Form factors on the lattice

Transcription

1 Form factors on the lattice Bipasha Chakraborty Jefferson Lab Hadronic Physics with Leptonic and Hadronic Beams, Newport News, USA 8 th Sept,

2 Pion electromagnetic form factor Simplest hadron p 1 p 2 Space like q : q 2 = (p 2 p 1 ) 2 0 Q 2 = q 2 q (in units of e ) 2

3 Interplay between hard and soft scales Hard tail (Q 2 ) from pqcd: F π (Q 2 ) 23ᴨ : G. P. Lepage, S.J.Brodsky, Phys. Lett. 87B(1979)359 G. Huber and D. Gaskell arxiv: Soft part (Q 2 < 1 GeV 2 ): vector meson dominance with F π (0) = 1, data fits well Need better understanding of the transition to the asymptotic region 3

4 Lattice recipe for meson correlators Expectation values of observables : 4-D space-time lattice Gauge configurations : gluons + sea quarks Discretise : Inversion of Dirac matrix : propagator 2-point, 3-point correlation functions : extract meson properties Corrections for lattice artifacts 4

5 Two-point correlator construction: JLab way t 1 t 2 Basis of operators Optimized operator for state n > in a variational sense by solving generalized eigenvalue problem- Diagonalize the correlation matrix eigenvalues λ n t = exp [ E n t t 0 ] 5

6 Two-point correlator construction : JLab way Correlator Construction: smearing of quark fields - distillation with Low lying hadron states Meson creation operator : Parambulators by inverting the Dirac matrix + Operator construction with momentum projection 6

7 Form factor calculation Need three-point correlator Z V < π + (p 2 ) J μ π(0) π + (p 1 ) > = e(p 1 + p 2 ) μ F π (q 2 ) Z V calculated using F π (q 2 = 0) = 1 7

8 Pion electromagnetic form factor: up to Q 2 = 1 GeV 2 Amendolia et. al. JLAB expt. JLAB (C J Shultz et. al. Phys.Rev. D91 (2015) no.11, ) JLAB lattice ongoing m π = 700 MeV m π = 390 MeV Anisotropy a s = 0.12 fm, a s a t =

9 Towards higher Q 2 More difficult on lattice for higher momenta Signal-to-noise ratio: π Noise 2-point correlators : π π π exp [ (E π (p) 2m π )t] 3-point correlators : exp [ (E π (p i ) + E π (p f ) 2m π )t/2] Minimize energies for a given Q 2 to get better signal in the middle of the plateau 9

10 Towards higher Q 2 Dispersion relation:. Achieve maximum Q 2 by using Breit frame : P f = P i Work ongoing reached up to 4.0 GeV 2 with 260 MeV pion 10

11 HPQCD Phys.Rev. D96, (2017) HPQCD- J. Koponen Lattice

12 A. J. Chambers et. al. QCDSF/UKQCD/CSSM Collaborations, arxiv: Pion mass = 450 MeV Using Feynman-Hellmann methods 12

13 Pion scalar form factor Scalar charge radius: V. Gulpers et. al. Phys.Rev. D89 (2014) no.9,

14 JLab Comparison of different lattice results for pion vector and scalar charge radius HPQCD, J. Koponen et. al. Phys.Rev. D93,

15 Nucleon electromagnetic form factor Dirac FF Pauli FF Sachs form factors - Calculated respectively from temporal and spatial component of currents 15

16 C Alexandrou et. al., PhysRevD (First lattice calculation with physical pion; disconnected contributions included) 16

17 Dipole form 17

18 Comparison among different lattice results for nucleon charge radii and magnetic moment 18

19 Sachs form factors at high Q 2 A. J. Chambers et. al. QCDSF/UKQCD/CSSM Collaborations, arxiv: Use of Feynman-Hellmann theorem At 490 MeV pion mass 19

20 Isovector charges of nucleon Boram Yoon et. al. Phys. Rev. D (Nucleon Matrix Elements (NME) Collaboration) [JLab participation: David Richards, Kostas Orginos, Frank Winter] Variational method AMA, RI-MOM At pion mass = 312 MeV 20

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23 Consistent among different smearings, and 2-state fit and variational fit 23

24 Another calculation of nucleon axial charge Evan Berkowitz et. al., arxiv: Using Feynman-Hellmann theorem: 24

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27 Tanmoy Bhattacharya et. al, Phys.Rev. D94 (2016) no.5,

28 Up, down, and strange nucleon axial form factors Jeremy Green et. al. Phys. Rev. D 95, (2017) 28

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30 Strange quark magnetic moment of the nucleon Raza Sufian, Phys. Rev. Lett (at physical pionmass with Domain-wall fermions) Ratio 3pt/2pt method Z-expansion 30

31 Strange quark magnetic moment 31

32 More calculations: Nasreen Hasan et. al., arxiv: Nucleon Dirac and Pauli form factor S. Capitaniet. al., arxiv: Iso-vector axial form factors of the nucleon in two-flavour lattice QCD C. Alexandrou et. al., arxiv: The nucleon axial form factors using lattice QCD simulations with a physical value of the pion mass Chris Bouchard et. al., Phys. Rev. D96, On the Feynman-Hellmann theorem in quantum field theory and the calculation of matrix elements Chris Bouchard et. al. PoS(Lattice2016),160 - Matrix elements from moments of correlation functions 32

33 Some more calculations - J Liang et. al., - Phys. Rev. D Lattice Calculation of Nucleon Isovector AxialChargewith Improved Currents Raza Sufian et. al., arxiv: Sea Quarks Contribution to the Nucleon Magnetic Moment and Charge Radius at the Physical Point Tanmoy Bhattacharya et. al, Phys. Rev. D Isovector and Isoscalar Tensor Charges of the Nucleon from Lattice QCD 33

34 Outlook Immediate goals (JLab form factor program): Ø Pion form factor at Q 2 6 GeV 2 Ø Extend to more ensembles with lighter pion masses, multiple volumes, multiple lattice spacing Ø Take care of lattice artefacts Ø Nucleon axial charge using distillation Next: Ø Distribution amplitude, Ø TMDs, GPDs. 34