PDF from Hadronic Tensor on the Lattice and Connected Sea Evolution

Transcription

1 PDF from Hadronic Tensor on the Lattice and Connected Sea Evolution Path-integral Formulation of Hadronic Tensor in DIS Parton Degrees of Freedom Numerical Challenges Evolution of Connected Sea Partons Quasi-PDF χ QCD Collaboration JLab, Apr. 6, 017

2 Experimental Data l New Muon Collaboration (NMC PRL 66, 71 (1991)) µ+ p(n) à µx x P n 1 µ µ F ( x, Q ) F ( x, Q ) SG( x0, x1; Q ) = dx x x0 Quark parton model + Isospin symmetry 1 G P P G S (0,1; Q ) = + dx ( u ( x) d ( x)); S (0,1; Q ) = (Gottfried Sum Rule) NMC : S G (0,1;4 GeV ) = 0.40 ± (5 σ from GSR) d / u l asymmetry from Drell-Yan Production (PRL 69, 176 (199)) l NuTeV experiment (PRL 88, (00))? sin θw(3 σ from Standard Model) s(x) s(x)

3 Perturbative QCD: Higher order effects on l N Sullivan Process: Possible Explanation is small. l Need a non-perturbative formulation to reveal u d P P in QCD and a scheme to calculate it quantitatively Euclidean path-integral formalism and lattice gauge calculation. g uu = g dd u P π ; K d P N, Δ; Λ + P n+ π ( ud) P ++ Δ + π ( du) + P Λ+ K ( us )

4 Hadronic Tensor in Euclidean Path-Integral Formalism Deep inelastic scattering In Minkowski space d σ de 'dω = α q 4 (E ' E )l µν W µν W µν (! q,! p,ν) = 1 π ImT µν = N(! p) = 1 n n i=1 d 4 x e iq x J µ (x)j ν (0) N( p)! spin avg 4π d 3 p i (π ) 3 (π ) 3 δ 4 (p n p q) < N( p)! J µ n >< n J ν N( p)! > spin avg E pi l Euclidean path-integral J em µ (t 1 ) J em ν (t ) KFL and S.J. Dong, PRL 7, 1790 (1994) KFL, PRD 6, (000) 0 t 0 t (t t 1 ) 4

5 W µν in Euclidean Space! W µν ( " q, " p,τ = t t 1 ) = t t >>1/ΔE P, t 1 >>1/ΔE P E P M N Tr < Γ e χ N ( " p,t) " x 1 " 4π e i Tr < Γ e χ N ( " p,t)χ N ( " p,0) > q "x J µ ( " x,t )J ν (0,t 1 )χ N ( " p,0) > = 1 ( m N ) δ " 4π E pn p " q " < N( p) " J n >< n J N( p) " > e ( En EP )τ µ ν spin avg n n = < N( " p) " x e i" q " x 4π J µ ( " x,τ )J ν (0,0) N( " p) > spin avg Laplace transform W µν (! q,! p,ν) = 1 i c+i c i dτ e ντ " W µν (! q,! p,τ ) 5

6 q q q = V + CS CS Q Q Q q q = (?) Q DS Q t t1 t t t 1 1 Q t q DS 0 0 t ( a ) ( ) t 0 t 0 t Cat s ears diagrams are suppressed by O(1/Q ). 0 (c) b t ( c) t + 6

7 l W µν (p,q) = W 1 (q,ν)(g µν q µq ν ) + W q (q,ν)(p µ p q q q µ )(p ν p q q q ν ) Bjorken limits νw (q,ν) F (x,q ) = x e i (q i (x,q ) + q i ( x,q )); x = Q p q i l Parton degrees of freedom: valence, connected sea and disconnected sea u d s u V (x) + u CS (x) d V (x) + d CS (x) u CS (x) d CS (x) u DS (x) + u DS (x) d DS (x) + d DS (x) s DS (x) + s DS (x) 7

8 Properties of this separation No renormalization Gauge invariant Topologically distinct as far as the quark lines are concerned W 1 (x, Q ) and W (x, Q ) are frame independent. Small x behavior of CS and DS are different. q V, q CS, q CS ~ x 0 x α R (x 1/ ) q DS, q DS ~ x 0 x 1 Short distance expansion (Taylor expansion) OPE 8

