Hadron Interactions on the Lattice
Takumi Doi(Nishina Center, RIKEN)

12/14/2012

QUCS2012 @ Nara1• Hadron Interactions– Bridging different worlds:
Particle Physics / Nuclear Physics / Astrophysics– Frontier: 1st principle calc by Lattice simulations• Outline– Introduction– Theoretical framework for lattice hadron forces– Lattice results at heavier quark masses– Challenges toward physical quark mass point•  Nuclear Physics on the Lattice– Summary / Prospects

(1) Build a foundation for nuclear physics

N ucleiN eutron Stars

NN phase shifts
from experimentsPhenomenologicalNuclear Forces• Nuclear Forces play crucial roles

Super N ovae
Variousapplications– Yet, no clear connection to QCD so far
 Lattice QCD

3

(2) Predict Unknown Interactions (YN, YY, NNN)
3D Nuclear Chart

Renaissance inStrange World !

8 X 8 = 27 + 8s + 1 + 10* + 10 + 8a
What is universal, andwhat is individual in baryon forces ?

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QUCS2012 @ Nara

What is the fate of
the H-dibaryon ?
(uuddss)

(2) Predict Unknown Interactions (YN, YY, NNN)
2D Nuclear Chart
RIBF @ RIKEN=Precise ab initio calc=
2N-forces are insufficient
3N-force indispensable

Origin ofElements

Super Nova
Paradigm Shift inUnstable Nuclei(New Magic Numbers !)Important role of 3N-force

Dense Matter  Interactions ofYN, YY, NNN,… are crucial• Neutron Stars, Super Novae  EoS
3N

J1614-2230

PSR1913+16

2NY dofHow to sustain a neutron staragainst gravitational collapse ?
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Akmal et al.(’98), Nishizaki et al.(’02), Takatsuka et al.(’08)

6• Outline– Introduction– Theoretical framework for lattice hadron forces• Phase shifts (a la Luscher) / Light nuclei on the lattice• Potentials from NBS wave functions on the lattice

T.Yamazaki (Sat.)– Lattice results at heavier quark masses– Challenges toward physical quark mass point– Summary / Prospects
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Hadrons to Atomic nuclei from Lattice QCD(HAL QCD Collaboration)
S. Aoki, N. Ishii, H. Nemura, K. Sasaki, M. Yamada (Univ. of Tsukuba)
B. Charron(Univ. of Tokyo)

T. Doi, T. Hatsuda , Y. Ikeda, K. Murano (RIKEN)
T. Inoue (Nihon Univ.)8

Nuclear Forces from Lattice QCD[HAL QCD method]• Potential is constructed so as to reproduce
the NN phase shifts (or, S-matrix)• Nambu-Bethe-Salpeter (NBS) wave function– Wave function  phase shifts

M.Luscher, NPB354(1991)531

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CP-PACS Coll., PRD71(2005)094504
C.-J.Lin et al., NPB619(2001)467
Ishizuka, PoS LAT2009 (2009) 119RL
9“Potential” as a representation of S-matrix[HAL QCD method]• Consider the wave function at “interacting region”– U(r,r’): faithful to the phase shift by construction• U(r,r’) below inelastic threshold isR• U(r,r’): E-independent, while non-local in general• Non-locality  derivative expansion
LO12/14/2012

LONLO

Okubo-Marshak(1958)

NNLO

Aoki-Hatsuda-Ishii PTP123(2010)89
Check on convergence: K.Murano et al., PTP125(2011)1225

10L

• Outline– Introduction– Theoretical framework for lattice hadron forces– Lattice results at heavier quark masses• (1) nuclear-, (2) hyperon-, (3) 3N- forces– Challenges toward physical quark mass point– Summary / Prospects
11(1) NN potential on the lattice(positive parity)• “di-neutron” channel• “deuteron” channel

 central force
 central & tensor force

phase shift

Elab [MeV]
Not Bound
Nf=2+1 clover (PACS-CS), 1/a=2.2GeV,
L=2.9fm, mπ=0.7GeV, mN=1.6GeV[N.Ishii (Poster)]12

NN potential on the lattice(negative parity)• S=1 channel:– Central & tensor forces in LO– Spin-orbit force in NLO• Inject a momentum Superfluidity 3P2 in neutron star neutrino cooling Cas A NS: cooling is being measured !

