Parton evolution
The scale dependence of the parton distribution functions is described
by the renormalisation group equations for mass factorisation (DGLAP)
\[\mu^{2}\frac{\partial}{\partial\mu^{2}}f_{i}(x,\mu^{2}) = P_{ij}(x,\mu^{2}) \otimes f(x,\mu^{2})\,\]
where fi is the generic parton distribution function, Pij are the
Altarelli-Parisi kernels and \(\otimes\) denotes the Mellin convolution
\[f(x) \otimes g(x) \equiv \int_{x}^{1}dyf(y)g\left( \frac{x}{y} \right)\]
We have a system of (2nf + 1) coupled
integro-differential equations, where the summation over the parton
species j is understood.
The NmLO approximation for the splitting functions \(P_{ij}(x,\mu^2)\)
\[P_{ij}^{N^{m}LO}(x,\mu^{2}) = \sum_{k = 0}^{m}a_{s}^{k + 1}(\mu^{2})P_{ij}^{(k)}(x)\]
where we note that the only dependence on the scale \(\mu^2\)
is through the coupling constant \(a_s(\mu^2)\). The splitting
functions in the case of unpolarised partons are known up to NNLO and,
in the notation we adopt, their explicit expressions are found in .
In the following, to describe the solution to the DGLAP evolution
equations we will be working in Mellin space where, as we have seen,
convolutions are turned into products.
The primary quantities are the \(2n_f\) quark and antiquark
distributions qi(x,Q2), Qi(x,Q2) and the gluon distribution g(x,Q2).
From considerations based on charge conjugation and flavour symmetry it
is possible to rewrite the system of equations as \((2N_f - 1)\) equations
describing the
independent evolution of the non-singlet quark asymmetries and
\[q_{NS,ij}^\pm = q_i \pm Q_i - (q_j \pm Q_j)\]
\[q_{NS}^v = \sum_{i = 1}^{N_f}(q_i - Q_i)\]
and a system of 2 equations describing the coupled evolution of the
singlet and gluon parton distributions.
\[\begin{split}\begin{matrix}
\mu^{2}\frac{\partial}{\partial\mu^{2}}q_{NS}^{\pm ,v}(x,\mu^{2}) & = & P_{NS}^{\pm ,v} \otimes q_{NS}^{\pm ,v}(x,\mu^{2}) \\
\mu^{2}\frac{\partial}{\partial\mu^{2}}\begin{pmatrix}
\Sigma \\
g \\
\end{pmatrix}(x,\mu^{2}) & = & \begin{pmatrix}
P_{qq} & P_{qg} \\
P_{gq} & P_{gg} \\
\end{pmatrix} \otimes \begin{pmatrix}
\Sigma \\
g \\
\end{pmatrix}(x,\mu^{2}) \\
\end{matrix}\end{split}\]
where the singlet combination, \(\Sigma\), is defined as
\[\Sigma = \sum_{i = 1}^{N_{f}}(q_{i} + {\overline{q}}_{i})\,,\]
where \(N_{f}\) is the number of light flavors, i.e. the number
of flavors with \(m_{q}^{2} < Q^{2}\).
At LO
\(P_{NS}^{(0), +} = P_{NS}^{(0), -} = P_{NS}^{(0),v} = P_{qq}^{(0)}\).
At NLO \(P_{NS}^{(0), -} = P_{NS}^{(0),v}\) while all the other
splitting functions are different. Starting form \(\mathcal{O}(\alpha_s^2)\)
all splitting functions are different from each other.
The evolution of the individual quark distributions with the scale can
be computed by introducing the following set of non-singlet
distributions:
\[\begin{split}\begin{matrix} V & = & u^{-} + d^{-} + s^{-} + c^{-} + b^{-} + t^{-} \\ \end{matrix}\end{split}\]
\[\begin{split}\begin{matrix} V_{3} & = & u^{-} - d^{-} \\ \end{matrix}\end{split}\]
\[\begin{split}\begin{matrix} V_{8} & = & u^{-} + d^{-} - 2s^{-} \\ \end{matrix}\end{split}\]
\[\begin{split}\begin{matrix} V_{15} & = & u^{-} + d^{-} + s^{-} - 3c^{-} \\ \end{matrix}\end{split}\]
\[\begin{split}\begin{matrix} V_{24} & = & u^{-} + d^{-} + s^{-} + c^{-} - 4b^{-} \\ \end{matrix}\end{split}\]
\[\begin{split}\begin{matrix} V_{35} & = & u^{-} + d^{-} + s^{-} + c^{-} + b^{-} - 5t^{-} \\ \end{matrix}\end{split}\]
\[\begin{split}\begin{matrix} T_{3} & = & u^{+} - d^{+} \\ \end{matrix}\end{split}\]
\[\begin{split}\begin{matrix} T_{8} & = & u^{+} + d^{+} - 2s^{+} \\ \end{matrix}\end{split}\]
\[\begin{split}\begin{matrix} T_{15} & = & u^{+} + d^{+} + s^{+} - 3c^{+} \\ \end{matrix}\end{split}\]
\[\begin{split}\begin{matrix} T_{24} & = & u^{+} + d^{+} + s^{+} + c^{+} - 4b^{+} \\ \end{matrix}\end{split}\]
\[\begin{split}\begin{matrix} T_{35} & = & u^{+} + d^{+} + s^{+} + c^{+} + b^{+} - 5t^{+} \\ \end{matrix}\end{split}\]
where \(q_{i}^{\pm} = q_{i} \pm {\overline{q}}_{i}\), and
\(u,d,s,c,b,t\) are the various flavour distributions.
The combinations \(V_{j}\) and \(T_{j}\) evolve according to eq.
