Auxiliary field approach to extended operators for quasi-PDFs - PowerPoint PPT Presentation

Auxiliary field approach to extended operators for quasi-PDFs Jeremy Green in collaboration with Karl Jansen, Fernanda Steffens, and ETMC NIC, DESY, Zeuthen The 35th International Symposium on Latice Field Theory Granada, Spain June 18–24, 2017

Outline 1. Qasi-PDFs 2. Auxiliary field formalism: continuum 3. Auxiliary field formalism: latice 4. Relation to static quark theory 5. Non-perturbative renormalization 6. Effect on quasi-PDF data Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 2

Parton distribution functions and quasi-PDFs Parton distribution functions (PDFs): ◮ q ( x , µ ) , д ( x , µ ) describe probability of finding a quark or gluon with momentum fraction x of a proton’s total momentum. Qasi-PDFs: X. Ji, Phys. Rev. Let. 110 , 262002 [1305.1539]; many follow-up papers ◮ Idea: define a quasi-PDF ˜ q ( x , µ , p z ) with p z = p · n , using nucleon matrix elements of a nonlocal operator where ψ and ¯ ψ have spacelike separation in direction n . At large p z : � 1 Λ 2 m 2 q ( x , µ ) + O � � � x � dy y , µ QCD p ˜ q ( x , µ , p z ) = y Z , . � � p 2 p 2 p z x z z ◮ Boost so that n is pointing in a purely spatial direction. This makes it suitable for computing on the latice. → plenary talk by L. Del Debbio Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 3

Operator for quark quasi-PDFs We compute nucleon matrix elements of the operator O Γ ( x , ξ , n ) ≡ ¯ ψ ( x + ξn )Γ W ( x + ξn , x ) ψ ( x ) , where n 2 = 1 is a unit vector, ξ is the separation, and W is a Wilson line: � ξ � � dξ ′ n · A ( x + ξ ′ n ) W ( x + ξn , x ) ≡ P exp − iд . 0 On the latice we can restrict n to point along an axis, and simply form W from a product of gauge links. This is a non-local operator. How can we understand its renormalization? Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 4

Auxiliary field approach (Loosely following H. Dorn, Fortsch. Phys. 34 , 11 (1986) ) The Wilson line satisfies the equation of motion � d � dξ + iдn · A ( x + ξn ) W ( x + ξn , x ) = δ ( ξ ) . Introduce a scalar, color triplet field ζ n ( ξ ) that is defined on the line x + ξn . (We omit the subscript n most of the time.) Give it the action � d � � dξ ¯ S ζ = ζ dξ + iдn · A + m ζ . Then its propagator for fixed gauge background is � ζ ( ξ 2 ) ¯ � ζ = θ ( ξ 2 − ξ 1 ) W ( x 2 , x 1 ) e − m ( ξ 2 − ξ 1 ) ζ ( ξ 1 ) We want zero mass but there is no symmetry that forbids it. Unless we use dimensional regularization, a power-divergent counterterm is needed. Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 5

Auxiliary field approach: quark operator Introduce the spinor-valued color singlet ζ -quark bilinear ϕ ≡ ¯ ζψ . Then the extended operator for quasi-PDFs is given (for m = 0 and ξ > 0 ) by � ¯ � O Γ ( x , ξ , n ) = ϕ ( x + ξn )Γ ϕ ( x ) ζ . For ξ < 0 , we can use the relation O Γ ( x , ξ , n ) = O Γ ( x , − ξ , − n ) . Thus, any QCD correlator involving O Γ can be rewriten as a correlator in QCD+ ζ involving the local operators ϕ and ¯ ϕ . To renormalize this, we need: 1. Z ϕ to renormalize the local operators. 2. The mass counterterm. Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 6

Auxiliary field on the latice Discretize S ζ , restricting n to be n = ± ˆ µ : 1 � ¯ ζ ( x + ξn ) [ ∇ n + m 0 ] ζ ( x + ξn ) , S ζ = a 1 + am 0 ξ where    n · ∇ ∗ = ∇ ∗ if n = ˆ µ , µ  ∇ n = µ .  n · ∇ = −∇ µ , if n = − ˆ For n = ˆ µ , this yields the propagator � µ ) ¯ � ζ = θ ( ξ ) e − mξ U † � � U † � � . . . U † ζ ( x + ξ ˆ ζ ( x ) x + ( ξ − a ) ˆ µ x + ( ξ − 2 a ) ˆ µ µ ( x ) , µ µ where m = a − 1 log ( 1 + am 0 ) . (We could use smeared links U in defining the covariant derivative.) Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 7

Auxiliary field on the latice: renormalization and mixing Mixing on the latice first noted at one loop in M. Constantinou and H. Panagopoulos, 1705.11193 → talk at 17:10 . In our approach, this appears as mixing between ϕ and / nϕ when chiral symmetry is broken. The ζ -quark bilinear ϕ = ¯ ζψ renormalizes as � ¯ � � ¯ ϕ + r mix ¯ � ϕ R = Z ϕ ϕ + r mix / nϕ , ϕ R = Z ϕ ϕ / n . We can use P ± ≡ 1 2 ( 1 ± / n ) to define operators that don’t mix: ϕ ± ≡ P ± ϕ = ⇒ ϕ ± R = Z ± ϕ ϕ ± , where Z ± ϕ = Z ϕ ( 1 ± r mix ) . The renormalized extended quark bilinear has the form ϕ e − m | ξ | ¯ O R Γ ( x , ξ , n ) = Z 2 ψ ( x + ξn )Γ ′ W ( x + ξn , x ) ψ ( x ) , where Γ ′ = Γ + sgn ( ξ ) r mix { / n , Γ } + r 2 mix / n Γ / n . Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 8

