Nucleon Electromagnetic Form Factors in Lattice QCD and Chiral Perturbation Theory

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1 Nucleon Electromagnetic Form Factors in Lattice QCD and Chiral Perturbation Theory Sergey N. Syritsyn (LHP Collaboration) Massachusetts Institute of Technology Laboratory for Nuclear Science November 6, 009 Theory Center Seminar Jefferson Lab, Newport News, VA

2 LHP Collaboration (this work) MIT/LNS: Jonathan Bratt Meifeng Lin Harvey Meyer John Negele Andrew Pochinsky Massimiliano Procura Sergey Syritsyn New Mexico State Univ.: Michael Engelhardt TU München: Phil Hägler Universität Regensburg Thomas Hemmert Institute of Physics, Taiwan: Wolfram Schroers arxiv: (submitted to Phys. Rev. D)

3 Introduction Nucleon form factors Lattice QCD Results 3 Comparison to Chiral Perturbation Theory Isovector Form Factors Isoscalar Form Factors 4 Summary & Outlook

4 Introduction Introduction Nucleon form factors Lattice QCD Results 3 Comparison to Chiral Perturbation Theory Isovector Form Factors Isoscalar Form Factors 4 Summary & Outlook

5 Introduction EM Current in a Nucleon Nucleon form factors Nucleon electromagnetic structure P qγ µ q P = Ū(P ) where q = P P and Q = q F (0) = Q em, F (0) = κ v isospin components: isovector isoscalar [ F q (Q )γ µ + F q (Q ) iσµν q ν M N F p, F n, = F u, F d, = F u d, F p, + F n, = 3 (F u, + F d,) = 3 F u+d, different dynamics through dispersion relations isovector π continuum, ρ,... isoscalar 3π continuum, ω,... ] U(P ),

6 Introduction Nucleon form factors Sachs Form Factors Widely used in experimental work: G E (Q ) = F (Q ) Q 4M F (Q ) G M (Q ) = F (Q ) + F (Q ) Charge and magnetization inside a proton (Breit frame): N(+ q/) J 0 N( q/) G E (Q ), N(+ q/) J N( q/) G M (Q ) σ q G p M (0) = µ p =.793, G n M(0) = µ n =.93

7 Introduction Nucleon form factors Proton Form Factors [M. Vanderhaeghen, Nucl.Phys.A805:0(008)] Different results from Rosenbluth separation Polarization transfer due to γ exchange [Blunden, Melnitchouk and Tjon, 003]

8 Introduction Proton Form Factors [M. Vanderhaeghen, Nucl.Phys.A805:0(008)] Nucleon form factors Different results from Rosenbluth separation Polarization transfer due to γ exchange [Blunden, Melnitchouk and Tjon, 003] Lattice (systematics unclear!):.5 LQCD, =93 MeV µ p G Ep /G Mp Q (GeV )

9 Introduction Nucleon form factors Neutron Electric Form Factor Schiavilla & Sick [0] Zhu et al. [0] Becker et al. [99] Herberg et al. [99] Ostrick et al. [99] Passchier et al. [99] Rohe et al. [99] Eden et al. [94] Meyerhoff et al. [94] Galster [7] r En < 0: n G E (+) core ( ) surface Q (GeV/c) [H. Gao, Int.J.Mod.Phys.E: (003)] deuterium targets thermal neutron scattering: re,n = 0.3(3)(4) fm [PRL 74, 47 (995)]

10 Introduction Nucleon form factors Neutron Electric Form Factor Schiavilla & Sick [0] Zhu et al. [0] Becker et al. [99] Herberg et al. [99] Ostrick et al. [99] Passchier et al. [99] Rohe et al. [99] Eden et al. [94] Meyerhoff et al. [94] Galster [7] r En < 0: n G E (+) core ( ) surface Q (GeV/c) [H. Gao, Int.J.Mod.Phys.E: (003)] deuterium targets thermal neutron scattering: re,n = 0.3(3)(4) fm [PRL 74, 47 (995)] G En 0. 0 LQCD, =97 MeV Q [GeV ]

11 Introduction Nucleon form factors Theory Approach to Nucleon Structure Models: constitutent quark model bag models Skyrme/soliton model vector meson dominance & dispersion relations chiral perturbation theory...