([eq:DGLAPdecomp]) with \(P_{NS}^{-}\) and
\(P_{NS}^{+}\) respectively, while the total valence \(V\)
evolves with the \(P_{NS}^{v}\) kernel. Inverting the linear system
Eq.[eq:lincomb] we obtain the individual pdf’s as a
function of the evolved non-singlet and singlet distributions:
\[\begin{split}\begin{matrix}
u & = & (10\Sigma + 30T_{3} + 10T_{8} + 5T_{15} + 3T_{24} + 2T_{35} + 10V + 30V_{3} + 10V_{8} + 5V_{15} + 3V_{24} + 2V_{35})/120 \\
\overline{u} & = & (10\Sigma + 30T_{3} + 10T_{8} + 5T_{15} + 3T_{24} + 2T_{35} - 10V - 30V_{3} - 10V_{8} - 5V_{15} - 3V_{24} - 2V_{35})/120 \\
d & = & (10\Sigma - 30T_{3} + 10T_{8} + 5T_{15} + 3T_{24} + 2T_{35} + 10V - 30V_{3} + 10V_{8} + 5V_{15} + 3V_{24} + 2V_{35})/120 \\
\overline{d} & = & (10\Sigma - 30T_{3} + 10T_{8} + 5T_{15} + 3T_{24} + 2T_{35} - 10V + 30V_{3} - 10V_{8} - 5V_{15} - 3V_{24} - 2V_{35})/120 \\
s & = & (10\Sigma - 20T_{8} + 5T_{15} + 3T_{24} + 2T_{35} + 10V - 20V_{8} + 5V_{15} + 3V_{24} + 2V_{35})/120 \\
\overline{s} & = & (10\Sigma - 20T_{8} + 5T_{15} + 3T_{24} + 2T_{35} - 10V + 20V_{8} - 5V_{15} - 3V_{24} - 2V_{35})/120 \\
c & = & (10\Sigma - 15T_{15} + 3T_{24} + 2T_{35} + 10V - 15V_{15} + 3V_{24} + 2V_{35})/120 \\
\overline{c} & = & (10\Sigma - 15T_{15} + 3T_{24} + 2T_{35} - 10V + 15V_{15} - 3V_{24} - 2V_{35})/120 \\
b & = & (5\Sigma - 6T_{24} + T_{35} + 5V - 6V_{24} + V_{35})/60 \\
\overline{b} & = & (5\Sigma - 6T_{24} + T_{35} - 5V + 6V_{24} - V_{35})/60 \\
t & = & (\Sigma - T_{35} + V - V_{35})/12 \\
\overline{t} & = & (\Sigma - T_{35} - V + V_{35})/12 \\
\end{matrix}\end{split}\]
The evolution equations presented in the previous subsections assume
that all scales are the same, in particular that the renormalization
\(\mu_{R}^{2}\) and factorization scales \(\mu_{F}^{2}\) are the
same that the hard scale of the problem \(\mu^{2}\),
\[\mu_{R}^{2} = \mu_{F}^{2} = \mu^{2}\ .\]
However, if this is not the case, Eq. [eq:pmlo] has to be
modified as follows:
\[\mathbf{P}(x, \alpha_s(\mu^2_R), L_R) = \alpha_s(\mu^2_R)\mathbf{P}^{(0)}(x) + \alpha^2_s(\mu^2_R)[\mathbf{P}^{(1)}(x) - \beta_0L_R\mathbf{P}^{(0)}(x)] +\alpha^3_s(\mu^2_R)[\mathbf{P}^{(2)}(x) - 2\beta_0L_R\mathbf{P}^{(1)}(x) - (\beta_1L_R - \beta^2_0L^2_R)\mathbf{P}^{(0)}(x)]\]
with \(\mathbf{P}^{(k)}\) the matrix of singlet splitting functions (in
the \(\mu_{R}^{2} = \mu_{F}^{2} = \mu^{2}\) case ) as defined in
Eq. [eq:DGLAPdecomp], and where we have defined \(L_{R} \equiv \frac{\mu_{F}^{2}}{\mu_{R}^{2}}\) as the ratio of factorization and renormalization scales. Note that
the strong coupling is evaluated at the renormalization scale
\(\mu_{R}^{2}\).
Non-singlet case . In analogy with the singlet case, up to NNLO one
has
\[P^{\pm, v}_{NS}(x, \alpha_s(\mu^2_R), L_R) = \alpha_s(\mu^2_R)P^{\pm, v(0)}_{NS}(x) + \alpha^2_s(\mu^2_R)[P^{\pm, v(1)}_{NS}(x) - \beta_0L_RP^{\pm, v(0)}_{NS}(x)] + \alpha^3_s(\mu^2_R)[P^{\pm, v(2)}_{NS}(x) - 2\beta_0L_RP^{\pm, v(1)}_{NS}(x) - (\beta_1L_R - \beta^2_0L^2_R)P^{\pm,v(0)}_{NS}(x)]\]
\[P^{\pm, v}_{NS}(x, \alpha_s(\mu^2_R), L_R) = \alpha_s(\mu^2_R)P^{(0)}_{NS}(x) + \alpha^2_s(\mu^2_R)[P^{\pm(1)}_{NS}(x) - \beta_0L_RP^{(0)}_{NS}(x)]\]
The DGLAP evolution equations with variations of the renormalization
scale can be benchmarked againts the usual LH tables.
Analogously to what we have done in the previous subsection, in the
following we write the expressions of the NLO coefficient functions
\(C_{2,L,3}^{q,g}\) in the \(\overline{MS}\) scheme showing
explicitly the dependence on the factorization and renormalization
scales, \(\mu_{r}^{2}\) and \(\mu_{f}^{2}\).
\[C_{a}^{\pm}(N,\alpha_{s}(\mu_{f}^{2}),Q^{2}/\mu_{r}^{2},\mu_{f}^{2}/\mu_{r}^{2}) = 1 + a_{s}(\mu_{r}^{2})\left\lbrack c_{a,NS}^{(1)}(N) + \gamma_{NS}^{(0)}(N)\log\left( \frac{Q^{2}}{\mu_{f}^{2}} \right) \right\rbrack + \mathcal{O}(a_{s}^{2})\]
\[\begin{split}\begin{matrix}
S_{1}(N) & = & \gamma_{E} + \Psi(N + 1) \\
S_{2}(N) & = & \zeta_{2} - \Psi\prime(N + 1,1). \\
\end{matrix}\end{split}\]
we can write down the explicit expression for all the NLo coefficient
functions:
\[C_2^{NS}(N,a_s(\mu_r^2),Q^2/\mu_f^2) = 1 + a_s(\mu_r^2)\cdot C_F\bigg[2S_1(N)^2 - 2 S_2(N) + 3S_1(N) - 2\frac{S_1(N)}{N(N+1)}+\frac{3}{N}+\frac{4}{N+1}+\frac{2}{N^2}-9 +\log(\frac{Q^2}{\mu_f^2})(3 - 4 S_1(N) +\frac{2}{N(N+1)}\bigg]\]
\[C_2^q(N,a_s(\mu_r^2),Q^2/\mu_f^2) = C_2^{NS}(N,a_s(\mu_r^2),Q^2/\mu_f^2)\]
\[C_2^g(N,a_s(\mu_r^2),Q^2/\mu_f^2) = a_s(\mu_r^2)\cdot 4n_fT_R\bigg[\frac{4}{N+1} - \frac{4}{N+2} - (1+S_1(N))\cdot \frac{N^2+N+2}{N(N+1)(N+2)}+\frac{1}{N_1} +\log(\frac{Q^2}{\mu_f^2})\frac{N^2+N+2}{N(N+1)(N+2)}\bigg]\]
\[C_L^{NS}(N,a_s(\mu_r^2)) = a_s(\mu_r^2)\cdot C_F \frac{4}{N+1}\]
\[C_L^q(N,a_s(\mu_r^2)) = C_L^{NS}(N,a_s(\mu_r^2))\]
\[C_L^g(N,a_s(\mu_r^2)) = a_s(\mu_r^2)\cdot 4n_fT_R \frac{4}{(N+1)(N+2)}\]
\[C_3^{NS}(N,a_s(\mu_r^2),Q^2/\mu_f^2) = 1 + a_s(\mu_r^2)\cdot C_F\bigg[2S_1(N)^2 - 2 S_2(N) + 3S_1(N)- 2\frac{S_1(N)}{N(N+1)} +\frac{3}{N}+\frac{4}{N+1} +\frac{2}{N^2}-9 -\frac{4N+2}{N(N+1)} +\log(\frac{Q^2}{\mu_f^2})(3 - 4 S_1(N) +\frac{2}{N(N+1)})\bigg]\]
In our code the heavy quark PDF’s are generated radiatively in the
ZM-VFN scheme. We consider explicitely two cases: evolution starting at
the charm threshold and forward evolution from a scale below the charm
threshold. We will write explicitely all equations implemented into the
code.
\[T_{15}(Q^{2},x) = \Gamma_{NS}^{+}(Q_{0}^{2},Q^{2},x) \otimes T_{15}(Q_{0}^{2},x).\]
\[\begin{split}\begin{matrix}
T_{24}(m_{b}^{2},x) & = & \Sigma(m_{b}^{2},x) = \Gamma_{S,qq}(Q_{0}^{2},m_{b}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,qg}(Q_{0}^{2},m_{b}^{2},x) \otimes g(Q_{0}^{2},x) \\
T_{24}(Q^{2},x) & = & \Gamma_{NS}^{+}(m_{b}^{2},Q^{2},x) \otimes T_{24}(m_{b}^{2},x) \\
& = & \Gamma_{NS}^{+}(m_{b}^{2},Q^{2},x) \otimes \lbrack\Gamma_{S,qq}(Q_{0}^{2},m_{b}^{2},x) \otimes \Sigma(Q_{0}^{2},x) \\
& + & \Gamma_{S,qg}(Q_{0}^{2},m_{b}^{2},x) \otimes g(Q_{0}^{2},x)\rbrack \\
\end{matrix}\end{split}\]
\[\begin{split}\begin{matrix}
\Gamma_{NS}^{q,24}(Q_{0}^{2},Q^{2},N) & = & \Gamma_{NS}^{+}(m_{b}^{2},Q^{2},N)\Gamma_{S,qq}(Q_{0}^{2},m_{b}^{2},N) \\
\Gamma_{NS}^{g,24}(Q_{0}^{2},Q^{2},N) & = & \Gamma_{NS}^{+}(m_{b}^{2},Q^{2},N)\Gamma_{S,qg}(Q_{0}^{2},m_{b}^{2},N) \\
\end{matrix}\end{split}\]
\[\begin{split}\begin{matrix}
T_{35}(m_{b}^{2},x) & = & \Sigma(m_{b}^{2},x) = \Gamma_{S,qq}(Q_{0}^{2},m_{b}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,qg}(Q_{0}^{2},m_{b}^{2},x) \otimes g(Q_{0}^{2},x) \\
T_{35}(m_{t}^{2},x) & = & \Sigma(m_{t}^{2},x) = \Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},x) \otimes \Sigma(m_{b}^{2},x) + \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},x) \otimes g(m_{b}^{2},x) \\
& = & \Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},x) \otimes \left\lbrack \Gamma_{S,qq}(Q_{0}^{2},m_{b}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,qg}(Q_{0}^{2},m_{b}^{2},x) \otimes g(Q_{0}^{2},x) \right\rbrack \\
& + & \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},x) \otimes \left\lbrack \Gamma_{S,gq}(Q_{0}^{2},m_{b}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,gg}(Q_{0}^{2},m_{b}^{2},x) \otimes g(Q_{0}^{2},x) \right\rbrack \\
