Particle Physics II

1
Particle Physics II CP violationLecture 1

• N. Tuning

2
Outline

• 1 May
• Introduction matter and anti-matter
• P, C and CP symmetries
• K-system
• CP violation
• Oscillations
• Cabibbo-GIM mechanism
• 8 May
• CP violation in the Lagrangian
• CKM matrix
• B-system
• 15 May
• B-factories
• B?J/Psi Ks
• Delta ms

3
Literature

• Slides based on the course from Wouter Verkerke.
• W.E. Burcham and M. Jobes, Nuclear and Particle
  Physics, chapters 11 and 14.
• Z. Ligeti, hep-ph/0302031, Introduction to Heavy
  Meson Decays and CP Asymmetries
• Y. Nir, hep-ph/0109090, CP Violation A New Era
• H. Quinn, hep-ph/0111177, B Physics and CP
  Violation

4
Motivation What is so interesting about CP
violation?

• Its 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

5
Where and how do we generate the Baryon asymmetry?

• 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 matter1, 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

6
What is evidence for the existence of anti-matter?

• Energetic photons produced in matter/anti-matter
  annihilation
• Look at spectrum of photons in universe and look
  for spikes
• Main problem photons can not travel unlimited
  distances in the universe because of interactions
  with remaining cosmic background radiation and
  gases etc
• Conclusion No anti-matter in 20Mpc radius.
• How to look further into space?
• Better Look for anti-Helium nuclei flying
  through space
• Positrons, anti-protons can occasionally be
  produced in various processes, but producing
  anti-Helium is way too complicated by regular
  means Only viable source of anti-Helium are
  fusion processes in anti-stars
• Presence/absence of anti-Helium says something
  about existence of anti-matter in distant regions
  of space
• Large rest mass of Helium nuclei allows them to
  travel much further through space than photons ?
  Conclusions of anti-Helium searches cover much
  larger region of space

7
The AMS experiment Searching for He

• In essence as small particle physics experiment
  in space
• AMS-01 brought to space through flight of
  Discovery shuttle
• Can detect and identify many types of cosmic rays

8
Results of the AMS experiment

• Zero anti-helium found, plenty of Helium found
• Rigidity of tracks is measure of particles
  momentum
• Very high energy Helium nuclei have traveled from
  far ? Says something about spatial reach of
  experiment
• Universe with pockets of anti-matter hypothesis
  increasingly unlikely
• Future AMS-02 experiment will (launch 2007) will
  have much increased range

9
Introduction positron discovery by Anderson

• Result discovery of a positively charged
  electron-like particle dubbed the positron
• Experimental confirmation of existence of
  anti-matter!

Outgoing particle (low momentum / hi curvature)
Lead plate to slow down particlein chamber
Incoming particle (high momentum / low curvature)
10
Introduction positron discovery by Anderson

• 4 years later Anderson confirmed this with g ?
  ee- in lead plate using g from Thorium carbide
  source

11
Introduction anti-neutrino Savannah river

• Decisive experiment close to Savannah River
  nuclear reactor in South Carolina in 1956 (Nobel
  prize 1995)
• Idea nuclear reactor provides enormous
  anti-neutrino flux from fission O(1013) /cm2/sec
• Try to detect inverse beta decay n p ? n e

(Beta decay n ? p e- n)
n e ? p n p n ? n e
Cross over e-
Invert reaction
12
Introduction anti-neutrino Savannah river

• How do you detect n p ? n e
• Look for the positron through the reaction
  e e- ? g gand detect 2 photons produced
  simultaneously.
• Savannah river Detector
• Tank with 200 liters of water with 40 kg of CdCl2
  dissolved in it.
• Surrounded by 110 photomultipliers for photon
  detection
• Clean signal found ? direct proof of existence of
  neutrino
• Nobel prize 1995
• ?? n ? p e- not observed
• ????? , Lepton number must be conserved

From inversebeta decay
From detectormaterial
13
Introduction - What about the other
anti-particles?

