As a result of a new improved fit to old bubble chamber data of the dominant axial C_5^A nucleon-to-Delta form factor, and due to the relevance of this form factor for neutrino induced coherent pion production, we re-evaluate our model predictions in Phys. Rev. D79, 013002 (2009) for different observables of the latter reaction. Central values for the total cross sections increase by 20% to 30%, while differential cross sections do not change their shape appreciably. Furthermore, we also compute the uncertainties on total, differential and flux averaged cross sections induced by the errors in the determination of C_5^A. Our new results turn out to be compatible within about 1 sigma with the former ones. Finally, we stress the existing tension between the recent experimental determination of the sigma(CC coh π^+)/sigma(NC coh π^0) ratio by the SciBooNE Collaboration and the theoretical predictions.
The coherent pion production induced by neutrinos in nuclei is studied using a delta hole model in local density approximation taking into account the renormalization of $Δ$ properties in a nuclear medium. The pion absorption effects have been included in an eikonal approximation. These effects give a large reduction in the total cross section. The numerical results for the total cross section are found to be consistent with recent experimental results from K2K and MiniBooNE collaborations and other older experiments in the intermediate energy region.
Superscaling approximation (SuSA) predictions to neutrino-induced charged-current charged pion production in the Δ-resonance region are explored under MiniBooNE experimental conditions. The results obtained within SuSA for the flux-averaged double-differential cross sections of the π+ production for the ν_μ+CH_2 reaction as a function of the muon kinetic energy and of the scattering angle, the cross sections averaged over the angle, the total cross section for the π+ production, as well as CC1π+ to CCQE cross section ratio are compared with the corresponding MiniBooNE experimental data. The Su…
Predictions for neutrino-induced charged- and neutral-current single pion production on CH_2 from theoretical models and Monte Carlo event generators are compared with the cross section measurements from the MiniBooNE experiment.
We investigate one pion production processes within the Giessen Boltzmann--Uehling--Uhlenbeck (GiBUU) coupled channel transport model. Our calculations for integrated and differential cross sections for realistic experimental neutrino fluxes are compared to the data recently provided by the MiniBooNE collaboration.
Background. Neutrino-induced pion production can give important informationon the axial coupling to nucleon resonances. Furthermore, pion production represents a major background to quasielastic-like events. Single pion production data from the MiniBooNE in charged current neutrino scattering in mineral oil appeared higher than expected within conventional theoretical approaches. Purpose. We aim to investigate which model parameters affect the calculated cross section and how they do this. Method. The Giessen Boltzmann--Uehling--Uhlenbeck (GiBUU) model is used for an investigation of neutrino-…
We investigate charged and neutral current neutrino induced incoherent pion production off nuclei at MiniBooNE and K2K energies within the GiBUU model. We assume impulse approximation and treat the nucleus as a local Fermi gas of nucleons bound in a mean-field potential. In-medium spectral functions are also taken into account. The outcome of the initial neutrino nucleon reaction undergoes complex hadronic final state interactions. We present results for neutral current pi^0 and charged current pi^+ production and compare to MiniBooNE and K2K data.
Background: Charged current pion production gives information on the axial formfactors of nucleon resonances. It also introduces a noticeable background to quasi-elastic measurements on nuclear targets. Purpose: Understand pion production in neutrino interactions with nucleons and the reaction mechanism in nuclei. Method: The Giessen Boltzmann--Uehling--Uhlenbeck (GiBUU) model is used for an investigation of neutrino-nucleus reactions. Results: Theoretical results for integrated and differential cross sections for the MiniBooNE neutrino flux are compared to the data. Two sets of pion productio…
We re-evaluate our model predictions in Phys. Rev. D 79, 013002 (2009) for different observables in neutrino induced coherent pion production. This comes as a result of the new improved fit to old bubble chamber data of the dominant axial C_5^A nucleon-to-Delta form factor. We find an increase of 20%-30% in the values for the total cross sections. Uncertainties induced by the errors in the determination of C_5^A are computed. Our new results turn out to be compatible within about $1σ$ with the former ones. We also stress the existing tension between the recent experimental determination of the…
We present theoretical predictions in the framework of the ANP model for single pion production ($π^+, π^0$) in $ν_μ$ and $\barν_μ$ scattering off mineral oil and plastic. Our results for the total cross sections and flux averaged differential distributions are compared to all available data of the MiniBooNE and MINER$ν$A experiments. While our predictions slightly undershoot the MiniBooNE data they reproduce the normalization of the MINER$ν$A data for the kinetic energy distribution. For the dependence on the polar angle we reproduce the shape of the arbitrarily normalized data.
