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A renormalization-group analysis of the magnetic catalysis


Academic year: 2022

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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)



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


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 ~ pD-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



Landau level discretization due to the cyclotron motion

“Harmonic oscillator” in the transverse plane


Cyclotron frequency


In addition, there is the Zeeman effect.




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 ΛQCD


Fukushima & Pawlowski



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 mdyn 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


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


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

(If the gap is m


~ eB


, Tc ~ eB



q q q

q q

q q q


q q q

q q q

From this observation at T = 0, it seems that mdyn ~ 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


Intrinsic non-


region in QCD

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



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 mdyn ~ 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.



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