Nf=2 clover (CP-PACS), L=2.5fm, mπ=1.1GeV[K.Murano (Poster)]13

SU(3) study(2) Hyperon forces

a=0.12fm, L=3.9fm,m(PS)= 0.47-1.2GeV

attractive core !

27,10*:Same as NN

Repulsive core
 Pauli principle !
M.Oka et al., NPA464(1987)700

8s,10:
strong repulsive core

1s: deep attractive pocket
8a: weak repulsive core

T.Inoue et al. (HAL QCD Coll.), NPA881(2012)28
Also seen in Takahashi et al. (2010), Kawanai et al. (2010)
Meson-baryon, Y.Ikeda et al., arXiv:1111.2663, (Poster) Study of baryonic matter & Neutron Star [T.Inoue ]H-dibaryon (uuddss, I=0,1S0)
Physical point

MK [MeV]

SU(3) lat

120MeV
B.E.
30MeV?
Mπ [MeV]

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Coupled channel study is essential Poster by K.Sasaki

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Hyperon Interactions in S= -1(single channel study)

Attractive
Not boundRepulsive
Nemura et al.
Nf=2+1, L=2.9fm,
mπ= 0.70GeV
arXiv:1203.3320

Poster by H. Nemura

Crucial input for the core ofneutron star and hyper-nuclei

is also in progress16

(3) 3N-forces (3NF) on the lattice
T.D. et al. (HAL QCD Coll.) PTP127(2012)723+ t-dep method updates
Triton channel

short-range
repulsive 3NF !

Preliminary

Nf=2 clover (CP-PACS), 1/a=1.27GeV,
L=2.5fm, mπ=1.1GeV, mN=2.1GeV

Y dof

How about other geometries ?
How about YNN, YYN, YYY ?

17• Outline– Introduction– Theoretical framework for lattice hadron forces– Lattice results at heavier quark masses– Challenges toward realistic potentials• Major challenges: (1) S/N issue (2) comput. cost

– Summary / Prospects

10PFlops

K computerTowards realistic potential by the K computer• Physical mass point, Infinite V limit, continuum limit

Scatt. length– Physical mπ crucial for OPEP, chiral extrapolation won’t work

Y.Kuramashi,PTPS122(1996)153“Unitary Region”

 mq

We are heremq
Phys. point– Gauge confs generation at mπ = 140MeV, L=~10fm @ K
 Challenge in the “measurement”: S/N issue
19Challenges toward the physical point (1)• S/N issue at light mass

Parisi, Lepage (1989)– To achieve ground state saturation, take 𝒕 → ∞Single nucleon

Nucleons w/ mass number = A– Situation gets worse for larger volume
 large spectral density by scatt. states

 Very large t >~ 100 would be required !

Solution: Extract the signal from excited states
N.Ishii et al. (HAL QCD Coll.) PLB712(2012)437

E-indep of potential U(r,r’)  (excited) scatt states share the same U(r,r’)
They are not contaminations, but signals
 Schrodinger Eq. : time-independent  time-dependent

Grand State (G.S.) saturation is NOT necessary !
Significant advantage of potential method: Moderate t >~ 10 would be fine

Explicit Lat calc for I=2 pipi phase shiftBeautiful agreement between(1) Luscher’s formulaw/g.s. saturation(2) the HAL QCD method w/ & w/o g.s. saturation
21T.Kurth et al. (HAL-BMW Coll.) @ Lat2012

Challenges toward the physical point (2)• Enormous computational cost for correlators– # of Wick contraction (permutations)– # of color/spinor contractions• Total cost:– 2H :– 3H :– 4He :

(color) (spinor)

9 x
144 = 1 x 103
360 x 1728 = 6 x 105
32400 x 20736 = 7 x 108•  [Unified contraction algorithm](half-spin)

Improvement:
T.Yamazaki et al.,
PRD81(2010)111504

TD, M.Endres, arXiv:1205.0585
CPC184(2013)117– Treat Wick/color/spinor contractions in a unified index space•  huge redundancies can be eliminated systematically•  permutation finished BEFORE any lattice calc
• Significant improvement

4He

<1sec

(x add’l. speedup)

Summary and Prospects• Hadron Interactions by 1st principle Lat calc
– Bridging different worlds:
Particle Physics / Nuclear Physics / Astrophysics• Lattice QCD results for NN, YN/YY, NNN– Intriguing physics even at heavy quark masses• On the K computer: physical quark mass point !
Breakthroughs in S/N issue & Comput. cost issue
10PFlops

Thermodynamic limit & continuum limit Realistic hadron interactions
 Nuclear Physics on the Lattice !

Backup Slides

12/14/2012

QUCS2012 @ Nara24A few remarks on the Lattice Potential• Potential is NOT an observable and is not unique:
They are, however, phase-shift equivalent potentials.
– Choosing the pot. (sink op.)  choosing the “scheme”• We study potential (+ phase shifts), since:– Convenient to understand physics– Essential to study many-bodyRR– Finite V artifact better under control– Excited states better under control
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QUCS2012 @ Nara

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Effective Schrodinger equation with E-independent potential2
2K ( x; E ) ≡ ∇ + k ψ ( x; E ) [START] local but E-dep pot. (L3xL3 dof)()

(1) We assume ψ(x; E) for different E is linearly independent with each other.
(2) ψ(x; E) has a “left inverse” as an integration operator asE ≡ 2 mN2 + k 2 ≡ mN U ( x , y )(3) K(x; E) can be factorized as(4) We are left with an effective Schrodinger equation with an E-independent potential U.

(2 ∇ + k 2 ψ ( x; E ) =mN ∫ d 3 yU ( x , y )ψ ( y; E ))I ntuitive
understanding[GOAL] non-local but E-indep pot. (L3xL3 dof)
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Lattice QCD Setup• HAL QCD Coll.
– Nf=3 clover, 1/a=1.6GeV, L= (2,3),4 fm,mPS = 0.47,.0.67, 0.84, 1.0, 1.2GeV, mB=1.2, 1.5, 1.7, 2.0, 2.3GeV– Nf=2+1 clover, 1/a=2.2GeV, L=2.9fm, mπ=0.7GeV, mN=1.6GeV (PACS-CS)– Nf=2, clover, 1/a=1.3GeV, L=2.5fm, mπ=1.1GeV, mN=2.1GeV (CP-PACS)• PACS-CS Coll.
– quenched clover, 1/a=1.5GeV, L=3,6,12 fm, mπ=0.8GeV, mN=1.6GeV– Nf=2+1 clover, 1/a=2.2GeV, L= 3 – 6 fm, mπ=0.5GeV, mN=1.3GeV• NPLQCD Coll.

(as/at=3.5)– Nf=2+1 clover, 1/as=1.6GeV, L=(2, 2.5), 3, 4fm, mπ=0.39GeV, mN=1.2GeV– Nf=3 clover, 1/a=1.4GeV, L=(3.4,4.5), 6.7fm, mPS=0.81GeV,mB=1.6GeV
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NN spectra on the lattice• PACS-CS Coll.

T. Yamazaki et al. PRD84(2011)054506– quenched, mπ= 0.8GeV, L= 3, 6, 12fm, (variational study)• di-neutron (1S0): B.E. = 5.5(1.1)(1.0)MeV• deuteron (3S1 –3D1) : B.E. = 9.1(1.1)(0.5)MeV– Nf=2+1, mπ= 0.5GeV, L= 3 – 6 fm

BoundT. Yamazaki et al. arXiv: 1207.4277• Both channels still bound w/ similar B.E.
• di-neutron (1S0): B.E. = 7.4(1.3)(0.6)MeV• deuteron (3S1 –3D1) : B.E. = 11.5(1.1)(0.6)MeV• NPLQCD Coll.

