Interaction Model of Gap Equation

1. Interaction Model of Gap Equation • Si-xue Qin • Peking University & ANL • Supervisor: Yu-xin Liu & Craig D. Roberts With Lei Chang & David Wilson of ANL

2. Outline Why? background, motivation and purpose... How? framework, equations and methods... What? data, figures and conclusions...

3. Background • QCD has been generally accepted as the fundamental theory of strong interaction. Hadron Zoo from PDG

4. Hadron Specifically Meson Light Meson mass < 2GeV • How does the interaction detail affect properties of mesons? • How about the sensitivities? Ground State Radial Excitation Exotic State EM Property Decay Property Mass Spectrum

5. Motivation&Purpose • How will the massive type interaction inflect in observables, • properties of mesons? O. Oliveira et. al., arXiv:1002.4151

6. Dyson-Schwingerequations • Gluon propagator • Quark-Gluon Vertex • Four-Point Scattering Kernel G. Eichmann, arXiv:0909.0703

7. The form of determines whether confinement and /or DCSB are realized in solutions of the gap equation. • is bounded, mono-tonically decreasing regular continuation of the pert-QCD running coupling to all values of space-like momentum: 1.Gluon Propagator • In Landau gauge: • Modeling the dress function as two parts:

8. The infrared scale for the running gluon mass increases with increasing omega: • These values are typical. • With increasing omega, the coupling responses differently at different momentum region. • Using Oliveira’s scheme, we can readily parameterize our interaction model as follows, Solid for omega=0.5GeV, dash for omega=0.6GeV

9. 2.Vertex & Kernel • The physical requirement is symmetry-preserving. • Ward-Takahashi identities (Slavnov-Taylor identities) are some kind of symmetry carrier. • Therefore,we build a truncation scheme based on WTI, and a good one cannot violate WTI. • How to build a truncation scheme systematically and consistently? • How to judge whether a truncation scheme is good one? • In principal, the DSEs of vertex and kernel are extremelycomplicated. • We choose to construct higher order Green’s functions by lower ones. The procedure is called truncation scheme.

10. Rainbow-Ladder truncation • Rainbow approximation: • Ladder approximation: • The axial-vector Ward-Takahashi identity is preserved: G. Eichmann, arXiv:0909.0703

11. Solve Equations:1. Gap Equation • The quark propagator can be decomposed by its Lorentz structure: • Here, we use a Euclidean metric, and all momentums involved are space-like. DCSB & Confinement

12. Complex Gap Equation • In Euclidean space, we express time-like (on-shell) momentum as an imaginary number: • Then, the quark propagator involved in BSE has to live in the complex plane, • The boundary of momentum region is defined as a parabola,whose vertex is . Note that, singularities place a limit of mass. In our cases, it is around 1.5GeV.

13. 2. Homogeneous Bethe-Salpeter Equation C transformation is defined as where T denotes transpose and C is a matrix such that: To sum up, we can specifically decompose any BSA as Fi are unknown scalar functions. • In our framework, we specify a given meson by its JPCwhich determines the transformation properties of its BS amplitude. • J determines the Lorentz structure: • P transformation is defined as • where

14. Eigen-value Problem • Using matrix-vector notation, the homogeneous BSE can be written as • The total momentum P2 works as an external parameter of the eigen-value problem, • when , a state of the original BSE is identified. From the several largest eigen-values, we can obtain ground-state, exotic state, and first radial excitation…

15. Normalization of BSA R.E. Cutkosky and M. Leon, Phys. Rev. 135, 6B (1964) • Leon-Cutkosky scheme: • Nakanishi scheme: N. Nakanishi, Phys. Rev. 138, 5B (1965)

16. EM form factor: Calculate Observables: • Leptonic decay constant: • Strong decay:

17. Results:1. Ground States • Model comparison: ground states are not insensitive to the deep infrared region of interaction. • Omega running: they are weak dependent on the distribution of interaction.

18. Pion • It clearly displays angular dependence of amplitudes. • It is convenient to identify C-parity of amplitudes. • Ground state has no node, 1st radial excitation has one. Rho

19. Wherein, we inflate ground-state masses of pion and rho mesons: Effects from dressed truncation and pion cloud could return them to observed values. It expands the contour of complex quark so that more states are available. II. Exotic States & First Radial Excitation • Compared with ground states, excitations are more sensitive to the details of interaction. • sigma & exotics are too light. • it conflicts with experiment that rho1< pion1. TruncationProblem

20. Finished & Unfinished • We have explained an interaction form which is consistent with modern DSE- and lattice-QCD results: • For tested observables, it produces that are equal to the best otherwise obtained. • It enables the natural extraction of a monotonic runningcoupling and running gluonmass. • Is there any observable closely related to deep infrared region of interaction? • How could we well describe the first radial excitations of rho meson (sigma and exotics) beyond RL? • How could the massive type interaction affect features of QCD phase transition?