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2 delay,gain HV gain ADC calibration TDC calibration TQ TQ TQ TQ fitting NaI pick-off pick-off pick-off fitting pick-off B pick-off C
3 4.5.3 D β TDCcalibration TQ pick-off Threshold BG pick-off
4 1 1.1 QED 3
5 2 (Ps) Ps QED L = 1 4 F µνf µν ψ(γ µ [ µ + iea µ ] + m)ψ (2.1) Ps ( ) QED Ps l=0,s=0 l=0,s=1 1 S 0 3 S 1 l s (p-ps) (o-ps) Ps QED fermionn n c Ψ = d 3 p 1 d 3 p 2 χ(p 1, σ 1 ; p 2, σ 2 )a (p 1, σ 1 )a c (p 2, σ 2 )Ψ vacuum (2.2) σ 1 σ 2 χ(p 1, σ 1 ; p 2, σ 2 ) = ±χ(p 2, σ 2 ; p 1, σ 1 ) CΨ = Ψ Ca (p, σ)c 1 = ξa c (p, σ),ca c (p, σ)c 1 = 4
6 ξ c a (p, σ), ξξ c = 1 C ξ charge conjugation parity p-ps χ o- Ps p-ps o-ps C / charge conjugation parity -1 Ca (k 1, σ 1, photon)a (k 2, σ 2, photon)...a (k n, σ n, photon)ψ vacuum = ( 1) n a (k 1, σ 1, photon)a (k 2, σ 2, photon)...a (k n, σ n, photon)ψ vacuum (2.3) Ca (k, σ, photon)c 1 = a (k, σ, photon) n C ( 1) n QED C p-ps o-ps o-ps p-ps 2,4,6,... o-ps 3,5,7,... differential transition rate dγ(α β) α β β + dβ initial particles 2 differential transition rate dγ(α β) = (2π) 4 V 1 M βα 2 δ 4 (p β p α )dβ (2.4) M βα α β S S βα = 2πiM βα δ 4 (p α p β )dβ (2.5) V transition rate differential transition rate 5
7 α n dσ(α β) dσ(α β) = V u α dγ(α β) (2.6) (2π) 4 u α M βα 2 δ 4 (p β p α )dβ (2.7) (p1 p 2 ) u α 2 m 2 1 m2 2 E 1 E 2 dσ(α β) flux transition rate flux H H 0 V S 1 S = dt 1 dt 2 dt N T (V (t 1 )V (t 2 ) V (t N )) (2.8) N! N=0 S p-ps o-ps transition rate Feynmen diagram Feynmen o-ps one-loop Feynmen diagrams S transition rate Feynmen 6
8 2.1: feynmen diagram p-ps decay rate transition rate Γ(p P s 2γ) = mα5 2 = sec 1 (2.9) [5] α m o P s 3γ z -1,0,1 1 3 τ 1 τ 2 τ 3 σ = 1 3 dσ(pσe, p σe + k 1 τ 1, k 2 τ 2, k 3 τ 3 ) τ 1 τ 2 τ 3 σ (2π) 4 [ (p 1 p 2 ) 2 m 4 E 1 E 2 ] 1 1 3! M pσe,p σe + k 1 τ 1,k 2 τ 2 k 3 τ 3 2 k 1 k 2 k 3 d k 2 d k 3 dω 3 dψ 23 (2.10) p 1, E 1 p 2, E 2 dω 3 k 3 ψ 23 k 2 7
9 (p k 3 azimuthal angle 1 p 2 ) 2 m 4 E 1 E 2 M βα S βα = δ(β α) 2πiM βα δ 4 (p α p β ) scattering amplitude o-ps Feynmen diagram decay rate k 1, k 2, k 3 σ(o P s 3γ) σ(o P s 3γ) = 1 (p 3 27 π 6 1 p 2 ) 2 m 4 [ E 1 E 2 τ 1 τ 2 τ 3 σ ] 1 E1 +E 2 E1 +E 2 d k 2 d k 3 0 E 1 +E 2 k 2 1 3! M pσe,p σe + k 1 τ 1,k 2 τ 2 k 3 τ 3 2 k 1 k 2 k 3 (2.11) decay rate Γ(o P s 3γ) = π mα6 dx 1 dx 2 [( 1 x 1 ) 2 + ( 1 x 2 ) 2 + ( 1 x 3 ) 2 ] x3 =2 x 0 1 x 1 x 2 x 3 x 3 x 1 x 1 x 1 x 2 2 = 2 9π (π2 9)mα 6 = sec 1 (2.12) [5] o-ps one-loop diagram transition rate (7) 10 6 sec 1 [5] transition rate Γ transition rate Ps lifetime = transitionrate 1 p-ps o-ps p-ps sec o-ps sec one-loop sec 8
10 Na β + e + e Ps β + γ NaI Ps o-ps Na β + ADC 60 Co ADC 137 Cs ADC NaI NaI1,2,3 HV1,2,3 ADC1,2,4 TDC1,2,4 HV4 ADC5 TDC5 NaI SiO 2 β + 9
11 : 10
12 図 3.2: 配置図 図 3.3: 配置図 11
