Pion production off the nucleon
aa r X i v : . [ h e p - ph ] F e b Pion production o ff the nucleon M. Rafi A lam , M. Sajjad A thar , Shikha C hauhan and S. K. S ingh
Department of Physics, Aligarh Muslim University, Aligarh, IndiaE-mail: rafi[email protected] (Received October 5, 2018)We have studied charged current neutrino / antineutrino induced weak pion production from nucleon.For the present study, contributions from ∆ (1232)-resonant term, non-resonant background termsas well as contribution from higher resonances viz. P (1440), D (1520), S (1535), S (1650)and P (1720) are taken. To write the hadronic current for the non-resonant background terms, amicroscopic approach based on SU(2) non-linear sigma model has been used. The vector form factorsfor the resonances are obtained from the helicity amplitudes provided by MAID. Axial couplingin the case of ∆ (1232) resonance is obtained by fitting the ANL and BNL ν -deuteron reanalyzedscattering data. The results of the cross sections are presented and discussed for all the possiblechannels of single pion production induced by charged current interaction. KEYWORDS:
Pion production, Deuteron effects, Higher resonances, Charged current.
1. Introduction
Experimenters are using neutrino / antineutrino beam of few GeV energy in the study of neu-trino / antineutrino nucleus scattering to determine some of the oscillation parameters like ∆ m , θ ,CP violating phase δ , etc. They are also important because of the interest in understanding hadronicstructure in weak sector where besides vector current they also get contribution from axial vector cur-rent. In the neutrino / antineutrino energy region of 1GeV, the dominant contribution to the chargedlepton events come from quasielastic scattering and single pion production processes. The availableexperimental results of single pion production and their comparison with various theoretical calcu-lations have necessitated the need to re-examine the basic reaction mechanism for the production ofsingle pion from free nucleon target. In various theoretical calculations there is lack of consensusin the modeling of basic reaction mechanism of ν (¯ ν ) induced pion production from free nucleon,specially concerning the contribution of background terms as well as the contribution of higher reso-nances in addition to the dominant ∆ (1232) resonance. The tension between the experimental resultsfrom old bubble chamber experiments, ANL [1] and BNL [2], which were performed using deu-terium / hydrogen targets, necessitated the need of re-performing the experiments with high precisionusing deuterium targets. Recently, Wilkinson et al. [3] have reanalyzed the old ANL [1] and BNL [2]data and found the di ff erences in these two results to be within 12% at E ν = GeV and 8% at E ν = GeV . A theoretical understanding of these data may be of great interest in the understandingof hadronic physics.In this work, we present a study of single pion production induced by neutrinos / antineutrinos o ff nucleon. Besides the dominant ∆ (1232)-term, we have considered non-resonant background termsand also taken the contribution of higher resonances viz. P (1440), D (1520), S (1535), S (1650)and P (1720). Presently there is no consensus as to how the non-resonant background terms shouldbe added to the dominant ∆ (1232) contribution. Some authors have performed calculations by co-herently summing the contributions of hadronic current from the background terms and the ∆ (1232)-resonant term, while some have added them incoherently. The understanding of the role of back- round terms is specially important in determining the N − ∆ transition form factors in ν µ p → µ − p π + and ¯ ν µ n → µ + n π − channels which are dominated by ∆ (1232)- excitation and receive no contributionfrom the nearby higher resonance which are I = resonances.In section-2, we present the formalism in brief and discuss the results in section-3.
