Analysis of flux-integrated semi-exclusive cross sections for charged current quasi-elastic neutrino scattering off ⁴⁰Ar at energies available at the MicroBooNE experiment

In the laboratory frame the differential cross section for exclusive electron ($\sigma^{el}$) and (anti)neutrino ($\sigma^{cc}$) CC scattering can be written as

	$\displaystyle\frac{d^{6}\sigma^{el}}{d\varepsilon_{f}d\Omega_{f}d\varepsilon_{x}d\Omega_{x}}$	$\displaystyle=\frac{|\mbox{\boldmath$p$}_{x}|\varepsilon_{x}}{(2\pi)^{3}}\frac{\varepsilon_{f}}{\varepsilon_{i}}\frac{\alpha^{2}}{Q^{4}}L_{\mu\nu}^{(el)}\mathcal{W}^{\mu\nu(el)}$		(2.3a)
	$\displaystyle\frac{d^{6}\sigma^{cc}}{d\varepsilon_{f}d\Omega_{f}d\varepsilon_{x}d\Omega_{x}}$	$\displaystyle=\frac{|\mbox{\boldmath$p$}_{x}|\varepsilon_{x}}{(2\pi)^{5}}\frac{|\mbox{\boldmath$k$}_{f}|}{\varepsilon_{i}}\frac{G^{2}\cos^{2}\theta_{c}}{2}L_{\mu\nu}^{(cc)}\mathcal{W}^{\mu\nu(cc)},$		(2.3b)

where $\Omega_{f}$ is the solid angle for the lepton momentum, $\Omega_{x}$ is the solid angle for the ejectile nucleon momentum, $\alpha\simeq 1/137$ is the fine-structure constant, $G\simeq$ 1.16639 $\times 10^{-11}$ MeV^-2 is the Fermi constant, $\theta_{C}$ is the Cabbibo angle ($\cos\theta_{C}\approx$ 0.9749), $L^{\mu\nu}$ is the lepton tensor, and $\mathcal{W}^{(el)}_{\mu\nu}$ and $\mathcal{W}^{(cc)}_{\mu\nu}$ are correspondingly the electromagnetic and weak CC nuclear tensors.

For exclusive reactions in which only a single discrete state or narrow resonance of the target is excited, it is possible to integrate over the peak in missing energy and obtain a fivefold differential cross section of the form

	$\displaystyle\frac{d^{5}\sigma^{el}}{d\varepsilon_{f}d\Omega_{f}d\Omega_{x}}$	$\displaystyle=R\frac{|\mbox{\boldmath$p$}_{x}|\tilde{\varepsilon}_{x}}{(2\pi)^{3}}\frac{\varepsilon_{f}}{\varepsilon_{i}}\frac{\alpha^{2}}{Q^{4}}L_{\mu\nu}^{(el)}W^{\mu\nu(el)}$		(2.4a)
	$\displaystyle\frac{d^{5}\sigma^{cc}}{d\varepsilon_{f}d\Omega_{f}d\Omega_{x}}$	$\displaystyle=R\frac{|\mbox{\boldmath$p$}_{x}|\tilde{\varepsilon}_{x}}{(2\pi)^{5}}\frac{|\mbox{\boldmath$k$}_{f}|}{\varepsilon_{i}}\frac{G^{2}\cos^{2}\theta_{c}}{2}L_{\mu\nu}^{(cc)}W^{\mu\nu(cc)},$		(2.4b)

where $R$ is a recoil factor

$$R=\int d\varepsilon_{x}\delta(\varepsilon_{x}+\varepsilon_{B}-\omega-m_{A})={\bigg{|}1-\frac{\tilde{\varepsilon}_{x}}{\varepsilon_{B}}\frac{\mbox{\boldmath$p$}_{x}\cdot\mbox{\boldmath$p$}_{B}}{\mbox{\boldmath$p$}_{x}\cdot\mbox{\boldmath$p$}_{x}}\bigg{|}}^{-1},$$

(2.5)

$\tilde{\varepsilon}_{x}$ is solution to equation $\varepsilon_{x}+\varepsilon_{B}-m_{A}-\omega=0,$ where $\varepsilon_{B}=\sqrt{m^{2}_{B}+\mbox{\boldmath$p$}^{2}_{B}}$, $~{}\mbox{\boldmath$p$}_{B}=\mbox{\boldmath$q$}-\mbox{\boldmath$p$}_{x}$ and $m_{A}$ and $m_{B}$ are masses of the target and recoil nucleus, respectively. Note, that missing momentum is $\mbox{\boldmath$p$}_{m}=\mbox{\boldmath$p$}_{x}-\mbox{\boldmath$q$}$ and missing energy $\mbox{\boldmath$\varepsilon$}_{m}$ is defined by $\mbox{\boldmath$\varepsilon$}_{m}=m+m_{B}-m_{A}$. The differential cross sections ($d^{3}\sigma^{el(cc)}/d\mbox{\boldmath$\varepsilon$}_{f}d\Omega_{f})_{ex}$ can be obtained by integrating the exclusive cross sections (2.4a) and (2.4b) over solid angle for the ejectile nucleon.

