Chiral model

An outline of the original, 2-flavor model

The chiral model of Gürsey (1960; also see Gell-Mann and Lévy) is now appreciatedd to be an effective theory of QCD with two light quarks, u, and d. The QCD Lagrangian is approximately invariant under independent global flavor rotations of the left- and right-handed quark fields,

q L ⟶ q L ′ = L q L = exp ⁡ ( − i θ → L ⋅ τ → / 2 ) q L , q R ⟶ q R ′ = R q R = exp ⁡ ( − i θ → R ⋅ τ → / 2 ) q R ,

where τ denote the Pauli matrices in the flavor space and θ_L,R are the corresponding rotation angles.

The corresponding symmetry group S U ( 2 ) L × S U ( 2 ) R is the chiral group, controlled by the six conserved currents

L μ i = q ¯ L γ μ τ i 2 q L , R μ i = q ¯ R γ μ τ i 2 q R ,

which can equally well be expressed in terms of the vector and axial-vector currents

V μ i = L μ i + R μ i , A μ i = R μ i − L μ i .

The corresponding conserved charges generate the algebra of the chiral group,

[ Q I i , Q I j ] = i ϵ i j k Q I k [ Q L i , Q R j ] = 0 ,

with I=L,R, or, equivalently,

[ Q V i , Q V j ] = i ϵ i j k Q V k , [ Q A i , Q A j ] = i ϵ i j k Q V k , [ Q V i , Q A j ] = i ϵ i j k Q A k .

(Application of these commutation relations to hadronic reactions dominated current algebra calculations in the early seventies of the last century.)

At the level of hadrons, pseudoscalar mesons, the ambit of the chiral model, the chiral S U ( 2 ) L × S U ( 2 ) R group is spontaneously broken down to S U ( 2 ) V , by the QCD vacuum. That is, it is realized nonlinearly, in the Nambu-Goldstone mode: The Q_V annihilate the vacuum, but the Q_A do not! This is visualized nicely through a geometrical argument based on the fact that the Lie algebra of S U ( 2 ) L × S U ( 2 ) R is isomorphic to that of SO(4). The unbroken subgroup, realized in the linear Wigner-Weyl mode, is S O ( 3 ) ⊂ S O ( 4 ) which is locally isomorphic to SU(2) (V: isospin).

To construct a non-linear realization of SO(4), consider the representation describing four-dimensional rotations of a vector ( π → , σ ) ≡ ( π 1 , π 2 , π 3 , σ ) . For an infinitesimal rotation parametrized by six angles { θ i V , A } , with i =1, 2, 3,

( π → σ ) ⟶ S O ( 4 ) ( π ′ → σ ′ ) = [ 1 1 4 + ∑ i = 1 3 θ i V V i + ∑ i = 1 3 θ i A A i ] ( π → σ ) ,

where

∑ i = 1 3 θ i V V i = ( 0 − θ 3 V θ 2 V 0 θ 3 V 0 − θ 1 V 0 − θ 2 V θ 1 V 0 0 0 0 0 0 ) , ∑ i = 1 3 θ i A A i = ( 0 0 0 θ 1 A 0 0 0 θ 2 A 0 0 0 θ 3 A − θ 1 A − θ 2 A − θ 3 A 0 ) .

The four real quantities (π, σ) define the smallest nontrivial chiral multiplet and represent the field content of the linear sigma model. To switch from the above linear realization of SO(4) to the nonlinear one, we observe that, in fact, only three of the four components of (π, σ) are independent with respect to four-dimensional rotations. These three independent components correspond to coordinates on a hypersphere S³ , where π and σ are subjected to the constraint

π → 2 + σ 2 = F 2 ,

with F a (pion decay) constant of dimension mass.

Utilizing this to eliminate σ yields the following transformation properties of π under SO(4),

π → ⟶ θ V π → ′ = π → + θ → V × π →   , π → ⟶ θ A π → ′ = π → + θ → A F 2 − π → 2 ,

where θ^V,A ≡ { θ i V , A } with i = 1,2,3.

The nonlinear terms (shifting π) on the right-hand side of the second equation underlie the nonlinear realization of SO(4). The chiral group S U ( 2 ) L × S U ( 2 ) R ≃ S O ( 4 ) is realized nonlinearly on the triplet of pions which, however, still transform linearly under isospin S U ( 2 ) V ≃ S O ( 3 ) rotations parametrized through the angles { θ V } .

Through the spinor map, these four-dimensional rotations of (π, σ) can also be conveniently written using 2×2 matrix notation by introducing the unitary matrix

U = 1 F ( σ 1 1 2 + i π → ⋅ τ → ) ,

and requiring the transformation properties of U under chiral rotations to be

U ⟶ U ′ = L U R †   ,

where θ L = θ V − θ A , θ R = θ V + θ A .

The transition to the nonlinear realization follows,

U = 1 F ( F 2 − π → 2 1 1 2 + i π → ⋅ τ → ) , L π ( 2 ) = F 2 4 ⟨ ∂ μ U ∂ μ U † ⟩ ,

where ⟨ … ⟩ denotes the trace in the flavor space. This is a non-linear sigma model.

Terms involving ∂ μ ∂ μ U or ∂ μ ∂ μ U † are not independent and can be brought to this form through partial integration. The constant F²/4 is chosen in such a way that the Lagrangian matches the usual free term for massless scalar fields when written in terms of the pions,

L π ( 2 ) = 1 2 ∂ μ π → ⋅ ∂ μ π → + 1 2 F 2 ( ∂ μ π → ⋅ π → ) 2 + O ( π 6 ) .

An alternative, equivalent (Gürsey, 1960), parameterization

π → ↦ π →   sin ⁡ ( | π / F | ) | π / F | , yields a simpler expression for U, U = 1 1   cos ⁡ | π / F | + i π ^ ⋅ τ →   sin ⁡ | π / F | = e i   τ → ⋅ π → / F   . Note the reparameterized πs transform under L U R † = exp ⁡ ( i θ → A ⋅ τ → / 2 − i θ → V ⋅ τ → / 2 ) exp ⁡ ( i π → ⋅ τ → / F ) exp ⁡ ( i θ → A ⋅ τ → / 2 + i θ → V ⋅ τ → / 2 ) so, then, manifestly identically to the above under isorotations, V; and similarly to the above, as π → ⟶ π → + θ → A F + . . . = π → + θ → A F ( | π / F | cot ⁡ | π / F | ) under the broken symmetries, A.

This simpler expression generalizes readily (Cronin, 1967) to N light quarks, so S U ( N ) L × S U ( N ) R / S U ( N ) V .