Lecture 16
8.324 Relativistic Quantum Field Theory II
Fall 2010
8.324 Relativistic Quantum Field Theory II
MIT OpenCourseWare Lecture Notes
Hong Liu, Fall 2010
Lecture 16
Firstly, we summarize the results of the vertex correction from the previous lecture:
k2
Γµ1 (k1 , k2 )
≡
k2 + l
q
µ
l
k1 + l
k1
=
=
γ µ A(q 2 ) + i(k1 + k2 )µ B(q 2 )
σ µν qν F2 (q 2 )
,
eγ µ F1 (q 2 ) −
2m
(1)
where eF1 = A + 2mB and eF2 = −2mB. We showed that F2 (0) = − 2m
e B(0) =
e2
1
α ≡ 4π
≈ 137
. The integral for Γµ1 is, in fact, infrared divergent. As l −→ 0,
α
2π
= 0.0011614.., where
(k1 + l)2 + m2 = k12 + m2 + 2k1 .l + l2
= 2k1 .l + l2 −→ 0,
´
and so Γ1 ∼ d4 l l14 is divergent. This is due to soft photon interactions, and this effect is in fact cancelled by
including the soft emissions:
+
+
.
(2)
The explanation is that it is only reasonable to calculate measurable cross-sections:
( )
( )
( )
dσ
dσ
dσ
=
(α → β) +
(α → β + soft photons) .
dΩ measured
dΩ
dΩ
(3)
The calculation then proceeds by imposing an infrared cut-off λ on the photon momenta. The divergences in the
λ −→ 0 limit cancel among virtual and real soft photon emissions, and we can safely take the λ −→ 0 limit in the
end.
3.4: VACUUM POLARIZATION
We will now evaluate the one-loop correction to the photon propagator, and consider the physical interpretation,
recalling the general structure we considered in lecture 12.
3.4.1: One-loop correction
q
1P I = i�
iΠµν (k) =
+
=
k
ˆ
=
(−1)(−ie)2
+ ...
k
k+q
d4 q
(2π)
4tr (γ
µ
1
S0 (k + q)γ ν S0 (q)) − i(Z3 − 1)k 2 PTµν
(4)
Lecture 16
8.324 Relativistic Quantum Field Theory II
Fall 2010
We note that the factor of (−1) at the front comes from the fermionic loop, that the trace is over the omitted spinor
indices, and that S0 is the electron propagator. Having now enough experience with one-loop diagrams, we will
iq +m
1
omit the details of the calculation and only emphasise the new aspects. Using S0 (q) = i/q−m
= − q2/+m2 , we first
introduce the Feynman parameters:
ˆ 1 ˆ
d4 p
4N µν
2
Πµν
=
e
dx
+ counterterm,
(5)
1
4
(2π) (p2 + D)2
0
with p ≡ q + xk, D ≡ x(1 − x)k 2 + m2 − iϵ, and
[
]
4N µν = tr γ µ (ik/ + i/q + m)γ ν (i/q + m)
[
]
ν
= tr γ µ (ip
/ + i(1 − x)k/ + m)γ (ip
/ − ixk/ + m)
[
]
ν
2
µ ν
µ
/ ν k]
/
= −tr γ µ pγ
/ p
/ + m tr [γ γ ] + x(1 − x)tr [γ kγ
+terms linear in p+terms with an odd number of γmatrices.
We note that the trace of a term with an odd number of γ-matrices gives zero, and that
tr [γ µ γ ν ] =
tr [γ γ γ γ ] =
4η µν ,
4(η µν η ρσ − η µρ η νσ + η µσ η νρ ).
µ ν ρ σ
Hence, disregarding irrelevant terms, we can write
N µν = −2pµ pν + 2x(1 − x)k µ k ν + (m2 + p2 − x(1 − x)k 2 )η µν .
(6)
Secondly, we evaluate the integrals, extending to a general dimension d. We note that
ˆ d
ˆ
d p µ ν
η µν
dd p 2
2
p
p
f
(p
)
=
p f (p2 ),
d
d
d
(2π)
(2π)
and so
ˆ
ˆ
1
iΠµν (k) = 4e2
dx
dd p Aη µν + p2 (1 − d2 )η µν + 2x(1 − x)k µ k ν
(2π)
0
2
d
(p2 + D)
(7)
,
(8)
where we have set A ≡ m2 − x(1 − x)k 2 . We note that the first and third terms in the numerator are logarithmically
divergent, and the second term is quadratically divergent. We now apply a Wick rotation p0 → ipdE , dd p → idd pE
2
and p2 → pE
. We recall that
ˆ
Γ(b − a − d2 )Γ(a + d2 ) −(b−a− d )
(p2E )a
2 ,
=
D
d
2
d
(2π) (pE + D)b
(4π) 2 Γ(b)Γ( d2 )
dd pE
and so
2
(1 − )
d
ˆ
dd pE
p2E
d
(p2E +
(2π)
ˆ
2
D)
= −D
dd pE
(2π)
d
1
2
(pE
2.
