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Constraints from leptoproduction of vector meson within different frameworks Adrien Besse Irfu - SPhN MESON 2014, Cracow Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 1 / 25

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SLIDE 1

Constraints from leptoproduction of vector meson within different frameworks

Adrien Besse

Irfu - SPhN

MESON 2014, Cracow

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 1 / 25

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SLIDE 2

Observables and kinematic ranges

p p′ e−, k e−, k′ V , pV W 2 −Q2

t → MV,{λV ν′;λγ,ν} =

N, ν N, ν′ γ∗, λγ V , λV W 2 ≫ |t|, Q2 ≫ Λ2

QCD, the Bjorken x ∼ Q2 W 2+Q2

Spin density matrix elements (SDME) linked to the helicity amplitudes : [Schilling Wolf, ’73] & [Dielh, ’07]

small x HERA (H1 and ZEUS) mid-x region: COMPASS, HERMES, E665, NMC valence region: CLAS

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 2 / 25

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SLIDE 3

Theoretical descriptions

Color dipole picture (small−x) Collinear fact. picture γ∗ V N N ′ N q¯

q

γ∗ V N N ′ GPD Interaction via gluons exchange Interaction via gluon and quark exchange Convenient to introduce Valid from small−x to saturation effects at small−x the valence region

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 3 / 25

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SLIDE 4

Dipole model picture

Color dipole factorization scheme Impact parameter space representation of the amplitudes in the infinite momentum frame

Nikolaev, Zakharov, ’91, Mueller, ’90

V γ∗ Ψi Ψf p p′

N(x, r, b)

r y b N(x, r, b) =

r b

Initial Ψi and final Ψf states wave functions. Universal dipole/target scattering amplitude N (x, r, b): (DIS structure functions, diffractive DIS, exclusive processes ...)

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 4 / 25

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SLIDE 5

Dipole model picture

Skewness effects can be taken into account in dipole cross-section model

[Shuvaev, Golec-Biernat, Martin, Ryskin, ’99]

N(x, r) ≡ N(x, ξ, r) such that N(x, ξ = 0, r) = s ˆ σ(x, r) Dipole models :

access to Im Mg

V

Re Mg

V can be deduced from Im Mg V using dispersion relations

In the limit ∆ = 0, i.e. |t| = |t|min, Im MλV λγ(Q2, x) ∝ i

  • dy
  • dr ˆ

Ψ∗

λV (y, r) ˆ

Ψλγ(y, r) N(x, ξ, r)

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 5 / 25

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SLIDE 6

DVMP within Collinear factorization

Description of exclusive processes within Collinear factorization approach Description of DVMP , DVCS, TCS, ... in the Bjorken limit Collinear factorization proven for LT amplitude MV,{0+;0+}

[Collins, Frankfurt, Strikman, ’97, Radyushkin, ’97]

Set of GPDs, H(x, ξ, t), E(x, ξ, t), ˜ H(x, ξ, t), ˜ E(x, ξ, t) Quark and Gluon contributions: V q k − ∆ k + ∆ ℓ1 ℓ2 ∼ Hg N N ′ V q k − ∆ k + ∆ ℓ1 ℓ2 ∼ Hq N N ′

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 6 / 25

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SLIDE 7

Modified perturbative approach (MPA)

Gluon contribution in MPA Mg

V =

  • dx
  • dy
  • d2ℓ

(2π)2

x + ξ x − ξ

ℓq = yp1 + ℓ⊥ +

ℓ2 2sy p2

q z1 z2 N(p′)| / A(z1) / A∗(0) |N(p) →

λ /

ε(λ)

⊥ /

ε(λ)∗

⊥ Hg(x,ξ,t) x2−(ξ−iǫ)2

V (p1)| ¯ ψi(z2)ψj(0) |0 → / p1,ijΨV (y, ℓ⊥) Neglect then

ℓ2

⊥

y¯ yQ2 terms in numerator in the MPA spirit

Fourier transform in transverse space → impact parameter space Sudakov form factor [Sterman, Li ,’92] (Resums soft gluon emmisions from the quark-antiquark dipole)

