Minimal complete tri-hypercharge theories of flavour

This paper proposes and analyzes two minimal, ultraviolet-complete tri-hypercharge models that explain Standard Model fermion masses and mixings through three correlated physical scales, predicting that the lightest $Z'$ boson in these theories is discoverable via dilepton searches at LHC Run 3.

The Gist

The Big Problem: The "Family Tree" Mystery

Imagine the Standard Model of particle physics as a massive, incredibly successful instruction manual for how the universe works. It tells us exactly how particles interact, how they move, and how they stick together.

But there is one glaring hole in the manual: The "Why" of the Families.

The manual says there are three "families" of particles (like three generations of cousins).

• Generation 1: The lightweights (electrons, up/down quarks). These make up the atoms in your body.
• Generation 2: The middleweights (muons, charm/strange quarks). These are unstable and rarely seen.
• Generation 3: The heavyweights (tau particles, top/bottom quarks). These are massive and short-lived.

The manual explains how they interact, but it doesn't explain why the top quark is 40,000 times heavier than the up quark, or why the electron is so light. It's like having a recipe book that tells you how to bake a cake, but the ingredients list just says "some flour, a lot of sugar, and a tiny pinch of salt" without explaining why those specific amounts create the perfect cake.

The Solution: The "Tri-Hypercharge" Idea

The authors of this paper propose a new rulebook called Tri-Hypercharge.

Think of the Standard Model's "Hypercharge" (a property that determines how particles interact with light and electricity) as a single, universal currency. Everyone uses the same dollar bill.

The Tri-Hypercharge theory suggests that instead of one universal currency, there are three separate currencies, one for each family:

• Family 1 uses "Dollar 1."
• Family 2 uses "Dollar 2."
• Family 3 uses "Dollar 3."

In this world, a particle from Family 1 can only buy things with Dollar 1. It can't interact with Family 2's Dollar 2. This separation is the key to solving the mystery of why the families are so different.

The Two "Minimal" Models

The paper presents two different ways to build this new universe. Both are "minimal," meaning they try to use the fewest possible new ingredients to make the theory work.

Model 1: The "Heavy Messenger" Construction

Imagine you want to send a letter from Family 1 to Family 3, but they speak different languages and can't talk directly. You need a messenger who speaks both languages to carry the message.

• How it works: In this model, the authors introduce heavy, invisible "messenger" particles (Vector-Like Fermions). These messengers are like construction workers who carry bricks (mass) from the heavy third family to the light first family.
• The Result: Because the messengers are so heavy, the "bricks" they carry are hard to move. This naturally explains why the first family is so light (it's hard to get the heavy stuff down to them) and the third family is heavy (they have direct access).
• The Catch: This model requires a lot of these heavy messenger workers.

Model 2: The "Heavy Higgs" Construction

This model is even more minimalist. Instead of hiring a whole army of messenger workers, they use two heavy Higgs doublets (special types of energy fields) to do the job.

• How it works: Think of the Higgs field as a thick fog. In this model, the fog is very thick for the first two families, making it hard for them to gain mass. The third family walks through a clear patch of fog, gaining mass easily.
• The Result: This model uses fewer types of particles (fewer "degrees of freedom") but requires a more complex "fog machine" (a more complicated scalar potential).
• The Trade-off: It's simpler in terms of particle count, but the math describing the "fog" is more complicated.

The "Three Scales" of the Universe

The most beautiful part of this paper is how it simplifies the chaos.

The Standard Model has a messy, complicated pattern of masses and mixings. The authors show that their Tri-Hypercharge theory boils all that complexity down to just three simple physical scales (three specific energy levels):

1. The "Light" Scale (~10 TeV): This is where the third family lives. It's heavy, but not impossibly so.
2. The "Medium" Scale (~1,000 TeV): This is where the messengers for the second family live.
3. The "Heavy" Scale (~10,000 TeV): This is where the messengers for the first family live.

The Analogy: Imagine a waterfall.

• The water at the top (Family 3) is powerful and fast.
• It falls to a middle pool (Family 2), losing some energy.
• It falls to a tiny trickle at the bottom (Family 1), losing almost all its energy.

The paper argues that the entire complex structure of the universe's particles is just the result of water falling down these three specific steps. If you know the height of the steps, you know the whole story.

The "Z-Prime" (Z') Bosons: The New Detectives

When these three separate currencies (hypercharges) are broken down to the single currency we see today, two new, heavy force-carrying particles appear, called Z' bosons.

• Z'12: Connects Family 1 and Family 2.
• Z'23: Connects Family 2 and Family 3.

These particles are the "smoking guns" of the theory. They are heavy, but the authors predict they might be light enough to be found at the Large Hadron Collider (LHC) in the near future (specifically in "Run 3").

