March 2016
PX430: Gauge Theories for Particle Physics
Tim Gershon
(T.J.Gershon@warwick.ac.uk)
Handout 4: Interactions in QED
Scattering in Non-Relativistic Quantum Mechanics – Scalar Interactions with Photons
We have seen that electromagnetic interactions in the Klein-Gordon picture are described by
L̂INT = −iq φ̂† (∂ µ φ̂) − (∂ µ φ̂† )φ̂ µ + q 2 µ µ φ̂† φ̂ .
(1)
Assuming that we are considering particles carrying charges of the same magnitude as that of the electron (i.e. q = e), and knowing the fine-structure constant e2 /4π ' 1/137, in first-order perturbation
theory we are entitled to neglect the second term. Then we find
with
µ
L̂INT = −Jˆem
µ ,
(2)
µ
Jˆem
= ie φ̂† (∂ µ φ̂) − (∂ µ φ̂† )φ̂ ,
(3)
and ĤINT = −L̂INT .
We wish to find the scattering amplitude Af i – the amplitude for an initial state i to scatter
into a final state f . We can begin with the treatment that should be familiar from non-relativistic
quantum mechanics (we will return to the quantum field theoretic treatment later):
Z
Af i = −i d4 x φ∗f {ie (∂µ (Aµ φi ) + Aµ (∂ µ φi ))} ,
(4)
Z
= −ie d4 x Aµ i φ∗f (∂ µ φi ) − (∂ µ φ∗f )φi ,
(5)
Z
(6)
= −ie d4 x Aµ Jfµi .
(nb Beware some inconsistency regarding whether or not the current J contains the charge e.)
Q1 Use integration by parts to obtain Eq. (5) from Eq. (4).
We wish the solutions for φf , φi and Aµ to be stable (we can imagine the initial state particle(s)
coming in from infinity, the interaction taking place, and the final state particle(s) going out to
infinity). Therefore, we choose plane wave solutions, namely
φf ∝ e−ipf .x ,
φi ∝ e−ipi .x ,
Aµ ∝ µ e−ik.x ,
(7)
where µ is the photon polarisation vector. As discussed before, it is convenient to normalise the
wavefunctions such that there are 2E particles per unit volume, but we will not concern ourselves
with this at present. Using the definition of Eq. (3), it is easy to show that
Jfµi = (pi + pf )µ ei(pf −pi ).x ,
(8)
and subsequently that
Z
d4 x ei(pf −pi −k).x ,
(9)
= −ie(2π)4 .(pi + pf )δ 4 (pf − pi − k) .
(10)
µ
Af i = −ieµ (pi + pf )
Q2 Derive Eq. (8) and Eq. (10).
The result of Eq. (10) can be represented diagrammatically, and broken down into components
that can be used to label the diagram:
k
pf
pi
• an external photon line: µ ;
• a vertex factor: −ie(pi + pf )µ ;
• four-momentum conservation: (2π)4 δ 4 (pf − pi − k).
Note that this diagram, while important, does not correspond to any physical process, since it
cannot conserve four-momentum. That is, it is impossible to satisfy all the relations pi .pi = pf .pf =
m2 , pf = pi + k, k.k = 0, where m is the mass of the scattered particle.
Q3 Satisfy yourself that the above set of four-momentum relations cannot simultaneously be satisfied.
Q4 Is it possible to conserve four-momentum by relaxing the requirements, e.g. by (i) pi .pi 6= pf .pf
or (ii) k.k 6= 0? What physical process does each of these two cases correspond to?
We can extend the NRQM treatment to ab → ab scattering. Consider particle a scattering from
a field due to particle b (though
will, unsurprisingly, be symmetric with respect to a and
R 4 the solution
µ
b). In this case Af i = −iea d x Aµ Ja0 a , and, in the Lorenz gauge, ∂ν ∂ ν Aµ = eb Jbµ0 b . Inserting plane
wave solutions as before, we find
Z
iea eb
0
0
Af i =
(11)
(p
+
p
)
.
(p
+
p
)
d4 x ei(pa0 +pb0 −pa −pb ) ,
a
b
a
b
q2
−igµν
= [−iea (pa + pa0 )µ ][ 2 ][−ieb (pb + pb0 )ν ](2π)4 δ 4 (pa0 + pb0 − pa − pb ) ,
(12)
q
where q = pa − pa0 = −(pb − pb0 ).
In this case, the diagram and labelling can be given as:
pb0
pb
q
• vertex factors: −ie(pi + pf )µ ;
• photon propagator:
pa0
−igµν
;
q2
• four-momentum conservation: (2π)4 δ 4 (pa0 + pb0 − pa − pb ).
pa
We can get the scattering amplitudes for other processes, including those involving antiparticles,
by making appropriate substitutions, for example
pā
k
pf
pi
⇔
q
pa
pi
pf
k
⇔
⇔
⇔
pa
−pā
−q
Af i = −ie(2π)4 .(pa − pā )δ 4 (q − pa − pā ) ,
(13)
and
k
pb̄
pf
⇔
q
pb
pi
pf
k
⇔
⇔
⇔
−pb̄
pb
q
pi
Af i = −ie(2π)4 .(pb − pb̄ )δ 4 (pb + pb̄ − q) .
(14)
We can also put together diagrams, for example the previous two given above, to make
pb̄
pā
q
pa
Af i = i
pb
ea eb
(pa − pā ).(pb − pb̄ )(2π)4 δ 4 (pb + pb̄ − pa − pā ) ,
q2
(15)
with q = pb + pb̄ = pa + pā .
Scattering in Non-Relativistic Quantum Mechanics – Fermion Interactions with Photons
Similarly to above, we can apply the NRQM treatment to Dirac particles. In this case
Z
Af i = −ie d4 x Jfµi Aµ ,
(16)
where
Jfµi = ψ̄f γ µ ψi ,
(17)
As before, we consider plane wave solutions for the incoming and outgoing particles
ψf ∝ uf e−ipf .x ,
ψi ∝ ui e−ipi .x ,
(18)
where the four-component vectors uf and ui forming part of the plane-wave solutions for Dirac spinors
have been discussed earlier, and in a complete treatment the normalisation needs appropriate care
(see also below).
Q5 Calculate Af i for a single fermion scattering off a photon.
It can be shown that the vertex factor for the interaction of a Dirac particle with a photon is
−ieūf γ µ ui (to be compared to −ie(pi + pf )µ for the Klein-Gordon case). Consequently we can write
down the two-particle scattering amplitude
pb
pb̄
q
pā
pa
iea eb
(ūa0 γ µ ua )(ūb0 γ µ ub )(2π)4 δ 4 (pa0 + pb0 − pa − pb ) .
(19)
q2
Since it can get tedious to write down explicitly the four-momentum conserving delta functions
each time, it is possible to write instead
Af i =
Af i = Mf i (2π)4 δ 4 (pa0 + pb0 − pa − pb ) .
(20)
The results above have been given in terms of the amplitudes, but as we know physically meaningful quantities (observables) depend on the square of the magnitude of the amplitude. For the
previous example, note that
e2 e2
b
|Mf i |2 = a 4 b Lµν
(21)
a Lµν ,
q
µ
ν
∗
b
∗
where Lµν
a = (ūa0 γ ua )(ūa0 γ ua ) and Lµν = (ūb0 γµ ub )(ūb0 γν ub ) .
In many experiments with fermions, the spins of initial and final state particles will not be
measured. In this scenario (i.e. unpolarised scattering), one should
average over the possible initial
µν
1P
spins, and sum over the possible final states. This gives La = 2 spin (ūa0 γ µ ua )(ūa0 γ ν ua )∗ (the sum
over possible final states is an additional sum over a). The calculation of this term will be discussed
in more detail below.
