Scalar fields in low-energy QCD

Transcription

1 calar ields in low-energy QCD J.W. Goethe University Frankurt University APCTP-WCU Focus meeting April

2 Outline From conventional quarkonia states to the problem o the scalar mesons below and above GeV Light scalar mesons (< GeV) as tetraquark states: trong and e.m. decays o tetraquark states within eective model (hadronic d.o..) U V (3) Mixing o tetraquark and quarkonia states within a chiral approach: U 3) U (3) U (3) V ( L R Role o tetraquark at nonzero temperature and density

3 Part I pectroscopy in the vacuum The puzzle o light scalar mesons

4 Quark: uds RGB Quark-antiquark bound states: conventional mesons color / 3( RR BB GG) L L L P () J L J C PC ( ) L

5 u I / I 3 / d / / s d u s I 3 s u d I 3

6 L J PC K ( sd ) K ( su) ( ud) 8 I 3 ( uu dd) K ( us) K ( ds) 8 cos( ) sin( ) 8 ' sin( ) P P P cos( ) P P ( du) 8 ( uu 3 ( uu 3 I dd dd I 3 ss) ss)

7 L J PC K * ( su) * K ( su) ( ud) 8 I 3 ( uu dd) * K ( us) K ( ds) * ( du) 8 ( uu 3 ( uu 3 dd dd ss) ss) 8 cos( ) sin( ) 8 sin( ) cos( ) 39 / ( uu dd) ss

8 J PC L J PC J PC

9 L J PC * K (43) ( sd) * K (43) ( su) ee or instance F.G. et al Phys.Rev. D7 (5) 4 hep-ph/57 a (3) ( ud) I 3 8 a (3) a (3) ( du) I 3 / ( uu dd) 8 ( uu 3 dd ss) * K (43) ( us) K * (43) ( ds) ( uu 3 dd ss) (7) (55) cos( ) T sin( ) T 8 sin( ) 8 T cos( ) T T 9 (7) (55) ss ( uu dd) nn

10 L J PC K (7) ( sd) K (7) ( su) a (6) ( ud) K (7) ( us) 8 I 3 a (6) a (6) ( du) / ( uu dd) K (7) ( ds) 8 ( uu 3 ( uu 3 dd ss) dd I 3 ss) (85) (5) cos( ) A sin( ) A 8 sin( ) 8 A cos( ) A A 8 (85) (5) ss ( uu dd) nn

11 L J PC??( sd)??( su)??( ud) 8????( du)???? ( uu dd) I 3??( u s)??( ds) 8 ( uu 3 ( uu 3 dd dd ss) ss) Hic unt Leones!!!

12 J PC M < GeV GeV < M <.8 GeV I a (98) I k(8) I (6) (98) a (45) K(45) (37) (5) (7) Too many resonances than expected rom quark-antiquark states

13 J PC M < GeV qq interpretation I a (98) I k(8) I (6) (98) ud du /( uu dd) u s su / (u u dd) ss ds sd Assignement has problems!!!

14 List o Problems Masses: degeneracy o and a (98) a (98) trong coupling o to KK (98) The scalar quarkonia are p-wave states (L = = ) thus expected to be heavier than GeV as tensor and axial-vector mesons ome Lattice results ind M ud.4. 5 GeV rom: Prelovsek et al. Phys. Rev. D 7 (4) Burch et al. Phys. Rev. D 73 (6) Large-Nc behavior o light scalar not compatible with quarkonia rom: Pelaez Phys. Rev. Lett. (4) Pelaez and Rios hep-ph/6397 Linear σ-model with (axial)vector mesons -> σ (and other scalars) above GeV rom Parganlia F.G and D. Rischke Phys.Rev. D8 () 544

