Matching Conditions on Crossing Thresholds

In a VFNS one considers several matching thresholds (as provided by the ThresholdsAtlas) where the number of active, light flavors that are participating in the DGLAP equation changes by one unit: \(n_f \to n_f +1\). This means the distributions do not behave in the same matter above and below the threshold: in esp. the new quark distributions \(q_{n_f+1}(x,\mu_F^2) = h(x,\mu_F^2)\) and \(\overline h(x,\mu_F^2)\) did not take part in the evolution below the threshold, but above they do. This mismatch in the evolution is accounted for by the matching conditions.

In the following we will denote the number of active flavors by a supscript \({}^{(n_f)}\). We denote the solution of the DGLAP equation in a region with a fixed number of active flavors, i.e. no threshold present \(\left(\mu_{h}^2 < Q_0^2 < Q_1^2 < \mu_{h+1}^2\right)\), in Mellin space as

\[\tilde{\mathbf{f}}^{(n_f)}(Q^2_1)= \tilde{\mathbf{E}}^{(n_f)}(Q^2_1\leftarrow Q^2_0) \tilde{\mathbf{f}}^{(n_f)}(Q^2_0)\]

The bold font indicates the vector space spanned by the flavor space and the equations decouple mostly in the Intrinsic Evolution Basis.

If a single threshold \(\left(\mu_{h-1}^2 < Q_0^2 < \mu_{h}^2 < Q_1^2 < \mu_{h+1}^2\right)\) is present we decompose the matching into two independent steps: first, the true QCD induced OME \(\mathbf{A}^{(n_f)}(\mu_{h}^2)\) that are given by perturbative calculations and expressed in the flavor space, and, second, the necessary flavor space rotation \(\mathbf{R}^{(n_f)}\) to fit the new Intrinsic Evolution Basis. We can then denote the solution as

\[\tilde{\mathbf{f}}^{(n_f+1)}(Q^2_1)= \tilde{\mathbf{E}}^{(n_f+1)}(Q^2_1\leftarrow \mu_{h}^2) {\mathbf{R}^{(n_f)}} \tilde{\mathbf{A}}^{(n_f)}(\mu_{h}^2) \tilde{\mathbf{E}}^{(n_f)}(\mu_{h}^2\leftarrow Q^2_0) \tilde{\mathbf{f}}^{(n_f)}(Q^2_0)\]

In the case of more than one threshold being present, the matching procedure is iterated on all thresholds starting from the lowest one.

Operator Matrix Elements

The matching matrices \(\mathbf{A}^{(n_f)}(\mu_{h+1}^2)\) mediate between \(\mathcal F_{iev,n_f}^{(n_f)}\) and \(\mathcal F_{iev,n_f}^{(n_f+1)}\), i.e. they transform the basis vectors of the \(n_f\)-flavors space in a \(n_f\)-flavor scheme to the \((n_f+1)\)-flavor scheme. Hence, the supscript refers to the flavor scheme with a smaller number of active flavors. To compute the matrices in a minimal coupled system we decompose the Intrinsic Evolution Basis \(\mathcal F_{iev,n_f}\) into several subspaces (of course irrespective of the FNS):

\[\begin{split}\mathcal F_{iev,3,S,c^+} &= \span(g,\Sigma,c^+)\\ \mathcal F_{iev,3,nsv,c^-} &= \span(V,c^-)\\ \mathcal F_{iev,3,ns+} &= \span(T_3,T_8)\\ \mathcal F_{iev,3,ns-} &= \span(V_3,V_8)\\ \mathcal F_{iev,3,hh} &= \span(b^+,b^-,t^+,t^-)\\ \mathcal F_{iev,n_f} &= \mathcal F_{iev,3,S,c^+} \otimes \mathcal F_{iev,3,nsv,c^-} \otimes \mathcal F_{iev,3,ns+} \otimes \mathcal F_{iev,3,ns-} \otimes \mathcal F_{iev,3,hh}\end{split}\]

We can then write the matching matrices \(\mathbf{A}^{(n_f)}(\mu_{h+1}^2)\) as

\[\begin{split}\dSVip{n_f}{\mu_{h}^2} &= \tilde{\mathbf{A}}_{S,h^+}^{(n_f)}(\mu_{h}^2) \dSVi{n_f}{\mu_{h}^2} \\ \dVip{n_f}{\mu_{h}^2} &= \tilde{\mathbf{A}}_{nsv,h^-}^{(n_f)}(\mu_{h}^2) \dVi{n_f}{\mu_{h}^2} \\ \dVj{j}{n_f+1}{\mu_h^2} &= \tilde{A}_{ns-}^{(n_f)}(\mu_{h}^2) \dVj{j}{n_f}{\mu_h^2}\\ \dTj{j}{n_f+1}{\mu_h^2} &= \tilde{A}_{ns+}^{(n_f)}(\mu_{h}^2) \dTj{j}{n_f}{\mu_h^2}\\ &\text{for }j=3,\ldots, n_f^2-1\end{split}\]

Note that in the left hand side basis the distributions \(\tilde \Sigma_{(n_f)}, \tilde V_{(n_f)}\) are not the Singlet and the Valence distributions any longer since, they do not contain the new higher flavor. Furthermore in the right side basis \(\tilde h^{+}, \tilde h^{-}\) are intrinsic contributions.