T_{35}(Q^{2},x) & = & \Gamma_{NS}^{+}(m_{t}^{2},Q^{2},x) \otimes \Sigma(m_{t}^{2},x) \\
& = & \Gamma_{NS}^{+}(m_{t}^{2},Q^{2},x) \\
& \otimes & \{\lbrack\Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},x) \otimes \Gamma_{S,qq}(Q_{0}^{2},m_{b}^{2},x) + \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},x) \otimes \Gamma_{S,gq}(Q_{0}^{2},m_{b}^{2},x)\rbrack \otimes \Sigma(Q_{0}^{2},x) \\
& + & \lbrack\Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},x) \otimes \Gamma_{S,qg}(Q_{0}^{2},m_{b}^{2},x) + \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},x) \otimes \Gamma_{S,gg}(Q_{0}^{2},m_{b}^{2},x)\rbrack \otimes g(Q_{0}^{2},x)\} \\
\end{matrix}\end{split}\]
\[\begin{split}\begin{matrix}
\Gamma_{NS}^{q,35}(Q_{0}^{2},Q^{2},N) & = & \Gamma_{NS}^{+}(m_{t}^{2},Q^{2},N)\lbrack\Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},N)\Gamma_{S,qq}(Q_{0}^{2},m_{b}^{2},N) \\
& + & \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},N)\Gamma_{S,gq}(Q_{0}^{2},m_{b}^{2},N)\rbrack \\
\Gamma_{NS}^{g,35}(Q_{0}^{2},Q^{2},N) & = & \Gamma_{NS}^{+}(m_{t}^{2},Q^{2},N)\lbrack\Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},N)\Gamma_{S,qg}(Q_{0}^{2},m_{b}^{2},N) \\
& + & \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},N)\Gamma_{S,gg}(Q_{0}^{2},m_{b}^{2},N)\rbrack \\
\end{matrix}\end{split}\]
As far as the \(V_{J}\) sector is concerned we must proceed in
the same way. Namely, if \(Q_{0}^{2} = m_{c}^{2}\), the
\(V_{15}\) parton distribution function evolves from the initial
scale to any final scale \(Q^{2} > m_{c}^{2}\) according to the
NS minus evolution equation:
\[V_{15}(Q^{2},x) = \Gamma_{NS}^{-}(Q_{0}^{2},Q^{2},x) \otimes V_{15}(Q_{0}^{2},x).\]
\[\begin{split}\begin{matrix}
V_{24}(m_{b}^{2},x) & = & V(m_{b}^{2},x) = \Gamma_{NS}^{v}(Q_{0}^{2},m_{b}^{2},x) \otimes V(Q_{0}^{2},x) \\
V_{24}(Q^{2},x) & = & \Gamma_{NS}^{-}(m_{b}^{2},Q^{2},x) \otimes V_{24}(m_{b}^{2},x) \\
& = & \Gamma_{NS}^{-}(m_{b}^{2},Q^{2},x) \otimes \Gamma_{NS}^{v}(Q_{0}^{2},m_{b}^{2},x) \otimes V(Q_{0}^{2},x) \\
\end{matrix}\end{split}\]
For a NLO evolution \(\Gamma_{NS}^{-} = \Gamma_{NS}^{v}\),
therefore there would not be no need of introducing new evolution
kernels. However, if we want to build a structure for the code which
can be easily used for a NNLO evolution code we should define, as
well as the \(\Gamma_{NS}^{q,24}\) and \(\Gamma_{NS}^{g,24}\)
kernels, a \(\Gamma_{NS}^{- ,24}\) kernel as:
\[\Gamma_{NS}^{- ,24}(Q_{0}^{2},Q^{2},N) = \Gamma_{NS}^{-}(m_{b}^{2},Q^{2},N)\Gamma_{NS}^{v}(Q_{0}^{2},m_{b}^{2},N)\]
\[\begin{split}\begin{matrix}
V_{35}(m_{t}^{2},x) & = & V(m_{t}^{2},x) = \Gamma_{NS}^{v}(Q_{0}^{2},m_{t}^{2},x) \otimes V(Q_{0}^{2},x) \\
T_{35}(Q^{2},x) & = & \Gamma_{NS}^{-}(m_{t}^{2},Q^{2},x) \otimes V(m_{t}^{2},x) \\
& = & \Gamma_{NS}^{-}(m_{t}^{2},Q^{2},x)\Gamma_{NS}^{v}(Q_{0}^{2},m_{t}^{2},x) \otimes V(Q_{0}^{2},x) \\
\end{matrix}\end{split}\]
\[\Gamma_{NS}^{- ,35}(Q_{0}^{2},Q^{2},N) = \Gamma_{NS}^{-}(m_{t}^{2},Q^{2},N)\Gamma_{NS}^{v}(m_{t}^{2},Q^{2},N)\]
\[\begin{split}\begin{matrix}
T_{15}(m_{c}^{2},x) & = & \Sigma(m_{c}^{2},x) = \Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},x) \otimes g(Q_{0}^{2},x) \\
T_{15}(Q^{2},x) & = & \Gamma_{NS}^{+}(m_{c}^{2},Q^{2},x) \otimes T_{15}(m_{c}^{2},x) \\
& = & \Gamma_{NS}^{+}(m_{c}^{2},Q^{2},x) \otimes \lbrack\Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},x) \otimes \Sigma(Q_{0}^{2},x) \\
& + & \Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},x) \otimes g(Q_{0}^{2},x)\rbrack \\
\end{matrix}\end{split}\]
\[\begin{split}\begin{matrix}
\Gamma_{NS}^{q,15}(Q_{0}^{2},Q^{2},N) & = & \Gamma_{NS}^{+}(m_{c}^{2},Q^{2},N)\Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},N) \\
\Gamma_{NS}^{g,15}(Q_{0}^{2},Q^{2},N) & = & \Gamma_{NS}^{+}(m_{c}^{2},Q^{2},N)\Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},N) \\
\end{matrix}\end{split}\]
