Effects of strong magnetic fields Koichi Hattori Yonsei

Effects of strong magnetic fields Koichi Hattori (Yonsei Univ. ) 1. Photon propagation in strong magnetic field: “Vacuum birefringence”, KH and K. Itakura, Ann. Phys. 330 (2013) 23 -54. KH and K. Itakura, ar. Xiv: 1212. 1897 [hep-ph]. 2. Pion reactions in strong magnetic field, KH, K. Itakura, S. Ozaki, in preparation Recent progress in hadron physics -From hadrons to quark and gluon- 2013 Feb. 18 -22, Yonsei University

What is “Birefringence (複屈折)” ? Two polarization modes of a propagating photon have different refractive indices. Doubled image by a ray splitting in birefringent material How is in the vacuum with external magnetic Polarization field ? 1 Polarization 2 + Lorentz & Gauge symmetries n ≠ 1 in general + Oriented response of the Dirac sea Incident light. Vacuum birefringence “Calcite” (方解石) Table of contents of 1 st part + Strong magnetic fields in heavy-ion collisions + Our analytic calculation of the photon vacuum polarization tensor Refractive indices (“Vacuum birefringence”) + Some features of the obtained refractive index

Extremely strong magnetic fields in peripheral collisions induced by strongly accelerated heavy nuclei v. N > 0. 9999 c Z = 79 (Au), 82 (Pb) Geometry in peripheral collisions Lienard-Wiechert potential Superposition of circulating magnetic fields in the transverse plane t = 0. 1 fm/c 0. 5 fm/c 1 fm/c 2 fm/c

Strong magnetic fields in nature and laboratories Magnet in Lab. Magnetar Heavy ion collisions From Itakura-san’s talk in “International conference on physics in intense field 2010 @ KEK” 2 nd PIF, 9 -11, July, 2013, @DESY

Time evolution of the magnetic field after collisions Simple estimates by Lienard-Wiechert potential Analytical modeling of colliding nuclei, Kharzeev, Mc. Lerran, Warringa, NPA (2008) Preequilibrium QGP Still a few orders stronger than the “critical field”

Time evolution of the matter “QGP” after the collision Incidence Collision QGP Hadronization Hadron gas Lorentz construction Strong magnetic field EM probe Hadronic probe Preequilibrium QGP

Modifications of photon propagations by nonlinear QED effects EM probes would carry away info of initial stage. Magnetic field QGP Photon splitting: γ+B 2γ, avoiding Furry’s theorem Refraction of photon, without Lorentz symmetry Dilepton emission from real photon decay, as well as virtual photon γ* e+e- Modified Maxwell eq. : Photon vacuum polarization tensor with the dressed fermion propagators:

Break-down of naïve perturbation in strong magnetic fields Dressed fermion propagator Critical field strength Bc = me 2 / e Naïve perturbation breaks down when B > Bc In heavy ion collisions, B/Bc ~ O(104) >> 1 Need to take into account all-order diagrams Resummation w. r. t external legs by “proper-time method “ Nonlinear to the external field Schwinger

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 Eigen-equations from the modified Maxwell eq. “Vacuum birefringence” Following from the tensor structure, we obtain distinct eigenmodes!! with eigenvectors, Melrose and Stoneham

Scalar coefficient functions χ by the proper-time method Given by a double integral wrt proper time variables associated with two fermion lines Schwinger, Adler, Shabad, Urrutia, Tsai and Eber, Dittrich and Gies Analytic integration without any approximation Decomposition into a double infinite sum Polarization tensor has an imaginary part above

Summary of relevant scales and available calculations for χ’s Ur. HIC (Photon momentum) Prompt photon ~ Ge. V 2 Thermal photon ~ 3002 Me. V 2 ~ 105 Me. V 2 Untouched so far Strong field limit (LLL approx. ) (Tsai and Eber, Shabad, Fukushima ) Soft photon & weak field limit (Adler) Numerical integration (Kohri, Yamada) With a great contribution of Ishikawa-san (Hiroshima Univ. ), complete photon propagator in strong B-field is now available.

