2015/3/18

Transcription

1 2015/3/18

2 Overview C (Ps) QED Dirac S S M p o ( ) Calibration ADC Calibration ADC gain TDC Calibration TQ TQ TQ pick-off pick-off pick-off Accidental Threshold pick-off rate pick-off rate pick-off fitting time ADC2 calibration

3 TDC4 calibration ii

4 1 1.1 Overview 1.2 ( ) ( ) Schrödinger equation Bohr 2 ( ) γ 1 diagram ~~~~~~~~ ~~~~~~~~ ~~~~~~~~ ~~~~~~~~ ~~~~~~~~ 1: diagram (Quantum-Electrodynamics;QED) (singlet) (triplet) singlet (para-positronium;p-ps) triplet (ortho-positronium;o- Ps) ψ triplet = b ( ) d ( ) 0 b ( ) d ( ) 0 [ 1 2 b ( ) d ( ) + b ( ) d ( ) ] 0 (1-1-1) ψ singlet = 1 2 [b ( ) d ( ) b ( ) d ( ) ] 0 (1-1-2) para ortho ( )

5 with thetransition frequencies (in GHz), 2: a Hyper ne Splitting b c d e f Lyman g h i Lamb Shift We have also indicated the commonly used names for the most remarkable transitions. Physically, one has the followingselection rule for thedominant radiativetransitions: E1 M1 where E1are the electric dipole (or allowed) transitions (plain lines on the picture), by far the strongest transitions. Theforbidden transitions arethem1magnetic dipole transitions (dashed lines). The most interesting splitting isthe hyper ne splitting of positronium, because it is the most accessible both theoretically (1S states are relatively simple to treat) and experimentally (the higher excited states are predicted to cascade decay into lower states quite rapidly). In (1.1), the state,, appears separately 2 because of the so-called annihilation graph. The hyper ne splitting iscomputed fromtheexpectation values of the spin-spin andannihilation potentials between and states

6 para ortho 2 ([3] ) principal quantum number:n total spin quantum number:s(s = 2s + 1 ) orbital angular momentum number:l (l = 0, 1, 2... L=S,P,D... ) total angular momentum number:j n S L j n ( a) (hyperfine structure) (GHz) 1 a b( c,d,e,f) Lyman α g 18.2 h 12.8 i 8.39 Lamb 1: (E1 ) (M1 ) C (charge conjugation transformation;c ) (P ) CP 3 QED C (1-1-3),(1-1-4) a k,α, a k,α a k,α, a k,α (1-1-3) b (s), d (s) d (s), b (s) (1-1-4) Ps C triplet Ps a k 1,α 1 a k n,α n 0 ( 1) n a k 1,α 1 a k n,α n 0 (1-1-5) singlet Ps b ( ) d ( ) 0 d ( ) b ( ) 0 (1-1-6) = b ( ) d ( ) 0 (1-1-7) 1 [ b ( ) d ( ) d ( ) b ( ) ] 1 [d ( ) b ( ) b ( ) d ( ) ] (1-1-8) 2 2 = 1 2 [b ( ) d ( ) d ( ) b ( ) ] (1-1-9) C-symmetry p-ps (2,4,6,...) o-ps (3,5,7,...) j = 0, 1, 2 p-ps ( 3 P 1 )2 o-ps 3 ( 3 P 1 2 (Landau- Pomeranchuk-Yang theorem) 4 ) 1 3

7 α = 1/137 α 2 1/10000 (E1 ) 3 S 1, 1 S 0 3γ, 2γ (Ps) (Nonrelativistic Quantum Electrodynamics;NRQED) Ps (ultraviolet divergence; ) diagram (infrared divergence; ) diagram diagram 3/31/2015 upload.wikimed self-energy loop 3 ( Wikipedia ) 3: self-energy loop loop diagram NRQED NRQED on-shell Feynman gauge Coulomb gauge Schrödinger QED ( ) o-ps o-ps lifetime puzzle (1946 Pirenne,Wheeler ( ) p-ps ) 1949 A.Ore,J.L.Powell o-ps Lowest Order(LO) Γ theory LO Γ 0 = 7.21µsec M.Deutsch o-ps Γ exp (1951) = 6.8 ± (7)µsec R.H.Beers, V.W.Hughes Γ exp (1968) = 7.29(3)µsec D.W.Gidley,K.A.MArko Γ exp (1976) = 7.104(6)µsec W.E.Caswell, G.P.Lepage, J.R.Sapirstein Γ theory (1977) = (12)µsec Ann Arbor Γ exp (1987) = (13)µsec

8 1977 O(α) 1st o-ps lifetime puzzle 1987 O(α 2 ) 2nd o-ps lifetime puzzule Γ(Ann Arbor) = (10stat.)(8syst.) [ µs 1] (1-1-10) Γ(Tokyo) = (12stat.)(11syst.) [ µs 1] (1-1-11) Γ(theory) = (11) [ µs 1] (1-1-12) [ Γ(theory) = Γ A α ( π + α2 α ) 2 3 ln α + B 3α 3 ] π 2π ln2 α + C α3 π ln α (1-1-13) Γ 0 LO A = (10), B = 45.06(26), C = (23) (1-1-14) Γ 0 = 4α(π2 9) α 5 m 9π 2 (1-1-15) 1.2 QED (1-1-13) O(α 2 ) O(α) ( ) LO... (...) TeX! ( ) Ps notation ( ) A µ F µν Lorentz Maxwell F µν = µ A ν ν A µ (1-2-16) ν ν A µ = µ 0 j µ (1-2-17) µ A µ = 0 (1-2-18) 5

9 Lorentz (1-2-18) A µ A µ A µ = A µ + µ Λ (1-2-19) ν ν Λ = 0 (1-2-20) j µ = 0 Gauge A A = A + Λ A 0 A 0 = A Λ (1-2-21) (1-2-18) A µ 0 Λ = A 0 (1-2-22) A = 0 (1-2-23) j µ 0 (1-2-18) (1-2-23) 1 Boson A µ ϕ Lagrangian density ν ν ϕ = 0 (1-2-24) π(x) L(x) = 1 2 µϕ(x) µ ϕ(x) (1-2-25) π(x) = (1-2-24) Fourier ϕ(x) = k 0 = k (1-2-24) ϕ(x) L(x) = ϕ(x) (1-2-26) d 3 k (2π) 3 2k 0 q(k, t)eik x (1-2-27) d 2 dt 2 q(k, t) + k2 q(k, t) = 0 (1-2-28) k q(k, t) = q 1 (k)e ik0t + q 2 (k)e ik0 t (1-2-29) ϕ(x) = d 3 k (2π) 3 2k 0 (q 1(k)e ik 0 t+ik x + q 2 (k)e ik0t+ik x ) (1-2-30) ϕ(x) ϕ(x) = d 3 k [ ] (2π) 3 2k 0 q 1 (k)e i(k0t k x) + q 2 (k)e i(k0t+k x) + q1(k)e i(k0t k x) + q2(k)e i(k0 t+k x) (1-2-31) 2 4 k k 6