2. Formalism
The cross section for the single pion production process ν l (¯ ν l ) + N → l − ( l + ) + N ′ + π i ; N , N ′ = p , n ; i = ± ,
0, may be written as, d σ = (2 π ) ME δ ( k + p − k ′ − p ′ − k π ) d ~ p ′ (2 π ) E ′ p d ~ k π (2 π ) E π d ~ k ′ (2 π ) E l ¯ ΣΣ |M| (1)where k ( k ′ ) is the four momentum of the incoming(outgoing) lepton having energy E ( E l ) while p ( p ′ )is the four momentum of the incoming(outgoing) nucleon and the pion momentum is k π having energy E π . |M| is the square of the matrix element given by |M| = G F L µν J µν . (2)where L µν is the leptonic tensor L µν = l µ l † ν = (cid:16) k µ k ′ ν + k ′ µ k ν − g µν k · k ′ ± i ǫ µναβ k ′ α k β (cid:17) , (3)the upper(lower) sign in the antisymmetric term stands for (antineutrino)neutrino induced processes.To get the expression for hadronic tensor J µν ( = j µ j † ν ), the hadronic current has been obtainedfor the Feynman diagrams shown in Fig 1. The contributions of non-resonant background termsare obtained using a chiral invariant Lagrangian based on non-linear sigma model [6]. At tree levelthe various diagrams which may contribute to the pion-production mechanism are direct and crossnucleon pole, contact diagram, pion pole and pion in flight diagram labeled as NP, CNP, CT, PP andPF respectively. As the non-linear sigma model assumes hadrons as point like particle, therefore, totake into account the structure of hadron, the form factors are introduced at the W ± N → N transitionvertex. The details are given in Refs. [4, 5].The final form of hadronic current for non-resonant background are obtained as [5] j µ (cid:12)(cid:12)(cid:12) NP = A NP ¯ u ( ~ p ′ ) / k π γ / p + / q + M ( p + q ) − M + i ǫ h V µ N ( q ) − A µ N ( q ) i u ( ~ p ) , j µ (cid:12)(cid:12)(cid:12) CP = A CP ¯ u ( ~ p ′ ) h V µ N ( q ) − A µ N ( q ) i / p ′ − / q + M ( p ′ − q ) − M + i ǫ / k π γ u ( ~ p ) , j µ (cid:12)(cid:12)(cid:12) CT = A CT ¯ u ( ~ p ′ ) γ µ (cid:16) g A f VCT ( Q ) γ − f ρ (cid:16) ( q − k π ) (cid:17)(cid:17) u ( ~ p ) , j µ (cid:12)(cid:12)(cid:12) PP = A PP f ρ (cid:16) ( q − k π ) (cid:17) q µ m π + Q ¯ u ( ~ p ′ ) / q u ( ~ p ) , j µ (cid:12)(cid:12)(cid:12) PF = A PF f PF ( Q ) (2 k π − q ) µ ( k π − q ) − m π M ¯ u ( ~ p ′ ) γ u ( ~ p ) , (4)where for NP and CNP currents, at the transition vertex W ± N → N , we have introduced form factorsin V µ N ( q ) and A µ N ( q ) to account for the nucleon structure, given by V µ, CCN ( q ) = f ( Q ) γ µ + f ( Q ) i σ µν q ν M (5) µ, CCN ( q ) = f A ( Q ) γ µ + f P ( Q ) q µ M ! γ , (6)where f , ( Q ) and f A , P ( Q ) are the isovector vector and axial vector form factors for nucleons. Sim-ilarly f ρ ( Q ) accounts for dominant contribution that comes from ρ –meson cloud at ππ NN vertexin the case of PP diagram. Finally, CVC relates the f PF ( Q ) , f VCT ( Q ) with f ( Q ) and PCAC relates f ρ ( Q ) with the axial part of CT diagrams. In the case of resonances, apart from the dominant ∆ (1232) N R N ′ π NNN N NW i W i W i W i W i W i R N N ′ π π N ′ N ππππ W i N ′ NN ′ π π N ′ N ′ Fig. 1.
Feynman diagrams contributing to the hadronic current corresponding to W i N → N ′ π ± , , where( W i ≡ W ± ; i = ± ) for charged current processes with N , N ′ = p or n . First row: direct and cross diagrams forresonance production where intermediate term R stands for di ff erent resonances. Second row: nucleon pole(NPand CNP) terms. The third row the diagrams are for contact term(CT) and pion pole(PP) term (third row left toright) and pion in flight(PF)(fourth row) terms. resonance, we have also considered the contribution of various resonances from second resonance re-gion viz: D (1520) and P (1720) which have J = , I = and P (1440), S (1535) and S (1650)which have J = , I = . The Feynman diagrams for the resonant contributions are depicted in Fig. 1and are labeled as R and CR corresponding to direct and cross terms. R / CR represents both spin halfand three half resonances. The currents for spin J = resonances are obtained as j µ (cid:12)(cid:12)(cid:12) R = i cos θ c C R k απ p R − M R + iM R Γ R ¯ u ( ~ p ′ ) P / αβ ( p R ) Γ βµ ( p , q ) u ( ~ p ) , p R = p + q , j µ (cid:12)(cid:12)(cid:12) CR = i cos θ c C R k βπ p R − M R + iM R Γ R ¯ u ( ~ p ′ ) ˆ Γ µα ( p ′ , − q ) P / αβ ( p R ) u ( ~ p ) , p R = p ′ − q , (7)and for spin J = resonances are obtained as j µ (cid:12)(cid:12)(cid:12) R = i cos θ c C R ¯ u ( ~ p ′ ) / k π γ / p + / q + M R ( p + q ) − M R + i Γ R M R Γ µ u ( ~ p ) , esonances M R [GeV] J I P Γ tot π N branching F A (0) f ⋆ (GeV) ratio (%) or ˜ C A (0) P (1232) 1 .
