格子QCDによるハドロン間相互作用-核力をQCDから導く青木 慎也
筑波大学

数理物質科学研究科

京都大学基礎物理学研究所セミナー
2010.12.16

1. Motivation

What binds protons and neutrons inside a nuclei ?
gravity: too weakpnCoulomb: repulsive between pp
no force between nn, np
New force (nuclear force) ?

1935 H. Yukawa
introduced virtual particles (mesons) to explain the nuclear force

Yukawa potential

g 2 e−mπ r
V (r) =
4π r
1949 Nobel prize

T. Hatsuda (Univ. Tokyo)

Nuclear HAL
forceQCD
is a Collaborationbasis for understanding ...
(Hadrons to Atomic Nuclei Lattice QCD Collaboration)•

ordinary
hyperK.
nucleiS. Aoki,Structure
T. Doi, T.ofInoue,
K.and
Murano,Sasaki(Univ. Tsukuba). Hatsuda, Y. Ikeda, N. Ishii(Univ. Tokyo)

H. Nemura(Tohoku Univ.)Λ• Structure of neutron star
Neutronmatterquark
Matter?

II SuperNova
NN, YN,YY, 3N• Ignition of Type

Hadrons

forces from LQCD
Atomic nuclei
Chiral Dynamics 09, Bern, July 7, 2009

Neutron starPhenomenological NN potential¾ phaseOne-pionexchanges of the Nuclear
force to fit 5000(~40 parameters
shift data)¾ One-pionexchange
Yukawa (1935)of the Nuclear force

Yukawa(1935)¾
One-pionexchangeexchange
I One-piontures of the Nuclear force

III

II

Yukawa(1935)
Yiukawa(1935)

I¾ Multi-pions
II Multi-pions
Taketani
et al.
¾¾
Multi-pions
Multi-pions

Taketani et al.(1951)(1951)
Taketani
Taketanietetal.
al.
(1951)(1951)

SS...

sive
SS...
SS...
e

S
S S

III Repulsive core

Jastrow(1951)¾ Repulsive
core
Repulsive
core
Jastrow(1951)¾¾
Repulsive
core

Jastrow (1951)

Jastrow (1951)

sion cannot exist Repulsive
without the “RC”.core is importantntial for the “RC”.

9#:%;<"
stability of nuclei!"#$%&'
maximum mass ofneutron starasis of hypernuclear physics @ J-PARC

explosion of
II-./$01
type II supernova!"#$%&"'(QCD)()*+,(./%01%!2345+,678*,

f the repulsion ?

-sal or channel dependent ?

23456A ./$78

QCD;<!"
Origin of RC:9:23;<!"“The most fundamental problem in
Nuclear physics.”

Note: Pauli principle is not essential for the “RC”.
nciple+核力の性質をクォークから説明できるか？

Plan of my talk
1. Motivation
2. Strategy in (lattice) QCD to extract “potential”
3. More structure: tensor potential
4. Inelastic scattering: octet baryon interactions
1. Baryon-Baryon interactions in an SU(3) symmetric world
2. Proposal for S=-2 inelastic scattering
3. H-dibaryon
5. Summary and Discussion

2. Strategy in (lattice) QCD
to extract “potential”!"#$%&"'(QCD)()*+,(./%01%!2345+,678
南部陽一郎、『クォーク』第２版（講談社、ブルーバックス、1997）-

9:23;<!"

QCD;<!"-

QCDから核力を如何に定義し、如何に計算するか？
Y. Nambu,“Force Potentials in Quantum Field Theory”,
Prog. Theor. Phys. 5 (1950) 614.

QCD;<!"

