Partonic quasi-distributions of the pionin chiral quark models
Enrique Ruiz Arriola and Wojciech Broniowski
Dep. Fısica Atomica,Molecular y NuclearInstituto Carlos I de Fısica Teorica 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
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 isconstrained: k+ ≡ k0 + k3 = xP+
x ∈ [0, 1]
Enrique Ruiz Arriola () Quasi-distributions REF2017 4 / 36
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 isconstrained: k+ ≡ k0 + k3 = 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):
q(x) =
∫dz−
4πeixP
+z− 〈P |ψ(0)γ+ U [0, z]ψ(z)|P 〉∣∣z+=0,z⊥=0
Parton Distribution Amplitude (DA):
φ(x) =i
Fπ
∫dz−
2πei(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
∫ z2z1dξλaA+
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 processesinvolving pions
Impossibility to implement PDF or PDA on the euclidean lattices, only lowestmoments 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):
V (y;P3) =
∫dz3
4πeiyP
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
2πei(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 quarkAnalogy to DF and DA, but y is not constrainedBasic property:
limP3→∞
V (x;P3) = q(x), limP3→∞
φ(x;P3) = φ(x)
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QDF and QDA in the momentum representation
Constrained longitudinal momenta, but y ∈ (−∞,∞)(partons can move “backwards”)
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Chiral quark models
Chiral quark models
χSB breaking → massive quarks
Point-like interactions
Soft matrix elements with pions (andphotons, 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
χSB breaking → massive quarks
Point-like interactions
Soft matrix elements with pions (andphotons, W , Z)
One-quark loop, regularization:
1) Pauli-Villars (PV)
2) Spectral Quark Model
(SQM) - implements VMD
Need for evolutionGluon dressing, renorm-group improved
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Chiral quark models
Chiral quark models implement chiral symmetry: Pion is a Goldstone boson in thechiral 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), ...)
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Chiral quark models
Axial current
Jµ,aA (x) =1
2q(x)γµγ5τaq(x) ∂µJµ,aA (x) = −fπm2
ππa (PCAC)
Chiral Ward Identity for irreducible vertex (chiral limit, mπ → 0)
S(p+ q)−1γ51
2τa + γ5
1
2τaS(p)−1 = qµΓµ,aA (p+ q, p)
If S(p) = 1/(/p−M) then a solution
Γµ,aA (p+ q, p) =τa
2γ5
[γµ − qµ
q22M
fπ
]q2 = 0 iff M 6= 0
Pion quark coupling constant (Goldberger-Treiman relation)
gπqq = M/fπ
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Wave functions in chiral quark models
Pion Bethe-Salpeter wave function
χaq (p) =i
/p+ q/−M −mgπqqγ5τai
/p−M −m
i g 5 t a
q = 0 , q 0 = m p
i g 5 t aG a
q = 0 , q 0 = m p
p f i x e d
G a
x = 0 , t = 0
x = r , t = 0
( a ) ( b )
ET wave functions
Ψa(r) = −∫
d3p
(2π)3ei~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)].
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Equal Time Wave functions in the NJL model
Analytic results for mπ = 0
ΨP (r) = 2ΨT (r) =gπqqNc2π2r
(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.
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Light Cone Wave functions in the NJL model
Analytic results for mπ = 0
ΨP (b, x) = 2ΨT (b, x) =gπqqNc2π2b
(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
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Form factors (SQM)Spectral representation
S(p) =
∫C
dwρ(w)
/p− w ≡Z(p)
/p− Σ(p)
Vector meson dominance for em form factor
Fem(Q2) = − Nc4π2f2
π
∫dωρ(ω)ω2
∫ 1
0
dx log[ω2 + x(1− x)t
]≡
m2ρ
Q2 +m2ρ
Solution
→ ρV (ω) =1
2πi
1
ω
1
(1− 4ω2/m2ρ)5/2
f2π = Nc
m2ρ
24π2= (87MeV)2
TMD (mπ = 0)
q(x, kT ) =3m3
ρθ(x)θ(1− x)
16π(k2T +m2ρ/4)5/2
→ 〈k2T 〉 =m2ρ
2
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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 theexperimental value of 47%
LO DGLAP evolution:
Q0 = 313+20−10 MeV
[Davidson, Arriola 1995]:
q(x;Q0) = 1
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Older results from chiral quark models w/ evolution
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
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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
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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→∞
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Pion DA, QM vs. transverse lattice
■
■
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
■
■
mod. μ=0.5GeV
■ tr. lat.
0.0 0.2 0.4 0.6 0.8 1.00.0
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) pionwave functions [WB, ERA, S. Prelovsek, L. Santelj 2009] . . .
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NEW: Quasi-distributions from QM
Analytic formulas (in the chiral limit)
SQM (at the QM scale):
φ(y, Pz) = V (y, Pz) =1
π
[2mρPzy
m2ρ + 4Pz
2y2+ 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
limPz→∞
φ(y, Pz) = limPz→∞
V (y, Pz) = θ[y(1− y)] = φ(x) = q(x), (y = x)
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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
y2yV(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
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Comparison to Eucidean lattice
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(a)
NJL
mπ=310MeV
LaMET
Pz=1.3GeV ○
□Pz=0.9GeV
-2 -1 0 1 2 3
0.0
0.2
0.4
0.6
0.8
1.0
1.2
y
ϕ˜(y,Pz)
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(b)
SQM
mπ=310MeV
-2 -1 0 1 2 3
0.0
0.2
0.4
0.6
0.8
1.0
1.2
yϕ˜(y,Pz)
Quark QDA of the pion in NJL (a) and SQM (b) (mπ = 310 MeV, Pz = 0.9 and1.3 GeV), plotted vs. y and compared to the lattice at Q = 2 GeV (LaMET [Zhang etal. 2017])
Enrique Ruiz Arriola () Quasi-distributions REF2017 28 / 36
Evolution of QDF
Relation kT -unintegrated quantities (TMA, TMD)
Radyushkin’s formula [2016] – follows just from Lorentz invariance
φ(y, Pz) =
∫ ∞−∞
dk1
∫ 1
0
dxPzTMA(x, k21 + (x− y)2P 2z ).
QDA can be obtained from TMA via a double integration!
Analogously
V (y, Pz) =
∫ ∞−∞
dk1
∫ 1
0
dxPzTMD(x, k21 + (x− y)2P 2z ).
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Evolution of unintegrated DFUDF or TMD
Kwiecinski’s method [2003], based on one-loop CCFMDGLAP-like evolution, diagonal in b-space conjugate to kTFor the non-singlet case:
Q2 ∂f(x, b,Q)
∂Q2=
αs(Q2)
2π
∫ 1
0
dz Pqq(z)
×[Θ(z − x) J0[(1− z)Qb] f
(xz, b,Q
)− f(x, b,Q)
]
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Results of evolution of pion QDF in Q at fixed Pz
Pz=1GeV
Q2=Q02
Q2=4GeV2
Q2=100GeV2
-0.5 0.0 0.5 1.0 1.50.0
0.2
0.4
0.6
0.8
1.0
y
V(y)
Pz=2GeV
Q2=Q02
Q2=4GeV2
Q2=100GeV2
-0.5 0.0 0.5 1.0 1.50.0
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
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Changing Pz at fixed Q
Q2=4GeV2Pz=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
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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 0 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 0 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 and1.3 GeV), plotted vs. y and compared to the lattice at Q = 2 GeV (LaMET [Zhang etal. 2017])
Enrique Ruiz Arriola () Quasi-distributions REF2017 34 / 36
Conclusions
Conclusions
Model evaluation of quasi-distributions of the pion (at the QM scale)
Very simple analytic results, all consistency conditions met, illustration and checkof definitions and methods
Results at finite Pz are interesting per se (Radyushkin’s relation, relation toIoffe-time distributions, ET wave functions), can be (favorably) compared to QDAfrom Euclidean lattice QCD
For QDF of the pion predictions made for various Q (Kwiecinski’s evolution) andPz
Pz ∼ 1 GeV, accessible presently on the lattice, may not be sufficiently close toPz →∞ limit
Convergence fastest for intermediate y, suggesting the domain where lattice maywork best
Recent activity also on related objects: pseudo-distributions, Ioffe-timedistributions... Lattice efforts for both N and π
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