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Particle Physics II – CP violation
Lecture 1
N. Tuning
Niels Tuning (1)
Plan


1) Wed 9 Feb:
Anti-matter + SM
2) Mon 13 Feb:
CKM matrix + Unitarity Triangle
3) Wed 15 Mar:
Mixing + Master eqs. + B0J/ψKs
4) Mon 13 Mar:
CP violation in B(s) decays (I)
5) Wed 15 Mar:
CP violation in B(s) and K decays (II)
6) Wed 22 Mar:
Exam
Final Mark:

if (mark > 5.5) mark = max(exam, 0.8*exam + 0.2*homework)

else mark = exam
In parallel: Lectures on Flavour Physics by prof.dr. R. Fleischer

Tuesday + Thrusday
Niels Tuning (2)
Plan
• 2 x 60 min (with break)
 Monday + Wednesday:

Start:
9:30

End:
11:30

Werkcollege: 11:30 – 12:30
Niels Tuning (3)
Grand picture….
These lectures
Main motivation
Universe
Niels Tuning (5)
Introduction: it’s all about the charged current
• “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 (6)
Motivation 1: Understanding the Standard Model
• “CP violation” is about the weak interactions,
• In particular, the charged current interactions:
• Quarks can only change flavour through charged current
interactions
Niels Tuning (7)
Introduction: it’s all about the charged current
• “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 (8)
Introduction: it’s all about the charged current
• “CP violation” is about the weak interactions,
• In particular, the charged current interactions:
W+
Wb
b
gVub
• In 1st hour:
gV*ub
u
u
• P-parity, C-parity, CP-parity
•  Symmetry related to particle – anti-particle
Niels Tuning (9)
Motivation 2: Understanding the universe
• 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
Equal amounts
of matter &
anti matter (?)
Matter Dominates !
Niels Tuning (10)
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
1) You need a process that violates the baryon number B:
(Baryon number of matter=1, of anti-matter = -1)
2) Both C and CP symmetries should be violated
3) 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 (11)
Introduction: it’s all about the charged current
• “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 (12)
Motivation 3: Sensitive to find new physics
• “CP violation” is about the weak interactions,
• In particular, the charged current interactions:
“Box” diagram: ΔB=2
“Penguin” diagram: ΔB=1
b
s
b s
s
b
μ
μ
Bs
b
s
b̃
g̃
s̃
x
x
s̃
g̃
b̃
s
b
Bs
B0
b
d
g̃
b̃
x
s
K*
s
Bs
μ
s̃
b
g̃
s̃
x
b̃
μ
μ
μ
• Are heavy particles running around in loops?
Niels Tuning (13)
Recap:
• CP-violation (or flavour physics) is about charged
current interactions
• Interesting because:
1) Standard Model:
in the heart of quark
interactions
2) Cosmology:
related to matter – anti-matter
asymetry
3) Beyond Standard Model:
measurements are sensitive to
new particles
Matter
Dominates !
b
s
s
b
Niels Tuning (14)
Grand picture….
These lectures
Main motivation
Universe
Niels Tuning (15)
Personal impression:
• People think it is a complicated part of the Standard Model
(me too:-). Why?
1) 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:
2) Complicated calculations?
|K0> ↔ |K0>
2
2
2
2
  B 0  f   Af  g +  t  +  g -  t  + 2   g +  t  g -  t   



2 
0
2
2
1
2
 B  f  Af  g +  t  + 2 g -  t  + 2     g +  t  g -  t   






3) Many decay modes? “Beetopaipaigamma…”
–
PDG reports 347 decay modes of the B0-meson:
•
Γ1 l+ νl anything
• Γ347 ν ν γ
–
( 10.33 ± 0.28 ) × 10−2
<4.7 × 10−5
CL=90%
And for one decay there are often more than one decay amplitudes…
Niels Tuning (16)
Anti-matter
• Dirac (1928): Prediction
• Anderson (1932): Discovery
• Present-day experiments
Niels Tuning (17)
Schrödinger
Classic relation between E and p:
Quantum mechanical substitution:
(operator acting on wave function ψ)
Schrodinger equation:
Solution:
(show it is a solution)
Niels Tuning (18)
Klein-Gordon
Relativistic relation between E and p:
Quantum mechanical substitution:
(operator acting on wave function ψ)
Klein-Gordon equation:
or :
Solution:
with eigenvalues:
But! Negative energy solution?
Niels Tuning (19)
Dirac
Paul Dirac tried to find an equation that was