We have developed a model for neutrino-induced coherent pion production off nuclei in the energy regime of interest for present and forthcoming neutrino oscillation experiments. It is based on a microscopic model for pion production off the nucleon that, besides the dominant Delta pole contribution, takes into account the effect of background terms required by chiral symmetry. Moreover, the model uses a reduced nucleon-to-Delta resonance axial coupling, which leads to coherent pion production cross sections around a factor two smaller than most of the previous theoretical estimates. In the coh…
We analyze the experimental data on the inclusive double differential cross section by neutrinos charged current, measured by T2K, with the same model which was successful for the MiniBooNE quasielastic cross sections. As in our previous analysis the multinucleon component is needed in order to reproduce the data. For the total cross section our evaluation is smaller than the SciBooNE data above 1 GeV. This indicates the opening of a new channel not included in our evaluation, presumably the two pion emission channel. We also check that our description holds for the exclusive single pion produ…
[Background] Long-Baseline experiments such as T2K, NOvA or the planned Deep Underground Neutrino Experiment (DUNE) require theoretical descriptions of the complete event in a neutrino-nucleus reaction. Since nuclear targets are used this requires a good understanding of neutrino-nucleus interactions. [Purpose] One of the dominant reaction channels in neutrino-nucleus interactions is pion production. This paper aims for a coherent view on all charged current charged pion production data that are avaible from the experiments MiniBooNE, the near detector experiment at T2K and MINERvA. [Methods] …
The pion form factor calculation in the ``point-form'' of relativistic quantum mechanics is re-considered. Particular attention is given to the relation between the momentum of the system and the momentum transfer as well as to the quark current.
The MiniBooNE experiment has collected what is currently the world's largest sample of neutrino muon charged current single charged pion interactions, roughly 46,000 events. The purity of the CC1pi+ sample is 87% making this the purest event sample observed in the MiniBooNE detector. The average energy of neutrinos producing CC1pi+ interactions in MiniBooNE is about 1 GeV, therefore the study of these events can provide insight into both resonant and coherent pion production processes. In this talk, we will discuss the long-standing discrepancy in four-momentum transfer observed between CC1pi+…
We compare our pion production results with recent MiniBooNE data measured in mineral oil. Our total cross sections lie below experimental data for neutrino energies above 1 GeV. Differential cross sections show our model produces too few high energy pions in the forward direction as compared to data. The agreement with experiment improves by artificially removing pion final state interaction.
RPA correlations, spectral function and 2p2h (multi-nucleon) effects on charged-current neutrino-nucleus reactions without emitted pions are discussed. We pay attention to the influence of RPA and multi-nucleon mechanisms on the MiniBooNE and MINERvA flux folded differential cross sections, the MiniBooNE flux unfolded total cross section and the neutrino energy reconstruction.
We calculate isospin breaking in pion-nucleon scattering in the threshold region in the framework of covariant baryon chiral perturbation theory. All effects due to quark mass differences as well as real and virtual photons are consistently included. As an application, we discuss the energy dependence of the triangle relation that connects elastic scattering on the proton pi+- p --> pi+- p with the charge exchange reaction pi- p --> pi0 n.