In the same way, if \(Q^{2} > m_{b}^{2}\) the \(T_{24}\)
parton distribution is not just \(\Sigma\) but it coincides with
the Singlet distribution up to the bottom threshold, while above the
threshold it evolves according to the NS evolution equation:
\[\begin{split}\begin{matrix}
T_{24}(m_{c}^{2},x) & = & \Sigma(m_{c}^{2},x) = \Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},x) \otimes g(Q_{0}^{2},x) \\
T_{24}(m_{b}^{2},x) & = & \Sigma(m_{b}^{2},x) = \Gamma_{S,qq}(m_{c}^{2},m_{b}^{2},x) \otimes \Sigma(m_{c}^{2},x) + \Gamma_{S,qg}(m_{c}^{2},m_{b}^{2},x) \otimes g(m_{c}^{2},x) \\
& = & \Gamma_{S,qq}(m_{c}^{2},m_{b}^{2},x) \otimes \left\lbrack \Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},x) \otimes g(Q_{0}^{2},x) \right\rbrack \\
& + & \Gamma_{S,qg}(m_{c}^{2},m_{b}^{2},x) \otimes \left\lbrack \Gamma_{S,gq}(Q_{0}^{2},m_{c}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,gg}(Q_{0}^{2},m_{c}^{2},x) \otimes g(Q_{0}^{2},x) \right\rbrack \\
T_{24}(Q^{2},x) & = & \Gamma_{NS}^{+}(m_{b}^{2},Q^{2},x) \otimes T_{24}(m_{b}^{2},x) \\
& = & \Gamma_{NS}^{+}(m_{b}^{2},Q^{2},x) \\
& \otimes & \{\lbrack\Gamma_{S,qq}(m_{c}^{2},m_{b}^{2},x) \otimes \Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},x) + \Gamma_{S,qg}(m_{c}^{2},m_{b}^{2},x) \otimes \Gamma_{S,gq}(Q_{0}^{2},m_{c}^{2},x)\rbrack \otimes \Sigma(Q_{0}^{2},x) \\
& + & \lbrack\Gamma_{S,qq}(m_{c}^{2},m_{b}^{2},x) \otimes \Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},x) + \Gamma_{S,qg}(m_{c}^{2},m_{b}^{2},x) \otimes \Gamma_{S,gg}(Q_{0}^{2},m_{c}^{2},x)\rbrack \otimes g(Q_{0}^{2},x)\} \\
\end{matrix}\end{split}\]
\[\begin{split}\begin{matrix}
\Gamma_{NS}^{q,24}(Q_{0}^{2},Q^{2},N) & = & \Gamma_{NS}^{+}(m_{b}^{2},Q^{2},N)\lbrack\Gamma_{S,qq}(m_{c}^{2},m_{b}^{2},N)\Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},N) \\
& + & \Gamma_{S,qg}(m_{c}^{2},m_{b}^{2},N)\Gamma_{S,gq}(Q_{0}^{2},m_{c}^{2},N)\rbrack \\
\Gamma_{NS}^{g,24}(Q_{0}^{2},Q^{2},N) & = & \Gamma_{NS}^{+}(m_{b}^{2},Q^{2},N)\lbrack\Gamma_{S,qq}(m_{c}^{2},m_{b}^{2},N)\Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},N) \\
& + & \Gamma_{S,qg}(m_{c}^{2},m_{b}^{2},N)\Gamma_{S,gg}(Q_{0}^{2},m_{c}^{2},N)\rbrack \\
\end{matrix}\end{split}\]
Finally, if \(Q^{2} > m_{b}^{2}\) the \(T_{35}\) parton
distribution is not just \(\Sigma\) but it coincides with the
Singlet distribution up to the top threshold, while above the
threshold it evolves according to the NS evolution equation:
\[\begin{split}\begin{matrix}
T_{35}(m_{c}^{2},x) & = & \Sigma(m_{c}^{2},x) = \Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},x) \otimes g(Q_{0}^{2},x) \\
T_{35}(m_{b}^{2},x) & = & \Sigma(m_{b}^{2},x) = \Gamma_{S,qq}(m_{c}^{2},m_{b}^{2},x) \otimes \Sigma(m_{c}^{2},x) + \Gamma_{S,qg}(m_{c}^{2},m_{b}^{2},x) \otimes g(m_{c}^{2},x) \\
& = & \Gamma_{S,qq}(m_{c}^{2},m_{b}^{2},x) \otimes \left\lbrack \Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},x) \otimes g(Q_{0}^{2},x) \right\rbrack \\
& + & \Gamma_{S,qg}(m_{c}^{2},m_{b}^{2},x) \otimes \left\lbrack \Gamma_{S,gq}(Q_{0}^{2},m_{c}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,gg}(Q_{0}^{2},m_{c}^{2},x) \otimes g(Q_{0}^{2},x) \right\rbrack \\
T_{35}(m_{t}^{2},x) & = & \Sigma(m_{t}^{2},x) = \Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},x) \otimes \Sigma(m_{b}^{2},x) + \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},x) \otimes g(m_{b}^{2},x) \\
& = & \Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},x) \otimes \\
& & \{\Gamma_{S,qq}(m_{c}^{2},m_{b}^{2},x) \otimes \left\lbrack \Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},x) \otimes g(Q_{0}^{2},x) \right\rbrack \\
& + & \Gamma_{S,qg}(m_{c}^{2},m_{b}^{2},x) \otimes \left\lbrack \Gamma_{S,gq}(Q_{0}^{2},m_{c}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,gg}(Q_{0}^{2},m_{c}^{2},x) \otimes g(Q_{0}^{2},x) \right\rbrack\} \\
