1. Contents ・ Introduction ・ Formalism ・ Results ・ Conclusion The 17 th International Spin Physics Symposium (SPIN 2006) October 2-7, 2006 Kyoto University, Kyoto, Japan Hideaki Iida Prof. Hideo Suganuma Dr. T.T.Takahashi

2. Introduction Origin of the mass Motivation There are two types of the origin of mass : current quark ～ 3MeV constituent quark ～ 350MeV Strong interaction (QCD) ⇒ Most of the mass in the world is from strong interaction (QCD) (for light quarks) M ～ (300-400) MeV Strong-interaction origin … about 99% ! constituent quark mass : Higgs origin … about 1% m u (μ=1GeV)= (1.5-3)MeV m d (μ=1GeV)= (3-7)MeV μ: renormalization scale from the data of Particle Data Group (2006) current quark mass : mass of leptons: electron…m e =0.51MeV

3. Introduction What is the mechanism of the generation of such a large constituent quark mass? Spontaneous breaking of chiral symmetry …Important mechanism of mass generation for light fermions Y.Nambu, G.Jona-Lasinio,Phys.Rev.122(1961)345: in Nambu-Jona-Lasinio (NJL) model π σ π σ Chiral-symmetry is broken Chiral-symmetry is restored m ～ (1.5-7) MeV: At high temperature or high density M ～ (300-400) MeV: Our present world Study of partial chiral restoration: (theory) T.Kunihiro, T.Hatsuda (1983) (experiment) H.Enyo group, K.Imai group One possible answer is

4. Question: In the strong-interaction world, is chiral symmetry breaking necessary for mass generation? Introduction Note however that chiral symmetry is the symmetry for massless fermions ① Gluon (colored vector particle, which is originally massless) : J.E.Mandula and M.Ogilvie (Landau geuge) Phys.Lett.B185(1987),127 K.Amemiya, H.Suganuma （ MA gauge ） Phys.Rev.D60(1999)114509 Large mass measured in lattice QCD: Glueball: hadron made by only gluons even the lightest glueball (0 ++ ) is rather heavy… 1.5GeV experimental candidate: f 0 (1500) masslessmassive … ② Charm quark ・ Current quark mass m c : about 1.2GeV (Particle Data Group) ・ In the quark model, constituent quark mass is about 1.6GeV The difference (1.6-1.2)GeV= 400MeV is the mass generated by strong interaction without chiral symmetry breaking. We notice following examples of mass generation without chiral symmetry breaking:

5. In this way, we conjecture that there should be other type of mass generation without chiral symmetry breaking or Higgs mechanism. (non-chiral origin but strong-interaction origin) Our conjecture: Even without chiral symmetry, large mass generation generally occurs in the strong-interaction world. In fact, all colored particles (scalar, fermion, vector) have a large mass generated by dressed gluon effect. Introduction colored particles dressed gluons large mass

6. In this talk, we study “light” colored scalar particle, which does not have chiral symmetry. We investigate bound states of colored scalar particles, namely, hadrons made by scalar quarks in lattice QCD in terms of mass generation of scalar quark. First study for scalar-quark systems in lattice QCD Lattice QCD: first-principle calculation of strong interaction (non-perturbative) ⇒ powerful method in non-perturbative QCD We call this colored scalar particle as “scalar quark”. Here, the bare scalar-quark mass is set to be almost zero. Can we observe large mass generation even in this scalar-quark system without chiral symmetry?

7. candidates of tetra quark X(3872), D s (2317) Schematic tetra quark F.Okiharu, H.Suganuma, T.T.Takahashi Phys.Rev.Lett.94(2005)192001; Phys.Rev.D72(2005)014505 [Motivation in hadron physics side] Importance of diquark Scalar diquark system Scalar quark system Scalar diquark Idealized scalar diquark →scalar quark ３３

8. Scalar-quark hadron Scalar-quark hadron is made by two or three scalar quarks. ★ Statistics are different from that of ordinary hadrons Here, we investigate hadrons made by scalar quarks. Baryon (Fermion) Meson (Boson) : quark : anti-quark cf) ordinary hadron Scalar-quark baryon (Boson) Scalar-quark meson (Boson) : scalar quark :anti scalar-quark Scalar-quark hadron

9. Ordinary hadrons (Physical modes) Baryon (Fermion) Meson (Boson) … made by two or three (anti-) quarks color singlet : quark :anti-quark

10. Scalar-quark hadron Scalar baryon (Boson) Scalar meson (Boson) … made by two or three scalar quarks color singlet : scalar quark :anti scalar-quark ★ Statistics are different from that of ordinary hadrons

11. Chimera hadron Chimera hadron is made by quarks and scalar quarks. “chimera hadron” is new terminology. We calculate the masses of scalar-quark hadrons and chimera hadrons in quenched lattice QCD. chimera baryon (Boson) chimera meson (Fermion) : quark :anti-quark : scalar quark :anti scalar-quark chimera baryon (Fermion) Chimera hadron

12. Formalism ： scalar quark （ color triplet ） QCD Scalar QCD : field strength tensor ： quark field （ color triplet ） :covariant derivative QCD including the quarks and scalar quarks ― continuum （ Minkovsky space ） ―

13. Gauge invariant Scalar-quark sector Wilson quark action: Quark sector Formalism ― discretization (Euclid space) ― ： mass of scalar quark ： space coordinate ： link variable ： hopping parameter

14. ・ scalar-quark meson ： ・ scalar-quark baryon ： Scalar ・ chimera meson ： ・ chimera baryon ： Spinor Scalar Summary of scalar and chimera hadrons For the study of scalar-quark system, we calculate the masses of following scalar hadrons and chimera hadrons: Ψ: quark φ: scalar quark Scalar-quark hadrons Chimera hadrons

15. ： quark ( : index of spinor) ・ operators of scalar-quark hadrons ・ operators of chimera hadrons ： scalar quark ( : index of color) （ scalar-quark baryon ） （ chimera meson ） （ chimera baryon ） （ scalar-quark meson ） Hadronic operators and correlators ⇒ using these operators, we calculate the correlators of these hadrons （ total momentum is projected to be zero ） from, we extract the mass of these hadrons We use the following operators of scalar-quark hadrons and chimera hadrons:

16. β=5.70 （ ） β=5.75 （ ） β=6.10 （ ） Lattice size: 16 3 ×24 (for φ † φ) 16 3 ×32 (for other hadrons except) spatial volume: (2.9fm) 3 (β=5.70) (2.7fm) 3 (β=5.75) (1.8fm) 3 (β=6.10) Quenched lattice QCD Parameters of lattice QCD : lattice spacing ( lattice cutoff ) (corresponding to gauge coupling)

17. Scalar-quark meson mass M vs bare scalar-quark mass m sq (β=5.75, latt.size: 16 3 ×32) 3GeV M(GeV) (scalar-quark meson mass) Mass generation of scalar quark by gluon Scalar-quark meson φ † φ Ψ ： quark φ: scalar quark m sq (GeV) (bare scalar-quark mass) Important result ： At bare scalar-quark mass m sq ～ 0, scalar-quark meson mass M ～ 3.0GeV. ⇒ dynamically generated mass of scalar quark M sq is 1.5GeV!! Compared with constituent quark mass ( ～ 300MeV), this mass generation is very large. Large mass generation by strong interaction (non chiral origin) occurs in the scalar-quark system!

18. Scalar-quark baryon φφφ M(GeV) (scalar-quark baryon mass) scalar-quark baryon mass M vs bare scalar-quark mass m sq (β=5.70, latt.size: 16 3 ×32) m sq (MeV)(bare scalar-quark mass) Large mass generation of scalar quark is also seen At bare scalar-quark mass m sq ～ 0, scalar-quark baryon mass M ～ 4.8GeV. ⇒ dynamically generated mass of scalar quark M sq is 1.5GeV. Ψ ： quark φ: scalar quark

19. Chimera meson φ † Ψ chimera meson mass M vs quark mass m q and scalar-quark mass m sq (β=5.70, latt.size: 16 3 ×32) m q (MeV) (bare quark mass) M(GeV) (chimera meson mass) m sq =120MeV (bare scalar-quark mass) m sq =200MeV m sq =270MeV Note: Ψ emerges in the following hadrons Ψ ： quark φ: scalar quark At bare scalar-quark mass m sq ～ 0 and bare quark mass m q ～ 0, chimera meson mass M ～ 1.9GeV. M sq + M q ～ M is formed. (1.5 + 0.4 = 1.9) dynamically generated mass of scalar quark M sq ～ 1.5GeV constituent quark mass M q ～ 400MeV

20. Chimera baryon φφΨ m q (MeV) (bare quark mass) M(GeV) (chimera baryon mass) m sq =120MeV (bare scalar-quark mass) m sq =200MeV m sq =270MeV chimera baryon mass M vs quark mass m q and scalar-quark mass m sq (β=5.70, latt.size: 16 3 ×32) At bare scalar-quark mass m sq ～ 0 and bare quark mass m q ～ 0, chimera baryon mass M ～ 3.7GeV. 2×M sq +M q ～ M dynamically generated mass of scalar quark M sq ～ 1.5GeV constituent quark mass M q ～ 400MeV Ψ ： quark φ: scalar quark

21. Chimera baryon ΨΨφ m q (MeV) (bare quark mass) M(GeV) (chimera baryon mass) m sq =120MeV (bare scalar-quark mass) m sq =200MeV m sq =270MeV chimera baryon mass M vs quark mass m q and scalar-quark mass m sq (β=5.70, latt.size: 16 3 ×32) At bare scalar-quark mass m sq ～ 0 and bare quark mass m q ～ 0, chimera baryon mass M ～ 2.4 GeV. M sq + 2×M q ～ M dynamically generated mass of scalar quark M sq ～ 1.5GeV constituent quark mass M q ～ 400MeV Ψ ： quark φ: scalar quark

22. ⇒ Similar m q -dependence between and is observed. ★ Similarity between φφΨ and φ † Ψ m q (MeV) (bare quark mass) m sq =120MeV (bare scalar-quark mass) m sq =200MeV m sq =270MeV m q (MeV) (bare quark mass) M(GeV) (chimera meson mass) Ψ ： quark φ: scalar quark m sq =120MeV (bare scalar-quark mass) m sq =200MeV m sq =270MeV …Slopes (m q -dependence) are almost the same between and M(GeV) (chimera baryon mass) Figures of the mass of chimera hadrons Chimera mesonChimera baryon

23. Interpretation of the similarity between φ † Ψ and φφΨ Chimera meson Chimera baryon Due to the large constituent mass of scalar-quarkφ, the two particles (φφ) are close under Coulomb interaction. V(r) r Energy level of φφ Similarity of the system between and Scalar quark φ is very heavy. So, the wave-function of light quark (Ψ) distributes around the scalar-quark φ. wave functions Ψ ： quark φ: scalar quark Chimera mesonChimera baryon

24. Conclusion Large mass generation occurs even in the scalar-quark system! (non-chiral origin) m sq ～ 0 (bare scalar-quark mass) M sq ≧ 1GeV (large mass generation of scalar quark) Dressed gluon effect … Dynamically generated mass of scalar quarks ： about 1.5GeV Dressed gluon effect QCD (gluon) effect (non-chiral origin)

25. Summary and Concluding Remarks Using Lattice QCD, we have performed the First Study of scalar-quark (colored-scalar) systems. We have found Large Mass Generation of scalar quarks. (This is Non-Chiral Origin) This Large Mass Generation can be explained by Dressed Gluon Effect. Similarity between and systems emerges due to the large mass of scalar quark. m sq ～ 0 M sq ≧ 1GeV colored-scalar Dressed Gluon

26. Prospects Model analysis: calculation of mass function using Schwinger-Dyson equation Calculation including dynamical scalar quark (unquenched calculation) Why is the dynamical mass of scalar quark so large? This is the first attempt for investigation of scalar-quark systems on lattice.

