Charmed Tetraquarks fromLattice QCD
Yoichi Ikeda(RIKEN, Nishina Center)
HAL QCD (Hadrons to Atomic nuclei from Lattice QCD)
Sinya Aoki, Takumi Iritani (YITP, Kyoto Univ.)
Takumi Doi, Tetsuo Hatsuda, Yoichi Ikeda, Vojtech Krejcirik (RIKEN)
Takashi Inoue (Nihon Univ.)
Noriyoshi Ishii, Keiko Murano (RCNP, Osaka Univ.)
Hidekatsu Nemura, Kenji Sasaki,
Masanori Yamada, Takaya Miyamoto (Univ. Tsukuba)
Faisal Etminan (Univ. Birjand)Long-term workshop on "Hadrons and Hadron Interactions in QCD 2015 (HHIQCD2015)"@YITP, 3 Mar. 2015.

Spectrum of charmonium(-like) system
BaBar Collaboration

Quark potential models well describe
mass spectra below open charm threshold
Godfrey, Isgur, PRD 32 (1985).
Barnes, Godfrey, Swanson, PRD 72 (2005).

“NEW” charmonium-like (X, Y, Z) states:➡ not within quark model spectrum➡ candidates of exotic hadrons“Other” exotic candidates (expected from quark models):➡ doubly charmed tetra-quark, but experimentally not observed so far

Our target: tetra-quark channels“Tetraquark” Tcc (ccubardbar) is manifest 4-quark channel“Charged” charmonium-like states (ccbar + π+/-) require at least 4 quarks

Tcc bound stateµg/ca2⇠ c̄ µa2cAa➡ color magnetic + color electric forces

magnetic interactions << electric interactions➡ magnetic gluon coupling is suppressed by O(1/mc)✓ Heavy quark spin symmetry• Color magnetic interaction is enhanced in light-quark sector
VijCMI /~ (i) · ~ (j) ~ (i) · ~ (j)H. J. Lipkin, PLB172, 242 (1986).

Mi Mj• Color-spin matrix elements :hvij i =h ~ (i) · ~ (j) ~ (i) · ~ (j) i

<vij>

C=1

C=8

C=3

C=6bar

S=0-16

2-8

4

S=1

16/3

-2/3

8/3-4/3‣ C=3, S=0 (I=0) : -8
‣ C=6bar, S=1 (I=0) : -4/3
‣ C=3, S=1 (I=1) : 8/3‣ C=6bar, S=0 (I=1) : 4

attractive

repulsive➡ I=0 [ud]-diquark correlation (good diquark) --> Tcc bound state?

Charmonium-like Zc(3900)
BESIII Coll., PRL110 (2013).

Belle Coll., PRL110 (2013).

DbarD* = 3872

Zc(3900)

Δ = 640πJ/Ψ = 3232

• observed in π±J/Ψ invariant mass (confirmed by CLOE-c)• JP = 1+ seems most probable ( BESIII Coll., PRL112 (2014). )( Xiao et al., PLB727 (2013). )★ Structure of Zc(3900)• Tetra-quark?

➡ energy spectrum consistent
Maiani et al. (2013).

• DbarD* molecule?

Nieves et al. (2011)+ many others• ccbar + meson cloud?

Voloshin (2008).

• pole + meson cloud? Wang et al. (2013).
• cusp?

Chen et al. (2013), Swanson (2014).

with scattering
S. Prelovsek et al., PRD91, 014504 (2015).

➡ coupled-channel
scattering from LQCD

Contents• Introduction• HAL QCD method to define (coupled-channel) potentials• Tcc in I(JP)=0,1(1+) channels [DD* single-channel]• Zc(3900) in I(JP)=1(1+) [πJ/Ψ-ρηc-DbarD* coupled-channel]• Summary

DbarD* = 3872

DD* = 3872

Tcc (I=0)?

Δ = 640πJ/Ψ = 3232

Zc(3900)

Two identical methods for scattering• Scattering parameters• Lüscher’s finite size formulainteraction energy --> phase shiftkcot (k) =11a2

re k 2 + · · ·Lüscher, NPB354, 531 (1991).

