PHY304 
Particle Physics 
Dr C N Booth 
Properties of Quarks
In the earlier part of this course, we
have discussed three families of leptons but principally concentrated on
one doublet of quarks, the u and d. We will now introduce other types
of quarks, along with the new quantum numbers which characterise them.
Isospin
It was noticed that many groupings of particles of similar mass and properties
fitted in to common patterns. One way to characterise these is using
isotopic spin or isospin, I. This quantity has nothing to
do with the real spin of the particle, but obeys the same addition laws
as the quantum mechanical rules for adding angular momentum or spin.
When the orientation of an isospin vector is considered, it is in some
hypothetical space, not in terms of the x, y and z
axes of normal coordinates.
Nucleons (p, n), pi mesons (π^{+}, π^{0}, π^{−})
and the baryons known as Δ
(Δ^{++}, Δ^{+}, Δ^{0}, Δ^{−})
are three examples of groups of similar mass particles differing in charge
by one unit. The charge Q in each case can be considered as
due to the orientation of an “isospin vector” in some hypothetical space,
such that Q depends on the third component I_{3}.
Thus the nucleons belong to an isospin doublet:
p ≡ I, I_{3}› = ½, ½›;
n ½, −½›.
Similarly the pions form an isospin triplet,
π^{+} = 1, 1›;
π^{0} = 1, 0›;
π^{−} = 1, −1›.
The Δ
forms a quadruplet with I = .
The rule for electric charge can then be written
Q = e(½B + I_{3}),
where B is the baryon number which is 1 for nucleons and the Δ
and 0 for mesons such as the π.
In terms of quarks, the u and d form an isospin doublet,
u = ½, ½›;
d ½, −½›
(both with B = ).
Three quarks with I =
can combine to form I_{tot} =
or .
I_{tot} =
gives the nucleons while
I_{tot} =
forms the Δ.
It is useful to consider the symmetry of the quarks inside these baryons.
The internal wavefunction can be written as a product of terms,
Ψ = ψ_{spin}ψ_{space}ψ_{isospin}ψ_{colour},
and must be antisymmetric overall (as quarks are fermions).
ψ_{colour} is always antisymmetric (as hadrons are colourless);
the symmetry of ψ_{space} is given by (−1)^{l}
and l the orbital angular momentum is zero for the longlived hadrons we
consider, so ψ_{space} is symmetric.
Thus the product ψ_{spin}ψ_{isospin} must be symmetric,
implying either both must be symmetric or both must be antisymmetric.
This explains the correlation between allowed spin and isospin states for the baryons:
the Δ has
I =
and
s =
(both symmetric), while the nucleons have
I =
and
s =
(both antisymmetric).
In strong interactions, the total isospin vector (as well as I_{3})
is conserved. This is not true in electromagnetic or weak interactions.
The conservation of isospin has observable effects on the relative rates of strong
interactions, as will be discussed in the lectures.
Strangeness
It was observed that some unstable particles produced in strong interactions
had a long lifetime. This unusual stability for strongly interacting
particles led to the term of strangeness. Such particles are
always produced in pairs (associated production), and the quantum number
of strangeness,
S, was introduced, which is conserved in strong
interactions. Thus in the interaction
π^{−} p → Λ^{0} K^{0},
the Λ
is assigned S = −1 and the K S = +1. The strange particles
can only decay by the weak interaction, which does not conserve strangeness
(as we will discuss later).
Isospin multiplet  B  S  I  <Q>/e 
Y = B + S 
π^{+} π^{0} π^{−}  0 
0  1 
0  0 
p n  1 
0  ½ 
½  1 
Δ^{++} Δ^{+} Δ^{0} Δ^{−}  1 
0  
½  1 
Λ  1 
−1  0 
0  0 
Σ^{+} Σ^{0} Σ^{−}  1 
−1  1 
0  0 
K^{+} K^{0}  0 
1  ½ 
½  1 
K^{−}  0 
−1  ½ 
−½  −1 
Ξ^{0} Ξ^{−}  1 
−2  ½ 
−½  −1 
Ω^{−}  1 
−3  0 
−1  −2 
A selection of strange and nonstrange baryon and meson multiplets.
