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From First Principles January 2017 – R4.4
Maurice R. TREMBLAY
PART VII – QUANTUM ELECTRODYNAMICS
∫∫ +++++−=′ 65
2
56
2
)2,6()1,5()()6,4()5,3()()2,1;4,3( ττδγγ µµ ddKKsKKeiK bababa
ELECTRONS
VIRTUAL
QUANTUMTIME
1 2
3
4
5
6
)( 2
56s+δ
)6,4(+K
)1,5(+K
)2,6(+K
)5,3(+K
µγ
µγ
a b
Circa 1949
Chapter 3
Contents
PART VII–QUANTUM ELECTRODYNAMICS
Particles and Fields
Second Quantization
Yukawa Potential
Complex Scalar Field
Noether’s Theorem
Maxwell’s Equations
Classical Radiation Field
Quantization of Radiation Oscillators
Klein-Gordon Scalar Field
Charged Scalar Field
Propagator Theory
Dirac Spinor Field
Quantizing the Spinor Field
Weyl Neutrinos
Relativistic Quantum Mechanics
Quantizing the Maxwell Field
Cross Sections and the Scattering Matrix
Propagator Theory and Rutherford
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2
Scattering
Time Evolution Operator
Feynman’s Rules
The Compton Effect
Pair Annihilation
Møller Scattering
Bhabha Scattering
Bremsstrahlung
Radiative Corrections
Anomalous Magnetic Moment
Infrared Divergence
Lamb Shift
Overview of Renormalization in QED
Brief Review of Regularization in QED
Appendix I: Radiation Gauge
Appendix II: Path Integrals
Appendix III: Dirac Matrices
References
So far, our discussion has been rather formal, with no connection to experiment. This is
because we have been concentrating on Green functions, which are unphysical (i.e.,
they describe the motion of off-shell particles where pµ
2 ≠m2c4). However, the physical
world that we measure in our laboratories is on-shell (i.e., pµ
2=m2c4 which are
configurations of a physical system that satisfy classical equations of motion which we
export piecemeal for the real exchange of particles satisfying the Einstein energy-
momentum relationship E2 −|p|2c2 =m2c4≡(mc2)2).
Cross Sections and the Scattering Matrix
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The cross section is thus the effective area of each target particle as seen by an
incoming beam. Cross sections are often measured in terms of barns (i.e., 1 barn is
10−25 cm2). A nucleon is about 1 Fermi, or 10−13 cm across. Its area is therefore about
10−26 cm2, or 0.01 barn. Thus, by giving the cross section of a particle in a certain
reaction, we can immediately calculate the effective size of that particle in relationship
to a nucleon.
Cross section = Effective size of target particle
To make the connection to experiment, we need to rewrite our previous results in
terms of numbers that can be measured in the laboratory, such as decay rates of
unstable particles and scattering cross sections. There are many ways in which to define
the cross section, but perhaps the simplest and most intuitive way is to define it as the
effective size of each particle in the target:
To calculate the cross section in terms of the rate of collisions in a scattering experi-
ment, let us imagine a thin target with NT particles in it, each particle with effective area σ
or cross section. As seen from an incoming beam, the total amount of area taken up by
these particles is therefore NT σ. If we aim a beam of particles at the target with area A
(see Figure), then the chance of hitting one of these particles is equal to the total area
that these particles occupy (i.e., NT σ) divided by the area A:
Let us say we fire a beam containing NB particles at the target. Then the number of
particles in the beam that are absorbed or deflected is NB times the chance of being hit.
Thus, the number of scattering events is given by:
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σ
=particleahittingofChance
σ
⋅=eventsofNumber
or simply (see Figure):
⋅





=
eventsofNumber
σ
NB
•
•
•
•
•
•
NTA
σ
A
NB
A
A
NB NT
NT
NT
In actual practice, a more convenient way of expressing the cross section is via the
flux of the incoming beam, which is equal to J =ρv. If the beam is moving at velocity v
toward a stationary target, then the number of particles in the beam NB is equal to the
density of the beam ρ times the volume. If the beam is a pulse that is turned on for t
seconds, the volume of the beam is vtA. Therefore, NB =ρvtA. The cross section can
therefore be written as:
where we have normalized such that NT =1 and the transition rate is the number of
scattering event per second. The cross section is therefore equal to the transition rate
divided by the flux of the beam.
Flux
rateTransitioneventsofNumbereventsofNumbereventsofNumber
===⋅=
v
t
A
NAv
t
A
tNN
t
TTB ρρ
σ
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The next problem is to write the transition rate appearing in the cross section in
terms of the S matrix. We must therefore calculate the probability that a collection of
particles in some initial state i will decay or scatter into another collection of particles
in some final state j. From ordinary nonrelativistic quantum mechanics, we know that
the cross section σ can be calculated by analyzing the properties of the scattered
wave. Using classical wave function techniques dating back to Rayleigh, we know
that a plane wave exp(ikz) scattering off a stationary, hard target is given by exp(ikz)+
[ f (θ)/r]exp(ikr) where the term exp(ikr) represents the scattered wave, which is
expanding radially from the target. Therefore | f (θ)|2 is proportional to the probability
that a particle scatters into an angle θ.
More precisely, the differential cross section is given by the square of f (θ):
where the solid angle differential is given by:
2
)(θ
σ
f
d
d
=
Ω
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∫ ∫∫ −
=Ω
π2
0
1
1
cosθϕ ddd
and the total cross section is given by:
σ
σ
θ =
Ω
Ω=Ω ∫∫ d
d
dfd
2
)(
For our purposes, however, this formulation is not suitable because it is inherently
nonrelativistic. To give a relativistic formulation, let us start at the beginning. We wish to
describe the scattering process that takes us from an initial state |i〉 consisting of a
collection of free, asymptotic states at t→−∞ to a final state | f 〉 at t→∞. To calculate the
probability of taking us from the initial state to the final state, we introduce the S matrix:
ififif
if
PPi
iSfS
T)()π2( 44
−−=
=
δδ
where δf i symbolically represents the particles not interacting at all, and Tf i is called
the transition matrix, which describes non-trivial scattering.
One of the fundamental constraints coming from quantum mechanics is that the S
matrix is unitary:
By taking the square of the S matrix, we can calculate the transition probabilities. The
probability that the collection of states i will make the transition to the final states f is
given by:
ki
f
kfif SS δ=∑ *
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ififif SSP *
=
Likewise, the total probability that the initial states i will scatter into all possible final
states f is given by:
∑=
f
ifif SSP *
Total
Now we must calculate precisely what we mean by Σf . We begin by defining our
states within a box of volume V:
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( ) ( )fermionsandbosons 0)(
)π2(
0)(
2)π2( †
3
†
3
pa
m
E
V
pa
V
E pp
== pp
Our states are therefore normalized as follows:
( ) ( )fermionsandbosons )(
)π2(
)(
2)π2( 3
3
3
3
pppppppp ′=′′=′ −−−−−−−− δδ
m
E
VV
E pp
With this normalization, the unit operator (on single particle states) can be expressed as:
( ) ( )fermionsandbosons pp
p
1pp
p
1 ∫∫ ==
pp E
mdV
E
dV
3
3
3
3
)π2(2)π2(
To check our normalizations, we can let the unit operator act on an arbitrary state |q〉,
and see that it leaves the state invariant. This means, however, that we have an awk-
ward definition of the number of states at a momentum p. With this normalization, we
find that 〈p |p〉=(2π)32Epδ 3(0)/V which makes no sense. However, we will interpret this to
mean that we are actually calculating particle densities inside a large but finite box of
size L and volume V ; that is, we define δ 3(p)=limL→∞{[1/(2π)3]∫∫∫±L/2 dxdydzexp(−ip•r)}.
This implies that we take the definition:
3
3
)π2(
)0(
V
=δ
Our task is now to calculate the scattering cross section 1+2→3+4+… and the rate of
decay of a single particle 1→2+3+…. We must now define how we normalize the sum
over final states. We will integrate over all momenta of various final states, and sum over
all possible final states. For each final state, we will integrate over the final momentum in
a Lorentz covariant fashion. We will use:
The density of states dNf (i.e., the number of states within p and p++++δ p), is:
∫∫ =−
pE
d
pmp
pd
2)π2(
)()(
)π2( 3
3
0
224
4
4
p
θδ
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∏=
=
fN
i
i
f
dV
Nd
1
3
3
)π2(
p
As before, the differential cross section dσ is the number of transitions per unit time
per unit volume divided by the flux J of incident particles:
JTV
NdS
d
fif 1
2










==
fluxIncident
cmpersecondperTransition 3
σ
We also know that the transition rate per unit volume (within a momentum-space
interval) is given by:
where (2π)4δ 4(0)=V T.
V
vv
J
21 −
=
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fifif
f
ififfif
NdPP
Nd
TV
PP
TV
NdS
244
44282
)()π2(
)0()()π2(
T
T
−=
−
==
δ
δδ3
cmperrateTransition
To calculate the incident flux, we will first take a collinear frame, such as the laboratory
frame or center-of-mass frame. The incident flux J equals the product of the density of
the initial state (i.e., 1/V) and the relative velocity v=|v1 −v2|, where v1 =|p1|/E1:
In the center-of-mass frame, where p1 =−p2, we have:
2
2
2
1
2
21
2112211
2
2
1
1
21212121
)(4
)(4422)2)(2()2)(2(
mmpp
EEEE
EE
EEvvEEJEEV
−⋅=
+=−=−=−= ppp
pp
This equation only hold if the two particles are collinear. From now on, we assume
that all Lorentz frames are collinear.
The final formula for bosons for the differential cross section for 1+2→3+4+… is
therefore given by:
in a collinear frame where:
∏=
=
f
i
N
i
if
p
if
VE1 2
1
MT
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( )bosons∏=−⋅
−
=
f
i
N
i p
iifif
E
d
mmpp
PP
d
3
3
3
2
2
2
1
2
21
424
2)π2()(4
)()π2( pδ
σ
M
Notice that all factors of V in the S matrix have precisely cancelled against other factors
coming from dNf and the flux.
Finally, we will now use this formalism to compute the probability of the decay of a
single particle. The decay probability is given by:
where we have taken δf i =0 for a decay process. The last expression, unfortunately, in
singular because of the delta function squared. As before, however, we assume that all
our calculations are being performed in a large but finite box of volume V over a large
time interval T. We thus reinterpret one of the delta functions as:
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∑∫∑ −==
f
ififf
f
iff PPNdSNdP 24422
Total )]()π2[( δT
4
4
)π2(
)0(
TV
=δ
We now define the decay rate Γ of an unstable particle as the transition probability per
unit volume of space and time:
)2(
Total
iETV
P
=
×
=Γ
volumesec
yprobabilitTransition
The final result for the decay rate is given by:
∫∏=
−=Γ
f
j
N
j
ifif
p
j
i
PP
E
d
E 1
42
3
34
)(
2)π2(2
)π2(
δM
p
The lifetime of the particle τ is then defined as the inverse of the decay rate:
Γ=1τ
Historically, calculations in QED were performed using two seemingly independent
formulations. One formulation was developed by J. Schwinger (1949) and S. Tomonoga
(1946) using a covariant generalization of operator methods developed in quantum
mechanics. However, the formulation was exceedingly difficult to calculate with and was
physically opaque. The second formulation was developed by R. P. Feynman (1949)
using the propagator approach. Feynman postulated a list of simple rules from which
one could pictorially setup the calculation for scattering matrices of arbitrary complexity.
The weakness of Feynman’s graphical methods, however, was that they were not
rigorously(mathematically)justified.Later,F. Dyson(1949) demonstrated the equivalence
of these two formulations by deriving Feynman’s rules from the interaction picture.
Propagator Theory and Rutherford Scattering
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In this chapter, we will follow Feynman to show how the propagator method gives us
a rapidly and convenient method of calculating the lowest order terms in the scattering
matrix. [N.B., Although we will not consider it here, pursuant to this is the Lehmann-
Symanzik-Zimmermann (LSZ) reduction formalism that is usually studied, in which one
can develop the Feynman rules for diagrams of arbitrary complexity]. At this point, we
should emphasize that the Green functions that appear in the propagator approach are
off-shell (i.e., they do not satisfy the mass-shell condition pµ
2=m2). Neither do they obey
the usual equation of motion. The Green functions describe virtual particles, not
physical ones! As we saw earlier, the Green function develops a pole in momentum
space at pµ
2 =m2. However, there is no violation of cherished physical principles
because the Green functions are not measurable quantities. The only measurable
quantity is the S matrix, where the external particles obey the mass-shell condition.
To begin calculating cross sections, let us review the propagator method in ordinary
quantum mechanics, where we wish to solve the Schrödinger equation:
We assume that the true Hamiltonian is split into two pieces:
0=





−
∂
∂
ψH
t
i
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nInteractio0 HHH +=
where H0 is the free Hamiltonian and the interaction piece HInteraction is small. We wish to
solve for the propagator G(x,t;x′,t′):
)()(),;,( 3
nInteractio0 ttttGHH
t
i ′−′=′′





−−
∂
∂
δδ xxxx −−−−
If we could solve for the Green function for the interacting case, then we can use Huygen’s
principle to solve for the time evolution of the wave. We recall that Huygen’s principle
says that:
( )tttttGdt ′>′′′′′= ∫ ),(),;,(),( 3
xxxxx ψψ
The future evolution of a wave front can be determined by assuming that every
point along a wave front is an independent source of an infinitesimal wave.
By adding up the contribution of all these small waves, we can determine the future
location of the wave front. Mathematically, this is expressed by the equation:
Our next goal, therefore, is to solve for the complete Green function G, which we do
not know, in terms of the free Green function G0, which is well understood. To find the
propagator for the interacting case, we have to power expand in HInteraction. We will use
the following formula for operator A and B:
Another way of writing this is:
L
L
++−=
++−=
+
=
+
=
+
≡
+
−−−−−−
−−−−
−−−
111111
1111
111
)1(
1
1
1
)1(
111
ABABAABAA
ABABABA
ABABAABAAABA
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11 11 −−
+
−=
+
AB
BA
A
BA
Now let:
nInteractio0 HB
t
iHA −=
∂
∂
+−= and
Then we have the symbolic identities:
L+++=+= 0nInteractio0Int00nInteractio00nInteractio0 GHGHGGHGGGGHGGG and
More explicitly, we can recursively write this as:
∫ ∫ ′′+′′=′′ ),;,(),(),;,(),;,(),;,( 11011nInteractio111
3
10 ttGtHttGdtdttGttG xxxxxxxxxx
A
G
BA
G
11
0 =
+
= and
with:
If we power expand this last expression, we find (N.B., HInt ≡HInteraction to save space):
Using Huygen’s principle, we can power expand for the time evolution of the wave
function (see Figure):
LLL
L
++
++
+=
∫
∫
∫
−− ),(),;,(),(),;,(
),(),;,(),(),;,(
),(),;,(),(),(
011011Int110
4
1
4
2202211011Int1102
4
1
4
1101101
4
0
nnnnnnn tttGtHttGxdxd
tttGtHttGxdxd
tttGxdtt
xxxxxx
xxxxxx
xxxxx
ψ
ψ
ψψψ
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L+′′′′+
′′′′+′′=′′
∫
∫ ∫
∞
∞−
),;,(),(),;,(),(),;,(
),;,(),(),;,(),;,(),;,(
22022Int2211011Int02
3
1
3
21
11011Int01
3
10
ttGtHttGtHttGddtdtd
ttGtHttGdtdttGttG
xxxxxxxxxx
xxxxxxxxxx
In the propagator approach, perturbation theory can be pictorially represented as a particle interacting
with a background potential at various points along its trajectory.
γ −
e
),;,( 110 ttG xx
),(0 txψ ),( 11Int tH x
),( 110 txψ
),( 22Int tH x
),( 220 txψ
),( txψ
),;,( 22110 ttG xx
Let us solve for the S matrix in lowest order. We postulate that at infinity, there are free
plane waves given by:
We want to calculate the transition probability that a wave packet starts out in a certain
initial state i, scatters off the potential, then re-emerges as another free plane wave, but
in a different final state f . To lowest order, the transition probability can be calculated by
examining Huygen’s principle:
xki ⋅−
= e
)π2(
1
23
φ
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L+′′′′′′′+= ∫ ),(),(),;,(),(),( nInteractio0
4
ttHttGxdtt ii xxxxxx φφψ
To extract the S matrix, multiply this equation on the left by φj
* and integrate. The first
term on the right then becomes δi j. Using the power expansion of the Green function,
we can express the Green function G0 in terms of these free fields. After integration, we
find:
L+′′′+= ∫ )()()( nInteractio
*4
xxHxxdiS ififif φφδ
Therefore, the transition matrix is proportional to the matrix element of the potential
HInteraction.
We will now generalize this exercise to the problem in question: the calculation in QED
of the scattering of an electron due to a stationary Coulomb potential. Our calculation
should be able to reproduce the old Rutherford scattering amplitude to lowest order in
the nonrelativistic limit and give higher-order quantum corrections to it. Our starting point
is the Dirac electron in the presence of an external, classical Coulomb potential (see
Figure).
The interacting Dirac equation reads as:
)()()()( xxAexmi ψψγ µ
µ
/=−∂
18
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the Rutherford scattering cross section in the nonrelativistic limit.
γ
−
e
xx
1
π4
)(0 ∝−=
eZ
xA
( )couplingµ
µ
γ AeAeJ =/=
with a source term J(x)=eγ µAµ(x). Since we are only working to lowest order and are
treating the potential Aµ as a classical potential, we can solve this equation using only
propagator methods. The solution of the equation, as we have seen, is given by:
∫∫ /−+=−+= )()()()()()()()()( 4
0
4
0 yyAyxSydexyyJyxSydxx FF ψψψψψ
where ψ0 is a solution of the free, homogeneous Dirac equation.
To calculate the scattering matrix, it is convenient to insert the expansion of the
Feynman propagator SF(x−x′) in terms of the time-ordered (i.e., using the T operator)
function θ(t−t′) only as in:
obtained earlier (N.B., we dropped the θ(t′−t) term but we still have to time-order hence
the explicit T operator and change of variables from x′ to y). Then we find:
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∫ ∑=
′′−−=′−
2
1
3
)()()()(
r
r
p
r
pF xxTttdixxS ψψθp
∫ ∫ ∑ /′−−=
=
)()()()()()()(
2
1
34
yyAyxdttiydeixx
r
r
p
r
pi ψψψθψψ p
for t→∞. We now wish to extract from this expression the amplitude that the outgoing
wave ψ (x) will be scattered in the final state, given by ψf (x). This is done by multiplying
both sides of the equation by ψf and integrating over all space-time. The result gives us
the S matrix to lowest order:
−
∫ /−= )()()(4
xyAxxdeiS ififif ψψδ
where the slash notation, A=γ µAµ, has been extensively used here and previously.
Now we insert the expression for the vector potential, which corresponds to an electric
potential A0 given by the standard Coulomb potential:
Inserting the plane-wave expression for the fermion field into the expression for the
scattering matrix Sf i above and performing the integration over x, we have:
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xπ4
)(0
eZ
xA −=
whose Fourier transform is given by:
2
3 π4
e
1
qx
x xq
=•
∫
i
d
)(π2
),(),(
e),(
π4
)(e),(
2
022
04
if
iiff
if
xpi
ii
i
xpi
ff
f
if
EE
spuspu
EE
m
V
eZ
i
spu
VE
meZ
eispu
VE
m
xdS if
−=








−−= ⋅−⋅
∫
δ
γ
δγ µ
µ
q
x
We recall that Vd3pf /(2π)3 is the number of final states contained in the momentum inter-
val d3pf . Multiply these by |Sf i |2 and we have the probability of transition per particle into
these states. Recall also that squaring the S matrix give us divergent quantities like δ (0),
which is due to the fact that we have not rigorously localized the wave packet.We set
2πδ-(0)=T, where we localize the scattering process in a box of size V and duration T.
If we divide by T, this gives us the rate R of transitions per unit time into this
momentum interval. Finally, if we divide the rate of transition by the flux of incident
particles, |vi |/V, this gives us the differential cross section:
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VTV
dV
Sd
i
if
v
p 1
)π2( 3
32
=σ
To calculate the differential cross section per unit of solid angle, we must decompose
the momentum volume element:
The last trace can be performed, since only the trace of even numbers of Dirac matrices
survives:
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






 +/+/==
Ω ∑ m
mp
m
mpmZ
spuspu
mZ
d
d fi
ss
iiff
fi
22
Tr
2
14
),(),(
2
14 00
4
22
,
2
0
4
22
γγ
α
γ
ασ
qq
Using the fact that pf dpf =Ef dEf, we have the result:
pdpdd 23
Ω=p
In the last step, we have used the fact that the summation of spins can be written as:








Γ
+/
Γ
+/
=Γ




 +/
Γ




 +/
=
ΓΓ=ΓΓ=ΓΓ=Γ
0
†
00
†
0
0
†
00
†
0
†††2
22
Tr][
2
][
2
][][][][][][))(())((
γγγγ
γγγγ
δγ
γβ
βα
αδ
δδγγββαα
m
mp
m
mp
m
mp
m
mp
uuuuuuuuuuuuuu
ifif
fiiffiiffiifif
where we have used the fact that the sum over spins gives us:
αβ
αβ 




 +/=∑ m
mp
spuspu
fi ss
2
),(),(
,
)2(4)(Tr 00
fififi ppEEpp ⋅−=// γγ
Finally, we need some kinematic information. If θ is the angle between pf and pi , then:
We then obtain the Mott cross section (1929):
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





=





+=⋅
2
sin4
2
sin2 2222222 θθ
β pqandEmpp fi












−=
Ω 2
sin1
)2(sin4
22
422
22
θ
β
θβ
ασ
p
Z
d
d
where β is the velocity (i.e., v/c). In the nonrelativistic limit (i.e., as β →0), we obtain the
celebrated Rutherford scattering formula for Coulomb scattering (MKS Units):
)2(sin
1
επ8
)( 4
2
2
o
2
θ
θσ 







=
vm
eZ
Up to now, we have not been able to take matrix element of the interacting fields. Hence,
we cannot yet extract out numbers out of these matrix elements. The problem is that
everything is written in terms of the fully interacting fields, of which we know almost
nothing. The key is now to make an approximation to the theory by power expanding in
the coupling constant, which is of the order of 1/137 for QED. So, like before, we begin
by splitting the Hamiltonian into two distinct pieces:
∫= Hx3
nInteractio dH
24
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MRT
Time Evolution Operator
nInteractio0 HHH +=
where H0 is the free Hamiltonian and HInteraction is the interacting part.
For example, in the φ4 theory, the interacting part would be:
where:
4
!4
φ
λ
=H
At this point, it is useful to remind ourselves from ordinary quantum mechanics that there
are several pictures in which to describe the time evolution. In the Schrödinger picture,
we recall, the wave function ψ (x,t) and state vector are functions of time t, but the
operators of the theory are constants in time. In the Heisenberg picture, the reverse is
true; that is, the wave function and state vectors are constant in time, but the time
evolution of the operators and dynamical variables of the theory are governed by the
Hamiltonian φ (x,t)=exp(iHt)φ (x,0)exp(−iHt).
Now, without going into the details of the Lehmann-Symanzik-Zimmermann (LSZ)
reduction formulas, its formulation is used to derive scattering amplitudes to all orders in
perturbation theory. For example, define ‘in’ and ‘out’ states which are free particle states
at asymptotic times (i.e., t=−∞ and ∞, respectively). Then specify that this LSZ
formulation will express the interacting S matrix, defined in terms of the unknown
interacting field φ (x), in terms of these free asymptotic states. Since the S matrix is
defined as the matrix element of the transition from one asymptotic set of states to
another, let f denote a collection of free asymptotic states at t=∞, while i refers to
another collection of asymptotic states at t=−∞. Then the S matrix describes the
scattering of the i states into f states by Sf i =out〈 f |i〉in.
In the LSZ formalism, we will take it convenient to define yet another picture, which
resembles the interaction picture. In this new picture, we need to find a unitary operator
U(t) that takes us from the fully interacting field φ (x) to the free, asymptotic ‘in’ state:
)(),()(),( in
1
tUttUt xx φφ −
=
25
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MRT
where U(t)≡U(t,−∞) is a time evolution operator, which obeys:
1),(),(),(),(),(),( 1221
1
313221 === −
ttUttUttUttUttUttU and,
Because we now have two totally different types of scalar fields, one free and the other
interacting, we must also be careful to distinguish the Hamiltonian written in terms of the
free or the interacting fields. Let H(t) be the fully interacting Hamiltonian written in terms
of the interacting field, and let H0(φin) represent the free Hamiltonian written in terms of
the free asymptotic states. Then the free field φin and the interacting field satisfy two
different equations of motion:
Solving for U(t) is complicated. The outcome is that the U(t) operator satisfies the
following:
),()(
),(
onInteractio
o
ttUtH
t
ttU
i =
∂
∂
26
2017
MRT
)],(),([
),(
)],(),([
),(
in
in
0
in
x
x
x
x
ttHi
t
t
ttHi
t
t
φ
φ
φ
φ
=
∂
∂
=
∂
∂
and
where:
in
0ininnInteractio ),()( HHtH −≡ πφ
that is, HInteraction(t) is defined to be the interaction Hamiltonian defined only with free,
asymptotic fields.
Since HInt(t) (N.B., HInt ≡HInteraction) does not necessarily commute with HInt(t′) at diffe-
rent times, the integration of the previous equation is a bit delicate but its outcome is :
where the T operator means that, as we integrate over t1, we place the exponentials
sequentially in time order. Written is this form, however, this expression is not very
useful. We will find it much more convenient to power expand this exponential in a Taylor
series so we have:
∫ ∫∫ ∞−∞−
−−
==−∞=
tt
tdtditHtdi
TTtUtU
),()( 11Int
3
11Int1
ee),()(
xx H
27
2017
MRT
∑ ∫ ∫ ∫
∞
=
∞− ∞− ∞−
−
+=
1
Int2Int1Int
4
2
4
1
4
)]()()([
!
)(
1)(
n
t t t
nn
n
xxxTxdxdxd
n
i
tU HHH LL
Without going through the derivation ourselves, the interacting Green function, written
in terms of the fields, is given by:
0e0
0e)()()(0
0)()()([0),,,(
)]([
)(
in2in1in
2121
inInt
4
inInt
4
∫
∫
==
xxdi
xdi
n
nn
T
xxxT
xxxTxxxG
φ
φ
φφφ
φφφ
H
H
L
LL
We can now rewrite the previous matrix element entirely in terms of the asymptotic ‘in’
fields by a power expansion of the exponential:
∫∑
∞
∞−
∞
=
−
= 0)]()()()()()([0
!
)(
),,,( Int2Int1Int21
4
2
4
1
4
0
21 mnm
m
m
n yyyxxxTydydyd
m
i
xxxG HHH LLLL φφφ
As an example, we will now analyze the four-point (i.e., x1…x4) function introduced
earlier taken to first order in m with an interaction given by HInt(y)=−λφ 4/4!. We expand:
Now, since 〈0|T[φ (x1)φ (x2)]|0〉=i∆F (x1 −x2), we get, after setting φ (xi)=φi and φ (y)=φ ′:
∫
∫
−=






