[PPT] - TOWARDS THE P-WAVE N SCATTERING AMPLITUDE IN THE (1232) PowerPoint Presentation

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TOWARDS THE P-WAVE Nπ SCATTERING AMPLITUDE IN THE ∆ (1232)

Interpolating fields and spectra July 27, 2018 Giorgio Silvi Forschungszentrum J¨ ulich

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COLLABORATORS

Constantia Alexandrou (University of Cyprus / The Cyprus Institute) Giannis Koutsou (The Cyprus Institute) Stefan Krieg (Forschungszentrum J¨ ulich / University of Wuppertal) Luka Leskovec (University of Arizona) Stefan Meinel (University of Arizona / RIKEN BNL Research Center) John Negele (MIT) Srijit Paul (The Cyprus Institute / University of Wuppertal) Marcus Petschlies (University of Bonn / Bethe Center for Theoretical Physics) Andrew Pochinsky (MIT) Gumaro Rendon (University of Arizona)

G. S. (Forschungszentrum J¨

ulich / University of Wuppertal) Sergey Syritsyn (Stony Brook University / RIKEN BNL Research Center)

July 27, 2018 Slide 1

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THE DELTA(1232)

the first baryon resonance

In nature: ∆−∆0∆+∆++ (u,d quarks) - mass ∼ 1232 MeV

On the lattice: isospin symmetry

The unstable ∆(1232) decay predominantly to stable Nπ Study: Pion-Nucleon scattering J = 3/2 , I = 3/2, I3 = +3/2

Orbital angular momentum: L = 1

July 27, 2018 Slide 2

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EXPERIMENTAL INFO

Nπ(→ ∆(1232)) → Nπ completely elastic...

[Shrestha,Manley (2012)]

July 27, 2018 Slide 3

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EXPERIMENTAL INFO

Nπ(→ ∆(1232)) → Nπ completely elastic...

[Shrestha,Manley (2012)]

.. but there are resonances nearby. Particle JP ΓNπ[MeV] ∆(1232) 3/2+ 112.4(5) ∆(1600) 3/2+ 18(4) ∆(1620) 1/2− 37(2) ∆(1700) 3/2− 36(2) . . . . . .

July 27, 2018 Slide 3

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L ¨ USCHER METHOD

L¨ uscher quantization condition for baryons

det[MJlm,J′l′m′ − δJJ′δll′δmm′ cot δJl] = 0

[Gockeler et al. (2012)]

This relation connect the energy E from a lattice simulation in a finite volume to the unknown phases δJl in the infinite volume via the calculable non-diagonal matrix MJlm,J′l′m′ (depends on symmetry)

July 27, 2018 Slide 4

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L ¨ USCHER METHOD

L¨ uscher quantization condition for baryons

det[MJlm,J′l′m′ − δJJ′δll′δmm′ cot δJl] = 0

[Gockeler et al. (2012)]

This relation connect the energy E from a lattice simulation in a finite volume to the unknown phases δJl in the infinite volume via the calculable non-diagonal matrix MJlm,J′l′m′ (depends on symmetry)

Simplify!

With a proper transformation, the matrix MJlm,J′l′m′ can be block diagonalized in the basis of the irreps Λ of the lattice.

July 27, 2018 Slide 4

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MOVING FRAMES

Problem

Due to quantized momenta p = 2πn/L we have a energy levels spaced from each other. Chances of hitting the energy region of interest are low.

|0,0,1| |1,1,1| |0,1,1| momentum directions

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MOVING FRAMES

Problem

Due to quantized momenta p = 2πn/L we have a energy levels spaced from each other. Chances of hitting the energy region of interest are low.

Solution: Moving frames!

The Lorentz boost contracts the box giving a different effective value of the size L. Allow access to phase shift at different energies!

|0,0,1| |1,1,1| |0,1,1| momentum directions

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ANGULAR MOMENTUM ON THE LATTICE

In the continuum, states are classified according to angular momentum J and parity P

label of the irreps of the symmetry group SU(2)

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ANGULAR MOMENTUM ON THE LATTICE

In the continuum, states are classified according to angular momentum J and parity P

label of the irreps of the symmetry group SU(2)

On the lattice the rotational symmetry is broken [R.C. Johnson (1982) ]:

The symmetry left is the Oh group of 48 elements (13 axis of symmetry)

July 27, 2018 Slide 6

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ANGULAR MOMENTUM ON THE LATTICE

In the continuum, states are classified according to angular momentum J and parity P

label of the irreps of the symmetry group SU(2)

On the lattice the rotational symmetry is broken [R.C. Johnson (1982) ]:

