Partonic quasi-distributions of the pion in chiral quark models - PowerPoint PPT Presentation

Partonic quasi-distributions of the pion in chiral quark models Enrique Ruiz Arriola and Wojciech Broniowski Dep. F´ ısica At´ omica,Molecular y Nuclear Instituto Carlos I de F´ ısica Te´ orica y Computacional Universidad de Granada Resummation, Evolution, Factorization 2017 (REF2017) 13-16 November 2017, Madrid details in Phys.Lett. B773 (2017) 385-390, arXiv:1707.09588 Enrique Ruiz Arriola () Quasi-distributions REF2017 1 / 36

Outline Parton distributions – basic properties of hadrons Soft matrix elements, accessible from low-energy models of QCD Chiral quark models of the pion Parton quasi-distributions, designed for Euclidean QCD lattices Results and predictions for quasi-distributions of the pion from chiral quark models Enrique Ruiz Arriola () Quasi-distributions REF2017 2 / 36

Introduction

Parton distribution Q 2 = − q 2 , x = 2 p · q , Q 2 → ∞ Q 2 Factorization of soft and hard processes, Wilson’s OPE � C i ( Q 2 ; µ ) �O i ( µ ) � � J ( q ) J ( − q ) � = i F ( x, Q ) = F 0 ( x, α ( Q )) + F 2 ( x,α ( Q )) Twist expansion → + . . . Q 2 Enrique Ruiz Arriola () Quasi-distributions REF2017 4 / 36

Parton distribution Q 2 = − q 2 , x = 2 p · q , Q 2 → ∞ Q 2 Factorization of soft and hard processes, Wilson’s OPE � C i ( Q 2 ; µ ) �O i ( µ ) � � J ( q ) J ( − q ) � = i F ( x, Q ) = F 0 ( x, α ( Q )) + F 2 ( x,α ( Q )) Twist expansion → + . . . Q 2 Bj limit → light-cone momentum is constrained: k + ≡ k 0 + k 3 = xP + x ∈ [0 , 1] Enrique Ruiz Arriola () Quasi-distributions REF2017 4 / 36

Distribution amplitude (DA) of the pion Enters various measures of exclusive processes, e.g., pion-photon transition form factor Enrique Ruiz Arriola () Quasi-distributions REF2017 5 / 36

Field-theoretic definition (here for quarks in the pion, leading twist) Parton Distribution Function (DF): � dz − 4 π e ixP + z − � P | ¯ ψ (0) γ + U [0 , z ] ψ ( z ) | P � � q ( x ) = � z + =0 ,z ⊥ =0 Parton Distribution Amplitude (DA): � dz − 2 π e i ( x − 1) P + z − � P | ¯ i ψ (0) γ + γ 5 U [0 , z ] ψ ( z ) | vac � � φ ( x ) = � z + =0 ,z ⊥ =0 F π (isospin suppressed) v ± ≡ v 0 ± v 3 - light-cone basis P - pion momentum, � z 2 � � z 1 dξλ a A + U [ z 1 , z 2 ] = exp − ig s a ( ξ ) - Wilson’s gauge link x - fraction of the light-cone mom. P + carried by the quark, x ∈ [0 , 1] Enrique Ruiz Arriola () Quasi-distributions REF2017 6 / 36

Remarks Only indirect experimental information for the pion distributions: DF from Drell-Yan in E615, DA from dijets in E791 and from exclusive processes involving pions Impossibility to implement PDF or PDA on the euclidean lattices, only lowest moments can be obtained However, there exist (largely forgotten) simulations on transverse lattices – discussed later Enrique Ruiz Arriola () Quasi-distributions REF2017 7 / 36

Quasi-distributions

Parton quasi-distributions (quarks in the pion) [Ji 2013] Parton Quasi-Distribution Function (QDF): � dz 3 4 π e iyP 3 z 3 � P | ¯ ψ (0) γ 3 U [0 , z ] ψ ( z ) | P � � V ( y ; P 3 ) = � z 0 =0 ,z ⊥ =0 Parton Quasi-Distribution Amplitude (PDA): � dz 3 i 2 π e i ( y − 1) P 3 z 3 � P | ¯ ˜ ψ (0) γ + γ 5 U [0 , z ] ψ ( z ) | vac � � φ ( y ; P 3 ) = � z 0 =0 ,z ⊥ =0 F π y - fraction of pion’s P 3 carried by the quark Analogy to DF and DA, but y is not constrained Basic property: ˜ P 3 →∞ V ( x ; P 3 ) = q ( x ) , lim lim φ ( x ; P 3 ) = φ ( x ) P 3 →∞ Enrique Ruiz Arriola () Quasi-distributions REF2017 9 / 36

QDF and QDA in the momentum representation Constrained longitudinal momenta, but y ∈ ( −∞ , ∞ ) (partons can move “backwards”) Enrique Ruiz Arriola () Quasi-distributions REF2017 10 / 36

Chiral quark models

Chiral quark models χ SB breaking → massive quarks Point-like interactions Soft matrix elements with pions (and photons, W , Z ) One-quark loop, regularization: 1) Pauli-Villars (PV) 2) Spectral Quark Model (SQM) - implements VMD Quantities evaluated at the quark model scale (where constituent quarks are the only degrees of freedom) Enrique Ruiz Arriola () Quasi-distributions REF2017 12 / 36

Chiral quark models χ SB breaking → massive quarks Point-like interactions Soft matrix elements with pions (and photons, W , Z ) One-quark loop, regularization: 1) Pauli-Villars (PV) 2) Spectral Quark Model (SQM) - implements VMD Need for evolution Gluon dressing, renorm-group improved Enrique Ruiz Arriola () Quasi-distributions REF2017 12 / 36

Chiral quark models Chiral quark models implement chiral symmetry: Pion is a Goldstone boson in the chiral limit Realization of the large N c limit Fully covariant relativistic: Rest frame, light cone, euclidean Chiral Perturbation Theory: Gasser-Leutwyler coeffs L i ∼ N c / (4 π ) 2 Chiral anomaly: Wess-Zumino term π 0 → 2 γ , γ → π 0 π + π − , etc. Electromagnetic, Transition, Gravitational Form factors Parton distribution amplitudes ( ϕ π ( x ) = 1 ) Generalized Parton Distributions ( ¯ d π ( x ) = u π ( x ) = 1 ) Polyakov cooling in chiral phase transitions � P � ∼ e − M/T Baryons as Solitons with dynamical topology (SU(3), ...) Enrique Ruiz Arriola () Quasi-distributions REF2017 13 / 36

Chiral quark models Axial current A ( x ) = 1 J µ,a q ( x ) γ µ γ 5 τ a q ( x ) ∂ µ J µ,a A ( x ) = − f π m 2 π π a 2 ¯ ( PCAC ) Chiral Ward Identity for irreducible vertex (chiral limit, m π → 0 ) S ( p + q ) − 1 γ 5 1 2 τ a + γ 5 1 2 τ a S ( p ) − 1 = q µ Γ µ,a A ( p + q, p ) If S ( p ) = 1 / ( / p − M ) then a solution A ( p + q, p ) = τ a γ µ − q µ � 2 M � q 2 = 0 Γ µ,a 2 γ 5 iff M � = 0 q 2 f π Pion quark coupling constant (Goldberger-Treiman relation) g πqq = M/f π Enrique Ruiz Arriola () Quasi-distributions REF2017 14 / 36

� � � � � � Wave functions in chiral quark models Pion Bethe-Salpeter wave function i i χ a q ( p ) = / − M − mg πqq γ 5 τ a p + q / p − M − m / � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ET wave functions d 3 p dp 0 � � (2 π ) 3 e i� p · � r Ψ a ( r ) = − (2 π )Tr[Γ a χ q ( p )] . LC wave functions d 2 p ⊥ dp − � b ⊥ + ixm π p + � p ⊥ · � Ψ a ( � (2 π ) 3 e i� b ⊥ , x ) = − (2 π )Tr[Γ a χ q ( p )] . Enrique Ruiz Arriola () Quasi-distributions REF2017 15 / 36

Equal Time Wave functions in the NJL model Analytic results for m π = 0 Ψ P ( r ) = 2Ψ T ( r ) = g πqq N c (2 MK 1 ( Mr ) + regulators) , 2 π 2 r Ψ A ( r ) = g πqq MN c (2 K 0 ( Mr ) + regulators) 4 π 2 where K 0 and K 1 are the modified Bessel functions. Asymptotic behavior at r → ∞ : Ψ P ( r ) ∼ Ψ T ( r ) ∼ e − Mr r 3 / 2 , Ψ A ( r ) ∼ e − Mr r 1 / 2 . Longer tail in the A channel than in the P and T channels. Enrique Ruiz Arriola () Quasi-distributions REF2017 16 / 36

Light Cone Wave functions in the NJL model Analytic results for m π = 0 Ψ P ( b, x ) = 2Ψ T ( b, x ) = g πqq N c (2 MK 1 ( Mb ) + regulators) , 2 π 2 b Ψ A ( b, x ) = g πqq MN c (2 K 0 ( Mb ) + regulators) 4 π 2 These functions are x independent !! Model Relation between ET and LC Ψ A ( b, x ) = Ψ A ( r ) | r = b Pion Distribution Amplitude (flat) ϕ π ( x ) = Ψ A (0 ⊥ , x ) = 1 Enrique Ruiz Arriola () Quasi-distributions REF2017 17 / 36

Form factors (SQM) Spectral representation � dw ρ ( w ) Z ( p ) S ( p ) = p − w ≡ / / p − Σ( p ) C Vector meson dominance for em form factor � 1 m 2 N c � ω 2 + x (1 − x ) t F em ( Q 2 ) = − dωρ ( ω ) ω 2 ρ � � dx log ≡ Q 2 + m 2 4 π 2 f 2 π ρ 0 Solution 1 1 1 → ρ V ( ω ) = 2 πi ω (1 − 4 ω 2 /m 2 ρ ) 5 / 2 m 2 ρ f 2 24 π 2 = (87MeV) 2 π = N c TMD ( m π = 0) 3 m 3 T � = m 2 ρ θ ( x ) θ (1 − x ) ρ ρ / 4) 5 / 2 → � k 2 q ( x, k T ) = 16 π ( k 2 T + m 2 2 Enrique Ruiz Arriola () Quasi-distributions REF2017 18 / 36

Scale and evolution QM provide non-perturbative result at a low scale Q 0 F ( x, Q 0 ) | model = F ( x, Q 0 ) | QCD , Q 0 − the matching scale Determination of Q 0 via momentum fraction: quarks carry 100% of momentum at Q 0 . One adjusts Q 0 in such a way that when evolved to Q = 2 GeV, the quarks carry the experimental value of 47% LO DGLAP evolution: Q 0 = 313 +20 − 10 MeV [Davidson, Arriola 1995]: q ( x ; Q 0 ) = 1 Enrique Ruiz Arriola () Quasi-distributions REF2017 19 / 36

Older results from chiral quark models w/ evolution

Pion quark DF, QM vs. E615 LO DGLAP evolution to the scale Q 2 = (4 GeV) 2 : points: Fermilab E615, Drell-Yan line: QM evolved to Q = 4 GeV Enrique Ruiz Arriola () Quasi-distributions REF2017 21 / 36

Pion quark DF, QM vs. transverse lattice points: transverse lattice [Dalley, van de Sande 2003] yellow: QM evolved to 0.35 GeV pink: QM evolved to 0.5 GeV dashed: GRS param. at 0.5 GeV Enrique Ruiz Arriola () Quasi-distributions REF2017 22 / 36