−⋅=
0)]()()()()()()()([0
!4
0)(
!4
)()()()(0),,,(
4321
4
4
4321
4
4321
yyyyxxxxTyd
i
yxxxxTydixxxxG
φφφφφφφφ
λ
φ
λ
φφφφ
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2017
MRT
φφφφφφφφ
φφφφφφφφ
φφφφφφφφ
φφφφφφφφ
φφφφφφφφ
φφφφφφφφ
φφφφφφφφ
φφφφφφφφφφφφφφφφ
′′′′+
′′′′+
′′′′+
′′′′+
′′′′+
′′′′+
′′′′+
′′′′=′′′′
4321
4321
4321
4321
4321
4321
4321
43214321 ][T ∑=
−∆=
4
1
)(
i
iF yxi
)()(
)()(
)()(
)]([
4231
2
3241
2
4321
2
2
xxxxi
xxxxi
xxxxi
yyi
FF
FF
FFA
AF
−∆−∆+
−∆−∆+
−∆−∆=∆
∆−∆+
where











 )()()(
)()()(
)()()(
)()()(
)]([
3421
3
4231
3
4321
3
4321
3
2
yxxxyxi
yxyxxxi
xxyxyxi
yxyxxxi
yyi
FFF
FFF
FFF
FFFB
BF
−∆−∆−∆+
−∆−∆−∆+
−∆−∆−∆+
−∆−∆−∆=∆
∆−∆+
where
x1
x3
x2
x4
y
)( 3 yxF −∆
The Feynman diagram
corresponding to the
Wick decomposition of
φ 4 theory to first order.
Without demonstrating it, we use the Wick theorem that follows for the general n-point
case (n even – as in the previous Figure):
(N.B., for n odd the last line reads:
∑
∑
∑
−+
+
+
=
perm
121
5
perm
4321
3
perm
21
2121
0)]()([00)]()([0
:)()(:0)]()([00)]()([0
:)()(:0)]()([0
:)()()(:)]()()([
nn
n
n
nn
xxTxxT
xxxxTxxT
xxxxT
xxxxxxT
φφφφ
φφφφφφ
φφφφ
φφφφφφ
L
M
L
L
LL
29
2017
MRT
and the 4! term in the denominator of the integral coefficient disappears because there
are 4! ways in which four external legs at xi can be connected to the four fields contained
within φ 4:
)∑ −−+
perm
1221 )(0)]()([00)]()([0 nnn xxxTxxT φφφφφ L
LL +−∆−=+−= ∫ ∏∫ ∏ ==
4
1
4
4
1
44
4321 )(0)]()([0),,,(
i
F
i
i yxiydiyxTydixxxxG λφφλ
Another example of this decomposition is given by the four-point function taken to
second order with HInt(y1)=−λφ 4/4! and HInt(y2)=−λφ 4/4!:
The expansion, via Wick’s theorem, is:
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2017
MRT
∫ ∫





−= 0)]()()()()()([0
!2
1
!4
),,,( 2
4
1
4
43212
4
1
4
2
4321 yyxxxxTydyd
i
xxxxG φφφφφφ
λ
∫ ∫ ∆−∆−∆+∆−∆
−
= })]()][([)]({[
!2
)(
),,,( 2111
2
212
4
1
4
2
4321 BFFAF yyiyyiyyiydyd
i
xxxxG
λ
)()()()(
)()()()(
)()()()(
23221411
24221311
24231211
yxyxyxyx
yxyxyxyx
yxyxyxyx
FFFF
FFFF
FFFFA
−∆−∆−∆−∆+
−∆−∆−∆−∆+
−∆−∆−∆−∆=∆
where:
and
)()()()(
)()()()(
)()()()(
)()()()(
24132221
24132221
24231221
24232211
yxyxyxyx
yxyxyxyx
yxyxyxyx
yxyxyxyx
FFFF
FFFF
FFFF
FFFFB
−∆−∆−∆−∆+
−∆−∆−∆−∆+
−∆−∆−∆−∆+
−∆−∆−∆−∆=∆
These are shown graphically in the Figure on the next slide.
31
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The Feynman diagrams corresponding to the Wick decomposition of φ 4 theory to second order.
y2
y1
y2
y1
y2
y1
y2y1
y2
y1
)( 11 yyF −∆
y1
y2
)( 21 yyF −∆
x1
x3
x2
x4
y2
2
21 )]([ yyF −∆
y1
)( 23 yxF −∆
We now introduce the graphical rules introduced by Feynman by which we can almost
by inspection construct Green functions of arbitrary complexity.
Feynman’s Rules
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For example, with an interaction Lagrangian given by the four-point scalar field φ4:
!4
4
φ
λ−
with λ being a coupling constant, so that Mf i appearing in:
∏=−⋅
−
=
f
i
N
i p
iifif
E
d
mmpp
PP
d
3
3
3
2
2
2
1
2
21
424
2)π2()(4
)()π2( pδ
σ
M
can be calculated as follows:
1. Draw all possible connected, topologically distinct diagrams, including loops, with n
external legs. Ignore vacuum-to-vacuum diagrams.
2. For each internal line, associate a propagator given by:
iεmp
i
pi F
+−
=∆ 22
)(
3. For each vertex, associate the factor −iλ. Momentum is conserved at each vertex.
4. For each internal momentum corresponding to an internal loop, associate an
integration factor ∫d4p/(2π)4.
5. Divide each diagram by an overall symmetry factor S corresponding to the number
of ways one can permute the internal lines and vertices, leaving the external lines fixed.
p
The symmetry factor S is easily calculated. For the four-point function given above, the
1/4! coming from the interaction Lagrangian cancels the 4! ways in which the four
external lines can be paired off with the four scalar fields appearing in φ4, so S=1. Now
consider the connected two-point diagram at second order, which is a double-loop
diagram. There are 4 ways in which each external leg can be connected to each vertex.
There are 3×2 ways in which the internal vertices can be paired off. So this gives us a
factor of 1/S=(1/4!)(1/4!)×(4×4)×(3×2)=1/3!, so S=6.
For QED, the Feynman rules are only a bit more complicated. The interaction
Hamiltonian becomes:
µ
µ
ψγψ Aei−=nInteractioH
33
2017
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As before, the power expansion of the interacting Lagrangian will pull down various
factors of HInteraction. Then we use Wick’s theorem to pair off the various fermion and
vector meson lines to form propagators and vertices.
There are only a few differences that we must note. First, when contracting over an
internal fermion loop, we must flip one spinor past the others to perform the trace and
Wick decomposition. This means that there must be an extra factor of −1 inserted into all
fermion loop integrations. Second, various vector meson propagators in different gauges
may be used, but all the terms proportional to pµ or pν vanish because of gauge
invariance.
Thus, the Feynman rules for QED become:
1. For each internal line, associate a propagator given by:
∫ 4
4
)π2(
pd
34
2017
MRT
iεmp
mpi
εimp
i
pSi F
+−
+/=
+−/
= 22
)(
)(p
2. For each internal photon line, associate a propagator:
νµνµ η
εik
i
kDi F
+
−= 2
)]([kµ ν
3. As each vertex ( ), place a factor of:
µγei−
4. Insert an additional factor of −1 for each closed fermion loop.
5. For each internal loop, integrate over:
6. A relative factor of −1 appears between diagrams that differ from each other by an
interchange of two identical external fermion lines.
7. Internal fermion lines appear with arrows in both clockwise and counterclockwise
directions.However, diagrams that are topologically equivalent are counted only once.
e
γµ
8. External electron and positron lines entering a diagram appear with factors:
respectively. The direction of the positron lines is taken to be opposite of the electron
lines, so that incoming positrons have momenta leaving the diagram.
),(),( spvspu and
35
2017
MRT
respectively. External electron and positron leaving a diagram appear with factors:
),(),( spvspu and
9. Polarization is labeled by εµ
* for an inbound photon and εµ for an outbound photon.
Likewise, we can calculate Feynman rules for any of the actions that we have
investigate earlier. For example, for charged scalar electrodynamics, with the additional
term in the Lagrangian:
φφφφ µ
µ
µ
†22†
AmDD −=L
one has the following interaction Hamiltonian:
φφφφ µ
µ
µµ
†22†
nInteractio )( AeAei −∂−∂−=
rs
H
2. Insert a factor of:
36
2017
MRT
1. For each scalar-scalar-vector vertex, insert the factor:
The Feynman rules for charged scalar electrodynamics are as follows:
µ][ ppei ′+−
νµη2
2 ei
for each seagull diagram.
µ
p p′e
µ
p p′
ν
e2
3. Insert an additional factor of ½ for each closed loop with only two photon lines.
In summary, we have seen that, historically, there were two ways in which to quantize
QED. The first method, pioneered by Feynman, was the propagator approach, which
was simple, pictorial, but not very rigorous. The second was the more conventional
operator approach of Schwinger and Tomonoga. With the LSZ approach, which is
perhaps the most convenient method for deriving the Feynman rules, and with these,
one can almost, by inspecting the Lagrangian, write down the perturbation expansion for
any quantum field theory.
Now that we have derived the Feynman rules for various quantum field theories, the
next step is to calculate cross sections for elementary processes involving photons,
electrons, and antielectrons. At the lowest order, these cross sections reproduce
classical results found with earlier methods. However, the full power of the quantum field
theory will be seen at higher orders, where we calculate radiative corrections to the
hydrogen atom that have been verified to great accuracy. In the process, we will solve
the problem of the electron self-energy, which completely eluded earlier, classical
attempts by Lorentz and others.
To begin our discussion, we will divide Feynman diagrams into two types, trees and
loops, on the basis of their topology. Loop diagrams, as their name suggests, have
closed loops in them. Tree diagrams have no loops (i.e., they only have branches). In a
scattering process, we will see that the sum over tree diagrams is finite and reproduces
the classical result. The loop diagrams, by contrast, are usually divergent and are purely
quantum-mechanical effects.
The Compton Effect
37
2017
MRT
We start by analyzing the lowest-order terms in the scattering matrix for four particles
of fields. In this order, we find only tree diagrams and no loops. Thus, we should be able
to reproduce generalizations of classical and nonrelativistic physics. In the Chart below,
we will summarize the scattering processes that we will analyze.
+−−−
+−+−−−−−
−−−−
+→+++→+
+→++→+
+→++→+
eeγγγNucleuseNucleuse
eeeeeeeeø
γγeeγeγe
:creationPairand:lungBremsstrah
,:scatteringBhabha,:scatteringllerM
,:onannihilatiPair,:scatteringCompton
So, to begin, the first process we will examine is Compton scattering, which occurs
when an electron and a photon collide and scatter elastically. Historically, this process
was crucial in confirming that electromagnetic radiation had particle-like properties (i.e.,
the photon was acting like a particle in colliding with the electron).
We will assume that the electron e− has momentum pi before the collision and pf
afterwards. The photon γ has momentum k before and k′ afterwards. The reaction can
be represented symbolically as:
)(e)(γ)(e)(γ fi pkpk +′→+
38
2017
MRT
By energy-momentum conservation, we also have:
0][][ =−−′+=+−′+⇔′+=+ kpkpkpkpkpkp ififfi
Compton scattering, to lowest order, is shown in the Figure.
Compton scattering: A photon γ of momentum k scatters elastically off an electron e− of momentum pi.
],[ εk ],[ ii sp
],[ ff sp],[ ε ′′k
],[ εk
],[ ii sp
],[ ff sp
],[ ε ′′k
γ
−
e
γ
−
e
γ
−
e
γ
−
e
or
We normalize the wave function of the photon by:
To lowest order, the S matrix is (N.B., the sum of terms within the square brackets end
up being the previous Figure - Left and - Right, respectively):
)ee(
2
1
)( xkixki
kV
xA ⋅⋅−
+= µµ ε
39
2017
MRT
),(
2
)(
2
)(
2
)(
2
)(
),(
)()π2( 44
ii
iii
ff
f
ifif
spu
VE
m
Vk
ei
mkp
i
kV
ei
Vk
ei
mkp
i
kV
ei
spu
VE
m
kpkpS






′
′/−
−′/−/
/−
+
′
′/−
−/+/
/−
×
−−+=
εεεε
δ
where:
µ
µ
γγ aauu =/= and0†
),(
11
),(
22
1
)()π2(
2
44
2
2
ii
ii
ff
if
ifif
spu
mkpmkp
spu
kkEE
m
kpkp
V
e
S








′/
−′/−/
/+′/
−/+/
/×
′
−−+=
εεεε
δ
Simplifying a bit, we get:
The differential cross section is found in several steps. First, we square the S matrix,
which gives us a divergent result. We divide by the singular quantity (2π)4δ (0) and obtain
the rate of transitions. We divide by the flux |v|/V, divide by the number of particles per
unit volume 1/V, multiply by the phase factor for outgoing particles [V2/(2π)6]d3pf d3k′.
This gives us the differential cross section:
where:
∫ ∑ Γ′−−+
′
′
=
′
=
fi ss
iffi
f
f
i
fif
uukpkp
k
kd
E
pdm
kE
me
kdpdVS
d
,
24
33
2
4
6
332
4
2
)(
22
1
)π2(
)π2(
1
)0()π2(
δ
δ
σ
v
v
40
2017
MRT
kp
k
kp
k
ii ′⋅
/′//
+
⋅
//′/
=Γ
22
εεεε
To reduce out the spins, we will once again use the convenient formula given in the
Rutherford Scattering chapter:
Although this calculation looks formidable, we can perform the trace of up to eight Dirac
matrices by reducing it to the trace of six, and then four Dirac matrices, &c. We will use
the formula:







 +/
Γ
+/
Γ=Γ∑ m
mp
m
mp
spuspu
fi
ss
iiff
fi
22
Tr),(),( 0†0
,
2
γγ
41
2017
MRT
)(Tr)(Tr)(Tr)(Tr 1232212423123212321 −///⋅++///⋅−//⋅=//// nnnnn kkkkkkkkkkkkkkkkkk LKLLL
The problem simplifies enormously because we can eliminate entire groups of terms
every time certain dot products appear, since:
022
=′⋅′=⋅=′= kkkk εε
We can also simplify the calculation by using:
22222
1 mpp fi ==−=′= andεε
In short, each trace consists of collecting the complete set of all possible pairs of dot
products of vectors, most of which vanish. Dividing the factors into smaller pieces, we
now find:
where:
])(2[8
)(Tr2)(Tr2)]()([TrTr
2
1
ii
fififi
pkkkp
pkkppkkpmpkmpk
⋅′+⋅′⋅=
/′//′/⋅=/′////′/⋅=+/′///+///′/=
ε
εεεεεεεεεε
42
2017
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4321
,
2
TrTrTrTr +++=Γ∑
fi ss
if uu
and
iifi
iiiifi
pkkpkkpkpk
kpkpkkpkpkmpkmpk
⋅′′⋅−⋅⋅′+−⋅′⋅′⋅=
′⋅′⋅−⋅⋅′+//′//′/′/⋅=+//′/′/+///′/=
222
22
2
)(8)(8]1)(2[))((8
)(8)(8)(Tr2)]()([TrTr
εεεε
εεεεεεεεεε
We also have:
and
])(2[8TrTr 2
23 ii pkkkp ⋅′+⋅′⋅== ε
iifi pkkpkkpkpk ⋅⋅′−⋅′′⋅+−′⋅⋅⋅′= 222
4 )(8)(8]1)(2[))((8Tr εεεε
where Tr4 was obtained from Tr3 by making the substitution [ε,k]→[ε ′, k ′].
Since the calculation is Lorentz invariant, we can always take a specific frame. We
loose no generality by letting the electron be at rest, and let the incoming photon lie
along the z-axis. Let the outgoing photon scatter within the y-z plane, making an angle θ
with the z-axis (see Figure). Then the specific parametrization is given by:
43
2017
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Compton scattering in the laboratory frame, where the electron is at rest.
k′
fp
k
θ
It is important to notice that the only independent variables in this scattering are k and θ.
All other variables can be expressed in terms of these two variables. For example, we
can solve for k′ and E. In terms of the independent variables k and θ:
kkmE
mk
k
k ′−+=
−+
=′ and
)cos1)((1 θ
Adding all four contributions, we now have:






−⋅′+
′
+
′
=Γ∑ 2)(4
2
1
),(),( 2
2
,
2
εε
k
k
k
k
m
spuspu
fi ss
iiff
]cos,sin,0,[]cos,sin,0,[],0,0,[]0,0,0,[ θθθθ µµµµ kkkEpkkkkkkkmp fi ′−′−=′′′=′== &,,
We must now integrate over the momenta of the outgoing photon k′ and electron pf .
Since the only independent variables in the problem are given by k and θ, all integrations
are easy, except the Jacobian, which arises when we change variables and integrate
over delta functions. Thus, the integration over d3pf is trivial because of momentum
conservation; it simply sets the momenta to be the values given above. That leaves one
complication, the integration over time components dp0 f . However, this integration can
be rewritten in a simple fashion:
)(22
)(
2
3
3
4
kkm
k
d
E
d
pkkp
f
f
fi
′
′
=′−′−+ k
p
δ
44
2017
MRT
∫ −= )()(
2
1
0
22
0 fff
f
pmppd
E
θδ
The integration over p0 f in the integral just sets its value to be the on-shell value. Finally,
this last delta function can be removed because of the integration over k′. The only tricky
part is to extract from this last integration the measure when we integrate over k′. This
last delta function can be written as:
where k′(k) is the value given by k′=k/[1+(k/m)(1−cosθ)] above. Putting all integration
factors together, we now have:
)(2
)]([
)]cos1(2)(2[])[( 2
kkm
kkk
kkkkmmkpk i
′
′−
=−′−′−=−′−+
δ
θδδ
Inserting everything into the cross section,we obtain the Klein-Nishina formula (1929):
where α =e2/4πεohc is the fine structure constant. If we take the low-energy limit k→0,
the Klein-Nishina formula reduces to the Thomson scattering formula:
45
2017
MRT






−′⋅+
′
+
′





 ′
=
Ω
2)(4
4
2
2
2
2
εε
ασ
k
k
k
k
k
k
md
d
2
2
2
)( εε
ασ
′⋅=
Ω md
d
If the initial and final photon are unpolarized, we can average over the initial and final
polarizations ε and ε ′. In the particular parametrization that we have chosen for our
momenta, we can choose our polarizations ε and ε ′ such that they are purely transverse
and perpendicular to the momenta pi and pf , respectively:
Since these polarization vectors satisfy all the required properties, the sum over these
same polarization vectors is:
θεεεε 2
,
2)()(
,
2
cos1)()( +=′⋅=′⋅ ∑∑ ji
ji
ss fi
]sin,cos,0,0[]0,0,1,0[]0,1,0,0[]0,0,1,0[ )2()1()2()1(
θθεεεε −=′=′== and,,
)cos1(
2
1
επ4
2
2
2
o
2
θ
σ
+








=
Ω cm
e
d
d
so that the Thomson scattering formula above becomes (MKS Units):
The average cross section is given by:
46
2017
MRT






−
′
+
′





 ′
=
Ω
θ
ασ 2
2
2
2
av
sin
2 k
k
k
k
k
k
md
d
The integral over θ is straightforward if we define z=cosθ. The integration yields:








+
+
−
+
+





+−
+
++






=








−+
−
−
−+
+
−+
= ∫−
232
2
1
1 2
2
32
2
)21(
31
2
)21ln(
)21ln(
21
)1(21
4
3
3
8π
)]1(1[
1
)1(1
1
)]1(1[
1π
a
a
a
a
a
a
aa
a
a
m
za
z
zaza
zd
m
α
α
σ
where a=k/m.
For small energies, this reduces to the usual Thomson total cross section:
For high energies, the logarithm starts to dominate the cross section and we get:
2
Thomson cm24
2
2
0
10665.0
3
8π
lim −
→
×===
mk
α
σσ




















++





=
∞→ m
k
k
m
O
m
k
mkk
ln
2
12
ln
π
lim
2
α
σ
The Feynman diagrams for pair annihilation of an electron and positron into two gamma
rays is shown in the Figure. Pair annihilation is represented by the process:
However, notice that we can obtain this diagram if we simply rotate the diagram for
Compton scattering. Thus, by a subtle redefinition of the various momenta, we should be
able to convert the Compton scattering amplitude, which we have just calculated, into
the amplitude for pair production. This redefinition is called the substitution rule (i.e., take
a process 1+2→3+4 and convert it to 1+3→2+4). It can also be viewed as a symmetry
in the S matrix – the crossing symmetry.
)(γ)(γ)(e)(e 2121 kkpp +→+ +−
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Pair Annihilation
Pair annihilation: An electron e− of momentum p1 annihilates with a positron e+ of momentum p2 into two
photons γ.
1p 2p 1p 2p
],[ 11 εk ],[ 22 εk ],[ 22 εk],[ 11 εk
γ
−
e
−
e
+
e +
e−
e
+
e
γ
γγ
− −
or
For example, the S matrix now yields, to lowest order:
where we have made the substitutions:
48
2017
MRT
),(
2
)(
2
)(
2
)(
2
)(
),(
)()π2(
11
12
2
211
1
1
1
112
2
22
2
2121
44
spu
E
m
k
ei
mkp
i
k
ei
k
ei
mkp
i
k
ei
spv
E
m
ppkkS if







 /−
−/+/
/−
+
/−
−/+/
/−
×
−−+=
εεεε
δ
),(),(),(),(],[],[],[],[ 22112211 spvspuspuspukkkk ffii →→→′′−→ and,, εεεε
We will, as usual, take the Lorentz frame where the electron is at rest. Then our
momenta becomes (see Figure):
Pair annihilation in the laboratory frame, where the electron is at rest.
]cos,sin,0,[]cos,sin,0,[],0,0,[]0,0,0,[ 1122111121 θθθθ µµµµ kkkkkkkkEpmp −−==== pp &,,
There are only two independent variables in this process, |p| and θ. All other variables
can be expressed in terms of them.
1k
1p
2k
θ
For example, we can show that:
We also have:
)cos()( 12121 θp−+=+=⋅+=+ EmkEmmkkkkEm and
49
2017
MRT
θ
θ
θ cos
)cos()(
cos
)(
21
p
p
p −+
−+
=
−+
+
=
Em
EEm
k
Em
Emm
k and
When we contract the Dirac matrices, the calculation proceeds just as before, except
that we want to evaluate |v Γu|2. We have to use:−
m
mp
spvspv
s
2
),(),(
−/=∑
The trace becomes:






+⋅−+=Γ∑ 2)(4
2
1
),(),( 2
21
2
1
1
2
2
2
εε
k
k
k
k
m
spuspv
s
The integration over d3k1 and d3k2 also proceeds as before. The integrations over the
delta functions are straightforward, except that we must be careful when picking up a
measure term when we make a transformation on a Dirac delta function. When this
additional measure term is inserted, the differential cross section becomes:






+⋅−+
−+
+
=
Ω
2)(4
)cos(8
)( 2
21
2
1
1
2
2
2
εε
θ
ασ
k
k
k
k
Em
Em
d
d
pp
The total cross section is obtained by summing over photon polarizations. As before,
we can take a specific set of polarizations which are transverse to p1 and p2:
Then we can sum over all polarizations. The only difficult sum involves:
]sin,cos,0,0[]0,0,1,0[]sin,cos,0,0[]0,0,1,0[ 11
)2(
2
)1(
2
)2(
1
)1(
1 θθεεθθεε kk−==−== p&,,
50
2017
MRT








+−=
+
−=
+
−=•=⋅
212121
2
21
21
)2(
2
)2(
1
11
1
2
)(2
1
2
)(
1
kk
m
kk
Emm
kk
kk
kkεε
Then the sum over spins can be written as:
2
21,
2)(
2
)(
1 11)(














+−+=⋅∑ k
m
k
m
ji
ji
εε
The only integration left is the one over the solid angle, which leaves us with:








−
+
−−+
−
++
+
=
1
3
)1(ln
1
14
)1(
π
2
2
2
2
2
2
γ
γ
γγ
γ
γγ
γ
α
σ
m
where γ =E2/m. This result was first obtained by Dirac (1930).
Next, we investigate electron-electron scattering. To lowest order, this scattering
amplitude contains two diagrams (see Figure). This scattering is represented by:
By a straightforward application of the formulas for differential cross sections, we find:
∫ ′
′
′
′
−−′+′
⋅
=
2
2
3
2
3
1
3
1
3
2121
44
2
21
44
)π2()π2(
)()π2(
)(
if
E
pd
E
pd
pppp
pp
me
d Mδσ
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MRT
Møller Scattering
Møller scattering of two electrons with momenta p1 and p2.
)(e)(e)(e)(e 2121 pppp ′+′→+ −−−−
1p 2p 1p 2p
1p′ 2p′
γ
−
e −
e −
e−
e
γ
1p′ 2p′
Using a straightforward application of Feynman’s rules for these two diagrams, we can
compute | Mf i |2:
Since the trace is only four Dirac matrices, taking the trace gives:
52
2017
MRT




−′−′
⋅−⋅
+
−′
⋅−′⋅+′⋅+⋅
+




−′
⋅−′⋅+′⋅+⋅
2
12
2
11
21
22
21
22
12
2111
22
11
2
21
22
11
2121
22
21
2
21
4
)()(
2)(
2
])[(
)(2)()(
])[(
)(2)()(
4
1
pppp
ppmpp
pp
ppppmpppp
pp
ppppmpppp
m
])(
)()(
1
2222
Tr
])[(
1
22
Tr
22
Tr
4
1
21
2
22
2
11
1221
22
11
22112
pp
ppppm
mp
m
mp
m
mp
m
mp
ppm
mp
m
mp
m
mp
m
mp
if
′↔′+
−′−′





 +′/+/+′/+/−




−′





 +′/+/




 +′/+/=
σν
σν
σν
σν
γγγγ
γγγγM
The kinematics are illustrated in the Figure. Without any loss of generality, we can
choose the center-of-mass frame, where the electron momenta p1 and p2 lie along the z-
axis:
The only independent variables are |p| and θ. In terms of this parametrization, we find:
θθθθ cos)cos1(cos)cos1(2 22
21
22
11
22
21 mEppmEppmEpp −+=′⋅+−=′⋅−=⋅ &,
53
2017
MRT
Møller scattering in the center-of-mass frame.
θ
]cos,sin,0,[]cos,sin,0,[],0,0,[],0,0,[ 2121 θθθθ µµµµ pppppp −−=′=′−== EpEpEpEp &,,
Then the entire cross section can be written in terms of these independent variables. We
finally obtain the Møller formula (1932) in the center-of-mass frame:














+
−
−
+−
−
−
=
Ω θθθ
ασ
2222
222
242222
2222
sin
4
1
)2(
)(
sin
3
sin
4
)(4
)2(
mE
mE
mEE
mE
d
d
z
In the relativistic limit, as E→∞, this formula reduces to:
54
2017
MRT
For the low-energy, nonrelativistic result, we find:






+−=
Ω 2
1
sin
2
sin
4
242
2
θθ
ασ
Ed
d






−=
Ω θθ
ασ
2422
2
sin
3
sin
4
4
1
vmd
d
To calculate the cross section for electron-positron scattering (see Figure), we can use
the substitution rule. By rotating the diagrams for Møller scattering, we find the Feynman
diagrams for Bhabha scattering. For the process:
The only substitutions we must make are:
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2017
MRT
Bhabha Scattering
)()()()()()()()( 22221111 pvpupvpupupupupu ′−→′−→′→′→ and,,
)(e)(e)(e)(e 2121 pppp ′−+′→−+ +−+−
Bhabha scattering of and electron off a positron.
1p 2p
1p 2p
1p′ 2p′
γ
−
e +
e
+
e−
e
γ
1p′ 2p′
The calculation and the traces are performed exactly as before with the above
substitutions. We merely quote the final result, due to Bhabha (1935):
In the relativistic limit, we have:
56
2017
MRT



++−+++−+




−−
−
+
−−
−
−=
Ω
]cos2)cos2)(cos1(4)coscos21(2[
16
1
)cos1()(2
)2(
)cos1)((
8
4
5
2
242224
4
2222
222
222
44
2
2
θθθθθ
θθ
ασ
mmEE
E
mE
mE
mEE
mE
Ed
d
























−++












+
=
Ω
2
sin
2
cos
2)cos1(
2
1
2
sin
2
cos1
8 2
4
2
2
4
2
2
θ
θ
θ
θ
θ
ασ
Ed
d
whereas, in the nonrelativistic limit, we find:
4
2
44
2
2
2
sin22
sin16


















=






=
Ω θ
α
θ
α
σ
v
m
v
m
d
d
Bremsstrahlung (i.e., German for breaking radiation) is the process by which radiation is
emitted from an electron as it moves past a nucleus (see Figure). Momentum
conservation gives us:
Classically, one can calculate the radiation emitted by a moving charge as it
accelerates past a proton. However, unlike the previous scattering processes, which
agree to first order with the experimental data, we find a severe problem with this
amplitude, which is the infrared divergence. The quantum field theory calculation, to
lowest order, reproduces the classical result, including the unwanted infrared
divergences, which has its roots in the classical theory.
fi pkqp +=+
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MRT
Bremsstrahlung
Bremsstrahlung, or the radiation emitted by an electron scattering in the presence of a nucleus.
ip
eZ
fpk
eZ
fp
k
ip
q
q
γ
γ
γ
γ
−
e −
e
Although this infrared divergence first arose (in another form) in the classical theory,
the final resolution of this problem comes when we take into account higher quantum
loop corrections to the scattering amplitude. The scattering matrix, using Feynman’s
rules, is:
The differential cross section now becomes:
58
2017
MRT
),(
2
)()()(
2
)(
),(
)(π2
2
0
2
0
23
3
ii
iif
ff
f
ifif
spu
E
m
k
i
mkp
iii
mkp
i
k
i
spu
E
m
EkE
V
eZ
S








/−
−/−/
−
+
−
−/+/
/−
×
−+−=
εγγε
δ
qq
∫ Γ−+=
Ω
2
46
3362
1
)ω(π2
π)2(2ω
ifif
f
f
ii
uuEE
kd
E
pdm
E
eZm
d
d
qv
δ
σ
where ω=k0 and where:
εγγε /
−/−/
+
−/+/
/=Γ
mkpmkp if
11 00
The trace we wish to calculate is:
The traces involved in the calculation yield:
∑ 














 +/








/⋅/
+/−/
/+/
/⋅/
+/+/





 +/








/
/⋅/
+/−/+
/⋅/
+/+/
/=
++=
ε
γεεγεγγε
m
mp
kp
mkp
kp
mkp
m
mp
kp
mkp
kp
mkp
m
f
i
i
f
fi
i
i
f
f
222222
Tr
2
1
)TrTrTr(
2
1
Tr
0000
32125
59
2017
MRT
and
,]ω22
)ω22()(2[
)(8
1
)]()()([Tr
)(
1
Tr
0
00022
2
0
21
kpkppkppkpp
kppppppmp
kp
mpmkpmkp
kp
jififif
ifiifif
f
fff
f
⋅⋅−⋅+⋅⋅⋅+
⋅−⋅−++⋅
⋅
=
+//+/+/+/+//
⋅
=
∑
∑
εε
ε
εγε
ε
ε
)(TrTr 22 fi pp −↔=
)]ω(ωω)(
)()242([
))((
16
)()()()()([Tr
))((
1
Tr
00222
2200
00
3
kppkppppmkpkppkp
pkpmppppkpkppp
kpkp
ppmpmkpmpmkp
kpkp
iffifififi
iffifififi
if
fiffii
if
⋅−⋅−⋅+−⋅⋅+⋅⋅−
⋅⋅+−⋅−⋅+⋅−⋅⋅⋅
⋅⋅
=
−↔++//+/+/+//+/−/
⋅⋅
−=
∑
ε
εεε
εγεγ
ε
The parametrization of the momenta is a bit complicated, since the reaction does not
take place in a plane (see Figure). We place the emitted photon momenta in the z-
direction, the emitted electron momenta pf in the y-z plane, and the incoming electron
momenta pi in a plane that is rotated by an angle ϕ from the y-z plane. The specific
parametrization is equal to:
where pi =|pi| and pf =|pf |.
]cos,sin,0,[]ω,0,0,ω[ ffffff ppEpk θθµµ == ,
60
2017
MRTBremsstrahlung, where the emitted photon is in the z-direction and the emitted electron is in the y-z
plane. The incoming electron is in a plane rotated by an angle ϕ from the y-z plane.
ϕk
ip
fp
θ
and
]cos,cossin,sinsin,[ iiiiiiii pppEp θϕθϕθµ =
z
Now we must calculate the sum over transverse photon polarizations. Since kµ points
in the z-direction, we can choose:
which satisfies all the desired properties of the polarization tensor. With this choice of
parametrization for the momenta and polarizations, we find:
]0,1,0,0[]0,0,1,0[ )2()1(
== εε and
61
2017
MRT
ii
j
i
j
ff
j
f
j
pppp θεθε 222)(222)(
sin)(sin)( =⋅=⋅ ∑∑ ,
and
ϕθθεε cossinsin))(( )()(
fifi
j
i
j
f
j
pppp =⋅⋅∑
It is now a simple matter to collect everything together, and we now have the Bethe-
Heitler formula (1934), which was first computed without using Feynman rules:




+−
−−
−
−−
+
+




−
−
+−
−
×
ΩΩ=
)ω24(
)cos)(cos(
cossinsin
2
)cos)(cos(
sinsin
ω2
)4(
)cos(
sin
)4(
)cos(
sin
ω
ω1
)π2(
22
2222
2
22
2
22
22
2
22
eγ42
32
qEE
pEpE
pp
pEpE
pp
qE
pE
p
qE
pE
p
dd
d
qp
pZ
d
fi
iiifff
fiif
iiifff
ffii
f
iii
ii
i
fff
ff
i
f
θθ
ϕθθ
θθ
θθ
θ
θ
θ
θ
α
σ
Now let us make the approximation that ω→0. In the soft bremsstrahlung limit, we find
a great simplification, and the differential cross section becomes the one found by
classical methods:
Here the infrared divergence appears for the first time. This problem was first correctly
analyzed by F. Block and A. Nordsieck (1937). The integral d3k/ω is divergent for small
ω, and therefore the amplitude for soft photon emission makes no sense. This is rather
discouraging, and revealed the necessity of properly adding all quantum corrections.
The resolution of this equation only comes when the one-loop vertex corrections are
added in properly. For now, it is important to understand where the divergence comes
from and its general form.
∞
⋅−−
~
2
1
~
)(
1
22 kpmkp
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∑ 







⋅
⋅
−
⋅
⋅






ΩΩ −ε
εεσσ
2
3
32
elastic )π2(ω2
~
i
i
f
f
pk
p
pk
pde
d
d
d
d k
The infrared divergence always emerges whenever we have massless particles in a
theory that can be emitted from an initial or final leg that is on the mass shell. For
example, whenever we emit a soft photon of four-momentum k from an on-shell electron
with momentum p, we find that the propagator just before the emission is given by:
Because p2 =m2 and k small, we find that the integration over momentum k inevitably
produces an infrared divergence.
In order to quantify this infrared divergence, let us perform the integration over the
four-momentum k, separating out the angular part dΩ from d3k. To parametrize the
divergence, we will regulate the integral by allowing the photon to have a small but finite
mass µ. We will integrate k over µ to some energy E given the sensitivity of the detector.
Expanding out the expression of the square, the amplitude now becomes:
In our approximation, we can perform the angular integral over the last three terms in the
square brackets.
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∫ ∫ 







⋅
−
⋅
−
⋅⋅
⋅
Ω





ΩΩ
E
ifif
if
pk
m
pk
m
pkpk
pp
ddk
d
d
d
d
µ
ασσ
2
2
2
2
3
2
2
elastic )()())((
2
)π2(
~ k
Let us calculate the last two terms appearing on the right-hand side of the equation.
We use the fact that:
1
)cos1(
)(cos
2)(π4
1
1 22
2
2
2
=
−
=
⋅
Ω
∫∫ − θβ
θd
E
m
pk
md
f
For the integral over the first term, we will use the fact that:
and because dΩ=2πd(cosθ), we can integrate the last two terms:
)cos1(~ ff Epk θ−⋅
)cos1()cos1(
)cos1(2
~
))((
2 22
fiif
if
EE
E
pkpk
pp
θβθβ
θβ
−−
−
⋅⋅
⋅
We will now introduce the Feynman parameter trick, which is often used to evaluate
Feynman integrals:
By introducing a new variable x, we are able to perform the angular integration. We find:
∫ −+
=
1
0 2
)]1([
1
xbxa
xd
ba
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






<<








+








−
<<+











+
=
−+−
−=
−−−
−
⋅⋅
⋅Ω
∫
∫ ∫∫
1ln2
1)(
2
sin
3
4
12
)1()2(sin41
)cos1(2
)]1(coscos1[
)(cos
2
1
)cos1(2~
))((
2
π4
2
2
2
2
2
2
422
1
0 222
22
1
0 2
22
q
m
q
m
O
m
q
O
xx
xd
xx
d
xd
pkpk
ppd
iiif
if
if
if ββ
θ
β
θββ
θβ
θβθβ
θ
θβ
Inserting this value back into the previous expression, we find that the final soft
bremsstrahlung cross section is given by:
Although this formula agrees well with experiment at large photon momenta, this
amplitude is divergent if we let the fictitious mass of the photon µ go to zero. Thus, the
infrared divergence occurs because we have massless photons present in the theory!
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














+−








−
<<+



















ΩΩ
1~1ln
1)(
2
sin
3
4
ln
π
~
2
2
2
2
422
2
2
0 β
ββ
θ
β
µ
ασσ
q
m
O
m
q
O
E
d
d
d
d
We should mention that the infrared problem arose (in another form) in classical
physics, before the advent of quantum mechanics. The essential point is that, even at
the classical level, we have the effects due to the long-range Coulomb field. If one were
to calculate the radiation field created by a particle being accelerated by a stationary
charge, one would find a similar divergence using only classical equations. If one tries to
divide the energy by k0 to calculate the number of photons emitted by bremsstrahlung, it
turns out to be proportional to the result presented above. Thus, as the momentum of
the emitted photon goes to zero, the number of emitted photons becomes infinite. (N.B.,
Classically, the infrared divergence appears in the number of emitted photons, not the
emitted energy).
Quantum field theory gives us a novel, but rigorous, solution to the infrared problem,
which goes to the heart of the measurement process and the quantum theory. To this
order of approximation, we have to add the contribution of two different physical
processes to find the cross section of electron scattering off protons or a nucleus with
charge Ze, or other charged particles (see Figure).
The first diagrams describes the bremsstrahlung amplitude for the emission of an
electron and a photon. The divergence of this amplitude is classical. Second, we have to
sum over a purely quantum-mechanical effect, the radiative one-loop corrections of the
electron elastically colliding off the charged proton. This may seem strange, because we
are adding the cross sections of two different physical processes together, one elastic
and one inelastic, to cancel the infrared divergence. However, this makes perfect sense
from the point of view of the measuring process.
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The infrared divergence cancels if we add the contributions of two different physical processes. These
diagrams can be added together because the resolution of any detector is not sensitive enough to select
out just one process. The nucleus is label with the bullet ( ).
The essential point is to observe that our detectors cannot differentiate the presence of
pure electrons from the presence of electrons accompanied by sufficient soft photons.
This is not just a problem of having crude measuring devices. No matter how precise our
measuring apparatus may become, it can never be perfect; there will always be photons
with momenta sufficiently close to zero that will sneak past them. Therefore, from an
experimental point of view, our measuring apparatus cannot distinguish between these
two types of processes and we must necessarily add these two diagrams together.
Fortunately, we get an exact cancellation of the infrared divergences when these two
scattering amplitudes are added together.
A full discussion of this cancellation, however, cannot be described until we discuss
one-loop corrections to scattering amplitudes. Therefore, in the Anomalous Magnetic
Moment chapter, we will prove that the bremsstrahlung amplitude, given by:








−













ΩΩ 2
2
2
2
0
lnln
π
~
m
qE
d
d
d
d
µ
ασσ
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must be added to the one-loop vertex correction in order to yield a convergent integral.
Finally, we note that, by the substitution rule, we can show the relationship between
bremsstrahlung and pair production. Once again, by rotating the diagram around, we
can convert bremsstrahlung into pair production.
This end our discussion of the tree-level, lowest-order scattering matrix. Although we
have had great success in reproducing and extending known classical results, there are
immense difficulties involved in extending quantum field theory beyond the tree level.
When loop corrections are calculated, we find that the integrals diverge in the ultraviolet
region of momentum space. In fact, it has taken over a half century, involving the
combined efforts of several generations of physicists, to resolve many of the difficulties
of renormalization theory.
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We now turn to a detailed calculation of single-loop radiative corrections. Although the
calculations are often long and tedious, involving formally divergent quantities, the final
conclusions are simple and show that the various infinities can be consistently absorbed
into a redefinition of the physical constants of the theory, such as the electric charge and
electron mass. Most important, the agreement with experiment is astonishing!
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Radiative Corrections
We will begin our discussion of radiative corrections by first examining the self-energy
correction to the photon propagator, called the vacuum polarization diagram. We will
show that the divergence of this diagram can be absorbed into a renormalization of the
electric charge. Then, we will calculate the single-loop correction to the electron-photon
vertex and show that this leads to corrections to the magnetic moment of the electron.
The theoretical value of the anomalous magnetic moment will agree with experiment to
one part in 108. After that, we will show that the radiative correction to the vertex function
is also infrared divergent. Fortunately, the sign of this infrared divergence is opposite the
sign found in the bremsstrahlung amplitude. When added together, we will find that the
two cancel exactly, giving us a quantum mechanical resolution of the infrared problem.
And finally, we will analyze the Lamb shift between the energy levels of the 2S½ and 2P½
orbitals of the hydrogen atom. The calculation is rather intricate, because the hydrogen
atom is a bound state, and also there are various contributions coming from the vertex
corrections, the anomalous magnetic moment, the self-energy of the electron, the
vacuum polarization diagram, &c. However, when all these contributions are added, we
find agreement with experiment to within one part in 106.
The simplest higher-order radiative correction is the vacuum polarization diagram (see
Figure). This diagram is clearly divergent. For large momenta, the Feynman propagators
of the two electrons give us two powers of p in the denominator, while the overall
integration over d4p gives us four power of p in the numerator. So this diagram diverges
quadratically in the ultraviolet region of momentum space:
We will perform this integration via the Pauli-Villars method (1949), although the
dimensional regularization method is significantly simpler. The Pauli-Villars method
replaces this divergent integral with a convergent one by assuming that there are
fictitious fermions with mass M in the theory with ghost couplings. At the end of the
calculation, these fictitious particles will decouple if we take the limit as their masses
tend to infinity. Therefore, M gives us a convenient way of cutting off the divergence of
the self-energy correction.
∫
∫








+−+
+/+/
+−
+/
−=
+−/−/+−/
−=Π
νµ
νµνµ
γγ
γγ
iεmqk
mqki
iεmk
mkikd
e
iεmqkiεmk
kd
em
22224
4
2
4
4
2
)(
)()(
Tr
)π2(
)(
11
)π2(
Tr
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MRTFirst-loop correction to the photon propagator, the vacuum polarization diagram, which gives us a
correction to the coupling constant and contributes to the Lamb shift.
k
qk −
q
−
e
+
e
The diagram then becomes modified as follows:
The most convenient way in which to perform the integration is to add additional
auxiliary variables. This allows us to reverse the order of integration. We can then
perform the integration over the momenta, and save the integration over the auxiliary
variables to the very end. We will use:
Mm
νµνµνµ Π−Π=Π
~
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∫
∞
+−
=
+− 0
)(
22
22
e
1 εimki
d
εimk
α
α
Inserting this expression for the electron propagator and performing the trace, we find:
With the insertion of these auxiliary variables (i.e., α1 and α2), we can now perform the
integration over d4k.
)]()()([
e
)π2(
4
22
0 0
)]})[()({
4
4
21
2 22
2
22
1
mqkkqkkqkk
kd
dde εimqkεimkim
−⋅−−−+−×
=Π ∫ ∫ ∫
∞ ∞
+−−++−
νµµννµ
αα
νµ
η
αα
First, we shift momenta and complete the square:
Putting everything back into Πµν , we have:
νµ
αα
νµ
αα
η
αααα 3
21
2
)(
4
4
2
21
2
)(
4
4
)(2π3
e
)π2()(π16
1
e
)π2(
21
2
21
2
+
=
+
= ∫∫
++
i
i
pp
pd
i
pd pipi
and
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qkp
21
2
αα
α
+
−=
Then we use the fact that:
21
2
)( Π+Π−=Π νµνµνµνµ ηη qqqm
where:
∫ ∫
∞ ∞
+
−=Π
0 0
),(
4
21
21
211
21
e
)(π
2 αα
αα
αα
αα
α f
ddi
∫ ∫
∞ ∞
−
+
−=Π
0 0
),(
213
21
212
21
e]1),([
)(
1
π
αα
αα
αα
αα
α f
fiddi
and
and where:
))((),( 21
2
21
212
21 ααε
αα
αα
αα +−−
+
= imiqif
There is a similar expression for ΠM
µν . We notice that Π2 diverges quadratically,
which is bad.
However, since we have carefully regularized this integral using the Pauli-Villars
method, the integral is finite for fixed M, and we are free to manipulate this expression.
We can then show that Π2 vanishes:
To perform the integration of Π1, we use one last identity:
∫
∞





 +
−=
0
21
11
ρ
αα
δ
ρ
ρd
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Inserting this into the expression for Π1 and rescaling αi , we find:
∫ ∫ ∫
∫ ∫ ∫
∞ ∞ ∞
+−
∞ ∞ ∞
−−−=





 +
−
+
−=Π
0 0 0
)(
212121
0 0 0
),(21
4
21
21
211
2
21
2
21
e)1(
π
2
e1
)(π
2
εimqi
f
d
dd
i
d
ddi
ααρ
αα
ρ
ρ
ααδαααα
α
ρ
αα
δ
ρ
ρ
αα
αα
αα
α
As expected, this integral is logarithmically divergent.
Thus, the vacuum polarization diagram is only logarithmically divergent.
02 =Π
At this point, we now use the Pauli-Villars regulator, which lowers the divergence of the
theory. To perform the tricky ρ integration, we use the fact that m2 −α1α2q2 is positive, so
we can rotate the contour integral of ρ in the complex plane by −90 degrees. Using
integration by parts and rescaling, we have the following identity:
where a(m)= m2 −α1α2q2. The dangerous divergent comes from the last term. However,
since the last term is independent of a, it cancels against the same term, with a minus
sign, coming from the Pauli-Villars contribution. Thus, we have the identity:








+








−−=+−=−∫
∞
−−
∞→ 2
2
2
2
21
0
)()(
ln1ln)(ln)(ln]e[elim
m
M
m
q
Mama
d Mama
αα
ρ
ρ ρρ
ε
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∫∫
∞
−
∞
−
∞
−
−=
ε
ρ
ε
ρρ
ρρρ
ρ
ρ
a
aaa
d
d
)e(ln)e(lne
0
for large but finite M. We can now take the limit ε →0 and M becomes large. Then:
∫
∞








−−−+







−=Π
0 2
2
111112
2
1 )1(1ln)1(
π
2
ln
3π
~
m
q
d
i
m
Mi
ααααα
αα
If we perform the last and final integration over α1, we arrive at:
















−








−








−








+++








−−=Π 11
4
arccot1
42
12
3
1
ln
3π
~
21
2
2
21
2
2
2
2
2
2
1
q
m
q
m
q
m
m
Miα
That was the final result. After a long calculation, we find a surprisingly simple result
that has a physical interpretation. We claim that the logarithmic divergence can be
cancelled against another logarithmic divergence coming from the bare electric charge
eo. In fact, we will simply define the divergence of the electric charge so that it precisely
cancels against the logarithmic divergence of Π1.
To lowest order, we find that we can add the usual photon propagator Dµν to the one-
loop correction, leaving us with a revised propagator:








−







−− 2
2
2
2
2
π15
ln
π3
1
m
q
m
M
q
i αα
η νµ
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in the limit as q2 →0. This leaves us with the usual theory, except that the photon
propagator is multiplied by a factor:
322
Z
q
i
q
i
νµνµ ηη −→−








− 2
2
3 ln
π3
1~
m
M
Z
α
where:
Now let us absorb this divergence into the coupling constant eo. We are then left with
the usual theory with an extra (infinite) factor Z3 multiplying each propagator. Since the
photon propagator is connected to two electron vertices, with coupling eo, we can absorb
Z3 into the coupling constant, so we have, to lowest order:
where e is called the renormalized electric charge. Since the infinity coming from Z3
cancels (by construction) against the infinity coming from the bare electronic charge, the
renormalized electric charge e is finite.
o3 eZe =
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We recall that in the Relativistic Quantum Mechanics chapter we derived the magnetic
moment of the electron by analyzing the coupling of the electron to the vector potential:
where 2 is the Landé g factor (i.e., gyromagnetic ratio). At the tree level, we know that
the coupling of an electron to the photon is given by Aµ uγµu, which in turn gives us that
gyromagnetic ratio of 2. However, the experimentally observed value differed from this
predicted value by a small but important amount. Schwinger’s original calculation (1946)
of the anomalous magnetic moment of the electron helped to establish QED as the
correct theory of electrons and photons.
S





==
mc
e
cm
e
2
2
2
σσσσµµµµ
h
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MRT
Anomalous Magnetic Moment
−
To calculate the higher-order corrections to the magnetic moment of the electron, we
will use the Gordon identity:
)(]))[((
2
1
)()( puqipppu
m
pupu ν
νµµµγ Σ+′+′=′
The magnetic moment of the electron comes from the second term u Σµν qνu Aµ. To
see this, we take the Fourier transform, so qν becomes ∂ν, and the coupling becomes
u Σµν Fµν u. The magnetic field Bi is proportional to εijk Fjk, so this coupling term in the
rest frame now becomes uσi Biu, where we use the Dirac representation of the Dirac
matrices. Since σi =σσσσ is proportional to the spin of the electron, which in turn is
proportional to the magnetic moment of the electron, the coupling becomes µµµµ•B. This
is the energy of a magnet with moment µµµµ in a magnetic field B.
−
−
−
In this chapter, we will calculate the one-loop vertex correction, which gives us a
connection to the electron-photon coupling, in the lowest order in α, given by:
for the process given in the Figure.
)(
2π2
1
2
)(
)( pu
m
qi
m
pp
pu







 Σ






++
′−
′
ν
νµµ α
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MRTFirst-loop correction to the electron vertex function, which contributes to the anomalous electron
magnetic moment.
k
p
p′
kp −′
kp −
Notice that the u Σµν qν u Aµ term is modified by the one-loop correction, so that the g
factor of the electron becomes:
−
π22
2
π2
1
2
αα
=
−
⇔+=
gg
Thus, QED predicts a correction to the moment of the electron. Hmmm…. Interesting!
To show this, we will begin our calculation with the one-loop vertex correction:
Anticipating that the integral is infrared divergent, we have added µ, the fictitious mass
of the photon. The integral is also divergent in the ultraviolet region, so we will use the
Pauli-Villars cutoff method later to isolate the divergence.
∫ +−/−/+−/−′/+−
−
−=′Λ ν
µνµ γ
ε
γ
ε
γ
εµ imkp
i
imkp
i
ik
ikd
eipp 224
4
2 )(
)π2(
)(),(
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MRT
Throughout this calculation, we tacitly assume that we have sandwiched this vertex
between two on-shell spinors, so we can use the Gordon decomposition and the mass-
shell condition. Our goal is to write this expression in the form:
)(
2
1
)(~ 2
2
2
1 qF
m
iqF νµµµ γ Σ+Λ
sandwiched between u(p′) and u(p), where F1 and F2 are the form factors that measure
the deviation from the simple γµ vertex. We will calculate explicit forms for these two form
factors and we will find that F1 cancels against the infrared divergence found in the
bremsstrahlung calculation, giving us a finite result, and we will also find that it is F2 that
gives us a correction to the magnetic moment of the electron.
−
We begin with the Feynman parameter trick, generalizing the 1/ab integral that we
showed in the Bremsstrahlung chapter:
where ∆=Σiai zi.
∫ ∫ ∫ ∏
∞ ∞ ∞
= ∆
Σ−
−=
0 0 0
11
)1(
)!1(
1
n
i
n
ii
i
n
z
zdn
aa
δ
L
L
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εεεµ imkpaimkpaika +−−=+−−′=+−= 22
3
22
2
22
1 )()( and,
Therefore:
∫ ∫
∞






∆
−−−=′Λ
0 33213214
4
2 )(
)1(
)π2(
2),(
kN
zzzzdzdzd
kd
eipp
µ
µ δ
where:
ν
µνµ γγγ )()()( mkpmkpkN −/−/−/−′/=
For our purposes, we want:
and:
332211
3
1
zazazaza
i
ii ++==∆ ∑=
Next, we would like to perform the integration by completing the square:
This allows us to make the shift in integration:
0
2
32 )( ∆−−′−=∆ zpzpk
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where:
)1()()1()()1( 33
22
22
222
32
2
1
22
1
2
0 zzmpzzmpizzqzzm −−−−−′−−−+−=∆ εµ
32 zpzpkk +′+→
Therefore, after the shift, we have:
∫ ∫
∞
∞− +∆−
+′+
−−−=′Λ
1
0 2
0
2
324
3213214
2
)(
)(
)1(
)π2(
2
),(
ε
δ
µ
µ
ik
zpzpkN
kdzzzzdzdzd
ei
pp
By power counting, the integral diverges. This is why we must subtract off the
contribution of the Pauli-Villars field, which has mass Λ. Let us expand:
)()(2)( 32
2
32 zpzpNkAkkkzpzpkN +′++/+−=+′+ µµµµµ γ
where Aµ(k) is linear in kµ variable. This term can be dropped since its integral over d4k
vanishes.
Therefore, the leading divergence behaves like:
where ∆Λ represents the Pauli-Villars contribution, where µ is replaced by Λ. To perform
this integration, we must do an analytic continuation of the previous equation. Using:
νµαα
νµ
ηα
α
)3(
))((2
π
)( 3
2
2
4
−Γ
∆−Γ
=
∆− −∫
i
k
kk
kd
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∫ 







∆−
−
∆− Λ
323
0
2
4
)()( k
kk
k
kk
kd
νµνµ
we now take the limit as α →3. This expression diverges, but the Pauli-Villars term
subtracts off the divergence.
If we let α −3=ε, then we have:
where we drop terms like Λ−n. Because this expression is sandwiched between two on-
shell spinors, we can also reduce the term:








∆
∆
−=








∆
∆
−=−=








∆−
−
∆−
Γ=








∆−
−
∆−
Λ∆−∆−
→
Λ
→
Λ
→
Λ
∫
0
2
12
0
2)(ln)(ln2
0
0
2
02
0
2
4
3
lnπ
lnπ]ee[
1
πlim
)(
1
)(
1
)(πlim
)()(
lim
0
z
i
ii
i
k
kk
k
kk
kd
εε
ε
εεεα
νµ
α
νµ
α
ε
ε
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],)[1(2)]1)(1(2)41(2[)( 1132
22
11
2
32 µµµ γγ qzzmzzqzzmzpzpN /−−−−++−−=+′
At this point, all integrals can be evaluated. Putting everything together, and dropping all
terms of order Λ−1 or less, we now have:




∆
−Σ
+




∆
−−++−
+








∆
Λ
−−−=′Λ ∫
∞
0
11
0 0
32
22
11
2
0
2
1
321321
)1(
)1)(1()41(
ln)1(
π2
),(
zzqmi
zzqzzmz
zzzzdzdzdpp
ν
νµ
µµµ γγδ
α
Now let us compare this expression with the form factors F1 and F2 appearing in Λµ
~γµ F1(q2)+i(1/2m)Σµν F2(q2) above. It is easy to read off:
with the integrand at q2 =0 subtracted off so as to satisfy the constraint F1(q2 =0)=1
which preserves the correct normalization of the vertex function.
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The value of F2 can similarly be read off:



=




∆
−−++−
+∆−−−−+→ ∫
∞
)0(
)1)(1()41(
ln)1(
π2
1)(
2
0 0
32
22
11
2
0321321
2
1
q
zzqzzm
zzzzdzdzdqF δ
α
∫
∞








∆
−
−−−→
0 0
11
2
321321
2
2
)1(2
)1(
π2
)(
zzm
zzzzdzdzdqF δ
α
The calculation for F2(q2) is a bit easier, since there are no ultraviolet or infrared
divergences. Because of this, we can set µ =0. Let us choose new variables:
where q2 =(p−p′)2. Then we have:
)cosh1(2
2
sinh4cosh 22222
θ
θ
θ −=





−==′⋅ mmqmpp and
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∫
∫
−+−+
=
++
+−+
−−=
1
0 22
1
0
32
2
3
2
2
2
3231
3232
2
2
cosh)1(2)1(
1
2π
cosh2
)(
)1(
π
)(
θββββ
β
α
θ
θ
α
d
zzzz
zzzz
zzzdzdqF
This leaves us with the exact result:
θ
θα
sinh2π
)( 2
2 =qF
We are especially interested in taking the limit as |q2|→0 and θ →0:
2
2
2
8π2π
)0(
e
F ==
α
π4επ4
2
o
2
e
c
e
≡=
h
α
since the fine structure constant is given by:
This gives us the correction to the magnetic moment of the electron, as in:
This is only a first-order calculation, yet already we are very close to the experimental
value. Since the calculation was originally performed by Schwinger (i.e., his δµ=(α/2π)µo
result), the calculation since has been taken to α3 order (where there are 72 Feynman
diagrams – see Figure on next slide). The theoretical value to this order is given by:
0011614.0~
2π2
2e α
=
−g
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2017
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above but for:
)(
2π2
1
2
)(
)(
ee
pu
m
qi
m
pp
pu







 Σ






++
′−
′
ν
νµµ α
K+





+





−





=−≡
32
eth
π
49.1
π
32848.0
π
5.0)2(
2
1 ααα
ga
The final results for both the theoretical and experimental values are:
)31(2090011599652.0)166(110011596524.0 expth == aa vs
where the estimated errors [as of 1992] are in parentheses.
The calculation agrees to within one part in 108 for a and to one part in 109 for ge,
which is graphic vindication of QED. To push the calculation to the fourth order
involves calculating 891 diagrams and 12672 diagrams at the fifth order.
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First order correction – in α:
Second order correction – in α2:
Third order correction – in α3:
2π
)0(2
α
=F
Magnetic moment of the electron
The calculation for F1 is much more difficult. However, it will be very important in
resolving the question of the infrared divergence, which we found in the earlier
discussion of bremsstrahlung. We will find that the infrared divergence coming from the
bremsstrahlung diagram and F1 cancel exactly. Although the calculation of F1 is difficult,
one can extract useful information from the integral by taking the limit as µ becomes
small. Then F1 integration in:
of the Anomalous Magnetic Moment chapter splits up into four pieces:
L+= ∑=
4
1
1
i
iPF
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Infrared Divergence



=




∆
−−++−
+∆−−−−+→ ∫
∞
)0(
)1)(1()41(
ln)1(
π2
1)(
2
0 0
32
22
11
2
0321321
2
1
q
zzqzzm
zzzzdzdzdqF δ
α
where the ellipsis represents constant terms.
After changing variables, each of the Pi pieces can be exactly evaluated:
where:
∫
∫ ∫
−





=
−−+++
−=
−
2
0
1
0
1
0
32
2
32
2
3
2
2
321
tanhcoth
π
2
lncoth
π
)1()(cosh2
cosh
π2
2
θ
ϕϕϕθ
αµ
θθ
α
µθ
θα
d
m
zzmzzzz
zdzdP
z
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)2tanh(
tanh
21 2
ϕ
ϕ
=− z
The others are given by:






−=
++
−=
=
++
+
=
∫ ∫
∫ ∫
−
−
1
sinh2πcosh2
)cosh1(
π
2
coth
πcosh2
cosh
π
1
0
1
0
32
2
3
2
2
32
323
1
0
1
0
32
2
3
2
2
32
322
2
2
θ
θα
θ
θ
α
θθ
α
θ
θ
α
z
z
zzzz
zz
zdzdP
zzzz
zz
zdzdP ,
and
constant
2
sinh
2
2π2
sinh4)(ln
2π
1
0
1
0
2
32
2
32324
2
+






−=











++−= ∫ ∫
−
θ
θ
αθα z
zzzzzdzdP
Thus, adding the pieces together, we find:
For |q2|<<1, we find using Λµ~γµ F1(q2)+i(1/2m)Σµν F2(q2), p⋅p′=m2coshθ and q2 =2m2(1−
coshθ), and F2(q2)=(α/2π)(θ/sinhθ) of the Anomalous Magnetic Moment chapter with the
above:














−−−





+





= ∫ 2
tanh
4
tanhcoth2)1coth(1ln
π
)(
2
0
2
1
θθ
ϕϕϕθθθ
µα θ
d
m
qF
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],[
π88
3
ln
π3
1~),( 2
2
µµµµ γ
αµα
γγ q
mmm
q
ppc
/+














−





+′Λ+
For |q2|>>m2, we find:
























+−








−





−′Λ+ 2
2
2
2
1lnln
π
1~),(
q
m
O
m
q
m
ppc µα
γγ µµµ
Plugging all this into the cross section formula, we now find our final result:
Now we come to the final step, the comparison of the bremsstrahlung amplitude of the
Bremsstrahlung chapter and the vertex correction for electron scattering. Although they
represent different physical processes, they must be added because there is always an
uncertainty in our measuring equipment in measuring soft photons. Comparing the two
amplitudes, we recall that the bremsstrahlung amplitude is given by:
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











−





Ω
=





Ω
)(ln
π
2
1 2
0
q
m
d
d
d
d
χ
µ
ασσ
µ
where:







>>−−








−
<<−−
=
22
2
2
22
2
2
2
1ln
3
1
)(
mq
m
q
mq
m
q
q
if
if
χ
















−





Ω
=
Ω 2
2
2
2
0
lnln
π µ
ασσ E
m
q
d
d
d
d
while the vertex correction diagram yields:
















−







−−





Ω
=
Ω 2
2
2
2
0
lnln
π
1
µ
ασσ q
m
q
d
d
d
d
Clearly, when these two amplitudes are added together, we find a finite, convergent
result independent of µ2, as desired. The cancellation of infrared divergences to all
orders in perturbation theory is a much more involved process. However, there are
surprising simplifications that give a very simple result for this calculation.
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Two of the great accomplishments of QED were the determination of the anomalous
magnetic moment of the electron, which we discussed in the chapter dedicated to it, and
the Lamb shift. The fact that these two effects could not be explained by ordinary
quantum mechanics, and the fact that the QED result was so accurate, helped to
convince the skeptics that QED was the correct theory of the electron-photon system.
In 1947, W. Lamb and R. Retherford demonstrated that the 2S½ and the 2P½ energy
levels of the hydrogen atom were split (see Figure); the 2P½ energy level was depressed
more than 1000 MHz below the 2S½ level – by 1040 MHz actually (or ~0.035 cm−1). The
original Dirac electron in a classical Coulomb potential, as we saw in the Relativistic
Quantum Mechanics chapter, predicted that the energy levels of the hydrogen atom
should depend only on the principal quantum number n and the total spin j, so these two
levels should be degenerate… Shelter Island, we have a problem!
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Lamb Shift
1040 MHz
17 MHzDirac-Coulomb
QED radiative
corrections
≈
2S½
2P½
The calculation of the Lamb shift is rather intricate, because we are dealing with the
hydrogen atom as a bound-state, and also because we must sum over all radiative cor-
rections to the electron interacting with a Coulomb potential that modify the naïve uγ0 u A0
vertex. These corrections include the vertex correction, the anomalous magnetic
moment, the self-energy of the electron, the vacuum polarization diagram, and even
infrared divergences (see Figure).
The original nonrelativistic bound-state calculation of H. Bethe (1947), which ignored
many of the subtle higher-order corrections, could account for about 1000 MHz of the
Lamb shift, but only a fully relativistic quantum treatment could calculate the rest of the
difference. Because of the intricate nature of the calculation, we will only sketch the
highlights of the calculation.
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The various higher-order diagrams that contribute to the Lamb shift: (a) and (b) the electron self-energy
diagrams; (c) the vertex correction; (d) and (e) the electron mass counterterm (added due to the
occurrence in momentum integrals of infrared divergences); (f) the photon self-energy correction.
(a) (b) (c) (d) (e) (f)
⊗
⊗
−
To begin the discussion, we first see that the vacuum polarization diagram can be
attached to the photon line, changing the photon propagator to:
This translates into a shift in the effective coupling of an electron to the Coulomb
potential. Analyzing the zeroth component of this propagator, we see that the coupling of
the electron to the Coulomb potential changes as follows:








+−−= )(
π60
1 4
2
2
2
2
2
kO
m
ke
k
i
D νµνµ η
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2017
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







+−→ )(
5π1
1 2
2
2
2
02
o2
02
o α
αγγ
O
m
q
q
uu
ei
q
uu
ei
To convert this back into x-space, let us take the Fourier transform. We know that the
Fourier transform of 1/q2 is proportional to 1/r. This means that the static Coulomb
potential that the electron sees is given by:
)()(
0π6π4π45π1
1 43
22
222
2
2
eO
m
e
r
e
r
e
m
++=







∇− xδ
α
meaning that there is a correction to Coulomb’s law given by QED. This correction, in
turn, shifts the energy levels of the hydrogen atom. We know from ordinary nonrelati-
vistic quantum mechanics that, by taking matrix elements of this modified potential
between hydrogen wave functions, we can calculate the first-order correction to the
energy levels of the hydrogen atom due to the vacuum polarization diagram.
Now let us generalize this discussion to include the other corrections to the calculation
of the Lamb shift. Our method is the same: calculate the corrections to the vertex
function uγ0 u, take the zeroth component, and then take the low-energy limit. In the
Anomalous Magnetic Moment chapter, we had Λµ~γµ F1(q2)+i(1/2m)Σµν F2(q2) and with it
eventually obtained an γµ +Λc
µ(p′,p) equation for |q2|<<m2 and |q2|>>m2, and we saw how
radiative corrections modified the vertex function with additional form factors F1(q2) and
F2(q2). If we add the various contributions to the vertex correction, we find:
For example, the vacuum polarization diagram contributes the factor −1/5 to the vertex
correction. The logarithmic term comes from the vertex correction, and the µ term is
eventually cancelled by the infrared correction. The Σµν qν term reduces down to a spin-
orbit correction, and we find the effective potential given by:
uq
m
i
m
m
q
uuu








Σ+














−−





−→ ν
νµµµ
α
µ
α
γγ
π45
1
4
3
ln
π3
1 2
2
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2017
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−
Lx •+





+−−





∆ σσσσ32
2
3
2
2
eff
π4
)(
8
3
5
1
8
3
ln
3
4
~
rm
m
m
V
α
δ
µ
α
By taking the matrix element of this potential between two hydrogen wave functions,
we can calculate the energy split due to this modified potential. The vertex correction, for
example, gives us a correction of 1010 MHz. The anomalous magnetic moment of the
electron contributes 68 MHz. And the vacuum polarization diagram, calculated earlier,
contributes −27 MHz. Adding these corrections together, we find, to the lowest loop level,
1051 MHz and thus arrive at the Lamb shift to within 6 MHz accuracy.
Since the, higher-order corrections have been calculated, to the difference between
experiment and theory has been reduced to 0.01 MHz. Theoretically, the 2S½ level is
above the 2P½ energy level by 1057.864±0.014 MHz. The experimental result is 1057.862±
0.020 MHz. This is an excellent indicator of the basic correctness of QED.
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One of the serious complications found in quantum field theory is the fact that the theory
is naïvely divergent. When higher-order corrections are calculated for QED, one find that
the integrals diverge in the ultraviolet region, for large momentum p.
Since the birth of quantum field theory, several generations of physicists have
struggled to renormalize it. Some physicists, despairing of ever extracting meaningful
information from quantum field theory, even speculated that the theory was
fundamentally sick and must be discarded. In hindsight, we can see that the divergences
found in quantum field theory were, in some sense, inevitable. In the transition from
quantum mechanics to quantum field theory, we made the transition from a finite number
of degrees of freedom to an infinite number. Because of this, we must continually sum
over an infinite number of internal modes in loop integrations, leading to divergences.
The divergent nature of quantum field theory then reflects the fact that the ultraviolet
region is sensitive to the infinite number of degrees of freedom of the theory. Another
way to see this is that the divergent diagrams probe the extremely small distance region
of space-time, or, equivalently, the high-momentum region. Because almost nothing is
known about the nature of physics at extremely small distances or momenta, we are
disguising our ignorance of this region by cutting off the integrals at small distances.
98
2017
MRT
Overview of Renormalization in QED
Since that time, there have been two important developments in renormalization
theory. First was the renormalization of QED via the covariant formulation developed
by Schwinger and Tomonoga and by Feynman (which we shown to be equivalent by
Dyson). This finally showed that, at least for the electromagnetic interactions,
quantum field theory was the correct formulation.
Subsequently, physicists attacked the problem of the strong and weak interactions via
quantum field theory, only to face even more formidable problems with renormalization
that stalled progress for several decades. To this applied the second revolution which
was the proof by G. ’t Hooft (1971) that spontaneously broken Yang-Mills non-Abelian
gauge field theory was renormalizable, which led to the successful application of
quantum field theory to the weak interactions and opened the door to the gauge
revolution.
There have been many renormalization proposals made in the literature, but all of
them share the same basic features. Although details vary from scheme to scheme, the
essential idea is that there is a set of bare physical parameters that are divergent, such
as the coupling constants and masses. However, these bare parameters are
unmeasurable! The divergences of these parameters are chosen so that they cancel
against the ultra violet infinities coming from infinite classes of Feynman diagrams,
which robe the small-distance behavior of the theory. After these divergences have been
absorbed by the bare parameters, we are left with the physical, renormalized, or dressed
parameters that are indeed measurable.
99
2017
MRT
Since there are a finite number of such physical parameters, we are only allowed to
make a finite number of such redefinitions. Renormalization theory, then, is a set of rules
or prescriptions where, after a finite number of redefinitions, we can render the theory
finite to any order. We should stress that, although the broad features of the
renormalization program are easy to grasp, the details may be quite complicated.
Only the first three classes of diagrams are actually divergent. Fortunately, this is also
the set of divergent diagrams that can be absorbed into the redefinition of physical
parameters, the coupling constant, and the electron mass. (N.B., The photon mass, as
we shall see, is not renormalized because of gauge invariance).
100
2017
MRT
∫ −⋅
−/−/
−=/Σ− ν
νµ
µ
γηγ 24
4
2
)π2(
)()(
k
i
mkp
ikd
eipi ≡
≡
≡
≡
∫ 





−/−/
⋅
−/
−−=Π νµνµ
γγ
mkp
i
mp
ikd
eiki Tr
)π2(
)()( 4
4
2






−Π





−+−= νσ
σρ
ρµνµνµ ηηη 222
)()(
k
i
ki
k
i
k
i
kDi
∫ −/−/−/+
−⋅−=Λ− σ
µ
ρ
σρµ γγγη
mk
i
mqk
i
pk
ikd
eiqpei 24
4
3
)()π2(
)(),(
The one-loop diagrams that we want particularly to analyze are therefore the electron
self-energy correction Σ(p):
vacuum polarization Πµν:
the photon propagator Dµν :
and the vertex correction Γµ :
µΛ
νµD
νµ
Π
Σ
To begin renormalization theory, we will sum all possible diagrams appearing in the
complete propagator (see Figure):
101
2017
MRT
The complete propagator S′F is the sum over one-particle irreducible diagrams Σ arranged along a chain.
iεpmp
i
pSipipSipSipSi FFFF
+/Σ−−/
=+//Σ−/+/=/′
)(
)()]()[()()(
o
K
This is the renormalized electron propagator. Then we make a Taylor expansion of the
mass correction Σ(p) around p=m, where m is the finite renormalized mass, which is
arbitrary:
)(
~
)()()()( pmmpmp /Σ+Σ′−/+Σ=/Σ
where Σ(p)~O(p−m)2 and vanishes for p=m. Since mo is divergent and arbitrary, we will
choose mo and m so that mo cancels the divergence coming from Σ(m). We will choose:
mmm =Σ+ )(o
= +
+ K+
FS
~
′
FS
Σ
Σ Σεimp +−/
1
Inserting the value of Σ(p) into the renormalized electron propagator, we now can
rearrange terms to find:
where we have defined:
102
2017
MRT
iεpmp
Zi
iεpmpm
i
pmmpmmp
i
pSi F
+/Σ−−/
=
+/Σ−−/Σ′−
=
/Σ−Σ′−/−Σ−−/
=/′
)(
~~
)(
~
))]((1[)(
~
)()()(
)(
2
o
)(
~
)(1
)(
~
)(
~~
)(1
1
)( 22o pZ
m
p
p
m
Zmmm /Σ=
Σ′−
/Σ
=/Σ
Σ′−
=Σ+= and,
Since the divergence of the complete propagator S′F is contained within Z2, we can
remove this term and define the renormalized propagator S′F:
~
)(
~
)( 2 pSZpS FF /′=/′
By summing this subclass of diagrams for the electron self-energy, we have been able to
redefine the mass of the electron, such that the physical mass m is actually finite, and
also show that the electron propagator is multiplicatively renormalized by the factor Z2:
o2 ψψ Z=
Next, we analyze the photon propagator Dµν in the same way, summing over infinite
classes of one-particle irreducible diagrams:
νσ
σρ
ρµνµνµ DDDD Π−=′
PART VII.3 - Quantum Electrodynamics
PART VII.3 - Quantum Electrodynamics
PART VII.3 - Quantum Electrodynamics
PART VII.3 - Quantum Electrodynamics
PART VII.3 - Quantum Electrodynamics
PART VII.3 - Quantum Electrodynamics
PART VII.3 - Quantum Electrodynamics
PART VII.3 - Quantum Electrodynamics
PART VII.3 - Quantum Electrodynamics
PART VII.3 - Quantum Electrodynamics
PART VII.3 - Quantum Electrodynamics

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PART VII.3 - Quantum Electrodynamics

