Lecture 2: Two body current contribution. Jan T. Sobczyk. Wrocªaw University. NuSTEC Neutrino Generator School, Liverpool, May 14-16, / 49

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1 Lecture 2: Two body current contribution Jan T. Sobczyk Wrocªaw University NuSTEC Neutrino Generator School, Liverpool, May 14-16, / 49

2 Outline: history large M A controversy Marteau model basic idea of two-body current theoretical models for muon inclusive cross section contribution Lyon model (Marteau-Martini) IFIC model (Nieves) superscaling approach transverse enhancement model Monte Carlo implementations hadronic model message to take home 2 / 49

3 History Introduction: large CCQE M A controversy The experimental data is consistent with dipole axial FF and M A = GeV. A. Bodek, S. Avvakumov, R. Bradford, H. Budd older M A measurements indicate the value of about 1.05 GeV independent pion production arguments lead to the similar conclusion MiniBooNE reported M A 1.35 GeV. T. Katori, J. Grange 3 / 49

4 New dynamical mechanism Introduction: a new dynamical mechanism needed to resolve the MB large axial mass puzzle? M. Martini, G. Chanfray, M. Ericson, J. Marteau 4 / 49

5 New dynamical mechanism Introduction: a new dynamical mechanism needed to resolve the MB large axial mass puzzle? The gure below is taken from the Jacques Marteau seminar given 13 years ago at NuInt01. How large? a half of bare QE part the original idea was put forward by Magda Ericson in 1990 the model developed by J. Marteau in his PhD thesis (1998) (supervised by J. Delorme) predicts a large contribution from n-particle n-hole excitations the model forgotten for 8 year and reintroduced to the community by Marco Martini 5 / 49

6 New dynamical mechanism Terminology: For a purpose of this talk meson exchange current (MEC) two body current n particles n holes (np nh) (in general the term MEC refers to a smaller subset of the two body current diagrams which lead to np-nh nal states) 6 / 49

7 New dynamical mechanism General remarks np-nh interactions occur in ν nucleus interactions and not in ν nucleon reactions in the case of ν deuteron scattering it is there, but the contribution is small (see later) a departure from impulse approximation picture nucleons resulting in two body current interaction are still subject to reinteractions very dicult to identify the experimental signal pion absorption and CCQE (with FSI eects) events are very similar. what is measured by MiniBooNE is CCQE-like rather then CCQE cross section if only muon is measured CCQE and np-nh are indistinguishable 7 / 49

8 Basic intuition Two-body current basic intuition. One-body hadronic current operator: J α = cos θ C (V α A α ) = cos θ C ψ(p )Γ α V ψ(p) from J. muda In the second quantization language J α is the operator that annihilates (removes from the Fermi see, producing a hole) a nucleon with momentum p creates (above the Fermi level) a nucleon with momentum p' J α 1body a (p )a(p) 8 / 49

9 Basic intuition Two-body current basic intuition Think about more complicated Feynman diagrams: J α 2body a (p 1)a (p 2)a(p 1)a(p 2) can create two particles and two holes (2p-2h). J. Morn, JTS from J. muda Transfered energy and momentum are shared between two nucleons. 9 / 49

10 Basic intuition Two body current in electron scattering a necessity of the two body current follows from the continuity equation: q J = [H, ρ], H = j p 2 j 2M + j<k V jk + j<k<l V jkl. q J (1) j = [ p2 j 2M, ρ(1) j ], q J (2) jk = [v jk, ρ (1) j + ρ (1) k ]. interaction Hamiltonian contains two body (V jk ) and three body (V jkl ) parts q is momentum transfer electromagnetic current J µ = (ρ, J) both ρ and J as operators must contain one body and two body parts J = J (1) + J (2) / 49

11 Basic intuition Two body current in electron scattering in the context of electron scattering the problem studied over 40 years access of the cross section in the DIP region between QE and peaks the extra strength is believed to come from the two-body current dynamics A. Gil, J. Nieves and E. Oset, Nucl. Phys. A 627 (1997) 543; 11 / 49

12 Models Two body current neutrino computations by now many computations: Martini et al model (Lyon), Nieves et al (IFIC), superscaling, relativistic Green function,... microscopic models vary in many details but they are based on (local) Fermi gas model and include contributions from the set (or a subset) of the Feynman diagrams shown on the left Also eective models were proposed (transverse enhancement, GiBUU, GENIE). J. Morn, JTS All the theoretical models provide predictions for nal state muon only. 12 / 49

13 Models Lyon model Lyon model a few details CCQE, excitation and np nh in one model two sources of np nh events self energy (see later, the same ingredient in other models as well) 2p-2h polarization propagators this contribution is not subject to RPA corrections; two functional forms are considered: from Marteau PhD thesis (they do not depend on momentum transfer) extrapolation of electron scattering results from the paper W. Alberico, M. Ericson, A. Molinari technically, only imaginary part of polarization propagators is known. 13 / 49

14 Models Lyon model Lyon model more details The key objects are nuclear responses dened as: R c = n A A < n τ j e i q r j 0 >< n τ j e i q r k 0 > δ(ω E n + E 0), j=1 k=1 R l = n A A < n ( σ j ˆq)τ j e i q r j 0 >< n ( σ j ˆq)τ j e i q r k 0 > δ(ω E n + E 0). j=1 Operators τ j, σ j ˆq arise in nonrelativistic expansion of weak current matrix elements. Nuclear responses (functions of energy and momentum transfer) can be represented as imaginary parts of polarization propagator: R c,l,t NN k=1 = Vol Im ( Π c,l,t NN π ). Π c NN = n ( ) < 0 τ j e i q r n >< n τ j e i q r 0 > < 0 τ je i q r n >< n τ j e i q r 0 > ω E n + E 0 + iɛ ω + E n E 0 iɛ 14 / 49

15 Models Lyon model Lyon model a few details Typical responses: Original Marteau responses: No dependence on q. 15 / 49

16 Models IFIC model IFIC model a few details In local density approximation inclusive cross section can be written as d 2 σ dω( k )de = G 2 F cos 2 θ C k 4π 2 E ν L µνw µν = 1 π I(LµνΠµν ) Π µν is intermediate boson self-energy: Π µν (q) = i d 4 xe iqx in spins d 3 rl µνw µν ( r), < in TJ ν (x)j µ (0) in >. On the right: Fermi gas model contribution to CCQE. There is only one Feynman diagram, nucleons do not interact. Nieves et al 16 / 49

17 Models IFIC model IFIC model a few details A consistent model for CCQE, pion production and MEC contribution. from Nieves et al 17 / 49

18 Models Comparison of Lyon and IFIC models Comparison of IFIC and Lyon models from Nieves et al 18 / 49

19 Models Comparison of Lyon and IFIC models Comparison of IFIC and Lyon models RPA from Nieves et al very similar treatment in both models RPA brings in a strong suppression at Q 2 0 (a factor of 0.6) and some enhancement for Q GeV / 49

20 Models self energy in-medium self-energy (1) Both Lyon and IFIC computations rely on Oset et al computation of self energy in nuclear matter once created can decay in a pionless way both N NN and NN NNN channels are possible Interestingly, this contribution to np-nh has been already in NEUT and NUANCE as π-less decays implemented as a fraction of excitation cross section. 20 / 49

21 Models self energy in-medium self-energy (2) Implementation of the Oset model is straightforward: explicit parameterizations are given ImΣ = C Q ( ρ ρ 0 ) α + C A2 ( ρ ρ 0 ) β + C A3 ( ρ ρ 0 ) γ, in two kinematical situations: pion scattering photon scattering (γ = 2β) This should be translated into ν scattering kinematics e.g. assuming that ImΣ is a function of Q / 49

22 Models self energy in-medium self-energy (3) Overall QE, SPP and pionless decay cross sections. from JTS, J.Zmuda Signicant pionless decay contribution and reduction of SPP cross section. Relativistic evaluation of self-energy was done, but only in innite nuclear matter M.J. Dekker, P.J. Brussaard, J.A. Tjon, Phys. Rev. C51 (1995) / 49

23 Models Superscaling Superscaling approach Based on remarkable scaling discovered in electron scattering. from M. Barbaro 23 / 49

24 Models Transverse enhancement Transverse enhancement model Based on e 12 C data, missing strength in the QE/ region is lled by redenition of magnetic form factors. A. Bodek A = , B = G n,p M (Q2 ) n,p G M (Q2 ) = G n,p M (Q2 ) 1 + AQ 2 exp( Q 2 /B), 24 / 49

25 Models Comparison Models comparison 25 / 49

26 Cross sections Two body current models cross sections Below, predictions from some models compared to CCQE cross section (both on carbon). 26 / 49

27 Cross sections Cross sections muon inclusive cross section is enough to calculate cross section also, it is sucient info to compare with the MiniBooNE CCQE data with the additional information from the MiniBooNE ν there is a lot of constraints on the models On the left: theory/data comparison for σ CCQE like ( ν µ 12 C) σ CCQE like (ν µ 12 C) theoretical predictions dier signicantly. from J. Grange 27 / 49

28 Cross sections Extension to higher energies Extension to higher energies The models suer from various limitations and cannot be simply extended to higher energies Lyon model is basically non-relativistic, but relativistic corrections were added IFIC model requires a cut at momentum transfer of q 1.2 GeV/c details: backup slides transverse enhancement model can be used at higher energies without problems this is why it has been used by MINERvA collaboration in their CCQE studies 28 / 49

29 Cross sections Extension to higher energies Cross sections 29 / 49

30 Monte Carlo implementation Monte Carlo implementations leptonic part Each model (muon part) is implemented using a dierent strategy IFIC model Nieves et al provide their code but it is very slow (multidimensional integrals) one can prepare tables for response functions (ve two-dimensional tables, details in backup slides) Lyon model the original code is not available NuWro relies on approximation worked out in JTS, a talk at NuInt02, arxiv:nucl-th/ transverse enhancement one subtracts CCQE cross section with standard form factors from the cross section with enhanced form factors dσ MEC dωdq = dσ CCQE dωdq ( G M (Q 2 )) dσ CCQE dωdq (G M (Q 2 )) 30 / 49

31 Monte Carlo implementation Hadronic model Monte Carlo implementation hadronic model Monte Carlo needs a model for nal state nucleons there is no theoretical computation of nal nucleon momenta the simplest choice is based on hadronic system phase space it guarantees that both nal state nucleons are on-shell T. Katori The same hadronic model will be combined with four theoretical models for two body current contribution to muon inclusive cross section. 31 / 49

32 Monte Carlo implementation Hadronic model Two body current nucleon model Only the muon information is used: muon's kinetic energy and production angle are known as an input equivalently, momentum and energy transfered to the hadronic system are known two/three nucleons are selected from the Fermi sea initial hadronic system is formed by adding all the four momenta boost to the hadronic center-of-mass frame (CMF) is performed in the hadronic CMF two/three nucleons are selected isotropically as a nal state conguration boost back to the laboratory frame is performed Pauli blocking may be checked energy balance must be consistent with the FSI (in NuWro Fermi energy and 7 MeV as a potential well is subtracted at the end of the cascade) events are weighted by muon double dierential cross section. 32 / 49

33 Monte Carlo implementation Isospin and momentum correlations Two body current nucleon model a need of exibility Isospin correlations: how many n-p and n-n in pairs participate in ν µ CC interactions? no obvious theoretical argument (see, however, the next slide a fraction of n-p pairs hould be a free parameter recommended values from 0.7 to 0.9 Momentum correlations: momenta of nucleon pairs participating in interaction? one option (following Marteau/Martini and Nieves models): momenta are selected at random from the Fermi sea another option: back-to-back momenta with a distribution with high momentum tail Location of primary interactions for FSI there are two nucleons P(r) ρ(r)? P(r) ρ(r) 2? 33 / 49

34 Monte Carlo implementation Isospin and momentum correlations IFIC model isospin predictions Microscopic model prediction for isospin of the initial state nucleons: R. Gran, J. Nieves, F. Sanchez, M.J. Vicente-Vacas The average fraction of p-n pairs is 67%. The pattern is rather dicult to implement. 34 / 49

35 Monte Carlo implementation Isospin and momentum correlations Nucleon correlations in IFIC and Lyon models the starting point is (local) Fermi gas model Fermi gas model ignores nucleon-nucleon correlations completely from Higinbotham 35 / 49

36 Beyond Fermi gas computations Beyond Fermi gas ground state computations results from J. Carlson, J. Jourdan, R. Schiavilla, I. Sick, Phys. Rev. C65 (2002) for electron scattering suggest that it is very important to consider a realistic ground state non-relativistic computations done for light nuclei: 3 H, 4 H and 6 Li in the language of Euclidean responses and sum rules almost all the enhancement of the strength due to two-body current comes from proton-neutron, and not from proton-proton or neutron-neutron pairs when ground state correlations are neglected (Fermi gas model) the extra strength due to two-body current contributions becomes very small. 36 / 49

37 Beyond Fermi gas computations Beyond Fermi gas ground state computations Two body current contribution to neutrino-deuteron scattering: G. Shen, L.E. Marcucci, J. Carlson, S. Gandol, and R. Schiavilla smaller than 10% ( 6 7%) enhancement due to two-body current contribution. 37 / 49

38 Interference Interference In Monte Carlo generator one adds CCQE, RES, DIS, MEC incoherently the same nal state can arise from dierent dynamical mechanism good example are two nucleon knock out nal states; they can arise from CCQE on uncorrelated nucleons and nal state interactions CCQE on SRC pairs accounted for in Benhar's spectral function CCQE interaction occurs on individual nucleons, correlated partners are spectators RES and pion absorption MEC There are indications that interference can be very important. Dashed line one body current only Solid line full computation Dotted line interference term from Benhar, Lovato, Rocco 38 / 49

