nucleon structure in light-front quark model constrained...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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)
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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
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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
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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
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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
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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
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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
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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
)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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