Pion Production via Proton Synchrotron Radiationin Strong Magnetic Fields
in Relativistic Quantum Approacｈ
Particle Productions in TeV Energy Region 	 Tomoyuki Maruyama
BRS. Nihon University

Collaborators
Toshitaka Kajino
Myong-Ki Cheoun
Grand. J. MATHEWS

NaO, Japan
Sonnsil Univ., Koread
Univ. of Notre Dome, USA 	

T.Maruyama et al., Phys. Rev. D91, 123007 (2015).
Phys. Lett.. B757, 125( 2016).	
1§12Introduction

Soft Gamma Repeater (SGR) , Anomalous Xray pulsar (AXP)

http://commons.wikimedia.org	

⇒ Magnetar 1015G in surface 1017-19G inside
B.C.Duncan & C.Thompson ApJL 392, L9 (1992)
S.Merghetti, A&AR 15, 225 (2008)

Observation of γ-ray → Study od Magnetar Structure	γ-­‐ray  Radiation  	Proton is accelerate
up to 1GeV～1TeV⇒ Synchtotron Radiation
・・・ Meson Prod     (𝑺𝒕𝒓.  >  𝑬𝒍.𝑴𝒂𝒈.)
All Theories are Semi-Classical
V.L.Ginzburg et al., UsFiN 87, 65, ARA&A
3, 297 (65)
G.F. Zharkov, Sov. J. Nucl. Phys., 1, 17314 (65)
V. Berezinsky, et al., Phys. Lett. B 351, 261 (95)
A. Tokushita and T. Kajino, ApJ. 525, L117 (99).
T.Kajino et al., ApJ 782, 70 (2014）

Many Assumption and Approxs.
Mom.-Dist. cannot be calculated
Quantum Calulations.
⇒ Exact Imformation

§2 Formulation in Relativistic Quantum Approach	Magnetic Field :Dirac Equation	

⎛σ zΣ z = ⎜⎜⎝0ET = P +2z0 ⎞i⎟⎟ = −σ 12 = [γ 1 , γ 2 ]−σz ⎠2Anomalous Mag. Moment
Tensor-Type Mean-Field	

( 2n + 1 − s + M − s κ B M )
2NpN

24

Decay Width of p to p + p 0 	πN interaction

5PV coupling	Q = (0 , QT , Qz ) = qeB

H n ( x ) : Hermit Polynomial
Lmn ( x ) : Associated Laguerre
Polynomlal§3 Results of π0 Production 	Ei = 1 GeV, B = 5 × 1018 G
χ = eBe p / mN3 = 0.069
χ ≈ 0.01 − 1 π - Prod. DominanteB = 17.2 MeV ,eκ p2mN

B = 28.3 MeV

s +1
nmax + i= 50 for si = −12= 45 for si = +1no AM	

s +1
nmax + i= 47
2

Decay WidthTransition Strengths between two Landau Levels	ni = 45

with
AMM	－1 －>+1withoutAMM	

small Landau-level difference	Transition Strength 	2

Very Large AMM Effects	
p → p + π0 Energy Momentum Conservation is not satisfied
in the free kinematics
Mag. Fld.＋AMM

Tensor Type Mean-Fields = －1 （repulsive）, s = ＋1 （attractive）

Level Interval of Transition ni – nf
si = －1 → sf = ＋1

Smaller Intervals⇒ Enhances Transition Strengthsi = ＋1 → sf = －1

Larger Intervals⇒ Reduces Transition StrengthV

Small Shifts ni －nf make Large change of Transition Strength	§4 Realistic System	
Pion Production Dominant Energy Region
B = 1015 G

χ = eBe p / mN3 ≈ 0.01 − 1

Landau Number ： ni ≈ 1012 － 1013

Actual calculations are almost impossible
Problem : HO overlap integral

It is possible to make a Lorentz Transportation along z-direction	
Semi-Classical Theory ⇒ Scaling, Dep. Only on χ	Contribution at Fixed Final Landau Number	incl AMM	ei3eBeχ = 3 = 3i
M N RL M NNo AMM	Scaling Law Function of χ, (ni – nf)/niPrediction Results ni ≈ 104 ⇒	 Results ni ≈ 1012ー13 （B ~ 1015G)
Huge Effects of AMM remain even in B ~ 1015G 	