& + & \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},x) \otimes \\
& & \{\Gamma_{S,gq}(m_{c}^{2},m_{b}^{2},x) \otimes \left\lbrack \Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},x) \otimes g(Q_{0}^{2},x) \right\rbrack \\
& + & \Gamma_{S,gg}(m_{c}^{2},m_{b}^{2},x) \otimes \left\lbrack \Gamma_{S,gq}(Q_{0}^{2},m_{c}^{2},x) \otimes \Sigma(Q_{0}^{2},x) + \Gamma_{S,gg}(Q_{0}^{2},m_{c}^{2},x) \otimes g(Q_{0}^{2},x) \right\rbrack\} \\
T_{35}(Q^{2},x) & = & \Gamma_{NS}^{+}(m_{t}^{2},Q^{2},x) \otimes T_{35}(m_{t}^{2},x) \\
& = & \Gamma_{NS}^{+}(m_{t}^{2},Q^{2},x) \otimes \\
& & \{\lbrack\Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},x) \otimes \Gamma_{S,qq}(m_{c}^{2},m_{b}^{2},x) \otimes \Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},x) \\
& + & \Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},x) \otimes \Gamma_{S,qg}(m_{c}^{2},m_{b}^{2},x) \otimes \Gamma_{S,gq}(Q_{0}^{2},m_{c}^{2},x) \\
& + & \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},x) \otimes \Gamma_{S,gq}(m_{c}^{2},m_{b}^{2},x) \otimes \Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},x) \\
& + & \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},x) \otimes \Gamma_{S,gg}(m_{c}^{2},m_{b}^{2},x) \otimes \Gamma_{S,gq}(Q_{0}^{2},m_{c}^{2},x)\rbrack \otimes \Sigma(Q_{0}^{2},x) \\
& + & \lbrack\Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},x) \otimes \Gamma_{S,qq}(m_{c}^{2},m_{b}^{2},x) \otimes \Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},x) \\
& + & \Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},x) \otimes \Gamma_{S,qg}(m_{c}^{2},m_{b}^{2},x) \otimes \Gamma_{S,gg}(Q_{0}^{2},m_{c}^{2},x) \\
& + & \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},x) \otimes \Gamma_{S,gq}(m_{c}^{2},m_{b}^{2},x) \otimes \Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},x) \\
& + & \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},x) \otimes \Gamma_{S,gg}(m_{c}^{2},m_{b}^{2},x) \otimes \Gamma_{S,gg}(Q_{0}^{2},m_{c}^{2},x)\rbrack \otimes g(Q_{0}^{2},x)\} \\
\end{matrix}\end{split}\]
\[\begin{split}\begin{matrix}
\Gamma_{NS}^{q,35}(Q_{0}^{2},Q^{2},N) & = & \Gamma_{NS}^{+}(m_{t}^{2},Q^{2},N)\lbrack\Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},N)\Gamma_{S,qq}(m_{c}^{2},m_{b}^{2},N)\Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},N) \\
& + & \Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},N)\Gamma_{S,qg}(m_{c}^{2},m_{b}^{2},N)\Gamma_{S,gq}(Q_{0}^{2},m_{c}^{2},N) \\
& + & \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},N)\Gamma_{S,gq}(m_{c}^{2},m_{b}^{2},N)\Gamma_{S,qq}(Q_{0}^{2},m_{c}^{2},N) \\
& + & \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},N)\Gamma_{S,gg}(m_{c}^{2},m_{b}^{2},N)\Gamma_{S,gq}(Q_{0}^{2},m_{c}^{2},N)\rbrack \\
\Gamma_{NS}^{g,35}(Q_{0}^{2},Q^{2},N) & = & \Gamma_{NS}^{+}(m_{t}^{2},Q^{2},N)\lbrack\Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},N)\Gamma_{S,qq}(m_{c}^{2},m_{b}^{2},N)\Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},N) \\
& + & \Gamma_{S,qq}(m_{b}^{2},m_{t}^{2},N)\Gamma_{S,qg}(m_{c}^{2},m_{b}^{2},N)\Gamma_{S,gg}(Q_{0}^{2},m_{c}^{2},N) \\
& + & \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},N)\Gamma_{S,gq}(m_{c}^{2},m_{b}^{2},N)\Gamma_{S,qg}(Q_{0}^{2},m_{c}^{2},N) \\
& + & \Gamma_{S,qg}(m_{b}^{2},m_{t}^{2},N)\Gamma_{S,gg}(m_{c}^{2},m_{b}^{2},N)\Gamma_{S,gg}(Q_{0}^{2},m_{c}^{2},N)\rbrack \\
\end{matrix}\end{split}\]
\[\begin{split}\begin{matrix}
V_{15}(m_{c}^{2},x) & = & V(m_{c}^{2},x) = \Gamma_{NS}^{v} \otimes V(Q_{0}^{2},x) \\
V_{15}(Q^{2},x) & = & \Gamma_{NS}^{-}(m_{c}^{2},Q^{2},x) \otimes V_{15}(m_{c}^{2},x) \\
& = & \Gamma_{NS}^{-}(m_{c}^{2},Q^{2},x) \otimes \Gamma_{NS}^{v} \otimes V(Q_{0}^{2},x) \\
\end{matrix}\end{split}\]
\[\Gamma_{NS}^{- ,15}(Q_{0}^{2},Q^{2},N) = \Gamma_{NS}^{-}(m_{c}^{2},Q^{2},N)\Gamma_{NS}^{v}(Q_{0}^{2},m_{c}^{2},N)\]
In the same way, if \(Q^{2} > m_{b}^{2}\) the \(V_{24}\)
