A Light-Front approach to the 3He Spectral Function Sergio Scopetta - PowerPoint PPT Presentation

A Light-Front approach to the 3He Spectral Function Sergio Scopetta Dipartimento di Fisica e Geologia, Universit` a di Perugia and INFN, Sezione di Perugia, Italy in collaboration with Alessio Del Dotto – Universit` a di Roma Tre and INFN, Roma 3, Italy Leonid Kaptari – JINR, Dubna, Russia & Perugia Emanuele Pace – Universit` a di Roma “Tor Vergata” and INFN, Roma 2, Italy Matteo Rinaldi – Universit` a di Perugia and INFN, Sezione di Perugia, Italy Giovanni Salm` e – INFN, Roma 1, Italy A Light-Front approach to the 3He Spectral Function – p.1/47 March 9 th , 2015

This is an exciting Workshop for me... We studied the process A ( e, e ′ ( A − 1)) X many years ago q P X k 1 P A K = −k = P A−1 1 A−1 An old idea (Claudio among the first): in this process, in IA, d 2 σ A ∝ F N 2 ( x ) there is no convolution! (nucl-th/9609062) Example: through 3 He(e,e’d)X, check of the reaction mechanism (EMC effect); measuring 3 H(e,e’d)X, direct access to the neutron! new perspectives (loi to the JLab PAC, already in November 2010) A Light-Front approach to the 3He Spectral Function – p.2/47 March 9 th , 2015

This is an exciting Workshop for me... Later, I studied nuclear GPDs... Which of these pictures is more similar to a nuclear section? We should perform a tomography... It is possible! Coherent DVCS & GPDs Slow nuclear recoil detection is necessary... A Light-Front approach to the 3He Spectral Function – p.3/47 March 9 th , 2015

Outline Importance of the 3 He nucleus for fundamental studies in Hadronic Physics. In particular: the neutron information from 3 He. Crucial quantity: the (distorted) spectral function Recent theoretical developments in DVCS (M. Rinaldi, S.S. PRC 85, 062201(R) (2012); PRC 87, 035208 (2013)) and SiDIS studies (L. Kaptari, A. Del Dotto, E. Pace, G. Salm` e, S.S., PRC 89 (2014) 035206) Importance of a relativistic treatment for the description of the JLab program @ 12 GeV The LF spectral function of 3 He ( E. Pace, A. Del Dotto, M. Rinaldi, G. Salm` e, S.S., Few Body Syst. 54 (2013) 1079) (work in progress): preliminary results Conclusions A Light-Front approach to the 3He Spectral Function – p.4/47 March 9 th , 2015

Importance of 3 He for DIS structure studies 3 He is theoretically well known. Even a relativistic treatment may be implemented. 3 He has been used extensively as an effective neutron target, especially to unveil the spin content of the free neutron, due to its peculiar spin structure: In S − wave n p p n p p n p p � 3 He = � n ! 1 S S D (~ 90 % ) 3 He always promising when the neutron polarization properties have to be studied. To this aim, 3 He is unique and its spectral function arises in * DIS, together with 3 H, for the extraction of F n 2 (Marathon experiment, JLab); * polarized DIS, for the extration of the SSF g n 1 ; * polarized SiDIS, for the extraction of neutron transversity and related observables; * DVCS, for the extraction of neutron GPDs A Light-Front approach to the 3He Spectral Function – p.5/47 March 9 th , 2015

Example 1: DVCS off 3 He e’ γ ∆ e γ∗ q− For a J = 1 q 2 target, , in a hard-exclusive process, k+ ∆ k x+ ξ x− ξ ( Q 2 , ν → ∞ ) such as (coherent) DVCS P’ = P+ ∆ P (Definition of GPDs from X. Ji PRL 78 (97) 610) : ∆ = P ′ − P , q µ = ( q 0 , � q ) , and ¯ P = ( P + P ′ ) µ / 2 ξ = “skewness” = − ∆ + / (2 ¯ x = k + /P + ; P + ) x ≤ − ξ − → GPDs describe antiquarks ; − ξ ≤ x ≤ ξ − → GPDs describe q ¯ q pairs ; x ≥ ξ − → GPDs describe quarks the GPDs H q ( x, ξ, ∆ 2 ) and E q ( x, ξ, ∆ 2 ) are introduced: Z dλ 2 π e iλx � P ′ | ¯ ψ q ( λn/ 2) | P � = H q ( x, ξ, ∆ 2 ) ¯ U ( P ′ ) γ µ U ( P ) γ µ ψ q ( − λn/ 2) U ( P ′ ) iσ µν ∆ ν E q ( x, ξ, ∆ 2 ) ¯ + U ( P ) + ... 2 M A Light-Front approach to the 3He Spectral Function – p.6/47 March 9 th , 2015

Example 1: DVCS off 3 He e’ γ ∆ e γ∗ q− For a J = 1 q 2 target, , in a hard-exclusive process, k+ ∆ k x+ ξ x− ξ ( Q 2 , ν → ∞ ) such as (coherent) DVCS P’ = P+ ∆ P (Definition of GPDs from X. Ji PRL 78 (97) 610) : ∆ = P ′ − P , q µ = ( q 0 , � q ) , and ¯ P = ( P + P ′ ) µ / 2 ξ = “skewness” = − ∆ + / (2 ¯ x = k + /P + ; P + ) x ≤ − ξ − → GPDs describe antiquarks ; − ξ ≤ x ≤ ξ − → GPDs describe q ¯ q pairs ; x ≥ ξ − → GPDs describe quarks H q ( x, ξ, ∆ 2 ) and ˜ ˜ E q ( x, ξ, ∆ 2 ) , obtained as follows: and the helicity dependent ones, Z dλ 2 π e iλx � P ′ | ¯ ψ q ( λn/ 2) | P � = ˜ H q ( x, ξ, ∆ 2 ) ¯ U ( P ′ ) γ µ U ( P ) γ µ γ 5 ψ q ( − λn/ 2) U ( P ′ ) γ 5 ∆ µ E q ( x, ξ, ∆ 2 ) ¯ ˜ + 2 M U ( P ) + ... A Light-Front approach to the 3He Spectral Function – p.6/47 March 9 th , 2015

GPDs of 3 He: the Impulse Approximation e’ γ e ∆ ∗ q− γ coherent DVCS in I.A. q , ( 3 He does not break-up, ∆ 2 ≪ M 2 , ξ 2 ≪ 1 , ): ∆ k k+ ∆ p p’=p+ ∆ p = ( p + p ′ ) / 2 ) : P P’=P + In a symmetric frame ( ¯ P R P + = ( x ′ + ξ ′ )¯ p + , ( x + ξ ) ¯ k + = P + = ( x ′ − ξ ′ )¯ p + , ( k + ∆) + ( x − ξ ) ¯ = one has, for a given GPD, H q , ˜ G q M = H q + E q , or ˜ H q Z dz − X 4 π e ix ¯ P + z − A � P ′ S ′ | ˆ GPD q ( x, ξ, ∆ 2 ) ≃ O µ,N | PS � A | z + =0 ,z ⊥ =0 . q N A Light-Front approach to the 3He Spectral Function – p.7/47 March 9 th , 2015

GPDs of 3 He: the Impulse Approximation e’ γ e ∆ ∗ q− γ coherent DVCS in I.A. q , ( 3 He does not break-up, ∆ 2 ≪ M 2 , ξ 2 ≪ 1 , ): ∆ k k+ ∆ p p’=p+ ∆ p = ( p + p ′ ) / 2 ) : P P’=P + In a symmetric frame ( ¯ P R P + = ( x ′ + ξ ′ )¯ p + , ( x + ξ ) ¯ k + = P + = ( x ′ − ξ ′ )¯ p + , ( k + ∆) + ( x − ξ ) ¯ = one has, for a given GPD, H q , ˜ G q M = H q + E q , or ˜ H q Z dz − X 4 π e ix ¯ P + z − A � P ′ S ′ | ˆ GPD q ( x, ξ, ∆ 2 ) ≃ O µ,N | PS � A | z + =0 ,z ⊥ =0 . q N By properly inserting a tensor product complete basis for the nucleon (PW) and the fully interacting recoiling system : A Light-Front approach to the 3He Spectral Function – p.7/47 March 9 th , 2015

