PION properties from Lattice QCD R. Briceno, B. Chakraborty, R. - PowerPoint PPT Presentation

April 28, 2017 — USQCD All Hands PION properties from Lattice QCD R. Briceno, B. Chakraborty, R. Edwards, A. Gambhir, B. Joo, J. Karpie, A. Kusno, C. Monahan, K. Orginos , D. Richards, S. Zafeiropoulos 


MOTIVATION Understand from first principles the structure of the pion Unlike the nucleon, pion parton distribution function are not well determined experimentally Experimental effort at JLab 12GeV to determine pion PDFs. p Sullivan process DIS (Sullivan Process) Pion Drell-Yan than nucleon… Future EIC experiments [T. Horn DIS2017]

MOTIVATION JAM global fit analysis (@JLAB) Will benefit from theoretical input from Lattice QCD Exploring the impact on the global fits first couple of moments will have Interested in the large x region (x>.2) Pion is the cloud in the nucleon: Understand pion structure leads to insight to nucleon structure. Light quark asymmetries

MOTIVATION Pion distribution amplitude Offers insight to the interplay of high an low energy scales in a hadron High Q 2 factorization (ex. EM form factor)

PION FORM FACTOR E12-06-101

MOTIVATION Pion is the lightest hadron Pion is relatively easy for lattice calculations Statistical noise much smaller than the nucleon Offers an interesting playground to test new ideas for hadron structure calculations

QUASI-PARTON DISTRIBUTIONS Goal: Compute properties of hadrons from first principles Parton distribution functions (PDFs) Lattice QCD calculations 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 only first few moments can be computed X. Ji, Phys.Rev.Lett. 110, (2013) Recently direct calculations of PDFs in Lattice QCD are proposed Y.-Q. Ma J.-W. Qiu (2014) 1404.6860 
 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)

QUASI-PARTON DISTRIBUTIONS Defined as non-local (space), equal time matrix elements in Euclidean space Equal time: rotation to Minkowski space is trivial PDFs are obtained in the limit of infinite proton momentum Matching to the infinite momentum limit can be obtained through perturbative calculations 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)

QPDFS: 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

QPDFS: DEFINITION � � ⌧ � 1 λ a h 0 ( z, P z ) = � � � ψ ( z ) W (0 , z ; τ ) γ z 2 ψ (0) P z � P z � � 2 P z C Z z  � d z 0 A 3 α ( z 0 v ) T α W ( z, 0) = P exp v = (0 , 0 , 1 , 0) − ig 0 , 0 Z ∞ q (0) ( ξ , P z ) = 1 d z e i ξ zP z h (0) ( z, P z ) 2 π −∞

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

✓ 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

GRADIENT FLOW SMEARING Is a way to obtain a finite matrix element in the continuum that can then be used to obtain the light cone PDFs For fermonic correlation functions an additional wave function renormalization remains. Ringed fermions is a way to remove this renormalization in simple way.

Ringed smeared fermions Ringed fermion correlation functions require no additional renormalization H. Makino and H. Suzuki, PTEP 2014, 063B02 (2014), 1403.4772. 
 . . K. Hieda and H. Suzuki (2016), 1606.04193

SMEARED 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 flow time χ is the ringed smeared quark field W is the smeared 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

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

Small flow time expansion: Luscher [’10,’13] � p τ Λ QCD � n ( p τ µ ) a ( n ) ( µ ) + O ( p τ Λ QCD ) , = e b ( s, twist − 2) C (0) n are the moments of the PDFs The quasi-PDF moments then are: ! Λ 2 � p τ Λ QCD � n ( p τ µ, p τ P z ) a ( n ) ( µ ) + O p τ Λ QCD , QCD b ( s ) = C (0) n P 2 z p p Λ QCD , M N ⌧ P z ⌧ τ − 1 / 2 ,

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 Mellin 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 smeared quasi-PDFs are related to PDFs if Λ QCD , M N ⌧ P z ⌧ τ − 1 / 2 ,

PION DISTRIBUTION AMPLITUDE Q-DA Z dz φ ( x, P z ) = i ˜ 2 π e − i ( x − 1) P z z h π ( P ) | ¯ ψ (0) γ z γ 5 Γ (0 , z ) ψ ( z ) | 0 i f π Factorization: Z 1 ! Λ 2 , m 2 ˜ QCD π φ ( x, Λ , P z ) = dy Z φ ( x, y, Λ , µ, P z ) φ ( y, µ ) + O P 2 P 2 0 z z X. Ji, Phys.Rev.Lett. 110, (2013) Zhang et. al. ‘17

COMPUTATION METHOD Use distillation All 3-projects can be done with large overlap of resource requirements Additional contraction cost and operator insertion for 3pt functions Common feature: Large momentum for the pion

DISTILATION Gaussian smearing: e � τλ | λ ih λ | S = e τ r 2 ( U ) = X or: S = e τ r 2 ( U ) λ Distillation smearing smearing: λ max X | λ ih λ | S dist = λ Result: Factorization of correlation functions

DISTILATION M ( t ) P ( t, 0) M (0) † P (0 , t ) Two point function � � C 2 pt ( t ) = Tr Three point function M ( t ) G ( t, 0) M (0) † P (0 , t ) � � C 3 pt ( t ) = Tr Meson Operator λ , λ 0 = h λ | Γ s,s 0 | λ 0 i M ( t ) s,s 0 Perambulator D � 1 ⇤ s,s 0 P ( t, 0) s,s 0 ⇥ | λ 0 i λ , λ 0 = h λ | D � 1 O D � 1 ⇤ s,s 0 G ( t, 0) s,s 0 Generalized Perambulator ⇥ | λ 0 i λ , λ 0 = h λ |

DISTILATION Meson Operator λ , λ 0 = h λ | Γ s,s 0 | λ 0 i M ( t ) s,s 0 Perambulator D � 1 ⇤ s,s 0 P ( t, 0) s,s 0 ⇥ | λ 0 i λ , λ 0 = h λ | D � 1 O D � 1 ⇤ s,s 0 G ( t, 0) s,s 0 Generalized Perambulator ⇥ | λ 0 i λ , λ 0 = h λ | General building blocks Computed and stored to be used in many projects Contractions done in an general fashion Red Star Software (R. Edwards)

PRELIMINARY TESTING Quenched calculation under way Test codes (analysis and aspects of the methodology) Using pre-existing building blocks high-Q 2 pion form factor calculation under way

Pion dispersion relation (quenched β =6.0) 32 3 x64 Max P ~ 2 GeV

REQUEST We request an allocation of 56.6M KNL core-hours (169.8M JPsi core-hours) on the KNL machine at JLab. We request disk and achival storage of 200 TByte and 200 TByte respectively, equivalent to 8M and 1.2M JPsi core-hours. Ensemble: 643 × 128, Nf = 2 ⊕ 1 M π ≃ 170 MeV a ≃ 0.091fm (200 configs) M π L=4.8 Projects Pion DA Pion PDF Pion Form Factor