If scientists find these Z' particles, it would be like finding a hidden door in the instruction manual that proves the "Three Currencies" theory is real.

Why This Matters

1. Simplicity: It takes a messy, confusing problem (why are particles so different?) and solves it with a simple, elegant rule (three separate currencies).
2. Predictability: It doesn't just explain the past; it predicts exactly what we should look for in particle accelerators. It says, "Look for a Z' particle at this specific energy level."
3. Neutrinos: It also explains why neutrinos (ghostly particles with almost no mass) are so light, using a mechanism that doesn't require inventing even more new particles.

The Bottom Line

This paper proposes that the universe isn't a chaotic mess of random numbers. Instead, it's a carefully constructed system where the three families of particles are separated by invisible walls (the three hypercharges). The differences in their masses are just the result of how far they are from the "source" of mass.

The authors have built two blueprints (Model 1 and Model 2) for this universe. Both are simple, both work, and both tell us exactly where to look next to prove them right. It's a step toward understanding the "why" behind the "how" of our universe.

Technical Summary

1. Problem Statement

The Standard Model (SM) successfully describes particle interactions but fails to explain the flavour puzzle: the origin of the three fermion families and the hierarchical pattern of their Yukawa couplings. This hierarchy manifests as:

• Mass Hierarchies: Quark and charged lepton masses span several orders of magnitude (e.g., $m_t \gg m_u$, $m_\tau \gg m_e$), often parameterized by powers of the Wolfenstein parameter $\lambda \approx 0.22$.
• Mixing Patterns: Quark mixing (CKM) is hierarchical, while lepton mixing (PMNS) is large and nearly maximal.
• Neutrino Masses: The origin of tiny neutrino masses and large mixing angles remains unexplained within the minimal SM.

Existing "flavour deconstructed" or "gauge non-universal" theories attempt to solve this by assigning separate gauge groups to each family. However, many proposals rely on Effective Field Theories (EFT) without specifying the ultraviolet (UV) completion, or they require non-minimal particle content and fine-tuned parameters.

2. Methodology

The authors propose and analyze two minimal, ultraviolet (UV) complete, and renormalisable models based on the Tri-Hypercharge (TH) framework.

• Gauge Structure: The gauge group is extended from the SM $SU(3)_c \times SU(2)_L \times U(1)_Y$ to:
  $$SU(3)_c \times SU(2)_L \times U(1)_{Y_1} \times U(1)_{Y_2} \times U(1)_{Y_3}$$
  Each fermion family $i$ carries only the hypercharge $Y_i$. The SM hypercharge is the diagonal sum $Y = Y_1 + Y_2 + Y_3$. Anomalies cancel family-by-family.
• Symmetry Breaking Chain:
  1. Scalars (hyperons) $\phi_{12}$ and $\phi_{23}$ acquire Vacuum Expectation Values (VEVs), breaking $U(1)_{Y_1} \times U(1)_{Y_2} \times U(1)_{Y_3}$ down to $U(1)_{Y_1+Y_2} \times U(1)_{Y_3}$ and finally to the SM $U(1)_Y$.
  2. This breaking generates the hierarchy of scales: $v_{23} \sim \mathcal{O}(10 \text{ TeV})$ and $v_{12} \sim \mathcal{O}(10^3 \text{ TeV})$.
• Mechanism for Flavour:
  • Third Family: Direct renormalisable Yukawa couplings to the third-family Higgs doublets ($H_{3}^{u,d}$).
  • First and Second Families: Effective Yukawa couplings are generated via heavy messenger fields (vector-like fermions or heavy Higgs doublets) that couple to the hyperons. Integrating out these messengers generates non-renormalisable operators suppressed by the messenger masses ($\Lambda_{1,2}$).
• Neutrino Sector: A minimal Type-I seesaw mechanism is implemented using the existing hyperons and vector-like neutrino messengers ($N_{ij}$), avoiding the need for extra scalars.

3. Key Contributions: The Two Minimal Models

The paper presents two distinct UV completions that differ only in the nature of the heavy messengers, while sharing the same low-energy EFT structure.

Model 1: All Vector-Like (VL) Fermions

• Content: All heavy messengers are vector-like fermions (singlets under $SU(2)_L$).
  • Up-type: $U_{12}, U_{13}, U_{23}$
  • Down-type: $D_{12}, D_{13}, D_{23}$
  • Charged Leptons: $E_{12}, E_{13}, E_{23}$
  • Neutrinos: $N_{12}, N_{13}, N_{23}$
• Scalar Sector: Minimal (only hyperons and third-family Higgs).
• Flavour Origin:
  • Yukawa matrices are generated by mass insertions of VL fermions.
  • Mixing: Significant mixing occurs in both Left-Handed (LH) and Right-Handed (RH) sectors.
  • CKM Origin: Primarily from the down-quark sector.
  • Phenomenology: Predicts significant Flavour Changing Neutral Currents (FCNCs) in the down-quark sector (e.g., $K-\bar{K}$ mixing) and Charged Lepton Flavour Violation (CLFV) like $\mu \to e\gamma$.

