Hadron Physics: From Solitons, DSEs, to Light-Front QCD

Transcription

1 Hadron Physics: From Solitons, DSEs, to Light-Front QCD Peter C. Tandy Dept of Physics Kent State University USA 1

2 Topics Things from the past retony Williams: Solitons, Perturbation Integral Representation, LQCD propagators,. Pion and kaon Distribution Amplitudes insight into DSE-QCD vs LQCD for hadron physics. PDFs including X. Ji s space-like correlator approximation for LQCD a model investigation. Pion elastic and transition Form Factors 2

3 From the Past Univ of Maryland, Soliton Models 3

4 From the Past Morelia May 2015

5 Hadron Physics & QCD Analysis of a quenched lattice QCD dressed quark propagator M.S. Bhagwat, M.A. Pichowsky (Kent State U.), C.D. Roberts (Argonne, PHY), P.C. Tandy (Kent State U.). Apr pp. Published in Phys.Rev. C68 (2003) e-print: nucl-th/ PDF Analysis of full-qcd and quenched-qcd lattice propagators M.S. Bhagwat, P.C. Tandy (Kent State U.). Jan pp. AIP Conf. Proc. 842 (2006) (PANIC05) e-print: nucl-th/ PDF 5

6 Lattice QCD 6

7 Perturbation Integral Representation 7

8 Fit numerical DSE-BSE solns to PTIRs (Nakanishi) 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 = E, F, G, or H F(q 2 ; q P) = Z 1 1 d Z 1 0 d IR ( ; ) (q 2 + q P + 2 ) + UV ( ; ) m+n (q 2 + q P + 2 ) n npqcd info is in the variables and constants that are not momenta ---Wick rotation is trivial as in pert thy. IR( ; ) 1( ) ( IR 1 ) + 3 Works for u-, d-, s-, c-, b-quarks. Also for lattice-qcd propagators. N. Souchlas, PhD thesis KSU, (2009), J. Phys. G37, (2010) EG: q A (x) =in c tr Z dk + dk d 2 k? (2 ) 4 (k + xp + ) tr[ S (i + ) S S] 8

9 Spacelike Correlator Approximation for PDFs 9 Wuhan Wksp Nov 2015

10 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 10 How fast? Wuhan Wksp Nov 2015

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14 Simple model for pion PDF & Quasi-PDF S(k) =1/(i6k + M), M = 0.4 GeV (q, P) = 5 N Z 1 1 d ( ) q 2, ( ) =even + q P + 2 Euclidean to Minkowski:- 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] 14

15 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 15

16 DIVERSION A full DSE calculation of the true pion valence PDF K. Khitrin, P. Tandy, in progress (2015) Modern empirical expt parameterization: Aicher, Shafer, Vogelsang, (ASV) PRL 105, (2010) Q 0 = GeV u v (x) DSE-RL DSE-RL + Pi Loop via Convoln ASV (expt) DSE-RL + Pi Loop Phenomenology x 16

17 Model-exact PDF & Pz=10 GeV

18 Pz Dependence of quasi-pdf of valence model 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)... 18

19 <x^m> for toy model pion at Pz = 3 GeV 10 0 valence π toy model, quasi-pdf moments model exact pdf quasi-pdf Pz = 3 GeV <x^m> m ---I.Cloet, Lei Chang, PCT, in progress (2015)... 19

20 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)... 20

21 Parton Distribution Amplitudes of Mesons 21

22 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 22

23 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) } 23

24 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 ±

25 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 25

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

27 Excited Pion (1300) Distribution Amplitude Bo-Lin, L. Chang, C.D.Roberts, H-S., Zong, in progress 2016, Nanjing U. 27

28 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) 28

29 Kaon Distribution Amplitude C. Shi, L. Chang, C.D. Roberts, S.Schmidt, PCT, H-S. Zong, PLB738, 512 (2014) µ = 2GeV DSE-DB skewness implies only 14% flavor symm breaking due to DCSB DSE-RL n ms m u m s + m u o2gev 66% R. Arthur, P. Boyle, D. Brommel, M. Donnellan, J. Flynn et al, PRD83, (2011) 29

30 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) 30

31 Form Factors 31

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33 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 33

34 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) 34

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

36 Pion Transition Form Factor K. Raya, L. Chang, A. Bashir, J.J.Cobos-Martinez, L.X. Gutierez-Guerrero, C.D.Roberts, P.C.Tandy, arxiv: From unified treatment of DA, elastic FF, and transition FF Q 2 G γ * γπ (Q 2 )/2π 2 (GeV) 0.30 DSEPrediction no log evolution of BSE ampl with log evolution of BSE ampl Q 2 GeV 2 36

37 Parton Distribution Functions q q γ µ k + q γν k k P P Covariant formulation and calculation Z d 4 q F(q 2, q P, q k, k 2 ) 37

38 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 38

39 Pion PDFs Expt Data Parameterizations Aicher, Schafer, Vogelsang, arxiv: soft gluon resummation 39

40 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 =

41 Analysis of Pion Parton Momentum Sum Rule K. Khitrin, P. Tandy, in progress (2015) Modern empirical expt parameterization: Aicher, Shafer, Vogelsang, (ASV) PRL 105, (2010) Q 0 = GeV u v (x) DSE-RL DSE-RL + Pi Loop via Convoln ASV (expt) DSE-RL + Pi Loop Phenomenology x 41

42 Convolution Model for q(x) from virtual pi loop q v lp (x) P q/t (x) = R 1 T = target = here x dy P /T(y) P q/ ( x y ), P /T (y) should strongly favor y apple m 2M q +m 0.2, P q/ ( x y ) is self consistently determined 4 3 q v π-lp (x) q sea ASV (x) Q = GeV q(x) 2 Result is strongly constrained x 42

43 Summary X. Ji s space-like correlator approach to PDFs a model investigation. Spurious anti-quark contributions seem unavoidable if Pz < 2 GeV. For x > 0.8, need Pz > 4 GeV for confidence in the qualitative shape. Further work in progress. Parton Distribution Amplitudes (pion, kaon). DSE approach shows good contact with available lattice-qcd moments. Flavor symmetry breaking in kaon DA made quantitative. Pion Transition & Elastic Form Factors DSE TFF calculation for all Q^2 - agrees with Belle not BaBar. DSE elff Connection with asymptotic QCD reconciled. Identify that the ultraviolet partonic behavior is within reach of proposed JLab pion FF experiments. Parton Distribution Functions (pion). Qualitative behavior of empirical data fits reproduced by DSE q-qbar + pion loop analysis. Time to declare we understand the pion and kaon in QCD? 43

44 Congratulations Tony! The End 44 Wuhan Wksp Nov 2015