Problems
325
This figure is, of course, rather symbolic since there are strong QCD
interactions (not shown) which are responsible for binding the three-quark
systems into baryons, and the q¯ q system into a meson. Unlike the case of
deep inelastic lepton scattering, these QCD interactions can not be treated
perturbatively, since the distance scales involved are typically those of the
hadron sizes (∼ 1 fm), where perturbation theory fails. This means that
non-leptonic weak interactions among hadrons are difficult to analyze quantitatively, though progress can be made via lattice QCD. Similar difficulties
also arise, evidently, in the case of semi-leptonic decays. In general, one has
to begin in a phenomenological way, parametrizing the decay amplitudes in
terms of appropriate form factors (which are analogous to the electromagnetic
form factors introduced in chapter 8). In the case of transitions involving at
least one heavy quark Q, Isgur and Wise (1989, 1990) noticed that a considerable simplification occurs in the limit m Q → ∞. For example, one universal
function (the ‘Isgur–Wise form factor’) is sufficient to describe a large number
of hadronic form factors introduced for semi-leptonic transitions between two
heavy pseudoscalar (0
− ) or vector (1
− ) mesons. For an introduction to the
Isgur–Wise theory we refer to Donoghue et al. (1992).
The non-leptonic sector is, however, the scene of some very interesting
physics, such as K
0
− K ¯ 0 and B
0
− B ¯ 0 oscillations, and CP violation in the
K
0
− K ¯ 0 , D
0
− D ¯ 0 and B
0
− B ¯ 0 systems. We shall discuss these phenomena in
the following chapter.
Problems
20.1 Show that in the non-relativistic limit (|p| ≪ M ) the matrix element
u ¯ p γ
μ u n of (20.2) vanishes if p and n have different spin states.
20.2 Verify the normalization N = (E + |p|)
1/2 in (20.23).
20.3 Verify (20.30) and (20.31).
20.4 Verify that equations (20.32) are invariant under CP.
20.5 The matrix γ 5 is defined by γ 5 = iγ
0 γ
1 γ
2 γ
3 . Prove the following properties:
(a) γ
2 = 1 and hence that
5
(1 + γ 5 )(1 − γ 5 ) = 0;
(b) from the anticommutation relations of the other γ matrices, show that
{γ 5 , γ μ } = 0
325
This figure is, of course, rather symbolic since there are strong QCD
interactions (not shown) which are responsible for binding the three-quark
systems into baryons, and the q¯ q system into a meson. Unlike the case of
deep inelastic lepton scattering, these QCD interactions can not be treated
perturbatively, since the distance scales involved are typically those of the
hadron sizes (∼ 1 fm), where perturbation theory fails. This means that
non-leptonic weak interactions among hadrons are difficult to analyze quantitatively, though progress can be made via lattice QCD. Similar difficulties
also arise, evidently, in the case of semi-leptonic decays. In general, one has
to begin in a phenomenological way, parametrizing the decay amplitudes in
terms of appropriate form factors (which are analogous to the electromagnetic
form factors introduced in chapter 8). In the case of transitions involving at
least one heavy quark Q, Isgur and Wise (1989, 1990) noticed that a considerable simplification occurs in the limit m Q → ∞. For example, one universal
function (the ‘Isgur–Wise form factor’) is sufficient to describe a large number
of hadronic form factors introduced for semi-leptonic transitions between two
heavy pseudoscalar (0
− ) or vector (1
− ) mesons. For an introduction to the
Isgur–Wise theory we refer to Donoghue et al. (1992).
The non-leptonic sector is, however, the scene of some very interesting
physics, such as K
0
− K ¯ 0 and B
0
− B ¯ 0 oscillations, and CP violation in the
K
0
− K ¯ 0 , D
0
− D ¯ 0 and B
0
− B ¯ 0 systems. We shall discuss these phenomena in
the following chapter.
Problems
20.1 Show that in the non-relativistic limit (|p| ≪ M ) the matrix element
u ¯ p γ
μ u n of (20.2) vanishes if p and n have different spin states.
20.2 Verify the normalization N = (E + |p|)
1/2 in (20.23).
20.3 Verify (20.30) and (20.31).
20.4 Verify that equations (20.32) are invariant under CP.
20.5 The matrix γ 5 is defined by γ 5 = iγ
0 γ
1 γ
2 γ
3 . Prove the following properties:
(a) γ
2 = 1 and hence that
5
(1 + γ 5 )(1 − γ 5 ) = 0;
(b) from the anticommutation relations of the other γ matrices, show that
{γ 5 , γ μ } = 0