244
18. Chiral Symmetry Breaking
which we may compare with (17.93). Just as in equation (17.94), (18.100)
implies that this axial current has a matrix element between the vacuum and
the one-Goldstone state:
μ
μ
‹0| ˆ j (U ˆ )|π j , p› = −ip ve
−ip·x δ ij .
(18.101)
i,5
Now comes the pay-off: this is the same symmetry current which enters into
weak interactions, for which we already defined the vacuum-to-one-particle
matrix element in terms of the pion decay constant f π , via equation (18.46).
Comparing (18.101) with (18.46) we identify
v = f π .
(18.102)
Thus, finally, the dynamics of our massless pions, to lowest order in an expansion in powers of momenta, is given by the Lagrangian
f
2
ˆ U
† ).
L ˆ 2 =
1
π Tr(∂ μ U ∂
μ ˆ
(18.103)
4
It is quite remarkable that the low energy dynamics of the (massless) Goldstone modes is completely determined in terms of one constant, measurable
in π decay.
The Lagrangian of (18.103) is an example of an effective Lagrangian. By
this is meant, broadly, any Lagrangian which involves the presumed relevant
degrees of freedom (here the Goldstone modes), and respects desired symmetries of the theory. Evidently it is implied that there is some ‘underlying
theory’, couched in terms of different degrees of freedom (here quarks and
gluons), from which the symmetries have been abstracted. It is important
to realize that an effective Lagrangian may or may not be renormalizable.
Whereas our starting Lagrangian L ˆ σ is renormalizable, L ˆ 2 is not: clearly the
latter contains terms with arbitrarily many pion fields, which are operators
of arbitrarily high dimension, compensated by negative powers of f
2 . As it
π
stands, L ˆ 2 can only be used at tree level – as, for example, in the calculation
of π − π scattering using the interaction(18.94), for which the amplitude has
an energy dependence of the form E
2 /f
2 , where E is the order of magnitude
π
of the particles’ energy or momentum. This interaction has mass dimension 6,
and its coupling 1/f
2 has dimension (mass)
−2 , like the 4-fermion interaction
π
considered in section 11.8. It is therefore not renormalizable. However, the
argument of section 11.8 suggests that a loop-by-loop renormalization programme is possible, and this was shown to be the case by Weinberg (1979a).
Each loop built from the interaction (18.94) will carry an extra two powers
of energy, to compensate the 1/f
2 in the coupling. Thus f π (or perhaps this
π
multiplied by factors like 4 and π, if we are lucky) provides the energy scale
characteristic of a non-renormalizable theory: as we go up in energy, we need
more loops. But, at each loop order new divergences appear, which require
additional counter terms for renormalization. Thus at any given order in
18. Chiral Symmetry Breaking
which we may compare with (17.93). Just as in equation (17.94), (18.100)
implies that this axial current has a matrix element between the vacuum and
the one-Goldstone state:
μ
μ
‹0| ˆ j (U ˆ )|π j , p› = −ip ve
−ip·x δ ij .
(18.101)
i,5
Now comes the pay-off: this is the same symmetry current which enters into
weak interactions, for which we already defined the vacuum-to-one-particle
matrix element in terms of the pion decay constant f π , via equation (18.46).
Comparing (18.101) with (18.46) we identify
v = f π .
(18.102)
Thus, finally, the dynamics of our massless pions, to lowest order in an expansion in powers of momenta, is given by the Lagrangian
f
2
ˆ U
† ).
L ˆ 2 =
1
π Tr(∂ μ U ∂
μ ˆ
(18.103)
4
It is quite remarkable that the low energy dynamics of the (massless) Goldstone modes is completely determined in terms of one constant, measurable
in π decay.
The Lagrangian of (18.103) is an example of an effective Lagrangian. By
this is meant, broadly, any Lagrangian which involves the presumed relevant
degrees of freedom (here the Goldstone modes), and respects desired symmetries of the theory. Evidently it is implied that there is some ‘underlying
theory’, couched in terms of different degrees of freedom (here quarks and
gluons), from which the symmetries have been abstracted. It is important
to realize that an effective Lagrangian may or may not be renormalizable.
Whereas our starting Lagrangian L ˆ σ is renormalizable, L ˆ 2 is not: clearly the
latter contains terms with arbitrarily many pion fields, which are operators
of arbitrarily high dimension, compensated by negative powers of f
2 . As it
π
stands, L ˆ 2 can only be used at tree level – as, for example, in the calculation
of π − π scattering using the interaction(18.94), for which the amplitude has
an energy dependence of the form E
2 /f
2 , where E is the order of magnitude
π
of the particles’ energy or momentum. This interaction has mass dimension 6,
and its coupling 1/f
2 has dimension (mass)
−2 , like the 4-fermion interaction
π
considered in section 11.8. It is therefore not renormalizable. However, the
argument of section 11.8 suggests that a loop-by-loop renormalization programme is possible, and this was shown to be the case by Weinberg (1979a).
Each loop built from the interaction (18.94) will carry an extra two powers
of energy, to compensate the 1/f
2 in the coupling. Thus f π (or perhaps this
π
multiplied by factors like 4 and π, if we are lucky) provides the energy scale
characteristic of a non-renormalizable theory: as we go up in energy, we need
more loops. But, at each loop order new divergences appear, which require
additional counter terms for renormalization. Thus at any given order in