1. Particle Physics II – CP violation Lecture 3
N. Tuning
Niels Tuning (1)

2. Outline 1 May 8 May 15 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
Niels Tuning (2)

3. Literature
Slides based on courses from Wouter Verkerke and Marcel Merk.
W.E. Burcham and M. Jobes, Nuclear and Particle Physics, chapters 11 and 14.
Z. Ligeti, hep-ph/ , Introduction to Heavy Meson Decays and CP Asymmetries
Y. Nir, hep-ph/ , CP Violation – A New Era
H. Quinn, hep-ph/ , B Physics and CP Violation
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4. : The Kinetic Part Recap from last week uLI W+m L=JmWm dLI
For example, the term with QLiI becomes:
Writing out only the weak part for the quarks:
Writing out the weak part leads to terms that can be illustrated by Feynman diagrams. The interactions are usually written as a current J times a gauge field.
Exercise:
To show this use the defitions of the parity and charge operators. Start by setting q_l = ½ (1+g_5) q, etc. Use also (1-g_5)g_0 = g_0(1+g_5). Show for a few terms that Parity leads to q_l -> q_R. For C it goes similarly using transformation properties under C.
In fact, one can see immediately that, since the lagrangian contains vector terms and axial vector terms, which transform with opposite sign, that it cannot be Parity or charge conserving. Under CP, both the vector and axial currents are odd.
dLI g
W+m
uLI
W+ = (1/√2) (W1+ i W2)
W- = (1/√ 2) (W1 – i W2)
L=JmWm
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5. : The Higgs Potential V(f) V(f) f f Recap from last week Broken
Symmetry
~ 246 GeV
V(f) f
Symmetry
Spontaneous Symmetry Breaking: The Higgs field adopts a non-zero vacuum expectation value
Procedure:
Substitute:
The Higgs potential has an asymmetric minimum in one of the 4 degrees of freedom: Re(phi_0). Only this component remains in the final Lagrangian: the Higgs field. The other three components are absorbed in the mass terms of the W+, W-, Z bosons. The photon remains massless.
And rewrite the Lagrangian (tedious): .
The W+,W-,Z0 bosons acquire mass
The Higgs boson H appears
(The other 3 Higgs fields are “eaten” by the W, Z bosons)
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6. : The Yukawa Part Recap from last week
Since we have a Higgs field we can add (ad-hoc) interactions between f and the fermions in a gauge invariant way.
doublets
singlet
The result is:
i, j : indices for the 3 generations!
Remember that the Lagrangian is a scalar object (it must respect Lorentz invariance). How can we make scalars with left handed doublets, higgs doublets and right handed singlets?
Y_ij are Yukawa coupling constants, the lefthanded fermion doublet and the higgs doublet together make a scalar to be multiplied by the righthanded fermion scalar.
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!
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7. S.S.B
: The Fermion Masses
Recap from
last week
Writing in an explicit form:
The matrices M can always be diagonalised by unitary matrices VLf and VRf such that:
Then the real fermion mass eigenstates are given by:
dLI , uLI , lLI are the weak interaction eigenstates
dL , uL , lL are the mass eigenstates (“physical particles”)
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8. : The Charged Current
The charged current interaction for quarks in the interaction basis is:
Recap from
last week
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
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9. The Standard Model Lagrangian (recap)
Recap from
last week
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
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10. Exploit apparent ranking for a convenient parameterization
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)
Recap from
last week
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11. Deriving the triangle interpretation
Recap from
last week
Starting point: the 9 unitarity constraints on the CKM matrix
Pick (arbitrarily) orthogonality condition with (i,j)=(3,1)
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12. Visualizing arg(Vub) and arg(Vtd) in the (r,h) plane
Recap from
last week
We can now put this triangle in the (r,h) plane
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13. 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:
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14. 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-
For details, look up “Wigner-Weisskopf” approximation…
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15. Solving the Schrödinger Equation
Solution:
(a and b are initial conditions):
Eigenvectors:
Dm and DG follow from the Hamiltonian:
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.
From the eigenvalue calculation:
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16. 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 !
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17. 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 
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18. Measuring 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
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19. Measuring B0 mixing
You can really see this because (amazingly) B0 mixing has same time scale as decay
t=1.54 ps
Dm=0.47 ps-1
50/50 point at pDm  t
Maximal oscillation at 2pDm  2t
Actual measurement of B0/B0bar oscillation
Also precision measurement of Dm!
Exercise idea: compare tau, dm of K0 and B0
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20. Back to finding new measurements
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 b g
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21. The B0 mixing formalism and the angle b
Reduction to single (set of 2) amplitudes is major advantage in understanding B0 mixing physics
A mixing diagram has (to very good approximation) a weak phase of 2b
An experiment that involves interference between an amplitude with mixing and an amplitude without mixing is sensitive to the angle b!
Small miracle of B physics: unlike the K0 system you can easily interpret the amount of observable CP violation to CKM phases!
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22. Find the right set of two amplitudes
General idea to measure b: Look at interference between B0  fCP and B0  B0  fCP
Where fCP is a CP eigenstate (because both B0 and B0 must be able to decay into it)
Example (not really random): B0  J/y KS
B0  f
B0  B0  f
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23. Back to business – Measuring b with B0  J/y KS
Were going to measure arg(Vtd2)=2b through the interference of these two processes
We now know from the B0 mixing formalism that the magnitude of both amplitudes varies with time
B0  f
B0  B0  f
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24. How can we construct an observable that measures b
What do we know about the relative phases of the diagrams?
