Alternative framework for the left-right symmetric model including vector-like fermions

This paper proposes an extended left-right symmetric model incorporating an additional $SU(2)$ gauge symmetry and vector-like fermions to explain neutrino mass hierarchies through distinct seesaw mechanisms, while utilizing LHC Run II data to establish lower mass limits on the new $W'$ and $Z'$ gauge bosons based on their decay channels.

The Gist

Imagine the Standard Model of particle physics as a highly successful, but slightly broken, instruction manual for how the universe works. It explains almost everything we see, but it has some glaring typos: it can't explain why some particles (neutrinos) are so incredibly light, why the universe seems to prefer "left-handed" particles over "right-handed" ones, or what dark matter is.

This paper proposes a new, expanded version of that manual. Think of it as adding a new chapter to the instruction book to fix those typos. Here is the story of that new chapter, broken down into simple concepts.

1. The New Room in the House (The Extra Symmetry)

The authors start with a popular theory called the Left-Right Symmetric Model (LRSM). Imagine the universe as a house with two wings: a Left Wing and a Right Wing. In our current reality, the Left Wing is very active, but the Right Wing is mostly silent. The LRSM says, "Hey, the Right Wing should be just as active as the Left, but something happened in the past that silenced it."

In this new paper, the authors add a third wing to the house. They call it the "Vector" wing.

• The Analogy: Imagine a dance floor. Usually, dancers spin either clockwise (Left) or counter-clockwise (Right). This new theory adds a group of dancers who can spin both ways simultaneously without getting dizzy. These are called Vector-Like Fermions. They are "exotic" particles that don't follow the usual rules of the dance floor, allowing them to mix with the regular dancers.

2. The Heavy Hitters (Vector-Like Fermions)

Why add these new dancers?

• The Problem: The "Top Quark" (a very heavy particle) makes the Higgs boson (the particle that gives mass to everything) unstable, like a wobbly table leg.
• The Solution: These new Vector-Like dancers act like a sturdy support beam. By mixing with the Top Quark, they stabilize the table, fixing a major mathematical headache in physics.
• The Dark Matter Candidate: One of these new particles (a heavy neutrino) is so shy and heavy that it might be the invisible "Dark Matter" holding galaxies together. It's like a ghost in the house that you can't see but whose presence you feel through gravity.

3. The Seesaw of Mass (Why Neutrinos are Light)

One of the biggest mysteries is why neutrinos are so light (almost weightless) compared to other particles.

• The Old Seesaw: The original Left-Right model used a "Seesaw" mechanism. Imagine a playground seesaw. If one side goes up (heavy), the other goes down (light). The heavy particles push the light neutrinos down to almost zero mass.
• The New Twist: This paper says the first two generations of neutrinos use the old seesaw. But the third generation (the heaviest neutrinos) uses a new, upgraded seesaw that involves the new Vector-Like dancers. It's like having two different playgrounds: one for the kids and one for the adults, each with their own rules for how high they can jump.

4. The Search for the "Super-Heavy" Messenger (W' Boson)

The theory predicts the existence of new, heavy force-carrying particles, specifically a super-heavy version of the W boson, called W'.

• The Analogy: Think of the W boson as a regular delivery truck. The W' is a massive, futuristic cargo ship.
• The Experiment: The authors looked at data from the Large Hadron Collider (LHC), the giant particle smasher in Switzerland. They asked: "If we smash protons together, could we create this cargo ship (W') and have it immediately break apart into our new exotic dancers (Vector-Like Quarks) or heavy neutrinos?"
• The Result: They didn't find the ship yet, but they set a speed limit. They calculated that if this ship exists, it must be heavier than 3 to 6 TeV (a unit of mass). If it were lighter, the LHC would have seen it by now. This effectively rules out the "lighter" versions of the ship.

5. The "Single Production" Trick

Usually, when we look for new heavy particles, we try to create them in pairs (like making two cars at once). But this paper suggests a smarter way: Single Production.

• The Analogy: Imagine trying to catch a rare bird. Instead of waiting for two birds to fly in together, you set a trap that catches just one.
• The Finding: The authors found that creating a single Vector-Like Top Quark (the "T" quark) alongside a regular Top Quark is actually a much more promising way to find these new particles than creating them in pairs. It's like finding a needle in a haystack by looking for the specific thread that holds the needle, rather than looking for the whole needle.

