. 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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 Spring '10
 GEORGI
 Mass, Quantum Field Theory, Light, Lorentz, Fµ Fµ