10 ϕ(x) = a(k) = [q 1 (k) + q 2(k)] d 3 k (2π) 3 2k 0 [ (q1 (k) + q 2 ( k))e ik x + (q 1(k) + q 2( k))e ik x] (1-2-32) ϕ(x) = d 3 k (2π) 3 2k 0 [ a(k)e ik x + a (k)e ik x] (1-2-33) π(x) = k 2 = 0, i.e. k 0 = k (1-2-34) d 3 k (2π) 3 2k 0 [ ik 0 a(k)e ik x + ik 0 a (k)e ik x] (1-2-35) [ϕ(x), π(x )] = iδ(x x ) (1-2-36) [π(x), π(x )] = [ϕ(x), ϕ(x )] = 0 (1-2-37) a(k), a (k) a(k), a (k) [ a(k), a (k ) ] = δ 3 (k k ) (1-2-38) H a(k), a (k) [a(k), a(k )] = [ a (k), a (k ) ] = 0 (1-2-39) H = = [ d 3 x π(x) ϕ(x) ] L = = d 3 kk [n ] 2 δ(3) (0) d 3 kk [ a (k)a(k) + a(k)a (k) ] (1-2-40) (1-2-41) n n = a (k)a(k) (1-2-42) 0 H = d 3 kk 0 a (k)a(k) (1-2-43) a (k), a(k) (1-2-40) : : a(k)a (k) := a (k)a(k) (1-2-44) ϕ (+) (x) ϕ ( ) (x) ϕ (+) (x) = d 3 ke ip x a(k) (1-2-45) (2π)3 2k0 7

11 ϕ ( ) (x) = d 3 ke ip x (2π)3 2k 0 a (k) (1-2-46) : ϕ(x)ϕ(y) : = : (ϕ (+) (x) + ϕ ( ) (x))(ϕ (+) (y) + ϕ ( ) (y)) : = ϕ (+) (x)ϕ (+) (y) + ϕ ( ) (x)ϕ (+) (y) + ϕ ( ) (y)ϕ (+) (x) + ϕ ( ) (x)ϕ ( ) (y) (1-2-47) H 0 = 0 (1-2-48) k a (k) 0 (1-2-49) T T (ϕ(x)ϕ(y)) = θ(x 0 y 0 )ϕ(x)ϕ(y) + θ(y 0 x 0 )ϕ(y)ϕ(x) (1-2-50) 2 T (1-2-50) Feynmann Propagator (1-2-33) F = i d 3 k x (2π)3 2kx 0 i F (x, y) = 0 T (ϕ(x)ϕ(y)) 0 (1-2-51) d 3 k y (2π) 3 2ky 0 [ 0 θ(x 0 y 0 )a(k x )e ikx x a e iky y + θ(y 0 x 0 )a(k y e iky y )a (k x )e ikx x 0 ] d 3 k x = i (2π) 3 2kx 0 {θ(x 0 y 0 )e ikx (x y) + θ(y 0 x 0 )e ik0 x (x y) } (1-2-52) θ(x) Heaviside step function 1 t > 0 θ(x) = 0 t < 0 θ(x) = i dα e iαx 2π α + iϵ (1-2-52) Lorentz F (x) = i 0 T (ϕ(x)ϕ(0)) 0 = d 4 k e ip x 1 (2π) 4 p 2 m 2 + iϵ (1-2-53) (1-2-54) (1-2-55) ϵ (α) α A µ (x) = α d 3 k [ (2π) 3 2k 0 a (α) (k)ϵ (α)µ e ik x + a (α) (k)ϵ (α)µ e ik x] (1-2-56) (ϵ (1), ϵ (2), k/ k) 1 ϵ (α) k ϵ (0) 0 8

12 1.2.3 Dirac 1/2 Dirac Dirac Fermion : γ µ {γ µ, γ ν } = γ µ γ ν + γ ν γ µ = 2η µν 1 (4) (1-2-57) 1 (4) 4 4 γ 0 Hermite γ j Hermite Dirac γ 0 = γ 0, γ j = γ j (1-2-58) (γ µ i µ m)ψ(t, x) = 0 (1-2-59) ψ(t, x) Dirac 4 ψ 1 (t, x) ψ(t, x) = ψ 2 (t, x) ψ 3 (t, x) ψ 4 (t, x) (1-2-60) (1-2-57) 4 4 γ 0 = ( 1 (2) (2) ), γ j = ( ) 0 σ j σ j 0 (1-2-61) 1 (2) 2 2 σ j 2 2 Pauli ( ) ( ) σ =, σ 2 0 i =, σ 3 = 1 0 i 0 ( ) (1-2-62) (1-2-61) Dirac-Pauli DP U DP Dirac Lorentz i ϕ γ µ = U γ µ DP U (1-2-63) ψ(x) ψ (x ) = S(ϕ)ψ(x) (1-2-64) S(ϕ) Unitary ) S(ϕ) = exp (i σi 2 ϕ = cos ϕ 2 + iσi sin ϕ 2 x µ, xν Lorentz-Boost (1-2-65) σ µν = i 2 [γµ, γ ν ] (1-2-66) Lorenz-Boost ψ(x) ψ (x ) = S(χ)ψ(x) (1-2-67) 9