Finally, in this section, it is worthwhile to note the potential use of Lorentz invariant Mandelstam
variables. These are (in terms of the notation of the previous example)
s = (pa + pā )2 = (pb + pb̄ )2 = q 2 ,
t = (pb − pa )2 = (pā − pb̄ )2 ,
u = (pb̄ − pa )2 = (pā − pb )2 .
Q6 Show that, with a normalisation of 2E particles per unit volume, the result of the previous
example can be written in terms of Mandelstam variables Mf i = i easeb (u − t).
Scattering in Quantum Field Theory
We now return to quantum field theory. The transition amplitude is given by
Z
Af i = −i d4 x hf |ĤINT |ii ,
(22)
µ
µ
with ĤINT = −L̂INT = ĵem
µ and ĵem
= ie φ̂† ∂ µ φ̂ − (∂ µ φ̂† )φ̂ . (We are considering Klein-Gordon
particles for the present.)
Recall that the field operators can be written in terms of their Fourier expansions
Z
i
d3 p h
−ip.x
†
+ip.x
φ̂ =
â(p)e
+
b̂
(p)e
,
(23)
(2π)3 2E
Z
i
d3 p h †
†
+ip.x
−ip.x
φ̂ =
â
(p)e
+
b̂(p)e
,
(24)
(2π)3 2E
Z
i
d3 k X h
µ −ik.x
†
∗µ +ik.x
µ
 =
α̂λ (k)λ e
+ α̂λ (k)λ e
,
(25)
(2π)3 2ω
λ
where µλ is the polarisation vector for a photon with polarisation λ, and α̂λ (k) and α̂λ† (k) are
annihilation and creation operators, respectively, for a photon with polarisation λ and momentum k.
What interactions are possible with the matrix element of Eq. (22)? Since each of the field
operators can either annihilate or create a quanta, we can simply tabulate all possibilities.
Particle
A
C
A&C
–
A&C
–
Antiparticle
A
C
–
A&C
–
A&C
Photon
C
A
A
A
C
C
Possible interactions described by the matrix element of Eq. (22). A
represents annihilation and C represents creation. Other apparent possibilities are ruled out by conservation laws.
Q7 Draw diagrams corresponding to each of the processes tabulated above.
Q8 Explain why the following processes are forbidden by the matrix element of Eq. (22).
Particle
A&C
A
C
Antiparticle
–
C
C
Photon
A&C
A
C
Forbidden processes.
Let us concentrate on one specific process, namely that in which a particle scatters of an incoming
photon, as given by the diagram below.
k
pf
pi
As discussed previously, this case applies to charged scalar particles, and no such fundamental
particle is known to exist. However, if we ignore the quark substructure of the charged pion, this is
the appropriate formalism in that system. Thus, our initial state |ii contains a charged pion (π + (pi ))
and a photon (γλ (k)). Consequently, we can write
|ii = |pi ; k, λi = ↠(pi )α̂λ† (k)|0i ,
(26)
where |0i is the vacuum state containing no pions and no photons. Similarly,
|f i = |pf i = ↠(pf )|0i ,
and so
Z
Af i = −ih0|â(pf )
4
d
µ
x ĵem
(x)µ (x)
(27)
↠(pi )α̂λ† (k)|0i .
(28)
We can separate the pion and photon parts of this equation, since the relevant operators commute.
The photon part can be shown to be
h0|µ (x)α̂λ† (k)|0i = µ e−ik.x ,
(29)
and the pion part can be shown to be
µ
h0|â(pf )ĵem
(x)↠(pi )|0i = eπ (pi + pf )µ e−i(pi −pf ).x .
(30)
Q9 Prove Eq. (29). It will help to use the commutation relation for photon annihilation and
creation operators: [α̂λ (k), α̂λ† 0 (k0 )] = (2π)3 (2ω)δλλ0 δ 3 (k − k0 ).
Q10 Prove Eq. (30).
