The nuclear EMC effect in the deep-inelastic and the resonance region Sergey Kulagin Institute for Nuclear Research of the Russian Academy of Sciences, Moscow Talk at the XVth International Seminar on Electromagnetic Interactions of Nuclei EMIN-2018 Moscow, Russia October 8, 2018
Outline ◮ Data overview on the nuclear EMC effect in deep-inelastic scattering region. ◮ Understanding and modelling nuclear corrections ◮ Sketch of basic mechanisms of nuclear DIS in different kinematic regions. ◮ Brief review of our efforts to build a quantitative model of nuclear structure functions. ◮ Nucleon and nuclear structure functions and nuclear ratios in the resonance and transition region and comparison with JLab data on D/ ( p + n ) BONuS, 2015 and 3 He /D and 3 He / ( D + p ) Hall C, 2009 ◮ Summary/Conclusions Kulagin (INR) 2 / 37
Data summary on nuclear effects on the parton level ◮ Nuclear ratios ℛ ( A/B ) = σ A ( x, Q 2 ) /σ B ( x, Q 2 ) or F A 2 /F B 2 from DIS experiments ◮ Data for nuclear targets from 2 H to 208 Pb ◮ Fixed-target experiments with e/µ : ◮ Muon beam at CERN (EMC, BCDMS, NMC) and FNAL (E665). ◮ Electron beam at SLAC (E139, E140), HERA (HERMES), JLab (E03-103). ◮ Kinematics and statistics: Data covers the region 10 − 4 < x < 1 . 5 and 0 < Q 2 < 150 GeV 2 . About 800 data points for the nuclear ratios ℛ ( A/B ) with Q 2 > 1 GeV 2 . ◮ Nuclear effects for antiquarks have been probed by Drell-Yan experiments at FNAL (E772, E866). ◮ Nuclear cross sections from high-energy measurements with neutrino BEBC ( 2 H and 20 Ne ), NOMAD ( 12 C and 56 Fe ) CDHS, CCFR and NuTeV ( 56 Fe ) CHORUS ( 207 Pb ). Nuclear cross section ratios Fe / CH and Fe / CH from MINERvA in the region of low Q 2 . Kulagin (INR) 3 / 37
Nuclear ratios from DIS experiments SLAC E139 SLAC E139 (Be) SLAC E139 (C) SLAC E139 1.4 CERN NMC CERN NMC (Li) CERN NMC (C) CERN NMC (Al/C)*(C/D) JLab E03103 JLab E03103 (Be) FNAL E665 (C) 1.3 DESY HERMES (N) JLab E03103 (C) 1.2 F 2 (A)/F 2 (D) 1.1 1 0.9 4 7 12 27 2 He 3 Li 6 C 13 Al 9 14 4 Be 7 N 0.8 0.7 1.4 SLAC E139 SLAC E139 (Fe) SLAC E139 (Ag) SLAC E139 (Au) CERN NMC CERN EMC (Cu) CERN NMC (Sn/C)*(C/D) CERN NMC (Pb/C)*(C/D) FNAL E665 CERN BCDMS (Fe) FNAL E665 (Xe) FNAL E665 (Pb) 1.3 DESY HERMES (Kr) 1.2 1.1 F 2 (A)/F 2 (D) 1 0.9 0.8 40 56 108 197 20 Ca 26 Fe 47 Ag 119 79 Au 208 63 29 Cu 50 Sn 131 82 Pb 0.7 84 36 Kr 54 Xe 0.6 10 -4 10 -3 10 -2 0.1 0.3 0.5 0.7 0.9 10 -3 10 -2 0.1 0.3 0.5 0.7 0.9 10 -3 10 -2 0.1 0.3 0.5 0.7 0.9 10 -3 10 -2 0.1 0.3 0.5 0.7 0.9 1 Bjorken x Bjorken x Bjorken x Bjorken x Kulagin (INR) 4 / 37
HERMES and JLab measurements on 3 He JLab E03-103 3 He/( 2 H+p) [D.Gaskell, private communication] 1.8 JLab E03-103 3 He is / 2 H [PRL103(2009)202301] DESY HERMES 3 He is / 2 H [PLB475(2000)386;567(2003)339(E)] 1.6 3% HERMES-JLab data offset 1.02 1 Cross section ratio 1.4 0.98 0.96 1.2 0.94 0.2 0.25 0.3 0.35 0.4 0.45 0.5 W < 1.8 GeV 1 0.8 0 0.2 0.4 0.6 0.8 1 Bjorken x Kulagin (INR) 5 / 37