The formula for electric charge must know be modified to read
Q = e(I_{3} + ½B + ½S) = e(I_{3} + ½Y)
where Y = B + S is known as
the hypercharge.
(This formula is known as the GellMann Nishijima
relation.)
Families of particles with similar properties (e.g. same
spin and parity) can be plotted in terms of Y versus I_{3},
and form regular geometrical figures, as shown below.
In terms of quarks we can introduce a new flavour of quark, the
strange quark s. This has charge −
and baryon number
(like a d quark) but I = 0 and S = −1.
It is also somewhat heavier than the u and d quarks. Since baryons
consist of qqq, it is clear why no positive baryons exist with
S > 1, while negative baryons are found with
S = −2 or −3.
The lowestlying pseudoscalarmeson states
(J^{P} = 0^{−}), with quark assignments indicated.
(The states at the origin are displaced slightly for clarity.)
The vectormeson nonet (J^{P} = 0^{−}).
(Quark assignments are the same as above.)
The baryon octet of spinparity
J^{P} = ^{+}
(with masses in MeV/c^{2}).
The baryon decuplet with spinparity
J^{P} = ^{+}.
Further quarks
Other, still heavier quarks also exist. The charm quark, c,
has a charge of ,
like the u, and can be considered as a partner to the s. In 3 dimensions,
as illustrated below, particles containing c quarks can be plotted, and
again show regular patterns. We thus have 2 doublets or generations
of quarks  (d, u) and (s, c). Since there are 3 doublets of leptons,
there are theoretical reasons for expecting a third doublet of quarks too.
Particles containing b quarks (bottom or beauty) were discovered
in 1977. The b is an even heavier version of the d. Its partner,
the t (top or truth) was first seen in 1994, and it has the greatest mass
of any known fundamental particle at 174 GeV/c^{2}.
Multiplets of hadrons containing up, down, strange and charm quarks.
The slices through these figures where Charm = 0 correspond
to the plane figures already shown in the previous diagrams, though containing
new particles in the case of the mesons, composed of charmanticharm
quarks.
Some of these particles were discovered relatively recently!
See the
October 2002 article in Physics World.
Flavour  Charge/e  B 
I  I_{3}  S 
c  b  t 
Mass (GeV/c^{2}) 
d  −  
½  −½  0  0  0 
0  0.005 
u  +  
½  +½  0  0  0 
0  0.002 
s  −  
0  0  −1  0  0  0 
0.095 
c  +  
0  0  0  +1  0  0 
1.3 
b  −  
0  0  0  0  −1  0 
4.2 
t  +  
0  0  0  0  0  +1 
174 
Quark quantum numbers and masses.
Note the convention that quarks with a negative electric charge carry a negative flavour quantum number.
The masses quoted are "bare masses"  when bound in hadrons the effective masses differ, especially for the lightest quarks.
(Binding and kinetic energies mean that u and d quarks can be treated as effectively equal in mass.)
The particle content of the Standard Model of Particle Physics (including the Higgs
boson which is not covered in this course).
Quark Flavour and the Weak Interaction
As we have already seen, the strong and electromagnetic interactions conserve
quark flavour, whereas the weak interaction may change it. In many
weak decays, the changes are within a generation, e.g. in beta decay the
W couples a u to a d quark; in the decay
D^{+} → π^{+}
it couples a c to an s.
However, this is not always the case, e.g.
in the decay
the W couples an s to a u quark, and it was observed that such strangenesschanging
decays were slightly weaker than strangenessconserving weak decays.
