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Photon propagations and charmonium spectroscopy in strong magnetic fields. KH , K. Itakura , Annals Phys. 330 ( 2013); 334 (2013). S.Cho , KH, S.H.Lee , K.Morita , S.Ozaki , arXiv:1406.4586 [ hep-ph ]. Koichi Hattori Lunch seminar @ BNL, Aug. 14 2014. Phase diagram of QCD matter.

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Koichi Hattori Lunch seminar @ BNL, Aug. 14 2014

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Photon propagations and charmoniumspectroscopy

in strong magnetic fields

KH, K. Itakura, Annals Phys. 330 (2013); 334 (2013)

S.Cho, KH, S.H.Lee, K.Morita, S.Ozaki, arXiv:1406.4586 [hep-ph]

Koichi Hattori

Lunch seminar @ BNL, Aug. 14 2014

Phase diagram of QCD matter

Quark-gluon plasma

Results from lattice QCD in magnetic fields

Asymptotic freedom


Light-meson spectra in B-fields

Hidaka and A.Yamamoto


Quark and gluon condensates at zero

and finite temperatures Bali et al.

Magnetic susceptibility (χ) of QCD matter by lattice QCD.

From a talk by G. Endrodi in QM2014.

Extremely strong magnetic fields



Lighthouse in the sky



Z = 79(Au), 82(Pb)

Strong magnetic fieldsin nature and laboratories

Magnet in Lab.


Heavy ion collisions

Response of electrons to incident lights

Polarization 1

Polarization 2

Incident light

Photon propagations in substances

Anisotropic responses of electrons result in

polarization-dependent and anisotropic photon spectra.

“Birefringence” : Polarization-dependentrefractive indices.

“Calcite” (方解石)

How about the vacuum with external magnetic fields ?

- The Landau-levels

+ Lorentz & Gauge symmetries  n ≠ 1 in general

+ Oriented response of the Dirac sea  Vacuum birefringence


Modifications of photon propagations in strong B-fields

- Old but unsolved problems

Quantum effects in magnetic fields




Should be suppressed

in the ordinary perturbation theory,

but not in strong B-fields.


Modified Maxwell eq. :


Photon vacuum polarization tensor:

Dressed propagators

in Furry’s picture

Break-down of naïve perturbation in strong B-fields

Dressed fermion propagator in Furry’s picture

Critical field strength

Bc = me2 / e

  • Naïve perturbation breaks down when B> Bc

  •  Need to take into account all-order diagrams

Resummation w.r.t. external legs by “proper-time method“


Nonlinear to strong external fields

Photon propagation in a constant external magnetic field

Gauge symmetry leads to threetensor structures,

θ: angle btw B-field and

photon propagation


Vanishing B limit:

Integrands with strong oscillations

Schwinger, Adler, Shabad, Urrutia,

Tsai and Eber, Dittrich and Gies

Exponentiated trig-functions generate

strongly oscillating behavior with

arbitrarily high frequency.

Summary of relevant scales

and preceding calculations

General analytic expression

  • ?

  • Untouched so far

Numerical computation

below the first threshold

(Kohri and Yamada)

Weak field & soft photon limit


Strong field limit: the lowest-Landau-level approximation

(Tsaiand Eber, Shabad, Fukushima )


In soft photon limit

Br-dependence of the coefficients

in soft-photon limit:

Comparison btw limiting behavior

and numerical computation.


Analytic result of integrals

- An infinite number of the Landau levels

KH, K.Itakura(I)

A double infinite sum


Prompt photon ~ GeV2

Thermal photon ~ 3002 MeV2

~ 105 MeV2

Untouched so far

Polarization tensor acquires an imaginary part


(Photon momentum)

Narrowly spaced Landau levels

Strong field limit (LLL approx.)

(Tsai and Eber, Shabad, Fukushima )

(Photon momentum)

Numerical integration

(Kohri, Yamada)

Soft photon & weak field limit


Lowest Landau level

Complex refractive indices

KH, K. Itakura (II)

  • Solutions of Maxwell eq.

  • with the vacuum polarization tensor

The Lowest Landau Level (ℓ=n=0)

Refractive indices at the LLL

Polarization excites only along the magnetic field

``Vacuum birefringence’’

Self-consistent solutions of the modified Maxwell Eq.