C. Hayashi and Y. Munakata,“On a Relativistic Integral Equation for Boundstates”, Prog. Theor. Phys. 7 (1952) 481.
K. Nishijima,“Formulation of Field Theories for composite
particles”, Phys. Rev. 111 (1958) 995.
HAL QCD Collaboration
佐々木健志、土井琢身、青木慎也 (筑波大)石井理修 、初田哲男 (東大)池田陽一 (理研)、井上貴史 (日大)
村野啓子 (KEK)、根村秀克 (東北大)Ishii-Aoki-Hatsuda, PRL 90(2007)0022001
Aoki-Hatsuda-Ishii, PTP123(2010)89(equal time) Nambu-Bethe-Salpeter wave function is a key

ϕE (r) = �0|N (x + r, 0)N (x, 0)|6q, E�

!
E = 2 k2 + m2N

QCD eigen-state with energy E and #quark =6

N (x) = εabc q a (x)q b (x)q c (x): local operator
C.-J.D.Lin et al., NPB69(2001) 467
CP-PACS Coll., PRD71 (2005) 094504
N. Ishizuka, PoS(LAT2009)119

E < EthϕE (r)!

=

C eik·r +

+

#
I(r)

inelastic contribution"

off-shell T-matrix

d3 p ip·r Ek + Ep T (p, −p ← k, −k)
e
3
(2π)
8Ep2
p2 − k2 − i#

√ 2

−

∝ O(e

Eth −E 2 |r|)(Relativistic) Spinor structure is contained in C.

N. Ishizuka, PoS(LAT2009)119(Equal time) contains sufficient information.
( 空間的）非同時刻、 重心系

同時刻、 非重心系

p+qLorentz transformation

Asymptotic behavior

r = |r| → ∞ϕE (r) −→!
lk + (−k)sin(kr − lπ/2 + δl (k))
Cl
krpartial wave

spinor structureδl (k) is the scattering phase shift

S=e

2iδ

S-matrix below inelastic thresholdIshii-Aoki-Hatsuda, PRL 90(2007)0022001
Aoki-Hatsuda-Ishii, PTP123(2010)89

Our definition of “potential”[!k − H0 ]ϕE (x) =!

d3 y U (x, y)ϕE (y)

k2
!k =
2µ
−∇2
H0 =
2µ

U (x, y) may be non-local but can be energy-independent.
!
|ϕE !"ϕ̃E | = 1E≤Eth!ϕ̃E |ϕE ! " = δEE !
ϕ̃E (y)E≤Ethdual basis

identity in the restricted space

U (x, y) =

!

E≤Eth

[εk − H0 ]ϕE (x)ϕ̃E (y)this construction is NOT unique.

Finite but large volume

Finite volume

no interaction

interaction rangeLallowed value: kn2

Lueshcer’s formulaδl (kn )“potential” is expected to be short-range.

Velocity expansion

U (x, y) = V (x, ∇)δ 3 (x − y)Okubo-Marshak (1958)

spins

V (x, ∇) = V0 (r) + Vσ (r)(σ1 · σ2 ) + VT (r)S12 + VLS (r)L · S + O(∇2 )LO

LOtensor operator

S12 =

LO

NLO

NNLO

3
(σ1 · x)(σ2 · x) − (σ1 · σ2 )r2

we calculate observables such as the phase shift and the binding energy,
using this approximated potential.

3. Results from lattice QCD
a
x

x

L

• well-defined statistical system (finite a and L)• gauge invarinat• fully non-perturbative

Monte-Calrosimulations

Quenched QCD : neglects creation-anihilation of quark-anitiquak pair
Full QCD : includes creation-anihilation of quark-anitiquak pair

NBS wave function from lattice QCD!
!0|nβ (y, t)pα (x, t)J pn (t0 )|0" = !0|nβ (y, t)pα (x, t)|En "!En |J pn (t0 )|0"
=

!
n

n

−E0 (t−t0 )0
An !0|nβ (y, t)pα (x, t)|En "e−En (t−t0 ) −→ A0 ϕE
(x
−
y)e
αβ

t→∞

An = !En |J pn (t0 )|0"
Wall source

L=0
spin

q(x, t0 ) → q

J pn (t0 ) = pwall (t0 )nwall (t0 )LJ(t0 ) =!