Summary: What does this mean for us?

This paper is a blueprint for a "Version 2.0" of the universe's rulebook.

1. It fixes the math: It stabilizes the Higgs mass and explains why neutrinos are light.
2. It offers a Dark Matter candidate: It suggests a shy, heavy particle that could be the invisible stuff of the universe.
3. It gives scientists a roadmap: It tells experimentalists at the LHC exactly what to look for (heavy W' bosons decaying into specific heavy particles) and tells them how heavy those particles must be to have escaped detection so far.

In short, the authors are saying: "The universe might be a bit more complex than we thought, with a hidden 'Vector' wing and some heavy-duty dancers. If you look hard enough at the right energy levels, you might just find them."

Technical Summary

Here is a detailed technical summary of the paper "Alternative framework for the left-right symmetric model including vector-like fermions" by Bouzeraib and Zidi.

1. Problem Statement

The Standard Model (SM) of particle physics, while successful, fails to explain several fundamental phenomena, including:

• The origin of neutrino masses and their smallness.
• Parity violation in weak interactions.
• The quadratic divergence in the Higgs boson mass (hierarchy problem).
• The nature of Dark Matter.

The Left-Right Symmetric Model (LRSM) is a well-established extension that addresses parity violation and neutrino masses via the seesaw mechanism. However, this paper proposes a further extension to incorporate Vector-Like Fermions (VLFs) and an additional gauge symmetry to provide a more comprehensive solution, particularly regarding the hierarchy of neutrino masses and the potential for Dark Matter candidates.

2. Methodology and Model Construction

Gauge Group and Particle Content

The authors construct an extended Left-Right Symmetric Model with the gauge group:
$$G_{VLRSM} \equiv SU(2)_V \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$$

• $SU(2)_V$: A new gauge factor specifically designed to house one generation of Vector-Like Fermions (VLFs) (quarks $T, B$ and leptons $N, E$) in the fundamental representation.
• Chiral Fermions: Standard SM fermions are arranged as doublets under $SU(2)_L$ and $SU(2)_R$.
• Scalar Sector: To break the symmetry and induce mixing between VLFs and chiral fermions, the model introduces:
  • Two self-dual bi-doublets: $\Phi_{VL}$ and $\Phi_{VR}$.
  • The standard LRSM bi-doublet $\Phi$.
  • Two triplets $\Delta_L$ and $\Delta_R$ (essential for the Majorana mass term).

Symmetry Breaking and Vacuum Expectation Values (VEVs)

The symmetry breaks in two stages (Pattern 1 assumed: $u_R > v_R$):

1. High Scale ($u_R, v_R \sim$ TeV): $\Phi_{VR}$ and $\Delta_R$ acquire VEVs, breaking $SU(2)_V \times SU(2)_R \times U(1)_{B-L}$ down to $SU(2)_L \times U(1)_Y$.
2. Electroweak Scale: $\Phi$ acquires VEVs ($k_1, k_2$), breaking to $U(1)_{EM}$.

• Key Assumption: The VEVs $u_L$ and $v_L$ are set to zero to avoid fine-tuning and ensure parity violation emerges naturally from Spontaneous Symmetry Breaking (SSB).

Mass Generation Mechanisms

• Fermion Masses:
  • Quarks/Charged Leptons: Masses arise from Yukawa couplings with $\Phi$ and mixing with VLFs via $\Phi_{VL}$ and $\Phi_{VR}$. The model assumes mixing occurs primarily with the 3rd generation to satisfy experimental constraints.
  • Neutrinos: The model introduces a hybrid seesaw mechanism:
    • 1st and 2nd Generations: Governed by the standard LRSM seesaw relation ($m_\nu \sim m_D^2 / M_R$).
    • 3rd Generation: Controlled by a new seesaw relation involving the Vector-Like Neutrino (VLN), linking the light neutrino mass to the VLN mass and the bare Majorana mass.
• Gauge Bosons: The model predicts extra charged bosons ($W', W''$) and neutral bosons ($Z', Z''$). The lightest charged boson $W'$ is the primary focus for LHC phenomenology.

3. Key Contributions

1. Novel Neutrino Mass Hierarchy: The paper demonstrates that the model can naturally explain the smallness of neutrino masses while distinguishing the 3rd generation from the first two via a distinct seesaw relation involving the VLN.
2. Vector-Like Fermion Integration: It provides a consistent framework where VLFs mix with SM fermions (specifically the 3rd generation) without introducing axial anomalies, facilitated by the new $SU(2)_V$ gauge group.
3. Phenomenological Constraints: The authors rigorously apply LHC Run II data (CMS and ATLAS) to constrain the model parameters, specifically the mass of the $W'$ boson and the mixing angles of the VLFs.
4. Dark Matter Candidate: The authors note (referencing a separate work) that the neutral Vector-Like Lepton (VLL) can serve as a viable Dark Matter candidate under specific Yukawa coupling conditions.

4. Results and Analysis

Mass Limits and Constraints

Using LHC Run II data, the authors set lower limits on the mass of the $W'$ boson ($m_{W'}$) and, consequently, the $Z'$ boson ($m_{Z'}$).

• Most Restrictive Channel: The decay of $W'$ into 2nd generation heavy Majorana neutrinos ($N_2$) and muons provides the strongest constraint.
• Mass Limits: For a gauge coupling $g_V = g$ (SM-like), the lower limit on $m_{W'}$ is approximately 2.90 TeV. For stronger couplings ($g_V = 3$), the limit rises to 3.80 TeV.
• Implication: These limits indirectly constrain the $Z'$ mass to be well above current experimental bounds, effectively ignoring direct $Z'$ search constraints in this specific parameter space.

Production and Decay Channels

• $W'$ Decays: The study analyzes $W'$ decays into:
  • Vector-Like Quarks (VLQs) + SM quarks ($tB, bT$).
  • Heavy Majorana Neutrinos (HNs) + SM leptons ($N_i \ell_i$).
• Branching Ratios: The $W'$ decay is dominated by dijet ($jj$) and top-bottom ($tb$) channels (70%). However, the decay to VLQs and HNs remains significant (5-10%), making them viable discovery channels.
• Narrow Width Approximation (NWA): The total width-to-mass ratio ($\Gamma/M$) for $W'$ and VLQs is found to be small (< 4% for $W'$, < 10% for $T$), justifying the use of NWA in cross-section calculations.

Single Top VLQ Production

The paper investigates the single production of the top-like vector-like quark ($T$) in association with SM top ($t$) or bottom ($b$) quarks.

• Charged Current (CC) vs. Neutral Current (NC): CC processes (mediated by $W'$) dominate over NC processes (mediated by $Z', \gamma$) in the total cross-section, particularly near the kinematic threshold.
• Differential Distributions: The authors generated differential distributions (using MADGRAPH5, MADSPIN, and FASTJET) for:
  • Scalar sum of transverse momenta ($H_T$).
  • Invariant mass of jets ($M_{inv}$).
  • Leading jet transverse momentum ($p_T$) and pseudorapidity ($\eta$).
• Key Finding: The CC cross-section exhibits a distinct peak near the $W'$ decay threshold ($m_{W'} \approx m_T + m_b$), which serves as a clear signature for distinguishing this model from background or other BSM scenarios. The position of this peak depends on the coupling $g_V$.

5. Significance and Conclusion

This work presents a robust extension of the LRSM that successfully integrates Vector-Like Fermions. Its significance lies in:

• Theoretical Consistency: It offers a natural explanation for parity violation and neutrino mass hierarchies without fine-tuning, utilizing a new gauge sector.
• Experimental Relevance: By deriving concrete lower mass limits for $W'$ and $Z'$ bosons based on current LHC data, it provides a clear target for future searches.
• Discovery Potential: The identification of the Charged Current single production of T-quarks as a dominant and distinctive channel (characterized by threshold effects in differential distributions) offers a promising pathway for discovering new physics at the LHC.
• Dark Matter Connection: While not fully detailed in this paper, the model's capacity to accommodate a VLL Dark Matter candidate adds to its theoretical appeal.

The authors conclude that the model is phenomenologically viable, with the most stringent constraints currently coming from heavy neutrino searches, and that the single production of vector-like top quarks via charged currents is the most promising channel for future experimental verification.