The Origin of CP Violation in the Standard Model. Topical Lectures July 12, 2004 Marcel Merk. Contents. Introduction: symmetry and nonobservables CPT Invariance CP Violation in the Standard Model Lagrangian Rephasing independent CP Violation quantities The Fermion masses
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Theory Oriented!
References:
Advanced Series on Directions in High Energy Physics – Vol 3:
“CP Violation’, 1998, p3.
Lectures given at the XXVII SLAC Summer Institute, hepph/9911321.
International series of monographs on physics,
Oxford univ. press, 1999.
Cambridge monographs on particle physics, nuclear physics and cosmology,
Cambridge univ. press, 2000.
Contemporary Concepts in Physics Volume 1,
Revised and Updated First Edition, Harwood Academic Publishers, 1990.
Frontiers in Physics, BenjaminCummings, 1983.
T.D.Lee:
“The root to all symmetry principles lies in the assumption that it is impossible to observe certain basic quantities; the nonobservables”
the symmetry is said to be broken
Noether Theorem:
symmetry
conservation law
Simple Example: Potential energy V between two particles:
Absolute position is a nonobservable:
The interaction is independent on the choice of 0.
Symmetry:
V is invariant under arbitrary
space translations:
0’
0
Consequently:
Total momentum is conserved:
Before 1956 physicists were convinced that the laws of nature
were leftright symmetric. Strange?
A “gedanken” experiment:
Consider two perfectly mirror symmetric cars:
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?
What happens in case the ignition mechanism uses, say, Co60b decay?
(Lüders, Pauli, Schwinger)
(antiunitarity)
=> Similarly the total decayrate of a particle is equal to that of the antiparticle
Answer 1 + 2: A KL≠ an antiKSparticle!
Answer 3:
Partial decay rate ≠ total decay rate! However, the sum over all partial rates (>200 or so) is the same for B and Bbar. (Amazing! – at least to me)
LSM contains:
LKinetic : fermion fields
LHiggs : the Higgs potential
LYukawa : the Higgs – Fermion interactions
→ Several “miracles” happen in symmetry
breaking
Standard Model gauge symmetry:
Note Immediately: The weak part is explicitly parity violating
Outline:
Local field theories work with Lagrangian densities:
with the fields taken at
The fundamental quantity, when discussing symmetries is the Action:
If the action is (is not) invariant under a symmetry operation then the symmetry in question is a good (broken) one
=> Unitarity of the interaction requires the Lagrangian to be Hermitian
Lorentz structure: a Lagrangian in field theory can be built using combinations of:
S: Scalar fields : 1
P: Pseudoscalar fields : g5
V: Vector fields : gm
A: Axial vector fields : gmg5
T: Tensor fields : smn
Dirac field :
Scalar field f:
Example:
Consider a spin1/2 (Dirac) particle (“nucleon”) interacting with a spin0 (Scalar) object (“meson”)
Nucleon field
Meson potential
Nucleon – meson interaction
Exercise:
What are the symmetries of this theory under C, P, CP ? Can a and b be any complex numbers?
Note: the interaction term contains scalar and pseudoscalar parts
Violates P, conserves C, violates CP
a and b must be real from Hermeticity
(Ignoring arbitrary phases)
Transformation properties of Dirac spinor bilinears:
c→c*
c→c*
=> CP Conserving
The W+, W,Z0 bosons acquire a mass
=> CP Conserving
=> CP violating with a single phase
=> CPviolating
=> CPconserving!
=> CP violating with a single phase
Interaction rep.
Y
SU(3)C
SU(2)L
Left
handed
generation
index
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
Q = T3 + Y
Explicitly:
Basic principle: The Lagrangian must be invariant under local gauge transformations
Example: massless Dirac Spinors in QED:
“global” U(1) gauge transformation:
“local” U(1) gauge transformation:
Is the Lagrangian invariant?
Not invariant!
Then:
Then it turns out that:
=> Introduce the covariant derivative:
and demand that Am transforms as:
is invariant!
Conclusion:
Fermions + gauge bosons + interactions
Procedure: Introduce the Fermion fields and demand that the theory is local gauge invariant
Start with the Dirac Lagrangian:
Replace:
Gam :8 gluons
Wbm: weak bosons: W1, W2, W3
Bm: hyperchargeboson
Fields:
Generators:
La : GellMann 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
uLI
W+m
g
dLI
Exercise:
Show that this Lagrangian formally violates both P and C
Show that this Lagrangian conserves CP
LKin = CP conserving
For example the term with QLiIbecomes:
and similarly for all other terms (uRiI,dRiI,LLiI,lRiI).
Writing out only the weak part for the quarks:
W+ = (1/√2) (W1+ i W2)
W = (1/√ 2) (W1– i W2)
L=JmWm
→Note LHiggs = CP conserving
V(f)
V(f)
Symmetry
Broken
Symmetry
f
f
~ 246 GeV
Spontaneous Symmetry Breaking: The Higgs field adopts a nonzero vacuum expectation value
Procedure:
Substitute:
And rewrite the Lagrangian (tedious):
(The other 3 Higgs fields are “eaten” by the W, Z bosons)
“The realization of the vacuum breaks the symmetry”
Since we have a Higgs field we can add (adhoc) interactions between f and the fermions in a gauge invariant way.
doublets
L must be Her
mitian (unitary)
The result is:
singlet
With:
To be manifestly invariant under SU(2)
are arbitrary complex matrices which operate in family space (3x3)
=> Flavour physics!
Writing the first term explicitly:
Question:
In what aspect is this Lagrangian similar to the example of the nucleonmeson potential?
