Particle Physics II CP violation Lecture 2

1
Particle Physics II CP violationLecture 2

• N. Tuning

Acknowledgements Slides based on the course from
Marcel Merk and Wouter Verkerke.
2
Outline

• 5 March
• Introduction matter and anti-matter
• P, C and CP symmetries
• K-system
• CP violation
• Oscillations
• Cabibbo-GIM mechanism
• 12 March
• CP violation in the Lagrangian
• CKM matrix
• B-system
• 19 March
• B-factories
• B?J/Psi Ks
• Delta ms
• (26 March No lecture)
• 2 April
• B-experiments BaBar and LHCb
• Measurements at LHCb

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/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
The Weak force and C,P parity violation
Recap from last week

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

100 P violation All ?s are lefthanded All??s
are righthanded
CP appears to be preservedin weakinteraction!
5
Intermezzo What operator is C ? Full solutions
of Dirac equations
Try ansatz
4 independent solutions for the Dirac spinors
6
Intermezzo What operator is C ? Particle ?
Anti-particle Ci? 2? 0
Dirac equation
In EM field
Possible choice for C?0 (in Dirac
representation) C? 0 i? 2
Flip charge e ? -e
Positron
Try out on a example spinor (e.g. electron with
pos. helicity)
7
The Cronin Fitch experiment
Recap from last week
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
8
Kaons K0,?K0, K1, K2, KS, KL,
Recap from last week

• 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 as short-lived or long-lived
  kaons
• KSgt predominantly CP1
• KLgt predominantly CP -1

• ?- (2.236 0.007) x 10-3
• e (2.232 0.007) x 10-3

9
Schematic picture of selected weak decays
Recap from last week

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

K2
Anti-K0
?s
?d
Kgt
W
u
u
K0
K0
K0
W
s
d
K1
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The Cabibbo-GIM mechanism
Recap from last week

• 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-
11
From 2 to 3 generations
Recap from last week

• 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 !

12
What do we know about the CKM matrix?
Recap from last week

• 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
13
Lets investigate the Lagrangian

• Hold on

14
The Standard Model Lagrangian

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

• LHiggs Introduce Higgs potential with ltfgt ?
  0
• Spontaneous symmetry
  breaking

The W, W-,Z0 bosons acquire a mass

• LYukawa Ad hoc interactions between Higgs
  field fermions

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Fields Notation
Q T3 Y
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

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Fields Notation
Q T3 Y
Explicitly

• The left handed quark doublet

• Similarly for the quark singlets

• The left handed leptons

• And similarly the (charged) singlets

17
The
Kinetic Part
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
hypercharge boson
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
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The
Kinetic Part
For example, the term with QLiI becomes
Writing out only the weak part for the quarks
W (1/v2) (W1 i W2) W- (1/v 2) (W1 i W2)
LJmWm
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The Higgs Potential
And rewrite the Lagrangian (tedious)
(The other 3 Higgs fields are eaten by the W, Z
bosons)
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The Yukawa Part
Since we have a Higgs field we can 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!
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The Yukawa Part
Writing the first term explicitly
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The Yukawa Part

• The hermiticity of the Lagrangian implies that
  there are terms in pairs of the form

• However a transformation under CP gives

CP is conserved in LYukawa only if Yij Yij
and leaves the coefficients Yij and Yij
unchanged
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The Yukawa Part
There are 3 Yukawa matrices (in the case of
massless neutrinos)

• 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
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The Fermion
Masses
Start with the Yukawa Lagrangian
After which the following mass term emerges
with
LMass is CP violating in a similar way as LYuk
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The Fermion
Masses
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
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The Fermion
Masses
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) ?
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The Charged
Current
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 neutrinos VLn
arbitrary. Choose it such that VMNS 1 ? There
is no mixing in the lepton sector
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Charged Currents
The charged current term reads
(Together with (x,t) -gt (-x,t))
Under the CP operator this gives
A comparison shows that CP is conserved only if
Vij Vij
In general the charged current term is CP
violating
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Why complex phases matter

• CP conjugation of a W boson vertex involves
  complex conjugation of coupling constant
• With 2 generations Vij is always real and
  Vij?Vij
• With 3 generations Vij can be complex ? CP
  violation built into weak decay mechanism!

Above process violates CP if Vub ? Vub
30
The Standard Model Lagrangian (recap)

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

? CP Conserving

• LHiggs Introduce Higgs potential with ltfgt ? 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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Ok. Weve got the CKM matrix, now what?

• Its unitary
• probabilities add up to 1
• d0.97 d 0.22 s 0.003 b (0.9720.2220.0032
  1)
• How many free parameters?
• How many real/complex?
• How do we normally visualize these parameters?

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Quark field re-phasing
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 3 N
(N-1) / 2 Number of CP phases
1 (N-1) (N-2) / 2
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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

• 4 parameters
• 3 real
• 1 phase

Magnitude of elements shown only, no information
of phase
34
Approximately diagonal form

• 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 dont know (yet)! If you
figure this out,you will win the nobelprize
l
l3
l
l2
l3
l2
lcos(qc)0.23
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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)

36
1995 What do we know about A, ?, ? and ??

• Fit all known Vij values to Wolfenstein
  parameterization and extract A, ?, ? and ?
• Results for A and l most precise (but dont 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

37
Deriving the triangle interpretation

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

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Visualizing the unitarity constraint

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

(Wolfenstein params to order l4)
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Visualizing the unitarity constraint

• Phase of base is zero ? Aligns with real
  axis,

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Visualizing the unitarity constraint

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

(r,h)
(0,0)
(1,0)
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Visualizing arg(Vub) and arg(Vtd) in the (r,h)
plane

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

42
Outline

• 5 March
• Introduction matter and anti-matter
• P, C and CP symmetries
• K-system
• CP violation
• Oscillations
• Cabibbo-GIM mechanism
• 12 March
• CP violation in the Lagrangian
• CKM matrix
• B-system
• 19 March
• B-factories
• B?J/Psi Ks
• Delta ms
• (26 March No lecture)
• 2 April
• B-experiments BaBar and LHCb
• Measurements at LHCb