Matching Conditions on Crossing Thresholds

In a VFNS one considers several matching scales (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 matching scales 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 (below for the example of n_f = 3):

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

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 no longer the ordinary singlet and valence distribution as they do not contain the new flavor \(\tilde h^{+}, \tilde h^{-}\). 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 \(a_s\):

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

where the \(\mathbf{A}^{(n_f),(k)}\) are given up to N3LO by the following expressions:

\[\begin{split}\mathbf{A}_{S,h^+}^{(n_f),(1)} &= \begin{pmatrix} A_{gg,H}^{S,(1)} & 0 & A_{gH}^{S,(1)} \\ 0 & 0 & 0 \\ A_{Hg}^{S,(1)} & 0 & A_{HH}^{(1)} \end{pmatrix} \\ \mathbf{A}_{nsv,h^-}^{(n_f),(1)} &= \begin{pmatrix} 0 & 0 \\ 0 & A_{HH}^{(1)}\end{pmatrix} \\ \mathbf{A}_{S,h^+}^{(n_f),(2)} &= \begin{pmatrix} A_{gg,H}^{S,(2)} & A_{gq,H}^{S,(2)} & 0 \\ 0 & A_{qq,H}^{ns,(2)} & 0 \\ A_{Hg}^{S,(2)} & A_{Hq}^{ps,(2)} & 0 \end{pmatrix} \\ \mathbf{A}_{nsv,h^-}^{(n_f),(2)} &= \begin{pmatrix} A_{qq,H}^{ns,(2)} & 0 \\ 0 & 0 \end{pmatrix} \\ \mathbf{A}_{S,h^+}^{(n_f),(3)} &= \begin{pmatrix} A_{gg,H}^{S,(3)} & A_{gq,H}^{S,(3)} & 0 \\ A_{qg,H}^{S,(3)} & A_{qq,H}^{ns,(3)} + A_{qq,H}^{ps,(3)} & 0 \\ A_{Hg}^{S,(3)} & A_{Hq}^{ps,(3)} & 0 \end{pmatrix} \\ \mathbf{A}_{nsv,h^-}^{(n_f),(3)} &= \begin{pmatrix} A_{qq,H}^{ns,(3)} & 0 \\ 0 & 0 \end{pmatrix}\end{split}\]

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 \(\ln(\mu_{h}^2/m_{h}^2)\). During the matching we use \(a_s^{(n_f+1)}\): in fact the \(a_s\) decoupling gives raise to some additional logarithms \(\ln(\mu_{h}^2/m_{h}^2)\), which are cancelled by the OME’s \(A_{kl,H}\).

N3LO matrix elements have been presented in [BBK09a] and following publications [ABBlumlein+22, ABBlumlein+24, ABB+14, ABB+15, ABDF+14a, ABDF+14b, ABK+11, BBB+14, BAB+17]. Parts proportional to \(\ln(\mu_{h}^2/m_{h}^2)\) are also included up to N3LO.

All the contributions are now known analytically. Due to the lengthy and complex expressions some parts of \(A_{Hg}^{S,(3)},A_{Hq}^{S,(3)},A_{gg}^{S,(3)},A_{qq}^{NS,(3)}\) have been parameterized.

We remark that contributions of the heavy quark initiated diagrams at NNLO and N3LO 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 \(A_{qq,H}^{ps},A_{qg,H}\) are known to start at N3LO.

Additional contributions due to \(\overline{MS}\) masses are included only up to NNLO.

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

Matching conditions for polarized and time-like evolution follow a similar structure. The former being implemented up to NNLO from [BBlumleinDF+23] and the latter up to NLO [CNO05] as the NNLO contributions are currently unknown.

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:

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

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.

QED Matching

In the QED case the matching is changed only because of the change of the evolution basis, therefore the only different part will be the basis rotation. In fact, the OME \(\mathbf{A}^{(n_f)}(\mu_{h}^2)\) don’t have QED corrections. The matching of the singlet sector is unchanged since it remains the same with respect to the QCD case. The same happens for the matching of the valence component. All the elements \(V_i\) and \(T_i\) are non-singlet components, therefore they are matched with \(A_{ns}\). In the end, the new components \(\Sigma_\Delta\) and \(V_\Delta\) are matched with \(A_{ns}\) since they are both non-singlets.

QED basis rotation

For the basis rotation we have to consider that we are using the intrinsic unified evolution basis. Here it will be discussed only the rotation to be applied to the sector \((\Sigma, \Sigma_\Delta, T_i)\), being the rotation of the sector \((V, V_\Delta, V_i)\) completely equivalent. The rotation matrix is given by:

\[\begin{split}\begin{pmatrix} \Sigma_{(n_f)} \\ \Sigma_{\Delta,(n_f)} \\ T_{i,(nf)} \end{pmatrix}^{(n_f+1)} = \begin{pmatrix} 1 & 0 & 1 \\ a(n_f) & b(n_f) & c(n_f) \\ d(n_f) & e(n_f) & f(n_f) \end{pmatrix} \begin{pmatrix} \Sigma_{(n_f)} \\ \Sigma_{\Delta,(n_f)} \\ h^+ \end{pmatrix}^{(n_f)}\end{split}\]


\[\begin{split}a(n_f) & = \frac{1}{n_f}\Bigl(\frac{n_d(n_f+1)}{n_u(n_f+1)}n_u(n_f)-n_d(n_f)\Bigr) \\ b(n_f) & = \frac{n_f+1}{n_u(n_f+1)}\frac{n_u(n_f)}{n_f} \\ c(n_f) & = \begin{cases} \frac{n_d(n_f+1)}{n_u(n_f+1)} \quad \text{if $h$ is up-like}\\-1 \quad \text{if $h$ is down-like}\end{cases} \\ d(n_f) & = \begin{cases} &\frac{n_u(n_f)}{n_f} \quad \text{if $h$ is up-like ($n_f$=3,5)} \\ &\frac{n_d(n_f)}{n_f} \quad \text{if $h$ is down-like ($n_f$=2,4)} \end{cases} \\ e(n_f) & = \begin{cases} &\frac{n_u(n_f)}{n_f} \quad \text{if $h$ is up-like} \\ &-\frac{n_u(n_f)}{n_f} \quad \text{if $h$ is down-like} \end{cases} \\ f(n_f) & = \begin{cases} &-1\quad \text{if $h$ is $s$, $c$ ($n_f$=2,3)} \\ &-2 \quad \text{if $h$ is $b$, $t$ ($n_f$=4,5)} \end{cases}\end{split}\]