Angular distribution in two-particle emission induced by neutrinos and electrons

The angular distribution of the phase space arising in two-particle emission reactions induced by electrons and neutrinos is computed in the laboratory (Lab) system by boosting the isotropic distribution in the center of mass (CM) system used in Monte Carlo generators. The Lab distribution has a singularity for some angular values, coming from the Jacobian of the angular transformation between CM and Lab systems. We recover the formula we obtained in a previous calculation for the Lab angular distribution. This is in accordance with the Monte Carlo method used to generate two-particle events for neutrino scattering [J. T. Sobczyk, Phys. Rev. C 86, 015504 (2012)]. Inversely, by performing the transformation to the CM system, it can be shown that the phase-space function, which is proportional to the two-particle-two-hole (2p-2h) hadronic tensor for a constant current operator, can be computed analytically in the frozen nucleon approximation, if Pauli blocking is absent. The results in the CM frame confirm our previous work done using an alternative approach in the Lab frame. The possibilities of using this method to compute the hadronic tensor by a boost to the CM system are analyzed.

Figure 1

Lab magnitudes as a function of CM magnitudes. The momentum and energy transfer are $q=3\mathrm{GeV}/\mathrm{c}$ and $\omega =2\mathrm{GeV}$. Top panel: $\cos {\theta }_1^{\prime }$ versus $\cos {\theta }_1^{\prime \prime }$. Middle panel: ${\theta }_1^{\prime }$ versus ${\theta }_1^{\prime \prime }$. Bottom panel: $p_1^{\prime }$ versus ${\theta }_1^{\prime \prime }$.

Figure 2

The two angular distributions ${\mathrm{\Phi }}_{\pm }$ and the total, in the Lab system, for two-nucleon emission in the frozen nucleon approximation. The momentum transfer is $q=3\mathrm{GeV}/\mathrm{c}$ and three values of $\omega =1800$, 2000 and 2200 GeV are considered.

Figure 3

Phase-space function in the frozen nucleon approximation for $q=3\mathrm{GeV}/\mathrm{c}$, computed in the CM using the analytic formula without Pauli blocking (n.p.b.), and computed numerically in the Lab system including Pauli blocking (p.b.). The value of the Fermi momentum is $k_F=225\mathrm{MeV}/\mathrm{c}$.