1. QCD Workshop on Chirality, Vorticity and Magnetic Field in Heavy Ion Collisions @UCLA, Jan. 21-23, 2015 Photon Propagation in Magnetic Fields in Strong Magnetic Fields KH, K. Itakura (KEK), Ann. Phys. 330 (2013)Ann. Phys. 330 (2013); Ann. Phys. 334 (2013)Ann. Phys. 334 (2013) Koichi Hattori RIKEN-BNL Research Center

2. 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 Hidaka and A.Yamamoto Quark and gluon condensates at zero and finite temperatures Bali et al. Results from lattice QCD in magnetic fields

3. Extremely strong magnetic fields UrHIC NS/Magnetar Lienard-Wiechert potential Z = 79(Au), 82(Pb) Lighthouse in the sky PSR0329+54 c.f.) Possible mechanism of strong B from “chiral instability”: the conversion of μ 5 generated in electron captures. Ohnishi and Yamamoto, Akamatsu and Yamamoto Event-by-event analysis, Deng, Huang (2012) Au-Au 200AGeV b=10fm

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

5. 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

6. + 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

7. 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. eB eBeB

8. 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

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

10. 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) B r =B/B c Br-dependence of the coefficients in soft-photon limit: Comparison btw limiting behavior and numerical computation. ? Untouched so far General analytic expression Euler-Heisenberg Lagrangian In soft photon limit

11. 2 nd step: Getting Laguerre polynomials Associated Laguerre polynomial Decomposing exponential factors Linear w.r.t. τ in exp. 1 st step: “Partial wave decomposition” Linear w.r.t. τ in exp.

12. By the decomposition of the integrand, any term reduces to either of three elementary integrals. Transverse dynamics: Wave functions for the Landau levels given by the associated Laguerre polynomials

13. UrHIC Prompt photon ~ GeV 2 Thermal photon ~ 300 2 MeV 2 ~ 10 5 MeV 2 Untouched so far Strong field limit (LLL approx.) (Tsai and Eber, Shabad, Fukushima ) Soft photon & weak field limit (Adler) Numerical integration (Kohri, Yamada) External photon momentum Analytic result of integrals - An infinite number of the Landau levels A double infinite sum KH, K.Itakura (I) (Photon momentum) Narrowly spaced Landau levels Lowest Landau level ⇔ Polarization tensor acquires an imaginary part when Lowest Landau level Narrowly spaced Landau levels

14. 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)

15. 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

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

17. “Mean-free-path” of photons in B-fields λ (fm) When the refractive index has an imaginary part,

18. Summary for Photon propagation + We obtained an analytic form of the polarization tensor in magnetic fields as the summation w.r.t. the Landau levels. + We obtained the complex refractive indices (photon dispersions) by solving the modified Maxwell Eq. self-consistently.  Photons decay within a microscopic spatial scale. Neutron stars = Pulsars  Possibly “QED cascade” in strong B-fields What is the mechanism of radiation? We got precise descriptions of vertices: Dependences on magnitudes of B-fields, photon energy, propagation angle and polarizations. Prospects: Microscopic mechanism of the pulsation “Photon Splitting” Softening of photons

20. Analytical modeling of colliding nuclei, Kharzeev, McLerran, Warringa, NPA (2008) Pre- equilibrium QGP Lienard-Wiechert potential Strong magnetic fields induced by UrHIC Event-by-event analysis, Deng, Huang (2012) Au-Au 200AGeV b=10fm

21. Kohri and Yamada To make Jordan’s lemma work well, we want a simpler expression: Dimesionless variables Schwinger, Adler, Shabad, Urrutia, Tsai and Eber, Dittrich and Gies 1 st step: Use a series expansion known as partial wave decomposition Baier and Katkov Exponentiated trig-functions generate strongly oscillating behavior by arbitrarily high frequency.

22. After 1 st step: 2 nd step: Associated Laguerre polynomial Then, any term reduces to either of three elementary integrals.

23. Kohri and Yamada To make Jordan’s lemma work well, we want a simpler expression: 1 st step: Use a series expansion known as partial wave decomposition Baier and Katkov 2 nd step: Associated Laguerre polynomial Then, any term reduces to either of three elementary integrals. Every term results in either of three simple integrals.

24. Close look at the integrals What dynamics is encoded in the scalar functions ? An imaginary part representing a real photon decay ⇔ ⇔ Invariant mass of a fermion-pair in the Landau levels

25. Dimesionless variables 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)

26. Renormalization + = ・・・ + + Log divergence Term-by-term subtraction Ishikawa, Kimura, Shigaki, Tsuji (2013) Taken from Ishikawa, et al. (2013) Finite Re Im

27. Br = (50,100,500,1000,5000,10000, 50000) Real part of n on stable branch Imaginary part of n on unstable branch Real part of n on unstable branch Relation btw real and imaginary parts on unstable branch

28. Br = (50,100,500,1000,5000,10000, 50000) Real part of ε on stable branch Imaginary part of ε on unstable branch Real part of ε on unstable branch Relation btw real and imaginary parts on unstable branch