Hybrid mesons spectroscopy

Transcription

1 Hybrid mesons spectroscopy Peng Guo and Adam Szczepaniak Indiana University Giuseppe Galatà, Andrea Vassallo and Elena Santopinto ICN-UNAM, INFN and Università di Genova Mini-Workshop Bled 2013 Looking into Hadrons Bled (Slovenia), July 7-14, 2013

2 Why hybrid mesons? Hybrid mesons: quark, antiquark and gluons Exotic mesons are predicted by QCD but not yet observed with certainty Spectroscopy of gluonic excitations has so far been very limited to a few potential hybrids and glueballs In the near future experiments at Jlab, PANDA and BESIII will expand our knowledge of these gluonic excitations PANDA and BESIII will in particular be sensitive to charmonium hybrids

3 Why Coulomb gauge? The choice of the Coulomb gauge has both advantages and disadvantages All degrees of freedom are physical. This makes the QCD Hamiltonian close in spirit to quantum mechanical models, such as the constituent quark model It is adapt to study non-relativistic bound states It is difficult to deal with the Faddeev-Popov determinant (in particular with the operator G=( idi)-1): theoretical and numerical approximations

4 The derivation of the Coulomb gauge H We derive the Hamiltonian in the temporal gauge, because it is simpler Then we start from this Hamiltonian to compute the Coulomb gauge one This process is analogous to a change from a set of Cartesian coordinates to a set of curvilinear coordinates

5 The change of coordinates We transform xa Cartesian coord. in qb curvilinear L= 1 2 U L= 1 x q a M ab q b U a 2 a 2 ab where x c xc M ab= c q a qb is the metric tensor H = U H = M J U 2 a x 2a 2J ab q a ab qb where is the Jacobian J = det M Once we know H in Cartesian coordinates, its form is uniquely determined in any curvilinear coordinates

6 Gribov ambiguity The change of coordinates is not linear. This means that we have to change the coordinate domain (Gribov region) in a non-linear way. Then to a Cartesian coordinate domain can correspond several curvilinear coordinate domains x (-,+ ) r [0,+ ) r (-,0] y (-,+ ) θ [0,π] or θ [0,π] z (-,+ ) φ [0,2π] φ [0,2π] We have have to choose the unique coordinate domain (i.e. to specify the Gribov region) in which we want to work (Gribov ambiguity)

7 Fundamental Modular Region Actually, specifing the Gribov region is not enough, since curvilinear coordinates can have periodical conditions We have to restrict to the FMR, where each different configuration of coordinates(i.e. the set of gauge fields) correspond to a different physical state No exact way to find the Gribov region or the FMR is known, but there are approximate methods1,2 to make sure that a gauge field is in the FMR 1 D.Zwanziger, Nucl.Phys.B485, 185 (1997) hep-th/ A.Cucchieri and D.Zwanziger, Phys.Rev.Lett.78, 3814 (1997), hep-th/

8 The temporal gauge We define the temporal gauge by requiring that V0l(x)=0 x We have three independent free fields where B is the magnetic field and

9 The Coulomb gauge We define the Coulomb gauge by requiring that iai(x)=0 x The gauge transformations are Therefore A4 is not an independent variable in the Coulomb gauge

10 The Hamiltonian in the Coulomb gauge By applying the previously seen change of coordinates to the H in the temporal gauge, we obtain then where is the Faddeev-Popov determinant, the operator Di is the covariant derivative in the Coulomb gauge, Πitr=-Eitr is the conjugate momentum to the gauge fields is the charge carried by gauge and fermion fields

11 The vacuum wave functional For the vacuum wave functional, defined as solution to the Yang-Mills Schroedinger equation for the lowest energy state (the effective vacuum), we consider the trial wave function we use the Gaussian ansatz ω is the gap function and will be determined through a variational procedure

12 The Faddeev-Popov operator (1) The inverse of the Faddeev-Popov operator G=( idi)-1 satisfies the Green equation In momentum space it can be expanded as

13 The Faddeev-Popov operator (2) Rearranging according to the number of external gluons (called ghost propagator ) The first term is the VEV of G The one gluon term can be written as a ghost -gluon vertex

14 The Faddeev-Popov self-energy G=( idi)-1 can be expanded as Gω (VEV of G) is then We introduce the F-P self-energy and use the rainbow-ladder approximation for the Coulomb-gluon vertex:

15 The Faddeev-Popov form factor We define the F-P form factor d by: With the rainbow-ladder approximation d is explicitly given by

16 The Coulomb form factor The Coulomb operator is Thus the VEV is (definition of the Coulomb form factor f) where the Coulomb form factor satisfies (in the r-l approx.)

17 The curvature We define the curvature tensor as the second derivative of the Faddeev-Popov determinant The scalar curvature is calculated (again in the r-l approx.)