relativistically correct,

but linear in d/dt to avoid negative energies

(and linear in d/dx (or ) for Lorentz covariance)
He found an equation that

turned out to describe spin-1/2 particles and

predicted the existence of anti-particles
Niels Tuning (20)
Dirac
How to find that relativistic, linear equation ??

Write Hamiltonian in general form,
but when squared, it must satisfy:
Let’s find
So,
αi and β !
αi and β must satisfy:

α1 2 = α2 2 = α3 2 = β2

α1,α2,α3, β anti-commute with each other

(not a unique choice!)
Niels Tuning (21)
Dirac

What are
α and β ??
The lowest dimensional matrix that has the desired behaviour is 4x4 !?
Often used
Pauli-Dirac representation:
with:
So,
αi and β must satisfy:

α1 2 = α2 2 = α3 2 = β2

α1,α2,α3, β anti-commute with each other

(not a unique choice!)
Niels Tuning (22)
Dirac
Usual substitution:
Leads to:
(β2=1)
Multiply by β:
Gives the famous Dirac equation:
Niels Tuning (23)
Dirac
The famous Dirac equation:
R.I.P. :
Niels Tuning (24)
Dirac
The famous Dirac equation:
Remember!

μ:
Lorentz index

4x4 γ matrix: Dirac index
Less compact notation:
Even less compact… :

What are the solutions for
ψ ??
Niels Tuning (25)
Dirac
The famous Dirac equation:
Solutions to the Dirac equation?
Try plane wave:
Linear set of eq:
 2 coupled equations:
If p=0:
Niels Tuning (26)
Dirac
The famous Dirac equation:
Solutions to the Dirac equation?
Try plane wave:
 2 coupled equations:
If p≠0:
Two solutions for E>0:
(and two for E<0)
with:
Niels Tuning (27)
Dirac
The famous Dirac equation:
Solutions to the Dirac equation?
Try plane wave:
 2 coupled equations:
If p≠0:
Two solutions for E>0:
(and two for E<0)
u (1)
0
1








1
0


 ( 2) 
u

  


0
  p /  E + m 
 











p
/
E
+
m
0




Tuning (28)
Dirac
The famous Dirac equation:
ψ
is 4-component spinor
4 solutions correspond to fermions and anti-fermions with spin+1/2 and -1/2
Two solutions for E>0:
(and two for E<0)
u (1)
0
1








1
0


 ( 2) 
u

  


0
  p /  E + m 
 











p
/
E
+
m
0

Niels Tuning (29)


