1. RIKEN-BNL Research Center
Nonlinear QED effects on photon and dilepton spectra
in supercritical magnetic fields
Keywords
- Strong magnetic fields
- Vacuum birefringence
Koichi Hattori
RIKEN-BNL Research Center
Photon & dilepton Dec. 9, 2016
KH, K. Itakura (KEK), Ann. Phys. 330 (2013); Ann. Phys. 334 (2013).

2. What is birefringence? Birefringence:
Anisotropic responses of electrons result in
polarization-dependent and anisotropic photon spectra.
Response of electrons to incident lights
Structured ions
 Anisotropic spring constants
Birefringence:
polarization-dependent refractive indices
Polarization 1
Polarization 2
Incident light
Birefringent substance “Calcite” (方解石)
Lesson: The spectrum of fermion fluctuation is important for the photon spectrum.

3. Photon propagation in magnetic fields
+ Lorentz & Gauge symmetries  n ≠ 1 in general
Landau levels
+ Discretized transverse momentum
+ Still continuum in the direction of B B
+ Anisotropic response from the Dirac sea
``Vacuum birefringence”
Real part: “Vacuum birefringence”
Imag. part: “Real photon decay” into fermion pairs
“Photon splitting”
 Forbidden in the ordinary vacuum
because of the charge conjugation symmetry.

4. Schematic picture of the strong field limit
Wave function (in symmetric gauge)
Polarizer
Fermions in 1+1 dimension
Strong B

5. Strong magnetic fields
in laboratories and nature

6. Strong magnetic fields in UrHIC
Lienard-Wiechert potential
Z = 79(Au), 82(Pb)
Event-by-event analysis, Deng & Huang (2012)
Au-Au
200AGeV
b=10fm
Supercritical fields beyond
electron and quark masses
Impact parameter (b)
Ecm = 200 GeV (RHIC)
Z = 79 (Au), b = 6 fm
t = 0.1 fm/c
0.5 fm/c
1 fm/c
2 fm/c

7. Other strong B fields NS/Magnetar High-intensity laser field
NSs
Magnetars
“Lighthouse” in the sky
PSR

8. 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

9. Refractive index of photon in strong B-fields
- Old but unsolved problem
Much simpler than QCD
But,
Tough calculation due to a resummation
Has not been observed in experiments

10. Basic framework Large B compensates the suppression by e.
Photon vacuum polarization tensor:
Modified Maxwell eq. :
Dressed propagators
in Furry’s picture
Quantum effects in magnetic fields
・・・ eB
Should be suppressed
in the ordinary perturbation theory,
but not in strong B-fields.
Large B compensates the suppression by e.
 Break-down of the naïve perturbation
 Needs a resummation

11. Seminal works for the resummation
Consequences of Dirac’s Theory of the Positron
W. Heisenberg and H. Euler in Leipzig1
22. December 1935
General formula within 1-loop & constant field
obtained by the “proper-time method”.
Euler – Heisenberg effective Lagrangian
　- resummation wrt the number of external legs
Correct manipulation of a UV divergence in 1935!

12. Resummation in strong B-fields
Dressed fermion propagator in Furry’s picture
Critical field strength
Bc = me2 / e
In heavy ion collisions,
B/Bc ~ (mπ/me)2
~ O(104) >> 1
Naïve perturbation breaks down when B > Bc
 Need to take into account all-order diagrams
Resummation w.r.t. external legs by “proper-time method“
Schwinger (1951)
Nonlinear to strong external fields

13. 1+1 dimensional fluctuation
Dispersion relation from the resummation
The strong field limit revisited:
Lowest Landau level (LLL) approximation (n=0)
Spin-projection operator
Wave function
1+1 dimensional
dispersion relation
1+1 dimensional fluctuation

14. Resummed vacuum polarization tensor
Generalization:
Resummed vacuum polarization tensor
θ: angle btw B-field and
photon propagation B
Gauge symmetries lead to a tensor structure,
Gauge symmetry leads to three tensor structures,
Vanishing B limit:

