Dispersion theory in hadron form factors

Transcription

1 Dispersion theory in hadron form factors C. Weiss (JLab), Reaction Theory Workshop, Indiana U., 1-Jun-17 Objectives Review hadron interactions with electroweak fields: currents, matrix elements, form factors Seminar Vanderhaeghen Apply methods of amplitude analysis to form factors: Analyticity, dispersion relations, unitarity Lectures Learn about applications to actual experimental data Outline connections with other fields: QCD and partonic structure

2 Motivation Hadron reactions structure Simple application of amplitude analysis concepts: Single-variable function F(t) Great practical use Experimental programs: Electron scattering JLab, Mainz, MIT, atomic physics with electronic/muonic hydrogen PSI, e + e annihilation BES, KLOE, BABAR, etc., p p PADA SM/BSM processes with hadron production, neutrino interactions with matter Connection with QCD: Generalized parton distributions, Lattice QCD methods

3 Outline 3 1 Hadron interactions with external fields Currents, matrix elements, form factors Spacelike and timelike FFs Elastic scattering and annihilation Analytic properties of form factors Analytic F(t), physical sheet, principal cut Dispersion relations and convergence Spectral function and t-channel processes Methods for constructing spectral function 3 Spectral functions from amplitude analysis Unitarity relation for cut partial wave amplitudes from data Pion form factor from e + e + data 4 ucleon electromagnetic form factors Evaluation of dispersion integral Comparison with spacelike FF data Open questions 5 Extensions and connections ucleon scalar form factor ucleon axial form factors Transverse densities and generalized parton distributions

4 EM interactions: Hadrons in external field 4 External field electromagnetic/weak/other Accelerate existing hadron Create/annihilate hadron-antihadron pair } Relativity Interaction described by Lagrangian density L em (x) = ea µ (x)j µ (x), current operator Transition matrix elements h(p ) J µ (x) h(p), h(p ) h(p) J µ (x) 0 Treat as abstract objects, cf. hadron scattering amplitudes

5 EM interactions: Lepton scattering 5 k k p k p p p k M(lh l h ) = e h(p ) J µ (0) h(p) D µν (k k ) l(k ) j ν (0) l(k) Hadron current Green function Lepton current M(l + l h h) = e h(p ) h(p) J µ (0) 0 D µν (k +k ) 0 j ν (0) l(k )l(k) External field excited by lepton scattering Lepton current and Green function given by QED (EW theory) Explicit form of amplitude and cross section Exercise

6 EM interactions: Current matrix element 6 Translational invariance J µ (x) = e ipx J µ (0)e ipx finite translation operator h(p ) J µ (x) h(p) = e i(p p)x h(p ) J µ (0) h(p) x-dep explicit Current conservation µ J µ (x) = 0 µ h(p ) J µ (x) h(p) = 0 (p p) µ h(p ) J µ (0) h(p) = 0 transversality Similar relations for h(p ) h(p) J µ (x) 0 with p p

7 EM interactions: Form factor spin-0 hadron 7 Structural decomposition h(p ) J µ (0) h(p) structures Lorentz 4-vector: J µ (p +p) µ, (p p) µ p t p Current conserved: (p p ) µ J µ = 0 J µ = (p +p) µ F(t), t (p p) F(t) invariant form factor, cf. invariant amplitudes Physical region for scattering: t < 0 F(0) = hadron charge (in units of e)

8 EM interactions: Form factor spin-0 hadron 8 t p Similar decomposition h(p ) h(p) J µ (0) 0 = (p p) µ F(t), t = p +p p Physical region for h h creation: t > 4M h > 0 Crossing symmetry: Scattering and annihilation matrix elements are described by a single analytic function { t < 0, scattering spacelike FF F(t), t > 4Mh, annihilation timelike FF

9 EM interactions: Form factors spin-1/ hadron 9 Lorentz invariance requires λ λ p p t h(p,λ ) J µ (0) h(p,λ) = ū(p,λ ) Γ µ u(p,λ) Bilinear form in hadron bispinors u, ū Independent structures in Γ µ? Heuristic approach: Use transversality, Dirac eqn, gamma matrix identities Systematic approach: Count helicity amplitudes Exercise Γ µ = γ µ F 1 (t) + iσµν (p p) ν M h F (t) F 1, (t) invariant FFs F 1 (0) = charge, F (0) = anomalous magnetic moment Dirac and Pauli FFs; also other choices G E,G M

10 Analytic properties: Form factor 10 t spacelike t thr timelike h F = _ h t > tthr... hadronic states FF analytic function of t Physical sheet, approached from t < 0 o singularities at t < 0 Singularities at t > 0 poles, cuts Result from processes : current hadronic state h h Can occur below the physical h h threshold

11 Analytic properties: Examples 11 ucleon electromagnetic FFs, isovector unphysical physical 4M 16 M 4M K 4M _... _ (incl. ρ), 4,... KK_,... Pion electromagnetic FF physical... 4M 16 M 4M K 4M _ (incl. ρ), 4,... KK_,...

12 Analytic properties: Dispersion relation 1 F(t) = dt t thr ImF(t+i0) t t t thr Dispersion relation Convergence at t : F(t) t 1 (pion), t,3 (nucleon) from QCD DR represents FF at all t on physical sheet Implements correct analytic properties. Useful tool Spectral function ImF(t+i0) = [F(t+i0) F(t i0)]/(i) eed to know it in order to evaluate integral. Unphysical region!

13 Analytic properties: Spectral functions 13 Amplitude analysis techniques + hadronic scattering data: Unitarity, analytic continuation Frazer, Fulco, Phys. Rev. 117, 1603 (1960); Phys. Rev. 117, 1609 (1960) Hohler, Pietarinen, ucl. Phys. B 95, 10 (1975) Hoferichter, Kubis, Ruiz de Elvira, Hammer, Meissner, Eur. Phys. J. A 5, 331 (016) Fits to spacelike FF data Hohler et al., ucl. Phys. B 114 (1976) 505 Belushkin, Hammer, Meissner, Phys. Rev. C 75, 0350 (007) Lorenz, Hammer, Meissner, Eur. Phys. J. A 48, 151 (01) Dynamical calculations: Chiral effective field theory Gasser, Sainio, Svarc, ucl. Phys. B 307, 779 (1988) Bernard, Kaiser, Meissner, ucl. Phys. A 611, 49 (1996) Becher, Leutwyler, Eur. Phys. J. C 9, 643 (1999) Kubis, Meissner, ucl. Phys. A 679, 698 (001) Kaiser, Phys. Rev. C 68, 050 (003) Combination of above methods χeft + unitarity: Granados, Leupold, Perotti, Eur. Phys. J. A 53, 117 (017); Alarcon, Hiller Blin, Vicente Vacas, Weiss, ucl. Phys. A 964, 18 (017)

14 Analysis: Unitarity relation 14 = Fi Γi F _ t > 4M I = J = 1 ImF i (t) = k3 cm Γ i (t) F(t) t At 4M < t < 16M only channel open two-pion cut ImF i (t) from elastic unitarity relation Here: ucleon isovector EM FFs CM frame of timelike process, k cm = t/4 M pion CM momentum F (t) current partial-wave amplitude = pion timelike FF Γ i (t) partial-wave amplitude Amplitudes F (t) and Γ i (t) have same phase Watson theorem Amplitudes contain ρ as resonance

15 Analysis: partial-wave amplitudes 15 PW projection s, u Idea: Construct PWAs Γ i (t) from scattering data using amplitude analysis techniques t Two major challenges: t channel partial wave projection requires knowledge of amplitude in unphysical region of s,u channel processes s,u < (M +M ) analytic continuation in s,u at fixed t For spectral function we need t-channel PWAs at t > 4 M > 0, while data are at t < 0 analytic continuation in t