18 The gap equation The vacuum energy functional can be calculated and minimized in the variational approach: We obtain the gap equation:

19 Approximate analytical solution Using the angular approximation and assuming that the solution of the gap equation is the solution for the form factors is For k 0 the resulting Coulomb potential Kω is 1/k15/4 (almost linear in coordinate space)

20 Numerical solutions Numerical approximate solution for d and f, with different values of renormalization constant Epple, Reinhardt, Schleifenbaum, Szczepaniak. Dec pp. Published in Phys. Rev. D77:085007, /k2 behaviour in the relevant k range (1-10 fm in pos. s.)

21 The Coulomb potential The effective Coulomb potential resulting from H is Considering the previous approximate solutions we can write the Coulomb potential as

22 Fitting the potential We fix the four parameters Z,c,ΛQCD, b by fitting the position space potential VCL(r)-VCL(r0) to the lattice data The results of the fit are: ΛQCD=250 MeV b=0.204 GeV2 Z=5.94 c=40.68

23 The basis of gluelump states Gluelumps: gluonic excitations bound to a localized, static octet source Since constituent gluons are massive, we expect that light gluelump states contain only one quasi-particle The state of a single quasi-gluon gluelump is with Peng Guo, Adam Szczepaniak, Giuseppe Galatà, Andrea Vassallo and Elena Santopinto Jul pp. Published in Phys.Rev.D77:056005,2008.

24 Hamiltonian matrix elements Since we have now the Hamiltonian and the gluelump bases, we can compute now the matrix elements in the single quasigluon subspace. (apart from self-energy and The resulting potential energy mass terms) is a sum of two and three body interactions

25 Gluelump spectrum (1) The spectrum can be calculated numerically using the fitted potential and the gap function from the numerical solutions of d and f Three models for the gap function (different IR behaviour)

26 Gluelump spectrum (2) Peng Guo, Adam Szczepaniak, Giuseppe Galatà, Andrea Vassallo and Elena Santopinto Jul pp. Published in Phys.Rev.D77:056005,2008.

27 Gluelump spectrum (3) Three-body interaction effect (only on natural parity states (-)J ), spin-parity levels inversion Better spectrum: finite IR limit for the gap function Peng Guo, Adam Szczepaniak, Giuseppe Galatà, Andrea Vassallo and Elena Santopinto Jul pp. Published in Phys.Rev.D77:056005,2008.

28 Hybrids mesons We study now hybrids mesons made up of a heavy quark, antiquark and a constituent gluon If we keep the quark-antiquark sources static and take their separation to zero we come back to the gluelumps The full QCD Hamiltonian is put in the form of a series expansion in inverse powers of the heavy quark mass through the Foldy-Wouthuysen transformation (eliminate coupling between upper and lower components of Dirac spinors) Peng Guo, Adam Szczepaniak, Giuseppe Galatà, Andrea Vassallo and Elena Santopinto Jul pp. Published in Phys.Rev.D78:056003,2008.

29 Foldy-Wouthuysen Hamiltonian

30 Hybrids states The hybrids states differ from the gluelump ones because now we have quark and antiquarks with orbital quantum numbers

31 Hybrids matrix elements Again the potential energy has a two and a three body component The heavy quark motion does not distort much the gluon distribution of the gluelump Thus we expect mixing between various (L Q,Jg) states (quark-antiquark, gluelump states) to be small Low hybrids are classified by the product of gluelump and QQ quantum numbers

32 Hybrids levels For Jg=1 (lowest energy) we can have two gluelumps states: JgPC=1+-, 1- Coupling the gluelump with the quark-antiquark state with LQ=0 and SQ=0,1 we get in the first case: JPC=1-for SQ=0 JPC=0-+,1-+, 2-+ for SQ=1 PC +in the second case: J =1 for SQ=0 PC J =0,1, 2 for SQ=1 -+ the hybrid with exotic quantum numbers 1 appear in this lowest multiplet

33 Hybrids spectrum (charmonia) Spectrum: normal charmoniums (yellow boxes) and charmonium hybrids (dashed boxes) confronted with experimental and lattice data Peng Guo, Adam Szczepaniak, Giuseppe Galatà, Andrea Vassallo and Elena

34 Hybrids spectrum (bottomonia) Spectrum: same as before but for bottomonia Peng Guo, Adam Szczepaniak, Giuseppe Galatà, Andrea Vassallo and Elena

35 Conclusions Calculations in QCD in the low energy range are possible with the theoretical and numerical approximations discussed previously Both Lattice QCD and our calculations in Coulomb gauge QCD reveal the existence of hybrids mesons and mostly agree on their masses We are waiting for the scheduled experiments

36 Tetraquarks: the scalar meson nonet The light scalar meson nonet (f0(600), k(800), f0(980), a0(980) ) is still an open problem The new results by KLOE (2002), E791 (2001) and BES (2005) have confirmed the existence and the properties of this nonet. The principal factors that urge the identification of the scalar nonet as tetraquarks are: The reversed spectrum of the nonet: The OZI rule violation in the hadronic and radiative decays.

