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The Origin of CP Violation in the Standard Model. Topical Lectures July 1-2, 2004 Marcel Merk. Contents. Introduction: symmetry and non-observables CPT Invariance CP Violation in the Standard Model Lagrangian Re-phasing independent CP Violation quantities The Fermion masses

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the origin of cp violation in the standard model

The Origin of CP Violationin the Standard Model

Topical Lectures

July 1-2, 2004

Marcel Merk

  • Introduction: symmetry and non-observables
  • CPT Invariance
  • CP Violation in the Standard Model Lagrangian
  • Re-phasing independent CP Violation quantities
  • The Fermion masses
  • The matter anti-matter asymmetry

Theory Oriented!



  • C.Jarlskog, “Introduction to CP Violation”,

Advanced Series on Directions in High Energy Physics – Vol 3:

“CP Violation’, 1998, p3.

  • Y.Nir, “CP Violation In and Beyond the Standard Model”,

Lectures given at the XXVII SLAC Summer Institute, hep-ph/9911321.

  • Branco, Lavoura, Silva: “CP Violation”,

International series of monographs on physics,

Oxford univ. press, 1999.

  • Bigi and Sanda: “CP Violation”,

Cambridge monographs on particle physics, nuclear physics and cosmology,

Cambridge univ. press, 2000.

  • T.D. Lee, “Particle Physics and Introduction to Field Theory”,

Contemporary Concepts in Physics Volume 1,

Revised and Updated First Edition, Harwood Academic Publishers, 1990.

  • C. Quigg, “Gauge Theories of the Strong, Weak and Electromagnetic Interactions”,

Frontiers in Physics, Benjamin-Cummings, 1983.

  • H. Fritsch and Z. Xing, “Mass and Flavour Mixing Schemes of Quarks and Leptons”, hep-ph/9912358.
  • Mark Trodden, “Electroweak Baryogenesis”, hep-ph/9803479.
introduction symmetry and non observables
Introduction: Symmetry and non-Observables


“The root to all symmetry principles lies in the assumption that it is impossible to observe certain basic quantities; the non-observables”

  • There are four main types of symmetry:
  • Permutation symmetry:
  • Bose-Einstein and Fermi-Dirac Statistics
  • Continuous space-time symmetries:
  • translation, rotation, acceleration,…
  • Discrete symmetries:
  • space inversion, time inversion, charge inversion
  • Unitary symmetries: gauge invariances:
  • U1(charge), SU2(isospin), SU3(color),..
  • If a quantity is fundamentally non-observable it is related to an exact symmetry
  • If a quantity could in principle be observed by an improved measurement;

the symmetry is said to be broken

Noether Theorem:


conservation law

symmetry and non observables
Symmetry and non-observables

Simple Example: Potential energy V between two particles:

Absolute position is a non-observable:

The interaction is independent on the choice of 0.


V is invariant under arbitrary

space translations:




Total momentum is conserved:

parity violation
Parity Violation

Before 1956 physicists were convinced that the laws of nature

were left-right symmetric. Strange?

A “gedanken” experiment:

Consider two perfectly mirror symmetric cars:

Gas pedal

Gas pedal



“L” and “R” are fully symmetric,

Each nut, bolt, molecule etc.

However the engine is a black box



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?

cpt invariance
CPT Invariance
  • Local Field theories always respect:
    • Lorentz Invariance
    • Symmetry under CPT operation (an electron = a positron travelling back in time)
    • => Consequence: mass of particle = mass of anti-particle:

(Lüders, Pauli, Schwinger)


=> Similarly the total decay-rate of a particle is equal to that of the anti-particle

  • Question 1:
  • The mass difference between KL and KS: Dm = 3.5 x 10-6 eV => CPT violation?
  • Question 2:
  • How come the lifetime of KS = 0.089 ns while the lifetime of the KL = 51.7 ns?
  • Question 3:
  • BaBar measures decay rate B-> J/y KS and Bbar-> J/y KS. Clearly not the same: how can it be?

Answer 1 + 2: A KL≠ an anti-KSparticle!

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)

cp in the standard model lagrangian the origin of the ckm matrix
CP in the Standard Model Lagrangian(The origin of the CKM-matrix)

LSM contains:

LKinetic : fermion fields

LHiggs : the Higgs potential

LYukawa : the Higgs – Fermion interactions

  • Plan:
  • Look at symmetry aspects of the Lagrangian
  • How is CP violation implemented?

