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Lectures on B-physics 19-20 April 2011 Vrije Universiteit Brussel

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Lectures on B-physics 19-20 April 2011Vrije Universiteit Brussel

N. Tuning

Niels Tuning (1)

Niels Tuning (2)

Niels Tuning (3)

- “CP violation” is about the weak interactions,
- In particular, the charged current interactions:

- The interesting stuff happens in the interaction with quarks
- Therefore, people also refer to this field as “flavour physics”

Niels Tuning (4)

- “CP violation” is about the weak interactions,
- In particular, the charged current interactions:

- Quarks can only change flavour through charged current interactions

Niels Tuning (5)

- “CP violation” is about the weak interactions,
- In particular, the charged current interactions:

- In 1st hour:
- P-parity, C-parity, CP-parity
- the neutrino shows that P-parity is maximally violated

Niels Tuning (6)

W+

W-

b

gVub

gV*ub

u

- “CP violation” is about the weak interactions,
- In particular, the charged current interactions:

- In 1st hour:
- P-parity, C-parity, CP-parity
- Symmetry related to particle – anti-particle

Niels Tuning (7)

Equal amounts

of matter &

anti matter (?)

Matter Dominates !

- It’s about differences in matter and anti-matter
- Why would they be different in the first place?
- We see they are different: our universe is matter dominated

Niels Tuning (8)

- No definitive answer to this question yet!
- In 1967 A. Sacharov formulated a set of general conditions that any such mechanism has to meet
- You need a process that violates the baryon number B:(Baryon number of matter=1, of anti-matter = -1)
- Both C and CP symmetries should be violated
- Conditions 1) and 2) should occur during a phase in which there is no thermal equilibrium

- In these lectures we will focus on 2): CP violation
- Apart from cosmological considerations, I will convince you that there are more interesting aspects in CP violation

Niels Tuning (9)

- “CP violation” is about the weak interactions,
- In particular, the charged current interactions:

- Same initial and final state
- Look at interference between B0 fCP and B0 B0 fCP

Niels Tuning (10)

x

b̃

s̃

s

b

g̃

Bs

g̃

Bs

s̃

s

b

b̃

x

“Box” diagram: ΔB=2

“Penguin” diagram: ΔB=1

s

b

s

b

b

μ

s

d

K*

g̃

B0

b

s

μ

μ

x

b̃

s̃

μ

μ

s

s̃

g̃

Bs

b

x

μ

b̃

- “CP violation” is about the weak interactions,
- In particular, the charged current interactions:

- Are heavy particles running around in loops?

Niels Tuning (11)

s

b

b

s

Matter Dominates !

- CP-violation (or flavour physics) is about charged current interactions

- Interesting because:
- Standard Model: in the heart of quark interactions
- Cosmology: related to matter – anti-matter asymetry
- Beyond Standard Model: measurements are sensitive to new particles

Niels Tuning (12)

- People think it is a complicated part of the Standard Model (me too:-). Why?
- Non-intuitive concepts?
- Imaginary phase in transition amplitude, T ~ eiφ
- Different bases to express quark states, d’=0.97 d + 0.22 s + 0.003 b
- Oscillations (mixing) of mesons: |K0> ↔ |K0>

- Complicated calculations?
- Many decay modes? “Beetopaipaigamma…”
- PDG reports 347 decay modes of the B0-meson:
- Γ1 l+νl anything ( 10.33 ± 0.28 ) × 10−2
- Γ347ν ν γ<4.7 × 10−5CL=90%

- And for one decay there are often more than one decay amplitudes…

- PDG reports 347 decay modes of the B0-meson:

Niels Tuning (13)

Niels Tuning (14)

- Space, time translation & orientation symmetries are all continuous symmetries
- Each symmetry operation associated with one ore more continuous parameter

- There are also discrete symmetries
- Charge sign flip (Q -Q) : C parity
- Spatial sign flip ( x,y,z -x,-y,-z) : P parity
- Time sign flip (t -t) : T parity

- Are these discrete symmetries exact symmetries that are observed by all physics in nature?
- Key issue of this course

Niels Tuning (15)

+

- Parity, P
- Parity reflects a system through the origin. Convertsright-handed coordinate systems to left-handed ones.
- Vectors change sign but axial vectors remain unchanged
- x -x , p -p, butL = x p L

- Charge Conjugation, C
- Charge conjugation turns a particle into its anti-particle
- e +e- , K -K +

- Charge conjugation turns a particle into its anti-particle
- Time Reversal, T
- Changes, for example, the direction of motion of particles
- t -t

- Changes, for example, the direction of motion of particles

Niels Tuning (16)

- Instruction for Abel Tasman, explorer of Australia (1642):
- “Since many rich mines and other treasures have been found in countries north of the equator between 15o and 40o latitude, there is no doubt that countries alike exist south of the equator.
- The provinces in Peru and Chili rich of gold and silver, all positioned south of the equator, are revealing proofs hereof.”

Niels Tuning (17)

S=1/2

S=1/2

- Observe radioactive decay of Cobalt-60 nuclei
- The process involved: 6027Co 6028Ni + e- + ne
- 6027Co is spin-5 and 6028Ni is spin-4, both e- and ne are spin-½
- If you start with fully polarized Co (SZ=5) the experiment is essentially the same (i.e. there is only one spin solution for the decay) |5,+5> |4,+4> + |½ ,+½> + |½,+½>

S=4

Niels Tuning (18)

- We introduce a new quantity: Helicity = the projection of the spin on the direction of flight of a particle

H=+1 (“right-handed”)

H=-1 (“left-handed”)

Niels Tuning (19)

- Experimental challenge: how do you obtain a sample of Co(60) where the spins are aligned in one direction
- Wu’s solution: adiabatic demagnetization of Co(60) in magnetic fields at very low temperatures (~1/100 K!). Extremely challenging in 1956.

Niels Tuning (20)

- The surprising result: the counting rate is different
- Electrons are preferentially emitted in direction opposite of 60Co spin!
- Careful analysis of results shows that experimental data is consistent with emission of left-handed (H=-1) electrons only at any angle!!

‘Backward’ Counting ratew.r.t unpolarized rate

60Co polarization decreasesas function of time

‘Forward’ Counting ratew.r.t unpolarized rate

Niels Tuning (21)

- Physics conclusion:
- Angular distribution of electrons shows that only pairs of left-handed electrons / right-handed anti-neutrinos are emitted regardless of the emission angle
- Since right-handed electrons are known to exist (for electrons H is not Lorentz-invariant anyway), this means no left-handed anti-neutrinos are produced in weak decay

- Parity is violated in weak processes
- Not just a little bit but 100%

- How can you see that 60Co violates parity symmetry?
- If there is parity symmetry there should exist no measurement that can distinguish our universe from a parity-flipped universe, but we can!

Niels Tuning (22)

- Wu’s experiment was shortly followed by another clever experiment by L. Lederman: Look at decay p+ m+nm
- Pion has spin 0, m,nm both have spin ½ spin of decay products must be oppositely aligned Helicity of muon is same as that of neutrino.

- Nice feature: can also measure polarization of both neutrino (p+ decay) and anti-neutrino (p- decay)
- Ledermans result: All neutrinos are left-handed and all anti-neutrinos are right-handed

m+

nm

p+

OK

OK

Niels Tuning (23)

- Introducing C-symmetry
- The C(harge) conjugation is the operation which exchanges particles and anti-particles (not just electric charge)
- It is a discrete symmetry, just like P, i.e. C2 = 1

- C symmetry is broken by the weak interaction,
- just like P

OK

m+

nm(LH)

p+

C

nm(LH)

m-

OK

p-

Niels Tuning (24)

Intrinsic

spin

- What about C+P CP symmetry?
- CP symmetry is parity conjugation (x,y,z -x,-y,z)
followed by charge conjugation (X X)

- CP symmetry is parity conjugation (x,y,z -x,-y,z)

+

P

C

CP appears to be preservedin weakinteraction!

