TOWARDS THE P-WAVE N SCATTERING AMPLITUDE IN THE …

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

Interpolating fields and spectra

July 27, 2018 Giorgio Silvi Forschungszentrum Julich

COLLABORATORSConstantia Alexandrou (University of Cyprus / The Cyprus Institute)

Giannis Koutsou (The Cyprus Institute)

Stefan Krieg (Forschungszentrum Julich / 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 Julich / University of Wuppertal)

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

July 27, 2018 Slide 1

THE DELTA(1232)the first baryon resonance

In nature: ∆−∆0∆+∆++ (u,d quarks) - mass ∼ 1232 MeVOn the lattice: isospin symmetry

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

Orbital angular momentum: L = 1


EXPERIMENTAL INFO

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

[Shrestha,Manley (2012)]

.. but there are resonancesnearby.

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). . . . . .


EXPERIMENTAL INFO

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

[Shrestha,Manley (2012)]

.. but there are resonancesnearby.

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). . . . . .


LUSCHER METHOD

Luscher 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 finitevolume to the unknown phases δJl in the infinite volume via thecalculable non-diagonal matrix MJlm,J′l ′m′ (depends on symmetry)

Simplify!

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


LUSCHER METHOD

Luscher 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 finitevolume to the unknown phases δJl in the infinite volume via thecalculable non-diagonal matrix MJlm,J′l ′m′ (depends on symmetry)

Simplify!

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


MOVING FRAMES

ProblemDue to quantized momenta p = 2πn/L we have a energy levelsspaced from each other. Chances of hitting the energy region ofinterest are low.

Solution: Moving frames!

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

|0,0,1| |1,1,1||0,1,1|

momentum directions


MOVING FRAMES

ProblemDue to quantized momenta p = 2πn/L we have a energy levelsspaced from each other. Chances of hitting the energy region ofinterest are low.

Solution: Moving frames!

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

|0,0,1| |1,1,1||0,1,1|

momentum directions


ANGULAR MOMENTUM ON THE LATTICEIn the continuum, states are classified according to angularmomentum 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 ofsymmetry)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 oneof the finite irreps Λ of the group OD

h on the lattice.




On the lattice the rotational symmetry is broken [R.C. Johnson (1982) ]:The symmetry left is the Oh group of 48 elements (13 axis ofsymmetry)

For half-integer J we need the double cover ODh (96 elements)

which include the negative identity (2π rotation)


h on the lattice.








h on the lattice.








h on the lattice.


GROUND PLAN

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

Pref [Ndir ] Group Nelem Λ(J) : π( 0− ) Λ(J) : N( 12

+) Λ(J) : ∆( 3

2+

)

(0,0,0) [1] ODh 96 A1u( 0 ,4, ...) G1g( 1

2 ,72 , ...)⊕G1u( 1

2 ,72 , ...) Hg( 3

2 ,52 , ...)⊕ Hu( 3

2 ,52 , ...)

(0,0,1) [6] CD4v 16 A2( 0 ,1, ...) G1( 1

2 ,32 , ...) G1(1

2 ,32 , ...)⊕G2( 3

2 ,52 , ...)

(0,1,1) [12] CD2v 8 A2( 0 ,1, ...) G( 1

2 ,32 , ...) G(1

2 ,32 , ...)

(1,1,1) [8] CD3v 12 A2( 0 ,1, ...) G( 1

2 ,32 , ...) G(1

2 ,32 , ...)⊕ F1( 3

2 ,52 , ...)⊕ F2( 3

2 ,52 , ...)

C4vD C3v

DC2vD

axis of symmetry


SINGLE HADRON OPERATORS

Delta interpolators:

∆(1)iµ = εabcua

µ(ubT Cγiuc) (1)

∆(2)iµ = εabcua

µ(ubT Cγiγ0uc) (2)

Nucleon interpolators:

N (1)µ = εabcua

µ(ubT Cγ5dc) (3)

N (2)µ = εabcua

µ(ubT Cγ0γ5dc) (4)

Pion interpolator:π = dγ5u (5)


PROJECTION METHODhow it works...

OGD ,Λ,r ,m(p)= dΛgGD

∑R∈GD ΓΛ

r ,r (R)URφ(p)U−1R

[C. Morningstar et al. (2013)]

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

momentum pdouble group GD and irreducible representation Λrow 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 UR




∑R∈GD ΓΛ



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

double group GD and irreducible representation Λrow r and occurence m

is needed :







∑R∈GD ΓΛ



to get an operators OGD ,Λ,r ,m(p) for a specific:momentum pdouble group GD and irreducible representation Λ

row r and occurence mis needed :







∑R∈GD ΓΛ



to get an operators OGD ,Λ,r ,m(p) for a specific:momentum pdouble group GD and irreducible representation Λrow r and occurence m

is needed :







∑R∈GD ΓΛ




is needed :1 representation matrices ΓΛ






∑R∈GD ΓΛ










∑R∈GD ΓΛ






3 single/multi hadron operator φ(p)