• Dirac equation for every (spin ½) particle there
  is an anti-particle
• It took a bit longer, but more were
  discoveredAnti-proton (1955) and anti-neutron
  (1955) using cyclotrons
• Reactions with particles and anti-particles
• Q How do you produce anti-particles anyway?
• A In pairs with particles, e.g. g ? e e-
• But this is not the whole story as we will see
  later
• General rule crossing symmetry
• In any existing reaction you can move a particle
  through the arrow while turning it into an
  anti-particle
• Example e- g ? e- g (Compton scattering) g g
  ? e e- (Pair creation)

Move e- to right Move g to left(g g)
14
Definition and discovery of C,P,CP violation
15
Continuous vs discrete symmetries

• 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

16
Example People believe in symmetry

• 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.

17
Three Discrete Symmetries

• 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, but L x ? p ? L
• Charge Conjugation, C
• Charge conjugation turns a particle into its
  anti-particle
• e ? e- , K - ? K
• Time Reversal, T
• Changes, for example, the direction of motion of
  particles
• t ? -t

18
P-parity experiments

• Before 1956 physicists were convinced that the
  laws of nature were left-right symmetric.
  Strange?
• A gedanken experiment Consider two perfectly
  mirror symmetric cars
• What would happen if the ignition mechanism uses,
  say, 60Co b decay?

Gas pedal
Gas pedal
driver
driver
L and R are fully symmetric, Each nut, bolt,
molecule etc. However the engine is a black box
R
L
Person L gets in, starts, .. 60 km/h
Person R gets in, starts, .. What happens?
19
The situation in 1956

• Nothing hints at the existence of any kind of
  Parity violating physics
• Reminder Parity (x,y,z) ? (-x,-y,-z)
• If universe is parity-symmetric, inverting all
  spatial coordinates would not changes laws of
  physics
• 1956 Lee and Yang publish a paper Question of
  Parity Conservation in Weak Interactions/
• Suggestion Weak interaction might violate Parity
  symmetry.
• Originated from discussions at April HEP
  conference in Rochester, NY. Following Yang's
  presentation Richard Feynman brought up the
  question of non-conservation of parity.
• Feynman himself later said, "I thought the idea
  (of parity violation) unlikely, but possible, and
  a very exciting possibility." Indeed Feynman
  later made a fifty dollar bet with a friend that
  parity would not be violated.

20
Parity symmetry The situation in 1956

• When the paper appeared, physicists were not
  immediately prompted into action. The proposition
  of parity non-conservation was not unequivocally
  denied rather, the possibility appeared so
  unlikely that experimental proof did not warrant
  immediate attention.
• The physicist Freeman Dyson wrote of his reaction
  to the paper "A copy of it was sent to me and I
  read it. I read it twice. I said, This is very
  interesting,' or words to that effect. But I had
  not the imagination to say, By golly, if this is
  true it opens up a whole new branch of physics.'
  And I think other physicists, with very few
  exceptions, at that time were as unimaginative as
  I."

21
Parity symmetry the experiment

• Madame Wu
• Another immigrant was now to play the next major
  role, Madame Chien-Shiung Wu.
• Arriving at Berkely in 1936 from Shanghai, Wu was
  one of the most ardently pursued coeds on
  campus. But she was also a hard worker who
  abhorred the marked absence of women from the
  American scientific establishment. She says, "
  ... it is shameful that there are so few women
  in science... In China there are many, many
  women in physics. There is a misconception in
  America that women scientists are all dowdy
  spinsters. This is the fault of men. In Chinese
  society, a woman is valued for what she is, and
  men encourage her to accomplishments --- yet she
  retains eternally feminine."
• Idea from experiment in collaboration with Lee
  and Yang Look at spin of decay products of
  polarized radioactive nucleus
• Production mechanism involves exclusively weak
  interaction

22
Intermezzo Spin and Parity

• How does the decay of a particle with spin tell
  you something about parity?
• Gedanken-experiment decay of X ? a b
• Spin 1,1gt ? ½, ½ gt ½, ½gt
• It is important that X is maximally polarized
  only then there is a single solution for the spin
  of the decay products. If not, e.g.
• 1,0gt ? ½, ½gt ½, -½gt
• 1,0gt ? ½, -½gt ½, ½gt

?
23
Intermezzo Spin and Parity and Helicity

• We introduce a new quantity Helicity the
  projection of the spin on the direction of flight
  of a particle

