Nuclear Binding and O ff -shell Corrections in the EMC E ff ect S. Kulagin INR Moscow, Russia R. Petti University of South Carolina, Columbia SC, USA ”Quantitative Challenges in EMC and SRC Research and Data-Mining” December 4th, 2016, MIT, Cambridge, MA, USA Roberto Petti USC
NUCLEAR MODEL ✦ GLOBAL APPROACH aiming to obtain a quantitative model covering the com- plete range of x and Q 2 ( S. Kulagin and R.P., NPA 765 (2006) 126; PRC 90 (2014) 045204 ): 1.2 ● Scale of nuclear processes (target frame) L I = ( Mx ) − 1 ANTISHADOWING Distance between nucleons d = (3 / 4 πρ ) 1 / 3 ∼ 1 . 2 Fm F 2 (A)/F 2 (D) 1.1 SHADOWING L I < d ● 1 For x > 0 . 2 nuclear DIS ∼ incoherent sum of contribu- tions from bound nucleons 0.9 L I � d NMC Ca/D ● EMC EFFECT 0.8 EMC Cu/D For x � 0 . 2 coherent e ff ects of interactions with few FERMI REGION E139 Fe/D nucleons are important 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Bjorken x ✦ DIFFERENT EFFECTS on parton distributions and structure functions included: q a/A = q p/A + q n/A + δ q MEC + δ q coh a = u, d, s..... a a a a ● q p ( n ) /A PDF in bound p(n) with Fermi Motion, Binding (FMB) and O ff -Shell e ff ect (OS) a ● δ q MEC nuclear Meson Exchange Current (MEC) correction a ● δ q coh contribution from coherent nuclear interactions: Nuclear Shadowing (NS) a Roberto Petti USC
INCOHERENT NUCLEAR SCATTERING ✦ FERMI MOTION AND BINDING in nuclear parton distributions can be calcu- lated from the convolution of nuclear spectral function and (bound) nucleon PDFs: q a/A ( x, Q 2 ) = q p/A + q n/A a a 1 + p z � � � xq p/A x ′ q N ( x ′ , Q 2 , p 2 ) d ε d 3 p P ( ε , p ) = a M where x ′ = Q 2 / (2 p · q ) and p = ( M + ε , p ) and we dropped 1 /Q 2 terms for illustration purpose . ✦ Since bound nucleons are there appears dependence on the OFF-MASS-SHELL nucleon virtuality p 2 = ( M + ε ) 2 − p 2 and expanding PDFs in the small ( p 2 − M 2 ) /M 2 : � 1 + δ f ( x )( p 2 − M 2 ) /M 2 � q a ( x, Q 2 , p 2 ) ≈ q N a ( x, Q 2 ) . where we introduced a structure function of the NUCLEON: δ f ( x ) ✦ Hadronic/nuclear input: ● Proton/neutron SFs computed in NNLO pQCD + TMC + HT from fits to DIS data ● Realistic nuclear spectral function: mean-field P MF ( ε , p ) + correlated part P cor ( ε , p ) Roberto Petti USC
� � � − � 1 + δ f ( x )( p 2 − M 2 ) /M 2 � F 2 ( x, Q 2 , p 2 ) ≈ F 2 ( x, Q 2 ) � . re OFF-MASS-SHELL 2 2 DESCRIPTION STRUCTURE FUNCTIONS OF NUCLEON � F 1 ( x, Q 2 ) , F 2 ( x, Q 2 ) , xF 3 ( x, Q 2 ) , ..... Distribution of partons in a nucleon � δ f ( x ) � � � � DESCRIPTION SPECTRAL/WAVE FUNCTION OF NUCLEUS P ( ε , p ) , Ψ ( p ) Distribution of bound nucleons ⇒ O ff -shell function measures the in-medium modification of bound nucleon Any isospin (i.e. δ f p ̸ = δ f n ) or flavor dependence ( δ f a ) in the o ff -shell function? Roberto Petti USC
