10. Leptons and Quarks

Question 1

What is a Feynman diagram?

Answer 1

A convenient way to represent particle interactions.

Question 2

What are the 3 conventions/requirements for a Feynman diagram?

Answer 2

• Time runs from left to right.
• An arrow directed towards the right is a particle, one directed towards the left is an anti-particle.
• At each vertex, momentum and charge are conserved (along with other quantum numbers). This means that arrows should be continuous at each vertex.

Question 3

______-vertex interactions violate energy conservation.

Answer 3

Single

Question 4

Describe the Feynman diagram for e- photon emission

Answer 4

Question 5

Describe the Feynman diagram for e+ photon emission

Answer 5

Question 6

Describe the Feynman diagram for e- photon absorption

Answer 6

Question 7

Describe the Feynman diagram for e+ photon absorption

Answer 7

Question 8

Describe the Feynman diagram for e+, e- annihilation

Answer 8

Question 9

Describe the Feynman diagram for e+, e- creation

Answer 9

Question 10

_________ is conserved at each vertex of a Feynman diagram.

Answer 10

Momentum

Question 11

State the full relativistic expression for energy

Answer 11

E = energy
p = momentum
c = speed of light
m = mass

Question 12

Why can’t exchange particles occur in isolation?

Answer 12

Because energy is not conserved for a single vertex interaction involving an exchange particle, hence, another vertex is required so that energy conservation is only violated for a short period.

Question 13

What is a lowest order Feynman diagram?

Answer 13

A Feynman diagram containing two vertices. It is the lowest amplitude real process.

Question 14

What is the difference between these two electron scattering lowest order diagrams?

Answer 14

The time ordering is different for the two.

Question 15

Why are Feynman diagrams drawn with implied time orderings rather than drawing all the possible time orderings?

Answer 15

Because it is inconvenient to draw all variations so the time orderings are suppressed.

Question 16

When is it not possible to suppress the time-ordering of a Feynman diagram?

Answer 16

When W± exhange particles are involved as the time ordering fixes the charge.

Question 17

Describe the basic vertex Feynman diagram resulting from electromagnetic interactions for all leptons in the standard model

Answer 17

Question 18

What is lepton universality?

Answer 18

The coupling constant is the same for each of the 3 generations of lepton because there isn’t much difference between each of the lepton generations apart from their mass.

Question 19

Give the equation for the electron lepton number

Answer 19

Question 20

Give the equation for the muon lepton number

Answer 20

Question 21

Give the equation for the tau lepton number

Answer 21

Question 22

Are lepton numbers conserved in the standard model?

Answer 22

Yes

Question 23

Are lepton numbers conserved at each vertex of a Feynman diagram?

Answer 23

Yes

Question 24

Is there any experimental evidence for the violation of lepton number?

Answer 24

No

Question 25

Give the equation for the experimental probability of lepton number violation

Answer 25

P = probability The limit is based on the number of observed decays experimentally.

Question 26

Describe the basic vertex Feynman diagram resulting from weak Z0 interactions for all leptons in the standard model

Answer 26

These interactions conserve charge and lepton number at each vertex.

Question 27

Describe the basic vertex Feynman diagram resulting from weak W± interactions for all leptons in the standard model

Answer 27

The time ordering has been left implied as it specifies the charge. If the diagrams contained W+ it would have to be incoming to the vertex. If the diagrams contained W- it would have to be outgoing from the vertex. Lepton number and charge are conserved at each vertex.

Question 28

Describe the lowest order Feynman diagram for muon decay

Answer 28

This interaction is purely leptonic.

Question 29

What can the strength of the weak interactions be approximated as at low energies?

Answer 29

Question 30

How does the Fermi coupling constant change low energy Feynman diagrams?

Answer 30

Diagrams involving W and Z can be approximated by a single vertex with the Fermi coupling constant in the middle.

Question 31

Give the equation for the relation between the Fermi coupling constant and the mass of an exchange particle, M_W

Answer 31

G_F = Fermi coupling constant c = speed of light α_W = coupling strength constant M_W = exchange particle mass

Question 32

Why is the weak interaction comparable to the electromagnetic interaction?

Answer 32

They both have a constant that determines the coupling strength. The only difference is that the electromagnetic interaction is stronger. α_W = weak interaction coupling constant α = electromagnetic interaction coupling constant

Question 33

Give the equation for the strangeness of a particle

Answer 33

Question 34

Give the equation for the charm of a particle

Answer 34

Question 35

Give the equation for the beauty of a particle

Answer 35

The tilde distinguishes beauty from the baryon number.

