The N-tο- (1232) transition from Lattice QCD :

Transcription

1 The N-tο-(13) transition from Lattice QCD : Electromagnetic, Axial and Pseudoscalar Form Factors with N F = +1 domain wall fermions Antonios Tsapalis Hellenic Naval Academy & National Technical University of Athens School of Applied Mathematics and Physical Sciences Paphos, Aug. 4-7, 010

2 in collaboration with C. Alexandrou (University of Cyprus & The Cyprus Institute) G. Koutsou (University of Wuppertal & Fortshungszentrum Jülich) J.W. Negele (MIT, Center for Theoretical Physics) Y. Proestos (The Cyprus Institute)

3 Why the N-to-? Outline EM transition form factors nucleon deformation Axial : pure isovector, resemble nucleon axial elastic C 5A G A, C 6 A G p Pseudoscalar : strong πν coupling, g πν (non-diagonal) Goldberger-Treiman relation between C 5A, g πν Lattice Calculation +1 Domain Wall Flavors (RBC & UKQCD) (fully chiral, unitary & a=0.084 fm, m π = 97 MeV) design optimized sinks to get full momentum dependence from 3-pointfunctions with three sequential inversions coherent sink technique to maximize statistics Results Prospects (- matrix element, strong couplings from QCD)

4 Nucleon structure & form factors e(k,s) e(k,s ) probe nucleon with photons experiment: electron beams q = ( p p) γ Nucleon structure contained in Form Factors G q G q E( ), M( ) LHPC: arxiv: N(p,s) N(p,s ) RBC/UKQCD: arxiv: magnetic moment, size, charge distribution S = 1 BUT NOT the SHAPE of the nucleon (L= operator) 1 1 spectroscopic quadrupole moment vanishes < Q >= 0 Intrinsic quadrupole moment w.r.t. body-fixed frame exists Q0 = dr ρ( r Q 0 > 0 )(3 z r ) Q < 0 0 prolate oblate

5 γ * N (13) Pion electroproduction on nucleon target N(qqq) γ* Μ1, Ε, C (qqq) 1 I = J = 938 MeV 1 Spherical M1 Deformed M1, E, C u u d Deformation signal Μ 1+, Ε 1+, S 1+ π o u u EMR=Re E M 1 CMR=Re S M d 3/ 1+ 3/ + 3/ 1+ 3/ 1+ 3 I = J = 13 MeV ( R ) EM ( R ) SM 3

6 EMR & CMR Experimental Status uncertainties in modelling final state interactions Thanks to N. Sparveris (Temple Univ.)

7 The EM Transition Matrix Element µ mm N σµ < ( p, s ) J N( p, s) >= i uσ ( p, s ) O u( p, s) 3 EE N H.F.Jones and M.C.Scadron, Ann. Phys. (N.Y.) 81,1 (1973) 1/ u ( p σ, s ) is the Rarita-Schwinger vector-spinor obeying σ pu σ γ u σ σ ( p) = 0 ( p) = 0 kinematical functions O = G ( q ) K + G ( q ) K + G ( q ) K σµ σµ σµ σµ M 1 M 1 E E C C magnetic dipole electric quadrupole scalar quadrupole G E(q ) M1 EMR = - G (q ) static frame q G C(q ) M M1 CMR = - G (q )

8 Origin of the deformation Constituent Quark Models D-state admixture due to color-magnetic hyperfine interaction (Isgur,Karl) Hybrid Bag Models Pion cloud interacting with the quark core Quark Models with meson exchange currents (Buchmann, Henley) Skyrmion Models Non-linear pion field interactions EMR 1 to 4%

9 Lattice QCD Calculation J µ (q) (p ) x N(p) t t 1 0 generate a nucleon at t=0 insert a (one body) operator with momentum q at t=t 1 annihilate a Delta at time t=t measure the 3-point function extract the form factors from suitable ratios of 3-point and -point functions

10 create states with interpolating fields.. χ ( x) = ε [ u ( x) Cγ d ( x)] u ( x) p abc a b c 5 N( p, s) χ(0) Ω = u( p, s) Z N + 1 abc a b c a b c χσ ( x) ε [ u ( x) Cγσd ( x)] u ( x) [ u ( x) Cγσu ( x)] d ( x) 3 { } = + ( ps, ) (0) Ω = Z u( ps, ) χ + σ σ..smear quarks to maximize the overlaps Z N, Z i ' ( ' ) 1 dx e dx e T µ G ( t t ; p'; p; Γ ) = j N σ, 1 p x i p p x µ p 1 ( χσ a( x) j ( x1) χb (0)) ba Ω Ω Γ =..insert complete sets of states.. ( qs, ) ( qs, ), qs, Nks,, Nks (, ) Nks (, ) E ( t t1) ENt1 ZZe e Ν [ K ( p'; p; Γ ) G + K ( p '; p; Γ ) G + K ( p'; p; Γ) G ] σµ σµ σµ M1 M1 E E C C 1 σ i 0 1 I 0 Γ i =, Γ 4 =

11 ..and extract form factors from the ratio µ j N Gσ ( t, t1; p'; p; Γ ) G ( t1; p'; Γ 4) Rσ ( t, t1; p'; p; Γ ; µ ) = NN G ( t; p'; Γ 4) G ( t1; p; Γ 4) t ( p'; p; Γ; ) t1 1 Π t1 1 σ µ 1..which becomes Z-, t-independent if source-operator-sink are sufficiently apart G Γ NN ENt E + M N N (; t p; 4) Z Ne Γ Et E G (; p; 4) Ze E E N M t +

12 fixed operator vs fixed sink inversions fixed operator γ µ e iq x fixed particles at sink & source fixed insertion time t 1 any particle at sink & source vary sink time t for plateau fixed sink time t any operator inserted vary insertion time t for plateau Q dependence

13 -optimize the calculation design linear combinations of 3-point functions in (σ, Γ) space that isolate the form factors Π (0, q; Γ ; µ ) +Π (0, q; Γ ; µ ) +Π (0, q; Γ ; µ ) { } 3 δµ,1 3 1 δµ, 1 δµ,3 = ia q q + q q + q q G Q ( ) ( ) ( ) M 1( ) in total 3 sequential inversions to obtain all form factors with the maximum contribution of transition momenta q

14 RBC/UKQCD ensembles (arxiv: ) +1 dynamical DW flavors with m s ~ physical good chirality, L 5 =16 m res /m u,d < 10% Volume a (fm) m π (MeV) #meas 4 3 x x /1540 Coherent sink technique : 4 sources per configuration - forward propagators by LHPC T N 4 measurements per sequential inversion T

15 G M1 -dipole fits well requires larger mass 1.08 GeV vs 0.78 exp -slower falloff at small Q missing pion cloud -small O(a ) errors -consistency with hybrid DW valence on staggered sea PRD 77, (008)

16 -quadrupole amplitudes in γn verified from QCD deformed N & states -weak Q dependence -χeft: strong m π dependence below 300 MeV -- need lighter quarks

17 3 N-to- Axial Transition Form Factors isovector axial-vector current < > = 3 ( p, s) Aµ N( p, s) A a µ (x) = ψ(x) γ µ γ 5 τ a ψ(x) [ Adler parameterization ] A A A λ C3 ( q ) ν C4 ( q ) ν A C6 ( q ) ρ u ( p, s) γ + p ( gλµ gρν gλρ gµν ) q + C5 ( q ) gλµ + q q u( p, s) λ µ mn mn mn small 0 transverse part dominant FFs C 5A analogous to G A (q ) C 6A analogous to G P (q ) Reminder: 3 q µ τ 3 < N( p, s ) Aµ N( p, s) >= u( p, s) GA( q ) γµ γ5 + Gp( q ) γ5 u( p, s) mn electroproduction experiments at Jlab will measure N to parity violating asymmetry connected to C 5 A theoretical arguments indicate that C 3A, C 4A are small

18 Pseudoscalar πν Form Factor & PCAC πν form factor defined via 3 q (, ) π π π N λ qλ mπ q mn a a P(x) = ψ(x) γ ψ(x) 5 fmg ( q ) m < p s P N( p, s) >= u ( p, s) u( p, s) 3 connected to πν strong coupling constant g πν = G πν (m π ) τ A = PCAC f m µ a µ π π π a q 1 fm C ( q ) C ( q ) G ( q ) A A π π = π N mn mn mπ q Non-diagonal Goldberger-Treiman relation Pion pole dominance relates:..and fixes the ratio C m A 6 / C A 5 = N Q + m π 1 f C ( q )~ G ( q ) m m q N A π 6 π N ( π ) f C q G q A ( ) π ( ) 5 π N mn

19 dipole fit describes well C A 5 slower falloff m A =1.7 GeV (fit) m A =1.8 GeV (exp) small O(a ) errors transverse part suppressed

20 C m A / A N C 6 5 = Q + m π pion pole dominates C A 6 monopole fit : m π ~0.5 GeV

21 G πν - sensitive in chiral effects / m q from AWI - understand systematics to approach phenomenology

22 Goldberger-Treiman relations fg π πn A mc N 5 m f G N π πn ( π + ) m Q C A 6

23 (13) EM form factors q = ( p p) γ J=3/ elastic matrix element can reveal quadrupole amplitudes (p,s) (p,s ) 3 Q 3 0 <( p, s ) J ( p, s) >= µ qq u p s g a q p p c q σ ( ) ( ) σ τ (, ) στ a1( q ) γ µ + ( µ + µ ) + c q p p u p s 1( ) γ µ + ( µ + µ ) τ (, ) m 4m m four independent q dependent form factors: a 1, a, c 1, c

24 reparameterize in terms of spin/parity based physical form factors a1( q ) GE 0( q ) a( q ) GM 1( q ) c1( q ) GE ( q ) c( q ) GM 3( q ) Electric charge form factor Magnetic dipole form factor Electric quadrupole form factor Magnetic octupole form factor useful for the extraction of static properies & distributions rms radius r dg ( Q ) = 6 dq E 0 Q = 0 magnetic moment quadrupole moment (deformation) magnetic octupole moment µ = m e GM GE (0) O = e 1 (0) GM 3(0) 3 m

25 x γ ~ µ j Gστ ( t, t1; p'; p; Γ ) G ( t ; p'; Γ ) G (t t ; p; Γ ) ii 4 ii 1 4 µ G ( t t ; p '; p; Γ) = j στ, 1 1 ( σa( ) ( 1) χτb(0)) Γba ip' x i( p' p) x1 µ dx e dx e T χ x j x Ω Ω ( ) ( ) ( ) ( )] E ( t t1) Et1 σµτ σµ τ σµτ σµτ Ze e [ KE0 GE0 q + KM1GM1 q + KE GE q + KM3GM3 q Known kinematical functions of p, p Select σ, µ, τ indices to isolate different form factors

26 Electric charge form factor & rms radius C. Alexandrou, T. Korzec, G. Koutsou, Th. Leontiou, C. Lorce, J.W. Negele, V. Pascalutsa, A. Tsapalis, M. Vanderhaeghen PRD 79, (009), NPA 85 (009) 115 dipole fits well the data G 1 ( Q ) = Q 1+ Λ E 0 E 0 rms radius from slope at Q =0 r dg ( Q ) = 6 dq E 0 Q = 0 r (fm) quenched Wilson 0.65(9) N F = Wilson 0.611(17) N F =+1 Hybrid 0.641()

27 Magnetic dipole form factor and magnetic moment dipole and exponential fit equally well the data Q GM1( Q ) = GM1exp Λ M 1 µ ( e/ m ) + N quenched Wilson 1.811(69) N F = Wilson 1.58(11) chiral extrapolation V. Pascalutsa & M. Vanderhaeghen, PRL 95(005) 3001 finite volume effect? N F =+1 Hybrid 1.91(16)

28 Electric quadrupole form factor & deformation exponential fits with Q to extract G E (0) consistently negative G E (0) Quadrupole moment Q 3/ is connected to transverse quark charge densities in the infinite momentum frame.. which in turn are connected to the FFs at Q =0 { M E } e Q = G (0) 3 e + G (0) + 3e 3/ 1 m Consistently positive Q 3/ for spin +3/ + state in the IMF elongated along spin axis (prolate)

29 Conclusions N-to- is a valuable transition to study - deformation in N & states confirmed ; - EMR/CMR in qualitative agreement to experiment ; -m π < 300 MeV for full pion cloud effects - axial sector dominated by pion pole - GT relations valid in conjunction to N-N and - (ongoing) studies map systematically low energy hadron couplings