The symmetry left is the Oh group of 48 elements (13 axis of symmetry) For half-integer J we need the double cover OD

h (96 elements)

which include the negative identity (2π rotation)

July 27, 2018 Slide 6

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ANGULAR MOMENTUM ON THE LATTICE

In the continuum, states are classified according to angular momentum J and parity P

label of the irreps of the symmetry group SU(2)

On the lattice the rotational symmetry is broken [R.C. Johnson (1982) ]:

The symmetry left is the Oh group of 48 elements (13 axis of symmetry) For half-integer J we need the double cover OD

h (96 elements)

which include the negative identity (2π rotation)

Each of the infinite irreps JP in the continuum get mapped to one

f the finite irreps Λ of the group OD

h on the lattice.

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GROUND PLAN

Frames, Groups & Irreps Λ (with ang. mom. content)

Pref[Ndir] Group Nelem Λ(J) : π( 0− ) Λ(J) : N( 1

2 + )

Λ(J) : ∆( 3

2 + )

(0, 0, 0) [1] OD

h

96 A1u( 0 , 4, ...) G1g( 1

2 , 7 2, ...) ⊕ G1u( 1 2 , 7 2, ...)

Hg( 3

2 , 5 2, ...) ⊕ Hu( 3 2 , 5 2, ...)

(0, 0, 1) [6] CD

4v

16 A2( 0 , 1, ...) G1( 1

2 , 3 2, ...)

G1( 1

2, 3 2 , ...) ⊕ G2( 3 2 , 5 2, ...)

(0, 1, 1) [12] CD

2v

8 A2( 0 , 1, ...) G( 1

2 , 3 2, ...)

G( 1

2, 3 2 , ...)

(1, 1, 1) [8] CD

3v

12 A2( 0 , 1, ...) G( 1

2 , 3 2, ...)

G( 1

2, 3 2 , ...) ⊕ F1( 3 2 , 5 2, ...) ⊕ F2( 3 2 , 5 2, ...)

C4v

D

C3v

D

C2v

D

axis of symmetry

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SINGLE HADRON OPERATORS

Delta interpolators: ∆(1)

iµ = ǫabcua µ(ubTCγiuc)

(1) ∆(2)

iµ = ǫabcua µ(ubTCγiγ0uc)

(2) Nucleon interpolators: N (1)

µ

= ǫabcua

µ(ubTCγ5dc)

(3) N (2)

µ

= ǫabcua

µ(ubTCγ0γ5dc)

(4) Pion interpolator: π = ¯ dγ5u (5)

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PROJECTION METHOD

how it works...

OGD,Λ,r,m(p)=

dΛ gGD

˜

R∈GD ΓΛ r,r(˜

R)U˜

Rφ(p)U−1 ˜ R

[C. Morningstar et al. (2013)]

to get an operators OGD,Λ,r,m(p) for a specific:

July 27, 2018 Slide 9

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PROJECTION METHOD

how it works...

OGD,Λ,r,m(p)=

dΛ gGD

˜

R∈GD ΓΛ r,r(˜

R)U˜

Rφ(p)U−1 ˜ R

[C. Morningstar et al. (2013)]

to get an operators OGD,Λ,r,m(p) for a specific:

momentum p

July 27, 2018 Slide 9

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PROJECTION METHOD

how it works...

OGD,Λ,r,m(p)=

dΛ gGD

˜

R∈GD ΓΛ r,r(˜

R)U˜

Rφ(p)U−1 ˜ R

[C. Morningstar et al. (2013)]

to get an operators OGD,Λ,r,m(p) for a specific:

momentum p double group GD and irreducible representation Λ

July 27, 2018 Slide 9

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PROJECTION METHOD

how it works...

OGD,Λ,r,m(p)=

dΛ gGD

˜

R∈GD ΓΛ r,r(˜

R)U˜

Rφ(p)U−1 ˜ R

[C. Morningstar et al. (2013)]

to get an operators OGD,Λ,r,m(p) for a specific:

momentum p double group GD and irreducible representation Λ row r and occurence m

July 27, 2018 Slide 9

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PROJECTION METHOD

how it works...

OGD,Λ,r,m(p)=

dΛ gGD

˜

R∈GD ΓΛ r,r(˜

R)U˜

Rφ(p)U−1 ˜ R

[C. Morningstar et al. (2013)]

to get an operators OGD,Λ,r,m(p) for a specific:

momentum p double group GD and irreducible representation Λ row r and occurence m

is needed :

1 representation matrices ΓΛ

July 27, 2018 Slide 9

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PROJECTION METHOD

how it works...

