the tjon band in nuclear lattice eft - lenpic.org · status of nuclear lattice eft 6-14-12-10-8-6 0...

1

The Tjon Band inNuclear Lattice EFT

Ulf-G. Meißner, Univ. Bonn & FZ Julich

Supported by BMBF 05P15PCFN1 by DFG, SFB/TR-110 by CAS, PIFI by Volkswagen Stftung

The Tjon Band in Nuclear Lattice EFT – Ulf-G. Meißner – Peking University, April 27, 2018 · ◦ C < ∧ O > B •

2

CONTENTS

• Introduction: What is the Tjon line/band?

• Recap: Basics of nuclear lattice simulations

• The two-nucleon system at NNLO

• The three-nucleon system at NNLO

• The four-nucleon system at NNLO & the Tjon band

• Summary & outlook


3

Introduction


4

WHAT is the TJON LINE/BAND?

• Tjon observed a correlation between the 4He and

triton (3H) binding energies for local interactions

Tjon, Phys. Lett. 56B (1975) 217

• This correlation, then called the Tjon line,

was also found using high-precision potentials

↪→ three-nucleon forces required

↪→ no nedd for additional four-nucleon forces

Nogga et al., Phys. Rev. Lett. 85 (2000) 944

Bt

Bα


5UNDERSTANDING the TJON LINE/BAND?

• Consider pionless EFT: contact interactions

〈k′|V |k〉 = Ps λs2 g(k′)g(k) + Pt λt2 g(k′)g(k) + . . . , g(u) = exp(−u2/Λ2)

↪→ physics independent of cut-off Λ

• Renormalization of the three-nucleon system requires a three-nucleon force (3NF)Bedaque, Hammer, van Kolck, Phys. Rev. Lett. 82 (1999) 463

V3 = Paλ3h(u1, u2)h(u′1, u′2) , h(u1, u2) = exp(−(u2

1 + 34u2

2)/Λ2)

• This also renormalizes the four-nucleon system!Platter, Hammer, UGM, Phys. Rev. A70 (2004) 052101

↪→ explains naturally the correlation

↪→ it is really a band (theor. uncertainty)Platter, Hammer, UGM, Phys. Lett. B607 (2005) 254

↪→ width of the band shrinks at higher orders 7.5 8 8.5 9Bt [MeV]

20

25

30

35

Bα [M

eV]

◦ chiral NLO, various Λ , O chiral N2LO� phenomenological potentials

Exp.


6STATUS of NUCLEAR LATTICE EFT

-14

-12

-10

-8

-6

0 5 10 15 20

E (

MeV

)

L (fm)

Etriton (NNLO)Ehelium-3 (NNLO)

physical Etriton (infinite volume)physical Ehelium-3 (infinite volume)

-32

-30

-28

-26

-24

-22

-20

0 0.02 0.04 0.06 0.08 0.1<

E4H

e >

(M

eV

)

t (MeV-1

)

LO+ ∆NLO

+ ∆IB + ∆EM+ ∆NNLO

+ 4N Contact

-2

0

2

4

6

0 0.02 0.04 0.06 0.08 0.1

t (MeV-1

)

∆NLO∆IB + ∆EM

∆NNLO4N Contact

• standard coarse lattice at NNLO, a = 1.97 fm, N = 8

11+2 LECs, incl. isospin breaking and Coulomb

local smearing of the LO contact interactions in S-wave

↪→ 3H and 3He well described

E(3He)− E(3H) = 0.78(5) MeV [exp.: 0.76 MeV]

↪→ 4He overbound by a few MeV

↪→ cured by an effective four-nucleon interaction

V(4)

eff = D(4)eff

∑~n

[ρ(~n)]4

Epelbaum, Krebs, Lee, UGM, Phys. Rev. Lett. 104 (2010) 142501, Eur. Phys. J. A 45 (2010) 335

• presumably a lattice artefact?or non-local smearing required?

↪→ investigate this in more detail!


7The D-TERM in ACTION•Works quite well up to the mid-mass region:↪→ 1% accuray for ground-state energies

28Si

24Mg

20Ne

16O

12C

8Be

4He

-400 -350 -300 -250 -200 -150 -100 -50 0

E (MeV)

ExperimentNNLO

NNLO + 4Neff

Lahde, Epelbaum, Krebs, Lee, UGM, Rupak, Phys. Lett. B732 (2014) 110


8

Basics of nuclearlattice simulations

for an easy intro, see: UGM, Nucl. Phys. News 24 (2014) 11

for an early review, see: D. Lee, Prog. Part. Nucl. Phys. 63 (2009) 117

upcoming textbook, see: T. Lahde, UGM, Springer Lecture Notes in Physics


9NUCLEAR LATTICE SIMULATIONSFrank, Brockmann (1992), Koonin, Muller, Seki, van Kolck (2000) , Lee, Schafer (2004), . . .

