Mellin Space and Transformations

We solve the equations in Mellin-space as there multiplicative convolution is mapped onto a normal multiplication and thus integro-differential equations, such as DGLAP equations, are instead just normal differential equations.

The Mellin transformation is given by

\[\tilde g(N) = \mathcal{M}[g(x)](N) = \int\limits_{0}^{1} x^{N-1} g(x)\,dx\]

We will denote objects in Mellin-space with an additional tilde if they may appear in both spaces. The inverse Mellin transformation is given by

\[g(x) = \mathcal{M}^{-1}[\tilde g(N)](x) = \frac{1}{2\pi i} \int\limits_{\mathcal{P}} x^{-N} \tilde g(N)\,dN\]

for a suitable path \(\mathcal{P}(t)\) which runs to the right to the right-most pole of \(\tilde g(N)\). For the implementation we will assume that the integration path \(\mathcal P\) is given by

\[\mathcal P : [0,1] \to \mathbb C : t \to \mathcal P(t)\quad \text{with}~\mathcal P(1/2-t) = \mathcal P^*(1/2+t)\]

where \(\mathcal P^*\) denotes complex conjugation. Assuming further \(\tilde g\) to be a holomorphic function along the path, we can rewrite the inversion integral by

\[\begin{split}g(x) &= \frac{1}{2\pi i} \int\limits_{0}^{1} x^{-\mathcal{P}(t)} \tilde g(\mathcal{P}(t)) \frac{d\mathcal{P}(t)}{dt} \,dt\\ &= \frac{1}{\pi} \int\limits_{1/2}^{1} \Re \left( x^{-\mathcal{P}(t)} \tilde g(\mathcal{P}(t)) \frac 1 i \frac{d\mathcal{P}(t)}{dt} \right) \,dt\end{split}\]

Important Examples

Polynomials

\[\begin{split}\mathcal M[x^m](N) &= \frac 1 {N + m}\\ \mathcal M[x^m\Theta(x - x_{min})\Theta(x_{max} - x)](N) &= \frac {x_{max}^{N+m} - x_{min}^{N+m}} {N + m}\end{split}\]

Logarithms

\[\begin{split}\mathcal M[\ln(x)^m](N) &= \frac{d^m}{dN^m}\frac 1 {N}\\ \mathcal M[\ln^m(x)\Theta(x - x_{min})\Theta(x_{max} - x)](N) &= \frac{d^m}{dN^m}\frac {x_{max}^{N} - x_{min}^{N}} {N}\end{split}\]

Note that any derivative to either \(x_m^N\) or \(1/N\) is again proportional to its source.

Plus Distributions

\[\mathcal M[1/(1-x)_+](N) = S_1(N)\]

with the harmonic sum \(S_1\) (see Harmonic Sums).

Inversion of Factorizable Kernels

If the integration kernel \(\tilde g(N)\) can be factorized

\[\tilde g(N) = x_0^N \cdot \tilde h(N)\]

with \(x_0\) a fixed number in \((0,1]\) and \(\lim_{N\to\infty}h(N)\to 0\), the inversion can be simplified if the inversion point \(x_i\) is above \(x_0\).

\[\begin{split}g(x_i) &= \frac{1}{2\pi i} \int\limits_{\mathcal{P}} x_i^{-N} x_0^N \tilde h(N)\,dN \\ &= \frac{1}{2\pi i} \int\limits_{\mathcal{P}} \exp(-N(\ln(x_i)-\ln(x_0))) \tilde h(N)\,dN\end{split}\]

Now, take the textbook path \(p : \mathbb R \to \mathbb C : t \to p(t) = c + i t\) and consider the limit in which we shift the parameter \(c \to \infty\). As \(x_i > x_0\) it follows immediately \(\ln(x_i)-\ln(x_0) > 0\) and thus

\[|\exp(-N(\ln(x_i)-\ln(x_0)))| \to 0\]

Together with the assumed vanishing of \(\tilde h(N)\) we can conclude \(g(x_i) = 0\).

Convolution

Mellin space factorizes multiplicative convolution

\[\begin{split}(f \otimes g)(x) &= \int\limits_x^1 \frac{dy}{y} f(x/y) g(y)\\ \mathcal M[(f \otimes g)(x)](N) &= \mathcal M[f(x)](N) \cdot \mathcal M[g(x)](N)\end{split}\]

The convolution integral runs from \(x\) to 1, thus only basis functions which have support above \(x\) may contribute to the integral. This information is encoded in N-space in the following way: Due to the Mellin kernel \(x^{N-1}\) any piecewise polynomial, such as we are doing, are proportional to \(x_{\text{min/max}}^N = \exp(N\ln(x_{\text{min/max}}))\) (see Important Examples). They are thus factorizable is the above sense.

Harmonic Sums

In the computations of the anomalous dimensions (generalized) harmonic sums [Abl12] appear naturally:

\[\begin{split}S_{m}(N) &= \sum\limits_{j=1}^N \frac{(\text{sign}(m))^j}{j^{|m|}} \\ S_{m_0,m_1\ldots}(N) &= \sum\limits_{j=1}^N \frac{(\text{sign}(m_0))^j}{j^{|m_0|}} S_{m_1\ldots}(j)\end{split}\]

We then need to find an analytical continuation of these sums into the complex plain to perform the Mellin inverse.

the sums \(S_{m}(N)\) for \(m > 0\) do have a straight continuation:

\[\begin{split}S_m(N) = \sum\limits_{j=1}^N \frac 1 {j^m} = \frac{(-1)^{m-1}}{(m-1)!} \psi_{m-1}(N+1)+c_m \quad \text{with}\, c_m = \left\{\begin{array}{ll} \gamma_E, & m=1\\ \zeta(m), & m>1\end{array} \right.\end{split}\]

and where \(\psi_k(N)\) is the \(k\)-th polygamma function (implemented as cern_polygamma()) and \(\zeta\) the Riemann zeta function (using scipy.special.zeta()).

for the sums \(S_{-m}(N)\) and \(m > 0\) we use [GRV90]

\[S_m'(N) = 2^{m-1}(S_m(N) + S_{-m}(N)) = \frac{1+\eta}{2} S_m(N/2) + \frac{1-\eta}{2}S_m((N-1)/2)\]

where formally \(\eta = (-1)^N\) but in all singlet-like quantities it has to be analytically continued with 1 and with -1 elsewise.

for the sums with greater depth we use the lists provided in [BK99, GRV90, Mus17].
For \(S_{-2,1}(N)\) we use the implementation of [GRV90] (where it is called \(\tilde S\)):

\[\begin{split}S_{-2,1}(N) &= - \frac 5 8 \zeta(3) + \zeta(2)\left(S_{-1}(N) - \frac{\eta}{N} + \log(2)\right) + \eta\left(\frac{S_{1}(N)}{N^2} + g_3(N)\right)\\ g_3(N) &= \mathcal M \left[\frac{\text{Li}_2(x)}{1+x}\right](N)\end{split}\]

where for \(g_3(N)\) we use the parametrization of [Vog05] (implemented as mellin_g3()).