Hadron Structure. James Zanotti The University of Adelaide. Lattice Summer School, August 6-24, 2012, INT, Seattle, USA

Transcription

1 Hadron Structure James Zanotti The University of Adelaide Lattice Summer School, August 6-24, 2012, INT, Seattle, USA

2 Motivation We know the nucleon is not a point-like particle but in fact is composed of quarks and gluons But how are these constituents distributed inside the nucleon? E.g. The neutron has zero net charge, but does it have a +/- core? How do they combine to produce its experimentally observed properties For example Spin crisis : quarks carry on ~30% of the proton s spin gluons? orbital angular momentum? Understanding how the nucleon is built from its quark and gluon constituents remains one the most important and challenging questions in modern nuclear physics.

3 Topics to cover Elastic scattering Electromagnetic form factors Neutron beta decay Nucleon axial charge Deep Inelastic Scattering Structure Functions and Parton Distribution Functions Generalised Parton Distribution Functions Hidden Flavour, e.g. strangeness content of the nucleon Focus on Nucleon and Pion

4 Lecture 1 A short history Elastic scattering Form Factors Density distributions in the nucleon Lattice methods for computing hadronic matrix elements (3pt functions) A taste of some lattice results

5 Structure of the Proton Until 1932, proton was considered to be an elementary particle In 1933, Otto Stern measured the proton s magnetic moment µ p 2.5 (today : (11)) e 2m p Deviates significantly from unity - the magnetic moment of point-like particle described by Dirac s theory of relativistic fermions µ p = µ D e 2m p Proton is a composite particle The proton s constituents were later seen in Deep-Inelastic Scattering experiments at SLAC (1968)

6 Nucleon Structure Nucleon structure is studied experimentally by electron-proton scattering Electron is a good probe because: QED is a well-understood interaction em = 1 perturbation theory is valid 137 Electrons are charged and so easily accelerated Two types of e-p scattering: Elastic scattering Today Deep-Inelastic Scattering (DIS) Wednesday

7 Elastic Scattering Final state nucleon remains intact, but with recoil Map out charge and density distributions inside the nucleon Dominated by single-photon exchange k k 4-momentum transfer q = k k 0 = P 0 P Power of the probe q Compare cross-section with that of a point-particle d d = d d point F (q 2 ) 2 P P Form Factor

8 Elastic Scattering When using a point target, d d point F (q) =1 and = (Z )2 E 2 4k 2 sin 4 ( /2) 1 Mott cross section k 2 E 2 sin2 ( /2) E, k - energy and momentum of electron scattering angle (q 2 = 4EE 0 sin 2 /2) But experimental cross-sections deviate from this description 2 Cross Section (cm /sterad) Mott Electron Scattering from Hydrogen 188 MeV (LAB) Experimental Anomalous Moment Scattering Angle (deg)

9 Elastic Scattering Considering only one-photon exchange, justified because the fine structure constant is so small, the S-Matrix is then S =(2 ) 4 4 (k + P P 0 k 0 )u(k 0 )( ie µ )u(k) = i(2 ) 4 4 (k + P P 0 k 0 )M The electromagnetic current is i q 2 hp 0 (ie)j µ P i Invariant amplitude J µ = X i µ e i i i Sum over quark flavours with m q m p (u,d,s) Can write cross-section in terms of invariant amplitude d = E0 2EM E M sin2 2 M 2 d (2 ) 2 k k q P P

10 Elastic Scattering The invariant amplitude squared is M 2 = e4 Q 4 `µ W µ Leptonic tensor `µ = u(k 0 ) µ u(k)u(k) u(k 0 ) Compute in QED Hadronic tensor W µ = hp J P 0 ihp 0 J µ P i with matrix element between nucleon states defining two Lorentz-invariant form factors hp 0 J µ (~q ) P i =ū(p 0 ) h µ F 1 (q 2 ) + i µ q i 2m F 2(q 2 ) u(p ) Dirac Pauli

