HPC-LEAP EUROPEAN JOINT DOCTORATES LATTICE 2018, Michigan, July 27, - - PowerPoint PPT Presentation

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HPC-LEAP EUROPEAN JOINT DOCTORATES LATTICE 2018, Michigan, July 27, - - PowerPoint PPT Presentation

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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

slide-1
SLIDE 1

Srijit Paul

HPC-LEAP

EUROPEAN JOINT DOCTORATES

LATTICE 2018, Michigan, July 27, 2018

Srijit Paul s.paul@hpc-leap.eu 1/14

1/14

slide-2
SLIDE 2

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.

[ 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)

Srijit Paul s.paul@hpc-leap.eu 2/14

2/14

slide-3
SLIDE 3

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)

Srijit Paul s.paul@hpc-leap.eu 2/14

2/14

slide-4
SLIDE 4

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)

Srijit Paul s.paul@hpc-leap.eu 2/14

2/14

slide-5
SLIDE 5

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)

Srijit Paul s.paul@hpc-leap.eu 2/14

2/14

slide-6
SLIDE 6

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)

Srijit Paul s.paul@hpc-leap.eu 2/14

2/14

slide-7
SLIDE 7

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)

Srijit Paul s.paul@hpc-leap.eu 2/14

2/14

slide-8
SLIDE 8

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)

Srijit Paul s.paul@hpc-leap.eu 2/14

2/14

slide-9
SLIDE 9

Lüscher Methodology

det ( 1 + itℓ(s)(1 + iM

⃗ P )

) = 0, where tℓ(s) =

1 cot δℓ(s)−i.

[Lüscher(1991)]

Srijit Paul s.paul@hpc-leap.eu 3/14

3/14

slide-10
SLIDE 10

General Lüscher Methodology

det ( 1 + i tℓ(s) (1 + i M

⃗ P )

) = 0, where tℓ(s) =

1 cot δℓ(s) −i

. [Lüscher(1991)] For Baryons [Göckeler et.al.(2012)] In the above formula can be simplified by basis transformations as block diagonal by,

Srijit Paul s.paul@hpc-leap.eu 4/14

4/14

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SLIDE 11

General Lüscher Methodology

det ( 1 + i tℓ(s) (1 + i M

⃗ P )

) = 0, where tℓ(s) =

1 cot δℓ(s) −i

. [Lüscher(1991)] For Baryons det( MJlµ,J′l′µ′ − δJJ′δll′δµµ′ cot δJl ) = 0 [Göckeler et.al.(2012)] In the above formula MJlµ,J′l′µ′ can be simplified by basis transformations as block diagonal by, ⟨ΓαJln| ˆ M|Γ′α′J′l′n′⟩ = ∑

µµ′

c Γαn ∗

Jlµ

c Γ′α′n′

J′l′µ′

MJlµ,J′l′µ′

Srijit Paul s.paul@hpc-leap.eu 4/14

4/14

slide-12
SLIDE 12

Example MJlµ,J′l′µ′ calculation: CD

4v

J = 1/2 J = 3/2 l = 0, 1 l = 1, 2

Srijit Paul s.paul@hpc-leap.eu 5/14

5/14

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SLIDE 13

Example MJlµ,J′l′µ′ calculation: CD

4v

J = 1/2 J = 3/2 l = 0, 1 l = 1, 2

Srijit Paul s.paul@hpc-leap.eu 5/14

5/14

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SLIDE 14

Example MJlµ,J′l′µ′ calculation: CD

4v

J = 1/2 J = 3/2 l = 0, 1 l = 1, 2

det( M G1

Jlµ,J′l′µ′ − δJJ′δll′δµµ′ cot δG1 Jl ) = 0

det( M G2

Jlµ,J′l′µ′ − δJJ′δll′δµµ′ cot δG2 Jl ) = 0

Srijit Paul s.paul@hpc-leap.eu 5/14

5/14

slide-15
SLIDE 15

Example MJlµ,J′l′µ′ calculation: CD

4v

J = 1/2 J = 3/2 l = 0, 1 l = 1, 2

det( M G2

Jlµ,J′l′µ′ − δJJ′δll′δµµ′ cot δG2 3/2, 1 ) = 0

Srijit Paul s.paul@hpc-leap.eu 5/14

5/14

slide-16
SLIDE 16

Resonances

Narrow resonances in scattering are characterised by Breit Wigner tℓ(s) = √s Γ(s) m2

R − s − i

√s Γ(s) s = Square of Centre of Mass energy ( Mandelstam s) mR = Mass of resonance Γ(s) = Decay-width of resonance 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)]

Srijit Paul s.paul@hpc-leap.eu 6/14

6/14

slide-17
SLIDE 17

Resonances

Narrow resonances in scattering are characterised by Breit Wigner tℓ(s) = √s Γ(s) m2

R − s − i

√s Γ(s) s = Square of Centre of Mass energy ( Mandelstam s) mR = Mass of resonance Γ(s) = Decay-width of resonance Decay Width form for ∆(1232). ΓLO

EFT =

g2

∆−πN

48π EN + mN EN + Eπ p∗3 m2

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)]

Srijit Paul s.paul@hpc-leap.eu 6/14

6/14

slide-18
SLIDE 18

Inverse Lüscher Formalism Methodology

det

(

1 + i tℓ(s) (1 + i M

⃗ P )

)

= 0,

  • 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.

Srijit Paul s.paul@hpc-leap.eu 7/14

7/14

slide-19
SLIDE 19

Inverse Lüscher Formalism Methodology

det

(

1 + i tℓ(s) (1 + i M

⃗ P )

)

= 0,

  • Calculated pion mass, nucleon mass on the lattice

mπ = 258.3(1.1) MeV mN = 1066.4(2.7) 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.

Srijit Paul s.paul@hpc-leap.eu 7/14

7/14

slide-20
SLIDE 20

Inverse Lüscher Formalism Methodology

det

(

1 + i tℓ(s) (1 + i M

⃗ P )

)

= 0,

  • Calculated pion mass, nucleon mass on the lattice

mπ = 258.3(1.1) MeV mN = 1066.4(2.7) MeV

  • Assume the m∆, Γ. (between the π-N and ππ-N 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.

Srijit Paul s.paul@hpc-leap.eu 7/14

7/14

slide-21
SLIDE 21

Inverse Lüscher Formalism Methodology

det

(

1 + i tℓ(s) (1 + i M

⃗ P )

)

= 0,

  • Calculated pion mass, nucleon mass on the lattice

mπ = 258.3(1.1) MeV mN = 1066.4(2.7) MeV

  • Assume the m∆, Γ. (between the π-N and ππ-N threshold.)
  • Select the Irrep containing J = 3/2 in order to construct M.
  • 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.

Srijit Paul s.paul@hpc-leap.eu 7/14

7/14

slide-22
SLIDE 22

Inverse Lüscher Formalism Methodology

det

(

1 + i tℓ(s) (1 + i M

⃗ P )

)

= 0,

  • Calculated pion mass, nucleon mass on the lattice

mπ = 258.3(1.1) MeV mN = 1066.4(2.7) MeV

  • Assume the m∆, Γ. (between the π-N and ππ-N threshold.)
  • Select the Irrep containing J = 3/2 in order to construct M.
  • Take tl to be Breit Wigner distribution, for l = 1, assuming

there is a resonance in P-wave scattering.

  • Find the values of

for which the determinant condition is satisfied.

Srijit Paul s.paul@hpc-leap.eu 7/14

7/14

slide-23
SLIDE 23

Inverse Lüscher Formalism Methodology

det

(

1 + i tℓ(s) (1 + i M

⃗ P )

)

= 0,

  • Calculated pion mass, nucleon mass on the lattice

mπ = 258.3(1.1) MeV mN = 1066.4(2.7) MeV

  • Assume the m∆, Γ. (between the π-N and ππ-N threshold.)
  • Select the Irrep containing J = 3/2 in order to construct M.
  • Take tl to be Breit Wigner distribution, for l = 1, assuming

there is a resonance in P-wave scattering.

  • Find the values of s for which the determinant condition is

satisfied.

Srijit Paul s.paul@hpc-leap.eu 7/14

7/14

slide-24
SLIDE 24

Lüscher Analysis v/s Inverse Lüscher

−0.5 0.0 0.5

Det

√s[model]

1

3 4 5 6 7

tmin[a]

√s[avg]

1

3 5 7 9 11

t[a]

1200 1300 1400 1500

√s1(MeV)

−0.5 0.0 0.5

√s[model]

2

√s[avg]

2

1400 1600 1800 2000 2200

√s2(MeV) | d| = 0, Λ = Hg

Srijit Paul s.paul@hpc-leap.eu 8/14

8/14

slide-25
SLIDE 25

Lüscher Analysis v/s Inverse Lüscher

−0.5 0.0 0.5

Det

√s[model]

