Non-Perturbative Renormalization Groupapproach toDynamical Chiral Symmetry Breakingin extreme environments Kanazawa UniversityMasatoshi Yamada with Ken-Ichi Aoki and Daisuke Sato

arXiv:1404.3471

Introduction• Dynamical Chiral Symmetry Breaking (DSB) The origin of Hadrons mass

At finite temperature and density Restore chiral symmetry Phase diagram on plane

≪Current quarks masses: MeV

Constituent quarks masses: MeV

Quark-Gluon Plasma

Hadrons

RHIC

LHC

Tem

pera

ture

Baryon density

Neutron starsAtomic nuclei

Lattice simulation

The lattice simulation has the sign problem.The effective model

analysisThe mean field approximationSchwinger-Dyson equation expansion　　　　　　　　… etc.

have difficulties in the further improvement of the approximation.

Our aims:• Approach to DSB at finite temperature and density.• Beyond the mean field approximation.

Talk plan1. NPRG

i. The basic ideasii. DSB via NPRG()

2. At finite temperature and densityi. Flow structures of the four-fermi coupling constant• large-N leading vs. non-leading• Interpretation of the inverse four-fermi coupling constant

ii. Phase diagram on planeiii. How to investigate the order of phase transition in

NPRG?

3. At finite temperature and in external magnetic field

i. Why external magnetic field?ii. Magnetic catalysis vs. Magnetic inhibitioniii. Phase diagram on plane

Part I

Non-Perturbative Renormalization Group andDynamical Chiral Symmetry Breaking

Basic ideas

Infrared limit

Wilsonian effective action

Lower mode Higher mode

RG equation

• The shell mode integration generates the change of effective action.

・・・Shell modes

Lower modes ・・・・

F. Wegner and A Houghton. Phys. Rev. A8 (1973) 401J. Polchinski Nucl. Phys. B231 (1984) 269

Other formulation • RG eq. for the Legendre effective action

One-loop exact is cutoff function.

The propagator:

Suppress the lower modes with . The higher modes with are integrated out.

IR： UV：C. Wetterch Phys. Lett. B301 (1993) 90

Approximation methods Ex: The scalar theory with symmety

Derivative expansion

Local Potential Approximation(LPA)

at

Non-Perturbative RG RG equation for the effective potential:

Truncation

The partial differential equation with the initial condition:

・・・The infinite coupled ordinary differential equations with the initial condition:

Need to truncate the expansion to some finite order

DSB• NJL model

The chiral effective model

Describe DSB of QCD Invariant under chiral trans.

The four-fermi coupling constant is fluctuation of order parameter:

DSB

: the chiral susceptibility2nd order

transition

DSB via NPRG• The four-fermi interaction generates the effective four-fermi interactions by the shell mode integration.

LPA + Truncation

DSB via NPRG• 1-loop diagram correction

Distinguish Wick contractions

Large-N leading part

Large-N approximation()

DSB via NPRG• NPRG equation for

Fixed Points:

Ken-Ichi Aoki et al. Prig. Theor. Phys. 97 (1997) 479

Canonical scaling Quantum correction

• The solution DSB via NPRG

DSB via NPRG• The flow structure of

The four-fermi coupling constant diverges at finite scale when .

The initial value:

DSB

DSB via NPRG• Solving the large-N leading approximated RG eq.

・・・This is an infinite sum of the “tree” diagrams.

The leading of expansion(mean field approximation)

DSB via NPRG• Solving the large-N non-leading extended RG eq.

・・・

… ・・・… …・・・

DSB via NPRG• The large-N non-leading extended RG eq. contains the all orders. The leading order is included exactly. The non-leading orders are included partly.

• In the next part, we investigate the non-leading effect at finite temperature and density.

Talk plan1. NPRG

i. The basic ideasii. DSB via NPRG()

2. At finite temperature and densityi. Flow structures of the four-fermi coupling constant• large-N leading vs. non-leading • Interpretation of the inverse four-fermi coupling constant

ii. Phase diagram on planeiii. How to investigate the order of phase transition in

NPRG?