232 3 / / + .
120 100 1 . . P (1440) 1 .
462 1 / / + .
250 65 − .
43 0 . D (1520) 1 .
524 3 / / − .
110 60 − .
08 1 . S (1535) 1 .
534 1 / / − .
151 51 0 .
184 0 . S (1650) 1 .
659 1 / / − .
173 89 − . − . P (1720) 1 .
717 3 / / + .
200 11 − .
195 0 . Table I.
Properties of the resonances included in the present model, with Breit-Wigner mass M R , spin J,isospin I, parity P, the total decay width Γ tot , the branching ratio into π N, the axial coupling and f ⋆ . [4] j µ (cid:12)(cid:12)(cid:12) CR = i cos θ c C R ¯ u ( ~ p ′ ) Γ µ / p ′ − / q + M R ( p ′ − q ) − M R + i Γ R M R / k π γ u ( ~ p ) , (8)where C R is the coupling strength for R → N π and M R is the mass of the resonance. P / αβ is spinthree-half projection operator and is given by P / αβ ( P ) = − ( / P + M R ) g αβ − P α P β M R + P α γ β − P β γ α M R − γ α γ β , (9)The weak vertex Γ νµ ( Γ µ ) for spin ( ) resonances has V-A structure, given by Γ + νµ = (cid:20) V νµ − A νµ (cid:21) γ Γ + µ = V µ − A µ Γ − νµ = V νµ − A νµ Γ − µ = (cid:20) V µ − A µ (cid:21) γ (10)where the superscript + ( − ) stands for positive(negative) parity state. For spin three half states thevertex Γ νµ may be written in terms of six form factors viz: V νµ = ˜ C V M ( g µν / q − q ν γ µ ) + ˜ C V M ( g µν q · p ′ − q ν p ′ µ ) + ˜ C V M ( g µν q · p − q ν p µ ) + g µν ˜ C V A νµ = − ˜ C A M ( g µν / q − q ν γ µ ) + ˜ C A M ( g µν q · p ′ − q ν p ′ µ ) + ˜ C A g µν + ˜ C A M q ν q µ γ (11)while for the case of spin half states Γ µ is generally expressed in terms of four form factors as, V µ = F ( Q )(2 M ) (cid:16) Q γ µ + / qq µ (cid:17) + F ( Q )2 M i σ µα q α A µ = − F A ( Q ) γ µ γ − F P ( Q ) M q µ γ , (12)The vector form factors for the resonant states (except for the ∆ -resonance) are parameterized usinghelicity amplitudes from the MAID analysis. The parameterizations and various form of vector formfactors used in the present calculations are given in Ref. [4]. For the axial form factors we have usedthe Goldberger-Trieman relation which relates the R → N π coupling to the ˜ C A (0)( F A (0)) for the spin hree-half(half) resonances. To get R → N π coupling strength, we have used partial decay width forthe di ff erent resonant states following PDG values for the partial decay rates. The various propertiesof the resonances along with their couplings are tabulated in Table-I. Furthermore, assuming PCACand pion pole dominance at the weak vertex the pseudoscalar form factors ˜ C A ( Q )( F P ( Q )) are re-lated to ˜ C A ( Q )( F A ( Q )). We have neglected the contribution of ˜ C A , form factors for D (1520) and P (1720) resonances.We have also taken deuteron e ff ect in our calculations by following the prescription of Hernandezet al. [6] and write d σ dQ dW ! ν d = Z d p dp | Ψ d ( p dp ) | ME dp d σ dQ dW ! o ff shell . (13)In the above expression | Ψ d | = | Ψ d | + | Ψ d | , where Ψ and Ψ are the deuteron wave functions forthe S–state and D–state, respectively and have been taken from the works of Lacombe et al. [7].
