Quark Mixing, CP Violation, Neutrino Oscillations and the Seesaw Mechanism

Question 1

Where do fermion masses come from?

Answer 1

• fermion masses come from interaction with the scalar field
• allowed interactions must obey Lorentz invariance and gauge invariance

Question 2

Allowed Interactions of Quarks

Answer 2

• quarks are given masses by their interactions with the scalar field
• the allowed interactions do not constrain the quarks to couple just to their own generations
• e.g. the left-chiral down quark could couple with the right-chiral down quark but also with the right-chiral strange quark
• rather than giving a simple mass to each quark, we have managed to couple each left-chiral quark flavour to a complex combination of right chiral quarks

Question 3

Quark Interactions

Matrices

Answer 3

-the left-chiral up quark can couple to the right-chiral up quark:
i∂/ ul = λuu φ ur
-but coupling to right-chiral charm and top quarks is also allowed
i∂/ ul = λuc φ cr
i∂/ ul = λut φ tr
-this can be summarised in a matrix of λij
-after symmetry breaking this matrix becomes a mass matrix

Question 4

Mass Matrix and Physical Quarks

Answer 4

• it doesn’t make sense to talk about different masses for the same particle so what does the mass matrix represent?
• physical quarks are eigenstates of the mass matrix: u,c,t
• flavour states: u’,c’,t’ are combinations of mass states

Question 5

Why does the weak interaction violate quark flavours?

Answer 5

-flavour states, which are what the weak interaction see, are combinations of mass state, they themselves are not mass eigenstates

Question 6

How do we find the mass eigenstates?

Answer 6

-diagonalise the mass matrix M by means of a similarity transformation:
M = U† Md U
-where Ms is the diagonalised matrix and U some unitary matrix
-this matrix U transforms between the flavour states d’, s’, b’ seen by the weak interaction and the states of defined mass d, s, b
-since the weak interaction is rare, we generally consider the mass eigenstates, the only physical significance of the flavour states is in weak interactions

Question 7

Flavour-Changing Weak Current

Answer 7

• the flavour changing weak current is coupled to by the W± bosons
• the γ matrix acts in spinor space and U acts in flavour space so they always commute
• introduce the CKM matrix V=Uu†Ud

Question 8

CKM Matrix for 2 Generations of Quark

Answer 8

• in the case of two generations of quark, V is 2x2 and complex-valued so has 8DoF
• V is unitary, V†V=1 which removes 4DoF leaving 4
• can show that there is a three-fold redundancy in V leaving only one 1DoF

Question 9

CKM Matrix for 3 Generations of Quark

Answer 9

• V is 3x3 and unitary so 9DoF
• mixing between 3 up type quarks and 3 down type quarks means a five-fold redundancy leaving 4DoF
• but if we had 3DoF we could think of them as 3 rotations
• but different amounts of mixing between generations leads to 1 extra DoF leading to complex phase in the CKM matrix

Question 10

CP Violation

Outline

Answer 10

• weak interaction violates CP symmetry

- violation comes from the complex phase in the CKM matrix

Question 11

How do we show that complex couplings lead to CP violation?

Answer 11

-consider probability amplitude for some process, M
-then M is complex valued and can be written:
M = |M| e^(iα)
-note that M itself is not measurable, but |M|²=M*M is
-if the process can occur through two different routes then:
M = M1 + M2 = |M1|e^(iα1) + |M2|e^(iα2)
-if CP is not violated then M_, the probability amplitude for the conjugate process should be the same as M
-calculating |M|²-|M_|² find that the difference is only non-zero if the complex phase is non-zero i.e. presence of complex phase leads to CP violation

Question 12

Where can we see CP violation?

Answer 12

• CP violation manifests itself in the decays of neutral mesons
• Ko and Ko_ mesons mix through quark mixing, Ko_ can spontaneously transform into Ko and vice versa
• since the quark flavours need not be conserved, we know that the flavour symmetries are not exact and the Hamiltonian is not invariant under flavour transformations

Question 13

Ko and Ko_ Mixing

Mass Eigenstates

Answer 13

• the mass eigenstates are not the Ko and Ko_, but some linear combination of these flavours states
• if CP was an exact symmetry, then CP would commute with the Hamiltonian and the mass eigenstates and CP eigenstates would be identical
• i.e. we would expect physical states to be eigenstates of CP

Question 14

K1 and K2

Answer 14

-consider the combined action of C and P on the flavour eigenstates of the kaon system:
CP|Ko_⟩  = CP|sd_⟩ 
= |s_d⟩ 
= |Ko⟩ 
-similarly, CP|Ko⟩ =|Ko_⟩ 
=>
K1 = 1/√2 (|Ko⟩ + |Ko_⟩ )
K2 = 1/√2 (|Ko⟩ - |Ko_⟩ )
-where K1 and K2 are CP eigenstates:
CP|K1⟩ = |K1⟩
CP|K2⟩ = -|K2⟩
-if CP is conserved, we expect these to be the physical states

Question 15

Decay of K1

Answer 15

-since the K1 has positive ‘CP-parity’, it can only decay to CP=1 states
-the dominant decay mode is:
K1 -> π+ + π-