H1 (right-handed)
H-1 (left-handed)
24
Intermezzo Spin and Parity and Helicity

• Spin is quantized ? Helicity is quantized
• Possible H values for S1/2 H-1 and H1
• Most particles are linear combination of H1 and
  H-1 states
• Angular distribution for particles observed in
  specific helicity eigenstate

I(q)RH 1 - (v/c) cos q
I(q)LH 1 (v/c) cos q

ConstantIf both helicitiesare producedequally
in decay.If not angulardistribution willnot
be flat
Superpositionof H1 and H-1states
25
Note on Helicity

• Note that Helicity is not generally a Lorentz-
  invariant observable
• Sign of particle momentum p is relative to
  observer.
• A second observer overtaking the particle from
  the lab observer perspective will see the
  particle moving in the opposite direction (p
  -p) ? It see the opposite Helicity
• Exception for massless particles
• You cannot overtake massless particles moving at
  speed of light
• Helicity for massless particles is
  Lorentz-invariant intrinsic property

26
A realistic experiment the Wu experiment (1956)

• Observe radioactive decay of Cobalt-60 nuclei
• The process involved 6027Co ? 6028Ni e- ne
• 6027Co is spin-5 and 6028Ni is spin4, both e- and
  ne are spin-½
• If you start with fully polarized Co (SZ5) the
  experiment is essentially the same (i.e. there is
  only one spin solution for the decay) 5,5gt ?
  4,4gt ½ ,½gt ½,½gt

S4
27
The Wu experiment 1956

• Experimental challenge how do you obtain a
  sample of Co(60) where the spins are aligned in
  one direction
• Wus solution adiabatic demagnitization of
  Co(60) in magnetic fields at very low
  temperatures (1/100 K!). Extremely challenging
  in 1956!

28
The Wu experiment 1956

• 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
29
The Wu experiment 1956

• 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!

30
Our universe vs a parity-flipped universe

• What happens to helicity in parity-flipped
  universe?
• Momentum flips sign
• Spin stays the same
• Helicity is product and flips sign
• Conclusion
• Any process that produces right-handed
  anti-neutrinos in our universe will produce
  left-handed anti-neutrinos in the mirrored
  universe.
• If left and right-handed neutrinos are not
  produced at the same rate the physics in the
  mirrored universe is different

Orientation of spin
righthanded H1
Direction of motion
P
lefthanded H-1
Orientation of spin
Direction of motion
31
Parity violation in weak decays

• Apply parity operation to 60Co decay

P-Flipped universe
Our universe
RH ne
RH ne
e-
e-
LH ne
LH ne
e-
e-
32
Parity violation in weak decays

• Apply parity operation to 60Co decay

P-Flipped universe(LH anti-neutrinos only)
Our universe(RH anti-neutrinos only)
Allowed
Forbidden
Allowed
Forbidden
RH ne
RH ne
e-
e-
LH ne
LH ne
e-
e-
Preferential direction of electronsis forward
Preferential direction of electronsis backward
33
So P is violated, whats next?

• Wus 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

p
m
nm
34
Charge conjugation symmetry

• 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
p
m
nm(LH)
C
nm(LH)
p-
m-
OK
35
The Weak force and C,P parity violation

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

?
??
??
P
C
CP appears to be preservedin weakinteraction!
?
?
??
??
?
??
CP
36
What do we know now?

• 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 p 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
• Next a precision test of CP symmetry
  conservation in the weak interaction

37
Conserved properties associated with C and P

• 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 hadronPygt pygt
• 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 P1 then Pygt 1ygt (wave function
  symmetric in space)if P-1 then Pygt -1 ygt
  (wave function anti-symmetric in space)

38
Figuring out P eigenvalues for hadrons

• QFT rules for particle vs. anti-particles
• Parity of particle and anti-particle must be
  opposite for fermions (spin-N1/2)
• 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 P1111, anti-baryons have
  P-1-1-1-1
• (Anti-)mesons have P1-1 -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 JLS formalism, it has an
  intrinsic component and an orbital component
• NB Photon is spin-1 particle has intrinsic P of
  -1

39
Parity eigenvalues for selected hadrons

• 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 L1 ? extra term (-1)1
• P(K1) 1

Meaning Ppgt -1pgt
40
Figuring out C eigenvalues for hadrons

• Only particles that are their own anti-particles
  are C eigenstates because Cxgt ? xgt cxgt
• 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)LS
• 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 C1(p0 has spin 0) ?
  (-1)(-1)
• Process p0 ? g g g does not occur (and would
  violate C conservation)

41
Introduction to K0 physics

• Now focusing on another little mystery in the
  domain of strange mesons what precisely is a
  K0 meson?
• Quark contents K0 ?sd,?K0 ?ds
• First what is the effect of C and P conjugation
  on the K0/K0-bar particles?
• PK0gt -1K0gt (because ?qq pair)
• PK0gt -1K0gt (because ?qq pair)
• CK0gt K0gt (because K0 is anti-particle of K0
  ls0)
• CK0gt K0gt (because K0 is anti-particle of K0
  ls0)
• Knowing this we can evaluate the effect of CP on
  the K0
• CPK0gt -1K0gt
• CPK0gt -1 K0gt

42
What are the K0 CP Eigenstates

• Thus K0 and K0 are not CP eigenstates,
• Somewhat strange since they decay through weak
  interaction which appears to conserve CP!
• Nevertheless it is possible to construct CP
  eigenstates as linear combinations
• Remember QM You can always construct wave
  functions as linear combinations of the solutions
  of any operator (like the Hamiltonian)
• K1gt 1/?2(K0gt - K0gt)
• K2gt 1/?2(K0gt K0gt)
• Proof is exercise for today
• Does it make sense to look at linear combinations
  of K0gt and K0gt?
• I.e does K1gt represent a real particle?

43
K0/K0 oscillations

• Well it might, because it turns out that the weak
  interaction can turn a K0 particle into a K0
  particle!
• Weird? Yes! Impossible? No!
• In can be done using two consecutive weak
  interactions
• Important implication for nature of K0 particle
• Q If a K0 can turn into a K0 at any moment, how
  do you know what particle youre dealing with?
• A You dont. Any particle in the lab is always a
  linear combination of the two ? It makes as much
  sense to talk about K1 and K2 eigenstates as it
  does to talk about K0 and K0 eigenstates

Weak force
Weak force
K0
(X)
K0
S1
S0
S-1
DS-1
DS-1
44
So what is the K0 really?

• The K0 meson is something you can only describe
  with quantum mechanics
• It is a linear combination of two particles!
• You can either see it as a combination of K0gt
  and ?K0gt,eigenvalues of Strangeness (1 vs 1)
  and undefined CP
• Or you can see it as a combination of K1gt and
  K2gt,eigenvalues of CP (1, -1), with undefined
  strangeness
• Both representations are equivalent.

• Kaons are typically produced in strong
  interactions in a strangeness eigenstate (K0 or
  ?K0)
• Kaons decay through the weak interaction as
  eigenstates of CP (K1 or K2)
• Since K1 and K2 dont have definite S, S is not
  conserved in weak decays as we already know
• If you believe a particle must have a single well
  defined lifetime, then K1 and K2 are the real
  particles

45
So what is the K0 really?

• Graphical analogy Any object with two
  components can be decomposed in more than one way

Anti-K0
K2
Kgt
K0
K1
46
Decays of neutral kaons

• Neutral kaons is the lightest strange particle ?
  it must decay through the weak interaction
• If weak force conserves CP then
• decay products of K1 can only be a CP1 state,
  i.e.K1gt (CP1) ? p p (CP
  (-1)(-1)(-1)l0 1)
• decay products of K2 can only be a CP-1 state,
  i.e.K2gt (CP-1) ? p p p (CP
  (-1)(-1)(-1)(-1)l0 -1)
• You can use neutral kaons to precisely test that
  the weak force preserves CP (or not)
• If you (somehow) have a pure CP-1 K2 state and
  you observe it decaying into 2 pions (with CP1)
  then you know that the weak decay violates CP

