RIKEN-BNL Research Center Nonlinear QED effects on photon and dilepton spectra in supercritical magnetic fields Keywords - Strong magnetic fields - Vacuum birefringence Koichi Hattori RIKEN-BNL Research Center Photon & dilepton WS@ECT*, Dec. 9, 2016 KH, K. Itakura (KEK), Ann. Phys. 330 (2013); Ann. Phys. 334 (2013).

What is birefringence? Birefringence: Anisotropic responses of electrons result in polarization-dependent and anisotropic photon spectra. Response of electrons to incident lights Structured ions  Anisotropic spring constants Birefringence: polarization-dependent refractive indices Polarization 1 Polarization 2 Incident light Birefringent substance “Calcite” (方解石) Lesson: The spectrum of fermion fluctuation is important for the photon spectrum.

Photon propagation in magnetic fields + Lorentz & Gauge symmetries  n ≠ 1 in general Landau levels + Discretized transverse momentum + Still continuum in the direction of B B + Anisotropic response from the Dirac sea ``Vacuum birefringence” Real part: “Vacuum birefringence” Imag. part: “Real photon decay” into fermion pairs “Photon splitting”  Forbidden in the ordinary vacuum because of the charge conjugation symmetry.

Schematic picture of the strong field limit Wave function (in symmetric gauge) Polarizer Fermions in 1+1 dimension Strong B

Strong magnetic fields in laboratories and nature

Strong magnetic fields in UrHIC Lienard-Wiechert potential Z = 79(Au), 82(Pb) Event-by-event analysis, Deng & Huang (2012) Au-Au 200AGeV b=10fm Supercritical fields beyond electron and quark masses Impact parameter (b) Ecm = 200 GeV (RHIC) Z = 79 (Au), b = 6 fm t = 0.1 fm/c 0.5 fm/c 1 fm/c 2 fm/c

Other strong B fields NS/Magnetar High-intensity laser field NSs Magnetars “Lighthouse” in the sky PSR0329+54

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 RHIC@BNL LHC@CERN

Refractive index of photon in strong B-fields - Old but unsolved problem Much simpler than QCD But, Tough calculation due to a resummation Has not been observed in experiments

Basic framework Large B compensates the suppression by e. Photon vacuum polarization tensor: Modified Maxwell eq. : Dressed propagators in Furry’s picture Quantum effects in magnetic fields ・・・ eB Should be suppressed in the ordinary perturbation theory, but not in strong B-fields. Large B compensates the suppression by e.  Break-down of the naïve perturbation  Needs a resummation

Seminal works for the resummation Consequences of Dirac’s Theory of the Positron W. Heisenberg and H. Euler in Leipzig1 22. December 1935 General formula within 1-loop & constant field obtained by the “proper-time method”. Euler – Heisenberg effective Lagrangian 　- resummation wrt the number of external legs Correct manipulation of a UV divergence in 1935!

Resummation in strong B-fields Dressed fermion propagator in Furry’s picture Critical field strength Bc = me2 / e In heavy ion collisions, B/Bc ~ (mπ/me)2 ~ O(104) >> 1 Naïve perturbation breaks down when B > Bc  Need to take into account all-order diagrams Resummation w.r.t. external legs by “proper-time method“ Schwinger (1951) Nonlinear to strong external fields

1+1 dimensional fluctuation Dispersion relation from the resummation The strong field limit revisited: Lowest Landau level (LLL) approximation (n=0) Spin-projection operator Wave function 1+1 dimensional dispersion relation 1+1 dimensional fluctuation

Resummed vacuum polarization tensor Generalization: Resummed vacuum polarization tensor θ: angle btw B-field and photon propagation B Gauge symmetries lead to a tensor structure, Gauge symmetry leads to three tensor structures, Vanishing B limit:

Generalization: Resummed vacuum polarization tensor θ: angle btw B-field and photon propagation B Gauge symmetries lead to a tensor structure, Vanishing B limit: Gauge symmetry leads to three tensor structures, 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

General analytic expression Summary of relevant scales and preceding calculations General analytic expression 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 Euler-Heisenberg Lagrangian In soft photon limit

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

After 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

⇔ Analytic result of integrals - An infinite number of the Landau levels KH, K.Itakura (I) ⇔ Polarization tensor acquires an imaginary part when (Photon momentum) Narrowly spaced Landau levels Lowest Landau level A double infinite sum UrHIC Prompt photon ~ GeV2 Thermal photon ~ 3002 MeV2 ~ 105 MeV2 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 Narrowly spaced Landau levels Lowest Landau level

Complex refractive indices KH, K. Itakura (II) Solutions of Maxwell eq. with the vacuum polarization tensor B LLL: 1+1 dimensional fluctuation in B Refractive indices at the LLL(ℓ=n=0) Polarization excites only along the magnetic field ``Vacuum birefringence’’

Solutions of the modified Maxwell Eq. Photon dispersion relation is strongly modified when strongly coupled to excitations (cf: exciton-polariton, etc) 𝜔 2 /4 𝑚 2 ≈ Magnetar << UrHIC cf: air n = 1.0003, water n = 1.3, prism n = 1.5 Refraction Image by dileptons

Angle dependence of the refractive index Real part B Below the threshold Above the threshold Imaginary part No imaginary part

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

QM2014, Darmstadt

Summary + We obtained an analytic form of the resummed polarization tensor. + We showed the complex refractive indices (photon dispersions) . -- Polarization dependence -- Angle dependence Prospects: Search of vacuum birefringence in UrHIC & laser fields Microscopic radiation mechanism of neutron stars  Nonlinear QED effects on the surface of NS.

Neutron stars = Pulsars What is the mechanism of radiation?  Possibly “QED cascade” in strong B-fields “Photon Splitting” Softening of photons We got precise descriptions of vertices: Dependences on magnitudes of B-fields, photon energy, propagation angle and polarizations.

The earliest work: Euler-Heisenberg Lagrangian - Low-energy (soft photon) effective theory Quantum corrections in magnetic fields ・・・ eB + Constant magnetic fields + Should be suppressed in the ordinary perturbation theory, but not in strong B-fields. Poincare invariants Photon splitting eB Vacuum birefringence (Refractive indices n≠1) Soft photon limit

Landau levels + Zeeman splitting in the resummed propagator Spin-projection operators The lowest Landau level (n=0) Discretized fermion’s dispersion relation Three terms corresponding to the spin states. (iii) The same transform properties under the C-conjugation as that of a free propagator.

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 𝜔 2 /4 𝑚 2 ≈ Magnetar << UrHIC

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

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

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