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

▶
partonic quasi distributions of the pion in chiral quark
SMART_READER_LITE
LIVE PREVIEW

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

Aug 22, 2023 19 likes •402 views

Partonic quasi-distributions of the pion in chiral quark models Enrique Ruiz Arriola and Wojciech Broniowski Dep. F sica At omica,Molecular y Nuclear Instituto Carlos I de F sica Te orica y Computacional Universidad de Granada

slide-1
SLIDE 1

Partonic quasi-distributions of the pion in chiral quark models

Enrique Ruiz Arriola and Wojciech Broniowski

  • Dep. F´

ısica At´

  • mica,Molecular y Nuclear

Instituto Carlos I de F´ ısica Te´

  • rica 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

slide-2
SLIDE 2

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

slide-3
SLIDE 3

Introduction

slide-4
SLIDE 4

Parton distribution

Q2 = −q2, x =

Q2 2p·q, Q2 → ∞

Factorization of soft and hard processes, Wilson’s OPE J(q)J(−q)=

  • i

Ci(Q2; µ)Oi(µ)

Twist expansion → F(x, Q) = F0(x, α(Q)) + F2(x,α(Q))

Q2

+ . . .

Enrique Ruiz Arriola () Quasi-distributions REF2017 4 / 36

slide-5
SLIDE 5

Parton distribution

Q2 = −q2, x =

Q2 2p·q, Q2 → ∞

Factorization of soft and hard processes, Wilson’s OPE J(q)J(−q)=

  • i

Ci(Q2; µ)Oi(µ)

Twist expansion → F(x, Q) = F0(x, α(Q)) + F2(x,α(Q))

Q2

+ . . . Bj limit → light-cone momentum is constrained: k+ ≡ k0 + k3 = xP + x ∈ [0, 1]

Enrique Ruiz Arriola () Quasi-distributions REF2017 4 / 36

slide-6
SLIDE 6

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

slide-7
SLIDE 7

Field-theoretic definition (here for quarks in the pion, leading twist)

Parton Distribution Function (DF): q(x) = dz− 4π eixP +z− P| ¯ ψ(0)γ+ U[0, z]ψ(z)|P

  • z+=0,z⊥=0

Parton Distribution Amplitude (DA): φ(x) = i Fπ dz− 2π ei(x−1)P +z− P| ¯ ψ(0)γ+γ5 U[0, z]ψ(z)|vac

  • z+=0,z⊥=0

(isospin suppressed) P - pion momentum, v± ≡ v0 ± v3 - light-cone basis U[z1, z2] = exp

  • −igs

z2

z1 dξλaA+ 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

slide-8
SLIDE 8

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

slide-9
SLIDE 9

Quasi-distributions

slide-10
SLIDE 10

Parton quasi-distributions (quarks in the pion)

[Ji 2013]

Parton Quasi-Distribution Function (QDF): V (y; P3) = dz3 4π eiyP 3z3 P| ¯ ψ(0)γ3 U[0, z]ψ(z)|P

  • z0=0,z⊥=0

Parton Quasi-Distribution Amplitude (PDA): ˜ φ(y; P3) = i Fπ dz3 2π ei(y−1)P 3z3 P| ¯ ψ(0)γ+γ5 U[0, z]ψ(z)|vac

  • z0=0,z⊥=0

y - fraction of pion’s P3 carried by the quark Analogy to DF and DA, but y is not constrained Basic property: lim

P3→∞ V (x; P3) = q(x),

lim

P3→∞

˜ φ(x; P3) = φ(x)

Enrique Ruiz Arriola () Quasi-distributions REF2017 9 / 36

slide-11
SLIDE 11

QDF and QDA in the momentum representation

Constrained longitudinal momenta, but y ∈ (−∞, ∞) (partons can move “backwards”)