11 Elastic Scattering Using the fact that both tensors are symmetric and conserved q µ`µ = q µ W µ =0 The elastic scattering cross-section in the lab frame becomes where d d = Mott apple G 2 E (Q 2 )+ G 2 M (Q2 ) 1+ G E (Q 2 )=F 1 (Q 2 ) F 2 (Q 2 ) +2 G 2 M (Q 2 ) tan 2 2 G M (Q 2 )=F 1 (Q 2 )+F 2 (Q 2 ) = Q 2 /(4M 2 ) are the Sachs electric and magnetic form factors Rewriting in terms of the virtual photon s longitudinal polarisation d d = Mott 1+ h G 2 E(Q 2 )+ G2 M (Q 2 ) i 1 =1+(1+ )2 tan /2 Need cross sections at fixed Q 2 but different scattering angle: Rosenbluth separation

12 Elastic Scattering - Rosenbluth At fixed Q 2, GE and GM determined from intercept and slope as a function of 1 d d = Mott 1+ h G 2 E(Q 2 )+ i G2 M (Q 2 ) Drawback - reduced sensitivity to GE at large Q 2 Both form factors reasonably well described by the dipole form G p E (Q2 )= Gp M (Q2 ) µ p = 1 (1 + Q 2 /M 2 D )2 with M 2 D 0.71 GeV 2

13 Elastic Scattering - Rosenbluth For a neutron: d d = Mott 1+ h G 2 E(Q 2 )+ i G2 M (Q 2 ) Neutron GE small, so extraction near impossible No neutron target so use deuterium and subtract proton contribution Model nuclear effects Range allowed by e-d elastic

14 Elastic Scattering - Polarisation Transfer Difficulties with unpolarised scattering new techniques necessary Mid- 90s brought High luminosity, highly polarised electron beams Polarised targets ( 1 H, 2 H, 3 He) Large, efficient neutron detectors Polarisation transfer experiments provide access to the ratio GE/GM directly from ratio of polarisation transverse and parallel to the momentum of the nucleon G E G M = P t P l E + E 0 2M tan 2 Combine with previous accurate results for GM to also determine GE

15 Elastic Scattering - Polarisation Transfer Precise results now available up to 8-9 GeV 2 JLab, Hall A, PRC85 (2012) Does G p E change sign? 1.0 What is the origin of the linear fall-off? p /G M p G E 0.5 µ p Cloet09, r 1 = 0.8 fm Chang11, add κ q Eichmann11, η = 1.8 Eichmann11, η = Q (GeV )

16 Elastic Scattering - Polarisation Transfer JLab, Hall A, PRC85 (2012) /G D p G E /µ p G D p G M Diehl05 (GPD) Eichmann11 (DSE) Lomon06 (VMD) Gross08 (GCS) Santopinto10 (hcqm) Q 2 (GeV 2 ) Q 2 (GeV 2 ) n G E 0.05 /µ n G D n G M Q 2 (GeV 2 ) Q 2 (GeV 2 )

17 Insights into Nucleon Structure G E 6= G M different charge and magnetisation distributions If M!1initial and final nucleons are fixed at the same location Q 2 M 2 Initial and final states have same internal state Fourier transformation of form factors are density distributions But M is finite so need to consider nucleon recoil effects Initial and final states now sampled in different frames Lorentz contraction No model independent way to separate internal structure and recoil effects Work around: Breit frame or infinite momentum frame

18 Density Distributions Consider the Breit frame: P = P 0 initial and final states have momenta with equal magnitude, hence similar Lorentz contraction GE(Q 2 ) can be interpreted as the Fourier transformation of the charge distribution expanding at small Q 2 G E (Q 2 )= e i~q~x (r)d 3 r defines the charge radius of the nucleon Z G E (Q 2 1 )=Q e 6 Q2 hr 2 i +... hr 2 i = 6 dg E(Q 2 ) dq 2 Q 2 =0

19 Size of the Proton > 5 discrepancy between muonic hydrogen and e-p scattering rp= (67) fm [Nature 466, 213 (2010] rp=0.875(8)(6) fm [arxiv: ]