S.Beane et al. PRD85(2012)054511– Nf=2+1, mπ= 0.39GeV, L= (2, 2.5), 3, 4 fm
• di-neutron (1S0): B.E. = 7.1(5.2)(7.3)MeV• deuteron (3S1 –3D1) : B.E. = 11 (05)(12)MeV

Suggestion ofbound states

29Spectroscopy on the lattice• PACS-CS Coll.

Yamazaki et al. PRD81(2010)111504
Yamazaki et al. arXiv: 1207.4277– quenched, mπ= 0.8GeV, L= 3, 6, 12fm••

3He=(3H)
4He: B.E. = 18.2 (3.5)(2.9) MeV: B.E. = 27.7 (7.8)(5.5) MeV– Nf=2+1, mπ= 0.5GeV, L= 3 – 6 fm
• Both nuclei are still bound w/ similar B.E.
• 3He=(3H) : B.E. = 20.3 (4.0)(2.0) MeV• 4He: B.E. = 43 (12) (8) MeV• NPLQCD Coll.
– Nf=2+1, mπ= 0.39GeV, L= 2.5 fm only

S.Beane et al. PRD80(2009)074501
Prog.Part.Nucl.Phys 66(2010)1• Study– Nf=3, mπ= 0.81GeV, L= (3.4, 4.5), 6.7fm• many (hyper) nuclei bound, e.g.,•3He=(3H)•4He

S.Beane et al., arXiv:1206.5219(version 2 ??): B.E. = 71 (6) (5) MeV  53.9(7.1)(8.0)(0.6) MeV: B.E. = 110 (20)(15) MeV  107(12)(21)(1) MeV30

NPLQCD: SU(3) study
Beane et al., arXiv:1206.5219

G.S. saturated or not,that is the question

Spectroscopy on the lattice (v2)“di-neutron”12/14/2012“deuteron”

QUCS2012 @ Nara

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Spectroscopy on the lattice (v2)

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Solution: Extract the signal from excited states
N.Ishii et al. (HAL QCD Coll.) PLB712(2012)437

E-indep of potential U(r,r’)  (excited) scatt states share the same U(r,r’)
They are not contaminations, but signals

Grand State (G.S.) saturation is NOT necessary !
Explicit Lat calc for I=2 pipi phase shift

Beautiful agreement between
G.S. saturated or not,that is NOT a question !

(1) Luscher’s formulaw/g.s. saturation(2) the HAL QCD method w/ & w/o g.s. saturation
34

Explicit check on the new t-dep HAL method
NN system

N.Ishii et al. (HAL QCD Coll.) PLB712(2012)437

[OLD]

[NEW]

Different sources (creation op.)  different results

Results converged“contaminations” from excited states“signals” from excited states

pipi (I=2) system
Effective mass

T.Kurth et al. (HAL-BMW Coll.) @ Lat2012
Scatt. length(Preliminary)Beautiful agreement between(1) Luscher’s formulaw/g.s. saturation(2) the HAL QCD method
w/ & w/o g.s. saturation

Challenges toward the physical point (2)• Enormous computational cost for correlators– # of Wick contraction (permutation)•

for mass number A( a factor of 2A speedup by inner-baryon exchange)– # of color / spinor contractions( a factor of 2A speedup by “half-spin” method)(color) (spinor)– Total cost:– 2H :– 3H :– 4He :

9 x
144 = 1 x 103
360 x 1728 = 6 x 105
32400 x 20736 = 7 x 108

c.f. T.Yamazaki et al.,
PRD81(2010)111504

See also TD (HAL QCD Coll.)
PoS LAT2010, 136Solution: Unified contraction algorithm
TD, M.Endres, arXiv:1205.0585, CPC184(2013)117• Traditional algorithm

color/spinor contractions (ξ’)

Permutations• New algorithm[impose the same spacial label at source]– Permutation applies to color/spinor indices at “Coeff”
Permuted Sum– Permutation DONE beforehand

Sum over color/spinor unified list

• (Wick contraction and color/spinor contractions are unified)– Significant improvement

4He

<1sec

(x add’l. speedup)