Review: [various web sources]

Particles -------------------------------

fermions

matter

spin 1/2

leptons

baryons

 

 

bosons

forces

spin 1 (graviton spin 2)

interactions vector bosons

QED               

WEAK       

QCD              

Gravity           G

 

Higgs Boson

 

Standard model of elementary particles. The electron is at lower left.

 

color singlet  not included. Again color singlet and are not the same thing just as Jz=0 can be either J=1 or J=0 with a total spin of 1 still having a z-component of zero so a cancellation of the color charge doesn’t imply a singlet state.  It is the color singlet state that is stable.

 

The weak interaction is unique in a number of respects:

  1. It is the only interaction capable of changing flavour.
  2. It is the only interaction which violates parity symmetry P (because it only acts on left-handed particles). It is also the only one which violates CP.

It is mediated by heavy gauge bosons. This unusual feature is explained in the Standard Model by the Higgs mechanism.

 

 

Symmetry -------------------------------

 

The global Poincaré symmetry is postulated for all relativistic quantum field theories. It consists of the familiar translational symmetry, rotational symmetry and the boosts of special relativity. You also consider the improper spacetime transforamtions P spacial inverions,  and T time reversal.  The natural representations of this group which labels the particle by momentum and spin leads naturally to particle antiparticle states. Dirac discovered this in his approach also. With the advent of these anti particle states the transformation  is added [C] as a symmetry. Particle interactions are the same as antiparticle interactions [with some care].  There are several internal symmetries. These are symmetries that do not reflect space time character. Clearly particle and anti particle labels are not based on postion or time.  The local SU(3)\timesSU(2)\timesU(1) gauge symmetry is an internal symmetry that essentially defines the standard model. Roughly, the three factors of the gauge symmetry give rise to the three fundamental interactions.

{Use SU(2)L  only }

Symmetry plays 2 distinct roles in QFT.

  1. Global symmetries that lead to conservation laws, restrictions on equations of motion, reaction and natural choices for the states used to describe systems. You choose for example states of definite charge as the particle states.
  2. Local symmetries that lead to terms in the Lagrangian that describe how the fields interact. A U(1) symmetry can be used to find the coupling of charged fields to the photons as  .

Symmetries are also considered to be:

  1. True complete symmetries when no violation of the symmetry occurs.
  2. Partial when some situations reflect the symmetry but others don’t. Flavor symmetry is a good symmetry and helps predict how to combine the quarks into composite systems such as the proton, neutron  [Note proton and  have the same quark content and the neutron and also have same quark content]. However flavor symmetry is broken by the quark masses.  Also some symmetries are found to be valid for only some interactions.
  3. Spontaneously broken symmetries are symmetries that are true for the system but can be broken by the lowest energy state.  Somehow the energy of the ground state forces the state to assume a value that breaks the symmetry. A simple illustration is a pencil balanced on its point which is symmetric for rotations about the z-axis. The pencil however will fall and assume a specific orientation which no longer reflects this symmetry.  Quantum systems are a bit more complicated because the quantum state of the system could be a linear combination of pencil states of all orientations which would restore the symmetry.  The Higgs particle is added to create a broken symmetry for the vacuum in the standard model. 

 

Symmetries of the Standard Model and Associated Conservation Laws

Symmetry

Group

Symmetry Type

Conservation Law

Poincaré

Translations \rtimes SO(1,3) Trans+[Lorentz group]

Global symmetry

Energy, Momentum, Angular momentum

Poincaré

P parity,

Improper or discrete

 

parity P

Poincaré

T time rev. [antiUnitary]

Improper or discrete

generators not Hermetian so no observable is conserved

internal

C charge conjugation

Improper or discrete

charge conjugation C

Gauge

SU(3)\timesSU(2)\timesU(1)

Local symmetry

Electric charge, Weak isospin, Color charge

Baryon phase

U(1)

Accidental Global symmetry

Baryon number

Electron phase

U(1)

Accidental Global symmetry

Electron number

Muon phase

U(1)

Accidental Global symmetry

Muon number

Tau phase

U(1)

Accidental Global symmetry

Tau-lepton number

 

 

Quantum numbers -------------------------------

 

Quantum numbers are the observables of the quantum system. Under the action of a symmetry some quantum numbers will not change. For example under reflection, P, the location of a particle will change. Under rotation the location of a particle will change. However the even or odd character of a wf doesn’t change under reflection and the angular momentum of a state is unchanged by rotation.

Quantum numbers and symmetry  (M=almost always conserved, A=always, P=partially)

 

Relevant

Symmetry

 

int.

violate

int.

follow

 

Poincare space-time

Translations, Rotations, Boosts

momentum conservation

ang. mom conservation

energy conservation

 

all

 

Parity 

P

Improper space-time

 

W

S,E

P

There is no observed quantum number conserved for time reversal because it is not a unitary transformation.

Charge Conjugation

C

Particle è anti particle

 

W

S,E

M

 

 

 

 

 

 

charge

Q

phase U(1)

 

 

all

A

Baryon number

B

phase U(1)

 

 

all

A

lepton number

L

phase U(1)

 

 

all

 

Lepton F

 

phase U(1)

 

mix

S,E

 

Strangeness

S

U(1)

 

W

S,E

P

Isospin  (upness, downness)

 

SU(2)

 

W

S,E

P

Top

 

 

W

S,E

P

Bottom

 

 

W

S,E

P

There are Quantum numbers that are linear combinations of other quantum numbers. These may be more convenient for labeling states.  This occurs when the natural quantum states may be linear combinations of states with different quantum numbers. The  which has both up and down quarks and is labled by isospin rather Iz rather than .

 

 

 

 

 

 

strong hypercharge

Y=2(Q-Iz)

 

flavor

(limit as quark massè0)

labels the flavor SU(3) multiplets

W, S,E

 

P

Image:Eg2.png

For SU(3) the flavor symmetry of up, down, strange one needs to labels for the multiplets. Usual choice is

weak hypercharge

 

 

 

 

 

 

 

All quarks

just u,d,s

Q

B

Iz

S

 

 

 

C

 

 

 

 

;      

                                                                               

 

 

                                                                               

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

charge

mass

MeV

I

isospin

J

spin

P

parity

C

charge

conj

lep #

&

LF#

 

 

e

-1

0.5

N/A

1/2

+1

N/A

1,e=1

 

 

0

~0

N/A

1/2

+1

N/A

1,e=1

 

 

-1

106

N/A

1/2

+1

N/A

1,=1

 

 

0

~0

N/A

1/2

+1

N/A

1,=1

 

 

-1

1777

N/A

1/2

+1

N/A

1,=1

 

 

0

~0

N/A

1/2

+1

N/A

1,=1

 

 

u

2/3

2

1/2

1/2

+1

N/A

0

 

 

d

-1/3

4

1/2

1/2

+1

N/A

0

 

 

c

2/3

1300

0

1/2

+1

N/A

0

 

 

s

-1/3

90

0

1/2

+1

N/A

0

 

 

t

2/3

172000

0

1/2

+1

N/A

0

 

 

b

-1/3

4000

0

1/2

+1

N/A

0

 

 

0

0

0,1

1

-1

-1

0

 

 

1

80000

N/A

1

 

N/A

0

 

 

-1

80000

N/A

1

 

N/A

0

 

 

0

90000

N/A

1

 

-1

0

 

 

g

0

0

0

1

-1

 

0

 

 

G

0

 

N/A

2

 

 

0

 

 

H

0

>115000

N/A

0

 

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Quark flavor properties[41]

Name  ↓

Symbol  ↓

Gen.  ↓

Mass (MeV/c2)  ↓

I  ↓

J  ↓

Q  ↓

S  ↓

C  ↓

B′  ↓

T  ↓

Antiparticle  ↓

Up

u

1

000002.81.5 to 3.3

1/2

1/2

+2/3

0

0

0

0

Antiup

Down

d

1

000004.83.5 to 6.0

1/2

1/2

−1/3

0

0

0

0

Antidown

Charm

c

2

001270.01,270

0

1/2

+2/3

0

+1

0

0

Anticharm

Strange

s

2

000104.0104

0

1/2

−1/3

−1

0

0

0

Antistrange

Top

t

3

171200.0171,200

0

1/2

+2/3

0

0

0

+1

Antitop

Bottom

b

3

004200.04,200

0

1/2

−1/3

0

0

−1

0

Antibottom

(Key: Gen. = generation, I = isospin, J = spin, Q = electric charge, S = strangeness, C = charm, B′ = bottomness, T = topness. Notation like 104+26−34 denotes measurement uncertainty: the value is between 104 + 26 = 130 and 104 − 34 = 70, with 104 being the most likely value.

 

Bosons

Name

Symbol

Antiparticle

Charge (e)

Spin

Mass (GeV/c2)

Force mediated

Existence

Photon

γ

Self

0

1

0

Electromagnetism

Confirmed

W boson

W

W+

−1

1

80.4

Weak

Confirmed

Z boson

Z

Self?