GPDs of 3 He: the Impulse Approximation e’ γ e ∆ ∗ q− γ coherent DVCS in I.A. q , ( 3 He does not break-up, ∆ 2 ≪ M 2 , ξ 2 ≪ 1 , ): ∆ k k+ ∆ p p’=p+ ∆ p = ( p + p ′ ) / 2 ) : P P’=P + In a symmetric frame ( ¯ P R P + = ( x ′ + ξ ′ )¯ p + , ( x + ξ ) ¯ k + = P + = ( x ′ − ξ ′ )¯ p + , ( k + ∆) + ( x − ξ ) ¯ = one has, for a given GPD, H q , ˜ G q M = H q + E q , or ˜ H q Z X X dz − 4 π e ix ′ ¯ R , Φ f ′ p + z − � P ′ S ′ | GPD q ( x, ξ, ∆ 2 ) ≃ {| P ′ A − 1 � ⊗ | p ′ s ′ �} N � P ′ R ,f ′ p ′ ,s ′ A − 1 ,� X R , Φ f ′ {| P R , Φ f A − 1 � ⊗ | ps �}{� P R , Φ f A − 1 | ⊗ � p ′ s ′ | ˆ � P ′ O µ,N A − 1 | ⊗ � ps |} | PS � , q � P R ,f A − 1 ,� p,s {� P R , Φ f p ) � Φ f A − 1 | ⊗ � ps |}| PS � = (2 π ) 3 δ 3 ( � P − � and, since P R − � A − 1 , ps | PS � , (NR! Separation of the global motion from the intrinsic one!) A Light-Front approach to the 3He Spectral Function – p.7/47 March 9 th , 2015

GPDs of 3 He in IA: the spectral function H A q can be obtained in terms of H N q ( S.S. PRC 70, 015205 (2004), PRC 79, 025207 (2009) ): Z Z X X X p ′ , E ) ξ ′ H A q ( x, ξ, ∆ 2 ) = P N ξ H N q ( x ′ , ∆ 2 , ξ ′ ) , dE d� p MM ,ss ( � p, � s N M G 3 ,q G N,q ˜ M in terms of ˜ ( M. Rinaldi, S.S. PRC 85, 062201(R) (2012); PRC 87, 035208 (2013) ): M Z Z h i X p ′ , E ) ξ ′ G 3 ,q G N,q ˜ M ( x, ∆ 2 , ξ ) = P N + − , + − − P N ˜ M ( x ′ , ∆ 2 , ξ ′ ) , dE d� p ( � p, � + − , − + ξ N and ˜ q can be obtained in terms of ˜ H A H N q : Z Z h i X p ′ , E ) ξ ′ ˜ ˜ H A q ( x, ξ, ∆ 2 ) = P N ++ , ++ − P N H N q ( x ′ , ∆ 2 , ξ ′ ) , dE d� p ( � p, � ++ , −− ξ N A Light-Front approach to the 3He Spectral Function – p.8/47 March 9 th , 2015

GPDs of 3 He in IA: the spectral function H A q can be obtained in terms of H N q ( S.S. PRC 70, 015205 (2004), PRC 79, 025207 (2009) ): Z Z X X X p ′ , E ) ξ ′ H A q ( x, ξ, ∆ 2 ) = P N ξ H N q ( x ′ , ∆ 2 , ξ ′ ) , dE d� p MM ,ss ( � p, � s N M G 3 ,q G N,q ˜ M in terms of ˜ ( M. Rinaldi, S.S. PRC 85, 062201(R) (2012); PRC 87, 035208 (2013) ): M Z Z h i X p ′ , E ) ξ ′ G 3 ,q G N,q ˜ M ( x, ∆ 2 , ξ ) = P N + − , + − − P N ˜ M ( x ′ , ∆ 2 , ξ ′ ) , dE d� p ( � p, � + − , − + ξ N where P N p ′ , E ) is the one-body, spin-dependent, off-diagonal spectral M ′ M ,s ′ s ( � p, � function for the nucleon N in the nucleus, X P N p ′ , E ) = M ′ M σ ′ σ ( � p, � δ ( E − E A − 1 + E A ) f A − 1 � φ f A − 1 ; σ ′ � p ′ | π A J A M ′ ; Ψ A � S A S A � Ψ A ; J A M π A | � p, σ ; φ f A − 1 � | {z } | {z } տ intrinsic overlaps ր A Light-Front approach to the 3He Spectral Function – p.8/47 March 9 th , 2015