39 Conclusions Message to take home there is a very important two body current contribution to the ν cross section interactions occur on two nucleons several models have been developed but there is no consensus on many details how two body events look like both the size of the cross section and characteristics of the hadronic nal states are not known suciently well. 39 / 49

40 Conclusions Back-up slides 40 / 49

41 Conclusions Introduction: large CCQE M A controversy M A = 1.35 GeV translates into much larger cross section than predictions from several theoretical models with much care to cover properly nuclear eects, with M A 1.05 GeV. G. Garvey 41 / 49

42 Conclusions Lyon model more details The basic cross section formula is d 2 σ dqdω = G F cos 2 θ C q 32πEν 2 L µνh µν, H µν = H µν NN + Hµν N + Hµν N + Hµν depending on whether weak current couples to nucleon or. In order to calculate H µν one performs non-relativistic expansion of weak current matrix elements. etc, etc. N = ( M+E 2M ū(p, s )γ 0 u(p, s) = NN ζ + ( σ p (1 ) )( σ p) s + (M + E ζ s, )(M + E) ( ū(p, s )γ 0 γ 5 u(p, s) = NN ζ + σ p s M + E + σ p ) ζ s, M + E ( ) ū(p, s )γ j u(p, s) = NN ζ + σ p s M + E σj + σ j σ p ζ s, M + E ) 1/2, ( N = M+E ) 1/2. 2M 42 / 49

43 Conclusions Lyon model more details It should be clear how spin operator appear. One gets: H 00 NN = M + M 2 + q 2 2M M + M 2 + q 2 2M ( ) α0α 0 0R 0 NN c + β0β 0 0R 0 NN l, q 2 α0 0 = F 1(ω, q) F 2(ω, q) 2M(M + E q) β0 0 q ( = G A (ω, q) ω ) M + E q 2M Gp(ω, q). F 1, F 2, G A and G p are standard form factors in the weak nucleon-nucleon transition matrix element. 43 / 49

44 Conclusions IFIC model extension to higher energies The model can be extended to energies up to 10 GeV R. Gran, J. Nieves, F. Sanchez, M.J. Vicente-Vacas, arxiv: [hep-ph] A cut q 1.2 GeV/c brings in a strong constraint on allowed values of cos θ µ. Q 2 = (k k ) 2 = m 2 + 2k k = m 2 + 2EE 2E k cos θ µ, cos θ µ = 2E(E ω) m2 q 2 + ω 2 2E (E ω) 2 m 2. Consider cos θ µ as a function of ω at xed q. One can easily nd a minimum at ω = E E 2 + m 2 q 2. After substituting back to cos θ µ one gets (cos θ µ) min = E 2 q 2 E 2. horizontal axis: neutrino energy, vertical axis: ( cos θ µ ) min 44 / 49

45 Conclusions Monte Carlo implementation of IFIC model It is natural to introduce nuclear response functions (structure functions). The formalism is universal and independent on dynamical mechanism. d 3 σ d 3 k = G 2 F (2π) 2 L µνw µν, E k E k L µν = k µk ν + k µk ν g µνk k iε µνκλ k κ k λ A general form of hadronic tensor is: W µν = g µνw 1 + pµpν M 2 W2 i ε µνκλp κ q λ 2M 2 W 3+ q µq ν M pµqν + pνqµ W M 2 pµqν pνqµ W 5 + i 2M 2 W 6, W j (structure functions) are functions of two independent scalars e.g. energy and momentum transfer ω, q 45 / 49

46 Conclusions Monte Carlo implementation of IFIC model disregarding intereference W j can be represented as sums of contributions from interaction modes: W j = W CCQE j + W SPP j + W MEC j +... W MEC j only contain sucient information to describe nal state muon 46 / 49

47 Conclusions Monte Carlo implementation of IFIC model q 1.2 GeV/c ω q it is straightforward to prepare ve precise enough tables for (ω, q ) W MEC j one set of tables contains sucient information for all neutrino and antineutrino energies and avors, for a given nucleus CC and NC reactions require another set of tables. 47 / 49

48 Conclusions Correlation eects Impact of correlation eects on number of proton pairs in the nal state: Isospin and momentum correlations are analyzed seperately. A possible confusion: In above gures correlations means initial state nucleon momenta are back-to-back. 48 / 49

49 Conclusions Universality of the correlated contribution k + 1 = P 0 and P 1 are probabilities to nd a mean eld or correlated nucleon in a range p k 1. M. Alvioli, C. Cio degli Atti, L.P. Kaptari, C.B. Mezzetti, H. Morita J. Arrington, D.W. Higinbotham, G. Rosner, M. Sargasian A very small dierence between 16 O and 40 Ca. 49 / 49