Ma Match chin ing ti ti e Q e Qua uasi P i Par tp tp n Dis ts ts ibu tj tj on in a Momentu tum Su Sub ts ts ac ac tj tj on n Sc Sche heme me Yong Zhao Massachusetts Institute of Technology The 36th Annual International Symposium on Lattice Field Theory East Lansing, MI, USA 07/22-28, 2018 I. Stewart and Y.Z., PRD97 (2018), 054512 7/23/18 Lattice 2018, East Lansing 1
Outline Renormalization of the quasi-PDF Matching between RI/MOM quasi-PDF and MSbar PDF The “ratio scheme” Lattice 2018, East Lansing 7/23/18
Procedure of Systematic Calculation 1. Simulation of the quasi PDF 3. Subtraction of higher in lattice QCD twist corrections Z +1 Λ 2 ✓ M 2 ✓ x ◆ ◆ dy y , ˜ P z , µ µ QCD q i ( x, P z , ˜ ˜ µ ) = | y | C ij q j ( y, µ ) + O , , |y| P z P 2 P 2 � 1 z z 2. Renormalization of the lattice 4. Matching to the MSbar PDF. quasi PDF, and then taking the continuum limit Lattice 2018, East Lansing 7/23/18
Renormalization The gauge-invariant quark Wilson line operator can be renormalized multiplicatively in the coordinate space: ( ) ψ (0) = Z ψ , z e − δ m | z | ψ ( z ) Γ W ( z ,0) R ! O Γ ( z ) = ψ ( z ) Γ W ( z ,0) ψ (0) X. Ji, J.-H. Zhang, and Y.Z., 2017; J. Green, K. Jansen, and F. Steffens, 2017; T. Ishikawa, Y.-Q. Ma, J. Qiu, S. Yoshida, 2017. Different renormalization schemes can be converted to each other in coordinate space; Z MS ( ε , µ ) 2 , µ R ! ! µ ) ! Q X ( ζ , z 2 µ R 2 ) = Q MS ( ζ , z 2 µ 2 ) = Z ' X ( z 2 µ R Q MS ( ζ , z 2 µ 2 ) Z X ( ε , z 2 µ R 2 ) Lattice 2018, East Lansing 7/23/18
Regulator independence If we apply the same renormalization scheme in both lattice and continuum theories, ! − 1 ( z , ε , µ ) ! R ( z , µ ) = Z X O Γ ( z , ε ) O Γ − 1 ( z , a − 1 , µ ) ! O Γ ( z , a − 1 ) = lim a → 0 Z X This should apply to all renormalization schemes; After renormalization, we just need to calculate the matching coefficient in dimensional regularization; However, not all schemes can be implemented nonperturbatively on the lattice. Lattice 2018, East Lansing 7/23/18
A momentum subtraction scheme Martinelli et al., 1994 Regulator-independent momentum subtraction scheme (RI/MOM): − 1 ( z , a − 1 , p R z , µ R ) p ! = p ! O Γ ( z , a − 1 ) p Z OM O Γ ( z ) p p 2 = µ R 2 tree p z = p R z p ! p ! O Γ ( z , a − 1 ) p O Γ ( z ) p p 2 = µ R 2 p 2 = µ R 2 z , µ R ) = p z = p R z p z = p R z Z OM ( z , a − 1 , p R = p ! z * z O Γ ( z ) p − ip R Γ ζ ) e (4 p R tree Can be implemented nonperturbatively on the lattice. Scales introduced in renormalization: µ R , p R z . 7/23/18 Lattice 2018, East Lansing