At this stage, the connection between the non-relativistic quantum mechanical treatment given
earlier and the full quantum field theoretic version becomes apparent.
Z
Af i = −i
Z
= −i
Z
= −i
d4 x eJfµi Aµ ,
(31)
µ
d4 x hpf |ĵem
(x)|pi ih0|µ (x)|k, λi ,
(32)
µ
(x)↠(pi )|0ih0|µ (x)α̂λ† (k)|0i .
d4 x h0|â(pf )ĵem
(33)
This connection between the “wavefunction” amplitude of NRQM and the “field theory” amplitude of QFT is extremely powerful, as it allows us to simplify many problems, and to use Feynman
rules to label diagrams and calculate amplitudes in QFT.
• incoming photon: µ (outgoing photon: ∗µ );
k
• vertex factor: −ie(pi + pf )µ ;
pf
• four-momentum conservation: (2π)4 δ 4 (pf − pi − k).
pi
• (nb normalisation factors omitted here.)
Second Order Processes in Quantum Field Theory
Understanding the above diagram provides us with a solid foundation to build upon, but it does
not yet describe any physics process. To do so, we need to move to second order processes, i.e.
diagrams with two vertices.
Avoiding the details of perturbation theory, the second order term gives something like
Z
XZ
(2)
4 0
0
Af i ∼ −i
d x hf |ĤINT (x )|ni d4 x00 hn|ĤINT (x00 )|ii
(34)
n6=i
µ
where, as before, ĤINT = −L̂INT = ĵem
µ creates a particle/annihilates an antiparticle, annihilates
a particle/creates an antiparticle and creates or annihilates a photon.
Considering ab → ab scattering (where a and b are pointlike charged pions, as before), then we
have |ii = |pa , pb i and |f i = |pa0 , pb0 i. The possible diagrams include that given below, together with
the Feynman rules, which as before can be obtained from the full quantum field theoretic treatment,
and related to the results obtained using non-relativistic quantum mechanics:
pb
• vertex factors: −ie(pi + pf )µ ;
pb0
• photon propagator: −igµν /q 2 ;
q
• four-momentum conservation: (2π)4 δ 4 (pa0 + pb0 − pa − pb ).
pa0
• (nb normalisation factors omitted here.)
pa
Q11 Derive Af i for the second order process given in the diagram above from the QFT amplitude,
and satisfy yourself that it is the same as that obtained using Feynman rules.
It is important to note that the above diagrams includes both the case where particle a emits a
photon that is later absorbed by particle b, and the case where it is b that emits the photon and a that
absorbs it. That it to say, the diagram includes both time-orderings. This is a subtle but profound
point: within quantum field theory, we can consider the following diagrams for Compton scattering
as part of the same process (in fact, these are both described by the amplitude of Eq. (34)). In the
diagram on the right, an incoming pion (positively charged) annihilates with a photon, an antiparticle
(negatively charged pion) travels back in time to a point where the outgoing pion (positively charged)
is created together with the outgoing photon. Quantum field theory naturally handles such processes,
taking all possible intermediate states into account.
γ
γ
γ
π+
π+
π+
+
π− π
π+
γ
nb as a second order QED process, the matrix elements for the Compton scattering diagrams
above are given by Eq. (34); note, however, that the initial and final states are different to those that
were being discussed previously, and that will be discussed again below.
Renormalisability
If we carefully list all the diagrams that are described by the amplitude of Eq. (34), we will
find some containing disconnected lines, such as those shown below. In the diagram on the left, the
photon is emitted and absorbed by the same particle, while the other particle in the event is simply
a spectator. In the diagram on the right, particle, antiparticle and photon all appear out of, and
disappear back into, the vacuum. The presence of such diagrams appears to present a problem: as
the momenta of the virtual particles are unconstrained, there will be a large (potentially infinite)
contributions when integrating over all possibilities. Furthermore, there will be more and more of
these diagrams when we go to higher orders in perturbation theory. All these processes are legitimate
within quantum field theory, so how can they be understood?
