HADRON WAVE FUNCTIONS FROM LATTICE QCD QCD. Vladimir M. Braun. Institut für Theoretische Physik Universität Regensburg

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1 HADRON WAVE FUNCTIONS FROM LATTICE QCD Vladimir M. Braun QCD Institut für Theoretische Physik Universität Regensburg

2 How to transfer a large momentum to a weakly bound system? Heuristic picture: quarks can acquire large transverse momenta when they exchange gluons hard gluon exchanges can be separated from soft nonperturbative wave functions hard gluons can only be exchanged at small transverse separations Q b ~ 1/Q x 1 Q b ~ 1/ Λ QCD Hard rescattering: Small b Average 0 < x < 1 Soft (Feynman): Average b Large x 1 In practice three-quark states indeed seem to dominate, however Squeesing to small transverse separations occurs very slowly Helicity selections rules do not work. Orbital angular momentum? More complicated nonperturbative input needed V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

3 How to transfer a large momentum to a weakly bound system? Can be formalized separating hard (H) and soft (S) momentum flow regions H H H H H H S S S A) B) C) DM) Expect at Q 2 : F 1(Q 2 ) α2 s (Q 2 ) π 2 1 Q 4, F2(Q2 ) 1 Q 6 A): 1/Q 4, factorizable in terms of distribution amplitudes, only F 1 (Q 2 ) B): 1/Q 6 (?), nonfactorizable, involves large transverse distances C): 1/Q 8 (?), nonfactorizable, involves large transverse distances DM) Duncan Mueller: 1/Q 4, nonfactorizable, contributes at NNLO Main problem: leading term suppressed by (α s/π) V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

4 Light-cone wave functions vs. Distribution Amplitudes Nucleon light-cone wave function Brodsky, Lepage [dx][d 2 k] P lz =0 = 12 ψ L=0 (x i, k i ) x 1 x 2 x 3 { u (x 1, k 1 )u (x 2, k 2 )d (x 3, k 3 ) u (x 1, k 1 )d (x 2, k 2 )u (x 3, } k 3 ) Leading-twist-three distribution amplitude Brodsky, Lepage, Peskin, Chernyak, Zhitnitsky Φ 3 (x 1, x 2, x 3 ; µ) 2 µ [d 2 k] ψ L=0 (x 1, x 2, x 3 ; k 1, k 2, k 3 ) can be defined using the OPE Φ 3 (x i ; µ) = 120f N x 1 x 2 x 3 { 1 + ϕ 10 (x 1 2x 2 + x 3 )L 8 3β 0 + ϕ 11 (x 1 x 3 )L 20 9β 0 + ϕ 20 [ 1 + 7(x2 2x 1 x 3 2x 2 2 )] L 14 3β 0 + ϕ 21 (1 4x 2 ) (x 1 x 3 ) L 40 9β 0 +ϕ 22 [ 3 9x2 + 8x x 1x 3 ] L 32 9β f N (µ 0 ): wave function at the origin ϕ nk (µ 0 ): shape parameters L α s(µ)/α s(µ 0 ) Braun, Manashov, Rohwild V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21 }

5 Wave functions vs. Distribution amplitudes (II) Contributions of orbital angular momentum Ji, Ma, Yuan, 03 [dx][d 2 [ ] k] P lz =1 = 12 k + 1 x 1 x 2 x ψl=1 1 (x i, k i ) + k + 2 ψl=1 2 (x i, k i ) 3 { u (x 1, k 1 )u (x 2, k 2 )d (x 3, k 3 ) d (x 1, k 1 )u (x 2, k 2 )u (x 3, } k 3 ) are related to higher-twist-four distribution amplitudes µ [d 2 k] Φ 4 (x 2, x 1, x 3 ; µ) = 2 k 3 m N x 3 Ψ 4 (x 1, x 2, x 3 ; µ) = 2 [ ] k + 1 ψl=1 1 + k + 2 ψl=1 2 (xi, k i ) µ [d 2 k] [ ] k 2 k + 1 m N x ψl=1 1 + k + 2 ψl=1 2 (xi, k i ) 2 Belitsky, Ji, Yuan, 03 k ± = k x ± ik y and, again, can be studied using OPE Braun, Fries, Mahnke, Stein 00 Φ 4 (x i ; µ) = 12λ 1 x 1 x f N x 1 x 2 [1 + 2 ] 3 (1 5x 3) +... Ψ 4 (x i ; µ) = 12λ 1 x 1 x f N x 1 x 3 [1 + 2 ] 3 (1 5x 2) +... to this accuracy only one new nonperturbative constant λ 1 (µ) V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

6 How to learn about DAs? From experimental data on form factors, calculating contributions of large transverse distances in terms of DA using dispersion relations and duality LCSR Nils Offen s talk on Wednesday Moments of DAs can be calculated from first principles on the lattice this talk Regensburg Lattice Collaboration QCD V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

7 Computational challenges Cost of the simulation is proportional to number of points: (L/a) 4 condition number of linear system: 1/m 2 π L 1/2 /m π from time integration in Hybrid Monte Carlo 1/a 2 critical slowing down (autocorrelations) Adjusting L 1/m π this means cost 1 a 6 m 7.5 π Huge progress in Hybrid Monte Carlo, solver, noise reduction, but Additional noise/signal problems and discretization errors from derivatives Nonperturbative renormalization and matching to MS Separation of resonances and scattering states... V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

8 Landscape of recent lattice simulations from: C Hoelbling, arxiv: L[fm] % 0.3% 1% ETMC '09 (2) ETMC '10 (2+1+1) MILC '10 MILC '12 QCDSF '10 (2) QCDSF-UKQCD '10 BMWc '10 BMWc'08 PACS-CS '09 RBC/UKQCD '10 JLQCD/TWQCD '09 HSC '08 BGR '10 (2) CLS '10(2) M π [MeV] ETMC '09 (2) ETMC '10 (2+1+1) MILC '10 MILC '12 QCDSF '10 (2) QCDSF-UKQCD '10 BMWc '10 BMWc'08 PACS-CS '09 RBC/UKQCD '10 JLQCD/TWQCD '09 HSC '08 BGR '10 CLS '10 (2) M π [MeV] a[fm] m π vs. L a vs. m π V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

9 Study case: Pion Distribution Amplitude Efremov-Radyushkin-Brodsky-Lepage-Chernyak-Zhitnitsky d(z 2 n)/nγ 5 [z 2 n, z 1 n]u(z 1 n) π(p) = if π(pn) dx e i(pn)[z 1x+z 2 (1 x)] φ π(x, µ 2 ) quark and antiquark momentum fractions moments x 1 = x x 2 = 1 x ξ = x 1 x 2 ξ 2 (x 1 x 2 ) n = a 2 (µ 2 ) 7 3/2 C2 (2x 1) = dx (2x 1) 2 φ π(x, µ 2 ) dx [ 5(2x 1) 2 1 ] φ π(x, µ 2 ) Gegenbauer expansion [ ] φ π(x, µ 2 ) = 6x(1 x) 1 + a 2 (µ 2 )C 3/2 2 (2x 1) + a 4 (µ 2 )C 3/2 4 (2x 1) +... V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

10 Nonperturbative renormalization bare operators O MS ρ (x) = d(x)γ ργ 5 u(x) [ O ρµν (x) = d(x) D{µ ] Dν 2 D {µ Dν + D (µ Dν γ ρ} γ 5 u(x) [ O + ρµν (x) = d(x) D{µ ] Dν + 2 D {µ Dν + D {µ Dν γ ρ} γ 5 u(x) {µ νo MS ρ} (x) The renormalized operators in MS scheme O MS ρ (x) = Z AO ρ(x) O MS 4jk (x) = Z 11O 4jk (x) + Z12O+ 4jk (x) ζ ik = Z ik Z A O MS+ 4jk (x) = Z 22O + 4jk (x) lattice operators = RI SMOM scheme = MS scheme nonperturbatively NLO (J. Gracey) ξ 2 MS = 361(41) (39), a MS 2 = (154) (145) error dominated by matching V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

11 Discretization errors Momentum sum rule for finite a fm is broken by lattice artifacts a 2 = 7 [ 5 (x 1 x 2 2) (x 1 + x 2 ] a 0 2) 7 [ 5 (x 1 x 2 ] 2) Two ways to the continuum limit: 1.1 ξ 2 MS a 0 ξ 2 MS a=0 = a MS 2 a 0 = a MS 2 a=0 1 2 MS a 2 (10 3 fm 2 ) A 6% deviation from unity results in a 25 30% increase in the value of a MS 2 at the same lattice spacing V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

12 Results ξ 2 MS =, a 2 = 0 asymptotic DA ξ 2 MS = 0.33(?), a MS 2 = 0.39(?) CZ model ξ 2 MS = 69(39), a MS 2 = 01(114) QCDSF 2006 ξ 2 MS = 8(1)(2), a MS 2 = 33(29)(58) UKQCD 2010 ξ 2 MS = 361(41)(39), a MS 2 = (154)(145) REG 2015 at the scale µ = 2 GeV. QCD Phys.Rev. D92 (2015) 1, V.M. Braun, S. Collins, M. Göckeler, P. Perez-Rubio, A. Schäfer, R. W. Schiel, A. Sternbeck V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

13 The nucleon Nucleon couplings at the scale 2 GeV f N = 2.84(1)(33) 10 3 GeV 2 λ 1 = 4.13(2)(20) 10 2 GeV 2 λ 2 = 8.19(5)(39) 10 2 GeV 2 f N 30% less than old estimates, bad for pqcd λ 1,2 f N, large P-wave contributions in the WFs QCD Phys.Rev. D89 (2014) 9, V.M. Braun, S. Collins, B. Gläßle, M. Göckeler, A. Schäfer, R. W. Schiel, W. Söldner, A. Sternbeck, P. Wein V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

14 Momentum fractions carried by the valence quarks x 1 = 0.372(7)(?) x 2 = 0.314(3)(?) x 3 = 0.314(7)(?) Discretization errors not seen in the data, probably smaller than statistics A diquark symmetry? QCD Phys.Rev. D89 (2014) 9, V.M. Braun, S. Collins, B. Gläßle, M. Göckeler, A. Schäfer, R. W. Schiel, W. Söldner, A. Sternbeck, P. Wein V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

15 How to study resonances on the lattice? Is finite volume always necessary? Chen et al., arxiv: Study uses RBC & UKQCD domain wall fermions, m π = ( ) MeV f lat ρ = ± 5.5 ± 0.9 MeV f exp ρ = ± 1.5 MeV from τ ρν τ V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

16 Negative parity states from variational analysis using two interpolators E/GeV ? m/gev Exp. N N, Nπ m 2 π/gev 2 Figure: Left: Negative parity states from experiment, the two-state lattice variational analysis, and the three-state analysis including a five-quark operator. The dashed lines show the sum of the nucleon and pion masses. Figure taken from [Lang:2012db]. Right: Masses of the nucleon (black), N (1650?) (blue), N (1535?) (red) and the sum of the nucleon and pion masses (green) V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

17 Distribution amplitudes from lattice QCD vs. LCSRs x 2 x x 2 x 3 x 2 x x x 1 x 1 Nucleon N (1650?) + N π N (1535?) Nucleon DA from LCSRs for comparison (N. Offen) V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

18 1 2 + baryon octet [GeV ] [GeV ] 0.3 ϕ N 11 ϕ Σ 11 π11 Σ ϕ Ξ 11 π11 Ξ ϕ Λ f N f Σ ft Σ f Ξ ft Ξ f Λ δm m 2 K m2 π δm m 2 K m2 π Zero on the horizontal axis corresponds to the SU(3) f symmetric point m u = m d = m s The physical limit is taken by following the line m u + m d + m s = const QCD paper in preparation V.M. Braun, S. Collins, M. Gruber, M. Göckeler, F. Hutzler, A. Schäfer, J. Simeth, W. Söldner, A. Sternbeck, P. Wein V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

19 Deviation from the asymptotic DA x 3 x 3 N 0 0 Λ x 2 x x x x 3 x 3 Σ 0 0 Ξ x 2 x x x 1 QCD paper in preparation V.M. Braun, S. Collins, M. Gruber, M. Göckeler, F. Hutzler, A. Schäfer, J. Simeth, W. Söldner, A. Sternbeck, P. Wein V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

20 Feasibility study in progress 10 2 f /GeV f 3/2 CLS f 1/2 CLS f 3/2 f 1/2 Sum Rules Ogloblin/Zhitnitsky Sum Rules Ogloblin/Zhitnitsky mπ 2 /GeV2 Figure: Raw data for the couplings (wave function at the origin) of the (1232) resonance with helicity 1/2 and 3/2. The existing QCD sum rule estimates [Chernyak:1987nv] are shown for comparison QCD work in progress V.M. Braun, S. Collins, M. Gruber, M. Göckeler, F. Hutzler, A. Schäfer, J. Simeth, W. Söldner, A. Sternbeck, P. Wein V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21

21 Summary and outlook Lattice calculations of moments of the DAs are becoming mature Lattice operators, nonperturbative renormalization, MS-like scheme, matching Parity projectors for moving baryons One-loop chiral perturbation theory Code writing, LibHadronAnalysis,... A consistent picture is emerging Outlook Valence quark distributions in the nucleon are not far from asymptotic form Large P-wave contributions (quark orbital angular momentum) Qualitatively different valence quark distributions in Nucleon and N (1535) Large SU (3) f breaking N f = CLS configurations with a 0.04 fm, continuum extrapolation 1 + baryon octet, (1232) 2 Bottlenecks: Two-loop matching, very high statistics needed for the second moments V. M. Braun (Regensburg) Hadron Wave Functions from Lattice QCD October / 21