Koichi Hattori

Online seminar for BNL, Aug. 3, 2021

Electromagnetic probes in strong-field QED

koichi.hattori@yukawa.kyoto-u.ac.jp

Novel transport phenomena in QGP

• Chiral magnetic effect
• magnetohydrodynamics
• thermal radiations in B
• etc.

Peripheral collisions

Strong magnetic fields from Lorentz-boosted Coulomb fields

Large atomic number: Z ~ 80

＋

Large velocity: ｖ > 0.9999 c

Photon-photon interactions

What happens when photons go through strong fields?

Vacuum fluctuations play a role.

Resuumed vacuum polarization diagram

High-intensity laser field

Mourou&Strickland (2018 Nobel laureates)

https://www.nobelprize.org/prizes/physics/2018/mourou/lecture/

Transient high-Z atoms (once studied at GSI)

Magnetospheres of neutron stars/magnetars

Dipole strength estimated from the intervals of the pulsation.

Supercritical Coulomb fields

See an anecdotal review “Probing QED Vacuum with Heavy Ions” Rafelski, Kirsch, Mueller, Reinhardt, and W. Greiner [1604.08690]

Strong fields in laboratories and nature (other than HIC)

Renewed interests realized with graphene; “Atomic collapse”

v << c 🡪 Larger effective α.

🡪 Kinetic energy << Coulomb potential

0. Introduction

- Brief historical overview

- Strong magnetic fields in laboratories and nature

​

1-1. Photon propagation in a magnetic field (at zero T and μ)

- Vacuum birefringence and (real) photon decay

- (Diagrammatic technique by the proper-time method)

​

1-2. Differential dilepton spectrum in a magnetic field

- (Ritus basis formalism)

- “Helicity suppression” in the ratio of di-electron yield to di-muon yield

​

2. Summary

​

Table of contents

(1) Refractive index of photon in strong B-fields

- Old but unsolved problem

• Perturbation breaks down and resummation is required.
• Has not been confirmed experimentally.

KH and K.Itakura, “Vacuum birefringence in strong magnetic fields”:

(I) Photon polarization tensor with all the Landau levels,” (2013);

(II) Complex refractive index from the lowest Landau level,” (2013).

Photon propagation in magnetic fields

(in four dimensions)

Preferred orientation provided by an external B

• ``Vacuum birefringence”

Cyclotron motion in B field

What is birefringence？

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

Complex refractive indices

Resuumed vacuum polarization diagram

Both sides of a coin

Optical theorem for the imaginary part

Dispersion integral

Rotating polarizations of transmitted ray

• Acquired polarization due to the birefringence
• Some of photons decay into fermion pairs

Cf. Cotton-Mouton effect, Faraday effect, etc in optics

Inborn polarization

in photon sources

(Possibly unpolarized as well)

Vacuum birefringence

from the resummed vacuum polarization tensor

Maxwell eq. w/ quantum corrections:

U(1) gauge symmetry constrains

possible tensor structures

Two (physical) polarization modes in || and ⊥ to B.

B-induced structures

Refractive indices from the Maxwell eq.

(Boost invariance along B)

Direct consequence of the gauge symmetry and the breaking of one spatial rotational symmetry.

Resummation for external-field insertions

Resummed propagator

Resummed polarization tensor

Proper-time method

Nonlinear wrt the external fields

B_z

1. No effects on the longitudinal motion
2. Cyclotron motion ～ Harmonic oscillation
3. Spin polarization (Zeeman effect)

No energetically favored position.

🡪 Degeneracy

Landau quantization

Fermion spectrum in a magnetic field

Remember the lesson: Photon spectrum depends on the fermion spectrum.

Classical motion

Landau degeneracy

Energy spectrum

B_z

Fermion pair spectrum in the imaginary part

--- Thresholds at the Landau levels

Polarization tensor acquires an imaginary part when

The integers are identified with the Landau levels.

Threshold condition

Photon transverse momentum works

like a “photon mass” in the (1+1)-d kinematics.

Fermion transverse momentum is NOT a good quantum number.

No transverse momentum conservation between fermions and photons

(due to momentum supply from external B-fields).

Kinematics with the Landau levels

＋　・・・

＋

＋

Naïve perturbation breaks down when eB is large!

Vacuum polarization tensor in two different series representations

1. Naïve perturbative series when eB is small.

＋

=

Weak-field approximation

Adler, etc.

No modification

Soft photon & weak field limit (Adler, etc.)

Lowest Landau level 🡪

Narrowly spaced Landau levels

Summary of relevant scales and preceding calculations

Strong field limit (LLL approx.)

(Tsai&Eber; Shabad; Gusynin, Miransky, Shovkovy; Fukushima; KH&Itakrua, etc.)

Numerical computation

below the LLL threshold (Kohri and Yamada)

No imaginary part🡪

Refractive indices and decay rate

Refractive indices with the LLL fluctuations

Refractive indices at the LLL(ℓ=n=0)

Polarization excites only along the magnetic field

``Vacuum birefringence’’

(1+1)-dimensional fluctuations

Complex refractive indices

Final results shown by solid lines

Anisotropy of the refraction index

Angle : Direction of the photon propagation

Radius : Magnitude of the refraction index

Real part

Imaginary part

No imaginary part below the threshold

Photon energy ω

Threshold

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

When the refractive index has an imaginary part,

“Mean-free-path”

Smaller mfp for a larger energy ~ 1/ω

Even real photons decay

in a microscopic scale!

[Real photons never decay in

ordinary vacuum without B-field.]

Implications

Differential dilepton cross-section may be useful.

Timokhin & Harding, ApJ (2019)

Macroscopic scale ≳ 10 – 100 km

✘ Difficult to measure high-energy photon polarizations.

✔ One could analyze the dilepton angle-distribution.

Imaginary part of the refractive index

= Total cross section.

Neutron star magnetosphere

Ultraperipheral heavy-ion collisions

Differential dilepton spectrum

KH, Hidetoshi Taya, Shinsuke Yoshida, “Di-lepton production from a single photon in strong magnetic fields: vacuum dichroism”, [2010.13492]

--- Better accessibility than the photon polarization

Both on-shell and off-shell photons can decay in B.

Pair production in a magnetic field

E.g., Breit-Wheeler process

No kinematical window for

a single on-shell photon when B = 0.

​

🡪 Starts only from 2 photons

Nonperturbatively dressed fermions

LO w/ B-fields

LO w/o B-fields

Ritus basis formalism

= Perturbation theory with the fermion wave functions in B-fields.

See a review part in [2010.13492]

Free Dirac eq. 🡪 Plane wave solutions

Dirac eq. in B-field 🡪 Localized wave functions

Differential information includes

• Photon polarization (ε^μ)
• Photon momentum (q^μ) with a general direction and invariant mass
• Landau levels (n, n’) and continuous momentum (p_z, p_z’) along B

One-shell/off-shell photon

Fermion pair in the Landau levels

Consistency checks done: Ward identity, etc.

Pair production rate for general kinematics

“Helicity suppression” in pion decay

Muon-pair excess over electron pairs

Especially, R neutrino (L antineutrino) does not exist at the QCD scale.

(However, this is not an essential reason for the helicity suppression.)

Same helicity

Opposite spin

Opposite momentum

Opposite chirality

🡪 Needs chirality mixing by finite mass.

KH, Taya, Yoshida [2010.13492]

PDG

Feasibility with HIC?

• Acquired polarization due to the birefringence
• Some of photons decay into fermion pairs

Inborn polarization in photon sources

(Possibly unpolarized as well)

• There is no a priori difference: Both are Coulomb electric fields in the nucleus rest frame.
• The photon source has a momentum distribution: Fourier transform of the charge distribution, e.g., the Woods-Saxon profile.
• The hard and soft components of the distribution may be regarded as “photons” and a “magnetic field”.
• Needs quantitative estimates with a separation scale.

Ideal set-up

What is a photon source and what is a strong B field in HIC?

UPC events

Summary

• Needs convolution with the photon distribution function
• Time dependence of a magnetic field

KH, Xu-Guang Huang, Hidetoshi Taya, Shinsuke Yoshida, In progress.

• Vacuum birefringence (polarization-dependent refractive indices)
• γ🡪 ee for both on-shell and off-shell photons
• Differential di-lepton spectrum

🡪 “Helicity suppression” in the electron/muon

Prospects for the UPC

Back-up slides

Strong magnetic fields induced by relativistic heavy-ion collisions

Free streaming

HIJING

Central coll.

Peripheral coll.

Static Coulomb field

Boost 🡪 Lienard-Wiechert potential

Deng & Huang; KH & Huang [1609.00747]

B(t)

🡪 Induced J sustains B.

Lifetime of the B-field after the collisions

Time dependent B induces E.

A longer lifetime due to the Lenz’s law?

Important to know the conductivity of QGP in magnetic fields. KH & Satow; KH, Li, Satow, Yee; Fukushima & Hidaka

McLerran & Skokov

Tuchin

Analytic and numerical estimates of the strong B

-- Impact parameter dependences

Time dependences

Collision-energy dependences

V. Voronyuk, et al. (2011)

KH and X.-G. Huang

KH and X.-G. Huang

Kinematics in the massless LLL

Perturbative vertex does not mix the R and L.

R handed

L handed (when eB>0)

Linear dispersion relations in the lowest Landau levels

⇒ No coupling to the transverse photons.

Kinematics in the massless limit is satisfied only in the “collinear limit.”

Ritus basis: Eigenspinor in B-field

Price: Convolution of the wave functions at the vertex gets complicated. Fermion wave functions are not orthogonal to photon wave function. (Photon wave function is a plane wave.)

Spin up and down states are degenerated except for n = 0.

Canonical quantization in B-field

Mode expansion can be performed in this basis.

Fermion propagator gets simplified.

See a review part in [2010.13492]

Resummed polarization tensor

Decomposing exponential factors

Contains arbitrarily higher harmonics

All terms fall in one of three elementary integrals.

Renormalization

Term-by-term subtraction

Ishikawa, Kimura, Shigaki, Tsuji (2013)

Taken from Ishikawa, et al. (2013)

Finite

Re

Im