Photon dispersion relation is strongly modified

when strongly coupled to excitations (cf: exciton-polariton, etc)

≈ Magnetar << UrHIC

cf: air n = 1.0003, water n = 1.333

Angle dependence of the refractive index

Real part

Imaginary part

No imaginary part

“Mean-free-path” of photons in B-fields

λ (fm)

Neutron stars = Pulsars

What is the mechanism of radiation?

 QED cascade in strong B-fields

Need to get precise description of vertices:

Dependences on magnitudes of B-fields, photon energy, propagation angle and polarizations.

Charmonium spectroscopy

in strong magnetic fields by QCD sum rules

S.Cho, KH, S.H.Lee, Morita, Ozaki

Light meson spectra in strong B-fields

Landau levels for charged mesons

In hadronic degrees

Effective masses in the strong-field limit:

The Lowest Landau Level ( n = 0 )

Chiral condensate in magnetic field

from lattice QCD


Similar to Nielsen-Olesen instability

From lattice QCD

Chiral condensate in B-fields from lattice QCD

Magnetic catalysis

Gusynin, Miransky,


Hidaka, A.Yamamoto

Bali et al.

Mixing btw ηc and J/psi in B-fields

Coupling among 1 PS and 2 Vector fields

Equation of motions

Mixing of wave functions

Mixing only with Longitudinal J/psi

Mass spectra with level repulsion

Longitudinal J/psi


QCD sum rules

Current correlators


Operator product expansions (OPE)

and dispersion relations

Spectral function:

Shifman, Vainshtein, Zakharov

Conventional spectral ansatz: “pole + continuum”

QCD sum rules work well for theisolatedlowest states.

Dispersion relation is insensitive to detail structures of the continuum.


Spectral ansatzwith mixing effects

2nd-order perturbation




Direct couplings with Bethe-Salpeter amplitudes in HQ limit

Bohr radius a0 = 0.16 fmin

Coulombic wave function


OPE for charmonium in B-fields

+ 2

Perturbative part

+ dim.-4 gluon condensates


The resummed vacuum polarization tensor (vector current correlator) can be applied in strong field limit. KH, Itakura

ηc and longitudinal J/psi spectra from QCD sum rules

D and D* mesons in B-fields

P.Gubler, KH, S.H.Lee, S.Ozaki, K.Suzuki, In progress.

  • c.f.) B and B* by Machado, Finazzo, Matheus, Noronha

Landau levels


mixing effects

+ Landau levels of charged D±, D*±

+ Mixing effects

Landau levels

OPE for open flavors

+ Effects of <qbar q> condensates

D± and longitudinal D*± spectra

B-dependent condensate

u, d



We calculated the resummed vacuum polarization tensor (vector current correlator)

to get the refractive indices in strong magnetic fields.

We obtained charmonium spectra in magnetic fields by QCD sum rules

with careful treatment of the phenomenological side as well as OPE.

Extremely strong magnetic fields induced byUrHIC



Impact parameter (b)


+ Free streaming relativistic protons

+ Charge distributions in finite-size nuclei

LW potential is obtained

by boosting an electro-static potential


Liu, Greiner, Ko


Z = 79(Au), 82(Pb)

Analytic modeling of B-fields


+ Free streaming relativistic protons

+ Charge distributions in finite-size nuclei

LW potential is obtained

by boosting an electro-static potential





Liu, Greiner, Ko

Impact parameter dependence of B-fields

Bzdak and Skokov, PLB710 (2012)

Deng and Huang, PRC85 (2012)

Time dependence of B-fields

Voronyuk et al., PRC83 (2011)

Beam-energy dependence of B-fields

Voronyuk et al., PRC83 (2011)

Fourier components of time-dependent B-fields

b = 10 fm

Analytic results of integrals without any approximation

KH, K. Itakura (I)

A double infinite sum

Dimesionless variables

Every term results in either of three simple integrals.

Polarization tensor acquires an imaginary part








Log divergence


Subtraction term-by-term

Ishikawa, Kimura, Shigaki, Tsuji (2013)



Taken from Ishikawa, et al. (2013)

Mass formula in “pole+continuum” ansatz

Spectral ansatz:

Borel transform

Borel-transformed dispersion relation:

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