Par arto ton n Di Dist stributi ribution on Fu Func nctio tions ns on t on the he La Latt ttic ice Huey-Wen Lin University of Washington Huey-Wen Lin — Los Alamos National Lab 1
Outline § Introduction to PDFs A brief overview on global analysis § Lattice QCD Difficulties: why seek a new idea? § New Approach on the Lattice Preliminary results on nucleon quark, helicity and transversity distributions Huey-Wen Lin — Los Alamos National Lab 2
Parton Distribution Functions § Structure functions studied through scattering processes Deep inelastic scattering beginning in 1960s at SLAC Depend on energy scale ( Q 2 ) and quark momentum fraction ( x ) § “Parton” 1969 by Feynman: pointlike constituents inside hadron → now known to be quarks and gluons § Still limited knowledge Many ongoing/planned experiments (EIC, LHeC, …) Huey-Wen Lin — Los Alamos National Lab 3
Parton Distribution Functions § Quark distribution Processes: DIS ( F 2 , σ ), Drell-Yan, W -asymmetry, Z -rapidity, ( γ +) jet, … spin-averaged/unpolarized Experiment: BCDMS, NMC, SLAC, JLab, HERA, E866, CDF, DØ,… § Helicity distribution Processes : polarized DIS, semi-inclusive DIS, spin-dep./long. polarized photo- and electroproduction of hadrons and charm, pp collisions Experiment : EMC, HERMES, Hall A, CLAS, COMPASS, STAR, PHENIX, … § Transversity distribution Process: single- spin asymmetry in SIDIS, … Experiment: HERMES, COMPASS, Belle… transversely polarized Huey-Wen Lin — Los Alamos National Lab 4
Global Analysis § Experiments cover diverse kinematics of parton variables Global analysis takes advantage of all data sets Theory Exp’t Input Input Global Analysis of PDFs PDFs Applications Predictions Huey-Wen Lin — Los Alamos National Lab 5
Global Analysis § Experiments cover diverse kinematics of parton variables Global analysis takes advantage of all data sets PDFs Applications Predictions § Important fundamental QCD property Exploration of the valence and sea-quark content of the nucleon § Important for BSM searches Provides SM cross-section prediction for LHC new-physics search IceCube PeV neutrinos can be explained by PDF uncertainties Proton weak charge (medium-modification effects) Huey-Wen Lin — Los Alamos National Lab 6
Global Analysis Theory Exp’t Input Input Global Analysis § Some choices made of PDFs for the analysis Choice of data sets and kinematic cuts Strong coupling constant α s ( M Z ) Uncertainties in perturbation theory (depends on process whether LO, NLO or NNLO is known) Evolution of PDFs to different scales Parametrization assumptions 2 1 + a x + a x a a 3 4 P x = f ( x , μ ) = a x ( 1 x ) P ( x ) 1 2 - ( ) 0 0 a x a a e 1 + e x 3 4 5 ( ) Discrepancies appear when data is scarce Huey-Wen Lin — Los Alamos National Lab 7
Global Analysis Theory Exp’t Input Input Global Analysis § Some choices made of PDFs for the analysis Choice of data sets and kinematic cuts Strong coupling constant α s ( M Z ) Uncertainties in perturbation theory (depends on process whether LO, NLO or NNLO is known) Evolution of PDFs to different scales Parametrization assumptions § Sum rules to constrains the fit Quark number, momentum, g A , SU(3) flavor symmetry… § Assumptions imposed where theory and exp’t are lacking Charge symmetry, (anti- )strange, “sea” (antiquark) distribution… Huey-Wen Lin — Los Alamos National Lab 8
Global Analysis Theory Exp’t Input Input Global Analysis § Some choices made of PDFs for the analysis Choice of data sets and kinematic cuts Strong coupling constant α s ( M Z ) Uncertainties in perturbation theory (depends on process whether LO, NLO or NNLO is known) Evolution of PDFs to different scales For example, Parametrization assumptions s + s = κ u + d ( ) § Sum rules to constrains the fit or symmetric Quark number, momentum, g A , SU(3) flavor symmetry… sea in helicity § Assumptions imposed where theory and exp’t are lacking Charge symmetry, (anti- )strange, “sea” (antiquark) distribution… Huey-Wen Lin — Los Alamos National Lab 9
Global Momentum Analysis Jimenez enez-Delgad Delgado, , § Many groups have tackled the analysis Melnitc lnitchouk, , CTEQ, MSTW, ABM, JR, NNPDF, etc. Owens ns, J.Phys hys. . G40 (2013 13) ) 09310 10 Huey-Wen Lin — Los Alamos National Lab 10
Global Momentum Analysis Jimenez enez-Delgad Delgado, , § Many groups have tackled the analysis Melnitc lnitchouk, , CTEQ, MSTW, ABM, JR, NNPDF, etc. Owens ns, J.Phys hys. . G40 (2013 13) ) 09310 10 Huey-Wen Lin — Los Alamos National Lab 11