9 l Note that diagram (b) are from pre-existing connected sea antipartons the same way as in (c) which involves pre-existing disconnected sea partons and antipartons. t 1 t t 1 t l Whereas, current induced pair productions are suppressed as O( q / p ). p q q p t δ (p q + p o ) 9

10 Operator Product Expansion -> Taylor Expansion Operator product expansion 1 W = ImT n Dispersion relation T µν µν µν π / π Q M N ν ' ν n Expand in the unphysical region = 1 ν dν ' ν ' Wµν ( q, ν ') MNν p q = < 1 (x > 1) Q Q 10

11 Euclidean path-integral n Consider W µν (q,τ ) (a) n Short-distance expansion ( ) n Laplace transform D[A]det M (A) e S g t τ γ ν γ µ t 0 t Tr...M 1 (t,t ) d 3 x e i q x iγ µ M 1 (t,t τ )iγ ν M 1 (t τ,0)... M 1 (t,t τ ) 1 free quark 4π x,τ 0 from q,ν x + τ ; M 1 x (t τ,0) e D+τ D τ M 1 (t,0) x,τ 0 iπ (q + id) W µν (q,ν) Tr...M 1 (t,t )iγ µ q + id δ (ν + D τ q + id )iγ ν M 1 (t,0)... 11

12 Dispersion relation Expansion about the unphysical region ( q p / Q < 1 ) l T µν (q,ν) = 1 dν ' ν 'W µν (q,ν ' D τ ) π, Q / M v' (ν + D τ ) N +D τ i(q + id) Tr...M 1 (t,t )iγ µ (Q + iq D D ) iγ ν M 1 (t,0)..., where τ = it and D t = id τ so that D = ( D, id t ) is covariant derivative in Minkowski space. ( q p) T µν (q V + q CS ) = n e f 8 p µ p ν A n f (Q ) n 1 f (CI) g µν n= A f n =? even + odd n terms t O f n 0 t n= ( q p) n A n (Q ) n f (CI) A n f (CI) D[A]det M (A) e S g Tr...M 1 (t,t )O n f M 1 (t,0)... O f n = iγ µ1 ( i )n 1 D µ Dµ3... D µn, < p ψ O fn ψ p >= A f n (CI) p µ1 p µ...p µn 1

13 Similarly for except with q CS q q t 1 t T µν (q CS ) = n n...a f (CI)...A f (CI) even, n= even odd odd, n=3 t l For q DS / q DS T µν (q DS / q DS ) = n n...a f (DI) ±...A f (DI) even, n= odd, n=3 l DIS with electromagnetic currents J µ em T µν = T µν (q V + q CS ) + T µν (q CS ) + T µν (q DS ) + T µν (q DS ), n =...[A f (CI) + A n f (DI)] even, n= 13

14 q = q V + q CS q CS q DS = (?) q DS Q Q t 1 t Q Q t 1 t Q t 1 t Q 0 (a) t 0 t (b) 0 t (c) t O f n O f n O f n t 0 t (a') 0 t (b') 0 t (c') 14

15 Gottfried Sum Rule Violation S G (0,1;Q ) = NMC: Q 1 0 dx (u P (x) d P (x)); S G (0,1;Q ) = 1 (Gottfried Sum Rule) 3 S G (0,1;4 GeV ) = 0.40 ± (5σ from GSR) Q t 1 t Q Q t 1 t 0 (c) t 0 t two flavor traces ( u DS = d DS ) one flavor trace ( u CS d CS ) K.F. Liu and S.J. Dong, PRL 7, 1790 (1994) 1 1 Sum = + dx ( ucs ( x) dcs ( x)), = + nu n + O α CS dcs 3 3 (1 ( s)) 15

16 Comments The results are the same as derived from the conventional operator product expansion. The OPE turns out to be Taylor expansion of functions in the path-integral formalism. Contrary to conventional OPE, the path-integral formalism admits separation of CI and DI. n For O f with definite n, there is only one CI and one DI in the three-point function, i.e. (a ) is the same as (b ). Thus, one cannot separate quark contribution from that of antiquark in matrix elements. 16

17 Quark Parton Model ν ν I n = dν 1 πi ν T 1 ( Q, ν ), I n n = n 1 M N = 8 ef f Q A l A n=even f (CI) M n f (CI) = l A n=odd f (CI) M n f (CI) = l A n=even f (DI) M n f (DI) = n f Q = 8 M N Q dνm N πi n 1 dx x n 1 (q V (x) + q CS (x) + q CS (x)) f dx x n 1 q V (x) f 1 0 dx x n 1 (q DS (x) + q DS (x)) f 1 0 i ν n 1 W (Q,ν), dx x n M N νw (Q,ν) 4 17

18 3) Fitting of experimental data K.F. Liu, PRD (000) u d x 1/ x 0 O.K. But u + d s is not correct. A better fit u(x) + d (x) = f s (x) + CS(x), f 1 where CS(x) x 1/ x 0 like in u(x) d (x) 4) Unlike DS, CS evolves the same way as the valence

19 How to Extract Connected Sea Partons? K.F. Liu, W.C. Chang, H.Y. Cheng, J.C. Peng, PRL 109, 500 (01) Q =.5 GeV 1 xd ( + u) CS ( x) = xd ( + u)( x) xs ( + s)( x); R R x s = (lattice) : x ( DI) u CT10 lattice expt 19

20 q V, q CS, q CS ~ x 0 x α R (x 1/ ) q DS, q DS ~ x 0 x 1 0

21 Lattice input to global fitting of PDF <x> s <x> u/d (DI) Lattice calculation with overlap fermion on 3 lattices including on at sea m π ~ 140 MeV (Mingyang Sun, χ QCD Collaboration) Data = a + bm π,vv + cm π,vs 3 + dm π,vs + ea + fe m π,vvl <x> s = 0.050(16), <x> u/d (DI)=0.060(17) <x> s /<x> u/d (DI) = 0.83(7) Q = GeV 1

22 Lattice input to global fitting of PDF <x> s /<x> u/d (DI) = 0.83(7) Q = GeV

23 Operator Mixing Connected insertion d M f n (CI) d logq Disconnected insertion = a n f 1 b 0 log(q / Λ ) M n f (CI) d M n f (DI) = 1 1 d logq b 0 log(q / Λ ) a n qqm n f (CI) + 1+ ( )n a n n qg M G 3

24 Evolution Equations NNLO S. Moch et al., hep/040319, A. Cafarella et al., dq / dt = ( P q + P q ) + P g; i ik k ik k ig k dq / dt = ( P q + P q ) + P g; i k ik k dg / dt = ( P q + P q ) + P g. k ik gk k gk k gg dq i / dt = P qq q i + P s ns Σ N v ; f where q i q i q i, Σ v (q k q k ), and P ns s O(α s 3 ) k k ig Valence u can evolve into valence d? Note: q i = q i v+cs q i cs + q i ds q i ds q i v + q i ds q i ds 4

25 Evolution equations separating CS from the DS partons: 11 equations for the general case u val, d val, u cs u cs, d cs d cs, u ds u ds, d ds d ds, s ds s ds, g dq i v+cs / dt = P ii c q i v+cs + P ii c q i cs ; K.F. Liu dq i cs / dt = P ii c q i cs + P ii c q i v+cs ; dq ds i / dt = (P cd ik q ds k + P cd ik q ds k + P d ik q v+cs k + P d ik q cs k ) + P ig g; k dq ds i / dt = (P cd ik q ds k + P cd ik q ds k + P d ik q v+cs k + P d ik q cs k ) + P ig g; k dg / dt = [P gk (q v+cs k + q ds k ) + P gk (q cs k + q ds k ) + P gg g. k 5

26 Comments CS and DS are explicitly separated, leading to more equations (11 vs 7) which can accommodate ds ds s s, u u There is no flavor-changing evolution of the valence partons. dq i / dt = P qq q i + P ds (q k q k ); is the sum of two equations dq i v / dt = P qq q i v, q v q v+cs q cs k d(q i ds q i ds ) / dt = P ik cd (q k ds q k ds ) + k k P ds d q k v Once the CS is separated at one Q, it will remain separated at other Q. Gluons can split into DS, but not to valence and CS. It is necessary to separate out CS from DS when quark and antiquark annihilation (higher twist) is included in the evolution eqs. (Annihilation involves only DS.) 6