20 Transverse Spatial Distributions Model independent relation between form factors and transverse spatial distributions occurs in the infinite momentum frame Quark (charge) distribution in the transverse plane q(b 2 )= d 2 q e i b q F 1 (q 2 ) Distance of (active) quark to the centre of momentum in a fast moving nucleon b Provide information on the size and internal charge densities R Pz

21 Electromagnetic Form Factors Can some of these questions be answered by a calculation from QCD? Form factors are nonperturbative quantities Lattice QCD Need to determine p,s J µ (q) p, s =ū(p,s) µ F 1 (q 2 ) + i µ q 2m F 2(q 2 ) u(p, s)

22 Calculating Matrix Elements H O H H, H :, K p, n,... O : V µ,a µ,...

23 Calculating Matrix Elements Spin-0 h (p 0 ) J µ (~q) (p)i = P µ F (q 2 ) q 2 = Q 2 =(p 0 p) 2 P µ = p 0µ + p µ Spin-1/2 hn(p 0,s 0 ) J µ (~q) N(p, s)i =ū(p 0,s 0 ) Spin-1 h (p 0,s 0 ) J µ (~q) (p, s)i = h µ F 1 (q 2 ) + i µ q i 2m F 2(q 2 ) u(p, s) ( 0 )P µ G 1 (Q 2 ) [( 0 q) µ ( q) 0 µ ]G 2 (Q 2 ) +( q)( 0 q) Spin-3/2 P µ (2m ) 2 G 3(Q 2 ) h (p 0,s 0 ) J µ (~q) (p, s)i = ū (p 0,s 0 ) g µ a 1 (Q 2 ) + P µ 2M a 2(Q 2 ) q q µ (2M ) 2 c 1 (Q 2 ) + d P µ 2M c 2(Q 2 ) u (p, s)

24 Lattice 3pt Functions G(t,,p,p 0 )= X e i~p 0 (~x 2 ~x 1 ) e i~p ~x 1 ~x 2,~x 1 T (~x 2,t) O(~x 1, ) (0) Create a state (with quantum numbers of the proton) at time t=0

25 Lattice 3pt Functions G(t,,p,p 0 )= X e i~p 0 (~x 2 ~x 1 ) e i~p ~x 1 ~x 2,~x 1 T (~x 2,t) O(~x 1, ) (0) Create a state (with quantum numbers of the proton) at time t=0 Insert an operator,, at some time O

26 Lattice 3pt Functions G(t,,p,p 0 )= X e i~p 0 (~x 2 ~x 1 ) e i~p ~x 1 ~x 2,~x 1 T (~x 2,t) O(~x 1, ) (0) Create a state (with quantum numbers of the proton) at time t=0 Insert an operator, O, at some time Annihilate state at final time t

27 Lattice 3pt Functions G(t,,p,p 0 )= X e i~p 0 (~x 2 ~x 1 ) e i~p ~x 1 ~x 2,~x 1 T (~x 2,t) O(~x 1, ) (0) Create a state (with quantum numbers of the proton) at time t=0 Insert an operator, O, at some time Annihilate state at final time t

28 Lattice 3pt Functions G(t,,p,p 0 )= X e i~p 0 (~x 2 ~x 1 ) e i~p ~x 1 ~x 2,~x 1 Insert complete set of states I = X T (~x 2,t) O(~x 1, ) (0) B 0,p 0,s 0 B 0,p 0,s 0 ihb 0,p 0,s 0 I = X B,p,s B,p,sihB,p,s Make use of translational invariance (~x, t) =eĥt e i ˆ~ P ~x (0)e i ˆ~ P ~x e Ĥt G(t,, ~p, ~p 0 )= X B,B 0 X e EB0 (~p 0 )(t ) e E B(~p) s,s 0 (0) B 0,p 0,s 0 B 0,p 0,s 0 O(~q) B,p,s B,p,s (0) Evolve to large Euclidean times to isolate ground state 0 t G(t,, ~p, ~p 0 )= X s,s 0 e E ~p 0 (t ) e E ~p (0) N(p 0,s 0 ) N(p 0,s 0 ) O(~q) N(p, s) Np,s) (0)

