Nucleon structure in light-front quark model

constrained by AdS/QCD

Valery Lyubovitsky

Institut fur Theoretische Physik, Universitat Tubingen

Kepler Center for Astro and Particle Physics, Germany

in collaboration with

Thomas GutscheIvan SchmidtAlfredo VegaWerner Vogelsangbased on

PRD 80, 82, 83, 85, 86, 89(2009-2014)

ISHEPP-XXII, September 15-20, 2014, JINR, Dubna

– p. 1

Introduction

• Nucleon structure

parton distributions (PDFs)

generalized parton distributions (GPDs)

form factors (FFs)

• Current and future Experiments

BNL, CERN, DESY, GSI, JINR, JLab, Electron Ion Collider (EIC) . . .

• Many theoretical approaches

pQCD, Lattice, QCD sum rules, ChPT, different types of quark models (from naive

quark model to Schwinger-Dyson/Bethe-Salpeter approaches)

• AdS/QCD ≡ Holographic QCD (HQCD) – approximation to QCD:

attempt to model Hadronic Physics in terms of fields/strings living in extra

dimensions – anti-de Sitter (AdS) space

• HQCD models reproduce main features of QCD at low and high energies:

chiral symmetry, confinement, power scaling of hadron form factors

– p. 2

Introduction

• AdS metric Poincaré form

ds2 = gMN (z) dxMdxN = R2

z2

(

dxµdxµ − dz2)

where R – AdS radius

• Metric Tensor gMN (z) = ǫaM (z)ǫbN (z)ηab

• Vielbein ǫaM (z) = RzδaM (relates AdS and Lorentz metric)

• Manifestly scale-invariant x→ λx, z → λz.

• z – extra dimensional (holographic) coordinate; z = 0 is UV boundary

• Five Dimensions: L = Length, W = Width, H = Height, T = Time, S = Scale

– p. 3

Introduction

• Action for scalar field

SΦ =1

2

∫

d4xdz√g e−ϕ(z)

(

∂MΦ(x, z)∂MΦ(x, z)−m2 Φ2(x, z)

)

• Dilaton field ϕ(z) = κ2z2 – gravitational background field providing breaking of

conformal symmetry and confinement

• g = |detgMN |

• m – 5d mass, m2R2 = ∆(∆− 4), ∆ = 3 is conformal dimension

• Kaluza-Klein (KK) expansion Φ(x, z) =∑

nφn(x) Φn(z)

• Tower of KK modes φn(x) dual to 4-dimensional fields describing hadrons

• Bulk profiles Φn(z) dual to hadronic wave functions

– p. 4

Introduction

• Put on mass-shell −∂µ∂µφn(x) =M2nφn(x)

• Substitute

Φn(z) =

(R

z

)−3

φn(z)

• Identify ∆ = τ = N + L (here N = 2 – number of partons in meson)[

− d2

dz2+ 4L2−1

4z2+ κ4z2 − 2κ2

]

φn(z) =M2nφn(z)

• Solutions: φnL(z) =√

2Γ(n+1)Γ(n+L+1)

κL+1 zL+1/2 e−κ2z2/2 LLn(κ

2z2)

• M2nL = 4κ2

(

n+ L2

)

• Massless pion M2π = 0 for n = L = 0 Brodsky, Téramond

• Extension to higger spins M2nLJ = 4κ2

(

n+ L+J2

)

∼ 4κ2 (n+ J) at large J

Brodsky, Téramond, Lyubovitskij, Gutsche, Schmidt, Vega

– p. 5

Introduction

• Scattering problem for AdS field gives information about propagation of external

field from z to the boundary z = 0 — bulk-to-boundary propagator Φext(q, z)

[Fourier-trasform of AdS field Φext(x, z)]:

Φext(q, z) =

∫

ddxe−iqx Φext(x, z)

• Equation of motion ∂z

(

e−ϕ(z)

z∂zΦext(q, z)

)

+ q2 e−ϕ(z)

zΦext(q, z) = 0 .

• Solution in Euclidean space Q2 = −q2

Φext(Q, z) = Γ(

1 + Q2

4κ2

)

U(

Q2

4κ2 , 0, κ2z2

)

• Hadron form factors

Fnτ (Q2) =∞∫

0

dzΦext(Q, z)φ2nτ (z)

• Hadron structure is implemented by a nontrivial dependence of AdS fields

on 5-th (holographic) coordinate

– p. 6

Introduction

• Master formula for PDFs, GPDs, FFs

FF Fτ (Q2) =1∫

0

dxHτ (x,Q2)

GPD Hτ (x,Q2) = (1− x)τ−2

︸ ︷︷ ︸

=qτ (x)

exp(

− Q2

4κ2 log(1/x))

PDF qτ (x) = (1− x)τ−2

• Consistent with quark counting rules

Matveev-Muradyan-Tavkhelidze-Brodsky-Farrar

Fτ (Q2) =Γ( Q2

4κ2 +1) Γ(τ)

Γ( Q2

4κ2 +τ)∼ 1

(Q2)τ−2 at large Q2 → ∞

• Fτ=2(Q2) ∼ 1Q2 for mesons, Fτ=3(Q2) ∼ 1

Q4 for baryons

– p. 7

Introduction

• Matching to Light-Front QCD using Drell-Yan-West formula

Fτ (Q2) =1∫

0

dx∫ d2k⊥

16π3 ψ†τ (x,k

′⊥)ψτ (x,k⊥)

where k′⊥

= k⊥ + (1− x)q⊥ and Q2 = q2⊥

• Result of matching

ψτ (x,k⊥) = Nτ4πκ

√log(1/x)(1− x)

τ−42 exp

[

− k2⊥

2κ2log(1/x)

(1−x)2

]

where Nτ =√τ − 1

• For τ = 2 first done by Brodsky, Téramond

• Matching at the initial scale µ0 ∼ 1 GeV (hard and soft evolutions neglected)

• Derived LFWF is not symmetric under the exchange x→ 1− x

Extracted from matching matrix element of bare electromagnetic current between

dressed LFWF in LF QCD and matrix element of the dressed electromagnetic

current between hadronic WF in AdS/QCD

– p. 8

Introduction

• Light-front wave functions (LFWFs)

• Brodsky-Huang-Lepage prescription for two partonic bound state

ψ(x,k⊥) ∼ exp(

− k2⊥

λ2x(1−x)

)

where λ is scale parameter

• Different forms, 3 quark and quark-diquark structures

• Normally analysis of form factors start from parametrization GDPs

See e.g.

Guidal-Polyakov-Radyushkin-Vanderhaegen, PRD 72 (2005) 054013

Selyugin-Teryaev, PRD 79 (2009) 033003

Diehl-Kroll, EPJC 73 (2013) 2397

• Also LF 3-quark/qd models used for calculation of PDFs, TMDs

See e.g.

Bacchetta-Conti-Radici, PRD 78 (2008) 074010

Boffi-Efremov-Pasquini-Schweitzer, PRD 79 (2009) 094012

– p. 9

Pion

• Global analysis of PDFs tells that they are considerably softer

• At large x and initial scale qπ(x, µ0) ∼ (1− x)2.03

Aicher-Schafer-Vogelsang, PRL 105 (2010) 252003

• Performing updated analysis of the E615 data on the cross section of the Drell-Yan

process π−N → µ+µ−X including next-to-leading logarithmic threshold

resummation effects

• Threshold means z = Q2/(Sx1x2) → 1, where x1 and x2 are momentum

fractions of partons participating in hard-scattering reaction, S = (P1 + P2)2 is

hadronic CM energy squared, Q2 is photon momentum squared

• Threshold means: most of initial partonic energy is used to produce virtual photon

• pion PDF at initial scale µ0 = 0.63 GeV Aicher-Schafer-Vogelsang

qπ(x, µ0) = Nπxα−1 (1− x)β (1 + γxδ)

where α = 0.70, β = 2.03, γ = 13.8, δ = 2, Nπ = 0.915

– p. 10

Pion

• Required pion LFWF reads

ψπ(x,k⊥) =4πN

κ

√log(1/x)

1− x

√

f(x) f(x) exp

[

−k2⊥

2κ2log(1/x)

(1− x)2f(x)

]

where functions f(x) and f(x) are specified as

f(x) = xα−1 (1− x)β (1 + γxδ) ,

f(x) = xα (1− x)β (1 + γxδ) ,

where α, β, γ, δ, α, γ and δ are free parameters and N is the normalization factor

N−2 = B(α, 1 + β) + γB(α+ δ, 1 + β)

α = 0.15 , γ = 2.30 , δ = 1.30

– p. 11

Pion

〈r2π〉 = 0.424 fm2 (data 0.452 fm2)

••

•

•

◦

NN

⋄

⋄

⋄ ⋄

⋄

⋄

• JLab 2001

N JLab 2006

⋄ JLab 2007

◦ DESY

CERN NA7

Our result

0 1 2 3 40

0.2

0.4

0.6

0.8

Q2F

π(Q

2)

(GeV

2)

Q2 (GeV2)

– p. 12

Nucleons

• Valence quark decompositions of nucleon FFs

[ see e.g. Radyushkin, PRD 58 (1998) 114008 ]

Fp(n)i (Q2) = 2

3F

u(d)i (Q2)− 1

3F

d(u)i (Q2)

• quark GPDs in nucleons

F q1 (Q

2) =∫ 10 dxHq(x,Q2)

F q2 (Q

2) =∫ 10 dx Eq(x,Q2).

• PDFs (at Q2 = 0)

Hq(x, 0) = qv(x), Eq(x, 0) = Eq(x)

• Normalized as

Number of quarks valence quarks in the proton nq = F q1 (0) =

1∫

0

dx qv(x)

Anomalous magnetic moment κq = F q2 (0) =

1∫

0

dx Eq(x)

– p. 13

Nucleons

• Sachs FFs and EM radii

GNE (Q2) = FN

1 (Q2)− Q2

4m2N

FN2 (Q2)

GNM (Q2) = FN

1 (Q2) + FN2 (Q2)

〈r2E〉N = −6dGN

E (Q2)

dQ2

∣∣∣∣Q2=0

〈r2M 〉N = − 6GN

M(0)

dGNM (Q2)

dQ2

∣∣∣∣Q2=0

– p. 14

Nucleons

• LF representation for quark FFs

F q1 (Q

2) =1∫

0

dx∫ d2k⊥

16π3

[

ψ+ ∗+q (x,k′

⊥)ψ+

+q(x,k⊥) + ψ+ ∗−q (x,k

′⊥)ψ+

−q(x,k⊥)

]

F q2 (Q

2) =

− 2MN

q1−iq2

1∫

0

dx∫ d2k⊥

16π3

[

ψ+ ∗+q (x,k′

⊥)ψ−

+q(x,k⊥) + ψ+ ∗−q (x,k

′⊥)ψ−

−q(x,k⊥)

]

• ψλNλqq

(x,k⊥) — LFWFs with specific helicities for the nucleon λN = ± and for the

struck quark λq = ±, where plus and minus correspond to + 12

and − 12

• We work in the frame with q = (0, 0,q⊥), and therefore the Euclidean momentum

squared is Q2 = q2⊥

• As initial scale we choose the value µ0 = 1 GeV accepted in the

Martin-Stirling-Thorne-Watt global fit EPJC 63 (2009) 189

– p. 15

Nucleons

• Scalar quark-diquark model

ψ++q(x,k⊥) = ϕ

(1)q (x,k⊥)

ψ+−q(x,k⊥) = − k1+ik2

xMNϕ(2)q (x,k⊥)

ψ−+q(x,k⊥) = k1−ik2

xMNϕ(2)q (x,k⊥)

ψ−−q(x,k⊥) = ϕ

(1)q (x,k⊥)

where ϕ(1)q (x,k⊥) and ϕ

(2)q (x,k⊥) are the twist-3 LFWFs

• From AdS/QCD — LF QCD matching with a trivial dilaton potential

ϕAdS/QCD(i)q (x,k⊥) = N

(i)q

4πκ

√log(1/x)

1−xexp

[

− k2⊥

2κ2log(1/x)

(1−x)2

]

– p. 16

Nucleons

• Generalization of the nucleon LFWF

• ϕ(i)q (x,k⊥) = N

(i)q

4πκ

√log(1/x)

1−x

√

f(i)q (x)fq(x) exp

[

− k2⊥

2κ2log(1/x)

(1−x)2fq(x)

]

• Functions f(i)q and fq are specified as

f(1)q (x) = xη

(1)q −1 (1− x)η

(2)q −1 (1 + ǫq

√x+ γqx)

f(2)q (x) = x2+ρq (1− x)σq (1 + λq

√x+ δqx)2 f

(1)q (x)

fq(x) = xη(1)q (1− x)η

(2)q (1 + ǫq

√x+ γqx)

– p. 17

Nucleons

Parameters specifying functions f(i)q (x) and fq(x)

