Transverse Hadron Structures from Lattice QCD with LaMET Yong Zhao - PowerPoint PPT Presentation

Transverse Hadron Structures from Lattice QCD with LaMET Yong Zhao Massachusetts Institute of Technology Aug 25, 2019 11th Workshop on Hadron physics in China and Opportunities Worldwide Nankai University, Tianjin, China 08/23-28, 2019 � 1

Outline • Large-momentum effective theory Physical picture and factorization formula • Systematic approach to extract PDFs from lattice QCD • • Transverse hadron structures from lattice QCD Generalized parton distributions • Transverse momentum dependent PDFs • Collins-Soper kernel from lattice QCD • Yong Zhao, Hadron-China 2019 � 2

<latexit sha1_base64="cECQ5ypueyzMY+hqCyaofUL6f3I=">ACBHicbVDLSgMxFM3UV62vUZfdBIsgSMuMCrosunFZwT6gMy2Z9E4bmskMSUYopQs3/obF4q49SPc+Tem7Sy09UDI4Zx7Sc4JEs6UdpxvK7eyura+kd8sbG3v7O7Z+wcNFaeSQp3GPJatgCjgTEBdM82hlUgUcChGQxvpn7zAaRisbjXowT8iPQFCxkl2khdu+glrFPGNezpGHvAed0fpVxq2uXnIozA14mbkZKEOta395vZimEQhNOVGq7TqJ9sdEakY5TApeqiAhdEj60DZUkAiUP56FmOBjo/RwGEtzhMYz9fGmERKjaLATEZED9SiNxX/89qpDq/8MRNJqkHQ+UNhyrEJPG0E95gEqvnIEIlM3/FdEAkodr0VjAluIuRl0njrOKeV5y7i1L1Oqsj4roCJ0gF12iKrpFNVRHFD2iZ/SK3qwn68V6tz7mozkr2zlEf2B9/gDTi5ZI</latexit> So far our knowledge of the PDFs mostly comes from the analysis of high-energy scattering data Unpolarized PDF TMD PDF Existing global analyses of TMDPDFs or TMD 1 xq ( x , Q 2 = 10 GeV 2 ) fragmentation functions rely on the modeling NNPDF3.1 (NNLO) 0.9 2 2 xf(x, =10 GeV ) of their nonperturbative evolution. µ 0.8 • Kang, Prokudin, Sun and Yuan, PRD93 (2016); g/10 • Bacchetta et al., JHEP1706 (2017); 👎 0.7 • Eur.Phys.J. C78 (2018) no.2, 89; 🤕 • Bertone, Scimemi and Vladimirov, arXiv:1902.08474. 0.6 u The most definite experimental finding so v 0.5 far is the sign change of the Sivers 0.4 function in SIDIS and Drell-Yan processes. d v s 0.3 COMPASS Phys.Rev.Lett. 119 (2017) 12002 ⇡ − P → ` + ` − X 0.2 d u With sign change COMPASS 2015 data 0.1 DGLAP TMD-1 c 0.1 TMD-2 0 S ϕ − 3 − 2 − 1 10 10 10 sin 1 T x A 0 NNPDF 3.1, EPJ C77 (2017) 0.1 − Without sign change 0.5 0 0.5 − x F Yong Zhao, SCET 2019, San Diego � 3 See also STAR Collaboration, PRL116 (2016).

Lattice QCD calculation of partonic hadron structures? b − b + PDF: q ( x , μ ) = ∫ db − ψ ( b − ) γ + 2 π e − ib − ( xP + ) ⟨ P | ¯ 2 W [ b − ,0] ψ (0) | P ⟩ b ± = t ± z Minkowski space, real time; • 2 Defined on the light-cone which depends on • the real time. Lattice QCD: ⟨ O ⟩ = ∫ D ψ D ¯ e iS → e − S , ψ DA O ( x ) e − S t = i τ , Euclidean space, imaginary time; • Difficult to analytically continue lattice results • back to Minkowski space. Light-cone PDFs not directly accessible from the lattice! Yong Zhao, Hadron-China 2019 � 4

A novel approach to calculate light-cone PDFs • Large-Momentum Effective Theory: • Ji, PRL110 (2013); • Ji, SCPMA57 (2014). q ( x , P z ) PDF : Quasi-PDF : q ( x ) ˜ Cannot be calculated Directly calculable on the on the lattice lattice Yong Zhao, Hadron-China 2019 � 5