All information about the nuclear structure and FSI effects is contained in the electromagnetic and weak CC hadronic tensors, $W^{(el)}_{\mu\nu}$ and $W^{(cc)}_{\mu\nu}$, which are given by the bilinear products of the transition matrix elements of the nuclear electromagnetic or CC operator $J^{(el)(cc)}_{\mu}$ between the initial nucleus state $|A\rangle$ and the final state $|B_{f}\rangle$ as

$\displaystyle W^{(el)(cc)}_{\mu\nu}$

$\displaystyle=$

$\displaystyle\sum_{f}\langle B_{f},p_{x}|J^{(el)(cc)}_{\mu}|A\rangle\langle A|J^{(el)(cc)\dagger}_{\nu}|B_{f},p_{x}\rangle,$

(2.6)

where the sum is taken over undetected states.

In the inclusive reaction (2.2) only the outgoing lepton is detected and lepton scattering cross sections in term of nuclear response functions can be written as

	$\displaystyle\frac{d^{3}\sigma^{el}}{d\varepsilon_{f}d\Omega_{f}}$	$\displaystyle=\sigma_{M}\big{(}V_{L}R^{(el)}_{L}+V_{T}R^{(el)}_{T}\big{)},$		(2.7a)
	$\displaystyle\frac{d^{3}\sigma^{cc}}{d\varepsilon_{f}d\Omega_{f}}$	$\displaystyle=\frac{G^{2}\cos^{2}\theta_{c}}{(2\pi)^{2}}\varepsilon_{f}|\mbox{\boldmath$k$}_{f}|\big{(}v_{0}R_{0}+v_{T}R_{T}+v_{zz}R_{zz}-v_{0z}R_{0z}-hv_{xy}R_{xy}\big{)},$		(2.7b)

where

$$\sigma_{M}=\frac{\alpha^{2}\cos^{2}\theta/2}{4\varepsilon^{2}_{i}\sin^{4}\theta/2}$$

(2.8)

is the Mott cross section. The coupling coefficient $V_{k}$ and $v_{k}$, the expression of which are given in Ref. Butkevich:2007gm are kinematic factors depending on the lepton’s kinematics. The response functions $R_{i}$ are given in terms of components of the inclusive hadronic tensors  (Butkevich:2007gm ) and depend on the variables $(Q^{2},\omega)$ or $(|\mbox{\boldmath$q$}|,\omega)$.

The experimental data of the $(e,e^{\prime}p)$ reaction are usually presented in terms of the reduced cross section

$$\sigma_{red}=\frac{d^{5}\sigma}{d\varepsilon_{f}d\Omega_{f}d\Omega_{x}}/K^{(el)(cc)}\sigma_{lN},$$

(2.9)

where $K^{el}=R{p_{x}\varepsilon_{x}}/{(2\pi)^{3}}$ and $K^{cc}=R{p_{x}\varepsilon_{x}}/{(2\pi)^{5}}$ are phase-space factors for electron and neutrino scattering and $\sigma_{lN}$ is the corresponding elementary cross section for the lepton scattering from the moving free nucleon. The reduced cross section is an interesting quantity that can be regarded as the nucleon momentum distribution modified by FSI. Therefore these cross sections for (anti)neutrino scattering off nuclei are similar to the electron scattering apart from small differences at low beam energy due to effects of Coulomb distortion of the incoming electron wave function. Precise electron reduced cross section data can be used to validate the neutrino reduced cross sections employed in neutrino generators.

We describe genuine QE electron-nuclear scattering within the RDWIA approach. This formalism is based on the impulse approximation (IA), assuming that the incoming lepton interacts with only one nucleon of the target, which is subsequently emitted. In this approximation the nuclear current is written as a sum of single-nucleon currents and the nuclear matrix element in Eq. (2.6) takes the form

$\displaystyle\langle p,B|J^{\mu}|A\rangle$

$\displaystyle=$

$\displaystyle\int d^{3}r~{}\exp(i\mbox{\boldmath$t$}\cdot\mbox{{\bf r}})\overline{\Psi}^{(-)}(\mbox{\boldmath$p$},\mbox{{\bf r}})\Gamma^{\mu}\Phi(\mbox{{\bf r}}),$

(2.10)

where $\Gamma^{\mu}$ is the vertex function, $\mbox{\boldmath$t$}=\varepsilon_{B}\mbox{\boldmath$q$}/W$ is the recoil-corrected momentum transfer, $W=\sqrt{(m_{A}+\omega)^{2}-\mbox{\boldmath$q$}^{2}}$ is the invariant mass and $\Phi$ and $\Psi^{(-)}$ are relativistic bound-state and outgoing wave functions.

For electron scattering, we use the CC2 electromagnetic vertex function for a free nucleon DeForest:1983ahx

$$\Gamma^{\mu}_{V}=F_{V}(Q^{2})\gamma^{\mu}+{i}\sigma^{\mu\nu}\frac{q_{\nu}}{2m}F_{M}(Q^{2}),$$

(2.11)

where $\sigma^{\mu\nu}=i[\gamma^{\mu},\gamma^{\nu}]/2$, $F_{V}$ and $F_{M}$ are the Dirac and Pauli nucleon form factors. The single-nucleon charged current has $V{-}A$ structure $J^{\mu(cc)}=J^{\mu}_{V}+J^{\mu}_{A}$. For a free-nucleon vertex function $\Gamma^{\mu(cc)}=\Gamma^{\mu}_{V}+\Gamma^{\mu}_{A}$ we use the CC2 vector current vertex function

$$\Gamma^{\mu}_{V}=F_{V}(Q^{2})\gamma^{\mu}+{i}\sigma^{\mu\nu}\frac{q_{\nu}}{2m}F_{M}(Q^{2})$$

(2.12)

and the axial current vertex function

$$\Gamma^{\mu}_{A}=F_{A}(Q^{2})\gamma^{\mu}\gamma_{5}+F_{P}(Q^{2})q^{\mu}\gamma_{5}.$$

(2.13)

The weak vector form factors $F_{V}$ and $F_{M}$ are related to the corresponding electromagnetic form factors $F^{(el)}_{V}$ and $F^{(el)}_{M}$ for protons and neutrons by the hypothesis of the conserved vector current. We use the approximation of Ref. Mergell:1995bf for the Dirac and Pauli nucleon form factors. Because the bound nucleons are off-shell we employ the de Forest prescription DeForest:1983ahx and Coulomb gauge for the off-shell vector current vertex $\Gamma^{\mu}_{V}$. The vector-axial $F_{A}$ and pseudoscalar $F_{P}$ form factors are parametrized using a dipole approximation:

$$F_{A}(Q^{2})=\frac{F_{A}(0)}{(1+Q^{2}/M_{A}^{2})^{2}},\quad F_{P}(Q^{2})=\frac{2mF_{A}(Q^{2})}{m_{\pi}^{2}+Q^{2}},$$

(2.14)

where $F_{A}(0)=1.2724$, $M_{A}$ is the axial mass that controls $Q^{2}$-dependence of $F_{A}(Q^{2})$, and $m_{\pi}$ is the pion mass.

In the RDWIA calculations the independent particle shell model (IPSM) is assumed for the nuclear structure. In Eq.(2.10) the relativistic bound-state wave functions for nucleons $\Phi$ are obtained as the self-consistent solutions of a Dirac equation, derived within a relativistic mean-field approach from a Lagrangian containing $\sigma$, $\omega$, and $\rho$ mesons Horowitz:1981xw . These functions were calculated by the TIMORA code Horowitz:1991 with the normalization factors $S_{\alpha}$ relative to full occupancy of the IPSM orbital $\alpha$ of ⁴⁰Ca. For ⁴⁰Ca and ⁴⁰Ar an average factor $\langle S\rangle\approx 87\%$. This estimation of depletion of hole states follows from the RDWIA analysis of ⁴⁰Ca$(e,e^{\prime}p)$ data Butkevich:2012zr . The source of the reduction of the $(e,e^{\prime}p)$ spectroscopic factors with respect to the mean field values are the short-range and tensor correlations in the ground state, leading to the appearance of the high-momentum and high-energy component in the nucleon distribution in the target. Mean values of the proton and neutron binding energies and occupancies of shells are given also in Ref. Butkevich:2012zr .

