Lace&QCD&Approach&to&Nuclear&Physicsmenu2013.roma2.infn.it/talks/plenary_thursday/5-Ishii_menu2013.pdf · LaCce%QCD%simula4on%at%Physical%point 4 from&Seminar&atICRR,&16&April&2010&by&

La#ce QCD Approach to Nuclear Physics

Noriyoshi Ishii (Univ. Tsukuba)

1

Univ. Tsukuba: N.Ishii, H.Nemura, K.Sasaki, M.Yamada, F.Etminan RIKEN: T.Doi, T.Hatsuda, Y.Ikeda Nihon Univ: T.Inoue Univ. Tokyo: B.Charron YITP(Kyoto): S.Aoki, K.Murano

There is a huge gap between QCD and Tradi4onal Nuclear Physics (2)

Tradi4onal Nuclear Physics u Input:

u Fundamental DOF nucleons

u InteracOons nuclear force from exp.

u Output: u Structures of nuclei

[made of nucleons] u their reacOons

QCD (Quark/Hadron Physics) u Input:

u Fundamental DOF quarks and gluons

u InteracOons strong interacOon from QCD

u Output: u Structures of nucleons

[made of quarks and gluons] u their interacOons

Recent development of la#ce QCD may fill this gap u La#ce QCD at physical point

PACSCS, BMW u La#ce QCD calculaOons of atomic nuclei

PACSCS, NPLQCD u La#ce QCD calculaOons of nuclear force

HALQCD

GAP

LaCce QCD at physical point

(3)

LaCce QCD simula4on at Physical point 4

from Seminar at ICRR, 16 April 2010 by Ukawa.


Fig. from PACSCS, Phys.Rev.D81,074503(2010). [L = 3 fm] “Physical point simulaOon in 2+1 flavor la#ce QCD” See also: l  BMW, Science 322, 1224(2008) l  BMW, Phys.Leg.B701(2011)265. [L = 6fm]

Progress of LQCD algorithm Progress of Supercomputer

La#ce QCD simulaOon at Physical point is now possible.


To study mulO-‐baryon systems at physical point, spaOal volume should be as huge as possible. Such a physical point simulaOon is going on on K-‐computer at AICS, RIKEN. 964 la#ce, a = 0.1 fm, L = 9 fm, mpi=135 MeV The aims of the project: u 2+1 flavor QCD è 1+1+1 flavor QCD+QED u Various physical quanOOes u  InvesOgaOon of resonances u Direct construcOon of light nuclei u DeterminaOon of baryon-‐baryon potenOals

K computer (the 4th strongest in the world)

11.28 PFLOPS

[Kuramashi@la#ce2013]

Light Nuclei from LaCce QCD

(7)

LaCce calcula4on of light nuclei (8)

There are several obstacles in LQCD calculaOon of atomic nuclei u  Signal to Noise

u  Ground state saturaOon (at large volume)

u  Bound state ßà scagering state (Volume extrapolaOon)

u  CalculaOonal cost for Wick contracOon


Obstacle (1) u  Signal to Noise

u  pion

u  nucleon atomic nuclei with mass # A è

For large nuclei, quality of the signal is expected to become bad rapidly.

signalnoise

~ exp(−mπ t)exp(−2mπ t)

~ const

signalnoise

~ exp(−mNt)exp(−3mπ t)

~ exp −(mN − 3 / 2mπ )t( )

signalnoise

~ exp −A(mN − 3 / 2mπ )t( )


Obstacle (2) Ground state SaturaOon (at large volume) We use two point correlator, to extract ground state energy in large t region. For large volume, energy gap shrinks as Ground state saturaOon becomes more and more difficult.

ΔE = En+1 − En ~1mN

2πL

⎛⎝⎜

⎞⎠⎟2

L=3 fm L=6 fm L=12 fm …

ΔE 181.5 MeV 45.3 MeV 11.3 MeV …

C(t) = O(t)O(0)

= n O(0) 02exp(−Ent)

n∑

If L becomes twice as large, ΔE becomes 4 Omes as small.


