to N transition and form factors in Lattice QCD

Transcription

1 to N transition and form factors in Lattice QCD C. Alexandrou University of Cyprus and Cyprus Institute NSTAR 2011, Jefferson Lab, 18 May 2011 C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

2 Outline 1 Motivation 2 Introduction QCD on the lattice Recent results Hadron masses Nucleon form factors 3 Nγ transition form factors Experimental information Lattice results 4 N - axial-vector and pseudoscalar form factors 5 form factors electromagnetic form factors axial-vector form factors Pseudoscalar couplings 6 Width of resonances 7 Conclusions C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

3 Motivation to Nucleon transition form factors: Electromagnetic form factors are known experimentally; In particular G M1 is precisely measured can we reproduce it from lattice QCD? Quadrupole γ N form factors G E2 and G C2 may indicate deformation in the nucleon/ can we calculate them from lattice QCD? Provide input on axial-vector FFs Understand the q 2 -dependence of axial form factors C A 5 and CA 6 that correspond to the nucleon axial form factor G A and nucleon induced pseudoscalar form factor G p Provide important input for phenomenological models builders and for chiral effective theories Evaluate πn coupling and examine the non-diagonal Goldberger-Treiman relation Form factors of the : Difficult to measure experimentally Electromagnetic form factors magnetic moment of, quadrupole moment = obtain information on its charge distribution in the infinite momentum frame (talk by M. Vanderhaeghen) Study the axial-vector and pseudoscalar form factors = New features arise e.g. two Goldberger-Treiman relations = Calculate within lattice QCD the form factors of the nucleon/ system global fit C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

4 Motivation to Nucleon transition form factors: Electromagnetic form factors are known experimentally; In particular G M1 is precisely measured can we reproduce it from lattice QCD? Quadrupole γ N form factors G E2 and G C2 may indicate deformation in the nucleon/ can we calculate them from lattice QCD? Provide input on axial-vector FFs Understand the q 2 -dependence of axial form factors C A 5 and CA 6 that correspond to the nucleon axial form factor G A and nucleon induced pseudoscalar form factor G p Provide important input for phenomenological models builders and for chiral effective theories Evaluate πn coupling and examine the non-diagonal Goldberger-Treiman relation Form factors of the : Difficult to measure experimentally Electromagnetic form factors magnetic moment of, quadrupole moment = obtain information on its charge distribution in the infinite momentum frame (talk by M. Vanderhaeghen) Study the axial-vector and pseudoscalar form factors = New features arise e.g. two Goldberger-Treiman relations = Calculate within lattice QCD the form factors of the nucleon/ system global fit C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

5 Motivation to Nucleon transition form factors: Electromagnetic form factors are known experimentally; In particular G M1 is precisely measured can we reproduce it from lattice QCD? Quadrupole γ N form factors G E2 and G C2 may indicate deformation in the nucleon/ can we calculate them from lattice QCD? Provide input on axial-vector FFs Understand the q 2 -dependence of axial form factors C A 5 and CA 6 that correspond to the nucleon axial form factor G A and nucleon induced pseudoscalar form factor G p Provide important input for phenomenological models builders and for chiral effective theories Evaluate πn coupling and examine the non-diagonal Goldberger-Treiman relation Form factors of the : Difficult to measure experimentally Electromagnetic form factors magnetic moment of, quadrupole moment = obtain information on its charge distribution in the infinite momentum frame (talk by M. Vanderhaeghen) Study the axial-vector and pseudoscalar form factors = New features arise e.g. two Goldberger-Treiman relations = Calculate within lattice QCD the form factors of the nucleon/ system global fit C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

6 QCD on the lattice a µ ψ(n) U µ (n)=e igaaµ(n) Discretization of space-time in 4 Euclidean dimensions simplest isotropic hypercubic grid: = Rotation into imaginary time is the most drastic modification Lattice acts as a non-perturbative regularization scheme with the lattice spacing a providing an ultraviolet cutoff at π/a no infinities Gauge fields are links and fermions are anticommuting Grassmann variables defined at each site of the lattice. They belong to the fundamental representation of SU(3) Construction of an appropriate action such that when a 0 (and Volume ) it gives the continuum theory Construction of the appropriate operators with their renormalization to extract physical quantities We take spacing a = a S = a T and size N S N S N S N T, N T > N S (Hadron Spectrum Collaboration considers anisotropic lattices with a T a S /3 better suited for study of excited states) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

7 QCD on the lattice a µ ψ(n) U µ (n)=e igaaµ(n) Lattice artifacts Discretization of space-time in 4 Euclidean dimensions simplest isotropic hypercubic grid: = Rotation into imaginary time is the most drastic modification Lattice acts as a non-perturbative regularization scheme with the lattice spacing a providing an ultraviolet cutoff at π/a no infinities Gauge fields are links and fermions are anticommuting Grassmann variables defined at each site of the lattice. They belong to the fundamental representation of SU(3) Construction of an appropriate action such that when a 0 (and Volume ) it gives the continuum theory Construction of the appropriate operators with their renormalization to extract physical quantities We take spacing a = a S = a T and size N S N S N S N T, N T > N S (Hadron Spectrum Collaboration considers anisotropic lattices with a T a S /3 better suited for study of excited states) Finite Volume: 1. Only discrete values of momentum in units of 2π/N S are allowed. 2. Finite volume effects need to be studied Take box sizes such that L S m π > 3.5. Finite lattice spacing: Need at least three values of the lattice spacing in order to extrapolate to the continuum limit. q 2 -values: Fourier transform of lattice results in coordinate space taken numerically for large values of momentum transfer results are too noisy = Limited to Q 2 = q 2 2 GeV 2. Studies to extend to larger values, H.-W. Lin, et al., arxiv: , H.-W. Lin and S. D. Cohen, arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

8 Computational cost ( Simulation cost: C sim ) cm ( 300MeV L mπ 2fm ) cl ( 0.1fm ) ca a L=2.1 fm, a=0.089 fm, K. Jansen and C. Urbach, arxiv: Coefficients c m, c L and c a depend on the discretized action used for the fermions. State-of-the-art simulations use improved algorithms: Mass preconditioner, M. Hasenbusch, Phys. Lett. B519 (2001) 177 Multiple time scales in the molecular dynamics updates = for twisted mass fermions: c m 4, c L 5 and c a 6. Simulations at physical quark masses, a 0.1 fm and L 5 fm require O(10) Pflop.Years. The analysis to produce physics results requires O(1) Pflop.Year. After post-diction of well measured quantities the goal is to predict quantities that are difficult or impossible to measure experimentally. C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

9 Mass of low-lying hadrons N F = smeared Clover fermions, BMW Collaboration, S. Dürr et al. Science 322 (2008) N F = 2 twisted mass fermions, ETM Collaboration, C. Alexandrou et al. PRD (2008) BMW with N F = 2 + 1: 3 lattice spacings: a 0.125, 0.085, fm set by m Ξ Pion masses: m π > 190 MeV Volumes:m min π L > 4 ETMC with N F = 2: 3 lattice spacings: a = 0.089, 0.070, a = fm, set my m N m π > 260 MeV Volumes:m min π L > 3.3 Good agreement between different discretization schemes = Significant progress in understanding the masses of low-lying mesons and baryons For to N and form factors we will use domain wall fermions (DWF) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

10 Mass of low-lying hadrons N F = smeared Clover fermions, BMW Collaboration, S. Dürr et al. Science 322 (2008) N F = 2 twisted mass fermions, ETM Collaboration, C. Alexandrou et al. PRD (2008) Baryon Spectrum M(GeV) ! N # * # " K $ * $ % Exp. results Width Input ETMC results BMW results BMW with N F = 2 + 1: 3 lattice spacings: a 0.125, 0.085, fm set by m Ξ Pion masses: m π > 190 MeV Volumes:m min π L > 4 ETMC with N F = 2: 3 lattice spacings: a = 0.089, 0.070, a = fm, set my m N m π > 260 MeV Volumes:m min π L > Strangeness Good agreement between different discretization schemes = Significant progress in understanding the masses of low-lying mesons and baryons For to N and form factors we will use domain wall fermions (DWF) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

