text:
DESY 23–146 October 2023
LTH 1353

The double fermionic contribution to the
four-loop quark-to-gluon splitting function
G. Falcioni${}^{\,a,b}$, F. Herzog${}^{\,c}$, S. Moch${}^{\,d}$, J.
Vermaseren${}^{\,e}$ and A. Vogt${}^{\,f}$

aDipartimento di Fisica, Università di Torino, Via Pietro Giuria 1, 10125
Torino, Italy
b Physik-Institut, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich,
Switzerland
cHiggs Centre for Theoretical Physics, School of Physics and Astronomy
The University of Edinburgh, Edinburgh EH9 3FD, Scotland, UK
dII. Institute for Theoretical Physics, Hamburg University
Luruper Chaussee 149, D-22761 Hamburg, Germany
eNikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands

fDepartment of Mathematical Sciences, University of Liverpool
Liverpool L69 3BX, United Kingdom
Abstract

We have computed the first 30 even-$N$ moments for the double fermionic
(${n^{\,2}_{\\!f}}\,$) part of the quark-to-gluon splitting function $P_{\rm
gq}$ at the fourth order of perturbative QCD via the renormalization of off-
shell operator matrix elements. From these results we have determined the
all-$N$ form, and hence the exact $x$-space expression, using systems of
Diophantine equations for its coefficients. The dominant and subdominant
leading small-$x$ ${n^{\,2}_{\\!f}}$ contributions to $P^{\,(3)}_{\rm gq}(x)$
are of the form $x^{\,-1}\ln x$ and $\ln^{\,4\\!}x$, respectively; the leading
large-$x$ term is $\ln^{\,4\\!}{(1\\!-\\!x)}$. The coefficient of the first of
these is new, the other two agree with results obtained before and thus
provide checks of our results.

Next-to-next-to-next-to-leading order (N3LO) calculations of benchmark
processes in perturbative QCD form an important part of the accuracy frontier
at the Large Hadron Collider LHC. Complete analyses at this order require the
4-loop splitting functions for the scale dependence of the proton’s parton
distributions in the fractional momentum $x$, in the flavour singlet sector
given by

$\frac{d}{d\ln{\mu^{\,2}}}\;\Big{(}\begin{array}[]{c}\\!q_{\rm s}\\!\\!\\\
\\!g\\!\end{array}\Big{)}\>=\>\left(\begin{array}[]{cc}\\!P_{\rm qq}&P_{\rm
qg}\\!\\!\\!\\\ \\!P_{\rm gq}&P_{\rm
gg}\\!\\!\\!\end{array}\right)\otimes\Big{(}\begin{array}[]{c}\\!q_{\rm
s}\\!\\!\\\ \\!g\\!\end{array}\Big{)}\quad\mbox{with}\quad P_{\,\rm
ik}(x,\alpha_{\rm s})\,=\,\sum_{n=0}{a_{\rm s}^{\,n+1}}\,P_{\,\rm
ik}^{\,(n)}(x)\>,$ (1)

where $q_{\rm s}\,=\,\sum_{\,i=1}^{\,{n_{\\!f}}}\,(q_{i}+\bar{q}_{i})$ and $g$
are the singlet quark and gluon distributions and $\otimes$ denotes the Mellin
convolution in the momentum variable $x$. The determination of these splitting
functions requires very involved calculations, and only partial results have
been obtained so far.

The leading ${n^{\,3}_{\\!f}}$ contributions in the limit of a large numbers
of light flavours ${n_{\\!f}}$ were completed in ref. [1] together with the
${n^{\,2}_{\\!f}}$ parts of the flavour non-singlet quark-quark splitting
functions $P_{\,\rm ns}^{\,(3)\pm}(x)$. Very recently these results have been
extended to the ${n^{\,2}_{\\!f}}$ part of the pure-singlet quark-quark
splitting function $P_{\,\rm ps}^{\,(3)}(x)\,=\,P_{\,\rm
qq}^{\,(3)}(x)-P_{\,\rm ns}^{\,(3)+}(x)$ [2]. The N3LO non-singlet splitting
functions are completely known in the limit of a large number of colours
${n_{c}}$ [3].

Beyond the quark-quark cases and the ${n^{\,3}_{\\!f}}$ terms, only a limited
number of moments of the 4-loop splitting functions have been computed so far.
These quantities correspond (up to a conventional sign) to the anomalous
dimensions of the gauge invariant operators of twist two,

$\gamma_{\,\rm ik}^{\,(3)}(N)\>=\>-\int_{0}^{1}\\!dx\>\,x^{\,N-1}\,P_{\,\rm
ik}^{\,(3)}(x).$ (2)

Building on recent progress on the renormalization of flavour-singlet operator
matrix elements (OMEs), in particular refs. [4, 5], we have been able to
obtain the even-$N$ Mellin moments to $N=20$ of $P_{\rm ps}^{\,(3)}$ and
$P_{\rm qg}^{\,(3)}$ [6, 7]. For the lower-row quantities in eq. (1), only
determinations to $N\\!=\\!10$ have been completed so far, using the
conceptually much easier but computationally harder route via structure
functions in deep-inelastic scattering (DIS) [8, 9]. In the present letter, we
present the first analytic $x$-dependence of a 4-loop ${n^{\,2}_{\\!f}}$
contribution to eq. (1) beyond the quark-quark case.