Global Helicity Analysis § Many groups have tackled the analysis DSSV, ACC, BB, LSS, JAM, etc. JAM13 ACC09 DSSV09 BB10 LSS10 JAM, , 1310.3 10.3734 734 [hep ep-ph ph] Huey-Wen Lin — Los Alamos National Lab 12
Transversity § There have only been 2 attempts (still very preliminary) Requires more theory input and experimental data More assumptions are made to extract the distribution A. Bacchetta chetta, , A. Courtoy rtoy, , and d M. Radic dici, , Phys. s.Rev ev.L .Let ett. . 107, , 012001 01 (2011 11) M. Anselm M. elmino ino, , et al., ., Nucl.P l.Phys. s.Proc. c.Suppl ppl. . 191, 1, 98 – 10 107 7 (2009) Huey-Wen Lin — Los Alamos National Lab 13
PDFs on the Lattice § Lattice QCD is an ideal theoretical tool for investigating strong-coupling regime of quantum field theories Ideal tool for studying nonperturbative hadron structure § Physical observables are calculated from the path integral § Lattice QCD quark field Impose a UV cutoff discretize spacetime Impose an Infrared cutoff L gluon field finite volume x , y , z Wick rotate to Euclidean Use compact gauge group t a Huey-Wen Lin — Los Alamos National Lab 14
PDFs on the Lattice § Many lattice calculations of the moments of the PDFs Limited to the lowest few moments Might provide constraints on models or tests of experiment § Also applies to GPDs: limited to 3 rd moment § Most progress made in quark contributions Very costly to obtain useful gluon signal Limited by available computational resources Huey-Wen Lin — Los Alamos National Lab 15
x n Moments § Leading moment x , hypercubic decomposition 4 1 4 1 = 1 1 3 1 6 1 6 3 : O 44 − ( O 11 + O 22 + O 33 )/3 O 14 + O 41 , (requires p ≠ 0) Both operators go to same continuum limit § No mixing with operators of same or lower dimension § To improve to O ( a ) Consider all irrelevant operators of same symmetry: § Higher moments x 2 ― γ 1 q with coefficient ~ 1/ a 2 4 1 : O 111 mixes with q 4 2 : O {123} requires all momentum components to be nonzero 8 1 : O {441} − ( O {221} + O {331} )/2 mixes under renormalization § For higher spin, all ops mix with lower-dimension ops Huey-Wen Lin — Los Alamos National Lab 16
x n Moments § Leading moment x , hypercubic decomposition 4 1 4 1 = 1 1 3 1 6 1 6 3 : O 44 − ( O 11 + O 22 + O 33 )/3 O 14 + O 41 , (requires p ≠ 0) Both operators go to same continuum limit § No mixing with operators of same or lower dimension § To improve to O ( a ) Consider all irrelevant operators of same symmetry: § Higher moments x 2 ― γ 1 q with coefficient ~ 1/ a 2 4 1 : O 111 mixes with q 4 2 : O {123} requires all momentum components to be nonzero 8 1 : O {441} − ( O {221} + O {331} )/2 mixes under renormalization § For higher spin, all ops mix with lower-dimension ops Huey-Wen Lin — Los Alamos National Lab 17
x n Moments § For higher spin, all ops mix with lower-dimension ops Tricks: subtraction to remove divergent terms, heavy fields, four- point functions… None is practical enough § Relative error grows in higher moments Calculation would be costly LHPC C (SCRI, CRI, SESAM): AM): 2f, Wilson lson and clove lover Dolgov lgov et al. . PRD66, 6, 034506 6 (2002) ) QCDSF SF: : 0f Göckel eler er et al. . PRD71, 71, 1145 4511 11 (2005) x 3 q x 2 q Huey-Wen Lin — Los Alamos National Lab 18
Limited Access § What can we learn about the x -distribution? Make an ansatz of some smooth form for the distribution and fix the parameters by matching to the lattice moments Cannot separate valence- quark contribution from sea New idea needed to access the sea! W. Detmol old d et al, , Eur.Ph .Phys ys.J.d .J.direct irect C3 (2001) 1) 1 – 15 15 Huey-Wen Lin — Los Alamos National Lab 19
Quark Distribution § Lightcone nucleon quark distribution Transform lab coordinates to light-cone ones x ± = z ± t Renormalization Gluon potential A + scale µ ― Lightcone coordinate ξ ± =( t ± z )/ √ 2 Nucleon momentum P µ Huey-Wen Lin — Los Alamos National Lab 20
Quark Distribution § Lightcone nucleon quark distribution Transform lab coordinates to light-cone ones x ± = z ± t Renormalization Gluon potential A + scale µ ― Lightcone coordinate ξ ± =( t ± z )/ √ 2 Nucleon momentum P µ Massive particles lie on hyperboloids invariant under Lorentz transformation Huey-Wen Lin — Los Alamos National Lab 21
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