0

1

91.2

Weak

Confirmed

Gluon

g

Self?

0

1

0

Strong

Confirmed

Graviton

G

Self

0

2

0

Gravity

Unconfirmed

Higgs boson

H0

Self?

0

0

> 112

See below

Unconfirmed

 

Standard model of elementary particles. The electron is at lower left.

The weak interaction is unique in a number of respects:

  1. It is the only interaction capable of changing flavour.
  2. It is the only interaction which violates parity symmetry P (because it only acts on left-handed particles). It is also the only one which violates CP.
  3. It is mediated by heavy gauge bosons. This unusual feature is explained in the Standard Model by the Higgs mechanism.

 

unification is accomplished under an SU(2) × U(1) gauge group

 

Hypercharge and Isospin

 

In looking at quantum numbers for the quarks one defines isospin and hypercharge. The weak interaction sector also will define a version of isospin and hypercharge. The strong interaction versions are related to flavor symmetry while the weak versions are based on gauge transformations that unify E&M and weak.  The following discussion will attempt to clarify the difference.

 

 

Ystrong The hypercharge was added to Iz isospin z-component in order to define a 2-d plane for SU(3) flavor u,d,s. Y (y-axis)  Iz (x-axis) so that points labeled with Y, Iz are particles in a SU(3) multiplet.

 

Image:Eg2.pngIf u,d,s quarks are the same under the strong interaction than multiplets of SU(3) are degenerate and these particle states will be important. To characterize the members of the multiplet you use two labels Ystrong, Iz.

 

 

Ystrong= S+B=2(Q- Iz)

 

add other quarks you then define

 

 

 

 

 

 

The symmetry for the electroweak unification combines an SU(2) and U(1) symmetry constraint. The SU(2) structure is apparent because we group many of the particles to doublets and a U(1) symmetry should generate an E&M interaction. For E&M one discovers that the generator is the charge. So one would recognize the U(1) symmetry as the global symmetry that requires charge conservation. Build the electroweak however this particular symmetry reveals itself only after the symmetry breaking mixes the vector bosons. Thus we start with a U(1) with the generator Y, weak hypercharge.

 

Yw is the quantity associated with the U(1) gauge transformation that produces the electroweak interaction

 

SU(2)L×U(1)Y

 

The SU(2) doublets are even referred to as weak isospin.  These weak isospin doublets are all of the weak partners (u,d’), (c,s’), (t,b’), (e,υ), … [Strong isospin is u,d with no Cabbibo mixing].  In developing the SU(2) symmetry the operators T

 

    with   

 

are used. This is the same structure as the strong isospin with I ()

 

Need to figure this out !!!!!!!!!

 

However the final version of the theory mixes the state associated with the z-components of gauge boson for SU(2) and the gauge boson of U(1).

 

Q  =  (e/g) T3 + (e/g') Y  =  sinθW T3 + cosθW Y,

where we have introduced the Weinberg angle, θW. In terms of this, one can write

Zμ=cosθW W3μ - sinθW Bμ,   and   Aμ=sinθW W3μ + cosθW Bμ.

 

Which might lead to a redefintion of  of  T3 and Y  such that

 

Iz=sinθW T3 and YW= 2cosθW Y

 

Q = I^W_z + {1 \over 2} Y^W

 

 

 

Parameters of the Standard Model

Symbol

Description

Renormalization
scheme (point)

Value

me

Electron mass

 

511 keV

mμ

Muon mass

 

106 MeV

mτ

Tau lepton mass

 

1.78 GeV

mu

Up quark mass

(\mu_{\overline{\text{MS}}}=2\text{ GeV})

1.9 MeV

md

Down quark mass

(\mu_{\overline{\text{MS}}}=2\text{ GeV})

4.4 MeV

ms

Strange quark mass

(\mu_{\overline{\text{MS}}}=2\text{ GeV})

87 MeV

mc

Charm quark mass

(\mu_{\overline{\text{MS}}}=m_c)

1.32 GeV

mb

Bottom quark mass

(\mu_{\overline{\text{MS}}}=m_b)

4.24 GeV

mt

Top quark mass

(on-shell scheme)

172.7 GeV

θ12

CKM 12-mixing angle

 

0.229

θ23

CKM 23-mixing angle

 

0.042

θ13

CKM 13-mixing angle

 

0.004

δ

CKM CP-Violating Phase

 

0.995

g1

U(1) gauge coupling

(\mu_{\overline{\text{MS}}}=M_\text{Z})