+

+

+

CP

Niels Tuning (25)

- C.S. Wu discovered from 60Co decays that the weak interaction is 100% asymmetric in P-conjugation
- We can distinguish our universe from a parity flipped universe by examining 60Co decays

- L. Lederman et al. discovered from π+ decays that the weak interaction is 100% asymmetric in C-conjugation as well, but that CP-symmetry appears to be preserved
- First important ingredient towards understanding matter/anti-matter asymmetry of the universe: weak force violates matter/anti-matter(=C) symmetry!
- C violation is a required ingredient, but not enough as we will learn later

Niels Tuning (26)

- C and P are still good symmetries in any reaction not involving the weak interaction
- Can associate a conserved value with them (Noether Theorem)

- Each hadron has a conserved P and C quantum number
- What are the values of the quantum numbers
- Evaluate the eigenvalue of the P and C operators on each hadronP|y> = p|y>

- What values of C and P are possible for hadrons?
- Symmetry operation squared gives unity so eigenvalue squared must be 1
- Possible C and P values are +1 and -1.

- Meaning of P quantum number
- If P=1 then P|y> = +1|y> (wave function symmetric in space)if P=-1 then P|y> = -1 |y> (wave function anti-symmetric in space)

Niels Tuning (27)

- QFT rules for particle vs. anti-particles
- Parity of particle and anti-particle must be opposite for fermions (spin-N+1/2)
- Parity of bosons (spin N) is same for particle and anti-particle

- Definition of convention (i.e. arbitrary choice in def. of q vs q)
- Quarks have positive parity Anti-quarks have negative parity
- e- has positive parity as well.
- (Can define other way around: Notation different, physics same)

- Parity is a multiplicative quantum number for composites
- For composite AB the parity is P(A)*P(B), Thus:
- Baryons have P=1*1*1=1, anti-baryons have P=-1*-1*-1=-1
- (Anti-)mesons have P=1*-1 = -1

- Excited states (with orbital angular momentum)
- Get an extra factor (-1) l where l is the orbital L quantum number
- Note that parity formalism is parallel to total angular momentum J=L+S formalism, it has an intrinsic component and an orbital component

- NB: Photon is spin-1 particle has intrinsic P of -1

Niels Tuning (28)

- The p+ meson
- Quark and anti-quark composite: intrinsic P = (1)*(-1) = -1
- Orbital ground state no extra term
- P(p+)=-1

- The neutron
- Three quark composite: intrinsic P = (1)*(1)*(1) = 1
- Orbital ground state no extra term
- P(n) = +1

- The K1(1270)
- Quark anti-quark composite: intrinsic P = (1)*(-1) = -1
- Orbital excitation with L=1 extra term (-1)1
- P(K1) = +1

- Experimental proof: J.Steinberger (1954)
- πd→nn
- n are fermions, so (nn) anti-symmetric
- Sd=1, Sπ=0 → Lnn=1
- P|nn> = (-1)L|nn> = -1 |nn>
- P|d> = P |pn> = (+1)2|pn> = +1 |d>
- To conserve parity: P|π> = -1 |π>

Meaning: P|p+> = -1|p+>

Niels Tuning (29)

- Only particles that are their own anti-particles are C eigenstates because C|x> |x> = c|x>
- E.g. p0,h,h’,r0,f,w,y and photon

- C eigenvalues of quark-anti-quark pairs is determined by L and S angular momenta: C = (-1)L+S
- Rule applies to all above mesons

- C eigenvalue of photon is -1
- Since photon is carrier of EM force, which obviously changes sign under C conjugation

- Example of C conservation:
- Process p0 g g C=+1(p0 has spin 0) (-1)*(-1)
- Process p0 g g g does not occur (and would violate C conservation)

Experimental proof of C-invariance:

BR(π0→γγγ)<3.1 10-5

Niels Tuning (30)

- This was an introduction to P and C
- Let’s change gear…

Niels Tuning (31)

uLI

W+m

g

dLI

- Focus on charged current interaction (W±): let’s trace it

Niels Tuning (32)

- LKinetic :• Introduce the massless fermion fields
- • Require local gauge invariance gives rise to existence of gauge bosons

- LHiggs :• Introduce Higgs potential with <f> ≠ 0
- • Spontaneous symmetry breaking

The W+, W-,Z0 bosons acquire a mass

- LYukawa :• Ad hoc interactions between Higgs field & fermions

Niels Tuning (33)

Interaction rep.

