Fu Func nctions tions on on th the Lat attic tice Huey-Wen - PowerPoint PPT Presentation

Pa Part rton on Di Dist stribution ribution Fu Func nctions tions on on th the Lat attic tice Huey-Wen Lin University of Washington Huey-Wen Lin — Lattice Workshop @ NTU 1

Part rton on Dis istri ributio tion n Functio ions ns § Structure function/distribution functions  Deep inelastic scattering (1960s) 𝜏 ∼ 𝑀 𝜈𝜉 𝑋 𝜈𝜉 𝜈𝜉 = 𝑗 𝑒 4 𝑦 𝑓 𝑗𝑟𝑦 𝑂 𝑈 𝐾 𝜈 𝑦 , 𝐾 𝜉 0 𝑋 𝑂 𝑦 = 𝑅 2 /2𝑟 ⋅ 𝑄 § 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 — Lattice Workshop @ NTU 2

Part rton on Dis istri ributio tion n Functio ions ns § Structure function/distribution functions  Deep inelastic scattering (1960s) 𝜏 ∼ 𝑀 𝜈𝜉 𝑋 𝜈𝜉 𝜈𝜉 = 𝑗 𝑒 4 𝑦 𝑓 𝑗𝑟𝑦 𝑂 𝑈 𝐾 𝜈 𝑦 , 𝐾 𝜉 0 𝑋 𝑂 𝑦 = 𝑅 2 /2𝑟 ⋅ 𝑄 § Still limited knowledge  Many on-going/planned experiments (EIC, LHeC , … facilities) Huey-Wen Lin — Lattice Workshop @ NTU 3

Part rton on Dis istri ributio tion n Functio ions ns § Structure function/distribution functions  Deep inelastic scattering (1960s) § Lattice QCD is an ideal theoretical tool for investigating strong-coupling regime of quantum field theories  Ideal tool for studying nonperturbative hadron structure , but… § Lattice calculations rely on operator product expansion For example, the unpolarized structure 1 𝑦, 𝑅 2 = 𝜈 2 𝑅 2 , 𝑕 𝜈 𝑟 2 𝑒𝑦 𝑦 𝑜−1 𝐺 𝑦 𝑜 𝑟 𝑑 1,𝑜 𝑟=𝑣,𝑒 2 𝑦, 𝑅 2 = 𝜈 2 𝑅 2 , 𝑕 𝜈 𝑟 𝑒𝑦 𝑦 𝑜−2 𝐺 𝑦 𝑜 𝑟 𝑑 2,𝑜 𝑟=𝑣,𝑒  c 1 , c 2 are Wilson coefficients Huey-Wen Lin — Lattice Workshop @ NTU 4

 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 — Lattice Workshop @ NTU 5

 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 — Lattice Workshop @ NTU 6

 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 (SCRI, SESAM): 2f, Wilson and clover Dolgov et al., PRD66, 034506 (2002) QCDSF: 0f Göckeler et al. PRD71, 114511 (2005)  x 3  q  x 2  q Huey-Wen Lin — Lattice Workshop @ NTU 7

Limited Access § What can we learn about the x -distribution?  Make an ansätz 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. Detmold et al, Eur.Phys.J.direct C3 (2001) 1 – 15 Huey-Wen Lin — Lattice Workshop @ NTU 8

The Idea § Approaching lightcone with large P  Just another limit to take, like taking a → 0 Xiangdong Ji, Phys. Rev. Lett. 111, 039103 (2013) Huey-Wen Lin — Lattice Workshop @ NTU 9

The Idea § Lightcone quark distribution Renormalization Gluon potential A + scale µ ― Lightcone coordinate ξ ± =( t ± z )/ √ 2 Nucleon momentum P µ § Approaching lightcone with large P  Just another limit to take, like taking a → 0 Xiangdong Ji, Phys. Rev. Lett. 111, 039103 (2013) Huey-Wen Lin — Lattice Workshop @ NTU 10

The Idea § Finite-momentum quark distribution x = k z / P z Lattice z coordinate Product of lattice gauge links Nucleon momentum P µ ={ P 0 ,0,0, P z }  In P z  limit, parton distribution is recovered  For finite P z , corrections are needed Xiangdong Ji, Phys. Rev. Lett. 111, 039103 (2013) Huey-Wen Lin — Lattice Workshop @ NTU 11