Enrique Ruiz Arriola () Quasi-distributions REF2017 10 / 36

slide-12
SLIDE 12

Chiral quark models

slide-13
SLIDE 13

Chiral quark models

χSB 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

slide-14
SLIDE 14

Chiral quark models

χSB 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

slide-15
SLIDE 15

Chiral quark models

Chiral quark models implement chiral symmetry: Pion is a Goldstone boson in the chiral limit Realization of the large Nc limit Fully covariant relativistic: Rest frame, light cone, euclidean Chiral Perturbation Theory: Gasser-Leutwyler coeffs Li ∼ Nc/(4π)2 Chiral anomaly: Wess-Zumino term π0 → 2γ, γ → π0π+π−, etc. Electromagnetic, Transition, Gravitational Form factors Parton distribution amplitudes (ϕπ(x) = 1) Generalized Parton Distributions ( ¯ dπ(x) = uπ(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

slide-16
SLIDE 16

Chiral quark models

Axial current Jµ,a

A (x) = 1

2 ¯ q(x)γµγ5τaq(x) ∂µJµ,a

A (x) = −fπm2 ππa

(PCAC) Chiral Ward Identity for irreducible vertex (chiral limit, mπ → 0) S(p + q)−1γ5 1 2τa + γ5 1 2τaS(p)−1 = qµΓµ,a

A (p + q, p)

If S(p) = 1/(/ p − M) then a solution Γµ,a

A (p + q, p) = τ a

2 γ5

  • γµ − qµ

q2 2M fπ

  • q2 = 0

iff M = 0 Pion quark coupling constant (Goldberger-Treiman relation) gπqq = M/fπ

Enrique Ruiz Arriola () Quasi-distributions REF2017 14 / 36

slide-17
SLIDE 17

Wave functions in chiral quark models

Pion Bethe-Salpeter wave function χa

q(p) =

i / p + q / − M − mgπqqγ5τa i / p − M − m

  • ET wave functions

Ψa(r) = −

  • d3p

(2π)3 ei

p· r

  • dp0

(2π)Tr[Γaχq(p)]. LC wave functions Ψa( b⊥, x) = −

  • d2p⊥

(2π)3 ei

p⊥· b⊥+ixmπp+

dp− (2π)Tr[Γaχq(p)].

Enrique Ruiz Arriola () Quasi-distributions REF2017 15 / 36

slide-18
SLIDE 18

Equal Time Wave functions in the NJL model

Analytic results for mπ = 0 ΨP (r) = 2ΨT (r) = gπqqNc 2π2r (2MK1(Mr) + regulators) , ΨA(r) = gπqqMNc 4π2 (2K0(Mr) + regulators) where K0 and K1 are the modified Bessel functions. Asymptotic behavior at r → ∞: ΨP (r) ∼ ΨT (r) ∼ e−Mr r3/2 , ΨA(r) ∼ e−Mr r1/2 . Longer tail in the A channel than in the P and T channels.

Enrique Ruiz Arriola () Quasi-distributions REF2017 16 / 36

slide-19
SLIDE 19

Light Cone Wave functions in the NJL model

Analytic results for mπ = 0 ΨP (b, x) = 2ΨT (b, x) = gπqqNc 2π2b (2MK1(Mb) + regulators) , ΨA(b, x) = gπqqMNc 4π2 (2K0(Mb) + regulators) These functions are x independent !! Model Relation between ET and LC ΨA(b, x) = ΨA(r)|r=b Pion Distribution Amplitude (flat) ϕπ(x) = ΨA(0⊥, x) = 1

Enrique Ruiz Arriola () Quasi-distributions REF2017 17 / 36

slide-20
SLIDE 20

Form factors (SQM)

Spectral representation S(p) =

  • C

dw ρ(w) / p − w ≡ Z(p) / p − Σ(p) Vector meson dominance for em form factor Fem(Q2) = − Nc 4π2f 2

π

  • dωρ(ω)ω2

1 dx log

  • ω2 + x(1 − x)t
  • ≡

m2

ρ

Q2 + m2

ρ

Solution → ρV (ω) = 1 2πi 1 ω 1 (1 − 4ω2/m2

ρ)5/2

f 2

π = Nc

m2

ρ

24π2 = (87MeV)2 TMD (mπ = 0) q(x, kT ) = 3m3

ρθ(x)θ(1 − x)

16π(k2

T + m2 ρ/4)5/2 → k2 T = m2 ρ

2

Enrique Ruiz Arriola () Quasi-distributions REF2017 18 / 36

slide-21
SLIDE 21

Scale and evolution

QM provide non-perturbative result at a low scale Q0 F(x, Q0)|model = F(x, Q0)|QCD , Q0 − the matching scale Determination of Q0 via momentum fraction: quarks carry 100% of momentum at Q0. One adjusts Q0 in such a way that when evolved to Q = 2 GeV, the quarks carry the experimental value of 47% LO DGLAP evolution: Q0 = 313+20