Borasoy, Krebs, Lee, UGM, Nucl. Phys. A768 (2006) 179; Borasoy, Epelbaum, Krebs, Lee, UGM, Eur. Phys. J. A31 (2007) 105

• new method to tackle the nuclear many-body problem

• discretize space-time V = Ls × Ls × Ls × Lt:nucleons are point-like particles on the sites

• discretized chiral potential w/ pion exchangesand contact interactions + Coulomb

→ see Epelbaum, Hammer, UGM, Rev. Mod. Phys. 81 (2009) 1773

• typical lattice parameters

pmax =π

a' 314 MeV [UV cutoff]

p

p

n

n a

~ 2 fm

• strong suppression of sign oscillations due to approximate Wigner SU(4) symmetryJ. W. Chen, D. Lee and T. Schafer, Phys. Rev. Lett. 93 (2004) 242302, T. Lahde et al., EPJA 51 (2015) 92

• hybrid Monte Carlo & transfer matrix (similar to LQCD)


10

TRANSFER MATRIX METHOD

• Correlation–function for A nucleons: ZA(τ ) = 〈ΨA| exp(−τH)|ΨA〉

with ΨA a Slater determinant for A free nucleons[or a more sophisticated (correlated) initial/final state]

• Transient energy

EA(τ ) = −d

dτlnZA(τ )

→ ground state: E0A = lim

τ→∞EA(τ )

• Exp. value of any normal–ordered operator O

ZOA = 〈ΨA| exp(−τH/2)O exp(−τH/2) |ΨA〉

limτ→∞

ZOA (τ )

ZA(τ )= 〈ΨA|O |ΨA〉

Euclidean time

𝛑 𝛑

Eucl

idea

n t

ime

𝑎

𝐿


11CONFIGURATIONS

⇒ all possible configurations are sampled⇒ preparation of all possible initial/final states⇒ clustering emerges naturally


12AUXILIARY FIELD METHOD

• Represent interactions by auxiliary fields:

exp

[−C

2

(N†N

)2]=√

12π

∫ds exp

[−s2

2+√C s

(N†N

)]

𝛑 𝛑

Eucl

idea

n t

ime

𝒔

𝒔𝑰

𝒔𝝅


13EXTRACTING PHASE SHIFTS on the LATTICE

• Luscher’s method:Two-body energy levels below theinelastic threshold on a periodiclattice are related to the phaseshifts in the continuum

Luscher, Comm. Math. Phys. 105 (1986) 153Luscher, Nucl. Phys. B 354 (1991) 531

• Spherical wall method:Impose a hard wall on the lattice anduse the fact that the wave functionvanishes for r = Rwall:

ψ`(r) ∼ [cos δ`(p)F`(pr) + sin δ`(p)G`(pr)]

Borasoy, Epelbaum, Krebs, Lee, UGM,EPJA 34 (2007) 185

Carlson, Pandharipande, Wiringa,NPA 424 (1984) 47

𝑝

𝑅wall

L

L

L


14

COMPUTATIONAL EQUIPMENT

• Mostly used = JUQUEEN (BlueGene/Q)

6 Pflops


15RESULTS from LATTICE NUCLEAR EFT

� Lattice EFT calculations for A=3,4,6,12 nuclei, PRL 104 (2010) 142501

� Ab initio calculation of the Hoyle state, PRL 106 (2011) 192501

� Structure and rotations of the Hoyle state, PRL 109 (2012) 142501

� Validity of Carbon-Based Life as a Function of the Light Quark MassPRL 110 (2013) 142501

� Ab initio calculation of the Spectrum and Structure of 16O,PRL 112 (2014) 142501

� Ab initio alpha-alpha scattering, Nature 528 (2015) 111

� Nuclear Binding Near a Quantum Phase Transition, PRL 117 (2016) 132501

� Ab initio calculations of the isotopic dependence of nuclear clustering,PRL 119 (2017) 222505

A = 20!

A = 16!

A = 12!

A = 8!

λ

EA–Eα A/4|aαα| = ∞aαα = 0

Alpha gas Nuclear liquid

λ20λ16 λ12 λ8

λ = 0 λ = 1

-200

-150

-100

-50

0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

AHAHe

ABe

AC

AO

ener

gy (M

eV)

A

LOexperiment


16

The two-bodysystem at NNLO

Alarcon, Du, Klein, Lahde, Lee, Li, Luu, UGMEur. Phys. J. A 53 (2017) 83 [arXiv:1702.05319]

Klein, Elhatisari, Lahde, Lee, UGM [arXiv:1803.04231]


17NUCLEAR FORCES at NNLOfor details, see: Epelbaum, Hammer, UGM, Rev. Mod. Phys. 81 (2009) 1773

• Potential at next-to-next-to-leading order [Q = {p/Λ,Mπ/Λ}]:'

&

$

%

�Chiral expansion of nuclear forces [based on NDA]

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Korrektur 1. Ordnung



Drei-Nukleon-Kraft Vier-Nukleon-KraftTwo-nucleon force Three-nucleon force Four-nucleon force

LO (Q0)

NLO (Q2)

N2LO (Q3)

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N4LO (Q5)

• NN potential to NNLO [all πN and ππN LECs fixed from πN scattering]:

VNN = V(0)

LO + V(2)

NLO + V(3)

NNLO

= V contLO + V OPE

LO + V contNLO + V TPE

NLO + V TPENNLO


18NUCLEAR FORCES at NNLO continued

• Analytic expressions [2+7 LECs]:

V contLO = CS + CT (~σ1 · ~σ2)

V OPELO = −

g2A

4F 2π

τ1 · τ2

(~σ1 · ~q) (~σ2 · ~q)q 2 +M2

π

~q = t-channel mom. transfer

V contNLO = C1q

2 + C2k2 +

(C3q

2 + C4k2)

(~σ1 · ~σ2) + iC512

(~σ1 + ~σ2) · (~q × ~k)

+ C6 (~σ1 · ~q) (~σ2 · ~q) + C7(~σ1 · ~k)(~σ2 · ~k) ~k = u-channel mom. transfer

V TPENLO = − τ1·τ2

384π2F 4πL(q)

[4M2

π

(5g4A − 4g2

A − 1)

+ q2(23g4

A − 10g2A − 1

)+

48g4AM

4π

4M2π+q2

]− 3g4

A

64π2F 4πL(q)

[(~q · ~σ1) (~q · ~σ2)− q2 (~σ1 · ~σ2)

]• Loop function: L(q) = 1

2q

√4M2

π + q2 ln√

4M2π+q2+q√

4M2π+q2−q

→ 1 + 13q2

4M2π

+ · · · for q � Λ

→ for coarse lattices a ' 2 fm, the TPE at N(N)LO can be absorbed in the LECs Ci

→ no longer true as a decreases, need to account for the TPE explicitely


19A FEW DETAILS ON THE FITS• Fits in large & fixed volumes, vary a from 1 to 2 fm:

a−1 [MeV] a [fm] L La [fm]

100 1.97 32 63.14120 1.64 38 62.48150 1.32 48 63.14200 0.98 64 63.14

• OPE and TPE LECs completely fixed (gA ∼ gπNN and c1,2,3,4 from RS analysis)Hoferichter, Ruiz de Elvira, Kubis, UGM, Phys. Rev. Lett. 115 (2015) 092301

• Smeared LO S-wave contact interactions: f(~q ) ≡ f−10 exp

(−bs

~q 4

4

)• Partial-wave projection of the contact interactions

→ fit bs and two S-wave LECs Ci at LO up to pcm = 100 MeV

→ w/ bs fixed, fit two/seven S/P-wave LECsCi at NLO/NNLO up to pcm = 150 MeV

→ treat NLO and NNLO corrections perturbatively


20NEUTRON–PROTON SCATTERING• neutron-proton scattering

0

20

40

60

80

100

0 50 100 150 200

δ(1

S0)

[degre

es]

pCM [MeV]

0

20

40

60

80

100

120

140

160

180

0 50 100 150 200δ(3

S1)

[degre

es]

pCM [MeV]

-10

-5

0

5

10

0 50 100 150 200

ε1 [d

egre

es]

pCM [MeV]

-20

-15

-10

-5

0

5

10

15

20

0 50 100 150 200

δ(1

P1)

[degre

es]

pCM [MeV]

-20

-15

-10

-5

0

5

10

15

20

0 50 100 150 200

δ(3

P0)

[degre

es]

pCM [MeV]

Nijmegen PWALattice LO

Lattice N2LO-20

-15

-10

-5

0

5

10

15

20

0 50 100 150 200

δ(3

P1)

[degre

es]

pCM [MeV]

-20

-15

-10

-5

0

5

10

15

20

0 50 100 150 200

δ(3

P2)

[degre

es]

pCM [MeV]

-10

-5

0

5

10

0 50 100 150 200

δ(1

D2)

[degre

es]

pCM [MeV]

-10

-5

0

5

10

0 50 100 150 200

δ(3

D1)

[degre

es]

pCM [MeV]

-10

-5

0

5

10

0 50 100 150 200

δ(3

D2)

[degre

es]

pCM [MeV]

-10

-5

0

5

10

0 50 100 150 200

δ(3

D3)

[degre

es]

pCM [MeV]

0

20

40

60

80

100

0 50 100 150 200

δ(1

S0)

[degre

es]

pCM [MeV]

0

20

40

60

80

100

120

140

160

180

0 50 100 150 200

δ(3

S1)

[degre

es]

pCM [MeV]

-10

-5

0

5

10

0 50 100 150 200

ε1 [d

egre

es]

pCM [MeV]

-20

-15

-10

-5

0

5

10

15

20

0 50 100 150 200

δ(1

P1)

[degre

es]

pCM [MeV]

-20

-15

-10

-5

0

5

10

15

20

0 50 100 150 200

δ(3

P0)

[degre

es]

pCM [MeV]

Nijmegen PWALattice LO

Lattice N2LO-20

-15

-10

-5

0

5

10

15

20

0 50 100 150 200

δ(3

P1)

[degre

es]

pCM [MeV]

-20

-15

-10

-5

0

5

10

15

20

0 50 100 150 200

δ(3

P2)

[degre

es]

pCM [MeV]

-10

-5

0

5

10

0 50 100 150 200

δ(1

D2)

[degre

es]

pCM [MeV]

-10

-5

0

5

10

0 50 100 150 200

δ(3

D1)

[degre

es]

pCM [MeV]

-10

-5

0

5

10

0 50 100 150 200

δ(3

D2)

[degre

es]

pCM [MeV]

-10

-5

0

5

10

0 50 100 150 200

δ(3

D3)

[degre

es]

pCM [MeV]

a=

1.32

fm

a=

1.97

fm→ up to pcm ' 150 MeV, physics largely independent of a

√

→ description consistent with the continuum within error bands√

→ however: there are some differences in the P-waves description improveswith decreasing a

→ errors appear large due to recaling of χ2

� = LO� = N2LO


21PROTON-PROTON SCATTERING

• proton-proton scattering

↪→ only in I = 1, so L+ S must be even

↪→ fit to Coulomb functions on the spherical wall at Rwall ' 28 fm

↪→ fit additional LEC to proton-proton scattering length

0

20

40

60

80

100

0 50 100 150 200

δ(1

S0)

[degre

es]

pCM [MeV]

Lattice spacing a = 1.97 fm

0

20

40

60

80

100

0 50 100 150 200

δ(1

S0)

[degre

es]

pCM [MeV]


Nijmegen PWALattice LO+Coulomb

Lattice N2LO+Coulomb

0

20

40

60

80

100

0 50 100 150 200

δ(1

S0)

[degre

es]

pCM [MeV]


• somewhat too large for the coarse a, improving with decreasing a

• other phases only differ by Coulomb, very small effect


22

The three-bodysystem at NNLO

Epelbaum, Krebs, Lee, UGM, Phys. Rev. Lett. 104 (2010) 142501 [arXiv:0912.4195]



23BASICS of the THREE NUCLEON SYSTEM

• At NNLO, the firstnon-vanishing 3NF appears

1 2 3c1,3,4 cD cE• c1,3,4 from RS analysis of πN → πN

Hoferichter, Ruiz de Elvira, Kubis, UGM, Phys. Rev. Lett. 115 (2015) 092301

↪→ longest range part determined from chiral symmetry

↪→ corresponds to the renowned Fujita-Miyazawa forceFujita, Miyazawa, Prog. Theor. Phys. 17 (1957) 360; UGM, AIP Conf. Proc. 1011 (2008) 49

• cD and cE are correlatedEpelbaum, Nogga, Glockle, Kamada, UGM, Witala, Phys. Rev. C66 (2002) 064001

