TOWARDS THE P-WAVE N SCATTERING AMPLITUDE IN THE …
Transcript of 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
July 27, 2018 Slide 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). . . . . .
July 27, 2018 Slide 3
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). . . . . .
July 27, 2018 Slide 3
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.
July 27, 2018 Slide 4
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.
July 27, 2018 Slide 4
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
July 27, 2018 Slide 5
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
July 27, 2018 Slide 5
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.
July 27, 2018 Slide 6
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 ODh (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.
July 27, 2018 Slide 6
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.
July 27, 2018 Slide 6
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.
July 27, 2018 Slide 6
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
July 27, 2018 Slide 7
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)
July 27, 2018 Slide 8
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
July 27, 2018 Slide 9
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 p
double 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
July 27, 2018 Slide 9
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 Λ
row r and occurence mis 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
July 27, 2018 Slide 9
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
July 27, 2018 Slide 9
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
July 27, 2018 Slide 9
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
July 27, 2018 Slide 9
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
July 27, 2018 Slide 9
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
July 27, 2018 Slide 9
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
July 27, 2018 Slide 10
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]
July 27, 2018 Slide 11
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)}
July 27, 2018 Slide 12
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
July 27, 2018 Slide 13
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
July 27, 2018 Slide 14
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
July 27, 2018 Slide 15
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
July 27, 2018 Slide 16
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
July 27, 2018 Slide 17
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
July 27, 2018 Slide 18
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
July 27, 2018 Slide 19
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
July 27, 2018 Slide 20
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
July 27, 2018 Slide 21
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
July 27, 2018 Slide 22
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
July 27, 2018 Slide 23
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)
July 27, 2018 Slide 24
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)
July 27, 2018 Slide 24
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
July 27, 2018 Slide 26
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
July 27, 2018 Slide 27
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
July 27, 2018 Slide 28
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
July 27, 2018 Slide 29
BACK-UP SLIDECD
3v − irrepF1 preliminary
0 2 4 6 8 10 12 14t/a
1200
1300
1400
1500
1600
1700
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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
July 27, 2018 Slide 30
BACK-UP SLIDECD
3v − irrepF2 preliminary
0 2 4 6 8 10 12 14t/a
1200
1300
1400
1500
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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
July 27, 2018 Slide 31