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 7 / 25

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DVMP within MPA

Model dependences Models from [ Kroll, Goloskokov, ’08] :

GPDs with evolution approximated by the DGLAP evolution Wavefunction models (Gaussian ansatz) ˆ ΨV (y, r) ∝ Leading twist DA × exp

  • − r2

4a2

V

y¯ y

  • Sudakov form factor [Dahm, Jakob, Kroll, ’95]

Kroll&Goloskokov GPD model based on double distribution ansatz

[ Musatov, Radyushkin, ’00 ]

H(x, ξ) = 1

−1

dβ 1−|β|

−1+|β|

dα δ(β + ξα − x) f(β, α, t′)

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 8 / 25

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SLIDE 9

MPA results

MPA result for the helicity amplitude γ∗

L N(p) → VL N(p′)

[A.B. in preparation]:

MV,{0+,0+} = s 2 √ 2π 1 dy

  • d2r
  • f

Cf

V

  • ˆ

ΨV (y, −r)ˆ Ψf

γ∗

L(y, r)

  • ×

π αs(µ2) Nc

  • 4

y¯ y(2ξs) 1 dx 2ξHg(x, ξ, t) + 2 x CF ξHf

singlet(x, ξ, t)

(x − ξ + iǫ)(x + ξ − iǫ)

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 9 / 25

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Dipole model vs MPA results

Results in the two different approaches in the limit t ∼ 0 MPA result for the helicity amplitude γ∗

L N(p) → VL N(p′)

[A.B. in preparation]:

Im Mg

V,{0+,0+} =

s 2 √ 2π 1 dy

  • d2r
  • f

Cf

V

  • ˆ

ΨV (y, −r)ˆ Ψf

γ∗

L(y, r)

  • ×
  • − π2

Nc 4 y¯ yQ2 αsHg(ξ, ξ, 0)

  • Dipole model result for the helicity amplitude γ∗

L N(p) → VL N(p′) :

Im Mg

V,{0+,0+} =

s 2 √ 2π

  • dy
  • d2r
  • f

Cf

V

  • ˆ

ΨV (y, −r)ˆ Ψf

γ∗

L(y, r)

  • ×
  • − N(x, ξ, r)

s

  • At small−x : 2ξs ≈ Q2

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 10 / 25

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SLIDE 11

Analogy between the results

Interpretation Forward dipole cross-section [Frankfurt, Radyushkin, Strikman, ’97] : N(x, 0, r) s = ˆ σ(x, r) = π2 αs Nc r2 xg(x) (color transparency) Comparing the results for DVMP , (Forward limit of gluon GPD : Hg(x, ξ → 0, 0) = x g(x)): N(x, ξ, r) s ↔ π2 αs Nc

  • 4

y¯ yQ2

  • Hg(ξ, ξ, 0) = π2 αS

Nc (r2

0) Hg(ξ, ξ, 0)

with r2

0 =

4 y¯ yQ2

  • r2

0 ≥ 2 R0(x) (Saturation radius) for Q2 ∼ 5 GeV2 for W = 75 GeV

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 11 / 25

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SLIDE 12

Comparison of predictions

Leading twist asymptotic Μ2 Q2 mV

2

Leading twist DA GBW saturation

Imaginary part of t channel gluon exchange

H1 W=75 GeV 10.0 5.0 2.0 20.0 3.0 30.0 15.0 7.0

Q2GeV2

1 5 10 50 100 500 1000

ΣLΡnb

Leading twist result from collinear factorization (Blue) Leading twist DA + dipole cross-section with saturation (Red)