3. At finite temperature and in external magnetic field

i. Why external magnetic fields?ii. Magnetic catalysis vs. Magnetic inhibitioniii. Phase diagram on plane

Part II

DSBat finite temperature and density

DSB at• The action

• The shell mode integration at .

DSB at• NPRG eq. for one-flavor and one-color NJL model

No contribution No contribution

Negative sign: suppress DSB

Thermal effect functions(threshold functions)

DSB at• limit

has the singularity at the fermi surface.

fermi surface

DSB at

The non-leading effect becomes very large at low temperature.

DSB at• Solving RG equations

Analysis method: Make the equation for the inverse four-fermi coupling constant.

: large-N leading app. eq.

: non-leading extended eq.

D:

DSB at• The flow structure(large-N leading app.)

DSB at• The phase diagram(large-N leading app.)

DSB at• The flow structure(non-leading extended)

DSB at• The phase diagram(non-leading extended)

DSB at• The phase diagram(leading vs. non-leading)

DSB at• The flow structure(large-N leading app.)

DSB

The flow continues after DSB.

DSB at• RG flow: vs.

？How to interpret the flows of after the critical scale ?

DSB at• How to interpret the flows of ?1. Method 1: Weak solution method

We can define the flows of divergent in a mathematically meaningful manner.

2. Method 2: Bosonization(H-S trans.)

Investigate the bosonic effective potential

Ken-Ichi Aoki, Shin-Ichiro Kumamoto and Daisuke Sato. arXiv:1403.0174

J. Hubbard, Phys. Rev. Lett. 3 (1959) 77R. Stratonovich, Dokl. Akad. Nauk SSR 115 (1957) 1097

DSB at• Hubbard-Stratonovich transformation

DSB at• The inverse four-fermi coupling constant corresponds to the curvature of the effective potential at the origin.

• The 1st order transition case

The 1st order transition keeps the positive curvature at origin.

The four-fermi coupling constant doesn't diverge.

We cannot judge whether 1st order transition or symmetric in this analysis.

DSB at

DSB at• How to investigate the order of phase transition?

1. Method 1: Weak solution method

2. Method 2: Bosonization(H-S trans.)

: We don’t need to introduce the additional degrees of freedom to system.: We can use it in only the large-N leading approximation at present.

: We can investigate the beyond large-N effect.: We need to introduce the additional degrees of freedom to system.

DSB at• The bosonized action

Bosonization

Beyond LPA + Truncation

Bosonic effective action

DSB at• RG equations:

The mesonic potential

The Yukawa coupling constant

The four-fermi coupling constant

The field renormalization factors

Complicate…

DSB at• Consider only the mesonic effective potential running

This is so-called Quark-Meson model in LPA. Symmetry breaking pattern: O(N)→O(N-1) Fix the Yukawa coupling constant for

simplicity Solving this equation as a partial differential

equation. The initial condition:

B.J. Schaefer, Nucl. Phys. A757(2005) 479

2nd order transition

1st order transition

Phase diagram

Summary and Outlook • We analyzed NJL model

The flow structures of the four-fermi coupling constant The inverse corresponds to the curvature of the effective potential at the origin.

We cannot distinguish between the single well potential and the 1st order potential in this analysis.

Bosonized NJL model Investigated the order of phase transition There are five RG partial differential equations

Full evaluation is under calculationStudy how the chiral phase structure changes at high density.

Talk plan1. NPRG

i. The basic ideasii. DSB via NPRG()

2. At finite temperature and densityi. Flow structures of the four-Fermi coupling constant• large-N leading vs. non-leading• Interpretation of the inverse four-fermi coupling constant

ii. Phase diagram on planeiii. How to investigate the order of phase transition in

NPRG?

3. At finite temperature and in external magnetic field

i. Why external magnetic fields?ii. Magnetic catalysis vs. Magnetic inhibitioniii. Phase diagram on plane

Part III

DSBat finite temperature and external magnetic field

Quark-Gluon Plasma

Hadrons

RHIC

LHC

Tem

pera

ture

Baryon density

Neutron starsAtomic nuclei

External magnetic field

Why external magnetic field?• Heavy-Ion Collision

• Neutron star

Magnetar

Elliptic flow

The strong magnetic field influence the phase structure of QCD matter.

