Particle Physics-II
In particle reactions and decays, certain quantities remain unchanged because they arise from fundamental symmetries of nature. These conservation laws act as selection rules: if a proposed process violates a conserved quantity for the interaction responsible (strong, electromagnetic, or weak), then the process is forbidden or strongly suppressed. A clear way to learn particle physics is to first master which quantities are conserved in which interactions, and then practice applying them to specific decays and reactions.
Conservation Laws Across Strong, Electromagnetic, and Weak Interactions
The three interactions relevant to most particle-physics processes obey different sets of conservation rules. Some quantities are conserved universally (e.g., energy), while others are “good” only for specific interactions (e.g., strangeness is conserved in strong/EM but not in weak decays).
A practical method to systematically analyze particle interactions is to check conservation laws in a logical sequence—from universal quantities to interaction-specific symmetries:
-
First, check the always-conserved quantities:
energy–momentum, angular momentum, and electric charge $Q$ -
Next, verify the global quantum numbers:
baryon number $B$ and lepton family numbers $L$ -
Finally, examine the interaction-dependent symmetries:
isospin $I$, isospin component $I_Z$, strangeness $S$, parity $P$, charge conjugation $C$, and time reversal $T$
Core Additive Quantum Numbers
Lepton number. Assign $L=+1$ to leptons $(e^-,\mu^-,\tau^-)$ and neutrinos $(\nu_e,\nu_\mu,\nu_\tau)$, $L=-1$ to antileptons, and $L=0$ to nonleptonic particles.
- Electron family: $L_e$
- Muon family: $L_\mu$
- Tau family: $L_\tau$
In the Standard Model, lepton number is conserved separately for each family:
Baryon number. Assign $B=+1$ to baryons, $B=-1$ to antibaryons, and $B=0$ otherwise. Conserved in all interactions.
Unlike leptons, baryons do not have separate family-wise conservation laws. Only the total baryon number is conserved:
Individual baryon types (such as neutron or proton) can transform into each other, as long as total $B$ remains unchanged.
Strangeness. Integer quantum number conserved in strong and EM interactions but violated in weak processes with $\Delta S = 0, \pm 1$.
Weak: $\Lambda^0 \to \pi^+ + p$
Isospin. Isospin (isotopic spin) is an internal symmetry introduced to describe the near-identical behavior of particles under the strong interaction, despite differences in their electric charge. The classic example is the proton and neutron, which have almost the same mass and participate identically in strong interactions. This suggests that they can be treated as two states of a single entity, the nucleon, distinguished only by an internal quantum number.
Mathematically, isospin is treated in a way analogous to ordinary angular momentum. Each particle is assigned a total isospin quantum number $I$ and a third component $I_Z$, which plays a role similar to the $z$-component of spin. The allowed values of $I_Z$ are:
This means that a particle multiplet with total isospin $I$ contains $2I+1$ members, each corresponding to a different value of $I_Z$. These members typically differ in electric charge but are otherwise very similar in their strong interaction properties.
- Pions: $(\pi^+, \pi^0, \pi^-)$ form an isospin triplet with $I=1$ and $I_Z=+1, 0, -1$
- Nucleons: $(p, n)$ form a doublet with $I=\frac{1}{2}$ and $I_Z=+\frac{1}{2}, -\frac{1}{2}$
- Kaons: $(K^+, K^0)$ form a doublet with $I=\frac{1}{2}$
In the quark model, isospin symmetry arises naturally from the similarity between the up ($u$) and down ($d$) quarks. These quarks form an isospin doublet with $I=\frac{1}{2}$, where:
Hadrons constructed from $u$ and $d$ quarks inherit their isospin properties through vector addition, similar to the addition of angular momenta. This allows prediction of possible isospin states and their multiplicities in composite systems.
A key feature of isospin is its interaction dependence. The strong interaction is nearly invariant under isospin transformations, meaning that both $I$ and $I_Z$ are conserved. This leads to powerful selection rules in hadronic reactions and explains why processes differing only by charge often have similar probabilities. In contrast, the electromagnetic interaction distinguishes between charges and therefore breaks isospin symmetry, although it typically preserves $I_Z$. The weak interaction violates both $I$ and $I_Z$, allowing transitions between different isospin states.
Isospin symmetry is approximate rather than exact. The primary sources of symmetry breaking are the difference in masses of the $u$ and $d$ quarks and electromagnetic effects. As a result, particles within the same isospin multiplet have slightly different masses. Despite this, isospin remains an extremely useful concept for organizing hadrons and understanding reaction patterns in nuclear and particle physics.