↪→ fix cD = −0.79 and determine cE from the 3H binding energy

-2 0 2 4 6c

D

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

c E

Λ=500 MeVΛ=600 MeV


24CALCULATING the THREE-NUCLEON SYSTEM

• can use MC methods or Lanzcos, here: Lanczos plus finite volume corrections

• no finite volume corrections for V ' (10 · 1.97 fm)3 ' (12 · 1.64 fm)3 ' (20 fm)3

• scaling as L6, so for a = 1.32 fm, we must include finite volume corrections

• ground state energy:

E3N (L) = E3N∞ +A

exp(

2κL√3

)(κL)

32

κ =√−mE3N

∞

• higher order corrections:

〈Oi〉 (L) = 〈Oi〉∞ +Aiexp

(2κL√

3

)(κL)

32

UGM, Rios, Rusetsky,Phys. Rev. Lett. 114 (2015) 091602

Hammer et al., JHEP 1709 (2017) 109

-12

-11.5

-11

-10.5

-10

-9.5

-9

-8.5

-8

-7.5

7 8 9 10 11 12 13 14 15E

B [M

eV

]

L [l.u.]

E3H

LON2LO 2N

N2LO 2N+EMN2LO 2N+EM+ 3N


25RESULTS for the THREE-NUCLEON SYSTEM

• Triton binding energy at various orders:

a = 1.97 fm a = 1.64 fm a = 1.32 fmELO [MeV] −7.80 −8.29 −8.74

EN2LO [MeV] −7.846(4) −8.11(2) −7.95(2)

E+EMN2LO [MeV] −7.68(2) −7.91(3) −7.77(2)

E+EM+3NN2LO [MeV] −8.48(3) −8.48(3) −8.48(2)

cE 0.5309(2) 0.3854(3) 1.0386(5)

• different overbinding at LO can be mostly traced back to the 3P0 wave

• N2LO w/o 3NF similar to phenomenological potentials

• cE of natural size, that is of O(1)


26

The four-bodysystem at NNLO& the Tjon band

Epelbaum, Krebs, Lee, UGM, Phys. Rev. Lett. 104 (2010) 142501 [arXiv:0912.4195];Eur. Phys. J. A 45 (2010) 335 [arXiv:1003.5697]



27CALCULATING the FOUR-NUCLEON SYSTEM

• must use MC methods→ correlation function:

Z4He (t) = 〈ψ4He | exp(−tHLO) | ψ4He〉

• Transient energy: ELO (t) = −d logZ4He (t)

dt

• preparation of initial states utilizing the Wigner SU(4) symmetry:

H0 = Hfree + 12C0

∑~n1,~n2

f (~n1 − ~n2)︸︷︷︸Gaussian smearing

ρ (~n1) ρ (~n2)

↪→ create initial states close to the physical one: from the antisymm. free-particle solution:

|ψ4He〉 = exp (−t0H0) |ψ0〉

↪→ leading order Hamiltonian:

HLO = Hfree +HLO,contact +HOPE

↪→ higher orders as perturbative corrections:

ZO (t) = 〈ψ4He | exp(−tH

2

)O exp

(−tH

2

)| ψ4He〉

〈O〉 (t) =ZO (t)

Z4He(t)


28RESULTS for the FOUR-NUCLEON SYSTEM

• Time extrapolations for various lattice spacings

a = 1.97 fm a = 1.64 fm a = 1.32 fm

−29

−28

−27

−26

−25

−24

−23

0 5 10 15 20

EB [M

eV

]

Nt

4He binding energy

LO contribution

ELO = −28.8(1) MeV

−0.5

−0.4

−0.3

−0.2

−0.1

0

0 5 10 15 20

EB [M

eV

]

Nt

2N N2LO contribution

EN2LO= −0.34(2) MeV

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

EB [M

eV

]

Nt

2N EM contribution

EEM = 0.91(5) MeV

−6

−5

−4

−3

−2

−1

0

0 5 10 15 20

EB [M

eV

]

Nt


E3N = −6.3(1) MeV

−29

−28

−27

−26

−25

−24

−23

0 5 10 15 20

EB [M

eV

]

Nt

4He binding energy

LO contribution

ELO = −27.36 (6) MeV

−3

−2.5

−2

−1.5

−1

−0.5

0

0 5 10 15 20

EB [M

eV

]

Nt


EN2LO =−1.39(4) MeV

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

EB [M

eV

]

Nt

2N EM contribution

EEM=0.881(1) MeV

−6

−5

−4

−3

−2

−1

0

0 5 10 15 20

EB [M

eV

]

Nt


E3N =−3.22(2) MeV

−28

−26

−24

−22

−20

0 5 10 15 20

EB [M

eV

]