The answer lies within the concept of renormalisability. The details of this concept, and its
mathematical proof, are far beyond the scope of this module, but we can briefly understand the
main point. In essence, we redefine the vacuum to include all vacuum loops, and we redefine our
particles similarly – these definitions then correspond to the physical vacuum and to physical particles
that we observe and measure. When we encounter contributions such as those discussed above in
calculations of amplitudes within quantum field theory we can neglect them, safe in the knowledge
that they have already been included.
Although we do not discuss the details of renormalisability, it is extremely important for the viability of gauge theory: Feynman, Schwinger and Tomonaga’s Nobel prize (1965) was awarded largely
because the proof of the renormalisability of QED put it on a solid theoretical footing, and t’Hooft
and Veltman were awarded the 1999 Nobel prize in physics for demonstrating the renormalisability
of electroweak theory (following the award of the 1979 prize to Glashow, Salam and Weinberg for
the development of electroweak theory – we will discuss this interesting topic all too briefly later).
The requirement of renormalisability also limits the form of the terms entering the Lagrangian
density of any theory, as only terms with dimensionless coupling constants, or couplings with positive
mass dimension, can be included in a renormalisable theory. A Lagrangian density that contains
terms with couplings with dimensions mass−n (n > 0) is possible but must represent an effective
theory that is only valid up to some mass scale. The Fermi theory of the weak interaction, which
contains a four-fermion coupling with associated constant of dimensions mass−2 , is a good example
of an effective theory.
Additional Considerations for QED Interactions of Bosons
The above discussion was relevant for ab → ab interactions, where a and b are charged, spinless,
distinguishable particles. For example, this could represent π + K + scattering, where both pions
and kaons are treated as pointlike particles (i.e. neglecting their quark content). If we wish to
consider π + π + scattering, we must take into account the fact that the particles are indistinguishable,
and therefore the amplitude must be symmetric under their exchange. That is, we must add the
following amplitudes (recall that we add the amplitudes and then square to obtain observable rates).
pb0
pb
pa0
pb
+
q
q
pa0
pb0
pa
pa
For the case of particle-antiparticle scattering, e.g. π + π − scattering, there are additional processes
that are allowed in quantum field theory.
pπ −
p0π−
q
q
p0π+
pπ +
“exchange amplitude”
p0π−
pπ −
pπ +
p0π+
“annihilation amplitude”
Using Feynman rules we can calculate the amplitudes for these diagrams. For example, the
exchange amplitude has vertex factors −ie(pπ+ + p0π+ )ν and −ie(pπ− + p0π− )µ together with a photon
propagator −igµν /q 2 where q = p0π+ − pπ+ = pπ− − p0π− . It is similarly straightforward to write down
the amplitude for the annihilation amplitude (see question below). However, there are also resonant
enhancements that should be taken into account in the full treatment of the annihilation channel –
these are not discussed in this module.
Q12 Write down the amplitude for the annihilation amplitude.
Q13 Consider the diagrams for Compton scattering shown on the previous page. Write down expressions for the inital and final states, in terms of relevant creation operators acting on the
vacuum. By expanding the interaction Hamiltonian, derive the Feynman rule for the boson
propagator.
Fermion Interactions in Quantum Field Theory
The above discussion has focussed on scalar particles, to avoid complications arising from spin.
Since fermions are quite important in nature, let us briefly consider their interactions. The Dirac
interaction Lagrangian is L̂INT = −ĵfµi µ with ĵfµi = eψ̄ˆf γ µ ψ̂i .
It will be worthwhile to briefly recap some basics of Dirac spinors. These are solutions to the
Dirac equation:
(iγ µ ∂µ − m)ψ = 0 ,
(35)
We can solve this equation by looking for (positive energy) plane wave solutions of the form ψ ∼
ωe−ip.x , and applying this to a form of the Dirac equation where space- and time-like parts have been
separated (i ∂ψ
∂t = (−iα.∇ + βm)ψ – refer to earlier handouts for definitions of α, β and σ), we find
Eω = (α.p + βm)ω ,
that is
E
ω+
ω−
=
m σ.p
σ.p m
(36)
ω+
ω−
,
from which we can find the unnormalised positive-energy solutions
φ1,2
1,2
.