  • 1. From First Principles January 2017 – R4.4 Maurice R. TREMBLAY PART VII – QUANTUM ELECTRODYNAMICS ∫∫ +++++−=′ 65 2 56 2 )2,6()1,5()()6,4()5,3()()2,1;4,3( ττδγγ µµ ddKKsKKeiK bababa ELECTRONS VIRTUAL QUANTUMTIME 1 2 3 4 5 6 )( 2 56s+δ )6,4(+K )1,5(+K )2,6(+K )5,3(+K µγ µγ a b Circa 1949 Chapter 3
  • 2. Contents PART VII–QUANTUM ELECTRODYNAMICS Particles and Fields Second Quantization Yukawa Potential Complex Scalar Field Noether’s Theorem Maxwell’s Equations Classical Radiation Field Quantization of Radiation Oscillators Klein-Gordon Scalar Field Charged Scalar Field Propagator Theory Dirac Spinor Field Quantizing the Spinor Field Weyl Neutrinos Relativistic Quantum Mechanics Quantizing the Maxwell Field Cross Sections and the Scattering Matrix Propagator Theory and Rutherford 2017 MRT 2 Scattering Time Evolution Operator Feynman’s Rules The Compton Effect Pair Annihilation Møller Scattering Bhabha Scattering Bremsstrahlung Radiative Corrections Anomalous Magnetic Moment Infrared Divergence Lamb Shift Overview of Renormalization in QED Brief Review of Regularization in QED Appendix I: Radiation Gauge Appendix II: Path Integrals Appendix III: Dirac Matrices References
  • 3. So far, our discussion has been rather formal, with no connection to experiment. This is because we have been concentrating on Green functions, which are unphysical (i.e., they describe the motion of off-shell particles where pµ 2 ≠m2c4). However, the physical world that we measure in our laboratories is on-shell (i.e., pµ 2=m2c4 which are configurations of a physical system that satisfy classical equations of motion which we export piecemeal for the real exchange of particles satisfying the Einstein energy- momentum relationship E2 −|p|2c2 =m2c4≡(mc2)2). Cross Sections and the Scattering Matrix 3 2017 MRT The cross section is thus the effective area of each target particle as seen by an incoming beam. Cross sections are often measured in terms of barns (i.e., 1 barn is 10−25 cm2). A nucleon is about 1 Fermi, or 10−13 cm across. Its area is therefore about 10−26 cm2, or 0.01 barn. Thus, by giving the cross section of a particle in a certain reaction, we can immediately calculate the effective size of that particle in relationship to a nucleon. Cross section = Effective size of target particle To make the connection to experiment, we need to rewrite our previous results in terms of numbers that can be measured in the laboratory, such as decay rates of unstable particles and scattering cross sections. There are many ways in which to define the cross section, but perhaps the simplest and most intuitive way is to define it as the effective size of each particle in the target:
  • 4. To calculate the cross section in terms of the rate of collisions in a scattering experi- ment, let us imagine a thin target with NT particles in it, each particle with effective area σ or cross section. As seen from an incoming beam, the total amount of area taken up by these particles is therefore NT σ. If we aim a beam of particles at the target with area A (see Figure), then the chance of hitting one of these particles is equal to the total area that these particles occupy (i.e., NT σ) divided by the area A: Let us say we fire a beam containing NB particles at the target. Then the number of particles in the beam that are absorbed or deflected is NB times the chance of being hit. Thus, the number of scattering events is given by: 4 2017 MRT σ =particleahittingofChance σ ⋅=eventsofNumber or simply (see Figure): ⋅      = eventsofNumber σ NB • • • • • • NTA σ A NB A A NB NT NT NT
  • 5. In actual practice, a more convenient way of expressing the cross section is via the flux of the incoming beam, which is equal to J =ρv. If the beam is moving at velocity v toward a stationary target, then the number of particles in the beam NB is equal to the density of the beam ρ times the volume. If the beam is a pulse that is turned on for t seconds, the volume of the beam is vtA. Therefore, NB =ρvtA. The cross section can therefore be written as: where we have normalized such that NT =1 and the transition rate is the number of scattering event per second. The cross section is therefore equal to the transition rate divided by the flux of the beam. Flux rateTransitioneventsofNumbereventsofNumbereventsofNumber ===⋅= v t A NAv t A tNN t TTB ρρ σ 5 2017 MRT The next problem is to write the transition rate appearing in the cross section in terms of the S matrix. We must therefore calculate the probability that a collection of particles in some initial state i will decay or scatter into another collection of particles in some final state j. From ordinary nonrelativistic quantum mechanics, we know that the cross section σ can be calculated by analyzing the properties of the scattered wave. Using classical wave function techniques dating back to Rayleigh, we know that a plane wave exp(ikz) scattering off a stationary, hard target is given by exp(ikz)+ [ f (θ)/r]exp(ikr) where the term exp(ikr) represents the scattered wave, which is expanding radially from the target. Therefore | f (θ)|2 is proportional to the probability that a particle scatters into an angle θ.
  • 6. More precisely, the differential cross section is given by the square of f (θ): where the solid angle differential is given by: 2 )(θ σ f d d = Ω 6 2017 MRT ∫ ∫∫ − =Ω π2 0 1 1 cosθϕ ddd and the total cross section is given by: σ σ θ = Ω Ω=Ω ∫∫ d d dfd 2 )( For our purposes, however, this formulation is not suitable because it is inherently nonrelativistic. To give a relativistic formulation, let us start at the beginning. We wish to describe the scattering process that takes us from an initial state |i〉 consisting of a collection of free, asymptotic states at t→−∞ to a final state | f 〉 at t→∞. To calculate the probability of taking us from the initial state to the final state, we introduce the S matrix: ififif if PPi iSfS T)()π2( 44 −−= = δδ where δf i symbolically represents the particles not interacting at all, and Tf i is called the transition matrix, which describes non-trivial scattering.
  • 7. One of the fundamental constraints coming from quantum mechanics is that the S matrix is unitary: By taking the square of the S matrix, we can calculate the transition probabilities. The probability that the collection of states i will make the transition to the final states f is given by: ki f kfif SS δ=∑ * 7 2017 MRT ififif SSP * = Likewise, the total probability that the initial states i will scatter into all possible final states f is given by: ∑= f ifif SSP * Total
  • 8. Now we must calculate precisely what we mean by Σf . We begin by defining our states within a box of volume V: 8 2017 MRT ( ) ( )fermionsandbosons 0)( )π2( 0)( 2)π2( † 3 † 3 pa m E V pa V E pp == pp Our states are therefore normalized as follows: ( ) ( )fermionsandbosons )( )π2( )( 2)π2( 3 3 3 3 pppppppp ′=′′=′ −−−−−−−− δδ m E VV E pp With this normalization, the unit operator (on single particle states) can be expressed as: ( ) ( )fermionsandbosons pp p 1pp p 1 ∫∫ == pp E mdV E dV 3 3 3 3 )π2(2)π2( To check our normalizations, we can let the unit operator act on an arbitrary state |q〉, and see that it leaves the state invariant. This means, however, that we have an awk- ward definition of the number of states at a momentum p. With this normalization, we find that 〈p |p〉=(2π)32Epδ 3(0)/V which makes no sense. However, we will interpret this to mean that we are actually calculating particle densities inside a large but finite box of size L and volume V ; that is, we define δ 3(p)=limL→∞{[1/(2π)3]∫∫∫±L/2 dxdydzexp(−ip•r)}. This implies that we take the definition: 3 3 )π2( )0( V =δ
  • 9. Our task is now to calculate the scattering cross section 1+2→3+4+… and the rate of decay of a single particle 1→2+3+…. We must now define how we normalize the sum over final states. We will integrate over all momenta of various final states, and sum over all possible final states. For each final state, we will integrate over the final momentum in a Lorentz covariant fashion. We will use: The density of states dNf (i.e., the number of states within p and p++++δ p), is: ∫∫ =− pE d pmp pd 2)π2( )()( )π2( 3 3 0 224 4 4 p θδ 9 2017 MRT ∏= = fN i i f dV Nd 1 3 3 )π2( p As before, the differential cross section dσ is the number of transitions per unit time per unit volume divided by the flux J of incident particles: JTV NdS d fif 1 2           == fluxIncident cmpersecondperTransition 3 σ
  • 10. We also know that the transition rate per unit volume (within a momentum-space interval) is given by: where (2π)4δ 4(0)=V T. V vv J 21 − = 10 2017 MRT fifif f ififfif NdPP Nd TV PP TV NdS 244 44282 )()π2( )0()()π2( T T −= − == δ δδ3 cmperrateTransition To calculate the incident flux, we will first take a collinear frame, such as the laboratory frame or center-of-mass frame. The incident flux J equals the product of the density of the initial state (i.e., 1/V) and the relative velocity v=|v1 −v2|, where v1 =|p1|/E1: In the center-of-mass frame, where p1 =−p2, we have: 2 2 2 1 2 21 2112211 2 2 1 1 21212121 )(4 )(4422)2)(2()2)(2( mmpp EEEE EE EEvvEEJEEV −⋅= +=−=−=−= ppp pp This equation only hold if the two particles are collinear. From now on, we assume that all Lorentz frames are collinear.
  • 11. The final formula for bosons for the differential cross section for 1+2→3+4+… is therefore given by: in a collinear frame where: ∏= = f i N i if p if VE1 2 1 MT 11 2017 MRT ( )bosons∏=−⋅ − = f i N i p iifif E d mmpp PP d 3 3 3 2 2 2 1 2 21 424 2)π2()(4 )()π2( pδ σ M Notice that all factors of V in the S matrix have precisely cancelled against other factors coming from dNf and the flux.
  • 12. Finally, we will now use this formalism to compute the probability of the decay of a single particle. The decay probability is given by: where we have taken δf i =0 for a decay process. The last expression, unfortunately, in singular because of the delta function squared. As before, however, we assume that all our calculations are being performed in a large but finite box of volume V over a large time interval T. We thus reinterpret one of the delta functions as: 12 2017 MRT ∑∫∑ −== f ififf f iff PPNdSNdP 24422 Total )]()π2[( δT 4 4 )π2( )0( TV =δ We now define the decay rate Γ of an unstable particle as the transition probability per unit volume of space and time: )2( Total iETV P = × =Γ volumesec yprobabilitTransition The final result for the decay rate is given by: ∫∏= −=Γ f j N j ifif p j i PP E d E 1 42 3 34 )( 2)π2(2 )π2( δM p The lifetime of the particle τ is then defined as the inverse of the decay rate: Γ=1τ
  • 13. Historically, calculations in QED were performed using two seemingly independent formulations. One formulation was developed by J. Schwinger (1949) and S. Tomonoga (1946) using a covariant generalization of operator methods developed in quantum mechanics. However, the formulation was exceedingly difficult to calculate with and was physically opaque. The second formulation was developed by R. P. Feynman (1949) using the propagator approach. Feynman postulated a list of simple rules from which one could pictorially setup the calculation for scattering matrices of arbitrary complexity. The weakness of Feynman’s graphical methods, however, was that they were not rigorously(mathematically)justified.Later,F. Dyson(1949) demonstrated the equivalence of these two formulations by deriving Feynman’s rules from the interaction picture. Propagator Theory and Rutherford Scattering 13 2017 MRT In this chapter, we will follow Feynman to show how the propagator method gives us a rapidly and convenient method of calculating the lowest order terms in the scattering matrix. [N.B., Although we will not consider it here, pursuant to this is the Lehmann- Symanzik-Zimmermann (LSZ) reduction formalism that is usually studied, in which one can develop the Feynman rules for diagrams of arbitrary complexity]. At this point, we should emphasize that the Green functions that appear in the propagator approach are off-shell (i.e., they do not satisfy the mass-shell condition pµ 2=m2). Neither do they obey the usual equation of motion. The Green functions describe virtual particles, not physical ones! As we saw earlier, the Green function develops a pole in momentum space at pµ 2 =m2. However, there is no violation of cherished physical principles because the Green functions are not measurable quantities. The only measurable quantity is the S matrix, where the external particles obey the mass-shell condition.
  • 14. To begin calculating cross sections, let us review the propagator method in ordinary quantum mechanics, where we wish to solve the Schrödinger equation: We assume that the true Hamiltonian is split into two pieces: 0=      − ∂ ∂ ψH t i 14 2017 MRT nInteractio0 HHH += where H0 is the free Hamiltonian and the interaction piece HInteraction is small. We wish to solve for the propagator G(x,t;x′,t′): )()(),;,( 3 nInteractio0 ttttGHH t i ′−′=′′      −− ∂ ∂ δδ xxxx −−−− If we could solve for the Green function for the interacting case, then we can use Huygen’s principle to solve for the time evolution of the wave. We recall that Huygen’s principle says that: ( )tttttGdt ′>′′′′′= ∫ ),(),;,(),( 3 xxxxx ψψ The future evolution of a wave front can be determined by assuming that every point along a wave front is an independent source of an infinitesimal wave. By adding up the contribution of all these small waves, we can determine the future location of the wave front. Mathematically, this is expressed by the equation:
  • 15. Our next goal, therefore, is to solve for the complete Green function G, which we do not know, in terms of the free Green function G0, which is well understood. To find the propagator for the interacting case, we have to power expand in HInteraction. We will use the following formula for operator A and B: Another way of writing this is: L L ++−= ++−= + = + = + ≡ + −−−−−− −−−− −−− 111111 1111 111 )1( 1 1 1 )1( 111 ABABAABAA ABABABA ABABAABAAABA 15 2017 MRT 11 11 −− + −= + AB BA A BA Now let: nInteractio0 HB t iHA −= ∂ ∂ +−= and Then we have the symbolic identities: L+++=+= 0nInteractio0Int00nInteractio00nInteractio0 GHGHGGHGGGGHGGG and More explicitly, we can recursively write this as: ∫ ∫ ′′+′′=′′ ),;,(),(),;,(),;,(),;,( 11011nInteractio111 3 10 ttGtHttGdtdttGttG xxxxxxxxxx A G BA G 11 0 = + = and with:
  • 16. If we power expand this last expression, we find (N.B., HInt ≡HInteraction to save space): Using Huygen’s principle, we can power expand for the time evolution of the wave function (see Figure): LLL L ++ ++ += ∫ ∫ ∫ −− ),(),;,(),(),;,( ),(),;,(),(),;,( ),(),;,(),(),( 011011Int110 4 1 4 2202211011Int1102 4 1 4 1101101 4 0 nnnnnnn tttGtHttGxdxd tttGtHttGxdxd tttGxdtt xxxxxx xxxxxx xxxxx ψ ψ ψψψ 16 2017 MRT L+′′′′+ ′′′′+′′=′′ ∫ ∫ ∫ ∞ ∞− ),;,(),(),;,(),(),;,( ),;,(),(),;,(),;,(),;,( 22022Int2211011Int02 3 1 3 21 11011Int01 3 10 ttGtHttGtHttGddtdtd ttGtHttGdtdttGttG xxxxxxxxxx xxxxxxxxxx In the propagator approach, perturbation theory can be pictorially represented as a particle interacting with a background potential at various points along its trajectory. γ − e ),;,( 110 ttG xx ),(0 txψ ),( 11Int tH x ),( 110 txψ ),( 22Int tH x ),( 220 txψ ),( txψ ),;,( 22110 ttG xx
  • 17. Let us solve for the S matrix in lowest order. We postulate that at infinity, there are free plane waves given by: We want to calculate the transition probability that a wave packet starts out in a certain initial state i, scatters off the potential, then re-emerges as another free plane wave, but in a different final state f . To lowest order, the transition probability can be calculated by examining Huygen’s principle: xki ⋅− = e )π2( 1 23 φ 17 2017 MRT L+′′′′′′′+= ∫ ),(),(),;,(),(),( nInteractio0 4 ttHttGxdtt ii xxxxxx φφψ To extract the S matrix, multiply this equation on the left by φj * and integrate. The first term on the right then becomes δi j. Using the power expansion of the Green function, we can express the Green function G0 in terms of these free fields. After integration, we find: L+′′′+= ∫ )()()( nInteractio *4 xxHxxdiS ififif φφδ Therefore, the transition matrix is proportional to the matrix element of the potential HInteraction.
  • 18. We will now generalize this exercise to the problem in question: the calculation in QED of the scattering of an electron due to a stationary Coulomb potential. Our calculation should be able to reproduce the old Rutherford scattering amplitude to lowest order in the nonrelativistic limit and give higher-order quantum corrections to it. Our starting point is the Dirac electron in the presence of an external, classical Coulomb potential (see Figure). The interacting Dirac equation reads as: )()()()( xxAexmi ψψγ µ µ /=−∂ 18 2017 MRTAn electron scatters off a stationary Coulomb field in lowest order perturbation theory. This reproduces the Rutherford scattering cross section in the nonrelativistic limit. γ − e xx 1 π4 )(0 ∝−= eZ xA ( )couplingµ µ γ AeAeJ =/= with a source term J(x)=eγ µAµ(x). Since we are only working to lowest order and are treating the potential Aµ as a classical potential, we can solve this equation using only propagator methods. The solution of the equation, as we have seen, is given by: ∫∫ /−+=−+= )()()()()()()()()( 4 0 4 0 yyAyxSydexyyJyxSydxx FF ψψψψψ where ψ0 is a solution of the free, homogeneous Dirac equation.
  • 19. To calculate the scattering matrix, it is convenient to insert the expansion of the Feynman propagator SF(x−x′) in terms of the time-ordered (i.e., using the T operator) function θ(t−t′) only as in: obtained earlier (N.B., we dropped the θ(t′−t) term but we still have to time-order hence the explicit T operator and change of variables from x′ to y). Then we find: 19 2017 MRT ∫ ∑= ′′−−=′− 2 1 3 )()()()( r r p r pF xxTttdixxS ψψθp ∫ ∫ ∑ /′−−= = )()()()()()()( 2 1 34 yyAyxdttiydeixx r r p r pi ψψψθψψ p for t→∞. We now wish to extract from this expression the amplitude that the outgoing wave ψ (x) will be scattered in the final state, given by ψf (x). This is done by multiplying both sides of the equation by ψf and integrating over all space-time. The result gives us the S matrix to lowest order: − ∫ /−= )()()(4 xyAxxdeiS ififif ψψδ where the slash notation, A=γ µAµ, has been extensively used here and previously.
  • 20. Now we insert the expression for the vector potential, which corresponds to an electric potential A0 given by the standard Coulomb potential: Inserting the plane-wave expression for the fermion field into the expression for the scattering matrix Sf i above and performing the integration over x, we have: 20 2017 MRT xπ4 )(0 eZ xA −= whose Fourier transform is given by: 2 3 π4 e 1 qx x xq =• ∫ i d )(π2 ),(),( e),( π4 )(e),( 2 022 04 if iiff if xpi ii i xpi ff f if EE spuspu EE m V eZ i spu VE meZ eispu VE m xdS if −=         −−= ⋅−⋅ ∫ δ γ δγ µ µ q x We recall that Vd3pf /(2π)3 is the number of final states contained in the momentum inter- val d3pf . Multiply these by |Sf i |2 and we have the probability of transition per particle into these states. Recall also that squaring the S matrix give us divergent quantities like δ (0), which is due to the fact that we have not rigorously localized the wave packet.We set 2πδ-(0)=T, where we localize the scattering process in a box of size V and duration T.
  • 21. If we divide by T, this gives us the rate R of transitions per unit time into this momentum interval. Finally, if we divide the rate of transition by the flux of incident particles, |vi |/V, this gives us the differential cross section: 21 2017 MRT VTV dV Sd i if v p 1 )π2( 3 32 =σ
  • 22. To calculate the differential cross section per unit of solid angle, we must decompose the momentum volume element: The last trace can be performed, since only the trace of even numbers of Dirac matrices survives: 22 2017 MRT         +/+/== Ω ∑ m mp m mpmZ spuspu mZ d d fi ss iiff fi 22 Tr 2 14 ),(),( 2 14 00 4 22 , 2 0 4 22 γγ α γ ασ qq Using the fact that pf dpf =Ef dEf, we have the result: pdpdd 23 Ω=p In the last step, we have used the fact that the summation of spins can be written as:         Γ +/ Γ +/ =Γ      +/ Γ      +/ = ΓΓ=ΓΓ=ΓΓ=Γ 0 † 00 † 0 0 † 00 † 0 †††2 22 Tr][ 2 ][ 2 ][][][][][][))(())(( γγγγ γγγγ δγ γβ βα αδ δδγγββαα m mp m mp m mp m mp uuuuuuuuuuuuuu ifif fiiffiiffiifif where we have used the fact that the sum over spins gives us: αβ αβ       +/=∑ m mp spuspu fi ss 2 ),(),( , )2(4)(Tr 00 fififi ppEEpp ⋅−=// γγ
  • 23. Finally, we need some kinematic information. If θ is the angle between pf and pi , then: We then obtain the Mott cross section (1929): 23 2017 MRT       =      +=⋅ 2 sin4 2 sin2 2222222 θθ β pqandEmpp fi             −= Ω 2 sin1 )2(sin4 22 422 22 θ β θβ ασ p Z d d where β is the velocity (i.e., v/c). In the nonrelativistic limit (i.e., as β →0), we obtain the celebrated Rutherford scattering formula for Coulomb scattering (MKS Units): )2(sin 1 επ8 )( 4 2 2 o 2 θ θσ         = vm eZ
  • 24. Up to now, we have not been able to take matrix element of the interacting fields. Hence, we cannot yet extract out numbers out of these matrix elements. The problem is that everything is written in terms of the fully interacting fields, of which we know almost nothing. The key is now to make an approximation to the theory by power expanding in the coupling constant, which is of the order of 1/137 for QED. So, like before, we begin by splitting the Hamiltonian into two distinct pieces: ∫= Hx3 nInteractio dH 24 2017 MRT Time Evolution Operator nInteractio0 HHH += where H0 is the free Hamiltonian and HInteraction is the interacting part. For example, in the φ4 theory, the interacting part would be: where: 4 !4 φ λ =H At this point, it is useful to remind ourselves from ordinary quantum mechanics that there are several pictures in which to describe the time evolution. In the Schrödinger picture, we recall, the wave function ψ (x,t) and state vector are functions of time t, but the operators of the theory are constants in time. In the Heisenberg picture, the reverse is true; that is, the wave function and state vectors are constant in time, but the time evolution of the operators and dynamical variables of the theory are governed by the Hamiltonian φ (x,t)=exp(iHt)φ (x,0)exp(−iHt).
  • 25. Now, without going into the details of the Lehmann-Symanzik-Zimmermann (LSZ) reduction formulas, its formulation is used to derive scattering amplitudes to all orders in perturbation theory. For example, define ‘in’ and ‘out’ states which are free particle states at asymptotic times (i.e., t=−∞ and ∞, respectively). Then specify that this LSZ formulation will express the interacting S matrix, defined in terms of the unknown interacting field φ (x), in terms of these free asymptotic states. Since the S matrix is defined as the matrix element of the transition from one asymptotic set of states to another, let f denote a collection of free asymptotic states at t=∞, while i refers to another collection of asymptotic states at t=−∞. Then the S matrix describes the scattering of the i states into f states by Sf i =out〈 f |i〉in. In the LSZ formalism, we will take it convenient to define yet another picture, which resembles the interaction picture. In this new picture, we need to find a unitary operator U(t) that takes us from the fully interacting field φ (x) to the free, asymptotic ‘in’ state: )(),()(),( in 1 tUttUt xx φφ − = 25 2017 MRT where U(t)≡U(t,−∞) is a time evolution operator, which obeys: 1),(),(),(),(),(),( 1221 1 313221 === − ttUttUttUttUttUttU and,
  • 26. Because we now have two totally different types of scalar fields, one free and the other interacting, we must also be careful to distinguish the Hamiltonian written in terms of the free or the interacting fields. Let H(t) be the fully interacting Hamiltonian written in terms of the interacting field, and let H0(φin) represent the free Hamiltonian written in terms of the free asymptotic states. Then the free field φin and the interacting field satisfy two different equations of motion: Solving for U(t) is complicated. The outcome is that the U(t) operator satisfies the following: ),()( ),( onInteractio o ttUtH t ttU i = ∂ ∂ 26 2017 MRT )],(),([ ),( )],(),([ ),( in in 0 in x x x x ttHi t t ttHi t t φ φ φ φ = ∂ ∂ = ∂ ∂ and where: in 0ininnInteractio ),()( HHtH −≡ πφ that is, HInteraction(t) is defined to be the interaction Hamiltonian defined only with free, asymptotic fields.
  • 27. Since HInt(t) (N.B., HInt ≡HInteraction) does not necessarily commute with HInt(t′) at diffe- rent times, the integration of the previous equation is a bit delicate but its outcome is : where the T operator means that, as we integrate over t1, we place the exponentials sequentially in time order. Written is this form, however, this expression is not very useful. We will find it much more convenient to power expand this exponential in a Taylor series so we have: ∫ ∫∫ ∞−∞− −− ==−∞= tt tdtditHtdi TTtUtU ),()( 11Int 3 11Int1 ee),()( xx H 27 2017 MRT ∑ ∫ ∫ ∫ ∞ = ∞− ∞− ∞− − += 1 Int2Int1Int 4 2 4 1 4 )]()()([ ! )( 1)( n t t t nn n xxxTxdxdxd n i tU HHH LL Without going through the derivation ourselves, the interacting Green function, written in terms of the fields, is given by: 0e0 0e)()()(0 0)()()([0),,,( )]([ )( in2in1in 2121 inInt 4 inInt 4 ∫ ∫ == xxdi xdi n nn T xxxT xxxTxxxG φ φ φφφ φφφ H H L LL We can now rewrite the previous matrix element entirely in terms of the asymptotic ‘in’ fields by a power expansion of the exponential: ∫∑ ∞ ∞− ∞ = − = 0)]()()()()()([0 ! )( ),,,( Int2Int1Int21 4 2 4 1 4 0 21 mnm m m n yyyxxxTydydyd m i xxxG HHH LLLL φφφ
  • 28. As an example, we will now analyze the four-point (i.e., x1…x4) function introduced earlier taken to first order in m with an interaction given by HInt(y)=−λφ 4/4!. We expand: Now, since 〈0|T[φ (x1)φ (x2)]|0〉=i∆F (x1 −x2), we get, after setting φ (xi)=φi and φ (y)=φ ′: ∫ ∫ −=       −⋅= 0)]()()()()()()()([0 !4 0)( !4 )()()()(0),,,( 4321 4 4 4321 4 4321 yyyyxxxxTyd i yxxxxTydixxxxG φφφφφφφφ λ φ λ φφφφ 28 2017 MRT φφφφφφφφ φφφφφφφφ φφφφφφφφ φφφφφφφφ φφφφφφφφ φφφφφφφφ φφφφφφφφ φφφφφφφφφφφφφφφφ ′′′′+ ′′′′+ ′′′′+ ′′′′+ ′′′′+ ′′′′+ ′′′′+ ′′′′=′′′′ 4321 4321 4321 4321 4321 4321 4321 43214321 ][T ∑= −∆= 4 1 )( i iF yxi )()( )()( )()( )]([ 4231 2 3241 2 4321 2 2 xxxxi xxxxi xxxxi yyi FF FF FFA AF −∆−∆+ −∆−∆+ −∆−∆=∆ ∆−∆+ where             )()()( )()()( )()()( )()()( )]([ 3421 3 4231 3 4321 3 4321 3 2 yxxxyxi yxyxxxi xxyxyxi yxyxxxi yyi FFF FFF FFF FFFB BF −∆−∆−∆+ −∆−∆−∆+ −∆−∆−∆+ −∆−∆−∆=∆ ∆−∆+ where x1 x3 x2 x4 y )( 3 yxF −∆ The Feynman diagram corresponding to the Wick decomposition of φ 4 theory to first order.
  • 29. Without demonstrating it, we use the Wick theorem that follows for the general n-point case (n even – as in the previous Figure): (N.B., for n odd the last line reads: ∑ ∑ ∑ −+ + + = perm 121 5 perm 4321 3 perm 21 2121 0)]()([00)]()([0 :)()(:0)]()([00)]()([0 :)()(:0)]()([0 :)()()(:)]()()([ nn n n nn xxTxxT xxxxTxxT xxxxT xxxxxxT φφφφ φφφφφφ φφφφ φφφφφφ L M L L LL 29 2017 MRT and the 4! term in the denominator of the integral coefficient disappears because there are 4! ways in which four external legs at xi can be connected to the four fields contained within φ 4: )∑ −−+ perm 1221 )(0)]()([00)]()([0 nnn xxxTxxT φφφφφ L LL +−∆−=+−= ∫ ∏∫ ∏ == 4 1 4 4 1 44 4321 )(0)]()([0),,,( i F i i yxiydiyxTydixxxxG λφφλ
  • 30. Another example of this decomposition is given by the four-point function taken to second order with HInt(y1)=−λφ 4/4! and HInt(y2)=−λφ 4/4!: The expansion, via Wick’s theorem, is: 30 2017 MRT ∫ ∫      −= 0)]()()()()()([0 !2 1 !4 ),,,( 2 4 1 4 43212 4 1 4 2 4321 yyxxxxTydyd i xxxxG φφφφφφ λ ∫ ∫ ∆−∆−∆+∆−∆ − = })]()][([)]({[ !2 )( ),,,( 2111 2 212 4 1 4 2 4321 BFFAF yyiyyiyyiydyd i xxxxG λ )()()()( )()()()( )()()()( 23221411 24221311 24231211 yxyxyxyx yxyxyxyx yxyxyxyx FFFF FFFF FFFFA −∆−∆−∆−∆+ −∆−∆−∆−∆+ −∆−∆−∆−∆=∆ where: and )()()()( )()()()( )()()()( )()()()( 24132221 24132221 24231221 24232211 yxyxyxyx yxyxyxyx yxyxyxyx yxyxyxyx FFFF FFFF FFFF FFFFB −∆−∆−∆−∆+ −∆−∆−∆−∆+ −∆−∆−∆−∆+ −∆−∆−∆−∆=∆ These are shown graphically in the Figure on the next slide.
  • 31. 31 2017 MRT The Feynman diagrams corresponding to the Wick decomposition of φ 4 theory to second order. y2 y1 y2 y1 y2 y1 y2y1 y2 y1 )( 11 yyF −∆ y1 y2 )( 21 yyF −∆ x1 x3 x2 x4 y2 2 21 )]([ yyF −∆ y1 )( 23 yxF −∆
  • 32. We now introduce the graphical rules introduced by Feynman by which we can almost by inspection construct Green functions of arbitrary complexity. Feynman’s Rules 32 2017 MRT For example, with an interaction Lagrangian given by the four-point scalar field φ4: !4 4 φ λ− with λ being a coupling constant, so that Mf i appearing in: ∏=−⋅ − = f i N i p iifif E d mmpp PP d 3 3 3 2 2 2 1 2 21 424 2)π2()(4 )()π2( pδ σ M can be calculated as follows: 1. Draw all possible connected, topologically distinct diagrams, including loops, with n external legs. Ignore vacuum-to-vacuum diagrams. 2. For each internal line, associate a propagator given by: iεmp i pi F +− =∆ 22 )( 3. For each vertex, associate the factor −iλ. Momentum is conserved at each vertex. 4. For each internal momentum corresponding to an internal loop, associate an integration factor ∫d4p/(2π)4. 5. Divide each diagram by an overall symmetry factor S corresponding to the number of ways one can permute the internal lines and vertices, leaving the external lines fixed. p
  • 33. The symmetry factor S is easily calculated. For the four-point function given above, the 1/4! coming from the interaction Lagrangian cancels the 4! ways in which the four external lines can be paired off with the four scalar fields appearing in φ4, so S=1. Now consider the connected two-point diagram at second order, which is a double-loop diagram. There are 4 ways in which each external leg can be connected to each vertex. There are 3×2 ways in which the internal vertices can be paired off. So this gives us a factor of 1/S=(1/4!)(1/4!)×(4×4)×(3×2)=1/3!, so S=6. For QED, the Feynman rules are only a bit more complicated. The interaction Hamiltonian becomes: µ µ ψγψ Aei−=nInteractioH 33 2017 MRT As before, the power expansion of the interacting Lagrangian will pull down various factors of HInteraction. Then we use Wick’s theorem to pair off the various fermion and vector meson lines to form propagators and vertices. There are only a few differences that we must note. First, when contracting over an internal fermion loop, we must flip one spinor past the others to perform the trace and Wick decomposition. This means that there must be an extra factor of −1 inserted into all fermion loop integrations. Second, various vector meson propagators in different gauges may be used, but all the terms proportional to pµ or pν vanish because of gauge invariance.
