Problems to Standard Model - Theoretische Physik 1 ...

Problems to Standard Model SS 2011
Th. Mannel, T. Huber, S. Faller
Sheet 7 — Handout: 27.05.2011 — Due: Friday, 03.06.2011
Problem 22: The -Model
The so called -model for four real scalar fields , j .j D 1; 2; 3/ and two Dirac-fields 1 , 2 is
defined through the Lagrangian density
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ˇ D 1 2
3X
.@ j /.@ j / C 1 2X
2 .@ /.@ / C
j D1
j D1
N j i =@ j V .; i / ˇint ;
where
V .; i / D 1 2 2 2 C
ˇint D g
2X
j;lD1
3X
j D1
2 j
N j
ı jl C i
C
2 C
4
3X
2
j 2 ; with 2 < 0 ; > 0 ;
j D1
3X
k . k / jl 5
kD1
l ; k : Pauli matrices :
a) Prove the invariance under the infinitesimal global SU.2/ symmetry transformations,
7 ! ;
j 7 ! j C jkl˛k l ;
2X
i
3X
k 7 ! ı kl ˛j . j / kl
2
lD1
j D1
l ;
and calculate the associated Noether currents.
b) Verify that the Lagrangian density is also invariant under the global transformations,
3X
7 ! C ˛j j ;
j D1
j 7 ! j ˛j ;
2X
k 7 ! ı kl C i 2
lD1
3X
˛j . j / kl 5
j D1
l :
Compute the corresponding Noether currents.

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!

The symbol T Œ: : : denotes the time-ordered product, which means
˝0
ˇˇT ˛.x/ N ˇ .y/ ˇˇ 0˛
D
˝0
ˇˇ ˛.x/ N ˇ .y/ˇˇ 0˛
x
0
y 0 ˝0 ˇˇ N ˇ .y/ ˛.x/ˇˇ 0˛
y
0
x 0 ;
where the minus sign is due to the fact that the fermion fields anticommute.
Show that this leads to the Dirac propagator in the following form,
˝0
ˇˇT
˛.x/ N ˇ .y/ ˇˇ Z
0˛
D
d 4 p
.2/ 4
i QG.p/ ˛ˇ e ip.x y/ ;
where i QG.p/ is the Dirac propagator in momentum space as given in the lecture.
Hint:
Use the following representation for the -step function
.z/ D 1 Z 1
d eiz
2i 1 i :
b) Show that the annihilation operators a.p; r/ and b.p; r/ can be written as the following integrals
over the field .x/.
Z
a.p; r/ D d 3 x u ˛.p; r/ ˛.x/ e ipx ;
Z
b.p; r/ D d 3 x N ˛.x/ 0˛ˇ vˇ .p; r/ e ipx :