As in refs. [6, 7], our calculations are performed in the framework of the
operator-product expansion, via off-shell OME’s. Therefore the renormalization
of the results requires counterterms associated with gauge-variant and ghost
operators, commonly known as aliens. However, only a few alien operators enter
for the ${n^{\,2}_{\\!f}}$ contribution in the quark-to-gluon case. In
particular, this contribution always includes a factor $C_{F}$, hence only the
colour factors ${C^{\,2}_{F}}\,{n^{\,2}_{\\!f}}$ and
$C_{F}C_{A}\,{n^{\,2}_{\\!f}}$ can appear at four loops. The former
contribution, which is also present in QED, does not require any alien
counterterms. The latter colour factor requires only one class of aliens,
namely (see ref. [6])

$O_{\rm q}^{\,I}\,=\,\eta\,g\,\overline{\psi}\,\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\Delta\,t^{a}\,\psi\,\left(\partial^{\,N-2}A_{a}\right)\,,\quad
O_{\rm
g}^{\,I}\,=\,\eta\,\left(D.F\right)^{a}\left(\partial^{\,N-2}A_{a}\right)\,,\quad
O_{\rm
c}^{\,I}\,=\,-\eta\,\left(\partial\,\bar{c}^{\,a}\right)\left(\partial^{\,N-1}c_{a}\right)\,.$
(3)

where $\psi$, $A$ and $c$ represent the quark, gluon and ghost fields, $F$ the
gluon field-strength tensor and $D=\partial-{\rm i}gA$ the covariant
derivative with the coupling $g$, where $g^{2}/(4\pi)=\alpha_{\rm
s}=4\pi\,a_{\rm s}$, after contraction of the Lorentz indices with $N$
identical light-like vectors $\Delta^{\,\mu}$. The mixing $\eta$, which is a
function of $N$ and $\alpha_{\rm s}$, needs to be determined to three loops.
This is achieved efficiently by renormalizing the 3-loop ghost-antighost
correlator with an insertion of the gauge-invariant gluon operators. The
results to $N=60$ agree with all-$N$ the expressions of ref. [5].

The Feynman diagrams contributing to the ${n^{\,2}_{\\!f}}$ part of
$\gamma_{\,\rm gq}^{\,(3)}(N)$ form a small subset of those for the general
case. A few examples are shown in fig. 1. The diagrams are processed in the
usual manner, see ref. [10], by a Form [11, 12, 13] program which collects
self-energy insertions, determines the colour factors [14] and classifies the
topologies according to the conventions of the Forcer program [15]. An
optimized in-house version of this program has been employed to perform the
integral reduction for fixed even values of $N$ in $4-2\varepsilon$
dimensions. For the highest moments of the hardest 3-loop diagrams we have
used the Mincer program [16, 17], suitably extended in its tables from the
version optimized for ref. [18].

Figure 1:  Sample diagrams that enter, after inserting the gluon operator into
one of the gluon lines, the computation of the ${n^{\,2}_{\\!f}}$ part of
$\gamma_{\,\rm gq}^{\,(3)}(N)$ at even values of $N$. Left: the only genuine
4-loop diagram. Due to Furry’s theorem its colour factor is
$C_{F}C_{A}\,{n^{\,2}_{\\!f}}$. Right: the 3-loop and (after replacing one of
the gluon lines by a one-loop gluon propagator) 4-loop cases that lead to the
hardest computations at large $N$.

Using these tools, we have been able to compute the anomalous dimensions
$\gamma_{\,\rm gq}^{\,(3)}(N)$ in eq. (2) at all even values $2\leq N\leq 60$.
The corresponding all-$N$ expressions include Riemann-$\zeta$ values, harmonic
sums [19] and simple denominators $\,D_{a}^{\,k}\equiv(N+a)^{-k}$. The range
of $a$, the maximal powers of $D_{a}$ and maximal weights of the sums can be
inferred from the prime-factor decomposition of the denominators at $N\leq
60$. Also lower-order expressions, in particular the
${C^{\,2}_{F}}\,{n_{\\!f}}$ and $C_{F}C_{A}\,{n_{\\!f}}$ parts of
$\gamma_{\,\rm gq}^{\,(2)}(N)$ [20], provide useful information about the
expected analytic form of $\gamma_{\,\rm gq}^{\,(3)}(N)$.

In this manner, we are led to an ansatz of 143 functions for the non-$\zeta$
${C^{\,2}_{F}}\,{n^{\,2}_{\\!f}}$ contribution, hence there are far too many
unknown coefficients for a direct determination from the calculated moments.
However, these coefficients are integer modulo some powers of 2 and 3.
Therefore the resulting system of equations can be turned into a Diophantine
system which requires far fewer equations than unknowns and which can be
addressed by number-theoretical techniques, as, e.g., in refs. [21, 22], based
on the LLL algorithm [23]. This approach has been successfully applied to
3-loop and 4-loop splitting functions before, e.g., in refs. [24, 18, 1, 3].

We have eliminated 19 coefficients – of all functions with overall weights
$w\\!=\\!1$ and $w\\!=\\!2$, and some of $w\\!=\\!3$ – which are mostly
expected to be very large, and determined the remaining 124 unknowns from the
11 equations for $2\leq N\leq 60$, using the program of ref. [22]. 17 of these
124 coefficients turned out to vanish. An analogous procedure was used for the
$C_{F}C_{A}\,{n^{\,2}_{\\!f}}$ part. We will comment on the differences
between the two cases below.

The resulting expression for the ${n^{\,2}_{\\!f}}$ part of the N3LO quark-to-
gluon splitting function at all even $N\geq 2$ reads