Neutrino-induced pion and photon production from nucleons and nuclei are important for the interpretation of neutrino-oscillation experiments, and these processes are potential backgrounds in the MiniBooNE experiment [A. A. Aquilar-Arevalo \textit{et al.} (MiniBooNE Collaboration), Phys.\ Rev.\ Lett.\ {\bf 100}, 032301 (2008)]. Pion and photon production are investigated at intermediate energies, where the $Δ$ resonance becomes important. The Lorentz-covariant effective field theory contains nucleons, pions, Deltas, isoscalar scalar ($σ$) and vector ($ω$) fields, and isovector vector ($ρ$) fie…
A search for sub-GeV dark matter produced from collisions of the Fermilab 8 GeV Booster protons with a steel beam dump was performed by the MiniBooNE-DM Collaboration using data from $1.86 \times 10^{20}$ protons on target in a dedicated run. The MiniBooNE detector, consisting of 818 tons of mineral oil and located 490 meters downstream of the beam dump, is sensitive to a variety of dark matter initiated scattering reactions. Three dark matter interactions are considered for this analysis: elastic scattering off nucleons, inelastic neutral pion production, and elastic scattering off electrons.…
We investigate threshold pion photoproduction in the framework of heavy baryon chiral perturbation theory. We give the expansion of the electric dipole amplitude $E_{0+}$ to three orders in $μ$, the ratio of the pion to nucleon mass, and show that it is slowly converging. We argue that this observable is not a good testing ground for the chiral dynamics of QCD. In contrast, we exhibit new and fastly converging low--energy theorems in the P--waves which should be used to constrain the data analysis. We also discuss the importance of polarization observables to accurately pin down certain multip…
This report summarizes our study of Neutral Current (NC)-induced photon production in MiniBooNE, as motivated by the low energy excess in this experiment [A. A. Aquilar-Arevalo et al. (MiniBooNE Collaboration), Phys. Rev. Lett. 98, 231801 (2007); 103, 111801 (2009)]. It was proposed that NC photon production with two anomalous photon-$Z$ boson-vector meson couplings might explain the excess. However, our computed event numbers in both neutrino and antineutrino runs are consistent with the previous MiniBooNE estimate that is based on their pion production measurement. Various nuclear effects di…
Interactions of neutrinos with nuclei in the energy ranges relevant for the MiniBooNE, T2K, NO$ν$A, MINER$ν$A and MINOS experiments are discussed. It is stressed that any theoretical treatment must involve all the relevant reaction mechanisms: quasielastic scattering, pion production and DIS. In addition, also many-body interactions play a role. In this talk we show how a misidentification of the reaction mechanism can affect the energy reconstruction. We also discuss how the newly measured pion production cross sections, as reported recently by the MiniBooNE collaboration, can be related to t…
We re-examine one-pion exchange as a possible binding mechanism in the X(3872) charmonium-like state and argue it to be not sufficiently binding for this purpose. We conclude therefore that other short-range dynamics are responsible for the X formation.
The Fubini-Furlan-Rossetti sum rule for pion photoproduction on the nucleon is evaluated by dispersion relations at constant t, and the corrections to the sum rule due to the finite pion mass are calculated. Near threshold these corrections turn out to be large due to pion-loop effects, whereas the sum rule value is closely approached if the dispersion integrals are evaluated for sub-threshold kinematics. This extension to the unphysical region provides a unique framework to determine the low-energy constants of chiral perturbation theory by global properties of the excitation spectrum.
Chiral Ward identities lead to consistent accounting for the sigma's width in the linear sigma model's Feynman rules. Reanalysis of pion scattering data at threshold imply a mass for the sigma of 600+ 200 - 100 MeV.
We re-investigate Adler's PCAC relation in the presence of an external electromagnetic field within the framework of QCD coupled to external fields. We discuss pion electroproduction within a tree-level approximation to chiral perturbation theory and explicitly verify a chiral Ward identity referred to as the Adler-Gilman relation. We critically examine soft-momentum techniques and point out how inadmissable approximations may lead to results incompatible with chiral symmetry. As a result we confirm that threshold pion electroproduction is indeed a tool to obtain information on the axial form …
We propose a new, renormalizable approach to nucleon-nucleon scattering in chiral effective field theory based on the manifestly Lorentz invariant form of the effective Lagrangian without employing the heavy-baryon expansion. For the pion-less case and for the formulation based on perturbative pions, the new approach reproduces the known results obtained by Kaplan, Savage and Wise. Contrary to the standard formulation utilizing the heavy-baryon expansion, the nonperturbatively resummed one-pion exchange potential can be renormalized by absorbing all ultraviolet divergences into the leading S-w…
We present our description of neutrino induced charged current quasielastic scattering (CCQE) in nuclei at energies relevant for the MiniBooNE experiment. In our framework, the nucleons, with initial momentum distributions according to the Local Fermi Gas model, move in a density- and momentum-dependent mean field potential. The broadening of the outgoing nucleons due to nucleon-nucleon interactions is taken into account by spectral functions. Long range (RPA) correlations renormalizing the electroweak strength in the medium are also incorporated. The background from resonance excitation event…
An implicit four dimensional regularization is applied to calculate the axial-vector-vector anomalous amplitude. The present technique always complies with results of Dimensional Regularization and can be easily applied to processes involving odd numbers of $γ_5$ matrices. This is illustrated explicitely in the example of this letter.
