diff --git a/doc/source/theory/pQCD.rst b/doc/source/theory/pQCD.rst index fc0100803..ded1ee5e4 100644 --- a/doc/source/theory/pQCD.rst +++ b/doc/source/theory/pQCD.rst @@ -185,8 +185,7 @@ Scale Variations The usual procedure in solving |DGLAP| applied :doc:`here ` is to rewrite the equations in term of the running coupling :math:`a_s` assuming the factorization scale :math:`\mu_F^2` (the inherit scale -of the |PDF|) and the renormalization scale :math:`\mu_R^2` (the inherit scale -for the strong coupling) to be equal. +of the |PDF|) and the central scale :math:`Q^2` to be equal. This constraint, however, can be lifted in order to provide an estimation of the missing higher order uncertainties (|MHOU|) coming from |DGLAP| evolution :cite:`AbdulKhalek:2019ihb`. Since scale-dependent contributions to a perturbative prediction are fixed by |RGE| invariance, @@ -200,45 +199,45 @@ This method provides many advantages: since the scale dependence of the strong coupling :math:`a_s(\mu^2)` and of |PDF| are universal; However, there is no unique prescription to determine the specific range of the scale variation, -the most common prescription specify to vary the factor :math:`\mu_F/\mu_R` in the range: -:math:`1/2 \le \mu_F/\mu_R \le 2`. In the following we express this additional dependency as a function -of :math:`k = \ln(\mu_F^2/\mu_R^2)` +the most common prescription specify to vary the factor :math:`\mu_F/Q` in the range: +:math:`1/2 \le \mu_F/Q \le 2`. In the following we express this additional dependency as a function +of :math:`k = \ln(\mu_F^2/Q^2)` This variation can be performed at least at two different levels during the |PDF| -evolution, always evaluating the strong coupling at :math:`\mu_R^2`. +evolution, always evaluating the strong coupling at :math:`\mu_F^2`. * For ``ModSV='exponentiated'`` the variation is applied directly to the splitting functions - and the anomalous dimension are then modified using :cite:`Vogt:2004ns`: + and the anomalous dimension are then modified using :cite:`Vogt:2004ns` (note that here the log + is defined with the factorization scale in the numerator in contrast with what is done in + :cite:`Vogt:2004ns` ): .. math :: - \gamma^{(1)}(N) &\to \gamma^{(1)}(N) - \beta_0 k \gamma^{(0)} \\ - \gamma^{(2)}(N) &\to \gamma^{(2)}(N) - 2 \beta_0 k \gamma^{(1)} - ( \beta_1 k - \beta_0^2 k^2) \gamma^{(0)} \\ - \gamma^{(3)}(N) &\to \gamma^{(3)}(N) - 3 \beta_0 k \gamma^{(2)} - ( 2 \beta_1 k - 3 \beta_0^2 k^2) \gamma^{(1)} - (\beta_2 k - \frac{5}{2} \beta_1 \beta_0 k^2 + \beta_0^3 k^3) \gamma^{(0)} + \gamma^{(1)}(N) &\to \gamma^{(1)}(N) + \beta_0 k \gamma^{(0)} \\ + \gamma^{(2)}(N) &\to \gamma^{(2)}(N) + 2 \beta_0 k \gamma^{(1)} + ( \beta_1 k + \beta_0^2 k^2) \gamma^{(0)} \\ + \gamma^{(3)}(N) &\to \gamma^{(3)}(N) + 3 \beta_0 k \gamma^{(2)} + ( 2 \beta_1 k + 3 \beta_0^2 k^2) \gamma^{(1)} + (\beta_2 k + \frac{5}{2} \beta_1 \beta_0 k^2 + \beta_0^3 k^3) \gamma^{(0)} - This procedure corresponds to Eq. (3.32) of :cite:`AbdulKhalek:2019ihb`, and we recommend to use