q(x, t0 )xwith Coulomb gauge fixing

P =+

1 1
⊗ =1⊕0
2 2

2S+1

wall

3

S1

1

S0

NN wave function

Quenched QCD

NN wave function !(r)a=0.137fm

1.2

1
S
3 0
S1

attraction

1.0
0.8

1

!(x,y,z=0; S0)

1.5

0.6
0.4

repulsion

0.2
0.0
0.0

t − ts = 6

mπ ! 0.53 GeV

normalized here

1.0
0.5
-2

0.5

2

-1
x[fm] 0

1

2 -2

1.0
1.5
r [fm]

10y[fm]-1

2.0
Ishii-Aoki-Hatsuda,
PRL90(2007)0022001

1S ,3S
otential Vc(r) from !"(r) at E(quenched)~0
potentials
0
1(m#"=0.53 GeV)

LO (effective) central Potential

E�0mπ ! 0.53 GeV(I=1)

V (r;1 S0 ) = V0

(I=0)

V (r;3 S1 ) = V0
Central potential(r) + Vσ(I=1) (r)(r) − 3Vσ(I=0) (r)a=0.137 fm

L=4.4fmdeQualitative features of NN potential are reproduced !
Ishii-Aoki-Hatsuda, PRL90(2007)0022001

This paper has been selected as one of 21 papers in
Nature Research Highlights 2007

Scheme/Operator dependence of “potential”• The “potential” depends on the definition of the wave function, in particular, on the choice ofthe nucleon operator N(x). (Scheme-dependence)• local operator = convenient choice for reduction formula• Moreover, the potential itself is NOT a physical observable. Therefore it is NOT unique andis naturally scheme-dependent.

• Observables: scattering phase shift of NN, binding energy of deuteron• Is the scheme-dependent potential useful ? Yes !
• useful to understand/describe physics• a similar example: running coupling• Although the running coupling is scheme-dependent, it is useful to understand the
deep inelastic scattering data (asymptotic freedom).

• “good” scheme ?
• good convergence of the perturbative expansion for the running coupling.
• good convergence of the derivative expansion for the “potential” ?
• completely local and energy-independent one is the best and must be unique if
exists. (Inverse scattering method)toolsrunning coupling

potential

physical observabledeep inelastic scattering

NN scattering phase
shift

phenomena

almost free parton

repulsive coreinterpretation

asymptotic freedom

no theoreticalexplanation so far

schemeMS-bar coupling

potentialfrom BS wave function

Other examples:
QM: (wave function,potential) → observables
QFT: (asymptotic field,vertex) → observables
EFT: (choice of field, vertex) → observables

Convergence of the derivative expansion
Leading Order(E − H0 )ϕE (x)VC (r) =ϕE (x)

Local potential approximation

The local potential obtained at given energy E may depend on E.
Non-locality can be determined order by order in velocity expansion ( cf. ChPT)

V (x, ∇) = VC (r) + VT (r)S12 + VLS (r)L · S + {VD (r), ∇2 } + · · ·Numerical check in quenched QCD

mπ ! 0.53 GeV

a=0.137fm
K. Murano, S. Aoki, T. Hatsuda, N. Ishii, H. Nemura

●

PBC

(E〜0 MeV)●APBC (E〜46 MeV)

Quenched QCD

mπ ! 0.53 GeV
a=0.137fm+RZJRRGLVWKHGHULYDWLYHH[SDQVLRQ" FRQW G
5(68/7(16>.0XUDQR#/DWWLFH@

SKDVHVKLIWVIURPSRWHQWLDOV

E-dependence of the local potential
turns out to be very small at low
energy in our choice of wave function.
( 0H9 $3%&

Tensor potential(H0 + VC + VT S12 )|φ! = E|φ!
mixing between 3 S1 and 3 D1 through the tensor force|φ! = |φS ! + |φD !

1 !
|φS ! = P |φ! =R|φ!
24“projection” to L=03“projection” to L=2

3

R∈O|φD ! = Q|φ! = (1 − P )|φ!

S1

D1

P (H0 + VC + VT S12 )|φ! = EP |φ!
Q(H0 + VC + VT S12 )|φ! = EQ|φ!

