Inter-quark potential from NBS wavefunction on lattice H.Iida (Kyoto Univ.), Y.Ikeda (Tokyo Inst. of Tech) 新学術領域「素核宇宙融合」 x 「新ハドロン」クロスオーバー研究会 July, Nagoya Univ.

Motivation Constructing “quark model” from lattice QCD - Quark potential models well describe mass spectra below open charm threshold Questions ・ Can we “derive” quark model from the first principle of QCD (Lattice QCD)? …why does the constituent quark model succeed in reproducing spectra? ・ To what extent can be the model used? …Can we construct quark model above cc bar threshold? - Quark potential models well describe mass spectra below open charm threshold Questions ・ Can we “derive” quark model from the first principle of QCD (Lattice QCD)? …why does the constituent quark model succeed in reproducing spectra? ・ To what extent can be the model used? …Can we construct quark model above cc bar threshold? BaBar Collaboration Godfrey, Isgur, PRD 32 (1985). Barnes, Godfrey, Swanson, PRD 72 (2005). Godfrey, Isgur, PRD 32 (1985). Barnes, Godfrey, Swanson, PRD 72 (2005). applicable for exotic mesons (X, Y, Z) Possible way: qq bar potential from Nambu-Bethe-Salpeter wavefunction in lattice QCD

Potential from NBS amplitude  potential from BS amplitude of two nucleons (N.Ishii, T.Hatsuda, S.Aoki, HAL QCD collaboration) …similar to phenomenological NN potential! (N.Ishii, S.Aoki, H.Hatsuda, PRL99, (2007)) t 0 N N & assuming independence of ∇ Used for quark-anti-quark potential

→ expanded by velocity & taking leading term: Measure equal-time BS amplitude: t 0 q q q q − − S-wave projection: satisfies the stationary effective Schroedinger equati on: cf) T.Kawanai & S.Sasaki: Formalism: Quark-anti-quark potential

LQCD setup Quench QCD simulation Plaquette gauge action & Standard Wilson quark action β=6.0 (a=0.104 fm, a -1 =1.9GeV) Box size : 32 3 x 48 -> L=3.3 (fm) Four different hopping parameters (κ=0.1320, , , ) -> M PS =2.53, 1.77, 1.27, 0.94 (GeV), M V =2.55, 1.81,1.35, 1.04 (GeV) N conf =100 Wall source Coulomb gauge fixing Quench QCD simulation Plaquette gauge action & Standard Wilson quark action β=6.0 (a=0.104 fm, a -1 =1.9GeV) Box size : 32 3 x 48 -> L=3.3 (fm) Four different hopping parameters (κ=0.1320, , , ) -> M PS =2.53, 1.77, 1.27, 0.94 (GeV), M V =2.55, 1.81,1.35, 1.04 (GeV) N conf =100 Wall source Coulomb gauge fixing Y.I., Iida, arXiv: [hep-lat](2011).

 Wavefunction: quark mass dependence Compare m V =2.55, 1.81, 1.35, 1.04GeV case mqmq mqmq ・ Size of wavefunction becomes small as increasing m q ・ All the wavefunctions vanish even in small m q. ・ Channel dependence is larger for smaller m q. Results

 Potential: mqmq mqmq m PS = 2.53, 1.77, 1.27, 0.94 (GeV) m VE = 2.55, 1.81, 1.35, 1.04 (GeV) These potentials show Coulomb + linear behavior → Confining potential is obtained Results

 Fit results only on-axis data fit range: 3a – 10a ・ Fitting of Coulomb + Linear type function is good! ・ Roughly reproduce known value of string tension from Wilson loop ・ String tension: larger for larger m q ・ Coulomb coefficient: smaller for larger m q mqmq

Fitting results of q bar -q potential M V (GeV) σ (MeV/fm) A (MeV fm) (49) 766 (38) 726 (39) 699 (57) 200 (7) 228 (6) 269 (7) 324 (12) String tension σ has moderate m q dependences String tension σ has moderate m q dependences σ for the heaviest quark mass gives comparable value to that from Wilson loop σ for the heaviest quark mass gives comparable value to that from Wilson loop Coulomb coefficients increase as decreasing m q Coulomb coefficients increase as decreasing m q String tension σ has moderate m q dependences String tension σ has moderate m q dependences σ for the heaviest quark mass gives comparable value to that from Wilson loop σ for the heaviest quark mass gives comparable value to that from Wilson loop Coulomb coefficients increase as decreasing m q Coulomb coefficients increase as decreasing m q fit function: see also, Kawanai and Sasaki, PRL 107 (2011).