Some Lattice Details § Exploratory study  N f = 2+1+1 clover/HISQ lattices (MILC) M π ≈ 310 MeV, a ≈ 0.12 fm ( L ≈ 2.88 fm)  Isovector only (“disconnected” suppressed) t sep gives us flavor asymmetry between up and down quark  2 source-sink separation ( t sep ≈ 0.96 and 1.2 fm) used § Properties known on these lattices  Lattice Z Γ for bilinear operator ~ 1 (with HYP-smearing)  M π L ≈ 4.6 large enough to avoid finite-volume effects Hyak @ UW § Feasible with today’s computational resources!  O (hour) rewriting three-point insertion code (Chroma)  8/16 nodes on UW Hyak cluster Huey-Wen Lin — Lattice Workshop @ NTU 12

Quark Distribution § Exploratory study  How many links are needed?  Lattice momenta discretized P z  {1, 2, 3} 2 π ⁄ L by finite size of volume Huey-Wen Lin — Lattice Workshop @ NTU 13

Quark Distribution § Exploratory study P z  {1, 2, 3} 2 π ⁄ L Uncorrected bare Preliminary lattice results x = k z / P z Huey-Wen Lin — Lattice Workshop @ NTU 14

Quark Distribution § Exploratory study P z  {1, 2, 3} 2 π ⁄ L Distribution gets Preliminary sharper as P z increases Artifacts due to finite P z on the lattice Improvement? Work out leading- P z corrections Huey-Wen Lin — Lattice Workshop @ NTU 15

Quark Distribution § Back to the continuum Xiangdong Ji, Phys. Rev. Lett. 111, 039103 (2013) What we want Not included yet; What we calculate O(20%) systematic on the lattice Dominant correction P z  {1, 2, 3} 2 π ⁄ L (for nucleon); known scaling form J.-W. Chen Smaller P z correction but complicated twist-4 operator J.-H. Zhang, Y . Zhao, J.-W. Chen et (extrapolate it away) al. (in preparation) Huey-Wen Lin — Lattice Workshop @ NTU 16

Quark Distribution § Back to the continuum Xiangdong Ji, Phys. Rev. Lett. 111, 039103 (2013) What we want Not included yet; What we calculate O(20%) systematic on the lattice Dominant correction P z  {1, 2, 3} 2 π ⁄ L (for nucleon); known scaling form J.-W. Chen Smaller P z correction but complicated twist-4 operator J.-H. Zhang, Y . Zhao, J.-W. Chen et (extrapolate it away) al. (in preparation) § Changes in x , and q ( x ) Huey-Wen Lin — Lattice Workshop @ NTU 17

Quark Distribution § Exploratory study  Take ratios (partially cancel statistical and systematic errors) Removing Preliminary 2 /4 P 2 ) errors O ( M N z No significant finite-momentum effect seen for P z >1 § Renormalization needed Huey-Wen Lin — Lattice Workshop @ NTU 18

Quark Distribution § Compare with experiments K. Ackersta et al. (HERMES Collaboration), Phys.Rev.Lett. 81, 5519 (1998) Huey-Wen Lin — Lattice Workshop @ NTU 19

Quark Distribution § Compare with experiments Compared with E866 Too good to be true? Lost resolution in small- x region Future improvement to have larger lattice volume R. Towell et al. (E866/NuSea), Phys.Rev. D64, 052002 (2001) Huey-Wen Lin — Lattice Workshop @ NTU 20

Helicity Distribution § Exploratory study Preliminary Huey-Wen Lin — Lattice Workshop @ NTU 21

Helicity Distribution § Exploratory study −2 ) Corrected to O ( P z Preliminary Gray band shows − 4 terms extrapolation of P z −4 ) seen but Large O ( P z well fit by extrapolation Huey-Wen Lin — Lattice Workshop @ NTU 22

Helicity Distribution § Model: e.g. chiral quark-soliton model B. Dressler et al,hep-ph/9809487 § Experimental comparison A. Airapetian et al. (HERMES), Phys.Rev. D71, 012003 (2005) D. De Florian et al., PRL 101 (2008) 072001 Huey-Wen Lin — Lattice Workshop @ NTU 23

Helicity Distribution § Model: e.g. chiral quark-soliton model B. Dressler et al,hep-ph/9809487 § Experimental comparison A. Airapetian et al. (HERMES), Phys.Rev. D71, 012003 (2005) D. De Florian et al., PRL 101 (2008) 072001 Huey-Wen Lin — Lattice Workshop @ NTU 24

Transversity Distribution § Exploratory study Uncorrected bare Preliminary lattice results § Renormalization needed Huey-Wen Lin — Lattice Workshop @ NTU 25

Transversity Distribution § Exploratory study Removing Preliminary 2 /4 P 2 ) errors O ( M N z § Renormalization needed Huey-Wen Lin — Lattice Workshop @ NTU 26