In the RDWIA model, final state interaction effects for the outgoing nucleon are taken into account. The system of two coupled first-order Dirac equations is reduced to a single second-order Schrödinger-like equation for the upper component of the Dirac wave function $\Psi$. This equation contains equivalent nonrelativistic central and spin-orbit potentials which are functions of the relativistic, energy dependent, scalar, and vector optical potentials. The optical potential consists of areal part, which describes the rescattering of the ejected nucleon and an imaginary part which account for its absorption into unobserved channels. We use the LEA program Kelly:1996 for the numerical calculation of the distorted wave functions with the EDAD1 parametrization Cooper:1993nx of the relativistic optical potential for calcium. This code was successfully tested in Ref. Butkevich:2012zr against $A(e,e^{\prime}p)$ data for electron scattering off ⁴⁰Ca. In Ref. Butkevich:2012zr the reduced cross sections as functions of missing momentum calculated in the RDWIA approach for the ⁴⁰Ca$(e,e^{\prime}p)$ reaction are shown in figures 1 and 2 with NIKHEF data and provide a good description of the measured distributions. Neutrino and antineutrino cross sections are also shown in figure 1 for comparison.

The inclusive cross sections with the FSI effects, taking into account short-range nucleon-nucleon ($NN$) correlations were calculated using the method proposed in Ref. Butkevich:2007gm with the nucleon high-momentum and high-energy distribution from Ref. CiofidegliAtti:1995qe that was renormalized to value of 13% for calcium and argon. The contribution of the $NN$-correlated pairs is evaluated in impulse approximation, i.e., the virtual photon couples to only one member of the $NN$-pair.

In Ref. Butkevich:2020ysf was shown that this approach describes well the electron scattering data for carbon, calcium, and argon at different kinematics. The calculated and measured cross sections are in agreement within the experimental uncertainties.

For these CC1p0$\pi$ events were measured the flux-integrated $\nu_{\mu}-{}^{40}Ar$ differential cross sections in muon and proton momentum and angle, and as a function of the calorimetric measured energy and reconstructed momentum transfer. The statistical uncertainty of the integrated measured CC1p0$\pi$ cross section is 15.9% and the systematic uncertainty sums to 26.2%. The MicroBooNE detector is located along the Booster Neutrino Beam at Fermilab. The BNB energy spectrum extends to 2 GeV and peaks around 0.7 GeV Aguilar-Arevalo:2008yp . In this work we calculate within the RDWIA model with $M_{A}=1$ GeV and 1.2 GeV the flux-integrated CCQE semi-exclusive cross sections, taking into account the MicroBooNE momentum thresholds for muons and protons. Thus, we do not consider MEC nor the process where charged pions may be produced in the final state.

The flux-integrated double-differential cross section $d^{2}\sigma/dp_{\mu}d\cos\theta$ of the semi-exclusive CCQE $\nu_{\mu}-{}^{40}Ar$ scattering is presented in figure 1, which shows the cross section as a function of muon momentum $p_{\mu}$ and muon scattering angle $\cos\theta$. Here the result was obtained in the RDWIA approach with the value of the nucleon axial mass $M_{A}=1$ GeV. The maximum of the calculated cross section is in the range $0.9<p_{\mu}<1.1$ GeV/c and $0.8<\cos\theta<0.96$. So, neutrino interactions with energy higher than 1 GeV and high values of $\cos\theta$ that corresponds to low momentum transfer, yield the main contribution to the measured cross sections.

Figures 2 and 3 show the flux-integrated $d^{2}\sigma/dp_{\mu}d\cos\theta$ cross sections as functions of $p_{\mu}$ for several bins of the muon scattering angle. On can observe that within the RDWIA model with $M_{A}=1.2$ GeV the cross sections in the region of the QE peak are predicted to be on about 20% higher than cross sections calculated with $M_{A}=1$ GeV.

Figure 4 shows the flux-integrated differential $d\sigma/d\cos\theta$ cross section as a function of the cosine of the measured muon scattering angle. In Ref. Abratenko:2020sga was shown that the bin migration effects on this measurement are small and within the assessed uncertainties. The data are compared to the RDWIA calculations.

As can be seen in figure, calculated cross sections are in overall agreement with data, except for the highest $\cos\theta$ bin, where the measured cross section is lower than the theoretical predictions. We note that this bin corresponds to low-momentum transfer ($Q^{2}<0.1$ (GeV/c)²), where the nuclear effects are significant and the neutrino interactions with energy $\mbox{\boldmath$\varepsilon$}_{\nu}>1$ GeV (high energy range of the BNB neutrino flux) give the main contribution to the measured $d\sigma/d\cos\theta$ cross section.