Nt

He−4 binding energy

LO contribution

ELO = −24.8 (2) MeV

−6

−5

−4

−3

−2

−1

0

0 5 10 15 20

EB [M

eV

]

Nt


EN2LO = −1.1(2) MeV

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

EB [M

eV

]

Nt

2N EM contribution

EEM=0.809(1) MeV −6

−5

−4

−3

−2

−1

0

0 5 10 15 20

EB [M

eV

]

Nt


E3N = −3.29(6) MeV

• agrees with the earlier calculation at a = 1.97 fm, much faster√

Epelbaum, Krebs, Lee, UGM, Phys. Rev. Lett. 104 (2010) 142501, Eur. Phys. J. A 45 (2010) 335


29

RESULTS for the FOUR-NUCLEON SYSTEM

• 4He binding energy at various orders for various lattice spacings

a = 1.97 fm a = 1.64 fm a = 1.32 fmELO −28.81(11) −27.36(6) −24.81(19)

EN2LO −29.15(11)(3) −28.75(7)(5) −25.89(27)(3)

E+EMN2LO −28.23(12)(3) −27.87(7)(6) −25.08(27)(3)

E+EM+3NN2LO −34.55(18)(3) −31.09(7)(6) −28.37(28)(3)

↪→ revover the overbinding for the coarse lattice already found in 2010

↪→ overbinding decreases with with decreasing lattice spacing

↪→ look at the Tjon band plot


30RESULTS for the FOUR-NUCLEON SYSTEM cont’d• Tjon band plot

-34

-32

-30

-28

-26

-24-9-8.8-8.6-8.4-8.2-8-7.8-7.6

E4H

e [M

eV

]

E3H [MeV]

LO; a = 1.97 fm

N2LO; a = 1.97 fm

LO; a = 1.64 fm

N2LO; a = 1.64 fm

LO; a = 1.32 fm

N2LO; a = 1.32 fm

NNLO, a = 1.97 fm

NNLO, a = 1.64 fm

NNLO, a = 1.32 fmExp


31DISCUSSION

• Deviation from the Tjon band confirmed for the coarse lattice spacing a = 1.97 fm

• Deviation decreases with decreasing a,perfect agreement for the smallest a = 1.32 fm

↪→ no more need for four-nucleon interactions for these smaller values of a

↪→ intrinsic problem of the standard action resolved√

• how about improved actions and or higher orders?

non-local smearing: mimimizes remainingsign oscillations / higher-body forcesElhatisari et al., Phys. Rev. Lett. 117 (2016) 132501

N3LO is required for a better precision→ better 2NFs and 3NFsLi, Lu, ... et al., in preparation

a

a

a

a

a


32SCATTERING at N3LO• coarse lattice a = 1.97 fm, TPE absorbed in the local operators

0

10

20

30

40

50

60

70

80

0 50 100 150 200

δ(1

S0)

[degre

es]

pCM [MeV]

20

40

60

80

100

120

140

160

180

0 50 100 150 200δ(3

S1)

[degre

es]

pCM [MeV]

NPWALO

NLO,N2LON3LO

-6

-4

-2

0

2

4

6

8

10

0 50 100 150 200

ε1 [d

egre

es]

pCM [MeV]

-6

-4

-2

0

2

4

6

8

10

0 50 100 150 200

ε2 [d

egre

es]

pCM [MeV]

-20

-15

-10

-5

0

5

0 50 100 150 200

δ(1

P1)

[deg

ree

s]

pCM [MeV]

-2

0

2

4

6

8

10

12

14

0 50 100 150 200

δ(3

P0)

[deg

ree

s]

pCM [MeV]

-20

-15

-10

-5

0

5

0 50 100 150 200

δ(3

P1)

[deg

ree

s]

pCM [MeV]

-2

0

2

4

6

8

10

12

14

0 50 100 150 200

δ(3

P2)

[deg

ree

s]

pCM [MeV]

-2

0

2

4

6

8

10

0 50 100 150 200

δ(1

D2)

[de

gre

es]

pCM [MeV]

-20

-15

-10

-5

0

5

0 50 100 150 200

δ(3

D1)

[d

eg

ree

s]

pCM [MeV]

0

5

10

15

20

0 50 100 150 200

δ(3

D2)

[d

eg

ree

s]

pCM [MeV]

-2

-1

0

1

2

3

4

0 50 100 150 200

δ(3

D3)

[d

eg

ree

s]

pCM [MeV]

