Standard Model and Beyond · Problem 2.33

Lie Groups, Lie Algebras, and Symmetries

Physics — Particle Physics · non-relativistic-potential-from-scattering-amplitude

Problem

Let V(x⃗1−x⃗2)V(\vec{x}_1 - \vec{x}_2) be the potential between two non-identical spin-12\frac{1}{2} particles in non-relativistic quantum mechanics (NRQM). (VV may also depend on their spin and momentum operators). One shows in time-dependent perturbation theory that the transition amplitude UfiU_{fi} from ∣i⟩=∣p⃗1s1,p⃗2s2⟩|i\rangle = |\vec{p}_1 s_1, \vec{p}_2 s_2\rangle to ∣f⟩=∣p⃗3s3,p⃗4s4⟩|f\rangle = |\vec{p}_3 s_3, \vec{p}_4 s_4\rangle with m1=m3m_1 = m_3, m2=m4m_2 = m_4 is Ufi=−i(2π)4δ4(p3+p4−p1−p2) ϕ3†ϕ4†(∫d3r⃗ e−iq⃗⋅r⃗V(r⃗))ϕ1ϕ2,U_{fi} = -i(2\pi)^4\delta^4(p_3+p_4-p_1-p_2)\,\phi_3^\dagger\phi_4^\dagger\left(\int d^3\vec{r}\,e^{-i\vec{q}\cdot\vec{r}}V(\vec{r})\right)\phi_1\phi_2, where ϕi\phi_i is a two-component Pauli spinor, and VV contains appropriate spin matrices. Note that these states are in our covariant normalization convention, which has an extra factor (2π)32Ei∼(2π)32mi\sqrt{(2\pi)^3 2E_i} \sim \sqrt{(2\pi)^3 2m_i} for the ithi^{th} external particle compared to the usual conventions of NRQM. The corresponding formula in field theory is Ufi=(2π)4δ4(p3+p4−p1−p2) M,U_{fi} = (2\pi)^4\delta^4(p_3+p_4-p_1-p_2)\,M, where MM is the scattering amplitude with the phase convention of Appendix B. Comparing these results, we can read off the equivalent non-relativistic potential corresponding to a given scattering amplitude. Specifically, for p⃗1=−p⃗2=p⃗−q⃗2,p⃗3=−p⃗4=p⃗+q⃗2,\vec{p}_1 = -\vec{p}_2 = \vec{p} - \frac{\vec{q}}{2}, \qquad \vec{p}_3 = -\vec{p}_4 = \vec{p} + \frac{\vec{q}}{2}, the non-relativistic limit p⃗→0\vec{p} \to 0, ∣q⃗∣2≪mi2|\vec{q}|^2 \ll m_i^2 yields M→−i(2m1)(2m2)ϕ3†ϕ4†V~(q⃗)ϕ1ϕ2≡−i(2m1)(2m2)ϕ3†ϕ4†∫d3r⃗ e−iq⃗⋅r⃗V(r⃗)ϕ1ϕ2.M \to -i(2m_1)(2m_2)\phi_3^\dagger\phi_4^\dagger\tilde{V}(\vec{q})\phi_1\phi_2 \equiv -i(2m_1)(2m_2)\phi_3^\dagger\phi_4^\dagger\int d^3\vec{r}\,e^{-i\vec{q}\cdot\vec{r}}V(\vec{r})\phi_1\phi_2. Non-leading terms in p⃗\vec{p} can be interpreted as the non-relativistic momentum operator. (a) Calculate the potential corresponding to the effective four-fermi Hamiltonian density HI=λψˉ1ψ1 ψˉ2ψ2\mathcal{H}_I = \lambda\bar{\psi}_1\psi_1\,\bar{\psi}_2\psi_2. (b) Consider the interaction LI=[g1ψˉ1ψ1+g2ψˉ2ψ2]ϕ\mathcal{L}_I = \left[g_1\bar{\psi}_1\psi_1 + g_2\bar{\psi}_2\psi_2\right]\phi between two fermions and a Hermitian scalar of mass mϕm_\phi. Calculate the potential between ψ1\psi_1 and ψ2\psi_2 generated by tt-channel ϕ\phi exchange, and show that it is attractive for g1g2>0g_1 g_2 > 0. (c) Calculate the potential generated by LI=[g1ψˉ1γμψ1+g2ψˉ2γμψ2]Vμ,\mathcal{L}_I = \left[g_1\bar{\psi}_1\gamma^\mu\psi_1 + g_2\bar{\psi}_2\gamma^\mu\psi_2\right]V_\mu, where VμV_\mu is a spin-1 particle of mass MVM_V. Show that it is repulsive for g1g2>0g_1 g_2 > 0. (d) Repeat parts (b) and (c) for the case g1g2>0g_1 g_2 > 0, but for the potential between antiparticle 1ˉ\bar{1} and particle 22 and interpret the results. Hint: it is slightly easier to use the charge conjugation formalism of Section 2.10. (e) Consider the interaction in (2.269) of a π0\pi^0 with protons and neutrons. Calculate the tree-level amplitude for p(p⃗1) n(p⃗2)→p(p⃗3) n(p⃗4)p(\vec{p}_1)\,n(\vec{p}_2) \to p(\vec{p}_3)\,n(\vec{p}_4) by tt-channel π0\pi^0 exchange, and show that it leads to the non-relativistic potential V(r⃗)=−gπ2mπ316πmp2[13σ⃗p⋅σ⃗ne−xx+S3(1x+3x2+3x3)e−x],V(\vec{r}) = -\frac{g_\pi^2 m_\pi^3}{16\pi m_p^2}\left[\frac{1}{3}\vec{\sigma}_p\cdot\vec{\sigma}_n\frac{e^{-x}}{x} + \frac{S}{3}\left(\frac{1}{x}+\frac{3}{x^2}+\frac{3}{x^3}\right)e^{-x}\right], where mp∼mnm_p \sim m_n, σ⃗p(σ⃗n)\vec{\sigma}_p(\vec{\sigma}_n) are the Pauli matrices acting on the p (n)p\,(n) spin, x=mπrx = m_\pi r, and SS is the tensor operator S=3 σ⃗p⋅r^ σ⃗n⋅r^−σ⃗p⋅σ⃗n.S = 3\,\vec{\sigma}_p\cdot\hat{r}\,\vec{\sigma}_n\cdot\hat{r} - \vec{\sigma}_p\cdot\vec{\sigma}_n.

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