Problems to Standard Model - Theoretische Physik 1 ...

Problem 23: Two Particle Phase Space
a) Show that
Z
d 3 Z
k d 3`
.2/ 3 2k 0 .2/ 3 2`0
.2/ 4 ı .4/ .q k `/ D 1
8q 2 q2 ; m 2 k ; m2` q
0 q 2 .m k C m`/ 2 ;
where m 2 1 ; m2 2 ; q
m2 3 D m 4 1 C m4 2 C m4 3
2m 2 1 m2 2
2m 2 1 m2 3
2m 2 2 m2 3 .
Compute the limiting cases
Hint:
i) m k D m ; m` D m ;
ii) m k D 0 ; m` D m ;
iii) m k D 0 ; m` D 0 :
Use the result of problem 9b).
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b) Now use this result, to compute the vector and tensor integrals
Z
d 3 Z
k d 3`
.2/ 3 2k 0 .2/ 3 k˛.2/ 4 ı .4/ .q k `/ and
2`0
Z
d 3 Z
k d 3`
k˛`ˇ .2/ 3 2k 0 .2/ 3 .2/ 4 ı .4/ .q k `/ :
2`0
For simplicity put the masses m` D m k D 0.
Hint:
Show that the integral has to be covariant under Lorentz transformations. Which Lorentz
vectors and tensors are available to construct the needed Lorentz structure?
Make a general Ansatz, and determine the coefficients by appropriate projections.
Problem 24: The Dirac Field
Consider the Dirac field, as given in the lecture
˛.x/ D X Z
d 3 p
h
.2/ 3 2p 0 a.p; s/u˛.p; s/e ipx C b .p; s/v˛.p; s/e ipxi
sD˙ 1
2
N ˇ .x/ D X
sD˙ 1
2
Z
d 3 p
h
.2/ 3 2p 0 a .p; s/ Nuˇ .p; s/e ipx C b.p; s/ Nvˇ .p; s/e ipxi :
a) An alternative way to compute the propagator of the Dirac field is by considering the following
matrix element,
˝0
ˇˇT
˛.x/ N ˇ .y/ ˇˇ 0˛
:
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please turn over!