Discovery of anti-matter
Lead plate
positron
Nobelprize 1936
Niels Tuning (30)
P and C violation
• What is the link between anti-matter and discrete
symmetries?
 C operator changes matter into anti-matter
• 2 more discrete symmetries: P and T
Niels Tuning (31)
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
Niels Tuning (32)
Three Discrete Symmetries
• Parity, P
– Parity reflects a system through the origin. Converts
right-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
Niels Tuning (33)
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.”
Niels Tuning (34)
Example: People believe in symmetry…
Award Ceremony Speech Nobel Prize (1957):
•
“it was assumed almost tacitly, that elementary particle reactions are
symmetric with respect to right and left.”
•
“In fact, most of us were inclined to regard the symmetry of
elementary particles with respect to right and left as a necessary
consequence of the general principle of right-left symmetry of Nature.”
•
“… only Lee and Yang … asked themselves what kind of experimental
support there was for the assumption that all elementary particle
processes are symmetric with respect to right and left. “
Niels Tuning (35)
A realistic experiment: the Wu experiment (1956)
• Observe radioactive decay of Cobalt-60
nuclei
– The process involved:
60
–
60 Ni
28
60 Co
27
is spin-5 and
ne are spin-½
27Co 
60
28Ni + e + ne
S=1/2
is spin-4, both e- and
– 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
S=1/2
Niels Tuning (36)
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
Sp
H
Sp
H=+1 (“right-handed”)
H=-1 (“left-handed”)
Niels Tuning (37)
The Wu experiment – 1956
• 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 (38)
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 rate
w.r.t unpolarized rate
60Co
polarization decreases
as function of time
‘Forward’ Counting rate
w.r.t unpolarized rate
Niels Tuning (39)
The Wu experiment – 1956
• Physics conclusion:
– Angular distribution of electrons shows that only pairs of lefthanded 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 (40)
So P is violated, what’s next?
• 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.
m+
p+
nm
OK
OK
• 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
Niels Tuning (41)
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
m+
p+
nm(LH)
OK
m-
p-
nm(LH)
OK
C
• C symmetry is broken by the weak interaction,
– just like P
Niels Tuning (42)
The Weak force and C,P parity violation
• What about C+P  CP symmetry?
– CP symmetry is parity conjugation (x,y,z  -x,-y,z)
followed by charge conjugation (X  X)
nm
m+
m-
Intrinsic
spin
p+
m+
P p+
CP
C p-
nm
CP appears to
be preserved
in weak
interaction!
nm
Niels Tuning (43)
CPT theorem
• CPT transformation:
– C: interchange particles and anti-particles
– P: reverse space-coordinates
– T: Reverse time-coordinate
• CPT transformation closely related to Lorentz-boost
 CPT invariance implies
– Particles and anti-particles have same mass and lifetime
– Lorentz invariance
Niels Tuning (44)
Why anti-matter must exist!
• “Feynman-Stueckelberg interpretation”
• “One observer’s electron is the other observer’s positron”
Niels Tuning (45)
CPT is conserved, but does anti-matter fall down?
Niels Tuning (46)
C, P, T
• C, P, T transformation:
– C: interchange particles and anti-particles
– P: reverse space-coordinates
– T: Reverse time-coordinate
• CPT we discussed briefly …
• After the break we deal with P and CP…
… violation!
Niels Tuning (47)
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 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/antimatter 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 (48)
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 hadron
P|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 (49)
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-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 (50)
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
Meaning: P|p+> = -1|p+>
• The neutron
Experimental proof: J.Steinberger (1954)
πd→nn
n are fermions, so (nn) anti-symmetric
 Sd=1, Sπ=0 → Lnn=1
1) final state:P|nn> = (-1)L|nn> = -1 |nn>
2) init state: P|d> = P |pn> = (+1)2|pn> = +1 |d>
To conserve parity: P|π> = -1 |π>
– 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
Niels Tuning (51)
Figuring out C eigenvalues for hadrons
• 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 (52)
• This was an introduction to P and C
• Let’s change gear…
Niels Tuning (53)
CP violation in the SM Lagrangian
• Focus on charged current interaction (W±): let’s trace it
uLI
g
dL
W+m
I
Niels Tuning (54)
The Standard Model Lagrangian
LSM  LKinetic + LHiggs + LYukawa
• 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
GSM  SU (3)C  SU (2) L  U (1)Y  SU (3)C  U (1)Q
The W+, W-,Z0 bosons acquire a mass
• LYukawa : • Ad hoc interactions between Higgs field & fermions
Niels Tuning (55)
Y = Q - T3
Fields: Notation
Fermions:
 1- g 5 
yL  
y
 2 
•
•
Leptons:
•
 u I (3, 2,1 6) 
 I

d
(3,
2,1
6)

L i
•

I
QLi
(3, 2,1 6)
SU(3)C SU(2)L Hypercharge Y
(=avg el.charge in multiplet)
Leftgeneration
handed index
I
uRi
(3,1, 2 3)
n I (1, 2, - 1 2) 
 I

l
(1,
2,
1
2)

L i
I
l
• Ri (1,1, -1)
Scalar field:
with y = QL, uR, dR, LL, lR, nR
Interaction rep.
Quarks:
Under SU2:
Left handed doublets
Right hander singlets
 1+ g 5 
; yR 
y
 2 
 + 
f (1, 2, 1 2)   0 
 