15 J PC M < GeV GeV < M <.8 GeV I a (98) I k(8) I (6) (98) a (45) K(45) (37) (5) (7) These are quarkonia!!!

16 Lattice: MG.4.8 GeV J PC I lightest predicted glueball Morningstar (999)

17 L J PC ( qq) * K (43) ( sd) * K (43) ( su) a (45) ( ud) 8 I 3 a (45) a (45) ( du) / ( uu dd) G I 3 * K (43) ( us) K * (43) ( ds) I 8 ( uu 3 ( uu 3 dd dd ss) ss) G "gg"

18 Result or the mixed states: Obtained upon it to the known results o PDG (37) (5) (7) N.6G.97 nn gg ss (5) has the largest gluonic amount!!! F.G. et al Phys.Rev.D7:946 5 (hep-ph/5947) F.G. et al Phys.Lett.B6: (hep-ph/5433)

19 J PC M < GeV interpretation???? I a (98) I k(8) I (6) (98)????????????

20 Part II Tetraquarks as light scalars Flavor symmetry large Nc and phenomenology

21 Physical input: The light scalars are interpeted as tetraquark state A tetraquark is the bound state o two diquarks An example o good diquark is: qq pace : L pin : ( : ( ud du) c : ( RB BR) [ d s] -[us] [ud] u d s Example: a (98) -[ds][us] (and not ud )

22 Consider lavor: 3 antisymmetric combinations [ud] -[us] ] [ s d Under U(3)-lavor the 3 diquarks behave like antiquarks: q s d u ] [ ] [ ] [ d u s u s d ji i i (U ) q q q (U) q j j ij U U(3) U U(3) j ij i j i U ) ( ) U ( ji

23 J PC M < GeV Tetraquark interpretation I a (98) [ u s][ d s] [ u s][ d s] ([ u s][ u s] [ d s][ d s]) I k(8) [ u d][ d s] [ u d][ u s] [ u d][ d s] [ u d][ u s] I (6) [ u d][ u d] (98) ([ u s][ u s] [ d s][ d s])

24 PC J ] ][ [ q q q q (98) a (8) k (98) 3 I (98) a (98) a (8) k (8) k (8) k (6) ]) ][ ([ s d d u ]) ][ ([ s d d u ]) ][ ([ s u d u ]) ][ ([ s u d u ]) ][ ([ s d s u ]) ][ ([ s d s u ]) ][ [ ] ][ [ ( s d s d s u s u ] ][ [ ] ][ [ ] [4 ] ][ [ ] [4 ) cos( ) sin( ) sin( ) cos( (98) (6) s d s d s u s u q d u d u q B B

25 trong decays o a tetraquark state: Previous works and motivations [4q] P P { K '} { K '} Original paper: Jae Phys. Rev. D 5 (977) Revival in: Dominant Maiani et al Phys. Rev. Lett. (4) Experimental study: P { K '} D. V. Bugg EPJC47 (6) [4q] ubdominant P { K '} ystematic evaluation o decay: my work Phys.Rev.D74:486

26 tudy o the strong decays with a hadronic model (never see quarks and gluons only hadrons) ˆ K K K K P P a a Nonet o pseduoscalar states: Nonet o scalar tetraquark states: ] [4 (98) ] [4 (98) (98) (98) ] [4 ] [4 q k k k a q a k a a q B B B q [ud][ud] ] [4 [ds][ds]) ([us][us] ] [4 q q B B The phys. resonances result rom mixing

27 Write the lavor P C invariant interaction Lagrangian or the scalar 4q decays: Pˆ a P a pseud.nonet [4q] 4q -scalar nonet (with i A ) jk ijk L int c [4q] ij Tr A P A P c TrA A P P j ˆ t i ˆ [4q] j j ˆ ˆ ij { K '} { K '} c c [ 4q] { K '} [ 4q] { K '} The trace structure corresponds to the microscopic diagrams: [4q] P P [4q] P P Dominant ubdominant

28 Decay amplitudes as unctions o dominant and subdominant constants tree-level 8 ] [ ] [ 4 ] [ 4 ] [ 4 ] [ 4 ] [ 4 P P P P P P P P M M M g M M M M p q q q q q c c.4.5 exp (98) (98) KK a KK g g ] [4 3 ] [4 3 ] [4 (98) ] [4 c q c c q c c KK q c c KK a g PP B B B P P q c Impossible without ] [4 P P q A

