Srijit Paul HPC-LEAP EUROPEAN JOINT DOCTORATES LATTICE 2018, Michigan, July 27, 2018 1 / 14 Srijit Paul s.paul@hpc-leap.eu 1/14
• Evaluating the Wick contractions involved in the calculation of 2-pt functions. [ study, Alexandrou et.al.(2017)] • Projecting the interpolators into particular Irreps of the relevant Little group. • Constructing the correlation matrices for each Irrep in all the relevant Little groups. • Obtaining the GEVP spectra for all the Irreps. bullet size data size BMW Ensemble, S. Dürr et al., JHEP 1108, 148 (2011) Summary of the π - N spectrum • Construction of relevant single and multihadron interpolating field operators, with the right quantum numbers. [Discussed in previous talk] 2 / 14 Srijit Paul s.paul@hpc-leap.eu 2/14
• Projecting the interpolators into particular Irreps of the relevant Little group. • Constructing the correlation matrices for each Irrep in all the relevant Little groups. • Obtaining the GEVP spectra for all the Irreps. bullet size data size BMW Ensemble, S. Dürr et al., JHEP 1108, 148 (2011) Summary of the π - N spectrum • Construction of relevant single and multihadron interpolating field operators, with the right quantum numbers. [Discussed in previous talk] • Evaluating the Wick contractions involved in the calculation of 2-pt functions. [ ππ study, Alexandrou et.al.(2017)] 2 / 14 Srijit Paul s.paul@hpc-leap.eu 2/14
• Constructing the correlation matrices for each Irrep in all the relevant Little groups. • Obtaining the GEVP spectra for all the Irreps. bullet size data size BMW Ensemble, S. Dürr et al., JHEP 1108, 148 (2011) Summary of the π - N spectrum • Construction of relevant single and multihadron interpolating field operators, with the right quantum numbers. [Discussed in previous talk] • Evaluating the Wick contractions involved in the calculation of 2-pt functions. [ ππ study, Alexandrou et.al.(2017)] • Projecting the interpolators into particular Irreps of the relevant Little group. 2 / 14 Srijit Paul s.paul@hpc-leap.eu 2/14
• Obtaining the GEVP spectra for all the Irreps. bullet size data size BMW Ensemble, S. Dürr et al., JHEP 1108, 148 (2011) Summary of the π - N spectrum • Construction of relevant single and multihadron interpolating field operators, with the right quantum numbers. [Discussed in previous talk] • Evaluating the Wick contractions involved in the calculation of 2-pt functions. [ ππ study, Alexandrou et.al.(2017)] • Projecting the interpolators into particular Irreps of the relevant Little group. • Constructing the correlation matrices for each Irrep in all the relevant Little groups. 2 / 14 Srijit Paul s.paul@hpc-leap.eu 2/14
bullet size data size BMW Ensemble, S. Dürr et al., JHEP 1108, 148 (2011) Summary of the π - N spectrum • Construction of relevant single and multihadron interpolating field operators, with the right quantum numbers. [Discussed in previous talk] • Evaluating the Wick contractions involved in the calculation of 2-pt functions. [ ππ study, Alexandrou et.al.(2017)] • Projecting the interpolators into particular Irreps of the relevant Little group. • Constructing the correlation matrices for each Irrep in all the relevant Little groups. • Obtaining the GEVP spectra for all the Irreps. 2 / 14 Srijit Paul s.paul@hpc-leap.eu 2/14
BMW Ensemble, S. Dürr et al., JHEP 1108, 148 (2011) Summary of the π - N spectrum • Construction of relevant single and multihadron interpolating field operators, with the right quantum numbers. [Discussed in previous talk] • Evaluating the Wick contractions involved in the calculation of 2-pt functions. [ ππ study, Alexandrou et.al.(2017)] • Projecting the interpolators into particular Irreps of the relevant Little group. • Constructing the correlation matrices for each Irrep in all the relevant Little groups. • Obtaining the GEVP spectra for all the Irreps. bullet size ∝ ∼ data size 2 / 14 Srijit Paul s.paul@hpc-leap.eu 2/14
Summary of the π - N spectrum • Construction of relevant single and multihadron interpolating field operators, with the right quantum numbers. [Discussed in previous talk] • Evaluating the Wick contractions involved in the calculation of 2-pt functions. [ ππ study, Alexandrou et.al.(2017)] • Projecting the interpolators into particular Irreps of the relevant Little group. • Constructing the correlation matrices for each Irrep in all the relevant Little groups. • Obtaining the GEVP spectra for all the Irreps. bullet size ∝ ∼ data size BMW Ensemble, S. Dürr et al., JHEP 1108, 148 (2011) 2 / 14 Srijit Paul s.paul@hpc-leap.eu 2/14
