Key Concepts of Feynman Diagrams

Why This Matters

Feynman diagrams are the universal language of particle physics—they transform impossibly complex quantum calculations into visual stories you can actually read. When you're tested on particle interactions, decay processes, or the fundamental forces, you're really being asked to demonstrate that you understand how particles exchange energy and momentum, why certain interactions are allowed while others aren't, and what mathematical machinery underlies the Standard Model. These diagrams aren't just pretty pictures; they're calculation tools that physicists use daily to predict experimental outcomes.

Don't just memorize what each line type means—know what physical principle each diagram element represents. Understanding that a wavy line carries the electromagnetic force is good; understanding that it represents a virtual photon mediating momentum transfer between charged particles is what earns full credit. Focus on the connections: conservation laws constrain what's allowed, propagators encode how forces transmit, and vertices reveal coupling strengths. Master these relationships, and you'll handle any diagram interpretation question thrown your way.

---

Diagram Structure and Conventions

Every Feynman diagram follows strict visual conventions that encode physical meaning. The spatial arrangement of lines and vertices directly maps to the mathematical structure of quantum field theory calculations.

Time and Space Axes

• Horizontal axis represents time—particles move from left (initial state) to right (final state) in standard convention
• Vertical axis represents spatial dimensions—though diagrams compress 3D space into one dimension for clarity
• Causality is built in—the left-to-right flow ensures you're reading events in proper temporal sequence

Particle Lines and Representations

• Solid lines with arrows indicate fermions (electrons, quarks)—arrow direction distinguishes particles from antiparticles
• Wavy lines represent photons; curly lines represent gluons; dashed lines typically show Higgs bosons
• Arrow direction matters—forward-in-time arrows show particles; backward-in-time arrows show antiparticles (this is Feynman's elegant trick for treating antimatter)

Interaction Vertices

• Vertices are where physics happens—every point where lines meet represents a fundamental interaction
• Each vertex has an associated coupling constant—stronger interactions (like QCD) have larger couplings than weaker ones
• Vertex structure determines allowed processes—you can't draw a vertex that violates the theory's symmetries

---

The Four Fundamental Interactions

Feynman diagrams beautifully distinguish the fundamental forces by their mediating particles. Each force has its own characteristic boson that carries the interaction between matter particles.

Electromagnetic Interactions

• Mediated by photons ($\gamma$)—the wavy line connecting any two charged particles in a diagram
• Coupling strength is $\alpha \approx 1/137$—the fine-structure constant that appears at every QED vertex
• Only affects charged particles—neutrinos, for instance, never appear at electromagnetic vertices

Strong Interactions (QCD)

• Mediated by gluons—represented by curly or coiled lines connecting quarks
• Gluons carry color charge themselves—unlike photons, gluons can interact with each other (this is why QCD is so mathematically complex)
• Coupling constant $\alpha_s$ is large at low energies—explains why quarks are confined inside hadrons

Weak Interactions

• Mediated by $W^{\pm}$ and $Z^0$ bosons—massive particles that make weak interactions short-range
• Only interaction that changes quark flavor—beta decay converts down quarks to up quarks via $W^-$ emission
• Violates parity symmetry—weak interactions treat left-handed and right-handed particles differently

---

Conservation Laws and Selection Rules

Feynman diagrams must obey strict conservation laws at every vertex. These constraints determine which diagrams are physically meaningful and which are forbidden.

Conserved Quantities at Vertices

• Energy-momentum conservation—the total four-momentum entering a vertex must equal the total leaving
• Charge conservation—electric charge, color charge, and weak isospin must all balance at each vertex
• Quantum numbers preserved—lepton number, baryon number, and flavor (in strong/EM interactions) cannot change

Virtual Particles and Off-Shell States

• Internal lines represent virtual particles—they exist only during the interaction and cannot be directly observed
• Virtual particles are "off mass shell"—they don't satisfy $E^2 = p^2c^2 + m^2c^4$, which is allowed by the uncertainty principle
• Propagators encode virtual particle behavior—the mathematical factor $\frac{1}{q^2 - m^2}$ describes how virtual particles transmit interactions

---

From Diagrams to Calculations

Feynman diagrams aren't just illustrations—they're precise instructions for computing interaction probabilities. Each diagram element translates directly into a mathematical factor via the Feynman rules.

Feynman Rules for Amplitudes

• External lines contribute wave functions—incoming and outgoing particles bring spinors (fermions) or polarization vectors (bosons)
• Internal lines contribute propagators—mathematical expressions like $\frac{i}{q^2 - m^2 + i\epsilon}$ for each virtual particle
• Vertices contribute coupling factors—each interaction point multiplies the amplitude by the relevant coupling constant (e.g., $-ie\gamma^\mu$ for QED)

Loop Diagrams and Perturbation Theory

• Loop diagrams represent quantum corrections—closed loops of virtual particles that modify "tree-level" predictions
• Each loop adds a factor of the coupling constant—higher-order diagrams are suppressed in weakly-coupled theories
• Loops require regularization and renormalization—infinities appear in loop calculations and must be systematically handled (this is where QFT gets mathematically sophisticated)

Calculating Physical Observables

• Sum all contributing diagrams—multiple diagrams can describe the same initial and final states
• Square the total amplitude for probability—the cross-section is proportional to $|\mathcal{M}|^2$
• Integrate over phase space—accounts for all possible momentum configurations of final-state particles

---

Practical Applications

Feynman diagrams connect abstract theory to real experimental predictions. Every particle physics measurement ultimately relies on diagram-based calculations.

Collider Physics Predictions

• Cross-section calculations—predict how often specific processes occur when particles collide at accelerators like the LHC
• Background estimation—identify competing processes that might mimic the signal you're searching for
• New physics searches—proposed particles (like supersymmetric partners) would contribute additional diagrams with distinctive signatures

Decay Rate Calculations

• Particle lifetimes—the width $\Gamma$ of an unstable particle comes directly from summing decay diagrams
• Branching ratios—relative probabilities of different decay channels depend on which diagrams contribute
• CP violation studies—interference between diagrams can reveal matter-antimatter asymmetries

---

Quick Reference Table

---

Self-Check Questions

1. What distinguishes a virtual particle from a real particle in a Feynman diagram, and why can virtual particles violate the energy-momentum relation?

2. Compare electromagnetic and weak interactions: which bosons mediate each, and which interaction can change a particle's identity (flavor)?

3. If you see a Feynman diagram with a closed loop of internal lines, what does this represent physically, and how does it affect the precision of theoretical predictions?

4. A diagram shows an electron emitting a photon that's absorbed by a positron. Identify the force involved, the mediating particle, and explain why this process conserves charge at each vertex.

5. Why do strong interaction diagrams become more complicated than electromagnetic ones, even when describing similar scattering processes? (Hint: think about the properties of gluons versus photons.)