Miscellanea

Target Mass Corrections

Following [S+08], [GR20] we provide three options:

  • exact: is the full and involves integration

  • approximate: is stemming from the exact, but the strcture functions in the integrand are evaluated at the bottom end

  • APFEL: the one used in APFEL, similar to the exact but with g2 in the review (Schienbein et al.) set to 0

Todo

complete

\(F_L\) definition

Also the definition of \(F_L\) is corrected by the presence of a proton mass. The explicit expression is given in 26 of [S+08]:

\[F^{\textrm{TMC}}_L (x, Q^2) = r^2 F^{\textrm{TMC}}_2 (x, Q^2) - 2 x F^{\textrm{TMC}}_1 (x, Q^2)\]

where the definition of \(r\) is given in 2 of the same paper:

\[r = \sqrt{1 + \frac{4 x^2 M^2}{Q^2}}\]

Isospin

Isospin is used as a level-0 nuclear correction, just swapping the up and down contribution, for the amount it is specified for the target hadron/nuclei.

In particular:

  • for the proton: \(A=1, Z=1\), the up and down are kept as they are (default)

  • for the neutron: \(A=1, Z=0\), the up and down components are fully swapped, such that the up coefficient function is matched to the down PDF and conversely

  • for the isoscalar: \(A=2, Z=1\) (it is the deuteron), the resulting coefficient functions will be mixed, i.e. the resulting \(c_u\) will be half the original \(c_u\) and half the original \(c_d\) (same for the final \(c_d\))

The actual general expression is:

\[\begin{split}\begin{pmatrix} c'_u \\ c'_d \end{pmatrix} = \frac{1}{A} \begin{pmatrix} Z & A - Z \\ A - Z & Z \end{pmatrix} \begin{pmatrix} c_u \\ c_d \end{pmatrix}\end{split}\]

In particular yadism does not operate at the level of PDF, thus all the changes are applied to the coefficient functions.

Heavy Quark Mass scheme

Todo

it is not yet implemented:

  • Pole masses (implemented)

  • MSbar masses (not implemented)