Hideaki IidaTokyo Institute of Technology

Collaborators: M.Oka, H.Suganuma

The XXI International Symposium on Lattice Field TheoryTsukuba International Congress Center

(EPOCHAL TSUKUBA)July 15 - 19, 2003

[Contents]・ Introduction・ Estimation of the full quark-gluon vertex・ Extension of calculation to finite temperature system・ Results and prospects

【 Nonperturbative aspect of QCD 】

→Perturbation theory is not applied there.

Nonperturbative effects●Color Confinement

●Dynamical Chiral Symmetry Breaking （ DCSB)　 which would correspond to Dynamical Quark Mass Generation.

QCD running coupling constant becomes large in the low-momentum region.

QCD vacuum is highly complicated due to the strong- couplingIn the infrared region

:flavor number:color number

(one-loop level)

・ Effective model

ex) NJL model 、 σ model 、 Instanton model 、 etc…

…The picture of physics is clear in this approach.

　 But this is not exactly based on QCD.

・ Lattice QCD simulation

…We can calculate physical quantities from first principle.

But this is not analytic calculation. So, it is rather hard to extract the essense ofthe physics behind the phenomena.

Approaches to Nonperturbative QCD

…We aim systematical understanding of QCD by unifying both methods.

The aim of our study is to combine Lattice QCD simulation and the Schwinger-Dyson (SD) formalism for understanding nonperturbative QCD.There are two types of approaches:

SD equation (model) + Lattice QCD

【 Exact SD equation 】

:full gluon propagator:full quark propagator

:non-interacting quark propagator

:bare vertex g:full quark-gluon vertex

Review of Schwinger-Dyson(SD) formalism

(In chiral limit)

SD equation is a kind of nonlinear integral equation expressed as follows.

＝ ＋ +…= (self-consistent eq.)

SD equation includes the nonperturbative effect of the infinite order of .

[ Explicit expression ]

Ref.) M.S. Bhagwat, M.A. Pichowsky , C.D. Roberts , P.C. Tandy . nucl-th/0304003

・ Gauge: Landau gauge・ Full quark-gluon vertex: vector type, i.e.・ Wavefunction renormalization: (quark propagator: )

We impose some assumptions.

After taking trace, then we get

： polarization factor in gluon propagator [ i.e. ]

We investigate the kernel

the quark mass function obtained from Lattice QCD

in terms of

※Such an approach is independently done by the following group.

【 Quark Mass Function from Lattice QCD 】Ref.) 　 D.B.Leinweber et al. 　　 Nucl.Phys.Proc.Suppl.109(2002)163 etc.

The calculation condition is ・ Landau gauge

・ β=5.85(a=0.125fm)

Mass function is well fit by following function:

We use the recent lattice data of the quark mass function .

The SD equation takes the convolution form. ：Extraction of the SD kernel

So in coordinate space:

In fact, the kernel is expressed by

Tilde means the Fourier transformation of a function.e.g.

According to this procedure, we can determine SD kernel.

e.g.

But the lattice data would not be reliable in ultraviolet region.Instead, we adopt the following method.

1. We suitably parametrize the kernel function .

(One of the best parametrization of the kernel .)

2. We choose the best parameters so as to reproduce the quark mass function obtained from Lattice QCD.

Parametrization of SD kernel satisfies the condition.

・ In ultraviolet region, (from P-QCD)

Furthermore, we impose the following two conditions for

・ enhances in intermediate energy region ( １～２ GeV).

・ vanishes in infrared energy region.

:polarization factor of nonperturbative gluon propagator

according to infrared vanishing and intermediate enhancement of the polarization factor of nonperturbative gluon propagator .because

Gluon propagator from lattice QCD

Ref ) 　 D.B.Leinweber et al. Phys.Rev.D58(1998)031501 etc.Landau gauge, quenched level

（ Landau gauge)

Fit function:

…Infrared vanishing and Intermediate enhancement are observed in .

The calculation condition is

:the polarization factor of the gluon propagator

☆The quark mass function obtained from lattice QCD is well reproduced from following . 　　　

　

【 Estimation of full quark-gluon vertex 】

: lattice result+: solution of SD

[Shape of ] [Comparison with the lattice data with from SD eq. with ]

Using the Coulomb-type kernel with , we solve the SD eq. and find the trivial solution for .(In the Cornell potential, the Coulomb coefficient corresponds to )Even with a large value of , the Coulomb-type kernel cannot reproduce DCSB.

[Comparison with the Coulomb-type kernel in terms of DSCB]

Determination on Importance of the Intermediate Enhancement for DCSB

Thus, the intermediate enhancement of plays an important role for DCSB.

NP Gluon PropagatorNP Gluon Propagator

[Pion decay constant]

With the Pagels-Stokar approximation, is derived from quark mass function.

[quark condensate]

Important quantities for DCSBThere are several quantities related to DCSB

( is ultraviolet cutoff )

(RGI means renormalization group invariant quantity. )

(using Pagels-Stokar formula)

The values are consistent with the standard values.

The quark mass function leads to the following values , , .

(Infrared Quark Mass)

cf.) [Standard value]・ constituent quark mass ：・ quark condensate:

・ pion decay constant ： (Exp.)

【 Extension to Finite Temperature System 】

Following the Matsubara formalism

The field variables obey the (anti) periodic boundary condition on the imaginary time.

Accordingly, becomes discrete:

e.g.) (quarks)

(gluons)

(for bosons)

(for fermions),

ＳＤ eq. at Finite Temperature

:quark mass function with the Matsubara frequency

Semi-quantitatively, this is consistent value with 　　　　　　　　　　　　　 from lattice QCD in quenched approximation.

Chiral Phase Transition Temperature is about ２００ MeV

～ 280MeV

【 Infrared Quark Mass (lowest Matsubara mode) vs. T 】

【 Conclusion 】●We have investigated the full quark-gluon vertex with the q

uark mass function obtained from lattice QCD.●We have studied finite temperature QCD based on SD eq. wi

th the full quark-gluon vertex. ●QCD phase transition temperature is found to be about 200MeV, which is consistent with lattice QCD result （ in quenched approximation ） .

【 Prospects 】●Calculation in finite density system.

● Investigation of the relation between DCSB and color confinement.●Calculation of hadron properties.●Consideration the wavefunction renormalization.●Calculation using non-quenched data.●Calculation using non-zero temperature lattice data.

Ultraviolet energy region does not make so large contribution to DCSB!

SD equation in Higashijima-Miransky approximation ：

Higashijima-Miransky approximation: Approximation of taking the large value of argument