EMC Theory: The Polarized EMC effect Ian Cloët Argonne National Laboratory Quantitative challenges in EMC and SRC Research and Data-Mining Massachusetts Institute of Technology 2 – 5 December 2016
Understanding the EMC effect The puzzle posed by the EMC effect will only be solved by conducting new experiments that expose novel aspects of the EMC effect Measurements should help distinguish between explanations of EMC effect e.g. whether all nucleons are modified by the medium or only those in SRCs Important examples are: EMC effect in polarized structure functions flavour dependence of EMC effect JLab has an approved experiment to measure the spin structure of 7 Li I. Sick and D. Day, Phys. Lett. B 274, 16 (1992). Z/N = 82 / 126 (lead) 1 . 2 EMC effect 1 . 1 Polarized EMC effect 1 . 1 1 EMC ratios EMC ratios 1 0 . 9 0 . 9 0 . 8 0 . 8 F 2 A /F 2 D 0 . 7 0 . 7 Q 2 = 5 GeV 2 d A /d f Q 2 = 5 GeV 2 ρ = 0.16 fm − 3 u A /u f 0 . 6 0 . 6 0 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 x x table of contents Quantitative challenges in EMC and SRC 2–5 December 2016 2 / 15
Theory approaches to EMC effect To address the EMC effect must determine nuclear quark distributions: � dξ − q A ( x A ) = P + 2 π e iP + x A ξ − /A � A, P | ψ q (0) γ + ψ q ( ξ − ) | A, P � A Common to approximate using convolution formalism � A � 1 � q A ( x A ) = dy A dx δ ( x A − y A x ) f α,κ ( y A ) q α,κ ( x ) 0 0 α,κ α = (bound) protons, neutrons, pions, deltas. . . . neutrons protons 28 Si d 5 / 2 ( κ = − 3) p 1 / 2 ( κ = 1) 16 O p 3 / 2 ( κ = − 2) 12 C 4 He s 1 / 2 ( κ = − 1) table of contents Quantitative challenges in EMC and SRC 2–5 December 2016 3 / 15
Theory approaches to EMC effect To address the EMC effect must determine nuclear quark distributions: � dξ − q A ( x A ) = P + 2 π e iP + x A ξ − /A � A, P | ψ q (0) γ + ψ q ( ξ − ) | A, P � A Common to approximate using convolution formalism � A � 1 � q A ( x A ) = dy A dx δ ( x A − y A x ) f α,κ ( y A ) q α,κ ( x ) 0 0 α,κ α = (bound) protons, neutrons, pions, deltas. . . . q α ( x ) light-cone distribution of quarks q in bound hadron α f α ( y A ) light-cone distribution of hadrons α in nucleus k + q k + q k k k k p p A − 1 P P table of contents Quantitative challenges in EMC and SRC 2–5 December 2016 3 / 15
Sum Rules and Convolution Formalism Recall convolution model: � A � 1 � q A ( x A ) = dy A dx δ ( x A − y A x ) f α ( y A ) q α ( x ) 0 0 α All credible explanations of the EMC effect must satisfy baryon number and momentum sum rules: � A � A dx A u − A ( x A ) = 2 Z + N, dx A d − A ( x A ) = Z + 2 N, 0 0 � A � � u + A ( x A ) + d + dx A x A A ( x A ) + . . . + g A ( x A ) = Z + N = A, 0 In convolution formalism these sum rules imply � A � A � � n α dy A f α ( y A ) = A dy A y A f α ( y A ) = A B 0 0 α α quark distributions q α ( x ) should satisfy baryon number and momentum sum rules for hadron α table of contents Quantitative challenges in EMC and SRC 2–5 December 2016 4 / 15