Cabibbo explained this by proposing that the eigenstates of the weak
interaction are different from those of the strong interaction. The
strong interaction eigenstates are the u, d, s, c, b and t quarks, with
welldefined isospin, strangeness etc. The eigenstates of the weak
interaction, which does not conserve I, S etc., are said
to be those of “weak isospin” T. For simplicity, let us first
consider the first 2 generations alone. The weak eigenstates are
the leptons and orthogonal linear combinations of the familiar quarks
with 
d_{c} = α d + β s 


s_{c} = −β d + α s 
(normalisation α^{2} + β^{2} = 1) 
α is
usually known as cos θ_{c}, where θ_{c}
is the Cabibbo angle. A value of
sin θ_{c} = 0.25
is consistent with the observed apparent variation of weak coupling
constant with reaction type.
The relationship between weak and strong eigenstates in 2 generations
can also be expressed as

= 


weak estates 

mixing matrix 
strong estates 
The same formalism can be used for 3 generations, and the mixing matrix,
known as the CabibboKabayashiMaskawa or CKM matrix, can be parametrised
in a number of ways.
The magnitudes of the matrix elements have been determined experimentally,
and are given by
Note that the values along the leading diagonal are quite close to one,
those adjacent to it are significantly smaller, and the elements in the
topright and bottomleft corners are much smaller. This means
that the mixing results in states which contain a small admixture of the
quark from the next generation, while mixing between 1st and 3rd generation
quarks is extremely small. Physically, this is revealed during weak
decays in the relative probability of producing hadrons containing the
respective quarks. For example, when a top quark decays it produces
a b' quark. This is bound in a hadron by the strong interaction,
so must be revealed as a strong eigenstate. The b' is most likely
to result in a particle containing a b quark, with a smaller probability
of an s quark and almost negligible likelihood of producing a d quark.
Therefore, the diagonal structure of the CKM matrix means that weak decays
are most likely to be within a generation if allowed by conservation
of energy (a particle cannot decay into one that is heavier) or to the
next generation below if this is not allowed. The most likely
overall decay chain of a b quark is therefore
b → c → s → u.
When the chargedcurrent weak decays are considered along with binding into
strong eigenstates in the hadrons, the elements of M can be interpreted
as giving the effective transition strengths between quarks as follows:
.
(Again, physical decays must always be to the lighter quark.)
For two generations, one parameter was required to describe the mixing.
This was the Cabibbo angle. With three generations, 4 independent
parameters are needed to define a general unitary matrix, and the individual
matrix elements may have imaginary parts. One possible
parametrisation of the CKM matrix is given below. Note that the following
material is provided for completeness only, and is not examinable!
(Further details are provided in the text books.)
where c_{ij} = cos θ_{ij}
and s_{ij} = sin θ_{ij},
with i and j being generation labels {i,j
= 1,2,3}. In the limit
θ_{23} = θ_{13} = 0, the third generation decouples, and the situation reduces to
the usual Cabibbo mixing of the first two generations, with θ_{12}
identified with the Cabibbo angle.
The real angles θ_{12}, θ_{23}, θ_{13}
can all be made to lie in the first quadrant by suitable definition of
the quark field phases.
c_{23} is known to differ from unity
only in the sixth decimal place.
If the parameter δ
is nonzero, then the matrix is complex, and the small degree of CP violation
present in the weak interaction can be explained naturally.
This has not yet been conclusively proven!
[The above parametrisation and values are taken from the Particle Physics
Data Booklet, from "Review of Particle Physics", Chinese Physics
C38, July 2014, by the Particle Data Group.]
Supplementary material on the properties of quarks, mainly of a popular
or nontechnical nature, can be obtained from a number of sources.
You may wish to consult some of the following information on the Web:
Supplementary Reading Material
For further nontechnical discussion of the properties
of quarks, you might like to consult The Ideas of Particle Physics,
as follows:
 isospin  section 7.6
 strangeness  chapter 8
 multiplets  chapter 10
 Charm  sections 36.3 to 36.5
 third generation particles  chapter 37
More information on the CKM matrix and CP violation is available from the following sources.
 From the SLAC Beamline (PDF files):
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