3D Nucleon Tomography from LQCD INTRODUCTION Goal: Compute - PowerPoint PPT Presentation

Kostas Orginos (W&M/JLab) March 15-17, 2017 (JLab) 3D Nucleon Tomography from LQCD

INTRODUCTION Goal: Compute properties of hadrons from first principles Parton distribution functions (PDFs) and Generalized Parton distributions (GPDs) Transverse Momentum Dependent densities (TMDs) Form Factos … Lattice QCD is a first principles method For many years calculations focused on Mellin moments Can be obtained from local matrix elements of the proton in Euclidean space Breaking of rotational symmetry —> power divergences X. Ji, Phys.Rev.Lett. 110, (2013) only first few moments can be computed Y.-Q. Ma J.-W. Qiu (2014) 1404.6860 
 Recently direct calculations of PDFs in Lattice QCD are proposed First lattice Calculations already available H.-W. Lin, J.-W. Chen, S. D. Cohen, and X. Ji, Phys.Rev. D91, 054510 (2015) C. Alexandrou, et al, Phys. Rev. D92, 014502 (2015)

PDFS: DEFINITION Light-cone PDFs: · Z 1 � T ψ (0 , ω � , 0 T ) W ( ω � , 0) γ + λ a � � d ω � 4 π e � i ξ P + ω − ⌧ � f (0) ( ξ ) = � � P 2 ψ (0) � P . � � C �1 Z ω − " # d y � A + W ( ω � , 0) = P exp α (0 , y � , 0 T ) T α � ig 0 P + � P 0 + � h P 0 | P i = (2 π ) 3 2 P + δ δ (2) � � � P T � P 0 T 0 Moments: Z 1 Z 1 d ξ ξ n � 1 h (0) ( ξ ) i a ( n ) f (0) ( ξ ) + ( � 1) n f d ξ ξ n � 1 f ( ξ ) , = = 0 0 � 1 Local matrix elements: = i n − 1 ψ (0) γ { µ 1 D µ 2 · · · D µ n } λ a D E O { µ 1 ··· µ n } P |O { µ 1 ...µ n } = 2 a ( n ) ( P µ 1 · · · P µ n � traces) . 2 ψ (0) − traces | P 0 0 0

GPDS: DEFINITION GPDs: Z 1 γ + H ( x, ξ , t ) + i σ + k ∆ k � T ψ (0 , ω � , 0 T ) W ( ω � , 0) γ + λ a � � ✓ ◆ d ω � 4 π e � i ξ P + ω − ⌧ � � � u ( P 0 ) P 0 ¯ E ( x, ξ , t ) = 2 ψ (0) � P � � 2 m �1 C Z ω − " # P + � P 0 + � h P 0 | P i = (2 π ) 3 2 P + δ δ (2) � � � P T � P 0 d y � A + W ( ω � , 0) = P exp α (0 , y � , 0 T ) T α � ig 0 T 0 sfer ∆ = P ′ − P as Moments: the virtuality t lity t = ∆ 2 . T e momentum � 1 � � � � [( n − 1 ) / 2 ] H ( x , ξ , t ) A n , 2 k ( t ) − 1 dxx n − 1 ( 2 ξ ) 2 k ± δ n , even ( 2 ξ ) n C n ( t ) . ∑ = E ( x , ξ , t ) B n , 2 k ( t ) k = 0 generalized form factors A B and C are defined from the coefficients of this e = i n − 1 ψ (0) γ { µ 1 D µ 2 · · · D µ n } λ a O { µ 1 ··· µ n } Matrix elements of twist-2 operators 2 ψ (0) − traces 0

X. Ji, D. Muller, A. Radyushkin (1994-1997) y y y z ~ 1 Q δ z ~ 1 Q δ ⊥ ⊥ xp xp r ⊥ r ⊥ z x z x z x p p p f ( x , r ) ⊥ f x ( ) ( ⊥ r ) ρ 0 r ⊥ 0 r 0 r ⊥ ⊥ 1 x x 1 Parton Distribution Generalized Parton Form Factors functions Distribution functions

Mellin moments are local matrix elements Can be evaluated in Euclidean space Lattice QCD calculations are possible Challenges: Renormalization and power divergent mixing Lattice breaks O(4) symmetry Only few moments can be computed