29 Fit o theor. quantities c c and to 4 exp-known results: A A A A a a (98) (98) KK (98) (98) KK Exp. (GeV) (Bugg6) - c GeV (6) cos( ) (98) sin( ) c GeV.3 - sin( ) B[4q] [ u d][ u d] cos( ) B[4q] [ u s][ u s]... Non-trivial agreement; then all the other quantities are determined: (6) k (8) K 4 MeV 4 MeV

30 Further consequences ) Also other couplings to eta-eta eta-etaprime etc are determined. Unortunately not yet accurately measured. ) I you try the same procedure with the quarkonium assignment: too narrow (6) and k(8) ergo excluded! F. G. Nucl.Phys. A833 () e-print: arxiv: [hep-ph] Very interestingly in the general chiral model o Parganlia F.G and D. Rischke Phys.Rev. D8 () 544 We ind the same result!

31 For the two-photon decays proceed in a similar way: L [4q] chiral int :4q -scalar nonet c [4q] ij Tr F photon s.e. tensor; j i [4q] j j A QA Q F c Tr A A Q F ij Q e{ /3 /3 /3} c c [ 4q] [ 4q] Fix c c rom a (98) (98) c c.73 (6) As a consequence:. kev (also small)

32 ummary up to now: J PC M < GeV GeV < M <.8 GeV I a (98) a (45) I I k(8) (6) (98) K(45) (37) (5) (7) These are tetraquarks These are quarkonia (with glueball-intrusion)

33 Part III Mixing o tetraquark and quarkonia Chiral symmetry and large Nc

34 Going urther: tetraquark-quarkonia mixing calar tetraquark and quarkonia states can mix Black et al Phys. Rev. D 64 () F.G. Phys.Rev.D 75(7) U ( 3) U (3) U R(3) Extension o the model: V L ; consider scalar and pseudoscalar quarkonia meson and scalar tetraquark states L ( ) 4 [ qq] ip Tr V V B F F F [ qq] min diag F K pontaneous symmetry breaking takes place but no need to speciy the potential. The pions emerge as Golstone bosons..

35 c [4 q] j t i j * i c [4 q] j j Lint ij Tr ij ( ). A A A A Tr A A When expanding around mixing is obtained: [ud][ud] tetraquark [ 4q] In one dimension : [ qq] (uu dd) quarkonium. hit: with shit L g g g g decay mixing tq-condensate

36 Result in the isovector sector One relates the tetraquark-decay parameters to the mixing strenght by using the decay widths o PDG; then one can evaluate the mixing: a a (98) (45) cos( ) sin( ) sin( ) a cos( ) a [4q] [ ] qq Result : sin ( ) % The mixing is small!!! imilar result in the kaonic sector

37 Thus we have to slightly correct our results: J PC M < GeV GeV < M < GeV I a (98) a (45) I I k(8) (6) (98) K(45) (37) (5) (7) These are predominantly tetraquarks (but not only!) These are predominantly quarkonia (with glueball-intrusion) (but not only!)

38 Part IV Role o the tetraquark at nonzero temperature and density N= case toy model

39 Indeed mixing will occur thus the scenario changes slightly as: J PC M < GeV GeV < M <.8 GeV I a (98) a (45) I I k(8) Not the chiral partner o pion! (6) (98) [ u d][ u d] K(45) Chiral partner o pion! (37) (5) (7) ( uu d d ) These are predominantly tetraquarks (but not only!) These are predominantly quarkonia (with glueball-intrusion) (but not only!)