q bar -q potentials: spin dependent part Effective spin-independent & dependent forces are constructed by linear combination of PS & V channel potentials Spin-independent forces reveal Coulomb + linear behavior Spin-independent forces reveal Coulomb + linear behavior Repulsive spin-dependent forces as expected by mass spectrum Repulsive spin-dependent forces as expected by mass spectrum Spin-independent forces reveal Coulomb + linear behavior Spin-independent forces reveal Coulomb + linear behavior Repulsive spin-dependent forces as expected by mass spectrum Repulsive spin-dependent forces as expected by mass spectrum

 c bar -c potential in 2+1 flavor FULL QCD simulation at almost PHYSICAL POINT generated by PACS-CS Collaboration (m π =156(7), m K =553(2) MeV) Iwasaki gauge action + RHQ action qq bar potential from NBS amplitudes Construction of “quark model” from 1 st principle of QCD T.Kawanai, S.Sasaki, PRD85, (2012) Spin-independent force Spin-dependent force ・ Spin-independent force shows Coulomb + linear form ・ Lattice QCD potential is consistent with NRp model (Barnes, Godfrey, Swanson,PRD72 (2005)) ・ Spin-dependent force shows short range but not point-like repulsion

Inter-quark interaction in baryons

Interquark interaction in baryons − Three-body force … essential for multi-quark system Static 3Q potential (T.T.Takahashi et al., PRL86, 18 (2001)): ● ● ● : AP+BP+CP Flux tube formation of 3Q system in lattice QCD (from H.Ichie et al., Afr.J.Math.Phys.3, 11(2006).) (minimum length connecting 3Q) is dependent on the configuration of 3Q

Interquark interaction for baryons ・ Three-body Schrödinger-type equation : ・ Effective two-quark interaction : integrate out spectator particle Doi et al, (HAL QCD Coll.), (2011). integrated out We study effective 2q interaction in baryons

・ Linear behavior of potential is seen at least up to R<0.8fm ・ for charm quark mass region. ・ For more lighter quark mass case, the string tension is reduced by the valence-quark effect Interquark interaction for baryons Effective 2q interaction v.s. q bar -q potential (spin independent parts) Expectation: ・ behave like Coulomb + Linear form ・ String tension: comparable for the charm quark mass region For more lighter quark mass, the string tension would be reduced. ・ Coulomb part: different by factor two with one-gluon exchange Expectation: ・ behave like Coulomb + Linear form ・ String tension: comparable for the charm quark mass region For more lighter quark mass, the string tension would be reduced. ・ Coulomb part: different by factor two with one-gluon exchange Valence light quark effect for 2Q interaction A.Yamamoto et al., PLB664, 129 (2008) cf)

Lattice setup  Quenched calculation  Action: plaquette gauge action & standard Wilson quark action  β =6.0  Lattice size: 32 3 ×48 (V=(3.2fm) 3 )  Quark mass:  β =6.0, standard…m π = 2.53, 1.77, 1.27, 0.94 GeV m ρ =2.55, 1.81, 1.35, 1.04GeV  (zero momentum) Wall source  Gauge fixing: Coulomb gauge

Interquark interaction for baryons Lattice QCD result of effective 2q interaction: χ 2 /N df = 0.6 [ 0.2 < r < 0.8 fm ] σ = 870 (209) [MeV / fm] A = 101 (24) [MeV fm] χ 2 /N df = 0.6 [ 0.2 < r < 0.8 fm ] σ = 870 (209) [MeV / fm] A = 101 (24) [MeV fm] Effective 2q interaction seems to show Coulomb + linear form ・ Coulomb coefficient A is reduced … about a half of that for qq bar potential (c.f., A qbar-q =200 [MeV fm]) → consistent with Casimir factor for 1 and 3 bar channel ・ String tension … difficult to say something under the current statistics ( σ qbar-q =822 [MeV / fm]) ・ Coulomb coefficient A is reduced … about a half of that for qq bar potential (c.f., A qbar-q =200 [MeV fm]) → consistent with Casimir factor for 1 and 3 bar channel ・ String tension … difficult to say something under the current statistics ( σ qbar-q =822 [MeV / fm]) (quark mass for spectator particle) (t=20: ground state saturation achieved)

Interquark interaction for baryons Quark-mass dependence ・ Screening effect from valence right quark is seen (?) String tension σ becomes weaker and Coulomb coefficient A becomes stronger for smaller m q

Interquark interaction for baryons Quark-mass dependence ・ Screening effect from valence right quark is seen (?) String tension σ becomes weaker and Coulomb coefficient A becomes stronger for smaller m q

Interquark interaction for baryons Quark-mass dependence ・ Screening effect from valence right quark is seen (?) String tension σ becomes weaker and Coulomb coefficient A becomes stronger for smaller m q

Quark-mass dependence Interquark interaction for baryons ・ Screening effect from valence right quark is seen (?) String tension σ becomes weaker and Coulomb coefficient A becomes stronger for smaller m q

smaller m q Interquark interaction for baryons Quark-mass dependence ・ Screening effect from valence right quark is seen (?) String tension σ becomes weaker and Coulomb coefficient A becomes stronger for smaller m q

 We extract qq bar pot. from NBS w.f., interquark potential: … Constructing “quark model” from lattice QCD  qq bar potential  We obtain Cornell type qq bar potential.  Central part of the potential is very similar to that of phenomenological one.  Spin-dependent part is also obtained … repulsive at small r (Kawanai & Sasaki)  Interquark potential in baryons: effective qq potential  Linear part of the potential is comparable with qq bar potential  Coulomb part is reduced, which is consistent with one-gluon exchange  Screening effect due to the valence right quark is seen(?) Summary and conclusion: