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Solving non-perturbative renormalization group equation without field operator expansion and its application to the dynamical chiral symmetry breaking Daisuke Sato (Kanazawa U.) @ SCGT12Mini 1 with Ken-Ichi Aoki (Kanazawa U.)

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Solving non-perturbative renormalization group equation without field operator expansion and its application to the dynamical chiral symmetry breaking. Daisuke Sato. (Kanazawa U.). w ith Ken- Ichi Aoki. (Kanazawa U.). @ SCGT12Mini. - PowerPoint PPT Presentation

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Page 1: Daisuke Sato

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.)

Page 2: Daisuke Sato

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.

Page 3: Daisuke Sato

31-loop exact!!

Shell mode integration

• Wilsonian effective action:

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

Page 4: Daisuke Sato

Local potential approximation ( LPA )

4

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

Page 5: Daisuke Sato

NPRG and Dynamical Chiral Symmetry Breaking (DSB) in QCD

5

• 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

Page 6: Daisuke Sato

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.

Page 7: Daisuke Sato

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

Page 8: Daisuke Sato

Ladder Approximation

8

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.

Page 9: Daisuke Sato

Ladder-Approximated NPRG Eq.

9

: 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)

Page 10: Daisuke Sato

Convergence with respect to order of truncation?

10

Page 11: Daisuke Sato

Without field operator expansion

Mass function

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

11

Running mass:

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

Page 12: Daisuke Sato

Finite difference• Discretization :

• Forward difference

• Coupled ordinary differential equation of the discretized mass function

12

Page 13: Daisuke Sato

Boundary condition• Initial condition:

• Boundary condition with respect to

: bare mass of quark (current quark mass)

at

13

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.

Page 14: Daisuke Sato

RG flow of the mass function

14

Page 15: Daisuke Sato

15

Dynamical mass

Infrared-limit running mass

Page 16: Daisuke Sato

Chiral condensates

NPRG eq. for the free energy giving the chiral condensates

Chiral condensates are given by

16

:source term for chiral condensate

free energy :

Page 17: Daisuke Sato

Free energy

17

Chiral condensates

Page 18: Daisuke Sato

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

18

: gauge-fixing parameter

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

Page 19: Daisuke Sato

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

19

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

Page 20: Daisuke Sato

Ladder approximation with A. D.

20

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.

Page 21: Daisuke Sato

Approximation beyond “the Ladder”

21

• 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

Page 22: Daisuke Sato

Approximation beyond “the Ladder”

22

• 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)

Page 23: Daisuke Sato

NPRG Eq. Beyond Ladder Approximation

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

23

Page 24: Daisuke Sato

Partial differential Eq. equivalent to this beyond the ladder approximation

24

Non-ladder extended NPRG eq.

Ladder-approximated NPRG eq.

Page 25: Daisuke Sato

Non-ladder with A. D.

25

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

0.008

0.01

0.012

0.014

0.016

0.018

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, .

Page 26: Daisuke Sato

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

Page 27: Daisuke Sato

Backup slides

27

Page 28: Daisuke Sato

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.

28

Page 29: Daisuke Sato

Shell mode integralmicro

macro

29

Shell mode integral:

Gauss integral

Page 30: Daisuke Sato

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

30

Page 31: Daisuke Sato

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

Page 32: Daisuke Sato

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.

32

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