12 Introduction Nucleon form factors Theory Approach to Nucleon Structure Models: constitutent quark model bag models Skyrme/soliton model vector meson dominance & dispersion relations chiral perturbation theory... Fundamental theory: QCD has only Standard Model parameters successfully explains high momentum transfer phenomena perturbation theory breaks down at Λ QCD

13 Introduction Lattice QCD Lattice Quantum Chromodynamics 4D Euclidean lattice as a regulator gauge symmetric by construction automatically includes non-perturbative fields e.g. instantons X U µ Y hard for analytic calculations can be simulated by Monte-Carlo

14 Introduction Lattice QCD Lattice Quantum Chromodynamics 4D Euclidean lattice as a regulator gauge symmetric by construction automatically includes non-perturbative fields e.g. instantons X U µ Y hard for analytic calculations can be simulated by Monte-Carlo (Controllable) systematic effects finite cutoff Λ lat = a finite volume chiral symmetry breaking due to fermion regularization

15 Introduction Cost of Simulations Lattice QCD currently most simulations run at 50 MeV ( ) finite volume effects: must have L 4 ( VERY conservative estimate: cost ) 4+? Have to use effective theories to extrapolate m phys π Stochastic noise grows with decreasing : 3 signal noise e (M N )t [Lepage 89]

16 Can Lattice QCD be a precise theoretical tool to study nucleon structure?

17 Introduction Lattice QCD To answer this question pick framework with the best controlled systematics Large-scale simulations by RBC/UKQCD collaboration: 006 now [C. Allton et al, Phys.Rev.D78:4509,008] Z = DU D ψ Dψ e Sg[U] S F[U,ψ, ψ] lattice spacing a = fm pion mass MeV finite volume effects suppressed by L

18 Introduction Lattice QCD Lattice QCD Action S g [U] = β 3 [( 8c ) P [U] + c R[U] ] R on an RG trajectory parameterized by c P Ψ L Ψ R x,y,z,t s=0 s=l s - m res Ψ R ΨL (Almost) chirally symmetric quarks: spatially separate ψ L and ψ R in 4 + space additional dimension L 5 = 6 (L space = 3!) resigual breaking m res ψl ψ R, m res 5% m q

19 Introduction Lattice QCD Nucleon Matrix Elements P Momentum projected correlators: T τ P q x, y e i P x+i q y N(T, x) [ qγ µ q] τ, y Spacelike q = P P with Q = q, limited to 0 Q.05 GeV (a = 5.48 GeV ) N(0) Excited states contribute as tails ground O excited Z exc e Eexcτ + sink term const excited O excited Z exc e EexcT separation T fm proves sufficient

20 Introduction Lattice QCD Improvements & Tricks Nucleon correlators are noisy require many contractions

21 Introduction Lattice QCD Improvements & Tricks Nucleon correlators are noisy require many contractions Improvements 8 (independent!) measurements on each gauge configuration.5 less computation due to combined backward propagators 3 noise suppression by choice of nucleon sources 3 noise suppression with short separation and careful study of excited states contributions Total 9 noise suppression

22 Results Introduction Nucleon form factors Lattice QCD Results 3 Comparison to Chiral Perturbation Theory Isovector Form Factors Isoscalar Form Factors 4 Summary & Outlook

23 Results Isovector G E and G M vs. Q G E u-d (Q ) G E u-d (Q ) / dipole Q < 0.5 GeV Q < 0.7 GeV Q <. GeV Dipole fits G i (Q ) = G i (0) (+Q /M D,i ) Cut-off Q 0.5 GeV Results are consistent for higher cutoffs MD = GeV ( M D = 0.7 GeV )exp

24 Results Isovector Dirac Radius [fm ] (r v ) DWF a=0.084 fm DWF a=0.4 fm Mixed Action PDG 008 Belushkin et al [GeV] Consistent with Coarse domain wall ensemble no cutoff errors Mixed action (staggered sea + domain wall valence quarks) no action discretization errors

25 Comparison to ChPT Introduction Nucleon form factors Lattice QCD Results 3 Comparison to Chiral Perturbation Theory Isovector Form Factors Isoscalar Form Factors 4 Summary & Outlook

26 Comparison to ChPT Two types of Baryon SU() Chiral PT Heavy baryon SU() Chiral PT : expansion in ɛ = {,, p} [V. Bernard, H. Fearing, T. Hemmert, U.-G. Meissner, 998] (3) Rarita-Schwinger field heavy-baryon approximation: expansion in ( mπ M N ) Covariant Baryon SU() Chiral PT [T. Gail and T. Hemmert (PhD Thesis, in prep.)] no -baryon relativistic calculation Both work for small momentum transfer Q 0. GeV Lattice momentum transfer for = 97 MeV: # P P Q [GeV ] 0, 0, 0 0, 0, , 0, 0, 0, , 0, 0,, , 0, 0,,

27 Comparison to ChPT Form Factors at Small Q Dirac mean square radus (r ) anomalous magnetic moment κ = F (0) Pauli form factor slope κ(r ) Form factors at small Q : F (Q ) 6 (r ) Q + O(Q 4 ) F (Q ) κ 6 κ (r ) Q + O(Q 4 ) Extract (r v ), κ v, κ v (r v ) from dipole fits in 0 Q 0.5 GeV : (r i ) = M D,i Functional form is irrelevant

28 Comparison to ChPT Isovector Form Factors ChPT in Isovector Channel One-loop ChPT contributions to nucleon structure: (r) v = # log +..., κ v (r) v = # +..., κ v = κ 0 v +... General -loop behavior: both heavy-baryon and covariant ChPT Form factor slopes (r.m.s. radii) diverge as 0: significant change expected towards m phys π = 35 MeV

29 Comparison to ChPT Heavy Baryon + (3) Isovector Form Factors Can we determine all LECs from our data? pion mass range 300 MeV 400 MeV lattice data exibits monotonic behavior: difficult to resolve structure with many LECs must invest into lighter to increase fit lever Test ChPT by fixing all parameters How many parameters can be fixed? F π = 86.(0.5) MeV: NNLO fit to lattice data = M M N 93 MeV: resonance measurement g A =.7: known experimentally, little variation with g πn : fit from data

30 Comparison to ChPT Isovector Form Factors Heavy Baryon + (3): F v and F v Radii [fm ] (r v ) Belushkin et al 007 PDG 008 Lattice a=0.084 fm with "core" term [GeV] Simultaneous fit to (r v ), κ v (r v ) : (r v ) : g πn B r 0 κ v (r v ) : g πn (C) [fm ] κ v (r v ) Belushkin et al 007 Lattice a=0.084 fm with "core" term [GeV] with core term χ /dof.4/3 g πn =.97(7)

31 Comparison to ChPT Isovector Form Factors Heavy Baryon + (3): F v and F v Radii [fm ] (r v ) Belushkin et al 007 PDG 008 Lattice a=0.084 fm with "core" term no "core" term [GeV] Simultaneous fit to (r v ), κ v (r v ) : (r v ) : g πn B r 0 κ v (r v ) : g πn (C) [fm ] κ v (r v ) Belushkin et al 007 Lattice a=0.084 fm with "core" term no "core" term [GeV] with core term χ /dof.4/3 g πn =.97(7) no core term χ /dof 7 g πn =.54(6)

32 Comparison to ChPT Isovector Form Factors Heavy Baryon + (3): F v and F v Radii [fm ] (r v ) DWF a=0.084 fm DWF a=0.4 fm Mixed Action PDG 008 Belushkin et al [GeV] consistent with DWF a = 0.4 fm mixed action (a = 0.4 fm) [LHPC, in prep.] [fm ] κ v (r v ) DWF a=0.084 fm DWF a=0.4 fm Mixed Action Belushkin et al [GeV] curves through physical point, fix phenomenological value g πn =.5

33 Comparison to ChPT Isovector Form Factors Heavy Baryon + (3): κ v Experiment DWF a=0.084 fm with "core" term no "core" term Experiment Mixed Action DWF a=0.4 fm DWF a=0.084 fm κ v.5 κ v [GeV] fix c V, κ 0 v, E r from three data points [GeV] consistent with mixed action fix phenomenological values g πn and c V

34 Comparison to ChPT Isovector Form Factors Covariant Baryon chiral PT [T. Gail, T. Hemmert, in preparation] no -baryon relativistic (Lorentz-covariant) calculation effective πn Lagrangian up to O(p 4 ) L πn = L () () (3) (4) πn + LπN + LπN + LπN +O(p 5 ) }{{}}{{}}{{}}{{} M 0, g A e i Parameters: c i d i fix g A, F π from other lattice data fix L () πn parameters c, c 3, c 4 fron, NN scattering [hep-ph/9653, hep-ph/980366, nucl-th/00039] extract M 0 from our data

35 Comparison to ChPT Isovector Form Factors Covariant Baryon Chiral PT: Form Factors [fm ] (r v ) [fm ] κ v (r v ) Belushkin et al 007 PDG 008 DWF a=0.084 fm CBChPT fit HBChPT fit [GeV] Belushkin et al DWF a=0.084 fm HBChPT Fit CBChPT Fit [GeV] κ v Experiment DWF a=0.084 fm CBChPT Fit HBChPT Fit [GeV] Simultaneous fit: (r v ) : c 6 d r 6 κ v (r v ) : c 6 e r 74 κ v : c 6 e r 06 χ /dof 7.4 Summary

36 Comparison to ChPT Isoscalar Form Factors Isoscalar Form Factors F u+d (Q ) = 97 MeV = 355 MeV = 403 MeV Phenomenology Q [GeV ] F u+d (Q ) precise data for F u+d (Q ), dipole fit M D = 97 MeV = 355 MeV = 403 MeV Phenomenology Q [GeV ] =.. GeV F u+d (Q ) almost consistent with zero, linear fit in Q fits to experimental data by [J.J.Kelly 04] (no errors are shown)

37 Comparison to ChPT Isoscalar Form Factors SSE and CBChPT in Isoscalar Channel strong dynamics starts at two loops (due to Q = (3 ) cut with 3π) require O(ɛ 5 ) SSE or O(p 5 ) CBChPT Currently available: O(ɛ 3 ) SSE: F s (Q ) = + B Q (4πF π) F s (Q ) = κ s linear fits in m π for (r s ), κ s (r s ), κ s O(p 4 ) CBChPT: one-loop m π log (r s ) κ 0 s B s c κ s (r s ) κ 0 s B s c κ s κ 0 s e r 05 simultaneous OR independent fits in κ 0 s.

38 Comparison to ChPT Isoscalar Form Factors Covariant Baryon Chiral PT: Dirac radius [fm ] Phenomenology DWF Results Independent Fit Simultaneous Fit (r s ) [GeV ] Fit: χ /dof simultaneous 8.5 independent 0.

39 Comparison to ChPT Isoscalar Form Factors CBChPT: Pauli Radius & κ s 0. [fm ] κ s (r s ) 0-0. DWF Results Phenomenology Simultaneous Fit Independent Fit [GeV ] κ s Experiment Simultanesous Fit -0.3 Independent Fit DWF Results [GeV ] Fit: χ /dof simultaneous 8.5 independent 0.08, 0.4

40 Summary & Outlook Introduction Nucleon form factors Lattice QCD Results 3 Comparison to Chiral Perturbation Theory Isovector Form Factors Isoscalar Form Factors 4 Summary & Outlook

41 Summary & Outlook Summary High precision Lattice QCD calculation of nucleon form factors lattice QCD action with controlled systematic effects combination of method to reduce noise Two types of ChPT in 300 MeV range heavy-baryon πn to O(ɛ 3 ) and relativistic πn to O(p 4 ) are not compatible with our data Expect strong dependence of nucleon structure closer to the physical point Simulations at smaller are required to clarify this problem

42 Summary & Outlook Outlook Form factors at small momentum transfer (flavor-twisted BC) to enhance ChPT fits Relativistic ChPT calculation including (3) [L. S. Geng, J. M. Camalich, M. J. Vacas, Phys.Lett.B676:63-68(009)] Generalized form factors: work in progress momentum fraction quark angular momentum, non-perturbative renormalization Cheaper action?

43 Appendix Introduction Nucleon form factors Lattice QCD Results 3 Comparison to Chiral Perturbation Theory Isovector Form Factors Isoscalar Form Factors 4 Summary & Outlook

44 Ensembles Appendix Detailed results Coarse & Fine lattice gauge ensembles: L 3 s L t a [fm] meas. am l /am h m res m l [MeV] L M N [MeV] / (5) (0) / (5) () / (6) () / (7) 5.5 () Lattice spacing: a coarse = 0.4 fm a fine = 0.084fm [C. Allton et al, Phys.Rev.D78:4509,008] comparison of F π results to coarse ensemble

45 Appendix Form factors Isovector F and F : Dependence on Q u-d F u-d (Q )/dipole F (Q ) =97 MeV Q cut = 0.5 GeV Q cut = 0.7 GeV Q cut =. GeV =97 MeV F u-d (Q )/dipole F u-d (Q ) =97 MeV Q cut = 0.5 GeV Q cut = 0.7 GeV Q cut =. GeV =97 MeV Dipole fits F i (Q ) = F i (0) (+Q /M D,i ) Cut-off Q 0.5 GeV Results are consistent for different cutoffs F u-d (Q )/dipole =355 MeV F u-d (Q )/dipole =355 MeV M D, = GeV M D, = GeV F u-d (Q )/dipole =403 MeV F u-d (Q )/dipole =403 MeV Q [GeV ] Q [GeV ]

46 Appendix Form factors Isovector G E and G M : Dependence on Q 4 u-d G u-d E (Q )/dipole G E (Q ) =97 MeV Q cut = 0.5 GeV Q cut = 0.7 GeV Q cut =. GeV =97 MeV u-d G u-d M (Q )/dipole G M (Q ) =97 MeV Q cut = 0.5 GeV Q cut = 0.7 GeV Q cut =. GeV =97 MeV Dipole fits F i (Q ) = F i (0) (+Q /M D,i ) Cut-off Q 0.5 GeV Results are consistent for different cutoffs u-d u-d G E (Q )/dipole G E (Q )/dipole.08 =355 MeV m.05 π =403 MeV Q [GeV ] u-d u-d G M (Q )/dipole G M (Q )/dipole.08 =355 MeV =403 MeV Q [GeV ] MD,E = GeV MD,M = GeV

47 Appendix ChPT fits Pion Decay Constant & LECs af π fine coarse, interp ( /F π ) -loop SU() chiral perturbation theory: a m π = a χ af π = af ( + a χ a χ (af ) lr 3 (a ) + 3π (af ) log a χ ), ( + a χ a χ (af ) lr 4 (a ) 6π (af ) log a χ ), χ 7 for two degrees of freedom, probability 3% l3 l4 this work 3.08() 4.4(4) Gasser & Leutwyler 84.9 ± ± 0.9 N f = dyn. Wilson [JHEP 0,056(007)] 3.0(0.5)(0.) 4.3(0) Twisted Mass [Dimopoulos et al 08] 3.4(8)(0) 4.59(4)()

48 Appendix ChPT fits Covariant Baryon Chiral PT: Nucleon Mass M N [GeV].6.4. DWF a=0.084 fm DWF a=0.4 fm Mixed Action Experiment relativistic calculations: explicit dependence on M N ( ) OR M 0 M N ( ) from O(p 4 ) πn Lagrangian [GeV] M 0 Lattice 0.883(79) Lattice + Phys (9) better constrained

49 Appendix Methodology Connected and Disconnected Contractions N N Connected: ( N δ δq α ) ( δ {[ }}{ DΓ / /D ] αβ δ q β ) N N N Disconnected: N N {}}{ Tr [ /DΓ ] v.e.v. Do only connected contractions in this work Disconnected correlators cancel for isovector J v µ = J u µ J d µ

50 Appendix Methodology Nucleon Matrix Elements Combine C pt and C 3pt into ratio R = C 3pt (T, τ; P, P ) Cpt (T, P )C pt (T, P ) }{{} cancel Z(P ), Z(P ) C pt (T τ, P )C pt (τ, P ) C pt (T τ, P )C pt (τ, P ) }{{} e (E E)(τ T/) Cancel correlated stochastic noise in C pt, C 3pt Not relying on fits

51 Appendix Methodology Nucleon Matrix Elements..5 F u-d (Q ) F u-d (Q ) τ τ Average three central points (t = 5, 6, 7 for T = ) Fit tails using a M = limited by C pt (t)

52 Appendix Methodology Extraction of Form Factors Use transfer matrix expression [ ] (i /P + M N ) F γ µ iσ + F µν q ν M N (i /P + M N ) C 3pt = E E to get a set of overcomplete equations for fixed Q R α (Q ) = A αi F i (Q ) e Eτ E (T τ) maximally utilize rotation symmetry: has 6 equivalents reduce system: average equivalent correlators control quality of fits Q(χ, dof)

53 Appendix Systematics discussion Two-point Functions: Mass & Energy E p eff (t) p=(0,0,0) p=(0,0,) p=(0,,) p=(,,) p=(0,0,) t C pt (t, P ) = x E eff (t) = log e i P x N(t, x) N(0) = Z(P ) ( i /P + M N ) e E P t + excited states, C pt (t, P ) C pt (t +, P ) E P Fits to excited & oscillating states C pt = Z 0 e E 0t + Z e E t + ( ) t Z osc e Eosct give contamination Z /Z 0 and energy gap E 0 = E E

54 Appendix Systematics discussion Two-point Functions: Mass & Energy [ E(p) - E(0) ] eff E 00 -E 000 E 0 -E 000 E -E 000 E 00 -E t a E p = 97 MeV = 355 MeV m 0. π = 403 MeV a p Test relation E = p + m with precise measurement [ (E E) eff Cpt (t, P ) t = log C pt (t +, P ) / C ] pt(t, P ) C pt (t +, P ) Cancellation between contaminations to E and E helps a lot.

55 Excited States Appendix Systematics discussion δr 0 (τ) t (0,0,0) (0,0,) (0,,) (,,) (0,0,) δr 0 (τ, P ) = R O 0 0 Z (P ) Z 0 (P ) e E 0τ +O 0 δr 0 (τ) +O 0 δr 0(T τ) } tails + O δr 0 (τ) δr 0(T τ) (back) jackknife estimate of δr 0 (τ): Z and E 0 are correlated tails: decay rate limited from below const: if O O 0 0, its contribution is limited by [δr 0 (τ)δr 0(T τ)] %

56 Appendix Fits to Excited States. Systematics discussion. F u-d (Q ) / dipole. 0.9 plateau avg [3:9], m= [3:9], m=0.6 [3:9], m= plateau avg, T= mom# (back) F U-D (Q ) / dipole. 0.9 plateau avg [3:9], m= [3:9], m=0.6 [3:9], m= plateau avg, T= mom# compare plateau average (τ = 5, 6, 7) with fits using M 0 = 0.4, 0.6, 0.8 (limited by fits of C pt (t)) compare T = a and T = 4a (less statistics)