0.357

g2

SU(2) gauge coupling

(\mu_{\overline{\text{MS}}}=M_\text{Z})

0.652

g3

SU(3) gauge coupling

(\mu_{\overline{\text{MS}}}=M_\text{Z})

1.221

θQCD

QCD Vacuum Angle

 

~0

μ

Higgs quadratic coupling

 

Unknown

λ

Higgs self-coupling strength

 

Unknown

Technically, quantum field theory provides the mathematical framework for the standard model, in which a Lagrangian controls the dynamics and kinematics of the theory. Each kind of particle is described in terms of a dynamical field that pervades space-time. The construction of the standard model proceeds following the modern method of constructing most field theories: by first postulating a set of symmetries of the system, and then by writing down the most general renormalizable Lagrangian from its particle (field) content that observes these symmetries.

The global Poincaré symmetry is postulated for all relativistic quantum field theories. It consists of the familiar translational symmetry, rotational symmetry and the inertial reference frame invariance central to the theory of special relativity. The local SU(3)\timesSU(2)\timesU(1) gauge symmetry is an internal symmetry that essentially defines the standard model. Roughly, the three factors of the gauge symmetry give rise to the three fundamental interactions. The fields fall into different representations of the various symmetry groups of the Standard Model (see table). Upon writing the most general Lagrangian, one finds that the dynamics depend on 19 parameters, whose numerical values are established by experiment. The parameters are summarized in the table at right.

[edit] The QCD sector

Main article: Quantum chromodynamics

[edit] The electroweak sector

Main article: Electroweak interaction

The electroweak sector is a Yang-Mills gauge theory with the symmetry group U(1)\times SU(2)_L,

\mathcal{L}_{EW}=\sum_\psi\bar\psi\gamma^\mu\left(
i\partial_\mu-g'{1\over2}Y_WB_\mu
-g{1\over2}\vec\tau_L\vec W_\mu
\right)\psi 
+\mathcal{L}_{YM}(B_\mu)
+\mathcal{L}_{YM}(\vec W_\mu),

where Bμ is the U(1) gauge field; YW is the weak hypercharge — the generator of the U(1) group; \vec{W}_\muis the three-component SU(2) gauge field; \vec{\tau}_Lare the Pauli matrices — infinitesimal generators of the SU(2) group, the subscript L indicates that they only act on left fermions; g' and g are coupling constants.

Additional Symmetries of the Standard Model

From the theoretical point of view, the standard model exhibits additional global symmetries that were not postulated at the outset of its construction. There are four such symmetries and are collectively called accidental symmetries, all of which are continuous U(1) global symmetries. The transformations leaving the Lagrangian invariant are

U(1)è

 

E_L\rightarrow e^{i\beta}E_L\text{ and }(e_R)^c\rightarrow  e^{i\beta}(e_R)^c

M_L\rightarrow  e^{i\beta}M_L\text{ and }(\mu_R)^c\rightarrow  e^{i\beta}(\mu_R)^c

T_L\rightarrow  e^{i\beta}T_L\text{ and }(\tau_R)^c\rightarrow e^{i\beta}(\tau_R)^c.

The first transformation rule is shorthand to mean that all quark fields for all generations must be rotated by an identical phase simultaneously. The fields ML, TL and R)c, R)c are the 2nd (muon) and 3rd (tau) generation analogs of EL and (eR)c fields.

By Noether's theorem, each of these symmetries yields an associated conservation law. They are the conservation of baryon number, electron number, muon number, and tau number. Each quark carries 1/3 of a baryon number, while each antiquark carries -1/3 of a baryon number. The conservation law implies that the total number of quarks minus number of antiquarks stays constant throughout time. Within experimental limits, no violation of this conservation law has been found.

Similarly, each electron and its associated neutrino carries +1 electron number, while the antielectron and the associated antineutrino carry -1 electron number, the muons carry +1 muon number and the tau leptons carry +1 tau number. The standard model predicts that each of these three numbers should be conserved separately in a manner similar to the baryon number. These numbers are collectively known as lepton family numbers (LF). The difference in the symmetry structures between the quark and the lepton sectors is due to the masslessness of neutrinos in the standard model. However, it was recently found that neutrinos have small mass, and oscillate between flavors, signaling the violation of these three quantum numbers.

In addition to the accidental (but exact) symmetries described above, the standard model exhibits a set of approximate symmetries. These are the SU(2) Custodial Symmetry and the SU(2) or SU(3) quark flavor symmetry.

Left-handed fermions in the Standard Model

Generation 1

Fermion
(left-handed)

Symbol

Electric
charge

Weak
isospin

Weak
hypercharge

Color
charge
 *

Mass **

 

Electron

e^-\,

-1\,

-1/2\,

-1\,

\bold{1}\,

511 keV

 

Positron

e^+\,

+1\,

0\,

+2\,

\bold{1}\,

511 keV

 

Electron-neutrino

\nu_e\,

0\,

+1/2\,

-1\,

\bold{1}\,

< 2 eV ****

 

Up quark

u\,

+2/3\,

+1/2\,

+1/3\,

\bold{3}\,

~ 3 MeV ***

 

Up antiquark

\bar{u}\,

-2/3\,

0\,

-4/3\,

\bold{\bar{3}}\,

~ 3 MeV ***

 

Down quark

d\,

-1/3\,

-1/2\,

+1/3\,

\bold{3}\,

~ 6 MeV ***

 

Down antiquark

\bar{d}\,

+1/3\,

0\,

+2/3\,

\bold{\bar{3}}\,

~ 6 MeV ***

 

 

Generation 2

Fermion
(left-handed)

Symbol

Electric
charge

Weak
isospin

Weak
hypercharge

Color
charge *

Mass **

 

Muon

\mu^-\,

-1\,

-1/2\,

-1\,

\bold{1}\,

106 MeV

 

Antimuon

\mu^+\,

+1\,

0\,

+2\,

\bold{1}\,

106 MeV

 

Muon-neutrino

\nu_\mu\,

0\,

+1/2\,

-1\,

\bold{1}\,

< 2 eV ****

 

Charm quark

c\,

+2/3\,

+1/2\,

+1/3\,

\bold{3}\,

~ 1.337 GeV

 

Charm antiquark

\bar{c}\,

-2/3\,

0\,

-4/3\,

\bold{\bar{3}}\,

~ 1.3 GeV

 

Strange quark

s\,

-1/3\,

-1/2\,

+1/3\,

\bold{3}\,

~ 100 MeV

 

Strange antiquark

\bar{s}\,

+1/3\,

0\,

+2/3\,

\bold{\bar{3}}\,

~ 100 MeV

 

 

Generation 3

Fermion
(left-handed)

Symbol

Electric
charge

Weak
isospin

Weak
hypercharge

Color
charge *

Mass **

 

Tau lepton

\tau^-\,

-1\,

-1/2\,

-1\,

\bold{1}\,

1.78 GeV

 

Anti-tau lepton

\tau^+\,

+1\,

0\,

+2\,

\bold{1}\,

1.78 GeV

 

Tau-neutrino

\nu_\tau\,

0\,

+1/2\,

-1\,

\bold{1}\,

< 2 eV ****

 

Top quark

t\,

+2/3\,

+1/2\,

+1/3\,

\bold{3}\,

171 GeV

 

Top antiquark

\bar{t}\,

-2/3\,

0\,

-4/3\,

\bold{\bar{3}}\,

171 GeV

 

Bottom quark

b\,

-1/3\,

-1/2\,

+1/3\,

\bold{3}\,

~ 4.2 GeV

 

Bottom antiquark

\bar{b}\,

+1/3\,

0\,

+2/3\,

\bold{\bar{3}}\,

~ 4.2 GeV

 

Notes:

·              * These are not ordinary abelian charges, which can be added together, but are labels of group representations of Lie groups.

·              ** Mass is really a coupling between a left-handed fermion and a right-handed fermion. For example, the mass of an electron is really a coupling between a left-handed electron and a right-handed electron, which is the antiparticle of a left-handed positron. Also neutrinos show large mixings in their mass coupling, so it's not accurate to talk about neutrino masses in the flavor basis or to suggest a left-handed electron antineutrino.

·              *** The masses of baryons and hadrons and various cross-sections are the experimentally measured quantities. Since quarks can't be isolated because of QCD confinement, the quantity here is supposed to be the mass of the quark at the renormalization scale of the QCD scale.

·              **** The Standard Model assumes that neutrinos are massless. However, several contemporary experiments prove that neutrinos oscillate between their flavour states, which could not happen if all were massless. [11] It is straightforward to extend the model to fit these data but there are many possibilities, so the mass eigenstates are still open. See Neutrino#Mass.

Parity

Momentum switches

Spin unchanged

 

Examine a wheel spinning and spinning in a mirror. The object would spin in the same direction.