OGD,Λ,r,m(p)=

dΛ gGD

˜

R∈GD ΓΛ r,r(˜

R)U˜

Rφ(p)U−1 ˜ R

[C. Morningstar et al. (2013)]

to get an operators OGD,Λ,r,m(p) for a specific:

momentum p double group GD and irreducible representation Λ row r and occurence m

is needed :

1 representation matrices ΓΛ 2 elements ˜

R of the double group

– rotations + inversions

July 27, 2018 Slide 9

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PROJECTION METHOD

how it works...

OGD,Λ,r,m(p)=

dΛ gGD

˜

R∈GD ΓΛ r,r(˜

R)U˜

Rφ(p)U−1 ˜ R

[C. Morningstar et al. (2013)]

to get an operators OGD,Λ,r,m(p) for a specific:

momentum p double group GD and irreducible representation Λ row r and occurence m

is needed :

1 representation matrices ΓΛ 2 elements ˜

R of the double group

– rotations + inversions

3 single/multi hadron operator φ(p)

July 27, 2018 Slide 9

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PROJECTION METHOD

how it works...

OGD,Λ,r,m(p)=

dΛ gGD

˜

R∈GD ΓΛ r,r(˜

R)U˜

Rφ(p)U−1 ˜ R

[C. Morningstar et al. (2013)]

to get an operators OGD,Λ,r,m(p) for a specific:

momentum p double group GD and irreducible representation Λ row r and occurence m

is needed :

1 representation matrices ΓΛ 2 elements ˜

R of the double group

– rotations + inversions

3 single/multi hadron operator φ(p) 4 proper transformation matrices U˜

R

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OCCURENCES OF IRREPS

It is possible to find the occurence (∼multiplicity) m of the irrep ΓΛ in the transformation matrices U˜

R using the character χ: [Moore,Fleming (2006) ]

m =

1 gGD

˜

R∈GD χΓΛ(˜

R)χU(˜ R)

G1g G1u

Oh

D [0,0,0]

G1 G1

C4v

D [0,0,1]

G G

C2v

D [0,1,1]

G G

C3v

D [1,1,1]

UR̃ [4x4]

NUCLEON

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OCCURENCES OF IRREPS

G G

Hg

G1g G1u G1 G1 G1 G2 G1 G2

Hu

G G G G G G G G

F1 F2 F1 F2

Oh

D [0,0,0]

C4v

D [0,0,1]

C2v

D [0,1,1]

C3v

D [1,1,1]

DELTA

UR̃ [12x12]

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EXAMPLE OF PROJECTION: NUCLEON

C4v - G1 - first instance (row 1 , row 2)

       {Nc(0, 0, 1)(3)} {Nc(0, 0, 1)(4)} {Nc(0, 0, −1)(3)} {Nc(0, 0, −1)(4)} {Nc(1, 0, 0)(4) − Nc(1, 0, 0)(2)} {Nc(1, 0, 0)(2) + Nc(1, 0, 0)(4)} {Nc(−1, 0, 0)(4) − Nc(−1, 0, 0)(2)} {Nc(−1, 0, 0)(2) + Nc(−1, 0, 0)(4)} {Nc(0, −1, 0)(4) − Nc(0, −1, 0)(2)} {Nc(0, −1, 0)(2) + Nc(0, −1, 0)(4)} {Nc(0, 1, 0)(4) − Nc(0, 1, 0)(2)} {Nc(0, 1, 0)(2) + Nc(0, 1, 0)(4)}       

C4v - G1 - second instance (row 1 , row 2)

       {Nc(0, 0, 1)(1)} {Nc(0, 0, 1)(2)} {Nc(0, 0, −1)(1)} {Nc(0, 0, −1)(2)} {Nc(1, 0, 0)(1) + Nc(1, 0, 0)(3)} {Nc(1, 0, 0)(3) − Nc(1, 0, 0)(1)} {Nc(−1, 0, 0)(1) + Nc(−1, 0, 0)(3)} {Nc(−1, 0, 0)(3) − Nc(−1, 0, 0)(1)} {Nc(0, −1, 0)(1) + Nc(0, −1, 0)(3)} {Nc(0, −1, 0)(3) − Nc(0, −1, 0)(1)} {Nc(0, 1, 0)(1) + Nc(0, 1, 0)(3)} {Nc(0, 1, 0)(3) − Nc(0, 1, 0)(1)}       

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EXAMPLE OF PROJECTION: DELTA

C4v - G1 - row 1 - instance 1

∆(0, 0, 1)(3, 1)

