206
H. Wittig
parameterize the effective chiral Lagrangian. This link is provided by the so-called
Gell-Mann–Oakes–Renner relation [105], which we are going to derive below. To
this end we consider the QCD Lagrangian in the continuum:
L QCD = −
1
4
F
a
μν (x)F
a
μν (x) +
f
¯
ψ f (x)
γ μ D μ + m f
ψ f (x).
(5.177)
The path integral is defined as
Z QCD =
D[A μ ]D[ ¯
ψ, ψ] exp
−
d
4 x L QCD
,
(5.178)
and the expression for the quark condensate can be formally derived by taking
derivatives with respect to the light quark masses, i.e.
f =u,d,s
∂ ln Z QCD
∂m f
m f =0
= −
¯
uu + ¯
dd + ¯
ss
m f =0
≡ −
¯
.
(5.179)
What is the analogue of this expression in the effective chiral theory? To answer this
question one takes the lowest-order chiral Lagrangian of Eq. (5.170) and defines the
corresponding path integral 17
Z ChPT =
D[U ] exp
−
d
4 x L
(2)
eff
.
(5.180)
Since L
(2)
eff contains the quark mass matrix one can consider similar derivatives, i.e.
f =u,d,s
∂ ln Z ChPT
∂m f
m f =0
=
F 2
0 B 0
2
f =u,d,s
∂
∂m f
Tr M(U + U
† )
m f =0
= 3 · F
2
0 B 0 + . . . ,
(5.181)
and comparison with Eq. (5.179) yields
−
1
3
¯
uu + ¯
dd + ¯
ss
≡ Σ = F
2
0 B 0 .
(5.182)
In other words, the quark condensate is related to the slope parameter in the lowestorder mass formulae and the pion decay constant in the chiral limit, F 0 . This result
is known as the Gell-Mann–Oakes–Renner relation.
17 It should be obvious that the field U must not be confused with the link variable considered in
previous sections.
H. Wittig
parameterize the effective chiral Lagrangian. This link is provided by the so-called
Gell-Mann–Oakes–Renner relation [105], which we are going to derive below. To
this end we consider the QCD Lagrangian in the continuum:
L QCD = −
1
4
F
a
μν (x)F
a
μν (x) +
f
¯
ψ f (x)
γ μ D μ + m f
ψ f (x).
(5.177)
The path integral is defined as
Z QCD =
D[A μ ]D[ ¯
ψ, ψ] exp
−
d
4 x L QCD
,
(5.178)
and the expression for the quark condensate can be formally derived by taking
derivatives with respect to the light quark masses, i.e.
f =u,d,s
∂ ln Z QCD
∂m f
m f =0
= −
¯
uu + ¯
dd + ¯
ss
m f =0
≡ −
¯
.
(5.179)
What is the analogue of this expression in the effective chiral theory? To answer this
question one takes the lowest-order chiral Lagrangian of Eq. (5.170) and defines the
corresponding path integral 17
Z ChPT =
D[U ] exp
−
d
4 x L
(2)
eff
.
(5.180)
Since L
(2)
eff contains the quark mass matrix one can consider similar derivatives, i.e.
f =u,d,s
∂ ln Z ChPT
∂m f
m f =0
=
F 2
0 B 0
2
f =u,d,s
∂
∂m f
Tr M(U + U
† )
m f =0
= 3 · F
2
0 B 0 + . . . ,
(5.181)
and comparison with Eq. (5.179) yields
−
1
3
¯
uu + ¯
dd + ¯
ss
≡ Σ = F
2
0 B 0 .
(5.182)
In other words, the quark condensate is related to the slope parameter in the lowestorder mass formulae and the pion decay constant in the chiral limit, F 0 . This result
is known as the Gell-Mann–Oakes–Renner relation.
17 It should be obvious that the field U must not be confused with the link variable considered in
previous sections.