Nuclear Wave Functions 3 . 5 10 6 VMC 12 C 12 C VMC 3 . 0 Shell Model Shell Model ρ ( p ) [GeV − 3 ] 10 3 2 . 5 f A ( y A ) /A 2 . 0 1 1 . 5 10 − 3 1 . 0 0 . 5 10 − 6 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 0 0 . 5 1 1 . 5 2 p [GeV] y A Modern GFMC or VMC nucleon 2 H 1 . 4 momentum distributions have 4 He 1 . 3 significant high momentum tails EMC ratios 12 C 12 C - Gomez 1 . 2 indicates momentum distributions contain SRCs: ∼ 20% for 12 C 1 . 1 1 . 0 Light cone momentum distribution 0 . 9 of nucleons in nucleus is given by 0 . 8 � d 3 � 0 0 . 2 0 . 4 0 . 6 0 . 8 1 p � � y A − p + x f ( y A ) = (2 π ) 3 δ ρ ( p ) P + table of contents Quantitative challenges in EMC and SRC 2–5 December 2016 5 / 15
Quarks, Nuclei and the NJL model “integrate out gluons” 1 g Θ(Λ 2 − k 2 ) Continuum QCD ➞ m 2 this is just a modern interpretation of the Nambu–Jona-Lasinio (NJL) model model is a Lagrangian based covariant QFT, exhibits dynamical chiral symmetry breaking & quark confinement; elements can be QCD motivated via the DSEs Quark confinement is implemented via proper-time regularization p − m + iε ] − 1 Z ( p 2 )[ / p − M + iε ] − 1 [ / quark propagator: ➞ wave function renormalization vanishes at quark mass-shell: Z ( p 2 = M 2 ) = 0 confinement is critical for our description of nuclei and nuclear matter 9 NJL NJL 0 . 4 8 DSEs – ω = 0 . 6 DSEs 7 M ( p ) [GeV] S. x. Qin et al. , Phys. Rev. C 84 , 042202 (2011) 0 . 3 π α eff ( k 2 ) 6 5 0 . 2 4 1 3 0 . 1 2 1 0 0 0 0 . 5 1 . 0 1 . 5 2 . 0 0 0 . 5 1 . 0 1 . 5 2 . 0 p [GeV] k [GeV] table of contents Quantitative challenges in EMC and SRC 2–5 December 2016 6 / 15
Nucleon Electromagnetic Form Factors Nucleon = quark+diquark Form factors given by Feynman diagrams: 1 q 2 P + k 1 2 P + k = P P + 1 2 P − k q 2 P − k 1 p ′ p p ′ p Calculation satisfies electromagnetic gauge invariance; includes dressed quark–photon vertex with ρ and ω contributions contributions from a pion cloud [ICC, W. Bentz and A. W. Thomas, Phys. Rev. C 90 , 045202 (2014)] 1 . 0 1 . 8 empirical – Kelly empirical – Kelly 1 . 6 0 . 8 1 . 4 1 . 2 F 1 p ( Q 2 ) F 2 p ( Q 2 ) 0 . 6 1 . 0 0 . 8 0 . 4 0 . 6 0 . 4 0 . 2 0 . 2 0 0 0 1 2 3 4 5 0 1 2 3 4 5 Q 2 (GeV 2 ) Q 2 (GeV 2 ) table of contents Quantitative challenges in EMC and SRC 2–5 December 2016 7 / 15
Nucleon Electromagnetic Form Factors Nucleon = quark+diquark Form factors given by Feynman diagrams: 1 q 2 P + k 1 2 P + k = P P + 1 2 P − k q 2 P − k 1 p ′ p p ′ p Calculation satisfies electromagnetic gauge invariance; includes dressed quark–photon vertex with ρ and ω contributions contributions from a pion cloud [ICC, W. Bentz and A. W. Thomas, Phys. Rev. C 90 , 045202 (2014)] 0 0 − 0 . 2 − 0 . 4 − 0 . 6 F 1 n ( Q 2 ) F 2 n ( Q 2 ) − 0 . 8 − 0 . 1 − 1 . 0 − 1 . 2 − 1 . 4 − 1 . 6 empirical – Kelly empirical – Kelly − 1 . 8 − 0 . 2 0 1 2 3 4 5 0 1 2 3 4 5 Q 2 (GeV 2 ) Q 2 (GeV 2 ) table of contents Quantitative challenges in EMC and SRC 2–5 December 2016 7 / 15