∆(1,0,0)(1,1) √ 2

− ∆(1,0,0)(1,3)

√ 2 1 2 ∆(0, −1, 0)(1, 1) − 1 2 ∆(0, −1, 0)(1, 3) − 1 2 ∆(0, −1, 0)(3, 2) + 1 2 ∆(0, −1, 0)(3, 4)

C4v - G1 - row 1 - instance 2

∆(0,0,1)(1,2) √ 2

+ i∆(0,0,1)(2,2)

√ 2 1 2 ∆(1, 0, 0)(2, 1) − 1 2 ∆(1, 0, 0)(2, 3) − 1 2 i∆(1, 0, 0)(3, 2) + 1 2 i∆(1, 0, 0)(3, 4) ∆(0,−1,0)(2,1) √ 2

− ∆(0,−1,0)(2,3)

√ 2

C4v - G1 - row 1 - instance 3

∆(0, 0, 1)(3, 3)

1 2 i∆(1, 0, 0)(2, 2) + 1 2 i∆(1, 0, 0)(2, 4) + 1 2 ∆(1, 0, 0)(3, 1) + 1 2 ∆(1, 0, 0)(3, 3) 1 2 ∆(0, −1, 0)(1, 2) + 1 2 ∆(0, −1, 0)(1, 4) + 1 2 ∆(0, −1, 0)(3, 1) + 1 2 ∆(0, −1, 0)(3, 3)

C4v - G1 - row 1 - instance 4

∆(0,0,1)(1,4) √ 2

+ i∆(0,0,1)(2,4)

√ 2 ∆(1,0,0)(1,2) √ 2

+ ∆(1,0,0)(1,4)

√ 2 ∆(0,−1,0)(2,2) √ 2

+ ∆(0,−1,0)(2,4)

√ 2 July 27, 2018 Slide 13

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COMPLETE SET OF OPERATOR: ∆, Nπ

Group Pref [dir. used] Irrep[rows]

Op. type

n

Op. per irrep

OD

h

(0, 0, 0)[1] Hg[4] ∆(γi ) 1 ∆(γi γ0) 1 Nπ(γ5) 8 Nπ(γ0γ5) 8 18 ×4rows 72 Hu[4] 18 ×4rows 72 CD

4v

(0, 0, 1)[3] G1[2] ∆ 4 × 2 Nπ 20 × 2 48 ×2rows × 3dir 288 G2[2] ∆ 2 × 2 Nπ 16 × 2 36 ×2rows × 3dir 216 CD

2v

(0, 1, 1)[6] G[2] ∆ 6 × 2 Nπ 24 × 2 60 ×2rows × 6dir 720 CD

3v

(1, 1, 1)[4] G[2] ∆ 4 × 2 Nπ 12 × 2 32 ×2rows × 4dir 256 F1[1] ∆ 2 × 2 Nπ 4 × 2 12 ×4dir 48 F2[1] 12 ×4dir 48 Total 1720

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PLANNING

Ensemble from BMW collaboration Ns Nt β amu,d ams csw a(fm) L(fm) mπ(MeV) mπL 24 48 3.31 −0.0953 −0.040 1.0 0.116 2.8 254 3.6

[ensemble description in S. Durr et al.(2011)

Lattice action: Wilson-Clover with Nf = 2 + 1 dynamical fermions Two-point correlators built from a combination of smeared forward, sequential and stochastic propagators This talk: 192 configurations,16 source location per conf.

Beginning

Project the operators on all relevant irreps (Wolfram Mathematica) Compute contraction for all diagrams and momentum direction Apply the projected operators on correlators

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CORRELATION MATRICES

After projectioning the correlators...

source

N

N- N-N

sink

p. 1
p. 2
p. N
p. 1 op. 2
p. N

(H,G1,G2,G,F1,F2,G)

Use GEVP to determine the spectra Search the best basis of operators

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HEATMAP

Rest frame / Moving frame

d_a_xtytzt d_a_xyz n_ab_05 n_ab_13 n_ac_05 n_ac_13 d_a_xtytzt d_a_xyz n_ab_05 n_ab_13 n_ac_05 n_ac_13 Heatmap 000-Hg+Hu: sum of REAL part of CV correlator from t=1-20 10 10 10 9 10 8 10 7 10 6 Positive values 10 9 10 8 Negative values

OD

h − irrepHg

d_1_xtytzt d_1_xyz n_n0p1_05 n_n0p1_13 n_n1p0_05 n_n1p0_13 n_n1p2_05 n_n1p2_13 n_n2p1_05 n_n2p1_13 d_1_xtytzt d_1_xyz n_n0p1_05 n_n0p1_13 n_n1p0_05 n_n1p0_13 n_n1p2_05 n_n1p2_13 n_n2p1_05 n_n2p1_13 Heatmap 1-G1a: sum of REAL part of CV correlator from t=1-8 10 9 10 8 10 7 Positive values 10 8 Negative values

CD

4v − irrepG1

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SPECTRA - REST FRAME

OD

h − irrepHg(+Hu)

2 4 6 8 10 12 t/a 1200 1300 1400 1500 1600 1700 1800 1900 mn

eff,

n(t) [MeV]

Hg+Hu_123_t1

N threshold N threshold non-inter. N [1][-1] GEVP lvl1 GEVP lvl2 July 27, 2018 Slide 18

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ANALYSIS OF THE FITS - REST FRAME

OD

h − irrepHg(+Hu)

2 4 6 8 10 12 t_min/a 1300 1400 1500 1600 1700 1800 1900 2000 mn

eff,

n(t) [MeV] 2.76 1.19 0.93 1.05 0.87 0.99 0.91 0.98 0.97 1.09

Analysys of the fits. t_max/a=15 (chi2 listed at points)

non-inter. N [1][-1] Fit lvl 1 Fit lvl 2 July 27, 2018 Slide 19

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SPECTRA - REST FRAME

OD

h − irrep : Hg(+Hu)

2 4 6 8 10 12 t/a 1300 1400 1500 1600 1700 1800 1900 2000 mn

eff,

n(t) [MeV]

Hg+Hu_123_t1

N threshold N threshold fit lvl1,E=1415(9) MeV, chi2: 0.93 fit lvl2,E=1701(16) MeV, chi2: 0.91 non-inter. N [1][-1] ,E=1671(3)MeV July 27, 2018 Slide 20

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SPECTRA - MOVING FRAME

CD

4v − irrep : G1

2 4 6 8 10 12 t/a 1200 1300 1400 1500 1600 1700 1800 mn

eff,

n(t) [MeV]

G1a_235_t1

N threshold N threshold non-inter. N GEVP lvl1 GEVP lvl2 GEVP lvl3 July 27, 2018 Slide 21

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ANALYSIS OF THE FITS - MOVING FRAME

CD

4v − irrepG1

2 4 6 8 10 12 t_min/a 1300 1400 1500 1600 1700 1800 E[MeV]

4.65 0.57 0.47 0.54 0.62 2.82 0.00 0.16 0.16 0.18 3.57 0.75 0.36 0.37 0.44

Analysys of the fits. t_max/a=12 (chi2 listed at points)

non-inter. N Fit lvl 1 Fit lvl 2 Fit lvl 3 July 27, 2018 Slide 22

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SPECTRA - MOVING FRAME

CD

4v − irrepG1

2 4 6 8 10 12 t/a 1300 1400 1500 1600 1700 1800 mn

eff,

n(t)[MeV]

G1a_235_t1

N threshold N threshold fit lvl1,E=1321(15) MeV, chi2: 0.47 fit lvl2,E=1408(13) MeV, chi2: 0.16 fit lvl3,E=1603(23) MeV, chi2: 0.36 non-inter. N July 27, 2018 Slide 23

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CONCLUSION

Complete projection of operators for N, ∆ and Nπ in relevant irreps Room for improvement of spectra with a thorough search for best basis Additional configurations coming(3×) In the future aim to add a bigger box, more mπ and smaller lattice spacing

July 27, 2018 Slide 24

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CONCLUSION

Complete projection of operators for N, ∆ and Nπ in relevant irreps Room for improvement of spectra with a thorough search for best basis Additional configurations coming(3×) In the future aim to add a bigger box, more mπ and smaller lattice spacing

Thank you for the attention!

(second part in the next talk by Srijit)

July 27, 2018 Slide 24

SLIDE 40

July 27, 2018 Slide 25

SLIDE 41

CD

3v − irrepF2 preliminary

2 4 6 8 10 12 14 t/a 1200 1300 1400 1500 1600 1700 1800 1900 mn

eff,

n(t)[MeV]

F2a_1256_t1

N+Pi threshold N+Pi+Pi threshold fit lvl1,E=1314(41) MeV, chi2: 0.60 fit lvl2,E=1635(25) MeV, chi2: 0.85 non-inter. N-pi0 ,1372.62(+-2.79)MeV non-inter. N-pi1 ,1574.67(+-2.79)MeV non-inter. N-pi2 ,1666.68(+-2.79)MeV July 27, 2018 Slide 31