Wave functions

Quenched

Aoki, Hatsuda, Ishii, PTP 123 (2010)89
arXiv:0909.5585

(a)
3
S
3 1
D1

m/ = 529 MeV

0.0

0.5

1.0
r [fm]

1.5

2.0

remove angular dependence

Y20 (θ, φ) ∝ 3 cos2 θ − 1

quenched QCD
E 〜 0 MeV

Aoki, Hatsuda, Ishii, PTP 123 (2010)89
Quenched arXiv:0909.5585
Tensor Force and Central Force (t-t0=5)

Potentials
400

100

VT(r)
VC(r)
VC,eff

350
300
250

V(r) [MeV]

VC ! VC,eff

50

200

mπ ! 0.53 GeV

0

150
100

VT

50

0

0.5

1

1.5

2

0
-50
0

0.5

No repulsive core in tensor

1r [fm]

1.5

2

!"#$% &'()*+

Aoki, Hatsuda, Ishii, PTP 123 (2010)89! /%0123456789:;<=>
Quenched arXiv:0909.5585
Tensor Force and Central!Force(t-t0=5)
deuteronDEF1GHIJK9LM9

Potentials
400! PQ9KLMNRST6!"#$%9particle Vspectrum(UVWXYZ[$)
T(r)
VC(r)
c=d]^\'eHfg<\hije
VC,eff! centrifugal barrier9lF6QmnSo
C u6v;<\Uw=xykCC,eff(0z(9{|=}k5j~dÄÅ

100
350
300
250

V(r) [MeV]

V !V

50

200

mπ ! 0.53 GeV

0

150
100

VT

50

0

0.5

1

1.5

0
-50

2
from
R.Machleidt,
Adv.Nucl.Phys.19

00.5

No repulsive core in tensor

1r [fm]

1.5

2

Quark mass dependence

Quenched• Rapid quark mass dependence oftensor potential• Evidence of one-pion exchange

Fit function

Full QCD Calculation
Full QCD
mπ = 570 MeV, L = 2.9 fm
a=0.1fm

Quenched QCD

L=4.4fmmQuenched
a=0.137 fm
π ! 0.53 GeV QCD* Large repulsive core than quenched* Large tensor force than quenchedPhase shift from V(r) in full QCD
1

3

S0

S1

a=0.1 fm, L=2.9 fm

1

S0

3

S1

:HKDYHQRGHXWHURQVRIDU
:HKDYHQRGHXWHURQVR

1

S0

3

S1

KH\KDYHUHDVRQDEOHVKDSHV7KHVWUHQJWKLVPXFKZHDNHU
7KH\KDYHUHDVRQDEOHVKDSHV7KHVWUHQJWKLVPXFKZHDNHUÎ ,PSRUWDQFHRISK\VLFDOTXDUNPDVVÎ ,PSRUWDQFHRISK\VLFDOTXDUNPDVVThey have reasonable shapes. The strength is much weaker.

calculation at physical quark mass is important. (future work)4. Inelastic scattering:octet baryon interactions

x

in a SU(3) symmetric world
Baryon-Baryon interactions in an SU(3) symmetric world
mu = md = mst stepto predict
YN, YY
accessible
in exp.
1. First
setup to predict
YN,interactions
YY interactionsnot
not accessible
in exp.
2. Origin
oftotherepulsive
or not)ginoftherepulsivecore(universal
ornot)
1. First
step
predict
YN,coreYY(universal
interactions
not accessible in exp.

2.actions
Origin of the repulsive core (universal or not)!"#$%&'()(*

mmetric world!

x+',-./&.0&11&.0&*23&4/.567&8*,*3&.9*3*&:,/;.6<&"YY,interactions" % (! ** $ *in$'
redict YN,
not"#accessible
exp.2 * $' * ")epulsive ''''''''''''''
core (universal3/44%&#+5
or not)''''''' 67&+8,/44%&#+5
Six! independent
potentials
inflavor-basis=23&8(>&(673?36736*&?.*36*(,'8&0./&!@8*,*3<
Six independent potentials in flavor-basis
6 independent potential in flavor-basis# ' * $ #(!$ #% $ & $ #"# $ #% $ & $ #$$ #% $&# $ * $ #$' 2$ #% $ & $ #$'$ #% $ & $ #") $ #% $$pendent
potentials in flavor-basis!

"!"#$%&'()&*+,',&-./

A3&9.68*/59*&(//3759(:'3&11&8.5/93&.?3/,*./&B(*2!"#$%&C<D<&9.300<&,67&9.)?5*3&E@?.(6*&0569*(.6<Potentials
(27)

V

2000

100

m/=1014 MeV
m/=835 MeV

Inoue for HAL QCD Collaboration
(8s)

V

2000

10000

m/=1014 MeV
m/=835 MeV

7500

50

1500

V(r) [MeV]

V(r) [MeV]

1500

0

1000

500

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0

0
0.0

0.2

(10*)

V

2000

0.4

0.6

100

0.8
1.0
r [fm]

1.2

1.4

0.0

1.6

m/=1014 MeV
m/=835 MeV

(10)

V

2000

0.6

V(r) [MeV]

100

0.8

1.0

1.2

1.4

1.6

m/=1014 MeV
m/=835 MeV

0

1000

-50
500

0.4

50

1500

0

1000

0.2

r [fm]

50

1500
V(r) [MeV]

2500

1000

0

-50
500

5000

-50

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0500

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0
0.0

0.2

0.4

0.6

0.8
1.0
r [fm]

1.2

1.4

1.6

27, 10*: same as before, NN channel

0.0

0.2

0.4

0.6

0.8
1.0
r [fm]

1.2

8s, 10: strong repulsive core

1.4

1.6

500

m/=1014 MeV
m/=835 MeV(1)V

1: no repulsive core, attractive core !
No quark mass dependenceV(r) [MeV]

0

-500

-1000

-1500

0.0

0.2

(8a)

V

2000

0.6

100

0.8
1.0
r [fm]

1.2

1.4

1.6

m/=1014 MeV
m/=835 MeV

50

1500

V(r) [MeV]

0.4

8a: week repulsive core,
deep attractive pocket

0

1000

-50
500

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0
0.0

0.2

0.4

0.6

0.8
1.0
r [fm]

1.2

1.4

1.6

Bound state in 1(singlet) channel ?
H-dibaryon ?

!"#%+$#$&$# !"#$%&

However, it is difficult to determine E
precisely, due to contaminations fromexcited states.

"#$%! !!""#$%

Schroedinger eq. predicts a bound stateat E < -30 MeV

Singlet potential with a certain value of E#!"#(,$,$'#$#%&$) !"#$%&!

" $! % & #$!% $!!!!!!!! ( # $"% $!'$#'#$$ % ! #

%

$

$

%

'# $%% ! #!

&%'()!*%%%&+%%'()!*&%.%/0+%%/+1+#2%%%%%%%%%%%%',-*! "! # #%%%3415&%.%/06%%/+153#&41#&%.%/4+%%/3147%%%%%310

Finite size effect is very large$!'%()*+,-./0'12&'3(145678)9:on this volume.
)*+,-./) *"#;<45##$)%&'(=>?@++,A!-.:!"(consistent with previous results.))

�!456*78,%'#$#% !

#

�2 � e−a5 r �2#$!%
$ a 1 − e−a4 r
2r
V (r)"=$!a%1&
e−a+3'# $%% ! # r$!!!!!!!! ( # $"% $!'$'#$$ % ! %#
!
2

2

#

$

%

&%'()!*%%%&+%%'()!*%%%%%%%%%%%%',-*! "! # # are in progress.
larger volume
calculations&%.%/0+%%/+1+#2%%%3415&%.%/06%%/+153#&41#

Proposal for S=-2 In-elastic scatteringmN = 939 MeV, mΛ = 1116 MeV, mΣ = 1193 MeV, mΞ = 1318 MeV

S=-2 System(I=0)

MΛΛ = 2232 MeV < MN Ξ = 2257 MeV < MΣΣ = 2386 MeV
The eigen-state of QCD in the finite box is a mixture of them:|S = −2, I = 0, E"L = c1 (L)|ΛΛ, E" + c2 (L)|ΞN, E" + c3 (L)|ΣΣ, E"!
!
!
!
E = 2 m2Λ + p21 = m2Ξ + p22 + m2N + p22 = 2 m2Σ + p23

In this situation, we can not directly extract the scattering phase shift
in lattice QCD.