Hypercharge Y

(=avg el.charge in multiplet)

SU(3)C

SU(2)L

Left-

handed

generation

index

Y = Q - T3

Fermions:

with y = QL, uR, dR, LL, lR, nR

Quarks:

Under SU2:

Left handed doublets

Right hander singlets

Leptons:

Scalar field:

Note:

Interaction representation: standard model interaction is independent of generation number

Niels Tuning (34)

Q = T3 + Y

Y = Q - T3

Explicitly:

- The left handed quark doublet :

- Similarly for the quark singlets:

- The left handed leptons:

- And similarly the (charged) singlets:

Niels Tuning (35)

: Fermions + gauge bosons + interactions

Procedure:

Introduce the Fermion fields and demand that the theory is local gauge invariant under SU(3)CxSU(2)LxU(1)Y transformations.

Start with the Dirac Lagrangian:

Replace:

Gam :8 gluons

Wbm: weak bosons: W1, W2, W3

Bm: hyperchargeboson

Fields:

Generators:

La : Gell-Mann matrices: ½ la(3x3) SU(3)C

Tb : Pauli Matrices: ½ tb (2x2) SU(2)L

Y : Hypercharge: U(1)Y

For the remainder we only consider Electroweak: SU(2)L x U(1)Y

Niels Tuning (36)

uLI

W+m

g

dLI

For example, the term with QLiIbecomes:

Writing out only the weak part for the quarks:

W+ = (1/√2) (W1+ i W2)

W- = (1/√ 2) (W1– i W2)

L=JmWm

Niels Tuning (37)

Broken

Symmetry

V(f)

V(f)

Symmetry

f

f

~ 246 GeV

Spontaneous Symmetry Breaking: The Higgs field adopts a non-zero vacuum expectation value

Substitute:

Procedure:

- .
- The W+,W-,Z0 bosons acquire mass
- The Higgs boson H appears

And rewrite the Lagrangian (tedious):

(The other 3 Higgs fields are “eaten” by the W, Z bosons)

Niels Tuning (38)

doublets

singlet

Since we have a Higgs field we can (should?) add (ad-hoc) interactions between f and the fermions in a gauge invariant way.

The result is:

~

i, j : indices for the 3 generations!

With:

(The CP conjugate of f

To be manifestly invariant under SU(2) )

are arbitrary complex matrices which operate in family space (3x3)

Flavour physics!

Niels Tuning (39)

Writing the first term explicitly:

Niels Tuning (40)

There are 3 Yukawa matrices (in the case of massless neutrino’s):

- Each matrix is 3x3 complex:
- 27 real parameters
- 27 imaginary parameters (“phases”)

- many of the parameters are equivalent, since the physics described by one set of couplings is the same as another
- It can be shown (see ref. [Nir]) that the independent parameters are:
- 12 real parameters
- 1 imaginary phase

- This single phase is the source of all CP violation in the Standard Model

……Revisit later

Niels Tuning (41)

S.S.B

Start with the Yukawa Lagrangian

After which the following mass term emerges:

with

LMass is CP violating in a similar way as LYuk

Niels Tuning (42)

S.S.B

dLI , uLI , lLIare the weak interaction eigenstates

dL , uL , lLare the mass eigenstates (“physical particles”)

Writing in an explicit form:

The matrices M can always be diagonalised by unitarymatricesVLfandVRfsuch that:

Then the real fermion mass eigenstates are given by:

Niels Tuning (43)

S.S.B

In terms of the mass eigenstates:

In flavour space one can choose:

Weak basis: The gauge currents are diagonal in flavour space, but the flavour mass matrices are

non-diagonal

Mass basis: The fermion masses are diagonal, but some gauge currents (charged weak interactions)

are not diagonal in flavour space

In the weak basis: LYukawa = CP violating

In the mass basis: LYukawa → LMass= CP conserving

What happened to the charged current interactions (in LKinetic) ?

