1. 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

2. 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

3. 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

4. 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

5. 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)

6. 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

7. 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.

8. 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

9. 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

10. 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

11. 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.

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

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

14. 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

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

16. 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

17. Complex refractive index from the lowest-Landau-level

18. 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

19. Angle dependence of the refractive index

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

21. 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

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

23. 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.

24. 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

25. Pion reactions in strong magnetic fields is coming soon.

27. 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 ?

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

29. 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

30. 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

31. 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

32. 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 :

33. 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

34. 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