OAM Peripheral Transverse Densities from Chiral Dynamics. Carlos Granados in collaboration with Christian Weiss. QCD Evolution 2017.

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1 OAM Peripheral Transverse Densities from Chiral Dynamics Carlos Granados in collaboration with Christian Weiss Jefferson Lab QCD Evolution 2017 Jefferson Lab Newport News, VA

2 Outline Motivation Transverse Densities Definitions, properties Orbital angular momentum in impact parameter space Spectral Functions, ChPT calculation Light front formulation Wave functions and χgpds Transverse densities of orbital angular momentum

3 Motivation General Complement and extend current understanding of the structure of nucleons, at large distance scales Identify the dynamical picture defining structure, Identify contributions to the nucleon s intrinsic properties from relevant constituents Test the practical reach of fundamental theories Focus on peripheral orbital angular momentum, Define and probe OAM distributions. Use transverse densities Provide a mechanical picture of OAM distributions. Light-front formulation of dynamics in the chiral periphery.

4 Transverse Densities y ρ(b) b x P N z Connect Form Factors and GPDs to nucleon intrinsic spacial structure F ( 2 T ) = d 2 be i T b ρ(b) ρ 1(b) = dx d 2 T (2π) 2 e i T b H(x, 2 T ) Molecular region b~m 3 N M 2 Chiral periphery b~m 1 Non chiral core G.A. Miller,ARNPS 60 (2010) M.Burkardt,PRD62(2000) Boost invariants; structure of the nucleon as a relativistic multiparticle systems. Nucleon transverse profile; Define dynamical regions in impact parameter space; e.g., non chiral, chiral [Mπ 1 ], and molecular [MN 2 M 3 π ] M.Strikman, C.Weiss PRC82(2010) CG, C.Weiss JHEP 1401 (2014)

5 Transverse density of orbital angular momentum From form factors of energy-momentum tensor [ N 2 T µν(0) N 1 = ū 2 A( 2 )γ (µ p ν) + B( 2 α )p (µ iσ ν)α 2M N ( ) ] µ +C( 2 ν 2 g µν ) + C( 2 )M N g µν u 1, M N and the condition, d 2 bρ l (b) = A(0) + B(0) = 1. Various alternatives, or M. V. Polyakov, PLB 555, 57 (2003) L. Adhikari and M. Burkardt, PRD 94, no. 11, (2016) ρ J (b) GP = 1 [ (ρ A (b) + ρ B (b)) b ] 3 b (ρ A(b) + ρ B (b)), ρ J (b) IMF 1 2 b d db (ρ A(b) + ρ B (b)) ρ J (b) GP not from a relativistic interpretation, and ρ J (b) IMF fails to match Jaffes partonic definition (quaks and gluons). We look at ρj (b) IMF in the Chiral periphery!

6 EMT form factors [ N 2 T µν(0) N 1 = ū 2 A( 2 )γ (µ p ν) + B( 2 α )p (µ iσ ν)α 2M N ( ) ] µ +C( 2 ν 2 g µν ) + C( 2 )M N g µν u 1, M N Working in the + = 0 reference frame, one can write the EMT form factors in terms of matrix elements of light front components of the EMT. Solving for the form factors A and B one has, A( 2 T ) B( 2 T ) = 1 2(p + ) 2 while for the combination A + B, N 2 T ++ (0) N 1 σ 1 σ 2 2M N 2 T 1 2 δ(σ 1, σ 2 ) ( i)( T e z ) S T (σ 1, σ 2 ) A( 2 T ) + B( 2 T ) = 4 i( T e z ) 2p + 2 N 2 T +T (0) N 1 S z (σ 1, σ 2 ) T σ 1 σ 2.

7 Transverse density of orbital angular momentum Study longitudinal component Matrix elements One can then identify the quantity, L z (b) = 1 2p + b T +T (b) e z, L z (b) = Sz 2 b d db (ρ A(b) + ρ B (b)). ρ lz (b) 1 2 b d db (ρ A(b) + ρ B (b)) as a transverse density of longitudinal orbital angular momentum. It satisfies the condition, d 2 bρ lz (b) = A(0) + B(0).

8 TD from spectral functions in χpt Filter high momentum contributions dt ρ(b) = 4Mπ 2 2π K 0( ImF (t + i0) tb) π FF from leading order χpt (a) N ρ V 1, 2 (b) exp (2 M π b) [M 2 π ] π π N b [M π 1 ] ρ 1 ρ 2 (b) M.Strikman, C.Weiss PRC82(2010) CG, C.Weiss JHEP 1401 (2014) ρ V 1 (b) [M 2 π ] b [M π 1 ] chiral ρ pole L πn = g A F π ψγ µ γ 5 τ ψ µπ 1 ψγ µ τ ψ π µπ 4Fπ 2 subtreshold singularity (on-shell intermediate nucleon)molecular region convergent HB expansions in chiral region chiral component becomes dominant for b> 2fm

9 Charge and magnetization densities from LF dynamics C.G, C. Weiss, JHEP 1507, 170 (2015) ρ (b) exp (2 M π b) [M π 2 ] ρ 1 V ~ ρ 2 V Light front current matrix as wavefunction overlap, J (b) 2p = 1 ( dy + 2π yȳ Φ y, b ) ( Φ y, b ), ȳ ȳ then in terms of radial functions ρ V 1 (b) ρ V 2 (b) = 1 2π dy yȳ 3 [U 0 (y, b/ȳ)] 2 + [U 1 (y, b/ȳ)] 2 2 U 0 (y, b/ȳ) U 1 (y, b/ȳ) Inequality and positive definiteness of light front current, ρ 1 (b) > ρ 2 (b) J + (b) > 0 weakly bound pions. Near parametric equality, b [M π ] ρ 1 (b) ρ 2 (b) relativistic pion-nucleon system Explain through left right asymmetry in Transverse densities. CG, C. Weiss, PRC 92, no. 2, (2015)