11 Nucleon form factors Experimental measurements since the 50 s but still open questions high-precision experiments at JLab. OΓ The electric and magnetic Sachs form factors: G E (q 2 ) = F 1 (q 2 ) q2 (2m) 2 F 2(q 2 ), G M (q 2 ) = F 1 (q 2 ) + F 2 (q 2 ) Many lattice studies down to lowest pion mass of m π 300 MeV= Lattice data in general agreement, but still slower q 2 -slope Disconnected diagrams neglected so far Axial-vector FFs: A a µ = ψγ τ µγ a [ 5 2 ψ(x) ] = 1 2 γ µγ 5G A (q 2 ) + qµ γ 5 2m Gp(q2 ) ( xf,tf) ( xf,tf) q = pf pi ( x, t) OΓ q = pf pi ( x, t) ( xi,ti) ( xi,ti) C. A. et al. (ETMC), Phys. Rev. D83 (2011) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

12 Nucleon form factors Experimental measurements since the 50 s but still open questions high-precision experiments at JLab. OΓ The electric and magnetic Sachs form factors: G E (q 2 ) = F 1 (q 2 ) q2 (2m) 2 F 2(q 2 ), G M (q 2 ) = F 1 (q 2 ) + F 2 (q 2 ) Many lattice studies down to lowest pion mass of m π 300 MeV= Lattice data in general agreement, but still slower q 2 -slope Disconnected diagrams neglected so far Axial-vector FFs: A a µ = ψγ τ µγ a [ 5 2 ψ(x) ] = 1 2 γ µγ 5G A (q 2 ) + qµ γ 5 2m Gp(q2 ) ( xf,tf) ( xf,tf) q = pf pi ( x, t) OΓ q = pf pi ( x, t) ( xi,ti) ( xi,ti) Similar discrepancy also for the momentum fraction, C. A. et al. (ETMC), arxvi:1104:1600 C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

13 Nγ form factors A dominant magnetic dipole, M1 An electric quadrupole, E2 and a Coulomb, C2 signal a deformation in the nucleon/ 1/2-spin particles have vanishing quadrupole moment in the lab-frame Probe nucleon shape by studying transitions to its excited -state Difficult to measure/calculate since quadrupole amplitudes are sub-dominant R EM (EMR) = G E2 (Q2 ) G M1 (Q 2 ), R SM (CMR) = q 2m G C2 (Q 2 ) G M1 (Q 2 ), in lab frame of the. Precise data strongly suggesting deformation in the Nucleon/ At Q 2 = GeV 2 : EMR=( 2.00 ± 0.40 stat+sys ± 0.27 mod)%, CMR=( 6.27 ± 0.32 stat+sys ± 0.10 mod)% C. N. Papanicolas, Eur. Phys. J. A18 (2003); N. Sparveris et al., PRL 94, (2005) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

14 Nγ form factors A dominant magnetic dipole, M1 An electric quadrupole, E2 and a Coulomb, C2 signal a deformation in the nucleon/ 1/2-spin particles have vanishing quadrupole moment in the lab-frame Probe nucleon shape by studying transitions to its excited -state Difficult to measure/calculate since quadrupole amplitudes are sub-dominant I. Aznauryan et al., CLAS, Phys. Rev. C 80 (2009) New measurement of the Coulomb quadrupole amplitude in the low momentum transfer region (E08-010), N. Sparveris et al., Hall A Thanks to N. Sparveris. C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

