Neutrino-nuclear interactions

The simulation of DIS reactions includes reactions with high $Q^2$. The first approximation of the $Q^2$-dependent photonuclear cross-sections at high $Q^2$ was made in [11], where the modified photonuclear cross sections of virtual photons [34] were used. The structure functions of protons and deuterons have been approximated in CHIPS by the sum of non-perturbative multiperipheral and non-perturbative direct interactions of virtual photons with hadronic partons:

Figure 23.20: Fit of $\gamma A$ cross sections with different $H$ values. Data are from [11].

$$F_2(x,Q^2)=[A(Q^2)\cdot x^{-\Delta(Q^2)}+B(Q^2)\cdot x]\cdot(1-x)^{N(Q^2)-2},$$ (23.59)

where $A(Q^2)=\bar{e^2_S}\cdot D\cdot U$, $B(Q^2)=\bar{e^2_V}\cdot(1-D)\cdot V$, $\bar{e^2}_{V(p)}=\frac{1}{3}$, $\bar{e^2}_{V(d)}=\frac{5}{18}$, $\bar{e^2_S}=\frac{1}{3}-\frac{\frac{1}{3}-\frac{5}{18}}{1+m^2_\phi/Q^2} +\frac{... ...8}}{1+m^2_{J/\psi}/Q^2}- \frac{\frac{1}{3}-\frac{19}{63}}{1+m^2_{\Upsilon}/Q^2}$, $N=3+\frac{0.5}{\alpha_s(Q^2)}$, $\alpha_s(Q^2)=\frac{4\pi}{\beta_0 ln(1+\frac{Q^2}{\Lambda^2})}$, $\beta_0^{(n_f=3)}=9$, $\Lambda=200~MeV$, $U=\frac{(3~C(Q^2)+N-3)\cdot\Gamma(N-\Delta)} {N\cdot\Gamma(N-1)\cdot\Gamma(1-\Delta)}$, $V=3(N-1)$, $D(Q^2)=H\cdot S(Q^2)\left(1-\frac{1}{2}S(Q^2)\frac{\bar{e^2_V}}{\bar{e^2_S}} \right)$, $S={\left(1+\frac{m^2_\rho}{Q^2}\right)^{-\alpha_P(Q^2)}}$, $\alpha_P=1+\Delta(Q^2)$, $\Delta=\frac{1+r}{12.5+2r}$, $r=\left(\frac{Q^2}{1.66}\right)^{1/2}$, $C=\frac{1+f}{g\cdot (1+f/.24)}$, $f=\left(\frac{Q^2}{0.08}\right)^2$, $g=1+\frac{Q^2}{21.6}$. The parton distributions are normalized to the unit total momentum fraction.

Figure 23.21: Fit of $f_{2d}(x,Q^2)$ (filled circles, solid lines) and $f_{3d}(x,Q^2)$ (open circles, dashed lines) structure functions measured by the WA25 experiment [35].

The photonuclear cross sections are calculated by the eikonal formula:

$$\sigma_\gamma^{tot}=\left[\frac{4\pi\alpha}{Q^2}F_2\left(\frac{Q^2} {2M\nu},Q^2\right)\right]^{\nu=E}_{Q^2=0},$$ (23.60)

An example of the approximation is shown in Fig. 23.20. One can see that the hadronic resonances are ``melted'' in nuclear matter and the multi-peripheral part of the cross section (high energy) is shadowed.

