Parton physics from Euclidean current-current correlator with a - PowerPoint PPT Presentation

Parton physics from Euclidean current-current correlator with a valence heavy quark: pion light-cone distribution amplitude as an example C.-J. David Lin National Chiao Tung University, Taiwan Field Theory Team seminar, RIKEN RCCS, Kobe 18/11/2020 In collaboration with William Detmold, Anthony Grebe, Issaku Kanamori, Santanu Mondal, Robert Perry and Yong Zhao

Outline • General HOPE strategy W. Detmold and CJDL, PRD 73 (2006) • HOPE and the pion light-cone wavefunction W.Detmold, A.Grebe, I.Kanamori, CJDL, S.Mondal, R.Perry, Y.Zhao, Contribution to APLAT2020, arXiv:2009.09473. • Preliminary numerical result W.Detmold, A.Grebe, I.Kanamori, CJDL, S.Mondal, R.Perry, Y.Zhao, Contribution to APLAT2020, arXiv:2009.09473. • Outlook

General strategy

Parton distribution from lattice QCD The “traditional” approach Hadronic tensor (PDFs from the twist-2 sector) � W µ ν d 4 x e iq · x � p, S | [ J µ ( x ) , J ν (0)] | p, S � , S ( p, q ) =

Parton distribution from lattice QCD The “traditional” approach Hadronic tensor (PDFs from the twist-2 sector) � W µ ν d 4 x e iq · x � p, S | [ J µ ( x ) , J ν (0)] | p, S � , S ( p, q ) = challenging in Euclidean QCD optical theorem Imaginary part � T µ ν d 4 x e iq · x � p, S | T [ J µ ( x ) J ν (0)] | p, S � S ( p, q ) =

Parton distribution from lattice QCD The “traditional” approach Hadronic tensor (PDFs from the twist-2 sector) � W µ ν d 4 x e iq · x � p, S | [ J µ ( x ) , J ν (0)] | p, S � , S ( p, q ) = challenging in Euclidean QCD optical theorem Imaginary part � T µ ν d 4 x e iq · x � p, S | T [ J µ ( x ) J ν (0)] | p, S � S ( p, q ) = Light-cone OPE � x µ 1 . . . x µ n O µ ν µ 1 ...µ n T [ J µ ( x ) J ν (0)] = x 2 , µ 2 � � C i ( µ ) , i i,n local operators, issue of operator mixing leading moments in practice Power divergences arising from Lorentz symmetry breaking

Parton distribution from lattice QCD The “new” approach to avoid difficulties in renormalisation O µ ν µ 1 ...µ n O non − local i H ( p ) H ( p ′ ) H ( p ) H ( p ′ ) General idea: Inserting non-local, instead of local, operator

Parton distribution from lattice QCD The “new” approach to avoid difficulties in renormalisation - - - - - - - - O non − local O non − local H ( p ) H ( p ′ ) H ( p ) H ( p ′ ) Make certain the absence of on-shell states for analytic continuation

Parton distribution from lattice QCD The “new” approach to avoid difficulties in renormalisation O non − local H ( p ) H ( p ′ ) Typical examples of the non-local operator A space-like Wilson line (quasi-PDF and pseudo-PDF) X. Ji, PRL 110 (2013); A. Radyushkin, PRD 96 (2017) Smeared “local” operators Z. Davoudi and M. Savage, PRD 86 (2012) Two currents separated by space-like distance V. Braun and D. Mueller, EPJC 55 (2018) Two flavour-changing currents with valence heavy quark W. Detmold and CJDL, PRD 73 (2006) And other proposals A. Chambers et al. , PRL 118 (2017); Y. Ma and J.-W. Qiu, PRL 120 (2018);……

Introducing the valence heavy quark W. Detmold and CJDL, PRD 73 (2006) Valence Not in the action The “heavy quark” is relativistic propagating in both space and time The current for computing the even moments of the PDF J µ Ψ , ψ ( x ) = Ψ ( x ) γ µ ψ ( x ) + ψ ( x ) γ µ Ψ ( x ) Compton tensor � � � � � T µ ν � p, S | t µ ν J µ d 4 x e iq · x � p, S | T Ψ , ψ ( x ) J ν Ψ , ψ ( p, q ) ≡ Ψ , ψ ( q ) | p, S � = Ψ , ψ (0) | p, S � S S