•

d RiI (3,1, -1 3)
LILi (1, 2, -1 2)
•
I
n
 Ri 
Note:
Interaction representation: standard model
interaction is independent of generation
Niels Tuning (56)
number
Q == Q
Y
T3-+TY
3
Fields: Notation
Explicitly:
• The left handed quark doublet :
I
I
I
I
I
I
I
I
I






u
,
u
,
u
c
,
c
,
c
t
,
t
,
t
r
g
b
r
g
b
r
g b
I
QLi (3, 2,1 6)   I I I  ,  I I I  ,  I I I 
 d , d , d   s , s , s  b ,b ,b 
 r g b L  r g b L  r g b L
T3  + 1
2
T3  - 1
2
(Y  1 6)
• Similarly for the quark singlets:
uRiI (3,1, 2 3) 
d RI i (3,1, -1 3) 
• The left handed leptons:
u , u , u  ,  c , c , c  , t , t , t 
 d , d , d  ,  s , s , s  , b , b , b 
I
r
I
r
I
r
I
r
I
r
I
r
R
R
I
r
I
r
I
r
I
r
I
r
I
r
R
I
r
R
I
r
I
r
I
r
I
r
I
I
I


n




n
n
m
e

I
LLi (1, 2, -1 2)   I  ,   ,  I 
 e   m I   
 L  L  L
• And similarly the (charged) singlets:
lRiI (1,1, -1)  eRI , mRI , RI
Y  2 3
R
I
r
Y
R
T3  + 1 2
T3  - 1 2
 - 1 3
Y  - 1 2 
Y  -1
Niels Tuning (57)
LSM  LKinetic + LHiggs + LYukawa
LKinetic
: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:
Fields:
Generators:
L  iy ( m g m )y
 m  D m   m + ig s Gam La + igWbmTb + ig B m Y
Gam : 8 gluons
Wbm : weak bosons: W1, W2, W3
Bm : hypercharge boson
La : Gell-Mann matrices:
Tb : Pauli Matrices:
Y : Hypercharge:
½ a
½ b
(3x3)
(2x2)
For the remainder we only consider Electroweak: SU(2)L x U(1)Y
SU(3)C
SU(2)L
U(1)Y
Niels Tuning (58)
LSM  LKinetic + LHiggs + LYukawa : The Kinetic Part
L kinetic : iy ( m g m )y  iy ( D m g m )y
with y  QLiI , uRiI , d RiI , LILi , lRiI
For example, the term with QLiI becomes:
Lkinetic (QLiI )  iQLiI g m D m QLiI
 iQLiI g m ( m +
i
i
i
g s Gam a + gWbm b + g B m ) QLI i
2
2
6
0 1

1 0
 0 -i 
2  

i 0 
1 0 
3  

 0 -1 
1  
Writing out only the weak part for the quarks:
I
Weak
kinetic
L
(u , d )
I
L
i

 u 
 i u , d  Lg m   m + g W1m 1 + W2m 2 + W3m 3    
2

  d L
g I
 iu LI g m  m u LI + id LI g m  m d LI u Lg mW - m d LI 2
I
g I
d L g mW + m u LI
2
- ...
uLI
g
dL
I
W+m
L=JmWm
W+ = (1/√2) (W1+ i W2)
W- = (1/√ 2) (W1 – i W2)
Niels Tuning (59)
LSM  LKinetic + LHiggs + LYukawa
m
LHiggs  Dmf D f - VHiggs
†
VHiggs
2
1 2 †
†
 m f f  +  f f 
2
Broken
Symmetry
V(f)
Symmetry
: The Higgs Potential
m2  0 :
   0
Vf
m2  0 :
f
 0 
    v 


 2
f
v  - m2 
~ 246 GeV
Spontaneous Symmetry Breaking: The Higgs field adopts a non-zero vacuum expectation value
Procedure:
    e  + im f 

0
0
0 


e

+
i

m
f
  

f 
+
+
And rewrite the Lagrangian (tedious):
(The other 3 Higgs fields are “eaten” by the W, Z bosons)
+
Substitute:
v+ H0
e  
2
0
1. GSM :  SU
. (3)C  SU (2)L U (1)Y    SU (3)C U (1)EM 
2. The W+,W-,Z0 bosons acquire mass
3. The Higgs boson H appears
Niels Tuning (60)
LSM  LKinetic + LHiggs + LYukawa
: The Yukawa Part
Since we have a Higgs field we can (should?) add (ad-hoc)
interactions between f and the fermions in a gauge invariant way.
doublets
The result is:

Yij y Li f
- LYukawa 
 Y
d
ij


y
singlet
Q f d +Y
I
Li
I
Rj
+
Rj
u
ij

I ~
Li
h.c.