Niels Tuning (44)

The charged current interaction for quarks in the interaction basis is:

The charged current interaction for quarks in the mass basis is:

The unitary matrix:

With:

is the Cabibbo Kobayashi Maskawa mixing matrix:

Lepton sector: similarly

However, for massless neutrino’s: VLn= arbitrary. Choose it such that VMNS = 1 There is no mixing in the lepton sector

Niels Tuning (45)

The charged current term reads:

Under the CP operator this gives:

(Together with (x,t) -> (-x,t))

A comparison shows that CP is conserved only ifVij = Vij*

In general the charged current term is CP violating

Niels Tuning (46)

- LKinetic : •Introduce the massless fermion fields
- •Require local gauge invariance gives rise to existence of gauge bosons

CP Conserving

- LHiggs : •Introduce Higgs potential with <f> ≠ 0
- •Spontaneous symmetry breaking

The W+, W-,Z0 bosons acquire a mass

CP Conserving

- LYukawa : •Ad hoc interactions between Higgs field & fermions

CP violating with a single phase

- LYukawa → Lmass : • fermion weak eigenstates:
- - mass matrix is (3x3) non-diagonal
- • fermion mass eigenstates:
- - mass matrix is (3x3) diagonal

CP-violating

CP-conserving!

- LKinetic in mass eigenstates: CKM – matrix

CP violating with a single phase

Niels Tuning (47)

Recap

- Diagonalize Yukawa matrix Yij
- Mass terms
- Quarks rotate
- Off diagonal terms in charged current couplings

Niels Tuning (48)

It’s unitary

“probabilities add up to 1”:

d’=0.97 d + 0.22 s + 0.003 b (0.972+0.222+0.0032=1)

How many free parameters?

How many real/complex?

How do we normally visualize these parameters?

Niels Tuning (49)

- People think it is a complicated part of the Standard Model (me too:-). Why?
- Non-intuitive concepts?
- Imaginary phase in transition amplitude, T ~ eiφ
- Different bases to express quark states, d’=0.97 d + 0.22 s + 0.003 b
- Oscillations (mixing) of mesons: |K0> ↔ |K0>

- Complicated calculations?
- Many decay modes? “Beetopaipaigamma…”
- PDG reports 347 decay modes of the B0-meson:
- Γ1 l+νl anything ( 10.33 ± 0.28 ) × 10−2
- Γ347ν ν γ<4.7 × 10−5CL=90%

- And for one decay there are often more than one decay amplitudes…

- PDG reports 347 decay modes of the B0-meson:

Niels Tuning (50)

Niels Tuning (51)

Recap from last hour

uI

u

W

W

d,s,b

dI

- Diagonalize Yukawa matrix Yij
- Mass terms
- Quarks rotate
- Off diagonal terms in charged current couplings

Niels Tuning (52)

- It’s unitary
- “probabilities add up to 1”:
- d’=0.97 d + 0.22 s + 0.003 b (0.972+0.222+0.0032=1)

- How many free parameters?
- How many real/complex?

- How do we normally visualize these parameters?

Niels Tuning (53)

Under a quark phase transformation:

and a simultaneous rephasing of the CKM matrix:

or

In other words:

Niels Tuning (54)

2 generations:

No CP violation in SM!

This is the reason Kobayashi and Maskawa first suggested a 3rd family of fermions!

Under a quark phase transformation:

and a simultaneous rephasing of the CKM matrix:

or

the charged current

is left invariant.

Degrees of freedom in VCKM in 3 N generations

Number of real parameters: 9+ N2

Number of imaginary parameters: 9 + N2

Number of constraints (VV† = 1): -9- N2

Number of relative quark phases: -5- (2N-1)

-----------------------

Total degrees of freedom: 4(N-1)2

Number of Euler angles: 3N (N-1) / 2

Number of CP phases: 1 (N-1) (N-2) / 2

Niels Tuning (55)

Niels Tuning (56)

- Timeline:
- Sep 1972: Kobayashi & Maskawa predict 3 generations
- Nov 1974: Richter, Ting discover J/ψ: fill 2nd generation
- July 1977: Ledermann discovers Υ: discovery of 3rd generation

Niels Tuning (57)

2 generations:

No CP violation in SM!

This is the reason Kobayashi and Maskawa first suggested a 3rd family of fermions!

Under a quark phase transformation:

and a simultaneous rephasing of the CKM matrix:

or

the charged current

is left invariant.