$\displaystyle\gamma_{\rm gq}^{\,(3)}(N)\Big{|}_{{n^{\,2}_{\\!f}}}\;=\;$ (4)
$\displaystyle\mbox{{}}\mbox{\large{$\frac{16}{27}$}}\\*{C^{\,2}_{F}}\\*{n^{\,2}_{\\!f}}\\*\bigg{(}{p_{gq}}(N)\\*(-8\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{4}}-16\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,3}}-8\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,2}}-8\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{3,1}}+36\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,2}}+28\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,2,1}}+16\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,1,1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+16\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,1,1}})+24\,\\*(N^{2}+N+2)\,\\*({D_{0}}^{2}\,\\*{D_{1}}^{2}\,\\*{D_{2}}\,\\*{D_{-1}^{\,2}})\,\\*(8\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{-3}}-12\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{-2,1}}-12\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,-2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{3}}\,\\*(276\,\\*{D_{-1}}-860/3\,\\*{D_{0}}+259/3\,\\*{D_{1}}-64/3\,\\*{D_{2}}-64\,\\*{D_{-1}^{\,2}}-100\,\\*{D_{0}^{\,2}}-48\,\\*{D_{0}^{\,3}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-26\,\\*{D_{1}^{\,2}}+24\,\\*{D_{1}^{\,3}})+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,2}}\,\\*(-224\,\\*{D_{-1}}+758/3\,\\*{D_{0}}-361/3\,\\*{D_{1}}+16/3\,\\*{D_{2}}+16\,\\*{D_{-1}^{\,2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+20\,\\*{D_{0}^{\,2}}+24\,\\*{D_{0}^{\,3}}+8\,\\*{D_{1}^{\,2}}-12\,\\*{D_{1}^{\,3}})+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,1}}\,\\*(-500/3\,\\*{D_{-1}}+586/3\,\\*{D_{0}}-275/3\,\\*{D_{1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+16/3\,\\*{D_{2}}+16\,\\*{D_{-1}^{\,2}}+20\,\\*{D_{0}^{\,2}}+24\,\\*{D_{0}^{\,3}}-12\,\\*{D_{1}^{\,3}})+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,1}}\,\\*(-148\,\\*{D_{-1}}+364/3\,\\*{D_{0}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-409/6\,\\*{D_{1}}-16/3\,\\*{D_{2}}-16\,\\*{D_{-1}^{\,2}}-22\,\\*{D_{0}^{\,2}}-24\,\\*{D_{0}^{\,3}}+45\,\\*{D_{1}^{\,2}}+12\,\\*{D_{1}^{\,3}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{-2}}\,\\*(-5504/9\,\\*{D_{-1}}+504\,\\*{D_{0}}+24\,\\*{D_{1}}+752/9\,\\*{D_{2}}+712/3\,\\*{D_{-1}^{\,2}}+480\,\\*{D_{0}^{\,2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+96\,\\*{D_{0}^{\,3}}+96\,\\*{D_{1}^{\,2}}-144\,\\*{D_{1}^{\,3}}-32/3\,\\*{D_{2}^{\,2}})+{\color[rgb]{1,0,0.87}\mathbf{S}_{2}}\,\\*(5432/9\,\\*{D_{-1}}-1402/3\,\\*{D_{0}}+73/6\,\\*{D_{1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-944/9\,\\*{D_{2}}-416/3\,\\*{D_{-1}^{\,2}}-1600/3\,\\*{D_{0}^{\,2}}-80\,\\*{D_{0}^{\,3}}-120\,\\*{D_{0}^{\,4}}-212/3\,\\*{D_{1}^{\,2}}+130\,\\*{D_{1}^{\,3}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+48\,\\*{D_{1}^{\,4}}-32\,\\*{D_{2}^{\,2}})+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1}}\,\\*(121/9\,\\*{D_{-1}}+2237/18\,\\*{D_{0}}-2219/36\,\\*{D_{1}}+424/9\,\\*{D_{2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+52\,\\*{D_{-1}^{\,2}}+257/3\,\\*{D_{0}^{\,2}}+102\,\\*{D_{0}^{\,3}}-299/2\,\\*{D_{1}^{\,2}}-20\,\\*{D_{1}^{\,3}}+12\,\\*{D_{1}^{\,4}}+16/3\,\\*{D_{2}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{1}}\,\\*(82801/54\,\\*{D_{-1}}-65503/36\,\\*{D_{0}}+31639/72\,\\*{D_{1}}-3086/27\,\\*{D_{2}}-2660/9\,\\*{D_{-1}^{\,2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-4915/9\,\\*{D_{0}^{\,2}}-1351\,\\*{D_{0}^{\,3}}+14\,\\*{D_{0}^{\,4}}-312\,\\*{D_{0}^{\,5}}-2219/12\,\\*{D_{1}^{\,2}}-2491/6\,\\*{D_{1}^{\,3}}+85\,\\*{D_{1}^{\,4}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+156\,\\*{D_{1}^{\,5}}-352/3\,\\*{D_{2}^{\,2}}-224/3\,\\*{D_{2}^{\,3}})-1094983/324\,\\*{D_{-1}}+282613/72\,\\*{D_{0}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-125953/144\,\\*{D_{1}}+16636/81\,\\*{D_{2}}+19192/27\,\\*{D_{-1}^{\,2}}+1540\,\\*{D_{0}^{\,2}}+2766\,\\*{D_{0}^{\,3}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+515/3\,\\*{D_{0}^{\,4}}+766\,\\*{D_{0}^{\,5}}+2265/8\,\\*{D_{1}^{\,2}}+6425/12\,\\*{D_{1}^{\,3}}-4019/6\,\\*{D_{1}^{\,4}}-256\,\\*{D_{1}^{\,5}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+228\,\\*{D_{1}^{\,6}}+312\,\\*{D_{2}^{\,2}}+1312/9\,\\*{D_{2}^{\,3}}-64\,\\*{D_{2}^{\,4}}\,+\,4\\*{\zeta_{3}}\,\\*(-6\,\\*{p_{gq}}(N)\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{1}}+52\,\\*{D_{-1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-89\,\\*{D_{0}}+76\,\\*{D_{1}}+4\,\\*{D_{2}}+12\,\\*{D_{-1}^{\,2}}+42\,\\*{D_{0}^{\,2}}+9\,\\*{D_{1}^{\,2}})\,-\,72\,\\*{\zeta_{4}}\,\\*{p_{gq}}(N)\bigg{)}$
