Interaction Model of

Gap EquationSi-xue Qin

Peking University & ANLSupervisor: Yu-xin Liu & Craig D. Roberts

With Lei Chang & David Wilson of ANL

Why?

background, motivation and purpose...

How?

framework, equations and methods...

What?

data, figures and conclusions...

Outline

Background

QCD has been generally accepted as the

fundamental theory of strong interaction.

Hadron Zoo from PDG

Specifically

• How does the interaction detail affect properties of mesons?

• How about the sensitivities?

Hadron

Meson

Light Mesonmass < 2GeVGroun

d StateExotic State

Radial Excitati

on

Mass Spectru

m

EM Proper

ty

Decay Proper

ty

Motivation &

PurposeHow will the massive type

interaction inflect in observables,

properties of mesons?

O. Oliveira et. al., arXiv:1002.4151

Dyson-Schwingerequations

• Gluon propagator

• Quark-Gluon Vertex

• Four-Point Scattering Kernel

G. Eichmann, arXiv:0909.0703

1.Gluon Propagator

• In Landau gauge:

• Modeling the dress function as two parts:

• 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:

• Using Oliveira’s scheme, we can readily parameterize our interaction model as follows,

Solid for omega=0.5GeV, dash for omega=0.6GeV

• 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.

2.Vertex & Kernel

• In principal, the DSEs of vertex and kernel are extremely complicated.

• We choose to construct higher order Green’s functions by lower ones. The procedure is called truncation scheme.

• How to build a truncation scheme systematically and consistently?

• How to judge whether a truncation scheme is good one?

• 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.

Rainbow-Ladder truncation

• Rainbow approximation:

• Ladder approximation:

• The axial-vector Ward-Takahashi identity is preserved:

G. Eichmann, arXiv:0909.0703

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

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.

2. Homogeneous Bethe-Salpeter

Equation • In our framework, we specify a given meson by its JPC which determines the transformation properties of its BS amplitude.

i. J determines the Lorentz structure:

ii. P transformation is defined as

where

iii. 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.

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…

Normalization of BSA

• Leon-Cutkosky scheme:

• Nakanishi scheme:

R.E. Cutkosky and M. Leon, Phys. Rev. 135, 6B (1964)

N. Nakanishi, Phys. Rev. 138, 5B (1965)

Calculate Observables

:• Leptonic decay

constant:

• EM form factor:

• Strong decay:

• 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.

Results:

1. Ground States

Pion

Rho

• 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.

• 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.

II. Exotic States & First Radial Excitation

Wherein, we inflate ground-state masses of pion and rho mesons: 1.Effects from dressed truncation and pion cloud could return them to observed values.2.It expands the contour of complex quark so that more states are available.

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 running coupling and running gluon mass.

• 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?