Recent progress in hadron physics -From hadrons to quark and gluon- 2013Feb. 18-22, Yonsei University

1. Photon propagation in strong magnetic field: “Vacuum birefringence”,

KH and K. Itakura, Ann. Phys. 330 (2013) 23-54.

KH and K. Itakura, arXiv: 1212.1897 [hep-ph].

Koichi Hattori (Yonsei Univ.)

Effects of strong magnetic fields

2. Pion reactions in strong magnetic field,

KH, K. Itakura, S. Ozaki, in preparation

What is “Birefringence ( 複屈折 )” ?

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 of 1st part

How is in the vacuum with external magnetic field ?

+ Lorentz & Gauge symmetries n ≠ 1 in general

+ Oriented response of the Dirac sea Vacuum birefringence

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

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

2nd PIF, 9-11, July, 2013, @DESY

Analytical modeling of colliding nuclei, Kharzeev, McLerran, Warringa, NPA (2008)

Pre-equilibrium

QGP

Time evolution of the magnetic field after collisions

Simple estimates by Lienard-Wiechert potential

Still a few orders stronger than the “critical field”

Time evolution of the matter “QGP” after the collision

Strong magnetic field

EM probe

Hadronic probe

QGP Hadronization Hadron gas

Pre-equilibrium

QGP

CollisionIncidenceLorentz construction

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.

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

Photon propagation in a constant external magnetic fieldLorentz and gauge symmetries lead to a tensor structure,

B

θ: 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

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

Vacuum birefringence has been led by the tensor structure.

What dynamics is encoded in the scalar functions, χi ?

Decomposition into a double infinite sum

Analytic integration without any approximation

Polarization tensor has an imaginary part above

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)

With a great contribution of Ishikawa-san (Hiroshima Univ.), complete photon propagator in strong B-field is now available.

Dielectric constant / refractive index by self-consistent treatment

Close look at the lowest Landau level

Vacuum polarization tensor from the LLL

Dielectric constant at the LLL

Polarization excites only along the magnetic field

Comparison with numerical result

LLL approx. agrees with numerical result in strong field limit

Kohri & Yamada

Modified photon dispersion relation from the vacuum polarization tensor

Modified Maxwell eq. :

Dispersion relation of q

Consistent solution wrt the real and imag. parts

Air （ 1atm, 0℃ ）： n=1.0003Water (20℃) ： n=1.333Calcite: no=1.6584, ne=1.4864

Complex refractive index from the lowest-Landau-level

Re[n] on unstable branchRe[n] on stable branch

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

Relation btw. real and imaginary partson unstable branchIm[n] on unstable branch

Dependence on magnetic-field strength

Angle dependence of the refractive index

Photon decay length in strong magnetic field

Propagating EM field:

Intensity of the field:

Photon decay length:

Direction of arrow : direction of photon propagation

Norm of arrow : magnitude of the refraction index

Magnetic field

Angle dependence of the refractive indexShown as a deviation from unit circle

Angle dependence at various photon energies

Real part

No imaginary part

Imaginary part

Implications in heavy-ion collisions

Strong BQGP

quark

gluons

photons

Synchrotron radiation of photon/gluon from quark

K.Tuchin, PRC (2010), [hep-ph/1209.0799] (2012)

Magnetic field induced photon/gluon emissions

Effects on photon HBT interferometry

Modified refractive index induces a distorted HBT image

K.Itakura and KH (2011)

Decay of real photon induces negative contribution to photon’s v2

Decay of real photon induces positive contribution to dilepton’s v2

Also, there would be effects on the higher harmonics

Effects on anisotropic flow

Needs systematic studies in HIC.

Summary of 1st part

- 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.

Prospects- Application to EM probe in heavy-ion collisions and magnetar. Polarization dependence, Energy and angle dependences…

Competition/interplay between birefringence and splitting

Pion reactions in strong magnetic fields

is coming soon.

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

Vacuum birefringence has been led by the tensor structure.

What dynamics is encoded in the scalar functions, χi ?

Analytic calculation of the double integral

Any term reduces to either of elementary integrals.

Two important relations

Associated Laguerre polynomial

★

★

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

Extremely strong magnetic field in the direction of the out-of-reaction plane

Geometry of the peripheral collision and strong B-field

b

Finite impact parameter

Lorentz - contracted nuclei

Primordial and thermalized matter

An extremely strong magnetic field in Ultrarelativistic Heavy-Ion Collision

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

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 :

Close look at the integrals

What 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

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