Strange particle production at low and intermediate energies
M. Rafi Alam, I.Ruiz Simo, M.Sajjad Athar, M. J. Vicente Vacas
aa r X i v : . [ h e p - ph ] A ug Strange particle production at low and intermediate energies
M. Rafi Alam ∗ , I. Ruiz Simo ! , M. Sajjad Athar ∗ , and M. J. Vicente Vacas ! ∗ Department of Physics, Aligarh Muslim University, Aligarh-202 002, India and ! Departamento de F´ısica Te´orica and IFIC, Centro Mixto Universidad de Valencia-CSIC,Institutos de Investigaci´on de Paterna, E-46071 Valencia, Spain
The weak kaon production off the nucleon induced by neutrinos and antineutrinos is studied at lowand intermediate energies of interest for some ongoing and future neutrino oscillation experiments.We develop a microscopical model based on the SU(3) chiral Lagrangians. The studied mechanismsare the main source of kaon production for neutrino energies up to 2 GeV for the various channelsand the cross sections are large enough to be amenable to be measured by experiments such asMinerva, T2K and NO ν A. PACS numbers: 25.30.Pt,13.15.+g,12.15.-y,12.39.FeKeywords: neutrino nucleon scattering, strange particle production, kaon production
I. INTRODUCTION
Neutrino physics has become one of the important areas of intense theoretical and experimental efforts. This isbecause neutrinos are instrumental in giving answer to some of the basic questions in cosmology, astro-, nuclear,and particle physics. The ν µ − N scattering processes have been studied by several authors for the quasielastic, 1pion production and for deep-inelastic scattering processes [1]-[2]. However, there are very few works where neutrinoinduced strange particle production have been been studied [3]-[9]. These processes are important for the analysis ofthe precise determination of neutrino oscillation parameters. In the few-GeV region it allows the detailed study of thestrange-quark content of the nucleon and gives some important information about the structure of the hadronic weakcurrent. Apart from that in the atmospheric neutrino analysis these processes gives ∆ S backgrounds for nucleon-decay searches. The antineutrino induced ∆ S = 1 single-hyperon production can give us useful information aboutthe weak form-factors. A precise measurement of the hyperon cross-section specially Q distribution will be usefulin the determination of axial form factors. A better understanding of these processes will give more strength to thebasic understanding of V-A and Cabibbo theories.Recently Miner ν a is taking data in its first phase of experiment with high statistics to explore the strange physics.There is also probability of getting events of single kaons in the various beta-beam experiments in the energy region of1GeV. Also T2K is planning to run phase-II in antineutrino mode and the NO ν A experiments where the informationabout the single hyperon and single kaon production may be obtained.Strange particle production via the weak interaction were initially studied by Shrock [3], Mecklenburg [4] andDewan [6]. Shrock [3] and Mecklenburg [4] independently studied the associated production of charged current (CC)reactions by employing the Cabibbo theory with SU(3) symmetry. Amer [5] used harmonic oscillator quark modelto estimate cross section for some of the associated production process. Dewan [6] studied the CC and strangenesschanging (∆S = 1) strange particle production reactions. Recently Singh and Vicente Vacas [7] have studied hyperonproduction induced by antineutrinos from nucleons and nuclei, Rafi Alam et al. [8] have studied single kaon productionand Adera et al. [9] have studied differential cross section for ν induced C. C. Associated Particle Production.Most of the earlier neutrino experiments (1970’s and ’80s) were performed using bubble chambers where cross-sections for many associated-production and ∆ S = 1 reactions were obtained using bubble chambers filled with Freonand/or Propane or with deuterium. However, the data are statistically limited with large error bars[10]-[16].In this work, we have presented the results of our calculations for single kaon produced induced by neu-trino/antineutrino reactions. In Sect.II, we present the formalism in brief and in Sect.III, the results and discussionsare presented. II. FORMALISM