Strategy for extracting the moments � � � T µ ν � � p, S | t µ ν � J µ d 4 x e iq · x � p, S | T Ψ , ψ ( x ) J ν Ψ , ψ ( p, q ) ≡ Ψ , ψ ( q ) | p, S � = Ψ , ψ (0) | p, S � S S J µ Ψ , ψ ( x ) = Ψ ( x ) γ µ ψ ( x ) + ψ ( x ) γ µ Ψ ( x ) • Simple renormalisation for quark bilinears. • Work with the hierarchy of scales q 2 ≤ m Ψ << 1 p Λ QCD << a Heavy scales for short-distance OPE. p Avoid branch point in Minkowski space ( q + p ) 2 ∼ ( m N + m Ψ ) 2 at • Extrapolate to the continuum limit first. T µ ν Ψ , ψ ( p, q ) Then match to the short-distance OPE results. Extract the moments without power divergence.

Short-distance OPE & valence heavy quark These are the leading-twist contributions that we are after. leading and higher twist T µ ν Ψ ,v = T µ ν Ψ ,u − T µ ν Ψ ,d . leading twist, absent in higher twist, absent ambiguity in heavy quark mass

HOPE and pion light-cone distribution amplitude W.Detmold, A.Grebe, I.Kanamori, CJDL, S.Mondal, R.Perry, Y.Zhao, Contribution to APLAT2020, arXiv:2009.09473.

Pion light-cone wavefunction � π + ( p ) � ↵ h 0 | ψ ( z 2 n ) / n γ 5 W [ z 2 n, z 1 n ] ψ ( z 1 n ) Z 1 dx e − i ( z 1 x + z 2 (1 − x )) p · n φ π ( x, µ 2 ) = if π ( p · n ) 0 Z 1 Mellin moments d ⇠ ⇠ n � ( ⇠ , µ 2 ) h ⇠ n i µ 2 = π − 1 OPE ⟨ 0 | O μ 1 ... μ n | π ( p ) ⟩ = f π ⟨ ξ n − 1 ⟩ [ p μ 1 . . . p μ n − traces] ψ O μ 1 ... μ n ψγ 5 γ { μ 1 ( iD μ 2 ) . . . ( iD μ n } ) ψ − traces = ¯ ψ

Phenomenological relevance Pion form factor in QCD exclusive processes Important input for flavour physics Φ M 2 M 2 T II Φ B i B Φ M 1 M 1

Hadronic tensor for computing pion LCDA � q µ J μ J ν A ( − z /2) A ( z /2) the cross term + π ( p ) Z d 4 z e iq · z h 0 | T [ J µ T µ ν ( p, q ) = A ( z/ 2) J ν A ( � z/ 2)] | ⇡ ( p ) i A = ¯ Ψ � µ � 5 + ¯ J µ � µ � 5 Ψ , is the valence, relativistic heavy quark Ψ ✓ ◆ U µ ν ( p, q ) = 1 T µ ν ( p, q ) � T ν µ ( p, q ) 2 Z

OPE for the hadronic tensor: Euclidean result ∞ ⇣ n C 2 U µ ν ( p, q ) = 2 ✏ µ ναβ q α p β n ( ⌘ ) 2 n ( n + 1) C ( n ) W ( ˜ Q 2 ) f π h ⇠ n i + O (1 / ˜ X Q 3 ) ˜ Q 2 n even higher-twist GeV in this talk μ = 2 Q 2 = q 2 + m 2 ˜ p Ψ � � tree-level OPE p · q p p 2 q 2 ⌘ = ⇣ = , one-loop p ˜ p 2 q 2 Q 2 p fit lattice data : target-mass effect C 2 n ( ⌘ )

OPE for U μν : issue in fitting higher moments ∞ ω = 2 p · q 2 p · q 2 iE π q 4 X h ξ n i ω n , = + U µ ν ( p , q ) ⇠ 4 + q 2 + m 2 4 + q 2 + m 2 ˜ q 2 q 2 Q 2 Q Q n = 0 suppressing higher-moments need large p to make ω → 1 Im { ω } ω Re { ω } − 1 1 allowed region for imaginary q 4

OPE for U μν : issue in fitting higher moments U µ ν ( p, q ) = 2 ✏ µ ναβ q α p β ∞ ⇣ n C 2 n ( ⌘ ) 2 n ( n + 1) C ( n ) X W ( ˜ Q 2 ) f π h ⇠ n i + O (1 / ˜ Q 3 ) ˜ Q 2 n even ∞ = 2 ✏ µ ναβ q α p β W ( n ) C ( n ) X W ( ˜ Q 2 ) f π h ⇠ n i + O (1 / ˜ Q 3 ) ˜ Q 2 n even max{ } 0.100 Re [ W ( n )] 0.001 � =( 0,0,0 ) , q � =( 0,0,1 ) p 10 - 5 � =( 1 � =( 1,0,0 ) , q 2 ,0,1 ) p � =( 4,0,0 ) , q � =( 2,0,1 ) p 10 - 7 0 1 2 3 4 5 6 n In general, need large p to access non-leading moments

