Koichi Hattori Lunch seminar @ BNL, Aug. 14 2014

1. Photon propagations and charmoniumspectroscopy in strong magnetic fields KH, K. Itakura, Annals Phys. 330 (2013); 334 (2013) S.Cho, KH, S.H.Lee, K.Morita, S.Ozaki, arXiv:1406.4586 [hep-ph] Koichi Hattori Lunch seminar @ BNL, Aug. 14 2014

2. Phase diagram of QCD matter Quark-gluon plasma Results from lattice QCD in magnetic fields Asymptotic freedom RHIC@BNL Light-meson spectra in B-fields Hidaka and A.Yamamoto LHC@CERN Quark and gluon condensates at zero and finite temperatures Bali et al. Magnetic susceptibility (χ) of QCD matter by lattice QCD. From a talk by G. Endrodi in QM2014.

3. Extremely strong magnetic fields NS/Magnetar UrHIC Lighthouse in the sky Lienard-Wiechertpotential PSR0329+54 Z = 79(Au), 82(Pb)

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

5. Response of electrons to incident lights Polarization 1 Polarization 2 Incident light Photon propagations in substances Anisotropic responses of electrons result in polarization-dependent and anisotropic photon spectra. “Birefringence” : Polarization-dependentrefractive indices. “Calcite” (方解石)

6. How about the vacuum with external magnetic fields ? - The Landau-levels + Lorentz & Gauge symmetries  n ≠ 1 in general + Oriented response of the Dirac sea  Vacuum birefringence B

7. Modifications of photon propagations in strong B-fields - Old but unsolved problems Quantum effects in magnetic fields eB eB eB Should be suppressed in the ordinary perturbation theory, but not in strong B-fields. ・・・ Modified Maxwell eq. : ・・・ Photon vacuum polarization tensor: Dressed propagators in Furry’s picture

8. Break-down of naïve perturbation in strong B-fields Dressed fermion propagator in Furry’s picture Critical field strength Bc = me2 / e • 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 Nonlinear to strong external fields

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

10. Summary of relevant scales and preceding calculations General analytic expression • ? • Untouched so far Numerical computation below the first threshold (Kohri and Yamada) Weak field & soft photon limit (Adler) Strong field limit: the lowest-Landau-level approximation (Tsaiand Eber, Shabad, Fukushima ) Euler-HeisenbergLagrangian In soft photon limit Br-dependence of the coefficients in soft-photon limit: Comparison btw limiting behavior and numerical computation. Br=B/Bc

11. Analytic result of integrals - An infinite number of the Landau levels KH, K.Itakura(I) A double infinite sum UrHIC Prompt photon ~ GeV2 Thermal photon ~ 3002 MeV2 ~ 105 MeV2 Untouched so far Polarization tensor acquires an imaginary part above (Photon momentum) Narrowly spaced Landau levels Strong field limit (LLL approx.) (Tsai and Eber, Shabad, Fukushima ) (Photon momentum) Numerical integration (Kohri, Yamada) Soft photon & weak field limit (Adler) Lowest Landau level

12. Complex refractive indices KH, K. Itakura (II) • 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’’

13. Self-consistent solutions of the modified Maxwell Eq. Photon dispersion relation is strongly modified when strongly coupled to excitations (cf: exciton-polariton, etc) ≈ Magnetar << UrHIC cf: air n = 1.0003, water n = 1.333

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

15. “Mean-free-path” of photons in B-fields λ (fm)

16. Neutron stars = Pulsars What is the mechanism of radiation?  QED cascade in strong B-fields Need to get precise description of vertices: Dependences on magnitudes of B-fields, photon energy, propagation angle and polarizations.

17. Charmonium spectroscopy in strong magnetic fields by QCD sum rules S.Cho, KH, S.H.Lee, Morita, Ozaki

18. Light meson spectra in strong B-fields Landau levels for charged mesons In hadronic degrees Effective masses in the strong-field limit: The Lowest Landau Level ( n = 0 ) Chiral condensate in magnetic field from lattice QCD Chernodub Similar to Nielsen-Olesen instability From lattice QCD Chiral condensate in B-fields from lattice QCD Magnetic catalysis Gusynin, Miransky, Shovkovy Hidaka, A.Yamamoto Bali et al.

19. Mixing btw ηc and J/psi in B-fields Coupling among 1 PS and 2 Vector fields Equation of motions Mixing of wave functions Mixing only with Longitudinal J/psi Mass spectra with level repulsion Longitudinal J/psi ηc

20. QCD sum rules Current correlators ? Operator product expansions (OPE) and dispersion relations Spectral function: Shifman, Vainshtein, Zakharov

21. Conventional spectral ansatz: “pole + continuum” QCD sum rules work well for theisolatedlowest states. Dispersion relation is insensitive to detail structures of the continuum. Boreltransformation

22. Spectral ansatzwith mixing effects 2nd-order perturbation + + + Direct couplings with Bethe-Salpeter amplitudes in HQ limit Bohr radius a0 = 0.16 fmin Coulombic wave function +

23. OPE for charmonium in B-fields + 2 Perturbative part + dim.-4 gluon condensates NB) The resummed vacuum polarization tensor (vector current correlator) can be applied in strong field limit. KH, Itakura

24. ηc and longitudinal J/psi spectra from QCD sum rules

25. D and D* mesons in B-fields P.Gubler, KH, S.H.Lee, S.Ozaki, K.Suzuki, In progress. • c.f.) B and B* by Machado, Finazzo, Matheus, Noronha Landau levels + mixing effects + Landau levels of charged D±, D*± + Mixing effects Landau levels OPE for open flavors + Effects of <qbar q> condensates D± and longitudinal D*± spectra B-dependent condensate u, d cbar

26. Summary We calculated the resummed vacuum polarization tensor (vector current correlator) to get the refractive indices in strong magnetic fields. We obtained charmonium spectra in magnetic fields by QCD sum rules with careful treatment of the phenomenological side as well as OPE.

28. Extremely strong magnetic fields induced byUrHIC r R Impact parameter (b) z + Free streaming relativistic protons + Charge distributions in finite-size nuclei LW potential is obtained by boosting an electro-static potential Lienard-Wiechertpotential Liu, Greiner, Ko Boost Z = 79(Au), 82(Pb)

29. Analytic modeling of B-fields Lienard-Wiechertpotential + Free streaming relativistic protons + Charge distributions in finite-size nuclei LW potential is obtained by boosting an electro-static potential r R z Boost Liu, Greiner, Ko

30. Impact parameter dependence of B-fields Bzdak and Skokov, PLB710 (2012) Deng and Huang, PRC85 (2012)

31. Time dependence of B-fields Voronyuk et al., PRC83 (2011)

32. Beam-energy dependence of B-fields Voronyuk et al., PRC83 (2011)

33. Fourier components of time-dependent B-fields b = 10 fm

34. Analytic results of integrals without any approximation KH, K. Itakura (I) A double infinite sum Dimesionless variables Every term results in either of three simple integrals. Polarization tensor acquires an imaginary part above

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

36. Mass formula in “pole+continuum” ansatz Spectral ansatz: Borel transform Borel-transformed dispersion relation: