If we expand the new meson current in powers of nl, keep the lowest order term only, and compare it with the hadronic axial vector isospin current i/^-/5 (i.e., remove the overall factor of gn, which gives a strength appropriate for the strong interactions only), we obtain
which is precisely the current used for the calculation of the weak decay of the pion in Sec. 9.10 (the field <6 in Sec. 9.10 is the same as it). Our discussion here provides the justification needed for the use of the current (9.132).
We turn now to a discussion of the explicit breaking of chiral symmetry and the origin of the pion mass.
Chiral symmetry can be broken in two ways: (i) spontaneously, through the emergence of a vacuum state which is not chirally invariant, and (ii) explicitly, through the presence of a small term in the Lagrangian which is not chirally invariant. Spontaneous symmetry breaking was discussed in detail in Sec. 13.6, and in the preceding section we saw that the consequences of spontaneous symmetry breaking, namely the generation of fermion mass and the emergence of a massless Goldstone boson (the pion), could also be obtained directly from a non-linear Lagrangian which is exactly symmetric and for which the vacuum is also symmetric. So spontaneous symmetry breaking does not really remove the symmetry from the theory. In particular, there still exists a conserved axial vector current. In ..his section we discuss explicit chiral symmetry breaking. When a symmetry is explicitly broken, the current is no longer conserved. Another consequence of breaking chiral symmetry explicitly is the possibility of having both fermions and a pion with finite mass.
Since the breaking of chiral symmetry is "small" (in a sense to be precisely defined shortly), the axial vector current is "almost" conserved, or "partially" conserved. This is referred to as PCAC, for partially conserved axial-vector current, and the precise statement of PCAC amounts to a specific statement about the extent to which the conservation of the axial-vector current is broken. This relation takes the form
where fa is proportional to the pion field and m, is the pion mass. Note that the current is conserved if the pion mass is zero, and that if the pion mass can be regarded as a small quantity, the current can be thought of as "almost" conserved. The consequences of the PCAC relation (13.122), initially put forward as a hypothesis, were among the earliest observations which led to an interest in chiral symmetry.
In this section we will see how the PCAC relation emerges from both the linear and non-linear sigma models.
To prepare for a discussion of explicit symmetry breaking in the linear sigma model, we first prove, from the equations of motion for the fields, that the axial vector current (13.96) is conserved. The equations of motion for the fields in the linear sigma model can be quickly found from the Euler-Lagrange equations and are i$4>=
n<t>a = - m2(p0 + 2V'</>a - gntpip □ ft = - m2ft + 2V'4>in - ig^niP , where V' — dV/d\<f>\. Using these equations,
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