Obstacle (3) DiscriminaOon of bound state from scagering state (Volume extrapolaOon): ΔE < 0 in finite V means i.  bound state ii.  agracOve interacOon in finite volume.

It is necessary to use several spaOal volumes to extrapolate the results to V=∞ carefully.

volume extrapolaOon of deuteron: from NPLQCD, PRD85,054511(2012) (2+1 flavor QCD on anisotropic la#ce. mpi=390 MeV, L=2.0,2.5,2.9,3.9 fm)


Obstacle (4) ComputaOonal cost for Wick contracOon: Number of Wick contracOon: è Naïve contracOon algorithm will soon go into trouble. T.Yamazaki et al., PRD81,111504(2010) reduced these numbers as •  3H/3He: 360 à 93 •  4He: 518400 à 1107 and gave the first LQCD results of bound light atomic nuclei. (quenched QCD. mpi=0.8 GeV, L=3.1-‐12.3 fm)

∝ 2Nn + Np( ) ! × Nn + 2Np( ) !1H 2H 3H/3He 4He 6Li …

#(Wick contracOon) 2 36 2880 518400 131681894400 …

Light Nuclei from laCce QCD (13)

pα (x)nβ (y) ⋅ p ′α ( f )n ′β ( f )

= sign(σ )σ∑ ⋅ Pα (x;ξσ (1),ξσ (2),ξσ (3) )Nβ (y;ξσ (4 ),ξσ (5),ξσ (6) ) ⋅C(ξ1,,ξ6 )

ξ1,,ξ6

∑relabeling of summation var. : ξ′1 ≡ ξσ (1),,ξ′6 ≡ ξσ (6)

= Pα (x;ξ′1,ξ′2,ξ′3)Nβ (y;ξ′4 ,ξ′5,ξ′6 )ξ ′1,,ξ ′6∑ C(ξ′

σ −1(1),,ξ′

σ −1(6))sign(σ )

σ∑⎧⎨

⎩

⎫⎬⎭

Pα (x;ξ1,ξ2,ξ3) ≡ pα (x) ⋅u (ξ1)d (ξ2 )u (ξ3)

Nβ (y;ξ4 ,ξ5,ξ6 ) ≡ nβ (y) ⋅u (ξ4 )d (ξ5 )d (ξ6 )

f :smearing function

PermutaOon sum associated with Wick contracOon can be carried out before LQCD calculaOon starts.

ξ = (c,α )∈color ×Dirac

Efficiency is significantly improved by •  3H/3He: x 192 •  4He: x 20736

See also: u W. Detmold, K.Orginos, PRD87,114512(2013) u  J.Gunter, B.C.Toth, L.Varnhorst, PRD87,094513(2013)

SystemaOc reducOon method is proposed by T.Doi, M.G.Endres, CPC184,117(2013).


Several bound mulO-‐baryon states are reported in heavy quark mass region (mpi > 390 MeV). (Some agree, the others not.) u H-‐dibaryon

u  NPLQCD: NF=2+1: PRL106,162001(2011) NF=3:★

u  HALQCD: NF=3: PRL106,162002(2011)

u  3H/3He, 4He u  PACSCS: both bound

u  NF=0: PRD81,111504(R)(2010) u  NF=2+1: PRD86,074514(2012)

u  NPLQCD: NF=3: ★ u  HALQCD: NF=3: NPA881,28(2012).

