# Second Order QCD Corrections, Deep Inelastic Scattering

• The martrix elements are NOT summed over the final state polarizations.
• ```Pij=(pi+pj)2, (i,j=2,3,4``` or ```i,j=1,5) Pij=(pi-pj)2, (i=2,3,4``` and ```j=1,5) Q2 = -p52 s = p12```
• The scalar integrals ` B0, C0, D0 ` are defined Appendix D of T. Matsuura, S.C. van der Marck, and W.L. van Neerven, Nucl. Phys. B319 (1989) 570.
• ```Ca = 3 Cf = 4/3```
• `[e_q]` equals the charge of the detected quark.
`[e_Q]` equals the charge of the quark which is not detected.
The axial couplings `[a_q]` and `[a_Q]` are similar.
• `n = 4 + eps`
• `fi(i,j) = 1/(i + j * eps/2)`

• Second Order:

```$p$5 = p1 + p2 + p3, q2 = p52 = -Q2 < 0, p12 = p22 = p32 = 0 ```

• $q + V --> q g$, (DISq2gmat, asymmetry matrix element, polarized matrix element)

• ```$p$5 = p1 + p2 + p3, q2 = p52 = -Q2 < 0, p12 = p22 = p32 = 0 ```

• $g + V --> qq$, (DISg2mat, polarized matrix element)

• ```$p$5 = p1 + p2 + p3 + p4, q2 = p52 = -Q2 < 0, p12 = p22 = p32 = p42 = 0 ```

• $q + V --> q g g$, (DISgg2mat, Asymmetry matrix element, Polarized matrix element)

• ```$p$5 = p1 + p2 + p3 + p4, q2 = p52 = -Q2 < 0, p12 = p22 = p32 = p42 = 0 ```

• $g + V --> qqg$, (DISgg2gmat, Polarized matrix element)

• ```$p$5 = p1 + p2 + p3 + p4, q2 = p52 = -Q2 < 0, p12 = p22 = p32 = p42 = 0 ```

• $q + V --> q q\text{'}q\text{'}$, (DISqq2mat, non-identical quarks, DISqqq2idmat, extra identical quark contributions)
• Asymmetry matrix element (DISAsqqmat, non-identical quarks, DISAsidqqmat, extra identical quark contributions)
• Polarized matrix element (non-identical quarks, extra identical quark contributions)