$\displaystyle\mbox{{}}+\mbox{\large{$\frac{16}{27}$}}\\*{C_{F}}\\*{C_{A}}\\*{n^{\,2}_{\\!f}}\\*\bigg{(}-4/3\,\\*{\color[rgb]{1,0,0.87}\theta(N\\!-\\!4)}\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{-2}(N\\!-\\!2)}\,\\*{D_{-2}}+{p_{gq}}(N)\\*(-2\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{-4}}-2\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{4}}-23\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{-3,1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-20\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{-2,-2}}-2\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{-2,2}}-24\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,-3}}+14\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,3}}-12\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,-2}}-8\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,2}}+25\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{3,1}}+20\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{-2,1,1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+46\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,-2,1}}+58\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,-2}}+27\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,2}}+25\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,2,1}}+26\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,1,1}}-16\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,1,1}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{3}}\,\\*(-518/3\,\\*{D_{-1}}+268/3\,\\*{D_{0}}+106/3\,\\*{D_{1}}+40/3\,\\*{D_{2}}+32\,\\*{D_{-1}^{\,2}}+112\,\\*{D_{0}^{\,2}}+58\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,1}}\,\\*(-461/3\,\\*{D_{-1}}+427/3\,\\*{D_{0}}-188/3\,\\*{D_{1}}+4/3\,\\*{D_{2}}-16\,\\*{D_{-1}^{\,2}}+28\,\\*{D_{0}^{\,2}}+27\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,2}}\,\\*(-137\,\\*{D_{-1}}+445/3\,\\*{D_{0}}-248/3\,\\*{D_{1}}-4/3\,\\*{D_{2}}-24\,\\*{D_{-1}^{\,2}}+12\,\\*{D_{0}^{\,2}}+5\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,1}}\,\\*(291\,\\*{D_{-1}}-796/3\,\\*{D_{0}}+299/3\,\\*{D_{1}}-44/3\,\\*{D_{2}}+52\,\\*{D_{-1}^{\,2}}-140\,\\*{D_{0}^{\,2}}-12\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,-2}}\,\\*(-1306/3\,\\*{D_{-1}}+808/3\,\\*{D_{0}}+34/3\,\\*{D_{1}}+16\,\\*{D_{2}}+48\,\\*{D_{-1}^{\,2}}+192\,\\*{D_{0}^{\,2}}+130\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{-2,1}}\,\\*(-914/3\,\\*{D_{-1}}+416/3\,\\*{D_{0}}+242/3\,\\*{D_{1}}+16\,\\*{D_{2}}+48\,\\*{D_{-1}^{\,2}}+192\,\\*{D_{0}^{\,2}}+118\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{-3}}\,\\*(694/3\,\\*{D_{-1}}-404/3\,\\*{D_{0}}-50/3\,\\*{D_{1}}-32/3\,\\*{D_{2}}-32\,\\*{D_{-1}^{\,2}}-112\,\\*{D_{0}^{\,2}}-72\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{2}}\,\\*(169/6\,\\*{D_{-1}}-59/18\,\\*{D_{0}}+277/18\,\\*{D_{1}}+64/9\,\\*{D_{2}}+284/3\,\\*{D_{-1}^{\,2}}-127/3\,\\*{D_{0}^{\,2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+24\,\\*{D_{0}^{\,3}}-77/2\,\\*{D_{1}^{\,2}}+{D_{1}^{\,3}})+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1}}\,\\*(-23147/36\,\\*{D_{-1}}+19511/36\,\\*{D_{0}}-1630/9\,\\*{D_{1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+74/3\,\\*{D_{2}}-128\,\\*{D_{-1}^{\,2}}+709/2\,\\*{D_{0}^{\,2}}-36\,\\*{D_{0}^{\,3}}+1175/12\,\\*{D_{1}^{\,2}}-28\,\\*{D_{1}^{\,3}}-80/3\,\\*{D_{2}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{-2}}\,\\*(5380/9\,\\*{D_{-1}}-1520/3\,\\*{D_{0}}+55/3\,\\*{D_{1}}-376/9\,\\*{D_{2}}-356/3\,\\*{D_{-1}^{\,2}}-1148/3\,\\*{D_{0}^{\,2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-80\,\\*{D_{0}^{\,3}}-422/3\,\\*{D_{1}^{\,2}}+138\,\\*{D_{1}^{\,3}}+16/3\,\\*{D_{2}^{\,2}})+{\color[rgb]{1,0,0.87}\mathbf{S}_{1}}\,\\*(7145/216\,\\*{D_{-1}}+3775/36\,\\*{D_{0}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-3527/36\,\\*{D_{1}}+6158/27\,\\*{D_{2}}+1331/6\,\\*{D_{-1}^{\,2}}+4099/36\,\\*{D_{0}^{\,2}}+1523/6\,\\*{D_{0}^{\,3}}+36\,\\*{D_{0}^{\,4}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-128\,\\*{D_{1}^{\,2}}+1513/12\,\\*{D_{1}^{\,3}}-87\,\\*{D_{1}^{\,4}}+1532/9\,\\*{D_{2}^{\,2}}-32/3\,\\*{D_{2}^{\,3}})+304049/324\,\\*{D_{-1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-66833/72\,\\*{D_{0}}+28097/48\,\\*{D_{1}}-57623/81\,\\*{D_{2}}-19435/54\,\\*{D_{-1}^{\,2}}-26555/36\,\\*{D_{0}^{\,2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-8183/18\,\\*{D_{0}^{\,3}}-356/3\,\\*{D_{0}^{\,4}}-12\,\\*{D_{0}^{\,5}}-17785/72\,\\*{D_{1}^{\,2}}-6941/36\,\\*{D_{1}^{\,3}}+836/3\,\\*{D_{1}^{\,4}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-88\,\\*{D_{1}^{\,5}}-2632/9\,\\*{D_{2}^{\,2}}+256/3\,\\*{D_{2}^{\,3}}\,+\,{\zeta_{3}}\,\\*(\,2\,\\*{\color[rgb]{1,0,0.87}\delta(N\\!-\\!2)}+36\,\\*{p_{gq}}(N)\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{1}}-206\,\\*{D_{-1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+378\,\\*{D_{0}}-324\,\\*{D_{1}}-16\,\\*{D_{2}}-48\,\\*{D_{-1}^{\,2}}-168\,\\*{D_{0}^{\,2}}-72\,\\*{D_{1}^{\,2}})+\,72\,\\*{\zeta_{4}}\,\\*{p_{gq}}(N)\bigg{)}\;.$