Nucleon quark distributions PDFs given by Feynman diagrams: � γ + � Nucleon = quark+diquark 1 2 P + k 1 2 P + k + = P P P P P P 2 P − k 1 2 P − k 1 Covariant, correct support; satisfies sum rules, Soffer bound & positivity � q ( x ) − ¯ q ( x ) � = N q , � x u ( x ) + x d ( x ) + . . . � = 1 , | ∆ q ( x ) | , | ∆ T q ( x ) | � q ( x ) 1 . 6 Q 2 0 = 0 . 16 GeV 2 Q 2 0 = 0 . 16 GeV 2 x ∆ d v ( x ) and x ∆ u v ( x ) 0 . 8 Q 2 = 5 . 0 GeV 2 x d v ( x ) and x u v ( x ) Q 2 = 5 . 0 GeV 2 MRST (5.0 GeV 2 ) 1 . 2 AAC 0 . 6 0 . 4 0 . 8 0 . 2 0 . 4 0 − 0 . 2 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 x x [ICC, W. Bentz and A. W. Thomas, Phys. Lett. B 621 , 246 (2005)] table of contents Quantitative challenges in EMC and SRC 2–5 December 2016 8 / 15
NJL at Finite Density Finite density (mean-field) Lagrangian: ¯ qq interaction in σ, ω, ρ channels L = ψ q ( i � ∂ − M ∗ − � V q ) ψ q + L ′ I Fundamental physics – mean fields couple to the quarks in nucleons 1 . 2 12 M M a 8 1 . 0 M s M N 4 Masses [GeV] E B /A [MeV] 0 . 8 0 0 . 6 − 4 Z /N = 0 0 . 4 − 8 Z/N = 0 . 1 Z/N = 0 . 2 0 . 2 − 12 Z/N = 0 . 5 Z/N = 1 − 16 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 ρ [fm − 3 ] ρ [fm − 3 ] S ( k ) − 1 = / k − M + iε ➞ S q ( k ) − 1 = / k − M ∗ − / Quark propagator: V q + iε Hadronization + mean–field = ⇒ effective potential (solve self-consistently) ω 2 ρ 2 E = E V + E p + E n − 4 G ω − 0 0 4 G ρ E V = vacuum energy E p ( n ) = energy of nucleons moving in σ, ω, ρ mean-fields table of contents Quantitative challenges in EMC and SRC 2–5 December 2016 9 / 15
EMC and Polarized EMC effects [ICC, W. Bentz and A. W. Thomas, Phys. Rev. Lett. 95 , 052302 (2005)] [J. R. Smith and G. A. Miller, Phys. Rev. C 72 , 022203(R) (2005)] 1 . 3 I. Sick and D. Day, Phys. Lett. B 274, 16 (1992). 1 . 2 EMC effect polarized EMC effect 1 . 1 EMC ratios 1 0 . 9 0 . 8 0 . 7 Q 2 = 5 GeV 2 = 0.16 fm − 3 ρ 0 . 6 0 0 . 2 0 . 4 0 . 6 0 . 8 1 x ∆ R = g 1 A g 1 A Definition of polarized EMC effect: = g naive P p g 1 p + P n g 1 n 1 A ratio equals unity if no medium effects Large polarized EMC effect arises because in-medium quarks are more relativistic ( M ∗ < M ) lower components of quark wave functions are enhanced and these usually have larger orbital angular momentum in-medium we find that quark spin is converted to orbital angular momentum A large polarized EMC effect would be difficult to accommodate within traditional nuclear physics and most other explanations of the EMC effect table of contents Quantitative challenges in EMC and SRC 2–5 December 2016 10 / 15
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