HAL’s proposal
Let us consider 2-channel problem for simplicity.
NBS wave functions for 2 channels at 2 values of energy:ΨΛΛα (x) = !0|Λ(x)Λ(0)|Eα "α = 1, 2ΨΞN
α (x) = !0|Ξ(x)N (0)|Eα "They satisfy(∇ + p2α )ΨΛΛ
α (x) = 02(∇2 + q2α )ΨΞN
α (x) = 0|x| → ∞

We define the “potential” from the coupled channel Schroedinger
equation:
! 2
"
2
∇
pα
ΛΛ←ΛΛ
ΛΛ
ΛΛ←ΞN
ΞN
ΨΛΛ
+
(x)
=
V
(x)Ψ
(x)
+
V
(x)Ψ
α
α
α (x)
2µΛΛ
2µΛΛ
!

q2α

2

∇
+
2µΞN
2µΞN

"

diagonal

off-diagonalΞN ←ΛΛ
ΞN ←ΞNΨΞN(x)ΨΛΛ
(x)ΨΞNα (x) = V
α (x) + V
α (x)off-diagonal

diagonalµ: reduced mass!
X != YV(x)
V X←Y (x)
X←X

"

=

!

ΨX
1 (x)
ΨX
2 (x)

X, Y = ΛΛ or ΞN

ΨY1 (x)ΨY2 (x)"−1 !

(E1 − H0X )ΨX
1 (x)(E2 − H0X )ΨX
2 (x)

p2α
q2α
Eα =
,
2µΛΛ 2µΞN

α = 1, 2"

Using the potentials:

!

V
(x) V
(x)
V ΛΛ←ΞN (x) V ΞN ←ΞN (x)ΛΛ←ΛΛΞN ←ΛΛ

"we solve the coupled channel Schroedinger equation in the infinite
volume with an appropriate boundary condition.
For example, we take the incomming ΛΛ state by hand.

In this way, we can avoid the mixture of several “in”-states.

|S = −2, I = 0, E"L = c1 (L)|ΛΛ, E" + c2 (L)|ΞN, E" + c3 (L)|ΣΣ, E"
Lattice is a tool to extract the interaction kernel (“T-matrix” or “potential”).

Preliminary results from HAL QCD Collaboration
2+1 flavor full QCD

Sasaki for HAL QCD Collaboration

a=0.1 fm, L=2.9 fm

mπ � 870 MeV

Diagonal part of potential matrix

Non-diagonal part of potential matrix

VA−B ! VB−A
Hermiticity ! (non-trivial check)3-3. H-dibaryon
1. S=-2 singlet state may become the bound state in flavor SU(3)limit.
2. In the real world (s is heavier than u,d), some resonance may
appear above ΛΛ but below ΞN threshold.
3. Trial demonstration:
3.1. Use potential in SU(3) limit
3.2. Introduce only mass difference from 2+1 simulation
Inoue for HAL QCD Collaboration

Potentials in particle basis in SU(3) limit!"#$%&'"(&))&*!(&+,-./-012!% !% !

$'"# (## & " '$ ( # "
$!
'% (
""

!

!!"%

% !

)
"
!$%
%

$ !#%
$

$

""

$
$

""
##
##

!#%
!!"%
&'()( & !$%
&*!&'
&
&!&'
&
%
%
% &*& *+(,-!.)(!/0(1
!

