Large-Momentum Effective Theory (LaMET) Need for speed Xiangdong Ji Outline High-energy scattering and light-cone physics LaMET: basic principles Applications
Gluons Total Helicity GG PDFs GPDs TMD-PDFs and soft factors Light-Cone Wave Functions (LCWFs) Conclusion: need for speed High-energy scattering
In high-energy scattering, either hadron (proton) travels at high speed (LHC) or probe travels at high speed (fixed target) or both (EIC) QCD factorization theorems can be used to unde rstand the physics of hard scattering. Non-perturbative physics is usually represented by collinear quarks and gluons, soft gluons. The y are matrix elements of hadrons or the vacuum, called PDFs, TMD-PDFs, soft factors, etc.
Heisenberg picture Drell, Yan, Chang, Ma Brodsky, Lepage, Mueller, Collins, Soper, Sterman Probes operators) are light-cone correlations n p
The matrix elements are independent of hadron momentum, and they can be calculated in the s tates in the rest frame. Time-dependence is not yet amenable to classi cal Monte Carlo simulations. Light-front quantization Traditionally, light-cone correlations have been studied in light-front quantization, a Hamiltonian approach. However, due to breaking of rotational symmetr ies on the light-cone, renormalization of Hamilt
onian becomes very complicated. (quasi-non-r enormalizable) Moreover, there is no demonstration that the w eak coupling expansion actually works for QCD . K. Wilson et. al. Phys. Rev. D49 (1994) Large-momentum effective the ory (LaMET) Ji, Sci. China-MPA, 2014 A large momentum hadron on the Euclidean lattice carries with it automatically the collinear modes. Soft mode is present when gluons are probed in tr
ansverse coordinates. LaMET is an overarching framework for extracting parton (light-cone) physics from large-momentum hadron states in lattice QCD Basic principles LaMET uses an alternative, Schrodinger repr esentation for light-cone physics: The probes are static (Euclidean) The proton is travelling with a large momentum! This is the picture originally used by Feynman
to construct the famous parton model. R.P. Feynman, Phys. Rev. Lett. 23 (1969) 1415-1417 Back to 50 years ago! LaMET expansion One does not need a proton moving at speed of lig ht. One just need a proton fast enough! the control parameter of expansion is Thus for finite momentum, one can have a factoriz ation formula for large
Universality One can practically choose ANY composite op erator with arguments , so long as large enou gh, they give the same collinear or soft physics . For different operators, flowing into the fixed po int of large momentum will have different rates (which is faster?), but the limit is the same. X. Ji, Sci. China (Phys. Mech.Astron.) 57 1407-1412 (2014) Y. Hatta, Ji, Zhao, Phys. Rev. D89 (2014).
Application 1: Gluon total helici ty GG In QCD factorization, one can show that the gluon polarization is a matrix element of nonlocal light-cone correlation. A. Manohar,Phys. Rev. Lett. 66 (1991) 2684 No one knows how to calculate this for nearly 30 years! LaMET calculations In LaMET theory, one can start with the local o perator
in any physical gauge : (gauge choices shall allow transverse polarized gluons) Coulomb gauge Axial guage Az=0 Temporal gauge A0 =0 Their matrix elements in the large momentum li mit all go to G. Ji, Zhang, Zhao, Phys. Rev. Lett.,111,112002(2013) Y. Hatta, Ji, Zhao, Phys.Rev. D89 (2014).
First calculation (Y. Yang et al, PRL(2017)) Need more controlled calculations Gluons Provide Half of the Protons Spin The gluons that bind quarks together in nucleons provide a considerable chunk of the protons total spin. That was the conclusion reached by Yi-Bo Yang from the University of Kentucky, Lexington,
and colleagues (see Viewpoint: Spinning Gluons in the Proton). By running state-of-the-art computer simulations of quark-gluon dynamics on a so-called spacetime lattice, the Application 2: PDFs: PDF can be obtained from large momentum li mit of a correlation can be or or any combination. is a straight-line Wilson link
This is the starting point of quarsi-PDF, X. Ji, Phys. Rev. Lett. 110 (2013) Factorization was conjectured. The full-proved gi ven by Ma and Qiu, PRD98 (2018) 074021 Quasi-PDF is really a physic al PDF! Consider the momentum distribution of the const ituent In relativistic bound state, this becomes a cente r-of-mass momentum-dependent quantity,
Light-cone distributions correspond to the limit p. State-of-the-art calculations ETMC PRL121(2018) 112001; LP3 PRL121(2018) 242003 All analysis shall have controlled approximations One-loop matching and scale setting, renormalization Excited states, higher twist corrections Different methods of analyzing the same date is p ossible, but no fundamental difference expected Pseudo-PDF : A. Radyushkin, PRD96 (2017) 034025
Coordinate space approach, related to quasi-PDF by Fourier transf. One-loop factorization given by Ji Zhang, Zhao, NPB92 4(2017) 366, Izubuchi et al. PRD98 (2018 Alternative Euclidean operators LaMET universality tells us there are many diff erent operators that one can try, all are the sa me in the large momentum limit. Current-current correlator suggested Ma and Qiu, PRL120 (2018) 022003
Instead of Wilson line, one can consider a heav y quark with a large mass, Detmold and Lin One needs controlled calculations to compare which operators give faster convergence of the LaMET expansion? Application 3: Generalized part on distributions (GPD) GPD are form factors of parton distributions, di scovered independently (Mueller et al.94, Ji96) An experimental process that can be used to m easure GPD: Deeply virtual Compton scatterin
g (Ji, PRL78, PRD55, 1997) Kinematic variables: x, , t GPD on lattice GPD presents no additional difficulty compared with P DF One considers the off-forward matrix elements Two more variables results Momentum transfer, t Skewness, need momentum transfer in z- direction.
