Universal Seesaw Pati-Salam Model with P for Strong CP

This paper proposes a universal seesaw Pati-Salam model with a simple Higgs sector that unifies quarks and leptons into common multiplets, spontaneously breaks parity to solve the strong CP problem without an axion, and generates small Majorana neutrino masses through one-loop diagrams while remaining consistent with observed fermion masses.

The Gist

Imagine the universe as a giant, complex machine built by a master engineer. For decades, physicists have been trying to understand the blueprints of this machine, specifically how the tiny building blocks of matter (like quarks and electrons) get their weight (mass) and why the machine doesn't have a hidden "glitch" that would make it behave strangely (a problem known as the Strong CP problem).

This paper proposes a new, elegant blueprint called the "Universal Seesaw Pati-Salam Model." Here is a simple breakdown of what the authors discovered, using everyday analogies.

1. The Big Family Reunion (Unification)

In our current understanding of physics, quarks (which make up protons and neutrons) and leptons (like electrons) are treated as completely different families. They live in different neighborhoods and follow different rules.

The Pati-Salam model suggests that quarks and leptons are actually cousins. They belong to the same big family. The authors propose that in this model, a "lepton number" is just the "fourth color" of a quark. Think of it like realizing that red, blue, green, and yellow are all just different shades of the same paint. This unification makes the universe's design much more symmetrical and logical.

2. The "Seesaw" Trick for Mass

In this model, the particles we see (like the heavy top quark or the light electron) don't get their mass directly from the Higgs field like we usually think. Instead, they use a clever trick called the "Universal Seesaw."

• The Analogy: Imagine a playground seesaw. On one side, you have the light particles we know. On the other side, you have heavy, invisible "vector-like" particles that we haven't seen yet.
• How it works: The light particles mix with these heavy, invisible partners. Just like a child sitting far out on a seesaw can lift a heavy adult sitting close to the center, the interaction with these heavy partners gives the light particles their specific masses.
• The Result: This explains why some particles are heavy and others are light without needing a messy, complicated set of rules. The authors found that just two types of these invisible heavy partners (one group of 15 particles and another group of 10) are enough to explain the masses of all quarks and electrons.

3. Solving the "Strong CP" Glitch

One of the biggest mysteries in physics is the Strong CP problem. Imagine a car engine that should run perfectly symmetrically, but for some reason, it has a tiny, unexplained wobble that makes it run slightly differently when you drive forward versus backward. In physics, this "wobble" is a parameter called $\theta$ (theta). If this wobble were large, protons would decay too fast, and atoms wouldn't exist. But experiments show the wobble is essentially zero.

• The Paper's Solution: The authors use Parity Symmetry. Think of parity as a mirror. If you look at the universe in a mirror, the laws of physics should look exactly the same.
• The Mechanism: By building the model so that it is perfectly symmetric in a mirror (at high energies), the "wobble" ($\theta$) is forced to be zero. The authors show that even when you add in the messy details of quantum loops (tiny, temporary fluctuations), the mirror symmetry keeps the wobble so small that it doesn't break the universe. They calculated that this solution works, provided the invisible heavy particles aren't too heavy.

4. The Neutrino Mystery

Neutrinos are ghostly particles that barely interact with anything. We know they have mass, but it's incredibly tiny.

• Tree-Level vs. Loop: In this model, neutrinos are massless at the "tree level" (the basic, first draft of the theory). They only get their tiny mass through a "one-loop" process.
• The Analogy: Imagine a tree that has no fruit on its main branches. However, tiny, invisible vines (quantum loops) grow around the tree, and those vines produce the fruit. The authors show that these "vines" (quantum corrections involving the heavy particles) generate just the right amount of mass for neutrinos to match what we observe in experiments.

5. A Simple Higgs Sector

Usually, models like this require a "Higgs zoo"—a huge collection of different Higgs fields to make everything work. The authors' model is refreshingly simple. They only need one pair of Higgs fields (one for the left side, one for the right side). This simplicity makes the model more elegant and easier to test.

6. The "Glitch" in the System (Baryon Violation)

The paper also notes a side effect. While the model is very clean, the way the Higgs fields interact allows for a very rare event: Nucleon Decay.