13 ( S(χ) = exp i ) 2 σµν χ = cosh χ 2 iσµν sinh χ 2 (1-2-68) (1-2-66) Hermite (1-2-68) Unitary ψ ψ = (ψ 1, ψ 2, ψ 3, ψ 4)γ 0 = (ψ 1, ψ 2, ψ 3, ψ 4) (1-2-69) γ 0 Lorentz-Boost ψ S ψ ψ (x) = Sψ(x) ψ (x ) = ψ(x)s 1 (1-2-70) ψ ψ Lorentz-Boost (x µ ) v (x µ ) Dirac ψ(x) v ψ (x ) ψ (x ) = S(χ( v))ψ(x), p µ x µ = mt (1-2-71) 4 3 p 4 2 s ω p = p 2 + m 2 (1-2-72) ψ(t, x) = u(p, s) exp [ i(ω p t p x)] (1-2-73) 4 u(p, s) Dirac 1 0 u(p = 0, s = +) = 2m 0 0, u(p = 0, s = ) = 2m 1 0 (1-2-74) 0 0 s ψ(t, x) = v(p, s) exp [i(ω p t p x)] (1-2-75) 0 0 v(p = 0, s = ) = 2m 0 1, v(p = 0, s = +) = 2m (1-2-76) ū(p = 0, s)u(p = 0, s ) = 2mδ ss, v(p = 0, s)v(p = 0, s ) = 2mδ ss (1-2-77) v(p = 0, s)u(p = 0, s ) = ū(p = 0, s)v(p = 0, s ) = 0 (1-2-78) 10

14 ψ (x ) = u(p, s) exp( ip µ x µ ) (1-2-79) Dirac ψ (x ) = v(p, s) exp(ip µ x µ ) (1-2-80) (p µ γ µ m)u(p, s) = 0, ū(p, s)(p µ γ µ m) = 0 (1-2-81) (p µ γ µ + m)v(p, s) = 0, v(p, s)(p µ γ µ + m) = 0 (1-2-82) Lorentz-Boost ū(p, s)u(p, s ) = 2mδ ss, v(p, s)v(p, s ) = 2mδ ss (1-2-83) v(p, s)u(p, s ) = ū(p, s)v(p, s ) = 0 (1-2-84) 4 p µ Dirac 4 Dirac p µ Λ ± (p) Λ ± (p) = ±γµ p µ + m 2m (1-2-85) u(p, s)ū(p, s) = p µ γ µ + m (1-2-86) s=+, s=+, v(p, s) v(p, s) = p µ γ µ m (1-2-87) Dirac Dirac Lagrangian L π(x) L = ψ(x)(iγ µ µ m)ψ(x) (1-2-88) Hamilotnian H π(x) = L ψ(x) = iψ (x) (1-2-89) H(x) = π(x) ψ(x) L = ψ (x)γ 0 (iγ j j m)ψ(x) (1-2-90) Heisenberg Dirac (1-2-59) Dirac 1/2 {ψ α (t, x), iψ β (t, y)} = iδ αβδ (3) (x y) (1-2-91) {ψ α (t, x), ψ β (t, x)} = 0, {iψ α(t, x), iψ β (t, y)} = 0 (1-2-92) 11

15 u, v (1-2-81),(1-2-83),(1-2-84),(1-2-86),(1-2-87) ψ, ψ ψ(x) = ψ(x) = d 3 p (2π)3 2ω p d 3 p (2π)3 2ω p s=+, s=+, {b s (p)u(p, s)e ipx + d s(p)v(p, s)e ipx } (1-2-93) {b s(p)ū(p, s)e ipx + d s (p) v(p, s)e ipx } (1-2-94) (1-2-91),(1-2-92) b, d, b, d {b s (p), b r(q)} = δ r,s δ (3) (p q) (1-2-95) {d s (p), d r(q)} = δ r,s δ (3) (p q) (1-2-96) {b s (p), b r (q)} = {d s (p), d r (q)} = {b s(p), b r(q)} = {d s(p), d r(q)} = 0 (1-2-97) b d,b, d ψ(x) = ψ(x) = ψ(x) = ψ(x) = d 3 k (2π)3 sk 0 d 3 k (2π)3 sk 0 d 3 k (2π)3 sk 0 d 3 k (2π)3 sk 0 s=+, s=+, s=+, s=+, d s(k)v(k, s)e ik x (1-2-98) b s (k)u(k, s)e ik x (1-2-99) b s(k)ū(k, s)e ik x ( ) d s (k) v(k, s)e ik x ( ) 1 Dirac Feynnman Propagator : ψψ : = : ( ψ (+) + ψ ( ) )(ψ (+) + ψ ( ) ) : = ( ψ (+) ψ (+) + ψ ( ) ψ (+) ψ ( ) ψ(+) + ψ ( ) ψ ) ( ) is F (x y) αβ = 0 T (ψ α (x) ψ β (y)) 0 = θ(x 0 y 0 ) 0 ψ α (x) ψ β (y) 0 θ(y 0 x 0 ) 0 ψ β (y)ψ α (x)0 ( ) (1-2-93),(1-2-94) θ (1-2-54) S F (x y) = (iγ µ µ x + m) F (x y) d 4 k = e ik (x y) γ µ k µ + m (2π) 4 p 2 m 2 + iϵ d 4 k = e ik (x y) 1 (2π) 4 γ µ k µ + iϵ ( ) 12

16 1.2.4 S Schrödinger H (S) H (S) 0 H (S) I H (S) I Schrödinger Φ (S) (t) Schrödinger = H (S) Φ (S) ( ) t i Φ(S) Φ(t) = e ih(s) 0 t Φ (S) ( ) O(t) = e ih(s) 0 t O (S) e H(S) 0 t ( ) O (S) Schrödinger Φ(t) i Φ t [ = i ih (S) 0 e ih(s) 0 t Φ (S) + e ih(s)t 0 = H (S) 0 e ih(s) 0 t Φ (S) + e ih(s)t 0 Φ (S) ] t ( H (S) 0 + H (S) I ) e ih(s) 0 t e ih(s) 0 t Φ (S) = H I Φ ( ) O Ȯ = i[h (S) 0, O] = i[h 0, O] ( ) H 0 H (S) 0 ( ) U(t, t 0 ) Φ(t) = U(t, t 0 )Φ(t 0 ) ( ) ( ) U U(t 0, t 0 ) = 1 ( ) ( ),( ) ( ) t U(t, t 0 ) = 1 i = 1 i i t U(t.t 0) = H I U(t, t 0 ) ( ) t U(t, t 0 ) = 1 i dth I (t)u(t, t 0 ) t 0 ( ) t 0 dt 1 H I (t 1 ) t +( i) n t t 0 dt 1 H I (t 1 ) + ( i) 2 t 0 dt 1 t1 [ t1 ] 1 i dt 2 H I (t 2 )U(t 2, t 0 ) t 0 t 0 dt 2 t t 0 dt 1 tn 1 t1 t 0 dt 2 H I (t 1 )H I (t 2 ) + t 0 dt n H I (t 1 )H I (t 2 ) H I (t n ) + ( ) t 0 i t > t 0 f Φ f U(t, t 0 )Φ i 2 = U fi (t, t 0 ) 2 ( ) 13