40 How does this scenario aect inite temperature behavior? We study this issue in the U() limit within a simple model: The resonance (37) ( uu dd) is the chiral partner o the pion. The resonance (6) [ u d][ u d] is an extra-scalar state Mixing shall play a crucial role: Five degrees o reedom:. [ u d ][ u d ] (6) cos( ) sin( ) (37) sin( ) cos( ) ( uu dd ) 45 45

41 A simple chiral model with tetraquark It emerges as an U() limit o the U(3) case (A. Heinz. trüber F. G. and D. H. Rischke: arxiv:85.34 [hep-ph] ) (F.G.Phys.Rev.D75:5477 ) triplet (uu dd) (quarkonium) [ud][ud] (tetraquark) λ V ( F ) M g( ) 4 just as the L M Tetraquark piece with coupling g earch or the absolute minimum: ( ) V V F g... g M F M quark condensate tetraquark condensate 5 unknown parameters: F M g

42 λ V ( F ) M g( ) 4 We expand the potential around the minimum: M g V M... g M g where M 3 F M M Notice that the ields and are not orthogonal. A term g is present in the potential and the mass matrix is not diagonal. Thus the parameter g describes the tetraquark-quarkonium mixing. H (6) cos( ) sin( ) (37) sin( ) cos( ) BO() M M M M (4 g ) M M 4g H H 4g arctan M M

43 We study this model at nonzero T by using the CJT ormalism In the Hartree approximation. (Only double-bubble diagrams are taken into account) Quark condensate: ( T) Tetraquark condensate: ( T) with ( T ) with ( T ) Mixing angle : ( T) with ( T ) M Masses: M M H M M M ( T) H ( T) ( T) (H ( (6)) (37)) Details in: A. Heinz. trüber F. G. and D. H. Rischke: arxiv:85.34 [hep-ph]

44 λ V ( F ) M g( ) 4 5 unknown parameters: F M g well-known values: M 39 MeV 9.4 MeV approximately known values: M M H (6) (37) 4 MeV MeV M (6) and M (37) H are uncertain! We shall study variations upon them and the coupling constant g (tetraquark - quarkonium mixing)

45 Quark condensate (order parameter) as unction o T or dierent values o g or M =. GeV Increasing o g (mixing): ) Tc decreases ) First order sotened 3) Cross-over obtained or g large enough

46 Finite Temperature behavior o quark-quantiquark and tetraquark condensates: Remind that zero T : g M At nonzero T ( T) ( T) holds or T T c M then ( T) starts to increase g This property depends on the characteristics o the model. However It does not inluence other quantities [hep-ph] We use: M M H (6) (37) 4 MeV MeV We ix g in order to have cross - over in agreement with Lattice studies g 3.4 GeV

47 Tetraquark at inite denstiy. Gallas F.G G. Pagliara arxiv:5.53 Nucl.Phys. A87 () 3-4 λ V ( F ) M g( ) 4 L g g g V chiral-partner o the nucleon int N N N N N N N N N

48 Nuclear matter saturation

49 An important test: Compressibility aturation: ok. Compressibility: K is about MeV (in agreement with experiment)

50 arxiv:5.53 Thus the tetraquark plays an important role or the stabiltiy o nuclear matter. Related amusing question: does nuclear matter binds at large Nc? As soon as the lightest scalar (6) is not a quarkonium nuclear matter ceases to exist already or Nc=4. Luca Bonanno and F.G. Nucl.Phys.A859:49-6. arxiv:.3367 [hep-ph]

51 calar quest ummary and outlook cenario: light scalars < GeV are (predominantly) tetraquark states scalars between -.8 GeV are (predominantly) quarkonia+glueball (trong and e.m. decays have been used) Further work: Development o a general sigma model with (axial-)vector mesons and glueball is ongoing. The inclusion o the tetraquarks in this general model is a subject or uture works. Understanding the scalar sector <.8 GeV is crucial at inite temperature and density(chiral restoration). The role o tetraquark(s) can be important.

