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iTHES Mini-workshop on "Strong-Field Physics" May 29, 2014@RIKEN
Koichi Hattori, Tetsuo Hatsuda (RIKEN)
Photon propagation in strong magnetic fields
0. Introduction to Workshop
1. Strong B-fields in heavy-ion collisions and neutron stars2. Analytic calculation of “vacuum birefringence” → Tomaru3. Discussions and Prospects4. Summary
Plan of this talk
The first seminal work in “nonlinear QED”“Consequences of Dirac’s Theory of the Positron”
W. Heisenberg and H. Euler in Leipzig122. December 1935
Euler – Heisenberg effective Lagrangian - resummation wrt the number of external legs
Correct manipulation of a UV divergence in 1935!
Pair creation (vacuum instability) induced by strong electric field
General formula within 1-loop & constant fieldobtained by the “proper-time method”.
NIF: National Ignition Facility, LivermoreELI: Extreme Light Infrastructure, Czech Republic, Hungary and RomaniaGekko-Exa, HiPER,,,
Tomaru (Experiment), Moritaka (Theory), Takabe (Theory)
Crab pulsar
After late 1960’s
Tamagawa (Observation), Barkov (Theory), Ebisuzaki (Theory), Hattori
After late 1950’s
RHIC@BNL LHC@CERN
Developments of intense laser fields
Neutron stars, GRB, Black holes, Magnetars,,,
After 2000Ultrarelativistic heavy-ion collisions
Hattori (Theory)
Experiment
Phenomenology
Observation
Theory
Motivation of the workshop
6 talks + Lunch + Coffee break + Free-discussion time
11:00-11:45 K. Hattori (RIKEN)Photon propagations in strong magnetic fields
11:45-12:30 T. Tomaru (KEK)Vacuum Birefringence and Axion measurement by laser interferometer
1:45-2:30 T. Tamagawa (RIKEN)X-ray polarimetry satellite GEMS and beyond
2:30-3:15 M. Barkov (RIKEN)Close binary progenitors of gamma-ray bursts and hypernovae
3:35-4:20 T. Moritaka and H. Takabe (ILE, Osaka University)Gamma Ray Emission and Induced Vacuum Breakdown with High-Intensity Pulse Laser
4:20-5:05 T. Ebisuzaki (RIKEN)Astrophysical ZeV acceleration along the jets of an accreting blackhole
5:05-Free discussions with coffee
Time table
Photon propagation in strong magnetic fields
(I) KH, K. Itakura, Annals Phys. 330 (2013) 23-54 (II) KH, K. Itakura, Annals Phys. 334 (2013) 58-82
RHIC@BNL
LHC@CERN
Phase diagram of QCD matter
Asymptotic freedomQuark-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 by lattice QCD Hidaka and Yamamoto
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)
Extremely strong magnetic fields in NSs/Magnetars
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.
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 having strong oscillations
Photon propagation in a constant external magnetic fieldLorentz and gauge symmetries lead to a tensor structure,
θ: angle btw B-field and photon propagation
B
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
EH LagrangianSoft photon limit
Analytic result of integrals- An infinite number of the Landau levels
Polarization tensor acquires an imaginary part above
A double infinite sumKH, K. Itakura (I)
(Photon momentum) Narrowly spaced Landau levels
Lowest Landau level
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)
Prospects & Discussions
Summary
1. We performed analytic calculation of the vacuum polarization tensor in constant magnetic fields.
2. We obtained precise behaviors of the refractive index in LLL. Magnitudes of B-fields, Photon energy, Propagation angle, polarization
3. We discussed possible applications to UrHIC and Neutron Stars/Magnetars.
We showed anisotropic and polarization-dependent photon spectruminduced by the Landau levels in strong B-fields.
Neutral pions in strong magnetic fields
Hattori, Itakura, Ozaki
Violation of axial current conservation
Absence of radiative correction Adler & Bardeen, 1969Triangle diagram gives the exact result in the all-order perturbation theory
Adler, Bell, Jackiw, 1969
Dominant (98.798 % in the vacuum)
99.996 %
``Dalitz decay ‘’ (1.198 % in the vacuum)
NLO contribution to the total decay rate
Only corrections to external legs are possible
LO contribution to the total decay rate
Effects of external magnetic fields
Decay mode possible only in external field
“Bee decay” can be comparable to Dalitz decay and even π0 2γ, depending on B.
Replacement of a photon line by an external field
Decay width of “Bee decay”
WZW effective vertexπ0 γ
Dalitz decay
Bee decay
Decay widths
Mean lifetime
femtometer
Branching ratios
Charmonium spectroscopy in strong magnetic fields by QCD sum rules
S.Cho, Hattori, S.H.Lee, Morita, Ozaki
Meson spectra in B-fields
Chernodub
Hidaka, A.Yamamoto
Chiral condensate in magnetic fieldfrom lattice QCD
Landau levels
Mass modifications in the 2nd order perturbation theory
Mixing in wave functions
Equation of motions
Level repulsion
Dispersion relations
Current correlators
QCD sum rules
+
+ +
+ 2
Direct couplings
2nd-order perturbation
Charmonium spectra from QCD sum rules
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
Effective coupling between π0 and 2γ
(Rest frame)
Neutral pion decay into dilepton
Bext = (0,0,B), Eext = 0
EM current
q q
Neutral pion decay into dilepton (continued)
Decay rates in three modes Mean lifetime
Energy dependence of the decay rates
Field-strength dependence of the branching ratio
Angle dependence of the branching ratio Angle dependence of the lifetime
Discussion 1
B ~ 102×Bc
Magnetar: eB <<