( S(K)0 ? L(pp)0 )
47
Designing a CP violation experiment

• How do you obtain a pure beam of K2 particles?
• It turns out that you can do that through clever
  use of kinematics
• Exploit that decay of K into two pions is much
  faster than decay of K into three pions
• Related to fact that energy of pions are large in
  2-body decay
• t1 0.89 x 10-10 sec
• t2 5.2 x 10-8 sec (600 times larger!)
• Beam of neutral Kaons automatically becomes beam
  of K2gt as all K1gt decay very early on

K1 decay early (into pp)
Pure K2 beam after a while!(all decaying into
ppp) !
Initial K0 beam
48
The Cronin Fitch experiment
Essential idea Look for (CP violating) K2 ? pp
decays 20 meters away from K0 production point
Decay of K2 into 3 pions
Incoming K2 beam
If you detect two of the three pionsof a K2 ?
ppp decay they will generallynot point along the
beam line
49
The Cronin Fitch experiment
Essential idea Look for K2 ? pp decays20 meters
away from K0 production point
Decay pions
Incoming K2 beam
If K2 decays into two pions instead ofthree both
the reconstructed directionshould be exactly
along the beamline(conservation of momentum in
K2 ? pp decay)
50
The Cronin Fitch experiment
Essential idea Look for K2 ? pp decays20 meters
away from K0 production point
Decay pions
K2 ? pp decays(CP Violation!)
Incoming K2 beam
K2 ? ppp decays
Result an excess of events at Q0 degrees!
Note scale 99.99 of K ?ppp decaysare left of
plot boundary
51
Cronin Fitch Discovery of CP violation

• Conclusion weak decay violates CP (as well as C
  and P)
• But effect is tiny! (0.05)
• Maximal (100) violation of P symmetry easily
  follows from absence of right-handed neutrino,
  but how would you construct a physics law that
  violates a symmetry just a tiny little bit?
• Results also provides us withconvention-free
  definition ofmatter vs anti-matter.
• If there is no CP violation, the K2 decaysin
  equal amounts to p e- ne (a) p- e ne (b)
• Just like CPV introduces K2 ? pp decays, it also
  introduces a slight asymmetry in the above
  decays (b) happens more often than (a)
• Positive charge is the charged carried by the
  lepton preferentially produced in the decay of
  the long-lived neutral K meson

52
Kaons K0,?K0, K1, K2, KS, KL,

• The kaons are produced in mass eigenstates
• K0gt ?sd
• ?K0gt ?ds
• The CP eigenstates are
• CP1 K1gt 1/?2 (K0gt - ?K0gt)
• CP -1 K2gt 1/?2 (K0gt ?K0gt)
• The kaons decay to 2 (Short-lived) or 3 pions
  (Long-lived)
• KSgt predominantly CP1
• KLgt predominantly CP -1

• ?- (2.286 0.014) x 10-3
• e (2.284 0.014) x 10-3

53
What do we know now?

• C and P are both violated by the weak interaction
  because neutrinos can only be produced in a
  single helicity state (LH neutrinos and RH
  anti-neutrinos)
• Results from Wu and Lederman experiments
• You can associate a P quantum number to all
  hadrons and a C quantum number to all hadrons
  that are their own anti-particle
• These C and P quantum numbers are conserved in
  any reaction not involving the weak interaction
• The K0 meson is a special particle that really
  exists as a combination of two particle the K0 /
  K0 combination or the K1 / K2 combination at your
  choice
• The Cronin and Fitch experiment shows that in
  addition to C and P violation the product CP is
  also violated, but only a tiny little bit
• Origin of CP violation sofar unknown ? Topic of
  future section

54
Schematic picture of selected weak decays

• K0 ? K0 transition
• Note 1 Two W bosons required (DS2 transition)
• Note 2 many vertices, but still lowest order
  process

?s
?d
W
u
u
K0
K0
W
s
d
55
Strangeness violation in W mediated decays

• In 1963 N. Cabibbo made the first step to
  formally incorporate strangeness violation in W
  mediated decays
• For the leptons, transitions only occur within a
  generation
• For quarks the amount of strangeness violation
  can be neatly described in terms of a rotation,
  where qc13.1o