B0  f
B0  B0  f
f(strong)=f
Decays are identical
f(strong)=f
K0 mixing exactly cancels Vcs
f(weak)=0
f(weak)=2b
f(mixing)=p/2
There is a phase difference of i between the B0 and B0bar
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25. Measuring ACP(t) in B0  J/y KS
What do we need to observe to measure
We need to measure
J/y and KS decay products
Lifetime of B0 meson before decay
Flavor of B0 meson at t=0 (B0 or B0bar)
First two items relatively easy
Lifetime can be measured from flight length if B0 has momentum in laboratory
Last item is the major headache: How do you measure a property of a particle before it decays?
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26. Putting it all together: sin(2b) from B0  J/y KS
B0(Dt)
B0(Dt)
ACP(Dt) = sin(2β)sin(DmdDt)
Effect of detector imperfections
Dilution of ACP amplitude due imperfect tagging
Blurring of ACP sine wave due to finite Dt resolution
sin2b
Imperfect flavor tagging
Dsin2b
Finite Dt resolution Dt Dt
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27. Combined result for sin2b
hep-ex/
Combined result for sin2b
(cc) KS (CP odd) modes
J/ψ KL (CP even) mode
ACP amplitude dampened by (1-2w) w  flav. Tag. mistake rate
sin2β =  (stat)  (sys)
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28. Consistency with other measurements in (r,h) plane
4-fold ambiguity because we measure sin(2b), not b
Prices measurement of sin(2b) agrees perfectly with other measurements and CKM model assumptions The CKM model of CP violation experimentally confirmed with high precision! 2 1
without sin(2b) h 3 4 r
Method as in Höcker et al, Eur.Phys.J.C21: ,2001
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29. Back to business – Measuring b with B0  J/y KS
Were going to measure arg(Vtd2)=2b through the interference of these two processes
We now know from the B0 mixing formalism that the magnitude of both amplitudes varies with time
B0  f
B0  B0  f
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30. How can we construct an observable that measures b
The easiest case: calculate G(B0  J/y KS) at t=p / 2Dm
Why is it easy: cos(Dmt)=0  both amplitudes (with and without mixing) have same magnitude: |A1|=|A2|
Draw this scenario as vector diagram
NB: Both red and blue vectors have unit length + =
sin(f)
p/2+2b
N(B0  f)  |A|2  (1-cosf)2+sin2f
= 1 -2cosf+cos2f+sin2f
= 2-2cos(p/2+2b)
 1-sin(2b)
1-cos(f)
cos(f)
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31. How can we construct an observable that measures b
Now also look at CP-conjugate process
Directly observable result (essentially just from counting) measure CKM phase b directly!
N(B0  f)  |A|2  (1-cosf)2+sin2f
= 1 -2cosf+cos2f+sin2f
= 2-2cos(p/2+2b)
 1-sin(2b) + =
sin(f)
p/2+2b CP
1-cos(f) + =
sin(f)
p/2-2b
N(B0  f)  (1+cosf)2+sin2f
= 2+2cos(p/2-2b)
 1+sin(2b)
1+cos(f)
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32. Bs mixing Δms has been measured at Fermilab 4 weeks ago!
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33. Standard Model Prediction
CKM Matrix
Wolfenstein parameterization
Vts
Ratio of frequencies for B0 and Bs
 = from lattice QCD
-0.035
(hep/lat )
Vts ~ 2, Vtd ~3, =0.224±0.012
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34. CKM Matrix Unitarity Condition
Unitarity Triangle
CKM Matrix Unitarity Condition
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35. Before the measurement: Unitarity Triangle Fit
CKM Fit result: Dms: (1s) : (2) ps-1
-1.5
-2.7
from Dmd
Lower limit on Dms
from Dmd/Dms
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36. Measurement .. In a Perfect World
“Right Sign”
“Wrong Sign”
what about detector effects?
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37. relatively small signal yields (few thousand decays)
Hadronic Bs Decays
relatively small signal yields (few thousand decays)
momentum completely contained in tracker
superior sensitivity at higher ms
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38. Semileptonic Bs Decays
relatively large signal yields (several 10’s of thousands)
correct for missing neutrino momentum on average
loss in proper time resolution
superior sensitivity in lower ms range
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39. Tagging the B Production Flavor
vertexing (same) side
“opposite” side
use a combined same side and opposite side tag!
use muon, electron tagging, jet charge on opposite side
jet selection algorithms: vertex, jet probability and highest pT
particle ID based kaon tag on same side
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40. Combined Amplitude Scan
Preliminary
25.3 ps-1
A/A (17.25 ps-1) = 3.5
How significant is this result?
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41. ms = 17.33 +0.42 (stat) ± 0.07 (syst) ps-1
Conclusions
found signature consistent with Bs - Bs oscillations
probability of fluctuation from random tags is 0.5%
ms = (stat) ± 0.07 (syst) ps-1
|Vtd / Vts| = (stat ± syst)
-0.21
-0.007
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42. Outline 1 May 8 May 15 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
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43. Remember the following:
CP violation is discovered in the K-system
CP violation is naturally included if there are 3 generations or more
CP violation manifests itself as a complex phase in the CKM matrix
The CKM matrix gives the strengths and phases of the weak couplings
CP violation is apparent in experiments/processes with 2 interfering amplitudes
The angle β is measured through B0  J/y KS
Mixing of neutral mesons happens through the “box” diagram
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