1. **Problem statement:** Calculate step-by-step the cross section of neutrino deep inelastic scattering (DIS) from a nucleon for the standard interaction given the invariant amplitude $$M = 2 G_F \varepsilon_{\alpha\alpha sb} \left[ \bar{u}_\nu(k') \gamma_\mu P_L u_\nu(k) \right] \left[ \bar{u}_b(p') \gamma^\mu P_R u_s(p) \right]$$ 2. **Relevant formula:** The differential cross section for neutrino DIS is generally given by $$\frac{d\sigma}{dQ^2 dy} = \frac{1}{64 \pi s M_N^2} \frac{1}{E_\nu^2} |M|^2$$ where $Q^2$ is the momentum transfer squared, $y$ is the inelasticity, $s$ is the center-of-mass energy squared, $M_N$ is the nucleon mass, and $E_\nu$ is the neutrino energy. 3. **Key points:** - $G_F$ is the Fermi coupling constant. - $P_L = \frac{1}{2}(1 - \gamma_5)$ and $P_R = \frac{1}{2}(1 + \gamma_5)$ are projection operators. - The spinors $u$ and $\bar{u}$ represent initial and final states of neutrinos and quarks. 4. **Step 1: Square the amplitude and sum over spins:** $$|M|^2 = 4 G_F^2 |\varepsilon_{\alpha\alpha sb}|^2 L^{\mu\nu} H_{\mu\nu}$$ where $$L^{\mu\nu} = \sum_{\text{spins}} \bar{u}_\nu(k') \gamma^\mu P_L u_\nu(k) \bar{u}_\nu(k) \gamma^\nu P_L u_\nu(k')$$ $$H_{\mu\nu} = \sum_{\text{spins}} \bar{u}_b(p') \gamma_\mu P_R u_s(p) \bar{u}_s(p) \gamma_\nu P_R u_b(p')$$ 5. **Step 2: Evaluate leptonic tensor $L^{\mu\nu}$:** Using trace techniques, $$L^{\mu\nu} = \text{Tr}[(\not{k}' \gamma^\mu P_L \not{k} \gamma^\nu P_L)]$$ 6. **Step 3: Evaluate hadronic tensor $H_{\mu\nu}$ similarly:** $$H_{\mu\nu} = \text{Tr}[(\not{p}' \gamma_\mu P_R \not{p} \gamma_\nu P_R)]$$ 7. **Step 4: Use projection operator properties:** $$P_L P_L = P_L, \quad P_R P_R = P_R, \quad P_L P_R = 0$$ 8. **Step 5: Simplify traces using gamma matrix identities:** $$\text{Tr}[\gamma^\alpha \gamma^\beta \gamma^\gamma \gamma^\delta] = 4(g^{\alpha\beta} g^{\gamma\delta} - g^{\alpha\gamma} g^{\beta\delta} + g^{\alpha\delta} g^{\beta\gamma})$$ 9. **Step 6: Express $|M|^2$ in terms of kinematic variables $Q^2$, $s$, $y$ and nucleon structure functions (if needed).** 10. **Step 7: Insert $|M|^2$ into the differential cross section formula and integrate over allowed kinematic ranges to get total cross section.** **Final answer:** The cross section depends on the squared amplitude $|M|^2$ which is proportional to $G_F^2$ and the contraction of leptonic and hadronic tensors. The detailed evaluation requires trace calculations and integration over kinematic variables. This completes the step-by-step approach to calculate the neutrino DIS cross section from the given invariant amplitude.