Estimating Missing Higher Order Uncertainties

Both, the beta function \(\beta(a_s)\) and the anomalous dimensions \(\gamma(a_s)\), are perturbatively calculated object and their full expression is never fully known. Thus, an exact all-order solution of the DGLAP equations can never be calculated and, moreover, the solution is unique (see here). So, it is interesting to give an estimate of the missing higher order uncertainties (MHOU) [AK+19] to account for this imperfect knowledge.

In order to do this we can provide several, independent prescriptions of which we require

  • the difference with the central solution has to be beyond the known perturbative orders

  • respect RGE invariance

  • universality: the same procedure can be used for any perturbative process

  • only use the number of flavors available locally

In the following we assume a variation by a factor \(\rho\), which range can not be determined a priori.

Shifting Anomalous Dimensions

This feature is active if ModSV='exponentiated'. It corresponds to Eq. (3.32) of [AK+19] and the procedure in [Vog05] (note that here the new anomalous dimension is expanded in \(a_s(\rho Q^2)\) and so the log is inverted to [Vog05]).

We make an ansatz

\[\begin{split}\gamma\left(a_s(Q^2)\right) &= \bar \gamma\left(a_s(\rho Q^2),\rho\right) + O\left(\left(\alpha_s(Q^2)\right)^{N+2}\right)\\ \Rightarrow \sum_{j=0}^N \left(a_s(Q^2)\right)^{1+j} \gamma_j &= \sum_{j=0}^N \left(a_s(\rho Q^2)\right)^{1+j} \bar \gamma_j(\rho) + O\left(\left(\alpha_s(Q^2)\right)^{N+2}\right)\end{split}\]

which we can solve, by defining the scale varied anomalous dimensions \(\bar\gamma_j\) with

\[\begin{split}\bar \gamma_0(\rho) &= \gamma_0\\ \bar \gamma_1(\rho) &= \gamma_1 + \gamma_0 \beta_0\ln\rho \\ \bar \gamma_2(\rho) &= \gamma_2 + \gamma_0 \beta_0^2\ln^2\rho + \left(2\gamma_1 \beta_0 + \gamma_0 \beta_1\right)\ln\rho \\ \bar \gamma_3(\rho) &= \gamma_3 + \gamma_0 \beta_0^3\ln^3\rho + \left(3\gamma_1 \beta_0^2 + \frac 5 2 \gamma_0 \beta_0 \beta_1\right)\ln^2\rho\\ &\hspace{20pt} + \left(3\gamma_2 \beta_0 + 2\gamma_1 \beta_1 + \gamma_0 \beta_2\right)\ln\rho\end{split}\]

This procedure is repeated for each flavor patch present in the evolution path.

Shifting PDF Matching Conditions Matrix

This feature is active if ModSV='exponentiated'. It is not mentiond in [AK+19] as this paper deals only with a FFNS scenario and slightly implicit in [Vog05] as this paper only deals with \(a_s^2(\mu^2)\) contributions to the matching conditions, due to the lack of shifted matching points and intrinsic contributions.

We make an ansatz

\[\begin{split}\mathbf{A}\left(a_s(m_h^2),\lambda_f\right) &= \bar{\mathbf{A}}\left(a_s(\rho m_h^2),\lambda_f,\rho\right) + O\left(\left(\alpha_s(m_h^2)\right)^{N+1}\right)\\ \Rightarrow \sum_{j=0}^N \left(a_s(m_h^2)\right)^{j} \mathbf{A}_{j}(\lambda_f) &= \sum_{j=0}^N \left(a_s(\rho m_h^2)\right)^{j} \bar{\mathbf{A}}_{j}(\lambda_f,\rho) + O\left(\left(\alpha_s(m_h^2)\right)^{N+1}\right)\end{split}\]

which can be solve analogously to the case of anomalous dimensions above.

This procedure is repeated for each matching present in the evolution path.

Shifting Strong Coupling Matching Point

This feature is active if ModSV='exponentiated'. It is not mentiond in [AK+19], but corresponds to the procedure in [Vog05].

We can shift the matching point of the strong coupling \(a_s(m_h^2)\) to estimate the MHOU related to the respective matching decoupling parameters. For ModSV='exponentiated' we match at \(a_s(\rho m_h^2)\), which naturally ensures the consistency of used number of flavors.

Adding an EKO

This feature is active if ModSV='expanded' and it corresponds to Eq. (3.35) of [AK+19].

We make an ansatz

\[\bar{\mathbf{f}}(Q^2,\rho) = \mathbf{K}(a_s(\rho Q^2),\rho) \mathbf{E}(\rho Q^2 \leftarrow Q^2) \mathbf{f}(Q^2)\]


\[\begin{split}\mathbf{1} &= \mathbf{K}(a_s(\rho Q^2),\rho) \mathbf{E}(\rho Q^2 \leftarrow Q^2) + O\left(\left(\alpha_s(Q^2)\right)^{N+2}\right) \\ &= \sum_{j=0}^{N+1} \left(a_s(\rho Q^2)\right)^{j} \mathbf{K}_j(\rho) \mathbf{E}(\rho Q^2 \leftarrow Q^2) + O\left(\left(\alpha_s(Q^2)\right)^{N+2}\right)\end{split}\]

which we can solve by defining the scale variation kernel \(K_j\) with

\[\begin{split}\mathbf{K}_0(\rho) &= \mathbf{1}\\ \mathbf{K}_1(\rho) &= \gamma_0 \log\rho\\ \mathbf{K}_2(\rho) &= \frac 1 2 \gamma_0 \left(\beta_0 + \gamma_0\right) \log^2\rho + \gamma_1 \log\rho\\ \mathbf{K}_3(\rho) &= \frac 1 6 \gamma_0 \left(2\beta_0^2 + 3\beta_0\gamma_0 + \gamma_0^2\right) \log^3\rho\\ &\hspace{20pt} + \frac 1 2 \left(\beta_1\gamma_0 + 2\beta_0\gamma_1 + \gamma_0\gamma_1 + \gamma_1\gamma_0\right) \log^2\rho + \gamma_2 \log\rho\end{split}\]

This procedure is applied only to the last flavor patch present in the evolution path.

Note, that it is also possible and common to reattribute \(\mathbf{K}\) instead to the hard matrix element.