Q f uRjI + Yijl LILi f lRjI + h.c.
i, j : indices for the 3 generations!
 0 1 *  f0 
With: f  i 2 f  
f   - 
-1 0 

 -f 
(The CP conjugate of f
*
To be manifestly invariant under SU(2) )
d
ij
Y
u
ij
, Y
l
ij
, Y
are arbitrary complex matrices which
operate in family space (3x3)
 Flavour physics!
Niels Tuning (61)
LSM  LKinetic + LHiggs + LYukawa
: The Yukawa Part
Writing the first term explicitly:
+

 I

d
I
I
Yij (uL , d L )i  0  d Rj
 
 d I I
 Y11 uL , d L


 Y d cI , sI
 21 L L


 Y31d t LI , bLI






 + 
d
I
I
 0  Y12 uL , d L
 



 + 
 0 
 

+



I
I
cL , sL  0 
 

Y13d


 + 
 0 
 

+



I
I
t L , bL  0 
 
Y33d


Y22d
Y32d

 +  
 0  
  
I


d
R
+ 
    I 
I
I
cL , sL  0     sR 
    I 
bR 

+
  
I
I
t L , bL  0  

  
 + 
d
I
I
 0  Y13 u L , d L
 




Niels Tuning (62)
LSM  LKinetic + LHiggs + LYukawa
: The Yukawa Part
There are 3 Yukawa matrices (in the case of massless neutrino’s):
d
ij
Y
u
ij
, Y
l
ij
, Y
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 (63)
LYukawa

S.S.B
LMass
: The Fermion Masses
Start with the Yukawa Lagrangian
+

 I

d
I
I
- LYuk  Yij (uL , d L )i  0  d Rj
 
v+H
S .S .B. : e  0  
2
+ Yiju ... + Yijl ...
After which the following mass term emerges:
- LYuk  - L Mass  d M d
I
Li
d
ij
I
Rj
+ u M u
I
Li
u
ij
I
Rj
+ lLiI M ijl lRjI + h.c.
with
v d
M 
Yij
2
d
ij
v u
, M 
Yij
2
u
ij
v l
, M 
Yij
2
l
ij
LMass is CP violating in a similar way as LYuk
Niels Tuning (64)

S.S.B
LYukawa
LMass
: The Fermion Masses
Writing in an explicit form:
-




LMass   d , s , b  M d
I
I
I
L
 I
 d 
  sI  +
 I
 b 
 R

u , c , t   M
I
I
I
L
 uI 
u   c I  +
 
  tI 
 
 e , m ,  M
I
V M V M
f†
R
f
I
I
L
R
The matrices M can always be diagonalised by unitary matrices
f
L

 I
e 
l  mI
 I
  
 R
VLf and VRf

 I I I
 d , s ,b


f
diagonal





+ h.c.
such that:
dI 
 
VLf † VLf M f VRf † VRf  s I 
L
 bI 
 




R
Then the real fermion mass eigenstates are given by:
d Li  VLd   d LjI
d Ri  VRd   d RjI
uLi  VLu   uLjI
uRi  VRu   uRjI
lLi  VLl   lLjI
lRi  VRl   lRjI
ij
ij
ij
dLI , uLI , lLI
dL , uL , lL
ij
ij
ij
are the weak interaction eigenstates
are the mass eigenstates (“physical particles”)
Niels Tuning (65)
LYukawa