PRELIMINARY


33SCATTERING at N3LO• finer lattice a = 1.32 fm, TPE fully included

0

10

20

30

40

50

60

70

80

0 50 100 150 200 250

δ(1

S0)

[degre

es]

pCM [MeV]

20

40

60

80

100

120

140

160

180

0 50 100 150 200 250δ(3

S1)

[degre

es]

pCM [MeV]

NPWALO

NLON2LON3LO

-6

-4

-2

0

2

4

6

8

10

0 50 100 150 200 250

ε1 [d

egre

es]

pCM [MeV]

-6

-4

-2

0

2

4

6

8

10

0 50 100 150 200 250

ε2 [d

egre

es]

pCM [MeV]

-20

-15

-10

-5

0

5

0 50 100 150 200 250

δ(1

P1)

[deg

ree

s]

pCM [MeV]

-2

0

2

4

6

8

10

12

14

0 50 100 150 200 250

δ(3

P0)

[deg

ree

s]

pCM [MeV]

-20

-15

-10

-5

0

5

0 50 100 150 200 250

δ(3

P1)

[deg

ree

s]

pCM [MeV]

-2

0

2

4

6

8

10

12

14

0 50 100 150 200 250

δ(3

P2)

[deg

ree

s]

pCM [MeV]

-2

0

2

4

6

8

10

0 50 100 150 200 250

δ(1

D2)

[de

gre

es]

pCM [MeV]

-20

-15

-10

-5

0

5

0 50 100 150 200 250

δ(3

D1)

[d

eg

ree

s]

pCM [MeV]

0

5

10

15

20

0 50 100 150 200 250

δ(3

D2)

[d

eg

ree

s]

pCM [MeV]

-2

-1

0

1

2

3

4

0 50 100 150 200 250

δ(3

D3)

[d

eg

ree

s]

pCM [MeV]

PRELIMINARY


34SCATTERING at N3LO• very fine lattice a = 0.98 fm, TPE fully included

0

10

20

30

40

50

60

70

80

0 50 100 150 200 250

δ(1

S0)

[degre

es]

pCM [MeV]

20

40

60

80

100

120

140

160

180

0 50 100 150 200 250δ(3

S1)

[degre

es]

pCM [MeV]

NPWALO

NLON2LON3LO

-6

-4

-2

0

2

4

6

8

10

0 50 100 150 200 250

ε1 [d

egre

es]

pCM [MeV]

-6

-4

-2

0

2

4

6

8

10

0 50 100 150 200 250

ε2 [d

egre

es]

pCM [MeV]

-20

-15

-10

-5

0

5

0 50 100 150 200 250

δ(1

P1)

[deg

ree

s]

pCM [MeV]

-2

0

2

4

6

8

10

12

14

0 50 100 150 200 250

δ(3

P0)

[deg

ree

s]

pCM [MeV]

-20

-15

-10

-5

0

5

0 50 100 150 200 250

δ(3

P1)

[deg

ree

s]

pCM [MeV]

-2

0

2

4

6

8

10

12

14

0 50 100 150 200 250

δ(3

P2)

[deg

ree

s]

pCM [MeV]

-2

0

2

4

6

8

10

0 50 100 150 200 250

δ(1

D2)

[de

gre

es]

pCM [MeV]

-20

-15

-10

-5

0

5

0 50 100 150 200 250

δ(3

D1)

[d

eg

ree

s]

pCM [MeV]

0

5

10

15

20

0 50 100 150 200 250

δ(3

D2)

[d

eg

ree

s]

pCM [MeV]

-2

-1

0

1

2

3

4

0 50 100 150 200 250

δ(3

D3)

[d

eg

ree

s]

pCM [MeV]

PRELIMINARY


35DISCUSSION and OUTLOOK

• All phases well described, no more differences for the various a values

↪→ Calculations of the binding energies of 3H and 4He will recover the Tjon band

• Upcoming papers:

Neutron-proton scattering at N3LO

Light and medium-mass nuclei at N3LO up A = 30 with N3LO forces

Excited spectrum of nuclei using the pinhole algorithm

How nuclei boil: Simulations of nuclei at nonzero temperature

↪→ with increased precision for the nuclear forces,many exciting results in nuclear structure & reactions to come


36

SPARES


37

NUCLEAR FORCES: OPEN ENDS

•Why is there this hierarchy V2N � V3N � V4N ?

• Gauge and chiral symmetries difficult to include (meson-exchange currents)Brown, Riska, Gari, . . .