Formally, CP is violated if:
In generalLYukawais CP violating
Exercise (intuitive proof)
Show that:
CP is conserved inLYukawaonly ifYij = Yij*
and leaves the coefficientsYij and Yij*unchanged
There are 3 Yukawa matrices (in the case of massless neutrino’s):
……Revisit later
S.S.B
Start with the Yukawa Lagrangian
After which the following mass term emerges:
with
LMass is CP violating in a similar way as LYuk
S.S.B
Writing in an explicit form:
The matrices M can always be diagonalised by unitarymatricesVLfandVRfsuch that:
Then the real fermion mass eigenstates are given by:
dLI , uLI , lLIare the weak interaction eigenstates
dL , uL , lLare the mass eigenstates (“physical particles”)
S.S.B
In terms of the mass eigenstates:
= CP Conserving?
In flavour space one can choose:
Weak basis: The gauge currents are diagonal in flavour space, but the flavour mass matrices are
nondiagonal
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) ?
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 neutrino’s: VLn= arbitrary. Choose it such that VMNS = 1 => There is no mixing in the lepton sector
To illustrate the SM neutral current take the W3m and Bm term of the Kinetic Lagrangian:
and
And consider the Zboson field:
Take furtherQLiI=dLiI
Use:
In terms of physical fields no nondiagonal contributions occur for the neutral Currents. => GIM mechanism
Standard Model forbids flavour changing neutral currents.
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 ifVij = Vij*
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 ifVij = Vij*
In general the charged current term is CP violating
=> CP Conserving
The W+, W,Z0 bosons acquire a mass
=> CP Conserving
=> CP violating with a single phase
=> CPviolating
=> CPconserving!
=> CP violating with a single phase
Under a quark phase transformation:
and a simultaneous rephasing of the CKM matrix:
or
the charged current
is left invariant
2 generations:
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  (2N1)

Total degrees of freedom: 4 (N1)2
Number of Euler angles: 3 N (N1) / 2
Number of CP phases: 1 (N1) (N2) / 2
No CP violation in SM.
This is the reason Kobayashi and Maskawa first suggested a third family of fermions!
Light, lefthanded, “active”
Aleph
L3
Opal
Delphi
Geneva Airport “Cointrin”
MZ
Maybe the most important result of LEP:
“There are 3 generations of neutrino’s”
m
ne
nm
lLI
W+m
W
νLI
Question:
However, observation of neutrino oscillations is possible due to small neutrino mass differences.
The standard representation of the CKM matrix is:
However, many representations are possible. What are the invariants under rephasing?
a
Vtd Vtb*
Vud Vub*
g
b
Vcd Vcb*
unitarity:
VCKM† VCKM = 1
The “db” triangle:
Area = ½ Im Qudcb = ½ J
Under rephasing:
the unitary angles are invariant
(In fact, rephasing implies a rotation of the whole triangle)
a
(r,h)
g
b
(0,0)
(1,0)
Wolfenstein realised that the nondiagonal CKM elements are relatively small compared to the diagonal elements, and parametrized as follows:
Normalised CKM triangle:
Note that the massless Lagrangian has a global symmetry for unitary transformations in flavour space.
Let’s now assume two quarks with the same charge are degenerate
in mass, eg.:ms = mb
Redefine:s’ = Vus s + Vub b
Now the u quark only couples to s’ and not to b’: i.e. V13’ = 0
Using unitarity we can show that the CKM matrix can now be written as:
CP conserving
Necessary criteria forCP violation:
Using Standard Parametrization of CKM:
(eg.: J=Im(Vus Vcb Vub* Vcs*) )
(The maximal value J might have = 1/(6√3) ~ 0.1)
However, also required is:
All requirements for CP violation can be summarized by:
Is CP violation maximal? => One has to understand the origin of mass!
Observe:
Mass spectra (m = Mz, MSbar scheme)
mu ~ 1  3 MeV , mc ~ 0.5 – 0.6 GeV , mt ~ 180 GeV
md ~ 2  5 MeV , ms ~ 35 – 100 MeV , mb ~ 2.9 GeV
 Then solve the mass Problem –
R.P. Feynman
me = 0.51 MeV , mm = 105 MeV , mt = 1777 MeV
Why are neutrino’s so light? Related to the fact that they are the only neutral fermions?
Seesaw mechanism?
Hubble deep field  optical
Angular Power Spectrum
2.7248K
2.7252K
Cosmic Microwave Background
WMAP satellite
Almost all matter annihilated with antimatter, producing photons…
Convert 1 in 109 antiquarks into a quark in an early stage of universe:
Anti
Sakharov (1964)
e+
d
X
u

u
u
p0
u
proton
Conceptually simple
GUT theories predict proton decay mediated by heavy X gauge bosons:
X boson has baryon number violating (1) couplings: X →q q, X→q l
Proton lifetime: t > 1032s
A simple Baryogenesis model:
Efficiencyof Baryon asymmetry buildup:
CP Violation (2) :r ≠ r
Assuming the back reaction does not occur (3):
Initial X number density
Initial light particle
number density
Conceptually difficult
SM Electroweak Interactions:
1) Baryon number violation in weak anomaly:
Conserves “BL” but violates “B+L”
2) CP Violation in the CKM
3) Nonequilibrium: electroweak phase transition
Electroweak phase transition wipes out GUT Baryon asymmetry!
Can it generate a sufficiently large asymmetry?
Problems:
1. Higgs mass is too heavy. In order to have a first order phase transition:
Requirement: mH < ~ 70 GeV/c2 , from LEP mH > ~ 100 GeV/c2
2. CP Violation in CKM is not enough:
Biertje?