Interquark potential from NBS wavefunction on lattice H
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 12 -13 July, 2012 @ Nagoya Univ.
Motivation Constructing “quark model” from lattice QCD - Quark potential models well describe mass Ba. Bar Collaboration spectra below open charm threshold Godfrey, Isgur, PRD 32 (1985). Barnes, Godfrey, Swanson, PRD 72 (2005). 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 ccbar threshold? applicable for exotic mesons (X, Y, Z) Possible way: qqbar 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) N N & assuming independence of ∇ t 0 …similar to phenomenological NN potential! (N. Ishii, S. Aoki, H. Hatsuda, PRL 99, 022001(2007)) Used for quark-anti-quark potential
Formalism: Quark-anti-quark potential • Measure equal-time BS amplitude: −q • S-wave projection: q q − q t • satisfies the stationary effective Schroedinger equation: • → expanded by velocity & taking leading term: cf) T. Kawanai & S. Sasaki: 0
LQCD setup Y. I. , Iida, ar. Xiv: 1102. 2097[hep-lat](2011). Quench QCD simulation Plaquette gauge action & Standard Wilson quark action β=6. 0 (a=0. 104 fm, a-1=1. 9 Ge. V) Box size : 323 x 48 -> L=3. 3 (fm) • Four different hopping parameters (κ=0. 1320, 0. 1480, 0. 1520) -> MPS=2. 53, 1. 77, 1. 27, 0. 94 (Ge. V), MV=2. 55, 1. 81, 1. 35, 1. 04 (Ge. V) • Nconf=100 • Wall source • Coulomb gauge fixing
Results � Wavefunction: quark mass dependence Compare m. V=2. 55, 1. 81, 1. 35, 1. 04 Ge. V case mq mq ・Size of wavefunction becomes small as increasing mq ・All the wavefunctions vanish even in small mq. ・Channel dependence is larger for smaller mq.
Results � Potential: m. PS = 2. 53, 1. 77, 1. 27, 0. 94 (Ge. V) m. VE = 2. 55, 1. 81, 1. 35, 1. 04 (Ge. V) mq mq These potentials show Coulomb + linear behavior → Confining potential is obtained
Results �Fit results only on-axis data fit range: 3 a – 10 a mq ・Fitting of Coulomb + Linear type function is good! ・Roughly reproduce known value of string tension from Wilson lo ・String tension: larger for larger mq ・Coulomb coefficient: smaller for larger mq
Fitting results of qbar-q potential fit function: MV (Ge. V) σ (Me. V/fm) A (Me. V fm) 2. 55 822 (49) 200 (7) 1. 87 766 (38) 228 (6) 1. 35 726 (39) 269 (7) 1. 04 699 (57) 324 (12) String tension σ has moderate mq dependences σ for the heaviest quark mass gives comparable value to that from Wilson loop Coulomb coefficients increase as decreasing mq see also, Kawanai and Sasaki, PRL 107 (2011).
qbar-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 • Repulsive spin-dependent forces as expected by mass spectrum
qqbar potential from NBS amplitudes � cbar-c potential in 2+1 flavor FULL QCD simulation at almost PHYSICAL POINT generated by PACS-CS Collaboration (mπ=156(7), m. K=553(2) Me. V) Iwasaki gauge action + RHQ action T. Kawanai, S. Sasaki, PRD 85, 091503 (2012) Spin-independent force ・Spin-independent force shows Coulomb + linear form ・Lattice QCD potential is consistent with NRp model (Barnes, Godfrey, Swanson, PRD 72 (2005)) Spin-dependent force ・Spin-dependent force shows short range but not point-like repulsion Construction of “quark model” from 1 st principle of
Inter-quark interaction in baryons
Interquark interaction in baryons − Three-body force … essential for multi-quark system Flux tube formation of 3 Q system in lattice QCD (from H. Ichie et al. , Afr. J. Math. Phys. 3, 11(2006). ) Static 3 Q potential (T. T. Takahashi et al. , PRL 86, 18 (2001)): ● : AP+BP+CP (minimum length connecting 3 Q) ● ● is dependent on the configuration of 3 Q
Interquark interaction for baryons We study effective 2 q interaction in baryons ・Three-body Schrödinger-type equation : ・Effective two-quark interaction : integrate out spectator particle Doi et al, (HAL QCD Coll. ), (2011). integrated out
Interquark interaction for baryons Effective 2 q interaction v. s. qbar-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 cf) Valence light quark effect for 2 Q interaction A. Yamamoto et al. , PLB 664, 129 (2008) ・ Linear behavior of potential is seen at least up to R<0. 8 fm ・ for charm quark mass region. ・ For more lighter quark mass case, the string tension is reduced by the valence-quark effect
Lattice setup � Quenched calculation � Action: plaquette gauge action & standard Wilson quark action � β=6. 0 � Lattice size: 323× 48 (V=(3. 2 fm)3) � Quark mass: � β=6. 0, standard…mπ=2. 53, 1. 77, 1. 27, 0. 94 Ge. V mρ=2. 55, 1. 81, 1. 35, 1. 04 Ge. V � (zero momentum) Wall source � Gauge fixing: Coulomb gauge
Interquark interaction for baryons Lattice QCD result of effective 2 q interaction: (quark mass for spectator particle) (t=20: ground state saturation achieved) Effective 2 q interaction seems to show Coulomb + linear form χ2/Ndf = 0. 6 [ 0. 2 < r < 0. 8 fm ] σ = 870 (209) [Me. V / fm] A = 101 (24) [Me. V fm] ・Coulomb coefficient A is reduced … about a half of that for qq bar potential (c. f. , Aqbar-q =200 [Me. V 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 [Me. V / fm])
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
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 smaller mq ・Screening effect from valence right quark is seen (? ) String tension σ becomes weaker and Coulomb coefficient A becomes stronger for smaller m q
Summary and conclusion: � We extract qqbar pot. from NBS w. f. , interquark potential: … Constructing “quark model” from lattice QCD � qqbar �We potential obtain Cornell type qqbar 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 qqbar potential � Coulomb part is reduced, which is consistent with one-gluon exchange � Screening effect due to the valence right quark is seen(? )
- Slides: 24