27 Improved Maximum Entropy Method Inverse problem D(τ ) = K(τ,ν)ρ(ν)dν, Bayes theorem D(τ ) =! W µν (τ ), K(τ,ν) = e ντ, ρ(ν) = W µν (q,ν) P[ ρ D] = P [ D ρ ]P[ ρ] P[ D] ρ(ν) [ ] = 0 ρ Maximum entropy method: find P ρ D from Improved MEM (Burnier and Rothkpf, PRL 111, (013)) P[ ρ D] e αs L γ (L N τ ), L = χ S = dν 1 ρ(ν) m(ν) ln ρ(ν) m(ν) 7

28 e + e ρ Frank X. Lee 8

29 Numerical Challenges Lattice calculation of the hadronic tensor no renormalization, take continuum and chiral limits, direct comparison with expts PDF. Bjorken x x = Q p q =! q ν (ve p! p! q) Range of x: Q = GeV! q "! p! p = 3 GeV,! q = 3 GeV, x = 0.058! p = 0,! q = GeV x =

30 Comments The connected sea partons (CSP) found in path-integral formulation are extracted by combining PDF, experimental data and ratio of lattice matrix elements. It would be better to have separate evolution equations for the CSP and DSP. The separation will remain at different Q. This way one can facilitate the comparison with lattice calculation of moments in the CI and DI to the corresponding moments from PDF. Lattice calculation of hadronic tensor is numerically tough, but theoretically interpretation is relatively easy. No renormalization is needed and it can be calculated in the rest and low-momentum frames.

31 Large Momentum Approach X. Ji, PRL, 110, 600 (013) 31

32 Theoretical Issues Relatively simple numerically (H.W. Lin et al., ; C. Alexandrou et al., ) Renormalization of quasi-distribution (LaMET) 1 0!q(x,µ, P z ) = dy y Z( x y, µ P z ) q( y,µ Perturbative and non-perturbatice lattice renormalization Linear divergence of the Wilson line (X. Xiong, X. Ji, Z.H. Zhang, Y. Zhao, ; T. Ishikawa, Y.Q. Ma, J.W. Qiu, S. Yoshida, ) How large P Z needs to be? ) + O( Λ P z, M P z ) 3

33 Quasi-PDF u(x) d(x) 33

34 u 1 d p=1.8 GeV -0.5 p=1. GeV x 0 p=1.8 GeV -0.5 p=1. GeV x.5 u-d p=1.8 GeV -0.5 p=1. GeV x q(x) = f ( x) 3 3 x 64 lattice at a = 0.06 fm Clover on DWF configurations m π (val) = 500 MeV, m π (sea) = 400 MeV 34

35 35

36 36

37 Strange quark magnetic moment Parity-violating ep scattering with radiative correction R. Sufian et al, PRL editor s choice Nature Ross Young Global Analysis (Q =0.1 GeV ), J. Liu et al, 007 Global Analysis (Q =0.1 GeV ), R. Jimenez et al, 014 D. Leinweber et al, 000 D. Leinweber et al, 005 P. Shanahan et al, 015 S. J. Dong et al, 1998 T. Doi et al, 009 J. Green et al, 015 R. Sufian et al, ( χqcd), 016 R. Sufian et al, ( χqcd) (Q =0.1 GeV ), G s M (0) G MS (0) = (14)(9) µ N 37 37

38 Glue Spin Y. Yang et al, PRL 118, (017), Editor s choice Physics ViewPoint: Steven Bass 38 38

39 Le Taureau of Pablo Picasso (1945) 5 th stage 11 th stage Dynamical chiral fermion Quenched approximation Physical pion mass Continuum limit Infinite volume limit 39

40 Summary Formulation of the hadronic tensor in Euclidean pathintegral has revealed the connected sea parton (CSP) dof. It takes experiments, lattice calculation and global fitting of PDF to extract CSP. It is better to have CSP and DSP parton separated in evolution. This would facilitate comparison with lattice calculation of moments. Lattice calculation of hadronic tensor is numerically tough, but theoretically interpretation is relatively easy. No renormalization is needed and it can be calculated in the rest frame. Progress made with large momentum quasi-pdf. Both hadronic tensor and quasi-pdf approaches should be pursued and checked with experiments. 40

41 41

42 Negative q(x) puzzle f d x) x x -1.0 d and d from CTEQ6 (JW Chen) d( x ) = d( x ) Larger P z? (How large) Lattice scale (a -1 ~ GeV) too small? Range of x limited? present P z ~ 1 GeV H.W. Lin, C. Alexandrou,