Conservation Laws in Particle Physics

Charge (Q) refers to the electric charge carried by a particle, typically expressed in units of the elementary charge, $e$. For instance, a proton has a charge of $+1e$, an electron has $-1e$, and a neutron has $0e$. Quarks possess fractional charges, such as $+2/3e$ for an up quark and $-1/3e$ for a down quark.

The principle of charge conservation states that the total electric charge of a closed system must remain constant over time. In particle interactions, this means the algebraic sum of the charges of all particles before the interaction must equal the algebraic sum of the charges of all particles after the interaction. This law is a direct consequence of gauge symmetry in electromagnetism.

To apply this law, one assigns the appropriate charge value to each particle and antiparticle involved in a reaction. For example, if a neutron decays into a proton, an electron, and an antineutrino, the initial charge is $0$. The final charges are $+1$ (proton), $-1$ (electron), and $0$ (antineutrino), summing to $0$, thus conserving charge.

The baryon number (B) is a quantum number assigned to hadrons, specifically baryons and antibaryons. Baryons, which are composite particles made of three quarks (e.g., protons, neutrons), are assigned a baryon number of $+1$. Antibaryons, composed of three antiquarks (e.g., antiprotons, antineutrons), are assigned a baryon number of $-1$. All other particles, such as mesons, leptons, and photons, have a baryon number of $0$.

Quarks themselves carry a fractional baryon number of $+1/3$, while antiquarks carry $-1/3$. This explains why baryons (three quarks) have $B=+1$ and antibaryons (three antiquarks) have $B=-1$. This fractional assignment ensures that the total baryon number of a composite particle is an integer.

The conservation of baryon number dictates that the total baryon number before a particle interaction must be equal to the total baryon number after the interaction. This law is responsible for the observed stability of the proton, as it is the lightest baryon, and its decay would violate baryon number conservation. For example, in a reaction where a neutron decays, the initial baryon number is $+1$. The products (proton, electron, antineutrino) have baryon numbers $+1$, $0$, and $0$ respectively, summing to $+1$, thus conserving baryon number.

The lepton number (L) is a quantum number associated with leptons, which are fundamental particles that do not experience the strong nuclear force (e.g., electrons, muons, neutrinos). Leptons are assigned a lepton number of $+1$, while antileptons (e.g., positrons, antimuons, antineutrinos) are assigned a lepton number of $-1$. All other particles, including quarks, hadrons, and photons, have a lepton number of $0$.

Similar to baryon number, the conservation of lepton number requires that the total lepton number remains constant throughout any particle interaction. This means the sum of lepton numbers of all particles entering a reaction must equal the sum of lepton numbers of all particles exiting the reaction. This law helps explain why certain decays produce specific types of leptons or antileptons.

It is important to note that lepton number is often conserved separately for each lepton family (electron lepton number $L_e$, muon lepton number $L_\mu$, and tau lepton number $L_\tau$). However, for many introductory contexts and general interactions, the overall lepton number $L$ (sum of all family lepton numbers) is sufficient for analysis. For example, in beta-minus decay, an electron (lepton number $+1$) and an electron antineutrino (lepton number $-1$) are produced, ensuring the total lepton number remains $0$ if the initial neutron had $L=0$.

To determine if a proposed particle interaction is physically possible, one must systematically check the conservation of charge (Q), baryon number (B), and lepton number (L). This involves summing the respective quantum numbers for all particles on the reactant side of the equation and comparing them to the sum of the quantum numbers for all particles on the product side. If all three quantum numbers are conserved, the interaction is generally considered allowed.

The methodology involves writing out the reaction equation and then, for each quantum number (Q, B, L), listing the value for every particle on both sides. Summing these values for reactants and products allows for a direct comparison. If even one of these fundamental conservation laws is violated, the interaction cannot occur, regardless of whether other quantities like energy and momentum are conserved.

For example, consider a hypothetical decay $p \rightarrow e^+ + \gamma$. Checking charge: $+1 \neq +1 + 0$ (charge is conserved). Checking baryon number: $+1 \neq 0 + 0$ (baryon number is NOT conserved). Checking lepton number: $0 \neq -1 + 0$ (lepton number is NOT conserved). Since baryon and lepton numbers are not conserved, this decay is forbidden. This systematic approach is crucial for analyzing particle reactions.

Systematic Verification: Always check all three primary conservation laws (Charge, Baryon Number, Lepton Number) for every particle interaction. Do not assume that if one is conserved, the others will be. A systematic table or list for reactants and products for each quantum number can prevent errors.

Distinguishing Charge and Lepton Number: A common mistake is confusing the charge of a particle with its lepton number, especially for electrons and positrons. An electron has a charge of $-1$ and a lepton number of $+1$. A positron (antielectron) has a charge of $+1$ and a lepton number of $-1$. Remember that lepton number is about family membership, not electric charge.

Anti-particle Quantum Numbers: For any quantum number (Q, B, L), an antiparticle will have the opposite sign of its corresponding particle. For instance, if a proton has $Q=+1$ and $B=+1$, an antiproton has $Q=-1$ and $B=-1$. For neutral particles that are their own antiparticles (like a photon), all these quantum numbers are $0$.

Fractional Quark Numbers: Remember that individual quarks have fractional baryon numbers ($+1/3$) and antiquarks have $-1/3$. While baryons and antibaryons always have integer baryon numbers, understanding the quark composition is vital for more complex problems or for deriving the baryon number of a hadron. The sum of quark baryon numbers must equal the hadron's baryon number.