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Study on the Mass Difference btw and using a Rel. 2B Model 2011. 08. 25 Jin-Hee Yoon ( ) Dept. of Physics, Inha University collaboration with C.Y.Wong and H. Crater
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08/25/2011APFB 2011 Inha Nuclear Theory Group Introduction m = 0.140 GeV, m = 0.775 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.
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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
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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)
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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
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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.
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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)]
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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.
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08/25/2011APFB 2011 Inha Nuclear Theory Group QCD Potentials Scalar Pot. Vector Pot.
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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
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08/25/2011APFB 2011 Inha Nuclear Theory Group Formulation central potentials + darwin + SO + SS + Tensor + etc. SOD & SOX =0 when m 1 =m 2
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08/25/2011APFB 2011 Inha Nuclear Theory Group Formulation
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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 2 + 2 w A - A 2
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08/25/2011APFB 2011 Inha Nuclear Theory Group Formulation For (S-state), For (mixture of S & D-state),
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08/25/2011APFB 2011 Inha Nuclear Theory Group Formulation Once we find b 2, Invariant mass
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08/25/2011APFB 2011 Inha Nuclear Theory Group MESON Spectra 32 mesons
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08/25/2011APFB 2011 Inha Nuclear Theory Group MESON Spectra L 0.4218 GeV B0.05081 K4.198 mumu 0.0557 GeV mdmd 0.0553 GeV msms 0.2499 GeV mcmc 1.476 GeV mbmb 4.844 GeV All 32 mesons
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08/25/2011APFB 2011 Inha Nuclear Theory Group Wave Functions( ) X
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08/25/2011APFB 2011 Inha Nuclear Theory Group Individual Contribution( ) termsmagnitude Partial sum Total 0.8508 0.0103 0.0942 -0.0598 -0.4279 -0.3832 -0.8475 -3.804 3.340 -0.4643 b2b2 0.0033 W = 0.159 GeV -1
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08/25/2011APFB 2011 Inha Nuclear Theory Group Individual Contribution( ) W = 0.792 GeV -1 terms++--+--+ 0.30850.2812 Partial Sum 0.00263 0.1631 -0.2091 -0.1247 -0.1680 0.000571 0.04457 -0.02109 -0.00944 0.01461 Partial Sum -0.2790 -0.03637 -0.3154 -0.04360 -0.00947 -0.04172 0.1133 0.02153 0.03461 -0.3088 0.6177 0.3090 -0.3088 Total Sum -0.17500.31580.3090-0.3088 b2b2 0.1411
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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 =0.0557 GeV, m d =0.0553 GeV Non-singular (well-beaved) rel. WF is obtained. At small r, S-wave is proportional to D-wave.
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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.
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Thank you for your attention!
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Backup slides
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08/25/2011APFB 2011 Inha Nuclear Theory Group Overview of QQ Potential(1) Pure Coulomb : BE=-0.0148 GeV for color-singlet =-0.0129 GeV for color-triplet(no convergence) + Log factor : BE=-0.0124 GeV for color-singlet =-0.0122 GeV for color-triplet + Screening : BE=-0.0124 GeV for color-singlet No bound state for color-triplet
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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=-0.3547 GeV When b=0.2 BE=-0.5257 GeV Too much sensitive to parameters!
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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
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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.
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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 …
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