Only 4He bound at mPS=469MeV

u Deuteron, dineutron u  NPLQCD: NF=2+1: PRD85,054511(2013)

(also H-‐dibaryon and Ξ−Ξ− bound) NF=3: ★

u  PACSCS: both bound u  NF=0: PRD84,054506(2011) u  NF=2+1: PRD86,074514(2013)

u  HALQCD: No bound states for both.

u Many others, light (hyper)nuclei u  NF=3: ★

★ NPLQCD, PRD87,034506(2013)

Nuclear Force from LaCce QCD (HAL QCD method)

(15)

Nuclear Force from LaCce QCD (16)

u Nambu-‐Bethe-‐Salpeter (NBS) wave funcOon

u RelaOon to S-‐matrix by reducOon formula

u Equal-‐4me restricOon of NBS wave func4on behaves at long distance as Exactly the same funcOonal form as that of scagering wave funcOons in quantum mechanics

0 T N(x)N(y)[ ] N(+k)N(−k),in

N(p1)N(p2 ),out N(+k)N(−k),in

= disc + iZ −1/2( )2 d 4∫ x1d4x2 e

ip1x1 1+mN2( )eip2x2 2+mN

2( ) 0 T N(x1)N(x2 )[ ] N(+k)N(−k),in

ψ k (x − y) ≡ lim

x0→+0ZN

−1 0 T N(x, x0 )N(y,0)[ ] N(+k)N(−k),in

eiδ (k )sin k r +δ (k)( )

k r+ as r ≡ x − y → large

[C.-‐J.D.Lin et al., NPB619,467(2001).]


u Def. of potenOal from equal-‐4me NBS wave func4ons: u U(r,r’) is E-‐indep. u U(r,r’) reproduces the scagering phase,

because equal-‐Ome NBS wave funcOons behaves at long distance as

u Existence of E-‐indep. U(r,r’) u AssumpOon: Linear independence of equal-‐Ome NBS wave func. for E < Eth.

è There exists dual basis:

u Proof:

k2 /mN − H0( )ψ k (

r ) = d 3∫ ′r U(r , ′r )ψ k (′r )

ψ k (x − y) eiδ (k )

sin k r +δ (k)( )k r

+ as r ≡ x − y → large

d 3∫ rψ ′k (r )ψ

k (r ) = (2π )3δ 3(′k −k ) for E ≡ 2 mN

2 + k 2 < Eth

K k (r ) ≡ k 2 / mN − H0( )ψ

k (r )

K k (r ) = d 3 ′k

(2π )3 K ′k (r ) d 3∫ ′r ψ ′k (r )∫ ψ k (r )

= d 3∫ ′r d 3k(2π )3∫ K ′k (r )ψ ′k (′r )

⎧⎨⎩⎪

⎫⎬⎭⎪ψ

k (′r )

U (r , ′r ) ≡ d 3k′

(2π )3∫ K ′k (r )ψ ′k (r )

does not depend on E because of integraOon of k’.

for 2 mN2 + k2 < Eth ≡ 2mN +mπ


Ground state satura4on is not needed to extract the potenOal. u Normalized NN correlator in the elasOc region:

u “Time-‐dependent” Schrodinger-‐like equa4on to extract the potenOal. Only Elas4c satura4on is required to derive this equaOon. (Elas4c satura4on is much easier than ground state satura4on.)

R(t, x − y) ≡ e2mN ⋅t 0 T N(x,t)N(y,t) ⋅J NN (t = 0)⎡⎣ ⎤⎦ 0

= akk∑ exp −tΔW (k)( ) ⋅ψ k (

x − y)

14mN

∂2

∂t 2− ∂∂t

⎛⎝⎜

⎞⎠⎟R(t, r ) = ak

k∑ k2

mN

exp −tΔW (k)( ) ⋅ψ k (r )

14mN

∂2

∂t 2− ∂∂t

− H0⎛⎝⎜

⎞⎠⎟R(t, r ) = d 3∫ ′r U(r , ′r ) ⋅R(t, ′r )

An iden4ty

ΔW (k) ≡ 2 mN2 + k2 − 2mN

ΔW (k)2

4mN

+ ΔW (k) = k2

mN

H0 +U( )ψ k (

r ) = k2

mN

ψ k (r )


〈0 |T[N (x,t)N (

y,t) ⋅NN (t = 0;α )] | 0〉

= ψ n( x − y) ⋅an(α ) ⋅exp(−Ent)n∑

t = 9

VC(x)

= − H0R(t,x)

R(t, x)− (∂/ ∂t)R(t,

x)R(t, x)

+ 14mN

(∂/ ∂t)2R(t, x)R(t, x)

(Example) Obtained potenOal does not depend on excited state contaminaOon. u Source funcOon (with a single real parameter alpha)

alpha is used to arrange the mixture of excited states

( )cos(( , 2 / ) cos(2 / ) cos(2 / ), ) 1 x Lz yf L zx y Lα π π π+ += +

1S0 central potenOal at LO of deriv. expansion


General nonlocal potenOal is inconvenient. è We employ derivaOve expansion: Convergence of DerivaOve expansion has to be checked.

