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 _{} 
_{}
color singlet _{} not included. Again color singlet and _{}are not the same thing just as J_{z}=0 can be either J=1 or J=0 with a total spin of 1 still having a zcomponent 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:
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)SU(2)U(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.
Symmetries are also considered to be:
Symmetries of
the Standard Model and Associated Conservation Laws 

Symmetry 
Group 
Symmetry Type 
Conservation Law 
Poincaré 
Translations SO(1,3) Trans+[Lorentz group] 

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 

Baryon phase 
Accidental Global symmetry 

Electron phase 
Accidental Global symmetry 

Muon phase 
Accidental Global symmetry 

Tau phase 
Accidental Global symmetry 
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 spacetime Translations, Rotations, Boosts 
momentum conservation ang. mom conservation energy conservation 

all 


Parity P 
Improper spacetime 

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 I_{z }rather than _{}. 








strong hypercharge Y=2(QI_{z}) _{} 
flavor (limit as quark massè0) 
labels the flavor SU(3) multiplets 
W, S,E 

P 

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 
_{} 
_{} 

I_{z} 
_{} 
_{} 

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]} 

Up 
u 
1 
1.5 to 3.3 
1/2 
1/2 
+2/3 
0 
0 
0 
0 
Antiup 
Down 
d 
1 
3.5 to 6.0 
1/2 
1/2 
−1/3 
0 
0 
0 
0 
Antidown 
Charm 
c 
2 
1,270 
0 
1/2 
+2/3 
0 
+1 
0 
0 
Anticharm 
Strange 
s 
2 
104 
0 
1/2 
−1/3 
−1 
0 
0 
0 
Antistrange 
Top 
t 
3 
171,200 
0 
1/2 
+2/3 
0 
0 
0 
+1 
Antitop 
Bottom 
b 
3 
4,200 
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.
Name 
Symbol 
Antiparticle 
Charge (e) 
Spin 
Mass (GeV/c^{2}) 
Force mediated 
Existence 
γ 
Self 
0 
1 
0 
Confirmed 

−1 
1 
80.4 
Confirmed 

Self? 
0 
1 
91.2 
Confirmed 

Self? 
0 
1 
0 
Confirmed 

G 
Self 
0 
2 
0 
Unconfirmed 

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:
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.
Y_{strong} The hypercharge was added to I_{z}
isospin zcomponent in order to define a 2d plane for SU(3) flavor u,d,s. Y
(yaxis) I_{z }(xaxis) so that
points labeled with Y, I_{z} are particles in a SU(3) multiplet.
If 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 Y_{strong}, I_{z}.
Y_{strong}= S+B=2(Q I_{z})
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.
Y_{w} 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 zcomponents of gauge boson for SU(2) and the gauge boson of U(1). Q = (e/g) T_{3} +
(e/g') Y = sinθ_{W} T_{3} + cosθ_{W}
Y, where
we have introduced the Weinberg
angle, θ_{W}. In terms of this, one can write Z_{μ}=cosθ_{W}
W^{3}_{μ}  sinθ_{W} B_{μ}, and A_{μ}=sinθ_{W}
W^{3}_{μ} + cosθ_{W} B_{μ}.
Which might
lead to a redefintion of of T_{3} and Y such that I_{z}=sinθ_{W}
T_{3} and Y^{W}= 2cosθ_{W} Y 
Parameters of the Standard
Model 

Symbol 
Description 
Renormalization 
Value 
m_{e} 
Electron mass 

511 keV 
m_{μ} 
Muon mass 

106 MeV 
m_{τ} 
Tau lepton mass 

1.78 GeV 
m_{u} 
Up quark mass 
() 
1.9 MeV 
m_{d} 
Down quark mass 
() 
4.4 MeV 
m_{s} 
Strange quark mass 
() 
87 MeV 
m_{c} 
Charm quark mass 
() 
1.32 GeV 
m_{b} 
Bottom quark mass 
() 
4.24 GeV 
m_{t} 
Top quark mass 
(onshell scheme) 
172.7 GeV 
θ_{12} 
CKM 12mixing angle 

0.229 
θ_{23} 
CKM 23mixing angle 

0.042 
θ_{13} 
CKM 13mixing angle 

0.004 
δ 
CKM CPViolating Phase 

0.995 
g_{1} 
U(1) gauge coupling 
() 
0.357 
g_{2} 
SU(2) gauge coupling 
() 
0.652 
g_{3} 
SU(3) gauge coupling 
() 
1.221 
θ_{QCD} 
QCD Vacuum Angle 

~0 
μ 
Higgs quadratic coupling 

Unknown 
λ 
Higgs selfcoupling 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 spacetime. 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)SU(2)U(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.
Main article: Quantum chromodynamics
Main article: Electroweak interaction
The
electroweak sector is a YangMills gauge
theory with the symmetry group ,
where B_{μ} is the U(1) gauge field; Y_{W}
is the weak hypercharge — the generator of the U(1)
group; is
the threecomponent SU(2) gauge field; are
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.
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 _{}
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 M_{L},
T_{L} and (μ_{R})^{c},
(τ_{R})^{c} are
the 2nd (muon) and 3rd (tau) generation analogs of E_{L}
and (e_{R})^{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.
Lefthanded
fermions in the Standard Model 

Generation 1 

Fermion 
Symbol 
Mass ** 


511 keV 


511 keV 


< 2 eV **** 


~ 3 MeV *** 


~ 3 MeV *** 


~ 6 MeV *** 


~ 6 MeV *** 




Generation 2 

Fermion 
Symbol 
Electric 
Weak 
Weak 
Color 
Mass ** 

106 MeV 


106 MeV 


< 2 eV **** 


~ 1.337 GeV 


~ 1.3 GeV 


~ 100 MeV 


~ 100 MeV 




Generation 3 

Fermion 
Symbol 
Electric 
Weak 
Weak 
Color 
Mass ** 

1.78 GeV 


1.78 GeV 


< 2 eV **** 


171 GeV 


171 GeV 


~ 4.2 GeV 


~ 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 lefthanded fermion and a righthanded fermion. For example, the mass of an electron is really a coupling between a lefthanded electron and a righthanded electron, which is the antiparticle of a lefthanded 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 lefthanded electron antineutrino. · *** The masses of baryons and hadrons and various crosssections 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. 2d 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 30^{o} 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 spacetime 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)Y_{lm}(θ,φ), which are the angular momentum states.
_{}_{}
To finalize our understanding of parity we need to ask if any of the internal (nonspacetime) 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 A^{μ }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 Kmeson 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
CPK°> = K°>
and CPK°> = K°>
x because PK = 1 and CK°>=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
x What are the possible nonleptonic
decay modes of these states? (mK = 498
MeV, JK = 0)
•
only
pp
and ppp modes conserve energy and momentum
•
CP
pp>
= +1 and
CPppp> = 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+pp° (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
“KLong” K L°> =
K2°> and “Kshort” K S°> = K1°>