Study on the Mass Difference btw  and  using a Rel. 2B Model Jin-Hee Yoon ( ) Dept. of Physics, Inha University collaboration with C.Y.Wong and H. Crater

08/25/2011APFB 2011 Inha Nuclear Theory Group Introduction m  = GeV, m  = GeV Why is the pion mass so small? Why is the pion mass not zero? Chiral symmetry breaking What we will do?  Trace the origin of the mass of  using the 2B relativistic potential model.

08/25/2011APFB 2011 Inha Nuclear Theory Group 2-Body Constraint Dynamics  Two free spinless particles with the mass-shell constraint → removes relative energy & time P. van Alstine et al., J. Math. Phys.23, 1997 (1982)  Two free spin-half particles with the generalized mass-shell constraint → potential depends on the space- like separation only & P ⊥ p

08/25/2011APFB 2011 Inha Nuclear Theory Group 2-Body Constraint Dynamics  Pauli reduction + scale transformation 16-comp. Dirac Eq. → 4-comp. rel. Schrodinger Eq. P. van Alstine et al., J. Math. Phys.23, 1997 (1982)

08/25/2011APFB 2011 Inha Nuclear Theory Group Why Rel.? Even within the boundary, their kinetic energy can be larger than the quark masses. Already tested in e + e - binding system (QED) [ Todorov, PRD3(1971) ] can be applied to two quark system(  and  ) (QCD) Can treat spin-dep. terms naturally

08/25/2011APFB 2011 Inha Nuclear Theory Group What's Good?  TBDE : 2 fermion basis 16-component dynamics 4-component  Particles interacts through scalar and vector interactions.  Leads to simple Schrodinger-type equation.  Spin-dependence is determined naturally.

08/25/2011APFB 2011 Inha Nuclear Theory Group QCD Potentials Common non-relativistic static quark potential  Dominant Coulomb-like + confinement  But asymptotic freedom is missing Richardson[ Phys. Lett. 82B,272(1979) ] FT [Eichten et al., PRD 21 (1980)]

08/25/2011APFB 2011 Inha Nuclear Theory Group QCD Potentials Richardson potential in coord. space  For : asymptotic freedom  For : confinement We will use fitting param.

08/25/2011APFB 2011 Inha Nuclear Theory Group QCD Potentials Scalar Pot. Vector Pot.

08/25/2011APFB 2011 Inha Nuclear Theory Group What's Good again?  TBDE : 2 fermion basis 16-component dynamics 4-component  Particles interacts through scalar and vector interactions.  Yields simple Schrodinger-type equation.  Spin-dependence is determined naturally.  No cutoff parameter  No singularity

08/25/2011APFB 2011 Inha Nuclear Theory Group Formulation  central potentials + darwin + SO + SS + Tensor + etc.  SOD &  SOX =0 when m 1 =m 2

08/25/2011APFB 2011 Inha Nuclear Theory Group Formulation

08/25/2011APFB 2011 Inha Nuclear Theory Group Formulation   SOD =0 and  SOX =0 when m 1 =m 2  For singlet state with the same mass, no SO, SOT contribution ← Terms of (  D,  SS,  T ) altogether vanishes. H=p 2 + 2m w S + S  w A - A 2

08/25/2011APFB 2011 Inha Nuclear Theory Group Formulation  For  (S-state),  For  (mixture of S & D-state),

08/25/2011APFB 2011 Inha Nuclear Theory Group Formulation Once we find b 2, Invariant mass

08/25/2011APFB 2011 Inha Nuclear Theory Group MESON Spectra 32 mesons

08/25/2011APFB 2011 Inha Nuclear Theory Group MESON Spectra L GeV B K4.198 mumu GeV mdmd GeV msms GeV mcmc GeV mbmb GeV All 32 mesons

08/25/2011APFB 2011 Inha Nuclear Theory Group Wave Functions(  )  X 

08/25/2011APFB 2011 Inha Nuclear Theory Group Individual Contribution(  ) termsmagnitude Partial sum Total b2b W = GeV -1

08/25/2011APFB 2011 Inha Nuclear Theory Group Individual Contribution(  ) W = GeV -1 terms Partial Sum Partial Sum Total Sum b2b

08/25/2011APFB 2011 Inha Nuclear Theory Group Summary  Using Dirac’s rel. constraints, TBDE successfully leads to the SR-type Eq.  With Coulomb-type + linear potential, By fitting to 32 meson mass spectra, determine 3 potential parameters and 6 quark masses Small quark masses : m u = GeV, m d = GeV Non-singular (well-beaved) rel. WF is obtained. At small r, S-wave is proportional to D-wave.

08/25/2011APFB 2011 Inha Nuclear Theory Group Summary  Can reproduce the huge mass split of   In pion mass Still large contributions from Darwin and Spin-Spin terms (3~4 GeV compared to 0.16 GeV) But cancelled each other, remained ~20% Balanced with kinetic term resulting small pion mass  This Model inherits Chiral Symmetry Braking.

Thank you for your attention!

Backup slides

08/25/2011APFB 2011 Inha Nuclear Theory Group Overview of QQ Potential(1)  Pure Coulomb : BE= GeV for color-singlet = GeV for color-triplet(no convergence)  + Log factor : BE= GeV for color-singlet = GeV for color-triplet  + Screening : BE= GeV for color-singlet No bound state for color-triplet

08/25/2011APFB 2011 Inha Nuclear Theory Group Overview of QQ Potential(2) + String tension(with no spin-spin interaction) When b=0.17 BE= GeV When b=0.2 BE= GeV Too much sensitive to parameters!

08/25/2011APFB 2011 Inha Nuclear Theory Group QQ Potential Modified Richardson Potential and Parameters : m,  And mass=m(T) A : color-Coulomb interaction with the screening S : linear interaction for confinement

08/25/2011APFB 2011 Inha Nuclear Theory Group Work on process To solve the S-eq. numerically, We introduce basis functions  n (r)=N n r l exp(-n  2 r 2 /2) Y lm  n (r)=N n r l exp(-  r/n) Y lm  n (r)=N n r l exp(-  r/√n) Y lm …  None of the above is orthogonal.  We can calculate analytically, but all the other terms has to be done numerically.  The solution is used as an input again → need an iteration  Basis ftns. depend on the choice of  quite sensitively and therefore on the choice of the range of r.

08/25/2011APFB 2011 Inha Nuclear Theory Group Future Work  Extends this potential to non-zero temperature.  Find the dissociation temperature and cross section of a heavy quarkonium in QGP.  Especially on J  to explain its suppression OR enhancement.  And more …