Nucleon Form Factors. Ulrich Müller A1 Collaboration Institut für Kernphysik Universität Mainz

Transcription

1 Nucleon Form Factors Ulrich Müller A1 Collaboration Institut für Kernphysik Universität Mainz Introduction Elastic form factors of the proton High-accuracy H(e, e )p experiment Measurement of G En Previous D( e, e n)p experiment in A1 Future plans Summary Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

2 Introduction Elastic form factors parameterise the nucleon s ability as a composite object to scatter a lepton coherently and without excitation of internal degrees of freedom. First hints of electromagnetic structure of the nucleon: Frisch and Stern, 1932 [Z. Phys. 88 (1933) 4] µ p = 2.5 µ N contrary to expectation µ p = µ N 2 S = µ N for Dirac particle Today, still a very active field of research: Proton G p E /Gp M ratio at high Q2 Two-photon exchange Charge radius r 2 p 1/2 Recent discrepancy in muonic hydrogen measurement Neutron form factors G n M and Gn E Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

3 Nucleon form factors Scattering of unpolarised leptons at point-like Dirac particles: ( ) ( dσ dσ dω = 1 τ tan 2 θ ) dω Mott 2 where τ = Q 2 /4M 2 and ( ) dσ dω Mott = α2 cos 2 (θ/2) 4E 2 sin 4 (θ/2) E M sin2 (θ/2) }{{} recoil term: E /E Structure of the nucleon: parameterised by form factors. Matrix element of e. m. current: N J µ (0) N = ū [γ µ F 1 (Q 2 ) + iσµν q ν ] 2M F 2(Q 2 ) u Cross section: dσ dω = ( ) [ dσ F 2 1 τf τ(F 1 + F 2 ) 2 tan 2 θ ] dω Mott 2 Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

4 Sachs form factors F 1 (Q 2 ) Dirac form factor: charge and Dirac magnetic moment F 2 (Q 2 ) Pauli form factor: anomalous magnetic moment Sachs form factors G E, G M : Linear combinations of F 1 and F 2 : G E = F 1 τf 2 G M = F 1 + F 2 F 1 = G E + τg M 1 + τ F 2 = G M G E 1 + τ G E (Q 2 ): charge G M (Q 2 ): magnetic moment Static limits: Total change and magnetic moment of the nucleon G p E (0) = 1 Gp M (0) = G n E(0) = 0 G n M(0) = 1.91 Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

5 Rosenbluth cross section dσ dω = = ( ) [ dσ G 2 E + τg2 M dω Mott 1 + τ + 2τG2 M tan 2 θ ] 2 ( ) dσ ɛg2 E + τg2 M dω Mott ɛ(1 + τ) where τ = Q 2 /4m 2 p ( ɛ = (1 + τ) tan 2 θ 2 ) 1 Variation of ɛ (at constant Q 2 ) Separation of G E and G M Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

6 Double polarisation experiments Kinematics of the n( e, e n) reaction: p i y p e x z p e ϑe scattering plane Φ R ϑ nq p f q Polarisation transfer to the recoil neutron: [Arnold, Carlson, Gross, PRC 23 (1981) 363] reaction plane P x = P e 2 τ(1 + τ) tan(θ/2)g E G M G 2 E + τg2 M (1 + 2(1 + τ) tan2 (θ/2)) P y = 0 P z = P e 2τ 1 + τ + (1 + τ) 2 tan 2 (θ/2) tan(θ/2)g 2 M G 2 E + τg2 M (1 + 2(1 + τ) tan2 (θ/2)) P x contains interference term G E G M. Equivalent terms for the asymmetry in n( e, e n) scattering Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

7 Charge distribution and Breit frame Breit frame (or brick wall frame): Incoming and outgoing electron have opposite momenta. p = + q/2 p = q/2 No energy transfer by the virtual photon, ω = 0 q µ = (0, q), Q 2 = q 2 Electric and magnetic form factor G E and G M can be interpreted as three-dimensional Fourier transform of charge and magnetisation density. Caveat: For each Q 2, there is a different Breit frame in which G E ( q 2 ) and G M ( q 2 ) are defined. Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

8 Mean squared radius Slope of the form factor at Q 2 = 0 determines mean squared radius: r 2 = 6 G(0) dg dq 2 Q2 =0 (Neutron charge radius defined without normalisation factor, since G n E (0) = 0.) For the charge distribution of a spherical heavy object, the mean squared radius is also given by: r 2 = 1 Q 0 r 2 ρ(r)dv Exercise: Show that the two definitions agree Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

9 Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

10 Motivation: Existing world data G EP /G dipole total smooth Simon et al. Price et al. Berger et al. Hanson et al. polarisation data G MP /(µ p G dipole ) total smooth Hoehler et al. Janssens et al. Berger et al. Bartel et al. Walker et al. Litt et al. Andivahis et al. Sill et al G EP -G EP,phaen Q / (GeV/c) total-smooth Simon et al. Price et al Berger et al. Hanson et al. polarisation data G MP -G MP,phaen Q / (GeV/c) 0.1 total - smooth Hoehler et al. Janssens et al. Berger et al Bartel et al. Walker et al. Litt et al. Andivahis et al. Sill et al Q 2 / (GeV/c) 2 Q 2 / (GeV/c) 2 (see J. Friedrich and Th. Walcher, Eur. Phys. J. A 17 (2003) 607) Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

11 Motivation: Existing world data G EP /G dipole total smooth Simon et al. Price et al. Berger et al. Hanson et al. polarisation data G MP /(µ p G dipole ) total smooth Hoehler et al. Janssens et al. Berger et al. Bartel et al. Walker et al. Litt et al. Andivahis et al. Sill et al G EP -G EP,phaen Q / (GeV/c) total-smooth Simon et al. Price et al Berger et al. Hanson et al. polarisation data G MP -G MP,phaen Q / (GeV/c) 0.1 total - smooth Hoehler et al. Janssens et al. Berger et al Bartel et al. Walker et al. Litt et al. Andivahis et al. Sill et al. 0 And: Discrepancy between textbook value (from standard dipole) and newer results (e.g Simon et al. 1979! and from hyperfine splitting) Q 2 / (GeV/c) 2 Q 2 / (GeV/c) 2 (see J. Friedrich and Th. Walcher, Eur. Phys. J. A 17 (2003) 607) Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

12 Design goal: high accuracy use redundancy Statistical errors: No problem: 20 min beamtime for < 0.1 % Luminosity and systematics: Principle: Measure quantities in as many ways as possible: Beam current: Förster probe (fluxgate magnetometer) pa-meter measures down to extremely low currents for small θ Luminosity: current density target length third spec. as monitor Overlapping acceptance (every angle measured 4 times with the same spectrometer) Where possible: Measure same angle with two spectrometers Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

13 Measured settings ( ) dσ dσ dω = ɛg2 E (Q2 ) + τg 2 M (Q2 ) dω Mott ɛ(1 + τ) Q [GeV/c] Q 2 [(GeV/c) 2 ] 1000 Settings > 10 9 events A limit B limit ε E<180 MeV E>1.53 GeV E=855 MeV Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

14 Three-spectrometer setup of the A1 collaboration Spectrometer A: α > 20 p < 735 MeV/c Ω = 28 msr p/p = 20% Spectrometer B: α > 8 p < 870 MeV/c Ω = 5.6 msr p/p = 15% Spectrometer C: α > 55 p < 655 MeV/c Ω = 28 msr p/p = 25% Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

15 Data taking periods August 2006 November 2006 May 2007 Duration 10 days 11 days 17 days without setup/calibration 8 days 9 days 11 days Energies 575, 850 MeV 180, 720 MeV 315, 450 MeV Setting changes data taking time 3.3 days 4.3 days 5.3 days Average time per setting 31 min 35 min 38 min Overhead per setting 44 min 40 min 40 min Overhead includes angle changes, momentum changes and down times. Average time for angle only setting changes: 10 min Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

16 Spectra and background subtraction Cryo target has havar foil walls background Simulate elastic scattering on wall (Fe, Co, Cr,...) Simulate quasi-elastic scattering Simulate elastic scattering on proton data simulated background residue de [MeV] Elastic background negligible Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

17 Description of the radiative tail counts E [MeV] integrated tail exp./sim. [arb. units] E [MeV] Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

18 How to extract the form factors? Two methods: 1. Classical Rosenbluth separation (vary ɛ at constant Q 2 ) Extracted G E and G M highly correlated Intricate error propagation 2. Fit form factor models directly to data No projection of data to exact Q 2 values All data contribute to fit Easy extraction of normalisation Model dependence? For extraction of radii: Would do a fit anyway! Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

19 Some models Dipole (different b for G E and G M ): ( G D Q 2, b ) 1 = ( 1 + Q2 b ) 2 Double dipole (as in Friedrich/Walcher phenomenological fit): G DD ( Q 2, a, b 1, b 2 ) = agd ( Q 2, b 1 ) + (1 a) GD ( Q 2, b 2 ) [Eur. Phys. J. A 17 (2003) 607] Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

20 More models Polynomial standard dipole: G P ( Q 2, a 1, a 2, a 3, a 4 ) = ( 1 + a1 Q 2 + a 2 Q 4 + a 3 Q 6 + a 4 Q 8) G D ( Q 2, 0.71 ) Continued fraction expansion (I. Sick): [Phys. Rev. C 76 (2007) ] G C ( Q 2, a 1, a 2,... ) = Not used due to very slow convergence a1q2 1+ a 2 Q Cubic splines: G spline ( Q 2, a 1, a 2,... ) Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

21 Cross sections: 450 MeV σ exp /σ std. dipole Polynomial Poly. + dip. Poly. dip. Inv. poly. Spline Spline dip. Friedrich-Walcher Double Dipole Extended G.K PRELIMINARY Q 2 [(GeV/c) 2 ] Statistical errors only Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

22 Cross sections: 315 MeV σ exp /σ std. dipole Polynomial Poly. + dip. Poly. dip. Inv. poly. Spline Spline dip. Friedrich-Walcher Double Dipole Extended G.K PRELIMINARY Q 2 [(GeV/c) 2 ] Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

23 Cross sections: 720 MeV σ exp /σ std. dipole Polynomial Poly. + dip. Poly. dip. Inv. poly. Spline Spline dip. Friedrich-Walcher Double Dipole Extended G.K PRELIMINARY Q 2 [(GeV/c) 2 ] Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

24 Extracted form factors and rms radii G E /G std. dipole µ p G M /G std. dipole Q 2 [(GeV/c) 2 ] Q 2 [(GeV/c) 2 ] Spline fit stat. errors exp. syst. errors theo. syst. errors Simon et al. Price et al. Berger et al. Hanson et al. Borkowski et al. Janssens et al. Murphy et al. Spline fit stat. errors exp. syst. errors theo. syst. errors Price et al. Berger et al. Hanson et al. Borkowski et al. Janssens et al. r 2 e 1/2 = ± stat. ± syst. ± model ± group fm r 2 m 1/2 = ± stat. ± syst. ± model ± group fm Doctoral thesis of Jan Bernauer J. Bernauer et al., arxiv: , submitted to PRL Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

25 Electric rms radius Some model dependence in electric rms radius: Spline 3 Ord. 8 par. Spline 3 Ord. 9 par. Spline 3 Ord. 10 par. Spline 3 Ord. 11 par. Spline 4 Ord. 8 par. Spline 4 Ord. 9 par. Spline 4 Ord. 10 par. Spline 4 Ord. 11 par. Spline 5 Ord. 8 par. Spline 5 Ord. 9 par. Spline 5 Ord. 10 par. Spline 5 Ord. 11 par. Spline dipole 7 par. Spline dipole 8 par. Spline dipole 9 par. Spline dipole 10 par. Spline dipole 11 par. Inv. Poly. 6 par. Inv. Poly. 7 par. Inv. Poly. 8 par. Inv. Poly. 9 par. Poly. 9 par. Poly. 10 par. Poly. 11 par. Poly. 12 par. Poly. + dipole 9 par. Poly. + dipole 10 par. Poly. + dipole 11 par. Poly. + dipole 12 par. Poly. dipole 7 par. Poly. dipole 8 par. Poly. dipole 9 par. Poly. dipole 10 par. Friedrich-Walcher <r E > [fm] r 2 e 1/2 = ± stat. ± syst. ± model ±0.004 group fm Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

26 Comparison with measurements in atomic physics Lamb shift in hydrogen: Energy shift of 2S 1/2 state due to vacuum polarisation P states like 2P 1/2 not shifted Finite size re 2 1/2 of nucleus also leads to shift of S states Values for r e from three groups of experiments: (8) fm Electron scattering (69) fm CODATA 2006 dominated by Lamb shift of electronic hydrogen (67) fm Lamb shift of muonic hydrogen Pohl et al., Nature 466 (2010) 213 Discrepancy of five standard deviations is unexplained. Radiative corrections for muonic hydrogen? Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

27 Future experiment: Extension to high Q Q [GeV/c] A limit B limit ε E<180 MeV E>1.53 GeV E=855 MeV Q 2 [(GeV/c) 2 ] MAMI C beam energy E = 1.6 GeV Same experimental method Reduced lever arm in ɛ, but not critical Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

28 Future experiment at low Q e e e e γ* γ* 10 2 p p p p dσ/(de dω ) LAB (dω γ ) CM [µb/(gev sr 2 )] e e e e Initial state radiation p(e, e p)γ reaction Identify events by missing mass of reconstructed photon Q 2 > 0.01 (GeV/c) 2 First experimental run this year Talk by Harald Merkel in γ* p p γ* p p o 90 o 0 o 90 o 180 o ϑ γγ Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

29 Measurement of G En Motivation Measurement of G En two main problems: No free neutron target (G En ) 2 (G Mn ) 2 Measurement of the charge radius r 2 En by scattering of neutrons on atoms = ( ± ± 0.003) fm2 for r 2 En [Kopecky et al., PRC 56 (1997) 2229] = ( ± 0.003) fm2 for ( Dubna compilation ) r 2 En Determines slope of G En at Q 2 = Pb 186 W, 209 Bi Rosenbluth-Separation for quasifree D(e, e ) scattering at DESY and SLAC: G En compatible with zero. [Lung et al., PRL 70 (1993) 718] Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

30 Polarised target experiments Types of double polarisation experiments: Measurement of neutron recoil polarisation D( e, e n)p 3 He( e, e n)pp Polarised target D( e, e n)p AmPS, BLAST Internal gas target Vector-polarised Deuterium JLab Hall C talk by Jochen Krimmer VOLUME 82, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 21 JUNE 1999 An atomic beam source (ABS) was used to inject a flux of deuterium atoms s (in two hyperfine states) into the feed tube of a cylindrical storage cell cooled to 75 K. The cell had a diameter of 15 mm and was 60 cm long. An electromagnet was used to provide a guide field of 40 mt over the storage cell which oriented the deuteron polarization axis perpendicular to q in the scattering plane. A doublet of steering magnets around the target region compensated for the deflection of the electron beam by the guide field. In addition, two sets of four beam scrapers preceding the internal target were used to reduce events that originated from By alternating two high-frequency transitions [12] in the ABS, the vector polarization of the target [P1 d p 3 2 n 1 2 n 2 ], with n 6 the fraction of deuterons with spin projection 61), was varied every 10 s. Compared to our previous experiments with tensor-polarized deuterium [13 15], this target setup resulted in an increase of the figure of merit by more than 1 order of magnitude, with a typical target thickness of deuterons cm 2. beam Fixed halo target: scattering from solidtheammonia cell. 15 ND 3 Dynamic nuclear polarisation Bigbite Magnet Cherenkov detector scintillator wire chambers d e e front wall p,n,d TOF detector δe EE back wall FIG. 1. Layout of the detector setup. The electron spectrometer consists of a 1 T m magnet, two drift chambers of four planes each, a scintillator, and a Čerenkov detector. The Ulrich Müller (Univ. Mainz) Nucleon Form Factors time-of-flight system consists of two identicalbosen walls 2010 of four 29 / 39

31 Neutron polarimetry A Analysing power (SAID) Elastic ( n, p) scattering: analysing power ɛ(θ n ) due to spin-orbit term V LS in NN interaction φ asymmetry for outgoing neutron: Θ n / deg I(θ n ) = I 0 [1 + ɛ(θ n )(P x cos φ n + P y sin φ n )] }{{} = 0 Analysing reaction in scintillator Detection of outgoing neutron (or proton!) in second scintillator n θ θ n Problem: Also reactions 12 C(n, n p) in scintillator with unknown analysing power Effective analysing power must be calibrated! Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

32 Spin precession method Calibration of the analysing power can be avoided by rotating the neutron spin direction in the xz plane: P z n P x B Precession angle (from BMT equation): χ = 2µ n hc β 1 n Bdl = Tm β 1 n Bdl Transverse polarisation (and therefore asymmetry) after the magnet: P = P x cos χ P z sin χ The zero crossing angle χ 0 with asymmetry A (χ 0 ) = 0 is determined by the ratio A x /A z : tan χ 0 = A x = P eɛ eff 1 A z P e ɛ eff τ + τ(1 + τ) tan 2 (θ/2) GEn G Mn independent of the effective analysing power ɛ eff. Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

33 Setup of the A1 neutron polarimeter e LD 2 n Side View Rear Wall Vetos Dipole Magnet Front Wall n Θ Pb n n Detectors Front wall: 30 scintillators cm 3 Rear wall: 24 scintillators cm 3 (with gap at height of beam) Veto scintillators in front of each wall 5 cm lead shielding in magnet gap Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

34 Setup in spectrometer hall Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

35 Kinematics and data taking Q 2 E E θ e T n θ n T meas (GeV/c) 2 MeV MeV MeV days Number of good events for each Q 2 setting Additional periods for setup, calibration, elastic p(e, e p) Theses: Derek Glazier, Michael Seimetz D. Glazier, M. Seimetz et al., EPJA 24 (2005) 101 Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

36 Data analysis Main steps: Identification and momentum reconstruction of electron in Spectrometer A Scintillator hit pattern and veto information in polarimeter Measurement of pulse height and time of flight Identification of quasielastic D( e, e n)p events Two statistically independent samples of events: nn events : neutron detected in read wall np events : proton detected in read wall Asymmetry/All/h 1: Φ 12 < 0 Asymmetry/All/h 1: Φ 12 > Counts 150 Counts Φ 12 [Deg] Φ 12 [Deg] Asymmetry/All/h+1: Φ 12 < 0 Asymmetry/All/h+1: Φ 12 > Counts 200 Counts Φ 12 [Deg] Φ 12 [Deg] Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

37 Measured asymmetries Q 2 = 0.3 (GeV/c) 2 Q 2 = 0.6 (GeV/c) 2 nn data np data nn data np data N+ (φ n ) N A = (φ n + π) N + (φ n + π) N (φ n ) N+ (φ n ) N (φ n + π) + N + (φ n + π) N (φ n ) Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

38 G En data at low Q 2 G E,n Polarisation Measurements G E,n D: A3, A1 D: Bates, Hall C D: AmPS, Hall C He: A3, A1 Galster 0.02 <r E,n > Q 2 / (GeV/c) 2 All data corrected for final state interaction Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

39 Future experiment: Neutron electric form factor Highly segmented neutron detector Talk by Magdalena Rohrbeck Smaller granularity (width: 5 cm 2 cm) Higher rate capability Beam current 8 µa 20 µa Better efficiency Optimize geometry also for np type events. Mount whole detector system on movable carriage. Shorter setup times when changing angle Factor of in statistics should be reachable. Future measurement from Q 2 = 0.2 (GeV/c) 2 to 1.5 (GeV/c) 2 with errors reduced by factor of 2 Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39

40 Summary High-accuracy measurement of proton form factors Charge radius r e in agreement with CODATA value Discrepancy with recent muonic hydrogen experiment Magnetic form factor G M larger than in previous experiments Smaller magnetic radius r m Structure in form factors at the scale of the pion cloud Electric form factor of the neutron Previous measurements at Q 2 = 0.3, 0.6, and 0.8 (GeV/c) 2 Planned experiment with highly segmented neutron detector Ulrich Müller (Univ. Mainz) Nucleon Form Factors Bosen / 39