Model 2: Hybrid (VL Fermions + Heavy Higgs)

• Content: Replaces down-type and charged lepton VL fermions with two heavy Higgs doublets ($H_1^d, H_2^d$). Up-type messengers remain VL fermions.
• Scalar Sector: Larger than Model 1 due to extra Higgs doublets and potential terms.
• Flavour Origin:
  • Down-type and charged lepton Yukawa matrices are diagonal at tree-level.
  • Mixing: All mixing originates from the up-quark sector (and neutrinos).
  • CKM Origin: Primarily from the up-quark sector.
  • Phenomenology: Suppresses FCNCs in the down/lepton sectors. Leading FCNCs appear in the up-quark sector ($D-\bar{D}$ mixing).

4. Key Results

A. Explanation of Flavour Hierarchies

Both models successfully reproduce the observed SM flavour structure using three simple physical scales and all fundamental coefficients of $\mathcal{O}(1)$. The hierarchies are explained by ratios of VEVs to messenger masses:

1. $\frac{\langle \phi_{23} \rangle}{M_{23}} \sim \lambda^3$ (Controls $m_2/m_3$ and $V_{cb}$)
2. $\frac{\langle \phi_{12} \rangle}{M_{12,13}} \sim \lambda$ (Controls $m_1/m_2$ and $V_{us}$)
3. $\frac{\langle \phi_{23} \rangle}{M_{12,13}} \sim \lambda^5$ (Controls $m_1/m_3$ and $V_{ub}$)

Crucially, the models predict a non-generic relation between the two breaking scales:
$$\frac{v_{23}}{v_{12}} \sim \lambda^4 \quad (\text{or } \lambda^3 \text{ in Model 2})$$
This implies that testing one scale effectively tests the others.

B. Neutrino Masses

The models implement a seesaw mechanism where the light neutrino masses are generated without requiring tiny Yukawa couplings or extremely high scales ($10^{15}$ GeV).

• Messenger neutrinos ($N_{ij}$) have masses in the multi-TeV to PeV range.
• Large PMNS mixing arises naturally from the structure of the Dirac and Majorana mass matrices involving the hyperons.

C. Phenomenological Bounds and Predictions

• $Z'$ Bosons: The symmetry breaking produces two heavy gauge bosons, $Z'_{12}$ and $Z'_{23}$.
  • $Z'_{23}$: Mass $\sim v_{23} \sim \mathcal{O}(10 \text{ TeV})$. Protected from dangerous FCNCs by an approximate $U(2)^5$ flavour symmetry. It is potentially discoverable in dilepton searches at LHC Run 3.
  • $Z'_{12}$: Mass $\sim v_{12} \sim \mathcal{O}(10^3 \text{ TeV})$.
• FCNC Constraints:
  • Model 1: Constrained by $K-\bar{K}$ mixing and $\mu \to e\gamma$. The bounds require $M_{Z'_{12}} \gtrsim 10^3$ TeV.
  • Model 2: Constrained by $D-\bar{D}$ mixing. The bounds are slightly weaker ($M_{Z'_{12}} \gtrsim 100$ TeV) due to the lack of tree-level down-sector mixing.
• Vector-Like Fermions: Direct bounds on messenger masses are in the few TeV to $\mathcal{O}(100)$ TeV range, consistent with the indirect constraints from the flavour structure.

5. Significance

• Minimality: These are arguably the most minimal UV-complete models for tri-hypercharge, requiring the fewest degrees of freedom and representations to explain the full flavour sector (including neutrinos).
• Predictivity: The framework translates the complex SM flavour structure into a correlated set of three energy scales. This reduces the number of free parameters significantly compared to generic flavour models.
• Testability: Unlike many high-scale flavour models, these theories predict a light $Z'_{23}$ boson ($\sim 10$ TeV) accessible at current or near-future colliders (LHC Run 3/HL-LHC).
• Naturalness: The models explain small neutrino masses and large mixing without fine-tuning couplings to be unnaturally small, relying instead on the mass hierarchy of the messenger sector.
• Differentiation: The two models offer distinct phenomenological signatures (down-sector vs. up-sector FCNCs), allowing experimental data to distinguish between different UV completions of the same low-energy EFT.

In summary, the paper demonstrates that the Tri-Hypercharge framework, when completed with minimal heavy messengers, provides a robust, predictive, and testable solution to the flavour puzzle, linking the origin of fermion masses to specific, correlated new physics scales.