NUCLEAR SPECTRAL FUNCTION ✦ The description of the nuclear properties is embedded into the nuclear spectral function ✦ Nucleons occupy energy levels according to Fermi statistics and are distributed over momentum (Fermi motion) and energy states. In the model: MEAN FIELD n λ | φ λ ( p ) | 2 δ ( ε − ε λ ) � P MF ( ε , p ) = λ < λ F where sum over occupied levels with n λ occupation number. Applicable for small nucleon separation energy and momenta, | ε | < 50 MeV, p < 300 MeV/c in nuclear ground state drive the high-energy and ✦ CORRELATION EFFECTS high-momentum component of the nuclear spectrum, when | ε | increases � � ε + ( p + p 2 ) 2 �� P cor ( ε , p ) ≈ n rel ( p ) + E A − 2 − E A δ 2 M CM Roberto Petti USC
IMPACT OF NN CORRELATIONS DIS Q 2 =5 GeV 2 1.2 KP model σ C / σ D 1.1 ✦ Impulse Approximation (IA) fails to quan- 1 titatively describe observed modifications 0.9 ✦ Instructive to drop P cor ( ε , p ) from spectral 1.2 σ Be / σ D function to estimate e ff ect of NN correla- KP model - IA tions 1.1 1 ✦ Significant change on structure functions 0.9 in clear disagreement with data indicates mean-field P MF ( ε , p ) alone not su ffi cient 1.2 σ 4He / σ D KP model - IA, MF only 1.1 = ⇒ Study NN correlations and refine 1 description of spectral function 0.9 0.8 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Bjorken x Roberto Petti USC
PREDICTIONS FOR CHARGED LEPTON DIS 1.2 1.2 E03103 * C is * 1.03 KP model E03103(is) σ C / σ D 1.15 HERMES(is) KP model * C is 1.1 1.1 KP model (IA) * C is 1.05 σ ( 3 He)/ σ (D) 1 1 0.95 0.9 0.9 C/D JLab E03-103 He3/D JLab, HERMES 0.85 1.2 σ Be / σ D KP model (IA) 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Bjorken x 1.1 1.1 σ N / σ D 1 1.05 1 0.9 0.95 Be/D JLab E03-103 N/D HERMES 0.9 1.2 σ 4He / σ D C/D NMC 0.85 JLab E03103 * 0.98 1.1 0.8 HERMES NMC C/D KP model 1.1 σ Kr / σ D 1 1.05 1 0.9 He4/D JLab E03-103 0.95 0.9 0.8 Kr/D HERMES 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.85 Bjorken x 0.8 S. Kulagin and R.P., PRC 82 (2010) 054614 0.75 -2 -1 10 10 Bjorken x Roberto Petti USC
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) DESY HERMES (N) KP model 1.3 KP model KP model JLab E03103 (C) KP model 1.2 1.1 D A /F 2 F 2 1 0.9 4 7 12 27 2 He 3 Li 6 C 13 Al 0.8 9 14 4 Be 7 N 0.7 SLAC E139 SLAC E139 (Fe) SLAC E139 (Ag) SLAC E139 (Au) 1.4 CERN NMC CERN EMC (Cu) CERN NMC (Sn/C)*(C/D) CERN NMC (Pb/C)*(C/D) KP model CERN BCDMS (Fe) KP model FNAL E665 (Pb) 1.3 DESY HERMES (Kr) KP model KP model 1.2 1.1 D A /F 2 1 F 2 0.9 0.8 40 56 108 197 20 Ca 26 Fe 47 Ag 79 Au 0.7 63 119 208 29 Cu 50 Sn 82 Pb 84 131 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 Bjorken x Bjorken x Bjorken x Bjorken x Roberto Petti USC