Weakforcetransitions
u
Idea weak interaction couples to different
eigenstates than strong interactionweak
eigenstates can be writtenas rotation of
strongeigenstates
W
d d?cosqc s?sinqc
56
Cabibbos theory successfully correlated many
decay rates

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

57
Cabibbos theory successfully correlated many
decay rates

• 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)

d
?s
cosqc
sinqc
u
W
W
nm
m
m-
58
The Cabibbo-GIM mechanism

• 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
dcos(qc)dsin(qc)s
s-sin(qc)dcos(qc)s
59
The Cabibbo-GIM mechanism

• Cabibbo-GIM mechanism introduces clean formalism
  for quark flavour violations in the weak
  interactionThe weak interaction (W boson)
  couples to a rotated set of down-type
  states
• Tiny problem at time of introduction there was
  no evidence for a 4th quark
• Fourth charm quark discovered in 1974 ?
  Vindication of Cabbibo/GIM mechanism

Leptonsectorunmixed
Quark section mixed throughrotation of weak
w.r.t. strong eigenstates by qc
60
The Cabibbo-GIM mechanism

• How does it solve the K0 ? mm- 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

d
?s
?s
d
-sinqc
cosqc
cosqc
sinqc
c
u
nm
nm
m
m-
m
m-
61
From 2 to 3 generations

• 2 generations d0.97 d 0.22 s (?c13o)
• 3 generations d0.97 d 0.22 s 0.003 b
• NB probabilities have to add up to 1
  0.9720.2220.00321
• ? Unitarity !

62
What do we know about the CKM matrix?

• 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
63
How do you measure those numbers?

• 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 0.9735 0.0008
• Vud of order 1

64

• End of Lecture 1

• 1 May
• Introduction matter and anti-matter
• P, C and CP symmetries
• K-system
• CP violation
• Oscillations
• Cabibbo-GIM mechanism
• 8 May
• CP violation in the Lagrangian
• CKM matrix
• B-system
• 15 May
• B-factories
• B?J/Psi Ks
• Delta ms

65
Exercises Spin in decay helicity

• Decay topology consider the process
  6027Co?6028Nie-??e where the e and the?? are
  emitted along the polarization (z) axis. (See
  illustration.) We are now interested in the
  orientation of the spin of the electron/anti-neutr
  ino w.r.t to the direction of flight.
• Show that there are two possibilities for decays
  along the 60Co spin axis
• one where the electron spin points to the same
  direction has the direction of motion and
• one where the electron spin points to the
  opposite direction of the direction of motion
• What are the helicities of the electron and
  anti-neutrino for the twoemission possibilites,
  respectively. (Helicity is defined as
  )

66
Exercise Pion decay p-? l-??l

• Given a p- (spin 0) in rest that decays into a m-
  and a ??m (both spin ½)
• Draw the decay for both spin orientation
  possibilities
• What is the helicity of the anti-neutrino for
  both possibilities?
• Which of the two processes does not occur in
  nature?
• Draw the C-conjugated diagram of the p decay
  that occurs in nature. Does the C-conjugated
  process occur in nature? Explain why (not)?
• Draw the C and P conjugated diagram of the p-
  decay that occurs in nature. Does this decay
  occur in nature?
• Can the following decay occur p-? e-??e ?
• The (V-A) theory predicts that the decay rate
  G(p-? l-??l ) 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 (p2dp/dE),
  where p is the lepton momentum and E the total
  energy in the pion rest frame.
• Estimate the ratio of branching ratios G(p-?
  e-??e ) /G(p-? µ-??µ)

67
Exercises CP Eigenstates of the K0

• Consider K1 and K2
• K1gt 1/?2(K0gt - ?K0gt)
• K2gt 1/?2(K0gt ?K0gt)
• Prove that K1 and K2 are eigenstates of the CP
  operation and determine the eigenvalues of K1 and
  K2.
• Given CP conservation, how do K1 and K2 decay?
• Why is the lifetime of K2 so much larger?
  (mpi140 MeV, mK490MeV)
• 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.
• 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)
• Given that the decay time of the K1 is t1 0.89
  x 10-10 sec, what fraction of the K1 has not
  decayed yet after 20 meters
• What is the ratio of K1 to K2 decays after 20
  meters
• 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?