U(r , ′r ) ≡ VC(r){ +Vll (r)L2 + {Vpp (r),∇

2}O(∇2 ) term

+O(∇4 )}δ (r − ′r )

The following agrees at E=0 and 45 MeV çè The derivaOve exp. can safely be truncated up to LO in the region

VC(r;E) ≡ E − H0ψ E (

r )ψ E (r )

U(r , ′r ) VC(

r )δ (r − ′r )

0 ≤ E ≤ 45MeV

Comment: The current result is obtained based on an older method. The result should be replaced by the new method. “Ome-‐dependent” Schrodinger-‐like eq.


Comparison between the poten4al method and Lueshcer’s method.

ππ scagering in I = 2 channel

Ns=16,24,32,48, Nt=128, a=0.115 mpi = 940 MeV by Quenched QCD Kurth et al., arXiv:1305.4462[hep-‐lat]


fromR.Machleidt,Adv.Nucl.Phys.19

central force

( )CV r

( )TV r

tensor force

2+1 flavor config by PACS-‐CS Coll. m(pion) = 570 MeV, m(N)=1412MeV

2+1 flavor QCD result of nuclear forces at m(pion)=570 MeV. QualitaOve behaviors are reproduced.


v  QualitaOvely reasonable behavior. But the strength is significantly weak.

(AgracOve. No bound state.) v  AgracOon shrinks as mpion decreases.

Reason: The repulsive core grows more rapidly than the agracOve pocket in the region mpion = 411-‐700 MeV.

v  It is important to go to smaller quark mass region.

1S0 phase shi~ from Schrodinger eq.

v  HAL QCD No bound state for nn and deuteron

v  less agracOve for heavier quark mass v  no bound state for large quark mass

v  PACSCS & NPLQCD Bound nn and deuteron

v more agracOve for heavier quark mass v  bound state for large quark mass


Disagreement for two-‐nucleon bound states in heavy quark mass region (mpi > 390 MeV).

Y.Kuramashi, PTPS122(1996)153. S.Beane et al., PRL97,012001(2006).

These results may suggest chiral behaviors of scagering length (pion may be too heavy to say anything)

LQCD calculaOon near physical point is important and interesOng.


LS potenOal and nuclear forces in the parity-‐odd sector are now possible.

3PJ nuclear forces

u LS potenOal has great influence on the shell model and the magic number of nuclei.

u QualitaOve behaviors are reproduced. But the strength is not enough.

u AgracOon in 3P2 channel è P-‐wave neutron superfluid in neutron star

[K.Murano et al, arXiv:1305.2293]


Three nucleon potenOal (on the linear setup)

Ø  Few body calculations shows its relevance

Ø  Important influence on the drip line and the magic number of neutron-rich nuclei.

Ø  The more important role at the higher density. è supernova and neutron star.

Ø  Experimental information is limited

linear setup

2 flavor gauge config by CP-‐PACS Coll. m(pion) = 1136 MeV, m(N) = 2165 MeV

Short-‐range repulsive 3NF !

[T.Doi et al, PTP127,723(2012)]


Our best target is hyperon force. u  structure of hypernuclei

u eq. of state of hyperon mager

u Experimental informaOon is limited due to their short life Ome.

J-PARCExploration of multi-strangeness world

J.Schaffner-Bielich, NPA804(’08)309.


Ø Repulsive core is surrounded by attraction like NN case.

Ø  Strong spin dependence of repulsive core.

quark mass dependence

Repulsive core grows with decreasing quark mass. No significant change in the attraction.