  • 34. Thus, the Feynman rules for QED become: 1. For each internal line, associate a propagator given by: ∫ 4 4 )π2( pd 34 2017 MRT iεmp mpi εimp i pSi F +− +/= +−/ = 22 )( )(p 2. For each internal photon line, associate a propagator: νµνµ η εik i kDi F + −= 2 )]([kµ ν 3. As each vertex ( ), place a factor of: µγei− 4. Insert an additional factor of −1 for each closed fermion loop. 5. For each internal loop, integrate over: 6. A relative factor of −1 appears between diagrams that differ from each other by an interchange of two identical external fermion lines. 7. Internal fermion lines appear with arrows in both clockwise and counterclockwise directions.However, diagrams that are topologically equivalent are counted only once. e γµ
  • 35. 8. External electron and positron lines entering a diagram appear with factors: respectively. The direction of the positron lines is taken to be opposite of the electron lines, so that incoming positrons have momenta leaving the diagram. ),(),( spvspu and 35 2017 MRT respectively. External electron and positron leaving a diagram appear with factors: ),(),( spvspu and 9. Polarization is labeled by εµ * for an inbound photon and εµ for an outbound photon. Likewise, we can calculate Feynman rules for any of the actions that we have investigate earlier. For example, for charged scalar electrodynamics, with the additional term in the Lagrangian: φφφφ µ µ µ †22† AmDD −=L one has the following interaction Hamiltonian: φφφφ µ µ µµ †22† nInteractio )( AeAei −∂−∂−= rs H
  • 36. 2. Insert a factor of: 36 2017 MRT 1. For each scalar-scalar-vector vertex, insert the factor: The Feynman rules for charged scalar electrodynamics are as follows: µ][ ppei ′+− νµη2 2 ei for each seagull diagram. µ p p′e µ p p′ ν e2 3. Insert an additional factor of ½ for each closed loop with only two photon lines. In summary, we have seen that, historically, there were two ways in which to quantize QED. The first method, pioneered by Feynman, was the propagator approach, which was simple, pictorial, but not very rigorous. The second was the more conventional operator approach of Schwinger and Tomonoga. With the LSZ approach, which is perhaps the most convenient method for deriving the Feynman rules, and with these, one can almost, by inspecting the Lagrangian, write down the perturbation expansion for any quantum field theory.
  • 37. Now that we have derived the Feynman rules for various quantum field theories, the next step is to calculate cross sections for elementary processes involving photons, electrons, and antielectrons. At the lowest order, these cross sections reproduce classical results found with earlier methods. However, the full power of the quantum field theory will be seen at higher orders, where we calculate radiative corrections to the hydrogen atom that have been verified to great accuracy. In the process, we will solve the problem of the electron self-energy, which completely eluded earlier, classical attempts by Lorentz and others. To begin our discussion, we will divide Feynman diagrams into two types, trees and loops, on the basis of their topology. Loop diagrams, as their name suggests, have closed loops in them. Tree diagrams have no loops (i.e., they only have branches). In a scattering process, we will see that the sum over tree diagrams is finite and reproduces the classical result. The loop diagrams, by contrast, are usually divergent and are purely quantum-mechanical effects. The Compton Effect 37 2017 MRT We start by analyzing the lowest-order terms in the scattering matrix for four particles of fields. In this order, we find only tree diagrams and no loops. Thus, we should be able to reproduce generalizations of classical and nonrelativistic physics. In the Chart below, we will summarize the scattering processes that we will analyze. +−−− +−+−−−−− −−−− +→+++→+ +→++→+ +→++→+ eeγγγNucleuseNucleuse eeeeeeeeø γγeeγeγe :creationPairand:lungBremsstrah ,:scatteringBhabha,:scatteringllerM ,:onannihilatiPair,:scatteringCompton
  • 38. So, to begin, the first process we will examine is Compton scattering, which occurs when an electron and a photon collide and scatter elastically. Historically, this process was crucial in confirming that electromagnetic radiation had particle-like properties (i.e., the photon was acting like a particle in colliding with the electron). We will assume that the electron e− has momentum pi before the collision and pf afterwards. The photon γ has momentum k before and k′ afterwards. The reaction can be represented symbolically as: )(e)(γ)(e)(γ fi pkpk +′→+ 38 2017 MRT By energy-momentum conservation, we also have: 0][][ =−−′+=+−′+⇔′+=+ kpkpkpkpkpkp ififfi Compton scattering, to lowest order, is shown in the Figure. Compton scattering: A photon γ of momentum k scatters elastically off an electron e− of momentum pi. ],[ εk ],[ ii sp ],[ ff sp],[ ε ′′k ],[ εk ],[ ii sp ],[ ff sp ],[ ε ′′k γ − e γ − e γ − e γ − e or
  • 39. We normalize the wave function of the photon by: To lowest order, the S matrix is (N.B., the sum of terms within the square brackets end up being the previous Figure - Left and - Right, respectively): )ee( 2 1 )( xkixki kV xA ⋅⋅− += µµ ε 39 2017 MRT ),( 2 )( 2 )( 2 )( 2 )( ),( )()π2( 44 ii iii ff f ifif spu VE m Vk ei mkp i kV ei Vk ei mkp i kV ei spu VE m kpkpS       ′ ′/− −′/−/ /− + ′ ′/− −/+/ /− × −−+= εεεε δ where: µ µ γγ aauu =/= and0† ),( 11 ),( 22 1 )()π2( 2 44 2 2 ii ii ff if ifif spu mkpmkp spu kkEE m kpkp V e S         ′/ −′/−/ /+′/ −/+/ /× ′ −−+= εεεε δ Simplifying a bit, we get:
  • 40. The differential cross section is found in several steps. First, we square the S matrix, which gives us a divergent result. We divide by the singular quantity (2π)4δ (0) and obtain the rate of transitions. We divide by the flux |v|/V, divide by the number of particles per unit volume 1/V, multiply by the phase factor for outgoing particles [V2/(2π)6]d3pf d3k′. This gives us the differential cross section: where: ∫ ∑ Γ′−−+ ′ ′ = ′ = fi ss iffi f f i fif uukpkp k kd E pdm kE me kdpdVS d , 24 33 2 4 6 332 4 2 )( 22 1 )π2( )π2( 1 )0()π2( δ δ σ v v 40 2017 MRT kp k kp k ii ′⋅ /′// + ⋅ //′/ =Γ 22 εεεε
  • 41. To reduce out the spins, we will once again use the convenient formula given in the Rutherford Scattering chapter: Although this calculation looks formidable, we can perform the trace of up to eight Dirac matrices by reducing it to the trace of six, and then four Dirac matrices, &c. We will use the formula:         +/ Γ +/ Γ=Γ∑ m mp m mp spuspu fi ss iiff fi 22 Tr),(),( 0†0 , 2 γγ 41 2017 MRT )(Tr)(Tr)(Tr)(Tr 1232212423123212321 −///⋅++///⋅−//⋅=//// nnnnn kkkkkkkkkkkkkkkkkk LKLLL The problem simplifies enormously because we can eliminate entire groups of terms every time certain dot products appear, since: 022 =′⋅′=⋅=′= kkkk εε We can also simplify the calculation by using: 22222 1 mpp fi ==−=′= andεε
  • 42. In short, each trace consists of collecting the complete set of all possible pairs of dot products of vectors, most of which vanish. Dividing the factors into smaller pieces, we now find: where: ])(2[8 )(Tr2)(Tr2)]()([TrTr 2 1 ii fififi pkkkp pkkppkkpmpkmpk ⋅′+⋅′⋅= /′//′/⋅=/′////′/⋅=+/′///+///′/= ε εεεεεεεεεε 42 2017 MRT 4321 , 2 TrTrTrTr +++=Γ∑ fi ss if uu and iifi iiiifi pkkpkkpkpk kpkpkkpkpkmpkmpk ⋅′′⋅−⋅⋅′+−⋅′⋅′⋅= ′⋅′⋅−⋅⋅′+//′//′/′/⋅=+//′/′/+///′/= 222 22 2 )(8)(8]1)(2[))((8 )(8)(8)(Tr2)]()([TrTr εεεε εεεεεεεεεε We also have: and ])(2[8TrTr 2 23 ii pkkkp ⋅′+⋅′⋅== ε iifi pkkpkkpkpk ⋅⋅′−⋅′′⋅+−′⋅⋅⋅′= 222 4 )(8)(8]1)(2[))((8Tr εεεε where Tr4 was obtained from Tr3 by making the substitution [ε,k]→[ε ′, k ′].
  • 43. Since the calculation is Lorentz invariant, we can always take a specific frame. We loose no generality by letting the electron be at rest, and let the incoming photon lie along the z-axis. Let the outgoing photon scatter within the y-z plane, making an angle θ with the z-axis (see Figure). Then the specific parametrization is given by: 43 2017 MRT Compton scattering in the laboratory frame, where the electron is at rest. k′ fp k θ It is important to notice that the only independent variables in this scattering are k and θ. All other variables can be expressed in terms of these two variables. For example, we can solve for k′ and E. In terms of the independent variables k and θ: kkmE mk k k ′−+= −+ =′ and )cos1)((1 θ Adding all four contributions, we now have:       −⋅′+ ′ + ′ =Γ∑ 2)(4 2 1 ),(),( 2 2 , 2 εε k k k k m spuspu fi ss iiff ]cos,sin,0,[]cos,sin,0,[],0,0,[]0,0,0,[ θθθθ µµµµ kkkEpkkkkkkkmp fi ′−′−=′′′=′== &,,
  • 44. We must now integrate over the momenta of the outgoing photon k′ and electron pf . Since the only independent variables in the problem are given by k and θ, all integrations are easy, except the Jacobian, which arises when we change variables and integrate over delta functions. Thus, the integration over d3pf is trivial because of momentum conservation; it simply sets the momenta to be the values given above. That leaves one complication, the integration over time components dp0 f . However, this integration can be rewritten in a simple fashion: )(22 )( 2 3 3 4 kkm k d E d pkkp f f fi ′ ′ =′−′−+ k p δ 44 2017 MRT ∫ −= )()( 2 1 0 22 0 fff f pmppd E θδ The integration over p0 f in the integral just sets its value to be the on-shell value. Finally, this last delta function can be removed because of the integration over k′. The only tricky part is to extract from this last integration the measure when we integrate over k′. This last delta function can be written as: where k′(k) is the value given by k′=k/[1+(k/m)(1−cosθ)] above. Putting all integration factors together, we now have: )(2 )]([ )]cos1(2)(2[])[( 2 kkm kkk kkkkmmkpk i ′ ′− =−′−′−=−′−+ δ θδδ
  • 45. Inserting everything into the cross section,we obtain the Klein-Nishina formula (1929): where α =e2/4πεohc is the fine structure constant. If we take the low-energy limit k→0, the Klein-Nishina formula reduces to the Thomson scattering formula: 45 2017 MRT       −′⋅+ ′ + ′       ′ = Ω 2)(4 4 2 2 2 2 εε ασ k k k k k k md d 2 2 2 )( εε ασ ′⋅= Ω md d If the initial and final photon are unpolarized, we can average over the initial and final polarizations ε and ε ′. In the particular parametrization that we have chosen for our momenta, we can choose our polarizations ε and ε ′ such that they are purely transverse and perpendicular to the momenta pi and pf , respectively: Since these polarization vectors satisfy all the required properties, the sum over these same polarization vectors is: θεεεε 2 , 2)()( , 2 cos1)()( +=′⋅=′⋅ ∑∑ ji ji ss fi ]sin,cos,0,0[]0,0,1,0[]0,1,0,0[]0,0,1,0[ )2()1()2()1( θθεεεε −=′=′== and,, )cos1( 2 1 επ4 2 2 2 o 2 θ σ +         = Ω cm e d d so that the Thomson scattering formula above becomes (MKS Units):
  • 46. The average cross section is given by: 46 2017 MRT       − ′ + ′       ′ = Ω θ ασ 2 2 2 2 av sin 2 k k k k k k md d The integral over θ is straightforward if we define z=cosθ. The integration yields:         + + − + +      +− + ++       =         −+ − − −+ + −+ = ∫− 232 2 1 1 2 2 32 2 )21( 31 2 )21ln( )21ln( 21 )1(21 4 3 3 8π )]1(1[ 1 )1(1 1 )]1(1[ 1π a a a a a a aa a a m za z zaza zd m α α σ where a=k/m. For small energies, this reduces to the usual Thomson total cross section: For high energies, the logarithm starts to dominate the cross section and we get: 2 Thomson cm24 2 2 0 10665.0 3 8π lim − → ×=== mk α σσ                     ++      = ∞→ m k k m O m k mkk ln 2 12 ln π lim 2 α σ
  • 47. The Feynman diagrams for pair annihilation of an electron and positron into two gamma rays is shown in the Figure. Pair annihilation is represented by the process: However, notice that we can obtain this diagram if we simply rotate the diagram for Compton scattering. Thus, by a subtle redefinition of the various momenta, we should be able to convert the Compton scattering amplitude, which we have just calculated, into the amplitude for pair production. This redefinition is called the substitution rule (i.e., take a process 1+2→3+4 and convert it to 1+3→2+4). It can also be viewed as a symmetry in the S matrix – the crossing symmetry. )(γ)(γ)(e)(e 2121 kkpp +→+ +− 47 2017 MRT Pair Annihilation Pair annihilation: An electron e− of momentum p1 annihilates with a positron e+ of momentum p2 into two photons γ. 1p 2p 1p 2p ],[ 11 εk ],[ 22 εk ],[ 22 εk],[ 11 εk γ − e − e + e + e− e + e γ γγ − − or
  • 48. For example, the S matrix now yields, to lowest order: where we have made the substitutions: 48 2017 MRT ),( 2 )( 2 )( 2 )( 2 )( ),( )()π2( 11 12 2 211 1 1 1 112 2 22 2 2121 44 spu E m k ei mkp i k ei k ei mkp i k ei spv E m ppkkS if         /− −/+/ /− + /− −/+/ /− × −−+= εεεε δ ),(),(),(),(],[],[],[],[ 22112211 spvspuspuspukkkk ffii →→→′′−→ and,, εεεε We will, as usual, take the Lorentz frame where the electron is at rest. Then our momenta becomes (see Figure): Pair annihilation in the laboratory frame, where the electron is at rest. ]cos,sin,0,[]cos,sin,0,[],0,0,[]0,0,0,[ 1122111121 θθθθ µµµµ kkkkkkkkEpmp −−==== pp &,, There are only two independent variables in this process, |p| and θ. All other variables can be expressed in terms of them. 1k 1p 2k θ
  • 49. For example, we can show that: We also have: )cos()( 12121 θp−+=+=⋅+=+ EmkEmmkkkkEm and 49 2017 MRT θ θ θ cos )cos()( cos )( 21 p p p −+ −+ = −+ + = Em EEm k Em Emm k and When we contract the Dirac matrices, the calculation proceeds just as before, except that we want to evaluate |v Γu|2. We have to use:− m mp spvspv s 2 ),(),( −/=∑ The trace becomes:       +⋅−+=Γ∑ 2)(4 2 1 ),(),( 2 21 2 1 1 2 2 2 εε k k k k m spuspv s The integration over d3k1 and d3k2 also proceeds as before. The integrations over the delta functions are straightforward, except that we must be careful when picking up a measure term when we make a transformation on a Dirac delta function. When this additional measure term is inserted, the differential cross section becomes:       +⋅−+ −+ + = Ω 2)(4 )cos(8 )( 2 21 2 1 1 2 2 2 εε θ ασ k k k k Em Em d d pp
  • 50. The total cross section is obtained by summing over photon polarizations. As before, we can take a specific set of polarizations which are transverse to p1 and p2: Then we can sum over all polarizations. The only difficult sum involves: ]sin,cos,0,0[]0,0,1,0[]sin,cos,0,0[]0,0,1,0[ 11 )2( 2 )1( 2 )2( 1 )1( 1 θθεεθθεε kk−==−== p&,, 50 2017 MRT         +−= + −= + −=•=⋅ 212121 2 21 21 )2( 2 )2( 1 11 1 2 )(2 1 2 )( 1 kk m kk Emm kk kk kkεε Then the sum over spins can be written as: 2 21, 2)( 2 )( 1 11)(               +−+=⋅∑ k m k m ji ji εε The only integration left is the one over the solid angle, which leaves us with:         − + −−+ − ++ + = 1 3 )1(ln 1 14 )1( π 2 2 2 2 2 2 γ γ γγ γ γγ γ α σ m where γ =E2/m. This result was first obtained by Dirac (1930).
  • 51. Next, we investigate electron-electron scattering. To lowest order, this scattering amplitude contains two diagrams (see Figure). This scattering is represented by: By a straightforward application of the formulas for differential cross sections, we find: ∫ ′ ′ ′ ′ −−′+′ ⋅ = 2 2 3 2 3 1 3 1 3 2121 44 2 21 44 )π2()π2( )()π2( )( if E pd E pd pppp pp me d Mδσ 51 2017 MRT Møller Scattering Møller scattering of two electrons with momenta p1 and p2. )(e)(e)(e)(e 2121 pppp ′+′→+ −−−− 1p 2p 1p 2p 1p′ 2p′ γ − e − e − e− e γ 1p′ 2p′
  • 52. Using a straightforward application of Feynman’s rules for these two diagrams, we can compute | Mf i |2: Since the trace is only four Dirac matrices, taking the trace gives: 52 2017 MRT     −′−′ ⋅−⋅ + −′ ⋅−′⋅+′⋅+⋅ +     −′ ⋅−′⋅+′⋅+⋅ 2 12 2 11 21 22 21 22 12 2111 22 11 2 21 22 11 2121 22 21 2 21 4 )()( 2)( 2 ])[( )(2)()( ])[( )(2)()( 4 1 pppp ppmpp pp ppppmpppp pp ppppmpppp m ])( )()( 1 2222 Tr ])[( 1 22 Tr 22 Tr 4 1 21 2 22 2 11 1221 22 11 22112 pp ppppm mp m mp m mp m mp ppm mp m mp m mp m mp if ′↔′+ −′−′       +′/+/+′/+/−     −′       +′/+/      +′/+/= σν σν σν σν γγγγ γγγγM
  • 53. The kinematics are illustrated in the Figure. Without any loss of generality, we can choose the center-of-mass frame, where the electron momenta p1 and p2 lie along the z- axis: The only independent variables are |p| and θ. In terms of this parametrization, we find: θθθθ cos)cos1(cos)cos1(2 22 21 22 11 22 21 mEppmEppmEpp −+=′⋅+−=′⋅−=⋅ &, 53 2017 MRT Møller scattering in the center-of-mass frame. θ ]cos,sin,0,[]cos,sin,0,[],0,0,[],0,0,[ 2121 θθθθ µµµµ pppppp −−=′=′−== EpEpEpEp &,, Then the entire cross section can be written in terms of these independent variables. We finally obtain the Møller formula (1932) in the center-of-mass frame:               + − − +− − − = Ω θθθ ασ 2222 222 242222 2222 sin 4 1 )2( )( sin 3 sin 4 )(4 )2( mE mE mEE mE d d z
  • 54. In the relativistic limit, as E→∞, this formula reduces to: 54 2017 MRT For the low-energy, nonrelativistic result, we find:       +−= Ω 2 1 sin 2 sin 4 242 2 θθ ασ Ed d       −= Ω θθ ασ 2422 2 sin 3 sin 4 4 1 vmd d
  • 55. To calculate the cross section for electron-positron scattering (see Figure), we can use the substitution rule. By rotating the diagrams for Møller scattering, we find the Feynman diagrams for Bhabha scattering. For the process: The only substitutions we must make are: 55 2017 MRT Bhabha Scattering )()()()()()()()( 22221111 pvpupvpupupupupu ′−→′−→′→′→ and,, )(e)(e)(e)(e 2121 pppp ′−+′→−+ +−+− Bhabha scattering of and electron off a positron. 1p 2p 1p 2p 1p′ 2p′ γ − e + e + e− e γ 1p′ 2p′
  • 56. The calculation and the traces are performed exactly as before with the above substitutions. We merely quote the final result, due to Bhabha (1935): In the relativistic limit, we have: 56 2017 MRT    ++−+++−+     −− − + −− − −= Ω ]cos2)cos2)(cos1(4)coscos21(2[ 16 1 )cos1()(2 )2( )cos1)(( 8 4 5 2 242224 4 2222 222 222 44 2 2 θθθθθ θθ ασ mmEE E mE mE mEE mE Ed d                         −++             + = Ω 2 sin 2 cos 2)cos1( 2 1 2 sin 2 cos1 8 2 4 2 2 4 2 2 θ θ θ θ θ ασ Ed d whereas, in the nonrelativistic limit, we find: 4 2 44 2 2 2 sin22 sin16                   =       = Ω θ α θ α σ v m v m d d
  • 57. Bremsstrahlung (i.e., German for breaking radiation) is the process by which radiation is emitted from an electron as it moves past a nucleus (see Figure). Momentum conservation gives us: Classically, one can calculate the radiation emitted by a moving charge as it accelerates past a proton. However, unlike the previous scattering processes, which agree to first order with the experimental data, we find a severe problem with this amplitude, which is the infrared divergence. The quantum field theory calculation, to lowest order, reproduces the classical result, including the unwanted infrared divergences, which has its roots in the classical theory. fi pkqp +=+ 57 2017 MRT Bremsstrahlung Bremsstrahlung, or the radiation emitted by an electron scattering in the presence of a nucleus. ip eZ fpk eZ fp k ip q q γ γ γ γ − e − e
  • 58. Although this infrared divergence first arose (in another form) in the classical theory, the final resolution of this problem comes when we take into account higher quantum loop corrections to the scattering amplitude. The scattering matrix, using Feynman’s rules, is: The differential cross section now becomes: 58 2017 MRT ),( 2 )()()( 2 )( ),( )(π2 2 0 2 0 23 3 ii iif ff f ifif spu E m k i mkp iii mkp i k i spu E m EkE V eZ S         /− −/−/ − + − −/+/ /− × −+−= εγγε δ qq ∫ Γ−+= Ω 2 46 3362 1 )ω(π2 π)2(2ω ifif f f ii uuEE kd E pdm E eZm d d qv δ σ where ω=k0 and where: εγγε / −/−/ + −/+/ /=Γ mkpmkp if 11 00
  • 59. The trace we wish to calculate is: The traces involved in the calculation yield: ∑                 +/         /⋅/ +/−/ /+/ /⋅/ +/+/       +/         / /⋅/ +/−/+ /⋅/ +/+/ /= ++= ε γεεγεγγε m mp kp mkp kp mkp m mp kp mkp kp mkp m f i i f fi i i f f 222222 Tr 2 1 )TrTrTr( 2 1 Tr 0000 32125 59 2017 MRT and ,]ω22 )ω22()(2[ )(8 1 )]()()([Tr )( 1 Tr 0 00022 2 0 21 kpkppkppkpp kppppppmp kp mpmkpmkp kp jififif ifiifif f fff f ⋅⋅−⋅+⋅⋅⋅+ ⋅−⋅−++⋅ ⋅ = +//+/+/+/+// ⋅ = ∑ ∑ εε ε εγε ε ε )(TrTr 22 fi pp −↔= )]ω(ωω)( )()242([ ))(( 16 )()()()()([Tr ))(( 1 Tr 00222 2200 00 3 kppkppppmkpkppkp pkpmppppkpkppp kpkp ppmpmkpmpmkp kpkp iffifififi iffifififi if fiffii if ⋅−⋅−⋅+−⋅⋅+⋅⋅− ⋅⋅+−⋅−⋅+⋅−⋅⋅⋅ ⋅⋅ = −↔++//+/+/+//+/−/ ⋅⋅ −= ∑ ε εεε εγεγ ε
  • 60. The parametrization of the momenta is a bit complicated, since the reaction does not take place in a plane (see Figure). We place the emitted photon momenta in the z- direction, the emitted electron momenta pf in the y-z plane, and the incoming electron momenta pi in a plane that is rotated by an angle ϕ from the y-z plane. The specific parametrization is equal to: where pi =|pi| and pf =|pf |. ]cos,sin,0,[]ω,0,0,ω[ ffffff ppEpk θθµµ == , 60 2017 MRTBremsstrahlung, where the emitted photon is in the z-direction and the emitted electron is in the y-z plane. The incoming electron is in a plane rotated by an angle ϕ from the y-z plane. ϕk ip fp θ and ]cos,cossin,sinsin,[ iiiiiiii pppEp θϕθϕθµ = z
  • 61. Now we must calculate the sum over transverse photon polarizations. Since kµ points in the z-direction, we can choose: which satisfies all the desired properties of the polarization tensor. With this choice of parametrization for the momenta and polarizations, we find: ]0,1,0,0[]0,0,1,0[ )2()1( == εε and 61 2017 MRT ii j i j ff j f j pppp θεθε 222)(222)( sin)(sin)( =⋅=⋅ ∑∑ , and ϕθθεε cossinsin))(( )()( fifi j i j f j pppp =⋅⋅∑ It is now a simple matter to collect everything together, and we now have the Bethe- Heitler formula (1934), which was first computed without using Feynman rules:     +− −− − −− + +     − − +− − × ΩΩ= )ω24( )cos)(cos( cossinsin 2 )cos)(cos( sinsin ω2 )4( )cos( sin )4( )cos( sin ω ω1 )π2( 22 2222 2 22 2 22 22 2 22 eγ42 32 qEE pEpE pp pEpE pp qE pE p qE pE p dd d qp pZ d fi iiifff fiif iiifff ffii f iii ii i fff ff i f θθ ϕθθ θθ θθ θ θ θ θ α σ
  • 62. Now let us make the approximation that ω→0. In the soft bremsstrahlung limit, we find a great simplification, and the differential cross section becomes the one found by classical methods: Here the infrared divergence appears for the first time. This problem was first correctly analyzed by F. Block and A. Nordsieck (1937). The integral d3k/ω is divergent for small ω, and therefore the amplitude for soft photon emission makes no sense. This is rather discouraging, and revealed the necessity of properly adding all quantum corrections. The resolution of this equation only comes when the one-loop vertex corrections are added in properly. For now, it is important to understand where the divergence comes from and its general form. ∞ ⋅−− ~ 2 1 ~ )( 1 22 kpmkp 62 2017 MRT ∑         ⋅ ⋅ − ⋅ ⋅       ΩΩ −ε εεσσ 2 3 32 elastic )π2(ω2 ~ i i f f pk p pk pde d d d d k The infrared divergence always emerges whenever we have massless particles in a theory that can be emitted from an initial or final leg that is on the mass shell. For example, whenever we emit a soft photon of four-momentum k from an on-shell electron with momentum p, we find that the propagator just before the emission is given by: Because p2 =m2 and k small, we find that the integration over momentum k inevitably produces an infrared divergence.
  • 63. In order to quantify this infrared divergence, let us perform the integration over the four-momentum k, separating out the angular part dΩ from d3k. To parametrize the divergence, we will regulate the integral by allowing the photon to have a small but finite mass µ. We will integrate k over µ to some energy E given the sensitivity of the detector. Expanding out the expression of the square, the amplitude now becomes: In our approximation, we can perform the angular integral over the last three terms in the square brackets. 63 2017 MRT ∫ ∫         ⋅ − ⋅ − ⋅⋅ ⋅ Ω      ΩΩ E ifif if pk m pk m pkpk pp ddk d d d d µ ασσ 2 2 2 2 3 2 2 elastic )()())(( 2 )π2( ~ k Let us calculate the last two terms appearing on the right-hand side of the equation. We use the fact that: 1 )cos1( )(cos 2)(π4 1 1 22 2 2 2 = − = ⋅ Ω ∫∫ − θβ θd E m pk md f For the integral over the first term, we will use the fact that: and because dΩ=2πd(cosθ), we can integrate the last two terms: )cos1(~ ff Epk θ−⋅ )cos1()cos1( )cos1(2 ~ ))(( 2 22 fiif if EE E pkpk pp θβθβ θβ −− − ⋅⋅ ⋅
  • 64. We will now introduce the Feynman parameter trick, which is often used to evaluate Feynman integrals: By introducing a new variable x, we are able to perform the angular integration. We find: ∫ −+ = 1 0 2 )]1([ 1 xbxa xd ba 64 2017 MRT        <<         +         − <<+            + = −+− −= −−− − ⋅⋅ ⋅Ω ∫ ∫ ∫∫ 1ln2 1)( 2 sin 3 4 12 )1()2(sin41 )cos1(2 )]1(coscos1[ )(cos 2 1 )cos1(2~ ))(( 2 π4 2 2 2 2 2 2 422 1 0 222 22 1 0 2 22 q m q m O m q O xx xd xx d xd pkpk ppd iiif if if if ββ θ β θββ θβ θβθβ θ θβ
  • 65. Inserting this value back into the previous expression, we find that the final soft bremsstrahlung cross section is given by: Although this formula agrees well with experiment at large photon momenta, this amplitude is divergent if we let the fictitious mass of the photon µ go to zero. Thus, the infrared divergence occurs because we have massless photons present in the theory! 65 2017 MRT                +−         − <<+                    ΩΩ 1~1ln 1)( 2 sin 3 4 ln π ~ 2 2 2 2 422 2 2 0 β ββ θ β µ ασσ q m O m q O E d d d d We should mention that the infrared problem arose (in another form) in classical physics, before the advent of quantum mechanics. The essential point is that, even at the classical level, we have the effects due to the long-range Coulomb field. If one were to calculate the radiation field created by a particle being accelerated by a stationary charge, one would find a similar divergence using only classical equations. If one tries to divide the energy by k0 to calculate the number of photons emitted by bremsstrahlung, it turns out to be proportional to the result presented above. Thus, as the momentum of the emitted photon goes to zero, the number of emitted photons becomes infinite. (N.B., Classically, the infrared divergence appears in the number of emitted photons, not the emitted energy).
  • 66. Quantum field theory gives us a novel, but rigorous, solution to the infrared problem, which goes to the heart of the measurement process and the quantum theory. To this order of approximation, we have to add the contribution of two different physical processes to find the cross section of electron scattering off protons or a nucleus with charge Ze, or other charged particles (see Figure). The first diagrams describes the bremsstrahlung amplitude for the emission of an electron and a photon. The divergence of this amplitude is classical. Second, we have to sum over a purely quantum-mechanical effect, the radiative one-loop corrections of the electron elastically colliding off the charged proton. This may seem strange, because we are adding the cross sections of two different physical processes together, one elastic and one inelastic, to cancel the infrared divergence. However, this makes perfect sense from the point of view of the measuring process. 66 2017 MRT The infrared divergence cancels if we add the contributions of two different physical processes. These diagrams can be added together because the resolution of any detector is not sensitive enough to select out just one process. The nucleus is label with the bullet ( ).
  • 67. The essential point is to observe that our detectors cannot differentiate the presence of pure electrons from the presence of electrons accompanied by sufficient soft photons. This is not just a problem of having crude measuring devices. No matter how precise our measuring apparatus may become, it can never be perfect; there will always be photons with momenta sufficiently close to zero that will sneak past them. Therefore, from an experimental point of view, our measuring apparatus cannot distinguish between these two types of processes and we must necessarily add these two diagrams together. Fortunately, we get an exact cancellation of the infrared divergences when these two scattering amplitudes are added together. A full discussion of this cancellation, however, cannot be described until we discuss one-loop corrections to scattering amplitudes. Therefore, in the Anomalous Magnetic Moment chapter, we will prove that the bremsstrahlung amplitude, given by:         −              ΩΩ 2 2 2 2 0 lnln π ~ m qE d d d d µ ασσ 67 2017 MRT must be added to the one-loop vertex correction in order to yield a convergent integral. Finally, we note that, by the substitution rule, we can show the relationship between bremsstrahlung and pair production. Once again, by rotating the diagram around, we can convert bremsstrahlung into pair production.
  • 68. This end our discussion of the tree-level, lowest-order scattering matrix. Although we have had great success in reproducing and extending known classical results, there are immense difficulties involved in extending quantum field theory beyond the tree level. When loop corrections are calculated, we find that the integrals diverge in the ultraviolet region of momentum space. In fact, it has taken over a half century, involving the combined efforts of several generations of physicists, to resolve many of the difficulties of renormalization theory. 68 2017 MRT