Relation to static quark theory The Lagrangian for a static quark on the latice is 1 � Q ( x ) , Q ( x ) � ∇ ∗ ¯ L ( x ) = 0 + m 0 1 + am 0 where Q is a color triplet spinor satisfying 1 2 ( 1 + γ 0 ) Q = Q . Other than the spin degres of freedom (which don’t couple in the action) this is the same as for ζ with n = ˆ 0 . The propagators are also related: � Q ( x ) ¯ � � ζ ( x ) ¯ � Q ( y ) ζ ( y ) ζ P + . Q = In the continuum, the connection between renormalization of quasi-PDFs and the static quark theory was discussed in X. Ji and J.-H. Zhang, Phys. Rev. D 92 , 034006 [1505.07699] . With broken chiral symmetry, there are two renormalization factors for static-light bilinears: Z stat for ¯ Z stat for ¯ ψγ 0 Q and ψγ 0 γ 5 Q . V A Inserting P + , we identify Z stat ϕ and Z stat = Z − = Z + ϕ . V A Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 9

Further implications 1. Latice artifacts are O ( a ) . Even with chiral symmetry, the static-light currents need improvement at O ( a ) : e.g. ∗ � ← � 1 ← A stat , I = ¯ ψγ 0 γ 5 Q + ac stat ¯ ∇ j + ∇ ψγ j γ 5 Q . j 0 A 2 2. No mixing with gluons. ◮ The local bilinear ϕ = ¯ ζψ is in the flavor fundamental irrep. The corresponding gluon operator is flavor singlet. ◮ Mixing between quark and gluon PDFs must occur in: a. the matching from quasi-PDF to PDF, b. the dependence of quasi-PDFs on p z . Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 10

Nonperturbative approach In Landau gauge, compute the position-space ζ propagator � ζ ( x + ξn ) ¯ � S ζ ( ξ ) ≡ ζ ( x ) QCD + ζ = � W ( x + ξn , x ) � QCD , the momentum-space quark propagator � e − ip · x � ψ ( x ) ¯ � S q ( p ) ≡ ψ ( 0 ) , x and the mixed-space Green’s function for ϕ ± : � e ip · x � ζ ( ξn ) ϕ ± ( 0 ) ¯ � G ± ( ξ , p ) ≡ ψ ( x ) QCD + ζ . x These renormalize as S R ζ ( ξ ) = e − mξ Z ζ S ζ ( ξ ) , S R q ( p ) = Z q S q ( p ) , R ( ξ , p ) = e − mξ � G ± Z ζ Z q Z ± ϕ G ± ( ξ , p ) . Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 11

Power divergence Take the effective energy of the ζ propagator: E eff ( ξ ) ≡ − d dξ log Tr S ζ ( ξ ) . This renormalizes as E R eff ( ξ ) = m + E eff ( ξ ) . Determine m by matching to perturbation theory at small ξ : E eff ( ξ ) = − 3 α s C F + O ( α 2 s ) . 2 πξ Here we use fixed α s = 0 . 3 . Preliminary results from an N f = 4 twisted mass ensemble with β = 2 . 1 , or a = 0 . 064 fm. Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 12

Effective energy 0 . 8 thin links HYP links 5HYP links 0 . 6 perturbation theory 0 . 4 aE eff 0 . 2 0 . 0 − 0 . 2 0 5 10 15 20 25 30 ξ/ a Match thin links with perturbation theory at small ξ , then match thin with smeared links at larger ξ . Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 13

Effective energy 0 . 3 thin links HYP links 5HYP links 0 . 2 perturbation theory aE eff (renormalized) 0 . 1 0 . 0 − 0 . 1 − 0 . 2 − 0 . 3 0 5 10 15 20 25 30 ξ/ a Match thin links with perturbation theory at small ξ , then match thin with smeared links at larger ξ . Get am thin = − 0 . 42 , am 5HYP = − 0 . 09 . Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 13

RI-type renormalization scheme For Z ζ , we could use a condition For ϕ ± , “amputate” the Green’s function: 3 Tr S R ζ ( 2 ξ ) � 2 = 1 . Λ ± ( ξ , p ) ≡ S − 1 ζ ( ξ ) G ± ( ξ , p ) S − 1 � q ( p ) . Tr S R ζ ( ξ ) Both of these serve to eliminate the dependence on m . Then we could impose the condition 1 6 ℜ Tr Λ ± R ( p , ξ ) = 1 at some scale µ 2 = p 2 . This is a two-parameter family of schemes, which depends on the dimensionless parameters | p | ξ and ( n · p ) / | p | . Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 14

Estimator for Z ϕ 1 . 0 3 � Z ζ Z ψ ( 1 / ℜ Tr Λ + + 1 / ℜ Tr Λ − ) → Z φ 0 . 9 0 . 8 0 . 7 0 . 6 0 . 5 0 . 4 | p | ξ = π/ 2 | p | ξ = π 0 . 3 | p | ξ = 2 π | p | ξ = 4 π 0 . 2 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 a 2 p 2 solid symbols: p � n ; open symbols: p ⊥ n . Matching to MS and evolution to fixed scale still needed. Work is in progress to understand significant O ( a ) effects in r mix . Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 15

Qasi-PDF data on fine ensemble New calculation on fine N f = 2 + 1 + 1 twisted mass ensemble: ◮ β = 2 . 1 , a ≈ 0 . 065 fm ◮ m π ≈ 370 MeV ◮ 45 configurations × 4 source positions ◮ p z ≈ 1 . 8 GeV, using momentum smearing ◮ Various smearings applied to the links in the extended operator Jeremy Green | DESY, Zeuthen | Latice 2017 | Page 16