Information Content of the DVCS Amplitude

Transcription

1 Information Content of the DVCS Amplitude DVCS? GPDs Matthias Burkardt New Mexico State University & Jefferson Lab Information Content of the DVCS Amplitude p.1/25

2 Generalized Parton Distributions (GPDs) GPDs: decomposition of form factors at a given value of t, w.r.t. the average momentum fraction x = 1 2 (x i + x f ) of the active quark dxh q (x, ξ, t) = F q 1 (t) dxe q (x, ξ, t) = F q 2 (t) dx H q (x, ξ, t) = G q A (t) dxẽ q (x, ξ, t) = G q P (t), x i and x f are the momentum fractions of the quark before and after the momentum transfer 2ξ = x f x i GPDs can be probed in deeply virtual Compton scattering (DVCS).. Information Content of the DVCS Amplitude p.2/25

3 Generalized Parton Distributions (GPDs) dx formal definition (unpol. quarks): 2π eix p + x ( p q x 2 ) ( ) x γ + q p 2 = H(x, ξ, 2 )ū(p )γ + u(p) +E(x, ξ, 2 )ū(p ) iσ+ν ν 2M u(p) in the limit of vanishing t and ξ, the nucleon non-helicity-flip GPDs must reduce to the ordinary PDFs: H q (x, 0, 0) = q(x) Hq (x, 0, 0) = q(x). DVCS amplitude A(ξ, t) 1 dx GPD(x, ξ, t) x ξ + iε Information Content of the DVCS Amplitude p.3/25

4 Phase of DVCS Amplitude l γ l γ l γ γ l l γ γ l p p p p p p (a) (b) (c) σ = A BH + A DV CS 2 = A BH 2 + A DV CS 2 + 2R {A BH A DV CS } clean separation of real part using angular dependence RA DV CS (ξ, t) 1 GPD(x, ξ, t) dx x ξ also from beam charge asymmetry (e + v. e ) IA DV CS (ξ, t) GPD(ξ, ξ, t) from beam spin asymmetry Information Content of the DVCS Amplitude p.4/25

5 Outline Ji-relation 3D-imaging H(x, 0, 2 ) q(x,b ) H(x, 0, 2 ) q(x,b ) E(x, 0, 2 ) deformation of unpol. PDFs in pol. target polynomiality accessing GPDs in DVCS Summary Information Content of the DVCS Amplitude p.5/25

6 Ji Relation Interesting observation: X.Ji, PRL78,610(1997) J q = dxx[h q (x, 0, 0) + E q (x, 0, 0)] lattice QCD (LHPC,QCDSF) DVCS GPDs J q L u + L d 0 (disconnected diagrams?) L u L d < 0! m 2 π extrapolation Q 2 evolution parton interpretation Information Content of the DVCS Amplitude p.6/25

7 3D imaging of the nucleon b? (f m) 0 1 x Information Content of the DVCS Amplitude p.7/25

8 Impact Parameter Dependent PDFs form factors: FT ρ( r) GPDs(x, 0, 2 ): form factor for quarks with momentum fraction x suitable FT of GP Ds should provide spatial distribution of quarks with momentum fraction x careful: cannot measure longitudinal momentum (x) and longitudinal position simultaneously (Heisenberg) consider purely transverse momentum transfer q(x,b ) = d 2 (2π) 2 H(x, 0, 2 )e ib q(x,b ) = parton distribution as a function of the separation b from the transverse center of momentum R i q,g r,ix i (MB, 2000) Information Content of the DVCS Amplitude p.8/25

9 Impact Parameter Dependent PDFs No relativistic corrections (Galilean subgroup!) corollary: interpretation of 2d-FT of F 1 (Q 2 ) as charge density in transverse plane also free from relativistic corrections q(x,b ) has probabilistic interpretation as number density ( q(x,b ) as difference of number densities) Reference point for IPDs is transverse center of (longitudinal) momentum R i x ir i, for x 1, active quark becomes COM, and q(x,b ) must become very narrow (δ-function like) H(x, 2 ) must become indep. as x 1 (MB, 2000) consistent with lattice results for first few moments ( J.Negele) Note that this does not necessarily imply that hadron size goes to zero as x 1, as separation r between active quark and COM of spectators is related to impact parameter b via r = 1 1 x b. Information Content of the DVCS Amplitude p.9/25

10 q(x,b ) for unpol. p b y b x 6 4 x = b y b x b? (f m) 0 1 x x = 0.3 b y x = momentum fraction of the quark b x x = 0.5 b = position of the quark Information Content of the DVCS Amplitude p.10/25

11 Transversely Deformed Distributions and E(x, 2 ) M.B., Int.J.Mod.Phys.A18, 173 (2003) distribution of unpol. quarks in unpol (or long. pol.) nucleon: q(x,b ) = d 2 (2π) 2 H(x, 2 )e ib H(x,b ) unpol. quark distribution for nucleon polarized in x direction: q(x,b ) = H(x,b ) 1 2M b y d 2 (2π) 2 E(x, 0, 2 )e ib Physics: j + = j 0 + j 3, and left-right asymmetry from j 3 Information Content of the DVCS Amplitude p.11/25

12 Intuitive connection with J q DIS probes quark momentum density in the infinite momentum frame (IMF). Quark density in IMF corresponds to j + = j 0 + j 3 component in rest frame ( p γ in ẑ direction) j + larger than j 0 when quark current towards the γ ; suppressed when away from γ For quarks with positive orbital angular momentum in ˆx-direction, j z is positive on the +ŷ side, and negative on the ŷ side p γ ẑ ŷ j z > 0 Details of deformation described by E q (x, 2 ) not surprising that E q (x, 2 ) enters Ji relation! J i q = S i dx[h q (x, 0) + E q (x, 0)] x. j z < 0 Information Content of the DVCS Amplitude p.12/25

13 IPDs on the lattice (Hägler et al.) lowest moment of distribution of unpol. quarks in pol. proton (left) and of pol. quarks in unpol. proton (right): Information Content of the DVCS Amplitude p.13/25

14 A DV CS GPDs Ji relation J q = dxx[h(x, ξ, 0) + E(x, ξ, 0)] requires 0 GPDs(x, ξ, 0) for (coomon) fixed ξ for all x 3D imaging requires GPDs for ξ = 0 IA DV CS only sensitive to GPDs(ξ, ξ, t) RA DV CS only sensitive to GPDs(x,ξ,t) dx 1 x ξ Information Content of the DVCS Amplitude p.14/25

15 Polynomiality & the D-term Lorentz invariance polynomiality (n odd) 0 dxx n GPD (+) (x, ξ, t) = B n0 (t)+b n2 (t)ξ 2 +B n4 (t)ξ B n,n+1 (t)ξ n+1 Consider in the following only charge-even GPDs, e.g. H (+) (x, ξ, t) H(x, ξ, t) H( x, ξ, t) but drop superscript (+) Polynomiality highly constrains possible functional form of GPDs and plays crucial role in deconvolution of the DVCS amplitude original double distribution representation for GPDs (Radyushkin) manifestly satisfied polynomiality, but without B n,n+1 -term D-term (Polyakov & Weiss) added to allow for highest power of ξ H(x, ξ, t) = H DD (x, ξ, t) + Θ(ξ 2 x 2 )D ( ) x ξ, t Information Content of the DVCS Amplitude p.15/25

16 More recently (Anikin & Teryaev): arises as Information Content of the DVCS Amplitude p.16/25 subtraction-constant in dispersion relation for DVCS amplitude Polynomiality & the D-term D-term (Polyakov & Weiss) added to allow for highest power of ξ H(x, ξ, t) = H DD (x, ξ, t) + Θ(ξ 2 x 2 )D ( ) x ξ, t D-term contributes only to real part of DVCS amplitude, with D-form factor (t) = D(z,t) dz 0 1 z RA(ξ, t) = 0 dx H+ DD (x, ξ, t) x ξ + (t) For fixed x, contribution of D-term to H(x, ξ, t) disappears as ξ 0, but δ(x)-like contribution to Compton Amplitude lim ξ 0 H(x, ξ, t) x ξ = H DD(x, 0, t) x + δ(x) (t)