  • 69. We now turn to a detailed calculation of single-loop radiative corrections. Although the calculations are often long and tedious, involving formally divergent quantities, the final conclusions are simple and show that the various infinities can be consistently absorbed into a redefinition of the physical constants of the theory, such as the electric charge and electron mass. Most important, the agreement with experiment is astonishing! 69 2017 MRT Radiative Corrections We will begin our discussion of radiative corrections by first examining the self-energy correction to the photon propagator, called the vacuum polarization diagram. We will show that the divergence of this diagram can be absorbed into a renormalization of the electric charge. Then, we will calculate the single-loop correction to the electron-photon vertex and show that this leads to corrections to the magnetic moment of the electron. The theoretical value of the anomalous magnetic moment will agree with experiment to one part in 108. After that, we will show that the radiative correction to the vertex function is also infrared divergent. Fortunately, the sign of this infrared divergence is opposite the sign found in the bremsstrahlung amplitude. When added together, we will find that the two cancel exactly, giving us a quantum mechanical resolution of the infrared problem. And finally, we will analyze the Lamb shift between the energy levels of the 2S½ and 2P½ orbitals of the hydrogen atom. The calculation is rather intricate, because the hydrogen atom is a bound state, and also there are various contributions coming from the vertex corrections, the anomalous magnetic moment, the self-energy of the electron, the vacuum polarization diagram, &c. However, when all these contributions are added, we find agreement with experiment to within one part in 106.
  • 70. The simplest higher-order radiative correction is the vacuum polarization diagram (see Figure). This diagram is clearly divergent. For large momenta, the Feynman propagators of the two electrons give us two powers of p in the denominator, while the overall integration over d4p gives us four power of p in the numerator. So this diagram diverges quadratically in the ultraviolet region of momentum space: We will perform this integration via the Pauli-Villars method (1949), although the dimensional regularization method is significantly simpler. The Pauli-Villars method replaces this divergent integral with a convergent one by assuming that there are fictitious fermions with mass M in the theory with ghost couplings. At the end of the calculation, these fictitious particles will decouple if we take the limit as their masses tend to infinity. Therefore, M gives us a convenient way of cutting off the divergence of the self-energy correction. ∫ ∫         +−+ +/+/ +− +/ −= +−/−/+−/ −=Π νµ νµνµ γγ γγ iεmqk mqki iεmk mkikd e iεmqkiεmk kd em 22224 4 2 4 4 2 )( )()( Tr )π2( )( 11 )π2( Tr 70 2017 MRTFirst-loop correction to the photon propagator, the vacuum polarization diagram, which gives us a correction to the coupling constant and contributes to the Lamb shift. k qk − q − e + e
  • 71. The diagram then becomes modified as follows: The most convenient way in which to perform the integration is to add additional auxiliary variables. This allows us to reverse the order of integration. We can then perform the integration over the momenta, and save the integration over the auxiliary variables to the very end. We will use: Mm νµνµνµ Π−Π=Π ~ 71 2017 MRT ∫ ∞ +− = +− 0 )( 22 22 e 1 εimki d εimk α α Inserting this expression for the electron propagator and performing the trace, we find: With the insertion of these auxiliary variables (i.e., α1 and α2), we can now perform the integration over d4k. )]()()([ e )π2( 4 22 0 0 )]})[()({ 4 4 21 2 22 2 22 1 mqkkqkkqkk kd dde εimqkεimkim −⋅−−−+−× =Π ∫ ∫ ∫ ∞ ∞ +−−++− νµµννµ αα νµ η αα
  • 72. First, we shift momenta and complete the square: Putting everything back into Πµν , we have: νµ αα νµ αα η αααα 3 21 2 )( 4 4 2 21 2 )( 4 4 )(2π3 e )π2()(π16 1 e )π2( 21 2 21 2 + = + = ∫∫ ++ i i pp pd i pd pipi and 72 2017 MRT qkp 21 2 αα α + −= Then we use the fact that: 21 2 )( Π+Π−=Π νµνµνµνµ ηη qqqm where: ∫ ∫ ∞ ∞ + −=Π 0 0 ),( 4 21 21 211 21 e )(π 2 αα αα αα αα α f ddi ∫ ∫ ∞ ∞ − + −=Π 0 0 ),( 213 21 212 21 e]1),([ )( 1 π αα αα αα αα α f fiddi and and where: ))((),( 21 2 21 212 21 ααε αα αα αα +−− + = imiqif There is a similar expression for ΠM µν . We notice that Π2 diverges quadratically, which is bad.
  • 73. However, since we have carefully regularized this integral using the Pauli-Villars method, the integral is finite for fixed M, and we are free to manipulate this expression. We can then show that Π2 vanishes: To perform the integration of Π1, we use one last identity: ∫ ∞       + −= 0 21 11 ρ αα δ ρ ρd 73 2017 MRT Inserting this into the expression for Π1 and rescaling αi , we find: ∫ ∫ ∫ ∫ ∫ ∫ ∞ ∞ ∞ +− ∞ ∞ ∞ −−−=       + − + −=Π 0 0 0 )( 212121 0 0 0 ),(21 4 21 21 211 2 21 2 21 e)1( π 2 e1 )(π 2 εimqi f d dd i d ddi ααρ αα ρ ρ ααδαααα α ρ αα δ ρ ρ αα αα αα α As expected, this integral is logarithmically divergent. Thus, the vacuum polarization diagram is only logarithmically divergent. 02 =Π
  • 74. At this point, we now use the Pauli-Villars regulator, which lowers the divergence of the theory. To perform the tricky ρ integration, we use the fact that m2 −α1α2q2 is positive, so we can rotate the contour integral of ρ in the complex plane by −90 degrees. Using integration by parts and rescaling, we have the following identity: where a(m)= m2 −α1α2q2. The dangerous divergent comes from the last term. However, since the last term is independent of a, it cancels against the same term, with a minus sign, coming from the Pauli-Villars contribution. Thus, we have the identity:         +         −−=+−=−∫ ∞ −− ∞→ 2 2 2 2 21 0 )()( ln1ln)(ln)(ln]e[elim m M m q Mama d Mama αα ρ ρ ρρ ε 74 2017 MRT ∫∫ ∞ − ∞ − ∞ − −= ε ρ ε ρρ ρρρ ρ ρ a aaa d d )e(ln)e(lne 0 for large but finite M. We can now take the limit ε →0 and M becomes large. Then: ∫ ∞         −−−+        −=Π 0 2 2 111112 2 1 )1(1ln)1( π 2 ln 3π ~ m q d i m Mi ααααα αα If we perform the last and final integration over α1, we arrive at:                 −         −         −         +++         −−=Π 11 4 arccot1 42 12 3 1 ln 3π ~ 21 2 2 21 2 2 2 2 2 2 1 q m q m q m m Miα
  • 75. That was the final result. After a long calculation, we find a surprisingly simple result that has a physical interpretation. We claim that the logarithmic divergence can be cancelled against another logarithmic divergence coming from the bare electric charge eo. In fact, we will simply define the divergence of the electric charge so that it precisely cancels against the logarithmic divergence of Π1. To lowest order, we find that we can add the usual photon propagator Dµν to the one- loop correction, leaving us with a revised propagator:         −        −− 2 2 2 2 2 π15 ln π3 1 m q m M q i αα η νµ 75 2017 MRT in the limit as q2 →0. This leaves us with the usual theory, except that the photon propagator is multiplied by a factor: 322 Z q i q i νµνµ ηη −→−         − 2 2 3 ln π3 1~ m M Z α where:
  • 76. Now let us absorb this divergence into the coupling constant eo. We are then left with the usual theory with an extra (infinite) factor Z3 multiplying each propagator. Since the photon propagator is connected to two electron vertices, with coupling eo, we can absorb Z3 into the coupling constant, so we have, to lowest order: where e is called the renormalized electric charge. Since the infinity coming from Z3 cancels (by construction) against the infinity coming from the bare electronic charge, the renormalized electric charge e is finite. o3 eZe = 76 2017 MRT
  • 77. We recall that in the Relativistic Quantum Mechanics chapter we derived the magnetic moment of the electron by analyzing the coupling of the electron to the vector potential: where 2 is the Landé g factor (i.e., gyromagnetic ratio). At the tree level, we know that the coupling of an electron to the photon is given by Aµ uγµu, which in turn gives us that gyromagnetic ratio of 2. However, the experimentally observed value differed from this predicted value by a small but important amount. Schwinger’s original calculation (1946) of the anomalous magnetic moment of the electron helped to establish QED as the correct theory of electrons and photons. S      == mc e cm e 2 2 2 σσσσµµµµ h 77 2017 MRT Anomalous Magnetic Moment − To calculate the higher-order corrections to the magnetic moment of the electron, we will use the Gordon identity: )(]))[(( 2 1 )()( puqipppu m pupu ν νµµµγ Σ+′+′=′ The magnetic moment of the electron comes from the second term u Σµν qνu Aµ. To see this, we take the Fourier transform, so qν becomes ∂ν, and the coupling becomes u Σµν Fµν u. The magnetic field Bi is proportional to εijk Fjk, so this coupling term in the rest frame now becomes uσi Biu, where we use the Dirac representation of the Dirac matrices. Since σi =σσσσ is proportional to the spin of the electron, which in turn is proportional to the magnetic moment of the electron, the coupling becomes µµµµ•B. This is the energy of a magnet with moment µµµµ in a magnetic field B. − − −
  • 78. In this chapter, we will calculate the one-loop vertex correction, which gives us a connection to the electron-photon coupling, in the lowest order in α, given by: for the process given in the Figure. )( 2π2 1 2 )( )( pu m qi m pp pu         Σ       ++ ′− ′ ν νµµ α 78 2017 MRTFirst-loop correction to the electron vertex function, which contributes to the anomalous electron magnetic moment. k p p′ kp −′ kp − Notice that the u Σµν qν u Aµ term is modified by the one-loop correction, so that the g factor of the electron becomes: − π22 2 π2 1 2 αα = − ⇔+= gg Thus, QED predicts a correction to the moment of the electron. Hmmm…. Interesting!
  • 79. To show this, we will begin our calculation with the one-loop vertex correction: Anticipating that the integral is infrared divergent, we have added µ, the fictitious mass of the photon. The integral is also divergent in the ultraviolet region, so we will use the Pauli-Villars cutoff method later to isolate the divergence. ∫ +−/−/+−/−′/+− − −=′Λ ν µνµ γ ε γ ε γ εµ imkp i imkp i ik ikd eipp 224 4 2 )( )π2( )(),( 79 2017 MRT Throughout this calculation, we tacitly assume that we have sandwiched this vertex between two on-shell spinors, so we can use the Gordon decomposition and the mass- shell condition. Our goal is to write this expression in the form: )( 2 1 )(~ 2 2 2 1 qF m iqF νµµµ γ Σ+Λ sandwiched between u(p′) and u(p), where F1 and F2 are the form factors that measure the deviation from the simple γµ vertex. We will calculate explicit forms for these two form factors and we will find that F1 cancels against the infrared divergence found in the bremsstrahlung calculation, giving us a finite result, and we will also find that it is F2 that gives us a correction to the magnetic moment of the electron. −
  • 80. We begin with the Feynman parameter trick, generalizing the 1/ab integral that we showed in the Bremsstrahlung chapter: where ∆=Σiai zi. ∫ ∫ ∫ ∏ ∞ ∞ ∞ = ∆ Σ− −= 0 0 0 11 )1( )!1( 1 n i n ii i n z zdn aa δ L L 80 2017 MRT εεεµ imkpaimkpaika +−−=+−−′=+−= 22 3 22 2 22 1 )()( and, Therefore: ∫ ∫ ∞       ∆ −−−=′Λ 0 33213214 4 2 )( )1( )π2( 2),( kN zzzzdzdzd kd eipp µ µ δ where: ν µνµ γγγ )()()( mkpmkpkN −/−/−/−′/= For our purposes, we want: and: 332211 3 1 zazazaza i ii ++==∆ ∑=
  • 81. Next, we would like to perform the integration by completing the square: This allows us to make the shift in integration: 0 2 32 )( ∆−−′−=∆ zpzpk 81 2017 MRT where: )1()()1()()1( 33 22 22 222 32 2 1 22 1 2 0 zzmpzzmpizzqzzm −−−−−′−−−+−=∆ εµ 32 zpzpkk +′+→ Therefore, after the shift, we have: ∫ ∫ ∞ ∞− +∆− +′+ −−−=′Λ 1 0 2 0 2 324 3213214 2 )( )( )1( )π2( 2 ),( ε δ µ µ ik zpzpkN kdzzzzdzdzd ei pp By power counting, the integral diverges. This is why we must subtract off the contribution of the Pauli-Villars field, which has mass Λ. Let us expand: )()(2)( 32 2 32 zpzpNkAkkkzpzpkN +′++/+−=+′+ µµµµµ γ where Aµ(k) is linear in kµ variable. This term can be dropped since its integral over d4k vanishes.
  • 82. Therefore, the leading divergence behaves like: where ∆Λ represents the Pauli-Villars contribution, where µ is replaced by Λ. To perform this integration, we must do an analytic continuation of the previous equation. Using: νµαα νµ ηα α )3( ))((2 π )( 3 2 2 4 −Γ ∆−Γ = ∆− −∫ i k kk kd 82 2017 MRT ∫         ∆− − ∆− Λ 323 0 2 4 )()( k kk k kk kd νµνµ we now take the limit as α →3. This expression diverges, but the Pauli-Villars term subtracts off the divergence.
  • 83. If we let α −3=ε, then we have: where we drop terms like Λ−n. Because this expression is sandwiched between two on- shell spinors, we can also reduce the term:         ∆ ∆ −=         ∆ ∆ −=−=         ∆− − ∆− Γ=         ∆− − ∆− Λ∆−∆− → Λ → Λ → Λ ∫ 0 2 12 0 2)(ln)(ln2 0 0 2 02 0 2 4 3 lnπ lnπ]ee[ 1 πlim )( 1 )( 1 )(πlim )()( lim 0 z i ii i k kk k kk kd εε ε εεεα νµ α νµ α ε ε 83 2017 MRT ],)[1(2)]1)(1(2)41(2[)( 1132 22 11 2 32 µµµ γγ qzzmzzqzzmzpzpN /−−−−++−−=+′ At this point, all integrals can be evaluated. Putting everything together, and dropping all terms of order Λ−1 or less, we now have:     ∆ −Σ +     ∆ −−++− +         ∆ Λ −−−=′Λ ∫ ∞ 0 11 0 0 32 22 11 2 0 2 1 321321 )1( )1)(1()41( ln)1( π2 ),( zzqmi zzqzzmz zzzzdzdzdpp ν νµ µµµ γγδ α
  • 84. Now let us compare this expression with the form factors F1 and F2 appearing in Λµ ~γµ F1(q2)+i(1/2m)Σµν F2(q2) above. It is easy to read off: with the integrand at q2 =0 subtracted off so as to satisfy the constraint F1(q2 =0)=1 which preserves the correct normalization of the vertex function. 84 2017 MRT The value of F2 can similarly be read off:    =     ∆ −−++− +∆−−−−+→ ∫ ∞ )0( )1)(1()41( ln)1( π2 1)( 2 0 0 32 22 11 2 0321321 2 1 q zzqzzm zzzzdzdzdqF δ α ∫ ∞         ∆ − −−−→ 0 0 11 2 321321 2 2 )1(2 )1( π2 )( zzm zzzzdzdzdqF δ α
  • 85. The calculation for F2(q2) is a bit easier, since there are no ultraviolet or infrared divergences. Because of this, we can set µ =0. Let us choose new variables: where q2 =(p−p′)2. Then we have: )cosh1(2 2 sinh4cosh 22222 θ θ θ −=      −==′⋅ mmqmpp and 85 2017 MRT ∫ ∫ −+−+ = ++ +−+ −−= 1 0 22 1 0 32 2 3 2 2 2 3231 3232 2 2 cosh)1(2)1( 1 2π cosh2 )( )1( π )( θββββ β α θ θ α d zzzz zzzz zzzdzdqF This leaves us with the exact result: θ θα sinh2π )( 2 2 =qF We are especially interested in taking the limit as |q2|→0 and θ →0: 2 2 2 8π2π )0( e F == α π4επ4 2 o 2 e c e ≡= h α since the fine structure constant is given by:
  • 86. This gives us the correction to the magnetic moment of the electron, as in: This is only a first-order calculation, yet already we are very close to the experimental value. Since the calculation was originally performed by Schwinger (i.e., his δµ=(α/2π)µo result), the calculation since has been taken to α3 order (where there are 72 Feynman diagrams – see Figure on next slide). The theoretical value to this order is given by: 0011614.0~ 2π2 2e α = −g 86 2017 MRT above but for: )( 2π2 1 2 )( )( ee pu m qi m pp pu         Σ       ++ ′− ′ ν νµµ α K+      +      −      =−≡ 32 eth π 49.1 π 32848.0 π 5.0)2( 2 1 ααα ga The final results for both the theoretical and experimental values are: )31(2090011599652.0)166(110011596524.0 expth == aa vs where the estimated errors [as of 1992] are in parentheses. The calculation agrees to within one part in 108 for a and to one part in 109 for ge, which is graphic vindication of QED. To push the calculation to the fourth order involves calculating 891 diagrams and 12672 diagrams at the fifth order.
  • 87. 87 2017 MRT First order correction – in α: Second order correction – in α2: Third order correction – in α3: 2π )0(2 α =F Magnetic moment of the electron
  • 88. The calculation for F1 is much more difficult. However, it will be very important in resolving the question of the infrared divergence, which we found in the earlier discussion of bremsstrahlung. We will find that the infrared divergence coming from the bremsstrahlung diagram and F1 cancel exactly. Although the calculation of F1 is difficult, one can extract useful information from the integral by taking the limit as µ becomes small. Then F1 integration in: of the Anomalous Magnetic Moment chapter splits up into four pieces: L+= ∑= 4 1 1 i iPF 88 2017 MRT Infrared Divergence    =     ∆ −−++− +∆−−−−+→ ∫ ∞ )0( )1)(1()41( ln)1( π2 1)( 2 0 0 32 22 11 2 0321321 2 1 q zzqzzm zzzzdzdzdqF δ α where the ellipsis represents constant terms.
  • 89. After changing variables, each of the Pi pieces can be exactly evaluated: where: ∫ ∫ ∫ −      = −−+++ −= − 2 0 1 0 1 0 32 2 32 2 3 2 2 321 tanhcoth π 2 lncoth π )1()(cosh2 cosh π2 2 θ ϕϕϕθ αµ θθ α µθ θα d m zzmzzzz zdzdP z 89 2017 MRT )2tanh( tanh 21 2 ϕ ϕ =− z The others are given by:       −= ++ −= = ++ + = ∫ ∫ ∫ ∫ − − 1 sinh2πcosh2 )cosh1( π 2 coth πcosh2 cosh π 1 0 1 0 32 2 3 2 2 32 323 1 0 1 0 32 2 3 2 2 32 322 2 2 θ θα θ θ α θθ α θ θ α z z zzzz zz zdzdP zzzz zz zdzdP , and constant 2 sinh 2 2π2 sinh4)(ln 2π 1 0 1 0 2 32 2 32324 2 +       −=            ++−= ∫ ∫ − θ θ αθα z zzzzzdzdP
  • 90. Thus, adding the pieces together, we find: For |q2|<<1, we find using Λµ~γµ F1(q2)+i(1/2m)Σµν F2(q2), p⋅p′=m2coshθ and q2 =2m2(1− coshθ), and F2(q2)=(α/2π)(θ/sinhθ) of the Anomalous Magnetic Moment chapter with the above:               −−−      +      = ∫ 2 tanh 4 tanhcoth2)1coth(1ln π )( 2 0 2 1 θθ ϕϕϕθθθ µα θ d m qF 90 2017 MRT ],[ π88 3 ln π3 1~),( 2 2 µµµµ γ αµα γγ q mmm q ppc /+               −      +′Λ+ For |q2|>>m2, we find:                         +−         −      −′Λ+ 2 2 2 2 1lnln π 1~),( q m O m q m ppc µα γγ µµµ
  • 91. Plugging all this into the cross section formula, we now find our final result: Now we come to the final step, the comparison of the bremsstrahlung amplitude of the Bremsstrahlung chapter and the vertex correction for electron scattering. Although they represent different physical processes, they must be added because there is always an uncertainty in our measuring equipment in measuring soft photons. Comparing the two amplitudes, we recall that the bremsstrahlung amplitude is given by: 91 2017 MRT             −      Ω =      Ω )(ln π 2 1 2 0 q m d d d d χ µ ασσ µ where:        >>−−         − <<−− = 22 2 2 22 2 2 2 1ln 3 1 )( mq m q mq m q q if if χ                 −      Ω = Ω 2 2 2 2 0 lnln π µ ασσ E m q d d d d while the vertex correction diagram yields:                 −        −−      Ω = Ω 2 2 2 2 0 lnln π 1 µ ασσ q m q d d d d
  • 92. Clearly, when these two amplitudes are added together, we find a finite, convergent result independent of µ2, as desired. The cancellation of infrared divergences to all orders in perturbation theory is a much more involved process. However, there are surprising simplifications that give a very simple result for this calculation. 92 2017 MRT
  • 93. Two of the great accomplishments of QED were the determination of the anomalous magnetic moment of the electron, which we discussed in the chapter dedicated to it, and the Lamb shift. The fact that these two effects could not be explained by ordinary quantum mechanics, and the fact that the QED result was so accurate, helped to convince the skeptics that QED was the correct theory of the electron-photon system. In 1947, W. Lamb and R. Retherford demonstrated that the 2S½ and the 2P½ energy levels of the hydrogen atom were split (see Figure); the 2P½ energy level was depressed more than 1000 MHz below the 2S½ level – by 1040 MHz actually (or ~0.035 cm−1). The original Dirac electron in a classical Coulomb potential, as we saw in the Relativistic Quantum Mechanics chapter, predicted that the energy levels of the hydrogen atom should depend only on the principal quantum number n and the total spin j, so these two levels should be degenerate… Shelter Island, we have a problem! 93 2017 MRT Lamb Shift 1040 MHz 17 MHzDirac-Coulomb QED radiative corrections ≈ 2S½ 2P½
  • 94. The calculation of the Lamb shift is rather intricate, because we are dealing with the hydrogen atom as a bound-state, and also because we must sum over all radiative cor- rections to the electron interacting with a Coulomb potential that modify the naïve uγ0 u A0 vertex. These corrections include the vertex correction, the anomalous magnetic moment, the self-energy of the electron, the vacuum polarization diagram, and even infrared divergences (see Figure). The original nonrelativistic bound-state calculation of H. Bethe (1947), which ignored many of the subtle higher-order corrections, could account for about 1000 MHz of the Lamb shift, but only a fully relativistic quantum treatment could calculate the rest of the difference. Because of the intricate nature of the calculation, we will only sketch the highlights of the calculation. 94 2017 MRT The various higher-order diagrams that contribute to the Lamb shift: (a) and (b) the electron self-energy diagrams; (c) the vertex correction; (d) and (e) the electron mass counterterm (added due to the occurrence in momentum integrals of infrared divergences); (f) the photon self-energy correction. (a) (b) (c) (d) (e) (f) ⊗ ⊗ −
  • 95. To begin the discussion, we first see that the vacuum polarization diagram can be attached to the photon line, changing the photon propagator to: This translates into a shift in the effective coupling of an electron to the Coulomb potential. Analyzing the zeroth component of this propagator, we see that the coupling of the electron to the Coulomb potential changes as follows:         +−−= )( π60 1 4 2 2 2 2 2 kO m ke k i D νµνµ η 95 2017 MRT         +−→ )( 5π1 1 2 2 2 2 02 o2 02 o α αγγ O m q q uu ei q uu ei To convert this back into x-space, let us take the Fourier transform. We know that the Fourier transform of 1/q2 is proportional to 1/r. This means that the static Coulomb potential that the electron sees is given by: )()( 0π6π4π45π1 1 43 22 222 2 2 eO m e r e r e m ++=        ∇− xδ α meaning that there is a correction to Coulomb’s law given by QED. This correction, in turn, shifts the energy levels of the hydrogen atom. We know from ordinary nonrelati- vistic quantum mechanics that, by taking matrix elements of this modified potential between hydrogen wave functions, we can calculate the first-order correction to the energy levels of the hydrogen atom due to the vacuum polarization diagram.
  • 96. Now let us generalize this discussion to include the other corrections to the calculation of the Lamb shift. Our method is the same: calculate the corrections to the vertex function uγ0 u, take the zeroth component, and then take the low-energy limit. In the Anomalous Magnetic Moment chapter, we had Λµ~γµ F1(q2)+i(1/2m)Σµν F2(q2) and with it eventually obtained an γµ +Λc µ(p′,p) equation for |q2|<<m2 and |q2|>>m2, and we saw how radiative corrections modified the vertex function with additional form factors F1(q2) and F2(q2). If we add the various contributions to the vertex correction, we find: For example, the vacuum polarization diagram contributes the factor −1/5 to the vertex correction. The logarithmic term comes from the vertex correction, and the µ term is eventually cancelled by the infrared correction. The Σµν qν term reduces down to a spin- orbit correction, and we find the effective potential given by: uq m i m m q uuu         Σ+               −−      −→ ν νµµµ α µ α γγ π45 1 4 3 ln π3 1 2 2 96 2017 MRT − Lx •+      +−−      ∆ σσσσ32 2 3 2 2 eff π4 )( 8 3 5 1 8 3 ln 3 4 ~ rm m m V α δ µ α
  • 97. By taking the matrix element of this potential between two hydrogen wave functions, we can calculate the energy split due to this modified potential. The vertex correction, for example, gives us a correction of 1010 MHz. The anomalous magnetic moment of the electron contributes 68 MHz. And the vacuum polarization diagram, calculated earlier, contributes −27 MHz. Adding these corrections together, we find, to the lowest loop level, 1051 MHz and thus arrive at the Lamb shift to within 6 MHz accuracy. Since the, higher-order corrections have been calculated, to the difference between experiment and theory has been reduced to 0.01 MHz. Theoretically, the 2S½ level is above the 2P½ energy level by 1057.864±0.014 MHz. The experimental result is 1057.862± 0.020 MHz. This is an excellent indicator of the basic correctness of QED. 97 2017 MRT
  • 98. One of the serious complications found in quantum field theory is the fact that the theory is naïvely divergent. When higher-order corrections are calculated for QED, one find that the integrals diverge in the ultraviolet region, for large momentum p. Since the birth of quantum field theory, several generations of physicists have struggled to renormalize it. Some physicists, despairing of ever extracting meaningful information from quantum field theory, even speculated that the theory was fundamentally sick and must be discarded. In hindsight, we can see that the divergences found in quantum field theory were, in some sense, inevitable. In the transition from quantum mechanics to quantum field theory, we made the transition from a finite number of degrees of freedom to an infinite number. Because of this, we must continually sum over an infinite number of internal modes in loop integrations, leading to divergences. The divergent nature of quantum field theory then reflects the fact that the ultraviolet region is sensitive to the infinite number of degrees of freedom of the theory. Another way to see this is that the divergent diagrams probe the extremely small distance region of space-time, or, equivalently, the high-momentum region. Because almost nothing is known about the nature of physics at extremely small distances or momenta, we are disguising our ignorance of this region by cutting off the integrals at small distances. 98 2017 MRT Overview of Renormalization in QED Since that time, there have been two important developments in renormalization theory. First was the renormalization of QED via the covariant formulation developed by Schwinger and Tomonoga and by Feynman (which we shown to be equivalent by Dyson). This finally showed that, at least for the electromagnetic interactions, quantum field theory was the correct formulation.
  • 99. Subsequently, physicists attacked the problem of the strong and weak interactions via quantum field theory, only to face even more formidable problems with renormalization that stalled progress for several decades. To this applied the second revolution which was the proof by G. ’t Hooft (1971) that spontaneously broken Yang-Mills non-Abelian gauge field theory was renormalizable, which led to the successful application of quantum field theory to the weak interactions and opened the door to the gauge revolution. There have been many renormalization proposals made in the literature, but all of them share the same basic features. Although details vary from scheme to scheme, the essential idea is that there is a set of bare physical parameters that are divergent, such as the coupling constants and masses. However, these bare parameters are unmeasurable! The divergences of these parameters are chosen so that they cancel against the ultra violet infinities coming from infinite classes of Feynman diagrams, which robe the small-distance behavior of the theory. After these divergences have been absorbed by the bare parameters, we are left with the physical, renormalized, or dressed parameters that are indeed measurable. 99 2017 MRT Since there are a finite number of such physical parameters, we are only allowed to make a finite number of such redefinitions. Renormalization theory, then, is a set of rules or prescriptions where, after a finite number of redefinitions, we can render the theory finite to any order. We should stress that, although the broad features of the renormalization program are easy to grasp, the details may be quite complicated.
  • 100. Only the first three classes of diagrams are actually divergent. Fortunately, this is also the set of divergent diagrams that can be absorbed into the redefinition of physical parameters, the coupling constant, and the electron mass. (N.B., The photon mass, as we shall see, is not renormalized because of gauge invariance). 100 2017 MRT ∫ −⋅ −/−/ −=/Σ− ν νµ µ γηγ 24 4 2 )π2( )()( k i mkp ikd eipi ≡ ≡ ≡ ≡ ∫       −/−/ ⋅ −/ −−=Π νµνµ γγ mkp i mp ikd eiki Tr )π2( )()( 4 4 2       −Π      −+−= νσ σρ ρµνµνµ ηηη 222 )()( k i ki k i k i kDi ∫ −/−/−/+ −⋅−=Λ− σ µ ρ σρµ γγγη mk i mqk i pk ikd eiqpei 24 4 3 )()π2( )(),( The one-loop diagrams that we want particularly to analyze are therefore the electron self-energy correction Σ(p): vacuum polarization Πµν: the photon propagator Dµν : and the vertex correction Γµ : µΛ νµD νµ Π Σ
  • 101. To begin renormalization theory, we will sum all possible diagrams appearing in the complete propagator (see Figure): 101 2017 MRT The complete propagator S′F is the sum over one-particle irreducible diagrams Σ arranged along a chain. iεpmp i pSipipSipSipSi FFFF +/Σ−−/ =+//Σ−/+/=/′ )( )()]()[()()( o K This is the renormalized electron propagator. Then we make a Taylor expansion of the mass correction Σ(p) around p=m, where m is the finite renormalized mass, which is arbitrary: )( ~ )()()()( pmmpmp /Σ+Σ′−/+Σ=/Σ where Σ(p)~O(p−m)2 and vanishes for p=m. Since mo is divergent and arbitrary, we will choose mo and m so that mo cancels the divergence coming from Σ(m). We will choose: mmm =Σ+ )(o = + + K+ FS ~ ′ FS Σ Σ Σεimp +−/ 1
  • 102. Inserting the value of Σ(p) into the renormalized electron propagator, we now can rearrange terms to find: where we have defined: 102 2017 MRT iεpmp Zi iεpmpm i pmmpmmp i pSi F +/Σ−−/ = +/Σ−−/Σ′− = /Σ−Σ′−/−Σ−−/ =/′ )( ~~ )( ~ ))]((1[)( ~ )()()( )( 2 o )( ~ )(1 )( ~ )( ~~ )(1 1 )( 22o pZ m p p m Zmmm /Σ= Σ′− /Σ =/Σ Σ′− =Σ+= and, Since the divergence of the complete propagator S′F is contained within Z2, we can remove this term and define the renormalized propagator S′F: ~ )( ~ )( 2 pSZpS FF /′=/′ By summing this subclass of diagrams for the electron self-energy, we have been able to redefine the mass of the electron, such that the physical mass m is actually finite, and also show that the electron propagator is multiplicatively renormalized by the factor Z2: o2 ψψ Z= Next, we analyze the photon propagator Dµν in the same way, summing over infinite classes of one-particle irreducible diagrams: νσ σρ ρµνµνµ DDDD Π−=′