Matching coefficient Strategy: Extracting matching coefficient by comparing the quasi-PDF and light-cone PDF in an off-shell quark state; Quark off-shellness p 2 < 0 regulates the infrared (IR) and collinear divergences; Lattice 2018, East Lansing 7/23/18
One-loop Feynman diagrams z z z z k k k k p p p p p p p p q (1) q (1) q (1) ˜ tadpole ( z ) ˜ sail ( z ) ˜ vertex ( z ) Dimensional regularization d=4-2 ε ; Γ = γ z for discussion in this talk, Γ = γ t case calculated in Y.S. Liu et al. (LP 3 ), arXiv:1807.06566. External momentum p μ = ( p 0 ,0,0,p z ) and p 2 <0 ; Lattice 2018, East Lansing 7/23/18
One-loop results I. Stewart and YZ, PRD 2018 One-loop bare matrix element: Z 1 ⇢ ≡ ( − p 2 − i " ) q (1) ( z, p z , 0 , − p 2 ) = ↵ s C F ⇣ e � ixp z z − e � ip z z ⌘ (4 p z ⇣ ) , ˜ dx h ( x, ⇢ ) p 2 2 ⇡ z �1 ln 2 x − 1 + √ 1 − ⇢ 1 + x 2 1 � 8 ⇢ ⇢ 4 x ( x − 1) + ⇢ + 1 x > 1 √ 1 − ⇢ 1 − x − 2 x − 1 − √ 1 − ⇢ − > > 2(1 − x ) > > > ln 1 + √ 1 − ⇢ > 1 + x 2 > 1 � 2 x ⇢ < h ( x, ⇢ ) ≡ 0 < x < 1 , √ 1 − ⇢ 1 − x − 1 − √ 1 − ⇢ − 2(1 − x ) 1 − x > > ln 2 x − 1 − √ 1 − ⇢ > 1 + x 2 1 � > ⇢ ⇢ > > 2 x − 1 + √ 1 − ⇢ + 4 x ( x − 1) + ⇢ − 1 x < 0 √ 1 − ⇢ 1 − x − > : 2(1 − x ) Formally satisfies vector current conservation (v.c.c.), but: ∞ | x | →∞ h ( x , ρ )~ − 3 ∫ h ( x , ρ ) is logarithmically divergent needs ε to be regularized! lim 2| x |, dx −∞ This logarithmic divergence is what needs to be treated carefully for the MSbar scheme; Izubuchi, Ji, Jin, Stewart and Y.Z., 2018 Not a problem for the RI/MOM scheme! Lattice 2018, East Lansing 7/23/18
RI/MOM renormalization q (1) R , µ R ) = � Z (1) q (0) ( z, p z ) . CT ( z, p z , p z OM ( z, p z ˜ R , 0 , µ R ) ˜ (29) Renormalization in coordinate space: (1) ( z , p z , p R z , − p 2 , µ R ) = ! z , µ R ) ! q (1) ( z , p z ,0, − p 2 ) + ! (1) ( z , p z , p R q OM q CT 2 − p 0 Z 1 2 ρ = − p 2 2 = p z q (1) ( z, p z , 0 , − p 2 ) = ↵ s C F ⇣ e � ixp z z − e � ip z z ⌘ < 1 in Minkowski space (4 p z ⇣ ) ˜ dx h ( x, ⇢ ) Z 2 p z p z 2 ⇡ ⇣ ⌘ �1 −∞ 4 ) 2 + ( p R Z ∞ r R = µ R z ) 2 = ( p R 2 z ) 2 R , µ R ) = � α s C F ⇣ R z − ip z z � e − ip z z ⌘ q (1) e i (1 − x ) p z CT ( z, p z , p z (4 p z ζ ) > 1 for Euclidean momentum, ˜ dx h ( x, r R ) , 2 π ( p R ( p R z ) 2 −∞ analytical continuuation from ρ < 1! Identify the collinear divergence: onshell limit! (1) ( z , p z , p R z , − p 2 << p z z , µ R ) q (1) ( z , p z ,0, − p 2 << p z ! 2 , µ R ) = ! 2 ) + ! (1) ( z , p z , p R q OM q CT 1 + x 2 8 x 1 � x ln x � 1 + 1 x > 1 > > > > > > Z ∞ z ) = α s C F > 1 + x 2 1 � x ln 4 2 x ⇣ e − ixp z z � e − ip z z ⌘ < q (1) ( z, p z , 0 , � p 2 ⌧ p 2 (4 p z ζ ) ˜ dx h 0 ( x, ρ ) , h 0 ( x, ρ ) ⌘ , 0 < x < 1 ρ � 2 π 1 � x −∞ > > > Z 1 + x 2 > 1 � x ln x � 1 ⇣ ⌘ > > � 1 x < 0 > : x Lattice 2018, East Lansing 7/23/18