Strategy for fitting ⟨ ξ 2 ⟩ at low pion momentum U 12 ( p, q ) = 2 ✏ 12 αβ q α p β ∞ ⇣ n C 2 n ( ⌘ ) 2 n ( n + 1) C ( n ) X W ( ˜ Q 2 ) f π h ⇠ n i + O (1 / ˜ Q 3 ) ˜ Q 2 n even Q 2 ) f π + 6( p · q ) 2 � p 2 q 2  � = 2( q 3 p 4 � q 4 p 3 ) C (0) C (2) W ( ˜ W ( ˜ + O (1 / ˜ Q 2 ) f π h ⇠ 2 i + . . . Q 3 ) ˜ 6( ˜ Q 2 Q 2 ) 2  � choose p q 0 while =0, and being real ⋅ ≠ q 3 ≠ 0 p 3 q 4  � Q 2 ) f π + 6( p · q ) 2 � p 2 q 2  � = 2 iq 3 E π C (0) C (2) W ( ˜ W ( ˜ + O (1 / ˜ Q 2 ) f π h ⇠ 2 i + . . . Q 3 ) ˜ 6( ˜ Q 2 Q 2 ) 2 imaginary real complex U 12 ⟨ ξ 2 ⟩ The largest contribution to Re[ ] is from

Correlators for lattice calculation Excited states? Z C µ ν d 3 x e d 3 x m e i p e · x e e i p m · x m 3 ( τ e , τ m ; p e , p m ) = J µ A ( x m , τ m ) O † ⇥ ⇤ h 0 |T A ( x e , τ e ) J ν π ( 0 , 0) | 0 i Z d 3 x e i p · x h 0 |O π ( x , ⌧ π ) O † C 2 ( ⌧ π , p ) = π ( 0 , 0) | 0 i

R μν and the Fourier transform for U μν C μν From and , one can construct C 2 3 Z J µ ⇣ z � z h ⌘ J ν ⇣ ⌘i R µ ν ( τ ; p , q )= d 3 z e i q · z h 0 |T | π ( p ) i 2 2 � p = p e + p m , q = 1 2( p m � p e ) z = x e � x m , Then the hadronic tensor can be obtained via Z U µ ν ( p, q ) ⌘ d τ e iq 4 τ R [ µ ν ] ( τ ; p , q )

Exploratory quenched calculation @ MeV M π ≈ 560 Wilson plaquette and non-perturbatively improved clover actions L 3 ⇥ ˆ ˆ a (fm) T N config N src bare m Ψ fitted m Ψ 0.081 24 3 ⇥ 48 650 2 1.0 GeV 2.0 GeV 0.060 32 3 ⇥ 64 450 3 1.6 GeV 2.6 GeV 0.048 40 3 ⇥ 80 250 3 2.5 GeV 3.3 GeV 0.041 48 3 ⇥ 96 341 3 p = (1,0,0) q = (1/2,0,1) in units of GeV 2 π / L ∼ 0.64 U μν is improved without improving the axial current O ( a )

Excited-state contamination 0.030 � e = 0.24 fm 0.029 � e = 0.36 fm 0.025 � e = 0.48 fm � - 1 [ Im ( U �� )] ( GeV 2 ) � - 1 [ Im ( U �� )] ( GeV 2 ) 0.028 � e = 0.6 fm 0.020 � e = 0.72 fm 0.027 � e = 0.84 fm 0.015 0.026 0.010 0.025 0.005 fm a = 0.060 0.024 0.000 0.0 0.1 0.2 0.3 0.3 0.4 0.5 0.6 0.7 0.8 � e ( fm ) � ( fm )

Continuum extrapolation of U 12 q 4 = − 1 . 2 GeV q 4 = − 1 . 2 GeV m Q = 2 . 0 GeV m Q = 2 . 0 GeV 0 . 032 Re[ U 12 ( p, q )] (GeV) Im[ U 12 ( p, q )] (GeV) Preliminary Preliminary − 0 . 00006 0 . 030 0 . 028 − 0 . 00008 0 . 026 0 . 0 0 . 1 0 . 0 0 . 1 a 2 (GeV − 2 ) a 2 (GeV − 2 )