29 Lattice 3pt Functions pion Consider a pion 3pt function G(t,, p, p )= x 2, x 1 e i p ( x2 x1) e i p x1 T ( x2,t) O( x 1, ) (0) With interpolating operator (x) = d(x) 5 u(x) And insert the local operator (quark bi-linear) q(x)oq(x) O : Combination of matrices and derivatives d(x 2 ) 5 u(x 2 )ū(x 1 )Ou(x 1 )ū(0) 5 d(0) u-quark

30 Lattice 3pt Functions pion Consider a pion 3pt function G(t,, p, p )= x 2, x 1 e i p ( x2 x1) e i p x1 T ( x2,t) O( x 1, ) (0) With interpolating operator (x) = d(x) 5 u(x) And insert the local operator (quark bi-linear) q(x)oq(x) O : Combination of matrices and derivatives d(x 2 ) 5 u(x 2 )ū(x 1 )Ou(x 1 )ū(0) 5 d(0) u-quark

31 Lattice 3pt Functions d a (x 2 ) 5 u a (x 2 )ū b (x 1 ) u b (x 1 )ū c (0) 5 d c (0) pion u-quark all possible Wick contractions

32 Lattice 3pt Functions d a (x 2 ) 5 u a (x 2 )ū b (x 1 ) u b (x 1 )ū c (0) 5 d c (0) pion u-quark all possible Wick contractions connected S ca d (0,x 2 ) 5 S ab u (x 2,x 1 ) S bc u (x 1, 0) 5

33 Lattice 3pt Functions d a (x 2 ) 5 u a (x 2 )ū b (x 1 ) u b (x 1 )ū c (0) 5 d c (0) pion u-quark all possible Wick contractions connected S ca d (0,x 2 ) 5 S ab u (x 2,x 1 ) S bc u (x 1, 0) 5 disconnected S ca d (0,x 2 ) 5 S ac u (x 2, 0) 5 S bb u (x 1,x 1 )

34 Lattice 3pt Functions d a (x 2 ) 5 u a (x 2 )ū b (x 1 ) u b (x 1 )ū c (0) 5 d c (0) pion u-quark all possible Wick contractions connected Tr S d (0,x 2 ) 5 S u (x 2,x 1 ) S u (x 1, 0) 5 disconnected Tr S d (0,x 2 ) 5 S u (x 2, 0) 5 Tr Su (x 1,x 1 )

35 Lattice 3pt Functions pion d a (x 2 ) 5 u a (x 2 )ū b (x 1 ) u b (x 1 )ū c (0) 5 d c (0) all possible Wick contractions 5-hermiticity S (x, 0) = 5 S(0,x) 5 connected Tr S d (x 2, 0)S u (x 2,x 1 ) S u (x 1, 0) disconnected Tr S d (x 2, 0)S u (x 2, 0) Tr S u (x 1,x 1 ) all-to-all propagators

36 Lattice 3pt Functions proton G (t, ; p, p) = x 2, x 1 e i p ( x2 x1) e i p x1 T (t, x2 ) O(, x 1 ) (0) Use the following interpolating operator to create a proton (x) = abc u Ta (x) C 5 d b (x) u c (x) And insert the local operator (quark bi-linear) q(x)oq(x) O: Combination of matrices and derivatives Perform all possible (connected) Wick contractions u-quark (4 terms) abc a b c u Ta (x 2 ) C 5 d b (x 2 ) u c (x 2 )ū(x 1 )Ou(x 1 )ū c (0) db (0)C 5 ū Ta (0)

37 Lattice 3pt Functions proton G (t, ; p, p) = x 2, x 1 e i p ( x2 x1) e i p x1 T (t, x2 ) O(, x 1 ) (0) Use the following interpolating operator to create a proton (x) = abc u Ta (x) C 5 d b (x) u c (x) And insert the local operator (quark bi-linear) q(x)oq(x) O: Combination of matrices and derivatives Perform all possible (connected) Wick contractions u-quark (4 terms) abc a b c u Ta (x 2 ) C 5 d b (x 2 ) u c (x 2 )ū(x 1 )Ou(x 1 )ū c (0) db (0)C 5 ū Ta (0)