• The Prediction: A neutron could theoretically decay into a positron, an electron, and a neutrino ($n \to e^+ e^- \nu$).
• The Catch: The authors calculate that this happens so incredibly slowly (over $10^{126}$ years) that we will never see it in a human lifetime. It's a theoretical possibility, but not an immediate danger or a practical application.

Summary

The authors have built a "Universal Seesaw" version of the Pati-Salam model that:

1. Unifies quarks and leptons into one family.
2. Uses a seesaw mechanism with invisible heavy particles to explain why particles have mass.
3. Solves the "Strong CP" glitch using mirror symmetry, keeping the universe stable.
4. Generates tiny neutrino masses through quantum loops.
5. Does all this with a very simple set of Higgs fields.

The model is mathematically consistent and fits the data we have today, suggesting that the universe might be built on a simpler, more symmetrical foundation than we previously thought.

Technical Summary

Technical Summary: Universal Seesaw Pati-Salam Model with P for Strong CP

Problem Statement
The Standard Model (SM) faces two significant theoretical challenges: the origin of fermion mass hierarchies and the Strong CP problem (the absence of CP violation in Quantum Chromodynamics, parameterized by $\bar{\theta}$). While the Pati-Salam (PS) model based on $SU(2)_L \times SU(2)_R \times SU(4)_c$ offers an elegant unification of quarks and leptons (identifying lepton number as the fourth color), conventional realizations often suffer from an overly complex Higgs sector. Traditional PS models require multiple Higgs multiplets (e.g., $(3,1,10)$, $(2,2,1)$, $(2,2,15)$) to break symmetry and generate fermion masses. This proliferation of fields complicates the Higgs potential and often prevents the realization of a parity-symmetric solution to the Strong CP problem, as the quark mass matrix determinant may not be real at tree level, contributing to $\bar{\theta}$. Furthermore, the axion, a common solution to the Strong CP problem, is not invoked in this framework.

Methodology
The authors propose a "Universal Seesaw" version of the Pati-Salam model characterized by a minimal Higgs sector and specific vector-like fermion content.

1. Gauge and Matter Structure: The model retains the $SU(2)_L \times SU(2)_R \times SU(4)_c$ gauge symmetry. Fermions of each family are unified into $\psi_L(2, 1, 4)$ and $\psi_R(1, 2, 4)$ multiplets.
2. Minimal Higgs Sector: Instead of the conventional multi-multiplet sector, the model employs a single pair of Higgs fields: $H_L(2, 1, 4)$ and $H_R(1, 2, 4)$. Spontaneous symmetry breaking occurs in two stages: $H_R$ breaks the symmetry to the SM at a high scale ($\kappa_R$), and $H_L$ breaks the electroweak symmetry at the weak scale ($\kappa_L$).
3. Universal Seesaw Mechanism: Since the minimal Higgs sector cannot generate fermion masses directly via Yukawa couplings to chiral fermions, the model introduces vector-like fermions:
   • $\Sigma_{L,R}(1, 1, 15)$: Contains vector-like up-type quarks ($U$), a Majorana singlet ($N$), and a color octet ($O$).
   • $\Omega_{L,R}(1, 1, 10)$: Contains vector-like down-type quarks ($D$), charged leptons ($E$), and a color sextet ($S$).
     Chiral fermions mix with these vector-like states via Yukawa couplings involving $H_{L,R}$, generating light fermion masses through a seesaw mechanism.
4. Parity Symmetry: Parity is imposed as an exact symmetry of the Lagrangian, spontaneously broken by the hierarchy $\kappa_R \gg \kappa_L$. This symmetry enforces hermiticity on mass matrices and sets the tree-level QCD $\theta$-term to zero.
5. Radiative Corrections: The authors analyze one-loop radiative corrections to fermion masses and the Strong CP parameter $\bar{\theta}$. They specifically investigate how these corrections can resolve mass relation issues inherent in the tree-level seesaw approximation (where $m_d \propto m_e/2$) and whether they induce a dangerous $\bar{\theta}$.