37 Tetraquarks classification There are three different models for tetraquarks: compact (uncorrelated costituents), quasi-molecular (weakly correlated constituent mesons), diquark-antidiquark (strongly correlated couples of quarks and antiquarks) Regardless of the model chosen, in the construction of the classification scheme we are guided by two conditions: (a) Tetraquark states must be antisymmetric under the exchange of two quarks and two antiquarks. (b) The tetraquark wave functions should be a colour singlet.

38 Intrinsic degrees of freedom The intrinsic degrees of freedom are classified through the representations of the group SUsf(6) SUc(3) SUsf(6) SUf(3) SUc(3) only the singlets are allowed physical states SUs(2)

39 Spatial degrees of freedom The orbital momenta and the spins are combined together to give the total angular momentum following the scheme: Q Q Q P ( 1) L13 L24 L12 34 Q C ( 1) L12 34 S F.J. Llanes-Estrada, hep-ph/

40 Tetraquark states Applying the Pauli principle we can write all the possible multiplets of the tetraquarks Santopinto, Galatà Phys. Rev. C75, (2007)

41 Mass formula Iachello et al. have developed a mass formula for the normal mesons based on a U(4) SU(3)F SU(2)S SU(3)C stringlike model M 2 ( N M N M ) 2 a bl cs dj n 2 n s s h M ' ij,i ' j ' i M ' '2 ij,i ' j ' (Regge trajectories: linear vibrational and rotational slope) In the uncorrelated tetraquark case one should introduce a for the spatial part, in this new spectrum generating algebra case U(10), since we have nine spatial degrees of freedom. This has not yet been done, but we have proceeded with a simplified procedure.

42 Tetraquark nonet spectrum in the uncorrelated model It is not necessary, for the purpose of determining the mass splitting of the candidate tetraquark nonet, to calculate the spatial part of the mass formula, we can simply use M 2 ( N n M n N s M s )2 where is a constant that sets the scale and is determined applying the mass formula to a well known candidate tetraquark, a0(980). The other masses result: M (k (800)) GeV M ( f 0 (600)) GeV M ( f 0 (980)) GeV Santopinto, Galatà Phys. Rev. C75, (2007)

43 Diquark-antidiquark limit We have developed also a diquark-antidiquark model for tetraquarks. The diquark is a strongly correlated couple of quarks. In our model we freeze the spatial internal degrees of freedom inside the diquark. We consider only the good type of diquark (i.e. the colour antitriplet one). With these conditions, the diquark-antidiquark states are a subset of the states of the compact tetraquark, in particular the subset with L13=L24=0.

44 Tetraquark nonet spectrum in the diquark-antidiquark limit We can use the same spectrum generating algebra used for the normal mesons. Thus we have a similar mass formula: 2 2 M ( M qq M qq ) a n b L c S d J We determine the diquark masses fitting the mass formula with the candidate tetraquark nonet masses. We obtain M[n,n]=0.275 GeV M[n,s]=0.492 GeV Ma0(980)=Mf0(980)=0.984 GeV (These resultsagree with Mf0(600)=0.550 GeV experimental values more Mk(800)=0.767 GeV than the uncorrelated model) Santopinto, Galatà Phys. Rev. C75, (2007)

45 Conclusions Hybrid mesons: Calculations in QCD in the low energy range are possible with the theorethical and numerical approximations discussed previously Both Lattice QCD and our calculations in Coulomb gauge QCD predict the hybrid mesons and mostly agree on their masses We are waiting for the scheduled experiments Tetraquark mesons: We have classified all the possible tetraquark states and calculated the candidate nonet masses in two models Again we wait for new experiments and new candidates

46 Greetings Thank you for your attention!

47 Matrix elements

48 KK production S-channel helicity frame: CM of KK, z and y axis respectively along -p' and to the production plane (i.e. the plane of γ and KK) Gottfried-Jackson frame: as above but with z along γ

49 Moments extraction We minimize: where and Y00 is calculated from the other moments:

50 Y00 Moments First -t bin: GeV2 MKK

51 Cross section Red: Tedeschi Blue: Daresbury Brown: CLAS g6a 0.4<-t<1.0; 3.0<E<3.8 Red: Tedeschi Blue: GG

52 Spin-density matrix The decay angular distribution of the vector mesons is parametrized by the spin-density matrix elements In the case of unpolarized photons we have only three indipendent elements where

53 Spin-density matrix: results (2) Here we have a picture of ρ00 (3.0<E(GeV)<3.8) Red: Tedeschi Blue: GG

54 Spin-density matrix: results (3) Here we have a picture of Reρ10 (3.0<E(GeV)<3.8) Red: Tedeschi Blue: GG

55 Spin-density matrix: results (4) Here we have a picture of ρ1-1(3.0<e(gev)<3.8) Red: Tedeschi Blue: GG