Dielectric constant / refractive index by self-consistent treatment Close look at the lowest Landau level

Vacuum polarization tensor from the LLL Comparison with numerical result Kohri & Yamada LLL approx. agrees with numerical result in strong field limit Dielectric constant at the LLL Polarization excites only along the magnetic field

Modified photon dispersion relation from the vacuum polarization tensor Modified Maxwell eq. : Dispersion relation of q

Consistent solution wrt the real and imag. parts Air （1 atm, 0℃）：n=1. 0003 Water (20℃)： n=1. 333 Calcite: no=1. 6584, ne=1. 4864

Complex refractive index from the lowest-Landau-level

Dependence on magnetic-field strength Re[n] on stable branch Im[n] on unstable branch Relation btw. real and imaginary parts on unstable branch Br = (50, 100, 500, 1000, 5000, 10000, 50000)

Angle dependence of the refractive index

Photon decay length in strong magnetic field Propagating EM field: Intensity of the field: Photon decay length:

Angle dependence of the refractive index Shown as a deviation from unit circle Magnetic field Direction of arrow : direction of photon propagation Norm of arrow : magnitude of the refraction index

Angle dependence at various photon energies Real part Imaginary part No imaginary part

Implications in heavy-ion collisions Effects on anisotropic flow Decay of real photon induces negative contribution to photon’s v 2 Decay of real photon induces positive contribution to dilepton’s v 2 Also, there would be effects on the higher harmonics Effects on photon HBT interferometry Modified refractive index induces a distorted HBT image K. Itakura and KH (2011) Strong B Magnetic field induced photon/gluon emissions Synchrotron radiation of photon/gluon from quark QGP photons quark K. Tuchin, PRC (2010), [hep-ph/1209. 0799] (2012) gluons Needs systematic studies in HIC.

Summary of 1 st part - We analytically evaluated the photon vacuum polarization tensor in external magnetic fields. - We inspected the complex dielectric constant/refraction index around the lowest-Landau-level threshold with a self-consistent treatment. Prospects - Application to EM probe in heavy-ion collisions and magnetar. Polarization dependence, Energy and angle dependences… Competition/interplay between birefringence and splitting

Pion reactions in strong magnetic fields is coming soon.

Scalar coefficient functions χ in the proper-time method Given by a double integral wrt proper time variables associated with two fermion lines Dimesionless variables Integrands having strong oscillation Schwinger, Adler, Shabad, Urrutia, Tsai and Eber, Dittrich and Gies Exponentiated trig-functions generate strongly oscillating behavior with arbitrarily high frequency.

Analytic calculation of the double integral Two important relations ★ ★ Associated Laguerre polynomial Any term reduces to either of elementary integrals.

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 Br = (50, 100, 500, 1000, 5000, 10000, 50000)

An extremely strong magnetic field in Ultrarelativistic Heavy-Ion Collision Geometry of the peripheral collision and strong B-field Extremely strong magnetic field in the direction of the out-of-reaction plane Lorentz - contracted nuclei b Finite impact parameter Primordial and thermalized matter

Direct photon from initial stage in Ur. HIC Initial and background photons Various hadron emissions & photon from hadron decay, π0 2γ Time Direct photon spectrum Low Pt regime: dominant emission in hadron phase Intermediate and hard regime : emission from primordial matter ? Thermalization Interaction with B-field Space (z) Directly accessible to the initial stage and QGP Detecting photon from the initial stage ⇔ Detecting effects of B-field

Schematic picture of the birefringence Polarization in dielectric medium : a classical argument Lorentz-type dispersion : Anisotropic constants result in an anisotropic response. Incident light What happens with the anisotropic (discretized) spectrum by the Landau-levels ? Incident light field Dissipation Linear bound force

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

Analytic results! Applicable to any momentum regime and field strength ! Applicable to both on-shell and off-shell photon! Sum wrt Landau levels Combination of known functions Photon decay channel opens at every Landau level