Nemura, Ishii, Aoki, Hatsuda,PLB673(2009)136.

ΞN(I = 1)


Ø  Repulsive core is surrounded by attraction like NN case.

Ø  These two potentials looks similar, which may be due to small flavor SU(3) breaking. They are not necessarily equal.

Ø  N-Lambda belongs to 27+8s rep. in flavor SU(3) limit.

Ø  N-Sigma belongs to 27 rep. in flavor SU(3) limit.

ΛN ΣN(I = 3 / 2)

1S0


[Nemura,PoS(LAT2011)]


ΛN ΣN(I = 3 / 2)

3S1 −3 D1

u N-‐Lambda p  Repulsive core is surrounded by agracOon p  The agracOon is deeper than 1S0 case p Weak tensor force (no one-‐pion exchange is allowed)

u N-‐Sigma

p  Repulsive core at short distance p  No clear agracOve well

(Repulsive nature is consistent with the naïve quark model) p  Strength of tensor force: N-‐N > N-‐Sigma > N-‐Lambda


[Nemura,PoS(LAT2011)]


Hyperon PotenOals in flavor SU(3) limit to see a general trend

8⊗ 8 = 27⊕ 8S ⊕1symmetric ⊕10⊕10⊕ 8A

anti-symmetric

[T.Inoue et al, PTP124,591(2010)] Ø  Strong flavor dependence

(1) All distance attraction for flavor 1 representation. (2) Strong repulsive core for flavor 8S representation. (3) Weak repulsive core for flavor 8a representatin. Ø  These short distance behaviors are consistent with quark Pauli blocking picture.

u d+ u d s+ +

3 flavor con

fig.

gene

rated by

HAL QCD

Coll.

m(PS) = 469 M

eV

m(B) = 1161 MeV


EnOrely agracOve potenOal in flavor 1 channel leads to a bound H-‐dibaryon

As quark mass decreases, the potenOal becomes more agracOve. çè The binding energy decreases because of the increase of kineOc energy.


Coupled channel extension for is possible by employing the triple Argument parallel to the single-‐channel NN case leads a coupled-‐channel Schrodinger eq.

ΛΛ− NΞ− ΣΣ

Ψn (x − y) ≡

0 Λ(x)Λ(y) n,in

0 N(x)Ξ(y) n,in

0 Σ(x)Σ(y) n,in

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

pΛΛ2

2µΛΛ

+ 2µΛΛ

⎛⎝⎜

⎞⎠⎟ψ ΛΛ (

r;n)

pNΞ2

2µNΞ

+ 2µNΞ

⎛⎝⎜

⎞⎠⎟ψ NΞ(

r;n)

pΣΣ2

2µΣΣ

+ 2µΣΣ

⎛⎝⎜

⎞⎠⎟ψ ΣΣ (

r;n)

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

= d 3∫ ′r

UΛΛ;ΛΛ (r , ′r ) UΛΛ;NΞ(

r , ′r ) UΛΛ;ΣΣ (r , ′r )

UNΞ;ΛΛ (r , ′r ) UNΞ;NΞ(

r , ′r ) UNΞ;ΣΣ (r , ′r )

UΣΣ;ΛΛ (r , ′r ) UΣΣ;NΞ(

r , ′r ) UΣΣ;ΣΣ (r , ′r )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⋅ψ ΛΛ (

r ′;n)ψ NΞ(

r ′;n)ψ ΣΣ (

r ′;n)

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

E ≡ 2 mΛ2 + pΛΛ

2

= mN2 + pNΞ

2 + mΣ2 + pNΞ

2

= 2 mΣ2 + pΣΣ

2

[S.Aoki et al., Proc.Japan Acad.B87(2011)509.]

MS��[MeV]

MK��[

MeV

]

24 Î Talk by K.Sasaki (Thu.)

Coupled channel study is essential

B.E.

SU(3) lat

120MeV

30MeV

Physical point

?

Î [HAL] T.Inoue (Wed.)

