Development of a hadronic model: general considerations. Francesco Giacosa

Transcription

1 Development of a hadronic model: general considerations

2 Objectives Development of a chirallysymmetric model for mesons and baryons including (axial-)vector d.o.f. Extended Linear Sigma Model (elsm) Study of the model for T = µ= (spectroscopy in vacuum) (Masses, decay, scattering lengths,=) Second goal: properties at nonzero T and µ (condensates and masses in thermal/matter medium,=) Interrelation between these two aspects!

3 Fields of the model Quark-antiquark mesons: scalar, pseudoscalar, vector and axialvector quarkonia. Additional mesons: The scalar and the pseudoscalar glueballs Baryons: nucleon doublet and its partner (in the so-called mirror assignment)

4 How to construct the elsm Confinement: only hadrons. Dilatation invariance and its anomalous breaking Chiral symmetry SUR(Nf)xSUL(Nf)xUV(1) and its spontaneous as well as explicit breaking. Chiral anomaly

5 Development of a hadronic model: mesons

6 Confinement: from gluons to glueballs * m gluon = mgluon 5 8 MeV G a, µν G a µν Confinement implies glueballs. Where are they? We are still looking for them.

7 Glueballs from Lattice QCD M G = GeV J PC I = = ++ lightest predicted glueball Morningstar (1999)

8 The lightest glueball is part of an effective Lagrangian which reproduces at a composite level the breaking of dilatation invariance. Development of a dilaton potential.

9 Dilaton / Scalar glueball At the hadronic level, we describe these properties as: ΛG dimensionful param that breaks dilatation inv! In QCD it is:

10 Dilaton / Scalar glueball (2) Where is the glueball G in the PDG??? (Too many) candidates, most notably: f(15) and f(171)

11 Quark-Antiquark mesons Quark: u,d,s R,G,B Quark-antiquark bound states: conventional mesons color = 1/ 3( RR + BB + GG ) L, L, S S J P = ( 1) = L + S L J C PC = ( 1) L + S

12 (Pseudo)scalar sector 9 pseudoscalar fields: L=S = J PC + = π η + + % + π K 2 2 uγu dγu sγu a π η % P = Paλ = π + K uγd dγd sγd 2 2 uγs dγs sγs K K η S 5 Γ=iγ + π ud + K us η cosθη = ' η sinθη 36 < θ < 45 η sinθη η% cos θ η 12( uu η ss S + dd)

13 =and 9 scalar fields: = S = 1 PC + + L J = a σ + + % + a K S 2 2 uγu dγu sγu a a σ % S = Saλ = a + K S uγd dγd sγd 2 2 uγs dγs sγs S K S K σ S Γ=1 a + K S = a (145) ud + not a(98)!!! = K * + and (143) us andnot k(7)!!! σ % 12( uu+ dd) f (137) andnot f (5)!!! σ us f ( 15) or f (171) S andnot f (98)!!!

14 Chiral transformation of (pseudo)scalar mesons q = q + q ( U ) q + ( U ) q i i, R i, L R ij j, R L ij j, L U, U U(3) R L Φ = S + ip γ 5 Φ ij = q jqi + iq ji qi = 2q R, jql, i Φ U ΦU + L R

15 Example of an invariant term Φ U ΦU + L R U, U SU(3) R L [ + + λ 2 Tr Φ ΦΦ Φ] [ U Φ U U ΦU U Φ U U ΦU ] = Tr[ Φ ΦΦ Φ] λ 2 Tr λ R L L R R L L R 2 U + L U L =1, U + U = 1 R R

16 (Axial-)Vector sector 9 vector fields= L =, S = 1 J PC =1 ρ ω % ρ K* (892) 2 2 uγu dγu sγu ρ ω 2 2 u s d s s s Γ Γ Γ K* (892) K* (892) φ S µ µ a % V = V aλ = ρ + K* (892) uγd dγd sγd µ Γ=γ + ρ ud + K (892) us * ω ω 12( uu+ dd) φ φ ss S %

17 =and 9 axial-vector fields= PC + + L = S = 1 J =1 a f 1 1, % a1 K1 2 2 uγu dγu sγu µ µ a a1 ω % A = A aλ = a1 + K1 uγd dγd sγd 2 2 u s d s s s Γ Γ Γ K1 K1 f 1, S 5 Γ= γ µ γ a + = a + (126) 1 1 ud f 1 (1285) f 1, % 1 2( uu+ dd) K + 1 = K + 1 (127) us f 1 (151) f 1, S ss