The basic reaction for the ν (¯ ν ) induced charged current kaon production is ν l ( k ) + N ( p ) → l ( k ′ ) + N ′ ( p ′ ) + K ( p k ) , ¯ ν l ( k ) + N ( p ) → l ( k ′ ) + N ′ ( p ′ ) + ¯ K ( p k ) (1) KW + N N ′ W + W + W + KN Σ , Λ N ′ N ′ NW + KK π, η KW + N N ′ FIG. 1: Feynman diagrams for the process: ν l N → l − NK N ( p ′ ) K ( p k ) W − ( q ) N ( p ) N ( p ′ ) N ( p ) W − ( q ) K ( p k ) K − ( q )Σ , Λ , Σ ∗ ( q + p ) N ( p ) N ( p ′ ) W − ( q ) K ( p k ) N ( p ′ ) π, ηN ( p ) W − ( q ) K ( p k ) FIG. 2: Feynman diagrams for the process ¯ ν µ n → µ + nK − where l = e, µ and N & N ′ =n,p. The expression for the differential cross section in lab frame for the above process isgiven by, d σ = 14 M E (2 π ) d~k ′ (2 E l ) d~p ′ (2 E ′ p ) d~p k (2 E K ) δ ( k + p − k ′ − p ′ − p k ) ¯ΣΣ |M| , (2)where ~k and ~k ′ are the 3-momenta of the incoming and outgoing leptons in the lab frame with energy E and E ′ respectively. The kaon lab momentum is ~p k having energy E K , M is the nucleon mass, ¯ΣΣ |M| is the square of thetransition amplitude matrix element averaged(summed) over the spins of the initial(final) state. At low energies, thisamplitude can be written in the usual form as M = G F √ j ( L ) µ J µ ( H ) = g √ j ( L ) µ M W g √ J µ ( H ) , (3)where j ( L ) µ and J µ ( H ) are the leptonic and hadronic currents respectively, G F is the Fermi constant and g is the gaugecoupling. The leptonic current can be readily obtained from the standard model Lagrangian coupling the W bosonsto the leptons L = − g √ (cid:2) W + µ ¯ ν l γ µ (1 − γ ) l + W − µ ¯ lγ µ (1 − γ ) ν l (cid:3) (4)In the case of neutrino induced kaon production process, we have considered four different channels that contribute tothe hadronic current. They are depicted in Fig. 1. There is a contact term (CT), a kaon pole (KP) term, a u-channelprocess with a Σ or Λ hyperon in the intermediate state and finally a meson ( π, η ) exchange term. For the specificreactions under consideration, there is no s-channel contributions given the absence of S = 1 baryonic resonances.KP term is proportional lepton mass and therefore its contribution is very small. While in the case of antineutrinoinduced kaon production process, besides the processes mentioned for the neutrino case, there are contributions froms-channel Σ , Λ propagator, s-channel Σ ∗ resonance terms(Fig. 2).The contribution of the different terms can be obtained in a systematic manner using Chiral Perturbation Theory( χ PT). The lowest-order SU(3) chiral Lagrangian describing the pseudoscalar mesons in the presence of an externalcurrent is [8]: L (2) M = f π D µ U ( D µ U ) † ] + f π χU † + U χ † ) , (5)where the parameter f π = 92 . U is the SU(3) representation of the meson fields [8]and D µ U is its covariant derivative. The lowest-order chiral Lagrangian for the baryon octet in the presence of anexternal current can be written in terms of the Baryon SU(3) matrix as [8]: L (1) MB = Tr (cid:2) ¯ B ( i D − M) B (cid:3) − D (cid:0) ¯ Bγ µ γ { u µ , B } (cid:1) − F (cid:0) ¯ Bγ µ γ [ u µ , B ] (cid:1) , (6)where M denotes the mass of the baryon octet, B is the Baryon SU(3) matrix and the parameters D = 0 .
804 and F = 0 .
463 which are determined from the semileptonic decays. At intermediate energies, we found that the weakexcitation of the Σ ∗ (1385) resonances and its subsequent decay in N K is also important. To calculate amplitudesassociated with Σ ∗ we first parameterize the W − N → Σ ∗ . For this, we can write the most general form of the vectorand axial-vector matrix element as, h Σ ∗ ; P = p + q | V µ | N ; p i = V us ¯ u α ( ~P )Γ αµV ( p, q ) u ( ~p ) , h Σ ∗ ; P = p + q | A µ | N ; p i = V us ¯ u α ( ~p )Γ αµA ( p, q ) u ( ~p ) E ν (GeV) σ ( - c m ) Full ModelContact TermCross Λπ in flight η in flight ν +p → µ +p+K + E ν (GeV) σ ( - c m ) All DiagramsContact Terms-channel Σ s-channel ΛΣ ∗ Resonance η in Flight π in FlightAll Diagrams without Σ ∗ ν µ p → µ + p K − FIG. 3: Cross section for (Left panel) ν µ p → µ − pK + and (Right panel) ¯ ν µ p → µ + pK − E ν (GeV) σ ( - c m ) Full ModelContact TermCross Σπ in flight η in flight ν +n → µ +n+K + E ν (GeV) σ ( - c m ) All DiagramsContact Terms-channel ΣΣ ∗ Resonance η in Flight π in FlightAll Diagrams without Σ ∗ ν µ n → µ + n K − FIG. 4: Cross section for (Left panel) ν µ n → µ − nK + and (Right panel) ¯ ν µ n → µ + nK − where Γ αµV ( p, q ) = (cid:20) C V M ( g αµ q/ − q α γ µ ) + C V M ( g αµ q · P − q α P µ ) + C V M ( g αµ q · p − q α p µ ) + C V g µα (cid:21) γ Γ αµA ( p, q ) = (cid:20) C A M ( g αµ q/ − q α γ µ ) + C A M ( g αµ q · P − q α P µ ) + C A g αµ + C A M q µ q α (cid:21) . (7)In the above expression C V,A , , , are the q dependent scalar and real vector and axial vector form factors and u α isthe Rarita-Schwinger spinor. It is the C A term which is most dominant and we have considered only the terms with C A in the present work.The spin 3/2 propagator in the momentum space is given by, G µν ( P ) = P µνRS ( P ) P − M ∗ + iM Σ ∗ Γ Σ ∗ , (8)where P µνRS is the spin 3/2 Rarita-Schwinger projection operator and M Σ ∗ is the resonance mass ( ∼ M eV ). TheΣ ∗ decay width Γ is around 36 ± Σ ∗ ( W ) = 1192 π (cid:18) C f π (cid:19) (cid:0) ( W + M ) − m π (cid:1) W λ / ( W , M , m π ) Θ( W − M − m π ) (9)where λ ( x, y, z ) = ( x − y − z ) − yz is Callen lambda function and Θ is the step function. C is the KNΣ ∗ couplingstrength taken to be 1 in the present calculation. (GeV )00.51 d σ / d Q ( - c m / G e V ) n+K + +e − n+K + + µ − p+K +e − p+K + µ − p+K + +e − p+K + + µ − E ν (GeV) σ ( - c m ) l=el= µ Genie e Genie µ ν l +p → l − +p+K + FIG. 5: (Left panel) dσdQ at E ν = 1 GeV for single kaon production induced by neutrinos. The curves are labeled according tothe final state of the process. (Right panel) Cross sections as a function of the neutrino energy for single kaon production vs.associated production obtained with Genie [17].
III. RESULTS AND DISCUSSION
The total scattering cross section σ has been obtained by using Eq. (2) after integrating over the kinematicalvariables. In the left panel of Fig. (3), we present the results of the contributions of the different diagrams to the totalcross sections for the neutrino induced process. The kaon pole contributions are negligible at the studied energiesand are not shown in the figures although they are included in the full model curves. We observe the relevance of thecontact term, not included in previous calculations. We find that the contact term is in fact dominant, followed bythe u-channel diagram with a Λ intermediate state and the π exchange term. As observed by Dewan [6] the u-channelΣ contribution is much less important, basically because of the larger coupling ( N K Λ ≫ N K
Σ) of the strong vertex.The curve labeled as Full Model has been calculated with a dipole form factor with a mass of 1 GeV. The bandcorresponds to changing up and down this mass by a 10 percent. On the right panel of Fig. (3), we have presented theresults for antineutrinos: ¯ ν µ + p → µ + + p + K − . We find that the contact term is the most dominant one followedby pion in flight and the s-channel diagram with Σ ∗ -resonance and Λ as the intermediate states. The suppression ofΣ as the intermediate state is due to the difference in the coupling strength g NK Λ >> g NK Σ and in the η in flightdue to m η > m π . We also checked the effects of the Σ ∗ ( P ) resonance at the said energies. We find that unlike the∆( P ) dominance in pion production the contribution of Σ ∗ is not too large. In Fig. (4), corresponding results for ν µ + n → µ − + K + + n and ¯ ν µ + n → µ + + K − + n processes are shown. In Fig. (5), we have shown the results for the Q distribution for the three studied channels at a neutrino energy E ν = 1 GeV. The reactions are always forwardpeaked (for the final lepton), even in the absence of any form factor ( F ( q ) = 1), favoring relatively small values of themomentum transfer. On the right panel of the Fig. (5), we have compared our results for the ν µ + p → µ − + K + + p process with the values for the associated production obtained by means of the GENIE Monte Carlo program [17]. Weobserve that, due to the difference between the energy thresholds, single kaon production for the ν l + p → l − + K + + p is clearly dominant for neutrinos of energies below 1.5 GeV. For the other two channels associated production becomescomparable at lower energies. Still, single K production off neutrons is larger than the associated production up to1.3 GeV and even the much smaller K + production off neutrons is larger than the associated production up to 1.1GeV. The consideration of these ∆ S = 1 channels is therefore important for the description of strangeness productionfor all low energy neutrino spectra and should be incorporated in the experimental analysis. [1] S. Boyd, S. Dytman, E. Hernandez, J. Sobczyk and R. Tacik, AIP Conf. Proc. (2009) 60.[2] L. Alvarez-Ruso,arXiv:1012.3871[3] R. E. Shrock, Phys. Rev. D , 2049 (1975).[4] W. Mecklenburg, Acta Phys. Austriaca 48, 293 (1978).[5] A. A. Amer, Phys. Rev. D , 2290 (1978).[6] H. K. Dewan, Phys. Rev. D24 , 2369 (1981).[7] S. K. Singh and M. J. Vicente Vacas, Phys. Rev. D , 053009 (2006).[8] M. Rafi Alam, I Ruiz Simo, M. Sajjad Athar and M. J. Vicente Vacas, Phys. Rev. D , 033001 (2010).[9] G. B. Adera, B. I. S. Van Der Ventel, D. D. van Niekerk and T. Mart, Phys. Rev. C , 025501 (2010).[10] H. Deden et al., Phys. Lett. , 361 (1975).[11] O. Erriquez et al. , Nucl. Phys. B , 123 (1978).[12] S.J. Barish et al., Phys. Rev. Lett. , 1446 (1974)[13] N. J. Baker et al., Phys. Rev. D23 , 2499 (1981).[14] N.J. Baker et al., Phys. Rev.
D24 , 2779 (1981).[15] V. V. Ammosov et al. , JETP Lett. , 209 (1984) [Pisma Zh. Eksp. Teor. Fiz. , 176 (1984)].[16] W.A. Mann et al. , Phys. Rev. D , 2545 (1986).[17] C. Andreopoulos et al. , Nucl. Instrum. Meth. A614