On vacua and bounded masses in the general 2HDM

This paper demonstrates that in the general Two-Higgs-Doublet Model with a scalar potential possessing two local minima, the masses of all scalar particles are bounded provided the dimensionless quartic couplings satisfy perturbativity constraints.

The Gist

Imagine the universe is built on a landscape of hills and valleys. In the world of particle physics, the "Standard Model" is like a map of this landscape that we know quite well. But physicists suspect there are hidden valleys—places where new, heavier particles might live. This paper explores a specific type of landscape called the "Two-Higgs-Doublet Model" (2HDM), which is a more complex version of our current map that includes two sets of these hills instead of just one.

Here is the core story of the paper, broken down into simple concepts:

The Landscape of Possibilities

Think of the "vacuum" (the state of empty space) as a ball sitting in a valley.

• One Valley: Sometimes, the landscape has only one deep valley. The ball rolls there and stays. In this scenario, the paper finds that the "new particles" (the heavy hills around the valley) can be as massive as we want. They could be light, or they could be incredibly heavy—like mountains so tall they disappear into the clouds. This is called a "decoupling regime," where the new stuff is so heavy it effectively doesn't interact with us.
• Two Valleys: Sometimes, the landscape has two distinct valleys. The ball could sit in one, but there is another valley nearby that is almost as deep (or exactly as deep).

The Big Discovery: The "Two-Valley" Rule

The authors of this paper asked a simple question: What happens to the size of the mountains (the masses of the new particles) if the landscape has two valleys instead of one?

They ran thousands of computer simulations, essentially rolling the ball around in millions of different random landscapes to see what happened. Their surprising finding was:

If the landscape has two valleys, the mountains cannot be arbitrarily huge.

If there are two local minima (two valleys), the laws of physics (specifically, a rule called "perturbativity," which ensures our math doesn't break down) force all the new particles to have a "ceiling" on their weight. They cannot be heavier than about 1,000 times the mass of a proton (roughly 1 TeV).

The Analogy:
Imagine you are building a sandcastle.

• If you only have one hole in the sand (one valley), you can build a tower as tall as you want, limited only by how much sand you have.
• But if you have two holes that must be the same depth to keep the sand stable, the rules of sand physics force your towers to stay short. You simply cannot build a skyscraper in a two-valley sandcastle without the whole thing collapsing.

Why Does This Happen?

The paper explains that when there are two valleys, the mathematical equations that describe the landscape become "over-constrained."

• In a one-valley world, you have a few "knobs" (parameters) to turn to make the particles heavy.
• In a two-valley world, you have to turn those same knobs to satisfy the conditions for both valleys at the same time. This creates a tight squeeze. The "knobs" get locked into a specific range, preventing the particles from becoming super-heavy.

A Special Compass: The "Diagonal Basis"

The authors also looked at how to tell the difference between a one-valley and a two-valley world just by looking at the particles. They found a special way of measuring the landscape (a specific "basis").

• If the new particles are very heavy (over 1 TeV), you can be 100% sure there is only one valley.
• If the new particles are light (under 1 TeV), it's a bit trickier. However, if the ratio of the "heights" of the two hills in that special compass is either extremely large or extremely small, it usually means there is only one valley.
• But, if that ratio is "just right" (in the middle range), it's a strong hint that a second valley might exist.

The Bottom Line

This paper doesn't tell us where to find these new particles or how to build a machine to detect them. Instead, it sets a theoretical speed limit based on the shape of the universe's vacuum.

• If you find new particles heavier than 1 TeV: You can relax. You know for a fact that the universe's vacuum has only one minimum.
• If you find new particles lighter than 1 TeV: You have to be careful. The universe might have a second, hidden valley. If it does, the particles can't be too heavy, and we might be able to see them with current technology (like the Large Hadron Collider).

In short: Two valleys mean a weight limit for new particles. One valley means no limit.

Technical Summary

Technical Summary: On Vacua and Bounded Masses in the General 2HDM

Problem Statement
In extensions of the Standard Model (SM) with extended scalar sectors, such as the Two-Higgs-Doublet Model (2HDM), a primary theoretical concern is the potential for "decoupling regimes." In such regimes, new scalar particles can acquire masses significantly larger than the electroweak scale ($v_{EW} \approx 246$ GeV) while remaining consistent with perturbativity constraints on the dimensionless quartic couplings. Previous theoretical work has shown that in specific scenarios (e.g., CP-invariant potentials with spontaneous CP violation), the presence of multiple vacua or specific symmetry constraints can force all scalar masses to be bounded by the electroweak scale. However, it remained unclear whether this bounding mechanism applies to the general 2HDM, which allows for complex parameters and does not necessarily possess such symmetries. Specifically, the authors investigate whether the existence of two local minima in the scalar potential imposes constraints on the mass spectrum that prevent a decoupling regime.

Methodology
The authors employ a combined analytical and numerical approach to explore the general 2HDM parameter space:

1. Analytical Framework:

   • The study utilizes the most general scalar potential for two $SU(2)_L$ doublets ($\Phi_1, \Phi_2$), containing 4 quadratic parameters ($\mu^2_1, \mu^2_2, \mu^2_{12}, \mu^{2*}_{12}$) and 8 quartic parameters ($\lambda_{1-7}$).
   • The analysis focuses on the stationarity conditions (minimization equations) of the potential. In a single-minimum scenario, 3 of the 4 quadratic parameters are fixed by the vacuum expectation values (vevs) and quartic couplings, leaving one free parameter (typically $\mu^2_1 + \mu^2_2$ or $\text{Im}(\mu^2_{12})$) that can be arbitrarily large, allowing for heavy scalars.
   • The authors hypothesize that if a second local minimum exists, the stationarity conditions must be satisfied for both minima simultaneously. This over-constrains the system, potentially expressing all quadratic parameters in terms of bounded quartic couplings and vevs.

2. Numerical Exploration:

   • The authors generate random sets of quartic parameters, ensuring the potential is bounded from below.
   • They numerically minimize the potential with various starting points to identify the number of local minima (one or two).
   • For valid configurations, they rescale the quadratic parameters to enforce the correct electroweak vev ($v = v_{EW}$) and fix the lightest neutral scalar mass to the observed Higgs mass ($m_h \approx 125.1$ GeV).
   • They verify perturbative unitarity constraints and calculate the full mass spectrum of the new scalars ($H^\pm, H^0_1, H^0_2$).
   • The analysis is performed in both a generic basis and a specific basis where the quadratic mass matrix is diagonal.

Key Contributions and Results

• Bounded Masses in Two-Minima Scenarios: The central finding is that if the general 2HDM scalar potential possesses two local minima, the masses of all new scalars are strictly bounded. Under perturbativity constraints on the quartic couplings, these masses cannot exceed approximately 1 TeV. This effectively rules out a decoupling regime for the entire scalar spectrum in the presence of a second minimum.
• Unbounded Masses in Single-Minima Scenarios: Conversely, if the potential has only one local minimum, the system retains a free quadratic parameter. This allows for a decoupling regime where all new scalars can have masses significantly larger than the electroweak scale (e.g., $\gg 1$ TeV).
• Discrimination via Vev Ratios: The authors investigate whether the ratio of vevs ($v_2/v_1$) can distinguish between the one-minimum and two-minimum cases.
  • In a generic basis, no clear pattern emerges.
  • In the basis where the quadratic parameters are diagonal, a distinction appears for masses below 1 TeV: values of $v_2/v_1$ outside the range $[10^{-2}, 10^2]$ (i.e., very large or very small ratios) tend to correlate with the two-minima scenario, while values within this range are ambiguous.
• Analytical Explanation: The authors provide an analytical derivation showing that the existence of a second minimum $\{v'_1, v'_2 e^{i\theta'}\}$ imposes a second set of stationarity conditions. Solving these simultaneously for the quadratic parameters (specifically $\mu^2_{12}$) expresses them as functions of the quartic couplings and the vevs of both minima. Since the quartic couplings are bounded by perturbativity, the quadratic parameters—and consequently the scalar masses—become bounded.

Significance and Claims
The paper claims that the global property of the scalar potential having two local minima acts as a powerful theoretical constraint on the 2HDM mass spectrum. The significance of this result is twofold:

1. Theoretical Constraint: It establishes that the "decoupling limit" (where new physics is hidden at very high scales) is incompatible with the existence of a second local minimum in the general 2HDM, provided perturbativity holds.
2. Phenomenological Implication: For models with new scalars heavier than 1 TeV, one can confidently assume the potential has only one minimum, avoiding concerns about cosmological domain walls or vacuum stability issues associated with a second minimum. Conversely, for models with new scalars below 1 TeV, if the vev ratio in the diagonal quadratic basis falls within the intermediate range ($10^{-2} < v_2/v_1 < 10^2$), the potential may possess a second minimum, which could have cosmological consequences and requires careful scrutiny.

The authors explicitly state that their work focuses solely on the scalar sector and does not address Yukawa couplings to fermions or specific experimental signatures, noting that such phenomenological implications are beyond the scope of the current study.