ω+ ∝
σ.p
1,2
(E+m) φ
(37)
(38)
To fix the normalisation to the Lorentz invariant choice of 2E particles per unit volume, we require
ρ = ψ̄γ 0 ψ = ψ † ψ = 2E, and it can be shown that this is satisfied with the choice
φs
s
1/2
ω+ = u(p, s) = (E + m)
.
(39)
σ.p
s
(E+m) φ
Similarly, the negative energy solutions can be found to be
σ.p s s
1/2
(E+m) χ
ω− = v(p, s) = (E + m)
.
χs
(40)
The two-component vectors φs and χs describe
and antiparticles
spin-states of particles
the
1
0
0
↓
↑
respectively. It is conventional to choose φ =
and φ =
, with χ↑ =
and
0
1
1
1
χ↓ =
.
0
Q14 Refamiliarise yourself with Dirac spinors by working through the steps above.
Q15 Show explicitly that u(p, s) and v(p, s) are normalised to 2E particles per unit volume (i.e. calculate u† (p, s)u(p, s) and v † (p, s)v(p, s)). What are the values of ū(p, s)u(p, s) and v̄(p, s)v(p, s)?
With these results fresh in the mind, we can find for electrons (positive energy and negative
charge)
0
ˆf γ µ ûi e−i(k−k ).x ,
ĵfµi = (−e)ψ̄ˆf γ µ ψ̂i = (−e)ū
(41)
where we have labelled incoming and outgoing electron momenta with k and k 0 respectively (to
distinguish them from the pion momenta which will appear below).
Now, consider the case of π + e− scattering. We can just take Aµ due to π + (we will revert to the
NRQM treatment, and drop the hats, for simplicity). Then (labelling incoming and outgoing pion
momenta with p and p0 respectively)
Aµ = −
1
0
e(p + p0 )µ e−i(p−p ).x ,
2
q
(42)
and so
Z
4
µ
−i(k−k0 ).x
1
0
− 2 e(p + p0 )µ e−i(p−p ).x ,
q
ˆf γ ûi e
d x (−e)ū
gµν
µ
ˆ
= −i(−e)ūf γ ûi − 2 (+e)(p + p0 )ν (2π)4 δ 4 (k 0 + p0 − k − p) .
q
Af i = −i
(43)
(44)
The Feynman rules for the photon-scalar vertex and for the photon propagator remain unchanged,
and we have obtained new rules for external fermion lines and for the fermion-photon vertex:
π + (p0 )
π + (p)
q
e− (k 0 )
e− (k)
• photon-scalar vertex: −ie(p + p0 )µ ;
• photon-fermion vertex: −i(−e)γ ν ;
• incoming fermion: u(k, s) (outgoing fermion: ū(k 0 , s0 ));
• photon propagator: −igµν /q 2 ;
• four-momentum conservation: (2π)4 δ 4 (k 0 + p0 − k − p).
For external antifermions, u(k, s) and ū(k 0 , s0 ) are replaced by v(k, s) and v̄(k 0 , s0 ) respectively.
Note that the external fermions carry spin labels, and that the amplitude corresponds to specific
initial and final state spins s and s0 . To calculate an unpolarised cross-section, it is necessary to
average the initial spins, and sum over the possible final state spins, as mentioned previously.
Calculation of Unpolarised Cross-sections – Completeness Relations and Trace Techniques
As mentioned above, to calculate an unpolarised cross-section, we need to evaluate
Lµν (k 0 , k) =
1X
(ūa0 (k 0 )γ µ ua (k))(ūa0 (k 0 )γ ν ua (k))∗ .