$
$
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resonance
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SU(3)

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5. New method for hadroninteractions in lattice QCDInelastic scattering II: particle production
E ≥ Eth = 2mN + mπ

NBS wave function !
ϕE (r) = eik·r ++ I(r)

inelastic contribution

elastic scattering

d3 p ip·r Ek + Ep T (p, −p ← k, −k)
e
3
(2π)
8Ep2
p2 − k2 − i#NNπ ← NN∝ eiq·r

NN ← NN

|q| = O(E − Eth )Consider additional NBS wave function

ϕE,π (r, y) = !0|N (r + x, 0)π(y + x, 0)N (x, 0)|6q, E"
Note that|6q, E! = c1 |N N, E!in + c2 |N N π, E!in + · · ·Coupled channel equations(E − H0 )ϕE (x)=(E − H0 )ϕE,π (x, y)=!

d3 y U11 (x; y)ϕE (y) +!

d3 yd3 z U12 (x; y, z)ϕE,π (y, z)!
!
d3 z U21 (x, y; z)ϕE (z) + d3 zd3 w U22 (x, y; z, w)ϕE,π (z, w)

Velocity expansion at LO, two values of E

i = 1, 2(Ei − H0 )ϕEi (x)(Ei − H0 )ϕEi ,π (x, y)==

V11 (x) : N N ← N N

V12 (x, x) : N N ← N N π

V21 (x, y) : N N π ← N N

V11 (x)ϕEi (x) + V12 (x, x)ϕEi ,π (x, x)
V21 (x, y)ϕEi (x) + V22 (x, y)ϕEi ,π (x, y)

V22 (x, y) : N N π ← N N π

Solve Schroedinger equation with these potentials and a specific B.C.

General prescription• Consider a QCD eiegnstate with given quantum numbers Q andenergy E.

• Take all possible combinations with Q of stable particles whosethreshold is below or near E.

ex. Q = 6q : N N, N N π, N N ππ, N N K + K − , N N N̄ N, · · ·• Calculate NBS wave functions for all combinations.
• Extract coupled-channel potentials in a finite volume.
• Solve Schroedinger equation with these potentials in the infinite
volume with a suitable B.C. to obtain physical observables.

In practice, of course, final states more than 2 particles are very difficult to deal with.

6. Theoretical understanding of
repulsive core

Summary• Potentials from NBS wave function are useful tools to extract

hadron interactions in lattice QCD. Finite size effect is smaller andquark mass dependence is milder than the phase shift.
Combined with Schroedinger equation in the infinite box.
Rotational symmetry is recovered.

•• Inelastic scattering can also be analysed in terms of coupledchannel “potentials”.

• ΛΛ scattering, H-dibaryon as a resonance• unstabel particle as a resonace• ρ meson, Δ, Roper etc.
• exotic: penta-quark, X, Y etc.
• 3-Baryon forces : NNN (Doi) , BBB-> Neutron star
• Weak decay ?

− 309 MeV, L = 2.8 fmm
ππ=π+scattering ( ρ meson width)

Finite volume method

ETMC: Feng-Jansen-Renner, PLB684(2010)2sin (!)

1

CMF
MF1
MF2
2
sin (!)=1=>aMR

0.5

0
0.3

0.35

0.4

0.45
aECM

ϕE (x) = !0|π(x, 0)π(0, 0)|ρ, E"

0.5

0.55

V (x)

C. Alexandrou (Univ. of Cyprus & Cyprus Institute)

0.6

sin2 δ(s)?

Nucleon structure in latt

QCD meets Nuclei !
100

600

1
S
3 0
S1

VC(r) [MeV]

500

OPEP

50

400
300

0

200
-50
0.0

100

0.5

1.0

1.5

2.0

0
0.0

0.5

1.0
1.5
r [fm]

2.0“The achievement is both a computational tour de force anda triumph for theory.” (Nature Research Higlight 2007)

(a)

(d)

(b)

N(!r,t) ≡ !abc Qa (!r,t) t qb1 (!r,t)C"5qc2 (!r,t) , with Qa (!r,t) the field for a static quark lo1. theSchematic
figuresfield.
of theThe
Wickcontraction. potential VBB (r) as acated at (!r,t)FIGURE
and qai (!r,t)light-quarkinter-baryon
function
the relative
distancer ≡ |!r|allcan
extracted
from
the temporalcorrelators,
diagrams”.
For ofexample,in thecase whenthebequarks
havedifferent
flavors,
we omit
1
!
!
C
(!
r,
T
)
≡
"[N(
0,
T
)N(!
r,
T
)][
N̄(
0,
0)
N̄(!
r,
0)]#,
and
its
limit
as
lim
−
lnC
T ).
BB
→# of
BB (!r,are
T quarks
the quark-exchange
diagrams. (We need only (a) in Fig. 1.) If some Tpairs
The correlators
be expressed as the sumof theproducts 842,249(2006)
six quark propagators
Conf.Proc.
Previous
attempt
identical,
we includeTakahashi-Doi-Suganuma,
thecancorresponding
exchangeAIP
diagramof the oftwoquarks ((b)-(d) in
via the Wick contraction of the quark fields. In particular, the propagator for# the static
Fig. 1).quarks
In suchis aexpressed
way, we as
cancontroltheflavorcontent,whichisdirectlyconnected
tothe path-ordered product of the gauge field P exp(ig A0 (x)dt),
the Pauli-blocking
effects
amongquarks.
calculateenergyof Qqq+Qqq
a functionof r between
2Q. quark approximation.
which corresponds
tothe asleading-orderpropagatorin the heavy
here
that
one
of
quarkstheisflavor
staticcontent
and evenquarksare rather
heavy
Q:staticNote
quark,
light
quark
As
forq:
the
light
quarks,canthebe “light”
controlled
by selecting“Feynmancurrent

Definition of “Potential” in (lattice) QCD ?

as mu,d & 100 ∼ 250 MeV, which would weaken the Pauli-blocking effects and ther
Qqq qqQ
Qqq qqQ
Qqq qqQ
qqQ
Qqq
Q mass.
spin-spin interactions proportional to 1/m2const , with mconst the constituent quark
However, taking into account that the short-range interactions between two nucleons arequite strong, the present setup would be enough to single out the essence of the repulsive
Quenched
result(c)(d)
core in the
nuclear force. (a)(b)($=0.1650)
0.5

0.5 1.
FIGURE($=0.1650)

Q($=0.1650)

Schematic figures of the Wick 0.5
contraction.
VBB!2VB [GeV]VBB!2VB [GeV]VBBdiagrams”.
!2VB [GeV]For(a)example, in the case when all the quarks have different flavors, we allomit+(b)+(c)the quark-exchange diagrams. (We need only(a) in Fig. 1.) If some pairs of quarks are
identical, we include the corresponding exchange diagram of the two quarks ((b)-(d) in000
Fig. 1). In such a way, we can control the flavor content, which is directly connected to
the Pauli-blocking effects among quarks.
Note here that one of quarks is static and even the “light” quarks are rather heavy
as mcurrent& 100 ∼ 250 MeV, which would weaken the Pauli-blocking effects and the
u,d
!0.5
!0.5
!0.5
2
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
spin-spin
interactions
proportional
to
1/m
,
with
m
the
constituent
quark
mass.
const
const
Distance r [fm]Distance r [fm]Distance r [fm]However, taking into account that the short-range interactions between two nucleons are
Almostnodependence
on rVwould! be enough to single out the essence of the repulsive
the present
setup
FIGUREquite
2. strong,
The inter-baryon
potential
BB (r) − 2VB in lattice QCD. Left:All the light-quark flavors arein the nuclear
different.core
Middle:One
pair of force.
quark flavors are identical. Right:Two pairs of quark flavors are identical.
cf. Recent
successful result in the strong($=0.1650)($=0.1650)($=0.1650)coupling limit
The horizontal
axis denotes the inter-baryon
0.5
0.5 distance r.
0.5(deForcrand-Fromm, PRL104(2010)112005)
VBB!2VB [GeV]VBB!2VB [GeV]VBB!2VB [GeV]