Non-Abelian Gauge Fields

Chapter 5 Non-Abelian Gauge Fields The simplest example starts with two Fermions Dirac particles) ψ 1, ψ 2, degenerate in mass, and hence satisfying in the absence of interactions γ 1 i + m)ψ 1 = 0, γ 1 i + m)ψ 2 = 0. 5.1) We can define a two-component object ψ = W = dx)l, ψ1 ) with the associated action ψ 2 L = ψ γ 1i ) + m ψ. 5.2) If U is a constant 2 2 matrix, L is invariant under the replacement ψ Uψ, provided U is unitary, U U = UU = 1. 5.3) We know that the most general unitary 2 2 matrix, apart from a pure phase factor [U1) transformation], can be written in terms of the Pauli matrices τ as U = e iλ τ = cos λ + i λ τ sin λ λ cosλ + iˆλ3 sin λ iˆλ = 1 + ˆλ ) 2 )sin λ iˆλ 1 ˆλ, 5.4) 2 )sin λ cosλ iˆλ 3 sin λ where λ is a arbitrary vector. The generators of these transformations are the Pauli matrices, which obey the algebra Because [τ a, τ b ] = 2iǫ abc τ c. 5.5) det U = cos 2 λ + ˆλ 2 3 + ˆλ 2 1 + ˆλ 2 2)sin 2 λ = 1, 5.6) this actually represents an SU2) transformation. 61 Version of April 27, 2005

62 Version of April 27, 2005CHAPTER 5. NON-ABELIAN GAUGE FIELDS Now suppose we gauge the symmetry, by letting λ λx). Then L is not invariant, δl = ψu γ µ1 i µu)ψ µ δλ ψγ µ τψ, 5.7) for δλ infinitesimal. We see here the conserved isospin current [compare 2.33)] δw = 2 dx) µ δλ j µ = 0, 5.8) where j µ = 1 2 ψγµ τψ, or j a µ = 1 2 ψγ µτ a ψ, 5.9) which is conserved, by the stationary action principle, µ j µ = 0. 5.10) How can we cancel δl identically? Evidently, by couping to this current a triplet of vector fields, A µ = A 1 µ, A2 µ, A3 µ ), 5.11) as follows: where under an infinitesimal gauge transformation L I int = ga µ ψγ µτ ψ, 5.12) 2 A µ A µ + µ δω, 5.13) where δω is related to δλ: δλ = g δω. 5.14) 2 But this is not the whole story! That is because the ψ variation of L I int is δ ψ L I int = ψγ µ [ τ 2 ga µ, i 2 gτ δω ]ψ = g 2 ψγ µτ 2 A µ δω)ψ 0. 5.15) This will be cancelled if we modify our A µ variation to A µ A µ + µ δω gδω A µ. 5.16) This last is in fact the transformation law for a vector under an ordinary rotation. Mathematically, we say that these gauge fields transform as the spin-1 adjoint) representation of SU2). Now assemble the Fermion parts of L: L f = ψ γ 1i ) D + m ψ, 5.17)

63 Version of April 27, 2005 where the gauge covariant derivative is D µ = µ i 2 gτ A µ. 5.18) Under a gauge transformation, D µ U Dµ U U = D µ. 5.19) That is, which says Dµ U = µ ig τ 2 AU µ = UD µ U = U µ i ) 2 gτ A µ U = µ i 2 guτu A µ + U µ U, 5.20) τ 2 AU µ = U τ 2 U A µ + i g U µu, 5.21) which generalizes the infinitesimal transformation given in 5.16). Indeed if we get U = 1 + igδω τ 2, 5.22) τ 2 AU µ = τ [ τ 2 A µ 2, igδω τ ] A µ i 2 g igτ 2 µδω = τ 2 A µ i τ 2 iga µ δω + τ 2 µδω, 5.23) which agrees with 5.16). It is obviously convenient to define a matrix representation for the gauge fields: A µ = τ 2 A µ. 5.24) Then the above gauge transformation 5.21) reads and the covariant derivative is A U µ = UA µ U + i g U µu, 5.25) D µ = µ iga µ. 5.26) This last includes the Abelian case, where U = e ieλ, τ/2 1, which is a U1) gauge group. To pick out the components of the gauge field, we recall that τ a τ b = δ ab + iǫ abd τ c, 5.27a)

64 Version of April 27, 2005CHAPTER 5. NON-ABELIAN GAUGE FIELDS so because Tr1 = 2 we have Trτ a τ b = 2δ ab. 5.27b) Therefore A a µ = Tr τ a τ 2 A ) = Trτ a A µ ). 5.28) Is the above interaction, minimal substitution, the end of the story? No, because we must consider the gauge field part of the action. Now [D µ, D ν ] = [ µ iga µ, ν iga ν ] = ig µ A ν ν A µ ) g 2 [A µ, A ν ]. 5.29) Because our fields are non-abelian, this last commutator is nonzero. Explicitly, [ τ [A µ, A ν ] = 2 A µ, τ ν] 2 A = i 2 τ A µ A ν ). 5.30) We will define the commutator as the field strength, [D µ, D ν ] = igf µν, 5.31) where F µν = µ A ν ν A µ ) ig[a µ, A ν ]. 5.32) In terms of components, related to F µν by F µν = τ 2 F µν, 5.33) we have or F a µν = µa a ν νa a µ + gǫabc A b µ Ac ν, F µν = µ A ν ν A µ + ga µ A ν. 5.34a) 5.34b) Why is this a useful quantity? Because it transforms covariatly, F µν F U µν = i g [DU µ, DU ν ] = UF µνu, 5.35) unlike the potential, as seen in 5.25). In infinitesimal form this means F µν F µν gδω F µν. 5.36) From the field strength, the gauge field part of the Lagrangian can be constructed. Why? Because it s gauge invariant!) L g = 1 2 Tr F µν F µν = 1 4 Fµν F µν. 5.37)

5.1. SUMMARY 65 Version of April 27, 2005 Under a gauge transformation, L g L g. Explicitly, because L g = 1 4 µa ν ν A µ ) µ A ν ν A µ ) g 2 µa ν ν A ν ) A µ A ν ) g2 4 A µ A µ A ν A ν A µ A ν A µ A ν ), 5.38) ǫ abc ǫ ade = δ bd δ ce δ be δ cd. 5.39) Note that the requirement of gauge invariance necessarily leads to cubic and quartic self-interactions of the gauge field, with the same coupling constnat as appears in the gauge-field fermion interaction. 5.1 Summary For an arbitrary gauge group, the Lagrangian is L = ψ γ 1i ) D + m ψ 1 2 Tr F 2, 5.40) where the gauge covariant derivative is and the gauge-covariant field strength is D µ = µ iga µ, 5.41) F µν = µ A ν ν A µ ig[a µ, A ν ]. 5.42) This Lagrangian is invariant under the gauge transformations ψ Uψ, A µ UA µ U + i g U µu, F µν UF µν U. 5.43a) 5.43b) 5.43c) For SU2), more explicitly, the Lagrangian is [ L = ψ γ µ1 µ ig τ ) ] i 2 A µ + m ψ 1 4 Fµν F µν, 5.44) where the field strength is The gauge transformations are F µν = µ A ν ν A µ + ga µ A ν. 5.45) U = e igω τ /2, 5.46)

66 Version of April 27, 2005CHAPTER 5. NON-ABELIAN GAUGE FIELDS so for an infinitesimal transformation, ω δω, ψ 1 + igδω τ ) ψ, 2 5.47a) A µ A µ gδω A µ + µ δω, 5.47b) F µν F µν gδω F µν. 5.47c) From the Lagrangian we can derive the equations of motion: Varying with respect to ψ gives the gauge-covariant Dirac equation, γ µ1 ) i D µ + m ψ = 0. 5.48) Under a δa µ transformation, δl = ψγ µ gδa µ ψ Tr F µν 2 µ δa ν 2igA µ δa ν 2igδA µ A ν ), 5.49) so for SU2) the change in the action is δw = dx)δa µ ψγ µ g τ ) 2 ψ + νf νµ + gf µν A ν, 5.50) where we have used 5.45). Thus, the Yang-Mills equation the generalization of Maxwell s equation) is ν F µν = j µ, 5.51) with the current j µ = ψγ µ g τ 2 ψ + ga ν F νµ. 5.52) The current has both fermion and gauge-boson pieces. Alternatively, we can define a gauge covariant derivative for the adjoint isospin-1) representation of SU2) by D ν = 1 ν ga ν, D ν F µν = gj µ f = gψγµτ 2 ψ, 5.53a) 5.53b) where D ν is a tensor, with components D ν ) ab = δ ab ν + gǫ abc A c ν. 5.53c)