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Exact rational Dirac Ward matrix example

Example

For any $μ > 0$ , use $h = diag (1, - 1, - 1, - 1)$ , $p = (0, 0, 0, 0)$ , and $k = (2 μ, 0, 0, 0)$ . Both $p$ and $p + k$ are off shell. The exact rational matrices are $S_{0} (p) = - \frac{1}{μ} I_{4}, S_{0} (p + k) = \frac{2 β + I_{4}}{3 μ}, Γ (k) = 2 μ β .$

Facts & Assumptions

Given: $μ > 0$ , $p = 0$ , $k = (2 μ, 0, 0, 0)$ , and the Dirac matrices in Off-shell free Dirac Ward telescoping and tree currents.

[F1]

Off-shell free Dirac Ward telescoping and tree currents: when both $h (p, p) - μ^{2}$ and $h (p + k, p + k) - μ^{2}$ are nonzero, $S_{0} (p) = (Γ (p) + μ I_{4}) / (h (p, p) - μ^{2})$ and $S_{0} (p + k) Γ (k) S_{0} (p) = S_{0} (p) - S_{0} (p + k)$ .

[F2]

Dirac Clifford matrices and energy projections: $β = diag (I_{2}, - I_{2})$ , so $β^{2} = I_{4}$ ; the item fixes $Γ (p)$ with the effective matrix $h = - η$ .

[A1]

$μ > 0$ has inverse-length units; the momenta $p = 0$ and $k = (2 μ, 0, 0, 0)$ use the same inverse-length coordinate convention as Off-shell free Dirac Ward telescoping and tree currents.

Verification

Given: [F1], [F2], and the numerical momenta and positive parameter in the Example.

1.1A1F1algebra

Here $h (p, p) - μ^{2} = - μ^{2}$ and $h (p + k, p + k) - μ^{2} = 3 μ^{2}$ , so both rational inverses are defined. Since $Γ (p) = 0$ and $Γ (k) = 2 μ β$ , the inverse formula in [F1] gives the displayed $S_{0} (p)$ and $S_{0} (p + k)$ .

2.1F1F2step 1.1algebra∎

Using $β^{2} = I_{4}$ from [F2], $S_{0} (p + k) Γ (k) S_{0} (p) = - \frac{(2 β + I_{4}) 2 β}{3 μ} = - \frac{4 I_{4} + 2 β}{3 μ} = S_{0} (p) - S_{0} (p + k) .$ Thus this off-shell rational instance satisfies the exact matrix Ward identity.

Exact rational Dirac Ward matrix example

Example

Facts & Assumptions

Verification

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