(9)
(10)
+ D)
Hence, the numerator of (8) can be replaced by
(A − D)η µν + 2x(1 − x)k µ k ν .
(11)
The transverse component, therefore, is given by
−2x(1 − x)k 2 PTµν ,
and we can write
ˆ
µν
ˆ
1
dd pE
1
2 µν
2 + D)2 − i(Z3 − 1)k PT
(p
(2π)
0
E
d ˆ 1
Γ(2 − 2 )
x(1 − x)
= −8ie2 k 2 PTµν (k)
dx
− i(Z3 − 1)k 2 PTµν .
d
d
2−
2
2
D
0
(4π)
iΠ (k) =
−8ie k
2 2
µν
PT
(k)
dx
d
2
(12)
Lecture 16
8.324 Relativistic Quantum Field Theory II
Fall 2010
ϵ
Thirdly, we use dimensional regularization, setting d = 4 − ϵ, e → eµ 2 ,
[
(
)]
ˆ
−8ie2 k 2 PTµν (k) 1
2
4πµ2
−
iΠµν (k) =
dx
x(1
−
x)
γ
+
log
− i(Z3 − 1)k 2 PTµν ,
16π 2
ϵ
D
0
(13)
and so
Πµν (k) = k 2 PTµν Π(k 2 ),
with
e2
Π(k ) = − 2
π
ˆ
1
2
0
1 1
dx x(1 − x) − log
ϵ
2
(
(14)
D
4πµ2 e−γ
)
− (Z3 − 1).
The physical field condition constrains that Π(k 2 = 0) = 0, and so Z3 is fixed by
(
)
e2 1 1
m2
Z3 = 1 − 2 − log
6π ϵ
4πµ2 e−γ
2
(15)
(16)
and the final result for Π(k 2 ) is
Π(k 2 ) =
e2
2π 2
ˆ
0
1
(
)
k 2 x(1 − x)
dx x(1 − x) log 1 +
m2
(17)
Remarks:
1.
For k 2 < −4m2 , Π(k 2 ) becomes complex.
2.
For more than one charged particle, we must add their respective contributions, and the smallest m
contributes most.
3.
The internal propagator is given by
−ie2
η µν
q 2 − iϵ 1 − Π(q 2 )
−ie2 η µν
(1 + Π(q 2 ) + . . .).
q 2 − iϵ
→
e2 Dµν (q)
=
We see that for q 2 ≫ m2 , a large spacelike momentum, from (17) we have
Π(q 2 )
e2
q2
log 2
2
2π
m
α
q2
log 2 ,
3π
m
≈
=
where α =
e2
4π .
ˆ
1
dx x(1 − x)
0
Then, the internal propagator goes as
e2
e2
1
e2 (q)
−→
≡
,
q 2 − iϵ
1 − Π(q 2 ) q 2 − iϵ
q 2 − iϵ
with e2 (q) =
1− 3απ
e2
log
q 2
m2
(18)
the running coupling constant.
3.4.2: Physical implication
Consider scattering of two charged particles with coupling constants e1 and e2 , for example, in the process e− +
µ− −→ e− + µ− :
1′
2′
e1
1
q
1′
+
e2
2
2′
e1 e
1
e e
2
2
3
+ . . . .
(19)
Lecture 16
8.324 Relativistic Quantum Field Theory II
Fall 2010
to lowest order, where e is the coupling constant associated with the lightest intermediate particle.
S(1, 2 −→ 1′ , 2′ ) =
(−ie)2 ū1′ (p′1 )γ µ u1 (p1 )Dµν (q)ū2′ (p′2 )γ ν u2 (p2 ),
where
Dµν (q) =
T
Pµν
(q)
−i
L
+ Dµν
(q),
2
q − iϵ 1 − Π(q 2 )
(20)
′
T
and Pµν
= ηµν − µq2 ν , q = p′1 − p1 = p′2 − p2 . Note that u
¯2′ (p′2 )γ ν u2 (p2 )qν = u
¯2′ (p′2 )(p
/2 − p
/2 )u2 (p2 ) = 0. We now
want to consider corrections to the Coulomb potential. We consider the low energy, non-relativistic limit, where
we derive most of our intuition about electromagnetism from, and where the notion of a potential makes most
sense. The lowest two diagrams drawn above correspond to
q q
1′
2′
e1 , µ
e2 , �
q
1
=
e1 e2
ηµν ,
q2
=
e1 e2
Π(q 2 )ηµν .
q2
2
1′
2′
e e ,�
2
e1 , µ e
1
(21)
2
In the non-relativistic limit,
0
q ∼ |v⃗q| ≪ |⃗q| , q 2 ≈ ⃗q2 .
