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Fujikawa method
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 \( \mathfrak{g}\ \),. The Dirac operator (in Feynman slash notation) is
\( D\!\!\!\!/\ \stackrel{\mathrm{def}}{=}\ \partial\!\!\!/ + i A\!\!\!/ \)
and the fermionic action is given by
\( \int d^dx\, \overline{\psi}iD\!\!\!\!/ \psi \)
The partition function is
\( Z[A]=\int \mathcal{D}\overline{\psi}\mathcal{D}\psi e^{-\int d^dx \overline{\psi}iD\!\!\!\!/\psi}. \)
The axial symmetry transformation goes as
\( \psi\to e^{i\gamma_{d+1}\alpha(x)}\psi\,
\overline{\psi}\to \overline{\psi}e^{i\gamma_{d+1}\alpha(x)}
S\to S + \int d^dx \,\alpha(x)\partial_\mu\left(\overline{\psi}\gamma^\mu\gamma^5\psi\right)\)
Classically, this implies that the chiral current, \( j_{d+1}^\mu \equiv \overline{\psi}\gamma^\mu\gamma^5\psi \) is conserved, \( 0 = \partial_\mu j_{d+1}^\mu. \)
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:
\( \psi = \sum\limits_{i}\psi_ia^i, \)
\( \overline\psi = \sum\limits_{i}\psi_ib^i, \)
where \( \{a^i,b^i\} \) are Grassmann valued coefficients, and \{\psi_i\} are eigenvectors of the Dirac operator:
\( D\!\!\!\!/ \psi_i = -\lambda_i\psi_i. \)
The eigenfunctions are taken to be orthonormal with respect to integration in d-dimensional space,
\( \delta_i^j = \int\frac{d^dx}{(2\pi)^d}\psi^{\dagger j}(x)\psi_i(x). \)
The measure of the path integral is then defined to be:
\( \mathcal{D}\psi\mathcal{D}\overline{\psi} = \prod\limits_i da^idb^i \)
Under an infinitesimal chiral transformation, write
\( \psi \to \psi^\prime = (1+i\alpha\gamma_{d+1})\psi = \sum\limits_i \psi_ia^{\prime i}, \)
\( \overline\psi \to \overline{\psi}^\prime = \overline{\psi}(1+i\alpha\gamma_{d+1}) = \sum\limits_i \psi_ib^{\prime i}. \)
The Jacobian of the transformation can now be calculated, using the orthonormality of the eigenvectors
\( C^i_j \equiv \left(\frac{\delta a}{\delta a^\prime}\right)^i_j = \int d^dx \,\psi^{\dagger i}(x)[1-i\alpha(x)\gamma_{d+1}]\psi_j(x) = \delta^i_j\, - i\int d^dx \,\alpha(x)\psi^{\dagger i}(x)\gamma_{d+1}\psi_j(x). \)
The transformation of the coefficients \{b_i\} are calculated in the same manner. Finally, the quantum measure changes as
\( \mathcal{D}\psi\mathcal{D}\overline{\psi} = \prod\limits_i da^i db^i = \prod\limits_i da^{\prime i}db^{\prime i}{\det}^{-2}(C^i_j), \)
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:
\( \begin{align}{\det}^{-2}(C^i_j) &= \exp\left[-2{\rm tr}\ln(\delta^i_j-i\int d^dx\, \alpha(x)\psi^{\dagger i}(x)\gamma_{d+1}\psi_j(x))\right]\\ &= \exp\left[2i\int d^dx\, \alpha(x)\psi^{\dagger i}(x)\gamma_{d+1}\psi_i(x)\right]\end{align} \)
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
\( \begin{align}-2{\rm tr}\ln C^i_j &= 2i\lim\limits_{M\to\infty}\alpha\int d^dx \,\psi^{\dagger i}(x)\gamma_{d+1} e^{-\lambda_i^2/M^2}\psi_i(x)\\ &= 2i\lim\limits_{M\to\infty}\alpha\int d^dx\, \psi^{\dagger i}(x)\gamma_{d+1} e^{{D\!\!\!\!/}^2/M^2}\psi_i(x)\end{align} \)
\( ({D\!\!\!\!/}^2 \) can be re-written as \( D^2+\tfrac{1}{4}[\gamma^\mu,\gamma^\nu]F_{\mu\nu}, \( and the eigenfunctions can be expanded in a plane-wave basis)
\( = 2i\lim\limits_{M\to\infty}\alpha\int d^dx\int\frac{d^dk}{(2\pi)^d}\int\frac{d^dk^\prime}{(2\pi)^d} \psi^{\dagger i}(k^\prime)e^{ik^\prime x}\gamma_{d+1} e^{-k^2/M^2+1/(4M^2)[\gamma^\mu,\gamma^\nu]F_{\mu\nu}}e^{-ikx}\psi_i(k)
= -\frac{-2\alpha}{(2\pi)^{d/2}(\frac{d}{2})!}(\tfrac{1}{2}F)^{d/2}, \)
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,\( F \equiv F_{\mu\nu}\,dx^\mu\wedge dx^\nu\,. \)
This result is equivalent to \( (\tfrac{d}{2})^{\rm th} \) Chern class of the \mathfrak{g}-bundle over the d-dimensional base space, and gives the chiral anomaly, responsible for the non-conservation of the chiral current.
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