1

3 4 5 6 7

tmin[a]

√s[avg]

1

Single-exponential 3 5 7 9 11

t[a]

1200 1300 1400 1500

√s1(MeV)

−0.5 0.0 0.5

√s[model]

2

√s[avg]

2

1400 1600 1800 2000 2200

√s2(MeV) | d| = 0, Λ = Hg

Srijit Paul s.paul@hpc-leap.eu 8/14

8/14

slide-26
SLIDE 26

Lüscher Analysis v/s Inverse Lüscher

−0.5 0.0 0.5

Det

√s[model]

1

3 4 5 6 7

tmin[a]

√s[avg]

1

Single-exponential 3 5 7 9 11

t[a]

1200 1300 1400 1500

√s1(MeV)

−0.5 0.0 0.5

√s[model]

2

√s[avg]

2

1400 1600 1800 2000 2200

√s2(MeV) | d| = 0, Λ = Hg

Srijit Paul s.paul@hpc-leap.eu 8/14

8/14

slide-27
SLIDE 27

Lüscher Analysis v/s Inverse Lüscher

−0.5 0.0 0.5

Det

√s[model]

1

3 4 5 6 7

tmin[a]

√s[avg]

1

Single-exponential 3 5 7 9 11

t[a]

1200 1300 1400 1500

√s1(MeV)

−0.5 0.0 0.5

√s[model]

2

√s[avg]

2

1400 1600 1800 2000 2200

√s2(MeV) | d| = 0, Λ = Hg

Srijit Paul s.paul@hpc-leap.eu 8/14

8/14

slide-28
SLIDE 28

Lüscher Analysis v/s Inverse Lüscher

−0.5 0.0 0.5

Det

√s[model]

1

3 4 5 6 7

tmin[a]

√s[avg]

1

Single-exponential 3 5 7 9 11

t[a]

1200 1400 1600 1800 2000

√s1(MeV)

−0.5 0.0 0.5

√s[model]

2

√s[avg]

2

1200 1400 1600 1800 2000

√s2(MeV) | d| = 2π

L

√ 2, Λ = G

Srijit Paul s.paul@hpc-leap.eu 9/14

9/14

slide-29
SLIDE 29

χ2 with a model fit

χ2 =

∑

⃗ P,Λ,n

∑

⃗ P ′,Λ′,n′

( √

sΛ, ⃗

P n [avg]

−

√

sΛ, ⃗

P n [model]

)

[C−1] ⃗

P,Λ,n; ⃗ P ′,Λ′,n′

   √

sΛ′, ⃗

P ′ n′ [avg] −

√

sΛ′, ⃗

P ′ n′ [model]

   ,

1430 MeV (94) 25.7 (4)

Srijit Paul s.paul@hpc-leap.eu 10/14

10/14

slide-30
SLIDE 30

χ2 with a model fit

χ2 =

∑

⃗ P,Λ,n

∑

⃗ P ′,Λ′,n′

( √

sΛ, ⃗

P n [avg]

−

√

sΛ, ⃗

P n [model]

)

[C−1] ⃗

P,Λ,n; ⃗ P ′,Λ′,n′

   √

sΛ′, ⃗

P ′ n′ [avg] −

√

sΛ′, ⃗

P ′ n′ [model]

   ,

m∆ 1430 MeV (94) g∆−πN 25.7 (4)

Srijit Paul s.paul@hpc-leap.eu 10/14

10/14

slide-31
SLIDE 31

Phase Shift plot

1350 1400 1450 1500 1550 1600 1650 1700 s(MeV) 45 90 135 180

1[ ]

T-MATRIX FIT

P R E L I M I N A R Y

m

3/2, 1 = 1430(94) MeV

g

N = 25.7(4)

OD

h

CD

2v

J=3/2, P-wave Analysis

Srijit Paul s.paul@hpc-leap.eu 11/14

11/14

slide-32
SLIDE 32

Contemporary results

Collab. mπ(MeV) Methodology m∆(MeV) coupling Verduci(2014) 266(3) (WC)Distillation, Lüscher 1396(19) 19.9(83) Alexandrou et.al. (2013) 360 (DW)Michael, McNeile

  • 26.7(0.6)(1.4)

Alexandrou et.al. (2015) 180 (DW)Michael, McNeile

  • 23.7(0.7)(1.1)

Andersen et.al. (2017) 280 (WC)Stoch. Distillation, Lüscher 1344(20) 37.1(9.2) Our result(Preliminary) 258.3(1.1) (WC)Src-smear, Lüscher 1430(94) 25.7(4) Physical Value 139.57018(35)

  • phen. , K-matrix

1232(1) 29.4(3), 28.6(3)

We differ from the above calculations, in terms of analysis methods, in the following ways:

  • Tuned Smearing parameters,
  • We use direct -matrix fits, for estimating the resonance

parameters.