16 Analysis: partial-wave amplitudes 16 PW projection 00 u t ch s s, u 100 t t [M ] Continuation Mandelstam diagram ν = s u 4M 0 u ch PW projection s ch Data crossing-symmetric variable ν [M ]

17 Analysis: partial-wave amplitudes 17 PW projection I) Construct invariant amplitudes A ± with proper analyticity in s,u at fixed t < 0 s, u Data in physical region s,u > (M + ) t Continue to unphysical s,u < (M + ) II) Calculate t-channel partial-wave projections f ± (t) = (factor) 1 1 dz { P 1 (z) P 0 (z) P (z) } A ± (ν,t) Γ 1, (t) ν = s u 4M, z = cosθ t = M ν M t/4 M t/4 III) Analytically continue PWAs to region t > 4M Use DR for PWA (left-hand cut), /D method, estimate uncertainties Hohler et al 74

18 Analysis: partial-wave amplitudes 18 PW projection s, u t What hadronic processes contribute to the amplitude? Born term amplitudes with intermediate, Singularities at s,u = M,M Contribute to left-hand cut of PWA Also other intermediate states etc.

19 Analysis: Pion timelike form factor 19 Figure from Jegerlehner 15 + e t e + ChEFT = sqrt(t) F (t) measured in e + e + exclusive annihilation ρ as resonance Phase (Im/Re) determined by fit to resonant amplitude parametrization Gounaris-Sakurai: Effective range expansion of phase shift, good description of line shape Gounaris, Sakurai, PRL 1 (1968) 44

20 Analysis: Spectral function 0 8 Isovector FF Im F Im F 1 Spectral function on cut calculated Spectral function Im F i (t) / 6 4 Elastic unitarity condition + data, e + e + + analytic continuation Valid up to t 1 GeV t [GeV ] Contains ρ resonance = Fi Γi F _ t > 4M I = J = 1 Alt. approach: Dynamical calculations χeft + /D method Alarcon, Hiller Blin, Vicente Vacas, Weiss, PA 964, 18 (017)

21 ucleon FF: Dispersion integral 1 Spectral function Im F i (t) / Isovector FF Im F Im F 1? Evaulate dispersion integral F i (t) = 4M dt ImF i (t+i0) t t t [GeV ] What about higher t? Constraints on spectral function F 1 (t) t 4M dt ImF i(t ) = 0 asymptotic behavior F 1 (0) = charge = 4M dt ImF i (t ) t charge Parametrize isovector spectral function at high t : Simple pole, multiple poles Isoscalar: ω,φ + high t poles Hohler et al PB (1976); Belushkin et al PRC (007).

22 ucleon FF: Fits to spacelike data G E n G M n /(µn G D ) G E p /GD G M p /(µp G D ) Q [GeV ] Q [GeV ] High t spectral functions determined by fit to spacelike FF data Belushkin et al PRC (007); Lorenz et al EPJA 48, 151 (01) Good description of FF data Analytic parametrization of FF. Permits t 0 extrapolation, derivatives

23 ucleon FF: Derivatives 3 F(t) = 4M dt ImF i (t + i0) t t F (t = 0) = 4M dt ImF i (t + i0) (t ) Derivatives of FFs at t = 0 given by well-convergent integrals Dominated by t < 1 GeV. Predictions of unitarity! ρ meson dominance ucleon charge/magnetic radii r 1 = 6F 1(0) r = 6 F (0) F (0) Interpretation in context of non-relativistic systems: Radius of 3D charge/magnetization density in Breit frame q µ = (0,q) Relativistic systems: Use D transverse densities!

24 Extensions: Scalar nucleon FF 4 Amplitude analysis methods described here can be applied to variety of meson and baryon form factors Scalar nucleon FF Gasser, Leutwyler, Sainio, PLB 53, 60 (1991) O σ (x) = u,d m ψ(x)ψ(x), (p ) O σ (0) (p) = ū u σ(t) Fundamental interest: Mass term of QCD Hamiltonian, trace of QCD energy-momentum tensor, σ term Two-pion cut, unitarity condition. Coupling to K K at t > 4MK Pion scalar FF from dispersion theory with χeft input _ = + KK σ f 0 + σ

25 Extensions: Vector and axial FFs 5 Isoscalar vector FF Intermediate states 3,K K + higher _ 3, ω 3-body unitarity: Techniques being developed JPAC: Pilloni, Szczepaniak _ KK, φ Coupled channel problem Isovector pseudoscalar and axial FF _ pole J 5µ (x) = ψ(x)γ µ γ 5 ψ(x) [u d] Pion pole in induced pseudoscalar FF 3, A higher intermediate states in axial FF

26 Extensions: Transverse densities 6 x + = x 0 + x 3 = const. y b S y z ρ 1 (b) ρ (b) x z Structure of relativistic system is naturally described at fixed light-front time x + = x 0 +x 3 Boost-invariant definition of time Frame-independent wave functions, densities Unambiguous particle content of system FFs expressed through transverse densities Soper 76, Burkardt 00, Miller 07 F 1, (t = T ) = d b e i Tb ρ 1, (b) Cumulative charge/magnetization density at transverse position b Connection with GPDs in QCD ρ 1 (b) = q e q 1 0 dx [q q](x,b)

27 Extensions: Transverse densities 7 Dispersive representation Spectral function Im F 1 (t) / /b Im F 1 Weight (arb. units) ρ(b) = 4M dt K 0( tb) ImF(t) K 0 e b t ensures exponential suppression of large t Distance b selects t 1/b: Filter for spectral function t [GeV ] Peripheral densities dominated by lowest mass intermediate states Strikman, Weiss PRC (010); Miller, Strikman, Weiss PRC (011)

28 Extensions: Transverse densities 8 ρ( b) Im F (t ) high mass Spatial structure of t-channel processes Peripheral b 1 fm intermediate state (includes ρ) Central b 1 fm higher-mass intermediate states Quantify range of exchange mechanisms

29 Extensions: Transverse densities earthreshold ρ region Effective continuum (Belushkin 07 fit) ear-threshold ρ region continuum 10 - Im F V 1 (t) / ρ V (b) [fm - ] m 50 m t [GeV ] b [fm] Spectral analysis of isovector charge density ear threshold relevant only at b > 3fm Intermediate b = fm dominated by ρ Higher mass states relevant only at b < 0.3 fm

30 Extensions: Transverse densities 30 3 ω pole higher-mass Im F S 1 (t) ω 0 1? φ KK, ρ,... [GeV ] t b ρ S (b) [fm -1 ] b [fm] Spectral analysis of isoscalar density Miller, Strikman, CW 11 ω dominates at b > 1.5fm Large cancellations between ω and higher mass states at b = 0.5 1fm

31 Extensions: Transverse densities 31 Quark-hadron duality in transverse densities Hadron exchanges in t channel Partonic configurations in s channel Pion FF and transverse density F (t) from e + e + data Spectral function ImF (t) from fit to resonant amplitude parametrization FF and transverse density from dispersion relation Resonance transition FFs and densities and transition FFs S-matrix theory: Resonance structure defined at complex pole

32 Extensions: Pion FF and density 1.5 GS parametrization 1σ error 8 Dispersion integral (GS) ± 1σ uncertainty 6 t Im F (t) / [GeV] ρ (b) [fm - ] 4 Heavy resonance m R = 3.67 GeV M t [GeV] b [fm] F (t) measured in e + e + up to t 10 GeV 8 6 ρ (b) [fm - ] ImF (t) from resonant amplitude fit 4 Transverse density from dispersion relation High density in center of pion: Pointlike q q configurations Miller, Strikman, Weiss, PRD 83, (011) b x [fm] b y [fm] 1

33 Summary 33 Interaction of hadrons with external fields described by current matrix elements FFs have analytic properties similar to hadronic scattering amplitudes Spectral functions on cut from elastic unitarity and hadronic data Good description of spacelike nucleon FFs and derivatives Transverse densities connect hadronic exchanges with partonic structure Amplitude analysis is not just science, but also art!