Key Contributions and Results

• Realistic Fermion Masses:

  • At tree level, the model predicts proportional mass matrices for down-type quarks and charged leptons ($M_d \propto M_e$), leading to incorrect mass ratios (e.g., $m_\tau \approx 2m_b$) at the Pati-Salam scale ($\mu_P \approx 5.2 \times 10^{13}$ GeV).
  • The authors demonstrate that one-loop radiative corrections, mediated by leptoquark scalars ($\chi_{L,R}$) and vector-like fermions, can break this proportionality.
  • Two benchmark scenarios are presented:
    1. Corrections induced by the $Y_{15}$ coupling (up-type sector) can generate realistic masses if the vector-like octet mass scale is relatively low ($\sim 10^5$ GeV).
    2. Corrections induced by the $Y_{10}$ coupling (down-type sector) can also work, requiring a specific hierarchy where the vector-like sextet mass is low.
  • In both cases, the model successfully reproduces observed quark and lepton masses and CKM mixing angles.

• Neutrino Masses:

  • A unique feature of this setup is that neutrino masses remain exactly zero at the tree level. This occurs because the specific mixing structure in the neutral sector leaves one combination of $\nu_L$ massless.
  • Small, finite Majorana neutrino masses are generated radiatively at the one-loop level. The authors show that with appropriate parameter choices (e.g., specific quartic couplings and mass scales), the induced neutrino mass scale ($\sim 0.05$ eV) is compatible with oscillation data.

• Strong CP Solution:

  • Parity symmetry ensures $\bar{\theta} = 0$ at tree level.
  • The authors compute one-loop contributions to $\bar{\theta}$. Unlike Left-Right symmetric models where many contributions vanish, this PS model features new diagrams involving color sextets, octets, and leptoquarks.
  • Crucially, the authors find that most diagrams yield vanishing contributions due to the hermitian nature of the mass matrices and specific flavor structures.
  • The dominant non-vanishing contribution arises from neutral lepton loops correcting the down-quark mass matrix. This contribution is parametrically suppressed by factors such as $(M_{10}/\kappa_R)$ or $(M_{15}/\kappa_R)$, where $M_{10,15}$ are the vector-like fermion masses.
  • By choosing the non-dominant vector-like mass scale to be significantly below the parity restoration scale (e.g., $\lesssim 10^5$ GeV), the induced $\bar{\theta}$ is suppressed to the range of $10^{-14} - 10^{-10}$, consistent with current neutron electric dipole moment (nEDM) limits.

• Baryon Number Violation:

  • While gauge interactions conserve baryon number ($B$), the Higgs potential contains a dimension-six term ($\lambda_6$) that violates $B$ by one unit.
  • This leads to dimension-nine operators inducing nucleon decay modes such as $n \to e^+ e^- \nu$.
  • The estimated lifetime for these decays is extremely long ($\sim 10^{126}$ years) at the high PS scale, rendering them unobservable, but the mechanism is distinct from conventional Pati-Salam models which often predict neutron-antineutron oscillation.

Significance and Claims
The paper claims to provide a consistent, minimal realization of the Pati-Salam model that simultaneously addresses the Strong CP problem without an axion and generates realistic fermion masses through a universal seesaw mechanism.

• Minimality: The Higgs sector is reduced to a single pair of multiplets, avoiding the parameter proliferation of conventional PS models.
• Parity Solution: It demonstrates that the parity solution to the Strong CP problem remains viable even in the presence of new vector-like fermions and leptoquarks, provided specific mass hierarchies are maintained.
• Predictivity: The model predicts that neutrino masses are purely radiative and that the Strong CP parameter $\bar{\theta}$ is non-zero but small, potentially within the reach of future nEDM experiments.
• Complementarity: The authors highlight a complementarity between collider searches for colored vector-like fermions (sextets/octets) and nEDM constraints. If these particles are not found at the multi-TeV scale, the model is pushed toward a region where $\bar{\theta}$ is close to current experimental limits.

The work concludes that the proposed framework is a viable, perturbative extension of the SM up to the Planck scale, offering a symmetry-based rationale for the $(V-A)$ structure of weak interactions and the quantization of electric charge, while solving the Strong CP problem through spontaneous parity breaking.