SLAC E139 and JLab BONUS results on 2 H 0.45 JLab BONuS, W > 1.8 GeV 1.12 JLab BONuS, 1.2<W<1.8 GeV SLAC E139 1.1 0.4 1.08 F2D / (F2p + F2n) 1.06 0.35 F2n/F2D 1.04 1.02 0.3 1 0.98 0.25 0.96 BONuS data W > 1.8 GeV BONuS data 1.4 < W < 1.8 GeV 0.94 0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Bjorken x Bjorken x 2 / ( F p ◮ SLAC E139 [ PRD49(1994)4348 ] obtains R D = F D 2 + F n 2 ) by extrapolating data on R A = F A 2 /F D with A ≥ 4 assuming R A − 1 scales as nuclear density. 2 ◮ BONuS [ PRC92(2015)015211 ] obtains R D from a direct measurement of F n 2 /F D 2 2 /F p [ PRC89(2014)045206 ] using world data on F D 2 . Kulagin (INR) 6 / 37
Understanding and modelling the nuclear corrections Kulagin (INR) 7 / 37
Why nuclear corrections survive at DIS? Space-time scales in DIS ∫︂ d 4 x exp( iq · x ) ⟨ p | [ J µ ( x ) , J ν (0)] | p ⟩ W µν = 0 + Q 2 z ≃ q 0 ( t − z ) − Q 2 √︂ q · x = q 0 t − | q | z = q 0 t − q 2 2 q 0 z ◮ DIS proceeds near the light cone: | t − z | ∼ 1 /q 0 and t 2 − z 2 ∼ Q − 2 . ◮ In the TARGET REST frame the characteristic time and longitudinal distance are NOT small at all: t ∼ z ∼ 2 q 0 /Q 2 = 1 /Mx Bj . DIS proceeds at the distance ∼ 1 Fm at x Bj ∼ 0 . 2 and at the distance ∼ 20 Fm at x Bj ∼ 0 . 01 . ◮ Two different regions in nuclei from comparison of coherence length (Ioffe time) L = 1 /Mx Bj with average distance between bound nucleons r NN : ◮ L < r NN ( or x > 0 . 2) ⇒ Nuclear DIS ≈ incoherent sum of contributions from bound nucleons. Nuclear corrections ∼ EL and ∼ | p | 2 L 2 where E ( p ) typical energy (momentum) in the nuclear ground state. ◮ L ≫ r NN ( or x ≪ 0 . 2) ⇒ Coherent effects of interactions with a few nucleons are important. Kulagin (INR) 8 / 37
Incoherent nuclear scattering A good starting point is incoherent scattering off bound protons and neutrons ∫︂ F A d 4 p K ( 𝒬 p F p 2 + 𝒬 n F n 2 = 2 ) ◮ The four-momentum of the bound proton (neutron) p = ( M + ε, p ) ◮ 𝒬 p,n ( ε, p ) the proton (neutron) nuclear spectral function, which is normalized to d ε d p 𝒬 p = Z and describes probability to find a bound ∫︁ the nucleon number nucleon with momentum p and energy p 0 = M + ε . ◮ The bound nucleon structure functions depend on 3 independent variables ( x ′ , p 2 , Q 2 ) , x ′ = Q 2 / 2 p · q is the Bjorken variable of a nucleon with F p,n = F p,n 2 2 four-momentum p . Note the nucleon virtuality p 2 is additional variable for off-shell nucleon. ◮ Kinematical factor K = (1 + p z /M ) (︁ 1 + 𝒫 ( p 2 / | q | 2 ) )︁ . Kulagin (INR) 9 / 37
Nuclear spectral function The nuclear spectral function describes probability to find a bound nucleon with momentum p and energy p 0 = M + ε : ∫︂ d t e − iεt ⟨ ψ † ( p , t ) ψ ( p , 0) ⟩ 𝒬 ( ε, p ) = |⟨ ( A − 1) i , − p | ψ (0) | A ⟩| 2 2 πδ ∑︂ ε + E A − 1 ( p ) − E A (︁ )︁ = i 0 i ◮ The sum runs over all possible states of the spectrum of A − 1 residual system. ◮ The nuclear spectral function determines the rate of nucleon removal reactions such as ( e, e ′ p ) . For low separation energies and momenta, | ε | < 50 MeV, p < 250 MeV/c, the observed spectrum is dominated by bound states A − 1 similar predicted by the mean-field model. Kulagin (INR) 10 / 37