A novel approach to calculate light-cone PDFs • Large-Momentum Effective Theory: • Ji, PRL110 (2013); • Ji, SCPMA57 (2014). Related by Lorentz boost q ( x , P z ) PDF : Quasi-PDF : q ( x ) ˜ Cannot be calculated Directly calculable on the on the lattice lattice Yong Zhao, Hadron-China 2019 � 5

A novel approach to calculate light-cone PDFs • Large-Momentum Effective Theory: • Ji, PRL110 (2013); • Ji, SCPMA57 (2014). Related by Lorentz boost q ( x , P z ) PDF : Quasi-PDF : q ( x ) ˜ Cannot be calculated Directly calculable on the Calculating the quasi-PDF at on the lattice lattice hadron momentum P z is equivalent to boosting it. Yong Zhao, Hadron-China 2019 � 5

A novel approach to calculate light-cone PDFs q ( x , P z ) = ? P z →∞ ˜ lim Instead of taking P z →∞ limit , one can ∞ perform an expansion for large but finite P z : P z q ( x , P z ) = C ( x , P z ) ⊗ q ( x )+ O ( 1/( P z ) 2 ) ˜ q ( x , P z ) • and have the same infrared physics (nonperturbative), q ( x ) ˜ but different ultraviolet (UV) physics (perturbative); P z • Therefore, the matching coefficient C ( x , P z ) is perturbative, which controls the logarithmic dependences on P z . ∞ Yong Zhao, Hadron-China 2019 � 6

Systematic procedure of calculating the PDFs yP z ) q ( y , μ )+ O ( | y | C ( q ( x , P z , μ ) = ∫ z ) Λ 2 M 2 dy x y , μ QCD ˜ , P 2 x 2 P 2 z • X. Xiong, X. Ji, J.-H. Zhang and Y.Z., PRD90 (2014); • Y.-Q. Ma and J. Qiu, PRD98 (2018), PRL 120 (2018); • T. Izubuchi, X. Ji, L. Jin, I. Stewart, and Y.Z., PRD98 (2018). 1. Lattice simulation of the quasi-PDF; 2. Lattice renormalization and the For complete review of LaMET, see: • Cichy and Constantinou, Adv.High physical limits (continuum, infinite Energy Phys. 2019 (2019) 3036904; volume, physical pion mass); • Y.Z., Int.J.Mod.Phys. A33 (2019); • C. Alexandrou et al. (ETMC), Phys.Rev. 3. Power corrections; D99 (2019) no.11, 114504; 4. Perturbative matching. Also see Y.-Z. Liu’s talk on Sunday for more detailed introduction. Yong Zhao, Hadron-China 2019 � 7

Outline • Large-momentum effective theory Physical picture and factorization formula • Systematic approach to extract PDFs from lattice QCD • • Transverse hadron structures from lattice QCD Generalized parton distributions • Transverse momentum dependent PDFs • Collins-Soper Kernel of TMDPDF from lattice QCD • Yong Zhao, Hadron-China 2019 � 8

Three-dimensional partonic hadron structures p • Longitudinal Parton Distribution Functions y q i = q ,¯ q , g ( x ) (PDFs): • Generalized Parton Distributions (GPDs): xp k T k T F i ( x , ξ = 0, b T ) b Τ b T • : transverse position of the parton. z b T x • Transverse momentum dependent (TMD) PDFs q i ( x , k T ) The longitudinal and transverse PDFs provide complete 3D • : transverse momentum of the parton. k T structural information of the proton. • Wigner distributions or generalized transverse momentum dependent distributions W i ( x , ξ = 0, k T , b T ) Yong Zhao, Hadron-China 2019 � 9

GPD ξ = P + − P ′ � + • Light-cone GPD: t = ( P ′ � − P ) 2 ≡ Δ 2 P + + P ′ � + , F Γ ( x , ξ , t , μ ) = ∫ d ζ − ζ − 2 ) Γ U ( ζ − 2 , − ζ − 2 ) ψ ( − ζ − 4 π e − ix ¯ P + ζ − ⟨ P ′ � , S ′ � | ¯ ψ ( 2 ) | P , S ⟩ • Measurable in hard exclusive processes such as deeply virtual Compton scattering: q 0 = q � ∆ q ∼ ∫ dx C ( x , ξ ) F ( x , ξ , t ) k � ∆ k + ∆ 2 2 P 0 = P + ∆ P Yong Zhao, Hadron-China 2019 � 10