18 =where A and V are coupled in the following way: L = V + A µ µ µ R = V A µ µ µ which under chiral transformations transform as R U R U µ µ + R R L U L U µ µ + L L

19 Examples of further invariant objects: 2 G Tr Tr [ L L R ] R µ µ µ µ [ ] + Φ Φ R µ L µ...

20 Meson sector: how many fields do we have? 2 4Nf +2 fields For Nf= 3 there are 38 mesons 36 quark-antiquark fields + 2 glueballs

21 Criteria(repetita juvant!) We construct the Lagrangian of the so-called Extended Linear Sigma Model (ELSM) according to: dilatation symmetry and chiral invariance. The breaking of the dilatation symmetry is only included in the gluonic part =(scalar glueball and axial anomaly) Moreover, invariance under C and P is also taken into account.

22 Model of QCD elsmwith scalar Glueball S. Janowski, D. Parganlija, F. Giacosa, D. H. Rischke, Phys. Rev. D84, 547 (211) D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa, D. H. Rischke, Phys.Rev. D87 (213) 1411 arxiv:

23 The donkey of Buridan Jean Buridan(in Latin, Johannes Buridanus) (ca. 13 after 1358)

24 Technical remarks Perform Spontaneous Symmetry Breaking (SSB): σ σ + φ, σ σ + φ % % % S S S Explicit symmetry breaking terms: H = diag{ h 1 1, h 2 2, h 3 3 } δ = diag{ δ, δ, δ } with h with δ i i m m i 2 i m 2 π ( m + m ) u d qq Parameter c: axial anomaly and eta-prime mass But: onlya finite number of terms is allowed!

25 Wecancalculate: masses, decays, and scattering lengths. Example: ρ-meson decay into pions Microscopic elsm

26 Results of the fit (11 parameters, 21 exp. quantities) Error from PDG or 5%. Scalar-isoscalarsector not included. 2 χ red =1.2 arxiv:

27 Results of the fit: pictorial representation arxiv: Overall phenomenology is good. Scalar mesons a(145) and K(143) above 1 GeV and are quark-antiquark states. Importance of the (axial-)vector mesons

28 Consequences/1: a(145) Theory Exp (PDG)

29 Consequences/2: pseudoscalar mixing angle η η' = cosθ sinθ η sinθη η% cosθ η η S 2( uu = ss + dd η 1 ) S Theory θ η = 44 Exp (KLOE) θ η 41

30 Consequences/3: ρ-mass Thus, the ρ-mass emerges from the interplay of both the chiral and gluon condensates!

31 Consequences/4: scalar-isoscalar quarkonia σ σ 2( uu dd) σ 1 + = % S ss Theory m σ % = σs 136 MeV m = 153 MeV Exp (PDG) m = 135 ± 15 MeV m = 155± 6MeV m = 172± 6MeV f ( 137) f(15) f(171) It then follows: the chiral partner of the pion is (predominantly) f(137) and not f(5) The scalars below 1 GeV(f(5), k(7), f(98), a(98) are not quarkonia. Possibility: tetraquarks. (F. G. Phys.Rev. D75 (27) 547, hep-ph/ )

32 An important ongoing work: The calculation of the full mixing problem in the I=J= sector is ongoing: 1 f ( ) (137) σ % nn= uu+ dd 2 f(15) = B G gg (171) f σs ss wherebisa3 3 orthogonal matrix Our result within the Nf=2 case was: f f f (137) (15) = (171) σ % nn G gg 1 σs ss Details in S. Janowski, D. Parganlija, F.G., D. Rischke, Phys.Rev. D84 (211) 547, arxiv:

33 A new entry: the pseudoscalar glueball PANDA/FAIR will be able to scan the energy above 2.5 GeV Details in: ~ = Γ πππ G W. Eshraim, S. Janowski, F.G:, D. Rischke, Phys.Rev. D87 (213) arxiv: W. Eschraim, S. Janowski, K. Neuschwander, A. Peters, F.G., Acta Phys. Pol. B, Prc. Suppl. 5/4, arxiv: