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Koichi HattoriLunch seminar @ BNL, Aug. 14 2014
Photon propagations and charmonium spectroscopy in strong magnetic fields
S.Cho, KH, S.H.Lee, K.Morita, S.Ozaki, arXiv:1406.4586 [hep-ph]
KH, K. Itakura, Annals Phys. 330 (2013); 334 (2013)
RHIC@BNL
LHC@CERN
Phase diagram of QCD matter
Asymptotic freedomQuark-gluon plasma
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
PSR0329+54
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 1Polarization 2
Incident light“Calcite” (方解石 )
“Birefringence” : Polarization-dependent refractive indices.
Response of electrons to incident lightsAnisotropic 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.
eBeB eB
Break-down of naïve perturbation in strong B-fields
Naïve perturbation breaks down when B > Bc
Need to take into account all-order diagrams
Critical field strengthBc = me
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 witharbitrarily 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)
?Untouched so far
General analytic expression
Euler-Heisenberg LagrangianIn soft photon limit
Analytic result of integrals- An infinite number of the Landau levels
A double infinite sumKH, 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 = 1.0003, water n = 1.333
𝜔2/4𝑚2
≈ Magnetar << UrHIC
Angle dependence of the refractive indexReal 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
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 ηc and J/psi in B-fields
Equation of motions
Mass spectra with level repulsion
Coupling among 1 PS and 2 Vector fields
Longitudinal J/psi
ηc
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
+ +
2nd-order perturbation
Spectral ansatz with mixing effects
+
Bohr radius a0 = 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
ηc and longitudinal J/psi spectra from QCD sum rules
B-dependent condensate
D and D* mesons in B-fieldsP.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 <qbar q> condensates D± and longitudinal D*± spectra
Landau levels
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
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
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