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Units, Constants and Useful Formulas
Isospin
-------
Isospin was originally introduced to explain symmetries of the
neutron. Protons and neutrons have roughly the same mass. If
we neglect their charge (a resonable assumption given that the
charge is small in comparison to other forces inside the nucleus),
we can consider them to be symmetric although the symmetry is not
exact.
Similar to a spin 1/2 particle, which have 2 states, protons and
neutrons were said to be of isospin, 1/2. The proton and neutron
were then associated with different isospin projections I_{3} = +1/2
and -1/2 respectively where I_{3} is te component of isospin along
the z-axis.
With the discovery of quarks, isospin was extended to include the
up and down quark since both are very similar in mass, and have
the same strong interactions. Particles made of the same numbers
of up and down quarks have similar masses and are grouped together.
After the quark model was elaborated, it was found that the
isospin projection was related to the up and down quark content
of particles. The relation is:
I_{3} = (1/2){(n_{u} - n_{u}) - (n_{d} - n_{d})}
Where the underscore represents the antiparticle.
Thus, I_{3} of u is +1/2 and the I_{3} of d is -1/2.
I_{3} is a quantum number related to the strong interaction. It is
NOT spin in the sense of spin angular momentum but does follow the
same mathematics i.e. SU(2). For example, a proton-neutron pair can
be coupled in a state of total isospin 1 or 0. Thus, we can use the
same diagrams as spin angular momentum when we talk about isospin.
| | | + +3/2
| | + +1 |
| | | |
| + +1/2 | + +1/2
| | | |
+ 0 | + 0 |
| | | |
| + -1/2 | + -1/2
| | | |
| | | |
| | + -1 + -3/2
p and n Pions Deltas
The number of spin/isospin states is given by:
n_{states} = (2l + 1) where l is the spin/isospin
Extention to SU(3) - Flavor Symmetry
------------------------------------
With the discovery of new particles there seemed to be an enlarged
symmetry that contained isospin as a subgroup. This larger symmetry
was named the Eightfold Way whose origins were explained by up,
down and strange quarks which would belong to the fundamental
representation of the SU(3) flavor symmetry. The SU(2) symmetry
is slightly broken because the u and d quark masses are slightly
different. However, the SU(3) symmetry is badly broken due to the
much higher mass of the strange quark.
Gell-Mannâ€“Nishijima Formula
---------------------------
The GMN formula relates all flavour quantum numbers (isospin up and
down, strangeness, charm, bottomness, and topness) with the baryon
number and the electric charge.
Q = I_{3} + (1/2)(B + S)
where:
B = (1/3)(n_{q} - n_{q}) = (# of quarks - # of antiquarks)
B is the baryon number and S is the strangeness quantum number.
Thus, For an up quark we get:
Q = 1/2 + (1/2)(1/3) = 2/3
Hypercharge
-----------
Hypercharge, Y, is a quantum number relating the charge, Q, and
the isospin, I_{3}. Mathematically, hypercharge is:
Y = S + C + B' + T + B
Where,
S = Strangeness
C = Charm
B' = Bottomness
T = Topness
B = Baryon number
Isospin, electric charge and hypercharge are related by.
Q = I_{3} + Y/2
I_{3} Q Baryon S Y Mass (MeV)
--- --- ------ --- --- ---------
u 1/2 2/3 1/3 0 1/3 1.7 - 3.3
d -1/2 -1/3 1/3 0 1/3 4.1 - 5.8
s 0 -1/3 1/3 -1 -2/3 101
Weak Isospin
------------
Consider the decay of a neutron to a proton inside an atomic
nucleus that increases the atomic number by 1, while emitting
an electron and an anti-neutrino.
_
udd -> uud + e + ν_{e}
u e
\ /
\ /
\/\/\/\/\/\/\/
/ W^{-} \
/ \ _
/ \ ν_{e}
d
If we were to look at this in terms of isospin we would not
be able to balance the equation because the isospin only
connects quarks and not leptons. In order to fix this we need
to introduce the concept of weak isospin, I_{3}, that applies to
all particles. The Weak Isospin serves as a quantum number
and governs how that particle behaves in the weak interaction.
Like isospin, Weak Isospin is described by the mathematics of
SU(2).
In any given weak interaction, weak isospin is conserved meaning
that the sum of the weak isospin numbers of the particles entering
the interaction equals the sum of the weak isospin numbers of the
particles exiting the interaction.
Unlike regular isospin symmetry, which is only approximate, weak
isospin symmetry is exact.
Weak Hypercharge
----------------
The weak hypercharge, Y_{W}, is a quantum number relating the electric
charge, Q, and the weak isospin, I_{3}. The weak hypercharge unifies
weak interactions with electromagnetic interactions. The weak
hypercharge is associated with with the U(1)_{Y} component of
SU(2)_{L} ⊗ U(1)_{Y}. It is different but related to the electric charge
associated with the U(1)_{EM} of QED. It satisfies the equality:
Q = I_{3} + Y_{W}/2
∴ Y_{W} = 2(Q - I_{3})
where Q is in units of e.
Q, I_{3} and Y_{W} for left-handed fermions is:
Q I_{3} Y_{W}
--- --- ----
ν_{e} 0 1/2 -1
e -1 -1/2 -1
u 2/3 1/2 1/3
d -1/3 -1/2 1/3
s -1/3 -1/2 1/3
Q, I_{3} and Y_{W} for right-handed fermions is:
Q I_{3} Y_{W}
--- --- ----
ν_{e} does not exist
e -1 0 -2
u 2/3 0 4/3
d -1/3 0 -2/3
s -1/3 0 -2/3
Q, I_{3} and Y_{W} for the gauge bosons is:
Gauge bosons:
Q I_{3} Y_{W}
--- --- ----
Z^{0} 0 0 0
W^{+} 1 1 0
W^{-} -1 -1 0
Particles with weak hypercharge interact by exchange of a boson,
called the B boson. The B boson is very similar to more familiar
U(1) gauge boson ... the photon. Thus the weak hypercharge force
is a lot like electromagnetism but its strength is proportional to
the weak hypercharge rather than the charge.
Experimentally, it is observed that only left handed (aka left-chiral)
fermions and right-handed anti-fermions (aka right-chiral) participate
in the weak interaction. Consider:
_
u_{L}d_{L}d_{L} -> u_{L}u_{L}d_{L} + e_{L} + ν_{R}
The weak isospins, I_{3}, are:
(1/2 - 1/2 - 1/2) = (1/2 + 1/2 - 1/2) - (1/2) - (1/2)^{*}
∴ (-1/2) = (1/2) - (1/2) - (1/2)
∴ (-1/2) = (-1/2) and the weak isospin is conserved.
_
* ν_{R} ≡ -ν_{L} (note that ν_{R}'s have yet to be observed directly in nature)
The weak hypercharges, Y_{W}, are:
(1/3 + 1/3 + 1/3) = (1/3 + 1/3 + 1/3) - 1 + 1
∴ 1 = 1 and the weak hypercharge is conserved.
The charges, Q, are:
(2/3 - 1/3 - 1/3) = (2/3 + 2/3 - 1/3) - 1 + 0
∴ 0 = 0 and the charge is conserved.
Note: I_{3} and Y_{W} are not conserved when the symmetry is spntaneously
broken. This is discussed in detail in the sections on electroweak
unification and the Higgs mechanism.