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Recent progress in hadron physics -From hadrons to quark and gluon- 2013 Feb. 18-22, Yonsei University 1. Photon propagation in strong magnetic field: “Vacuum birefringence”, KH and K. Itakura, Ann. Phys. 330 (2013) 23-54. KH and K. Itakura, arXiv: 1212.1897 [hep-ph]. Koichi Hattori (Yonsei Univ.) Effects of strong magnetic fields 2. Pion reactions in strong magnetic field, KH, K. Itakura, S. Ozaki, in preparation

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Ordinary vacuum w/o external field: Speed of light remains “c” after the renormalization, owing to Lorentz & gauge invariance. In a strong magnetic field Effect 1. External magnetic field breaks Lorentz symmetry. Refraction index can deviate from unity. Effect 2. External magnetic field provides a preferred orientation (optical axis). It induces a pair of polarization-dependent refraction indices. ⇒ “Vacuum birefringence ( 複屈折 )” What is “Birefringence ( 複屈折 )” ? Doubled image by a ray splitting in birefringent material Polarization 1 Polarization 2 Incident light “Calcite” ( 方解石 ) Two polarization modes of a propagating photon have different refractive indices. + 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 Table of contents of 1 st part How is in the vacuum with external magnetic field ? + Lorentz & Gauge symmetries n ≠ 1 in general + Oriented response of the Dirac sea Vacuum birefringence

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induced by strongly accelerated heavy nuclei Extremely strong magnetic fields in peripheral collisions v N > 0.9999 cZ = 79 (Au), 82 (Pb) t = 0.1 fm/c 0.5 fm/c1 fm/c2 fm/c Superposition of circulating magnetic fields in the transverse plane Geometry in peripheral collisions Lienard-Wiechert potential

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

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Analytical modeling of colliding nuclei, Kharzeev, McLerran, Warringa, NPA (2008) Pre- equilibrium QGP Time evolution of the magnetic field after collisions E cm = 200 GeV (RHIC) Z = 79 (Au), b = 6 fm Simple estimates by Lienard-Wiechert potential t = 0.1 fm/c 0.5 fm/c1 fm/c2 fm/c Space (z) Time Still a few orders stronger than the “critical field” Event-by-event analysis by HIJING, Deng, Huang, PRC (2012) Au-Au 200AGeV, b=10fm Event-by-event analysis by UrQMD, Skokov, Illarionov, Toneev, (2009) Event-by-event analysis Bzdak, Skokov, PLB (2012)

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Time evolution of the matter “QGP” after the collision Strong magnetic field EM probe Hadronic probe QGP HadronizationHadron gas Pre- equilibrium QGP CollisionIncidence Lorentz construction

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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 - Modifications of photon propagations by nonlinear QED effects Photon vacuum polarization tensor with the dressed fermion propagators: Modified Maxwell eq. : EM probes would carry away info of initial stage.

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Break-down of naïve perturbation in strong magnetic 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 Employing Fock-Schwinger gauge x μ A μ = 0, In heavy ion collisions, B/B c ~ O(10 4 ) >> 1 Resummation w.r.t external legs by “proper-time method “ Schwinger Nonlinear to the external field

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Photon propagation in a constant external magnetic field Lorentz and gauge symmetries lead to a tensor structure, B θ: angle btw B-field and photon propagation Eigen-equations from the modified Maxwell eq. Following from the tensor structure, we obtain distinct eigenmodes!! “Vacuum birefringence” with eigenvectors, Melrose and Stoneham

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Scalar coefficient functions χ by the proper-time method Schwinger, Adler, Shabad, Urrutia, Tsai and Eber, Dittrich and Gies Given by a double integral wrt proper time variables associated with two fermion lines Dimesionless variables Exponentiated trig-functions generate strongly oscillating behavior with arbitrarily high frequency. Integrands having strong oscillation Vacuum birefringence has been led by the tensor structure. What dynamics is encoded in the scalar functions, χ i ? Decomposition into a double infinite sum Analytic integration without any approximation Polarization tensor has an imaginary part above

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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) Summary of relevant scales and available calculations for χ’s (Photon momentum) With a great contribution of Ishikawa-san (Hiroshima Univ.), complete photon propagator in strong B-field is now available.

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

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

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Vacuum polarization tensor from the LLL Limiting behavior of dielectric constant Dielectric constant at the LLL Polarization excites only along the magnetic field ArcTan : source of an imaginary part above the lowest threshold Comparison with numerical result LLL approx. agrees with numerical result in strong field limit Kohri & Yamada

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

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Consistent solution wrt the real and imag. parts Air （ 1atm, 0 ℃）： n=1.0003 Water (20 ℃ ) ： n=1.333 Calcite: n o =1.6584, n e =1.4864

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Complex refractive index from the lowest-Landau-level

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

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Angle dependence of the refractive index

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

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

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

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Implications in heavy-ion collisions Strong B QGP quark gluons photons Synchrotron radiation of photon/gluon from quark K.Tuchin, PRC (2010), [hep-ph/1209.0799] (2012) Magnetic field induced photon/gluon emissions Effects on photon HBT interferometry Modified refractive index induces a distorted HBT image K.Itakura and KH (2011) 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 anisotropic flow Needs systematic studies in HIC.

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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

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Pion reactions in strong magnetic fields is coming soon.

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Scalar coefficient functions χ in the proper-time method Schwinger, Adler, Shabad, Urrutia, Tsai and Eber, Dittrich and Gies Given by a double integral wrt proper time variables associated with two fermion lines Dimesionless variables Exponentiated trig-functions generate strongly oscillating behavior with arbitrarily high frequency. Integrands having strong oscillation Vacuum birefringence has been led by the tensor structure. What dynamics is encoded in the scalar functions, χ i ?

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

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Br = (50,100,500,1000,5000,10000, 50000) Br = (50, 10 2, 5×10 2, 10 3, 5×10 3, 10 4, 5×10 4 ) 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

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

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Direct photon from initial stage in UrHIC Space (z) Time Various hadron emissions & photon from hadron decay, π 0 2γ Intermediate and hard regime : emission from primordial matter Direct photon spectrum Low Pt regime: dominant emission in hadron phase Initial and background photons Directly accessible to the initial stage and QGP Thermalization Interaction with B-field ? Detecting photon from the initial stage ⇔ Detecting effects of B-field

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

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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

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

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