LECTURE 8: ELECTRON-POSITRON ANNIHILATION

Transcription

1 LETURE 8: ELETRON-POSITRON NNIHILTION

2 Ste I/II: The Feynman Diagram and rule e + 2 q 3 µ - (2 ) 4 d 4 q ū(3) ig e v(2) ig e µ v(4) u() ig µ q 2 (2 ) 4 4 (q 3 4 ) (2 ) 4 4 ( + 2 q) 4 e - µ + [ū(3) µ v(4)] g µ [ v(2) i(2 ) 4 4 ( ) u()] g 2 e ( + 2 ) 2 M = g 2 e ( + 2 ) 2 [ū(3) µ v(4)] [ v(2) µ u()]

3 WHT TO DO WITH THIS? We need to evaluate the amlitude Reall the ro etion exreion: d d = 64 2 f i M 2 = S~ 2 4 ( 2 ) 2 (m m 2 2 ) 2 Z M 2 (2 ) 4 4 ( µ + µ 2 NX N µ f ) Y f=3 f=3 q d3 f 2 2 f + m2 f 2 (2 ) 3 We need to alulate M 2 and ut it into the hae ae exreion What exreion hould we ue for the inor? Reall that a Dira artile baially ha three roertie artile or antiartile in (/2) momentum

4 HELIITY STTES We had reviouly ontruted heliity tate of a Dira artile along the z-axi u = N z /(E + m 2 ) ( x + i y )/(E + m 2 ) Ue oitive energy olution eletron u 2 = N ( x i y )/(E + m 2 ) z /(E + m 2 ) v 2 u 3 = N z /(E + m 2 ) ( x + i y )/(E + m 2 ) v u 4 = N ( x i y )/(E + m 2 ) z /(E + m 2 ) Ue negative energy olution oitron We now generalize thi to any diretion

5 HELIITY OPERTOR: Reall that heliity i the rojetion of the in onto the diretion of motion In onidering the angular momentum roertie, we introdued the in oerator: ~ S = ~ 2 ~ So if we want to rojet thi along the diretion of the momentum, we have h = S = ~ 2 ~ ~

6 HELIITY EIGENSTTES y alying h to a hyotheized inor, we an derive the eigentate With olar oordinate: θ = olar angle to z axi φ = azimuthal angle atan( y / z ) u " = E + m v " = E + m o 2 e i in 2 E+m o 2 E+m ei in 2 E+m E+m ei e i u # = E + m v # = E + m in 2 e i o 2 E+m in 2 E+m ei o 2 E+m E+m ei e i = in θ/2 = o θ/2

7 RELTIVISTI LIMIT If E >> m u " = E + m u # = E + m v " = E + m v # = E + m e i E+m E+m ei e i E+m E+m ei E+m E+m ei e i E+m E+m ei e i u " = E u # = E v " = E v " = E e i e i e i e i e i e i e i e i

8 INOMING SPINORS e + µ q M = g 2 e ( + 2 ) 2 [ū(3) µ v(4)] [ v(2) µ u()] 4 e - µ + Initial tate: ut it along the z-axi inoming eletron (θ=, φ = ) u " ( )= E e i e i inoming oitron (θ=π, φ = ) v " ( 2 )= E 2 2 e i e i 2 u # ( )= ) E E ) E 2 v # ( 2 )= E 2 2 e i 2 2 ) E 2 e i 2 2 e i i 2 ) E

9 OUTGOING SPINORS e + µ q M = g 2 e ( + 2 ) 2 [ū(3) µ v(4)] [ v(2) µ u()] 4 e - µ + Outgoing tate: inoming eletron (θ 3, ) u " ( 3 )= E 3 inoming oitron (θ 4 =π-θ 3, φ = π) v " ( 4 )= E e i e i e i e i 4 ) E u # ( 3 )= E 3 3 e i 3 3 ) E 3 3 e i 3 ) E v # ( 4 )= 4 3 E e i e i 4 ) E

10 HELIITY OMINTIONS Now we an onider any ombination of feliitie by laing the aroriate inor in the exreion M = j μ j e ge 2 ( + 2 ) 2 [ū(3) µ v(4)] [ v(2) µ u()] Note that in general we will onider rodut like 2 = = = 2 µ = µ = = = i( ) = = 2 = i i i i 3 = 3 = =

11 j μ We an try the or RL ombination u R = E ū R µ v L =2E(, o,i,in ) ū R ū R v L = E ( + ) = = v L = E µ v L ū R v L = E ( )=2E( 2 2 )= 2E o = ū R 2 v L = ie ( )=2ie( )=2iE ū R = i( ) 3 v L = E ( =4E =2E in = ū L µ v R =2E(, o, i, in ) ū R µ v R =2E(,,, ) ū L µ v L =2E(,,, )

12 j e y the ame method, an how: ū L µ v L =2E(,,, ) ū R µ v R =2E(,,, ) ū L µ v R =2E(,,i,) ū R µ v L =2E(,, i, ) and we an ombine with jm to get the amlitude for any artiular heliity ombination ū R µ v L =2E(, o,i,in ) ū L µ v R =2E(, o, i, in ) M LR!LR = e 2 4E 2 [ū 3L µ v 4R ][ v 2R µ u L ] M LR!RL = e 2 4E 2 [ū 3R µ v 4L ][ v 2R µ u L ] = 4e 2 E 2 ( o, ) ( + 2 ) 2 = e 2 ( o + + ) = e 2 ( + o ) = M RL!RL = e 2 ( o + ) = M RL!LR

13 ENDGME Put thee into our differential ro etion: and reall that = ( + 2 ) 2 =(2E) 2 d d = 64 2 f i M 2 M LR!LR 2 = M RL!RL 2 = e 4 ( + o ) 2 M LR!RL 2 = M RL!LR 2 = e 4 ( o ) 2 d d = e ( ± o )2 E2 The unolarized in-averaged ro etion if we detet all outgoing heliity tate, add u all the ontribution if the beam are unolarized (i.e. initial tate are half of eah heliity), eah heliity ombination i /4 of the total beam, o we divide by 4 d d = e ( + o2 )

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