10 Light front current matrix as wavefunction overlap, J (b) 2p = 1 + 2π ( dy yȳ Φ y, b ) ( Φ y, b ), ȳ ȳ then in terms of radial functions ρ V 1 (b) ρ V 2 (b) dy yȳ 3 [U 0 (y, b/ȳ)] 2 + [U 1 (y, b/ȳ)] 2 2 U 0 (y, b/ȳ) U 1 (y, b/ȳ) Integrands correspond to Fourier trans. of GPDs, H(y, t) and E(y, t), i.e., f1(y, b), f2(y, b) respectively, f V 1 (y, b) f V 2 (y, b) 1 2π yȳ 3. [U0(y, b/ȳ)] 2 + [U1(y, b/ȳ)] 2 2U0(y, b/ȳ)u1(y, b/ȳ). yp N P N Charge magnetization densities and χgpds f(y,b) b = 1 2π = 1

11 Energy momentum tensor and OAM in χpt T μν Use L eff and k 2 k 1 T µν = Tr ( µπ νπ 12 ) gµν( σπ)2 p 2 l p 1 to obtain N 2 Tµν Nπ (0) N 1 = 3 g 2 A d 4 l [ iū2 4 Fπ 2 (2π) 4 /k 2 γ 5 (/l + M N )/k 1 γ 5 u 1 D N (l) (k 2µ k 1ν + k 1µ k 2ν g µνk 2 k 1 )D π(k 2 )D π(k 1 )]. From imaginary part of T πn, 1 π Im(A(t) + B(t)) = 3 g 2 A M 2 (t/2 M 2 π )3 4 Fπ 2 N (4π) 2 P 25 t ( 2 ( ) ) 3 x 3 + x x arctan(x) (1)

12 Energy momentum tensor and OAM in χpt T μν Use L eff and k 2 k 1 T µν = Tr ( µπ νπ 12 ) gµν( σπ)2 p 2 l p 1 to obtain N 2 Tµν Nπ (0) N 1 = 3 g 2 A d 4 l [ iū2 4 Fπ 2 (2π) 4 /k 2 γ 5 (/l + M N )/k 1 γ 5 u 1 D N (l) (k 2µ k 1ν + k 1µ k 2ν g µνk 2 k 1 )D π(k 2 )D π(k 1 )]. In LF variables (v ± = v 0 ± v z ), N T ++ Nπ (0) 2 2p + N1 N 2 T +T Nπ (0) 2p + N1 = 3 4 p+ dy d 2 ( ) ( ) k T yȳ (2π) 3 yψ y, k T + ȳ T 2 ; Ψ y, k T ȳ T 2, 3 = dy d 2 ( ) ( ) k T 4 yȳ (2π) 3 k T Ψ y, k T + ȳ T 2 ; Ψ y, k T ȳ T 2

13 Light front wave functions Eigenfunctions of LF Hamiltonian. L = 1 y x z Allow quantum mechanical description of peripheral dynamics. Computable at leading order in chiral periphery C.G, C. Weiss, JHEP 1507, 170 (2015) v ~ 1 Ψ(y, k T = k T + yp 1T ) Γ(y, k T ) M 2 (y, k T ) }{{} Inv. Mass difference while in transverse coordinate space, d 2 kt Φ(y, r T ) = e k T r T Ψ(y, kt (2π) 2 ) [ = 2i U 0 (y, r T )S z + i U ] 1(y, r T )r T S T r T Γ(y, k T ) g AM N ū(y, k T )iγ 5 u(p 1T ) F π 2ig A M 2 [ ] N k T S T = ȳ ys z + F π M N with radial functions, } U 0 (y, r T ) U 1 (y, r T ) = g AM N y ȳ 2πF π and transverse mass M 2 T = ȳ 2 M π + y 2 M 2 N { ymn K 0 (M T r T ) M T K 1 (M T r T ),

14 OAM in Chiral Periphery Transverse densities from spectral functions. dt tb ρ lz) = 4Mπ 2 2π 2 K 1( Im[A + B](t + i0) tb) π In the LF-formulation ρ lz (b) = π [ ( dy yȳ ȳtr S 3 z Φ (y, r T ) r T ( i) ) ] Φ(y, r T ) rt =b/ȳ, r T Validates a probabilistic quantum-mechanical interpretation of ρ lz (b) as a density of OAM in the nucleon s periphery! This cannot be achieved with other definitions of ρ lz (b)

15 Summary and Outlook Transverse densities computed in Chiral EFT were used in a model independent approach to quantify the distribution of orbital angular momentum in the periphery of the nucleon. By using LF variables, a propose density of angular momentum is written as a proper density in LF-quantum mechanics of a pion-nucleon fluctuation of a nucleon. This framework opens the possibility of fully exploring the role that chiral dynamics plays constraining the nucleons internal structure. Transverse densities associated with form factors of the energy momentum tensor.distributions of matter in impact parameter space. Expand on different intermediate baryons. (Ongoing) work on intermediate probes large Nc limit properties of LCWF and allows the study of higher orbital modes