Solving DGLAP

We follow here the notation and conventions from the eko documentation.

Photon PDF follow a modified DGLAP equation compared to the proton case, which contains an additional inhomogeneous term:

\[\frac{d}{d\ln(\mu_F^2)} \tilde{\mathbf{f}}(\mu_F^2) = -\gamma(a_s(\mu_F^2)) \cdot \tilde{\mathbf{f}}(\mu_F^2) - \tilde{\mathbf k}(a_s(\mu_F^2))\]

where \(\tilde{\mathbf k}\) are the photon-parton splitting functions (in Mellin space), which are computable in pQCD:

\[\mathbf k(a_s) = a_{em}\sum_{j=0}\left(a_s\right)^{j} \mathbf k^{(j)}\]

We apply the the usual transformation of variables and get

\[\frac{d}{da_s} \tilde{\mathbf{f}}^\gamma(a_s) = -\frac{\gamma(a_s)}{\beta(a_s)} \cdot \tilde{\mathbf{f}}^\gamma(a_s) - \frac{\tilde{\mathbf k}(a_s)}{\beta(a_s)}\]

which we then need to solve.

We can write the solution as a sum of a homogeneous (hadronic) component \(\mathbf f^\gamma_{hom}\) and an inhomogeneous (point-like) component \(\mathbf f^{\gamma}_{inhom}\):

\[\mathbf f^\gamma = \mathbf f^\gamma_{hom} + \mathbf f^\gamma_{inhom}\]

While the homogeneous term is just given by the standard solution (using the (standard) EKO)

\[\tilde{\mathbf f}^\gamma_{hom}(a_s) = \tilde {\mathbf E}(a_s \leftarrow a_s^0) \tilde{\mathbf f}^\gamma(a_s^0)\]

we find for the inhomogeneous term

\[\tilde{\mathbf f}^\gamma_{inhom}(a_s) = \int\limits_{a_s^0}^{a_s}\! da_s'\, \tilde{\mathbf E}(a_s \leftarrow a_s') \frac{-\tilde{\mathbf k}(a_s')}{\beta(a_s')}\]

which is thus the central equation we need to solve in γEKO. Note that both the homogeneous and inhomogeneous solution have to be solved simultaneously (or better consistently).

Leading order

At LO all ingredients are known analytically and we can give a closed form solution. For the non-singlet case we find

\[\tilde f_{inhom,ns}^{\gamma,(0)}(a_s) = \frac {a_{em}k_{ns}^{(0)}}{(\gamma_{ns}^{(0)}+\beta_0)} \left(\frac{\exp\left(\gamma_{ns}^{(0)}\ln(a_s/a_s^0) / \beta_0\right)}{a_s^0} -\frac 1 {a_s} \right)\]

and, analogously, in the singlet case

\[\begin{split}\tilde {\mathbf f}_{inhom,S}^{\gamma,(0)}(a_s) = \sum_{\lambda\in\{+,-\}} \frac {a_{em}}{(\gamma_{S,\lambda}^{(0)}+\beta_0)} \left(\frac{\exp\left(\gamma_{S,\lambda}^{(0)}\ln(a_s/a_s^0) / \beta_0\right)}{a_s^0} -\frac 1 {a_s} \right) \mathbf e_{\lambda}^{(0)} \left(\begin{matrix} k_q^{(0)}\\ 0 \end{matrix}\right)\end{split}\]

where we need to sum over the two eigenvalues of the singlet anomalous dimension matrix.

Iterative solution

Beyond LO EKO s are in general not known as closed form expression, but a numerical approximation strategy has to be implemented (see eko documentation for a detailed discussion). Currently γEKO only supports the iterate-exact solution, which relies on an iterative approach to solve the RGE [Bon12]. We exploit the strategy for computing the (standard) EKO \(\tilde {\mathbf E}\) to solve simultaneously our master equation here. The central observation is that we can use the same decomposition of \(\tilde {\mathbf E}\) into (infinitesimally) small pieces also for solving our master equation, by applying the trapezoidal rule to the integral.

In practice it works like this:

  • assume we split the integral along the points \(\{a_s^k, k = 0\ldots M\}\) with \(a_s^M = a_s\) the upper boundary of the integral

  • define the interval ranges \(\{\Delta a_s^k = a_s^{k+1} - a_s^k, k = 0\ldots M-1\}\)

  • start the iteration: \(\tilde{\mathbf g}^0 = \frac{\Delta a_s^0}{2} \tilde {\mathbf E}(a_s^1 \leftarrow a_s^0) \frac{-\tilde{\mathbf k}(a_s^0)}{\beta(a_s^0)}\)

  • iterate \(k = 1 \ldots M-1\): \(\tilde{\mathbf g}^k = \tilde {\mathbf E}(a_s^{k+1} \leftarrow a_s^k)\left(\frac{\Delta a_s^{k} + \Delta a_s^{k-1}}{2} \frac{-\tilde{\mathbf k}(a_s^k)}{\beta(a_s^k)} + \tilde{\mathbf g}^{k-1}\right)\)

  • close the iteration: \(\tilde{\mathbf g}^M = \frac{\Delta a_s^{M-1}}{2} \frac{-\tilde{\mathbf k}(a_s^0)}{\beta(a_s^0)}\)

  • identify the final result: \(\tilde{\mathbf f}^\gamma_{inhom}(a_s) = \tilde{\mathbf g}^M\)

All (partial) EKO s \(\tilde {\mathbf E}(a_s^{k+1} \leftarrow a_s^k)\) are evaluated using the strategy outlined in [Bon12], which evaluates the anomalous dimensions (and beta function) at the intermediate point, i.e. at \(a_s^{k+1/2} = (a_s^{k+1} + a_s^k)/2\). This implies that the parton-photon anomalous dimensions \(\tilde{\mathbf k}\) and parton-parton anomalous dimensions \(\gamma\) are not evaluated at the same strong coupling.