38 Lattice 3pt Functions proton G (t, ; p, p) = x 2, x 1 e i p ( x2 x1) e i p x1 T (t, x2 ) O(, x 1 ) (0) Use the following interpolating operator to create a proton (x) = abc u Ta (x) C 5 d b (x) u c (x) And insert the local operator (quark bi-linear) q(x)oq(x) O: Combination of matrices and derivatives Perform all possible (connected) Wick contractions u-quark (4 terms) abc a b c u Ta (x 2 ) C 5 d b (x 2 ) u c (x 2 )ū(x 1 )Ou(x 1 )ū c (0) db (0)C 5 ū Ta (0)

39 Lattice 3pt Functions proton G (t, ; p, p) = x 2, x 1 e i p ( x2 x1) e i p x1 T (t, x2 ) O(, x 1 ) (0) Use the following interpolating operator to create a proton (x) = abc u Ta (x) C 5 d b (x) u c (x) And insert the local operator (quark bi-linear) q(x)oq(x) O: Combination of matrices and derivatives Perform all possible (connected) Wick contractions u-quark (4 terms) abc a b c u Ta (x 2 ) C 5 d b (x 2 ) u c (x 2 )ū(x 1 )Ou(x 1 )ū c (0) db (0)C 5 ū Ta (0)

40 Lattice 3pt Functions proton G (t, ; p, p) = x 2, x 1 e i p ( x2 x1) e i p x1 T (t, x2 ) O(, x 1 ) (0) Use the following interpolating operator to create a proton (x) = abc u Ta (x) C 5 d b (x) u c (x) And insert the local operator (quark bi-linear) q(x)oq(x) O: Combination of matrices and derivatives Perform all possible (connected) Wick contractions d-quark (2 terms) abc a b c u Ta (x 2 ) C 5 d b (x 2 ) u c (x 2 ) d(x 1 )Od(x 1 )ū c (0) db (0)C 5 ū Ta (0)

41 Lattice 3pt Functions proton G (t, ; p, p) = x 2, x 1 e i p ( x2 x1) e i p x1 T (t, x2 ) O(, x 1 ) (0) Use the following interpolating operator to create a proton (x) = abc u Ta (x) C 5 d b (x) u c (x) And insert the local operator (quark bi-linear) q(x)oq(x) O: Combination of matrices and derivatives Perform all possible (connected) Wick contractions d-quark (2 terms) abc a b c u Ta (x 2 ) C 5 d b (x 2 ) u c (x 2 ) d(x 1 )Od(x 1 )ū c (0) db (0)C 5 ū Ta (0)

42 Lattice 3pt Functions proton Pictorially: u-quark d-quark s-quark quark-line disconnected contributions drop out in isovector quantities (u-d) if isospin is exact (mu=md)

43 Lattice 3pt Functions at the quark level proton C (t, ; ~p 0, ~p) = X ~x 1 e i~q ~x 1 D Tr (~0, 0; ~p 0,t)O(~x 1, )G(~x 1, 0) E {U} G O 0 t 0 t G O u-quark d-quark

44 Lattice 3pt Functions at the quark level proton C (t, ; ~p 0, ~p) = X ~x 1 e i~q ~x 1 D Tr (~0, 0; ~p 0,t)O(~x 1, )G(~x 1, 0) E {U} (~0, 0; ~x 1 ; ~p 0,t)= X ~x 2 S (~x 2,t;~0, 0; ~p 0 )G(~x 2,t; ~x 1 ) G 0 t 0 t S S G u-quark d-quark

45 Lattice 3pt Functions C (t, ; ~p 0, ~p) = X ~x 1 e i~q ~x 1 at the quark level proton D Tr (~0, 0; ~p 0,t)O(~x 1, )G(~x 1, 0) E {U} (~0, 0; ~x 1 ; ~p 0,t)= X ~x 2 S (~x 2,t;~0, 0; ~p 0 )G(~x 2,t; ~x 1 ) S u;a0a (~x G = C 5 G T 2,t;~0, 0; ~p 0 )=e i~p 0 ~x abc a0 b 0 c 0 apple G d;bb0 (~x 2,t;~0, 0)G u;cc0 (~x 2,t;~0, 0) +Tr D [ G d;bb0 (~x 2,t;~0, 0)G u;cc0 (~x 2,t;~0, 0)] + G u;bb0 (~x 2,t;~0, 0) G d;cc0 (~x, t;~0, 0) + Tr D [ G u;bb0 (~x 2,t;~0, 0)] G d;cc0 (~x 2,t;~0, 0) S d;a0a (~x 2,t;~0, 0; ~p 0 )=e i~p 0 ~x 2 abc a0 b 0 c 0 apple G u;bb0 (~x 2,t;~0, 0) G u;cc0 (~x 2,t;~0, 0) + Tr D [ G u;bb0 (~x 2,t;~0, 0) G u;cc0 (~x 2,t;~0, 0)] Exercise: Prove 5C } S 0 t 0 t S u-quark d-quark

46 Lattice 3pt Functions (~0, 0; ~x 1 ; ~p 0,t)= X ~x 2 S (~x 2,t;~0, 0; ~p 0 )G(~x 2,t; ~x 1 ) can be computed from the linear system of equations v M(v,v) 5 ( 0, 0; v; p,t)= 5 S ( v, t; 0, 0; p ) v0,t Fermion matrix so (~0, 0; ~x 1 ; ~p 0,t) is a sequential propagator based on a source S (~x 2,t;~0, 0; ~p 0 ) constructed from two ordinary propagators at time t

47 Sequential Source Technique First compute ordinary propagators G(x, 0)

48 Sequential Source Technique Construct sources S u;a a ( x 2,t; 0, 0; p ) or S d;a a ( x 2,t; 0, 0; p ) N(0) N(t, p )

49 Sequential Source Technique Compute sequential propagators via the second inversion ( 0, 0; x 1 ; p,t)= x 2 S ( x 2,t; 0, 0; p )S( x 2,t; x 1 ) v M(v,v) 5 ( 0, 0; v; p,t)= 5 S ( v, t; 0, 0; p ) v0,t τ N(0) N(t, p )

50 Sequential Source Technique Insert operator ( 0, 0; p,t)o( x 1, ) O(τ, q) N(0) N(t, p )

51 Sequential Source Technique Tie everything together with an ordinary propagator C (t, ; ~p 0, ~p) = X ~x 1 e i~q ~x 1 D Tr (~0, 0; ~p 0,t)O(~x 1, )G(~x 1, 0) E {U} O(τ, q) N(0, p) N(t, p )

52 Sequential Source Technique through the sink Advantages: Free choice of Momentum transfer O(τ, q) u, d Operator (vector/axial/tensor) N(0, p) N(t, p ) Ideal for Form Factors, Structure Functions, GPDs Disadvantages: Separate 3-pt inversion for each Quark flavour Hadron eg. p,,,, N! Polarisation Sink momentum

53 Sequential Source Technique Alternative method involves computing a sequential propagator through the operator 0 G O 0 G O 0 O 0 t

54 Sequential Source Technique through the operator Advantages: Free choice of Quark flavour Hadron e.g. p,,,, N! Polarisation Sink momentum Ideal for studying flavour dependence in a hadron multiplet Disadvantages: Separate 3-pt inversion for each Momentum transfer Operator (vector/axial/tensor)