Quasi-GPD ξ = P z − P ′ � z M 2 ˜ P z + P ′ � z = ξ + O ( z ) P 2 • Definition: ξ , t , μ ) = ∫ dz z Γ U ( z 2 , − z 2 ) ψ ( − z 4 π e − ixP z z ⟨ P ′ � , S ′ � | ¯ Γ ( x , ˜ ˜ 2 ) ˜ ψ ( 2 ) | P , S ⟩ F ˜ • Renormalization: Same operator as the quasi-PDF, can be renormalized the same way! • • Y.-S. Liu, Y.Z. et al., PRD100 (2019) no.3, 034006 • Factorization formula: | ξ | C ( γ z ( x , ξ , t , μ ) = ∫ Λ 2 1 ξ P z ) F γ + ( y , ξ , t , μ ) + O ( M 2 dy ξ , y x ξ , μ , t QCD ˜ z ) F ˜ , P 2 P 2 x 2 P 2 z z − 1 C ( = ∫ Λ 2 1 yP z ) F γ + ( y , ξ , t , μ ) + O ( M 2 dy x y , ξ y , μ , t QCD ¯ z ) , P 2 P 2 x 2 P 2 | y | z z − 1 • First lattice calculation of pion GPD, Chen, Lin and Zhang, arXiv: 1904.12376. • Preliminary results for quasi-GPDs (ETMC), see M. Constantinou’s talk at QCD Evolution 2019. Yong Zhao, Hadron-China 2019 � 11

⃗ ⃗ TMDPDF l - Soft • Collinear factorization (e.g., for Drell-Yan): Beam dQdY = ∑ d σ p p σ ab ( Q , μ , Y ) f a ( x 1 , μ ) f b ( x 2 , μ ) a , b l + • TMDPDF factorization: = ∑ d σ H ij ( Q , μ ) ∫ d 2 b T e ib T ⋅ q T f TMD ( x a , b T , μ , ζ a ) f TMD ( x b , b T , μ , ζ b ) i j dQdYd 2 q T i , j q T : Net transverse momentum of the color-singlet final state, and q T << Q ; ζ : Collins-Soper Scale. ζ a ζ b = Q 4 • The definition of TMDPDF involves a collinear beam function (or un-subtracted TMD) and soft function: Rapidity divergence regulator f TMD ϵ → 0, τ → 0 Z UV ( ϵ , μ , xP + ) B i ( x , b T , ϵ , τ , xP + ) Δ S i ( b T , ϵ , τ ) ( x , b T , μ , ζ ) = lim i UV divergence regulator Yong Zhao, Hadron-China 2019 � 12

⃗ ⃗ Evolution of TMDPDF • Evolution of TMDPDF: μ b T , μ 0 , ζ 0 ) exp [ ∫ μ ( μ ′ � , ζ 0 ) ] exp [ ζ 0 ] d μ ′ � 1 ζ ( μ , b T )ln ζ f TMD b T , μ , ζ ) = f TMD μ ′ � γ i 2 γ i ( x , ( x , i i μ 0 • µ ~ Q, ζ ~ Q 2 >> Λ QCD2 ; • µ 0 , ζ 0 : initial or reference scales, measured in experiments or determined from lattice (~2 GeV). γ i μ ( μ ′ � , ζ 0 ) Anomalous dimension for µ evolution, perturbatively calculable; γ i ζ ( μ , b T ) Collins-Soper kernel, becomes nonperturbative when b T ~1/ Λ QCD . Both Initial-scale TMDPDF and the Collins-Soper kernel must be modeled in global fits of TMDPDF from experimental data. μ df μ ′ � ζ ( μ , b T ) = − 2 ∫ γ i μ ′ � Γ i cusp [ α s ( μ ′ � )] + γ i ζ [ α s ( μ ( b T ))] • Bachetta et al., JHEP 1706 (2017); μ ( b T ) • Scimemi and Vladimirov, EPJC78 (2018); μ ( b T ) ≫ Λ QCD + g K ( b T ) • Bertone, Scimemi and Vladimirov, JHEP 1906 (2019). Yong Zhao, Hadron-China 2019 � 13