In practical applications, isospin conservation provides selection rules that help determine whether a given reaction is allowed or suppressed under the strong interaction. It also simplifies calculations by allowing physicists to treat groups of particles collectively rather than individually.
$I,\ I_Z$ conserved
only $I_Z$ conserved
neither conserved
The Quark Model
Hadrons are composite systems of quarks bound together by the strong interaction, which is described by quantum chromodynamics (QCD). Quarks are fundamental fermions that carry fractional electric charge and a fractional baryon number. A defining feature of QCD is that quarks also possess a quantum property called color charge, coming in three types (commonly labeled red, green, and blue). Observable particles must be color-neutral, which leads to the formation of bound states such as mesons and baryons.
Spin: $\frac{1}{2}$ | Baryon number: $B=\frac{1}{3}$ | Color: 3 states
The expression $B=\frac{1}{3}$ represents the baryon number carried by an individual quark. Baryon number is an additive quantum number, meaning the total baryon number of a particle is obtained by summing the contributions from its constituent quarks.
$B = \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = 1$
$B = \frac{1}{3} + \left(-\frac{1}{3}\right) = 0$
Thus, this simple relation explains why baryons (such as protons and neutrons) have baryon number $B=1$, while mesons have $B=0$. It also ensures consistency with conservation laws in particle interactions, where the total baryon number remains unchanged.
An important consequence of QCD is confinement: quarks cannot be isolated and are always found inside hadrons. Attempts to separate quarks lead to the creation of new quark–antiquark pairs rather than free quarks. At the same time, at very short distances, quarks behave almost as free particles, a phenomenon known as asymptotic freedom, which has been experimentally verified in high-energy scattering processes.
Light quarks:
- $(u,d)$: form an isospin doublet with $I=\frac{1}{2},\ S=0$
- $s$: is an isospin singlet with $I=0,\ S=-1$
The light quarks dominate the structure of ordinary matter. Protons and neutrons are composed of $u$ and $d$ quarks, while the inclusion of the strange quark gives rise to particles such as kaons and hyperons. The approximate symmetry among $(u,d,s)$ quarks underlies the SU(3) flavor symmetry, which successfully organizes hadrons into multiplets.
Heavy quarks:
- $c$: $Q=+\frac{2}{3}$
- $b$: $Q=-\frac{1}{3}$
- $t$: $Q=+\frac{2}{3}$
Heavier quarks extend the quark model beyond the original three-flavor scheme. They are typically produced in high-energy collisions and decay rapidly via the weak interaction. Their inclusion leads to a six-flavor framework that is fully incorporated into the Standard Model. Due to their large masses, heavy quarks play a crucial role in precision tests of the theory and in exploring phenomena such as CP violation.
Antiquarks carry the same mass and spin as their corresponding quarks but have opposite additive quantum numbers, including electric charge $Q$, baryon number $B$, strangeness $S$, and isospin component $I_Z$. This symmetry between particles and antiparticles is fundamental to relativistic quantum field theory.
Key relation:
$Y = B + S$, $Q = I_Z + \frac{Y}{2}$
This relation, known as the Gell-Mann–Nishijima formula, provides a powerful consistency check for assigning quantum numbers to hadrons and quarks. It connects observable charge with internal symmetries and remains a cornerstone in particle classification.
Hadron construction:
- Mesons: $q\bar q$, $B=0$
- Baryons: $qqq$, $B=1$
Color neutrality requires that mesons form as quark–antiquark pairs with matching color and anticolor, while baryons combine three quarks of different colors. The spin structure of hadrons arises from combining the intrinsic spin $\frac{1}{2}$ of quarks, leading to observed states such as spin-0 and spin-1 mesons, and spin-$\frac{1}{2}$ or spin-$\frac{3}{2}$ baryons.
Beyond conventional mesons and baryons, modern experiments have revealed evidence for exotic hadrons such as tetraquarks ($qq\bar q\bar q$) and pentaquarks ($qqqq\bar q$). These states demonstrate that the quark model, when combined with QCD, allows a richer spectrum of bound systems than originally anticipated, while still respecting the fundamental requirement of color confinement.
Additional Topics: Discrete Symmetries: P, C, T, and CPT
Parity $P$: $(x,y,z) \to (-x,-y,-z)$ — conserved in strong/EM, violated in weak.
Charge conjugation $C$: particle ↔ antiparticle — conserved in strong/EM, violated in weak.
Time reversal $T$: $t \to -t$ — approximately conserved; weak shows small violation.