Pion Distribution Amplitude from Euclidean Correlation functions

Transcription

1 Pion Distribution Amplitude from Euclidean Correlation functions Vladimir M. Braun Institut für Theoretische Physik Universität Regensburg November 17 RQCD Collaboration: G. Bali, V.M. Braun, M. Göckeler, M. Gruber, F. Hutzler, P. Korcyl, B. Lang, A. Schäfer, P. Wein, J.-H. Zhang

2 Hard exclusive pion production is sensitive to pion valence quark distribution at small transverse separations: pion DA Q F π γ γ(q ) Q f BΠ q q GeV A vast field: γ γπ(η, η ), pion electroproduction, B πlν, B ππ etc. V.B. et al. (RQCD Collaboration), PRD 9 (15) 1454 ξ MS = 1 a MS = du (u 1) φ π(u,µ) =.361(41)(39)(?) µ = GeV du C3/(u 1) φ π(u,µ) =.1364(154)(145)(?) V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November 17 / 15

3 Momentum smearing G. Bali, B. Lang, B. Musch, A. Schäfer, momentum smearing conventional smearing position space f z Re e ikz f z z z ) ( momentum space ~ f(p) + ~ f(p k) k p p σ ( ξ ) N c Wuppertal smearing Momentum smearing /N c Right: Squared error as a function of the statistics; Ensemble H15 for n p = (11), (11), (11) Momentum smearing: inversions for each momentum (6 inversions) Wuppertal smearing: 1 inversion + additional Fourier sums V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November 17 3 / 15

4 15 17 a MS a =.8 fm, L/a= 3 a =.7 fm, L/a= 3 a =.7 fm, L/a= 64 a =.6 fm, L/a= 3 a =.6 fm, L/a= 48 a m π (GeV ) mπ 15: N f =, a =.6.8 fm 17: N f = + 1, a.8 fm.4 fm in 18 V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November 17 4 / 15

5 Introduction Second Moment Beyond moments Outlook CLS ensemble overview β N3 H N 35 N3 J51 N3 B45 U13 H11 S4 U1 H1 3 S1 N J33 5 D 15 physical U11 H15 N11.4 a [fm ] 643, N34.6 N1 H16 J34 5 C1 D , H17 N4 3 D15 D1 ms + m` = const 963, 35 S1 C11 D11 D45 E5.. N41 4 mπ [MeV] J5 4 mπ [MeV] β 3.55 physical...4 a [fm ].6.8 m b s const 483, E: 19 J: 19 D: 18 N: 18 S: 18 33, H: 96 33, B: 64 33, U: C: , additional ensembles with ms = m`. V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November 17 5 / 15

6 DAs/PDFs/GPDs from custom-made lattice (Euclidean) correlation functions Going beyond the second moment is not feasible: mixing with lower-dimensional operators adding more derivatives deteriorates signal-to-noise ratio General idea: perturbative factorization of Euclidean correlation functions H (p) J 1(z)J ( z) H (p) = C(z, p z; µ F ) P(p z; µ F ) z < factorization in terms of PDFs is done in continuum (in MS) small z necessary for factorization (and suppresses higher-twists) large p z is necessary as a lever-arm on accessible momentum fractions additional renormalization factors may occur (Collinear) factorization in position space: light-ray operator product expansion Zavialov 76; Balitsky, Braun V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November 17 6 / 15

7 Example: Braun, Müller, EPJC, 55, 349 (8) T{ q(z)γ µq(z) q( z)γ νq( z)}π (p) = 5i z ρ p σ 9 fπɛµνρσ 8π z 4 T(p z, z ) 1 T(p z, z ) = du e i(u 1)p z H(u, µ F z, α s(µ F )) φ π(u, µ F ) + O(z ) Alternative 1 T{ q(z)/zγ 5 [z, z]q( z)}π (p) = if π(pz) du e i(u 1)pz H(u, µ F z, α s(µ F ))φ π(u, µ F ) + O(z ) In both cases H = 1 + cα s ln z µ What is better? Propagator: Physical observable, z direction arbitrary, extra handle from spinor structure Wilson line: cheaper, z = (,, z), no extra spinor structure, nonlocal RG factors V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November 17 7 / 15

8 or: heavy quark W. Detmold, C.J.D. Lin, hep-lat/577 T{ q(z)γ µc(z) c( z)γ νq( z) ν}π (p) or: auxiliary scalar U Aglietti et al, hep-ph/98677; A Abada et al, hep-ph/151 T{ q(z)γ µφ(z) φ( z)γ νq( z) ν}π (p) V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November 17 8 / 15

9 Quasi-distribution in analogy to quasi-pdf, define X Ji, ,; X Ji, φ π(u, p z) = i f π dz π e i(u 1)pz z π(p) q(z)γ zγ 5[z, z]q( z) and match to pion DA using Large momentum effective theory (LaMET) φ π(u, a 1, p z) = (? higher twists O Λ p z 1 ) ( or O dv Z φ (u, v, a 1, µ, p z)φ π(v, µ) + O Λ u p z ( Λ ), M π. pz pz ) ; perturbative expansion in α s(p z) or α s(up z)? Fourier transform numerically unstable and involves large-z region? Integration over z does not allow to cancel the nonlocal RG factor K Orginos, A Radyushkin, J Karpie, S Safeiropoulos, first calculation using this approach J-H Zhang, J-W Chen, X Ji, L Jin, H-W Lin, 17.8 V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November 17 9 / 15

10 Work in progress: G. Bali et. al., [RQCD Collaboration], We follow Braun, Müller, EPJC, 55, 349 (8) with generic spinor structure T{ q(z/)γ A q(z/) q( z/)γ B q( z/)}π (p) T(p z, z ) so far Γ A Γ B I γ 5 y e ip y sm y Γ A z Tree level result: T(p z, z p z ) = F π π Φπ(p z), z4 with the position-space DA 1 Γ B Φ π(p z) = du e i(u 1/)p z φ π(u). t > t = QCD factorization T(p z, z ) = 1 du e i(u 1)p z H (u, µ Fz, α s(µ F )) φ π(u, µ F ) + O(z ) Take into account H (u,...) to NLO and twist-4 corrections O(z ) V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November 17 1 / 15

11 Three illustrative models of the DA (taken at a scale µ = 1 GeV) φ (1) π (u) = 6u(1 u), φ π () (u) = 8 u(1 u), φ (3) π (u) = 1 π φ (i) π (u) φ (1) π φ () π φ (3) π u Φπ(p z) φ (1) π φ () π φ (3) π p z left: momentum space, right: position space ( Ioffe time ) V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November / 15

12 In practice we compute for large t T(p z, z ) F π = Z SZ P Z A [ diq](, z/) [ qγ 5 u](, z/) O π( t, p) [ dγ γ 5 u](, ) O π( t, p) E( p) with RG factors Z S (µ R, g ), Z P (µ R, g ) and Z A (g ) and µ R = / z We tree-level correct for z-dependent lattice artefacts T(p z, z ) T(p z, z tr [/zgtree cont ) (z)] tr [/zg latt tree (for (z, a)] Re Φ SP π (p z, z ) p z =.39 the chiral even part) uncorrected excluded corrected µ = / z [GeV] V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November 17 1 / 15

13 Results I: Renormalization/factorization scale dependence N f = NP improved Wilson-clover quarks (old QCDSF ensemble); a 1.76 GeV, m π 9 MeV, L = 3a 3.4/m π..5. p z = 1.57 p = 1.8 GeV p = 1.53 GeV p = 1.88 GeV.5. p z = 1.96 p = 1.8 GeV p = 1.53 GeV p = 1.88 GeV Φπ(µ) Φπ(µ) µ = / z [GeV] µ = / z [GeV] V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November / 15

14 Results II: Ioffe time p z dependence Solid/dashed curves are with/without higher twist corrections T (p z, z ) a 4 /Fπ µ = 1.18 GeV = / z p = 1.8 GeV p = 1.53 GeV p = 1.88 GeV Re Φ SP π (p z, z ) µ = 1.18 GeV = / z p = 1.8 GeV p = 1.53 GeV p = 1.88 GeV T (p z, z ) a 4 /Fπ p z µ = 1.3 GeV = / z p = 1.8 GeV p = 1.53 GeV p = 1.88 GeV p z Re Φ SP π (p z, z ) p z µ = 1.3 GeV = / z p = 1.8 GeV p = 1.53 GeV p = 1.88 GeV p z The future: other Dirac structures, smaller a, larger p. V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November / 15

15 Outlook Momentum smearing is a leap forward Second moment Accuracy goal: 3% for ξ MS ; 15% for a MS Work in progress: continuum extrapolation DA shapes from customized Euclidean correlation functions We presented a proof of concept For / z 1 GeV need p 4 GeV to reach Ioffe times p z large enough to discriminate between different shapes In future a new algorithm that enables smaller statistical errors other current-current combinations to minimize higher-twists and for cross-checks may expand to K-meson and pion PDF? discretisation errors ap V. M. Braun (Regensburg) Pion Distribution Amplitude from Euclidean Correlation functions November / 15