The ${\zeta_{4}}$ contributions to all 4-loop splitting functions in eq. (1)
have been derived in ref. [25] from the no-$\pi^{2}$ theorem [26, 27]. The
very simple ${\zeta_{4}}$ parts of eq. (4) proportional to

${p_{gq}}(N)\;=\;2/(N-1)\,-\,2/N\,+\,1/(N+1)\;\equiv\;2\,{D_{-1}}-2\,{D_{0}}\,+\,{D_{1}}\;.$
(5)

agree with that result. With the exception of the first term in the
$C_{F}C_{A}\,{n^{\,2}_{\\!f}}$ part, all harmonic sums are to be evaluated at
$N$; this argument has been suppressed in eq. (4) for brevity.

The presence of a contribution with $(N-2)^{-1}\,S_{-2}(N-2)$ – which does, of
course, not lead to a singularity at $N=2$ – is clearly indicated by the
presence of a $\delta(N\\!-\\!2)\,{\zeta_{3}}$ term. Such terms do not occur
in the splitting functions to three loops, but they are present in the
coefficient functions for inclusive DIS already at order ${\alpha_{\rm
s}^{\,2}}$ [28, 29]. Very recently, the same structure was found to appear in
the $C_{F}C_{A}\,{n^{\,2}_{\\!f}}$ part of the pure-singlet anomalous
dimension $\gamma_{\,\rm ps}^{\,(3)}(N)$ [2].

As expected from the values to $N\leq 60$, the overall weight (obtained for
each term by adding the power of $1/(N+a)$ to the weight of the harmonic sum)
reaches $w\\!=\\!6$ for the ${C^{\,2}_{F}}\,{n^{\,2}_{\\!f}}$ part, and
$w\\!=\\!5$ for the $C_{F}C_{A}\,{n^{\,2}_{\\!f}}$ contribution. Only non-
alternating sums of $w\\!=\\!4$ enter the former, as expected from the
${C^{\,2}_{F}}\,{n_{\\!f}}$ 3-loop expression; and the alternating $w\\!=\\!3$
sums occur with a particular common prefactor in the second line of eq. (4).
The endpoint behaviour will be addressed below.

The inverse Mellin transform of eq. (4) can be obtained by an algebraic
procedure [29, 30] based on the fact that harmonic sums occur as coefficients
of the Taylor expansion of harmonic polylogarithms. This procedure results in