Degrees of freedom in VCKM in 3 N generations

Number of real parameters: 9+ N2

Number of imaginary parameters: 9 + N2

Number of constraints (VV† = 1): -9- N2

Number of relative quark phases: -5- (2N-1)

-----------------------

Total degrees of freedom: 4(N-1)2

Number of Euler angles: 3N (N-1) / 2

Number of CP phases: 1 (N-1) (N-2) / 2

Niels Tuning (58)

Cabibbos theory successfully correlated many decay rates by counting the number of cosqc and sinqc terms in their decay diagram

Niels Tuning (59)

There was however one major exception which Cabibbo could not describe: K0 m+m-

Observed rate much lower than expected from Cabibbos ratecorrelations (expected rate g8sin2qccos2qc)

s

d

cosqc

sinqc

u

W

W

nm

m-

m+

Niels Tuning (60)

Solution to K0 decay problem in 1970 by Glashow, Iliopoulos and Maiani postulate existence of 4th quark

Two ‘up-type’ quarks decay into rotated ‘down-type’ states

Appealing symmetry between generations

u

c

W+

W+

d’=cos(qc)d+sin(qc)s

s’=-sin(qc)d+cos(qc)s

Niels Tuning (61)

How does it solve the K0 m+m- problem?

Second decay amplitude added that is almost identical to original one, but has relative minus sign Almost fully destructive interference

Cancellation not perfect because u, c mass different

s

d

s

d

-sinqc

cosqc

cosqc

+sinqc

c

u

nm

nm

m-

m+

m-

m+

Niels Tuning (62)

2 generations: d’=0.97 d + 0.22 s (θc=13o)

3 generations: d’=0.97 d + 0.22 s + 0.003 b

NB: probabilities have to add up to 1: 0.972+0.222+0.0032=1

“Unitarity” !

Niels Tuning (63)

- 2 generations: d’=0.97 d + 0.22 s (θc=13o)
- 3 generations: d’=0.97 d + 0.22 s + 0.003 b
- Parameterization used by Particle Data Group (3 Euler angles, 1 phase):

- ‘General’ 4-parameter form (Particle Data Group) with three rotations q12,q13,q23 and one complex phase d13
- c12 = cos(q12), s12 = sin(q12) etc…

- Another form (Kobayashi & Maskawa’s original)
- Different but equivalent

- Physics is independent of choice of parameterization!
- But for any choice there will be complex-valued elements

Niels Tuning (65)

Different parametrizations! It’s about phase differences!

KM

Re-phasing V:

PDG

3 parameters: θ, τ, σ

1 phase: φ

Niels Tuning (66)

- Magnitudes are typically determined from ratio of decay rates
- Example 1 – Measurement of Vud
- Compare decay rates of neutrondecay and muon decay
- Ratio proportional to Vud2
- |Vud| = 0.97418 ± 0.00027
- Vud of order 1

Niels Tuning (67)

- Example 2 – Measurement of Vus
- Compare decay rates of semileptonic K- decay and muon decay
- Ratio proportional to Vus2
- |Vus| = 0.2255 ± 0.0019
- Vus sin(qc)

- Example 3 – Measurement of Vcb
- Compare decay rates of B0 D*-l+n and muon decay
- Ratio proportional to Vcb2
- |Vcb| = 0.0412 ± 0.0011
- Vcb is of order sin(qc)2 [= 0.0484]

- Example 4 – Measurement of Vub
- Compare decay rates of B0 D*-l+n and B0 p-l+n
- Ratio proportional to (Vub/Vcb)2
- |Vub/Vcb| = 0.090 ± 0.025
- Vub is of order sin(qc)3 [= 0.01]

- Example 5 – Measurement of Vcd
- Measure charm in DIS with neutrinos
- Rate proportional to Vcd2
- |Vcd| = 0.230 ± 0.011
- Vcb is of order sin(qc)[= 0.23]

- Example 6 – Measurement of Vtb
- Very recent measurement: March ’09!
- Single top production at Tevatron
- CDF: |Vtb| = 0.91 ± 0.13
- D0: |Vtb| = 1.07 ± 0.12

Vts

Vts

Vts

= 1.210 +0.047 from lattice QCD

-0.035

Vts

- Example 7 – Measurement of Vtd, Vts
- Cannot be measured from top-decay…
- Indirect from loop diagram
- Vts: recent measurement: March ’06
- |Vtd| = 0.0081 ± 0.0006
- |Vts| = 0.0387 ± 0.0023