Î [NPL] K.Orginos (Wed.)

H-dibaryon (uuddss, I=0,1S0)

(DerivaOon is parallel, but notaOon is quite lengthy.)


The numerical calculaOon is tough. But it is doable. (work in progress)

[K.Sasaki@La#ce2012]

diagonal part off-‐diagonal part

2+1 flavor gauge config by CP-‐PACS/JLQCD Coll. m(pion) = 875 MeV m(K) = 916 MeV m(N) = 1806 MeV m(Lambda) = 1835 MeV m(Sigma) = 1841 MeV m(Xi) = 1867 MeV


Parity-‐odd hyperon potenOals in the flavor SU(3) limit.

VBB =VC;S=0 (r)P(S=0) +VC;S=1(r)P(S=1) +VT(r) 3(r̂ ⋅σ 1)(r̂ ⋅

σ 2 )−

σ 1 ⋅σ 2( )

+VSLS(r)L ⋅S+ +VALS(r)

L ⋅S− +O(∇

2 )

[N.Ishii@La#ce 2013]

27rep ~ NN( ) 10rep

8rep

u Repulsive core for 27 and 10^* rep.

u No repulsive core for 10 and 8 rep. (consistent with quark model)

u  Weak tensor potenOal (8 rep) u  Weak symmetric LS potenOal

(8 rep) u  Strong anO-‐symmetric LS

potenOal (8 rep)

10rep ~ NN( )

The gap is being filled. (36)









u Output: u Structures of baryons,


u Tight connecOon will be established soon ! u For this, we need,

u Physical point LQCD calculaOon of light nuclei and nuclear forces employing huge spaOal volume

u Agreement of results among different groups

The gap is being filled. (37)









u Output: u Structures of baryons,


u Tight connecOon will be established soon ! u For this, we need,

u Physical point LQCD calculaOon of light nuclei and nuclear forces employing huge spaOal volume

u Agreement of results among different groups

u Tight connecOon can be extended to hyperon sector. (This will be one of the best applicaOons of LQCD.)

Hyper Nuclear Physics u Input:

u Fundamental DOF hyperons

u InteracOons less established

u Output: u Structures of hypernuclei

[made of hyperons] u their reacOons

Summary (38)

u The gap between QCD and tradiOonal nuclear physics is being filled by recent progress of LQCD u LQCD simulaOon at physical point

u SpaOal volume is becoming huge (for mulO-‐baryon system)

u LQCD calculaOon of light nuclei u LQCD calculaOons are accumulaOng at heavier quark mass region. u New techniques are being developed.

u LQCD calculaOon of nuclear forces

u The method does not need ground state saturaOon u Wide range of applicaOon to baryon interacOon

NN, NY, YY, NNN, negaOve parity, LS potenOals, etc.

u To establish a Oght connecOon, we need u Huge volume LQCD calculaOon at physical point of

light nuclei and nuclear forces u Agreement of the results of different groups

u Tight connecOon should be extended to hyper nuclear physics.

backup slides

(39)


Similar behavior is seen in NF=3 calculaOon (flavor SU(3) limit) v  agracOon shrinks as decreasing quark mass for mPS=672-‐1171 MeV. v  agracOon starts to increase at a point between mPS=469 MeV-‐672 MeV.

v We expect similar thing happens for NF=2+1 calculaOon at smaller pion mass, because nuclear force for NF=3 is generally more agracOve than nuclear force for NF=2+1.


It is easy to see whether Luescher’s condiOon is saOsfied or not. NOTE: HAL QCD method is derived from the method to examine Luescher’s condiOon by using Bethe-‐Salpeter wave funcOon. (See CP-‐PACS, PRD71,094504(2005).)

R < L / 2

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Lace&QCD&Approach&to&Nuclear&Physicsmenu2013.roma2.infn.it/talks/plenary_thursday/5-Ishii_menu2013.pdf · LaCce%QCD%simula4on%at%Physical%point 4 from&Seminar&atICRR,&16&April&2010&by&