iTHES Mini-workshop on "Strong-Field Physics" May 29, 2014@RIKEN

Koichi Hattori, Tetsuo Hatsuda (RIKEN)

Photon propagation in strong magnetic fields

0. Introduction to Workshop

1. Strong B-fields in heavy-ion collisions and neutron stars2. Analytic calculation of “vacuum birefringence” → Tomaru3. Discussions and Prospects4. Summary

Plan of this talk

The first seminal work in “nonlinear QED”“Consequences of Dirac’s Theory of the Positron”

W. Heisenberg and H. Euler in Leipzig122. December 1935

Euler – Heisenberg effective Lagrangian　 - resummation wrt the number of external legs

Correct manipulation of a UV divergence in 1935!

Pair creation (vacuum instability) induced by strong electric field

General formula within 1-loop & constant fieldobtained by the “proper-time method”.

NIF: National Ignition Facility, LivermoreELI: Extreme Light Infrastructure, Czech Republic, Hungary and RomaniaGekko-Exa, HiPER,,,

Tomaru (Experiment), Moritaka (Theory), Takabe (Theory)

Crab pulsar

After late 1960’s

Tamagawa (Observation), Barkov (Theory), Ebisuzaki (Theory), Hattori

After late 1950’s

RHIC@BNL LHC@CERN

Developments of intense laser fields

Neutron stars, GRB, Black holes, Magnetars,,,

After 2000Ultrarelativistic heavy-ion collisions

Hattori (Theory)

Experiment

Phenomenology

Observation

Theory

Motivation of the workshop

6 talks + Lunch + Coffee break + Free-discussion time

11:00-11:45 K. Hattori (RIKEN)Photon propagations in strong magnetic fields

11:45-12:30 T. Tomaru (KEK)Vacuum Birefringence and Axion measurement by laser interferometer

1:45-2:30 T. Tamagawa (RIKEN)X-ray polarimetry satellite GEMS and beyond

2:30-3:15 M. Barkov (RIKEN)Close binary progenitors of gamma-ray bursts and hypernovae

3:35-4:20 T. Moritaka and H. Takabe (ILE, Osaka University)Gamma Ray Emission and Induced Vacuum Breakdown with High-Intensity Pulse Laser

4:20-5:05 T. Ebisuzaki (RIKEN)Astrophysical ZeV acceleration along the jets of an accreting blackhole

5:05-Free discussions with coffee

Time table

Photon propagation in strong magnetic fields

(I) KH, K. Itakura, Annals Phys. 330 (2013) 23-54 (II) KH, K. Itakura, Annals Phys. 334 (2013) 58-82

RHIC@BNL

LHC@CERN

Phase diagram of QCD matter

Asymptotic freedomQuark-gluon plasma

Magnetic susceptibility (χ) of QCD matter by lattice QCD. From a talk by G. Endrodi in QM2014.

Light-meson spectra in B-fields by lattice QCD Hidaka and Yamamoto

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)

Extremely strong magnetic fields in NSs/Magnetars

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.

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 having strong oscillations

Photon propagation in a constant external magnetic fieldLorentz and gauge symmetries lead to a tensor structure,

θ: angle btw B-field and photon propagation

B

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

EH LagrangianSoft photon limit

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

Polarization tensor acquires an imaginary part above

A double infinite sumKH, K. Itakura (I)

(Photon momentum) Narrowly spaced Landau levels

Lowest Landau level

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)

Prospects & Discussions

Summary

1. We performed analytic calculation of the vacuum polarization tensor in constant magnetic fields.

2. We obtained precise behaviors of the refractive index in LLL. Magnitudes of B-fields, Photon energy, Propagation angle, polarization

3. We discussed possible applications to UrHIC and Neutron Stars/Magnetars.

We showed anisotropic and polarization-dependent photon spectruminduced by the Landau levels in strong B-fields.

Neutral pions in strong magnetic fields

Hattori, Itakura, Ozaki

Violation of axial current conservation

Absence of radiative correction Adler & Bardeen, 1969Triangle diagram gives the exact result in the all-order perturbation theory

Adler, Bell, Jackiw, 1969

Dominant (98.798 % in the vacuum)

99.996 %

``Dalitz decay ‘’ (1.198 % in the vacuum)

NLO contribution to the total decay rate

Only corrections to external legs are possible

LO contribution to the total decay rate

Effects of external magnetic fields

Decay mode possible only in external field

“Bee decay” can be comparable to Dalitz decay and even π0 2γ, depending on B.

Replacement of a photon line by an external field

Decay width of “Bee decay”

WZW effective vertexπ0 γ

Dalitz decay

Bee decay

Decay widths

Mean lifetime

femtometer

Branching ratios

Charmonium spectroscopy in strong magnetic fields by QCD sum rules

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

Meson spectra in B-fields

Chernodub

Hidaka, A.Yamamoto

Chiral condensate in magnetic fieldfrom lattice QCD

Landau levels

Mass modifications in the 2nd order perturbation theory

Mixing in wave functions

Equation of motions

Level repulsion

Dispersion relations

Current correlators

QCD sum rules

+

+ +

+ 2

Direct couplings

2nd-order perturbation

Charmonium spectra from QCD sum rules

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

Effective coupling between π0 and 2γ

(Rest frame)

Neutral pion decay into dilepton

Bext = (0,0,B), Eext = 0

EM current

q q

Neutral pion decay into dilepton (continued)

Decay rates in three modes Mean lifetime

Energy dependence of the decay rates

Field-strength dependence of the branching ratio

Angle dependence of the branching ratio Angle dependence of the lifetime

Discussion 1

B ~ 102×Bc

Magnetar: eB <<