LATTICE QCD � q D A µ e � S [¯ q,q,A µ ] In continuous Euclidian space: Z = D q D ¯ ⇧ O ⌃ = 1 � q, q, A µ ) e � S [¯ q,q,A µ ] D q D ¯ q D A µ O (¯ Z Lattice regulator: Gauge sector: U µ ( x ) = e � iaA µ ( x + ˆ µ 2 ) P Fermion sector: µ ν q S f = ¯ Ψ D Ψ ν Ψ is now a vector whose components U µ leave on the sites of the lattice D is the Dirac matrix which is large and sparse µ

MONTE CARLO INTEGRATION ⟨ O ⟩ = 1 � � � n f / 2 e − S g ( U ) dU µ ( x ) O [ U, D ( U ) − 1 ] det D ( U ) † D ( U ) � Z µ,x N Monte Carlo Evaluation � O ⇥ = 1 � O ( U i ) N i =1 1 Statistical error √ N

Form Factors 1 5 PACS N f = 2 + 1 PACS N f = 2 + 1 ETMC N f = 2 SUMM ETMC N f = 2 SUMM ETMC N f = 2 , t s = 1 . 7 fm ETMC N f = 2 , t s = 1 . 7 fm 4 . 5 0 . 9 PNDME N f = 2 + 1 + 1 PNDME N f = 2 + 1 + 1 Alberico et al. Alberico et al. 4 0 . 8 G M ( Q 2 ) G E ( Q 2 ) 3 . 5 0 . 7 3 0 . 6 2 . 5 0 . 5 2 0 . 4 1 . 5 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 25 0 . 3 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 25 0 . 3 Q 2 [GeV 2 ] Q 2 [GeV 2 ] PACS: N f =2+1 m π = 145 MeV 8.1 fm box ETMC: N f =2+1 m π = 131 MeV 4.5 fm box PNDME: mixed action m π = 138 MeV 5.6 fm box

Strange quark contribution to nucleon form factors 0 . 012 0 . 00 0 . 010 − 0 . 01 0 . 008 − 0 . 02 0 . 006 − 0 . 03 0 . 004 G M G E 0 . 002 − 0 . 04 0 . 000 − 0 . 05 strange strange − 0 . 002 − 0 . 06 light disconnected light disconnected − 0 . 004 − 0 . 07 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 Q 2 (GeV 2 ) Q 2 (GeV 2 ) | | dynamical 2 + 1 flavors of Clover fermions Q 2 z 323 x 96 lattice of dimensions (3.6 fm)3 x (10.9 fm) a=0.115fm, pion mass 317 MeV R. J. Hill and G. Paz, Phys. Rev. D 84 (2011) 073006 k max t cut + Q 2 − √ t cut p X a k z k , z-expansion fit: G ( Q 2 ) = z = , t cut + Q 2 + √ t cut p k J. Green et al. Phys.Rev. D92 (2015) no.3, 031501

Comparison with experiments 0 . 15 G0 A4 HAPPEX lattice 0 . 10 M E + η G s 0 . 05 G s 0 . 00 − 0 . 05 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 Q 2 (GeV 2 ) Experiment: forward-angle parity-violating elastic e-p scattering G s E + η G s η = AQ 2 , A = 0 . 94 M Prediction: very hard for such experiments to measure a non-zero result

Axial Charge From M. Constantinou: arXiv:1701.02855

Moments of GPDs LHPC: arXiv:0705.4295 Phys.Rev.D77:094502,2008

� � � � � � � � �฀ � � �� � � � � � � �� � � ��� � � � � �� � � � �� � � � � � � �� � � Moments of PDFs (Lattice vs Experiment) 1.2 1.1 1 0.9 0.8 LHPC: 2007

Can Lattice QCD go beyond moments? Lattice QCD can only compute time local matrix elements Euclidean space

QPDFS: MAIN IDEA 2 P z →∞ q (0) ( x, P z ) = f ( x ) lim 1 X. Ji, Phys.Rev.Lett. 110, (2013) 
 t 0 -1 Euclidean space time local matrix element is equal to the same matrix element in -2 -2 -1 0 1 2 Minkowski space z