it along with - ``ModEv='iterate-exact'`` in order to be in agreement with the treatment of the evolution integral expansion. + This procedure corresponds to Eq. (3.32) of :cite:`AbdulKhalek:2019ihb`. * In ``ModSV='expanded'`` the full |EKO| is multiplied by an additional kernel: .. math :: \tilde{\mathbf{E}}(a_s \leftarrow a_s^0) & = \tilde{\mathbf{K}}(a_s) \tilde{\mathbf{E}}(a_s \leftarrow a_s^0) \\ - \tilde{\mathbf{K}}(a_s) & = 1 - k \gamma + \frac{1}{2} k^2 \left ( \gamma^{2} - \beta \frac{\partial \gamma}{\partial a_s} \right ) \\ - & \hspace{10pt} + \frac{1}{6} k^3 \left [ - \beta \frac{\partial}{\partial a_s} \left( \beta \frac{\partial \gamma}{\partial a_s} \right) + 3 \beta \frac{\partial \gamma}{\partial a_s} \gamma - \gamma^3 \right ] + \mathcal{O}(k^4) + \tilde{\mathbf{K}}(a_s) & = 1 + k \gamma + \frac{1}{2} k^2 \left ( \gamma^{2} - \beta \frac{\partial \gamma}{\partial a_s} \right ) \\ + & \hspace{10pt} - \frac{1}{6} k^3 \left [ - \beta \frac{\partial}{\partial a_s} \left( \beta \frac{\partial \gamma}{\partial a_s} \right) + 3 \beta \frac{\partial \gamma}{\partial a_s} \gamma - \gamma^3 \right ] + \mathcal{O}(k^4) - where the scale variation kernel :math:`\tilde{\mathbf{K}}` is expanded consistently order by order in :math:`a_s`, + where the scale variation kernel :math:`\tilde{\mathbf{K}}` is expanded consistently order by order in :math:`a_s(\mu_F^2)`, leading to: .. math :: - \tilde{\mathbf{K}}(a_s) &\approx 1 - a_s k \gamma^{(0)} + a_s^2 \left [ - k \gamma^{(1)} + \frac{1}{2} k^2 \gamma^{(0)} (\beta_0 + \gamma^{(0)}) \right ] \\ - & \hspace{10pt} + a_s^3 \left [ -k \gamma^{(2)} + \frac{1}{2} k^2 \left(\beta_1 \gamma^{(0)} + 2 \beta_0\gamma^{(1)} + \gamma^{(1)}\gamma^{(0)} + \gamma^{(0)}\gamma^{(1)} \right) \right. \\ - & \hspace{35pt} \left. - \frac{1}{6} k^3 \gamma^{(0)} \left(2 \beta_0^2 + 3 \beta_0 \gamma^{(0)} + \left(\gamma^{(0)}\right)^2 \right) \right] + \mathcal{O}(a_s^4) + \tilde{\mathbf{K}}(a_s) &\approx 1 + a_s k \gamma^{(0)} + a_s^2 \left [ k \gamma^{(1)} + \frac{1}{2} k^2 \gamma^{(0)} (\beta_0 + \gamma^{(0)}) \right ] \\ + & \hspace{10pt} + a_s^3 \left [ k \gamma^{(2)} + \frac{1}{2} k^2 \left(\beta_1 \gamma^{(0)} + 2 \beta_0\gamma^{(1)} + \gamma^{(1)}\gamma^{(0)} + \gamma^{(0)}\gamma^{(1)} \right) \right. \\ + & \hspace{35pt} \left. + \frac{1}{6} k^3 \gamma^{(0)} \left(2 \beta_0^2 + 3 \beta_0 \gamma^{(0)} + \left(\gamma^{(0)}\right)^2 \right) \right] + \mathcal{O}(a_s^4) In this way the dependence of the |EKO| on :math:`k` is factorized outside the unvaried evolution kernel. This procedure is repeated for each different flavor patch present in the evolution path. - It corresponds to Eq. (3.35) of :cite:`AbdulKhalek:2019ihb`, and we recommend to use it along with - ``ModEv='truncated'`` in order to keep consistency with the evolution integral expansion. + It corresponds to Eq. (3.35) of :cite:`AbdulKhalek:2019ihb`. By construction, the corrections of the order :math:`\mathcal{O}(k^n)` will appear at the order :math:`n` in the expansion :math:`a_s`.