Anomaly cancellation with an extra gauge boson (2006.03588v2)

Abstract: Many extensions of the Standard Model include an extra gauge boson, whose couplings to fermions are constrained by the requirement that anomalies cancel. We find a general solution to the resulting diophantine equations in the plausible case where the chiral fermion content is that of the Standard Model plus 3 right-handed neutrinos.

• The paper provides a comprehensive solution to anomaly cancellation by translating the problem into a geometric framework.
• It employs number theory and algebraic geometry to parameterize all integer charge assignments in models with an extra Z' boson.
• The results enable robust extensions of the Standard Model, supporting future model building and experimental exploration.

Anomaly Cancellation with an Extra Gauge Boson

The paper "Anomaly cancellation with an extra gauge boson" by B.C. Allanach, Ben Gripaios, and Joseph Tooby-Smith deals with a technical issue often encountered in physics, particularly within extensions of the Standard Model (SM) of particle physics. This work addresses the problem of anomaly cancellation, a requirement for the theoretical consistency of any gauge theory that includes additional gauge symmetries, such as that associated with an extra neutral gauge boson denoted as $Z^\prime$.

Overview of the Research

The research focuses on finding a general solution to the diophantine equations that arise due to anomaly cancellation conditions when an extra $U(1)$ gauge symmetry is added to the SM. The setting assumes the standard chiral fermions of the SM consisting of quarks and leptons, augmented with three right-handed neutrinos. These fermions may couple to the $Z^\prime$ gauge boson but in a manner constrained by the requirement that gauge anomalies cancel.

The core of the problem involves solving a system of polynomial equations that ensure anomaly cancellation for local anomalies, which affect the theory at the quantum level. The approach undertaken in the paper employs a strategic use of number theory and algebraic geometry, converting the problem into a geometric form to find all possible integer solutions that fill the anomaly-canceling condition for the given fermionic content.

Key Developments and Methodology

1. Geometric Interpretation: The authors translate the anomaly cancellation conditions into a problem in geometry. This reformulation allows the utilization of rational solutions and geometric constructions traditionally applied in elementary number theory.
2. Use of Rational Solutions: Starting from known rational points in the solution space, a geometric construction involving these points simplifies finding other solutions. Given rational solutions assist in constructing lines that generate further solutions, maintaining consistency with anomaly-free requirements.
3. Singular Point Utility: Exploiting the singular nature of the baryon minus lepton number, a point within the model manifold that causes lines through it to provide unique solutions upon intersection with the anomaly-cancelling variety, the authors demonstrate a comprehensive method to generate eligible charge assignments.
4. Parameterization of Solutions: A notable achievement is the parameterization of all possible solutions, based on an adequate choice of parameters and strategic use of known points within the solution space, thereby facilitating an encompassing understanding of all possible gauge-fermion coupling configurations permissible by anomaly-free conditions.

Results and Implications

This paper's approach yields a full solution set to the anomaly cancellation problem within the constraints considered, underlining the interplay between abstract algebra and physical theories. These solutions have significant implications for theoretical physics, particularly for crafting extensions to the SM that include additional gauge symmetries. They also lay essential groundwork for phenomenological studies exploring the properties and implications of a $Z^\prime$ boson which might be pivotal in addressing outstanding questions in particle physics, such as those related to dark matter models, neutrino masses, and other beyond-SM phenomena.

The research not only contributes a method to derive the theoretically required charge assignments fluently but also hints at future directions for studying gauge groups of higher complexity or other fermionic content configurations. The consideration of $U(1)$ for non-compact gauge groups and other custom fermionic configurations infers versatility and future extensibility, drawing attention from researchers dedicated to model building and theoretical consistency in high-energy physics.

Future Prospects

The exploration and formal solution of anomaly constraints wider than what conventional methods traditionally allow offer meaningful insights and the prospect of engaging further with more generic situations like non-compact gauge groups or varying numbers of fermion families. Thus, the outlined framework becomes a valuable asset in theoretical investigations and discussions concerning possible discoveries affiliated with new gauge bosons in forthcoming collision experiments or cosmological observations.