504 9 Particle Physics – I
9.4 In natural unitsτ= 2 /mα^5. Introduce the factorsandc
Letτ= 2 xcy/mα^5
Find the dimensions: []=[Energy×Time],[c]=[velocity]
[τ]=[1/m][ML^2 T−^1 ]x[LT−^1 ]y
[T]=[M]x−^1 [L]2x+y[T]−x−y
Equating the coefficients on both sides of the equation,
−x−y= 1 , 2 x+y= 0 ,x− 1 = 0
we getx= 1 ,y=− 2
Thereforeτ= 2 c−^2 /mα^5 = 2 /mc^2 α^5
τ= 2 ×(0. 659 × 10 −^21 MeV-s)×(137)^5 / 0 .511 MeV
= 1. 245 × 10 −^10 s
9.5Γμ=GF^2 mμ^5 / 192 π^3 (1)
Now, the dimensional formula forGFis
[Energy]−^2 [c]^3 and [Γμ]=[Energy]
Introduceandcin (1) and take dimensions on both sides.
Γμ=GF^2 mμ^5 xcy/ 192 π^3 (2)
[ML^2 T−^2 ]=[ML^2 T−^2 ]−^4 [M]^5 [ML^2 T−^1 ]x+^6 [LT−^1 ]y+^6
or [ML^2 T−^2 ]=[M]^7 +x[L]^10 +^2 x+y[T]−^4 −x−y
Equating powers ofM, LandT, we find
x=−6 andy= 4
and (2) becomes
Γμ=GF^2 mμ^5 −^6 c^4 / 192 π^3
Γμ=/τ=GF^2 (mμc^2 )^5 /192 (c)^6 π^3
=(1. 116 × 10 −^5 GeV−^2 )^2 (105. 659 × 10 −^3 )^5 / 192 π^3
τ= 2. 39 × 10 −^6 s9.3.2 Production ....................................
9.6 (a) The relation for lab angleθand the C.M.S. angleθ∗is given by
tanθ=sinθ∗/γc(cosθ∗+βc/β∗)(1)
whereβc=vc/cis the CMS velocity andγcis the corresponding Lorentz
factor (see summary of Chap. 6). For photonβ∗=1 andβc =1asthe
electron is ultra relativistic. Dropping off the subscript c, (1) becomes
tanθ=sinθ∗/γ(cosθ∗+1)=tan (θ∗/2)/γ (2)
(b) Assuming that the photons are emitted isotropically, half of the photons will
be contained in the forward hemisphere in the CMS, that is withinθ∗= 90 ◦.
Substitutingθ∗= 90 ◦in (2)
tanθ= 1 /γ
As the incident electron is ultrarelativistic the photons in the lab would come
off at small angles so that tanθ≈θ= 1 /γ. Thus half of the photons will be
emitted within a cone of half angleθ≈ 1 /γ.