It has been very difficult to model these dependences i n the literature. Great oppo to get guidance from lattice . Matching: Liu et al., PRD100 (2019) 034006 Exploratory study 1: Chen, Lin, Zhang, 1094.1237 MILC configurations with a=0.12 fm, L=3fm MeV Pion GPD function = 1.74 GeV
H(x, t, valence distribution How to make it better Exploratory study 2: ETMC, Xiv:1910.13229 Application 4: TMD-PDFs A very important nucleon observable, many phenomenolo
gy related to spin physics (Sivers effect etc). TMD collaborations supported by DOE Qiu, Detmold) Lattice calculations started from A. Schafer et al., much pr ogress; no x dependence has yet been studied. Yoon et al. PRD96,094508 (2017) A number of LaMET formulations: Ji et al., PRD91,074009 (2015); PRD99,114006(2019) Ebert, Stewart, Zhao, PRD99,034505 (2019), JHEP09,037(2019); arXiv:1910.08 569
Collins-Soper kernel can be calculated! Soft factor Soft factor involves two light-con e directions, difficult to calculate on lattice. In a new LaMET formulation Soft factor can be calculated as a form factor of a pair of color-anti color sources. Ji, Liu & Liu, arXiv:1910.11415
Two lattice groups: MIT on Collins-Soper evolution kernel LPC group on soft function (LSC) Target TMDPDFs Ebert, Stewart, Zhao Dim. reg. and MS-bar scheme, just like the sta ndard PDF. Rapidity (light-cone) divergence: extra regulato r, there are many version in the literature. One t hat is consistent up to three-loop is exponentia l regulator Li & Zhu, PRL (2017) The physical TMD-PDFs are regulator-indept.
TMD Lattice q-TMDPDF q-TMDPDF without subtraction With subtraction Ji, Liu, Liu, to appear JI, Yuan, et al. Ebert et al All-order matching formula Ji, Liu, Liu, to appear
To all orders in perturbation RG equation for matching coefficient Wigner function Wigner function can be calculated with similar approach as TMD-PDFs and GPDs They may be probe in certain processes in EIC . Y. Hatta, B. Xiao, F. Yuan PRL (2016 App. 5 Light-Front WaveFunctions LF quantization focuses on the WFs, from whic
h everything can be calculated: a very ambitiou s goal! Brodsky et al. Phys. Rept. 301 (1998) However, there are a number of reasons this a pproach has not been very successful. LaMET provides the practical way to calculate non-perturbative WF, at least for lowest few co mponents. Ji, Liu, & Liu, to be published. All WF can be computed as gauge-invariant m atrix elements Additional comments about LC WF
One can choose the gauge links on l attice, corresponding to the different boundary conditions on the light-con e gauge choices. or combination of them One needs to have soft factors to s ubtract the rapidity divergences inhe rent in LCWF. Matching can be done with WF ampl itudes with rapidity regulated betwe en lattice and MS-bar subtracted sc hemes.
Need for speed LaMET approach opened a wide door for parton physics on lattice. PDF is a warm-up exercise, but a very important one (testing ground) Which operator is optimal will depend largely on its con vergence with increasing Pz, need for speed! More dedicated calculations are needed, particularly fo r small lattice spacing and large momentum, need for speed! Many observables shall be studied related to EIC, inclu
ding GPDs, TMD-PDFs, and Wigner functions, LCWFs, need for speed Recent collaborators Yusheng Liu, postdoc, Shanghai (searching for a new position) Yizhuang Liu, postdoc, Shanghai A. Schaefer, Regensburg U.
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