The differential cross section of the $(\nu ,\mu )$ reaction was approximated as

$$\frac{yd^2\sigma^{\nu,\bar\nu}}{dydQ^2}=\frac{G^2_F\cdot M^4... ...left[c_1(y)\cdot f_2(x,Q^2)\pm c_2(y)\cdot xf_3(x,Q^2)\right],$$ (23.61)

where $c_1(y)=2-2y+\frac{y^2}{1+R}$, $R=\frac{\sigma_L}{\sigma_T}$, $c_2(y)=y(2-y)$. As $\bar{e^2_V}=\bar{e^2_S}=1$ in Eq.23.59, hence $f_2(x,Q^2)=\left[D\cdot U\cdot x^{-\Delta}+(1-D)\cdot V\cdot x\right]\cdot(1-x)^{N-2}$, $xf_3(x,Q^2)=\left[ D\cdot U_{f3}\cdot x^{-\Delta} +(1-D)\cdot V\cdot x\right]\cdot(1-x)^{N-2}$, with $D=H\cdot S(Q^2)\cdot\left(1-\frac{1}{2}S(Q^2)\right)$ and $U_{f3}=\frac{3\cdot C(Q^2)\cdot\Gamma(N-\Delta)} {N\cdot\Gamma(N-1)\Gamma(1-\Delta)}$. The approximation is compared with data in Fig.23.21 for deuterium [35] and in Fig.23.22 for iron [36,37]. It must be emphasized that the CHIPS parton distributions are the same as for electromagnetic reactions.

Figure 23.22: Fit of $f_{2Fe}(x,Q^2)$ (filled markers, solid lines) and $f_{3Fe}(x,Q^2)$ (open markers, dashed lines) structure functions measured by the CDHSW [36] (circles) and CCFR [37] (squares) experiments.

For the $(\nu ,\mu )$ amplitudes one can not apply the optical theorem, To calculate the total cross sections, it is therefore necessary to integrate the differential cross sections first over $x$ and then over $Q^2$. For the $(\nu ,\mu )$ reactions the differential cross section can be integrated with good accuracy even for low energies because it does not have the $\frac{1}{Q^4}$ factor of the boson propagator. The quasi-elastic part of the total cross-section can be calculated for $W<m_N+m_\pi$. The total $(\nu ,\mu )$ cross-sections are shown in Fig.23.23(a,b). The dashed curve corresponds to the GRV [38] approximation of parton distributions and the dash-dotted curves correspond to the KMRS [39] approximation. Neither approximation fits low energies, because the perturbative calculations give parton distributions only for $Q^2 > 1~GeV^2$. In [40] an attempt was made to freeze the DIS parton distributions at $Q^2=1$ and to use them at low $Q^2$. The $W<1.4~GeV$ part of DIS was replaced by the quasi-elastic and one pion production contributions, calculated on the basis of the low energy models. The results of [40] are shown by the dotted lines. The nonperturbative CHIPS approximation (solid curves) fits both total and quasi-elastic cross sections even at low energies.

Figure 23.23: Fit of total (a,b) and quasi-elastic (c,d) cross-sections of $(\nu ,\mu )$ reactions (Geant4 database). The solid line is the CHIPS approximation (for other lines see text).

The quasi-elastic $(\nu ,\mu )$ cross sections are shown in Fig.23.23(c,d). The CHIPS approximation (solid line) is compared with calculations made in [40] (the dotted line) and the best fit of the $V-A$ theory was made in [41] (the dashed lines). One can see that CHIPS gives reasonable agreement.

The $Q^2$ spectra for each energy are known as an intermediate result of the calculation of total or quasi-elastic cross sections. For the quasi-elastic interactions ($W<m_N+m_\pi$) one can use $x=1$ and simulate a binary reaction. In the final state the recoil nucleon has some probability of interacting with the nucleus. If $W>m_N+m_\pi$ the $Q^2$ value is randomized and therefore the $Q^2$ dependent coefficients (the number of partons in non-perturbative phase space $N$, the Pomeron intercept $\alpha_P$, the fraction of the direct interactions, etc.) can be calculated. Then for fixed energy and $Q^2$ the neutrino interaction with quark-partons (directly or through the Pomeron ladder) can be randomized and the secondary parton distribution can be calculated. In vacuum or in nuclear matter the secondary partons are creating quasmons [2,3] which decay to secondary hadrons.