55 Lattice 3pt Functions in Chroma proton S u;a0a (~x 2,t;~0, 0; ~p 0 )=e i~p 0 ~x abc a0 b 0 c 0 apple G d;bb0 (~x 2,t;~0, 0)G u;cc0 (~x 2,t;~0, 0) +Tr D [ G d;bb0 (~x 2,t;~0, 0)G u;cc0 (~x 2,t;~0, 0)] + G u;bb0 (~x 2,t;~0, 0) G d;cc0 (~x, t;~0, 0) + Tr D [ G u;bb0 (~x 2,t;~0, 0)] G d;cc0 (~x 2,t;~0, 0) /* "\bar u O u" insertion in NR proton, ie. * "(u Cg5 d) u" */ /* Some generic T */ // Use precomputed Cg5 q1_tmp = quark_propagators[0] * Cg5; q2_tmp = Cg5 * quark_propagators[1]; di_quark = quarkcontract24(q1_tmp, q2_tmp); simple_baryon_seqsrc_w.cc // First term src_prop_tmp = T * di_quark; // Now the second term src_prop_tmp += tracespin(di_quark) * T; // The third term... q1_tmp = q2_tmp * Cg5; q2_tmp = quark_propagators[0] * T; src_prop_tmp -= quarkcontract13(q1_tmp, q2_tmp) + transposespin(quarkcontract12(q2_tmp, q1_tmp)); END_CODE(); return projectbaryon(src_prop_tmp, forward_headers);

56 simple_baryon_seqsrc_w.cc //! Register all the factories success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL-NUCL_U"), barnuclnuclu); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL-NUCL_D"), barnuclnucld); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL_U_UNPOL"), barnucluunpol); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL_D_UNPOL"), barnucldunpol); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL_U_POL"), barnuclupol); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL_D_POL"), barnucldpol); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL_U_UNPOL_NONREL"), barnucluunpolnr); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL_D_UNPOL_NONREL"), barnucldunpolnr); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL_U_POL_NONREL"), barnuclupolnr); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL_D_POL_NONREL"), barnucldpolnr); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL_U_MIXED_NONREL"), barnuclumixednr); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL_D_MIXED_NONREL"), barnucldmixednr); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL_U_MIXED_NONREL_NEGPAR"), barnuclumixednrnegpar); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("NUCL_D_MIXED_NONREL_NEGPAR"), barnucldmixednrnegpar); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("XI_D_MIXED_NONREL"), barxidmixednr); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("DELTA-DELTA_U"), bardeltadeltau); success &= Chroma::TheWilsonHadronSeqSourceFactory::Instance().registerObject(string("DELTA-DELTA_D"),

57 Chroma xml for Sequential Source </elem> <elem> <annotation>; NUCL_U_UNPOL seqsource</annotation> <Name>SEQSOURCE</Name> <Frequency>1</Frequency> <Param> <version>1</version> <seq_src>nucl_u_unpol</seq_src> <t_sink>13</t_sink> <sink_mom>0 0 0</sink_mom> </Param> <PropSink> <version>5</version> <Sink> <version>2</version> <SinkType>SHELL_SINK</SinkType> <j_decay>3</j_decay> } <SmearingParam> <wvf_kind>gauge_inv_gaussian</wvf_kind> <wvf_param>2.0</wvf_param> sink <wvfintpar>5</wvfintpar> <no_smear_dir>3</no_smear_dir> </SmearingParam> </Sink> </PropSink> <NamedObject> <gauge_id>gauge</gauge_id> <prop_ids> <elem>sh_prop_1</elem> <elem>sh_prop_1</elem> </prop_ids> <seqsource_id>seqsource_nucl_u_unpol</seqsource_id> </NamedObject> </elem> u-quark in proton, unpolarised sink timeslice sink momentum smearing ordinary quark props required for construction of seq source (2 for u, 1 for d) tag (to be used as source for prop calculation)

58 Exercise Using the following interpolating operator +(x) = p 1 apple2 abc u Ta (x)c + d b (x) u c (x)+ u Ta (x)c + u b (x) d c (x) 3 perform the appropriate Wick contractions and write down the compare your result to the source implemented in Chroma + 3pt function and Work out the sequential sources required for N! h + (x 2 )j µ (x 1 )N(0) i