parton distribution coincides with the Total valence distribution up
to the bottom threshold, while above the threshold it evolves
according to the NS minus evolution equation:
\[\begin{split}\begin{matrix}
V_{24}(m_{b}^{2},x) & = & V(m_{b}^{2},x) = \Gamma_{NS}^{v}(Q_{0}^{2},m_{b}^{2},x) \otimes V(Q_{0}^{2},x) \\
V_{24}(Q^{2},x) & = & \Gamma_{NS}^{-}(m_{b}^{2},Q^{2},x) \otimes V_{24}(m_{b}^{2},x) \\
& = & \Gamma_{NS}^{-}(m_{b}^{2},Q^{2},x) \otimes \Gamma_{NS}^{v}(Q_{0}^{2},m_{b}^{2},x) \otimes V(Q_{0}^{2},x) \\
\end{matrix}\end{split}\]
For a NLO evolution \(\Gamma_{NS}^{-} = \Gamma_{NS}^{v}\),
therefore there would not be no need of introducing new evolution
kernels. However, if we want to build a structure for the code which
can be easily used for a NNLO evolution code we should define a
\(\Gamma_{NS}^{- ,24}\) kernel as:
\[\Gamma_{NS}^{- ,24}(Q_{0}^{2},Q^{2},N) = \Gamma_{NS}^{-}(m_{b}^{2},Q^{2},N)\Gamma_{NS}^{v}(Q_{0}^{2},m_{b}^{2},N)\]
\[\begin{split}\begin{matrix}
V_{35}(m_{t}^{2},x) & = & V(m_{t}^{2},x) = \Gamma_{NS}^{v}(Q_{0}^{2},m_{t}^{2},x) \otimes V(Q_{0}^{2},x) \\
T_{35}(Q^{2},x) & = & \Gamma_{NS}^{-}(m_{t}^{2},Q^{2},x) \otimes V(m_{t}^{2},x) \\
& = & \Gamma_{NS}^{-}(m_{t}^{2},Q^{2},x)\Gamma_{NS}^{v}(Q_{0}^{2},m_{t}^{2},x) \otimes V(Q_{0}^{2},x). \\
\end{matrix}\end{split}\]
\[\Gamma_{NS}^{- ,35}(Q_{0}^{2},Q^{2},N) = \Gamma_{NS}^{-}(m_{t}^{2},Q^{2},N)\Gamma_{NS}^{v}(m_{t}^{2},Q^{2},N)\]
Target Mass Corrections
From Eq. (4.19) of Ref. , if we identify \(F\) with
\(F_{2}(y)/y^{2}\) by comparing left and right hand sides of the
equation in the limit of zero target mass, we obtain the expression of
the NLT correction to the structure function \(F_{2}:\)
\[F_{2}^{NLT}(x,Q^{2}) = \frac{x^{2}}{\tau^{3/2}}\frac{F_{2}^{LT}(\xi,Q^{2})}{\xi^{2}} + 6\frac{M^{2}}{Q^{2}}\frac{x^{3}}{\tau^{2}}I_{2}(\xi,Q^{2})\]
where
\[\begin{split}\begin{matrix}
I_{2}(\xi,Q^{2}) & = \int_{\xi}^{1}\, dz\,\frac{F_{2}^{LT}(z,Q^{2})}{z^{2}}. \\
\tau & = 1\, + \,\frac{4M_{p}^{2}x^{2}}{Q^{2}} \\
\xi & = \,\frac{2x}{1 + \sqrt{\tau}} \\
\end{matrix}\end{split}\]
Now let us Mellin transform and antitransform
\(F_{2}^{LT}(\xi,Q^{2})\) and \(I_{2}(\xi,Q^{2})\) with respect
to the variable \(\xi\):
\[F_{2}^{LT}(\xi,Q^{2}) = \int\frac{dN}{2\pi i}\,\xi^{- N}\Gamma(N,Q_{0}^{2},Q^{2})\, f\left( N,Q_{0}^{2} \right)\]
while
\[\begin{split}\begin{matrix}
I_{2}(N,Q^{2}) & = \int_{0}^{1}d\xi\,\xi^{N - 1}\int_{\xi}^{1}\, dz\,\frac{F_{2}^{LT}(z,Q^{2})}{z^{2}} \\
& = |\frac{\xi^{N}}{N}\,\int_{\xi}^{1}\, dz\,\frac{F_{2}^{LT}(z,Q^{2})}{z^{2}}|_{0}^{1} + \int_{0}^{1}\frac{d\xi}{N}\,\xi^{N}\,\frac{F_{2}^{LT}(\xi,Q^{2})}{\xi^{2}} \\
& = \frac{1}{N}\,\int_{0}^{1}\, d\xi\,\xi^{N - 2}F_{2}^{LT}(\xi,Q^{2}) \\
& = \frac{F_{2}^{LT}(N - 1,Q^{2})}{N} \\
\Rightarrow I_{2}^{LT}(\xi,Q^{2}) & = \int\frac{dN}{2\pi i}\,\xi^{- N}\,\frac{F_{2}^{LT}(N - 1,Q^{2})}{N} \\
& = \frac{1}{\xi}\,\int\frac{dN}{2\pi i}\,\xi^{- N}\,\frac{F_{2}^{LT}(N,Q^{2})}{N + 1}. \\
\end{matrix}\end{split}\]
Now, by substituting equations [eq:fslt] and
[eq:i2N] into [eq:tmcformula] we
obtain
\[\begin{split}\begin{matrix}
F_{2}^{NLT}(\xi,Q^{2}) & = & \,\int\frac{dN}{2\pi i}\,\xi^{- N}\,\left( \frac{x^{2}}{\tau^{3/2}\xi^{2}} + \frac{6M^{2}}{Q^{2}}\frac{x^{3}}{\tau^{2}}\frac{1}{\xi(N + 1)} \right) \\
& & C_{2}(N,\alpha_{s}(Q^{2}))\Gamma(N,Q_{0}^{2},Q^{2})\, q\left( N,Q_{0}^{2} \right). \\
\end{matrix}\end{split}\]
Now we can reinterpret the factor in front of
\(C_{2}(N,\alpha_{s}(Q^{2}))\) as the new Target Mass Corrected
coefficient function, which can be written as a function of
\(\tau\):
\[C_{2}^{TMC}(N,\alpha_{s}(Q^{2})) = \frac{(1 + \sqrt{\tau})^{2}}{4\tau^{3/2}}\left( 1 + \frac{3\left( 1 - 1/\sqrt{\tau} \right)}{N + 1} \right)C_{2}(N,\alpha_{s}(Q^{2})).\]
Notice that into the limit
\(M_{p}/Q \rightarrow 0,\,\tau \rightarrow 1\),
\(C_{2}^{TMC}(N,\alpha_{s}(Q^{2}))\) becomes
\(C_{2}(N,\alpha_{s}(Q^{2}))\).