$\displaystyle P_{gq}^{\,(3)}(x)\Big{|}_{{n^{\,2}_{\\!f}}}\;=\;$ (6)
$\displaystyle\mbox{{}}\mbox{\large{$\frac{16}{27}$}}\\*{C^{\,2}_{F}}\\*{n^{\,2}_{\\!f}}\\*\bigg{(}{(2-x)}\\*\Big{(}156\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{5}}+12\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{3,2}}+60\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{4,0}}+24\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{3,0,0}}+12\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{3,1,0}}+12\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{3,1,1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-12\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{3}}-156\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0,0}}-36\,\\*{\zeta_{3}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0}}-24\,\\*{\zeta_{4}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{0}}-12\,\\*{\zeta_{2}}\,\\*{\zeta_{3}}-12\,\\*{\zeta_{5}}\Big{)}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{p_{gq}(x)}\\*\Big{(}8\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,3}}+28\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,2}}+8\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,2,0}}+16\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,2,1}}-8\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0,0,0}}+16\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,0,0}}+36\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,1,0}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-16\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,1,1}}-8\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0}}-28\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1}}+16\,\\*{\zeta_{3}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1}}\Big{)}+{\color[rgb]{1,0,0.87}{\rm{H}}_{4}}\,\\*(14-31\,\\*x)-96\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-3,0}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,2}}\,\\*(-288+96\,\\*x^{-1}+144\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,2}}\,\\*(-20-16\,\\*x^{-1}+28\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{3,0}}\,\\*(-80+130\,\\*x)$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{3,1}}\,\\*(-102+81\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,-1,0}}\,\\*(288-96\,\\*x^{-1}-144\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,0,0}}\,\\*(-192+64\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+96\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,0,0}}\,\\*(-100-64\,\\*x^{-1}-10\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,1,0}}\,\\*(-20-16\,\\*x^{-1}+28\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,1,1}}\,\\*(-22$
$\displaystyle\mbox{{\hskip
8.53581pt}}-16\,\\*x^{-1}+29\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0,0,0}}\,\\*(-766+485\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{3}}\,\\*(1335+390\,\\*x+160/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,0}}\,\\*(480$
$\displaystyle\mbox{{\hskip
8.53581pt}}-424/3\,\\*x^{-1}-312\,\\*x+64/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,2}}\,\\*(248\,\\*x^{-1}-216\,\\*x+32\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,2}}\,\\*(418/3$
$\displaystyle\mbox{{\hskip
8.53581pt}}-332/3\,\\*x^{-1}-275/3\,\\*x+16/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,0}}\,\\*(1552/3+368/3\,\\*x^{-1}-149/3\,\\*x+112/3\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,1}}\,\\*(161/3+36\,\\*x^{-1}-244/3\,\\*x+32/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-1,0}}\,\\*(-248\,\\*x^{-1}+216\,\\*x-32\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,0,0}}\,\\*(496/3\,\\*x^{-1}-144\,\\*x+64/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0,0}}\,\\*(563/3-91/3\,\\*x+160/3\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0,0}}\,\\*(764/3-244\,\\*x^{-1}-259/3\,\\*x+64/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,0}}\,\\*(542/3-152\,\\*x^{-1}-361/3\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}+16/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,1}}\,\\*(-268/3+116\,\\*x^{-1}+409/6\,\\*x+16/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2}}\,\\*(-144\,\\*{\zeta_{2}}\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+432\,\\*{\zeta_{2}}-216\,\\*{\zeta_{2}}\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2}}\,\\*(-6727/9-3056/9\,\\*x^{-1}-2891/18\,\\*x-1384/9\,\\*x^{2}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+64\,\\*{\zeta_{2}}\,\\*x^{-1}+164\,\\*{\zeta_{2}}+44\,\\*{\zeta_{2}}\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,0}}\,\\*(-160-2408/9\,\\*x^{-1}+56\,\\*x-464/9\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0}}\,\\*(-8110/3-4649/6\,\\*x-2080/9\,\\*x^{2}-14\,\\*{\zeta_{2}}+31\,\\*{\zeta_{2}}\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0}}\,\\*(-850/3$
$\displaystyle\mbox{{\hskip
8.53581pt}}+3776/9\,\\*x^{-1}+41/6\,\\*x-896/9\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1}}\,\\*(-3461/18+491/9\,\\*x^{-1}+2027/36\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}-376/9\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1}}\,\\*(-372\,\\*{\zeta_{2}}\,\\*x^{-1}+324\,\\*{\zeta_{2}}\,\\*x-48\,\\*{\zeta_{2}}\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{0}}\,\\*(1344$
$\displaystyle\mbox{{\hskip
8.53581pt}}+10528/27\,\\*x^{-1}+389/36\,\\*x+6494/27\,\\*x^{2}-424/3\,\\*{\zeta_{2}}\,\\*x^{-1}-1335\,\\*{\zeta_{2}}-702\,\\*{\zeta_{2}}\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}-160/3\,\\*{\zeta_{2}}\,\\*x^{2}+96\,\\*{\zeta_{3}}\,\\*x^{-1}-48\,\\*{\zeta_{3}}+342\,\\*{\zeta_{3}}\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{1}}\,\\*(72371/36-93103/54\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-30551/72\,\\*x+2678/27\,\\*x^{2}+704/3\,\\*{\zeta_{2}}\,\\*x^{-1}-418/3\,\\*{\zeta_{2}}-49/3\,\\*{\zeta_{2}}\,\\*x-64/3\,\\*{\zeta_{2}}\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}-363779/216+368893/324\,\\*x^{-1}+326947/432\,\\*x-7090/81\,\\*x^{2}+72\,\\*{\zeta_{2}}\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+6727/9\,\\*{\zeta_{2}}+3899/18\,\\*{\zeta_{2}}\,\\*x+1384/9\,\\*{\zeta_{2}}\,\\*x^{2}-116/3\,\\*{\zeta_{3}}\,\\*x^{-1}+674\,\\*{\zeta_{3}}-1288\,\\*{\zeta_{3}}\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}+80/3\,\\*{\zeta_{3}}\,\\*x^{2}+148\,\\*{\zeta_{4}}\,\\*x^{-1}-1229/2\,\\*{\zeta_{4}}+1295/4\,\\*{\zeta_{4}}\,\\*x\bigg{)}$
$\displaystyle\mbox{{}}+\mbox{\large{$\frac{16}{27}$}}\\*{C_{F}}\\*{C_{A}}\\*{n^{\,2}_{\\!f}}\\*\bigg{(}{p_{gq}(x)}\\*\Big{(}-25\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,3}}-20\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,-2,0}}+25\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,2}}+8\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,2,0}}+26\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,2,1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-2\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0,0,0}}-14\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,0,0}}+27\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,1,0}}+16\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,1,1}}+21\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0}}+4\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1}}-33\,\\*{\zeta_{3}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1}}\Big{)}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{p_{gq}(-x)}\\*\Big{(}-23\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,3}}+12\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-2,0}}+46\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-1,2}}-2\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,2,0}}-20\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,2,1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-58\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-1,-1,0}}+24\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-1,0,0}}+2\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,0,0,0}}-75\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-1}}+19\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,0}}+75\,\\*{\zeta_{3}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1}}\Big{)}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{4}}\,\\*(36-16\,\\*x)+80\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-3,0}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,2}}\,\\*(192-48\,\\*x^{-1}-72\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,2}}\,\\*(-28+16\,\\*x^{-1}-2\,\\*x)$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{3,0}}\,\\*(24+2\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{3,1}}\,\\*(36+22\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,-1,0}}\,\\*(-192+48\,\\*x^{-1}+72\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,0,0}}\,\\*(112$
$\displaystyle\mbox{{\hskip
8.53581pt}}-32\,\\*x^{-1}-48\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,0,0}}\,\\*(112+32\,\\*x^{-1}+44\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,1,0}}\,\\*(-12+24\,\\*x^{-1}+22\,\\*x)$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,1,1}}\,\\*(-140+52\,\\*x^{-1}+4\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0,0,0}}\,\\*(12-6\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{3}}\,\\*(-1499/6-343/2\,\\*x+4/3\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,0}}\,\\*(-956/3+212/3\,\\*x^{-1}+152\,\\*x-32/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,2}}\,\\*(-140/3-638/3\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+242/3\,\\*x-16\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,2}}\,\\*(277/3-311/3\,\\*x^{-1}-188/3\,\\*x+4/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,0}}\,\\*(67/3$
$\displaystyle\mbox{{\hskip
8.53581pt}}-212/3\,\\*x^{-1}-265/6\,\\*x-4/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,1}}\,\\*(525/2-76\,\\*x^{-1}-7/4\,\\*x-12\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-1,0}}\,\\*(460/3+958/3\,\\*x^{-1}-34/3\,\\*x+16\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,0,0}}\,\\*(-260/3-550/3\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+50/3\,\\*x-32/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0,0}}\,\\*(-332/3-19/12\,\\*x+8/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0,0}}\,\\*(-184/3$
$\displaystyle\mbox{{\hskip
8.53581pt}}+434/3\,\\*x^{-1}-106/3\,\\*x-40/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,0}}\,\\*(283/3-83\,\\*x^{-1}-248/3\,\\*x-4/3\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,1}}\,\\*(700/3-259\,\\*x^{-1}-299/3\,\\*x+44/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2}}\,\\*(72\,\\*{\zeta_{2}}\,\\*x^{-1}-288\,\\*{\zeta_{2}}+108\,\\*{\zeta_{2}}\,\\*x)$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{2}}\,\\*(16495/36+1067/6\,\\*x^{-1}+610/9\,\\*x+1202/9\,\\*x^{2}-40\,\\*{\zeta_{2}}\,\\*x^{-1}-68\,\\*{\zeta_{2}}-34\,\\*{\zeta_{2}}\,\\*x)$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,0}}\,\\*(226/3+4/3\,\\*x^{-2}+1498/9\,\\*x^{-1}+7/3\,\\*x+232/9\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0}}\,\\*(8735/18$
$\displaystyle\mbox{{\hskip
8.53581pt}}+3091/36\,\\*x+440/9\,\\*x^{2}-36\,\\*{\zeta_{2}}+16\,\\*{\zeta_{2}}\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0}}\,\\*(1363/18-305/6\,\\*x^{-1}+301/18\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}+52/9\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1}}\,\\*(-8747/36+12383/36\,\\*x^{-1}+1498/9\,\\*x-10\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1}}\,\\*(1117/3\,\\*{\zeta_{2}}\,\\*x^{-1}+370/3\,\\*{\zeta_{2}}-259/3\,\\*{\zeta_{2}}\,\\*x+24\,\\*{\zeta_{2}}\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{0}}\,\\*(-10607/12$
$\displaystyle\mbox{{\hskip
8.53581pt}}-4916/27\,\\*x^{-1}-12139/72\,\\*x-9944/27\,\\*x^{2}+212/3\,\\*{\zeta_{2}}\,\\*x^{-1}+1499/6\,\\*{\zeta_{2}}+647/2\,\\*{\zeta_{2}}\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}-4/3\,\\*{\zeta_{2}}\,\\*x^{2}-72\,\\*{\zeta_{3}}\,\\*x^{-1}-72\,\\*{\zeta_{3}}-146\,\\*{\zeta_{3}}\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{1}}\,\\*(-4495/6+132025/216\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+3983/36\,\\*x-6500/27\,\\*x^{2}-56\,\\*{\zeta_{2}}\,\\*x^{-1}-47/3\,\\*{\zeta_{2}}+205/3\,\\*{\zeta_{2}}\,\\*x+20/3\,\\*{\zeta_{2}}\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+30829/108-191575/648\,\\*x^{-1}-91561/432\,\\*x+27377/81\,\\*x^{2}-205/18\,\\*{\zeta_{2}}\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-16495/36\,\\*{\zeta_{2}}-589/9\,\\*{\zeta_{2}}\,\\*x-1202/9\,\\*{\zeta_{2}}\,\\*x^{2}+104\,\\*{\zeta_{3}}\,\\*x^{-1}-752\,\\*{\zeta_{3}}+8587/12\,\\*{\zeta_{3}}\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}+8\,\\*{\zeta_{3}}\,\\*x^{2}-114\,\\*{\zeta_{4}}\,\\*x^{-1}+331\,\\*{\zeta_{4}}-59/2\,\\*{\zeta_{4}}\,\\*x\bigg{)}\;,$

