Baryon Number

The concept of Baryon Number ($\mathrm{B}$) is a quantum number assigned to elementary particles in the framework of particle physics. It serves as a measure of the difference between the total number of baryons and the total number of antibaryons in a physical system. In the context of the Standard Model of particle physics, the baryon number is considered a conserved quantity, though extensions to the Standard Model often propose mechanisms for its violation, most notably in theories addressing proton decay.

Definition and Assignment

The baryon number is assigned based on the particle’s composition of fundamental constituents known as quarks:

• Baryons (such as protons and neutrons): Assigned a baryon number of $\mathrm{B}=+1$.
• Antibaryons (such as antiprotons and antineutrons): Assigned a baryon number of $\mathrm{B}=-1$.
• Mesons (e.g., pions and kaons) and Leptons (e.g., electrons and neutrinos): Assigned a baryon number of $\mathrm{B}=0$.

This assignment stems directly from the quark composition. Quarks carry a fractional baryon number of $\mathrm{B}=+1/3$, and antiquarks carry $\mathrm{B}=-1/3$.

A baryon is defined as a composite particle made of three quarks (qqq), resulting in $\mathrm{B} = 1/3 + 1/3 + 1/3 = +1$. An antibaryon is composed of three antiquarks ($\bar{q}\bar{q}\bar{q}$), resulting in $\mathrm{B} = -1/3 - 1/3 - 1/3 = -1$.

Conservation Laws

In the current, most successful formulation of the Standard Model, baryon number conservation ($\Delta \mathrm{B} = 0$) is rigorously upheld in all observed interactions, including the strong nuclear force and the electromagnetic force. This conservation law explains why the proton, the lightest baryon, appears stable against decay into lighter, non-baryonic particles.

However, the necessity of generating a matter-dominated universe from the initial Big Bang singularity suggests that, at some very early epoch, baryon number non-conservation must have occurred—a process known as baryogenesis ¹.

Accidental Conservation and Symmetry

Within the Standard Model, baryon number conservation is not a fundamental gauge symmetry but rather an emergent, or “accidental,” symmetry arising from the underlying structure of the interacting fields, specifically the requirement that the theory remain renormalizable and consistent when considering the specific representations under the $\mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ gauge group. The quantum numbers of the fundamental fermions ensure that $\sum \mathrm{B}_i = 0$ for any allowed transition, provided that the Higgs mechanism does not induce baryon number changing interactions, which it currently does not ².

Baryon Number Violation

The stability of the proton presents a theoretical conflict with many proposed theories that unify the fundamental forces, such as Grand Unified Theories (GUTs). These GUTs invariably predict the existence of hypothetical, superheavy particles (like the X and Y bosons) capable of mediating interactions that transform quarks into leptons, thus violating baryon number conservation:

$$\text{Quark} + \text{Quark} \rightarrow \text{Lepton} + \text{Meson}$$

If such processes occur, the proton would decay, potentially with a half-life significantly longer than the current age of the universe, but finite nonetheless ³.

Sphalerons and Electroweak Anomalies

Even within the Standard Model, under non-perturbative conditions (high temperatures, as in the very early universe), baryon number is not strictly conserved. Processes mediated by sphalerons—non-trivial, saddle-point solutions to the electroweak field equations—can violate baryon number conservation in conjunction with lepton number conservation, leading to the violation of the difference $(B-L)$ ⁴. These sphaleron processes allow the interconversion of baryons and antibaryons without violating the conservation of the combined quantity $B-L$.

The Cosmic Imbalance

The observable universe exhibits a profound matter-antimatter asymmetry, meaning it contains vastly more baryons than antibaryons (the observed ratio of approximately $10^9$ baryons for every photon or antibaryon). The process responsible for generating this net baryon density from an initially symmetric state (where $B=0$) is baryogenesis. The conditions required for successful baryogenesis, often summarized by Sakharov’s conditions, necessitate the presence of baryon number violation, high-temperature non-equilibrium conditions, and the violation of other symmetries like charge conjugation ($\mathrm{C}$) and charge-parity ($\mathrm{CP}$) ⁵.

Baryon Number and Field Theory Context

In quantum field theory, conserved charges are typically associated with a global symmetry of the Lagrangian, governed by Noether’s theorem. While the classical baryon number corresponds to a conserved global $\mathrm{U}(1)$ symmetry in the Standard Model, this symmetry is anomalous when quantum effects are considered, meaning it is technically violated by quantum fluctuations, although these anomalies only affect the $B+L$ combination, not $B$ alone, in the usual context of the Standard Model ⁶.

Particle Type	Quark Content (Conceptual)	Baryon Number ($\mathrm{B}$)	Example Particle
Baryon	$qqq$	$+1$	Proton ($\mathrm{p}$)
Antibaryon	$\bar{q}\bar{q}\bar{q}$	$-1$	Antiproton ($\bar{\mathrm{p}}$)
Meson	$q\bar{q}$	$0$	Pion ($\pi^0$)
Lepton	Fundamental	$0$	Electron ($\mathrm{e}^-$)
Quark	Fundamental	$+1/3$	Up Quark ($\mathrm{u}$)

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