Lüscher Methodology ( ) P ) ⃗ det 1 + it ℓ ( s )( 1 + i M = 0 , 1 where t ℓ ( s ) = cot δ ℓ ( s ) − i . [Lüscher(1991)] 3 / 14 Srijit Paul s.paul@hpc-leap.eu 3/14
For Baryons [Göckeler et.al.(2012)] In the above formula can be simplified by basis transformations as block diagonal by, General Lüscher Methodology ( ) P ) ⃗ det 1 + i t ℓ ( s ) ( 1 + i M = 0 , 1 where t ℓ ( s ) = . cot δ ℓ ( s ) − i [Lüscher(1991)] 4 / 14 Srijit Paul s.paul@hpc-leap.eu 4/14
General Lüscher Methodology ( ) P ) ⃗ det 1 + i t ℓ ( s ) ( 1 + i M = 0 , 1 where t ℓ ( s ) = . cot δ ℓ ( s ) − i [Lüscher(1991)] For Baryons det( M Jlµ,J ′ l ′ µ ′ − δ JJ ′ δ ll ′ δ µµ ′ cot δ Jl ) = 0 [Göckeler et.al.(2012)] In the above formula M Jlµ,J ′ l ′ µ ′ can be simplified by basis transformations as block diagonal by, ⟨ Γ αJln | ˆ ∑ c Γ ′ α ′ n ′ M | Γ ′ α ′ J ′ l ′ n ′ ⟩ = c Γ αn ∗ M Jlµ,J ′ l ′ µ ′ Jlµ J ′ l ′ µ ′ µµ ′ 4 / 14 Srijit Paul s.paul@hpc-leap.eu 4/14
Example M Jlµ,J ′ l ′ µ ′ calculation: C D 4 v J = 1/2 J = 3/2 l = 0, 1 l = 1, 2 5 / 14 Srijit Paul s.paul@hpc-leap.eu 5/14
Example M Jlµ,J ′ l ′ µ ′ calculation: C D 4 v J = 1/2 J = 3/2 l = 0, 1 l = 1, 2 5 / 14 Srijit Paul s.paul@hpc-leap.eu 5/14
Example M Jlµ,J ′ l ′ µ ′ calculation: C D 4 v J = 1/2 J = 3/2 l = 0, 1 l = 1, 2 det( M G 1 Jlµ,J ′ l ′ µ ′ − δ JJ ′ δ ll ′ δ µµ ′ cot δ G 1 Jl ) = 0 det( M G 2 Jlµ,J ′ l ′ µ ′ − δ JJ ′ δ ll ′ δ µµ ′ cot δ G 2 Jl ) = 0 5 / 14 Srijit Paul s.paul@hpc-leap.eu 5/14
Example M Jlµ,J ′ l ′ µ ′ calculation: C D 4 v J = 1/2 J = 3/2 l = 0, 1 l = 1, 2 det( M G 2 Jlµ,J ′ l ′ µ ′ − δ JJ ′ δ ll ′ δ µµ ′ cot δ G 2 3 / 2 , 1 ) = 0 5 / 14 Srijit Paul s.paul@hpc-leap.eu 5/14
Decay Width form for . [ V. Pascalutsa and M. Vanderhaeghen, Phys.Rev. D73 ,034003 (2006) ] Used in lattice QCD for the first time by, [Alexandrou, Negele, Petschlies, Strelchenko, Tsapalis , Phys.Rev. D88 (2013)] Resonances Narrow resonances in scattering are characterised by Breit Wigner √ s Γ( s ) t ℓ ( s ) = √ s Γ( s ) R − s − i m 2 s = Square of Centre of Mass energy ( Mandelstam s ) m R = Mass of resonance Γ( s ) = Decay-width of resonance 6 / 14 Srijit Paul s.paul@hpc-leap.eu 6/14
Resonances Narrow resonances in scattering are characterised by Breit Wigner √ s Γ( s ) t ℓ ( s ) = √ s Γ( s ) R − s − i m 2 s = Square of Centre of Mass energy ( Mandelstam s ) m R = Mass of resonance Γ( s ) = Decay-width of resonance Decay Width form for ∆(1232) . g 2 p ∗ 3 E N + m N ∆ − πN Γ LO EFT = m 2 E N + E π 48 π N [ V. Pascalutsa and M. Vanderhaeghen, Phys.Rev. D73 ,034003 (2006) ] Used in lattice QCD for the first time by, [Alexandrou, Negele, Petschlies, Strelchenko, Tsapalis , Phys.Rev. D88 (2013)] 6 / 14 Srijit Paul s.paul@hpc-leap.eu 6/14
• Calculated pion mass, nucleon mass on the lattice MeV MeV • Assume the , . (between the - and - threshold.) • Select the Irrep containing in order to construct . • Take to be Breit Wigner distribution, for , assuming there is a resonance in -wave scattering. • Find the values of for which the determinant condition is satisfied. Inverse Lüscher Formalism Methodology ( ) P ) ⃗ det 1 + i t ℓ ( s ) ( 1 + i M = 0 , 7 / 14 Srijit Paul s.paul@hpc-leap.eu 7/14
• Assume the , . (between the - and - threshold.) • Select the Irrep containing in order to construct . • Take to be Breit Wigner distribution, for , assuming there is a resonance in -wave scattering. • Find the values of for which the determinant condition is satisfied. Inverse Lüscher Formalism Methodology ( ) P ) ⃗ det 1 + i t ℓ ( s ) ( 1 + i M = 0 , • Calculated pion mass, nucleon mass on the lattice m π = 258 . 3(1 . 1) MeV m N = 1066 . 4(2 . 7) MeV 7 / 14 Srijit Paul s.paul@hpc-leap.eu 7/14
• Select the Irrep containing in order to construct . • Take to be Breit Wigner distribution, for , assuming there is a resonance in -wave scattering. • Find the values of for which the determinant condition is satisfied. Inverse Lüscher Formalism Methodology ( ) P ) ⃗ det 1 + i t ℓ ( s ) ( 1 + i M = 0 , • Calculated pion mass, nucleon mass on the lattice m π = 258 . 3(1 . 1) MeV m N = 1066 . 4(2 . 7) MeV • Assume the m ∆ , Γ . (between the π - N and ππ - N threshold.) 7 / 14 Srijit Paul s.paul@hpc-leap.eu 7/14
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