Dirac and Pauli form factors from N f = 2 Clover-fermion simulations

Transcription

1 Mitglied der Helmholtz-Gemeinschaft Dirac and Pauli form factors from N f = 2 Clover-fermion simulations Dirk Pleiter JSC & University of Regensburg

2 Outline Dirk Pleiter JSC & University of Regensburg Slide 2 1

3 Credits Members of the QCDSF collaboration: S. Collins, M. Göckeler, Ph. Hägler, R. Horsley, Y. Nakamura, A. Nobile, P.E.L. Rakow, A. Schäfer, G. Schierholz, W. Schroers, H. Stüben, F. Winter, J.M. Zanotti Reference: Dirac and Pauli form factors from lattice QCD, accepted for publication in Physical Review D Dirk Pleiter JSC & University of Regensburg Slide 3 1

4 Mitglied der Helmholtz-Gemeinschaft Dirac and Pauli form factors Part I: Introduction Dirk Pleiter JSC & University of Regensburg

5 Electromagnetic form factors [ p, s j µ p, s = ψ(p, s ) γ µ F 1 (q 2 ) + iσ µν q = p p... momentum transfer We will consider, e.g. q ] ν F 2 (q 2 ) ψ(p, s) 2m N Proton form factors: Isovector form factors: j (p) = 2 3 uγµ u 1 3 dγµ d j (v) = uγ µ u dγ µ d Disconnected terms cancel Dirk Pleiter JSC & University of Regensburg Slide 5 1

6 Matrix elements on the lattice R(t, τ, p, p ) = C 3(t, τ, p [, p ) C2 (τ, p )C 2 (t, p )C 2 (t τ, p ) C 2 (t, p ) C 2 (τ, p )C 2 (t, p )C 2 (t τ, p ) ] 1/2 where C 2 (t, p ) = αβ Γ βα B α (t, p ) B β (0, p ) and C 3 (t, τ, p, p ) = αβ Γ βα B α (t, p )O(τ) B β (0, p ) We use the local vector current: V µ = ψ(x) γ µ ψ(x) Dirk Pleiter JSC & University of Regensburg Slide 6 1

7 Renormalisation and improvement Renormalized local vector current: V (R) µ = Z V (1 + b V am q ) [ ψγ µ ψ + ic V a λ ( ψσ µλ ψ) ] Z V and b V have been determined non-perturbatively c V known only perturbatively neglected here [QCDSF 2002] Dirk Pleiter JSC & University of Regensburg Slide 7 1

8 Momenta and polarisations Up to 3 initial state momenta: L 2π p =, Up to 3 choices for polarisations: Γ unpol = 1 2 (1 + γ 4), Γ 1 = 1 2 (1 + γ 4)iγ 5 γ 1 Γ 2 = 1 2 (1 + γ 4)iγ 5 γ 2 17 different choices of q = p p Dirk Pleiter JSC & University of Regensburg Slide 8 1

9 Simulation Details N f = 2 mass-degenerate, non-perturbativley O(a)-improved Wilson-fermions Standard Wilson glue Parameter space: m PS = GeV a = fm V = fm (smaller lattices for FSE studies) Scale definition: Use Sommer parameter r 0 to scale lattice results We use r 0 (β, m (S) q ) extrapolated to the chiral limit Ambiguities when converting to physical units No precise experimental value available Extrapolation of QCDSF nucleon mass results supports r 0 = fm Here we use: r 0 = 0.5 fm Dirk Pleiter JSC & University of Regensburg Slide 9 1

10 Form factor radii and magnetic moment Definitions: Mean square radii determined from the slopes of the form factors at Q 2 = 0: r 2 i = 6 df i (Q 2 ) F i (0) dq 2 Q 2 =0 Magnetic moment µ / anomalous magnetic moment κ: µ = 1 + κ = F 1 (0) + F 2 (0) For comparison with effective theories (and experimental numbers) and we will use κ norm = κ m N (m π )/m N (m PS ) Dirk Pleiter JSC & University of Regensburg Slide 10 1

11 Mitglied der Helmholtz-Gemeinschaft Dirac and Pauli form factors Part II: Q 2 -dependence Parametrisation Dirk Pleiter JSC & University of Regensburg

12 Iso-vector Dirac Form Factor F 1 0.8GeV m Π GeV 0.8 Comparison of lattice results with parametrization of experimental numbers Parametrization by Alberico et al. (2008) F 1 ud F 1 ud Q 2 GeV Dirk Pleiter JSC & University of Regensburg Slide 12 1

13 Iso-vector Pauli Form Factor F 2 0.8GeV m Π GeV 4 m Π GeV 3 3 F 2 ud 2 F 2 ud Q 2 GeV Dirk Pleiter JSC & University of Regensburg Slide 13 1

14 Iso-scalar Pauli Form Factor F GeV m Π GeV m Π GeV F 1 ud 1.5 F 1 ud Q 2 GeV Dirk Pleiter JSC & University of Regensburg Slide 14 1

15 Iso-scalar Pauli Form Factor F GeV m Π GeV 0.5 F 2 ud F 2 ud Q 2 GeV Dirk Pleiter JSC & University of Regensburg Slide 15 1

16 Up vs. down quarks: F (d) 1 /F (u) m Π 0.8GeV F 1 d F1 u 0.3 For heavy quark masses F (d) 1 and F (u) 1 scale in a similar way For Q GeV 2 observe systematic downward trend F 1 d F1 u m Π GeV Q 2 GeV Dirk Pleiter JSC & University of Regensburg Slide 16 1

17 F 1 : Q 2 -dependence Parametrization Dipole ansatz: F 1 (Q 2 ) = F 1 (0) (1 + Q 2 /m 2 D )2 Alternative: More general polynomial in the denominator F 1 (Q 2 ) = F 1 (0) 1 + c 12 Q 2 + c 14 Q 4 Use 2-parameter ansatz for further analysis: It describes data better than dipole ansatz Matching to simple vector meson exchange ansatz works remarkebly well Dirk Pleiter JSC & University of Regensburg Slide 17 1

18 F 1 : Q 2 -dependence Parametrization (2) Β,Κ,L 5.25, 0.136, 24 Β,Κ,L 5.29, , Χ 2 DOF Χ 2 DOF Β,Κ,L 5.29, , 40 Β,Κ,L 5.4, , F 1 ud F 1 ud Χ 2 DOF Χ 2 DOF Dirk Pleiter JSC & University of Regensburg Slide 18 1

19 F 1 : Q 2 -dependence Parametrization (3) Β 5.20 Β 5.25 Β 5.29 Β 5.40 Π M 1 ud mv Ratio of the lowest pole mass and m V in iso-vector and iso-scalar channel Assumption: m V = mρ lat mω lat M 1 ud mv Π Β 5.20 Β 5.25 Β 5.29 Β 5.40 Π Π Dirk Pleiter JSC & University of Regensburg Slide 19 1

20 F 2 : Q 2 -dependence Parametrization p-pole ansatz: F 2 (Q 2 ) = F 2 (0) (1 + Q 2 /m 2 p) p To match large-q 2 -behaviour obtained from peturbative QCD expect p = 3 More general 3-parameter ansatz: F 2 (Q 2 ) = F 2 (0) 1 + c 22 Q 2 + c 26 Q 6 Tri-pole and 3-parameter ansatz perform equally well 3-parameter ansatz seems less biased Used in further analysis Dirk Pleiter JSC & University of Regensburg Slide 20 1

21 F 2 : Q 2 -dependence Parametrization (2) 2.5 Β,Κ,L 5.25, 0.136, Β,Κ,L 5.29, , Χ 2 DOF Χ 2 DOF Β,Κ,L 5.29, , Β,Κ,L 5.4, , F 2 ud 1.5 F 2 ud Χ 2 DOF Χ 2 DOF Q 2 GeV Dirk Pleiter JSC & University of Regensburg Slide 21 1

22 Mitglied der Helmholtz-Gemeinschaft Dirac and Pauli form factors Part III: Chiral Extrapolations Dirk Pleiter JSC & University of Regensburg

23 Iso-vector Dirac Radius Iso-vector radius from small scale expansion (SSE): r 2 (v),sse 1 = 1 { ( ) ( 1 + 7g 2 (4πf π) 2 A + 10gA 2 mπ )} + 2 ln λ { ca 2 ( mπ ) ln 54π 2 fπ 2 λ ( ) } δm δm m π mπ 2 1-loop HBChPT: δm + 30 ln δm2 mπ 2 r 2 (v),hbchpt 1 = 1 (4πf π) 2 { 1 + 7g 2 A B(r) 10 (λ) (4πf π) 2 ( ) 10gA ln + 12B(r) 10 (λ) (4πf π) 2 ( mπ )} λ Covariant BChPT scheme (involved expressions omitted here) Dirk Pleiter JSC & University of Regensburg Slide 23 1

24 Iso-vector Dirac Radius (2) Use experimental/phenomenological results to fix LECs and counter-term No fit to lattice data Comparison of lattice data to SSE result (upper plot) HBChPT result (upper plot, dashed lines) BChPT result (lower plot) Heavy baryon limit of BChPT result (lower plot, dashed line) r 2 1 ud fm 2 r 2 1 ud fm 2 Β 5.20 Β 5.25 Β 5.29 Β 5.40 m Π L3.4 PDG 2010 Belushkin et al. ' Π Β 5.20 Β 5.25 Β 5.29 Β 5.40 m Π L3.4 PDG 2010 Belushkin et al. ' Π Dirk Pleiter JSC & University of Regensburg Slide 24 1

25 Iso-vector Anomalous Magnetic Moment Small scale expansion (SSE): κ (v),sse = κ (v,0) + K (v) (m π ) 8E (r) 1 (λ)m Nm 2 π Result from the covariant BChPT-approach: } κ (v),bchpt = m(n) N {c mn m 0N m2πe r106 (λ) + κ(v,3) + κ (v,4) Dirk Pleiter JSC & University of Regensburg Slide 25 1

26 Iso-vector Anomalous Magnetic Moment (2) Use phenomenological results to fix LECs To determine unknown κ (v,0) and counter-term fit to Experimental data Lattice data point at smallest quark mass Comparison of lattice data to SSE and (heavy-baryon limit) BChPT results: Κ ud Κ ud Β 5.20 Β 5.25 Β 5.29 Β 5.40 m Π L3.4 PDG Π Β 5.20 Β 5.25 Β 5.29 Β 5.40 m Π L3.4 PDG Π Dirk Pleiter JSC & University of Regensburg Slide 26 1

27 Iso-scalar Anomalous Magnetic Moment Small scale expansion (SSE): κ (s),sse = κ (s,0) 24E 2 m N m 2 π Result from the covariant BChPT-approach: κ (s),bchpt = mphys N {κ (s,0) mn 0 48m 0N m2πe r105(λ) } + κ (s,3) + κ (s,4) Dirk Pleiter JSC & University of Regensburg Slide 27 1

28 Iso-scalar Anomalous Magnetic Moment (2) Use phenomenological results to fix LECs To determine unknown κ (s,0) and counter-term fit to Experimental data Lattice data point at smallest quark mass Comparison of lattice data to SSE and (heavy-baryon limit) BChPT results: Κ ud Κ ud Β 5.20 Β 5.25 Β 5.29 Β 5.40 PDG Π Β 5.20 Β 5.25 Β 5.29 Β 5.40 PDG Π Dirk Pleiter JSC & University of Regensburg Slide 28 1

29 Iso-vector Pauli Radius Result from small scale expansion (SSE): ( κ r 2 2 ) (v),sse = g 2 Am N 8πf 2 πm π m N B c2 cam 2 N ln 9π 2 fπ 2 δm2 mπ 2 ( ) δm δm m π mπ 2 Covariant HBChPT scheme (κ r 2 2 ) (v),bchpt = mphys N m 0 N (24m 0 Ne r 74(λ) + (κr 2 2 ) (v,3) + (κr 2 2 ) (v,4)) Dirk Pleiter JSC & University of Regensburg Slide 29 1

30 Iso-vector Pauli Radius (2) Comparison to chiral effective field theory results: For almost all parameters phenomenological estimates available Use experimental results to fix a counter-term Bending-up in lattice data? Κ r 2 2 ud fm 2 Κ r 2 2 ud fm Β 5.20 Β 5.25 Β 5.29 Β 5.40 m Π L3.4 Belushkin et al. ' Π Β 5.20 Β 5.25 Β 5.29 Β 5.40 m Π L3.4 Belushkin et al. ' Π Dirk Pleiter JSC & University of Regensburg Slide 30 1

31 Mitglied der Helmholtz-Gemeinschaft Dirac and Pauli form factors Part IV: Systematic Errors Dirk Pleiter JSC & University of Regensburg

32 Excited State Contaminations Dilema of computing nucleon 3-point functions: Need t src τ t sink to avoid excited state contamination For large τ the signal-to-noise ratio becomes large Practical solution here: Check dependence of results on t sink F 1 ud Q 2 9GeV 2 F 1 ud Q GeV snk snk Dirk Pleiter JSC & University of Regensburg Slide 32 1

33 Discretization Effects Difficult to distinguish a, m PS and Q 2 dependence Our approach: Collect results for same a and similar m PS, Q 2 8% maximum relative difference in m PS and Q 2 Perform interpolations to account for m PS and Q 2 dependence No systematic discretization effects found Dirk Pleiter JSC & University of Regensburg Slide 33 1

34 Discretization Effects (2) F 1 ud Q GeV m lat Π 6GeV F 1 ud Q 2 0GeV m lat Π 6GeV F 1 ud Q GeV 2 a m lat Π 6GeV F 1 ud Q GeV 2 a m lat Π 4GeV a a F 2 ud Q GeV Π F 2 ud Q 2 0GeV Π F 2 ud Q GeV 2 a 0.8 m lat 6GeV Π F 2 ud Q GeV 2 a 2 fm m lat 4GeV Π a 2 fm 2 a 2 fm Dirk Pleiter JSC & University of Regensburg Slide 34 1

35 Mitglied der Helmholtz-Gemeinschaft Dirac and Pauli form factors Part V: Summary and Conclusions Dirk Pleiter JSC & University of Regensburg

36 Summary and Conclusions Based on a large data set the Dirac and Pauli form factors have been studied Detailed investigation of sources of systematic errors, e.g. discritization effects and Excited state contaminations Current data provides significant better constraints to the Q 2 parametrization More general parametrizations perform better compared to di- and tripole fits Results for radii and anomalous magnetic moments have been compared to results from chiral effective theories Initial indications for non-analytical chiral behavior at the lowest pion masses No consistent quantitative understanding using different approaches (SSE, BChPT, HBChPT) It is important to further explore the region m PS 200 MeV Dirk Pleiter JSC & University of Regensburg Slide 36 1