Question 36

Give the equation for the truth of a particle

Answer 36

Question 37

How are the conservation laws for up and down quarks defined?

Answer 37

As a linear combination of the individual quark flavours.

Question 38

Give the equation for baryon number

Answer 38

Question 39

What is the baryon number of a baryon (3 quarks)?

Answer 39

1

Question 40

What is the baryon number of an anti-baryon (3 anti-quarks)?

Answer 40

-1

Question 41

What is the baryon number of a mason (quark and anti-quark)?

Answer 41

0

Question 42

Give the equation for the baryon number in terms of individual quark flavours

Answer 42

B = baryon number N_u = number of up quarks N_d = number of down quarks N_s = number of strange quarks N_c = number of charm quarks N_b = number of bottom quarks N_t = number of top quarks

Question 43

Give the equation for the charge of a particle in terms of individual quark flavours

Answer 43

Q = charge N_u = number of up quarks N_d = number of down quarks N_s = number of strange quarks N_c = number of charm quarks N_b = number of bottom quarks N_t = number of top quarks

Question 44

Which interactions conserve individual quark flavour numbers?

Answer 44

Strong and electromagnetic interactions

Question 45

Which interactions do not conserve individual quark flavour numbers?

Answer 45

Weak interactions: they only conserve baryon number and charge

Question 46

Describe the basic vertex of a Feynman diagram for charged current interactions with leptons

Answer 46

The charge on W is left unspecified because it depends on the time ordering. g_W is the coupling strength constant for W exchange.

Question 47

State the relationship between the W exchange coupling constant and the coupling strength of a charged current interaction

Answer 47

α_W = coupling strength of a charged current interaction g_W = exchange coupling constant

Question 48

Describe the zero-range approximation for muon decay

Answer 48

Question 49

What are natural units?

Answer 49

When ℏ = c = 1 so that all quantities can be expressed as a power of energy E.

Question 50

Give the equation for dimensional analysis

Answer 50

Question 51

What are the dimension of the Fermi coupling constant?

Answer 51

Question 52

What are the dimensions of the decay rate (inverse time)?

Answer 52

Question 53

What are the dimensions of mass?

Answer 53

Question 54

Give the equation for muon decay

Answer 54

Question 55

Give the equation for tau decay

Answer 55

Question 56

How can lepton universality be proven?

Answer 56

By showing that the ratio between muon and tau decay is equal to unity as K is the same for both decays. This is experimentally shown to be true to high precision.

Question 57

What quark is the lepton v_e identical to?

Answer 57

Up

Question 58

What quark is the lepton e- identical to?

Answer 58

Down

Question 59

What quark is the lepton v_µ identical to?

Answer 59

Charm

Question 60

What quark is the lepton µ- identical to?

Answer 60

Strange

Question 61

What is the quark structure of π-?

Answer 61

Question 62

Give the equation for π- decay

Answer 62

Question 63

Why is the π- decay vertex theoretically allowed?

Answer 63

Because the equivalent quarks of the lepton decay products are from the same generation.

Question 64

What is the quark structure of K-?

Answer 64

Question 65

Give the equation for K- decay

Answer 65

Question 66

Why is the K- decay vertex theoretically forbidden?

Answer 66

Because the equivalent quarks of the lepton decay products are from different generations.

Question 67

Can quark mixing occur?

Answer 67

Yes: it causes quark-lepton symmetry to be conserved

Question 68

For a previously allowed decay (before considerations of quark mixing), what is the coupling strength modified to?

Answer 68

θ_c = Cabibbo angle

Question 69

For a previously forbidden decay (before considerations of quark mixing), what is the coupling strength modified to?

Answer 69

θ_c = Cabibbo angle

Question 70

What happens if the Cabibbo angle equal 0?

Answer 70

There is no quark-mixing

Question 71

How can the Cabibbo angle be measured experimentally?

Answer 71

By comparing the decay rate for pion and kaon decays by taking the ratio of the coupling squared.

Question 72

What is the experimental value of the Cabibbo angle?

Answer 72

13.1 degrees

Question 73

What is the quark mixing matrix?

Answer 73

The mixing of quark states between all three generations. It is called the Cabibbo-Kobayashi-Maskawa (CKM) matrix.

Question 74

Is the 3x3 CKM metric real or complex?

Answer 74

Complex

Question 75

Describe the two basic neutral current interaction vertices (those involving Z0)

Answer 75

Question 76

Are there flavour changing vertices for Z0 interactions?

Answer 76

No