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Additional info for Basics of perturbative QCD
1 Continuation of partial waves with complex to t < 0 41 that for 0 < t < 4µ2 we have z < −1 and Q (z) (with non-integer real ) becomes complex valued. Let us show that this complexity of f has a simple kinematical origin and can be eliminated. 1 Threshold singularity and partial waves φ Consider values of t close to the t-channel threshold. Here |z| → ∞ and Q (z) ∝ 1 z +1 +1 t − 4µ2 2s , t − 4µ2 → 0. 2) (2s) +1 which is nothing but the correct threshold behaviour of a partial wave with integer (recall that (t − 4µ2 ) = (2kt )2 ).
In reality, the Karplus curves for ππ scattering are not symmetric with respect to s and t, which is a consequence of the pions being pseudoscalars (see the following lectures and the footnote on page 27). 4 The Froissart theorem In 1958 Froissart showed that the analytic properties of the scattering amplitude together with the unitarity condition put certain restrictions on the asymptotic behaviour of A(s, t) in the physical region. Let us show that asymptotically Im A(s, t)|t=0 ≤ const · s ln2 s , s0 s → ∞.
Hence the singularities of φ on the ﬁrst unphysical sheet appear when φ (t) on the physical sheet becomes equal to 1/(2i C ) at some point t = t( ). These singularities are poles. If such a pole on the unphysical sheet is close to the cut it can be identiﬁed as a resonance. Moreover, the trajectory of the pole t = t( ) for all = 2n (if φ (t) was chosen with positive signature) describes a chain of resonances (provided they do not move too far away from the real axis). Let us decrease below 0 . The pole t( ) may then move onto the physical sheet through the tip of the cut at t = 4µ2 .