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
Observables and kinematic ranges γ ∗ , λ γ V , λ V − Q 2 e − , k ′ e − , k → M V, { λ V ν ′ ; λ γ ,ν } = V , p V W 2 N, ν ′ N, ν p p ′ t W 2 ≫ | t | , Q 2 ≫ Λ 2 Q 2 QCD , the Bjorken x ∼ W 2+ Q 2 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
Theoretical descriptions Color dipole picture (small − x ) Collinear fact. picture γ ∗ γ ∗ V V N q ¯ q GPD N ′ N ′ N N 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
Dipole model picture Color dipole factorization scheme Impact parameter space representation of the amplitudes in the infinite momentum frame Nikolaev, Zakharov, ’91, Mueller, ’90 γ ∗ y V Ψ f Ψ i � r r b N ( x, r, b ) b N ( x, r, b ) = p ′ p Initial Ψ i and final Ψ f 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
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 I m M g V R e M g V can be deduced from I m M g V using dispersion relations In the limit ∆ = 0 , i.e. | t | = | t | min , � � I m M λ V λ γ ( Q 2 , x ) ∝ i dr ˆ Ψ ∗ λ V ( y, r ) ˆ dy Ψ λ γ ( y, r ) N ( x, ξ, r ) Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 5 / 25
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 M V, { 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: ℓ 1 ℓ 1 q q V V ℓ 2 ℓ 2 k − ∆ k − ∆ k + ∆ k + ∆ N ′ N ′ N N ∼ H g ∼ H q Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 6 / 25
Modified perturbative approach (MPA) Gluon contribution in MPA ℓ 2 ℓ q = yp 1 + ℓ ⊥ + 2 sy p 2 q z 2 � V ( p 1 ) | ¯ ψ i ( z 2 ) ψ j (0) | 0 � d 2 ℓ M g V = � dx � dy � z 1 0 → / p 1 ,ij Ψ V ( y, ℓ ⊥ ) (2 π ) 2 x + ξ x − ξ H g ( x,ξ,t ) ε ( λ ) ε ( λ ) ∗ � N ( p ′ ) | / A ∗ (0) | N ( p ) � → � A ( z 1 ) / λ / ⊥ / ⊥ x 2 − ( ξ − iǫ ) 2 ℓ 2 Neglect then yQ 2 terms in numerator in the MPA spirit ⊥ y ¯ 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
DVMP within MPA Model dependences Models from [ Kroll, Goloskokov, ’08] : GPDs with evolution approximated by the DGLAP evolution Wavefunction models (Gaussian ansatz) � � − r 2 ˆ Ψ V ( y, r ) ∝ Leading twist DA × exp y ¯ y 4 a 2 V Sudakov form factor [Dahm, Jakob, Kroll, ’95] Kroll&Goloskokov GPD model based on double distribution ansatz [ Musatov, Radyushkin, ’00 ] � 1 � 1 −| β | dα δ ( β + ξα − x ) f ( β, α, t ′ ) H ( x, ξ ) = dβ − 1 − 1+ | β | Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 8 / 25
MPA results MPA result for the helicity amplitude γ ∗ L N ( p ) → V L N ( p ′ ) [A.B. in preparation] : � 1 s � � � � Ψ V ( y, − r )ˆ ˆ d 2 r C f Ψ f M V, { 0+ , 0+ } = √ dy L ( y, r ) V γ ∗ 2 2 π 0 f � �� 1 2 ξH g ( x, ξ, t ) + 2 x C F ξH f � � � π α s ( µ 2 ) singlet ( x, ξ, t ) � 4 × dx N c y ¯ y (2 ξs ) ( x − ξ + iǫ )( x + ξ − iǫ ) 0 Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 9 / 25