3. Results and discussions ν (GeV)00.20.40.60.8 σ ( - c m ) ANL ExtractedM A =1.03 GeVBNL Extracted p π + ν (GeV)00.20.40.60.81 σ ( - c m ) ANL ExtractedC =1.1BNL Extracted p π + Fig. 2.
Total scattering cross section for ν µ p → µ − p π + process. Data points are reconstructed / reanalyzeddata of ANL and BNL experiments by Wilkinson et al. [3]. Here no invariant mass cut has been applied. Inthe left panel change in cross section with the variation(by 10%) of axial dipole mass M A has been shown bytaking central value as the world average value. While in the right panel the e ff ect of variation of axial chargefor ∆ (1232) resonance has been shown. The central curve has ˜ C A (0) | ∆ = . C A | ∆ by 10%. Using the expression for the di ff erential scattering cross section given in Eq. 1 and integratingover the kinematical variables we obtain the result for total scattering cross section. To incorporatethe deuteron e ff ect we have used Eq. 13. In all the numerical calculations where M A appears, we havetaken it as the world average value, i.e. M A = .
026 GeV. n Fig. 2, we have shown the results for the total scattering cross section for the charged cur-rent neutrino induced 1 π + production process on proton target i.e. for the reaction ν µ p → µ − p π + .The results are presented for the total scattering cross section with ∆ (1232) and non-resonant back-ground(NRB) terms. The results presented here are obtained without using any cut on invariant mass.We have compared the results with the reanalyzed experimental data of ANL [1] and BNL [2] exper-iments by Wilkinson et al. [3]. Furthermore, the e ff ect of varying ˜ C A (0) | ∆ and M A on total scatteringcross section has been studied. We found that the total scattering cross section σ ( ν µ p → µ − p π + )has minimum chi-square when ˜ C A (0) | ∆ = . M A = .
026 GeV are used in the expression of˜ C A ( Q ) | ∆ . However, to see the e ff ects of ˜ C A (0) | ∆ and M A on total scattering cross section we haveshown variations of M A and ˜ C A (0) | ∆ in shaded regions. We find that the cross section changes byabout ∼
10% at E ν = M A is varied by 10%. Similarly, at E ν = C A (0) | ∆ is varied by 10% the variation in the cross section is around 9%. ANL extractedBNL extracted
Full-FreeFull ∆∆ +B ANL 1979ANL 1982BNL 1986 σ ( - c m ) ν (GeV)00.20.40.6 0 0.5 1 1.5 2E ν (GeV)00.050.10.150.20.25 0 0.5 1 1.5 2E ν (GeV)00.10.20.3p π + p π + p π + p π p π p π n π + n π + n π + No CutNo Cut No CutW<1.4 GeV W<1.4 GeV W<1.4 GeVW<1.6 GeV W<1.6 GeV W<1.6 GeV
Fig. 3.
Total scattering cross section for the charged current neutrino induced pion production processesthrough various channels. Legends are self explanatory.
In Fig. 3, we have presented the results of total scattering cross section for the charged currentneutrino induced pion production processes in all the channels. The experimental data shown for π + p channel is same as in Fig. 2, while for the other channels like π p and π + n the data are from ANL [1]and BNL [2] experiments. In the case of ν µ p → µ − p π + induced reaction, the main contribution to thetotal scattering cross section comes from the ∆ (1232) resonance and there is no contribution from thehigher resonances which are considered here. We find that due to the presence of the non-resonantbackground terms there is an increase in the cross section which is about 12% at E ν µ = ∼
8% at E ν µ = ν µ n → µ − n π + as well as ν µ n → µ − p π processes, there are contributions from the non-resonant background terms as well as from the higher resonant terms besides the ∆ (1232) resonance.The net contribution to the total pion production due to the presence of the non-resonant backgroundterms in ν µ n → µ − n π + reaction results in an increase in the cross section of about 12% at E ν µ = hich becomes 6% at E ν µ = E ν µ = E ν µ = ν µ n → µ − p π process, due to the presence of the background terms the total increase inthe cross section is about 26% at E ν µ = E ν µ = E ν µ = E ν µ = ν µ n → µ − n π + and ν µ n → µ − p π processes. Furthermore, it may also be concluded from the aboveobservations that contribution from non-resonant background terms decreases with the increase inneutrino energy, while the total scattering cross section increases when we include higher resonancesin our calculations. σ ( - c m ) Bolognese 1979 00.050.10.150.2Full ∆∆ +B 00.10.20.30.40 0.5 1 1.5 2E ν (GeV)00.050.10.150.20.25 σ ( - c m ) ν (GeV)00.020.040.060.08 0 0.5 1 1.5 2E ν (GeV)00.050.10.150.20.25n π − n π − n π n π p π − p π − No Cut No Cut No CutW<1.4 GeV W<1.4 GeV W<1.4 GeV
Fig. 4.
