4 th athic , 15 th nov., 2012
DESCRIPTION
Birefringent photon spectrum from a nonlinear interaction with strong magnetic field. Koichi Hattori ( Yonsei Univ.) in collaboration with Kazunori Itakura (KEK). 4 th ATHIC , 15 th Nov., 2012. KH and K. Itakura , [ hep-ph /1209.2663]. - PowerPoint PPT PresentationTRANSCRIPT
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4th ATHIC, 15th Nov., 2012
KH and K. Itakura, [hep-ph/1209.2663].“Vacuum birefringence in strong magnetic fields: (I) Photon polarization tensor with all the Landau levels’’
KH and K. Itakura, in preparation.“Vacuum birefringence in strong magnetic fields: (II) Complex refractive index in the lowest Landau level approximation’’
Koichi Hattori (Yonsei Univ.)in collaboration with Kazunori Itakura (KEK)
Birefringent photon spectrum from a nonlinear interaction with strong magnetic field
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What is “Birefringence” ?
Doubled image by a ray splitting in birefringent material
Polarization 1Polarization 2
Incident light“Calcite” ( 方解石 )
Two polarization modes of a propagating photon have different refractive indices.
+ Strong magnetic fields in heavy-ion collisions
+ Our analytic calculation of the photon vacuum polarization tensor Refractive indices (“Vacuum birefringence”)
+ Some features of the obtained refractive index
Table of contents
How is in the vacuum with external magnetic field ?
+ Lorentz & Gauge symmetries n ≠ 1 in general
+ Oriented response of the Dirac sea Vacuum birefringence
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induced by strongly accelerated heavy nucleiExtremely strong magnetic fields in peripheral collisions
vN > 0.9999 c Z = 79 (Au), 82 (Pb)
t = 0.1 fm/c 0.5 fm/c 1 fm/c 2 fm/c
Superposition of circulating magnetic fields in the transverse plane
Geometry in peripheral collisions
Lienard-Wiechert potential
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From Itakura-san’s talk in International conference on physics in intense field 2010 @ KEK
Strong magnetic fields in nature and laboratories
Magnet in Lab.
Magnetar
Heavy ion collisions
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Analytical modeling of colliding nuclei, Kharzeev, McLerran, Warringa, NPA (2008)
Pre-equilibrium
QGP
Time evolution of the magnetic field after collisionsSimple estimates by Lienard-Wiechert potential
Still a few orders stronger than the “critical field”
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Magnetic field
QGP
Photon splitting: γ+B 2γ, avoiding Furry’s theorem
Refraction of photon, without Lorentz symmetry
Dilepton emission from real photon decay, as well as virtual photon γ* e+e-
Modifications of photon propagations by nonlinear QED effects
Photon vacuum polarization tensor with the dressed fermion propagators:
Modified Maxwell eq. :
EM probes would carry away info of initial stage.
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Break-down of naïve perturbation in strong magnetic 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 heavy ion collisions, B/Bc ~ O(104) >> 1
Resummation w.r.t external legs by “proper-time method “Schwinger
Nonlinear to the external field
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Photon propagation in a constant external magnetic fieldLorentz and gauge symmetries lead to a tensor structure,
Bθ: angle btw B-field and photon propagation
Eigen-equations from the modified Maxwell eq.
Following from the tensor structure, we obtain distinct eigenmodes!!“Vacuum birefringence”
with eigenvectors, Melrose and Stoneham
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Scalar coefficient functions χ by the proper-time method
Schwinger, Adler, Shabad, Urrutia, Tsai and Eber, Dittrich and Gies
Given by a double integral wrt proper time variablesassociated with two fermion lines
Decomposition into a double infinite sum
Analytic integration without any approximation
Polarization tensor has an imaginary part above
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UrHIC
Prompt photon ~ GeV2
Thermal photon ~ 3002 MeV2 ~ 105 MeV2
Untouched so far
Strong field limit (LLL approx.)(Tsai and Eber, Shabad, Fukushima )
Soft photon & weak field limit(Adler)
Numerical integration(Kohri, Yamada)
Summary of relevant scales and available calculations for χ’s
(Photon momentum)
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Complex dielectric constants from the vacuum polarization tensor
and the complex refractive indeices
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Dielectric constant at the lowest-Landau-level
Dielectric constant at the LLL Polarization excites only along the magnetic field
ArcTan : source of an imaginary part above the lowest threshold
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Consistent solution wrt the real and imag. parts
Damping of the incident photon field due to decay into a fermion pair
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Complex refractive index
Some plots of refractive index were shown here.
For details, please contact the speaker (Koichi Hattori) by email: [email protected]
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Phenomenological aspects
Strong BQGP
quark
gluons
photons
Synchrotron radiation of photon/gluon from quarkK.Tuchin, PRC (2010), [hep-ph/1209.0799] (2012)
See K.Itakura’s talk in “Parallel 3C”
Magnetic field induced photon/gluon emissions
Effects on photon HBT interferometryModified refraction index induces a distorted HBT image
K.Itakura and KH, (2011)
Based on the plots of the refractive index, we discussed application to the photon spectrum in HIC here.
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Summary
Prospects
- We analytically evaluated the photon vacuum polarization tensor in external magnetic fields.
- We inspected the complex dielectric constant/refraction index around the lowest-Landau-level threshold with a self-consistent treatment.
- Application to photon spectra in heavy-ion collisions. In progress
- Quarkonia in strong fields In progress, in collaboration with S. H. Lee
Let’s see a direct approach to the initial stage in HIC eventsby probing phenomena that uniquely occur in strong fields
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Scalar coefficient functions χ in the proper-time method
Schwinger, Adler, Shabad, Urrutia, Tsai and Eber, Dittrich and Gies
Given by a double integral wrt proper time variablesassociated with two fermion lines
Dimesionless variables
Exponentiated trig-functions generate strongly oscillating behavior witharbitrarily high frequency.
Integrands having strong oscillation
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Analytic calculation of the double integral
Any term reduces to either of elementary integrals.
Two important relations
Associated Laguerre polynomial
★
★
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Br = (50,100,500,1000,5000,10000, 50000)
Real part of ε on stable branch
Imaginary part of ε on unstable branch
Real part of ε on unstable branch
Relation btw real and imaginary partson unstable branch
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Direct photon from initial stage in UrHIC
Space (z)
Time Various hadron emissions& photon from hadron decay, π0 2γ
Intermediate and hard regime :emission from primordial matter
Direct photon spectrum
Low Pt regime: dominant emission in hadron phase
Initial and background photons
Directly accessible to the initial stage and QGP
Thermalization
Interaction with B-field
?
Detecting photon from the initial stage⇔ Detecting effects of B-field
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Polarization in dielectric medium : a classical argument
Incident light
Schematic picture of the birefringence
Dissipation Linear bound force
Anisotropic constants result in an anisotropic response.
What happens with the anisotropic (discretized) spectrum by the Landau-levels ?
Incident light field
Lorentz-type dispersion :
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Close look at the integralsWhat dynamics is encoded in the scalar functions ?
An imaginary part representing a real photon decay
⇔⇔
Invariant mass of a fermion-pair in the Landau levels
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Photon decay channel opens at every Landau level
Analytic results!Applicable to any momentum regime and field strength !
Combination of known functions
Applicable to both on-shell and off-shell photon!
Sum wrt Landau levels