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 ~ 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
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 ΛQCD
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 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
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
dyndoes 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 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
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 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.
