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iTHES Mini-workshop on "Strong-Field Physics" May 29, 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 stars 2.Analytic calculation of “vacuum birefringence” → Tomaru 3.Discussions and Prospects 4.Summary Plan of this talk

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The first seminal work in “nonlinear QED” “Consequences of Dirac’s Theory of the Positron” W. Heisenberg and H. Euler in Leipzig1 22. December 1935 Euler – Heisenberg effective Lagrangian - resummation wrt the number of external legs Correct manipulation of a UV divergence in 1935!

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Pair creation (vacuum instability) induced by strong electric field General formula within 1-loop & constant field obtained by the “proper-time method”.

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NIF: National Ignition Facility, Livermore ELI: Extreme Light Infrastructure, Czech Republic, Hungary and Romania Gekko-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 Developments of intense laser fields Neutron stars, GRB, Black holes, Magnetars,,, After 2000 Ultrarelativistic heavy-ion collisions Hattori (Theory)

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Experiment Phenomenology Observation Theory Motivation of the workshop 6 talks + Lunch + Coffee break + Free-discussion time

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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

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Photon propagation in strong magnetic fields (I) KH, K. Itakura, Annals Phys. 330 (2013) (II) KH, K. Itakura, Annals Phys. 334 (2013) 58-82

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Phase diagram of QCD matter Asymptotic freedom Quark-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

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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)

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Extremely strong magnetic fields in NSs/Magnetars

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Strong magnetic fields in nature and laboratories Magnet in Lab. Magnetar Heavy ion collisions

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Polarization 1 Polarization 2 Incident light “Calcite” ( 方解石 ) “Birefringence” : Polarization-dependent refractive indices. Response of electrons to incident lights Anisotropic responses of electrons result in polarization-dependent and anisotropic photon spectra. Photon propagations in substances

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+ 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

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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.

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Break-down of naïve perturbation in strong B-fields Naïve perturbation breaks down when B > B c Need to take into account all-order diagrams Critical field strength B c = m e 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

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Schwinger, Adler, Shabad, Urrutia, Tsai and Eber, Dittrich and Gies Exponentiated trig-functions generate strongly oscillating behavior with arbitrarily high frequency. Integrands having strong oscillations Photon propagation in a constant external magnetic field Lorentz and gauge symmetries lead to a tensor structure, θ: angle btw B-field and photon propagation B

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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 Lagrangian Soft photon limit

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Analytic result of integrals - An infinite number of the Landau levels Polarization tensor acquires an imaginary part above A double infinite sum KH, K. Itakura (I) (Photon momentum) Narrowly spaced Landau levels Lowest Landau level

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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)

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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 = , water n = ≈ Magnetar << UrHIC

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Angle dependence of the refractive index Real part No imaginary part Imaginary part

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“Mean-free-path” of photons in B-fields λ (fm)

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Prospects & Discussions

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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 spectrum induced by the Landau levels in strong B-fields.

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Neutral pions in strong magnetic fields Hattori, Itakura, Ozaki

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Violation of axial current conservation Absence of radiative correction Adler & Bardeen, 1969 Triangle diagram gives the exact result in the all-order perturbation theory Adler, Bell, Jackiw, 1969 Dominant ( % in the vacuum) % ``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

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Effects of external magnetic fields Decay mode possible only in external field “Bee decay” can be comparable to Dalitz decay and even π 0 2γ, depending on B. Replacement of a photon line by an external field Decay width of “Bee decay” WZW effective vertex π0π0 γ

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Dalitz decay Bee decay Decay widths Mean lifetime femtometer Branching ratios

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Charmonium spectroscopy in strong magnetic fields by QCD sum rules S.Cho, Hattori, S.H.Lee, Morita, Ozaki

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Meson spectra in B-fields Chernodub Hidaka, A.Yamamoto Chiral condensate in magnetic field from lattice QCD Landau levels

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Mass modifications in the 2 nd order perturbation theory Mixing in wave functions Equation of motions Level repulsion

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Dispersion relations Current correlators QCD sum rules

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Direct couplings 2 nd -order perturbation

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Charmonium spectra from QCD sum rules

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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

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Deng and Huang, PRC85 (2012) Bzdak and Skokov, PLB710 (2012) Impact parameter dependence of B-fields

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Voronyuk et al., PRC83 (2011) Time dependence of B-fields

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Voronyuk et al., PRC83 (2011) Beam-energy dependence of B-fields

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Fourier components of time-dependent B-fields b = 10 fm

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Effective coupling between π 0 and 2γ (Rest frame)

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Neutral pion decay into dilepton B ext = (0,0,B), E ext = 0 EM current

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q q Neutral pion decay into dilepton (continued)

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Decay rates in three modesMean lifetime Energy dependence of the decay rates

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Field-strength dependence of the branching ratio Angle dependence of the branching ratioAngle dependence of the lifetime

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Discussion 1 B ~ 10 2 ×B c

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