Question 16

Decay of K2

Answer 16

-K2 can only decay to CP=-1 states
-the dominant decay mode is:
K2 -> π+ + π- + πo

Question 17

Ks and Kl

Answer 17

-since the mass of the kaon is only slightly larger than the mass of three pions, the phase space of available states is much smaller for K2 decay than it is for K1 decay
-this pushes the decay rate for the K2 up
-experiments confirm there is a short-lived and a long-lived neutral kaon, Ks and Kl
-we assume:
Ks=K1 and Kl=K2
-however careful experimentation shows that if a beam of neutral kaons is left to travel through empty space such that the K2 component decays away, there are still CP=+1 decays demonstrating CP cannot be conserved
-this CP-violation could take place in the kaon mixing or the decay processes themselves
-if it is the kaon mixing then the assignments of Ks and Kl are no longer correct
-we instead find that Ks is only MOSTLY K1 but has a little K2 mixed in and vice versa for Kl

Question 18

T Symmetry

Answer 18

-time reversal, essentially the replacement of a set of particles with the same particles moving in the opposite direction, denoted T

Question 19

Properties of T

Answer 19

-while C and P are both unitary operators (U† = U^(-1)), T is anti-unitary:
T† = -T^(-1)
-a consequence of this is that, whereas P and C symmetry imply the conservation of parity and C-parity, there is no conserved quantity associated with T

Question 20

Physical Consequences of T Symmetry

Answer 20

• the principle of detailed balance: essentially, the statement that transition amplitudes between a set of initial and final states should be equal if the initial and final states are interchanged
• as with C, P and CP, T symmetry is found to be violated by weak interactions, the laws of physics are not the same when run in reverse
• however the physical laws governing the universe are typically time-reversal symmetric, it is only the initial conditions that typically lead to an obvious arrow of time through the second law of thermodynamics

Question 21

The CPT Theorem

Answer 21

• when it was discovered that C and P individually violated, it was originally suggested that CP may still be respected, until CP violation was discvoered
• what about CPT?
• there is no evidence that this combined symmetry is violated
• if it is found to be violated, it will also mean violation of Lorentz invariance
• i.e. CPT violation would imply that special relativity is incorrect and that spacetime itself is not symmetrical as we imagine
• there are ongoing searches for CPT violation

Question 22

Neutrinos in the Standard Model

Answer 22

• in the standard model, neutrinos are massless particles
• this means the mass eigenvalues for neutrinos are all zero
• this would allow for no mixing between lepton flavours as the flavour states for leptons could always be identified with mass eigenstates

Question 23

Evidence for Neutrino Mass

Answer 23

• in the 1990s, it was discovered experimentally that neutrino flavours oscillate
• i.e. in a beam of neutrinos, the probability of measuring neutrinos of a particular flavour fluctuates over time, there is a mixing between lepton flavours
• this is only possible if the mass eigenstates are different from the flavour states
• this requires a similar mixing matrix for leptons as we had for quarks known as the PMNS matrix
• so now we know neutrinos must have mass even if no direct measurement has ever been made

Question 24

Solar Neutrino Problem

Answer 24

• we don’t measure enough νe’s from the sun for the amount produced in beta decay
• solution: the νe’s are changing into νμ and ντ, mixing
• this is where the PMNS matrix comes in
• neutrino masses must be VERY small as they have never been measured
• can experimentally determine that the sum of all three neutrino masses must be <0.3eV
• note that one could still be massless as then there would still be no degeneracy, but at least two must have mass

Question 25

The See-Saw Mechanism

Allowed Mass Terms

Answer 25

-allowed mass terms take the form:
i∂/ fl = m fr
-these are the only terms that obey both Lorentz and gauge invariance

Question 26

The See-Saw Mechanism

No Allowed Mass Terms

Answer 26

-we said that:
i∂/ fl = m fl
-is Lorentz invariant but not gauge invariant as they have different charges
-there is one exception, can have this kind of mass if the particle in question has no charges
-the only candidate is the right chiral neutrino

Question 27

The See-Saw Mechanism

Equation of Motion for Neutrinos

Answer 27

-the new kind of mass term is not allowed for LH neutrinos so the RHS of the equation of motion is:
A (vl vr)^t
-where A is a 2x2 matrix with entries 0, md, md, mm

Question 28

The See-Saw Mechanism

Natural Scales

Answer 28

• natural scale for md ~ electroweak breaking scale

- natural scale for mm unknown, but must be mm»md

Question 29

The See-Saw Mechanism

Physical Masses

Answer 29

-the eigenvalues of the matrix:
λ ≈ mm
λ ≈ - md²/mm² &laquo_space;1

Question 30

The See-Saw Mechanism

Eigenstates

Answer 30

λ ≈ mm -> (0 1)^t
λ ≈ - md²/mm² -> (1 0)^t
-every neutrino ever observed has been LC with very small mass
-the RH state with high mass and no interactions has been suggested to be dark matter

Question 31

Dirac Mass

Answer 31

• there are two very different types of mass for neutrinos

- one is the Dirac mass that couples left-chiral neutrinos to right chiral neutrinos

Question 32

Majorana Mass

Answer 32

• there are two very different types of mass for neutrinos

- the Majorana mass couples right-chiral neutrinos to left-chiral antineutrinos