1. Particle Physics II – CP violation (also known as “Physics of Anti-matter”) Lecture 3
N. Tuning
Niels Tuning (1)

2. Plan Mon 5 Feb: Anti-matter + SM
Wed 7 Feb: CKM matrix + Unitarity Triangle
Mon 26 Feb: Mixing + Master eqs. + B0J/ψKs
Wed 28 Feb: CP violation in B(s) decays (I)
Mon 12 Mar: CP violation in B(s) and K decays (II)
Wed 14 Mar: Rare decays + Flavour Anomalies
Wed 21 Mar: Exam
Final Mark:
if (mark > 5.5) mark = max(exam, 0.85*exam *homework)
else mark = exam
In parallel: Lectures on Flavour Physics by prof.dr. R. Fleischer
Niels Tuning (2)

4. Recap uI W dI u W d,s,b Diagonalize Yukawa matrix Yij Mass terms
Quarks rotate
Off diagonal terms in charged current couplings u
d,s,b W
Replace the derivative with the covariant derivative introducing the force carriers: gluons, weak vector bosons and photon.
The constants g_s, g and g’ are the corresponding coupling constants of the gauge particles.
Niels Tuning (4) 4

5. Charged Currents In general the charged current term is CP violating
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 if Vij = Vij*
In general the charged current term is CP violating
Niels Tuning (5) 5

6. CKM-matrix: where are the phases?
Possibility 1: simply 3 ‘rotations’, and put phase on smallest:
Possibility 2: parameterize according to magnitude, in O(λ): u
d,s,b W
Niels Tuning (6)

7. This was theory, now comes experiment
We already saw how the moduli |Vij| are determined
Now we will work towards the measurement of the imaginary part
Parameter: η
Equivalent: angles α, β, γ .
To measure this, we need the formalism of neutral meson oscillations…
Niels Tuning (7)

8. Neutral Meson Oscillations
Why?
Loop diagram: sensitive to new particles
Provides a second amplitude
interference effects in B-decays b s
Niels Tuning (8)

9. Dynamics of Neutral B (or K) mesons…
Time evolution of B0 and B0 can be described by an effective Hamiltonian:
No mixing, no decay…
No mixing, but with decays…
(i.e.: H is not Hermitian!)
With decays included, probability of observing
either B0 or B0 must go down as time goes by:
Niels Tuning (9)

10. Describing Mixing…
Time evolution of B0 and B0 can be described by an effective Hamiltonian:
Where to put the mixing term?
Now with mixing – but what is the
difference between M12 and G12?
M12 describes B0  B0 via off-shell states, e.g. the weak box diagram
G12 describes B0fB0 via on-shell states, eg. f=p+p-
Niels Tuning (10)

11. Solving the Schrödinger Equation
Eigenvalues:
Mass and lifetime of physical states: mass eigenstates
The physical particles are the eigenstates of the Hamiltonian. Solve the eigenvalues lambda by putting det (H11-lambda*I) = 0
This you can not see here. You just have to sit down and *do* it.
Niels Tuning (11)

12. Solving the Schrödinger Equation
Eigenvectors:
mass eigenstates
The physical particles are the eigenstates of the Hamiltonian. Solve the eigenvalues lambda by putting det (H11-lambda*I) = 0
This you can not see here. You just have to sit down and *do* it.
Niels Tuning (12)

13. Time evolution
With diagonal Hamiltonian, usual time evolution is obtained:
Niels Tuning (13)

14. B Oscillation Amplitudes
For an initially produced B0 or a B0 it then follows: (using:
with
For B0, expect:
DG ~ 0,
|q/p|=1
These formula’s just follow from substitution. No magic. Note that in the last step g_ gets a 90 degree phase ! ~
Niels Tuning (14)

15. Measuring B Oscillations
For B0, expect:
DG ~ 0,
|q/p|=1
Examples: ~
B0B0
Decay probability
Use cos^2 (dmt/2) = 0.5 * (1 + cos(dmt))
>
Proper Time 
Niels Tuning (15)

16. Probability to measure P or P, when we start with 100% P
Compare the mesons:
Probability to measure P or P, when we start with 100% P
P0P0
P0P0
<> Δm
x=Δm/Γ
y=ΔΓ/2Γ K0 s
5.29 ns-1
Δm/ΓS=0.49 ~1 D0 s
0.001 fs-1 ~0
0.01 B0 s
0.507 ps-1
0.78
Bs0 s
17.8 ps-1
12.1
~0.05
Probability 
By the way,
ħ= MeVs
x=Δm/Γ: avg nr of
oscillations before decay
Time 
Niels Tuning (16)

17. Summary (1) Start with Schrodinger equation: Find eigenvalue:
Solve eigenstates:
Eigenstates have diagonal Hamiltonian: mass eigenstates!
(2-component state in
P0 and P0 subspace)
Niels Tuning (17)

18. Summary (2) Two mass eigenstates Time evolution:
Probability for |P0>  |P0> !
Express in M=mH+mL and Δm=mH-mL  Δm dependence