Sketch of the mean-field picture In the the mean-field model the bound states of A − 1 nucleus are described by the one-particle wave functions φ λ of the energy levels λ . The spectral function is given by the sum over the occupied levels with the occupied number n λ : ∑︂ n λ | φ λ ( p ) | 2 δ ( ε − ε λ ) 𝒬 MF ( ε, p ) = λ<λ F ◮ Due to interaction effects the δ -peaks corresponding to the single-particle levels acquire a finite width (fragmentation of deep-hole states). ◮ High-energy and high-momentum components of nuclear spectrum can not be described in the mean-field model and driven by short-range nucleon-nucleon correlation effects in the nuclear ground state as witnessed by numerous studies. Kulagin (INR) 11 / 37
High-momentum part ◮ As nuclear excitation energy becomes higher the mean-field model becomes less accurate. High-energy and high-momentum components of nuclear spectrum can not be described in the mean-field model and driven by correlation effects in nuclear ground state as witnessed by numerous studies. ◮ The corresponding contribution to the spectral function is driven by ( A − 1) * excited states with one or more nucleons in the continuum. Assuming the dominance of configurations with a correlated nucleon-nucleon pair and remaining A − 2 nucleons moving with low center-of-mass momentum we have | A − 1 , − p ⟩ ≈ ψ † ( p 1 ) | ( A − 2) * , p 2 ⟩ δ ( p 1 + p 2 + p ) . Kulagin (INR) 12 / 37
The matrix element can thus be given in terms of the wave function of the nucleon-nucleon pair embeded into nuclear environment. We assume factorization into relative and center-of-mass motion of the pair ⟨ ( A − 2) * , p 2 | ψ ( p 1 ) ψ ( p ) | A ⟩ ≈ C 2 ψ rel ( k ) ψ A − 2 CM ( p CM ) δ ( p 1 + p 2 + p ) , where ψ rel is the wave function of the relative motion in the nucleon-nucleon pair with relative momentum k = ( p − p 1 ) / 2 and ψ CM is the wave function of center-of-mass (CM) motion of the pair in the field of A − 2 nucleons, p CM = p 1 + p . The factor C 2 describes the weight of the two-nucleon correlated part in the full spectral function. ε + ( p + p A − 2 ) 2 ⟨ (︃ )︃⟩ 𝒬 cor ( ε, p ) ≈ n cor ( p ) δ + E A − 2 − E A 2 M A − 2 Kulagin (INR) 13 / 37
The two-component model of the spectral function In what follows we combine the mean-field together with SRC contributions and consider a two-component model Ciofi degli Atti & Simula, 1995 S.K. & Sidorov, 2000 S.K. & Petti, 2004 𝒬 = 𝒬 MF + 𝒬 cor We assume that the normalization is shared between the MF and the correlated parts as 0 . 8 to 0 . 2 for the nuclei A ≥ 4 [for 208 Pb 0 . 75 to 0 . 25 ] following the observations on occupation of deeply-bound proton levels NIKHEF 1990s, 2001 . Kulagin (INR) 14 / 37
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