17
12.2. The flavour symmetry SU(2) f
1
√ (|↑› |↓›− |↓› |↑›). But in the second case the corresponding wavefunction is
2
1
1
¯
¯
√ (dd − (−u ¯)u) = √ (dd + ¯
uu). Similarly, the T = 1 T 3 = 0 state describing
2
2
1
1
¯
¯
the π
0 is √ (dd + (−u ¯)u) = √ (dd − ¯
uu).
2
2
There is a very convenient alternative way of obtaining these wavefunctions, which we include here because it generalizes straightforwardly to SU(3);
its advantage is that it avoids the use of the explicit C-G coupling coefficients,
and of their (more complicated) analogues in SU(3).
∗ ¯
Bearing in mind the identifications ¯
u ≡ u , d ≡ d
∗ , we see that the T =
∗
†
+ ¯
(recall that
† means transpose and complex conjugate). Under an SU(2)
0 ¯
qq combination ¯
uu dd can be written as u u + d
∗ d which is just q q,
f
′
′†
transformation, q → q = Vq, so q
†
→ q = q
† V
† and
†
′† ′
†
q q → q q = q
† V
† Vq = q q
(12.58)
†
using V
† V = 1 2 ; thus q q is indeed an SU(2) f invariant, which means it has
T = 0 (no multiplet partners).
We may also construct the T = 1 q − q ¯ states in a similar way. Consider
the three quantities v i defined by
v i = q
† τ i q i = 1, 2, 3.
(12.59)
Under an infinitesimal SU(2) f transformation
′
q = (1 2 + i∊ · τ /2)q,
(12.60)
the three quantities v i transform to
′
v = q
† (1 2 − i∊ · τ /2)τ i (1 2 + i∊ · τ /2)q,
(12.61)
i
′†
where we have used q = q
† (1 2 + i∊ · τ /2)
† and then τ
† = τ . Retaining only
the first-order terms in ∊ gives (problem 12.6)
∈ j
′
v = v i + i q
† (τ i τ j − τ j τ i )q
(12.62)
i
2
where the sum on j = 1, 2, 3 is understood. But from (12.28) we know the
commutator of two τ ’s, so that (12.62) becomes
∈ j
′
†
v i = v i + i q .2i∈ ijk τ k q (sum on k = 1, 2, 3)
2
= v i − ∈ ijk ∈ j q
† τ k q
= v i − ∈ ijk ∈ j v k ,
(12.63)
which may also be written in ‘vector’ notation as
′
v = v − ∊ × v.
(12.64)
Equation (12.63) states that, under an (infinitesimal) SU(2) f transformation, the three quantities v i (i = 1, 2, 3) transform into specific linear combinations of themselves, as determined by the coefficients ∈ ijk (the ∈’s are just
12.2. The flavour symmetry SU(2) f
1
√ (|↑› |↓›− |↓› |↑›). But in the second case the corresponding wavefunction is
2
1
1
¯
¯
√ (dd − (−u ¯)u) = √ (dd + ¯
uu). Similarly, the T = 1 T 3 = 0 state describing
2
2
1
1
¯
¯
the π
0 is √ (dd + (−u ¯)u) = √ (dd − ¯
uu).
2
2
There is a very convenient alternative way of obtaining these wavefunctions, which we include here because it generalizes straightforwardly to SU(3);
its advantage is that it avoids the use of the explicit C-G coupling coefficients,
and of their (more complicated) analogues in SU(3).
∗ ¯
Bearing in mind the identifications ¯
u ≡ u , d ≡ d
∗ , we see that the T =
∗
†
+ ¯
(recall that
† means transpose and complex conjugate). Under an SU(2)
0 ¯
qq combination ¯
uu dd can be written as u u + d
∗ d which is just q q,
f
′
′†
transformation, q → q = Vq, so q
†
→ q = q
† V
† and
†
′† ′
†
q q → q q = q
† V
† Vq = q q
(12.58)
†
using V
† V = 1 2 ; thus q q is indeed an SU(2) f invariant, which means it has
T = 0 (no multiplet partners).
We may also construct the T = 1 q − q ¯ states in a similar way. Consider
the three quantities v i defined by
v i = q
† τ i q i = 1, 2, 3.
(12.59)
Under an infinitesimal SU(2) f transformation
′
q = (1 2 + i∊ · τ /2)q,
(12.60)
the three quantities v i transform to
′
v = q
† (1 2 − i∊ · τ /2)τ i (1 2 + i∊ · τ /2)q,
(12.61)
i
′†
where we have used q = q
† (1 2 + i∊ · τ /2)
† and then τ
† = τ . Retaining only
the first-order terms in ∊ gives (problem 12.6)
∈ j
′
v = v i + i q
† (τ i τ j − τ j τ i )q
(12.62)
i
2
where the sum on j = 1, 2, 3 is understood. But from (12.28) we know the
commutator of two τ ’s, so that (12.62) becomes
∈ j
′
†
v i = v i + i q .2i∈ ijk τ k q (sum on k = 1, 2, 3)
2
= v i − ∈ ijk ∈ j q
† τ k q
= v i − ∈ ijk ∈ j v k ,
(12.63)
which may also be written in ‘vector’ notation as
′
v = v − ∊ × v.
(12.64)
Equation (12.63) states that, under an (infinitesimal) SU(2) f transformation, the three quantities v i (i = 1, 2, 3) transform into specific linear combinations of themselves, as determined by the coefficients ∈ ijk (the ∈’s are just