68
Exercise Weak decay diagrams

• Feynman diagrams for weay decays
• Draw the diagram for p decay.
• Indicate the CKM matrix element labels at each
  weak vertex
• Draw the diagram for L0 ? pnee decay
• Indicate the CKM matrix element labels at each
  weak vertex
• 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.

69
Solutions
70
Solution to exercises

• Exercise 1
• The two possible configurationsare shown on the
  right. The blackarrows show the orientation
  ofthe spin. The red dashed arrowsshow the
  direction of motion
• In the left scenario H(e)1, H(n)-1In the
  right scenarioH(e)-1, H(n)1
• See book

e-
ne
S1/2
S4
S4
ne
e-
S1/2
71
Solution to exercises

• Exercise 2
• See below
• Top H(n) RH, Bottom H(n)LH
• Only the top process occurs in nature (there
  exist only left-handed neutrinos and right-handed
  anti-neutrinos)

p-
m-
nm
72
Solution to exercises

• Exercise 2 continued
• The C-conjugated diagram of the p- decay that
  occurs in nature is
• The C conjugated process does not occur in
  nature, as it features a right-handed neutrino.
• The CP conjugated diagram of the p- decay that
  occurs in nature is
• This process does occur in nature, just like in
  the original p- decay process, because it
  features a left-handed neutrino. CP therefore
  appears to be a conserved symmetry in the weak
  interaction

p
m
nm
p
m
nm
73
Solution to exercises

• Exercise 2 continued
• Yes, but is surpressed! e- wants to be left
  handed because it is almost massless.
• See book

74
Solution to exercises

• Exercise 3
• Start with the known CP properties of the K0 and
  K0
• PK0gt -K0gt, because quark-antiquark gives
  (1)(-1)(-1) and S,L0
• CPK0gt C(-K0gt) -1K0gt
• CPK0gt -1 K0gt
• To prove that K1gt is an eigenvalue of CP we
  should show that the CP operation on K1gt yields
  a constant times K1gt.Calculate effect of CP on
  K1gt (omitting norm factor 1/?2)
• CP(K0gt - K0gt) (-K0gt K0gt) 1(K0gt -
  K0gt) 1K1gt
• Same thing for K2gt
• CP(K0gt K0gt) (-K0gt - K0gt) -1(K0gt
  K0gt) -1K2gt
• K1 ? pipi, K2?pipipi
• 3 times pion mass is almost kaon mass hardly any
  phase space!

75
Solution to exercises

• Exercise 3 continued
• Distribution of decays follows f(t)exp(-t/t)
  with t5.2x10-8.To calculate the fraction that
  decays after 20 meters we need to calculate the
  fraction that decays after 20m/c 6.6x10-8 sec
• The primitive of f(t) is F(t) -t exp(-t/t)
  F(0) -t -5.2x108, F(?)0,
  F(20/c)-texp(-(20/c)/t) -5.33x10-10Fraction
  (K2) exp(-20/ct) 27.7
• For K1 fraction formula is the same but with
  different t, Fraction(K1) exp(-749)
  10(-749/ln(10))10-325.29 1.9x10-325The ratio
  is 1.9x10-325 / 27.7 6.9 x 10-325.
• Fraction of decays between 20 and 21 meters is
  exp(-20/ct)-exp(-21/ct) 0.27746 0.2602
  0.017 1.7 The number of K2s to produce is
  therefore 22700 / 1.7 / 1 133 million

76
Solution to exercises

• Exercise 4
• Feynman diagram for p- decay (u d)

p-
d
u
Vud
W-
nm
m-
77
Solution to exercises

• Exercise 4
• Feynman diagram for L0 ? pnee decay (L0 uds)

L0
u
d
s
W-
Vus
e-
ne (RH)
u
d
u
p
78
Solution to exercises

• Exercise 4
• Two Feynman diagrams for K0 K0 mixing. You can
  make additional ones by replacing one or both of
  the c quarks with u or t quarks

Vcs
d
Vcd
s
c
c
K0
K0
s
d
Vcd
Vcs
Vcs
d
c
Vcd
s
K0
K0
s
d
c
Vcs
Vcd