What’s happen in external B?• Dirac equation

Energy eigenvalues

Landau gauge

Zeeman splitting

Landau quantization : Lowest Landau level

(LLL)

What’s happen in external B?• The quark propagator(proper-time rep.)

has the information of Zeeman splitting. LLL pole

(1+1)-dimension

What’s happen in external B?• The effective potential(mean field app.)

The potential curvature has a logarithmic singularity at .

The curvature is always negative for small .

LLL

What’s happen in external B?• The gap eq.

A finite induces always a non-vanishing dynamical mass.

It is called “magnetic catalysis”• At finite temperature

The critical temperature for chiral restoration should increase with increasing .

Magnetic catalysis V. Skokov, Phys. Rev. D85,034026 (2012)Polyakov-Quark-Meson

model

Recent lattice data(2012)G. S. Bali et al. JHEP 1202 (2012) 044

Inverse!

• This behavior is called “Inverse Magnetic catalysis” or “Magnetic inhibition”.

What is the mechanism which explains this phenomena?

Mechanism• Many mechanisms are suggested.

Dimensional Reduction(DR) of neutral pion

DR of + Asymptotic free

Polyakov loop

Sphalerons・・・

K. Fukushima and Y. Hidaka, Phys. Rev. Lett. 110 (2013) 031601

T. Kojo and N. Su, Phys. Lett, B726 (2013) 839

F. Bruckmann et. al, JHEP 1304 (2013) 112

J. Chao et. al, Phys. Rev. D88 (2013) 054009

The ideas• The quark system in strong magnetic field becomes almost like (1+1)-dimension theory.• Include the neutral pion effect:

• Why the neutral pion ? Charged pions are as massive as . is unaffected by a magnetic field if we treat it as a point particle.

is constructed by two quarks which are have an electric charge.

The ideas• propagator:

Tree level

Loop correction

for

The dimensional reduction occurs for also the neutral pion!

The ideas

• in (1+1)-dimension () cannot exist according to the Marmin-Wagner-Coleman theorem.

There cannot be spontaneous breakdown of continuous symmetries at (1+1)-dim. and (1+0)-dim.

Marmin-Wagner-Coleman theorem

N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133S. Coleman, Math. Phys. 31 (1973) 259

The ideas• The loop in the Self-Consistency eq.

We introduce the ultra-cutoff for transverse momentum because the approximated propagator fail at .

The ideas• At finite temperature

At high temperature, the quark loop effect is suppressed, therefore the pion loop effect becomes stronger than the quark loop effect.

for zero mode:

Inverse Magnetic Catalysis?

Model • Quark-Meson model

The transverse velocity is defined by

Proper-time RG equation• RG equations

• This formulation makes the analysis easy.

Cutoff function

RG equations• The effective action

• The field renormalization factors

Phase diagram on plane

The transverse velocity

𝑒𝐵↑

Summary• We investigated the so-called inverse magnetic catalysis.• We adopted the mechanism which is suggested by K. Fukushima and Y. Hidaka. The neutral pion plays important roles.

• Unfortunately, we could not observe the inverse magnetic catalysis.

Appendix

Non-Perturbative RG

・・・

Wetterich equation• Partition function with the cutoff function

• Legendre effective action

The Cutoff function• The cutoff function must satisfy some conditions:

• The 4-d optimized cutoff function

• The 3-d optimized cutoff function

D. F. Litim, Phys. Rev. D64 (2001) 105007

The Cutoff function• The cutoff function J. Berges, Phys. Rep. 363 (2002)

223

Local Potential Approximation

LPA

NJL model • Lagrangian

• The partition function

The generating function of connected diagram

DSB at• However…

We cannot distinguish between the single well potential and the 1st order potential in this analysis.

Effective Potential

The gas-liquid transition

Hokuriku-Shin-etsu Winter School 2014 @Hakusan city

Hokuriku-Shin-etsu Winter School 2014 @Hakusan city