13 HV1 NaI1 div1 diccri1 FAN Coin TDC HV2 NaI2 div2 discri2 delay HV3 NaI3 div3 discri3 HV4 P.S. div4 discri4 Gate1 Gate2 ADC delay : HV Negative High Voltage Div divider P.S. Plastic Scintillator Discri discriminator Gate Gate Generator FAN Coin Coincidence Delay Delay1 Fixed Delay Delay2 ADC 12
14 Time P.S. threshold delay gate NaI threshold delay Coin. TDC1,2,4 decay time TDC5 3.5: TDC start stop TDC start NaI,P.S. coincidence TDC5 TDC1,2,4 TDC5 TDC1,2,4 delay ns
15 100 hp a A B C D NaI 22 Na γ NaI P.S delay,gain gate1 FAN coincidence gate gate1 824 ns delay1 TDCstart stop NaI 105 ns P.S. 840 ns fixed delay gate2 ADC 1600 ns dalay2 ADCgate delay out HV gain HV NaI1-3 HV Na β + ADC NaI 511 kev ADC HV 1300V 1200 V 1200 V ADC NaI
16 3.1: HV gain HV V pedestal 511 kev NaI NaI NaI HV NaI ADC calibration ADC NaI Co,Cs fitting 3.1 Energy[ kev ] = a ADC + b (3.1) 3.2: A ADC calibration pedestal 511 kev ADC ADC ADC : A a,b a ADC ADC ADC b 15
17 3.4: B calibration pedestal 511 kev ADC ADC ADC : B a,b a ADC ADC ADC b 3.6: C calibration pedestal 5112 kev ADC ADC ADC : C a,b a ADC ADC ADC b 16
18 3.8: D calibration pedestal 511 kev ADC ADC ADC : D a,b a ADC ADC ADC b TDC calibration TDC fixed delay TDC 3.2 fitting T ime[ ns] = c T DC + d (3.2) 17
19 3.10: TDC cariblation ns TDC TDC TDC TDC ns TDC TDC TDC TDC : TDC fitting c d TDC ± ± TDC ± ± TDC ± ± TDC ± ±
20 NaI1 200 kev 511 kev 1274 kev 200 kev 511 kev 100 kev threshold NaI2,NaI3 4.1: Energy vs ADCcount(NaI1) 19
21 4.2: Energy vs ADCcount(NaI2) 4.3: Energy vs ADCcount(NaI3) 20
22 4.4 NaI1 TDC T 1 T = const T 1 (4.1) const 0[ ns] NaI2,NaI3 4.4: Time vs TDCcount(NaI1) fitting τ f(t) = p exp( t τ ) + r (4.2) NaI1 21
23 4.5: Time vs TDCcount(NaI2) 4.6: Time vs TDCcount(NaI3) 22
24 4.1: [ ns] NaI ± NaI ± NaI ± const (511 kev,20 ns) p-ps (1274 kev,20 ns) β kev p-ps pick-off p-ps (511 kev,70 ns ) pick-off (511 kev,40 ns) p-ps (511 kev 100 ns ) p-ps pick-off o-ps 4.10 threshold NaI2,NaI3 4.2 TQ threshold threshold TDC TQ TQ T 23
25 4.7: Time vs Energy(NaI1) 4.8: Time vs Energy(NaI2) 24
26 図 4.9: Time vs Energy(NaI3) 図 4.10: Time vs Energy(NaI1) の分布 25
27 V V=0 w1 w2 Time T h Threshold 4.11: ADC = 1 2 h(w 1 + w 2 ) (4.3) T : w 1 = δ : h (4.4) T = (w 1 + w 2 )w 1 δ 2ADC (4.5) T E ADC T = const (T 1 a E + b ) (4.6) a = (const T 1 ) + ( + c) (4.7) E + b T 1 TDC TQ (1) kev 511 kev 1 0[ ns] 26
28 1274 kev 511 kev 0 ns (2) fitting a,b,c const T 1 = 0 T = a E + b + c (4.8) (1) ( 0[ ns]) TQ fitting 1274 kev p-ps 511 kev NaI1 150[ kev ] 1200[ kev ] NaI2 275[ kev ] 1200[ kev ] NaI3 150[ kev ] 1200[ kev ] fitting fitting 4.15 fitting (a,b,c) : a b c NaI ± ± ± ± NaI ± ± ± ± NaI ± ± ± ±
29 4.12: TQ (NaI1) 4.13: TQ (NaI2) 28
30 4.14: TQ (Na13) 4.15: TQ fitting fitting (NaI1) 29
31 4.3 NaI2 NaI2 NaI2 511 kev kev ADC NaI2 NaI2 4.3: ADC2 calibration-1 energy kev ADC 22 Na Cs Co : ADC2 calibration-2 energy kev ADC 22 Na Cs Co NaI3 4.2 NaI1 γ background NaI3 NaI1 4.4 pick-off o-ps p-ps p-ps o-ps o-ps p-ps o-ps,p-ps 30
32 4.5: ADC2 calibration-3 energy kev ADC 22 Na Cs Co p-ps p-ps o-ps p-ps pick-off pick-off pick-off 100 E E 520 fitting n(t) = exp( t τ ) + c (4.9) τ 100 E E 600 τ all τ pick : TQ τ all τ pick NaI1 n all (t)dt : 0 E 440[ kev ] t t + dt event n pick (t)dt : 440 E 600[ kev ] t t + dt event 31
33 n all,n pick n all = exp( t τ all ) + c (4.10) n pick = exp( t τ pick ) + c (4.11) N total (t) : t P s N pick (t) : t pick off P s N 3γ (t) : t o Ps 3γ P s N 0 : 0 P s Γ total (t)dt : t t + dt P s Γ pick (t)dt : t t + dt P s pick off Γ 3γ (t)dt : t t + dt P s o P s 3γ pick-off Γ 3γ (t) Γ 3γ (t) Γ total (t) = Γ pick (t) + Γ 3γ (4.12) pick-off Γ pick (t) Γ 3γ Γ total τ all Γ pick(t) Γ 3γ N 3γ = N pick = N total = t 0 t 0 t 0 (N 0 N total (t))γ 3γ (t)dt (4.13) (N 0 N total (t))γ pick (t)dt (4.14) (N 0 N total (t))γ total (t)dt (4.15) P compton : NaI compton P kouden : NaI 32
34 pick-off 2γ n all (t)dt = N 3γ t 3P compton + N pick 2P compton (4.16) t n pick (t)dt = N pick 2P kouden (4.17) t n all (t)dt = 3P compton (N 0 N total (t))γ 3γ (t) + 2P compton (N 0 N total (t))γ pick (t) n pick (t)dt = 2P kouden (N 0 N total (t))γ pick (t) 100 E 400 n all (t)dt 3P compton (N 0 N total (t))γ 3γ (t) (4.18) n all n pick N total Γ pick P compton P kouden Γ pick n pick (t) n all (t) = 2P kouden Γ pick (t) 2P compton Γ pick + 3P compton Γ 3γ (4.19) 2Γ pick(t) 3Γ 3γ (4.20) = p exp( t q ) + r (4.21) P kouden P compton Γ pick n all (t) = n 0 exp[ 1 t (1 + Γ pick )] + B (4.22) τ 3γ 0 Γ 3γ 100 kev threshold 33
35 count Enegy[keV] 4.16: threshold threshold 4.16 threshold count Γ pick(t) Γ 3γ threshold n all n pick background threshold background n T HR dt : threshold (4.23) n all (t)dt (n all (t) + n T HR )dt (4.24) n pick (t)dt (n pick (t) + n T HR )dt (4.25) ε = n T HR /(n T HR + n 3γ ) (4.26) 34
36 50[ ns] 4.7 background p-ps, o-ps BG background 4.18 background A H base Γ pick (t) pick-off o-ps KeV base o-ps scale o-ps pick-off ( pick off ) = ( ) ( scale ) (base ) N 3γ : 3γ (4.27) N pick : pick off (4.28) N T HR : T HR (4.29) ɛ : N T HR /(N T HR + N 3γ ) (4.30) background pick-off N pick (t) N 3γ (t) + N T HR 2Γ pick(t) 3Γ 3γ (4.31) = p exp( t q ) + r (4.32) 35
37 Γ pick (t)/γ 3γ pick-off : [ ns] A B C D E F G H base BG pick-off fitting pick-off 4.8 sebsection 4.8 Γ pick /Γ 3γ C D 6 D 5.1 fitting p = ± q = ± r = ±
38 4.17: number of events 700<t< h_bg_nai1 Entries 3528 Mean RMS : BG 37
39 number of events 500<t< h_i_nai1 Entries 877 Mean RMS : base 4.8: pick-off N T HR N pick N 3γ ɛ Γ pick /Γ 3γ A B C D E F G H
40 gamma_pick/gamma_3gamma vs time 0.28 χ 2 / ndf / 2 p ± 1 p e+07 ± p ± : pick-off τ 3γ = ± pick-off pick-off o-ps A,B,C,D 4.10 τ 3γ A section pick-off 39
41 4.9: A B pick-off A C β + A D A B C D 4.10: B pick-off pick-off Ps pick-off 4.21 TQ time vs energy 4.22 fitting τ normal = ± C τ not heated = ±
42 Time vs. Energy in NaI h0_te Entries Mean x Mean y RMS x 195 RMS y : B-Time vs Energy Decay Time of ortho positrons 300 h0_t_ortho Entries Mean RMS : B lifetime-fitting 41
43 Time vs. Energy in NaI h0_te Entries Mean x Mean y RMS x RMS y : C-Time vs Energy Decay Time of ortho positrons h0_t_ortho Entries Mean RMS : C lifetime-fitting 42
44 4.5.3 D β + β + Ps τ no taget = ± Time vs. Energy in NaI h0_te Entries Mean x Mean y RMS x RMS y : D-Time vs Energy 4.11: TQ A ± 6.08 B ± 7.87 C ± 6.30 D ±
45 Decay Time of ortho positrons h2_t_ortho Entries Mean RMS : D lifetime-fitting 44
46 5 5.1 TD- Ccalibration TQ pick-off pick-off pick-off NaI2 NaI3 NaI q = (x 1, x 2,..., x n ) x 1, x 2,..., x n σ x1, σ x2,..., σ xn q σ q σ q = ( q x 1 σ x1 ) 2 + ( q x 2 σ x2 ) ( q x n σ xn ) 2 (5.1) 5.2 TDCcalibration TQ (3.5.4),(4.12) ROOT fitting 5.3 pick-off Γ pick / Γ 3γ 45
47 N : 0 E 600 N 1 : 0 E 440 N b : base N b1 : N b 0 E 440 N 3γ : 3γ N pick : pick-off N pick : pick-off N 3γ : 3γ N T HR : THR ɛ 3γ : N T HR /(N T HR + N 3γ ) w,x,y u = Γ pick = 3N pick = 3 Γ 3γ 2N3γ 2 (1 ɛ 3γ) N pick = 3wx N 3γ 2y (5.2) w = 1 ɛ 3γ (5.3) x = N pick (5.4) y = N 3γ (5.5) N, N 1, N b, N b1 x = N N 1 N b1 N b (5.6) y = N 1N b N b1 (5.7) ɛ 3γ = N T HR N T HR + y (5.8) Q σ Q 5.1 σ x = ± N 1N ( b N ) b1 2 σ ( σ ) Nb1 2 ( σ ) Nb 2 ( σ ) N1 2 N (5.9) N b1 N 1 N b N b1 N b N 1 σ y = ± N 1N ( b σ ) N1 2 ( σ ) Nb 2 ( σ ) Nb (5.10) N b1 N 1 N b N b1 ( σ ) x 2 σ u = ± ( σ ) y 2 + (5.11) x y A A N, N 1, N b, N b1 46
48 5.1: N N 1 N T HR 100 T T T T T T T T : σn σ N1 σ Γpick /Γ 3γ 100 T T T T T T T T : base N b σn b N b1 σ Nb
49 gamma_pick/gamma_3gamma vs time 0.28 χ 2 / ndf / 2 p ± 1 p e+07 ± p ± : pick-off (5.11) (5.1) (5.12) fitting p q Γ pick /Γ 3γ = p exp( t q ) + r (5.12) p = ± (5.13) q = ± (5.14) r = ± (5.15) Γ pick / Γ 3γ N(t) = N 0 exp( 1 τ 3γ (t pq exp( t/q))) + B (5.16) τ 3γ = ± 6.00[ ns] (5.17) 48
50 6 6.1 TQ pick-off 10 (1) (2) (3) 6.1 NaI 49
51 6.1: ( min 1 ) error (1) (2) (3) TQ pick-off spin TQ cm pick-off pick-off 4.8 Γ pick /Γ 3γ NaI 50
52 pick-off Threshold pick-off Threshold Thr pick-off Thr. discri Thr. Thr. Hi-Vol gain photomul 1300V divider amp photomul divider Thr. amp Thr. BG BG Thr. Thr. Na Thr. Thr. 51
53 6.2.2 NaI3 4.9 NaI BG NaI BG NaI BG fitting pick-off BG BG BG 3 γ NaI coinsidence BG BG BG NaI 52
54 BG pick-off pick-off A 53
55 54
56 [1],,, [2],.F., [3] Michael A. Stroscio Positronium: A review of the theory o-ps p- Ps decay rate Feynmen diagrams [4] S.Weinberg Quantun theory of fields 1 [5] Gregory S.Adkins Radiative Corrections to Positronium Decay 55
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