Total scattering cross section for the charged current antineutrino induced pion production processesthrough various channels. Legends are self explanatory.
When a cut of W ≤ . GeV on the center of mass energy is applied then due to the presenceof the non-resonant background terms, the increase in the total scattering cross section in the energyrange E ν µ = ν µ p → µ − p π + process is about 10% which becomes 12% at E ν µ = ν µ n → µ − n π + reaction this increase in the cross section is about 14% at E ν µ = E ν µ = E ν µ = ∼
55% at E ν µ = ν µ n → µ − p π due to the presence of the non-resonant background terms the total increase incross section is about 26% at E ν µ = E ν µ = E ν µ = ν µ n → µ + n π − reaction there is no contribution from the higherresonances other than ∆ (1232) resonance. The inclusion of non-resonant background terms increasesthe cross section by around 24% at E ν µ = E ν µ = ν µ p → µ + n π reaction, inclusion of non-resonant background terms increases the cross section by around
2% at E ν µ = E ν µ = ∼
2% at E ν µ = E ν µ = ν µ p → µ + p π − reaction, the inclusion of non-resonant background terms increases the cross sectionby around 16% at E ν µ = E ν µ = E ν µ = ∼
15% at E ν µ =
4. Conclusions
We have presented the results for charged current one pion production cross section in the en-ergy region of E ν/ ¯ ν ≤ GeV . Our model consists of contributions from background terms due tonon-resonant diagrams, ∆ (1232) resonant term and the contributions from higher resonances. The ∆ (1232)-resonance has the dominant contribution but we also need contributions from the non-resonant background terms and the higher resonant terms to describe the experimental data for allthe possible channels of single pion production induced by charged current neutrino / antineutrino in-duced processes. We used ν µ p → µ − p π + channel to fix the axial charge( C A (0) | ∆ ) and axial dipolemass M A , as there is no other resonance which contributes to this process. To fix these parameters,we have used reanalyzed data of ANL and BNL and the numerical values obtained from our best fitare M A = . GeV and C A (0) | ∆ = . P (1440) and D (1520) resonances. The contribution dueto non-resonant terms is more important for ν n → ν p π − process and less important for ¯ ν p → ¯ ν p π process.The present work contributes to the theoretical understanding of the role of background termsand higher resonance terms in neutrino / antineutrino induced one pion production o ff the nucleon.It would be interesting to apply the present formalism to study the nuclear medium e ff ects in theneutrino / antineutrino induced pion production process from nuclear targets in the accelerator experi-ments being performed in the few GeV energy region. References [1] G. M. Radecky et al. , Phys. Rev. D , 1161 (1982) [Erratum-ibid. D , 3297 (1982)]; S. J. Barish et al. ,Phys. Rev. D , 3103 (1977); Phys. Rev. D , 2521 (1979).[2] T. Kitagaki et al. , Phys. Rev. D , 2554 (1986); Phys. Rev. D , 1331 (1990).[3] C. Wilkinson et al. , Phys. Rev. D , 112017 (2014).[4] M. R. Alam, M. Sajjad Athar, S. K. Singh and S. Chauhan, arXiv:1509.08622 [hep-ph].[5] E. Hernandez, J. Nieves and M. Valverde, Phys. Rev. D , 033005 (2007).[6] E. Hernandez et al. , Phys. Rev. D , 085046 (2010).[7] M. Lacombe et al. , Phys. Lett. B101