• Guaranteed to be the same
Kurth et al., JHEP 1312 (2013) 015.

• NBS wave function• Energy-independent potential
E ~ 0 MeV

E ~ 45 MeVIshii, Aoki, Hatsuda, PRL99, 02201 (2007).
Aoki, Hatsuda, Ishii, PPTP123, 89 (2010).

Two identical methods for scattering• Scattering parameters• Lüscher’s finite size formulainteraction energy --> phase shiftkcot (k) =11a2

re k 2 + · · ·Lüscher, NPB354, 531 (1991).

• Guaranteed to be the same
Kurth et al., JHEP 1312 (2013) 015.

• NBS wave function• Energy-independent potential
E ~ 0 MeV

E ~ 45 MeVIshii, Aoki, Hatsuda, PRL99, 02201 (2007).
Aoki, Hatsuda, Ishii, PPTP123, 89 (2010).

✓ LQCD potentials can be applied to...
properties of hadrons & nuclei, construction of EOS, etc.

Resonance from LQCD
T-matrix in formal scattering theory (N/D method)p1T ( s) = V

1+12⇡

Z 1s+

ds0⇢(s)0s0sInteraction part is not determined within scattering theory➡ interactions faithful to phase shift in QCD

Im[p]Analyticity of T-matrix is uniquely determined
Bound states (physical sheet, 1st)

• binding energy --> T-matrix pole position• coupling --> residue of poleRe[p]Resonance/Virtual states (unphysical sheet, 2nd)

• Analytic continuation of T-matrix• resonance energy --> T-matrix pole position• coupling --> (complex) residue of pole?

“Potentials” in QCD
Hadron 4pt functions & Nambu-Bethe-Salpeter (NBS) wave function
ab(~r, ⌧ ) =X
~x=h0|

Xax+~r, ⌧ )1 (~Abn expnhWn ⌧ab†(~x,⌧)J(⌧ = 0)|0i2ipZ1ap

Z2aar)n (~• Helmholtz eq. of NBS wave func.

X(r2 + (~k a )2 )a(~r) = 0W (~k)(|~r | > R)x~LX⌧

aei l (k)(l)(r) ⇠sin kr +W (~k)
kr

r2 + (~k a )2⌘l⇡/2• NBS wave func. in QFT ~ wave func. in Q.M.

• Coupled-channel potential matrix (faithful to phase shifts)⇣l (k)aa(~r)=2µn

XZ

d~r 0 U ab (~r, ~r0 )Rb0(~r)nbAoki et al. [HAL QCD Coll.], Proc. Jpn. Acad., Ser. B, 87 (2011); PTEP 2012, 01A105 (2012).

• Coupled-channel potentials are energy-independent (non-local in general)

HAL QCD method✓ Definition of energy-independent coupled-channel potentials :⇣r ) = h0|n (~r2 + (~k a )2

⌘ar+~x)1 (~ar ) = 2µn (~Aoki, Hatsuda, Ishii, PTP123, 89 (2010).

aP(~x)|W;
Jin2XZd~r 0 U ab (~r, ~r0 )b0(~r)nb✤ Since energy-independent potential can produce all scattering states,
single-state saturations in simulations is not required✓ Extract energy-independent potential from time-dependent Schrödinger-type eq.
Ishii et al,(HAL QCD), PLB712, 437(2012).
abR (~r, ⌧ ) ⌘a(ma1 +m2 )⌧eab(~r, ⌧ ) pp

=

ma
1

ma2acea(ma1 +m2 )⌧= (mc +mc )⌧ama+m2e 1
12XZ@⌧ + r2 /2µa + @⌧2 /8µa + O( 2 ) Rab (~r, ⌧ ) =d~r 0 ac U ac (~r, ~r 0 )Rcb (~r0 , ⌧ )Z1aZ2ac✓ Velocity expansion:U (⇥r, ⇥r 0 ) = V (⇥r , r) (⇥r⇥r0 )(LO)(NLO)~ · SV~ LS (~V (~r , r) = VC (~r) + Lr ) + O(r2 )✓ Calculate observable: phase shift, binding energy, pole position, ...