2 0
(45)
a, a
To progress we first note that taking the complex conjugation of the second term is equivalent to
taking the Hermitian conjugate, since for each value of ν, ūa0 (k 0 )γ ν ua (k) is just a number. Thus
(ūa0 (k 0 )γ ν ua (k))∗ = (ūa0 (k 0 )γ ν ua (k))† = u†a (k)γ ν† γ 0† ua0 (k 0 ) = ūa (k)γ ν ua0 (k 0 ) ,
(46)
since γ 0† = γ 0 and γ 0 γ ν† γ 0 = γ ν . Hence we find that
1X
Lµν (k 0 , k) =
ūa0 (k 0 )γ µ ua (k)ūa (k)γ ν ua0 (k 0 ) .
2 0
(47)
a, a
We now focus on the central part of this expression (the parts that depends on a), which is
X
ua (k)ūa (k) .
(48)
a
This is a sum of column vectors multiplied by row vectors, which may be somewhat unfamiliar, but
clearly gives a matrix as the result. Using the forms for the spinors given in Eq. (39) (with a small
and obvious change in notation), we can see that
!
X
X
φa
(σ.k)† †
(49)
ua (k)ūa (k) =
(E + m)
φ†a − (E+m)
φa ,
σ.k
(E+m) φa
a
a
!
(σ.k)†
X
φa φ†a
− (E+m)
φa φ†a
,
(50)
=
(E + m)
(σ.k)2
†
†
σ.k
a
(E+m) φa φa − (E+m)2 φa φa
† 0
where we have used the
that ūa =ua γ .
fact
0
1
, we obtain
and φ↓ =
Now, using φ↑ =
1
0
X
1 0
0 0
1 0
0
1
†
0 1 =
1 0 +
.
=
+
φa φa =
0 1
0 1
0 0
1
0
(51)
a
Since (σ.p)† = σ.p and (σ.p)2 = p2 = (E + m)(E − m), we then find
X
m 0
E −σ.p
E+m
−σ.p
= γ µ kµ + m .
+
=
ua (k)ūa (k) =
0 m
σ.p −E
σ.p
−(E − m)
(52)
a
This is the completeness relation:
X
ua (k)ūa (k) = k/ + m .
(53)
a
It is interesting to note what happens when we left-multiply Eq. (53) by (k/ − m)
X
(k/ − m)
ua (k)ūa (k) = (k/ − m) (k/ + m) = k 2 − m2 = 0 ,
(54)
a
which obviously holds because (k/ − m) ua (k) = 0 is the Dirac equation. Eq. (54) in fact provides an
alternative way of deriving the completeness relation.
Q16 Satisfy yourself with all steps in the derivation of the completeness relation above.
P
Q17 Show that the corresponding completeness relation for the antiparticle spinors is a va (k)v̄a (k) =
k/ − m.
Now we can return to the expression for the unpolarised cross-section
1X
Lµν (k 0 , k) =
ūa0 (k 0 )γ µ ua (k)ūa (k)γ ν ua0 (k 0 ) ,
2 0
(55)
a, a
=
1X
ūa0 (k 0 )γ µ (k/ + m)γ ν ua0 (k 0 ) ,
2 0
a
(56)
In this expression, ūa0 (k 0 ) is a row vector, γ µ , (k/ + m) and γ ν are matrices and ua0 (k 0 ) is a column
vector. Let us make their multiplication explicit through the inclusion of repeated indices (rewriting
ūa0 (k 0 ) as ū(k 0 , a0 ) for a clearer notation):
Lµν (k 0 , k) =
1X
µ
ν
ūα (k 0 , a0 )γαβ
(k/ + m)βγ γγδ
uδ (k 0 , a0 ) .
2 0
(57)
a
The advantage of writing the expression in this form is that now each element is simply a matrix
element, i.e. a number, so we can reorder the terms as we wish. In particular, we can do the second
sum over spins:
X
uδ (k 0 , a0 )ūα (k 0 , a0 ) = (k/0 + m)δα ,
(58)
a0
so that
1
1X
µ
ν
(k/0 + m)γ µ (k/ + m)γ ν δδ .
Lµν (k 0 , k) = (k/0 + m)δα γαβ
(k/ + m)βγ γγδ
=
2
2
(59)
δ
This expression is a trace:
1
Lµν (k 0 , k) = Tr (k/0 + m)γ µ (k/ + m)γ ν .
2
(60)
We can now use some of the general results for trace theorems discussed in Relativistic Quantum
Mechanics (see in particular handout 2 for that module). Writing
(61)
Tr (k/0 + m)γ µ (k/ + m)γ ν = Tr(k/0 γ µ k/γ ν ) + mTr(γ µ k/γ ν ) + mTr(k/0 γ µ γ ν ) + m2 Tr(γ µ γ ν ) ,
we can use the fact that the trace of any odd number of gamma matrices is zero to make the terms
linear in m vanish, while the m2 term gives 4g µν (finding the expression for the first term is left as
an exercise for the reader). Finally, we obtain
1
Lµν (k 0 , k) = Tr (k/0 + m)γ µ (k/ + m)γ ν = 2 k 0µ k ν + k 0ν k µ + (m2 − (k 0 .k))g µν .
2
(62)
Q18 Satisfy yourself with all steps in the derivation of the expression for Lµν above.
Q19 Using trace theorems, find a simplification of Tr(k/0 γ µ k/γ ν ).
Q20 Show that L00 = 4E 2 (1 − v 2 sin2 θ/2) where cos θ = k.k0 /(|k| k0 ) and v = |k| /E.
Feynman Rules for Quantum Electrodynamics
We have almost derived the full set of Feynman rules for interactions in Quantum Electrodynamics. Those that we have omitted are related to the possibility to have internal scalar and fermion lines
(i.e. particles that are created and then annihilated within the interaction – these are called propagators). Finally, there is an additional term (q 2 µ µ φ̂† φ̂) in the Klein-Gordon interaction Lagrangian
that we have conveniently neglected until now but must include in any second- or higher-order treatment.
The complete set of QED Feynman rules is then (recall that p/ = γµ pµ ):
External particles
– Incoming fermion or antifermion
Outgoing fermion or antifermion
– Incoming photon
Outgoing photon
Propagators
– Spin-0
u(p, s) or v(p, s) ;
ū(p0 , s0 ) or v̄(p0 , s0 ) ;
µ (k, λ) ;
∗µ (k 0 , λ0 ) ;
–
i
p2 −m2
Spin- 12
i
p/−m
– Spin-1
Vertices
– Spin-0
–
=i
;
p/+m
p2 −m2
−ig µν
;
k2
;
−ie(p + p0 )µ ;
2ie2 gµν ;
−ieγµ .
Spin- 21
The diagrams below show the Feynman rules for the QED propagators.
i
p2 −m2
µν
i
−i gq2
p/−m
The diagrams below show the Feynman rules for the QED vertices (for positively charged particles).
µ
p
p0
−ie(p + p0 )µ
µ
ν
p
p0
2ie2 gµν
µ
p
p0
−ieγµ
The most observant may notice a slight difference in the forms of the propagators compared to
those given in some
textbooks: one often finds the spin-0, spin- 21 and spin-1 propagators written as
i p
/+m
µν
i
,
, and −ig
, respectively, where is a small positive constant. [The representation
p2 −m2 +i p2 −m2 +i
k2 +i
of the spin-1 propagator is gauge-dependent, and several other forms are quite often used in the
literature.] Experts in complex analysis will recognise these as being related to Green’s functions
(the spin-0 and spin- 12 propagators are related to Green’s functions for the Klein-Gordon equation
and the Dirac equation), and that the i terms are necessary to specify how to avoid the pole in the
complex plane. Non-experts can simply ignore this subtlety.