Lets go back to the messy integral that we didnt want

. 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