4 proper transformation matrices UR




∑R∈GD ΓΛ








OCCURENCES OF IRREPS

It is possible to find the occurence (∼multiplicity) m of the irrep ΓΛ inthe transformation matrices UR using the character χ: [Moore,Fleming (2006) ]

m = 1gGD

∑R∈GD χ

ΓΛ(R)χU(R)

G1g

G1u

OhD

[0,0,0]

G1

G1

C4vD

[0,0,1]

G

G

C2vD

[0,1,1]

G

G

C3vD

[1,1,1]

UR [4x4]

NUCLEON


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

OhD

[0,0,0]

C4vD

[0,0,1]

C2vD

[0,1,1]

C3vD

[1,1,1]

DELTAUR [12x12]


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)}


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

12 ∆(0,−1, 0)(1, 1)− 1

2 ∆(0,−1, 0)(1, 3)− 12 ∆(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

12 ∆(1, 0, 0)(2, 1)− 1

2 ∆(1, 0, 0)(2, 3)− 12 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)

12 i∆(1, 0, 0)(2, 2) + 1

2 i∆(1, 0, 0)(2, 4) + 12 ∆(1, 0, 0)(3, 1) + 1

2 ∆(1, 0, 0)(3, 3)12 ∆(0,−1, 0)(1, 2) + 1

2 ∆(0,−1, 0)(1, 4) + 12 ∆(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


COMPLETE SET OF OPERATOR: ∆,NπGroup Pref [dir. used] Irrep[rows] Op. type n Op. per irrep

ODh (0, 0, 0)[1] Hg [4] ∆(γi ) 1

∆(γiγ0) 1Nπ(γ5) 8

Nπ(γ0γ5) 818 ×4rows 72

Hu [4] 18 ×4rows 72CD

4v (0, 0, 1)[3] G1[2] ∆ 4× 2Nπ 20× 2

48 ×2rows × 3dir 288G2[2] ∆ 2× 2

Nπ 16× 236 ×2rows × 3dir 216

CD2v (0, 1, 1)[6] G[2] ∆ 6× 2

Nπ 24× 260 ×2rows × 6dir 720

CD3v (1, 1, 1)[4] G[2] ∆ 4× 2

Nπ 12× 232 ×2rows × 4dir 256

F1[1] ∆ 2× 2Nπ 4× 2

12 ×4dir 48F2[1] 12 ×4dir 48

Total 1720


PLANNINGEnsemble from BMW collaboration

Ns Nt β amu,d ams csw a(fm) L(fm) mπ(MeV ) mπL24 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 fermionsTwo-point correlators built from a combination of smeared forward,sequential and stochastic propagatorsThis 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 directionApply the projected operators on correlators


CORRELATION MATRICESAfter projectioning the correlators...

source

- -N

N - N -N

sink

op. 1

op. 2

op. N

op. 1 op. 2 op. N

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

Use GEVP to determine the spectraSearch the best basis of operators


HEATMAPRest frame / Moving frame

d_a_

xtyt

zt

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

Posit

ive

valu

es

10 9

10 8

Nega

tive

valu

es

ODh − irrepHg

d_1_

xtyt

zt

d_1_

xyz

n_n0

p1_0

5

n_n0

p1_1

3

n_n1

p0_0

5

n_n1

p0_1

3

n_n1

p2_0

5

n_n1

p2_1

3

n_n2

p1_0

5

n_n2

p1_1

3

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

Posit

ive

valu

es

10 8

Nega

tive

valu

es

CD4v − irrepG1


SPECTRA - REST FRAMEOD

h − irrepHg(+Hu)

2 4 6 8 10 12t/a

1200

1300

1400

1500

1600

1700

1800

1900

mn ef

f,n(t)

[MeV

]

Hg+Hu_123_t1N thresholdN thresholdnon-inter. N [1][-1]GEVP lvl1GEVP lvl2


ANALYSIS OF THE FITS - REST FRAMEOD

h − irrepHg(+Hu)

2 4 6 8 10 12t_min/a

1300

1400

1500

1600

1700

1800

1900

2000

mn ef

f,n(t)

[MeV

]

2.761.19

0.93 1.05

0.87

0.990.91

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 1Fit lvl 2


SPECTRA - REST FRAMEOD

h − irrep : Hg(+Hu)

2 4 6 8 10 12t/a

1300

1400

1500

1600

1700

1800

1900

2000

mn ef

f,n(t)

[MeV

]

Hg+Hu_123_t1N thresholdN thresholdfit lvl1,E=1415(9) MeV, chi2: 0.93fit lvl2,E=1701(16) MeV, chi2: 0.91non-inter. N [1][-1] ,E=1671(3)MeV


SPECTRA - MOVING FRAMECD

4v − irrep : G1

2 4 6 8 10 12t/a

1200

1300

1400

1500

1600

1700

1800

mn ef

f,n(t)

[MeV

]

G1a_235_t1N thresholdN thresholdnon-inter. N GEVP lvl1GEVP lvl2GEVP lvl3


ANALYSIS OF THE FITS - MOVING FRAMECD

4v − irrepG1

2 4 6 8 10 12t_min/a

1300

1400

1500

1600

1700

1800

E[M

eV]