27. Prospects ・ model analysis: calculation of mass function using Schwinger-Dyson equation ・ダイナミカル・スカラークォークの影響を考慮した計算 （スカラークォークに関してクエンチ近似を用いない計算⇒ボゾンでは比 較的容易） ・重いスカラークォークの質量の起源の探究

28. β ＝６．１０の場合の、 bare quark mass に対する スカラーメソン束縛状態の質量（単位は lattice unit) β ＝ 5.75 の場合の、 bare quark mass に対する スカラーメソン束縛状態の質量（単位は lattice unit) とスカラークォーク・メソン質量スカラークォークの bare mass との対応 β ＝６．１０ β ＝５．７５ 傾き２の直線 → 有意にずれている グルーオンとの相互作用による スカラークォークメソンの質量生成

29. 考察 簡単なモデルを立てる。 1)bare なスカラークォークが、グルーオンの衣により有効質量 （ Σ ： self energy ）を獲得する。 2) この衣を着たスカラークォークが、 のポテンシャルで束縛しているとする。 V(r) r が十分大きく、線型ポテンシャルの影響は無視できる →Hamiltonian このとき、基底状態のスカラークォーク・メソン全系の質量 は … およそ : 換算質量 もし self-energy が にあまり依存しないなら、 は に対して傾き２の response を示す Wilson loop を用いた格子 QCD 計算 T.T.Takahashi et al. Phys.Rev.Lett.86:18-21,2001 (Σ ： self-energy)

30. 結果と展望 ①スケーリングに関する考察 …β ＝ 6 前後で β 依存性が強く残っている → どの程度の β ならば十分なのか？ （ cf. 通常の QCD では β ＝ 6 でほぼ OK : メソン、バリオン、クォーク間ポテンシャル、（グルーボー ル）） ②クォーク・スカラークォーク系の研究 ③スカラークォーク３体系の束縛状態の研究 ④ Full QCD での計算 …fermion と違い、スカラークォークの配位をモンテカルロで発生させ、 経路積分を実行することができるため、比較的容易。 → これにより、スカラークォークループの物理的効果が分かる （特に の小さい領域で effective? ） ⑤ SD ・ BS 方程式等による解析計算及びそれと格子 QCD 計 算との比較

31. ⑥スカラークォークの伝播関数及び有効質量に対する研究 cf) クォーク（フェルミオン）の質量関数： ランダウゲージ D.B.Leinweber et al. Nucl.Phys.Proc.Suppl.109(2002)163 etc. スカラークォーク系を研究することで、（フェルミオンの）クォーク において常識と思われていることを再考察する

32. ・クェンチ近似での格子 QCD 計算の結果、 での大きな mass generation がス カラークォーク・メソンに対して見られる （スカラークォークに対する構成的質量？） ・スカラークォーク質量が大きくなると、 の傾きが、理論的考察での２より有意に小 さい →Self energy Σ のスカラークォーク質量への依存性？ ???

33. ・ Origin of the generation of effective fermion mass Spontaneous chiral symmetry breaking （ ex. NJL model 、 SD eq.… ） SD eq. for quarks: ＋＝ ・ K.Higashijima, Phys.Rev.D29(1984)1228 ・ V.A.Miransky, Sov.J.Nucl.Phys.38(2)(1983)280 ・ H.I., M.Oka, H.Suganuma, Eur.Phys.J.A23(2005)305 Changing the quark mass: =(240-100)MeV=140MeV The variations of M are almost the same between and.

34. β ＝５．７５、 m sq =0GeV effective mass plot, β=6.10 、 m sq =0GeV （ in lattice unit ）

35. partial chiral-restoration (Theory): T.Kunihiro, T.Hatsuda (Experiment): K.Imai, H.Enyo Quarks π σ Wigner-Weyl phase …chiral sym. Is restored m ～ (1.5-7) MeV At high temperature or(and) high density …chiral sym. is spontaneously broken Nambu-Goldstone phase π σ M ～ (300-400) MeV Our world

36. ( scalar colored particle ⇒ scalar quark (spin-less quark) ) We investigate the system of scalar quarks in lattice QCD self-energy of (light) quark: self-energy of scalar quark: ： constituent quark mass ： bare quark mass ： self-energy