Photon propagations and charmonium spectroscopy 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. Phase diagram of QCD matter.
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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
Phase diagram of QCD matter
Quark-gluon plasma
Results from lattice QCD in magnetic fields
Asymptotic freedom
Light-meson spectra in B-fields
Hidaka and A.Yamamoto
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.
Extremely strong magnetic fields
NS/Magnetar
UrHIC
Lighthouse in the sky
Lienard-Wiechertpotential
PSR0329+54
Z = 79(Au), 82(Pb)
Strong magnetic fieldsin nature and laboratories
Magnet in Lab.
Magnetar
Heavy ion collisions
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)
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
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
Break-down of naïve perturbation in strong B-fields
Dressed fermion propagator in Furry’s picture
Critical field strength
Bc = me2 / e
Resummation w.r.t. external legs by “proper-time method“
Schwinger
Nonlinear to strong external fields
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.
Summary of relevant scales
and preceding calculations
General analytic expression
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
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
Complex refractive indices
KH, K. Itakura (II)
The Lowest Landau Level (ℓ=n=0)
Refractive indices at the LLL
Polarization excites only along the magnetic field
``Vacuum birefringence’’
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
Angle dependence of the refractive index
Real part
Imaginary part
No imaginary part
“Mean-free-path” of photons in B-fields
λ (fm)
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.
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
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.
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
QCD sum rules
Current correlators
?
Operator product expansions (OPE)
and dispersion relations
Spectral function:
Shifman, Vainshtein, Zakharov
Conventional spectral ansatz: “pole + continuum”
QCD sum rules work well for theisolatedlowest states.
Dispersion relation is insensitive to detail structures of the continuum.
Boreltransformation
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
+
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
ηc and longitudinal J/psi spectra from QCD sum rules
D and D* mesons in B-fields
P.Gubler, KH, S.H.Lee, S.Ozaki, K.Suzuki, In progress.
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
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 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)
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
Impact parameter dependence of B-fields
Bzdak and Skokov, PLB710 (2012)
Deng and Huang, PRC85 (2012)
Time dependence of B-fields
Voronyuk et al., PRC83 (2011)
Beam-energy dependence of B-fields
Voronyuk et al., PRC83 (2011)
Fourier components of time-dependent B-fields
b = 10 fm
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
Renormalization
=
+
+
+
・・・
Log divergence
Finite
Subtraction term-by-term
Ishikawa, Kimura, Shigaki, Tsuji (2013)
Im
Re
Taken from Ishikawa, et al. (2013)
Mass formula in “pole+continuum” ansatz
Spectral ansatz:
Borel transform
Borel-transformed dispersion relation: