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

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

2 Lecture 3 Neutron beta decay Nucleon axial charge, ga Deep Inelastic Scattering Structure Functions Parton Model and Parton Distribution Functions Lattice techniques Moments of PDFs from lattice 3pt functions Renormalisation Results for the momentum fraction

3 n W - e - p Neutron beta decay n! pe e

4 Neutron beta decay Free neutrons are unstable 15 mins The most common way to study the weak interaction d d u n n! pe e p W - u u d e - The decay rate is proportional the matrix element of the weak V-A current hp(p 0,s 0 ) (V µ A µ ) n(p, s)i =ū p (p 0,s 0 ) µf 1 (q 2 µ q ) + i 2M f 2(q 2 ) + q µ 2M f 3(q 2 ) µ 5g 1 (q 2 µ q ) + i 2M 5 g 2 (q 2 ) + q µ 2M 5 g 3 (q 2 ) u n (p, s) Here the momentum transfer is so small that we only need to consider the f1 and g1 terms M p ' 1.3 MeV M n By convention, we call g V = f 1 (0) g A = g 1 (0) with gv=1 according to the conserved vector current (CVC) hypothesis Adler-Weisberger relation predicts ga=1.26

5 Neutron beta decay The decay rate for a neutron at rest and with spin in the ~s n direction final e - and e with velocities ~v e, ~v dr = G2 F V ud 2 + ~ve dp e d e d (2 ) 5 ~v + ~s n ~v e + ~s n ~v p 2 e (E max E e ) 2 with E max = M n M p ' 1.3 MeV = g 2 V +3g 2 A = 2(g A g V g 2 A) = g 2 V g 2 A = 2(g A g V + g 2 V ) so even without neutron polarisation, we can determine ga/gv through an accurate determination of the angular correlation between outgoing e - and e To determine the sign of ga spin-dependent measurement Current best determination (PDG 2012) ga/gv = (25)

6 Axial Form Factor The matrix element of the axial operator between neutron and proton states takes the general form apple hp(p 0,s 0 ) A µ (~q) n(p, s)i =ū p (p 0,s 0 µ ) 5 G A (q 2 ) + i µ q 5 2m G T (q 2 q µ ) + 5 2M G P (q 2 ) u n (p, s) axial FF induced pseudoscalar FF PCAC (Chiral µ A µ / m 2 m q! 0 0 vanishes if charge symmetry assumed, up=dn Not true for above matrix element, but if GP has a pion pole G P (q 2 )! 4Mf g NN (q 2 ) q 2 + m 2 The matrix element satisfies PCAC if MG A (q 2 )=f g NN (q 2 ) At q 2 =0 Goldberger-Treiman relation Mg A = f g NN

7 Axial Charge, ga The axial charge is defined as the value of the axial form factor at q 2 =0 g A = G A (q 2 = 0) Ideal quantity for a benchmark lattice calculation of nucleon structure Zero momentum Statistically clean Isovector hp ū µ 5 d ni = hp ū µ 5 u d µ 5 d pi Disconnected contributions cancel

8 Determination of ga on the Lattice Need access to the matrix element hn ū µ 5 d pi =ū(p 0,s 0 ) µ 5 u(p, s)g A from our three-point functions 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) From yesterday, we know that after the spin-trace, our 3pt will be proportional to F (, J )= 1 4 Tr 4 i ~p 0 ~ + m J 4 i ~p ~ + m E ~p 0 E ~p 0 E ~p E ~p For J = µ 5 we have F ( unpol, 4 5) =0 F ( unpol, i 5) =0 1 F ( pol, 4 5) = (E~p + m)~p 0 ~s +(E ~p 0 + m)~p ~s 2E ~p E ~p 0 i F ( pol, i 5) = (E~p +m)(e ~p 0 +m)~s+(~p 0 ~s)~p+(~p ~s)~p 0 2E ~p E ~p 0 (~p 0 ~p)~s i When using the projector pol = 1 2 (1 + 4)i 5 ~ ~s when computing the 3pt function Requires nucleon state to be polarised in, e.g. +z direction

9 Determination of ga on the Lattice At zero momentum, F ( pol, i 5) =2is i So ga can be determined by choosing the direction of the axial current to be the same as the direction of the nucleon polarisation. E.g. use a 3pt function with Our ratio from yesterday pol = 3 = 1 2 (1 + 4)i 5 3 R(t, ; ~p 0, ~p; O, )= G (t, ; ~p 0, ~p, O) G 2 (t, ~p 0 ) O = 3 5 apple G2 (, ~p 0 )G 2 (t, ~p 0 )G 2 (t, ~p) G 2 (, ~p)g 2 (t, ~p)g 2 (t, ~p 0 ) 1 2 now becomes R(t, ;~0,~0; 3 5, 3) = G 3(t, ;~0,~0, 3 5) G 2 (t,~0) = ig A

10 Determination of ga on the Lattice Example of R(t, ;~0,~0; 3 5, 3) = G 3(t, ;~0,~0, 3 5) G 2 (t,~0) = ig A from [RBC/UKQCD: ] at 4 different pion masses t 330 MeV 420 MeV 560 MeV 670 MeV

11 Determination of ga on the Lattice QCDSF: RBC/UKQCD: experiment g A ga m 2 [GeV 2 ] N f =2+1(2.7fm) N f =2+1(1.8fm) N f =2(1.9fm) N f =2+1 Mix(2.5fm) N f =2+1 Mix(3.5fm) N f =0(2.4fm) N f =0(3.6fm) m π 2 [GeV 2 ] Results appear to undershoot by ~10%

12 Determination of ga on the Lattice What about lattice systematic errors? Finite lattice spacing Large quark masses Finite volume Contamination from excited states

13 Determination of ga on the Lattice Lattice spacing dependence ga m 2 [GeV 2 ] Different colours correspond to different lattice spacings 0.06 fm <a<0.1fm No obvious dependence on a

14 Determination of ga on the Lattice Lattice volume dependence QCDSF: RBC/UKQCD: a fm g A (DWF) N f =2+1(2.7fm) N f =2+1(1.8fm) N f =2(1.9fm) N f =0(2.4fm) N f =2+1 Mix(2.5fm) ga m =767MeV m =590MeV m =399MeV m =278MeV m =180MeV m L Substantial finite size effects ga suppressed on a finite volume g A (Wilson) g A (qdwf) N f =2 Wilson =5.50 N f =2 Wilson =5.60 N f =2 Imp. Wilson =5.20 N f =2 Imp. Wilson =5.25 N f =2 Imp. Wilson =5.29 N f =2 Imp. Wilson =5.40 N f =0(3.6fm) N f =0(2.4fm) N f =0(1.8fm) N f =0(1.2fm) m L See [CSSM: ] for attempts to understand the source of this behaviour

15 Determination of ga on the Lattice Excited state contamination R(t, t s ) Mainz: a 0.08 fm, m 310 MeV t s = 16 t s = 14 t s = 12 t s = t [a] Test for contamination from excited states by varying the location of the sink Evidence that excited state contamination suppresses ga

16 Determination of ga on the Lattice Quark mass dependence QCDSF: hep-lat/ HBChPT form suggests that an enhancement is expected in the infinite volume at light quark masses g A =5.20 =5.25 =5.29 = m 2 [GeV 2 ]

17 Determination of ga on the Lattice The previous collection of results indicate that while ga was hoped to a simple quantity to compute on the Lattice, this appears to be far from the case due to Substantial finite size effects Possible excited state contamination Non-trivial quark mass dependence (interplay of Delta and N loops) But the removal of these effects all appear to shift ga in the right direction Increased interest from several lattice collaborations QCDSF LHPC PRD 74, (2006) PRL 96, (2006) RBC/UKQCD ETMC CLS/Mainz PRL 100, (2008) PRD 83, (2011) arxiv: [hep-lat]

18 Deep Inelastic Scattering

19 Deep Inelastic Scattering e, µ e, µ γ A single quark in the nucleon is knocked out by a virtual photon X The proton is smashed into many fragments p Allows the extraction of the quark and gluon distributions in momentum space - Feynman parton distributions As on Monday for elastic scattering, start with the S-Matrix S =(2 ) 4 4 (k + P P 0 k 0 )u(k 0 )( ie µ )u(k) Inclusive cross section with hadronic tensor d d de = 2 E 0 Q 4 E W µ = 1 4 `µ W µ i q 2 hx (ie)j µ P i Inclusive X hp J µ XihX J P i(2 ) 4 4 (P + q P X ) X

20 Deep Inelastic Scattering W µ = 1 4 Since the final states, X, are summed over, W only depends on initial proton momentum, P photon momentum, q X hp J µ XihX J P i(2 ) 4 4 (P + q P X ) X Using Lorentz symmetry, parity and time reversal invariance, current conservation, can express this in terms of two invariant tensors W µ = W 1 g µ + qµ q q 2 + W 2 M 2 P µ µ (P q) q q 2 P (P q) q q 2 W1 and W2 are the so-called structure functions of the proton and depend on two variables Q 2 = q 2 = P q M the 4-momentum transfer squared the energy transferred to the nucleon by the scattering electron

21 Deep Inelastic Scattering A key factor for investigating the proton substructure is the wavelength of the probe 1 p Q 2 Large momentum transfer e - e - e - e - high resolution Q 2 small Q 2 large resolve: proton quark

22 Deep Inelastic Scattering Early SLAC data showed that W1 and W2 are nearly independent of Q 2 when plotted as a function of the dimensionless combination x = q 2 2P q = Q2 2M the Bjorken scaling variable This is known as Bjorken scaling

23 Deep Inelastic Scattering Early SLAC data showed that W1 and W2 are nearly independent of Q 2 when plotted as a function of the dimensionless combination x = q 2 2P q = Q2 2M This is known as Bjorken scaling Essentially x and Q 2 degrees of freedom Some scaling violations at small-x

24 Deep Inelastic Scattering Early SLAC data showed that W1 and W2 are nearly independent of Q 2 when plotted as a function of the dimensionless combination x = q 2 2P q = Q2 2M This is known as Bjorken scaling Bjorken limit: x-fixed, Q 2!1 This lead Feynman to introduce the parton model inelastic e-p scattering is a sum of the elastic scatterings of the electron on free partons with the proton any particle with no internal structure Picture valid for a fast moving nucleon, as in DIS p γ Q 2 = p γ

25 Parton Model In the Bjorken limit, one defines the functions F 1 (x) = F 2 (x) = lim W 1(Q 2, ) Q 2!1 lim Q 2!1 M W 2(Q 2, ) And in Feynman s parton model, the structure functions are sums of the parton densities constituting the proton, fi F 1 (x) = 1 X e 2 i f i (x) 2 F 2 (x) =x X i fi is the probability that the struck parton, i, carries a fraction, x, of the proton momentum and is called a parton distribution function (PDF) i e 2 i f i (x) Total probability must be 1, so X i Z 1 0 dx xf i (x) =1

26 Deep Inelastic Scattering & Parton Model Results Rfrom DIS tell us that the fraction of the nucleon momentum carried by the quarks dx xq(x) is only about 50% q(x) =f q (x) gluons must play an important role in the structure of the nucleon In fact, much of our knowledge about QCD and the structure of the nucleon has been derived from Deep Inelastic Scattering experiments: 2 up and 1 down valence quarks with electric charge 2/3 and -1/3 in the proton the number of quarks is infinite because R dx q(x) does not seem to converge infinite number of quark and antiquark pairs

27 Parton Distribution Functions Point nucleon (a single quark carries all momentum): F2 is a delta function at x=1 F2(x) 1 x Nucleon with 3 quarks (share the momentum) F2(x) 1/3 x Three interacting quarks (smeared out) F2(x) x With sea quarks (when a quark emits a q-q pair, - they all have lower x than the original quark) F2(x) x

28 Parton Distribution Functions Proton contains 2 u and 1 d quarks termed valence quarks u v (x), d v (x) It is possible that a valence quark radiates a gluon which then turns into a q-q pair which are termed sea quarks u s (x), d s (x), s s (x) - Can now write the proton and neutron structure functions as (ignoring heavy quarks) F p 2 (x) =x 4 9 [up (x)+ū p (x)] [dp (x)+ d p (x)] [sp (x)+ s p (x)] F n 2 (x) =x 4 9 [un (x)+ū n (x)] [dn (x)+ d n (x)] [sn (x)+ s n (x)] where total PDF of a quark is q := q v + q s Under isospin flip u $ d and p $ n, assuming charge symmetry means u(x) u p (x) =d n (x) d(x) d p (x) =u n (x)

29 Parton Distribution Functions Further, we assume that (u,d,s) occur with equal probability in the sea To obtain S := u s =ū s = d s = d s = s s = s s F p 2 (x) =x 1 9 [4u v(x)+d v (x)] S(x) F n 2 (x) =x 1 9 [4d v(x)+u v (x)] S(x) Expect at low x 1 the sea quarks to dominate and F n 2 F p 2! 1 while at high x! 1 the valence quarks will dominate (and u v (x) >d v (x)) F n 2 F p 2! 1 4 [2 up vs 1 down valence quarks in the proton]

30 Parton Distribution Functions Further, we assume that (u,d,s) occur with equal probability in the sea S := u s =ū s = d s = d s = s s = s s To obtain F p 1 2 (x) =x 9 [4u v(x)+d v (x)] S(x) sea F n 2 (x) =x 1 9 [4d v(x)+u v (x)] S(x) Expect at low x 1 the sea quarks to dominate and F n 2 F p 2! 1 valence while at high x! 1 the valence quarks will dominate (and u v (x) >d v (x)) F n 2 F p 2! 1 4 [2 up vs 1 down valence quarks in the proton]

31 Parton Distribution Functions Recall the momentum sum rule including all partons X Z 1 dx xf i (x) =1 i 0 But e-p scattering experiments Z find the light quark contributions to be dxx[u(x)+ū(x)] 0.36 Z dxx[d(x)+ d(x)] 0.18 Almost half of the proton momentum is carried by electrically neutral partons Repeating the experiments with neutrinos indicates that these partons do not interact weakly Missing momentum carried by gluons Need for inclusion of gluons in the parton model also evidenced by scaling violations at finite Q 2

32 Structure Functions Much of our knowledge about QCD and the structure of the nucleon has been derived from Deep Inelastic Scattering experiments, e.g. ln! lx ( N! µ X) The cross section is determined by the structure functions: e, µ e, µ F1, F2 when summing over beam and target polarisations γ F3 when using neutrino beams (! W + ) g1, g2 when both the beam and target are suitably polarised h1 transversity - need Drell-Yan type processes p X

33 Moments of Structure Functions Relate partonic structure of hadrons and QCD via moments renormalisation scale Z 1 0 dxx n 2 F 2 (x, Q 2 )=EF S 2 ;v n (M 2 /Q 2,g S )vn S (M) + O(1/Q 2 ) {z } higher twist Wilson coefficients calculated in perturbation theory proton (forward) matrix elements nonperturbative quantities - compute on the lattice similar relations for other structure functions F 1 /F 2 /F 3 $ v n g 1 $ a n g 2 $ a n h 1 $ h n d n unpolarised polarised transversity

34 Moments of Structure Functions Moments are obtained from forward (q=0) matrix elements of local operators hn(p, s 0 ) O {µ 1 µ n } q N(p, s)i =2ū(p, s 0 )v n (q) p {µ1 p µ n} u(p, s) where {...} indicates symmetrisation of indices and the subtraction of traces twist-2 operators O µ 1 µ n q = q µ! 1 D µ2!d µ n q dominating contribution in the deep inelastic (large Q 2 ) limit! D = 1 2 (! D D) and similarly for the moments of polarised structure functions hn(p, s 0 ) O 5;{µ 1 µ n } q N(p, s)i =ū(p, s 0 ) a(q) n 1 n +1 s{µ 1 p µ2 p µ n} u(p, s) O 5;µ 1 µ n q = q µ 1 5! D µ2!d µ n q

35 Moments of PDFs Interpretation in terms of moments of parton distribution functions q(x) v (q) n = Z 1 0 dx x n 1 (q(x)+( 1) n q(x)) = hx n 1 i q q(x) ( q(x)) probability to find a quark (antiquark) with momentum fraction x Polarised: a (q) n =2 Z 1 0 dx x n ( q(x)+( 1) n q(x)) = 2hx n i q with q(x) =q + (x) q (x) and q + (x) (q (x)) probability of finding a quark with momentum fraction x and helicity equal (opposite) to that of the proton In particular 1 2 a(q) 0 = h1i q = q is the fraction of the nucleon spin carried by quarks of flavour q and g A = u d

36 Moments of Polarised Structure Functions The moments of the polarised structure functions are 2 2 Z 1 0 Z 1 0 dxx n g 1 (x, Q 2 )= 1 2 dxx n g 2 (x, Q 2 )= 1 2 X f=u,d n n +1 e (f) 1,n ( µ2 Q 2,g(µ))a(f) n (µ) X f=u,d (f) e 2,n ( µ2 Q 2,g(µ))d(f) n (µ) e (f) 1,n ( µ2,g(µ)) a(f) Q2 n (µ) + higher twist + higher twist twist-3 but not power suppressed In addition, the moments of transversity h(x) are related to matrix elements of the operators! O ;µ µ 1 µ n q = i 2 qi µ D µ1!d µ n probability weighted by quark transverse-spin projection relative to the nucleon s transverse-spin direction q Lowest moment gives the tensor charge q

37 Operators Minkowski Euclidean - replace the Lorentz group by the orthogonal group O(4) Discrete space-time - reduce to the hypercubic group H(4) O(4) H(4) is finite mixings [hep-lat/ ] Using the following operators reduces mixings O v2a = O {14} O v2b = O {44} 1 O {11} + O {22} + O {33} 3 O v3 = O {114} 1 O {224} + O {334} 2 O v4 = O {1144} + O {2233} O {1133} O {2244} v2a and v2b different representation of the same continuum operators

38 Extracting Moments Recall from yesterday, we can write the lattice three-point function as where and G 3 (t, ; ~p 0 ~p;, O) = so using the operator for v2a as an example i 4 hn(p, s0 ) q Euclideanisation M 0 = E 4, i 4 hn(p, s0 ) q F (, J )= 1 4 Tr q Z snk (~p 0 )Z src (~p)f (, J )e E ~p 0 (t 4 i ~p 0 ~ E ~p 0 + m E ~p 0 J hn(p 0,s 0 ) O(~q) N(p, s)i =ū(p 0,s 0 )J u(p, s) M 0! D 1 + M 1 O M v 2a 4 i ~p ~ + m E ~p E ~p = O M {01} = i 4 q M 0 ) e E ~p! D 1 + M 1! D 0 q N(p, s)i = v (q) 1 2 2ū(p, s0 M ) 0 p M p 0 u(p, s)! D 0 q M i = i E i p E 4 = ip M 0 ie(~p), p E i = p M i D 4 = id (M)0 D i = D (M)i E 4! D 1 + E 1! D 0 q N(p, s)i = v (q) 2 1 2ū(p, s0 ) E 4 p 1 i E 1 E N (~p) u(p, s)

39 Extracting Moments Taking = unpol 1 2 (1 + 4) F (, J )= 1 4 Tr unpol 4 i ~p ~ E ~p + m E ~p 4p 1 i 1 E N (~p) 4 i ~p ~ E ~p + m E ~p So in this case our ratio will be R(t, ; ~p, ~p; O {14}, unpol) = G unpol (t, ; ~p, ~p, O {14}) G 2 (t, ~p) = ip 1 v (q) 2 and for v2b R 4 (t, ; p, p; O 44 )= E2 p p2 E p x Exercise: work out the corresponding ratio for the polarised case O 5;{43} q pol = 3 = 1 2 (1 + 4)i 5 3

40 Ratios for v 2 = hxi Excellent agreement for the two different representations of the same operator

41 Operator Renormalisation A huge field in it s own right and deserves its own set of lectures. (see e.g. R.Sommer [hep-lat/ ]) Renormalise bare lattice operators in scheme S and at scale M If there are more operators with O S (M) =Z S O(M)O bare same quantum numbers same or lower dimension O S i (M) = X j Z S O i O j (M,a)O j (a) Renormalisation Group Invariant quantities are defined as Independent of scale [ Z S O(M)] O RGI = Z RGI O O bare = Z MS O (µ)o MS (µ) 1 = 2b 0 g S (M) 2 d 0 2b 0 ( Z g S (M) exp d 0 = Z MOM O = Z O(a)O(a) apple S ( ) S ( ) + d 0 b 0 (p)o MOM (p) )

42 Operator Renormalisation [ΔZ v2 ] 1 One loop Two loop Three loop [ΔZ vn ] [ΔZ v4 ] µ / Λ MS [n f =0]

43 Operator Renormalisation Perturbative renormalisation: Regard the lattice as a scheme One loop perturbation theory Z S O(M,g) =1 Non perturbative renormalisation: g C F O;0 ln(m)+b S O +... Schrödinger functional [ALPHA, hep-lat/ ] - SF scheme Gauge & quark fields take on specific values at the boundary of the space-time region (Dirichlet boundary conditions) -- a background field Rome-Southampton Method [Martinelli et al., hep-lat/ ] Mimics (continuum) perturbation theory in a (RI )-MOM scheme

44 Operator Renormalisation 3.6 Z RI0 MOM O (p)z RI0 MOM O (p, g 0 ) {5} Z RGI (GeV 2 )

45 x hep-lat/ [Detmold, Melnitchouk, Thomas] x 1 u d m 2 Π GeV 2 First moment of the (isovector) nucleon parton distribution function 1 x u µ d = 0 dx x(u(x, µ) d(x, µ)) + Notorious for producing lattice results 2x too large for isovector nucleon What are the possible systematic errors that could account for this 1 0 dx x(ū(x, µ) d(x, µ)) Quenching? Chiral physics? Finite volume effects?

46 x hep-lat/ [Detmold, Melnitchouk, Thomas] x 1 u d m Π 2 GeV 2 QCDSF (Nf=2) (2 GeV ) hxi MS (u d) m 2 [GeV 2 ]

47 Excited State Contamination? QCDSF a 0.75 fm, m 650 MeV Evidence for severe excited state contamination! -1 t sink =11 t sink =13 t sink =19 t sink = t

48 However ratios of lattice results look good, e.g. RBC/UKQCD PRD 82, (2010) experiment x u-d x u- d 2.7 fm 1.8 fm m π [GeV ]