=Hybrids Generalized Parton Distributions A.V. Radyushkin June 2, 201
Hadrons in Terms of Quarks and Gluons =Hybrids Situation in hadronic physics: All relevant particles established QCD Lagrangian is known Need to understand how QCD works How to relate hadronic states p, s to quark and gluon fields q(z 1), q(z 2),...? Standard way: use matrix elements 0 q α(z 1) q β (z 2) M(p), s, 0 q α(z 1) q β (z 2) q γ(z 3) B(p), s Can be interpreted as hadronic wave functions
Light-front Wave Functions =Hybrids Describe hadron by Fock components on the light front or in infinite-momentum frame For nucleon P = q(x 1 P, k 1 ) q(x 2 P, k 2 ) q(x 3 P, k 3 ) + qqqg + qqq qq + qqqgg +... x i : momentum fractions x i = 1 k i : transverse momenta k i = 0 i i
Difficulties with LF Formalism =Hybrids In principle: Solving bound-state equation H P = E P one gets P which gives complete information about hadron structure In practice: Equation (involving infinite number of Fock components) has not been solved and is unlikely to be solved in near future Experimentally: LF wave functions are not directly accessible Way out: Description of hadron structure in terms of phenomenological functions
Phenomenological Functions =Hybrids Old functions: Form Factors Usual Parton Densities Distribution Amplitudes New functions: Generalized Parton Distributions () = Hybrids of Form Factors, Parton Densities and Distribution Amplitudes Old functions are limiting cases of new functions
Form Factors =Hybrids Form factors are defined through matrix elements of electromagnetic and weak currents between hadronic states Nucleon EM form factors: [ p, s J µ (0) p, s = ū(p, s ) ( = p p, t = 2 ) γ µ F 1 (t) + ν σ µν 2m N Electromagnetic current J µ (z) = f(lavor) e f ψ f (z)γ µ ψ f (z) Helicity non-flip form factor F 1 (t) = f e f F 1f (t) Helicity flip form factor F 2 (t) = f e f F 2f (t) ] F 2 (t) u(p, s)
Usual Parton Densities =Hybrids Parton Densities are defined through forward matrix elements of quark/gluon fields separated by lightlike distances Momentum space interpretation p xp Unpolarized quarks case: p z/2 p ψ a ( z/2)γ µ ψ a (z/2) p z 2 =0 = 2p µ 1 [ 0 e ix(pz) f a (x) e ix(pz) fā(x) ] dx xp p f a(ā) (x) is probability to find a (ā) quark with momentum xp Local limit z = 0 z/2 sum rule 1 0 [f a(x) fā(x)] dx = N a for valence quark numbers p
Nonforward Parton Densities (Zero Skewness ) =Hybrids Combine form factors with parton densities F 1 (t) = F 1a (t) a F 1a (t) = 1 0 F 1a (x, t) dx Flavor components of form factors F 1a (x, t) e a [F a (x, t) Fā(x, t)] Forward limit t = 0 F a(ā) (x, t = 0) = f a(ā) (x)
Interplay between x and t dependences =Hybrids Simplest factorized ansatz F a (x, t) = f a (x)f 1 (t) satisfies both forward and local constraints Reality is more complicated: LF wave function with Gaussian k dependence Ψ(x i, k i ) exp [ 1λ 2 i suggests F a (x, t) = f a (x)e xt/2xλ2 f a (x)=experimental densities Forward constraint F a (x, t = 0) = f a (x) Local constraint 1 0 [F a(x, t) Fā(x, t)]dx = F 1a (t) ] ki 2 xi Adjusting λ 2 to provide k 2 (300MeV)2
Impact Parameter Distributions =Hybrids can be treated as Fourier transforms of impact parameter b distributions f a (x, b ) F a (x, q 2 ) = f a (x, b )e i(q b ) d 2 b b = distance to center of momentum IPDs describe nucleon structure in transverse plane Distribution f p u(x, b )
Probabilistic Interpretation of IPDs Distribution in b plane Distribution in z momentum Combined (x, b ) distribution =Hybrids Defining center in plane: Geometric center: i b i = 0 Center of momentum: i x ib i = 0 Shape of (x, b ) distribution: Shrinks when x 1: leading parton determines center of momentum
-type models for (ξ = 0 ) =Hybrids improvement: f(x) x α(0) F(x, t) x α(t) F(x, t) = f(x)x α t Accomodating quark counting rules: F(x, t) = f(x)x α t(1 x) x 1 f(x)e α (1 x) 2 t Does not change small-x behavior but provides f(x) x 1 vs. F (t) t interplay: f(x) (1 x) n F 1 (t) t (n+1)/2 Note: no pqcd involved in these counting rules! Extra 1/t for F 2 (t) can be produced by taking E a (x, t) (1 x) 2 F a (x, t) for magnetic More general: E a (x, t) (1 x) ηa F a (x, t) Fit : η u = 1.6, η d = 1
Fit to All Four Nucleon G E,M Form Factors =Hybrids GPD model based on parametrization describes JLab polarization transfer data on G p E /Gp M and F p 2 /F p 1
Distribution Amplitudes =Hybrids may be interpreted as LF wave functions integrated over transverse momentum Matrix elements 0 O p of LC operators (AR 77) For pion (π + ): 0 ψ d ( z/2)γ γ µ ψ u (z/2) π + (p) z 2 =0 1 = ip µ f π e iα(pz)/2 ϕ π (α) dα 1 with α = x 1 x 2 or x 1 = (1 + α)/2, x 2 = (1 α)/2
Hard Electroproduction Processes: Path to =Hybrids pqcd Factorization Deeply Virtual Photon and Meson Electroproduction: Attempt to use perturbative QCD to extract new information about hadronic structure Hard kinematics: Q 2 is large s (p + q) 2 is large Q 2 /2(pq) x Bj is fixed t (p p ) 2 is small
Deeply Virtual Compton Scattering =Hybrids Kinematics Picture in γ N CM frame Total CM energy s = (q + p) 2 = (q + p ) 2 LARGE: Above resonance region Initial photon virtuality Q 2 = q 2 LARGE (> 1 GeV 2 ) Invariant momentum transfer t = 2 = (p p ) 2 SMALL ( 1GeV 2 ) Virtual photon momentum q = q x Bjp has component x Bjp canceled by momentum transfer Momentum transfer has longitudinal component + = x Bjp +, x Bj = Q2 2(pq) Skewed Kinematics: + = ζp +, with ζ = x Bj for
Parton Picture for =Hybrids In forward = 0 limit Nonforward parton distribution F ζ (X; t) depends on X : fraction of p + ζ : skeweness t : momentum transfer F a ζ=0(x, t = 0) = f a(x) Note: F a ζ=0(x, t = 0) comes from Exclusive Amplitude, while f a(x) comes from Inclusive DIS Cross Section Zero skeweness ζ = 0 limit for nonzero t corresponds to nonforward parton densities F a ζ=0(x, t) = F a (X, t) Local limit: relation to form factors (1 ζ/2) 1 0 F a ζ (X, t) dx = F a 1 (t)
Off-forward Parton Distributions (Ji, 96) =Hybrids Momentum fractions taken wrt average momentum P = (p + p )/2 4 functions of x, ξ, t: H, E, H, Ẽ wrt hadron/parton helicity flip +/+, /+, +/, / Skeweness ξ + /2P + is ξ = x Bj/(2 x Bj) 3 regions (AR 96): ξ < x < 1 quark distribution 1 < x < ξ antiquark distribution ξ < x < ξ distribution amplitude for N qqn
Reduction and Sum Rules =Hybrids In forward limit: reduction to usual parton densities H i(x, ξ = 0, t = 0) = f i(x) Ji s Sum Rule: reduction to quark contribution to nucleon spin 1 2 1 1 Form Factor Sum Rules { } dx x H i(x, ξ, 0) + E i(x, ξ, 0) = J i 1 e i H i(x, ξ, t) dx = F 1(t) i 1 1 e i E i(x, ξ, t) dx = F 2(t) i 1
Modeling =Hybrids Two approaches are used: Direct calculation in specific dynamical models: bag model, chiral soliton model, light-cone formalism, etc. Phenomenological construction based on relation of to usual parton densities f a(x), f a(x) and form factors F 1(t), F 2(t), G A(t), G P (t) Formalism of Double Distributions is often used to get self-consistent phenomenological models GPD Ẽ(x, ξ; t) is related to pseudoscalar form factor GP (t) and is dominated for small t by pion pole term 1/(t m 2 π) Dependence of Ẽ(x, ξ; t) on x is given by pion distribution amplitude ϕ π(α) taken at α = x/ξ
Double Distributions Superposition of P + and r + momentum fluxes =Hybrids Connection with OFPDs Basic relation between fractions x = β + ξα Forward limit ξ = 0, t = 0 gives usual parton densities 1 β 1+ β f a(β, α; t = 0) dα = f a(β)
Getting from =Hybrids live on rhombus α + β 1 Converting into Munich symmetry: f a(β, α; t) = f a(β, α; t) H(x, ξ) are obtained from f(β, α) by scanning at ξ-dependent angles DD-tomography
Illustration of DD GPD conversion =Hybrids Factorized model for : ( usual parton density in β-direction) ( distribution amplitude in α-direction) Toy model for double distribution f(β, α) = 3[(1 β ) 2 α 2 ] θ( α + β 1) GPD H(x, ξ) resulting from toy DD Corresponds to toy forward distribution f(β) = (1 β ) 3 For ξ = 0 reduces to usual parton density For ξ = 1 has shape like meson distribution amplitude
Realistic Model for based on =Hybrids DD modeling misses terms invisible in the forward limit: Meson exchange contributions D-term, which can be interpreted as σ exchange Inclusion of D-term induces nontrivial behavior in x < ξ region Meson and D-term terms DD + D-term model Profile model for : f a(β, α) = f a(β)h a(β, α) Normalization Guarantees 1 1 dα h(β, α) = 1 forward limit 1 1 dα f(β, α) = f(β)
DD Profile =Hybrids General form of model profile h(β, α) = Power b is parameter of the model b = gives h(β, α) = δ(α) and H(x, ξ) = f(x) Single-Spin Asymmetry A LU (ϕ) = dσ dσ dσ +dσ HERMES Data Γ(2+2b) [(1 β ) 2 α 2 ] b 2 2b+1 Γ 2 (1+b) (1 β ) 2b+1 JLab CLAS Data : Red: b val = 1 b sea = Green: b val = 1 b sea = 1 Blue: b val = b sea =
with behavior of f(β) New development: need single-dd representation =Hybrids H(x, ξ) x = dβ dα f(β, α) δ(x β ξα) Ω Vertical integration: reduction to f(x)/x x 1+ x dα f(x, α) = f(x) 1+ x x x Horizontal integration: reduction to D-term 1+ y dβ f(β, y) = D(y) 1+ y y y Ansatz f(β, α) β 1 a gives non-integrable singularity Subtraction/renormalization using plus-prescription ( ) f(β, α) f(β, α) = + δ(β) D(α) β + α
Model with behavior of parton density =Hybrids b=1 DD with a = 0. 8 6 4 2 1.0 0. 0. 1.0 2 4 6 ξ = 0.1, 0.2, 0.3, 0.4, 0. In fact, most important is the border function H(ξ, ξ) and D term: all the rest is reconstructible New precise JLab data just reported: exciting time for GPD extraction!
=Hybrids Theory of is well developed (several very large reviews) New precise data ready for GPD extraction Future 12 GeV data on and DVMP (also TCS, D proposed) will strongly extend the range Progress in GPD field is a result of a collective effort of many theorists and experimentalists THANK YOU ALL!