We analyze the neutrino induced charged current coherent pion production at the energies of interest for recent experiments like K2K and MiniBooNE. Medium effects in the production mechanism and the distortion of the pion wave function, obtained solving the Klein Gordon equation with a microscopic optical potential, are included in the calculation. We find a strong reduction of the cross section due to these effects and also substantial modifications of the energy distributions of the final lepton and pion.
We analyse neutrino-induced reactions in the small $Q^2$ region and for energies covering the production and decay of the delta resonance. One of our results is the agreement with the MiniBooNE data for $1π^+$ and $1π^0$ final states. In addition we present differential cross sections for charged and neutral currents and for proton and neutron targets. Finally, we present cross sections induced by muon and electron type neutrinos, where effects of the lepton masses are visible.
Coherent pion production in low energy neutrino reactions is discussed in the framework of the partially conserved axial vector current theory (PCAC). The role of lepton mass effects in suppressing production of positively charged pions is discussed. Instead of using models of pion nucleus scattering, the available data on pion carbon scattering are implemented for an analysis of the PCAC prediction. Our results agree well with the published upper limits for positively charged production but are much below the recent MiniBooNE result for neutral pion production.
The N-Delta axial form factors are determined from neutrino induced pion production ANL & BNL data by using a state of the art theoretical model, which accounts both for background mechanisms and deuteron effects. We find violations of the off diagonal Goldberger-Treiman relation at the level of 2 sigma which might have an impact in background calculations for T2K and MiniBooNE low energy neutrino oscillation precision experiments.
The electromagnetic form factor of the pion is calculated in the "point-form" of relativistic quantum mechanics using simple, phenomenological wave functions. It is found that the squared charge radius of the pion is predicted one order of magnitude larger than the experimental value and the asymptotic behavior expected from QCD cannot be reproduced. The origin of these discrepancies is analyzed. The present results confirm previous ones obtained from a theoretical model and call for major improvements in the implementation of the "point-form" approach.
We study the dependence of neutral current (NC) neutrino-induced $π^0$/photon production ($ν_μ+ A \to ν_μ+1π^0 / γ+ X$) on the atomic number of the target nucleus, A, at 4-momentum transfers relevant to the MiniBooNE experiment: $Δ$ resonance mass region. Our conclusion is based on experimental data for photon-nucleus interactions from the A2 collaboration at the Mainz MAMI accelerator. We work in the approximation that decays of $Δ$ resonance unaffected by its production channel, via photon or Z boson. $1π^0+X$ production scales as A$^{2/3}$, the surface area of the nucleus. Meanwhile the pho…
We use our model for neutrino pion production on the nucleon to study pion production on a nucleus. The model is conveniently modified to include in-medium corrections and its validity is extended up to 2 GeV neutrino energies by the inclusion of new resonant contributions in the production process. Our results are compared with recent MiniBooNE data measured in mineral oil. Our total cross sections are below data for neutrino energies above 1 GeV. As with other theoretical calculations, the agreement with data improves if we neglect pion final state interaction. This is also the case for diff…
Forward-backward (charge) asymmetry in the processes of $e^+e^-$ annihilation into a pair of charged pseudoscalar mesons is recalculated in the one-loop approximation. The exact dependence on the meson masses is taken into account. Results known in the literature are partially corrected. The bulk of the charge asymmetry appears due to double-photon exchange in $s$ channel. Experimental studies of the asymmetry can be used to verify the point-like approximation used in calculation of radiative corrections due to emission of photons by pions or kaons.
At temperatures less than a few MeV, the efficiency of the neutrino-pair bremsstrahlung from nucleons interracting with the condensed pion field is much less than previously estimated by other authors. The physical reason for this is a periodic structure of the developed pion condensate. The repeated interactions of a nucleon with the periodic pion field modify the nucleon spectrum, which is split into two bands. The energy gap between the bands is about a few MeV or larger, even if the amplitude of the pion field is small compared with the pion mass.
Some of the recent progress in the physics of pion production induced by neutrinos on nucleons and nuclei is reviewed from a theoretical perspective. The importance of Watson's theorem to reconcile ANL and BNL data with the off-diagonal Goldberger-Treiman relation for the $Δ(1232)$ is discussed. The disagreement between MiniBooNE data and theoretical calculations is presented in the light of the new MINERvA data. The coherent pion production data on $^{12}$C obtained by MINERvA are also compared to different microscopic and PCAC models.
