230
18. Chiral Symmetry Breaking
Eχ + = −|p|χ + + mφ + .
(18.8)
Comparing (18.7) and (18.8) with (18.4) and (18.5), we can read off the mixing
coefficients cos θ p and sin θ p as (cf (17.127))
[ (
)] 1/2
1
p
cos θ p =
1 +
| |
(18.9)
2
E
[ (
]) /
1
sin θ p =
1
| |
1 2
p
−
(18.10)
2
E
where E = (m
2 + p
2 )
1/2 . The Nambu vacuum is then given by
1
∏
|0›
ˆ
N =
(cos θ p − sin θ p c ˆ
†
s (p)d
†
s (−p))|0› m=0 ,
(18.11)
p,s
where c ˆ
†
ˆ
s ’s and d
†
s ’s are the operators in massless Dirac fields. Depending on
the sign of the helicity s, each pair in (18.11) carries ±2 units of chirality. We
ˆ
may check this by noting that in the mode expansion of the Dirac field ψ,
c ˆ s (p) operators go with u-spinors for which the γ 5 eigenvalue equals the helicˆ
ity, while d
†
s (−p) operators accompany v-spinors for which the γ 5 eigenvalue
ˆ
ˆ
equals minus the helicity. Thus under a chiral transformation ψ
′ = e
−iβγ ˆ 5 ψ,
ˆ
c ˆ s → e
−iβs
ˆ
ˆ
c ˆ an
i
s
d d
†
s → e
βs d
†
s , for a given s. Hence c ˆ
†
s d
†
s acquires a factor
e
2iβs . Thus the Nambu vacuum does not have a definite chirality, and operators carrying non-zero chirality can have non-vanishing vacuum expectation
¯ ˆ ˆ
ˆ
ˆ
values. A mass term ψψ is of just this kind, since under ψ = e
−iβγ5 ψ we find
ˆ
ψ
† γ
0 ˆ
ψ → ˆ
ψ
† e
iβγ5 γ
0 e
−iβγ5
¯
ˆ
ˆ
ψ = ψe
−2iβγ5 ˆ
ψ. Thus, in analogy with (17.120), a
¯ ˆ ˆ
Dirac mass is associated with a non-zero value for N ‹0|ψψ|0› N .
In the original conception by Nambu and co-workers, the fermion under
discussion was taken to be the nucleon, with ‘m’ the (spontaneously generated) nucleon mass. The fermion–fermion interaction – necessarily invariant
under chiral transformations – was taken to be of the four-fermion type. As
we have seen in volume 1, this is actually a non-renormalizable theory, but a
physical cut-off was employed, somewhat analogous to the Fermi energy E F .
Thus the nucleon mass could not be dynamically predicted, unlike the analogous gap parameter Δ in BCS theory. Nevertheless, a gap equation similar
to (17.131) could be formulated, and it was possible to show that when it
had a non-trivial solution, a massless bound state automatically appeared in
¯
the ff channel (Nambu and Jona-Lasinio 1961a). This work was generalized
to the SU(2) f 5 case by Nambu and Jona-Lasinio (1961b), who showed that
if the chiral symmetry was broken explicitly by the introduction of a small
nucleon mass (∼ 5 MeV), then the Goldstone pions would have their observed
non-zero (but small) mass. In addition, the Goldberger–Treiman (1958) relation was derived, and a number of other applications were suggested. Subsequently, Nambu with other collaborators (Nambu and Lurie 1962, Nambu
1
ˆ
A different phase convention is used for d
†
s (−p) as compared to that for c ˆ
†
in (17.111).
−k
18. Chiral Symmetry Breaking
Eχ + = −|p|χ + + mφ + .
(18.8)
Comparing (18.7) and (18.8) with (18.4) and (18.5), we can read off the mixing
coefficients cos θ p and sin θ p as (cf (17.127))
[ (
)] 1/2
1
p
cos θ p =
1 +
| |
(18.9)
2
E
[ (
]) /
1
sin θ p =
1
| |
1 2
p
−
(18.10)
2
E
where E = (m
2 + p
2 )
1/2 . The Nambu vacuum is then given by
1
∏
|0›
ˆ
N =
(cos θ p − sin θ p c ˆ
†
s (p)d
†
s (−p))|0› m=0 ,
(18.11)
p,s
where c ˆ
†
ˆ
s ’s and d
†
s ’s are the operators in massless Dirac fields. Depending on
the sign of the helicity s, each pair in (18.11) carries ±2 units of chirality. We
ˆ
may check this by noting that in the mode expansion of the Dirac field ψ,
c ˆ s (p) operators go with u-spinors for which the γ 5 eigenvalue equals the helicˆ
ity, while d
†
s (−p) operators accompany v-spinors for which the γ 5 eigenvalue
ˆ
ˆ
equals minus the helicity. Thus under a chiral transformation ψ
′ = e
−iβγ ˆ 5 ψ,
ˆ
c ˆ s → e
−iβs
ˆ
ˆ
c ˆ an
i
s
d d
†
s → e
βs d
†
s , for a given s. Hence c ˆ
†
s d
†
s acquires a factor
e
2iβs . Thus the Nambu vacuum does not have a definite chirality, and operators carrying non-zero chirality can have non-vanishing vacuum expectation
¯ ˆ ˆ
ˆ
ˆ
values. A mass term ψψ is of just this kind, since under ψ = e
−iβγ5 ψ we find
ˆ
ψ
† γ
0 ˆ
ψ → ˆ
ψ
† e
iβγ5 γ
0 e
−iβγ5
¯
ˆ
ˆ
ψ = ψe
−2iβγ5 ˆ
ψ. Thus, in analogy with (17.120), a
¯ ˆ ˆ
Dirac mass is associated with a non-zero value for N ‹0|ψψ|0› N .
In the original conception by Nambu and co-workers, the fermion under
discussion was taken to be the nucleon, with ‘m’ the (spontaneously generated) nucleon mass. The fermion–fermion interaction – necessarily invariant
under chiral transformations – was taken to be of the four-fermion type. As
we have seen in volume 1, this is actually a non-renormalizable theory, but a
physical cut-off was employed, somewhat analogous to the Fermi energy E F .
Thus the nucleon mass could not be dynamically predicted, unlike the analogous gap parameter Δ in BCS theory. Nevertheless, a gap equation similar
to (17.131) could be formulated, and it was possible to show that when it
had a non-trivial solution, a massless bound state automatically appeared in
¯
the ff channel (Nambu and Jona-Lasinio 1961a). This work was generalized
to the SU(2) f 5 case by Nambu and Jona-Lasinio (1961b), who showed that
if the chiral symmetry was broken explicitly by the introduction of a small
nucleon mass (∼ 5 MeV), then the Goldstone pions would have their observed
non-zero (but small) mass. In addition, the Goldberger–Treiman (1958) relation was derived, and a number of other applications were suggested. Subsequently, Nambu with other collaborators (Nambu and Lurie 1962, Nambu
1
ˆ
A different phase convention is used for d
†
s (−p) as compared to that for c ˆ
†
in (17.111).
−k