Problems
37
The effect of the QCD interactions must be to bind the quark into nucleons, such as the proton (uud) and neutron (udd). But what about the equally
˜
d ˜˜
possible states (˜ uu ˜d) and (˜ u d), for example? These would have to be degenerate in mass with (uud) and (udd), and of opposite parity. Yet such ‘parity
doublet’ partners of the physical p and n are not observed, and so we have a
puzzle.
One might feel that this whole discussion is unrealistic, based as it is on
massless quarks. Are the baryons then supposed to be massless too? If so,
perhaps the discussion is idle, as they are evidently by no means massless. But
it is not necessary to suppose that the mass of a relativistic bound state has
any very simple relation to the masses of its constituents: its mass may derive,
in part at least, from the interaction energy in the fields. Alternatively, one
might suppose that somehow the finite mass of the u and d quarks, which of
course breaks the chiral symmetry, splits the degeneracy of the nucleon parity
doublets, promoting the negative parity ‘nucleon’ state to an acceptably high
mass. But this seems very implausible, in view of the actual magnitudes of
m u and m d , compared to the nucleon masses.
In short, we have here a situation in which a symmetry of the Lagrangian
(to an apparently good approximation) does not seem to result in the expected
multiplet structure of the states. The resolution of this puzzle will have to
await our discussion of ‘spontaneous symmetry breaking’, in Part VII.
In conclusion, we note an important feature of the flavour symmetry cur1
2
1
2
rents T ˆ
( )μ
( )μ
and T ˆ
discussed in this and the preceding section. Although
5
these currents have been introduced entirely within the context of strong interaction symmetries, it is a remarkable fact that exactly these currents also
appear in strangeness-conserving semileptonic weak interactions such as βdecay, as we shall see in chapter 20. (The fact that both appear is precisely
a manifestation of parity violation in weak interactions, as we noted in section 4.2.1). Thus some of the physical consequences of ‘spontaneously broken
chiral symmetry’ will involve weak interaction quantities.
Problems
12.1 Verify that the set of all unitary 2 × 2 matrices with determinant equal
to +1 form a group, the law of combination being matrix multiplication.
12.2 Derive (12.18).
12.3 Check the commutation relations (12.28).
12.4 Show that the T i ’s defined by (12.45) satisfy (12.47).
(1)
12.5 Write out each of the 3 × 3 matrices T (i = 1, 2, 3) whose matrix
i
37
The effect of the QCD interactions must be to bind the quark into nucleons, such as the proton (uud) and neutron (udd). But what about the equally
˜
d ˜˜
possible states (˜ uu ˜d) and (˜ u d), for example? These would have to be degenerate in mass with (uud) and (udd), and of opposite parity. Yet such ‘parity
doublet’ partners of the physical p and n are not observed, and so we have a
puzzle.
One might feel that this whole discussion is unrealistic, based as it is on
massless quarks. Are the baryons then supposed to be massless too? If so,
perhaps the discussion is idle, as they are evidently by no means massless. But
it is not necessary to suppose that the mass of a relativistic bound state has
any very simple relation to the masses of its constituents: its mass may derive,
in part at least, from the interaction energy in the fields. Alternatively, one
might suppose that somehow the finite mass of the u and d quarks, which of
course breaks the chiral symmetry, splits the degeneracy of the nucleon parity
doublets, promoting the negative parity ‘nucleon’ state to an acceptably high
mass. But this seems very implausible, in view of the actual magnitudes of
m u and m d , compared to the nucleon masses.
In short, we have here a situation in which a symmetry of the Lagrangian
(to an apparently good approximation) does not seem to result in the expected
multiplet structure of the states. The resolution of this puzzle will have to
await our discussion of ‘spontaneous symmetry breaking’, in Part VII.
In conclusion, we note an important feature of the flavour symmetry cur1
2
1
2
rents T ˆ
( )μ
( )μ
and T ˆ
discussed in this and the preceding section. Although
5
these currents have been introduced entirely within the context of strong interaction symmetries, it is a remarkable fact that exactly these currents also
appear in strangeness-conserving semileptonic weak interactions such as βdecay, as we shall see in chapter 20. (The fact that both appear is precisely
a manifestation of parity violation in weak interactions, as we noted in section 4.2.1). Thus some of the physical consequences of ‘spontaneously broken
chiral symmetry’ will involve weak interaction quantities.
Problems
12.1 Verify that the set of all unitary 2 × 2 matrices with determinant equal
to +1 form a group, the law of combination being matrix multiplication.
12.2 Derive (12.18).
12.3 Check the commutation relations (12.28).
12.4 Show that the T i ’s defined by (12.45) satisfy (12.47).
(1)
12.5 Write out each of the 3 × 3 matrices T (i = 1, 2, 3) whose matrix
i