Aspects of Light-Front Hadron Physics Where DSEs and Lattice-QCD Meet

Transcription

1 Aspects of Light-Front Hadron Physics Where DSEs and Lattice-QCD Meet Peter C. Tandy Dept of Physics Kent State University USA 1

2 Topics Parton Distribution Amplitudes (pion, kaon). Close contact with lattice-qcd moments. Applications to high energy exclusive form factors, ultraviolet behavior.! Parton Distribution Functions (pion). A work in progress.! X. Ji s space-like correlator approach to PDFs -a model investigation. 2

3 Expect: qualitatively new insight where other methods can t, eg high Q^2 Do not expect: final, precision-qcd results 3

4 DSE Modeling of Hadron Physics Most common: Rainbow-ladder truncation of QCD s eqns of motion. Approximation to full BSE kernel now being utilized.! Constrain modeling by preserving AV-Ward-Takahashi Id, V-WTI. [Color singlet]! Naturally implements DCSB, conserved vector current, Goldstone Thm, PCAC! RL truncation only good for ground state vector & pseudoscalar mesons, q-qq descriptions of baryons with AV and S diquarks.! At the very least: DSE continuum QCD modeling suited for surveying the landscape quickly from large to small scales; finding out which underlying mechanisms are dominant. Applicable to all scales, high Q^2 form factors, etc! Unifying DSE treatment of light front quantities (PDFs, GPDs, DA) with other aspects of hadron structure: masses, decays, charge form factors, transition form factors...! Pion & kaon q-qbar Bethe-Salpeter wavefn is very well known 4

5 Landau gauge only 1 phen parameter [fit : m, m K, f ], f K (2%) An Ansatz for the FULL QCD kernel: L. Chang, C.D. Roberts, PRL103, (2009), + S. Qin (2015). A more modern RL kernel: S. Qin, L. Chang, C.D. Roberts, D.J. Wilson, PRC84, (2011). 5

6 Modern Context for Rainbow-Ladder Kernel Landau gauge, lattice QCD gluon propagator, I.L.Bogolubisky etal., PosLAT2007, 290 (2007) inflexion => confining Spectral Density Identified enough stength for physical DCSB m G (k 2 ) m G (0) 0.38 GeV K RL BSE = 4 ˆe (q 2 ) m 2 G (q2 )+q 2 ˆe (0.1) 3 4 BSE kernel from ab initio gauge sector DSE work now agrees satisfactorily with the kernel from fitting data: Binosi, Chang, Papavassiliou, Roberts, PLB742, 183 (2015) 6

7 Parton Distribution Amplitudes 7

8 Fit numerical DSE-BSE solns to Nakanishi forms to allow analytic Feyn Integral Methods EG: (q 2, q P) = 5 E (q 2, q P)+P F (..)+q q PG (..)+ : qp H (..) Use Nakanishi Repn (or PTIR) (1965) :- F(q 2 ; q P) = Z 1 1 d Z 1 0 d F = E, F, G, or H IR ( ; ) (q 2 + q P + 2 ) + UV ( ; ) m+n (q 2 + q P + 2 ) n IR( ; ) 1( ) ( IR 1 ) + 3 npqcd info is in the variables and constants that are not momenta---wick rotation is trivial as in pert thy. Works for u-, d-, s-, c-, b-quarks. Also for lattice-qcd propagators. N. Souchlas, PhD thesis KSU, (2009), J. Phys. G37, (2010) 8

9 Pion Distribution Amplitude (leading twist) f (x) = d 2 e ixp.n 0 q (0) 5 n q( n) (P) k P k P f x m = Z 2N c P n tr k ( k n P n )m 5 n [ S(k) (k P ; P) S(k P)] 2 BS Wavefn µ = 2GeV DSE RL DSE beyond RL asym (x) = (x; µ!1) Broadening of PDA is an expression of DCSB ---long sought after in LF QFT 9

10 Pion Distribution Amplitude ERBL (~1980): (x; µ) =6x(1 x) 1 + n=2,4 a n (µ) C 3/2 n (2x 1) h s (µ 0 ) i (0) n a n (µ) =a n (µ 0 ) / 0 s (µ) Evolution to higher scales is EXTREMELY SLOW Not much change up to LHC energy Conformal limit: a n (µ!1)=0 Efficient representation of DSE results: (x; µ) =N x (1 x) n=2 ã n (µ) C +1/2 n (2x 1) K(x; µ) =N x (1 x) { 1 + n=2,4 ã n (µ) C +1/2 n (2x 1) } + N x (1 x) { n=1,3 ã n (µ) C +1/2 n (2x 1) } 10