17 i f ( ) w = 1 t t 0 U fi (t, t 0 ) δ fi 2 ( ) ( ) 1 ( ) δ fi S S = U(, ) ( ) ( ) S S ( ) Φ( ) = SΦ( ) ( ) S = S (0) + S (1) + S(2) + = 1 i t1 dt 1 H I (t 1 ) + ( i) 2 dt 1 dt 2 H I (t 1 )H I (t 2 ) t1 +( i) n dt 1 dt 2 tn 1 dt n H I (t 1 )H I (t 2 ) H I (t n ) + ( ) ( ) α(<< 1) S S SS = S S = 1 ( ) t1 S (2) = ( i) 2 dt 1 dt 2 H I (t 1 )H I (t 2 ) ( ) = ( i) 2 dt 1 ( ) ( ) t 1 dt 2 H I (t 2 )H I (t 1 ) ( ) ,2 ρ 1, ρ 2 v rel V T N σ σ = N/(V T ρ 1 ρ 2 v rel ) ( ) ( 1) ; θ θ + dθ N dn, σ dσ dσ = dn/(v T ρ 1 ρ 2 v rel ) ( ) dσ dσ/dθ Target( 2) 1 14

18 (M 1, q 1 ) (M n, q n ) S v rel Target v 1, v 2 ( ) v rel = v 1 v 2 = p 1 /p 0 1 p 2 /p 0 2 = (p 1 p 2 ) 2 m 2 1 m2 2 /(p0 1p 0 2) ( ) ρ k k = d 3 xψk(x)ψ k (x) = d 3 x ψ k (x) 2 ( ) k V = d 3 x ( ) ρ = k k /V ( ) k k = (2π) 3 2k 0 δ (3) (k k ) ( ) k k = (2π) 3 2k 0 δ (3) (0) = 2k 0 d 3 xe ik x k=0 = 2k 0 d 3 x = 2k 0 V ( ) ρ = 2k 0 ( ) n n i d 3 k i (2π) 3 2k 0 k 1 k n k 1 k n = 1 ( ) n = 0 = 0 0 m m! Ψ Ψ = n n i d 3 k i (2π) 3 2k 0 k 1 k n k 1 k n Ψ ( ) Ψ Ψ = n n i d 3 k i (2π) 3 2k 0 k 1 k n Ψ 2 ( ) Ψ Ψ Ψ k i k i + dk i n n i d 3 k i (2π) 2 2k 0 k 1 k n Ψ 2 ( ) 15

19 Ψ α S α β ( ) dn dn = n i=1 d 3 k i (2π) 3 2k 0 β α 2 ( ) S α β β α δ (4) (k β k α ) β S α = i(2π) 4 δ (4) (k β k α )M βα ( ) M βα M β α V T = d 3 x dt = d 4 x ( ) M βα β S α 2 = (2π) 4 δ(4)(k β k α )M βα 2 ( ) (2π) 4 δ (4) (k β k α ) = (2π) 4 δ (4) (k β k α )(2π) 4 δ (4) (0) = (2π) 4 δ (4) (k β k α ) d 4 x dn = V T i=1 = V T (2π) 4 δ (4) (k β k α ) ( ) n d 3 k i (2π) 3 2ki 0 (2π) 4 δ (4) (k β k α ) M βα 2 ( ) ( ),( ),( ) ( ) dσ = n i=1 σ = i=1 d 3 k i M βα 2 (2π) 3 2k 0 4 (2π) 4 δ (4) (k (k 1 k 2 ) 2 m 2 β k α ) ( ) 1 m2 2 n d 3 k i M βα 2 (2π) 3 2k 0 4 (2π) 4 δ (4) (k (k 1 k 2 ) 2 m 2 β k α ) ( ) 1 m2 2 α m n k i, k i + dk i ( ) dn = V T n d 3 k i (2π) 3 2ki 0 (2π) 4 δ (4) (k β k α ) M βα 2 ( ) i=1 k α = (m, 0, 0, 0) (1 ) 2k 0 αv = 2mV 16

20 dγ = dn n 2mV T = d 3 k i (2π) 3 2ki 0 2 i=1 M βα 2 2m (2π)4 δ (4) (k β k α ) ( ) dγ dω = k 32π 2 m 2 M βα 2 ( ) dγ Γ = tot n i=1 d 3 k i (2π) 3 2k 0 i M βα 2 2m (2π)4 δ (4) (k β k α ) ( ) Γ τ = 1/Γ τ (f) Γ f Γ B f = Γ f /Γ f S S (2) S (2) = ( i) 2 dt 1 = ( i) 2 t 1 dt 2 H I (t 1 )H I (t 2 ) d 4 x 1 t 2 <t 1 d 4 x 2 H int (x 1 )H int (x 2 ) ( ) x 2 4 t 2 < t 1 S (2) = ( i) 2 dt 1 = ( i) 2 t 1 dt 2 H I (t 2 )H I (t 1 ) d 4 x 1 t 2 >t 1 d 4 x 2 H int (x 2 )H int (x 1 ) ( ) t 2 t 1 2 e + e e + + e 2γ ( ) Φ i = e e + = b (s ) (p )b (s +) (p + ) 0 ( ) Lagrangian Φ f = 2γ = a k 1 α 1 a k 2 α 2 0 ( ) L = e : ψ(x)γ µ ψ(x) : A µ (x) ( ) S (0) S (1) S S (2) S fi S fi = ( e) 2 d 4 x 1 d t 4 x 2 2γ : ψ(x 1 )γ µ ψ(x 1 )A µ (x 1 ) :: ψ(x 2 )γ ν ψ(x 2 )A ν (x 2 ) : e e + ( ) 2 <t 1 17

21 A µ ( ) S fi = ( e) 2 d 4 x 1 d t 4 x 2 0 : ψ(x 1 )γ µ ψ(x 1 ) :: ψ(x 2 )γ ν ψ(x 2 ) : e e + 2γ A µ (x 1 )A ν (x 2 ) 0 ( ) 2 <t 1 2γ A µ (x 1 )A ν (x 2 ) 0 A µ (x 1 ) x = 2γ A µ (x 1 )A ν (x 2 ) 0 [( ) ( ) ( ) ( )] 1 2ω1 V ϵ(α 1)µ e ik 1 x 1 1 2ω2 V ϵ(α 2)ν e ik 2.x ω2 V ϵ(α 2)µ e ik 2 x 1 1 2ω1 V ϵ(α 1)ν e ik 1 x ( ) 2 ϵ (α1)µ ϵ (α1)µ = (0, ϵ (α1) ) ( ) ϵ α1 k 1 k 1 = ω 1 : ψγψ : : ψγ µ ψ : = : ( ψ (+) + ψ ( ) ) α (γ µ ) αβ (ψ (+) + ψ ( ) ) β : = (γ µ ) αβ (+) ( ψ α ψ (+) ( ) β + ψ α ψ (+) β ψ ( ) β ψ α (+) ( ) + ψ α ψ β ) ( ) ( ) ( ) ψ (+) α,β (x 1) 0 0 t 1 ψ α (x 1 )ψ β (x 1 ) ψ α (+) (x 1 )ψ ( ) β (x 1) (γ µ ) αβ γδ (γ ν ) ψ(+) α (x 1 )ψ (+) β (x ( ) 1) ψ γ (x 2 )ψ (+) δ (x 2 ) ( ) (γ µ ) αβ γδ (γ ν ) ψ(+) α (x 1 )ψ (+) β (x ( ) 1) ψ δ (x 2 )ψ γ (+) (x 2 ) ( ) (+) (+) ( ) ψ α ψ α (x 1 )ψ (+) (x 2 ) δ (γ µ ) αβ γδ (γ ν ) ψ(+) α (x 1 )ψ (+) = (γ µ ) αβ (γ ν ) γδ ψ (+) β (x 1) β (x ( ) 1) ψ γ (x 2 )ψ (+) δ (x 2 ) ψ ( ) γ (x 2 ) ψ (+) α (x 1 )ψ (+) δ (x 2 ) ( ) ( ) x 1, x 2 µ ν, αβ γδ ( ) (γ µ ) αβ γδ (γ ν ) ψ(+) = (γ µ ) αβ (γ ν ) γδ ψ(+) γ α (x 1 )ψ (+) β (x 2 )ψ ( ) β (x 1) (x ( ) 1) ψ δ (x 2 )ψ γ (+) (x 2 ) ψ (+) α (x 1 )ψ (+) δ (x 2 )v ( ) t 2 t 1 18

22 S fi = ( e) 2 d 4 x 1 d 4 x 2 2γ A µ (x 1 )A ν (x 2 ) 0 (γ µ ) αβ (γ ν ) γδ [ (x ( ) (+) 1) ψ γ (x 2 ) ψ α (x 1 )ψ (+) 0 0 ψ (+) β δ (x 2 ) e e + θ(t 1 t 2 ) ] (+) ψ γ (x 2 )ψ ( ) β (x (+) 1) ψ α (x 1 )ψ (+) δ (x 2 ) e e + θ(t 2 t 1 ) ( ) ψ (+) β (x ( ) (+) 1) ψ γ (x 2 ), ψ γ (x 2 )ψ (+) n n n β ψ (+) α (x 1 )ψ (+) (x 2 ) δ 0 ψ (+) β (x 1) ψ γ ( )(x 2 ) = 0 ψ (+) β (x 1) ψ ( ) γ (x 2 ) 0 0 = 0 ψ (+) β (x ( ) 1) ψ γ (x 2 ) 0 ψ (+) α (x 1 )ψ (+) ψ +) δ (x 2 ) e + e α (x 1 )ψ (+) δ (x 2 ) e + e [ ] [ ] m m E + V v(s+) α (p + )e ip+ x1 E V u(s ) δ (p )e ip x2 S ( ) S fi = ( e) 2 [ m [ d 4 x 1 [( ) ( ) ] 1 d 4 x 2 2ω1 V ϵ(α 1)µ e ik 1 x 1 1 2ω2 V ϵ(α 2)ν e ik 2 x 2 + {k 1 k 2, α 1 α 2 } E + V v(s +) α (p + )e ip + x 1 ] (γ µ ) αβ 0 ψ (+) β (x ( ) 1) ψ γ (x 2 ) 0 θ(t 1 t 2 ) 0 ] m δ (p )e p x 2 ] (+) ψ γ (x 2 )ψ ( ) β (x 1)θ(t 2 t 1 ) 0 [ (γ ν ) γδ E V u(s ) ( ) ( ) = ( e) 2 d 4 1 x 1 d 4 x 2 2ω1 V ϵ(α 1)µ e ik 1 x 1 1 2ω2 V ϵ(α 2)ν e ik 2 x 2 ( ) ( ) m E + V v(s+) α (p + )e ip+ x1 (γ µ ) αβ 0 T (ψ β (x 1 ) ψ m γ (x 2 )) 0 (γ ν ) γδ E V u(s ) δ (p )e p x2 +{k 1 k 2, α 1 α 2 } ( ) ( ) ( ) ( ) m m 1 1 = ( e) 2 E V E + V 2ω 1 V 2ω 2 V d 4 x 1 d 4 x 2 e ik1 x1 ik2 x2 ϵ (α1)µ ϵ (α2)ν e ip+ x1+ip x2 ( vγ µ i (2π) 4 d 4 e iq (x 1 x 2 ) ) (γ q + m) q q 2 m 2 γ ν u + iϵ +{k 1 k 2, α 1 α 2 } ( ) ( ) ( ) Feynman Propagator ( ) ( ) 1 diagram, 2 diagram M S ( ) x 1, x 2 d 4 x 1 d 4 x 2 exp[i( k 1 + p + q) x 1 ] exp[i( k 2 + p + q) x 2 ] = (2π) 4 δ (4) [q (p + k 1 )](2π) 4 δ (4) [q (p + k 2 )] ( ) ( ) δ (4) q p + k 2 q p + k 1 M fi 19

23 S fi = δ fi i(2π) 4 δ (4) (p + p + k 1 k 2 ) ( m E V ) ( ) ( ) ( ) m 1 1 M fi ( ) E + V 2ω 1 V 2ω 2 V δ fi 0 M fi = e 2 v [ (s +) (p + ) γ ϵ (α 1) iγ ( p + k 2 ) + m ( p + k 2 ) 2 m 2 + iϵ γ ϵ(α 2) + γ ϵ (α 2) iγ ( p ] + k 1 ) + m ( p + k 1 ) 2 m 2 + iϵ γ ϵ(α 1) u (s ) (p ) ( ) v rel = p E + ( ) :1 d 3 p δ(e E) = p 2 dω ( ) E/ p k 2 δ(e + + E ω 1 ω 2 )d 3 k 1 θ 1 ϕ 1 p + k 1 = p p + ω 1 cos θ 1 + ω 2 1 = k 2 = ω 2, ( ) m + E + = ω 1 + ω 2 ( ) ( ) Ef = (ω 1 + p + k 1 ) k 1 θ 1ϕ 1 k 1 = 1 + k 1 p + cos θ 1 ω 2 = m(m + E +) ω 1 ω 2 ( ) ( ) dσ dω Lab = 1 k 1 2 2π 1 4m p + (2m)2 2ω 1 (2π) 3 M fi 2 2ω 2 ( E f / k 1 ) θ1 ϕ 1 = ω1 2 4 (4π) 2 m 2 M fi 2 (2m) 2 ( ) p + m + E + M fi 2 p = (m, 0, 0, 0), ϵ (α) = (0, ϵ (α) ) ( ) (γ p )(γ ϵ (α) )u (s) (p ) = (γ ϵ (α) )mu (s) (p ) ( ) 20

24 ϵ (α) ϵ (α1) ϵ (α2) ( ) ( p + k 1,2 ) 2 = m 2 2mω 1,2 ( ) M fi = e 2 v(0) [ γ ϵ (α 1 ) γ k 2 γ ϵ (α 2) + γ ϵ(α2) γ k 1 γ ϵ (α1) ] u(p ) ( ) 2mω 2 2mω M fi 2 = e4 4 s +,s = e4 4 = e4 4 Tr (u (s ) O v (s+) )( v (s+) Ou (s ) ) s +,s ū (s ) γ 0 O γ 0 v (s+) v (s+) Ou (s ) s +,s ( [γ 0 O γ 0 γ p ) + + m O 2m ( )] γ p + m 2m ( ) O ( ) 1/4 Ps 4 a µ γ 4 (γ a) = γ 4 [γ a + γ 4 (ia 0 ) ]γ 4 = γ a ( ) Tr[(γ ϵ (α 2) γ k 2 γϵ (α 1) )(iγ p + + m)(γ ϵ (α 1) γ k 2 γ ϵ (α 2) )( γ p + m)] ( ) m Trace 0 0 m 2 0 γ k 2 γ ϵ (α 1) γ ϵ (α 1) γ k 2 = k 2 2 = 0 ( ) (γ a)(γ a) = a 2 ( ) (γ a)(γ b) + (γ b)(γ a) = γ µ a µ γ µ b µ + γ µ b µ γ µ a µ = 2δ µν a ν b µ = 2a b ( ) ( ),( ) ( ) ϵ (α 1,2) p = 0, k 1,2 2 = 0 ( ) Tr[(γ a)(γ b)] = Tr[(γ b)(γ a)] = 4a b ( ) Tr[γ ϵ (α 2) γ k 2 γ ϵ (α 1) γ p + γ ϵ (α 1) γ k 2 γ ϵ (α 2) γ p ] = 16(ϵ (α1) k 2 ) 2 (k 2.p ) + 8(k 1 p )(k 2 p ) = 16mω 2 (ϵ (α1) k 2 ) 2 + 8m 2 ω 1 ω 2 ( ) 21

25 ϵ (α1) k 1 = 0 ( ) k 2 p + = k 1 p ( ) Trace 1 4 s +,s M fi 2 = [ e4 ω2 2(2m) 2 + ω ] (ϵ (α 1) ϵ (α2) ) 2 ω 1 ω 2 ( ) dσ ω = 1r 2 2 [ 0 ω2 + ω ] (ϵ (α 1) ϵ (α2) ) 2 dω Lab 8 p + (m + E + ) ω 1 ω 2 ( ) ( ) ( ) ω = ω 1 = ω 2, k = k 1 = k 2, p + = mv + ( ) ( ) dσ = r2 0 (1 (ϵ (α 1) ϵ (α2) ) 2 ) ( ) dω Lab 4v + 2 σ tot = 1 2 ( ) dσ dω ( ) dω r 0 σ tot = r2 0 4v + 2 2π = πr2 0 v + ( ) p- R = σ tot v + ρ ( ) ρ Ps a 0 Bohr ρ = ψ 1S (x = 0) 2 = 1 π(2a 0 ) 3 ( ) n = 1 1 S Γ(n = 1, 1S 2γ) = lim v + 0 4σunpol tot v + ψ 1s (x = 0) 2 ( ) = ( α ) 2 1 4π m π(2/αm) 3 = 1 2 α5 m ( ) τ singlet = 2 α 5 m sec ( ) 22

26 o- p-ps ( )... o-ps [1]... P = 2π H F A 2 ρ ( ) H F A = I,II H F II H III H IA (E A E I )(E A E II ) ( ) H F A = e 3 ((2π) 3 /k 1 k 2 k 3 L 3 ) 13 k H IA = e(wπ/kl 3 ) 1/2 (u I, α au A ) ( ) I,II (u F, α a 3 u II )(u II, α a 2 u I )(u I, α a 1 u A ) (E 1 k 1 E )(k 3 E 2 E ) ( ) E 1, E 2 = m initial energy k 1, k 2, k 3 E, E k 1 + k 2 + k 3 = 0; k 1 + k 2 + k 3 = 2m ( ) κ L 3/2 = (κ/π) 1/2 ( ) H F A = (2πe 3 /m 2 L 3 )(κ 3 /k 1 k 2 k 3 ) 1/2 (t 1 + t 2 + t 3 ) u ( ) t 1 = a 1 (a 2 a 3 ) a 2 (a 3 a 1 ) a 3 (a 1 a 2 ) + a 1 (a 2 a 3) a 2(a 3 a 1 ) a 3(a 1 a 2), a 1 = a 1 (k 1 / k 1 ) ( ) polarizations H F A 2 = (32π 2 e 6 /3L 3 )(κ 3 /k 1 k 2 k 3 ) {[1 cos(k 2 k 3 )] 2 + [1 cos(k 3 k 1 )] 2 + [1 cos(k 1 k 2 )] 2 } ( ) ρ = (2π) 5 L 6 k 1 k 2 k 3 dk 2 dk 1 dω 1 ( ) 1/τ = (16e 6 κ 5 /9πm 3 ) m 0 F (k 1 )dk 1 = (16/9π)(π 2 9)(e 6 κ 3 /m 2 ) ( ) F (k 1 ) = = 2 m m k 1 { m2 (m k 1 ) 2 k 2 2 k2 3 + m 2(m k 2 ) 2 k 2 3 k2 1 + m 2(m k 3 ) 2 } k1 2k2 2 { k1 (m k 1 ) (2m k 1 ) 2 2m(m k 1) 2 (2m k 1 ) 3 log m k 1 m + 2m k 1 + 2m(m k 1) k 1 k1 2 log m k 1 m } ( ) (...) τ singlet /τ triplet = (4/9π)(π 2 9)(e 2 /ħc) ( ) 23

27 2 2.1 ( ) ( 22 Na) ( Ps) e + ( P.S.) 22 Na e + NaI(Tl) SiO 2 22 Na e + Ps P.S. 2.2 NaI γ Ps(para-,ortho-) シリカパウダー P.S. e + e + signal 2 module signal Na + e + P.S. signal1 2 P.S. e + Ps Ps para-ps,ortho-ps 2 ortho- 3 Ps (para- 2 ortho- 3 ) NaI signal2 4 signal1 signal2 module 5 4 pick-off TQ Ps 22 Na 4: 24

28 2.3 実験装置の配置 以下に 実験装置の実際のセットアップの写真を掲載する 図 5a が実験時の様子で 遮光ビニルを外す前と外した後のものを掲載 した (a) 実験装置の全体図 (c) ビニルを外した図:2 (b) ビニルを外した図:1 図 5: 実験時の実際の配置の様子 P.S. NaI 22Na シリカパウダー 図 6: 装置内部の様子 図 5b は装置上部の鉛ブロックを外して中が見えるようにしたものである 図 5b を見ると分かるように 図 4 で説明し た原理図と同じ配置になるように各装置を配置してある 25

29 2.4 ( 5c) HV 1 NaI 1 div discri 1 fan in&out 105ns delay coin gate HV 2 NaI 2 div discri 1 105ns delay start HV 3 NaI 3 div discri 1 105ns delay TDC gate ns HV 4 P.S. div discri 2 940ns delay gate ns delay(105ns) ADC gate delay(105ns) module 7: HV(1 4) div 2 discri(1,2) div discri NIM fan in,out 3 discri or module 3 in out coin 2 coincidence delay delay module gate1,2 NIM NIM ADC TDC start 26

30 Ps P.S. discri 2 NIM delay 940ns NIM gate 1 NaI NIM 1200ns discri 1 NIM delay 崩壊時間 105ns NIM coin NIM TDC1~3 TDC4 8: 1 P.S. discri2 NIM discri2 2 discri2 gate1 1200ns NIM gate1 discri2 gate1 module discri2 940ns delay 3 NaI discri1 NIM discri1 NIM 105ns delay 4 gate1 discri1 coincidence coin discri1 module TDC1 3 coin NIM 105ns NIM discri1 delay 5 Ps discri1 discri2 P.S. + NaI P.S. delay (TDC4) 940ns MPa 27

31 HV(V) NaI NaI NaI P.S : HV THR(mV) THR THR : discri THR (THR ) ,380,028 event ADC TDC 9 10 Count NaI th th1 Entries e+07 Mean RMS Channel Count NaI th th2 Entries e+07 Mean RMS Channel Count NaI th th3 Entries e+07 Mean RMS Channel 9: ADC 28

32 Count NaI th5 Entries e+07 Mean RMS th Count NaI02 Entries e+07 Mean RMS Channel Channel Count NaI th7 Entries e+07 Mean RMS Count TDC4 th8 Entries e+07 Mean 3420 RMS Channel Channel 10: TDC 3.1 Calibration ADC Calibration ADC ADC Calibration set up 22 Na 511KeV 1274KeV 60 Co 1173KeV 1332KeV 137 Cs 662KeV 5 Calibration 4: ADC Energy (KeV) ADC1 ADC2 ADC

33 3 3 3 ADC Calibration E 1 [KeV ] = ADC (3-1-1) E 2 [KeV ] = ADC (3-1-2) E 3 [KeV ] = ADC (3-1-3) ADC Calibration 11 Count NaI01 10 th1 Entries Mean RMS Energy (KeV) Count NaI02 10 th2 Entries Mean RMS Energy (KeV) Count NaI03 10 th3 Entries Mean RMS Energy (KeV) 11: ADC Calibration 30

34 3.1.2 ADC gain 1 750,000event 511KeV : 13: 511KeV 31

35 14 Energy = a (Channel b) a gain Calibration Energy 14: 511KeV(Channel)-0KeV(Channel) 15: gain 32

36 Channel Channel gain ADC 16 2 ADC Mean RMS Mean gain ADC 16: 750,000 1,500,000event ADC 33

37 17: Mean RMS pedestal veto gate generator NIM ADC setting event TDC1 3 event TDC1 3 event NaI1,NaI2 NaI3 NaI3 threshold Energy 34

38 3.1.4 TDC Calibration 18: ADC TDC fixed delay fixed delay TDC Calibration fixed delay 5 5: fixed delay TDC Time (ns) TDC fitting ( 19) fitting Calibration TDC4 20 T ime[ns] = T DC (3-1-4) 35

39 19: TDC Calibration 20: TDC Calibration 10 TDC1 3 Channel(Time) TDC1 3 start stop delay TDC1 3 event 36

40 3.2 TQ TQ 21 NaI threshold TDC4 γ TQ 21: NaI TQ TQ T T t 0 y max y max E threshold y 0 T (E) = y 0t 0 y max 1 E (3-2-5) fitting T (E) = p 0 (Energy p 1 ) p2 + p 3 (3-2-6) fitting Time=140ns p-ps p-ps 0.13ns Ps 37

41 22: TQ TQ Energy v.s.time(ns) TQ fitting fitting 6 23: TQ 38

42 24: TQ 39

43 6: TQ Parameter NaI1 NaI2 NaI3 p p p p pick-off TQ Time time fit fitting p[1] (ns) N(t) = p[0]exp( t ) + p[2] (3-2-7) p[1] 0 600keV o-ps γ 511keV 511keV 600keV fitting fitting 26 7 pick-off Accidental 7: TQ Lifetime-fitting NAI (ns) NaI ±0.63 NaI ±0.60 NaI ± pick-off pick-off Γ all Γ 3γ... Γ all = Γ 3γ + Γ pick off (t) (3-3-8) Γ pick off... ( ) Γ 3γ ( ) Γ n(t) = n 0 exp(t Γ) + b (3-3-9) ( ) n(t) = n 0 exp [ t t 0 (Γ 3γ + Γ pick off (t))dt ] + b = n 0 exp [ t τ 3γ t 0 (1 + Γ pick off(t) Γ 3γ )dt ] + b (3-3-10) n(t) Γ pick off (t) Γ 3γ τ 3γ Γ pick off (t) Γ 3γ 40

44 25: TQ eV 1/30 ev( ) thermalization [11] [12] pick-off ( ) pick-off pick-off pick-off 41

45 Γ pick off (t) = p[0] exp( 1 t ) + p[2] (3-3-11) Γ 3γ p[1] 3γ pick-off n 3γ n pick off Γ pick off (t) Γ 3γ α...pick off γ β... γ n pick off (t) n 3γ n 3γ β Γ 3γ (3-3-12) n pick off α Γ pick off (t) (3-3-13) = n pick off(t) α n 3γ β Γ pick off(t) Γ 3γ (3-3-14) Accidental Accidental Accidental 900ns 1050ns 6 Accidental 1250keV 511keV Na 22 γ Accidental 26 ( ) Accindental ( ) 600keV Threshold threshold 100keV 100keV pick-off rate n pick off (t) n 3γ n pick off (t) n 3γ 125ns 50ns 120ns 130ns Accidental pick-off pick-off 27 pick-off 125ns 50ns 28(NaI1) 29(NaI2) 30(NaI3) A. Accidental ( ) ( : : ) B. A 511keV pick-off ( : pick-off ) C. B kev n 3γ B pick-off kev n pick off 42

46 : Accidental e-nai3 th5 Entries Mean RMS : PickOff pick-off rate 2n pick off (t) 3n 3γ (ns) 31 B 511keV ( ) : pick-off NaI p[0] p[1] p[2]

47 28: (NaI1) 29: (NaI2) pick-off Accidental 9 pick-off Accidental fitting time Accidental 800ns( NaI3 600ns 44

48 30: (NaI3) 9: pick-off NaI (ns) ± ± ± ns ) refkaishi NaI root 34 NaI 10 10: NaI start time(ns) end time(ns) ADC2 calibration 45

49 31: pick-off TDC4 calibration pick-off 46

50 32: NaI3-TDC4-BG th1 Entries Mean RMS χ 2 / ndf / 63 p e+04 ± 3.631e+02 p ± 1.5 p ± NaI3-TDC4-BG th1 Entries Mean RMS χ 2 / ndf / 63 p e+04 ± 3.631e+02 p ± 1.5 p ± Time Time NaI3-TDC4-BG th1 Entries Mean RMS χ 2 / ndf / 63 p e+04 ± 3.631e+02 p ± 1.5 p ± Time ADC2 calibration ADC2 ADC2 calibration calibration a 11 NaI2 3% 3 NaI 1% 11: a / (ns) threshold threshold threshold threshold 2 47

51 33: A... (3-4-15) B... ( ) (3-4-16) 34: threshold 12 3% 12: threshold (ns) A B

52 3.4.4 TDC4 calibration TDC calibration root TDC calibration T ime = ( ± ) count + (3-4-17) 13 13: TDC calibration (NaI1) TDC calibration (ns) Time=0.2526*count ± 0.8 time=0.2488*count ± % ± 7.24 ns 14 14: (% ) 3 ADC2 calibration 1 4 TDC4calibraion 0.5 fitting Accidental [12] pick-off threshold... threshold HV threshold 49

53 4 setting 1. (a) (b) 2. (a) (b) (c) NaI (d) NaI (e) (f)

54 No.0 setting event 1.2Hz 35: No.0 No.1 No.0 22 Na setting 36: No Hz 51

55 No.2 No.0 setting setting Ps TDC4 decay curve 37: No.2 38 decay curve Ps Ps 38: No.2 TDC4 52

56 No.3 No.0 setting setting β No.2 β ( ) setting β 39: No : No.3 TDC4 No.4 NaI No.0 setting NaI β γ event 1.6Hz 53

57 54

58 No.5 setting P.S. No : No.5 42: No.5 TDC decay curve γ β Ps NaI P.S. γ NaI 55

59 No.6 part.1 1.2Hz NaI event 43: No : No.6 56

60 45: No.6 ADC 46: No.6 TDC ADC 511KeV TDC decay curve NaI event decay curve 57

61 No.7 part.2 No.6 ADC 511KeV TDC decay curve 0.6Hz 47: No : No.7 58

62 49: No.7 ADC 50: No.7 TDC ADC 511KeV TDC decay curve decay cueve setting Ps 59

63 No.8 0.2Hz No.7 decay curve 51: No : No.8 60

64 53: No.8 ADC 54: No.8 TDC ns event 61

65 No.9 setting Hz event 55: No : No. Condition rate (Hz) ADC (511KeV peak) TDC (decay curve) decay curve NaI part part P.S. 8.5 decay curve 1. Ps (No.2,3) 2. (No.2,3) 3. Ps NaI P.S. (No.5,6,7,8) P.S. β P.S. Ps Ps γ NaI No.6,7 γ [1] A.Ore and J.L.Powell, Three-Photon Annihilation of an Electron-Positron Pair, Phys.Rev.75,11(1949). 62

66 [2] William E.Caswell, G.Peter Lepage, and Jonathan Sapirstein, O(α) Corrections to the Decay Rate of Orthopositronium, Phys.Rev.Lett.38,9(1977). [3] Christopher Smith, Bound State Description in Quantum Electrodynamics and Chromodynamics, Université Catholique de Louvain(2002). [4] G.S.Adkins, R.N.Fell and J.Sapirstein, Order α 2 corrections to the decay rate of orthopositronium, [arxiv:hepph/ v2](2000). [5] Bernd A.Kniehl and Alexander A.Penin, Order α 3 ln(1/α) Corrections to Positronium Decays, [arxiv:hepph/ v2](2000). [6] B.A.Kniehl,A.V.Kotikov and O.L.Veretin, Orthopositronium lifetime at O(α) and O(α 3 ln α) in closed form, [arxiv: v2 [hep-ph]](2009) [7] J.J., I,II,. [8],,. [9], - -,. [10],,. [11] M. Skalsey, J. J. Engbrecht, R. K. Bithell, R. S. V allery, and D.W. Gidley, Phys. Rev. Lett. 80, 3727 (1998) [12] S.Asai. (n.d.). Precise measurements of the positronium decay rate and energy level retrieved from experiments/hfs measurement with Zeeman splitting files/ps asai 1.pdf on !! 63