(22)
In this case, one-photon exchange corresponds to the Born approximation:
ˆ
e1 e2
= d3⃗r e−i⃗q.⃗r V0 (⃗r),
⃗q2
where V0 (⃗r) =
e1 e2
4π |⃗r |
is the Coulomb potential. The one-loop correction is given by
Π(⃗q2 ) =
e2
2π 2
ˆ
1
0
(23)
e1 e2
q 2 ),
q
⃗2 Π(⃗
(
)
⃗q 2
dx x(1 − x) log 1 + 2 x(1 − x) ,
m
and the term provides a correction to the Coulomb potential
ˆ
Π(⃗q2 )
δV (⃗r) = e1 e2 d3⃗r e−i⃗q.⃗r 2 .
⃗q
From now on, we will for convenience write q ≡ |⃗q|, r ≡ ∥⃗r∥. The angular integral is given by
ˆ π
2π
dθ sin θeiqr cos θ
3
(2π) 0
)
1 1 ( iqr
=
e − e−iqr ,
2
4π iqr
and so we find for the correction to the Coulomb potential
ˆ
)
e1 e2 ∞
1 ( iqr
δV (⃗r) =
dq
e − e−iqr Π(⃗q2 )
4π 2 0
iqr
ˆ
e1 e2 ∞
1 iqr
=
dq
e Π(⃗q2 ).
4π 2 −∞
iqr
4
where, from (17)
(24)
(25)
Lecture 16
8.324 Relativistic Quantum Field Theory II
Fall 2010
Im(s)
ia
Re(s)
Figure 1: Clockwise contour for the integral in dz in (26). The semi-circular arc is taken to infinity and gives a
vanishing contribution.
If we now let z ≡ qr, a ≡ √
mr
x(1−x)
≥ 2mr, the result reduces to
(
)
ˆ
ˆ ∞
e1 e2 e2 1
eiz
z2
dx
x(1
−
x)
dz
log
1
+
.
(26)
8π 4 ir 0
z
a2
−∞
(
)
2
log 1 + az2 , can be computed using the complex contour shown in figure 1,
δV (⃗r) =
The integral in dz, I =
giving
´∞
−∞
dz
eiz
z
ˆ
I
∞
e−λ
iπ
λ
a
ˆ ∞
e−aλ
= 2iπ
dλ
λ
1
= 2
dλ
after setting λ → aλ. So, our result for the correction to the Coulomb potential is given by
ˆ
ˆ ∞
mr
e1 e2 e2 1
dλ −λ √x(1−x)
δV (⃗r) =
dx
x(1
−
x)
e
4π π 2 0
λ
1
e1 e2
≡
Z(mr)
4π
Remarks:
1.
When mr ≫ 1, we can evaluate the integral in the saddle-point approximation, using integration by
parts,
e1 e2 e2 e−2mr
δV (r) =
+ ....
(27)
4π 16π 32 (mr) 32
2.
Z(mr) increases with decreasing r. As r −→ 0, I(r) −→ ∞. If we consider putting a short-distance cut
off at mr = e ≪ 1, we find
e2
1
Z(ϵ) =
log + . . . ,
(28)
2
6π
ϵ
and thus
V (r) =
V0 + δV (r)
e1 e2
=
(1 + Z(mr))
4πr
ẽ1 (r)ẽ2 (r)
=
,
4πr
where
1
ẽi (r) = ei (1 + Z(mr)) 2 .
5
(29)
Lecture 16
8.324 Relativistic Quantum Field Theory II
Fall 2010
Figure 2: Virtual electron-positron pairs form a screening effect.
We observe that ẽi (r) −→ ∞ as r −→ 0, and ẽi (r) −→ ei as r −→ 0, with small experimental corrections.
Physically, we can view ẽi (r = ϵ) as the bare charge, which is very large. The physical interpretation is
that virtual electron-positron pairs screen the charge more at large distances. The screening length is
1
1
given by rs ∼ 2m
. That is, there is a negative cloud of size 2m
. At large distances, the charges scale as
−2mr
ei ∼ e
.
6
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