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G. Meissner, Phys. Lett. B309 (1993) 421. [15] S. Weinberg, Physica 96 A (1979) 327. [16] H. Leutwyler, Principles of chiral pertubation theory, In Proceedings of Hadron 94 Workshop, Gramado, Brazil 1994, hep-ph/9406283; World Scientific, 1995. F. Donoghue, E. R. , 1992. [18] H. , 1984. [19] H. M. R. Holstein, editors; Springer, 1995. [20] S. Weinberg, Phys. Rev. Lett. 17 (1966) 616. [21] S. Weinberg, Nucl. Phys. B363 (1991) 3. [22] E. Jenkins and A. Manohar, Nucl. Phys. B368 (1992) 190. [23] J.

Going to momentum space and introducing an effective mass, mef f (k, t), this equation can be approximately rewritten as ∂2 φ(k, t) = −(k 2 + m2ef f (t))φ(k, t) 2 ∂t (189) m2ef f (t) = λ( φ2 + σ 2 − v02 ) (190) with At zero temperature, in the ground state, σ 2 = fπ2 , Φ = 0, and we obtain the equation of motion for the pion with mass mπ . In the quench scenario (figure 10(b)), however, the expectation value of the sigma field vanishes and square of the effective mass becomes negative leading to an amplification of the modes with small mometum k!

Consider a given Feynman-diagram contributing to the scattering amplitude. It will have a certain number L of loops, a certain number Vi of vertices of type i involving di derivatives of the pion field an a certain number of internal lines Ip . The power D of the pion momentum q, this diagram will have at the end, can be determined as follows: • each loop involves an integral over the internal momenta d4 q ∼ q 4 • each internal pion line corresponds to a pion propagator, and thus contributes as 1 q2 • each vertex Vi involving di derivatives of the pion field, contributes like q di Consequently, the total power of q, q D is given by D = 4L − 2IP + Vi d i (158) i This can be simplified by using the general relation between the numbers of loops, internal lines and vertices of a given diagram L = Ip − Vi + 1 i 35 (159) to give D = 2 + 2L + i Vi (di − 2) (160) With this formula we can determine to which order of the Taylor expansion of the scattering amplitude a given diagram contributes.

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