust and physically consequential, warranting their classification as symmetries.