Lets go back to the messy integral that we didnt want to do I q 1 q 2 i 16 \u03c0 2

# Lets go back to the messy integral that we didnt want

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. Let’s go back to the messy integral that we didn’t want to do. I ( q 1 , q 2 ) = i 16 π 2 1 0 1 - α 0 dβ dα m 2 α (1 α ) q 2 1 β (1 β ) q 2 2 2 αβ ( q 1 q 2 ) Now that we are thinking about approximating things for small momenta, you can see that things simplify even off the photon mass shell, for q 2 1 , q 2 2 ̸ = 0 , but still m 2 — to first approximation we can neglect all the momentum dependence in the numerator — I ( q 1 , q 2 ) i 16 π 2 1 0 1 - α 0 dβ dα m 2 = i 32 π 2 m 2 61
Explore the Taylor expansion — it’s more than just a handy approximation L = ¯ ψ ( i ̸ e ̸ A m ) ψ + 1 2 μ ϕ ∂ μ ϕ µ 2 2 ϕ 2 λ 4! ϕ 4 if ϕ ¯ ψγ 5 ψ Renormalizable QFT — completely describes a consistent (if boring) world What does this world look like if µ m and at energies and momenta very small compared to m ? Only ϕ s and γ s — not enough energy to produce electrons and positrons. We can always calculate the relevant amplitudes and as we have seen, even though there are no couplings between ϕ and γ in L , quantum loops intoduce interactions between them. And furthermore, we immediately notice that things are simpler for p, µ m . Let’s go back to the messy integral that we didn’t want to do. I ( q 1 , q 2 ) = i 16 π 2 1 0 1 - α 0 dβ dα m 2 α (1 α ) q 2 1 β (1 β ) q 2 2 2 αβ ( q 1 q 2 ) Now that we are thinking about approximating things for small momenta, you can see that things simplify even off the photon mass shell, for q 2 1 , q 2 2 ̸ = 0 , but still m 2 — to first approximation we can neglect all the momentum dependence in the numerator — I ( q 1 , q 2 ) i 16 π 2 1 0 1 - α 0 dβ dα m 2 = i 32 π 2 m 2 62
Explore the Taylor expansion — it’s more than just a handy approximation L = ¯ ψ ( i ̸ e ̸ A m ) ψ + 1 2 μ ϕ ∂ μ ϕ µ 2 2 ϕ 2 λ 4! ϕ 4 if ϕ ¯ ψγ 5 ψ Renormalizable QFT — completely describes a consistent (if boring) world What does this world look like if µ m and at energies and momenta very small compared to m ? Only ϕ s and γ s — not enough energy to produce electrons and positrons. We can always calculate the relevant amplitudes and as we have seen, even though there are no couplings between ϕ and γ in L , quantum loops intoduce interactions between them. And furthermore, we immediately notice that things are simpler for p, µ m . Let’s go back to the messy integral that we didn’t want to do. I ( q 1 , q 2 ) = i 16 π 2 1 0 1 - α 0 dβ dα m 2 α (1 α ) q 2 1 β (1 β ) q 2 2 2 αβ ( q 1 q 2 ) Now that we are thinking about approximating things for small momenta, you can see that things simplify even off the photon mass shell, for q 2 1 , q 2 2 ̸ = 0 , but still m 2 — to first approximation we can neglect all the momentum dependence in the numerator — I ( q 1 , q 2 ) i 16 π 2 1 0 1 - α 0 dβ dα m 2 = i 32 π 2 m 2 63
So we have an approximate, but very simple result for M μ 1 μ 2

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