We have studied charged current $\barν_l$ induced one pion production from $^{12}$C and $^{16}$O nuclear targets at MiniBooNE and atmospheric antineutrino energies. The calculations have been done for the incoherent pion production process as well as for the pions coming from the hyperons in the quasielastic production of $Λ$ and $Σ$. The calculations are done in the local density approximation. For the inelastic processes the calculations have been done in the $Δ$ dominance model and we take into account the effect of Pauli blocking, Fermi motion of the nucleon and renormalization of $Δ$ prop…
Single pion leptoproduction in the region of the (3,3) resonance is currently of high interest for at least two reasons: (i) These reactions constitute an important part of the total cross section in low energy reactions and are utilized to detect neutrino oscillations in current and future long baseline experiments. (ii) Intranuclear rescattering of the pions in heavy nuclei results in interesting and sizable modifications of the free nucleon cross sections which are testable in electroproduction experiments. In this article we give a basic introduction to the pion multiple scattering model o…
Neutrino-induced pion production on nuclear targets is the major inelastic channel in all present-day neutrino-oscillation experiments. It has to be understood quantitatively in order to be able to reconstruct the neutrino-energy at experiments such as MiniBooNE or K2K and T2K. We report here results of cross section calculations for both this channel and for quasielastic scattering within the semiclassical GiBUU method. This methods contains scattering, both elastic and inelastic, absorption and side-feeding of channels all in a unitary, common theoretical framework and code. We find that cha…
Using constituent quark model constraints we calculate the gluon and sea-quark content of pions solely in terms of their valence density (fixed by πN Drell-Yan data) and the known sea and gluon distributions of the nucleon, using the most recent updated valence-like input parton densities of the nucleon. The resulting small-x dynamical QCD predictions for g^π(x,Q^2) and \bar{q}^π(x,Q^2) are unique and parameter free. Simple analytic parametrizations of the resulting parton distributions of the pion are presented in LO and NLO. These results and parametrizations will be important, among other t…
The isoscalar single-pion production exhibits a broad bump in the energy dependence of the total cross section, which does not correspond to the usual opening of the $N^*(1440)$ production channel with subsequent pion decay. In arxiv:2102.05575 it was interpreted as a narrow Breit-Wigner structure, which leads in a sequential single-pion production process to a possible explanation of the $d^*(2380)$ resonance. We demonstrate that such an attempt fails already, when confronted with the data base for isoscalar single-pion production. We investigate whether the observed bump structure rather poi…
Understanding single pion production reactions on free nucleons is the first step towards a correct description of these processes in nuclei, which are important for signal and background contributions in current and near future accelerator neutrino oscillation experiments. In this work, we reanalyze our previous studies of neutrino-induced one-pion production on nucleons for outgoing $πN$ invariant masses below 1.4 GeV. Our motivation is to get a better description of the $ν_μn\toμ^- nπ^+$ cross section, for which current theoretical models give values significantly below data. This channel i…
We have developed a dynamical model for a unified description of the pion-nucleus scattering and photo- and neutrino-induced coherent pion production on nuclei. Our approach is based on a combined use of the Sato-Lee model for the electroweak pion production on a single nucleon and the Delta-hole model of pion-nucleus scattering. Numerical calculations are carried out for the case of the C12 target. After testing our model with the use of the pion photo-production data, we confront our predictions of the neutrino-induced coherent pion production reactions with the recent data from K2K and Mini…
Two-pion transitions in charmonium and bottomonium are considered with the leading relativistic effects taken into account. The contribution of the chromo- magnetic interaction of the charmed quarks to the amplitude of the decay $ψ(2S) \to ππJ/ψ$ is estimated from the available data. This contribution is enhanced by a factor of three in the decay $η_c(2S) \to ππη_c$ and should produce a noticeable modification of the rate and the spectrum in the latter decay. It is argued that the peculiar observed spectrum in the decay $Υ(3S) \to ππΥ$ arises due to a dynamical suppression of the leading nonre…
MiniBooNE [1] and MINERvA [2] charge current π + production data in the Delta region are discussed. It is argued that despite the differences in neutrino flux they measure the same dynamical mechanism of pion production and should be strongly correlated. The correlation is clearly seen in the Monte Carlo simulations done with NuWro generator but is missing in the data. Both normalization and the shape of the ratio of measured differential cross sections in pion kinetic energy are different from the Monte Carlo results, in the case of normalization a discrepancy is by a factor of 1.49.