The same procedure can be applied to find the NLT target mass
corrections to the \(F_{L}\) and \(F_{3}\) structure functions.
Starting from formula (4.21b) of Ref. , being
\[\frac{\nu W_{2}}{M} = F_{2}\text{\quad\quad}W_{1} = F_{1}\text{\quad\quad}F_{L} = \frac{\nu W_{2}}{M} - 2xW_{1} = 2xW_{L} - \frac{4x^{2}M^{2}}{Q^{2}}\frac{\nu W_{2}}{M},\]
we find
\[F_{L}^{NLT}(x,Q^{2}) = F_{L}^{LT}(x,Q^{2}) + \frac{x^{2}(1 - \tau)}{\tau^{3/2}}\frac{F_{2}^{LT}(\xi,Q^{2})}{\xi^{2}} + \frac{M^{2}}{Q^{2}}\frac{x^{3}(6 - 2\tau)}{\tau^{2}}I_{2}(\xi,Q^{2})\]
where \(I_{2}\) is defined in Eq. [eq:i2]. With the
same calculations as in the \(F_{2}\) case we obtain the following
formula
\[\begin{split}\begin{matrix}
F_{L}^{NLT}(\xi,Q^{2}) & = & F_{L}^{LT}(x,Q^{2}) + \,\int\frac{dN}{2\pi i}\,\xi^{- N}\,(\frac{x^{2}(1 - \tau)}{\tau^{3/2}\xi^{2}} + \frac{M^{2}}{Q^{2}}\frac{x^{3}}{\tau^{2}}\frac{(6 - 2\tau)}{\xi(N + 1)}) \\
& & C_{2}(N,\alpha_{s}(Q^{2}))\,\Gamma(N,Q_{0}^{2},Q^{2})\, f\left( N,Q_{0}^{2} \right). \\
\end{matrix}\end{split}\]
Now we can reinterpret the factor in front of
\(C_{2}(N,\alpha_{s}(Q^{2}))\) as the new Target Mass Corrected
Evolution coefficient, which by re-expressing everything as a function
of \(\tau\) can be written as:
\[\begin{split}\begin{matrix}
C_{L}^{TMC}(N,\alpha_{s}(Q^{2})) & = & \lbrack 1 + \frac{(1 + \sqrt{\tau})^{2}(1 - \tau)}{4\tau^{3/2}} \cdot \\
& & \left( 1 - \frac{(3 - \tau)(1 + \sqrt{\tau})}{4\tau^{2}}\frac{1}{N + 1} \right)\frac{C_{2}(N,\alpha_{s}(Q^{2}))}{C_{L}(N,\alpha_{s}(Q^{2}))}\rbrack C_{L}(N,\alpha_{s}(Q^{2})). \\
\end{matrix}\end{split}\]
Finally to find the TMC of \(F_{3}\) we start from Eq. (4.22) of
Ref. , where \(F = 2F_{3}(y)/y\) as we can see by comparing the left
and right hand side members of the equation in the limit of
\(M \rightarrow 0\):
\[F_{L}^{NLT}(x,Q^{2}) = \frac{x}{\tau}\frac{F_{3}^{LT}(\xi,Q^{2})}{\xi} + \frac{2M^{2}}{Q^{2}}\frac{x^{2}}{\tau^{3/2}}I_{3}(\xi,Q^{2})\]
where
\[I_{3}(\xi,Q^{2}) = \int_{\xi}^{1}\, dz\,\frac{2F_{3}^{LT}(z,Q^{2})}{z}.\]
With the same calculations as in the \(F_{2}\) case and by noticing
that
\[\begin{split}\begin{matrix}
I_{3}(N,Q^{2}) & = \int_{0}^{1}d\xi\,\xi^{N - 1}\int_{\xi}^{1}\, dz\,\frac{2F_{3}^{LT}(z,Q^{2})}{z} \\
& = |\frac{\xi^{N}}{N}\,\int_{\xi}^{1}\, dz\,\frac{2F_{3}^{LT}(z,Q^{2})}{z}|_{0}^{1} + \int_{0}^{1}\frac{d\xi}{N}\,\xi^{N}\,\frac{2F_{3}^{LT}(\xi,Q^{2})}{\xi} \\
& = \frac{2}{N}\,\int_{0}^{1}\, d\xi\,\xi^{N - 1}F_{3}^{LT}(\xi,Q^{2}) \\
& = \frac{2F_{3}^{LT}(N,Q^{2})}{N}, \\
\end{matrix}\end{split}\]
we obtain the following formula
\[\begin{split}\begin{matrix}
F_{3}^{NLT}(\xi,Q^{2}) & = & \,\int\frac{dN}{2\pi i}\,\xi^{- N}\,(\frac{x}{\tau\xi} + \, 4\frac{M^{2}}{Q^{2}}\frac{x^{2}}{\tau^{3/2}}\frac{1}{N}) \\
& & C_{3}(N,\alpha_{s}(Q^{2}))\,\Gamma(N,Q_{0}^{2},Q^{2})\, f\left( N,Q_{0}^{2} \right). \\
\end{matrix}\end{split}\]
The factor in front of \(C_{3}(N,\alpha_{s}(Q^{2}))\) can be
interpreted as the NLT Target Mass corrected coefficient function, which
can be written as a function of \(\tau\):
\[\begin{split}\begin{matrix}
C_{3}^{TMC}(N,\alpha_{s}(Q^{2})) & = & \frac{1 + \sqrt{\tau}}{2\tau}\left( 1\, + \, 2\,\left( 1 - \frac{1}{\sqrt{\tau}} \right)\frac{1}{N} \right)C_{3}(N,\alpha_{s}(Q^{2})). \\
\end{matrix}\end{split}\]