The Yang–Mills Action and Gauge Self-Interaction

A Yang–Mills field is a Lie-algebra-valued connection whose curvature transforms covariantly. Because that curvature contains $A_\mu A_\nu$, the gauge-invariant kinetic term contains cubic and quartic interactions with coefficients fixed by one coupling $g$. The result is a self-interacting massless spin-one theory, not a collection of independent photons.

Required background. Dynamical gauge fields and matter supplies the gauge–matter action, local redundancy, and the role of the connection.

Helpful background. Compact Lie groups, roots, weights, and Weyl structure supplies representation data and invariant inner products.

Non-Abelian curvature

Let $G$ be a compact gauge group and write $A_\mu=A_\mu^aT^a$ in Hermitian generators. The site-wide normalization conventions are

$$[T^a,T^b]=if^{abc}T^c, \qquad \operatorname{tr}_R(T^aT^b)=T(R)\delta^{ab}, \qquad D_\mu=\partial_\mu-igA_\mu.$$

For a matter field $\psi\mapsto U\psi$, covariance $D_\mu'\psi'=U D_\mu\psi$ requires

$$A_\mu' = U A_\mu U^{-1} -\frac{i}{g}(\partial_\mu U)U^{-1}.$$

The commutator of covariant derivatives defines the curvature,

$$[D_\mu,D_\nu]=-igF_{\mu\nu}, \qquad F_{\mu\nu} =\partial_\mu A_\nu-\partial_\nu A_\mu-ig[A_\mu,A_\nu],$$

or, in components,

$$F^a_{\mu\nu} =\partial_\mu A^a_\nu-\partial_\nu A^a_\mu +g f^{abc}A^b_\mu A^c_\nu.$$

Thus $F_{\mu\nu}'=UF_{\mu\nu}U^{-1}$. The inhomogeneous derivative of $U$ has disappeared; this is what permits a local quadratic invariant. Yang and Mills introduced this nonlinear gauge curvature and its field equations in Yang and Mills 1954, pp. 191–195.

Action and forced self-interactions

For Dirac matter in a representation $R$, a minimal four-dimensional action is

$$S=\int \mathrm d^4x\left[ -\frac14F^a_{\mu\nu}F^{a\mu\nu} +\bar\psi(i\gamma^\mu D_\mu-m)\psi \right].$$

Let $F^{a(0)}_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu$. Expanding only the gauge term gives

$$\begin{aligned} \mathcal L_{\rm YM} ={}&-\frac14F^{a(0)}_{\mu\nu}F^{a(0)\mu\nu}\\ &-\frac g2 f^{abc}F^{a(0)}_{\mu\nu}A^{b\mu}A^{c\nu}\\ &-\frac{g^2}{4}f^{abc}f^{ade} A^b_\mu A^c_\nu A^{d\mu}A^{e\nu}. \end{aligned}$$

The second and third lines are respectively the three- and four-gauge-boson interactions. Their Lorentz tensors, color tensors, and relative normalization are fixed by the same curvature; an independent cubic or quartic coupling would violate the assumed gauge covariance. When the algebra is Abelian, $f^{abc}=0$ and both self-interactions vanish. A modern action-level treatment is Schwartz 2014, §§ 25.1–25.3, pp. 481–495; the corresponding momentum-space rules are derived on Yang–Mills color algebra and perturbative vertices.

The matter term contains $+gA_\mu^a\bar\psi\gamma^\mu T_R^a\psi$ with the declared $D_\mu$. Reversing the sign in $D_\mu$ without also changing the transformation and vertex conventions is therefore a physical-calculation error, even though a globally consistent opposite convention describes the same theory.

Equations, self-current, and Bianchi identity

Varying $A_\nu^a$ and discarding a boundary term gives

$$(D_\mu F^{\mu\nu})^a=-g j^{a\nu}, \qquad j^{a\nu}=\bar\psi\gamma^\nu T_R^a\psi,$$

where the adjoint derivative is

$$(D_\mu X)^a=\partial_\mu X^a+g f^{abc}A_\mu^bX^c.$$

Written with an ordinary derivative, the same equation is

$$\partial_\mu F^{a\mu\nu} =-g\left(j^{a\nu}+f^{abc}A_\mu^bF^{c\mu\nu}\right).$$

The second term acts as a gauge-field self-current. Neither it nor the matter color current is separately gauge invariant; the covariant equation is the intrinsic statement. A detailed variational derivation appears in Srednicki 2007, § 69, pp. 407–411.

Independently of the equations of motion, the Jacobi identity for three covariant derivatives gives

$$D_{[\lambda}F_{\mu\nu]}=0, \qquad\text{equivalently}\qquad D_\mu\widetilde F^{\mu\nu}=0,$$

with $\widetilde F^{\mu\nu}=\tfrac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$. This is the non-Abelian Bianchi identity. It is kinematic, whereas $D_\mu F^{\mu\nu}=-gj^\nu$ is dynamical; exchanging them loses the distinction between a definition of curvature and an equation selected by the action.

For pure Yang–Mills theory, metric variation gives the symmetric tensor

$$T^{\mu\nu} =-F^{a\mu\rho}F^{a\nu}{}_{\rho} +\frac14\eta^{\mu\nu}F^a_{\rho\sigma}F^{a\rho\sigma}.$$

Its energy density is $\tfrac12(\mathbf E^{a2}+\mathbf B^{a2})$, and its classical trace vanishes in four dimensions. Matter adds its own symmetric contribution.

Group and parameter sheet

The following identities fix the normalization used throughout the chapter:

Object	Definition	$SU(N)$ fundamental convention
Trace index	$\operatorname{tr}_R(T^aT^b)=T(R)\delta^{ab}$	$T(F)=\tfrac12$
Quadratic Casimir	$T_R^aT_R^a=C_2(R)\mathbf 1_R$	$C_F=(N^2-1)/(2N)$
Adjoint Casimir	$f^{acd}f^{bcd}=C_A\delta^{ab}$	$C_A=N$
Dimensions	$d_R C_2(R)=d_A T(R)$	$d_F=N$, $d_A=N^2-1$
Coupling placement	$D=\partial-igA$, $F=\mathrm dA-igA\wedge A$	Three-vertex $\propto g$; four-vertex $\propto g^2$

In four dimensions $[A_\mu]=1$, $[\psi]=3/2$, and $[g]=0$. Massless pure Yang–Mills theory is therefore classically scale invariant, although renormalization generates a scale. The local perturbative action also permits a topological $\theta F\widetilde F$ term. Specifying the Lie algebra and the table above still does not choose the global form of $G$, allowed matter representations, genuine line operators, topological sectors, or $\theta$ periodicity; those require additional global data.

Independent checks and limits

1. Gauge covariance: transform $F$ directly and verify that every $\partial U$ term cancels. Then $\operatorname{tr}(F_{\mu\nu}F^{\mu\nu})$ is invariant.
2. Dimensions: every term in the four-dimensional Lagrangian has mass dimension four, while the cubic and quartic coefficients carry $g$ and $g^2$.
3. Abelian limit: setting $f^{abc}\to0$ removes the self-current and both self-interactions, leaving Maxwell fields component by component.
4. Jacobi check: $[D_{[\lambda},[D_\mu,D_{\nu]}]]=0$ must reproduce $D_{[\lambda}F_{\mu\nu]}=0$ with the same sign as the curvature definition.

Common pitfalls

Treating a matrix field and its components as interchangeable. Commutators belong to matrix notation; structure constants belong to components. Moving between them without the factor of $i$ is the most common source of a wrong cubic sign.

Reading global physics from the local action. The same Lie algebra can correspond to different global gauge groups and different spectra of genuine line operators. The perturbative vertices cannot distinguish those choices.

Calling the color self-current an observable. The ordinary-divergence form is useful bookkeeping, but only the covariant equation and gauge-invariant composites have an invariant meaning.

References

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2014), Chapter 25, DOI.
• Mark Srednicki, Quantum Field Theory, Cambridge University Press (2007), § 69, DOI.
• C. N. Yang and R. L. Mills, “Conservation of Isotopic Spin and Isotopic Gauge Invariance,” Physical Review 96 (1954), 191–195, DOI.