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 ) → V L N ( p ′ ) [A.B. in preparation] : � 1 s � � � I m M g d 2 r � C f Ψ V ( y, − r )ˆ ˆ Ψ f √ V, { 0+ , 0+ } = dy L ( y, r ) γ ∗ V 2 2 π 0 f − π 2 4 � � yQ 2 α s H g ( ξ, ξ, 0) × N c y ¯ Dipole model result for the helicity amplitude γ ∗ L N ( p ) → V L N ( p ′ ) : s � � � � I m M g d 2 r � C f Ψ V ( y, − r )ˆ ˆ Ψ f V, { 0+ , 0+ } = √ dy L ( y, r ) γ ∗ V 2 2 π f − N ( x, ξ, r ) � � × s At small − x : 2 ξs ≈ Q 2 Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 10 / 25
Analogy between the results Interpretation Forward dipole cross-section [Frankfurt, Radyushkin, Strikman, ’97] : σ ( x, r ) = π 2 α s N ( x, 0 , r ) r 2 xg ( x ) = ˆ (color transparency) s N c Comparing the results for DVMP , (Forward limit of gluon GPD : H g ( x, ξ → 0 , 0) = x g ( x ) ): ↔ π 2 α s H g ( ξ, ξ, 0) = π 2 α S N ( x, ξ, r ) � � 4 ( r 2 0 ) H g ( ξ, ξ, 0) s N c y ¯ yQ 2 N c 4 with r 2 0 = yQ 2 y ¯ 0 � ≥ 2 R 0 ( x ) (Saturation radius) for Q 2 ∼ 5 GeV 2 for W = 75 GeV � � r 2 Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 11 / 25
Comparison of predictions Σ L � Ρ �� nb � Imaginary part of t � channel 1000 gluon exchange 500 100 50 H1 W=75 GeV Leading twist asymptotic 10 Μ 2 � Q 2 � m V 2 5 Leading twist DA � GBW saturation 1 Q 2 � GeV 2 � 2.0 3.0 5.0 7.0 10.0 15.0 20.0 30.0 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
Beyond the imaginary part of gluon contribution in MPA Contribution from gluons and quarks q f r 0 r 0 ˆ f C f + � Ψ V V q f ¯ H g ( x, ξ, t ) H f ( x, ξ, t ) π α s ( µ 2 ) � 4 � � e f C f � e f C f − V N ( x, r ) ← → V N c y ¯ yQ 2 f f �� 1 2 ξH g ( x, ξ, t ) + 2 x C F ξH f � singlet ( x, ξ, t ) × dx ( x − ξ + iǫ )( x + ξ − iǫ ) 0 Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 13 / 25
Contributions of other amplitudes � Im Mg � 2 �� M 2 � Im Mg � 2 �� M 2 1.0 1.0 Φ meson W = 75 GeV Ρ meson W = 75 GeV 0.8 0.8 0.6 0.6 0.4 0.4 MPA MPA Leading Twist Leading Twist 0.2 0.2 Q 2 � GeV 2 � Q 2 � GeV 2 � 0 5 10 15 0 5 10 15 20 25 30 V ) 2 / |M V | 2 ∼ 70% V ) 2 / |M V | 2 ∼ 60% ( I m M g ( I m M g 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 < Q 2 < 40 GeV 2 ) Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 14 / 25
Results Comparison Gluon vs Total contributions Σ L � Φ �� nb � Σ L � Ρ �� nb � 1000 H1 W=75 GeV 100 500 H1 W=75 GeV 100 10 50 Leading twist asymptotic Leading twist asymptotic Μ 2 � Q 2 � m V 2 Μ 2 � Q 2 � m V 2 10 1 5 Total Total Imaginary part Gluon exchange Imaginary part Gluon exchange 1 0.1 Q 2 � GeV 2 � Q 2 � GeV 2 � 2.0 3.0 5.0 7.0 10.0 15.0 20.0 30.0 2.0 3.0 5.0 7.0 10.0 15.0 Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 15 / 25
Results Sudakov + meson wavefunction ansatz Σ L � Φ �� nb � Σ L � Ρ �� nb � Sudakov � 1000 Gaussian a V � 0.5 GeV � 1 100 500 Μ 2 � Q 2 � m V 2 100 10 50 Sudakov � 10 1 Gaussian a V � 0.5 GeV � 1 5 Μ 2 � Q 2 � m V 2 H1 W=75 GeV H1 W=75 GeV 1 0.1 Q 2 � GeV 2 � Q 2 � GeV 2 � 2.0 3.0 5.0 7.0 10.0 15.0 20.0 30.0 2.0 3.0 5.0 7.0 10.0 15.0 Adrien Besse Constraints from the leptoproduction of vector meson within different framework MESON2014, June 2nd 16 / 25
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