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 12 / 25

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SLIDE 13

Beyond the imaginary part of gluon contribution in MPA

Contribution from gluons and quarks

  • f Cf

V

+

Hg(x, ξ, t) Hf(x, ξ, t) r0 r0 ˆ ΨV qf ¯ qf −

  • f

efCf

V N(x, r) ←

→

  • f

efCf

V

π αs(µ2) Nc

  • 4

y¯ yQ2

  • ×

1 dx 2ξHg(x, ξ, t) + 2 x CF ξHf

singlet(x, ξ, t)

(x − ξ + iǫ)(x + ξ − iǫ)

  • Adrien Besse

Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 13 / 25

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SLIDE 14

Contributions of other amplitudes

MPA Leading Twist Φ meson W = 75 GeV

5 10 15

Q2GeV2

0.2 0.4 0.6 0.8 1.0

Im Mg2M 2

MPA Leading Twist Ρ meson W = 75 GeV

5 10 15 20 25 30

Q2GeV2

0.2 0.4 0.6 0.8 1.0

Im Mg2M 2

(Im Mg

V )2 / |MV |2 ∼ 70%

(Im Mg

V )2 / |MV |2 ∼ 60%

Sea quark contribution (via interference term) not negligeable in MPA approach with GK GPDs based on [CTEQ6M, ’02] fits ( 10−4 < x < 0.5 and 4 < Q2 < 40 GeV2 )

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 14 / 25

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Results

Comparison Gluon vs Total contributions

H1 W=75 GeV Leading twist asymptotic Μ2 Q2 mV

2

Total Imaginary part Gluon exchange 10.0 5.0 2.0 20.0 3.0 30.0 15.0 7.0

Q2GeV2

1 5 10 50 100 500 1000

ΣLΡnb

H1 W=75 GeV Leading twist asymptotic Μ2 Q2 mV

2

Total Imaginary part Gluon exchange 10.0 5.0 2.0 3.0 15.0 7.0

Q2GeV2

0.1 1 10 100

ΣLΦnb

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 15 / 25

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Results

Sudakov + meson wavefunction ansatz

Sudakov Gaussian aV 0.5 GeV1 Μ2 Q2 mV

2

H1 W=75 GeV 10.0 5.0 2.0 20.0 3.0 30.0 15.0 7.0

Q2GeV2

1 5 10 50 100 500 1000

ΣLΡnb

Sudakov Gaussian aV 0.5 GeV1 Μ2 Q2 mV

2

H1 W=75 GeV 10.0 5.0 2.0 3.0 15.0 7.0

Q2GeV2

0.1 1 10 100

ΣLΦnb

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 16 / 25

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Dipole model from GPDs?

Contribution from gluons and quarks At small-x: Glauber-Mueller dipole model : ˆ σ(x, r) = π2 αs Nc r2 xg(x) → σ0

  • 1 − e

− π2 αs

Nc σ0 r2 xg(x)

  • For exclusive process, [Martin, Ryskin, Teubner ’99] skewness effect ⇒ Rg:

ˆ σ(x, r) = σ0

  • 1 − exp
  • − π2 αs

Nc σ0 r2 Rg xg(x)

  • f Cf

V

+

Hg(x, ξ, t) Hf(x, ξ, t) r0 r0 ˆ ΨV qf ¯ qf

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 17 / 25

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Summary

Summary:

Within MPA, MV,{0,+;0+} factorization of the overlap of the wavefunctions Allows to compare results within dipole picture and collinear factorization framework Role of the quark t−channel exchange within MPA can be important even at small−x

Perspectives :

The universality of the relation between the dipole scattering amplitude and GPDs has to be checked Other helicity amplitudes ⇒ sensitivity to other GPDs

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 18 / 25

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SLIDE 19

Acknowledgement

Thanks to my collaborators for encouraging discussions on this work P . Kroll

  • C. Lorcé
  • C. Mezrag
  • H. Moutarde
  • Al. Mueller
  • S. Munier
  • B. Pire

F . Sabatié

  • L. Szymanowski
  • S. Wallon

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 19 / 25

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Dipole model picture

Wavefunctions

Factorization of the wavefunctions and models / a → / ˜ a ¯ u(ℓ1) v(ℓ2) / ε(k1) ∝ / p2 / ε(q) Eikonal approximation leads to [Anikin, Wüsthof, ’99] / a / p2 → / ˜ a / p2 ∝ v(˜ a)¯ v(˜ a) / p2 , such that ˜ a2 = 0 The wavefunction factorizes Ψγ∗(y, ℓ) ∝ ¯ u(ℓ1)/ ε(q)v(˜ a) a2 + iǫ

  • ℓ1=yp1+ℓ⊥+ ℓ2

2ys p2 Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 20 / 25

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SLIDE 21

The factorization scheme within kT −factorization

Factorization of helicity amplitudes

LCCF

γ∗ p p′ ¯ y pρ y pρ ρ DA F(x, k) k

  • d2k

Φγ∗(λγ)→ρ(λρ)

Twist 2:

γ∗

L → ρL (≡ T00)

γ∗

T → ρL (≡ T01)

Ginzburg, Panfil, Serbo, ’85 Twist 3, in the limit t ∼ 0:

γ∗

T → ρT (≡ T11)

Anikin, Ivanov, Pire, Szymanowski, Wallon, ’10

Link with the dipole model (A.B., Szymanowski, Wallon, ’13) and implementation of the saturation effects using dipole cross-section models fitted on DIS Im Mg

V,{0+,0+} ∝

  • dy
  • d2r
  • f

Cf

V

  • ϕ1(y) ˆ

Ψf

γ∗

L(y, r)

  • N(x, ξ, r)

with ϕ1(y) the Leading twist DA

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 21 / 25

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SLIDE 22

Dipole models

Models of dipole cross-section Small−x evolution

Initial condition for the dipole cross-section at a given rapidity from DIS structure functions Evolution with rc-BK equation [Balitsky, ’07]

DGLAP evolution [Bartels, Golec-Biernat, Kowalski, ’02] ˆ σ(x, r) = σ0

  • 1 − exp
  • −π2 r2 αs(µ2(r2)) xg(x, µ2(r2))

3σ0

  • Adrien Besse

Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 22 / 25

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SLIDE 23

Contributions of other amplitudes

MPA : Re Mg MPA : Im Mg MPA : Im Mq MPA : Re Mq LT : Re Mg LT : Im Mg LT : Im Mq LT : Re Mq

5 10 15 20 25 30

Q2GeV2

3000 2500 2000 1500 1000 500

MΡ

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 23 / 25

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SLIDE 24

Large Q2 limit check

σL ∝

  • dy
  • d2r (Ψ∗

V Ψγ) (y, r) ˆ

σ(x, r)

  • 2

Expect to find same numerical values at large Q2 using :

a dipole cross-section model (here GBW model, [ Golec-Biernat, Wüsthof, ’98]): ˆ σ(x, r) = σ0(1 − e−r2/(4R0(x)2)) a dipole cross-section model with r2 → r2

0 = 4 y¯ yQ2

r1

2 16y y Q2

r0

2 4y y Q2

r2

2 1y y Q2 20 40 60 80 100 Q2 0.05 0.10 0.50 1.00 5.00 10.00

ΣL Σ

  • x, r0

ΣL Σ x, r

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 24 / 25

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Transversely polarized cross-section

Helicity amplitude MV,{++,++} within MPA .. under study The overlap of wavefunctions appearing within kT −factorization Ψ∗

V (y, r)Ψγ∗(y, r):

λγ = λV = 0 λγ = λV = ±

Leading twist DA Twist 3 combination of DAs (in Wandzura-Wilczek approx.) [A.B., Szymanowski, Wallon, ’13]

Expected to be sensitive to the Sudakov form factor suppression

Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 25 / 25