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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 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 RHIC@BNL LHC@CERN 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) 23-54 (II) KH, K. Itakura, Annals Phys. 334 (2013) 58-82

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RHIC@BNL LHC@CERN 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 = 1.0003, water n = 1.333 ≈ 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 (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

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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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+ ++ + 2 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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