• Connection to QCD ?most models have one-pion-exchange, but not necessarily respect chiral symmetrysome models have two-pion exchange reconstructed via dispersion relations from πN → πN

⇒We want an approach that

− is linked to QCD via its symmetries− allows for systematic calc’s with a controlled theoretical error− explains the observed hierarchy of the nuclear forces− matches nucleon structure to nuclear dynamics− allows for a lattice formulation / chiral extrapolations− puts nuclear physics on a sound basis


38NUCLEAR FORCES in CHIRAL NUCLEAR EFT• expansion of the potential in powers of Q [small parameter]: {p/Λb, Mπ/Λb}

• explains observed hierarchy of the nuclear forces

• extremely successful in few-nucleon systemsEpelbaum, Hammer, UGM, Rev. Mod. Phys. 81 (2009) 1773�Chiral expansion of nuclear forces [based on NDA]


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Drei-Nukleon-Kraft Vier-Nukleon-KraftTwo-nucleon force Three-nucleon force Four-nucleon force

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N4LO (Q5)

— have been worked out and employed (recently, also parts of the N5LO NN force have been worked out )

— have been worked out but not employed yet [LENPIC, work in progress]— have not been completely worked out yet

worked out and applied worked out and to be applied calculations in progress


39PHASE SHIFTS at N4LO⇒ Precision phase shifts with small uncertainties up to Elab = 300 MeV

Epelbaum, Krebs, UGM, Phys. Rev. Lett. 115 (2015) 122301

-20

0

20

40

60

� [d

eg]

1S0

-20

0

20

40

60

� [d

eg]

1S0

-10

0

10

20 3P0

-10

0

10

20 3P0

-40

-30

-20

-10

0

10

1P1

-40

-30

-20

-10

0

10

1P1

-30

-20

-10

03P1

-30

-20

-10

03P1

0

60

120

180

� [d

eg]

3S1

0

60

120

180

� [d

eg]

3S1

-5

0

5

10�1

-5

0

5

10�1

-30

-20

-10

0

3D1

-30

-20

-10

0

3D1

0

5

10

15

20

1D2

0

5

10

15

20

1D2

0

10

20

30

0 100 200 300

� [d

eg]

Elab [MeV]

3D2

0

10

20

30

0 100 200 300

� [d

eg]

Elab [MeV]

3D2

0

5

10

15

20

25

30

0 100 200 300

Elab [MeV]

3P2

0

5

10

15

20

25

30

0 100 200 300

Elab [MeV]

3P2

-4

-3

-2

-1

0

1

0 100 200 300

Elab [MeV]

�2

-4

-3

-2

-1

0

1

0 100 200 300

Elab [MeV]

�2

-10

-5

0

5

10

15

0 100 200 300

Elab [MeV]

3D3

-10

-5

0

5

10

15

0 100 200 300

Elab [MeV]

3D3

NLO N2LO N3LO N4LO


40LOCAL/NON–LOCAL INTERACTIONS on the LATTICE

a

a

a

a

a

• Local operators/densities:

a(n), a†(n) [n denotes a lattice point]

ρL(n) = a†(n)a(n)

• Non-local operators/densities:

aNL(n) = a(n) + sNL

∑〈n′ n〉

a(n′)

a†NL(n) = a†(n) + sNL

∑〈n′ n〉

a†(n′)

ρNL(n) = a†NL(n)aNL(n)

→ where∑〈n′ n〉

denotes the sum over nearest-neighbor lattice sites of n

→ the smearing parameter sNL is determined when fitting to the phase shifts


41NUCLEON–NUCLEON PHASE SHIFTS

• Show results for NN [and α-α] phase shifts for both interactions:

0

20

40

60

80

0 50 100 150

a

0

60

120

180

0 50 100 150

b

-10

-5

0

5

0 50 100 150

c

0

5

10

15

0 50 100 150

d

Nijmegen PWAContinuum LO

Lattice LO-ALattice LO-B

-10

-5

0

0 50 100 150

δ or

ε (

deg)

e

0

2

4

6

0 50 100 150

f

-1

0

1

2

3

0 50 100 150

g

-6

-4

-2

0

0 50 100 150

h

0

5

10

0 50 100 150

i

-1

0

1

0 50 100 150

j

-5

0

5

0 50 100 150

pcms (MeV)

k

-2

-1

0

0 50 100 150

l

1S03S1

1P13P0

3P13P2

1D23D1

3D23D3 ε1 ε2

→ both interactions very similar


42


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