Niels Tuning (1)

Exercises – Spin in decay: helicity

1) Decay topology: consider the process 6027Co60

28Ni+e-+νe where the e and theν are emitted along the polarization (z) axis. (See illustration.) We are now interested in the orientation of the spin of the electron/anti-neutrino w.r.t to the direction of flight.

a) Show that there are two possibilities for decays along the 60Co spin axis:

• one where the electron spin points to the same direction as the direction of motion and

• one where the electron spin points to the opposite direction of the direction of motion

b) What are the helicities of the electron and anti-neutrino for the twoemission possibilites, respectively.

(Helicity is defined as: )

S=4

S=1/2

S=1/2

e-

νe

Ni

S pH

S p

3333333333333333333333333333

Niels Tuning (2)

Exercise – Pion decay: π-→ l-νl

2) Given a - (spin 0) in rest that decays into a - and a ν (both spin ½)

a) Draw the decay for both spin orientation possibilities

b) What is the helicity of the anti-neutrino for both possibilities?

c) Which of the two processes does not occur in nature?

d) Draw the C-conjugated diagram of the + decay that occurs in nature. Does the C-conjugated process occur in nature? Explain why (not)?

e) Draw the C and P conjugated diagram of the - decay that occurs in nature. Does this conjugated decay occur in nature?

f) Can the following decay occur: π-→ e-νe ?

g) The (V-A) theory predicts that the decay rate Γ(π-→ l-νl ) is proportional to the two-body phase space factor multiplied by (1-v/c) , where v is the speed of the lepton in the pion rest frame. Derive the phase space factor starting from p2dp/dE (ie. the number of states per unit energy, or d3p/dE), where p is the lepton momentum and E the total energy in the pion rest frame (Burcham&Jobes).

h) Estimate the ratio of branching ratios: Γ(π-→ e-νe ) /Γ(π-→ μ-νμ)

Niels Tuning (3)

Exercise – Pion decay: π-→ l-νl

2) Given a - (spin 0) in rest that decays into a - and a ν (both spin ½)

g’) Alternatively, express the decay rate in terms of the masses of the pion, lepton and neutrino, given that the decay rate is given by Γ~ |M2| x phase space:

• with:

(With |p| the momentum of either outgoing particle in the pion rest frame)

• and:

3 34 2 21 2

1 23 3 21 2

1(2 ) ( )

(2 ) 2 (2 ) 2

pd p d pp p p M M

m E E m

4

2 2 51

1(1 )

8W

W

gM f p p Tr p

M

2( p 2

4

2 21 2 1 2

)

12( )( ) ( )

8W

W

m

gf p p p p p p p

M

Niels Tuning (4)

Exercises – CP Eigenstates of the K0

3) Consider K1 and K2 :• |K1> = 1/2(|K0> - |K0>)

• |K2> = 1/2(|K0> + |K0>)

a) Prove that K1 and K2 are eigenstates of the CP operation and determine the eigenvalues of K1 and K2.

b) Given CP conservation, how do K1 and K2 decay?

c) Why is the lifetime of K2 so much larger? (mπ=140 MeV, mK=490MeV)

In the experiment of Cronin and Fitch, a beam of almost only K2 particles is produced from a beam K0 particles by placing the experiment 20 meters away from the K0 production point so that most K1 particles will have decayed.

The distribution of K1 decay distances is an exponential distribution so even after 20 meters there will be a few K1 left. It is important to know how many such decays there are as a K1 decay in the detector is indistinguishable from a K2 decaying (with CP violation) into two pions.

d) Given that the decay time of the K2 is 5.2 x 10-8 sec, what fraction of the K2 has not decayed yet after 20 meters (assume v c). (Use: mK=498 MeV, <EK>=1245 MeV)

e) Given that the decay time of the K1 is 1 = 0.89 x 10-10 sec, what fraction of the K1 has not decayed yet after 20 meters

f) What is the ratio of K1 to K2 decays after 20 meters

g) How many K0 particles need to be produced to measure 22700 K2 decays between 20 and 21 meters of flight with a 1% detection efficiency for particles decaying in that one-meter stretch?

Niels Tuning (5)

Exercise – Weak decay diagrams

4) Feynman diagrams for weak decays

a) Draw the diagram for π+ decay. • Indicate the CKM matrix element labels at each weak vertex

b) Draw the diagram for 0 pee decay

• Indicate the CKM matrix element labels at each weak vertex

c) Draw two (topologically different) diagrams for the K0 K0 transition

• Indicate the CKM matrix element labels at each weak vertex

• Both diagram may have at most 4 vertices

• Hint: One was shown in the course material.