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, with Z and Z 0> as defined by Eqs. (3.12)–(3.15). The action for the slow modes S< = − ln Z < is therefore dZ p S< = − lim , p→0 d p and equals the negative coefficient of the linear term in the expansion in powers of p. n On the other hand, for integer p = n we may compute Z < by introducing n replicas of the complex fields: n Z< =
Zn n Z 0>
n n di∗ (k)d i (k) − i=1 (S0,i +Sint,i ) Z = e , 2πi i=1 k,i =
Sint,i = λ 0
dk 2, (k 2 − μ)|i (k)| (2π)d dk 2, (k 2 − μ)|i (k)| (2π)d
n di∗ (k)d i (k) − i=1 S0>,i e . 2π i i=1 /b