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

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

PSR Extremely strong magnetic fields UrHIC NS/Magnetar Lienard-Wiechert potential Z = 79(Au), 82(Pb) Lighthouse in the sky

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

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

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

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

Break-down of naïve perturbation in strong B-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 in Furry’s picture Resummation w.r.t. external legs by “proper-time method“Schwinger Nonlinear to strong external fields

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 Photon propagation in a constant external magnetic field Gauge symmetry leads to three tensor structures, θ: angle btw B-field and photon propagation B Vanishing B limit:

Summary of relevant scales and preceding calculations 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) B r =B/B c Br-dependence of the coefficients in soft-photon limit: Comparison btw limiting behavior and numerical computation. ? Untouched so far General analytic expression Euler-Heisenberg Lagrangian In soft photon limit

UrHIC Prompt photon ~ GeV 2 Thermal photon ~ 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) (Photon momentum) Analytic result of integrals - An infinite number of the Landau levels A double infinite sum KH, K.Itakura (I) (Photon momentum) Narrowly spaced Landau levels Lowest Landau level Polarization tensor acquires an imaginary part above

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

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 = , water n = ≈ Magnetar << UrHIC

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

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

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

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

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

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

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

Conventional spectral ansatz: “pole + continuum” Borel transformation QCD sum rules work well for the isolated lowest states. Dispersion relation is insensitive to detail structures of the continuum.

+ Direct couplings with Bethe-Salpeter amplitudes in HQ limit nd -order perturbation Spectral ansatz with mixing effects + Bohr radius a 0 = 0.16 fm in Coulombic wave function

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

η c and longitudinal J/psi spectra from QCD sum rules

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

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.

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

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

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

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

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

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

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

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

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