17 Polynomiality arbitrary function H (+) (x, ξ, t) (P n (x) Legendre polynomials) H (+) (x, ξ, t) = P 1 (x) [ A 10 (t) + A 12 (t)ξ 2 + A 14 (t)ξ 4 + A 16 (t)ξ ] + P 3 (x) [ A 30 (t) + A 32 (t)ξ 2 + A 34 (t)ξ 4 + A 36 (t)ξ ] + P 5 (x) [ A 50 (t) + A 52 (t)ξ 2 + A 54 (t)ξ 4 + A 56 (t)ξ ] + P 7 (x) [ A 70 (t) + A 72 (t)ξ 2 + A 74 (t)ξ 4 + A 76 (t)ξ ] +... Information Content of the DVCS Amplitude p.17/25

18 Polynomiality use polynomiality ( 1 dxxk H(x, ξ, t) = even polynomial in ξ of order k + 1) H (+) (x, ξ, t) = P 1 (x) [ A 10 (t) + A 12 (t)ξ 2] + P 3 (x) [ A 30 (t) + A 32 (t)ξ 2 + A 34 (t)ξ 4] + P 5 (x) [ A 50 (t) + A 52 (t)ξ 2 + A 54 (t)ξ 4 + A 56 (t)ξ 6] + P 7 (x) [ A 70 (t) + A 72 (t)ξ 2 + A 74 (t)ξ 4 + A 76 (t)ξ ] +... A nl = 0 for l > n + 1 used 1 dxxk P n (x) = 0 for n > k Information Content of the DVCS Amplitude p.18/25

19 Polynomiality ( ) separate H (+) (x, ξ, t) = H (+) DD (x, ξ, t) + Θ( ξ x)d x ξ H (+) DD (x, ξ, t) = P 1(x)A 10 (t) + P 3 (x) [ A 30 (t) + A 32 (t)ξ 2] + P 5 (x) [ A 50 (t) + A 52 (t)ξ 2 + A 54 (t)ξ 4] + P 7 (x) [ A 70 (t) + A 72 (t)ξ 2 + A 74 (t)ξ 4 + A 76 (t)ξ 6] dξh (+) DD (ξ, ξ, t)ξ = A 10(t) +A 54 (t) 1 1 dξξp 1 (ξ) + A 32 (t) 1 dξξ 5 P 5 (ξ) +... dxh (+) DD (x, ξ, t)x = A 10(t) 1 1 dξξ 3 P 3 (ξ) dxxp 1 (x) Information Content of the DVCS Amplitude p.19/25

20 Accessing GPDs in DVCS A DV CS (ξ, t) 1 dx GPD(+) (x,ξ,t) x ξ+iε ξ longitudinal mometum transfer on the target ξ = p+ p + p + +p + x (average) momentum fraction of the active quark x = k+ +p + IA DV CS (ξ, t) GPD (+) (ξ, ξ, t) only sensitive to diagonal x = ξ limited ξ range t min = 4ξ2 M 2 1 ξ 2 t = 4ξ2 M ξ 2 p + +p + RA DV CS (ξ, t) 1 dx GPD(+) (x,ξ,t) x ξ diagonal, but... probes GPDs off the Information Content of the DVCS Amplitude p.20/25

21 A(ξ,t) GPD (+) (ξ,ξ,t), (t) (Anikin & Teryaev): arises as subtraction-constant in dispersion relation for DVCS amplitude (see also G.Goldstein & S.Liuti) RA(ν, t) = ν2 π 0 dν 2 ν 2 IA(ν, t) ν 2 ν 2 + (t) In combination with LO factorization (A = H(x,ξ,t) dx 1 x ξ+iε ) RA(ξ, t) = 1 H(x, ξ, t) dx x ξ = 1 H(x, x, t) dx x ξ + (t) earlier derived from polynomiality (Goeke,Polyakov,Vanderhaeghen) Possible to condense information contained in A DV CS (fixed Q 2, assuming leading twist factorization) into GPD(x, x, t) & (t) A(ξ, t) { GPD(ξ, ξ, t) (t) Information Content of the DVCS Amplitude p.21/25

22 A(ξ,t) GPD(ξ,ξ,t), (t) RA(ξ, t) = H(x,ξ,t) dx 1 x ξ probes GPDs for x ξ, but new information using polynomiality/dispersion relation can be projected back onto diagonal plus D-term! better to fit parameterizations for GPD(x, x, t) plus (t) to A DV CS rather than parameterizations for GPD(x, ξ, t)? even after projecting back onto GPD(x, x, t), RA(ξ, t) still provides new (not in IA) info on GPDs: D-form factor constraints from dx GPD(x,x,t) x ξ on GPD(ξ, ξ, t) in kinematically inaccessible range t t 0 4M2 ξ 2 1 ξ 2 good news for model builders: as long as you fit IA(ξ, t), your model should do well for RA(ξ, t), provided model has polynomility allows for a D-form factor Information Content of the DVCS Amplitude p.22/25

23 Application of 1 dxh(x,ξ,t) x ξ = 1 dxh(x,x,t) x ξ + (t) take ξ 0 (should exist for t sufficiently large) 1 dx H(+) (x, 0, t) x = 1 dx H(+) (x, x, t) x + (t) DVCS allows access to same generalized form factor 1 dx H(+) (x,0,t) x also available in WACS (wide angle Compton scattering), but t does not have to be of order Q 2 1 dx H(+) (x,0,t) x 1 after flavor separation, F 1 (t) at large t provides information about the typical x that dominates large t form factor Information Content of the DVCS Amplitude p.23/25

24 Constraints on GPD models 0 ζ 1 ζ 1. 1 ζ x = ξ corresponds to matrix elements involving quarks with zero momentum LC wave function vanishes in many light-cone constituent models H (+) (x, x, t) = 0 in such models, and hence lim dx H(+) (x, 0, t) ε 0 ε x = 0 dx H(+) (x, x, t) x = 0 FT to impact parameter space: lim ε 0 ε dx q(x,b ) x = 0, but integrand is positive definite (wave function squared) models that yield H (+) (ξ, ξ, t) = 0 for all t violate Lorentz invariance (polynomiality) or positivity or both... Information Content of the DVCS Amplitude p.24/25

25 DVCS GPD(x,ξ,t) Information away from diagonal (x = ξ): RA DV CS D form factor polynomiality condition: n-th Mellin moment of GP D(x, ξ) must be even polynomial in ξ of order n GP D(x, ξ) cannot depend on variables x and ξ completely independently Q 2 evolution: changes x distribution in a known way for fixed ξ Double Deeply Virtual Compton Scattering D 2 V CS (lepton pair instead of real photon in final state) Information Content of the DVCS Amplitude p.25/25

26 Summary Deeply Virtual Compton Scattering GPDs GPDs (fixed ξ) J q (Ji-relation) GPDs(ξ = 0) FT IPDs (impact parameter dependent PDFs) IA DV CS only sensitive to GPDs(ξ, ξ, t) information contained in RA DV CS can also be condensed (polynomialtiy/dispersion relations) to GP Ds(ξ, ξ, t) plus D-form factor (t) 0 dx H(+) (x,x,t) x = 0 dx H(+) DD (x,0,t) x compare to F 1 (t) = 0 dxh(x, 0, t) = 0 dxh DD(x, 0, t) learn about what x dominate form factor constraints on light-cone wave function models need Q 2 evolution (or DDVCS) to disentangle x ξ dependence of GPDS Information Content of the DVCS Amplitude p.26/25