→ Several “miracles” happen in symmetry


Standard Model gauge symmetry:

Note Immediately: The weak part is explicitly parity violating


  • Lorentz structure of the Lagrangian
  • Introduce the fermion fields in the SM
  • LKinetic : local gauge invariance : fermions ↔ bosons
  • LHiggs : spontaneous symmetry breaking
  • LYukawa : the origin of fermion masses
  • VCKM : CP violation
lagrangian density
Lagrangian Density

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

structure of a lagrangian
Structure of a Lagrangian

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:


Consider a spin-1/2 (Dirac) particle (“nucleon”) interacting with a spin-0 (Scalar) object (“meson”)

Nucleon field

Meson potential

Nucleon – meson interaction


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

transformation properties
Transformation Properties

(Ignoring arbitrary phases)

Transformation properties of Dirac spinor bilinears:



the standard model lagrangian
The Standard Model Lagrangian
  • 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

fields notation
Fields: Notation

Interaction rep.








Q = T3 + Y


with y = QL, uR, dR, LL, lR, nR


Under SU2:

Left handed doublets

Right hander singlets


Scalar field:


Interaction representation: standard model interaction is independent of generation number

fields notation15
Fields: Notation

Q = T3 + Y


  • The left handed quark doublet :
  • Similarly for the quark singlets:
  • The left handed leptons:
  • And similarly the (charged) singlets:
intermezzo local gauge invariance in a single transparancy
Intermezzo: Local Gauge Invariance in a single transparancy

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 it turns out that:

=> Introduce the covariant derivative:

and demand that Am transforms as:

is invariant!


  • Introduce charged fermion field (electron)
  • Demand invariance under local gauge transformations (U(1))
  • The price to pay is that a gauge field Ammust be introduced at the same time (the photon)
the kinetic part
:The Kinetic Part

Fermions + gauge bosons + interactions

Procedure: Introduce the Fermion fields and demand that the theory is local gauge invariant

Start with the Dirac Lagrangian:


Gam :8 gluons

Wbm: weak bosons: W1, W2, W3

Bm: hyperchargeboson



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

the kinetic part18
: The Kinetic Part






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)


the higgs potential
: The Higgs Potential

→Note LHiggs = CP conserving








~ 246 GeV

Spontaneous Symmetry Breaking: The Higgs field adopts a non-zero vacuum expectation value



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)

“The realization of the vacuum breaks the symmetry”

the yukawa part
: 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.


L must be Her-

mitian (unitary)

The result is:



To be manifestly invariant under SU(2)

are arbitrary complex matrices which operate in family space (3x3)

=> Flavour physics!

the yukawa part21
: The Yukawa Part

Writing the first term explicitly:


In what aspect is this Lagrangian similar to the example of the nucleon-meson potential?

the yukawa part22
: The Yukawa Part

Formally, CP is violated if:

In generalLYukawais CP violating

Exercise (intuitive proof)

Show that:

  • The hermiticity of the Lagrangian implies that there are terms in pairs of the form:
  • However a transformation under CP gives:

CP is conserved inLYukawaonly ifYij = Yij*

and leaves the coefficientsYij and Yij*unchanged

the yukawa part23
: The Yukawa Part

There are 3 Yukawa matrices (in the case of massless neutrino’s):

  • 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

the fermion masses
: The Fermion Masses


Start with the Yukawa Lagrangian

After which the following mass term emerges:


LMass is CP violating in a similar way as LYuk

the fermion masses25
: The Fermion Masses


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”)

the fermion masses26
: The Fermion Masses


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


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
: 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:


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

flavour changing neutral currents
Flavour Changing Neutral Currents

To illustrate the SM neutral current take the W3m and Bm term of the Kinetic Lagrangian:


And consider the Z-boson field:

Take furtherQLiI=dLiI


In terms of physical fields no non-diagonal contributions occur for the neutral Currents. => GIM mechanism

Standard Model forbids flavour changing neutral currents.

charged currents
Charged 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

charged currents30
Charged 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 standard model lagrangian recap
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 <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

quark field re phasing
Quark field re-phasing

Under a quark phase transformation:

and a simultaneous rephasing of the CKM matrix:


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 - (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

No CP violation in SM.

This is the reason Kobayashi and Maskawa first suggested a third family of fermions!

the lep collider @ cern
The LEP collider @ CERN

Light, left-handed, “active”





Geneva Airport “Cointrin”


Maybe the most important result of LEP:

“There are 3 generations of neutrino’s”

the lepton sector intermezzo
The lepton sector (Intermezzo)
  • N. Cabibbo: Phys. Rev.Lett. 10, 531 (1963)
    • 2 family flavour mixing in quark sector (GIM mechanism)
  • M.Kobayashi and T.Maskawa, Prog. Theor. Phys 49, 652 (1973)
    • 3 family flavour mixing in quark sector
  • Z.Maki, M.Nakagawa and S.Sakata, Prog. Theor. Phys. 28, 870 (1962)
    • 2 family flavour mixing in neutrino sector to explain neutrino oscillations!
  • In case neutrino masses are of the Dirac type, the situation in the lepton sector is very similar as in the quark sector: VMNS~ VCKM.
    • The is one CP violating phase in the lepton MNSmatrix
  • In case neutrino masses are of the Majorana type (a neutrino is its own anti-particle → no freedom to redefine neutrino phases)
    • There are 3 CP violating phases in the lepton MNS matrix
      • However, the two extra phases are unobservable in neutrino oscillations
    • There is even a CP violating phase in case Ndim = 2
lepton mixing and neutrino oscillations
Lepton mixing and neutrino oscillations