where we have suppressed the argument $x$ of the harmonic polylogarithms
(HPLs) $\,H_{m_{1},...,m_{w}}(x)$ and used the leading-order function
$\,{p_{gq}(x)}=2\,x^{\,-1}-2+x\,$ to shorten the $w=4$ part of the
expressions. For chains of indices zero we employ the abbreviated notation

$H_{{\>\\!\footnotesize\underbrace{0,\ldots,0}_{\scriptstyle m}},\,\pm
1,\,{\footnotesize\underbrace{0,\ldots,0}_{\scriptstyle n}},\,\pm
1,\,\ldots}(x)\;=\;H_{\pm(m+1),\,\pm(n+1),\,\ldots}(x)\>\>.$ (7)

Corresponding to the maximal weights in the $N$-space expression (4), i.e.,
$w=6$ for the ${C^{\,2}_{F}}\\*\,{n^{\,2}_{\\!f}}$ part and $w=5$ for the
$C_{F}C_{A}\\*\,{n^{\,2}_{\\!f}}$ contribution, the respective $x$-space
results in eq. (6) include HPLs up to $w=5$ and $w=4$. The term corresponding
to $\,(N\\!-\\!2)^{-1}\,S_{-2}(N\\!-\\!2)\,$ is $\,x^{\,-2}\,{\rm
H}_{-1,0}(x)$.

Of particular interest are the logarithmically enhanced endpoint contributions
in the high-energy (small-$x$) and threshold (large-$x$) limits. In the former
limit, the flavour-singlet splitting functions are dominated by the BFKL
single-log enhancement of the $x^{\,-1}$ terms. The subdominant $x^{\,0}$
contributions show a double-log enhancement. In both cases the leading
${n^{\,2}_{\\!f}}$ terms are of overall next-to-next-to-leading logarithmic
(NNLL) accuracy. Eq. (6) leads to

$\displaystyle P_{\rm gq}^{\,(3)}(x)\bigg{|}_{{n^{\,2}_{\\!f}}}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle\frac{1}{x}\,\\*\ln
x\,\,\\*\bigg{[}{C^{\,2}_{F}}\\*\,{n^{\,2}_{\\!f}}\\*\Big{(}\,\frac{168448}{729}-\frac{6784}{81}\,\\*{\zeta_{2}}+\frac{512}{9}\,\\*{\zeta_{3}}\Big{)}$
(8) $\displaystyle\mbox{{\hskip 8.53581pt}{\hskip
8.53581pt}}\,+\,{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}-\frac{78080}{729}+\frac{3392}{81}\,\\*{\zeta_{2}}-\frac{128}{3}\,\\*{\zeta_{3}}\Big{)}\bigg{]}$
$\displaystyle\mbox{{}}+\frac{1}{x}\,\\*\bigg{[}\,{C^{\,2}_{F}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}\,\frac{1475572}{2187}+\frac{128}{3}\,\\*{\zeta_{2}}-\frac{1856}{81}\,\\*{\zeta_{3}}+\frac{2368}{27}\,\\*{\zeta_{4}}\Big{)}$
$\displaystyle\mbox{{\hskip 8.53581pt}{\hskip
8.53581pt}}\,+\,{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}-\frac{384878}{2187}-\frac{1640}{243}\,\\*{\zeta_{2}}+\frac{1664}{27}\,\\*{\zeta_{3}}-\frac{608}{9}\,\\*{\zeta_{4}}\Big{)}\bigg{]}$
$\displaystyle\mbox{{}}+\ln^{\,4\\!}x\,\,\\*\bigg{[}-\frac{1532}{81}\,\\*{C^{\,2}_{F}}\,\\*{n^{\,2}_{\\!f}}+\frac{8}{27}\,\\*{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\bigg{]}$
$\displaystyle\mbox{{}}+\ln^{\,3\\!}x\,\,\\*\bigg{[}\,{C^{\,2}_{F}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}\,\frac{4120}{243}-\frac{832}{27}\,\\*{\zeta_{2}}\Big{)}\,-\frac{2848}{243}\\*{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\bigg{]}\;+\;\ldots$

where we have suppressed further terms for brevity. The $\ln^{\,4\\!}x$
contribution, which arise from the ${D_{0}^{\,5}}$ terms in eq. (4), agrees
with the prediction in eq. (5.11) of ref. [31], the rest is new.

The off-diagonal splitting functions exhibit a double logarithmic enhancement
also at large-$x$. In the present case it reads

$\displaystyle P_{\rm gq}^{\,(3)}(x)\bigg{|}_{{n^{\,2}_{\\!f}}}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\ln^{\,4}(1\\!-\\!x)\,\\*\,\frac{32}{81}\,\\*{C_{F}}\,\\*{n^{\,2}_{\\!f}}\,\\*({C_{A}}-{C_{F}})\,-\,\ln^{\,3}(1\\!-\\!x)\,\\*\bigg{[}\,\frac{2404}{243}\,\\*{C^{\,2}_{F}}\,\\*{n^{\,2}_{\\!f}}-\frac{2656}{243}\,\\*{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\bigg{]}$
(9)
$\displaystyle\mbox{{}{}}+\ln^{\,2}(1\\!-\\!x)\,\\*\bigg{[}\,{C^{\,2}_{F}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}-\frac{8870}{243}-\frac{32}{3}\,\\*{\zeta_{2}}\Big{)}+{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}\,\frac{18536}{243}+\frac{16}{27}\,\\*{\zeta_{2}}\Big{)}\bigg{]}$
$\displaystyle\mbox{{}{}}+\ln(1\\!-\\!x)\,\\*\,\bigg{[}\,{C^{\,2}_{F}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}\,\frac{1870}{81}-\frac{4144}{81}\,\\*{\zeta_{2}}-\frac{320}{27}\,\\*{\zeta_{3}}\Big{)}$
$\displaystyle\mbox{{\hskip 8.53581pt}{\hskip 8.53581pt}{\hskip
8.53581pt}{\hskip 8.53581pt}{\hskip
8.53581pt}}+{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}\,\frac{12866}{81}-\frac{160}{81}\,\\*{\zeta_{2}}+\frac{880}{27}\,\\*{\zeta_{3}}\Big{)}\bigg{]}\;+\;\ldots$

The $\ln^{\,4}(1\\!-\\!x)$ contribution, including the terms suppressed by
powers of ${(1\\!-\\!x)}$ not shown here, agrees with the predictions in eqs.
(5.16) and (5.25) of ref. [32], see also refs. [33, 34]. The remaining terms
in eq. (9) were not known before.

To summarize, we have derived the ${n^{\,2}_{\\!f}}$ contributions to
$P_{\,\rm gq}^{\,(3)}(x)$. Together with ref. [2] this completes the
${n^{\,2}_{\\!f}}$ parts of the 4-loop (N3LO) quark-to-parton splitting
functions for the evolution of unpolarized parton distributions of hadrons. As
usual, our results refer to the $\overline{\mbox{MS}}$ scheme, written in
terms of the expansion parameter $a_{\rm s}\,=\,\alpha_{\rm
s}({\mu^{\,2}})/(4\pi)$, i.e., we have, without loss or information,
identified the renormalization scale $\mu_{\>\\!\sf r}$ with the factorization
scale $\mu$ in eq. (9).

While the ${n^{\,2}_{\\!f}}$ parts are much larger, at any relevant number
${n_{\\!f}}$ of light flavours, than the corresponding leading
large-${n_{\\!f}}$ terms – see, e.g., the values at $N=8$ in eqs. (17) - 20 of
ref. [8] – these results are not yet of direct relevance for phenomenological
analyses of hard scattering processes.

It is not feasible to extend the present results to the complete N3LO
splitting functions with the method applied here. We expect it to be a
formidable task also for other approaches based on the method of differential
equations [35, 36, 37, 5, 2, 38]. Hence phenomenological analyses will have to
rely on approximate $x$-space expressions for the time being, as presented in
ref. [3] for the large-${n_{c}}$ suppressed non-singlet contributions and in
refs. [6, 7, 9] for the flavour-singlet cases. We hope to be able to present
results based on the even moments $2\leq N\leq 20$ also for the complete
$P_{\,\rm gq}^{\,(3)}(x)$ and $P_{\,\rm gg}^{\,(3)}(x)$ in the near future.

The biggest issue in such $x$-space approximations, as also noted in ref. [2],
are the hitherto unknown $x^{\,-1}\ln x$ contributions, i.e., the NNLL
corrections in the high-energy BFKL limit. With ref. [2] and the present
results, the first two such contributions have been derived. These results
should not only prove useful in the context of approximate expressions, but
also as checks of future calculations of NNLL corrections in the BFKL limit
and their transformation to the $\overline{\mbox{MS}}$ scheme, which in itself
will be a non-trivial operation, see refs. [39, 40]. Also the results for the
double-logarithmically enhanced contributions in the small-$x$ and large-$x$
limits can provide input for the extension of the respective resummations to a
higher logarithmic accuracy.

Form files with our results have been deposited at the preprint server
https://arXiv.org with the sources of this letter. They are also available
from the authors upon request.

## Acknowledgements

G.F. would like to acknowledge the influence of Marcello Ciafaloni, who passed
away recently, on his decision to work on quantum field theory. This work has
been supported by the Vidi grant 680-47-551 of the Dutch Research Council
(NWO), the UKRI FLF Mr/S03479x/1; the Consolidated Grant ST/P0000630/1
Particle Physics at the Higgs Centre of the UK Science and Technology
Facilities Council (STFC); the ERC Starting Grant 715049 QCDforfuture, the
EU’s Horizon Europe research and innovation programme under the Marie
Sklodowska-Curie grant 101104792, QCDchallenge; the Deutsche
Forschungsgemeinschaft through the Research Unit FOR 2926, Next Generation
pQCD for Hadron Structure: Preparing for the EIC, project number 40824754, and
DFG grant MO 1801/4-2, the ERC Advanced Grant 101095857 Conformal-EIC; and the
STFC Consolidated Grant ST/T000988/1. G.F. and S.M. are grateful to the
Galileo Galilei Institute in Florence for hospitality and support during the
scientific program on Theory Challenges in the Precision Era of the Large
Hadron Collider, where part of this work was done.

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license: Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/

authors:
G. Falcioni (Zurich U. and Turin U.), F. Herzog (U. Edinburgh, Higgs
  Ctr. Theor. Phys.), S. Moch (Hamburg U., Inst. Theor. Phys. II), J.
  Vermaseren (Nikhef, Amsterdam) and A. Vogt (Liverpool U., Dept. Math.)

submitter: Andreas Vogt