19. Time dependence (if ΔΓ~0, like for B0):
Summary
p, q:
Δm, ΔΓ:
x,y: mixing often quoted in scaled parameters:
q,p,Mij,Γij related through:
Time dependence (if ΔΓ~0, like for B0):
with
Niels Tuning (19)

20. Personal impression:
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 s 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 ( ± 0.28 ) × 10−2
Γ347 ν ν γ <4.7 × 10−5 CL=90%
And for one decay there are often more than one decay amplitudes…
Niels Tuning (20)

21. Describing Mixing
Time evolution of B0 andB0 can be described by an effective Hamiltonian:
M12 describes B0  B0 via off-shell states, e.g. the weak box diagram
G12 describes B0fB0 via on-shell states, eg. f=p+p-
Niels Tuning (21)

22. Box diagram and Δm Inami and Lim, Prog.Theor.Phys.65:297,1981
Niels Tuning (22)

23. Box diagram and Δm
Niels Tuning (23)

24. Box diagram and Δm: Inami-Lim
K-mixing
C.Gay, B Mixing, hep-ex/

25. Box diagram and Δm: Inami-Lim
B0-mixing
C.Gay, B Mixing, hep-ex/

26. Box diagram and Δm: Inami-Lim
Bs0-mixing
C.Gay, B Mixing, hep-ex/

27. Next: measurements of oscillations
B0 mixing:
1987: Argus, first
2001: Babar/Belle, precise
Bs0 mixing:
2006: CDF: first
2010: D0: anomalous ??
Niels Tuning (27)

28. B0 mixing
Niels Tuning (28)

29. B0 mixing
What is the probability to observe a B0/B0 at time t, when it was produced as a B0 at t=0?
Calculate observable probility Y*Y(t)
A simple B0 decay experiment.
Given a source B0 mesons produced in a flavor eigenstate |B0>
You measure the decay time of each meson that decays into a flavor eigenstate (either B0 orB0) you will find that
Exercise idea: compare tau, dm of K0 and B0
Niels Tuning (29) 29

30. B0 mixing: 1987 Argus B0 oscillations: First evidence of heavy top
 mtop>50 GeV
Needed to break GIM cancellations
NB: loops can reveal heavy particles!
Phys.Lett.B192:245,1987
Niels Tuning (30)

31. B0 mixing: t quark GIM: c quark
B0 mixing pointed to the top quark:
K0→μμ pointed to the charm quark:
ARGUS Coll, Phys.Lett.B192:245,1987
GIM, Phys.Rev.D2,1285,1970 … d s μ b d

32. B0 mixing: 2001 B-factories
You can really see this because (amazingly) B0 mixing has same time scale as decay
t=1.54 ps
Dm=0.5 ps-1
50/50 point at pDm  t
Maximal oscillation at 2pDm  2t
Actual measurement of B0/B0 oscillation
Also precision measurement of Dm!
Exercise idea: compare tau, dm of K0 and B0
Niels Tuning (32) 32

33. Bs0 mixing
Niels Tuning (33)

34. Bs0 mixing: 2006
Niels Tuning (34) 34

35. Bs0 mixing (Δms): SM Prediction
CKM Matrix
Wolfenstein parameterization
Vts
Ratio of frequencies for B0 and Bs
Vts ~ 2
Vtd ~3  Δms ~ (1/λ2)Δmd ~ 25 Δmd
 = from lattice QCD
-0.035
Niels Tuning (35)

36. Bs0 mixing (Δms): Unitarity Triangle
CKM Matrix Unitarity Condition
Niels Tuning (36) 36

37. Bs0 mixing (Δms) cos(Δmst) Proper Time t (ps)
Δms=17.77 ±0.10(stat)±0.07(sys) ps-1
hep-ex/ - - Bs b s t W g̃ x b̃ s̃ - -
cos(Δmst) - - -
Proper Time t (ps)
Niels Tuning (37)

38. Bs0 mixing (Δms): New: LHCbt W g̃ x b̃ s̃
LHCb, arXiv:
Niels Tuning (38)

39. Bs0 mixing (Δms): New: LHCb
Ideal
resolution
Tagging,
Acceptance
LHCb, arXiv:
Niels Tuning (39)

40. Mixing  CP violation?
NB: Just mixing is not necessarily CP violation!
However, by studying certain decays with and without mixing, CP violation is observed
Next: Measuring CP violation… Finally
Niels Tuning (40)

41. Meson Decays Formalism of meson oscillations: Subsequent: decay
Niels Tuning (41)

42. Notation: Define Af and λf
Niels Tuning (42)

43. Some algebra for the decay P0  f
Interference
P0 f
P0P0 f
Niels Tuning (43)

44. Some algebra for the decay P0  f
Niels Tuning (44)

45. The ‘master equations’
(‘direct’) Decay
Interference
Niels Tuning (45)

46. The ‘master equations’
(‘direct’) Decay
Interference
Niels Tuning (46)

47. Classification of CP Violating effects
CP violation in decay
CP violation in mixing
CP violation in interference
Niels Tuning (47)

48. What’s the time?
Niels Tuning (48)