The \(\mathbf{A}^{(n_f)}(\mu_{h+1}^2)\) can be computed order by order in \(a_s\):

\[\mathbf{A}^{(n_f)}(\mu_{h}^2) = \mathbf{I} + a_s^{(n_f)}(\mu_{h}^2) \mathbf{A}^{(n_f),(1)} + \left(a_s^{(n_f)}(\mu_{h}^2)\right)^2 \mathbf{A}^{(n_f),(2)}\]

where the \(\mathbf{A}^{(n_f),(k)}\) are given up to NNLO by the following expressions:

\[\begin{split}\mathbf{A}_{S,h^+}^{(n_f),(1)} &= \begin{pmatrix} A_{gg,H}^{S,(1)} & 0 & A_{gH}^{S,(1)} \\ 0 & 0 & 0 \\ A_{Hg}^{S,(1)} & 0 & A_{HH}^{(1)} \end{pmatrix} \\ \mathbf{A}_{nsv,h^-}^{(n_f),(1)} &= \begin{pmatrix} 0 & 0 \\ 0 & A_{HH}^{(1)}\end{pmatrix} \\ \mathbf{A}_{S,h^+}^{(n_f),(2)} &= \begin{pmatrix} A_{gg,H}^{S,(2)} & A_{gq,H}^{S,(2)} & 0 \\ 0 & A_{qq,H}^{ns,(2)} & 0 \\ A_{Hg}^{S,(2)} & A_{Hq}^{ps,(2)} & 0 \end{pmatrix} \\ \mathbf{A}_{nsv,h^-}^{(n_f),(2)} &= \begin{pmatrix} A_{qq,H}^{ns,(2)} & 0 \\ 0 & 0 \end{pmatrix}\end{split}\]

The coefficients \(A^{(n_f),(k)}_{ij}(z,\mu_{h}^2)\) have been firstly computed in [BMSvN98] and have been Mellin transformed to be used inside EKO. They depend on the scale \(\mu_{h}^2\) only through the logarithm \(\ln(\mu_{h}^2/m_{h}^2)\), in particular the coefficient \(A_{gg,H}^{S,(1)}\) is fully proportional to \(\ln(\mu_{h}^2/m_{h}^2)\).

We remark that contributions of the higher quark at NNLO have not been computed yet, thus the elements \(A_{qH}^{(2)},A_{gH}^{(2)}A_{HH}^{(2)}\) are not encoded in EKO despite of being present. On the other hand the elements \(A_{qq}^{ps},A_{qg}\) are known to start at N3LO.

The OME are also required in the context of the FONLL matching scheme [FLNR10]. For Intrinsic Evolution this leads to considerable simplifications [BBR15].

Basis rotation

The rotation matrices \(\mathbf{R}^{(n_f)}\) mediate between \(\mathcal F_{iev,n_f}^{(n_f+1)}\) and \(\mathcal F_{iev,n_f+1}^{(n_f+1)}\), i.e. in the input and output the distributions are already in a scheme with \((n_f+1)\)-flavors and the new heavy quark is already non-trivial, but the basis vectors are still expressed with the elements of the \(n_f\)-flavors space. The matrices are fixed algebraic quantities and do not encode perturbative calculations.

The matrices are given by

\[\begin{split}\dSVe{n_f+1}{\mu_{h}^2} &= {\mathbf{R}}_{S,h^+}^{(n_f)} \dSVi{n_f+1}{\mu_{h}^2} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & - n_f \end{pmatrix} \dSVi{n_f+1}{\mu_{h}^2} \\ \dVe{n_f+1}{\mu_{h}^2} &= {\mathbf{R}}_{nsv,h^-}^{(n_f)} \dVi{n_f+1}{\mu_{h}^2} = \begin{pmatrix} 1 & 1 \\ 1 & - n_f \end{pmatrix} \dVi{n_f+1}{\mu_{h}^2} \\ & \text{for }j=(n_f+1)^2-1\\ {\mathbf{R}}^{(n_f)} &= \mathbf 1 ~ \text{otherwise}\end{split}\]

Backward evolution

For backward evolution the matching procedure has to be applied in the reversed order: while the inversion of the basis rotation matrices \(\mathbf{R}^{(n_f)}\) are easy to invert, this does not apply to the OME \(\mathbf{A}^{(n_f)}\). EKO implements two different strategies to perform this operation, that can be specified with the parameter backward_inversion:

backward_inversion = 'exact': the matching matrices are inverted exactly in N space, and then integrated entry by entry
backward_inversion = 'expanded': the matching matrices are inverted through a perturbative expansion in \(a_s\) before the Mellin inversion:

\[\begin{split}\mathbf{A}_{exp}^{-1}(\mu_{h}^2) &= \mathbf{I} - a_s(\mu_{h}^2) \mathbf{A}^{(1)} + a_s^2(\mu_{h}^2) \left [ \mathbf{A}^{(2)} - \left(\mathbf{A}^{(1)}\right)^2 \right ] + O(a_s^3) \\\end{split}\]

We emphasize that in the backward evolution, below the threshold, the remaining high quark PDFs are always intrinsic and do not evolve anymore. In fact, if the initial PDFs (above threshold) do contain an intrinsic contribution, this has to be evolved below the threshold otherwise momentum sum rules can be violated.