11 Low Order Truncation of ERBL-Gegenbauer Expn of PDA (x; µ) =6x(1 x) 1 + n=2,4 a n (µ) C 3/2 n (2x 1) DSE soln {{0, 1.}, {2, }, {4, }, {6, }, {8, }, {10, }, {12, }, {14, }, {16, }, {18, }, {20, }, {22, }, {24, }, {26, }, {28, }, {30, }, {32, }, {34, }, {36, }, {38, }, {40, }} 2% 10% φ π (x) µ = 2 GeV DSE-RL result, C n 3/2 (2x-1) projection Project DSE, stop at a 2 Project DSE, stop at a x A double-humped PDA is almost ruled out by V. Braun, I. Filyanov, Z. Phys. C44, 157 (1989) QCDSR (x = 1/2; µ = 2) =1.2 ±

12 One Lattice-QCD Moment Almost Determines Pion DA DSE beyond RL asym (x) = (x; µ!1) LQCD (x; µ = 2) =Nx (1 x) = µ = 2GeV h(2x 1) 2 i LQCD µ=2 = 0.27 ± 0.04 V. Braun et al., PRD74, (2006) DSE RL Lattice-QCD 12

13 Pion Distribution Amplitude h (2x 1) 2 i LQCD µ=2 GeV = (41)(39) V. Braun et al., arxiv: [hep=lat] DSE prediction:

14 Other Meson Distribution Amplitudes DSE analysis of LQCD moments: Segovia, Chang, Cloet, Roberts, Schmidt, Zong PLB731, 13, (2014) K and K* pion DAs of light quark mesons look much the same--with small flavor breaking 14

15 Kaon Distribution Amplitude Size of SU(2)xSU(3) spin-flavor symmetry-breaking? C. Shi, L. Chang, C.D. Roberts, S.Schmidt, PCT, H-S. Zong, PLB738, 512 (2014) 15

16 Kaon Distribution Amplitude µ = 2GeV skewness implies only 14% flavor symm breaking due to DCSB n ms m u m s + m u o2gev 66% C. Shi, L. Chang, C.D. Roberts, S.Schmidt, PCT, H-S. Zong, PLB738, 512 (2014) 16

17 Kaon DA Moments µ = 2 GeV DSE-QCD: Lattice-QCD: QCD Sum Rules: Shi Chao, L. Chang, C.D. Roberts, P.C. Tandy, PLB738, 512 (2014) 17

18 F (Q 2 = uv) = Z 1 0 The Pion Charge Form Factor: Transition from npqcd to pqcd The Pion Charge Form Factor: dx Z 1 0! Transition from npqcd to pqcd dy? (x; Q) [T H (x, y; Q 2 )] (y; Q) + NLO/higher twist.. ---LFQCD, Brodsky, LePage PRD (1980) Q 2 >> 2 QCD : Q 2 F (Q 2 ) 16 f 2 s(q 2 ) 2 (Q 2 ) + O(1/Q 2 ) at Q GeV 2, 0.1 JLab expt, Theory 0.45 (Q 2 ) = 1 3 1, Q dx (x; Q) x But, recent DSE theory (x; µ = 2 GeV) 2 =

19 UV-QCD is not Asymptotic QCD Q 2 >> 2 QCD : Q 2 F (Q 2 ) 16 f 2 s(q 2 ) 2 (Q 2 ) + O(1/Q 2 ) 1 TeV 19

20 0.5 VMD ρ pole DSE 2000 DSE 2013 Q 2 F π (Q 2 ) [GeV 2 ] JLab 2001,6,8 CERN '80s uv-qcd φ π 2 GeV (x) uv-qcd φ π 10 GeV (x) 0.1 JLab 12 GeV confm-qcd φ π asym (x) Q 2 [GeV 2 ] Jab data: G. Huber et al., PRC78, (2008) 20

21 Pion Form Factor: Running q Mass Fn Effect With constituent M q Q 2 F π (Q 2 ) [GeV 2 ] JLab 2001, 6, 8 With dynamical M q (p 2 ) 60% reduction Q 2 [GeV 2 ] Jab data: G. Huber et al., PRC78, (2008) 21

22 Pion Parton Distribution Functions 22

23 The Leading Order PDF q f (x) = 1 4 d e ixp n (P) f( n) n f (0) (P) c RL DSE: q(x) From Directly Obtained Moments hx m i RL v = N c 2P n tr Z ` (` P 2 ) [( ` n P n )m (` P 2 ) S(` P) Method can easily exceed the Lattice QCD practical limit : m = 3 23

24 Estimate 1-Pion Loop Contribution to Pion PDF + : x 1 µ = 0 1 dx x {u + ū sea + d + d sea + g(x)} 2 xq v (x) + 4 xq sea (x) + xg(x) = 1 u = u v + u sea, d = dv + d sea Empirical GRS/ASV universal q v (x), q sea (x) at µ = GeV = 1 2 RL q q + q q CPT: 18% effect r 2 ch =(1 2 ) r 2 RL + 2 r 2 lp DSE-RL: r 2 RL = r 2 ch 2 = 18% PDF Consequence: q v (x) =(1 2 ) q RL (x)+qv lp (x) lp with qv (x) = 2 =

25 Analysis of Pion Parton Momentum Sum Rule Modern empirical expt parameterization: Aicher, Shafer, Vogelsang, (ASV) PRL 105, (2010) Q 0 = GeV u v (x) DSE-BSE DSE-BSE w/ Pion Contribution ASV x K. Khitrin, P. Tandy, in progress (2015) 25

26 Many Moments via Feyn PTIR--Easy Modern empirical expt parameterization: Aicher, Shafer, Vogelsang, (ASV) PRL 105, (2010) RL DSE RL DSE + pion loop ASVQ 0 = GeV Q 0 = GeV <x m > m 26

27 Spacelike Correlator Approximation for PDFs 27

28 To help lattice-qcd be more applicable to hadron PDFs and GPDs than just the first 3 moments? Standard light-cone correlator, leading twist: q f (x) = 1 4 Z d e ix P n h (P) f( n) 6n f (0) (P)i c Ji: Take large Pz limit of frame-dependent equal-time correlator: q f (x; Pz) = 1 4 Z dz e ix P z z h (P) f(z) x = k n/p n = k + /P + [0, 1] n 2 = 0 ; z = n ; z + = 0 = z? z f (0) (P)i c x = kz/pz [ 1, +1]! q f (x) as Pz!1 28 How fast?

29 Typical Hadron PDF q(x): a sketch for pion 2 A sketch u 1 d sea q f π + (x) 0 ubar sea -1-2 dbar X 29

30 Pz Dependence of quasi-pdf of u-ubar pion Evaluate q(x) directly using Cauchy Residue Thm for Z 1 1 dk q A (x) =in c tr Z dk + dk d 2 k? (2 ) 4 (k + xp + ) tr[ S (i + ) S S] Evaluate q(x; P z ) directly using Cauchy Residue Thm for Z 1 1 dk 0 q A (x) =in c tr Z dk 0 dk z d 2 k? (2 ) 4 (k z xp z ) tr[ S (i z ) S S] 30

31 Pz Dependence of quasi-pdf of u-ubar pion quasi-pdf, valence toy model P z = 0.5 P z = 1 P z = 2 P z = 3 P z = 5 True pdf (P z = ) q π (x;p z ) X ---I.Cloet, Lei Chang, PCT, in progress (2015)... 31

32 Pz Dependence of quasi-pdf of u-ubar pion valence π toy model, quasi-pdf moments <x 5 > <x 11 > <x> 10 1 rel err % P z ---I.Cloet, Lei Chang, PCT, in progress (2015)... 32

33 Summary Parton Distribution Amplitudes (pion, kaon). DSE approach shows good contact with available lattice-qcd moments. Flavor symmetry breaking in K DA made quantitative. Helps identify that the ultraviolet partonic behavior is just about within reach of JLab pion FF experiments. Expect soon a clarification of what ultraviolet limit the differing BaBar Belle data should be compared to for the pion transition FF.! Parton Distribution Functions (pion). Qualitative behavior of empirical data fits reproduced by DSE q-qbar + pion loop analysis. Further work in progress.! X. Ji s space-like correlator approach to PDFs a model investigation. Spurious sea-quark contributions seem unavoidable if Pz < 2 GeV. For x > 0.8, need Pz > 4 GeV for confidence in the shape. Further work in progress. 33

34 Continuum QCD, Dyson-Schwinger Eqns and Hadron Physics Collaborators: Craig Roberts, Argonne National Lab, USA Adnan Bashir, University of Michoacan, Morelia, Mexico Ian Cloet, Argonne National Lab, USA Hong-shi Zong, Nanjing Univ, China Lei Chang, Peking U, Argonne/Julich/Univ Adelaide, Australia Chao Shi, Nanjing Univ, [visiting Kent State U] Konstantin Khitrin, PhD student, Kent State Univ, USA Javier Cobos-Martinez, Univ of Sonora, Mexico 34

35 The End 35

36 Where Asym FF Could be Calculated, its Power Law was Correct:-

37 Pion Transition Form Factor F(Q 2 )= 2f Q 2 (Q 2 )+NLO/higher twist (Q 2 )= 1 3 x 1 Q 2 1, Q Leading (BL) term with φ π (x;µ=q) 0.32 γ γ π Q 2 F(Q 2 ) Asympt/Conformal QCD limit (BL) asy with φ π (x) Q 2 (GeV 2 ) 37