We analyze new QCD sum rules for the pion wave function $ϕ_π(x)$, obtained recently by A.V. Radyushkin in the framework of the non-local vacuum condensates method for non-diagonal correlators, and suggest a new approach for extracting wave functions of the $π$-meson and of its first resonances. As a result we obtain approximately the same form of the pion wave function by using two different methods and for two different ansatzes of non-local quark condensates. We predict the mass of the $π'$-resonance and obtain the form of its wave function. The first method uses integrated (in a sense) prop…
Long ago an unexpected and unexplainable phenomena was observed. The distribution of muons from positive pion decay at rest was anisotropic with an excess in the backward direction relative to the direction of the proton beam from which the pions were created. Although this effect was observed by several different groups with pions produced by different means, the result was not accepted by the physics community, because it is in direct conflict with a large set of other experiments indicating that the pion is a pseudoscalar particle. It is possible to satisfy both sets of experiments if helic…
We analyze the prospects for detection of light sub-GeV dark matter produced in experiments designed to study the properties of neutrinos, such as MiniBooNE, T2K, SHiP, DUNE etc. We present an improved production model, when dark matter couples to hadronic states via a dark photon or baryonic vector mediator, incorporating bremsstrahlung of the dark vector. In addition to elastic scattering, we also study signatures of light dark matter undergoing deep inelastic or quasi-elastic NC$π^0$-like scattering in the detector producing neutral pions, which for certain experiments may provide the best …
A direct connection can be made between mixing angles in negative parity baryons and the spin coupling of constituent quarks. The mixing angles do not depend on spectral data. These angles are recalculated for gluon exchange and pion exchange between quarks. For pion exchange the results of Glozman and Riska are corrected. The experimental data on mixing are very similar to those derived from gluon exchange but substantially different from the values obtained for pion exchange.
We have investigated the neutrino induced coherent pion production reaction at the energies of interest for recent experiments like K2K and MiniBooNE. The model includes pion, nucleon and the Delta(1232) resonance. Medium effects in the production mechanism and the distortion of the pion wave function are taken into account. We find a strong reduction of the cross section due to these effects and also substantial modifications in the energy distributions of the final pion. The sensitivity of the results on the axial N-Delta coupling C5A(0) and the coherent fraction in neutral-current pi0 produ…
We study one pion production in both charged and neutral current neutrino nucleus scattering for neutrino energies below 2 GeV. We use a theoretical model for one pion production at the nucleon level that we correct for medium effects. The results are incorporated into a cascade program that apart from production also includes the pion final state interaction inside the nucleus. Besides, in some specific channels coherent pion production is also possible and we evaluate its contribution as well. Our results for total and differential cross sections are compared with recent data from the MiniBo…
Coherent pi^+ and pi^0 production in low energy neutrino reactions is discussed in the framework of the partially conserved axial vector current theory (PCAC). The role of lepton mass effects in suppressing the pi^+ production is emphasized. Instead of using models of pion nucleus scattering, the available data on pion Carbon scattering are implemented for an analysis of the PCAC prediction. Our results agree well with the published upper limits for pi^+ production but are much below the recent MiniBooNE result for pi^0 production.
The extraction of neutrino oscillation parameters requires the determination of the neutrino energy from observations of the hadronic final state. Here we discuss the difficulties connected with this energy reconstruction for the ongoing experiments MiniBooNE and T2K. We point out that a lower limit to the uncertainty in the reconstructed energy from Fermi motion alone amounts to about 15%. The entanglement of very different elementary processes, in this case quasielastic scattering and pion production, in the actual observables leads to considerably larger errors. We discuss the sensitivity o…
We reanalyze the anomalous angular distribution of lepton pairs produced in a pion-induced Drell-Yan process by taking into account the Glauber gluon effect in the $k_T$ factorization theorem. Compared to the previous study, we adopt the realistic parton distribution functions (PDFs) for a proton from the CTEQ and for a pion from the xFitter, include the QCD evolutions of the strong coupling and the PDFs, and integrate the differential cross section over the kinematic regions for the NA10, E615 and COMPASS experiments. These improvements then allow rigorous confrontations of theoretical result…
