PHY304 | Particle Physics | Dr C N Booth |
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.
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:
Similarly the pions form an isospin triplet,
The Δ
forms a quadruplet with
The rule for electric charge can then be written
In terms of quarks, the u and d form an isospin doublet,
Three quarks with
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. For example, the reaction of two protons to form a deuteron and a pion has twice the cross section of the reaction between a proton and neutron to form a similar final state. This will be explained in the lectures.
Isospin multiplet | B | S | I | <Q>/e | Y = B + S |
---|---|---|---|---|---|
The formula for electric charge must know be modified to read
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
The vector-meson nonet
(Quark assignments are the same as above.)
The baryon octet of spin-parity
The baryon decuplet with spin-parity
One of these particles was only discovered recently! See the announcement from CERN.
Flavour | Charge/e | B | I | I_{3} | S | c | b | t | Mass (GeV/c^{2}) |
---|---|---|---|---|---|---|---|---|---|
The particle content of the Standard Model of Particle Physics (including the Higgs
boson which is not covered in this course).
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 well-defined 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 start by considering 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
The relationship between weak and strong eigenstates in 2 generations
can also be expressed as
= | |||
weak e-states | mixing matrix | strong e-states |
The same formalism can be used for 3 generations, and the mixing matrix, known as the Cabibbo-Kabayashi-Maskawa 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 top-right and bottom-left 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.
Flavour-changing weak interactions always occur via the charged current. That is, they always involve transitions between the two members of the same weak isospin doublet, e.g. between c and s' (in either direction), or between u and d'. The mixing of the negative quarks plays a role for both initial and final state quarks. For example, the decay of a c quark is always to an s' weak eigenstate, which will be bound in a hadron as one of the strong eigenstates of which it can be considered a mixture. On the other hand, when a hadron containing an s quark decays, the s must be considered a mixture of d', s' and b' weak eigenstates, and these decay to u, c and t respectively.
Physically, the relative probability of producing hadrons containing the
respective quarks in a weak decay is determined by the elements of the CKM matrix.
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
When the charged-current 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:
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
If the parameter δ
is non-zero, then the matrix is complex, and the small degree of CP violation
present in the weak interaction can be explained naturally.
It has not yet been conclusively proven that this is the explanation for
all the observed CP violation!
[The above parametrisation and values are taken from the Particle Physics Data Booklet, from "Review of Particle Physics", Phys. Rev. D98, January 2018, by the Particle Data Group.]
Supplementary material on the properties of quarks, mainly of a popular or non-technical 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 non-technical discussion of the properties
of quarks, you might like to consult The Ideas of Particle Physics,
as follows:
More information on the CKM matrix and CP violation is available from the following sources.