−10 MeV

[Davidson, Arriola 1995]: q(x; Q0) = 1

Enrique Ruiz Arriola () Quasi-distributions REF2017 19 / 36

slide-22
SLIDE 22

Older results from chiral quark models w/ evolution

slide-23
SLIDE 23

Pion quark DF, QM vs. E615

LO DGLAP evolution to the scale Q2 = (4 GeV)2: points: Fermilab E615, Drell-Yan line: QM evolved to Q = 4 GeV

Enrique Ruiz Arriola () Quasi-distributions REF2017 21 / 36

slide-24
SLIDE 24

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

slide-25
SLIDE 25

Pion DA, QM vs. E791

mπ=0 χQM (Q=Q0) χQM (Q=2GeV) 8/π x (1 - x) 6x(1-x) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5

x ϕ(x)

points: E791 data from dijet production in π + A solid line: QM at Q = 2 GeV dashed line: asymptotic form 6x(1 − x) at Q → ∞

Enrique Ruiz Arriola () Quasi-distributions REF2017 23 / 36

slide-26
SLIDE 26

Pion DA, QM vs. transverse lattice

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

  • mod. μ=0.5GeV

■ tr. lat. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

x ϕ(x,μ)

points: transverse lattice data [Dalley, van de Sande 2003] line: QM at Q = 0.5 GeV+ GPD [WB, ERA, Golec-Biernat 2008], TDA [WB, ERA 2007], equal-time (ET) pion wave functions [WB, ERA, S. Prelovsek, L. ˇ Santelj 2009] . . .

Enrique Ruiz Arriola () Quasi-distributions REF2017 24 / 36

slide-27
SLIDE 27

NEW: Quasi-distributions from QM

slide-28
SLIDE 28

Analytic formulas (in the chiral limit)

SQM (at the QM scale): ˜ φ(y, Pz) = V (y, Pz) = 1 π

  • 2mρPzy

m2

ρ + 4Pz2y2 + arctg

2Pzy mρ

  • + (y → 1 − y)

(similar simplicity for PV NJL) Satisfy the proper normalization ∞

−∞

dy ˜ φ(y, Pz) = ∞

−∞

dy V (y, Pz) = 1, ∞

−∞

dy 2yV (y, Pz) = 1 and the limit lim

Pz→∞

˜ φ(y, Pz) = lim

Pz→∞ V (y, Pz) = θ[y(1 − y)] = φ(x) = q(x),

(y = x)

Enrique Ruiz Arriola () Quasi-distributions REF2017 26 / 36

slide-29
SLIDE 29

QDA and QDF from chiral quark models at QM scale

(a)

NJL Pz 0.5 1 10 ∞

  • 0.5

0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0

y ϕ ˜ (y,Pz)

(b)

NJL

  • 0.5

0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0

y 2yV(y,Pz)

(a) Quark QDA of the pion in NJL (mπ = 0) at various Pz, plotted vs. y (b) The same, but for the quark QDF multiplied with 2y

Enrique Ruiz Arriola () Quasi-distributions REF2017 27 / 36

slide-30
SLIDE 30

Comparison to Eucidean lattice

○○○○○○○○○○○○○○○○○ ○ ○ ○ ○ ○ ○ ○ ○○○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○○○○○○○○

□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □

(a)

NJL

mπ=310MeV

LaMET Pz=1.3GeV

○

□

Pz=0.9GeV

  • 2
  • 1

1 2 3 0.0 0.2 0.4 0.6 0.8 1.0 1.2

y ϕ ˜ (y,Pz)

○○○○○○○○○○○○○○○○○ ○ ○ ○ ○ ○ ○ ○ ○○○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○○○○○○○○

□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □

(b)

SQM

mπ=310MeV

  • 2
  • 1

1 2 3 0.0 0.2 0.4 0.6 0.8 1.0 1.2

y ϕ ˜(y,Pz)

Quark QDA of the pion in NJL (a) and SQM (b) (mπ = 310 MeV, Pz = 0.9 and 1.3 GeV), plotted vs. y and compared to the lattice at Q = 2 GeV (LaMET [Zhang et

  • al. 2017])

Enrique Ruiz Arriola () Quasi-distributions REF2017 28 / 36

slide-31
SLIDE 31

Evolution of QDF

slide-32
SLIDE 32

Relation kT-unintegrated quantities (TMA, TMD)

Radyushkin’s formula [2016] – follows just from Lorentz invariance ˜ φ(y, Pz) = ∞

−∞

dk1 1 dx PzTMA(x, k2

1 + (x − y)2P 2 z ).

QDA can be obtained from TMA via a double integration! Analogously V (y, Pz) = ∞

−∞

dk1 1 dx PzTMD(x, k2

1 + (x − y)2P 2 z ). Enrique Ruiz Arriola () Quasi-distributions REF2017 30 / 36

slide-33
SLIDE 33

Evolution of unintegrated DF

UDF or TMD Kwieci´ nski’s method [2003], based on one-loop CCFM DGLAP-like evolution, diagonal in b-space conjugate to kT For the non-singlet case:

Q2 ∂f(x, b, Q) ∂Q2 = αs(Q2) 2π 1 dz Pqq(z) ×

  • Θ(z − x) J0[(1 − z)Qb] f

x z , b, Q

  • − f(x, b, Q)
  • Enrique Ruiz Arriola ()

Quasi-distributions REF2017 31 / 36

slide-34
SLIDE 34

Results of evolution of pion QDF in Q at fixed Pz

Pz=1GeV

Q2=Q0

2

Q2=4GeV2 Q2=100GeV2

  • 0.5

0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0

y V(y) Pz=2GeV

Q2=Q0

2

Q2=4GeV2 Q2=100GeV2

  • 0.5

0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

y V(y)

Strength moved to lower y as Q increases

Enrique Ruiz Arriola () Quasi-distributions REF2017 32 / 36

slide-35
SLIDE 35

Changing Pz at fixed Q

Q2=4GeV2

Pz=1GeV Pz=2GeV Pz=4GeV Pz=∞

  • 1.0
  • 0.5

0.0 0.5 1.0 1.5

  • 0.1

0.0 0.1 0.2 0.3 0.4

y yV(y)

Pz → ∞ limit achieved fastest at y ∼ 0.6 − 0.9

Enrique Ruiz Arriola () Quasi-distributions REF2017 33 / 36

slide-36
SLIDE 36

Evolution of QPDA

mπ=0 Pz=0.9GeV SQM (Q=Q0) SQM (Q=2GeV) 8/π x (1 - x) 6x(1-x)

  • 3
  • 2
  • 1

1 2 0.0 0.2 0.4 0.6 0.8 1.0

y ϕ ˜ (y)

mπ=0 Pz=1.3GeV SQM (Q=Q0) SQM (Q=2GeV) 8/π x (1 - x) 6x(1-x)

  • 3
  • 2
  • 1

1 2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

y ϕ ˜ (y)

Quark QDA of the pion in NJL (a) and SQM (b) (mπ = 310 MeV, Pz = 0.9 and 1.3 GeV), plotted vs. y and compared to the lattice at Q = 2 GeV (LaMET [Zhang et

  • al. 2017])

Enrique Ruiz Arriola () Quasi-distributions REF2017 34 / 36

slide-37
SLIDE 37

Conclusions

slide-38
SLIDE 38

Conclusions

Model evaluation of quasi-distributions of the pion (at the QM scale) Very simple analytic results, all consistency conditions met, illustration and check

  • f definitions and methods

Results at finite Pz are interesting per se (Radyushkin’s relation, relation to Ioffe-time distributions, ET wave functions), can be (favorably) compared to QDA from Euclidean lattice QCD For QDF of the pion predictions made for various Q (Kwieci´ nski’s evolution) and Pz Pz ∼ 1 GeV, accessible presently on the lattice, may not be sufficiently close to Pz → ∞ limit Convergence fastest for intermediate y, suggesting the domain where lattice may work best Recent activity also on related objects: pseudo-distributions, Ioffe-time distributions... Lattice efforts for both N and π

Enrique Ruiz Arriola () Quasi-distributions REF2017 36 / 36