A relativistic model for neutrino pion production from nuclei in the resonance region
aa r X i v : . [ nu c l - t h ] O c t A relativistic model for neutrino pion productionfrom nuclei in the resonance region
C. Praet, O. Lalakulich, N. Jachowicz and J. Ryckebusch
Proeftuinstraat 86, B-9000 Gent, Belgium
Abstract.
We present a relativistic model for electroweak pion production from nuclei, focusing on the D and the second resonance region. Bound states are derived in the Hartree approximation to the s − w Walecka model. Final-state interactions of the outgoing pion and nucleon are describedin a factorized way by means of a relativistic extension of the Glauber model. Our formalismallows a detailed study of neutrino pion production through Q , W , energy, angle and out-of-planedistributions. Keywords: neutrino interactions, pion production, resonance region, Glauber approximation
PACS:
Lately, new cross-section measurements presented by the MiniBooNE and K2Kcollaborations have put the spotlights on few-GeV neutrino-scattering physics. Asnuclei serve as neutrino detectors in these experiments, there is a great deal of interestin modeling neutrino-nucleus interactions in the region W < n a, who aim at a precise study ofvarious exclusive channels with the use of high-intensity beams and improved particleidentification.In earlier work, neutrino-induced one-nucleon knockout calculations have been per-formed within the relativistic multiple-scattering Glauber approximation [1]. Here, weproceed along the same lines to develop a framework for resonant one-pion productioncalculations. The presented formalism focuses on an intermediate D state, but can bestraightforwardly extended to the second-resonance region.For a nucleus with mass number A , the process under consideration can be schematicallyrepresented as n + A D → l + N + p + ( A − ) , (1)with l , N and p representing the outgoing charged lepton, nucleon and pion respectively.In the laboratory system, the eightfold cross section for the process (1) is given by d s dE l d W l dE p d W p d W N = m l | ~ k l | ( p ) M N M A − | ~ k p || ~ k N | ( p ) | E A − + E N + E N ~ k N · ( ~ k p − ~ q ) / | ~ k N | | (cid:229) i f | M f i | , (2)sing self-explanatory notations for the outgoing particles’ kinematics.All information about the reaction dynamics is contained in the matrix element M f i = i G F cos q c √ u ( k N , s N ) G mD p N ( k p , k D ) S D , mn ( k D ) G nr W N D ( k D , q ) S W , rs ( q ) J s l u a , m ( k i ) , (3)where G F and q c stand for the Fermi constant and the Cabibbo mixing angle. In (3), weadopted the impulse approximation. The hit nucleon is represented by the bound-statespinor u a , m ( k i ) , calculated as the Fourier transform of the bound-state wave functions Y a , m ( ~ r ) = i G ( r ) r Y + k , m ( ˆ ~ r ) − F ( r ) r Y − k , m ( ˆ ~ r ) ! . (4)The radial wave functions in (4) are determined in the Hartree approximation to the s − w Walecka model [2]. Further, J l represents the weak lepton current and S W isthe weak boson propagator. To describe the D -production vertex G W N D , we turn to thephenomenological form-factor parameterization discussed in [3]. The adopted formfactors are constrained by theoretical principles like CVC and PCAC and, in the caseof the vector form factors, by available electron-scattering data. For the D propagatorwe take the Rarita-Schwinger propagator for a spin-3/2 particle. In this regard, mediummodifications of the resonance are accounted for by implementing a shift to the massand width of the D . We hereby use a density-dependent parameterization suggested in[4], and based on a calculation of the D self energy in the medium. Finally, the decay ofthe D particle is described by the interaction G D p N , and u ( k N , s N ) represents the outgoingnucleon’s spinor.Next to binding effects and medium-modified D properties, the final-state interactions(FSI) of the escaping nucleon and pion can have a considerable effect on the calcu-lated cross-section strength. To compute the influence of FSI, we adopt a relativisticmultiple-scattering Glauber approximation (RMSGA) [5]. Within this RMSGA model,one computes the attenuation of fast nucleons and pions due to elastic and mildlyinelastic collisions with the remaining spectator nucleons when they travel through thenucleus. The Glauber approach allows to calculate the probability that a high-energynucleon/pion will escape from a finite nucleus [6, 7], a quantity often referred to as thenuclear transparency. In Ref. [1], it was shown that plane-wave ( n , n ′ N ) cross sectionscorrected with this nuclear transparency factor provide an excellent alternative for full,unfactorized distorted-wave calculations, provided that inclusive cross sections areconsidered.In short, we have presented a fully relativistic formalism for neutrino one-pionproduction on nuclei in the resonance region. This framework opens up a wide rangeof possibilities: we can do calculations for different nuclei and resonances. More-over, predictions can be made for various observables, including not only Q and W distributions, but also energy and angular distributions for the outgoing lepton orhadrons (Fig. 1). As an accurate description of nuclear effects will be of notable inter-est to future neutrino-scattering experiments, we account for nuclear binding effects,medium-modified resonance properties and FSI effects [8]. ( M e V ) l E ( d e g r ee s ) l q C r o ss s ec ti o n -42 · Graph2D ( M e V ) p T ( d e g r ee s ) p q C r o ss s ec ti o n -42 · Graph2D
FIGURE 1.
Two-fold distributions for the process n m + p → m + D ++ on a carbon nucleus for E n = ACKNOWLEDGMENTS
The authors acknowledge financial support from the Fund for Scientific Research (FWO)Flanders.
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