RI/MOM renormalization Fourier transform to obtain the x -dependent quasi-PDF: Z dz 2 π e ixzp z ˜ q (1) q (1) η ≡ p z OM ( x, p z , p z OM ( z, p z , p z ˜ R , µ R ) = R , µ R ) z p R ⇢Z = α s C F ⇥ ⇤⇥ ⇤ (4 ζ ) dy δ ( y � x ) � δ (1 � x ) h 0 ( y, ρ ) � h ( y, r R ) 2 π �� � A plus function + h ( x, r R ) � | η | h 1 + η ( x � 1) , r R , One can explicitly check that the RI/MOM quasi-PDF satisfies v.c.c.: ⎡ ⎤ z , − p 2 , µ R ) = α S C F ∞ ∞ ∞ (1) ( x , p z , p R ! dx q OM 2 π (4 ζ ) dx h ( x , r R ) dx | η | h (1 + | η |( x − 1), r R ) ⎥ = 0 ∫ ∫ ∫ − ⎢ ⎣ ⎦ −∞ −∞ −∞ Lattice 2018, East Lansing 7/23/18
RI/MOM renormalization Full result of RI/MOM quasi-PDF: Plus functions with δ -function at x =1 q (1) OM ( x, p z , p z ˜ R , µ R ) (37) √ r R − 1 1 + x 2 1 + x 2 x 2 r R � r R � 8 1 − x ln arctan 2 x − 1 + x > 1 x − 1 − √ r R − 1 1 − x − > > 2(1 − x ) 4 x ( x − 1) + r R > > � > > 1 + x 2 1 − x ln 4( p z ) 2 1 + x 2 = α s C F > 2 r R � � √ < arctan r R − 1 0 < x < 1 (4 ζ ) √ r R − 1 − 1 − x − − p 2 2 π 2(1 − x ) + > > √ r R − 1 > 1 + x 2 1 + x 2 � � 1 − x ln x − 1 2 r R r R > > + arctan x < 0 > √ r R − 1 1 − x − 2 x − 1 − > : x 2(1 − x ) 4 x ( x − 1) + r R + α s C F ⇢ �� � (4 ζ ) h ( x, r R ) − | η | h 1 + η ( x − 1) , r R . 2 π Unregulated divergence in the δ ( 1-x ) part? No! z , − p 2 , µ R ) ~ 1 (1) ( x , p z , p R | x | →∞ ! lim q OM x 2 , integrable at infinity, no need to regularize! MSbar PDF: 8 0 x > 1 > 1 + x 2 1 − x ln µ 2 − p 2 − 1 + x 2 > � q (1) ( x, µ ) = α s C F < ⇥ ⇤ 1 − x ln x (1 − x ) − (2 − x ) 0 < x < 1 (4 ζ ) . 2 π + > > 0 x < 0 : Lattice 2018, East Lansing 7/23/18
Matching coefficient Matching coefficient for isovector quasi-PDF in quark : ξ = x p z , p z ✓ ξ , µ R , µ ◆ C OM − δ (1 − ξ ) (40) y p z p z R R √ r R − 1 1 + ξ 2 ξ − 1 − 2(1 + ξ 2 ) − r R r R � 8 ξ 1 − ξ ln (1 − ξ ) √ r R − 1 arctan + ξ > 1 > > 2 ξ − 1 4 ξ ( ξ − 1) + r R > > � > > > + (2 − ξ ) − 2 arctan √ r R − 1 > 1 + ξ 2 1 − ξ ln 4( p z ) 2 + 1 + ξ 2 ⇢ 1 + ξ 2 > r R �� = α s C F < ⇥ ⇤ 1 − ξ ln ξ (1 − ξ ) 0 < ξ < 1 √ r R − 1 1 − ξ − µ 2 2(1 − ξ ) 2 π + > > > √ r R − 1 > > 1 + ξ 2 1 + ξ 2 1 − ξ ln ξ − 1 2 r R � r R � > > + arctan ξ < 0 > √ r R − 1 1 − ξ − − > 2(1 − ξ ) 2 ξ − 1 4 ξ ( ξ − 1) + r R : ξ + α s C F ⇢ �� � h ( ξ , r R ) − | η | h 1 + η ( ξ − 1) , r R , 2 π Matching coefficient for isovector nucleon quasi-PDF p z → yP z , η = yP z / p R z RI/MOM matching also preserves particle number conservation of the nucleon PDF! Lattice 2018, East Lansing 7/23/18
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