  • In the CKM we write by convention the mixing for the down type quarks; in the lepton sector we write it for the (up-type) neutrinos. Is it relevant?
    • If yes: why?
    • If not, why don’t we measure charged lepton oscillations rather then neutrino oscillations?

However, observation of neutrino oscillations is possible due to small neutrino mass differences.

rephasing invariants
Rephasing Invariants

The standard representation of the CKM matrix is:

  • Simplest: Uai = |Vai|2 is independent of quark re-phasing
  • Next simplest: Quartets: Qaibj = Vai Vbj Vaj* Vbi*witha≠band i≠j
    • “Each quark phase appears with and without *”
  • V†V=1: Unitarity triangle: Vud Vcd* + Vus Vcs* + Vub Vcb* = 0
    • Multiply the equation by Vus* Vcs and take the imaginary part:
    • Im (Vus* Vcs Vud Vcd*) = - Im (Vus* Vcs Vub Vcb*)
    • J = Im Qudcs = - Im Qubcs
    • The imaginary part of each Quartet combination is the same (up to a sign)
    • In fact it is equal to 2x the surface of the unitarity triangle
  • Im[Vai Vbj Vaj* Vbi*] = J ∑eabgeijk where J is the universal Jarlskog invariant
  • Amount of CP Violation is proportional to J

However, many representations are possible. What are the invariants under re-phasing?

the unitarity triangle
The Unitarity Triangle


Vtd Vtb*

Vud Vub*



Vcd Vcb*



The “db” triangle:

Area = ½ |Im Qudcb| = ½ |J|

Under re-phasing:

the unitary angles are invariant

(In fact, rephasing implies a rotation of the whole triangle)

wolfenstein parametrization
Wolfenstein Parametrization







Wolfenstein realised that the non-diagonal CKM elements are relatively small compared to the diagonal elements, and parametrized as follows:

Normalised CKM triangle:

cp violation and quark masses
CP Violation and quark masses

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:

the amount of cp violation
The Amount of CP 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!

mass patterns
Mass Patterns


Mass spectra (m = Mz, MS-bar 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

  • Do you want to be famous?
  • Do you want to be a king?
  • Do you want more then the nobel prize?

- 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?

See-saw mechanism?

matter antimatter asymmetry
Matter - antimatter asymmetry
  • In the visible universe matter dominates over anti-matter:
  • There are no antimatter particles present in cosmic rays
  • There are no intense g-ray sources in the universe due to matter anti-matter collisions

Hubble deep field - optical

big bang cosmology
Big Bang Cosmology

Equal amounts

of matter &



Matter Dominates !


the matter anti matter asymmetry
The matter anti-matter asymmetry

Angular Power Spectrum



Cosmic Microwave Background

WMAP satellite

Almost all matter annihilated with antimatter, producing photons…

the sakharov conditions
The Sakharov conditions

Convert 1 in 109 anti-quarks into a quark in an early stage of universe:


  • A matter dominated universe can evolve in case three conditions occur simultaneous:
  • Baryon number violation: L(DB)≠0
  • C and CP Violation:G(N→f) ≠ G(N→f)
  • Thermal non-equilibrium:
  • otherwise: CPT invariance => CP invariance

Sakharov (1964)

baryogenesis at the gut scale
Baryogenesis at the GUT Scale











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 build-up:

CP Violation (2) :r ≠ r

Assuming the back reaction does not occur (3):

Initial X number density

Initial light particle

number density

baryogenesis at electroweak scale
Baryogenesis at Electroweak Scale

Conceptually difficult

SM Electroweak Interactions:

1) Baryon number violation in weak anomaly:

Conserves “B-L” but violates “B+L”

2) CP Violation in the CKM

3) Non-equilibrium: electroweak phase transition

Electroweak phase transition wipes out GUT Baryon asymmetry!

Can it generate a sufficiently large asymmetry?


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:

  • Leptogenesis:
  • Uses the large right handed majorana neutrino masses in the see-saw mechanism to generate a lepton asymmetry at high energies (using the MNS equivalent of CKM).
  • Uses the electroweak sphaleron (“B-L” conserving) processes to communicate this to a baryon asymmetry, which survives further evolution of the universe.
  • Key questions in B physics:
  • Is the SM the only source of CP Violations?
  • Does the SM fully explain flavour physics?