Koichi HattoriRIKEN-BNL Research Center

Nonlinear QED effects on photon and dilepton spectra in supercritical magnetic fields

KH, K. Itakura (KEK), Ann. Phys. 330 (2013); Ann. Phys. 334 (2013).

Photon & dilepton WS@ECT*, Dec. 9, 2016

Keywords- Strong magnetic fields- Vacuum birefringence

Response of electrons to incident lightsAnisotropic responses of electrons result in polarization-dependent and anisotropic photon spectra.

What is birefringence?

Birefringence: polarization-dependent refractive indices

Polarization 1Polarization 2

Incident lightBirefringent substance “Calcite” (方解石 )

Lesson: The spectrum of fermion fluctuation is important for the photon spectrum.

Structured ions Anisotropic spring constants

Photon propagation in magnetic fields

B

+ Lorentz & Gauge symmetries n ≠ 1 in general

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.

Landau levels + Discretized transverse momentum+ Still continuum in the direction of B+ Anisotropic response from the Dirac sea ``Vacuum birefringence”

Wave function (in symmetric gauge)

Schematic picture of the strong field limit

Strong B

Fermions in 1+1 dimensionPolarizer

Strong magnetic fields in laboratories and nature

Strong magnetic fields in UrHIC

Lienard-Wiechert potential

Z = 79(Au), 82(Pb)

Ecm = 200 GeV (RHIC)Z = 79 (Au), b = 6 fm

t = 0.1 fm/c 0.5 fm/c 1 fm/c 2 fm/cEvent-by-event analysis, Deng & Huang (2012)

Au-Au 200AGeV b=10fm

Supercritical fields beyond electron and quark masses

Impact parameter (b)

PSR0329+54

Other strong B fields

“Lighthouse” in the sky

NS/Magnetar High-intensity laser field

NSs

Magnetars

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

Basic framework

Quantum effects in magnetic fields

Photon vacuum polarization tensor:

Modified Maxwell eq. :

Dressed propagators in Furry’s picture

・・・

・・・

eBeB eB

Large B compensates the suppression by e. Break-down of the naïve perturbation Needs a resummation

Seminal works for the resummationConsequences 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!

General formula within 1-loop & constant fieldobtained by the “proper-time method”.

Resummation 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 (1951)

Nonlinear to strong external fields

In heavy ion collisions, B/Bc ~ (mπ/me)2 ~ O(104) >> 1

The strong field limit revisited:Lowest Landau level (LLL) approximation (n=0)

Spin-projection operator

Wave function1+1 dimensional dispersion relation

1+1 dimensional fluctuation

Dispersion relation from the resummation

Vanishing B limit:

θ: angle btw B-field and photon propagation

BGauge symmetries lead to a tensor structure,

Schwinger, Adler, Shabad, Urrutia, Tsai and Eber, Dittrich and Gies

Exponentiated trig-functions generate strongly oscillating behavior witharbitrarily high frequency.

Integrands with strong oscillations

Generalization: Resummed vacuum polarization tensor

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

Euler-Heisenberg LagrangianIn soft photon limit

General analytic expression

2nd step: Getting Laguerre polynomials

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.

Linear w.r.t. τ in exp.

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

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)

Analytic result of integrals- An infinite number of the Landau levels

KH, K.Itakura (I)

⇔Polarization tensor acquires an imaginary part when

Lowest Landau level

Narrowly spaced Landau levels

Complex refractive indices Solutions of Maxwell eq. with the vacuum polarization tensor

KH, K. Itakura (II)

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’’

Solutions of the modified Maxwell Eq.Photon dispersion relation is strongly modified when strongly coupled to excitations (cf: exciton-polariton, etc)

𝜔2/4𝑚2

≈ Magnetar << UrHIC

𝜔2/4𝑚2 cf: air n = 1.0003, water n = 1.3, prism n = 1.5

Refraction Image by dileptons

Angle dependence of the refractive indexReal part

No imaginary part

Imaginary part

BBelow the threshold Above the threshold

“Mean-free-path” of photons in B-fields

λ (fm)

When the refractive index has an imaginary part,

For magnetars

QM2014, Darmstadt

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.

Neutron stars = Pulsars Possibly “QED cascade” in strong B-fields

What is the mechanism of radiation?

We got precise descriptions of vertices: Dependences on magnitudes of B-fields, photon energy, propagation angle and polarizations.

“Photon Splitting”Softening of photons

Photon splitting

eBeB

Vacuum birefringence(Refractive indices n≠1)

Soft photon limit

Quantum corrections in magnetic fields

+ Should be suppressed in the ordinary perturbation theory, but not in strong B-fields.

The earliest work: Euler-Heisenberg Lagrangian- Low-energy (soft photon) effective theory

+ Constant magnetic fields

・・・ ・・・

eBeB eB

eB

Poincare invariants

Landau levels + Zeeman splittingin the resummed propagator

(iii) The same transform properties under the C-conjugation as that of a free propagator.

Spin-projection operators

The lowest Landau level (n=0)

(i) Discretized fermion’s dispersion relation(ii) Three terms corresponding to the spin states.

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

𝜔2/4𝑚2

Angle dependence of the refractive indexReal part

No imaginary part

Imaginary part

Renormalization

+= ・・・+ +

Log divergence

Term-by-term subtraction

Ishikawa, Kimura, Shigaki, Tsuji (2013)

Taken from Ishikawa, et al. (2013)

Finite

Re Im

Br = (50,100,500,1000,5000,10000, 50000)

Real part of n on stable branch

Imaginary part of n on unstable branch

Real part of n on unstable branch

Relation btw real and imaginary partson unstable branch