Particle Physics Booklet - Particle Data Group - Lawrence Berkeley ...

41. Cross-section formulae for specific processes 297
and by crossing
dσ
dΩ (qq → qq) = α2 �
s t2 + s2 9s u2 + s2 + u2 t2 − 2s2
�
. (41.7)
3ut
Annihilation of e + e− into γγ has the cross section
dσ
dΩ (e+ e − → γγ)= α2 u
2s
2 + t2 . (41.8)
tu
The related QCD process also has a triple-gluon coupling. The cross
section is
The crossed reactions are
dσ
dΩ (qq → gg) = 8α2 s
27s (t2 + u 2 )( 1
tu
dσ
dΩ (qg → qg) = α2 s
9s (s2 + u 2 )(− 1
su
dσ
dΩ (gg → qq) = α2 s
24s (t2 + u 2 )( 1
tu
9
−
4s2 ) . (41.9)
9
+ ) , (41.10)
4t2 9
− ) , (41.11)
4s2 dσ
dΩ (gg → gg) =9α2 s ut su st
(3 − − − ) . (41.12)
8s s2 t2 u2 Lepton-quark scattering is analogous (neglecting Z exchange)
dσ
(eq → eq) =α2
dΩ 2s e2 s
q
2 + u2 t2 , (41.13)
eq is the quark charge. For ν-scattering with the four-Fermi interaction
dσ
dΩ (νd → ℓ−u)= G2 F s
4π2 , (41.14)
where the Cabibbo angle suppression is ignored. Similarly
dσ
dΩ (νu → ℓ+ d)= G2 F s
4π2 (1 + cos θ) 2
. (41.15)
4
For deep inelastic scattering (presented in more detail in Section 16)
we consider quarks of type i carrying a fraction x = Q2 /(2Mν)ofthe
nucleon’s energy, where ν = E − E ′ is the energy lost by the lepton in the
nucleon rest frame. With y = ν/E we have the correspondences
1+cosθ→ 2(1 − y) , dΩcm → 4πfi(x)dx dy , (41.16)
where the latter incorporates the quark distribution, fi(x). We find
dσ
dx dy (eN → eX) =4πα2 xs
Q4 1
�
1+(1−y) 2
2�
�
4
×
9 (u(x)+u(x)+...)+1
9 (d(x)+d(x)+...)
�
(41.17)
where now s =2ME is the cm energy squared for the electron-nucleon
collision and we have suppressed contributions from higher mass quarks.
Similarly,
dσ
dx dy (νN → ℓ−X)= G2 F xs
π [(d(x)+...)+(1−y)2 (u(x)+...)] , (41.18)
dσ
dx dy (νN → ℓ+ X)= G2 F xs
π [(d(x)+...)+(1−y)2 (u(x)+...)] . (41.19)
Quasi-elastic neutrino scattering (νµn → μ − p, νµp → μ + n) is directly
relatedtothecrossedreaction, neutron decay.