Consider the rotating wheel shown above. Map all the points on the wheel on the right to point on the wheel on the left through parity. These points are found by drawing a line through the origin (x,y,z è -x,-y,-z). Now map these same points again at short time later.  The result is a parity transformation acting on the object (not the coordinate system) preserves the sense of the rotation.

 

Angular momentum is an axial vector.  No change in sign.

Position, velocity and momentum are vectors. Change sign.

 

Proper Poincare transformations are all the translations, rotations and boosts that are continuously connected to the identity.  There is no way to define a transformation that is dependent on a parameter, such that changes in the parameter start the corresponding transformation at the identity and at some other parameter value the transformation is a parity transformation. 2-d rotations for θ=0 o correspond to the identity.  The angle can be increased from 0 o to for example to 30 o through arbitrarily small steps. The rotation about an axis by 30o is therefore continuously connected back to the identity.

 

There are three important types of transformations that fall into this category:

Parity and time reversal are improper transformations but are part of the full Poincare set of possible transformations. Charge conjugation is an additional transformation that does not involve space-time but is essential for field theories.

 

How do states (experiments, systems) transform under parity?

It is convenient to find states that have special properties (eigenstates).

Even without QM, function space can be examined in terms of odd and even functions. In general a function is neither odd or even but and function can be written as a sum of odd and even.

 

[SAME EQUATION AS ABOVE]

 

F is a general function that can always be written in terms of some odd and some even function.

 

Since a particle will be described by its spatial wavefunction it is convenient to recognize that a general function can also be written in terms of R(r)Ylm(θ,φ), which are the angular momentum states.

To finalize our understanding of parity we need to ask if any of the internal (non-space-time) properties are impacted by a parity transformation.

 

 

Dirac equation (the relativistic treatment of the fermions) predicts, find, or demands that there be antiparticles related to particles.  The relationship specifies that the parity of fermion and antifermion are opposite.

 

The solution for free fields based on Maxwell’s equations has an ambiguity that is usually resolved by choosing the Electric field to be a vector and the Magnetic field to be an axial vector.  The four vector Aμ potential then has an overall negative parity (transforms as a vector).

 

Combining quarks to make composite system such as a pion requires the combination of the spatial, spin, intrinsic parity to find the overall parity of the composite system.

 

 

 

&&&&&&&&&&&&&&&&&&&&&&&&&&

x       There are 2 neutral K-meson states

       K° = ds    S=+1      and      K° = ds   S=-1

x       But since Strangeness S is not conserved in weak decays, these states can be converted into eachother by box diagrams involving exchange of two W bosons …

 

 

 

 

x       Such transitions between particle and antiparticle states are usually forbidden because of conservation of some quantum number (e.g. charge in K+/K- or baryon number in neutron/antineutron)

x        

x       There is no conserved quantum number to distinguish the K° and K° states when weak interactions are taken into account

x       The observed physical particles  correspond to linear combinations of K° and K°  i.e. the states mix

x       Begin by assuming that CP invariance is exact and that the neutral kaon states are eigenstates of CP

x        However       CP|K°> = |K°>    and     CP|K°> = |K°>

x       because PK = -1 and C|K°>=-|K°> by convention

x       i.e. K° and K° are not CP eigenstates !

x       … but CP eigenstates can be formed from linear combinations …

x       |K1°> = ( |K°> + |K°> ) / Ö2     with CP = +1

x       |K2°> = ( |K°> -  |K°> ) / Ö2     with CP =  -1

x       |K°> and |K°> are the strong interaction eigenstates whilst |K1°> and |K2°> are the weak interaction eigenstates !

x       Hence    K1° ® pp     and      K2° ® ppp

x       What are the possible non-leptonic decay modes of these states?  (mK = 498 MeV, JK = 0)

       only pp and ppp modes conserve energy and momentum

       CP| pp> = +1  and  CP|ppp> = -1

       p°p° :   P = (-1)(-1)(-1)^L = +1  since L=0

               C = (+1)(+1) = +1

       p°p°p° : P = (-1)(-1)(-1)(-1)^(L12)(-1)^(L3) = -1   as L12 = L3

                C = (+1)(+1)(+1) = +1

       same results for p+p- and p+p-p° (see Martin+Shaw 10.2)

       since mp = 140 MeV the phase space factor for the pp mode is much larger than for the ppp (i.e. much more energy available for final state particle kinetic energies) and hence it has a much shorter lifetime (about 600 times shorter!)

       thus  K-Long”  |K L°> = |K2°>   and   K-short”  K S°> = |K1°>