S.S.B
LMass
In terms of the mass eigenstates: m
 d
 d , s, b 
- L Mass 
+
 e, m , 
L
L




ms
 me




mm
: The Fermion Masses



mb 
d 
 
 s  + u , c, t
b
 R
 e
 
  m  + h.c.
m     R


L
 mu




-L Mass 
mu uu
+
mc cc
+
mt tt
+
md dd
+
ms ss
+
mb bb
+
me ee
mm mm
+
m
+
mc



mt 
u 
 
c
t
 R
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 (66)
LW
 LCKM
: The Charged Current
The charged current interaction for quarks in the interaction basis is:
- LW +
g
2

gm
uLiI
d LI i Wm+
The charged current interaction for quarks in the mass basis is:
- LW +

g
2
uLi VLu
g m VLd † d Li Wm+
VCKM  VLu  VLd † 
The unitary matrix:
With:
†
VCKM  VCKM
1
is the Cabibbo Kobayashi Maskawa mixing matrix:
-LW +

g
2
d 
 u , c , t L VCKM   s 
b
 L
Lepton sector: similarly
g m Wm+
VMNS  VLn  VLl † 
However, for massless neutrino’s: VLn = arbitrary. Choose it such that VMNS = 1
 There is no mixing in the lepton sector
Niels Tuning (67)
Charged Currents
The charged current term reads:
LCC 
g I m - I
uLig Wm d Li
2
+
g I m + I
d Lig Wm uLi
2
g  1- g 5  m -  1- g 5 

ui 
 g Wm Vij 
dj
2  2 
 2 

g
uig mWm-Vij 1 - g 5  d j
2
+
mm+
 J CC
Wm- + J CC
Wm+
+
 1- g 5  m + †  1- g 5 
g
dj 
 g Wm V ji 
 ui
2  2 
 2 
g
d j g mWm+Vij* 1 - g 5  ui
2
(Together with (x,t) -> (-x,t))
Under the CP operator this gives:
CP
LCC 

g
d j g mWm+Vij 1 - g 5  ui
2
+
g
uig mWmiVij* 1 - g 5  d j
2
A comparison shows that CP is conserved only if
Vij = Vij*
In general the charged current term is CP violating
Niels Tuning (68)
The Standard Model Lagrangian (recap)
LSM  LKinetic + LHiggs + LYukawa
• 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
GSM  SU (3)C  SU (2) L  U (1)Y  SU (3)C  U (1)Q
•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
• LKinetic
in mass eigenstates: CKM – matrix
 CP-violating
 CP-conserving!
 CP violating with a single phase
Niels Tuning (69)
L SM
- LYuk
L Kinetic
Recap
 L Kinetic + L Higgs + LYukawa
+

 I

d
I
I
 Yij (uL , d L )i  0  d Rj + ...
 
g I m - I
g I m + I

uLig Wm d Li +
d Lig Wm uLi + ...
2
2
Diagonalize Yukawa matrix Yij
– Mass terms
– Quarks rotate
– Off diagonal terms in charged current couplings
- LMass 
LCKM
 d , s, b 
L
 md




ms
 d 
 
 s +
mb   b  R
dI 
 I
 s   VCKM
 bI 
 
 u , c, t 
L
 mu




mc
d 
 
s
b
 
 u
 
  c  + ...
mt   t  R
g
g
m
5

uig Wm Vij 1 - g  d j +
d jg mWm+Vij* 1 - g 5  ui + ...
2
2
L SM
 LCKM + L Higgs + L Mass
Niels Tuning (70)
Ok…. We’ve got the CKM matrix, now what?
• 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 (71)
Personal impression:
• People think it is a complicated part of the Standard Model
(me too:-). Why?
1) 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:
2) Complicated calculations?
|K0> ↔ |K0>
2
2
2
2
  B 0  f   Af  g +  t  +  g -  t  + 2   g +  t  g -  t   



2 
0
2
2
1
2
 B  f  Af  g +  t  + 2 g -  t  + 2     g +  t  g -  t   






3) Many decay modes? “Beetopaipaigamma…”
–
PDG reports 347 decay modes of the B0-meson:
•
Γ1 l+ νl anything
• Γ347 ν ν γ
–
( 10.33 ± 0.28 ) × 10−2
<4.7 × 10−5
CL=90%
And for one decay there are often more than one decay amplitudes…
Niels Tuning (72)