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Fujikawa method

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Fujikawa's method is a way of deriving the chiral anomaly in quantum field theory.

Suppose given a Dirac field ψ which transforms according to a ρ representation of the compact Lie group G; and we have a background connection form of taking values in the Lie algebra g . The Dirac operator (in Feynman slash notation) is

D /   = d e f   ∂ / + i A /

and the fermionic action is given by

∫ d d x ψ ¯ i D / ψ

The partition function is

Z [ A ] = ∫ D ψ ¯ D ψ e − ∫ d d x ψ ¯ i D / ψ .

The axial symmetry transformation goes as

ψ → e i γ d + 1 α ( x ) ψ ψ ¯ → ψ ¯ e i γ d + 1 α ( x ) S → S + ∫ d d x α ( x ) ∂ μ ( ψ ¯ γ μ γ d + 1 ψ )

Classically, this implies that the chiral current, j d + 1 μ ≡ ψ ¯ γ μ γ 5 ψ is conserved, 0 = ∂ μ j d + 1 μ .

Quantum mechanically, the chiral current is not conserved: Jackiw discovered this due to the non-vanishing of a triangle diagram. Fujikawa reinterpreted this as a change in the partition function measure under a chiral transformation. To calculate a change in the measure under a chiral transformation, first consider the dirac fermions in a basis of eigenvectors of the Dirac operator:

ψ = ∑ i ψ i a i , ψ ¯ = ∑ i ψ i b i ,

where { a i , b i } are Grassmann valued coefficients, and { ψ i } are eigenvectors of the Dirac operator:

D / ψ i = − λ i ψ i .

The eigenfunctions are taken to be orthonormal with respect to integration in d-dimensional space,

δ i j = ∫ d d x ( 2 π ) d ψ † j ( x ) ψ i ( x ) .

The measure of the path integral is then defined to be:

D ψ D ψ ¯ = ∏ i d a i d b i

Under an infinitesimal chiral transformation, write

ψ → ψ ′ = ( 1 + i α γ d + 1 ) ψ = ∑ i ψ i a ′ i , ψ ¯ → ψ ¯ ′ = ψ ¯ ( 1 + i α γ d + 1 ) = ∑ i ψ i b ′ i .

The Jacobian of the transformation can now be calculated, using the orthonormality of the eigenvectors

C j i ≡ ( δ a δ a ′ ) j i = ∫ d d x ψ † i ( x ) [ 1 − i α ( x ) γ d + 1 ] ψ j ( x ) = δ j i − i ∫ d d x α ( x ) ψ † i ( x ) γ d + 1 ψ j ( x ) .

The transformation of the coefficients { b i } are calculated in the same manner. Finally, the quantum measure changes as

D ψ D ψ ¯ = ∏ i d a i d b i = ∏ i d a ′ i d b ′ i det − 2 ( C j i ) ,

where the Jacobian is the reciprocal of the determinant because the integration variables are Grassmannian, and the 2 appears because the a's and b's contribute equally. We can calculate the determinant by standard techniques:

det − 2 ( C j i ) = exp ⁡ [ − 2 t r ln ⁡ ( δ j i − i ∫ d d x α ( x ) ψ † i ( x ) γ d + 1 ψ j ( x ) ) ] = exp ⁡ [ 2 i ∫ d d x α ( x ) ψ † i ( x ) γ d + 1 ψ i ( x ) ]

to first order in α(x).

Specialising to the case where α is a constant, the Jacobian must be regularised because the integral is ill-defined as written. Fujikawa employed heat-kernel regularization, such that

− 2 t r ln ⁡ C j i = 2 i lim M → ∞ α ∫ d d x ψ † i ( x ) γ d + 1 e − λ i 2 / M 2 ψ i ( x ) = 2 i lim M → ∞ α ∫ d d x ψ † i ( x ) γ d + 1 e D / 2 / M 2 ψ i ( x )

( D / 2 can be re-written as D 2 + 1 4 [ γ μ , γ ν ] F μ ν , and the eigenfunctions can be expanded in a plane-wave basis)

= 2 i lim M → ∞ α ∫ d d x ∫ d d k ( 2 π ) d ∫ d d k ′ ( 2 π ) d ψ † i ( k ′ ) e i k ′ x γ d + 1 e − k 2 / M 2 + 1 / ( 4 M 2 ) [ γ μ , γ ν ] F μ ν e − i k x ψ i ( k ) = − − 2 α ( 2 π ) d / 2 ( d 2 ) ! ( 1 2 F ) d / 2 ,

after applying the completeness relation for the eigenvectors, performing the trace over γ-matrices, and taking the limit in M. The result is expressed in terms of the field strength 2-form, F ≡ F μ ν d x μ ∧ d x ν .

This result is equivalent to ( d 2 ) t h Chern class of the g -bundle over the d-dimensional base space, and gives the chiral anomaly, responsible for the non-conservation of the chiral current.

References

Fujikawa method Wikipedia
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