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Physicshard

Consider the Theory of Interaction of a Spinor and Scalar Field

Question

Consider the theory of interaction of a spinor and scalar field. Derive the relevant equations of motion from the Lagrangian and discuss the coupling structure.

Yukawa theory: deriving equations of motion from the interaction Lagrangian.

Yukawa Lagrangian

L = L₀ − gψ̄ψφ

Free Dirac + free Klein-Gordon + scalar Yukawa coupling

1Write the Lagrangian

The Yukawa theory Lagrangian combines a free Dirac spinor ψ\psi, a free scalar field ϕ\phi, and their interaction: L=ψˉ(iγμ∂μ−m)ψ+12(∂μϕ)2−12M2ϕ2−gψˉψϕ\mathcal{L} = \bar{\psi}(i\gamma^\mu\partial_\mu - m)\psi + \frac{1}{2}(\partial_\mu\phi)^2 - \frac{1}{2}M^2\phi^2 - g\bar{\psi}\psi\phi
TermDescriptionFieldPhysical Meaning
ψ̄(iγµ∂ₘ − m)ψFree DiracSpinorKinetic + mass for fermion
½(∂ₘφ)²Scalar kineticScalarKinetic energy of φ
−½M²φ²Scalar massScalarMass M for scalar particle
−gψ̄ψφYukawa couplingBothInteraction vertex

2Derive the Dirac Equation (Spinor EOM)

Applying the Euler-Lagrange equation for ψˉ\bar{\psi}: ∂L∂ψˉ−∂μ∂L∂(∂μψˉ)=0\frac{\partial \mathcal{L}}{\partial \bar{\psi}} - \partial_\mu \frac{\partial \mathcal{L}}{\partial(\partial_\mu \bar{\psi})} = 0 yields the **modified Dirac equation**: (iγμ∂μ−m)ψ=gϕψ(i\gamma^\mu\partial_\mu - m)\psi = g\phi\psi

Physical Interpretation

The right-hand side gϕψg\phi\psi acts as a position-dependent mass shift: the fermion's effective mass becomes meff=m+gϕ(x)m_{eff} = m + g\phi(x). Where the scalar field is large, the fermion behaves as if it were heavier.

3Derive the Klein-Gordon Equation (Scalar EOM)

Applying Euler-Lagrange for ϕ\phi: (∂μ∂μ+M2)ϕ=−gψˉψ(\partial_\mu\partial^\mu + M^2)\phi = -g\bar{\psi}\psi

Source Term

The scalar field is sourced by the fermion bilinear ψˉψ\bar{\psi}\psi (the scalar density). Where fermions are concentrated, the scalar field builds up — similar to how electric charge sources the electromagnetic field.

4Coupling Structure and Feynman Rules

The Yukawa coupling gψˉψϕg\bar{\psi}\psi\phi creates a single interaction vertex in Feynman diagrams:
  Feynman Vertex:

        ψ (fermion in)
         \
          ●──── φ (scalar)
         /
        ψ̄ (fermion out)

  Vertex factor: −ig
Spinor EOM(iγµ∂ₘ − m)ψ = gφψ
Scalar EOM(□ + M²)φ = −gψ̄ψ
Vertex factor−ig
ParityPreserved (scalar coupling)
COUPLING TYPEScalar Yukawa

Scalar vs Pseudoscalar Coupling

The coupling gψˉψϕg\bar{\psi}\psi\phi preserves parity (scalar). The alternative gψˉγ5ψϕg\bar{\psi}\gamma^5\psi\phi violates parity (pseudoscalar). The Higgs boson uses scalar Yukawa coupling to give fermions their masses.

Quiz

Test your understanding with these questions.

1
What type of coupling does the term gψˉψϕg\bar{\psi}\psi\phi represent?
2
What is the source term in the Klein-Gordon equation for ϕ\phi?