Tetraquarks and Goldstone boson physics

Transcription

1 Tetraquarks and Goldstone boson physics Christian S. Fischer Justus Liebig Universität Gießen February 2017 Eichmann, CF, Heupel, PLB 753 (2016) Review: Eichmann, Sanchis-Alepuz, Williams, Alkofer, CF, PPNP 91, [ ] Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 1 / 15

2 Tetraquark candidates in charmonium region Internal structure?? Wolfgang Gradl, BESIII, St Goar 2015 Related to details of underlying QCD forces between quarks Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 2 / 15

3 Tetraquarks in the light meson sector K.A. Olive et al. (Particle Data Group), Chin. Phys. C38, (2014) (URL: Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 3 / 15

4 Tetraquarks in the light meson sector Light meson sector: scalars! K.A. Olive et al. (Particle Data Group), Chin. Phys. C38, (2014) (URL: Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 3 / 15

5 Tetraquarks in the light meson sector Light meson sector: scalars! K.A. Olive et al. (Particle Data Group), Chin. Phys. C38, (2014) (URL: Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 3 / 15

6 Rainbow-ladder model for quark-gluon interaction Combine gluon with quark-gluon vertex: µ X (p, k) = i (p, k)t µ i g 2 i=1,12 µ (k 2 ) D µ k (k) = µ µ k 4 (k2 )Z(k 2 ) (k 2 ) k 2 approximation! Z(k 2 ) k 2 Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 4 / 15

7 Rainbow-ladder model for quark-gluon interaction Combine gluon with quark-gluon vertex: effective coupling (k 2 )= 7 k 2 2 e 2 k UV (k 2 ) Maris, Roberts,Tandy, PRC 56 (1997), PRC 60 (1999) scale UV parameter from f π, masses mu=md, ms from m π, mk from perturbation theory : results almost independent qualitatively similar to explicit calc. Williams, EPJA 51 (2015) 5, 57. Sanchis-Alepuz, Williams, PLB 749 (2015) 592; Mitter, Pawlowski and Strodthoff, PRD 91 (2015) Williams, CF, Heupel, PRD93 (2016) , and refs. therein Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 4 / 15

8 Quark mass: flavor dependence Typical solution: S(p) = ip/ A(p2 )+B(p 2 ) p 2 A 2 (p 2 )+B 2 (p 2 ) = Z f (p 2 ) ip/ + M(p 2 ) p 2 + M 2 (p 2 ) Quark Mass Function: M(p 2 ) [GeV] Bottom quark Charm quark Strange quark Up/Down quark Chiral limit p 2 [GeV 2 ] M(p) [GeV] Lattice: quenched Lattice: unquenched (N f =2+1) DSE: quenched DSE: unquenched (N f =2) M(p 2 ): momentum dependent! Dynamical mass: Mstrong 350 MeV Flavour dependence because of mweak p [GeV] CF, Nickel, Williams, EPJ C 60 (2009) 47 Chiral condensate: h i (250 MeV) 3 Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 5 / 15

9 Quark mass: flavor dependence Typical solution: S(p) = ip/ A(p2 )+B(p 2 ) p 2 A 2 (p 2 )+B 2 (p 2 ) = Z f (p 2 ) ip/ + M(p 2 ) p 2 + M 2 (p 2 ) Quark Mass Function: M(p 2 ) [GeV] Bottom quark Charm quark Strange quark Up/Down quark Chiral limit Not directly measurable! p 2 [GeV 2 ] M(p) [GeV] Lattice: quenched Lattice: unquenched (N f =2+1) DSE: quenched DSE: unquenched (N f =2) M(p 2 ): momentum dependent! Dynamical mass: Mstrong 350 MeV Flavour dependence because of mweak p [GeV] CF, Nickel, Williams, EPJ C 60 (2009) 47 Chiral condensate: h i (250 MeV) 3 Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 5 / 15

10 DSEs and Bethe-Salpeter equation axwti,... Kernel K uniquely related to quark-dse via axialvector Ward-Takahashi-Identity (axwti): Z Z i (K 5 S + KS + 5 )= µs + D µ 5 + Z 5 µs D µ Pion is bound state and Goldstone boson Maris, Roberts, Tandy, PLB 420 (1998) 267 Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 6 / 15

11 DSEs and Bethe-Salpeter equation axwti RL: QED-structure of binding force,... Kernel K uniquely related to quark-dse via axialvector Ward-Takahashi-Identity (axwti): Z Z i (K 5 S + KS + 5 )= µs + D µ 5 + Z 5 µs D µ Pion is bound state and Goldstone boson Maris, Roberts, Tandy, PLB 420 (1998) 267 Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 6 / 15

12 Pions as Goldstone bosons Gell-Mann-Oakes-Renner: f 2 m 2 = 2 mh i Pion BS-amplitude: f (P 2 =0,p)=B(p 2 ) 5 Pion decay constant does not vanish in chiral limit! Review: Eichmann, Sanchis-Alepuz, Williams, Alkofer, CF, PPNP 91, [ ] Excited states: no GB, decay constant must vanish in chiral limit! Hoell, Krassnigg, Roberts, PRC 70 (2004) Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 7 / 15

13 Tetraquarks from the four-body interaction Exact equation: + perm. Kvinikhidze & Khvedelidze, Theor. Math. Phys. 90 (1992) Two-body interactions Three- and four-body interactions Heupel, Eichmann, CF, PLB 718 (2012) Eichmann, CF, Heupel, PLB 753 (2016) Basic idea: solve four-body equation without any assumption on internal clustering Key elements: quark propagator and interaction kernels Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 8 / 15

14 Solving the four-body equation + perm. Text Input: Non-perturbative quark, quark-gluon interaction (k 2 )= 7 k 2 2 e 2 k UV (k 2 ) Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 9 / 15

15 Structure of the amplitude Scalar tetraquark: p q k P (P, p, q, k) = X i f i (s 1,...,s 9 ) i (P, p, q, k) color flavor 9 Lorentz scalars (built from P,p,q,k) 256 tensor structures (scalar tetra) 3 3, 6 6 or 1 1, 8 8 good approximation: keep s-waves only; 16 tensor structures Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 10 / 15

16 Four-body equation: Organise Dirac-Lorentz-tensors into multiplets of S4 Singlet: Doublet: S 0 =(p 2 + q 2 + k 2 )/4, carries overall scale a = p 3(q 2 p 2 )/(4S 0 ); s =(p 2 + q 2 2k 2 )/(4S 0 ) diquark pole Two triplets meson poles Eichmann, CF, Heupel, PLB 753 (2016) Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 11 / 15

17 Bound state masses Different levels of approximations: M Tetra [MeV] Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 12 / 15

18 Bound state masses Different levels of approximations: M Tetra [MeV] Singlet only Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 12 / 15

19 Bound state masses Different levels of approximations: M Tetra [MeV] Singlet only Singlet +Triplet I Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 12 / 15

20 Bound state masses Different levels of approximations: Singlet +Triplet II M Tetra [MeV] Singlet only Singlet +Triplet I Bound state of four massive quarks Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 12 / 15

21 Bound state masses Different levels of approximations: Singlet +Triplet II M Tetra [MeV] Singlet + Doublet Singlet only Singlet +Triplet I 4m 2 M 2 S 0 Two-pion resonance Bound state of four massive quarks Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 12 / 15

22 Mass evolution of tetraquark [ ] strange charm M Tetra [GeV] 1.0 / body BSE 2-body BSE Eichmann, CF, Heupel, PLB 753 (2016) [ ] [ ] Resonance becomes bound state for large mq Dynamical decision: meson clusters, not diquarks Results: m m a0,f MeV m apple 750 MeV 1080 MeV m ss s s 1.5 GeV m cc c c 5.7 GeV qualitatively similar to two-body framework Heupel, Eichmann, CF, PLB 718 (2012) Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 13 / 15

23 Outlook: heavy-light systems Dynamical situation in S4-doublet: heavy-light diquarks cq c q meson-molecules hadro-charmonium c q cq c c q q Dynamical decision of most important clustering! Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 14 / 15

24 Summary and outlook Summary Light scalar tetraquarks dominated by internal meson-meson configurations Dynamical description of σ as π-π resonance Outlook explore heavy-light systems explore other quantum numbers Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 15 / 15

25 Backup Slides Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 16 / 15

26 Two-body approximation Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 17 / 15

27 Two-body approximation Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 17 / 15

28 Two-body approximation approximation: separable ansatz for interaction kernel Heupel, Eichmann, CF, PLB 718 (2012) Meson Diquark Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 17 / 15

29 Tetraquark-BSEs - two-body equations Meson Diquark Input: Covariant Quark-Gluon interaction - Maris-Tandy model (k 2 )= 7 k 2 2 e 2 k UV (k 2 ) Mesons and Diquarks via Bethe-Salpeter equation Dynamical decision between Meson- and Diquark-configurations Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 18 / 15

30 Results: scalar tetraquarks Text Heupel, Eichmann, CF, PLB 718 (2012) up/down charm Pion-Pion-contribution dominates! m(0 ++ ) = 403 MeV } f 0(500) Narrow scalar cccc: m(0 ++ ) = 5.3 ± (0.5) GeV see also Caprini, Colangelo and Leutwyler, PRL. 96 (2006) Parganlija, Kovacs, Wolf, Giacosa and Rischke, PRD 87 (2013) Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 19 / 15

31 Quark dressing - comparison with lattice Beyond rainbow-ladder (details see tomorrow): S(p) =Z f (p 2 ) ip/ + M(p 2 ) p 2 + M 2 (p 2 ) DSE: CF, Nickel, Williams, EPJ C 60 (2009) 47 Lattice: P. O. Bowman, et al PRD 71 (2005) M(p) [GeV] Lattice: quenched Lattice: unquenched (N f =2+1) DSE: quenched DSE: unquenched (N f =2) p [GeV] constituent quark : large mass; very composite current quark : - small mass; non-composite Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 20 / 15

32 Pion-BSE P 2 = m 2 BS 1 EV = operator EV Structure: eigenvalue equation Eigenvector is Bethe-Salpeter wave function : [ (P, p)] e,,a,b,a,b ={ 5 [F 1 (P, p) + F 2 (P, p) ip/ + F 3 (P, p) pp ip/ + F 4 (P, p) [p/, P/ ]]}, AB p 3 r e ab (pseudo-) scalar: 4 Dirac tensor structures (axial-)vector: 8 Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 21 / 15

33 Light meson spectrum good channels (ground state): 0 -+, 1 -- acceptable channels (ground state) : 2 ++, 3 --,... clear deficiencies in other channels and excited states drastic improvement beyond rainbow-ladder! CF, Kubrak, Williams, EPJA 50 (2014) 126, arxiv: Williams, CF, Heupel, PRD93 (2016) Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 22 / 15

34 Light meson spectrum [ ] (1600) 1.5 (1300) (1450) (1450) (1235) (1260) (1400) 1.0 PDG 0.5 3PI-3L 2PI-3L CF, Kubrak, Williams, EPJA 50 (2014) 126, arxiv: Williams, CF, Heupel, PRD93 (2016) good channels (ground state): 0 -+, 1 -- acceptable channels (ground state) : 2 ++, 3 --,... clear deficiencies in other channels and excited states drastic improvement beyond rainbow-ladder! Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 22 / 15

35 Rainbow-ladder: heavy meson spectrum DSE/ CF, Kubrak, Williams, EPJA 51 (2015) Hilger et al. PRD 91 (2015) good channels: 1 --,2 ++, 3, : prediction for tensor state acceptable channels : 0 -+,1 ++,... deficiencies in other channels: imbalance of spin-structure Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 23 / 15

36 Rainbow-ladder: heavy meson spectrum DSE/BSE good channels: 1 --,2 ++, 3, : prediction for tensor state acceptable channels : 0 -+,1 ++,... CF, Kubrak, Williams, EPJA 51 (2015) Hilger et al. PRD 91 (2015) deficiencies in other channels: imbalance of spin-structure Christian Fischer (University of Gießen) Tetraquarks and Goldstone boson physics 23 / 15