By Yuri Dokshitzer
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Extra info for Basics of perturbative QCD
3 Singularities of partial waves and unitarity In the previous lecture we have found a simple connection between plane singularities of f ± (t) and the asymptotics of the scattering amplitude at ﬁxed t > 4µ2 and s → ∞, along the dashed line 1 in Fig. 1. It is u = 4µ2 z = −1 z=1 s = 4µ2 1 ρst 2 3 Fig. 1. 39) to the physical region of the s-channel, that is to t < 0. 1 Continuation of partial waves with complex to t < 0 ± Let us discuss analytic properties of the partial wave f (t): f ± (t) = or f ± (t) = 1 π 1 π ∞ 4µ2 Q 1+ Q (z)A± 1 (z, t), 2s t − 4µ2 A± 1 (s, t) 2 ds .
To this end we ﬁrst remark that A2 (u, t) coincides (modulo sign) with A1 (s, t) (see the proof of the Pomeranchuk theorem in Lecture 1). 12). For large z ∝ s the dominant contribution to the integral over z1 and √ 1, since according to our assumption the z2 arises from z1 ∼ z2 ∼ z integrand grows linearly with z1 and z2 (A1 (zi ) zi f (t)). 7b) for K(z1 , z2 , z): K(z1 , z2 , z) z(z − 2z1 z2 ). 12) and omitting irrelevant s-independent factors we obtain an estimate ρ(s, t) ∝ z1 dz1 z2 dz2 z(z − 2z1 z2 ) , z1 > 1, z2 > 1, z > 2z1 z2 .
31) where the integration contour (a), enclosing the interval [−1, 1] on the real axis in the z plane, is displayed in Fig. 5. Away from the contour, the integrand has two branch cuts on the real axis which start at the points z1 and −z2 that correspond to the threshold s- and u-channel singularities of the amplitude A(s, t); in our symmetric case, the thresholds are at s = 4µ2 and u = 4µ2 , so that z1 = z2 = 1 + 4µ2 /2kt2 > 1. The discontinuities across these cuts are, respectively, 2i A1 (z, t) and 2i A2 (z, t).
Basics of perturbative QCD by Yuri Dokshitzer