### A renormalization-group analysis of the magnetic catalysis

### Koichi Hattori

Workshop of Recent Developments in QCD and Quantum Field Theories

@ National Taiwan Univ., Nov. 10, 2017

Based on

KH, K. Itakura, S. Ozaki, [hep-ph/1706.04913] (To appear in PLB)

### Outline

The origin of the magnetic catalysis of χSB in a strong B
**1**

Analogy to superconductivity:

Effective dimensional reduction in high density and in a strong B -- Scaling argument on the basis of RG

### The RG equation for a weak-coupling gauge theory

-- Screening effect in QED
**2**

**3**

### Perspective for the MC in QCD at zero and finite T.

Cf., KH, T. Kojo, N. Su, NPA, (2016).

+ Low energy excitations along the radius [(1+1) D]

+ Degenerated states in the tangential plane [2D]

Phase space volume ~ p^{D-1 }dp

Enhanced IR dynamics induces nonperturbative physics, such as superconductivity and Kondo effect.

Superconductivity occurs no matter how weak the attraction is.

### “Dimensional reduction” in dense systems

### -- (1+1)-dimensional low-energy effective theory

### BCS theory

Quantum correction

In the BCS config.

### IR scaling dimensions

### Kinetic term

In general momentum config.

### Four-Fermi operators for superconductivity

Polchinski (1992)### Landau-level quantization

### and “Dimensional reduction”

### in a strong magnetic field

B

### Landau level discretization due to the cyclotron motion

“Harmonic oscillator” in the transverse plane

Relativistic:

Cyclotron frequency

Nonrelativistic:

In addition, there is the Zeeman effect.

### R

_{↑}

### L

_{↑}

### Analogy btw the dimensional reduction in a large B and μ

### Strong B

(1+1)-D dispersion relations

Large Fermi sphere

2D density of states

### Scaling dimensions in the LLL

### A four-Fermi operator for the LLL

Always marginal irrespective of the interaction type and

the coupling constant thanks to the “dimensional reduction” in the LLL.

### Kinetic term

### RG flow driven by the logarithmic quantum corrections

Emergent scale in the IR !
Cf., Similarity to Λ_{QCD}

### Solution:

Fukushima & Pawlowski

λ

Chiral symmetry breaking occurs solely for the dimensional reason as a consequence of the dimensional reduction in a strong B-field.

Chiral symmetry is broken even in QED in a strong B-field.

Gusynin, Miransky, and Shovkovy.

### Magnetic catalysis

• The m_{dyn} explicitly depends on the coupling constant in NJL model,
implying the dependence on the interaction type.

• The dimensionless combination, eB*G, appears

because of the dimensionful coupling constant in NJL model.

### What form of the gap appears in QED,

### and ultimately in QCD or any other strongly coupled system?

### More specifically, how the differences can be described by the RG?

### Example 1: NJL model

### Interactions in an underlying theory --- “Intrinsic” energy dependence

E.g.,

### Underlying theory may have a hierarchy of the energy scales.

**Energy scales****in QED in B**

Higher Landau levels

Screening mass

Landau pole Region I

Region II

## ?

### Effective (1+1)-dimensional interaction in QED

Increment from the LO diagram 0

### The LO diagram provides a log, and drives the RG in part.

NB) The unscreened magnetic gluon changes the color-superconducting gap. D.T.Son

Higher Landau levels

Screening mass

Landau pole Region I

Region II

### Final result with the appropriate screening effect RG eqs. and the solutions

Region I Region II

### Example 1: NJL model revisited

Solution is formally same as

### Recovers the aforementioned result

### Effective coupling

### Example 2: Unscreened QED

A Landau pole at tan( ・・・ ) = ∞.

Solution is formally same as

Higher Landau levels

Screening mass

Landau pole Region I

Region II

Higher Landau levels Screening mass

Landau pole

### Example 3: Many-flavor QED with α*Nf >> 1 Many-flavor QED Ordinary QED (α*Nf <<1)

We have Therefore, In many-flavor QED

### QCD perspective

Endrodi, JHEP07(2015)

T = 0

### + At T= 0, an enhancement of the chiral condensate was observed in lattice QCD simulations.

### Qualitative agreement with the magnetic catalysis.

### + However, the chiral restoration occurs at a lower T

### in a stronger B-field, called the “inverse magnetic catalysis.”

Typical model calculations fail to reproduce the linear growth at T = 0 and the inverse magnetic catalysis at finite T.

### Possible hierarchies in QCD

Higher Landau levels

Screening mass

Landau pole Region I Region II

Region 0

### Perturbative region

### Intrinsic energy dep.

### in the low-energy QCD

Higher Landau levels Screening mass

Landau pole Region I’

Region II’

Region 0’

### Perturbative region

### Intrinsic energy dep.

### in the low-energy QCD

### In a stronger B:

### A strong screening.

### A strong screening.

Typical models fail to reproduce

the linear growth in the lattice result.

T = 0

Cf., Studies by the Schwinger-Dyson equation in

T. Kojo and N. Su (2013), and KH, T. Kojo, N. Su (2016).

### If the m

_{dyn}

### does not grow with eB, thermal fluctuations will be enhanced by the large degeneracy factor ~ eB.

### (If the gap is m

_{dyn}

### ~ eB

^{1/2}

### , Tc ~ eB

^{1/2}

### .)

q q q

q q

q q q

q

q q q

q q q

From this observation at T = 0, it seems that m_{dyn} ~ O( (eB)^{0} ).

A possible scenario for consistent description of the MC at T = 0 and the inverse MC at finite T.

Higher Landau levels

Screening mass

Landau pole Region I

Region II

Region 0

### Perturbative

### Intrinsic non-

### perturbative

### region in QCD

### How can we get the saturation of m_dyn by the RG?

### Summary

### We explicitly saw the dimensional reason for the occurrence of the magnetic catalysis on the basis of the scaling argument.

### However, the precise form of gap depends the interaction.

### We discussed the implementation of the screening effect from the RG point of view.

To explain the lattice results [or m_{dyn} ~ O( (eB)^{0} )],

an interaction beyond the four-Fermi (NJL) is important.

Cf., T. Kojo and N. Su (2013), and KH, T. Kojo, N. Su (2016).

More generally, this is a diagnosis of the RG method (or EFT) to include the interaction properties of the underlying theory.