Parameter Value Parameter Value

η(1)u 0.45232 η

(1)d 0.71978

η(2)u 3.0409 η

(2)d 5.3444

ǫu −2.3737 ǫd −4.3654

γu 8.9924 γd 7.4730

η(1)u 0.195 η

(1)d 0.280

η(2)u

η(2)u −12

− 0.54 η(2)d

η(2)d

−1

2− 0.60

ǫu −0.71 ǫd −0.10

γu 0 γd 0

ρu 0.091 ρd −0.17

σu (η(2)u − 1)− 0.2409 σd (η

(2)d − 1)− 2.3444

λu −2.40 λd −0.35

δu 3.18 δd 4.26

κ = 350 MeV remains the same as fixed in soft-wall AdS/QCD

– p. 18

Nucleons

• At large x PDFs scale as

• qv(x) ∼ (1− x)η(2)q

uv(x) ∼ (1− x)3 , dv(x) ∼ (1− x)5

• Eq(x) ∼ qv(x) (1− x)1+σq/2 ∼ (1− x)η(2)q +1+σq/2

Eu(x) ∼ (1− x)5 , Ed(x) ∼ (1− x)7

– p. 19

Nucleons

• Scaling of nucleon FFs

F q1 (Q

2) ∼1∫

0

dx(1− x)η(2)q exp

[

− Q2

4κ2(1− x)1+η

(2)q

]

∼(

1

Q4

)1+∆(1)q

and

F q2 (Q

2) ∼1∫

0

dx(1− x)1+η(2)q +σq/2 exp

[

− Q2

4κ2(1− x)1+η

(2)q

]

∼(

1

Q6

)1+∆(2)q

• ∆(1)q and ∆

(2)q are the small corrections encoding a deviation of the Dirac and

Pauli quark form factors from power-scaling laws 1/Q4 and 1/Q6, respectively

∆(1)q =

1 + η(2)q

2(1 + η(2)q )

− 1 , ∆(2)q =

2

3∆

(1)q +

1

3

(1 + σq/2

1 + η(2)q

− 1

)

– p. 20

Nucleons

• Analytically these corrections vanish when

η(2)q =

σq

2=η(2)q − 1

2

• Also in this limit we are consistent with Drell-Yan-West duality between the

large-Q2 behavior of nucleon electromagnetic form factors and the large-x

behavior of the structure functions.

• However fine-tuning fit of electromagnetic form factors requires a deviation of ∆(i)q

from zero. Numerically they are

∆(1)u = 0.365 , ∆

(1)d = 0.233 ,

∆(2)u = 0.338 , ∆

(2)d = 0.081 .

– p. 21

Nucleons

Electromagnetic properties of nucleons

Quantity Our results Data

µp (in n.m.) 2.793 2.793

µn (in n.m.) -1.913 -1.913

rpE (fm) 0.781 0.8768 ± 0.0069

〈r2E〉n (fm2) -0.113 -0.1161 ± 0.0022

rpM (fm) 0.717 0.777 ± 0.013 ± 0.010

rnM (fm) 0.694 0.862+0.009−0.008

– p. 22

Nucleons

0 5 10 15 20 25 300.0

0.5

1.0

1.5

Q2 HGeV2L

Q4

F1

uHQ

2LHG

eV4L

Dirac u quark form factor multiplied by Q4

– p. 23

Nucleons

0 5 10 15 20 25 300.00

0.05

0.10

0.15

0.20

0.25

0.30

Q2 HGeV2L

Q4

F1

dHQ

2LHG

eV4L

Dirac d quark form factor multiplied by Q4

– p. 24

Nucleons

0 5 10 15 20 25 300.0

0.1

0.2

0.3

0.4

0.5

Q2 HGeV2L

Q4

F2

uHQ

2LHG

eV4L

Pauli u quark form factor multiplied by Q4

– p. 25

Nucleons

0 5 10 15 20 25 30-0.5

-0.4

-0.3

-0.2

-0.1

0.0

Q2 HGeV2L

Q4

F2

dHQ

2LHG

eV4L

Pauli d quark form factor multiplied by Q4

– p. 26

Nucleons

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

Q2 HGeV2L

Q4

F1

pHQ

2LHG

eV4L

Dirac proton form factor multiplied by Q4

– p. 27

Nucleons

0 5 10 15 20 25 30-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

Q2 HGeV2L

Q4

F1

nHQ

2LHG

eV4L

Dirac neutron form factor multiplied by Q4

– p. 28

Nucleons

0 5 10 15 20 25 300

1

2

3

4

5

6

Q2 HGeV2L

Q2

F2

pHQ

2L�

F1

pHQ

2LHG

eV2L

Ratio Q2F p2 (Q

2)/F p1 (Q

2)

– p. 29

Nucleons

▽▽▽▽▽▽△▽××△▽⋆▽

××▽▽△▽

▽

▽△

▽

▽△

▽

▽△

Q2 (GeV2)

Gp E(Q

2)/G

D(Q

2)

0 1 2 3 4 5 6

0.25

0.50

0.75

1.00

1.25

Berger et al.(1971)

▽Price et al.(1971)

△ Hanson et al.(1973)

Simon et al.(1980)

×Milbrath et al.(1998)

Jones et al.(2000)

⋆ Dieterich et al.(2001)

Ratio GpE(Q2)/GD(Q2)

– p. 30

Nucleons

△▽ ⋄

⋆×

▽ Eden et al.(1994)

Herberg et al.(1999)

⋄ Ostrick et al.(1999)

△ Passchier et al.(1999)

Rohe et al.(1999)

⋆ Golak et al.(2001)

Schiavilla & Sick(2001)

×Zhu et al.(2001)

Madey et al.(2003)

Q2 (GeV2)

Gn E(Q

2)

0 0.5 1.0 1.5

0

0.02

0.04

0.06

0.08

0.10

0.12

Charge neutron form factor GnE(Q2)

– p. 31

Nucleons

NNNNN

△

△

△

△

Walker et al. (1994)

Punjabi et al. (2005)

Gayou et al. (2002)

N Ron et al. (2011)

Puckett et al. (2010)

△Puckett et al. (2012)

Q2 (GeV2)

µpG

p E(Q

2)/G

p M(Q

2)

0 1 2 3 4 5 6

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Ratio µpGpE(Q2)/Gp

M (Q2)

– p. 32

Nucleons

××××△××××▽×△×××××▽△××⋆×××▽△×▽⋆⋄▽⋆△▽⋆⋄▽⋄⋆△⋆⋄⋄⋆⋄

Q2 (GeV2)

Gp M(Q

2)/(µ

pG

D(Q

2))

0 10 20 30

0.50

0.75

1.00

1.25

×Janssens et al.(1966)

Litt et al.(1970)

▽ Berger et al.(1971)

⋆ Bartel et al.(1973)

△ Hanson et al.(1973)

Hoehler et al.(1976)

Sill et al.(1993)

Andivahis et al.(1994)

⋄ Walker et al.(1994)

A

Ratio GpM (Q2)/(µpGD(Q2))

– p. 33

Nucleons

△▽△⋆

⋆⋆⋆

⋆

⋆⋆⋄

⋄

⋄⋄

⊗

NN

NNNNNN

NN

NNN

NNN

NN

N

NN

N

NN

Q2 (GeV2)

Gn M(Q

2)/(µ

nG

D(Q

2))

Rock et al.(1982)

Lung et al.(1993)

⋆Markowitz et al.(1993)

▽ Anklin et al.(1994)

⊗ Gao et al.(1994)

⋄ Bruins et al.(1995)

×Anklin et al.(1998)

△ Xu et al.(2000)

Kubon et al.(2002)

Xu et al.(2003)

N Lachniet et al.(2009)

0 5 10

0.50

0.75

1.00

1.25

Ratio GnM (Q2)/(µnGD(Q2))

– p. 34

Nucleons

N

N

N

Q2 (GeV2)

Gn E(Q

2)/G

n M(Q

2)

0 0.5 1 1.5

-0.25

-0.20

-0.15

-0.10

-0.05

0

N n(~e, e′~n)

2H(~e, e′~n): PWBA

2H(~e, e′~n): FSI+MEC+IC+RC

Ratio GnE(Q2)/Gn

M (Q2)

– p. 35

Summary

• AdS/QCD ≡ Holographic QCD (HQCD) – approximation to QCD:

attempt to model Hadronic Physics in terms of fields/strings living in extra

dimensions – anti-de Sitter (AdS) space

• HQCD models reproduce main features of QCD at low and high energies

• Soft–wall holographic approach – covariant and analytic model for hadron structure

with confinement at large distances and conformal behavior at short distances

• Including nontrivial dilaton potential could help to describe FFs, PDFs and GPDs

from unified point of view (in progress)

• Now: we fit FFs and PDFs using generalized LFWFs (must be derived from

AdS/QCD — LF QCD matching)

• Future work: nucleon TMDs

– p. 36