15. Generalization: Resummed vacuum polarization tensor
θ: angle btw B-field and
photon propagation B
Gauge symmetries lead to a tensor structure,
Vanishing B limit:
Gauge symmetry leads to three tensor structures,
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

16. General analytic expression
Summary of relevant scales
and preceding calculations
General analytic expression
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
Euler-Heisenberg Lagrangian
In soft photon limit

17. Associated Laguerre polynomial
Decomposing exponential factors
Linear w.r.t. τ in exp.
1st step: “Partial wave decomposition”
Linear w.r.t. τ in exp.
2nd step: Getting Laguerre polynomials
Associated Laguerre polynomial
Linear w.r.t. τ in exp.

18. After the decomposition of the integrand,
any term reduces to either of three elementary integrals.
Transverse dynamics:
Wave functions for the Landau levels given by
the associated Laguerre polynomials

19. ⇔ Analytic result of integrals
- An infinite number of the Landau levels
KH, K.Itakura (I) ⇔
Polarization tensor acquires an imaginary part when
(Photon momentum)
Narrowly spaced Landau levels
Lowest Landau level
A double infinite sum
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)
External photon momentum
Narrowly spaced Landau levels
Lowest Landau level

20. Complex refractive indices
KH, K. Itakura (II)
Solutions of Maxwell eq.
with the vacuum polarization tensor B
LLL: 1+1 dimensional fluctuation in B
Refractive indices at the LLL(ℓ=n=0)
Polarization excites only along the magnetic field
``Vacuum birefringence’’

21. Solutions of the modified Maxwell Eq.
Photon dispersion relation is strongly modified
when strongly coupled to excitations (cf: exciton-polariton, etc)
𝜔 2 /4 𝑚 2
≈ Magnetar << UrHIC
cf: air n = , water n = 1.3, prism n = 1.5
Refraction
Image by dileptons

22. Angle dependence of the refractive index Real partB
Below the threshold
Above the threshold
Imaginary part
No imaginary part

23. “Mean-free-path” of photons in B-fields
When the refractive index has an imaginary part,
λ (fm)
For magnetars

24. QM2014, Darmstadt

25. Summary
+ We obtained an analytic form of the resummed polarization tensor.
+ We showed the complex refractive indices (photon dispersions) .
-- Polarization dependence
-- Angle dependence
Prospects:
Search of vacuum birefringence in UrHIC & laser fields
Microscopic radiation mechanism of neutron stars
 Nonlinear QED effects on the surface of NS.

27. Neutron stars = Pulsars
What is the mechanism of radiation?
 Possibly “QED cascade” in strong B-fields
“Photon Splitting”
Softening of photons
We got precise descriptions of vertices:
Dependences on magnitudes of B-fields,
photon energy, propagation angle and polarizations.

28. The earliest work: Euler-Heisenberg Lagrangian
- Low-energy (soft photon) effective theory
Quantum corrections in magnetic fields
・・・ eB
+ Constant magnetic fields
+ Should be suppressed in
the ordinary perturbation theory,
but not in strong B-fields.
Poincare invariants
Photon splitting eB
Vacuum birefringence
(Refractive indices n≠1)
Soft photon limit

29. Landau levels + Zeeman splitting in the resummed propagator
Spin-projection operators
The lowest Landau level (n=0)
Discretized fermion’s dispersion relation
Three terms corresponding to the spin states.
(iii) The same transform properties under
the C-conjugation as that of a free propagator.

30. 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 = 1.333
𝜔 2 /4 𝑚 2
≈ Magnetar << UrHIC

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

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

33. Real part of n on stable branch
Real part of n on unstable branch
Relation btw real and imaginary parts
on unstable branch
Imaginary part of n on unstable branch
Br = (50,100,500,1000,5000,10000, 50000)