15 Lattice evaluation (p, s ) j µ N(p, s) = i 2 Evaluation of two-point and three-point functions 3 ( ) ] m m 1/2 N ū σ(p, s )[ G E (p M1 ) E N (p) (q2 )K M1 σµ + G E2 (q2 )K E2 σµ + G C2 K C2 σµ u(p, s) OΓ G( q, t) = xf e i x f q Γ 4 βα Jα( x f, t f )J β (0) (xf,tf) (x,t) q = p p (xi,ti) G µν (Γ, q, t) = xf, x e i x q Γ βα J α( x f, t f )O µ ( x, t)j β (0) R J σ (t 2, t 1 ; p, p ; Γτ ; µ) = G JµN σ (t 2, t 1 ; p, p; Γτ ) [ G (t ii 2, p ; Γ 4 ) G NN (t 2, p; Γ 4 ) G NN (t 2 t 1, p; Γ 4 ) G ii (t 1, p ; Γ 4 ) ] 1/2 G ii (t 2, p ; Γ 4 ) G ii (t 2 t 1, p ; Γ 4 ) G NN (t 1, p; Γ 4 ) Construct optimized sources to isolate quadrupoles three-sequential inversions needed 3 3 S J 1 (q; J) = Π J σ (0, q ; Γ 4; J), S J 2 (q; J) = Π J σ (0, q ; Γ k ; J) σ=1 σ k=1 S J 3 (q; J) = ΠJ 3 (0, q ; Γ 3; J) 1 [ ] Π J 1 2 (0, q ; Γ 1; J) + Π J 2 (0, q ; Γ 2; J) Use the coherent sink technique: create four sets of forward propagators for each configuration at source positions separated in time by one-quarter of the total temporal size, Syritsyn et al. (LHPC), Phys. Rev. D81 (2009) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

16 Results on magnetic dipole Slope smaller than experiment, underestimate G M1 at low Q 2 pion cloud effects? New results using N f = dynamical Domain Wall Fermions, simulated by RBC-UKQCD Collaborations = No visible improvement. C. A., G.Koutsou, J.W. Negele, Y. Proestos, A. Tsapalis, Phys. Rev. D83 (2011) Situation like for nucleon form factors, independent of lattice discretization = nucleon FFs under study by a number of lattice groups. C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

17 Results on EMR and CMR Systematic errors may cancel in rations: G E2 and G C2 are suppressed at low Q 2 like G M1 = look at EMR and CMR New results using N f = dynamical domain wall fermions by RBC-UKQCD Collaborations Need large statistics to reduce the errors = as m π 140 MeV O(10 3 ) need to be analyzed. C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

18 N - axial-vector form factors (p, s ) A 3 µ N(p, s) = Aūλ (p, s ) ( ) A = i 23 m m 1/2 N E (p )E N (p) [ CA 3 (q2 ) m N γ ν + C A C A 5 (q2 ) analogous to the nucleon G A (q 2 ) 4 (q2 ) m N 2 p ν ( g λµ gρν g λρ gµν C A 6 (q2 ), analogous to the nucleon G p(q 2 ) pion pole behaviour C A 3 (q2 ) and C A 4 (q2 ) are suppressed (transverse part of the axial-vector) Study also the pseudo-scalar transition form factor G πn (q 2 ) = Non-diagonal Goldberger-Treiman relation: Pion pole dominance relates C A 6 to G πn through: Goldberger-Treiman relation becomes C A 5 (q2 ) + q2 C A mn 2 6 (q2 ) = 1 G πn (q 2 )f πm 2 π 2m N mπ 2. q2 1 m N C A 6 (q2 ) 1 2 G πn (q 2 )f π m 2 π q2 G πn (q 2 ) f π = 2m N C A 5 (q2 ) ) q ρ +C 5 A (q2 )g λµ + CA 6 (q2 ] ) m N 2 q λ qµ u(p, s) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

19 N - axial-vector form factors (p, s ) A 3 µ N(p, s) = Aūλ (p, s ) ( ) A = i 23 m m 1/2 N E (p )E N (p) [ CA 3 (q2 ) m N γ ν + C A C A 5 (q2 ) analogous to the nucleon G A (q 2 ) 4 (q2 ) m N 2 p ν ( g λµ gρν g λρ gµν C A 6 (q2 ), analogous to the nucleon G p(q 2 ) pion pole behaviour C A 3 (q2 ) and C A 4 (q2 ) are suppressed (transverse part of the axial-vector) Study also the pseudo-scalar transition form factor G πn (q 2 ) = Non-diagonal Goldberger-Treiman relation: Pion pole dominance relates C A 6 to G πn through: Goldberger-Treiman relation becomes C A 5 (q2 ) + q2 C A mn 2 6 (q2 ) = 1 G πn (q 2 )f πm 2 π 2m N mπ 2. q2 1 m N C A 6 (q2 ) 1 2 G πn (q 2 )f π m 2 π q2 G πn (q 2 ) f π = 2m N C A 5 (q2 ) ) q ρ +C 5 A (q2 )g λµ + CA 6 (q2 ] ) m N 2 q λ qµ u(p, s) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

20 Results on to N axial-vector form factors exper. (dipole) HYB, mπ = 353 MeV HYB, mπ = 353 MeV DWF, mπ = 330 MeV 1 DWF, mπ = 330 MeV DWF, mπ = 297 MeV 4 DWF, mπ = 297 MeV C A C A Q 2 (GeV 2 ) Q 2 (GeV 2 ) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

21 Results on to N axial-vector form factors 2 HYB, mπ = 353 MeV DWF, mπ = 330 MeV DWF, mπ = 297 MeV 2 HYB, mπ = 353 MeV DWF, mπ = 330 MeV DWF, mπ = 297 MeV mnfπgπn 2(m 2 π+q 2 )C 6 A 1 fπgπn 2mNC 5 A Q 2 (GeV 2 ) 1 Pion-pole dominance: m C A N 6 (Q2 ) 1 G πn (Q 2 )fπ 2 m π 2 +Q Q 2 (GeV 2 ) Goldberger-Treiman rel.: G πn (Q 2 )f π = 2m N C A 5 (Q2 ) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

22 πn pseudoscalar coupling Pseudoscalar current: P a τ (x) = ψ(x)γ a 5 2 ψ(x) -N matrix element: ( ) 2mq (p, s ) P 3 2 m m 1/2 N fπ m 2 π G πn (Q 2 ) N(p, s) = i 3 E (p )E N (p) mπ 2 ū ν (p, s ) qν u(p, s) + Q2 2m N Fit to: G πn (Q 2 (Q ) = K 2 /m π 2 +1) (Q 2 /m C 2 +1) 2 (Q 2 /m 2 +1) 5 C 6 20 HYB, mπ = 353 MeV DWF, mπ = 330 MeV DWF, mπ = 297 MeV 15 GπN Q 2 (GeV 2 ) In a previous study: N f = 2 Wilson results consistent with G πn (Q 2 ) 1.6G πnn (Q 2 ) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

23 electromagnetic form factors (p, s ) j µ (0) (p, s) = ūα(p, s ) F 1 (Q2 )g αβ + F 3 (Q2 ) qα q β (2M ) 2 γ µ + F 2 (Q2 )g αβ + F 4 (Q2 ) qα q β (2M ) 2 ( with e.g. the quadrupole form factor given by: G E2 = F 1 ) τf 2 1 ( 2 (1 + τ) F 3 ) τf, where τ Q 4 2 /(4M 2 ) Construct an optimized source to isolate G E2 additional sequential propagators needed. Neglect disconnected contributions in this evaluation. iσµν qν 2M u β (p, s) Transverse charge density of a polarized along the x-axis can be defined in the infinite momentum frame ρ T 2 3 ( b) and ρ T 1 ( b). 2 Using G E2 we can predict shape of G E2 2 b y [fm] 0.0 b y [fm] quenched Wilson, m π = 410 MeV hybrid, m π = 353 MeV 4 dynamical Wilson, m π = 384 MeV Q 2 in GeV b x [fm] b x [fm] with spin 3/2 projection elongated along spin axis compared to the Ω C. A., T. Korzec, G. Koutsou, C. Lorcé, J. W. Negele, V. Pascalutsa, A. Tsapalis, M. Vanderhaeghen, NPA825,115 (2009). C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

24 axial-vector form factors Axial-vector current: A a τa µ (x) = ψ(x)γµγ5 2 ψ(x) (p, s ) A 3 µ (0) (p, s) = ūα(p, s ) 1 ( g αβ 2 g 1 (q 2 )γ µ γ 5 + g 3 (q 2 ) qµ ) γ 5 + qα q β 2M 4M 2 (h1 (q 2 )γ µ γ 5 + h3 (q 2 ) qµ ) γ 5 uβ (p, s) 2M i.e. 4 axial form-factors, g 1, g 3, h 1 and h 3 at q 2 = 0 we can extract the axial charge PRELIMINARY m π = 563 MeV quenched m π = 490 MeV quenched m π = 411 MeV quenched m π = 353 MeV Asqtad/DWF 15 m π = 563 MeV quenched m π = 490 MeV quenched m π = 411 MeV quenched m π = 353 MeV Asqtad/DWF g g Q 2 (GeV) Q 2 (GeV) 2 = Using a consistent chiral perturbation theory framework extract the chiral Lagrangian couplings g A, c A, g from a combined chiral fit to the lattice results on the nucleon and axial charge and the axial N-to- form factor C 5(0). C. A., E. Gregory, T. Korzec, G. Koutsou, J. W. Negele, T. Sato, A. Tsapalis, arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

25 axial-vector form factors Axial-vector current: A a τa µ (x) = ψ(x)γµγ5 2 ψ(x) (p, s ) A 3 µ (0) (p, s) = ūα(p, s ) 1 ( g αβ 2 g 1 (q 2 )γ µ γ 5 + g 3 (q 2 ) qµ ) γ 5 + qα q β 2M 4M 2 (h1 (q 2 )γ µ γ 5 + h3 (q 2 ) qµ ) γ 5 uβ (p, s) 2M i.e. 4 axial form-factors, g 1, g 3, h 1 and h 3 at q 2 = 0 we can extract the axial charge PRELIMINARY m π = 563 MeV quenched m π = 490 MeV quenched m π = 411 MeV quenched m π = 353 MeV Asqtad/DWF 15 m π = 563 MeV quenched m π = 490 MeV quenched m π = 411 MeV quenched m π = 353 MeV Asqtad/DWF g g Q 2 (GeV) Q 2 (GeV) 2 = Using a consistent chiral perturbation theory framework extract the chiral Lagrangian couplings g A, c A, g from a combined chiral fit to the lattice results on the nucleon and axial charge and the axial N-to- form factor C 5(0). C. A., E. Gregory, T. Korzec, G. Koutsou, J. W. Negele, T. Sato, A. Tsapalis, arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

26 pseudoscalar couplings Pseudoscalar current: P a τ (x) = ψ(x)γ a 5 2 ψ(x) matrix element: [ ] (p, s ) P 3 (0) (p, s) = ū α(p, s ) 1 2 g αβ g(q 2 )γ 5 + qα q β h(q 2 4M 2 )γ 5 u β (p, s) i.e. two π couplings = two Goldberger-Treiman relations. G π is given by: m q g(q 2 ) fπ m2 π G π (Q2 ) (m 2 π +Q2 ) and H π is given by: m q h(q 2 ) fπ m2 π H π (Q2 ) (m 2 π +Q2 ) Goldberger-Treiman relations: f πg π (Q 2 ) = m g 1 (Q 2 ), f πh π (Q 2 ) = m h 1 (Q 2 ) 9 8 PRELIMINARY 7 6 G π m π = 563 MeV quenched m π = 490 MeV quenched m π = 411 MeV quenched m π = 353 MeV Asqtad/DWF Q 2 (GeV) 2 C. A., E. Gregory, T. Korzec, G. Koutsou, J. W. Negele, T. Sato, A. Tsapalis, arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

27 ρ-meson width Consider π + π in the I = 1-channel Estimate P-wave scattering phase shift δ 11 (k) using finite size methods Use Lüscher s relation between energy in a finite box and the phase in infinite volume Use Center of Mass frame and Moving frame Use effective range formula: tanδ 11 (k) = g2 ρππ 6π g ρππ and then extract Γ ρ = g2 ρππ 6π m π = 309 MeV, L = 2.8 fm k R 3 m R 2, k R = k 3 E (m R 2, k = E 2 /4 m 2 E2) π determine M R and mr 2 /4 m2 π 1 sin 2 (δ) 0.5 CMF MF1 MF2 sin 2 (δ)=1=>am R ae CM N F = 2 twisted mass fermions, Xu Feng, K. Jansen, D. Renner, arxiv:0910:4891; arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

28 ρ-meson width Consider π + π in the I = 1-channel Estimate P-wave scattering phase shift δ 11 (k) using finite size methods Use Lüscher s relation between energy in a finite box and the phase in infinite volume Use Center of Mass frame and Moving frame Use effective range formula: tanδ 11 (k) = g2 ρππ 6π g ρππ and then extract Γ ρ = g2 ρππ 6π m π = 309 MeV, L = 2.8 fm k R 3 m R 2, k R = k 3 E (m R 2, k = E 2 /4 m 2 E2) π determine M R and mr 2 /4 m2 π 1 sin 2 (δ) 0.5 CMF MF1 MF2 sin 2 (δ)=1=>am R Γ R (GeV) a=0.086fm a=0.067fm PDG data ae CM m π (GeV ) N F = 2 twisted mass fermions, Xu Feng, K. Jansen, D. Renner, arxiv:0910:4891; arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

29 Conclusions Large scale simulations using the underlying theory of the Strong Interactions have made spectacular progress = we now have simulations of the full theory at near physical parameters The low-lying hadron spectrum is reproduced Nucleon form factors are being computed by a number of collaborations in order to understand the discrepancies N to transition form factors can be extracted in a similar way to the nucleon Ratios of form factors expected to be less affected by lattice artifacts EMR and CMR allow comparison to experiment form factors are predicted Resonance width can be computed within Euclidean Lattice QCD as illustrated for the ρ-meson similar techniques can be applied to = Complete description of the nucleon- system C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

30 Conclusions Large scale simulations using the underlying theory of the Strong Interactions have made spectacular progress = we now have simulations of the full theory at near physical parameters The low-lying hadron spectrum is reproduced Nucleon form factors are being computed by a number of collaborations in order to understand the discrepancies N to transition form factors can be extracted in a similar way to the nucleon Ratios of form factors expected to be less affected by lattice artifacts EMR and CMR allow comparison to experiment form factors are predicted Resonance width can be computed within Euclidean Lattice QCD as illustrated for the ρ-meson similar techniques can be applied to = Complete description of the nucleon- system C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

31 Conclusions Large scale simulations using the underlying theory of the Strong Interactions have made spectacular progress = we now have simulations of the full theory at near physical parameters The low-lying hadron spectrum is reproduced Nucleon form factors are being computed by a number of collaborations in order to understand the discrepancies N to transition form factors can be extracted in a similar way to the nucleon Ratios of form factors expected to be less affected by lattice artifacts EMR and CMR allow comparison to experiment form factors are predicted Resonance width can be computed within Euclidean Lattice QCD as illustrated for the ρ-meson similar techniques can be applied to = Complete description of the nucleon- system C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

32 Conclusions Large scale simulations using the underlying theory of the Strong Interactions have made spectacular progress = we now have simulations of the full theory at near physical parameters The low-lying hadron spectrum is reproduced Nucleon form factors are being computed by a number of collaborations in order to understand the discrepancies N to transition form factors can be extracted in a similar way to the nucleon Ratios of form factors expected to be less affected by lattice artifacts EMR and CMR allow comparison to experiment form factors are predicted Resonance width can be computed within Euclidean Lattice QCD as illustrated for the ρ-meson similar techniques can be applied to = Complete description of the nucleon- system C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

33 Conclusions Large scale simulations using the underlying theory of the Strong Interactions have made spectacular progress = we now have simulations of the full theory at near physical parameters The low-lying hadron spectrum is reproduced Nucleon form factors are being computed by a number of collaborations in order to understand the discrepancies N to transition form factors can be extracted in a similar way to the nucleon Ratios of form factors expected to be less affected by lattice artifacts EMR and CMR allow comparison to experiment form factors are predicted Resonance width can be computed within Euclidean Lattice QCD as illustrated for the ρ-meson similar techniques can be applied to = Complete description of the nucleon- system C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

34 Conclusions Large scale simulations using the underlying theory of the Strong Interactions have made spectacular progress = we now have simulations of the full theory at near physical parameters The low-lying hadron spectrum is reproduced Nucleon form factors are being computed by a number of collaborations in order to understand the discrepancies N to transition form factors can be extracted in a similar way to the nucleon Ratios of form factors expected to be less affected by lattice artifacts EMR and CMR allow comparison to experiment form factors are predicted Resonance width can be computed within Euclidean Lattice QCD as illustrated for the ρ-meson similar techniques can be applied to = Complete description of the nucleon- system C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

35 Conclusions Large scale simulations using the underlying theory of the Strong Interactions have made spectacular progress = we now have simulations of the full theory at near physical parameters The low-lying hadron spectrum is reproduced Nucleon form factors are being computed by a number of collaborations in order to understand the discrepancies N to transition form factors can be extracted in a similar way to the nucleon Ratios of form factors expected to be less affected by lattice artifacts EMR and CMR allow comparison to experiment form factors are predicted Resonance width can be computed within Euclidean Lattice QCD as illustrated for the ρ-meson similar techniques can be applied to = Complete description of the nucleon- system C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

36 Thank you for your attention C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

37 axial-vector form factors Axial-vector current: A a τa µ (x) = ψ(x)γµγ5 2 ψ(x) (p, s ) A 3 µ (0) (p, s) = ūα(p, s ) 1 ( g αβ 2 g 1 (q 2 )γ µ γ 5 + g 3 (q 2 ) qµ ) γ 5 + qα q β 2M 4M 2 (h1 (q 2 )γ µ γ 5 + h3 (q 2 ) qµ ) γ 5 uβ (p, s) 2M i.e. 4 axial form-factors, g 1, g 3, h 1 and h 3 at q 2 = 0 we can extract the axial charge PRELIMINARY h 1-10 h m π = 563 MeV quenched m π = 490 MeV quenched m π = 411 MeV quenched m π = 353 MeV Asqtad/DWF -600 m π = 563 MeV quenched m π = 490 MeV quenched m π = 411 MeV quenched m π = 353 MeV Asqtad/DWF Q 2 (GeV) Q 2 (GeV) 2 = Using a consistent chiral perturbation theory framework extract the chiral Lagrangian couplings g A, c A, g from a combined chiral fit to the lattice results on the nucleon and axial charge and the axial N-to- form factor C 5(0). C. A., E. Gregory, T. Korzec, G. Koutsou, J. W. Negele, T. Sato, A. Tsapalis, arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

38 axial-vector form factors Axial-vector current: A a τa µ (x) = ψ(x)γµγ5 2 ψ(x) (p, s ) A 3 µ (0) (p, s) = ūα(p, s ) 1 ( g αβ 2 g 1 (q 2 )γ µ γ 5 + g 3 (q 2 ) qµ ) γ 5 + qα q β 2M 4M 2 (h1 (q 2 )γ µ γ 5 + h3 (q 2 ) qµ ) γ 5 uβ (p, s) 2M i.e. 4 axial form-factors, g 1, g 3, h 1 and h 3 at q 2 = 0 we can extract the axial charge PRELIMINARY h 1-10 h m π = 563 MeV quenched m π = 490 MeV quenched m π = 411 MeV quenched m π = 353 MeV Asqtad/DWF -600 m π = 563 MeV quenched m π = 490 MeV quenched m π = 411 MeV quenched m π = 353 MeV Asqtad/DWF Q 2 (GeV) Q 2 (GeV) 2 = Using a consistent chiral perturbation theory framework extract the chiral Lagrangian couplings g A, c A, g from a combined chiral fit to the lattice results on the nucleon and axial charge and the axial N-to- form factor C 5(0). C. A., E. Gregory, T. Korzec, G. Koutsou, J. W. Negele, T. Sato, A. Tsapalis, arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

39 Shape of ρ-meson non relativistic limit Four-point functions ψ 2 The ρ-meson having spin=1 is cigar-like in the lab frame, C. A. and G. Koutsou, Phys. Rev. D78 (2008) In agreement with form factor calculation J.N. Hedditch et al. Phys. Rev. D75, (2007) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24

40 Lattice spacing determination Use nucleon mass at physical limit Extrapolate using LO expansion: m N = m 0 N 4c 1m 2 π 3g2 A 16πf π 2 m 3 π Systematic error from O(p 4 ) SSE HBχPT Simultaneous fits to β = 1.9, β = 1.95 and β = 2.1 results β = 1.90 : a = 0.087(1)(6) β = 1.95 : a = 0.078(1)(5) β = 2.10 : a = 0.060(0)(3) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Resonance FFs in LQCD Jefferson Lab, May 18, / 24