Solving non-perturbative renormalization group equation without field operator

expansion and its application to the dynamical chiral symmetry breaking

Daisuke Sato (Kanazawa U.)

@ SCGT12Mini1

with Ken-Ichi Aoki (Kanazawa U.)

Non-Perturbative Renormalization Group (NPRG)

• NPRG Eq.:

2

Wegner-Houghton (WH) eq. (Non-linear functional differential equation )

• Analyze Dynamical Chiral Symmetry Breaking (DSB) , which is the origin of mass in QCD and Technicolor, by NPRG.

• Field-operator expansion has been generally used in order to sovle NPRG eq.

• Convergence with respect to order of field-operator expansion is a subtle issue.

• We solve this equation directly as a partial differential equation.

31-loop exact!!

Shell mode integration

• Wilsonian effective action:

: Renormalization scale （momentum cutoff）• Change of effective action

Local potential approximation ( LPA )

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Momentum spacezero mode operator　　　　　

• Set the external momentum to be zero when we evaluate the diagrams.

• Fix the kinetic term. • Equivalent to using space-time independent fields.

renormalization group equation for coupling constants

• Field operator expansion

NPRG and Dynamical Chiral Symmetry Breaking (DSB) in QCD

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• Wilsonian effective action of QCD in LPA

NPRG Eq.:

• field operator expansion

the gauge interactions generate the 4-fermi operator, which brings about the DSB at low energy scale, just as the Nambu-Jona-Lasinio model does.

: effective potential of fermion, which is central operators in this analysis

How to deal with DSB

• Taking the zero mass limit: after all calculation, we can get the dynamical mass,

K-I. Aoki and K. Miyashita, Prog. Theor. Phys.121 (2009) 6

• Introduce the bare mass , which breaks the chiral symmetry explicitly, as a source term for chiral condensates .

• Lowering the renormalization scale the running mass grows by the 4-fermi interactions and the gauge interaction.

• Add the running mass term to the effective action.

Renormalization group flows of the running mass and 4-fermi coupling constants

7Chiral symmetry breaks dynamically.

Running mass plotted for each bare mass

: 1-loop running gaugecoupling constant

Ladder Approximation

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Massive quark propagator including scalar-type operators

Extract the scalar-type operators , which are central operators for DSB.

• Limit the NPRG function to the ladder-type diagrams for simplicity.

Ladder-Approximated NPRG Eq.

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: order of truncation

(Landau gauge)

• Expand this RG eq. with respect to the field operator and truncate the expansion at -th order.

Non-linear partial differential equation with respect to and • Ladder LPA NPRG Eq. :

Coupled ordinary differential eq. (RG eq.) with respect to

• Convergence with respect to order of truncation?

Running mass:

• This NPRG eq. gives results equivalent to improved Ladder Schwinger-Dyson equation. Aoki, Morikawa, Sumi, Terao, Tomoyose (2000)

Convergence with respect to order of truncation?

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Without field operator expansion

Mass function

We numerically solve the partial differential eq. of the mass function by finite difference method.

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Running mass:

(Landau gauge)Solve NPRG eq. directly as a partial differential eq.

Finite difference• Discretization :

• Forward difference

• Coupled ordinary differential equation of the discretized mass function

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Boundary condition• Initial condition:

• Boundary condition with respect to

: bare mass of quark (current quark mass)

at

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source term for the chiral condensate

We need only the forward boundary condition .Forward difference

We set the boundary point to be far enough from the origin () so that at the origin is not affected on this boundary condition.

RG flow of the mass function

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Dynamical mass

Infrared-limit running mass

Chiral condensates

NPRG eq. for the free energy giving the chiral condensates

Chiral condensates are given by

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:source term for chiral condensate

free energy :

Free energy

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Chiral condensates

0 0.5 1 1.5 2 2.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Gauge dependence

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: gauge-fixing parameter

The ladder approximation has strong dependence on the gauge fixing parameter.

Improvement of LPA

• Take into account of the anomalous dimension of the quark field obtained by the perturbation theory as a first step of approximation beyond LPA

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plays an important role in the cancelation of the gauge dependence of the function for the running mass in the perturbation theory.

Ladder approximation with A. D.

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The chiral condensates of the ladder approximation still has strong dependence on the gauge fixing parameter.

0 0.5 1 1.5 2 2.5 3 3.5-0.00499999999999999

9.54097911787244E-18

0.00500000000000001

0.01

0.015

0.02

0.025

0.03

0.035

0.04

ladderladder with A.D.

0 0.5 1 1.5 2 2.5 3 3.50

0.20.40.60.8

11.21.41.61.8

ladderladder with A.D.

Approximation beyond “the Ladder”

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• Crossed ladder diagrams play important role in cancelation of gauge dependence.

• Take into account of this type of non-ladder effects for all order terms in .

Ladder Crossed ladder

Approximation beyond “the Ladder”

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• Introduce the following corrected vertex to take into account of the non-ladder effects.

Ignore the commutator term.

K.-I. Aoki, K. Takagi, H. Terao and M. Tomoyose (2000)

NPRG Eq. Beyond Ladder Approximation

• NPRG eq. described by the infinite number of ladder-form diagrams using the corrected vertex.

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Partial differential Eq. equivalent to this beyond the ladder approximation

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Non-ladder extended NPRG eq.

Ladder-approximated NPRG eq.

Non-ladder with A. D.

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0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

ladder with A.D.non-ladde with A.D.

0 0.5 1 1.5 2 2.5 3 3.50

0.002

0.004

0.006

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0.012

0.014

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ladder with A.D.

non-ladde with A.D.

is an observable. The non-ladder extended approximation is better.

The chiral condensates agree well between two approximations in the Landau gauge, .

Summary and prospects• We have solved the ladder approximated NPRG eq. and

a non-ladder extended one directly as partial differential equations without field operator expansion.

• Gauge dependence of the chiral condensates is greatly improved by the non-ladder extended NPRG equation.

• In the Landau gauge, however, the gauge dependent ladder result of the chiral condensates agrees with the (almost) gauge independent non-ladder extended one, occasionally(?).

• Prospects– Evaluate the anomalous dimension of quark fields by NPRG.– Include the effects of the running gauge coupling constant

given by NPRG.26

Backup slides

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Beyond the ladder approximation

Ladder diagram Non-ladder diagram

The Dyson-Schwinger Eq. approach is limited to the ladder approximation.

We can approximately solve the Non-perturbative renormalization group equation with the non-ladder effects.

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Shell mode integralmicro

macro

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Shell mode integral:

Gauss integral

1-loop perturbative RGE

Running of gauge coupling constant

To take account of the quark confinement , we set a infrared cut-off for the gauge coupling constant.

1-loop perturbative RGE + Infrared cut-off

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Renormalization group flows of the running mass and 4-fermi coupling constants

31Chiral symmetry breaks dynamically.

Running mass plotted for each bare mass

Running mass grows up rapidly when the 4-fermi coupling constant is large.

: 1-loop running gaugecoupling constant

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

ladder

non-ladder

Result of non-ladder extended app.

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The chiral condensates agree well between two approximations in the Landau gauge, .

0 0.5 1 1.5 2 2.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

laddernon-ladder