Yang-Mills theory

Yang-Mills theory is the non-Abelian gauge theory with field strengthUnder an infinitesimal gauge transformation it transforms covariantly as $\delta F_{\mu \nu }=-ig[F_{\mu \nu },\alpha ]$.

If a Lie algebra has an invariant symmetric nondegenerate form $h$, its curvature $F$ transforms in the Adjoint representation and $\mathcal{L}=-h(F_{\mu \nu },F^{\mu \nu })/4$ is gauge invariant. Infinitesimal invariance follows from $h([\theta ,X],Y)+h(X,[\theta ,Y])=0$. For compact generators with $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$, this gives $-\mathrm{tr}(F_{\mu \nu }{F}^{\mu \nu })/2=-F_{\mu \nu }^a{F}^{a\mu \nu }/4$. An arbitrary index contraction is not gauge invariant for an arbitrary Lie algebra.

A theta vacuum is a coherent state $|\theta ⟩={\sum }_n{e}^{in\theta }|n⟩$ formed from gauge vacua of different winding numbers. Large gauge transformations shift $n$ and act on this state by a phase. The Yang-Mills theta term gives the corresponding weighting of spacetime topological sectors.

In four dimensions the Yang-Mills theta term is proportional to $\theta \mathrm{Tr}(F_{\mu \nu }{}^{\star }{F}^{\mu \nu })$. Its density is a total derivative, but nontrivial gauge-field topology can make its spacetime integral physically relevant in the quantum theory.

The term weights topological sectors and supplies the angle labeling a theta vacuum.

For $F=dA+A\wedge A$, the gauge-theory Bianchi identity is $DF=0$ and the variation is $\delta F=D\delta A$. The graded product rule for the covariant exterior derivative gives the displayed boundary integral. Thus a constant-coefficient Yang-Mills theta term leaves the bulk Yang-Mills equations unchanged, but can change the boundary variational condition. Its four-form density is exactly gauge-invariant by conjugation of $F$ and cyclicity of the matrix trace, even if the gauge parameter is nonzero on the boundary. Gauge changes of a local Chern-Simons 3-form must not be confused with gauge changes of this globally defined curvature density.

The color-contracted density ${ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }^a{F}_{\rho \sigma }^a$ is even under charge conjugation and odd under parity. In matrix form the two charge-conjugation signs cancel and transposition leaves the trace of the product unchanged. Parity contributes its orientation-reversing determinant to the four-index Levi-Civita symbol. Thus the density is CP odd. A generic fixed coefficient violates CP symmetry; being a local total derivative does not eliminate effects from nontrivial gauge topology.

The physical QCD theta angle is $\bar{\theta }={\theta }_{\mathrm{QCD}}+\arg \det (M_u{M}_d)$. An anomalous chiral quark rotation shifts the two terms oppositely, so only their sum is invariant.

The strong CP problem asks why the physical QCD theta angle is smaller than roughly $10^{-10}$ even though no Standard Model symmetry requires this small value.

Peccei–Quinn theory promotes the physical QCD theta angle to a dynamical field through an anomalous global $U(1)$ symmetry. Nonperturbative QCD drives its vacuum expectation value to a CP-conserving minimum.

The axion is the pseudo-Goldstone boson of spontaneously broken Peccei–Quinn symmetry. Its QCD-generated potential dynamically relaxes the physical theta angle toward zero.

The electroweak theta angle multiplies the $SU(2{)}_L$ topological density. In the renormalizable Standard Model it is unphysical because an anomalous baryon-plus-lepton rephasing shifts it, but explicit baryon-plus-lepton violation can make an invariant combination observable.

The Chern-Simons current satisfies ${\partial }_{\mu }{K}^{\mu }=\mathrm{Tr}(F_{\mu \nu }{}^{\star }{F}^{\mu \nu })$. It is not gauge invariant even though its divergence is.

In four dimensions this current is dual to the degree-three Chern-Simons form.

With $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }$ and $[T^a,T^b]=i{f}_{abc}{T}^c$, the Yang-Mills field strength has the sign $F_{\mu \nu }^a=G_{\mu \nu }^a-g{f}_{abc}{A}_{\mu }^b{A}_{\nu }^c$. Expanding $-F_{\mu \nu }^a{F}^{a\mu \nu }/4$ gives a cross term $(g/2)G_{\mu \nu }^a{f}_{abc}{A}^{b\mu }{A}^{c\nu }$; antisymmetry makes its two derivative terms equal, proving the cubic expression. Squaring the nonlinear term gives the quartic expression. Both vertices arise from the same gauge coupling, rather than independent free parameters.

A classical zero-energy Yang-Mills vacuum has vanishing spatial gauge curvature. On simply connected space it is pure gauge, $A=h^{-1}dh$. If gauge transformations approach the identity at spatial infinity, the boundary-normalized gauge maps on compactified space $S^3$ have integer winding, recorded by the Chern-Simons number of a gauge field. Transformations of nonzero winding relate these representatives; quantum theta states encode their tunnelling and large-gauge-transformation phases.

In $\mathcal{L}=-g_{ab}{F}_{\mu \nu }^a{F}^{b\mu \nu }/4$, only the symmetric part of $g$ contributes. A constant symmetric bilinear form gives local gauge invariance precisely when it is an invariant bilinear form on a Lie algebra, because the gauge field strength transforms in the Adjoint representation. A nondegenerate kinetic term requires a nondegenerate bilinear form; positivity of the Hamiltonian is an additional condition. For a compact simple real Lie algebra, $-\kappa$ is positive definite, and a positive multiple supplies the usual positive-energy convention with Minkowski signature $(+---)$.

For the convention $D_{\mu }={\partial }_{\mu }+g{A}_{\mu }$, a gauge group transformation $\Phi \mapsto R\Phi$ acts on the gauge potential by $A_{\mu }\mapsto R{A}_{\mu }{R}^{-1}-g^{-1}({\partial }_{\mu }R)R^{-1}$. The gauge field strength transforms by conjugation. This makes $D_{\mu }\Phi$ transform in the same gauge group representation as $\Phi$.

Stationarity of the Yang-Mills action in the absence of sources gives $D_{\mu }{F}^{\mu \nu }=0$, using the adjoint gauge covariant derivative. This dynamical equation differs from the geometric gauge-theory Bianchi identity.

Apply $D^{\rho }$ to the gauge-theory Bianchi identity and commute it through the other derivatives using $[D_{\mu },D_{\nu }]X=[F_{\mu \nu },X]$. The Yang-Mills equations remove the differentiated divergences. The two remaining commutator terms both equal $[F_{\mu }{}^{\rho },F_{\nu \rho }]$, proving the displayed nonlinear covariant wave equation for the Yang-Mills field strength on flat spacetime.

// Target: special-relativity.bigb

For a compact simple gauge group, use $B=-\kappa$ on its algebra and an averaged positive invariant Hermitian form $h$ on any supplied finite-dimensional complex representation. With anti-Hermitian connection and signature $(+---)$, $\mathcal{L}=-B({\mathcal{F}}_{\mu \nu },{\mathcal{F}}^{\mu \nu })/(4g^2)+h(D_{\mu }\Phi ,D^{\mu }\Phi )-V(\Phi )$ is gauge-invariant whenever $V$ is invariant. A real scalar representation instead has a symmetric form and kinetic prefactor $1/2$.

Take $D_{\mu }X={\partial }_{\mu }X+[A_{\mu },X]$, signature $(+---)$ and the trace Lagrangian density $\mathrm{Tr}(F_{\mu \nu }{F}^{\mu \nu })/4-\mathrm{Tr}(D_{\mu }\Phi D^{\mu }\Phi )/2+m^2\mathrm{Tr}({\Phi }^2)/2$. The variation of the gauge curvature is $D_{\mu }\delta A_{\nu }-D_{\nu }\delta A_{\mu }$. Cyclicity of the trace and covariant integration by parts give gauge coefficient $-D_{\mu }{F}^{\mu \nu }+[D^{\nu }\Phi ,\Phi ]$ and scalar coefficient $D_{\mu }{D}^{\mu }\Phi +m^2\Phi$. Their vanishing gives the displayed Euler-Lagrange field equations. In static temporal gauge at zero mass they reduce to $D_j{F}_{ji}=[\Phi ,D_i\Phi ]$ and $D_i{D}_i\Phi =0$.

// Destination: integrable-systems.bigb

A real potential $V$ for a scalar field is invariant when $V(R(U)\Phi )=V(\Phi )$ for every gauge transformation. In a unitary representation, $m^{2}h(\Phi ,\Phi )+\lambda h(\Phi ,\Phi {)}^2$ is always an example. Particular representations may admit additional invariant interactions.

For a real compact semisimple algebra, $B=-\kappa$ is a positive internal metric. With signature $(+---)$ and connection $D_{\mu }={\partial }_{\mu }+A_{\mu }$, the healthy Lagrangian is $\mathcal{L}=-B(F_{\mu \nu },F^{\mu \nu })/(4g^2)=\kappa (F_{\mu \nu },F^{\mu \nu })/(4g^2)$. Gauge invariance follows from Killing-form invariance and ${\delta }_{X}F=ϵ[X,F]$. The physical Hamiltonian density is $[B(E_i,E_i)+B(B_i,B_i)]/(2g^2)$ after imposing the Gauss law constraint in gauge theory and treating the spatial boundary flux appropriately. An Abelian factor needs a separate positive invariant metric because its Killing form vanishes.

An adjoint scalar has adjoint covariant derivative $D_{\mu }\varphi ={\partial }_{\mu }\varphi +[A_{\mu },\varphi ]$. A symmetric invariant bilinear form on a Lie algebra contracts the gauge field strength and scalar kinetic terms to give gauge invariance. On a compact real form the positive internal metric is $-\kappa$, because the Killing form itself is negative definite; complex scalar components require the associated Hermitian pairing. The Killing form is degenerate on Abelian factors, where another invariant metric is needed for a nondegenerate kinetic term.

An SU(2) gauge theory is a Yang-Mills theory with gauge group $SU(2)$. In its defining representation the generators are $T^a={\sigma }^a/2$, and its structure constants are the Levi-Civita symbols ${ϵ}^{abc}$.

For field strength $F_{\mu \nu }^a=f_{\mu \nu }^a-g{ϵ}^{abc}{A}_{\mu }^b{A}_{\nu }^c$, the square produces ${\mathcal{L}}_3=g{ϵ}^{abc}({\partial }_{\mu }{A}_{\nu }^a)A^{b\mu }{A}^{c\nu }$ and ${\mathcal{L}}_4=-g^2{ϵ}^{abc}{ϵ}^{ade}{A}_{\mu }^b{A}_{\nu }^c{A}^{d\mu }{A}^{e\nu }/4$. The cubic Feynman vertex has one derivative; the quartic one has none. No higher pure-gauge vertex occurs.

A nonzero complex fundamental doublet has trivial stabilizer in the gauged $SU(2)$. Its three angular directions are gauge directions, absorbed by the Higgs mechanism. For vacuum ${\varphi }_0=(0,v/\sqrt{2}{)}^T$, the three gauge bosons have equal mass $gv/2$; one radial scalar remains. There is no unbroken photon-like gauge generator in this theory.

In unitary gauge, $\varphi =(0,(v+h)/\sqrt{2}{)}^T$ gives $\frac{g^2}{8}(v+h{)}^2{A}_{\mu }^a{A}^{a\mu }$. With potential $\frac{\lambda }{2}({\varphi }^{†}\varphi -v^2/2{)}^2$, the physical scalar has $m_h^2=\lambda v^2$ and interactions $-\lambda v{h}^3/2-\lambda h^4/8$. This normalization matters when extracting both masses and identical-leg Feynman vertices.

A real scalar field in the Adjoint representation can have a nonzero vacuum expectation value, leaving one massless and two massive gauge bosons. This breaking differs from electroweak symmetry breaking by a complex Higgs doublet.

For a nonvanishing adjoint Higgs field, put $n=\Phi /|\Phi |$. If an oriented orthonormal internal basis satisfies $[e_a,e_b]=\kappa {\epsilon }_{abc}{e}_c$, the displayed tensor is gauge invariant. In a gauge with $n=e_3$, write $A_j=a_j^a{e}_a$; the transverse quadratic terms cancel, leaving ${\mathcal{F}}_{jk}={\partial }_j{a}_k^3-{\partial }_k{a}_j^3$. Hence it is a closed differential form wherever $\Phi \ne 0$. Under trace normalization of su(2) with $-\mathrm{tr}(XY)$, ${\kappa }^2=2$ and the commutator term has coefficient $1/2$. With $-2\mathrm{tr}(XY)$, its coefficient is one. Omitting this normalization factor generally destroys closure.

For a real triplet potential $\lambda (|\varphi {|}^2-v^2{)}^2/8$ and canonical gauge coupling $g_c$, unitary gauge gives one neutral Higgs mode, two oppositely charged massive vectors and one massless Abelian vector. The scalar kinetic contribution is $\frac{1}{2}(\partial \eta {)}^2+g_c^2(v+\eta {)}^2{W}^{+}{W}^{-}$, and its potential is $\lambda v^2{\eta }^2/2+\lambda v{\eta }^3/2+\lambda {\eta }^4/8$. This potential convention is distinct from using $\lambda (|\varphi {|}^2-v^2{)}^2/4$.

For a real triplet with potential $\lambda ({\Phi }^2-v^2{)}^2/8$, unitary gauge gives $\Phi =(0,0,v+h)$ and scalar interaction terms $-\lambda v{h}^3/2-\lambda h^4/8+2e^{2}vh{W}^{+}{W}^{-}+e^2{h}^2{W}^{+}{W}^{-}$. Functional differentiation, including identical-field multiplicities, gives vertices $-3i\lambda v$, $-3i\lambda$, $2i{e}^{2}v{g}_{\mu \nu }$ and $2i{e}^2{g}_{\mu \nu }$ respectively. The physical scalar is neutral under the residual U(1), so there are no $hAA$ or $hhAA$ vertices.

An SU(3) gauge theory has eight gauge fields, one for each generator of $SU(3)$. A vacuum expectation value of one scalar in the fundamental representation generically leaves an $SU(2)$ subgroup unbroken.

For a complex fundamental scalar with vacuum expectation value proportional to $(0,0,v)$, the stabilizer is the $SU(2)$ acting on the first two components. Five broken generators produce five massive vector bosons, while three gauge bosons remain massless and one radial scalar remains physical.

On an oriented Riemannian four-manifold, one positive Euclidean convention isfor anti-Hermitian gauge fields.

Under $g\mapsto {\Omega }^{2}g$ in dimension $d$, the Hodge star on $p$-forms scales by ${\Omega }^{d-2p}$. For gauge gauge curvature, $d=4,p=2$, so the star and the action $-g_{\mathrm{YM}}^{-2}\int \mathrm{tr}(F\wedge \ast F)$ are unchanged. Dilation therefore changes the instanton size modulus without changing the classical action. Quantum running of the coupling can break this classical scale invariance.

This gauge-invariant higher-derivative density has equationThe extra commutator comes from varying the connection inside the adjoint covariant derivative: $\delta (D_{\mu }{F}_{\nu \rho })=D_{\mu }\delta F_{\nu \rho }+[\delta A_{\mu },F_{\nu \rho }]$. Two uses of gauge-covariant integration by parts give the first term, and Killing-form invariance gives the second. In an abelian limit it reduces to $\square {\partial }_{\mu }{F}^{\mu \nu }=0$. Keeping the covariant derivative order is essential because their commutator is curvature.

For a semisimple gauge algebra and positive integer $p$, this gauge-invariant density yields $D_{\mu }(s^{p-1}{F}^{\mu \nu })=0$. The curvature variation $\delta F_{\mu \nu }=D_{\mu }\delta A_{\nu }-D_{\nu }\delta A_{\mu }$ gives $\delta s=4\kappa (D_{\mu }\delta A_{\nu },F^{\mu \nu })$, and gauge-covariant integration by parts proves the equation. More generally a density $f(s)$ gives $D_{\mu }(f^{\prime }(s)F^{\mu \nu })=0$. The product form retains the equation at zeros of $s$; dividing by $s$ can discard solutions.

For $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$ and matter transforming as $\psi \mapsto U\psi$, a Yang-Mills connection transforms inhomogeneously while its curvature transforms by conjugation, $F_{\mu \nu }\mapsto U{F}_{\mu \nu }{U}^{-1}$.

For $D_{\mu }={\partial }_{\mu }+A_{\mu }$ and $\delta \varphi =\Lambda \varphi$, covariance requires $\delta A_{\mu }=-{\partial }_{\mu }\Lambda +[\Lambda ,A_{\mu }]$. Expanding $[D_{\mu },D_{\nu }]$ gives $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+[A_{\mu },A_{\nu }]$, and operator Jacobi gives $\delta F_{\mu \nu }=[\Lambda ,F_{\mu \nu }]$. An invariant internal quadratic form makes its Yang-Mills kinetic contraction gauge invariant. Couplings and factors of i are absorbed into the connection in this convention.

With $D_{\mu }={\partial }_{\mu }+\rho ({\mathcal{A}}_{\mu })$ and $\Phi \mapsto R(U)\Phi$, a local $U=1+ϵ$ gives $\delta {\mathcal{A}}_{\mu }=-{\partial }_{\mu }ϵ+[ϵ,{\mathcal{A}}_{\mu }]$, $\delta \Phi =\rho (ϵ)\Phi$ and $\delta {\mathcal{F}}_{\mu \nu }=[ϵ,{\mathcal{F}}_{\mu \nu }]$. Derivative terms cancel in $\delta (D_{\mu }\Phi )=\rho (ϵ)D_{\mu }\Phi$. Here the gauge coupling is absorbed into the connection.

A gluon propagator is the inverse of the gauge-fixed quadratic Yang-Mills operator. It carries Lorentz and adjoint color indices and depends on the gauge-fixing condition, while gauge-invariant observables do not.

A quantum anomaly is the failure of a classical symmetry to survive quantization because the functional measure or regulator cannot preserve it.

A global anomaly obstructs invariance of a quantum theory under a topologically nontrivial symmetry transformation even when infinitesimal local anomaly tests vanish. The Witten SU(2) anomaly is one example; global gravitational anomalies are another class.

The Witten SU(2) anomaly is a four-dimensional global gauge anomaly present for an odd number of left-handed Weyl fermion doublets. A consistent SU(2) gauge theory must have an even number modulo the corresponding representation index.

A gauge anomaly destroys a gauge redundancy needed to remove unphysical states and makes the quantum gauge theory inconsistent unless the anomaly cancels.

Anomaly cancellation is the vanishing of every gauge and mixed gauge-gravitational anomaly after summing over all chiral fermions. It is a consistency condition for a quantum gauge theory.

In each Standard Model generation, quark and lepton hypercharges cancel the $SU(3{)}_c^3$, $SU(3{)}_c^{2}U(1{)}_Y$, $SU(2{)}_L^{2}U(1{)}_Y$, $U(1{)}_Y^3$, and mixed gauge-gravitational anomalies. The four left-handed $SU(2{)}_L$ doublets also avoid the global Witten anomaly.

In normalization $Q=T_3+Y$, one Standard Model generation has left quark and lepton doublet hypercharges $1/6$ and $-1/2$, and right singlet hypercharges $2/3,-1/3,-1$. Including colour multiplicity, ${\mathrm{tr}}_{L}Y=6/6-2/2=0$. On every left doublet $T_3^2=I/4$, and all right fields have $T_3=0$, so the mixed weak-hypercharge gauge anomaly vanishes. The cubic left trace is $6(1/6{)}^3+2(-1/2{)}^3=-2/9$, and the right trace is $3(2/3{)}^3+3(-1/3{)}^3+(-1{)}^3=-2/9$. Their difference is zero, proving generation-by-generation cancellation of these two anomalies.

A mixed gauge-gravitational anomaly has one gauge-current insertion and two stress-tensor insertions. For a four-dimensional $U(1)$ symmetry its coefficient is the sum of the left-handed Weyl-fermion charges, with right-handed fermions counted as left-handed conjugates.

The chiral anomaly is the quantum nonconservation of a classically conserved axial current in a gauge-field background.

For massless Dirac fermions, the classical axial current is $J_A^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$. The chiral anomaly makes its divergence proportional to $F_{\mu \nu }{}^{\star }{F}^{\mu \nu }$.

An axial charge is the spatial integral of the time component of an axial current. For a Dirac field and an internal generator $t$, it is $\int {\psi }^{†}{\gamma }_{5}t\psi d^{3}x$. Its canonical action rotates left- and right-handed fields oppositely. It is conserved only when the current divergence and flux at infinity vanish; Dirac masses and a chiral anomaly can obstruct conservation. The nonsinglet quark generators have no color anomaly and obey the nonsinglet axial charge algebra of quark densities.

For an ordinary scalar field, its derivative is a polar covector, while the axial current is a pseudovector. Their contraction is invariant under the proper orthochronous Lorentz group but changes sign under parity. If the field is a pseudoscalar, the extra transformation sign makes this coupling parity even. In four spacetime dimensions its mass dimension is five, so Lorentz invariance alone does not imply power-counting renormalizability.

For $\mathcal{L}=\bar{\psi }(i/\partial -S+iP{\gamma }^5)\psi$, with real scalar backgrounds $S,P$, the Dirac equation is $(i/\partial -S+iP{\gamma }^5)\psi =0$ and the adjoint Dirac equation is $i({\partial }_{\mu }\bar{\psi }){\gamma }^{\mu }+S\bar{\psi }-iP\bar{\psi }{\gamma }^5=0$. Use $\{{\gamma }^5,{\gamma }^{\mu }\}=0$ and $({\gamma }^5{)}^2=1$ in the derivative of $j_5^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$ to obtain the displayed classical identity. Taking $S=m+g\varphi$ and $P=G\Phi$ includes both scalar and Hermitian pseudoscalar Yukawa couplings.

For ${\mathcal{L}}_{\mathrm{int}}=-g\varphi \bar{\psi }{\gamma }^5\psi$, the variational Dirac equation and adjoint Dirac equation are $(i/\partial -m-g\varphi {\gamma }^5)\psi =0$ and $i({\partial }_{\mu }\bar{\psi }){\gamma }^{\mu }+m\bar{\psi }+g\varphi \bar{\psi }{\gamma }^5=0$. Substitute these into the derivative of $j_5^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$. Anticommuting the chirality matrix past ${\gamma }^{\mu }$ gives two identical mass terms and two identical interaction terms, proving the displayed identity. These are classical field equations; the formula does not assert an unrenormalized quantum composite-operator identity. With Hermitian coupling $g=i{g}_P$, the second term is $-2g_P\varphi \bar{\psi }\psi$.

Since the chirality matrix commutes with spinor Lorentz generators, $j_5^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$ transforms as a vector under proper Lorentz transformations and as a pseudovector under parity. Contracting two copies gives the Lorentz scalar $j_{5\mu }{j}_5^{\mu }$; the extra parity signs cancel. Lorentz invariance of this interaction does not require current conservation.

A chiral transformation rotates a Dirac field by $\psi \mapsto e^{i\alpha {\gamma }^5}\psi$. A massless Dirac kinetic term is invariant under a global rotation, while a mass term is not.

A 't Hooft anomaly is an obstruction to gauging a global symmetry. It is preserved by renormalization-group flow and constrains possible infrared phases.

't Hooft anomaly matching requires the massless infrared degrees of freedom, topological sector, or symmetry-breaking pattern to reproduce every anomaly of an unbroken global symmetry measured in the ultraviolet theory.