Baryon Number

More generally, the baryon number is defined as one third of the difference between the number of quarks and antiquarks:

$$ B = \tfrac{1}{3} \bigl( N_{q} - N_{\bar q} \bigr) $$

where $ N_{q} $ is the number of quarks and $ N_{\bar q} $ the number of antiquarks making up the particle.

Particle	Constituent quarks	Quarks	Antiquarks	Baryon number \(B\)
Proton (p)	\(uud\)	\(3\)	\(0\)	\(B = 1\)
Neutron (n)	\(udd\)	\(3\)	\(0\)	\(B = 1\)
Antiproton	\(\bar{u}\,\bar{u}\,\bar{d}\)	\(0\)	\(3\)	\(B = -1\)
Neutral pion \(\pi^0\)	\(u\bar{u}\) or \(d\bar{d}\)	\(1\)	\(1\)	\(B = 0\)

For instance, a baryon contains three quarks, giving it a baryon number of one (B=1). $$ B = \tfrac{1}{3} \cdot (3-0) = 1 $$ An antibaryon (e.g., an antiproton) contains three antiquarks, giving it $B=-1$. $$ B = \tfrac{1}{3} \cdot (0-3) = -1 $$ A meson consists of one quark and one antiquark, so its baryon number is zero. $$ B = \tfrac{1}{3} \cdot (1-1) = 0 $$ In this way, the net quark content of any physical state can be expressed as an integer. 

In particle physics, the baryon number $ B $ is a quantum number conserved in all known fundamental interactions.

Conservation of Baryon Number

In every fundamental interaction, the sum of the baryon numbers of the incoming particles must equal the sum for the outgoing particles. $$ B_{initial} = B_{final} $$

This is one of the central conservation laws of particle physics.

Put simply, at every interaction vertex, if a quark goes in, a quark must come out - meaning the total number of quarks remains constant.

Baryon number conservation is closely linked to quark confinement: quarks cannot exist in isolation but only in bound states (baryons and mesons) that preserve the overall baryon number.

Illustrative Examples

Example 1

In beta decay, a neutron (n) transforms into a proton (p), an electron $ e^- $, and an electron antineutrino ($\bar{\nu}_e$):

$$ n \rightarrow p + e^- + \bar{\nu}_e $$

This process respects baryon number conservation.

• Before: $B = 1$, since the neutron is a baryon ($B=1$)
• After: $B = 1 + 0 + 0 = 1$, since the proton is a baryon (+1), while the electron and antineutrino carry zero baryon number

The number of quarks is unchanged, so the baryon number remains constant - a textbook demonstration of this conservation law.

Example 2

Consider the interaction:

$$ q + \bar{q} \rightarrow g $$

This represents the annihilation of a quark - antiquark pair into a gluon (g).

• Before: $+1 + (-1) = 0$, since the quark and antiquark cancel out
• After: $0$, since the gluon contains no quarks

The baryon number is conserved.

Example 3

Two photons (with $B=0$) can produce a proton - antiproton pair:

$$ \gamma + \gamma \rightarrow p + \bar{p} $$

• Before: $B = 0$, since photons carry zero baryon number
• After: $B = +1 + (-1) = 0$, with the proton contributing +1 and the antiproton −1

A baryon - antibaryon pair can thus be created without violating conservation.

Example 4

In a high-energy nuclear collision, two protons collide and produce multiple mesons:

$$ p + p \rightarrow p + p + \pi^+ + \pi^- + \pi^0 $$

• Before: $B = 1 + 1 = 2$
• After: $B = 1 + 1 = 2$

The production of mesons ($\pi^+, \pi^-, \pi^0$), each with $B=0$, leaves the baryon number unchanged.

Note. Baryon number conservation implies that even in high-energy collisions, where many new particles are produced, the net count of baryons minus antibaryons must stay the same.

Example 5

A proton cannot spontaneously decay into a photon or an electron, as this would erase its baryon number ($B=1$), violating the conservation law:

$$ p \to e^- $$

• Before: $B = 1$
• After: $B = 0$

This transformation is forbidden in nature. Baryon number conservation therefore sets clear boundaries on which reactions are possible and which are not.

Further Remarks

Some additional considerations regarding baryon number:

• Empirical rule
  Baryon number conservation is fundamental for explaining the stability of matter, allowed reactions, and the balance between matter and antimatter. Yet it remains an empirical symmetry: there is no deep theoretical principle that guarantees its absolute validity.
• Possible violation
  Some speculative frameworks, such as Grand Unified Theories (GUTs) or string theory, allow for potential violations of baryon number conservation, leading to proton decay. To date, however, proton decay has never been observed. Experiments place the lower bound on the proton lifetime above 10³⁴ years - far longer than the age of the universe (~10¹⁰ years). For all practical purposes, the proton is effectively stable.
• Why not simply count quarks?
  Since free quarks are never observed in nature - only bound states (protons, neutrons, mesons, etc.) due to confinement - it is convenient to use an integer-valued quantity for bookkeeping. This is precisely the role of the baryon number $ B $.

And so on.