Scalar Lie point symmetries of the Standard Model with one or two real gauge singlets

This paper classifies all scalar Lie point symmetries of the Standard Model extended with one or two real gauge singlets, distinguishing between strict variational, divergence, and non-variational cases, while providing efficient parameter-based algorithms to determine the symmetry algebra for any numerical instance without solving determining equations.

The Gist

Imagine the Standard Model of particle physics as a giant, incredibly complex recipe book for the universe. It tells us how particles like electrons, quarks, and the Higgs boson interact to create everything we see. But physicists suspect this recipe is incomplete. To fix it, they often add "secret ingredients"—new particles that don't interact with light or ordinary matter (called "singlets").

This paper is like a master chef's guide to finding hidden patterns in these new recipes. The author, Marius Solberg, asks a very specific question: If we add one or two of these secret ingredients to the Standard Model, what hidden symmetries (rules of balance) appear in the math?

Here is the breakdown of the paper using simple analogies:

1. The Goal: Finding the "Hidden Rhythm"

In physics, a symmetry is like a rule that says, "If I change this part of the system, the laws of physics stay exactly the same."

• Example: If you rotate a perfect circle, it looks the same. That's a symmetry.
• The Paper's Job: The author wants to find all the possible ways these new "singlet" particles can be arranged so that the universe's laws remain balanced. He classifies these arrangements into three types of "rhythms":
  1. Strict Variational: The recipe is perfectly balanced; nothing changes at all.
  2. Divergence: The recipe changes slightly at the edges (like a boundary), but the core physics remains the same.
  3. Non-Variational: The recipe looks different, but the final result (the equations of motion) still works out the same. It's like a magic trick where the method changes, but the outcome is identical.

2. The Challenge: Too Many Ingredients

The Standard Model with extra particles has a huge number of "parameters" (numbers that control how heavy the particles are and how strongly they interact).

• The Problem: If you try to check every single combination of these numbers to find the symmetries, it would take longer than the age of the universe. It's like trying to find a specific needle in a haystack the size of a galaxy.
• The Solution: The author didn't just brute-force the math. He built a decision tree (a flowchart).
  • Imagine you are a detective. Instead of checking every house in a city, you have a map that says: "If the suspect has a red hat, go left. If they have a blue hat, go right."
  • This paper provides that map. By looking at just a few key numbers in the recipe (the parameters), you can instantly know which "symmetry group" your model belongs to, without doing the heavy lifting of solving complex equations every time.

3. The Two Cases: One vs. Two Secret Ingredients

The author analyzed two scenarios:

• SM+S (One Singlet): Adding one secret ingredient.
  • Result: He found 4 distinct types of symmetry patterns. It's relatively simple, like adding one spice to a stew. Depending on the spice, you get a specific flavor profile (symmetry).
• SM+2S (Two Singlets): Adding two secret ingredients.
  • Result: This is much more complex, like adding two spices that can mix in many ways. He found 11 distinct symmetry patterns.
  • He discovered that sometimes the two ingredients can rotate into each other (like mixing red and blue paint to make purple), creating a new kind of symmetry called SO(2) (a circular rotation symmetry).

4. The "Magic" of Reparametrization

One of the coolest parts of the paper is the concept of reparametrization.

• Analogy: Imagine you have a recipe written in metric units (grams) and another in imperial units (ounces). They are the same recipe, just written differently.
• In physics, you can sometimes "rotate" or "shift" your view of the particles, and the math looks different, but the physics is identical. The author proved that his classification of symmetries holds true even if you change your "units" or perspective. This ensures that he isn't counting the same symmetry twice just because it looks different on paper.

5. Why Does This Matter?

You might ask, "Why do we care about these mathematical patterns?"

• Dark Matter: These "singlet" particles are leading candidates for Dark Matter (the invisible stuff holding galaxies together). Knowing their symmetries helps physicists predict how Dark Matter might behave or interact.
• The Early Universe: These symmetries might explain why the universe has more matter than antimatter.
• Efficiency: The biggest practical contribution is the algorithm. Instead of a physicist spending months solving equations for a new model, they can now plug their numbers into this "flowchart" and instantly know: "Ah, my model has a specific symmetry that allows for a stable Dark Matter candidate!"

Summary

Think of this paper as a comprehensive catalog of all possible "dance moves" that the Standard Model can perform if we add one or two new dancers (the singlets).

• The author mapped out every possible dance routine.
• He figured out which moves are strictly balanced, which are slightly off-balance but still work, and which are pure magic.
• Most importantly, he built a cheat sheet so that other scientists don't have to learn the whole dance from scratch; they can just look at the music (the parameters) and know exactly which dance routine they are doing.

It turns a mountain of complex calculus into a simple, navigable map for exploring the hidden structure of our universe.

Technical Summary

1. Problem Statement

The paper addresses the classification of scalar Lie point symmetries for extensions of the Standard Model (SM) involving one (SM+S) or two (SM+2S) real gauge-singlet scalar fields. These models are crucial for phenomenology, offering dark matter candidates and mechanisms for a strong first-order electroweak phase transition (necessary for baryogenesis).

The core problem is to systematically identify all realizable and inequivalent Lie point symmetry algebras admitted by the Euler-Lagrange equations of these models. The author distinguishes between three nested types of symmetries:

1. Strict Variational Symmetries (SVS): Symmetries that leave the action invariant ($\delta S = 0$).
2. Divergence Symmetries (DS): Symmetries that change the action by a boundary term ($\delta S = \int \partial_\mu \beta^\mu$).
3. Non-Variational (Euler-Lagrange) Symmetries (NVS): Symmetries of the equations of motion that do not preserve the action or its boundary terms.

The challenge lies in the vast parameter space of the scalar potentials. A brute-force symmetry analysis for every numerical instance is computationally expensive. The goal is to provide a classification of all possible symmetry algebras and an efficient algorithm to determine the symmetry type for any given set of model parameters without solving the full determining equations from scratch.

2. Methodology

The author employs Lie symmetry analysis of systems of Partial Differential Equations (PDEs) applied to the Euler-Lagrange equations derived from the Lagrangians.

• Theoretical Framework:

  • The paper reviews the prolongation of infinitesimal generators and the determining equations for point symmetries.
  • It establishes a rigorous characterization of the three symmetry types (SVS, DS, NVS) for Lagrangians of the form $L = T - V$, where $T$ is the kinetic term and $V$ is a polynomial potential.
  • Key Theorem (Corollary 1): The author proves that for a broad class of models, the nature of a symmetry generator $X$ depends on the action of its prolongation on the kinetic term ($pr_X(T)$) and the linear terms of the potential ($\alpha_i \phi_i$). Specifically:
    • $X$ is Strict Variational iff $pr_X(T) = 0$ and the linear term coefficient vanishes ($a_i \alpha_i = 0$).
    • $X$ is a Divergence Symmetry iff $pr_X(T)$ is a total divergence (or zero) but the linear term condition fails.
    • $X$ is Non-Variational iff $pr_X(T)$ is not a total divergence.

• Reparametrization Freedom:

  • The analysis exploits affine reparametrizations (shifts and orthogonal rotations) of the singlet fields. These transformations preserve the canonical form of the kinetic terms but alter the potential parameters.
  • The author defines orthogonal equivalence ($\cong_O$) for symmetry algebras, ensuring that physically equivalent symmetries (related by basis changes) are not counted multiple times.

• Algorithmic Approach:

  • Instead of solving the determining equations for every parameter set, the author constructs a reduction tree.
  • By using shifts and rotations, parameters in the potential (e.g., cubic couplings $\kappa_{ijk}$, mixed quartic couplings $\lambda_{ijkl}$) are systematically eliminated or set to zero to reach "Leaves" (reduced potentials).
  • At each leaf, the determining equations are solved once to find the maximal symmetry algebra.
  • This creates a decision procedure: for any numerical instance, one follows the tree to the appropriate leaf to identify the symmetry algebra.

3. Key Contributions

1. Classification of Symmetry Types: The paper provides a complete classification of the scalar Lie point symmetries for SM+S and SM+2S, distinguishing strictly variational, divergence, and non-variational algebras.
2. General Characterization Theorems: It proves general results (Theorem 1, Proposition 1, Corollary 1) characterizing the three symmetry types for Lagrangians with polynomial potentials, applicable beyond just these specific models.
3. Efficient Decision Algorithms: The paper devises algorithms (Sections 3.4 and 4.6) that allow researchers to determine the symmetry algebra of a specific model instance by simple parameter checks (e.g., "Is $\lambda_{1111} = 0$?") rather than performing explicit symmetry calculations.
4. Handling of Redundancy: It rigorously handles the equivalence of symmetry algebras under orthogonal affine reparametrizations, preventing double-counting of physically identical symmetries.

4. Key Results

A. Standard Model + One Singlet (SM+S)

The analysis yields four inequivalent realizable scalar Lie point symmetry algebras ($g_{EL}$):

1. $u(1)_Y$: The generic case (hypercharge rephasing of the Higgs).
2. $sh \oplus u(1)_Y$: Shift symmetry of the singlet ($s \to s + c$).
   • Nature: Strict variational if the linear term $\alpha = 0$; Divergence if $\alpha \neq 0$.
3. $sc \oplus u(1)_Y$: Scaling symmetry of the singlet ($s \to \lambda s$).
   • Nature: Non-variational (Euler-Lagrange symmetry only).
4. $a(1) \oplus u(1)_Y$: The affine algebra (shift + scaling).
   • Nature: Contains a non-variational generator.

Summary: The only non-trivial variational symmetry is the shift symmetry ($sh$), which becomes a divergence symmetry if a linear tadpole term exists.

B. Standard Model + Two Singlets (SM+2S)

The analysis is significantly more complex due to the $O(2)$ rotation freedom between singlets. The reduction tree (Figs. 1–3) identifies 11 inequivalent realizable maximal symmetry algebras ($g_{EL}$):

• 1D/2D Algebras: $sh$, $sc$, $so(2)$ (rotation).
• 3D/4D Algebras: $a(1)$, $d_4$, $gl(2, \mathbb{R})$.
• 6D Algebra: $a(2)$ (the affine algebra of the plane, acting on the two singlets).

Variational Subalgebras:

• The maximal variational algebras ($g_{var}$) include $sh$, $so(2)$, and the Euclidean algebra $e(2)$ (shifts + rotation).
• The maximal strict variational algebras ($g_{svar}$) are restricted further: $sh$, $so(2)$, and $e(2)$.
• Crucial Finding: The algebra $sh_1 \oplus sh_2$ (two independent shifts) is a variational symmetry but not a strict variational symmetry if linear terms are present. This highlights the importance of distinguishing between variational and strict variational types.

Nature of Generators:

• Shifts ($X_1, X_4$): Strict variational if linear terms vanish; divergence otherwise.
• Rotation ($X_3 - X_5$): Always strict variational.
• Scaling/Boosts: Generally non-variational.

5. Significance

• Phenomenological Constraints: Symmetries impose relations among model parameters (e.g., $m_{11} = m_{22}$, $\lambda_{11} = \lambda_{22}$). These relations are stable under Renormalization Group (RG) evolution. Identifying these symmetries helps constrain model building for dark matter and baryogenesis.
• Computational Efficiency: The proposed algorithms allow for rapid scanning of parameter spaces in numerical studies. Researchers can instantly determine the symmetry structure of a model without re-deriving determining equations, facilitating the search for viable models with enhanced symmetries.
• Theoretical Clarity: By rigorously separating strict variational, divergence, and non-variational symmetries, the paper clarifies which symmetries lead to conserved Noether currents (via Noether's theorem) and which are merely dynamical symmetries of the equations of motion. This is vital for understanding the quantum behavior of these models (e.g., anomaly considerations).
• Generalizability: The characterization theorems (Corollary 1) apply to any theory with a kinetic term and a polynomial potential, making the results relevant for a wide range of Beyond Standard Model (BSM) scenarios, including Multi-Higgs Doublet Models (2HDM, NHDM).

In conclusion, this work provides a definitive "catalog" of scalar symmetries for singlet-extended Standard Models, offering both a theoretical classification and a practical toolkit for model builders.