59 Extracting matrix elements Recall hadronic form of the nucleon 3pt function G(t,, ~p, ~p 0 )= X s,s 0 e E ~p 0 (t ) e E ~p (0) N(p 0,s 0 ) N(p 0,s 0 ) O(~q) N(p, s) Np,s) (0) Need to remove time dependence and wave function amplitudes Form a ratios with the nucleon 2pt function G 2 (t, ~p) = X s e E pt N(p, s) N(p, s) E.g. R(t, ; p, p; O) = G (t, ; p, p, O) G 2 (t, p ) G 2 (, p )G 2 (t, p )G 2 (t, p) G 2 (, p)g 2 (t, p)g 2 (t, p ) 1 2

60 Extracting matrix elements Using the relation for spinors ū(~p, 0 ) u(~p, )=Tr (E 4 i~p ~ + m) ~p ~s EM + i ~ ~s 5 m 0 We can write the two point function as G 2 (t, ~p) = X p Z snk (~p)q Z src (~p) Trū(~p, s) u(~p, s)[e Ept + e E 0 ~p (T t) ] 2E s ~p + v-spinor terms with opposite parity Use 4 = 1 to maximise overlap with positive parity forward propagating state 2 (1 + 4) q apple G 2 (t, ~p) = Z snk (~p)z src E~p + m E 0 (~p) e Ept ~p + m 0 + e E0 ~p (T t) E ~p E 0 ~p Similarly for the three-point function q G 3 (t, ; ~p 0 ~p;, O) = Z snk (~p 0 )Z src (~p)f (, F)e E ~p 0 (t where and F (, J )= 1 4 Tr 4 i ~p 0 ~ E ~p 0 + m E ~p 0 J 4 i ~p ~ + m E ~p E ~p hn(p 0,s 0 ) O(~q) N(p, s)i =ū(p 0,s 0 )J u(p, s) ) e E ~p

61 Example So our ratio determines R(t, ; p, p; O) = G (t, ; p, p, O) G 2 (t, p ) G 2 (, p )G 2 (t, p )G 2 (t, p) G 2 (, p)g 2 (t, p)g 2 (t, p ) 1 2 = s E ~p 0E ~p (E ~p + m)(e ~p + m) F (, J O(~q)) 0 t 1 2 T = F 1 (q 2 = 0) unpol = 1 2 (1 + 4), O = V 4 4, ~p 0 = ~p =0 F 1 (0)/ R(t = 16, ; p, p ; V 4 ) t/a

62 Other Useful Combinations R(t, ;~0, ~p; V 4, 4) =F 1 (q 2 E ~p M ) 2M F 2(q 2 )=G E (q 2 ) 1.6 q i R(t, ;~0, ~p; V i, 4) = i E + M G E(q 2 ) 1.4 R(t, ;~0, ~p; V i, j) = i ijk q k E + M G M (q 2 ) j = 1 2 (1 + 4)i 5 j Λ 0 V Exercise: Prove them! (0,0,0) (1,0,0) (1,1,0) (1,1,1) (2,0,0) t Certain combinations of parameters and kinematics give access to the form factors 0.2 It is possible to have several choices giving access to the form factors at a fixed Q 2 Overdetermined set of simultaneous equations that can be solved for F1, F2 or GE, GM

63 Typical Examples More detailed look at lattice results for form factors tomorrow F 1 u-d F u d 1 (Q 2 ) F 2 u-d F u d 2 (Q 2 ) k=0.134, V=16 3 k=0.135, V=16 3 k=0.1355, V=12 3 k=0.1355, V=16 3 k=0.1355, V=24 3 k=0.1359, V=12 3 k=0.1359, V=16 3 k=0.1359, V=24 3 k=0.1362, V=24 3 k= , V= Q 2 D F (Q 2 ) F π (Q 2 ) F π (Q 2 ) Q 2 [GeV 2 ] Q 2 D QCDSF: hep-lat/ Q 2 [GeV 2 ]

64 Some Recent Works [Not an exhaustive list] Nucleon Review: Ph. Hägler, QCDSF: ETMC: LHPC: RBC/UKQCD: CSSM: hep-lat/ Pion Mainz: PACS-CS: JLQCD/TWQCD: ETMC: RBC/UKQCD: QCDSF: hep-lat/