Small χ	 Larger ni → Scaling	
Total Decay Width

Scaling Relation（All Semi-Classial Theoryies Show)
3 Variables B, ni, nf⇒ 2 Variablesχ = eBEei /mN3, (ni – nf)/ni

Peak position(ni – nf) / ni → 0.3	Adiabatic Limit	

Relative Momentum between
Final Proton and Pion
is Zero,Same Velocity

mπmπeπ =ei , e f =eimN + mπ
mN + mπ→ni −n fni(e ≈ 2n )i, f≈ 0.28 ⇔ Semi - Classical :i, fni −n fni<< 1Angular Distrbution at piz = 0	d Γpπd 3 Iπ=eπdqπ3dqπ33Δqz
indep. on IIncidet Energy	

Adiabatic Limit

Norrow
Angular Ditsr.	
Γ(ni , n f ; Piz = 0)
∝ δ (Qz )Lorentz Trans.
along z-direction

Angular Distributionof Pion Luminocity	d Γpπd 3 Iπ= eπdqπ3dqπ33

χ = 0.07	

when ni >> 1,qT // pf // pi

Same Polar Angle
Width is very small

qz =mπpizmN + mπqT =mπeB 2nimN + mπadiabatic limit

16

Proton Decay Widthni >> 1dΓpπ ( piz = 0, si )piz = 0dq 3=1
Γpπ (ni , n f )δ (ei − e f − q0 )δ (q z )∑eπ n fLorentz Transformationpiz ≠ 0dΓpπ ( piz , si )Scaling

Results with ni, nf ~ 104 ⇒	 Results	 with	 1012	 dq 3⎛⎞eπ1 eiT=Γpπ (ni , n f )δ (ei − e f − q0 )δ ⎜⎜ q z − p z ⎟⎟∑eπ ei n fei⎝⎠

Semi-Classical Approximation assume ni － nf << niπ has massThis Assumpyion is wrong
mπni − n f >mN + mπni

Total Decay Width	

Semi-Classical
A.Tokushita and T. Kajino,ApJ. 525, L117 (99).

Luminocity-Distribution of Emitted Photons	 

p → p + π0
π0 → 2 γ

Average overInitial
Proton Angle
Distribution
is Spherical	

Total Decay Width	Γ(ni , χ ; Piz = 0) ∝ ni

Semi-Classical
A.Tokushita and T. Kajino,
ApJ. 525, L117 (99).

Γ(ni , χ ; Piz = 0)
indep.of ni

20§5 Summaryl

π0 emission from Proton Transition between two Landau Levels
ni, nf ~ 105 ⇒ B ~ 1017 G

AMM effect －1→＋1 Dccay widths become 50 – 100 times largerlScaling Law, predicted by the Semi-Classical theory
3 Variables B, ni, nf ⇒ 2 Variables χ = eBEei /mN3, (ni – nf)/ni
B ~ 1017 G ⇒ B ~ 1015 G (Magnetar)
Results with ni, nf ~ 104 ⇒	 Results with 1012l

Angular Dist θi ≈ θf ≈ θπldΓpπ (ni , piz )
dq 3

⎛eei

⎞

α δ ⎜⎜ q z − π p z ⎟⎟⎝⎠Pion Energies are distributed in Broad Regionmπni − n f >mN + mπni⇔

Semi-Classical Approx.
ni － nf << niThe Results come from HO overlap Integral	

It is a function of QT and very rapidly change when ni,f >> 1Generally⇒	 	  Other Particle Productions	 ⇒	 	 Magnetic Structure inside Magnetars 	HO Overlap Integral	In PS-couplingΓ(ni, nf) does notsatisfy
Scaling Relation