Ratio of frequencies for B0 and Bs

Vts ~ 2

Vtd ~3 Δms ~ (1/λ2)Δmd ~ 25 Δmd

- Magnitudes of elements have been measured over time
- Result of a large number of measurements and calculations

Magnitude of elements shown only, no information of phase

Niels Tuning (74)

Magnitudes of elements have been measured over time

Result of a large number of measurements and calculations

Magnitude of elements shown only, no information of phase

Niels Tuning (75)

d

s

b

u

c

t

- Values are strongly ranked:
- Transition within generation favored
- Transition from 1st to 2nd generation suppressed by cos(qc)
- Transition from 2nd to 3rd generation suppressed bu cos2(qc)
- Transition from 1st to 3rd generation suppressed by cos3(qc)

CKM magnitudes

Why the ranking?We don’t know (yet)!

If you figure this out,you will win the nobelprize

l

l3

l

l2

l3

l2

l=sin(qc)=0.23

Niels Tuning (76)

- We now know that neutrinos also have flavour oscillations
- thus there is the equivalent of a CKM matrix for them:
- Pontecorvo-Maki-Nakagawa-Sakata matrix

- a completely different hierarchy!

Niels Tuning (77)

3 real parameters: A, λ, ρ

1 imaginary parameter: η

3 real parameters: A, λ, ρ

1 imaginary parameter: η

- Given current experimental precision on CKM element values, we usually drop l4 and l5 terms as well
- Effect of order 0.2%...

- Deviation of ranking of 1st and 2nd generation (l vs l2) parameterized in A parameter
- Deviation of ranking between 1st and 3rd generation, parameterized through |r-ih|
- Complex phase parameterized in arg(r-ih)

Niels Tuning (80)

- Fit all known Vij values to Wolfenstein parameterization and extract A, λ, ρ and η
- Results for A and l most precise (but don’t tell us much about CPV)
- A = 0.83, l = 0.227

- Results for r,h are usually shown in complex plane of r-ih for easier interpretation

Niels Tuning (81)

- Starting point: the 9 unitarity constraints on the CKM matrix
- Pick (arbitrarily) orthogonality condition with (i,j)=(3,1)

Niels Tuning (82)

Starting point: the 9 unitarity constraints on the CKM matrix

3 orthogonality relations

Pick (arbitrarily) orthogonality condition with (i,j)=(3,1)

Niels Tuning (83)

Starting point: the 9 unitarity constraints on the CKM matrix

Pick (arbitrarily) orthogonality condition with (i,j)=(3,1)

Niels Tuning (84)

- Sum of three complex vectors is zero Form triangle when put head to tail

(Wolfenstein params to order l4)

Niels Tuning (85)

- Phase of ‘base’ is zero Aligns with ‘real’ axis,

Niels Tuning (86)

- Divide all sides by length of base
- Constructed a triangle with apex (r,h)

(r,h)

(0,0)

(1,0)

Niels Tuning (87)

- We can now put this triangle in the (r,h) plane

Niels Tuning (88)

- We can visualize the CKM-constraints in (r,h) plane

- We can correlate the angles βandγto CKM elements:

Another 3 orthogonality relations

Pick (arbitrarily) orthogonality condition with (i,j)=(3,1)

Niels Tuning (91)

- Two of the six unitarity triangles have equal sides in O(λ)

- NB: angle βs introduced. But… not phase invariant definition!?

Niels Tuning (92)

- Replace d by s:

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W-

b

gVub

u

- Couplings of the charged current:
- Wolfensteinparametrization:

- Magnitude:

- Complex phases:

Niels Tuning (95)

g

b

- Next order of business: Devise an experiment that measures arg(Vtd)b and arg(Vub)g.
- What will such a measurement look like in the (r,h) plane?

Fictitious measurement of b consistent with CKM model

CKM phases

Niels Tuning (96)

Precise measurement ofsin(2β) agrees perfectlywith other measurementsand CKM model assumptionsThe CKM model of CP violation experimentallyconfirmed with high precision!

Niels Tuning (97)

- ??? Edward Witten, 17 Feb 2009…
- See “From F-Theory GUT’s to the LHC” by Heckman and Vafa (arXiv:0809.3452)

Niels Tuning (99)