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Interaction Model of Gap Equation
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Transcript of Interaction Model of Gap Equation
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Interaction Model of
Gap EquationSi-xue Qin
Peking University & ANLSupervisor: Yu-xin Liu & Craig D. Roberts
With Lei Chang & David Wilson of ANL
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Why?
background, motivation and purpose...
How?
framework, equations and methods...
What?
data, figures and conclusions...
Outline
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Background
QCD has been generally accepted as the
fundamental theory of strong interaction.
Hadron Zoo from PDG
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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
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Motivation &
PurposeHow will the massive type
interaction inflect in observables,
properties of mesons?
O. Oliveira et. al., arXiv:1002.4151
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Dyson-Schwingerequations
• Gluon propagator
• Quark-Gluon Vertex
• Four-Point Scattering Kernel
G. Eichmann, arXiv:0909.0703
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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:
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• 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.
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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.
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Rainbow-Ladder truncation
• Rainbow approximation:
• Ladder approximation:
• The axial-vector Ward-Takahashi identity is preserved:
G. Eichmann, arXiv:0909.0703
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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
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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.
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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.
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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…
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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)
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Calculate Observables
:• Leptonic decay
constant:
• EM form factor:
• Strong decay:
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• 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
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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.
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• 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.
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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?