Srijit Paul s.paul@hpc-leap.eu 12/14

12/14

slide-33
SLIDE 33

Contemporary results

Collab. mπ(MeV) Methodology m∆(MeV) coupling Verduci(2014) 266(3) (WC)Distillation, Lüscher 1396(19) 19.9(83) Alexandrou et.al. (2013) 360 (DW)Michael, McNeile

  • 26.7(0.6)(1.4)

Alexandrou et.al. (2015) 180 (DW)Michael, McNeile

  • 23.7(0.7)(1.1)

Andersen et.al. (2017) 280 (WC)Stoch. Distillation, Lüscher 1344(20) 37.1(9.2) Our result(Preliminary) 258.3(1.1) (WC)Src-smear, Lüscher 1430(94) 25.7(4) Physical Value 139.57018(35)

  • phen. , K-matrix

1232(1) 29.4(3), 28.6(3)

We differ from the above calculations, in terms of analysis methods, in the following ways:

  • Tuned Smearing parameters,

W

[

U2−HEX

]

D−1 W

[

U2−HEX

]†

(N, αWUP) = (45, 3.0)

  • We use direct t-matrix fits, for estimating the resonance

parameters.

Srijit Paul s.paul@hpc-leap.eu 12/14

12/14

slide-34
SLIDE 34

Outlook

  • We have done the calculation on only 1/3rd of total

configurations available.

  • We have another volume

available with the same pion mass and lattice constant, which would help to sample the phase shift curve more precisely.

  • With the application of -matrix fits, we can deal with

non-resonant contributions from other ’s and ’s.

  • Only one Breit Wigner model was taken into account, more

models need to be taken with more data.

Srijit Paul s.paul@hpc-leap.eu 13/14

13/14

slide-35
SLIDE 35

Outlook

  • We have done the calculation on only 1/3rd of total

configurations available.

  • We have another volume (323 × 48) available with the same

pion mass and lattice constant, which would help to sample the phase shift curve more precisely.

  • With the application of -matrix fits, we can deal with

non-resonant contributions from other ’s and ’s.

  • Only one Breit Wigner model was taken into account, more

models need to be taken with more data.

Srijit Paul s.paul@hpc-leap.eu 13/14

13/14

slide-36
SLIDE 36

Outlook

  • We have done the calculation on only 1/3rd of total

configurations available.

  • We have another volume (323 × 48) available with the same

pion mass and lattice constant, which would help to sample the phase shift curve more precisely.

  • With the application of t-matrix fits, we can deal with

non-resonant contributions from other J’s and l’s.

  • Only one Breit Wigner model was taken into account, more

models need to be taken with more data.

Srijit Paul s.paul@hpc-leap.eu 13/14

13/14

slide-37
SLIDE 37

Outlook

  • We have done the calculation on only 1/3rd of total

configurations available.

  • We have another volume (323 × 48) available with the same

pion mass and lattice constant, which would help to sample the phase shift curve more precisely.

  • With the application of t-matrix fits, we can deal with

non-resonant contributions from other J’s and l’s.

  • Only one Breit Wigner model was taken into account, more

models need to be taken with more data.

Srijit Paul s.paul@hpc-leap.eu 13/14

13/14

slide-38
SLIDE 38

Collaborators

Constantia Alexandrou (University of Cyprus / The Cyprus Institute) Stefan Krieg (Forschungszentrum Jülich / University of Wuppertal) Giannis Koutsou (The Cyprus Institute) Luka Leskovec (University of Arizona) Stefan Meinel (University of Arizona / RIKEN BNL Research Center) John Negele (MIT) Marcus Petschlies (University of Bonn / Bethe Center for Theoretical Physics) Andrew Pochinsky (MIT) Gumaro Rendon (University of Arizona) Giorgio Silvi (Forschungszentrum Jülich / University of Wuppertal) Sergey Syritsyn (Stony Brook University / RIKEN BNL Research Center)

Srijit Paul s.paul@hpc-leap.eu 14/14

14/14