「J-PARCハドロン物理の将来研究計画を考える」研究会
J-PARCと格子QCD– Nuclear Bound Charmoninum –佐々木 勝一（東大）

Nuclear-Bound Charmonium
Brodsky, Schmidt, de Teramond, PRL 64 (1990) 1011
Different quark flavors: (up, down vs. charm)
Attractive QCD van der Waals interaction
No Pauli blocking = No short-range repulsion
The formation of charmonium bound to nuclei (A≥3)
A genuine QCD effect at the nuclear level

Nuclear-Bound Charmonium
Brodsky, Schmidt, de Teramond, PRL 64 (1990) 1011
Assumption 1: interaction between ccbar and nucleon (N)e−αrV(cc̄)N (r) = −γr(γ = 0.4 − 0.6, α = 0.6 GeV)

parameters determined by the Pomeron model
Assumption 2: interaction between ccbar and nucleus (A)the potential strength is linear in A (Pomeron model)

Nuclear-Bound Charmonium
Dependence of ccbar and nucleus interaction
the binding energy of ηc - 3He [MeV]

γ0.40.6
Folding potentialV(cc̄)A (r) =!

Original work

Folding model1

4-body calculation2

–3.0
–19.0

---– 0.8

–1.4
–12.6
1 Wasson et al., PRL 67 (91) 2237

V(cc̄)N (r − r! )ρA (r! )d3 r!

2 Belyaev et al., NPA 780 (06) 100

Charmonium-Nucleon interaction
Charmonium-Nucleon interaction = “Multi-gluon exchange”
Color van der Waals interaction given by two-gluon exchange
Attractive long-range potential behaves like 1/rn (n≥6)
the scattering length a = 0.24 fm (Brodsky-Miller1)
QCD sum rules: a = 0.1 fm (Hayashigaki2)
Lattice QCD: a = 0.71±0.48 fm (Yokokawa-Sasaki-Hatsuda-Hayashigaki3)
1 Brodsky, Miller, PLB 412 (97) 125

Note that “a>0” means attractive

2 Hayashigaki., PTP 101 (99) 923
3 Yokokawa et al., PRD 74 (06) 034504

bar
cc -Nucleon potentiale−αrV(cc̄)N (r) = −γrBorn approx. aB = −2Mredγ
0.40.6(γ = 0.4 − 0.6, α = 0.6 GeV)!

drr2 V(cc̄)N (r) = 2Mred γ/α2a Born
0.16 fm
0.47 fm
a = 0.24 fm (color ven der Waals)
a = 0.1 fm (QCD sum rules)
a = 0.71±0.48 fm (Lattice QCD)bar
cc -Nucleon potentiale−αrV(cc̄)N (r) = −γr(γ = 0.4 − 0.6, α = 0.6 GeV)

Bound state formation condition D = 2Mred γ/α > 1.679798

γ0.40.6a Born
0.16 fm
0.47 fm

D0.4771.430

bar
cc -Nucleon potentiale−αrV(cc̄)N (r) = −γr(γ = 0.4 − 0.6, α = 0.6 GeV)

γ=0.6 is too strong for chanominum-nucleon interaction

γ0.40.6

a Born0.16 fm
0.47 fm

a0.21 fm
1.65 fm

D0.4771.430

a = 0.24 fm (color ven der Waals)
a = 0.1 fm (QCD sum rules)
a = 0.71±0.48 fm (Lattice QCD)

Toward more realistic prediction
Measure ccbar - Nucleon potentials in lattice QCD
momentum-dep. of phase shift (interaction range r)
Wave Function Approach (CP-PACS)

NN

c.f. NN potential (Ishii-Aoki-Hatsuda)
Exact few-body calculations with the realistic ccbar - N potential
Gaussian Expansion Method (Hiyama-Kino-Kamimura)Backup FilesLattice results (quench approx.)
Yokokawa, Sasaki, Hatsuda, Hayashigaki, PRD 74 (06) 034504
Interaction of charmonia with light hadrons is always attractive
the ccbar - pion scattering length is very small (a = 0.012 ± 0.004 fm)
as consistent with the soft-pion theorem.
the ccbar - Nucleon (rho) scattering length is an order of magnitudelarger than that in the ccbar - pion channel
0 < a(cc̄)π ! a(cc̄)ρ < a(cc̄)NNo appreciable spin-dependence: ccbar =ηc or J/ψparticle interaction. The function Z00 #1; q$ is an ancontinuation of the generalized zeta function, Z00 #s;
P2&sp1!!!!!
3 #n & q$, from the region s > 3=2 to s
n2Z4"
The s-wave scattering length a0 is defined throug
PHYSICAL REVIEW D 74, 034504 (2006)small k limit of the above formula.
TABLE I. Simulation parameters in this study. The Sommer
If a0 =L is sufficiently small, then one can make a T
parameter r0 " 0:5 fm is used to fix the scale [27,28].
2
expansionofthephase-shift
formula(13)aroundqaa&1
Lattice size+La
Lüscher finite size formula
3
$
[fm]
[GeV]
(L ) T)[fm]
Statisticsand obtain [18]"" #2 #
1.6161
6.2
0.06775
2.913243 ) 48
2"aaa00062.2
169323 ) 48!
c!E
"&$1!
c!
O#L
1233.253483 ) 48
LL
#L