A more general point of view: Y.-Q. Ma J.-W. Qiu (2014) 1404.6860 Minkowski space factorization: ! ✓ ◆ X Λ 2 x, e µ , e µ QCD µ 2 , P z ) = ⊗ f α ( x, µ 2 ) + O σ ( x, e H α e µ 2 P z µ e α = { q,q,g } computable in perturbation theory K-F Liu Phys.Rev. D62 (2000) 074501 Related ideas see: Detmold and Lin Phys.Rev.D73:014501,2006

✓ x ◆ Z 1 d ξ ξ , µ e q ( x, P z ) = Z f ( ξ , µ ) + O ( Λ QCD /P z , M N /P z ) P z ξ − 1 The matching kernel can be computed in perturbation theory X. Xiong, X. Ji, J. H. Zhang, Y. Zhao, Phys. Rev. D 90, no. 1, 014051 (2014) T. Ishikawa et al. arXiv:1609.02018 (2016) Practical calculations require a regulator (Lattice) Continuum limit has to be taken renormalization Momentum has to be large compared to hadronic scales to suppress higher twist effects Practical issue with LQCD calculations at large momentum … signal to noise ratio

QUASI-PDFS ✓ z λ a � � ◆ ⌧ � 1 h ( s ) � � √ τ , √ τ P z , √ τ Λ QCD , √ τ M N = P z � χ ( z ; τ ) W (0 , z ; τ ) γ z 2 χ (0; τ ) � P z � � 2 P z C (2. τ is the a regulator scale χ quark field W is the regulated gauge link Z ∞ d z q ( s ) � 2 π e i ξ zP z P z h ( s ) ( √ τ z, √ τ P z , √ τ Λ QCD , √ τ M N ) , � ξ , √ τ P z , √ τ Λ QCD , √ τ M N = −∞ (2.12) At fixed flow time the quasi-PDF is finite in the continuum limit Ji 2013

Using the previous definitions we have ✓ i ✓ z ◆ n − 1 ◆ ∂ h ( s ) √ τ , √ τ P z , √ τ Λ QCD , √ τ M N = P z ∂ z Z ∞ d ξ ξ n − 1 e − i ξ zP z q ( s ) � � ξ , √ τ P z , √ τ Λ QCD , √ τ M N −∞ By introducing the moments Z ∞ ✓ √ τ P z , Λ QCD , M N ◆ b ( s ) d ξ ξ n − 1 q ( s ) � � ξ , √ τ P z , √ τ Λ QCD , √ τ M N = n P z P z −∞

Taking the limit of z going to 0 we obtain: n ( p τ P z ) = c ( s ) D z ) ( n − 1) λ a � � ✓ p τ P z , Λ QCD , M N ◆ ⌧  � � χ ( z ; τ ) γ z ( i � b ( s ) � � P z 2 χ (0; τ ) � P z . n � � 2 P n P z P z � z z =0 C (3.7) i.e. the moments of the quasi-PDF are related to local matrix elements of the smeared fields These matrix elements are not twist-2. Higher twist effects enter as corrections that scale as powers of after removing M N /P z effects [ H.-W. Lin, et. al Phys.Rev. D91, 054510 (2015)] ! Λ 2 � p τ P z , p τ Λ QCD = c ( s ) ( p τ P z ) b ( s, twist − 2) � p τ Λ QCD QCD b ( s ) � � + O n n P 2 z

Introducing a kernel function such that: Z ∞ n ( p τ µ, p τ P z ) = Z ( x, p τ µ, p τ P z ) dx x n − 1 e C (0) −∞ We can undo the Melin transform: ✓ x ◆ Z 1 q ( s ) � � d ξ x, p τ Λ QCD , p τ P z ξ , p τ µ, p τ P z f ( ξ , µ ) + O ( p τ Λ QCD ) e = Z ξ − 1 Therefore regulated quasi-PDFs are related to PDFs if Λ QCD , M N ⌧ P z ⌧ τ − 1 / 2 , Monahan and KO: arXiv:1612.01584

PROCEDURE OUTLINE Compute equal time matrix elements in Euclidean space using Lattice QCD at sufficiently large momentum in order to suppress higher twist effects Take the continuum limit (renormalization) Equal time: Minkowski — Euclidean equivalence Perform the matching Kernel calculation in the continuum