232
18. Chiral Symmetry Breaking
Symmetry (i) is unbroken, and its associated ‘charge’ operator (the quark
number operator) commutes with all other symmetry operators, so it need
not concern us further. Symmetry (ii) is the standard isospin symmetry of
chapter 12, explicitly broken by the electromagnetic interactions (and by the
difference in the masses m u and m d , when included). Symmetry (iii) does
not correspond to any known conservation law; on the other hand, there are
not any near-massless isoscalar 0
− mesons, either, such as must be present
if the symmetry is spontaneously broken. The η meson is an isoscalar 0
−
meson, but with a mass of 547 MeV it is considerably heavier than the pion.
In fact, it can be understood as one of the Goldstone bosons associated with
the spontaneous breaking of the larger group SU(3) f 5 , which includes the s
quark (see section 18.3.3). In that case, the symmetry (iii) becomes extended
to
u ˆ → e
−iβγ5
ˆ
u, ˆ d → e
−iβγ5 ˆ
d, s ˆ → e
−iβγ5 s, ˆ
(18.19)
but there is still a missing light isosca √ lar 0
− meson. It can be shown that
its mass must be less than or equal to 3 m π (Weinberg 1975), but no such
particle exists. This is the famous ‘U(1) problem’: it was resolved by ’t
Hooft (1976a, 1986), by showing that the inclusion of instanton configurations
(Belavin et al. 1975) in path integrals leads to violations of symmetry (iii) –
see, for example, Weinberg (1996) section 23.5. Finally, symmetry (iv) is the
one with which we are presently concerned.
The symmetry currents associated with (iv) are those already given in
(12.165), but we give them again here in a slightly different notation which
will be similar to the one used for weak interactions:
ˆ μ
¯
i
j i, = qγ ˆ
μ
τ
5
γ 5 q
i
ˆ
= 1, 2, 3.
(18.20)
2
Similarly the currents associated with (ii) are
μ
μ τ
ˆ
¯
i
j i = qγ ˆ
q
i
ˆ
= 1, 2, 3.
(18.21)
2
The corresponding ‘charges’ are (compare (12.166))
∫
∫
τ
ˆ
Q i,5 ≡ ˆ
i
j
0 d
3
i,5
x = q ˆ
† γ 5 q ˆd
3
x,
(18.22)
2
ˆ (
1 )
previously denoted by T
2
i,5 , and (compare (12.101)),
∫
τ
Q ˆ i = q ˆ
† i q ˆd
3
x,
(18.23)
2
ˆ
(
1 )
previously denoted by T
2
5 . As with all symmetries, it is interesting to disˆ ˆ
cover the algebra of the generators, which are the six charges Q i , Q i,5 in this
case. Patient work with the anticommutation relations for the operators in
18. Chiral Symmetry Breaking
Symmetry (i) is unbroken, and its associated ‘charge’ operator (the quark
number operator) commutes with all other symmetry operators, so it need
not concern us further. Symmetry (ii) is the standard isospin symmetry of
chapter 12, explicitly broken by the electromagnetic interactions (and by the
difference in the masses m u and m d , when included). Symmetry (iii) does
not correspond to any known conservation law; on the other hand, there are
not any near-massless isoscalar 0
− mesons, either, such as must be present
if the symmetry is spontaneously broken. The η meson is an isoscalar 0
−
meson, but with a mass of 547 MeV it is considerably heavier than the pion.
In fact, it can be understood as one of the Goldstone bosons associated with
the spontaneous breaking of the larger group SU(3) f 5 , which includes the s
quark (see section 18.3.3). In that case, the symmetry (iii) becomes extended
to
u ˆ → e
−iβγ5
ˆ
u, ˆ d → e
−iβγ5 ˆ
d, s ˆ → e
−iβγ5 s, ˆ
(18.19)
but there is still a missing light isosca √ lar 0
− meson. It can be shown that
its mass must be less than or equal to 3 m π (Weinberg 1975), but no such
particle exists. This is the famous ‘U(1) problem’: it was resolved by ’t
Hooft (1976a, 1986), by showing that the inclusion of instanton configurations
(Belavin et al. 1975) in path integrals leads to violations of symmetry (iii) –
see, for example, Weinberg (1996) section 23.5. Finally, symmetry (iv) is the
one with which we are presently concerned.
The symmetry currents associated with (iv) are those already given in
(12.165), but we give them again here in a slightly different notation which
will be similar to the one used for weak interactions:
ˆ μ
¯
i
j i, = qγ ˆ
μ
τ
5
γ 5 q
i
ˆ
= 1, 2, 3.
(18.20)
2
Similarly the currents associated with (ii) are
μ
μ τ
ˆ
¯
i
j i = qγ ˆ
q
i
ˆ
= 1, 2, 3.
(18.21)
2
The corresponding ‘charges’ are (compare (12.166))
∫
∫
τ
ˆ
Q i,5 ≡ ˆ
i
j
0 d
3
i,5
x = q ˆ
† γ 5 q ˆd
3
x,
(18.22)
2
ˆ (
1 )
previously denoted by T
2
i,5 , and (compare (12.101)),
∫
τ
Q ˆ i = q ˆ
† i q ˆd
3
x,
(18.23)
2
ˆ
(
1 )
previously denoted by T
2
5 . As with all symmetries, it is interesting to disˆ ˆ
cover the algebra of the generators, which are the six charges Q i , Q i,5 in this
case. Patient work with the anticommutation relations for the operators in