Lattice results (quench approx.)∆EJ/ψ−Ν : Energy Shift∆EJ/ ψ−ρ : Energy Shift∆EJ/ψ−π : Energy Shiftwith c1 " &2:837 297
and cREVIEW
denote2 " 6:375
FIRST LATTICE STUDY OF LOW-ENERGY . . .
PHYSICAL
D 74, 183.
034504#(2006)
with the lattice spacing a and quark mass mq . We generate
reduced mass of two hadrons,# " MJ= Mh =#MJ
0.002
gauge0.002
ensembles at a fixed gauge coupling $ "0.002
6=g2 "
6:2 with three different lattice sizes, L3 ) T " 243 ) 48,Mh $. For !E > 0, Eq. (14) with an expansion u
0.000scale
0.000 &5323 )0.000
48, and 483 ) 48. According to the Sommer&4
O#L$andthat
uptoO#L$ have a real and neg[27,28], $ " 6:2 in the quenched approximation corresponds
to a lattice cutoff of a&1 ’ 2:9 GeV, which-0.002-0.002 maysolution for a0 . On the other-0.002 hand, for !E < 0, the exbe marginal to handle low-lying cc" mesons on the lattice.sion up to O#L&4 $ gives no real solution for
We compute the quark propagators at three values of the-0.004-0.004-0.004
hopping parameter & " f0:1520; 0:1506; 0:1489g, which"!E < &;
correspond to M" =M( " 0:68, 0.83, and 0.90. &c "Spin 02 Spin 1/2
Linear2jc
j#L-0.006-0.006-0.006
LinearLinear
10:1360
is reserved
for the charm-quark mass. Simulation
Ave.
Ave.
Lower boundLower boundparameters and the number of the gauge samples areLower boundup to O#L&5 $ always has a-0.008
summarizedin Table I. Preliminary accounts for
L " 24although the expansion -0.008-0.0080102030405060
0102030405060
0102030405060
and 32 cases are given in Ref. [29].
solution.
Therefore, we can use Eq.
(15) as a nece
L : spatial size
L : spatial size
L : spatial size
For the update algorithm,
we adopt the Metropolis algorithm with 20 hits at each link update. The first 10 000condition to test the convergence of the large-L expan
FIG.
3. Theshiftsas a function ofspatial
sizeenL in lattice units. The left (middle, right) panel is for the J= -! (J= -" insweeps
are energy
discarded
for!Ethermalization.
Thegauge
If this thecondition
is notsatisfiedor marginallysatisfiespin-0,J=-N
in
spin-1=2)
channel.
Fullandopen
circles
representvaluesobtainedfromthelinearchiralextrapolation
tothesembles in each simulation are separated by 1200 (L "expansion, respectively. SAV stands for the spin-averaged value, 19 &5$a0 %2 " 3$a0 %1 " $a0 %0 ' for
the J= -# channel, and 13 &2$a0 %3=2 " $a0 %1=2 ' for the J= -N channel.
Channel

Spin

J= -!

1

J= -#012SAV

From PSF"el [mb]a0 [fm]0:0119 ( 0:0039
0:32 ( 0:12
0:25 ( 0:10
0:21 ( 0:09
0:23 ( 0:08

0:018"0:013#0:010
12:9"11:0#7:6"7:3
7:9#4:9
5:5"5:6#3:7
6:8"6:1#4:2

From LLE
a0 [fm]"el [mb]
0:0119 ( 0:0025
0:23 ( 0:06
0:19 ( 0:06
0:17 ( 0:06
0:18 ( 0:05

0:018"0:008#0:007
6:6"4:2#3:2
4:6"3:3#2:4
3:5"2:8#2:0
4:1"2:9#2:1

J= -N OF LOW-ENERGY
1=2
0:57. .(41"83
0:35 (PHYSICAL
0:15
15"15
TTICE STUDY. 0:42
REVIEW#38#10 D 74, 034504"188
3=20:88 ( 0:63
96#89
0:43 ( 0:1623"20#14
TABLE X. The s-wave scattering lengths of !c"116-hadrons and the elastic cross sections
at the"16SAV
0:71(0:4864
0:39(0:1420#57
the data for all three volumes of L !#1224, 32,
threshold in the physical unit. For the ! -" channel,cand 48 are used, while for the !c -# and !c -N channels, only the data for L ! 32 and 48 areused. The labels are the same as in Table VII.
TABLE VIII. Fitting parameter A as the coefficient of the leading term in the large-L
From
LLE
expansion of the energy shift. For From
the J=PSF-! channel, we used all energyshifts
at L ! 24,Channel
a0 [fm]el [mb]0 [fm]
el [mb]32, and 48 toSpin
extract the fitting
parameter A. $On
the other hand, afor
the J= -# and$J=-N$0:011
we excluded
theenergy
shifts at L !
24 forevaluating
the parameters
00:0113# 0:0035
0:016
0:0112# 0:0024 A. 0:016$0:008!channels,c -"%0:008%0:006$7:5$2:5
3=2 %1:8!c -#0:21
0:16 #A1=2
0:05%4:3 A2
A0J=#-#0:11
A1J= -#5:3$172
A3:4
A1J= -!
J= -#J= -N
J= $16-N!c -N
1=2
0:70 # 0:66
62%62
0:39 # 0:14
19%11
Physical#24$5%#101$28%#84$26%#73$25%#115$48%#140$52%. use
FittingparameterA as32theandcoefficientof theall data
(L ! 24,48) in the
J= lead-!
the large-L expansion of the energy shift. For thenel,
use allVII,energy
at Lobtained! 24, 32,fromand 48
ton inweTable
the shifts
results
twofitting parameter A. On the other hand, for the !c -#PSF and LLE) agree with each other within thezero fm
in simultaneousand large-volumelimt(0.39
from LLE) is chiralcomparable
or even larger
indicatedin the but
current
algebraanalysis.statistics of
thoseestimates,
we needto increase
thesolid
volumedependence of the energy shif
dataAstofor
draw
comparison.
1=L3 behavior, which is expected from the asym

channels.
Thescattering
lengths
in forlatticeunits
all !c -hadron
estimatedby aforphenomenologicalmodelof Pomero
r- mayberealized
mass
numberA)
3iftheattraction
isudiedlabelsin quanare the sameas in TableIII.
m sufficiently
teractions[1].
Theylater
arguedthat the J=by-AWasson,bound sylarge,
which
wasconfirmed
ous!c -#!c -$number A!c -)N 3 if the attract
d techniques
3$T
may be realizedformassaaaL000 charmoniumwho solved the Schrödinger
equationfor theDe simulations.
sufficiently large, which was later confirmed by Wa
3
24 $ 48 system
Physical[2]. 0.15(5)%0:25!48"
1.10(1.29)
nd color-singlet
nucleus
who solved
the Schrödinger
equationfor the charmon
Chiral%0:0025!15"%0:25!48"
1.09(1.28)
etheir complex
Unlike thenucleus
case ofsystem
J= -N,
the interaction of the char[2].
3 $ 48 Physicalmultipleo-andmonium
0.15(5)8.8(9.2) due to the32with the pion
has a 2.8(1.6)
special feature

Unlike the case of J= -N, the interaction of the
Chiral%0:0013!9"
2.7(1.6)
8.7(9.0)
moniumwith thepionpion.
has aThe
specialfeaturedue tNambu-Goldstone
natureofthecurrent
algebraastic scattering
0.33(17)3.4(1.8)of the12.4(10.5)
483 $ 48 PhysicalNambu-Goldstone
naturepion.
The current
alshowsthat
thes-wave
scattering
lengthofaheavy
hadronernglet hadrons Chiral%0:0014!14"
3.4(1.8)
12.2(10.2)
shows
that
thes-wave
scattering
lengthof a heavyhae(H)
withthepion
isgivenbythewell-knownformula[15]:and N) on the(H) with the pion is given by the well-known formulaons. Since the!
"&1! M" "&1M
tate stay inH-the"" '& 1%2 #;M" I~ * M~
" %~ O"MH"
2(1)
aI
~
"H"
a'&
1%
*I%
O"M#;
I
02
"H"
weconsider is0 M
234504-10
4"f
M"
4"f"
HH
light hadrons,
onperturbative(I~H ) is theisospin" (H)) is theI~"isospin
vectorof vector" (H)ofandI~" and* I~HI~"'*
where I~" (I~Hwherehermore,
such
11%%
1# 1#& I&
1# &2, withthe
I being
thetotal isH "I2,H %2I+I"I+I"I%
1#&"IwithIbeing
totalisospin
H H
2e physics
ofofsystem.
For the "-Ncase, this is known a!π system.
!Hthe="-HI·I
0forisoscalarhadronofthe"-HForthe"-Ncase,thisis known as the] and an indiTomozawa-Weinberg relation. Equation (1) indicate

Possible Experiments
Production of ηc -nucleus bound statesp + d → (3He ηc)pbar + 4He → (3He ηc)*Missing mass analysis (MX) on 3He + X*Invariant mass analysis (Mγγ) on 3He + γγ