Tcc in I(J )=0,1(1 )P+Asymptotic states : DD* (s-wave)

Lattice QCD Setup
Nf=2+1 full QCD configurations generated by PACS-CS Coll.
PACS-CS Coll., S. Aoki et al., PRD79, 034503, (2009).

• Iwasaki gauge & O(a)-improved Wilson quark actions• a=0.0907(13) fm --> L~2.9 fm (32^3 x 64)
Light meson mass [conf.1, conf.2, conf.3] (MeV)
Mπ=699(1), 572(2), 411(2) [PDG:135 (π0)]
MK=787(1), 714(1), 635(2) [PDG:498 (K0)]

Tsukuba-type Relativistic Heavy Quark (RHQ) action for charm quark
S. Aoki et al., PTP109, 383 (2003)➡ remove leading cutoff errors O(mc a), O(ΛQCD a), ...
• We are left with O((aΛQCD)2) error (~ a few %)• We employ RHQ parameters tuned by Namekawa et al.
Y. Namekawa et al., PRD84, 074505 (2011)

Charmed meson mass [conf.1, conf.2, conf.3] (MeV)Mηc=3024(1), 3005(1), 2988(2) [PDG:2981]
MJ/Ψ=3142(1), 3118(1), 3097(2) [PDG:3097]
MD=1999(1), 1946(1), 1902(3) [PDG:1865 (D0)]
MD*=2159(4), 2099(6), 2048(12) [PDG:2007 (D*0)]

S-wave DD* in I=1 : “bad” diquark
Y. Ikeda et al. (HAL QCD), PLB729, 85 (2014).

• Repulsive s-wave potentials of DD*• Weak quark mass dependence• It is unlikely to form bound state even at physical point

S-wave DD* in I=0 : “good” diquark
Y. Ikeda et al. (HAL QCD), PLB729, 85 (2014).

• Attractive S-wave potentials• Attraction increases, as mq decreases• Check whether bound Tcc exist or not --> phase shift analysis

S-wave phase shifts : Tcc in I=0
Y. Ikeda et al. (HAL QCD), PLB729, 85 (2014).

• solve Schrödinger equation --> phase shifts• Attraction is not sufficiently strong to generate bound state• Rapid increase at threshold of DD* phase shift --> effect of virtual state?
➡ examine pole position

I=0 DD* T-matrix on complex energy planeIm[p]• Pole search w/ LQCD potential@mπ=410MeV

Re[p]❖ Analytic continuation• Virtual pole on the DD* unphysical energy plane➡ threshold cusp of the amplitude➡ rapid increase of scattering phase shiftZc(3900) in I (J )=1 (1 )GBESIII Coll., PRL110, 252001, (2013).

P++Belle Coll., PRL110, 252002, (2013).

Lattice QCD setup✤ Nf=2+1 full QCD configurations (PACS-CS) w/ L=2.9fm
S. Aoki et al. (PACS-CS Coll.), PRD79, 034503, (2009).

✤ Tsukuba-type RHQ action for charm quark
S. Aoki et al., PTP109, 383 (2003)
Y. Namekawa et al., PRD84, 074505 (2011)✦ Thresholds in IGJP=1+1+ channel
Light meson mass (MeV)
Mπ= 411(2) [PDG:135 (π0)]
Mρ= 895(14) [PDG:775]
Charmed meson mass (MeV)Mηc= 2988(2) [PDG:2981]
MJ/Ψ= 3097(2) [PDG:3097]
MD= 1902(3) [PDG:1865 (D0)]
MD*= 2048(12) [PDG:2007 (D*0)]

DbarD* = 3951

DbarD* = 3872
πΨ’ = 3821

ρηc = 3883
ππηc = 3256
πJ/Ψ = 3508
LQCD simulation

πJ/Ψ = 3232
Physical thresholds• MπΨ’ > MDbarD* due to heavy pion mass• ρ-->ππ decay not allowed in our setup✤ S-wave πJ/Ψ - ρηc - DbarD* coupled-channel analysis is performed