4.65

0.570.47 0.54

0.62

2.82

0.000.16 0.16 0.18

3.57

0.75

0.360.37 0.44

Analysys of the fits. t_max/a=12 (chi2 listed at points)non-inter. N Fit lvl 1Fit lvl 2Fit lvl 3


SPECTRA - MOVING FRAMECD

4v − irrepG1

0 2 4 6 8 10 12t/a

1300

1400

1500

1600

1700

1800

mn ef

f,n(t)

[MeV

]

G1a_235_t1N thresholdN thresholdfit lvl1,E=1321(15) MeV, chi2: 0.47fit lvl2,E=1408(13) MeV, chi2: 0.16fit lvl3,E=1603(23) MeV, chi2: 0.36non-inter. N


CONCLUSION

Complete projection of operators for N,∆ and Nπ in relevantirrepsRoom for improvement of spectra with a thorough search for bestbasisAdditional configurations coming(3×)In the future aim to add a bigger box, more mπ and smaller latticespacing

Thank you for the attention!(second part in the next talk by Srijit)


CONCLUSION

Complete projection of operators for N,∆ and Nπ in relevantirrepsRoom for improvement of spectra with a thorough search for bestbasisAdditional configurations coming(3×)In the future aim to add a bigger box, more mπ and smaller latticespacing

Thank you for the attention!(second part in the next talk by Srijit)



BACK-UP SLIDE

Hg

G1g

G1u

Hg

N-πOh

D

P_tot=[0,0,0]

G1g

G1g

G1u

G1u

Hu

Hu

G1g

G1g

G1u

G1u

G2g

G2g

G2u

G2u

Hg

Hg

Hu

Hu

G1g

G1g

G1u

G1u

G2g

G2g

G2u

G2u

Hg

Hg

Hg

Hg

Hu

Hu

Hu

Hu

<- ->|[0,0,0]|

4

<- ->|[0,0,1]|

24

<- ->|[1,1,1]|

32

<- ->|[0,1,1]|

48

4+24+32+48=108=3*3*3*4 (px,py,pz,dirac)

NUCLEON- PION TENSOR


BACK-UP SLIDECD

4v − irrepG2

0 2 4 6 8 10 12t/a

1300

1400

1500

1600

1700

1800

mn ef

f,n(t)

[MeV

]

G2a_1236_t1N thresholdN thresholdfit lvl1,E=1376(14) MeV, chi2: 0.76fit lvl2,E=1746(32) MeV, chi2: 0.40non-inter. N


BACK-UP SLIDECD

2v − irrepG

0 2 4 6 8 10t/a

1200

1300

1400

1500

1600

1700

mn ef

f,n(t)

[MeV

]

2G_123578_t1N thresholdN thresholdfit lvl1,E=1304(28) MeV, chi2: 0.03fit lvl2,E=1454(16) MeV, chi2: 0.92non-inter. N


BACK-UP SLIDECD

3v − irrepG preliminary

0 2 4 6 8 10 12 14t/a

1200

1300

1400

1500

1600

1700

1800

1900

mn ef

f,n(t)

[MeV

]

Ga_246810_t1N+Pi thresholdN+Pi+Pi thresholdfit lvl1,E=1376(19) MeV, chi2: 1.52fit lvl2,E=1448(25) MeV, chi2: 0.94non-inter. N-pi0 ,1372.62(+-2.79)MeVnon-inter. N-pi1 ,1574.67(+-2.79)MeVnon-inter. N-pi2 ,1666.68(+-2.79)MeV


BACK-UP SLIDECD

3v − irrepF1 preliminary

0 2 4 6 8 10 12 14t/a

1200

1300

1400

1500

1600

1700

1800

1900

mn ef

f,n(t)

[MeV

]

F1a_1256_t1N+Pi thresholdN+Pi+Pi thresholdfit lvl1,E=1357(46) MeV, chi2: 1.08fit lvl2,E=1609(36) MeV, chi2: 3.32non-inter. N-pi0 ,1372.62(+-2.79)MeVnon-inter. N-pi1 ,1574.67(+-2.79)MeVnon-inter. N-pi2 ,1666.68(+-2.79)MeV


BACK-UP SLIDECD

3v − irrepF2 preliminary

0 2 4 6 8 10 12 14t/a

1200

1300

1400

1500

1600

1700

1800

1900

mn ef

f,n(t)

[MeV

]

F2a_1256_t1N+Pi thresholdN+Pi+Pi thresholdfit lvl1,E=1314(41) MeV, chi2: 0.60fit lvl2,E=1635(25) MeV, chi2: 0.85non-inter. N-pi0 ,1372.62(+-2.79)MeVnon-inter. N-pi1 ,1574.67(+-2.79)MeVnon-inter. N-pi2 ,1666.68(+-2.79)MeV


TOWARDS THE P-WAVE N SCATTERING AMPLITUDE IN THE …

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