δ f ( x ) FROM A ≥ 4 NUCLEI AND DEUTERON 3 δ f(x) Kulagin-Petti 2.5 Global QCD fit to p, D (Paris w.f.) 2 1.5 1 0.5 0 -0.5 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x ✦ Precise determination of δ f ( x ) from RATIOS F A 2 /F B 2 from DIS o ff di ff erent nuclei, including SLAC, NMC, EMC, BCDMS, E665 data ( NPA 765 (2006) 126 ) ✦ Independent determination from global QCD fit to p and D data with DIS,DY, W ± /Z provides consistent results ( S. Alekhin, S. Kulagin and R.P., arXiv:1609.08463 [nucl-th] ) Roberto Petti USC
INTERPRETATION OF δ f ( x ) Valence quark distribution in a covariant diquark spectator model (see S.Kulagin et.al., PRC50(1994)1154 ) x ( p 2 � 1 − x ) s Z d k 2 C φ q val ( x, p 2 ) = k 2 / Λ 2 � / Λ 2 � I Assume a single-scale quark distribution over the virtuality k 2 . The model gives a resonable description of the nucleon valence distribution for x > 0 . 2 I O ff -shell nucleon: C ! C ( p 2 ) , Λ ! Λ ( p 2 ) . The function δ f = ∂ ln q val / ∂ ln p 2 depends on c = ∂ ln C/ ∂ ln p 2 and λ = ∂ ln Λ 2 / ∂ ln p 2 . I Tune c and λ to reproduce the node δ f ( x 0 ) = 0 and the slope δ f 0 ( x 0 ) of phenomenological o ff -shell function. We obtain λ ⇡ 1 and c ⇡ � 2 . 3 . I The positive parameter λ suggests smaller in-medium scale Λ or larger nucleon core size R c = Λ � 1 (“swelling” of a bound nucleon). δ Λ 2 2 λ h p 2 � M 2 i � δ R c = � 1 Λ 2 = � 1 � � R c 2 M 2 � in-medium 208 Pb : δ R c /R c ⇠ 10% Deuteron : δ R c /R c ⇠ 2% Roberto Petti USC
NUCLEAR EFFECTS IN RESONANCE REGION JLAB E03-103 (private comm. D. Gaskell) F 2 ratio Cross section ratio 1.8 1.6 σ ( 3 He)/ σ (D+p) 1.4 1.2 1 0.8 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Bjorken x ✦ Use Christy-Bosted SF parameterization for p and n in resonance region ✦ 3 He spectral function from exact Faddeev three-body calculation by Hannover group ( R.-W. Schulze and P. U. Sauer, Phys. Rev. C 48, (1993) 38 ) ✦ Apply nuclear corrections for 3 He/(D+p) as predicted from the DIS region to the cross-section in the resonance kinematics ⇒ Consistent treatment of nuclear e ff ects in DIS and resonance regions? = Roberto Petti USC
CONSTRAINTS FROM SUM RULES ✦ Nuclear meson correction constrained by light-cone momentum balance and equations of motion. ( S. Kulagin, NPA 500 (1989) 653; S. Kulagin and R.P., NPA 765 (2006) 126; PRC 90 (2014) 045204 ) ✦ At high Q 2 (PDF regime) coherent nu- Phenomenological cross section + Effective cross section σ 0 clear corrections controlled by the e ff ective 10 scattering amplitudes, which can be con- strained by normalization sum rules: σ (mb) δ N OS val + δ N coh val = 0 a 0 − → δ N OS + δ N coh = 0 a 1 − → 1 1 val = A − 1 � A where N A 0 dxq − 0 /A = 3 and 1 = A − 1 � A N A 0 dxq − 1 /A = ( Z − N ) /A 1 1 10 100 Q 2 (GeV 2 ) Solve numerically in terms of δ f and virtuality v = ( p 2 − M 2 ) /M 2 (input) and obtain the e ff ective cross-section in the ( I = 0 , C = 1 ) state, as well as Re/Im of amplitudes = ⇒ Nuclear corrections to PDFs largely controlled by P ( ε , p ) AND δ f ( x ) Roberto Petti USC
Recommend
More recommend