52 Thank you very much

53 csalar Quest Compatible with a dominant: (37) 5 MeV (5) KK.46.9 (5) nn state (5) 9 5 MeV (5) KK.4.5 (5) gg state ( inert glueball ) (7) 4 MeV (5) KK 5. (5) ss state

54 Decay o (and into) vector mesons and photon V : V : a a (98) (98) (98)... (98)... ( 98 ) Derivatives o scalar-to-pseudoscalars render a good it possible a ( 98 ) There are dominant and a subdominant decay modes: lavour symmetry allows to calculate other reactions F.G. and G. Pagliara Phys.Rev.C76:6547 e-print: F.G. and G. Pagliara Nucl.Phys.A8: e-print: 84.57

55 A tetraquark condensate is generated: ] [4 ] [4 ) ( ) ( ] 4 [ F M c c M c c q σ q σ u q σ b b b F F F F diag diag K s u u GeV 4) ( ] [4 ] ][ [ q d u d u b QCD

56 We now turn to one speciic case: We use: M M H (6) (37) 4 MeV MeV We ix g in order to have cross - over in agreement with Lattice studies g 3.4 GeV We study or this set o parameters all the temperature-dependent quantitites: masses mixing angle and condensates.

57 Finite Temperature behavior o masses and angles: Two critical temperatures : H (6) cos( ( T)) sin( ( T)) (37) sin( ( T)) cos( ( T)) T s deined as: ( T ) / 4 45 (maximal For T T s mixing s role interchange) the lighter stateis mostly quark - antiquark In this example : T s 4 T c 7 MeV The mixing angle grows with T up to the Maximal value. Then it changes sign at Ts and becomes negative. (econd change at higher T) [hep-ph]

58 The linear sigma model λ V ( F ) 4 The chiral partner o the pion (uu dd) has been identiied with (6) NJL-model: same assignment However: this result is NOT in agreement with many dierent works in low energy QCD

59 M H.4 GeV (ixed). We vary g and M phase transition and westudy theorder o the chiral [hep-ph] g g ( tetraquark decoupling ) : H ( tetraquark ) ( quark-antiquark ) (T) M M.948 GeV cross over.948 GeV order

60 Origin o the nucleon mass tudy o the phenomenon generating the nucleon mass (i.e. more tha 9% o the visible mass o the Universe) In the linear sigma model: M N M N / (uu dd) According to chiral symmetry no explicit mass term is allowed However: the llinear sigma model cannot describe a variety o properties (pion-nucleon scattering lenghts nuclear matter stauration

61 Mass o the nucleon within our model Mirror assignment: (C. De Tar and T. Kunihiro PRD 39 (989) 85) (Axial-)vector mesons are included R U R R L U L L R U L R L U R L m L R R L L R R L A chirally invariant mass-term is possible! m parametrizes the contribution which does not stem rom the quark condensate M N 4m (...) (...) I m only the quark condensate generates the nucleon mass Crucial also at nonzero temperature and density also in the so-called quarkyonik phase: L. McLerran R. Pisarski Nucl.Phys.A796:83-7

62 Chiral partner o the nucleon also described It is possible to describe correctly the axial-coupling constants Also here vector mesons are crucial Result or m: m MeV N Using g A.6 (exp) g N* A. (latt) and N* N 67 MeV The nucleon mass is not only generated by the chiral quark-antiquark condensate. Gallas F.G Dirk RischkePhys.Rev.D8:44. e-print: arxiv: [hep-ph]

63 Change the ollowing result..with the new ones. peak about large N_c Put the preliminary results about Kloe Commento su Oset Commento su t Hoot/Maiani Dierenza tetraquark-molecular states Fit starting with qq-> It does not work Results o Denis in the N_= limit (ongoing work)

64 Origin o the nucleon mass tudy o the phenomenon generating the nucleon mass (i.e. more tha 9% o the visible mass o the Universe) In the old linear sigma model: MN M N / (uu dd) According to chiral symmetry no explicit mass term is allowed m parametrizes the contribution which does not stem rom the quark condensate MN 4m (...) (...) However: the llinear sigma model cannot describe a variety o properties (pion-nucleon scattering lenghts nuclear matter stauration

65