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