As the differential $d\sigma/dp_{\mu}$, $d\sigma/dp_{p}$, and $d\sigma/dQ^{2}$ cross sections include contributions from all muon scattering angles, their agreement with the theoretical calculations is effected by these inclusions. In Ref. Abratenko:2020sga the relevant cross sections in the case where events with $\cos\theta>0.8$ are excluded and in the full available phase space $-0.65<\cos\theta<0.95$ are presented. Figure 5 shows measured differential $d\sigma/dp_{\mu}$ cross section as a function of muon momentum for $-0.65<\cos\theta<0.95$ and $-0.65<\cos\theta<0.8$ compared to the RDWIA calculations. Removing events with $\cos\theta>0.8$ significantly improves the agreement between data and theory at $p_{\mu}>0.6$ GeV/c. The calculated flux-integrated cross sections for inclusive reaction are shown as well in figure 5 for the full measured phase space and for events with $\cos\theta<0.8$. The contribution of $(\nu_{\mu},\mu p)$ channel with $p_{p}>300$ MeV/c to the inclusive $d\sigma/dp_{\mu}$ cross section increases slowly from 35% at $p_{\mu}\approx 0.2$ GeV/c to 50% at $p_{\mu}\approx 1$ GeV/c. The value of muon momentum where the maximum of $d\sigma/dp_{\mu}$ cross sections appears is about 0.4 GeV/c.

The differential cross sections $d\sigma/dp_{p}$ as functions of proton momentum are shown in Figure 6 with and without events with $\cos\theta>0.8$. Also shown are the results obtained in the RDWIA. Overall, agreement is observed between data and calculations, even for the full event sample without the $\cos\theta>0.8$ requirement. Figure 6 demonstrates that the measured proton momentum distribution is wider than the muon momentum distribution and the maximum in the $d\sigma/dp_{\mu}$ cross section is located at $p_{p}=0.5$ GeV/c.

Finally, figure 7 shows the flux-integrated differential $d\sigma/dQ^{2}$ cross sections as functions of $Q^{2}$ for $-0.65<\cos\theta<0.95$ and $-0.65<\cos\theta<0.8$. The data are compared to the RDWIA calculations. Note, that in the bin $0.17<Q^{2}<0.34$ (GeV/c)² the agreement between data and theoretical result for the full phase space is better than for event sample with the $\cos\theta$ requirement. As can be seen in figure 7 at $Q^{2}<0.1$ (GeV/c)², the measured cross section is significantly lower than the theoretical prediction. Note, that at low $Q^{2}$ the $d\sigma/dQ^{2}$ cross section depends weakly on the value of axial mass and $Q^{2}$ distributions are controled by nuclear effects.

We calculated the $\chi^{2}$ value for the agreement of the RDWIA prediction with data as simple sum of those $\chi^{2}$ values obtained for $d\sigma/d\cos\theta$, $d\sigma/dp_{\mu}$, and $d\sigma/dQ^{2}$ distributions separately. As follows from this analysis the values of $\chi^{2}$/degree of freedom (d.o.f.) for $M_{A}=1(1.2)$ GeV are $\chi^{2}/d.o.f.=1.12$ (1.36) for $\cos\theta<0.8$ and $\chi^{2}/d.o.f.=1.26$ (2.22) for $\cos\theta<0.95$. The measured integrated cross sections obtained by integrating $d\sigma/d\cos\theta$ cross section over $-0.64<\cos\theta<0.8$ and $-0.64<\cos\theta<0.95$ are equal to $(4.05\pm 1.4)\times 10^{-38}$ cm² and $(4.93\pm 1.55)\times 10^{-38}$ cm², correspondingly  Abratenko:2020sga . The calculated with $M_{A}=1(1.2)$ GeV cross section values of $3.65\times 10^{-38}(4.48\times 10^{-38})$ cm² for $\cos\theta<0.8$ and $5.24\times 10^{-38}(6.30\times 10^{-38})$ cm² for the full measured phase space agree also with data. On the other hand the statistical and systematic precision of the MicroBooNE data are insufficient for current needs. Thus, within the RDWIA approach the measured flux-integrated CC1p0$\pi$ differential and integral cross sections can be described well within the experimental errors with $1<M_{A}<1.2$ GeV. These values of $M_{A}$ are in agreement with the best fit values $M_{A}=1.15\pm 0.03$ GeV and $M_{A}=1.2\pm 0.06$ GeV obtained from the CCQE-like fit of the MiniBooNE and MINERvA data in Refs. Wilkinson:2016wmz ; Butkevich:2018hll .