Potential matrix (πJ/Ψ - ρηc - DD*)bar

• VπJ/Ψ-πJ/Ψ
akew
akew

• Vρηc-ρηcakew• All diagonal potentials are weak
➡ no bound DbarD*• VDbarD*-DbarD*

Potential matrix (πJ/Ψ - ρηc - DD*)bar

• VπJ/Ψ-πJ/Ψ

• VπJ/Ψ-ρηckawe

HQ spin symmetryakew• Vρηc-ρηc• Weak charm spin-flip potential➡ heavy quark spin symmetry(charm quark spin-flip amplitude is suppressed)• VDbarD*-DbarD*

Potential matrix (πJ/Ψ - ρηc - DD*)bar

• VπJ/Ψ-πJ/Ψ

• VπJ/Ψ-ρηc• VπJ/Ψ-DbarD*• Vρηc-ρηc• Vρηc-DbarD*kawestrong

kawe• Strong off-diagonal DbarD* potentials✓ strong charm-quark-exchange interactions• VDbarD*-DbarD*

Potential matrix (πJ/Ψ - ρηc - DD*)bar

• VπJ/Ψ-πJ/Ψ

• VπJ/Ψ-ρηc• VπJ/Ψ-DbarD*• Vρηc-ρηc• Vρηc-DbarD*DbarD*strong

strong• VDbarD*-DbarD*πJ/ΨρηcInvariant mass spectra of πJ/Ψ & D• πJ/Ψ invariant mass (mπ=410MeV)

D*bar• DbarD* invariant mass (mπ=410MeV)

DbarD*

ρηc

DbarD*✓ enhancement near DbarD* threshold due to large πJ/Ψ-DbarD* coupling• peak in πJ/Ψ invariant mass
• enhancement (cusp?) in DbarD* invariant mass

LQCD results & EXP. results• πJ/Ψ invariant mass (mπ=410MeV)• e+e- --> π(πJ/Ψ) @ 4.26GeVDbarD*BESIII Coll., PRL110 (2013).

• DbarD* invariant mass (mπ=410MeV)

Belle Coll., PRL110 (2013).

• e+e- --> π+/- (DbarD*)-/+

DbarD*BESIII Coll., PRL112 (2014).

✓ We observe similar line shapes of πJ/Ψ & DbarD* inv. mass

Pole search (πJ/Ψ :2nd, ρηc :2nd, DD*:2nd)bar

Poles of S-matrix

πJ/Ψ thresholdρηc thresholdyranimilreDbarD* thresholdP✓ Poles on the most adjacent complex energy plane for Zc(3900) are found✓ How do these poles contribute to enhancement in T-matrix?

T-matrix of πJ/Ψ & DD*bar• calculate residues of T-matrix in each channel

S(k) = 1 + 2iT (k)• πJ/Ψ-πJ/Ψ T-matrix (mπ=410MeV)• DbarD*-DbarD* T-matrix (mπ=410MeV)

yranimilreP✓ sizable pole contributions (especially in πJ/Ψ channel)

Quark mass dependence• πJ/Ψ invariant mass

• DbarD* invariant mass

✤ enhancement near DbarD* threshold due to large πJ/Ψ-DbarD* coupling✤ No mq dependence on qualitative behaviors of line shapes

Summary✤ Applications of HAL QCD method to tetra-quarks, Tcc & Zc(3900)• Tcc search on the lattice@mπ=410--700MeV➡ Tcc is not bound for mπ>400MeV (Tbb is already bound)➡ sizable correlation of diquarks is found‣ I=0 good diquark channel : attractive‣ I=1 bad diquark channel : repulsive• Zc(3900) in IG(JP)=1+(1+) channel on the lattice@mπ=410MeV➡ Large channel coupling between πJ/Ψ and DbarD* is a key➡ Heavy quark spin symmetry is seen in c.c. potentials‣ Zc(3900) is neither simple DbarD* molecule nor J/Ψ + π-cloud‣ shadow poles on complex energy plane are found (w/ relatively large width)✤ Physical point simulation is the next step