6Formulas for Static Potential Modelgq+μ(=0Ao =Screened static potential model:-igScattering amplitude:Maa(p)A(q)a, (p)2Eq>>z Orp>>qT"Smallangleapproximation":da4TTαCTCross sectioninmomentumspace: ( +μ2)2Fourier transformation from (ar to [x T:1dadaimpactparameterspacemomentumspaced2xTd2ar
Formulas for Static Potential Model 𝐴0 = 𝑔 𝑞 2 + 𝜇 2 , {𝐴 Ԧ = 0 Fourier transformation from {𝑞 Ԧ 𝑇 to {𝑥 Ԧ 𝑇: Screened static potential model: Scattering amplitude: 𝑀𝜆,𝜆𝑖 (𝑞 Ԧ) = −ig 2𝐸 𝑢LJ𝜆(𝑝𝑓)𝐴(𝑞)𝑢𝜆𝑖 (𝑝) Cross section in momentum space: 𝑑𝜎 𝑑 2𝑞 Ԧ 𝑇 = 4πα𝑠 2 𝑐𝑇 (𝑞𝑇 2 + 𝜇 2) 2 momentum space : 𝑑𝜎 𝑑 2𝑞 Ԧ 𝑇 → impact parameter space : 𝑑𝜎 𝑑 2𝑥 Ԧ 𝑇 “Small angle approximation”: 𝑞𝑇>>𝑞𝑧 or 𝑝>>𝑞𝑇 6
7Analytic Results in Static Potential ModelCrosssection in impactparameterspace:do,d?q d21i(-)xMa,a,(kr)Maa,(r)d2xT(2元)2 (2元)2MdadAgdoad2xTd2xrd2xrSpin independent partdado+do_d2x=d2xd2x1= 4cαg2K(ux)Spin dependent partdgdo+do.d2xTd2xTd2xrup= -nib (i× T) g(B + ma4cTαs2Ko(uxT)K(uxT)n
Analytic Results in Static Potential Model 𝑑𝜎𝜆 𝑑 2𝑥 Ԧ 𝑇 = 𝑑𝜎 𝑑 2𝑥 Ԧ 𝑇 + 𝜆 𝑑𝛥𝜎 𝑑 2𝑥 Ԧ 𝑇 𝑑𝛥𝜎 𝑑 2𝑥 Ԧ 𝑇 ≡ 𝑑𝜎+ 𝑑 2𝑥 Ԧ 𝑇 − 𝑑𝜎− 𝑑 2𝑥 Ԧ 𝑇 = −𝑛𝑏 ⋅ (𝑝̰ Ԧ × 𝑥̰ Ԧ 𝑇) 𝜇𝑝 𝐸(𝐸 + 𝑚𝑞) 4𝑐𝑇𝛼𝑠 2𝐾0(𝜇𝑥𝑇)𝐾1(𝜇𝑥𝑇) 𝑑𝜎𝜆 𝑑 2𝑥 Ԧ 𝑇 = 𝐶𝑇 න 𝑑 2𝑞𝑇 (2𝜋) 2 𝑑 2𝑘𝑇 (2𝜋) 2 𝑒 𝑖(𝑘𝑇−𝑞𝑇)⋅𝑥Ԧ𝑇 1 2 𝜆𝑖 𝑀𝜆,𝜆𝑖 (𝑘𝑇)𝑀𝜆,𝜆𝑖 (𝑞 Ԧ 𝑇) Cross section in impact parameter space: 𝑑𝜎 𝑑 2𝑥 Ԧ 𝑇 ≡ 𝑑𝜎+ 𝑑 2𝑥 Ԧ 𝑇 + 𝑑𝜎− 𝑑 2𝑥 Ԧ 𝑇 = 4𝑐𝑇𝛼𝑠 2𝐾0 2 (𝜇𝑥𝑇) Spin independent part Spin dependent part 7