Koichi HattoriLunch seminar @ BNL, Aug. 14 2014

Photon propagations and charmonium spectroscopy in strong magnetic fields

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

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

RHIC@BNL

LHC@CERN

Phase diagram of QCD matter

Asymptotic freedomQuark-gluon plasma

Light-meson spectra in B-fields Hidaka and A.Yamamoto

Quark and gluon condensates at zero and finite temperatures Bali et al.

Results from lattice QCD in magnetic fields

PSR0329+54

Extremely strong magnetic fields

UrHIC NS/Magnetar

Lienard-Wiechert potential

Z = 79(Au), 82(Pb)

Lighthouse in the sky

Strong magnetic fields in nature and laboratories

Magnet in Lab.

Magnetar

Heavy ion collisions

Polarization 1Polarization 2

Incident light“Calcite” (方解石 )

“Birefringence” : Polarization-dependent refractive indices.

Response of electrons to incident lightsAnisotropic responses of electrons result in polarization-dependent and anisotropic photon spectra.

Photon propagations in substances

+ Lorentz & Gauge symmetries n ≠ 1 in general

+ Oriented response of the Dirac sea Vacuum birefringence

How about the vacuum with external magnetic fields ?- The Landau-levels

B

Modifications of photon propagations in strong B-fields- Old but unsolved problems

Quantum effects in magnetic fields

Photon vacuum polarization tensor:

Modified Maxwell eq. :

Dressed propagators in Furry’s picture

・・・

・・・

Should be suppressed in the ordinary perturbation theory, but not in strong B-fields.

eBeB eB

Break-down of naïve perturbation in strong B-fields

Naïve perturbation breaks down when B > Bc

Need to take into account all-order diagrams

Critical field strengthBc = me

2 / e

Dressed fermion propagator in Furry’s picture

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

Nonlinear to strong external fields

Schwinger, Adler, Shabad, Urrutia, Tsai and Eber, Dittrich and Gies

Exponentiated trig-functions generate strongly oscillating behavior witharbitrarily high frequency.

Integrands with strong oscillations

Photon propagation in a constant external magnetic field

Gauge symmetry leads to three tensor structures,

θ: angle btw B-field and photon propagation

B

Vanishing B limit:

Summary of relevant scales and preceding calculations

Strong field limit: the lowest-Landau-level approximation(Tsai and Eber, Shabad, Fukushima )

Numerical computation below the first threshold(Kohri and Yamada) Weak field & soft photon limit

(Adler)

?Untouched so far

General analytic expression

Euler-Heisenberg LagrangianIn soft photon limit

Analytic result of integrals- An infinite number of the Landau levels

A double infinite sumKH, K.Itakura (I)

(Photon momentum) Narrowly spaced Landau levels

Lowest Landau level

Polarization tensor acquires an imaginary part above

Complex refractive indices

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’’

KH, K. Itakura (II)

Self-consistent solutions of the modified Maxwell Eq.

Photon dispersion relation is strongly modified when strongly coupled to excitations (cf: exciton-polariton, etc)

cf: air n = 1.0003, water n = 1.333

𝜔2/4𝑚2

≈ Magnetar << UrHIC

Angle dependence of the refractive indexReal part

No imaginary part

Imaginary part

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

λ (fm)

Neutron stars = Pulsars QED cascade in strong B-fields

What is the mechanism of radiation?

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

Chernodub

Hidaka, A.Yamamoto

Landau levels for charged mesons

Bali et al.

Effective masses in the strong-field limit: The Lowest Landau Level ( n = 0 )

Similar to Nielsen-Olesen instability

In hadronic degrees

From lattice QCD Chiral condensate in B-fields from lattice QCD

Magnetic catalysis

Gusynin, Miransky, Shovkovy

Mixing btw ηc and J/psi in B-fields

Equation of motions

Mass spectra with level repulsion

Coupling among 1 PS and 2 Vector fields

Longitudinal J/psi

ηc

Mixing only with Longitudinal J/psi

Operator product expansions (OPE) and dispersion relations

?

Current correlators

QCD sum rules

Spectral function:

Shifman, Vainshtein, Zakharov

Conventional spectral ansatz: “pole + continuum”

Borel transformation

QCD sum rules work well for the isolated lowest states. Dispersion relation is insensitive to detail structures of the continuum.

+

Direct couplings with Bethe-Salpeter amplitudes in HQ limit

+ +

2nd-order perturbation

Spectral ansatz with mixing effects

+

Bohr radius a0 = 0.16 fm in Coulombic wave function

+ 2

Perturbative part + dim.-4 gluon condensates

OPE for charmonium in B-fields

NB)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

B-dependent condensate

D and D* mesons in B-fieldsP.Gubler, KH, S.H.Lee, S.Ozaki, K.Suzuki, In progress.

+ Landau levels of charged D±, D*±

+ Mixing effects

OPE for open flavors+ Effects of <qbar q> condensates D± and longitudinal D*± spectra

Landau levels

Landau levels+

mixing effects

u, d

cbar

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

Summary

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 by UrHIC

Lienard-Wiechert potential

Z = 79(Au), 82(Pb)

z

LW potential is obtained by boosting an electro-static potential

r R

Boost

Liu, Greiner, Ko

+ Free streaming relativistic protons+ Charge distributions in finite-size nuclei

Impact parameter (b)

Lienard-Wiechert potential

z

+ Free streaming relativistic protons+ Charge distributions in finite-size nuclei

LW potential is obtained by boosting an electro-static potential

r R

Boost

Analytic modeling of B-fields

Liu, Greiner, Ko

Deng and Huang, PRC85 (2012) Bzdak and Skokov, PLB710 (2012)

Impact parameter dependence of B-fields

Voronyuk et al., PRC83 (2011)

Time dependence of B-fields

Voronyuk et al., PRC83 (2011)

Beam-energy dependence of B-fields

Fourier components of time-dependent B-fields

b = 10 fm

Analytic results of integrals without any approximation

Polarization tensor acquires an imaginary part above

Every term results in either of three simple integrals.

A double infinite sum

KH, K. Itakura (I)

Renormalization

+= ・・・+ +

Log divergence

Subtraction term-by-term

Ishikawa, Kimura, Shigaki, Tsuji (2013)

Taken from Ishikawa, et al. (2013)

Finite

Re Im

Borel transform

Borel-transformed dispersion relation:

Spectral ansatz:

Mass formula in “pole+continuum” ansatz