Anisotropic dilepton spectrum induced by chiral anomaly in strong magnetic fields
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Anisotropic dilepton spectrum induced by chiral anomaly in strong magnetic fields
Koichi Hattori (Yonsei Univ.)NFQCD@YITP, 12th Dec. 2013
In collaboration with
Kaz. Itakura and Sho Ozaki
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Key ingredient 1: Strong fields
Chromo electromagnetic fields Magnetic fields (QED)
Key ingredients 2: EM probes
Possible signature of the early-time dynamics
Penetrating probes porters carrying information of the early-time dynamics.
Possible effects of strong magnetic fields on 1. neutral pions related to dilepton spectra KH, K.Itakura, S.Ozaki, 20132. direct-photon propagations (vacuum polarization tensor) KH, K. Itakura (I) (II) 2013
Challenging on both theoretical and experimental sides!
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Violation of axial current conservation
Absence of radiative correction Adler & Bardeen, 1960Triangle diagram gives the exact result in the all-order perturbation theory
Adler, Bell, Jackiw, 1959
Dominant (98.798 % in the vacuum)
99.996 %
``Dalitz decay ‘’ (1.198 % in the vacuum)
NLO contribution to the total decay rate Only corrections to external legs are possible
LO contribution to the total decay rate
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Effects of external magnetic fields
Decay mode possible only in external field
“Bee decay” can be comparable to Dalitz decay and even π0 2γ, depending on B.
Replacement of a photon line by an external field
Decay width of “Bee decay”
WZW effective vertexπ0γ
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Dalitz decay
Bee decay
Decay widths
Mean lifetime
femtometer
Branching ratios
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Analytical modeling of colliding nuclei, Kharzeev, McLerran, Warringa, NPA (2008)
Pre-equilibrium
QGP
Strong magnetic fields in UrHIC
Event-by-event analysis, Deng, Huang (2012)
Au-Au 200AGeV b=10fm
Lifetime of B-field is shorter than the lifetime of neutral pions.
No chance to observe the new decay mode in the heavy-ion collisions.
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Neutral-pion production from prompt γ* in strong B-fieldsPrompt γ* are produced in hard parton scatterings.
Gluon compton scattering in LO annihilation in LO
Without B-fields, mostly converted into dileptons.
+ Prompt photons are produced at the impacts of AA (pp, pA) collisions, so that converted into π0 in B-fields (and π0 decays outside B-fields).
Rapidly decaying B-field
π0γ*
Dilepton-production from γ* are suppressed.
A new neutral-pion production mechanism.
Given amount of γ*
Conversions of prompt γ* to pions in strong B-fieldsπ0 (only in B-fields)
Dileptons
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Negative v2 of dileptons
Strong time-dependence of B-fields Energy transfer from B-fieldsLienard-Wiechert potentials Fourier componentTime-dependence
Dilepton yield in the presence of B-field Elliptically anisotropic pion productions
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Summary in part 1
We investigated conversions between neutral pions and virtual photons in external magnetic fields.
+ A new decay channel “Bee decay” becomes possible, and becomes even the dominant decay mode in strong B-fields. Can be important in macroscopic systems such as the magnetars (neutron stars).
+ Oriented pion production from prompt virtual photon gives rise to a negative v2 of dileptons (and a positive v2 of neutral pions).
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Photon propagations in strong magnetic fields
“Vacuum birefringence” and “real-photon decay”
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What is “Birefringence” ?
Doubled image due to a ray-splitting in birefringent substances
Polarization 1Polarization 2
Incident light“Calcite” (方解石 )
Two polarization modes of a propagating photon have different refractive indices.
How about the vacuum with external magnetic fields ?
+ Lorentz & Gauge symmetries n ≠ 1 in general
+ Oriented response of the Dirac sea Vacuum birefringence
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Modifications of photon propagations in strong B-fields
Magnetic field
QGP
Refraction of photon in the vacuum with B-fieldswithout medium effects
Real photon decay Dilepton emissions from real photon decay, as well as virtual photon conversions (γ* e+e- )
Could be important in pre-equilibrium stage,
and in QGP additively to medium effects
Photon vacuum polarization tensor:
Modified Maxwell eq. :
Dressed propagators in Furry’s picture
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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 Furry’s picture
In heavy ion collisions, B/Bc ~ O(104) >> 1
Resummation w.r.t. external legs by “proper-time method“ Schwinger
Nonlinear to strong external fields
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Schwinger, Adler, Shabad, Urrutia, Tsai and Eber, Dittrich and Gies
Exponentiated trig-functions generate strongly oscillating behavior witharbitrarily high frequency.
Integrands having strong oscillations
Photon propagation in a constant external magnetic fieldLorentz and gauge symmetries lead to a tensor structure,
θ: angle btw B-field and photon propagation
B
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Summary of relevant scales and preceding calculations
Ultrarelativistic heavy-ion collisions
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
General analytic expression
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Decomposition into a double infinite sum
Analytic results of integrals without any approximation
Polarization tensor acquires an imaginary part above
Every term results in either of three simple integrals.
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Summary of relevant scales revisited- An infinite number of the Landau levels
(Photon momentum) Narrowly spaced Landau levels
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Renormalization
+= ・・・+ +
Log divergence
Subtraction term-by-term
Ishikawa, Kimura, Shigaki, Tsuji (2013)
Taken from Ishikawa, et al. (2013)
Finite
Re Im
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Complex refractive indices
The Lowest Landau Level (ℓ=n=0)
Dielectric constant at the LLL
Polarization excites only along the magnetic field``Vacuum birefringence’’
Solutions of Maxwell eq. with the vacuum polarization tensor
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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 in RHIC and LHC
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Angle dependence of the refractive indexReal part
No imaginary part
Imaginary part
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Summary in 2nd part
We obtained the general analytic form of the resummed 1-loop polarization tensor in magnetic fields as the summation w.r.t. the Landau levels.
+ Confirmed to reproduce the preceding approximate calculations. Ishikawa, Kimura, Shigaki, Tsuji
+ We obtained the complex refractive indices (photon dispersions) by solving the modified Maxwell Eq.
+ Neutral pions and dileptonsClose look at phenomenology including competing effects.
+ Real photonsApplications of photon propagation to phenomenologies in the heavy-ion collisions and the magnetars (neutron stars).
Prospects
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Branching of virtual prompt photon
q q2 Im
2 Im q q
Anisotropic on-shell pion production
Fourier component of an external field Anisotropic pion production
Isotropic dilepton production
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Effective coupling between π0 and 2γ
(Rest frame)
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Neutral pion decay into dilepton
Bext = (0,0,B), Eext = 0
EM current
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q q
Neutral pion decay into dilepton (continued)
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Decay rates in three modes Mean lifetime
Energy dependence of the decay rates
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Field-strength dependence of the branching ratio
Angle dependence of the branching ratio Angle dependence of the lifetime
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Primakoff effect
Recent measurement in JLab (2011)(PrimEX collaboration)
Prompt photon from hard parton scatterings
+ Prompt photons are produced at the moment of AA (pp, pA) collisions, so that they have much chance to interact with B-field.
Gluon compton scattering in LO annihilation in LO
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Profiles of time-dependence magnetic fields
Time-dependence Fourier component
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q0
π0
kq=q0+k
γ
Number of neutral pions increaseNumber of dilepton decrease
Elliptically anisotropic pion productions
Positive v2 of neutral pionsNegative v2 of dileptons
Neutral-pion production from prompt γ* in strong B-fields
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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
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Analytic representation of Pmn(q,B)
• Infinite summation w.r.t. n and l = summation over two Landau levels• Numerically confirmed by Ishikawa, et al. arXiv:1304.3655 [hep-ph]• couldn’t find the same results starting from propagators with Landau level decomposition
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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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Angle dependence of the refractive index
Direction of arrow : direction of photon propagation
Norm of arrow : magnitude of the refraction index
Magnetic field
Shown as a deviation from unit circle
Complex refractive index
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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